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0704.0211

Linkedness and ordered cycles in digraphs

LINKEDNESS AND ORDERED CYCLES IN DIGRAPHS DANIELA KÜHN AND DERYK OSTHUS Abstract. Given a digraph D, let δ(D) := min{δ+(D), δ−(D)} be the min- imum degree of D. We show that every sufficiently large digraph D with δ(D) ≥ n/2 + ℓ − 1 is ℓ-linked. The bound on the minimum degree is best possible and confirms a conjecture of Manoussakis [16]. We also determine the smallest minimum degree which ensures that a sufficiently large digraph D is k-ordered, i.e. that for every sequence s1, . . . , sk of distinct vertices of D there is a directed cycle which encounters s1, . . . , sk in this order. 1. Introduction The minimum degree δ(D) of a digraph D is the minimum of its minimum outdegree δ+(D) and its minimum indegree δ−(D). When referring to paths and cycles in digraphs we always mean that these are directed without mentioning this explicitly. A digraph D is ℓ-linked if |D| ≥ 2ℓ and if for every sequence x1, . . . , xℓ, y1, . . . , yℓ of distinct vertices there are disjoint paths P1, . . . , Pℓ in D such that Pi joins xi to yi. Since this is a very strong and useful property to have in a digraph, the question of course arises how it can be forced by other properties. In the case of (undirected) graphs, much progress has been made in this di- rection. In particular, linkedness is closely related to connectivity: Bollobás and Thomason [2] showed that every 22k-connected graph is k-linked (this was recently improved to 10k by Thomas and Wollan [17]). However, for digraphs the situa- tion is quite different: Thomassen [18] showed that for all k there are strongly k-connected digraphs which are not even 2-linked. Our first result determines the minimum degree forcing a (large) digraph to be ℓ-linked, which confirms a conjecture of Manoussakis [16] for large digraphs. Theorem 1. Let ℓ ≥ 2. Every digraph D of order n ≥ 1600ℓ3 which satisfies δ(D) ≥ n/2 + ℓ− 1 is ℓ-linked. It is not hard to see that the bound on minimum degree in Theorem 1 is best possible (see Proposition 3). It is also easy to see that for ℓ = 1 the correct bound is δ(D) ≥ ⌊n/2⌋. The cases ℓ = 2, 3 of Theorem 1 were proved by Heydemann and Sotteau [9] and Manoussakis [16] respectively. Manoussakis [16] also determined the number of edges which force a digraph to be ℓ-linked. A discussion of these and related results can be found in the monograph by Bang-Jensen and Gutin [1]. Note that it does not make sense to ask for the minimum outdegree of a di- graph D which ensures that D is ℓ-linked (or similarly, to ask for the minimum indegree). Indeed, the digraph obtained from a complete digraph A of order n− 1 by adding a new vertex x which sends an edge to every vertex in A has minimum outdegree n− 2 but is not even 1-linked. A slightly weaker notion is that of a k-ordered digraph: a digraph D is k-ordered if |D| ≥ k and if for every sequence s1, . . . , sk of distinct vertices ofD there is a cycle http://arxiv.org/abs/0704.0211v1 2 DANIELA KÜHN AND DERYK OSTHUS which encounters s1, . . . , sk in this order. It is not hard to see that every ℓ-linked digraph is also ℓ-ordered. Conversely, every 2ℓ-ordered digraph D is also ℓ-linked: if x1, . . . , xℓ, y1, . . . , yℓ is a sequence of vertices as in the definition of ℓ-linkedness then a cycle which encounters x1, y1, x2, y2, . . . , xℓ, yℓ in this order would yield the paths required for the linking. The next result says that as far as the minimum degree is concerned it is no harder to guarantee the 2ℓ paths forming such a cycle than to guarantee just the ℓ paths required for the linking. In particular, note that Theorem 2 immediately implies Theorem 1. Theorem 2. Let k ≥ 2. Every digraph D of order n ≥ 200k3 which satisfies δ(D) ≥ (n+ k)/2 − 1 is k-ordered. Again, the bound on the minimum degree is best possible (see Proposition 4). Moreover, it is easy to see that if k = 1 then the correct bound is δ(D) ≥ n/2− 1. The proof of Theorem 2 yields paths between the k ‘special’ vertices whose length is at most 6 and it is also easy to translate the proof into an algorithm which finds these paths in polynomial time (see the remarks after the end of the proof). Somewhat surprisingly, the minimum degree in both theorems is not quite the same as in the undirected case: Kawarabayashi, Kostochka and Yu [12] proved that the smallest minimum degree which guarantees a graph on n vertices to be ℓ-linked is ⌊n/2⌋ + ℓ − 1 for large n. (Egawa et al. [4] independently determined the smallest minimum degree which guarantees the existence of ℓ disjoint cycles containing ℓ specified independent edges, which is clearly a very similar property.) Kierstead, Sarközy and Selkow [13] proved that the smallest minimum degree which guarantees a graph on n vertices to be k-ordered is δ(D) ≥ ⌈n/2⌉ + ⌊k/2⌋ − 1 for large n. So in the undirected case the ‘2ℓ-ordered’ result does not quite imply the ‘ℓ-linked’ result. The proofs in [4, 12, 13] do not seem to generalize to digraphs. 2. Further work and open problems In a sequel to this paper, we hope to apply Theorem 2 to obtain the following stronger results, which would also generalize the theorem of Ghouila-Houri [6] that any digraph D on n vertices with δ(D) ≥ n/2 contains a Hamilton cycle: we aim to apply Theorem 2 to show that if k ≥ 2 and D is a sufficiently large digraph whose minimum degree is as in Theorem 2 then D is even k-ordered Hamiltonian, i.e. for every sequence s1, . . . , sk of distinct vertices of D there is a Hamilton cycle which encounters s1, . . . , sk in this order. One can use this to prove that the minimum degree condition in Theorem 1 already implies that the digraph D is Hamiltonian ℓ-linked, i.e. the paths linking the pairs of vertices span the entire vertex set of D. Note that this in turn would immediately imply that D is ℓ-arc ordered Hamiltonian, i.e. D has a Hamilton cycle which contains any ℓ disjoint edges in a given order. Note that in each case the examples in Section 3 show that the minimum degree condition would be best possible. Undirected versions of these statements were first obtained by Kierstead, Sarközy and Selkow [13] and Egawa et al. [4] respectively (and a common generalization of these in [3]). For graphs, the concepts ‘ℓ-linked’ and ‘k-ordered’ were generalized to ‘H-linked’ by Jung [11]: a graph G isH-linked ifG contains a subdivision ofH with prescribed branch vertices (so G is k-ordered if and only if it is Ck-linked). The minimum LINKEDNESS AND ORDERED CYCLES IN DIGRAPHS 3 degree which forces a graph to be H-linked for an arbitrary H was determined in [5, 14, 15, 7]. Clearly, one can ask similar questions also for digraphs. Finally, we believe that the bound on n which we require in Theorem 2 (and thus in Theorem 1) can be reduced to one which is linear in k. 3. Notation and extremal examples Before we discuss the examples showing that the bounds on the minimum degree in Theorems 1 and 2 are best possible, we will introduce the basic notation used throughout the paper. A digraph D is complete if every pair of vertices of D is joined by edges in both directions. The order |D| of a digraph D is the number of its vertices. We write N+(x) for the outneighbourhood of a vertex x and d+(x) := |N+(x)| for its outdegree. Similarly, we write N−(x) for the inneighbourhood of a vertex x and d−(x) := |N−(x)| for its indegree. We set d(x) := min{d+(x), d−(x)}. Given a set A of vertices of D, we write N+ (x) for the set of all outneighbours of x in A. N− (x), d+ (x) and d− (x) are defined similarly. Given two vertices x, y of a digraph D, an x-y path in D is a directed path which joins x to y. Given two disjoint vertex sets A and B of D, an A-B edge is an edge ab where a ∈ A and b ∈ B. The following proposition shows that the bound on the minimum degree in Theorem 1 cannot be reduced. Proposition 3. For every ℓ ≥ 2 and every n ≥ 2ℓ there exists a digraph D on n vertices with minimum degree ⌈n/2⌉+ ℓ− 2 which is not ℓ-linked. Proof. We will distinguish the following cases. Case 1. n is even. Let D be the digraph which consists of complete digraphs A and B of order n/2+ ℓ − 1 which have precisely 2ℓ − 2 vertices in common. To see that D is not ℓ- linked let x1, . . . , xℓ−1, y1, . . . , yℓ−1 denote the vertices in A∩B. Pick some vertex xℓ ∈ A \ B and some vertex yℓ ∈ B \ A. Then D does not contain disjoint paths between xi and yi for all i = 1, . . . , ℓ. The minimum degree of D is attained by the vertices in (A \B) ∪ (B \A) and thus is as desired. Case 2. n is odd. In this case, we define D as follows. Let A and B be disjoint complete digraphs of order ⌈n/2⌉ − ℓ − 1. Add a complete digraph X of order 2ℓ − 3 and join all vertices in X to all vertices in A ∪ B with edges in both directions. Add a set S := {x1, x2, y1, y2} of 4 new vertices such that each vertex in S is joined to each vertex in X with edges in both directions. Moreover, we add all the edges between different vertices in S except for −−→x1y1 and −−→x2y2. Finally, we connect the vertices in S to the vertices in A ∪ B as follows. Both x1 and y1 receive edges from every vertex in B and send edges to every vertex in A. Additionally, x1 will receive an edge from every vertex in A and y1 will send an edge to every vertex in B. Both x2 and y2 receive edges from every vertex in A and send edges to every vertex in B. Additionally, x2 will receive an edge from every vertex in B and y2 will send an edge to every vertex in A (see Figure 1). To check that D has the required minimum degree, consider first any vertex a ∈ A. As a sends edges to 3 vertices in S and receives edges from 3 such vertices, 4 DANIELA KÜHN AND DERYK OSTHUS PSfrag replacements Figure 1. The digraph D in Case 2 of Proposition 3. The dashed arrows indicate the missing edges between x1 and y1 and between x2 and y2. we have that d(a) = |A| − 1 + |X| + 3 = ⌈n/2⌉ + ℓ − 2. It follows similarly that the vertices in B have the correct degree. Thus consider any vertex s ∈ S. Then s sends edges to all vertices in A or to all vertices in B (or both) and s receives edges from all vertices in A or from all vertices in B (or both). Thus d(s) = |A| + |X| + 2 = ⌈n/2⌉ + ℓ − 2. It is easy to check that the vertices in X have the required degree and thus δ(D) = ⌈n/2⌉+ ℓ− 2. To see that D is not ℓ-linked, let x, x3, . . . , xℓ, y3, . . . , yℓ denote the vertices in X. Then we cannot link xi to yi for each i = 1, . . . , ℓ since every x1-y1 path must meet X ∪ {x2, y2} (and thus would contain x) and the analogue is true for every x2-y2 path. � We conclude this section with the examples showing that the bound on the minimum degree in Theorem 2 is best possible. Proposition 4. For every k ≥ 2 and every n ≥ 2k there exists a digraph D on n vertices with minimum degree ⌈(n+ k)/2⌉ − 2 which is not k-ordered. Proof. We will distinguish the following cases. Case 1. k ≥ 3 is odd and n is even. In this case, we define D as follows. Let A and B be disjoint complete digraphs of order n/2 − k + 1. Add a complete digraph X of order k − 2 and join all its vertices to all vertices in A ∪ B with edges in both directions. Add new vertices s1, . . . , sk such that every si is joined to all vertices in X with edges in both directions. Moreover, we add all the edges −−→sisj for j 6= i, i + 1 where sk+1 := s1. We also add the edge −−→s1s2. Finally, we connect the si to the vertices in A ∪ B as follows. Both s1 and s2 receive edges from every vertex in B and send edges to every vertex in B. Additionally, s1 will send an edge to every vertex in A and s2 will receive an edge from every vertex in A. Each of s3, s5, . . . , sk receives an edge from every vertex in A and sends an edge to every vertex in A. Each of s4, s6, . . . , sk−1 receives an edge from every vertex in B and sends an edge to every vertex in B (see Figure 2). Let us now check that the minimum degree of the digraph D thus obtained is as required. Let S := {s1, . . . , sk}. Note that each LINKEDNESS AND ORDERED CYCLES IN DIGRAPHS 5 PSfrag replacements Figure 2. The digraph D for k = 5 in Case 1 of Proposition 4. The dashed arrows indicate missing edges between the vertices si. vertex v ∈ A ∪ B sends edges to precisely (k + 1)/2 vertices in S and receives edges from precisely that many vertices. Since |A| = |B|, it follows that d(v) = |A| − 1 + |X|+ (k + 1)/2 = n/2− 2 + (k + 1)/2 = ⌈(n + k)/2⌉ − 2. Now consider any si ∈ S. Then si receives edges from either all vertices in A or all vertices in B (or both) and si sends edges to either all vertices in A or all vertices in B (or both). Hence d(si) ≥ |A|+ |X| + |S| − 2 = n/2− 1 + k − 2 ≥ ⌈(n + k)/2⌉ − 2. It is easy to check that the degree of the vertices in X is > ⌈(n+ k)/2⌉ − 2. To see that D is not k-ordered note that every cycle in D which encounters s1, . . . , sk in this order would use at least one vertex from X between si and si+1 for every i 6= 1 (see Figure 2). But since |X| = k − 2 this is impossible. Case 2. k is even. LetD be the digraph which consists of a complete digraph A of order ⌈n/2⌉+k/2−1 and a complete digraph B of order ⌊n/2⌋+ k/2 which has precisely k − 1 vertices in common with A. It is easy to check that δ(D) = |A| − 1 = ⌈(n + k)/2⌉ − 2. To see that D is not k-ordered, pick vertices s1, s3, . . . , sk−1 in A\B and s2, s4, . . . , sk in B \ A. Then every cycle in D which encounters s1, . . . , sk in this order would meet A∩B when going from si to si+1, i.e. it would meet A∩B k times, which is impossible. Case 3. k ≥ 3 is odd and n is odd. This time we take D to be the digraph which consists of two complete digraphs A and B of order (n + k)/2 − 1 having k − 2 vertices in common. Then δ(D) = |A| − 1 = (n+ k)/2− 2. To see that D is not k-ordered, pick vertices s1, s3, . . . , sk in A \B and s2, s4, . . . , sk−1 in B \ A. � Note that in the proof of Proposition 4 we could have omitted the (easy) case when k is even as Proposition 3 already gives a digraph of the required minimum degree which is not k/2-linked and thus not k-ordered. 6 DANIELA KÜHN AND DERYK OSTHUS 4. Proof of Theorem 2 We first prove Theorem 2 for the case when k = 2. So suppose that D is a digraph of minimum degree at least ⌈n/2⌉. Let s1 and s2 be the vertices which our cycle has to encounter. If −−→s1s2 is not an edge then s1, s2 /∈ N +(s1) ∪ N −(s2) and so |N+(s1) ∩N −(s2)| ≥ 2δ(D) − (n− 2) ≥ 2. Similarly, if −−→s2s1 is not an edge then |N−(s1) ∩N +(s2)| ≥ 2. Altogether this shows that there is a cycle of length at most 4 which contains both s1 and s2. Thus we may assume that k ≥ 3 and that D is a digraph of minimum degree at least ⌈(n + k)/2⌉ − 1. Let S := (s1, . . . , sk) be the given sequence of vertices of D which our cycle has to encounter. We will call these vertices special and will sometimes also use S for the set of these vertices. We set sk+1 := s1. Given a set I ⊆ [k] and a family T := (ti)i∈I of positive integers, an (S, I, T )-system is a family (Pi)i∈I where each Pi is a set of ti paths joining si to si+1 and each path in Pi has length at most 6 and is internally disjoint from S, from all other paths in Pi and from the paths in all the other Pj . An (S, I)-system is an (S, I, T )-system where ti = 1 for all i ∈ I. Thus to prove Theorem 2 we have to show that there exists an (S, [k])-system. Let I be the set of all those indices i ∈ [k] for which D does not contain at least 6k internally disjoint si-si+1 paths of length at most 6. Claim 1. It suffices to show that D contains an (S, I)-system. Indeed, suppose that (Pi)i∈I is an (S, I)-system in D. So each Pi contains precisely one path Pi. We will show that for every i ∈ [k]\I we can find an si-si+1 path Pi of length at most 6 which meets S only in si and si+1 such that all the paths P1, . . . , Pk are internally disjoint. We will choose such a path Pi for every i ∈ [k] \ I in turn. Suppose that next we want to find Pj . Recall that since j ∈ [k] \ I the digraph D contains a set P of at least 6k internally disjoint sj-sj+1 paths of length at most 6. Since at most 5(k − 1) + k < 6k vertices of D lie in S or in the interior of some of the other paths Pi, one of the paths in P must be internally disjoint from S and all the other paths Pi, and so we can take this path to be Pj . This proves Claim 1. In order to prove the existence of an (S, I)-system, choose an (S, J, T )-system (Pj)j∈J in D such that J ⊆ I is as large as possible and subject to this j∈J tj is maximal. Note that tj < 6k for all j ∈ J since J ⊆ I. Assume that |J | < |I|. By relabelling the special vertices, we may assume that k ∈ I \ J . So we would like to extend (Pj)j∈J by a suitable sk-s1 path. Let X ′ be the set of all those vertices which lie in the interior of some path belonging to (Pj)j∈J . Note that |S ∪X ′| < 6k · 5(k − 1) + |S| < 30k2 =: k0. Let A := N+(sk) \ (S ∪X ′) and B := N−(s1) \ (S ∪X ′). Then (1) |A|, |B| ≥ δ(D) − |S ∪X ′| ≥ n/2− k0. Moreover, A ∩ B = ∅ as otherwise we could extend our (S, J, T )-system (Pj)j∈J by adding the path Pk := skxs1 where x ∈ A ∩B, a contradiction to the choice of (Pj)j∈J . In particular, this shows that the set X ′′ of all vertices outside A ∪ B ∪ S ∪X ′ has size at most 2k0 and thus, setting Y := S ∪X ′ ∪X ′′, we have that |Y | ≤ 3k0. LINKEDNESS AND ORDERED CYCLES IN DIGRAPHS 7 Note that D does not contain an edge ab with a ∈ A and b ∈ B. Indeed, otherwise we could extend (Pj)j∈J by adding the path Pk := skabs1. We will often use the following claim. Claim 2. Let a ∈ A and let A′ ⊆ A be a set of size at least k0. Then N +(a)∩A′ 6= ∅. Similarly, if b ∈ B and B′ ⊆ B is a set of size at least k0 then N −(b) ∩B′ 6= ∅. Suppose that N+(a) ∩ A′ = ∅. Then (1) together with the fact that D does not contain an A-B edge implies that d+(a) ≤ n − |B| − k0 ≤ n/2, a contradiction. The proof of the second part of the claim is similar. We say that a special vertex si has out-type A if si sends at least k0 edges to A. Similarly we define when si has out-type B, in-type A and in-type B. As |Y |+2k0 ≤ 5k0 ≤ δ(G), it follows that each si has out-type A or out-type B (or both) and in-type A or in-type B (or both). Note that s1 has in-type B but not in-type A whereas sk has out-type A but not out-type B. Claim 3. Let j ∈ J . If sj has out-type A then sj+1 has in-type B but not in-type A. Similarly, if sj has out-type B then sj+1 has in-type A but not in-type B. Suppose that sj has out-type A and sj+1 has in-type A. Let a ∈ N (sj). Claim 2 implies that a sends an edge to one of the at least k0 vertices in N (sj+1). Let a′ ∈ N− (sj+1) be such a neighbour of a. Then we could extend our (S, J, T )- system by adding the path sjaa ′sj+1, a contradiction. The proof of the second part of Claim 3 is similar. Claim 4. No vertex in B sends an edge to A. Suppose that b∗a∗ is an edge of D, where a∗ ∈ A and b∗ ∈ B. Given vertices a ∈ A and b ∈ B, put Nab := N +(a) ∩ N−(b). Note that Nab ⊆ Y and a, b /∈ N+(a) ∪N−(b) as D does not contain an A-B edge. Thus (2) |Nab| ≥ 2δ(D) − (n− 2) = − (n− 2) ≥ k. Let us now show that no special vertex si with i ∈ J has out-type B. So suppose i ∈ J and si has out-type B. Then Claim 3 implies that si+1 has in-type A. By Claim 2 some of the at least k0 vertices in N (si+1) receives an edge from a ∗. Let a′ be such a vertex. Similarly, some of the vertices in N+ (si) sends an edge to b Let b′ be such a vertex. Then we could extend our (S, J, T )-system by adding the path sib ′b∗a∗a′si+1, a contradiction. This shows that whenever si is a special vertex of out-type B then i /∈ J . Let Q denote the set of such vertices si. Note that sk 6∈ Q as sk does not have out-type B. Thus each special vertex in Q forbids one index in J . Altogether this shows that (3) |J | ≤ k − 1− |Q|. Let SA be the set of all those special vertices si with 1 ≤ N (si) < k0. Let SB be the set of all those special vertices si with 1 ≤ N (si) < k0. Let A ∗ be the set of all those vertices in A which do not send an edge to some vertex in SA. Then |A∗| > |A| − kk0. Similarly, let B ∗ be the set of all those vertices in B which do not receive an edge from some vertex in SB. Then |B ∗| > |B| − kk0. 8 DANIELA KÜHN AND DERYK OSTHUS PSfrag replacements Figure 3. Modifying our (S, J, T )-system in the proof of Claim 4. Consider any pair a, b with a ∈ A∗ and b ∈ B∗. As each special vertex in Nab belongs to Q, it follows that (4) |Nab \ S| ≥ |Nab| − |Q| ≥ k − |Q| > |J |. Suppose first that J 6= ∅. Given j ∈ J , let X ′′j be the union of X ′′ with the set of all vertices lying in the interior of paths in Pj. As Nab ⊆ Y there must be an index jab ∈ J such that Nab contains at least two vertices in X . Note that |A∗|, |B∗| > 2k0k. Thus there are 2k0+1 disjoint pairs a, b for which this index jab must be the same. Let aq, bq (q = 0, . . . , 2k0) denote these pairs and let j ∈ J denote the common index. Note that sj has out-type A since we have seen before that no special vertex si with i ∈ J has out-type B. Claim 3 now implies that sj+1 has in-type B. Pick vertices a ∈ N+ (sj) and b ∈ N (sj+1) such that a 6= a0 and b 6= b0. Claim 2 implies that there are indices q1, . . . , qk0 such that a sends an edge to each aqr . Apply Claim 2 again to find an index r ≤ k0 such that b receives an edge from bqr . Let x ∈ Naqr bqr and y ∈ Na0b0 be distinct vertices such that x, y ∈ X j . We can now modify our (S, J, T )-system to obtain an (S, J ∪ {k}, T ′)-system in D by replacing Pj with the single path sjaaqrxbqrbsj+1 and adding the sk-s1 path ska0yb0s1 (see Figure 3). If J = ∅ then we just add the sk-s1 path (which is still guaranteed by (4)). In both cases this contradicts the choice of our (S, J, T )-system and completes the proof of Claim 4. Claim 5. Let a ∈ A and let A′ ⊆ A be a set of size at least k0. Then N −(a)∩A′ 6= ∅. Similarly, if b ∈ B and B′ ⊆ B is a set of size at least k0 then N +(b) ∩B′ 6= ∅. Using Claim 4, this can be shown similarly as Claim 2. Let S+ be the set of all those special vertices which send an edge to A and let S− be the set of all those special vertices which receive an edge from A. Define S+ and S− similarly. Note that these sets are not disjoint. The proof of the next claim LINKEDNESS AND ORDERED CYCLES IN DIGRAPHS 9 is similar to that of Claim 3. (To prove the second and third part of Claim 6 we use Claim 5 instead of Claim 2.) Claim 6. If j ∈ J and sj ∈ S then sj+1 cannot have in-type A. If j − 1 ∈ J and sj ∈ S then sj−1 cannot have out-type A. If j ∈ J and sj ∈ S then sj+1 cannot have in-type B. Finally, if j − 1 ∈ J and sj ∈ S then sj−1 cannot have out-type B. Let q+ := |S+ | and define q− and q− similarly. Let Ȳ := V (D) \ Y = A ∪ B and X := X ′ ∪X ′′ = Y \ S. Consider any pair a, b with a ∈ A and b ∈ B. Then − 1 ≤ |N+(a)| ≤ q− + |N+ (a)|+ |N+ − 1 ≤ |N−(b)| ≤ q+ + |N− (b)|+ |N− (b)|. Since N+ (a)∩N− (b) = ∅ (as D does not contain an A-B edge) and a, b /∈ N+ (b) we have (a)|+ |N− (b)| ≤ |Ȳ | − 2 = n− |X| − k − 2. Adding (5) and (6) together now gives − 2 ≤ q− + |N+ (a)|+ |N− (b)|+ n− |X| − k − 2. Hence (a)∩N− (b)| ≥ |N+ (a)|+|N− (b)|−|X| ≥ 2 −n+k−q− ≥ 2k−q− Similarly, using Claim 4, one can show that (8) |N− (a) ∩N+ (b)| ≥ 2k − q+ Consider any j ∈ J . Recall that by Claim 3 we have that either sj has out- type A and sj+1 has in-type B or sj has out-type B and sj+1 has in-type A. Let JAB denote the set of all those indices j ∈ J for which the former holds and let JBA be the set of all those j ∈ J for which the latter holds. Our next aim is to estimate jAB := |JAB | and jBA := |JBA|. Note that Claim 6 implies that if sj ∈ S then j /∈ JAB . As sk /∈ S and k /∈ J , this shows that jAB ≤ k − 1− |S \ {sk}| = k − 1− |S | = k − 1− q+ Also, if sj ∈ S then j − 1 /∈ JAB by Claim 6. As s1 /∈ S , this shows that jAB ≤ k − 1− |S \ {s1}| = k − 1− |S | = k − 1− q− Adding these two inequalites gives (9) jAB ≤ k − 1− In order to give an upper bound for jBA, note that if sj ∈ S then j /∈ JBA by Claim 6. Thus jBA ≤ k − 1− |S \ {sk}| ≤ k − q 10 DANIELA KÜHN AND DERYK OSTHUS Also, if sj ∈ S then j − 1 /∈ JBA by Claim 6. Thus jBA ≤ k − 1− |S \ {s1}| ≤ k − q Adding these two inequalites gives (10) jBA ≤ k − Our next aim is to show that D contains a (S, J ∪{k})-system. This will complete the proof of Theorem 2 since it contradicts the choice of our (S, J, T )-system. Pick distinct vertices a0 ∈ A, aj ∈ N (sj) for all j ∈ JAB , a j ∈ N (sj+1) for all j ∈ JBA, b0 ∈ B, bj ∈ N (sj+1) for all j ∈ JAB and b j ∈ N (sj) for all j ∈ JBA. Choose a vertex x0 ∈ N (a0) ∩ N (b0) and link sk to s1 by the path Qk := ska0x0b0s1. (This can be done since the right hand side of (7) is at least 2.) To find the other paths, we distinguish two cases. Case 1. jBA ≤ jAB For all j ∈ JBA we pick a vertex xj ∈ N (a′j) ∩N (b′j) such that all these xj are pairwise distinct and distinct from x0. Inequalities (8) and (10) together imply that this can be done. Inequality (7) together with the fact that 2k − q− − 1− jBA ≥ 2jAB + 1− jBA ≥ jAB + 1, implies that for all j ∈ JAB we can now pick a vertex xj ∈ N (aj) ∩ N such that x0 and all the xj (j ∈ J) are pairwise distinct. If j ∈ JAB we link sj to sj+1 by the path Qj := sjajxjbjsj+1. If j ∈ JBA we link sj to sj+1 by the path Qj := sjb jsj+1. The paths Qj (j ∈ J) and Qk are internally disjoint and have length 4, so they form an (S, J ∪ {k})-system, as required. Case 2. jBA > jAB We proceed similiarly as in Case 1, but this time we choose the vertices xj ∈ (aj) ∩N (bj) for all j ∈ JAB first. As 2k − q+ − 1− jAB ≥ 2jBA − 1− jAB > jBA − 1, inequality (8) implies that we can then pick the vertices xj ∈ N (a′j) ∩ N (b′j) for all j ∈ JBA. The paths Qj (j ∈ J) and Qk are then defined as before. This completes the proof of Theorem 2. Note that throughout the proof, the paths we constructed always had length at most 6 (the only case where they had length exactly 6 was in the proof of Claim 4). This means that the proof can easily be translated into polynomial algorithm so that the exponent of the running time does not depend on k: We simply start with any (S, J, T )-system with J ⊆ I. Now we go through the steps of the proof and find a ‘better’ (S, J ′, T ′)-system with J ′ ⊆ I. Claim 1 implies that for fixed k we only need to do this a bounded number of times. Since the paths we need have length at most 6 and there are only a bounded number of cases to consider in the proof, it is clear that one can find the better system in polynomial time with exponent independent of k. Altogether this means that the problem of finding a cycle encountering a given sequence of k vertices is fixed parameter tractable for digraphs whose minimum degree satisfies the conditon in Theorem 2 (where k is LINKEDNESS AND ORDERED CYCLES IN DIGRAPHS 11 the fixed parameter). The same applies to the problem of linking ℓ given pairs of vertices. In general, even the problem of deciding whether a digraph is 2-linked is already NP-complete [10]. For a survey on fixed parameter tractable digraph problems, see [8]. 5. Acknowledgement We are grateful to Andrew Young for reading through the manuscript. References [1] J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications, Springer, 2000. [2] B. Bollobás and A. Thomason, Highly linked graphs, Combinatorica 16 (1996), 313–320. [3] G. Chen, R.J. Faudree, R.J. Gould, L. Lesniak and M.S. Jacobson, Linear forests and ordered cycles, Discussiones Mathematicae – Graph theory 24 (2004), 47–54. [4] Y. Egawa, R. Faudree, E. Györi, Y. Ishigami, R. Schelp and H. Wang, Vertex-disjoint cycles containing specified edges, Graphs and Combinatorics 16 (2000), 81–92. [5] M. Ferrara, R. Gould, G. Tansey and T. Whalen, On H-linked graphs, Graphs and Combi- natorics 22 (2006), 217–224. [6] A. Ghouila-Houri, Une condition suffisante d’existence d’un circuit Hamiltonien, C. R. Acad. Sci. Paris 251 (1960), 495–497. [7] R. Gould, A.V. Kostochka and G. Yu, On minimum degree implying that a graph is H-linked, SIAM J. on Discrete Math., to appear. [8] G. Gutin and A. Yeo, Some parameterized problems on digraphs, preprint 2007. [9] M.C. Heydemann and D. Sotteau, About some cyclic properties in digraphs, J. Combinatorial Theory B 38 (1985), 261–278. [10] S. Fortune, J.E. Hopcroft, J. Wyllie, The directed subgraph homeomorphism problem, The- oretical Computer Science 10 (1980), 111–121. [11] H.A. Jung, Eine Verallgemeinerung des k-fachen Zusammenhangs für Graphen, Math. An- nalen 187 (1970), 95–103. [12] K. Kawarabayashi, A. Kostochka and G. Yu, On sufficient degree conditions for a graph to be k-linked, Combinatorics, Probability and Computing, to appear. [13] H. Kierstead, G. Sarközy and S. Selkow, On k-ordered Hamiltonian graphs, J. Graph The- ory 32 (1999), 17–25. [14] A. Kostochka and G. Yu, An extremal problem for H-linked graphs, J. Graph Theory 50 (2005), 321–339. [15] A. Kostochka and G. Yu, Minimum degree conditions for H-linked graphs, Discrete Applied Math., to appear. [16] Y. Manoussakis, k-linked and k-cyclic digraphs, J. Combinatorial Theory B 48 (1990), 216– [17] R. Thomas and P. Wollan, An improved extremal function for graph linkages, European Journal of Combinatorics 26 (2005), 309–324. [18] C. Thomassen, Note on highly connected non-2-linked digraphs, Combinatorica 11 (1991), 393–395. Daniela Kühn, Deryk Osthus School of Mathematics University of Birmingham Edgbaston Birmingham B15 2TT E-mail addresses: {kuehn,osthus}@maths.bham.ac.uk 1. Introduction 2. Further work and open problems 3. Notation and extremal examples 4. Proof of Theorem ?? 5. Acknowledgement References

0704.0212

Curvature and isocurvature perturbations in two-field inflation

Curvature and iso urvature perturbations in two-�eld in�ation Z. Lalak , D. Langlois , S. Pokorski , K. Turzy«ski Institute of Theoreti al Physi s, Warsaw University, ul. Ho»a 69, 00-681 Warsaw, Poland; APC (Astroparti ules et Cosmologie, CNRS-UMR 7164), Université Paris 7 10, rue Ali e Domon et Léonie Duquet, 75205 Paris Cedex 13, Fran e; GReCO, Institut d'Astrophysique de Paris, CNRS, 98bis Boulevard Arago, 75014 Paris, Fran e; Physi s Department, University of Mi higan, 450 Chur h St., Ann Arbor, MI-48109, USA O tober 30, 2018 Abstra t We study osmologi al perturbations in two-�eld in�ation, allowing for non- standard kineti terms. We al ulate analyti ally the spe tra of urvature and iso urvature modes at Hubble rossing, up to �rst order in the slow-roll parameters. We also ompute numeri ally the evolution of the urvature and iso urvature modes from well within the Hubble radius until the end of in�ation. We show expli itly for a few examples, in luding the re ently proposed model of `roulette' in�ation, how iso urvature perturbations a�e t signi� antly the urvature perturbation between Hubble rossing and the end of in�ation. http://arxiv.org/abs/0704.0212v2 1 Introdu tion In�ation provides a simple and elegant s enario for the early Universe (see e.g. [1℄ for a re ent textbook presentation). Although single �eld in�ation models are perfe tly om- patible with the present osmologi al data, many early universe models based on high energy physi s, in parti ular derived from supergravity or string theory, usually involve many s alar �elds whi h an have non-standard kineti terms. This is why multi-�eld in�ationary s enarios, where several s alar �elds play a dynami al role during in�ation, have re eived some attention in the literature (see e.g. [2℄-[13℄). However, ex ept for a few spe i� models, the predi tions for the spe tra of primordial perturbations are, in general, a nontrivial task, in ontrast with single-�eld models. The main reason is that the urvature (or adiabati ) perturbation, whi h is generated during in�ation and eventually observed, an evolve on super-Hubble s ales in multi-�eld in�ation whereas it remains frozen in single-�eld in�ation. This is due to the presen e of additional perturbation modes, often alled iso urvature (or entropy) modes, orrespond- ing to relative perturbations between the various s alar �elds, whi h a t as a sour e term in the evolution equation for the urvature perturbation. This property was �rst pointed out in [6℄ in the ontext of Brans-Di ke in�ation. This phenomenon o urs during in�a- tion and a�e ts the �nal urvature perturbation at the end of in�ation, independently whether iso urvature modes survive or not after in�ation. The purpose of the present work is to study in detail how the iso urvature perturba- tions, present during in�ation, a�e t the urvature perturbations, both at Hubble rossing and in the subsequent evolution on super-Hubble s ales. Sin e our intention is to stress some qualitative properties spe i� to multi-�eld in�ation, we have hosen to restri t our study to the ase of two s alar �elds. Moreover, we onsider models where a non-standard kineti term is allowed for one of the s alar �elds. This in ludes, in parti ular, s enarios motivated by supergravity and string theory, whi h have been re ently proposed (see, e.g., [14℄-[18℄). The produ tion of adiabati and iso urvature modes for two-�eld in�ation with a generi potential, in the slow-roll approximation, was studied in [19℄ where a de ompo- sition into adiabati and iso urvature modes was introdu ed. Models with non-standard kineti terms for in�atons have been studied in the slow-roll approximation in [9℄ and [11℄, and the adiabati -iso urvature de omposition te hnique of [19℄ was later extended to su h two-�eld models in [20℄ and [21℄. Very re ently, two-�eld in�ation with standard kineti terms was investigated in [22℄ at next-to-leading order orre tion in a slow-roll expansion and it was also shown that the adiabati and iso urvature modes at Hubble rossing are orrelated at �rst order in slow-roll parameters. In parallel to these analyti al studies, a numeri al study of the evolution of adiabati and iso urvature was presented in [23℄ and [24℄. In the present work, we extend the previous analyses in the following dire tions. First, we present a detailed analysis of the orrelation of adiabati and iso urvature just after Hubble rossing, both analyti ally and numeri ally. This orrelation was in general sup- posed to vanish in most previous works, ex ept in the numeri al study [23℄ and the analyti al work [22℄, both in the ontext of anoni al kineti terms. Here, we extend the analysis with non-standard kineti terms. We ompute analyti ally the spe tra and orrelation at Hubble rossing in the slow-roll approximation, by taking spe ial are of the time-dependen e shortly after Hubble rossing. Se ond, we study numeri ally the whole evolution of adiabati and iso urvature per- turbations from within the Hubble radius until the end of in�ation. This allows us to go beyond the slow-roll approximation whi h is needed to derive analyti al results. Our nu- meri al study enables us to see pre isely how iso urvature perturbations an be transferred into adiabati perturbations during the in�ationary phase depending on the ba kground traje tory in �eld spa e. We illustrate this behaviour by studying numeri ally three mod- els. The last one is the so- alled 'roulette' in�ation model, whi h has been proposed re ently [14℄. The plan of the paper is the following. The next se tion presents the lass of models we onsider and gives the homogeneous equations of motion as well as the equations governing the perturbations. The third se tion is devoted to the study of the perturbations from deep inside the Hubble radius until a few e-folds after Hubble rossing. We then dis uss, in se tion 4, analyti al methods to determine the evolution of the perturbations on super-Hubble s ales. Se tion 5 is devoted to the numeri al study of the evolution of the perturbations, whi h is ompared with the analyti al estimates of the previous se tions. We �nally draw our on lusions in the last se tion. 2 The model In this paper, we study models with two s alar �elds, in whi h one of the s alar �elds has a non-standard kineti term, des ribed by an a tion of the form (∂µφ)(∂ µφ)− e 2b(φ) (∂µχ)(∂ µχ)− V (φ, χ) , (1) where MP is the redu ed Plan k mass, MP ≡ (8πG)−1/2. This type of a tion usually appears when χ orresponds to an axioni omponent. It is also motivated by generalized Einstein theories [6, 11℄. When b(φ) = 0 one re overs standard kineti terms for the two �elds. In this se tion, we give the equations of motion for the homogeneous �elds and then for the linear perturbations, following the results (and notation) of [20℄ and [21℄, where the same type of models was onsidered. 2.1 Homogeneous equations Let us start with the homogeneous equations of motion. We assume a spatially �at FLRW (Friedmann-Lemaître-Robertson-Walker) geometry, with metri ds2 = −dt2 + a(t)2dx2, (2) where t is the osmi time. One an also de�ne the omoving time τ = dt/a(t). The equations of motion for the s ale fa tor and the homogeneous �elds read φ̈+ 3Hφ̇+ Vφ = bφe 2bχ̇2 , (3) χ̈+ (3H + 2bφφ̇)χ̇+ e −2b Vχ = 0 , (4) φ̇2 + χ̇2 + V , (5) Ḣ = − 1 φ̇2 + e2bχ̇2 , (6) where H ≡ ȧ/a and a dot stands for a derivative with respe t to the osmi time t and a subs ript index φ or χ denotes a derivative with respe t to the orresponding �eld. It is also useful to introdu e the following slow-roll parameters ǫφφ = 2M2PH , ǫφχ = e b φ̇χ̇ 2M2PH , ǫχχ = e 2b χ̇ 2M2PH , (7) ηIJ = ǫ = ǫφφ + ǫχχ = − . (9) 2.2 Linear perturbations We now dis uss the linear perturbations of our model (one an �nd a detailed presentation of the theory of osmologi al perturbations in e.g. [1, 25, 26℄ and a pedagogi al introdu - tion in e.g. [27℄). For simpli ity, we shall dire tly work in the longitudinal gauge. In the absen e of anistropi stress (the o�-diagonal spatial omponents of the stress-energy ten- sor), whi h is the ase when matter onsists of s alar �elds, the metri in the longitudinal gauge is of the form ds2 = −(1 + 2Φ)dt2 + a2(1− 2Φ)dx2 . (10) where only s alar perturbations are taken into a ount. We now de ompose the s alar �elds into their homogeneous (ba kground) parts and the perturbations: φ(t,x) = φ(t) + δφ(t,x) and χ(t,x) = χ(t) + δχ(t,x) . (11) We shall work with the Fourier omponents of the perturbations, δφk(t) and δχk(t), routinely omitting the subs ript k to shorten the expressions. The perturbed Klein- Gordon equations read δ̈φ + 3H ˙δφ+ + Vφφ − (bφφ + 2b2φ)χ̇2e2b δφ+ Vφχδχ− 2bφe2bχ̇ ˙δχ = 4φ̇Φ̇− 2VφΦ (12) δ̈χ + (3H + 2bφφ̇) ˙δχ+ + e−2bVχχ δχ+ 2bφχ̇ ˙δφ+ e−2b (Vχφ − 2bφVχ) + 2bφφφ̇χ̇ δφ = 4χ̇Φ̇− 2e−2bVχΦ . (13) The energy and the momentum onstraints, given by Einstein equations, are, respe - tively: 3H(Φ̇ +HΦ) + ḢΦ+ Φ = − 1 φ̇ ˙δφ+ e2bχ̇ δ̇χ+ bφe 2bχ̇2δφ+ Vφδφ+ Vχδχ Φ̇ +HΦ = φ̇ δϕ+ e2bχ̇ δχ . (15) It is onvenient, instead of using the perturbations δφ and δχ, de�ned here in the longitudinal gauge, to introdu e the so- alled gauge-invariant Mukhanov-Sasaki variables: Qφ ≡ δφ+ Φ and Qχ ≡ δχ+ Φ , (16) whi h an be identi�ed with the s alar �eld perturbations in the �at gauge. Substituting (16) into (12)-(13) and using the ba kground equations of motion as well as the energy and momentum onstraints, one �nds Q̈φ + 3HQ̇φ − 2e2bbφ χ̇ Q̇χ + + Cφφ Qφ + CφχQχ = 0 (17) Q̈χ + 3HQ̇χ + 2bφ φ̇ Q̇χ + 2bφ χ̇ Q̇φ + + Cχχ Qχ + CχφQφ = 0 , (18) where we have de�ned the following ba kground-dependent oe� ients: Cφφ = −2e2bb2φχ̇2 + e2bφ̇2χ̇2 2M4PH 2M4PH − e2bbφφχ̇2 + 2φ̇Vφ + Vφφ (19) Cφχ = 3e2bφ̇χ̇ 4bφ̇χ̇3 2M4PH 2bφ̇3χ̇ 2M4PH e2bχ̇Vφ + Vφχ (20) Cχχ = 3e2bχ̇2 4bχ̇4 2M4PH 2bφ̇2χ̇2 2M4PH 2χ̇Vχ + e−2bVχχ (21) Cχφ = 3φ̇χ̇ e2bφ̇χ̇3 2M4PH φ̇3χ̇ 2M4PH + 2bφφφ̇χ̇− 2e−2bbφVχ + e−2bφ̇Vχ + e−2bVφχ (22) The two equations (17) and (18) form a losed system for the two gauge-invariant quan- tities Qφ and Qχ. 2.3 De omposition into adiabati and entropy omponents As originally proposed in [19℄, in order to fa ilitate the interpretation of the evolution of osmologi al perturbations, it an be useful to de ompose the s alar �eld perturbations along the two dire tions respe tively parallel and orthogonal to the homogeneous traje - tory in �eld spa e. The proje tion parallel to the traje tory is usually alled the adiabati , or urvature, omponent while the orthogonal proje tion orresponds to the entropy, or iso urvature, omponent. Note that there was a semanti shift in the terminology sin e one used to all adiabati and entropy modes during in�ation the two parti ular solutions for the perturbations that would mat h after in�ation, respe tively, to the adiabati and iso urvature modes de�ned in the radiation era. This terminology is used for example in the papers on double in�ation su h that [5℄ and [10℄. This de omposition into instantaneous adiabati and entropy omponents, introdu ed in [19℄, has re ently been extended [28℄ to fully nonlinear perturbations in the ontext of the ovariant nonlinear formalism introdu ed in [29, 30℄. Here, we onsider only the de omposition at the linear level, but sin e we allow for non-standard kineti terms, we will need to generalize the equations to su h a ase, as was done in [20℄. Let us re all here the main results. The essential idea is to introdu e the linear ombinations δσ ≡ cos θ δφ+ sin θ eb δχ and δs ≡ − sin θ δφ+ cos θ eb δχ , (23) where cos θ ≡ φ̇ , sin θ ≡ χ̇ e with σ̇ ≡ φ̇2 + e2bχ̇2 . (24) The notations σ̇ and δσ are just used for onvenien e; they do not refer to any s alar �eld Instead of δσ, it is in fa t more onvenient to work dire tly with the Mukhanov-Sasaki variables and therefore to de�ne Qσ ≡ cos θ Qφ + sin θ ebQχ and δs ≡ − sin θ Qφ + cos θ ebQχ , (25) by noting that Qσ ≡ δσ + Φ. (26) In the so- alled omoving gauge, the perturbation Qσ is dire tly related to the three- dimensional urvature of the onstant time spa e-like sli es. This gives the gauge-invariant quantity refered to as the omoving urvature perturbation: Qσ. (27) The perturbation δs, alled the iso urvature perturbation, is automati ally gauge-invariant. It is sometimes onvenient, by analogy with the urvature perturbation, to introdu e a renormalized entropy perturbation whi h is de�ned as S ≡ H δs. (28) In �eld spa e, Qσ orresponds to perturbations parallel to the velo ity ve tor (φ̇, e bχ̇), while δs to the orthogonal ones. Introdu ing the adiabati and entropy �ve tors� in �eld spa e, respe tively EIσ = cos θ, e−b sin θ , EIs = − sin θ, e−b cos θ , I = {φ, χ} , (29) one an de�ne various derivatives of the potential with respe t to the adiabati and entropy dire tions. Assuming an impli it summation on the indi es I (and J), the �rst order derivatives are de�ned as Vσ = E σVI , Vs = E sVI , (30) whereas the se ond order derivatives are Vσσ = E σVIJ , Vσs = E s VIJ , Vss = E s VIJ . (31) By ombining the two Klein-Gordon equations for the ba kground �elds, (3) and (4), one gets the ba kground equations of motion along the adiabati and entropy dire tions, respe tively, σ̈ + 3Hσ̇ + Vσ = 0, (32) θ̇ = −Vs − bφσ̇ sin θ, (33) while the equations of motion for the perturbations read: Q̈σ + 3HQ̇σ + + Cσσ δ̇s+ Cσs δs = 0 (34) δ̈s+ 3Hδ̇s+ + Css δs− 2Vs Q̇σ + CsσQσ = 0 , (35) Cσσ = Vσσ − 2M4PH s2θcθVσ + (c θ + 1)sθVs Cσs = 6H 2VσVs + 2Vσs + + 2bφ(s θVσ − c3θVs) (37) Css = Vss − + bφ(1 + s θ)cθVσ + bφc θsθVs − σ̇2(bφφ + b2φ) (38) Csσ = −6H − 2VσVs where sθ ≡ sin θ and cθ ≡ cos θ. The above system onsists (34-35) of two oupled se ond order di�erential equations involving only Qσ and δs. In order to relate these variables to the metri perturbation Φ de�ned in the longitudinal gauge, it is useful to use the Poisson-like onstraint, whi h follows from the energy and momentum onstraints (14) and (15), Φ = − 1 ǫm (40) where ǫm is the omoving energy density and an be expressed as ǫm = σ̇Q̇σ + σ̇Qσ + VσQσ + 2Vsδs. (41) 2.4 Perturbation spe tra The in�ationary observables are ustomarily expressed in terms of power spe tra and orrelation fun tions. Given their origin as quantum �u tuations, the perturbations an be represented as random variables. We introdu e the power spe tra of the adiabati and entropy perturbations, respe tively 〈Qσ∗kQσk′〉 = PQσ(k)δ(k− k′), 〈δs∗k δsk′〉 = Pδs(k)δ(k− k′), (42) as well as the orrelation spe trum 〈Qσ∗k δsk′〉 = CQσ δs(k)δ(k− k′). (43) 3 Evolution of perturbations inside the Hubble radius In order to study the generation of perturbations from va uum �u tuations, we start, for a given omoving wave number k, at an instant ti (or τi) during in�ation when the physi al wave number k/a is mu h bigger than the Hubble parameter H . Our initial onditions are given, as usual, by the Minkowski-like va uum at τi, Qσ(τi) ≃ e−ıkτi a(τi) and δs(τi) ≃ e−ıkτi a(τi) for initial adiabati and iso urvature �u tuations, respe tively. These two initial �u tu- ations are statisti ally independent be ause the orresponding equations of motion are de oupled in the limit k ≫ aH . Although the adiabati and entropy �u tuations are initially, i.e. deep inside the Hub- ble radius, statisti ally independent, this is, in general, no longer the ase at Hubble rossing. In the ontext of two-�eld in�ation with anoni al kineti terms, this point has been stressed in the numeri al analysis of [23℄ and studied analyti ally in [22℄. In the following, we extend the analysis of [22℄ to non- anoni al kineti terms. 3.1 Equations in the slow-roll approximation We start with the perturbations Qσ and δs, whose dynami s is des ribed by eqs. (34) and (35). Using the onformal time τ and introdu ing the variables uσ = aQσ, us = a δs, (45) these equations an be rewritten in the form u′′σ + au′s + k2 − a + a2Cσσ a′ + a2Cσs us = 0, (46) u′′s − au′σ + + a2Css a′ + a2Csσ uσ = 0, (47) where the four oe� ients CIJ are given in eqs. (36)-(39) and a prime denotes a derivative with respe t to the onformal time τ . Let us now dis uss the slow-roll approximation. The only di�eren e with respe t to the ase with anoni al kineti terms will arise from some of the terms depending on the derivatives of b. In the slow-roll approximation, one an use the relation = Hησs − bφσ̇s3θ , (48) and the various oe� ients in the above system of equations simplify to yield + k2 − 2 + 3ǫ 1+ 2E = 0 (49) where the matri es E and M are given by 0 −ησs ησs 0 0 ξs3θ −ξs3θ 0 −6ǫ+ 3ησσ 4ησs 2ησs 3ηss 3ξs2θcθ −4ξs3θ −2ξs3θ −3ξcθ(1 + s2θ) 2bφMP ǫ. (52) In eqs. (50) and (51), we have kept only the terms linear in bφ, i.e. proportional to ξ, and negle ted the terms quadrati in bφ as well as the terms proportional to bφφ. We thus treat ξ on the same footing as the other slow-roll parameters. In order to emphasize the di�eren e between generalized kineti terms and anoni al kineti terms, we have however separated the terms proportional ξ from the others. The system of equations (49) that we have obtained is of the form u′′ + 2Lu′ +Qu = 0 . (53) The matrix oe� ient for the �rst order time derivative is 2L = 2E/τ , where E is an an- tisymmetri matrix, linear in the slow-roll parameters. Let us introdu e a time-dependent orthogonal matrix R whi h satis�es R′ = −LR. Note that this is possible only if L is an antisymmetri matrix. Reexpressing the above equation (53) in terms of a new matrix ve tor v, de�ned by u = Rv, one an eliminate the terms proportional to the �rst order time derivative and we obtain the following equation v′′ +R−1 −L2 − L′ +Q Rv = 0. (54) At linear order in the slow-roll parameters, one �nds − L2 − L′ ≃ E. (55) Therefore, apart from the trivial part proportional to the identity matrix, the ombination −L2 − L′ +Q ontains (E+M) = −2ǫ+ ησσ + ξs2θcθ ησs − ξs3θ ησs − ξs3θ ηss − ξcθ(1 + s2θ) , (56) whi h is a symmetri matrix. We now assume that the slow-roll parameters vary su� iently slowly during the few e-folds when the given s ale rosses out the Hubble radius. We thus repla e the time- dependent matrix on the right hand side of (56) by the same matrix evaluated at Hubble rossing, i.e. for k = aH , and the only remaining time dependen e appears in the global oe� ient 3/τ 2. One an now diagonalize this matrix by introdu ing the time-independent rotation matrix R̃∗ = cosΘ∗ − sin Θ∗ sinΘ∗ cosΘ∗ , (57) so that ∗ (M+ E) R̃∗ = Diag(λ̃1, λ̃2). (58) In parti ular, one an easily ompute the following linear ombinations, whi h will be useful later: λ̃1 + λ̃2 = 3 (ησσ + ηss − 2ǫ− ξcθ) , (59) (λ̃1 − λ̃2) sin 2Θ∗ = 6 ησs − ξs3θ , (60) (λ̃1 − λ̃2) cos 2Θ∗ = 3 ησσ − ηss − 2ǫ+ ξcθ(1 + 2s2θ) , (61) where the right hand sides of the three above equations are evaluated at k = aH . Similarly, the rotation matrix R is slowly varying per efold, sin e (dR/d ln a)RT = E, where E is linear in slow-roll parameters. Around Hubble rossing, one an thus repla e R by its value R∗ at Hubble rossing. By introdu ing w = R̃−1∗ R∗v, (62) the system of equations an be written as two independent equations of the form w′′A + (2 + 3λA) wA = 0, (A = 1, 2) (63) λA = ǫ− λ̃A. (64) De�ning + 3λA (65) the solution of (63) with the proper asymptoti behaviour an be written as ei(µA+1/2)π/2 −τH(1)µA (−kτ)eA(k), (66) where H µ is the Hankel fun tion of the �rst kind of order µ and the eA are two indepen- dent normalized Gaussian random variables so that 〈eA(k)〉 = 0, 〈eA(k)e∗B(k′)〉 = δABδ(3)(k − k′). (67) Using the independen e of the variables w1 and w2, one an express the orrelations for the variables uσ and us around Hubble rossing time as a2〈Q†σQσ〉 = cos2Θ∗〈w 1w1〉+ sin2Θ∗〈w 2w2〉 (68) a2〈δs†Qσ〉 = sin 2Θ∗ 〈w†1w1〉 − 〈w a2〈δs†δs〉 = sin2Θ〈w†1w1〉+ cos2Θ〈w 2w2〉 (70) where one an substitute 〈w†AwA〉 = (−τ)|H(1)µA (−kτ)| 2 ≡ 1 (kτ)2 FA(−kτ). (71) This yields PQσ = (1− 2ǫ∗) cos2Θ∗ F1(−kτ) + sin2Θ∗ F2(−kτ) CQσδs = (1− 2ǫ∗) sin 2Θ∗ [F1(−kτ)− F2(−kτ)] (73) Pδs = (1− 2ǫ∗) sin2 Θ∗ F1(−kτ) + cos2Θ∗ F2(−kτ) , (74) where we have used a ≃ −1 + ǫ∗ . (75) At this stage, it is worth noting that our derivation is still valid if the parameter ηss, whi h orresponds to the urvature of the potential along the dire tion orthogonal to the �eld traje tory, is not small. In this ase, the entropy �u tuations are e�e tively suppressed and only adiabati �u tuations are generated. As far as the perturbations are on erned, this parti ular situation is similar to the single �eld ase. A further simpli� ation o urs when λA ≪ 1, in whi h ase µA ≃ 32+λA. The fun tions FA(x) an be expanded as FA(x) = x3|H3/2(x)|2 (1 + 2λAf(x)) = (1 + x2) (1 + 2λAf(x)) , (76) f(x) = Re µ (x) µ=3/2  . (77) Using the relations (59-61) and (64), we �nally get for the urvature and entropy perturbations de�ned in (27) and (28), the following expressions 2πσ̇∗ (1 + k2τ 2) 1− 2ǫ∗ + 6ǫ∗ − 2ησσ∗ − 2ξ∗s2θ∗cθ∗ CRS = 2πσ̇∗ (1 + k2τ 2) θ∗ − 2ησs∗ 2πσ̇∗ (1 + k2τ 2) 1− 2ǫ∗ + 2ǫ∗ − 2ηss∗ + 2ξ∗(1 + s2θ∗)cθ∗ .(80) Let us omment these results. First, one an verify that, for anoni al kineti terms (i.e. ξ = 0), we re over the results of [22℄ if we repla e the fa tor (1 + k2τ 2) by 1 and the fun tion f(−kτ) by the number C = 2 − ln2 − γ ≃ 0.7296, where γ ≃ 0.5772 is the Euler-Mas heroni onstant. Our �nal expression depends expli itly on τ and allows us a more pre ise estimate of the spe tra around the time of Hubble rossing. As we will see expli itly later, there are some in�ationary s enarios where the amplitude of the urvature perturbation spe trum evolves very qui kly after Hubble rossing and never rea hes its asymptoti value ( orresponding to the limit kτ → 0). In these ases, one needs to evaluate more pre isely the amplitude around Hubble rossing, whi h our more detailed formula enables to do. Another di�eren e with [22℄ is that we derived the adiabati and iso urvature spe tra by working dire tly with the equations for the adiabati and entropy omponents, instead of working with the initial s alar �elds. 4 Evolution of perturbations on super-Hubble s ales When the iso urvature modes are suppressed, for instan e if the e�e tive mass along the iso urvature dire tion is large with respe t to the Hubble parameter the �nal adiabati spe trum an be omputed simply by taking the usual single-�eld result applied to the adiabati dire tion: PsfR(k) ≃ 4π2σ̇2 8π2Lkin , (81) where all the quantities are evaluated at Hubble rossing. The simpli� ation works be- ause, in this parti ular situation where iso urvature �u tuations are absent, the urvature perturbation remains frozen on super-Hubble s ales. If iso urvature modes are present however, they will a�e t the super-Hubble evolution of the adiabati perturbations be ause they will a t as a sour e term on the right hand side of the equation governing the evolution of the urvature perturbation [7℄ (and [28℄ for the non-linear generalisation). In order to obtain the �nal power spe tra and orrelations, and to ompare the pre- di tions of a multi-in�aton model with observations, one must then solve the oupled system of di�erential equations (34-35). In general, a numeri al approa h is ne essary and will be onsidered in the next se tion. In some parti ular ases, within the slow-roll approximation, one an solve analyti ally the equations of motion on super-Hubble s ales. We now dis uss these ases in the rest of this se tion. Following [21℄, we an then write eqs. (34) and (35) in the slow-roll approximation as: Q̇σ ≃ AHQσ +BHδs and δ̇s ≃ DHδs , (82) where: A = −ησσ + 2ǫ− ξcθs2θ (83) B = −2ησs + 2ξs3θ ≃ 2 − 2ξsθ (84) D = −ηss + ξcθ(1 + s2θ) . (85) Qualitatively, it is lear that if the iso urvature perturbations do not de ay very fast, there is a strong intera tion between the adiabati and iso urvature perturbations, whenever the oe� ient B be omes large, i.e. when the lassi al traje tory makes a sharp turn in the �eld spa e or when ξ is relatively large and in�ation is driven at least partially by the �eld χ (sθ 6= 0). For onstant A,B,D, the equations (82) an be solved expli itly to give Qσ(N) ≃ eANQσ∗ + (eDN − eAN )δs∗ and δs(N) ≃ eDNδs∗ (86) where N stands for the number of efolds after Hubble rossing. Taking into a ount that (H/σ̇) ≃ (H∗/σ̇∗)e−AN , we an express the power spe tra and orrelations as: P(a)R (N) ≃ P̄R∗ + P̄S∗ eγN − 1 + 2C̄RS∗ eγN − 1 C(a)RS(N) ≃ C̄RS∗ e γN + P̄S∗ eγN − 1 P(a)S (N) ≃ P̄S∗ e 2γ∗N , (89) where γ = D−A. The quantities P̄R∗, C̄RS∗ and P̄S∗ orrespond to the asymptoti limit, i.e. when kτ → 0, of the expressions (78-80). The use of this approximation is in pra ti e rather limited be ause it relies on the assumption that the slow-roll parameters are time-independent between Hubble rossing and the �nal time. In most ases, this approximation, whi h we all onstant slow- roll approximation, holds only for a few e-folds and breaks down long before the end of in�ation. In some simple in�ationary models, there exists an analyti al approa h to ompute analyti ally the evolution of the perturbations on super-Hubble s ales. This is the ase for double in�ation with anoni al kineti terms (i.e. b = 0) and potential V (φ, χ) = 2 , (90) where the equations of motion for the metri perturbation Φ and the two s alar �eld perturbations an be integrated expli itly in the slow-roll approximation and on super- Hubble s ales [5℄ (this an be seen as a parti ular ase within a more general ontext dis ussed in [9℄). One �nds Φ ≃ −C1Ḣ + 2C3 (m2χ −m2φ)m2χχ2m2φφ2 3(m2χχ 2 +m2φφ , (91) − 2C3 Hm2χχ 2 +m2φφ + 2C3 Hm2φφ 2 +m2φφ , (92) where C1 and C3 are time-independent onstants of integration. The urvature and iso urvature perturbations during in�ation are respe tively R = Φ +H χ̇ δχ + φ̇ δφ χ̇2 + φ̇2 , S = H φ̇ δχ− χ̇ δφ χ̇2 + φ̇2 . (93) By plugging the solutions (91-92) into the above expressions, one obtains the expli it evolution of the adiabati and iso urvature perturbations, knowing that the ba kground evolution is given by χ = 2MP s sinα, φ = 2MP s cosα, s = s0 (sinα)2/(R (cosα)2R 2/(R2−1) where s = − ln(a/ae) is the number of e-folds between a given instant and the end of in�ation, and R ≡ mχ/mφ. Note that the iso urvature perturbation S that we have de�ned during in�ation, fol- lowing other works, is proportional but does not oin ide with the iso urvature pertur- bation Srad de�ned during the radiation era. In the s enario dis ussed in [10℄, where the heavy �eld χ de ays into dark matter, the iso urvature perturbation Srad = δcdm−(3/4)δγ is related to the perturbations during in�ation via the relation Srad = − m2χC3 = − . (95) Another method to al ulate the �nal urvature perturbations is the so- alled δN formalism [31, 32, 33℄. In pra ti e however, this method requires the expression of the number of e-folds as a fun tion of the initial values of the s alar �elds and ex ept in a few simple ases where this expression an be determined analyti ally, a numeri al approa h is also needed in the general ase. Moreover, the approa h we have adopted allows to follow not only the evolution of the urvature perturbation but also that of the iso urvature perturbation. This is important if some iso urvature perturbations survive after the end of in�ation. Their evolution then depends on the details of the pro esses whi h o ur at the end in�ation and after, in parti ular the reheating (or preheating) pro esses, whi h goes beyond the s ope of this study. 5 Numeri al analysis In Se tion 3, we studied the spe tra and orrelations of the perturbations in the vi inity of the Hubble rossing and we obtained analyti approximations (78)-(80). The aim of the present se tion is to onfront these expressions with the result of a numeri al integration of the equations of motion (34)-(35). We would also like to study numeri ally the super-Hubble dynami s of the perturbations, whi h is often the only way to al ulate the in�ationary observables su h as the spe tral index ns with a pre ision required by the present and forth oming observations. 5.1 Numeri al pro edure Our numeri al pro edure, similar to that of [23℄, is the following. In order to take into a ount the statisti al independen e of the adiabati and iso urvature perturbations deep inside the Hubble radius, we integrate eqs. (34)-(35) twi e: �rst with the initial value of Qσ orresponding to the Minkowski-like va uum and δs = 0, then with the initial value of δs orresponding to the Minkowski-like va uum and Qσ = 0. Unless stated otherwise, we impose the initial onditions 8 efolds before the Hubble rossing. The initial onditions also in lude the slow-roll for the ba kground �elds. The evolution pro eeds along a ba kground traje tory whi h provides a su� ient number of efolds before the end of in�ation (50-60, depending on the model). We identify the end of in�ation, at whi h we terminate the evolution, when ǫ = 1. As the out ome of the �rst (se ond) run, we obtain the urvature and entropy perturbations, R1 and S1 (R2 and S2). We then al ulate the spe tra and orrelations as: |R1|2 + |R2|2 |S1|2 + |S2|2 CRS = R†1S1 +R . (98) We shall sometimes des ribe the orrelations using the relative orrelation oe� ient: C̃ = |CRS |√ The value of C̃ lies between 0 and 1, and it indi ates to what extent the �nal urvature perturbations result from the intera tions with the iso urvature perturbations. Figure 1: Examples of lassi al in�ationary traje tories for double in�ation with anoni al kineti terms (left), double in�ation with non- anoni al kineti terms ( enter) and roulette in�ation (right). The details of the models are des ribed in Se tion 5.2. Subsequent tens of efolds are indi ated along the urves. 5.2 Examples of in�ationary models There is an enormous number of examples of in�ationary models. Here we restri t our analysis to just three ases des ribed below. We will use these examples to he k the analyti al results of the pre eding Se tions and to illustrate some generi features in the evolution of adiabati and iso urvature perturbations. 5.2.1 Double in�ation with anoni al kineti terms Double in�ation (with b = 0) is ertainly the most thoroughly studied example of multi- �eld in�ation. It employs the potential V (φ, χ) = 2 . (100) In order to make de�nite al ulations, we set 7mφ = mχ and we hoose the initial ondi- tions φi = χi (later we shall also omment on the ase φi = 50χi), 8 efolds before the s ale we onsider leaves the Hubble radius, after whi h in�ation goes on for about ∼ 60 efolds. The lassi al traje tory in �eld spa e is shown in Figure 1. In our example, the traje tory is strongly bent roughly 35th efolds after the Hubble rossing. As we shall see, it is at this moment when the adiabati perturbations are strongly fed by the iso urvature ones. 5.2.2 Double in�ation with non- anoni al kineti terms In order to on entrate just on the e�e ts due to the non- anoni al nature of the kineti terms, we onsider a very simple generalization of the previous example by taking b(φ) = −φ/MP and mφ = mχ in (100). We hoose the initial onditions so that φi = 0. Then it is almost exlusively the �eld χ whi h slides down to the minimum of the potential during in�ation, but due to non- anoni ality of the kineti terms, the intera tion of χ with φ drives the latter slightly away from zero. The lassi al traje tory in the �eld spa e is shown in Figure 1. 5.2.3 Roulette in�ation Re ently, in�ation in the large volume ompa ti� ation s heme in the type IIB string theory model has been investigated in [14℄ (see also [34℄). In our notation, this model an be e�e tively des ribed by b(φ) = b0 − (101) V (φ, χ) = V0 + V1 ψ(φ)e−2β1ψ(φ) + V2ψ(φ)e −β1ψ(φ) cos(β2χ) , (102) where ψ(φ) = (φ/MP ) and b0, Vi, βi are fun tions of the parameters of the underlying string model. A generi feature of the potential (102) is that it has an in�nite number of minima arranged periodi ally in χ and a plateau for large values of φ, admitting a large variety of in�ationary traje tories, whi h may end at di�erent minima even if they originate from neighboring points in the �eld spa e � hen e the model has been dubbed roulette in�ation. In this work, we adopt the parameter set no. 1 (in Plan k units: b0 = −11; V0 = 9.0 × 10−14; V1 = 3.2 × 10−4; V2 = 1.1 × 10−5; and β1 = 9.4 × 105; β2 = 2π/3) from [14℄ and hoose the parti ular in�ationary traje tory shown in Figure 1. For this traje tory, the fa tor bφMP is rather large, of the order 10 , but the e�e t of the non- anoni al kineti terms is strongly suppressed by a very small value of ǫ on the plateau of the potential. The smallness of ǫ also suppresses the energy s ale of in�ation and one needs a smaller number of efolds than in the models des ribed above. For de�niteness, we assumed that there are ∼ 50 efolds between the moment that the s ale of interest rosses the Hubble radius and the end of in�ation. 5.3 Numeri al results for the perturbations For the three in�ationary models des ribed in Se tion 5.2, we performed the numeri al analysis, as des ribed in Se tion 5.1. Here, we dis uss the out ome for the spe tra and orrelations in Figures 2-4. In the right panel of ea h Figure we plot the evolution of PR and PS , normalized to the single-�eld result PsfR given in eq. (81), as well as the evolution of the orrelation oe� ient C̃ de�ned in (99) and the parameter B, de�ned in (84), whi h is the oupling between the iso urvature and the urvature perturbations. These quantities are plotted as fun tions of the number of efolds N after Hubble rossing. Left panels of ea h Figures 2-4 are basi ally lose-ups of the right ones to the vi inity of the Hubble rossing. There, we plot the evolution of PR, PS and CRS , normalized to the single- �eld result PsfR given in eq. (81). These are shown as fun tions of the variable (k/aH)−1 whi h allows us to ompare dire tly the numeri al results with the predi tions of the eqs. (78)-(80). Note that in the leading order in the slow-roll parameters ln(aH/k) = N , hen e the logarithmi s ale in the left panels dire tly orresponds to the linear s ale in the right panels. In Figures 2-4, we also plot the evolution of the spe tra, denoted by the supers ript (a), when one assumes the onstant slow-roll approximation after Hubble rossing, i.e. when one uses eqs. (87), (88) and (89). For ompleteness, we also dis uss brie�y the parti ular ase where the generation of iso urvature modes is e�e tively suppressed, situation whi h applies to some of the models dis ussed in the literature for spe i� parameters and/or initial onditions. 5.3.1 Double in�ation with anoni al kineti terms In this example, the �eld χ initially dominates the energy density of the Universe and drives the �rst part of in�ation, and only when it is almost at its minimum, in�ation is further driven by φ. All the slow-roll parameters are small at the Hubble rossing, whi h makes eqs. (78)-(80) an ex ellent approximation of the numeri al solutions of the equations of motion. Due to the smallness of B = −2ησs ≈ 2dθ/dN right after the Hubble rossing, the urvature perturbations be ome pra ti ally onstant during the χ-domination. At the transition to φ-dominated in�ation B be omes large, whi h leads to a sizable in rement in PR be ause of intera tion with the iso urvature perturbations. During φ-dominated in�ation the traje tory is almost straight again, the iso urvature perturbations de ay qui kly and the urvature perturbations are frozen at the value a quired at the transition. This model has the advantage that the numeri al results an be dire tly ompared to analyti al ones, as the time evolution of the perturbations an be solved without assuming onstan y of the slow-roll parameters [10℄, and we �nd a good agreement between two approa hes. Figure 2: Predi tions for the spe tra and orrelations of the perturbations in double in- �ation with anoni al kineti terms. Thi k lines show the numeri al results for PR, PS and CRS or C̃ normalized to the single-�eld result (81), respe tively. Cir les, stars and squares indi ate the predi tions of eqs. (78), (79) and (80), respe tively. Thin dashed lines indi ate the predi tions of eqs. (87), (88) and (89), respe tively. The oupling B between the urvature and iso urvature perturbations is also shown. 5.3.2 Double in�ation with non- anoni al kineti terms In this example, the ba kground traje tory is almost straight. However, the slow-roll parameter ǫ is around 0.1, whi h makes the oupling B large throughout the entire in�a- tionary era. Figure 3 shows that eqs. (78)-(80) are a good approximation for the spe tra and orrelations at the Hubble rossing, k/aH = 1, but it is no longer true at super- Hubble s ales, k/aH = 0.1 or 0.01, be ause the iso urvature perturbations already start feeding the urvature ones sizably. As a result, the �nal urvature perturbations originate almost ex lusively from the intera tions with the iso urvature modes, not from the �u - tuations along the in�ationary traje tory, whi h makes the relative orrelation oe� ient very lose to 1. Figure 3: Predi tions for the spe tra and orrelations of the perturbations in double in- �ation with non- anoni al kineti terms. Thi k lines show the numeri al results for PR, PS and CRS or C̃ normalized to the single-�eld result (81), respe tively. Cir les, stars and squares indi ate the predi tions of eqs. (78), (79) and (80), respe tively. Thin dashed lines indi ate the predi tions of eqs. (87), (88) and (89), respe tively. The oupling B between the urvature and iso urvature perturbations is also shown. 5.3.3 Roulette in�ation As we already argued in Se tion 5.2, most of the in�ationary traje tory in this example lies on the plateau of the potential (102), the slow-roll parameter ǫ is very small, whi h makes the dire t impa t of the non- anoni ality negligible. The traje tory is, however, strongly urved in the �eld spa e and the intera tion between the iso urvature and urvature modes is still important. Again, eqs. (78)-(80) a urately predi t the spe tra and orrelations in the vi inity of the Hubble rossing, with deviations on super-Hubble s ales resulting from the sour ing of the urvature perturbations by the iso urvature ones. Eventually, most of the urvature perturbations arise through this e�e t. Figure 4: Predi tions for the spe tra and orrelations of the perturbations in roulette in�ation. Thi k lines show the numeri al results for PR, PS and CRS or C̃ normalized to the single-�eld result (81), respe tively. Cir les, stars and squares indi ate the predi tions of eqs. (78), (79) and (80), respe tively. Thin dashed lines indi ate the predi tions of eqs. (87), (88) and (89), respe tively. The oupling B between the urvature and iso urvature perturbations is also shown. 5.3.4 Impa t of the non- anoni al terms In order to estimate the impa t of the non anoni al kineti terms, one an separate ea h of the oe� ients (36-39) into two parts: the terms that depend expli itly on the derivatives of non- anoni al fun tion b and those whi h do not. For example, the oe� ient that is dire tly responsible for the transfer of iso urvature modes into urvature modes, Cσs is de omposed into a ` anoni al' omponent C σs and a 'non- anoni al' omponent C Cσs = C σs + C σs , (103) C(c)σs = 6H 2VσVs + 2Vσs + , C(nc)σs = 2bφ(s θVσ − c3θVs). (104) Figure 5: Evolution of the oe� ients C σs and C σs de�ned in (104), parametrizing the oupling between the urvature and iso urvature modes: (a) for double in�ation with non- anoni al kineti terms and (b) for roulette in�ation. The other oe� ients an be de omposed similarly For the two models with non- anoni al kineti terms, we have ompared in Fig. 5 the anoni al and non- anoni al ontributions of the various oe� ients. For double in�ation with non- anoni al kineti terms, the non- anoni al ontribution in Cσs is dominant and therefore plays a ru ial role in the evolution of the urvature mode. For roulette in�ation, as already noti ed earlier, the non- anoni al ontributions turn out to be negligible. 5.3.5 E�e tively single-�eld ases Many supergravity- or string-inspired models aim at des ribing supersymmetry breaking and in�ation in a uni�ed framework. Often, despite the presen e of many s alar �elds, one an �nd model parameters and initial onditions su h that only for one ombination of the Note that there is some arbitrariness in this de omposition: the de omposition would be di�erent if one expresses V in terms of θ̇ via the ba kground equation (33). Figure 6: Predi tions for the spe tra and orrelations of the perturbations in the e�e tive single-�eld ase. The lines show the numeri al results for PR and PS , ir les orrespond to predi tion of eq. (78) and stars orrespond to the analyti al ulation outlined in Se tion 5.3.5. �elds a potential is su� iently �at to support in�ation, e.g. in pseudo-Goldstone in�ation [15℄, �better ra etra k� s enario [16℄, no-s ale supergravity models with moduli stabilised through D-terms [17℄ or in D-term uplifted supergravity of [18℄. In these works, small values of the �eld velo ity (i.e. ǫ ≪ 1) have been ensured by setting the initial onditions in the vi inity of the saddle point of the potential, while small urvature of the potential along one dire tion has been obtained by a �ne-tuning of the parameters. It has been assumed that the iso urvature perturbations de ay fast after the Hubble radius rossing and do not a�e t the urvature perturbations even though the traje tory in the �eld spa e an have sharp turns at later stages of the in�ationary evolution. Consequently, the single �eld approximation (81) has been used in these works to a ount for the spe tra of the urvature perturbation. In the two-�eld language developed in the present paper, the situation des ribed above orresponds to ηss ∼ O(1) ≫ |ησσ|, |ησs|, ǫ. This justi�es setting Θ∗ = 0 in eqs. (68)-(70), whi h is equivalent to the assumption that the urvature and iso urvature modes evolve independently. While we an still apply the slow-roll expansion that led to eq. (78) for the power spe trum of the urvature perturbations, we now have to use the full result (66) to des ribe the spe trum of the iso urvature modes. For ηss ∼ O(1), the latter result de ays very fast for k|τ | → 0 and we on lude that the iso urvature modes be ome irrelevant for the evolution of the urvature perturbations soon after the Hubble radius rossing. We he ked numeri ally that the above on lusion applies for the models des ribed in [17℄, whi h are easily redu ed to the two-�eld ase and their in�ationary traje tories are urved in the �eld spa e. For simpli ity, we would like, however, to illustrate this point with the model of double in�ation with standard kineti terms, des ribed in Se tion 5.2.1, for whi h we set the initial ondition φi = 50χi. Then the heavy �eld χ ontributes negligibly to the potential energy and ηss∗ ≃ 0.4. In Figure 6, we plot the numeri ally al ulated spe tra of the urvature and iso urvature perturbations and ompare them with the analyti approximations outlined above. The de ay of the iso urvature modes agrees with the solution (66), for whi h we show two ases: the small solid stars orrespond to the onstant value of ηss, whereas the large empty stars show the result orresponding to adjusting the index of the Hankel fun tion in eq. (66) to the value of ηss at a given instant, i.e. ηss = 0.40, 0.42, 0.44 for (k/aH) −1 = 1, 10, 100, respe tively. With the iso urvature modes absent, the urvature perturbations are ex ellently des ribed by the single-�eld result, whi h justi�es the use of the single-�eld approximations in the situations des ribed above. 5.3.6 Closing dis ussion The three examples presented here show that in multi-�eld in�ationary models, a large part of the urvature perturbations an originate from intera tions between the urvature and iso urvature perturbations on super-Hubble s ales, not only from quantum �u tua- tions along the traje tory at the Hubble exit. In su h ases, the single-�eld result (81) does not provide a orre t predi tion either for the normalization of the power spe trum or for its spe tral index ns = 1 + d lnPR/d ln k . (105) There are te hniques whi h allow relating the spe tral index of the urvature perturba- tions, ns, to the spe tral indi es of the entropy perturbations and the urvature-entropy orrelations through a set of onsisten y relations [19, 22, 13℄, but all these quantities separately depend on the super-Hubble evoulution of the perturbations. Again, we �nd it the most straightforward to al ulate the spe tral index ns for ea h model numeri ally. In Table 1, we ompare naive estimate ns ∼ 1−6ǫ∗+2ησσ∗ and the predi tions of eq. (81) for the spe tral index ns with the numeri al results of Se tion 5.3. In our three examples, one an see that the single-�eld result signi� antly overestimates the orre t spe tral index. The dis repan ies that we �nd follow from the fa t that the two types of perturbations ns 1− 6ǫ∗ + 2ησσ∗ single-�eld result full result double in�ation ( anoni al) 0.929 0.982 0.967 double in�ation (non- anoni al) 0.953 0.968 0.934 roulette in�ation 1.017 1.019 0.932 Table 1: A omparison between the predi tions for the spe tral index ns in the three examples of in�ationary models des ribed in Se tion 5.2. The third olumn ontains result derived from the single-�eld approximation (81); the result of full numeri al al ulations are shown in the fourth olumn. experien e the slow-roll of the ba kground �elds and the urvature of the in�ationary potential in a di�erent way. Then, if the �nal urvature perturbations originate mainly from the iso urvature ones, they inherit the features of the iso urvature power spe tra at the Hubble rossing. 6 Con lusion In this paper, we have studied two-�eld in�ation and extended several previous results on urvature and iso urvature perturbations to the ase of non-standard kineti terms. First, we have al ulated analyti ally the urvature and iso urvature spe tra, as well as the orrelation, just after Hubble rossing for two-�eld in�ation models, in luding next-to- leading order orre tions in the slow-roll approximation. Our results (78)-(80) generalize those of Byrnes and Wands, who assumed only standard kineti terms. We have also given a re�ned analyti al treatment of the spe tra around Hubble rossing, whi h is important when the perturbations still evolve after Hubble rossing. Se ond, we have studied numeri ally the evolution of the urvature and iso urvature perturbations after the Hubble rossing. This type of analysis is important sin e in multi- �eld in�ation, in ontrast with single-�eld in�ation, the urvature perturbation spe trum after in�ation is in general di�erent from the urvature perturbation spe tra at Hubble rossing be ause of iso urvature perturbations, as �rst emphasized in [6℄. This well- known result applies to multi-�eld in�ation with either standard or non-standard kineti terms. This e�e t has been studied numeri ally for standard kineti terms, using the de omposition into instantaneous urvature and iso urvature, in the analysis of [23℄. We have done a similar analysis here for non-standard kineti terms. In parti ular, we have ompared the numeri al evolution of the perturbations with an analyti al approximation, whi h we denoted the onstant slow-roll approximation: this approximation assumes not only that the slow-roll approximation is valid, but also that the slow-roll parameters remain almost onstant during the subsequent evolution where iso urvature perturbations are signi� ant. This approximation has been used in [21℄ to ompute the ��nal� spe tra in the ontext of non-standard kineti terms. In most ases, this approximation is however not very realisti as we learly show in our numeri al study. We have also estimated the impa t of non-standard kineti terms on the oupled evolution of the urvature and iso urvature modes. There has been a re ent interest in onstru ting in�ationary models in the ontext of string theory. These models naturally lead to s alar �elds with non-standard kineti terms, to whi h our analysis an apply. In this work, we have studied a very re ent model, alled `roulette� in�ation, and omputed quantitatively the �nal urvature spe trum, thus showing that the in�uen e of iso urvature modes on super-Hubble s ales turns out to be very important. Note that the authors of [14℄ omputed the �nal urvature spe trum by using a �single in�aton approximation�. They stress however that iso urvature e�e ts � ould produ e big e�e ts�, whi h we indeed on�rm in our analysis. As a message of aution for the readers who are not familiar with the e�e t of iso urva- ture modes in multi-�eld in�ation, we have also omputed the spe tral index for urvature perturbations in a few models and ontrasted it with the value that one would naively obtain by using the urvature perturbation at Hubble radius like in single �eld in�ation. Finally, an interesting question, whi h goes beyond the s ope of this paper, would be to investigate in whi h ir umstan es these multi-�eld models ould produ e iso urvature perturbations, after in�ation and the reheating phase. These �primordial� iso urvature perturbations are today severely onstrained by CMB data. A knowledgements We would like to thank V. Mukhanov for stimulating dis us- sions. D.L. would like to thank the Institute of Theoreti al Physi s of Warsaw for their warm hospitaly and for their �nan ial support via a �Marie Curie Host Fellowship for Transfer of Knowledge� , proje t MTKD-CT-2005-029466. Z.L. was partially supported by TOK proje t MTKD-CT-2005-029466, by the EC 6th Framework Programme MRTN- CT-2006-035863, and by the grant MEiN 1 P03D 014 26. S.P. was partially supported by TOK proje t MTKD-CT-2005-029466 and by the grant MEiN 1 P03B 099 29. The work of K.T. is partially supported by the US Department of Energy. 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Quevedo, JHEP 0601, 146 (2006) [arXiv:hep-th/0509012℄. http://arxiv.org/abs/hep-th/0702103 http://arxiv.org/abs/astro-ph/0009131 http://arxiv.org/abs/astro-ph/0211276 http://arxiv.org/abs/astro-ph/0505198 http://arxiv.org/abs/astro-ph/0605679 http://arxiv.org/abs/astro-ph/0210322 http://arxiv.org/abs/astro-ph/0701247 http://arxiv.org/abs/hep-th/0405053 http://arxiv.org/abs/astro-ph/0610064 http://arxiv.org/abs/astro-ph/0503416 http://arxiv.org/abs/astro-ph/0509078 http://arxiv.org/abs/astro-ph/9507001 http://arxiv.org/abs/gr-qc/9801017 http://arxiv.org/abs/hep-th/0509012 Introduction The model Homogeneous equations Linear perturbations Decomposition into adiabatic and entropy components Perturbation spectra Evolution of perturbations inside the Hubble radius Equations in the slow-roll approximation Evolution of perturbations on super-Hubble scales Numerical analysis Numerical procedure Examples of inflationary models Double inflation with canonical kinetic terms Double inflation with non-canonical kinetic terms Roulette inflation Numerical results for the perturbations Double inflation with canonical kinetic terms Double inflation with non-canonical kinetic terms Roulette inflation Impact of the non-canonical terms Effectively single-field cases Closing discussion Conclusion

0704.0214

A schematic model of scattering in PT-symmetric Quantum Mechanics

arXiv:0704.0214v1 [quant-ph] 2 Apr 2007 A schematic model of scattering in PT −symmetric Quantum Mechanics Miloslav Znojil Ústav jaderné fyziky AV ČR, 250 68 Řež, Czech Republic e-mail: [email protected] Abstract One-dimensional scattering problem admitting a complex, PT −symmetric short- range potential V (x) is considered. Using a Runge-Kutta-discretized version of Schrödinger equation we derive the formulae for the reflection and transmission coef- ficients and emphasize that the only innovation emerges in fact via a complexification of one of the potential-characterizing parameters. PACS 03.65.Ge http://arxiv.org/abs/0704.0214v1 1 Introduction Standard textbooks describe the stationary one-dimensional motion of a quantum particle in a real potential well V (x) by the ordinary differential Schrödinger equation + V (x) ψ(x) = E ψ(x) , x ∈ (−∞,∞) (1) which may be considered and solved in the bound-state regime at E < V (∞) ≤ +∞ or in the scattering regime with, say, E = κ2 > V (∞) = 0. In this way one either employs the boundary conditions ψ(±∞) = 0 and determines the spectrum of bound states or, alternatively, switches to the different boundary conditions, say, ψ(x) = Aeiκx +B e−iκx , x≪ −1 , C eiκx , x≫ 1 . Under the conventional choice of A = 1 the latter problem specifies the reflection and transmission coefficients B and C, respectively [1]. The conventional approach to the quantum bound state problem has recently been, fairly unexpectedly, generalized to many unconventional and manifestly non- Hermitian Hamiltonians H 6= H† which are merely quasi-Hermitian, i.e., which are Hermitian only in the sense of an identity H† = ΘH Θ−1 which contains a nontrivial “metric” operator Θ = Θ† > 0 as introduced, e.g., in ref. [2]. The key ideas and sources of the latter new development in Quantum Mechanics incorporate the so called PT −symmetry of the Hamiltonians and have been summarized in the very fresh review by Carl Bender [3]. This text may be complemented by a sample [4] of the dedicated conference proceedings. In this context we intend to pay attention to a very simple PT −symmetric scat- tering model where V (x) = Z(x) + i Y (x) , Z(−x) = Z(x) = real , Y (−x) = −Y (x) = real and where the ordinary differential equation (1) is replaced by its Runge-Kutta- discretized, difference-equation representation ψ(xk−1)− 2ψ(xk) + ψ(xk+1) + V (xk)ψ(xk) = E ψ(xk) (3) xk = k h , h > 0 , k = 0,±1, . . . as employed, in the context of the bound-state problem, in refs. [5]. 2 Runge-Kutta scattering Once we assume, for the sake of simplicity, that the potential in eq. (3) vanishes beyond certain distance from the origin, V (x±j) = 0 j =M,M + 1, . . . , we may abbreviate ψk = ψ(xk), Vk = h 2V (xk) and 2 cosϕ = 2− h 2E in eq. (3), −ψk−1 + (2 cosϕ+ Vk) ψk − ψk+1 = 0 . (4) In the region of |k| ≥ M with vanishing potential Vk = 0 the two independent solutions of our difference Schrödinger eq. (4) are easily found, via a suitable ansatz, as elementary functions of the new “energy” variable ϕ, ψk = const · ̺ k =⇒ ̺ = ̺± = exp(±i ϕ) . This enables us to replace the standard boundary conditions (2) by their discrete scattering version ψ(xm) = Aeimϕ +B e−imϕ , m ≤ −(M − 1) , C eimϕ , m ≥ M − 1 with a conventional choice of A = 1. Two comments may be added here. Firstly, one notices that the condition of the reality of the new energy variable ϕ imposes the constraint upon the original energy itself, −2 ≤ 2 − h2E ≤ 2, i.e., E ∈ (0, 4/h2). At any finite choice of the lattice step h > 0 this inequality is intuitively reminiscent of the spectra in relativistic quantum systems. Via an explicit display of the higher O(h4) corrections in eq. (3), this connection has been given a more quantitative interpretation in ref. [6]. The second eligible way of dealing with the uncertainty represented by the O(h4) discrepancy between the difference- and differential-operator representation of the Schrödinger’s kinetic energy is more standard and lies in its disappearance in the limit h → 0. This is a purely numerical recipe known as the Runge-Kutta method [7]. In the present context of scattering one has to keep in mind that the two “small” parameters h and 1/M may and, in order to achieve the quickest convergence, should be chosen and varied independently. 3 The matching method of solution 3.1 The simplest model of the scattering with M = 1 Once we are given the boundary conditions (5) the process of the construction of the solutions is straightforward. Let us first illustrate its key technical ingredients on the model with the first nontrivial choice of the cutoff M = 1. In this case our difference Schrödinger eq. (4) degenerates to the mere three nontrivial relations, −ψ−2 + 2 cosϕψ−1 − ψ 0 = 0 −ψ−1 + (2 cosϕ+ Z0) ψ0 − ψ1 = 0 0 + 2 cosϕψ1 − ψ2 = 0 where we may insert, from eq. (5), ψ−1 = e −i ϕ +B ei ϕ , ψ 0 = 1 +B , ψ 0 = C , ψ1 = C e i ϕ (7) and where we have to demand, subsequently, 0 = 1 +B = ψ 0 = C = ψ0 , −e−i ϕ −B ei ϕ + (2 cosϕ + Z0) C − C e i ϕ = 0 . Thus, at an arbitrary “energy” ϕ one identifies B = C − 1 and gets the solution 2i sinϕ 2i sinϕ− Z0 , B = 2i sinϕ− Z0 Of course, as long as we deal just with the real “interaction term” Z0, our M = 1 toy problem remains Hermitian since no PT −symmetry has entered the scene yet. 3.2 PT −symmetry and the scattering at M = 2 In the next, M = 2 version of our model we have to insert the four known quantities ψ−2 = e −2 i ϕ +B e2 i ϕ , ψ−1 = e −i ϕ +B ei ϕ , ψ1 = C e i ϕ , ψ2 = C e 2 i ϕ in the triplet of relations −ψ−2 + (2 cosϕ+ Z−1 − i Y−1) ψ−1 − ψ 0 = 0 −ψ−1 + (2 cosϕ+ Z0) ψ0 − ψ1 = 0 0 + (2 cosϕ+ Z−1 + i Y−1) ψ1 − ψ2 = 0 where the three symbols ψ0, ψ 0 and ψ 0 defined by these respective equations should represent the same quantity and must be equal to each other, therefore. Having this in mind we introduce ξ 0 = 1 +B and ξ 0 = C and decompose 0 = ξ 0 + χ 0 , ψ 0 = ξ 0 + χ This enables us eliminate 0 = V−1 ψ−1 , χ 0 = V1 ψ1 and eq. (9) becomes reduced to the pair of conditions, 1 +B + V−1 ψ−1 = C + V1 ψ1 = ψ0 , −ψ−1 + (2 cosϕ+ Z0) ψ0 − ψ1 = 0 They lead to the two-dimensional linear algebraic problem which defines the reflection and transmission coefficients B and C at any input energy ϕ. The same conclusion applies to all the models with the larger M . 4 The matrix-inversion method of solution Let us now re-write our difference Schrödinger eq. (4) as a doubly infinite system of linear algebraic equations . . . . . . . . . . . . S−1 −1 0 . . . . . . −1 S0 −1 . . . . . . 0 −1 S1 . . . . . . . . . . . . = 0 , (11) where Sk ( ≡ S −k) = 2 cosϕ+ Zk + i Yk sign k , |k| < M , 2 cosϕ , |k| ≥M and where the majority of the elements of the “eigenvector” are prescribed, in ad- vance, by the boundary conditions (5). Once we denote all of them by a different symbol, ψ(xm) = Aeimϕ +B e−imϕ ≡ ξ(−)m , m ≤ −(M − 1) , C eimϕ ≡ ξ(+)m , m ≥M − 1 , we may reduce eq. (11) to a finite-dimensional and tridiagonal non-square-matrix problem −1 S∗(M−1) −1 . . . . . . . . . −1 S∗1 −1 −1 S0 −1 −1 S1 −1 . . . . . . . . . . . . . . . −1 S(M−1) −1 −(M−1) ψ−(M−2) = 0 (14) or, better, to a non-homogeneous system of 2M − 1 equations −(M−1) ψ−(M−2) where the (2M − 1)−dimensional square-matrix of the system can be partitioned as follows, S∗(M−1) −1 −1 S∗(M−2) −1 . . . . . . . . . S0 . . . . . . . . . −1 −1 S(M−2) −1 −1 S(M−1) . (16) Whenever this matrix proves non-singular, it may assigned the inverse matrixR=T−1, the knowledge of which enables us to re-write eq. (15), in the same partitioning, as follows, −(M−1) = R · , ~Ψ = ψ−(M−2) . (17) In the next step we deduce that the matrix R has the following partitioned form α∗ ~tT β ~u Q ~v β ~wT α We may summarize that in the light of the overall partitioned structure of eq. (17), the knowledge of the (2M − 3)−dimensional submatrix Q as well as of the two (2M − 3)−dimensional row vectors ~tT and ~wT (where T denotes transposition) is entirely redundant. Moreover, the knowledge of the other two column vectors ~u and ~v only helps us to eliminate the “wavefunction” components ψ−(M−2), ψ−(M−3), . . . , ψM−3, ψM−2. In this sense, equation (15) degenerates to the mere two scalar relations −(M−1) − α −(M) − β ξ M = 0 , M−1 − β ξ −M − α ξ M = 0 . Once we insert the explicit definitions from eq. (13) we get the final pair of linear equations e−i (M−1)ϕ +B ei (M−1)ϕ − α∗ e−iM ϕ +B eiM ϕ − C β eiM ϕ = 0 , C ei (M−1)ϕ − β e−iM ϕ +B eiM ϕ − C α eiM ϕ = 0 which are solved by the elimination of B = −e−2iMϕ + e−iϕ − α and, subsequently, of 2iβe−2iMϕ sinϕ β2 − (e−iϕ − α∗) (e−iϕ − α) . (21) This is our present main result. 5 Coefficients α and β Our final scattering-determining formulae (20) and (21) indicate that the complex coefficient α and the real coefficient β carry all the “dynamical input” information. At any given energy parameter ϕ these matrix elements are, by construction, rational functions of our 2M − 1 real coupling constants Z0, Z1, . . . , ZM−1 and Y1, . . . , YM−1. In particular, β is equal to 1/ detT and α has the same denominator of course. An explicit algebraic determination of the determinant detT and of the numerator (say, γ) of α is less easy. Let us illustrate this assertion on a few examples. 5.1 M = 2 once more detT = Z0 Z1 2 − 2Z1 + Y1 Re γ = Z0 Z1 − 1 Im γ = −Z0 Y1 5.2 M = 3 detT = Z0 Z1 2 − 2Z0 Z1 Z2 − 2Z1 Z2 Z0 Z2 2 + 2Z2 + Y2 Z0 Z1 2 − 2Y2 Z1 + Y2 Z0 + 2Z0 Y1 Y2 + Z0 Re γ = Z0 Z1 Z2 − 2Z1 Z2 + Y1 Z0 Z2 − Z1 Z0 + 1 Im γ = −Z0 Z1 Y2 + 2Z1 Y2 − Y1 Z0 Y2 −Y1 Z0 5.3 M = 4 The growth of complexity of the formulae occurs already at the dimension as low as M = 4. The determinant detT and the real and imaginary parts of γ are them represented by the sums of 15 and 14 and 32 products of couplings, respectively. A simplification is only encountered in the weak coupling regime where one finds just two terms in the determinant which are linear in the couplings, detT = −2Z3 − 2Z1 + . . . being followed by the 10 triple-product terms, . . .+ Y1 Z0 + 4Z1 Z2 Z3 − 4Z1 Y2 Y3 + Z1 Z0 + 2Y3 Z2 + 2Y1 Z0 Y3+ +2Z1 Z0 Z3 + Z3 Z0 + Y3 Z0 + 2Z3 Z2 + . . . etc. Similarly, we may decompose, in the even-number products, Re γ = −1 + 2Z2 Z3 + Z0 Z1 + Z0 Z3 + 2Z2 Z1 + . . . and continue . . .+−2Z0 Z1 Z2 Z3 + 2Z0 Y1 Y2 Z3− −Z2 Y1 Z0 − 2Z2 Z1 Z3 − Z2 Z1 Z0 − 2Y2 Z1 Z3 + . . . etc, plus Im γ = −2Z2 Y3 + 2Y2 Z1 − Z0 Y1 − Z0 Y3 − . . . with a continuation . . .− Y2 Y1 Z0 −Y2 Z1 Z1 Y3 + 2Z0 Z1 Z2 Y3 + 2Z2 Z1 Y3 − 2Z0 Y1 Y2 Y3 + . . . etc. Symbolic manipulations on a computer should be employed at all the higher dimensions M ≥ 4 in general. 6 Discussion The main inspiration of the activities and attention paid to the PT −symmetry originates from the pioneering 1998 letter by Bender and Boettcher [8] where the operator P meant parity and where the (antilinear) T represented time reversal. Its authors argued that the complex model V (x) = x2 (ix)δ seems to possess the purely real bound-state spectrum at all the exponents δ ≥ 0. After a rigorous mathematical proof of this conjecture by Dorey, Duncan, Tateo and Shin [9] and after the (crucial) clarification of the existence of a nontrivial, “physical” Hilbert space H where the Hamiltonian remains self-adjoint [2, 10, 11, 12], the bound-state version of eq. (1) may be considered more or less well understood, especially after it has been clarified that the physics-inspired concept of PT −symmetry of a Hamiltonian H should in fact be understood, in the language of mathematics, as a P−pseudo-Hermiticity of H specified by the property H† = P H P−1 [10, 11, 13]. In the spirit of the latter generalization, current literature abounds in the studies of the potentials which are analytically continued [14], singular and multisheeted [15], multidimensional [16], manybody [17], relativistic [18], supersymmetric [19] and channel-coupling [20]. Among all these developments, a comparatively small number of papers has been devoted to the problem of the scattering. For a sample one might recollect the key reviews [21] and various Kleefeld’s conceptual conjectures [22] as well as a very explicit study of the scattering by the separable PT −symmetric potentials of rank one [23] or by the rectangular or reflectionless barriers [24], or the motion considered along the so called tobogganic (i.e., complex and topologically nontrivial) integration contours [25]. In this context our present difference-equation-based study may be understood just as another attempt to fill the gap. Technically we felt inspired by our old Runge-Kutta-type discretization of the PT −symmetric Schrödinger equations [5] as well as by our recent chain-model ap- proximations of bound states in a finite-dimensional Hilbert space [26]. In a certain unification of these two approaches we succeeded here in showing that there exists a close formal parallelism between the description of the (one-dimensional, Runge- Kutta-approximated) scattering by a real (i.e., Hermitian) potentials and by their complex, PT −symmetric generalizations. We showed that in both these contexts, the definition of the transmission and reflection coefficients has the same form [cf., e.g., eq. (21)], with all the differences represented by the differences in the form of the “dynamical input” information. It has been shown to be encoded, in both the Hermitian and non-Hermitian cases, in the two functions α and β of the lattice po- tentials, with the vanishing or non-vanishing coefficients Yk, respectively (cf. a few samples of the concrete form of α and β in section 5). On the level of physics we would like to emphasize that one of the main dis- tinguishing features of the scattering problem in PT −symmetric quantum mechan- ics lies in the manifest asymmetry between the “in” and “out” states [22]. In its present solvable exemplification we showed that such an asymmetry is merely formal and that the problem remains tractable by the standard, non-matching and non- recurrent techniques of linear algebra. A key to the success proved to lie in the partitioning of the Schrödinger equation which enabled us to separate its essential and inessential components and to reduce the construction of the amplitudes to the mere two-dimensional matrix inversion [cf. eq. (18)] where all the dynamical input is represented by the four corners of the inverse matrix R = T−1 [cf. the definition (16)]. We believe that the merits of the present discrete model were not exhausted by its present short analysis and that its further study might throw new light, e.g., on the non-Hermitian versions of the inverse problem of scattering. Acknowledgement Work supported by the MŠMT “Doppler Institute” project Nr. LC06002, by the In- stitutional Research Plan AV0Z10480505 and by the GAČR grant Nr. 202/07/1307. References [1] S. Flügge, Practical Quantum Mechanics (Springer, Berlin, 1971), p. 42. [2] F. G. Scholtz, H. B. Geyer and F. J. W. Hahne, Ann. Phys. (NY) 213 (1992) [3] C. M. Bender, Rep. Prog. Phys., submitted (hep-th/0703096). [4] cf., e.g., the dedicated issues of J. Phys. A: Math. Gen. 39 (2006), Nr. 32 (H. Geyer, D. Heiss and M. Znojil, editors, pp. 9963 - 10261), and Czechosl. J. Phys. 56 (2006), Nr. 9 (M. Znojil, editor, pp. 885 - 1064). [5] M. Znojil, Phys. Lett. A 223 (1996) 411; F. M. Fernández, R. Guardiola R, J. Ros and M. Znojil, J. Phys. A: Math. Gen. 32 (1999) 3105; U. Guenther, F. Stefani and M. Znojil, J. Math. Phys. 46 (2005) 063504. [6] M. Znojil, Phys. Lett. A 203 (1995) 1. [7] S. Acton, Numerical Methods that Work (Harper, New York, 1970). [8] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 24 (1998) 5243. [9] P. Dorey, C. Dunning and R. Tateo, J. Phys. A: Math. Gen. 34 (2001) L391; P. Dorey, C. Dunning and R. Tateo, J. Phys. A: Math. Gen. 34 (2001) 5679; K. C. Shin, Commun. Math. Phys. 229 (2002) 543. [10] M. Znojil, J. Nonlin. Math. Phys. 9, suppl. 2 (2002) 122 (quant-ph/0103054); M. Znojil, Rendiconti del Circ. Mat. di Palermo, Ser. II, Suppl. 72 (2004) 211 (math-ph/0104012); B. Bagchi, C. Quesne and M. Znojil, Mod. Phys. Lett. A 16 (2001) 2047 (quant- ph/0108096). [11] A. Mostafazadeh, J. Math. Phys. 43 (2002) 205 and 2814 and 3944; H. Langer and Ch. Tretter, Czech. J. Phys. 54 (2004) 1113. [12] C. M. 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0704.0215

The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts

The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts. Zbigniew Pucha la1,2,3 and Tomasz Rolski1,2 November 4, 2018 Abstract In this note we consider the time of the collision τ for n independent Brownian motions X1t , . . . , X t with drifts a1, . . . , an, each starting from x = (x1, . . . , xn), where x1 < . . . < xn. We show the exact asymptotics of IPx(τ > t) = Ch(x)t −αe−γt(1 + o(1)) as t → ∞ and identify C, h(x), α, γ in terms of the drifts. Keywords: Brownian motion with drift, collision time. AMS 2000 Subject Classification: Primary: 60J65. 1 Mathematical Institute, University of Wroc law, pl. Grunwaldzki 2/4, 50-384 Wroc law, Poland 2This work was partially supported by a Marie Curie Transfer of Knowledge Fellowship of the European Community’s Sixth Framework Programme: Programme HANAP under contract number MTKD-CT-2004- 13389. 3This work was partially supported by This work was partially supported by KBN Grant N201 049 31/3997 (2007). http://arxiv.org/abs/0704.0215v1 1 Introduction and results Let W = {y : y1 < . . . < yn} be the Weyl chamber. Consider Xt = (X1t , . . . ,Xnt ), wherein coordinates are independent Brownian motions with unit variance parameter, drift vector a = (a1, . . . , an) and starting point X0 = x ∈ W . In this paper we study the collision time τ , which is the exit time of Xt from the Weyl chamber, i.e. τ = inf{t > 0 : Xt /∈ W} . For identical drifts a1 = . . . = an , say ai ≡ 0, the celebrated Karlin-McGregor formula states (see [7]) IP(τ > t;X t ∈ dy) = det [pt(xi, yj)] dy , (1.1) where pt(x, y) = (x−y)2 2t , which yields the tail distribution of τ : IPx(τ > t) = det [pt(xi, yj)] dy . For the use of Karlin-McGregor formula it is essential that processes X1t , . . . ,X t are inde- pendent copies of the same strong Markov, with skip-free realizations process, starting at t = 0 from x ∈ W . In this case the asymptotic of IPx(τ > t) was first studied by Grabiner [5] (for the Brownian case) (see also proofs by Doumerc and O’Connell [4] and Pucha la [9]) Later Pucha la & Rolski [10]) showed that this asymptotic is also true for the Poisson and continuous time random walk case. The above mentioned asymptotics is: IPx(τ > t) ∼ D∆(x)t−n(n−1)/4, (1.2) where ∆(x) = det (j−1) i,j=1 is the Vandermonde determinant, and D = (2π) 2 ∆ (y) dy , (1.3) for t → ∞. Here and below 1/cn = j=1 j!. In this note we study the same problem, however for Brownian motions with different drifts. For this we derive first, in Section 2, a formula for IPx(τ > t) by the change of measure. It is apparent that possible results must depend on the form of drift vector a. For example we can analyze all cases for n = 2, because in this case the collision equals to the first passage to zero of the Brownian process X2t − X1t , for which the density function is known (see e.g. [3]). Hence IPx(τ > t) = 2πs3/2 −(x + as) where x = x2 − x1 and a = a2 − a1. This yields IPx(τ > t) = xeaxt−3/2e−ta 2/4 (1 + o(1)) , a1 > a2 2 (1 + o(1)) , a1 = a2 1 − e−ax + o(1) a1 < a2 . For general n the situation is much more complex and different scenarios are possibles. For example the drifts can be diverging and then IPx(τ > t) tends to a positive constant, which the situation was analyzed by Biane et al [2]. Another case is when all drifts are equal, in which the case the probability IPx(τ > t) is polynomially decaying, as it was found by Grabiner [5]. However there are various situations when the probabilities are exponentially decaying with polynomial prefactors. The full characterization depends on a concept of the stable partition of the drift vector, which the notion is introduced in Section 3. In Section 4 we state the main theorem, which shows all possible exact asymptotics of IPx(τ > t) in from of Ch(x)t−αe−γt, where formulas for C,α and γ are given in terms of the stable partition of the drift vector. 2 Formula for IPx(τ > t). We note our basic probabilistic space with natural history filtration (Ω,F , (Ft), IPx) and consider on it process Xt as defined in the Introduction. Unless otherwise stated we tacitly assume that x ∈ W . We start off a lemma on the change of measure for the Brownian case, which the proof can be found for example in Asmussen [1], Theorem 3.4. Let Mt = e<α,Xt>/IEe<α,Xt> be a Wald martingale. For a probability measure IPx its restriction to Ft we denote by IPx|t. Let ĨPx be a probability measure obtained by the change of measure IPx with the use of martingale Mt, that is defined by a family of measures ĨPx|t = Mt dIPx|t, t ≥ 0. For the theory we refer e.g. to Section XIII.3 in [1] Lemma 2.1 If Xt is a Brownian motion with drift a under IPx, then this process is a Brownian motion with drift a + α under ĨPx. The sought for formula for the tail distribution of the collision time is given in the next proposition. Proposition 2.2 IPx(τ > t) = = (2π)−n/2e−<a,x>−||x|| e−||y−a t||2/2 det[exiyj/ t] dy . (2.4) Proof. We use α = −a to eliminate the drift under ĨPx. Thus IPx(τ > t) = ĨEx[M−1t ; τ > t] . Now by Karlin-McGregor formula (1.1) we write IPx(τ > t) = ĨEx[e <a,Xt>IExe <−a,Xt>; τ > t] = e<−a,x>−||a|| e<a,y> det[pt(xi, yj)] dy , and next, algebraic manipulations yield (2.4). In the paper we use the following vector notations. For a vector a ∈ IRn we denote a[i,j] = (ai, ai+1, . . . , aj) and ā[i,j] = (ai+ai+1+ . . .+aj)/(j−i+1). We also use a(i,j] = (ai+1, . . . , aj) and a(i,j) = (ai+1, . . . , aj−1). By z k, where z = (z1, . . . , zm) and k = (k1, . . . , km) we denote j=1 z 3 Stable partition of a. Let a ∈ IRn. Our aim is to make a suitable partition (a1, . . . , aν1)(aν1+1, . . . , aν1+ν2), . . . , (aν1+...+νq−1+1, . . . , aν1+...+νq ) . (3.5) of a, where νi > 0. For short we denote m1 = ν1,m2 = ν1 + ν2, . . . ,mq = ν1 + . . . + νq = n. We also set m0 = 0. We say that sequence a is irreducible if ā[1;1] > ā[2;n] ā[1;2] > ā[3;n] ā[1;n−1] > ā[n;n] . (3.6) Suppose we have a partition defined by m1, . . . ,mq . The mean of the i th sub-vector is denoted by f i = ā(mi−1;mi]. Furthermore we define a vector f = (f1, . . . , fn) by fi = f k, if mk−1 < i ≤ mk . It is said that partition (3.5) of vector a is stable if f1 ≤ f2 ≤ . . . ≤ f q (3.7) and each vector a(mi−1,mi] is irreducible (i = 1, . . . , q). Remark that a stable partition is defined if we know m = (m1, . . . ,mq) for which (3.7) hold and each a(mi−1,mi] is irreducible (i = 1, . . . , q). In the sequel, for a given stable partition of a, characters q,f ,m are reserved for it. Consider now fm1 , fm2 , . . . , fmq and define a subsequence m = (m′1, . . . ,m q′) of m = (m1,m2, . . . ,mq) as follows. Let q ′ be the number of strict inequalities in f1 ≤ f2 ≤ . . . ≤ f q plus 1. Furthermore we define inductively by m′0 = 0 and for i = 1, . . . , q ′ − 1 m′i = inf{mj > m′i−1 : mj ∈ m, fmj < fmj+1} . and finally we set mq′ = n. We also define a subsequence of indices i0, i1, . . . , iq′ inductively by i0 = 0 and ik = inf{j > ik−1 : fmj < fmj+1}. Hence we have fm′1 < fm < · · · < fm′ In this case we say that (m′1, . . . ,m q′) is a strong representation of the stable partition of a and q′, (m′1, . . . ,m q′) are characters reserved for it. Set ν i = m i −m′i−1, (i = 1, . . . , q′). Example 1 Suppose that a = (3, 1, 2, 5, 1). Then q = 3 and m1 = 2,m2 = 3,m3 = 5 define the stable partition (3, 1)(2)(5, 1) with means f1 = 2, f2 = 2, f3 = 3. furthermore q′ = 2, m′1 = 3,m 2 = 5 and i1 = 2, i2 = 5. Proposition 3.1 For each vector a, there exists its unique stable partition. Before we state a proof of Proposition 3.1 we prove few lemmas. Lemma 3.2 If a = (a1, . . . , an) is irreducible, then ā[1;1] > fn > ā[2;n] ā[1;2] > fn > ā[3;n] ā[1;n−1] > fn > ā[n;n] . (3.8) Proof. fn is a nontrivial weighted mean of every pair ā[1;i] and ā[i+1;n]. Lemma 3.3 In a stable partition, for each element a(mi−1;mi] ā(mi−1;mi−1+k] ≥ fmi . Proof. The case k ≤ mi −mi−1 follows from Lemma 3.2. Clearly for k = mi −mi−1 we have equality. Consider now k > mi −mi−1. Than ā(mi−1;mi−1+k] is a weighted mean of fmi and ā(mi;mi+k−(mi−mi−1)] and the later term is greater or equal than fmi by (3.7) and (3.8). In the next lemma we consider two vectors a1 ∈ IRn1 and a2 ∈ IRn2 . The corresponding f -s are fn1 and fn2 respectively. We consider a situation of creating a new vector (a1,a2) = (a1 . . . , an1+n2) ∈ IRn1+n2 . Lemma 3.4 Suppose that a1 and a2 are irreducible and fn1 > fn2. Then vector (a1,a2) is irreducible. Proof. Recall that (a1,a2) = (a1 . . . , an1+n2) ∈ IRn1+n2 . Suppose 1 ≤ k ≤ n1. By Lemma 3.2 we have ā[1;k−1] > fn1 > ā[k;n1], also ā[1;k−1] > fn1 > fn2 . Hence ā[1;k−1] > ā[k;n1+n2] becasue ā[k;n1+n2] is a weighted mean of ā[k;n1] and fn2 . Suppose now n1 < k. Then ā[1;k−1] is a weighted mean of fn1 and ā[n1+1,k] and both by Lemma 3.2 are greater than ā[k;n1+n2], which completes the proof. Proof of Proposition 3.1. The existence part is by induction with respect n. For n = 2 we have two situations 1. if a1 ≤ a2, than q = 2 with m1 = 1, m2 = 2 is a stable partition, 2. if a1 > a2, than q = 1 with m1 = 2 is a stable partition. Assume that there exists a stable partition with q partition vectors of a vector a ∈ Rn. We add a new element an+1 at the end of vector a to create new one (a, an+1) = (a1, . . . , an+1). We have two situations. 1. If an+1 ≥ f q than in a stable partition an+1 is alone in the q + 1 partition vector. 2. If an+1 < f q, than we proceed inductively as follow. We use Lemma 3.4 with a1 = a[mq−1;mq] and a2 = (an+1) and let f q and f q+1 = an+1 are means of these partition vectors. In result (a(mq−1;mq], an+1) form an irreducible vector, for which we have to check whether condition (3.7) holds. If yes, then we end with a stable partition, other- wise we join the q − 1 partition vector with the new q partition vectors and repeat the procedure. In the worst case we end up with one partition vector. For the uniqueness proof , suppose that we have two different stable partitions: m11 < m12 < · · · < m1q1 and m 1 < m 2 < · · · < m2q2 . The means of fs are (f 1)1, . . . , (f1)q1 for the first partition vector and (f2)1, . . . , (f2)q2 for the second respectively. Since parti- tions are supposely different, there exists i such that m1i 6= m2i . We take the minimal i with this property and without loss of generality we can assume m2i > m i . Set k = m i −m1i . We have to analaze the following cases. 1. (m2i = m i+1). We have [m2i−1+1;k] > ā2 [k+1;m2i ] On the other hand (f1)i = ā[m1i−1+1;m = ā[m2i−1+1;m > ā[m1i+1;m = ā[m1i+1;m (f2)i and this contradics with (f1)i ≤ (f1)i+1. 2. (m2i > m i+1). We have ā[m2i−1+1;m i+1;m and by Lemma 3.3 (f1)i = ā[m2i−1+1;m > ā[m1i+1;m ≥ (f1)i+1 , which is a contradiction. 3. (m2i < m i+1). We have by Lemma 3.3 (f1)i = ā[m1i−1+1;m > ā[m1i+1;m ≥ (f1)i+1 , which is a contradiction. The proof is completed. Remark The stable partition can be obtained by considering the following simple determin- istic dynamical system. We have n particles starting from x1 < x2 < · · · < xn. The ith particle has speed ai. Each particle moves with a constant speed on the real line until it collides with one of its neighboring particle (if it happens). Then both the particles coalesce and from this time on they move with the proportional speed which is the mean of speed of colliding particles, and so on. Ultimately the particles will form never colliding groups, which are the same as in the stable partition of a. Notice that resulted grouping do not depend on a starting position x. 4 The theorem and examples. We begin introducing some notations. Suppose that a has a stable partition with character- istics q, (mi), q ′, (m′i) respectively. In the sequel we will use the following notations: ml−1<u<v≤ml (au − av)2  , (4.9) + (n− q) +  , (4.10) h(x) = e−<x,a> det exifjx (j−m′ l−1−1)1I{m′ l−1<j≤m . (4.11) Moreover we define a function I(a, t) |z|2e ml−1<u<v≤ml (zu−zv)(av−au) ∆(z(m′j−1;m ) d z . (4.12) Remark that from Lemma 5.1 it will follow ml−1<u<v≤ml (au − av)2 = ml−1<u<v≤ml (au − āl)2, where āl = u=ml−1+1 Using this notation we now state a proposition which is useful for calculations in some cases. Proposition 4.1 IPx(τ > t) = (2π) e−γtt− j=1 ( ×e−<x,a> det exkfjx (j−mil−1−1)1I{mil−1<j≤mil} ×I(a, t) (1 + o(1)). (4.13) Remark that formula (4.13) does not give us straightforward asymptotic because inte- gral I(a, t) depends on t. However in some cases this dependence vanishes and this is why Proposition 4.1 can be sometimes useful. The next theorem gives us asymptotic for all cases. Theorem 4.2 For some C given below, as t → ∞ IPx(τ > t) = Ch(x)t −αe−γt(1 + o(1)), γ, α, and h(x) are defined in (4.9),(4.10),(4.11) respectively. To show C we need few more definitions. Let H(s1, . . . , sℓ) = 1≤i≤j≤ℓ+1 (si + . . . + sj−1). (4.14) Define now C = A1 ×A2 ×A3 , (4.15) where A1 = (2π) −n/2√2πn · · · ξi>0:i/∈{m1,...,mq} ml−1<u<v≤ml (ξu+···+ξv−1)(au−av) ξ(mi−1;mi−1) i/∈{m1,...,mq} dξi , · · · ξi>0:i∈{m1,...,mq}\{ml1 ,...,mlq′ } · · · ξi>−∞:i∈{ml1 ,...,mlq′ } k,l∈{m1,...,mq} Sklξkξl i:{i,i+1,...,i+k} ∈{1,...,q}\{l1,...,lq′ } ξmi+j νiνi+k+1 i∈{m1,...,mq} dξi , where Skl = (n− 2)k for k ≤ l and Skl = Slk. In the remaining part of this section we diplay some special cases. Example 2 (a1 = a2 = · · · = an) This is no drift case. Here q = n and m1 = 1,m2 = 2, . . . ,mn = n, also q ′ = 1 and m′1 = n. In result fm1 = a1, . . . , fmn = an. Let a be the common value of the drift. Using Proposition 4.1 we have IPx(τ > t) = (2π) −n/2cne −<x,a> det exkfjx 2 ∆n(z[1;n]) dz (1 + o(1)). First we notice that since all the coordinates in vector f are the same, we have exkfjx = e<x,f> det = e<x,a>∆(x). Furthermore W −f t = W because y1 < y2 < · · · < yn if and only if y1 + a t < y2 + a · · · < yn + a t. Finally we write IPx(τ > t) = C h(x) t −α (1 + o(1)), where h(x) = ∆n(x), C = (2π)−n/2cn 2 ∆(z) dz. Before we state the next example we prove the following lemma. Lemma 4.3 If a ∈ W , then {W − at} → IRn as t → ∞. Proof. Let a ∈ W . We show that for all y ∈ IRn there exists s > 0, such that for all t > s, y ∈ {W − at}. Let y ∈ IRn. We note bi = yi+1 − yi and di = ai+1 − ai. Condition a ∈ W implies di > 0 for all i = 1, 2, . . . , n − 1. We take s = max{−bi, 0}/min{di} and t > s. Set zi = yi + tai, then we get that z ∈ W , because zi+1 − zi = yi+1 + tai+1 − yi − tai = bi + tdi > bi + sdi ≥ bi + max{−bi, 0} ≥ 0. Thus for t > s we have y = z − ta, where z ∈ W , and so y ∈ {W − ta} for all t > s. Example 3 (a1 < a2 < · · · < an) This is the case of non-colliding drifts. Here q = q′ = n, m1 = m 1 = 1, . . . ,mn = m n = n, fm1 = a1, . . . , fmn = an. Using Proposition 4.1 we have IPx(τ > t) = (2π) −n/2e−<x,a> det [exkaj ] |z|2 dz (1 + o(1)). By Lemma 4.3 we have that |z|2 dz = |z|2 dz = (2π)n/2. Finally we write IPx(τ > t) = e −<x,a> det [exkaj ] . This result was derived earlier by Biane et al [2] Example 4 Case when q = q′ = 1. This is the case of a one irreducible drift vector. Here m1 = m 1 = n, f1 = f2 · · · = fn = ā[1;n] = a1+···+an . Using Proposition 4.1 we have IPx(τ > t) = Ch(x)t −αe−γt(1 + o(1)), where 0<u<v≤n (au − av)2 (n− 1)(n + 1) h(x) = e−<x,a> det exifjx C = (2π)−n/2 2πncn · · · ξi>0:i=1,2,...,n−1 ml−1<u<v≤ml (ξu+···+ξv−1)(au−av) ×H(ξ[1;n−1]) dξi . We now analyze a remaining situation for n = 3. Example 5 (a1 > a2 and a1+a2 < a3). This is the case of two subsequences. Thus q = 2, q′ = 2 and m1 = m 1 = 2, m2 = m 2 = 3. By Theorem 4.2 we have IPx(τ > t) = Ch(x)e (a2−a1)2t− where (a2 − a1)2 h(x) = e−<x,a> a1+a2 2 ex1 a1+a2 2 x1 e a1+a2 2 ex2 a1+a2 2 x2 e a1+a2 2 ex3 a1+a2 2 x3 e C = (2π)−3/2 (a1 − a2)2 5 Auxiliary results. For the proof we need a set of lemmas and propositions, presented in subsections below. 5.1 Useful lemmas. We need a few technical lemmas, which we state without proofs. Lemma 5.1 For a ∈ IRm ā[1;m] − ai 1≤u<v≤m (au − av)2. Lemma 5.2 For a,z ∈ IRm ā[1;m] − ai (zv − zu)(au − av). The proof of the following lemma follows easily from Lemmas 5.1 and 5.2. Lemma 5.3 For a,f ∈ Rn such that f is is a vector obtained from the stable partition of a, and z ∈ Rn, we have t + z|2 = |z|2 + ml−1<u<v≤ml (au − av)2 ml−1<u<v≤ml (zv − zu)(au − av) Lemma 5.4 −1 1 0 . . . 0 0 0 −1 1 . . . 0 0 . . . 0 0 0 . . . −1 1 1 1 1 . . . 1 1 (A−1)TA−1 = n− 2 2(n − 2) n− 3 2(n − 3) 3(n − 3) . . . 1 2 3 . . . n− 1 0 0 0 . . . 0 1 Note that (A−1)TA−1 is symmetric. By Proposition 2.2 we have IPx(τ > t) = (2π) −n/2e−<a,x>−||x|| e−||y−a t||2/2 det(exiyj/ t) dy . We now introduce new variable z by y = f t + z, where f = (f1, . . . , fn) is a vector obtained from the stable partition of a. Finally we rewrite formula (2.4) in new variables by the use of Lemma 5.3: Lemma 5.5 IPx(τ > t) = (2π) −n/2e−<a,x>−||x|| 2/2te−γt ml−1<u<v≤ml (zv−zu)(au−av) exi(zj/ t+fj) dz. (5.16) 5.2 Asymptotic behavior of determinant. The following lemma is an extension of Lemma 2 from Pucha la [9] . We define functions gk(z) = det[z det[z for k = (k1, . . . , kn) ∈ Zn and 0 ≤ k1 < · · · < kn Functions g corresponds to Schur functions gk = sk−(0,1,...,n); see e.g. Macdonald [8], Ch. 1.3. Lemma 5.6 Let k0 = exi(zj/ t+fj) t−k/2Tk , (5.17) where z(m′j−1,m k1+···+kn=k k1<···<km′ ...... <···<k gk(m′ (z(m′0,m k1! . . . km′1 ! . . . gk(m′ (z(m′ q′−1,m q′−1+1 ! . . . km′ exifjx In particular as t → ∞ exi(zj/ t+fj) j=1 ( ∆(z(m′ j−1,m × det exkfjx (j−mil−1−1)1I{mil−1<j≤mil} (1 + o(1)). Proof. By Sn we denote the group of permutations on n-set. We write exi(zj/ t+fj) (−1)σe xifσ(i)e xizσ(i)/ (−1)σe xifσ(i) t−k/2(x1zσ(1) + · · · + xnzσ(n))k/k! t−k/2 (−1)σe xifσ(i)(x1zσ(1) + · · · + xnzσ(n))k. Now the coefficient at t−k/2 is equal to Tk = Tk(z) = (−1)σe xifσ(i)(x1zσ(1) + · · · + xnzσ(n))k (−1)σe xifσ(i) k1+···+kn=k k1! . . . kn! (x1zσ(1)) kσ(1) . . . (xnzσ(n)) kσ(n) k1+···+kn=k k1! . . . kn! (−1)σe xifσ(i)(x1zσ(1)) kσ(1) . . . (xnzσ(n)) kσ(n) k1+···+kn=k k1! . . . kn! det[exifjx Recall that f1 = · · · = fm′1 < fm′1+1 = · · · = fm′2 < · · · < fm′q′−1+1 = · · · = fm′q′ . If ki = kj and fi = fj, then the determinant det exifjx is 0. Thus we have non-zero determinant if ki are different for those i such that fi are equal. Thus index k such that Tk is non-zero must be at least kj ≥ k0 = Moreover we get all nonzero det exifjx putting in each subsequence (k(m′0,m , . . . ,k(m′ q′−1,m all possible permutations of strictly ordered numbers from Z+ such that all sum up to k. Thus we have k1+···+kn=k k1<···<km′ ...... <···<k σ1∈Sν′ · · · σq′∈Sν′ σ1(k(m′ (m′0,m k1! . . . km′1 ! . . . σ1(k(m′ q′−1,m q′−1+1 ! . . . km′ × det exifjx σl(kj)1m′ l−1<j≤m Again we notice that permutations in the determinant influence only by the change of sign. These signs and sums over the group of permutations form determinants, thus we have k1+···+kn=k k1<···<km′ ...... <···<k i,j=1 k1! . . . km′1 . . . i,j=m′ q′−1+1 q′−1+1 ! . . . km′ exifjx Remark. Using Itzykson–Zuber integral (see e.g. [6]) we can write exi(zj/ t+fj) ∆(x)∆(z/ t + f) eTrdiag(x)Udiag(z/ t+f)U∗µ( dU) , where µ( dU) is (normalized) Haar measure on the unitary group U(n). Now letting t → ∞, eTrdiag(x)Udiag(z/ t+f)U∗µ( dU) → eTr(diag(x)Udiag(f)U ∗)µ( dU) exifj ∆(x)∆(f) t + f) = t− i=1 ( ∆(z(m′i−1;m 1≤k<l≤n (f l − fk)ν′kν′l(1 + o(1)) . Hence, as t → ∞ exi(zj/ t+fj) i=1 ( ∆(z(m′i−1;m 1≤k<l≤n (f l − fk)ν′kν′l exifj This is a less detailed version of the formula from Lemma 5.6. 6 Proof of the Theorem. Using (5.17) and formula (5.16) we write IPx(τ > t) = = (2π)−n/2e−||x|| 2/2te−<x,a>e−γt |z|2e ml−1<u<v≤ml (zu−zv)(av−au) t−k/2Tk(z) dz ,(6.18) First we will analyze above expression by taking only the first term in the sum (6.18), and then we show that it gives the right asymptotic. Thus the first term equals to (2π)−n/2e−||x|| 2/2te−γt ml−1<u<v≤ml (zu−zv)(av−au) ×e−<x,a> ∆(z(m′j−1;m × det exkfjx (j−mil−1−1)1I{mil−1<j≤mil } k0 dz = (2π)−n/2e−||x|| 2/2te−γt ×e−<x,a> det exkfjx (j−mil−1−1)1I{mil−1<j≤mil} I(a, t), where I(a, t) was introduced in (4.12). 6.1 Asymptotic behavior of integral. If s = Az, where −1 1 0 . . . 0 0 0 −1 1 . . . 0 0 . . . 0 0 0 . . . −1 1 1 1 1 . . . 1 1 than zu − zv = sv + sv+1 + · · · + su−1 and |z|2 = zTz = (A−1s)T (A−1s) = sT (A−1)TA−1s. Hence by Lemma 5.4 we have |z|2 = s2n + s (n)((A −1)TA−1)(n)s(n), where s(n) is obtained from s by deleting the n th coordinate and A(n) is matrix A without nth row and nth column. After substitution s = Az, integral I(a, t) is I(a, t) = · · · si>(fi−fi+1) for i=1,...,n−1 ((A−1)TA−1)(n)s(n)) ml−1<u<v≤ml (su+···+sv−1)(au−av) (6.19) H(s(m′ k−1;m ) ds(n) dsn · · · si>(fi−fi+1) for i=1,...,n−1 ((A−1)TA−1)(n)s(n)) ml−1<u<v≤ml (su+···+sv−1)(au−av) (6.20) H(s(m′ k−1,m )) ds(n). It is important to notice that the second exponent in integral I(a, t) in (6.19) depends only on those si, where i /∈ {m1, . . . mq}. We also see that if mi−1 < k < mi, then the coefficient at sk in (6.19) is mi −mi−1 (mi − k)(k −mi−1) ami−1+1 + · · · + ai i−mi−1 ai+1 + · · · + ami mi − i and it is strictly negative by the definition of the stable partition. Note also that polynomials H in integral I(a, t) depends only on sj, where j /∈ {m′1, . . . ,m′q′}. We now introduce new variables ξ = (ξ1, . . . , ξn−1) by tsj, for j 6= mi, j = 1, . . . , n− 1, i = 1, . . . , q − 1 sj, for j = mi, j = 1, . . . , n− 1, i = 1, . . . , q − 1. (6.21) We define function K by K k−1,m k−1,m Consider now H s(1;m′1) . Since m′ is a subsequence of m, we recall that i1 is such that mi1 = m 1. Similarly are defined i1, . . . , iq′ . We now factorize H s(1;m′1) into parts in which there in none of mi, where is exactly one mi, exactly two and so on. Thus s(m0;m′1) s(mk−1;mk) mk−1<i≤mk mk<j≤mk+1 (si + · · · + sj−1) mk−1<i≤mk mk+1<j≤mk+2 (si + · · · + sj−1) m0<i≤m1 mi1−1<j≤mi1 (si + · · · + sj−1). We make analogous factorization for other H(s(mk−1;mk)). Lemma 6.1 As t → ∞ K(ξ(m′ k−1,m ), t) = t i=1 ( H(ξ(mi−1;mi)) i:{i,i+1,...,i+k} ∈{1,...,q}\{i1,...,iq′ } ξmi+j νiνi+k+1 (1 + o(1)) Proof. After the substitution we get K(ξ(m′ k−1,m ), t) = ξ(mk−1;mk)/ mk−1<i≤mk mk<j≤mk+1 r=i,r 6=mk t + ξmk mk−1<i≤mk mk+1<j≤mk+2 r=i,r /∈{mk ,mk+1} t + ξmk + ξmk+1 m0<i≤m1 mi1−1<j≤mi1 r=i,r /∈{m1,...,mi1−1} It is not difficult to see that asymptotic behavior of the above expression is K(ξ(m′ k−1,m ), t) = t l=1 ( ξ(mk−1;mk) (ξmk) νkνk+1 (ξmk + ξmk+1) νkνk+2 )ν1νi1 (1 + o(1)). In result the whole polynomial is asymptotically K(ξ(m′ k−1;mk) , t) = t− i=1 ( ξ(mi−1;mi) i:{i,i+1,...,i+k} ∈{1,...,q}\{i1,...,iq′ } ξmi+j νiνi+k+1 (1 + o(1)). For substitution (6.21), we have ds(n) = t −(n−q)/2 dξ. Note that fk+1 = fk for k 6= mi, and hence the integration on the kth coordinate starts from 0. On the other hand if k = mi for some i, and k 6= m′j for every j, then we also have fk+1 = fk and therefore the integration starts from 0. Finally if k = mij for some j, then fk+1 > fk and the integrations starts from (fk − fk+1) t. Hence we have after the substitution I(a, t) = t−(n−q)/2 · · · ξj>(fj−fj+1) for i=1,...,n−1 k,l∈{m1,...,mq} Sklξkξl) k,l/∈{m1,...,mq} Sklξkξl/t+2 k∈{m1,...,mq},l/∈{m1,...,mq} Sklξkξl/ ml−1<u<v≤ml (ξu+···+ξv−1)(au−av) K(ξ(m′ k−1,m ), t) dξ . So we can clearly see that k=1K(ξ(m′k−1,m ), t) depends only on ξi’s such that i /∈ {ml1 , . . . ,mlq′} and it can be factorized into a part which depends only on i /∈ {m1, . . . ,mq} and a part that depends on i ∈ {m1, . . . ,mq} \ {ml1 , . . . ,mlq′}. Thus finally we can write I(a, t) = t−(n−q)/2 · · · ξi>(fi−fi+1) for i=1,...,n−1 k,l∈{m1,...,mq} Sklξkξl) k,l/∈{m1,...,mq} Sklξkξl/t+2 k∈{m1,...,mq},l/∈{m1,...,mq} Sklξkξl/ ml−1<u<v≤ml (ξu+···+ξv−1)(au−av) i=1 ( H(ξ(mi−1;mi), t) i:{i,i+1,...,i+k} ∈{1,...,q}\{i1,...,iq′ } ξmi+j νiνi+k+1 dξ (1 + o(1)) Hence I(a, t) = t−(n−q)/2t− i=1 ( · · · ξi>0:i=1,...,n−1 i/∈{m1,...,mq} ml−1<u<v≤ml (ξu+···+ξv−1)(au−av) H(ξ(mi−1;mi)) i/∈{m1,...,mq} · · · ξi>0:i∈{m1,...,mq}\{ml1 ,...,mlq′ } · · · ξi>−∞:i∈{ml1 ,...,mlq′ } k,l∈{m1,...,mq} Sklξkξl i:{i,i+1,...,i+k} ∈{1,...,q}\{l1,...,lq′ } ξmi+j νiνi+k+1 i∈{m1,...,mq} dξi (1 + o(1)) . (6.22) Concluding we have I(a, t) = C1t −(n−q)/2t− i=1 ( 2 )(1 + o(1)), where C1 depends only on drift vector a. 6.2 Proof of Theorem 4.2. Following considerations of Section 6.1, notice first that it suffices to take the first term from the sum (6.18) for asymptotic analysis because next terms consists of positive rank polynomials of variable z and therefore they will tend to zero faster after substitution (6.21). For the proof of the main theorem we have to plug the asymptotics (6.22) to integral (6.18). References [1] S. Asmussen (2003) Applied Probability and Queues. Second Ed., Springer , New York. [2] Ph. Biane, Ph. Bougerol and N. O’Connell (2005) Littelmann paths and Brownian paths. Duke Math. J. 130, 127–167. [3] A.N. Borodin and P. Salminen (2002) Handbook of Brownian Motion - Facts and For- mulae. Birkhäuser Verlag, Basel. [4] Y. Doumerc, N. O’Connell (2005) Exit problems associated with finite reflection groups. Probability Theory and Related Fields 132, 501 - 538 [5] D.J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices. Ann. Inst. H. Poincaré. Probab. Statist.35 (1999), 177-204. [6] C. Itzykson and J.-B. Zuber (1980) The planar approximation II. J. Math. Phys. 21, 411–421 . [7] S. Karlin and J. McGregor (1959) Coincidence probabilities. Pacific J. Math. 1141–1164. [8] I.G. Macdonald, Symetric Functions and Hall Polynomials. Clarendon Press (1979), Oxford. [9] Z. Pucha la (2005) A proof of Grabiner theorem on non-colliding particles. Probability and Mathematical Statistics 25, 129–132. [10] Z. Pucha la and T. Rolski (2005) The exact asymptotics of the time to collison. Electronic Journal of Probability 10, 1359–1380. Introduction and results Formula for IPbold0mu mumu xxxxxx(>t). Stable partition of bold0mu mumu aaaaaa. The theorem and examples. Auxiliary results. Useful lemmas. Asymptotic behavior of determinant. Proof of the Theorem. Asymptotic behavior of integral. Proof of Theorem ??.

0704.0216

Ab initio Study of Graphene on SiC

Ab initio Study of Graphene on SiC Alexander Mattausch∗ and Oleg Pankratov Theoretische Festkörperphysik, Universität Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany (Dated: October 30, 2018) Employing density-functional calculations we study single and double graphene layers on Si- and C-terminated 1 × 1 - 6H-SiC surfaces. We show that, in contrast to earlier assumptions, the first carbon layer is covalently bonded to the substrate, and cannot be responsible for the graphene- type electronic spectrum observed experimentally. The characteristic spectrum of free-standing graphene appears with the second carbon layer, which exhibits a weak van der Waals bonding to the underlying structure. For Si-terminated substrate, the interface is metallic, whereas on C-face it is semiconducting or semimetallic for single or double graphene coverage, respectively. PACS numbers: 68.35.Ct, 68.47.Fg, 73.20.-r The last years have witnessed an explosion of inter- est in the prospect of graphene-based nanometer-scale electronics [1, 2, 3, 4]. Graphene, a single hexagonally ordered layer of carbon atoms, has a unique electronic band structure with the conic “Dirac points” at two in- equivalent corners of the two-dimensional Brillouin zone. The electron mobility may be very high and lateral pat- terning with standard lithography methods allows de- vice fabrication [1]. Two ways of obtaining graphene samples have been used up to now. In the first “me- chanical” method, the carbon monolayers are mechani- cally split off the bulk graphite crystals and deposited onto a SiO2/Si substrate [4]. This way an almost “free- standing” graphene is produced, since the carbon mono- layer is practically not coupled to the substrate. The sec- ond method uses epitaxial growth of graphite on single- crystal silicon carbide (SiC). The ultrathin graphite layer is formed by vacuum graphitization due to Si depletion of the SiC surface [5]. This method has apparent techno- logical advantages over the “mechanical” method, how- ever it does not guarantee that an ultrathin graphite (or graphene) layer is electronically isolated from the sub- strate. Moreover, one expects a covalent coupling be- tween both which may strongly modify the electronic properties of the graphene overlayer. Yet, experiments show that the transport properties of the interface are dominated by a single epitaxial graphene layer [1, 2]. Most surprisingly, the electronic spectrum seems not to be affected much by the substrate. As in free-standing graphene one observes the “Dirac points” with the linear dispersion relation around them. The electron dynamics is governed by a Dirac-Weyl Hamiltonian with the Fermi velocity of graphene replacing the speed of light. This leads to an unusual sequence of Landau levels in a mag- netic field and hence to peculiar features in the quantum Hall effect [1, 4]. The growth of high-quality graphene layers on both Si-terminated or C-terminated SiC{0001} surfaces oc- curs in vacuum at annealing temperatures above 1400◦C. The geometric structure of the interface is unclear. For- beaux et al. [5] proposed that on the Si-face the graphite FIG. 1: (Color online) Side view (a) and top view (b) of a graphene layer on the SiC(0001) surface. The 3R30◦ surface unit cell is highlighted. layer is loosely bound by van der Waals-forces to the√ 3R30◦-reconstructed substrate. On the con- trary, combining STM and LEED data with DFT cal- culations Chen et al. [6] came to the conclusion that the graphite sheet is formed on a complex 6 × 6-structure, from which originates the observed 6 3 × 6 3R30◦ re- construction that precedes the graphite formation. On the C-terminated SiC(0001̄) face, graphite growth on top of a 2 × 2 reconstruction was reported [5, 7]. Berger et al. [1, 2] observed the formation of large high-quality graphene islands on top of a 1×1 C-terminated SiC sub- strate with a 3R30◦ interface reconstruction. In this work we employ an ab initio density-functional theory approach to study the bonding and electronic structure of graphene on SiC. We find that a strong co- valent bonding of the first carbon layer to the substrate removes the graphene-type electronic features from the energy region around the Fermi level. However, these features reappear with the second carbon layer. We also compare the electronic properties of graphene on Si- and C-terminated surfaces. Our calculations were performed with the density- functional theory program package VASP [8, 9, 10, 11] in the local spin density approximation (LSDA). Projector augmented wave (PAW) pseudopotentials [12] were used. A special 7× 7× 1 k-point sampling was applied for the Brillouin-zone integration. The plane wave basis set was http://arxiv.org/abs/0704.0216v2 restricted by a cut-off energy of 400 eV. We have chosen a 6H-SiC polytype, which is most often used in experimen- tal studies. The supercell was constructed of 6 bi-layers of SiC in the S3-structure [13], one or two carbon mono- layers and a vacuum interval needed to separate the slabs. The vacuum separation varied, depending on the carbon coverage, between 10 to 15 Å. The graphene layer was placed on top of the unreconstructed 6H-SiC substrate such that the structure had a lateral 3R30◦ el- ementary cell (Fig. 1a). Due to the lattice mismatch of 8% between SiC and graphite, this requires stretching the graphene layer. We verified that for the free-standing graphene layer the stretch reduces the total bandwidth from 19.1 eV to 17.3 eV but does not affect the electronic spectrum close to the Fermi energy. The elastic energy is 0.8 eV per graphene unit cell. The interface unit cell (cf. Fig. 1b) contains three sur- face atoms of the substrate and four elementary unit cells of graphene. The dangling bonds of the substrate atoms at the corners of the unit cell are unsaturated, while the other surface atoms bind to two carbon atoms of the hexagonal graphene ring. In case of the Si-terminated SiC(0001) surface, we find that the graphene layer is sep- arated by 2.58 Å from the SiC substrate. The carbon atoms covalently bonded to the substrate relax towards the SiC surface, such that the bond length is 2.0 Å. This is only slightly longer than the bond length 1.87 Å in SiC. The graphene bonding releases 0.72 eV per graphene unit cell. For the C-terminated SiC(0001̄) face, the graphene layer is somewhat closer (2.44 Å) to the substrate and the bond length between the bonding carbon atoms reduces to 1.87 Å. The energy gain is 0.60 eV per graphene unit cell. On both interfaces, the bonding atom of the sub- strate relaxes outwards, whereas the partner graphene atom moves towards the substrate. The bonding ener- gies are quite close but somewhat smaller than the elastic deformation energy of the graphene layer. However, the latter can be drastically lessened by defects which result from the lattice mismatch. For a second graphene layer placed in the graphite- type AB stacking, we find a weak bonding at a distance of 3.3 Å, very close to the bulk graphite value 3.35 Å. This conforms to the fact that LSDA, despite the lack of long-range non-local correlations, produces reason- able interlayer distances in van der Waals crystals like graphite [14, 15] or h-BN [16]. As shown by Marini et al. [16], a delicate error cancellation between exchange and correlation underlies this apparent performance of the LSDA. The semilocal GGA, which violates this bal- ance, fails to generate the interplanar bonding in both graphite [15] and h-BN [16], while producing a band structure identical to LSDA [15]. It is thus natural to assume that in our situation the bonding between the graphene layers is the same as in bulk graphite with the same interplanar distance. To reduce the calculational cost, we fixed the interplanar distance at this value. The first graphene layer, which is covalently bonded to the substrate, thus serves as a buffer separating the SiC crystal and the van der Waals bonded second graphene sheet. Most probably, the 6 3R30◦ reconstructed carbon-rich Si-terminated surface observed as a precur- sor of graphitization is a natural realization of this buffer layer in the epitaxial process. The 6 3 structure is practically commensurate with graphene since 13 times the graphene lattice constant almost precisely fits 6 times the SiC lattice parameter. In any case, there is no stress in the second carbon layer. Even placed on a strongly stretched buffer layer, the upper layer relaxes to its natural lattice constant due to the weak interlayer in- teraction. For the C-face Berger et al. [1] found graphene formation on a 1× 1 substrate with a 3 interface unit cell. This structure is the same as we used in our calculations. Figs. 2a and 2b show the electronic energy spectrum of a single graphene layer on the two SiC surfaces. The shaded regions are the projected conduction and valence energy bands of SiC. The Kohn-Sham energy gap of 1.98 eV is smaller than the optical band gap (3.02 eV) of the bulk 6H-SiC, which is a common consequence of LSDA. The covalent bonding drastically changes the graphene electron spectrum at the Fermi energy. The “Dirac cones” are merged into the valence band, whereas the upper graphene bands overlap with the SiC conduc- tion band. Hence a wide energy gap emerges in the graphene spectrum. A similar gap opening due to hy- drogen absorption on a single graphene sheet was pre- dicted in Ref. 17. The weakly dispersive interface states visible in Figs. 2a and 2b result from the interaction of the graphene layer with the three dangling orbitals of the substrate. Two of them make covalent bonds, while the third one in the center of the graphene ring remains unsaturated (cf. Fig. 1b). A projection analysis of the wave functions reveals that the gap states close to the Fermi energy originate from the remaining dangling bonds of the substrate. On the Si-face we find a half-filled metallic state, whereas on the C-face the interface state is split into a singly occupied (spin polarized) and an empty state, making the interface insulating. In contrast, on both clean SiC surfaces LSDA predicts a substantial splitting of the surface states (0.86 eV for SiC(0001) and 0.45 eV for SiC(0001̄), see Table I). Actually, the gap separating a singly occupied and an empty state is larger due to the Hubbard repulsion of the electrons (about 2 eV for the 3R30◦ reconstructed surface [18, 19]), but already LSDA correctly reproduces the insulating char- acter of both surfaces. The reason for the striking difference between the two graphene-covered surfaces becomes clear if one compares the planar localization of the two gap states. As seen in Fig. 3a for the Si-face the interface state electron density is strongly delocalized. As the projection analysis shows, this results from the hybridization with the graphene- Γ Κ Μ Γ SiC(0001)/Graphene Γ Κ Μ Γ SiC(0001)/Graphene Γ ΓΚ Μ SiC(0001)/2 graph. lay. Γ ΓΚ Μ SiC(0001)/2 graph. lay. c) d) FIG. 2: Energy spectrum of the interface states of a) the SiC(0001)/graphene interface, b) the SiC(0001̄)/graphene interface, c) SiC(0001) with two layers of graphene and d) SiC(0001̄) with two layers of graphene. The Fermi energy is indicated by the dashed line. K̄ and M̄ are the high-symmetry points of the surface Brillouin zone of the 3R30◦ surface unit cell. FIG. 3: (Color online) Charge density of the interface states at the Fermi energy for a single graphene layer on a) SiC(0001) and b) SiC(0001̄). induced electron states overlapping with the conduction band (see Fig. 2a). Given the delocalized nature of the interface state we expect the influence of Hubbard cor- relations to be small. In contrast, at the C-terminated substrate the electron state retains its localized charac- ter, although it is smeared over a carbon ring just above the unsaturated C-dangling bond. The localization fa- vors the spin polarization and thus the splitting of the gap state, whereas the interface state at the Si-face re- mains spin-degenerate. In the former case, Hubbard cor- relations may lead to a further splitting of the interface state. Figures 2c and 2d show that the second carbon layer indeed possesses an electronic structure similar to free- standing graphene. The characteristic conic point ap- pears on the Γ̄ − K̄ line (note that since the Brillouin zone corresponds to the 3R30◦ unit cell, the conic point is not located at the K̄-point). The interface states of the buffer layer remain practically unchanged since the interaction of the carbon layers is very small. The metallic interface state on the Si-terminated substrate pins the Fermi level just above the conic point, making the second graphene layer n-doped. On C-terminated substrate the Fermi level runs exactly through the conic point. Hence the interface is semimetallic just as for free- standing graphene. Indeed, for a graphene-covered C- face Berger et al. [1] found that the thin graphite layers possess electronic properties of free-standing graphene. The parameters of the electron states for the different interfaces are summarized in Table I. For clean unre- constructed surfaces we find work functions of 4.75 eV (Si-terminated surface) and 5.75 eV (C-terminated sur- face). The former value is practically the same as the work function of the reconstructed SiC(0001) [20]. The first graphene layer reduces this value to 3.75 eV, which is 1.3 eV lower than the work function of free-standing graphene. The drastic reduction of the work function is caused by charge flow from graphene to the interface region, which induces a dipole layer. On the C-face the graphene overlayer also reduces the work function, but to a lesser extent such that it remains above the graphene value. Adding the second graphene layer makes the work function closer to that of graphene for both faces. The Fermi level pinning close to the conduction band makes the graphitized Si-face especially suitable for Ohmic contacts on n-type SiC, because it guarantees a low Schottky barrier. Indeed, Lu et al. [21] find a very low resistance for thermally treated SiC contacts with TABLE I: Parameters of the unreconstructed and graphene-covered SiC{0001} surfaces in eV: work function φ, positions of the occupied and the unoccupied surface and interface states above the valence band edge (Eo, Eu) and their corresponding bandwidths (Bo, Bu). Work function φ Eo Bo Eu Bu SiC(0001) 1×1 4.75 Ev + 0.92 0.45 Ev + 1.78 0.53 SiC(0001)/Graphene 3.75 Ev + 1.64 0.35 − − SiC(0001)/2 Graphene 4.33 Ev + 1.64 0.40 − − SiC(0001̄) 1×1 5.75 Ev + 0.05 0.75 Ev + 0.50 0.45 SiC(0001̄)/Graphene 5.33 Ev + 0.43 0.13 Ev + 1.19 0.14 SiC(0001̄)/2 Graphene 5.31 Ev + 0.44 0.10 Ev + 1.19 0.15 Graphene (single layer) 5.11 nickel and cobalt, while other metals, which form car- bides and thereby remove the graphitic inclusions, were rectifying. Recently Seyller et al. measured the Schot- tky barrier between n-type 6H-SiC(0001) and graphite by photoelectron spectroscopy and found a low value of 0.3 eV [22]. On the contrary, the C-terminated face has the Fermi level close to the middle of the band gap and is semiconducting or semimetallic. In conclusion, we investigated the interface between 1 × 1 - 6H-SiC{0001} surfaces and carbon layers em- ploying ab initio density-functional theory. We find that graphene overlayers on SiC(0001) and SiC(0001̄) faces possess qualitatively different electronic structures. While the former is metallic, the latter has semiconduct- ing properties. The conic points at the Fermi energy, which are specific for graphene, appear only with the sec- ond layer. The first carbon sheet is covalently bound to the substrate and plays the role of a transition region be- tween a covalent SiC crystal and a van der Waals bonded stack of graphene layers. This work was supported by Deutsche Forschungsge- meinschaft within the SiC Research Group. We are grate- ful to L. Magaud and F. Varchon for communicating to us similar results on the SiC/graphene system [23] and fruitful discussions. ∗ Electronic address: [email protected] [1] C. Berger, Z. Song, X. Li, X. Wu, N. Brown, C. Naud, D. Mayou, T. Li, J. Hass, A. N. Marchenkov, et al., Sci- ence 312, 1191 (2006). [2] J. Hass, C. A. Jeffrey, R. Feng, T. Li, X. Li, Z. Song, C. Berger, W. A. de Heer, P. N. First, and E. H. Conrad, Appl. Phys. Lett. 89, 143106 (2006), cond-mat/0604206. [3] T. Seyller, K. V. Emtsev, K. Gao, F. Speck, L. Ley, A. Tadich, L. Broekmann, J. D. Riley, R. C. G. Leckey, O. Rader, et al., Surf. Sci. 600, 3906 (2006). [4] Y. Zhang, Z. Jiang, J. P. Small, M. S. Purewal, Y.-W. Tan, M. Fazlollahi, J. D. Chudow, J. A. Jaszczak, H. L. Stormer, and P. Kim, Phys. Rev. Lett. 96, 136806 (2006). [5] I. Forbeaux, J.-M. Themlin, and J.-M. Debever, Phys. Rev. B 58, 16396 (1998). [6] W. Chen, H. Xu, L. Liu, X. Gao, D. Qi, G. Peng, S. C. Tan, Y. Feng, K. P. Loh, and A. T. S. Wee, Surf. Sci. 596, 176 (2005). [7] I. Forbeaux, J.-M. Themlin, A. Charrier, F. Thibaudau, and J.-M. Debever, Appl. Surf. Sci. 162-163, 406 (2000). [8] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993). [9] G. Kresse, Ph.D. thesis, Technische Universität Wien, Austria (1993). [10] G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996). [11] G. Kresse and J. Furthmüller, Comput. Mat. Sci 6, 15 (1996). [12] G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). [13] U. Starke, J. Schardt, J. Bernhardt, M. Franke, and K. Heinz, Phys. Rev. Lett. 82, 2107 (1999). [14] J.-C. Charlier, X. Gonze, and J. P. Michenaud, Carbon 32, 289 (1994). [15] N. Ooi, A. Rairkar, and J. B. Adams, Carbon 44, 231 (2006). [16] A. Marini, P. Garćıa-González, and A. Rubio, Phys. Rev. Lett. 96, 136404 (2006). [17] E. J. Duplock, M. Scheffler, and P. J. D. Lindan, Phys. Rev. Lett. 92, 225502 (2004). [18] M. Rohlfing and J. Pollmann, Phys. Rev. Lett. 84, 135 (2000). [19] V. I. Anisimov, A. E. Bedin, M. A. Korotin, G. San- toro, S. Scandolo, and E. Tosatti, Phys. Rev. B 61, 1752 (2000). [20] M. Wiets, M. Weinelt, and T. Fauster, Phys. Rev. B 68, 125321 (2003). [21] W. Lu, W. C. Mitchel, G. R. Landis, T. R. Crenshaw, and W. E. Collins, J. Appl. Phys. 93, 5397 (2003). [22] T. Seyller, K. V. Emtsev, F. Speck, K.-Y. Gao, and L. Ley, Appl. Phys. Lett. 88, 242103 (2006). [23] F. Varchon, L. Magaud, and V. Olevano, private com- munication; F. Varchon, R. Feng, J. Hass, X. Li, B. N. Nguyen, C. Naud, P. Mallet, J. Y. Veuillen, C. Berger, E. H. Conrad, L. Magaud, arXiv:cond-mat/0702311. mailto:[email protected]

0704.0217

Capacity of a Multiple-Antenna Fading Channel with a Quantized Precoding Matrix

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 1 Capacity of a Multiple-Antenna Fading Channel With a Quantized Precoding Matrix Wiroonsak Santipach, Member, IEEE, and Michael L. Honig, Fellow, IEEE Abstract—Given a multiple-input multiple-output (MIMO) channel, feedback from the receiver can be used to specify a transmit precoding matrix, which selectively activates the strongest channel modes. Here we analyze the performance of Random Vector Quantization (RVQ), in which the precoding ma- trix is selected from a random codebook containing independent, isotropically distributed entries. We assume that channel elements are i.i.d. and known to the receiver, which relays the optimal (rate-maximizing) precoder codebook index to the transmitter using B bits. We first derive the large system capacity of beamforming (rank-one precoding matrix) as a function of B, where large system refers to the limit as B and the number of transmit and receive antennas all go to infinity with fixed ratios. RVQ for beamforming is asymptotically optimal, i.e., no other quantization scheme can achieve a larger asymptotic rate. We subsequently consider a precoding matrix with arbitrary rank, and approximate the asymptotic RVQ performance with optimal and linear receivers (matched filter and Minimum Mean Squared Error (MMSE)). Numerical examples show that these approximations accurately predict the performance of finite-size systems of interest. Given a target spectral efficiency, numerical examples show that the amount of feedback required by the linear MMSE receiver is only slightly more than that required by the optimal receiver, whereas the matched filter can require significantly more feedback. Index Terms—Beamforming, large system analysis, limited feedback, Multi-Input Multi-Output (MIMO), precoding, vector quantization. I. INTRODUCTION G IVEN a multi-input multi-output (MIMO) channel, pro-viding channel information at the transmitter can in- crease the achievable rate and simplify the coder and decoder. Namely, this channel information can specify a precoding matrix, which aligns the transmitted signal along the strongest channel modes (i.e., singular vectors corresponding to the Manuscript recieved December 22, 2006; revised July 17, 2008. This work was supported by the U.S. Army Research Office under grant DAAD19-99-1- 0288 and the National Science Foundation under grant CCR-0310809, and was presented in part at IEEE Military Communications (MILCOM), Boston, MA, USA, October 2003, IEEE International Symposium on Information Theory (ISIT), Chicago, IL, USA, June 2004, and IEEE International Symposium on Spread Spectrum Techniques and Applications (ISSSTA), Sydney, Australia, August 2004. W. Santipach was with the Department of Electrical Engineering and Computer Science; Northwestern University, Evanston, IL 60208 USA. He is currently with the Department of Electrical Engineering; Faculty of En- gineering; Kasetsart University, Bangkok, 10900 Thailand (email: wiroon- [email protected]). M. L. Honig is with the Department of Electrical Engineering and Com- puter Science; Northwestern University, Evanston, IL 60208 USA (email: [email protected]). Communicated by H. Boche, Associate Editor for Communications. Digital Object Identifier 10.1109/TIT.2008.2011437 largest singular values). In practice, the precoding matrix must be quantized at the receiver, and relayed to the transmitter via a feedback channel. The corresponding achievable rate is therefore limited by the accuracy of the quantizer. The design and performance of quantized precoding matri- ces for multi-input single-output (MISO) and MIMO channels has been considered in numerous references, including [1]– [12]. In those references, and in this paper, the channel is assumed to be stationary, known at the receiver, and the perfor- mance is evaluated as a function of the number of quantization bits B. (This is in contrast with other work, which models estimation error at the receiver, but does not explicitly account for quantization error (e.g., [13], [14]), and which assumes a time-varying channel with feedback of second-order statistics [15]–[18].) Optimization of vector quantization codebooks is discussed in [1]–[3], [5], [6] for beamforming, and in [4], [7], [12] for MIMO channels with precoding matrices that provide multiplexing gain (i.e., have rank larger than one). It is shown in [2], [3] that this optimization can be interpreted as maximizing the minimum distance between points in a Grassmannian space. (See also [9].) The performance of this class of Grassmannian codebooks is also studied in [8]–[10]. In this paper, we evaluate the performance of a Random Vector Quantization (RVQ) scheme for the precoding matrix. Namely, given B feedback bits, the precoding matrix is selected from a random codebook containing 2B matrices, which are independent and isotropically distributed. RVQ has been analyzed in other source coding contexts (e.g., see [19] and the related discussion in [20]), and achieves the rate- distortion bound for ergodic Gaussian sources. This work is motivated by prior work [21] in which RVQ is considered for signature quantization in Code-Division Multiple Access (CDMA). In that scenario, limited feedback is used to select a signature for a particular user, which maximizes the received Signal-to-Interference-Plus-Noise-Ratio (SINR). RVQ has the attractive properties of being tractable and asymptotically optimal. Namely, in [21] the received SINR with RVQ is evaluated in the asymptotic (large system) limit as processing gain, number of users, and feedback bits all tend to infinity with fixed ratios. Furthermore, it is shown that no other quantization scheme can achieve a larger asymptotic SINR. Here we assume an i.i.d. block Rayleigh fading channel model with independent channel gains, and take ergodic capacity as the performance criterion. The receiver relays B bits to the transmitter (per codeword) via a reliable feedback channel (i.e., no feedback errors) with no delay. We start by evaluating the capacity of MISO and MIMO channels with a 0018–9448/$25.00 c© 2009 IEEE http://arxiv.org/abs/0704.0217v2 2 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 quantized beamformer, i.e., rank-one precoding matrix. Our results are asymptotic as the number of transmit antennas Nt and feedback bits B both tend to infinity with fixed B/Nt (feedback bits per degree of freedom). For the MIMO channel the number of receive antennas Nr also tends to infinity in proportion with Nt and B. The asymptotic expressions accurately predict the performance of finite-size systems of interest as a function of normalized feedback and background Signal-to-Noise Ratio (SNR). In analogy with the optimality result shown in [21], RVQ is also asymptotically optimal in this scenario, i.e., no other quantization scheme can achieve a larger asymptotic rate. Furthermore, numerical examples for small Nt show that RVQ performance averaged over code- books is essentially the same as that obtained from codebooks optimized via the Lloyd-Max algorithm [1], [5], [6]. (See also the numerical examples in [22], which compare RVQ performance with the optimized (Grassmannian) codebooks in [3].) We then consider quantization of a precoding matrix with arbitrary rank. Namely, a rank K precoding matrix multiplexes K independent streams of transmitted information symbols onto the Nt transmit antennas. In that case, the capacity with limited feedback is approximated in the limit as B, Nt, Nr, and K all tend to infinity with fixed ratios K/Nt, Nr/Nt, and B/N2r . That is, the number of feedback bits again scales linearly with the number of degrees of freedom, which is pro- portional to N2r . Although our results for beamforming suggest that RVQ is also asymptotically optimal in this scenario, this remains an open question. The asymptotic results for a precoder matrix with arbitrary rank K can be used to determine the normalized rank, or multiplexing gain K/Nt, which maximizes the capacity. This optimized rank in general depends on the normalized feedback, the ratio of antennas Nt/Nr, and the SNR. For example, if Nt/Nr ≥ 1 and the SNR is sufficiently large, then as the feedback increases from zero to infinity, the optimized rank decreases from one to Nr/Nt. Numerical results are presented, which illustrate the effect of normalized rank on achievable rate, and also show that the asymptotic results accurately predict simulated results for finite-size systems of interest. We also evaluate the performance of RVQ with linear receivers (i.e., the matched filter and linear Minimum Mean Squared Error (MMSE) receivers), and compare their perfor- mance with the optimal (capacity-achieving) receiver. With the optimal precoding matrix, corresponding to infinite feedback, both linear receivers are optimal. With limited feedback the two linear receivers are simpler than the optimal receiver, but require more feedback to achieve a target rate. Numerical results show that this additional feedback required by the linear MMSE receiver is quite small, whereas the additional feedback required by the matched filter can be significant (e.g., about one bit per precoding matrix element). In addition to quantizing the optimal precoding matrix, power for each data stream can also be optimized, quantized, and fed back to the transmitter (e.g., see [23], [24]). Asymptot- ically, the amount of feedback required to specify the power is negligible compared to the feedback required for the precoding matrix. Furthermore, uniform power over the set of activated channel typically performs close to the optimal (water-filling) performance [8]. We therefore only consider quantization of the precoding matrix. Other related work on RVQ for MIMO channels has been presented in [25]–[27]. Namely, exact expressions for the ergodic capacity with beamforming and RVQ for a finite-size MISO channel are derived in [25]. The performance of RVQ for precoding over a broadcast MIMO channel is analyzed in [26], [27]. A closely related random beamforming scheme for the multiuser MIMO broadcast channel was previously presented in [28]. In that work, the growth in sum capacity is characterized asymptotically as the number of users becomes large with a fixed number of antennas. (Random beamforming was previously proposed in [29], although there the main focus is to improve fairness among users.) The paper is organized as follows. Section II describes the channel model, Section III considers the capacity of beamforming with limited feedback, and sections IV and V examine the capacity of a quantized precoding matrix with optimal and linear receivers, respectively. Derivations of the main results are given in the appendices. II. CHANNEL MODEL We consider a point-to-point, flat Rayleigh fading channel with Nt transmit antennas and Nr receive antennas. Let x = [xk] be a K×1 vector of transmitted symbols with covariance matrix IK , where IK is the K ×K identity matrix, and K is the number of independent data streams. The received Nr × 1 vector is given by HV x+ n (1) where H = [hnr ,nt ] is an Nr × Nt channel matrix, V = [v1 v2 . . . vK ] is an Nt ×K precoding matrix, and n is a complex Gaussian noise Nr×1 vector with covariance matrix σ2nINr . Assuming rich scattering and Rayleigh fading, the elements of H are independent, and the channel coefficient be- tween the ntth transmit antenna and the nrth receive antenna, hnr,nt , is a circularly symmetric complex Gaussian random variable with zero mean and unit variance (E[|hnr,nt |2] = 1). We assume i.i.d. block fading, i.e., the channel is static within a fading block, and the channels across blocks are independent. The ergodic capacity is achieved by coding the transmitted symbols across an infinitely large number of fading blocks. With perfect channel knowledge at the receiver and a given precoding matrix V , the ergodic capacity is the mutual information between x and y with a complex Gaussian distributed input, averaged over the channel, given by I(x;y) = EH log det where ρ = 1/σ2n is the background SNR. We wish to specify the precoding matrix V that maximizes the mutual information, subject to a power constraint ‖vk‖ ≤ 1, for 1 ≤ k ≤ K . With unlimited feedback, the columns of the optimal pre- coding matrix, which maximizes (2), are eigenvectors of the channel covariance matrix H†H . With B feedback bits per SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 3 fading block, we can specify the precoding matrix from a quantization set or codebook V = {V1, · · · ,V2B} known a priori to both the transmitter and receiver. The receiver chooses the Vj that maximizes the sum mutual information, and relays the corresponding index back to the transmitter. Of course, the performance (ergodic capacity) depends on the codebook V . III. BEAMFORMING WITH LIMITED FEEDBACK We start with a rank-one precoding matrix, corresponding to a single data stream (K = 1). In that case, the precoding matrix is specified by an Nt×1 beamforming vector v, which ideally corresponds to the strongest channel mode. That is, the optimal v, which maximizes the ergodic capacity in (2), is the eigenvector of H†H corresponding to the largest eigenvalue. This vector is computed at the receiver and a quantized version is relayed back to the transmitter. Let V = {v1, . . . ,v2B} denote the quantization codebook for v, given B feedback bits. Optimization of this codebook has been considered in [2], [3], [6] with outage capacity and ergodic capacity as performance metrics. The performance of an optimized codebook is difficult to evaluate exactly, and is approximated in [2], [3], [6], [9]–[11]. Here we consider RVQ in which v1, · · · ,v2B are independent, isotropically distributed random vectors, each with unit norm. This is motivated by the observation that given a channel matrix H with i.i.d. elements, the eigenvectors of H†H are isotropi- cally distributed [30], hence the codebook entries should be uniformly distributed over the space of beamforming vectors. A. MISO Channel We first consider a MISO channel, corresponding to a single receive antenna (Nr = 1). In that case, H is an Nt×1 channel vector, which we denote as h. The optimal beamformer, which maximizes the mutual information in (2), is the normalized channel vector h/‖h‖ and the corresponding mutual informa- tion is Eh[log(1 + ρh †h)]. The receiver selects the quantized precoding vector to maximize the mutual information, i.e., v̂ = arg max 1≤j≤2B Ij = log(1 + ρ|h†vj |2) and the corresponding achievable rate is INtrvq , max 1≤j≤2B Ij . (4) where the superscript Nt denotes the system size. The achiev- able rate depends on the codebook V and the channel vector h, and is therefore random. Rather than averaging INtrvq over V and h to find the ergodic capacity, we instead evaluate the limiting performance as Nt and B tend to infinity with fixed B̄ = B/Nt (feedback bits per transmit antenna). In this limit, INtrvq converges to a deterministic constant. This is illustrated in Fig. 1, which shows the pdf of |h†v̂|2/‖h‖2 for different Nt with no feedback (B̄ = 0), and for RVQ with B̄ = 2. The figure shows that convergence of the pdf to a point mass is faster with feedback than without. As Nt → ∞, (h†h)/Nt → 1 almost surely, so that log(1 + ρh†h)− log(ρNt) → 0. That is, with perfect channel 0 0.2 0.4 0.6 0.8 1 Pdf for |h†v̂j| 2/ ‖h‖2 with RVQ codebook = 25 = 15 B̄ = 0 B̄ = 2 Fig. 1. pdf of |h†v̂|2/‖h‖2 with RVQ for different values of Nt. knowledge at the transmitter, the ergodic capacity increases as log(ρNt). With finite feedback there is a rate loss, which is defined as rvq = I rvq − log(ρNt). (5) For finite Nt, I rvq is random; however, in the large system limit I△rvq converges to a deterministic constant. Theorem 1. As (Nt, B) → ∞ with fixed B̄ = B/Nt, the rate difference I△rvq converges in the mean square sense to I△rvq = log(1− 2−B̄). (6) The proof is given in Appendix A. For B̄ > 0, the rate loss due to finite feedback is a constant. As B̄ → 0, this rate loss tends to infinity, since with B̄ = 0, the capacity tends to a constant as Nt → ∞, whereas the capacity grows as logNt for B̄ > 0. Of course, as B̄ → ∞ (unlimited feedback), the rate loss vanishes. RVQ is asymptotically optimal in the following sense. Suppose that {VNt} is an arbitrary sequence of codebooks for the beamforming vector where VNt = 2 , . . . ,v is the codebook for a particular Nt and ‖vNtj ‖2 = 1 for each j. The associated rate is given by IVNt = max 1≤j≤2B log(1 + ρ|h†vNtj | 2) (8) and the rate difference I△VNt = IVNt − log(ρNt). Theorem 2. For any sequence of codebooks {VNt}, lim sup (Nt,B)→∞ ] ≤ I△rvq. (9) The proof is given in Appendix B. Although the optimality of RVQ holds only in the large system limit, numerical results in Section III-C show that for finite-size systems of interest RVQ performs essentially the same as optimized quantization codebooks. 4 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 B. Multi-Input Multi-Output (MIMO) Channel We now consider quantized beamforming for a MIMO channel, i.e., with multiple transmit and receive antennas. Taking the rank K = 1 maximizes the diversity gain [31], but the corresponding capacity grows only as logNt instead of linearly with Nt, which is the case when K grows pro- portionally with Nt. (This is true with both unlimited and limited feedback, assuming a fixed number of feedback bits per precoder element.) Also, a beamformer is significantly less complex than a matrix precoder with K > 1, and requires less feedback to specify. We again consider an RVQ codebook V with 2B indepen- dent unit-norm vectors, where each vector is uniformly dis- tributed over the Nt-dimensional unit sphere. The achievable rate is EH [I rvq ], where INtrvq = EV 1≤j≤2B 1 + ρ‖Hvj‖2 log(1 + ρ max 1≤j≤2B ‖Hvj‖2) . (11) As for the MISO channel, with unlimited feedback the achiev- able rate increases as log(ρNt). We again define the rate difference due to quantization as rvq , I rvq − log(ρNt) = EV where Hvj . (13) Evaluating the expectation in (12) is difficult for finite Nt, Nr, and B, so that we again resort to a large system analysis. Namely, we let Nt, Nr, and B each tend to infinity with fixed B̄ = B/Nt and N̄r = Nr/Nt. For each Nt andNr the channel matrix H is chosen as the Nr × Nt upper-left corner of a matrix H̄ with an infinite number of rows and columns, and with i.i.d. complex Gaussian entries. The received power in this large system limit is given by γ∞rvq = lim (Nt,Nr,B)→∞ 1≤j≤2B where convergence to the deterministic limit can be shown in the mean square sense. Conditioned on H̄ , the γj’s are i.i.d. since the beamforming vectors vj are i.i.d., and applying [32, Theorem 2.1.2], it can be shown that γ∞rvq = lim (Nt,Nr,B)→∞ 1− 2−B where Fγ|H̄(·) is the cdf of γj given H̄ . Analogous results for the interference power in CDMA with quantized signatures have been presented in [21], so that we omit the proofs of (14) and (15). Note that N̄r ≤ γ∞rvq ≤ (1+ 2, where the lower and upper bounds correspond to B̄ = 0 and B̄ = ∞, respectively. That is, (1+ 2 is the asymptotic maximum eigenvalue of the channel covariance matrix 1 H†H [33]. The asymptotic rate difference is given by I∆rvq = lim (Nt,Nr,B)→∞ I∆rvq = log(γ rvq) (16) The limit in (15) can be explicitly evaluated, and is inde- pendent of the channel realization H̄ . Theorem 3. For 0 ≤ B̄ ≤ B̄∗, γ∞rvq satisfies γ∞rvq rvq = 2−B̄ and for B̄ ≥ B̄∗, γ∞rvq = (1 + 2 − exp N̄r log(N̄r) − (N̄r − 1) log(1 + N̄r) + N̄r − B̄ log(2) where B̄∗ = log(2) N̄r log . (19) The proof is given in Appendix C and is motivated by an analogous result for CDMA, presented in [34]. As stated in Theorem 3, γ∞rvq depends only on B̄ and N̄r. Letting N̄r → 0 gives the the asymptotic capacity of the MISO channel with RVQ. As for the MISO channel, RVQ is asymptotically optimal. Theorem 4. As (Nt, Nr, B) → ∞ with fixed N̄r = Nr/Nt and B̄ = B/Nt, lim sup (Nt,Nr,B)→∞ INtVNt − log(ρNt) ≤ I∆rvq (20) for any sequence of codebooks {VNt}. The proof is similar to the proof of Theorem 2 in [21] and is therefore omitted. C. Numerical Results Figs. 2 and 3 show I△rvq for MISO and MIMO channels, respectively, with beamforming and RVQ versus normalized feedback bits (B̄) with ρ = 5 and 10 dB. Also shown for comparison are achievable rates with a quantization code- book optimized via the Lloyd-Max algorithm [1], [5], [6], and the capacity with perfect beamforming, corresponding to unlimited feedback. The results for RVQ are averaged over codebook realizations, and are essentially the same as those shown for the optimized Lloyd-Max codebooks. For the MISO channel, the asymptotic capacity (6) accurately predicts the simulated results shown even with a relatively small number of transmit antennas (Nt = 3 and 6). For the MIMO results N̄r = 1.5, and simulation results are shown for 4 × 6 and 16 × 24 channels. The asymptotic results accurately predict the performance for the larger channel, and are somewhat less accurate for the smaller channel. Comparing finite feedback with perfect beamforming, the results show that one feedback bit per complex entry (B̄ = 1) provides more than 50% of the potential gain due to feedback. For both the MISO and MIMO examples shown, the perfect beamforming capacity is nearly achieved with two feedback bits per complex coefficient. SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 5 0 0.5 1 1.5 2 2.5 3 MISO; SNR = 5 dB RVQ, simulation w/ N = 3, N RVQ, simulation w/ N = 6, N Optimal beamforming RVQ, large system Lloyd−Max, simulation w/ N = 6, N Lloyd−Max, simulation w/ N = 3, N Fig. 2. Asymptotic and simulated rate differences versus feedback bits for a MISO channel with beamforming. 0 0.5 1 1.5 2 2.5 3 3.5 4 = 1.5; SNR = 10 dB Optimal Beamforming RVQ, simulation w/ N = 4, N RVQ, simulation w/ N = 16, N = 24 RVQ, large system Lloyd−Max, simulation w/ N = 4, N Lloyd−Max, simulation w/ N = 16, N = 24 Fig. 3. Asymptotic and simulated rate differences versus feedback bits for a MIMO channel with beamforming (N̄r = 1.5). IV. PRECODING MATRIX WITH ARBITRARY RANK In this section we consider the performance of a single-user MIMO channel with precoding matrix V having rank K > 1. We wish to determine the asymptotic capacity with RVQ as in the previous section. Here we consider the large system limit as (Nt, Nr, B,K) → ∞ with fixed ratios N̄r = Nr/Nt, B̂ = B/N2r , and K̄ = K/Nt. That is, we scale the rank of the precoding matrix with Nt. The number of feedback bits is normalized by N2r , instead of Nr, since the feedback must scale linearly with degrees of freedom (in this case the number of channel elements NrNt).) Given a fixed number of feedback bits per channel coefficient, the capacity grows linearly with the number of antennas (Nt or Nr). Given a rank K ≤ Nt, the precoding matrix is chosen from the RVQ set V = {Vj , 1 ≤ j ≤ 2B}, (21) where the entries are independent Nt × K random unitary matrices, i.e., V †j Vj = IK . This codebook is an extension of the RVQ codebook for beamforming. Letting JNrj = log det INr + , (22) the receiver again selects the quantized precoding matrix, which maximizes the mutual information V̂ = arg max 1≤j≤2B JNrj . (23) For finite Nr, we define INrrvq = EV [ max 1≤j≤2B JNrj |H̄ ] (24) log det INr + HV̂ V̂ and the average sum mutual information per receive antenna with B feedback bits is then EH [I rvq ]. Here the power allocation over channel modes is “on- off”. Namely, active modes are assigned equal powers. This simplifies the analysis, and it has been observed that the additional gain due to an optimal power allocation (water pouring) is quite small [8]. Since the entries of the RVQ codebook are i.i.d., the mutual informations JNrj , j = 1, . . . , 2 B, are also i.i.d. for a given H . In principle, the large system limit of INrrvq can be evaluated, in analogy with (15), given the cdf of JNrj given H , denoted as FJ;Nr|H . This cdf appears to be difficult to determine in closed-form for general (Nr, Nt,K), so that we are unable to derive the exact asymptotic capacity with RVQ. Still, we can provide an accurate approximation for this large system limit. Before presenting this approximation, we first compare the capacity with no channel information at the transmitter (B̂ = 0) to the capacity with perfect channel information (B̂ = ∞). If B̂ = 0, then the optimal transmit covariance matrix V V † = INt and K = Nt [35]. That is, all channel modes are allocated equal power. As (Nt, Nr) → ∞ with fixed N̄r = Nr/Nt, the capacity per receive antenna is given by log det INr + log (1 + ρλ) g(λ) dλ = Irvq(B̂ = 0) (27) where convergence is in the almost sure sense, and g(λ) is the asymptotic probability density function for a randomly chosen eigenvalue of 1 HH†, and is given by [33] g(λ) = (λ − a)(b− λ) 2πλN̄r for a ≤ λ ≤ b, (28) and b = for N̄r ≤ 1. The integral in (26) has been evaluated in [36], which gives the closed-form expression Irvq(B̂ = 0) = log ρy + 1− N̄r 6 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 where 1 + N̄r + 1 + N̄r + − 4N̄r  (31) 1 + N̄r + 1 + N̄r + − 4N̄r  . (32) If B̂ = ∞, then the K columns of the optimal V are the eigenvectors of the channel covariance matrix corresponding to the K largest eigenvalues. As (Nt, Nr, B) → ∞, we have Irvq(B̂ = ∞) = g(λ) dλ (33) where η satisfies g(λ) dλ = min{1, } (34) for N̄r ≤ 1. We emphasize that this corresponds to a uniform allocation of power over the set of K active eigenvectors. (This result has also been presented in [8].) The rank of the optimal V , or optimal multiplexing gain, is at most min{Nt, Nr} and can be obtained by differentiating (33) with respect to K̄. It can be verified that Irvq(B̂ = 0) ≤ Irvq(B̂ = ∞). To illustrate the increase in capacity with feedback, in Fig. 4 we plot the rate ratio Irvq(B̂ = ∞)/Irvq(B̂ = 0) versus SNR for different values of N̄r, where Irvq(B̂ = ∞) is optimized over rank K . For large SNR ρ, we can expand Irvq(B̂ = 0) = log(ρ) + o(log(ρ)) (35) Irvq(B̂ = ∞) = log(ρ) g(λ) dλ+ o(log(ρ)). (36) Therefore Irvq(B̂ = ∞) Irvq(B̂ = 0) g(λ) dλ = min{1, K̄ } (37) which implies that the optimal rank K∗ = min{Nt, Nr}, and the corresponding asymptotic rate ratio is one. The increase in achievable rate from feedback is small in this case, since for large SNRs, the transmitter excites all channel modes, and the uniform power allocation asymptotically gives the same capacity as water pouring. Of course, although the increase in rate is small, feedback can simplify coding and decoding. For small ρ, we can expand log(1+ρλ) and log(1+ρλ/K̄) in Taylor series. Taking ρ→ 0 gives Irvq(B̂ = ∞) Irvq(B̂ = 0) λg(λ) dλ ≤ λg(λ) dλ g(λ) dλ g(λ) dλ g(λ) dλ where b is the asymptotic maximum eigenvalue given by (29). The inequality in (38) follows from (34), which implies K̄ ≥ g(λ) dλ. Note that (39) corresponds to allocating all transmission power to the strongest channel mode, which is known to maximize capacity at low SNRs. The maximal rate ratio (39) can also be obtained from Theorem 3. The rate increase due to feedback is substantial when N̄r is small, and the rate ratio tends to infinity as N̄r → 0. This is because the channel becomes a MISO channel, in which case the capacity is a constant with B̄ = 0 and increases as log(ρNt) with B̄ = ∞. −10 −5 0 5 10 15 20 25 ρ (dB) = 0.1 = 0.5 = 0.75 Fig. 4. The rate ratio Irvq(B̂ = ∞)/Irvq(B̂ = 0) versus SNR (dB) for various values of N̄r . To evaluate the asymptotic capacity with arbitrary B̂, we approximate JNrj given H̄ as a Gaussian random variable. This is motivated by the fact that JNrj is Gaussian in the large system limit [37], since HV is i.i.d. Conditioning on H introduces dependence among the elements of HV ; however, numerical examples indicate that the Gaussian assumption is still valid for large Nt and Nr. Alternatively, if we do not condition on H , then the rates {JNrj } are dependent. Application of the results from extreme statistics, assuming the rates {JNrj } are independent, gives an upper bound on the asymptotic achievable rate (e.g., see the proof of Theorem 2 in [21]). This is illustrated by subsequent numerical results. Evaluating the large system limit of INrrvq , assuming that the cdf of JNrj is Gaussian, gives the approximate rate Ĩrvq = µJ + σJ 2B̂ log 2 (40) independent of the channel realization, where µJ and σ are the asymptotic mean of JNrj , and variance of N respectively. The derivation of (40) is a straightforward exten- sion of [32, Sec. 2.3.2] and is not shown here. As B̂ → 0, this approximation becomes exact. However, as B̂ → ∞, the approximate rate Ĩrvq → ∞, whereas the actual rate Irvq(B̂ = ∞) is finite, and can be computed from (33) and (34). This is because JNrj is bounded for all Nr, whereas a Gaussian random variable can assume arbitrarily large values. Therefore the Gaussian approximation gives an inaccurate estimate of Irvq for large B̂. (This implies that we should approximate Irvq as min{Ĩrvq, Irvq(B̂ = ∞)}.) SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 7 The asymptotic mean and variance of JNrj are computed in Appendix D. The asymptotic mean is given by ρ− N̄r + log 1 + ρ− N̄r where 2N̄rρ − 4K̄ The asymptotic variance is approximated for 0 ≤ K̄ = N̄r ≤ 1 and small SNR (ρ ≤ −5 dB) as σ2J ≈ ρ2(1 − N̄r). (43) The asymptotic variance for moderate SNRs and normalized rank K̄ 6= N̄r can be computed easily via numerical simula- tion.1 In contrast with the beamforming results in the preceding section, we are unable to show that RVQ is asymptotically optimal when the precoding matrix has arbitrary rank. The corresponding argument for beamforming relies on the evalu- ation of the asymptotic rate difference I∆rvq. Since here we are unable to evaluate Irvq exactly, we cannot apply that argument. Nevertheless, numerical results have indicated that the performance of RVQ matches that of optimized codebooks (e.g., see [22]). Fig. 5 shows Ĩrvq with normalized rank K̄ = N̄r versus B̂ for ρ = −5, 0, 5 dB and N̄r = 0.5. The dashed lines show the unlimited feedback capacity Irvq(B̂ = ∞), which is computed from (33) with optimized K̄. The asymptotic rate with RVQ is computed from (40), where σJ for ρ = −5 dB is approximated by (43), and σJ is determined from simulation with Nt = 20 for ρ = 0 and 5 dB. Also shown in Fig. 5 are simulation results for Irvq with Nt = 8 and Nr = 4. Because the size of the RVQ codebook increases exponentially with B̂, it is difficult to generate simulation results for moderate to large values of B̂. Hence simulation results are shown only for B̂ ≤ 0.8. The asymptotic results accurately approximate the simulated results shown. The accuracy increases as the feedback B̂ decreases. Since Ĩrvq is a function of both rank K̄ and feedback B̂, for a given B̂, we can select K̄ to maximize Ĩrvq. Fig. 6 shows mutual information per receive antenna versus normalized rank from (40) with N̄r = 0.2, ρ = 5 dB, and different values of B̂. (σJ is obtained from numerical simulations.) The maximal rates are attained at K̄ = 1, 0.3, and 0.2 for B̂ = 0, 0.5, and 2, respectively. In general, the optimal rank is approximately N̄r for large enough B̂ and SNR. The results in Fig. 6 indicate that taking K̄ = N̄r achieves near-optimal performance, independent of B̂ when B̂ > 0. As B̂ increases, the rate increases and the difference between the rate with optimized rank and full-rank (K̄ = 1) also increases. For the 1We note that the simulation needed to compute this variance is much simpler than the simulation, which would be required to obtain the RVQ rate directly, especially with a moderate to large number of feedback bits. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 = 0.5 SNR = −5 dB SNR = 0 dB SNR = 5 dB (B/N2 = ∞) w/ (K = N Sim. I = 8) Fig. 5. Sum mutual information per receive antenna with RVQ and an optimal receiver versus normalized feedback. The asymptotic approximation is shown along with Monte Carlo simulation results for an 8× 4 channel. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 = 0.2; N = 10; SNR = 5 dB 2 = 0 2 = 0.5 2 = 2 Fig. 6. Mutual information per receive antenna versus normalized rank with different normalized feedback. Discrete points correspond to simulation with Nt = 10. example shown, the rate increase from selecting the optimal rank is as high as 50% when B̂ = 2. V. QUANTIZED PRECODING WITH LINEAR RECEIVERS In this section we evaluate the performance of a quantized precoding matrix with linear receivers (matched filter and MMSE), and compare with the performance of the opti- mal receiver. As B̂ → ∞, the optimal precoding matrix eliminates the cross-coupling among channel modes, and the optimal receiver becomes the linear matched filter. Hence the corresponding achievable rates should be the same in this limit. However, for finite B̂ the optimal receiver is expected to perform better than the linear receiver. Given a target rate, increasing the feedback therefore enables a reduction in receiver complexity. We again assume that there are K independent data streams, which are multiplexed by the linear precoder onto Nt transmit antennas. To detect the transmitted symbols in data stream k, the received signal y is passed through the Nr × 1 receive 8 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 filter ck. The matched filter is given by Hvk (44) where vk is the kth column of the precoding matrix V , and the linear MMSE filter is given by † + σ2nINr Hvk. (45) The SINR at the output of the linear filter ck is SINRk = |c†kHvk|2 i6=k Hviv † +Kσ2nINr . (46) Of course, the interference among data streams can signifi- cantly decrease the channel capacity. The performance measure is again mutual information be- tween the transmitted symbol xk and the output of the filter ck, denoted by x̂k. In what follows, we assume independent coders and decoders for each data stream. Assuming that the interference plus noise at the output of the linear filter has a Gaussian distribution, which is true in the large system limit to be considered, the sum mutual information of all data streams per receive antenna is given by RNr = I(xk, x̂k) (47) log(1 + γk). (48) where γk is the SINR for the kth data stream. Given a channel matrix H , the sum rate RNr depends on the precoding matrix V . We are interested in maximizing RNr subject to the power constraint ‖vk‖ ≤ 1, ∀k, assuming that the power is allocated equally across streams. Given the codebook of precoding matrices V = {Vj , 1 ≤ j ≤ 2B}, the receiver selects the precoding matrix V̂ = arg max 1≤j≤2B RNr(Vj). (49) We again consider RVQ, in which the Vj’s are i.i.d. unitary matrices. A. Matched filter Substituting (44) into (46), the SINR at the output of the matched filter is given by γk;mf = †Hvk) Kσ2n(v †Hvk) + i=1,i6=k |v †Hvi|2 where subscript k denotes the kth data stream. The average sum rate per receive antenna is given by 1≤j≤2B {RNrmf (Vj) = log(1 + γk;mf)} where the expectation is over the channel matrix and code- book. Since the pdf of RNrmf is unknown for finite (Nt, Nr,K), we are unable to evaluate (51). Motivated by the central limit theorem,2 in what follows we approximate the cdf of RNrmf as 2The terms in the sum in (51) are not i.i.d., which prevents a direct application of the central limit theorem. 0 0.2 0.4 0.6 0.8 1 (nat) PDF for R = 10; N = 1; K/N = 0.3; SNR = 5dB Empirical pdf Gaussian approximation Fig. 7. Comparison of the empirical pdf for Rmf;Nr with the Gaussian approximation. Gaussian. The mean is taken to be the asymptotic limit µmf = lim (Nt,Nr,K)→∞ RNrmf = K̄(1 + σ2n) This limit follows from the fact that γk;mf converges almost surely to [K̄(1 + σ2n)/N̄r] −1 as (Nt, Nr,K) → ∞ with fixed N̄r and K̄ . As for the optimal receiver, N2r var[R mf |Vj ] → σ mf (53) where σ2mf can be easily obtained by numerical simulations. The accuracy of the Gaussian approximation for RNrmf is illustrated in Fig. 7, which compares the empirical pdf with the Gaussian approximation for Nr = 10, N̄r = 1, K/Nr = 0.3 and SNR = 5dB. The difference between the empirical and asymptotic means vanishes as (Nt, Nr,K) → ∞. We wish to apply the theory of extreme order statistics [32] to evaluate the large system limit Rrvq;mf = lim (Nt,Nr,K,B)→∞ [ max 1≤j≤2B RNrmf (Vj)|V ]. (54) Given V , the sum rates {RNrmf (V1), . . . , R mf (V2B )} are identi- cally distributed. However, the RNrmf (Vj)’s are not independent since each depends on H . This makes an exact calculation of the asymptotic rate difficult. Nevertheless, for a small number of entries in the codebook (small B), assuming that the rates for a given codebook are independent leads to an accurate approximation. We therefore replace the rates RNrmf (Vj), j = 1, · · · , 2 B , with i.i.d. Gaussian variables with mean µmf and variance σ r . In analogy with the analysis of the optimal receiver in the preceding section, this gives the approximate asymptotic rate R̃rvq;mf = µmf + σmf 2B̂ log 2. (55) Numerical results, to be presented, show that this asymptotic approximation is very accurate for small to moderate values of normalized feedback B̂. As B̂ → 0, this approximation becomes exact. However, as B̂ → ∞, R̃rvq;mf → ∞, whereas Rrvq;mf with B̂ = ∞ is the same as the asymptotic rate with SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 9 RVQ and an optimal receiver, given by (33) and (34). Hence Rrvq;mf with B̂ = ∞ is finite. As for the analysis of the optimal receiver, this discrepancy is again due to the fact that the cdf of RNrmf , which has compact support, is being approximated by a Gaussian cdf with infinite support, and also because the dependence among the sum rates RNrmf (Vj) is being ignored. B. MMSE receiver Substituting (45) into (46) gives the SINR at the output of MMSE receiver for the kth symbol stream γk;mmse = v † +Kσ2nINr As for the matched filter receiver, given a codebook V , we approximate the pdf of the instantaneous sum rate RNrmmse = log(1 + γk;mmse) (57) as a Gaussian pdf with mean µmmse = lim (Nt,Nr)→∞ RNrmmse = log(1 + γmmse) (58) where the large system SINR is given by [30] γmmse = 1− K̄/N̄r (1 − K̄/N̄r)2 1 + K̄/N̄r As for the matched filter, the asymptotic variance σ2mmse can be obtained via numerical simulation. In analogy with (55), the asymptotic rate with RVQ and the MMSE receiver is given by Rrvq;mmse ≈ R̃rvq;mmse = µmmse + σmmse 2B̂ log 2. (60) As for the matched filter receiver, when B̂ is large, R̃mmse over-estimates Rmmse. For B̂ = ∞, Rmmse = Rmf = Irvq with the optimal receiver, given by (33). C. Numerical Results Fig. 8 compares the approximation for asymptotic RVQ performance with a matched filter receiver from (55) with simulated results for Nt = 12, N̄r = 0.75, K/Nt = 1/2, and SNR = 5 dB. Also shown for comparison are the asymptotic rate for RVQ with an optimal receiver, derived in Section IV, the water-filling capacity (B̂ = ∞), and the rate achieved with a scalar quantizer for each coefficient.3 For the case shown, the analytical approximation gives an accurate estimate of the performance of the finite size system with limited feedback. The capacity with the water-filling power allocation is only slightly greater than that achieved with the on-off power allo- cation. The optimal receiver requires B̂ ≈ 0.6 bit/dimension 3For the scalar quantization results the available bits are spread evenly over the corresponding fraction of precoding coefficients. The remaining coefficients are set to one. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 = 0.75; K/N = 0.5; SNR = 5 dB Water−filling capacity (large sys.) RVQ w/ Opt Rx. (large sys.) RVQ w/ MF Rx. (large sys.) Scalar quantization (N = 12) RVQ w/ MF Rx. = 12) Fig. 8. Sum rate per receive antenna versus normalized feedback bits with a matched filter receiver. Results are shown for RVQ (asymptotic and Nt = 12) and scalar quantization. Also shown are results for the optimal receiver, and the water-filling capacity with infinite feedback. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 = 0.75; K/N = 0.5; SNR = 5 dB RVQ w/ MMSE Rx. = 12) Scalar quantization (N = 12) RVQ w/ MMSE Rx. (large sys.) Water−filling capacity (large sys.) RVQ w/ Opt Rx. (large sys.) Fig. 9. Sum rate per receive antenna versus normalized feedback bits with a linear MMSE receiver. Results are shown for RVQ (asymptotic and Nt = 12) and scalar quantization. Also shown are results for the optimal receiver, and the water-filling capacity with infinite feedback. to achieve the capacity corresponding to unlimited feedback (33), whereas the matched filter requires 1.2 feedback bits per dimension to reach that capacity. For other target rates, these curves illustrate the trade-off between feedback and receiver complexity. Fig. 9 shows the same set of results as those shown in Fig. 8, but with an MMSE receiver. These results show that for the parameters selected, the MMSE receiver performs nearly as well as the optimal receiver, and requires substantially less feedback than the matched filter to achieve a target rate. Again the asymptotic approximation accurately predicts the performance of a system with a relatively small number of antennas. VI. CONCLUSIONS We have studied the capacity of single-user MISO and MIMO fading channels with limited feedback. The feedback 10 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 specifies a transmit precoding matrix, which can be optimized for a given channel realization. We first considered the perfor- mance with a rank-one precoding matrix (beamformer), and showed that the RVQ codebook is asymptotically optimal. Exact expressions for the asymptotic mutual information for MISO and MIMO channels were presented, and reveal how much feedback is required to achieve a desired performance. For the cases considered, one feedback bit for each precoder coefficient can achieve close to the water-filling capacity. Perhaps more important than the increase in capacity provided by this feedback is the associated simplification in the coding and decoding schemes that can achieve a rate close to capacity. The performance of a precoding matrix with rank K > 1 was also evaluated with RVQ. Although numerical examples and our beamforming results (K = 1) suggest that RVQ is also asymptotically optimal in this case, proving this is an open problem. To compute the asymptotic achievable rate for RVQ with both optimal and linear receivers, the achievable rate with a random channel and fixed precoding matrix is approximated as a Gaussian random variable. The asymptotic rate then depends on the asymptotic mean and variance of this random variable. Although the asymptotic variance appears to be difficult to compute analytically, it can be easily obtained by simulation. Numerical results have shown that the resulting approximation accurately estimates the achievable rate with limited feedback for finite-size systems of interest. Numerical examples comparing the performance of opti- mal and linear receivers have shown that the linear MMSE receiver requires little additional feedback, relative to the optimal receiver, to achieve a target rate close to the water- filling capacity. The matched filter requires significantly more feedback than the MMSE receiver (more than 0.5 bit per degree of freedom for the cases shown). At low feedback rates the achievable rate with RVQ is generally much greater than that associated with scalar quantization. Of course, this comes at a price of high complexity, since the receiver is assumed to compute the performance metric for every entry in the codebook. Other reduced complexity schemes for quantizing a beamforming vector are presented in [21], [38], [39]. Key assumptions for our results are that the channel is stationary and known at the receiver, and that the channel elements are i.i.d. Depending on user mobility and associated Doppler shifts, the channel may change too fast to allow reli- able channel estimation and feedback. In that case, feedback of channel statistics, as proposed in [15]–[18], [40], can exploit correlation among channel elements. The design of quanti- zation codebooks for precoders, which takes correlation into account, is addressed in [40]. The effect of channel estimation error on the performance of limited feedback beamforming with finite coherence time (i.e., block fading) is presented in [41], [42]. We have also assumed that the channel gains are not frequency-selective. Limited feedback schemes for frequency- selective scalar channels are discussed in [43], and could be combined with the quantization schemes considered here. Finally, the approach presented here for a single-user MIMO channel can also be applied to multi-user models. Quantization of beamformers for the MIMO downlink have been considered in [26]–[28]. In that scenario the potential capacity gain due to feedback is generally much more than for the single-user channel considered here. The benefits of limited feedback for related models (e.g., frequency-selective MIMO downlink) are currently being studied. APPENDIX A. Proof of Theorem 1 Given h, the receiver selects the quantized beamforming vector v̂ to maximize the instantaneous rate in (3). Since log is monotonically increasing, the quantized beamforming vector is given by v̂ = arg max 1≤j≤2B Yj = |v†jh| 2/‖h‖2 . (61) Since the codebook entries vj , j = 1, · · · , 2B, are i.i.d., the Yj’s, given h, are also i.i.d. with the cdf [25] FY |h(y) = 1− (1 − y)Nt−1, 0 ≤ y ≤ 1. (62) We wish to determine the distribution of maxj Yj given h. From [32, Theorem 2.1.2] it follows that maxj Yj − an D−→ Y (63) where Y is a Weibull random variable having distribution Hγ(x) = 1, x ≥ 0 exp(−(−x)γ), x < 0 , (64) D denotes convergence in distribution, and an and bn are nor- malizing sequences, where n = 2B. Specifically, the theorem requires that ω(FY |h) = sup{y : FY |h(y) < 1} be finite, and that the distribution function F ∗ Y |h(y) = FY |h(ω(FY |h)− 1/y), y > 0 satisfies, for all y > 0, 1− F ∗ Y |h(ty) 1− F ∗ Y |h(t) = y−γ (65) where the constant γ > 0. Substituting the expression for FY |h in (62) into (65), where ω(FY |h) = 1, gives 1− F ∗ Y |h(ty) 1− F ∗ Y |h(t) = lim (1/(ty)) (1/t) Nt−1 (66) = y−(Nt−1) (67) so that [32, Theorem 2.1.2] applies when Nt > 1. Further- more, the normalizing constants are given by an = ω(FY |h) = 1 (68) bn = ω(FY |h)− inf y : 1− FY |h(y) ≤ = 1− F−1 . (70) SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 11 To take the limit as Nt → ∞, we will assume that the channel vector h contains the first Nt elements of an infinite- length i.i.d. complex Gaussian vector h̄. Rearranging terms in (63) and taking the large system limit gives (Nt,n)→∞ EV [max Yj |h̄] (71) = lim (Nt,n)→∞ an + bnE[Y] (72) = 1− lim (Nt,n)→∞ Nt − 1 = 1− lim (Nt,B)→∞ Nt−1 (74) = 1− 2−B̄ (75) where the gamma function Γ(z) = tz−1e−t dt and we have used the fact that E[Y] = −Γ(1− 1/(Nt − 1)) [44]. From (63), as (n,Nt) → ∞, var[max Yj |h̄]− b2nvar[Y] → 0. (76) Since var[Y] = Γ(1−2/(Nt−1))−Γ2(1−1/(Nt−1)) → 0, it follows that var[maxj Yj |h] → 0. This establishes that given Yj → 1− 2−B̄ (77) in the mean square sense. The asymptotic rate difference is given by I△rvq = lim (Nt,B)→∞ log(1 + ρ‖h‖2 max 1≤j≤2B − log(ρNt) = lim (Nt,B)→∞ ‖h‖2 max 1≤j≤2B = log(1− 2−B̄) (80) in the mean square sense, since ‖h‖2/Nt → 1 almost surely. B. Proof of Theorem 2 The rate difference associated with codebook VNt is = max 1≤j≤2B |h†vNtj | = log + max 1≤j≤2B |h†vNtj | . (82) Taking expectation of the rate difference with respect to h and applying Jensen’s inequality, we obtain ] ≤ log |h†v̂Nt |2] = log ‖h‖2]E[µ] where the optimal beamforming vector Nt = arg max 1≤j≤2B |h†vNtj | 2, (85) µ = |h†v̂Nt |2/‖h‖2, and (84) follows from the fact that ‖h‖2 and µ are independent [25]. We now derive an upper bound for E[|h†v̂Nt |2/‖h‖2] . From (30) in [2] we have Pr{|h†v̂Nt |2 > γs | ‖h‖2 = γ} 1, 0 ≤ s < s∗ 2B(1− s)Nt−1, s∗ ≤ s ≤ 1 where s∗ = 1− 2− Nt−1 . (87) Since the right-hand side of (86) is independent of γ, averaging over γ gives Pr{µ > s} ≤ 1, 0 ≤ s < s∗ 2B(1− s)Nt−1, s∗ ≤ s ≤ 1 . (88) Integrating by parts, we have that E[µ] = Pr{µ > x} dx (89) Pr{µ > x} dx+ Pr{µ > x} dx. (90) Substituting (88) into (90) and evaluating both integrals gives E[µ] ≤ 1− 2− Nt−1 + Nt−1 . (91) Substituting E[‖h‖2] = Nt and (91) into (84) gives ] ≤ log 1− 2− Nt−1 + Nt−1 + and taking the large system limit gives (Nt,B)→∞ ] ≤ log(1− 2−B̄). (93) Theorem 1 states that RVQ achieves this upper bound, and therefore upper bounds the asymptotic rate difference corre- sponding to any quantization scheme. C. Proof of Theorem 3 We first prove the theorem for N̄r ≥ 1. Let z = F−1γ|H̄(1− 2−B). Rearranging (15) gives (Nt,Nr)→∞ z→γ∞rvq 1− Fγ|H̄(z) Nt = 2−B̄. (94) Next, we derive upper and lower bounds for the left-hand side of (94) and show that they are the same. The derivation of the upper bound is motivated by the evaluation of a similar bound for CDMA signature optimization in [34]. That is, 1− Fγ|H̄(z) = Pr γj > z| H̄ vj > z|Λ,U where γj = †Hvj and we have applied the singular value decomposition 1 H†H = UΛU† , where U is an Nt × Nt unitary matrix, Λ = diag{λ1, · · · , λNt}, and the eigenvalues are ordered as λ1 ≥ λ2 ≥ · · · ≥ λNt . Also, wj 12 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 is an Nt × 1 vector with independent, circularly symmetric, zero-mean and unit-variance Gaussian elements. Both vj and wj/‖wj‖ are isotropically distributed, i.e., Uwj/‖wj‖ and wj/‖wj‖ have the same distribution, so that 1− Fγ|H̄(z) = 1− Fγ|Λ(z) (98) i=1 λiw i=1 w (z − λi)w2i > 0 (100) (z − λi)w2i > 0 , ∀ρ > 0 (101) (z − λi)w2i (102) where {wi} are elements of wj . (We omit the index j to simplify the notation.) Applying Markov’s inequality and the independence of the wi’s gives 1− Fγ|Λ(z) ≤ E{wi} (z − λi)w2i (103) −ρ(z − λi)w2i (104) exp {−ρ(z − λi)x} e−xdx (105) exp {−(1 + ρ(z − λi))x} dx (106) 1 + ρ(z − λi) (107) = exp log(1 + ρ(z − λi)) (108) when 1 + ρ(z − λi) > 0 for all i, or ρ < 1/(λ1 − z). Taking the large system limit, we obtain (Nt,Nr)→∞ z→γ∞rvq 1− Fγ|Λ(z) Nt ≤ exp{−Φ(γ∞rvq, ρ)} (109) for 0 < ρ < 1 −γ∞rvq , where Φ(γ∞rvq, ρ) , log(1 + ρ(γ∞rvq − λ))g(λ)dλ, (110) g(λ) is given by (28)-(29), and λ∞max = lim(Nt,Nr)→∞ λ1 = 2. To tighten the upper bound, we minimize (109) with respect to ρ, i.e., (Nt,Nr)→∞ z→γ∞rvq 1− Fγ|Λ(z) Nt ≤ exp{−Φ(γ∞rvq, ρ∗)} (111) where ρ∗ = arg max 0<ρ< 1 −γ∞rvq Φ(γ∞rvq, ρ). (112) A similar expression for RVQ performance when used to quantize signatures for CDMA is derived in [34]. To derive the lower bound, we use a change of measure. (A similar approach was used in [45, Section 1.2].) Let yi , (λi − z)w2i , which is a scaled exponential random variable with cdf Fi(·). We define the new distribution Gi(x) , Mi(ρ∗) ∗y dFi(y), (113) so that ∗)dGi(x) = e ρ∗xdFi(x), (114) where ρ∗ is given in (112), and the moment generating function for yi is Mi(θ) , E[e θyi] (115) eθ(λi−z)xe−x dx (116) 1 + θ(z − λi) . (117) Applying the change of measure (114), we have 1− Fγ|Λ(z) = Pr{ yi > 0} (118) · · · yi > 0] dF1(y1) · · · dFNt(yNt) (119) · · · yi > 0]e ∗y1dF1(y1) (120) · · · eρ ∗yNtdFNt(yNt) · · · yi > 0]e yi dG1(y1) (121) · · · dGNt(yNt) where 1[x > 0] = 1 : x > 0 0 : x ≤ 0 . (122) For any ǫ > 0, 1− Fγ|Λ(z) · · · 1[ǫNt ≥ yi > 0]e (123) dG1(y1) · · ·dGNt(yNt) ∗)e−ρ · · · 1[ǫNt ≥ yi > 0] (124) dG1(y1) · · ·dGNt(yNt) ∗)e−ρ ∗ǫNt Pr{ǫNt ≥ ỹi > 0} (125) SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 13 where the ỹi’s are independent random variables with cdf Gi(·), and the second inequality follows since ρ∗ > 0. To determine the probability on the right-hand side of (125), we first compute the mean of ỹi, y dGi(y) (126) Mi(ρ∗) ∗ydFi(y) (127) = (1 + ρ∗(z − λi)) (λi − z)xeρ ∗(λi−z)xe−x dx (128) λi − z 1 + ρ∗(z − λi) . (129) Therefore λi − z 1 + ρ∗(z − λi) (130) and the asymptotic mean m∞ = lim (Nt,Nr)→∞ z→γ∞rvq mi (131) λ− γ∞rvq 1 + ρ∗(γ∞rvq − λ) g(λ) dλ <∞. (132) Similarly, since ỹi is exponentially distributed, the variance of ỹi is σ2i = λi − z 1 + ρ∗(z − λi) <∞ (133) and the asymptotic variance σ2∞ = lim (Nt,Nr)→∞ z→γ∞rvq σ2i (134) λ− γ∞rvq 1 + ρ∗(γ∞rvq − λ) g(λ) dλ <∞. (135) Both the asymptotic mean and variance are finite. Since the ỹi’s are independent with finite mean and variance, the central limit theorem implies that the cdf for i=1 ỹi − i=1mi i=1 σ (136) converges to a Gaussian cdf with zero mean and unit variance. Therefore we have Pr{0 < ỹi ≤ ǫNt} = Pr{− NtaNt < T ≤ − NtaNt + } (137) = FT (− NtaNt + )− FT (− NtaNt) (138) where FT (·) is the cdf for T and aNt , i=1mi/Nt i=1 σ i /Nt , (139) bNt , σ2i /Nt → σ∞. (140) Let φ(·) denote a Gaussian cdf with zero mean and unit variance. We can rewrite (138) as Pr{0 < ỹi ≤ ǫNt} = φ(− NtaNt + NtaNt) + ζNt − ξNt (141) where ζNt , FT (− NtaNt + − φ(− NtaNt + ), (142) ξNt , FT (− NtaNt)− φ(− NtaNt). (143) Applying the Berry-Esséen theorem [46], we can bound both ζNt and ξNt for large Nt as |ζNt |, |ξNt | ≤ (144) where C is a positive constant that depends on the variance and third moment of ỹ. Similar to the mean and variance, we can show that the third moment is also finite. We can now evaluate NtaNt + )− φ(− NtaNt) NtaNt+ NtaNt 2/2 dt (145) e−Nt(ant−ǫ/bNt) . (146) Substituting (144) and (146) into (141), we have Pr{0 < ỹi ≤ ǫNt} = O(1/ Nt) (147) Taking the large system limit and applying L’Hopital’s rule, it follows that (Nt,Nr)→∞ z→γ∞rvq [Pr{0 < ỹi ≤ ǫNt}] Nt = 1. (148) Taking the Ntth root and large system limit on both sides of (125) gives (Nt,Nr)→∞ z→γ∞rvq 1− Fγ|Λ(z) Nt ≥ exp{−Φ(γ∞rvq, ρ∗)} (149) where we use (148) and let ǫ→ 0. 14 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 The lower bound in (149) is exactly the upper bound (111). Therefore, (Nt,Nr)→∞ z→γ∞rvq 1− Fγ|Λ(z) Nt = exp{−Φ(γ∞rvq, ρ∗)} (150) = 2−B̄ (151) and the asymptotic RVQ received power satisfies the fixed- point equation Φ(γ∞rvq, ρ ∗) = B̄ log(2) (152) where ρ∗ is given by (112). The goal of the rest of the proof is to simplify (152). To determine ρ∗, we first compute ∂Φ(γ∞rvq, ρ) γ∞rvq − λ 1 + (γ∞rvq − λ)ρ g(λ) dλ (153) + γ∞rvq ︸ ︷︷ ︸ g(λ) dλ (154) SΛ(y) (155) where SΛ(·) is the Stieltjés Transform of the asymptotic eigenvalue distribution of Λ. Setting the derivative to zero and solving for ρ gives SΛ(y) = −ρ = γ∞rvq − y (156) as the only valid solution. Substituting the expression for SΛ given in [33] into (156) gives (−1 + N̄r − y)± y2 − 2(N̄r + 1)y + (N̄r − 1)2 γ∞rvq − y (157) which simplifies to the quadratic equation (N̄r − γ∞rvq)y2 + [γ∞rvq + N̄rγ∞rvq + (γ∞rvq)2]y = 0. (158) Solving for y gives y = 0 or y = γ∞rvq[1 + 1/(γ rvq − N̄r)], or equivalently, ρ = −1/γ∞rvq or ρ = (γ∞rvq − N̄r)/γ∞rvq. Since ρ > 0, we must have γ∞rvq − N̄r γ∞rvq . (159) Since ∂2Φ(γ∞rvq, ρ) λ− γ∞rvq 1 + ρ∗(γ∞rvq − λ) g(λ) dλ (160) < 0, (161) therefore ρ∗ achieves a maximum. By also evaluating Φ(γ∞rvq, ρ) at the boundary points ρ = 0 and ρ = 1/(λ∞max − γ∞rvq), we have γ∞rvq−N̄r γ∞rvq , N̄r ≤ γ∞rvq ≤ N̄r + N̄r)2−γ∞rvq , N̄r + N̄r ≤ γ∞rvq < (1 + (162) To evaluate Φ(γ∞rvq, ρ ∗), we re-write (110) as Φ(γ∞rvq, ρ log(ρ∗) + log + γ∞rvq g(λ) dλ (163) = log(ρ∗) + + γ∞rvq g(λ) dλ. (164) To evaluate the integral in (164), we apply the following Lemma. Lemma 1. For x ≥ (1 + Θ(x) , log(x− λ)g(λ) dλ (165) = log(w(x)) + N̄ru(x)− (N̄r − 1) log (166) where w(x) = (x− 1− N̄r) + (x− 1− N̄r)2 − 4N̄r , (167) u(x) = (x− 1− N̄r)− (x− 1− N̄r)2 − 4N̄r . (168) The proof of this Lemma is similar to that given in [36] and is therefore omitted here. For N̄r + N̄r ≤ γ∞rvq < (1 + 2, we substitute ρ∗ = [(1 + 2 − γ∞rvq]−1 into (164) to obtain Φ(γ∞rvq, [(1 + 2 − γ∞rvq]−1) (169) = − log[(1 + 2 − γ∞rvq] + Θ (170) = − log[(1 + 2 − γ∞rvq] + N̄r log(N̄r) − (N̄r − 1) log(1 + N̄r) + N̄r (171) = B̄ log(2). (172) Solving for γ∞rvq gives (18). Taking γ rvq = N̄r + N̄r and solving for B̄ gives B̄∗ in (19). For N̄r ≤ γ∞rvq < N̄r+ N̄r, or 0 ≤ B̄ ≤ B̄∗, we substitute γ∞rvq−N̄r γ∞rvq into (164) to obtain γ∞rvq, γ∞rvq − N̄r γ∞rvq = log(γ∞rvq − N̄r)− log(γ∞rvq) γ∞rvq + γ∞rvq γ∞rvq − N̄r (173) To simplify (173), we let ψ , γ∞rvq − N̄r and re-write (173) as ψ − N̄r, ψ + N̄r = log(ψ)− log(ψ + N̄r) 1 + N̄r + ψ + (174) After some manipulation we have 1 + N̄r + ψ + , (175) 1 + N̄r + ψ + , (176) SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 15 1 + N̄r + ψ + = log(N̄r)− log(ψ) −(N̄r − 1) log (177) Substituting (177) into (174), we obtain ψ − N̄r, ψ + N̄r = ψ − N̄r log (178) = γ∞rvq − N̄r − N̄r log(γ∞rvq) + N̄r log(N̄r). (179) Setting this to B̄ log(2) and simplifying gives (17). For N̄r < 1, the asymptotic eigenvalue density of is given by g(λ) = (1 − N̄r)δ(λ) + (λ− a)(b − λ) . (180) where a and b are given by (28)-(29). Following the same steps again from (110) gives (18) and (17). This completes the proof of Theorem 3. D. Derivation of (41)-(43) To compute µJ , we first write JNrj = 1 + ρ (181) where υk is the kth eigenvalue of Υ = As (Nt, Nr,K) → ∞, the empirical eigenvalue distribution converges to a deterministic function FΥ(t). The asymptotic mean is given by µJ = lim (Nt,Nr,K)→∞ E[JNrj ] = 1 + ρ dFΥ(t). (182) A similar integral has been evaluated in [36, Eq. (6)], and the result can be directly applied to (182), giving (41). To compute the variance, we express JNrj differently by first performing the singular value decomposition H = VHΣHU , where VH is the Nr ×Nr left singular matrix, UH is the Nt × Nr right singular matrix, and ΣH is an Nr × Nr diagonal matrix. Here we assume that Nt ≥ Nr. (The result for Nt < Nr can be shown by a similar approach.) We therefore have JNrj = log det (INr + ρΛLj) (183) log (1 + ρηi) (184) where Λ = 1 , Lj = U j UH , and ηi is the ith eigenvalue of ΛLj . To compute var[J j ], correlations be- tween pairs of ηi’s are needed. Although the joint distribution of eigenvalues is known, it is complicated, so that computing the variance appears intractable. To approximate the variance of JNrj , we substitute a Taylor series expansion for log(1 + δx) into (184) to write JNrj = η2i + η3i + . . . (185) tr{ΛL} − ρ tr{(ΛL)2}+ ρ tr{(ΛL)3} + . . . (186) for ρηmax < 1, where ηmax = maxi ηi, and is the maximum eigenvalue of HV V †H†/K . Since HV is Nr×K and i.i.d., ηmax has asymptotic value (1 + N̄r/K̄) 2. If K̄/N̄r = 1, then the condition asymptotically becomes ρ < 1/4 (-6 dB). Ignoring the terms of order ρ3 and higher, we can approximate the variance of JNrj at low SNR as var[JNrj ] ≈ ρ tr{ΛL} . (187) Letting Λ = diag{λi} and lij denote the (i, j)th element of L, the first term in (187) can be expanded as var[tr{ΛL}|Λ] = E[l2ii]− E2[lii] λiλj (E[liiljj ]− E[lii]E[ljj ]) . (188) For a given UH and random unitary V with K = Nr, The- orem 3 in [7] states that L has a multivariate beta distribution with parameters Nr and Nt − Nr. (The distribution of L is not known for general K .) From Theorem 2 in [47], we have E[lii] = Nr + 1 Nt + 2 (189) E[l2ii] = (Nr + 1)(Nr + 3) (Nt + 2)(Nt + 4) (190) E[liiljj ] = Nr(Nr + 1)(Nt + 4) + (Nr + 1)(Nt −Nr + 1) (Nt + 1)(Nt + 2)(Nt + 4) i 6= j, (191) for 1 ≤ i, j ≤ Nr. Substituting (189)-(191) into (188) gives var[tr{ΛL}|Λ] = N̄2r (1 − N̄r) +O (N̄r − 1)N̄3r +O (192) Taking expectation with respect to Λ, and the large system limit, we have EΛ (var[tr{ΛL}|Λ]) → 1− N̄r. (193) Also, in the large system limit λ2i → t2dFΛ(t) = (194) λiλj → tdFΛ(t) (195) 16 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 55, NO. 3, MARCH 2009 where FΛ(t) is the asymptotic distribution for the diagonal elements of Λ or, equivalently, the asymptotic eigenvalue distribution of HH†/Nr. Substituting (193) into (187), we have σ2J = lim (Nr,Nt)→∞ N2r var[J j ] (196) ≈ ρ2(1− N̄r). (197) ACKNOWLEDGEMENT The authors thank the anonymous reviewers for their de- tailed comments and for pointing out mistakes in the proofs of Theorems 2 and 3, which appeared in an earlier draft. REFERENCES [1] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, “Efficient use of side information in multiple antenna data transmission over fading channels,” IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1423– 1436, Oct. 1998. [2] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On beamforming with finite rate feedback in multiple antenna systems,” IEEE Trans. Info. Theory, vol. 49, no. 10, pp. 2562–2579, Oct. 2003. [3] D. J. Love and R. W. Heath, Jr., “Grassmannian beamforming for multiple-input multiple-output wireless systems,” IEEE Trans. Info. Theory, vol. 49, no. 10, pp. 2735–2745, Oct. 2003. 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Silverstein, “CLT for linear spectral statistics of large dimensional sample covariance matrices,” Annals of Probability, vol. 32, no. 1A, pp. 553–605, 2004. [38] W. Santipach and M. L. Honig, “Signature optimization for DS-CDMA with limited feedback,” in Proc. IEEE Int. Symp. on Spread-Spectrum Tech. and Appl. (ISSSTA), Prague, Czech Republic, Sept. 2002, pp. 180– [39] D. J. Ryan, I. V. L. Clarkson, I. B. Collings, D. Guo, and M. L. Honig, “QAM and PSK codebooks for limited feedback MIMO beamforming,” to appear in IEEE Trans. on Commun., Feb. 2009. [40] V. Raghavan, R. W. Heath, Jr., and A. M. Sayeed, “Systematic codebook designs for quantized beamforming in correlated MIMO channels,” IEEE J. Select. Areas Commun., vol. 25, no. 7, pp. 1298–1310, Sept. 2006. [41] W. Santipach and M. L. Honig, “Capacity of beamforming with limited training and feedback,” in Proc. IEEE Int. Symp. on Info. Theory (ISIT), Seattle, WA, July 2006. [42] ——, “Optimization of training and feedback for beamforming over a MIMO channel,” in Proc. IEEE Wireless Commun. and Networking Conf. (WCNC), Hong Kong, China, Mar. 2007. [43] Y. Sun and M. L. Honig, “Asymptotic capacity of multicarrier transmis- sion with frequency-selective fading and limited feedback,” IEEE Trans. Info. Theory, vol. 54, no. 7, pp. 2879–2902, July 2008. [44] E. Castillo, Extreme Value Theory in Engineering. Academic Press, 1988. [45] A. Shwartz and A. Weiss, Large Deviations for Performance Analysis. Queues, Communications, and Computing. London, UK: Chapman & Hall, 1995. SANTIPACH AND HONIG: CAPACITY OF A MULTIPLE-ANTENNA FADING CHANNEL WITH A QUANTIZED PRECODING MATRIX 17 [46] J. K. Patel and C. B. Read, Handbook of the Normal Distribution, 2nd ed., ser. Statistics: a Series of Textbooks and Monographs. New York: Marcel Dekker, 1996. [47] C. G. Khatri and K. C. S. Pillai, “Some results on the non-central multivariate Beta distribution and moments of traces of two matrices,” Annals of Mathematical Statistics, vol. 36, no. 5, pp. 1511–1520, Oct. 1965. Wiroonsak Santipach (S’00-M’06) received the B.S. (summa cum laude), M.S., and Ph.D. degrees all in electrical engineering from Northwestern University, Illinois, USA in 2000, 2001, and 2006, respectively. He is currently a lecturer at the Department of Electrical Engineering, Faculty of Engineering, Kasetsart University in Bangkok, Thailand. His research interests are in wireless communications, and include performance evaluation of CDMA and MIMO system. Michael L. Honig (S’80-M’81-SM’92-F’97) received the B.S. degree in elec- trical engineering from Stanford University in 1977, and the M.S. and Ph.D. degrees in electrical engineering from the University of California, Berkeley, in 1978 and 1981, respectively. He subsequently joined Bell Laboratories in Holmdel, NJ, where he worked on local area networks and voiceband data transmission. In 1983 he joined the Systems Principles Research Division at Bellcore, where he worked on Digital Subscriber Lines and wireless communications. Since the Fall of 1994, he has been with Northwestern University where he is a Professor in the Electrical Engineering and Computer Science Department. He has held visiting scholar positions at the Technical University of Munich, Princeton University, the University of California, Berkeley, Naval Research Laboratory (San Diego), and the University of Sydney. He has also worked as a free-lance trombonist. Dr. Honig has served as an editor for the IEEE Transactions on Information Theory (1998-2000), the IEEE Transactions on Communications (1990-1995), and was a guest editor for the European Transactions on Telecommunications and Wireless Personal Communications. He has also served as a member of the Digital Signal Processing Technical Committee for the IEEE Signal Processing Society, and as a member of the Board of Governors for the Information Theory Society (1997-2002). He is the recipient of a Humboldt Research Award for Senior U.S. Scientists, and the co-recipient of the 2002 IEEE Communications Society and Information Theory Society Joint Paper Award. Introduction Channel Model Beamforming with Limited Feedback MISO Channel Multi-Input Multi-Output (MIMO) Channel Numerical Results Precoding Matrix with Arbitrary Rank Quantized Precoding with Linear Receivers Matched filter MMSE receiver Numerical Results Conclusions Appendix Proof of Theorem ?? Proof of Theorem ?? Proof of Theorem ?? Derivation of (??)-(??) References Biographies Wiroonsak Santipach Michael L. Honig

0704.0218

On Almost Periodicity Criteria for Morphic Sequences in Some Particular Cases

On Almost Periodicity Criteria for Morphic Sequences in Some Particular Cases Yuri Pritykin∗ November 4, 2018 Abstract In some particular cases we give criteria for morphic sequences to be almost periodic (=uni- formly recurrent). Namely, we deal with fixed points of non-erasing morphisms and with auto- matic sequences. In both cases a polynomial-time algorithm solving the problem is found. A result more or less supporting the conjecture of decidability of the general problem is given. 1 Introduction Different problems of decidability in combinatorics on words are always of great interest and dif- ficulty. Here we deal with two main types of symbolic infinite sequences — morphic and almost periodic — and try to understand connections between them. Namely, we are trying to find an algorithmic criterion which given a morphic sequence decides whether it is almost periodic. Though the main problem still remains open, we propose polynomial-time algorithms solving the problem in two important particular cases: for pure morphic sequences generated by non-erasing morphisms (Section 3) and for automatic sequences (Section 4). In Section 5 we say a few words about connections with monadic logics. In particular, in a curious result of Corollary 4 we give a reason why the main problem may be decidable. Some attempts to solve the problem were already done. In [3] A. Cobham gives a criterion for automatic sequence to be almost periodic. But even if his criterion gives some effective procedure solving the problem (which is not clear from his result, and he does not care about it at all), this procedure could not be fast. We construct a polynomial-time algorithm solving the problem. In [5] A. Maes deals with pure morphic sequences and finds a criterion for them to belong to a slightly different class of generalized almost periodic sequences (but he calls them almost periodic — see [9] for different definitions). And again, his algorithm does not seem to be polynomial-time. All the results of this paper can be found in [10]. 2 Preliminaries Denote the set of natural numbers {0, 1, 2, . . . } by N and the binary alphabet {0, 1} by B. Let A be a finite alphabet. We deal with sequences over this alphabet, i. e., mappings x : N → A, and denote the set of these sequences by AN. ∗Moscow State University, Russia, http://lpcs.math.msu.su/~pritykin/, [email protected]. The work was par- tially supported by RFBR grants 06-01-00122, 05-01-02803, Kolmogorov grant of Institute of New Technologies, and August Möbius grant of Independent University of Moscow. http://arxiv.org/abs/0704.0218v1 Denote by A∗ the set of all finite words over A including the empty word Λ. If i ≤ j are natural, denote by [i, j] the segment of N with ends in i and j, i. e., the set {i, i + 1, i + 2, . . . , j}. Also denote by x[i, j] a subword x(i)x(i + 1) . . . x(j) of a sequence x. A segment [i, j] is an occurrence of a word u ∈ A∗ in a sequence x if x[i, j] = u. We say that u 6= Λ is a factor of x if u occurs in x. A word of the form x[0, i] for some i is called prefix of x, and respectively a sequence of the form x(i)x(i+ 1)x(i+ 2) . . . for some i is called suffix of x and is denoted by x[i,∞). Denote by |u| the length of a word u. The occurrence u = x[i, j] in x is k-aligned if k|i. A sequence x is periodic if for some T we have x(i) = x(i+ T ) for each i ∈ N. This T is called a period of x. We denote by P the class of all periodic sequences. Let us consider an extension of this class. A sequence x is called almost periodic1 if for every factor u of x there exists a number l such that every factor of x of length l contains at least one occurrence of u (and therefore u occurs in x infinitely many times). Obviously, to show almost periodicity of a sequence it is sufficient to check the mentioned condition only for all prefixes but not for all factors (and even for some increasing sequence of prefixes only). Denote by AP the class of all almost periodic sequences. Let A, B be finite alphabets. A mapping φ : A∗ → B∗ is called a morphism if φ(uv) = φ(u)φ(v) for all u, v ∈ A∗. A morphism is obviously determined by its values on single-letter words. A morphism is non-erasing if |φ(a)| > 1 for each a ∈ A. A morphism is k-uniform if |φ(a)| = k for each a ∈ A. A 1-uniform morphism is called a coding. For x ∈ AN denote φ(x) = φ(x(0))φ(x(1))φ(x(2)) . . . Further we consider only morphisms of the form A∗ → A∗ (but codings are of the form A → B, which in fact does not matter, they can be also of the form A→ A without loss of generality). Let φ(s) = su for some s ∈ A, u ∈ A∗. Then for all natural m < n the word φn(s) begins with the word φm(s), so φ∞(s) = limn→∞ φ n(s) = suφ(u)φ2(u)φ3(u) . . . is well-defined. If ∀n φn(u) 6= Λ, then φ∞(s) is infinite. In this case we say that φ is prolongable on s. Sequences of the form h(φ∞(s)) for a coding h : A→ B are called morphic, of the form φ∞(a) are called pure morphic. Notice that there exist almost periodic sequences that are not morphic (in fact, the set of almost periodic sequences has cardinality continuum, while the set of morphic sequences is obviously countable), as well as there exist morphic sequences that are not almost periodic (you will find examples later). Our goal is to determine whether a morphic sequence is almost periodic or not given its constructive definition. First of all, observe the following Lemma 1. A sequence φ∞(s) is almost periodic iff s occurs in this sequence infinitely many times with bounded distances. Proof. In one direction the statement is obviously true by definition. Suppose now that s occurs in φ∞(s) infinitely many times with bounded distances. Then for every m the word φm(s) also occurs in φ∞(s) infinitely many times with bounded distances. But every word u occurring in φ∞(s) occurs in some prefix φm(s) and thus occurs infinitely many times with bounded distances. For a morphism φ : {1, . . . , n} → {1, . . . , n} we can define a corresponding matrix M(φ), such that M(φ)ij is a number of occurrences of symbol i into φ(j). One can easily check that for each l we have M(φ)l =M(φl). Morphism φ is called primitive if for some l all the numbers in M(φl) are positive. 1It was called strongly or strictly almost periodic in [7, 8]. Let us construct an oriented graph G corresponding to a morphism. Let its set of vertices be A. In G edges go from b ∈ A to all the symbols occurring in φ(b). For φ∞(s) it can easily be found using the graph corresponding to φ which symbols from A really occur in this sequence. Indeed, these symbols form the set of all vertices that can be reached from s. So without loss of generality from now on we assume that all the symbols from A occur in φ∞(s). A morphism is primitive if and only if its corresponding graph is strongly connected, i. e., there exists an oriented path between every two vertices. This reformulation of the primitiveness notion seems to be more appropriate for computational needs. By Lemma 1 (and the observation that codings preserve almost periodicity) morphic sequences obtained by primitive morphisms are always almost periodic. Moreover, in the case of increasing morphisms (such that |φ(b)| > 2 for each b) this sufficient condition is also necessary (and this is a polynomial-time algorithmic criterion). However when we generalize this case even on non- erasing morphisms, it is not enough to consider only the corresponding graph or even the matrix of morphism (which has more information), as it can be seen from the following example. Let φ1 be as follows: 0 → 01, 1 → 120, 2 → 2, and φ2 be as follows: 0 → 01, 1 → 210, 2 → 2. Then these two morphisms have identical matrices of morphism, but φ∞1 (0) is almost periodic, while φ∞2 (0) is not. Indeed, in φ 2 (0) there are arbitrary long segments like 222. . . 22, so φ 2 (0) /∈ AP . There is no such problem in φ∞1 (0). Since 0 occurs in both φ1(0) and φ1(1), and 22 does not occur in φ∞1 (0), it follows that 0 occurs in φ 1 (0) with bounded distances. Thus φ 1 (0) for every m > 0 occurs in φ∞1 (0) with bounded distances, so φ 1 (0) ∈ AP. See Theorem 1 for a general criterion of almost periodicity in the case of fixed points of non-erasing morphisms. To introduce a bit the notion of almost periodicity, let us formulate an interesting result on this topic. It seems to be first proved in [3], but also follows from the results of [9]. For x ∈ AN, y ∈ BN define x× y ∈ (A×B)N such that (x× y)(i) = 〈x(i), y(i)〉. Proposition 1. If x is almost periodic and y is periodic, then x× y is almost periodic. 3 Pure Morphic Sequences Generated by Non-erasing Morphisms Here we consider the case of morphic sequence of the form φ∞(s) for non-erasing φ. We present an algorithm that determines whether a morphic sequence φ∞(s) is almost periodic given an alphabet A, a morphism φ and a symbol s ∈ A. Suppose we have A, φ and s ∈ A, such that |A| = n, maxb∈A |φ(b)| = k, s begins φ(s). Remember that we suppose that all the symbols from A appear in φ∞(s). Divide A into two parts. Let I be the set of all symbols b ∈ A such that |φm(b)| → ∞ as m→ ∞. Denote F = A \ I, it is the set of all symbols b such that |φm(b)| is bounded. Also define E ⊆ F to be the set of all symbols b such that |φ(b)| = 1. We can find a decomposition A = I ⊔ F in poly(n, k)-time as follows. Find E. Then find all the cycles in G with all the vertices lying in E. Join all the vertices of all these cycles in a set D. This set is stabilizing: F is the set of all vertices in G such that all infinite paths starting from them stabilize in D. Polynomiality can be checked easily. Construct “a graph of left tails” L with marked edges. Its set of vertices is I. From each vertex b exactly one edge goes off. To construct this edge, find a representation φ(b) = uv, where c ∈ I, u is the maximal prefix of φ(b) containing only symbols from F . It follows from the definitions of I and F that u does not coincide with φ(b), that is why this representation is correct. Then construct in L an edge from b to c and write u on it. Analogously we construct “a graph of right tails” R. (In this case we consider representations φ(b) = vu where u ∈ F ∗, c ∈ I.) Now we formulate a general criterion. Theorem 1. A sequence φ∞(s) is almost periodic iff 1) G restricted to I is strongly connected; 2) in graphs L and R on each edge of each cycle an empty word Λ is written. It seems that full and detailed proof of this theorem can only confuse a reader, rather than a proof sketch. Proof sketch. By Lemma 1 for almost periodicity it is necessary and sufficient to check whether symbol s occurs infinitely many times with bounded distances. For every symbol b ∈ I the symbol s should occur in some φl(b), that is what the 1st part of the criterion says. Furthermore, in the sequence φ∞(s) all the segments of consecutive symbols from F should be bounded. Indeed, every such segment consists only of symbols from F , but s /∈ F . That is what the 2nd part of the criterion means, let us explain why. Consider some v = buc occurring somewhere in φ∞(a), where b, c ∈ I, u ∈ F ∗. Every element of sequence of words v, φ(v), φ2(v), φ3(v), . . . occurs in φ∞(s). Somewhere in the middle of φl(v) = φl(b)φl(u)φl(c) a word φl(u) occurs. As l increases, some words from F ∗ might stick to φl(u) from left or right for these words can come from φl(b) or φl(c). These words exactly correspond to those written on edges of L or R. The 2nd part of the criterion exactly says that this situation can happen only finitely many times, until we get to some cycle in L or R. Let us consider examples with φ1 and φ2 from the end of Section 2. In both cases I = {0, 1}, F = {2}. On every edge of R in both cases Λ is written. Almost the same is true for L: the only difference is about the edge going from 1 to 1. In the case of φ1 an empty word is written on this edge, while in the case of φ2 a word 2 is written. That is why φ 1 (0) is almost periodic, while φ 2 (0) is not. Corollary 1. If for all b ∈ A we have |φ(b)| > 2, then φ∞(s) is almost periodic iff φ is primitive. Proof. Follows from Theorem 1. In that case A = I, and on all the edges of L and R the empty word is written. Corollary 2. There exists a poly(n, k)-algorithm that says whether φ∞(s) is almost periodic. Proof. Conditions from Theorem 1 can be checked in polynomial time. It also seems useful to formulate an explicit version of the criterion for the binary case. We do it without any additional assumptions, opposite to the previous. Corollary 3. For non-erasing φ : B → B that is prolongable on 0 a sequence φ∞(0) is almost periodic iff one of the following conditions holds: 1) φ(0) contains only 0s; 2) φ(1) contains 0; 3) φ(1) = Λ; 4) φ(1) = 1 and φ(0) = 0u0 for some word u. 4 Uniform Morphisms Now we deal with morphic sequences obtained by uniformmorphisms. Again we present a polynomial- time algorithm for solving the problem in this situation. Suppose we have an alphabet A, a morphism φ : A∗ → A∗, a coding h : A→ B, and s ∈ A, such that |A| = n, |B| 6 n, ∀b ∈ A |φ(b)| = k, s begins φ(s). We are interested in whether h(φ∞(s)) is almost periodic. Sequences of the form h(φ∞(s)) with φ being k-uniform are also called k-automatic (see [1]). 4.1 Equivalence Relations and Uniform Morphisms For each l ∈ N define an equivalence relation on A: b ∼l c iff h(φ l(b)) = h(φl(b)). We can easily continue this relation on A∗: u ∼l v iff h(φ l(u)) = h(φl(v)). In fact, this means |u| = |v| and u(i) ∼l v(i) for all i, 1 6 i 6 |u|. Let Bm be the Bell number, i. e., the number of all possible equivalence relations on a finite set with exactly m elements, see [13]. As it follows from this article, we can estimate Bm in the following way. Lemma 2. 2m 6 Bm 6 2 Cm logm for some constant C. Thus the number of all possible relations ∼l is not greater than Bn = 2 O(n logn). Moreover, the following lemma gives a simple description for the behavior of these relations as l tends to infinity. Lemma 3. If ∼r equals ∼s, then ∼r+p equals ∼s+p for all p. Proof. Indeed, suppose ∼r equals ∼s. Then b ∼r+1 c iff φ(b) ∼r φ(c) iff φ(b) ∼s φ(c) iff b ∼s+1 c. So if ∼r equals ∼s, then ∼r+1 equals ∼s+1, which implies the lemma statement. This lemma means that the sequence (∼l)l∈N turns out to be ultimately periodic with a period and a preperiod both not greater than Bn. Thus we obtain the following Lemma 4. For some p, q 6 Bn we have for all i and all t > p that ∼t equals ∼t+iq. 4.2 Criterion Now we are trying to get a criterion which we could check in polynomial time. Notice that the situation is much more difficult than in the pure case because of a coding allowed. In particular, the analogue of Lemma 1 for non-pure case does not hold. We will move step by step to the appropriate version of the criterion reformulating it several times. This proposition is quite obvious and follows directly from the definition of almost periodicity since all h(φm(a)) are the prefixes of h(φ∞(a)). Proposition 2. A sequence h(φ∞(s)) is almost periodic iff for all m the word h(φm(s)) occurs in h(φ∞(s)) infinitely often with bounded distances. And now a bit more complicated version. Proposition 3. A sequence h(φ∞(s)) is almost periodic iff for all m the symbols that are ∼m- equivalent to s occur in φ∞(s) infinitely often with bounded distances. Proof. ⇐. If the distance between two consecutive occurrences in φ∞(s) of symbols that are ∼m- equivalent to s is not greater than t, then the distance between two consecutive occurrences of h(φm(s)) in h(φ∞(s)) is not greater than tkm. ⇒. Suppose h(φ∞(s)) is almost periodic. Let ym = 012 . . . (k m − 2)(km − 1)01 . . . (km − 1)0 . . . be a periodic sequence with a period km. Then by Proposition 1 a sequence h(φ∞(s))×ym is almost periodic, which means that the distances between consecutive km-aligned occurrences of h(φm(s)) in h(φ∞(s)) are bounded. It only remains to notice that if h(φ∞(s))[ikm, (i+1)km−1] = h(φm(s)), then φ∞(s)(i) ∼m s. Let Ym be the following statement: symbols that are ∼m-equivalent to s occur in φ ∞(s) infinitely often with bounded distances. Suppose for some T that YT is true. This implies that h(φ T (s)) occurs in h(φ∞(s)) with bounded distances. Therefore for all m 6 T a word h(φm(s)) occurs in h(φ∞(s)) with bounded distances since h(φm(s)) is a prefix of h(φT (s)). Thus we do not need to check the statements Ym for all m, but only for all m > T for some T . Furthermore, it follows from Lemma 4, that we are sufficient to check the only one such state- ment as in the following Proposition 4. For all r > Bn: a sequence h(φ ∞(s)) is almost periodic iff the symbols that are ∼r-equivalent to s occur in φ ∞(s) infinitely often with bounded distances. And now the final version of our criterion. Proposition 5. For all r > Bn: a sequence h(φ ∞(s)) is almost periodic iff for some m the symbols that are ∼r-equivalent to s occur in φ m(b) for all b ∈ A. Indeed, if the symbols of some set occur with bounded distances, then they occur on each km-aligned segment for some sufficiently large m. 4.3 Polynomiality Now we explain how to check a condition from Proposition 5 in polynomial time. We need to show two things: first, how to choose some r > Bn and to find in polynomial time the set of all symbols that are ∼r-equivalent to s (and this is a complicated thing keeping in mind that Bn is exponential), and second, how to check whether for some m the symbols from this set for all b ∈ A occur in φm(b). Let us start from the second. Suppose we have found the set H of all the symbols that are ∼r-equivalent to s. For m ∈ N let us denote by P m the set of all the symbols that occur in φm(b). Our aim is to check whether exists m such that for all b we have P m ∩H 6= ∅. First of all, notice that if ∀b P m ∩H 6= ∅, then ∀b P ∩H 6= ∅ for all l > m. Second, notice that the sequence of tuples of sets ((P m )b∈Σ) m=0 is ultimately periodic. Indeed, the sequence (P m=0 is obviously ultimately periodic with both period and preperiod not greater than 2n (recall that n is the size of the alphabet Σ). Thus the period of ((P m )b∈Σ) m=0 is not greater than the least common divisor of that for (P m=0, b ∈ A, and the preperiod is not greater than the maximal that of m=0. So the period is not greater than (2 n)n = 2n and the preperiod is not greater than 2n. Third, notice that there is a polynomial-time-procedure that given a graph corresponding to some morphism ψ (see Section 2 to recall what is the graph corresponding to a morphism) outputs a graph corresponding to morphism ψ2. Thus after repeating this procedure n2 + 1 times we obtain a graph by which we can easily find (P )b∈Σ, since 2 n2+1 > 2n + 2n. Similar arguments, even described with more details, are used in deciding our next problem. Here we present a polynomial-time algorithm that finds the set of all symbols that are ∼r-equivalent to s for some r > Bn. We recursively construct a series of graphs Ti. Let its common set of vertices be the set of all unordered pairs (b, c) such that b, c ∈ A and b 6= c. Thus the number of vertices is n(n−1) . The set of all vertices connected with (b, c) in the graph Ti we denote by Vi(b, c). Define a graph T0. Let V0(b, c) be the set {(φ(b)(j), φ(c)(j)) | j = 1, . . . , k, φ(b)(j) 6= φ(c)(j)}. In other words, b ∼l+1 c if and only if x ∼l y for all (x, y) ∈ V0(b, c). Thus b ∼2 c if and only if for all (x, y) ∈ V0(b, c) for all (z, t) ∈ V0(x, y) we have z ∼0 t. For the graph T1 let V1(b, c) be the set of all (x, y) such that there is a path of length 2 from (b, c) to (x, y) in T0. The graph T1 has the following property: b ∼2 c if and only if x ∼0 y for all (x, y) ∈ V1(b, c). And even more generally: b ∼l+2 c if and only if x ∼l y for all (x, y) ∈ V1(b, c). Now we can repeat operation made with T0 to obtain T1. Namely, in T2 let V2(b, c) be the set of all (x, y) such that there is a path of length 2 from (b, c) to (x, y) in T1. Then we obtain: b ∼l+4 c if and only if x ∼l y for all (x, y) ∈ V2(b, c). It follows from Lemma 2 that log2Bn 6 Cn logn. Thus after we repeat our procedure r = [Cn log n] times, we will obtain the graph Tl such that b ∼2r c if and only if x ∼0 y for all (x, y) ∈ V2(b, c). Recall that x ∼0 y means h(x) = h(y), so now we can easily compute the set of symbols that are ∼2r -equivalent to s. 5 Monadic Theories Combinatorics on words is closely connected with the theory of second order monadic logics. Here we just want to show some examples of these connections. More details can be found, e. g., in [11, 12]. We consider monadic logics on N with the relation “<”, that is, first-order logics where also unary finite-value function variables and quantifiers over them are allowed. We also suppose that we know some fixed finite-value function x : N → Σ and can use it in our formulas. Such a theory is denoted by MT〈N, <, x〉 and is called monadic theory of x. The main question here can be the question of decidability, that is, does there exist an algorithm that given a sentence in a theory says whether this sentence is true of false. The criterion of decidability for monadic theories of almost periodic sequences can be formulated in terms of some their very natural characteristic, namely, almost periodicity regulator. An almost periodicity regulator of an almost periodic sequence x is a function f : N → N such that every factor u of x of length n occurs in each factor of x of length f(n). So an almost periodicity regulator somehow regulates how periodic a sequence is. Notice that an almost periodicity regulator of a sequence is not unique: every function greater than regulator is also a regulator. Theorem 2 (Semenov 1983 [12]). If x is almost periodic, then MT〈N, <, x〉 is decidable iff x and some its almost periodicity regulator are computable. The following result was obtained recently, but uses the technics already used in [11, 12]. Theorem 3 (Carton, Thomas 2002 [2]). If x is morphic, then MT〈N, <, x〉 is decidable. A curious result can be implied from two these theorems. Corollary 4. If x is both morphic and almost periodic, then some its regulator is computable. Proof. Indeed, if x is morphic, then by Theorem 3 the theory MT〈N, <, x〉 is decidable. Since x is almost periodic, from Theorem 2 it follows that some almost periodicity regulator of x is computable. Notice that Corollary 4 does not imply the existence of an algorithm that given a morphic sequence computes some almost periodicity regulator of this sequence whenever it is almost periodic (but probably this algorithm can be constructed after deep analyzing the proofs of Theorems 2 and 3 and showing uniformity in a sense). And it also does not imply the decidability of almost periodicity for morphic sequences. This decidability also does not imply Corollary 4. By the way, Corollary 4 allows us to hope that these algorithms exist. Though the formulation of this statement uses only combinatorics on words, the proof also involves the theory of monadic logics. Of course, it would be interesting to find a simple combinatorial proof of the result. And the last remark here is that Corollary 4 (and its probable uniform version) seems to be the best progress that we can obtain by this monadic approach. One could try to express in the monadic theory of morphic sequence (which is decidable by Theorem 3) the property of almost periodicity, but it turns out to be impossible. 6 In General Case We have described two polynomial-time algorithms, but without any precise bound for their working time. Of course, it can be done after deep analyzing of all the previous, but is probably not so interesting. It is not still known whether the problem of determining almost periodicity of arbitrary morphic sequence is decidable. Corollary 4 somehow supports the conjecture of decidability (but even does not follow from this conjecture!). Theorem 7.5.1 from [1] allows us to represent an arbitrary morphic sequence h(φ∞(s)) as g(ψ∞(b)) where ψ is non-erasing. So it is sufficient to solve our main problem for h(φ∞(s)) with non-erasing φ. It seems that the general problem is tightly connected with a particular case of h(φ∞(a)) where |φ(b)| > 2 for each b ∈ A. There is no strict reduction to this case but solving problem in this case can help to deal with general situation. The problem of finding an effective periodicity criterion in the case of arbitrary morphic se- quences is also of great interest, as well as criteria for variations with periodicity and almost pe- riodicity: ultimate periodicity, generalized almost periodicity, ultimate almost periodicity (see [9] for definitions). If one notion is a particular case of another, it does not mean that corresponding criterion for the first case is more difficult (or less difficult) than for the second. Acknowledgements The author is grateful to An. Muchnik and A. Semenov for their permanent help in the work, to A. Frid, M. Raskin, K. Saari and to all the participants of Kolmogorov seminar, Moscow [4], for fruitful discussions, and also to anonymous referees for very useful comments. References [1] J.-P. Allouche, J. Shallit. Automatic Sequences. Cambridge University Press, 2003. [2] O. Carton, W. Thomas. The Monadic Theory of Morphic Infinite Words and Generalizations. Information and Computation, vol. 176, pp. 51–76, 2002. [3] A. Cobham. Uniform tag sequences. Math. Systems Theory, 6, pp. 164–192, 1972. [4] Kolmogorov Seminar: http://lpcs.math.msu.su/kolmogorovseminar/eng/. [5] A. Maes. More on morphisms and almost-periodicity. Theoretical Computer Science, vol. 231, N 2, pp. 205–215, 2000. [6] M. Morse, G. A. Hedlund. Symbolic dynamics. American Journal of Mathematics, 60, pp. 815– 866, 1938. [7] An. Muchnik, A. Semenov, M. Ushakov. Almost periodic sequences. Theoretical Computer Science, vol. 304, pp. 1–33, 2003. [8] Yu. L. Pritykin. Finite-Automaton Transformations of Strictly Almost-Periodic Se- quences. Mathematical Notes, vol. 80, N 5, pp. 710–714, 2006. Preprint on http://arXiv.org/abs/cs.DM/0605026. [9] Yu. Pritykin. Almost Periodicity, Finite Automata Mappings and Related Effectiveness Issues. Proceedings of WoWA’06, St. Petersburg, Russia (satellite to CSR’06). To appear in ”Izvestia VUZov. Mathematics”, 2007. Preprint on http://arXiv.org/abs/cs.DM/0607009. [10] Yu. Pritykin. On Almost Periodicity Criteria for Morphic Sequences in Some Particular Cases. Accepted to Developments in Language Theory, Turku, Finland, 2007. To appear in Lecture Notes in Computer Science. [11] A. L. Semenov. On certain extensions of the arithmetic of addition of natural numbers. Math. of USSR, Izvestia, vol. 15, pp. 401–418, 1980. [12] A. L. Semenov. Logical theories of one-place functions on the set of natural numbers. Math. of USSR, Izvestia, vol. 22, pp. 587–618, 1983. [13] Eric W. Weisstein. Bell Number. From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/BellNumber.html Introduction Preliminaries Pure Morphic Sequences Generated by Non-erasing Morphisms Uniform Morphisms Equivalence Relations and Uniform Morphisms Criterion Polynomiality Monadic Theories In General Case

0704.0219

The Radio Emission, X-ray Emission, and Hydrodynamics of G328.4+0.2: A Comprehensive Analysis of a Luminous Pulsar Wind Nebula, its Neutron Star, and the Progenitor Supernova Explosion

Draft version November 4, 2018 Preprint typeset using LATEX style emulateapj v. 3/25/03 THE RADIO EMISSION, X-RAY EMISSION, AND HYDRODYNAMICS OF G328.4+0.2: A COMPREHENSIVE ANALYSIS OF A LUMINOUS PULSAR WIND NEBULA, ITS NEUTRON STAR, AND THE PROGENITOR SUPERNOVA EXPLOSION Joseph D. Gelfand, Patrick O. Slane, and Daniel J. Patnaude Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 B. M. Gaensler∗ Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 and School of Physics, The University of Sydney, NSW 2006, Australia John P. Hughes Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854-8019 Fernando Camilo Columbia Astrophysics Lab, Columbia University, New York, NY 10027 Draft version November 4, 2018 ABSTRACT We present new observational results obtained for the Galactic non-thermal radio source G328.4+0.2 to determine both if this source is a pulsar wind nebula or supernova remnant, and in either case, the physical properties of this source. Using X-ray data obtained by XMM, we confirm that the X-ray emission from this source is heavily absorbed and has a spectrum best fit by a power law model of photon index Γ = 2 with no evidence for a thermal component, the X-ray emission from G328.4+0.2 comes from a region significantly smaller than the radio emission, and that the X-ray and radio emission are significantly offset from each other. We also present the results of a new high resolution (7′′) 1.4 GHz image of G328.4+0.2 obtained using the Australia Telescope Compact Array, and a deep search for radio pulsations using the Parkes Radio Telescope. By comparing this 1.4 GHz image with a similar resolution image at 4.8 GHz, we find that the radio emission has a flat spectrum (α ≈ 0; Sν ∝ ν α), though some areas of the eastern edge of G328.4+0.2 have a steeper radio spectral index of α ∼ −0.3. Additionally, we searched without success for a central radio pulsar, and obtain a luminosity limit of L1400 <. 30mJykpc 2, assuming a distance of 17 kpc. In light of these observational results, we test if G328.4+0.2 is a pulsar wind nebula (PWN) or a large PWN inside a supernova remnant (SNR) using a simple hydrodynamic model for the evolution of a PWN inside a SNR. As a result of this analysis, we conclude that G328.4+0.2 is a young (. 10000 years old) pulsar wind nebula formed by a low magnetic field (. 1012 G) neutron star born spinning rapidly (. 10 ms) expanding into an undetected SNR formed by an energetic (& 1051 ergs), low ejecta mass (Mej . 5M⊙) supernova explosion which occurred in a low density (n ∼ 0.03 cm −3) environment. If correct, the low magnetic field and fast initial spin period of this neutron star poses problems for models of magnetar formation which require fast initial periods. Subject headings: stars: neutron, stars: pulsars: general, ISM: supernova remnants, radio continuum: ISM, X-rays: individual 1. introduction Stars with initial masses between ∼9 and 25 M⊙ are expected to end their lives in a giant supernova (SN) ex- plosion during which neutron stars are created. The fast moving ejecta from the SN create a supernova remnant (SNR), while the particle wind produced by the neu- tron star as it loses rotational energy inflates a pulsar wind nebula (PWN; Gaensler & Slane 2006). Initially, the PWN is inside the SNR – and when the PWN is detected inside the SNR the system is called a “compos- ite” SNR (Helfand & Becker 1987). The evolution of the central neutron star and the outer SNR affect the PWN, Electronic address: [email protected] ∗Alfred P. Sloan Research Fellow, Australian Research Council Federation Fellow Electronic address: [email protected] and as a result the PWN goes through several evolution- ary phases while it is inside the SNR. This evolution is determined by the physical properties of the neutron star (specifically the initial period P0, the braking index p and the strength of the dipole component to the surface magnetic field Bns), the SN explosion (explosion energy Esn and ejecta mass Mej), and the surrounding medium (ambient number density n). As a result, by measur- ing the properties of the PWN inside a SNR at a given time one is able to constrain these physical parameters which allows one to study the mechanisms behind both core-collapse SNe and massive star evolution. With this in mind, we present results of new radio 1 The braking index is defined as Ω̇ ∝ Ωp, where Ω is the angular velocity of the neutron star’s surface. http://arxiv.org/abs/0704.0219v1 mailto:[email protected] mailto:[email protected] observations of this source with the Australia Telescope Compact Array (ATCA), as well as a new X-ray (X-ray Multi-mirror Mission; XMM) observation of G328.4+0.2 (MSH 15-57; Mills et al. 1961). This source is a distant (d ≥ 17.4 ± 0.9 kpc; Gaensler et al. 2000), radio bright (flux density Sν = 14.3 ± 0.1 Jy at ν =1.4 GHz), po- larized, extended (diameter D ≃ 5.0′) radio source with a relatively flat spectral index (α ≃ −0.12± 0.03 where Sν ∝ ν α; Gaensler et al. 2000). Based on these radio properties, and the discovery of non-thermal X-ray emis- sion from this source by ASCA (Hughes et al. 2000), this source was classified as a PWN – the largest and most radio-luminous PWN in the Galaxy. In this interpre- tation, the expectation is that G328.4+0.2 is ∼ 7000 years old and powered by an extremely energetic neu- tron star (Gaensler et al. 2000). However, follow up ra- dio polarimetry work (Johnston et al. 2004) implied that G328.4+0.2 is an older composite SNR in which the PWN is just a small fraction of the total volume, and as a result is powered by a significantly less energetic neutron star then argued by Gaensler et al. (2000). In this paper, we analyze new observations of this source in order to determine the age of G328.4+0.2, the energet- ics of the neutron star and the progenitor SN, and the density of its environment. In §2 we present new X-ray and radio observations of G328.4+0.2. In §3, we first discuss the expected evolu- tionary sequence for PWNe inside SNRs (§3.1), making general comments regarding the expected observational signature of each phase. In §3.2 use the observational results presented in §2 to draw some initial conclusions about the nature of G328.4+0.2. In §4, we present a sim- ple hydrodynamical model for the evolution of a PWN inside a SNR, which we apply to G328.4+0.2 assuming it is a composite SNR (§4.1.1) or a PWN (§4.1.2). Finally, in §5 we summarize our results. 2. observations In this Section, we present the data gathered in a XMM observation (§2.1), a 1.4 GHz ATCA observation of G328.4+0.2 (§2.2), and a search for a radio pulsar in this source (§2.3). 2.1. X-ray Observations On 2003 March 9–10, G328.4+0.2 was observed for ∼50 ks by XMM. During this observation, the pn cam- era was operated in Small Window Mode, and the Mos1 and Mos2 camera were operated in Full Frame Mode. The “Thick” optical filter was used due to the presence of numerous bright stars in the field-of-view of G328.4+0.2. The data were reduced with the software package xmm- sas v 6.0.0 with calibration files current throughXMM- CCF-REL-174, using the standard procedure for re- ducing XMM data outlined in the XMM-Newton ABC Guide2 and the Birmingham XMM Guide3. 2.1.1. Image Analysis A vignetting corrected 0.2–12 keV image from the Mos1 and Mos2 instruments4, is shown in Fig. 1, in 2 Available at http://heasarc.gsfc.nasa.gov/docs/xmm/abc/ 3 Available at http://www.sr.bham.ac.uk/xmm2/guide.html 4 We did not use the pn data due to the substantially larger pixel size of this instrument. which we observe three spatial components to the X-ray emission: a bright, compact feature located along the SW edge of the X-ray emission (“Clump 1”), a fainter, slightly extended feature located NE of the compact fea- ture described above (“Clump 2”), and extended diffuse emission, roughly 1′ in diameter, surrounding the two features described above (“Diffuse”). From the “Clump 1” region we detected 120 ± 12 counts above the back- ground between 0.5–10 keV in the Mos1 detector and 136± 12 in the Mos2 detector, from the “Clump 2” re- gion we detected 66 ± 9 counts in both the Mos1 and Mos2 detectors, and from the “Diffuse” regions we de- tected 360± 20 and 380± 20 counts from the Mos1 and Mos2 detectors, respectively. We determined the spatial properties of these com- ponents using the Sherpa modeling software package (Freeman et al. 2001). Due to the low number of counts per pixel, we used the simplex fitting method and min- imized the cash statistic (Cash 1979). We attempted to model Clump 1 and Clump 2 as circular 2D Gaus- sians, elliptical 2D Gaussians, or circular 2D Lorentzians, the latter of which is a good model for XMM’s point spread function (PSF; Ghizzardi & Molendi 2002). We attempted to model the Diffuse region as a circular or elliptical 2D Gaussian, and assumed a constant back- ground. We fit the observed image to all model com- binations of Clump 1, Clump 2, and Diffuse (attempts to eliminate one of these components resulted in signif- icantly worse fits), and the best fit was obtained for a model in which Clump 1 is a 2D Lorentzian while Clump 2 and Diffuse are elliptical 2D Gaussians. The fit parameters for this model are given in Table 1, and the model and residuals are shown in Fig. 1, and from this conclude that the emission from Clump 1 is consis- tent with the PSF of XMM. Additionally, from this fit we estimate that Clump 1 and Clump 2 are separated by ∼ 10′′, while the centers of Clump 1 and Diffuse are separated by ∼ 15′′. 2.1.2. Spectral Results In generating the spectra, only events with flag = 0 were used. Additionally, the event files were screened for background flares by binning the 10–15 keV light curve of each instrument by 50 s and then recursively flagging all bins with a count rate > 3σ above the average. This procedure removed 3.0 ks, 2.3 ks, and 0.6 ks of data from the Mos1, Mos2, and pn detectors, respectively. Spec- tra were extracted for the regions shown in Fig. 1, and the resulting spectra were binned into a minimum of 25 counts per channel and modeled using Xspec v12.2.0. The background regions used are also shown in Fig. 1. To determine the composite spectra of G328.4+0.2, we jointly fit the spectrum obtained by the Mos1, Mos2, and pn detectors – shown in Fig. 2. The background- subtracted observed 0.5–10 keV count rate of G328.4+0.2 was 0.012 ± 0.001 counts s−1 in the the Mos1 detector (555± 29 counts), 0.012± 0.001 counts s−1 in the Mos2 detector (580±30 counts), and 0.045±0.001 counts s−1 in the pn detector5 (1576±63 counts). We fit the spectra to seven different models separately – a power-law, a black- body, bremsstrahlung, a Raymond-Smith plasma, and a 5 The pn detector count rate does not account for 29% dead time since this instrument was operated in Small Window Mode. http://heasarc.gsfc.nasa.gov/docs/xmm/abc/ http://www.sr.bham.ac.uk/xmm2/guide.html power-law plus one of these three thermal models – all attenuated for interstellar absorption. Only the single- component models produced reasonable fits (reduced χ2 ∼ 1), and the fitted parameters are presented in Table 2. Both the blackbody (kTBB ∼ 1.7 keV, TBB ∼ 20 MK) and the bremsstrahlung models (kT ∼ 9 keV) require unrealistically high temperatures, especially if the X-rays are from a PWN as argued by Hughes et al. (2000). The derived parameters for the power law model are similar to that observed in other PWN (e.g. Gotthelf 2003), and agree well with the results obtained for G328.4+0.2 by Hughes et al. (2000). There is no evidence for thermal X-ray emission, which one would expect from a SNR, in this source. Since the individual regions discussed in §2.1.1 did not have enough counts for spectral fitting, we measured their hardness ratio (HR), defined as: H − S H + S where H is the number of counts in the Hard (higher energy) band and S is the number of counts in the Soft (lower energy) band, of the regions discussed in §2.1.1 to determine if there were any spatial variations in the X- ray spectrum of G328.4+0.2. Using the X-ray spectrum as a guide, we calculate HR with H as the number of counts between 4 and 8 keV and S as the number of counts between 2 and 4 keV. Since the pixels on the pn detector are sufficiently large that it is not possible to separate the emission from these regions, we only use data from the mos1 and mos2 detectors. The calculated, background subtracted HR of G328.4+0.2 is 0.25± 0.04, of the Clump 1 region is 0.29±0.07, of the Clump 2 region is 0.31 ± 0.11, and of the diffuse region is 0.24 ± 0.05 (1σ errors). As a result, we conclude that there is no significant change in the X-ray spectrum of G328.4+0.2 between these features. 2.1.3. Timing Results A clear signature for the presence of a neutron star would be the detection of X-ray pulsations in the emis- sion from G328.4+0.2. We searched for this using the Z2n test defined by Buccheri et al. (1983), where n is the har- monic number of periodic signal, for n = 1, 2, 3, 4. The maximum frequency searched was νmax = 1/(2n∆t), the minimum frequency searched was 1/20 Hz, and the fre- quency step was 1/4tobs (5× 10 −6 Hz, oversampling the Nyquist rate by a factor of 2), where ∆t is the time res- olution of the dataset (5.7 ms) and tobs is the length of the observation. We only used events from the pn in- strument (5.7 ms since it was operated in Small Win- dow Mode) due to the poor time resolution (2.6s) of the mos data. Additionally, we only used events from the Clump 1 region because only only emission from the central NS should be pulsed and as the brightest X-ray region of G328.4+0.2, this region is the most probable location of any neutron star. Unfortunately, due to the large pixel size of the pn instrument this region is con- taminated by emission from Clump 2 and the Diffuse re- gion. The event times from the resultant event list were barycentered to the Solar System reference frame, and we searched for a periodicity over multiple energy ranges in order to maximize the sensitivity of our search. The most significant period detected was in the dataset which only included photons between 5 and 10 keV (159 photons), in which for n = 4 a signal with period P = 336 ms had Z23 = 49.5 in 2074786 independent trials for this value of n and energy range, which has a 12% chance of being a false positive, a < 2σ result. Statistically, the most significant sinusoidal (n = 1) pulse was in the 1–20 keV dataset (340 photons), and had a period P = 73.1 ms with a Z21 = 34.4 in 8365396 trials, a 34% chance of being a false positive. Assuming that this signal is not significant, we derive an upper limit on the pulse fraction of 45% for a sinusoidal pulse profile and 22.5% for a δ- function pulse profile (Leahy et al. 1983). Using the pn count rate and size of the Clump 2 and Diffuse regions, ∼ 1/3 of the counts in the Clump 1 region is contamina- tion from these regions. Accounting for this, we are only able to put an upper limit on the pulse fraction of 67% for a δ-function pulse profile. This upper limit is consis- tent with the pulse fraction observed from other young neutron stars. 2.2. Australia Telescope Compact Array Observations Based on previous radio observations of G328.4+0.2, there was a dispute in the literature as to whether this source is a PWN, as argued by Gaensler et al. (2000), or a composite SNR, as argued by Johnston et al. (2004). The argument for this source being a PWN centered on the flat spectrum of the radio emission, as well as the high degree of polarization observed from the center (Gaensler et al. 2000), while the radial polariza- tion angles observed at the edge of this is more consis- tent with a SNR (Johnston et al. 2004). If there is a SNR component in G328.4+0.2, we expect that some of the radio emission from this source should have a steep (α < −0.3) spectrum. To search for such emis- sion, we observed G328.4+0.2 for 12 hours at 1.4 GHz with the Australia Telescope Compact Array (ATCA) on 2005 June 25. Flux density calibration was carried out using an observation of PKS B1934-638, and phase calibration was carried out with regular observations of PMN J1603-4904. The observation was carried out using two 128 MHz bands, one centered at 1.344 GHz and the other at 1.432 GHz, and the data reduction was done using the miriad software package. The observation was conducted when the ATCA was in the 6B configu- ration, which has a longest baseline of ∼6000 m (∼ 6.′′9) and a shortest baseline of ∼200 m (∼ 3.′4). As a re- sult, this dataset alone is not sensitive to large-scale emission from G328.4+0.2. To improve the sensitivity to diffuse emission, we combined this dataset with the 1.4 GHz data used by Gaensler et al. (2000) as well as continuum data gathered in the Southern Galactic Plane Survey (McClure-Griffiths et al. 2005). Total intensity images from this combined dataset were formed using natural weighting, multi-frequency synthesis, and maxi- mum entropy deconvolution. The final image, shown in Fig. 3, has a resolution of 7.′′0×5.′′8, and an rms noise of ∼ 0.15 mJy beam−1. The measured 1.4 GHz flux density of G328.4+0.2 is 13.8±0.4 Jy – consistent with the value measured by Gaensler et al. (2000). For a 4.5 GHz flux of 12.5± 0.2 Jy (Gaensler et al. 2000), this implies that G328.4+0.2 has a radio spectral index α = −0.03± 0.03. As seen in Fig. 3, the radio emission from G328.4+0.2 is very complicated, and contains multiple morphological features. The major features are: • Central Bar: The Central Bar is the bright- est morphological feature in G328.4+0.2, and has been previously detected at both 4.8 GHz (Gaensler et al. 2000) and 19 GHz (Johnston et al. 2004). The Central Bar runs roughly E-W, and the western edge of the bar is bifurcated, first noticed by Johnston et al. (2004). The length of the bar is ∼ 1.′75 long, and inside the bar there are three peaks in the radio emission. • Filamentary Structure A: These are the curved filaments near the center of G328.4+0.2 which ap- pear to be connected to the Central Bar, the bright- est of which is the “Y” shaped structure NE of the eastern edge of the Central Bar. In general, these filaments are more prominent on the eastern side of G328.4+0.2 and appear confined to the central region of G328.4+0.2, not extending much beyond the inner half. • Filamentary Structure B: These are the faint, radial filaments predominantly found on the west- ern side of G328.4+0.2, as shown in Fig. 4. The inner parts of these filaments are located ∼ 2.′5 from the center of G328.4+0.2, and their length varies across G328.4+0.2 – in the southern half, two filaments appear to extend to the edge of the source while in the northern and western parts of G328.4+0.2 they are substantially shorter. While some of these filaments are kinked or curved, most are fairly straight and radial in orientation. These features were not detected by Gaensler et al. (2000) and Johnston et al. (2004) due to the insufficient u–v coverage of those observations. • Filamentary Structure C: As shown in Fig. 3, these are features located near the outer edge of G328.4+0.2 that are parallel to the outer edge. These features are more prominent and more plen- tiful in the eastern half of G328.4+0.2. • Outer Protrusions: The Outer Protrusions are faint features – the two most prominent of which are in the NE quadrant of G328.4+0.2 – that ex- tend beyond the outer boundary of G328.4+0.2. Several of these structures have bow-shock mor- phologies. The physical interpretation of these features will be pre- sented in §3.2. It is worthwhile to note here that several of these morphological features (e.g. the Central Bar and the internal filamentary structures) have been observed in other PWNe such as MSH 15-56 (Dickel et al. 2000) and 3C58 (Slane et al. 2004), while others (e.g. the Fil- amentary Structure C and Protrusions) are more char- acteristic of SNRs, such as the Vela SNR (Bock et al. 1998). This XMM observation also allows, for the first time, a comparison between the radio and X-ray morphology of G328.4+0.2. As shown in Fig. 3, there is a signifi- cant offset between the X-ray and the center of the radio emission, with Clump 1 located ∼ 80′′ from the center of the radio emission. Additionally, the extent of the X-ray emission is significantly smaller than that of the radio emission. A physical interpretation of the X-ray morphology and its relation to the radio emission will be discussed in §3.2. 2.2.1. Spectral Index Map The previous 1.4 GHz dataset had a resolution (∼ 20′′) significantly worse than that of the 4.5 GHz data, and therefore is not suitable for determining if there are small scale changes in α inside G328.4+0.2. With these new, high-resolution 1.4 GHz observations, it was pos- sible to make a spectral index map of G328.4+0.2 us- ing our new 1.4 GHz data and the 4.5 GHz data pre- sented by Gaensler et al. (2000) since these datasets have comparable u–v coverage. To detect any variation in α, we made a spectral tomography map of G328.4+0.2 (Katz-Stone & Rudnick 1997). To do this, we first produced a 1.4 GHz image of G328.4+0.2 from data matched in u–v coverage with the 4.5 GHz data, and then smoothed both the new 1.4 GHz image and the 4.5 GHz image to a resolution of 8′′ to account for the poorer reso- lution of the 1.4 GHz data. Finally, we produced a series of difference images (Idiff,α) using the following formula: Idiff,α= I1.4 − I4.5 where I1.4 and I4.5 are the 1.4 and 4.5 GHz images produced above. In this method, the spectral index of a region is determined by the spectral index at which it disappears from the difference image. As shown in Fig. 5, most of the radio emission from G328.4+0.2 has a spectral index between α ∼ −0.1 and α ∼ +0.1, while the outer edges of G328.4+0.2 have a steeper spectrum (α ∼ −0.4) than the center, particularly the western edge of G328.4+0.2. This steeper spectrum material is coin- cident with some of the Filamentary Structure C dis- cussed in §2.2, but there are no spectral features associ- ated with any of the other radio morphological features in G328.4+0.2 or with the X-ray emission. A physical interpretation of these results will be discussed in §3.2. 2.3. Search for the radio pulsar at Parkes As part of a project to search for pulsar counterparts to all Galactic PWNe (e.g. Camilo et al. 2002a), on 2005 October 13 we observed G328.4+0.2 using the ATNF Parkes telescope in NSW, Australia. As for similar such work, we employed the central beam of the Parkes multibeam receiver at a central frequency of 1374MHz, with 96 frequency channels across a total bandwidth of 288MHz in each of two polarizations. The integration time was 24ks, during which total-power samples were recorded every 0.25ms for off-line analysis. We analyzed the data with standard pulsar searching techniques using PRESTO (Ransom et al. 2002). We searched the dispersion measure range 0–2600cm−3 pc (twice the maximum Galactic DM predicted for this line of sight by the Cordes & Lazio 2002 electron density model), while maintaining close to optimal time reso- lution. In our search we were sensitive to pulsars whose spin period could have changed moderately during the observation due to very large intrinsic spin-down. The search followed very closely that described in more detail in Camilo et al. (2006). We did not identify any promis- ing pulsar candidate in this search. Applying the standard modification to the radiometer equation, for an assumed pulsation duty cycle of 10%, and accounting for a sky temperature at this location of 15K, we were nominally sensitive to long-period pul- sars having a period-averaged flux density at 1.4GHz of S1400 > 0.05mJy. In fact, this limit applies only to such long-period pulsars as to not be of practical interest for us: for a distance of ∼ 17 kpc along this line of sight, the expected DM is ≈ 1200 cm−3 pc, and the scatter-broadening of the radio pulses due to multi- path propagation is expected to be ∼ 50ms at 1.4GHz (Cordes & Lazio 2002). This would likely render pulsa- tions undetectable from any short-period pulsar, such as we expect to power G328.4+0.2, regardless of average radio flux. We therefore repeated the search at a higher radio frequency ν, since the scattering timescale is ap- proximately ∝ ν−4. On 2007 January 4 we observed G328.4+0.2 at Parkes at a central frequency of 3078MHz, with 288 channels spanning a bandwidth of 864MHz in each of two polar- izations. The total-power samples were recorded every 1ms for 30 ks, and analyzed in a manner analogous to that described previously for the 1.4GHz data. This time a few somewhat-promising candidates were identified in the analysis, and a second 3GHz observation was made, on 2007 March 19, for 36 ks. Analysis of this second ob- servation did not confirm the original candidates, and we have therefore not detected any radio pulsar counterpart for the PWN in G328.4+0.2. The sensitivity of our 3GHz observations was about 0.03mJy for long-period pulsars (P & 20ms) and decreasing gradually for shorter peri- ods. For the predicted DM, at this frequency the scat- tering timescale is expected to be ∼ 2ms, comparable to the dispersion smearing across each individual channel, ∼ 1ms. Propagation effects should therefore not have prevented the detection of signals with P & 5ms. Converting the 3GHz flux density limit to a frequency of 1.4GHz, using a typical pulsar spectral index of –1.6 (Lorimer et al. 1995), results in S1400 . 0.1mJy. For a distance of ∼ 17 kpc, this corresponds to a pseudo- luminosity limit of L1400 ≡ S1400d 2 . 30mJykpc2. This is comparable to L1400 of the very young pulsars B1509– 58, J1119–6127, and Crab (Camilo et al. 2002b), but a factor of about 60 greater than for the young pulsar in 3C58 (Camilo et al. 2002c), which has the smallest known radio luminosity among young pulsars. Based on these results, it is therefore entirely possible that G328.4+0.2 harbors an as-yet undetected young pulsar beaming toward the Earth with an ordinary radio lumi- nosity. 3. interpretation of x-ray and radio observations of g328.4+0.2 The X-ray spectrum of G328.4+0.2 is characteristic of PWNe (see Gotthelf 2003 for a compilation of the X- ray properties of PWNe), and therefore conclude that the X-ray emission from G328.4+0.2 comes from a PWN and not from a SNR. The same is true for the polarized, flat-spectrum radio emission detected from the center of G328.4+0.2 which are also characteristic of PWNe. Therefore, in the following discussion we assume that the X-ray and flat spectrum radio emission are both pro- duced by a PWN. In §3.1, we discuss the evolutionary sequence of PWNe in SNRs and the observational signa- tures of each stage. In §3.2, we use the results from the observations presented in §2.1 and §2.2 to draw general conclusion about the properties of G328.4+0.2. 3.1. Evolution of PWN in SNRs Both PWNe and SNRs are dynamic objects, and when the PWN is inside the SNR its evolution is affected by the behavior of both the central neutron star and the sur- rounding SNR. While the PWN is inside the SNR, it typ- ically goes through three evolutionary phases (Chevalier 1998; van der Swaluw et al. 2004): • The Free-Expansion Phase – In this phase, the PWN freely expands into the cold material inside the SNR, sweeping up and shocking the surround- ing ejecta into a thin shell (Chevalier & Fransson 1992; van der Swaluw et al. 2001). Since the PWN is confined only by the shock wave its expansion drives into the surrounding SNR, and the velocity of this shock wave is much larger than the neutron star velocity, it is free to move inside the SNR with the neutron star. • Collision with the Reverse Shock – As the SN sweeps up and shocks the surrounding a am- bient material, a reverse shock (RS) is driven into the ejecta. Eventually, the PWN will en- counter the RS, and as a result, can not con- tinue to freely expand inside the SNR because it is no longer in an essentially pressureless environ- ment. Initially, the pressure behind the RS is higher than the pressure inside the PWN, and as result the PWN is compressed. As it contracts, the pressure inside the PWN increases adiabati- cally and eventually will be higher than its sur- rounding, and will as a result re-expand inside the SNR (Blondin et al. 2001; Bucciantini et al. 2003; Reynolds & Chevalier 1984; van der Swaluw et al. 2001). Once the PWN encounters the RS, the ex- pansion velocity of the PWN decreases signficantly and falls below that of the neutron star, which is unaffected by this collision. As a result, the neu- tron star can detach itself from its PWN. • Relic PWN Phase – When the neutron star detaches from the relic nebula, it forms a new PWN from the relativistic e+/e− plasma it contin- ues to inject into the SNR (van der Swaluw et al. 2004). The PWN around the neutron star and relic nebula evolve differently. The relic nebula continues to contract/expand inside the SNR un- til it achieves pressure equilibrium with its sur- roundings, a process that can take many tens of thousands of years. The new PWN initially ex- pands sub-sonically, but when the neutron star is ∼ 2/3 of the way to the SNR shell, its velocity will become supersonic relative to the surrounding material and the PWN will take on a bow-shock morphology (van der Swaluw et al. 2004). Eventu- ally, the neutron star will leave the SNR, and as it passes through the SNR shell it may re-energize the surrounding SNR material, as possibly observed in SNRs G5.4-1.2 and CTB80 (Shull et al. 1989). As the PWN evolves inside the SNR, its appearance changes radically. During the Free-Expansion phase, the morphology of the PWN is determined by the properties of the particle wind expelled by the neutron star. In these cases, the neutron star is in the center of the PWN, and there is no significant offset between the radio and X-ray emission from the PWN. In general, during this stage the PWN is located near the center of the observed SNR shell. Examples of PWNe in this evolutionary stage are the PWNe in SNR 0540-693 in the LMC (Reynolds 1985), as well as those in the Milky Way SNRs G11.2- 0.3 (Roberts et al. 2003; Tam et al. 2002) and G21.5-0.9 (Matheson & Safi-Harb 2005). Due to the offset in the PWN’s position with respect to the SNR’s center as a result of the neutron star’s veloc- ity or inhomogeneities in the ISM, it is expected that one side of the PWN will encounter the RS before the other side (Blondin et al. 2001; van der Swaluw et al. 2004). As a result, the PWN will no longer be symmetrically oriented around the neutron star (van der Swaluw et al. 2004), leading to an offset between the radio and X-ray emission from the PWN. Because the cooling time for X- ray producing electrons is very short compared to that of radio emitting electrons, the X-ray emission of a PWN is expected to be brightest at the current location of the neutron star while the radio emission reflects the effect of the RS on the PWN. The compression/re-expansion cycle triggered by the PWN/RS collision will also affect the appearance of the PWN. According to a spherically symmetric MHD simulation of a PWN in this phase, compression of the PWN by the RS leads to an over- pressurized region forming in the center of the PWN. Material injected by the neutron star after this point is then confined to the small region, leading to the forma- tion of a radio/infrared “hot spot” in the center of the PWN (Bucciantini et al. 2003). The PWN/RS interaction also leads to the forma- tion of hydrodynamic, primarily Rayleigh-Taylor (R-T), instabilities at the PWN/SNR interface (Blondin et al. 2001). During the PWN’s initial free-expansion phase, the shell of material swept up by the PWN is subject to both thin shell and R-T instabilities (Bucciantini et al. 2004; Jun 1998), but the growth rate of these features is expected to be sufficiently small that the PWN is not disrupted, especially if even a small percentage of the total energy of the pulsar’s wind is in magnetic fields (Bucciantini et al. 2004). However, during the PWN/RS interaction, rapid mixing of pulsar wind and SNR ma- terial is expected when the PWN re-expands into the SNR (Blondin et al. 2001). In fact, numerical simula- tions suggest that the PWN is disrupted after its first re-expansion into the SNR as a result of these instabil- ities (Blondin et al. 2001). It is important to note that these instabilities are only expected to affect the relic nebula and not the new PWN formed by the neutron star further from the SNR’s center. Once the composite SNR enters the Relic PWN phase, the X-ray emission is expected to be dominated by the new PWN since it con- tains the high energy particles recently injected by the neutron star, while the radio emission is dominated by the relic nebula which contains most of the older parti- cles that are expected to contribute at low frequencies. As a result, during this phase the radio-emitting elec- trons are expected to be dominated by electrons injected during the free-expansion phase, while the X-ray is from electrons injected after the passage of the RS. 3.2. Observational Results In this Section, we use the results from the observa- tions presented in §2.1 and §2.2, as well as the basic evo- lutionary sequence for PWN in SNRs described above in §3.1 to make some initial statements on the nature, evolutionary state, and properties of G328.4+0.2. The discussion given below is a very general interpretation of observed radio and X-ray features in G328.4+0.2, it pro- vides a framework in which to test the various scenarios for G328.4+0.2 discussed in §4.1.1 and §4.1.2. As mentioned in §1, there is a debate in the literature as to whether G328.4+0.2 is a PWN (Gaensler et al. 2000; Hughes et al. 2000) or a composite SNR (Johnston et al. 2004). In neither of the X-ray or radio observations presented above is there clear evidence (e.g. thermal X-ray emission or a bright, steep spec- trum, radio shell) for a SNR component. If G328.4+0.2 is a composite SNR, then the outer boundary of the ra- dio emission likely marks the outer radius of the SNR component, while the observed flat-spectrum radio emis- sion and power law X-ray emission are emitted by the PWN. Using the extent of the flat-spectrum radio emis- sion shown in Fig. 5 to estimate the size of the PWN component, we obtain that the radius of the PWN in this source must be & 2/3RG328, the radius of G328.4+0.2. If G328.4+0.2 is a composite SNR, then Filamentary Structure 3, which as mentioned in §2.2.1 might have a steeper spectral index than the rest of the radio emis- sion in G328.4+0.2, would be emission from the SNR. This emission is then possibly analogous to the corru- gated structures seen in the NE part of the Tycho’s SNR (Velazquez et al. 1998). Additionally, in this case the Outer Protrusions mentioned in §2.2 maybe are ejecta “bullets”, similar to those observed in the Vela SNR (Aschenbach et al. 1995) and SNR N63A (Warren et al. 2003). If G328.4+0.2 is a PWN, then the outer boundary of the radio emission is the outer radius of the PWN and the SNR in which it resides is undetected, similar to the case for the Crab Nebula. As a result, the steep spectrum radio emission seen at the edge of G328.4+0.2, as well as Filamentary Structure C and the Outer Protrusions observed in the radio, correspond to material swept-up by the PWN. This is because radio emission from the pulsar wind is observed to have a flat spectrum, un- like this material. As a result, Filamentary Structure C, as well as the radial component seen by Johnston et al. (2004) in the polarization angle along the outer edge of G328.4+0.2, are the result of the hydrodynamical (HD) instabilities at the PWN/SNR interface. Since these instabilities only occur when the PWN is accelerating the shell of swept-up material surrounding it, he current pressure inside the PWN (Ppwn) must be higher than that of the SNR material just of the PWN [Psnr(Rpwn)]. Additionally, in this case the Outer Protrusions would be the result of the PWN currently expanding into a clumpy medium (e.g. the PWN analog of the process described for young SNR by Jun et al. 1996), which requires that the expansion speed of the PWN, vpwn, is currently posi- tive. In this case, it is possible that the SNR surrounding G328.4+0.2 will be detected at a later date, as was the case for G21.5-0.9 (Matheson & Safi-Harb 2005). Regardless of whether G328.4+0.2 is a composite SNR or a PWN, the offset between the radio and X-ray emission from the PWN component implies that the PWN/RS collision has already occurred. The Central Bar is then the remains of the over-pressurized region created when the PWN was compressed by the RS, with the bar-like shape of this region the result of either an asymmetric RS or the anisotropic wind emitted by the neutron star. The Central Bar should then consist of pulsar wind material, which accounts for the α ∼ 0 spec- tral index of this region as shown in Fig. 5. Additionally, the flat spectral index observed from Filamentary Struc- tures A implies that this feature is emitted by pulsar wind material. As mentioned in §3.1, when the PWN re-expands after the initial compression by the RS, nu- merical simulations suggest that the rapid mixing of the SNR ejecta and pulsar wind material can then occur. In their 2D simulations of this process, Blondin et al. (2001) observe features similar to that of Filamentary Structure A, and as a result we conclude that the existence of these features requires that the PWN has re-expanded at least once after its initial compression by the RS. Since the ra- dius of flat-spectrum radio emission from G328 is larger than the outer radius of Filamentary Structure A, the instabilities formed during this expansion must not have completely disrupted the PWN. This differs from the re- sults of Blondin et al. (2001), in which the PWN is dis- rupted during the first re-expansion. This is most likely due to the damping effect of the PWN’s magnetic field on Raleigh-Taylor instabilities, which is not accounted for by Blondin et al. (2001). Finally, we discuss the X-ray emission seen from G328.4+0.2 which, based on its X-ray spectrum, is emit- ted by pulsar wind material. The observed offset between Clump 1, Clump 2, and the Diffuse regions of the X-ray emission implies that the PWN is not freely expanding, consistent with the explanation that the PWN has col- lided with the RS. We identify Clump 1 as the current location of the neutron star since it is the brightest X-ray feature, Clump 2 as the location of the termination shock in the PWN (Kennel & Coroniti 1984), and the Diffuse emission is produced by recently injected plasma stream- ing away from the neutron star. In this case, the extent of the Diffuse component depends on the synchrotron lifetime of the X-ray emitting particles in the PWN. The X-ray emission also provides an estimate of the physical properties of the central neutron star, namely a measure of the neutron star’s rotational spin-down en- ergy, Ė. A comparison of observed X-ray luminosity Lx and Ė shows a trend that neutron stars with a higher Lx have a higher Ė, and that the relationship between these two quantities is (Possenti et al. 2002): logLX,(2−10)=1.34 log Ė − 15.34 (3) where LX,(2−10) is the X-ray luminosity of the source between 2 and 10 keV, albeit with a significant scatter (Possenti et al. 2002). For the absorbed power-law fit to the X-ray emission from G328.4+0.2 given in Table 2, the unabsorbed 2–10 keV flux of G328.4+0.2 is ∼ 1 × 10−12 ergs cm−2 s−1. For a distance to G328.4+0.2 of d = 17d17 kpc, we obtain that: LX,(2−10)∼ 3.5d 17 × 10 34 ergs s−1, (4) which, using Eq. (3), gives us an estimate of Ė: Ė ∼ 1.7d1.4917 × 10 37 ergs s−1. (5) This number is somewhat less than estimate obtained byGaensler et al. (2000) (Ė = 8.3× 1038 ergs s−1), who assumed that Ė = 4 × 10−4LR, where LR is the radio luminosity of G328.4+0.2. Our estimate of Ė is similar to the estimate by Hughes et al. (2000) (Ė ∼ 1037 − 2× 1038 ergs s−1), who also used the Lx − Ė relation. 4. simple hydrodynamic model for the evolution of a pwn inside a snr In order to determine what neutron star, SN, and am- bient density properties are required to produce a system with these properties described in §3.2, we have devel- oped a simple hydrodynamic (HD) model for the evo- lution of a PWN inside of a SNR, which we then ap- ply to G328.4+0.2 in §4.1. This model is based largely on the models developed by Blondin et al. (2001) and van der Swaluw et al. (2001). The main goal of this model is to determine the radius of the PWN, Rpwn, as it progresses through the evolutionary sequence de- scribed in §3.1. In this model, we assume that the PWN can be treated as a perfect gas with adiabatic index γ = 4/3 and that is expanding into a SNR filled with a perfect gas with adiabatic index γ = 5/3. We also as- sume that the material swept-up by the PWN initially lies in a thin shell with inner radius R = 23/24 Rpwn (van der Swaluw et al. 2001), as shown in Fig. 6. The dynamics of this mass shell are determined by the dif- ference in pressure between the PWN and SNR, and by calculating the radius of this mass shell we determine Rpwn(t). Once the PWN enters the Relic PWN phase of its evolution, this model only determines the proper- ties of the relic nebula. What follows is a brief qualitative description of the model used, while the full suite of equa- tions used to implement it quantitatively can be found in Appendix A. As mentioned above, we model Rpwn(t) by calculating the outer radius of the mass shell swept up by the PWN, ignoring the effect of any instabilities which could dis- rupt this shell. We solve for Rpwn(t) by assuming that we know the values for the relevant quantities at a time t − ∆t, and then calculate them for a time t, since the relevant equations can not be solved at all times analyt- ically. we wrote a program in IDL to implement this numerically using the following procedure: 1. Calculate Rpwn(t+∆t) by assuming that the mass shell around the PWN between t and t+∆t moves with a constant velocity vpwn(t). 2. Calculate the internal energy of the PWN, Epwn(t+ ∆t), using the first law of thermodynamics: ∆Epwn= Ėt− Ppwn∆Vpwn (6) which takes into account energy losses from the adi- abatic expansion/contraction of the PWN as well as any energy input from neutron star into the PWN between t and t + ∆t if the neutron star is still inside the PWN. Since we assume the PWN is filled with a γ = 4/3 perfect gas, Epwn ∝ PpwnVpwn from the ideal gas law, where Ppwn is the internal pressure of the PWN and Vpwn is the volume of the PWN, as defined in Equations (A5 and (A6). Since Vpwn ∝ R 3, and γ = 4/3 requires that Ppwn ∝ V pwn , we derive that Ppwn ∝ R −4. As a result, if there is no input from the neutron star, then Epwn ∝ R pwn. The energy input from the neutron star (∆Epsr) is calculated by integrating Eq. (A7) between t and t+∆t. 3. Calculate Ppwn(t + ∆t) using Equations (A5) and (A6). 4. Calculate the pressure inside the SNR (Psnr), the density inside the SNR (ρej), the velocity of the material inside the SNR (vej), and the sound speed of the material inside the SNR (cs), at the outer radius of the PWN, R = Rpwn(t + ∆t) using a model for the evolution and structure of a SNR, as described in Appendix A. 5. If the PWN is expanding faster than the SNR ma- terial around it, increase the mass of the shell sur- rounding the PWN, Msw,pwn(t+∆t), accordingly, as described in Appendix A. 6. Calculate the force on the mass shell surrounding the PWN, Fpwn(t + ∆t), using Eqs. (A14) and (A15). During the initial free-expansion of the PWN inside the SNR, these equations reduce to Equation A4 of van der Swaluw et al. (2001) and Equation 14 of Chevalier (2005). 7. Calculate the new velocity of the mass shell around the PWN, vpwn(t + ∆t), assuming that any mass swept up by the PWN between t and t+∆t is done so inelastically: vpwn(t+∆t)= Msw(t)vpwn(t) + Fpwn(t+∆t)∆t Msw(t+∆t) This is believed to be a reasonable approximation because the newly swept-up material is shocked by the mass shell and, as a result, its pre-existing mo- mentum is transferred to the internal energy of the mass shell. This model is assumes that both the SNR and PWN are spherically symmetric, the PWN remains centered on the center of the SNR at all times, the PWN has no effect on the evolution of the SNR, and that the ma- terial swept-up by the PWN is incompressible and has a negligible internal pressure. Additionally, this model ignores the effects of magnetic field (Bucciantini et al. 2003) and RT instabilities at the PWN/SNR interface (Blondin et al. 2001; van der Swaluw et al. 2004) on the properties of the PWN. Despite these simplifications, our model does a reasonably good job of reproducing the re- sults for Model A in Blondin et al. (2001), as shown in Fig. 7. In general, relative to results of Blondin et al. (2001) and other authors, our model tends to result in larger oscillations in Rpwn/Rsnr and a larger initial com- pression. The first discrepancy results from neglecting the effect of instabilities at the PWN/SNR interface that damp these oscillations, and the second from not includ- ing the effect of reflected shocks that enter the PWN at the time of the PWN/RS collision (Blondin et al. 2001). Additionally, we find that scenarios with the same to- tal amount of energy deposited by the neutron star into the PWN, Epsr but with different neutron star properties (e.g. different values of P0 and Bns) produce the same behavior of Rpwn(t). This is different than the conclu- sion of Blondin et al. (2001), and believe that this dis- crepancy is the result of using a more realistic expression for Ė, Eq. (A7), than a step function, the form used by Blondin et al. (2001). 4.1. Application of Model to G328.4+0.2 In the following discussion, we use the model given in §4 for a PWN’s evolution inside a SNR to examine the different possibilities for the nature of G328.4+0.2 given in §3.2. We first analyze the possibility that G328.4+0.2 is a composite SNR (§4.1.1), and then he possibility that G328.4+0.2 is a PWN (§4.1.2). This model requires six inputs: the characteristic timescale of the neutron star’s spin-down, τ0, initial spin-down power of the neutron star Ė0, the velocity of the neutron star, vns, the kinetic energy of the SN ejecta Esn, the mass of the SN ejecta Mej, and the number density of the surrounding material n. In order to calculate these values, we use the following information: • The distance to G328.4+0.2 is 17 kpc (d17 ≡ 1), which is the lower limit on the distance to this source as determined by Gaensler et al. (2000) us- ing Hi absorption. This implies that the current radius of G328.4+0.2 is RG328 ≡ 12.5 pc. • The neutron star inside G328.4+0.2 is spinning down with a braking index p = 3, the braking in- dex produced by a pure dipole surface magnetic field. Additionally, we assume that the neutron star’s moment of inertia is I = 1045 g cm2, the value derived for standard equations of state for a neutron star (Shapiro & Teukolsky 1983). Both of assumptions are standard in the literature (e.g. Blondin et al. 2001). • The offset between the Clump 1, which as described in §3.2 is believed to be the location of the neu- tron star powering the PWN, and the center of the radio emission is due to neutron star’s spatial ve- locity, vns. Using §2.1.1, we determine that this observed offset corresponds to physical distance of ∼ 6.6d17 pc. Since the observed offset is due only to the neutron star’s velocity in the plane of the sky, it is a lower limit on the true distance the neu- tron star has traveled since the SN explosion, rns. If we assume that vns = rns/tnow, where tnow is the age of G328.4+0.2, the observed offset allows us to estimate the minimum spatial velocity of the neutron star, vminns , equal to: vminns = 6.6d17 pc . (8) • Using Equation (A19), we determine that for stan- dard initial periods (P0 ∼ 5− 20 ms) and magnetic field strengths (Bns = 5× 10 11 − 1013 G), τ0 varies from ∼ 100 − 2000 years. To cover this range, we assume that τ0 of the neutron star in G328.4+0.2 can have one of three different values: τ0 = 430, 770, and 1730 years. (9) which respectively correspond to a neutron star with B = 1012 G and P0 = 5 ms, B = 3 × 10 and P0 = 20 ms, or B = 5 × 10 11 G and P0 = 5 ms. This range of τ0 is similar to those used by Blondin et al. (2001), van der Swaluw et al. (2001), and Bucciantini et al. (2003). • G328.4+0.2 is expanding into a uniform medium (s = 0 in the notation of Chevalier 1982). This assumes G328.4+0.2 is much larger than either the main sequence and late-stage wind bubble formed by its progenitor, both of which are ex- pected to have an interior ρ ∝ r−2 density struc- ture. While the typical size of these structures is smaller than RG328, this is not the the case for very massive stars (M & 15 M⊙) for which these bubbles can reach sizes of ∼ 100 pc or larger in low density (n . 1 cm−3) environments (Chevalier & Emmering 1989; Chevalier & Liang 1989). In this case, our assumption of a constant density medium would not be correct. However, the effect of G328.4+0.2 still being inside a stellar wind bubble since this does not significantly mod- ify the evolution of the SNR. Since their no a priori information on the density around G328.4+0.2, we assume it is one of the assume it has one of the following values: logn = −1.5,−1.0, 0, 0.5 cm−3. (10) which cover the range of densities in the warm ion- ized medium and the warm neutral medium. • The ejecta mass of the SN explosion that formed G328.4+0.2, Mej has one of the following values: Mej = 1, 5, 10M⊙ (11) and that the kinetic energy of the ejecta, Esn is: log(Esn/10 51ergs) = −0.5, 0, 0.5. (12) This range of Mej and Esn incorporate the range inferred from observations of “normal” SNe, but do not include hypernovae. • We assume that the Lx − Ė relationship used in §3.2 is accurate to better than two orders of mag- nitude. As a result, in §4.1.1, we assume that the current spin-down luminosity of the neutron star in G328.4+0.2 is one of the following: Ė=0.017, 0.17, 1.7, 17, and 170× 1037 ergs s−1,(13) and in §4.1.2 assume that 1.7 × 1035 < Ė < 1.7 × 1039 ergs s−1. These are the initial conditions used in both §4.1.1 and §4.1.2. The remaining input parameters into the model are the initial spin-down luminosity Ė0 and space ve- locity vns of the neutron star, and the method for deter- mining the possible values of these parameters is given in §4.1.1 and §4.1.2. Finally, for all trials discussed in §4.1.1 and §4.1.2 the model begins at a time t = 0.5 years, with ∆t = 0.5 years. 4.1.1. G328.4+0.2 as a Composite SNR In this Section, we evaluate the possibility that G328.4+0.2 is a composite SNR. To do this, we first assume that the outer edge of the radio emission from this source denotes the edge of the SNR, and therefore RG328 ≡ Rsnr. As a result, for a given value of Esn, Mej, and n, we use the model for the evolution of a SNR discussed in Appendix A to calculate the current age of G328.4+0.2, tnow. With this value of tnow and assumed values for the current spin-down luminosity of the neu- tron star, Ė, and τ0, we are able to calculate both the initial spin-down luminosity Ė0 and initial period P0 of the neutron star in G328.4+0.2 using Eqs. (A7) and (A9), respectively. Using this procedure, we ran our model using all pos- sible combinations of the input parameters (τ0, Ė, Esn, Mej, and n) given in §4.1, for a total of 540 different com- binations. To see which combination of these input pa- rameters provide a plausible explanation for G328.4+0.2, we require the following: • Criterion 1: vminns < 2000 km s −1, since a neu- tron star with a higher velocity than this is ex- tremely implausible based on pulsar observations (Hobbs et al. 2005). • Criterion 2: P0 > 2 ms, the minimum rota- tion period of a young proto-neutron star before it breaks up (Goussard et al. 1998). • Criterion 3: The PWN is smaller than the SNR for all t < tnow. While it is possible that PWN could expand to fill the entire SNR, it is not con- sidered likely, and is contrary to the model as- sumption that the PWN does not affect the evolu- tion of the SNR. Additionally, we also require that Rpwn(tnow) ≥ 0.67Rsnr(tnow), due to the large ob- served size of the flat-spectral index radio emission which is produced from the PWN, as described in §3.2. • Criterion 4: G328.4+0.2 is currently in the Free- Expansion or Sedov-Taylor phase of its expansion, i.e. tnow < trad, where trad, the age when a SNR goes radiative, is defined in Eq. A4. Once a SNR has entered its Radiative phase, it is expected that the radio emission from the SNR be confined to thin, bright, filaments like those observed in SNR G6.4-0.1 (Mavromatakis et al. 2004) – which are not observed in G328.4+0.2, or that the SNR is radio-quiet. • Criterion 5: The PWN/RS collision has already occurred, as described in §3.2, and the PWN has been compressed as a result of its collision with the RS. This is required to explain the Central Bar, as described in §3.2. This requires that vpwn < 0 at some point in the past – which can only occur at a time t > tcol, the time when the PWN and RS collide. • Criterion 6: The Central Bar created by the compression is still observable. This is satisfied if either the PWN is currently being compressed, vpwn(tnow) < 0, or if the compression ended suffi- ciently recently such that it can be observed. The Central Bar is believed to be formed by both a pressure and a magnetic field enhancement at the center of the PWN (Bucciantini et al. 2003). As a result its observable lifetime is the synchrotron life- time of electrons accelerated by the magnetic field enhancement. Therefore, we assume that the life- time of the central bar is the synchrotron age of the accelerated electrons, τsynch, equal to: τsynch = 3× 10 4ν−1/2B−3/2pwn years, (14) where ν is the observed frequency, in units of Hz, and Bpwn is the strength of the magnetic field in- side of the PWN, in units of G. With this infor- mation, in the case that vpwn(tnow) > 0 we deter- mine if the central bar is still observable by eval- uating τsynch at 22 GHz (since this is the highest frequency at which the Central Bar is observed; Johnston et al. 2004) at the time when the com- pression ends, assuming that Bpwn can be derived using the minimum energy estimate. This criterion is satisfied if τsynch > tnow − tre−exp, where tre−exp is the time when the compression phase ends. That the above criteria fall into two categories: criteria required for physical plausibility (Criteria 1–3) and those which depend on our interpretation of the radio and X- ray properties of G328.4+0.2 (Criteria 4–6). Of the 540 possible combinations of the input parame- ters, only one passes all six criteria. The predicted SNR, PWN, and neutron star properties of this scenario are given in Table 3, and the behavior of Rpwn as a function of time is given in Figure 8. In this scenario, G328.4+0.2 is quite young, ∼ 4900 years old, and the energy in- jected into the SNR by the neutron star is similar to the kinetic energy of the SN explosion (∼ 1051 ergs). However, in this scenario, as shown in Fig. 8, the pre- dicted compression is very small; when the compression begins, Rpwn = 3.838 pc, and when the PWN begins to re-expand into the SNR, Rpwn = 3.834 pc. This small decrease is not surprising given that, as shown in Table 3, the total energy inputed into the PWN by the pulsar (Epsr) is very close to the kinetic energy of the SN ejecta (Esn). This negligible decrease in the volume of the PWN is unlikely to form a central bar as prominent as the one observed (Fig. 3), and therefore we feel is unlikely to be the correct explanation for G328.4+0.2. 4.1.2. G328.4+0.2 as a PWN In order to evaluate if G328.4+0.2 is a PWN, we use the model presented in §4 to determine the earliest time6 (tnow) at which a PWN powered by a neutron star with a given initial period P0 reaches the observed size of G328.4+0.2 (Rpwn = 12.5 pc) if it is expanding into as yet unseen SNR formed by ejecta with initial mass Mej and kinetic energy Esn which exploded in a constant- density ambient medium with number density n. To consider all reasonable cases, we ran our model using all combinations of the values of τ0, Esn, Mej, and n given in §4.1, as well as P0 = 5, 10, 25, 100 ms for a total of 432 different trials. In this scenario, since it is not pos- sible to determine an independent estimate of the age of the system, it is necessary to assume a value of P0. To determine which of these combinations are possible explanations for G328.4+0.2, we required that: 6 Due to oscillations in radius the PWN undergoes after its col- lision with the RS, it can reach the current size at multiple times. • Criterion 1: vminns < 2000 km s −1. Since, in this scenario, we have no prior estimate of the age of G328.4+0.2, when we run our model we as- sume that vns = 0. This does not effect our re- sults because rns, as measured in §4.1, is less than Rpwn ≡ RG328, and therefore the neutron star is always injecting energy into the PWN as it does if vns = 0. Once, for a given set of input parameters we have determined tnow, we calculate v ns using Equation (8). • Criterion 2: The current spin-down energy of the neutron star in G328.4+0.2 is between 0.017 ≤ ˙E,37 ≤ 170, where ˙E,37 ≡ Ė/10 37 ergs. This is based on the work done in §3.2, and is consistent with the initial values of Ė used in §4.1.1. Since in this scenario we have no estimate of the age of G328.4+0.2, we are unable to assume a value for Ė of the central neutron star and then calculate its initial spin-down luminosity, as we did in §4.1.1. • Criterion 3: Rpwn < Rsnr for all times t < tnow, as explained in §4.1.1. • Criterion 4: The PWN has already collided, and has been compressed by, the RS, as explained in §3.2. • Criterion 5: The Central Bar created by the compression of the PWN is still observable. The method of determining if this is satisfied is the same as the one used in §4.1.1. • Criterion 6: The PWN must have been able to form RT instabilities after the PWN/RS collision. As in §4.1.1, we implement this requirement by re- quiring that Ppwn > Psnr(Rpwn) for some t > tcol. Additionally, as explained in §3.2, in order for the PWN to create Filamentary Structure C it must currently be unstable to R-T instabilities – requir- ing that Ppwn > Psnr(Rpwn) now. • Criterion 7: As explained in §3.2, the ob- served Outer Protrusions in the radio require that the PWN currently be expanding into the SNR, vpwn(tnow) > 0. • Criterion 8: G328.4+0.2, must have only under- gone one compression/re-expansion cycle. As ex- plained in §3.2, numerical simulations of PWN in- side SNRs finds that the PWN is disrupted after the first such cycle (Blondin et al. 2001). It is important to note that, if G328.4+0.2 is a PWN, then the radio and X-ray observations provide little in- formation on the evolutionary phase of the (unseen) SNR and no information on the current ratio of the PWN and SNR radii. Out of the 432 possible combinations of the input pa- rameters, only five satisfy all ten of the above criteria, as listed in Table 4. While the neutron star appears to be inside the PWN, it is possible that this is just a projec- tion effect. To evaluate the possibility that the PWN in G328.4+0.2 has already entered the Relic PWN phase of its evolution, we calculated vmin,IIns , defined as: vmin,IIns = RG328 . (15) If vns > v min,II ns , then the G328.4+0.2 is a Relic PWN, if not, then it is still in the Collision with the RS phase of its evolution. As shown in this Table, the predicted properties of G328.4+0.2 vary substantially if G328.4+0.2 is inside a Sedov or Radiative SNR. In the Sedov case, G328.4+0.2 is quite young, and progenitor SN explosion had a normal explosion energy but a low ejecta mass (Mej ∼ 1M⊙), and it occurred in a low density environment. Ad- ditionally, the neutron star formed in this explosion was spinning rapidly, has a low surface magnetic field strength (Bns < 10 12 G), and a high space velocity (vns & 800 km s −1). In the Radiative Case, G328.4+0.2 is substantially older, and the progenitor SN explosion was a low kinetic energy (Esn ∼ 3 × 10 50 ergs) and high ejecta mass expanding. The neutron star in this case was born spinning somewhat slower and has a normal magnetic field strength for a young neutron star. With the information presented in Table 5, it is possi- ble to further refine the expected SNR and PWN prop- erties. As argued in §4.1.1, the prominence of the central bar argues that, during the compression stage, the vol- ume of the PWN decreased significantly. Though it is not possible at this time to quantify the compression needed, an examination of Table 5 shows that for only two mod- els, ST 2 and Rad 1, did the volume of the PWN decrease by more than 10% – and therefore these two models are the most probable descriptions of G328.4+0.2. In the case of Rad 1, vmin,IIns ∼ 100 km s −1 is significantly less then the average neutron star velocity, v ∼ 400 km s−1 (Faucher-Giguère & Kaspi 2006; Hobbs et al. 2005), im- plying that the PWN is in the Relic PWN phase of its evolution. Since the sound speed inside a Radiative SNR is quite low, ∼ 100 km s−1, we expect that the PWN in the Rad 1 scenario would have a bow-shock morphology. Since there is no clear evidence for this in the X-ray or radio emission from G328.4+0.2, this suggests that ST 2 is a better fit to the data. The conclusion that ST 2 is an accurate description of G328.4+0.2 is supported by circumstantial evidence as well. For this scenario, the expected radius of the termination shock around the neutron star, rts, defined as (Slane et al. 2004): r2ts = 4πcPpwn , (16) assuming a spherical wind, is rts ∼ 0.6 pc, which corre- sponds to an angle of θts ∼ 8d17 ′′ – a distance which is comparable to the offset between Clump 1 and Clump 2 derived in §2.1.1. Another interesting feature for this model is that, as shown in Fig. 4, the radius of the PWN at the time of re-expansion is similar to that of the outer parts of the central filamentary structures discussed in §2.2. While this correlation might be coincidental, this could imply that the hydrodynamic instabilities formed at the PWN/SNR interface during the re-expansion dis- rupted the shell of material swept up by the PWN – consistent with the simulation of Blondin et al. (2001). While not definitive, these two pieces of evidence argue that ST 2 is a reasonable description of G328.4+0.2. The properties of ST 2 are given in Table 6, and the evolution of Rpwn is shown in Fig. 9. It is interesting to note that in this scenario, the PWN collides with the RS at a time tcol < τ0 (tcol ≈ 850 years) so energy injection by the neutron star into the PWN after the PWN/RS collision is important to the PWN’s evolution during this stage. It is important to note that this model predicts that the neutron star powering G328.4+0.2 is the most energetic neutron star in the Milky Way, as well as one of the fastest. Given that G328.4+0.2 is largest and has the highest radio luminosity of any known PWN, it is not surprising that it was formed by such a powerful neutron star. Finally, the age and Ė predicted by this method are similar to those predicted by Gaensler et al. (2000). In order to better understand the limitations of the approach in determining the properties of the neutron star and SNR in G328.4+0.2, we have run the model presented in §4 over a finer grid of parameters and eval- uated the resulting PWN evolution using the same cri- teria as above. In Fig. 10, we show which values of P0 and Bns pass all of this criteria for three different kind of SN explosions: Esn = 10 51 ergs and Mej = 1 M⊙, Esn = 3 × 10 51 ergs and Mej = 1 M⊙, and Esn = 4 × 1051 ergs and Mej = 3.25 M⊙, assuming an ambi- ent density with n = 0.03. The first set of SN parame- ters corresponds to ST 1, the second to ST 2, and third to a higher ejecta mass SN explosion is compatible with a neutron star with the same parameters as ST 2. For the first case, we find that a wide range of P0 values are allowed but that Bns . 10 12 G. In fact, for this set of SN parameters a P0 ∼ 10 ms, Bns ∼ 8 × 10 11 G neu- tron star results in a PWN which is compressed a similar amount as in ST 2. In the second case, we find that P0 and Bns are tightly constrained around P0 ≈ 5 ms and Bns ≈ 5× 10 11 G. In the third set of SN parameters, we find that P0 . 6 ms, but that Bns spans a wide range of values, ∼ 1011 − 2× 1012 G. To determine the allowed values of Esn and Mej, we followed the same procedure as above using two different sets of neutron star parameters: P0 = 5 ms and Bns = 5× 1011 G (the neutron star parameters in the ST 1 and ST 2 scenarios), and P0 = 10 ms and Bns = 8 × 10 11 G. For the first case, only models with Esn ∼ 1−4×10 51 ergs andMej ∼ 0.5−3.5M⊙ satisfy the criteria above – though a substantial compression of the PWN requires Esn & 2 × 1051 ergs. In the second case, we find that only models with Esn . 10 51 ergs and Mej ∼ 0.5− 3.5M⊙ are allowed. While this error analysis shows that the method used above to determine the properties of the neutron star and SN explosion which formed G328.4+0.2 is unable to do so to much better than an order of magnitude, the different combinations values of P0, Bns, Esn, and Mej which are allowed predict different physical properties for G328.4+0.2 which are testable with further observations. For example, in the case of P0 = 10 ms, Bns = 8×10 11 G, Esn = 1 × 10 51 ergs, and Mej = 1 M⊙, G328.4+0.2 is ∼ 13, 000 years old, twice the age predicted in the ST 2 model as shown in Table 6, and as a result the required velocity of the neutron star is significantly lower, vminns ∼ 500 km s−1. The predicted period for the neutron star in this scenario is also significantly slower than required by the ST 2 scenario, P ∼ 24 ms, with a value of Ė approximately an order of magnitude lower than that in the ST 2 scenario. The termination shock radius for this set of parameters is ∼ 5′′, considerable smaller than the ∼ 8′′ for the ST 2 scenario and detectable with the Chandra X-ray Observatory. 5. conclusions In this paper, we first presented new X-ray (§2.1) and radio (§ §2.2, 2.3) and observations of Galactic non- thermal radio and X-ray source, G328.4+0.2, from which we infer the current properties and evolutionary history of this source (§3.2). We then presented a simple hydro- dynamic model for the evolution of a PWN inside a SNR (§4), which is used to determine which values of Esn, Mej, n, P0, and Bns are able to reproduce the properties discussed in §3.2 if G328.4+0.2 was a Composite SNR (§4.1.1) or a PWN (§4.1.2). As a result of this analysis, we determine the G328.4+0.2 is a PWN inside an unde- tected SNR. Though we are not able to precisely deter- mine the properties of the SN explosion and the neutron star which have created this system, our analysis implies that the neutron star in G328.4+0.2 was born with an ini- tial period P0 . 10 ms, has a lower than average surface dipole magnetic field strength, and has a higher than av- erage spatial velocity vns & 400 km s −1. We assume de- termining that the SN explosion which created the neu- tron star had a normal explosion energy, Esn ∼ 10 51 ergs, but a relative low ejecta mass, Mej . 4M⊙. Future X- ray and radio observations can significantly decrease this uncertainty, particularly if they are able to either detect pulsations from the neutron star or continuum X-ray or radio emission from the currently undetected SNR in this system. While we are not able to definitely determine the ini- tial period (P0) or surface magnetic field strength (Bns) of the neutron star, nor the kinetic energy (Esn) or ejecta mass (Mej) of the progenitor SN explosion, the estimates quoted above are of interest. Our non-detection of the pulsar via radio pulsations is not particularly constrain- ing, due to the very large distance of the PWN. The low magnetic field but rapid initial period predicted for the neutron star in G328.4+0.2 has implications for mod- els concerning the origin of neutron star magnetic fields. For example, according to the α − Ω dynamo model of (Thompson & Duncan 1993), neutron star born spinning rapidly (P0 . 5 ms) should have a strong dipole compo- nent to their surface magnetic fields (Bns ≫ 10 12 G). If the ST 2 scenario proves to be correct, then the low magnetic field of the neutron star in this system would be a problem for such a model. Additionally, the low ejecta mass inferred in this scenario requires that the progenitor of this system was either a single, massive star (M & 35M⊙) which exploded in a Type Ib/c SN (Woosley et al. 1995), or was initially in a binary sys- Finally, the method used in this paper to study G328.4+0.2 is complementary to other methods used (e.g. Chevalier 2005) to infer the initial period and mag- netic field strength of other neutron stars in young PWN as well as the properties of the SN explosion in which they were formed, and is easily applicable to other such systems. JDG would like to thank Niccolo Bucciantini, Shami Chatterjee, Roger Chevalier, Tracey DeLaney, David Ka- plan, Kelly Korreck, Cara Rakowski, and John Ray- mond for useful discussions, and the anonymous referee for many useful comments. We are extremely grateful to John Reynolds for prompt scheduling and observing assistance at Parkes in 2007. The Australia Telescope is funded by the Commonwealth of Australia for oper- ation as a National Facility managed by CSIRO. JDG and BMG were supported in this work by XMM grant NAG5-13202 and LTSA grant NAG5-13032. REFERENCES Aschenbach, B., Egger, R., & Trumper, J. 1995, Nature, 373, 587 Bandiera, R. 1984, A&A, 139, 368 Blondin, J. M., Chevalier, R. A., & Frierson, D. M. 2001, ApJ, 563, Blondin, J. M., Wright, E. B., Borkowski, K. J., & Reynolds, S. P. 1998, ApJ, 500, 342 Bock, D. C.-J., Turtle, A. J., & Green, A. J. 1998, AJ, 116, 1886 Buccheri, R., Bennett, K., Bignami, G. F., Bloemen, J. B. G. M., Boriakoff, V., Caraveo, P. 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Clump 1 was fitted to a 2D Lorentzian, defined as f(r) = A 1 + r where f(r) is the expected number of counts at radius r away from the center of the source, A is given in counts, and the core-radius r0 is in arc-seconds. The background component is given in counts pixel−1. Both Clump 2 and Diffuse were modeled with elliptical 2D Gaussians, where the full width, half maximum (FWHM) is given in arc-seconds, the ellipticity is e, defined as 1− b/a, where b and a are, respectively, the major and minor axis of the source, the position angle θ is given in degrees counterclockwise from north, and the amplitude A is given in counts. Table 2 Spectral Fits to Mos1 + Mos2 + pn Data Parameter Value Model phabs * pow phabs * bbodyrad phabs * bremss phabs * ray NH 11.6 × 1022 6.4 × 1022 10.4 × 1022 7.8 × 1022 Γ or kT 2.0 +>900 Absorbed Flux 4.6 × 10−13 4.4 × 10−13 4.5 × 10−13 4.9 × 10−13 Unabsorbed Flux 1.9 × 10−12 6.5 × 10−13 1.2 × 10−12 9.6 × 10−13 χ2 103.3/131 98.4/131 101.7/131 122.8/131 Reduced χ2 0.81/128 0.77/128 0.79/128 0.96/128 Note – Results from joint fits to the mos1, mos2, and pn spectrum between 0.5 and 10 keV. The model phabs * pow refers to a power-law attenuated by interstellar absorption, phabs * bbodyrad refers to a blackbody attenuated by interstellar absorption, phabs * bremss refers to a Bremsstrahlung source spectrum attenuated by interstellar absorption, and phabs * ray refers to a Raymond-Smith thermal plasma spectrum attenuated by interstellar absorption. In the table, NH is given in cm −2, kT in keV, and flux in ergs cm−2 s−1, both the absorbed and unabsorbed flux are calculated between 0.5 and 10 keV, and the errors represent the 90% confidence level. Table 3 Expected Properties of G328.4+0.2 if it is a Composite SNR Supernova Remnant Properties Pulsar Wind Nebula Properties Neutron Star Properties Parameter Value Parameter Value Parameter Value Esn 1× 10 51 ergs Epwn 4.8× 10 50 ergs Ė0 2.5× 10 40 ergs s−1 Mej 1 M⊙ Rpwn 8.7 parsecs P0 3.8 ms Phase Sedov-Taylor M sw 5.3 M⊙ τ0 1730 years Psnr(Rpwn) 1.4× 10 −9 dynes Ppwn 2.0× 10 −9 erg cm−4 Bns 1.1× 10 vej(Rpwn) 400 km s −1 vpwn 700 km s −1 vminns 1300 km s tnow 4900 years rts 0.5 parsecs Ė 1.7× 10 39 ergs s−1 n 0.32 cm−3 θts 6. ′′7 P 7.5 ms · · · · · · · · · · · · Ṗ 1.8× 10−14 s/s · · · · · · · · · · · · τc 6700 years · · · · · · · · · · · · Epsr 1.0× 10 51 ergs Note. – The values in bold are model assumptions, while the others are predicted by the model presented in §4. Psnr(Rpwn) is the pressure inside of the SNR just outside of the PWN, vej(Rpwn) is the velocity of material inside the SNR just outside of the PWN, Epwn is the internal energy of the PWN, Rpwn is the radius of the PWN, M sw is the mass of material swept up by the PWN, Ppwn is the internal pressure of the PWN, vpwn is the expansion velocity of the PWN, rts is the radius of the termination shock around the neutron star inside the PWN, calculated using Equation 16, θts is the angular size of this feature assuming d17 ≡ 1, Bns is the dipole magnetic field of the neutron star in G328.4+0.2 according to this model, P is the period of the neutron star in G328.4+0.2, Ṗ is the period-derivative of the neutron star, τc is the characteristic age of the neutron star, defined as τc = P/(2Ṗ ), and Epsr is the total amount of energy injected by the neutron star into G328.4+0.2 for t < tnow. All values are given for t = tnow unless otherwise noted. Table 4 Scenarios for G328 as a PWN inside an Undetected SNR Scenario # τ0 P0 Esn,51 Mej n tnow SNR Phase Rsnr Bns,12 v min,II ST 1 1730 5.0 1.00 1 0.03 5100 Sedov-Taylor 19.8 0.5 1300 2400 ST 2 1730 5.0 3.16 1 0.03 6500 Sedov-Taylor 28.0 0.5 1000 1900 Rad 1 430 10.0 0.32 10 0.32 84200 Radiative 28.4 2.0 100 100 Rad 2 770 10.0 0.32 10 0.32 52400 Radiative 24.8 1.5 100 200 Rad 3 770 10.0 0.32 10 1.00 101400 Radiative 22.1 1.5 100 100 Note. – Values for τ0, P0, Esn, Mej, and n that satisfy the criteria listed in §4.1.2 for G328 being a PWN inside an undetected SNR. The value of τ0 is given in years, P0 in ms, Esn,51 ≡ Esn/10 51 ergs, Mej in solar masses, n in cm −3, tnow in years, Rsnr in parsecs, Bns = Bns,12 × 10 12 G, vminns is in km s −1, and v min,II ns is also in km s Table 5 Compression/Expansion properties of G328.4+0.2 if it is a PWN Scenario # t(vpwn = 0) Central Bar Lifetime Rpwn(vpwn = 0) re−exp compres ST 1 1648.5, 2432.0 63930 8.20, 7.96 0.91 ST 2 969.0, 1837.0 35250 8.36, 6.34 0.44 Rad 1 12772.0, 26541.0 243480 8.53, 7.55 0.69 Rad 2 13336.5, 21112.5 256230 8.48, 8.27 0.93 Rad 3 9761.5, 11035.0 102640 5.72, 5.72 1.00 Note. – In this table, values Scenario # correspond to the values of τ0, P0, Esn, Mej, and n given in Table 4. t(vpwn = 0) and the Central Bar lifetime are given in years, and Rpwn(vpwn = 0) is given in parsecs. re−exp compres is the ratio of the volume of the PWN at re-expansion and compression. Table 6 Expected Properties of G328.4+0.2 if it is a PWN (ST 2 scenario in Table 4) Supernova Remnant Properties Pulsar Wind Nebula Properties Neutron Star Properties Parameter Value Parameter Value Parameter Value Esn 3.2× 10 51 ergs Epwn 3.2× 10 50 ergs Ė0 1.5× 10 40 ergs s−1 Mej 1 M⊙ Rpwn 12.5 parsecs P0 5 ms Phase Sedov-Taylor M sw 0.4 M⊙ τ0 1730 years Psnr(Rpwn) 3.6× 10 −10 dynes Ppwn 4.4× 10 −10 dynes Bns 5× 10 vej(Rpwn) 460 km s −1 vpwn 790 km s −1 vminns 990 km s Age (tnow) 6500 years rts 0.6 parsecs Ė 6.4× 10 38 ergs s−1 n 0.03 cm−3 θts 7. ′′7 P 10.9 ms · · · · · · · · · · · · Ṗ 2.1× 10−14 s/s · · · · · · · · · · · · τc 8200 years · · · · · · · · · · · · Epsr 6.3× 10 50 ergs Note. – The model assumptions are given in bold. Psnr(Rpwn) is the pressure inside just outside of the PWN, vej(Rpwn) is the velocity of material inside just outside of the PWN, Epwn is the internal energy of the PWN, Rpwn is the radius of the PWN, M sw is the mass of material swept up by the PWN, Ppwn is the internal pressure of the PWN, vpwn is the expansion velocity of the PWN, rts is the radius of the termination shock around the neutron star inside the PWN, θts is the predicated angular radius of this feature assuming d17 ≡ 1, Bns is the predicted dipole magnetic field of the neutron star in G328.4+0.2, P is the predicted period of the neutron star in G328.4+0.2, Ṗ is the predicted period-derivative of the neutron star, τc is characteristic age of the neutron star, and Epsr is the total amount of energy injected by the neutron star into G328.4+0.2 for t < tnow. All values are given for t = tnow unless otherwise noted. Point Source Background Background Region for Spectral Analysis Source Region for Spectral Analysis Clump 1 DiffuseClump 2 Clump 2 Clump 1 Diffuse Fig. 1.— Top: Exposure normalized, vignette corrected mos1 + mos2 image of G328.4+0.2 smoothed by a 5′′ Gaussian. The white contours indicate 20, 40, 50, 70, and 90% of the peak X-ray flux in the smoothed image, while the boxes indicate the background and source regions used for the spectral analysis described in §2.1.2. The background point source labeled in the image was excluded from the background region. Additionally, the labels in this plot point to the morphological features discussed in §2.1.1. Bottom: Unsmoothed normalized, vignette corrected mos1 and mos2 image of G328.4+0.2 overlaid with the regions used for the Hardness Ratio analysis discussed in §2.1.2. Fig. 2.— Mos1, Mos2, and pn spectrum of G328.4+0.2 overlaid with the absorbed power-law model whose parameters are given in Table 2. Fig. 3.— 1.4 GHz image of G328.4+0.2, overlaid with X-ray contours in green which represent 20%, 35%, ..., 90% of the peak flux in the smooth X-ray image shown in Fig. 1. The beam size of this image is 7.′′0×5.′′8, and is shown in the lower left-hand corner of the image. The labels indicate examples of the different radio morphological features discussed in §2.2. Fig. 4.— 20cm radio image of G328.4+0.2 (same data as shown in Fig. 3), with a color scale chosen to enhance the visibility of Filamentary Structure B discussed in §2.2. The yellow circle indicates the size of PWN predicted in the ST 2 model listed in Table 5 when it re-expanded after the initial compression by the SNR reverse shock, and the white cross indicates the center of G328.4+0.2 (15h55m33s,−53◦17′00′′; J2000) as determined by Gaensler et al. (2000). Fig. 5.— Spectral tomography images of G328.4+0.2, as described in §2.2.1. The spectral index α is given in the upper left hand corner of each image, where Sν ∝ ν Contact Discontinuity SNR Forward Shock SNR Reverse Shock Outer Boundary of PWN Neutron Star Shocked ISM Ambient ISM Unshocked Ejecta Material Swept Up by PWN Pulsar Shocked Ejecta Fig. 6.— Diagram of a Composite SNR in the Free Expansion stage of its evolution. In this image, the ratio between the thickness of the mass shell surrounding the PWN and the radius of the PWN is 1/24, as determined by van der Swaluw et al. (2001), and the radio of the SNR Forward Shock, Contact Discontinuity, and Reverse Shock radii are equal to the values given in Chevalier (1982) for his n = 9, s = 0 case. The colors denote the nature of the material within each region. 1000 10000 100000 Time (years) Fig. 7.— Rpwn/Rsnr for Models A (solid), B (long-dashed line), & C (short-dashed line) in Blondin et al. (2001). The top plot shows the result of the model presented in §4, while the bottom is a reproduction of Fig. 3 by Blondin et al. (2001), reproduced by permission of the AAS. Fig. 8.— The radius of the PWN, SNR, and SNR reverse shock as well as the location of the neutron star as a function of time if G328.4+0.2 is a composite SNR. The vertical line indicates the current age of the system, and the properties of this system are given in Table 3. Fig. 9.— The radius of the PWN, SNR, and SNR reverse shock as well as the location of the neutron star as a function of time for the favored (ST 2; Table 4) scenario if G328.4+0.2 is a PWN. The vertical line indicates the current age of the system. Fig. 10.— The results for varying P0 and Bns for a Esn = 10 51 ergs, Mej = 1M⊙ (top), Esn = 3× 10 51 ergs, Mej = 1M⊙ (middle), and Esn = 4×10 51 ergs, Mej = 3.25M⊙ (bottom) SN explosion, assuming n = 0.03 cm −3. The small black squares indicate models which failed the criteria described in §4.1.2, while the colored circles indicate scenarios which passed. The color represents the Compression Fraction of the PWN, defined as the ratio of the PWN’s volume at the beginning and end of the compression stage. The Compression Fraction of the ST 2 case given in Table 4 is 0.44 (light blue on this color scale), and lower values correspond to a more substantial compression. The star indicates the position of a P0 = 5 ms, Bns = 5 × 10 11 G neutron star, while the large square indicates the position of a P0 = 10 ms, Bns = 8× 10 11 G neutron star – the two neutron stars used in Fig. 11. Fig. 11.— The results of varying Esn and Mej for a P0 = 5 ms, Bns = 5 × 10 11 G (top) and P0 = 10 ms, Bns = 8 × 10 11 G (bottom) neutron star, assuming n = 0.03 cm−3. The black squares indicate which scenarios failed the criteria described in §4.1.2, while the colored circles indicate those that passed, with the color representing the Compression Fraction (defined as the ratio of the PWN’s volume at the beginning and end of the compression stage) of the PWN. The star indicates a Esn = 1× 10 51 ergs, Mej = 1 M⊙ SN explosion, the square indicates a Esn = 3× 10 51 ergs, Mej = 1 M⊙ SN explosion, and the circle indicates a Esn = 4× 10 51 ergs, Mej = 3.25 M⊙ SN explosion – the SN explosion parameters used in Fig. 10. APPENDIX equations for the hd model for the evolution of a pwn inside a snr In this Appendix, we provide many of the details concerning the properties of the neutron star, PWN, and SNR needed to simulate the HD model for the evolution of a PWN inside a SNR described in §4. For the SNR, we assume that the initial ejecta density profile consists of a constant density core surrounded by a ρ ∝ r−9 envelope – the standard assumption for a SNR produced by a Type-II SNR (Blondin et al. 2001; Chevalier 1982) – and that the ejecta is expanding ballistically (vej ≡ rej/t). The boundary between the constant density core and the outer ejecta envelope has a velocity vcore, defined as (Blondin et al. 2001): vcore= where Esn is the explosion energy of the SN and Mej is the ejecta mass. As a result, the density ρcore of the ejecta core is (Blondin et al. 2001): ρcore(t)= Esn v core t −3. (A2) As the SNR expands, it sweeps up and shocks the surrounding interstellar medium (ISM). This swept-up material has a higher pressure than the cold ejecta driving the expansion of the SNR, and as a result drives a reverse shock (RS) into the SN ejecta. In between the outer edge of the SNR, which marks the location of the forward shock (FS) and the RS is a contact discontinuity which separates the shocked ISM from the ejecta shocked by the RS. The pressure inside the SNR at r < rrs, where rrs is the radius of the RS, is assumed to be zero. A diagram of this is shown in Fig. 6. Since we assume that both the SN ejecta and the shocked ISM behave as a γ = 5/3 perfect gas, the sound speed cs of this material is: When the RS is still in the ejecta envelope, we determine the pressure, velocity, and density profiles on the material between the RS and FS using the self-similar equations given by Chevalier (1982), evaluating them for the n = 9, s = 0 case. However, when the RS enters the constant density ejecta core, it is no longer possibly to apply the self-similar solution of Chevalier (1982), and we use the work of Truelove & McKee (1999) to determine the radius of the RS and the results given by Bandiera (1984) to determine the pressure, velocity, and density profiles between the RS and FS. It is also necessary to model the radius of the FS (Rsnr), which we do using the work of Truelove & McKee (1999). This is valid while the SNR is in the Free Expansion and Sedov-Taylor phases of its evolutions. After the SNR goes radiative, which occurs at a time t = trad defined as (Blondin et al. 1998): trad≈ 2.9E 17 × 104 yr. (A4) After this point, Rsnr ∝ t 2/7. An analytic model for the pressure, velocity, and density distribution of a SNR in this phase does not currently exist, and therefore it is difficult to extend our model to this phase, though if one assumes that the interior of the SNR evolves adiabatically, Rpwn/Rsnr ∝ t 0.075 for t > trad (Blondin et al. 2001). For the PWN, as mentioned in §4, we assume that it is a bubble filled with a γ = 4/3 perfect gas. As a result, the internal pressure of the PWN, Ppwn is equal to: Ppwn= 3Vpwn , (A5) where Epwn is the internal energy of the PWN and Vpwn is the volume of the PWN, defined as: Vpwn= πR3pwn (A6) where Rpwn is the radius of the PWN. The internal energy of the PWN is determined by the rate of energy injected into the PWN by the neutron star (Ė), and energy loss due to its expansion inside the SNR ( ˙Eadpwn). For Ė, we use the standard assumption that it is equal to: Ė= Ė0 where t is the age of the neutron star, p is the pulsar braking index (p = 3 for a magnetic dipole), τ0 is the characteristic timescale of pulsar spin-down, and Ė0 is the initial spin-down energy of the neutron star. Both τ0 and Ė0 depend on the physical properties of the neutron star, with τ0 defined as (Blondin et al. 2001; Shapiro & Teukolsky 1983): 3c3 IP 20 4π2B2nsR ns sin where I is the neutron star’s moment of inertia, P0 is the initial spin period, Bns is the magnetic field of the neutron star, Rns is the radius of the neutron star, α is the angle between the neutron star’s rotation axis and magnetic field, and (Blondin et al. 2001): Ė0= I τ0(p− 1) . (A9) Since the PWN expands adiabatically, ˙Eadpwn is equal to: ˙Eadpwn=− . (A10) As a result, the change in the internal energy of the PWN over time ( ˙Epwn) is equal to: ˙Epwn=− + Ė0 (A11) assuming that p = 3. This equation can be solved analytical, and we result that Epwn(t) can be expressed as: Epwn(t)= Ė0τ0 ln(1 + t/τ0) t/τ0 − 1 . (A12) When we run our model, we use Eq. (A12) to determine the initial value of Epwn, but determine Epwn at later times using the procedure described in Step 2 in §4. During its free-expansion, the PWN is moving faster than its surroundings, and the mass of the shell surrounding the PWN (Msw,pwn) is simply: Msw,pwn(t)= ∫ Rpwn 4πR2ρej(r, t)dr (A13) where ρej(r) is the density profile of the SNR. After the collision with the reverse shock, if the PWN is moving faster than its surroundings we determine the mass of the ejecta shell recently swept up by the PWN and add it to the value of Msw,pwn calculated at the time of the reverse shock collision. If the PWN is moving slower than its surroundings, we assume that Msw,pwn remains constant, even if the PWN is being compressed by the surrounding SNR. Due to the difference in pressure between the PWN interior to the mass shell and the SNR exterior to the mass shell (Psnr(r = Rpwn)), the mass shell is subject to a force F∆P, defined as: F∆P=4πR pwn[Ppwn − Psnr(Rpwn)]. (A14) In this notation, F∆P > 0 means that the PWN interior has a higher pressure than inside the surrounding SNR. If the PWN has not yet encountered the RS, we assume that Psnr(r = Rpwn) = 0. If the mass shell is moving faster than the sound speed of the surrounding material (vpwn > cs(Rpwn)), which is the case before the PWN interacts with the RS (Chevalier & Fransson 1992), the mass shell is decelerated by ram pressure, and the total force on the mass swept-up by the PWN, Fpwn, is: Fpwn=F∆P − 4πR pwnρej(Rpwn)[vpwn − vej(Rpwn)] 2. (A15) If vpwn < cs, then Fpwn = F∆P. For t ≪ τ0, analytical solutions to these equations give Rpwn ∝ t 6/5 if the PWN is still inside the central constant-density core – a result which is reproduced by our numerical implementation of the model described in §4. In this framework, the period P of a neutron star evolves as: P =P0 , (A16) the period-derivative Ṗ evolves as: (A17) and the surface magnetic field Bns of the neutron star, assuming p = 3, is: Bns=1.5 ˙E0,37 2P 20,ms × 109 G (A18) where ˙E0,37 = Ė0/10 37 ergs s−1, P0,ms is the initial period in ms, and R14 is the radius of the neutron star R/14 km. Additionally, for p = 3 τ0 is equal to: τ0=17.3 B212R years (A19) where I = 1045I45 g cm 2 and the angle between the spin and magnetic field axes of the neutron star is α = 45◦.

0704.0220

Three Particle Correlations from STAR

November 4, 2018 14:16 WSPC/INSTRUCTION FILE IWCF˙v3 International Journal of Modern Physics E c© World Scientific Publishing Company THREE PARTICLE CORRELATIONS FROM STAR Jason Glyndwr Ulery Department of Physics, Purdue University, 525 Northwestern Avenue West Lafayette, IN 47907, USA [email protected] Received (received date) Revised (revised date) Two-particle correlations have shown modification to the away-side shape in central Au+Au collisions relative to pp, d+Au and peripheral Au+Au collisions. Different sce- narios can explain this modification including: large angle gluon radiation, jets deflected by transverse flow, path length dependent energy loss, Cerenkov gluon radiation of fast moving particles, and conical flow generated by hydrodynamic Mach-cone shock-waves. Three-particle correlations have the power to distinguish the scenarios with conical emission, conical flow and Cerenkov radiation, from other scenarios. In addition, the dependence of the observed shapes on the pT of the associated particles can be used to distinguish conical emission from a sonic boom (Mach-cone) and from QCD-Čerenkov radiation. We present results from STAR on 3-particle azimuthal correlations for a high pT trigger particle with two softer particles. Results are shown for pp, d+Au and high statistics Au+Au collisions at sNN=200 GeV. An important aspect of the analysis is the subtraction of combinatorial backgrounds. Systematic uncertainties due to this sub- traction and the flow harmonics v2 and v4 are investigated in detail. The implications of the results for the presence or absence of conical flow from Mach-cones are discussed. 1. Introduction Though relativistic heavy ion collisions a medium is created that may be the quark gluon plasma (QGP). We can study this medium though the use of jets and jet- like correlations. Jets make good probes because their properties in vacuum can be calculated with perturbative quantum chromodynamics (pQCD). Previous results on two-particle azimuthal jet-like correlations have revealed a broadened away-side shape in central Au+Au collisions relative to pp, d+Au and peripheral Au+Au collisions, or even double humped 1,2,3,4. The away-side shape is consistent with many different physics mechanisms including: large angle gluon radiation 5,6, jets deflected by radial flow or preferential selection of particles due to path-length dependent energy loss, hydrodynamic conical flow generated by Mach-cone shock waves 7,8, and Čerenkov gluon radiation 9,10. 3-particle correlations can be used to differentiate mechanisms with conical emission, Mach-cone and Ĉerenkov radiation, from the other mechanisms. Additionally, the dependence of the conical emission angle on associated particle pT can be used to differentiate between Mach-cone and http://arxiv.org/abs/0704.0220v1 November 4, 2018 14:16 WSPC/INSTRUCTION FILE IWCF˙v3 2 Jason Glyndwr Ulery For the STAR Collaboration QCD-Čerenkov radiation. 2. Analysis Procedure The 3-particle correlation analysis method has been rigorously described in refer- ence 11. Results are reported here for trigger particles of 3 < pT < 4 Gev/c and associated particles of 1 < pT < 2 GeV/c, except where otherwise noted. Results are from pp, d+Au, and Au+Au collisions at sNN = 200 GeV/c. All particles are charged particles measuremented in the STAR time projection chamber (TPC). φ-φ=φ∆ -1 0 1 2 3 4 5 -1 0 1 2 3 4 5 -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 (a) (b) (c) (d) Fig. 1. (color online) (a) Raw 2-particle correlation (points), background from mixed events with flow modulation added-in (solid) and scaled by ZYA1 (dashed), and background subtracted 2- particle correlation (insert). (b) Raw 3-particle correlation, (c) soft-soft background, βα2Binc and (d) hard-soft background + trigger flow, Ĵ2 ⊗ αBinc2 + βα2B inc,TF . See text for detail. Results are from ZDC-triggered 0-12% Au+Au collisions at sNN=200 GeV/c. Figure 1a shows the 2-particle azimuthal distribution (J2), its background (B and the background subtracted 2-particle correlations (Ĵ2). The background is con- structed from mixed events where the trigger particle and the associated particles are from different events within the same centrality window. The flow modulation is added in pairwise using the average v2 values from the measurements based on the reaction plane and 4-particle cumulant methods 1 and the v4 contribution uses the parameterization v4 = 1.15v from the data12. The background is normalized (with scale factor α) to the signal within 0.8 < |∆φ| < 1.2 (zero yield at 1 radian or ZYA1). Figure 1b shows the 3-particle azimuthal distribution (J3) in ∆φaT = φa − φTrigger and ∆φbT = φb −φT where φT , φa, and φb are the azimuthal angles of the trigger particle and the two associated particles respectively. Combinatorial back- grounds must be removed to extract the genuine jet-like 3-particle signal. Events are treated as composed of two components, particles that are jet-like correlated with the trigger particle and background particles. One source of background, the hard- soft background, results when one of the associated particles is jet-like correlated with the trigger particle and the other uncorrelated, other than the correlation due to flow. It is constructed from the 2-particle jet-like correlations, Ĵ2 folded with the normalized 2-particle background, αBinc . We shall refer to the hard-soft November 4, 2018 14:16 WSPC/INSTRUCTION FILE IWCF˙v3 Three-Particle Correlations from STAR 3 background as Ĵ2 ⊗ αBinc2 . Another source of background, the soft-soft background, results from correla- tions between the two associated particles which are independent of the trigger particle. This background is obtained from mixing the trigger particle with a dif- ferent event of the same centrality. We shall refer to this background as Binc . Since the two associated particle are from the same event all correlations between them that are independent of the trigger are preserved. This may include contribution from minijets, other jets in the event and flow. The soft-soft background is shown in figure 1c. Although the flow correlations between the two associated particles is accounted for by the soft-soft term, those between the associated particles and the trigger par- ticle are not. Those correlations are added in triplet-wise from mixed events where the trigger and the associated particles are all from different events in the same centrality window. The v2 and v4 values are obtained from the same measurements as used in the 2-particle background. The total number of triplets is determined from the soft-soft. We shall refer to this background as B inc,TF The total background is then, Ĵ2 ⊗ αBinc2 + βα2(Binc3 + B inc,TF ). Binc inc,tf are scaled by βα2. The normalization factor α is determined from 2-particle correlations and deviates from unity due to the combined effects of trigger bias and centrality definition. If the events are poisson then α2 is the correct multiplicity scaling in 3-particle correlations. The normalization factor β corrects for the effect of non-poisson multiplicity distributions and is obtained such that the number of triplets in the background subtracted jet-like three-particles correlation equals the square of the number of pairs in the background subtracted jet-like 2-particle correlation. Figure 1d shows Ĵ2 ⊗ αBinc2 + βα2B inc,TF 3. Results Figure 2 shows background subtracted 3-particle jet-like correlation signals for dif- ferent collisions and centralities. The pp and d+Au results are similar with peaks clearly visible for the near-side, (0,0), away-side, (π,π), and the two cases of one particle on the near-side and the other on the away-side, (0,π) and (π,0). The away- side peak displays on-diagonal elongation which is consistent with kT broadening. Additional on-diagonal elongation is present in the Au+Au results, possibly due to deflected jets or large angle gluon radiation. The more central Au+Au colli- sions display an off-diagonal structure, at about π± 1.45 radians, that is consistent with conical emission. This structure increases in magnitude with centrality and is prominent in the high statistics top 12% data provided by the online zero degree calorimeter (ZDC) trigger. Figure 3 shows away-side projections of on-diagonal strips to (∆φaT+∆φbT )/2− π and off-diagonal strips to (∆φaT − ∆φbT )/2. In d+Au collisions only a strong central peak is present for both on-diagonal and off-diagonal projections. The on- diagonal projection is broader than the off-diagonal projection, likely due to kT November 4, 2018 14:16 WSPC/INSTRUCTION FILE IWCF˙v3 4 Jason Glyndwr Ulery For the STAR Collaboration -1 0 1 2 3 4 5-1 -0.02 -1 0 1 2 3 4 5-1 -0.02 -1 0 1 2 3 4 5-1 -0.05 -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 STAR Preliminary Fig. 2. (color online) Background subtracted jet-like 3-particle correlations for pp (top left), d+Au (top middle), and Au+Au 50-80% (top right), 30-50% (bottom left), 10-30% (bottom center), and ZDC triggered 0-12% (bottom right) collisions at sNN=200 GeV/c. -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.2 STAR Preliminary Fig. 3. (color online) Away-side projections of a strip of width 0.7 radians for (left) d+Au and (right) 0-12% ZDC Triggered Au+Au. Off-diagonal projection (solid) is (∆φaT − ∆φbT )/2 and on-diagonal projection (open) is (∆φaT +∆φbT )/2 − π. Shaded bands are systematic errors. broadening. In central Au+Au collisions strong peaks are seen in the off-diagonal projection, as expected for conical emission. The on-diagonal projection is similar to the off-diagonal but with additional contribution between the peaks likely due to deflected jets and/or large angle gluon radiation. The fitted angle of the side peaks in the off-diagonal projection is about 1.45 radians. Figure 4 shows the centrality dependence of the average signal strengths in different regions. The right panel shows the away-side signal, average singal centered November 4, 2018 14:16 WSPC/INSTRUCTION FILE IWCF˙v3 Three-Particle Correlations from STAR 5 0 1 2 3 4 5 6 -0.05 0 1 2 3 4 5 6 -0.05 0 1 2 3 4 5 6 -0.05 Deflected + Cone partN( Fig. 4. (color online) Average signals in 0.7 × 0.7 boxes at (0,0), left, (π ± 1.45,π ∓ 1.45), center, and (π ± 1.45,π ± 1.45). Solid error bars are statistical and shaded are systematic. Npart is the number of participants. The ZDC 0-12% points (open symbols) are shifted to the left for clarity. at (π,π). It increases with centrality in pp, d+Au and perpherial Au+Au and then seems to level off for mid-central and central Au+Au collisions. The middle panel shows the average signal where we only expect conical emission, at (π ± 1.45,π ∓ 1.45). It increases with centrality and significantly deviates from zero in central Au+Au collisions. The right panel shows the average signal were conical emissions deflected jets, and large angle gluon radiation could all contribute, at π ± 1.45,π± 1.45). This signal is similar to what we see where we only expect conical emission. Figure 5 shows the difference between on-diagonal signals, where conical emis- sion, deflected jets, and large angle gluon radiation could contribute, and off- diagonal signals, where only conical emission contributes. Since conical emission signals are expected to be of equal magnitude on-diagonal and off-diagonal, the difference may indicate the contribution from deflected jets and large angle gluon radiation. The centrality dependence is shown for three different angles. The differ- ence decreases with distance from (π,π). If we have Mach-cone emission, the emission angle is expected to be indepen- dent of the associated particle momentum; however, the Čerenkov radiation model in Ref. 10 predicts an emission angle that is sharply decreasing with associated particle momentum. For this reason we shall look at the associated particle pT dependence of our signal. Figure 6 shows the background subtracted 3-particle cor- relations for different associated pT bins. The angle is determined from fitting the off-diagonal projection, (∆φaT − ∆φbT )/2, to a central Gaussian and two symet- ric side Gaussians. The strength of the off-diagonal signal decrease with increasing November 4, 2018 14:16 WSPC/INSTRUCTION FILE IWCF˙v3 6 Jason Glyndwr Ulery For the STAR Collaboration partN 0 1 2 3 4 5 6 1.00)±πDeflected ( 1.30)±πDeflected ( 1.45)±πDeflected ( STAR Preliminary Fig. 5. (color online) Differences in average signals, between (π±1.45,π±1.45) and (π±1.45,π∓1.45) (triangle), between (π± 1.3,π± 1.3) and (π± 1.3,π∓ 1.3) (square), and between (π± 1.0,π± 1.0) and (π ± 1.0,π ∓ 1.0) (circle). Solid error bars are statistical and shaded are systematic. Npart is the number of participants. The ZDC 0-12% points (open symbols) are shifted to the left for clarity. -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 -0.15 -0.05 -1 0 1 2 3 4 5-1 -0.05 -1 0 1 2 3 4 5-1 -0.02 (a) (b) (c) (d) Fig. 6. (color online) Background subtracted jet-like 3-particle correlations for 0.75 < p < 1.0 GeV/c (left), 1.0 < p < 1.5 GeV/c (left center), 1.5 < p < 2.0 GeV/c (right center), and 2.0 < p < 3.0 GeV/c (right). Trigger particle pT is 3 < p < 4 GeV/c. Results are from ZDC triggered top 12% central Au+Au collisons at sNN=200 GeV. pT and is almost gone in the highest pT bin. This is not surprising since we need two away-side particles each with a pT that is a significant fraction of the trigger particle pT . Figure 7 (left) shows the dependence of the angle of the off-diagonal peaks on associated particle pT . The angle is consistent with remaining constant as a function of associated particle pT . Figure 7 (right) shows the centrality dependence of the angle of the off-diagonal peaks obtained from fits to the projections as done for the pT dependence. The angle is consistent with remaining constant from mid-central to central Au+Au collisions. If we have Mach-cone emission this likely implies the speed of sound in the medium does not greatly vary from mid-central to central Au+Au collisions. The solid line at 1.46 on the plot is from a fit to a constant. November 4, 2018 14:16 WSPC/INSTRUCTION FILE IWCF˙v3 Three-Particle Correlations from STAR 7 Assoc (GeV/c) 0 0.5 1 1.5 2 2.5 0.02±Au+Au 0-12 1.41 0.02±Au+Au 0-50 1.45 Statistical Systematic STAR Preliminary partN( 3 3.5 4 4.5 5 5.5 6 Au+Au 0-12% (shifted) Au+Au 30-50%, 10-30% and 0-10% 0.03±1.46 STAR Preliminary Fig. 7. (color online) Emission angles from double Gaussian fits. (left) Angle as a function of associated particle pT for Au+Au 0-12% ZDC triggered (filled) and Au+Au 0-50% from minimum bias (open). Numbers on the plot are results from a fit to a constant for the data points with the fit errors displayed. (right) Angle as a function of centrality for Au+Au 0-12% ZDC triggered data (circle) and Au+Au 30-50%, 10-30% and 0-10% from minimum bias data (square). The 0-12% point has been shifted for clarity. The number is from a fit to a constant for the points, shown with the solid line. The dashed line is at π/2. Solid error bars are statistical and shaded are systematic. 4. Systematics The major sources of systematic error are from the elliptic flow measurements and the background normalization. Our default v2 is the average of measurements from the reaction plane and 4-particle cumulant methods. The systematic uncer- tainty due to the v2 has been determined by varying it between the reaction plane and 4-particle cumulant results. Figure 8a and b show the background subtracted 3-particle correlation for the reaction plane and 4-particle v2, respectively. Even though the hard-soft background and trigger flow backgrounds individually vary a great deal with the change in elliptic flow, the variations cancel out to first order. Therefore the signal, as seen in Fig. 8 is robust with respect to the variation in elliptic flow. -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 (a) (b) (c) Fig. 8. (color online) 0-12% Au+Au ZDC triggerd data for different systematic checks: (a) reaction plane v2, (b) 4-particle cumulant v2, and (c) normalization region for α of 0.6 < |∆φ| < 1.4 November 4, 2018 14:16 WSPC/INSTRUCTION FILE IWCF˙v3 8 Jason Glyndwr Ulery For the STAR Collaboration To study the effect of the background normalization the size of the normalization window used to determine α was doubled to 0.6 < |∆φ| < 1.4. The signal is robust with this change in normalization. Figure 8 shows the background subtracted 3- particle correlation using this larger normalization window. Other sources of systematic error include the effect on the trigger particle flow from requiring a correlated particle (a 20% change on trigger particle v2 is applied), uncertainity in the v4 parameterization, and multiplicity bias effects on the soft- soft background. The systematic errors shown in Figures 3, 4, 5, and 7 reflect the quadratic sum of all the systematic uncertainties mentioned. 5. Conclusion Three-particle azimuthal correlations have been studied for trigger particles of 3 < pT < 4 GeV/c and associated particles of 1 < pT < 2 GeV/c in pp, d+Au, and Au+Au collisions at sNN=200 GeV/c by STAR. This analysis treats events as the sum of two components, particles that are jet-like correlated with the trigger and background particles. On-diagonal broadening has been observed in pp and d+Au collisions that is consistent with kT broadening. Additional on-diagonal broadening has been observed in heavy ion collisions that may be due to contributions from deflected jets and/or large angle gluon radiation. Off-diagonal peaks have been detected in central Au+Au collisions that are consistent with conical emission. A study of the angle as a function of associated particle pT was performed to discriminate between hydrodynamic conical flow and Čerenkov gluon radiation. No strong dependence on associated particle pT was beheld. This result is consistent with Mach-cone emission. References 1. J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 95, 152301 (2005). 2. S.S. Adler et al. (PHENIX Collaboration), Phys. Rev. Lett. 97, 052301 (2006). 3. J. Ulery et al. (STAR Collaboration), Nuc. Phys. A774, 581-584 (2006). 4. M. Horner et al. (STAR Collaboration), QM2006 Talk Proceedings to appear in Jour. Phys. G. 5. I. Vitev, Phys. Lett. B 630, 78 (2005). 6. A.D. Polosa and C.A. Salgado, hep-ph/0607295. 7. H. Stoecker, Nucl. Phys. A750, 121 (2005). 8. J. Casalderrey-Solana, E. Shuryak and D. Teaney, J. Phys. Conf. Ser. 27, 23 (2005). 9. I.M. Dremin, Nucl. Phys. A767 233 (2006). 10. V. Koch, A. Majumder and X.-N. Wang, Phys. Rev. Lett. 96, 172302 (2006). 11. J. Ulery and F. Wang, nucl-ex/0609016. 12. J. Adams et al. (STAR Collaboration), Phys. Rev. C 72, 014904 (2004). http://arxiv.org/abs/hep-ph/0607295 http://arxiv.org/abs/nucl-ex/0609016 Introduction Analysis Procedure Results Systematics Conclusion

0704.0221

The Return of a Static Universe and the End of Cosmology

The Return of a Static Universe and the End of Cosmology Lawrence M. Krauss1,2 and Robert J. Scherrer2 1Department of Physics, Case Western Reserve University, Cleveland, OH 44106; email: [email protected] and 2Department of Physics & Astronomy, Vanderbilt University, Nashville, TN 37235; email: [email protected] (Dated: February 1, 2008) Abstract We demonstrate that as we extrapolate the current ΛCDM universe forward in time, all evi- dence of the Hubble expansion will disappear, so that observers in our “island universe” will be fundamentally incapable of determining the true nature of the universe, including the existence of the highly dominant vacuum energy, the existence of the CMB, and the primordial origin of light elements. With these pillars of the modern Big Bang gone, this epoch will mark the end of cosmology and the return of a static universe. In this sense, the coordinate system appropriate for future observers will perhaps fittingly resemble the static coordinate system in which the de Sitter universe was first presented. http://arXiv.org/abs/0704.0221v3 Shortly after Einstein’s development of general relativity, the Dutch astronomer Willem de Sitter proposed a static model of the universe containing no matter, which he thought might be a reasonable approximation to our low density universe. One can define a coor- dinate system in which the de Sitter metric takes a static form by defining de Sitter space- time with a cosmological constant Λ as a four dimensional hyperboloid SΛ : ηABξ AξB = −R2, R2 = 3Λ−1 embedded in a 5d Minkowski spacetime with ds2 = ηABdξ AdξB, and (ηAB) = diag(1,−1,−1,−1,−1), A, B = 0, · · · , 4. The static form of the de Sitter metric is then ds2s = (1 − r 2)dt2s − 1 − r2s/R − r2sdΩ which can be obtained by setting ξ0 = (R2 − r2s) 1/2 sinh(ts/R), ξ 1 = rs sin θ cos ϕ, ξ rs sin θ sin ϕ, ξ 3 = rs cos θ, ξ 4 = (R2 − r2s) 1/2 cosh(ts/R). In this case the metric only corre- sponds to the section of de Sitter space within a cosmological horizon at R = r − s. In fact de Sitter’s model wasn’t globally static, but eternally expanding, as can be seen by a coordinate transformation which explicitly incorporates the time dependence of the scale factor R(t) = exp(Ht). While spatially flat, it actually incorporated Einstein’s cosmological term, which is of course now understood to be equivalent to a vacuum energy density, leading to a redshift proportional to distance. The de Sitter model languished for much of the last century, once the Hubble expansion had been discovered, and the cosmological term abandoned. However, all present observa- tional evidence is consistent with a ΛCDM flat universe consisting of roughly 30% matter (both dark matter and baryonic matter) and 70% dark energy [1, 2, 3], with the latter having a density that appears constant with time. All cosmological models with a non-zero cosmological constant will approach a de Sitter universe in the far future, and many of the implications of this fact have been explored in the literature [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. Here we re-examine the practical significance of the ultimate de Sitter expansion and point out a new eschatological physical consequence: from the perspective of any observer within a bound gravitational system in the far future, the static version of de Sitter space outside of that system will eventually become the appropriate physical coordinate system. Put more succinctly, in a time comparable to the age of the longest lived stars, observers will not be able to perform any observation or experiment that infers either the existence of an expanding universe dominated by a cosmological constant, or that there was a hot Big Bang. Observers will be able to infer a finite age for their island universe, but beyond that cosmology will effectively be over. The static universe, with which cosmology at the turn of the last century began, will have returned with a vengeance. Modern cosmology is built on integrating general relativity and three observational pillars: the observed Hubble expansion, detection of the cosmic microwave background radiation, and the determination of the abundance of elements produced in the early universe. We describe next in detail how these observables will disappear for an observer in the far future, and how this will be likely to affect the theoretical conclusions one might derive about the universe. A. The disappearance of the Hubble Expansion The most basic component of modern cosmology is the expansion of the universe, firmly established by Hubble in 1929. Currently, galaxies and galaxy clusters are gravitationally bound and have dropped out of the Hubble flow, but structures on larger length scales are observed to obey the Hubble expansion law. Now consider what happens in the far future of the universe. Both analytic [7] and numerical [10] calculations indicate that the Local Group remains gravitationally bound in the face of the accelerated Hubble expansion. All more distant structures will be driven outside of the de Sitter event horizon in a timescale on the order of 100 billion years ([4], see also Refs. [8, 9]). While objects will not be observed to cross the event horizon, light from them will be exponentially redshifted, so that within a time frame comparable to the longest lived main sequence stars all objects outside of our local cluster will truly become invisible [4]. Since the only remaining visible objects will in fact be gravitationally bound and decou- pled from the underlying Hubble expansion, any local observer in the far future will see a single galaxy (the merger product of the Milky Way and Andromeda and other remnants of the Local Group) and will have no observational evidence of the Hubble expansion. Lacking such evidence, one may wonder whether such an observer will postulate the correct cosmo- logical model. We would argue that in fact, such an observer will conclude the existence of a static “island universe,” precisely the standard model of the universe c. 1900. This will be true in spite of the fact that the dominant energy in this universe will not be due to matter, but due to dark energy, with ρM/ρΛ ∼ 10 −12 inside the horizon volume [9]. The irony, of course, is that the denizens of this static universe will have no idea of the existence of the dark energy, much less of its magnitude, since they will have no probes of the length scales over which Λ dominates gravitational dynamics. It appears that dark energy is undetectable not only in the limit where ρΛ ≪ ρM , but also when ρΛ ≫ ρM . Even if there were no direct evidence of the Hubble expansion, we might expect three other bits of evidence, two observational and one theoretical, to lead physicists in the future to ascertain the underlying nature of cosmology. However, we next describe how this is unlikely to be the case. B. Vanishing CMB The existence of a Cosmic Microwave Background was the key observation that convinced most physicists and astronomers that there was in fact a hot big bang, which essentially implies a Hubble expansion today. But even if skeptical observers in the future were inclined to undertake a search for this afterglow of the Big Bang, they would come up empty- handed. At t ≈ 100 Gyr, the peak wavelength of the cosmic microwave background will be redshifted to roughly λ ≈ 1 m, or a frequency of roughly 300 MHz. While a uniform radio background at this frequency would in principle be observable, the intensity of the CMB will also be redshifted by about 12 orders of magnitude. At much later times, the CMB becomes unobservable even in principle, as the peak wavelength is driven to a length larger than the horizon [4]. Well before then, however, the microwave background peak will redshift below the plasma frequency of the interstellar medium, and so will be screened from any observer within the galaxy. Recall that the plasma frequency is given by where ne and me are the electron number density and mass, respectively. Observations of dispersion in pulsar signals give [14] ne ≈ 0.03 cm −3 in the interstellar medium, which corresponds to a plasma frequency of νp ≈ 1 kHz, or a wavelength of λp ≈ 3 × 10 7 cm. This corresponds to an expansion factor ∼ 108 relative to the present-day peak of the CMB. Assuming an exponential expansion, dominated by dark energy, this expansion factor will be reached when the universe is less than 50 times its present age, well below the lifetime of the longest-lived main sequence stars. After this time, even if future residents of our island universe set out to measure a universal radiation background, they would be unable to do so. The wealth of information about early universe cosmology that can be derived from fluctuations in the CMB would be even further out of reach. C. General Relativity Gives No Assistance We may assume that theoretical physicists in the future will infer that gravitation is described by general relativity, using observations of planetary dynamics, and ground-based tests of such phenomena as gravitational time dilation. Will they then not be led to a Big Bang expansion, and a beginning in a Big Bang singularity, independent of data, as Lemaitre was? Indeed, is not a static universe incompatible with general relativity? The answer is no. The inference that the universe must be expanding or contracting is dependent upon the cosmological hypothesis that we live in an isotropic and homogeneous universe. For future observers, this will manifestly not be the case. Outside of our local cluster, the universe will appear to be empty and static. Nothing is inconsistent with the temporary existence of a non-singular isolated self-gravitating object in such a universe, governed by general relativity. Physicists will infer that this system must ultimately collapse into a future singularity, but only as we presently conclude our galaxy must ultimately coalesce into a large black hole. Outside of this region, an empty static universe can prevail. While physicists in the island universe will therefore conclude that their island has a finite future, the question will naturally arise as to whether it had a finite beginning. As we next describe, observers will in fact be able to determine the age of their local cluster, but not the nature of the beginning. D. Polluted Elemental Abundances The theory of Big Bang Nucleosynthesis reached a fully-developed state [15] only after the discovery of the CMB (despite early abortive attempts by Gamow and his collaborators [16]). Thus, it is unlikely that the residents of the static universe would have any motivation to explore the possibility of primordial nucleosynthesis. However, even if they did, the evidence for BBN rests crucially on the fact that relic abundances of deuterium remain observable at the present day, while helium-4 has been enhanced by only a few percent since it was produced in the early universe. Extrapolating forward by 100 Gyr, we expect significantly more contamination of the helium-4 abundance, and concomitant destruction of the relic deuterium. It has been argued [17] that the ultimate extrapolation of light elemental abundances, following many generations of stellar evolution, is a mass fraction of helium given by Y = 0.6. The primordial helium mass fraction of Y = 0.25 will be a relatively small fraction of this abundance. It is unlikely that much deuterium could survive this degree of processing. Of course, the current “smoking gun” deuterium abundance is provided by Lyman-α absorption systems, back-lit by QSOs (see, e.g., Ref. [18]). Such systems will be unavailable to our observers of the future, as both the QSOs and the Lyman-α systems will have redshifted outside of the horizon. Astute observers will be able to determine a lower limit on the age of their system, however, using standard stellar evolution analyses of their own local stars. They will be able to examine the locus of all stars and extrapolate to the oldest such stars to estimate a lower bound on the age of the galaxy. They will be able to determine an upper limit as well, by determining how long it would take for all of the observed helium to be generated by stellar nucleosynthesis. However, without any way to detect primordial elemental abundances, such as the aforementioned possibility of measuring deuterium in distant intergalactic clouds that currently absorb radiation from distant quasars and allow a determination of the deuterium abundance in these pre-stellar systems, and with the primordial helium abundance dwarfed by that produced in stars, inferring the original BBN abundances will be difficult, and probably not well motivated. Thus, while physicists of the future will be able to infer that their island universe has not been eternal, it is unlikely that they will be able to infer that the beginning involved a Big Bang. E. Conclusion The remarkable cosmic coincidence that we happen to live at the only time in the history of the universe when the magnitude of dark energy and dark matter densities are comparable has been a source of great current speculation, leading to a resurgence of interest in possible anthropic arguments limiting the value of the vacuum energy (see, e.g., Ref. [19]). But this coincidence endows our current epoch with another special feature, namely that we can actually infer both the existence of the cosmological expansion, and the existence of dark energy. Thus, we live in a very special time in the evolution of the universe: the time at which we can observationally verify that we live in a very special time in the evolution of the universe! Observers when the universe was an order of magnitude younger would not have been able to discern any effects of dark energy on the expansion, and observers when the universe is more than an order of magnitude older will be hard pressed to know that they live in an expanding universe at all, or that the expansion is dominated by dark energy. By the time the longest lived main sequence stars are nearing the end of their lives, for all intents and purposes, the universe will appear static, and all evidence that now forms the basis of our current understanding of cosmology will have disappeared. Note added in proof: After this paper was submitted we learned of a prescient 1987 paper [20], written before the discovery of dark energy and other cosmological observables that are central to our analysis, which nevertheless raised the general question of whether there would be epochs in the Universe when observational cosmology, as we now understand it, would not be possible. 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[15] R.V. Wagoner, W.A. Fowler, and F. Hoyle, Astrophys. J. 148, 3 (1967). [16] R.A. Alpher, H. Bethe, and G. Gamow, Phys. Rev. 73, 803 (1948); R.A. Alpher, J.W. Follin, and R.C. Herman, Phys. Rev. 92, 1347 (1953). [17] F.C. Adams and G. Laughlin, Rev. Mod. Phys. 69, 337 (1997). [18] D. Kirkman, D. Tytler, N. Suzuki, J.M. O’Meara, and D. Lubin, Ap.J. Suppl. 149, 1 (2003). [19] S. Weinberg, Phys. Rev. Lett. 59, 2607 (1987); J. Garriga, M. Livio, and A. Vilenkin, Phys. Rev. D 61, 023503 (2000). [20] T. Rothman and G. F. R. Ellis, The Observatory, 107, 24 (1987) The disappearance of the Hubble Expansion Vanishing CMB General Relativity Gives No Assistance Polluted Elemental Abundances Conclusion Acknowledgments References

0704.0222

Dark Matter annihilation in Draco: new considerations on the expected gamma flux

Dark Matter annihilation in Draco: new considerations of the expected gamma flux 1 Miguel A. Sánchez-Conde Instituto de Astrofisica de Andalucia (CSIC), E-18008, Granada, Spain Abstract. A new estimation of the γ-ray flux that we expect to detect from SUSY dark matter annihilation from the Draco dSph is presented using the DM density profiles compatible with the latest observations. This calculation takes also into account the important effect of the Point Spread Function (PSF) of the telescope. We show that this effect is crucial in the way we will observe and interpret a possible signal detection. Finally, we discuss the prospects to detect a possible gamma signal from Draco for MAGIC and GLAST. Keywords: cosmology: dark matter — galaxies: dwarf — gamma-rays: theory PACS: 95.35.+d; 95.55.Ka; 95.85.Pw; 98.35.Gi; 98.52.Wz INTRODUCTION The Draco galaxy, a satellite of the Milky Way, represents one of the best suitable candidates to search for DM outside our galaxy [1], since it is near (80 kpc) and it has probably more observational constraints than any other known DM dominated system. This fact becomes crucial when we want to make realistic predictions of the expected observed γ-ray flux due to DM annihilation. The expected total number of continuum γ-ray photons received per unit time and per unit area, from a circular aperture on the sky of width σt (which represents the resolution of the telescope) observing at a given direction Ψ0 relative to the centre of the dark matter halo is given by: F(E > Eth) = fSUSY ·U(Ψ0), with fSUSY = Nγ 〈σv〉 , U(Ψ0) = J(Ψ)B(Ω)dΩ (1) where the factor fSUSY encloses all the particle physics, and the factor U(Ψ0) involves all the astrophysical properties such as the dark matter distribution, geometry considerations and telescope performances like the PSF, this one directly related to the angular resolution (for a detailed explanation of each of these terms, see e.g. [2]). DRACO γ-RAY FLUX PROFILES AND THE EFFECT OF THE PSF We assumed that the dark matter distribution in Draco can be approximated by the formula ρd(r) = Cr−α exp(− rrb ) proposed by [3], which was found to fit the density distribution of a simulated dwarf dark matter halo stripped during its evolution in the potential of a giant galaxy. Here we will consider two cases, the profile with a cusp α = 1 and a core α = 0. The calculated γ-ray flux profiles are shown in left panel of Fig.1. For both cusp and core DM density profiles, the flux values should be very similar for the inner region of the dwarf, where signal detection would be easier. It is also necessary to take into account the role of the PSF in the calculations. Its effect on gamma ray flux profiles, usually neglected, may be crucial to correctly interpret a possible signal in the telescope, as it can be clearly seen in right panel of Fig. 1. 1 Poster presented at the First GLAST Symposium, Stanford University, USA, 5-8 February 2007 http://arxiv.org/abs/0704.0222v1 FIGURE 1. Left panel: Draco flux predictions for the core (red line) and cusp (blue line) DM density profiles, computed using a PSF= 0.1◦. A value of fSUSY = 10 −33GeV−2cm3s−1 was used (the most optimistic case given by particle physics at 100 GeV). Right panel: Flux predictions for the cusp density profile and using a PSF=0.1◦ (blue line), PSF=1◦ (red) and without PSF (green). FIGURE 2. Draco flux profile detection prospects for GLAST (red lines) and MAGIC (blue lines), and for the cusp density profile using a PSF=0.1◦. Values of fSUSY = 10 −33GeV−2cm3s−1 at 100 GeV and fSUSY = 4 10 −33GeV−2cm3s−1 at 10 GeV were used for MAGIC and GLAST respectively (the most optimistic scenarios given by particle physics at those energies. DETECTION PROSPECTS We carried out some calculations concerning the possibility to detect a γ-ray signal coming from DM annihilation in Draco for MAGIC and GLAST (Fig.2). According to these calculations, a detection of the gamma ray flux profiles seems to be very hard. We computed also the prospects for an excess signal detection, i.e. we are not interested in the shape of the gamma ray flux profile, only in detectability (Table. 1). According to the results we reached there is no chance to detect a γ-ray signal (flux profiles or just an excess) coming from Draco with current experiments, at least with the preferred particle physics and astrophysics models (for a more detailed study, see the complete work [4]). It will be necessary go a step further with IACTs that join a large field of view with a high sensitivity. TABLE 1. Prospects of an excess signal detection for MAGIC and GLAST. For FDraco, the most optimistic and pessimistic values are given in the form FDraco,min - FDraco,max. Fmin represents the minimum detectable flux for each instrument. All values refer to the inner 0.5◦ of the dwarf. FDraco (ph cm 2 s−1) Fmin (ph cm 2 s−1) MAGIC 1.6 10−19 - 4.0 10−16 4.4 10−11 (250 h, 5σ ) GLAST 1.6 10−19 - 1.6 10−15 3.9 10−10 (1 yr, 5σ ) REFERENCES 1. Evans N. W., Ferrer F. and Sarkar S., 2004, Physical Review D, 69, 123501 2. Prada F., Klypin A., Flix J., Martínez M. and Simonneau E., 2004, Phys. Rev. Letters, 93, 241301 3. Kazantzidis S., Mayer L., Mastropietro C., Diemand J., Stadel J., Moore B., 2004, ApJ, 608, 663 4. Sánchez-Conde M.A., Prada F. Łokas E.L., Gómez M.E., Wojtak R. Moles M., 2007, submitted to JCAP, [astro-ph/0701429] http://arxiv.org/abs/astro-ph/0701429 Introduction Draco -ray flux profiles and the effect of the PSF Detection prospects

0704.0223

Magnetohydrodynamic Rebound Shocks of Supernovae

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 4 November 2018 (MN LATEX style file v2.2) Magnetohydrodynamic Rebound Shocks of Supernovae Yu-Qing Lou1,2,3 ⋆ and Wei-Gang Wang1 1Physics Department and Tsinghua Centre for Astrophysics (THCA), Tsinghua University, Beijing, 100084, China; 2Department of Astronomy and Astrophysics, the University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637, USA; 3National Astronomical Observatories, Chinese Academy of Sciences, A20, Datun Road, Beijing 100012, China. 4 November 2018 ABSTRACT We construct magnetohydrodynamic (MHD) similarity rebound shocks joining ‘quasi- static’ asymptotic solutions around the central degenerate core to explore an MHD model for the evolution of random magnetic field in supernova explosions. This pro- vides a theoretical basis for further studying synchrotron diagnostics, MHD shock ac- celeration of cosmic rays, and the nature of intense magnetic field in compact objects. The magnetic field strength in space approaches a limiting ratio, that is comparable to the ratio of the ejecta mass driven out versus the progenitor mass, during this self-similar rebound MHD shock evolution. The intense magnetic field of the remnant compact star as compared to that of the progenitor star is mainly attributed to both the gravitational core collapse and the radial distribution of magnetic field. Key words: magnetohydrodynamics (MHD) – shock waves – stars: neutron – stars: winds, outflows – supernova remnants – white dwarfs 1 INTRODUCTION Self-similar evolution of a spherical gas flow under self- gravity and thermal pressure has been studied over past four decades: from simulations and the discovery of Larson- Penston (L-P) type solutions (Bodenheimer & Sweigart 1968; Larson 1969a, b; Penston 1969a, b), to the construc- tion of the expansion-wave collapse solution (EWCS) using the central free-fall asymptotic solution (Shu 1977) as well as to the application of phase-match techniques for construct- ing infinite series of discrete global solutions including L-P type solutions (Hunter 1977) and solutions for envelope ex- pansion with core collapse (EECC; Lou & Shen 2004). Prop- erties of eigensolutions crossing the sonic critical line were examined (Jordan & Smith 1977; Shu 1977; Whitworth & Summers 1985; Hunter 1986). Self-similar shocks were stud- ied and applied to various astrophysical settings by Tsai & Hsu (1995), Shu et al. (2002), Shen & Lou (2004), and Bian & Lou (2005). While these major results were obtained for an isothermal gas, the counterpart problem with a poly- tropic equation of state (EoS) was also studied by Cheng (1978), Goldreich & Weber (1980), Yahil (1983), Suto & Silk (1988), McLaughlin & Pudritz (1997), Fatuzzo et al. (2004) and Lou & Gao (2006). In most cases, the polytropic results share a feature that by setting the polytropic index γ = 1 in the isothermal limit, all asymptotic behaviours ap- ⋆ E-mail: [email protected] and [email protected]; [email protected] proach the isothermal counterpart solutions. However, Lou & Wang (2006) reported new ‘quasi-static’ asymptotic so- lutions unique to a polytropic gas with γ > 1.2 and con- structed self-similar rebound shocks for supernovae (SNe). Chiueh & Chou (1994) studied a self-similar MHD problem by including the magnetic pressure gradient force in the momentum equation. Yu & Lou (2005) improved their formulation and provided a more detailed analysis (see Zel’dovich & Novikov 1971 for a discussion of random mag- netic field). Wang & Lou (2006) studied this MHD problem for a polytropic gas and derived the ‘quasi-static’ asymp- totic solutions. Self-similar MHD shocks were explored by Yu et al. (2006). As magnetic field is inevitably involved in SNe and is crucial for synchrotron radiation and cosmic ray acceleration, we construct here rebound MHD shocks with ‘quasi-static’ asymptotic solutions to model magnetic field evolution in SN explosions. Type II, Ib, Ic SNe are thought to be caused by grav- itational core collapse due to an insufficient nuclear fuel; such collapse creates an over-dense core, which rebounds abruptly initiating a powerful rebound shock. The energetics of sustaining such a rebound shock has been an outstanding problem. We approach this issue in the following perspec- tive. Triggered by such a core collapse, the rebound shock is essentially supported by the neutrino-driven mechanism, and several complicated physical processes are involved in the stellar interior: all four elementary forces and the cou- pling of various fluids and matters such as baryons, neutri- nos, photons etc. (e.g., Janka et al. 2006). We approximate c© 0000 RAS http://arxiv.org/abs/0704.0223v1 2 Y.-Q. Lou & W.-G. Wang such a dynamic system in terms of a single fluid with a polytropic EoS, and treat the shock as an energy-conserved self-similar shock. Conceptually, the ‘rebound shock’ here refers to a neutrino-driven shock, as opposed to the ‘prompt shock’ mentioned in Janka et al. (2006). We constructed such a rebound shock (Lou & Wang 2006) to model a SN explosion followed by a self-similar evolution leading to a quasi-static configuration. In reference to the hydrodynamic model of Lou & Wang (2006), the main thrust of this Let- ter is to construct approximately a self-similar model of a quasi-spherically symmetric rebound MHD shock for a SN explosion, providing the profile and evolution of magnetic field to facilitate future studies of synchrotron radiation and MHD shock acceleration of cosmic rays, and to probe the nature of intense magnetic field of compact stellar objects left behind. 2 FORMULATION AND ANALYSIS 2.1 The Self-Similar MHD Formulation A quasi-spherical similarity MHD flow embedded with a completely random magnetic field on small scales is formu- lated the same as in Yu & Lou (2005) and Yu et al. (2006); the key difference here is the polytropic EoS p = κργ instead of an isothermal gas, where p is the pressure, ρ is the mass density, and κ is constant. Using the magnetic flux frozen-in condition, the ideal MHD equations, viz., the mass conserva- tion equation, the radial momentum equation, the magnetic induction equation and the polytropic EoS, can be reduced to two nonlinear ordinary differential equations (ODEs) (n− 1)v + nx− v 3n− 2 2(x− v)(nx− v) α(nx− v)2 − (γαγ + hα2x2) , (1) (n− 1) αv(nx− v) + 2hα2x2 (nx− v)2 (3n− 2) − 2γαγ (x− v) α(nx− v)2 − (γαγ + hα2x2) along with a useful relation m = αx2(nx−v) by the following MHD self-similar transformation in a polytropic gas flow r = k x, u = k v, ρ = 4πGt2 , p = kt2n−4 k3/2t3n−2 (3n− 2)G (nx−v) , < B2t >= kt2n−4 , (3) where G is the gravitational constant, M is the enclosed mass at time t within radius r, u is the radial flow speed, < B2t > is the mean square of random transverse magnetic field Bt, x is the independent self-similar variable, v(x) is the reduced flow speed, α(x) is the reduced density, m(x) is the reduced enclosed mass, the prime ′ stands for the first derivative d/dx, k and n are two parameters, and h is a parameter for the strength of < B2t > 1/2. We expedi- ently take γ = 2 − n for a polytropic EoS with a constant κ ≡ k(4πG)γ−1 = pρ−γ . The magnetosonic critical curve is determined by the simultaneous vanishing of the numerator and denominator on the RHS of eq (1) or (2). The two eigen- solutions of v′ across the magnetosonic critical curve can be derived by using the L’Hôspital rule (Lou & Wang 2006; Yu & Lou 2005; Yu et al. 2006). The solutions are obtained for v(x) and α(x), and the magnetic field < B2t > 1/2 is then known from transformation (3). 2.2 Analytic Asymptotic MHD Solutions For h < hc ≡ n2/[2(1− n)(3n− 2)], eqs (1) and (2) give the magnetostatic solution of a magnetized singular polytropic sphere (MSPS) with v = 0 and 2γ(4− 3γ) (1− γ) ]−1/n , (4) t >= h 2γ(4− 3γ) (1− γ) ]−2/n 2−4/n . (5) There exists an asymptotic MHD solution approaching this limiting form at small x (referred to as the type I ‘quasi- static’ asymptotic MHD solution), viz., v = LxK and 2γ(4− 3γ) (1− γ) ]−1/n (K + 2− 2/n)L n(K − 1) n2/(2γ) 4− 3γ ]−1/n K−1−2/n , (6) where K is the root of quadratic equation /2 + n(3n− 2)h]K2 − (4− 3n)[n/2 + (3n− 2)h]K + γ(2/n− 2)(3n− 2)h = 0 . (7) When 12 − 8 2 < n < 0.8 and h0 < h < hc, where h0 ≡ (3 + 2 2)n − 4 4 − (3 − 2 /[2n(3n − 2)], or when 2/3 < n < 12−8 2 for h < hc, eq (7) gives two roots K > 1, corresponding to two possible ‘quasi-static’ solutions. The asymptotic MHD solution at large x is α = −2/n + · · · and v = B0x 1−1/n − (3n− 2) 2h(n− 1) 1−2/n n[2(n+ γ)− 3] (2−2γ−n)/n + · · · , (8) where A0 and B0 are two constants. Solution (8) at large x can be connected to ‘quasi-static’ asymptotic MHD solution (6) and (7) at small x by a Runge-Kutta integration (Press et al. 1986), crossing the magnetosonic critical curve either smoothly or with an MHD shock (Yu et al. 2006). 2.3 MHD Shock Jump Conditions MHD shock conditions (Yu et al. 2006; Lou & Wang 2006) include conservations of mass, momentum, energy and mag- netic flux, and in self-similar forms, they appear as = 0 , (9) = 0 , (10) c© 0000 RAS, MNRAS 000, 000–000 MHD Rebound Shocks in Supernovae 3 (γ − 1) αγ−1s + 2hαs = 0 , (11) where quantities in square brackets with superscript ‘1’ (up- stream) and subscript ‘2’ (downstream) remain conserved across the MHD shock front indicated by a subscript s. The parameter k changes according to k2 = k1x s2 on two sides of a shock. For the specific entropy to increase from upstream to downstream sides, xs1 > xs2 is necessary. MHD shock conditions (9)−(11) lead to a quadratic equation (Lou & Wang 2006); once we specify physical conditions on one side of a chosen shock location, the corresponding quantities α, v, x on the other side are readily computed. 3 REBOUND MHD SHOCKS IN SUPERNOVA EXPLOSIONS Various rebound MHD shocks are constructed numerically, parallel to Lou & Wang (2006). With chosen inner and outer radii, e.g., ri = 10 6cm and ro = 10 12cm for neutron star formation, and when the k parameter in transformation (3) is specified, we apply our solutions to a physical rebound MHD shock scenario for SNe (Lou & Wang 2006). 3.1 Final and Initial Configurations Similar to the hydrodynamic rebound shock model of Lou & Wang (2006), the final configuration (small x) of our re- bound MHD shock solutions gradually evolves to a MSPS and is regarded as a remnant compact object after the re- bound MHD shock ploughing through stellar ejecta; the initial configuration (large x) marks the onset of gravity- induced core collapse with outer inflows or outflows such as stellar winds or stellar oscillations. We define the outer initial mass Mo,ini and the inner ul- timate mass Mi,ult the same way as in Lou & Wang (2006) and regard them as rough estimates for the masses of the progenitor star and the remnant compact object. The ratio of the two masses isMo,ini/Mi,ult = λ1(ro/ri) (3−2/n) where λ1 ≡ A0(k1/k2)1/n n2/[2γ(4−3γ)]+(1−γ)h/γ involves parameters of the rebound MHD shock and is equal to the ratio of enclosed masses at the same r. Similar to the result of Lou & Wang (2006), we find numerically that λ1 > 1 de- pends on the choice of solutions, clearly indicating that a rebound MHD shock drives out stellar materials. By eq (3), the final magnetostatic configuration gives t,ult > k2γ(4− 3γ) ]−1/n 1−2/n The ratio of initial to final magnetic fields at the same r is t,ini > 1/2 / < B2 t,ult > 1/2= λ1, where λ1 > 1 by numer- ical exploration. Thus a rebound MHD shock breakout pro- cess reduces the magnetic field by the same ratio of enclosed masses at the same r; yet this decrease in magnetic field is insignificant as compared to the radial variation of magnetic field, i.e., the r1−2/n dependence. As γ approaches 4/3 or n → 2/3, this scaling approaches r−2, while the dependence of enclosed mass on r goes to r0. For a ∼ 10G (0.1G) surface magnetic field at ro = 10 12cm, we estimate a magnetic field in the interior of the final configuration (ri = 10 6cm) to be ∼ 1013G (1011G), sensible for magnetized neutron stars; if we take ri = 10 9cm, then the final interior magnetic field is estimated to be ∼ 107G (105G), fairly close to relevant magnetic field strengths of white dwarfs (e.g., Euchner et al. 2005, 2006; Schmidt et al. 2003). 3.2 Evolution of Rebound MHD Shocks Time evolution of density, velocity and enclosed mass are similar to those described by Lou & Wang (2006). We focus here on the magnetic field evolution. Figure 1 shows a typical time evolution of < B2t > 1/2 / < B2 t,ult >1/2 to complement the r1−2/n behaviour. Magnetic field increases at first, and gradually decreases until reaching the magnetostatic config- uration much smaller than the initial configuration in size. In short, magnetic field changes moderately. The crucial point is that the magnetic field varies significantly in r within a star. If we take the magnetic field at the outer boundary to be the surface magnetic field of the progenitor star and take the magnetic field at the inner boundary as the surface magnetic field of the remnant compact star, then a large ratio of ∼ 1012 appears in forming a neutron star (Lou & Wang 2006). This model feature may explain the intense magnetic field of neutron stars inferred from spin-down ob- servations of radio pulsars. In our scenario, after the passage of such a rebound MHD shock, stellar ejecta detach from the central degenerate neutron star which is thus exposed with a surface magnetic field of 1013∼11G. In the same spirit of Lou & Wang (2006), we also suggest the formation of magnetic white dwarfs from the end of main-sequence stars with 6 ∼ 8M⊙; in this scenario, the surface magnetic field of an exposed central white dwarf is in a plausible range of ∼ 107∼5G (e.g., Euchner et al. 2005, 2006; Schmidt et al. 2003). The major point to be emphasized is that random mag- netic field preexists inside progenitor stars through various dynamo processes. We detect magnetic field strengths of or- der 10−2∼3G on stellar surface and this corresponds to a much stronger magnetic field in the stellar interior with a scaling of ∼ r1−2/n shortly after the initiation of core col- lapse. In addition, the interior magnetic field can be consid- erably strengthened by the free-fall core collapse preceding the emergence of a rebound MHD shock (see Lou & Wang 2006 for descriptions of the rebound shock scenario and the core collapse process), according to the frozen-in flux and accretion shock conditions. In reality, these two processes happen concurrently to produce the resultant self-similar distribution of magnetic field. In short, the interior magnetic field would be much stronger than the surface magnetic field and can be further enhanced to reach a high-field regime. The origin of stellar magnetic field was argued by several authors to come from various processes, includ- ing dynamo effects and thermomagnetic instabilities (e.g., Reisenegger et al. 2005). Our MHD scenario of interior core collapse and rebound shock appears to grossly match with observational facts. From our MSPS configuration with a random magnetic field strength scaled as Bt ∝ r1−2/n in a polytropic gas, we see a real possibility that the interior magnetic field can be actually much stronger than the sur- c© 0000 RAS, MNRAS 000, 000–000 4 Y.-Q. Lou & W.-G. Wang r(cm) t = ∞ Typical Magnetic Field Evolution During Shock Breakout Figure 1. The ratio Bt/Bt,ult ≡< B 1/2 / < B2 t,ult >1/2 is the rms magnetic field strength divided by the corresponding rms magnetic field strength of the final magnetostatic configuration at the same r. This example is constructed by integrating inward from (x0 , v0 , α0) on the magnetosonic critical curve and using an eigensolution to match with a quasi-static solution as x → 0+; we use the solution portion within xs2 < x0 for the downstream. We then obtain the upstream point (xs1, vs1, αs1) by the MHD shock jump conditions from the values of (xs2, vs2, αs2) obtained in the former integration and further integrate outward to determine the upstream solution. The relevant parameters are γ = 1.32, n = 0.68, h = 0.01, k1 = 7.7 × 10 16 cgs units, k2 = 4 × 10 cgs units, x0 = 1.778 , v0 = 0.4620 , α0 = 0.067, and xs2 = 1.1. Here, t1 = 6.61× 10 −5s is the time when the MHD shock crosses the inner boundary and is the initial time of application; t2 = 4.40 × 104s is the time when the MHD shock crosses the outer boundary; tm1 = 0.1s and tm2 = 1 × 10 8s are two intermediate times between t1 and t2 and t2 and t = ∞. face magnetic field. Once the onset of a gravitational col- lapse has been initiated within a magnetized progenitor star and following subsequent free-fall core collapse and accretion shock, an eventual emergence of a rebound MHD shock can evolve in a quasi-spherical self-similar manner and can end up to a MSPS configuration with a high-density compact degenerate object left behind. 4 CONCLUSIONS AND DISCUSSION We outline and propose the model scenario of a quasi- spherical rebound MHD shock to form high-density com- pact stars after a gravity-induced collapse in the core of a progenitor star that runs out of nuclear fuels. The stellar in- terior magnetic field is expected to be enhanced during the core collapse before the eventual emergence of a rebound MHD shock; also the interior magnetic field should be much stronger than the stellar surface magnetic field prior to the onset of a core collapse and during the outward propagation of the rebound MHD shock. Once the magnetostatic config- uration of a remnant degenerate star appears, stellar ejecta gradually detach from the compact object, exposing intense surface magnetic fields of ∼ 1013∼11G for neutron stars or ∼ 107∼5G for magnetic white dwarfs. Formally, MSPS solution (4) for density diverges as x → 0+. Conceptually, this can be readily reconciled by the onset of degeneracy in core materials at a nuclear mass density. In our model, there are two parameters for magnetic field: index n for radial variation and ratio h. While it ap- pears in this Letter that n depends on the stiffness (i.e., γ) of EoS, as discussed below it is in fact a parameter free from the stiffness (i.e., γ). Meanwhile, ratio h represents an ideal MHD approximation that dictates the magnitude variation of random transverse magnetic field; other factors, such as metalicity, differential rotation, convective motions, buoy- ancy etc. (Janka 2006), are important in generating random magnetic fields inside a star prior to the onset of the core collapse. In contexts of SN explosions, two-shock models, i.e., models involving a ‘forward shock’ for the SN remnant shock after the powerful rebound shock crashing into the inter- stellar medium and a ‘reverse shock’ produced by the same impact process (see, e.g., Chevalier et al. 1992 and Truelove & McKee 1999), have been studied earlier. The major for- mulation difference between these earlier works and ours is that they ignored self-gravity of stellar ejecta. By estimates, the self-gravity cannot be obviously dropped and thus these models including forward and reverse shocks would be ap- plicable in the limit of extremely strong shocks in order to ignore self-gravity. Another major difference is that these earlier models focus on circumstellar interactions, while we focus on a rebound MHD shock as it travels within the mag- netized stellar interior. Our polytropic model is currently restricted to γ = 2−n for a constant κ merely for expediency. This constraint can be actually removed if we consistently allow the reduced pressure to be ∝ αγmq where index parameter q ≡ 2(n + γ − 2)/(3n − 2) 6= 0 in general and m = αx2(nx− v) is the reduced enclosed mass. It is then possible for 1 < γ < 2 while n → 2/3. This more general case will be reported separately (Wang & Lou 2006). Numerical MHD simulations and observations are needed to further test our scenario for rebound MHD shocks in SNe, such as direct or indirect observation of density and flow speed profiles (Lou & Wang 2006) as well as diagnostics of synchrotron emissions caused by relativistic electrons in random magnetic field generated by MHD shocks. ACKNOWLEDGMENTS This research has been supported in part by the ASCI Center for Astrophysical Thermonuclear Flashes at the Univ. of Chicago, by THCA, by the NSFC grants 10373009 and 10533020 at the Tsinghua Univ., and by the SRFDP 20050003088 and the Yangtze Endowment from the Min- istry of Education at the Tsinghua Univ. REFERENCES Bian F.-Y., Lou Y.-Q., 2005, MNRAS, 363, 1315 Bodenheimer P., Sweigart A., 1968, ApJ, 152, 515 Boily C. M., Lynden-Bell D., 1995, MNRAS, 276, 133 Chandrasekhar S., 1957, Stellar Structure. Dover Publica- tions, New York c© 0000 RAS, MNRAS 000, 000–000 MHD Rebound Shocks in Supernovae 5 Cheng A. F., 1978, ApJ, 221, 320 Chevalier R. A., Blondin J. M., Emmering R. T., 1992, ApJ, 392, 118 Chiueh T., Chou J.-K., 1994, ApJ, 431, 380 Euchner F., et al., 2005, A&A, 442, 651 Euchner F., et al., 2006, A&A, 451, 671 Fatuzzo M., Adams F. C., Myers P. C., 2004, ApJ, 615, Goldreich P., Weber S. V., ApJ, 1980, 238, 991 Hu J., Shen Y., Lou Y.-Q., Zhang S.N., 2006, MNRAS, 365, 345 Hunter C., 1977, ApJ, 218, 834 Hunter C., 1986, MNRAS, 223, 391 Jordan D. W., Smith P., 1977, Nonlinear Ordinary Differ- ential Equations, Oxford University Press. Oxford Janka H.-Th., Langanke K., Marek A., Mart́ınez-Pinedo G., Müeller B., 2006, arXiv:astro-ph/0612072 Landau L. D., Lifshitz E. M., 1959, Fluid Mechanics, Perg- amon Press, NY Larson R. B., 1969a, MNRAS, 145, 271 Larson R. B., 1969b, MNRAS, 145, 405 Lattimer J. M., Prakash M., 2004, Science, 304, 536 Lou Y.-Q., Shen Y., 2004, MNRAS, 348, 717 Lou Y.-Q., Gao Y., 2006, MNRAS, 373, 1610 Lou Y.-Q., Wang W. G., 2006, MNRAS, 372, 885 McLaughlin D. E., Pudritz R. E., 1997, ApJ, 476, 750 Penston M. V., 1969a, MNRAS, 144, 425 Penston M. V., 1969b, MNRAS, 145, 457 Press W. H., et al., 1986, Numerical Recipes (Cambridge University Press) Reisenegger A., et al., 2005, AIP Conf. Proc., 784, 263 Schmidt G. D., et al., ApJ, 595, 1101 Shapiro S. L., Teukolsky S. A., 1983, Black Holes, White Dwarfs and Neutron Stars, John Wiley & Sons, Inc. Shen Y., Lou Y. Q., 2004, ApJL, 611, L117 Shu F. H., 1977, ApJ, 214, 488 Shu F. H., Lizano S., Galli D., Cantó J., Laughlin G., 2002, ApJ, 580, 969 Suto Y., Silk J., 1988, ApJ, 326, 527 Terebey S., Shu F. H., Cassen P., 1984, ApJ, 286, 529 Truelove J. K., McKee C. F., 1999, ApJS, 120, 299 Tsai J. C., Hsu J. J. L., 1995, ApJ, 448, 774 Wang W. G., Lou Y.-Q., 2006, in preparation Whitworth A., Summers D., 1985, MNRAS, 214, 1 Yahil A., 1983, ApJ, 265, 1047 Yu C., Lou Y.-Q., 2005, MNRAS, 364, 1168 Yu C., et al., 2006, MNRAS, 370, 121 (astro-ph/0604261) Zel’dovich Ya. B., Novikov I. D., 1971, Stars and Relativ- ity – Relativistic Astrophysics, Vol. 1, The University of Chicago Press, Chicago c© 0000 RAS, MNRAS 000, 000–000 http://arxiv.org/abs/astro-ph/0612072 http://arxiv.org/abs/astro-ph/0604261 INTRODUCTION Formulation and Analysis

0704.0224

Are There Mach Cones in Heavy Ion Collisions? Three-Particle Correlations from STAR

November 4, 2018 14:16 WSPC/INSTRUCTION FILE QM2006PosterProceedings˙V4 International Journal of Modern Physics E c© World Scientific Publishing Company ARE THERE MACH CONES IN HEAVY ION COLLISIONS? THREE-PARTICLE CORRELATIONS FROM STAR JASON GLYNDWR ULERY FOR THE STAR COLLABORATION Department of Physics, Purdue University, 525 Northwestern Avenue West Lafayette, Indiana 47907, USA [email protected] Received (received date) Revised (revised date) We present results from STAR on 3-particle azimuthal correlations for a 3 < pT < 4 GeV/c trigger particle with two softer 1 < pT < 2 GeV/c particles. Results are shown for pp, d+Au and high statistics Au+Au collisions at sNN = 200GeV . We observe a 3-particle correlation in central Au+Au collisions which may indicate the presence of conical emission. In addition, the dependence of the observed signal angular position on the pT of the associated particles can be used to distinguish conical flow from simple QCD-Čerenkov radiation. An important aspect of the analysis is the subtraction of combinatorial backgrounds. Systematic uncertainties due to this subtraction and the flow harmonics v2 and v4 are investigated in detail. 1. Introduction Heavy ion collisions create a medium that may be the quark gluon plasma (QGP). This medium can be studied through jets and jet-correlations. Jets make a good probe because their properties can be calculated in the vacuum with perturba- tive quantum chromodynamics (pQCD). Two-particle jet-like azimuthal correla- tions have shown the away-side shape in central Au+Au collisions to be broadened with respect to pp and peripheral Au+Au collisions or even double humped 1,2 (see Fig. 1a). The away-side structure is consistent with many different physics mechanisms including: large angle gluon radiation 3,4, jets deflected by radial flow or preferential selection of particles due to path-length dependent energy loss, hy- drodynamic conical flow generated by Mach-cone shock waves 5,6, and Čerenkov radiation 7,8. Three-particle correlations can be used to differentiate conical flow and Čerenkov radiation, which have the characteristic of conical emission, from other mechanisms. In addition, the associated particle pT dependence of the coni- cal emission angle can be used to differentiate between hydrodynamic conical flow and simple Čerenkov radiation. http://arxiv.org/abs/0704.0224v1 November 4, 2018 14:16 WSPC/INSTRUCTION FILE QM2006PosterProceedings˙V4 2 Jason Glyndwr Ulery For the STAR Collaboration 2. Analysis Procedure The 3-particle correlation analysis method is rigorously described in 9. The results reported here are for charged trigger particles of 3 < pT < 4 GeV/c and two charged associated particles of 1 < pT < 2 GeV/c (except where otherwise noted). The data were all taken in the STAR time projection chamber for pp, d+Au and Au+Au collisions at sNN=200 GeV/c. Figure 1b shows the raw 3-particle azimuthal distribution in ∆φaT = φa − φT and ∆φbT = φb − φT where φT , φa, and φb are the azimuthal angles of the trigger particle and the two associated particles respectively. Combinatorial backgrounds must be removed to obtain the genuine 3-particle correlation signal. The analysis is performed by treating the events as composed of two components, particles that are jet-like correlated with the trigger particle and background particles. One source of background, the hard-soft background, results when one of the associated particles has a jet-like correlation with the trigger particle and the other is uncorrelated, except for the correlation due to flow. The background is constructed from the 2- particle jet-like correlation, Ĵ2, folded with the normalized 2-particle background, , Fig. 1a. The 2-particle background is constructed by mixing events with the flow modulation added in pairwise from the average v2 values from the measure- ments based on the reaction plane and 4-particle cumulant methods 1. For the v4 contribution we use the parameterization v4 = 1.15v from the data10. The back- ground is normalized (with scale factor α) to the signal within 0.8 < |∆φ| < 1.2 (zero yield at 1 radian or ZYA1). We shall refer to the hard-soft background as Ĵ2 ⊗ αBinc2 . φ-φ=φ∆ -1 0 1 2 3 4 5 -1 0 1 2 3 4 5 -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 (a) (b) (c) (d) Fig. 1. (color online) (a) Raw 2-particle correlation (points), background from mixed events with flow modulation added-in (solid) and scaled by ZYA1 (dashed), and background subtracted 2- particle correlation (insert). (b) Raw 3-particle correlation, (c) soft-soft background, βα2Binc and (d) hard-soft background + trigger flow, Ĵ2⊗αBinc2 + βα2B inc,TF . See text for detail. Plots are from ZDC-triggered 0-12% Au+Au collisions at sNN=200 GeV/c. Another source of background, the soft-soft background, results from correla- tions between the two associated particles which are independent of the trigger particle. This background is obtained from mixed events, where the trigger particle and the associated particles are from different events in the same centrality win- dow. We shall refer to the soft-soft background as Binc . Since the two associated November 4, 2018 14:16 WSPC/INSTRUCTION FILE QM2006PosterProceedings˙V4 ARE THERE MACH CONES IN HEAVY ION COLLISIONS? THREE-PARTICLE CORRELATIONS FROM STAR 3 particles are from the same event, this background contains all of the correlations between the two associated particles that are independent of the trigger particle, including correlations from minijets, other jets in the event, and flow. The flow between the two associated particles that is independent of the trigger particle was accounted for in the soft-soft term, but particles are also correlated with the trigger particle via flow. The trigger flow is added in triplet-wise from mixed events, where the trigger and associated particles are all from different events in the same centrality window. The v2 and v4 values are obtained the same way as for the 2-particle background. The number of triplets is determined from the inclusive events. We shall refer to the backgrounds from trigger flow as B inc,TF . The total background is then, Ĵ2 ⊗ αBinc2 + βα2(Binc3 + B inc,TF ). Both Binc and B inc,tf are scaled by βα2. The normalization α2 corrects for the multiplicity bias from requiring a trigger particle. The factor β accounts for the effect of non-poission multiplicity distributions and is obtained such that the number of associated pairs in the background subtracted jet-like three-particle correlation signal equals the square of the number of associated particles in the background subtracted jet-like two-particle correlation signal. Figure 1c and d show βα2Binc and Ĵ2 ⊗ αBinc2 + inc,TF , respectively. 3. Results -1 0 1 2 3 4 5-1 -0.02 -1 0 1 2 3 4 5-1 -0.02 -1 0 1 2 3 4 5-1 -0.05 -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 STAR Preliminary Fig. 2. (color online) Background subtracted 3-particle correlations for pp (top left), d+Au (top middle), and Au+Au 50-80% (top right), 30-50% (bottom left), 10-30% (bottom center), and ZDC triggered 0-12% (bottom right) collisions at sNN=200 GeV/c. Figure 2 shows background subtracted 3-particle jet-like correlation signals. The pp and d+Au results are similar. Peaks are clearly visible for the near-side, (0,0), the November 4, 2018 14:16 WSPC/INSTRUCTION FILE QM2006PosterProceedings˙V4 4 Jason Glyndwr Ulery For the STAR Collaboration away-side, (π,π) and the two cases of one particle on the near-side and the other on the away-side, (0,π) and (π,0). The away-side peak displays diagonal elongation that is consistent with kT broadening. The perpherial Au+Au results show additional on-diagonal elongation of the away-side peak which may be due to contribution from deflected jets. The additional on-diagonal broadening persists into the more central Au+Au collisions. In addition, the more central Au+Au collisions display an off-diagonal structure, at about π± 1.45 radians, that is consistent with conical emission. The structure increases in magnitude with centrality and is prominent in the high statistics top 12% central data provided by the on-line ZDC trigger. partN 0 1 2 3 4 5 6 0.5 Near 1.45)±πDeflected ( 1.45)±πCone ( STAR Preliminary partN 0 1 2 3 4 5 6 1.00)±πDeflected-Cone ( 1.30)±πDeflected-Cone ( 1.45)±πDeflected-Cone ( STAR Preliminary Fig. 3. (color online) (left) Average signals in 0.7 × 0.7 boxes at (0,0) (triangle), (π,π) (star), (π± 1.45,π± 1.45) (square), and (π± 1.45,π∓ 1.45) (circle). (right) Differences in average signals, between (π ± 1.45,π ± 1.45) and (π ± 1.45,π ∓ 1.45) (triangle), between (π ± 1.3,π ± 1.3) and (π± 1.3,π∓ 1.3) (square), and between (π± 1.0,π± 1.0) and (π± 1.0,π∓ 1.0) (circle). Solid error bars are statistical and shaded are systematic. Npart is the number of participants. The ZDC 0-12% points (open symbols) are shifted to the left for clarity. Figure 3 (left) shows the centrality dependence of the average signal strengths in different regions. The off-diagonal signals (circle) increase with centrality and significantly deviate from zero in central Au+Au collisions. The locations of the off-diagonal signals were determined from a double Gaussian fit to a strip projected to the off-diagonal, Fig. 4, and were found to be 1.45 radians from π. The differ- ences between on-diagonal signals, where both conical emission and deflected jets may contribute, and off-diagonal signals, where only conical emission contributes is shown in figure 3 (right). Since conical emission signals are expected to be of equal magnitude on-diagonal as off-diagonal, the difference may indicate the contribution from deflected jets. The difference decreases with distance from (π,π). The Mach cone emission angle is expected to be independent of the associated particle momentum 6, while the Čerenkov radiation model in Ref. 7 predicts an emission angle that is sharply decreasing with increasing associated particle mo- mentum. Figure 5 (left) shows the dependence the off-diagonal peak angle on asso- ciated particle pT . The angle is consistent with constant as a function of associated particle pT . Figure 5 (right) shows the centrality dependence the off-diagonal peak angle. November 4, 2018 14:16 WSPC/INSTRUCTION FILE QM2006PosterProceedings˙V4 ARE THERE MACH CONES IN HEAVY ION COLLISIONS? THREE-PARTICLE CORRELATIONS FROM STAR 5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.2 STAR Preliminary Fig. 4. (color online) Away-side projections of a strip of width 0.7 radians for (left) d+Au and (right) 0-12% ZDC Triggered Au+Au. Off-diagonal projection (solid) is (∆φ1 − ∆φ)/2 and on- diagonal projection (open) is (∆φ1 +∆φ)/2− π. Shaded bands are systematic errors. Assoc (GeV/c) 0 0.5 1 1.5 2 2.5 0.02±Au+Au 0-12 1.41 0.02±Au+Au 0-50 1.45 Statistical Systematic STAR Preliminary partN( 3 3.5 4 4.5 5 5.5 6 Au+Au 0-12% (shifted) Au+Au 30-50%, 10-30% and 0-10% 0.03±1.46 STAR Preliminary Fig. 5. (color online) Emission angles from double Gaussian fits. (left) Angle as a function of associated particle pT for Au+Au 0-12% ZDC triggered data (filled) and Au+Au 0-50% from minimum bias data (open). (right) Angle as a function of centrality for Au+Au 0-12% ZDC triggered data (circle) and Au+Au 30-50%, 10-30% and 0-10% from minimum bias data (square). The 0-12% point has been shifted for clarity. Corresponding off-diagonal peak values from fits to a constant are indicated in the legends. The dashed line is at π/2. Solid error bars are statistical and shaded are systematic. The angle is consistent with remaining constant as a function of centrality for mid- central and central Au+Au collisions. The solid line at 1.46 on the plot is from a fit to a constant. 4. Systematics The major sources of systematic error are the elliptic flow measurement and the normalization. The default v2 used is the average those measured by the reaction plane and 4-particle cumulant methods. We use the reaction plane and 4-particle cumulant v2 as the upper and lower bounds to estimate the systematic uncertainty of the v2 subtraction. Figure 6a and b show the background subtracted 3-particle correlation for reaction plane and 4-particle v2 respectively. The signal is robust with respect to this variation. The hard-soft background and trigger flow backgrounds individually vary a great deal with the change in elliptic flow but the variations cancel to first order in the sum. November 4, 2018 14:16 WSPC/INSTRUCTION FILE QM2006PosterProceedings˙V4 6 Jason Glyndwr Ulery For the STAR Collaboration To study the effect of the normalization the size of the normalization window was doubled to 0.6 < |∆φ| < 1.4. The signal is robust with respect to this change in normalization. Other sources of systematic error include the effect on the trigger particle flow from requiring a correlated particle (20% on trigger particle v2), un- certainty in the v4 parameterization, and multiplicity bias effects on the soft-soft background. The systematic errors shown in figures include all sources mentioned. -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 -1 0 1 2 3 4 5-1 (a) (b) (c) Fig. 6. (color online) 0-12% Au+Au ZDC triggerd data for different systematic checks: (a) reaction plane v2, (b) 4-particle cumulant v2, and (c) normalization region for α of 0.6 < |∆φ| < 1.4 5. Conclusion Three-particle azimuthal correlations have been studied for trigger particles of 3 < pT < 4 GeV/c and associated particles of 1 < pT < 2 GeV/c in pp, d+Au, and Au+Au collisions at sNN=200 GeV/c by STAR. This analysis treats events as the sum of two components, particles that are jet-like correlated with the trigger and background particles. On-diagonal broadening has been observed in pp and d+Au consistent with kT broadening. Additional on-diagonal broadening has been seen in Au+Au collisions possibly due to deflected jets. Off-diagonal peaks consistent with conical emission are present in central Au+Au collisions. To discriminate between Mach cone emission and simple Čerenkov gluon radiation, a study of the associated particle pT dependence was performed. No strong pT dependence of the angles on associated particle pT is observed. This result is consistent with Mach cone emission. References 1. J. Adams et al. (STAR Collaboration), Phys. Rev. Lett. 95, 152301 (2005). 2. S.S. Adler et al. (PHENIX Collaboration), Phys. Rev. Lett. 97, 052301 (2006). 3. I. Vitev, Phys. Lett. B 630, 78 (2005). 4. A.D. Polosa and C.A. Salgado, hep-ph/0607295. 5. H. Stoecker, Nucl. Phys. A750, 121 (2005). 6. J. Casalderrey-Solana, E. Shuryak and D. Teaney, J. Phys. Conf. Ser. 27, 23 (2005). 7. I.M. Dremin, Nucl. Phys. A767 233 (2006). 8. V. Koch, A. Majumder and X.-N. Wang, Phys. Rev. Lett. 96, 172302 (2006). 9. J. Ulery and F. Wang, nucl-ex/0609016. 10. J. Adams et al. (STAR Collaboration), Phys. Rev. C 72, 014904 (2004). http://arxiv.org/abs/hep-ph/0607295 http://arxiv.org/abs/nucl-ex/0609016 Introduction Analysis Procedure Results Systematics Conclusion

0704.0225

Exploring First Stars Era with GLAST

Microsoft Word - kashlinsky_band.doc Exploring First Stars Era with GLAST A. Kashlinsky1,2 and D. Band1,3,4 1 Observational Cosmology Lab, Goddard Space Flight Center, Greenbelt MD 20771, 2SSAI, 3UMBC, 4CRESST Abstract. Cosmic infrared background (CIB) includes emissions from objects inaccessible to current telescopic studies, such as the putative Population III, the first stars. Recently, strong direct evidence for significant CIB levels produced by the first stars came from CIB fluctuations discovered in deep Spitzer images. Such CIB levels should have left a unique absorption feature in the spectra of high-z GRBs and blazars as suggested in [4]. This is observable with GLAST sources at z>2 and measuring this absorption will give important information on energetics and constituents of the first stars era. Keywords: Cosmology - background radiation, cosmic – gamma-rays, bursts – gamma-rays, astronomical observations PACS: 98.80.-k,98.70.Vc, 98.70.Rz, 95.85.Pw Cosmic infrared background (CIB) is a repository of emission throughout the entire history of the Universe, including from epochs containing objects inaccessible to current telescopic studies (see [1] for review). One such epoch is when the first stars are thought to have been formed corresponding to z>10. If the first stars (commonly called Population III – Pop III) were massive, they should have produced significant near-IR (NIR) CIB which should be cut-off by Lyman absorption at < 1 µm [2]. CIB derived from DIRBE- and IRTS-based analyses suggest higher fluxes at ~1-5 µm than that obtained by integrating galaxy counts [1]. The net CIB flux produced by these observed populations saturates at mAB~20-21 with fainter galaxies contributing little to the total budget. The excess flux at the near-IR (NIR) is commonly referred to as the NIRBE. The NIRBE, if extragalactic, must thus originate in still fainter systems, likely located at very early cosmic times. It is important to emphasize that e.g. K-band galaxies at mAB~19 are only at z~1 (e.g. [3]) with ordinary galaxies at earlier times contributing very little CIB; any modeling must account for the CIB evolution as observed in galaxy counts. The entire NIRBE correspond ~30 nW/m2/sr at 1- 4 µm of and this can be accounted for if only ~2% of the baryons have been processed through Pop III by z~10 [4]. FIGURE 1. Right: Squares show net CIB in MJy/sr from integrating galaxy counts and the dashed line corresponds to the CIB in the absence of NIRBE. Comoving number density of these photons is also shown. Circles show the NIRBE levels as discussed in [1]. Left: HESS measured spectra for the two blazars [5] are shown with crosses. Circles show the spectra corrected for absorption due to CIB photons produced by Pop III at z>10 and with total amplitude normalized to the shown value of ∆NIRBE. Recent HESS observations of absorption in the spectra of two z~0.2 blazars were taken to suggest that most of the NIRBE is not extragalactic for otherwise the unabsorbed spectra of the blazars would have to be too steep [5]. The situation is not as straightforward because the analysis in [5] used a template CIB which extends to <1 µm and is unsuitable for modeling CIB from z>7-10 sources, such as Pop III, whose CIB component, with a Lyman cutoff around 1 µm, should be used in any proper modeling. With this the constraints become significantly less severe and substantial CIB is still allowed by the data. This is shown in Fig.2, where we reconstructed the unabsorbed HESS blazar spectra using a more appropriate modeling of the CIB from Pop III. The model for the Pop III CIB component assumed: 1) CIB with Iν∝ν -2, 2) CIB with a cutoff at the present wavelength of 1 µm, corresponding to the Lyman cutoff at z~10, 3) the amplitude normalized to the net NIRBE of amplitude ∆NIRBE shown in the figure, and 4) the contribution from ordinary galaxies is given by the observed values derived from deep galaxy counts. The right panels show the uncorrected spectra for various levels of NIRBE. Using the limit on the hardness parameter (Γ>-1.5) from [5], at least half of the claimed NIRBE levels can still be produced by Pop III (∆NIRBE<15 nW/m 2/sr). The most direct evidence for substantial energy release during the first stars’ era comes from the recent measurements of CIB fluctuations at 3.6-8 µm in deep Spitzer images, which remain after removing intervening ordinary galaxies to very faint levels [6,7]. The fluctuations are produced by populations that have significant clustering component (from clustering of the emitters), but only a small shot noise level (from sources occasionally entering the beam). The results indicate that 1) the CIB fluxes from cosmological populations producing these fluctuations are substantial with > 1-2 nW/m2/sr at 3.6 and 4.5 µm, and 2) these CIB fluctuations are produced by sources with individual fluxes of only <10-20 nJy, which likely places them at z>10 [8]. If early stars produced even a fraction of the NIRBE, they would provide a source of abundant photons at high z. The present-day value of Iν=1 MJy/sr corresponds to the comoving number density of photons per logarithmic energy interval of 4π/c IνhPlanck=0.6 cm -3; if these photons come from high z, their number density would increase as (1+z)3 at early times. These photons would also have higher (blue-shifted) energies in the past and would thus provide abundance of absorbers for sources of sufficiently energetic photons at high z via two-photon absorption. FIGURE 2. Left: shows the range of CIB photons which affects gamma-rays at given energy and the marked redshifts. Right: Optical depth vs γ-ray energy due to 2-photon absorption by the NIRBE photons for GRBs and other sources at marked redshifts. Left panel of Fig.2 shows the range of the CIB wavelengths where the two-photon absorption operates for different z; GLAST/LAT observations of GRB’s and blazars at z>1-2 should thus provide a test of emissions from the Pop III era. Fig. 2 shows the optical depth for sources at high z (e.g., GRBs) assuming the entire NIRBE originated at z>10 [4]. If so, then γ-rays with energies > 260(1+z)-2 Gev from sources at z>1-2 should be completely absorbed even if only a faction of the NIRBE originated from the first stars era. GLAST and the Pop III era parameters have thus “conspired” very fortunately to uncover the CIB produced during the latter: 1) there should be a sharp cutoff at the Lyman limit at z>10, so CIB coming from these epochs is cut-off below the present-day wavelength of ~1µm; and 2) the threshold on the two-photon absorption process is such that with the GLAST energy limit of 300 GeV such measurement is not sensitive to the CIB beyond ~ 5 µm. For our purposes GRBs are cosmological sources with smooth spectra that extend up to GeV energies and are ideal for detecting absorption in the intervening IGM. As discussed, a source at z>1-2 should have a spectral cutoff at Ec=260(1+z) -2 GeV due to significantly energetic Pop III era emission. While GRBs are known to emit up to 18 GeV [9] and perhaps to TeV energies [10], the >100 MeV spectrum that the LAT will detect must be extrapolated from a few detections by the CGRO/EGRET. At the lower energies (10 keV to 1 MeV), where many bursts have been detected and characterized, the GRB spectrum can be described by a smoothly broken power law (the ‘Band’ function [11]); in 6 bursts the EGRET observations [12] are consistent with an extrapolation of the simultaneously observed lower energy emission with a high energy spectral index of β ~ -2 (where N(E)∝Eβ). In a few bursts EGRET observed emission that lasted longer than the low energy emission (for GRB 940217 burst photons were observed up to 90 minutes after the 3 minute long lower energy emission [9]). Milagrito may have detected TeV emission from GRB 970717a [10]. In GRB 941017 EGRET detected an additional hard power law component simultaneous with the lower energy ‘Band’ function component [13]. Thus the EGRET observations demonstrate the existence of >1 GeV emission. In some cases the lower energy spectral components extrapolate to this energy band, but additional spectral and temporal components, expected on theoretical grounds [14,15] exist. Despite these uncertainties, we can estimate the number of LAT GRBs where absorption by Pop III era photons might be observed by extrapolating the <1 MeV (synchrotron) spectra up to >1 GeV for a GRB population based on the BATSE observations. The GRB rate as a function of intensity, the burst spectrum, and the LAT's effective area as a function of photon energy have been convolved to predict the LAT's burst detection rate (see [16] on the GRB rate per year). FIGURE 3. Distribution of z of Swift-detected GRBs (solid histogram) and the empirical function (dashes) used in calculations. GRBs originate over a wide redshift range, and therefore we can expect the cutoff resulting from the Pop III era photon field to range over a wide energy range. As a result of a broad intrinsic energy distribution and possibly evolution, GRB fluence correlates poorly with z, and consequently GRBs that are bright in the LAT will not necessarily originate at lower z than dim LAT bursts. The z-distribution of the bursts the LAT will detect is currently uncertain: this distribution can be calculated assuming GRBs are correlated with other astrophysical phenomena (e.g., the star formation rate), or can be estimated empirically. We use the observed distribution from the Swift mission (Fig. 3), which detects and rapidly localizes ~100 GRBs per year [17]. By convolving the LAT detection rate of bursts in which Ec can be observed, the relationship between Ec and z, and the z-probability distribution, we estimate that the LAT will detect ~7 GRBs/year with observable Pop III era absorption cutoffs. The stronger case for absorption by Pop III era photons will result from determining Ec and z for the same burst. The fraction of LAT-observed bursts for which redshifts will be determined is uncertain. The Swift mission is projected to last well into the GLAST era, and Swift should observe ~1/6 of the bursts the LAT detects; currently redshifts have been determined for ~1/3 of the Swift bursts. GLAST will rapidly localize bursts onboard and on the ground, permitting follow-up ground-based observations that might result in burst redshifts; however, the localization by GLAST's GBM (degrees) and LAT (tenths of a degree) detectors may be too large to result in many redshifts. In addition, various relations between burst characteristics have been proposed that could turn bursts into ‘standard candles.’ For example, a tight relationship between the burst’s peak luminosity, a measure of its duration, and the average photon energy has been found [18]. The two GLAST instruments will observe the burst lightcurve (from which the duration can be measured) and spectrum (from which the average photon energy can be measured). The predicted ratio between the observed peak flux and the peak luminosity gives an estimate of the redshift that is akin to a photometric redshift. Thus we estimate that enough bursts with LAT-determined cutoffs will have redshifts enabling robust determination of the Pop III era spectral absorption over the course of the GLAST mission. AK acknowledges support from the National Science Foundations grant AST-0406587. REFERENCES 1. A. Kashlinsky., Phys. Rep., 409, 361-438 (2005) 2. M. Santos, V. Bromm & M. Kamionkowski, MNRAS, 336, 1082-1092 (2002) 3. M. Cirasuolo et al. astro-ph/0609287 4. A. Kashlinsky, Ap.J.(Letters), 633, L5-L8 (2005) 5. F. Aharonian et al., Nature, 440, 1018-1021 (2006) 6. A. Kashlinsky, R. Arendt, J. Mather & H. Moseley, Nature, 438, 45-50 (2005) 7. A. Kashlinsky, R. Arendt, J. Mather & H. 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0704.0226

Correlated modulation between the redshifted Fe K alpha line and the continuum emission in NGC 3783

Astronomy & Astrophysics manuscript no. 6713 c© ESO 2018 October 31, 2018 Correlated modulation between the redshifted Fe Kα line and the continuum emission in NGC 3783 F. Tombesi1,2, B. De Marco3, K. Iwasawa4, M. Cappi1, M. Dadina1, G. Ponti1,2, G. Miniutti5, and G.G.C. Palumbo2 1 INAF-IASF Bologna, Via Gobetti 101, I-40129 Bologna, Italy 2 Dipartimento di Astronomia, Università degli Studi di Bologna, Via Ranzani 1, I-40127 Bologna, Italy 3 International School for Advanced Studies (SISSA), Via Beirut 2-4, I-34014 Trieste, Italy 4 Max-Planck-Institut fur Extraterrestrische Physik, Giessenbachstrasse, D-85748 Garching, Germany 5 Institute of Astronomy, Madingley Road, Cambridge CB3 OHA, United Kingdom Received 8 November 2006 / Accepted 20 March 2007 ABSTRACT Aims. It has been suggested that X-ray observations of rapidly variable Seyfert galaxies may hold the key to probe the gas orbital motions in the innermost regions of accretion discs around black holes and, thus, trace flow patterns under the effect of the hole strong gravitational field. Methods. We explore this possibility by re-analyzing the multiple XMM-Newton observations of the Seyfert 1 galaxy NGC 3783. A detailed time-resolved spectral analysis is performed down to the shortest possible time-scales (few ks) using “excess maps” and cross-correlating light curves in different energy bands. Results. In addition to a constant core of the Fe Kα line, we detected a variable and redshifted Fe Kα emission feature between 5.3–6.1 keV. The line exhibits a modulation on a time-scale of ∼27 ks that is similar to and in phase with a modulation of the 0.3–10 keV source continuum. The two components show a good correlation. Conclusions. The time-scale of the correlated variability of the redshifted Fe line and continuum agrees with the local dynamical time-scale of the accretion disc at ∼10 rg around a black hole with the optical reverberation mass ∼10 7M⊙. Given the shape of the redshifted line emission and the overall X-ray variability pattern, the line is likely to arise from the relativistic region near the black hole, although the source of the few cycles of coherent variation remains unclear. Key words. Line: profiles – Relativity – Galaxies: active – X-rays: galaxies – Galaxies: individual: NGC 3783 1. Introduction The fluorescent iron K (Fe Kα) emission line is considered to be a useful probe of the accretion flow around the central black hole of an active galaxy. In particular, due to the high orbital velocity and strong gravitational field in the innermost regions of an accretion disc, its profile shall be deformed by the con- currence of Doppler and relativistic shifts. The resulting line is therefore broadened, with a red-wing extending towards lower energies (e.g. Fabian et al. 2000; Reynolds & Nowak 2003). Detailed modelling of time-averaged spectra have been used to obtain important estimates of the disc ionization state, its cover- ing factor and, at least for the brightest and best cases, its emis- sivity law, inner radius and BH spin (Brenneman & Reynolds 2006; Miniutti et al. 2006; Guainazzi et al. 2006; Nandra et al. 2006). It is also well established that time-resolved spec- tral analysis is a fundamental tool if we want to understand not only the geometry and kinematics of the inner accretion flow but also its dynamics. Early attempts (i.e. Iwasawa et al. Send offprint requests to: F. Tombesi e-mail: [email protected] 1996; Vaughan & Edelson 2001; Ponti et al. 2004; Miniutti et al. 2004) have clearly shown that the redshifted component of the Fe Kα line is indeed variable and that complex geometrical and relativistic effects should be taken into account (Miniutti et al. 2004). More recently, results of iron line variability have been reported (Iwasawa, Miniutti & Fabian 2004; Turner et al. 2006; Miller et al. 2006). These are consistent with theoretical studies on the dynamical behaviour of the iron emission arising from local- ized hot spots on the surface of an accretion disc (e.g. Dovčiak et al. 2004). Iwasawa, Miniutti & Fabian (2004), for example, measured a ∼25 ks modulation in the redshifted Fe Kα line flux in NGC 3516, which suggests that the emitting region is very close to the central black hole. However, these are likely to be transient phenomena, since such spots are not expected to sur- vive more than a few orbital revolutions. For this reason, it is inherently difficult to establish the observational robustness of these type of models, if not by accumulating further observa- tional data. Here we present results on the iron line variability in the bright Seyfert galaxy NGC 3783 (z≃0.01) based on XMM- http://arxiv.org/abs/0704.0226v1 2 F. Tombesi et al.: Correlated modulation between the redshifted Fe Kα line and the continuum emission in NGC 3783 Fig. 1. The X-ray light curves of NGC 3783 in the 0.3–10 keV band. Left panel: light curve of the 2000 observation. Right panel: light curves of the 2001a and 2001b observations. Newton data. This object has been taken as an example in which multiple warm absorbers can mimic the broad iron line feature (Reeves et al. 2004), contrary to the initial claim of the presence of a broad iron line emission using ASCA data (Nandra et al. 1997). However, the recent study by De Marco et al. (2006) found evidence for a transient excess feature in the 5–6 keV energy band, interpreted as a redshifted component of the Fe K line. This result is also supported by a variability study by O’Neill & Nandra (2006), who examined rms vari- ability spectra of a sample of bright active galaxies observed with XMM-Newton. Given the above considerations, we re- examined all the XMM-Newton observations of NGC 3783 to perform a comprehensive study of the iron line temporal evo- lution, on the shortest possible time-scale. 2. XMM-Newton observations XMM-Newton observed NGC 3783 on 2000 December 28– 29 and on 2001 December 17–21. The first observation (ID 0112210101) has a duration of ∼40 ks while the second (ID 0112210201 and ID 0112210501, hereafter observation 2001a and 2001b respectively) lasts over two complete orbits for a total duration of ∼270 ks. Only the EPIC pn data are used in the following analysis because of the high sensitivity in the Fe K band. The EPIC pn camera was operated in the “Small Window” mode with the Medium filter both during the 2000 and the 2001 observations. The live time fraction is thus 0.7. The data were reduced using the XMM-SAS v. 6.5.0 software while the analysis was carried on using the lheasoft v. 5.0 pack- age. High background time intervals were excluded from the analysis. The useful exposure time intervals are listed in Tab. 1, together with the mean 0.3–10 keV count rate for each obser- vation. Only single and double events were selected. Source photons were collected from a circular region of 56 arcsec ra- dius, while the background data were extracted from rectangu- Table 1. Date, duration, useful exposure and mean EPIC pn 0.3–10 keV count rate for each XMM-Newton observation of NGC 3783. Obs. ID Date Duration Exposure 〈CR〉 (ks) (ks) (c/s) 0112210101 2000 Dec 28–29 40.412 35 8.5 0112210201 2001 Dec 17–19 137.818 115 6.5 0112210501 2001 Dec 19–21 137.815 120 8.5 lar, nearly source-free regions on the detector. The background is assumed to be constant throughout the useful exposure. The 0.3–10 keV light curves are shown in Fig. 1 for each observa- tion. 3. Data analysis 3.1. Spectral features of interest and selection of the energy resolution The time-averaged spectrum was analyzed using the XSPEC v. 11.2 software package. For simplicity, we limited the anal- ysis to the 4–9 keV band. In this energy band, we checked that the complex and highly ionized warm absorber (with logξ and NH up to ∼2.9 erg cm s −1 and 5×1022 cm−2, Reeves et al. 2004) shall not affect our conclusions below. The resid- uals against a simple power-law plus cold absorption contin- uum model for the 2001b observation, the longest continuous dataset available, are shown in Fig. 2. In this fit we excluded the Fe K energy band (i.e. 5–7 keV) and the best fit parameters are (2.5 ± 0.6) × 1022 cm−2 and 1.81 ± 0.04, for the absorber column density and power-law slope respectively. Identified are four excess emission features: the main Fe Kα core at ∼6.4 keV, a wing to the line core at around 6 keV, a peak at ∼7 keV (possibly Fe Kβ) and a narrow peak at F. Tombesi et al.: Correlated modulation between the redshifted Fe Kα line and the continuum emission in NGC 3783 3 Fig. 2. The 4–9 keV residuals against a simple power-law plus cold absorption continuum model for the spectrum of NGC 3783 during the 2001b observation. The data are obtained from EPIC pn. ∼5.4 keV. Moreover we identified two absorption features at ∼6.7 keV and ∼7.6 keV. When fitted with Gaussian emission and absorption lines, all these features are significant at more than ∼99% confidence level. Similar results were also obtained by a detailed analysis with more complex models (Reeves et al. 2004). We will focus here on the analysis of the features vari- ability properties. The application of the excess map technique to the identified absorption features did not give significant re- sults, thus, in the following, we will focus on the analysis of emission features variability only. The 2001a observation has been divided into two parts because of the gap in the data be- tween t∼5×104 s and t∼6×104 s. Since all the selected spec- tral features are comparable to the CCD spectral resolution, we chose 100 eV for the energy resolution of the excess maps. 3.2. Selection of the time resolution In choosing the time resolution for the excess maps we looked for the best trade-off between getting a sufficiently short time- scale, in order to oversample variability, and keeping enough counts in each energy resolution bin. We first considered the 2001b observation, having the longest and continuous expo- sure. Spectra were extracted during different time intervals (1 ks, 2.5 ks and 5 ks) around the local minimum flux state at t∼215 ks. The required condition is that each 100 eV energy bin in the 4–9 keV band has to contain at least 50 counts. At the time resolution of 2.5 ks we got ∼90 counts per energy bin at the energies of the “red” feature (5.3–5.4 keV), and ∼80 counts per energy bin in the “wing” feature energy band (5.8– 6.1 keV). Moreover, for a 107 M⊙ black hole we expect the Keplerian orbital period to be ∼104 s at a radius of 10 rg. Thus, selecting 2.5 ks as the excess maps time resolution, enables us to completely oversample this typical time-scale. This choice of time resolution, optimized for the 2001b observation, was extended to the 2000 and 2001a data. 4. Excess emission maps Energy spectra for a duration of the chosen exposure time (2.5 ks) are extracted in time sequence. 14 spectra are obtained Table 2. Spectral features of interest in the 4–9 keV band with the selected band-passes and mean intensity. Feature Energy band 〈I〉 (keV) (10−5 ph s−1cm−2) red 5.3–5.4 0.6 wing 5.8–6.1 2 core (Kα) 6.2–6.5 5.3 Kβ 6.8–7.1 1.2 Red+Wing 5.3–6.1 3.2 from the 2000 observation, 46 and 48 spectra are obtained from 2001a and 2001b observations respectively. For each spectrum the continuum is determined and subtracted. The residuals in counts unit are corrected for the detector response and put together in time sequence to construct an image in the time- energy plane. 4.1. Continuum subtraction The continuum model is assumed to be always a simple ab- sorbed power-law, throughout all the observations. For each spectrum the energy band of the observed spectral features (i.e. 5–7 keV) is excluded during the continuum fit. The 4–5 keV and 7–9 keV data are rebinned so that each channel contains more than 50 counts to enable the use of the χ2 minimization process when performing spectral fitting and to ensure that the high energy end of the data (7–9 keV) have enough statistical weight. Because of the chosen low energy bound (4 keV), the fit is not sensitive to cold absorption. Thus the cold absorption column density is fixed to the time-averaged spectrum value (NH ≃ 2.5× 10 22 cm−2). Each 4–9 keV spectrum at 100 eV en- ergy resolution is then fitted with its best-fit continuum model and residuals are used to construct the excess emission map in the time-energy plane. Once all the continuum spectral fits have been done, we checked if continuum changes could affect our measurements of the line fluxes. The mean power-law slopes during the three observations are 1.85, 1.75 and 1.79 respectively, with stan- dard deviations of 0.08, 0.1 and 0.09. The power-law slopes are indeed quite constant, consistent with values obtained from the mean spectrum (§3.1), which result in a very marginal ef- fect (< 0.1%) in the flux measurement of the narrow features we found here. These power-law continuum slopes are also in agreement with those previously found in observations us- ing other instruments with overlapping spectral coverage, like BeppoSAX (De Rosa et al. 2002), ASCA, RXTE and Chandra (Kaspi et al. 2001). 4.2. Image smoothing As discussed in Iwasawa, Miniutti & Fabian (2004), if the data are acquired continuously and the characteristic time-scale of any variation in a feature of interest is longer than the sam- pling time (i.e. the time resolution), it is possible to suppress random noise between neighboring pixels by applying a low- pass filter. A circular Gaussian filter is used with σ=0.85 pixel 4 F. Tombesi et al.: Correlated modulation between the redshifted Fe Kα line and the continuum emission in NGC 3783 Time (ks) Energy (keV) 0 0.05 Excess (ph/2500s/cm^2) 0 50 100 150 200 250 Time (ks) Energy (keV) 0 0.02 0.04 0.06 Excess (ph/2500s/cm^2) Fig. 3. The excess emission maps of the 4–9 keV band in the time-energy plane at 2.5 ks time resolution. The images have been smoothed. Since the 6.4 keV line core is very strong and stable, the color map is adjusted to saturate the line core and allow lower surface brightness features to be visible. Left panel: excess emission map from the 2000 observation. Right panel: excess emission map from the 2001a and 2001b observations. (200 eV in energy and 5 ks in time, FWHM). The excess map Gaussian-filtered images for each observation are shown in Fig. 3. Systematic variations are observed in the 5.3–5.4 keV and in the 5.8–6.1 keV energy bands of the 2001b observation. However, the image filtering can slightly smear these narrow features and reduce their intensity. 5. Results 5.1. Light curves of the individual spectral features Light curves of the four emission features are extracted from the excess map filtered images. The selected band-passes are listed in Tab. 2. During the image filtering process individ- ual pixels lose their independence to the neighbouring ones. This means that a simple counting statistics may be inappro- priate for estimating the features light curves errors. For this reason the estimation of the errors has been done by exten- sive Monte Carlo simulations. We implemented 1000 simula- tions following the same procedure in making the excess map images. In the simulations all the spectral features parameters and the power-law slope are assumed to be constant, while let- ting the power-law normalization vary according to the 0.3–10 keV light curve. Light curves of individual spectral features have been extracted from each simulation and their mean val- ues and variances recorded. The square root of the mean of the variances (i.e. the dispersion) has been regarded as the light curves error. In Fig. 4 the emission features light curves for the 2001a and 2001b observations are shown. The 2000 observa- tion light curves are not reported because they do not show any sign of variability. The most intense variations are registered in the light curves of the “red” (E=5.3–5.4 keV) and “wing” (E=5.8–6.1 keV) features during the 2001b observation. The observed peaks seem to follow the same kind of variability pattern and, as shown in more details below, appear to be in phase with the continuum emission. In order to check the sig- nificance of the observed variability we extracted both real and simulated data light curves in the entire 5.3–6.1 keV band, i.e. of the “red+wing” structure. Then we compared the χ2 values against a constant hypothesis for the real data and the 1000 simulations; equivalent results can be derived comparing the variances directly. Only 73 of the simulations show variability at the same level or greater than the real data, therefore we get a variability confidence level of 93%. The light curves of the excess emission features in the 5–6 keV energy band (red, wing and red+wing) seem to show a variabil- ity pattern with a recurrence of the flux peaks on time-scales of 27 ks. We further investigated how it is likely to occur by chance applying a method that makes use of the 1000 Monte Carlo simulations. We folded the real data light curve with the interval of 27 ks and we fitted it with a constant. We obtained a χ2r = 88 for 19 degrees of freedom. We did the same to the simu- lated red+wing light curves but, this time, folding in n = 9 trial periods, from 17 to 37 ks at intervals of 2.5 ks, and recorded the χ2i values. If N is the total number of simulated red+wing light curves for which χ2i ≥ χ r , the confidence level can be de- F. Tombesi et al.: Correlated modulation between the redshifted Fe Kα line and the continuum emission in NGC 3783 5 /s Continuum 8 Core 0 105 2×105 Time (s) Kbeta Fig. 4. The light curves of the total 0.3–10 keV continuum flux and of the four spectral features (Tab. 2) extracted from the excess maps of the 2001a and 2001b observations (Fig. 3, Right panel), with errors computed from simulations. The time resolution is 2.5 ks. rived as (1 − N1000·n ). Only N = 54 of the simulated red+wing light curves folded at the trial periods show chi-square values greater than the real one. Therefore, we could derive a confi- dence level for the recurrence pattern on the 27 ks time scale of 99.4%. Finally, we checked that the continuum above 7 keV (which carries the photons eventually responsible for the Fe line pro- duction) varies following the same pattern of the 0.3–10 keV band (Fig. 4, Top panel). 5.2. Correlation with the continuum light curve In the 0.3–10 keV light curve of observation 2001b (Fig. 5, Upper panel) flux variations of ∼30% are visible with four peaks separated by approximately equal time intervals. Given such peculiar time series shape, we focused on this observa- tion and searched for some typical time-scale in the variability pattern. Thus, we applied the efsearch task (in Xronos), which searches for periodicities in a time series calculating the maxi- mum chi-square of the folded light curve over a range of peri- ods. We found a typical time-scale for variability of 26.6±2.2 ks. We then removed the underlying long-term variability trend by subtracting a 4th degree polynomial to the 0.3–10 keV con- tinuum light curve (see Fig. 5, Middle panel). The polynomial has been determined using the lcurve task (in Xronos), which makes use of the least-square technique. Applying again the efsearch task we found a typical time-scale for short-term vari- ability of ∼27.4 ks1. The peaks observed in the continuum light curve seem to appear at the same times at which those ob- served in the “wing” and “red” light curves do. In order to look 1 It should be noted that the variability PSD study of this XMM- Newton dataset by Markowitz (2005) suggests an excess of power around 4×10−5 Hz (corresponding to about 25 ks) during the 2001b observation (square symbols in his Fig. 3). 0.3−10 keV 0.3−10 keV 0 5×104 105 Time (s) Red+Wing Fig. 5. Upper panel: The 0.3–10 keV light curve of NGC 3783 during 2001b observation at 2.5 ks time resolution. Middle panel: The 0.3–10 keV light curve of NGC 3783 during the 2001b observation after subtraction of a 4th degree polynomial (long-term variations) at 2.5 ks time resolution. Lower panel: The 5.3–6.1 keV (“red+wing” energy band) light curve ex- tracted from the excess emission map of the 2001b observation (Fig. 3, Right panel) at 2.5 ks time resolution. for some correlation between the continuum and the 5.3–6.1 keV (“red+wing”) feature flux we computed the cross corre- lation function (CCF) between the two time series, where the input continuum light curve is the “de-trended” one. It is re- ported in Fig. 6 as a function of time delay, measured with respect to the continuum flux variations. No delay is evident, with an estimated error at the peak of 2.5 ks. The continuum and “red+wing” fluxes seem to show a correlation, with peak value 0.7. To estimate the significance of the correlation we computed the CCFs between the continuum and the simulated “red+wing” light curves. If N is the number of simulated light curves which have a higher cross correlation than the real one, the significance of the correlation is (1 − N/1000). Applying this method we found a confidence level greater than 99.9%. 5.3. High/Low flux state line profiles Looking at the “red+wing” light curve (Fig. 5, Lower panel) we constructed two spectra from the integrated high and low flux intervals to verify the variability in this energy band. The line profiles are shown in Fig. 7 where the ratio between the data and a simple power-law plus cold absorption model is shown. While the 6.4 keV and 6.9 keV features remain the same, a small increase of counts is visible in the 5.3–5.4 keV and 5.8–6.1 keV bands in the high flux state. Adding an emis- sion Gaussian model to the simple power law plus Gaussian line (at the Fe Kα energy) model in the high flux state inte- grated spectra improves the χ2 of 18. Thus the significance of the excess is ∼99.9%. 6 F. Tombesi et al.: Correlated modulation between the redshifted Fe Kα line and the continuum emission in NGC 3783 −104 0 104− Time Delay (s) Fig. 6. The cross correlation function calculated between the de-trended 0.3–10 keV continuum light curve and the 5.3–6.1 keV feature (“red+wing”) light curve. .4 HIGH 6 7 8 Energy (keV) Fig. 7. The Fe K line profile during the High flux (open squares) and the Low flux (solid circles) phases of the 5.3–6.1 keV feature. The ratios are computed against the best-fitting continuum model. The energies are in the observer frame. 6. Discussion In the 2001b observation, NGC 3783 clearly exhibits contin- uum emission with two different time-scales: a long-term mod- ulation with variations up to a factor 2 on intervals greater than 60 ks is superimposed to a shorter one on a time-scale of 27 ks, with modulations of 30% of the average value. According to an estimated black hole mass of MBH = (3.0±0.5)×10 (Peterson et al. 2004), the latter modulation occurs with a characteristic time-scale corresponding to the orbital period at ∼9–10 rg (e.g. Bardeen, Press & Teukolsky (1972)). It should be noted, however, that a previous mass estimate by Onken & Peterson (2002) gave the value of MBH = (8.7± 1.1)× 10 6 M⊙, that would correspond to ∼20 rg. These mass discrepancies are mainly due to a different scaling of the virial relation in per- forming reverberation mapping measurements. However, the former estimate is more accurate because it has been calibrated to the MBH −σ relation and thus we will adopt this value in the following discussion. The power spectral density of the source is consistent with a red-noise shape (Markowitz 2005) where variability mainly occurs on intervals of the order of days. Here we identify an additional (additive) shorter time-scale compo- nent (see the footnote 1), most likely produced within the in- nermost accretion flow/corona system. The most remarkable result of our analysis is the detec- tion of redshifted (5.3–6.1 keV) Fe K emission and of its vari- ability. The redshifted emission appears to respond only to the shorter ∼27 ks time-scale modulation and shows a good cor- relation with the continuum with a time-lag consistent with zero within the errors (∆τ ∼ 1.25 ks). This indicates that the continuum modulation on this time interval is likely to induce Fe K emission from dense material close to the black hole (which explains the observed redshift of the emission feature); moreover the lack of time-lags implies that the distance be- tween the sites of continuum and line production is smaller than c∆τ ∼ 4×1013 cm ∼ 8 rg (for the black hole mass given above). We are therefore most likely observing emission from the in- nermost accretion flow in both the continuum and line emission (corona and disc). As discussed above, the variability time-scale suggests we are looking at emission from around ∼9–10 rg. As a consis- tency check, we fitted the time-averaged spectrum by including a diskline component to account for the redshifted features. We forced the emission region to be an annulus of ∆r= 0.5 rg with uniform emissivity because the purpose of this test is to as- sess the approximate location of the line-emitting region. We obtain good fits with an almost face-on disc (i = 11 ± 4◦) and an annulus at 9–15 rg depending on the assumed Fe line rest-frame energy (from neutral at 6.4 keV to highly ionized at 6.97 keV). For all cases we tested, the statistical improve- ment is of ∆χ2 ∼ 16 for 3 additional degrees of freedom, cor- responding to a confidence level of 99.7%. This fit shows that the redshifted Fe line emission we detected is indeed consistent in shape with being produced around 10 rg, where the disc or- bital period is of the order of 27 ks, which agrees well with the correlated (and zero-lag) variability of the two components. The interpretation of our results is however not straightfor- ward. The quasi-sinusoidal modulations of the continuum and line emission (see Fig. 5) would suggest the presence of a local- ized co-rotating flare above the accretion disc which irradiates a small spot on its surface. The intensity modulations we see (Fig. 5) would then be produced by Doppler beaming effects (acting on both the flare and spot emission, i.e. on both contin- uum and line) and the characteristic time-scale of 27 ks would be identified with the orbital period (because of gravitational time dilation, the period measured by an observer on the disc at 10 rg would be shorter by ∼10%). As demonstrated above, a flare/spot system orbiting the black hole at ∼10 rg would also produce a time-averaged line profile in agreement with the ob- servations. However, such a model makes definite prediction on the Fe line energy modulation within one orbital period. In this framework, the orbiting spot on the accretion disc would also give rise to energy modulations of the Fe line due to the Doppler effect and such energy modulation is barely seen in the data (see Fig. 3). We stress that the adopted time resolu- tion (2.5 ks) is good enough to detect energy modulations with a characteristic 27 ks time-scale. This has been demonstrated with XMM-Newton in the case of NGC 3516, where the modu- lation occurs on a very similar time-scale (Iwasawa et al. 2004; F. Tombesi et al.: Correlated modulation between the redshifted Fe Kα line and the continuum emission in NGC 3783 7 see also Dovčiak et al. 2004 for theoretical models). Therefore, the lack of Fe line energy modulation disfavours the orbiting flare/spot interpretation for NGC 3783. However, a variability time-scale of the order of the orbital one at a given radius does not necessarily imply the motion of a point-like X-ray source. In fact, since the orbital time-scale is the fastest at a given disc radius, and since the observed time- scale of ∼27 ks corresponds to the orbital period at ∼10 rg, one could argue that the data only imply that the X-ray variability likely originates from within ∼10 rg. The apparent recurrence in the X-ray continuum modulation may not necessarily be re- lated to a real physical periodicity, especially considering the limited length of the observation (only four putative cycles are detected). A possible explanation for the observed behaviour is that the X-ray continuum source(s) (located within ∼10 rg from the center) irradiates the whole accretion disc, but only a ring- like structure around 10 rg is responsible for the fluorescent Fe emission. This is possible if the bulk of the accretion disc is so highly ionized that little Fe line is produced, while an over- dense (and therefore lower ionization) structure is responsible for the fluorescent emission. Such an over-dense region could have an approximate ring-like geometry if it is for instance as- sociated with a spiral-wave density perturbation. In this case, the Fe emitting region is extended in the azimuthal direction and we do not expect strong energy modulation with time, whatever the origin of the continuum modulation. We point out that spiral density distributions could result from the ordered magnetic fields in the inner region of the disc and that the en- ergy dissipation (via e.g. magnetic reconnection) could be en- hanced there, thereby providing a common site for the produc- tion of the X-ray continuum and the Fe line (e.g. Machida & Matsumoto 2003). On the other hand, if the apparent recurrence is in fact real, it is worth noting that the continuous theoretical effort in understanding the origin of quasi-periodic oscillations (QPO) in neutron star and black hole systems provides a wealth of mechanisms inducing quasi-periodic variability, although none is firmly established (Psaltis 2001; Kato 2001; Rezzolla et al. 2003; Lee et al. 2004; Zycki & Sobolewska 2005 and many others). A connection between QPO phase and Fe line inten- sity has been previously claimed in the Galactic black hole GRS 1915+105 (Miller & Homan 2005). Although in our case, the presence of a QPO cannot be claimed because of the very small number of detected cycles, the analogy is suggestive. In the case of GRS 1915+105, Miller & Homan consider that a warp in the inner disc, possibly due to Lense-Thirring preces- sion, may produce the observed QPO-Fe line connection (e.g. Markovic & Lamb 1998). However, to produce the observed ∼15% rms variability in the X-ray lightcurve, the black hole spin axis should be inclined with respect to the line of sight by at least 60◦ (which is at odds with our inclination estimate of 11±4◦ and with the Seyfert 1 nature of NGC 3783), and the tilt precession angle should be larger than 20◦–30◦ (Schnittman, Homan, Miller 2006). Both requirements make it highly un- likely that Lense-Thirring precession can successfully account for the observed modulations in NGC 3783. While the origin of the coherent intensity modulation still remains unclear, the correlated variation of the continuum and line emission and the Fe line shape are consistent with an emis- sion site at ∼10 rg. Moreover, the fact that the iron line variabil- ity responds to the 27 ks time-scale modulation only implies that this short time-scale variation is somehow detached from the long-term variability. The latter may be associated with per- turbations in the accretion disc propagating inwards from outer radii and modulating the X-ray emitting region (Lyubarskii 1997), while the former seems to genuinely originate in the inner disc. Further observational data may help to clarify the complex phenomena related to the relativistic Fe line temporal evolu- tion in Seyfert 1 galaxies. Our work makes it clear that higher quality data in the Fe band will be able to probe the inner- most regions of accretion flows with high accuracy. Next gen- eration of large collecting area X-ray missions such as XEUS and Constellation-X or even very long observations with XMM- Newton will be crucial to fully exploit such potential. Acknowledgements. This paper is based on observations obtained with the XMM-Newton satellite, an ESA funded mission with con- tributions by ESA Member States and USA. We thank A. Müller, K. Nandra, L. Nicastro, P. O’Neill and M. Orlandini for useful discus- sions. MC, MD and GP acknowledge financial support from ASI un- der contract ASI/INAF I/023/05/0. The authors thank the anonymous referee for suggestions that led to improvements in the paper. References Bardeen, J. M., Press, W. H., & Teukolsky, S. A. 1972, ApJ, 178, 347 Brenneman, L. W., & Reynolds, C. S. 2006, ArXiv Astrophysics e- prints, arXiv:astro-ph/0608502 De Rosa, A., Piro, L., Fiore, F. et al. 2002, A&A, 387, 838 De Marco, B., Cappi, M., Dadina, M., & Palumbo, G.G.C., 2006, Astron. Nach., astro-ph/0610882 Dovčiak, M., Bianchi, S., Guainazzi, M., Karas, V., & Matt, G. 2004, MNRAS, 350, 745 Fabian, A. C., Iwasawa, K., Reynolds, C. S., & Young, A. J. 2000, PASP, 112, 1145 Guainazzi, M., Bianchi, S., & Dovciak, M. 2006, ArXiv Astrophysics e-prints, arXiv:astro-ph/0610151 Iwasawa, K., et al. 1996, MNRAS, 282, 1038 Iwasawa, K., Miniutti, G., & Fabian, A. 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J. 2006, ArXiv Astrophysics e-prints, arXiv:astro-ph/0610585 O’Neill, P. & Nandra, K., 2006, in preparation http://arxiv.org/abs/astro-ph/0608502 http://arxiv.org/abs/astro-ph/0610882 http://arxiv.org/abs/astro-ph/0610151 http://arxiv.org/abs/astro-ph/0609521 http://arxiv.org/abs/astro-ph/0610585 8 F. Tombesi et al.: Correlated modulation between the redshifted Fe Kα line and the continuum emission in NGC 3783 Onken, C. A., & Peterson, B. M. 2002, ApJ, 572, 746 Peterson, B. M., et al. 2004, ApJ, 613, 682 Ponti, G., Cappi, M., Dadina, M., & Malaguti, G. 2004, A&A, 417, Psaltis D., 2001, Adv Space Res., 28, 481 Reeves, J. N., Nandra, K., George, I. M. et al. 2004, ApJ, 602, 648 Reynolds, C. S., & Nowak, M. A. 2003, Phys. Rep., 377, 389 Rezzolla L., Yoshida S., Maccarone T.J., Zanotti O., 2003, MNRAS, 344, L37 Schnittman J.D., Homan J., Miller J.M., 2006, ApJ, 642, 420 Turner, T. J., Miller, L., George, I. M., & Reeves, J. N. 2006, A&A, 445, 59 Vaughan, S., & Edelson, R. 2001, ApJ, 548, 694 Zycki P.T. & Sobolewska M.A., 2005, MNRAS, 364, 891 Introduction XMM-Newton observations Data analysis Spectral features of interest and selection of the energy resolution Selection of the time resolution Excess emission maps Continuum subtraction Image smoothing Results Light curves of the individual spectral features Correlation with the continuum light curve High/Low flux state line profiles Discussion

0704.0227

Isotopic Effects in Nuclear Reactions at Relativistic Energies

Temperature and Density in Heavy Ion Reactions at Intermediate Energies ISOTOPIC EFFECTS IN NUCLEAR REACTIONS AT RELATIVISTIC ENERGIES C.Sfienti(1), M. De Napoli(2), S. Bianchin(1), A.S. Botvina(1)(6), J. Brzychczyk(4), A. Le Fèvre(1), J. Lukasik(1)(9), P. Pawlowski(4), W. Trautmann(1), P. Adrich(1), T. Aumann(1), C.O. Bacri(3), T. Barczyk(4), R. Bassini(5), C. Boiano(5), A. Boudard(7), A. Chbihi(8), J. Cibor(9), B. Czech(9), J.-E. Ducret(7), H. Emling(1), J. Frankland(8), M. Hellström(1), D. Henzlova(1), K. Kezzar(1), G. Immè(2), I. Iori(5), H. Johansson(1), A. Lafriakh(3), E. Le Gentil(7), Y. Leifels(1), W.G. Lynch(10), J. Lühning(1), U. Lynen(1), Z. Majka(4), M. Mocko(10), W.F.J.Müller(1), A. Mykulyak(11), H. Orth(1), A.N. Otte(1), R. Palit(1), A. Pullia(5), G. Raciti(2), E. Rapisarda(2), H. Sann(1)†, C. Schwarz(1), H. Simon(1), K. Sümmerer(1), C. Volant(7), M. Wallace(10), H. Weick(1), J. Wiechula(1), A. Wieloch(4) and B. Zwieglinski(11) The ALADiN2000 Collaboration (1) Gesellschaft für Schwerionenforschung, D-64291 Darmstadt, Germany (2) Dipartimento di Fisica dell'Università and LNS-INFN, I-95126 Catania, Italy (3) Institut de Physique Nuclèaire, IN2P3-CNRS et Universitè, F-91406 Orsay, France (4) M. Smoluchowski Institute of Physics, Jagiellonian Univ., Pl-30059 Krakòw, Poland (5) Istituto di Scienze Fisiche, Università degli Studi and INFN, I-20133 Milano, Italy (6) Inst. Nucl. Res., Russian Accademy of Science, Ru-117312 Moscow, Russia (7) DAPNIA/SPhN, CEA/Saclay, F-91191 Gif-sur-Yvette, France (8) GANIL, CEA et IN2P3-CNRS, F-14076 Caen, France (9) H. Niewodniczanski Institute of Nuclear Physics, Pl-31342 Krakòw,Poland (10) Department of Physics and Astronomy and NSCL, MSU, East Lansing, MI 48824, USA (11) A. Soltan Institute for Nuclear Studies,Pl-00681 Warsaw, Poland Abstract A systematic study of isotopic effects in the break-up of projectile spectators at relativistic energies has been performed at the GSI laboratory with the ALADiN spectrometer coupled to the LAND neutron detector. Besides a primary beam of 124Sn, also secondary beams of 124La and 107Sn produced at the FRS fragment separator have been used in order to extend the range of isotopic compositions. The gross properties of projectile fragmentation are very similar for all the studied systems but specific isotopic effects have been observed in both neutron and charged particle production. The breakup temperatures obtained from the double ratios of isotopic yields have been extracted and compared with the limiting-temperature expectation. 1. Introduction Challenging motivations for isotopic studies in nuclear multifragmentation are derived from the importance of the density dependence of the symmetry-energy term of the nuclear equation of state for astrophysical applications and for effects linked to the manifestation of the nuclear liquid-gas phase transition. Müller and Serot, in their seminal paper [1], have demonstrated that the two-fluid nature of nuclear matter has very specific consequences for the phase behavior in the coexistence region. Different isotopic compositions are predicted for the coexisting liquid and gas phases, with the gas being more neutron rich than the liquid in asymmetric (N ≠ Z) matter. This difference stems from the decrease in the symmetry energy in nuclear matter as the density is decreased. The expected magnitude of this density dependence, however, is model dependent and very poorly constrained by existing data [2]. The calculations of Müller and Serot are restricted to infinite matter with no Coulomb force included. In addition, the isotopic composition, in the calculations, is typically varied within a range of proton fractions ρp/ρ = 0.3 to 0.5 whose limits are not easily accessible in experiments with heavy nuclei. Theoretical studies for finite systems also indicate that the sequential decay of excited reaction products has a tendency to reduce some of the expected effects [3]. If the phase transition for asymmetric nuclei still manifests itself as a plateau in the caloric curve [5] (i.e. the correlation between the temperature of the system and its excitation energy), this could be influenced by the degree of asymmetry as well as by the mass of the system undergoing the fragmentation. Figure 1. Location of the four studied projectiles in the plane of atomic number Z versus neutron number N. The contour lines represent the limiting temperatures according to [4] , the dashed line gives the valley of stability, and the full line corresponds to the N/Z = 1.49 of 197Au. It has been shown [4] that, due to the Coulomb pressure, there exists a limiting temperature which represents the maximum temperature at which nuclei are found to exist as self-bound objects in Hartree-Fock calculations. The dependence of the breakup temperature on the excitation energy could therefore be governed by the limiting temperature. Figure 1 shows the calculated limiting temperature as a function of the neutron and proton number [4]. As it is possible to see from the picture, in case of proton- rich nuclei, the phenomenon of vanishing limiting temperature is predicted, the limiting temperature decreasing with increasing proton fraction. Clearly, new experiments exploring such phenomena are mandatory for having a better knowledge of the thermodynamics of a finite nucleus and its decay. Recently a systematic investigation of projectile-spectator fragmentation has been undertaken at the ALADiN spectrometer at the GSI [6,7]: four different projectiles, 124Sn, 197Au, 124La and 107Sn, all with an incident energy of 600 AMeV on 116Sn and 197Au targets, have been studied. The two latter beams have been delivered by the FRagment Separator (FRS) of the GSI as products of the fragmentation of a primary 142Nd beam at 890 AMeV on a 9Be production target [7]. The necessity of low beam intensities for the best operational condition of the ALADiN setup (≅2000 particles/sec), and the possibility of using a thick target in order to achieve high interaction rates are indeed conditions compatible with radioactive-ion-beam experiments. Moreover, the inverse kinematics offers the possibility of a threshold-free detection of all heavy fragments and residues and thus gives a unique access to the breakup dynamics. The measurement of the charge and the momentum vector of all projectile fragments with Z≥2 has been performed, with high efficiency and high resolution, with the TP-MUSIC IV detector [8]. Using the reconstructed values for the rigidity and pathlength, the charge of the particle measured by the TP-MUSIC IV, and the time-of-flight given by the TOF- Wall, the velocity and the momentum vector can be calculated for each detected charged particle. The knowledge of velocity and momentum allows then the calculation of the particle's mass. Neutrons emitted in directions close to θlab = 0 o , are detected with the Large-Area Neutron Detector (LAND) which covers about half of the solid angle required for neutrons from the spectator decay. 2. Gross Properties of multifragment decay The gross properties of projectile fragmentation are very similar for all the studied systems. The fragments emerging from the decay of the projectile spectators are well localised in rapidity. The distributions are peaked around a rapidity value very close to be beam rapidity and become narrower with increasing mass of the fragment. The observed independence of the Rise and Fall, i.e. of the correlation between the multplicity of intermediate mass fragments with Zbound, also for the unstable systems [6], confirms the hypothesis of equilibrium at freeze-out. The shape of the charge distributions are as well similar for all the systems. For larger values of Zbound, the charge distribution exhibits a U-shape. The heavy fragments are the residues of the lowly-excited projectile-spectators after the evaporation of light fragments and nucleons. In semi-central reactions, i.e. smaller values of Zbound, the distribution broadens and flattens over nearly the full charge distribution. For still lower values of Zbound, the charge distribution becomes steep. This is consistent with the system disassembling into predominantly lighter fragments. The charge distributions have been fitted with a power-law parameterization, σ(Z) ∝ Z-τ, in the charge range 3≤Z≤15. The power-law parameters τ (Figure 2 – left panel) allow to follow the transition from the U-shape to a pure exponential spectrum and reach the minimum value in the Zbound range which corresponds to the maximum production of IMF, i.e. in the multifragmentation region. They follow a nearly universal curve almost independent of the isotopic composition of the original spectator system. Figure 2. (Left Panel) The extracted τ parameters as a function of normalized Zbound for 124La, 124Sn and 107Sn at 600 AMeV, compared with earlier data for 197Au obtained at the same energy. (Right Panel) Neutron multiplicities as obtained from the LAND detector for all the studied systems (colored symbols are SMM predictions). Specific isotopic effects, even though small, can nevertheless be observed: in particular, the hierarchy of τ for the neutron-poor 124La and 107Sn and neutron-rich 124Sn and 197Au systems for Zbound/Zproj>0.5 is opposite to the standard predictions of the Statistical Multifragmentation Model SMM [9]. It can, however, be explained with a weak isotopic dependence of the surface-term coefficient in the liquid-drop description of the fragment masses at low excitation energy which gradually disappears with increasing excitation of the fragmenting system [10]. Specific isotopic effects are as well found in the reconstructed neutron multiplicities measured with the LAND detector [11] (Figure 2 - right panel). For peripheral collisions the values of the multiplicity depend on the number of available neutrons in the entrance channel. Going towards central collisions the number of emitted neutrons is progressively determined by the N/Z in the entrance channel. In a preliminary comparison with SMM calculation (colored points in Figure 2) a promising agreement has been observed. Neutrons will be important for establishing the mass and energy balance and in particular for calorimetry. In this respect it is crucial to identify the spectator neutrons and to distinguish them from the fireball ones. From a preliminary analysis of their rapidity distributions, the corresponding spectator sources have been identified. They are characterized by temperatures up to 4 MeV, possibly caused by large contributions from evaporation. 2. Structure and Memory Effects in particle production The mass resolution obtained for projectile fragments entering into the acceptance of the ALADiN spectrometer is about 3% for fragments with Z=3 and decreases to 1.5% for Z≥6 [7]. Masses are thus individually resolved for fragments with atomic number Z≤10. The elements are resolved over the full range of atomic numbers up to the projectile Z with a resolution of ΔZ≤0.2 obtained with the TP-MUSIC IV detector. The mean N/Z of the mass distributions of light fragments in the range 3 ≤Z ≤ 13 for two different Zbound cuts is presented in Figure 3. Figure 3. Mean values <N>/Z of light fragments with 3 ≤ Z ≤ 13 produced in the fragmentation of 124Sn and 124La at 600 A MeV for two different bins in Zbound. The values obtained for 124Sn are larger than those for 124La or 107Sn (not shown) as expected from the different N/Z of the original projectiles. Their odd-even variation is, however, much more strongly pronounced for the neutron-poor cases. The strongly bound α-type nuclei (even-even N=Z) attract a large fraction of the product yields during the secondary evaporation stage. This effect is, apparently, larger if already the hot fragments are close to N=Z symmetry, as it is expected for the fragmentation of 124La and 107Sn [12]. Inclusive data obtained with the FRS fragment separator at GSI for 238U [13] and 56Fe [14] fragmentations on titanium targets at 1 A GeV bombarding energy confirm that the observed patterns are very systematic, exhibiting at the same time nuclear structure effects characteristic for the isotopes produced and significant memory effects of the isotopic composition of the excited system by which they are emitted. This has the consequence that, because of its strong variation with Z, the neutron-to-proton ratio <N>/Z is not a useful observable for studying nuclear matter properties. For this purpose, techniques, such as the isoscaling [7], will have to be used which cause the nuclear structure effects to cancel out. A precise modeling of these secondary processes is, therefore, necessary for quantitative analyses. Another interesting feature of the mass distributions predicted by SMM is their dependence, in the case of the proton-rich systems, on the excitation energy of the system [10]. This difference arises from the dependence of the number of neutrons in light fragments on the Z spectrum. For the neutron-rich 124Sn system, on the other hand, the distributions should be independent of the excitation energy. From a first qualitative comparison with the model prediction of the mass distributions for different Zbound cuts, however, we have not observed any noticeable variation in the mean N/Zs for the 124La system (Figure 3). Also in this case it will be crucial to precisely model the sequential decay in order to clarify whether it has washed out the expected effect. 3. Limiting temperature As previously mentioned, for proton-rich systems a rapid drop of the limiting temperature has been theoretically predicted (Figure 1) because of the increasing Coulomb pressure [4]. On the other hand, SMM calculations predict nearly isospin-invariant temperatures for the coexistence region [10]. Therefore, in order to distinguish whether the breakup temperature is determined by the binding properties of the excited hot nuclear system or by the phase space accessible to it by fragmentation we have analyzed the temperature for the studied neutron-poor and neutron-rich systems. The temperatures used in the caloric curve studies [5,15] were deduced from a double ratio of isotopic yields. Under the assumptions of low density and chemical equilibrium a grand canonical treatment [16] yields for the double ratio: ZAZA exp ),(),( ),(),( with ΔB being the double difference of the binding energies of the chosen isotopes and a a statistical factor containing spin and mass terms. Furthermore, for best sensitivity the binding energy difference ΔB should be comparable or larger than the temperatures to be measured [17]. The thermometry with the 3,4He isotope pairs used in the nuclear caloric curve [5] benefits from the large difference ∆B = 20.6 MeV of the binding energies of these two nuclei but it is not the only choice. There are also other combinations of isotopes which can be expected to provide the necessary sensitivity for the measurement of temperatures in the MeV range. In a recent systematic study of different isotopic thermometers for spectator fragmentation [18] TBeLi, TCLi, and TCC have been analyzed in detail. In particular, it has been found that the rise at small Zbound, i.e. high excitation energy, previously observed with THeLi is well reproduced by most thermometers, including TBeLi which is derived from the 7,9Be and 6,8Li isotope ratios. This is an indication that the rise is not necessarily related to a particular behavior of either 3He or 4He. An overall good agreement between the different temperature observables has been obtained with the exception of those containing carbon isotopes. In the latter cases, the apparent temperature values remain approximately constant with values between 4 and 5 MeV. Measured isotope ratios as a function of Zbound for the 124Sn, 124La and 107Sn systems are shown in Figure 4. A very interesting dependence on the isotopic composition of the Figure 4. Measured isotopic yield ratios as a function of Zbound/Zproj for the neutron-rich 124Sn and the neutron-poor 124La and 107Sn systems. spectator is observed for all analyzed ratios with the exception of the 3He/4He ratio, exhibiting a reduced difference between the neutron-rich and neutron-poor systems. The strong sequential decay into α particles could explain the difference between these two behaviors. Therefore, also in this case, the structure effects responsible for the observed odd-even variation in the N/Z/s of intermediate-mass fragments (Figure 3) play the major role. For the 124Sn projectile spectator, moreover, the ratios, with the exception of 3He/4He, are almost independent of Zbound. For the mentioned ratio the decrease between the most central and the most peripheral collisions is about a factor 4 whereas for example, for the 9Be/8Li ratio the total variation amounts hardly to a factor of 2. There seems to be a clear correlation between the difference of the binding energies of the involved isotopes and Zbound, i.e. excitation energy of the system [18]: the higher the first the stronger the variation of the corresponding ratio with Zbound. On the other hand, in the case of the proton-rich systems all ratios exhibit a strong dependence on the excitation energy deposited in the system. From the measured ratios the isotopic temperatures have been extracted and an overall agreement with the previous systematics [18] have been obtained for the THeLi, TBeLi, TCLi, and TCC thermometers. In Figure 5 in particular, the apparent THeLi and TBeLi temperatures for the 124Sn and 124La, Figure 5. Isotopic temperatures THeLi and TBeLi as a function of Zbound/Zproj for 124Sn and 124La, 107Sn projectile spectators. 107Sn spectator systems are reported. By comparing the two systems with the same mass but different isospin-content, the average difference of the obtained THeLi temperatures is 0.7±0.1 MeV whereas almost no difference (hardly 0.1 MeV in average) between the two systems is observed in the case of the TBeLi thermometer. The small difference observed in the case of THeLi is caused by the fact that the dependence of the 6Li/7Li ratio (Figure 4) on the isotopic composition of the system is not compensated by the weak dependence of the 3He/4He ratio. In the case of TBeLi both isotopic ratios 7,9Be and 6,8Li exhibit a dependence on the N/Z of the original projectiles, which cancels out in the double ratio. The invariance of the isotopic temperature with the isotopic composition of the system is inconsistent with the limiting temperature predictions [4] of 2 MeV for the 124Sn and 124La systems (Figure 1) and seems to favor a statistical interpretation [10]. 4. Conclusions Isotopic effects in the break-up of projectile spectators at relativistic energies have been reported. The gross properties of projectile fragmentation are very similar for all the studied systems. Specific isotopic effects, even though small, can nevertheless be observed: in particular, the inversion in the hierarchy of the τ exponential parameter of the charge distribution can be explained with a weak isotopic dependence of the surface- term coefficient of the nuclear equation of state [10]. The mean N/Zs of the isotope distributions of light fragments exhibit as well specific isotopic effects. In particular, the observed odd-even variation is much more strongly pronounced for the neutron-poor cases and could be explained in terms of a simultaneous concurrence of both structure effects characteristic for the isotopes produced and memory effects of the isotopic composition of the excited system from which they are emitted. From the double ratios of Z≤4 isotopes, the isotopic temperatures have been determined. The small dependence (of about 0.7 MeV) observed in the THeLi thermometer could be due to the influence of structure effects in the sequential decay. The invariance with the isotopic composition in the entrance channel is inconsistent with the limiting-temperature predictions and seems to favors the statistical interpretation. On the other hand, the limiting-temperature concept reproduces nicely the mass dependence of the caloric curves [15] and only seems to fail when applied to the isospin degree of freedom. It should be noted that most of the limiting-temperature calculations are made for beta stable nuclei [19] while it was not explicitly tested whether the studied systems are still near the stability at breakup. The weak N/Z dependence of the breakup temperatures measured in this experiment shows that this is not a major point of concern. The same observation, on the other hand, is not compatible with the strong Coulomb effect predicted by Besprosvany and Levit [4]. This open point in the connection between limiting temperatures for heavy nuclei and breakup temperatures of fragmenting nuclear systems will require an improved understanding. C.Sf. acknowledges the receipt of an Alexander-von-Humboldt fellowship. This work was supported by the European Community under contract No. RII3-CT-2004-506078 and HPRI-CT-1999-00001 and by the Polish Scientific Research Committee under contract No. 2P03B11023. Bibliography [1] H. Müller and B.D. Serot, Phys. Rev. C 52 (1995) 2072. [2] for reviews see, e.g., C. Fuchs and H.H. Wolter Eur. Phys. J. A 30 (2006) 5. [3] A.B. Larionov et al., Nucl. Phys. A 658 (1999) 375. [4] J. Besprosvany and S. Levit, Phys. Lett. B 217 (1989) 1. [5] J. Pochodzalla et al., Phys. Rev. Lett. 75 (1995) 1040. [6] C. Sfienti et al., Nucl. Phys. A 749 (2005) 83c. [7] S. Bianchin et al., Contribution to this proceedings. [8] C.Sfienti et al., Proceeding of the XLI International Winter Meeting on Nuclear Physics, Bormio, 26 Jan. – 1 Feb. 2003, Ricerca Scientifica ed Educazione Permanente, Supplemento N.120, p.323. [9] R. Ogul and A. S. Botvina, Phys. Rev. C 66 (2005) 051601R. [10] A. S. Botvina et al., Phys. Rev. C 74 (2006) 044609. [11] P. Pawlowski et al, to be published. [12] N. Buykcizmeci et al., Eur. Phys. J A 25 (2005) 57. [13] M. V. Ricciardi et al.,Nucl. Phys. A 733 (2004) 299. [14] P. Napolitani et al., Phys. Rev. C 70 (2004) 054607. [15] J. Natowitz et al., Phys. Rev. C 65 (2002) 034618. [16] S. Albergo et al., Il Nuovo Cimento 89A (1985) 1. [17] M.B. Tsang et al., Phys. Rev. Lett 78 (1997) 3836. [18] W. Trautmann et al., to be published. [19] P. Wang et al., Nucl. Phys. A 748 (2005) 226.

0704.0228

Einstein vs Maxwell: Is gravitation a curvature of space, a field in flat space, or both?

arXiv:0704.0228v1 [gr-qc] 2 Apr 2007 epl draft Einstein vs Maxwell: Is gravitation a curvature of space, a field in flat space, or both? Theo M. Nieuwenhuizen Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands PACS 04.20.Cv – Fundamental problems and general formalism PACS 04.20.Fy – Canonical formalism, Lagrangians, and variational principles PACS 98.80.Bp – Origin and formation of the Universe Abstract. - Starting with a field theoretic approach in Minkowski space, the gravitational energy momentum tensor is derived from the Einstein equations in a straightforward manner. This allows to present them as acceleration tensor = const. × total energy momentum tensor. For flat space cosmology the gravitational energy is negative and cancels the material energy. In the relativistic theory of gravitation a bimetric coupling between the Riemann and Minkowski metrics breaks general coordinate invariance. The case of a positive cosmological constant is considered. A singularity free version of the Schwarzschild black hole is solved analytically. In the interior the components of the metric tensor quickly die out, but do not change sign, leaving the role of time as usual. For cosmology the ΛCDM model is covered, while there appears a form of inflation at early times. Here both the total energy and the zero point energy vanish. It is said that in introducing the general theory of rela- tivity (GTR), Einstein made the step that Lorentz and Poincaré had failed to make: to go from flat space to curved space. Technically, this arises from the group of general coordinate transformations [1, 2]. One fundamen- tal difficulty is then how to deal with the physics of gravi- tation itself, since there is only a quasi energy-momentum tensor [3]. For gravitational wave detection, e.g., this leaves open the question as to how energy can be faithfully transferred from the wave to the detector. The proper en- ergy momentum tensor of gravitation was derived only recently by Babak and Grishchuk [4], who start with a field theoretic approach to gravitation, in terms of a ten- sor field hµν in a Minkowski background space-time. The metric of the latter, ηµν = diag(1,−1,−1,−1), is denoted in arbitrary coordinates by γµν = (γ µν)−1. The Riemann metric tensor gµν = (g µν)−1, is then defined by gµν = γµν + hµν ≡ kµν , g det(gµν) det(γµν) . (1) It is just a way to code the gravitational field, allowing to expresses distances by ds2 = gµνdx µdxν . Such a non- linear way to code distances in a flat space is not uncom- mon. For diffuse light transport through clouds, one may express distances in the optical thickness, the number of extinction lengths. If the cloud is not homogeneous, points at the same physical distance are described by a different optical distance and, vice versa. The Maxwell view that gravitation is a field in flat space, was actually the starting point for Einstein, and reappeared regularly. Nathan Rosen [5], coauthor of the Einstein-Podolsky-Rosen paper that led the basis for quantum information, considers a bimetric theory, in- volving the Minkowski metric and the Riemann metric. Bimetrism is quite natural, with ηµν entering e.g. particle physics, and gµν e.g. cosmology. Rosen considers covari- ant derivatives Dµ of Minkowski space, with Christoffel symbols γλ ·µν vanishing in Cartesian coordinates. When re- placing in the Riemann Christoffel symbols partial deriva- tives by Minkowski covariant ones, ·µν = gλσ(∂µgνσ + ∂νgµσ − ∂σgµν) 7→ ·µν = gλσ(Dµgνσ +Dνgµσ −Dσgµν), (2) the obtained Christoffel-type symbols Gλ ·µν are tensors in Minkowski space. Inspired by the Landau-Lifshitz and Babak-Grishchuk results, we may define the acceleration tensor Aµν = DαDβ(k µνkαβ − kµαkνβ), (3) where kµν = γµν + hµν and in which the γγ terms do not http://arxiv.org/abs/0704.0228v1 Th.M. Nieuwenhuizen contribute. Then we can calculate the combination τµν = Aµν − (Rµν − 1 gµνR) . (4) In doing so, we make use of Rosen’s observation that Rµν remain unchanged if one replaces all partial derivatives by covariant ones in Minkowski space [5]. It appears that all second order derivatives drop out from (4), leaving a bilinear form in first order covariant derivatives, τµν = · · :λh · · :ρ − · · :λh · · :ρ hµλ:ρhν ·λ:ρ + kµνhλρ:σhλσ:ρ − hλρ:µhν hµλ:ρh · · :νλρ + hλρ:µh · · :νλρ − kµνhλρ:σhλρ:σ + kµνh ·ρ:λρ h . (5) in which X:µ ≡ DµX and raising (lowering) of indices · · :ρ is performed with k µν (kµν). τ µν is a tensor in Minkowski space. For Cartesian coordinates, it coin- cides with the Landau-Lifshitz quasi-tensor. In general, it coincides with the Babak-Grishchuk tensor γtµν/g. Inclu- sion of matter is now much easier than in [4]. Inserting the Einstein equations in the right hand side of (4), we may write the Einstein equations in the Newton shape: acceleration=mass−1×force, Aµν = Θµν , Θµν = θµν , θµν ≡ τµν + T µν . (6) Θµν is the total energy momentum tensor of gravitation and matter. It is conserved, DνΘ µν = 0, since Eq. (3) im- plies DνA µν = 0, because covariant Minkowski derivatives commute. As an application, let us consider cosmology, described by the Friedman-Lemaitre-Robertson-Walker (FLRW) metric, ds2 = U(t)c2dt2 − V (t) 1− kr2 + r2dΩ2 , (7) dΩ2 = dθ2 + sin2 θdφ2. Let us consider flat space, k = 0, and U = 1, V (t) = a2(t) with a the scale factor. Then ds2 = c2dt2 − a2(t)dr2 is space-independent, implying that A00 = 0, due to the shape (3). According to (6) it then follows that the total energy density is zero, because the gravitational energy density, τ00 = −3c4ȧ2/(8πGa2), is negative and cancels the one of matter, T 00 = ρ, due to the Friedman equation. In other words, such a universe contains no overall energy. So far we have discussed an alternative, field theoretic formulation of GTR. If we consider a local energy mo- mentum density as a sine qua non property, then we are led to consider Minkowski space as a fixed “pre-space”, that exist already without matter, just as a region of space ahead of the earth’s orbit is right now almost empty (Minkowskian), and when the earth arrives, there will be more gravitational and matter fields, but, in our view, no change of space. Also for cosmology there is a different interpretation. In GTR coordinates are fixed to clusters of galaxies, this is called “coordinate space”, but due to the increasing scale factor galaxies are said to move away from each other: physical space (i.e. Riemann space) is said to expand. Here we are led to another view: Coordi- nate space is physical space, so clusters of galaxies do not move away from each other in time. [6] However, the cos- mic speed of light dr/dt = c/a(t), which was very large at early times, keeps on decreasing, thus causing a redshift, till a is infinite, when galaxies are invisible. Relativistic Theory of Gravitation, RTG. Let us move on to an extension of GTR, giving up general coordinate invariance. Discarding a total derivative of the Hilbert- Einstein action, Rosen expresses the gravitational action d3xdt −g LR in terms of [5] c4gµν ·λσ −Gλ·µσGσ·νλ) = 128πG × (2hµν:ρhµν:ρ − 4hµν:ρhµρ:ν − hν·ν:µh ·ρ:µρ ). Involving only Minkowski covariant first order derivatives, it is close to general approaches in field theory. Logunov and coworkers continue on this [6]. The subgroup of gauge transformations that transform hµν but leave coordinates invariant, allows three extra terms [6], Lg = LR − ρΛ + ρbiγµνg µν − ρ0 γ/g. (9) Here ρΛ is the familiar energy related to a cosmological constant. The ρ0 term describes a harmless shift of the zero level of energy, δS = − d3xdt −γρ0. The bimetric term ρbi couples the Minkowski and the Riemann metrics. It acts like a mass term, because it breaks general coor- dinate invariance, and has some analogy to a mass term in massive electrodynamics. Logunov then imposes the relation ρΛ = ρbi = ρ0, (10) which, in the absence of matter, keeps space flat, hµν = 0, gµν = γµν and also Lg = 0. Thus one free parameter re- mains. Logunov’s choice ρbi ≡ −m2c4/(16πG) < 0 leads to an inverse length m and, in quantum language, a gravi- ton mass h̄m/c. The negative cosmological constant can be counteracted by an inflaton field [7]. The obtained theory has some drawbacks, such as self-repulsive prop- erties for matter falling onto a black hole, and a minimal and a maximal size of the scale factor in cosmology [6] [7]. For a related approach to finite range gravity, based on a generalized Fierz-Pauli coupling, see [8]. Einstein vs Maxwell We shall focus on the opposite choice, a positive cosmo- logical constant Λ, [9] [10] Ωv,0H 9.78Gyr ρbi ≡ = ρΛ. (11) Now the graviton has an “imaginary mass”, m = −2Λbi/c, it is a “tachyon”: Gravitational waves are unstable at today’s Hubble scale. But this is of no con- cern, since on that scale, not single gravitational waves but the whole Universe matters, being unstable (expand- ing) anyhow. Though we take ρbi = ρΛ, Λbi = Λ, our further notation is valid for the general case ρbi 6= ρΛ, Λbi 6= Λ. The Einstein equations that couple the Riemann metric to matter read Rµν − gµνR = tot , (12) tot = T µν + ρΛg µν + ρbiγρσ(g µρgσν − 1 gµνgρσ). Conservation of energy momentum, T tot;ν = 0, imposes a constraint due to the ρbi terms, [6] = 0, or Dνh µν = 0, (13) which for Cartesian coordinates coincides with the GTR harmonic condition ∂ν( −ggµν) = 0 [2]. Thus the the- ory automatically demands the harmonic constraint for gµν , or, equivalently, the Lorentz gauge for hµν , thereby severely reducing the gauge invariance of GTR. Changes of Einstein’s GTR have mostly met deep trou- bles with one or another established property, though not all proposals are ruled out [1,11]. The present one is rather subtle and promising. For most applications, the Hubble- size ρΛ = ρbi terms in Eq. (11,12) are too small to be relevant, so known results from general relativity can be reproduced. Indeed, viewed from a GTR standpoint, Eq. (13) is only a particular gauge, and actually often con- sidered, while the cosmological constant only plays a role in cosmology. Logunov checked a number of effects in the solar system: deflection of light rays by the sun, the delay of a radio signal, the shift of Mercury’s perihelion, the precession of a gyroscope, and the gravitational shift of spectral lines. [6] Likewise, we expect agreement for binary pulsars. [11] Differences between GTR and RTG may arise, though, for large gravitational fields, that we consider now. Black holes. It is known that true black holes, ob- jects that have a horizon, do not occur in the RTG with ρΛ, ρbi → 0. [5] But there are solutions very similar to it, that might be named “grey holes”, but we just call them “black holes”. The Minkowski line element in spherical coordinates is simply γµνdx µdxν = c2dt2 − dr2 − r2dΩ2. The one of Riemann space is ds2 = gµνdx µdxν = U(r)c2dt2 − V (r)dr2 −W 2(r)dΩ2. (14) In harmonic coordinates, the Schwarzschild black hole is described by [2] r − rh r + rh , Ws = r + rh, rh = . (15) The horizon radius rh equals half the Schwarzschild radius. Let us scale r → rrh, and define U = eu, V = ev, W = 2rhe w, (16) so that w is small near the horizon. The dimensionless small parameter arising from ρbi = ρΛ, is very small, λ̄ ≡ rh 2Λ = 2.38 10−23 . µ̄ ≡ rh 2Λbi = λ̄. (17) The sum and difference of the (t, t) and (r, r) Einstein equations give ev−2w − w′(u′ − v′ + 4w′)− 2w′′ = ev(λ̄2 − 1 µ̄2r2e−2w) + 8πGr2h ev(ρ− p), (18) w′(u′ + v′ − 2w′)− 2w′′ µ̄2(ev−u − 1) + 8πGr ev(ρ+ p), (19) respectively. The harmonic condition imposes u′ − v′ + 4w′ = r exp(v − 2w). In the Schwarzschild black hole of GTR, there is no matter outside the origin. We shall focus on that situation. A parametric solution of these equations then reads 1 + η(eξ + ξ + log η + r0) 1− η(eξ + ξ + log η + r0) , (20) u = ξ + log η, v = ξ − ln η − 2 log(eξ + 1), (21) w = ηeξ + µ̄2(ξ + log η + w0). where ξ is the running variable and η is a small scale. Corrections of next order in η can be expressed in diloga- rithms, but they are not needed since µ̄ is very small. To fix the scale η, we note that energy momentum con- servation implies, as in GTR, (ρ + p)u′ + 2p′ = 0. In the stationary state all matter is located at the origin, which is only possible if p(r) ≡ 0, implying ρ(r)u′(r) = 0. This is obeyed for r 6= 0 since ρ = 0 there, but since ρ(0) > 0 (it is infinite), we have to demand u′(0) = 0. Let us define a factor α by α = µ̄2/η. The above solu- tion brings w′(r) = ∂ξw/∂ξr = (e ξ + α)/[2(eξ + 1)], so in the interior w′ = 1 α. Since ev ≪ 1 there, Eqs. (18,19) confirm that w′′ = 0, and with w(1) = O(η) this solves Th.M. Nieuwenhuizen 1.510.5 Fig. 1: Black hole functions U(r), V (r) and W (r), scaled by factors 10, (bold lines) for µ̄ = 0.1, compared to the Schwarzschild solution (thin lines; the part V < 0 for r < 1 is not shown). Inside the horizon, U and V decay very rapidly. Since they remain positive, time keeps its role in the interior. w(r) = 1 α (r − 1). Moreover, from the harmonic con- straint (20) we have in the interior u(r)− v(r) + 4w(r) = const = 2 ln η, implying that Eq. (19) yields in the interior u′(r) = {exp[2α(r − 1)]− η2 − µ̄2}/(2η). From u′(0) = 0 we can now solve α, α = log , η = ln 1/µ̄ . (22) As seen in fig. 1, our solution (16,20,-22) coincides with Schwarzschild’s for ξ ≫ 1. In the regime ξ = O(1), there is a transition towards the interior ξ ≪ −1, where ex- ponential corrections can be neglected. Both U = ηeξ and V = eξ/η are very small there, but, contrary to the Schwarzschild case, they remain positive: The behavior in the interior of the RTG black hole is not qualitatively dif- ferent from usual, be it that the gravitational field is large. Width of the brick wall. The transition layer ξ = O(1) acts like ’t Hooft’s brick wall, [12] of characteristic width ℓ⋆ = ηrh. Comparing to the Planck length ℓP = h̄G/c3, we get 0.977 10−9 1 + 0.019 log(M/M⊙) . (23) If quantum physics sets in at the Planck scale, our ap- proach makes sense only for M > 103 M⊙. Motion of test particles. For RTG with a negative cos- mological constant, [6] it was claimed that an incoming spherical shell of matter is scattered off from a black hole, a counter-intuitive finding. Let us reconsider this issue. The motion of a test body occurs along a geodesic + Γµνρv νvρ = 0, vµ = . (24) For spherical shells of in-falling matter one needs Γ001 = U ′/(2U). This brings dt/ds = v0 = 1/(CiU), for some Ci. Solving v 1 = dr/ds from gµνv µvν = 1, we then get dr/dt = (ds/dt)(dr/ds) = −c U(1− C2i U)/V . We can now fix Ci at the initial position r = ri, where the spherical shell is assumed to have a speed dri/dt = vi = βic Ui/Vi, viz. Ci = (1 − β2i )/Ui, with |βi| ≤ 1. The differential proper time dτ = Udt and length dℓ = V dr bring in the particle’s rest frame dℓ/dτ = V/Udr/dt, yielding 1− U(r(τ)) U(ri) (1 − β2i ). (25) The extreme case is when βi = 0 at ri = ∞, dℓ/dτ = 1− U. To have |dℓ/dτ | < c, it thus suffices that 0 < U ≤ 1, which is the case. Near the horizon, |dℓ/dτ | is almost equal to c and the more the shell penetrates the interior, the closer its speed gets to c. For an outside observer, the time to see it hit the center of the hole, dr/|ṙ| equals (rh/c) dr exp[ 1 (v − u)]. It is finite and predominantly comes from the horizon, T = rh/cµ̄ 2.74× 1032M/M⊙ yr. The approaches [6–8] have a similar a black hole. While [8] properly has U ′(0) = 0, in Logunov’s case one has µ̄2 < 0, so w′ = 1 α < 0 in the interior. This seems to solve the paradox of “matter reflected by the black hole”: In-falling matter just enters, but the Logunov coordinate x = exp(w) − 1 is non-monotonic (x′ < 0 in the interior). However, the situation is more severe: For α < 0, the theory does not allow a solution with u′(0) = 0, depriving that theory of a proper black hole. This condition can neither be obeyed in GTR: If the central mass is slightly smeared, the Schwarzschild black hole cannot obey energy- momentum conservation in GTR. Cosmology. Starting from the FLRW metric, the har- monic condition brings two relations: U ∼ V 3 and k = 0: Minkowski space filled homogeneously with matter re- mains flat [6]. We may thus put U = a6(t)/a4 , V = a2(t). Going from cosmic time t to conformal time τ = a3a−2 yields the familiar Einstein equations, extended by Λbi terms, − Λbi , (26) = −4πG (ρ+ 3p) + − Λbi The first is the modified Friedman equation, the second corresponds to the first law d(ρtota 3) = −ptotda3 provided we define ρtot = ρ+ρΛ+ρ2+ρ6 and ptot = p−ρΛ− 13ρ2+ρ6, with ρ2 = −3ρbi/2a2 and ρ6 = ρbia4∗/2a6. Note that ρ2 acts as a positive curvature term. The scale factor has an absolute meaning. If we as- sume that a ≫ 1 and a ≫ a2/3∗ , Eq. (26) just coin- cides with the ΛCDM model (cosmological constant plus cold dark matter), that gives the best fit of the observa- tions [9] [10]. The ρ2 term allows a positive curvature- type contribution. At large times, there is the exponential growth a(τ) = C exp(H∞τ) with H∞ = c Λ/3. In cos- mic time this reads a(t) = a ∗ [3H∞(t0 − t)]−1/3, where Einstein vs Maxwell t0 is “the end of time”, the moment where the scale factor has become infinite. The minimal scale factor is zero: in this theory a big bang can occur since ρbi > 0. With- out including an inflaton field, Eq. (26) yields an initial growth of the expansion a = (a2 3Λbi/2) 1/3. In cos- mic time this reads a = a1 exp(ct Λbi/6), i. e., a certain inflation scenario starting at t = −∞. Also in RTG the gravitational energy precisely compen- sates the other energy contributions at all times. The vacuum energy also vanishes: In empty space, the cosmo- logical constant energy ρΛ cancels the ρbi terms, due to Eq. (10). See Eq. (12) for gµν = γµν . In conclusion, we have first written the Einstein equa- tion in a form that involves the gravitational energy momentum tensor. An underlying Minkowski space is needed, in which gravitation is a field. The metric ten- sor is a way to deal with it, but the equations for the field itself exist too, see Eq. (6). For flat cosmology it follows that the total energy vanishes. Next we have broken general coordinate invariance by going to the bimetric theory of Logunov, called Relativis- tic Theory of Gravitation. We have shown that the choice of a positive bimetric constant allows to regularize the interior of the Schwarzschild black hole: time keeps its standard role and escape is, in principle, possible. While neither the Schwarzschild nor the Logunov black hole sur- vives smearing of the central mass by a tiny pressure in the equation of state, ours does. Our modification of the Ein- stein equations involves the cosmological constant, so it is of Hubble size, immaterial for solar problems. In cosmol- ogy, the theory directly leads to the ΛCDM model, while it could accommodate a positive curvature-like term. At short times, there is a form of inflation. The gravitational energy exactly compensates the material energy. The zero point energy vanishes (“again”), though the cosmological constant is finite and positive: It is canceled by the bimet- ric terms. Euclidean space, a special case of Riemann geometry, seems to be invoked by Nature, at least far away from bod- ies and in cosmology. Our approach supports the follow- ing space-time interpretation: curvature is a geometric de- scription of the gravitational field in flat space. Clusters of galaxies do not move away from each other, but the speed of light changes with cosmic time, dr/dt = [a(t)/a∗] while the conformal speed is dr/dτ = c/a(τ) as usual. An empirical way to establish the Minkowski metric is to present the Einstein equations as (c4/8πG)Rµν −Tµν + gµνT + ρΛgµν = ρbiγµν , and to measure the left hand side, which in the geometric view is considered to consist of curved space properties alone. [6] As in the standard model of elementary particles, the separation of curved space into flat space and the gravi- tational field has the following implication: the quantum version of RTG – if it exists – will involve quantization of fields, but not of space. Finally we answer the question posed in the title. The field theoretic approach to gravitation is by itself equiva- lent to a curved space description, so both views apply, de- scribing the same physics from a different angle. But when the theory is extended to the relativistic theory of grav- itation, the bimetrism forces to describe the Minkowski metric separately, and then we see it as most natural to view gravitation as a field in flat space, which is Maxwell’s view. Topics such as a realistic equation of state for black holes and classical tunneling of its radiation, regularization of other singularities, as well as aspects of the inflation and of inhomogeneous cosmology are under study. ∗ ∗ ∗ Discussion with Martin Nieuwenhuizen and Armen Al- lahverdyan is gratefully remembered. REFERENCES [1] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, (Freeman, San Francisco, 1973). [2] S. Weinberg, Gravitation and Cosmology: Principles and Application of the General Theory of Relativity, (Wiley, New York, 1972). [3] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, (Pergamon, Oxford, U.K., 1951; revised 1979). [4] S. V. Babak and L. P. Grishchuk, Phys. Rev. D61, 024038 (1999). [5] N. Rosen, Phys. Rev. 57, 147 (1940); ibid 150; Ann. Phys. 22, 11 (1963). [6] A.A. Logunov, The Theory of Gravity, (Nauka, Moscow, 2001). [7] S.S. Gershtein, A.A. Logunov and M.A.Mestvirishvili, gr- qc/0602029. [8] S. V. Babak and L. P. Grishchuk, Int. J. Mod. Phys. D12, 1905 (2003). [9] M. Tegmark, A. Aguirre, M.J. Rees, and F. Wilczek, Phys. Rev D73, 023505 (2006). [10] D. N. Spergel, et al., Astrophys. J. Suppl. 148, 175 (2003). [11] C.M. Will, Theory and Experiment in Gravitational Physics, (Cambridge Univ. Press, New York, 1993), chap- ter 12.3, discusses that gravitational radiation of binary pulsars rules out a more recent bimetric theory of Rosen, in which black holes do not have the Schwarzschild shape. [12] G. ’t Hooft, Nucl. Phys. B256, 727 (1985).

0704.0229

Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry

Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry Dedicated to Sri Ramakrishna Ketan D. Mulmuley The University of Chicago http://ramakrishnadas.cs.uchicago.edu (Technical report TR 2007-04, Comp. Sci. Dept., The University of Chicago, May, 2007) Revised version October 23, 2018 http://arxiv.org/abs/0704.0229v4 http://ramakrishnadas.cs.uchicago.edu Abstract This article belongs to a series on geometric complexity theory (GCT), an approach to the P vs. NP and related problems through algebraic geometry and representation theory. The basic principle behind this approach is called the flip. In essence, it reduces the negative hypothesis in complexity theory (the lower bound problems), such as the P vs. NP problem in characteristic zero, to the positive hypothesis in complexity theory (the upper bound prob- lems): specifically, to showing that the problems of deciding nonvanishing of the fundamental structural constants in representation theory and algebraic geometry, such as the well known plethysm constants [Mc, FH], belong to the complexity class P . In this article, we suggest a plan for implementing the flip, i.e., for showing that these decision problems belong to P . This is based on the reduction of the preceding complexity-theoretic positive hypotheses to mathematical positivity hypotheses: specifically, to showing that there exist positive formulae–i.e. formulae with nonnegative coefficients–for the struc- tural constants under consideration and certain functions associated with them. These turn out be intimately related to the similar positivity proper- ties of the Kazhdan-Lusztig polynomials [KL1, KL2] and the multiplicative structural constants of the canonical (global crystal) bases [Kas2, Lu2] in the theory of Drinfeld-Jimbo quantum groups. The known proofs of these positivity properties depend on the Riemann hypothesis over finite fields (Weil conjectures proved in [Dl]) and the related results [BBD]. Thus the reduction here, in conjunction with the flip, in essence, says that the validity of the P 6= NP conjecture in characteristic zero is intimately linked to the Riemann hypothesis over finite fields and related problems. The main ingradients of this reduction are as follows. First, we formulate a general paradigm of saturated, and more strongly, positive integer programming, and show that it has a polynomial time al- gorithm, extending and building on the techniques in [DM2, GCT3, GCT5, GLS, KB, KTT, Ki, KT1]. Second, building on the work of Boutot [Bou] and Brion (cf. [Dh]), we show that the stretching functions associated with the structural constants under consideration are quasipolynomials, generalizing the known result that the stretching function associated with the Littlewood-Richardson coefficient is a polynomial for type A [Der, Ki] and a quasi-polynomial for general types [BZ, Dh]. In particular, this proves Kirillov’s conjecture [Ki] for the plethysm constants. Third, using these stretching quasi-polynomials, we formulate the math- ematical saturation and positivity hypotheses for the plethysm and other structural constants under consideration, which generalize the known sat- uration and conjectural positivity properties of the Littlewood-Richardson coefficients [KT1, DM2, KTT]. Assuming these hypotheses, it follows that the problem of deciding nonvanishing of any of these structural constants, modulo a small relaxation, can be transformed in polynomial time into a saturated, and more strongly, positive integer programming problem, and hence, can be solved in polynomial time. Fourth, we give theoretical and experimental results in support of these hypotheses. Finally, we suggest an approach to prove these positivity hypotheses motivated by the works on Kazhdan-Lusztig bases for Hecke algebras [KL1, KL2] and the canonical (global crystal) bases of Kashiwara and Lusztig [Lu2, Lu4, Kas2] for representations of Drinfeld-Jimbo quantum groups [Dri, Ji]. Steps in this direction are taken [GCT4, GCT7, GCT8]. Specifically, in [GCT4, GCT7] are constructed nonstandard quantum groups, with compact real forms, which are generalizations of the Drinfeld- Jimbo quantum group, and also associated nonstandard algebras, whose re- lationship with the nonstandard quantum groups is conjecturally similar to the relationship of the Hecke algebra with the Drinfeld-Jimbo quantum group. The article [GCT8] gives conjecturally correct algorithms to con- struct canonical bases of the matrix coordinate rings of the nonstandard quantum groups and of nonstandard algebras that have conjectural posi- tivity properties analogous to those of the canonical (global crystal) bases, as per Kashiwara and Lusztig, of the coordinate ring of the Drinfeld-Jimbo quantum group, and the Kazhdan-Lusztig basis of the Hecke algebra. These positivity conjectures (hypotheses) lie at the heart of this approach. In view of [KL2, Lu2], their validity is intimately linked to the Riemann hypothesis over finite fields and the related works mentioned above. Contents 1 Introduction 4 1.1 The decision problems . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Deciding nonvanishing of Littlewood-Richardson coefficients . 12 1.3 Back to the general decision problems . . . . . . . . . . . . . 16 1.4 Saturated and positive integer programming . . . . . . . . . . 16 1.5 Quasi-polynomiality, positivity hypotheses, and the canonical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.6 The plethysm problem . . . . . . . . . . . . . . . . . . . . . . 20 1.7 Towards PH1, SH, PH2,PH3 via canonial bases and canonical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.8 Basic plan for implementing the flip . . . . . . . . . . . . . . 29 1.9 Organization of the paper . . . . . . . . . . . . . . . . . . . . 30 1.10 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 Preliminaries in complexity theory 34 2.1 Standard complexity classes . . . . . . . . . . . . . . . . . . . 34 2.1.1 Example: Littlewood-Richardson coefficients . . . . . 35 2.2 Convex #P . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.1 Littlewood-Richardson coefficients . . . . . . . . . . . 38 2.2.2 Littlewood-Richardson cone . . . . . . . . . . . . . . . 38 2.2.3 Eigenvalues of Hermitian matrices . . . . . . . . . . . 39 2.3 Separation oracle . . . . . . . . . . . . . . . . . . . . . . . . . 39 3 Saturation and positivity 41 3.1 Saturated and positive integer programming . . . . . . . . . . 41 3.1.1 A general estimate for the saturation index . . . . . . 45 3.1.2 Extensions . . . . . . . . . . . . . . . . . . . . . . . . 47 3.1.3 Is there a simpler algorithm? . . . . . . . . . . . . . . 47 3.2 Littlewood-Richardson coefficients again . . . . . . . . . . . . 47 3.3 The saturation and positivity hypotheses . . . . . . . . . . . 49 3.4 The subgroup restriction problem . . . . . . . . . . . . . . . . 52 3.4.1 Explicit polynomial homomorphism . . . . . . . . . . 53 3.4.2 Input specification and bitlengths . . . . . . . . . . . . 55 3.4.3 Stretching function and quasipolynomiality . . . . . . 57 3.5 The decision problem in geometric invariant theory . . . . . . 58 3.5.1 Reduction from Problem 1.1.3 to Problem 1.1.4 . . . . 59 3.5.2 Input specification . . . . . . . . . . . . . . . . . . . . 59 3.5.3 Stretching function and quasi-polynomiality . . . . . . 60 3.5.4 Positivity hypotheses . . . . . . . . . . . . . . . . . . . 61 3.5.5 G/P and Schubert varieties . . . . . . . . . . . . . . . 62 3.6 PH3 and existence of a simpler algorithm . . . . . . . . . . . 63 3.7 Other structural constants . . . . . . . . . . . . . . . . . . . . 63 4 Quasi-polynomiality and canonical models 65 4.1 Quasi-polynomiality . . . . . . . . . . . . . . . . . . . . . . . 65 4.1.1 The minimal positive form and modular index . . . . 68 4.1.2 The rings associated with a structural constant . . . . 69 4.2 Canonical models . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.2.1 From PH0 to PH1,3 . . . . . . . . . . . . . . . . . . . 70 4.2.2 On PH0 in general . . . . . . . . . . . . . . . . . . . . 72 4.3 Nonstandard quantum group for the Kronecker and the plethysm problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4 The cone associated with the subgroup restriction problem . 75 4.5 Elementary proof of rationality . . . . . . . . . . . . . . . . . 78 5 Parallel and PSPACE algorithms 84 5.1 Complex semisimple Lie group . . . . . . . . . . . . . . . . . 85 5.2 Symmetric group . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3 General linear group over a finite field . . . . . . . . . . . . . 92 5.3.1 Tensor product problem . . . . . . . . . . . . . . . . . 93 5.4 Finite simple groups of Lie type . . . . . . . . . . . . . . . . . 94 6 Experimental evidence for positivity 95 6.1 Littlewood-Richardson problem . . . . . . . . . . . . . . . . . 95 6.2 Kronecker problem, n = 2 . . . . . . . . . . . . . . . . . . . . 95 6.3 G/P and Schubert varieties . . . . . . . . . . . . . . . . . . . 96 6.4 The ring of symmetric functions . . . . . . . . . . . . . . . . 97 7 On verification and discovery of obstructions 111 7.1 Obstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 7.2 Decision problems . . . . . . . . . . . . . . . . . . . . . . . . 113 7.3 Verification of obstructions . . . . . . . . . . . . . . . . . . . 113 7.4 Robust obstruction . . . . . . . . . . . . . . . . . . . . . . . . 115 7.5 Verification of robust obstructions . . . . . . . . . . . . . . . 116 7.6 Arithemetic version of the P#P vs. NC problem in charac- teristric zero . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 7.6.1 Class varieties . . . . . . . . . . . . . . . . . . . . . . . 117 7.6.2 Obstructions . . . . . . . . . . . . . . . . . . . . . . . 118 7.6.3 Robust obstructions . . . . . . . . . . . . . . . . . . . 119 7.6.4 Verification of robust obstructions . . . . . . . . . . . 121 7.6.5 On explicit construction of obstructions . . . . . . . . 122 7.6.6 Why should robust obstructions exist? . . . . . . . . . 123 7.6.7 On discovery of robust obstructions . . . . . . . . . . 124 7.7 Arithmetic form of the P vs NP problem in characteristic zero126 Chapter 1 Introduction This article belongs to a series of papers, [GCT1] to [GCT11], on geomet- ric complexity theory (GCT), which is an approach to the P vs. NP and related problems in complexity theory through algebraic geometry and rep- resentation theory. We assume here that the underlying field of computation is of characteristic zero. The usual P vs. NP problem is over a finite field. The characteristic zero version is its weaker, formal implication, and philo- sophically, the crux. The basic principle underlying GCT is called the flip [GCTflip]. The flip, in essence, reduces the negative hypotheses (lower bound problems) in complexity theory, such as the P 6=?NP problem in characteristic zero, to positive hypotheses in complexity theory (upper bound problems): specifi- cally, to the problem of showing that a series of decision problems in rep- resentation theory and algebraic geometry belong to the complexity class P . Each of these decision problem is of the form: Given a nonnegative structural constant in representation theory or geometric invariant theory, such as the well known plethysm constant, decide if it is nonzero (nonvan- ishing), or rather, if is nonzero after a small relaxation. This flip from the negative to the positive may be considered to be a nonrelativizable form of the flip–from the undecidable to the decidable–that underlies the proof of Gödel’s incompleteness theorem. But the classical diagonalization technique in Gödel’s result is relativizable [BGS], and hence, not applicable to the P vs. NP problem. The flip, in contrast, is nonrelativizable. It is furthermore nonnaturalizable [GCT10]); i.e., it crosses the natural proof barrier [RR] that any approach to the P vs. NP problem must cross. We suggest here a plan for implementating the flip; i.e., for showing that the decision problems above belong to P . This is based on the reduction in this paper of the complexity-theoretic positivity hypotheses mentioned above to mathematical positivity hypotheses: specifically, to showing that there exist positive formulae for the structural constants under consideration and certain functions associated with them. We also give theoretical and experimental evidence in support of the latter hypotheses. Here we say that a formula is positive if its coefficients are nonegative. The problem finding the positive formulae as above turns out be intimately related to the analogous problem for the Kazhdan-Lusztig polynomials [KL1] and the multiplicative structural constants of the canonical (global crystal) bases [Kas2, Lu2] in the theory of Drinfeld-Jimbo quantum groups. The known solution to the latter problem [KL2, Lu2] depends on the Riemann hypothesis over finite fields, proved in [Dl], and the related results in [BBD]. Thus the flip and the reduction here together roughly say that the valid- ity of the P 6= NP conjecture in characteristic zero is intimately linked to the Riemann hypothesis over finite fields and related problems. This is illustrated in Figure 1.1; the question marks there indicate unsolved prob- lems. It seems that substantial extension of the techniques related to the Riemann hypothesis over finite fields may be needed to prove the required mathematical positivity hypotheses here. We do not have the necessary mathematical expertize for this task. But it is our hope that the experts in algebraic geometry and representation theory will have something to say on this matter. It may be conjectured that the flip paradigm would also work in the context of the usual P vs. NP problem over F2 (the boolean field) or the finite field Fp. But implementation of the flip over a finite field is expected to be much harder than in characteristic zero. That is why we focus on characteristic zero here, deferring discussion of the problems that arise over finite field to [GCT11]. Now we turn to a more detailed exposition of the main results in this paper and of Figure 1.1. Acknoledgements We are grateful to the authors of [BOR] for pointing out an error in the saturation hypothesis (SH) in the earlier version of this paper. It has been corrected in this version with appropriate relaxation without affecting the overall approach of GCT (cf. Section 1.6 and also [GCT6erratum]). We are also grateful to Peter Littelmann for bringing the reference [Dh] to our Complexity theoretic negative hypotheses (lower bound problems) The flip Complexity theoretic positive hypotheses (upper bound problems) The reduction in this paper| Mathematical positivity hypotheses | (?) The Riemann hypothesis over finite fields, related problems and their extensions Figure 1.1: Pictorial depiction of the basic plan for implementing the flip attention, to H. Narayanan for suggesting the use of [KB] in the proof of Theorem 3.1.1 and bringing the positivity conjecture in [DM2] to our atten- tion, and to Madhav Nori for a helpful discussion. The experimental results in Chapter 6 were obtained using Latte [DHHH]. 1.1 The decision problems We now describe the relevant decision problems in representation theory and algebraic geometry. The actual decision problems that arise in the flip (cf. the second box in Figure 1.1) are relaxed versions of these problems described later (cf. Hypothesis 1.1.6). Problem 1.1.1 (Decision version of the Kronecker problem) Given partitions λ, µ, π, decide nonvanishing of the Kronecker coefficient . This is the multiplicity of the irreducible representation (Specht mod- ule) Sπ of the symmetric group Sn in the tensor product Sλ ⊗ Sµ. Equivalently [FH], let H = GLn(C) × GLn(C) and ρ : H → G = GL(Cn ⊗ Cn) = GLn2(C) the natural embedding. Then k is the multi- plicity of the H-module Vλ(GLn(C))⊗Vµ(GLn(C)) in the G-module Vπ(G), considered as an H-module via the embedding ρ. Here Vλ(GLn(C)) denotes the irreducible representation (Weyl module) of GLn(C) corresponding to the partition λ; Vπ(G) is the Weyl module of G = GLn2(C). Problem 1.1.1 is a special case of the following generalized plethysm problem. Problem 1.1.2 (Decision version of the plethysm problem) Given partitions λ, µ, π, decide nonvanishing of the plethysm constant . This is the multiplicity of the irreducible representation Vπ(H) of H = GLn(C) in the irreducible representation Vλ(G) of G = GL(Vµ), where Vµ = Vµ(H) is an irreducible representation H. Here Vλ(G) is considered an H- module via the representation map ρ : H → G = GL(Vµ). (Decision version of the generalized plethysm problem) The same as above, allowing H to be any connected reductive group. This is a special case of the following fundamental problem of represen- tation theory (characteristic zero): Problem 1.1.3 (Decision version of the subgroup restriction problem) Let G be connected reductive group, H a reductive group, possibly discon- nected, and ρ : H → G an explicit, polynomial homomorphism (as defined in Section 3.4). Here H will generally be a subgroup of G, and ρ its em- bedding. Let Vπ(H) be an irreducible representation of H, and Vλ(G) an irreducible representation of G. Here π and λ denote the classifying labels of the irreducible representations Vπ(H) and Vλ(G), respectively. Let m the multiplicity of Vπ(H) in Vλ(G), considered as an H-module via ρ. Given specifications of the embedding ρ and the labels λ, π, as described in Section 3.4, decide nonvanishing of the multiplicity mπ All reductive groups in this paper are over C. The reductive groups that arise in GCT in characteristic zero are: the general and special linear groups GLn(C) and SLn(C), algebraic tori, the symmetric group Sn, and the groups formed from these by (semidirect) products. The reader may wish to focus on just these concrete cases, since all main ideas in this paper are illustrated therein. Problem 1.1.3 is, in turn, a special case of the following most general problem. Problem 1.1.4 (Decision problem in geometric invariant theory) Let H be a reductive group, possibly disconnected, X a projective H- variety (H-scheme), i.e., a variety with H-action. Let ρ denote this H- action. Let R = ⊕dRd be the homogeneous coordinate ring of X. Assume that the singularities of spec(R) are rational. We assume that X and ρ have special properties (as described in Sec- tion 3.5), so that, in particular, they have short specifications. Let Vπ(H) be an irreducible representation of H. Let sπd be the multiplicity of Vπ(H) in Rd, considered as an H-module via the action ρ. Given d, π, the specifications of X and ρ, decide nonvanishing of the multiplicity sπ This last problem is hopeless for general X. Indeed the usual specifi- cation of X, say in terms of the generators of the ideal of its appropriate embedding, is so large as to make this problem meaningless for a general X. But the instances of this decision problem that arise in GCT are for the following very special kinds of projective H-varieties X, which, in particular, have small specifications (Section 3.5): 1. G/P , where G is a connected, reductive group, P ⊆ G its parabolic subgroup, and H ⊆ G a reductive subgroup with an explicit polyno- mial embedding. Problem 1.1.3 reduces to this special case of Prob- lem 1.1.4; cf. Section 3.5. 2. Class varieties [GCT1, GCT2], which are associated with the funda- mental complexity classes such as P and NP . They are very special like G/P , with conjecturally rational singularities [GCT10]. Each class variety is specified by the complexity class and the parameters of the lower bound problem under consideration. Briefly, the P vs. NP prob- lem in characteristic zero is reduced in [GCT1, GCT2] to showing that the class variety corresponding to the complexity class NP and the pa- rameters of the lower bound problem (such as the input size) cannot be embedded in the class variety corresponding to the complexity class P and the same parameters. Efficient criteria for the decision prob- lems stated above are needed to construct explicit obstructions [GCT2] to such embeddings, thereby proving their nonexistence. Specifically, Problems 1.1.3 and 1.1.4 are the decision problems associated with Problems 2.5 and 2.6 in [GCT2], respectively. See Sections 7.6-7.7 for a brief review of this story. For these varieties Problem 1.1.4 turns out to be qualitatively similar to Problem 1.1.3 (cf. Section 3.5 and [GCT2, GCT10]). For this reason, the Kronecker and the plethysm problems, which lie at the heart of the subgroup restriction problem, can be taken as the main prototypes of the decision problems that arise here. One can now ask: Question 1.1.5 Do the decision problems above (Problems 1.1.1-1.1.3 and Problem1.1.4, when X therein is G/P or a class variety) belong to P? That is, can the nonvanishing of any of structural constants in these problems be decided in poly(〈x〉) time, where x denotes the input-specification of the structural constant and 〈x〉 its bitlength? For Problem 1.1.2, the input specification for the plethysm constant aπλ,µ is given in the form of a triple x = (λ, µ, π). Here the partition λ is specified as a sequence of positive integers λ1 ≥ λ2 ≥ · · ·λk > 0 (the zero parts of the partition are suppressed); k is called the height or length of λ, and is denoted by ht(λ). The bitlength 〈λ〉 is defined to be the total bitlength of the integers λr’s. The bitlength 〈x〉 is defined to be 〈λ〉 + 〈µ〉 + 〈π〉. A detailed specification of the input specification x and its bitlength 〈x〉 for the other problems is given in Section 3.3. For the reasons described in Section 1.6, Question 1.1.5 may not have an affirmative answer in general; i.e., these problems may not be in P in their strict form stated above. The following main conjectural complexity- theoretic positivity hypothesis governing the flip says that the relaxed forms of these decision problems described in Section 3.3 belong to P . As we shall see in Chapter 7, these relaxed forms suffice for the purposes of the flip. Hypothesis 1.1.6 (PHflip) The relaxed forms (cf. Section 3.3) of Prob- lems 1.1.1, 1.1.2, 1.1.3, and the special cases of Problem 1.1.4, when X therein is G/P or a class variety–which together include all decision prob- lems that arise in the flip–belong to the complexity class P . This means nonvanishing of any of these structural constants, modulo a small relaxation (as described in Section 3.3), can be decided in poly(〈x〉) time, where x denotes the input-specification of the structural constant and 〈x〉 its bitlength. The phrase “modulo a small relaxation” in the relaxed form of the plethysm problem means the following: (a) Let h = dimG + htλ + htπ, where dim(G) is the dimension of the group G in Problem 1.1.2. Then there exist absolute nonnegative constants c and c′, independent of λ, µ and π, such that nonvanishing of the relaxed (stretched) plethysm constant abπ bλ,bµ , for any positive integral relaxation parameter b > chc , can be decided in O(poly(〈λ〉, 〈µ〉, 〈π〉, 〈b〉)) time, where 〈b〉 denotes the bitlength b. The notation poly(〈λ〉, 〈µ〉, 〈π〉, 〈b〉) here means bounded by a polynomial of constant degree in 〈λ〉, 〈µ〉, 〈π〉 and 〈b〉. In particular, the time is O(poly(〈λ〉, 〈µ〉, 〈π〉) if the relaxation parameter b is small; i.e. if its bitlength 〈b〉 is O(poly(〈λ〉, 〈µ〉, 〈π〉)). (Observe that the bit length of h is O(poly(〈λ〉, 〈µ〉, 〈π〉)).) (b) There exists a polynomial time algorithm for deciding nonvanishing of , which works correctly on almost all λ, µ and π. Here polynomial time means O(poly(〈λ〉, 〈µ〉, 〈π〉) time. The meaning of “correctly on almost all” is specified in Hypothesis 1.6.5 below. A detailed specification of the relaxation, i.e., the meaning of the phrase “modulo a small relaxation” for the other problems is given in Section 3.3. The structural constants in Problems 1.1.1-1.1.3 are of fundamental im- portance in representation theory. The kronecker and the plethysm con- stants in Problems 1.1.1 and 1.1.2, in particular, have been studied inten- sively; see [FH, Mc, St4] for their significance. There are many known formulae for these structural constants based on on the character formulae in representation theory. Several formulae for the characters of connected, reductive groups are known by now [FH], starting with the Weyl character formula. For the symmetric group, there is the Frobenius character formula [FH], for the general linear group over a finite field, Green’s formula [Mc], and for finite simple groups of Lie type, the character formula of Deligne- Lusztig [DL], and Lusztig [Lu1]. (Finite simple groups of Lie type, other than GLn(Fq), are not needed in GCT.) One obvious method for deciding nonvanishing of the structural con- stants in Problems 1.1.1-1.1.4 is to compute them exactly. But all known al- gorithms for exact computation of the structural constants in Problems 1.1.1- 1.1.3 take exponential time. This is expected, since this problem is #P - complete. In fact, even the problem of exact computation of a Kostka number, which is a very special case of these structural constants, is #P - complete [N]. This means there is no polynomial time algorithm for com- puting any of them, assuming P 6= NP . Of course, there are #P -complete quantities–e.g. the permanent of a nonnegative matrix [V]–whose nonvanishing can still be decided in polyno- mial time [Sc]. But the decision problems above are of a totally different kind and, at the surface, appear to have inherently exponential complexity. This is because the dimensions of the irreducible representations that occur in their statements can be exponential in the ranks of the groups involved and the bit lengths of the classifying labels of these representations. For example, the dimension of the Weyl module Vλ(GLn(C)) can be exponen- tial in n and the bit length of the partition λ. Furthermore, the number of terms in any of the preceding character formulae is also exponential. All these decisions problems ask if one exponential dimensional representation can occur within another exponential dimensional representation. To solve them, it may seem necessary to take a detailed look into these representa- tions and/or the character formulae of exponential complexity. Hence, it seemed hard to believe that nonvanishing of these structural constants can, nevertheless, be decided in polynomial time (modulo a small relaxation). This constituted the main philosophical obstacle in the course of GCT. 1.2 Deciding nonvanishing of Littlewood-Richardson coefficients The first result, which indicated that this obstacle may be removable, came in the wake of the saturation theorem of Knutson and Tao [KT1]. This concerns the following special case of Problem 1.1.3, with G = H ×H, the embedding ρ : H → G being diagonal. Problem 1.2.1 (Littlewood-Richardson problem) Given a complex semisimple, simply connected Lie group H, and its dominant weights α, β, λ, decide nonvanishing of a generalized Littelwood- Richardson coefficient cλ . This is the multiplicity of the irreducible repre- sentation Vλ(H) of H in the tensor product Vα(H)⊗ Vβ(H). It was shown in [GCT3, KT2, DM2] independently that nonvanishing of the Littlewood-Richardson coefficient of type A can be decided in polyno- mial time; i.e., polynomial in the bit lengths of α, β, λ. Furthermore, the algorithm in [GCT3] works in strongly polynomial time in the terminology of [GLS]; cf. Section 2.1. The three main ingradients in this result are: 1. PH1: The Littlewood-Richarson rule, which goes back to 1940’s, and whose most important feature is that it is positive–i.e., it involves no alternating signs as in character-based formulae–and its strengthening in [BZ], which gives a positive, polyhedral formula for the Littlewood- Richardson coefficient as the number of integer points in a polytope; this can be the BZ-polytope [BZ] or the hive polytope [KT1]. We shall refer to this positivity property as the first positivity hypothesis (PH1). 2. The polynomial and strongly polynomial time algorithms for linear programming [Kh, Ta], and 3. SH: The saturation theorem of Knutson and Tao [KT1]. This says that cλ is nonzero if cnλ nα,nβ is nonzero for any n ≥ 1. We shall refer to this saturation property as the saturation hypothesis (SH). Brion [Z] observed that the verbatim translation of the saturation prop- erty in [KT1] fails to hold for the the generalized Littlewood-Richardson coefficients of types B, C, D (it also fails for the Kronecker coefficients, as well as the plethysm constants [Ki]). Hence, the algorithms in [GCT3, KT2, DM2] do not work in types B, C and D. Fortunately, this situation can be remedied. It is shown in [GCT5] that nonvanishing of the generalized Littewood-Richardson coefficient cλα,β of arbitrary type can be decided in (strongly) polynomial time, assuming the positivity conjecture of De Loera and McAllister [DM2]. This conjectural hypothesis, based on considerable experimental evidence, is as follows. Let c̃λα,β(n) = c nα,nβ (1.1) be the stretching function associated with the Littlewood-Richardson co- efficient cλα,β . It is known to be a polynomial in type A [Der, Ki], and a quasi-polynomial, in general [BZ, Dh, DM2]. Recall that a fuction f(n) is called a quasi-polynomial if there exist l polynomials fj(n), 1 ≤ j ≤ l, such that f(n) = fj(n) if n = j mod l. Here l is supposed to be the smallest such integer, and is called the period of f(n). The period of c̃λ (n) for types B,C,D is either 1 or 2 [DM2]. In general, it is bounded by a fixed constant depending on the types of the simple factors the Lie algebra. Definition 1.2.2 We say that the quasi-polynomial f(n) is strictly posi- tive, if all coefficients of fj(n), for all j, are nonnegative; i.e., the nonzero coefficients are positive. In general, we define the positivity index p(f) of f to be the smallest nonnegative integer such that f(n + p(f)) is strictly positive. We also say that f(n) is positive with index p(f). Thus f(n) is strictly positive, iff its positivity index is zero. With this terminology, the hypothesis mentioned above is the following. We say a connected reductive group H is classical, if each simple factor of its Lie algebra H is of type A,B,C or D. We also say that the type of H or H is classical. Hypothesis 1.2.3 (PH2): [KTT, DM2] Assume that H in Problem 1.2.1 is classical. Then the Littlewood-Richardson stretching quasi-polynomial (n) is strictly positive. We shall refer to this as the second positivity hypothesis (PH2). This was conjectured by King, Tollu and Toumazet [KTT] for type A, and De Loera and McAllister for types B,C,D. Since the stretching function above is a polynomial in type A, the positivity conjecture of King et al clearly implies the saturation theorem of Knutson and Tao. That is, PH2 implies SH for type A. We can formulate an analogue of SH for a Lie algerbra of arbitrary clas- sical type so that PH2 implies SH for an arbitrary type. For this, we need to formulate the notion of a saturated quasi-polynomial, which is not con- tradicted by the counterexamples, mentioned above, to verbatim translation of the saturation property in [KT1, Ki] to the setting of quasi-polynomials. Specifically, the notion of saturation in [KT1, Ki] works well if the stretching function is a polynomial, but not so if it is a quasipolynomial. Let f(n) be a quasi-polynomial with period l. Let fj(n), 1 ≤ j ≤ l, be the polynomials such that f(n) = fj(n) if n = j mod l. The index of f , index(f), is defined to be the smallest j such that the polynomial fj(n) is not identically zero. If f(n) is identically zero, we let index(f) = 0. If f(1) 6= 0, then clearly index(f) = 1. Definition 1.2.4 We say that f(n) is strictly saturated if for any i: fi(n) > 0 for every n ≥ 1 whenever fi(n) is not identically zero. The saturation in- dex s(f) of f is defined to be the smallest nonnegative integer such that f(n + s(f)) is strictly saturated. We also say that f(n) is saturated with index s(f). Thus f(n) is strictly saturated iff its saturation index is zero. Clearly the saturation idex is bounded above by the positivity index. Thus if f(n) is strictly positive, it is strictly saturated. Hence, PH2 (Hypothesis 1.2.3) implies: Hypothesis 1.2.5 (SH): The Littlewood-Richardson stretching quasi-polynomial (n) of arbitary classical type is strictly saturated. The polynomial time algorithm in [GCT5] works assuming SH as well. For the Littlewood-Richardson coefficient of type A, the notion of strict saturation here coincides with the notion of saturation in [KT1] since cλα,β(n) is a polynomial in that case. Knutson and Tao [KT1] also conjectured a generalized saturation property for arbitrary types. But that property, unlike the one defined above, is only conjectured to be sufficient, but not claimed to be, or expected to be necessary. For this reason, it cannot be used in the complexity-theoretic applications in this paper. There is another positivity conjecture for Littlewood-Richardson coeffi- cients that also implies the saturation theorem of Knutson and Tao. For this consider the generating function Cλα,β(t) = c̃λα,β(n)t n. (1.2) It is a rational function since c̃λ (n) is a quasi-polynomial [St1]. For type A, if c̃λ (n) is not identically zero, then Cλ (t) is a rational function of d + · · ·+ h0 (1− t)d+1 , (1.3) since c̃λα,β(n) is a polynomial [St1]. It is conjectured in [KTT] that: Hypothesis 1.2.6 (PH3:) The coefficients hi’s in eq.(1.3) are nonnegative (and h0 = 1). We shall call this the third positivity hypothesis (PH3). It clearly implies SH for Littlewood-Richardson coefficients of type A. To describe its analogue for arbitrary classical type we need a definition. Let F (t) = n f(n)t n be the generating function associated with the quasi-polynomial f(n). It is a rational function [St1]. Definition 1.2.7 We say that F (t) has a positive form, if, when f(n) is not identically zero, it can be expressed in the form F (t) = d + · · ·+ h0 i=0(1− t ai)di , (1.4) where (1) h0 = 1, and hi’s are nonnegative integers, (2) ai’s and di’s are positive integers, (3) i di = d+ 1, where d = max deg(fj(n)) is the degree of f(n). We define the modular index of this positive form to be max{ai}. If F (t) has a positive form with a0 = 1, then f(n) is strictly saturated (Definition 1.2.4); this easily follows from the power series expansion of the right hand side of eq.(1.4). The analogue of Hypothesis 1.2.6 for arbitrary classical type is: Hypothesis 1.2.8 (PH3:) The rational function Cλ (t) has a positive form, with a0 = 1, of modular index bounded by a constant depending only on the types of the simple factors of the Lie algebra of H. This too implies SH for arbitrary classical type. For types B,C,D, the constant above is 2. Experimental evidence for this hypothesis is given in Section 6.1. The analogue of the PH3, even in the more general q-setting, is known to hold for the generating function of the Kostant partition function of type A, and more generally, for a parabolic Kostant partition function; cf. Kirillov [Ki]. This also gives a support for the PH3 above, given a close relationship between Littlewood-Richardson coefficients and Kostant partition functions [FH]. 1.3 Back to the general decision problems It may be remarked that the Littlewood-Richardson problem actually never arises in the flip. It is only used as a simplest proptotype of the actual (much harder) problems that arise–namely relaxed forms of Problems 1.1.1-1.1.4. Now we turn to these problems. The goal is to generalize the preced- ing results and hypotheses for the Littlewood-Richardson coefficients to the structural constants that arise in these problems. The problem of finding a positive, combinatorial formula for the plethysm constant (Problem 1.1.2), akin to the positive Littlewood-Richardson rule, has already been recog- nized as an outstanding, classical problem in representation theory [St4]– the known formulae based on character theory mentioned in Section 1.1 are not positive, because they involve alternating signs. Indeed, existence of such a formula is a part of the first positivity hypothesis (PH1) below for the plethysm constant, and this problem is the main focus of the work in [GCT4, GCT7, GCT8, GCT9]. In view of the intensive work on the plethym constant in the literature, it has now become clear that the com- plexity of the plethysm problem (Problem 1.1.2) is far higher than that of the Littlewood-Richardson problem (Problem 1.2.1). This gap in the com- plexity is the main source of difficulties that has to be addressed. We now state the main ingradients in the plan in this paper to show that the relaxed forms of Problems 1.1.1, 1.1.2, 1.1.3, and 1.1.4, with X = G/P or a class variety, belong to P . 1.4 Saturated and positive integer programming First, we formulate a general algorithmic paradigm of saturated and positive integer programming that can be applied in the context of these problems. Let A be an m×n integer matrix, and b an integral m-vector. An integer programming problem asks if the polytope P : Ax ≤ b contains an integer point. In general, it is NP-complete. We want to define its relaxed version, which will turn out to have a polynomial time algorithm. We allow m, the number of constraints, to be exponential in n. Hence, we cannot assume that A and b are explicitly specified. Rather, it is assumed that the polytope P is specified in the form of a (polynomial-time) separation oracle in the spirit of Grötschel, Lovász and Schrijver [GLS]; cf. Section 2.3. Given a point x ∈ Rn, the separation oracle tells if x ∈ P , and if not, gives a hyperplane that separates x from P . Let fP (n) be the Ehrhart quasi-polynomial of P [St1]. By definition, fP (n) is the number of integer points in the dilated polytope nP . An integer programming problem is called saturated, if 1. The specification of P also contains a number sie(P ), called the sat- uration index estimate, with the guarantee that the saturation in- dex s(fP ) ≤ sie(P ); cf. Definition 1.2.4. In particular, this means fP (n+ sie(P )) is strictly saturated. 2. the goal of the problem is to give an efficient algorithm to decide if, given an integral relaxation parameter c > sie(P ), if cP contains an integer point. The algorithm has to work only for relaxation parameters c > sie(P ). In particular, if sie(P ) ≥ 1, the algorithm problem does not have to determine if P contains an integer point. An integer programming problem is called positive, if 1. the specification of P also contains a number pie(P ), called the pos- itivity index estimate, with the guarantee that the positivity index p(fP ) ≤ pie(P ); cf. Definition 1.2.2. In particular, this means fP (n+ pie(P )) is strictly positive. 2. the goal of the problem is to give an efficient algorithm to decide if, given an integral relaxation parameter c > pie(P ), if cP contains an integer point. Again, the algorithm has to work only for relaxation parameters c > pie(P ). Since s(fP ) ≤ p(fP ), a positive integer programming problem is also satu- rated. The following is the main complexity-theoretic result in this paper. Theorem 1.4.1 (cf. Section 3.1) 1. Index of the Ehrhart quasi-polynomial fP (n) of a polytope P presented by a separation oracle can be computed in oracle-polynomial time, and hence, in polynomial time, assuming that the oracle works in polyno- mial time. 2. A saturated, and hence positive, integer programming problem has a polynomial time algorithm. 3. Suppose the polytopes P ’s that arise in a specific decision problem have the following property: whenever P is nonempty, the Ehrhart quasi- polynomial fP (n) is “almost always” strictly saturated. Then there exists a polynomial time algorithm for deciding if P contains an integer point that works correctly “almost always”. The meaning of the phrase “almost always” in the context of the decision problems in this paper will be specified later (cf. Theorem 3.1.1). It may be remarked that the index as well as the period of the Ehrhart quasi-polynomial can be exponential in the bit length of the specification of P . In contrast to the polynomial time algorithm above to compute the index, the known algorithms to compute the period (e.g. [W]) take time that is exponential in the dimension of P . It may be conjectured that one cannot do much better: i.e., the period, unlike the index here, cannot be computed in polynomial time, in fact, even in 2o(dim(P )) time. The algorithm in Theorem 1.4.1 is based on the separation-oracle-based linear programming algorithm of Grötschel, Lovász and Schrijver [GLS], and a polynomial time algorithm for computing the Smith normal form [KB]. The paradigm of saturated integer programming is useful when one knows, a priori, a good estimate for the saturation index of the polytope under consideration, or when the saturation index is almost always zero. For example, if P is the hive polypolype for the Littlewood-Richardson co- efficient (type A), then sie(P ) = 0, by the saturation theorem [KT1], and pie(P ) = 0, by PH2 (Hypothesis 1.2.3). For the polytopes P that would arise in this paper, sie(P ) and pie(P ) would in general be nonzero, but con- jecturally always small, and sie(P ) would conjecturally be almost always zero. 1.5 Quasi-polynomiality, positivity hypotheses, and the canonical models The basic goal now is to use Theorem 1.4.1 to get polynomial time algorithms to decide nonvanishing, modulo small relaxation, of the structural constants in Problems 1.1.1, 1.1.2, 1.1.3 and 1.1.4, with X = G/P or a class variety. The main results in this paper which go towards this goal are as follows. Quasi-polynomiality We associate stretching functions with the structural constants in Prob- lems 1.1.1-1.1.4, akin to the stretching function c̃λα,β(n) in eq.(1.1) associ- ated with the Littlewwod-Richardson coefficient, and show that they are quasipolynomials; cf. Chapter 4. (But their periods need not be constants, as in the case of Littlewood-Richardson coefficients; in fact, they may be exponential in general.) In particular, this proves Kirillov’s conjecture [Ki] for the plethysm constants. The proof is an extension of Brion’s remarkable proof (cf. [Dh]) of quasi-polynomiality of the stretching function associ- ated with the Littlewood-Richardson coefficient. The main ingradient in the proof is Boutot’s result [Bou] that singularities of the quotient of an affine variety with rational singularities with respect to the action of a re- ductive group are also rational. This is a generalization of an earlier result of Hochster and Roberts [Ho] in the theory of Cohen-Macauley rings. Saturation and positivity hypotheses Using the stretching quasipolynomials above, we formulate (cf. Section 3.3) analogues of the saturation and positivity hypotheses SH, PH1,PH2,PH3 in Section 1.2 for the structural constants in Problems 1.1.1-1.1.3 and Prob- lem 1.1.4, with X = G/P or a class variety. As for Littlewood-Richardson coefficients, it turns out that PH2 implies SH. The hypotheses PH1 and SH (more strongly, PH2) together imply that the problem of deciding nonvan- ishing of the structural constant in any of these problems, modulo a small relaxation, can be transformed in polynomial time into a saturated (more strongly, positive) integer programming problem, and hence, can be solved in polynomial time by Theorem 1.4.1. In particular, this shows that all the relaxed decision problems that arise in flip (cf. Hypothesis 1.1.6) have polynomial time algorithms, assuming these positivity hypotheses. Though these algorithms are elementary, the positivity hypotheses on which their correctness depends turn out to be nonelementary. They are intimately linked to the fundamental phenomena in algebraic geometry and the theory of quantum groups, as we shall see. We also give theoretical and experimental results in support of these hypotheses; cf. Chapter 4-6. Canonical models The proofs of quasi-polynomiality mentioned above also associate with each structural constant under consideration a projective scheme, called the canon- ical model, whose Hilbert function coicides with the stretching quasi-polynomial associated with that structural constant, akin to the model associated by Brion [Dh] with the Littlewood-Richardson coefficient. These canonical models play a crucial role in the approach to the posivity hypotheses sug- gested in Section 1.7. 1.6 The plethysm problem We now give precise statements of these results and hypotheses for the plethysm problem (Problem 1.1.2). It is the main prototype in this paper, which illustrates the basic ideas. Precise statements for the more general Problems 1.1.3 and 1.1.4 appear in Section 3.3. As for the Littlewood-Richardson coefficients (cf.(1.1)), Kirillov [Ki] as- sociates with a plethysm constant aπ a stretching function ãπλ,µ(n) = a nλ,µ, (1.5) and a generating function Aπλ,µ(t) = anπnλ,µt (Note that µ is not stretched in these definitions.) He conjectured that Aπ (t) is a rational function. This is verified here in a stronger form: Theorem 1.6.1 (a) (Rationality) The generating function Aπλ,µ(t) is ratio- (b) (Quasi-polynomiality) The stretching function ãπ (n) is a quasi-polynomial function of n. This is equivalent to saying that all poles of Aπλ,µ(t) are roots of unity, and the degree of the numerator of Aπλ,µ(t) is strictly smaller than that of the denominator. (c) There exist graded, normal C-algebras S = S(aπ ) = ⊕nSn, and T = T (aπ ) = ⊕nTn such that: 1. The schemes spec(S) and spec(T ) are normal and have rational singu- larities. 2. T = SH , the subring of H-invariants in S, where H = GLn(C) as in Problem 1.1.2, 3. The quasi-polynomial ãπλ,µ(n) is the Hilbert function of T . In other words, it is the Hilbert function of the homogeneous coordinate ring of the projective scheme Proj(T ). (d) (Positivity) The rational function Aπλ,µ(t) can be expressed in a positive form: Aπλ,µ(t) = h0 + h1t+ · · ·+ hdt j(1− t a(j))d(j) , (1.6) where a(j)’s and d(j)’s are positive integers, j d(j) = d + 1, where d is the degree of the quasi-polynomial ãπ (n), h0 = 1, and hi’s are nonnegative integers. The specific rings S(aπλ,µ) and T (a λ,µ) constructed in the proof of The- orem 1.6.1 are very special. We call them canonical rings associated with the plethysm constant aπ . We call Y (aπ ) = Proj(S(aπ )), and Z(aπ Proj(T (aπ )) the canonical models associated with aπ . The canonical rings are their homogenous coordinate rings. It may be remarked that the analogue of Theorem 1.6.1 (b) for Littlewood- Richardson coefficients has an elementary polyhedral proof. Specifically, the Littlewood-Richardson stretching function c̃λα,β(n) of any type is a quasi- polynomial since it coincides with the Ehrhart quasi-polynomial of the BZ- polytope [BZ]. Similarly, the analogue of Theorem 1.6.1 (d) for Littlewood- Richardson coefficients follows from Stanley’s positivity theorem for the Ehrhart series of a rational polytope (which is implicit in [St3]). These polyhderal proofs cannot be extended to the plethysm constant at this point, since no polyhedral expression for them is known so far–in fact, this is a part of the conjectural positivity hypothesis PH1 below. In contrast, Brion’s proof in [Dh] of quasi-polynomiality of c̃λα,β(n) can be extended to prove Theorem 1.6.1 since it does not need a polyhedral interpretation for aπλ,µ. But Boutot’s result [Bou] that it relies on is nonelementary (because it needs resolution of singularities in characteristic zero [Hi], among other things). We also give an elementary (nonpolyhedral proof) for Theorem 1.6.1 (a) (ra- tionality). But this does not extend to a proof of quasipolynomiality for all n, which turns out to be a far delicate problem. It is crucial in the context of saturated integer programming. Theorem 1.6.2 (Finitely generated cone) For a fixed partition µ, let Tµ be the set of pairs (π, λ) such that the irreducible representation Vπ(H) of H = GLn(C) occurs in the irreducible representation Vλ(G) of G = GL(Vµ(H)) with nonzero multiplicity. Then Tµ is a finitely generated semigroup with respect to addition. This is proved by an extension of Brion and Knop’s proof of the analogous result for Littlewood-Richardson coefficients based on invariant theory. In the case of Littlewood-Richardson coefficients, this again has an elementary polyhedral proof [Z]. Theorem 1.6.3 (PSPACE) Given partitions λ, µ, π, the plethysm constant aπ can be computed in poly(〈λ〉, 〈µ〉, 〈π〉) space. The main observation in the proof of Theorem 1.6.3 is that the oldest algorithm for computing the plethysm constant, based on the Weyl character formula, can be efficiently parallelized so as to work in polynomial parallel time using exponentially many processors. After this, the result follows from the relationship between parallel and space complexity classes. It may be remarked that the known algorithms for computing aπ in the literature– e.g., the one based on Klimyk’s formula [FH]–take exponential time as well as space. Theorems 1.6.1, 1.6.2 and 1.6.3 lead to the following conjectural sat- uration and positivity hypotheses for the plethysm constant. These are analogues of PH1,PH2,PH3, SH in Section 1.2 for Littlewood-Richardson coefficients. Hypothesis 1.6.4 (PH1) For every (λ, µ, π) there exists a polytope P = P π ⊆ Rm such that: (1) The Ehrhart quasi-polynomial of P coincides with the stretching quasi- polynomial ãπ (n) in Theorem 1.6.1. (This means P is given by a linear system of the form Ax ≤ b, (1.7) where A does not depend on λ and π and b depends only on λ and π in a homogeneous, linear fashion.) In particular, aπλ,µ = φ(P ), (1.8) where φ(P ) is equal to the number of integer points in P . (2) The dimension m of the ambient space, and hence the dimension of P as well, and the bitlength of every entry in A are polynomial in the bitlength of µ and the heights of λ and π. (3) Whether a point x ∈ Rm lies in P can be decided in poly(〈λ〉, 〈µ〉, 〈π〉, 〈x〉) time. That is, the membership problem belongs to the complexity class P . If x does not lie in P , then this membership algorithm also outputs, in the spirit of [GLS], the specification of a hyperplane separating x from P . The first statement here, in particular, would imply a positive, polyhedral formula for a , in the spirit of the known positive polyhedral formulae for the Littlewood-Richardson coefficients in terms of the BZ- [BZ], hive [KT1] or other types of polytopes [Dh]. It would also imply polyhedral proofs for Theorem 1.6.1 (a), (b), (d), and Theorem 1.6.2. Conversely, Theorem 1.6.1 (a), (b), (d), and Theorem 1.6.2 constitute a theoretical evidence for exis- tence of such a positive polyhedral formula. The second statement in PH1 is justified by Theorem 1.6.3. Specifi- cally, it should be possible to compute the number of integer points in P in PSPACE in view of Theorem 1.6.3. If dim(P ) and m were exponential, then the usual algorithms for this problem, e.g. Barvinok [Bar], cannot be made to work in PSPACE. Indeed, it may be conjectured that the number of integer points in a general polytope P ⊆ Rm can not be computed in o(m) space. The number of constraints in the hive [KT1] or the BZ-polytope [BZ] for the Littlewood-Richardson coefficient cλ is polynomial in the number of parts of α, β, λ. In contrast, the number of constraints defining P π be exponential in the 〈µ〉 and the number of parts of λ and π. But this is not a serious problem. As long as the faces of the polytope P have a nice description, the third statement in PH1 is a reasonable assumption. This has been demonstrated in [GLS] for the well-behaved polytopes in combina- torial optimization with exponentially many constraints. The situation in representation theory should be similar, or even better. For example, the facets of the hive polytope [KT1] are far nicer than the facets of a typical polytope in combinatorial optimization. It is known that membership in a polytope is a “very easy” problem. Formally, if a polytope has polynomially many constraints, this problem belongs to the complexity class NC ⊆ P [KR], the subclass of problems with efficient parallel algorithms, which is very low in the usual complexity hierarchy. Even if the number of constraints of P πλ,µ in PH1 is exponen- tial, the membership problem may still be conjectured to be in NC (cf. Remarkrnc)–which would be “very easy” compared to the decision problem we began with (Problem 1.1.2). For this reason, PH1 is primarily a mathe- matical positivity hypothesis as against PHflip (Hypothesis 1.1.6), and the positive, polyhedral formula for aπ in (1.8) is its main content. The remaining positivity hypotheses are purely mathematical. They generalize SH,PH2 and PH3 for the Littlewood-Richardson coefficients to the plethysm constants. We turn their specification next. We can begin by asking if the stretching quasipolynomial ãπ (n) is strictly saturated or positive. This need not be so. The recent article [Ro] shows that strict saturation need not hold for the Kronecker coefficients, as was conjectured in the earlier version of this paper. A similar phenomenon was also reported in [GCT7, GCT8], where it was observed that the structural constants of the nonstandard quantum groups associated with the plethysm problem (of which the Kronecker problem is a special case) need not satisfy an analogue of PH2. But it was observed there that the positivity (and hence saturation) indices of these structural constants are small, though not always zero; eg. see Figures 30,33,35 in [GCT8]. The same can be expected here. This is also supported by the experimental evidence in [BOR] where too it may be observed that the positivity index is small. Furthermore, in the special case (n = 2) of the Kronecker problem analysed in [BOR], the saturation index is zero for almost all Kronecker coefficients. These considerations suggest: Hypothesis 1.6.5 (SH) (a): The saturation index (Definition 1.2.4) of ãπ (n) is bounded by a poly- nomial in the dimension of G in Problem 1.1.2 and the heights of λ and π. This means there exist absolute nonnegative constants c and c′, independent of n, λ, µ and π, such that the saturation index is bounded above by chc where h = dimG+ htλ+ htπ. (b): The quasi-polynomial ãπ (n) is strictly saturated, i.e. the saturation index is zero, for almost all λ, µ, π. Specifically, the density of the triples (λ, µ, π) of total bit length N with nonzero aπ for which the saturation index is not zero is less than 1/N c , for any positive constant c′′, as N → ∞. A stronger form of (a) is: Hypothesis 1.6.6 (PH2) The positivity index (Definition 1.2.2) of the stretching quasi-polynomial ãπλ,µ(n) is bounded by a polynomial in the di- mension of G and the heights of λ and π. The following is another stronger form of SH (a). For this, we observe that the positive rational form in Theorem 1.6.1 (d) is not unique. Indeed, there is one such form for every h.s.o.p. (homogeneous sequence of param- eters) of the homogenenous coordinate ring S; the a(j)’s in eq.(1.6) are the degrees of these parameters. Kirillov asked if the only possible pole of Aπ is at t = 1–i.e. if a (n) is a polynomial. This is not so (cf. Section 6.2). But it may be conjectured that the structural constants a(j)’s are small. Specifcally, consider an h.s.o.p. of S with a (lexicographically) minimum degree sequence, and call the (unique) positive rational form in Theorem 1.6.1 (d) associated with such an h.s.o.p. minimal. The modular index χ(aπ ) of the plethysm constant is defined to be the modular index (Definition 1.2.7) of this minimal positive form. Then: Hypothesis 1.6.7 (PH3) The function Aπ (t) associated with aπ has a positive rational form with modular index bounded by a polynomial in the dimension of G and the heights of λ and π. More specifically, this is so for the minimial positive rational form of (t) as above; i.e., the modular index χ(aπ ) is itself bounded by a poly- nomial in the dimension of G and the heights of λ and π. This is a conjectural analogue of a stronger form of PH3 for Littlewood- Richardson coefficients (Hypothesis 1.2.6), which says that the modular in- dex of a Littlewood-Richardson coefficient, defined similarly, is one. PH3 here would imply that the period of Aπ (t) is smooth–i.e. has small prime factors–though it may be exponential in the heights of λ, µ, π. It can be shown that PH3 implies SH (a) (Section 3.3). The following result addresses the second arrow in Figure 1.1 in the context of the relaxed decision problem for the plethysm constant: Theorem 1.6.8 The complexity theoretic positivity hypothesis PHflip (Hy- pothesis 1.1.6) for the plethysm constant is implied by the mathematical positivity hypotheses PH1 and SH above. Specifically, assuming PH1 and (a) Nonvanishing of abπbλ,bµ for any b > ch c′ , with c, c′, h as in SH, can be decided in O(poly(〈λ〉, 〈µ〉, 〈π〉, 〈b〉)) time. (b) There is an O(poly(〈λ〉, 〈µ〉, 〈π〉)) time algorithm for deciding if aπ nonvanishing, which works correctly on almost all λ, µ and π; almost all means the same as in SH. Here (a) follows by applying Theorem 1.4.1 (2) to the polytope P π PH1, and letting the positivity index estimate for this polytope be chc ; (b) follows from Theorem 1.4.1 (3). Evidence for the positivity hypotheses in special cases Littlewood-Richardson coefficients are special cases of (generalized) plethym constants. We have already seen that PH1 holds in this case, and that there is considerable experimental evidence for PH2 and PH3 (Section 1.2). An- other crucial special case of the plethym problem is the Kronecker prob- lem (Problem 1.1.1)–in fact, this may be considered to be the crux of the plethysm problem. It follows from the results in [GCT9] that PH1 holds for the Kronecker problem when n = 2; the earlier known formulae [RW, Ro] for the Kronecker coefficient in this case are not positive. It can also be seen from the experimental evidence in [BOR] that the saturation and positivity indices of the Kronecker coefficient, for n = 2, are very small, and almost always zero. We also give in Chapter 6 additional experimental evidence for PH2 for another basic special case of Problem 1.1.3, with H therein being the symmetric group. 1.7 Towards PH1, SH, PH2,PH3 via canonial bases and canonical models In this section, we suggest an approach to prove PH1, SH, PH2 and PH3 for the plethysm constant and the analogous hypotheses for the other structural constants in Problems 1.1.3, and 1.1.4, with X = G/P or a class variety. In the case of Littlewood-Richardson coefficients of type A, PH1 and SH have purely combinatorial proofs. But it seems unrealistic to expect such proofs of the saturation and positivity hypotheses for the plethysm and other structural constants under consideration here given their substantially higher complexity. The approach that we suggest is motivated by the proof of PH1 for Littlewood-Richardson coefficients of arbitrary types based on the canonical (local/global crystal) bases of Kashiwara and Lusztig for representations of Drinfeld-Jimbo quantum groups [Dh, Kas2, Li, Lu2, Lu4]. By a Drinfeld- Jimbo quantum group we shall mean in this paper quantization Gq of a complex, semisimple group G as in [RTF] that is dual to the Drinfeld-Jimbo quantized enveloping algebra [Dri]. Canonical bases for representions of a Drinfeld-Jimbo quantum group in type A are intimately linked [GrL] to the Kazhdan-Lusztig basis for Hecke algebras [KL1, KL2]. A starting point for the approach suggested here is: Observation 1.7.1 (PH0) The homogeneous coordinate rings of the canon- ical models associated by Brion with the Littlewood-Richardson coefficients have quantizations endowed with canonical bases as per Kashiwara and Lusztig. This is a consequence of the work of Kashiwara [Kas3] and Lusztig [Lu3, Lu4]; see Proposition 4.2.1 for its precise statment. This is why we call the models here canonical models. We shall refer to the property above as the zeroeth positivity hypothesis PH0. Positivity here refers to the deep characteristic positivity property of the canonical basis proved by Lusztig: namely its multiplicative and comul- tiplicative structure constants are nonnegative. For this reason, we say that a canonical basis is positive. Similar positivity property is also known for the Kazhdan-Lusztig basis [KL2]. The proofs of these positivity properties are based on the Riemann hypothesis over finite fields (Weil conjectures) [Dl] and the related work of Beilinson, Bernstein, Deligne [BBD]. The property above is called PH0 because it implies PH1 for Littlewood- Richardson coefficients of arbitrary types. Specifically, the latter is a formal consequence of the abstract properties of these canonical bases and is inti- mately related to their positivity; cf. Section 4.2.1, and [Dh, Kas2, Li, Lu4]. The saturation hypothesis SH in type A [KT1] is a refined property of the polyhedral formulae in PH1. In Section 4.2 we suggest an approach to prove SH, PH2 and PH3 for arbitrary types based on the properties of these canon- ical bases. All this indicates that for the Littlewood-Richardson problem PH1, SH, PH2 and PH3 are intimately linked to PH0. This suggests the following approach for proving PH1, SH, PH2 and PH3 for the plethysm and other structural constants under consideration in this paper (cf. Section 4.2.2): 1. Construct quantizations of the homogeneous coordinate rings of the canonical models associated with these structural constants, 2. Show that they have canonical bases in some appropriate sense thereby extending PH0 to this general setting. 3. Prove PH1, SH, PH2, and PH3 by a detailed analysis and study of these canonical bases as per this extended PH0, just as in the case of Littlewood-Richardson coefficients. Pictorially, this is depicted in Figure 1.2. Quantizations of the homogeneous coordinate rings of the canonical models associated with Littlewood-Richardson coefficients and their posi- tive canonical bases are constructed using the theory Drinfeld-Jimbo quan- tum group. In type A, it is intimately related to the theory of Hecke al- gebras. But, as expected, the theories of Drinfeld-Jimbo quantum groups and Hecke algebras do not work for the plethysm problem. What is needed is a quantum group and a quantized algebra that can play the same role in the plethysm problem that the Drinfeld-Jimbo quantum group and the Hecke algebra play in the Littlewood-Richardson problem. These have been constructed in [GCT4] for the Kronecker problem (Problem 1.1.1) and in [GCT7] for the generalized plethysm problem (Problem 1.1.2). We shall call them nonstandard quantum groups and nonstandard quantized algebras; cf. Section 4.3 for their brief overview. In the special case of the Littlewood- Richardson problem, these specialize to the Drinfeld-Jimbo quantum group and the Hecke algebra, respectively. The article [GCT8] gives conjecturally correct algorithms to construct canonical bases of the matrix coordinate rings of the nonstandard quantum groups and of nonstandard algebras that have conjectural positivity properties analogous to those of the canonical Construction of quantizations of the coordinate rings of canonical models Construction of canonical bases for these quantizations (PH0) Positivity and saturation hypotheses PH1, SH | Polynomial-time algorithms for the relaxed decision problems Figure 1.2: Pictorial depiction of the approach (global crystal) bases, as per Kashiwara and Lusztig, of the coordinate ring of the Drinfeld-Jimbo quantum group, and the Kazhdan-Lusztig basis of the Hecke algebra. These conjectures lie at the heart of the approach suggested here, since they are crucial for the extension of PH0 (cf. Figure 1.2) to the general setting here. Their verification seems to need substantial extension of the work surrounding the Riemann hypothesis over finite fields mentioned above. 1.8 Basic plan for implementing the flip The main application of the results and hypotheses in this paper in the context of the flip is the following result. As mentioned in Section 1.1, and described in more detail in Sections 7.6-7.7, each lower bound problem, such as the P vs. NP problem over C, is reduced in [GCT1, GCT2] to the prob- lem of proving obstructions to embeddings among the class varieties that arise in the problem. In Chapter 7 we define a robust obstruction, which is an obstruction that is well behaved with respect to relaxation, and whose validity (correctness) depends only on an appropriate PH1 but not SH. It is conjectured that in each of the lower bound problems under consideration, robust obstructions exist (Section 7.6.6). In the lower bound problems un- der consideration, ultimately one is only interested in proving existence of obstructions. So one may as well search for only robust obstructions. Theorem 1.8.1 (cf. Chapter 7) Consider the P vs. NP or the NC vs. P#P problem over C [GCT1]. Assume that the homogeneous coordinate rings of the relevant class varieties [GCT1, GCT2] in this context have ra- tional singularities. Also assume that the structural constants associated with these class varieties satisfy analogous PH1 as specified in Chapter 7. Then: (a) The problem of verifying a robust obstruction in each of these problems belongs to P , so also the relaxed form of the problem of verifying any ob- struction (not necessarily robust). (b) There exists an explicit family of robust obstructions in each of these problems assuming an additional hypothesis OH specified in Chapter 7; the meaning of the term explicit is also given there. (b) The problem of deciding existence of a geometric obstruction also belongs to P , assuming a stronger form of PH1 specified in Chapter 7. Here geomet- ric obstruction is a simpler type of robust obstruction, defined in Chapter 7, which is conjectured to exist in the lower bound problems under considera- tion. For a precise statement of this theorem, see Chapter 7. This theorem needs only PH1, but not SH, which is only needed to argue why robust obstructions should exist (Section 7.6.6), and furthermore, it is only needed for Problems 1.1.1-1.1.3 and not for the GIT Problem 1.1.4. Thus PH1 is the main positivity hypothesis of GCT in the context proving existence of (robust) obstructions for the lower bound problems under consideration. A basic plan for implementing the flip suggested by the considerations above is summarized in Figure 1.3. It is an elaboration of Figure 1.1. Ques- tion marks in the figure indicate open problems. 1.9 Organization of the paper The rest of this paper is organized as follows. Negative hypotheses in complexity theory (Lower bound problems) The flip Positive hypotheses in complexity theory (Upper bound problems) Saturated and positive integer programming, and the quasi-polynomiality results in this paper Mathematical saturation and positivity hypotheses: PH1,SH (PH2,3) Construction of the canonical models in this paper, and construction of the quantum groups in GCT4,7 (PH0): Construction of quantizations of the coordinate rings of the canonical models and their canonical bases (?): Problems related to the Riemann Hypothesis over finite fields, and their generalizations Figure 1.3: A basic plan for implementing the flip In Chapter 2 we describe the basic complexity theoretic notions that we need in this paper and describe their significance in the context of represen- tation theory. In Chapter 3, we give a polynomial time algorithm for saturated integer programming (Theorem 1.4.1), and give precise statements of the results and positivity hypotheses for Problems 1.1.3 and 1.1.4 (with X = G/P or a class variety) mentioned in Section 1.5. These generalize the ones given in Section 1.6 for the plethysm constant. The framework of saturated in- teger programming in this paper may be applicable to many other struc- tural constants in representation theory and algebraic geometry, such as the Kazhdan-Lusztig polynomials (cf. Sections 3.7). In Chapter 4, we prove the basic quasi-polynomiality results–Theorem 1.6.1 and its generalizations for Problems 1.1.3 and 1.1.4. We also define canonical models for the structural constants under consideration, and briefly describe the relevance of the nonstandard quantum groups and the related results in [GCT4, GCT7, GCT8] in the context of quantizing the coordinate rings of these canonical models and extending PH0 to them (Figure 1.2). In Chapter 5, we prove the basic PSPACE results–Theorem 1.6.3 and its extensions for the various cases of Problem 1.1.3. In Chapter 6, we give experimental evidence for the positivity hypotheses PH2 and PH3 in some special cases of the Problems 1.1.1-1.1.4. In Chapter 7, we describe an application (Theorem 1.8.1) of the re- sults and positivity hypotheses in this paper to the problem of verifying or discovering a robust obstruction, i.e., a “proof of hardness” [GCT2] in the context of the P vs. NP and the permanent vs. determinant problems in characteristic zero. 1.10 Notation We let 〈X〉 denote the total bitlength of the specification of X. Here X can be an integer, a partition, a classifying label of an irreducible representation of a reductive group, a polytope, and so on. The exact meaning of 〈X〉 will be clear from the context. The notation poly(n) means O(na), for some constant a. The notation poly(n1, n2, . . .) similarly means bounded by a polynomial of a constant degree in n1, n2, . . .. Given a reductive group H, Vλ(H) denotes the irreducible representation of H with the classifying label λ. The meaning depends on H. Thus if H = GLn(C), λ is a partition and Vλ(H) the Weyl module indexed by λ, if H = Sm, then λ is a partition of size |λ| = m, and Vλ(H) the Specht module indexed by λ, and so on. Chapter 2 Preliminaries in complexity theory In this chapter, we recall basic definitions in complexity theory, introduce additional ones, and illustrate their significance in the context of represen- tation theory. 2.1 Standard complexity classes As usual, P , NP and PSPACE are the classes of problems that can be solved in polnomial time, nondeterministic polynomial time, and polyno- mial space, respectively. The class of functions that can be computed in polynomial time (space) is sometimes denoted by FP (resp. FPSPACE). But, to keep the notation simple, we shall denote these classes by P and PSPACE again. Let SPACE(s(N)) denote the class of problems that can be solved in O(s(N)) space on inputs of bit length N ; by convention s(N) counts only the size of the work space. In other words, the size of the input, which is on the read-only input tape, and the output, which is on the write-only output tape is not counted. Hence s(N) can be less than the size of the input or the output, even logarithmic compared to these sizes. The class space(log(N)) is denoted by LOGSPACE. An algorithm is called strongly polynomial [GLS], if given an input x = (x1, . . . , xk), 1. the total number of arithmetic steps (+, ∗,− and comparisones) in the algorithm is polynomial in k, the total number of input parameters, but does not depend 〈x〉, where 〈x〉 = i〈xi〉 denotes the bitlength of 2. the bit length of every intermediate operand in the computation is polynomial in 〈x〉. Clearly, a strongly polynomial algorithm is also polynomial. let strong P ⊆ P denote the subclass of problems with strongly polynomial time algorithms. The counting class associated with NP is denoted by #P . Specifically, a function f : Nk → N, where N is the set of nonnegative integers, is in #P if it has a formula of the form: f(x) = f(x1, · · · , xk) = χ(x, y), (2.1) where χ is a polynomial-time computable function that takes values 0 or 1, and y runs over all tuples such that 〈y〉 = poly(〈x〉). The formula (2.1) is called a #P -formula. An important feature of a #P -formula in the context of representation theory is that it is positive; i.e., it does not contain any alternating signs. The formula (2.1) is called a strong #P -formula, if, in addition, l is polynomial in k and χ is a strongly polynomial-time computable function. Let strong #P be the class of functions with strong #P -fomulae. It is known and easy to see that #P ⊆ PSPACE. (2.2) 2.1.1 Example: Littlewood-Richardson coefficients By the Littlewood-Richardson rule [FH], the coefficient cλ (cf. Prob- lem 1.2.1) in type A is given by: cλα,β = χ(T ), (2.3) where T runs over all numbering of the skew shape λ/α, and χ(T ) is 1 if T is a Littlewood-Richardson skew tableau of content β, and zero, other- wise. The total number of entries in T is quadratic in the total number of nonzero parts in α, β, λ, and the number of arithmetic steps needed to com- pute χ(T ) is linear in this total number. Hence (2.3) is a strong #P -formula, and Littlewood-Richardson function c(α, β, λ) = cλα,β belongs to strong #P . It may be remarked that the character-based formulae for the Littlewood- Richardson coefficients are not #P -formulae, since they involve alternat- ing signs. But the algorithms based on the these formulae for computing Littlewood-Richardson coefficients run in polynomial space. Thus, from the perspective of complexity theory, the main significance of the Littlewood- Richardson rule is that it puts the problem, which at the surface is only in PSPACE, in its smaller subclass (strong) #P . Though the Littlewood-Richardson rule is often called efficient in the representation theory literature, it is not really so from the perspective of complexity theory. Because computation of cλ using this formula takes time that is exponential in both the total number of parts of α, β and λ, and their bit lengths. This is inevitable, since this problem is #P -complete [N]. Specifically, this means there is no polynomial time algorithm to compute , assuming P 6= NP . As remarked in earlier, nonzeroness (nonvanishing) of cλ can be decided in poly(〈α〉, 〈β〉, 〈λ〉) time; [DM2, GCT3, KT1]. Furthermore, the algorithm in [GCT3] is strongly polynomial; i.e., the number of arithemtic steps in this algorithm is a polynomial in the total number of parts of α, β, λ, and does not depend on the bit lengths of α, β, λ. Hence the problem of deciding nonvanishing of cλ (type A) belongs to strong P . The discussion above shows that the Littlewood-Richardson problem is akin to the problem of computing the permanent of an integer matrix with nonnegative coefficients. The latter is known to be #P -complete [V], but its nonvanishing can be decided in polynomial time, using the polynomial- time algorithm for finding a perfect matching in bipartite graphs [Sc]. If the positivity hypotheses in this paper hold, the situation would be similar for many fundamental structural constants in representation theory and algebraic geometry in a relaxed sense. 2.2 Convex #P Next we want to introduce a subclass of #P called convex #P . Given a polytope P ⊆ Rl, let χP denote the characteristic (membership) function of P : i.e., χP (y) = 1, if y ∈ P , and zero otherwise. We say that f = f(x) = f(x1, . . . , xk) has a convex #P -formula if, for every x ∈ Z there exists a convex polytope (or, more generally, a convex body) Px ⊆ R such that 1. The membership function χPx(y) can be computed in poly(〈x〉, 〈y〉) time, each integer point in Px has O(poly(〈x〉)) bitlength, and f(x) = φ(Px), (2.4) where φ(Px) denotes the number of integer points in Px. Equivalently, f(x) = χPx(y), (2.5) where y runs over tuples in Zl of poly(〈x〉) bitlength, and χPx denotes the membership function of the polytope Px. Equation (2.5) is similar to eq.(2.1). The main difference is that χ is now the membership function of a convex polytope. Clearly, eq.(2.5), and hence, eq.(2.4) is a #P -formula, when χPx can be computed in polynomial time. Let convex #P be the subclass of #P consisting of functions with convex #P -formulae. We say that eq.(2.4) is a strongly convex #P -formula, if the character- istic function of Px is computable in strongly polynomial time. Let strongly convex #P be the subclass of #P consisting of functions with strongly con- vex #P -formulae. We do not assume in eq.(2.4) that the polytope Px is explicitly specified by its defining constraints. Rather, we only assume, following [GLS], that we are given a computer program, called a membership oracle, which, given input parameters x and y, tells whether y ∈ Px in poly(〈x〉, 〈y〉) time. If the number of constraints defining Px is polynomial in 〈x〉, then it is possible to specify Px by simply writing down these constraints. In this case the membership question can be trivially decided in polynomial time–in fact, even in LOGSPACE–by verifying each constraint one at a time. This would not work if Px has exponentially many constraints. In good cases, it is possible to answer the membership question in polynomial time even if Px has exponentially many facets. Many such examples in combinatorial optimization are given in [GLS]. One such illustrative example in repre- sentation theory is given in Section 2.2.2. The polytopes that would arise in the plethysm and other problems of main interest in this paper are also expected to be of this kind. We now illustrate the notion of convex #P with a few examples in rep- resentation theory. 2.2.1 Littlewood-Richardson coefficients A geeneralized Littlewood-Richardson coefficient cλα,β for arbitrary semisim- ple Lie algebra (Problem 1.2.1) has a strong, convex #P -formula, because cλα,β = φ(P α,β), where P λα,β is the BZ-polytope [BZ] associated with the triple (α, β, λ). It is easy to see from the description in [BZ] that the number of defin- ing constraints of P λα,β is polynomial in the total number of parts (coor- dinates) of α, β, λ. Given α, β, λ, these constraints can be computed in strongly polynomial time. Hence, the membership problem for P λ belongs to LOGSPACE ⊆ P . It follows that the Littlewood-Richardson function c(α, β, λ) = cλ belongs to strongly convex #P . 2.2.2 Littlewood-Richardson cone We now give a natural example of a polytope in representation theory, the number of whose defining constraints is exponential, but whose membership function can still be computed in polynomial time. Given a complex, semisimple, simply connected groupG, let the Littlewood- Richardson semigroup LR(G) be the set of all triples (α, β, λ) of dominant weights of G such that the irreducible module Vλ(G) appears in the tensor product Vα(G) ⊗ Vβ(G) with nonzero multiplicity [Z]. Brion and Knop [El] have shown that LR(G) is a finitely generated semigroup with respect to addition. This also follows from the polyhedral expression for Littlewood- Richardson coefficients in terms of BZ-polytopes [Z]. Let LRR(G) be the polyhedral cone generated by LR(G). When G = GLn(C), the facets of LRR(G) have an explicit description by the affirmative solution to Horn’s conjecture in [Kl, KT1]. But their number can be quite large (possibly exponential). Nevertheless, membership of any rational (α, β, λ) (not necessarily integral) in LRR(G) can be decided in strongly polynomial time. This is because LRR(G) is the projection of a polytope P (G), the num- ber of whose constraints is polynomial in the heights of α, β, λ [Z]. If φ : P (G) → LR(G) is this projection, we can choose P (G) so that for any integral (α, β, λ), φ−1(α, β, λ) is the BZ-polytope associated with the triple (α, β, λ). To decide if (α, β, λ) ∈ LR(G), we only have to decide if the polytope φ−1(α, β, λ) is nonempty. This can be done in strongly polynomial time using Tardos’ linear programming algorithm [Ta]. 2.2.3 Eigenvalues of Hermitian matrices Here is another example of a polytope in representation theory with expo- nentially many facets, whose membership problem can still belong to P . For a Hermitian matrix A, let λ(A) denote the sequence of eigenvalues of A arranged in a weakly decreasing order. Let HEr be the set of triple (α, β, λ) ∈ Rr such that α = λ(A + B), β = λ(A), λ = λ(B) for some Hermitian matrices A and B of dimension r. It is closely related to the Littlewood-Richardson semisgroup LRr = LR(GLr(C)): HEr ∩ P r = LRr, where Pr is the semigroup of partitions of length ≤ r. I. M. Gelfand asked for an explicit description of HEr. Klyachko [Kl] showed that HEr is a convex polyhedral cone. An explicit description of its facets is now known by the affirmative answer to Horn’s conjecture. But their number may be exponential. Hence, membership in HEr is still not easy to check using this explicit description. This leads to the following complexity theoretic variant of Gelfand’s question: Question 2.2.1 Does the memembership problem for HEr belong to P? Given that the answer is yes for the closely related LRr = LR(GLr(C)) (Section 2.2.2), this may be so. If HEr were a projection of some polytope with polynomially many facets, this would follow as in Section 2.2.2. But this is not necessary. For example, Edmond’s perfect matching polytope for non-bipartite graphs is not known to be a projection of any polytope with polynomially many constraints. Still the associated membership problem belongs to P [Sc]. 2.3 Separation oracle Suppose P ⊆ Rl is a convex polytope whose membership function χP is polynomial time computable. If χP (y) = 0 for some y ∈ R r, it is natural to ask, in the spirit of [GLS], for a “proof” of nonmembership in the form of a hyperplane that separates y from P . In this paper, we assume that all polytopes are specified by the separation oracle. This is a computer program, which given y, tells if y ∈ P , and if y 6∈ P , returns such a separting hyperplane as a proof of nonmembership. We assume that the hyperplane is given in the form l = 0, where a linear function l such that P is contained in the half space l ≥ 0, but l(y) < 0. Furthermore. we assume that P is a well-described polyhedron in the sense of [GLS]. This means P is specified in the form of a triple (χP , n, φ), where P ⊆ R n, χP is a program for computing the membership function given y ∈ Rn, and there exists a system of inequalities with rational coefficients having P as its solution set such that the encoding bit length of each inequality is at most φ. We define the encoding length 〈P 〉 of P as n+ φ. We also assume that the separation oracle works in O(poly(〈P 〉, 〈y〉) time. For example, the polynomial time algorithm for the membership function of the Littlewood-Richardson cone (cf. Section 2.2.2) can be easily modified to return a separating hyperplane as a proof of nonmembership. In what follows, we shall assume, as a part of the definition of a convex #P -formula, that Px in (2.4) is a well-described polyhedron specified by a separation oracle that works in polynomial time with 〈Px〉 = poly(〈x〉). These additional requirements are needed for the saturated integer program- ming algorithm in Chapter 3. Chapter 3 Saturation and positivity In this chapter we describe (Section 3.1) a polynomial time algorithm for saturated and positive integer programming (Theorem 1.4.1). In Section 3.3 we state the main results and positivity hypotheses for the relaxed forms of Problem 1.1.3 and Problem 1.1.4, with X = G/P or a class variety therein. Together they say that these relaxed decision problems can be efficiently transformed into saturated (more strongly, positive) integer programming problems, and hence can be solved in polynomial time. 3.1 Saturated and positive integer programming We begin by proving Theorem 1.4.1. Let P ⊆ Rn be a polytope given by a separation oracle (Section 2.3). Let 〈P 〉 be the encoding length of P as defined in Section 2.3. An oracle- polynomial time algorithm [GLS] is an algorithm whose running time is O(poly(〈P 〉)), where each call to the separation oracle is computed as one step. Thus if the separation oracle works in polynomial time, then such an algorithm works in polynomial time in the usual sense. Let φ(P ) be the number of integer points in P . Let fP (n) = φ(nP ) be the Ehrhart quasi-polynomial [St1] of P . Let l(P ) be the least period of fP (n), if P is nonempty. Let fi,P (n), 1 ≤ i ≤ l(P ), be the polynomials such that fP (n) = fi,P (n) if n = i modulo l(P ). Let FP (t) = n≥0 fP (n)t n denote the Ehrhart series of P . It is a rational function. Theorem 3.1.1 (a) The index of fP (n), index(fP ), can be computed in oracle-polynomial time, and hence, in polynomial time, assuming that the oracle works in polynomial time. Furthermore, if index(fP ) 6= 0 (i.e. if P is nonempty), then fi,P (n) is not an identically zero polynomial for every i divisible by index(fP ). (b) The saturated, and hence, positive integer programming problem, as de- fined in Section 1.4, can be solved in oracle-polynomial time. Here it is assumed that the specification of P also contains the saturation index esti- mate sie(P ), or the positivity index estimate pie(P ), and that the bitlength of this estimate is O(poly(〈P 〉)). Given a relaxation parameter c > sie(P ) (or pie(P )), the problem is to determine if cP contains an integer point in O(poly(〈P 〉, 〈c〉)) time. (c) Suppose {Px} is a family of polytopes, indexed by some parameter x, with the following property: wherenver Px is nonempty, the Ehrhart quasi- polynomial fPx(n) is “almost always” strictly saturated. Almost always means, the density of x’s of bitlength ≤ N , with nonempty Px for which fPx(n) is not strictly saturated is less than 1/N c′′ , for any positive c′′, as N → 0. We also assume that Px is given by a separation oracle that works in O(poly(〈x〉)) time, where 〈x〉 is the bitlength of x, and 〈Px〉 = O(poly(〈x〉)). Then there exists a O(poly(〈x〉)) time algorithm for deciding if Px con- tains an integer point that works correctly “almost always”; i.e., on almost all x. Proof: Nonemptyness of P can be decided in oracle-polynomial time using the algorithm of Grötschel, Lovász and Schrijver [GLS] (cf. Theorem 6.4.1 therein). An extension of this algorithm, furthermore, yields a specifica- tion of the affine space span(P ) containing P if P is nonempty (cf. Theo- rems 6.4.9, and 6.5.5 in [GLS]). Specifically, it outputs an integral matrix C and an integral vector d such that span(P ) is defined by Cx = d. This final specification is exact, even though the first part of the algorithm in [GLS] uses the ellipsoid method. Indeed, the use of simultneous diophan- tine approximation based on basis reduction in lattices is precisely to ensure this exactness in the final answer. This is crucial for the next step of our algorithm. If P is empty, index(fP ) = 0. So assume that it is nonempty. Let C̄ be the Smith normal form of C; i.e., C̄ = ACB for some unimodular matrices A and B, where the leftmost principal submatrix of C̄ is a diagonal, integral matrix, and all other columns are zero. The matrices C̄, A and B can be computed in polynomial time using the algorithm in [KB]. After a unimodular change of coordinates, by letting z = B−1x, span(P ) is specified by the linear system C̄z = d̄ = Ad. The equations in this system are of the form: c̄izi = d̄i, (3.1) i ≤ codim(P ), for some integers c̄i and d̄i. By removing common factors if necessary, we can assume that c̄i and d̄i are relatively prime for each i. Let c̃ be the l.c.m. of c̄i’s. The statement (a) follows from: Claim 3.1.2 index(fP ) = c̃ and fi,P (n) is not an identically zero polyno- mial for every i divisible by c̃. Proof of the claim: Indeed, nP = {nz | z ∈ P} contains no integer point unless c̃ divides n. Hence, it is easy to see that FP (t) = FP̄ (t c̃), where FP̄ (x) is the Ehrhart series of the dilated polytope P̄ = c̃P . By eq.(3.1), the equations defining P̄ are: zi = d̄i(c̃/c̄i), (3.2) Clearly, c̃ divides the least period l(P ) of fP , and l(P̄ ) = l(P )/c̃ is the period of the Ehrhart quasipolynomial fP̄ (n). It suffices to show that the index of fP̄ (n) is one and that fj,P̄ (n) is not an identically zero polynomial for every 1 ≤ j ≤ l(P̄ ). This is equivalent to showing that P̄ contains a point z with with zi = ai/b, for some integers ai’s and b such that b = j modulo l(P̄ ). Let us call such a point j-admissible. Because of the form of the equations (3.2) defining span(P̄ ), we can assume, without loss of generality, that P̄ is full dimensional. This means the system (3.2) is empty. Then this follows from denseness of the set of j-admissible points. This proves the claim, and hence (a). (b): Let s = sie(P ) be the given saturation index estimate. This means fP (n + s) is strictly saturated. This in conjunction with (a) implies that, given a relaxation parameter c > s, cP contains an integer point, iff c is divisible by index(fP ) (by letting n = c − s). This can be checked in O(poly(〈P 〉, 〈c〉)) time since index(fP ) can be computed in polynomial time by (a). (c) The algorithm computes index(fPx) and says “Probably Yes”if the index is one, and “No” otherwise. Since the saturation index of fPx(n) is zero almost always, by the argument in (b) with s = 0 and c = 1, “Probably Yes” really means “Yes” almost always. Q.E.D. The algorithm in (c) has one drawback. If the answer is “Probably Yes”, we have no easy way of checking if Px really contains an integer point. Ideally, we would like an algorithm that says “Yes”, with an integer point in Px as a proof certificate, or “No”, or “Unsure”, and the density of x’s on which it says “Unsure” should be very small. This problem can be overcome if the family {Px} has the following stronger property, akin to the family of hive polytopes [KT1]: there is a linear function lx such that, for almost all x, if {Px} is nonempty, then the lx-optimum of Px is integral (this is stronger than saying that fPx(n) is strictly saturated). In this case, the algorithm in (c) can be extended to yield the integral lx-optimum as a proof certificate. If the lx-optimum is not integeral, the algorithm says “Unsure”. PH1 and SH (Section 1.6) for the plethysm (and more generally, the subgroup restriction) problem may be strengthened by stipulating that the polytopes therein have this property. But this is not needed in this paper. We note down one corollary of the proof of Theorem 3.1.1 (this should be well known): Proposition 3.1.3 The rational function FP (t) = FP̄ (t c̃), where FP̄ (x) is the Ehrhart series of the dilated polytope P̄ = c̃P , and c̃ is the index of fP (n). If P is explicitly specified in the form a linear system Ax ≤ b, (3.3) where A is an m × n matrix, b an m vector and m = poly(n), then the following stronger version of Theorem 3.1.1 holds. Let 〈A〉 and 〈A, b〉 denote the bitlength of the specification of A and of the linear system (3.3). Theorem 3.1.4 Suppose P is specified in terms of an explicit linear system (3.3). Then the index of the Erhart quasi-polynomial fP (n) can be computed in poly(〈A, b〉) time, using poly(〈A〉) arithmetic operations. Thus, saturated, and hence, positive integer programming problem spec- ified in the form (3.3) can be solved in in poly(〈A, b, c〉) time, where c is the relaxation parameter, using poly(〈A〉) arithmetic operations. Proof: This is proved exactly as Theorem 3.1.1, but with Tardos’ strongly polynomial time algorithm for combinatorial linear programming [Ta] used in place of the algorithm in [GLS]. Q.E.D. 3.1.1 A general estimate for the saturation index Now we give a general estimate for the saturation index of any polytope P with a specification of the form Ax ≤ b, (3.4) where A is an m × n matrix, m possibly exponential. Let ‖P‖ = n + ψ, where ψ is the maximum bitlength of any entry of A. Trivially, ‖P‖ ≤ 〈P 〉. We do not assume that we know the specification (3.4) of P explicitly. We only assume that it exists, and that we are told ‖P‖. Then: Theorem 3.1.5 The saturation index of P is O(2poly(‖P‖)). Thus the bitlength of the saturation index is O(poly(‖P‖)). Conjecturally, this also holds for the positivity index. This estimate is very conservative, but useful when no better estimate is available. Proof: There exists a triangulation of P into simplices such that every vertex of any simplex is also a vertex of P . Then fP (n) = f∆(n), where ∆ ranges over all open simplices in this triangulation; a zero-dimensional open simplex is a vertex. The saturation index of fP (n) is clearly bounded by the maximum of the saturation indices of f∆(n). Hence, we can assume, without loss of generality, that P is an open sim- plex. Let v0, . . . , vn be its vertices. Then, by Ehrhart’s result (cf. Theorem 1.3 in [st5]), FP (t) = i hit j=0(1− t , (3.5) where h0 = 1, hi’s are nonnegative, and aj is the least positive integer such that ajvj is integral. By Cramer’s rule, the bit length of each aj is poly(‖P‖). Without loss of generality, we can also assume that aj’s are relatively prime. Otherwise, the estimate on the saturation index below has to be multiplied by the g.c.d. of aj ’s. Then the result follows by applying the following lemma to FP (t), since 〈aj〉 = O(poly(‖P‖)). Q.E.D. Lemma 3.1.6 Let f(n) be a quasipolynomial whose generating function F (t) has a positive form F (t) = i hit j=0(1− t , (3.6) where h0 = 1, hi’s are nonnegative, and aj’s are positive and relatively prime. Let a = max{aj}. Then the saturation index s(f) of f(n) is O(poly(a, n)). Proof: Let g(n) be the quasi-polynomial whose generating function G(t) = g(n)tn is 1/ j=0(1− t aj ). It is known that this is the Ehrhart quasipoly- nomial of the polytope N(a0, . . . , an) defined by the linear system ajxj = 1, xj > 0. The saturation index s(g) of g(n) is bounded by the Frobenius number associated with the set of integers {aj}–this is the largest positive integer m such that the diophantine equation ajxj = m has no positive integeral solution (x0, . . . , xn). It is known (e.g. [BDR]) that the Frobenius number is bounded by a0a1a2(a0 + a1 + a2) = O(poly(a)), assumming that a0 ≤ a1 . . .. Hence, s(g) = O(poly(a)). Since f(n) is a quasi-polynomial, the degree of the numerator of F (t) is less than the degree of the denominator. Thus the maximum value of i that occurs in (3.6) is an. Let gi(n), i ≤ an, be the quasi-polynomial whose generating function is j=0(1− t aj ). Then s(gi) ≤ i+ s(g) = O(poly(a, n)). Since, hi’s in (3.6) are nonnegative, s(f) = max s(gi). The result follows. Q.E.D. 3.1.2 Extensions We now mention a few straightforard extensions of Theorem 3.1.1. First, it is not necessary that P be a closed polytope. We can allow it to be half-closed. Specifically, it can be a solution set of a system of inequalitites of the form: A1x ≤ b1 and A2x < b2, (3.7) where we have allowed strict inequalities. The function FP (n) = φ(nP ), the number of integer points in nP , is again a quasi-polynomial. Hence, the notions of saturation and positivity can be generalized to this setting in a natural way. Second, the algorithm in Theorem 3.1.1 (b) only needs a nonnegative number s(P ) such that, for any positive integer c > s(P ): Saturation guarantee: If the affine span of cP , contains an integer point, then cP is guaranteed to contain an integer point. If s(P ) = sie(P ), then this guarantee holds, as can be seen from the proof of Theorem 3.1.1. 3.1.3 Is there a simpler algorithm? Though the algorithm for saturated integer programming in Theorem 3.1.1 is conceptually very simple, in reality it is quite intricate, because the work of Grötschel, Lovász and Schrijver [GLS] needs a delicate extension of the el- lipsoid algorithm [Kh] and the polynomial-time algorithm for basis reduction in lattices due to Lenstra, Lenstra and Lovász [LLL]. As has been empha- sized in [GLS], such a polynomial-time algorithm should only be taken as a proof of existence of an efficient algorithm for the problem under consider- ation. It may be conjectured that for the problems under consideration in this paper such simple, combinatorial algorithms exist. But for the design of such algorithms, saturation alone does not suffice. The stronger property (PH3), and more, is necessary. We shall address this issue in Section 3.6. 3.2 Littlewood-Richardson coefficients again Theorem 3.1.4 applied to the BZ-polytope [BZ], with saturation index esti- mate equal to zero, specializes to the following in the setting of the Littlewood- Richardson problem (Problem 1.2.1): Theorem 3.2.1 [GCT5] Assuming SH (Hypothesis 1.2.5), nonvanishing of , given α, β, λ, can be decided in strongly polynomial time (Section 2.1) for any semisimple classical Lie algebra G. It is assumed here that α, β, λ are specified by their coordinates in the basis of fundamental weights. For type A, this reduces to the result in [GCT3], which holds unconditionly. The saturation conjecture for type A arose [Z] in the context of Horn’s conjecture and the related result of Klyachko [Kl]. We now turn to implica- tions of Theorem 3.2.1 in this context. Given a complex, semisimple, simply connected, classical group G, let LR(G) be the Littlewood-Richardson semigroup as in Section 2.2.2. The following is a natural generalization of the problem raised by Zelevinsky [Z] to this general setting: Problem 3.2.2 Give an efficient description of LR(G). Zelevinsky asks for a mathematically explicit description. This is a com- puter scientist’s variant of his problem. Let LRR(G) be the polyhedral convex cone generated by LR(G). For G = GLn(C), by the saturation theorem, a triple (α, β, λ) of dominant weights belongs to LR(G) iff it belongs to LRR(G). Assuming SH (Hy- pothesis 1.2.5), Theorem 3.2.1 provides the following efficient description for LR(G) in general. Recall that the period of the Littlewood-Richardson stretching polynomial c̃λ (n) divides a fixed constant d(G), which only de- pends on the types of simple factors of G [DM2, GCT5]. Let αi’s denote the coordinates of α in the basis of fundamental weights. Corollary 3.2.3 (a) Assuming SH, whether a given (α, β, λ) belongs to LR(G) can be determined in strongly polynomial time. (b) There exists a decomposition of LRR(G) into a set of polyhedral cones, which form a cell complex C(G), and, for each chamber C in this complex, a set M(C) of O(rank(G)2) modular equations, each of the form aiαi + biβi + ciλi = 0 (mod d), for some d dividing d(G), such that 1. SH (Hypothesis 1.2.5) is equivalent to saying that: (α, β, λ) ∈ LR(G) iff (α, β, λ) ∈ LRR(G) and (α, β, λ) satisfies the modular equations in the set M(Cα,β,λ) associated with the cone Cα,β,λ containing α, β, λ. 2. Given (α, β, λ), whether (α, β, λ) ∈ LRR(G) can be determined in strongly polynomial time (cf. Section 1.2.5). 3. If so, the cone Cα,β,λ and the associated set M(C ) of modular equa- tions can also be determined in strongly polynomial time. After this, whether (α, β, λ) satisfies the equations in M(Cλα,β) can be trivially determined in strongly polynomial time. Proof: (a) is a consequence of Theorem 3.2.1. (b) follows from a careful analysis of the algorithm therein; see the proof of a more general result (Theorem 4.4.2) later. Q.E.D. We call the labelled cell complex C(G), in which each cell C ∈ C(G) is labelled with the set of modular equations M(C), the modular complex, associated with LRR(G). When G = SLn(C), the modular complex is trivial: it just consists of the whole cone LRR(G) with only one obvious modular equation attached to it. But, for general G, the modular complex and the map C → M(C) are nontrivial. We do not know their explicit description. Corollary 3.2.3 says that, given x = (α, β, λ), whether x ∈ LRR(G), and whether the relevant modular equations are satisfied can be quickely verified on a computer, though the modular equations cannot be easily determined and verified by hand, as in type A. This is the main difference between type A and general types. This naturally leads to: Question 3.2.4 Is there a mathematically explicit description of the mod- ular complex C(G) for a general G? 3.3 The saturation and positivity hypotheses Now let f(x), x ∈ Nk, be a counting function associated with a structural constant in representation theory or algebraic geometry. Here x denotes the sequence of parameters associated with the constant. Let 〈x〉 denote the bitlength of x. Let ‖x‖ and rank(x) denote its combinatorial size and combinatorial rank–these measure complexity of the nonstretchable part in the specification of x and will be specified later for the f ’s of interest in this paper. For example, in the Littlewood-Richardson problem, x is the triple (α, β, λ), f(x) = f(α, β, λ) = cλ , 〈x〉 is the total bitlength of the coordinates of α, β, λ, ‖x‖ is the total number of coordinates of α, β and λ, and rank(x) = ‖x‖. The number of coordinates does not change during stretching, and hence, constitute the nonstretchable part of the input specification here. Assume that f(x) is nonnegative for all x ∈ Nk, Then we can successively ask the following questions: 1. Does f ∈ PSPACE? That is, can f(x) be computed in poly(〈x〉) space? 2. Does f ∈ #P? (cf. Section 2.1) 3. Does f ∈ convex#P? (cf. Section 2.2) 4. Can a stretching function f̃(x, n) be associated with f(x) intrinsically so that f̃(x, n) is quasi-polynomial? 5. (PH1?): Is there a polytope Px, for every x, with 〈Px〉 = O(poly(〈x〉)) and ‖Px‖ = O(poly(‖x‖)), such that f̃(x, n) = fPx(n)? 6. Are there good analogues of SH and/or PH2, PH3 for f̃(x, n)? If so, nonvanishing of f(x), modulo small relaxation, can be decided in O(poly(〈Px〉)) time by Theorem 3.1.1. In the rest of this paper, we study these questions when f = f(x) is a nonnegative function associated with a structural constant in any of the deci- sion problems in Section 1.1. Exact specifications of x, 〈x〉, ‖x‖, rank(x), f(x), and f̃(x, n) for these decision problems are given in Sections 3.4-3.5. It is shown in Chapter 5 that f(x) ∈ PSPACE for Problem 1.1.2 and the special cases of Problem 1.1.3 that arise in the flip. This may be conjectured to be so for the f ’s in Problem 1.1.4, with X therein a class variety; cf [GCT10] for its justification. Quasipolynomiality of f̃(x, n) is addressed in Chapter 4. The hypotheses PH1, SH, PH2, and PH3 in these cases have the following unified form. Hypothesis 3.3.1 (PH1) Let f = f(x) be the function associated with a structural constant in 1. Problem 1.1.1, or 2. 1.1.2, or 3. Problem 1.1.3, or 4. Problem 1.1.4, with X being a class variety therein. Then the function f(x) has a convex #P -formula (cf. (2.4)) f(x) = φ(Px), such that: 1. for every fixed x, the Ehrhart quasi-polynomial fPx(n) of Px coincides with f̃(x, n). 2. 〈P 〉 = O(poly(P )) and ‖P‖ = O(poly(‖x‖)). Hypothesis 3.3.2 (SH) (a) Suppose f(x) is a structural constant as in PH1 above. Then for every x, the saturation index s(f̃) of f̃(x, n) is O(poly(rank(x))). This means there exist absolute nonnegative constants c, c′ such that s(f̃) ≤ c(rank(x))c (b) For f(x) in Problems 1.1.1-1.1.3, the saturation index of f̃(x, n) is zero– i.e., f̃(x, n) is strictly saturated–for almost all x. This means the density of x, with 〈x〉 ≤ N and f(x) nonzero, for which the saturation index s(f̃) is nonzero is ≤ 1/N c , for any positive costant c′′, as N → ∞. More strongly than (a), Hypothesis 3.3.3 (PH2) For f(x) as in PH1, the positivity index of f̃(x, n) is O(poly(rank(x))). Hypothesis 3.3.4 (PH3) For f(x) as in PH1, the generating function F (x, t) = n f̃(x, n)t n has a positive rational form of modular index O(poly(rank(x))). More specifically, the modular index of f̃(x, n), as defined in Section 4.1.1 for f ’s that arise in this paper, is O(poly(rank(x))). PH3 implies SH (a); this follows from Lemma 3.1.6. The following conservative bound follows from Theorem 3.1.5. Theorem 3.3.5 (Weak SH) Assuming PH1 (Hypothesis 3.3.1), the saturation index of f̃(x, n) is bounded by 2O(poly(‖x‖)); hence its bitlength is bounded by O(poly(‖x‖)). The following result addresses the relaxed forms of the decision problems for the structural constants under consideration (cf. Section 1.1). Theorem 3.3.6 Suppose f(x) is a structural constant as in PH1 above. Then PH1 (Hypothesis 3.3.1) and SH (Hypothesis 3.3.2) imply Hypothe- sis 1.1.6 (PHflip) in this case. Specifically: (a) For f(x) in Problems 1.1.1-1.1.4, nonvanishing of f̃(x, a), for a given x and a relaxation parameter a > c(rank(x))c , with c, c′ as in Hypothesis 3.3.2, can be decided in poly(〈x〉, 〈a〉) time. (b) For f(x) as in Problems 1.1.1-1.1.3, there is a poly(〈x〉) time algorithm for deciding nonvanishing of f(x) that works correctly on almost all x. This follows from Theorem 3.1.1. The following sections give precise descriptions of x, 〈x〉, ‖x‖, rank(x) and f̃(x, n) for the structural constants under consideration. 3.4 The subgroup restriction problem In this section we consider the subgroup restriction problem (Problem 1.1.3). The Kronecker and the plethysm problems (Problems 1.1.1, 1.1.2) are its special cases. Let G,H, ρ, λ, π,mπλ be as in Problem 1.1.3. We shall define below an ex- plicit polynomial homomorphism ρ : H → G, as needed in the statement of Problem 1.1.3, and also the precise specifications [H], [ρ], [λ], [π] of H, ρ, λ, π, respectively. We shall also define the bitlengths 〈H〉, 〈ρ〉, 〈λ〉, 〈π〉 and the combinatorial bit lengths ‖λ‖, ‖π‖. We let ‖H‖ = 〈H〉 and ‖ρ‖ = 〈ρ〉, since H and ρ belong to the nonstretchable part of the input. On the other hand, λ and π will be stretched in the definition of f̃(x, n), and hence their com- binatorial bit lengths will differ from the usual bit lengths. The input x in the subgroup restriction problem is the tuple ([H], [ρ], [λ], [π]). Its bitlength 〈x〉 is defined to be the sum of the bitlengths 〈H〉, 〈ρ〉, 〈λ〉, 〈π〉, and ‖x‖ is defined to be the sum of ‖H‖, ‖ρ‖, ‖λ‖ and ‖π‖. Finally rank(x) is defined to the sum of the ranks of H and G and ‖λ‖ and ‖π‖. Here that rank of a (reductive) group is defined in a standard way. For example, the rank of the symmentric group Sn is n, that of GLn(C) is n. The rank of a general finite or connected simple group can be defined similarly, and the rank of a more complex reductive group is defined to be the sum of the ranks of its simple components. With this terminology, we let f(x) = mπ , with x as defined here in Hypotheses 3.3.1-3.3.4 and Theorem 3.3.6 for the subgroup restriction problem. Here H and ρ are implicit in the definition of mπ For example, in the plethym problem (Problem 1.1.2), these specifi- cations are as follows. The specification [H] is just the root system for H = GLn(C). Its bitlength 〈H〉 is n. The specification [ρ] of the repre- sentation map ρ : H → G = GL(Vµ(H)) consists of just the partition µ specified in terms of its nonzero parts. Its bitlength 〈ρ〉 = 〈µ〉. The ranks of H and G are as usual. The partitions λ and µ are specified in terms of their nonzero parts. Their bitlength is the total bitlength of the parts, and the combinatorial bit length is the total number of parts (the height). It is crucial here that only nonzero parts of λ are specified, because the rank of G can be exponential in the rank of H and the bitlength of µ. Hence, the bitlength of this compact representation of λ can be polynomial in the rank of H and the bitlength of µ, even if the dimension of G is exponential. The main difference between 〈x〉 and ‖x‖ is that the stretchable data λ and π contribute their bitlengths to the former, and their heights to the latter. The plethysm problem is the main prototype of the subgroup restriction problem. If the reader wishes, (s)he can skip the rest of this subsection and jump to Section 3.4.3 in the first reading. In general, we assume that H in Problem 1.1.3 is a finite simple group, or a complex simple, simply connected Lie group, or an algebraic torus (C∗)k, or a direct product of such groups. The results and hypotheses in this paper are also applicable if we allow simple types of semidirect products, such as wreath products, which is all that we need for the sake of the flip. But these extensions are routine, and hence, for the sake of simplicity, we shall confine ourselves to direct products. 3.4.1 Explicit polynomial homomorphism Now let us define an explicit polynomial homomorphism. This will be done by defining basic explicit homomorphisms, and composing them functorially. Basic explicit homomorphisms: Let V be an irreducible polynomial representation of H (character- istic zero), or more generally, an explicit polynomial representation that is constructed functorially from the irreducible polynomial representations using the operations ⊕ and ⊗. Then the corresponding homomorphism ρ : H → G = GL(V ) is an explicit polynomial homomorphism. The iden- tity map H → H is also an explicit polynomial homomorphism. The polynomiality restriction here only applies to the torus component of H. If H is a finite simple group, or a complex semisimple group, then any irreducible representation of H is, by definition, polynomial. In general, a representation is polynomial if its restriction to the torus component is polynomial; i.e., a sum of polynomial (one dimensional) characters. To see why the polynomiality restriction is essential, let H be a torus, V its rational representation, and G = GL(V ). Let Vλ(G) = Sym d(V ), the symmetric representation of G, and let π be the label of the trivial character of H. Then the multiplicity mπ is the number of H-invariants in Symd(V ). This is easily seen to be the number of nonnegative solutions of a system of linear diophontine equations. But the problem of deciding whether a given system of linear diophontine equations has a nonnegative solution is, in general, NP -complete. Though the system that arises above is of a special form, it is not expected to be in P if V is allowed to be any rational representation; the associated decision problem may be NP -complete even in this special case. If V is a polynomial representation of a torus H, then all coefficients of the system are nonnegative, and the decision problem is trivially in P . Composition: We can now compose the basic explicit (polynomial) homomorphisms above functorially: 1. If ρi : H → Gi are explicit, the product map ρ : H → iGi is also explicit. 2. If ρi : Hi → Gi are explicit, the product map ρ : Gi is also explicit. Instead of products, we can also allow simple semi-direct products such as wreath products here. We may also allow other functorial constructions such as induced representations and restrictions. For example, if ρ : H → G is an explicit polynomial homomorphism, and G′ ⊆ G is an explicit subgroup of G such that ρ(H) ⊆ G′, then the restricted homomorphism ρ′ : H → G′ can also be considered to be an explicit polynomial homomorphism. But for the sake of simplicity, we shall confine ourselves to the simple functorial constructions above. 3.4.2 Input specification and bitlengths Now we describe the specifications [H], [ρ], [λ], [µ], their bitlengths. These are very similar to the ones in the plethysm problem. The specification [H]: We assume that H is specified as follows. (1) IfH is a complex, simple, simply connected Lie group, then the specifica- tion [H] consists of the root system of H or the Dynkin diagram. Let 〈H〉 be the bitlength of this specification. Thus, if H = SLn(C), then 〈H〉 = O(n). (2) If H is a simple group of Lie type (Chevalley group) then it has a similar specification [Ca]. The only finite groups of Lie type that arise in GCT are SLn(Fpk) and GLn(Fpk). In this case the specification [H] is easy: we only have to specify n, p, k. We define 〈H〉 in this case to be n + k + log2 p; not log2 n+ log2 k + log2 n. As a rule, 〈H〉 is defined to be the sum of the rank parameters (such as n and k here) and bit lengths of the weight parameters (such as p here) in the specification. This is equivalent to assuming that the rank parameters are specified in unary. (3) If H is the alternating group An, we only specify n. Let 〈H〉 = n. (4) The torus is specified by its dimension. We define 〈H〉 to be the dimen- sion. (5) If H is a product of such groups, its specification is composed from the specifications of its factors, and the bitlength 〈H〉 is defined to be the sum of the bitlengths of the constituent specifcations. The specification [ρ]: Let us first assume that ρ is a basic explicit polynomial homomorphism. In this case the specification of ρ : H → G = GL(V ) is a pair [ρ] = ([H], [V ]) consisting of the specification [H] of H as above, and the combinatorial specification [V ] of the representation V as defined below: (1) If H is a semisimple, simply connected Lie group, and V = Vµ(H) its irrreducible representation for a dominant weight µ of H, then V is specified by simply giving the coordinates of µ in terms of the fundamental weights of H. Thus [V ] = µ, and its bitlength 〈V 〉 is the total bitlength of all coordinates of µ, and the combinatorial bit length ‖V ‖ is the total number of coordinates of µ. (2) If H = Sn, and V = Sγ its irreducible representation (Specht module), then [V ] is the partition γ labelling this Specht module. We define 〈V 〉 to be the bitlength of this partition, and ‖V ‖ = 〈V 〉. (3) If H is a finite general linear group GLn(Fpk), and V its irreducible rep- resentation, as classified by Green [Mc], then [V ] is the combinatorial clas- sifying label of V as given in [Mc]. It is a certain partition-valued function, which can be specified by listing the places where the function is nonzero and the nonzero partition values at these places. Let 〈V 〉 be the bitlength of this specification; it is O(poly(n, k, 〈p〉)). We let ‖V ‖ = 〈V 〉. More gener- ally, if H is a finite group of Lie type, and V its irreducible representation, then [V ] is the combinatorial classifying label of V as given by Lusztig [Lu1]. (4) If H is a torus and V is a polynomial character, then [V ] is the speci- fication of the character. Its bitlength is the bitlength of the specification, and combinatorial bit length is the dimension of H. (5) If V is composed from irreducible representations, then [V ] is composed from the specifications of the irreducible representations in an obvious way. Bitlengths and combinatorial bitlengths are defined additively. The bitlength 〈ρ〉 is defined to be 〈H〉+ 〈V 〉, where 〈V 〉 is the bitlength of [V ]. If ρ is a composite homomorphism, its specification [ρ] is composed from the specifications of its basic constituents in an obvious way. The bitlength 〈ρ〉 is defined to be the sum of the bitlengths of these basic specifications. The specifications [λ] and [π]: Vπ(H) is the tensor product of the irreducible representations of the factors of H. We let [π] be the tuple of the combinatorial classifying labels of each of these irreducible representations, as specified above. Let 〈π〉 be their total bit length, and ‖π‖ the total combinatorial bit length. Similarly, Vλ(G) is the tensor product of the irreducible representations of the factors of G. When G = GLm(C), λ is a partition, which we specify by only giving its nonzero parts, whose number is equal to the height of λ. This is crucial since the height of λ can be much less than than the rank m of G, as in the plethysm problem (Problem 1.1.2). We shall leave a similar compact specification [λ] for a general connected, reductive G to the reader. Let 〈λ〉 be its bitlength and ‖λ‖ its combinatorial bit length. 3.4.3 Stretching function and quasipolynomiality Let f(x) = mπ as above, with x = ([H], [ρ], [λ], [π]). Here λ is the dominant weight of G. First, assume that H is connected, reductive. Then π is the dominant weight of H. For a given x, let us define the stretching function f̃(x, n) = m̃πλ(n) = m nλ, (3.8) which is the multiplicity of Vnπ(H) in Vnλ(G), considered as an H-module via ρ : H → G. Let Mπ (t) = n≥0 m̃ (n)tn be the generating function of this stretching quasi-polynomial. The following is the generalization of Theorem 1.6.1 in this setting. Theorem 3.4.1 (a) (Rationality) The generating function Mπ (t) is ratio- (b) (Quasi-polynomiality) The stretching function m̃π (n) is a quasi-polynomial function of n. (c) There exist graded, normal C-algebras S = S(mπ ) = ⊕nSn and T = T (mπ ) = ⊕nTn such that: 1. The schemes spec(S) and spec(T ) are normal and have rational singu- larities. 2. T = SH , the subring of H-invariants in S. 3. The quasi-polynomial m̃π (n) is the Hilbert function of T . (d) (Positivity) The rational function Mπ (t) can be expressed in a positive form: Mπλ (t) = h0 + h1t+ · · ·+ hdt j(1− t a(j))d(j) , (3.9) where a(j)’s and d(j)’s are positive integers, j d(j) = d+1, where d is the degree of the quasi-polynomial, h0 = 1, and hi’s are nonnegative integers. The specific rings S(mπ ) and T (mπ ) constructed in the proof of this result are called the canonical rings associated with the structrural con- stant mπ . The projective schemes Y (mπ ) = Proj(S(mπ )), and Z(mπ Proj(T (mπλ)) are called the canonical models associated with m Theorem 3.4.1 and its generalization, when H can be disconnected, is proved in Chapter 4; cf. Theorem 4.1.1. Finitely generated semigroup The following is an analogue of Theorem 1.6.2. Theorem 3.4.2 Assume that H is connected. For a fixed ρ : H → G, let T (H,G) be the set of pairs (µ, λ) of dominant weights of H and G such that the irreducible representation Vπ(H) of H occurs in the irreducible repre- sentation Vλ(G) of G with nonzero multiplicity. Then T (H,G) is a finitely generated semigroup with respect to addition. This is proved in Section 4.4. PSPACE The following is a generalization of Theorem 1.6.3. Theorem 3.4.3 Assume that H in Problem 1.1.3 is a direct product, whose each factor is a complex simple, simply connected Lie group, or an alternat- ing (or symmetric) group, or SLn(Fpk) (or GLn(Fpk)), or a torus. Then f(x) = mπ can be computed in poly(〈x〉) space, with x as specified above. This is proved in Chapter 5. It may be conjectured that Theorem 3.4.3 holds even when the composition factors of H are allowed to be general finite simple groups of Lie type. This will be so if Lusztig’s algorithm [Lu5] for computing the characters of finite simple groups of Lie type can be parallelized; cf. Section 5.4. Positivity hypotheses Theorem 3.4.1-3.4.3, along with the experimental results in special cases (cf. Chapter 6), constitute the main evidence in support of the positivity Hypotheses 3.3.1-3.3.4 for the subgroup restrition problem. 3.5 The decision problem in geometric invariant theory Finally, let us turn to the most general Problem 1.1.4. 3.5.1 Reduction from Problem 1.1.3 to Problem 1.1.4 First, let us note that the subgroup restriction problem (Problem 1.1.3) is a special case of Problem 1.1.4. To see this, let H, ρ and G be as in Problem 1.1.3, and let X be the closed G-orbit of the point vλ corresponding to the highest weight vector of Vλ(G) in the projective space P (Vλ(G)). Then X = Gvλ ∼= G/Pλ, (3.10) where the P = Pλ = Gvλ is the parabolic stabilizer of vλ. We have a natural action of H on X via ρ. Let R be the homogeneous coordinate ring of X. By [Ha, MR, Rm, Sm], the singularities of spec(R) are rational. By Borel-Weil [FH], the degree one component R1 of the homogeneous coordinate ring R of X is Vλ(G). Hence, s 1 in this special case of Problem 1.1.4 is precisely m in Problem 1.1.3. The results in Section 3.4 for sπ1 generalize in a natural way for sπd . 3.5.2 Input specification The variety X in the above example is completely specified by H, ρ and λ. Hence its specification [X] can be given in the form a tuple ([H], [ρ], [λ]), where [H], [ρ] and [λ] are the specifications of H, ρ and λ as in Section 3.4, The input specification x for Problem 1.1.4 in the special case above is the tuple ([X], d, [π]) = ([H], [ρ], [λ], d, [π]), where [π] is the specification of π as in Section 3.4. We now describe a class of varieties X which have similar compact spec- ifications. Let G be a connected, reductive group, H a reductive, possibly discon- nected, reductive group, and ρ : H → G an explicit polynomial homomor- phism as in Section 3.4. Let V = Vλ(G) be an irreducible representation of G for a dominant weight λ. Let P (V ) be the projective space associated with V . It has a natural action of H via ρ. Let v ∈ P (V ) be a point that is char- acterized by its stabilizer Gv ⊆ G. This means it is the only point in P (V ) that is stabilized by Gv. For example, the point vλ above is characterized by its parabolic stabilier. We assume that we know the Levi decompositioon of Gv explicity, and its compact specification [Gv ], like that of H, and also an explicit compact specification of the embedding ρ′ : Gv → G, aking to that of the explicit homomorphism ρ : H → G. Let X ⊆ P (V ) be the projective closure of the G-orbit of v in P (V ). Then X as well as the action of H on X are completely specified by λ,H, ρ,Gv and ρ ′. Hence, we can let [X] be the tuple (λ, [H], [ρ], [Gv ], [ρ ′]). The input specification x for Problem 1.1.4 with the X of this form is the tuple ([X], d, [π]). The bitlengths 〈x〉 and ‖x‖ are defined additively. The rank(x) is defined to be the sum of the ranks of H and G, dim(V ) and ‖π‖. Since the point vλ above is characterized by its stabilizer, G/P is a variety of this form. The class varieties [GCT1, GCT2] are either of this form, or a slight ex- tension of this form, and admit such compact specifications. The algebraic geometry of an X of the above form is completely determined by the repre- sentation theories of the two homomorphisms ρ : H → G and ρ′ : Gv → G. Furthermore, the results in [GCT2] say that Problem 1.1.4 for a class variety is intimately linked with the subgroup restriction problem and its variants for the homomomorphisms ρ and ρ′. Hence it is qualitatively similar to the subgroup restriction problem in this case; cf. [GCT10] for further elabora- tion of the connection between these two problems. 3.5.3 Stretching function and quasi-polynomiality Now let H,X,R and sπ be as in Problem 1.1.4, with H therein assumed to be connected. We associate with f(x) = sπd the following stretching fucntion: f̃(x, n) = s̃πd (n) = s nd , (3.11) where snπ is the multiplicity of the irrreducible representation Vnπ(H) of H in Rnd, the componenent of the homogeneous coordinate ring R of X with degree nd. Let S(t) = n≥0 s̃ (n)tn. Theorem 3.5.1 Assume that the singularities of spec(R) are rational. (a) (Rationality) The generating function Sπ (t) is rational. (b) (Quasi-polynomiality) The stretching function s̃πd(n) is a quasi-polynomial function of n. (c) There exist graded, normal C-algebras S = S(sπ ) = ⊕nSn and T = T (sπ ) = ⊕nTn such that: 1. The schemes spec(S) and spec(T ) are normal and have rational singu- larities. 2. T = SH , the subring of H-invariants in S. 3. The quasi-polynomial s̃πd(n) is the Hilbert function of T . (d) (Positivity) The rational function Sπ (t) can be expressed in a positive form: Sπd (t) = h0 + h1t+ · · ·+ hkt j(1− t a(j))k(j) , (3.12) where a(j)’s and k(j)’s are positive integers, j k(j) = k + 1, where k is the degree of the quasi-polynomial s̃πd(n), h0 = 1, and hi’s are nonnegative integers. This is proved in Chapter 4. Theorem 3.4.1 is a special case of this theorem, in view of the reduction in Section 3.5.1. Theorem 3.5.1 is applicable when X is a class variety, assuming that its singularities are rational. 3.5.4 Positivity hypotheses Even though Theorem 3.5.1 holds for any X, with spec(R) having ratio- nal singularities, the positivity hypotheses PH1, SH, PH2, and PH3 can be expected to hold for only very special X’s. In general, characterizing the X’s with compact specification for which these hypotheses hold is a delicate problem. Hypotheses 3.3.1-3.3.4 say that these hold when X in Problem 1.1.4 is G/P (as in Section 3.5.1) or a class variety, with the input specification x as described above. For future reference, we shall reformulate these hypotheses purely in geometric terms. For this we need a definition. Let T = n Tn be a graded complex C-algebra so that the singularities of spec(T ) rational. Let Z = Proj(T ). Assume that Z has a compact specification [Z]; we shall specify it below for the Z’s of interest to us. We let [T ], the specification of T , to be [Z]. This will play the role of the input in the definition below. Let 〈T 〉 denote its bitlength, and ‖T‖ combinatorial bit length. Let hT (n) = dim(Tn) be its Hilbert function, which is a quasipolynomial, since the singularities of spec(T ) are rational; cf. Lemma 4.1.3. Definition 3.5.2 We say that PH1 holds for T (or Z) if the Hilbert quasi- polynomial hT (n) is convex. This means there exists a polytope P = PT depending on the input [T ], whose Ehrhart quasipolynomial fP (n) coincides with the Hilbert function hT (n), and whose membership function χP (y) can be computed in poly(〈T 〉, y) time. We assume that a separating hyperplane can also be computed in polynomial time if y 6∈ P (Section 2.3). If PH1 holds we can also ask if analogues of SH, PH2, and PH3–whose formulation is similar and hence omitted–hold. 3.5.5 G/P and Schubert varieties Let us illustrate this definition with an example. Let X ∼= G/Pλ be as in Section 3.5.1 and R its homogeneous coordinate ring. We have already seen that it has a compact specification: namely [X] = λ. Since singularities of spec(R) are rational, PH1 makes sense. For G/P it follows from the Borel-Weil theorem. The Hilbert series of R is of the form h0 + · · ·+ hdt (1− t)d+1 with h0 = 1 and hi’s nonnegative. This is so because R is Cohen-Macauley [Rm] and is generated by its degree one component. Hence, the modular index of the Hilbert function is one (PH3). PH2 turns out to be nontriv- ial. Experimental evidence in its support for the classical G/P is given in Section 6.3. Considerations for the Schubert subvarieties are similar. Ex- perimental evidence for PH2 for the classical Schubert varieties is also given in Section 6.3. Now let s = sπ be the multiplicity as Problem 1.1.4, with X having a compact specification [X] as above. Let T = T (s) be the ring associated with s as in Theorem 3.5.1 (c). Let Z = Z(s) = Proj(T ). We let the specification [Z] = ([X], d, π). Let 〈Z〉 be its bitlength. So Theorem 3.1.1 in this context implies: Theorem 3.5.3 If PH1 and SH holds for Z(s) then nonvanishing of s, modulo small relaxation, can be decided in poly(〈Z〉) time. We also have the following reformulation: Proposition 3.5.4 Hypotheses 3.3.1-3.3.4 are equivalent to PH1,SH,PH2,PH3 for Z(s), where s is a stucture constant that corresponds the structure con- stant f(x) in Hypotheses 3.3.1. Thus, in the case of the subgroup restriction problem, s = sπ1 = m λ as in Section 3.5.1. This is just a consequence of definitions. 3.6 PH3 and existence of a simpler algorithm As we remarked in Section 3.1.3, the use of the ellipsoid method and basis reduction in lattices makes the the algorithm for saturated integer program- ming (cf. Theorem 3.1.1) fairly intricate. For the flip (cf. [GCTflip] and Chapter 7), it is desirable to have simpler algorithms for the relaxed forms of the decision problems under consideration, akin to the the polynomial time combinatorial algorithms in combinatorial optimization [Sc] that do not rely on the elliposoid method or basis reduction. We briefly examine in this section the role of PH3 in this context. The simple combinatorial algorithms in combinatorial optimization work only when the problem under consideration is unimodular–in which case the vertices of the underlying polytope P are integral–or almost unimodular– e.g. when the vertices of P are half integral. Edmond’s algorithm for finding minimum weight perfect matching in nonbipartite graphs [Sc] is a classic example of the second case. In the unimodular case, Stanley’s positivity result [St1] implies that the rational function FP (t) has a positive form FP (t) = h(d)td + · · ·+ h(0) (1− t)d+1 If PH3 (Hypothesis 3.3.4) holds for a structural function f(x) under con- sideration then the Ehrhart series FPx(t) of the polytope Px associated with x in PH1 (Hypothesis 3.3.1) has a minimal positive form in which each root of the denominator has O(poly(‖x‖)) order. Roughly, this says that the situation is “close” to the unimodular case. Hence, in such a case we can expect a purely combinatorial polynomial-time algorithm for deciding non- vanishing of f(x), modulo small relaxation, that does not need the ellipsoid method or basis reduction. 3.7 Other structural constants The paradigm of saturated and positive integer programming in this paper, along with appropriate analogues of PH1,SH,PH2,PH3, may be applicable several other fundamental structural constants in representation theory and algebraic geometry, in addition to the ones in Problems 1.1.1-1.1.4 treated above, such as 1. the value of a Kazhdan-Lusztig polynomial at q = 1, [KL1]; 2. the values at q = 1 of the well behaved special cases of the parabolic Kostka polynomials and their q-analogues [Ki]; 3. the structural coefficients of the multiplication of Schubert polynomi- als, and so on. Chapter 4 Quasi-polynomiality and canonical models In this chapter we prove quasipolynomiality of the stretching functions as- sociated with the various structural constants under consideration (Sec- tion 4.1), describe the associated canonical models (Section 4.2), describe the role of nonstandard quantum groups in [GCT4, GCT7, GCT8] in the deeper study of these models (Section 4.3), prove finite generation of the semigroup of weights (Theorem 3.4.2) (Section 4.4), and give an elementary proof of rationality in Theorem 3.4.1 (a) (Section 4.5). 4.1 Quasi-polynomiality Here we prove Theorem 3.5.1; Theorems 1.6.1 and 3.4.1 are its special cases in view of the reduction in Section 3.5.1. This, in turn, follows from the following more general result. Let R = ⊕kRd be a normal graded C-algebra with an action of a reduc- tive group H. Assume that spec(R) has rational singularities. Let H0 be the connected component of H containing the identity. Let HD = H/H0 be its discrete component. Given a dominant weight π of H0, we consider the module Vπ = Vπ(H0), an H-module with trivial action of HD. Let s denote the multiplicity of the H-module Vπ in Rd. Let s̃ (n) be the multiplicity of the H-module Vnπ in Rnd. This is a stretching function associated with the mulitplicity sπ . Let Sπ (t) = n≥0 s̃ (n)tn. Theorem 4.1.1 (a) (Rationality) The generating function Sπ (t) is ratio- (b) (Quasi-polynomiality) The stretching function s̃π (n) is a quasi-polynomial function of n. (c) There exist graded, normal C-algebras S = S(sπ ) = ⊕nSn and T = T (sπd ) = ⊕nTn such that: 1. The schemes spec(S) and spec(T ) are normal and have rational singu- larities. 2. T = SH , the subring of H-invariants in S. 3. The quasi-polynomial s̃π (n) is the Hilbert function of T . (d) (Positivity) The rational function Sπd (t) can be expressed in a positive form: Sπd (t) = h0 + h1t+ · · ·+ hkt j(1− t a(j))k(j) , (4.1) where a(j)’s and k(j)’s are positive integers, j k(j) = k + 1, where k is the degree of the quasi-polynomial s̃π (n), h0 = 1, and hi’s are nonnegative integers. Theorem 3.5.1 follows from this by letting R be the homogeneous coordinate ring of X. More generally, if W is an irreducible representation of HD, we can consider the H-module Vπ ⊗ W . Let s be its multiplicity in Rd. Let (n) be the multiplicity of the trivial H-representation in the H-module Rnd ⊗ V nπ ⊗ Sym n(W ∗). Then Theorem 4.1.2 Analogue of Theorem 4.1.1 holds for s̃ For the purposes of the flip, Theorem 4.1.1 suffices. Proof: We shall only prove Theorem 4.1.1, the proof of Theorem 4.1.2 be- ing similar. The proof is an extension of M. Brion’s proof (cf. [Dh]) of quasi-polynomiality of the stretching function associated with a Littlewood- Richardson coefficient of any semisimple Lie algebra. Clearly (a) follows from (b); cf. [St1]. (b) and (c): Let Cd be the cyclic group generated by the primitive root ζ of unity of order d. It has a natural action on R: x ∈ Cd maps z ∈ Rk to x kz. Let B = RCd = n≥0Rnd ⊆ R be the subring of Cd-invariants. By Boutot [Bou], B is a normal C-algebra and spec(B) has rational singularities. Assume thatH0 is semisimple; extension to the reductive case being easy. Let π∗ be the dominant weight of H0 such that V π = Vπ∗ . By Borel-Weil [FH], Cπ∗ = ⊕n≥0V nπ = ⊕n≥0Vnπ∗ , is the homogeneous coordinate ring of theH0-orbit of the point vπ∗ ∈ P (Vπ∗) corresponding to the highest weight vector. This H0-orbit is isomorphic to H0/Pπ∗ , where Pπ∗ ⊆ H0 is the parabolic stabilizer of vπ∗ . Hence Cπ∗ is normal and spec(Cπ∗) has rational signularities; cf. [Ha, MR, Rm, Sm]. It follows that B ⊗ Cπ∗ is also normal, and spec(B ⊗ Cπ∗) has rational singularities. Consider the action of C∗ on B ⊗ Cπ∗ given by: x(b⊗ c) = (x · b)⊗ (x−1 · c), where x ∈ C∗ maps b ∈ Bn to x nb, the action on Cπ∗ being similar. Consider the invariant ring S = (B ⊗ Cπ∗) = ⊕nSn = ⊗n≥0Rnd ⊗ V nπ. (4.2) By Boutot [Bou], it is a normal, and spec(D) has rational singularities. Since Vnπ is an H-module, the algebra S has an action of H. Let T = T (sπd ) = S H = ⊕n≥0Tn (4.3) be its subring of H-invariants. By Boutot [Bou], it is normal, and spec(T ) has rational singularities–this is the crux of the proof. By Schur’s lemma, the multiplicity of the trivial H-representation in Sn = Rnd⊗V nπ is precisely the multiplicity s̃πd (n) of the H-module Vnπ in Rnd. Hence, the Hilbert function of T , i.e., dim(Tn), is precisely s̃ d (n), and the Hilbert series n≥0 dim(Tn)t is Sπ (t). Quasipolynomiality of s̃π (n) follows by applying the following lemma: Lemma 4.1.3 (cf. [Dh]) If T = ⊕∞n=0Tn is a graded C-algebra, such that spec(T ) is normal and has rational simgularites, then dim(Tn), the Hilbert function of T , is a quasi-polynomial function of n. (d) Since spec(T ) has rational singularities, T is Cohen-Macaualey. Let t1, . . . , tu be its homogeneous sequence of parameters (h.s.o.p.), where u = k + 1 is the Krull dimension of T . By the theory of Cohen-Macauley rings [St2], it follows that its Hilbert series Sπd (t) is of the form h0 + h1t+ · · ·+ hkt i=1 (1− t , (4.4) where (1) h0 = 1, (2) di is the degree of ti, and (3) hi’s are nonnegative integers. This proves (d). Q.E.D. Remark 4.1.4 A careful examination of the proof above shows that ratio- nality of Sπd (t), and more strongly, asymptotic quasi-polynomiality of s̃ d (n) as n → ∞, can be proved using just Hilbert’s result on finite generation of the algebra of invariants of a reductive-group action. Boutot’s result is nec- essary to prove quasi-polynomiality for all n. This is crucial for saturated and positive integer programming (Chapter 3). 4.1.1 The minimal positive form and modular index The form (4.4) of Sπd (t) is not unique because it depends on the degrees di’s of the paramters ti’s. For future use, let us record the following consequences of the proof. Let T be the ring constructed in the proof above. Corollary 4.1.5 Suppose T has an h.s.o.p. t = (t1, . . . , tu) with di = deg(ti). Then S (T ) has a positive rational form (4.4) with di = deg(ti) therein. The proof above is lets us define a minimal positive form of the rational function Sπ (t) associated with a structural constant s. For this, let us or- der h.s.o.p.’s of T lexicographically as per their degree sequences. Here the degree seqeunce of an h.s.o.p. t = (t1, . . . , tu) is defined to be (d1, . . . , du), where di = deg(ti). The form (4.4) is the same for any h.s.o.p. of lexi- cographically minimum degree sequence. We call it the minimal positive form of Sπ (t). The modular index of sπ is defined to be max{di}, where (d1, . . . , du) is the degree sequence of a lexicographically minimal h.s.o.p. Since Problems 1.1.1, 1.1.2,1.1.3, 1.2.1 are special cases of Problem 1.1.4, this defines minimal positive forms of the rational generating functions of the stretching quasi-polynomials (cf. Theorem 3.4.1) associated with the struc- tural constants in these problems, and also the modular indices of these structural constants. 4.1.2 The rings associated with a structural constant The preceding proof also associates with the structural constant s a few rings which will be important later. Specifically, let S = S(s) and T = T (s) be the rings as in Theorem 4.1.1 (c) associated with the structural constant s = sπ . Let R = R(s) be the homogeneous coordinate ring of X as in Theorem 4.1.1. We call R(s), S(s) and T (s) the rings associated with the structure constant s. When s = mπ , as in the subgroup restriction problem (Problem 1.1.3), X ∼= G/P as given in eq.(3.10. Then these rings are explicitly as follows: ) = ⊕n≥0Vnλ(G), ) = ⊕n≥0Vnλ(G)⊗ Vnπ(H) T (mπ ) = ⊕n≥0(Vnλ(G) ⊗ Vnπ(H) ∗)H . (4.5) By specializing the subgroup restriction problem further to the Littlewood- Richardson problem (Problem 1.2.1), we get the following rings associated by Brion (cf. [Dh]) with the Littlewood-Richardson coefficient cλ R(cλα,β) = ⊕n≥0Vnα(H)⊗ Vnβ(H), ) = ⊕n≥0Vnα(H)⊗ Vnβ(H)⊗ Vnλ(H) T (cλ ) = ⊕n≥0(Vnα(H)⊗ Vnβ(H)⊗ Vnλ(H) ∗)H . (4.6) 4.2 Canonical models There are several rings other than T (cλ ) whose Hilbert function coincides with the Littlewood-Richardson stretching quasi-polynomial c̃λ (n). For example, let P = P λ be the BZ-polytope [BZ] whose Ehrhart quasi- polynomial coincides with c̃λ (n). We can associate with P a ring TP as in Stanley [St3] whose Hilbert function coincides with c̃λ (n). There are many other choices for P . For example, in type A, we can consider a hive polytope or a honeycomb polytope [KT1] instead of the BZ-polytope. The rings TP ’s associated with different P ’s will, in general, be different, and there is nothing canonical about them. In contrast, the ring T (cλα,β) is special because: Proposition 4.2.1 (PH0) The rings R(cλ ), S(cλ ), T (cλ ) have quan- tizations Rq(c ), Sq(c ), Tq(c ) endowed with canonical bases in the ter- minology of Lusztig [Lu4]. Furthermore, the canonical bases of Rq(c ), Sq(c are compatible with the action of the Drinfeld-Jimbo quantum group associ- ated with H = GLn(C), and the canonical basis of Sq(c α,β) is an extension of the canonical basis of Tq(c α,β) in a natural way. This follows from the work of Lusztig (cf. [Lu3], Chapter 27 in [Lu4]) and Kashiwara (cf. Theorem2 in [Kas3]). Specializations of these canonical bases at q = 1 will be called canonical bases of R(cλα,β), S(c α,β), T (c α,β). Lusztig [Lu4] has conjectured that the structural constants associated with the canonical bases in Proposition 4.2.1 are polynomials in q with nonnega- tive integral coefficients as in the case of the canonical basis of the (negative part of the) Drinfeld-Jimbo enveloping algebra. We refer to Proposition 4.2.1 as PH0 in view of this (conjectural) positivity property. In view of this proposition, we call the rings R(cλα,β), S(c α,β) and T (c the canonical rings associated with the Littlewood-Richardson coefficient , and X = Proj(R(cλ )), Y = Proj(S(cλ )) and Z = Proj(T (cλ )) the canonical models associated with cλα,β. 4.2.1 From PH0 to PH1,3 Now we study the relevance of PH0 above in the context of PH1,SH,PH2, and PH3 for Littlewood-Richardson coefficients (Section 1.2). As already remarked in Section 1.7, PH1 for Littlewood-Richardson coeffi- cients is a formal consequence of the properties of Kashiwara’s crystal oper- ators on the canonical bases in PH0 (Proposition 4.2.1); [Dh, Kas2, Li, Lu4]. Specifically, the canonical basis of the ring Rq(c ) also yields a canon- ical basis for the tensor product Vq,α ⊗ Vq,β of the irreducible Hq modules with highest weights α and β. The Littlwood-Richardson rule for arbitrary types follows from the study of Kashiwara’s crystal operators on this canon- ical basis for the tensor product; [Lu4]. This rule is equivalent to the one in [Li] based on combinatorial interpretation of the crystal operators in the path model therein. The article [Dh] derives a convex polyhedral formula for Littlewood-Richardson coefficients (of arbitrary type) using this com- binatorial interpretation. Though the complexity-theoretic issues are not addressed in [Dh], it can be verified that the polyhedral formula therein is a convex #P -formula. This yields PH1 for Littlewood-Richardson coefficients of arbitrary types using PH0. Now let us see the relevance of PH0 in the context of SH for Littlewood- Richardson coefficients of arbitrary type. The polytope in [Dh], mentioned above, for type A is equivalent to the hive polytope in [KT1] in the sense that the number integer points in both the polytopes is the same. Knutson and Tao prove SH for type A by show- ing that the hive polytope always has in integral vertex. To extend this proof to an arbitrary type, one has to convert the polytope in [Dh] into a polytope that is guaranteed to contain an integral vertex if the index of the stretching quasipolynomial c̃λ (n) is one. The main difficulty here is that we do not have a nice mathematical interpretation for the index. Algorithm in Theorem 3.1.1 applied to the polytope in [Dh] computes this index in polynomial time. But it does not give a nice interpretation that can be used in a proof as above. This index is simply the largest integer dividing the degrees of all ele- ments in any basis of the canonical ring T (cλ )–in particular, the canon- ical basis. This follows by applying Proposition 3.1.3 to the polytope in [Dh]. This leads us to ask: is there an interpretation for the index based on Lusztig’s topological construction of the canonical basis in Proposition 4.2.1? If so, this may be used to extend the known polyhedral proof for SH in type A to arbitrary types. Alternatively, it may be possible to prove SH using topological properties of the canonical basis in the spirit of the topological (intersection-theoretic) proof [Bl] of SH in type A. Now let us see the relevance of PH0 in the context of PH3 for Littlewood- Richardson coefficients. First, let us consider the minimal positive form (Section 4.1.1) associated with a Littlewood-Richardson coefficient cλ of type A. Let T = T (cλ denote the ring that arises in this case; cf. eq.(4.6). Now we can ask: Question 4.2.2 Are all di’s occuring in the minimal positive form (cf. (4.4)) one in this special case? This is equivalent to asking if the ring T = T (cλα,β) in this case is integral over T1, the degree one component of T . If so, this would provide an explanation for the conjecture of King at al [KTT] (cf. eq.(1.3)) in the theory of Cohen-Macauley rings: Proposition 4.2.3 Assuming yes, the conjecture of King et al [KTT] (Hy- pothesis 1.2.6) holds. Remark 4.2.4 In contrast, the ring TP associated with the hive polytope (cf. beginning of Section 4.2) need not be integral over its degree one compo- nent, in view of the fact that the hive polytope can have nonintegral vertices [DM1]. Remark 4.2.5 T = T (cλ ) need not be generated by its degree one compo- nent T1. If this were always so, the h-vector (hd, · · · , h0) in eq.(1.3) would be an M-vector (Macauley-vector) [St2]. But one can construct α, β and λ for which this does not hold. Proof: (of the proposition) Since T is integral over T1, it has an h.s.o.p., all of whose elements have degree 1. By Theorem 3.4.1, the singularities of spec(T ) are rational. Hence T is Cohen-Macaulay. Now the result immediately follows from the theory of Cohen-Macauley rings [St2]. Q.E.D. In view of this Proposition, the conjecture of King et al will follow if all canonical basis elements of T (cλ ) can be shown to be integral over the basis elements of degree one. This requires a further study of the multiplicative structure of this canonical basis. Considerations for PH3 (Hypothesis 1.2.8) for Littlewood-Richardson coefficients of arbitrary type are similar. Similarly, the positivity property (PH2) of the stretching quasipolynomial associated with Littlewood-Richardson coefficients may possibly follow from a deep study of the multiplicative structure of the canonical basis as per PH0 (Proposition 4.2.1), just as positivity of the multiplicative structural coefficients of the canonical basis for the (negative part of the) Drinfeld- Jimbo enveloping algebra follows from a deep study of the multiplicative structure of this basis [Lu4]. 4.2.2 On PH0 in general The discussion above indicates that for Littlewood-Richardson coefficients PH1,SH,PH3, and plausibly PH2 as well are intimately related to PH0 (Proposition 4.2.1). This leads us to ask if the rings associated in Sec- tion 4.1.2 with other structural constants under consideration in this paper have quantizations which satisfy appropriate forms of PH0. If so, this PH0 may be used to derive PH1, SH, PH3, and PH2 (Hypotheses 3.3.1-3.3.4) for these structural constants. Note that SH (a) follows from PH3 (see the remark after Hypothesis 3.3.4); PH2 may also follow from PH3. Thus PH1 and PH3 are the ones to focus on. To formalize this, let s be a structural constant which is either the Kro- necker coefficient as in Problem 1.1.1, or the plethysm constant as in Prob- lem 1.1.2, or the multiplicity mπ in Problem 1.1.3, or the multiplicity sπ in Problem 1.1.4, when X therein is a class variety. Let R(s), S(s), T (s) be the rings associated with s (Section 4.1.2). Let X(s) = Proj(R(s)), Y (s) = Proj(S(s)) and Z(s) = Proj(R(s)). We call R = R(s), S = S(s), T = T (s) the canonical rings associated with s, and X(s), Y (s), Z(s) the canonical models associated with s, because we expect these rings and models to be special as in the case of the Littlewood-Richardson coefficients. Let H be as in Problem 1.1.3 or Problem 1.1.4. Assume that H is connected. Let Hq denote the Drifeld-Jimbo quantization of H. Now we Question 4.2.6 (PH0??) Are there quatizations Rq, Sq of R,S, with Hq- action, and a quantization Tq of T with “canonical” bases (in some appro- priate sense) B(Rq), B(Sq), B(Tq), where B(Rq) and B(Sq) are compatible with the Hq-action and B(Sq) is an extension of B(Tq)? Furthermore, do these canonical bases have appropriate positivity properties? In other words, are there quantizations of R,S and T for which PH0 (Proposition 4.2.1) can be extended in a natural way? If so, this extended PH0 may be used to prove PH1 and SH for s just as in the case of Littlewood-Richardson coefficients (of type A). 4.3 Nonstandard quantum group for the Kronecker and the plethysm problems We now consider this question when s is the kronecker or the plethysm constant (cf. Problems 1.1.1 and 1.1.2). PH0 for Littlewood-Richardson coefficients (Proposition 4.2.1) depends critically on the theory of Drinfeld- Jimbo quantum groups. This is intimately related (in type A) [GrL] to the representation theory of Hecke algebras. To extend PH0 in the context of the kronecker and the plethysm constants, one needs extensions of these theories in the context of Problems 1.1.1-1.1.2. In this section, we briefly review the results in [GCT4, GCT8, GCT7] in this direction and the theoretical and experimental evidence it provides in support of PH0–that is, affirmative answer to Question 4.2.6–in this context. So let us consider the generalized plethysm problem (Problem 1.1.2). As expected, the representation theory of Drinfeld-Jimbo quantum groups and Hecke algebras does not work in the context of this general problem. Briefly, the problem is that if H is a connected, reductive group and V its representation, then the homomorphism H → G = GL(V ) does not quan- tize in the setting of Drinfeld-Jimbo quantum groups. That is, there is no quantum group homomorphism from Hq, the Drinfeld-Jimbo quantization of H, to Gq, the Drinfeld-Jimbo quantization of G. In [GCT4, GCT7], a new nonstandard quantization GHq of G– called a nonstandard quantum group–is constructed so that there is a quantum group homomorphism Hq → G When H = G, GHq coincides with the Drinfeld-Jimbo quantum group. The article [GCT8] gives a conjectural scheme for constructing a nonstandard canonical basis for the matrix coordinate ring of GHq that is akin to the canonical basis for the matrix coordinate ring of the Drinfeld-Jimbo quan- tum group [Lu4, Kas3]. It is known that the Drinfeld-Jimbo quantum group Gq = GLq(V ) and the Hecke algebra Hn(q) are dually paired: i.e., they have commuting ac- tions on V ⊗nq from the left and the right that determine each other, where Vq denotes the standard quantization of V . Furthermore, the Kazhdan-Lusztig basis forHn(q) is intimately related to the canonical basis for Gq [GrL]. Sim- ilarly, [GCT7] constructs a nonstandard generalization BHn (q) of the Hecke algebra which is (conjecturally) dually paired to GHq . The article [GCT8] gives a conjectural scheme for constructing a nonstandard canonical basis of BHn (q) akin to the Kazhdan-Lusztig basis of the Hecke algebra Hn(q). The nonstandard quantum groupGHq and the nonstandard algebraB n (q) turn out to be fundamentally different from the standard Drinfeld-Jimbo quantum group Gq and the Hecke algebra Hn(q). For example, the non- standard quantum group GHq is a nonflat deformation of G in general. This means the Poincare series of the matrix coordinate ring of GHq is different from the Poincare series of the matrix coordinate ring of G. Specifically, the terms of the first series can be smaller than the respective terms of the second series. Similarly, BHn (q) is a nonflat deformation of the group algebra C[Sn] of the symmetric group Sn; i.e., its dimension can be bigger than that of C[Sn]. Nonflatness of GHq intuitively means that it is “smaller” than G in gen- eral. Hence, it may seem that there is a loss of information when one goes from G to GHq . Fortunately, there is none, as per the reciprocity conjecture in [GCT7]. This roughly says that the information which is lost in the tran- sition from G to GHq simply gets transfered to B n (q), which is bigger than Hn(q). In other words, there is no information loss overall. Hence analogues of the properties in the standard setting should also hold in the nonstandard setting, though in a far more complex way. That is what seems to happen to positivity. Specifically, experimental ev- idence suggests that the conjectural nonstandard canonical bases in [GCT8] have nonstandard positivity properties which are complex versions of the positivity properties in the standard setting. See [GCT7, GCT8, GCT10] for a detailed story. 4.4 The cone associated with the subgroup restric- tion problem In this section, we prove Theorem 3.4.2, by extending the proof of Brion and Knop (cf. [El]) for the Littlewood-Richardson problem. The proof is in the spirit of the proof of quasipolynomiality in Section 4.1. Let G be a connected, reductive group, H a connected, reductive sub- group, and ρ : H → G a homomorphism. Theorem 3.4.2 has the following equivalent formulation. Let S(H,G) be the set of pairs (µ, λ) such that Vµ(H)⊗ Vλ(G) has a nonzero H-invariant. Then, Theorem 4.4.1 The set S(H,G) is a finitely generated semigroup with re- spect to addition. When G = H ×H and the embedding H ⊆ G is diagonal, this special- izes to the Brion-Knop result mentioned above. The proof follows by an extension the technique therein. Proof: Let B be a Borel subgroup of G, U the unipotent radical of B and T the maximal torus in B. Similarly, let B′ be a Borel subgroup of H, U ′ the unipotent radical of B′ and T ′ the maximal torus in B′. Without loss of generality, we can assume that B′ ⊆ B, U ′ ⊆ U , T ′ ⊆ T . Let A = C[G]U be the algebra of regular functions on G that are invariant with respect to the right multiplication by U . It is known to be finitely generated [El]. The groups G and T act on A via left and right multiplication, respectively. As a G× T -module, A = ⊕λVλ(G), (4.7) where the torus T acts on Vλ(G) via multiplication by the highest weight λ∗ of the dual module. Similarly, A′ = C[H]U = ⊕λVµ(H), (4.8) where the torus T ′ acts on Vµ(H) via multiplication by the highest weight µ∗ of the dual module. Now A⊗A′ is finitely generated since A and A′ are. Let X = (A⊗A′)H be the ring of invariants of H acting diagonally on A⊗A′. The torus T ×T ′ acts on X from the right. Since H is reductive, X is finitely generated [PV]. Hence, the semigroup of the weights of the right action of T × T ′ on X is finitely generated. We have X = (A⊗A′)H = ((⊕Vλ(G)) ⊗ (⊕Vµ(H))) H = ⊕(Vλ(G)⊗ Vµ(H)) and the weights of the algebra X are of the form (λ∗, µ∗) such that Vλ(G)⊗ Vµ(H) contains a nontrivial H-invariant. Therefore these pairs form a finitely generated semigroup. Q.E.D. For the sake of simplicity, assume that G and H are semisimple in what follows. Let TR(H,G) denote the polyhedral convex cone in the weight space of H ×G generated by T (H,G), as defined in Theorem 3.4.2. This is a generalization of the Littlewood-Richardson cone (Section 2.2.2). The following generalization of Corollary 3.2.3 is a consequence of The- orem 3.1.1 and its proof. Theorem 4.4.2 Assume that the positivity hypothesis PH1 (Section 3.3) holds for the subgroup restriction problem for the pair (H,G), where both H and G are classical. Given dominant weights µ, λ of H and G, the polytope Pµ,λ as in PH1 has a specification of the form Ax ≤ b (4.9) where A depends only on H and G, but not on µ or λ, and b depends homogeneously and linearly on µ, λ. Let n be the total number of columns in A. Then, there exists a decomposition of TR(H,G) into a set of polyhedral cones, which form a cell complex C(H,G), and, for each chamber C in this complex, a set M(C) of O(n) modular equations, each of the form aiµi + biλi = 0 (mod d), such that 1. Saturation hypothesis SH is equivalent to saying that: (µ, λ) ∈ T (H,G) iff (µ, λ) ∈ TR(H,G) and (µ, λ) satisfies the modular equations in the set M(Cµ,λ) associated with the smallest cone Cµ,λ ∈ C(H,G) contain- ing (µ, λ). 2. Given (µ, λ), whether (µ, λ) ∈ TR(H,G) can be determined in polyno- mial time. 3. If so, whether (µ, λ) satisfies the modular equations associated with the smallest cone in C(H,G) containing it can also be determined in polynomial time. Proof: Given a point p = (µ′, λ′) in the weight space of H×G, where µ′ and λ′ are arbitrary rational points, let S(p) denote the constraints (half-spaces) in the sytem (4.9) whose bounding hyperplanes contain the polytope Pµ′,λ′ . We can decompose TR(H,G) into a conical, polyhedral cell complex, so that given a cone C in this complex, and a point p in its interior, the set S(p) does not depend on p. We shall denote this set by S(C). Thus the affine span of Pµ,λ, for any (µ, λ) ∈ C, is determined by the linear system A′x = b′, where [A′, b′] consists of the rows of [A, b] in (4.9) corresponding to the set S(C). By finding the Smith normal form of A′, we can associate with C a set of modular equations that the entries of b′ must satisfy for this affine span to contain an integer point; see the proof of Theorem 3.1.1. Since the entries of A′ depend only on H and G, these equations depend only on C. If (µ, λ) ∈ T (H,G), then (µ, λ) is integral, and hence these equations are satisfied. Conversely, if (µ, λ) ∈ TR(H,G) and these equations are satisfied, then the saturation property implies that (µ, λ) ∈ T (H,G), as seen by examining the proof of Theorem 3.1.1. Furthermore, given (µ, λ), the algorithm in the proof of Theorem 3.1.1 implicitly determines if (µ, λ) ∈ TR(H,G) and if these modular equations are satisfied in polynomial time. Q.E.D. 4.5 Elementary proof of rationality In this section we give an elementary proof of rationality in Theorem 3.4.1 (a), when H therein is connected–actually of a slightly stronger statement: namely, the stretching function m̃π (n) is asymptotically a quasipolynomial, as n → ∞; cf. Remark 4.1.4. But this proof cannot be extended to prove quasipolynomiality for all n. The proof here is motivated by the work of Rassart [Rs], De Loera and McAllister on the stretching function associated with a Littlewood-Richardson coefficient. First, we recall some standard results that we will need. Vector partition functions Given an integral s×n matrix B and integral n-vector c, consider the vector paritition function φB(c), which is the number of integer solutions to the integer programming problem By = c, y ≥ 0. (4.10) For a fixed c, b, let φB,c(n) = φB(nc) φB,c,b(n) = φB(nc+ b). (4.11) By Sturmfels [Stm] and Szenes-Vergne residue formula [SV], φB(c) is a piecewise quasipolynomial function of c. That is, Rn can be decomposed into polyhedral cones, called chambers, so that the restriction of φB(c) to each chamber R is a multivariate quasipolynomial function of the coordinates of c. This implies that φB,c(n) is a quasipolynomial function of n. It also implies that the function φB,c,b(n) is asymptotically a quasipolynomial function of n, as n→ ∞, because the points nc+ b, as n→ ∞, lie in just one chamber. The Szenes-Verne residue formula [SV] for vector partition functions also implies that there is a constant d(B), depending only on B, such that the period of φB,c(n), for any c, divides d(B). Klimyk’s formula Let H ⊆ G and mπλ be as in Theorem 3.4.1 (a), with H connected. Let us assume that H is semisimple, the general case being similar. Let H and G be the Lie algebras of H and G respectively. We recall Klimyk’s formula for . Without loss of generality, we can assume that the Cartan subalgebra C ⊆ H is a subalgebra of the Cartan subalgebra D ⊆ G. So we have a restriction from D∗ to C∗, and we assume that the half-spaces determining positive roots are compatible. We denote weights of H by symbols such as µ and of G by symbols such as µ̄. To be consistent, we shall use the notation instead of mπ in this proof. We write µ̄ ↓ µ if the weight µ̄ of G restricts to the weight µ of H. We denote a typical element of the Weyl group of H by W , and a typical element of the Weyl group of G by W̄ . Given a dominant weight π of G and a weight µ̄ of G, let nµ̄(λ̄) denote the dimension of the weight space for µ̄ in Bλ̄ = Vλ̄(G). We assume that: (A): For any weight µ of H, the number of µ̄’s such that µ̄ ↓ µ is finite. For example, this is so in the plethysm problem (Problem 1.1.2). We shall see later how this assumption can be removed. By Klimyk’s formula (cf. page 428, [FH]), (−1)W µ̄↓π−ρ−W (ρ) nµ̄(Vλ̄), (4.12) where ρ is half the sum of positive roots of H. We allow µ̄ in the inner sum to range over all weights µ̄ of G such that µ̄ ↓ π − ρ −W (ρ) by defining nµ̄(Vλ̄) to be zero if µ̄ does not occur in Vλ̄. Proof of Theorem 3.4.1 (a) The goal is to express m̃π (n) as a linear combination of vector partition functions φB,c,b(n)’s, for suitable B, c, b’s, using Klimyk’s formula for m After this, we can deduce asymptotic quasipolynomiality of m̃π (n) from asymptotic quasipolynomiality of φB,c,b(n)’s. By Kostant’s multiplicity formula (cf. page 421 [FH]), nµ̄(Vλ̄) = (−1)W̄P (W̄ (λ̄+ ρ̄)− (µ̄+ ρ̄)), (4.13) where P (λ̄), for a weight λ̄ of G, denotes the Kostant partition function; i.e., the number of ways to write λ̄ as a sum of positive roots of G. It is important for the proof that Kostant’s formula (4.13) holds even if µ̄ is not a weight that occurs in the representation Vλ̄–in this case, nµ̄(Vλ̄) = 0, and the right hand side of (4.13) vanishes. By eq.(4.12) and (4.13), (−1)W (−1)W̄ µ̄↓π−ρ−W (ρ) P (W̄ (λ̄+ ρ̄)− (µ̄+ ρ̄)). (4.14) Let D denote the dominant Weyl chamber in the weight space of G. Let C denote the Weyl chamber complex associated with the weight space of G. The cells in this complex are closed polyhedral cones. Each cone is either the chamber W̄ (D), for some Weyl group element W̄ , or a closed face of W̄ (D) of any dimension. Using Möbius inversion, the inner sum µ̄↓π−ρ−W (ρ) P (W̄ (λ̄+ ρ̄)− (µ̄+ ρ̄)) in eq.(4.14) can be written as a linear combination µ̄∈C:µ̄↓π−ρ−W (ρ) P (W̄ (λ̄+ ρ̄)− (µ̄+ ρ̄)), where C ranges over chambers in the Weyl chamber complex C, a(C) is an appropriate constant for each C. Hence, (−1)W (−1)W̄ µ̄∈C:µ̄↓π−ρ−W (ρ) P (W̄ (λ̄+ ρ̄)− (µ̄+ ρ̄)). (4.15) Now think of π and λ̄ as variables. But H and G are fixed, and hence also the quantities such as ρ and ρ̄. Claim 4.5.1 For fixed Weyl group elements W, W̄ and a fixed C, the sum µ̄∈C:µ̄↓π−ρ−W (ρ) P (W̄ (λ̄+ ρ̄)− (µ̄ + ρ̄)) (4.16) can be expressed as a vector partition function associated with an appropriate linear system By = c, y ≥ 0, (4.17) where the matrix B = BH,G,C , depends only on C and the root systems of H and G, but not on π and λ̄, and the coordinates of the vector c = mW,W̄,C(λ̄, π, ρ, ρ̄), depend on W, W̄ ,C, ρ, ρ̄, π, π, and furthermore, their dependence on π, λ̄, ρ, ρ̄ is linear. Here assumption (A) is crucial. Without it, the sum (4.16) can diverge. Of course, without assumption (A), we can still make the sum finite, by requir- ing that µ̄ lie within the convex hull Hλ̄ generated by the points {W̄ (λ̄)}, where W̄ ranges over all Weyl group elements. This means we have to add constraints to the system (4.17) corresponding to the facets of Hλ̄. But the entries of the resulting B would depend on λ̄, and the theory of vector partition functions will no longer apply. Proof of the claim: Let µ̄i’s denote the integer coordinates of µ̄ in the basis of fundamental weights. We denote the integer vector (µ̄1, µ̄2, · · · ) by µ̄ again. The Kostant partition function P (ν) is a vector partition function associated with an integer programming problem: BP v = ν, v ≥ 0, where the columns of BP correspond to positive roots of G. The sum in (4.16) is equal to the number of integral pairs (µ̄, v) such that 1. µ̄ ∈ C, 2. µ̄ ↓ π − ρ−W (ρ), 3. BP v = W̄ (λ̄+ ρ̄)− (µ̄ + ρ̄), v ≥ 0. The first two condititions here can be expressed in terms of linear con- straints (equalities and inequalities) on the coordinates µ̄i’s. Thus the three conditions together can be expressed in terms of linear constraints on (µ̄, v). By the finiteness assumption (A), the polytope determined by these con- straints is a bounded polytope. The number of integer points in such a polytope can be expressed as a vector partition function (cf. [BBCV]). This proves the claim. Let us denote the vector partition associated with the integer program- ming problem (4.17) in the claim by φW,W̄ ,C(c(λ̄, π, ρ, ρ̄)). Then (−1)W (−1)W̄ a(C)φW,W̄ ,C(c(λ̄, π, ρ, ρ̄)). (4.18) Hence, (n) = mnλ̄nπ = (−1)W (−1)W̄ a(C)φW,W̄ ,C(c(nλ̄, nπ, ρ, ρ̄)). (4.19) It follows from Claim 4.5.1 and the standard results on vector partition functions mentioned in the begining of this section that gW,W̄ ,C(n) = φW,W̄ ,C(c(nλ̄, nπ, ρ, ρ̄)), is asymptitically a quasipolynomial function of n. Hence, m̃π (n) is also asymptotically a quasipolynomial function of n. This implies (cf. [St1]) (t) = (n)tn (4.20) is rational function of t. This proves Theorem 3.4.1 (a) under the finiteness assumption (A). It remains to remove the assumption (A). Let G′ ⊇ H be the smallest Levi subalgebra of G containing H. Then mππ′ , (4.21) where π′ ranges over dominant weights of G′, mπ denotes the multiplicity of Vπ′(G ′) in Vλ̄(G), and m the multiplicity of Vπ(H) in Vπ′(G ′). Furthermore, 1. the finiteness asssumption (A) is now satisfied for the pair (G′,H): i.e., for any weight µ of H, the number of weights µ′’s of G′ such that µ′ ↓ µ is finite. 2. There is a polyhedral expression for mπ ; this follows from [Li, Dh]. By the first condition and the argument above, we get an expression for akin to (4.18). Substituting this expression and the polyhedral expres- sion for mπ in (4.21), leads to a formula for m̃π (n) as a linear combination of φB,c,b(n)’s for appropriate B, c, b’s. After this, we proceed as before. This proves Theorem 3.4.1 (a). Q.E.D. We also note down the following consequence of the proof. Proposition 4.5.2 There is a constant D depending only G and H, such that for any λ̄, π, orders of the poles of Mπ (t) (cf. (4.20), as roots of unity, divide D. A bound onD provided by the proof below is very weak: D = O(2O(rank(G))). Proof: It suffices to to bound the period of the quasipolynomial m̃π (n). For this, it suffices to let n→ ∞. For a fixed W, W̄ ,C, the chamber containing c(nλ̄, nπ, ρ, ρ̄)) is completely determined by λ̄ and π as n→ ∞. Under these conditions, the degree of φW,W̄ ,C(c(nλ̄, nπ, ρ, ρ̄)) is equal to the dimension of the polytope associated with this vector partition function. This dimension is clearly O(rank(G)2). By Szenes-Vergne residue formula [SV], there is a constant D depending on only G,H,W, W̄ , C, such that the period of the quasipolynomial h(n) = φW,W̄ ,C(c(nλ̄, nπ, 0, 0)) divides D for every λ̄, π; here we are putting ρ and ρ̄ equal to zero, since we are interested in what happens as n→ ∞. Q.E.D. Chapter 5 Parallel and PSPACE algorithms In this chapter we give PSPACE algorithms (cf. Theorem 3.4.3) for com- puting the various structural constants under consideration . We shall only prove Theorem 3.4.3, when H is therein is either a complex, semisimple group, or a symmetric group, or a general linear group over a finite field, the extension to the general case being routine. We recall two standard results in parallel complexity theory [KR], which will be used repeatedly. Let NC(t(N), p(N) denote the class of problems that can be solved in O(t(N)) parallel time using O(p(N)) processors, where N denotes the bitlength of the input. Let NC = ∪iNC(log i(N),poly(N)). This is the class of problems having efficient parallel algorithms. Proposition 5.0.3 [Cs, KR] Let A be an n × n-matrix with entries in a ring R of characteristic zero. Then the determinant of A, and A−1, if A is nonsingular, can be computed in O(log2 n) parallel steps using poly(n) processors; here each operation in the ring is considered one step. Hence, if R = Q, the problems of computing the determinant, the inverse and solving linear systems belong to NC. Proposition 5.0.4 The class NC(t(N), 2t(N)) ⊆ SPACE(O(t(N))). In particular, NC(poly(N), 2O(poly(N))) ⊆ PSPACE. 5.1 Complex semisimple Lie group In this section we prove a special case of Theorem 3.4.3 for the general- ized plethym problem (Problem 1.1.2). Accordingly, let H be a complex, semisimple, simply connected Lie group, G = GL(V ), where V = Vµ(H) is an irreducible representation of H with dominant weight µ, ρ : H → G the homomorphism corresponding to the representation, and mπ the multiplic- ity of Vπ(H) in Vλ(G), considered as an H-module via ρ; cf. Problem 1.1.3. Then: Theorem 5.1.1 The multiplicitymπ can be computed in poly(〈λ〉, 〈µ〉, 〈π〉,dim(H)) space. Here it is assumed that the partition λ = λ1 ≥ λ2 ≥ · · ·λr > 0 is rep- resented in a compact form by specifying only its nonzero parts λ1, . . . , λr. This is important since dim(G) can be exponential in dim(H) and 〈µ〉. A compact representation allows 〈λ〉 to be small, say poly(dim(H), 〈µ〉), in this case. We begin with a simpler special case. Proposition 5.1.2 If dim(V ) = poly(dim(H)), then mπ can be computed in PSPACE; i.e., in poly(〈λ〉, 〈µ〉, 〈π〉,dim(H)) space. This implies that the Kronecker coefficient (Problem 1.1.1) can be computed in PSPACE. Proof: Let us use the notation λ̄ instead of λ to be consistent with the notation used in Klimyk’s formula (4.12). By the latter,mπ can be computed in PSPACE if nµ̄(Vλ̄) in that formula can be computed in PSPACE for every µ̄ and λ̄. In type A, this is just the number of Gelfand-Tsetlin tableau with the shape λ̄ and weight µ̄. If dim(V ) = poly(dim(H)), the size of such a tableau is O(dim(V )2) = poly(dim(H)). So we can count the number of such tableu in PSPACE as follows: Begin with a zero count, and cycle through all tableaux of shape λ̄ in polynomial space one by one, increasing the count by one everytime the tableau satisfies all constraints for Gelfand- Tsetlin tableau and has weight µ̄. In general, the role of Gelfand-Tsetlin tableaux is played by Lakshmibai-Seshadri (LS) paths [Li, Dh]. Q.E.D. The argument above does not work if dim(V ) is not poly(dim(H)), as in the plethym problem (Problem 1.1.2), where dim(V ) = dim(Vµ) can be exponential in n = dim(H) and the bitlength of µ. In this case, the algorithm cannot even afford to write down a tableau since its size need not be polynomial. Next we turn to Theorem 5.1.1. For the sake of simplicity, we shall prove it only for H = SLn(C), or rather GLn(C)–i.e., the usual plethysm problem. This illustrates all the basic ideas. The general case is similar. We shall prove a slightly stronger result in this case: Theorem 5.1.3 The plethysm constant aπ can be can be computed in poly(〈λ〉, 〈µ〉, 〈π〉) space. Here the dependence on n = dim(H) is not there. This makes a difference if the heights of µ and π are less than n = dim(H)–remember that we are using a compact representation of a partition in which only nonzero parts are specified. This is really not a big issue. Because aπ depends only on the partitions λ, µ, π and not n. Hence, without loss of generality, we can assume that n is the maximum of the heights of µ and π. It is possible to strengthen Theorem 5.1.1 similarly. To prove Theorem 5.1.3, we shall give an efficient parallel algorithm to compute ãπλ,µ that works in poly(〈λ〉, 〈µ〉, 〈π〉) parallel time usingO(2 poly(〈λ〉,〈µ〉,〈π〉)) processors. This will show that the problem of computing ãπλ,µ is in the com- plexity class NC(poly(〈λ〉, 〈µ〉, 〈π〉), 2poly(〈λ〉,〈µ〉,〈π〉)), which is contained in PSPACE by Proposition 5.0.4. The basic idea is to parallelize the classical character-based algorithm for computing aπ by using efficient parallel algo- rithm for inverting a matrix and solving a linear system (Proposition 5.0.3). We begin by recalling the standard facts concerning the characters of the general linear group. Given a representation W of GLm(C), let ρ : GLm(C) → GL(W ) be the representation map. Let χρ(x1, . . . , xm) de- note the formal character of this representation W . This is the trace of ρ(diag(x1, . . . , xm)), where diag(x1, . . . , xn) denotes the generic diagonal ma- trix with variable entries x1, . . . , xm on its diagonal. If W is an irreducible representation Vλ(GLm(C)), then χρ(x1, . . . , xm) is the Schur polynomial Sλ(x1, . . . , xm). By the Weyl character formula, λi+m−i |xm−ij | , (5.1) where |aij | denotes the determinant of anm×m-matrix a. The Schur polyno- mials form a basis of the ring of symmetric polynomials in x1, . . . , xm. The simplest basis of this ring consists of the complete symmetric polynomials Mβ(x1, . . . , xm) defined by Mβ(x1, . . . , xm) = where γ ranges over all permutations of β and tγ = i . Schur polyno- mials are related to Mβ by: Mβ , (5.2) where k is the Kostka number. This is the number of semistandard tableau of shape λ and weight β. If the representation W is reducible, its decomposition into irreducibles is given by: m(π)Vπ(GLn(C)), (5.3) where m(π)’s are the coefficients of the formal character χρ(x1, . . . , xm) in the Schur basis: m(π)Sπ. Proof of Theorem 5.1.3 Let λ, µ, π be as in Theorem 5.1.3. Let H = GLn(C), V = Vµ(H), G = GL(V ). Let sλ(x1, . . . , xm) be the formal character of the representation Vλ(G) of G. Here m = dim(Vµ) can be exponential in n and 〈µ〉. The basis of Vµ(H) is indexed by semistandard tableau of shape µ with entries in [1, n]. Let us order these tableau, say lexicographically, and let Ti, 1 ≤ i ≤ m, denote the i-th tableau in this order. With each tableau T , we associate a monomial t(T ) = wi(T ) where wi(T ) denotes the number of i’s in T . Given a polynomial f(x1, . . . , xm), let us define fµ = fµ(t1, . . . , tn) to be the polynomial obtained by substi- tuting xi = t(Ti) in f(x1, . . . , xm). Then the formal character of Vλ(G), considered as an H-representation of via the homomorphism H → G = GL(Vµ(H)), is the symmetric polynomial Sλ,µ(t1, . . . , tn) = (Sλ)µ. The plethysm constant aπλ,µ is defined by: Sλ,µ(t1, . . . , tn) = aπλ,µSπ(t1, . . . , tn). (5.4) An efficient parallel algorithm to compute aπλ,µ is as follows. Here by an efficient parallel algorithm, we mean an algorithm that works in poly(〈λ〉, 〈µ〉, 〈π〉) time using 2poly(〈λ〉,〈µ〉,〈π〉) processors. We will repeatedly use Proposi- tion 5.0.3. Algorithm (1) Compute Sλ,µ(t1, . . . , tn). By the Weyl character formula (5.1), Sλ,µ(t1, . . . , tn) = Aλ,µ(t1, . . . , tn) Bλ,µ(t1, . . . , tn) where Aλ(x1, . . . , xm) and Bλ(x1, . . . , xm) denote the numerator and denom- inator in (5.1), and Aλ,µ = (Aλ)µ, and Bλ,µ = (Bλ)µ. Let R = C[t1, . . . , tn]. Aλ,µ(t1, . . . , tn) = |t(Tj) λi+m−i|. This is the determinant of an m×m matrix with entries in R, where m = dim(V ) can be exponential in n and 〈µ〉. It can be evaluated in O(log2m) parallel ring operations using poly(m) processors. Each ring element that arises in the course of this algorithm is a polynomial in t1, . . . , tn of total degree O(|λ|m), where |λ| denotes the size of λ. The total number of its coefficients is r = O((|λ|m)n). Hence each ring operation can be carried out efficiently in O(log2(r)) parallel time using poly(r) processors. Since logm = poly(n, 〈µ〉) and log r = poly(n, 〈λ〉, 〈µ〉), it follows that Aλ,µ can be evaluated in poly(n, 〈µ〉, 〈λ〉) parallel time using 2poly(n,〈µ,λ〉) processors. The determinant Bλ,µ can also be computed efficiently in parallel in a similar fashion. To compute Sλ,µ, we have to divide Aλ,µ by Bλ,µ. This can be done by solving an r × r linear system, which, again, can be done efficiently in parallel. This computation yields representation of Sλ,µ in the monomial basis {Mβ} of the ring of symmetric polynomials in t1, . . . , tn. (2) To get the coefficients aπλ,µ, we have to get the representation of Sλ,µ(t) in the Schur basis. This change of basis requires inversion of the matrix in the linear system (5.2). The entries of the matrix K occuring in this linear system are Kostka numbers. Each Kostka number can be computed efficiently in parallel. Hence, all entries of this matrix can be computed efficiently in parallel. After this, the matrix can be inverted efficiently in parallel, and the coefficients aπλ,µ’s of Sλ,µ in the Schur basis can be computed efficiently in parallel. Finally, we use Proposition 5.0.4 to conclude that aπ can be computed in PSPACE. Q.E.D. 5.2 Symmetric group Next we prove Theorem 3.4.3 when H = Sm. Let X = Vµ(Sm) be an irreducible representation (the Specht module) of Sm corresponding to a partition µ of size m. Let ρ : H → G = GL(X) be the corresponding homomorphism. Theorem 5.2.1 Given partitions λ, µ, π, where µ and π have size m, the multiplicity mπ of the Specht module Vπ(Sm) in Vλ(G) can be computed in poly(m, 〈λ〉) space. Remark 5.2.2 The bitlengths 〈µ〉 and 〈π〉 are not mentioned in the com- plexity bound because they are bounded by m. For this, we need three lemmas. Lemma 5.2.3 The character of a symmetric group can be computed in PSPACE. Specifically, given a partition π of size m, and a sequence i = (i1, i2, . . .) of nonnegative integers such that jij = m, the value of the character χπ of Sm on the conjugacy class Ci of permutations indexed by i can be computed in poly(m) parallel time using 2poly(m) processors. Hence it can be computed in poly(m) space (cf. Proposition 5.0.4). Here the conjugacy class Ci consists of those permutations that have i1 1-cycles, i2 2-cycles, and so on. Proof: Let k be the height of the partition π. Let x = (x1, . . . , xk) be the tuple of variables xi’s. Given a formal series f(x) and a tuple (l1, . . . , lk) of nonnegative integers, let [f(x)](l1,...,lk) denote the coefficient of x 1 · · · x By the Frobenius character formula [FH], χλ(Ci) = [f(x)](l1,...,lk), (5.5) where l1 = π1 + k − 1, l2 = π2 + k − 2, . . . , lk = πk, f(x) = ∆(x) Pj(x) ∆(x) = i<j(xi − xj), Pj(x) = x 1 + · · ·+ x (5.6) Since deg(f) = poly(m) and k ≤ m, the total number of coefficients of f(x) is 2poly(m). Hence, we can evaluate f(x) in PSPACE by setting up appropriate recurrence relations. Alternatively, we can easily evaluate f(x) in poly(m) parallel time using 2poly(m) processors, and then extract its required coefficient. After this, the result follows from Proposition 5.0.4. Q.E.D. Lemma 5.2.4 Suppose φ is a character of Sm whose value on any conju- gacy class Ci can be computed in O(s) space, for some parameter s. Then, the multiplicity of the representation Vπ(Sm) in the representation Vφ(Sm) corresponding φ can be computed in O(poly(m) + s) space. Proof: The multiplicity is given by the inner product 〈φ, χπ〉 = ¯φ(σ)χπ(σ). (5.7) By assumption, φ(σ) can be computed in O(s) space, and by Lemma 5.2.3, χπ(σ) can be computed in poly(m) space. Hence, the result follows from the preceding formula. Q.E.D. Given an irreducible representation X = Vµ(Sm) and an irreducible rep- resentation W = Vλ(G) of G = GL(X)), let ρµ denote the representation map Sm → G, ρλ the representation map G→ GL(W ), and ρ : Sm → G→ GL(W ) their composition. This is a representation of Sm. Let χρ be the character of ρ. Lemma 5.2.5 For any σ ∈ Sm, χρ(σ) can be computed in poly(m, 〈λ〉) in poly(m, 〈λ〉) space. The bitlength 〈µ〉 is not mentioned in the complexity bound because it is bounded by m. Proof: Let r = dim(X). The formal character of the representation Vλ(G) of G = GL(X) is the Schur polynomial Sλ(x1, . . . , xr), r = dim(X). Hence, χρ(σ) = Sλ(α) where α = (α1, . . . , αr) is the tuple of eigenvalues of ρµ(σ). We shall compute the right hand side fast in parallel–i.e., in poly(m, 〈λ〉) parallel time using 2poly(m,〈λ〉) processors–and then use Proposition 5.0.4 to conclude the proof. This is done as follows. (1) Let χµ denote the character of the representation ρµ. Let pi(α) = α · · · + αir denote the i-th power sum of the eigenvalues. For any i, pi(α) = χµ(σ We can compute σi, for i ≤ |λ|, where |λ| denotes the size of λ, in poly(log i,m) = poly(m, 〈λ〉) time using repeated squaring. After this χµ(σ i) can be com- puted fast in parallel in poly(m) time using Lemma 5.2.3. Thus each pi(α) can be computed in poly(m, 〈λ〉) time in parallel using 2poly(m,〈λ〉) proces- sors. We calculate pi(α) in parallel for all i ≤ |λ|, and all pγ(α) = j pγj (α) in parallel for all partitions γ of size at most m. (2) After this, we calculate the complete symmetric function hi(α), for each i ≤ |λ|, fast in parallel, by using the relation [Mc]: |γ|=i z−1γ pγ , where zγ = i≥1 i mimi!, and mi = mi(γ) denotes the number of parts of γ equal to i. Thus we can calculate hγ(α) = j hγj (α), for all partitions γ of size m, fast in parallel. (3) To compute Sλ(α), we recall that the transition matrix between the Schur basis {Sλ} and the complete symmetric basis {hγ} of the ring of symmetric functions is K∗, the transpose inverse of the Kostka matrix K = [Kλ,γ ], where Kλ,γ denote the Kostka number; cf. [Mc]. As we noted in the proof of Theorem 5.1.3, each Kostka number can be computed in fast in parallel. Hence, K can be computed fast in parallel. After this, its inverse K−1 can be computed fast in parallel by Proposition 5.0.3–this is the crux of the proof–and finally K∗ as well. Thus Sλ(α) can be computed fast in parallel, since each hγ(α) can be computed fast in parallel. Q.E.D. Theorem 5.2.1 follows from Lemma 5.2.3,5.2.4 and 5.2.5. Q.E.D. 5.3 General linear group over a finite field In this section we prove Theorem 3.4.3, when H therein is the general lin- ear group GLn(Fpk) over a finite field Fpk . Irreducible representations of H = GLn(Fpk) have been classified by Green [Mc]. They are labelled by certain partition-valued functions. See [Mc] for a precise description of these labelling functions. It is clear from the description therein that each labelling function has a compact representation of bitlength O(n + k + 〈p〉), where 〈p〉 = log2 p; we specify a function by giving its partition values at the places where it is nonzero. Let µ denote any such label. Let X = Vµ(H) be the corresponding irreducible representation of H, and ρ : H → G = GL(X) the corresponding homomorphism. Theorem 5.3.1 Given a partition λ and labelling functions µ and π as above, the multiplicity mπλ,µ of the irreducible representation Vπ(H) in Vλ(G) can be computed in poly(n, k, 〈p〉, 〈λ〉) space. The proof is similar to that of Theorem 5.2.1 for the symmetric group with the following result playing the role of Lemma 5.2.3: Lemma 5.3.2 Given a label γ of an irreducible character χγ of H = GLn(Fpk) and a label δ of a conjugacy class in H, the value χγ(δ) can be computed in poly(n, k, 〈p〉) parallel time using 2poly(n,k,〈p〉) processors, and hence by Proposition 5.0.4, in poly(n, k, 〈p〉) space. The label δ of a conjugacy class in H is also a partition valued function [Mc], which admits a compact representation of bitlength poly(n, k, 〈p〉). Proof: We shall parallelize Green’s algorithm [Mc] for computing the charac- ter values, and then conclude by Proposition 5.0.3. Green shows that χγ(δ)’s are entries of a transition matrix between a two polynomial bases: the first constructed using Hall-Littlewood polynomials, and the second using Schur polynomials. We have construct this transition matrix fast in parallel. We shall only indicate here how the transition matrix between the basis of Hall- Littlewood polynomials and the Schur basis for the ring symmetric functions over Z[t] can be constructed fast in parallel. This idea can then be easily extended to complete the proof. First, we recall the definition of the Hall-Littlewood polynomial Pπ(x; t) = Pπ(x1, . . . , xk; t) [Mc]. This is a symmetric polynomial in xi’s with co- efficients in Z[t]. It interpolates between the Schur function sπ(x) and the monomial symmetric function mπ(x) because Pπ(x; 0) = sπ(x) and Pπ(x; 1) = mπ(x). The formal definition is as follows: For a given partition π, let vπ(t) = i≥0 vmi(t), where mi is the number of parts of π equal to i, and vm(t) = 1− ti Pπ(x; t) = Aπ(x, t) Bπ(x, t) , (5.8) where Aπ(x, t) = sgn(σ)σ(xπ11 · · · x i<j xi − txj , Bπ(x, t) = vπ(t) i<j(xi − xj). (5.9) Here sgn(σ) denotes the sign of σ. Let wπ,α(t)’s be the coeffcients of Pπ(x, t) in the Schur basis: Pπ(x; t) = wπ,α(t)sα(x). (5.10) We want to calculate the matrix W = [wπ,α] fast in parallel. Using formula (5.9), we calculate Aπ(x; t) fast in parallel; i.e., we calculate the coefficients of Aπ(x; t) in the basis of monomials in x and t. We calculate Bπ(x; t) similarly. After this the division in (5.8) can be carried out by solving a an appropriate linear system. This can be done fast in parallel by Proposition 5.0.3. Since, Pπ(x; t) is symmetric in xi’s, this yields its coefficients in the monomial symmetric basis {mα(x)} with the coefficients being in Z[t]. The transition matrix [Mc] from the monomial symmetric basis to the Schur basis is given by the inverse of the Kostka matrix. This inverse can be computed fast in parallel by Proposition 5.0.3. After this, the coefficients wπ,α’s of Pπ(x; t) in the Schur basis can be computed fast in parallel. Furthermore, the inverse of W = [Wπ,α] can also be computed fast in parallel by Proposition 5.0.3. Q.E.D. 5.3.1 Tensor product problem Analogue of the Kronecker problem (Problem 1.1.1) for H = GLn(Fpk) is: Problem 5.3.3 Given partition valued functions λ, µ, π, decide if the mul- tiplicity bπλ,µ of Vπ(H) in the tensor product Vλ(H)⊗ Vµ(H) is nonzero. In this context: Theorem 5.3.4 The multiplicity mπ can be computed in PSPACE; i.e., in poly(n, k, 〈p〉) space. Proof: This follows from Lemma 5.3.2 and analogues of Lemmas 5.2.4 and 5.2.5 in this setting. Q.E.D. A possible canditate for a stretching function assoociated with bπ b̃πλ,µ(n) = b nλ,nµ, where nλ denotes the stretched partition-valed function obtained by stretch- ing each partition value of λ by a factor of n. In other words b̃π (n) is the multiplicity of Vnπ(H(n)) in Vnλ(H(n)) ⊗ Vnµ(H(n)), where H(n) = GLnm(Fpk) is the stretched group. Is it a quasi-polynomial? If so, we can also ask for a good bound on its saturation and positivity indices. 5.4 Finite simple groups of Lie type The work of Deligne-Lusztig [DL] and Lusztig[Lu5] yield an algorithm for computing the character values for finite simple groups of Lie type. Question 5.4.1 Can this algorithm be parallelized? If so, Lemma 5.3.2, and hence Theorem 5.3.1, can be extended to finite simple groups of Lie type. Chapter 6 Experimental evidence for positivity In this chapter we give experimental evidence for positivity (PH2,3). 6.1 Littlewood-Richardson problem Experimental evidence for PH2 in the context of the Littlewood-Richardson problem (Problem 1.2.1) has been given in [DM2], and for PH3 in type A in [KTT]. We give experimental evidence for PH3 in types B,C,D here. Let Cλ (t) be as in eq.(1.2). Its reduced positive form for various values of α, β, λ is shown in Figure 6.1 for type B, in Figure 6.2 for type C, and Figure 6.3 for type D. The rank of the Lie algebra is three in all cases. In these types, the period of the stretching quasipolynomial c̃λ (n) is at most two. Accordingly, the period of every pole of Cλ (t) is at most two. The tables were computed from the tables in [DM2] for the stretching quasi- polynomial c̃λ (n) in these cases. 6.2 Kronecker problem, n = 2 Let kπλ,µ be the Kronecker coefficient in Problem 1.1.1. Let k̃ λ,µ(n) = k̃ nλ,nµ be the associated stretching quasi-polynomial, and Kπλ,µ(t) = k̃πλ,µ(n)t the associated rational function. An explicit formula (with alternating signs) for the Kronecker coefficient, when n = 2, has given by Remmel and White- head [RW] and Rosas [Ro], and a positive formula in [GCT9]. This case turns out to be nontrivial. For example, the number of chambers (domains) of quasi-polynomiality in this case turns out to be more than a million. Their explicit description can be found out using the formula for the Kronecker coefficient in [RW]. We implemented Rosas’ formula to check PH2 for the quasipolynomial (n) for a few thousand values of µ, ν and λ with the help of a computer. A large number of samples was chosen to ensure that a significant fraction of the chambers were sampled. The quasi-polynomial k̃πλ,µ(n) and a positive form of the rational function Cπλ,µ(t) are shown Figures 6.4 and 6.5 for few sample values of λ = (λ1, λ2), µ = (µ1, µ2), and π = (π1, π2, π3, π4). It may be noted that k̃π (n) need not be a polynomial; this answers Kirillov’s question [Ki] in the negative. But its period is at most two for n = 2. This follows from the formula in [RW]. For the λ, µ and π that we sampled, positivity index of k̃π (n) is always zero. But it turns out [BOR] that there are some λ, µ and π for which the saturation and positivity indices of (n) are nonzero (one), but very small and thus consistent with SH and PH2 (Hypothesis1.6.6) in this paper; in the earlier version of this paper, SH and PH2 stipulated that the saturation and positivity indices are always zero. These (λ, µ, π) escaped our random sampling, because their density is extremely small [BOR]. 6.3 G/P and Schubert varieties Let V = Vλ(G) be an irreducible representation of G = SLk(C) correspond- ing to a partition λ. Let vλ be the point in P (V ) corresponding to the highest weight vector, and X = Gvλ ∼= G/Pλ its closed orbit. Let hk,λ(n) be the Hilbert function of the homogeneous coordinate ring R of X. It is a quasipolynomial since spec(R) has rational singularities. In fact, it is a polynomial, since t = 1 is the only pole of the Hilbert series Hk,λ(t) = hk,λ(n)t Figure 6.6 gives experimental evidence for strict positivity (PH2) of hk,λ(n) (as discussed in Section 3.5.5) for a few sample values of k and λ. Figure 6.7 gives experimental evidence for strict positivity of the Hilbert polynomial of the Schubert subvarieties of the Grassmanian; there Gn,k denotes the Grassmannian of k-planes in V = Cn, and Ωa, a = (a(1), . . . , a(d)) its Schubert subvariety consisting of the k-subspaces W such that dim(W ∩ Vn−k+i−a(i)) ≥ i for all i, where V = Vn ⊃ · · ·V1 ⊃ 0 is a complete flag of subspaces in V . The Hilbert polynomials were computed using the explicit polyhedral interpretation for them deduced from the theory of algebras with straightening laws (Hodge algebras) [DEP2]. 6.4 The ring of symmetric functions Let V = Ck, G = GL(V ), H = Sk, with the natural embedding H → G. Let us consider the spacial case of the subgroup restriction problem (Problem 1.1.3), with Vλ(G) = V , and Vπ(H) the trivial representation of H. Then s = mπ , the multiplicity of the trivial representation in V , is one. Though the decision problem (Problem 1.1.3) is trivial in this case, the canonical model associated with s is nontrivial. The canonical rings R = R(s) and S = S(s) associated with s in this case coincide with C[V ] = C[x1, . . . , xk]. The ring T = T (s) = S C[x1, . . . , xk] Sk is the subring of symmetric functions. Its Hilbert function h(n) is a quasipolynomial. PH1 and PH3 for Z = Proj(T ), as per Defini- tion 3.5.2, follow easily, the latter from the well known rational generating function for the partition function [St1]. But PH2 turns out to be nontriv- ial. Figures 6.8-6.13 give experimental evidence for strict positivity of h(n) (PH2). In these figures, the i-th row of the table for a given k shows hi(n), where hi(n), 1 ≤ i ≤ l, are such that h(n) = hi(n), when n = i modulo the period l of h(n). α β λ Cλ (0, 15, 5) (12, 15, 3) (6, 15, 6) 350 t 8+19121 t7+123576 t6+297561 t5+342064 t4+192779 t3+46992 t2+2641 t+1 (1−t) (1−t2) (4, 8, 11) (3, 15, 10) (10, 1, 3) 1+5 t+6 t (1−t2) (8, 1, 3) (11, 13, 3) (8, 6, 14) 2 t 8+45 t7+259 t6+591 t5+773 t4+522 t3+198 t2+29 t+1 (1−t)3(1−t2) (8, 9, 14) (8, 4, 5) (1, 5, 15) 136 t 9+3422 t8+20204 t7+53608 t6+76076 t5+60986 t4+26674 t3+5568 t2+345 t+1 (1−t)3(1−t2) (10, 5, 6) (5, 4, 10) (0, 7, 12) 219 t 8+12135 t7+79231 t6+193003 t5+223919 t4+127907 t3+31704 t2+1870 t+1 (1−t)6(1+t)3 Figure 6.1: Cλ (t) for B3 α β λ Cλα,β(t) (1, 13, 6) (14, 15, 5) (5, 11, 7) 18145 t 8+267151 t7+1070716 t6+1917716 t5+1735692 t4+778184 t3+144596 t2+5538 t+1 (1−t)4(1−t2) (4, 15, 14) (12, 12, 10) (4, 9, 8) 2280 t 9+267658 t8+2746131 t7+9276935 t6+14682332 t5+11903923 t4+4746803 t3+751126 t2+21249 t+1 (1−t)4(1−t2) (9, 0, 8) (8, 12, 9) (7, 7, 3) 3 t 2+4 t+1 (1−t)6 (10, 2, 7) (8, 10, 1) (7, 5, 5) 8984 t 8+132826 t7+534183 t6+960491 t5+873227 t4+394045 t3+74067 t2+2941 t+1 (1−t)4(1−t2) (10, 10, 15) (11, 3, 15) (10, 7, 15) 7162 t 9+736327 t8+7178960 t7+23540366 t6+36359642 t5+28788904 t4+11166361 t3+1693696 t2+43515 t+1 (1−t)7(1+t)3 Figure 6.2: Cλα,β(t) for C3 α β λ Cλ (0, 15, 5) (12, 15, 3) (6, 15, 6) 633 t 7+24259 t6+142236 t5+252113 t4+168220 t3+36131 t2+1414 t+1 (1−t) (1−t2) (4, 8, 11) (3, 15, 10) (10, 1, 3) 7962 t 8+503679 t7+4525372 t6+11944350 t5+12218255 t4+4879052 t3+586370 t2+10862 t+1 (1−t) (1−t2) (8, 1, 3) (11, 13, 3) (8, 6, 14) 81 t 7+19407 t6+211964 t5+513585 t4+426652 t3+110317 t2+4609 t+1 (1−t) (1−t2) (8, 9, 14) (8, 4, 5) (1, 5, 15) 9 t 2+8 t+1+2 t3 (1−t) (10, 5, 6) (5, 4, 10) (0, 7, 12) 3647 t 7+111208 t6+570739 t5+920201 t4+560336 t3+106748 t2+3435 t+1 (1−t) (1+t) Figure 6.3: Cλ (t) for D3 λ1 λ2 µ1 µ2 π1 π2 π3 π4 k̃ λ,µ(n); n odd k̃ λ,µ(n); n even K λ,µ(t) 87 62 97 52 64 39 24 22 1/2 + 4n+ 11/2n2 1 + 4n+ 11/2n2 1+8 t+11 t 2+2 t3 (1−t)2(1−t2) 104 95 149 50 95 78 15 11 1/2 + 13/2n + 18n2 1 + 13/2n + 18n2 1+23 t+36 t 2+12 t3 (1−t)2(1−t2) 101 85 102 84 78 72 24 12 17/2n + 71 n2 1 + 17/2n + 71 n2 1+42 t+72 t 2+27 t3 (1−t) (1−t2) 79 63 93 49 88 37 14 3 3/4 + 27 n+ 303 n2 1 + 27 n+ 303 n2 1+88 t+151 t 2+63 t3 (1−t) (1−t2) 97 93 114 76 77 66 47 0 1/2 + 15/2n + 21n2 1 + 15/2n + 21n2 1+27 t+42 t 2+14 t3 (1−t)2(1−t2) 88 56 113 31 99 35 7 3 1/2 + 11/2n + 10n2 1 + 11/2n + 10n2 1+14 t+20 t 2+5 t3 (1−t)2(1−t2) 134 82 140 76 91 72 49 4 3/4 + 21n + 669 n2 1 + 21n + 669 n2 1+187 t+334 t 2+147 t3 (1−t)2(1−t2) 133 69 149 53 98 55 43 6 1 + 6n + 8n2 1 + 6n + 8n2 15 t 2+13 t+1+3 t3 (1−t)3 80 63 111 32 88 38 10 7 1 1 1+t 118 69 151 36 95 63 20 9 1 + 4n + 4n2 1 + 4n + 4n2 7 t 2+7 t+1+t3 (1−t)3 96 51 103 44 90 53 3 1 1/2 + 39 n+ 36n2 1 + 39 n+ 36n2 1+54 t+72 t 2+17 t3 (1−t)2(1−t2) 117 72 133 56 82 57 41 9 1 + 9n+ 18n2 1 + 9n + 18n2 35 t 2+26 t+1+10 t3 (1−t) 72 63 77 58 49 38 28 20 1/2 + 7n + 55 n2 1 + 7n+ 55 n2 1+33 t+55 t 2+21 t3 (1−t) (1−t2) 48 37 49 36 34 24 16 11 1/2 + 6n + 37 n2 1 + 6n+ 37 n2 1+23 t+37 t 2+13 t3 (1−t) (1−t2) 108 56 113 51 73 50 29 12 1 + 4n + 4n2 1 + 4n + 4n2 7 t 2+7 t+1+t3 (1−t)3 Figure 6.4: The quasipolynomial k̃π and the rational function Kπ (t) for the Kronecker problem, n = 2. λ1 λ2 µ1 µ2 π1 π2 π3 π4 k̃ (n); n odd k̃π (n); n even Kπ 77 40 78 39 58 29 24 6 1 + 19/2n + 57 n2 1 + 19/2n + 57 n2 56 t 2+37 t+1+20 t3 (1−t)3 153 81 157 77 96 63 61 14 1 + 3n+ 2n2 1 + 3n + 2n2 3 t 2+4 t+1 (1−t)3 90 89 102 77 90 42 30 17 1/2 + 13/2n + 6n2 1 + 13/2n + 6n2 1+11 t+12 t (1−t)2(1−t2) 145 102 160 87 96 84 39 28 1 + 10n + 25n2 1 + 10n + 25n2 49 t 2+34 t+1+16 t3 (1−t)3 109 95 136 68 78 60 46 20 1 + 3n+ 2n2 1 + 3n + 2n2 3 t 2+4 t+1 (1−t)3 100 42 104 38 85 27 27 3 1 + 8n 1 + 8n 8 t+1+7 t (1−t)2 74 51 86 39 52 34 26 13 1 1 1+t 98 90 124 64 92 67 22 7 1/2 + 23/2n + 60n2 1 + 23/2n + 60n2 1+70 t+120 t 2+49 t3 (1−t)2(1−t2) 57 38 75 20 52 25 17 1 1 + 3n+ 2n2 1 + 3n + 2n2 3 t 2+4 t+1 (1−t) 159 140 170 129 89 82 73 55 1 + 3/2n + 1/2n2 1 + 3/2n + 1/2n2 1+t (1−t)3 144 122 157 109 88 86 74 18 3/4 + n+ 1/4n2 1 + n+ 1/4n2 1 (1−t)2(1−t2) 90 68 92 66 88 37 23 10 1/4 + 12n + 351 n2 1 + 12n+ 351 n2 1+98 t+176 t 2+76 t3 (1−t) (1−t2) 89 42 100 31 76 28 19 8 1 + 6n+ 8n2 1 + 6n + 8n2 15 t 2+13 t+1+3 t3 (1−t) 88 56 107 37 71 39 20 14 1 + 9/2n + 9/2n2 1 + 9/2n + 9/2n2 8 t 2+8 t+1+t3 (1−t)3 124 111 133 102 98 89 27 21 1/2 + 7n+ 53 n2 1 + 7n+ 53 n2 1+32 t+53 t 2+20 t3 (1−t)2(1−t2) Figure 6.5: Continuation of Figure 6.4 k λ hk,λ(n) 3 (21, 19) 399n3 + 35527969472513 137438953472 n2 + 4329327034365 137438953472 5 (21, 19) 3700378042361 4194304 n7 + 575575719967 524288 n6 + 2157156441 n5 + 266554253 n4 + 4643843 n3 + 1468423 n2 + 7619 3 (21, 9, 6) 270n3 + 40819369181185 274877906944 n2 + 3092376453119 137438953472 3 (12, 9, 5) 42n3 + 40132174413825 1099511627776 n2 + 11544872091645 1099511627776 3 (21, 9, 6) 27396522639355 536870912 n6 + 463063744509 8388608 n5 + 6265700353 262144 n4 + 5577375771 1048576 n3 + 84246529 131072 n2 + 20971505 524288 n+ 1048573 1048576 3 (21, 19, 16) 15n3 + 81363860455425 4398046511104 n2 + 8246337208319 1099511627776 4 (9, 7, 5) 7215545057279 17179869184 n6 + 4183298146289 4294967296 n5 + 247765925897 268435456 n4 + 1914699777 4194304 n3 + 4160749567 33554432 n2 + 587202553 33554432 n+ 67108863 67108864 4 (21, 12, 9) 16437913583613 268435456 n6 + 132498063359 2097152 n5 + 109509083155 4194304 n4 + 1462763527 262144 n3 + 171442179 262144 n2 + 10485755 262144 n+ 524287 524288 4 (21, 9, 5) 32469952757755 536870912 n6 + 129805320191 2097152 n5 + 108129157137 4194304 n4 + 2926313487 524288 n3 + 86638593 131072 n2 + 10616825 262144 n+ 262143 262144 4 (21, 9, 6) 27396522639355 536870912 n6 + 463063744509 8388608 n5 + 6265700353 262144 n4 + 5577375771 1048576 n3 + 84246529 131072 n2 + 20971505 524288 n+ 1048573 1048576 4 (31, 19, 5) 35969680015355 33554432 n6 + 1424674346311 2097152 n5 + 22705493343 131072 n4 + 46973953 n3 + 3423915 n2 + 65365 n+ 16383 16384 Figure 6.6: Hilbert polynomial for G/Pλ, G = SLk(C). There is a slight rounding error caused by interpolation– e.g., the constant term of each polynomial should be one. n k a 7 3 (1, 3, 5) 1/3n3 + 59373627899905 39582418599936 n2 + 28587302322173 13194139533312 7 3 (1, 2, 4) n+ 1 7 3 (1, 4, 6) 22265110462465 534362651099136 n5 + 4638564679679 11132555231232 n4 + 13 n3 + 105942526633 34359738368 n2 + 146028888073 51539607552 n+ 34359738361 34359738368 6 2 (1, 4, 5) 15637498706143 2251799813685248 n6 + 3665038759245 35184372088832 n5 + 1389660529559 2199023255552 n4 + 272014595421 137438953472 +230973796809 68719476736 n2 + 100215903571 34359738368 6 2 (1, 4, 6) 69578470195 25048249270272 n7 + 1217623228439 25048249270272 n6 + 372534725887 1043677052928 n5 + 30953963537 21743271936 n4 + 12044363351 3623878656 +683671553 150994944 n2 + 1335466297 402653184 n+ 268435457 268435456 7 3 (1, 4, 6) 23456248059223 562949953421312 n5 + 7330077518505 17592186044416 n4 + 1786706395137 1099511627776 n3 + 423770106525 137438953472 n2 + 24338148015 8589934592 n+ 17179869169 17179869184 6 3 (1, 3, 6) 1/8n4 + 16126170540715 17592186044416 n3 + 19 n2 + 710101259605 274877906944 8 3 (1, 3, 6) 171798691840001 1374389534720000 n4 + 31496426837333 34359738368000 n3 + 4080218931199 1717986918400 n2 + 443813287253 171798691840 Figure 6.7: Hilbert polynomial of the Schubert subvariety Ωa, a = (a(1), . . . , a(k)), of the Grassmannian Gn,k. k = 2 1/2n + 1/2 1/2n + 1 k = 3 1/12n2 + 1/2n + 5 1/12n2 + 1/2n + 2/3 1/12n2 + 1/2n + 3/4 1/12n2 + 1/2n + 46912496118443 70368744177664 1/12n2 + 1/2n + 58640620148053 140737488355328 1/12n2 + 1/2n + 1 k = 4 n3 + 5 n2 + 61572651155457 140737488355328 n+ 15881834623431 35184372088832 n3 + 117281240296107 1125899906842624 n2 + 140737488355325 281474976710656 n+ 19 n3 + 234562480592215 2251799813685248 n2 + 123145302310909 281474976710656 n+ 19791209299969 35184372088832 n3 + 234562480592215 2251799813685248 n2 + 70368744177667 140737488355328 n+ 62549994824587 70368744177664 n3 + 5 n2 + 61572651155453 140737488355328 n+ 748278746681 2199023255552 n3 + 117281240296107 1125899906842624 n2 + 70368744177665 140737488355328 n+ 26388279066621 35184372088832 n3 + 117281240296107 1125899906842624 n2 + 7 n+ 7940917311717 17592186044416 n3 + 117281240296107 1125899906842624 n2 + 35184372088831 70368744177664 n+ 6841405683939 8796093022208 n3 + 29320310074027 281474976710656 n2 + 30786325577729 70368744177664 n3 + 58640620148053 562949953421312 n2 + 35184372088831 70368744177664 n+ 5619726097523 8796093022208 n3 + 117281240296105 1125899906842624 n2 + 7 n+ 2993114986727 8796093022208 n3 + 58640620148055 562949953421312 n2 + 1/2n + 1 Figure 6.8: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk] Sk ; k = 2, 3, 4. n4 + 46912496118441 4503599627370496 n3 + 3787206717893 35184372088832 n2 + 469583091025 1099511627776 n+ 499743305817 1099511627776 n4 + 46912496118445 4503599627370496 n3 + 3787206717891 35184372088832 n2 + 503942829399 1099511627776 n+ 310001195055 549755813888 n4 + 46912496118441 4503599627370496 n3 + 3787206717897 35184372088832 n2 + 469583091031 1099511627776 n+ 30279519437 68719476736 400319966877379 1152921504606846976 n4 + 23456248059221 2251799813685248 n3 + 3787206717893 35184372088832 n2 + 503942829403 1099511627776 n+ 47340083975 68719476736 400319966877379 1152921504606846976 n4 + 5864062014805 562949953421312 n3 + 3787206717895 35184372088832 n2 + 117395772755 274877906944 n+ 89955703917 137438953472 400319966877379 1152921504606846976 n4 + 5864062014805 562949953421312 n3 + 3787206717893 35184372088832 n2 + 31496426837 68719476736 n+ 92771293595 137438953472 400319966877379 1152921504606846976 n4 + 46912496118447 4503599627370496 n3 + 1893603358947 17592186044416 n2 + 234791545515 549755813888 n+ 5661005505 17179869184 n4 + 23456248059223 2251799813685248 n3 + 1893603358949 17592186044416 n2 + 62992853675 137438953472 n+ 94680167945 137438953472 400319966877379 1152921504606846976 n4 + 46912496118441 4503599627370496 n3 + 3787206717891 35184372088832 n2 + 117395772757 274877906944 n+ 38869454029 68719476736 n4 + 46912496118441 4503599627370496 n3 + 3787206717897 35184372088832 n2 + 62992853675 137438953472 n+ 52494044729 68719476736 n4 + 5864062014805 562949953421312 n3 + 946801679473 8796093022208 n2 + 234791545515 549755813888 n+ 2830502753 8589934592 400319966877379 1152921504606846976 n4 + 23456248059219 2251799813685248 n3 + 1893603358949 17592186044416 n2 + 251971414695 549755813888 n+ 27487790695 34359738368 400319966877379 1152921504606846976 n4 + 23456248059223 2251799813685248 n3 + 1893603358945 17592186044416 n2 + 234791545509 549755813888 n+ 31233956605 68719476736 n4 + 11728124029611 1125899906842624 n3 + 946801679473 8796093022208 n2 + 251971414699 549755813888 n+ 19375074691 34359738368 n4 + 11728124029611 1125899906842624 n3 + 473400839737 4398046511104 n2 + 29348943189 68719476736 n+ 11005853695 17179869184 400319966877379 1152921504606846976 n4 + 23456248059219 2251799813685248 n3 + 473400839737 4398046511104 n2 + 251971414699 549755813888 n+ 5917510497 8589934592 n4 + 11728124029611 1125899906842624 n3 + 473400839737 4398046511104 n2 + 234791545511 549755813888 n+ 15616978311 34359738368 n4 + 23456248059223 2251799813685248 n3 + 1893603358947 17592186044416 n2 + 62992853673 137438953472 n+ 23192823403 34359738368 n4 + 11728124029611 1125899906842624 n3 + 946801679475 8796093022208 n2 + 117395772757 274877906944 n+ 2830502755 8589934592 400319966877379 1152921504606846976 n4 + 11728124029609 1125899906842624 n3 + 1893603358949 17592186044416 n2 + 31496426837 68719476736 n+ 30541989663 34359738368 400319966877379 1152921504606846976 n4 + 11728124029611 1125899906842624 n3 + 946801679473 8796093022208 n2 + 117395772759 274877906944 n+ 9717363505 17179869184 n4 + 2932031007403 281474976710656 n3 + 1893603358945 17592186044416 n2 + 125985707353 274877906944 n+ 4843768673 8589934592 400319966877379 1152921504606846976 n4 + 23456248059223 2251799813685248 n3 + 59175104967 549755813888 n2 + 117395772755 274877906944 n+ 2830502755 8589934592 400319966877379 1152921504606846976 n4 + 23456248059221 2251799813685248 n3 + 946801679475 8796093022208 n2 + 62992853675 137438953472 n+ 3435973837 4294967296 400319966877379 1152921504606846976 n4 + 23456248059223 2251799813685248 n3 + 1893603358947 17592186044416 n2 + 58697886379 137438953472 n+ 11244462985 17179869184 400319966877379 1152921504606846976 n4 + 11728124029611 1125899906842624 n3 + 946801679475 8796093022208 n2 + 15748213419 34359738368 n+ 9687537337 17179869184 n4 + 11728124029609 1125899906842624 n3 + 473400839737 4398046511104 n2 + 117395772753 274877906944 n+ 1892469965 4294967296 n4 + 23456248059221 2251799813685248 n3 + 946801679473 8796093022208 n2 + 7874106709 17179869184 n+ 11835020991 17179869184 400319966877379 1152921504606846976 n4 + 11728124029611 1125899906842624 n3 + 473400839737 4398046511104 n2 + 117395772753 274877906944 n+ 3904244579 8589934592 400319966877379 1152921504606846976 n4 + 11728124029611 1125899906842624 n3 + 946801679473 8796093022208 n2 + 125985707349 274877906944 n+ 1879048193 2147483648 Figure 6.9: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk] Sk , k = 5; the first 30 rows. n4 + 11728124029609 1125899906842624 n3 + 946801679473 8796093022208 n2 + 58697886375 137438953472 n+ 88453211 268435456 n4 + 11728124029611 1125899906842624 n3 + 946801679473 8796093022208 n2 + 15748213419 34359738368 n+ 2958755247 4294967296 400319966877379 1152921504606846976 n4 + 23456248059223 2251799813685248 n3 + 946801679475 8796093022208 n2 + 58697886375 137438953472 n+ 4858681755 8589934592 400319966877379 1152921504606846976 n4 + 11728124029609 1125899906842624 n3 + 473400839737 4398046511104 n2 + 62992853673 137438953472 n+ 151367771 268435456 n4 + 23456248059221 2251799813685248 n3 + 946801679473 8796093022208 n2 + 58697886375 137438953472 n+ 2274244835 4294967296 n4 + 11728124029611 1125899906842624 n3 + 236700419869 2199023255552 n2 + 62992853671 137438953472 n+ 858993459 1073741824 400319966877379 1152921504606846976 n4 + 11728124029611 1125899906842624 n3 + 946801679473 8796093022208 n2 + 14674471595 34359738368 n+ 3904244571 8589934592 400319966877379 1152921504606846976 n4 + 23456248059219 2251799813685248 n3 + 59175104967 549755813888 n2 + 7874106709 17179869184 n+ 2421884337 4294967296 400319966877379 1152921504606846976 n4 + 11728124029611 1125899906842624 n3 + 59175104967 549755813888 n2 + 29348943189 68719476736 n+ 3784939927 8589934592 400319966877379 1152921504606846976 n4 + 23456248059223 2251799813685248 n3 + 473400839737 4398046511104 n2 + 62992853673 137438953472 n+ 1908874355 2147483648 400319966877379 1152921504606846976 n4 + 5864062014805 562949953421312 n3 + 946801679473 8796093022208 n2 + 29348943189 68719476736 n+ 1952122291 4294967296 400319966877379 1152921504606846976 n4 + 11728124029609 1125899906842624 n3 + 946801679471 8796093022208 n2 + 62992853675 137438953472 n+ 2899102929 4294967296 400319966877379 1152921504606846976 n4 + 11728124029611 1125899906842624 n3 + 473400839735 4398046511104 n2 + 58697886381 137438953472 n+ 707625689 2147483648 400319966877379 1152921504606846976 n4 + 11728124029613 1125899906842624 n3 + 59175104967 549755813888 n2 + 15748213419 34359738368 n+ 2958755253 4294967296 n4 + 11728124029609 1125899906842624 n3 + 473400839737 4398046511104 n2 + 58697886377 137438953472 n+ 3288334339 4294967296 100079991719345 288230376151711744 n4 + 5864062014805 562949953421312 n3 + 59175104967 549755813888 n2 + 62992853675 137438953472 n+ 1210942169 2147483648 n4 + 11728124029611 1125899906842624 n3 + 946801679477 8796093022208 n2 + 29348943193 68719476736 n+ 1415251375 4294967296 n4 + 5864062014805 562949953421312 n3 + 59175104967 549755813888 n2 + 31496426839 68719476736 n+ 3435973835 4294967296 100079991719345 288230376151711744 n4 + 1466015503701 140737488355328 n3 + 473400839737 4398046511104 n2 + 29348943195 68719476736 n+ 976061147 2147483648 100079991719345 288230376151711744 n4 + 11728124029611 1125899906842624 n3 + 473400839737 4398046511104 n2 + 31496426839 68719476736 n+ 3280877793 4294967296 n4 + 5864062014805 562949953421312 n3 + 946801679473 8796093022208 n2 + 29348943191 68719476736 n+ 946234983 2147483648 100079991719345 288230376151711744 n4 + 2932031007403 281474976710656 n3 + 946801679475 8796093022208 n2 + 15748213419 34359738368 n+ 1479377623 2147483648 n4 + 5864062014805 562949953421312 n3 + 59175104967 549755813888 n2 + 29348943195 68719476736 n+ 122007643 268435456 100079991719345 288230376151711744 n4 + 2932031007403 281474976710656 n3 + 473400839737 4398046511104 n2 + 3937053355 8589934592 n+ 1449551461 2147483648 100079991719345 288230376151711744 n4 + 2932031007403 281474976710656 n3 + 236700419869 2199023255552 n2 + 29348943193 68719476736 n+ 71070151 134217728 n4 + 1466015503701 140737488355328 n3 + 59175104967 549755813888 n2 + 15748213419 34359738368 n+ 739688813 1073741824 100079991719345 288230376151711744 n4 + 11728124029611 1125899906842624 n3 + 473400839739 4398046511104 n2 + 14674471597 34359738368 n+ 607335219 1073741824 n4 + 2932031007403 281474976710656 n3 + 236700419869 2199023255552 n2 + 3937053355 8589934592 n+ 605471085 1073741824 100079991719345 288230376151711744 n4 + 11728124029611 1125899906842624 n3 + 473400839737 4398046511104 n2 + 29348943193 68719476736 n+ 88453211 268435456 100079991719345 288230376151711744 n4 + 1466015503701 140737488355328 n3 + 473400839737 4398046511104 n2 + 3937053355 8589934592 n+ 2147483647 2147483648 Figure 6.10: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk] Sk , k = 5; the last 30 rows. 53375995583651 4611686018427387904 n5 + 21892498188609 36028797018963968 n4 + 418085902907 35184372088832 n3 + 115486898397 1099511627776 n2 + 26847522421 68719476736 n+ 8448724291 17179869184 53375995583651 4611686018427387904 n5 + 10946249094305 18014398509481984 n4 + 836171805815 70368744177664 n3 + 7396888121 68719476736 n2 + 15302809397 34359738368 n+ 9853503363 17179869184 106751991167299 9223372036854775808 n5 + 21892498188599 36028797018963968 n4 + 1672343611625 140737488355328 n3 + 57743449207 549755813888 n2 + 14060052663 34359738368 n+ 975427339 2147483648 13343998895913 1152921504606846976 n5 + 10946249094301 18014398509481984 n4 + 1672343611641 140737488355328 n3 + 59175104961 549755813888 n2 + 15302809423 34359738368 n+ 610309549 1073741824 106751991167305 9223372036854775808 n5 + 21892498188611 36028797018963968 n4 + 1672343611623 140737488355328 n3 + 115486898411 1099511627776 n2 + 13423761203 34359738368 n+ 4461910959 8589934592 13343998895913 1152921504606846976 n5 + 21892498188605 36028797018963968 n4 + 1672343611631 140737488355328 n3 + 29587552483 274877906944 n2 + 15939100865 34359738368 n+ 1927366573 2147483648 213503982334599 18446744073709551616 n5 + 10946249094305 18014398509481984 n4 + 418085902909 35184372088832 n3 + 57743449199 549755813888 n2 + 13423761217 34359738368 n+ 835973461 2147483648 106751991167299 9223372036854775808 n5 + 10946249094305 18014398509481984 n4 + 836171805821 70368744177664 n3 + 59175104963 549755813888 n2 + 7651404713 17179869184 n+ 320001575 536870912 106751991167299 9223372036854775808 n5 + 10946249094299 18014398509481984 n4 + 1672343611637 140737488355328 n3 + 115486898395 1099511627776 n2 + 14060052667 34359738368 n+ 255936429 536870912 213503982334597 18446744073709551616 n5 + 2736562273577 4503599627370496 n4 + 1672343611629 140737488355328 n3 + 7396888121 68719476736 n2 + 7651404703 17179869184 n+ 2859997491 4294967296 106751991167299 9223372036854775808 n5 + 342070284197 562949953421312 n4 + 104521475727 8796093022208 n3 + 57743449199 549755813888 n2 + 1677970153 4294967296 n+ 1790721319 4294967296 86400 n5 + 10946249094299 18014398509481984 n4 + 836171805815 70368744177664 n3 + 59175104959 549755813888 n2 + 3984775219 8589934592 n+ 987842475 1073741824 86400 n5 + 5473124547153 9007199254740992 n4 + 209042951453 17592186044416 n3 + 57743449193 549755813888 n2 + 3355940307 8589934592 n+ 884291849 2147483648 106751991167303 9223372036854775808 n5 + 10946249094301 18014398509481984 n4 + 836171805817 70368744177664 n3 + 118350209919 1099511627776 n2 + 1912851173 4294967296 n+ 264972307 536870912 213503982334605 18446744073709551616 n5 + 2736562273577 4503599627370496 n4 + 836171805827 70368744177664 n3 + 57743449201 549755813888 n2 + 7030026327 17179869184 n+ 1233125369 2147483648 53375995583651 4611686018427387904 n5 + 10946249094299 18014398509481984 n4 + 836171805819 70368744177664 n3 + 118350209933 1099511627776 n2 + 1912851177 4294967296 n+ 1478317113 2147483648 106751991167297 9223372036854775808 n5 + 1368281136787 2251799813685248 n4 + 836171805817 70368744177664 n3 + 57743449195 549755813888 n2 + 1677970151 4294967296 n+ 117959881 268435456 1667999861989 144115188075855872 n5 + 10946249094303 18014398509481984 n4 + 104521475727 8796093022208 n3 + 59175104961 549755813888 n2 + 3984775215 8589934592 n+ 109722991 134217728 106751991167297 9223372036854775808 n5 + 342070284197 562949953421312 n4 + 836171805815 70368744177664 n3 + 28871724597 274877906944 n2 + 1677970153 4294967296 n+ 332087389 1073741824 213503982334607 18446744073709551616 n5 + 10946249094307 18014398509481984 n4 + 836171805821 70368744177664 n3 + 59175104963 549755813888 n2 + 7651404709 17179869184 n+ 384426083 536870912 Figure 6.11: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk] Sk , k = 6; the first 20 rows. 213503982334615 18446744073709551616 n5 + 10946249094307 18014398509481984 n4 + 836171805813 70368744177664 n3 + 28871724597 274877906944 n2 + 7030026337 17179869184 n+ 640721865 1073741824 106751991167309 9223372036854775808 n5 + 10946249094297 18014398509481984 n4 + 209042951455 17592186044416 n3 + 29587552479 274877906944 n2 + 1912851179 4294967296 n+ 157275009 268435456 26687997791827 2305843009213693952 n5 + 342070284197 562949953421312 n4 + 836171805825 70368744177664 n3 + 57743449199 549755813888 n2 + 3355940301 8589934592 n+ 90445245 268435456 13343998895913 1152921504606846976 n5 + 5473124547153 9007199254740992 n4 + 104521475729 8796093022208 n3 + 29587552481 274877906944 n2 + 1992387605 4294967296 n+ 450971557 536870912 106751991167303 9223372036854775808 n5 + 10946249094307 18014398509481984 n4 + 836171805827 70368744177664 n3 + 28871724597 274877906944 n2 + 209746269 536870912 n+ 17843591 33554432 106751991167303 9223372036854775808 n5 + 10946249094297 18014398509481984 n4 + 836171805819 70368744177664 n3 + 924611015 8589934592 n2 + 239106397 536870912 n+ 5146825 8388608 1667999861989 144115188075855872 n5 + 5473124547153 9007199254740992 n4 + 418085902911 35184372088832 n3 + 7217931151 68719476736 n2 + 54922081 134217728 n+ 265331665 536870912 1667999861989 144115188075855872 n5 + 10946249094291 18014398509481984 n4 + 836171805825 70368744177664 n3 + 59175104963 549755813888 n2 + 478212795 1073741824 n+ 326629601 536870912 1667999861989 144115188075855872 n5 + 10946249094297 18014398509481984 n4 + 418085902909 35184372088832 n3 + 14435862303 137438953472 n2 + 3355940305 8589934592 n+ 24121261 67108864 53375995583655 4611686018427387904 n5 + 10946249094291 18014398509481984 n4 + 209042951453 17592186044416 n3 + 59175104973 549755813888 n2 + 3984775217 8589934592 n+ 503316475 536870912 26687997791827 2305843009213693952 n5 + 10946249094285 18014398509481984 n4 + 836171805825 70368744177664 n3 + 57743449207 549755813888 n2 + 3355940305 8589934592 n+ 115234099 268435456 106751991167303 9223372036854775808 n5 + 2736562273573 4503599627370496 n4 + 209042951459 17592186044416 n3 + 7396888121 68719476736 n2 + 3825702363 8589934592 n+ 42684549 67108864 106751991167301 9223372036854775808 n5 + 10946249094305 18014398509481984 n4 + 209042951453 17592186044416 n3 + 28871724605 274877906944 n2 + 1757506587 4294967296 n+ 277411267 536870912 106751991167303 9223372036854775808 n5 + 10946249094299 18014398509481984 n4 + 418085902905 35184372088832 n3 + 924611015 8589934592 n2 + 3825702353 8589934592 n+ 33950041 67108864 106751991167307 9223372036854775808 n5 + 1368281136787 2251799813685248 n4 + 52260737865 4398046511104 n3 + 28871724601 274877906944 n2 + 209746269 536870912 n+ 61328751 134217728 53375995583651 4611686018427387904 n5 + 2736562273575 4503599627370496 n4 + 104521475727 8796093022208 n3 + 29587552487 274877906944 n2 + 249048451 536870912 n+ 128849017 134217728 106751991167301 9223372036854775808 n5 + 5473124547145 9007199254740992 n4 + 209042951459 17592186044416 n3 + 7217931151 68719476736 n2 + 1677970157 4294967296 n+ 30318473 67108864 106751991167303 9223372036854775808 n5 + 342070284197 562949953421312 n4 + 418085902919 35184372088832 n3 + 14793776243 137438953472 n2 + 1912851181 4294967296 n+ 143223581 268435456 53375995583651 4611686018427387904 n5 + 5473124547153 9007199254740992 n4 + 209042951459 17592186044416 n3 + 1804482787 17179869184 n2 + 878753295 2147483648 n+ 13898875 33554432 106751991167303 9223372036854775808 n5 + 5473124547147 9007199254740992 n4 + 418085902907 35184372088832 n3 + 29587552479 274877906944 n2 + 956425585 2147483648 n+ 195527051 268435456 Figure 6.12: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk] Sk , k = 6; the middle 20 rows. 13343998895913 1152921504606846976 n5 + 5473124547145 9007199254740992 n4 + 418085902919 35184372088832 n3 + 7217931149 68719476736 n2 + 209746269 536870912 n+ 64348647 134217728 53375995583651 4611686018427387904 n5 + 5473124547143 9007199254740992 n4 + 104521475729 8796093022208 n3 + 14793776237 137438953472 n2 + 498096901 1073741824 n+ 28772927 33554432 106751991167301 9223372036854775808 n5 + 5473124547141 9007199254740992 n4 + 209042951457 17592186044416 n3 + 3608965575 34359738368 n2 + 1677970153 4294967296 n+ 11719905 33554432 13343998895913 1152921504606846976 n5 + 5473124547151 9007199254740992 n4 + 418085902905 35184372088832 n3 + 3698444059 34359738368 n2 + 478212797 1073741824 n+ 74631683 134217728 106751991167311 9223372036854775808 n5 + 5473124547147 9007199254740992 n4 + 418085902907 35184372088832 n3 + 28871724591 274877906944 n2 + 878753299 2147483648 n+ 10682367 16777216 106751991167313 9223372036854775808 n5 + 5473124547151 9007199254740992 n4 + 104521475729 8796093022208 n3 + 14793776237 137438953472 n2 + 478212797 1073741824 n+ 84006215 134217728 106751991167307 9223372036854775808 n5 + 2736562273573 4503599627370496 n4 + 209042951459 17592186044416 n3 + 28871724589 274877906944 n2 + 104873135 268435456 n+ 3161957 8388608 53375995583655 4611686018427387904 n5 + 5473124547149 9007199254740992 n4 + 209042951451 17592186044416 n3 + 29587552485 274877906944 n2 + 996193803 2147483648 n+ 118111589 134217728 106751991167309 9223372036854775808 n5 + 1368281136787 2251799813685248 n4 + 418085902907 35184372088832 n3 + 28871724601 274877906944 n2 + 419492539 1073741824 n+ 49899533 134217728 106751991167305 9223372036854775808 n5 + 2736562273573 4503599627370496 n4 + 52260737863 4398046511104 n3 + 29587552479 274877906944 n2 + 1912851173 4294967296 n+ 87717907 134217728 106751991167309 9223372036854775808 n5 + 1368281136787 2251799813685248 n4 + 104521475729 8796093022208 n3 + 28871724595 274877906944 n2 + 878753303 2147483648 n+ 8962701 16777216 106751991167305 9223372036854775808 n5 + 2736562273571 4503599627370496 n4 + 209042951453 17592186044416 n3 + 29587552499 274877906944 n2 + 956425585 2147483648 n+ 10878263 16777216 53375995583651 4611686018427387904 n5 + 5473124547157 9007199254740992 n4 + 104521475727 8796093022208 n3 + 3608965575 34359738368 n2 + 838985083 2147483648 n+ 53611225 134217728 13343998895913 1152921504606846976 n5 + 5473124547145 9007199254740992 n4 + 418085902907 35184372088832 n3 + 14793776249 137438953472 n2 + 498096903 1073741824 n+ 104354293 134217728 106751991167299 9223372036854775808 n5 + 2736562273575 4503599627370496 n4 + 209042951447 17592186044416 n3 + 3608965575 34359738368 n2 + 838985087 2147483648 n+ 7873221 16777216 106751991167303 9223372036854775808 n5 + 5473124547151 9007199254740992 n4 + 418085902899 35184372088832 n3 + 14793776243 137438953472 n2 + 956425599 2147483648 n+ 90737805 134217728 53375995583655 4611686018427387904 n5 + 5473124547155 9007199254740992 n4 + 104521475727 8796093022208 n3 + 14435862303 137438953472 n2 + 878753295 2147483648 n+ 18680385 33554432 106751991167305 9223372036854775808 n5 + 2736562273573 4503599627370496 n4 + 104521475725 8796093022208 n3 + 3698444063 34359738368 n2 + 478212799 1073741824 n+ 36634403 67108864 106751991167311 9223372036854775808 n5 + 684140568395 1125899906842624 n4 + 209042951463 17592186044416 n3 + 7217931153 68719476736 n2 + 52436567 134217728 n+ 19926951 67108864 26687997791827 2305843009213693952 n5 + 1368281136789 2251799813685248 n4 + 6532592233 549755813888 n3 + 14793776243 137438953472 n2 + 498096903 1073741824 n+ 67108871 67108864 Figure 6.13: The Hilbert quasipolynomial of Tk = C[x1, . . . , xk] Sk , k = 6; the last 20 rows. Chapter 7 On verification and discovery of obstructions In this chapter we give applications of the results and positivity hypotheses in this paper to the problem of verifying or discovering an obstruction, i.e., a “proof of hardness” [GCT2] in the context of the P vs. NP and the permanent vs. determinant problems in characteristic zero. 7.1 Obstruction An obstruction in an abstract setting of Problem 1.1.4 is defined as follows. Let X and Y be H-varieties with compact specifications (Section 3.5), H a connected reductive group. Let 〈X〉 and 〈Y 〉 denote the bit lengths of their specifications (Section 3.5). Suppose we wish to show that X cannot be embedded as an H-subvariety of Y . Pictorially: X 6 →֒ Y. (7.1) For example, in the context of the P vs. NP problem in characteristic zero [GCT1, GCT2], X is a class variety XNP (n, l) associated with the complexity class NP for the given input size parameter n and the circuit size parameter l. The variety Y is the class variety XP (l) associated with the class P for given l. And H is SLl(C). If NP ⊆ P (over C) to the contrary, then it would turn out that XNP (n, l) →֒ XP (l) as an H-subvariety, for every l = poly(n). The goal is to show that this embedding cannot exist when l = poly(n) and n→ ∞. Let R(X) and R(Y ) be the homogeneous coordinate rings of X and Y , respectively. Let R(X)d and R(Y )d denote their degree d-compoenents. Suppose to the contrary that an H-embedding as in (7.1) exists. Then there exists a degre preserving H-equivariant surjection from R(Y )d to R(X)d for every d, and hence, a degree-preserving H-equivariant injection from R(X)∗ to R(Y )∗ . Hence, every irreducible H-module Vλ(G) that occurs in R(X) also occurs in R(Y )∗ . This leads to: Definition 7.1.1 An irreducible representation Vλ(H) is called an obstruc- tion for the pair (X,Y) if it occurs (as an H-submodule) in R(X)∗ but not in R(Y )∗ , for some d. We say that Vλ(H) is an obstruction of degree d. Remark 7.1.2 The obstruction as defined here is dual to the obstrution as defined in [GCT2]. Existence of such an obstruction implies that X cannot be embedded in Y as an H-subvariety. Let us assume that X and Y are H-subvarieties of P (V ), where V is an H-module, and that we are given a point y ∈ Y ⊆ P (V ) that is distiguished in the following sense. Let Hy ⊆ H be the stabilizer of y. Then Cy, the line in V corresponding to y, is invariant under Hy. Let [y] be the set of points in P (V ) stabilized by Hy. We say that y is characterized by its stabilizer Hy if y = [y]; i.e., y is the only point in P (V ) stabilized by Hy. Let H[y] = ∪z∈[y]Hz be the union of the H-orbits of all points in [y]. We say that y is a distin- guished point of Y if Y equals the projective closure of H[y] in P (V ). If y is characterized by its stabilizer, this means Y is the projective closure of the orbit Hy of y. If Vλ(H) occurs in R(Y ) , then it can be shown (cf. Proposition 4.2 in [GCT2]) that Vλ(H) contains an Hy-submodule isomorphic to (Cy) d, the d-th tensor power of Cy. This leads to the following stronger notion of obstruction: Definition 7.1.3 [GCT2] We say that Vλ(H) is a strong obstruction for the pair (X,Y ) if, for some d, it occurs in R(X)∗ , but it does not contain an Hy-module isomorphic to (Cy) Existence of a strong obstruction also implies that X cannot be embedded in Y as an H-subvariety. The results in [GCT2] suggest that strong obstruc- tions exist in the context of the lower bound problems under cosideration. The goal then is to show their existence. 7.2 Decision problems The decisions problems that arise in this context are the following. Let sλ be the multiplicity of Vλ(H) in R(X) , and md the multiplicity of the Hy- module (Cy)d in Vλ(H), considered an Hy-module via the the embedding ρ : Hy →֒ H. Thus λ is a strong obstruction of degree d iff s is nonzero and md is zero. Problem 7.2.1 (Decision Problems) (a) Given d, λ and the specification of X, decide if sλ is nonzero. (b) Given d, λ and the specifications of H,Hy and ρ, decide if m λ is nonzero. (c) Given d, λ and the specifications of X,H,Hy and ρ, decide if λ is a strong obstruction of degree d. The first is an instance of the decision Problem 1.1.4 in geometric in- variant theory, and the second of the subgroup restriction Problem 1.1.3. By the results in Chapter 3-4, relaxed forms of the decision problems in (a) and (b) belong to P assuming appropriate PH1 and SH; this implies that a relaxed form of the decision problem in (c) also belongs to P assumming PH1 and SH. We will only need a weak relaxed form of (c), for which the weak form of SH that is implied by PH1 (cf. Theorem 3.3.5) will suffice. 7.3 Verification of obstructions The relevant PH1 are as follows. Assume that that singularities of spec(R(X)) are rational. By Theo- rem 3.5.1, the stretching function s̃λ (k) = skλ is a quasipolynomial. Hence PH1 for sλ (Hypothesis 3.3.1, or rather its slight variant obtained by replac- ing R(X)d with R(X) ) is well defined. It is: Hypothesis 7.3.1 (PH1): There exists a polytope P λ such that: 1. The number of integer points in P λ is equal to sλ 2. The Ehrhart quasi-polynomial of P λd coincides with the stretching quasi- polynomial s̃λd(n) (cf. Theorem 3.5.1). 3. The polytope P λ is given by a separating oracle, as in Section 2.3. Its encoding bitlength 〈P λ 〉 is poly(〈d〉, 〈λ〉, 〈X〉), and the combinatorial size ‖P λ ‖ is poly(ht(λ), ‖X‖), where ‖X‖ is the combinatorial size of X (Section 3.5), and ht(λ) is the height of λ. Similarly by Theorem 3.4.1 (or rather its slight variant which can be proved similarly), the stretching function mkdkλ is a quasipolynomial. Hence PH1 for mdλ (cf. Hypothesis 3.3.1 and Section 3.4) is also well defined. It is: Hypothesis 7.3.2 (PH1:) There exists a polytope Qd such that: 1. The number of integer points in Qd is equal to md 2. The Ehrhart quasi-polynomial of Qd coincides with the stretching quasi- polynomial m̃d (n) (Theorem 3.5.1). 3. The polytope Qdλ is given by a separating oracle. Its encoding bitlength 〈Qdλ〉 is poly(〈d〉, 〈λ〉, 〈ρ〉, 〈Hy 〉, 〈H〉), and the combinatorial size ‖Q is O(poly(ht(λ), 〈H〉, 〈Hy〉, 〈ρ〉)). Here 〈H〉, 〈Hy〉 and 〈ρ〉 denote the bitlengths of H,Hy and ρ (Section 3.4). Theorem 7.3.3 (Weak SH:) (a) Assuming PH1 (Hypothesis 7.3.1), the saturation index of s̃λ (n) is at most apoly(‖P ‖), for some explicit constant a > 0. (b) Assuming PH1 (Hypothesis 7.3.2), the saturation index of m̃dλ(n) is at most bpoly(‖Q ‖), for some explicit constant b > 0. This follows from Theorem 3.3.5. Theorem 7.3.4 Assume PH1 (Hypotheses 7.3.1-7.3.2). Then, given d, λ, the specifications of X,H,Hy and ρ, and a relaxation parameter c greater than the explicit bounds on the saturation indices in Theorem 7.3.3, whether cλ is an obstruction of degree d can be decided in poly(〈d〉, 〈λ〉, 〈X〉, 〈H〉, 〈Hy 〉, 〈ρ〉, 〈c〉) time. This follows by applying Theorem 3.1.1 to the polytopes P λ and Qd the saturation index estimates in Theorem 7.3.3. 7.4 Robust obstruction We now define a notion of obstruction that is well behaved with respect to relaxation. Definition 7.4.1 Assume PH1 for both sλd andm λ (Hypotheses 7.3.1-7.3.2). We say that Vλ(H) is a robust obstruction for the pair (X,Y ) if one of the following hold: 1. Qd is empty, and P λ is nonempty. 2. Both Qdλ and P λ are nonempty, the affine span of Qd does not contain an integer point and the affine span of P λ contains an integer point. If Vλ(H) is a robust obstruction, so is Vlλ(H), for all or most positive integral l, hence the name robust. Proposition 7.4.2 Assume PH1 for both sλd and m λ as above. If Vλ(H) is a robust obstruction for the pair (X,Y ), then for some positive integer k–called a relaxation parameter–Vkλ(H) is a strong obstruction for (X,Y ). In fact, this is so for most large enough k. Proof: (1) Suppose Qd is empty, and P λ is nonempty. Let k be a large enough positive integer k such that kP λ = P kλ contains an integer point. Then skλ is nonzero. But mkd is zero since Qkd = kQd is empty. Thus kλ is a strong obstruction. (2) Suppose both Qdλ and P λd are nonempty, the affine span of Q λ does not contain an integer point and the affine span of P λd contains an integer point. We can choose a positive integer k such that the affine span of kQdλ = Q does not contain an integer point, but kP λ = P kλ contains an integer point; most large enough k have this property. This means skλ is nonzero, butmkd is zero. Thus kλ is a strong obstruction. Q.E.D. 7.5 Verification of robust obstructions Theorem 7.5.1 Assume that the singularities of spec(R(X)) are rational. Assume PH1 for both sλ and md as above. Then, given λ, d and the speci- fications of ρ : Hy →֒ H and X, whether Vλ(H) is a robust obstruction can be verified in poly(〈ρ〉, 〈Hy〉, 〈H〉, 〈X〉, 〈d〉, 〈λ〉) time. Furthermore, a positive integral relaxation parameter k such that Vkλ(G) is a strong obstruction can also be found in the same time. The crucial result used implicitly here is the quasipolynomiality theorem (Theorem 4.1.1) because of which PH1 for both sλ and md are well defined. Proof: By linear programming [GLS], whether Qd is nonempty or not can be determined in poly(〈Qdλ〉) = poly(〈ρ〉, 〈Hy〉, 〈H〉, 〈d〉, 〈λ〉) time. If it is nonempty, the linear programming algorithm also gives its affine span. Whether this contains an integer point can be determined in polynomial time, using the polynomial time algorithm for computing the Smith normal form, as in the proof of Theorem 3.1.1. Similarly, whether P λ is nonempty or not can be determined in poly(〈P λ poly(〈X〉, 〈d〉, 〈λ〉) time. If it is nonempty, whether its affine span contains an integer point can be determined in polynomial time similarly. Further- more, the algorithm can also be made to return a vertex v of the polytope P λd if it is nonempty. Using these observations, whether Vλ(G) is a robust obstruction can be determined in polynomial time. As far as the computation of the relaxation parameter k is concerned, let us consider the second case in Definition 7.4.1–when both Qdλ and P d are nonempty, the affine span of Qdλ does not contain an integer point and the affine span of P λd contains an integer point–the first case being simpler. In this case, by examining the Smith normal forms of the defining equations of the affine spans of P λ and Qd and the rational coordinates of a vertex v ∈ P λ , we can find a large enough k so that the affine span of Qkd does not contain an integer point, the affine span of P kλ contains an integer point, and P kλ contains an integer point that is some multiple of v. Q.E.D. The value of the relaxation parameter k computed above is rather con- servative. One may wish to compute as small value of k as possible for which Vkλ(G) is a strong obstruction (though in our application this is not necessary). If SH for holds for the structural constant sλ (cf. Hypothe- sis 3.3.2 and Section 3.5), then we can let k be the smallest integer larger than the saturation index (estimate) for P λ such that affine span of Qkd nonempty) does not contain an integer point (as can be ensured by looking at the Smith normal of the defining equations of the affine span). 7.6 Arithemetic version of the P#P vs. NC prob- lem in characteristric zero We now specialize the discussion in the preceding sections in the context of the arithmetic form of the P#P vs. NC problem in characteristric zero [V]. In concrete terms, the problem is to show that the permanent of an n×n complex matrix X cannot be expressed as a determinant of an m×m complex matrix, whose entries are (possibly nonhomogeneous) linear com- binations of the entries of X. 7.6.1 Class varieties The class varieties in this context are as follows [GCT1]. Let Y be an m×m variable matrix, which can also be thought of as a variable l-vector, l = m2. Let X be its, say, principal bottom-right n × n submatrix, n < m, which can be thought of as a variable k-vector, k = n2. Let V = Symm(Y ) be the space of homogeneous forms of degree m in the variable entries of Y . The space V , and hence P (V ), has a natural action of G = GL(Y ) = GLl(C) given by (σf)(Y ) = f(σ−1Y ), for any f ∈ V , σ ∈ G, and thinking of Y as an l-vector. Let W = Symn(X) be the space of homogeneous forms of degree n in the variable entries of X. The space W , and also P (W ), has a similar action of K = GL(X) = GLk(C). We use any entry y of Y not in X as the homogenizing variable for embedding W in V via the map φ : W → V defined by: φ(h)(Y ) = ym−nh(X), (7.2) for any h(X) ∈W . We also think of φ as a map from P (W ) to P (V ). Let g = det(Y ) ∈ P (V ) be the determinant form, and f = φ(h), where h = perm(X) ∈ P (W ). Let ∆V [g],∆V [f ] ⊆ P (V ) be the projective closures of the orbits Gg and Gf , respectively, in P (V ). Let ∆W [h] ⊆ P (W ) be the projective closure of the K-orbit Kh of h in P (W ). Then ∆V [g] is called the class variety associated with NC and ∆V [f ] the class variety associated with P#P ; ∆W [h] is called the base class variety associated with P #P . (The base class variety is not used in what follows. Rather its variant, called a reduced class variety defined below, will be used.) These class varieties depend on the lower bound parameters n and m. If we wish to make these explicit, we would write ∆V [f, n,m] and ∆V [g,m] instead of ∆V [f ] and ∆V [g]. The class varieties ∆V [g] = ∆V [g,m] and ∆V [f ] = ∆V [f, n,m] are G-subvarieties of P (V ), and their homogeneous coordinate rings RV [g] = RV [g,m] and RV [f ] = RV [f, n,m] have natural degree-preserving G-action. It is conjectured in [GCT1] that, if m = poly(n) and n → ∞, then f 6∈ ∆V [g]; this is equivalent to saying that the class variety ∆V [f, n,m] cannot be embedded in the class variety ∆V [g,m] (as a subvariety). This implies the arithmetic form of the P#P 6= NC conjecture in characteristic zero. 7.6.2 Obstructions The obstruction in this context is defined as follows. A G-module Vλ(G) is called an obstruction for the pair (f, g) if it occurs in RV [f, n,m] d but not RV [g,m] for some d. It is called a strong obstruction if, for some d, it occurs in RV [f, n,m] but it does not contain (Cg)d as a Gg-submodule, where (Cg) ⊆ V denotes the one dimensional line corresponding to g, and Gg ⊆ G is the stabilizer of g = det(Y ) ∈ P (V ). If Vλ(G) is a (strong) obstruction of degree d, then the size |λ| = dm; hence d is completely determined by λ and m. Existence of an obstruction or a strong obstruction implies that the class variety ∆V [f, n,m] cannot be embedded in the class variety ∆V [g,m], as sought. The main algebro-geometric results of [GCT1, GCT2] suggest that strong obstructions should indeed exist for all n → ∞, assuming m = poly(n); cf. Section 4, Conjecture 2.10 and Theorem 2.11 in [GCT2]. The goal then is to prove existence of strong obstructions for all n. The definition of a strong obstruction can be simplified further as follows. Let X ′ denote the set of variables, which consists of the variable entries in X and the homogenizing variable y above. Let W ′ = Symm(X ′) ⊆ V = Symm(Y ) be the space of homogeneous forms of degree m in the variables of X ′. We have a natural action of H = GL(X ′) = GLn2+1(C) on W and hence on P (W ′). We have a natural map φ′ : W → W ′ given by φ′(h)(X ′) = ym−nh(X). The map φ in (7.2) is φ′ followed by the inclusion from W ′ to V . We also think of φ′ as a map from P (W ) to P (W ′). Let f ′ = φ′(h), for h = perm(X) ∈ P (W ). Let ∆W ′ [f ′] ⊆ P (W ′) be the orbit closure of Hf ′. It is an H-subvariety of P (W ′), and hence its homogeneous coordinate ring RW ′ [f ′] has the natural degree preserving H- action. We call ∆W ′ [f ′] the reduced class variety for P#P . It is known (cf. Theorem 8.2 in [GCT2]) that Vλ(G) occurs in RV [f ] iff Vλ(H) occurs in RW ′ [f . Here the dominant weight λ of G is considered a dominant weight of H by restriction from G to H. Hence Vλ(G) is a strong obstruction for the pair (f, g), iff for some d, Vλ(H) occurs in RW ′ [f as an H-submodule and Vλ(G) does not contain (Cg)d as a Gg-submodule. In particular, we can assume without loss of generality that the height of the Young diagram for λ is at most n2 + 1; otherwise Vλ(H) would be zero. 7.6.3 Robust obstructions It is known that the stabilizer Gg of g = det(Y ) ∈ P (V ) consists of lin- ear transformations in G of the form Y → AY ∗B−1, thinking of Y as an m×m matrix, where Y ∗ is either Y or Y T , A,B ∈ GLm(C). Thus the con- nected component of Gg is essentially GLm(C)×GLm(C) ⊆ G = GLl(C) = GLm2(C). This means the subgroup restriction problem for the embedding ρ : Gg →֒ G is essentially the Kronecker problem (Problem 1.1.1). Assume PH1 (Hypothesis 7.3.2) for the subgroup restriction ρ : Gg →֒ G; which is essentially PH1 for the Kronecker problem. It now assumes the following concrete form. Let md denote the multiplicity of the Gg-module (Cg)d in Vλ(G). Assume that the height of λ is at most n 2+1 for the reasons give above. Hypothesis 7.6.1 (PH1:) There exists a polytope Qd such that: 1. The number of integer points in Qd is equal to md 2. The Ehrhart quasi-polynomial of Qdλ coincides with the stretching quasi- polynomial m̃d (n) (cf. Theorem 3.5.1). 3. The polytope Qd is given by a separating oracle, and its encoding bitlength 〈Qdλ〉 is poly(n, 〈m〉, 〈d〉, 〈λ〉) time. We have to explain why 〈Qd 〉 is stipulated to depend polynomially on n and 〈m〉, rather than m. After all, the bitlengths 〈G〉, 〈Gg〉 and 〈ρ〉 are O(poly(m2)) as per the definitions in Section 3.4. So, as per PH1 for sub- group restriction in Section 3.4.3, 〈Qdλ〉 should depend polynomially on m. We are stipulating a stronger condition for the following reason. First, as we already mentioned, the above hypothesis is essentially PH1 for the Kronecker problem, which is obtained by specializing PH1 for the plethysm problem (Hypothesis 1.6.4). In Hypothesis 1.6.4, the encoding bitlength of the poly- tope depends polynomially on the bitlengths of the various partition param- eters λ, π, µ of the plethysm constant aπ , but is independent of the rank of the group G therein. (As explained in the remarks after Hypothesis 1.6.4, this is justified because the bound in Theorem 1.6.3 is also independent of the rank of G). For the same reason, the encoding bitlength of the polytope here should be independent of the rank of G (which is m2), but should de- pend polynomiallly on the total bit length of the partitions parametrizing the representations Vλ(G) and (Cy) d. This is O(n+ 〈m〉+ 〈d〉+ 〈λ〉). (Note that the one dimensional representation (Cy)d of Gg is essentially the d-th power of the determinant representation of Gg, since the connected compo- nent of Gg is isomorphic to GLm(C)×GLm(C). The Young diagram for the partition corresponding to the d-th power of the determinant representation of GLm(C) is a rectangle of height m and width d. It can be specified by simply giving m and d–this specification has bit length 〈m〉+ 〈d〉.) Next let us specialize PH1 as per Hypothesis 7.3.1. The class variety ∆V [f ] = ∆[f, n,m] will now play the role of X in Hypothesis 7.3.1. But, for the reasons explained in the proof of Proposition 7.6.4 below, we shall instead specialize Hypothesis 7.3.1 to the (simpler) reduced class variety Z = ∆W ′[f ′]. It now assumes that following concrete form. Let sλ denote the multiplicity of Vλ(H) in RW ′ [f . Putting Z in place of X in Hypothe- sis 7.3.1, we get: Hypothesis 7.6.2 (PH1): There exists a polytope P λ such that: 1. The number of integer points in P λ is equal to sλ 2. The Ehrhart quasi-polynomial of P λ coincides with the stretching quasi- polynomial s̃λ (n) (cf. Theorem 3.5.1). 3. The polytope P λd is given by a separating oracle, and its encoding bitlength 〈P λd 〉 is poly(〈d〉, 〈λ〉, 〈Z〉) = poly(〈d〉, 〈λ〉, n, 〈m〉). (7.3) Here (7.3) follows because 〈Z〉 = n+〈m〉. To see why, let us observe that Z = ∆W ′ [f ′] is completely specified once the point f ′ = ym−nh ∈ P (W ′) is specified. To specify f ′, it sufficies to specify m,n and the point h ∈ P (W ). It is known [GCT2] that the point h = perm(X) ∈ P (W ) is completely characterized by its stabilizer Kh ⊆ K = GL(X) = GLk(C). Furthermore, Kh is explicitly known [Mc]. It is generated by the linear transformation in K of the form X → λXµ−1, thinking of X as an n×n matrix, where λ and µ are either diagonal or permutation matrices. So to specifiy h, it suffices to specify Kh,K and the embedding ρ ′ : Kh →֒ K. The bit length of this specification is O(n) (cf. Section 3.4). To specify f ′, and hence Z, it suffices to specify m,n,K,Kh and ρ ′. The total bit length of this specification is O(n+ 〈m〉). Assume PH1 for both md and sλ , i.e., Hypotheses 7.6.1 and 7.6.2. Definition 7.6.3 We say that Vλ(G) is a robust obstruction for the pair (f, , g) if one of the following hold: 1. Qdλ is empty, and P d is nonempty. 2. Both Qdλ and P d are nonempty, the affine span of Q λ does not contain an integer point and the affine span of P λd contains an integer point. If the first condition holds, we say that Vλ(G) is a geometric obstruction. If the second condition holds, it is called a modular obstruction. Proposition 7.6.4 Assume PH1 for both mdλ and s d (Hypotheses 7.6.1 and 7.6.2). If Vλ(G) is a robust obstruction for the pair (f, g), then for some positive integral relaxation parameter k, Vkλ(G) is a strong obstruction for (f, g). In fact, this is so for most large enough k. Proof: This essentially follows from Proposition 7.4.2. It only remains to clarify why we can use PH1 for the reduced class variety ∆W ′[f ′]–as we are doing here– in place of PH1 for the class variety ∆V [f ]. This is because, as already mentioned, Vλ(G) occurs in RV [f ] d iff Vλ(H) occurs in RW ′ [f ′]∗d. Q.E.D. 7.6.4 Verification of robust obstructions Theorem 7.6.5 Assume that the singularities of spec(RW ′ [f ′]) are ratio- nal. Assume PH1 for both md and sλ as above (Hypotheses 7.6.1 and 7.6.2). Then, given n,m, λ and d, whether Vλ(H) is a robust obstruction can be verified in poly(n, 〈m〉, 〈d〉, 〈λ〉) time. Furthermore, a positive integral relax- ation parameter k such that kλ is a strong obstruction can also be computed in this much time. Once n andm are specified, the various class varieties andK,Kh, ρ ′, G,Gg, ρ above are automatically specified implicitly. Proof: This follows from Theorem 7.5.1; cf. also the remark following its proof. Q.E.D. Theorem 7.3.4 can be similarly specialized in this context; we leave that to the reader. 7.6.5 On explicit construction of obstructions Theorem 7.6.6 Assume that m = poly(n) or even 2polylog(n), and: 1. (RH) [Rationality Hypothesis]: The singularities of spec(RW ′ [f ′]) are rational. 2. PH1 for both md and sλ (Hypotheses 7.6.1 and 7.6.2). 3. OH [Obstruction Hypothesis]: For every (large enough) n, there exists λ of poly(n) bit length such that |λ| is divisible by m and one of the following holds (with d = |λ|/m): (a) Qd is empty, and P λ is nonempty. (b) Both Qdλ and P λ are nonempty, the affine span of Qd does not contain an integer point and the affine span of P λ contains an integer point. Then there exists an explicit family {λn} of robust obstructions. Here we say that {λn} is an explicit family of robust obstructions if each λn is short and easy to verify. Short means 〈λn〉 is O(poly(n)). Easy to verify means whether λn is a robust obstruction can be verified in O(poly(n)) time. The poly(n) bound here and in OH is meant to be independent of m, as long as m << 2n; i.e., it should hold even when m = 2polylog(n). In other words, the family {λn} should continue to remain an explicit robust obstruction family, as we vary m over all values ≤ 2polylog(n), and perhaps even values ≤ 2o(n), but will cease to be an obstruction family for some large enough m = 2Ω(n). This is an important uniformity condition. Proof: OH basically says that there exists a short robust obstruction λn for every n. By Theorem 7.6.5, it is easy to verify. Q.E.D. 7.6.6 Why should robust obstructions exist? The main question now is: why should OH hold? That is, why should (short) robust obstructions exist? As we already mentioned, the results in [GCT1, GCT2] indicate that strong obstructions should exist for every n, assuming m = poly(n). We shall give a heuristic argument for existence of robust obstructions assuming that strong obstructions exist. This will crucially depend on the following SH formd , which is essentially SH for the Kronecker problem (i.e. specialization of Hypothesis 1.6.5 to the Kronecker problem), good experimental evidence for which is provided in [BOR]. Hypothesis 7.6.7 (SH:) (a): The saturation index of m̃d (k) is bounded by a polynomial in m. (Observe that the rank of G is poly(m) and the height of λ is at most n2 + 1). (b): The quasi-polynomial m̃d (n) is strictly saturated, i.e. the saturation index is zero, for almost all λ (and d). If Vλ(G) is a strong obstruction, s is nonzero but md is zero. Thus, assumming PH1, there are three possibilities: 1. Qd is empty, and P λ is nonempty and contains an integer point. 2. Both Qdλ and P λd are nonempty, the affine span of Q λ does not contain an integer point and P λd contains an integer point. 3. Both Qd and P λ are nonempty. The affine span of Qd contains an integer point, but Qd does not. And P λ contains an integer point. In the first two cases, λ is a robust obstruction. As per SH (Hypothe- sis 7.6.7), for almost all λ, the Ehrhart quasipolynomial of Qd is saturated: this means (cf. the proof of Theorem 3.1.1), if the affine span of Qd contains an integer point then Qd also contains an integer point. And hence, with a high probability, the third case should not occur. In other words, strong obstructions can be expected to be robust with a high probability. Let us call a strong obstruction λ fragile if it is not robust; this means the affine span of Qdλ contains an integer point, but Q λ does not. By SH (Hypothesis 7.6.7), if λ is fragile, then for some k = poly(m), Qkdkλ contains an intger point, and hence, kλ is not obstruction. Thus fragile obstructions are close to not being obstructions, and furthermore, are expected to be rare, as argued above. This is why we are focussing on robust obstructions. It may be remarked that the only SH needed in the argument above is the one (Hypothesis 7.6.7) for the structural constant md . This is a special case of the SH for the subgroup restriction problem (cf. Section 3.4) specialized to the embedding Gg →֒ G. In particular, we do not need SH for the structural constant sλd ; i.e., for the more difficult decision problem in geometric invariant theory (cf. Problem 1.1.4 and Section 3.5). 7.6.7 On discovery of robust obstructions It may be conjectured that not just the verification (cf. Theorem 7.6.5) but also the discovery of robust obstructions is easy for the problem under consideration. In this section we shall give an argument in support of this conjecture for geometric (robust) obstructions (which may be conjectured to exist in the problem under consideration). For this we need to reformulate the notions of strong and robust obstructions (Definition 7.6.3) as follows. Let TZ be the set of pairs (d, λ) such that s d is nonzero and SZ the set of pairs (d, λ) such that mdλ is nonzero. Proposition 7.6.8 Assuming PH1 above (Hypotheses 7.6.1 and 7.6.2), TZ and SZ are finitely generated semigroups with respect to addition. These semi-groups are analogues of the Littlewood-Richardson semigroup (Section 2.2.2) in this setting. Proof: The proof is similar to that for the Littlewood-Richardson semigroup For given d and λ, the polytope P λ in PH1 for sλ (Hypothesis 7.6.2) has a specification of the form Ax ≤ b (7.4) where A depends only the variety Z = ∆W ′ [f ′], but not on d or λ, and b depends homogeneously and linearly on d and λ. Let P be the polytope defined by the inequalities (7.4) where both d and λ are treated as variables. Then P is a polyhedral cone (through the origin) in the ambient space containing P with the coordinates x, d and λ. Let PZ be the set of integer points in P . It is a finitely generated semigroup since P is a polyhedral cone. Let TR be the orthogonal projection of P on the hyperplane corresponding to the coordinates d and λ. Now TZ is simply the projection of PZ. Hence it is a finitely generated semigroup as well. The proof for SZ is similar, with SR defined similarly. Q.E.D. The polyhedral cones TR and SR here are analogues of the Littlewood- Richardson cone (Section 2.2.2) in this setting. Note that (d, λ) ∈ TR iff P is nonempty; similarly for SR. A Weyl module Vλ(G) is a strong obstruction for the pair (f, g) of degree d iff (d, λ) occurs in TZ but not in SZ. It is a robust obstruction iff it occurs in TR but not in SZ. It is a geometric obstruction iff it occurs in TR but not in SR. It is a modular obstruction iff it occurs in TR and also in SR but not in SZ. Assuming PH1 (Hypothesis 7.6.2), whether (d, λ) belongs to TR can be determined in polynomial time by linear programming, since (d, λ) ∈ TR iff P λ is nonempty. Similarly, assuming PH1 (Hypothesis 7.6.1), whether (d, λ) ∈ SR can be determined in polynomial time. The following is a stronger complement to PH1. Hypothesis 7.6.9 (PH1*) Whether TR\SR is nonempty can be determined in polynomial time; i.e., poly(n, 〈m〉) time. If so, the algorithm can also output (d, λ) ∈ TR \ SR of polynomial bit length. Proposition 7.6.10 Assuming PH1*, given n and m, the problem of decid- ing if a geometric obstruction exists for the pair (f, g), and finding one if one exists, belongs to the complexity class P ; i.e., it can be done in poly(n, 〈m〉) time. This immediately follows from Hypothesis 7.6.9 since (d, λ) is a geometric obstruction iff (d, λ) ∈ TR \ SR. Hypothesis 7.6.9 is supported by the following: Proposition 7.6.11 Assuming PH1 (Hypotheses 7.6.1 and 7.6.2), Hypoth- esis 7.6.9 holds if TR and SR have polynomially many explicitly given con- straints with the specification of polynomial bit length; here polynomial means poly(n, 〈m〉). The proposition holds even if the polytope SR has exponentially many constraints, as long as it is given by a separation oracle that works in poly- nomial time. Proof: It suffices to check if SR satisfies each constraint of TR. This can be done in polynomial time using the linear programming algorithm in [GLS]. Specifically, let l(y) ≥ 0 be a constraint of TR. Then we just need to minimize l(y) on SR and check if the minimum exceeds zero. Q.E.D. But this method does not work when the number of constraints of TR is exponential, as expected in the context of the lower bound problems under consideration. In fact, no generic black-box-type algorithm, like the one in [GLS] based on just a membership or separation oracle for TR, can be used to prove (4) when the number of constraints of TR is exponential. Fortunately, this is not a serious problem. A basic principle in combi- natorial optimization, as illustrated in [GLS], is that a complexity theoretic property that holds for polytopes with polynomially many constraints will also hold for polytopes with exponentially many constraints, provided these constraints are sufficiently well-behaved. For example, Edmond’s perfect matching polytope for nonbipartite graphs has complexity-theoretic proper- ties similar to the perfect matching polytope for bipartite graphs, though it can have exponentially many constraints. We have already remarked that TR and SR are analogues of the Littlewood-Richardson cones. The facets of the Littlewood-Richardson cone have a very nice explicit description [Kl, Z]. The cones TR, SR here are expected to have similar nice explicit descrip- tion. This is why Hypothesis 7.6.9 can be expected to hold even if the number of constraints of TR is exponential, just as it holds even when SR has exponentially many constraints. But a polynomial-time algorithm as in Hypothesis 7.6.9 would have to depend crucially on the specific nature of the facets (constraints) of TR in the spirit of the linear-programming-based algorithm for the construction of a maximum-weight perfect matching in nonbipartite graphs [Ed], where too the number of constraints is exponen- tial but the algorithm still works because of the structure theorems based on the specific nature of the constraints. 7.7 Arithmetic form of the P vs NP problem in characteristic zero We turn now to the arithemetic form of the P vs. NP problem in character- istic zero. The arguments are essentially verbatim translations of those for the arithmetic form of the P#P vs. NC problem in the preceding section. Hence we shall be brief. In the preceding section h(X) was perm(X) and g(Y ) was det(Y ). Now h(X) and g(Y ) would be explicit (co)-NP-complete and P -complete func- tions E(X) and H(Y ) constructed in [GCT1]. They can be thought of as points in suitable W = Symk(X) and V = Syml[Y ], k = O(n2), l = O(m2), with the natural action of GL(X) and G = GL(Y ), where n denotes the number of input parameters and m denotes the circuit size parameter in the lower bound problem. These functions are extremely special like the determinant and the permanent in the sense that they are “almost” char- acterized by their stabilizers as explained in [GCT1]–and this is enough for our purposes. We again have a natural embedding φ : P (W ) → P (V ), which lets us define f = φ(h). The class variety for NP is defined to be ∆V [f ] ⊆ P (V ), the projective closure of the orbit Gf . The class variety for P is ∆̃V [g] ⊆ P (V ), which is defined to be the projective closure of G[g], where [g] denotes the set of points in P (V ) that are stabilized by Gg ⊆ G, the stabilizer of g. An explicit description of Gg is given in [GCT1]; cf. Section 7 therein. To show P 6= NP in characteristic zero, it suffices to show that ∆V [f ] is not a subvariety of ∆̃V [g] for all large enough n, if m = poly(n) (cf. Conjecture 7.4. in [GCT1]). For this, in turn, it suffices to show existence of strong obstructions, defined very much as in Section 7.6, for all n, assumming m = poly(n). We can then formulate PH1 for the new h(X) and g(Y ) just as in Hy- potheses 7.6.1 and 7.6.2, and the notion of a robust obstruction as in Defi- nition 7.6.3. We then have: Theorem 7.7.1 (Verification of obstructions) Analogues of Theorems 7.6.5 and 7.6.6 holds for h(X) = E(X) and g(Y ) = H(Y ). Furthermore, even discovery of robust obstructions can be conjectured to be easy (poly-time)–this would follow from the obvious analogue of Hy- pothesis 7.6.9 here. Heuristic argument for existence of robust obstructions is very similar to the one in Section 7.6.6. It needs SH for the special case of the subgroup restriction problem for the embedding Gg →֒ G. 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Zelevinsky, Littlewood-Richardson semigroups, arXiv:math.CO/9704228 v1 30 Apr 1997. http://arxiv.org/abs/math/9704228 Introduction The decision problems Deciding nonvanishing of Littlewood-Richardson coefficients Back to the general decision problems Saturated and positive integer programming Quasi-polynomiality, positivity hypotheses, and the canonical models The plethysm problem Towards PH1, SH, PH2,PH3 via canonial bases and canonical models Basic plan for implementing the flip Organization of the paper Notation Preliminaries in complexity theory Standard complexity classes Example: Littlewood-Richardson coefficients Convex #P Littlewood-Richardson coefficients Littlewood-Richardson cone Eigenvalues of Hermitian matrices Separation oracle Saturation and positivity Saturated and positive integer programming A general estimate for the saturation index Extensions Is there a simpler algorithm? Littlewood-Richardson coefficients again The saturation and positivity hypotheses The subgroup restriction problem Explicit polynomial homomorphism Input specification and bitlengths Stretching function and quasipolynomiality The decision problem in geometric invariant theory Reduction from Problem 1.1.3 to Problem 1.1.4 Input specification Stretching function and quasi-polynomiality Positivity hypotheses G/P and Schubert varieties PH3 and existence of a simpler algorithm Other structural constants Quasi-polynomiality and canonical models Quasi-polynomiality The minimal positive form and modular index The rings associated with a structural constant Canonical models From PH0 to PH1,3 On PH0 in general Nonstandard quantum group for the Kronecker and the plethysm problems The cone associated with the subgroup restriction problem Elementary proof of rationality Parallel and PSPACE algorithms Complex semisimple Lie group Symmetric group General linear group over a finite field Tensor product problem Finite simple groups of Lie type Experimental evidence for positivity Littlewood-Richardson problem Kronecker problem, n=2 G/P and Schubert varieties The ring of symmetric functions On verification and discovery of obstructions Obstruction Decision problems Verification of obstructions Robust obstruction Verification of robust obstructions Arithemetic version of the P#P vs. NC problem in characteristric zero Class varieties Obstructions Robust obstructions Verification of robust obstructions On explicit construction of obstructions Why should robust obstructions exist? On discovery of robust obstructions Arithmetic form of the P vs NP problem in characteristic zero

0704.0230

Two new basaltic asteroids in the Outer Main Belt?

Two new V-type asteroids in the outer Main Belt?1 R. Du�ard∗2 Instituto de Astrofísica de Andalucía, C/ Bajo de Huetor, 50. 18008. Granada, Spain, and Max Planck Institute for Solar System Research, Max Planck Str. 2, Katlenburg-Lindau, Germany [email protected] F. Roig Observatório Nacional, Brazil [email protected] ABSTRACT The identi�cation of basaltic asteroids in the asteroid Main Belt and the de- scription of their surface mineralogy is necessary to understand the diversity in the collection of basaltic meteorites. Basaltic asteroids can be identi�ed from their visible re�ectance spectra and are classi�ed as V-type in the usual tax- onomies. In this work, we report visible spectroscopic observations of two candi- date V-type asteroids, (7472) Kumakiri and (10537) 1991 RY16, located in the outer Main Belt (a > 2.85 UA). These candidate have been previously identi�ed by Roig and Gil-Hutton (2006, Icarus 183, 411) using the Sloan Digital Sky Sur- vey colors. The spectroscopic observations have been obtained at the Calar Alto Observatory, Spain, during observational runs in November and December 2006. The spectra of these two asteroids show the steep slope shortwards of 0.70 µm and the deep absorption feature longwards of 0.75 µm that are characteristic of V-type asteroids. However, the presence of a shallow but conspicuous absorption band around 0.65 µm opens some questions about the actual mineralogy of these 1Based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC). 2corresponding author � 2 � two asteroids. Such band has never been observed before in basaltic asteroids with the intensity we detected it. We discuss the possibility for this shallow absorption feature to be caused by the presence of chromium on the asteroid sur- face. Our results indicate that, together with (1459) Magnya, asteroids (7472) Kumakiri and (10537) 1991 RY16 may be the only traces of basaltic material found up to now in the outer Main Belt. Subject headings: Asteroids, composition 1. Introduction Basaltic asteroids are small bodies connected to the processes of heating and melting that may have led to the mineralogical di�erentiation in the interiors of the largest asteroids. Therefore, a precise knowledge of the inventory of basaltic asteroids may help to estimate how many di�erentiated bodies actually formed in the asteroid Main Belt, and this in turn may provide important constraints to the primordial conditions of the solar nebula. In the visible wavelengths range, the re�ectance spectrum of basaltic asteroids is char- acterized by a steep slope shortwards of 0.70 µm and a deep absorption band longwards of 0.75 µm. Asteroids showing this spectrum are classi�ed as V-type in the usual taxonomies (e.g. Bus & Binzel, 2002). A few years ago, most of the known V-type asteroids were members of the Vesta dy- namical family, located in the inner asteroid belt �semi-major axis a < 2.5 AU�. This family formed by the excavation of a large crater (Thomas et al., 1997; Asphaug, 1997) on the surface of asteroid (4) Vesta, which is the only known large asteroid �diameter D ∼ 500 km� to show a basaltic crust (McCord et al., 1970). Nowadays, however, several V-type asteroids have been identi�ed in the inner belt but outside the Vesta dynamical family (Burbine et al., 2001; Florczak et al., 2002; Alvarez- Candal et al., 2006). Basaltic asteroids have also been found in the middle Main Belt �2.5 < a < 2.8 AU� (Binzel et al., 2006; Roig et al., 2007), as well as among the Near Earth Asteroids (NEA) population (McFadden et al., 1985; Cruikshank et al., 1991; Binzel et al., 2004; Du�ard et al., 2006). Recent works (Carruba et al., 2005, 2007; Nesvorný et al., 2007; Roig et al., 2007) provide evidence that many of these V-type asteroids may be former members of the Vesta family, that reached their present orbits due to long term dynamical evolution. The exception is asteroid (1459) Magnya, the only basaltic object so far discovered in the outer belt �a > 2.8 AU� (Lazzaro et al., 2000). This asteroid is too far away from the Vesta family and it is also too big �D = 20-30 km� to have a real probability � 3 � of being a fragment from the Vesta's crust (Michtchenko et al., 2002). Beyond the existence of (4) Vesta and the V-type asteroids related to the Vesta dy- namical family, the paucity of intact di�erentiated asteroids and of their fragments observed today in the main belt is an strong constraint to the formation scenario of basaltic mate- rial. The sample of iron meteorites collected in the Earth indicates that they would come from the iron core of dozens of di�erentiated parent bodies. However, there are very few olivine-rich asteroids (classi�ed as A-type) which are assumed to come from the mantle of di�erentiated bodies, and only one asteroid, (1459) Magnya, is known to sample the basaltic crust of a di�erentiated parent body other than (4) Vesta. Finally, the other Main Belt asteroid families, which formed from the disruption of over �fty 10 < D < 400 km asteroids, show little spectroscopic evidence that their parent bodies were heated enough to produce a distinct core, mantle and crust (Cellino, 2003). Aiming to establish if other V-type asteroids might be found together with (1459) Mag- nya in the outer belt, thus giving support to the existence of a di�erentiated parent body in that part of the belt, Roig & Gil-Hutton (2006) used the �ve band photometry from the 3rd release of the Sloan Digital Sky Survey Moving Objects Catalog (SDSS-MOC; Ivezi¢ et al., 2001; Juri¢ et al., 2002) to identify candidate V-type asteroids. Among 263 candidates that are not members of the Vesta dynamical family, Roig & Gil-Hutton found �ve possible V-type asteroids in the outer belt: (7472) Kumakiri, (10537) 1991 RY16, (44496) 1998 XM5, (55613) 2002 TY49, and (105041) 2000 KO41. However, these �ndings need to be con�rmed by accurate spectroscopic observations. The aim of this work is to describe the visible spectroscopic observations of two of these candidates: (7472) Kumakiri and (10537) 1991 RY16. Our goal is to provide a more reliable taxonomic classi�cation of these asteroids indicating that they would the second and third basaltic asteroids discovered up to now in the outer belt. Our observations also reveal certain peculiarities of their spectra that deserve special attention in future studies. Last but not least, our results help to validate the approach of Roig & Gil-Hutton (2006) to predict V-type asteroids. It is worth recalling that a similar study has been performed by Roig et al. (2007), who used visible spectroscopic observations taken at the Gemini Observatory to con�rm the classi�cation of two candidate V-type asteroids in the middle belt: (21238) 1995 WV7 and (40521) 1999 RL95. In Sect. 2, we describe the observations and the reduction procedures. In Sect. 3, we present and discuss the results obtained. Finally, Sect. 4 is devoted to conclusions. � 4 � 2. Observations Low resolution spectroscopy of (7472) Kumakiri and (10537) 1991 RY16 were obtained on November 14, 2006, as part of a 4 nights observational run, using the Calar Alto Faint Object Spectrograph (CAFOS) at the 2.2m telescope in Calar Alto Observatory, Spain. The prime aim of the run was to characterize V-type asteroids inside and outside the Vesta family. Asteroid (7472) Kumakiri was observed again on December 29, 2006, using the same instrument and telescope, under Director's Discretionary Time (DDT). Table 1 summarizes the observational circumstances. CAFOS1 is equipped with a 2048×2048 CCD detector SITe-1d (pixel size 24 µm/pixel, plate scale 0.53"/pixel). We used the R400 grism allowing to obtain an observable spectral range between 0.50 and 0.92 µm. To remove the solar component of the spectra and obtain the re�ectance spectra, the solar analog stars HD 191854, HD 20630 and HD 28099 (Hardorp, 1978) were also observed at similar airmasses as the asteroids. In order to estimate the quality of each night, at least two solar analogs were observed per night and we veri�ed that the ratios between the corresponding spectra show no signi�cant variations. Bias frames, spectral dome �at �elds and calibration lamps spectra were also taken in each night to allow reduction of the science images. Spectrum exposures for each asteroid were splitted in two exposures at two di�erent slit positions, A and B, separated by 20" (the width of the slit was 2.0"). The observations were performed with the telescope tracking at the proper motion of the asteroid. Hence by subtracting A from B and B from A, a very accurate background removal is achieved. Finally, standard methods for spectra extraction were applied. 3. Results and discussion The re�ectance spectra of (7472) Kumakiri and (10537) 1991 RY16 are shown in Figs. 1 and 2. Both spectra show a steep slope shortwards of 0.70 µm and a deep absorption band longwards of 0.75 µm. Using the algorithm of Bus (1999), we determine that the spectra can be classi�ed as V-type. Figure 1 show that our observations are compatible with the spectra of previously known V-type asteroids (gray lines) taken from the SMASS survey (Bus & Binzel, 2002) and the S3OS2 survey (Lazzaro et al., 2004). Figure 2 show the good agreement between the �ve band photometry of the SDSS-MOC (black lines) and the observed spectra. It is worth noting that the values of maximum and minimum re�ectance prevents to attribute to these spectra other taxonomic classi�cation, like R-, O- or Q-type. 1See http://www.caha.es/alises/cafos/cafos22.pdf for more details. � 5 � In view of this, (7472) Kumakiri and (10537) 1991 RY16 may be considered, together with (1459) Magnya, the only V-type asteroids discovered up to now in the outer belt. Notwithstanding, the spectra of (7472) Kumakiri and (10537) 1991 RY16 show a shal- low absorption feature around 0.60-0.70 µm that has never been reported before in V-type asteroids. This feature is more evident in the spectrum of (10537) 1991 RY16. After the its identi�cation in the November 14 observations, and excluding possible reduction artifacts or solar analogs problems, we requested Director Discretionary Time (DDT) for another obser- vational run on December 29. Only the spectra of (7472) Kumakiri was able to be observed during this run, con�rming the presence of the absorption band. Nevertheless, the band in the spectrum of (10537) 1991 RY16 has also been observed independently by Moskovitz et al. (2007). To analyze this band, we recti�ed the spectra by subtracting a linear continuum in the interval 0.55 and 0.75 µm and then �tted several polynomials of di�erent degrees. This allowed to determine the center of the band at 0.63± 0.01 µm and the FWHM of ∼ 0.1 µm (e.g. Fig. 3). The origin of this absorption band is unclear. Such kind of bands are usually believed to arise from the Fe2+ → Fe3+ charge transfer absorptions in phyllosilicate (hydrated) minerals (Vilas & Ga�ey, 1989; Vilas et al., 1993). However, it is di�cult to explain the presence of a hydrated mineral in the surface of a basaltic object, because the heating and melting that produce the basalt also eliminate any traces of water. It is known that pyroxene crystals Fe2+ cations do not show any absorption bands in the spectral region from 0.56 to 0.72 µm. Therefore, the origin of the observed band might be related to other impurity cations like Mn2+, Cr3+, and Fe3+, usually located in the M1 site of terrestrial and meteorite orthopyroxenes (Shestopalov et al., 2007). In particular, broad spin-allowed bands of trivalent chromium around 0.430-0.455 µm and 0.620-0.650 µm have been observed in both re�ectance and transmitted spectra of Cr-containing terrestrial ortho and clinopyroxenes (see Cloutis, 2002), as well as in diogenite re�ectance spectra (McFadden et al., 1982). Cr3+ cations also give spin-forbidden bands near 0.480, 0.635, 0.655, and 0.670 µm but they do not give absorptions near 0.57 µm. Cloutis (2002) speci�cally found that Cr3+ gives rise to an absorption band near 0.455 µm and a more complex absorption feature in the 0.65 µm region. However, changes in the grain size of the pyroxenes may have an e�ect on the depth of these absorption bands (Cloutis and Ga�ey 1991; Sunshine and Pieters 1993). Therefore, the presence of speci�c absorption bands can be taken as an evidence for the presence of a particular cation, but the characteristics of these bands (depth and width) are probably not reliable enough to constrain � 6 � the cation abundance (Cloutis 2002). For example, the grain size may be responsible of the di�erent band depth observed between the spectra of (7472) Kumakiri and (10537) 1991 RY16. The slight di�erences in the band pro�le between the November and December spectra of (7472) Kumakiri might be attributed to di�erent rotational phases2. Another interesting feature observed in our spectra is that the band center of the major absorption feature at 0.90 µm is displaced to larger wavelenghts. In our spectra, this region is the noisiest but using di�erent polynomial �ts it was possible to estimate the center of the band nearer to 0.92-0.93 µm. This behavior may also be attributed to the presence of chromium on the surface. Actually, Cloutis and Ga�ey (1991) suggested that the Cr-rich pyroxene samples in their study have the two major absorption features (i.e. the one centered at 0.9 and the one centered at 1.9 µm, respectively) displaced to larger wavelengths than expected, relative to their Fe contents. These authors also presented the predicted versus actual wavelength position of the major Fe2+ absorption band center in the 1 µm region, and this center is closer to 0.92 µm than to 0.90 µm. Therefore, the observational evidence points to a possible Cr-rich basaltic composition on the surfaces of (7472) Kumakiri and (10537) 1991 RY16. Concerning the dynamical behavior of these two asteroids, Table 2 lists their proper elements and diameters, as well as those of (1459) Magnya. The three asteroids are too small to be di�erentiated bodies by themselves, they are quite spread in proper elements space and do not belong to any of the asteroid dynamical families identi�ed in the outer belt. Therefore, they are likely to be fragments from more than one di�erentiated parent bodies. Nevertheless, at variance with (1459) Magnya, (7472) Kumakiri and (10537) 1991 RY16 evolve very close to the non linear secular resonance de�ned by the combination g0+s0−g5−s7 ' 0, where gi and si represent the frequencies of the perihelion $ and node Ω, respectively (i = 0 for asteroid, i = 5 for Jupiter, i = 7 for Uranus; see Milani & Kneºevi¢, 1992). A 50 My simulation of the orbits of these two asteroids, including gravitational perturbations from the four major planets, indicate that they have quite stable orbits showing a slow circulation of the angle $0+Ω0−$5−Ω7. Although this may be just a coincidence, a dynamical connection between (7472) Kumakiri and (10537) 1991 RY16 cannot be ruled out and should be addressed by more detailed studies. 2We have veri�ed that these di�erences cannot be related to observation/reduction problems, since we do not �nd any di�erences between the spectra of the solar analog stars used in the di�erent nights. � 7 � 4. Conclusions We presented visible spectroscopic observations of two asteroids, (7472) Kumakiri and (10537) 1991 RY16, located in the outer belt. The main goal of our work was to show that these observations are compatible with the V-type taxonomic class. Therefore, these bodies would constitute the second and third basaltic asteroids discovered up to now in that part of the Main Belt. However, the presence of a shallow absorption band in the spectra around 0.65 µm opens some questions about the actual mineralogy of these two asteroids. This band is likely to be related to the presence of Cr3+ cations, and provides evidence for a possible a Cr-rich basaltic surface. The spectroscopic similarities among the two asteroids, together with some shared dy- namical properties, point to the idea of a common origin from the breakup of a di�erentiated parent body in the outer belt. Further studies, including near infrared (NIR) spectroscopic observations, are mandatory to better address these issues. We thank Calar Alto Observatory for allocation of Director's Discretionary Time to this programme. Fruitful discussions with D. Nesvroný are also highly appreciated. Based on observations collected at the Centro Astronómico Hispano Alemán (CAHA) at Calar Alto, operated jointly by the Max- Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC). RD acknowledges �nancial support from the MEC (contract Juan de la Cierva). REFERENCES Alvarez-Candal, A., Du�ard, R., Lazzaro, D., & Michtchenko, T.A. 2006, A&A, 459, 969 Asphaug, E. 1997, Meteor. & Planet. Sci., 32, 965 Binzel, R., Rivkin, A., Stuart, S., et al. 2004, Icarus, 170, 259 Binzel, R.P., Masi, G., & Foglia, S. 2006, Bull. Amer. Astron. Soc., 38, 627 Burbine, T.H., Buchanan, P.C, Binzel, R.P., et al. 2001, Meteor. & Planet. Sci., 36, 761 Bus, S.J. 1999, PhD Thesis, MIT Bus, S.J., & Binzel, R.P. 2002, Icarus, 158, 146 � 8 � Carruba, V., Michtchenko, T.A., Roig, F., et al. 2005, A&A, 441, 819 Carruba, V., Roig, F., Michtchenko, T.A., et al. 2007, A&A, 465, 315 Cellino, A., Bus, S. J., Doressoundiram, A. & Lazzaro, D. in Asteroids III (eds Bottke, W. F. et al.) 632�643 (Univ. Arizona Press, Tucson, 2003) Cloutis, E. A.; Ga�ey, M. J. Journal of Geophysical Research, vol. 96, 1991, p. 22809-22826. Cloutis, E.A. Journal of Geophysical Research (Planets), Volume 107, Issue E6, pp. 6-1. Cruikshank, D.P., Tholen, D.J., Bell, J.F., et al. 1991, Icarus, 89, 1 Delbo, M., Gai, M., Lattanzi, M.G., et al. 2006, Icarus, 181, 618 Du�ard, R., de León, J., Licandro, J., et al. 2006, A&A, 456, 775 Florczak, M., Lazzaro, D., & Du�ard, R. 2002, Icarus, 159, 178 Hardorp, J. 1978, A&A, 63, 383 Ivezi¢, �., Tabachnik, S., Ra�kov, R., et al. 2001, AJ, 122, 2749 Juri¢, M., Ivezi¢, �., Lupton, R.H., et al. 2002, AJ, 124, 1776 Lazzaro, D., Michtchenko, T.A., Carvano, J.M., et al. 2000, Science, 288, 2033 Lazzaro, D., Angeli, C.A., Carvano, J.M., et al. 2004, Icarus, 172, 179 McCord, T.B., Adams, J.B., & Johnson, T.V. 1970, Science, 168, 1445 McFadden, L.A., Ga�ey, M.J., Takeda, H., Jackowski, T.L., Reed, K.L., 1982. Mem. Nat. Inst. Polar. Res. 25, 188-206. McFadden, L., Ga�ey, M.J., & McCord, T. 1985, Science, 229, 160 Michtchenko, T.A., Lazzaro, D., Ferraz-Mello, S., et al. 2002, Icarus, 158, 343 Milani, A., & Kneºevi¢, Z. 1992, Icarus, 98, 211 Moskovitz, N.A., Willman, M., Lawrence, S.J., et al. 2007, LPI Conf., 38, 1663 Nesvorný, D., Roig, F., Gladman, B., et al., Icarus (2007), doi:10.1016/j.icarus.2007.08.034 Roig, F., & Gil-Hutton, R. 2006, Icarus, 183, 411 � 9 � Roig, F., Nesvorný, D., Gil-Hutton, R., & Lazzaro, D., Icarus (2007), doi:10.1016/j.icarus.2007.10.004 Shestopalov, D. I.; McFadden, L. A.; Golubeva, L. F., Icarus, Volume 187, Issue 2, p. 469-481. Sunshine, J. M.; Pieters, C. M. Journal of Geophysical Research, vol. 98, no. E5, p. 9075- 9087. Thomas, P.C., Binzel, R.P., Ga�ey, M.J., et al. 1997, Science, 277, 1492 Vilas, F., & Ga�ey, M.J. 1989, Science, 246, 790 Vilas, F., Larson, S.M., Hatch, E.C., et al. 1993, Icarus, 105, 67 This preprint was prepared with the AAS LATEX macros v5.2. � 10 � Table 1: Observational circumstances for the targets: Universal Time (UT), heliocentric distance (r), geocentric distance (∆), phase angle (φ), visual magnitude (V ), airmass and total exposure time (Texp). Asteroid UT r [AU] ∆ [AU] φ [deg] V [mag] airmass Texp November 14 (7472) Kumakiri 03:25:00 2.920 2.123 13.5 16.6 1.037 3400 sec (10537) 1991 RY16 00:05:08 3.040 2.053 1.8 17.1 1.091 4000 sec December 29 (7472) Kumakiri 21:48:52 2.873 1.901 3.8 15.9 1.094 3600 sec Table 2: Proper elements and sizes of V-type asteroids in the outer belt. For (1459) Magnya, the last column gives the diameter from Delbo et al. (2006). For (7472) Kumakiri and (10537) 1991 RY16, the diameter was computed assuming an albedo of 0.40. Asteroid ap [AU] ep sin Ip D [km] (1459) Magnya 3.14986 0.2183 0.2651 17.0± 1.0 (7472) Kumakiri 3.01033 0.1372 0.1562 8.5 (10537) 1991 RY16 2.84958 0.1023 0.1101 7.3 � 11 � Fig. 1.� Re�ectance spectra of (7472) Kumakiri and (10537) 1991 RY16 (black lines) compared to the spectra of several known V-type asteroids taken from the SMASS and S3OS2 surveys (gray lines). The spectra are normalized to 1 at 0.55 µm and shifted by 0.5 units in re�ectance for clarity. To remove the solar contribution, we have used the solar analog HD 191854 in the November 14 observations and the solar analog HD 28099 in the December 29 observation. � 12 � Fig. 2.� Re�ectance spectra of (7472) Kumakiri and (10537) 1991 RY16 (gray lines) com- pared to the photometric observations of the SDSS-MOC (black lines). The spectra are normalized to 1 at 0.477 µm (i.e. the center of the g band in the SDSS photometric system), and shifted by 0.5 units in re�ectance for clarity. The errors in the SDSS-MOC �uxes are less than 3%. � 13 � (10537) 1991 RY16 -0.15 -0.05 0.54 0.59 0.64 0.69 0.74 Fig. 3.� Re�ectance spectrum of (10537) 1991 RY16 in the 0.55-0.75 µm interval. The spectrum has been recti�ed by subtracting a linear continuum in this interval. From a polynomial �t (thick line), the center of the absorption band is detected at 0.63 µm with a FWHM of 0.1 µm. Introduction Observations Results and discussion Conclusions

0704.0231

Interpolating and sampling sequences in finite Riemann surfaces

arXiv:0704.0231v1 [math.CV] 2 Apr 2007 INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES JOAQUIM ORTEGA-CERDÀ Abstract. We provide a description of the interpolating and sampling sequences on a space of holomorphic functions with a uniform growth restriction defined on finite Riemann surfaces. 1. Introduction and statement of the results Let S be an open finite Riemann surface endowed with the Poincaré (hyperbolic) met- ric. We will study some properties of holomorphic functions in the Riemann surface with uniform growth control. Namely we will deal with the Banach space Aφ(S) of holomorphic functions in S such that ‖f‖ := supS |f |e −φ <∞ where φ is a given subharmonic function that controls the growth of the functions in the space. The fact that φ is subharmonic is a natural assumption on the weight that limits the growth since any other growth control given by a weight ψ, ‖f‖∗ = supS |f |e −ψ can be replaced by an equivalent subharmonic function because φ = sup‖f‖∗≤1 log |f | is a subhar- monic function and Aψ(S) = Aφ(S) with equality of norms, supS |f |e −ψ = supS |f |e We have fixed a metric. It is then natural to restrict the possible weights φ, in a way that the functions in Aφ oscillate in a controlled way when the points are nearby in the Poincaré metric. This is achieved for instance by assuming that φ has bounded Laplacian (the Laplace-Beltrami operator with respect to the hyperbolic measure). That is, if in a local coordinate chart the Poincaré metric is of the form ds2 = e2ν(z)|dz|2, then we assume that ∆φ = 4e−2ν(z) ∂ ∂z∂z̄ satisfies C−1 ≤ ∆φ ≤ C. If we want to deal with other weights then it is possible to introduce a natural metric associated to the weight as it is done in the plane in [MMOC03]. In this work we will only consider the Poincaré metric and bounded Laplacian since it already covers many interesting cases and it is technically simpler. The problems that we will consider are the following: (A) The description of the interpolating sequences for Aφ(S): i.e. the sequences Λ ⊂ S such that it is always possible to find an f ∈ Aφ(S) such that f(λ) = vλ for all λ ∈ Λ whenever the data {vλ}Λ, satisfies the compatibility condition supΛ |vλ|e −φ(λ) < +∞ (B) The description of sampling sets for Aφ(S): i.e. the sets E ⊂ S such that there is a constant C > 0 that satisfies |f |e−φ ≤ C sup |f |e−φ, ∀f ∈ Aφ(S). Date: Working draft: July 20, 2021. Supported by DGICYT grant MTM2005-08984-C02-02 and the CIRIT grant 2005SGR00611. http://arxiv.org/abs/0704.0231v1 2 JOAQUIM ORTEGA-CERDÀ In the solution of these problems the Poincaré distance and the potential theory in the surface play a key role. This has already been observed by A. Schuster and D. Varolin in [SV04], where they provide sufficient conditions for a sequence to be interpolating/sampling for functions in a slightly different context where the weighted uniform control of the growth of the functions is replaced by a weighted L2 control. Their condition basically coincides with the description that we reach so our work can be considered as the counterpart of their theorems, although we will give a different proof of their results as well. We will rely on the well-known case of the disk and some simplifying properties of finite Riemann surfaces. Their method of proof looks more promising if one wants to extend the result to Riemann surfaces with more complicated topology. When the surface is a disk, which will be our model situation, the corresponding problems have been solved in [BOC95], [OCS98] and in a different way in [Sei98]. Of course, the more basic problem of describing the interpolating sequences for bounded holomorphic functions in finite Riemann surfaces (in our notation φ ≡ 0), has been known for a long time, see [Sto65]). We introduce now some definitions that will be needed to state our results. For any point z ∈ S and any r > 0 we denote by D(z, r) the domain in the surface S that consits of points at hyperbolic distance from z less than r. They are topological disks if the center z is outside a big compact of S, or if r is small enough, as we will see in Section 2. A sequence Λ of points in S is hyperbolically separated if there is an ε > 0 such that the domains {D(λ, ε)}λ∈Λ are pairwise disjoint. Let gr(z, w) be the Green function associated to the surface D(z, r) with pole at the “center” z and g(z, w) = g∞(z, w) be the Green function associated to the surface S. We define the densities D+φ (Λ) := lim sup 1/2<d(z,λ)<r gr(z, λ) D(z,r) gr(z, w)i∂∂̄φ(w) D−φ (Λ) := lim inf 1/2<d(z,λ)<r gr(z, λ) D(z,r) gr(z, w)i∂∂̄φ(w) The main result is Theorem 1. Let S be a finite Riemann surface and let φ be a subharmonic function with bounded Laplacian. (A) A sequence Λ ⊂ S is an interpolating sequence for Aφ(S) if and only if it is hyper- bolically separated and D+φ (Λ) < 1. (B) A set E ⊂ S is a sampling set for Aφ(S) if and only if it contains an hyperbolically separated sequence Λ ⊂ E such that D−φ (Λ) > 1. INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 3 In Section 2 we will prove some key properties of finite Riemann surfaces. In particular we need to study the behavior of the hyperbolic metric as we approach the boundary of the surface. We will also prove some weighted uniform estimates for the inhomogeneous Cauchy-Riemann equation in the surface, Theorem 6, that has an interest by itself. In the next section, we use the tools and Lemmas proved in Section 2 to reduce the interpolating and sampling problem in S to a problem near the boundary that can be reduced to the known case of the disk. Finally in Section 4 we show how our results can be extended to other Banach spaces of holomorphic functions where the uniform growth is replaced by weighted Lp spaces. A final word on notation. By f . g we mean that there is a constant C independent of the relevant variables such that f ≤ Cg and by f ≃ g we mean that f . g and g . f . 2. Basic properties of finite Riemann surfaces We start by the definition and then we collect some properties of S that follow from the restrictions that we are assuming on the topology of S. Definition 2. A finite Riemann Surface is the interior of a smooth bordered compact Riemann surface. Our surface is an open Riemann surface and it is in fact an open subset of a compact surface (the double, see [SS54]). See Figure 1 for a typical representation. Observe that the genus is finite and the border of the surface consists of a finite number of smooth closed Jordan curves. In most of what follows the particular case of a smooth finitely connected open set in C has all the difficulties of the general case. The following claim follows from instance from [Sch78, Prop 7.1-7.4] Lemma 3. For any (0, 1)-form ω there is a solution u to the inhomogeneous Cauchy- Riemann equation ∂̄u = ω. Moreover since S has an essential extension to a compact Riemann surface if the data is a smooth form with compact support K in S then there is a bounded linear solution u = T [w] with the bound |u| ≤ CK〈ω〉. In this statement and in the following 〈ω〉 is the Poincaré length of the (0, 1)-form ω. In the disk we have Blaschke factors that are very convenient to divide out zeros of holomorphic functions without changing essentially the norm. The analogous functions that provide us with the same property in the case of finite Riemann surfaces are given by the next proposition: Proposition 4. There is a constant C = C(S) > 0 such that for any point z ∈ S there is a function hz ∈ H(S) with | log |hz(w)| − g(z, w)| < C. In particular hz(w) is a bounded holomorphic function that vanishes only on the point z and for any ε > 0 K > |hz(w)| > C(ε) if d(z, w) > ε. 4 JOAQUIM ORTEGA-CERDÀ Figure 1. A finite Riemann surface with three funnels Proof. The obstruction for an harmonic function u to have an harmonic conjugate is that for a set of generators {γi} i=1 of the homology we have ∗du = 0, i = 1, . . . , m. If we want u = log |f | for an f ∈ H(S), we just need that ∗du ∈ Z. Being a finite Riemann surface there are {hj} j=1 functions in the algebra of S without zeros such that ∗d log |hj| = δij , see [Wer64, Lemma 1]. Thus the function v(z) = u(z)− log |hi(z)| is the logarithm of an holomorphic function log |f | = v. Therefore there is a constant C such that any harmonic function u in S admits an holomorphic function f with |u−log |f || < C. Take a point z ∈ S and any holomorphic function kz ∈ H(S) that vanishes only on z. Then g(z, w)− log |kz(w)| is harmonic in S and therefore there is a holomorphic function fz such that |g(z, w)− kz(w)− log |fz|| < C. Thus we may define hz(w) = fz(w)kz(w) and it has the estimate |g(z, w)− log |hz|| < C. The estimate |g(z, w)| > C(ε) when d(z, w) > ε holds in finite Riemann surfaces, see for instance [Dil95, Theorem 5.5]. � 2.1. The hyperbolic metric in a finite Riemann surface. The open ends of the Riemann surface can be parametrized as follows: The border of the Riemann surface S is a finite union of smooth closed curves γ̃i, i = 1, . . . , n. Near each γ̃i there is a closed geodesic γi that is homotopic to γ̃i. The subdomain of S bounded by γi and γ̃i is denoted a “funnel” following the terminology of [DPRS87] and [Dil01]. We need to be more precise about the hyperbolic metric in the funnel. There are nice coordinates in the funnel that provide good estimates. These are given by the collar theorem. Let D be the universal holomorphic cover of S and let Tγ ∈ Aut(D) be the deck transformation corresponding to the closed loop γ. Consider the surface Y = D/{T nγ }n∈Z. This an annulus since π1(Y ) = Z. If we quotient it by the rest of the deck transformations of the universal cover we get an holomorphic covering map πγ from Y → S which is a local isometry (in Y and S we consider the Poincaré metric inherited from D). In fact Y = {e−R < |z| < eR}, where R = π2/Length(γ), and πγ maps the unit circle isometrically to γ. Moreover πγ is an isometric injection of the outer part of the annulus {1 < |z| < e INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 5 ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� ������������� |z|=eR |z|=e −R |z|=1 �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� �������� Figure 2. Standard coordinates on the funnel onto the funnel. These will be called the standard coordinates of the funnel. See [Dil01] and [Bus92] for details. The Poincare metric in the the funnel is explicit in the standard coordinates and it is comparable to the hyperbolic metric on the disk in the coordinate disk |z| < eR when restricted to |z| > 1. We denote by Ai, i = 1, . . . , n the funnels of S bounded by γi and γ̃i. 2.2. The inhomogeneous Cauchy-Riemann equation on the surface. We want to solve the inhomogeneous Cauchy-Riemann equation on S with weighted uniform estimates. In order to get good estimates it is useful to find functions f ∈ H(S) with precise size control, i.e., |f | ≃ eφ outside a neighborhood of the zero set of f . With this function we can later modify an integral formula to get a bounded solution to the ∂̄-equation when the data has compact support. The following Lemma provides such a function that in other context has been termed a “multiplier”: Lemma 5. Let S be a finite Riemann surface and let φ be a subharmonic function with bounded Laplacian. Then there is a function f with hyperbolically separated zero set Σ such that |f | ≃ eφ whenever d(z,Σ) > ε. Moreover if we fix any compact K in S it is possible to find f with the above properties and without zeros in K. Proof. In any of the funnels Ai we transfer the subharmonic weight φ to the standard coordinate chart 1 < |z| < eRi . We define a weight φi on the disk |z| < e Ri in such a way that φi has bounded invariant Laplacian and moreover |φ − φi| < C on the region 1 < |z| < eRi . One way to do so is the following: we assume from the very beginning that φ is smooth (this is no restriction since otherwise it can be approximated by a smooth function). Define (2) φi(z) = φ(z)χ(z) +Mi‖z‖ where χ is a cutoff function such that χ ≡ 1 in eRi/2 < |z| < eRi , χ ≡ 0 in |z| < 1 and Mi is taken big enough such that φi is subharmonic and the invariant Laplacian of φi is bounded above and below. 6 JOAQUIM ORTEGA-CERDÀ We are under the hypothesis of the result from [Sei95] that states that there is an holomorphic function in the disk fi with separated zero set Z(fi) (in the hyperbolic metric of the disk) such that |fi| ≃ e φi whenever d(z, Z(fi)) > ε. Since the hyperbolic metric of the disk is comparable to the hyperbolic metric in the funnel, we have found a function fi ∈ H(Ai) with separated zero set such that |fi(z)| ≃ e φ(z) if d(z, Z(fi)) > ε. Moreover dividing out fi by a finite Blaschke product we can assume that fi is zero free in any prefixed compact of the disk. We consider the “core” of S to be S \ Ãi, where Ãi are the outer part of the funnels mapped by eSi < |z| < eRi . The values of the Si are taken so big as to make sure that the compact K in the hypothesis of the Lemma is contained in the core of S. We adjust the fi i = 1, . . . , n as mentioned before to make sure that they are zero free in the inner part of the funnels 1 < |z| < eSi . We finally define f0 ≡ 1 in the core of S. To patch the different fi together we will need to solve a Cousin II problem with bounds. Our data is fi defined on the inner parts of the funnels mapped by 1 < |z| < e Si. The data are bounded above and below in the inner parts of the funnels (because φ is bounded above and below in any compact of S and fi have no zeros there). We want to find functions gi ∈ H(Ai) and g0 holomorphic on the core of S such that fi = g0/gi in the inner part of the funnel. If moreover gi and g0 are bounded (above and below) then the function f defined as figi in each of the funnels Ai and g0 on the core of S is holomorphic on S and has the desired growth properties. To find the functions gi observe that since the intersection of the funnel Ai with the core of S strictly separates the outer part of the funnel from the inner part of the core we can reduce the Cousin II problem to solving a ∂̄-equation with bounded estimates of the solution on S when the data is bounded and with compact support (the support is in the inner part of the funnels). This can be achieved by Lemma 3. � With this function we can then obtain the following result which is interesting by itself: Theorem 6. Let S be a finite Riemann surface and let φ be a subharmonic function with a bounded Laplacian. There is a constant C > 0 such that for any (0, 1)-form ω on S there is a solution u to the inhomogeneous Cauchy-Riemann equation ∂̄u = ω in S with the estimate |u(z)|e−φ(z) ≤ C sup 〈ω(z)〉e−φ(z), whenever the right hand is finite. Recall that the notation 〈ω(z)〉 means the hyperbolic norm of ω at the point z. Proof. Let wi be the form w restricted to the funnel Ai. We take a standard coordinate chart and we may think of wi as a (0, 1)-form defined on the disk |z| < e Ri and with support in 1 < |z| < eRi. Consider as in the proof of Lemma 5 a subharmonic function φi in the disk with bounded laplacian and such that |φ− φi| < C if 1 < |z| < e By the results in [OC02, Thm 2] there is a solution ui to the problem ∂̄ui = wi in the disk |z| < eRi with the estimate |z|<eRi |ui|e −φi ≤ Ci sup 1<|z|<eRi 〈wi〉e INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 7 Observe that the hyperbolic metric of the disk and of the surface S in the funnel are equivalent. We consider ũi = uiχi, where χi is a cutoff function with support in 1 < |z| < eRi and such that χi ≡ 1 if |z| > e Ri/2. The function ũi is extended by 0 to the remaining of S and it has the estimate supS |ũi|e −φ ≤ Ci supS〈w〉e −φ. Now ∂̄ũi coincides with w on the outer part of the funnel Ai. Thus the (0, 1)-form wk = w − i ∂̄ũi has compact support in S and it satisfies supS〈wk〉e −φ ≤ supS〈w〉e −φ. The desired solution is then ũi + v, where v is such that ∂̄v = wk. We must then solve ∂̄v = wk with weighted uniform estimates but with the advantage that wk has compact support K. Let T (ωk) be a solution operator for ∂ū = ωk. We take the operator T given by Lemma 3 the estimate supS |T [wk](z)| ≤ CK supK〈wk〉 holds. Take f with |f | ≃ e φ and without zeros in K as given in Lemma 5. Then we define R as (3) R[ωk](z) = f(z)T [ωk/f ](z), It solves ∂̄R[ωk] = ωk with the estimate |R[ωk]|e −φ ≤ CK sup 〈ωk〉e The solution is thus v = R[wk]. � 3. The main results Proposition 7. A separated sequence Λ ⊂ S is interpolating for Aφ(S) if and only if the sequences Λi = Λ ∩Ai are interpolating in Aφ(Ai). Proof. We only need to prove that we can pass from the local to the global interpolation property. We split the proof in two steps (1) From a funnel Ai to global S: We need to prove that there are finite sets Fi ⊂ Λi such that ∪ni=1(Λi \ Fi) is interpolating globally. (2) Filling up the remainder. We shall prove that by adding a finite number of points to an interpolating sequence we still get an interpolating sequence. Thus Λ is interpolating if (Λ1 \ F1) ∪ · · · ∪ (Λn \ Fn) is interpolating. Let γ̃ be one of the closed curves on the boundary. Take a funnel A with outer end curve in γ̃ and inner end curve in γ. The constant of interpolation in the funnel A is K > 0. Take a cutoff function χε with support in the funnel such that 〈∂̄χε〉 < ε/(KC) (where C is the constant in Theorem 6), the support is in a thick annulus of hyperbolic thickness M = M(ε,K, C). We consider a smaller funnel where χε ≡ 1. The sequence Λ in this smaller funnel has still at most interpolation constant K. We can interpolate arbitrary values on Λ being small near the inner curve γ of A in the following way. Take some values vλ with norm one. Take a function in the funnel f with norm at most K that solves the interpolation problem. We are going to approximate it by a function in A that is small near γ. Cut it off by χε and correct via the following inhomogeneous Cauchy-Riemann equation: 8 JOAQUIM ORTEGA-CERDÀ ∂̄u = f∂̄χ The function h = u − fχ is holomorphic. By using Theorem 6 on it is possible to solve the equation with a solution u such that sup |u|e−φ ≤ ε. The function h does not solve the problem directly but it almost does. We reiterate the procedure (interpolating the error vλ − h(λ) and with a convergent series we get finally a function g such that h(λ) = vλ, supA |h|e −φ ≤ 2 and moreover in the inner half of the funnel that we denote by Ã, supà |h|e −φ ≤ ε. Now it is easier to make it global. Take a new cutoff function χ with support in the funnel A and that is one on the outer part of (i.e. A \ Ã. Then we need to solve ∂̄u = h∂̄χ, with good global estimates in S. These are given by Theorem 6. We have solved the interpolation problem when the sequence lies in the funnels. For the general situation we only need to add a finite number of points. The existence of “Blaschke”-type factors hλ(z) provided by Theorem 4 shows that Λ ∪ λ is interpolating if Λ is interpolating (it is immediate to build functions in the space such that f |Λ ≡ 0 and f(λ) 6= 0). � For the sampling part we need the following definition Definition 8. Given the pair (S, φ) of a finite Riemann surface and a subharmonic function defined on it, we associate to it the pairs: (Di, φi)i=1,...n of disks Di and subharmonic functions φi defined on the disks as follow: If Ai = {1 < |z| < e Ri}, i = 1, . . . , n are the standard charts of the funnels of S we define Di = {|z| < e Ri} and φi is any subharmonic function in Di such that |φi − φ| < C in the region 1 < |z| < e Ri, ∆φi = ∆φ in e Ri/2 < |z| < eRi and ∆φi ≃ 1 in |z| < e Ri/2. They can be defined similarly as in (2), but to make sure ∆φi = ∆φ we may take instead φi(z) = φ(z)χ(z) +Miψ(z), where ψ is any bounded subharmonic function in Di such that ∆ψ(z) = 1 if |z| < e and 0 elsewhere. The funnels Ai can be considered funnels of S and they are subdomains of Di too. We will exploit this double nature in the following theorem Theorem 9. Let S be a finite Riemann surface and let φ be a subharmonic function with bounded Laplacian. A separated sequence Λ is sampling for Aφ(S) if and only if all the sequences in the funnels Λi = Λi ∩ Ai ⊂ Di are sampling sequences for Aφi(Di), where (Di, φi) are the associated pairs to S given by Definition 8. Thus this Theorem and Proposition 7 show that the properties of sampling and inter- polation only depend on the behavior of the sequence and the weight near the boundary pieces. To prove Theorem 9 we need some previous results INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 9 Lemma 10. Let S be a finite Riemann surface and let φ be a subharmonic function with bounded Laplacian. A sequence Λ ⊂ S is a uniqueness sequence for Aφ(S) if and only if all the sequences in the funnels Λi = Λi ∩ Ai ⊂ Di are uniqueness sequences for Aφi(Di), where (Di, φi) are the associated pairs to S given by Definition 8. Proof. It is easier to deal by negation. Let Λ be contained in the zero set of a function f ∈ Aφ(S). Therefore Λi is in the zero set of f ∈ Aφ(Ai). We divide by a finite number of zeros Ei and we obtain a new function g ∈ Aφ(Ai) without zeros in 1 < |z| ≤ e Ri/2 and such that Λi \Ei ⊂ Z(g). Take the disk Di and consider the cover by two open sets |z| > 1 and |z| < eRi/2. On the first set we have the function g and on the second the function 1. The quotient is bounded above and below in the intersection of the sets. This defines a bounded Cousin II in the disk Di problem that can be solved with bounded data. We get a new function h ∈ Aφi(Di) that vanishes in Z(g). We can now add the finite number of zeros Ei without harm. The reciprocal implication follows with the same argument. � The next result is inspired by a result of Beurling ([Beu89, pp. 351–365]) that relates the property of sampling sequence to that of uniqueness for all weak limits of the sequence. In the context of the Bernstein space (in the original work by Beurling) the space was fixed (it was C, the space of functions was fixed, the Bernstein class, and he considered translates and limits of it of the sampling sequence). Here we need to move and take limits of the sequence (by zooming on appropriate portions of it) but we also need to change the support space (portions of S near the funnel that look like the unit disk) and we will also move the space of functions by changing the weights. We need some definitions: Definition 11. We consider triplets (Dn, φn,Λn) where Dn are disks Dn = D(0, rn) ⊂ D, φn are subharmonic functions defined in a neighborhood of Dn and Λn is a finite collection of points in Dn. We say that (Dn, φn,Λn) converges weakly to (D, φ,Λ) (where D is the unit disk, φ a subharmonic function in D and Λ a discrete sequence in D) if the following conditions are fullfilled: • The domains Dn tend to D, i.e.: rn → 1, • The weights φn tend to the weight φ in the sense that ∆φn as measures converges weakly to ∆φ. • The sequences Λn converge weakly to Λ, i.e, the measure δλ converges weakly to the measure λ∈Λ δλ. Let us fix a point p ∈ S. If a sequence of points zn ∈ S goes to ∞, i.e. d(zn, p) → ∞, from a point n0 on it will eventually belong to the union of the funnels A1 ∪ · · · ∪ An. If we take the set of points Dn = {z ∈ S; d(z, zn) < d(zn, p)/2} then Dn is an hyperbolic disk contained in the funnels if n is big enough. In each of the Dn we consider the function φn = φ|Dn and Λn = Λ ∩ Dn. Thus for any sequence of points zn with d(p, zn) → ∞ we build a triplet (Dn, φn,Λn) for n big enough. Definition 12. Let W (S, φ,Λ) be the set of all triplets (D, φ∗,Λ∗) which are weak limits of triplets (Dn, φn,Λn) associated to any sequence zn such that d(p, zn) → ∞. The theorem of Beurling on our context is 10 JOAQUIM ORTEGA-CERDÀ Theorem 13. Let S be Riemann surface of finite type and let φ be a subharmonic function with bounded Laplacian. A separated sequence Λ is sampling for Aφ(S) if and only if • The sequence Λ is a uniqueness set for Aφ(S) • For any triplet (D, φ∗,Λ∗) ∈ W (S, φ,Λ), the sequence Λ∗ is a uniqueness set for Aφ∗(D). Proof. Let us prove that the uniqueness conditions imply that Λ is a sampling sequence. If it were not, there would be a sequence of functions fn ∈ Aφ(S) such that supΛ |fn|e −φ ≤ 1/n and supS |fn|e −φ = 1. Take a sequence of points zn with |fn(zn)|e −φ ≥ 1/2. If zn are bounded we can take a subsequence of points that we still denote zn convergent to z ∗ ∈ S and by a normal family argument there is a partial of fn convergent to f ∈ Aφ, such that f |Λ ≡ 0, f(z ∗) 6= 0 and this is not possible. Thus zn must be unbounded. Then we take the triplets (Dn, φn,Λn) associated to zn and Dn → D because zn → ∞ and the hyperbolic radius of Dn is d(zn, p)/2. Since φn has bounded Laplacian, the mass of ∆φn restricted to any compact K in D is bounded, thus we can take a subsequence that converges weakly to a positive measure µ in D which satisfies (1−|z|)2µ ≃ 1 because all the mesures ∆φn satisfy this inequalities with uniform constants. Let φ be such that ∆φ∗ = µ. Since Λn = Dn ∩ Λ are all separated with uniform bound, there is a weak limit Λ∗. The functions fn in the disks can be modified by a factor egn in such a way that hn = fne gn satisfies hn(0) = 1 and |hn| ≤ e φn+Re(gn), if n big enough and supΛn |hn|e −φn+Re(gn) ≤ 1/n. We can add an harmonic function v to φ∗ in such a way that φn+Re(gn) → v+ φ ∗ uniformly on compact sets. Thus hn has a partial convergent to h ∈ Aφ∗ , h(0) = 1 and h|Λ∗ ≡ 0 which was not possible by assumption. In the other direction, we assume that Λ is a sampling sequence forAφ(S), and (D, φ ∗,Λ∗) ∈ W (S, φ,Λ). We want to prove that any f ∈ A∗φ(D) that vanishes in Λ ∗ is identically 0. Take a sequence of points zn that escapes to infinity and (Dn, φn,Λn) the associated triple that converges weakly to (D, φ∗,Λ∗). As φn → φ ∗ and Λn → Λ ∗ uniformly on compact sets we can take a sequence of radii sn such that d(Λ ∩ D(zn, sn),Λ ∗ ∩ D(0, sn)) < 1/n, D(zn, sn) ⊂ D(zn, rn) and |φn − φ ∗| ≤ 1/n. If f vanishes in Λ∗ that means that f is very small in D(zn, sn)∩Λ. Assume that f(0) = 1. Take a cutoff function χn such that χn ≡ 0 outside D(zn, sn), χ(zn) = 1, and 〈dχ〉 < εn. Define gn = fχn − un, where ∂̄u = f∂̄χn is the solution estimates by Theorem 6. Clearly gn is small in all points of Σ and it has at least norm 1. Thus we are contradicting the fact that Λ is sampling. Observe that one particular instance of finite Riemann surface, where we can apply the result are the disks Di associated to the funnels with the metric φi. The final piece for the proof of Theorem 9 is then Lemma 14. If S is a finite Riemann surface, φ a subharmonic function with bounded Laplacian and Λ is a uniformly separated sequence, then all possible weak limits coincide with the weak limits of the disks associated to the surface, i.e, W (S, φ,Λ) =W (D1, φ1,Λ1) ∪ · · · ∪W (Dn, φn,Λn). INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 11 Proof. The proof amounts to the observation that the metric in Di converges uniformly to the metric in S as z → ∂Di, and in the definition of weak limits we only consider uniform convergence over compacts. � Theorem 9 follows now immediately from Theorem 13 and Lemmas 10 and 14. � Now Theorem 9 and Theorem 7 show that the property of being a sampling/interpolating sequence are determined by the behavior near the boundary, more precisely in the associ- ated disks. In these disks there is a precise description of the interpolating and sampling sequences (see [BOC95] and [OCS98]) that can be transported to the surface. If we rewrite it we get the density conditions of Theorem 1, but the disks are not hyperbolic disks on the surface, they correspond to hyperbolic disks in disks Di, but since the condition is only relevant near the boundary, then the disks in both metrics look more an more sim- ilar. Moreover the difference between the corresponding Green functions converge to 0 uniformly as we go to the boundary. Finally, as the sequence is uniformly discrete and the Laplacian of the weight is bounded above and bellow, the small difference is absorbed by the fact that the inequalities are strict and this proves Theorem 1. In fact it is possible to replace in the definition of the density, (1) the Green function gr of D(z, r) by the Green function g of S, because as before supw∈D(z,r) |gr(z, w)− g(z, w)| → 0 as z approaches the boundary. 4. Some Lp-variants We have considered up to now pointwise growth restrictions. It is possible to obtain from our Theorem other results in different Banach spaces of holomorphic functions. Consider for instance the weighted Bergman spaces φ(S) = {f ∈ H(S); |f |pe−φ dA < +∞}, where dA is s the hyperbolic area measure in S and p ∈ [1,∞). The natural problem in this context is the following: Definition 15. Let S be a finite Riemann surface, and let φ be a subharmonic function with bounded Laplacian bigger than one, i.e., 1 + ε < ∆φ < M . • A sequence Λ ⊂ S is interpolating for A φ(S) if for any values vλ such that pe−φ(λ) <∞ there is a function f ∈ A φ(S) such that f(λ) = vλ. The spaces A φ can be empty if we only ask φ to be with positive bounded Laplacian. It is then natural to require that the Laplacian is strictly bigger than one so that the Laplacian plus the curvature of the metric in the manifold is strictly positive and there are functions in the space (consider the case of the disk S = D for instance). Let φ0 be a subharmonic function in S such that ∆φ0 = 1. The corresponding theorem will be 12 JOAQUIM ORTEGA-CERDÀ Theorem 16. Let S be a finite Riemann surface, and let φ be a subharmonic function with bounded Laplacian strictly bigger than one. Let p ∈ [1,+∞) and Λ be a separated sequence. • The sequence Λ is interpolating for A φ(S) if and only if D (φ−φ0) (Λ) < 1/p. In the case of the unit disk dA(z) = (1 − |z|)−2 this description is well-known, see for instance [Sei98, Thm 2,3]. Proof. The proof of the theorem is the same mutatis-mutandi as in the L∞ setting. The basic tool that allows us to glue the pieces together is the next theorem which is the generalization of Theorem 6 and it is proved in the same way: Theorem 17. Let S be a finite Riemann surface, let φ be a subharmonic function with a bounded Laplacian strictly bigger than one and let p ∈ [1,∞). There is a constant C = C(p, S) > 0 such that for any (0, 1)-form ω on S there is a solution u to the inhomogeneous Cauchy-Riemann equation ∂̄u = ω in S with the estimate |u(z)|pe−φ(z)dA(z) ≤ C 〈ω(z)〉pe−φ(z)dA(z), whenever the right hand is finite. The proof of this result is again the same as in Theorem 6. We can separately solve the C-R equation in each funnel using Theorem 2 from [OC02]. We glue them together with a C-R equation with data that has compact support that can be solved with the operator (3). � References [BOC95] Bo Berndtsson and Joaquim Ortega Cerdà, On interpolation and sampling in Hilbert spaces of analytic functions, J. Reine Angew. Math. 464 (1995), 109–128. MR 96g:30070 [Beu89] Arne Beurling, The collected works of Arne Beurling. Vol. 1, Contemporary Mathematicians, Birkhäuser Boston Inc., Boston, MA, 1989, Complex analysis, Edited by L. Carleson, P. Malliavin, J. Neuberger and J. Wermer. MR 92k:01046a [Bus92] Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston Inc., Boston, MA, 1992. MR 93g:58149 [Dil95] Jeffrey Diller, A canonical ∂ problem for bordered Riemann surfaces, Indiana Univ. Math. J. 44 (1995), no. 3, 747–763. MR 96m:30063 [Dil01] , Green’s functions, electric networks, and the geometry of hyperbolic Riemann surfaces, Illinois J. Math. 45 (2001), no. 2, 453–485. MR 2003j:30064 [DPRS87] Jozef Dodziuk, Thea Pignataro, Burton Randol, and Dennis Sullivan, Estimating small eigen- values of Riemann surfaces, The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985), Contemp. Math., vol. 64, Amer. Math. Soc., Providence, RI, 1987, pp. 93–121. MR 88h:58119 [MMOC03] N. Marco, X. Massaneda, and J. Ortega-Cerdà, Interpolating and sampling sequences for entire functions, Geom. Funct. Anal. 13 (2003), no. 4, 862–914. MR 2004j:30073 [OC02] Joaquim Ortega-Cerdà, Multipliers and weighted ∂-estimates, Rev. Mat. Iberoamericana 18 (2002), no. 2, 355–377. MR 2003j:32046 [OCS98] Joaquim Ortega-Cerdà and Kristian Seip, Beurling-type density theorems for weighted Lp spaces of entire functions, J. Anal. Math. 75 (1998), 247–266. MR 2000k:46030 INTERPOLATING AND SAMPLING SEQUENCES IN FINITE RIEMANN SURFACES 13 [Sch78] Stephen Scheinberg, Uniform approximation by functions analytic on a Riemann surface, Ann. of Math. (2) 108 (1978), no. 2, 257–298. MR 58 #17111 [Sei95] Kristian Seip, On Korenblum’s density condition for the zero sequences of A−α, J. Anal. Math. 67 (1995), 307–322. MR 97c:30044 [Sei98] , Developments from nonharmonic Fourier series, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), no. Extra Vol. II, 1998, pp. 713–722 (electronic). MR 99h:42023 [SS54] Menahem Schiffer and Donald C. Spencer, Functionals of finite Riemann surfaces, Princeton University Press, Princeton, N. J., 1954. MR 16,461g [Sto65] E. L. Stout, Bounded holomorphic functions on finite Riemann surfaces, Trans. Amer. Math. Soc. 120 (1965), 255–285. MR 32#1358 [SV04] A. Schuster and D. Varolin, Interpolation and Sampling for generalized Bergman spaces on finite Riemann Surfaces., Preprint, 2004. [Wer64] John Wermer, Analytic disks in maximal ideal spaces, Amer. J. Math. 86 (1964), 161–170. MR 28 #5355 Dept. Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08071 Barcelona, Spain E-mail address : [email protected]

0704.0232

New algebraic aspects of perturbative and non-perturbative Quantum Field Theory

New algebraic aspects of perturbative and non-perturbative Quantum Field Theory Christoph Bergbauer1,4 and Dirk Kreimer2,3 1Freie Universität Berlin, Institut für Mathematik II Arnimallee 3, 14195 Berlin, Germany 2CNRS at Institut des Hautes Etudes Scientifiques 35 route de Chartres, 91440 Bures-sur-Yvette, France 3Boston University, Center for Mathematical Physics 111 Cummington Street, Boston, MA 02215, USA 4Erwin-Schrödinger-Institut Boltzmanngasse 9, 1090 Wien, Austria [email protected], [email protected] April 1, 2007 Abstract In this expository article we review recent advances in our understand- ing of the combinatorial and algebraic structure of perturbation theory in terms of Feynman graphs, and Dyson-Schwinger equations. Starting from Lie and Hopf algebras of Feynman graphs, perturbative renormalization is rephrased algebraically. The Hochschild cohomology of these Hopf al- gebras leads the way to Slavnov-Taylor identities and Dyson-Schwinger equations. We discuss recent progress in solving simple Dyson-Schwinger equations in the high energy sector using the algebraic machinery. Finally there is a short account on a relation to algebraic geometry and number theory: understanding Feynman integrals as periods of mixed (Tate) mo- tives. 1 Introduction As elements of perturbative expansions of Quantum field theories, Feynman graphs have been playing and still play a key role both for our conceptual understanding and for state-of-the-art computations in particle physics. This http://arxiv.org/abs/0704.0232v2 article is concerned with several aspects of Feynman graphs: First, the com- binatorics of perturbative renormalization give rise to Hopf algebras of rooted trees and Feynman graphs. These Hopf algebras come with a cohomology the- ory and structure maps that help understand important physical notions, such as locality of counterterms, the beta function, certain symmetries, or Dyson- Schwinger equations from a unified mathematical point of view. This point of view is about self-similarity and recursion. The atomic (primitive) elements in this combinatorial approach are divergent graphs without subdivergences. They must be studied by additional means, be it analytic methods or algebraic geom- etry and number theory, and this is a significantly more difficult task. However, the Hopf algebra structure of graphs for renormalization is in this sense a sub- structure of the Hopf algebra structure underlying the relative cohomology of graph hypersurfaces needed to understand the number-theoretic properties of field theory amplitudes [6, 5]. 2 Lie and Hopf algebras of Feynman graphs Given a Feynman graph Γ with several divergent subgraphs, the Bogoliubov recursion and Zimmermann’s forest formula tell how Γ must be renormalized in order to obtain a finite conceptual result, using only local counterterms. This has an analytic (regularization/extension of distributions) and a combinatorial aspect. The basic combinatorial question of perturbative renormalization is to find a good model which describes disentanglement of graphs into subdiver- gent pieces, or dually insertion of divergent pieces one into each other, from the point of view of renormalized Feynman rules. It has been known now for several years that commutative Hopf algebras and (dual) Lie algebras provide such a framework [26, 14, 15] with many ramifications in pure mathematics. From the physical side, it is important to know that, for example, recovering aspects of gauge/BRST symmetry [39, 37, 30, 38] and the transition to nonperturbative equations of motion [12, 28, 29, 36, 3, 35, 32, 34, 4] are conveniently possible in this framework, as will be discussed in subsequent sections. In order to introduce these Lie and Hopf algebras, let us now fix a renormaliz- able quantum field theory (in the sense of perturbation theory), given by a local Lagrangian. A convenient first example is massless φ3 theory in 6 dimensions. We look at its perturbative expansion in terms of 1PI Feynman graphs. Each 1PI graph Γ comes with two integers, |Γ| = rankH1(Γ), its number of loops, and sdd(Γ), its superficial degree of divergence. As usual, vacuum and tadpole graphs need not be considered, and the only remaining superficial divergent graphs have exactly two or three external edges, a feature of renormalizability. Graphs without subdivergences are called primitive. Here are two examples. Both are superficially divergent as they have three external edges. The first one has two subdivergences, the second one is primitive. Note that there are infinitely many primitive graphs with three external edges. In particular, for every n ∈ N one finds a primitive Γ such that |Γ| = n. Let now L be the Q-vector space generated by all the superficially divergent (sdd ≥ 0) 1PI graphs of our theory, graded by the number of loops | · |. There is an operation on L given by insertion of graphs into each other: Let γ1, γ2 be two generators of L. Then γ1 ⋆ γ2 := n(γ1, γ2,Γ) where n(γ1, γ2,Γ) is the number of times that γ1 shows up as a subgraph of Γ and Γ/γ1 ∼= γ2. Here are two examples: ⋆ = + + ⋆ = 2 This definition is extended bilinearly onto all of L. Note that ⋆ respects the grading as |γ1 ⋆ γ2| = |γ1| + |γ2|. The operation ⋆ is not in general associative. Indeed, it is pre-Lie [14, 17]: (γ1 ⋆ γ2) ⋆ γ3 − γ1 ⋆ (γ2 ⋆ γ3) = (γ1 ⋆ γ3) ⋆ γ2 − γ1 ⋆ (γ3 ⋆ γ3). (1) To see that (1) holds observe that on both sides nested insertions cancel. What remains are disjoint insertions of γ2 and γ3 into γ1 which do obviously not depend on the order of γ2 and γ3. One defines a Lie bracket on L : [γ1, γ2] := γ1 ⋆ γ2 − γ2 ⋆ γ1. The Jacobi identity for [·, ·] is satisfied as a consequence of the pre-Lie property (1) of ⋆. This makes L a graded Lie algebra. The bracket is defined by mutual insertions of graphs. As usual, U(L), the universal envelopping algebra of L is a cocommutative Hopf algebra. Its graded dual, in the sense of Milnor-Moore, is therefore a commutative Hopf algebra H. As an algebra, H is free commutative, generated by the vector space L and an adjoined unit I. By duality, one expects the coproduct ofH to disentangle its argument into subdivergent pieces. Indeed, one finds ∆(Γ) = I⊗ Γ + Γ⊗ I+ γ ⊗ Γ/γ. (2) The relation γ ( Γ refers to disjoint unions γ of 1PI superficially divergent subgraphs of Γ.Disjoint unions of graphs are in turn identified with their product in H. For example, = I⊗ + ⊗ I+ 2 ⊗ . The coproduct respects the grading by the loop number, as does the product (by definition). Therefore H = n=0 Hn is a graded Hopf algebra. Since H0 it is connected. The counit ǫ vanishes on the subspace n=1 Hn, called aug- mentation ideal, and ǫ(I) = 1. As usual, if ∆(x) = I⊗ x+ x⊗ I, the element x is called primitive. The linear subspace of primitive elements is denoted PrimH. The interest in H and L arises from the fact that the Bogoliubov recursion is essentially solved by the antipode of H. In any connected graded bialgebra, the antipode S is given by S(x) = −x− S(x′)x′′, x /∈ H0 (3) in Sweedler’s notation. Let now V be a C-algebra. The space of linear maps LQ(H, V ) is equipped with a convolution product (f, g) 7→ f ∗ g = mV (f ⊗ g)∆ where mV is the product in V. Relevant examples for V are suggested by regularization schemes such as the algebra V = C[[ǫ, ǫ−1] of Laurent series with finite pole part for dimensional regularization (space-time dimensionD = 6+2ǫ.) The (unrenormalized) Feynman rules provide then an algebra homomorphism φ : H → V mapping Feynman graphs to Feynman integrals in 6+2ǫ dimensions. On V there is a linear endomorphism R (renormalization scheme) defined, for example minimal subtraction R(ǫn) = 0 if n ≥ 0, R(ǫn) = ǫn if n < 0. If Γ is primitive, as defined above, then φ(Γ) has only a simple pole in ǫ, hence (1−R)φ(Γ) is a good renormalized value for Γ. If Γ does have subdivergences, the situation is more complicated. However, the map S R : H → V R(Γ) = −R φ(Γ)− ′)φ(Γ′′) provides the counterterm prescribed by the Bogoliubov recursion, and (S φ)(Γ) yields the renormalized value of Γ. The map S R is a recursive deforma- tion of φ ◦ S by R, compare its definition with (3). These are results obtained by one of the authors in collaboration with Connes [26, 14, 15]. For S R to be an algebra homomorphism again, one requires R to be a Rota- Baxter operator, studied in a more general setting by Ebrahimi-Fard, Guo and one of the authors in [20, 22, 21]. The Rota-Baxter property is at the algebraic origin of the Birkhoff decomposition introduced in [15, 16]. In the presence of mass terms, or gauge symmetries etc. in the Lagrangian, φ, S R and S R ⋆φ may contribute to several form factors in the usual way. This can be resolved by considering a slight extension of the Hopf algebra containing projections onto single structure functions, as discussed for example in [15, 32]. For the case of gauge theories, a precise definition of the coefficients n(γ1, γ2,Γ) is given in [30]. The Hopf algebra H arises from the simple insertion of graphs into each other in a completely canonical way. Indeed, the pre-Lie product determines the co- product, and the coproduct determines the antipode. Like this, each quantum field theory gives rise to such a Hopf algebra H based on its 1PI graphs. It is no surprise then that there is an even more universal Hopf algebra behind all of them: The Hopf algebra Hrt of rooted trees [26, 14]. In order to see this, imagine a purely nested situation of subdivergences like which can be represented by the rooted tree To account for each single graph of this kind, the tree’s vertices should actually be labeled according to which primitive graph they correspond to (plus some gluing data) which we will suppress for the sake of simplicity. The coproduct on Hrt – corresponding to the one (2) of H – is ∆(τ) = I⊗ τ + τ ⊗ I+ adm.c Pc(τ) ⊗Rc(τ) where the sum runs over all admissible cuts of the tree τ. A cut of τ is a nonempty subset of its edges which are to be removed. A cut c(τ) is defined to be admissible, if for each leaf l of τ at most one edge on the path from l to the root is cut. The product of subtrees which fall down when those edges are removed is denoted Pc(τ). The part which remains connected with the root is denoted Rc(τ). Here is an example: ⊗ I+ I⊗ + 2 • ⊗ + • • ⊗ ✁ ⊗ •. Compared to Hrt, the advantage of H is however that overlapping divergences are resolved automatically. To achieve this in Hrt requires some care [27]. 3 From Hochschild cohomology to physics There is a natural cohomology theory on H and Hrt whose non-exact 1-cocycles play an important ”operadic” role in the sense that they drive the recursion that define the full 1PI Green’s functions in terms of primitve graphs. In order to introduce this cohomology theory, let A be any bialgebra. We view A as a bicomodule over itself with right coaction (id ⊗ ǫ)∆. Then the Hochschild cohomology of A (with respect to the coalgebra part) is defined as follows [14]: Linear maps L : A → A⊗n are considered as n-cochains. The operator b, defined bL := (id⊗ L)∆ + (−1)i∆iL+ (−1) n+1L⊗ I (4) furnishes a codifferential: b2 = 0. Here ∆ denotes the coproduct of A and ∆i the coproduct applied to the i-th factor in A⊗n. The map L ⊗ I is given by x 7→ L(x) ⊗ I. Clearly this codifferential encodes only information about the coalgebra (as opposed to the algebra) part of A. The resulting cohomology is denoted HH•ǫ (A). For n = 1, the cocycle condition bL = 0 is simply ∆L = (id⊗ L)∆ + L⊗ I (5) for L a linear endomorphism of A. In the Hopf algebraHrt of rooted trees (where things are often simpler), a 1-cocycle is quickly found: the grafting operator B+, defined by B+(I) = • B+(τ1 . . . τn) = τ1 . . . τn for trees τi joining all the roots of its argument to a newly created root. Clearly, B+ reminds of an operad multiplication. It is easily seen that B+ is not exact and therefore a generator (among others) of HH1ǫ (Hrt). Foissy [23] showed that L 7→ L(I) is an onto map HH1ǫ(Hrt) → PrimHrt. The higher Hochschild cohomology (n ≥ 2) of Hrt is known to vanish [23]. The pair (Hrt, B+) is the universal model for all Hopf algebras of Feynman graphs and their 1-cocycles [14]. Let us now turn to those 1-cocycles of H. Clearly, every primitve graph γ gives rise to a 1-cocycle + defined as the operator which inserts its argument, a product of graphs, into γ in all possible ways. Here is a simple example: See [30] for the general definition involving some combinatorics of insertion places and symmetries. It is an important consequence of the B + satisfying the cocycle condition (5) R ∗ φ)B+ = (1−R)B̃+(S R ∗ φ) (6) where B̃+ is the push-forward of B+ along the Feynman rules φ. In other words, + is the integral operator corresponding to the skeleton graph γ. This is the combinatorial key to the proof of locality of counterterms and finiteness of renor- malization [13, 28, 2, 3]. Indeed, equation (6) says that after treating all subdi- vergences, an overall subtraction (1−R) suffices. The only analytic ingredient is Weinberg’s theorem applied to the primitive graphs. In [2] it is emphasized that H is actually generated (and determined) by the action of prescribed 1-cocycles and the multiplication. A version of (6) with decorated trees is available which describes renormalization in coordinate space [2]. The 1-cocycles B + give rise to a number of useful Hopf subalgebras of H. Many of them are isomorphic. They are studied in [3] on the model of decorated rooted trees, and we will come back to them in the next section. In [30] one of the authors showed that in nonabelian gauge theories, the existence of a certain Hopf subalgebra, generated by 1-cocycles, is closely related to the Slavnov- Taylor identities for the couplings to hold. In a similar spirit, van Suijlekom showed that, in QED, Ward-Takahashi identities, and in nonabelian Yang-Mills theories, the Slavnov-Taylor identities for the couplings generate Hopf ideals I of H such that the quotients H/I are defined and the Feynman rules factor through them [37, 38]. The Hopf algebra H for QED had been studied before in [11, 33, 39]. 4 Dyson-Schwinger equations The ultimate application of the Hochschild 1-cocycles introduced in the previous section aims at non-perturbative results. Dyson-Schwinger equations, reorga- nized using the correspondence PrimH → HH1ǫ (H), become recursive equations inH[[α]], α the coupling constant, with contributions from (degree 1) 1-cocycles. The Feynman rules connect them to the usual integral kernel representation. We remain in the massless φ3 theory in 6 dimensions for the moment. Let Γ� be the full 1PI vertex function, Γ� = I+ res Γ=� (normalized such that the tree level contribution equals 1). This is a formal power series in α with values in H. Here res Γ is the result of collapsing all internal lines of Γ. The graph res Γ is called the residue of Γ. In a renormalizable theory, res can be seen as a map from the set of generators of H to the terms in the Lagrangian. For instance, in the φ3 theory, vertex graphs have residue �, and self energy graphs have residue −. The number SymΓ denotes the order of the group of automorphisms of Γ, defined in detail for example in [30, 38]. Similarly, the full inverse propagator Γ− is represented by Γ− = I− res Γ=− . (8) These series can be reorganized by summing only over primitive graphs, with all possible insertions into these primitive graphs. In H, the insertions are afforded by the corresponding Hochschild 1-cocycles. Indeed, Γ� = I+ γ∈PrimH,res γ=� α|γ|B �Q|γ|) Sym γ Γ− = I− γ∈PrimH,res γ=− α|γ|B −Q|γ|) Sym γ . (9) The universal invariant charge Q is a monomial in the Γr and their inverses, where r are residues (terms in the Lagrangian) provided by the theory. In φ3 theory we have Q = (Γ�)2(Γ−)−3. In φ3 theory, the universality of Q (i. e. the fact that the same Q is good for all Dyson-Schwinger equations of the theory) comes from a simple topological argument. In nonabelian gauge theories how- ever, the universality of Q takes care that the solution of the corresponding system of coupled Dyson-Schwinger equations gives rise to a Hopf subalgebra and therefore amounts to the Slavnov-Taylor identities for the couplings [30]. The system (9) of coupled Dyson-Schwinger equations has (7,8) as its solution. Note that in the first equation of (9) an infinite number of cocycles contributes as there are infinitely many primitive vertex graphs in φ36 theory – the second equation has only finitely many contributions – here one. Before we describe how to actually attempt to solve equations of this kind analytically (application of the Feynman rules φ), we discuss the combinatorial ramifications of this con- struction in the Hopf algebra. It makes sense to call all (systems of) recursive equations of the form X1 = I± . . . Xs = I± combinatorial Dyson-Schwinger equations, and to study their combinatorics. Here, the Bdn+ are non-exact Hochschild 1-cocycles and the Mn are monomials in the X1 . . .Xs. In [3] we studied a large class of single (uncoupled) combinatorial Dyson-Schwinger equations in a decorated version of Hrt as a model for vertex insertions: X = I+ αnwnB where the wn ∈ Q. For example, X = I+αB+(X 2)+α2B+(X 3) is in this class. It turns out [28, 3] that the coefficients cn of X, defined by X = n=0 α generate a Hopf subalgebra themselves: ∆(cn) = Pnk ⊗ ck. The Pnk are homogeneous polynomials of degree n−k in the cl, l ≤ n. These poly- nomials have been worked out explicitly in [3]. One notices in particular that the Pnk are independent of the wn and B + , and hence that under mild assump- tions (on the algebraic independence of the cn) the Hopf subalgebras generated this way are actually isomorphic. For example, X = I+αB+(X 2) +α2B+(X and X = I + αB+(X 2) yield isomorphic Hopf subalgebras. This is an aspect of the fact that truncation of Dyson-Schwinger equations – considering only a finite instead of an infinite number of contributing cocycles – does make (at least combinatorial) sense. Indeed, the combinatorics remain invariant. Similar results hold for Dyson-Schwinger equations in the true Hopf algebra of graphs H where things are a bit more difficult though as the cocycles there involve some bookkeeping of insertion places. The simplest nontrivial Dyson-Schwinger equation one can think of is the linear X = I+ αB+(X). Its solution is given by X = n=0 α n(B+) n(I). In this case X is grouplike and the corresponding Hopf subalgebra of cns is cocommutative [25]. A typical and important non-linear Dyson-Schwinger equation arises from propagator insertions: X = I− αB+(1/X), for example the massless fermion propagator in Yukawa theory where only the fermion line obtains radiative corrections (other corrections are ignored). This problem has been studied and solved by Broadhurst and one of the authors in [12] and revisited recently by one of the authors and Yeats [35]. As we now turn to the analytic aspects of Dyson-Schwinger equations, we briefly sketch the general approach presented in [35] on how to successfully treat the nonlinearity of Dyson-Schwinger equations. Indeed, the linear Dyson-Schwinger equations can be solved by a simple scaling ansatz [25]. In any case, let γ be a primitive graph. The following works for amplitudes which depend on a single scale, so let us assume a massless situation with only one non-zero external momentum – how more than one external momentum (vertex insertions) are incorporated by enlarging the set of primitive elements is sketched in [32]. The grafting operator + associated to γ translates to an integral operator under the (renormalized) Feynman rules +)(I)(p 2/µ2) = (Iγ(k, p)− Iγ(k, µ))dk where Iγ is the integral kernel corresponding to γ, the internal momenta are denoted by k, the external momentum by p, and µ is the fixed momentum at which we subtract: R(x) = x|p2=µ2 . In the following we stick to the special case discussed in [35] where only one internal edge is allowed to receive corrections. The integral kernel φ(B +) defines a Mellin transform F (ρ) = Iγ(k, µ)(k where ki is the momentum of the internal edge of γ at which insertions may take place (here the fermion line). If there are several insertion sites, obvious multiple Mellin transforms become necessary. The case of two (propagator) in- sertion places has been studied, at the same example, in [35]. The function F (ρ) has a simple pole in ρ at 0. We write F (ρ) = We denote L = log p2/µ2. Clearly φR(X) = 1 + n γnL n. An important result of [35] is that, even in the difficult nonlinear situation, the anomalous dimension γ1 is implicitly defined by the residue r and Taylor coefficients fn of the Mellin transform F. On the other hand, all the γn for n ≥ 2, are recursively defined in terms of the γi, i < n. This last statement amounts to a renormalization group argument that is afforded in the Hopf algebra by the scattering formula of [16]. Curiously, for this argument only a linearized part of the coproduct is needed. We refer to [35] for the actual algorithm. For a linear Dyson-Schwinger equation, the situation is considerably simpler as the γn = 0 for n ≥ 2 since X is grouplike [25]. Let us restate the results for the high energy sector of non-linear Dyson-Schwinger equations [12, 35]: Primitive graphs γ define Mellin transforms via their integral kernels B̃ +. The anomalous dimension γ1 is implicitly determined order by order from the coefficients of those Mellin transforms. All non-leading log coefficients γn are recursively determined by γ1, thanks to the renormalization group. This reduces, in principle, the problem to a study of all the primitive graphs and the intricacies of insertion places. Finding useful representations of those Mellin transforms – even one-dimensional ones – of higher loop order skeleton graphs is difficult. However, the two-loop primitive vertex in massless Yukawa theory has been worked out by Bierenbaum, Weinzierl and one of the authors in [4], a result that can be applied to other theories as well. Combined with the algebraic treatment [12, 3, 35] sketched in the previous paragraphs and new geometric insight on primitive graphs (see section 5), there is reasonable hope that actual solutions of Dyson-Schwinger equations will be more accessible in the future. Using the Dyson-Schwinger analysis, one of the authors and Yeats [34] were able to deduce a bound for the convergence of superficially divergent ampli- tudes/structure functions from the (desirable) existence of a bound for the su- perficially convergent amplitudes. 5 Feynman integrals and periods of mixed (Tate) Hodge structures A primitive graph Γ ∈ PrimH defines a real number rΓ, called the residue of Γ, which is independent of the renormalization scheme. In the case that Γ is massless and has one external momentum p, the residue rΓ is the coefficient of log p2/µ2 in φR(Γ) = (1 − R)φ(Γ). It coincides with the coefficient r of the Mellin transform introduced in the previous section. One may ask what kind of a number r is, for example if it is rational or algebraic. The origin of this ques- tion is that the irrational or transcendental numbers that show up for various Γ strongly suggest a motivic interpretation of the rΓ. Indeed, explicit calcula- tions [9, 10, 8] display patterns of Riemann zeta and multiple zeta values that are known to be periods of mixed Tate Hodge structures – here the periods are provided by the Feynman rules which produce Γ 7→ rΓ. By disproving a related conjecture of Kontsevich, Belkale and Brosnan [1] have shown that not all these Feynman motives must be mixed Tate, so one may expect a larger class of Feynman periods than multiple zeta values. Our detailed understanding of these phenomena is still far from complete, and only some very first steps have been made in the last few years. However, techniques developed in recent work by Bloch, Esnault and one of the authors [7] do permit reasonable insight for some special cases which we briefly sketch in the following. Let Γ be a logarithmically divergent massless primitive graph with one ex- ternal momentum p. It is convenient to work in the ”Schwinger” parametric representation [24] obtained by the usual trick of replacing propagators dae−ak and performing the loop integrations (Gaussian integrals) first which leaves us with a (divergent) integral over various Schwinger parameters a. It is a classical exercise [24, 7, 6] to show that in four dimensions, up to some powers of i and φ(Γ) = da1 . . . dan e−QΓ(a,p 2)/ΨΓ(a) Ψ2Γ(a) where n is the number of edges of Γ. QΓ and ΨΓ are graph polynomials of Γ, where ΨΓ, sometimes called Symanzik or Kirchhoff polynomial, is defined as follows: Let T (Γ) be the set of spanning trees of Γ, i. e. the set of connected simply connected subgraphs which meet all vertices of Γ. We think of the edges e of Γ as being numbered from 1 to n. Then t∈T (Γ) This is a homogeneous polynomial in the ai of degree |H1(Γ)|. It is easily seen (scaling behaviour of QΓ and ΨΓ) that rΓ = ∂φR(Γ) ∂ log p2/µ2 is extracted from φ(Γ) by considering the ai as homogeneous coordinates of P n−1(R) and evaluating at p2 = 0 : σ⊂Pn−1(R) where σ = {[a1, . . . , an] : all ai can be choosen ≥ 0} and Ω is a volume form on Pn−1. Let XΓ := {ΨΓ = 0} ⊂ P n−1. If |H1(Γ)| = 1, the integrand in (10) has no poles. If |H1(Γ)| > 1, poles will show up on the union ∆ = γ(Γ,H1(γ) 6=0 of coordinate linear spaces Lγ = {ae = 0 for e edge of γ} – these need to be separated from the chain of integration by blowing up. The blowups being understood, the Feynman motive is, by abuse of notation, Hn−1(Pn−1 −XΓ,∆−∆ ∩XΓ) with Feynman period given by (10). See [7, 6] for details. Some particularly accessible examples are the wheel with n spokes graphs Γn := studied extensively in [7]. The corresponding Feynman periods (10) yield ratio- nal multiples of zeta values [9] rΓn ∈ ζ(2n− 3)Q. Due to the simple topology of the Γn, the geometry of the pairs (XΓn ,∆Γn) are well understood and the corresponding motives have been worked out explicitly [7]. The methods used are however nontrivial and not immediately applicable to more general situations. When confronted with non-primitive graphs, i. e. graphs with subdivergences, there are more than one period to consider. In the Schwinger parameter picture, subdivergences arise when poles appear along exceptional divisors as pieces of ∆ are blown up. This situation can be understood using limiting mixed Hodge structures [6], see also [31, 36] for a toy model approach to the combinatorics involved. In [6] it is also shown how the Hopf algebra H of graphs lifts to the category of motives. For the motivic role of solutions of Dyson-Schwinger equa- tions we refer to work in progress. Finally we mention that there is related work by Connes and Marcolli [18, 19] who attack the problem via Riemann-Hilbert correspondences and motivic Galois theory. Acknowledgements. We thank Spencer Bloch and Karen Yeats for discus- sion on the subject of this review. The first named author (C. B.) thanks the organizers of the ICMP 2006 and the IHES for general support. His research is supported by the Deutsche Forschungsgemeinschaft. The IHES, Boston Univer- sity and the Erwin-Schrödinger-Institute are gratefully acknowledged for their kind hospitality. At the time of writing this article, C. B. is visiting the ESI as a Junior Research Fellow. References [1] P. Belkale and P. Brosnan. Matroids, motives, and a conjecture of Kontse- vich. Duke Math. J. 116, (1):147–188, 2003. math.AG/0012198. [2] C. Bergbauer and D. Kreimer. The Hopf algebra of rooted trees in Epstein– Glaser renormalization. Ann. Henri Poincare, 6:343–367, 2004. hep- th/0403207. [3] C. Bergbauer and D. Kreimer. 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Introduction Lie and Hopf algebras of Feynman graphs From Hochschild cohomology to physics Dyson-Schwinger equations Feynman integrals and periods of mixed (Tate) Hodge structures

0704.0233

Many-body interband tunneling as a witness for complex dynamics in the Bose-Hubbard model

Many-body interband tunneling as a witness for complex dynamics in the Bose-Hubbard model Andrea Tomadin,1 Riccardo Mannella,1 and Sandro Wimberger1,2 Dipartimento di Fisica, Unversità degli Studi di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy CNISM, Dipartimento di Fisica del Politecnico, C. Duca degli Abruzzi 24, 10129 Torino, Italy (Dated: November 4, 2018) A perturbative model is studied for the tunneling of many-particle states from the ground band to the first excited energy band, mimicking Landau-Zener decay for ultracold, spinless atoms in quasi-one dimensional optical lattices subjected to a tunable tilting force. The distributions of the computed tunneling rates provide an independent and experimentally accessible signature of the regular-chaotic transition in the strongly correlated many-body dynamics of the ground band. PACS numbers: 03.65.Xp,32.80.Pj,05.45.Mt,71.35.Lk The experimental advances in atom and quantum op- tics allow the experimentalist to directly study a plethora of minimal models which have been developed to de- scribe usually much more complex phenomena occurring in solid states [1, 2, 3]. Bose-Einstein condensates loaded into optical lattices, which perfectly realize spatially peri- odic potentials, are used, e.g., to implement the Wannier- Stark problem [4, 5, 6] as a paradigm of quantum trans- port where atoms move in a tilted lattice. Up to now all experiments on the Wannier-Stark system with ultracold atoms have been performed in a regime where atom-atom interactions are either negligible [4] or reduce to an effec- tive mean-field description [5, 7]. State-of-the-art setups are, however, capable to achieve small filling factors of the order of one atom per lattice site [2]. Moreover, the atom-atom interactions can be tuned by the transversal confinement and by Feshbach resonances [3, 8], resulting in strong interaction-induced correlations. The regime of strong correlations in the Wannier-Stark system was addressed in [9, 10], revealing the sensitive dependence of the system’s dynamics on the Stark force F . The single-band Bose-Hubbard model of [9, 10] is defined by the following Hamiltonian with the creation l,1, annihilation âl,1, and number operators n̂ l,1 for the first band of a lattice l = 1 . . . L: Fl n̂ l,1− â l+1,1 † â l,1 + h.c. n̂ l,1 ( n̂ l,1 − 1) . A transition from a regular dynamical (dominated by F ) to a quantum chaotic regime (with comparable values of J1, U1 , F ) was found [9, 10]. The transition was quan- titatively studied using the distribution of the spacings between next nearest eigenenergies of the Hamiltonian (1). This analysis [9, 10] verifies that the normalized level spacings s ≡ ∆E/∆E obey a Poisson (P(s) = exp(−s) ) and a Wigner-Dyson (WD: P(s) = sπ/2 exp(−πs2/4) ) distribution in the regular and chaotic case, respectively [11]. P(s) and the cumulative distribution functions (CDF: C(s) ≡ ds′ P(s′)) are shown for typical cases in Fig. 1, where we scanned F to emphasize the crossover 0 1 2 3 4 5 1.8 2 2.2 2.4 2.6 2.8 3 (2π/F) 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 5 10 15 20 25 FIG. 1: (a,b) CDF (stairs) and P(s) (stairs in insets) for N = 5 atoms, L = 8, lattice depth V = 10 recoil energies (fixing J1 = 0.038), U1 = 0.032, F ≃ 0.063 (a) and 0.021 (b), with WD (solid) and Poisson distributions (dashed). (c) χ2 test with values close to zero for good WD statistics. The dashed line marks the transition to quantum chaos as F is tuned. (d) variance of the number of levels in intervals of length dE (with normalized mean spacing), for the cases of (a) (squares) and (b) (circles), with the random matrix predictions for Poisson (dashed) and WD (solid) [11]. between the regular and the chaotic regime. Statistical tests are also shown which confirm the analysis of [9, 10] in a more systematical manner [12]. As shown in [9], the strong correlations in the quantum chaotic regime induce a fast and irreversible decay of the Bloch oscillations, which otherwise would persist in the ideal, non-interacting case. Therefore, the crossover be- tween the two regimes discussed above could be measured in experiments by observing just the mean momentum as a function of time. Here we introduce a new, robust and hence also experimentally accessible prediction for this crossover. In the presence of strong interactions parame- terized by U1 , the single-band model should be extended to allow for interband transitions [13], as recently realized http://arxiv.org/abs/0704.0233v1 at F = 0 in experiments with fermionic interacting atoms [3]. Instead of using a numerically hardly tractable com- plete many-bands model, we introduce a perturbative de- cay of the many-particles modes in the ground band to a second energy band. Our novel approach to study the Landau-Zener-like tunneling between the first and the second band [1, 5, 7, 14, 15] leads to predictions for the expected decay rates and their statistical distributions. As we will show, the latter are drastically affected by the dynamics in the ground band, and they therefore provide a measurable witness for the regular-chaotic transition. We first derive the individual decay rates of the dom- inating interband coupling channels. These decay rates will serve to effectively open the single-band model (1) for mimicking losses arising from the interband coupling. Our analysis starts from the following “unperturbed” Hamiltonian for the first two bands: ε1 n̂ l,1 + ε2 n̂ l,2 − J22 ( â l+1,2 † â l,2 + h.c.) +Fl ( n̂ l,1 + n̂ l,2) + n̂ l,1 ( n̂ l,1 − 1) . (2) For a moment, we neglect the hopping in the lower band, where the single-particle Wannier functions [14] are more localized than in the upper band. In the latter we neglect the interactions, since initially only a few particles pop- ulate the excited levels. A closer analysis of the full two- bands system [12] shows that there are two dominating mechanisms that promote particles to the second band. The first one is a single-particle dipole coupling arising from the force term: H1 = F · D â l,2 â l,1 + â l,1 â l,2 , (3) where D depends only on the lattice depth V (measured in recoil energies according to the definition in [7]). The second one is a many-body effect, describing two particles of the first band entering the second band together: â l,2 â l,2 â l,1 â l,1 + (1 ↔ 2) . (4) The cross-band interaction is characterized by the pa- rameter U× ≡ ãs dxχ21χ 2 ≃ 0.5U1 (for V = 3 . . . 10) [12], for U1 ≡ ãs dxχ41, with renormalized scattering length ãs [8, 12] and the Wannier functions χ1,2 local- ized in each well for the first or second band. To justify the following perturbative approach, it is crucial to real- ize that the terms (3) and (4) must be small compared with the band gap ∆ ≡ ε2 − ε1 (not necessarily small with respect to the single band terms in (1)), and indeed FD,U×, U1 ≪ ∆ for the parameters considered here. For the first perturbation, the decay channel of a given unperturbed Fock state labelled |b〉 (with a total number of atoms N and nh atoms in an arbitrary well h) is |b;N〉 ⊗ |vac〉 → |b′;N − 1〉 ⊗ |w〉 , n′h = nh − 1. (5) Here, |w〉 = m=−∞ Jm−w(|J2|/F ) âm,2 †|vac〉 is the single-particle eigenstate for the Wannier-Stark problem, localized around the site w in the second band, with the Bessel function of the first kind Jm(x) [14]. The expectation value of (3) for |b;N〉 of the first band, equal to the first-order δE(b), is zero because the opera- tor does not conserve the number of particles within the bands. The decay width at first-order is given by the ma- trix element of the perturbation between the initial and final state according to Fermi’s Golden Rule, and only the first term in (3) gives a nonzero contribution [12]: 〈k|〈b′| l=1 â l,2 † â l,1 |b〉|vac〉 = l=1 Jl−w(|J2|/F ) ·δ(n′ , nl − 1) m 6=l δ(n m, nm). (6) The δ(·, ·) functions act as a selection rule for the Fock states that are coupled by the perturbation. The tun- neling mechanism does not include any income of en- ergy from an external source, so the initial and final energies E0(b) = 〈vac|〈b|H0|b〉|vac〉 and E0(b′, w) = 〈w|〈b′|H0|b′〉|w〉, respectively, must be equal as required by the Golden Rule. The condition on the energy conser- vation is, however, relaxed to account for the uncertainty ∆E(b) of the unperturbed energy levels of the initial and final states in the lower band arising from the hopping in this band initially neglected in (2). A detailed derivation is given in [12], and here we only state the result: ∆E(b) = 2π (J1 /2) ∆E(b → b′) = 2π (J1 /2) ∆l=±1 n2l δ(nl+∆l + 1, nl). (7) The level density ρ(E, b) around the unperturbed en- ergy E0(b) of a Fock state |b〉 is then approximated by a rectangular profile, of width ∆E(b) and unit area: ρ(E, b) = χ {|E − E0(b)| ≤ ∆E(b)/2} / ∆E(b). The re- laxed energy conservation rule selects from (5) the set K of permitted decay channels (h,w) parameterized by the two indices h,w such that: ′, w)− E0(b) = ∆− F (h− w)− U1 (nh − 1) −∆E(b)+∆E(b ∆E(b)+∆E(b′) . (8) Hence the energy ∆ required to promote a particle to the second band is supplied by the decrease of the interaction (∝ U1 ) and by the work of the force (∝ F ) exerted on the promoted particle. The total width Γ1(b) for the decay via the allowed channels K, is proportional to the square of the matrix element and to the level density ρ(E, b): Γ1(b) = 2π(FD) (h,w)∈K Jh−w( |J2|F ) · ∆E(b)∆E(b′) . (9) Jm(x) significantly contributes only for |m| <∼ |x|. If U1 ,∆E(b) ≪ ∆, the energy conservation is roughly given by |∆| ≃ F (h − w). Requiring that the Bessel function in (9) is substantially larger than zero, we ob- tain the inequality |∆| ≤ |J2|. The last condition does not depend on F , since a twofold effect is at work: a stronger force produces a larger energy gain when a par- ticle moves along the lattice, but the extension |J2/F | of the single-particle state shrinks. Therefore, increasing F results in an increased energy matching and a strongly reduced “geometrical” matching. For 3 < V < 26, we have |∆|− |J2| > 1.0 [12], such that the energy matching cannot be realized by just tuning the lattice depth. The decay can, however, be activated by an increase of the in- teractions, which can be experimentally achieved by act- ing on the transversal confining potential of a quasi-one dimensional lattice, or by a Feshbach resonance [8]. In the calculations presented below, we augmented U1 used in [9, 10] by a factor of order 10, and as noted in the in- troduction, a similar increase of the interaction strength was used in the experiment to promote fermions to higher bands [3], in close analogy to the here described field- and interaction-induced interband coupling of bosons. The second term (4) is treated in a similar way, with the difference that two particles are promoted to the sec- ond band, and the position of the second single-particle state |w′〉 is an additional degree of freedom for the tran- sition. The decay channels are: |b,N〉⊗ |vac〉 → |b′, N − 2〉 ⊗ |w,w′〉 ; n′h = nh − 2. (10) The energy matching selects a set K of decay channels, parameterized by the three site indices h,w,w′: (h,w,w′) ∈ K s.t. E0(b′, w, w′)− E0(b) = = 2∆− F (2h− w − w′)− U1 (2nh − 3) ∆E(b) + ∆E(b′) ∆E(b) + ∆E(b′) . (11) The computation of the matrix element yields [12]: Γ2(b) = 2π (h,w,w′)∈K Jh−w( |J2|F ) · Jh−w′( |J2|F ) · 4nh (nh − 1) 1∆E(b)∆E(b′) . (12) With respect to (9), the additional degree of freedom w′ results in a summation over all possible values of w−w′. This follows from the possibility to conserve the energy even if a particle is pushed far, if the other parti- cle is pushed almost equally far in the opposite direction. Since the decay widths in (12) depend on the product of two (rapidly decaying) Bessel functions – again a “geo- metrical” matching condition – we apply the truncation |w−w′| ≤ |J2/F |, to reduce the formula to a finite form. We can now compute the total width ΓF(b) = Γ1(b) + Γ2(b) defined by the two analyzed coupling processes for 0 1 2 3 4 5 -6.5 -6 -5.5 0 1 2 3 4 5 -5 -4.5 -4 -3.5 Log10Γ 0 1 2 3 4 5 -4 -3.5 -3 -2.5 -6.5 -6 -4 -3.5 FIG. 2: (a,c,e) CDF from Re {Ej} (stairs), together with WD (solid) and Poisson predictions (dashed). (b,d,f) distri- butions of the logarithm of the rates. In (a,b), (c,d), (e,f) F ≃ 0.17, 0.31, 0.47, respectively, with (N,L) = (7, 6), V = 3, U1 = 0.2 (fixing U× ≃ 0.1). In the regular regime (f), a log-normal distribution (dotted) well fits the data, with a scaling P(Γ) ∝ Γ−x for the largest Γ (dashed line in the inset of (f) with x = 1). In the chaotic case, a global power-law behavior with x ≈ 2 is found (dashed line in the inset of (b)). each basis state |b〉 of the single-band problem given in (1). The ΓF(b) are inserted as complex potentials in the diagonal of the single-band Hamiltonian matrix. After a gauge transform that recovers the translational invari- ance of the problem (see [10, 12] for details), the latter matrix is used to compute the evolution operator over one Bloch period TB, which is finally diagonalized to obtain its eigenphases exp (−iEj TB). Along with the statistics of the level spacings defined by Re {Ej}, the Figs. 2 and 3 analyze the statistical distributions of the tunneling rates Γj = −2Im{Ej} for some paradigmatic cases. All rates are much smaller than unity, which a posteriori is fully consistent with our perturbative approach. To observe what happens at the regular-chaotic tran- sition (c.f. Fig. 1), we scan F in Fig. 2, and as F in- creases, the average decay increases by orders of mag- nitude, while the distributions broaden. The large in- crease of the rates is due to an improved energy match- ing, when F supplies the necessary energy to promote particles to the second band. For the parameters of Fig. 2, the single-particle Landau-Zener formula [14] gives ΓLZ = F/(2π) exp −π2∆2/(8F ) ∼ 10−23, 10−12, 10−8 for (b,d,f). This huge variation, typical of semiclassical formulae, implies that there are possibly parameters for which our results are comparable to the single-particle prediction, but, in general, the many-particle effects can- not be neglected. Moreover, mean-field treatments of the Landau-Zener tunneling at best predict a shift of Γ [7, 15], but cannot account for their distributions. In the chaotic regime, the Fock states are strongly mixed by the dynamics [9, 10, 12] and a fast decaying -12 -11 -10 -9 -8 -5.5 -5 -4.5 -4 10a) b) c) d) FIG. 3: (a,c) rate distributions in the chaotic regime with F ≃ 0.17, U1 = 0.2 (U× ≃ 0.1), together with the corresponding unscaled P(Γ) in (b,d). In (a,b) (N,L) = (7, 6), V = 4, and in (c,d) (N,L) = (9, 8), V = 3. Power-laws P(Γ) ∝ Γ−x are found with x ≈ 2 (dashed lines in (b,d)). Fock state can act as a privileged decay channel for many eigenstates. Many states then share similar rates, lead- ing to thinner distributions. Therefore, the thinner dis- tribution of Fig. 2 (b) is a direct signature of the chaotic dynamics evidenced in (a), as compared with the regular case in (e,f). In Fig. 2 (f), we found a good agreement with the expected log-normal distribution of decay rates [16] (or of the similarly behaving conductance [17]) in the regular regime. There the system shows nearly per- fect Bloch oscillations [9], and the motion of the atoms is localized along the lattice [14]. We can even detect a qualitative crossover to a power-law P(Γ) ∝ Γ−1 in the right tail of the distribution, as predicted from localiza- tion theory [16, 18, 19]. The distributions in Figs. 2 (b) and 3 follow the expected power-law for open quantum chaotic systems in the diffusive regime [18]. The expo- nents x are, however, nonuniversal and depend on the opening of the system. In our case, the decay channels are defined by the interband coupling, which in a sense attaches “leads” to all lattice sites within the sample. Going along with the regular-to-chaotic transition in the lower band of our model (from Fig. 2 (f) to (b), or to Fig. 3) the Γ distributions transform from a log-normal to a power-law with x ≈ 2, in close analogy to the tran- sition from Anderson-localized to diffusive dynamics in open disordered systems [18, 20]. In summary, our perturbative opening of the single- band Wannier-Stark system allows one to study Landau- Zener-like interband tunneling within a many-body de- scription of the dynamics of ultracold atoms. The statis- tical characterization of the tunneling rates (mean values and form of the distributions) provides clear and robust signatures of the regular-to-chaotic transition for future experiments. A more detailed analysis of the interband coupling in a full-blown model, in which at least two bands are completely included, calls for huge computa- tional resources to access the complete quantum spec- tra. Nonetheless, our results are a first step in the di- rection of studies for which “horizontal” and “vertical” quantum transport along the lattice are simultaneously present and influence each other in a complex manner. We thank the Centro di Calcolo, Dipartimento di Fisica, Università di Pisa, for providing CPU, and the Humboldt Foundation, MIUR-PRIN, and EU-OLAQUI for support. [1] O. Morsch et al., Rev. Mod. Phys. 78, 179 (2006). [2] M. Greiner et al., Nature 415, 39 (2002); T. Stöferle et al., Phys. Rev. Lett. 92, 130403 (2004); S. Fölling et al., ibid. 97, 060403 (2006). [3] M. Köhl et al., Phys. Rev. Lett. 94, 080403 (2005). [4] M. BenDahan et al., Phys. Rev. Lett. 76, 4508 (1996); S.R. Wilkinson et al., ibid. 4512; B.P. Anderson et al., Science 282, 1686 (1998). [5] O. Morsch et al., Phys. Rev. Lett. 87, 140402 (2001). [6] G. Roati et al., Phys. Rev. Lett. 92, 230402 (2004). [7] S. Wimberger et al., Phys. Rev. A 72, 063610 (2005). [8] T. Bergeman et al., Phys. Rev. Lett. 91, 163201 (2003). [9] A. Buchleitner et al., Phys. Rev. Lett. 91, 253002 (2003). [10] A.R. Kolovsky et al., Phys. Rev. E 68, 056213 (2003). [11] M.L. Mehta, Random Matrices and the Statistical Theory of Energy Levels (Academic Press, New York, 1991). [12] A. Tomadin, Master’s thesis, Università di Pisa (2006). [13] V.W. Scarola et al., Phys. Rev. Lett. 95, 033003 (2005). [14] M. Glück et al., Phys. Rep. 366, 103 (2002). [15] B. Wu et al., Phys. Rev. A 61, 023402 (2000); O. Zobay et al., ibid. 61, 033603 (2000); D. Witthaut et al., ibid. 75, 013617 (2007); S. Wimberger et al., J. Phys. B 39, 729 (2006). [16] M. Terraneo et al., Eur. Phys. J. B 18, 303 (2000). [17] C.W.J. Beenakker, Rev. Mod. Phys. 69, 731 (1997). [18] T. Kottos, J. Phys. A 38, 10761 (2005). [19] G. Casati et al., Phys. Rev. Lett. 82, 524 (1999); S. Wim- berger et al., ibid. 89, 263601 (2002). [20] S. Wimberger et al., J. Phys. A 34, 7181 (2001).

0704.0234

Comments on ``Are Swift Gamma-Ray Bursts consistent with the Ghirlanda relation?", by Campana et al.(astro--ph/0703676)

7 Comments on “Are Swift Gamma-Ray Bursts consistent with the Ghirlanda relation?”, by Campana et al.(astro–ph/0703676) G. Ghirlanda a, L. Navaa b, G. Ghisellini a, C. Firmania c aOsservatorio Astronomico di Brera, via Bianchi 46, I–20387, Merate, Italy bUniv. degli Studi dell’Insubria, via Valleggio 11, I–22100, Como, Italy c Instituto de Astronomı́a, U.N.A.M., A.P. 70-264, 04510, México, D.F., México ABSTRACT In their recent paper, Campana et al. (2007) found that 5 bursts, among those detected by Swift, are outliers with respect to the Epeak–Eγ (“Ghirlanda”) correlation. We instead argue that they are not. 1. Introduction Campana et al. (2007, C07 hereafter) investi- gate the Epeak–Eγ (so called “Ghirlanda”) cor- relation, including all GRBs detected by Swift for which we know the redshift, the peak energy Epeak and we have information on the presence of the jet break, necessary to estimate the jet open- ing angle, and therefore to calculate the collima- tion corrected bolometric energy, Eγ . In a similar study performed by us (Ghirlanda et al. 2007, G07 hereafter), we concluded that there was no new outlier with respect to the Epeak–Eγ correla- tion (besides GRB 980425 and GRB 031203, but see Ghisellini et al. 2006), while C07 claim that there are 5 Swift bursts which do not obey the correlation. The sample of GRBs studied by C07 and G07 is the same. In the following we give arguments contrasting the claim of C07. 2. GRB 060526 This burst is the second most important out- lier (in term of contribution to the χ2) presented by C07. Both C07 and G07 used the same source of data: Schaefer (2007) for the fluence and Epeak, and Dai et al. (2006) for tjet. Using the listed bolometric fluence one obtains Eγ,iso = 2.53 × 1052 erg. We recomputed the bolometric fluence from the spectral parameters reported by Schaefer (2007), obtaining Eγ,iso = 2.58 × 10 erg, which is the value we used. Instead C07 list an isotropic energy Eγ,iso = 1.07 +0.16 −0.14 × 10 erg. We remind that the isotropic energy is found through Eγ,iso = Sbol 4πd2L (1 + z) where Sbol is the bolometric fluence and the (1+z) term accounts for the cosmological time di- lation. Neglecting the (1+ z) term, and using the bolometric fluence Sbol = 1.17×10 −7, as listed by Schaefer (2007), one obtains Eγ,iso = 1.07× 10 erg, which is the value reported in C07. We there- fore suggest that C07, for this burst, missed the (1 + z) = 4.21 term when calculating Eγ,iso. The Eγ value used by C07 is therefore larger than the value found by G07 because of the larger Eγ,iso (tjet is the same). A separate problem concerns the values of Epeak and bolometric fluence for this burst re- ported by Schaefer (2007). In fact, this burst showed two main peaks in BAT, separated by ∼200 seconds, with the second peak having http://arxiv.org/abs/0704.0234v1 http://arxiv.org/abs/astro--ph/0703676 a slightly larger fluence than the first, with a softer spectrum. The spectral behaviour of this burst is thus complex, and the value of Epeak = 25 ± 5 keV reported by Schaefer (with a fluence corresponding to the first peak only) may be controversial. For this reason we have analyzed the available Swift data for this burst. Our results and the consequences for the Ghirlanda correlation can be found at: www.brera.inaf/utenti/gabriele/060526/060526.html 3. GRB 050922C and GRB 060206 These two bursts lack optical data at times late enough to encompass the jet break time predicted by the Ghirlanda relation. The fact that there is indeed an early break in the optical does not guar- antee that this is a jet break, since we now know that there is the possibility of multiple breaks in the optical. In these cases only a lower limit on the break time can be taken, corresponding to the latest optical observations, as discussed in G07. 4. GRB 050401 and GRB 050416A Several authors published a partial coverage of the optical afterglow of these two bursts, but none of them discussed the results which can be ob- tained by collecting all the available data (at least in one band). Therefore, the claim that in these GRBs there is no apparent break refers to the par- tial coverage presented in each paper. Because of that, in G07 we constructed the light curves with data from different sources. In GRB 050401 the result of the fitting is some- what dependent from the (yet unknown) assumed magnitude of the host galaxy, which can con- tribute to the late photometric points. Further- more, there is a large uncertainty in the normal- isation of the De Pasquale et al. (2006) points, because they used a different reference star for their differential photometry. What we plotted in Fig. 1 of G07 assumes the maximum possible displacement (–0.5 mag): assuming a lower one would inevitably worsen a single power law decay For GRB 050416A, it is true that Soderberg et al. (2006) stated that a single power law decay plus a 1998bw–like supernova light curve can fit the data, but also in this paper there is no com- plete collection of points coming from the avail- able literature. Anyway, SN 2006aj associated with GRB 060218 is by far the best studied at early times, so using this as a template should give a more reliable result. In this case the pres- ence of a break in the optical light curve is clearly required. Given all the above, we think that in these two GRBs there exists a margin of subjectivity for judging the presence or not of a possible jet break (this margin is however small for GRB 050416A). But just because of this, it is not appropriate to declare that they are outliers, and treat them as such in the fits. At the very least, one should consider them having a lower limit in Eγ corre- sponding to the jet break time we have derived. 5. Additional comments The pre–Swift data plotted in the figures of C07 are the values of Eγ calculated taking Eγ,iso from Firmani et al. (2006) and the jet angles from Nava et al. (2006), who reported slightly differ- ent values of Eγ,iso. Since the derived jet angle depends upon Eγ,iso, this procedure is not cor- rect. When calculating the χ2 value for the bursts in the sample of Nava et al. (2006), C07 find agree- ment in the case of an homogeneous circumburst medium, and a larger χ2 in the case of a wind profile. We instead confirm the original value re- ported in Nava et al. (2006). We note that the χ2 values given in Table 2 of C07 for the “Swift data achromatic breaks” and “Swift data pure breaks” cases, do not cor- respond to the values given in the text. GRB 061121 is plotted as a lower limit in Eγ , and lies to the left of the Ghirlanda correlation. It should not be included in the fit as instead done in C07. A symmetric error on a linear quantity trans- forms into an asymmetric error in the logarithm. We believe that C07 underestimated the error on Eγ,iso due to the systematic choice of the smallest error in the logarithmic quantity. In G07, instead, we propagated the errors in the logarithmic space. Finally, in Fig. 2 of C07 (wind case) there is an additional pre–Swift burst which is not present in Fig. 1. 6. Conclusions We would like to stress that we are not willing to defend the Epeak–Eγ correlation to death. As any other scientific result, it must be the object of severe scrutiny from the scientific community. This is even more true since its potential cosmo- logical use makes this correlation very important [as well as the related, model independent and assumption free, Liang & Zhang (2005) correla- tion]. Furthermore, its existence can flag some crucial property of GRB physics which are not yet fully understood (but some attempts have already been done, see Thompson 2006 and Thompson, Meszaros & Rees 2007). Therefore to demon- strate that this correlation is the result of some selection effects (or not), or that its dispersion is much larger than what it is now (or not), or that there are outliers (or not) is a mandatory task, that must be pursued carefully. REFERENCES 1. 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Thompson, C., Meszaros, P. & Rees, M.J., 2007, subm to ApJ, (astro–ph/0608282) http://arxiv.org/abs/astro--ph/0703676 http://arxiv.org/abs/astro-ph/0609269 http://arxiv.org/abs/astro--ph/0702352 http://arxiv.org/abs/astro-ph/0612285 http://arxiv.org/abs/astro--ph/0607511 http://arxiv.org/abs/astro--ph/0608282 Introduction GRB 060526 GRB 050922C and GRB 060206 GRB 050401 and GRB 050416A Additional comments Conclusions

0704.0235

The Determination of the Helicity of $W'$ Boson Couplings at the LHC

SLAC-PUB-12392 The Determination of the Helicity of W ′ Boson Couplings at the LHC ∗ † Thomas G. Rizzoa Stanford Linear Accelerator Center, 2575 Sand Hill Rd., Menlo Park, CA, 94025 Abstract Apart from its mass and width, the most important property of a new charged gauge boson, W ′, is the helicity of its couplings to the SM fermions. Such particles are expected to exist in many extensions of the Standard Model. In this paper we explore the capability of the LHC to determine the W ′ coupling helicity at low integrated luminosities in the ℓ+ Emiss discovery channel. We find that measurements of the transverse mass distribution, reconstructed from this final state in the W −W ′ interference region, provides the best determination of this quan- tity. To make such measurements requires integrated luminosities of ∼ 10(60) fb−1 assuming MW ′ = 1.5(2.5) TeV and provided that the W ′ couplings have Standard Model magnitude. This helicity determination can be further strengthened by the use of various discovery channel leptonic asymmetries, also measured in the same interference regime, but with higher integrated luminosities. ∗Work supported in part by the Department of Energy, Contract DE-AC02-76SF00515 †e-mail: [email protected] http://arxiv.org/abs/0704.0235v4 1 Introduction The ATLAS and CMS experiments at the LHC will begin taking data in a few months and it is widely believed that new physics beyond the Standard Model(SM) will be discovered in the coming years. There are many expectations as to what this new physics may be and in what form it will manifest itself, but it is likely that we will be in for a surprise. Once this new physics is discovered our primary goal will be to understand its essential nature and how the specific discoveries, such as the production and observed properties of new particles, fit into a broader theoretical framework. The existence of a new charged gauge boson, W ′, or a W ′-like object, is now a relatively common prediction which results from many new physics scenarios. These possibilities include the Little Higgs(LH) model[1], the Randall-Sundrum(RS)[2] model with bulk gauge fields[3], Universal Extra Dimensions(UED)[4], TeV scale extra dimensions[5, 6, 7], as well as many different extended electroweak gauge models, such as the prototypical Left-Right Symmetric Model(LRM)[8, 9]. Al- though the physics of a new Z ′ has gotten much attention in the literature[10], the detailed study of a possible W ′ has fared somewhat less well[11]. Perhaps the most important property of a W ′, apart from its mass and width, is the helicity of its couplings to the fermions in the SM. For all of the models discussed in the literature above, these couplings are either purely left- or right-handed, apart from some possible small mixing effects. Determining the helicity of the couplings of a newly discovered W ′ is thus the first major step in opening up the underlying physics as it is an order one discriminator between different classes of models.‡ As will be discussed below, there have been many suggestions over the last 20-plus years as to how to measure the helicity of W ′ couplings, all of which have their own strengths and weaknesses. These analyses have generally relied upon the use of the narrow width approximation. However, in employing this approximation much valuable information about the properties of the W ′ can be lost, in particular, that obtained from W −W ′ interference. The goal of this paper will ‡This is similar in nature to determining whether the known light neutrinos are Dirac or Majorana particles. be to explore the effects of this interference on the transverse mass dependent distributions of the W ′. As we will see the rather straightforward measurement of the transverse mass distribution itself will allow us obtain the necessary W ′ helicity information. Furthermore, we will demonstrate that such measurements will require only relatively low integrated luminosities for W ′ masses which are not too large, and will employ the traditional ℓ+EmissT W ′ discovery channel. Section II of the paper contains some background material and a historically-oriented overview of previous ideas that have been suggested to address the W ′ helicity issue including a discussion of their various strengths and weaknesses. Section III will present an analysis of the W ′ transverse mass distribution and its helicity dependence for a range of W ′ masses, cou- pling strengths and LHC integrated luminosities. The use of various asymmetries evaluated in the W −W ′ interference region in order to assist with the W ′ helicity determination will also be discussed. Section IV contains a final summary and discussion of our results. 2 Background and History Let us begin by establishing some notation; since much of this should be fairly familiar we will be rather sketchy and refer the interested reader to Ref.[10] for details. We denote the couplings of the SM fermions to the Wi = (W = WSM ,W ′) as Vff ′C i f̄γµ(1− hiγ5)f i + h.c. , (1) where for the case of Wi = WSM , the coupling strength(for leptons and quarks, respectively) and helicity factors are given by C i , hi = 1 and Vff ′ is the CKM(unit) matrix when f, f ′ are quarks(leptons); note that the helicity structure for both leptons and quarks is assumed to be the same as in all the model cases above.§ Following the notation given in Ref.[10], with some obvious §For simplicity in what follows we will further assume that the corresponding RH and LH CKM matrices are identical up to phases and we will generally neglect any possible small effects arising from W − W ′ mixing. In the case of RH couplings, we will further assume that the SM neutrinos are Dirac fields. modifications, the inclusive pp → W+i → ℓ+ν +X differential cross section can be written as dτ dy dz |Vqq′ |2 (1 + z2) + 2AG− , (2) where K is a kinematic/numerical factor that accounts for NLO and NNLO QCD corrections[12] as well as leading electroweak corrections[13] and is roughly of order ≃ 1.3 for suitably defined couplings, τ = M2/s ( s = 14 TeV at the LHC) with M2 being the lepton pair invariant mass. Furthermore, Pij(CiCj) ℓ(CiCj) q(1 + hihj) 2 (3) Pij(CiCj) ℓ(CiCj) q(hi + hj) where the sums extend over all of the exchanged particles in the s-channel. Here Pij = ŝ (ŝ−M2i )(ŝ−M2j ) + ΓiΓjMiMj [(ŝ −M2i )2 + Γ2iM2i ][i → j] , (4) with ŝ = M2 being the square of the total collision energy and Γi the total widths of the exchanged Wi particles. Note that we have employed z = cos θ, the scattering angle in the CM frame defined as that between the incoming u-type quark and the outgoing neutrino (both being fermions as opposed to being one fermion and one anti-fermion). Furthermore, the following combinations of parton distribution functions appear: q(xa,M 2)q̄′(xb,M 2)± q(xb,M2)q̄′(xa,M2)] , (5) where q(q′) is a u(d)−type quark and xa,b = τe±y are the corresponding parton momentum fractions. Analogous expressions can also be written in the case of W−i exchange by taking z → −z and interchanging initial state quarks and anti-quarks. In most cases of interest one usually converts the distribution over z above into one over the transverse mass, MT , formed from the final state lepton and the missing transverse energy associated with the neutrino; at fixed M , one has z = (1 −M2T /M2)1/2. The resulting transverse mass distribution can then be written as dy J(z → MT ) dτ dy dz , (6) where Y = min(ycut,−1/2 log τ) allows for a rapidity cut on the outgoing leptons and J(z → MT ) is the appropriate Jacobian factor[15]. In practice, ycut ≃ 2.5 for the two LHC detectors. Note that dσ will only pick out the z-even part of dσ dτ dy dz as well as the even combination of terms in the product of the parton densities, G+ . In the usual analogous fashion to the Z ′ case[10], as we will see in our discussion below, one can also define the forward-backward asymmetry as a function of the transverse mass, in principle prior to integration over the rapidity y, AFB(MT , y), whose numerator now picks out the z-odd terms in dσ dτ dy dz as well as the odd combination of terms in the parton densities G− To be complete, we note that historically when discussing new gauge boson production, particularly when dealing with states which are weakly coupled as will be the case in what fol- lows, use is often made of the narrow width approximation(NWA). In the W ′ case of relevance here, the NWA essentially replaces the integration over dτ ∼ dM by a δ function, i.e., the W ′ is assumed to be produced on-shell. Thus, for any smooth function f(M), essentially, dM f(M) dM f(M) π ΓW ′δ(M −MW ′) → π2ΓW ′f(MW ′), apart from some overall factors. Note that use of the NWA implies that we evaluate quantities on the ‘peak’ of the W ′ mass distribution, i.e., at M = MW ′ . This approximation is usually claimed to be valid up to O(ΓW ′/MW ′) corrections(at worst), but there are occasions, e.g., when W −W ′ interference is important, when its use can lead to a loss of valuable information and may even lead to wrong conclusions[16]. Unfortunately, in the W ′ case, the quantity M itself is not a true observable due to the missing longitudinal momentum of the neutrino. Given this background, let us now turn to an historical discussion of the determination of the W ′ coupling helicity. To be concrete, we will consider two different W ′ models; we will assume for simplicity that C = 1 in both cases and that only the value of hW ′ = ±1 distinguishes them. In this situation, employing the NWA, the cross section for on-shell W ′ production (followed by its leptonic decay) is proportional to ∼ (1+h2W ′) and is trivially seen to be independent of the helicity of the couplings. We would thus conclude that cross section measurements are not useful helicity discriminants. More interestingly, as was noted long ago[17], we find that the rapidity integrated value of AFB, given in the NWA by AFB ∼ h2W ′ (1 + h2W ′) , (7) also has the same value for either purely LH or RH couplings¶. Thus, in the NWA, AFB provides no help in determining the W ′ coupling helicity structure for the cases we consider here. However, we note that if the quark and leptonic coupling helicities of the W ′ are opposite, then the value of AFB will flip sign in comparison to the above expectation. It is apparent from this result that some other observable(s) must be used to distinguish these two cases. Keeping the NWA assumption, the first suggestion[18] along these lines was to examine the polarization of τ ’s originating in the decay W ′ → τν. In that paper it was explicitly shown that the the energy spectrum of the final state particle in the decay τ → ℓ, π or ρ (in the τ rest frame) was reasonably sensitive to the original W ′ helicity since the τ itself effectively decays only through the SM LH couplings of the W (provided the W ′ is sufficiently massive as we will assume here). The difficulty with this method is that the observation of this decay mode at the LHC is not all that straightforward and even the corresponding Z ′ → ττ mode, which is somewhat easier to observe, is just beginning to be studied by the LHC experimental collaborations[19]. ¶This follows immediately from the fact that we have assumed that both the hadronic and leptonic couplings of the W ′ have to have the same helicity. Clearly, measuring the polarization of the τ ’s in W ′ → τν will be reasonably difficult in the LHC detector environment and may, at the very least, require large integrated luminosities even for a relatively light W ′. The results of detailed studies by the LHC collaborations to address this issue are anxiously awaited. In the early 90’s, two important NWA-based methods for probing the helicity of the W ′ were suggested[20]. The first of these is an examination of the rare decay mode W ′ → ℓ+ℓ−W (with the W decaying into jets); in particular, one makes a measurement of the ratio of branching fractions B(W ′ → ℓ+ℓ−W ) B(W ′ → ℓν) , (8) obtained by employing the NWA. RW is expected to be roughly ∼O(0.01) or so after suitable cuts. One of the main SM backgrounds, i.e., WZ production, can essentially be removed by demanding that the dileptons do not form a Z, demanding that the mass of the jjℓℓ system be not far from the (already known) value of MW ′ and that of the dijets reconstructs to the W mass. Even after there requirements, however, some background from the continuum would remain. Furthermore, as the energy of the final state W increases it is more likely that the resulting dijets will coalesce into a single jet depending on the jet cone definition which is employed. In this case, at the very least, a very large additional background from single jets may appear; it is also possible that the events with a final state W would be completely lost without the dijet mass reconstruction. The 3ℓ+EmissT final state, with suitable cuts, would be obviously cleaner and would avoid some of these issues but at the price of an overall suppression due to ratio of branching fractions of ≃ 1/3 thus reducing the mass range over which this process would be useful. In a general gauge model, the amplitude for this process is the sum of two graphs. In the first graph, W ′− → ℓ−ν̄∗, i.e., the production of a virtual neutrino followed by the ‘decay’ ν̄∗ → ℓ+W−. Clearly, if the W ′ couples in a purely RH manner to the SM leptons then this graph will vanish in the limit of massless neutrinos due to the presence of two opposite helicity projection operators. This graph will, of course, be non-zero only if the W ′ couples in an at least partially LH manner. The second graph involves the presence of the trilinear couplings W ′ZW and W ′Z ′W ; recall that in any model with a W ′, a Z ′ will also appear just based on gauge invariance. In this case, the decay proceeds as W ′ → WZ/Z ′∗ → Wℓ+ℓ−, noting that the on-shell SM Z contribution can be removed by a suitable cut on the dilepton invariant mass. The main issue is the size of the W ′Z ′W (and W ′ZW ) couplings and this can involve such things such as, e.g., the detailed electroweak symmetry breaking patterns of the given model under study. Generically in extra dimensional models[3, 4, 5, 6, 7], these couplings are absent in the limit of small mixing due the orthogonality of the Kaluza-Klein wavefunctions of the states. In models where the SM SU(2)L arises from a diagonal breaking of the form G1 ⊗ G2 → SU(2)Diag , such as in LH models[1], the W ′Z ′W coupling is of order the SM weak coupling, g, while the W ′ZW coupling is either of order g or can be mixing angle suppressed. In other cases, such as in the LRM[8], where SU(2)L⊗SU(2)R just breaks to SU(2)L, the W ′ZW,WZ ′W couplings are only generated by mixings and for the diagrams of interest are not longitudinally enhanced. Since the amplitude associated with the pure leptonic graphs are absent in this case, the entire amplitude is mixing angle suppressed so that this process has an unobservably small rate. In fact, there are no known models where the W ′ helicity is RH and the W ′ZW,WZ ′W couplings are not mixing angle suppressed‖. Thus, based on known models, it appears that the observation of the rare decay W ′ → ℓ+ℓ−W would be a compelling indication that the W ′ is at least partially coupled in a LH manner with apparently most of the serious SM backgrounds being removable by conventional cuts. However, in making a truly model-independent analysis one must exercise care in the use of this result. A detailed analysis of the signal and backgrounds, including that for the jjℓ+ℓ− final state, for such decays including realistic detector effects would be very useful in addressing all these issues and should be performed. However, it also seems clear that is unlikely that a reliable measurement of RW can be made with relatively low integrated luminosities. ‖In a fundamental UV complete theory, this may follow directly from arguments based solely on gauge invariance and the requirement of high energy unitarity. A second, imaginative possibility is to observeWW ′ associated production[20] withW → jj for the same reasons as above. Many of the arguments made in the previous paragraph will also apply in this case as well since the diagrams responsible for this process are quite similar to previously discussed. Essentially these graphs are obtained by crossing, with the final state leptons now replaced by an initial state qq̄. In this case one looks for the jjℓEmissT final state with the ℓEmissT transverse mass peaking near MW ′ . One would anticipate this cross section to be of order ∼ 0.01 of that of the W ′ discovery channel. The main issues here are, as above, the SM backgrounds and the nature of the triple gauge vertices. It is not likely that a reliable measurement of this cross section will be performed with low luminosities that could be interpreted in a model-independent way until all of the background and detector issues are dealt with. Again, a detailed analysis including detector effects should be performed. 3 W −W ′ Interference as a Function of MT What we have learned from the previous discussion is that tools which employ the NWA are not particularly useful when we are trying to determine the W ′ coupling helicity with relatively low luminosities in an easily examined final state. One of the key reasons for this is that the use of NWA does not allow us to examine the influence of W − W ′ interference to which we now turn[21]∗∗. To be specific, in the analysis that follows, we will employ the CTEQ6M parton densities[25] and will restrict our attention only to the ℓ = e final state since it is better measured at these energies[23] yielding a better MT resolution. Furthermore, we will assume that only SM particles are accessible in the decay of theW ′ so that the total width can be straightforwardly calculated from the assumptions described above and its assumed mass value; for example, we obtain Γ(W ′) = 51.9 GeV assuming a W ′ mass of 1.5 TeV including QCD corrections. NLO QCD modifications to the distributions we discuss below have been ignored but those distributions we consider are rather ∗∗We note in passing that the usual experimental analyses at LHC[23] performed by both the ATLAS and CMS collaborations (as well as those at the Tevatron by CDF and D0[22]) ignore the effects of W −W ′ interference since these contributions are absent from default versions of stand-alone PYTHIA[24]. Figure 1: Transverse mass distribution for the production of a 1.5 TeV W ′ including interference effects at the LHC displayed on both log and linear scales assuming an integrated luminosity of 300 fb−1. The lowest histogram is the SM continuum background. The upper blue(middle red) histogram at MT = 600 GeV corresponds to the case of hW ′ = −1(1). robust against large corrections. Figure 2: Same as in the previous figure but now on a linear scale with lower luminosities and smeared by the detector resolution. In the top(bottom) panel an integrated luminosity of 30(10) fb−1 has been assumed. Detector smearing has now been included assuming δMT /MT = 2%. The most obvious distribution to examine first is dσ itself; for the moment let us restrict ourselves to the two cases where C = 1 and hW ′ = ±1. Fig. 1 shows this distribution for a large integrated luminosity, assuming MW ′ = 1.5 TeV[22], as well as the SM continuum background ††. In ††Note that we would expect to see many excess events for such W ′ masses as only ≃ 25 pb−1 of luminosity would obtaining these and other MT -deprndent distributions below, a cut on the lepton rapidity, |ηℓ ≤ 2.5, has been applied. Several things are immediately clear: (i) In the region near the Jacobian peak both distributions are quite similar; this is not surprising as this is the region where the NWA is most applicable since now MT ≃ M and W −W ′ interference is minimal. In this limit we would indeed recover our earlier result that the cross section is helicity independent. (ii) In the lower MT region where interference effects are important the two models lead to quite different distributions. In particular, for the LH case with hW ′ = 1, we observe a destructive interference with the SM amplitude producing a distribution that lies below that of the pure SM continuum background. (This is not surprising as the overall signs of the W and W ′ contributions are the same but we are at values of ŝ that are above MW yet below MW ′ so that the relevant propagators have opposite signs.) However, for the RH case with hW ′ = −1, there is no such interference and therefore the resulting distribution always lies above the SM background. It is fairly obvious that these two distributions are trivially distinguishable at these large integrated luminosities. Note that other contributions to the SM background, e.g., those from the decay of top quarks as well as guage boson pairs, have been shown to be rather small at these masses at the detector level [23], at the level of a few percent, and will be ignored in the analysis that follows. Fig. 2 shows the same dσ distribution on a linear scale but now for far smaller integrated luminosities that may be obtained during early LHC running; here we include the effects of detector smearing, with δMT /MT ≃ 2%, which is somewhat less important in the very large statistics sample cases shown above. It is immediately apparent that even with only ∼ 10 fb−1 of luminosity the two cases remain quite distinct; however, it also appears unlikely that much smaller luminosities would be very useful in this regard. This result is a significant improvement over previous attempts to determine the W ′ coupling helicity with low luminosities in clean channels. At this point there are several important questions one might ask: (i) What happens for a more massive W ′, i.e., how much luminosity will be needed in such cases to distinguish W ′ be needed to discover(5σ) such as state at the LHC. Figure 3: High luminosity plot of the transverse mass distribution assuming MW ′ = 2.5(3.5) for the upper(lower) pair of histograms along with the SM continuum background. In the interference region near ≃ 0.5MW ′ the upper(lower) member of the pair corresponds to the case of hW ′ = −1(1). Detector smearing has now been included assuming δMT /MT = 2%. couplings of opposite helicities? (ii) What if the the W ′ couplings are weaker than our canonical choice above? (iii) Do other observables, e.g., AFB, measured in the interference region below the Jacobian peak assist us in model separation? (iv) In the case where the W ′ is a KK excitation, does the presents of the additional W KK tower members alter these results? (v) In the discussion above we have assumed that CℓW ′ = C W ′ ; what would happen, e.g., if their signs were opposite thus modifying the interference bewteen the W and W ′? (vi) What if the W ′ couplings are not purely chiral and are an admixture of LH and RH helicities? It is to these issues that we now turn. Fig. 3 provides us with a high luminosity overview for the more massive cases where MW ′ = 2.5 or 3.5 TeV. In the MW ′ = 2.5 TeV case, Fig. 4 demonstrates that the full 300 fb −1 luminosity is not required to distinguish the two possibly helicities; ∼ 60fb−1 seems to be the approximate minimum luminosity that appears to be necessary. For higher masses, distinguishing the two cases becomes far more difficult due to the smaller production cross section as we see from Fig. 5 for the case of MW ′ = 3.5 TeV assuming a luminosity of 300 fb −1; essentially the full luminosity is Figure 4: Transverse mass distribution assuming a mass of 2.5 TeV for the W ′ along with the SM continuum background; the upper(lower) panel corresponds to a luminosity of 300(75) fb−1. In the interference region near ≃ 0.5MW ′ the upper(lower) histogram corresponds to the case of hW ′ = −1(1). required for model distinction in this case. Figure 5: High luminosity plot of the transverse mass distribution assuming MW ′ = 3.5 TeV along with the SM continuum background. In the interference region near ≃ 0.5MW ′ the upper(lower) histogram corresponds to the case of hW ′ = −1(1). What if the W ′ couplings are weaker? Clearly if they are too weak there will be insufficient statistics to discriminate the two possible coupling helicity assignments for any fixed value of MW ′ . In order to examine a realistic example of this situation, we consider the case of the second W KK excitation in the UED model[4, 26] with a conserved KK-parity. In such a scenario the LH couplings of this field to SM fermions vanish at tree level but are induced by one loop effects. In this case one finds that the effective values of Cℓ,q are distinct but are qualitatively of order ∼ 0.05 though we employ the specific values obtained in Ref.[4, 26] below in the actual calculations. Fig. 6 shows the transverse mass distributions in this case assuming that MW ′ =1 TeV for the second level KK state. The signal for this W KK state is clearly visible above the SM background. However, we also see that for even for these high luminosities and low masses the two helicity choices are not distinguishable. Clearly, one cannot determine the W ′ coupling helicity for such very weak interaction strengths. Semi-quantitatively, we find that that this breakdown in the discriminating power occurs when (CℓCq)1/2 ∼ 0.1 at these luminosities and masses. Figure 6: High luminosity plot of the transverse mass distribution assuming MW ′ = 1 TeV for the second W KK level in the UED model smeared by detector resolution as above. As usual the lower histogram is the SM background while the other two correspond to the signal cases with hW ′ = −1(1) and are essentially indistinguishable. We now turn to the next question we need to address: can asymmetries be useful in strength- ening our ability to determine the W ′ coupling helicity? We know from the discussion above that the answer is apparently ‘no’ in the NWA limit, i.e., when MT ≃ M . Thus we must focus our at- tention on the MT region below the peak where W−W ′ interference is strongest or, more generally, examine the asymmetries’ MT -dependence directly. The most obvious quantity to begin with is the y-integrated value of AFB for both W ′± channels. To make such a measurement, we need to know several things in addition to the sign of the lepton (which we assume can be done with ≃ 100% efficiency). At the parton level, in the case of W ′− for example, the relevant angle used to define AFB lies between the incoming d-type quark and the outgoing ℓ −. Reconstructing this direction presents us with two problems: first, since the longitudinal momentum of the ν is unknown there is an, in principle, two-fold ambiguity in the motion of the center of mass in the lab frame; this can cause a serious dilution of the observed asymmetry but can be corrected for statistically using Monte Carlo once the W ′ mass is known. Second, even when it is determined, the direction of motion of the center of mass is not necessarily that of the d-type quark though it is likely to be so when the boost of the center of mass frame is large. The later problem also arises for the case of a Z ′ and has also been shown to be mostly correctable in detailed Monte Carlo studies[27]. For the moment, let us forget these issues and ask what the y-integrated AFB(MT ) looks like in both ℓ± channels; the results are shown in Fig. 7 assuming high luminosities and MW ′ = 1.5 TeV. Here we see that these integrated quantities, even for luminosities of 300 fb−1, are essentially useless in distinguishing the two coupling helicity cases. Furthermore, we also see that the two coupling helicities lead to essentially identical results when MT ≃ MW ′ as would be expected based on the NWA. A short analysis indicates that approximately ten times more integrated luminosity would be required before some separation in the two cases becomes possible[28]. Clearly this situation would only become worse if we were to raise the mass of the W ′ or reduce its coupling strength. It is perhaps possible that some information is lost by only using the integrated quantity AFB and we need to consider instead AFB(yW ), where yW is the rapidity of the center of mass frame. This distribution is odd under the interchange yW → −yW at the LHC so we can simply fold this distribution over the yW = 0 boundary to double the statistics. Furthermore, by integrating over a wide MT range in the interference region below the W ′ peak, e.g., 0.4 ≤ MT ≤ 1 TeV in the case of a 1.5 TeV W ′, further statistics can be gained. Fig 8 shows the resulting AFB(yW ) distributions for a W ′± with mass of 1.5 TeV assuming a luminosity of 300 fb−1 for hW ′ = ±1. At these large luminosities, the AFB(yW ) distributions for the two helicity choices are clearly distinguishable but this will certainly become more difficult for lower luminosities or for larger masses. We find that we essentially loose all coupling helicity information when the luminosity falls much below ≃ 100fb−1 for this W ′ mass. The next observable we consider is the charge asymmetry, AWQ(yW ): AWQ(yW ) = N+(yW )−N−(yW ) N+(yW ) +N−(yW ) , (9) where N±(yW ) are the number of events with charged leptons of sign ± in a given bin of rapidity. Note that at the LHC, AWQ(yW ) is symmetric under yW → −yW so that we can again fold the Figure 7: The y-integrated value of AFB, as a function of the transverse mass, assuming a mass of 1.5 TeV for the W ′+(W ′−) in the top(bottom) panel. Here an integrated luminosity of 300 fb−1 has been assumed. The two essentially indistinguishable histograms correspond to the two possible choices of the helicity, hW ′ = ±1. Figure 8: The value of AFB as a function the center of mass rapidity, yW , integrated over the transverse mass bin 400-1000 GeV assuming a mass of 1.5 TeV for theW ′−) in the top(bottom) panel. An integrated luminosity of 300 fb−1 has been assumed and the distribution has been folded around yW = 0. The upper(lower) set of data points in the top(lower) panel for small values of yW corresponds to the choice of hW ′ = −1. Note that we have chosen signs to make the ranges of AFB comparable in both cases. distribution around yW = 0. Fig. 9 shows this distribution, integrated over the interference region 0.4 ≤ MT ≤ 1 TeV, assuming MW ′ = 1.5 TeV and a luminosity of 300 fb−1. It is clear that at this level of integrated luminosity the two distributions are reasonably distinguishable. However, as we lower the luminosity or raise the mass of the W ′ the quality of the separation degrades significantly. Certainly for luminosities less that ≃ 100 fb−1, this asymmetry measurement would not be very helpful. Thus AWQ(yW ) is not a very useful tool for coupling helicity determination until high luminosities become available. Figure 9: The W − W ′ induced charge asymmetry, assuming MW ′ = 1.5 TeV, as a function the center of mass rapidity, yW , integrated over the transverse mass bin 400-1000 GeV. An integrated luminosity of 300 fb−1 has been assumed and the distribution has been folded around yW = 0. The upper set of data points at low values of yW corresponds to the choice of hW ′ = 1. A last asymmetry possibility to consider is the rapidity asymmetry for the final state charged leptons themselves, Aℓ(yℓ): Aℓ(yℓ) = N+(yℓ)−N−(yℓ) N+(yℓ) +N−(yℓ) , (10) which is also an even function of yℓ so the distribution can again be folded around yℓ = 0. The resulting distribution can be seen in Fig. 10 for large integrated luminosities. Here we again see reasonable model differentiation at low values of yℓ <∼ 1 but this fades in utility as integrated luminosities drop much below ≃ 100 fb−1 as the two curves are generally rather close. From this general discussion of possibly asymmetries that one can form employing this final state we can thus conclude that their usefulness in coupling helicity determination will require ≃ 100fb−1. Figure 10: The W − W ′ induced lepton asymmetry, assuming MW ′ = 1.5 TeV, as a function the lepton’s rapidity, yℓ, integrated over the transverse mass bin 400-1100 GeV. An integrated luminosity of 300 fb−1 has been assumed and the distribution has been folded around yℓ = 0. The upper set of data points at low values of yℓ corresponds to the choice of hW ′ = 1. In the case of extra dimensions we know that an entire tower of W ′-like KK states is expected to exist. Do the presence of these additional states modify the results we have obtained above for an ordinary W ′? To address this, consider the simplified case of a second W ′-like KK state which have the same coupling strength as the SM W and is twice as heavy as the W ′ discussed above, i.e.,3 TeV. Now imagine that the coupling helicity of this second state is uncorrelated with that of the W ′; in the MT distribution in the W −W ′ interference region influenced by this state? The upper panel in Fig. 11 addresses this issue for modest luminosities including the effects of smearing. The upper(lower) set of three histograms corresponds to the case where hW ′ = −1(1) and either there is no W ′′, as above, or hW ′′ = ±1. This demonstrates that the existence of the extra KK states has little influence on the results we obtained above independent of their coupling helicities. Up to now we have assumed that CℓW ′ = C W ′ ; what if this was no longer true? How would the MT distribution and our ability to determine coupling helicity be modified? The simplest case to examine is CℓW ′ = −C = 1 with hW ′ = ±1. (Note that interchanging the signs of these two couplings, i.e., which one of these two couplings we choose to be negative, has no physical effect on the MT distribution or on any of the asymmetries discussed earlier.) The result of this investigation is shown in the lower panel of Fig. 11. Here the red(green) histograms correspond to the cases analyzed above where CℓW ′ = C W ′ = 1 and hW ′ = 1(−1) whereas the blue(magenta) histograms corresponds to the cases where CℓW ′ = −C = 1 with hW ′ = 1(−1). It is clear from this Figure that the MT distribution distinguishes only three of these cases with the C W ′ = ±C = 1, hW ′ = −1 possibilities being degenerate. The reason for this is that in both these cases there is no interference with the SMW ′ exchange and in the pureW ′ term in the cross section this sign change is irrelevant; these two degenerate cases are, of course, separable using the information obtained from AFB as they produce values with opposite sign. Lastly, and to be more general, we must at least consider possible scenarios where the couplings of the W ′ to SM fermions are a substantial admixture of both LH and RH helicities, though obvious examples of such kinds of models are apparently absent from the existing literature. To get a feel for such a possibility, we perform two analyses: first, we set Cℓ,q = 1 as before and vary the values of hW ′ between pairs of positive and negative values. As we do this, the helicity of the couplings of the W ′ will vary as will its total decay width which behaves as ∼ 1 + h2W . In a second analysis, we can rescale the values of the Cℓ,q so that the W ′ width is held fixed. In this case, as we will see, the resulting histograms for the transverse mass distribution lie especially close to one another. The results of these two sets of calculations are shown in Fig. 12 in the case of large integrated luminosities assuming the default value of MW ′ = 1.5 TeV. In the first analysis shown in the top panel, we see that at these assumed luminosities all of the different histograms Figure 11: SmearedMT distributions for several scenarios; the top panel, the lower(upper) compares the single W ′ case discussed above to that where a second KK state, W ′′, exists with coupling helicities uncorrelated to that of the W ′. Details are given in the text. In the lower panel, we compare the cases for hW ′ = ±1 allowing for the possibility that CℓW ′ = ±C with the signs uncorrelated with the coupling helicity; the details are discussed in the text. are distinguishable and not just the two pairs of cases with opposite helicities. This result generally remains true down to luminosities ∼ 75fb−1 or so. If we are only interested in separating opposite helicity pairs then we find that the cases hW ′ = ±0.8(0.6, 0.4, 0.2) can be distinguished down to luminosities of order ∼ 10(25, 50, 75)fb−1 , respectively. In the second analysis, as seen in the lower panel of the figure, the histograms for hW ′ = 0.8, 0.6 and 0.4 (as well as for their corresponding opposite helicity partners) are very close to one another and are essentially inseparable even at these high luminosities. However, the two sets of opposite helicity histograms remain distinguishable and this will remains true down to luminosities of order 30 − 75 fb−1. It would seem from these analyses that the transverse mass distribution will play the dominant role in W ′ coupling helicity determination in all possible cases although somewhat higher integrated luminosities may be required in some scenarios. 4 Summary and Conclusions Apart from its mass and width, the most important property of a new charged gauge boson, W ′, is the helicity of its couplings to the SM fermions. Such particles are predicted to exist in the TeV mass range in many new physics models and this coupling helicity is an order one discriminator between the various classes of models. The main difficulties with the existing techniques for determining this helicity are potentially threefold: (i) they require rather high integrated luminosities even for a relatively light W ′, and/or (ii) they are sufficiently intricate as to require a detailed background and detector study to determine their feasibility, and/or (iii) they make use of more complex final states other than the standard ℓ + EmissT discovery channel. Some of these techniques also suffer from employing the narrow width approximation which can result in loss of valuable information regarding the effects of W − W ′ interference. In this paper we propose a simple technique for making this helicity determination at the LHC. In order to attempt to circumvent all of these difficulties, we have examined the W −W ′ interference region of the transverse mass distribution Figure 12: Same as the linear plot shown in Fig.1, but now for other values of the coupling helicities. From top to bottom the pairs of histograms in the upper panel correspond to h(W ′) = ±0.8,±0.6,±0.4and±0.2, respectively. The next lowest single histogram corresponds to the case of pure vector couplings, i.e., h(W ′) = 0. In producing these results we have assumed that the values of the Cℓ,q=1. In the lower panel, we show the same result now but with the overall couplings rescaled so as to keep the W ′ width a constant. for the ℓ + EmissT discovery mode. We have found that this distribution is particularly sensitive to the helicity of the W ′ couplings. In particular, using this technique we have shown that such helicity differentiation requires only ∼ 10(60, 300) fb−1 assuming MW ′ = 1.5(2.5, 3.5) TeV and provided that the W ′ has Standard Model strength couplings. 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0704.0236

Curvature flows in semi-Riemannian manifolds

CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS CLAUS GERHARDT Abstract. We prove that the limit hypersurfaces of converging curva- ture flows are stable, if the initial velocity has a weak sign, and give a survey of the existence and regularity results. Contents 1. Introduction 1 2. Notations and preliminary results 2 3. Evolution equations for some geometric quantities 4 4. Essential parabolic flow equations 9 5. Stability of the limit hypersurfaces 15 6. Existence results 25 7. The inverse mean curvature flow 39 8. The IMCF in ARW spaces 41 Transition from big crunch to big bang 45 References 47 1. Introduction In this paper we want to give a survey of the existence and regularity results for extrinsic curvature flows in semi-Riemannian manifolds, i.e., Riemannian or Lorentzian ambient spaces, with an emphasis on flows in Lorentzian spaces. In order to treat both cases simultaneously terminology like spacelike, timelike, etc., that only makes sense in a Lorentzian setting should be ignored in the Riemannian case. The general stability result for the limit hypersurfaces of converging curva- ture flows in Section 5 is new. The regularity result in Theorem 6.5—especially the time independent Cm+2,α-estimates—for converging curvature flows that are graphs is interesting too. Date: October 23, 2018. 2000 Mathematics Subject Classification. 35J60, 53C21, 53C44, 53C50, 58J05. Key words and phrases. semi-Riemannian manifold, mass, stable solutions, cosmological spacetime, general relativity, curvature flows, ARW spacetime. This research was supported by the Deutsche Forschungsgemeinschaft. http://arxiv.org/abs/0704.0236v4 2 CLAUS GERHARDT 2. Notations and preliminary results The main objective of this section is to state the equations of Gauß, Co- dazzi, and Weingarten for hypersurfaces. In view of the subtle but important difference that is to be seen in the Gauß equation depending on the nature of the ambient space—Riemannian or Lorentzian—, which we already men- tioned in the introduction, we shall formulate the governing equations of a hypersurface M in a semi-Riemannian (n+1)-dimensional space N , which is either Riemannian or Lorentzian. Geometric quantities in N will be denoted by (ḡαβ), (R̄αβγδ), etc., and those in M by (gij), (Rijkl), etc. Greek indices range from 0 to n and Latin from 1 to n; the summation convention is always used. Generic coordinate systems in N resp. M will be denoted by (xα) resp. (ξi). Covariant differentiation will simply be indicated by indices, only in case of possible ambiguity they will be preceded by a semicolon, i.e. for a function u in N , (uα) will be the gradient and (uαβ) the Hessian, but e.g., the covariant derivative of the curvature tensor will be abbreviated by R̄αβγδ;ǫ. We also point out that (2.1) R̄αβγδ;i = R̄αβγδ;ǫx with obvious generalizations to other quantities. Let M be a spacelike hypersurface, i.e. the induced metric is Riemannian, with a differentiable normal ν. We define the signature of ν, σ = σ(ν), by (2.2) σ = ḡαβν ανβ = 〈ν, ν〉. In case N is Lorentzian, σ = −1, and ν is timelike. In local coordinates, (xα) and (ξi), the geometric quantities of the spacelike hypersurface M are connected through the following equations (2.3) xαij = −σhijνα the so-called Gauß formula. Here, and also in the sequel, a covariant derivative is always a full tensor, i.e. (2.4) xαij = x ,ij − Γ kijxαk + Γ̄αβγx The comma indicates ordinary partial derivatives. In this implicit definition the second fundamental form (hij) is taken with respect to −σν. The second equation is the Weingarten equation (2.5) ναi = h where we remember that ναi is a full tensor. Finally, we have the Codazzi equation (2.6) hij;k − hik;j = R̄αβγδναxβi x and the Gauß equation (2.7) Rijkl = σ{hikhjl − hilhjk}+ R̄αβγδxαi x CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 3 Here, the signature of ν comes into play. 2.1. Definition. (i) Let F ∈ C0(Γ̄ ) ∩ C2,α(Γ ) be a strictly monotone cur- vature function, where Γ ⊂ Rn is a convex, open, symmetric cone containing the positive cone, such that (2.8) F |∂Γ = 0 ∧ F |Γ > 0. Let N be semi-Riemannian. A spacelike, orientable1 hypersurface M ⊂ N is called admissible, if its principal curvatures with respect to a chosen normal lie in Γ . This definition also applies to subsets of M . (ii) Let M be an admissible hypersurface and f a function defined in a neighbourhood of M . M is said to be an upper barrier for the pair (F, f), if (2.9) F |M ≥ f (iii) Similarly, a spacelike, orientable hypersurfaceM is called a lower barrier for the pair (F, f), if at the points Σ ⊂M , where M is admissible, there holds (2.10) F |Σ ≤ f. Σ may be empty. (iv) If we consider the mean curvature function, F = H , then we suppose F to be defined in Rn and any spacelike, orientable hypersurface is admissible. One of the assumptions that are used when proving a priori estimates is that there exists a strictly convex function χ ∈ C2(Ω̄) in a given domain Ω. We shall state sufficient geometric conditions guaranteeing the existence of such a function. The lemma below will be valid in Lorentzian as well as Riemannian manifolds, but we formulate and prove it only for the Lorentzian case. 2.2. Lemma. Let N be globally hyperbolic, S0 a Cauchy hypersurface, (xα) a special coordinate system associated with S0, and Ω̄ ⊂ N be compact. Then, there exists a strictly convex function χ ∈ C2(Ω̄) provided the level hypersur- faces {x0 = const} that intersect Ω̄ are strictly convex. Proof. For greater clarity set t = x0, i.e., t is a globally defined time function. Let x = x(ξ) be a local representation for {t = const}, and ti, tij be the covariant derivatives of t with respect to the induced metric, and tα, tαβ be the covariant derivatives in N , then (2.11) 0 = tij = tαβx j + tαx and therefore, (2.12) tαβx j = −tαx ij = −h̄ijtανα. Here, (να) is past directed, i.e., the right-hand side in (2.12) is positive definite in Ω̄, since (tα) is also past directed. 1A hypersurface is said to be orientable, if it has a continuous normal field. 4 CLAUS GERHARDT Choose λ > 0 and define χ = eλt, so that (2.13) χαβ = λ 2eλttαtβ + λe λttαβ . Let p ∈ Ω be arbitrary, S = {t = t(p)} be the level hypersurface through p, and (ηα) ∈ Tp(N). Then, we conclude (2.14) e−λtχαβη αηβ = λ2|η0|2 + λtijηiηj + 2λt0jη0ηi, where tij now represents the left-hand side in (2.12), and we infer further (2.15) e−λtχαβη αηβ ≥ 1 λ2|η|02 + [λǫ− cǫ]σijηiηj λ{−|η0|2 + σijηiηj} for some ǫ > 0, and where λ is supposed to be large. Therefore, we have in Ω̄ (2.16) χαβ ≥ cḡαβ , c > 0, i.e., χ is strictly convex. � 3. Evolution equations for some geometric quantities Curvature flows are used for different purposes, they can be merely vehicles to approximate a stationary solution, in which case the flow is driven not only by a curvature function but also by the corresponding right-hand side, an external force, if you like, or the flow is a pure curvature flow driven only by a curvature function, and it is used to analyze the topology of the initial hypersurface, if the ambient space is Riemannian, or the singularities of the ambient space, in the Lorentzian case. In this section we are treating very general curvature flows2 in a semi- Riemannian manifold N = Nn+1, though we only have the Riemannian or Lorentzian case in mind, such that the flow can be either a pure curvature flow or may also be driven by an external force. The nature of the ambient space, i.e., the signature of its metric, is expressed by a parameter σ = ±1, such that σ = 1 corresponds to the Riemannian and σ = −1 the Lorentzian case. The parameter σ can also be viewed as the signature of the normal of the spacelike hypersurfaces, namely, (3.1) σ = 〈ν, ν〉. Properties like spacelike, achronal, etc., however, only make sense, when N is Lorentzian and should be ignored otherwise. We consider a strictly monotone, symmetric, and concave curvature F ∈ C4,α(Γ ), homogeneous of degree 1, a function 0 < f ∈ C4,α(Ω), where Ω ⊂ N is an open set, and a real function Φ ∈ C4,α(R+) satisfying (3.2) Φ̇ > 0 and Φ̈ ≤ 0. For notational reasons, let us abbreviate (3.3) f̃ = Φ(f). 2We emphasize that we are only considering flows driven by the extrinsic curvature not by the intrinsic curvature. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 5 Important examples of functions Φ are (3.4) Φ(r) = r, Φ(r) = log r, Φ(r) = −r−1 (3.5) Φ(r) = r k , Φ(r) = −r− 1k , k ≥ 1. 3.1. Remark. The latter choices are necessary, if the curvature function F is not homogeneous of degree 1 but of degree k, like the symmetric polynomials Hk. In this case we would sometimes like to define F = Hk and not H k , since (3.6) F ij = is then divergence free, if the ambient space is a spaceform, cf. Lemma 5.8 on page 24, though on the other hand we need a concave operator for technical reasons, hence we have to take the k-th root. The curvature flow is given by the evolution problem (3.7) ẋ = −σ(Φ− f̃)ν, x(0) = x0, where x0 is an embedding of an initial compact, spacelike hypersurfaceM0 ⊂ Ω of class C6,α, Φ = Φ(F ), and F is evaluated at the principal curvatures of the flow hypersurfaces M(t), or, equivalently, we may assume that F depends on the second fundamental form (hij) and the metric (gij) of M(t); x(t) is the embedding of M(t) and σ the signature of the normal ν = ν(t), which is identical to the normal used in the Gaussian formula (2.3) on page 2. The initial hypersurface should be admissible, i.e., its principal curvatures should belong to the convex, symmetric cone Γ ⊂ Rn. This is a parabolic problem, so short-time existence is guaranteed, cf. [18, Chapter 2.5] There will be a slight ambiguity in the terminology, since we shall call the evolution parameter time, but this lapse shouldn’t cause any misunderstand- ings, if the ambient space is Lorentzian. At the moment we consider a sufficiently smooth solution of the initial value problem (3.7) and want to show how the metric, the second fundamental form, and the normal vector of the hypersurfaces M(t) evolve. All time derivatives are total derivatives, i.e., covariant derivatives of tensor fields defined over the curve x(t), cf. [17, Chapter 11.5]; t is the flow parameter, also referred to as time, and (ξi) are local coordinates of the initial embedding x0 = x0(ξ) which will also serve as coordinates for the the flow hypersurfaces M(t). The coordinates in N will be labelled (xα), 0 ≤ α ≤ n. 3.2. Lemma (Evolution of the metric). The metric gij of M(t) satisfies the evolution equation (3.8) ġij = −2σ(Φ− f̃)hij . 6 CLAUS GERHARDT Proof. Differentiating (3.9) gij = 〈xi, xj〉 covariantly with respect to t yields (3.10) ġij = 〈ẋi, xj〉+ 〈xi, ẋj〉 = −2σ(Φ− f̃)〈xi, νj〉 = −2σ(Φ− f̃)hij , in view of the Codazzi equations. � 3.3.Lemma (Evolution of the normal). The normal vector evolves according (3.11) ν̇ = ∇M (Φ− f̃) = gij(Φ− f̃)ixj . Proof. Since ν is unit normal vector we have ν̇ ∈ T (M). Furthermore, differ- entiating (3.12) 0 = 〈ν, xi〉 with respect to t, we deduce �(3.13) 〈ν̇, xi〉 = −〈ν, ẋi〉 = (Φ− f̃)i. 3.4. Lemma (Evolution of the second fundamental form). The second fun- damental form evolves according to (3.14) ḣ i = (Φ− f̃) i + σ(Φ− f̃)h k + σ(Φ− f̃)R̄αβγδν γxδkg (3.15) ḣij = (Φ− f̃)ij − σ(Φ − f̃)hki hkj + σ(Φ − f̃)R̄αβγδναx γxδj . Proof. We use the Ricci identities to interchange the covariant derivatives of ν with respect to t and ξi (3.16) (ναi ) = (ν̇ α)i − R̄αβγδνβx = gkl(Φ− f̃)kixαl + gkl(Φ− f̃)kxαli − R̄αβγδνβx For the second equality we used (3.11). On the other hand, in view of the Weingarten equation we obtain (3.17) D (ναi ) = (hki x k ) = ḣ k + h Multiplying the resulting equation with ḡαβx j we conclude (3.18) ḣki gkj − σ(Φ − f̃)hki hkj = (Φ− f̃)ij + σ(Φ − f̃)R̄αβγδναx or equivalently (3.14). To derive (3.15), we differentiate (3.19) hij = h i gkj with respect to t and use (3.8). � CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 7 We emphasize that equation (3.14) describes the evolution of the second fundamental form more meaningfully than (3.15), since the mixed tensor is independent of the metric. 3.5. Lemma (Evolution of (Φ− f̃)). The term (Φ− f̃) evolves according to the equation (3.20) (Φ− f̃) − Φ̇F ij(Φ− f̃)ij =σΦ̇F ijhikhkj (Φ− f̃) + σf̃ανα(Φ− f̃) + σΦ̇F ijR̄αβγδν γxδj(Φ − f̃), where (3.21) (Φ− f̃)′ = d (Φ− f̃) (3.22) Φ̇ = Φ(r). Proof. When we differentiate F with respect to t we consider F to depend on the mixed tensor h i and conclude (3.23) (Φ− f̃)′ = Φ̇F ij ḣ i − f̃αẋ The equation (3.20) then follows in view of (3.7) and (3.14). � 3.6. Remark. The preceding conclusions, except Lemma 3.5, remain valid for flows which do not depend on the curvature, i.e., for flows (3.24) ẋ = −σ(−f)ν = σfν, x(0) = x0, where f = f(x) is defined in an open set Ω containing the initial spacelike hypersurface M0. In the preceding equations we only have to set Φ = 0 and f̃ = f . The evolution equation for the mean curvature then looks like (3.25) Ḣ = −∆f − σ{|A|2 + R̄αβνανβ}f, where the Laplacian is the Laplace operator on the hypersurfaceM(t). This is exactly the derivative of the mean curvature operator with respect to normal variations as we shall see in a moment. But first let us consider the following example. 3.7. Example. Let (xα) be a future directed Gaussian coordinate system in N , such that the metric can be expressed in the form (3.26) ds̄2 = e2ψ{σ(dx0)2 + σijdxidxj}. Denote by M(t) the coordinate slices {x0 = t}, then M(t) can be looked at as the flow hypersurfaces of the flow (3.27) ẋ = −σ(−eψ)ν̄, 8 CLAUS GERHARDT where we denote the geometric quantities of the slices by ḡij , ν̄, h̄ij , etc. Here x is the embedding (3.28) x = x(t, ξi) = (t, xi). Notice that, if N is Riemannian, the coordinate system and the normal are always chosen such that ν0 > 0, while, if N is Lorentzian, we always pick the past directed normal. Hence the mean curvature of the slices evolves according to (3.29) ˙̄H = −∆eψ − σ{|Ā|2 + R̄αβ ν̄αν̄β}eψ. We can now derive the linearization of the mean curvature operator of a spacelike hypersurface, compact or non-compact. 3.8. Let M0 ⊂ N be a spacelike hypersurface of class C4. We first assume that M0 is compact; then there exists a tubular neighbourhood U and a cor- responding normal Gaussian coordinate system (xα) of class C3 such that ∂ is normal to M0. Let us consider in U of M0 spacelike hypersurfaces M that can be writ- ten as graphs over M0, M = graphu, in the corresponding normal Gaussian coordinate system. Then the mean curvature of M can be expressed as (3.30) H = {−∆u+ H̄ − σv−2uiujh̄ij}v, where σ = 〈ν, ν〉, and hence, choosing u = ǫϕ, ϕ ∈ C2(M0), we deduce (3.31) H |ǫ=0 = −∆ϕ+ ˙̄Hϕ = −∆ϕ− σ(|Ā|2 + R̄αβνανβ)ϕ, in view of (3.29). The right-hand side is the derivative of the mean curvature operator applied to ϕ. If M0 is non-compact, tubular neighbourhoods exist locally and the relation (3.31) will be valid for any ϕ ∈ C2c (M0) by using a partition of unity. The preceding linearization can be immediately generalized to a hypersurface M0 solving the equation (3.32) F |M0 = f, where f = f(x) is defined in a neighbourhood of M0 and F = F (hij) is curva- ture operator. 3.9. Lemma. Let M0 be of class C m,α, m ≥ 2, 0 ≤ α ≤ 1, satisfy (3.32). Let U be a (local) tubular neighbourhood of M0, then the linearization of the operator F − f expressed in the normal Gaussian coordinate system (xα) cor- responding to U and evaluated at M0 has the form (3.33) − F ijuij − σ{F ijhki hkj + F ijR̄αβγδναx γxδj + fαν CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 9 where u is a function defined in M0, and all geometric quantities are those of M0; the derivatives are covariant derivatives with respect to the induced metric of M0. The operator will be self-adjoint, if F ij is divergence free. Proof. For simplicity assume thatM0 is compact, and let u ∈ C2(M0) be fixed. Then the hypersurfaces (3.34) Mǫ = graph(ǫu) stay in the tubular neighbourhood U for small ǫ, |ǫ| < ǫ0, and their second fundamental forms (hij) can be expressed as (3.35) v−1hij = −(ǫu)ij + h̄ij , where h̄ij is the second fundamental form of the coordinate slices {x0 = const}. We are interested in (3.36) (F − f)|ǫ=0 . To differentiate F with respect to ǫ it is best to consider the mixed form i ) of the second fundamental form to derive (3.37) (F − f) = F ij ḣ u = −F ijuij + F ij ˙̄h where the equation is evaluated at ǫ = 0 and ˙̄h i is the derivative of h̄ i with respect to x0. The result then follows from the evolution equation (3.14) for the flow (3.27), i.e., we have to replace (Φ− f̃) in (3.14) by −1. � 4. Essential parabolic flow equations From (3.14) on page 6 we deduce with the help of the Ricci identities a parabolic equation for the second fundamental form 4.1. Lemma. The mixed tensor h i satisfies the parabolic equation (4.1) i − Φ̇F i;kl = σΦ̇F klhrkh i − σΦ̇Fhrih rj + σ(Φ− f̃)hki h − f̃αβxαi x kj + σf̃αν i + Φ̇F kl,rshkl;ih + Φ̈FiF j + 2Φ̇F klR̄αβγδx − Φ̇F klR̄αβγδxαmx rj − Φ̇F klR̄αβγδxαmx + σΦ̇F klR̄αβγδν γxδl h i − σΦ̇F R̄αβγδν γxδmg + σ(Φ− f̃)R̄αβγδναxβi ν γxδmg + Φ̇F klR̄αβγδ;ǫ{ναxβkx mj + ναx 10 CLAUS GERHARDT Proof. We start with equation (3.14) on page 6 and shall evaluate the term (4.2) (Φ− f̃)ji ; since we are only working with covariant spatial derivatives in the subsequent proof, we may—and shall—consider the covariant form of the tensor (4.3) (Φ− f̃)ij . First we have (4.4) Φi = Φ̇Fi = Φ̇F klhkl;i (4.5) Φij = Φ̇F klhkl;ij + Φ̈F klhkl;iF rshrs;j + Φ̇F kl,rshkl,;ihrs;j . Next, we want to replace hkl;ij by hij;kl. Differentiating the Codazzi equation (4.6) hkl;i = hik;l + R̄αβγδν where we also used the symmetry of hik, yields (4.7) hkl;ij = hik;lj + R̄αβγδ;ǫν + R̄αβγδ{ναj x i + ν i + ν i + ν To replace hkl;ij by hij;kl we use the Ricci identities (4.8) hik;lj = hik;jl + hakR ilj + haiR and differentiate once again the Codazzi equation (4.9) hik;j = hij;k + R̄αβγδν To replace f̃ij we use the chain rule (4.10) f̃i = f̃αx f̃ij = f̃αβx j + f̃αx Then, because of the Gauß equation, Gaussian formula, and Weingarten equation, the symmetry properties of the Riemann curvature tensor and the assumed homogeneity of F , i.e., (4.11) F = F klhkl, we deduce (4.1) from (3.14) on page 6 after reverting to the mixed representa- tion. � 4.2. Remark. If we had assumed F to be homogeneous of degree d0 instead of 1, then we would have to replace the explicit term F—occurring twice in the preceding lemma—by d0F . If the ambient semi-Riemannian manifold is a space of constant curvature, then the evolution equation of the second fundamental form simplifies consid- erably, as can be easily verified. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 11 4.3. Lemma. Let N be a space of constant curvature KN , then the second fundamental form of the curvature flow (3.7) on page 5 satisfies the parabolic equation (4.12) i − Φ̇F i;kl = σΦ̇F klhrkh i − σΦ̇Fhrih rj + σ(Φ − f̃)hki h − f̃αβxαi x kj + σf̃αν i + Φ̇F kl,rshkl;ih + Φ̈FiF +KN{(Φ− f̃)δji + Φ̇F δ i − Φ̇F klgklh Let us now assume that the open set Ω ⊂ N containing the flow hyper- surfaces can be covered by a Gaussian coordinate system (xα), i.e., Ω can be topologically viewed as a subset of I × S0, where S0 is a compact Riemannian manifold and I an interval. We assume furthermore, that the flow hypersur- faces can be written as graphs over S0 (4.13) M(t) = { x0 = u(xi) : x = (xi) ∈ S0 }; we use the symbol x ambiguously by denoting points p = (xα) ∈ N as well as points p = (xi) ∈ S0 simply by x, however, we are careful to avoid confusions. Suppose that the flow hypersurfaces are given by an embedding x = x(t, ξ), where ξ = (ξi) are local coordinates of a compact manifoldM0, which then has to be homeomorphic to S0, then (4.14) x0 = u(t, ξ) = u(t, x(t, ξ)), xi = xi(t, ξ). The induced metric can be expressed as (4.15) gij = 〈xi, xj〉 = σuiuj + σklxki xlj , where (4.16) ui = ukx i.e., (4.17) gij = {σukul + σkl}xki xlj , hence the (time dependent) Jacobian (xki ) is invertible, and the (ξ i) can also be viewed as coordinates for S0. Looking at the component α = 0 of the flow equation (3.7) on page 5 we obtain a scalar flow equation (4.18) u̇ = −e−ψv−1(Φ− f̃), which is the same in the Lorentzian as well as in the Riemannian case, where (4.19) v2 = 1− σσijuiuj, and where (4.20) |Du|2 = σijuiuj 12 CLAUS GERHARDT is of course a scalar, i.e., we obtain the same expression regardless, if we use the coordinates xi or ξi. The time derivative in (4.18) is a total time derivative, if we consider u to depend on u = u(t, x(t, ξ)). For the partial time derivative we obtain (4.21) = u̇− ukẋki = −e−ψv(Φ− f̃), in view of (3.7) on page 5 and our choice of normal ν = (να) (4.22) (να) = σe−ψv−1(1,−σui), where ui = σijuj. Controlling the C1-norm of the graphs M(t) is tantamount to controlling v, if N is Riemannian, and ṽ = v−1, if N is Lorentzian. The evolution equations satisfied by these quantities are also very important, since they are used for the a priori estimates of the second fundamental form. Let us start with the Lorentzian case. 4.4. Lemma (Evolution of ṽ). Consider the flow (3.7) in a Lorentzian space N such that the spacelike flow hypersurfaces can be written as graphs over S0. Then, ṽ satisfies the evolution equation (4.23) ˙̃v − Φ̇F ij ṽij =− Φ̇F ijhikhkj ṽ + [(Φ− f̃)− Φ̇F ]ηαβνανβ − 2Φ̇F ijhkjxαi x kηαβ − Φ̇F ijηαβγx − Φ̇F ijR̄αβγδναxβi x − f̃βxβi x k ηαg where η is the covariant vector field (ηα) = e ψ(−1, 0, . . . , 0). Proof. We have ṽ = 〈η, ν〉. Let (ξi) be local coordinates for M(t). Differenti- ating ṽ covariantly we deduce (4.24) ṽi = ηαβx α + ηαν (4.25) ṽij = ηαβγx α + ηαβx + ηαβx j + ηαβx i + ηαν The time derivative of ṽ can be expressed as (4.26) ˙̃v = ηαβ ẋ βνα + ηαν̇ = ηαβν ανβ(Φ − f̃) + (Φ− f̃)kxαk ηα = ηαβν ανβ(Φ − f̃) + Φ̇F kxαk ηα − f̃βx ikηα, where we have used (3.11) on page 6. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 13 Substituting (4.25) and (4.26) in (4.23), and simplifying the resulting equa- tion with the help of the Weingarten and Codazzi equations, we arrive at the desired conclusion. � In the Riemannian case we consider a normal Gaussian coordinate system (xα), for otherwise we won’t obtain a priori estimates for v, at least not without additional strong assumptions. We also refer to x0 = r as the radial distance function. 4.5. Lemma (Evolution of v). Consider the flow (3.7) in a normal Gaussian coordinate system where the M(t) can be written as graphs of a function u(t) over some compact Riemannian manifold S0. Then the quantity (4.27) v = 1 + |Du|2 = (rανα)−1 satisfies the evolution equation (4.28) v̇ − Φ̇F ijvij = −Φ̇F ijhikhkj v − 2v−1Φ̇F ijvivj + rαβν ανβ [(Φ− f̃)− Φ̇F ]v2 + 2Φ̇F ijhki rαβxαkx + Φ̇F ijR̄αβγδν + Φ̇F ijrαβγν 2 + f̃αx mkrβx Proof. Similar to the proof of the previous lemma. � The previous problems can be generalized to the case when the right-hand side f is not only defined in N or in Ω̄ but in the tangent bundle T (N) resp. T (Ω̄). Notice that the tangent bundle is a manifold of dimension 2(n+1), i.e., in a local trivialization of T (N) f can be expressed in the form (4.29) f = f(x, ν) with x ∈ N and ν ∈ Tx(N), cf. [17, Note 12.2.14]. Thus, the case f = f(x) is included in this general set up. The symbol ν indicates that in an equation (4.30) F |M = f(x, ν) we want f to be evaluated at (x, ν), where x ∈M and ν is the normal of M in The Minkowski problem or Minkowski type problems are also covered by the present setting, though the Minkowski problem has the additional property that the problem is transformed via the Gauß map to a different semi-Riemannian manifold as a dual problem and solved there. Minkowski type problems have been treated in [5], [23], [16] and [21]. 4.6. Remark. The equation (4.30) will be solved by the same methods as in the special case when f = f(x), i.e., we consider the same curvature flow, the evolution equation (3.7) on page 5, as before. 14 CLAUS GERHARDT The resulting evolution equations are identical with the natural exception, that, when f or f̃ has to be differentiated, the additional argument has to be considered, e.g., (4.31) f̃i = f̃αx i + f̃νβν i = f̃αx i + f̃νβx (4.32) f = f̃αẋ α + f̃νβ ν̇ β = −σ(Φ− f̃)f̃ανα + f̃νβgij(Φ− f̃)ix The most important evolution equations are explicitly stated below. Let us first state the evolution equation for (Φ − f̃). 4.7. Lemma (Evolution of (Φ− f̃)). The term (Φ− f̃) evolves according to the equation (4.33) (Φ− f̃) − Φ̇F ij(Φ− f̃)ij = σΦ̇F ijhikhkj (Φ− f̃) + σf̃αν α(Φ− f̃)− f̃ναxαi (Φ− f̃)jgij + σΦ̇F ijR̄αβγδν γxδj(Φ− f̃), where (4.34) (Φ− f̃)′ = d (Φ− f̃) (4.35) Φ̇ = Φ(r). Here is the evolution equation for the second fundamental form. 4.8. Lemma. The mixed tensor h i satisfies the parabolic equation (4.36) i − Φ̇F = σΦ̇F klhrkh i − σΦ̇Fhrih rj + σ(Φ − f̃)hki h − f̃αβxαi x kj + σf̃αν i − f̃ανβ (x kj + xαl x − f̃νανβxαl x lj − f̃νβx i;l g lj + σf̃ναν αhki h + Φ̇F kl,rshkl;ih rs; + 2Φ̇F klR̄αβγδx − Φ̇F klR̄αβγδxαmx rj − Φ̇F klR̄αβγδxαmx + σΦ̇F klR̄αβγδν γxδl h i − σΦ̇F R̄αβγδν γxδmg + σ(Φ− f̃)R̄αβγδναxβi ν γxδmg mj + Φ̈FiF + Φ̇F klR̄αβγδ;ǫ{ναxβkx mj + ναx CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 15 The proof is identical to that of Lemma 4.1; we only have to keep in mind that f now also depends on the normal. If we had assumed F to be homogeneous of degree d0 instead of 1, then, we would have to replace the explicit term F—occurring twice in the preceding lemma—by d0F . 4.9. Lemma (Evolution of ṽ). Consider the flow (3.7) in a Lorentzian space N such that the spacelike flow hypersurfaces can be written as graphs over S0. Then, ṽ satisfies the evolution equation (4.37) ˙̃v − Φ̇F ij ṽij =− Φ̇F ijhikhkj ṽ + [(Φ− f̃)− Φ̇F ]ηαβνανβ − 2Φ̇F ijhkjxαi x kηαβ − Φ̇F ijηαβγx − Φ̇F ijR̄αβγδναxβi x − f̃βxβi x k ηαg ik − f̃νβx ikxαi ηα, where η is the covariant vector field (ηα) = e ψ(−1, 0, . . . , 0). The proof is identical to the proof of Lemma 4.4. In the Riemannian case we have: 4.10. Lemma (Evolution of v). Consider the flow (3.7) in a normal Gauss- ian coordinate system (xα), where the M(t) can be written as graphs of a func- tion u(t) over some compact Riemannian manifold S0. Then the quantity (4.38) v = 1 + |Du|2 = (rανα)−1 satisfies the evolution equation (4.39) v̇ − Φ̇F ijvij =− Φ̇F ijhikhkj v − 2v−1Φ̇F ijvivj + [(Φ− f)− Φ̇F ]rαβνανβv2 + 2Φ̇F ijhkjx krαβv 2 + Φ̇F ijrαβγx + Φ̇F ijR̄αβγδν + f̃βx k rαg ikv2 + f̃νβx ikxαi rαv where r = x0 and (rα) = (1, 0, . . . , 0). 5. Stability of the limit hypersurfaces 5.1. Definition. Let N be semi-Riemannian, F a curvature operator, and M ⊂ N a compact, spacelike hypersurface, such that M is admissible and F ij , evaluated at (hij , gij), the second fundamental form and metric of M , is divergence free, then M is said to be a stable solution of the equation (5.1) F |M = f, 16 CLAUS GERHARDT where f = f(x) is defined in a neighbourhood of M , if the first eigenvalue λ1 of the linearization, which is the operator in (3.33) on page 8, is non-negative, or equivalently, if the quadratic form (5.2) F ijuiuj − σ {F ijhki hkj + F ijR̄αβγδναx γxδj + fαν is non-negative for all u ∈ C2(M). It is well-known that the corresponding eigenspace is then onedimensional and spanned by a strictly positive eigenfunction η (5.3) − F ijηij − σ{F ijhki hkj + F ijR̄αβγδναx γxδj + fαν α}η = λ1η. Notice that F ij is supposed to be divergence free, which will be the case, if F = Hk, 1 ≤ k ≤ n, and the ambient space has constant curvature, as we shall prove at the end of this section. If k = 1, then F ij = gij and N can be arbitrary, while in case k = 2, we have (5.4) F ij = Hgij − hij , hence N Einstein will suffice. To simplify the formulation of the assumptions let us define: 5.2. Definition. A curvature function F is said to be of class (D), if for every admissible hypersurfaceM the tensor F ij , evaluated at M , is divergence free. We shall prove in this section that the limit hypersurface of a converging curvature flow will be a stable stationary solution, if the initial flow velocity has a weak sign. 5.3. Theorem. Suppose that the curvature flow (3.7) on page 5 exists for all time, and that the leaves M(t) converge in C4 to a hypersurface M , where the curvature function F is supposed to be of class (D). Then M is a stable solution of the equation (5.5) F |M = f provided the velocity of the flow has a weak sign (5.6) Φ− f̃ ≥ 0 ∨ Φ− f̃ ≤ 0 at t = 0 and M(0) is not already a solution of (5.5). Proof. Convergence of a subsequence of the M(t) would actually suffice for the proof, however, the assumption (5.6) immediately implies that the flow converges, if a subsequence converges and a priori estimates in C4,α are valid. The starting point is the evolution equation (3.20) on page 7 from which we deduce in view of the parabolic maximum principle that Φ− f̃ has a weak sign CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 17 during the evolution, cf. [18, Proposition 2.7.1], i.e., if we assume without loss of generality that at t = 0 (5.7) Φ− f̃ ≥ 0, then this inequality will be valid for all t. Moreover, there holds (5.8) (Φ− f̃) > 0 ∀ 0 ≤ t <∞ if this relation is valid for t = 0, as we shall prove in the lemma below. On the other hand, the assumption (5.9) (Φ− f̃) > 0 is a natural assumption, for otherwise the initial hypersurface would already be a stationary solution which of course may not be stable. Notice also that apart from the factor Φ̇ the equation (3.20) looks like the parabolic version of the linearization of (F − f). If the technical function Φ = Φ(r) is not the trivial one Φ(r) = r, then we always assume that f > 0 and that this is also valid for the limit hypersurface M . Only in case Φ(r) = r and F = H , we allow f to be arbitrary. Thus, our assumptions imply that in any case (5.10) Φ̇ > ǫ0 > 0 ∀ t ∈ R+. Furthermore, we derive from (3.20) that not only the elliptic part converges to 0 but also (5.11) (Φ− f̃)′ = Φ̇Ḟ + σΦ̇(f)fανα(Φ− f̃), i.e., (5.12) lim Ḟ = 0. Suppose now that M is not stable, then the first eigenvalue λ1 is negative and there exists a strictly positive eigenfunction η solving the equation (5.3) evaluated atM . Let U be a tubular neighbourhood ofM with a corresponding future directed normal Gaussian coordinate system (xα) and extend η to U by setting (5.13) η(x0, x) = η(x), where, by a slight abuse of notation, we also denote (xi) by x. Thus there holds (5.14) ηαν α = 0 in M , and choosing U sufficiently small, we may assume (5.15) |ηανα| < η for all hypersurfaces M(t) ⊂ U . Now consider the term (5.16) Φ̇−1(Φ− f̃)η 18 CLAUS GERHARDT for large t, which converges to 0. Since it is positive, in view of (5.8), there must exist a sequence of t, not explicitly labelled, tending to infinity such that (5.17) 0 ≥ d Φ̇−1(Φ− f̃)η −Φ̇−2Φ̈Ḟ (Φ− f̃)η + Φ̇−1(Φ− f̃)′η Φ̇−1ηαν α(Φ− f̃)2 − σ Φ̇−1(Φ− f̃)2Hη, where we used the relation (3.8) on page 5 to derive the last integral. The rest of the proof is straight-forward. Multiply the equation (3.20) by Φ̇−1η and integrate over M(t) for those values of t satisfying the preceding inequality to deduce (5.18) Φ̇−1(Φ− f̃)′ = −F ijηij(Φ− f̃) {F ijhki hkj + F ijR̄αβγδναx γxδj + Φ̇ −1Φ̇(f)fαν α}η(Φ− f̃), and conclude further that the right-hand side can be estimated from above by (5.19) η(Φ− f̃) for large t, while the left-hand side can be estimated from below by (5.20) − ǫ(t) (Φ− f̃)η such that (5.21) ǫ(t) > 0 ∧ lim ǫ(t) = 0 in view of (5.17), where we used (5.12), (5.15) as well as (5.22) lim(Φ− f̃) = 0; a contradiction because of (5.8). � 5.4. Lemma. Let M(t) be a solution of the curvature flow (3.7) on page 5 defined on a maximal time interval [0, T ∗), 0 < T ∗ ≤ ∞, and suppose that Φ− f̃ has a weak sign at t = 0, e.g., (5.23) (Φ− f̃) ≥ 0 and suppose furthermore that (5.24) (Φ− f̃) > 0, (5.25) (Φ− f̃) > 0 ∀ 0 ≤ t < T ∗. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 19 Proof. Let M0 be an abstract compact Riemannian manifold that is being isometrically embedded inN with imageM(0). Let (ξi) be a generic coordinate system for M0 and abbreviate (Φ− f̃ ) by u. The evolution equation (3.20) can then be looked at as a linear parabolic equation for u = u(t, ξ) on M0 with time dependent coefficients and time dependent Riemannian metric gij(t, ξ). By assumption u doesn’t vanish identically at t = 0, i.e., there exists a ball Bρ = Bρ(ξ0) such that (5.26) u(0, ξ) > 0 ∀ ξ ∈ B̄ρ(ξ0). Let C be the cylinder (5.27) C = [0, T ∗)× B̄ρ and assume that there exists a first t0 > 0 such that (5.28) inf u(t0, ·) = 0 = u(t0, ξ1). We shall show that this is not possible: If ξ1 ∈ Bρ, then this contradicts the strong parabolic maximum principle, cf. [18, Lemma 2.7.1], and if ξ1 ∈ ∂Bρ, then we deduce from [18, Lemma 2.7.4] (a parabolic version of the Hopf Lemma) (5.29) (t0, ξ1) < 0, where ν is the exterior normal of the ball Bρ in ξ1, contradicting the fact that the gradient of u(t0, ·) vanishes in ξ1 because it is a minimum point; notice that we already know u ≥ 0 in [0, T ∗)×M0. � For some curvature operators one can prove a priori estimates for the second fundamental form only for stationary solutions and not for the leaves of a corresponding curvature flow. In order to use a curvature flow to obtain a stationary solution one uses ǫ-regularization“, i.e., instead of the curvature function F one considers (5.30) F̃ (hij) = F (hij + ǫHgij) for ǫ > 0, and starts a curvature flow with F̃ and fixed ǫ > 0. A priori estimates for the regularized flow are usually fairly easily derived, since (5.31) F̃ ij = F ij + ǫF klgklg but of course the estimates depend on ǫ. Having uniform estimates one can deduce that the flow—or at least a subsequence—converges to a limit hyper- surfaces Mǫ satisfying (5.32) F̃ |Mǫ = f. Then, if uniform C4,α-estimates for the Mǫ can be derived, a subsequence will converge to a solution M of (5.33) F |M = f, 20 CLAUS GERHARDT cf. [13], where this method has been used to find hypersurfaces of prescribed scalar curvature in Lorentzian manifolds, see also Theorem 6.9 on page 36. We shall now show that the solutions M obtained by this approach are all stable, if F is of class (D) and the initial velocities of the regularized flows have a weak sign. Notice that the curvature functions F̃ are in general not of class 5.5. Theorem. Let F be of class (D), then any solution M of (5.34) F |M = f obtained by a regularized curvature flow as described above is stable, provided the initial velocity of the regularized flow has a weak sign, i.e., it satisfies (5.35) Φ− f̃ ≥ 0 ∨ Φ− f̃ ≤ 0 at t = 0 and the flow hypersurfaces converge to the stationary solution in C4. Proof. Let Mǫ be the limit hypersurfaces of the regularized flow for ǫ > 0, and assume that the Mǫ satisfy uniform C 4,α-estimates such that a subsequence, not relabelled, converges in C4 to a compact spacelike hypersurface M solving the equation (5.36) F |M = f. Assume that M is not stable so that the first eigenvalue of the linearization is negative and there exists a strictly positive eigenfunction η satisfying (5.3). Extend η in a small tubular neighbourhood U of M such that (5.15) is valid for all Mǫ, if ǫ is small, ǫ < ǫ0. For those ǫ we then deduce (5.37) −F̃ ijηij − F̃ ij;ijη − 2F̃ ;j ηi − σ{F̃ ijhki hkj + F̃ ijR̄αβγδναx γxδj + fαν α}η < λ1 where the inequality is evaluated at Mǫ and where we used the convergence in Now, fix ǫ, ǫ < ǫ0, then the preceding inequality is also valid for the flow hypersurfaces M(t) converging to Mǫ, if t is large, and the same arguments as at the end of the proof of Theorem 5.3 lead to a contradiction. Hence, M has to be a stable solution. � Knowing that a solution is stable often allows to deduce further geometric properties of the underlying hypersurface like that it is either strictly stable or totally geodesic especially if the curvature function is the mean curvature, cf. e.g., [29], where the stability property has been extensively used to deduce geometric properties. We want to prove that a neighbourhood of stable solutions can be foliated by a family of hypersurfaces satisfying the equation modulo a constant. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 21 5.6. Theorem. Let M ⊂ N be compact, spacelike, orientable and a stable solution of (5.38) G|M ≡ (F − f)|M = 0, where F is of class (D) and M as well as F , f are of class Cm,α, 2 ≤ m ≤ ∞, 0 < α < 1, then a neighbourhood of M can be foliated by a family (5.39) Λ = {Mǫ : |ǫ| < ǫ0 } of spacelike Cm,α-hypersurfaces satisfying (5.40) G|Mǫ = τ(ǫ), where τ is a real function of class Cm,α. The Mǫ can be written as graphs over M in a tubular neighbourhood of M (5.41) Mǫ = { (u(ǫ, x), x) : x ∈M } such that u is of class Cm,α in both variables and there holds (5.42) u̇ > 0. Proof. (i) Let us assume that M is strictly stable. Consider a tubular neigh- bourhood of M with corresponding normal Gaussian coordinates (xα) such that M = {x0 = 0}. The nonlinear operator G can then be viewed as an elliptic operator (5.43) G : Bρ(0) ⊂ Cm,α(M) → Cm−2,α(M) where ρ is so small that all corresponding graphs are admissible. In a smaller ball DG is a topological isomorphism, sinceM is strictly stable, and hence G is a diffeomorphism in a neighbourhood of the origin, and there exist smooth unique solutions (5.44) Mǫ = { u(ǫ, x) : x ∈M } |ǫ| < ǫ0 of the equations (5.45) G|Mǫ = ǫ such that u ∈ Cm,α((−ǫ0, ǫ0)×M). Differentiating with respect to ǫ yields (5.46) DGu̇ = 1. Let us consider this equation for ǫ = 0, i.e., on M , and define (5.47) η = min(u̇, 0). Then we deduce (5.48) 0 ≤ 〈DGη, η〉 = η ≤ 0, and hence there holds (5.49) u̇ ≥ 0, because of the strict stability of M . 22 CLAUS GERHARDT Applying then the maximum principle to (5.46), we deduce further (5.50) inf u̇ > 0, hence the hypersurfaces form a foliation if ǫ0 is chosen small enough such that (5.51) inf u̇(ǫ, ·) > 0 ∀ |ǫ| < ǫ0. (ii) Assume now thatM is not strictly stable. After introducing coordinates corresponding to a tubular neighbourhood U of M as in part (i) any function u ∈ Cm,α(M) with |u|m,α small enough defines an admissible hypersurface (5.52) M(u) = graphu ⊂ U such that G|M(u) can be expressed as (5.53) G|M(u) = G(u). (5.54) A = DG(0), then A is self-adjoint, monotone (5.55) 〈Au, u〉 ≥ 0 ∀u ∈ H1,2(M) and the smallest eigenvalue of A is equal to zero, the corresponding eigenspace spanned by a strictly positive eigenfunction η. Similarly as in [2, p. 621] we consider the operator (5.56) Ψ(u, τ) = (G(u) − τ, ϕ(u)) defined in Bρ(0) × R, Bρ(0) ⊂ Cm,α(M) for small ρ > 0, where ϕ is a linear functional (5.57) ϕ(u) = Ψ is of class Cm,α and maps (5.58) Ψ : Bρ(0)× R → Cm−2,α(M)× R, such that (5.59) DΨ = DG −1 evaluated at (0, 0) is bijective as one easily checks. Indeed let (u, ǫ) satisfy (5.60) DΨ(u, ǫ) = (0, 0), (5.61) Au = DGu = ǫ ∧ ηu = 0, hence (5.62) ǫ η = 〈Au, η〉 = 〈u,Aη〉 = 0 CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 23 and we conclude ǫ = 0 as well as u = 0. To prove the surjectivity, let (w, δ) ∈ Cm−2,α(M)×R be arbitrary. Choosing (5.63) ǫ = − we deduce (5.64) (ǫ+ w)η = 0, hence there exists ū ∈ Cm,α(M) solving (5.65) Aū = ǫ+ w (5.66) u = ū+ λη (5.67) λ = δ − then satisfies (5.68) ηu = ǫ, i.e., (5.69) DΨ(u, ǫ) = (w, δ). Applying the inverse function theorem we conclude that there exists ǫ0 > 0 and functions (u(ǫ, x), τ(ǫ)) of class Cm,α in both variables such that (5.70) G(u(ǫ)) = τ(ǫ) ∧ ηu(ǫ) = ǫ ∀ |ǫ| < ǫ0; τ(ǫ) is constant for fixed ǫ. The hypersurfaces (5.71) Λ = {Mǫ =M(u(ǫ)) : |ǫ| < ǫ0 } will form a foliation, if we can show that (5.72) u̇ 6= 0. Differentiating the equations in (5.70) with respect to ǫ and evaluating the result at ǫ = 0 yields (5.73) Au̇(0) = τ̇(0) ∧ ηu̇(0) = 1 and we deduce further (5.74) τ̇ (0) η = 〈Au̇(0), η〉 = 〈u̇(0), Aη〉 = 0 and thus (5.75) τ̇ (0) = 0 ∧ u̇(0) = η > 0, 24 CLAUS GERHARDT if η is normalized such that 〈η, η〉 = 1, i.e., we have u̇(ǫ) > 0, if ǫ0 is chosen small enough. � 5.7. Remark. Let M be a stable solution of (5.76) G|M = 0 as in the preceding theorem, but not strictly stable and let Mǫ be a foliation of a neighbourhood of M such that (5.77) G|Mǫ = τ(ǫ) ∀ |ǫ| < ǫ0. If M is the limit hypersurface of a curvature flow as in Theorem 5.3, then (5.78) τ(ǫ) > 0 ∀ 0 < ǫ < ǫ0, if the flow hypersurfacesM(t) converge to M from above, which is tantamount (5.79) Φ(F )− f̃ ≥ 0, or we have (5.80) τ(ǫ) < 0 ∀ − ǫ0 < ǫ < 0, (5.81) Φ(F )− f̃ ≤ 0, in which case the flow hypersurfaces converge to M from below. The direction above“ is defined by the region the normal σν of M points Proof. Let us assume that the flow hypersurfaces satisfy (5.79) and fix 0 < ǫ < ǫ0. We may also suppose that the initial hypersurface M(0) doesn’t intersect the tubular neighbourhood of M which is being foliated by Mǫ. Now, fix 0 < ǫ < ǫ0, then there must be a first t > 0 such that M(t) touches Mǫ from above which yields, in view of the maximum principle, (5.82) G|Mǫ = τ(ǫ) > 0, since τ(ǫ) ≤ 0 would imply τ(ǫ) = 0 and Mǫ = M(t), cf. [18, Theorem 2.7.9], i.e.,M(t) would be a stationary solution, which is impossible as we have proved in Lemma 5.4. � Finally, let us show that the symmetric polynomials Hk, 1 ≤ k ≤ n, are of class (D), if the ambient space has constant curvature. 5.8. Lemma. Let N be a semi-Riemannian space of constant curvature, then the symmetric polynomials F = Hk, 1 ≤ k ≤ n, are of class (D). In case k = 2 it suffices to assume N Einstein. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 25 Proof. We shall prove the result by induction on k. First we note that the cones of definition Γk ⊂ Rn of the Hk form an ordered chain (5.83) Γk ⊂ Γk−1 ∀ 1 < k ≤ n, cf. [7], so that a hypersurface admissible for Hk is also admissible for Hk−1. For k = 1 we have (5.84) F ij = gij and the result is obviously valid for arbitrary N . Thus let us assume that the result is already proved for 1 ≤ k < n. Set F = Hk+1, F̂ = Hk and let M be an admissible hypersurface for F with principal curvatures κi. From the definition of the Hk’s we immediately deduce (5.85) F̂ = for fixed i, no summation over i, or equivalently, (5.86) F̂ gij = F ij + F̂ jmhim, notice that the last term is a symmetric tensor, since for any symmetric cur- vature function F F ij and hij commute, cf. [18, Lemma 2.1.9]. Thus there holds (5.87) F ij = F̂ gij − F̂ jmhim and we deduce, using the induction hypothesis, (5.88) ;j = F̂ i − F̂ jmhim;j = F̂ i − F̂ jmh imj; = F̂ i − F̂ i = 0, where we applied the Codazzi equations at one point. If F = H2, then (5.89) F ij = Hgij − hij and the assumption N Einstein suffices to conclude that F ij is divergence free. � 6. Existence results From now on we shall assume that ambient manifold N is Lorentzian, or more precisely, that it is smooth, globally hyperbolic with a compact, connected Cauchy hypersurface. Then there exists a smooth future oriented time function x0 such that the metric in N can be expressed in Gaussian coordinates (xα) as (6.1) ds̄2 = e2ψ{−(dx0)2 + σijdxidxj}, where x0 is the time function and the (xi) are local coordinates for (6.2) S0 = {x0 = 0}. 26 CLAUS GERHARDT S0 is then also a compact, connected Cauchy hypersurface. For a proof of the splitting result see [4, Theorem 1.1], and for the fact that all Cauchy hyper- surfaces are diffeomorphic and hence S0 is also compact and connected, see [3, Lemma 2.2]. One advantage of working in globally hyperbolic spacetimes with a compact Cauchy hypersurface is that all compact, connected spacelike Cm-hypersurfaces M can be written as graphs over S0. 6.1. Lemma. Let N be as above and M ⊂ N a connected, spacelike hyper- surface of class Cm, 1 ≤ m, then M can be written as a graph over S0 (6.3) M = graphu|S0 with u ∈ Cm(S0). We proved this lemma under the additional hypothesis that M is achronal, [10, Proposition 2.5], however, this assumption is unnecessary as has been shown in [25, Theorem 1.1]. We are looking at the curvature flow (3.7) on page 5 and want to prove that it converges to a stationary solution hypersurface, if certain assumptions are satisfied. The existence proof consists of four steps: (i) Existence on a maximal time interval [0, T ∗). (ii) Proof that the flow stays in a compact subset. (iii) Uniform a priori estimates in an appropriate function space, e.g., C4,α(S0) or C∞(S0), which, together with (ii), would imply T ∗ = ∞. (iv) Conclusion that the flow—or at least a subsequence of the flow hypersur- faces—converges if t tends to infinity. The existence on a maximal time interval is always guaranteed, if the data are sufficiently regular, since the problem is parabolic. If the flow hypersurfaces can be written as graphs in a Gaussian coordinate system, as will always be the case in a globally hyperbolic spacetime with a compact Cauchy hypersurface in view of Lemma 6.1, the conditions are better than in the general case: 6.2. Theorem. Let 4 ≤ m ∈ N and 0 < α < 1, and assume the semi- Riemannian space N to be of class Cm+2,α. Let the strictly monotone curvature function F , the functions f and Φ be of class Cm,α and let M0 ∈ Cm+2,α be an admissible compact, spacelike, connected, orientable3 hypersurface. Then the curvature flow (3.7) on page 5 with initial hypersurface M0 exists in a maximal time interval [0, T ∗), 0 < T ∗ ≤ ∞, where in case that the flow hypersurfaces cannot be expressed as graphs they are supposed to be smooth, i.e, the conditions should be valid for arbitrary 4 ≤ m ∈ N in this case. 3Recall that oriented simply means there exists a continuous normal, which will always be the case in a globally hyperbolic spacetime. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 27 A proof can be found in [18, Theorem 2.5.19, Lemma 2.6.1]. The second step, that the flow stays in a compact set, can only be achieved by barrier assumptions, cf. Definition 2.1. Thus, let Ω ⊂ N be open and precompact such that ∂Ω has exactly two components (6.4) ∂Ω =M1 where M1 is a lower barrier for the pair (F, f) and M2 an upper barrier. More- over, M1 has to lie in the past of M2 (6.5) M1 ⊂ I−(M2), cf. [18, Remark 2.7.8]. Then the flow hypersurfaces will always stay inside Ω̄, if the initial hyper- surface M0 satisfies M0 ⊂ Ω, [18, Theorem 2.7.9]. This result is also valid if M0 coincides with one the barriers, since then the velocity (Φ− f̃) has a weak sign and the flow moves into Ω for small t, if it moves at all, and the arguments of the proof are applicable. In Lorentzian manifolds the existence of barriers is associated with the pres- ence of past and future singularities. In globally hyperbolic spacetimes, when N is topologically a product (6.6) N = I × S0, where I = (a, b), singularities can only occur, when the endpoints of the interval are approached. A singularity, if one exists, is called a crushing singularity, if the sectional curvatures become unbounded, i.e., (6.7) R̄αβγδR̄ αβγδ → ∞ and such a singularity should provide a future resp. past barrier for the mean curvature function H . 6.3. Definition. Let N be a globally hyperbolic spacetime with compact Cauchy hypersurface S0 so that N can be written as a topological product N = I × S0 and its metric expressed as (6.8) ds̄2 = e2ψ(−(dx0)2 + σij(x0, x)dxidxj). Here, x0 is a globally defined future directed time function and (xi) are lo- cal coordinates for S0. N is said to have a future resp. past mean curvature barrier, if there are sequences M+k resp. M k of closed, spacelike, admissible hypersurfaces such that (6.9) lim = ∞ resp. lim (6.10) lim sup inf x0 > x0(p) ∀ p ∈ N 28 CLAUS GERHARDT resp. (6.11) lim inf sup x0 < x0(p) ∀ p ∈ N, If one stipulates that the principal curvatures of the M+k resp. M k tend to plus resp. minus infinity, then these hypersurfaces could also serve as barriers for other curvature functions. The past barriers would most certainly be non- admissible for any curvature function except H . 6.4.Remark. Notice that the assumptions (6.9) alone already implies (6.10) resp. (6.11), if either (6.12) lim sup inf x0 > a resp. (6.13) lim inf sup x0 < b where (a, b) = x0(N), or, if (6.14) R̄αβν ανβ ≥ −Λ ∀ 〈ν, ν〉 = −1. where Λ ≥ 0. Proof. It suffices to prove that the relation (6.10) is automatically satisfied under the assumptions (6.12) or (6.14) by switching the light cone and replacing x0 by −x0 in case of the past barrier. Fix k, and let (6.15) τk = inf then the coordinate slice (6.16) Mτk = {x0 = τk} touches Mk from below in a point pk ∈ Mk where τk = x0(pk) and the maxi- mum principle yields that in that point (6.17) H |Mτk ≥ H |Mk , hence, if k tends to infinity the points (pk) cannot stay in a compact subset, i.e., (6.18) lim supx0(pk) → b (6.19) lim supx0(pk) → a. We shall show that only (6.18) can be valid. The relation (6.19) evidently contradicts (6.12). CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 29 In case the assumption (6.14) is valid, we consider a fixed coordinate slice M0 = {x0 = const}, then all hypersurfaces Mk satisfying (6.20) H |M0 < infMk nΛ < inf have to lie in the future of M0, cf. [18, Lemma 4.7.1], hence the result. � A future mean curvature barrier certainly represents a singularity, at least if N satisfies the condition (6.21) R̄αβν ανβ ≥ −Λ ∀ 〈ν, ν〉 = −1 where Λ ≥ 0, because of the future timelike incompleteness, which is proved in [1], and is a generalization of Hawking’s earlier result for Λ = 0, [24]. But these singularities need not be crushing, cf. [15, Section 2] for a counterexample. The uniform a priori estimates for the flow hypersurfaces are the hardest part in any existence proof. When the flow hypersurfaces can be written as graphs it suffices to prove C1 and C2 estimates, namely, the induced metric (6.22) gij(t, ξ) = 〈xi, xj〉 where x = x(t, ξ) is a local embedding of the flow, should stay uniformly positive definite, i.e., there should exist positive constants ci, 1 ≤ i ≤ 2, such (6.23) c1gij(0, ξ) ≤ gij(t, ξ) ≤ c2gij(0, ξ), or equivalently, that the quantity (6.24) ṽ = 〈η, ν〉, where ν is the past directed normal of M(t) and η the vector field (6.25) η = (ηα) = e ψ(−1, 0, . . . , 0), is uniformly bounded, which is achieved with the help of the parabolic equation (4.37) on page 15, if it is possible at all. However, in some special situations C1-estimates are automatically satisfied, cf. Theorem 6.11 at the end of this section. For the C2-estimates the principle curvatures κi of the flow hypersurfaces have to stay in a compact set in the cone of definition Γ of F , e.g., if F is the Gaussian curvature, then Γ = Γ+ and one has to prove that there are positive constants ki, i = 1, 2 such that (6.26) k1 ≤ κi ≤ k2 ∀ 1 ≤ i ≤ n uniformly in the cylinder [0, T ∗)×M0, whereM0 is any manifold that can serve as a base manifold for the embedding x = x(t, ξ). The parabolic equations that are used for these curvature estimates are (4.36) on page 14, usually for an upper estimate, and (4.33) on page 14 for the lower estimate. Indeed, suppose that the flow starts at the upper barrier, then (6.27) F ≥ f 30 CLAUS GERHARDT at t = 0 and this estimate remains valid throughout the evolution because of the parabolic maximum principle, use (4.36). Then, if upper estimates for the κi have been derived and if f > 0 uniformly, then we conclude from (6.27) that the κi stay in a compact set inside the open cone Γ , since (6.28) F |∂Γ = 0. To obtain higher order estimates we are going to exploit the fact that the flow hypersurfaces are graphs over S0 in an essential way, namely, we look at the associated scalar flow equation (4.21) on page 12 satisfied by u. This equation is a nonlinear uniformly parabolic equation, where the operator Φ(F ) is also concave in hij , or equivalently, convex in uij , i.e., the C 2,α-estimates of Krylov and Safonov, [26, Chapter 5.5] or see [28, Chapter 10.6] for a very clear and readable presentation, are applicable, yielding uniform estimates for the standard parabolic Hölder semi-norm (6.29) [D2u]β,Q̄T for some 0 < β ≤ α in the cylinder (6.30) Q = [0, T )× S0, independent of 0 < T < T ∗, which in turn will lead to Hm+2+α, m+2+α 2 (Q̄T ) estimates, cf. [18, Theorem 2.5.9, Remark 2.6.2]. Hm+2+α, m+2+α 2 (Q̄T ) is a parabolic Hölder space, cf. [27, p. 7] for the original definition and [18, Note 2.5.4] in the present context. The estimate (6.29) combined with the uniform C2-norm leads to uniform C2,β(S0)-estimates independent of T . These estimates imply that T ∗ = ∞. Thus, it remains to prove that u(t, ·) converges in Cm+2(S0) to a stationary solution ũ, which is then also of class Cm+2,α(S0) in view of the Schauder theory. Because of the preceding a priori estimates u(t, ·) is precompact in C2(S0). Moreover, we deduce from the scalar flow equation (4.21) on page 12 that u̇ has a sign, i.e., the u(t, ·) converge monotonely in C0(S0) to ũ and therefore also in C2(S0). To prove that graph ũ is a solution, we again look at (4.21) and integrate it with respect to t to obtain for fixed x ∈ S0 (6.31) |ũ(x) − u(t, x)| = e−ψv|Φ− f̃ |, where we used that (Φ− f̃) has a sign, hence (Φ− f̃)(t, x) has to vanish when t tends to infinity, at least for a subsequence, but this suffices to conclude that graph ũ is a stationary solution and (6.32) lim (Φ− f̃) = 0. Using the convergence of u to ũ in C2, we can then prove: CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 31 6.5. Theorem. The functions u(t, ·) converge in Cm+2(S0) to ũ, if the data satisfy the assumptions in Theorem 6.2, since we have (6.33) u ∈ Hm+2+β, m+2+β 2 (Q̄), where Q = Q∞. Proof. Out of convention let us write α instead of β knowing that α is the Hölder exponent in (6.29). We shall reduce the Schauder estimates to the standard Schauder estimates in Rn for the heat equation with a right-hand side by using the already estab- lished results (6.29) and (6.34) u(t, ·) → C2(S0) ũ ∈ Cm+2,α(S0). Let (Uk) be a finite open covering of S0 such that each Uk is contained in a coordinate chart and (6.35) diamUk < ρ, ρ small, ρ will be specified in the proof, and let (ηk) be a subordinate finite partition of unity of class Cm+2,α. Since (6.36) u ∈ Hm+2+α, m+2+α 2 (Q̄T ) for any finite T , cf. [18, Lemma 2.6.1], and hence (6.37) u(t, ·) ∈ Cm+2,α(S0) ∀ 0 ≤ t <∞ we shall choose u0 = u(t0, ·) as initial value for some large t0 such that (6.38) |aij(t, ·)− ãij |0,S0 < ǫ/2 ∀ t ≥ t0, where (6.39) aij = v2Φ̇F ij and ãij is defined correspondingly for M̃ = graph ũ. However, making a variable transformation we shall always assume that t0 = 0 and u0 = u(0, ·). We shall prove (6.33) successively. (i) Let us first show that (6.40) Dxu ∈ H2+α, 2 (Q̄). This will be achieved, if we show that for an arbitrary ξ ∈ Cm+1,α(T 1,0(S0)) (6.41) ϕ = Dξu ∈ H2+α, 2 (Q̄), cf. [18, Remark 2.5.11]. Differentiating the scalar flow equation (4.21) on page 12 with respect to ξ we obtain (6.42) ϕ̇− aijϕij + biϕi + cϕ = f, 32 CLAUS GERHARDT where of course the symbol f has a different meaning then in (4.21). Later we want to apply the Schauder estimates for solutions of the heat flow equation with right-hand side. In order to use elementary potential estimates we have to cut off ϕ near the origin t = 0 by considering (6.43) ϕ̃ = ϕθ, where θ = θ(t) is smooth satisfying (6.44) θ(t) = 1, t > 1, 0, t ≤ 1 This modification doesn’t cause any problems, since we already have a priori estimates for finite t, and we are only concerned about the range 1 ≤ t < ∞. ϕ̃ satisfies the same equation as ϕ only the right-hand side has the additional summand wθ̇. Let η = ηk be one of the members of the partition of unity and set (6.45) w = ϕ̃η, then w satisfies a similar equation with slightly different right-hand side (6.46) ẇ − aijwij + biwi + cw = f̃ but we shall have this in mind when applying the estimates. The w(t, ·) have compact support in one of the Uk’s, hence we can replace the covariant derivatives of w by ordinary partial derivatives without changing the structure of the equation and the properties of the right-hand side, which still only depends linearly on ϕ and Dϕ. We want to apply the well-known estimates for the ordinary heat flow equa- (6.47) ẇ −∆w = f̂ where w is defined in R × Rn. To reduce the problem to this special form, we pick an arbitrary x0 ∈ Uk, set z0 = (0, x0), z = (t, x) and consider instead of (6.46) (6.48) ẇ − aij(z0)wij = f̂ = [aij(z)− aij(z0)]wij − biwi − cw + f̃ , where we emphasize that the difference (6.49) |aij(z)− aij(z0)| can be made smaller than any given ǫ > 0 by choosing ρ = ρ(ǫ) in (6.35) and t0 = t0(ǫ) in (6.38) accordingly. Notice also that this equation can be extended into R × Rn, since all functions have support in {t ≥ 1 Let 0 < T <∞ be arbitrary, then all terms belong to the required function spaces in Q̄T and there holds (6.50) [w]2+α,QT ≤ c[f̂ ]α,QT , CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 33 where c = c(n, α). The brackets indicate the standard unweighted parabolic semi-norms, cf. [18, Definition 2.5.2], which are identical to those defined in [27, p. 7], but there the brackets are replaced by kets. Thus, we conclude (6.51) [w]2+α,QT ≤ c sup Uk×(0,T ) |aij(z)− aij(z0)|[D2w]α,QT + c[f ]α,QT + c1{[D2u]α,Q+ + [Du]α,QT + [u]α,QT + |w|0,QT + |D2w|0,QT }, where c1 is independent of T , but dependent on ηk. Here we also used the fact that the lower order coefficients and ϕ,Dϕ are uniformly bounded. Choosing now ǫ > 0 so small that (6.52) cǫ < 1 and ρ, t0 accordingly such the difference in (6.49) is smaller than ǫ, we deduce (6.53) [w]2+α,QT ≤ 2c[f ]α,QT + 2c1{[D2u]α,Q+ + [Du]α,QT + [u]α,QT + |w|0,QT + |D2w|0,QT }. Summing over the partition of unity and noting that ξ is arbitrary we see that in the preceding inequality we can replace w by Du everywhere resulting in the estimate (6.54) [Du]2+α,QT ≤ c1[f ]α,QT + c1{[D2u]α,QT + [Du]α,QT + [u]α,QT + |Du|0,QT + |D3u|0,QT }, where c1 is a new constant still independent of T . Now the only critical terms on the right-hand side are |D3u|0,QT , which can be estimated by (6.57), and the Hölder semi-norms with respect to t (6.55) [Du]α ,t,QT + [u]α2 ,t,QT . The second one is taken care of by the boundedness of u̇, see (4.21) on page 12, while the first one is estimated with the help of equation (6.42) revealing (6.56) |Du̇| ≤ c{sup [0,T ] |u|3,S0 + |f |0,QT }, since for fixed but arbitrary t we have (6.57) |u|3,S0 ≤ ǫ[D3u]α,S0 + cǫ|u|0,S0 , where cǫ is independent of t. Hence we conclude (6.58) |Du|2+α,QT ≤ const uniformly in T . (ii) Repeating these estimates successively for 2 ≤ l ≤ m we obtain uniform estimates for (6.59) [Dlxu]2+α,QT , 34 CLAUS GERHARDT which, when combined with the uniform C2-estimates, yields (6.60) |u(t, ·)|m+2,α,S0 ≤ const uniformly in 0 ≤ t <∞. Looking at the equation (4.21) we then deduce (6.61) |u̇(t, ·)|m,α,S0 ≤ const uniformly in t. (iii) To obtain the estimates for Drtu up to the order (6.62) [m+2+α we differentiate the scalar curvature equation with respect to t as often as necessary and also with respect to the mixed derivatives DrtD x to estimate (6.63) 1≤2r+s<m+2+α using (6.60), (6.61) and the results from the prior differentiations. Combined with the estimates for the heat equation in R×Rn these estimates will also yield the necessary a priori estimates for the Hölder semi-norms in Q̄, where again the smallness of (6.49) has to be used repeatedly. � 6.6. Remark. The preceding regularity result is also valid in Riemannian manifolds, if the flow hypersurfaces can be written as graphs in a Gaussian coordinate system. In fact the proof is unaware of the nature of the ambient space. With the method described above the following existence results have been proved in globally hyperbolic spacetimes with a compact Cauchy hypersurface. Ω ⊂ N is always a precompact domain the boundary of which is decomposed as in (6.4) and (6.5) into an upper and lower barrier for the pair (F, f). We also apply the stability results from Section 5 and the just proved regularity of the convergence and formulate the theorems accordingly. By convergence of the flow in Cm+2 we mean convergence of the leaves M(t) = graphu(t, ·) in this norm. 6.7.Theorem. LetM1, M2 be lower resp. upper barriers for the pair (H, f), where f ∈ Cm,α(Ω̄) and the Mi are of class Cm+2,α, 4 ≤ m, 0 < α < 1, then the curvature flow (6.64) ẋ = (H − f)ν x(0) = x0, where x0 is an embedding of the initial hypersurface M0 = M2 exists for all time and converges in Cm+2 to a stable solution M of class Cm+2,α of the equation (6.65) H |M = f, provided the initial hypersurface is not already a solution. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 35 The existence result was proved in [11, Theorem 2.2], see also [18, Theorem 4.2.1] and the remarks following the theorem. Notice that f isn’t supposed to satisfy any sign condition. For spacetimes that satisfy the timelike convergence condition and for func- tions f with special structural conditions existence results via a mean curvature flow were first proved in [6]. The Gaussian curvature or the curvature functions F belonging to the larger class (K∗), see [10] for a definition, require that the admissible hypersurfaces are strictly convex. Moreover, proving a priori estimates for the second fundamental form of a hypersurface M in general semi-Riemannian manifolds, when the curvature function is not the mean curvature, or does not behave similar to it, requires that a strictly convex function χ is defined in a neighbourhood of the hypersur- face, see Lemma 2.2 on page 3 where sufficient assumptions are stated which imply the existence of strictly convex functions. Furthermore, when we consider curvature functions of class (K∗), notice that the Gaussian curvature belongs to that class, then the right-hand side f can be defined in T (Ω̄) instead of Ω̄, i.e., in a local trivialization of the tangent bundle f can be expressed as (6.66) f = f(x, ν) ∧ ν ∈ Tx(N). We shall formulate the existence results with this more general assumption, though of course any stability claim only makes sense for f = f(x). 6.8. Theorem. Let F ∈ Cm,α(Γ+), 4 ≤ m, 0 < α < 1, be a curvature function of class (K∗), let 0 < f ∈ Cm,α(T (Ω̄)), and let M1, M2 be lower resp. upper barriers for (F, f) of class Cm+2,α. Then the curvature flow (6.67) ẋ = (Φ− f̃)ν x(0) = x0 where Φ(r) = log r and x0 is an embedding of M0 =M2, exists for all time and converges in Cm+2 to a stationary solution M ∈ Cm+2,α of the equation (6.68) F |M = f provided the initial hypersurface M2 is not already a stationary solution and there exists a strictly convex function χ ∈ C2(Ω̄). When f = f(x) and F is of class (D), then M is stable. The theorem was proved in [10] when f is only defined in Ω̄ and in the general case in [18, Theorem 4.1.1]. When F = H2 is the scalar curvature operator, then the requirement that f is defined in the tangent bundle and not merely in N is a necessity, if the scalar curvature is to be prescribed. To prove existence results in this case, f 36 CLAUS GERHARDT has to satisfy some natural structural conditions, namely, 0 < c1 ≤ f(x, ν) if 〈ν, ν〉 = −1,(6.69) |||fβ(x, ν)||| ≤ c2(1 + |||ν|||2),(6.70) |||fνβ (x, ν)||| ≤ c3(1 + |||ν|||),(6.71) for all x ∈ Ω̄ and all past directed timelike vectors ν ∈ Tx(Ω), where ||| · ||| is a Riemannian reference metric. Applying a curvature flow to obtain stationary solutions requires to approx- imate F and f by functions Fǫ and fk and to use these functions for the flow. The Fǫ are the ǫ-regularizations of F , which we already discussed before, cf. (5.30) on page 19. Let us also write F̃ instead of Fǫ as before. The functions fk have the property that |||fkβ ||| only grows linearly in |||ν||| and |||fkνβ (x, ν)||| is bounded. To simplify the presentation we shall therefore assume that f satisfies (6.72) |||fβ(x, ν)||| ≤ c2(1 + |||ν|||), (6.73) |||fνβ (x, ν)||| ≤ c3, and also (6.74) 0 < c1 ≤ f(x, ν) ∀ ν ∈ Tx(N), 〈ν, ν〉 < 0, although the last assumption is only a minor point that can easily be dealt with, see [13, Remark 2.6], and [13, Section 7 and 8] for the other approximations of The barriers Mi, i = 1, 2, for (F, f) satisfy the barrier condition of course only weakly, i.e., no strict inequalities; however, because of the ǫ-regularization we need strict inequalities, so that the Mi’s are also barriers for (F̃ , f), if ǫ is small. In [13, Remark 2.4 and Lemma 2.5] it is shown that strict inequalities for the barriers may be assumed without loss of generality. Now, we can formulate the existence result for the scalar curvature operator F = H2 under these provisions. 6.9. Theorem. Let f ∈ Cm,α(T (Ω̄)), 4 ≤ m, 0 < α < 1, satisfy the con- ditions (6.72), (6.73) and (6.74), and let M1, M2 be strict lower resp. upper barriers of class Cm+2,α for (F, f). Let F̃ be the ǫ-regularization of the scalar curvature operator F , then the curvature flow for F̃ (6.75) ẋ = (Φ− f̃) x(0) = x0 where Φ(r) = r 2 and x0 is an embedding of M0 = M2, exists for all time and converges in Cm+2 to a stationary solution Mǫ ∈ Cm+2,α of (6.76) F̃ |Mǫ = f CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 37 provided there exists a strictly convex function χ ∈ C2(Ω̄) and 0 < ǫ is small. The Mǫ then converge in C m+2 to a solution M ∈ Cm+2,α of (6.77) F |M = f. If f = f(x) and N Einstein, then M is stable. These statements, except for the stability and the convergence in Cm+2, are proved in [13]. 6.10. Remark. Let us now discuss the pure mean curvature flow (6.78) ẋ = Hν with initial spacelike hypersurface M0 of class C m+2,α, m ≥ 4 and 0 < α < 1. From the corresponding scalar curvature flow (4.21) on page 12 we immediately infer that the flow moves into the past of M0, if (6.79) H |M0 ≥ 0 and into its future, if (6.80) H |M0 ≤ 0. Let us only consider the case (6.79) and also assume that M0 is not maximal. From the a priori estimates in [11, Section 3 and Section 4] we then deduce that the flow remains smooth as long as it stays in a compact set of N , and if a compact, spacelike hypersurface M1 of class C 2 satisfying (6.81) H |M1 ≤ 0 lies in the past of M0, then the flow will exist for all time and converge in Cm+2 to a stable maximal hypersurface M , hence a neighbourhood of M can be foliated by CMC hypersurfaces, where those in the future ofM have positive mean curvature, in view of Remark 5.7 on page 24. Thus, the flow will converge if and only if such a hypersurfaceM1 lies in the past of M0. Conversely, if there exists a compact, spacelike hypersurface M1 in N sat- isfying (6.81), and there is no stable maximal hypersurface in its future, then this is a strong indication that N has no future singularity, assuming that such a singularity would produce spacelike hypersurfaces with positive mean curvature. An example of such a spacetime is the (n + 1)-dimensional de Sitter space which is geodesically complete and has exactly one maximal hypersurface M which is also totally geodesic but not stable, and the future resp. past of M are foliated by coordinate slices with negative resp. positive mean curvature. To conclude this section let us show which spacelike hypersurfaces satisfy C1-estimates automatically. 38 CLAUS GERHARDT 6.11. Theorem. Let M = graphu|S0 be a compact, spacelike hypersurface represented in a Gaussian coordinate system with unilateral bounded principal curvatures, e.g., (6.82) κi ≥ κ0 ∀ i. Then, the quantity ṽ = 1√ 1−|Du|2 can be estimated by (6.83) ṽ ≤ c(|u|,S0, σij , ψ, κ0), where we assumed that in the Gaussian coordinate system the ambient metric has the form as in (6.1). Proof. We suppose as usual that the Gaussian coordinate system is future oriented, and that the second fundamental form is evaluated with respect to the past directed normal. We observe that (6.84) ‖Du‖2 = gijuiuj = e−2ψ |Du|2 hence, it is equivalent to find an a priori estimate for ‖Du‖. Let λ be a real parameter to be specified later, and set (6.85) w = 1 log‖Du‖2 + λu. We may regard w as being defined on S0; thus, there is x0 ∈ S0 such that (6.86) w(x0) = sup and we conclude (6.87) 0 = wi = ‖Du‖2 j + λui in x0, where the covariant derivatives are taken with respect to the induced metric gij , and the indices are also raised with respect to that metric. Expressing the second fundamental form of a graph with the help of the Hessian of the function (6.88) e−ψv−1hij = −uij − Γ̄ 000uiuj − Γ̄ 00iuj − Γ̄ 00jui − Γ̄ 0ij . we deduce further (6.89) λ‖Du‖4 = −uijuiuj = e−ψ ṽhiju iuj + Γ̄ 000‖Du‖4 + 2Γ̄ 00ju j‖Du‖2 + Γ̄ 0ijuiuj . Now, there holds (6.90) ui = gijuj = e −2ψσijujv and by assumption, (6.91) hiju iuj ≥ κ0‖Du‖2, CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 39 i.e., the critical terms on the right-hand side of (6.89) are of fourth order in ‖Du‖ with bounded coefficients, and we conclude that ‖Du‖ can’t be too large in x0 if we choose λ such that (6.92) λ ≤ −c|||Γ̄ 0αβ||| − 1 with a suitable constant c; w, or equivalently, ‖Du‖ is therefore uniformly bounded from above. � Especially for convex graphs over S0 the term ṽ is uniformly bounded as long as they stay in a compact set. 7. The inverse mean curvature flow Let us now consider the inverse mean curvature flow (IMCF) (7.1) ẋ = −H−1ν with initial hypersurfaceM0 in a globally hyperbolic spacetimeN with compact Cauchy hypersurface S0. N is supposed to satisfy the timelike convergence condition (7.2) R̄αβν ανβ ≥ 0 ∀ 〈ν, ν〉 = −1. Spacetimes with compact Cauchy hypersurface that satisfy the timelike con- vergence condition are also called cosmological spacetimes, a terminology due to Bartnik. In such spacetimes the inverse mean curvature flow will be smooth as long as it stays in a compact set, and, if H |M0 > 0 and if the flow exists for all time, it will necessarily run into the future singularity, since the mean curvature of the flow hypersurfaces will become unbounded and the flow will run into the future of M0. Hence the claim follows from Remark 6.4 on page 28. However, it might be that the flow will run into the singularity in finite time. To exclude this behaviour we introduced in [15] the so-called strong volume decay condition, cf. Definition 7.2. A strong volume decay condition is both necessary and sufficient in order that the IMCF exists for all time. 7.1. Theorem. Let N be a cosmological spacetime with compact Cauchy hy- persurface S0 and with a future mean curvature barrier. Let M0 be a closed, connected, spacelike hypersurface with positive mean curvature and assume fur- thermore that N satisfies a future volume decay condition. Then the IMCF (7.1) with initial hypersurface M0 exists for all time and provides a foliation of the future D+(M0) of M0. The evolution parameter t can be chosen as a new time function. The flow hypersurfaces M(t) are the slices {t = const} and their volume satisfies (7.3) |M(t)| = |M0|e−t. Defining a new time function τ by choosing (7.4) τ = 1− e− 40 CLAUS GERHARDT we obtain 0 ≤ τ < 1, (7.5) |M(τ)| = |M0|(1− τ)n, and the future singularity corresponds to τ = 1. Moreover, the length L(γ) of any future directed curve γ starting from M(τ) is bounded from above by (7.6) L(γ) ≤ c(1− τ), where c = c(n,M0). Thus, the expression 1 − τ can be looked at as the radius of the slices {τ = const} as well as a measure of the remaining life span of the spacetime. Next we shall define the strong volume decay condition. 7.2.Definition. Suppose there exists a time function x0 such that the future end of N is determined by {τ0 ≤ x0 < b} and the coordinate slicesMτ = {x0 = τ} have positive mean curvature with respect to the past directed normal for τ0 ≤ τ < b. In addition the volume |Mτ | should satisfy (7.7) lim |Mτ | = 0. A decay like that is normally associated with a future singularity and we simply call it volume decay. If (gij) is the induced metric of Mτ and g = det(gij), then we have (7.8) log g(τ0, x)− log g(τ, x) = 2eψH̄(s, x) ∀x ∈ S0, where H̄(τ, x) is the mean curvature ofMτ in (τ, x). This relation can be easily derived from the relation (3.8) on page 5 and Remark 3.6 on page 7. A detailed proof is given in [12]. In view of (7.7) the left-hand side of this equation tends to infinity if τ approaches b for a.e. x ∈ S0, i.e., (7.9) lim eψH̄(s, x) = ∞ for a.e. x ∈ S0. Assume now, there exists a continuous, positive function ϕ = ϕ(τ) such that (7.10) eψH̄(τ, x) ≥ ϕ(τ) ∀ (τ, x) ∈ (τ0, b)× S0, where (7.11) ϕ(τ) = ∞, then we say that the future of N satisfies a strong volume decay condition. 7.3. Remark. (i) By approximation we may assume that the function ϕ above is smooth. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 41 (ii) A similar definition holds for the past of N by simply reversing the time direction. Notice that in this case the mean curvature of the coordinate slices has to be negative. 7.4. Lemma. Suppose that the future of N satisfies a strong volume decay condition, then there exist a time function x̃0 = x̃0(x0), where x0 is the time function in the strong volume decay condition, such that the mean curvature H̄ of the slices x̃0 = const satisfies the estimate (7.12) eψ̃H̄ ≥ 1. The factor eψ̃ is now the conformal factor in the representation (7.13) ds̄2 = e2ψ̃(−(dx̃0)2 + σijdxidxj). The range of x̃0 is equal to the interval [0,∞), i.e., the singularity corre- sponds to x̃0 = ∞. A proof is given in [15, Lemma 1.4]. 7.5. Remark. Theorem 7.1 can be generalized to spacetimes satisfying (7.14) R̄αβν ανβ ≥ −Λ ∀ 〈ν, ν〉 = −1 with a constant Λ ≥ 0, if the mean curvature of the initial hypersurface M0 is sufficiently large (7.15) H |M0 > cf. [25]. In that thesis it is also shown that the future mean curvature barrier assumption can be dropped, i.e., the strong volume decay condition is sufficient to prove that the IMCF exists for all time and provides a foliation of the future of M0. Hence, the strong volume decay condition already implies the existence of a future mean curvature barrier, since the leaves of the IMCF define such a barrier. 8. The IMCF in ARW spaces In the present section we consider spacetimes N satisfying some structural conditions, which are still fairly general, and prove convergence results for the leaves of the IMCF. Moreover, we define a new spacetime N̂ by switching the light cone and using reflection to define a new time function, such that the two spacetimes N and N̂ can be pasted together to yield a smooth manifold having a metric singularity, which, when viewed from the region N is a big crunch, and when viewed from N̂ is a big bang. The inverse mean curvature flows in N resp. N̂ correspond to each other via reflection. Furthermore, the properly rescaled flow in N has a natural smooth extension of class C3 across the singularity into N̂ . With respect to this natural diffeomorphism we speak of a transition from big crunch to big bang. 42 CLAUS GERHARDT 8.1. Definition. A globally hyperbolic spacetime N , dimN = n+1, is said to be asymptotically Robertson-Walker (ARW) with respect to the future, if a future end of N , N+, can be written as a product N+ = [a, b) × S0, where S0 is a Riemannian space, and there exists a future directed time function τ = x0 such that the metric in N+ can be written as (8.1) ds̆2 = e2ψ̃{−(dx0)2 + σij(x0, x)dxidxj}, where S0 corresponds to x0 = a, ψ̃ is of the form (8.2) ψ̃(x0, x) = f(x0) + ψ(x0, x), and we assume that there exists a positive constant c0 and a smooth Riemann- ian metric σ̄ij on S0 such that (8.3) lim eψ = c0 ∧ lim σij(τ, x) = σ̄ij(x), (8.4) lim f(τ) = −∞. Without loss of generality we shall assume c0 = 1. Then N is ARW with respect to the future, if the metric is close to the Robertson-Walker metric (8.5) ds̄2 = e2f{−dx02 + σ̄ij(x)dxidxj} near the singularity τ = b. By close we mean that the derivatives of arbitrary order with respect to space and time of the conformal metric e−2f ğαβ in (8.1) should converge to the corresponding derivatives of the conformal limit metric in (8.5) when x0 tends to b. We emphasize that in our terminology Robertson- Walker metric does not imply that (σ̄ij) is a metric of constant curvature, it is only the spatial metric of a warped product. We assume, furthermore, that f satisfies the following five conditions (8.6) − f ′ > 0, there exists ω ∈ R such that (8.7) n+ ω − 2 > 0 ∧ lim |f ′|2e(n+ω−2)f = m > 0. Set γ̃ = 1 (n+ ω − 2), then there exists the limit (8.8) lim (f ′′ + γ̃|f ′|2) (8.9) |Dmτ (f ′′ + γ̃|f ′|2)| ≤ cm|f ′|m ∀m ≥ 1, as well as (8.10) |Dmτ f | ≤ cm|f ′|m ∀m ≥ 1. If S0 is compact, then we call N a normalized ARW spacetime, if (8.11) det σ̄ij = |Sn|. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 43 8.2. Remark. (i) If these assumptions are satisfied, then the range of τ is finite, hence, we may—and shall—assume w.l.o.g. that b = 0, i.e., (8.12) a < τ < 0. (ii) Any ARW spacetime with compact S0 can be normalized as one easily checks. For normalized ARW spaces the constantm in (8.7) is defined uniquely and can be identified with the mass of N , cf. [20]. (iii) In view of the assumptions on f the mean curvature of the coordinate slices Mτ = {x0 = τ} tends to ∞, if τ goes to zero. (iv) ARW spaces with compact S0 satisfy a strong volume decay condition, cf. Definition 7.2 on page 40. (v) Similarly one can define N to be ARW with respect to the past. In this case the singularity would lie in the past, correspond to τ = 0, and the mean curvature of the coordinate slices would tend to −∞. We assume that N satisfies the timelike convergence condition and that S0 is compact. Consider the future end N+ of N and let M0 ⊂ N+ be a spacelike hypersurface with positive mean curvature H̆ |M0 > 0 with respect to the past directed normal vector ν̆—it will become apparent in a moment why we use the symbols H̆ and ν̆ and not the usual ones H and ν. Then, as we have proved in the preceding section, the inverse mean curvature flow (8.13) ẋ = −H̆−1ν̆ with initial hypersurface M0 exists for all time, is smooth, and runs straight into the future singularity. If we express the flow hypersurfaces M(t) as graphs over S0 (8.14) M(t) = graphu(t, ·), then we have proved in [14] 8.3. Theorem. (i) Let N satisfy the above assumptions, then the range of the time function x0 is finite, i.e., we may assume that b = 0. Set (8.15) ũ = ueγt, where γ = 1 γ̃, then there are positive constants c1, c2 such that (8.16) − c2 ≤ ũ ≤ −c1 < 0, and ũ converges in C∞(S0) to a smooth function, if t goes to infinity. We shall also denote the limit function by ũ. (ii) Let ğij be the induced metric of the leaves M(t), then the rescaled metric (8.17) e tğij converges in C∞(S0) to (8.18) (γ̃m) γ̃ (−ũ) γ̃ σ̄ij . 44 CLAUS GERHARDT (iii) The leaves M(t) get more umbilical, if t tends to infinity, namely, there holds (8.19) H̆−1|h̆ji − 1nH̆δ i | ≤ ce −2γt. In case n+ ω − 4 > 0, we even get a better estimate (8.20) |h̆ji − 1nH̆δ i | ≤ ce (n+ω−4)t. To prove the convergence results for the inverse mean curvature flow, we con- sider the flow hypersurfaces to be embedded in N equipped with the conformal metric (8.21) ds̄2 = −(dx0)2 + σij(x0, x)dxidxj . Though, formally, we have a different ambient space we still denote it by the same symbol N and distinguish only the metrics ğαβ and ḡαβ (8.22) ğαβ = e 2ψ̃ ḡαβ and the corresponding geometric quantities of the hypersurfaces h̆ij , ğij , ν̆ resp. hij , gij , ν, etc., i.e., the standard notations now apply to the case when N is equipped with the metric in (8.21). The second fundamental forms h̆ i and h i are related by (8.23) eψ̃h̆ i = h i + ψ̃αν and, if we define F by (8.24) F = eψ̃H̆, (8.25) F = H − nṽf ′ + nψανα, where (8.26) ṽ = v−1, and the evolution equation can be written as (8.27) ẋ = −F−1ν, since (8.28) ν̆ = e−ψ̃ν. The flow exists for all time and is smooth, due to the results in the preceding section. Next, we want to show how the metric, the second fundamental form, and the normal vector of the hypersurfaces M(t) evolve by adapting the general evolution equations in Section 3 on page 4 to the present situation. CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 45 8.4. Lemma. The metric, the normal vector, and the second fundamental form of M(t) satisfy the evolution equations (8.29) ġij = −2F−1hij , (8.30) ν̇ = ∇M (−F−1) = gij(−F−1)ixj , (8.31) ḣ i = (−F i + F −1hki h k + F −1R̄αβγδν γxδkg (8.32) ḣij = (−F−1)ij − F−1hki hkj + F−1R̄αβγδναx γxδj . Since the initial hypersurface is a graph over S0, we can write (8.33) M(t) = graphu(t)|S0 ∀ t ∈ I, where u is defined in the cylinder R+×S0. We then deduce from (8.27), looking at the component α = 0, that u satisfies a parabolic equation of the form (8.34) u̇ = where we emphasize that the time derivative is a total derivative, i.e. (8.35) u̇ = + uiẋ Since the past directed normal can be expressed as (8.36) (να) = −e−ψv−1(1, ui), we conclude from (8.34) (8.37) For this new curvature flow the necessary decay estimates and convergence results can be proved, which in turn can be immediately translated to corre- sponding convergence results for the original IMCF. Transition from big crunch to big bang With the help of the convergence results in Theorem 8.3, we can rescale the IMCF such that it can be extended past the singularity in a natural way. We define a new spacetime N̂ by reflection and time reversal such that the IMCF in the old spacetime transforms to an IMCF in the new one. By switching the light cone we obtain a new spacetime N̂ . The flow equation in N is independent of the time orientation, and we can write it as (8.38) ẋ = −H̆−1ν̆ = −(−H̆)−1(−ν̆) ≡ −Ĥ−1ν̂, where the normal vector ν̂ = −ν̆ is past directed in N̂ and the mean curvature Ĥ = −H̆ negative. 46 CLAUS GERHARDT Introducing a new time function x̂0 = −x0 and formally new coordinates (x̂α) by setting (8.39) x̂0 = −x0, x̂i = xi, we define a spacetime N̂ having the same metric as N—only expressed in the new coordinate system—such that the flow equation has the form (8.40) ˙̂x = −Ĥ−1ν̂, where M(t) = graph û(t), û = −u, and (8.41) (ν̂α) = −ṽe−ψ̃(1, ûi) in the new coordinates, since (8.42) ν̂0 = −ν̆0 ∂x̂ (8.43) ν̂i = −ν̆i. The singularity in x̂0 = 0 is now a past singularity, and can be referred to as a big bang singularity. The unionN∪N̂ is a smooth manifold, topologically a product (−a, a)×S0— we are well aware that formally the singularity {0}×S0 is not part of the union; equipped with the respective metrics and time orientation it is a spacetime which has a (metric) singularity in x0 = 0. The time function (8.44) x̂0 = x0, in N, −x0, in N̂ , is smooth across the singularity and future directed. N ∪ N̂ can be regarded as a cyclic universe with a contracting part N = {x̂0 < 0} and an expanding part N̂ = {x̂0 > 0} which are joined at the singularity {x̂0 = 0}. It turns out that the inverse mean curvature flow, properly rescaled, defines a natural C3- diffeomorphism across the singularity and with respect to this diffeomorphism we speak of a transition from big crunch to big bang. Using the time function in (8.44) the inverse mean curvature flows in N and N̂ can be uniformly expressed in the form (8.45) ˙̂x = −Ĥ−1ν̂, where (8.45) represents the original flow in N , if x̂0 < 0, and the flow in (8.40), if x̂0 > 0. Let us now introduce a new flow parameter (8.46) s = −γ−1e−γt, for the flow in N, γ−1e−γt, for the flow in N̂ , CURVATURE FLOWS IN SEMI-RIEMANNIAN MANIFOLDS 47 and define the flow y = y(s) by y(s) = x̂(t). y = y(s, ξ) is then defined in [−γ−1, γ−1]× S0, smooth in {s 6= 0}, and satisfies the evolution equation (8.47) y′ ≡ d −Ĥ−1ν̂ eγt, s < 0, Ĥ−1ν̂ eγt, s > 0. In [14] we proved: 8.5. Theorem. The flow y = y(s, ξ) is of class C3 in (−γ−1, γ−1)×S0 and defines a natural diffeomorphism across the singularity. The flow parameter s can be used as a new time function. 8.6. Remark. The regularity result for the transition flow is optimal, i.e., given any 0 < α < 1, then there is an ARW space such that the transition flow is not of class C3,α, cf. [19]. 8.7. Remark. Since ARW spaces have a future mean curvature barrier, a future end can be foliated by CMC hypersurfaces the mean curvature of which can be used as a new time function., see [9] and [22]. In [8] we study this folia- tion a bit more closely and prove that, when writing the CMC hypersurfaces as graphs Mτ = graphϕ(τ, ·) in the special coordinate system of the ARW space, where τ is the mean curvature, of Mτ then (8.48) τ(−ϕ)1+γ̃ → const > 0, notice that ϕ < 0, and hence (8.49) lim ϕ(τ, x) ϕ(τ, y) = 1 ∀x, y ∈ S0. Moreover, the new time function (8.50) s = −τ−q, q = γ̃ 1 + γ̃ can be extended to the mirror universe N̂ by odd reflection as a function of class C3 across the singularity with non-vanishing gradient. References [1] Lars Andersson and Gregory Galloway, Ds/cft and spacetime topology, Adv. Theor. Math. Phys. 68 (2003), 307–327, hep-th/0202161, 17 pages. [2] Robert Bartnik, Remarks on cosmological spacetimes and constant mean curvature sur- faces, Comm. Math. Phys. 117 (1988), no. 4, 615–624. [3] Antonio N. Bernal and Miguel Sanchez, On smooth Cauchy hypersurfaces and Geroch’s splitting theorem, Commun. Math. Phys. 243 (2003), 461–470, arXiv:gr-qc/0306108. [4] , Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Comm. Math. Physics 257 (2005), 43–50, arXiv:gr-qc/0401112. [5] Shiu Yuen Cheng and Shing Tung Yau, On the regularity of the solution of the n- dimensional Minkowski problem, Comm. Pure Appl. Math. 29 (1976), no. 5, 495–516. [6] Klaus Ecker and Gerhard Huisken, Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes., Commun. Math. Phys. 135 (1991), no. 3, 595–613. http://arXiv.org/pdf/hep-th/0202161 http://arXiv.org/abs/gr-qc/0306108 http://arXiv.org/abs/gr-qc/0401112 48 CLAUS GERHARDT [7] Lars Gȧrding, An inequality for hyperbolic polynomials, J. Math. Mech. 8 (1959), 957– [8] Claus Gerhardt, Properties of the CMC foliation of ARW spaces, in preparation. [9] , H-surfaces in Lorentzian manifolds, Commun. Math. Phys. 89 (1983), 523–553. [10] , Hypersurfaces of prescribed curvature in Lorentzian manifolds, Indiana Univ. Math. J. 49 (2000), 1125–1153, arXiv:math.DG/0409457. [11] , Hypersurfaces of prescribed mean curvature in Lorentzian manifolds, Math. Z. 235 (2000), 83–97, arXiv:math.DG/0409465. [12] , Estimates for the volume of a Lorentzian manifold, Gen. Relativity Gravitation 35 (2003), 201–207, math.DG/0207049. [13] , Hypersurfaces of prescribed scalar curvature in Lorentzian manifolds, J. reine angew. Math. 554 (2003), 157–199, math.DG/0207054. [14] , The inverse mean curvature flow in ARW spaces - transition from big crunch to big bang, 2004, arXiv:math.DG/0403485, 39 pages. [15] , The inverse mean curvature flow in cosmological spacetimes, 2004, arXiv:math.DG/0403097, 24 pages. [16] , Minkowski type problems for convex hypersurfaces in the sphere, 2005, arXiv:math.DG/0509217, 30 pages. [17] , Analysis II, International Series in Analysis, International Press, Somerville, MA, 2006, 395 pp. [18] , Curvature Problems, Series in Geometry and Topology, vol. 39, International Press, Somerville, MA, 2006, 323 pp. [19] , The inverse mean curvature flow in Robertson-Walker spaces and its applica- tion to cosmology, Methods Appl. Analysis 13 (2006), no. 1, 19–28, gr-qc/0404112. [20] , The mass of a Lorentzian manifold, Adv. Theor. Math. Phys. 10 (2006), 33–48, math.DG/0403002. [21] , Minkowski type problems for convex hypersurfaces in hyperbolic space, 2006, arXiv:math.DG/0602597, 32 pages. [22] , On the CMC foliation of future ends of a spacetime, Pacific J. Math. 226 (2006), no. 2, 297–308, math.DG/0408197. [23] Bo Guan and Pengfei Guan, Convex hypersurfaces of prescribed curvatures., Ann. Math. 156 (2002), 655–673. [24] S. W. Hawking and G. F. R. Ellis, The large scale structure of space-time, Cambridge University Press, London, 1973. [25] Heiko Kröner, Der inverse mittlere Krümmungsfluß in Lorentz Mannigfaltigkeiten, 2006, Diplomthesis, Heidelberg University. [26] N. V. Krylov, Nonlinear elliptic and parabolic equations of the second order, Mathe- matics and its Applications (Soviet Series), vol. 7, D. Reidel Publishing Co., Dordrecht, 1987. [27] O.A. Ladyzhenskaya, V.A. Solonnikov, and N.N. Ural’tseva, Linear and quasi-linear equations of parabolic type. Translated from the Russian by S. Smith, Translations of Mathematical Monographs. 23. Providence, RI: American Mathematical Society (AMS). XI, 648 p. , 1968 (English). [28] Oliver C. Schnürer, Partielle Differentialgleichungen 2, 2006, pdf file, Lecture notes. [29] Shing-Tung Yau, Geometry of three manifolds and existence of Black Hole due to bound- ary effect, arXiv:math.DG/0109053. Ruprecht-Karls-Universität, Institut für Angewandte Mathematik, Im Neuen- heimer Feld 294, 69120 Heidelberg, Germany E-mail address: [email protected] URL: http://www.math.uni-heidelberg.de/studinfo/gerhardt/ http://arXiv.org/abs/math.DG/0409457 http://arXiv.org/abs/math.DG/0409465 http://arXiv.org/pdf/math.DG/0207049 http://arXiv.org/pdf/math.DG/0207054 http://arXiv.org/pdf/math.DG/0403485 http://arXiv.org/abs/math.DG/0403097 http://arXiv.org/abs/math.DG/0509217 http://arXiv.org/pdf/gr-qc/0404112 http://arXiv.org/pdf/math.DG/0403002 http://arXiv.org/abs/math.DG/0602597 http://arxiv.org/abs/math.DG/0408197 http://page.mi.fu-berlin.de/~schnuere/skripte/pde.pdf http://arXiv.org/abs/math.DG/0109053 1. Introduction 2. Notations and preliminary results 3. Evolution equations for some geometric quantities 4. Essential parabolic flow equations 5. Stability of the limit hypersurfaces 6. Existence results 7. The inverse mean curvature flow 8. The IMCF in ARW spaces Transition from big crunch to big bang References

0704.0237

Hydrodynamic and Spectral Simulations of HMXB Winds

Hydrodynamic and Spectral Simulations of HMXB Winds Christopher W. Mauche1,∗), Duane A. Liedahl1,∗∗), Shizuka Akiyama1,∗∗∗), and Tomek Plewa2,†) 1Lawrence Livermore National Laboratory, Livermore, CA 94550, USA 2University of Chicago, 5640 South Ellis Avenue, RI-475, Chicago, IL 60615, USA We describe preliminary results of a global model of the radiatively-driven photoionized wind and accretion flow of the high-mass X-ray binary Vela X-1. The full model combines FLASH hydrodynamic calculations, XSTAR photoionization calculations, HULLAC atomic data, and Monte Carlo radiation transport. We present maps of the density, temperature, ve- locity, and ionization parameter from a FLASH two-dimensional time-dependent simulation of Vela X-1, as well as maps of the emissivity distributions of the X-ray emission lines. §1. Introduction As described by Castor, Abbott, and Klein (hereafter CAK),1) mass loss in the form of a high velocity wind is driven from the surface of an OB star by radiation pressure on a multitude of resonance transitions of intermediate charge states of cosmically abundant elements. The wind is characterized by a mass-loss rate Ṁ ∼ 10−6–10−5 M⊙ yr −1 and a velocity profile V (R) ∼ V∞(1− ROB/R) β , where β ≈ 1 the terminal velocity V∞ ∼ 3Vesc = 3 (2GMOB/ROB) 1/2 ∼ 1500 km s−1, R is the distance from the OB star, and MOB and ROB are respectively the mass and radius of the OB star. In a detached high-mass X-ray binary (HMXB), a compact object, typically a neutron star, captures a fraction f ∼ πR2BH/4πa 2 of the OB star wind, where a is the binary separation, RBH = 2GMNS/[V (a) 2 + c2s ] is the Bondi-Hoyle radius, cs ∼ 10 (T/10 4)1/2 km s−1 is the sound speed, and T is the wind temperature. Accretion of this material onto the neutron star powers an X-ray luminosity LX ∼ fGṀMNS/RNS ∼ 10 36–1037 erg s−1, where MNS and RNS are respectively the mass and radius of the neutron star. The resulting X-ray flux photoionizes the wind and reduces its ability to be radiatively driven, both because the higher ionization state of the plasma results is a reduction in the number of resonance transitions, and because the energy of the transitions shifts to shorter wavelengths where the overlap with the stellar continuum is lower. To first order, the lower radiative driving results in a reduced wind velocity near the neutron star V (a), which increases the Bondi- Hoyle radius RBH, which increases the accretion efficiency f , which increases the X-ray luminosity LX. In this way, the X-ray emission of HMXBs is the result of a complex interplay between the radiative driving of the wind of the OB star and the photoionization of the wind by the neutron star. Known since the early days of X-ray astronomy, HMXBs have been extensively ∗) E-mail: [email protected] ∗∗) E-mail: [email protected] ∗∗∗) E-mail: [email protected] †) E-Mail: [email protected] typeset using PTPTEX.cls 〈Ver.0.9〉 http://arxiv.org/abs/0704.0237v1 2 C. W. Mauche, D. A. Liedahl, S. Akiyama, and T. Plewa studied observationally, theoretically,2)–4) and computationally.5)–8) They are excel- lent targets for X-ray spectroscopic observations because the large covering fraction of the wind and the moderate X-ray luminosities result in large volumes of photoionized plasma that produce strong recombination lines and narrow radiative recombination continua of H- and He-like ions, as well as fluorescent lines from lower charge states. §2. Vela X-1 Vela X-1 is the prototypical detached HMXB, having been studied extensively in nearly every waveband, particularly in X-rays, since its discovery as an X-ray source during a rocket flight four decades ago. It consists of a B0.5 Ib supergiant and a magnetic neutron star in an 8.964-day orbit. From an X-ray spectroscopic point of view, Vela X-1 distinguished itself in 1994 when Nagase et al.,9) using ASCA SIS data, showed that, in addition to the well-known 6.4 keV emission line, the eclipse X-ray spectrum is dominated by recombination lines and continua of H- and He-like Ne, Mg, Si, S, Ar, and Fe. These data were subsequently modeled in detail by Sako et al.,10) using a kinematic model in which the photoionized wind was characterized by the ionization parameter ξ ≡ LX/nr 2, where r is the distance from the neutron star and n is the number density, given by the mass-loss rate and velocity law of an undisturbed CAK wind. Vela X-1 was subsequently observed with the Chandra HETG in 2000 for 30 ks in eclipse11) and in 2001 for 85, 30, and 30 ks in eclipse and at binary phases 0.25 and 0.5, respectively.12), 13) Watanabe et al.,13) using very similar assumptions as Sako et al. and a Monte Carlo radiation transfer code, produced a global model of Vela X-1 that simultaneously fit the HETG spectra from the three binary phases with a wind mass-loss rate Ṁ ≈ 2 × 10−6 M⊙ yr −1 and terminal velocity V∞ = 1100 km s −1. One of the failures of this model was the velocity shifts of the emission lines between eclipse and phase 0.5, which were observed to be ∆V ≈ 400–500 km s−1, while the model simulations predicted ∆V ∼ 1000 km s−1. In order to resolve this discrepancy, Watanabe et al. performed a 1D calculation to estimate the wind velocity profile along the line of centers between the two stars, accounting, in an approximate way, for the reduction of the radiative driving due to photoionization. They found that the velocity of the wind near the neutron star is lower by a factor of 2–3 relative to an undisturbed CAK wind, which was sufficient to explain the observations. However, these results were not fed back into their global model to determine the effect on the X-ray spectra. §3. Hydrodynamic Simulations To make additional progress in our understanding of the wind and accretion flow of Vela X-1 in particular and HMXBs in general — to bridge the gap between the detailed hydrodynamic models of Blondin et al. and the simple kinetic-spectral models of Sako et al. and Watanabe et al. — we have undertaken a project to develop improved models of radiatively-driven photoionized accretion flows, with the goal of producing synthetic X-ray spectral models that possess a level of detail commensurate with the grating spectra returned by Chandra and XMM-Newton. This project combines (1) XSTAR14) photoionization calculations, (2) HUL- Hydrodynamic and Spectral Simulations of HMXB Winds 3 (a) (b) (c) (d) x x x Fig. 1. Color-coded maps of (a) log T (K) = [4.4, 8.3], (b) log n (cm−3) = [7.4, 10.8], (c) log V (km s−1) = [1.3, 3.5], and (d) log ξ (erg cm s−1) = [1.1, 7.7] in the orbital plane of Vela X-1. The positions of the OB star and neutron star are shown by the circle and the “×,” respectively. The horizontal axis x = [−5, 7]× 1012 cm, and the vertical axis y = [−4, 8]× 1012 cm. LAC15) emission models appropriate to X-ray photoionized plasmas, (3) improved models of the radiative driving of the photoionized wind, (4) FLASH16) three- dimensional time-dependent adaptive-mesh hydrodynamics calculations, and (5) a Monte Carlo radiation transport code.17) Radiative driving of the wind is accounted for via the force multiplier formalism,1) accounting for X-ray photoionization and non-LTE population kinetics using HULLAC atomic data for 2× 106 lines of 35,000 energy levels of 166 ions of the 13 most cosmically abundant elements. In addi- tion to the usual hydrodynamic quantities, the FLASH calculations account for (a) the gravity of the OB star and neutron star, (b) Coriolis and centrifugal forces, (c) radiative driving of the wind as a function of the local ionization parameter, temper- ature, and optical depth, (d) photoionization and Compton heating of the irradiated wind, and (e) radiative cooling of the irradiated wind and the “shadow wind” be- hind the OB star. To demonstrate typical results of our simulations, we show in Fig. 1 color-coded maps of the log of the (a) temperature, (b) density, (c) velocity, and (d) ionization parameter of a FLASH simulation with parameters appropriate to Vela X-1. This is a 2D simulation in the binary orbital plane, has a resolution of ∆l = 9.4 × 1010 cm, and, at the time step shown (t = 100 ks), the relatively slow (V ≈ 400 km s−1)∗) irradiated wind has reached just ∼ 2 stellar radii from the stellar surface. The various panels show (1) the effect of the Coriolis and centrifugal forces, which cause the flow to curve clockwise, (2) the cool, fast wind behind the OB star, (3) the hot, slow irradiated wind, (4) the hot, low density, high velocity flow downstream of the neutron star, and (5) the bow shock and two flanking shocks formed where the irradiated wind collides with the hot disturbed flow in front and downstream of the neutron star. Given these maps, it is straightforward to determine where in the binary the X-ray emission originates. To demonstrate this, we show in Fig. 2 color-coded maps of the log of the emissivity of (a) SiXIV Lyα, (b) SiXIII Heα, (c) FeXXVI Lyα, and (d) FeXXV Heα. The gross properties of these maps agree with Fig. 24 of Watanabe et al., but they are now (1) quantitative rather than qualitative and (2) specific to individual transitions of individual ions. The maps also capture features that otherwise would not have been supposed, such as the excess emission in the H- ∗) Note that this velocity reproduces the value that Watanabe et al. found was needed to match the velocity of the emission lines in the Chandra HETG spectra of Vela X-1. 4 C. W. Mauche, D. A. Liedahl, S. Akiyama, and T. Plewa (a) (b) (c) (d) x x x x Fig. 2. Color-coded maps of the log of the X-ray emissivity of (a) SiXIV Lyα, (b) SiXIII Heα, (c) FeXXVI Lyα, and (d) FeXXV Heα. In each case, two orders of magnitude are plotted. and He-like Si lines downstream of the flanking shocks. Combining these maps with the velocity map (Fig. 1c), these models make very specific predictions about (1) the intensity of the emission features, (2) where the emission features originate, and (3) their velocity widths and amplitudes as a function of binary phase. The next step in our modeling effort is to feed the output of the FLASH simula- tions into the Monte Carlo radiation transfer code, to determine how the spatial and spectral properties of the X-ray emission features are modified by Compton scatter- ing, photoabsorption followed by radiative cascades, and line scattering. This work is underway. Acknowledgements This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48. T. Plewa’s contribution to this work was supported in part by the U.S. Department of Energy under Grant No. B523820 to the Center for Astrophysical Thermonuclear Flashes at the University of Chicago. References 1) J. I. Castor, D. C. Abbott, and R. I. Klein, Ap.J. 195 (1975), 157, CAK. 2) S. Hatchett and R. McCray, Ap.J. 211 (1977), 552. 3) R. McCray, T. R. Kallman, J. I. Castor, and G. L. Olson, Ap.J. 282 (1984), 245. 4) I. R. Stevens and T. R. Kallman, Ap.J. 365 (1990), 321. 5) J. M. Blondin, T. R. Kallman, B. A. Fryxell, and R. E. Taam, Ap.J. 356 (1990), 591. 6) J. M. Blondin, I. R. Stevens, and T. R. Kallman, Ap.J. 371 (1991), 684. 7) J. M. Blondin, Ap.J. 435 (1994), 756. 8) J. M. Blondin and J. W. Woo, Ap.J. 445 (1995), 889. 9) F. Nagase, G. Zylstra, T. Sonobe, T. Kotani, H. Inoue, and J. Woo, Ap.J. 436 (1994), L1. 10) M. Sako, D. A. Liedahl, S. M. Kahn, and F. Paerels, Ap.J. 525 (1999), 921. 11) N. S. Schulz, C. R. Canizares, J. C. Lee, and M. Sako, Ap.J. 564 (2002), L21. 12) G. Goldstein, D. P. Huenemoerder, and D. Blank, A.J. 127 (2004), 2310. 13) S. Watanabe, et al., Ap.J. 651 (2006), 421. 14) T. Kallman and M. Bautista, Ap.J.S. 133 (2001), 221 15) A. Bar-Shalom, M. Klapisch, and J. Oreg, Phys. Rev. A38 (1988), 1773 16) B. Fryxell, et al., Ap.J.S. 131 (2000), 273. 17) C. W. Mauche, D. A. Liedahl, B. F. Mathiesen, M. A. Jimenez-Garate, and J. C. Raymond, Ap.J. 606 (2004), 168. Introduction Vela-0.25 X-1 Hydrodynamic-0.25 Simulations

0704.0238

Radio Astrometric Detection and Characterization of Extra-Solar Planets: A White Paper Submitted to the NSF ExoPlanet Task Force

Radio Astrometric Detection and Characterization of Extra-Solar Planets: A White Paper Submitted to the NSF ExoPlanet Task Force Geoffrey C. Bower1, Alberto Bolatto1, Eric Ford2, Paul Kalas1, Jim Ulvestad3 ABSTRACT The extraordinary astrometric accuracy of radio interferometry creates an important and unique opportunity for the discovery and characterization of exo- planets. Currently, the Very Long Baseline Array can routinely achieve better than 100 µas accuracy, and can approach 10 µas with careful calibration. We describe here RIPL, the Radio Interferometric PLanet search, a new program with the VLBA and the Green Bank 100 m telescope that will survey 29 low-mass, active stars over 3 years with sub-Jovian planet mass sensitivity at 1 AU. An upgrade of the VLBA bandwidth will increase astrometric accuracy by an order of magnitude. Ultimately, the colossal collecting area of the Square Kilometer Array could push astrometric accuracy to 1 microarcsecond, making detection and characterizaiton of Earth mass planets possible. RIPL and other future radio astrometric planet searches occupy a unique volume in planet discovery and characterization parameter space. The parameter space of astrometric searches gives greater sensitivity to planets at large radii than radial velocity searches. For the VLBA and the expanded VLBA, the targets of radio astrometric surveys are by necessity nearby, low-mass, active stars, which cannot be studied efficiently through the radial velocity method, coronagraphy, or optical interferometry. For the SKA, detection sensitivity will extend to solar- type stars. Planets discovered through radio astrometric methods will be suitable for characterization through extreme adaptive optics. The complementarity of radio astrometric techniques with other methods demonstrates that radio astrometry can play an important role in the roadmap for exoplanet discovery and characterization. 1Astronomy Department & Radio Astronomy Laboratory, University of California, Berkeley, CA 94720; gbower,bolatto,[email protected] 2Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138; ericb- [email protected] 3National Radio Astronomy Observatory, P.O. Box 0, Socorro NM 87801, U.S.A. ; [email protected] http://arxiv.org/abs/0704.0238v1 – 2 – 1. Radio Astrometry and Extra-Solar Planets Radio astrometry has long been the gold standard for definition of celestial reference frames (Fey et al. 2004, AJ 127, 3587) and has been used to obtain the most accurate geometric measurements of any astronomical technique. Astrometric results include mea- surement of the deflection of background sources due to the gravitational fields of the Sun and Jupiter (Fomalont & Kopeikin 2003, ApJ, 598, 704), the parallax and proper motion of pulsars at distances greater than 1 kpc (Chatterjee et al. 2005, ApJ, 630, L61), an upper limit to the proper motion of Sagittarius A* of a few km s−1 (Reid & Brunthaler 2004, ApJ, 616, 872), the rotation of the disk of M33 (Brunthaler et al. 2005, Science, 307, 1440), and a < 1% distance to the Taurus star-forming cluster (Loinard 2006, BAAS, 209, 1080). The Very Long Baseline Array (VLBA) images nonthermal radio emission and can routinely achieve 100 µas astrometric accuracy, but has achieved an accuracy as high as 8 µas under favorable circumstances (Fomalont & Kopeikin 2003). Nonthermal stellar radio emission has been detected from many stellar types (Güdel 2002, ARA&A, 40, 217), including brown dwarfs (Berger et al. 2001, Nat, 410, 338), proto-stars (Bower et al. 2003, ApJ, 598, 1140) , massive stars with winds (Dougherty et al. 2005, ApJ, 623, 447), and late-type stars (Berger et al. 2006, ApJ, 648, 629). Only late-type stars are sufficiently bright, numerous, nearby, and low mass to provide a large sample of stars suitable for large-scale astrometric exoplanet searches. Radio astrometric searches can determine whether or not M dwarfs, the largest stellar constituent of the Galaxy, are surrounded by planetary systems as frequently as FGK stars and if the planet mass-period relation varies with stellar type. The population of gas giants at a few AU around low mass stars is an important discriminant between planet formation models. Radio astrometric searches have a number of unique qualities: • Opportunity to discover planets around nearby active M dwarfs at large radii; • Ability to fully characterize orbits of detected planets, without degeneracies in mass, inclination, and longitude of ascending node; • Sensitivity to long-period planets with sub-Jovian masses currently and Earth masses ultimately; • Complementary with existing planet searching techniques: most targets cannot be explored through other methods; • Ability to follow-up detected planets with imaging and spectroscopy; and, • Absolute astrometric positions within the radio reference frame for stars and planets. The quality and uniqueness of radio astrometry for planet searches are the result of two factors: – 3 – Fig. 1.— Sensitivity of different methods in planet mass and semi-major axis space for radio astrometric surveys and other methods. “Exp. VLBA” refers to the upgraded VLBA described in § 4. The semi-major axis at the minimum in the astrometric search curves is determined by the search duration, which is 3 years for RIPL and the Exp. VLBA campaign. • High precision of radio astrometry: The VLBA can routinely achieve 100 µas accuracy through relative astrometry. This precision is an order of magnitude better than obtained from laser-guide star adaptive optics (e.g., Pravdo et al. 2005). Future instruments will have one to two orders of magnitude more accurate astrometry, comparable to the best accuracy achievable with the proposed SIM spacecraft. • Active stars are difficult to study in optical programs: Our target stars are active M dwarfs, which have radio fluxes on the order of 1 mJy. These radio stars are difficult to study through optical radial velocity techniques because they are faint and because the activity in these stars distorts line profiles, reducing the accuracy of radial velocity measurements. We give a sketch of the parameter space for RIPL, future radio astrometric searches, the Space Interferometric Mission, radial velocity searches, and coronagraphic searches in Figure 1. A comparison of the radial velocity and astrometric amplitudes indicates that astrometric techniques are favored over radial velocity techniques for long period ( ∼> 1 year) planets for these faint objects, for an astrometric accuracy of ∼ 100 µas (Ford 2006, PASP, 118, 364). – 4 – In Section 2, we describe the sensitivity and methods of radio astrometry. In Section 3, we describe a new program with the VLBA and the Green Bank 100m telescope to search for planets around nearby M dwarfs. In Section 4, we demonstrate that a bandwidth upgrade for the VLBA will increase astrometric accuracy or stellar sample sizes by an order of magnitude. In Section 5, we discuss the role that the Square Kilometer Array can play with its three order of magnitude increase in sensitivity over the VLBA. 2. Radio Astrometry Sensitivity and Methods Astrometric exoplanet searches must be able to detect an astrometric signal that has an amplitude of θ = 2 = 1400 µas ∗ ∗Mp/MJ ∗ 0.2M⊙ , (1) for a planet of mass Mp orbiting a star of mass M∗ with a semimajor axis a at a distance D from the Sun (a mass of 0.2 M⊙ corresponds to a M5 dwarf). To robustly detect a planet, observations must span at least a significant fraction of a period T = 2.2 yr ∗ 0.2M⊙ . (2) The ultimate accuracy that can be obtained through a radio astrometric technique is σast = σbeam/SNR, (3) where σbeam = b/λ is the synthetic beam size for an array with maximum baseline b, λ is the observing wavelength, and SNR is the signal to noise ratio of the target source detection. For the VLBA σast ≈ 500 µas/SNR. The astrometric position is defined relative to nearby (∼ 1◦) compact radio sources. Typical observations include switching on minute timescales between the calibrator and the target sources, with less frequent observations of secondary calibrators. The use of mul- tiple calibrators is intended to determine the differential delay in position on the sky due to varying path length from tropospheric water vapor. The extent to which this cannot be calibrated sets the final astrometric accuracy in observations that are not SNR-limited. The nearer the calibrators and the greater sensitivity at which they can be detected typ- ically determines this error. The error decreases linearly with decreasing calibrator-target separation. The increased sensitivity of future arrays will increase the calibrator density and therefore decrease the typical separation from calibrator to target and the uncalibrated astrometric error. For sufficiently small target to calibrator separation, the calibrator will be in the primary beam of the antenna, enabling simultaneous observations of the target and calibrator that also remove temporal dependence of tropospheric variations. – 5 – RIGHT ASCENSION (J2000) 22 01 13.3054 13.3052 13.3050 13.3048 13.3046 13.3044 13.3042 13.3040 28 18 25.142 25.140 25.138 25.136 25.134 25.132 25.130 25.128 25.126 25.124 RIGHT ASCENSION (J2000) 22 01 13.3054 13.3052 13.3050 13.3048 13.3046 13.3044 13.3042 13.3040 28 18 25.142 25.140 25.138 25.136 25.134 25.132 25.130 25.128 25.126 25.124 RIGHT ASCENSION (J2000) 22 01 13.3054 13.3052 13.3050 13.3048 13.3046 13.3044 13.3042 13.3040 28 18 25.142 25.140 25.138 25.136 25.134 25.132 25.130 25.128 25.126 25.124 Fig. 2.— Images of GJ4247 in three separate epochs on 23, 25, and 26 March 2006 (right to left) from the VLBA Precursor Astrometric Survey. Contour levels are -3, 3, 4, 5, 6, 7, 8 times the rms noise of 95 µJy. The synthesized beam is shown in the lower left hand corner of each image. 3. RIPL: Radio Interferometric Planet Search RIPL is a 1400-hour, 3-year VLBA and GBT program to search for planets around 29 nearby, low-mass, active stars. The program will achieve sub-Jovian planet mass sensitivity. The observing program will be completed in 2009. The most serious limitation to astrometric accuracy may be from stellar activity that jitters the apparent stellar position. Most evidence, however, indicates that this radio astro- metric jitter is small. For instance, White, Lim and Kundu (1994, ApJ, 422, 293) model the radio emission from dMe stars as originating within ∼ 1 stellar radius of the photosphere. At a distance of 10 pc for a M5 dwarf a stellar radius is ∼ 100 µas, roughly an order of magnitude smaller than the astrometric signature of a Jupiter analog. We conducted the VLBA Precursor Astrometric Survey (VPAS) in Spring 2006 to assess the effect of stellar jitter on astrometric accuracy (Bower et al. 2007, in prep.). For each star, three VLBA epochs were spread over fewer than 10 days. Seven stars were detected in at least one epoch and four were detected in all three epochs (Figure 2). All stars have proper motions and parallaxes determined by Hipparcos or other optical methods with a precision of a few mas per year, yielding predicted relative positions accurate to ∼ 100 µas during the length of the study. For all stars detected with multiple epochs, the motions match the results of Hipparcos astrometry well with rms in each coordinate ranging from 0.08 to 0.26 µas. Deviations in the positions appear to be limited by our sensitivity; i.e., the effect of stellar activity on their positions is unimportant. In fact, the small differences in the fitted proper motion and the Hipparcos proper motion already eliminate brown dwarfs as companions to these objects (Figure 3). The measured differences are consistent with noise in the VLBA astrometry (200 µas /3day ∼ 20 mas/yr). The typical reflex motion due to a long period brown dwarf is ∼ 100 mas/yr, which would be apparent. The much longer time baseline and better sensitivity of RIPL will reduce proper motion errors by ∼ 2 orders of magnitude. – 6 – r (AU) GJ4247: 0.144 AU y−2 2 4 6 8 10 r (AU) GJ896A: 0.070 AU y−2 2 4 6 8 10 Fig. 3.— Region of planetary mass and semi-major axis phase-space rejected by acceleration upper limits based on combination of 3 epochs of radio astrometric measurements and optical astrometry, primarily from Hipparcos. Different contours indicate confidence intervals for excluded regions. 3.1. Synergy with other Planet Searches RIPL is synergistic with the existing and future planet-search programs, as well as cur- rent ground-based planet searches (including radial velocities, transits, adaptive optics, and interferometry). RIPL provides an opportunity to search for planetary systems in a unique area of parameter space that will not be targeted by other planet searches until the launch of NASA SIM - Planetquest. Ground based transit searches are most sensitive for very short periods (P ∼ days), and the Kepler mission aims to detect planets with orbital periods of slightly more than a year. Thus, RIPL will make a valuable contribution to our understanding of the frequency of long-period planets around M stars. Further, unlike transits and radial velocity observations astrometric measurements directly measure the planet mass, which is important for testing models of planet formation. While the unknown inclination is less of an issue for studying large samples of planets, measuring individual inclinations will be particularly valuable for planets around M dwarfs, since a relatively modest number of M dwarfs are being surveyed by RIPL (∼ 30 vs ∼ 3000 stars by radial velocities). Ground-based optical and near-infared interferometers (e.g., PTI, NPOI) require bright stars and are not appropriate for faint low-mass stars. The RIPL astrometric accuracy is an order of magnitude better than the astrometric error from Keck Laser Guide Star Adaptive Optics astrometry (Pravdo et al. 2005, ApJ, 630, 528). Thus, RIPL is the best means for an astrometric search of M dwarfs until SIM launches (now estimated for no earlier than 2016). – 7 – A long-period planet detected by RIPL would enable exciting scientific investigations such as photometric and spectroscopic observations to determine the planets physical prop- erties. While space based missions such as TPF-C and TPF-I are expected to be extremely powerful and aim to directly detect terrestrial mass planets, these missions are not expected to launch for at least a decade in the future. Knowing which stars have giant planets suitable for direct imaging would enable direct probes of an extrasolar planet. 4. VLBA Upgrade and Planet Detection The VLBA is presently being upgraded from a typical data rate of 256 Mbit/s to 4 Gbit/s, with project completion estimated by 2010. This will result in a sensitivity increase by a factor of 4, or about a factor of 8 increase in areal density of reference sources on the sky. Thus, the typical distance between a target star and its nearest reference source will decrease by a factor of ∼ 3. A few years later we expect a data rate of 16 Gbit/s, yielding a target-calibrator separation more than 10 times smaller than current values. Since in the limit of infinite SNR the astrometric error depends linearly on the separation from the reference source, relative astrometric errors of . 10 µas should be fairly routine; in principle, this would permit detection of a planet with a mass of less than 10% of the mass of Jupiter. The sensitivity increase afforded by these upgrades will also permit a sizable increase of the late-type dwarf sample. 5. Square Kilometer Array The Square Kilometer Array (SKA; Carilli & Rawlings 2004, New AR, 48, 979) is a proposed future radio telescope that would have a collecting area of a square kilometer, approximately 200 times the collecting area of the VLBA. The SKA would be built toward the end of the next decade; it is planned to cover the frequency range from 0.1 to 25 GHz, with the 5–10 GHz range being most useful for astrometric planet detection. If 25% of the SKA area at ∼ 8 GHz is constructed on baselines of 1000-5000 km, it will supply revolutionary astrometric accuracy (Fomalont & Reid 2004, New AR, 48, 1473). With dish antennas of 12m diameter, the combination of sensitivity and wide field of view often will enable many astrometric reference sources to be found in the same antenna field of view as the target star, allowing all temporal variations in Earth’s atmosphere to be removed. In such a case, the relative astrometric accuracy may reach ∼ 1 µas, competitive with SIM and enabling astrometric detection of Earth-mass planets. The sensitivity of the SKA will enable astrometric detection of thermal emission from stars. The Sun, for instance, would be detectable to a distance of 10 pc with the SKA. Thus, the SKA will be capable of detecting and characterizing planets around Sun-like stars. Radio Astrometry and Extra-Solar Planets Radio Astrometry Sensitivity and Methods RIPL: Radio Interferometric Planet Search Synergy with other Planet Searches VLBA Upgrade and Planet Detection Square Kilometer Array

0704.0239

Interface dynamics of microscopic cavities in water

Interface dynamics of microscopic cavities in water Joachim Dzubiella1, ∗ Physics Department, Technical University Munich, 85748 Garching, Germany (Dated: November 4, 2018) An analytical description of the interface motion of a collapsing nanometer-sized spherical cavity in water is presented by a modification of the Rayleigh-Plesset equation in conjunction with ex- plicit solvent molecular dynamics simulations. Quantitative agreement is found between the two approaches for the time-dependent cavity radius R(t) at different solvent conditions while in the con- tinuum picture the solvent viscosity has to be corrected for curvature effects. The typical magnitude of the interface or collapse velocity is found to be given by the ratio of surface tension and fluid vis- cosity, v ≃ γ/η, while the curvature correction accelerates collapse dynamics on length scales below the equilibrium crossover scales (∼1nm). The study offers a starting point for an efficient implicit modeling of water dynamics in aqueous nanoassembly and protein systems in nonequilibrium. I. INTRODUCTION Hydrophobic hydration in equilibrium is a phe- nomenon which exhibits qualitatively different behavior at small and large length scales.1,2 While small solutes (radii R .1nm) are accommodated by water with only minor perturbations, larger solutes (R &1nm) induce ma- jor rearrangements of water interfacial structure. As a consequence the solvation free energy G(R) of small hy- drophobic cavities scales with solute volume while for larger cavities it grows with surface area (as a good approximation near liquid-vapor coexistence) accompa- nied by weak solvent dewetting at extended restrain- ing hydrophobic surfaces.3 Furthermore, water, which is close to the liquid-vapor transition at normal condi- tions, can minimize interface area by locally evaporat- ing and forming a ’nanobubble’ within hydrophobic con- finement. Evidence of bubble formation in confined ge- ometry has been given early by computer simulations of smooth plate-like solutes,4 but more recently it has been demonstrated in varying degrees in atomistically resolved plate-like solutes,5,6 hydrophobic tubes and ion channels,7,8 and in the collapse of proteins,9,10 suggesting that it plays a key role in the stabilization and folding dynamics of certain classes of biomolecules.11,12 Experi- mental evidence of nanobubbles in strong confinement (in contrast to bubbles at a single planar surface3) has been given for instance in studies of water between hydropho- bic surfaces,13 in zeolites and silica nanotubes,14,15 and on a subnanometer scale in nonpolar protein cavities.16 The dewetting induced change in solvation energy is typically estimated using simple macroscopic arguments as known from capillarity theory, e.g. by describing in- terfaces with Laplace-Young (LY) type of equations.14,17 Recently an extension of the LY equation has become available which extrapolates to microscopic scales by in- cluding a curvature correction to the interface tension and considering atomistic dispersion and electrostatic potentials of the solvated solute explicitly.18 Although those macroscopic considerations (e.g,. the concept of surface tension) are supposed to break down on atom- istic scales they show surprisingly good results for the solvation energy of microscopic solutes, e.g. alkanes and noble gases, and quantitatively account for dewetting effects in nanometer-sized hydrophobic confinement.19 While we conclude that the equilibrium location of the solute-solvent interface seems to be well described by those techniques, nothing is known about the interface dynamics of evolution and relaxation. In this study we address two fundamental questions: First, what are the equations which govern the interface motion on atomistic (∼1nm) scales? Secondly, does the dynamics exhibit any signatures of the length scale crossover found in equilib- rium? On macroscopic scales the collapse dynamics of a (va- por or gas) bubble is related to the well-known phe- nomenon of sonoluminescence.20 The governing equa- tions can be derived from Navier-Stokes and capillarity theory and are expressed by the Rayleigh-Plesset (RP) equation.21 We will show that the RP equation simpli- fies in the limit of microscopic cavities and can be ex- tended to give a quantitative description of cavity inter- face dynamics on nanometer length scales. We find a qualitatively different dynamics than the typical “mean- curvature flow” description of moving interfaces,22 in par- ticular a typical magnitude of interface or collapse veloc- ity given by the ratio of surface tension and fluid viscosity, v ≃ γ/η. Our study is restricted to the generic case of the collapse of a spherical cavity and is complemented by explicit solvent molecular dynamics (MD) computer simulations. We note here that recently, Lugli and Zer- betto studied nanobubble collapse in ionic solutions by MD simulations on similar length scales.23 While their MD data compares favorably with our results their in- terpretation and conclusions in terms of the RP equa- tion are different. We will resume this discussion in the conclusion section. In this study we show that a simple analytical approach quantitatively describes microscopic cavity collapse for a variety of different solvent situations while the sim- ulations suggest that the solvent viscosity needs to be corrected for curvature effects. Our study might offer a simple starting point for an efficient implicit modeling of water dynamics in aqueous nanoassembly and protein http://arxiv.org/abs/0704.0239v1 systems in nonequilibrium. II. THEORY The Rayleigh-Plesset equation for the time evolution of a macroscopic vapor bubble with radius R(t) can be written as21 = ∆P + 4η , (1) where ρm is the solvent mass density, ∆P = P − Pv the difference in liquid and vapor pressures, η the dynamic viscosity, and γ the liquid-vapor interface tension. While for macroscopic bubble radii the inertial terms (left hand side) control the dynamics, for decreasing radii the fric- tional and pressure terms (right hand side) grow in rel- ative magnitude and eventually dominate, so that com- pletely overdamped dynamics can be assumed on atom- istic scales: Ṙ ≃ − . (2) A rough estimate for the threshold radius Rt below which friction dominates is given when the Reynolds number R = vRρm/η becomes unity and viscous and inertial forces are balanced. With a typical initial interface ve- locity of the order of v ∼ γ/η [from R̈(0) = 0 in eq. (1)] we obtain Rt = η 2/(ρmγ) (3) which is ≃ 10nm for water at normal conditions. Note that this threshold value can deviate considerably for a fluid different than water and that the viscosity typically has a strong temperature (T ) dependence which implies that Rt can change significantly with T . In equilibrium (Ṙ = 0) the remaining expression in eq. (2) is the (spherical) LY equation ∆P + 2γ/R = 0. Thus eq. (2) describes a linear relationship between cap- illary pressure and interface velocity where R/(4η) plays the role of an interface mobility (inverse friction).22 In- terestingly, the mobility is linear in bubble radius which leads to a constant velocity driven by surface tension in- dependent of radius (assuming P ≃ 0); this has to be con- trasted to the typically used capillary dynamics which is proportional to the local mean curvature ∝ 1/R.22 Generalizations of the LY equation to small scales are available by adding a Gaussian curvature term (∼ 1/R2) as shown by Boruvka and Neumann24; that has been demonstrated to be equivalent to a first order curvature correction in surface tension, i.e. γ(R) = γ∞(1−δT/R), where δT is the Tolman length 25 and γ∞ the liquid-vapor surface tension for a planar interface (R = ∞). The Tolman length has a magnitude which is usually of the order of the size of a solvent molecule. Furthermore, it has been observed experimentally that the viscosity of strongly confined water can depend on the particular na- ture of the surface/interface.26 We conclude that in gen- eral one has to anticipate that - analogous to the surface tension - the effective interface viscosity obeys a curva- ture correction in the limit of small cavities due to water restructuring in the first solvent layers at the hydropho- bic interface. In the following we make the simple first order assumption that the correction enters eq. (2) also linear in curvature (∼ 1/R) yielding Ṙ = − ∆PR+∆Pδvis + 2γ∞ + , (4) where the constant δvis is the coefficient for the first order curvature correction in viscosity and η∞ the macroscopic bulk viscosity. Additionally, we define δ = δvis − δT and second order terms in curvature are neglected. We note that the choice of the 1/R-scaling of the viscosity curva- ture correction has no direct physical justification and is arbitrary. We think however, that a curvature correction based on an expansion in orders of mean curvature is the simplest and most natural way for such a choice. In water at normal conditions the pressure terms in (4) are negligible so that for large radii (R ≫ δ) the in- terface velocity is constant and R(t) = R0 − γ∞/(2η∞)t. This leads to a collapse velocity of about v ≃0.4Å/ps (40m/s) which is 6% of the thermal velocity of water vth = 3kBT/m showing that dissipative heating of the system is relatively weak on these scales. A rough esti- mate for the dissipation rate can be made by the released interfacial energy dG(R, t)/dt ≃ d(4πR(t)2γ∞)/dt = −4πγ2 R(t)/η∞ yielding for instance dG(R, t = 0)/dt ≃ −35kBT/ps for a bubble with R0 =2nm. At small radii (R ≃ δ) the solution of (4) goes as R(t) ∼ const− (δγ∞/η∞)t decreasing or increasing the ve- locity depending on the sign of δ = δvis − δT, i.e. the acceleration depends on the particular sign and mag- nitude of the curvature corrections to surface tension and viscosity. For large pressures and radii the first term dominates which gives rise to an exponential de- cay R(t) ∼ exp[−∆P/(4η∞)t]. While extending to small scales we have assumed that the time scale of internal in- terface dynamics, i.e. hydrogen bond rearrangements,27 is much faster than the one of bubble collapse. III. MD SIMULATION In order to quantify our analytical predictions we complement the theory by MD simulations using ex- plicit SPC/E water.28 The liquid-vapor surface tension of SPC/E water has been measured and agrees with the experimental value for a wide range of temperatures.29 For T = 300K and P = 1bar we have γ∞ = 72mN/m. The Tolman length has been estimated to be δT ≃ 0.9Å from equilibrium measurements of the solvation energy of spherical cavities.30 At the same conditions the dynamic viscosity of SPC/E water has been found to be η∞ = 6.42 · 10 −4Pa·s,31 ∼24% smaller than for real water. In experiments in nanometer hydrophobic con- finement and at interfaces however, the viscosity shows deviations from the bulk value but remains comparable.26 We proceed by treating the viscosity η∞ as an adjustable parameter together with its curvature correction coeffi- cient δvis. The MD simulations are carried out with the DLPOLY2 package32 using an integration time step of 2fs. The simulation box is cubic and periodic in all three dimensions with a length of L = (61.1 ± 0.2)Å in equi- librium involving N = 6426 solvent molecules. Electro- static interactions are calculated by the smooth-particle mesh Ewald summation method. Lennard-Jones inter- actions are cut-off and shifted at 9Å. Our investigated systems are at first equilibrated in the NPT ensemble with application of an external spherical potential of the form βV (r) = [Å/(r −R′0)] 12 and all molecules removed with r < R′0 since vapor can safely be neglected on these scales. This stabilizes a well-defined spherical bubble of radius R0 ≃ R 0 + 1Å. We define the cavity radius by the radial location where the water density ρ(r) drops to half of the bulk density ρ0/2. Thirty independent config- urations in 20ps intervals are stored and serve as initial configurations for the nonequilibrium runs. We employ a Nosé-Hoover barostat and thermostat with a 0.2ps relax- ation time to maintain the solvent at a pressure P and a temperature T . Other choices of relaxation times in the reasonable range between 0.1 and 0.5ps do not alter our results. In the nonequilibrium simulations the con- straining potential is switched off and the relaxation to equilibrium is averaged over the thirty runs. IV. RESULTS system P/bar T/K cNaCl/M Q/e η∞/(10 −4Pa·s) I 1 300 0 0 5.14 II 1 300 1.5 0 5.94 III 1 277 0 0 8.48 IV 2000 300 0 0 4.56 V 1000 300 0 0 4.72 VI 1 300 0 +2 5.14 TABLE I: Investigated system parameters: pressure P , tem- perature T , and salt (NaCl) concentration c. In system VI a fixed ion with charge Q = +2e is placed at the center of the collapsing bubble. The viscosity η∞ is a fit-parameter in systems I-V (see text). We perform simulations of six different systems I-VI whose features are summarized in Tab. I and differ in thermodynamic parameters T and P (I, III, IV, and V) but also inclusion of dispersed salt (II), and the influ- ence of a charged particle in the bubble center (VI) are considered. Note that the exact value of the crossover length scale (however defined) can depend on the detailed thermodynamic or solvent condition but remains close to 1nm.2 0 5 10 15 20 25 30 t=1ps t=5ps t=10ps t=14ps t=17ps t=19ps t=23ps 0 5 10 15 20 τ(R)/ "10-90"- thickness FIG. 1: Interface density profiles ρ(r)/ρ0 for system I are plotted vs. the radial distance r from the bubble center for different times t/ps=1,5,10,14,17,19,23. Symbols denote MD simulation data and lines are fits using 2ρ(r)/ρ0 = erf{[r − R(t)]/d}+1. The bubble radius R(t) is defined by the distance at which the density is ρ0/2 (dotted line). The inset shows the “10-90” thickness τ = 1.8124 d of the interface vs. R for initial radii R0 = 19.83Å (pluses) and R0 = 25.6Å (crosses). System I is at normal conditions (T=300K, P=1bar) and consists of pure SPC/E water. Fig. 1 shows the observed interface profiles in the nonequilibrium situa- tion at different times t/ps=1, 5, 10, 14, 17, 19, and 23 starting from an initial radius R0 = 19.83Å. The liquid- vapor interface stays relatively sharp in the process of relaxation but broadens noticeably for smaller radii. At t ≃ 23ps the system is completely relaxed to a homoge- neous density distribution. The same time scale of bubble collapse has been found in explicit water computer simu- lations of dewetting in nanometer-sized paraffin plates,17 polymers,11 and atomistically resolved proteins.9,10 We find that the interface profiles can be fitted very well with a functional form 2ρ(r)/ρ0 = erf{[r−R(t)]/d}+ 1, where d is a measure of the interface thickness. The interface fits are also shown in Fig. 1 together with the MD data. The experimentally accessible “10-90” thick- ness τ of an interface is the thickness over which the density changes from 0.1ρ0 to 0.9ρ0 and is related to the parameter d via τ = 1.8124 d. While experimental values of τ for the planar water liquid-vapor interface vary be- tween ∼ 4 and 8Å the measured values for SPC/E water in the finite simulation systems are τ∞ =3 to 4Å. 29 We find a strongly radius-dependent function τ(R) plotted in the inset to Fig. 1 for initial radii R0 = 19.83Å and R0 = 25.6Å. For R ≃ R0 the thickness increases during the following 5ps from the equilibrium value τ ≃ 3Å to FIG. 2: Time evolution of the cavity radius R(t) for parame- ters as defined in systems I-VI. The solution of the modified RP equation (4) (lines) is plotted vs. MD data (symbols). The inset shows the solution of the modified RP equation in- cluding inertia terms, cf. lhs of (1), (dashed lines) compared to eq. (4) for system I with initial radii R0 = 19.83Å and R0 = 10.0Å. about τ ≃ 4.5− 5Å independent of R0. While the exact equilibrium thickness at t = 0 depends on the particu- lar choice of the confining potential V (r) (e.g., a softer potential might lead to a broader initial interface) this suggest that 4.5-5Å is the typical interface thickness for a bubble of 1nm size. Regarding the slope of the curve one might speculate that τ(R → ∞) saturates to the thickness τ∞ of the measured planar interface for R0 → ∞. For R . 10Å the thickness increases twofold during the relaxation to equilibrium. This broadening might be attributed to increased density fluctuations and the structural change of interfacial water in the system when crossing from large to small length scales which has been shown to happen in equilibrium at ∼ 1nm.1,2 In Fig. 2 we plot the time evolution of the bubble ra- dius R(t) for all investigated systems. Let us first focus on the simulation data of system I (circles). As antic- ipated the bubble radius decreases initially in a linear fashion while for smaller radii (R(t) . 10Å) the velocity steadily increases. From the best fit of eq. (4) we find a viscosity η∞ = 5.14 · 10 −4Pa·s and its curvature cor- rection coefficient δvis = 4.4Å. Although investigating a confined system with large interfaces the viscosity value differs only 20% from the SPC/E bulk value. Further- more, from our macroscopic point of view the MD data show that high curvature decreases the viscosity and the latter has to be curvature-corrected with a (positive) co- efficient larger than the Tolman length δT. If the surface tension decreased in a stronger fashion with curvature than viscosity the collapse velocity would drop in qual- itative disagreement with the simulation. The overall behavior of R(t) and the collapse velocity of about ∼ 1Å/ps agrees very well with the recent MD data of Lugli and Zerbetto, who simulated the collapse of a 1nm sized bubble in SPC water.23 The inset to Fig. 2 shows the solution of eq. (4) in- cluding inertial terms [left hand side of (1)] to check the assumption of overdamped dynamics. While inertial ef- fects are indeed small but not completely negligible for an initial radius R0 = 19.83Å they basically vanish for R0 = 10Å. Interestingly, the inertial effects are not visible in the MD simulation data at all. We attribute this obser- vation to the finite and periodic simulation box which is known to suppress long-ranged inertial (hydrodynamic) effects.33 In the following we assume δvis to be independent of the other parameters and treat only η∞ as adjustable variable. In system II we add 175 salt pairs of sodium chloride (NaCl) into the aqueous solution resulting in a concentration of c ≃1.5M. The ion-SPC/E interaction parameters are those used by Bhatt et al.34 who mea- sured a linear increase of surface tension with NaCl con- centration in agreement with experimental data. While this increment for c = 1.5M is about small 2-3%, the vis- cosity has been measured experimentally to increase by approximately 18% at 298.15K.35 Indeed by comparing the simulation data to the theory we find a 16% larger viscosity η∞ = 5.94 ·10 −4Pa·s. A slower collapse velocity has been found also in the MD simulations of Lugli and Zerbetto in concentrated LiCl and CsCL solutions when compared to pure water.23 In system III we investigate the effect of lowering the temperature by simulating at T = 277K.While only a 5% increase of the water surface tension (SPC/E and real water) is estimated from available data29 the viscosity depends strongly on temperature: the relative increase has been reported to be between 55 − 75% for SPC/E water (85% for real water).36 Inspecting the MD data and considering the surface tension increase we find in- deed a large decrease in viscosity of 65% with a best-fit η∞ = 8.48 · 10 −4Pa·s. Both systems, II and III, show that solvent viscosity has a substantial influence on bub- ble dynamics as quantitatively described by our simple analytical approach. In systems IV and V we return to T = 300K but increase the pressure P by a factor of 2000 and 1000, respectively. Best fits provide viscosities which are around 10% smaller than at normal conditions in agreement with the very weak pressure dependence of the viscosity found in experiments37,38 at T=300K. The major contribution to the faster dynamics comes explic- itly from the pressure terms in eq. (4). Although mov- ing away from liquid-vapor coexistence by increasing the pressure up to 2000bar we assume (and verify hereby) that the bubble interface tension can still be described by γ∞. In system VI we investigate the influence of a hy- drophilic solute on the bubble interface motion in order to make connection to cavitation close to molecular (pro- tein) surfaces. As a simple measure we fix a divalent ion at the center of the bubble so that we retain spherical symmetry. The ion is modeled by a Lennard-Jones (LJ) potential ULJ(r) = 4ǫ[(σ/r) 12 − (σ/r)6] with Q = +2e point charges and uses the LJ parameters of the SPC/E oxygen-oxygen interaction. As demonstrated recently the LY equation can be modified to include dispersion and electrostatic solute-solvent interactions explicitly,18 which extends (4) to Ṙ = − − ρ0ULJ(R) + 32πǫ0R4 The last term in (5) is the Born electrostatic energy den- sity of a central charge Q in a spherical cavity with ra- dius R with low dielectric vapor ǫv = 1 surrounded by a high dielectric liquid (1/ǫl ≃ 0). The electric field around the ionic charge and the dispersion attracts the surrounding dipolar water what accelerates and eventu- ally completely governs the bubble collapse below a ra- dius R(t) . 13Å (t & 7ps) as also shown in Fig. 2. The theoretical prediction (5) agrees very well without any fitting using the viscosity from system I. We find that the acceleration is mainly due to the electrostatic attraction; the dispersion term plays just a minor role while the excluded volume repulsion eventually deter- mines the final (equilibrium) radius of the interface with R(t = ∞) ≃ 2Å. V. CONCLUSIONS In conclusion, we have presented a simple analytical and quantitative description of the interface motion of a microscopic cavity by modifying the macroscopic RP equation. Based on our MD data we find for the macro- scopic description that analogous to the surface tension the viscosity has to be corrected for curvature effects, a prediction compelling to investigate further in detail and probably related to the restructuring of interfacial water for high curvatures (small R). The viscosity correction accelerates collapse dynamics markedly below the equi- librium crossover scale (∼1nm) in contrast to the pure equilibrium picture where surface tension decreases what slows down the collapse. Further, we find that the dy- namics is curvature-driven due to the corrections to sur- face tension and viscosity, not due surface tension as often postulated.22 As a simple estimate, the interface velocity is typically given by the ratio of surface tension and fluid viscosity, v ≃ γ/η. A comment has to be made regarding the recent work of Lugli and Zerbetto on MD simulations of nanobubble collapse in ionic solutions. While their MD data of the collapse velocity for a 1nm bubble agree very well with our results their interpretation in terms of the RP equa- tion is different. They fit the ’violent regime’ solution of the RP equation to the data [which is the solution of only the inertial part, left hand side of (1)] and argue that the violent regime still holds on the nm scale. As demonstrated in this work, we arrive to a different con- clusion: the collapse is friction dominated, the collapse driving force is mainly capillary pressure, and we suggest that the microscopic viscosity has to be curvature cor- rected to explain the high curvature collapse behavior in the MD simulations. The good agreement between our modified RP equation and the MD data for different sol- vent conditions, leading for instance to an altered solvent surface tension or viscosity, support our view. We finally note that extensions of the LY equation are based on minimizing an appropriate free energy G(R) or free energy functional18,24 so that we can write in a more general form Ṙ ∼ [∂G(R)/∂R]/[η(R)R]. It is highly de- sirable to generalize this simple dynamics further to ar- bitrary geometries with which a wide field of potential applications might open up, i.e. an efficient implicit mod- eling of the water interface dynamics in the nonequilib- rium process of hydrophobic nanoassembly, protein dock- ing and folding, and nanofluidics. Acknowledgements J. D. thanks Lyderic Bocquet for pointing to the RP equation, Bo Li (Applied Math, UCSD), Roland R. Netz, Rudi Podgornik, and Dominik Horinek for stimu- lating discussions, and the Deutsche Forschungsgemein- schaft (DFG) for support within the Emmy-Noether- Programme. ∗ e-mail address:[email protected] 1 D. Chandler, Nature 437, 640 (2005). 2 S. Rajamani, T. M. Truskett, and S. Garde, PNAS 102, 9475 (2005). 3 A. Poynor, L. Hong, I. K. Robinson, S. Granick, Z. Zhang, and P. A. Fenter, Phys. Rev. Lett. 97, 266101 (2006). 4 A. Wallquist and B. J. Berne, J. Phys. Chem. 99, 2893 (1995). 5 T. Koishi, Phys. Rev. Lett. 93, 185701 (2004). 6 N. Giovambattista, P. J. Rossky, and P. D. Debenedetti, Phys. Rev. E 73, 041604 (2006). 7 O. Beckstein and M. S. P. Sansom, Proc. Nat. Acad. Sci. 100, 7063 (2003). 8 A. Anishkin and S. Sukharev, Biophys. J. 86, 2883 (2004). 9 R. Zhou, X. Huang, C. Margulis, and B. J. Berne, Science 305, 1605 (2004). 10 P. Liu, X. Huang, R. Zhou, and B. J. Berne, Nature 437, 159 (2005). 11 P. R. ten Wolde and D. Chandler, Proc. Natl. Acad. Sci. 99, 6539 (2002). 12 D. M. Huang and D. Chandler, PNAS 97, 8324 (2000). 13 A. Carambassis, L. C. Jonker, P. Attard, and M. W. Rut- mailto:[email protected] land, Phys. Rev. Lett. 80, 5357 (1998). 14 R. Helmy, Y. Kazakevich, C. Ni, and A. Y. Fadeev, J. Am. Chem. Soc. 127, 12446 (2005). 15 K. Jayaraman, K. Okamoto, S. J. Son, C. Luckett, A. H. Gopalani, S. B. Lee, and D. S. English, J. Am. Chem. Soc. 127, 17385 (2005). 16 M. D. Collins, G. Hummer, M. L. Quillin, B. W. Matthews, and S. M. Gruner, PNAS 102, 16668 (2005). 17 X. Huang, C. J. Margulis, and B. J. Berne, PNAS 100, 11953 (2003). 18 J. Dzubiella, J. M. J. Swanson, and J. A. McCammon, Phys. Rev. Lett. 96, 087802 (2006). 19 J. Dzubiella, J. M. J. Swanson, and J. A. McCammon, J. Chem. Phys. 124, 084905 (2006). 20 C. E. Brennen, Cavitation and bubble dynamics (Oxford U. Press, New York, 1995). 21 M. S. Plesset and A. Prosperetti, Annu. Rev. Fluid. Mech. 25, 577 (1977). 22 H. Spohn, J. Stat. Phys. 71, 1081 (1993). 23 F. Lugli and F. 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0704.0240

Viscosity, Black Holes, and Quantum Field Theory

INT PUB 07-02 Viscosity, Black Holes, and Quantum Field Theory Dam T. Son Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA Andrei O. Starinets Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada Key Words AdS/CFT correspondence, hydrodynamics Abstract We review recent progress in applying the AdS/CFT correspondence to finite-temperature field theory. In particular, we show how the hydrodynamic behavior of field theory is reflected in the low-momentum limit of correlation functions computed through a real-time AdS/CFT prescription, which we formulate. We also show how the hydrodynamic modes in field theory correspond to the low-lying quasinormal modes of the AdS black p-brane metric. We provide a proof of the universality of the viscosity/entropy ratio within a class of theories with gravity duals and formulate a viscosity bound conjecture. Possible implications for real systems are mentioned. CONTENTS INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 HYDRODYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Kubo’s Formula For Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Hydrodynamic Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Viscosity In Weakly Coupled Field Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 AdS/CFT CORRESPONDENCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Review Of AdS/CFT Correspondence At Zero Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Black Three-Brane Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 REAL-TIME AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Prescription For Retarded Two-Point Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Calculating Hydrodynamic Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 THE MEMBRANE PARADIGM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 THE VISCOSITY/ENTROPY RATIO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Universality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 The Viscosity Bound Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 http://arXiv.org/abs/0704.0240v2 2 Son, Starinets 1 INTRODUCTION This review is about the recently emerging connection, through the gauge/gravity correspondence, between hydrodynamics and black hole physics. The study of quantum field theory at high temperature has a long history. It was first motivated by the Big Bang cosmology when it was hoped that early phase transitions might leave some imprints on the Universe [1]. One of those phase transitions is the QCD phase transitions (which could actually be a crossover) which happened at a temperature around Tc ∼ 200 MeV, when matter turned from a gas of quarks and gluons (the quark-gluon plasma, or QGP) into a gas of hadrons. An experimental program was designed to create and study the QGP by colliding two heavy atomic nuclei. Most recent experiments are conducted at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory. Although significant circumstantial evidence for the QGP was accumulated [2], a theoretical interpretation of most of the experimental data proved difficult, because the QGP created at RHIC is far from being a weakly coupled gas of quarks and gluons. Indeed, the temperature of the plasma, as inferred from the spectrum of final particles, is only approximately 170 MeV, near the confinement scale of QCD. This is deep in the nonperturbative regime of QCD, where reliable theoretical tools are lacking. Most notably, the kinetic coefficients of the QGP, which enter the hydrodynamic equations (reviewed in Sec. 2), are not theoretically computable at these temperatures. The paucity of information about the kinetic coefficients of the QGP in particular and of strongly coupled thermal quantum field theories in general is one of the main reasons for our interest in their computation in a class of strongly coupled field theories, even though this class does not include QCD. The necessary technological tool is the anti–de Sitter–conformal field theory (AdS/CFT) correspondence [3, 4, 5], discovered in the investigation of D-branes in string theory. This correspondence allows one to describe the thermal plasmas in these theories in terms of black holes in AdS space. The AdS/CFT correspondence is reviewed in Sec. 3. The first calculation of this type, that of the shear viscosity in N = 4 supersymmetric Yang- Mills (SYM) theory [6], is followed by the theoretical work to establish the rules of real-time finite-temperature AdS/CFT correspondence [7, 8]. Applications of these rules to various special cases [9, 10, 11, 12] clearly show that even very exotic field theories, when heated up to finite temperature, behave hydrodynamically at large distances and time scales (provided that the number of space-time dimensions is 2+1 or higher). This development is reviewed in Sec. 4. Moreover, the way AdS/CFT works reveals deep connections to properties of black holes in classical gravity. For example, the hydrodynamic modes of a thermal medium are mapped, through the correspondence, to the low-lying quasi-normal modes of a black-brane metric. It seems that our understanding of the connection between hydrodynamics and black hole physics is still incomplete; we may understand more about gravity by studying thermal field theories. One idea along this direction is reviewed in Sec. 5. From the point of view of heavy-ion (QGP) physics, a particularly interesting finding has been the formulation of a conjecture on the lowest possible value of the ratio of viscosity and volume density of entropy. This conjecture was motivated by the universality of this ratio in theories with gravity duals. This is reviewed in Sec. 6. Viscosity, Black Holes, and QFT 3 This review is written primarily for readers with a background in QCD and QGP physics who are interested in learning about AdS/CFT correspondence and its applications to finite-temperature field theory. Some parts of this review (for example, the section about hydrodynamics) should be useful for readers with a string theory or general relativity background who are interested in the connection between string theory, gravity, and hydrodynamics. The perspectives here are shaped by our personal taste and therefore may appear narrow, but the authors believe that this review may serve as the starting point to explore the much richer original literature. In this review we use the “mostly plus” metric signature − + ++. 2 HYDRODYNAMICS From the modern perspective, hydrodynamics [13] is best thought of as an effective theory, describ- ing the dynamics at large distances and time-scales. Unlike the familiar effective field theories (for example, the chiral perturbation theory), it is normally formulated in the language of equations of motion instead of an action principle. The reason for this is the presence of dissipation in thermal media. In the simplest case, the hydrodynamic equations are just the laws of conservation of energy and momentum, µν = 0 . (1) To close the system of equations, we must reduce the number of independent elements of T µν . This is done through the assumption of local thermal equilibrium: If perturbations have long wave- lengths, the state of the system, at a given time, is determined by the temperature as a function of coordinates T (x) and the local fluid velocity uµ, which is also a function of coordinates uµ(x). Because uµu µ = −1, only three components of uµ are independent. The number of hydrodynamic variables is four, equal to the number of equations. In hydrodynamics we express T µν through T (x) and uµ(x) through the so-called constitutive equations. Following the standard procedure of effective field theories, we expand in powers of spatial derivatives. To zeroth order, T µν is given by the familiar formula for ideal fluids, T µν = (ǫ + P )uµuν + Pgµν , (2) where ǫ is the energy density, and P is the pressure. Normally one would stop at this leading order, but qualitatively new effects necessitate going to the next order. Indeed, from Eq. 2 and the thermodynamic relations dǫ = TdS, dP = sdT , and ǫ+P = Ts (s is the entropy per unit volume), one finds that entropy is conserved [14] ∂µ(su µ) = 0 . (3) Thus, to have entropy production, one needs to go to the next order in the derivative expansion. At the next order, we write T µν = (ǫ + P )uµuν + Pgµν − σµν , (4) where σµν is proportional to derivatives of T (x) and uµ(x) and is termed the dissipative part of T µν . To write these terms, let us first fix a point x and go to the local rest frame where ui(x) = 0. 4 Son, Starinets In this frame, in principle one can have dissipative corrections to the energy-momentum density T 0µ. However, one recalls that the choice of T and uµ is arbitrary, and thus one can always redefine them so that these corrections vanish, σ00 = σ0i = 0, and so at a point x, T 00 = ǫ, T 0i = 0 . (5) The only nonzero elements of the dissipative energy-momentum tensor are σij. To the next-to- leading order there are extra contributions whose forms are dictated by rotational symmetry: σij = η ∂iuj + ∂jui − δij∂ku + ζδij∂ku k . (6) Going back to the general frame, we can now write the dissipative part of the energy-momentum tensor as σµν = PµαP νβ ∂αuβ + ∂βuα − gαβ∂λu + ζgαβ∂λu , (7) where Pµν = gµν + uµuν is the projection operator onto the directions perpendicular to uµ. If the system contains a conserved current, there is an additional hydrodynamic equation related to the current conservation, µ = 0 . (8) The constitutive equation contains two terms: jµ = ρuµ − DPµν∂να , (9) where ρ is the charge density in the fluid rest frame and D is some constant. The first term corresponds to convection, the second one to diffusion. In the fluid rest frame, j = −D∇ρ, which is Fick’s law of diffusion, with D being the diffusion constant. 2.1 Kubo’s Formula For Viscosity As mentioned above, the hydrodynamic equations can be thought of as an effective theory describing the dynamics of the system at large lengths and time scales. Therefore one should be able to use these equations to extract information about the low-momentum behavior of Green’s functions in the original theory. For example, let us recall how the two-point correlation functions can be extracted. If we couple sources Ja(x) to a set of (bosonic) operators Oa(x), so that the new action is S = S0 + Ja(x)Oa(x) , (10) then the source will introduce a perturbation of the system. In particular, the average values of Oa will differ from the equilibrium values, which we assume to be zero. If Ja are small, the perturbations are given by the linear response theory as 〈Oa(x)〉 = − GRab(x − y)Jb(y) , (11) where GRab is the retarded Green’s function iGRab(x − y) = θ(x0 − y0)〈[Oa(x), Ob(y)]〉 . (12) Viscosity, Black Holes, and QFT 5 The fact that the linear response is determined by the retarded (and not by any other) Green’s function is obvious from causality: The source can influence the system only after it has been turned on. Thus, to determine the correlation functions of T µν , we need to couple a weak source to T µν and determine the average value of T µν after this source is turned on. To find these correlators at low momenta, we can use the hydrodynamic theory. So far in our treatment of hydrodynamics we have included no source coupled to T µν . This deficiency can be easily corrected, as the source of the energy-momentum tensor is the metric gµν . One must generalize the hydrodynamic equations to curved space-time and from it determine the response of the thermal medium to a weak perturbation of the metric. This procedure is rather straightforward and in the interest of space is left as an exercise to the reader. Here we concentrate on a particular case when the metric perturbation is homogeneous in space but time dependent: gij(t,x) = δij + hij(t), hij ≪ 1 (13) g00(t,x) = −1, g0i(t,x) = 0 . (14) Moreover, we assume the perturbation to be traceless, hii = 0. Because the perturbation is spatially homogeneous, if the fluid moves, it can only move uniformly: ui = ui(t). However, this possibility can be ruled out by parity, so the fluid must remain at rest all the time: uµ = (1, 0, 0, 0). We now compute the dissipative part of the stress-energy tensor. The generalization of Eq. 7 to curved space-time is σµν = PµαP νβ η(∇αuβ + ∇βuα) + ζ − 2 gαβ∇ · u . (15) Substituting uµ = (1, 0, 0, 0) and gµν from Eq. 13, we find only contributions to the traceless spatial components, and these contributions come entirely from the Christoffel symbols in the covariant derivatives. For example, σxy = 2ηΓ xy = η∂0hxy . (16) By comparison with the expectation from the linear response theory, this equation means that we have found the zero spatial momentum, low-frequency limit of the retarded Green’s function of T xy: GRxy,xy(ω,0) = dt dx eiωtθ(t)〈[Txy(t,x), Txy(0,0)]〉 = −iηω + O(ω2) (17) (modulo contact terms). We have, in essence, derived the Kubo’s formula relating the shear viscosity and a Green’s function: η = − lim ImGRxy,xy(ω,0) . (18) There is a similar Kubo’s relation for the charge diffusion constant D. 2.2 Hydrodynamic Modes If one is interested only in the locations of the poles of the correlators, one can simply look for the normal modes of the linearized hydrodynamic equations, that is, solutions that behave as e−iωt+ik·x. Owing to dissipation, the frequency ω(k) is complex. For example, the equation of charge diffusion, ∂tρ − D∇2ρ = 0, (19) 6 Son, Starinets corresponds to a pole in the current-current correlator at ω = −iDk2. To find the poles in the correlators between elements of the stress-energy tensor one can, without loss of generality, choose the coordinate system so that k is aligned along the x3-axis: k = (0, 0, k). Then one can distinguish two types of normal modes: 1. Shear modes correspond to the fluctuations of pairs of components T 0a and T 3a, where a = 1, 2. The constitutive equation is T 3a = −η∂3ua = − ǫ + P 0a , (20) and the equation for T 0a is 0a − η ǫ + P 0a = 0 . (21) That is, it has the form of a diffusion equation for T 0a. Substituting e−iωt+ikx into the equation, one finds the dispersion law ω = −i η ǫ + P k2 . (22) 2. Sound modes are fluctuations of T 00, T 03, and T 33. There are now two conservation equations, and by diagonalizing them one finds the dispersion law ω = csk − η + ζ ǫ + P , (23) where cs = dP/dǫ. This is simply the sound wave, which involves the fluctuation of the en- ergy density. It propagates with velocity cs, and its damping is related to a linear combination of shear and bulk viscosities. In CFTs it is possible to use conformal Ward identities to show that the bulk viscosity vanishes: ζ = 0. Hence, we shall concentrate our attention on the shear viscosity η. 2.3 Viscosity In Weakly Coupled Field Theories We now briefly consider the behavior of the shear viscosity in weakly coupled field theories, with the λφ4 theory as a concrete example. At weak coupling, there is a separation between two length scales: The mean free path of particles is much larger than the distance scales over which scatterings occur. Each scattering event takes a time of order T−1 (which can be thought of as the time required for final particles to become on-shell). The mean free path ℓmfp can be estimated from the formula ℓmfp ∼ , (24) where n is the density of particles, σ is the typical scattering cross section, and v is the typical particle velocity. Inserting the values for thermal λφ4 theory, n ∼ T 3, σ ∼ λ2T−2, and v ∼ 1, one finds ℓmfp ∼ . (25) The viscosity can be estimated from kinetic theory to be η ∼ ǫℓmfp , (26) Viscosity, Black Holes, and QFT 7 where ǫ is the energy density. From ǫ ∼ T 4 and the estimate of ℓmft, one finds η ∼ T . (27) In particular, the weaker the coupling λ, the larger the viscosity η. This behavior is explained by the fact that the viscosity measures the rate of momentum diffusion. The smaller λ is, the longer a particle travels before colliding with another one, and the easier the momentum transfer. It may appear counterintuitive that viscosity tends to infinity in the limit of zero coupling λ → 0: At zero coupling there is no dissipation, so should the viscosity be zero? The confusion arises owing to the fact that the hydrodynamic theory, and hence the notion of viscosity, makes sense only on distances much larger than the mean free path of particles. If one takes λ → 0, then to measure the viscosity one has to do the experiment at larger and larger length scales. If one fixes the size of the experiment and takes λ → 0, dissipation disappears, but it does not tell us anything about the viscosity. As will become apparent below, a particularly interesting ratio to consider is the ratio of shear viscosity and entropy density s. The latter is proportional to T 3; thus . (28) One has η/s ≫ 1 for λ ≪ 1. This is a common feature of weakly coupled field theories. Extrapolat- ing to λ ∼ 1, one finds η/s ∼ 1. We shall see that theories with gravity duals are strongly coupled, and η/s is of order one. More surprisingly, this ratio is the same for all theories with gravity duals. To compute rather than estimate the viscosity, one can use Kubo’s formula. It turns out that one has to sum an infinite number of Feynman graphs to even find the viscosity to leading order. Another way that leads to the same result is to first formulate a kinetic Boltzmann equation for the quasi-particles as an intermediate effective description, and then derive hydrodynamics by taking the limit of very long lengths and time scales in the kinetic equation. Interested readers should consult Refs. [15,16] for more details. 3 AdS/CFT CORRESPONDENCE 3.1 Review Of AdS/CFT Correspondence At Zero Temperature This section briefly reviews the AdS/CFT correspondence at zero temperature. It contains only the minimal amount of materials required to understand the rest of the review. Further information can be found in existing reviews and lecture notes [17,18]. The original example of AdS/CFT correspondence is between N = 4 supersymmetric Yang-Mills (SYM) theory and type IIB string theory on AdS5×S5 space. Let us describe the two sides of the correspondence in some more detail. The N = 4 SYM theory is a gauge theory with a gauge field, four Weyl fermions, and six real scalars, all in the adjoint representation of the color group. Its Lagrangian can be written down explicitly, but is not very important for our purposes. It has a vanishing beta function and is a conformal field theory (CFT) (thus the CFT in AdS/CFT). In our further discussion, we frequently use the generic terms “field theory” or CFT for the N = 4 SYM theory. 8 Son, Starinets On the string theory side, we have type IIB string theory, which contains a finite number of massless fields, including the graviton, the dilaton Φ, some other fields (forms) and their fermionic superpartners, and an infinite number of massive string excitations. It has two parameters: the string length ls (related to the slope parameter α ′ by α′ = l2s) and the string coupling gs. In the long-wavelength limit, when all fields vary over length scales much larger than ls, the massive modes decouple and one is left with type IIB supergravity in 10 dimensions, which can be described by an action [19] SSUGRA = 2κ210 −g e−2Φ (R + 4 ∂µΦ∂µΦ + · · ·) , (29) where κ10 is the 10-dimensional gravitational constant, κ10 = 8πG = 8π7/2gsl s , (30) and · · · stay for the contributions from fields other than the metric and the dilaton. One of these fields is the five-form F5, which is constrained to be self-dual. The type IIB string theory lives is a 10-dimensional space-time with the following metric: ds2 = (−dt2 + dx2) + R dr2 + R2dΩ25 . (31) The metric is a direct product of a five-dimensional sphere (dΩ25) and another five-dimensional space-time spanned by t, x, and r. An alternative form of the metric is obtained from Eq. (31) by a change of variable z = R2/r, ds2 = (−dt2 + dx2 + dz2) + R2dΩ25 . (32) Both coordinates r and z are known as the radial coordinate. The limiting value r = ∞ (or z = 0) is the boundary of the AdS space. It is a simple exercise to check that the (t,x, r) part of the metric is a space with constant negative curvature, or an anti de-Sitter (AdS) space. To support the metric (31) (i.e., to satisfy the Einstein equation) there must be some background matter field that gives a stress-energy tensor in the form of a negative cosmological constant in AdS5 and a positive one in S 5. Such a field is the self-dual five-form field F5 mentioned above. Field theory has two parameters: the number of colors N and the gauge coupling g. When the number of colors is large, it is the ’t Hooft coupling λ = g2N that controls the perturbation theory. On the string theory side, the parameters are gs, ls, and radius R of the AdS space. String theory and field theory each have two dimensionless parameters which map to each other through the following relations: g2 = 4πgs, (33) g2Nc = . (34) Equation (33) tells us that, if one wants to keep string theory weakly interacting, then the gauge coupling in field theory must be small. Equation (34) is particularly interesting. It says that the large ’t Hooft coupling limit in field theory corresponds to the limit when the curvature radius of Viscosity, Black Holes, and QFT 9 space-time is much larger than the string length ls. In this limit, one can reliably decouple the massive string modes and reduce string theory to supergravity. In the limit gs ≪ 1 and R ≫ ls, one has classical supergravity instead of string theory. The practical utility of the AdS/CFT correspondence comes, in large part, from its ability to deal with the strong coupling limit in gauge theory. One can perform a Kaluza-Klein reduction [20] by expanding all fields in S5 harmonics. Keeping only the lowest harmonics, one finds a five-dimensional theory with the massless dilaton, SO(6) gauge bosons, and gravitons [21]: S5D = 8π2R3 R5D − 2Λ − ∂µΦ∂µΦ − F aµνF aµν + · · · . (35) In AdS/CFT, an operator O of field theory is put in a correspondence with a field φ (“bulk” field) in supergravity. We elaborate on this correspondence below; here we keep the operator and the field unspecified. In the supergravity approximation, the mathematical statement of the correspondence Z4D[J ] = e iS[φcl] . (36) On the left is the partition function of a field theory, where the source J coupled to the operator O is included: Z4D[J ] = Dφ exp iS + i d4xJO . (37) On the right, S[φcl] is the classical action of the classical solution φcl to the field equation with the boundary condition: φcl(z, x) = J(x) . (38) Here ∆ is a constant that depends on the nature of the operator O (namely, on its spin and dimension). In the simplest case, ∆ = 0, and the boundary condition becomes φcl(z=0) = J . Differentiating Eq. (36) with respect to J , one can find the correlation functions of O. For example, the two-point Green’s function of O is obtained by differentiating Scl[φ] twice with respect to the boundary value of φ, G(x − y) = −i〈TO(x)O(y)〉 = − δ 2S[φcl] δJ(x)δJ(y) φ(z=0)=J . (39) The AdS/CFT correspondence thus maps the problem of finding quantum correlation functions in field theory to a classical problem in gravity. Moreover, to find two-point correlation functions in field theory, one can be limited to the quadratic part of the classical action on the gravity side. The complete operator to field mapping can be found in Refs. [5, 17]. For our purpose, the following is sufficient: • The dilaton Φ corresponds to O = −L = 1 F 2µν + · · ·, where L is the Lagrangian density. • The gauge field Aaµ corresponds to the conserved R-charge current Jaµ of field theory. • The metric tensor corresponds to the stress-energy tensor T µν . More precisely, the partition function of the four-dimensional field theory in an external metric g0µν is equal to Z4D[g µν ] = exp(iScl[gµν ]) , (40) 10 Son, Starinets where the five-dimensional metric gµν satisfies the Einstein’s equations and has the following asymptotics at z = 0: ds2 = gµνdx µdxν = (dz2 + g0µνdx µdxν) . (41) From the point of view of hydrodynamics, the operator 1 F 2 is not very interesting because its correlator does not have a hydrodynamic pole. In contrast, we find the correlators of the R-charge current and the stress-energy tensor to contain hydrodynamic information. We simplify the graviton part of the action further. Our two-point functions are functions of the momentum p = (ω,k). We can choose spatial coordinates so that k points along the x3-axis. This corresponds to perturbations that propagate along the x3 direction: hµν = hµν(t, r, x 3). These perturbations can be classified according to the representations of the O(2) symmetry of the (x1, x2) plane. Owing to that symmetry, only certain components can mix; for example, h12 does not mix with any other components, whereas components h01 and h31 mix only with each other. We assume that only these three metric components are nonzero and introduce shorthand notations φ = h12, a0 = h 0, a3 = h 3 . (42) The quadratic part of the graviton action acquires a very simple form in terms of these fields: Squad = 8π2R3 d4x dr gµν∂µφ∂νφ − 4g2eff gµαgνβfµνfαβ , (43) where fµν = ∂µaν − ∂νaµ, and g2eff = gxx. In deriving Eq. (43), our only assumption about the metric is that it has a diagonal form, ds2 = gttdt 2 + grrdr 2 + gxxdx 2 , (44) so it can also be used below for the finite-temperature metric. As a simple example, let us compute the two-point correlation function of T xy, which corresponds to φ in gravity. The field equation for φ is −g gµν∂νφ) = 0 . (45) The solution to this equation, with the boundary condition φ(p, z = 0) = φ0(p), can be written as φ(p, z) = fp(z)φ0(p) , (46) where the mode function fp(z) satisfies the equation fp = 0 (47) with the boundary condition fp(0) = 1. The mode equation (47) can be solved exactly. Assuming p is spacelike, p2 > 0, the exact solution and its expansion around z = 0 is fp(z) = (pz)2K2(pz) = 1 − (pz)2 − 1 (pz)4 ln(pz) + O((pz)4) . (48) The second solution to Eq. (47), (pz)2I2(pz), is ruled out because it blows up at z → ∞. Viscosity, Black Holes, and QFT 11 We now substitute the solution into the quadratic action. Using the field equation, one can perform integration by parts and write the action as a boundary integral at z = 0. One finds φ(x, z)φ′(x, z)|z→0 = (2π)4 φ0(−p)F(p, z)φ0(p)|z→0 , (49) where F(p, z) = N f−p(z)∂zfp(z) . (50) Differentiating the action twice with respect to the boundary value φ0 one finds 〈TxyTxy〉p = −2 lim F(p, z) = N p4 ln(p2) . (51) Note that we have dropped the term ∼ p4 ln z, which, although singular in the limit z → 0, is a contact term [i.e., a term proportional to a derivative of δ(x) after Fourier transform]. Removing such terms by adding local counter terms to the supergravity action is known as the holographic renormalization [22]. It is, in a sense, a holographic counterpart to the standard renormalization procedure in quantum field theory, here applied to composite operators. For time-like p, p2 < 0, there are two solutions to Eq. (47) which involve Hankel functions H(1)(z) and H(2)(z) instead of K2(z). Neither function blows up at z → ∞, and it is not clear which should be picked. Here we encounter, for the first time, a subtlety of Minkowski-space AdS/CFT, which is discussed in great length in subsequent sections. At zero temperature this problem can be overcome by an analytic continuation from space-like p. However, this will not work at nonzero temperatures. 3.2 Black Three-Brane Metric At nonzero temperatures, the metric dual to N = 4 SYM theory is the black three-brane metric, ds2 = (−fdt2 + dx2) + R dr2 + R2dΩ25 , (52) with f = 1 − r40/r4. The event horizon is located at r = r0, where f = 0. In contrast to the usual Schwarzschild black hole, the horizon has three flat directions x. The metric (52) is thus called a black three-brane metric. We frequently use an alternative radial coordinate u, defined as u = r20/r 2. In terms of u, the boundary is at u = 0, the horizon at u = 1, and the metric is ds2 = (πTR)2 (−f(u)dt2 + dx2) + R 4u2f(u) du2 + R2dΩ25 . (53) The Hawking temperature is determined completely by the behavior of the metric near the horizon. Let us concentrate on the (t, r) part of the metric, ds2 = −4r0 (r − r0)dt2 + 4r0(r − r0) dr2 . (54) Changing the radial variable from r to ρ, r = r0 + , (55) 12 Son, Starinets and the metric components become nonsingular: ds2 = dρ2 − 4r ρ2dt2 . (56) Note also that after a Wick rotation to Euclidean time τ , the metric has the form of the flat metric in cylindrical coordinates, ds2 ∼ dρ2 + ρ2dϕ2, where ϕ = 2r0R−2τ . To avoid a conical singularity at ρ = 0, ϕ must be a periodic variable with periodicity 2π. This fact matches with the periodicity of the Euclidean time in thermal field theory τ ∼ τ + 1/T , from which one finds the Hawking temperature: . (57) One of the first finite-temperature predictions of AdS/CFT correspondence is that of the ther- modynamic potentials of the N = 4 SYM theory in the strong coupling regime. The entropy is given by the Bekenstein-Hawking formula S = A/(4G), where A is the area of the horizon of the metric (52); the result can then be converted to parameters of the gauge theory using Eqs. (30), (33), and (34). One obtains N2T 3 , (58) which is 3/4 of the entropy density in N = 4 SYM theory at zero ’t Hooft coupling. We now try to generalize the AdS/CFT prescription to finite temperature. In the Euclidean formulation of finite-temperature field theory, field theory lives in a space-time with the Euclidean time direction τ compactified. The metric is regular at r = r0: If one views the (τ, r) space as a cigar-shaped surface, then the horizon r = r0 is the tip of the cigar. Thus, r0 is the minimal radius where the space ends, and there is no point in space with r less than r0. The only boundary condition at r = r0 is that fields are regular at the tip of the cigar, and the AdS/CFT correspondence is formulated as Z4D[J ] = Z5D[φ]|φ(z=0)→J . (59) 4 REAL-TIME AdS/CFT In many cases we must find real-time correlation functions not given directly by the Euclidean path- integral formulation of thermal field theory. One example is the set of kinetic coefficients expressed, through Kubo’s formulas, via a certain limit of real-time thermal Green’s functions. Another related example appears if we want to directly find the position of the poles in the correlation functions that would correspond to the hydrodynamic modes. In principle, some real-time Green’s functions can be obtained by analytic continuation of the Euclidean ones. For example, an analytic continuation of a two-point Euclidean propagator gives a retarded or advanced Green’s function, depending on the way one performs the continuation. However, it is often very difficult to directly compute a quantity of interest in that way. In particular, it is very difficult to get the information about the hydrodynamic (small ω, small k) limit of real-time correlators from Euclidean propagators. The problem here is that we need to perform an analytic continuation from a discrete set of points in Euclidean frequencies (the Matsubara frequencies) ω = 2πin, where n is an integer, to the real values of ω. In the hydrodynamic limit, we are interested in real and small ω, whereas the smallest Matsubara frequency is already 2πT . Viscosity, Black Holes, and QFT 13 Therefore, we need a real-time AdS/CFT prescription that would allow us to directly compute the real-time correlators. However, if one tries to naively generalize the AdS/CFT prescription, one immediately faces a problem. Namely, now r = r0 is not the end of space but just the location of the horizon. Without specifying a boundary condition at r = r0, there is an ambiguity in defining the solution to the field equations, even as the boundary condition at r = ∞ is set. As an example, let us consider the equation of motion of a scalar field in the black hole back- ground, ∂µ(g µν∂νφ) = 0. The solution to this equation with the boundary condition φ = φ0 at u = 0 is φ(p, u) = fpφ0(p), where fp(u) satisfies the following equation in the metric (53): f ′′p − 1 + u2 f ′p + fp = 0 . (60) Here the prime denotes differentiation with respect to u, and we have defined the dimensionless frequency and momentum: , q = . (61) Near u = 0 the equation has two solutions, f1 ∼ 1 and f2 ∼ u2. In the Euclidean version of thermal AdS/CFT, there is only one regular solution at the horizon u = 1, which corresponds to a particular linear combination of f1 and f2. However, in Minkowski space there are two solutions, and both are finite near the horizon. One solution termed fp behaves as (1−u)−iw/2, and the other is its complex conjugate f∗p ∼ (1 − u)iw/2. These two solutions oscillate rapidly as u → 1, but the amplitude of the oscillations is constant. Thus, the requirement of finiteness of fp allows for any linear combination of f1 and f2 near the boundary, which means that there is no unique solution to Eq. (60). 4.1 Prescription For Retarded Two-Point Functions Physically, the two solutions fp and f p have very different behavior. Restoring the e −iωt phase in the wave function, one can write e−iωtfp ∼ e−iω(t+r∗) , (62) e−iωtf∗p ∼ eiω(t−r∗) , (63) where the coordinate ln(1 − u) was introduced so that Eqs. (62) and (63) looked like plane waves. In fact, Eq. (62) corresponds to a wave that moves toward the horizon (incoming wave) and Eq. (63) to a wave that moves away from the horizon (outgoing wave). The simplest idea, which is motivated by the fact that nothing should come out of a horizon, is to impose the incoming-wave boundary condition at r = r0 and then proceed as instructed by the AdS/CFT correspondence. However, now we encounter another problem. If we write down the classical action for the bulk field, after integrating by parts we get contributions from both the boundary and the horizon: (2π)4 φ0(−p)F(p, z)φ0(p) . (65) 14 Son, Starinets If one tried to differentiate the action with respect to the boundary value φ0, one would find G(p) = F(p, z)|zH0 + F(−p, z)| 0 . (66) From the equation satisfied by fp and from f p = f−p, it is easy to show that the imaginary part of F(p, z) does not depend on z; hence the quantity G(p) in Eq. (66) is real. This is clearly not what we want, as the retarded Green’s functions are, in general, complex. Simply throwing away the contribution from the horizon does not help because F(−p, z) = F∗(p, z) owing to the reality of the equation satisfied by fp. A partial solution to this problem was suggested in Ref. [7]. It was postulated that the re- tarded Green’s function is related to the function F by the same formula that was found at zero temperature: GR(p) = −2 lim F(p, z) . (67) In particular, we throw away all contributions from the horizon. This prescription was established more rigorously in Ref. [8] (following an earlier suggestion in Ref. [23]) as a particular case of a general real-time AdS/CFT formulation, which establishes the connection between the close-time- path formulation of real-time quantum field theory with the dynamics of fields in the whole Penrose diagram of the AdS black brane. Here we accept Eq. (67) as a postulate and proceed to extract physical results from it. It is also easy to generalize this prescription to the case when we have more than one field. In that case, the quantity F becomes a matrix Fab, whose elements are proportional to the retarded Green’s function Gab. 4.2 Calculating Hydrodynamic Quantities As an illustration of the real-time AdS/CFT correspondence, we compute the correlator of Txy. First we write down the equation of motion for φ = hxy : φ′′p − 1 + u2 φ′p + w2 − q2f φp = 0 . (68) In contrast to the zero-temperature equation, now ω and k enter the equation separately rather than through the combination ω2 − k2. Thus the Green’s function will have no Lorentz invariance. The equation cannot be solved exactly for all ω and k. However, when ω and k are both much smaller than T , or w, q ≪ 1, one can develop series expansion in powers of w and q. There are two solutions that are complex conjugates of each other. The solution that is an incoming wave at u = 1 and normalized to 1 at u = 0 is fp(z) = (1 − u2)−iw/2 + O(w2, q2) . (69) The kinetic term in the action for φ is S = −π 2N2T 4 φ′2 . (70) Applying the general formula (67), one finds the retarded Green’s function of Txy, GRxy,xy(ω, k) = − π2N2T 4 iw , (71) Viscosity, Black Holes, and QFT 15 and, using Kubo’s formula for η, the viscosity, N2T 3 . (72) It is instructive to compute other correlators that have poles corresponding to hydrodynamic modes. As a warm-up, let us compute the two-point correlators of the R-charge currents, which should have a pole at ω = −iDk2, where D is the diffusion constant. We first write down Maxwell’s equations for the bulk gauge field. Let the spatial momentum be aligned along the x3-axis: p = (ω, 0, 0, k). Then the equations for A0 and A3 are coupled: wA′0 + qfA 3 = 0 , (73) A′′0 − (q2A0 + wqA3) = 0 , (74) A′′3 + A′3 + (w2A3 + wqA0) = 0 . (75) One can eliminate A3 and write down a third-order equation for A0, A′′′0 + (uf)′ A′′0 + w2 − q2f A′0 = 0 . (76) Near u=1 we find two independent solutions, A′0 ∼ (1− u)±iw/2, and the incoming-wave boundary condition singles out (1 − u)−iw/2. One can substitute A′0 = (1 − u)−iw/2F (u) into Eq. (76). The resulting equation can be solved perturbatively in w and q2. We find A′0 = C(1 − u)−iw/2 1 + u + q2 ln 1 + u . (77) Using Eq. (74) one can express C through the boundary values of A0 and A3 at u = 0: q2A0 + wqA3 iw − q2 . (78) Differentiating the action with respect to the boundary values, we find, in particular, 〈J0J0〉p = iω − Dk2 , (79) where . (80) The correlator given by Eq. (79) has the expected hydrodynamic diffusive pole, and D is the R-charge diffusion constant. Similarly, one can observe the appearance of the shear mode in the correlators of the metric tensor. We note that the shear flow along the x1 direction with velocity gradient along the x3 direction involves T01 and T31, hence the interesting metric components are a0 = h 0 and a3 = h Two of the field equations are a′0 − a′3 = 0 , (81) a′′3 − 1 + u2 a′3 + (w2a3 + wqa0) = 0 . (82) 16 Son, Starinets They can be combined into a single equation: a′′′0 − a′′0 + 2uf − q2f + w2 a′0 = 0 . (83) Again, the solution can be found perturbatively in w and q: a′0 = C(1 − u)−iw/2 u − iw 1 − u − u 1 + u (1 − u) . (84) Applying the prescription, one finds the retarded Green’s functions. For example, Gtx,tx(ω, k) = iω −Dk2 , (85) where N2T 3, D = 1 . (86) Thus, we found that the correlator contains a diffusive pole ω = −iDk2, just as anticipated from hydrodynamics. Furthermore, the magnitude of the momentum diffusion constant D also matched our expectation. Indeed, if one recalls the value of η from Eq. (72) and the entropy density from Eq. (58), one can check that D = η ǫ + P . (87) 5 THE MEMBRANE PARADIGM Let us now look at the problem from a different perspective. The existence of hydrodynamic modes in thermal field theory is reflected by the existence of the poles of the retarded correlators computed from gravity. Are there direct gravity counterparts of the hydrodynamic normal modes? If the answer to this question is yes, then there must exist linear gravitational perturbations of the metric that have the dispersion relation identical to that of the shear hydrodynamic mode, ω ∼ −iq2, and of the sound mode, ω = csq− iγq2. It turns out that one can explicitly construct the gravitational counterpart of the shear mode. (It should be possible to find a similar construction for the sound mode, but it has not been done in the literature; for a recent work on the subject, see [24].) Our discussion is physical but somewhat sketchy; for more details see Ref. [25]. First, let us construct a gravity perturbation that corresponds to a diffusion of a conserved charge (e.g., the R-charge in N = 4 SYM theory). To keep the discussion general, we use the form of the metric (44), with the metric components unspecified. Our only assumptions are that the metric is diagonal and has a horizon at r = r0, near which g00 = −γ0(r − r0), grr = r − r0 . (88) The Hawking temperature can be computed by the method used to arrive at Eq. (57), and one finds T = (4π)−1(γ0/γr) We also assume that the action of the gauge field dual to the conserved current is Sgauge = 4g2eff FµνFµν , (89) Viscosity, Black Holes, and QFT 17 where geff is an effective gauge coupling that can be a function of the radial coordinate r. For simplicity we set geff to a constant in our derivation of the formula for D; it can be restored by replacing √−g → √−g/g2eff in the final answer. The field equations are g2eff −g Fµν = 0 . (90) We search for a solution to this equation that vanishes at the boundary and satisfies the incoming- wave boundary condition at the horizon. The first indication that one can have a hydrodynamic behavior on the gravity side is that Eq. (90) implies a conservation law on a four-dimensional surface. We define the stretched horizon as a surface with constant r just outside the horizon, r = rh = r0 + ε, ε ≪ r0 , (91) and the normal vector nµ directed along the r direction (i.e., perpendicularly to the stretched horizon). Then with any solution to Eq. (90), one can associate a current on the stretched horizon: jµ = nνF . (92) The antisymmetry of Fµν implies that jµ has no radial component, jr = 0. The field equation (90) and the constancy of nν on the stretched horizon imply that this current is conserved: ∂µj µ = 0. To establish the diffusive nature of the solution, we must show the validity of the constitutive equation ji = −D∂ij0. Such constitutive equation breaks time reversal and obviously must come from the absorptive boundary condition on the horizon. The situation is analogous to the propagation of plane waves to a non-reflecting surface in classical electrodynamics. In this case, we have the relation B = −n×E between electric and magnetic fields. In our case, the corresponding relation is Fir = − r − r0 , (93) valid when r is close to r0. This relates ji ∼ Fir to the parallel to the horizon component of the electric field F0i, which is one of the main points of the “membrane paradigm” approach to black hole physics [26, 27]. We have yet to relate ji to j0 ∼ F0r, which is the component of the electric field normal to the horizon. To make the connection to F0r, we use the radial gauge Ar = 0, in which F0i ≈ −∂iA0 . (94) Moreover, when k is small the fields change very slowly along the horizon. Therefore, at each point on the horizon the radial dependence of the scalar potential A0 is determined by the Poisson equation, −g grrg00∂rA0) = 0 , (95) whose solution, which satisfies A0(r = ∞) = 0, is A0(r) = C0 g00(r ′)grr(r −g(r′) . (96) 18 Son, Starinets This means that the ratio of the scalar potential A0 and electric field F0r approaches a constant near the horizon: g00grr g00grr√−g (r) . (97) Combining the formulas ji ∼ F0i ∼ ∂iA0, and A0 ∼ F0r ∼ j0, we find Fick’s law ji = −D∂ij0, with the diffusion constant √−g00grr −g00grrg2eff√ −g (r) . (98) Thus, we found that for a slowly varying solution to Maxwell’s equations, the corresponding charge on the stretched horizon evolves according to the diffusion equation. Therefore, the gravity solution must be an overdamped one, with ω = −iDk2. This is an example of a quasi-normal mode. We also found the diffusion constant D directly in terms of the metric and the gauge coupling geff . The reader may notice that our quasinormal modes satisfy a vanishing Dirichlet condition at the boundary r=∞. This is different from the boundary condition one uses to find the retarded propagators in AdS/CFT, so the relation of the quasinormal modes to AdS/CFT correspondence may be not clear. It can be shown, however, that the quasi-normal frequencies coincide with the poles of the retarded correlators [28,29]. We can now apply our general formulas to the case of N = 4 SYM theory. The metric components are given by Eq. (52). For the R-charge current geff = const, Eq. (98) gives D = 1/(2πT ), in agreement with our AdS/CFT computation. For the shear mode of the stress-energy tensor we have effectively g2eff = gxx, so D = 1/(4πT ), which also coincides with our previous result. In both cases, the computation is much simpler than the AdS/CFT calculation. 6 THE VISCOSITY/ENTROPY RATIO 6.1 Universality In all thermal field theories in the regime described by gravity duals the ratio of shear viscosity η to (volume) density of entropy s is a universal constant equal to 1/(4π) [h̄/(4πkB), if one restores h̄, c and the Boltzmann constant kB ]. One proof of the universality is based on the relationship between graviton’s absorption cross section and the imaginary part of the retarded Green’s function for Txy [31]. Another way to prove the universality [32] is via the direct AdS/CFT calculation of the correlation function in Kubo’s formula (18). We, however, follow a different method. It is based on the formula for the viscosity derived from the membrane paradigm. A similar proof was given by Buchel & Liu [30]. The observation is that the shear gravitational perturbation with k = 0 can be found exactly by performing a Lorentz boost of the black-brane metric (52). Consider the coordinate transformations r, t, xi → r′, t′, x′i of the form r = r′ , t′ + vy′√ 1 − v2 ≈ t′ + vy′ , Viscosity, Black Holes, and QFT 19 y′ + vt′√ 1 − v2 ≈ y′ + vt′ , xi = x i , (99) where v < 1 is a constant parameter and the expansion on the right corresponds to v ≪ 1. In the new coordinates, the metric becomes ds2 = g00dt ′ 2 + grrdr ′ 2 + gxx(r) (dx′ i)2 + 2v(g00 + gxx)dt ′dy′ . (100) This is simply a shear fluctuation at k = 0. In our language, the corresponding gauge potential is a0 = vg xx(g00 + gxx) . (101) This field satisfies the vanishing boundary condition a0(r = ∞) = 0 owing to the restoration of Poincaré invariance at the boundary: g00/gxx → −1 when r → ∞. This clearly has a much simpler form than Eq. (96) for the solution to the generic Poisson equation. The simple form of solution (101) is valid only for the specific case of the shear gravitational mode with g2eff = gxx. We have also implicitly used the fact that the metric satisfies the Einstein equations, with the stress-energy tensor on the right being invariant under a Lorentz boost. Equation (97) now becomes = −1 + g xxg00 ∂r(gxxg00) gxx(r0) . (102) The shear mode diffusion constant is D = a0 gxx(r0) . (103) Because D = η/(ǫ + P ), and ǫ + P = Ts in the absence of chemical potentials, we find that . (104) In fact, the constancy of this ratio has been checked directly for theories dual to Dp-brane [25], M -brane [11], Klebanov-Tseytlin and Maldacena-Nunez backgrounds [30], N = 2∗ SYM theory [33] and others. Curiously, the viscosity to entropy ratio is also equal to 1/4π in the pre-AdS/CFT “membrane paradigm” hydrodynamics [34]: there, for a four-dimensional Schwarzschild black hole one has η = 1/16πGN , while the Bekenstein-Hawking entropy is s = 1/4GN . As remarked in Sec. 2, the ratio η/s is much larger than the one for weakly coupled theories. The fact that we found the ratio to be parametrically of order one implies that all theories with gravity duals are strongly coupled. In N = 4 SYM theory, the ratio η/s has been computed to the next order in the inverse ’t Hooft coupling expansion [35] 135ζ(3) 8(g2N)3/2 . (105) The sign of the correction can be guessed from the fact that in the limit of zero ’t Hooft coupling g2N → 0, the ratio diverges, η/s → ∞. 20 Son, Starinets 6.2 The Viscosity Bound Conjecture From our discussion above, one can argue that (106) in all systems that can be obtained from a sensible relativistic quantum field theory by turning on temperatures and chemical potentials. The bound, if correct, implies that a liquid with a given volume density of entropy cannot be arbitrarily close to being a perfect fluid (which has zero viscosity). As such, it implies a lower bound on the viscosity of the QGP one may be creating at RHIC. Interestingly, some model calculations suggest that the viscosity at RHIC may be not too far away from the lower bound [36,37]. One place where one may think that the bound should break down is superfluids. The ability of a superfluid to flow without dissipation in a channel is sometimes described as “zero viscosity”. However, within the Landau’s two-fluid model, any superfluid has a measurable shear viscosity (together with three bulk viscosities). For superfluid helium, the shear viscosity has been measured in a torsion-pendulum experiment by Andronikashvili [38]. If one substitutes the experimental values, the ratio η/s for helium remains larger than h̄/4πkB ≈ 6.08 × 10−13 K s for all ranges of temperatures and pressures, by a factor of at least 8.8. As discussed in Sec. 2.3, the ratio η/s is proportional to the ratio of the mean free path and the de Broglie wavelength of particles, ∼ ℓmfp . (107) For the quasi-particle picture to be valid, the mean free path must be much larger than the de Broglie wavelength. Therefore, if the coupling is weak and the system can be described as a collection of quasi-particles, the ratio η/s is larger than 1. We have found is that, within the N = 4 SYM theory and, more generally, theories with gravity duals, even in the limit of infinite coupling the ratio η/s cannot be made smaller than 1/(4π). 7 CONCLUSION In this review, we covered only a small part of the applications of AdS/CFT correspondence to finite-temperature quantum field theory. Here we briefly mention further developments and refer the reader to the original literature for more details. In addition to N = 4 SYM theory, there exists a large number of other theories whose hydrody- namic behavior has been studied using the AdS/CFT correspondence, including the worldvolume theories on M2- and M5-branes [11], theories on Dp branes [25], and little string theory [39]. In all examples the ratio η/s is equal to 1/(4π), which is not surprising because the general proofs of Sec. 6 apply in these cases. We have concentrated on the shear hydrodynamic mode, which has a diffusive pole (ω ∼ −ik2). One can also compute correlators which have a sound-wave pole from the AdS/CFT prescription [10]. One such correlator is between the energy density T 00 at two different points in space-time. The result confirms the existence of such a pole, with both the real part and imaginary part having exactly the values predicted by hydrodynamics (recall that in conformal field theories the bulk viscosity is zero and the sound attenuation rate is determined completely by the shear viscosity). Viscosity, Black Holes, and QFT 21 Some of the theories listed above are conformal field theories, but many are not (e.g., the Dp- brane worldvolume theories with p 6= 3). The fact that η/s = 1/(4π) also in those theories implies that the constancy of this ratio is not a consequence of conformal symmetry. Theories with less than maximal number of supersymmetries have been found to have the universal value of η/s, for example, the N = 2∗ theory [40], theories described by Klebanov-Tseytlin, and Maldacena-Nunez backgrounds [30]. A common feature of these theories is that they all have a gravitational dual description. The bulk viscosity has been computed for some of these theories [41,39]. Besides viscosity, one can also compute diffusion constants of conserved charges by using the AdS/CFT correspondence. Above we presented the computation of the R-charge diffusion constant in N = 4 SYM theory; for similar calculations in some other theories see Ref. [11,25]. Recently, the AdS/CFT correspondence was used to compute the energy loss rate of a quark in the fundamental representation moving in a finite-temperature plasma [42,43,44,45]. This quantity is of importance to the phenomenon of “jet quenching” in heavy-ion collisions. So far, the only quantity that shows a universal behavior at the quantitative level, across all theories with gravitational duals, is the ratio of the shear viscosity and entropy density. Recently, it was found that this ratio remains constant even at nonzero chemical potentials [46,47,48,49,50]. What have we learned from the application of AdS/CFT correspondence to thermal field theory? Although, at least at this moment, we cannot use the AdS/CFT approach to study QCD directly, we have found quite interesting facts about strongly coupled field theories. We have also learned new facts about quasi-normal modes of black branes. However, we have also found a set of puzzles: Why is the ratio of the viscosity and entropy density constant in a wide class of theories? Is there a lower bound on this ratio for all quantum field theories? Can this be understood without any reference to gravity duals? With these open questions, we conclude this review. Acknowledgments The work of DTS is supported in part by U.S. Department of Energy under Grant No. DE-FG02- 00ER41132. Research at Perimeter Institute is supported in part by the Government of Canada through NSERC and by the Province of Ontario through MEDT. Literature Cited 1. Kirzhnits DA, Linde AD. Macroscopic consequences of the Weinberg model. Phys. Lett. B 42:471 (1972) 2. Gyulassy M, and McLerran L, New forms of QCD matter discovered at RHIC. Nucl. Phys. A 750:30 (2005) 3. Maldacena JM. The large N limit of superconformal field theory and supergravity. Adv. Theor. Math. Phys. 2:231 (1998) 4. Gubser SS, Klebanov IR, Polyakov AM. Gauge theory correlators from noncritical string theory. Phys. Lett. B 428:105 (1998) 5. Witten E. Anti de Sitter space and holography. Adv. Theor. Math. Phys. 2:253 (1998) 6. Policastro G, Son DT, Starinets AO. Shear viscosity of strongly coupled N = 4 supersymmetric Yang-Mills plasma. Phys. Rev. Lett. 87:081601 (2001) 22 Son, Starinets 7. Son DT, Starinets AO. Minkowski-space correlators in AdS/CFT correspondence: Recipe and applications. J. High Energy Phys. 0209:042 (2002) 8. Herzog CP, Son DT. Schwinger-Keldysh propagators from AdS/CFT correspondence. J. High Energy Phys. 0303:046 (2003) 9. Policastro G, Son DT, Starinets AO. From AdS/CFT correspondence to hydrodynamics. J. High Energy Phys. 0209:042 (2002) 10. Policastro G, Son DT, Starinets AO. From AdS/CFT correspondence to hydrodynamics. II: Sound waves. J. High Energy Phys. 0212:054 (2002) 11. Herzog CP. The hydrodynamics of M-theory. J. High Energy Phys. 0212:026 (2002) 12. Herzog CP. The sound of M-theory. Phys. Rev. D 68:024013 (2003) 13. Forster D. Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions, Reading: Benjamin (1975) 14. Landau LD, Lifshitz EM. Fluid Mechanics, Oxford: Pergamon Press (1987) 15. Jeon S. Hydrodynamic transport coefficients in relativistic scalar field theory. Phys. Rev. D 52:3591 (1995) 16. Jeon S, Yaffe LG. 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B 708:451 (2005) http://arXiv.org/abs/hep-th/0009139 http://arXiv.org/abs/hep-th/0703170 Viscosity, Black Holes, and QFT 23 34. Damour T. Surface effects in black hole physics. Proceedings of the Second Marcel Grossmann Meeting on General Relativity. Ed. R. Ruffini, Amsterdam: North Holland (1982) 35. Buchel A, Liu JT, Starinets AO. Coupling constant dependence of the shear viscosity in N = 4 supersymmetric Yang-Mills theory. Nucl. Phys. B 707:56 (2005) 36. Teaney D. Effect of shear viscosity on spectra, elliptic flow, and Hanbury Brown-Twiss radii. Phys. Rev. C 68:034913 (2003) 37. Shuryak E. Why does the quark gluon plasma at RHIC behave as a nearly ideal fluid? Prog. Part. Nucl. Phys. 53:273 (2004) 38. Andronikashvili E. Zh. Eksp. Teor. Fiz. 18:429 (1948) 39. Parnachev A, Starinets A. The silence of the little strings. J. High Energy Phys. 0510:027 (2005) 40. Buchel A, Liu JT, Thermodynamics of the N = 2∗ flow. J. High Energy Phys. 0311:031 (2003) 41. Benincasa P, Buchel A, Starinets AO. Sound waves in strongly coupled non-conformal gauge theory plasma. Nucl. Phys. B 733:160 (2006) 42. Herzog CP et al. Energy loss of a heavy quark moving through N = 4 supersymmetric Yang- Mills plasma. J. High Energy Phys. 0607:013 (2006) 43. Liu H, Rajagopal K, Wiedemann UA. Calculating the jet quenching parameter from AdS/CFT, Phys. Rev. Lett. 97:182301 (2006) 44. Casalderrey-Solana J, Teaney D. Heavy quark diffusion in strongly coupled N = 4 Yang Mills. Phys. Rev. D 74:085012 (2006) 45. Gubser, SS. Drag force in AdS/CFT. Phys. Rev. D 74:126005 (2006) 46. Son DT, Starinets AO. Hydrodynamics of R-charged black holes. J. High Energy Phys. 0603:052 (2006) 47. Mas J, Shear viscosity from R-charged AdS black holes. J. High Energy Phys. 0603:016 (2006) 48. Maeda K, Natsuume M, Okamura T. Viscosity of gauge theory plasma with a chemical potential from AdS/CFT. Phys. Rev. D 73:066013 (2006) 49. Saremi O. The viscosity bound conjecture and hydrodynamics of M2-brane theory at finite chemical potential. J. High Energy Phys. 0610:083 (2006) 50. Benincasa P, Buchel A, Naryshkin R. The shear viscosity of gauge theory plasma with chemical potentials. Phys. Lett. B 645:309 (2007) INTRODUCTION HYDRODYNAMICS Kubo's Formula For Viscosity Hydrodynamic Modes Viscosity In Weakly Coupled Field Theories AdS/CFT CORRESPONDENCE Review Of AdS/CFT Correspondence At Zero Temperature Black Three-Brane Metric REAL-TIME AdS/CFT Prescription For Retarded Two-Point Functions Calculating Hydrodynamic Quantities THE MEMBRANE PARADIGM THE VISCOSITY/ENTROPY RATIO Universality The Viscosity Bound Conjecture CONCLUSION

0704.0241

Superconducting states of the quasi-2D Holstein model: Effects of vertex and non-local corrections

7 Superconducting states of the quasi-2D Holstein model: Effects of vertex and non-local corrections J.P.Hague Dept. of Physics and Astronomy, University of Leicester, Leicester, LE1 7RH Abstract. I investigate superconducting states in a quasi-2D Holstein model using the dynamical cluster approximation (DCA). The effects of spatial fluctuations (non-local corrections) are examined and approximations neglecting and incorporating lowest-order vertex corrections are computed. The approximation is expected to be valid for electron-phonon couplings of less than the bandwidth. The phase diagram and superconducting order parameter are calculated. Effects which can only be attributed to theories beyond Migdal– Eliashberg theory are present. In particular, the order parameter shows momentum dependence on the Fermi-surface with a modulated form and s- wave order is suppressed at half-filling. The results are discussed in relation to Hohenberg’s theorem and the BCS approximation. [Published as: J. Phys.: Condens. Matter vol. 17 (2005) 5663-5676] PACS numbers: 71.10.-w, 71.38.-k, 74.20.-z, 74.62.-c 1. Introduction The discovery of large couplings between electrons and the lattice in the cuprate superconductors has led to a call for more detailed theoretical studies of electron- phonon systems in low dimensions [1, 2, 3]. One of the best-known traditional approaches to the electron-phonon problem is attributed to Migdal and Eliashberg [4, 5]. In a bulk 3D system, the perturbation theory may be sharply truncated at 1st order and momentum dependence neglected if the phonon frequency is much less than the Fermi energy [4]. In physical terms, Migdal’s approach requires that there is a very high probability that emitted phonons are reabsorbed in a last-in-first-out order. The typical materials of interest at the time were bulk metallic superconductors where electron-phonon coupling is relatively weak, and the phonon frequency small compared to the Fermi energy. For this reason, the application of Migdal–Eliashberg (ME) theory has been very successful and remains highly regarded. Strong electron-phonon coupling and large phonon frequencies in low dimensional systems are outside the limits of validity of the Migdal–Eliashberg approach. Therefore, the aim of this paper is to evaluate and discuss the effects of both vertex corrections (VC) and spatial fluctuations on the theory of coupled electron-phonon systems in the superconducting state. This follows on from the work by Hague treating the normal (non-superconducting) state of the Holstein model using DCA [6]. Initial attempts to include vertex corrections were carried out by Engelsberg and Schrieffer [7]. Other previous attempts to extend ME theory include the introduction of vertex corrections into the Eliashberg equations by Grabowski and Sham [8], and http://arxiv.org/abs/0704.0241v1 Superconducting states of the quasi-2D Holstein model 2 an expansion to higher order in the Migdal parameter by Kostur and Mitrović to investigate the 2D electron-phonon problem [9]. Grimaldi et al. generalised the Eliashberg equations to include momentum dependence and vertex corrections [10]. An anomalous hardening of the phonon mode was seen by Alexandrov and Schrieffer [11]. A discussion of the applicability of these and other approximations to the vertex function can be found in reference [12]. The current paper uses DCA to introduce a fully self-consistent momentum- dependent self-energy. DCA extends DMFT by introducing short-range fluctuations in a controlled manner [13]. It is particularly good at describing the electron-phonon problem, due to the limited momentum dependence of the self-energy, and in this case, the self-consistent DCA can be viewed as an expansion about the Eliashberg equations (in which momentum dependence is effectively coarse grained in a manner similar to DMFT) [5]. In contrast to the Eliashberg equations, the full form of the Green’s function is considered here, rather than the renormalised weak coupling Green’s function (which has the form G−1(ǫk, iωn) = Ziωnσ0 − (ǫk + χ)σ3 −∆σ1). Two approximations for the electron and phonon self energies are applied in this paper. The first neglects vertex corrections, but incorporates non-local fluctuations. The second incorporates lowest order vertex and non-local corrections. The vertex corrections allow the sequence of phonon absorption and emission to be reordered once, and therefore introduce exchange effects. The DCA result is compared to the corresponding DMFT result and in this way low-dimensional effects are isolated. It should be noted that in the extreme strong coupling limit, the Holstein model forms a bipolaronic ground state, and perturbative methods in the electron and phonon Green’s functions break down [14]. In the dilute limit, the Holstein model forms a polaronic liquid. There are significant differences between the weak and strong coupling limits of the polaron problem. In the strong-coupling limit, the Lang–Firsov approximation may be applied, and physical properties have very different behaviour. For example the effective mass is reduced by an exponential factor of coupling. Exact numerical results show that the crossover between weak and strong coupling regimes occurs rapidly at λ ∼ 1 [15]. For this reason, the present approximation should be considered valid for |U | < W . The paper is organised as follows. In section 2, the DCA is introduced. In section 3, the Holstein model of electron-phonon interactions is described, and the perturbation theory and the full algorithm used in this work are detailed. In section 4, the results are presented. The momentum dependence of the superconducting order parameter is examined through the density of superconducting pairs. The phase diagram is then computed and comparison is made with analytical results. A summary of the major findings of this research is provided in section 5. 2. The dynamical cluster approximation The dynamical cluster approximation [13, 16] is an extension to the dynamical mean- field theory. DMFT has been applied as an approximation to models of 3D materials [17, 18, 19]. However, application of DMFT to one- and two-dimensional models gives an incomplete description of the physics. An example of significant differences between two- and three-dimensional physics comes from quantum spin-systems. In 3D, the Heisenberg model orders at a transition temperature, TN . Significant non-local fluctuations in two dimensions reduce the Néel temperature to zero (Mermin–Wagner theorem), and the mean-field approach fails completely. Superconducting states of the quasi-2D Holstein model 3 ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� ������� Figure 1. A schematic representation of the reciprocal-space coarse graining scheme for a 4 site DCA. Within the shaded areas, the self-energy is assumed to be constant. There is a many to one mapping from the crosshatched areas to the points at the centre of those areas. The coarse-graining procedure corresponds to the mapping to a periodic cluster in real space, with spatial extent N Also shown are the high symmetry points Γ, W and X, and lines connecting the high symmetry points. An infinite number of k states are involved in the coarse-graining step, so the approximation is in the thermodynamic limit. DMFT corresponds to NC = 1. Conceptually, DCA is similar to DMFT. The Brillouin zone is divided up into NC subzones consistent with the lattice symmetry (see figure 1). Within each of these zones, the self-energy is assumed to be momentum independent. For a system in the normal state, the Green’s function is determined as, G(Ki, z) = Di(ǫ) dǫ z + µ− ǫ− Σ(Ki, z) where Di(ǫ) is the non-interacting Fermion density of states for subzone i, and the vectors Ki represent the average k for each subzone (plotted as the large dots in figure 1). The theory deals with the thermodynamic limit, and introduces non-local fluctuations with a characteristic length scale of N C . For NC = 1, DCA is equivalent to DMFT. Since superconducting states are to be considered, DCA is extended within the Nambu formalism [20] in a similar manner to DMFT [17]. Green’s functions and self- energies are described by 2 × 2 matrices, with off-diagonal anomalous terms relating to the superconducting states. Note that in the following equations 4-vectors are used, i.e. K ≡ (iωn,K). The Green’s function and self-energy matrices have the components, G(K) = G(K) F (K) F ∗(K) −G(−K) Σ(K) = Σ(K) φ(K) φ∗(K) −Σ(−K) The coarse graining step is generalised to the superconducting state as, G(K, iωn) = Di(ǫ)(ζ(Ki, iωn)− ǫ) |ζ(Ki, iωn)− ǫ|2 + φ(Ki, iωn)2 Superconducting states of the quasi-2D Holstein model 4 Figure 2. Diagrammatic representation of the approximation used in this paper. Series (a) represents the vertex-neglected theory which corresponds to the Migdal– Eliashberg approach. This is valid when there is a high probability that the last emitted phonon is the first to be reabsorbed, which is true if the phonon energy ω0 and electron-phonon coupling U are small compared to the Fermi energy. Series (b) represents additional diagrams for the vertex corrected theory. The inclusion of the lowest order vertex correction allows the order of absorption and emission of phonons to be swapped once. For moderate phonon frequency and electron-phonon coupling, these additions to the theory, in combination with non- local corrections are expected to improve the theory to sufficient accuracy. The phonon self energies are labeled with Π, and Σ denotes the electron self-energies. Lines represent the full electron Green’s function and wavy lines the full phonon Green’s function. F (K, iωn) = − Di(ǫ)φ(Ki, iωn) |ζ(Ki, iωn)− ǫ|2 + φ(Ki, iωn)2 where ζ(Ki, iωn) = iωn + µ− Σ(Ki, iωn). The symmetry of the problem was constrained using the pm3m planar point group suitable for a 2D square lattice [21]. The partial DOS used in the self-consistent condition were calculated using the analytic tetrahedron method to ensure very high accuracy [22]. 3. The Holstein model A simple, yet non-trivial, model of electron-phonon interactions treats phonons as nuclei vibrating in a time-averaged harmonic potential (representing the interactions between all nuclei) i.e. only one frequency ω0 is considered. The phonons couple to the local electron density via a momentum-independent coupling constant g. The resulting Holstein Hamiltonian [23] is written as, H = − t<ij>σc iσcjσ + niσ(gri−µ)+ Mω20r The first term in this Hamiltonian represents a tight binding model with hopping parameter t. Its Fourier transform takes the form ǫk = −2t i=1 cos(ki). The second term connects the local ion displacement, ri to the local electron density. Finally the last term can be identified as the bare phonon Hamiltonian, which is a simple harmonic oscillator. The creation and annihilation of electrons is represented by c and ci respectively, pi is the ion momentum and M the ion mass. t = 0.25 in this paper, corresponding to a bandwidth of W = 2. A small interplanar hopping of Superconducting states of the quasi-2D Holstein model 5 t⊥ = 0.01 is included to reduce the strength of the logarithmic singularity at ǫ = 0 in Dπ,0(ǫ) and D0,π(ǫ) and stabilise the solution. This is only expected to modify the results at very low temperature for large clusters, and gives the problem a quasi-2D character. It is possible to find an expression for the effective interaction between electrons by integrating out phonon degrees of freedom [24]. In Matsubara space, this interaction has the form, U(iωs) = ω2s + ω Here, ωs = 2πsT represent the Matsubara frequencies for Bosons and s is an integer. A variable U = −g2/Mω20 is defined to represent the effective electron-electron coupling in the remainder of this paper. When phonon frequency and coupling are small, Migdal’s theorem applies. Migdal’s approach allows vertex corrections to be neglected and becomes exact when U → 0−, ω0 → 0 + and is 1st order in U . In the limit of huge phonon frequency, the model maps onto an attractive Hubbard model, so the weak coupling limit of the Holstein model is only obtained by considering all second-order diagrams in U , and ME theory fails. The vertex-corrected theory described in this paper has the appropriate weak coupling behaviour for both large and small ω0. In this paper, perturbation theory to 2nd order in U is used [19] (figure 2). The derivation of the perturbation theory in Ref. [19] made use of the conserving approximations of Bahm and Kadanoff [25, 24], which Miller et al. then simplified by applying the dynamical mean-field theory (or local approximation). Here the theory has been extended to include partial momentum dependence through the application of the DCA. The electron self-energy has two terms, ΣME(K, iωn) neglects vertex corrections (figure 2(a)), and ΣVC(K, iωn) corresponds to the vertex corrected case (figure 2(b)). ΠME(K, iωs) and ΠVC(K, iωs) correspond to the equivalent phonon self energies. The diagrams translate as follows: ΣME(K) = UT G(Q)D(K−Q) (8) φME(K) = −UT F (Q)D(K−Q) (9) ΠME(K) = −2UT [G(Q)G(K+Q)− F (Q)F ∗(K+Q)] (10) ΣVC(K) = (UT ) Q1,Q2 [G(Q1)G(Q2)G(K−Q2 −Q1) − F (Q1)G(Q2)F ∗(K−Q2 −Q1) − F ∗(Q1)G(Q2)F (K−Q2 −Q1) −G∗(Q1)F (Q2)F ∗(K−Q2 −Q1)] ×D(K−Q2)D(Q1 −Q2) (11) φVC(K) = (UT ) Q1,Q2 [F ∗(Q1)F (Q2)F (K−Q2 +Q1) −G(Q1)F (Q2)G(K−Q2 +Q1) Superconducting states of the quasi-2D Holstein model 6 −G∗(Q1)F (Q2)G ∗(K−Q2 +Q1) − F (Q1)G(Q2)G ∗(K−Q2 +Q1)] ×D(K−Q2)D(Q1 −Q2) (12) ΠVC(K) = −(UT ) Q1,Q2 Tr {σ3G(Q2 +K)σ3G(Q2)σ3G(Q1)σ3G(K+Q1)} ×D(Q2 −Q1) (13) where σ3 is the third Pauli matrix. Σ = ΣME +ΣVC and Π = ΠME + ΠVC. The coarse-grained phonon propagator D(K, iωs) is calculated from, D(K, iωs) = ω2s + ω 0 −Π(K, iωs) since the bare dispersion of the Holstein model is flat. The time taken to perform the double integration over momentum and Matsubara frequencies is the main barrier to performing vertex-corrected calculations, and this limits the cluster size. Since the Holstein model with ω0, |U | ≪ W (W is the bandwidth) has fluctuations which are almost momentum independent, the DCA has especially fast convergence in NC for the parameter regime where ω0, |U | < W , and calculations with relatively small cluster size accurately reflect the physics [6]. In this respect, finite size calculations take too long to compute, and the application of DCA to this problem is essential. 4. Results In this section, I discuss results from the self-consistent scheme. Calculations are carried out along the Matsubara axis, with sufficient Matsubara points for an accurate calculation. The vertex corrected self-energies drop off more quickly with Matsubara frequency, so it is possible to increase efficiency by calculating for less frequencies. Typically, 256 Matsubara frequencies are used for the vertex neglected diagrams, and 64 for the vertex corrected diagrams, which reach asymptotic behaviour at smaller Matsubara frequencies. The scheme was iterated until the normal and anomalous self-energies had converged to an accuracy of approximately 1 part in 104. This corresponds to a very high accuracy for the Green’s function. Obtaining superconducting solutions involves an additional step, which is not obvious at the outset. Since the anomalous Green’s function is proportional to the anomalous self energy, initialising the problem with the non-interacting Green’s function leads to a non-superconducting (normal) state. Also, the non-interacting Green’s function is consistent with an ungapped state and opening a gap in the electron spectrum can lead to limit cycles during self-consistency, which are damped in the normal way [17]. To induce superconductivity, a constant superconducting field is applied to the whole system, leading to a non-zero anomalous Green’s function, and automatically opening a gap in the normal-state Green’s function. The procedure of applying a fictitious superconducting field is analogous to the application of a magnetic field to a spin system to induce a moment (the order parameter in that case). The superconducting field is applied by adding a constant term to the anomalous self- energy in equation 9. With the field applied, equations 4,5 and 8-14 are solved self- consistently until convergence is reached. Once satisfactory convergence is reached, Superconducting states of the quasi-2D Holstein model 7 0.05 0.15 0.25 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 n=1.12 (0,0) (π,0) (π,π) 0.005 0.01 0.015 0.02 0.025 0.03 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 n=1.54 (0,0) (π,0) (π,π) 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 n=1.12, VC (0,0) (π,0) (π,π) 0.005 0.01 0.015 0.02 0.025 0.03 0.035 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 n=1.54, VC (0,0) (π,0) (π,π) Figure 3. Real part of the anomalous self-energy at various fillings: (a) n = 1.12, no vertex corrections (b) n = 1.54, no vertex corrections (c) n = 1.12, vertex corrections, (d) n = 1.54, vertex corrections. Calculations were carried out at T = 0.005 with U = 0.6 and ω0 = 0.4. Momentum dependence corresponding to non-local corrections is clearly visible at half-filling, but drops off as the edge of the superconducting phase is reached. Vertex corrections are also most important at half-filling. There is a dip in the anomalous self-energy because the vertex corrections drop off more quickly in ωn, with opposite sign to φME, indicating that the approximation is close to breakdown at half-filling. The slight increase in the anomalous self-energy at n = 1.54 due to vertex corrections arises from a change in the form of the electronic Green’s function. For half filling, the Green’s function at the van-Hove points is pure imaginary, whereas for the dilute system, it is mostly real, so sums over products of Green’s functions in the vertex can change sign. the fictitious field is completely removed. Iteration then continues until the true superconducting state is reached. This procedure corresponds to initialising the self- consistent cycle with a superconducting solution; note that similar techniques are used for obtaining Mott insulating solutions in the Hubbard model using DMFT [17]. By following this procedure, a superconducting state may be found below the transition temperature, TC . Green’s functions and self-energies computed in the superconducting state can then be used to initialise the self-consistent equations for similar couplings, fillings, temperatures (with an appropriate rescaling of the Matsubara frequencies) and phonon frequencies. Above the transition temperature, the magnitude of the anomalous Green’s function tends to zero during self-consistency as expected. It is possible to see the generic effects of vertex and non-local corrections by examining the anomalous self energy. In figure 3, the anomalous self energy is shown for n = 1.12 and n = 1.54 for a cluster size of NC = 4 with parameters of U = 0.6, T = 0.005 and ω0 = 0.4 with and without vertex corrections. The panels are (a) n = 1.12, no vertex corrections (b) n = 1.54, no vertex corrections (c) n = 1.12, vertex corrections, (d) n = 1.54, vertex corrections. In panel (a), the momentum dependence of the vertex neglected theory is clearly visible, and φ(K, iωn) has a much Superconducting states of the quasi-2D Holstein model 8 larger value at the (π, 0) point. Momentum dependence is significantly reduced as the system moves away from half-filling (panel b), indicating that Midgal-Eliashberg theory is more accurate in dilute systems. This is expected, since in very dilute systems, the electron density is sufficiently low that electrons meet very infrequently, and therefore the crossed diagrams of figure 2b have extremely small contributions. By scanning vertically, the effect of including vertex corrections can be seen. Corrections are strongest close to half-filling, and drop off as the edge of the superconducting phase is reached. Migdal–Eliashberg theory is clearly quite accurate for dilute 2D systems, but it consistently fails close to half-filling. Initially, it seems as though vertex corrections are larger than the vertex neglected results at n = 1.12. In fact, this is not the case. As discussed in ref. [6], at half-filling vertex corrections act to reduce the magnitude of the phonon self-energy, so there is much less renormalisation of the phonon propagator. A smaller phonon propagator means that the effective coupling is smaller, stabilising the expansion in λeff . There is a dip in the anomalous self-energy because the vertex corrections drop off more quickly in ωn, with opposite sign to φME, which is an indication that the approximation is close to breakdown at half-filling. The slight increase in the anomalous self-energy at n = 1.54 due to vertex corrections comes about from a change in form of the electronic Green’s function. For half filling, the Green’s function at the van-Hove points is pure imaginary, whereas for the dilute system, it is mostly real, so sums over products of Green’s functions can change sign with respect to the Migdal–Eliashberg result. This sign change is also seen in DMFT simulations of the 3D Holstein model [19]. At this stage, it is appropriate to examine the size of the parameter λeff that defines the vertex correction expansion. Since the expansion in this case is in the full phonon Green’s function, the expansion parameter is renormalised by the phonons, and reads λeff = UD(µ)D(iωs = 0), where D(iωs = 0) is the phonon propagator at zero Matsubara frequency. For dilute systems, D(µ) is typically small, and so λeff is small (N.B. Unlike in 3D, D(µ) is never zero in 2D, because of the discontinuity in the band edge of the non-interacting DOS). Close to half-filling, the DOS in 2D is divergent, and this parameter is expected to be large. In the current approximation, a small interplanar hopping was applied to stabilise the solution, so λeff is smaller than expected in a pure 2D system. As noted in ref. [6], D(0) is reduced by vertex corrections as compared to the Migdal–Eliashberg result. For most energies, the bare density of states in 2D is smaller than the bare density of states in 3D, since the divergence drops off logarithmically close to half filling. Therefore, λeff is only really large for n = 1 within the current parameter range. For the mid to dilute limits, the relative magnitude of the second order vertex correction goes like λ2 ∼ 0.04. At half filling, with the current parameters, λ2 ∼ 0.5, so the approximation can only be considered to be qualitatively correct. Nonetheless, the current approximation has features appropriate to Hohenberg’s theorem (discussed later) and the bare DOS drops off so quickly moving away from half-filling, that results are expected to be accurate for most n. How do the differences in the self-energy relate to observable quantities? One of the big questions in unconventional superconductivity concerns the possible forms that the order parameter can take, and a large discussion has grown up around issues such as the existence of unconventional order parameters such as extended s-wave and higher harmonics. To examine this idea, I demonstrate the evolution of the shape of the anomalous pairing density (ns(k) = |T n F (k, iωn)|), which is related to the order parameter. In this paper, the superconducting order parameter is treated in Superconducting states of the quasi-2D Holstein model 9 0.05 0.15 U=0.6, ω0=0.4, T=0.005, Nc=1 0.05 0.15 0.05 0.15 U=0.6, ω0=0.4, T=0.005, Nc=4 0.05 0.15 0.05 0.15 U=0.6, ω0=0.4, T=0.005, Nc=64 0.05 0.15 0.14 0.15 0.16 0.17 0.18 0.19 0.21 0.22 (0,π)(π,0) NC= 1 NC= 4 NC=64 Figure 4. Variation of superconducting (anomalous) pairing density across the Brillouin zone. U = 0.6, ω0 = 0.4, n = 1 and T = 0.005. Cluster sizes are increased from NC = 1 to NC = 64. Pairing occurs between electrons close to the “Fermi-surface” at k and the opposite face of the surface at −k. Also shown is the pairing density at the Fermi surface for the 3 different cluster sizes (bottom right). For a cluster size of Nc = 1 corresponding to DMFT, the pairing is uniform around the “Fermi-surface”, demonstrating that momentum dependence has been neglected. Momentum dependence favours additional pairing along the kx = ky line, and a peak can clearly be seen. The expansion in spherical harmonics contains even momentum states with m = 0. For Nc = 64, additional peaks can be seen, suggesting that the order parameter also contains extra higher-order harmonics. The Fermi-surface is not very clearly defined, with the mobile electrons spread out over a significant range of momentum states. T = 0.005 << W , so the spread should be very small in all locations in the Brillouin zone, except at the van Hove points, (π, 0) and (0, π), indicating that the spreading is due to the low dimensionality. a fully self-consistent manner within the lattice symmetry and no assumptions have been made in advance about its form. Figure 4 shows the variation of superconducting pairing across the Brillouin zone. In all of the panels, U = 0.6, ω0 = 0.4, n = 1 and T = 0.005. A range of cluster sizes is shown. In the dynamical mean-field theory which corresponds to the Eliashberg solution (cluster size of Nc = 1) the pairing is uniform around the Fermi-surface, as is expected when momentum-dependence is neglected. The inclusion of non-local momentum dependent fluctuations has a small, but significant effect on the ordering. Pairing is reduced most at the (π, 0) and (0, π) points, leading to a visible peak at (π/2, π/2). This demonstrates that the order parameter must necessarily include higher harmonics. For Nc = 64 additional peaks are also seen. The additional features may be examined by determining the parameters of an expansion in spherical harmonics, clm = (2π)3 Ylm(θ, φ)ns(k) (figure 5). This shows that the anomalous density can be thought of as m = 0 harmonics with s, d, g... character, and additional harmonics with m = ±4 in the g channel. The harmonics can be quite large, especially away from half-filling, and undoubtedly need to be included if the superconductivity is to be described correctly. Superconducting states of the quasi-2D Holstein model 10 -0.025 -0.02 -0.015 -0.01 -0.005 0.005 0.01 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 s, m=0 d, m=0 g, m=0 g, m=4,-4 Figure 5. Harmonic decomposition of the anomalous density computed for Nc = 64 as chemical potential is varied. It can be seen that pure s-wave states are the largest contributors to the anomalous density, followed by g and then d states with m = 0. Owing to the hump at the (π/2, π/2) point, there are also g states with m = ±4. The m = ±4 states have equal magnitude, so mtot = 0. Note that the relative contribution of higher harmonics is greatest away from half filling. 0.05 0.15 U=0.9, ω0=0.05, T=0.005, Nc=4 0.05 0.15 0.05 0.15 U=0.9, ω0=0.4, T=0.005, Nc=4 0.05 0.15 0.05 0.15 U=0.6, ω0=0.05, T=0.005, Nc=4 0.05 0.15 0.05 0.15 U=0.6, ω0=0.4, T=0.005, Nc=4 0.05 0.15 Figure 6. Variation of superconducting (anomalous) pairing density across the Brillouin zone. T = 0.005 and NC = 4. Changes in the order parameter are shown as coupling and phonon frequency are changed. As the phonon frequency is increased, the momentum dependence also increases, and the Fermi-surface is less well defined. For U = 0.9, ω0 = 0.05, the order parameter is almost flat along the Fermi-surface, indicating that DMFT is a good approximation for those parameters. For the largest coupling and phonon frequencies (at the edge of applicability for the current approximation), the Fermi-surface is practically destroyed. Superconducting states of the quasi-2D Holstein model 11 Figure 6 shows the variation of superconducting (anomalous) pairing density across the Brillouin zone as coupling and phonon frequency are changed. T = 0.005, n = 1 and NC = 4 with vertex corrections excluded. As the phonon frequency is increased, the momentum dependence also increases. For U = 0.9, ω0 = 0.05, the order parameter is almost flat along the Fermi-surface, indicating that DMFT is a good approximation for those parameters. Typically, additional coupling makes the order more uniform in the Brillouin zone. Note that for very strong coupling, the DMFT solution is expected to become exact, even in 2D, since the bare dispersion is then essentially flat, and the problem is completely local. For weak coupling and phonon frequency, there is a well defined Fermi-surface. For the largest coupling and phonon frequencies (at the edge of applicability for the current approximation), the Fermi-surface is practically destroyed. It is of clear interest to map the phase diagram associated with superconducting order. First, the superconductivity arising from DMFT is investigated in the absence of vertex corrections. Figure 7 shows the resulting phase diagram. Note that in the DMFT solution, the superconductivity is strongest at half-filling and the order drops off monotonically as the filling increases. Assuming a form for the density of states in 2D (with small interplane hoppnig) of D(ǫ) = (1− t log[(ǫ2 + t2⊥)/16t 2])/tπ2 (for |ǫ| < 4t) [26], the BCS result may be calculated using the expression TC(n) = 2ω0 exp(−1/|U |D(µ(n)))/π, with the chemical potential taken from the self-consistent solution for a given n. This result also drops off monotonically. Results in the dilute limit are in good agreement with the BCS result. Closer to half-filling, the DMFT result is significantly smaller than the BCS result (which predicts TC(n = 1) > 0.07). The difference in results between the two mean-field theories at half-filling is due to the self-consistency in the DMFT. For small U , the self-consistent equations converge on the first iteraction, but for larger U , the phonon and electron Green’s functions are significantly renormalised, thus reducing the transition temperature. To show the differences induced by spatial fluctuations, the phase diagram is computed for a cluster size of NC = 4. Figure 8 shows the total density of superconducting pairs for U = 0.6, ω0 = 0.4 and various temperatures and fillings, without vertex corrections. Of most interest is an anomalous bump centred about n = 1.25, indicating that the strongest superconductivity occurs away from half filling, and that this is due to non-local fluctuations in 2D. Assuming a material with a non- interacting band width of 1eV, the highest transition temperature would correspond to 145K. This value is higher that that eventually expected in real materials. For instance, the effect of a Coulomb pseudopotential UC will be a reduction of the transition temperature. The standard BCS result is modified by Coulomb repulsion in the following way, TC = 2ω0 exp(−1/(λ− UC)))/π. In addition to the TC reduction due to Coulomb repulsion, a fundamental limit on the transition temperature in pure 2D is the Hohenberg theorem [27]. This is closely related to the effects of spatial fluctuations. The basis of Hohenberg’s proof is the divergence of certain quantities (which are known to be finite) in d ≤ 2 at k = 0 when anomalous expectation values (e.g. the superconducting order parameter) are non zero. In the DMFT solution, there are no specific k = 0 states due to the coarse graining, and so the divergence in the correlation functions that led Hohenberg to determine that the order parameter must be zero for zero momentum pairing in d ≤ 2 is washed out, leading to a finite transition temperature for the 2D local approximation. In DCA, partial momentum dependence is restored. Therefore, the effects of the washed out divergences are stronger. There is still a finite transition Superconducting states of the quasi-2D Holstein model 12 0.02 0.04 0.06 0.08 0.01 0.02 0.03 0.04 1 1.2 1.4 1.6 1.8 0.02 0.04 0.06 0.08 U=0.6, ω0=0.4, Nc=1 Figure 7. Superconducting phase diagram showing the total number of superconducting states. U = 0.6, ω0 = 0.4 and various T . A cluster size of Nc = 1 has been used, and no vertex corrections are included. The superconductivity is strongest at half-filling and the order drops off monotonically as the filling increases. Results in the dilute limit are in good agreement with the BCS result (the transition temperature from BCS is shown as the line with points in the ns = 0 plane). Closer to half-filling, the DMFT result is significantly smaller than the BCS result (which predicts TC(n = 1) > 0.07). The difference in results between the two mean-field theories at half-filling is due to the self-consistency in the DMFT. temperature, but it is reduced wherever there is strong momentum dependence. This is demonstrated by the drop in superconducting order at and close to half-filling in figure 8, where the momentum dependence is strongest. As the number of cluster points increases, the momentum resolution becomes superior, and the divergences of Hohenberg’s theorem are expected to emerge in a systematic manner. In real materials with quasi-2D character, some interplane hopping remains. In that case, the results from small cluster DCA are expected to be more reliable. Finally, I demonstrate the effects of vertex corrections on the superconducting phase diagram. Figures 9 and 10 show the total number of superconducting states as a function of filling and temperature for cluster sizes of NC = 1 and NC = 4 respectively. An electron-phonon coupling of U = 0.6 and phonon frequency of ω0 = 0.4 have been used. Vertex corrections do not appear to make a large difference to the DMFT result in figure 9. For NC = 4, the bulge that was seen in the non vertex-corrected phase diagram is very clearly enhanced. Superconductivity at half- filling is completely suppressed in the vertex corrected solution. This is the precursor to Hohenberg’s theorem applying across the entire phase diagram, and both the effects of spatial fluctuations and the lowest order vertex correction were essential to obtain that agreement. Superconducting states of the quasi-2D Holstein model 13 0.02 0.04 0.06 0.08 U=0.6, ω0=0.4, Nc=4 0.01 0.02 0.03 0.04 1 1.2 1.4 1.6 1.8 0.02 0.04 0.06 0.08 Figure 8. Superconducting phase diagram showing the total number of superconducting states. U = 0.6, ω0 = 0.4 and various T . A cluster size of Nc = 4 has been used, and no vertex corrections are included. There is an anomalous bump centred about n = 1.25, indicating that the strongest superconductivity occurs away from half filling. The highest transition temperature occurs for T = 0.025. The reduction in the transition temperature close to half-filling shows the onset of Hohenberg’s theorem. The largest superconductivity coincides with the increase in components without pure s-wave character (see figure 5). 0.02 0.04 0.06 0.08 0.12 U=0.6, ω0=0.4, Nc=1, VC 0.01 0.02 0.03 0.04 1 1.2 1.4 1.6 1.8 0.02 0.04 0.06 0.08 0.12 Figure 9. Superconducting phase diagram showing the total number of superconducting states. U = 0.6, ω0 = 0.4 and various T . A cluster size of Nc = 1 has been used, and vertex corrections are included. As in figure 7, the DMFT result falls off monotonically with increased filling. Superconducting states of the quasi-2D Holstein model 14 0.02 0.04 0.06 0.08 U=0.6, ω0=0.4, Nc=4, VC 0.01 0.02 0.03 0.04 1 1.2 1.4 1.6 1.8 0.02 0.04 0.06 0.08 Figure 10. Superconducting phase diagram showing the total number of superconducting states. U = 0.6, ω0 = 0.4 and various T . A cluster size of Nc = 4 has been used, and vertex corrections are included. Superconducting states are suppressed at half-filling, and there is a significant bulge away from half-filling, with a maximum transition temperature of 0.015W. It is significant that the transition temperature is reduced to zero at half-filling and supressed close to half filling, since reduction of transition temperatures is expected in 2D due to Hohenberg’s theorem. 5. Summary In this paper I have carried out DCA calculations of a quasi-2D Holstein model in the superconducting state with large in plane hopping and small out of plane hopping (t = 0.25, t⊥ = 0.01). Several approximations to the self-energy were considered, including the neglect of vertex corrections (which corresponds to a momentum- dependent extension to the Eliashberg theory), the inclusion of vertex corrections as a corrected approximation for stronger couplings, and the introduction of spatial fluctuations. The anomalous self energy, superconducting order parameter and phase diagram were calculated. The superconducting order parameter was found to modulate around the fermi surface, and is not pure s-wave. Analysis of the harmonics showed that the state is a conbination of s, d, f etc. states with m = 0, and other states with integer value of m = ±4n. The total angular momentum is always zero. The contribution of the m 6= 0 states is considerably larger away from half filling. Increases in bare phonon frequency tended to increase the strength of the superconducting order, and contributed to a degeneration of the Fermi-surface. The phase diagram was analysed for small cluster sizes of NC = 1, 4. For NC = 1, the phase diagram was shown to agree qualitatively with the BCS theory. When spatial fluctuations are included, the superconducting order is suppressed at half- filling, leading to a characteristic hump at a doping of approximately δn = 0.25. Vertex corrections completely suppress superconductivity at half-filling, which is believed to be a manifestation of Hohenberg’s theorem. In particular, the states with the largest momentum dependence showed the strongest reduction in the transition temperature, Superconducting states of the quasi-2D Holstein model 15 indicating that spatial fluctuations as well as vertex corrections contribute to the supression of superconducting order in pure 2D materials. Acknowledgments JPH would like to thank the University of Leicester for hospitality and use of facilities while carrying out this research. [1] A.Lanzara, P.V.Bogdanov, X.J.Zhou, S.A.Kellar, D.L.Feng, E.D.Lu, T.Yoshida, H.Eisaki, A.Fujimori, K.Kishio, J.-I.Shimoyama, T.Noda, S.Uchida, Z.Hussa, and Z.-X.Shen. Nature, 412:6846, 2001. [2] R.J.McQueeny, Y.Petrov, T.Egami, M.Yethiraj, G.Shirane, and Y.Endoh. Phys. Rev. Lett., 82:628, 1999. [3] G.M.Zhao, M.B.Hunt, H.Keller, and K.A.Müller. Nature, 385:236, 1997. [4] A.B.Migdal. JETP letters, 7:996, 1958. [5] G.M.Eliashberg. Interactions between electrons and lattice vibrations in a superconductor. JETP letters, 11:696, 1960. [6] J.P.Hague. Electron and phonon dispersions of the two dimensional Holstein model: Effects of vertex and non-local corrections. J. Phys. Condens. Matt, 15:2535, 2003. [7] S.Engelsberg and J.R.Schrieffer. Phys. Rev., 131:993, 1963. [8] M.Grabowski and L.J.Sham. Superconductivity from nonphonon interactions. Phys. Rev. B, 29:6132, 1984. [9] V.N.Kostur and B.Mitrović. Electron-phonon interaction in two dimensions: Variation of ImΣ(ǫp, ω) with increasing ωD/EF . Phys. Rev. B, 48:16388, 1993. [10] C.Grimaldi, L.Pietrono, and S.Strässler. Nonadiabatic superconductivity: Electron-phonon interaction beyond Migdal’s theorem. Phys. Rev. Lett., 75:1158, 1995. [11] A.S.Alexandrov and J.R.Schrieffer. Phys. Rev. B, 56:13731, 1997. [12] O.V.Danylenko and O.V.Dolgov. Nonadiabatic contribution to the quasiparticle self-energy in systems with strong electron-phonon interaction. Phys. Rev. B, 63:094506, 2001. [13] M.Hettler, A.N.Tahvildar-Zadeh, M.Jarrell, T.Pruschke, and H.R.Krishnamurthy. Nonlocal dynamical correlations of strongly interacting electron systems. Phys. Rev. B, 58:7475, 1998. [14] A.S.Alexandrov and N.F.Mott. Rep. Prog. Phys., 57:1197, 1994. [15] P.E.Kornilovitch. Phys. Rev. Lett., 81:5382, 1998. [16] M.Hettler, M.Mukherjee, M.Jarrell, and H.R.Krishnamurthy. Phys. Rev. B, 61:12739, 2000. [17] A.Georges, G.Kotliar, W.Krauth, and M.Rozenburg. Rev. Mod. Phys, 68:13, 1996. [18] A.N.Tahvildar-Zadeh, J.K.Freericks, and M.Jarrell. Magnetic phase diagram of the hubbard model in three dimensions: the second-order local approximation. Phys. Rev. B, 55:942, 1997. [19] P.Miller, J.K.Freericks, and E.J.Nicol. Possible experimentally observable effects of vertex corrections in superconductors. Phys. Rev. B, 58(21):14498, 1998. [20] Thomas Maier, Mark Jarrell, Thomas Pruschke, and Matthias H. Hettler. Quantum cluster theories. cond-mat/0404055. [21] T.Hahn, editor. International tables for crystallography. Volume A: Space group symmetry. Kluwer Academic Publishers, Dordrecht, 1996. [22] Ph. Lambin and J.P.Vigneron. Computation of crystal green’s functions in the complex-energy plane with the use of the analytical tetrahedr on method. Phys. Rev. B, 29:3430, 1984. [23] T.Holstein. Ann. Phys., 8:325–342, 1959. [24] N.E.Bickers and D.J.Scalapino. Ann. Phys., 193:206, 1989. [25] G.Bahm and L.P.Kadanoff. Phys. Rev., 124:287, 1961. [26] L.S.Macarie and N.d’Ambrumenil. J. Phys. Condens. Matt, 7:3237, 1995. [27] P.C.Hohenberg. Phys. Rev., 158:383, 1967. http://arxiv.org/abs/cond-mat/0404055 Introduction The dynamical cluster approximation The Holstein model Results Summary

0704.0242

Masers and star formation

Astrophysical masers and their environments Proceedings IAU Symposium No. 242, 2007 J. Chapman & W. Baan, eds. c© 2007 International Astronomical Union DOI: 00.0000/X000000000000000X Masers and star formation Vincent L. Fish Jansky Fellow, National Radio Astronomy Observatory, 1003 Lopezville Rd., Socorro, NM 87801, USA email: [email protected] Abstract. Recent observational and theoretical advances concerning astronomical masers in star forming regions are reviewed. Major masing species are considered individually and in combination. Key results are summarized with emphasis on present science and future prospects. Keywords. masers — stars: formation — radio lines: ISM — ISM: molecules — ISM: jets and outflows — ISM: kinematics and dynamics 1. Introduction This review summarizes maser results pertinent to star formation appearing in the literature since the last maser meeting (IAU Symposium 206). References are drawn from recent literature when possible. 2. Masing species 2.1. Water (H2O) The 22.235 GHz water line is the predominant water maser line. Masers in this transition are very bright, easily observable, and inverted under a wide range of conditions (e.g., Babkovskaia & Poutanen 2004). Several millimeter and submillimeter transitions of wa- ter are also seen as masers. Discussion of these transitions can be found in the section of these proceedings devoted to millimeter and submillimeter masers. Water masers are frequently seen in outflows from both high-mass and low-mass YSOs (Honma et al. 2005; Moscadelli, Cesaroni, & Rioja 2005; Goddi & Moscadelli 2006; Moscadelli et al. 2006). These jets are seen in deceleration (Imai et al. 2002) and often have substructure on AU scales (Torrelles et al. 2003; Furuya et al. 2005; Uscanga et al. 2005). Water masers are sometimes believed to trace disks as well as outflows (Seth, Greenhill, & Holder 2002; Gallimore et al. 2003), possibly excited by an expanding shock wave. Indeed, shocks likely are responsible for arc-like maser distributions (Honma et al. 2004) and may ex- cite masers in accreting material as well (Menten & van der Tak 2004). Water masers appear in Bok globules (Gómez et al. 2006), likely tracing bipolar molecular outflows (de Gregorio-Monsalvo et al. 2006), as well as bright rimmed clouds (Valdettaro et al. 2005), again likely associated with outflows (Urquhart et al. 2006). The location of water masers near the ionization front of large H ii regions provides evidence in support of triggered star formation (Healy, Hester, & Claussen 2004). The common thread of all these en- vironments is the existence of energetic shocks, which fits with conventional wisdom regarding water maser pumping. Despite the small Zeeman splitting coefficient of water, line-of-sight magnetic fields of tens to hundreds of milligauss have been measured in star forming regions via water maser Zeeman splitting (Sarma et al. 2002; Vlemmings et al. 2006), although direct in- terpretation of the Stokes V profile as a magnetic field strength may be in error by up to a http://arxiv.org/abs/0704.0242v1 2 Fish factor of two depending on local velocity and magnetic field gradients (Vlemmings 2006). Linear polarization observations of water masers can provide information on the orien- tation of the magnetic field in the plane of the sky. The hourglass morphology of the magnetic field in W3 IRS 5 appears to be due to the processes of collapse rather than a result of the outflow traced by water masers (Imai et al. 2003). Further interpretation of water maser polarization can be found in the review by Wouter Vlemmings. Water masers have a fractal distribution over 4 orders of magnitude in spatial scale, possibly indicating that they appear at the turbulent dissipation scale (Strelnitski et al. 2002; Ripman & Strelnitski 2006; see also Vladimir Strelnitski’s contribution to these pro- ceedings). Turbulence may also be responsible for variability, including changes in the line-of-sight velocity of individual components, of the water masers in some sources (Lekht et al. 2006a). Larger-scale variations may contribute as well, such as changes in outflow parameters or cyclic variability of the central star (Pashchenko & Lekht 2005; Lekht et al. 2006b). An ordered structure is inferred in W31(2) from successive flaring of features at different velocities (Lekhn, Munitsyn, & Tolmachev 2005). 2.2. Methanol (CH3OH) Methanol masers divide into two categories, known as class I and class II, based on their propensity for certain transitions to produce masers while others are seen in ab- sorption. Traditionally, class I and class II masers do not mix (e.g., Ellingsen 2005); however, fine-tuned conditions may rarely excite lines from both classes simultaneously, as appears to be the case in OMC-1 (Voronkov et al. 2005). For purposes of this review, class I and class II methanol masers will be treated separately. Improved laboratory data for rest frequencies have been obtained for many methanol transitions in both classes (Müller, Menten, & Mäder 2004). 2.2.1. Class I Class I masers are primarily collisionally pumped. They are typically found in younger sources that are Class II masers and may trace distant parts of outflows interacting with dense molecular gas (Beuther et al. 2005; Ellingsen 2006). Comparison of interferometric maps of the 9.9 and 104.3 GHz transitions with H2 data confirms the outflow association in IRAS 16547−4247 (Voronkov et al. 2006). Linear polarization suggests that Class I masers may appear in oblique shocks parallel to the outflow axis and perpendicular to the magnetic field in OMC-2 (Wiesemeyer, Thum, & Walmsley 2004), although interpre- tation of methanol polarization may be complicated (e.g., Elitzur 2002). Many different Class I transitions have been observed. Weak masers at 84.5 and 95.2 GHz to the southwest of W3(OH) provide strong evidence in support of colli- sional pumping and allow for physical conditions to be inferred (Sutton et al. 2004). The latter transition is commonly seen as a maser in both Class I and Class II sources (Minier & Booth 2002). Strong 36 GHz maser emission is believed to be an indicator of an early evolutionary stage, as may line ratios of other transitions (Gillis, Pratap, & Strelnitski 2005; Hoffmann, Pratap, & Strelnitski 2006). The intensity ratio of highly-excited 146.6 and 156.8 GHz methanol masers may be a sensitive probe of density and temperature in mas- ing regions (Lemonias, Strelnitski, & Pratap 2006). Short-timescale variability is seen in the 44 and 146.6 GHz transitions (Pratap, Hoffmann, & Strelnitski 2006). 2.2.2. Class II The key Class II maser transitions are at 6668 and 12178 MHz. Class II masers are often found in an earlier evolutionary stage than ultracompact (UC) H ii regions (e.g., Minier et al. 2005), but observations of methanol masers cospatial with both millimeter Masers and star formation 3 and centimeter continuum emission indicate that Class II masers appear over a wide range of evolutionary stages (Pestalozzi et al. 2006). While the lower stellar mass limit for methanol masers is still a subject of research, 6.7 GHz masers do not appear below approximately 3 solar masses (Minier et al. 2003). Linear structures of masers with organized velocity structures have led some to con- clude that methanol masers often trace edge-on disks (Norris et al. 1993, 1998). Obser- vations of maser proper motions (Minier et al. 2000), shocked H2 (De Buizer 2003), and SiO (De Buizer et al. 2006) indicate that, in the majority of cases at least, methanol masers are aligned with an outflow, not a disk. These structures may be explained by propagation of a shock front into a region with large-scale velocity structure, such as rota- tion (Dodson, Ojha, & Ellingsen 2004). Disk candidate sources remain (Slysh et al. 2002b; Pestalozzi et al. 2004; Pillai et al. 2006), although these, too, may turn out to be associ- ated with outflows when studied at high resolution in the mid infrared (e.g., De Buizer & Minier 2005). The conclusion to be drawn is that a linear distribution of masers with a velocity gra- dient does not by itself present convincing evidence that the masers trace an edge-on disk. An intriguing variant is the possibility of methanol masers tracing a face-on disk in G23.657−0.127 (Bartkiewicz, Szymczak, & van Langevelde 2005). There is evidence to support the hypothesis that most methanol masers are tracing shocked regions, often in the presence of outflows. Methanol masers appear preferentially near radio sources with a spectral index indicative of an outflow (Zapata et al. 2006). Mid-infrared images of some sources indicate that masers are found along the shocked material on the surface of an outflow cavity (De Buizer 2006, 2007). In some sources, methanol masers appear near but offset from UCH ii regions, suggesting that they appear in the shocked molecular gas outside the ionization front, similar to hydroxyl masers (Phillips & van Langevelde 2005). The 6.7 GHz transition is a popular line for Galactic maser surveys. Several surveys were reported on during IAU Symposium 206. Details of the Arecibo and Parkes multi- beam 6.7 GHz surveys can be found in the section of these proceedings devoted to Galac- tic maser surveys. Unsurprisingly, their distribution correlates well with Galactic struc- ture (Pestalozzi, Minier, & Booth 2005; Pestalozzi et al. 2007). The 12.2 GHz line, when it occurs, is almost always weaker than 6.7 GHz emission (B laszkiewicz & Kus 2004). Both lines have also been the subject of monitoring studies (e.g., Goedhart, Gaylard, & van der Walt 2005), which find variability in a large fraction of sources including periodic variability and a time delay between features possibly due to light travel time (Goedhart, Gaylard, & van der Walt 2005; Goedhart et al. 2005). Based on comparison of spectra over a period of a decade, the life- time of an individual 6.7 GHz maser feature is about 150 years (Ellingsen 2007), while the lifetime of the 6.7 GHz maser phase in a source is a few× 104 years (Codella et al. 2004; van der Walt 2005), similar to the lifetime of the OH maser phase (e.g., Fish & Reid 2006). Numerous other Class II transitions have been observed. New maser sources have been found in rare transitions at 85.5, 86.6, and 107.0 GHz (Minier & Booth 2002; Ellingsen et al. 2003) and a torsionally-excited line at 44.9 GHz (Voronkov, Austin, & Sobolev 2002). Several weak maser lines near 165 GHz have also been detected (Salii & Sobolev 2006). Emission in the 19.9 GHz transition is usually weak and correlates well with 6035 MHz OH masers (Ellingsen et al. 2004). A search for 23.1 GHz emission resulted in no new detections beyond the previously known maser in NGC 6334F (Cragg et al. 2004). Ob- servations of these less common methanol maser transitions can help constrain physical parameters in maser models. Improved collisional rate data has also allowed refinement of methanol models, which slightly affects predicted excitation conditions and bright- ness temperatures but not which transitions are expected to produce detectable Class II masers (Cragg, Sobolev, & Godfrey 2005). 4 Fish 2.3. Hydroxyl (OH) Hydroxyl masers are usually studied in sources with associated UCH ii regions (e.g., Fish et al. 2005) but are also found toward less evolved massive protostellar objects (Edris, Fuller, & Cohen 2007). A different class of OH masers is seen at the ends of the jet in the W3 TW object (Argon, Reid, & Menten 2003). Hydroxyl masers around more evolved sources are usually seen in expansion (some- times very rapid; see Stark et al. 2007) ahead of the ionization front of a UCH ii region (Fish & Reid 2006). Sometimes the masers appear to trace a molecular disk or torus (Slysh et al. 2002a; Hutawarakorn & Cohen 2005; Edris et al. 2005; Nammahachak et al. 2006). Masers are often seen along arcs or filaments (Cohen et al. 2006), with extended filamen- tary emission especially common at 4.7 GHz (Palmer, Goss, & Devine 2003). Multitransition overlaps are of special interest because of their ability to constrain physical conditions in models. The 4765 MHz line is observed to be the strongest line of the 6 cm triplet but is usually only weakly inverted and often spatially extended (e.g., Palmer, Goss, & Whiteoak 2004; Harvey-Smith & Cohen 2005). A histogram of 18 cm emission resembles 4.7 GHz lineshapes in W49A, suggesting that the high-gain 18 cm emission and low-gain 6 cm emission have similar velocity distributions, even if the 4660 MHz emission is spatially separate (Palmer & Goss 2005). Much is made of over- laps between 4765 and 1720 MHz masers (Palmer, Goss, & Devine 2003; Niezurawska et al. 2004, 2005). It should be noted that 6035 MHz maser emission correlates more strongly with 4765 MHz than does 1720 MHz, even if the velocities do not always agree (Dodson & Ellingsen 2002; Smits 2003). Masers in the 1720 MHz transition also appear to correlate with 1665 and 6035 MHz OH masers and 6.7 GHz methanol masers, at least to arcsecond accuracy (Caswell 2004a). Masers in the 6030 MHz transition are almost always accompanied by stronger emission at 6035 MHz (Caswell 2003), with excellent spatial coincidence and agreement of magnetic field strengths with each other and with 1665 MHz masers (Etoka, Cohen, & Gray 2005). Masers in the highly-excited 13441 MHz transition are rare but are always accompanied by 6035 MHz masers at the same velocity, although the intensities in the two transitions do not show a high degree of correlation (Baudry & Desmurs 2002; Caswell 2004c). The ground-state, satellite line transitions at 1612 and 1720 MHz are usually conjugate with respect to absorption and maser emis- sion (Szymczak & Gérard 2004). Sources do exist in which both transitions are inverted, though not in direct spatial overlap (e.g., Wright, Gray, & Diamond 2004). Magnetic fields as strong as 40 mG are seen in OH masers (Slysh & Migenes 2006; Fish & Reid 2007). Magnetic fields are highly ordered in star forming regions (Fish & Reid 2006) and support pictures in which the processes of star formation do not tangle field lines significantly. While magnetic field strengths are usually stable from epoch to epoch, monotonic decay of the field in a Zeeman group in Cep A continues to be observed (Bartkiewicz et al. 2005). High spectral resolution observations support the conventional assumption that the Zeeman splitting coefficient appropriate for σ±1 components should be assumed when measuring magnetic fields at 1612 and 1720 MHz (Fish, Brisken, & Sjouwerman 2006). Linear polarization is of limited usefulness in determining the full, three-dimensional orientation of the magnetic field, likely due to a combination of Faraday rotation and anisotropic magnetohydronamic turbulence (Watson et al. 2004; Fish & Reid 2006). Extreme variability is occasionally seen in OH. The 1665 MHz maser in W75 N briefly flared to nearly 1 kJy to become the brightest OH maser in the sky (Alakoz et al. 2005). The 4765 MHz transition is highly time-variable (Palmer, Goss, & Whiteoak 2004): the maser in Mon R2 flared to nearly 80 Jy before disappearing (Smits 2003) and reap- Masers and star formation 5 pearing (Fish et al. 2006b). Short-timescale variability is seen in the ground-state lines (Ramachandran, Deshpande, & Goss 2006; also Miller Goss in these proceedings). 2.4. Formaldehyde (H2CO) Formaldehyde masers are seen near a handful of several massive YSOs, with several new detections in recent years (Araya et al. 2005, 2006). They have both a compact and an ex- tended component with velocity gradients (Hoffman et al. 2003; Hoffman, Goss, & Palmer 2007). A short-duration flare has been detected toward one source (Araya et al. 2007). Further details can be found in the review by Esteban Araya. 2.5. Silicon monoxide (SiO) While SiO masers are commonly seen in evolved stars, they are rare in star form- ing regions. They are seen in bipolar outflows in W51 IRS 2 and Orion KL source I and appear much closer to the central source than do water masers (Eisner et al. 2002; Greenhill et al. 2004). As is the case in evolved stars, the maser species SiO, water, and OH occur at progressively larger distances from source I (Cohen et al. 2006). While OH masers are ubiquitous throughout Orion, there is a “zone of avoidance” associated with source I in which they do not appear but SiO and water masers do. Interestingly, the v = 1, J = 2 → 1 masers are found closer to the protostar in source I than are the J = 1 → 0 masers, a finding that is difficult to understand in the context of SiO maser pumping models (Doeleman et al. 2004). 2.6. Other species Few other new maser species or transitions have been reported in the literature since the last meeting. The first (J,K) = (6, 6) ammonia (NH3) maser has been detected centered on a millimeter peak in NGC 6334 I (Beuther et al. 2007). Weakly inverted acetaldehyde (CH3CHO) has been detected in the 111 → 110 transition at 1065.075 MHz toward Sgr B2 (Chengalur & Kanekar 2003). 3. Multi-species associations Methanol, OH, and water masers are frequently in the same source, although water and methanol masers usually originate in different regions (Beuther et al. 2002; Caswell 2004b; Edris et al. 2005; Szymczak, Pillai, & Menten 2005). Most 6.7 GHz methanol maser sources have associated OH masers, almost always at 1665 MHz and frequently at 1667 MHz as well (Szymczak & Gérard 2004), while the correlation between 6.7 GHz methanol masers and 22 GHz water masers is less strong (e.g., Breen et al. 2007). The distribu- tions of 6.7 GHz methanol masers and 6.0 GHz OH masers in W3(OH) are very similar, although direct overlap of the two species is rare (Etoka, Cohen, & Gray 2005). Simi- lar phenomena are also seen between 6.7 GHz methanol and 1.6/4.7 GHz OH masers (Harvey-Smith & Cohen 2006). All three species correlate more strongly with mid infrared emission than centimeter or near infrared emission, and all are frequently found in linear groupings (De Buizer et al. 2005). However, the luminosity of water masers correlates less strongly with far infrared lumi- nosity than is the case for methanol and OH masers, likely because water masers are not predominantly pumped by infrared photons (Szymczak, Pillai, & Menten 2005), al- though they may not be pumped entirely by collisions either (Liu, Forster, & Sun 2005; Liu et al. 2007). In any case, the existence of either masers (water and methanol) or out- flows towards a UCH ii region is an excellent predictor of the other (Codella et al. 2004), indicating that both masers and outflows are usually detectable in the UCH ii phase. 6 Fish 4. Observational advances 4.1. Proper motions and geometric distances Evidence in support of the kinematic interpretation of maser motions continues to pile up, with reports of the persistence of spot shapes in methanol (Moscadelli et al. 2002) and water masers (Goddi et al. 2006) and the inferred average overdensity of masers as compared to non-masing material (Fish, Reid, & Menten 2005; Fish et al. 2006a). In ad- dition to tracing internal source motions, maser proper motions can be used to obtain geometric parallax distances and measurements of Galactic rotation. Recent years have seen this technique used for both methanol and water masers in W3(OH) to obtain dis- tances accurate to a few percent (Xu et al. 2006; Hachisuka et al. 2006). Further details can be found in proceedings in the Galactic structure session. 4.2. Spectral resolution Very high spectral line observations at VLBI (very long baseline interferometry) spa- tial resolution have been obtained toward masers in several species in star forming re- gions. Detailed line profile analyses of water masers conclude that the near-Gaussian lineshapes indicate that they occur in hot (∼ 1200 K) gas with small beaming angles (Watson, Sarma, & Singleton 2002). Similar observations of 12.2 GHz methanol masers (Moscadelli et al. 2003) and the ground-state quartet of OH in W3(OH) (Fish, Brisken, & Sjouwerman 2006) find similarly Gaussian spectral profiles as well as maser spot positional gradients as a function of velocity (equivalently, velocity gradients). The positional gradients show no clear large-scale spatial organization but have similar magnitudes in methanol and three of the four OH transitions. It is possible that these positional gradients represent turbu- lent motions on very small spatial scales. If so, it is important to understand the char- acteristics of the turbulence, since observed maser properties, including variability, can be highly sensitive to turbulence in the masing region (Böger, Kegel, & Hegmann 2003; Sobolev, Watson, & Okorokov 2003; Silant’ev et al. 2006). Similar polarization charac- teristics in Class II methanol features at different velocities may indicate that velocity gra- dients induce velocity redistribution (Wiesemeyer, Thum, & Walmsley 2004), which may play a critical role in preventing saturated rebroadening (e.g., Nedoluha & Watson 1988). 4.3. Infrared pumping lines Molecular infrared transitions observations, particularly of OH, are essential for measur- ing maser pump efficiencies and may help place observational constraints on the radiative pump cycles of some models (e.g., Gray 2007). Archival Infrared Space Observatory data have been searched for the 34.6 and 53.3 µm pumping lines of OH with limited success, due to the low spectral resolution of the instruments (He & Chen 2004; He 2005). Her- schel and SOFIA will have the spectral resolution and frequency coverage required to observe pumping lines of OH as well as the critical 560 µm line of methylidyne (CH). 5. Further remarks Two quotes from De Buizer et al. (2005) serve to summarize the common themes of the observations over the past five years. The first is that “maser emission in general can trace a variety of phenomena associated with massive stars including shocks, outflows, infall, and circumstellar disks. No one maser species is linked exclusively to one particular process or phenomenon.” Indeed, while certain maser species may preferentially turn up in a particular context (possibly a result of observational biases), the set of all observa- tions of any one maser species resist being pigeonholed into a particular phenomenon. Masers and star formation 7 The second quote is that water, OH, and methanol masers “do not seem to be associated with different early evolutionary stages of massive stars. Instead it appears that they all trace a variety of stellar phenomena throughout many early stages of massive stellar evolution.” As is clear from §3, different maser species are commonly found together, independent of the evolutionary stage of the source. While proposed sequences in which certain masers turn on before others may be useful for statistical evaluation of evolu- tionary phases, important exceptions to such sequences exist. Those oddball masers that do not seem to fit present standard paradigms should be studied in especial detail, since we cannot predict beforehand what will be thereby learned about their environment or about maser processes in general. It was only a few years ago that Ellingsen (2004) referred to masers as “the Bart Simp- son of star formation research,” noting that they are “under-achievers” in comparison with masers in other environments due to the lack of sensitive, high resolution observa- tions at complementary wavelengths. While this may once have been true, recent maser observations have made great advancements in probing a wide range of dynamic struc- tures relevant to star formation. Maser VLBI allows observations of small- and large-scale morphologies, magnetic fields, and motions on AU scales and is showing great promise as a tool to trace Galactic structure. Maser models for some species are becoming suf- ficiently refined to provide good constraints on physical conditions. The community is beginning to appreciate the role of turbulence and the ability to probe its properties using maser observations. Synergies with mid infrared instruments have clarified many of the mysteries of linear structures with velocity gradients. We have entered the era of greatly improved far infrared instrumentation, and the ALMA (Atacama Large Millime- ter Array) era, with unprecedented sensitivity and angular resolution at submillimeter wavelengths, will begin in a few years. Further advancements in radio instrumentation, including new space VLBI missions and the SKA (Square Kilometre Array), will provide even greater insights. It is perhaps more correct to state that maser observations are at the vanguard of star formation research: yesterday’s observations can be explained by complementary data and theory today, and today’s observations lay the groundwork for the breakthroughs that will be achieved in the context of tomorrow. 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0704.0243

Renormalized quasiparticles in antiferromagnetic states of the Hubbard model

EPJ manuscript No. (will be inserted by the editor) Renormalized quasiparticles in antiferromagnetic states of the Hubbard model J. Bauer and A.C. Hewson Department of Mathematics, Imperial College, London SW7 2AZ, United Kingdom November 4, 2018 Abstract. We analyze the properties of the quasiparticle excitations of metallic antiferromagnetic states in a strongly correlated electron system. The study is based on dynamical mean field theory (DMFT) for the infinite dimensional Hubbard model with antiferromagnetic symmetry breaking. Self-consistent solutions of the DMFT equations are calculated using the numerical renormalization group (NRG). The low energy behavior in these results is then analyzed in terms of renormalized quasiparticles. The parameters for these quasiparticles are calculated directly from the NRG derived self-energy, and also from the low energy fixed point of the effective impurity model. From these the quasiparticle weight and the effective mass are deduced. We show that the main low energy features of the k-resolved spectral density can be understood in terms of the quasiparticle picture. We also find that Luttinger’s theorem is satisfied for the total electron number in the doped antiferromagnetic state. PACS. 7 1.10.Fd,71.27.+a 1 Introduction The nature of the metallic antiferromagnetically ordered state in strongly correlated systems has been subject of study for over two decades, but still remains to be fully un- derstood. Interest in this topic has been stimulated by the fact that the high temperature superconductivity of the cuprates emerges from the doping of an antiferromagnetic insulating compound, such as La2CuO4 [1,2]. The sim- plest models to describe the electrons in the CuO2 planes of the cuprates are the two dimensional t-J-model and Hubbard model. Much of the initial effort went into the study of a single hole state of these models in an antifer- romagnetic background. For the motion of this hole, there is a competition between the gain in kinetic energy from the hopping and its disruptive effect on the antiferromag- netic order, and consequent loss of potential energy. As a result a hole excitation becomes a quasiparticle or mag- netic polaron, heavily dressed by antiferromagnetic spin fluctuations (see review article by Dagotto [3] and refer- ences therein). Much of this work, however, relied on exact diagonalization or quantum Monte Carlo methods, which are limited to small clusters and very few hole excitations, and cannot be readily extended to study the many-hole, finite doping situation. More recent studies capable of describing finite dop- ing have concentrated on the relation between the antifer- romagnetic fluctuations and superconducting order (for a review see [4] and the references therein). One of the main motivations is to understand whether the exchange of these types of fluctuations can provide a purely elec- tronic mechanism for inducing superconductivity. Here, in this paper, we focus on the metallic antiferromagnetism, the doped state with long range antiferromagnetic order. Our interest is to examine how well the low energy ex- citations in this ordered state can be described in terms renormalized quasiparticles. To tackle this problem we use the infinite dimensional Hubbard model. The simplification in the infinite dimensional limit is that the electron self-energy becomes local in character, with no wavevector dependence [5,6]. The self-energy then depends only on the frequency, as is the case for impurity models, allowing the lattice problem to be cast in the form of a self-consistent impurity model. There are several rea- sonably accurate techniques for solving this effective im- purity problem, a very accurate one for the zero and low temperature regime being the numerical renormalization group approach (NRG) [7]. Recently we studied the effect of a magnetic field on the quasiparticle excitations in the strong correlation regime of the infinite dimensional Hubbard model using the NRG method [8]. We also extended a form of renormalized per- turbation theory (RPT), originally developed for impu- rity models [9], to this model, and used it to calculate the local dynamic spin susceptibilities, obtaining results in good overall agreement with those from the NRG. In this paper we extend this combination of renormalization techniques, NRG and RPT within dynamical mean field theory, to look at the low energy excitations of the infi- nite dimensional Hubbard model in a staggered field, and in antiferromagnetic broken symmetry states. Extensive calculations of the antiferromagnetic states in the Hub- http://arxiv.org/abs/0704.0243v2 2 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model bard model using the DMFT-NRG approach have already been reported in the paper of Zitzler, Pruschke and Bulla [10]. We confirm their results for the phase diagram and extend the calculations and analysis to the description of the low energy renormalized excitations, and how these can be described within the framework of a renormalized perturbation theory. 2 Antiferromagnetic Broken Symmetry in In considering the response of the Hubbard model [11] to a staggered magnetic field and antiferromagnetic order, we take the case of a bipartite lattice, which consists of two sublattices A and B such that the nearest neighbors of a site in the A sublattice are on the B sublattice and vice versa. The Hamiltonian for the Hubbard model can be written in the form, i,j,σ (tijc A,i,σcB,j,σ + h.c.) + U nα,i,↑nα,i,↓ A,i,σcA,i,σ + µ−σc B,i,σcB,i,σ), (1) where the hopping matrix element is taken as tij = −t between nearest sites i and j only, and zero otherwise, and α = A,B. A staggered field His His = H for i ∈ A sublattice −H for i ∈ B sublattice (2) has been included so that µσ = µ+σh, where h = gµBH/2 with the Bohr magneton µB. The non-interacting part of the Hamiltonian H0,µ can be diagonalized in terms of Bloch states and then expressed in the form, H0,µ = k,σMk,σCk,σ. (3) where C k,σ = (c A,k,σ, c B,k,σ), and the matrix Mk,σ is given by Mk,σ = −µσ εk εk −µ−σ . (4) The k sums run over a reduced Brillouin zone, and the energy of the Bloch state is εk = j tije i(Ri−Rj)·k. The free Green’s function matrix G0k,σ(ω) is given by (ω − Mk,σ) −1. The poles of the free Green’s function give the elementary single particle excitations, which are given by E0k,±(U = 0) = −µ0(h)± h2 + ε2 , (5) where µ0(h) is the chemical potential of the noninteracting system in a staggered field. This illustrates that the elec- tronic excitations are split into two subbands for a finite staggered field. Notice that we have adopted a special choice of ba- sis {cA,k,σ, cB,k,σ} here [12,10]. Another common basis to study antiferromagnetic and spin density wave symmetry (SDW) breaking is {ck,σ, ck+q0,σ}, where q0 is the recip- rocal lattice vector for commensurate SDW ordering. The bases can be related by a linear transformation, k+q0,σ cA,k,σ cB,k,σ . (6) For the latter basis the matrix Mk,σ would be diagonal in the kinetic energy term and the symmetry breaking is offdiagonal. For our study in the DMFT framework the A−B-sublattice basis is, however, more convenient and we will use it throughout the rest of this paper. It is possible, of course, to relate the obtained quantities with the help of (6) to the {ck,σ, ck+q0,σ} basis. We can generalize the equations to the interacting prob- lem by introducing a self-energy Σα,k,σ(ω), so that the matrix Green’s function can be written in the form Gk,σ(ω)= ζA,k,σ(ω)ζB,k,σ(ω)− ε2k ζB,k,σ(ω) −εk −εk ζA,k,σ(ω) where ζα,k,σ(ω) = ω + µσ − Σα,k,σ(ω). As we are dealing with the infinite dimensional limit of the model, we take the self-energy to be local so we can drop the k index. This is the reason why the self-energy has a single site index α = A,B and no offdiagonal terms appear in equation (7). The symmetry of the bipartite lattice gives ΣB,σ(ω) = ΣA,−σ(ω) ≡ Σ−σ(ω) and hence ζB,−σ(ω) = ζA,σ(ω) ≡ ζσ(ω), where we have simplified the notation. To determine these quantitiesΣσ(ω) it is sufficient to focus on the A sublattice only. Summing the first component in the Green’s function in equation (7) over k we obtain the Green’s function for a site on the A sublattice, Glocσ (ω), Glocσ (ω) = ζ−σ(ω) ζσ(ω)ζ−σ(ω) ρ0(ε) ζσ(ω)ζ−σ(ω)− ε , (8) where ρ0(ε) is the density of states of the non-interacting system in the absence of the staggered field. In the DMFT this local Green’s function, and the self- energy Σσ(ω), are identified with the corresponding quan- tities for an effective impurity model [12]. This implies that the Green’s function G0,σ(ω) for the effective impu- rity in the absence of an interaction at the impurity site is given by G−10,σ(ω) = Glocσ (ω)−1 +Σσ(ω). (9) We can take the form of this impurity model to correspond to an Anderson model [13] in a magnetic field, HAM = εd,σd σdσ + Und,↑nd,↓ (10) (Vk,σd σck,σ + V k,σdσ) + εk,σc k,σck,σ, J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 3 where εd,σ = εd−σh is the energy of the localized level at an impurity site in a magnetic fieldH , U the interaction at this local site, and Vk,σ the hybridization matrix element to a band of conduction electrons of spin σ with energy εk,σ. As we are focusing on an A site as the impurity we take H = Hs. The one-electron Green’s function for the impurity site of this model is given by Gimpσ (ω) = ω − εdσ −Kσ(ω)−Σσ(ω) , (11) where Kσ(ω) = |Vk,σ|2 ω − εk,σ . (12) If this impurity Green’s function is equated to the local lattice Green’s function Glocσ (ω), we identify εdσ = −µσ and from equation (9), Kσ(ω) is given by Kσ(ω) = ω + µσ − G−10,σ(ω). (13) The function Kσ(ω) plays the role of the effective medium and has to be calculated self-consistently. The self-consistent calculations for Kσ(ω) can usually be performed iteratively. Starting from a conjectured form for Kσ(ω), the NRG method is used to calculate the self- energy of the effective Anderson model, from which the impurity Green’s function Gimpσ (ω) in (11) and the local Green’s function for the lattice Glocσ (ω) in (8) can be de- duced. If these two Green’s functions do not agree, then equation (9) is used to derive a new starting value for Kσ(ω) and the process continued until self-consistency is achieved. To find antiferromagnetic solutions, we calculated self- consistent solutions for a decreasing sequence of staggered magnetic fields to see if broken symmetry solutions of this type exist as the staggered field is reduced to zero. For the non-interacting density of states ρ0(ε) we take the Gaussian form ρ0(ε) = e −(ε/t∗)2/ πt∗, corresponding to an infinite dimensional hypercubic lattice. It is useful to define an effective bandwidth W = 2D for this density of states via D, the point at which ρ0(D) = ρ0(0)/e 2, giving 2t∗ corresponding to the choice in reference [14]. In all the results we present here we take the value W = 4. In the NRG calculations we have used the improved method [15,16] of evaluating the response functions with the complete Anders-Schiller basis [17], and also determine the self-energy from a higher order Green’s function [18]. In figure 1 we show the self-consistently calculated lo- cal spectral density for the spin-up (upper panel) and spin- down electrons (lower panel) at an A site with U = 3 and 5% hole doping (from the state at half-filling) for various values of an applied staggered field. The staggered mag- netic field induces a sublattice magnetization, (nA,↑ − nA,↓), (14) so that these spectra are quite different. For this set of parameters, this difference persists as the staggered field −4 −2 0 2 4 h =0.05 h =0.1 −4 −2 0 2 4 h =0.05 h =0.1 Fig. 1. (Color online) The spectral densities for the spin-up electrons (upper panel) and spin-down electrons (lower panel) at the A site for various values of the applied staggered field for U = 3 and x = 0.95 is reduced to zero so that we have a spontaneous sub- lattice magnetization corresponding to spontaneous anti- ferromagnetic order. For the case away from half filling, δ 6= 0, we have to keep adjusting the chemical potential when iterating for a self-consistent solution. It shows a slightly oscillatory behavior when iterating for a specific filling x, and we follow the procedure described in refer- ence [10]. This feature is related to the fact that the calcu- lations are for a metastable ground state and instabilities to more complicated ground states for antiferromagnetic ordering than the homogeneous, commensurate Néel state, which forms the basis for these DMFT calculations, can occur [19,20,21,22,23,24,25,26,10]. As far as phase sep- aration in the ground state is concerned, the results of our calculations are generally in line with the conclusions in [10] as they are carried out within the same frame- work. The focus of this work is, however, the analysis of generic quasiparticle properties in a doped antiferromag- netic state. We consider the approach as a valid, approxi- mate starting point for this endeavor, but modifications to the results presented here can occur for calculations based 4 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model on a more complicated ground states not accessible within the DMFT framework. For a more extensive discussion of the applicability of the DMFT in this situation we refer to the earlier work [10]. From results of this type of calculation, we have built up a global antiferromagnetic/paramagnetic phase dia- gram as a function of the doping δ and the on-site inter- action U . This phase diagram is shown in figure 2, where the value of the corresponding sublattice magnetization is shown in a false color plot. We have added a line separat- ing the spontaneously ordered and paramagnetic regimes. 0 0.05 0.1 0.15 0.2 0.1 0.2 0.3 0.4 Fig. 2. (Color online) Phase diagram showing the doping and the U dependence of the sublattice magnetization mA as de- duced from the DMFT-NRG calculations. At half filling (δ = 0 axis) the spontaneous magnetiza- tion increases with U . We can see that the antiferromag- netic order from the half filled case persists when holes are added. The value of the critical doping δc at which the antiferromagnetism disappears depends on the on-site interaction U . We expect that for small U the critical dop- ing δc will increase with U since a tendency to order only appears when an on-site interaction is present. From the mapping to the t−J model we also expect that for large U the antiferromagnetic coupling J decreases and therefore the order is destroyed more easily. The values of U are, however, not large enough to display this trend. If we compare these results with the phase diagram given by Zitzler et al. [10] we see that they are in very good agreement. In their case the antiferromagnetic region was mapped out to values of U ≃ 4.5. As the iterations tend to oscillate, as discussed before, there is a problem of ob- taining a self-consistent antiferromagnetic solution in the large U regime. We have managed to extend the diagram to somewhat larger values of U by stabilising the calcula- tions by averaging the effective medium over a number of iterations. 3 Local Quasiparticle Parameters To examine the nature of the low energy excitations, we will assume that the self-energy Σσ(ω) is non-singular at ω = 0 so that, at least asymptotically, it can be expanded in powers of ω. This assumption is not expected to be valid close to the quantum critical point when the magnetic order sets in, but to be a reasonable assumption otherwise. We also assume that the imaginary part of the self-energy vanishes which is confirmed by the numerical results of the DMFT-NRG calculations. We will retain terms to order ω only for the moment. The higher order corrections will be considered later. We then find for ζσ(ω), ζσ(ω) = ω(1−Σ′σ(0)) + µσ −Σσ(0) (15) = z−1σ (ω + µ̃0,σ), (16) where µ̃0,σ = zσ(µ−Σσ(0)), and z−1σ = 1−Σ′σ(0). (17) The interacting Green’s function (7) has poles at the roots of the quadratic equation, ζσ(ω)ζ−σ(ω)− ε2k = 0. (18) The solutions of this equation are E0k,± = −µ̃± +∆µ̃2, (19) where ε̃k = z↑z↓εk, ∆µ̃ = (µ̃0,↑ − µ̃0,↓)/2, and µ̃ = (µ̃0,↑ + µ̃0,↓)/2. This has the same form as for the non- interacting system in a staggered field (5), so we can in- terpret these excitations as quasiparticles coupled to an effective staggered magnetic field h̃s = ∆µ̃/gµB, with µ̃ playing the role of a quasiparticle chemical potential. This equation gives the dispersion relation for these single par- ticle excitations, which can be regarded as constituting a renormalized band, or bands as there are two branches. The term magnetic polaron is sometimes used to describe these single particle excitations in states of magnetic or- der, because of the analogy with the motion of a particle in a lattice to which it is strongly coupled, where the ex- citation is termed a polaron. The corresponding density of states of these free local quasiparticles on the sublattice is ρ̃0,σ(ω)= ω + µ̃− σ∆µ̃ ω + µ̃+ σ∆µ̃ (ω + µ̃)2 −∆µ̃2 for |ω + µ̃| > |∆µ̃|, and is zero otherwise. In the case of a half-filled band µ̃ = 0 and there is a gap at the Fermi level εF = 0. To determine this local quasiparticle density of states in the presence of the symmetry breaking staggered mag- netic field we need to calculate zσ and µ̃0,σ for each spin type. Using the NRG we can do this in two ways. As the DMFT-NRG calculations give us the self-energyΣσ(ω) di- rectly, we only need its value, and that of its first derivative J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 5 at ω = 0, to deduce both zσ and µ̃0,σ using equation (17). However, because the model is solved using an effective impurity model, we can also deduce these quantities indi- rectly from the many-body energy levels of the impurity on approaching the low energy fixed point [27]. This sec- ond method gives us not only a check on the results of the direct method, but also allows to deduce some informa- tion about the quasiparticle interactions, as we shall show in the next section. 3.1 Calculation of Renormalized Parameters To describe how the renormalized parameters are deduced from the energy levels of the NRG calculation, we need to outline how the NRG calculations are carried out. Fol- lowing the procedure introduced by Wilson [28], the con- duction band is logarithmically discretized and the model then converted into the form of a one dimensional tight binding chain, coupled via an effective hybridization Vσ to the impurity at one end. In this representation Kσ(ω) = |Vσ|2g(N)0,σ (ω), where g 0,σ (ω) is the one-electron Green’s function for the first site of the isolated conduction elec- tron chain of length N . The impurity Green’s function for this discretized model then takes the form, Gimpσ (ω) = ω − εdσ − |Vσ|2g(N)0,σ (ω)−Σσ(ω) . (21) We can find the quasiparticle excitations of this model by expanding the self-energy Σσ(ω) in the denominator of this equation to first order in ω, and write the result in the form, Gimpσ (ω) = ω − ε̃dσ − |Ṽσ |2g(N)0,σ (ω) +O(ω2) , (22) where ε̃dσ = zσ[εdσ +Σσ(0)], |Ṽσ|2 = zσ|Vσ|2. (23) We can then define a free quasiparticle propagator, G̃0,σ(ω), 0,σ (ω) = ω − ε̃dσ − |Ṽσ |2g(N)0,σ (ω) , (24) and interpret zσ as the local quasiparticle weight. In the NRG calculation the many-body excitations are calculated iteratively, starting at the impurity site, and increasing the chain length N by one site with each itera- tion. When the matrices become too large to handle, only the lowest 500-1500 states are kept at each iteration. The many-body energy levels for the Nth iteration and the set of quantum numbers M , EM (N), depend on the chain length N and the discretization parameter Λ > 1. When N becomes large these energy levels go to zero as Λ−N/2. We now conjecture that the lowest single particle Eσp (N) and single hole excitations Eσh (N) determined from the NRG many-body excitations correspond to quasiparticle excitations. If this is the case then they should correspond to the poles of the quasiparticle Green’s function given in equation (24), with values of Ṽσ and ε̃dσ, which are inde- pendent of N as N → ∞. We can test this hypothesis by substituting the values, ω = Eσp (N) and ω = E h (N), into the equation, ω − ε̃dσ − |Ṽσ|2g(N)0,σ (ω) = 0, (25) and deduce values of Ṽσ and ε̃dσ, which will in general depend upon N . From these we can deduce zσ = |Ṽσ/Vσ|2 and µ̃0,σ = −ε̃dσ, which will also depend upon N , but if the lowest single particle excitations of the system do cor- respond to free quasiparticles, the values of zσ and µ̃0,σ will become independent of N for large N . It should be noted that we need both the particle and hole excitations for each spin to determine the four renormalized parame- ters. The parameters corresponding to spin up involve the particle excitations with spin up and the hole excitations with spin down. That parameters can be found, which are independent ofN for largeN , can be seen in figure 3, where we take the results of a Kσ(ω) and µσ from the antiferromagnetic self- consistent solution for the Hubbard model with U = 3 and 10% doping, using a value for the discretization parameter Λ = 1.8. 0 10 20 30 40 50 Fig. 3. (Color online) The N-dependence of the renormalized parameters zσ and µ̃0,σ for U = 3 and x = 0.9. It can be seen that after about 25 iterations all the values deduced for zσ and µ̃0,σ become independent of N . In the next section, where we compare these results with the cor- responding values deduced directly from the self-energy, we get further confirmation that the values deduced re- ally do describe the quasiparticle excitations of the lattice model. When two or more quasiparticles are excited from the interacting ground state, there will be an interaction be- tween them. For the Anderson impurity model this inter- action will be local and can be expressed as Ũ , a renor- malized value of the original interaction of the ‘bare’ par- 6 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model ticles. The value of Ũ can be deduced by looking at low- est lying two-particle excitations derived from NRG cal- culation. These could either be two-particle excitations, E↑,↓pp (N), two-hole excitations, E hh (N) or a particle-hole excitation E ph (N). By looking at the difference between a two-particle excitation and two single particle excita- tions, E↑,↓pp (N) − E↑p(N) − E↓p(N), as a function of N we can deduce an effective interaction Ũ↑,↓pp (N) between these two quasiparticles, as has been described fully ear- lier for the standard Anderson model [27]. In a similar way we can deduce an effective interaction between two holes, hh (N), or a particle and hole, −Ũ ph (N). To be able to define a single quasiparticle interaction Ũ , not only must Ũ↑,↓pp (N), Ũ hh (N) and Ũ ph (N), give values which are in- dependent of N for large N , these values must be equal so Ũ↑,↓pp = Ũ hh = Ũ ph = Ũ . 0 10 20 30 40 50 Fig. 4. (Color online) The N-dependence of the renormal- ized particle-particle, particle-hole and hole-hole interactions for U = 6 and x = 0.9, showing that they converge to a unique value Ũ . In figure 4 we give the values of Ũ↑,↓pp (N), Ũ hh (N) and ph (N) as deduced from DMFT-NRG calculation for the Hubbard model in an antiferromagnetic state with U = 6, 10% doping and Λ = 1.8. It can be seen that the three sets of results settle down to a common value Ũ . We can go further and identify Ũ with the local quasi- particle 4-vertex interaction for the effective impurity model, Ũ = z↑z↓Γ↑,↓,↓,↑(0, 0, 0, 0), (26) where Γ↑,↓,↓,↑(ω1, ω2, ω3, ω4) is the total 4-vertex at the impurity site, which is equal to the same quantity for a site in the lattice model. With this interpretation it is possible to identify these parameters with those used in a renormalized perturbation expansion. The parameters, V , εd,σ and U , together with g 0,σ(ω), specify the effective im- purity model. The renormalized parameters, Ṽ , ε̃d,σ and Ũ , together with gN0,σ(ω), can be used as an alternative way of specifying this model. The renormalized perturba- tion theory (RPT) is set up by expanding the self-energy to order ω, as earlier, but retaining all the higher order correction terms in a remainder term, Σσ(ω) = Σσ(0) + ωΣ σ(0) +Σ σ (ω), (27) where Σremσ (ω) is the remainder term. On substituting this into the equation for the impurity Green’s function in equation (11), we can deduce a general expression for the quasiparticle Green’s function in the form, G̃impσ (ω) = ω − ε̃dσ − K̃σ(ω)− Σ̃σ(ω) , (28) where K̃σ(ω) = zσKσ(ω) and Σ̃σ(ω) = zσΣ σ (ω) plays the role of a renormalized self-energy. A diagrammatic perturbation theory can then be carried out for Σ̃σ(ω) in terms of the free quasiparticle propagators, with ad- ditional diagrams arising from counter terms, which are required to prevent over-counting (renormalization condi- tions) [29,9,30]. This form of perturbation theory is valid for all energy scales but is particularly effective for calcu- lating the low energy terms arising from the quasiparticle interactions. For the symmetric Anderson impurity model it has been shown that this perturbation theory taken to second order in Ũ , gives the exact spin and charge suscep- tibilities at T = 0, and the exact T 2 contribution to the conductivity [9]. Because, within DMFT, the self-energy for the lat- tice is the same as that for the effective impurity, we can equally well use the effective impurity model to calculate it. This means that we can use the renormalized pertur- bation theory for the effective impurity model to estimate the correction terms to the free quasiparticle picture aris- ing from the quasiparticle interactions. 3.2 Local Quasiparticle Weight We now consider the values of the local quasiparticle weight factor zσ, commonly known also as the wavefunction renor- malization factor. This is an important factor in deter- mining the parameters needed to describe the low energy behavior of the system. When there is no k-dependence of the self-energy, as is the case for infinite dimensional models and DMFT, the effective mass of the quasiparti- cles in the paramagnetic state is proportional to 1/zσ. We show later that in the antiferromagnetic state the expres- sion is more complicated and depends both on zσ and the renormalized chemical potential µ̃0,σ. We have determined zσ from the NRG results by the two methods described and give the values of zσ deduced for both spin types as a function of doping in figure 5. The results are for the case U = 3, where there is antiferromagnetic order and the ex- ternal staggered field has been set to zero. It can be seen that there is a reasonable agreement between the values obtained by the two different methods of calculation. Visi- ble differences can be attributed to the inaccuracies when J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 7 0 0.05 0.1 0.15 0.2 from Σ from Σ from FP from FP Fig. 5. (Color online) The local quasiparticle weight zσ as de- duced directly from the self-energy and also from the impurity fixed point (FP) for U = 3 and various dopings. numerically computing the derivative of the self-energy, whose calculation involves a broadening procedure. When the system is doped but still ordered we have z↑ 6= z↓, and the renormalization effects are stronger for the minority (down) spin particles on the sublattice. This is similar to the results we found for a doped Hubbard model in a para- magnetic state in the presence of a strong uniform mag- netic field [8]. For a certain range of dopings the values of z↑ and z↓ do not vary much. The tendency is that z↓ first decreases and later increases, whereas z↑ decreases over the whole range until both of them merge at the doping point where the antiferromagnetic order disappears. The results for the corresponding case with U = 6, a value which is larger than the bandwidth, are shown in figure 6. 0 0.05 0.1 0.15 0.2 from Σ from Σ from FP from FP Fig. 6. (Color online) The local quasiparticle weight zσ as deduced directly from the self-energy and from the impurity fixed point (FP) for U = 6 and various dopings. On the whole the behavior is quite similar to that for the case U = 3, only that the renormalization effects are more pronounced. For a range of dopings the local quasiparticle weights do not change much and have the same tendency as described above. The implications for the spectral quasiparticle weight and the effective mass enhancement will be discussed in detail later. 3.3 Renormalized chemical potential In figure 7 we give the results for the renormalized chemi- cal potential, µ̃0,σ [defined in equation (17) and (23)], for the two spin types in the spontaneously ordered antifer- romagnetic states for U = 3 and U = 6 for a range of dopings. 0 0.05 0.1 0.15 from Σ from Σ from FP from FP 0 0.05 0.1 0.15 from Σ from Σ from FP from FP Fig. 7. (Color online) The renormalized chemical potential µ̃0,σ as deduced directly from the self-energy and from the impurity fixed point (FP) for various dopings for U = 3 (upper panel) and U = 6 (lower panel). The values calculated by the two different methods can be seen to be in good agreement here, as well. We have added the values for the half filled case. These were calculated from the self-energy in the gap at ω = 0. The general 8 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model behavior of the values for µ̃0,σ for the case with U = 6 is very similar to the case with smaller U The renormalized chemical potential µ̃0,σ is an impor- tant parameter in specifying the form of the local sublat- tice quasiparticle spectral density ρ̃0σ(ω). From equation (20) it can be seen that, as ω → −µ̃0,σ, ρ̃0,σ(ω) behaves asymptotically as ρ̃0,σ(ω) ∼ ω + µ̃0,σ , (29) so the quasiparticle density of states has a square root singularity at ω = −µ̃0,σ. On the other hand, however, as ω → −µ̃0,−σ, ρ̃0,σ(ω) behaves as ρ̃0,σ(ω) ∼ ω + µ̃0,−σ, (30) so the quasiparticle density of states goes to zero at ω = −µ̃0,−σ. Between the two points, ω = −µ̃0,σ and ω = −µ̃0,−σ, the quasiparticle density of states has a gap of magnitude 2∆µ̃. As can be seen in figure 7 this free quasi- particle gap decreases with increasing doping and closes in the paramagnetic state. If we take into account the values at half filling we see a strong reduction of 2∆µ̃, when doping the system. We also see that µ̃0,↑ drops to small negative values for finite hole doping, which corre- sponds to the fact that the Fermi level then lies within the lower band. These features will be seen clearly in the fig- ures presented in the next section, where we compare the quasiparticle densities of states with the full local spectral densities calculated from the DMFT-NRG. 3.4 The Quasiparticle Interaction The quasiparticles can be further characterized by an ef- fective interaction Ũ as described before. In figure 8 we plot the doping dependence of the renormalized interac- tion over a range of dopings and U = 3 and U = 6. We can see that in both cases the values decrease with increasing doping. Hence, the effective quasiparticle inter- action is stronger for a smaller hole density. For a certain range of dopings, however, Ũ does not vary much. We can also see that the ratio Ũ/U for the effective interaction as- sume smaller values the larger the bare U becomes. Also the absolute value of Ũ , i.e. without the scaling with U as in figure 8, is smaller for larger bare U for the full range of dopings. This effect of smaller quasiparticle interactions for the stronger coupling case can be seen as sharper quasi- particle peaks for larger U as will be discussed in the next section. 4 Spectra and Quasiparticle Bands 4.1 Local Spectra In this section we examine how well the local sublattice quasiparticle density of states ρ̃0,σ(ω), evaluated from equa- tion (20) with the renormalised parameters, describes the 0.05 0.1 0.15 U / U for U =3 U / U for U =6 Fig. 8. (Color online) The renormalised quasiparticle interac- tion Ũ/U as deduced from the impurity fixed point for various dopings and U = 3, 6. low energy features seen in the local spectral density ρσ(ω) calculated from the DMFT-NRG. At half filling there is a gap at the Fermi level, so there are no single particle excitations in the immediate neighbourhood of the Fermi level, and this is not a very interesting case to consider. We look in detail at the case of 10% doping where the Fermi level lies at the top of the lower band, and consider the two cases U = 3 and U = 6. In the upper panel of figure 9 we compare the spectral density ρ↑(ω) with the corresponding quantity z↑ρ̃0,↑(ω), from the quasiparticle density of states. We see that the behavior near the Fermi level (ω = 0), and the singular feature seen in the lower branch of ρ↑(ω), are well reproduced by the quasiparticle density of states. Above the Fermi level there is a peak in the quasiparti- cle density of states similar to that in the full spectrum but somewhat more pronounced. Above the Fermi level and below the upper peak there is a pseudo-gap region. In the free quasiparticle spectrum it is a definite gap. In the spectrum calculated from the direct NRG evaluation it ap- pears as a pseudo-gap, with rather small spectral weight just above the Fermi level. From the direct DMFT-NRG calculations, due to the broadening features introduced to obtain a continuous spectrum, it is not always possible to say definitively whether there is a true gap above the Fermi level or not. To resolve this question we can ap- peal to the renormalised perturbation theory to look at the corrections to the quasiparticle density of states aris- ing from the quasiparticle interactions. A calculation of the imaginary part of the renormalised self-energy Σ̃σ(ω) to order Ũ2 should be sufficient to settle this issue. The imaginary part of the second order diagram for the renor- malised self-energy in the limit T → 0 for ω > 0 is given J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 9 −2 −1 0 1 2 3 4 −2 −1 0 1 2 3 4 Fig. 9. (Color online) The free local quasiparticle spectrum (dashed line) in comparison with DMFT-NRG spectrum for x = 0.9 and U = 3 for the spin-up electrons (upper panel) and spin-down electrons (lower panel). ImΣ̃(2)σ (ω) = πŨ dε2 ρ̃0,σ(ε1)ρ̃0,−σ(ω − ε1 + ε2) × ρ̃0,−σ(ε2)θ(ω − ε1 + ε2), (31) where ρ̃0,σ(ε) is the free quasiparticle density of states. The integration area is a triangle in the (ε1, ε2)-plane as shown in figure 10. To analyze the behavior of ImΣ̃ σ (ω) in the regime |µ̃0,↑| < ω < |µ̃0,↓| we have to study where the integrand is non- zero taking into account that ρ̃0,σ(ε) = 0 for |µ̃0,↑| < ε < |µ̃0,↓|. The only non-zero contribution comes from the small shaded region in figure 10, which leads to the estimate, ImΣ̃(2)σ (ω) ≃ πŨ2ρ̃0,σ(0)ρ̃0,−σ(−ω)ρ̃0,−σ(0)µ̃20,↑. (32) When µ̃0,↑ is small, which occurs when the lower edge of the gap in the quasiparticle density of states is very near the Fermi level, this contribution to the imaginary part PSfrag replacements −ω + |µ̃0,↑| |µ̃0,↑| |µ̃0,↓| Fig. 10. (Color online) Integration region in the (ε1, ε2)-plane for the imaginary part of the self-energy. The original triangle region (0, ω,−ω) for integration in equation (31) is reduced in the gap region, |µ̃0,↑| < ω < |µ̃0,↓|, to the small shaded region shown in the figure. of the renormalized self-energy will be finite but small. It decreases with ω due the behavior of ρ̃0,−σ(−ω). Based on this argument we conclude that there is a small, but finite imaginary part of the self-energy in the free quasiparticle gap 2∆µ̃, when it lies above the Fermi level, giving rise to a finite spectral weight there. However, this spectral weight is very small close to the lower edge of the free quasiparticle density of states, when this edge lies only just above the Fermi level. In the lower panel of figure 9 we compare the ρ↓(ω) with z↓ρ̃0,↓(ω). We see that in this case also the quasi- particle density of states reproduces well the spectrum in the region of the Fermi level and the peak structure in the lower band, which is non-singular in this case. The position of the peak above the Fermi level is also well re- produced, but the peak in the free quasiparticle density of states is singular, whereas that in the DMFT-NRG re- sults is not. We would expect to lose this singularity in the free quasiparticle density of states once the quasiparticle scattering is taken into account and the renormalized self- energy is included. It is possible also that the peak above the Fermi level in the DMFT-NRG spectrum should be sharper, as there is some tendency for the broadening in- troduced in this approach to flatten peaked features in regions away from the Fermi level. The spectral weight in the pseudo-gap is even smaller than in the case for the spin-up electrons, particularly in the region of the gap that lies closest to the Fermi level. This is qualitatively in line with the conclusions based on the renormalized pertur- bation theory estimate of the effects of the quasiparticle scattering. We see very similar features in the spectra for the case U = 6 and also 10% doping shown in figure 11. Here, the peaks near the Fermi level are a bit sharper. The observations made on the comparison of the quasiparticle and DMFT-NRG spectra apply equally well to this case. In addition to the low energy features charge peaks cor- responding to the Hubbard bands appear. The lower one can be identified in the full spectra, whereas the upper 10 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model −2 −1 0 1 2 3 −2 −1 0 1 2 3 Fig. 11. (Color online) The free quasiparticle spectrum (dashed line) in comparison with DMFT-NRG spectrum for x = 0.9 and U = 6 for the spin-up electrons (upper panel) and spin-down electrons (lower panel). Hubbard peak is not seen on the energy scale shown. The quasiparticle density of states does not contain informa- tion about these features at higher energy. 4.2 k-resolved Spectra We can learn more about the low energy single parti- cle excitations by looking at the spectral density of the Green’s function Gk,σ(ω) in equation (7) for a given wave- vector k. With the self-energies Σσ(ω) calculated within the DMFT-NRG approach all elements of this matrix can be evaluated. The local spectra and self-energies are spin- dependent in the doped broken symmetry state, however, the free quasiparticle bands E0 k,± [equation (19)] do not depend on the spin. Here, we focus on the diagonal part of Gk,σ(ω) corresponding to the A sublattice, Gk,σ(ω) = ζ−σ(ω) ζσ(ω)ζ−σ(ω)− ε2k . (33) The weights of the quasiparticle excitations in this case depend on the spin corresponding to the sublattice prop- erties. We note that one can also analyze the quasiparticle bands differently, for instance, from the k-resolved spectra and the diagonal form of Gk,σ(ω). The form of the quasi- particle bands remains unchanged then, but the weights differ and do not depend on the spin σ in that case. We first of all look at the Fermi surface which is the locus of the k-points at the Fermi level (ω = 0) where the Green’s function has poles. The conduction electron energy εkF at these point is given by ε2kF = (µ↑ −Σ↑(0))(µ↓ −Σ↓(0)). (34) By Luttinger’s theorem, the volume of the Fermi sur- face for the interacting system must equal that for the non-interacting system with the same density. As the self- energy depends only on ω, the two Fermi surfaces must also have the same shape, and therefore must be identical. The Fermi surface of the non-interacting system is given by εkF = µ0, where µ0 is the chemical potential of the non-interacting system in the absence of any applied field for the given density. For this to be identical with that given in equation (34), (µ↑ −Σ↑(0))(µ↓ −Σ↓(0)) = µ20. (35) We can check that this relation indeed holds from our results for Σσ(ω) and µσ, independent of the value of U , or in the case of an applied staggered field, independent of the field value. This relation implies that the total number of electrons per site n can be calculated from an integral over the non-interacting density of states, n = 2 ρ0(ω)dω, (36) where in the hole doped case µ0 = − µ̄↑µ̄↓ and µ̄σ = µσ −Σσ(0). To relate this result to the quasiparticle picture, we expand the self-energy in equation (33) to first order in ω, but retain the remainder term, ΣRσ (ω) as in equation (27). The Green’s function can be rewritten in the form, G̃k,σ(ω) = ζ̃−σ(ω) ζ̃σ(ω)ζ̃−σ(ω)− ε̃2k , (37) where ζ̃σ(ω) = ω + µ̃0,σ − Σ̃σ(ω). We define a quasipar- ticle Green’s function G̃k,σ(ω) via zσG̃k,σ(ω) = Gk,σ(ω). The renormalized self-energy vanishes, Σ̃σ(ω) = 0, for the free quasiparticle Green’s function G̃ k,σ(ω), which can be separated into two independent branches of free quasipar- ticles, k,σ(ω) = uσ+(εk) ω − E0 uσ−(εk) ω − E0 , (38) where E0 k,± was defined in equation (19) and the weights are given by uσ±(εk) = 1∓ σ ∆µ̃√ ∆µ̃2 + ε̃2 . (39) J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 11 This is similar in form to mean field theory, which would correspond to putting zσ = 1, and ∆µ̃ = Ummf , where mmf is the mean field sublattice magnetization. The spin dependent contribution in (39) which arises from the sec- ond term is most marked in the region near the Fermi level. It should be noted that the quasiparticle excitations k,± and weights u ±(εk) here are defined by expanding the self-energy at ω = 0. This is so that they correspond to the free quasiparticles in the renormalized perturbation theory which have an infinite lifetime. The spectral density ρ̃ (ω) for this free quasiparticle Green’s function is a set of delta-functions, k,σ(ω) = u +(εk)δ(ω−E0k,+)+uσ−(εk)δ(ω−E0k,−). (40) On the Fermi surface E0k,− = 0, which is consistent with the result for the Fermi surface given in equation (34). Summing over k gives the local quasiparticle density of states in equation (20). We define the quasiparticle num- ber ñ as the integral of the sum of the spin up and spin down quasiparticle density of states up to the Fermi level, dω(ω + µ̃) (ω + µ̃)2 −∆µ̃2 (ω + µ̃)2 −∆µ̃2 If we change the variable of integration to ω′, where z↑z↓ = (ω + µ̃)2 −∆µ̃2, the integration can be shown to be identical with that in equation (36), using the fact that µ0 = − µ̄↑µ̄↓. We then have an alternative statement of Luttinger’s theorem in the form ñ = n. This can also be found by summing both spin components in (40), integrating over ω and then converting the k-summation to an integral over the free electron density of states ρ0(ω). We can check in our nu- merical results that the relation in this form holds. The occupation number n can be calculated both from a direct evaluation of the number operator in the ground state, and also by integrating the sum of the spectral densities ρσ(ω) of the full local Green’s function to the Fermi level. The value of ñ is similarly determined from the integral over the total quasiparticle density of states, ρ̃σ(ω). All three results were found to be in good agreement, to within one or two percent deviation at the most. Before discussing the k-resolved spectra in detail we would like to ask what the spectral weight wqp of a quasi- particle excitation at the Fermi level in the lower band is, such that the Green’s function reads there Gqp(ω) = ω − E0 . (42) To calculate wqp, we can not focus on the spin depen- dent local sublattice quantities, but have to sum over both sublattices or equivalently the two spin components. The reason for this is that the antiferromagnetically ordered state does not possess any net magnetization and has on average as many spin up polarized as spin down electrons. The division in the A and B sublattices is convenient for the DMFT calculations but somewhat artificial. In our case with hole doping the Fermi level lies within the lower band, which for the free quasiparticles is denoted by E0 The corresponding weight on the Fermi surface defined by (34) is then given by wqp = −(εkF) = z↑ + z↓ (z↑ − z↓)∆µ̃ 2|µ̃| , (43) where the average of the renormalized chemical potential µ̃ and the difference ∆µ̃ were defined below equation (19). From the definition of ∆µ̃ we can see that the second term in (43) is spin rotation invariant. The spectral quasipar- ticle weight wqp on the Fermi surface depends not only on the renormalization factors zσ, but also on the renor- malized chemical potentials µ̃0,σ. The same result for the weight (43) can be obtained from the diagonal form of Gk,σ(ω) and the spectral weight of the lower band. The weight wqp corresponds to the spectral weight Z at the Fermi level as for example given in references [3,31,32]. The first term of the result for wqp is like the arithmetic average of zσ. From figures 5 and 6 we can see that z↑ > z↓ and from figure 7 that µ̃0,↓ < µ̃0,↑ < 0. Therefore the sec- ond term in (43) gives a positive contribution to the spec- tral weight. At the end of the section in figure 18 we show values of wqp in comparison with the arithmetic average of zσ. In order to understand better the properties of the quasiparticle bands, we now compare the quasiparticle spectrum with the k-resolved spectral density ρk,σ(ω) de- rived from the DMFT-NRG results. In figure 12 we make a comparison for the case of 12.5% doping with U = 3 for the Green’s function Gk,σ(ω) given in equation (33), ρk,σ(ω) = −ImGk,σ(ω+)/π, where ω+ = ω+ iη, with η → 0, with that derived for the free quasiparticles, zσ ρ̃ k,σ(ω) from equation (40). The delta-functions of the free quasi- particle results are indicated by arrows with the height of the arrow indicating the value of the corresponding spec- tral weight. The plots as a function of ω are shown for a sequence values of εk and, where the peaks in ρk,σ(ω) get very narrow and high in the vicinity of the Fermi level, they have been truncated. It can be seen that the free quasiparticle results give a reasonable picture of the form of ρk,σ(ω), particularly in the immediate region of the Fermi level. There is considerable variation along the curves in the way the overall spectral weight is distributed between the excitations below and above the pseudo-gap as a function of εk. This is most marked in the region near the Fermi level for the spin-up electrons (upper panel) where most of the spectral weight is in the lower band and it is much reduced in the upper band, whereas the opposite is the case for the spin-down electrons. This is reflected in the expression of the quasiparticle weights uσ±(εk) in equation (39). For instance, u −(εk) corresponding to the lower band E0 k,− becomes maximal near the Fermi en- ergy, whereas u +(εk) goes to zero there. The finite width of the quasiparticle peaks in ρk,σ(ω) can be described by a RPT, when we take into account the renormalized self- 12 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model −1 −0.5 0 0.5 1 −1.125 −0.75 −0.375 0.375 1.125 −1 −0.5 0 0.5 1 −1.125 −0.75 −0.375 0.375 1.125 Fig. 12. (Color online) The spectral density ρk,σ(ω) for the spin-up electrons (upper panel) and spin-down (lower panel) plotted as a function of ω and a sequence of values of εk for U = 3 and 12.5% doping. Also shown with arrows are the positions of the free quasiparticle excitations, with the height of the arrow indicating the corresponding weight. energy Σ̃σ(ω) in equation (37). If we, for instance, use the the second order approximation in Ũ , which was illus- trated in the last section (31), we get a similar behavior for small ω as seen for ρk,σ(ω) in figure 12. From the positions of the peaks in the ρk,σ(ω) spectra we can deduce two branches of an effective dispersion Ek,± for single particle excitations and compare it with the ones for the free quasiparticles E0 k,±. We give the results for U = 3 in figure 13. It can be seen that E0 k,− tracks the peak in the lower band closely over a wide range of εk, −1.5 < εk < 1.5 (note the bandwidth W = 4). This is not the case in the upper band, where E0 k,+ tracks the peak closely only in the lowest section that lies closest to the Fermi level. As one can see from the dotted line the Fermi level lies in the lower band and intersects the lower band twice. This corresponds to the two values with opposite sign ε± can be see from equation (34). The corresponding results for the k-resolved spectra for U = 6 and also 12.5% doping are shown in figure 14. −1.5 −1 −0.5 0 0.5 1 1.5 Fig. 13. (Color online) A plot of the peak positions Ek,± in the spectral density ρk,σ(ω) (full line) as a function of εk for U = 3 and 12.5% doping compared with the free quasiparticle dispersion E0k (dashed line). −1 −0.5 0 0.5 1 −1.125 −0.75 −0.375 0.375 1.125 −1 −0.5 0 0.5 1 −1.125 −0.75 −0.375 0.375 1.125 Fig. 14. (Color online) The spectral density ρk,σ(ω) for the spin-up electrons (upper panel) and spin-down (lower panel) plotted as a function of ω and a sequence of values of εk for U = 6 and 12.5% doping. Also shown with arrows are the positions of the free quasiparticle excitations, with the height of the arrow indicating the corresponding weight. J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 13 In order to compare well with the case U = 3 we have chosen an identical range for ω and εk, although the large spectral peaks near the energy are very close together in this presentation. It can be seen that the overall features are very similar to those seen for U = 3. For the spin up spectrum (upper panel) the peaks for the lower band have most of the weight near the Fermi energy, whereas the upper band is suppressed there, and vice versa for the opposite spin direction. The lower bands are tracked well by the free quasiparticles, and we can see that the bands for the larger value of U are significantly flatter. This is also clearly visible in the following figure 15, where we again compare the quasiparticle band with the peak position of the full spectra. On the range shown the lower band Ek,− completely coincides with the free quasiparticle band E0 −1.5 −1 −0.5 0 0.5 1 1.5 Fig. 15. (Color online) A plot of the peak positions Ek,± in the spectral density ρk,σ(ω) (full line) as a function of εk for U = 6 and 12.5% doping compared with the free quasiparticle disper- sion E0k (dashed line). On the range shown the lower band Ek,− completely coincides with the free quasiparticle band E0k,−. From the k-resolved spectra in figures 12 and 14 we can also extract the width of the quasiparticle peak ∆qp in the spectral density ρk,σ(ω) (majority spin σ =↑). Its in- verse 1/∆qp gives a measure of the quasiparticle lifetime. The results for ∆qp for the lower band Ek,− for the two cases U = 3, 6 and 12.5% doping are shown in figure 16 as function of εk. This plot brings out more clearly the feature that can be seen already in figures 12 and 14 (upper panel) that the width increases sharply when we move away from the Fermi level and the values for the width ∆qp for U = 6 are significantly smaller than those for U = 3. This is in line with the fact that the local quasiparticle interaction Ũ is smaller for the larger value of the bare interaction U as commented on earlier. The free quasiparticle picture is therefore even more appropriate in the case with stronger interaction. To numerical accuracy the width vanishes at and is finite for the interval ε− < εk < ε which lies within the lower band but above the Fermi level. −1 −0.5 0 0.5 1 Fig. 16. (Color online) A plot of the width of the peaks ∆qp in the spectral density of the majority spin ρk,↑(ω) as a function of εk for U = 3 (dashed line) and U = 6 (full line) and 12.5% doping. Another quasiparticle property, the effective mass en- hancement m∗/m, can be extracted by calculating the derivative of E0k,− in (19) with respect to εk, which yields when evaluated at the Fermi energy (34), µ̃0,↑µ̃0,↓ . (44) The effective mass enhancement therefore does not only depend on zσ, but also on the renormalized chemical po- tentials µ̃0,σ. The general trend for m ∗/m as function of U can be seen in figure 17 for the case of 7.5% doping. 0.2 0.4 0.6 0.8 1 Fig. 17. The ratio m∗/m according to (44) plotted over a range of t2/U for 7.5% doping. The effective mass increases sharply for large U as the hole motion is energetically more costly in the ordered background. The fact that the lower band for U = 6 seen in figure 15 is flatter than in the case U = 3 in figure 13 14 J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model can be attributed to the larger effective mass. We find a similar behavior of m∗/m as function of U for different filling factors from the ones shown in figure 16. The trend is that the effective mass enhancement is less pronounced for larger doping, which is intuitively understandable by the quasiparticle motion in an ordered background. In the DMFT framework for the paramagnetic state as well as the case with homogeneous magnetic field, the quasiparticle spectral weight wqp and the inverse of the ef- fective mass enhancement m/m∗ can be described simply by the renormalization factor zσ. In figure 18 we show a comparison of the spectral quasiparticle weight wqp (43), the arithmetic, (z↑ + z↓)/2, and geometric, z↑z↓, aver- age of the renormalization factors, and the inverse of the effective mass, m/m∗, (44) for U = 3 for various dopings. 0 0.05 0.1 0.15 m/ m* Fig. 18. (Color online) Comparison of the spectral quasiparti- cle weight wqp from equation (43), the arithmetic, (z↑+ z↓)/2, and geometric, z↑z↓, average of the renormalization factors, and the inverse of the effective mass, m/m∗ from equation (44), for U = 3 and a range of dopings. As seen in this case with antiferromagnetic symmetry break- ing these quantities take a different form (43) and (44) and have distinct values. For different values of U the behaviour is qualitatively similar. As a first approxima- tion the quasiparticle spectral weight wqp corresponds to the arithmetic average of the renormalization factors zσ, whilst m/m∗ relates to the geometric average. In general, one can, however, not omit the dependence on the renor- malized chemical potential as it gives a significant contri- bution as can be seen in figure 18. This can be understood for example for the limit of zero doping. The system then becomes an antiferromagnetically ordered insulator with spectral gap. The weights zσ tend to finite values, but the effective mass must diverge. This is found in equation (44) since µ̃0,↑ → 0 for δ → 0, and the trend can be seen in figure 18. 5 Conclusions We have studied the field induced and spontaneous anti- ferromagnetic ordering in the hole doped Hubbard model with DMFT-NRG calculations at T = 0. A phase diagram separating antiferromagnetic and paramagnetic solutions for different values of doping and interactions U ranging from zero to about 1.5 times the bandwidth W has been established and is in agreement with earlier results by Zitzler et al. [10]. Our main objective has been to ana- lyze the properties of the quasiparticle excitations in the metallic antiferromagnetic state. We presented two differ- ent ways of calculating the parameters zσ and µ̃0,σ, which define the renormalized quasiparticles, and the two sets of results have been shown to be in agreement. We have also been able to deduce the effective on-site quasiparticle interaction Ũ from the NRG low lying excitations. The low energy properties of the local spectral function can be understood in terms of the free quasiparticle picture. We have used the second order perturbation expansion in powers of Ũ to estimate the spectral weight in the pseudo- gap region above the Fermi level. We have been able to compare the position of the peaks found in the k-dependent spectral functions with the dispersion relation for the free quasiparticles. The free quasiparticle dispersion gives a very good fit to the posi- tion of these peaks in the lower band which intersects the Fermi level. The quasiparticle lifetime, as deduced from the widths of the peaks in the spectrum, increases for stronger interactions. This is consistent with the fact that the on-site quasiparticle interaction Ũ , which gives the quasiparticles a finite lifetime, decreases with increase of U in the same range. We have also shown how the spec- tral quasiparticle weight at the Fermi level wqp and the effective mass can be deduced from the parameters zσ and µ̃0,σ. The effective mass is found to increase with the in- teraction, and it diverges in the limit of zero doping whilst wqp remains finite. We have found that Luttinger’s theorem for the total electron density in the antiferromagnetically ordered state holds within the numerical accuracy for the range of dop- ings and interactions studied. This is a further indication that many aspects of Fermi liquid description may hold in situations with symmetry breaking. It is not easy to make a direct comparison of our re- sults with earlier work [3] analyzing the quasiparticle exci- tations in an metallic antiferromagnet as these have been mainly based on the t − J-model for one or two holes in a finite cluster. However, at a semiquantitative level, the overall trend in our results seems to be similar to the results surveyed by Dagotto, where the effective quasipar- ticle bandwidth Weff is found to decrease with decreasing J . This is line with our results if we identify Weff ∼ m/m∗ and J ∼ t2/U (see figure 17). Our values for the spectral quasiparticle weight wqp are qualitatively similar to those presented as the wavefunction renormalization Z in the review article by Dagotto (see fig 27 [3]), and also the ones reported more recently [32]. J. Bauer and A.C. Hewson: Renormalized quasiparticles in antiferromagnetic states of the Hubbard model 15 Acknowledgment We wish to thank N. Dupuis, D.M. Edwards, W. Koller, D. Meyer and A. Oguri for helpful discussions and W. Koller and D. Meyer for their contributions to the development of the NRG programs. We also acknowledge stimulating discussions with G. Sangiovanni. One of us (J.B.) thanks the Gottlieb Daimler and Karl Benz Foundation, the Ger- man Academic exchange service (DAAD) and the EPSRC for financial support. References 1. P. W. Anderson, Science 235, 1196 (1987). 2. P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006). 3. E. Dagotto, Rev. Mod. Phys. 66, 763 (1994). 4. A.-M. S. Tremblay, B. Kyung, and D. Senechal, Low Tem- perature Physics 32, 424 (2006). 5. W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324 (1989). 6. E. Müller-Hartmann, Z. Phys. B 74, 507 (1989). 7. R. Bulla, T. Costi, and T. Pruschke, cond-mat/0701105 (unpublished). 8. J. Bauer and A. C. Hewson, cond-mat/0705.3824, to be published in Phys. Rev. B (unpublished). 9. A. C. Hewson, Phys. Rev. Lett. 70, 4007 (1993). 10. R. Zitzler, T. Pruschke, and R. Bulla, Eur. Phys. J. B 27, 473 (2002). 11. J. Hubbard, Proc. R. Soc. London, Ser. A 276, 238 (1963). 12. A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). 13. P. W. Anderson, Phys. Rev. 124, 41 (1961). 14. R. Bulla, Phys. Rev. Lett. 83, 136 (1999). 15. R. Peters, T. Pruschke, and F. B. Anders, Phys. Rev. B 74, 245114 (2006). 16. A. Weichselbaum and J. von Delft, cond-mat/0607497 (un- published). 17. F. B. Anders and A. Schiller, Phys. Rev. Lett. 95, 196801 (2005). 18. R. Bulla, A. C. Hewson, and T. Pruschke, J. Phys.: Cond. Mat. 10, 8365 (1998). 19. B. I. Shraiman and E. D. Siggia, Phys. Rev. Lett. 62, 1564 (1989). 20. M. Kato, K. Machida, H. Nakanishi, and M. Fujita, J. Phys. Soc. Japan 59, 1047 (1990). 21. V. J. Emery, S. A. Kivelson, and H. Q. Lin, Phys. Rev. Lett. 64, 475 (1990). 22. P. G. J. van Dongen, Phys. Rev. Lett. 74, 182 (1995). 23. P. G. J. van Dongen, Phys. Rev. B 54, 1584 (1996). 24. H. J. Schulz, Phys. Rev. Lett. 64, 1445 (1990). 25. J. K. Freericks and M. Jarrell, Phys. Rev. Lett. 74, 186 (1995). 26. V. J. Emery, S. A. Kivelson, and J. M. Tranquada, Proc. Natl. Acad. Sci. (USA) 96, 8814 (1999). 27. A. C. Hewson, A. Oguri, and D. Meyer, Eur. Phys. J. B 40, 177 (2004). 28. H. R. Krishna-murthy, J. W. Wilkins, and K. G. Wilson, Phys. Rev. B 21, 1003 (1980). 29. L. H. Ryder, Quantum Field Theory (Cambridge Univer- sity Press, Cambridge, 1996). 30. A. C. Hewson, J. Phys.: Cond. Mat. 13, 10011 (2001). 31. G. Sangiovanni, A. Toschi, E. Koch, K. Held, M. Capone, C. Castellani, O. Gunnarsson, S.-K. Mo, J. W. Allen, H.-D. Kim, A. Sekiyama, A. Yamasaki, S. Suga, and P. Metcalf, Phys. Rev. B 73, 205121 (2006). 32. G. Sangiovanni, O. Gunnarsson, E. Koch, C. Castellani, and M. Capone, Phys. Rev. Lett. 97, 046404 (2006). Introduction Antiferromagnetic Broken Symmetry in DMFT Local Quasiparticle Parameters Spectra and Quasiparticle Bands Conclusions

0704.0244

Comparison of exact-exchange calculations for solids in current-spin-density- and spin-density-functional theory

arXiv:0704.0244v2 [cond-mat.mtrl-sci] 4 Jun 2007 Comparison of exact-exchange calculations for solids in current-spin-density- and spin-density-functional theory S. Sharma1,2,∗ S. Pittalis2, S. Kurth2, S. Shallcross3, J. K. Dewhurst4, and E. K. U. Gross2 1 Fritz Haber Institute of the Max Planck Society, Faradayweg 4-6, D-14195 Berlin, Germany. 2 Institut für Theoretische Physik, Freie Universität Berlin, Arnimallee 14, D-14195 Berlin, Germany 3 Department of Physics, Technical University of Denmark, Building 307, DK-2800 Kgs. Lyngby and 4 School of Chemistry, The University of Edinburgh, Edinburgh EH9 3JJ. The relative merits of current-spin-density- and spin-density-functional theory are investigated for solids treated within the exact-exchange-only approximation. Spin-orbit splittings and orbital magnetic moments are determined at zero external magnetic field. We find that for magnetic (Fe, Co and Ni) and non-magnetic (Si and Ge) solids, the exact-exchange current-spin-density functional approach does not significantly improve the accuracy of the corresponding spin-density functional results. PACS numbers: 71.15 Mb, 71.15 Rf, 75.10 Lp In the past 30 years, several generalizations of den- sity functional theory (DFT) have been proposed. In the early 70’s, DFT was extended to spin-DFT (SDFT) [1] by including the spin magnetization as basic quantity in ad- dition to the density. This allows for coupling of the spin degrees of freedom to external magnetic fields and pro- duces better results for spontaneously spin-polarized sys- tems using approximate functionals. Adding yet another density, the paramagnetic current, leads to the frame- work of current-SDFT (CSDFT) [2, 3]. CSDFT includes the coupling of the external magnetic field, through its corresponding vector potential, to the orbital-degrees of freedom [3]. SDFT has been enormously successful in predicting the magnetic properties of materials. This success can be attributed to the availability of exchange correlation (xc) functionals which, even though originally designed for non-magnetic systems, could be systematically ex- tended to the spin polarized case. The most popular of these functionals are the local spin density approxima- tion (LSDA) and the generalized gradient approximation (GGA). CSDFT, on the other hand, has not enjoyed the same attention mainly because of problems which arise in the extension of LSDA and GGA to include the paramag- netic current density [4–6]. Exposing the homogeneous electron gas to an external magnetic field leads to the appearance of Landau levels which, in turn, give rise to derivative discontinuities in the resulting xc energy den- sity. Using this quantity to construct (semi-)local func- tionals then automatically leads to local discontinuities in the corresponding xc potentials, which are then awkward to use in practical calculations. Such problems can be avoided with the use of orbital functionals and this fact, coupled with the success of these functionals for SDFT calculations, has led to re- cent interest in orbital functionals for CSDFT [7–11]. The results from these works have shown mixed suc- cess. A modified version of the original CSDFT [12] lead to promising results for spin-orbit induced splittings of bands in solids, such as Si and Ge [9]. In contrast it was found that for open-shell atoms and quantum dots the difference between SDFT and CSDFT results was mini- mal [7, 8]. Similarly, calculations for solids using a local vorticity functional [13] and for quantum dots using a LSDA-type xc functional [14] could not establish the su- periority of CSDFT over SDFT. In this work we present a systematic comparison of the relative merits of CSDFT and SDFT for solids. Since the Kohn-Sham (KS) system in CSDFT reproduces the current of the interacting system, one would expect dif- ferences between SDFT and CSDFT results for orbital magnetic moments (which can be directly derived from the current). With this in mind we calculate the orbital magnetic moment of the spontaneous magnets Fe, Co and Ni. Since CSDFT is believed to improve the spin-orbit induced band splitting in the non-magnetic semiconduc- tors Si and Ge [9] it makes these materials interesting candidates for a study of the differences between the two approaches. Following Vignale and Rasolt [2, 3], the ground state energy of a (non-relativistic) system of interacting elec- trons in the presence of an external magnetic field B0(r) = ∇×A0(r) can be written as functional of three independent densities the particle density ρ(r), the mag- netization density m(r) and the paramagnetic current density jp(r). This functional is given by http://arxiv.org/abs/0704.0244v2 E[ρ,m, jp] = Ts[ρ,m, jp] + U [ρ] + Exc[ρ,m, jp] + ρ(r)v0(r) d 3r (1) m(r) ·B0(r) d jp(r) ·A0(r) d ρ(r)A20(r) d where Ts[ρ,m, jp] is the kinetic energy functional of non-interacting electrons, U [ρ] is the Hartree energy, and Exc[ρ,m, jp] is the exchange-correlation energy. Minimization of Eq. (1) with respect to the three basic densities leads to the Kohn-Sham (KS) equation which reads As(r) + vs(r)− µBσ ·Bs(r) Φj(r) = εjΦj(r) . (2) Here σ is the vector of Pauli matrices and the Φi are spinor valued wave functions. The effective potentials vs, Bs and As are such that the ground-state densities ρ, m and jp of the interacting system are reproduced. These effective potentials are given by vs(r) = v0(r) + vH(r) + vxc(r) + 0(r)−A s (r) , Bs(r) = B0(r) +Bxc(r), As(r) = A0(r) +Axc(r). (3) Here, v0 is the external electrostatic potential and vH(r) = ρ(r′)/|r − r′| d3r′ is the Hartree potential. The xc potentials are given as functional derivatives of the xc energy with respect to the corresponding conjugate densities which can be obtained from KS wave functions using the following relations ρ(r) = i (r)Φi(r), m(r) = −µB i (r)σΦi(r), jp(r) = i (r)∇Φi(r)− i (r) Φi(r) where the sum runs over the occupied orbitals. For practical calculations, an approximation for the xc energy functional Exc[ρ,m, jp] has to be adopted. Here we concentrate on approximations of the xc functional which explicitly depend on the KS orbitals and therefore only implicitly on the densities. Such orbital functionals are usually treated within the framework of the so-called Optimized Effective Potential (OEP) method [15–18] where the xc potential is obtained as solution of the OEP integral equation. Recently, the OEP method has been generalized to non-collinear SDFT [19] and CSDFT [7, 8]. Another generalization of the OEP method in the context of a spin-current DFT (SCDFT) based on a different choice of densities has also been put forward [11]. In the present work the formalism of Refs. (7) and (8) is used and the corresponding OEP equations can be put in a compact form as (r)Ψk(r) + h.c. = 0, −µB (r)σΨk(r) + h.c. = 0, (r)∇Ψk(r) − Ψk(r) + h.c. = 0 , where the so-called orbital shifts [17, 20] are defined as Ψk(r) = ∑unocc Φj(r)Λkj εk−εj , here the summation runs over the unoccupied states and Λkj = vxc(r ′)ρkj(r Axc(r ′) · jpkj(r ′)−Bxc(r ′) ·mkj(r ′)− Φ where ρkj(r) = Φ (r)Φk(r), mkj(r) = −µBΦ (r)σΦk(r) and jpkj(r) = (r)∇Φk(r− Φk(r) Eq. (5) has a structure very similar to the OEP equa- tions for non-collinear SDFT differing only by the redefi- nition of the matrix Λ, which now also contains an extra term depending upon the current density and its con- jugate field. Due to their similar structure the CSDFT OEP equations are solved by generalizing the ‘residue al- gorithm’, successfully applied to solve the non-collinear SDFT equations [19–21]. The only difference in the case of CSDFT is introduction of an additional residue coming from the third OEP equation in Eq. (5). In the present work we have used the exchange-only exact-exchange (EXX) functional to solve the OEP equations. The EXX (gauge invariant) energy functional is the Fock exchange energy but evaluated with KS spinors EEXXx [{Φi}] ≡ − ∫ ∫ occ i (r)Φj(r)Φ ′)Φi(r |r− r′| d3r d3r′ . In order to keep the numerical analysis as accurate as possible, in the present work all calculations are per- formed using the state-of-the-art full-potential linearized augmented plane wave (FPLAPW) method [22], imple- mented within the EXCITING code [23]. The single- electron problem is solved using an augmented plane wave basis without using any shape approximation for the effective potential. Likewise, the magnetization and current densities and their conjugate fields are all treated as unconstrained vector fields throughout space. The deep lying core states (3 Ha below the Fermi level) are treated as Dirac spinors and valence states as Pauli spinors. To obtain the Pauli spinor states, the Hamilto- nian containing only the scalar fields is diagonalized in the LAPW basis: this is the first-variational step. The scalar states thus obtained are then used as a basis to set up a second-variational Hamiltonian with spinor de- grees of freedom, which consists of the first-variational eigenvalues along the diagonal, and the matrix elements obtained from the external and effective vector fields in Eq. (2). This is more efficient than simply using spinor LAPW functions, but care must be taken to ensure there are a sufficient number of first-variational eigenstates for convergence of the second-variational problem. Spin- orbit coupling is also included at this stage. As was shown above for CSDFT, the magnetic field couples not only to spin but also to the orbital degrees of freedom through the vector potential. This makes CS- DFT specifically important for magnetic materials and particularly interesting for their orbital properties. By analogy with SDFT, one might expect that the introduc- tion of the paramagnetic current density gives an im- provement in properties such as orbital moments and spin-orbit induced band splitting, which are related to this new basic variable. However, within the framework of existing functionals it is yet to be established conclu- sively that CSDFT performs better than SDFT for these properties. The recent development of the OEP method both for SDFT and CSDFT allows for a direct compar- ison of these two approaches for the same xc functional, namely EXX. The orbital moments of spontaneous magnets Fe, Co and Ni, in the absence of external magnetic fields and with spin-orbit coupling included, are presented in Ta- ble I. For SDFT, the LSDA, GGA and EXX functionals are used, while for CSDFT the values are obtained us- ing the EXX functional. It is clear from Table I that there is no difference between the results obtained us- ing EXX-CSDFT and EXX-SDFT. Formally, the jp de- termined from SDFT does not correspond to the true paramagnetic current density of the fully interacting sys- SDFT CSDFT Solid Exp. LSDA GGA EXX EXX Fe 0.08 0.053 0.051 0.034 0.034 Co 0.14 0.069 0.073 0.013 0.013 Ni 0.05 0.038 0.037 0.029 0.029 36.2 36.7 63.4 63.4 TABLE I: Orbital magnetic moments for bulk Fe, Co and Ni in µB . The experimental data are taken from Ref. (24). The final row lists the average percentage deviation of the numerical results from the experimental value. tem. Nevertheless, it is standard practice to compute the orbital magnetic moment L, like those listed in Table I, which is related to jp from the KS orbitals by the relation L = 1 r×jp(r)d 3r. The fact the EXX-SDFT and EXX- CSDFT orbital moments are so close may be viewed as a post-hoc justification of this practice for magnetic metals. It should also be noted that in comparison to experiments the EXX results are significantly worse than their LSDA and GGA counterparts. One reason, of course, is the fact that LSDA and GGA also include correlation in an ap- proximate way which is neglected completely within the EXX framework. In a recent work [9] it is shown that the use of the EXX functional in the framework of SCDFT, improves the spin-orbit induced splitting of the bands in semicon- ductors. Unfortunately, it is not clear if this improve- ment is due to the use of different functionals (going from LSDA to EXX), or due to the use of an extra den- sity when going from SDFT to CSDFT. This has moti- vated us to compare CSDFT and SDFT results for this quantity using the same functional in both cases. We have determined the value of this splitting for solid Si and Ge and the results are presented in Table II. While the EXX functional significantly improves the agreement with experimental values, there is almost no change on going from SDFT to CSDFT. Thus the improvement is solely due to the orbital based functional. We also note that the EXX-CSDFT results of Ref. (9) are significantly different from ours, and in much worse agreement with experiments. This might be due to the use of pseudopo- tentials in the previous work. In this respect it is worth noting that EXX derived KS energy gaps also show sig- nificant differences depending on whether an all-electron full-potential or pseudopotential method is used [25]. The paramagnetic current density of Ge for LSDA, GGA, EXX-SDFT and EXX-CSDFT is plotted in Fig. 1. Ge is chosen as an example since the spin-orbit in- duced splitting is largest for this system and, unlike in the case of metallic orbital moments, this quantity does show some difference on going from SDFT to CSDFT. We immediately notice that there is no significant qual- itative difference between the LSDA and GGA currents. There are, however, pronounced differences in the cur- Symmetry SDFT CSDFT point Exp. LSDA GGA EXX EXXp EXXo Ge Γ7v−8v 297 311 296 291.3 289 258.1 Ge Γ6c−8c 200 229.7 220 201.3 199 173.3 Si Γ25v 44 50 58 42.5 45.5 42.5 9.5 14.0 2.0 2.2 10.5 TABLE II: Spin-orbit induced splittings for bulk Ge and Si in meV. The experimental data is taken from Ref. (26). EXXp are results of the present work and EXXo are results from Ref. (9). The final row lists the average percentage deviation of numerical results from the experimental value. rent density between LSDA/GGA and EXX-(C)SDFT: the current in the latter case being smaller and more homogeneous than that of the former. This is an inter- esting finding since it indicates the tendency of (semi-) local functionals towards higher values of the paramag- netic current density. It is worthwhile noting previous EXX-(C)SDFT results for open-shell atoms in which it was found [7] that this effect was even more pronounced and lead to vanishing currents. Even though the EXX-SDFT current is considerably lower in magnitude than that of EXX-CSDFT and also has a less symmetric structure, the spin-orbit splittings for the two cases are almost the same. Similar conclusions regarding the total energies were also drawn for quantum dots in external magnetic fields studied using EXX [8] and other functionals of the current density [14]. From Fig. 1 it is also clear that one of the major effects of using the OEP method and of using jp as an extra density is to change the local structure of the paramagnetic current, which in turn suggests that quantities depending on local properties of the currents, such as chemical shifts, might exhibit larger differences in the two approaches. Such calculations [27] of chemical shifts, performed using local functionals, found that for molecules this is not the case. The effect of the EXX functional on these shifts may be an interesting subject for future investigations. To summarize, in this work we have presented EXX- SDFT and CSDFT calculations for solids. The orbital magnetic moments of Fe, Co and Ni and the spin-orbit induced band splitting of Si and Ge are computed. Our analysis shows only minor differences between EXX- CS- DFT and SDFT results. The spin-orbit induced band splittings in EXX calculations are in rather good agree- ment with experiments, while the results for the orbital moments are worse than the LSDA or GGA values. This highlights the importance of proper treatment of corre- lations for the accurate determination of the orbital mo- ments. We acknowledge Deutsche Forschungsgemeinschaft (SPP-1145) and NoE NANOQUANTA Network (NMP4- CT-2004-50019) for financial support. FIG. 1: (Color online)Paramagnetic current density for Ge, in the [110] plane, calculated using the SDFT and CSDFT. Arrows indicate the direction and information about the mag- nitude (in atomic units) is given in the colour bar. ∗ Electronic address: sangeeta.sharma@physik. fu-berlin.de [1] U. von Barth and L. Hedin, J. Phys. C 5, 1629 (1972). [2] G. Vignale and M. Rasolt, Phys. Rev. Lett. 59, 2360 (1987). [3] G. Vignale and M. Rasolt, Phys. Rev. B 37, 10685 (1988). [4] P. Skudlarski and G. Vignale, Phys. Rev. B 48, 8547 (1993). [5] Y. Takada and H. Goto, J. Phys. Condens. Matter 10, 11315 (1998). [6] K. Higuchi and M. Higuchi, Phys. Rev. B 74, 195122 (2006). [7] S. Pittalis, S. Kurth, N. Helbig, and E.K.U. Gross, Phys. Rev. A 74, 062511 (2006). [8] N. Helbig, S. Kurth, S. Pittalis, E. Räsänen, and E.K.U. Gross, cond-mat/0605599 (2006). [9] S. Rohra, E. Engel, and A. Görling, cond-mat/0608505 (2006). [10] T. Heaton-Burgess, P. Ayers, and W. Yang, Phys. Rev. Lett. 98, 036403 (2007). [11] S. Rohra and A.Görling, Phys. Rev. Lett. 97, 013005 (2006). [12] K. Bencheikh, J. Phys. A: Math. Gen. 36, 11929 (2003). [13] H. Ebert, M.Battocletti, and E.K.U. Gross, Euro- phys. Lett. 40, 545 (1997). [14] H. Saarikoski, E. Räsänen, S. Siljamäki, A. Harju, and R. M. Nieminen, Phys. Rev. B 67, 205327 (2003). [15] R. Sharp and G. Horton, Phys. Rev. 90, 317 (1953). [16] J.D. Talman and W.F. Shadwick, Phys. Rev. A 14, 36 (1976). [17] T. Grabo, T. Kreibich, S. Kurth, and E.K.U. Gross, in Strong Coulomb Correlations in Electronic Structure Cal- culations: Beyond Local Density Approximations, edited by V. Anisimov (Gordon and Breach, Amsterdam, 2000), p. 203. [18] E. Engel and S. H. Vosko, Phys. Rev. A 47, 2800 (1993). [19] S. Sharma, J.K. Dewhurst, C. Ambrosch-Draxl, S. Kurth, N. Helbig, S. Pittalis, E.K.U. Gross, S. Shallcross, and L. Nordström, cond-mat/0510800 (2006). [20] S. Kümmel and J.P. Perdew, Phys. Rev. Lett. 90, 043004 (2003). [21] S. Kümmel and J.P. Perdew, Phys. Rev. B 68, 035103 (2003). [22] D.J. Singh, in Planewaves, pseudopotentials and the LAPW method (Kluwer, Dordrecht, 1994). [23] J. K. Dewhurst, S. Sharma, and C. Ambrosch-Draxl, 2004, http://exciting.sourceforge.net/. [24] M. B. Stearns, in Magnetic properties of 3d, 4d, and 5d elements, alloys and compounds, edited by Landolt- Boernstein (Springer, Berlin, 1987), Vol. Vol. III/19a. [25] S. Sharma, J.K. Dewhurst, and C. Ambrosch-Draxl, Phys. Rev. Lett. 95, 136402 (2005). [26] O.Madelung, Semiconductors: Data Handbook (Springer- Verlag, Berlin, 2004). [27] A. M. Lee, N. C. Handy, and S. M. Colwell, J. Chem. Phys. 103, 10095 (1995).

0704.0245

One-loop MHV Rules and Pure Yang-Mills

QMUL-PH-07-09 One-loop MHV Rules and Pure Yang-Mills Andreas Brandhuber, Bill Spence, Gabriele Travaglini and Konstantinos Zoubos1 Centre for Research in String Theory Department of Physics Queen Mary, University of London Mile End Road, London, E1 4NS United Kingdom Abstract It has been known for some time that the standard MHV diagram formulation of perturbative Yang-Mills theory is incomplete, as it misses rational terms in one-loop scattering amplitudes of pure Yang-Mills. We propose that certain Lorentz violating counterterms, when expressed in the field variables which give rise to standard MHV vertices, produce precisely these missing terms. These counterterms appear when Yang-Mills is treated with a regulator, introduced by Thorn and collaborators, which arises in worldsheet formulations of Yang- Mills theory in the lightcone gauge. As an illustration of our proposal, we show that a simple one-loop, two-point counterterm is the generating function for the infinite sequence of one-loop, all-plus helicity amplitudes in pure Yang-Mills, in complete agreement with known expressions. 1{a.brandhuber, w.j.spence, g.travaglini, k.zoubos}@qmul.ac.uk http://arxiv.org/abs/0704.0245v2 Contents 1 Introduction 1 2 Background 3 2.1 The classical MHV Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 A four–dimensional regulator for lightcone Yang–Mills . . . . . . . . . . . . . 6 2.3 The one–loop (++++) amplitude . . . . . . . . . . . . . . . . . . . . . . . . 11 3 The all-plus amplitudes from a counterterm 12 3.1 Mansfield transformation of LCT . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 The four–point case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 The general all–plus amplitude . . . . . . . . . . . . . . . . . . . . . . . . . 22 4 Discussion 27 A Notation 29 B Details on the four–point calculation 31 1 Introduction One of the success stories arising from twistor string theory [1] (see [2] for a review) has been the development of new techniques in perturbative quantum field theory. These include recursion relations [3, 4], generalised unitarity [5] and MHV methods (see [6] for a review). One of the key motivations of this work is to provide new approaches to study and derive phenomenologically relevant scattering amplitudes. In particular, this requires that one be able to deal with non-supersymmetric theories, and to include fermions, scalars, and particles with masses. A vital first step is to apply these new methods to pure Yang-Mills (YM) theory, and indeed, some of the first new results inspired by twistor string theory involved pure YM amplitudes at tree- [7, 8, 9, 10, 11, 12, 13, 14] and one-loop [15] level. A recalcitrant issue in this work is the derivation of rational terms in quantum amplitudes. Unitarity-based techniques [16] and loop MHV methods [17] are successful in obtaining the cut-constructible parts of amplitudes; essentially this is because at some level they are dealing with four-dimensional cuts. In principle performing D-dimensional cuts generates all parts of amplitudes [18, 19, 20, 21] as long as only massless particles are involved, however these techniques still appear to be relatively cumbersome. Combinations of recursive techniques and unitarity have led to important progress recently [22, 23, 24, 25, 26, 27, 28, 29, 30, 31], but it would be preferable to have a more powerful prescriptive formulation, particularly keeping in mind that applications to more general situations are sought. A promising development from this point of view is the Lagrangian approach [32, 33, 34]. Here it has been argued that lightcone Yang-Mills theory, combined with a certain change of field variables, yields a classical action which comprises precisely the MHV vertices. A full Lagrangian description of MHV techniques would in principle give a prescription for applying such methods to diverse theories. The next step in developing this is to understand the quantum corrections in this Lagrangian approach. If one directly uses in a path integral the classical MHV action, containing only purely four-dimensional MHV vertices, then it is immediately clear that this cannot yield all known quantum amplitudes. For example, there is no way to construct one-loop amplitudes where the external gluons all have positive helicities, or where only one gluon has negative helicity, as all MHV vertices contain two negative helicity particles (this issue has been recently discussed in [35]). These amplitudes are particular cases where the entire amplitude consists of rational terms. More generally, it seems clear that the vertices of the classical MHV Lagrangian will not yield the rational parts of amplitudes, but only the cut-constructible terms [15]. Important insights into this question can be obtained from the study of self-dual Yang-Mills theory, which has the same all-plus one-loop amplitude of full YM [36, 37, 38] as its sole quantum correction.1 An example, relevant to the discussion in this paper, is given in [35] where it was shown how these amplitudes might be obtained from the Jacobian arising from a Bäcklund-type change of variables which takes the self-dual Yang-Mills theory to a free theory. A discussion of the full Yang-Mills theory in the lightcone gauge has recently been given by Chakrabarti, Qiu and Thorn (CQT) in [39, 40, 41]. These papers employ an interesting regularisation which, importantly, does not change the dimension of spacetime. For this reason, we find it particularly suitable for setting the scene for the MHV diagram method, which is inherently four-dimensional in current approaches. The regularisation of CQT requires the introduction of certain counterterms, which prove to be rather simple in form. What we will show in this paper is that these simple counterterms provide a very compact and powerful way to represent the rational terms in gauge theory amplitudes; specifically, we will demonstrate that the single two-point counterterm contains all the n-point all-plus amplitudes. The way this happens is through the use of the new field variables of [32, 33, 34]. Other counterterms will combine with vertices from the Lagrangian and should generate the rational parts of more general amplitudes. Based on the discussion in this paper, we propose 1In real Minkowski space, this is in fact its single non-vanishing amplitude. that the counterterms, expressed in the field variables which give rise to standard MHV vertices, in combination with Lagrangian vertices, generate the rational terms previously missing from MHV diagram formulations. The rest of the paper is organised as follows. After giving some background material in section 2, we explicitly derive in section 3 the four point all-plus amplitude from the two- point counterterm of CQT. We follow this by showing that the n-point expression, obtained by writing the counterterm in new variables, has precisely the right collinear and soft limits required for it to be the correct all-plus n-point amplitude. We present our conclusions in section 4, and our notation and derivations of certain identities have been collected in two appendices. 2 Background In this section, we first review the classical field redefinition from the lightcone Yang– Mills Lagrangian to the MHV–rules Lagrangian. We then move on to motivate the four– dimensional regularisation scheme we will employ, and argue that it leads directly to the introduction of a certain Lorentz–violating counterterm in the Yang–Mills Lagrangian. We close the section with the remarkable observation that this counterterm provides a simple way to calculate the four–point all-plus one–loop amplitude using only tree–level combinatorics. 2.1 The classical MHV Lagrangian It seemed clear from the beginning that the MHV diagram approach to Yang-Mills theory must be closely related to lightcone gauge theory. This idea was substantiated by Mansfield [33] (see also [32]). The starting point of [33] is the lightcone gauge-fixed YM Lagrangian for the fields corresponding to the two physical polarisations of the gluon. It was argued convincingly in [33] that a certain canonical change of the field variables re-expresses this lightcone Lagrangian as a theory containing the infinite series of MHV vertices. Some of the arguments in [33] were rather general; these were reviewed in [34], where the change of variables was discussed in more detail, and in particular it was shown how the four- and five-point MHV vertices arise from the change of variables. In this paper we will mainly follow the notation of [34]. The general structure of the lightcone YM Lagrangian, after integrating out unphysical degrees of freedom, is (see appendix A for more details) LYM = L+− + L++− + L−−+ + L++−− , (2.1) where the gauge condition is ηµAµ = 0 with the null vector η = (1/ 2, 0, 0, 1/ 2). Since this Lagrangian contains a + +− vertex, it is not of MHV type. In [33], Mansfield proposed to eliminate this vertex through a suitably chosen field redefinition. Specifically, he performed a canonical change of variables from (A, Ā) to new fields (B, B̄), in such a way that L+−(A, Ā) + L++−(A, Ā) = L+−(B, B̄) . (2.2) The remarkable result is that upon inserting this change of variables into the remaining two vertices, the Lagrangian, written in terms of (B, B̄), becomes a sum of MHV vertices, LYM = L+− + L+−− + L++−− + L+++−− + . . . . (2.3) The crucial property of Mansfield’s transformation that makes this possible is that, while both A and Ā are series expansions in the new B fields, A has no dependence on the B̄ fields while Ā turns out to be linear in B̄. Thus, since the remaining vertices are quadratic in the B̄, the new interaction vertices have the helicity configuration of an MHV amplitude. Mansfield was also able to show that the explicit form of the vertices coincides with the CSW off-shell continuation of the Parke-Taylor formula for the MHV scattering amplitudes, as proposed by [7]. One of the main results of [34] was the derivation of an explicit, closed formula for the expansion of the original fields (A, Ā) in terms of the new fields (B, B̄). This was then used to verify that the new vertices are indeed precisely the MHV vertices of [7], at least up to the five-point level. We will now briefly review these results. First, recall that the positive helicity field A is a function of the positive helicity B field only. It is expanded as follows: A(~p ) = (2π)3 ∆(~p , ~p , . . . ~p ) Y(~p ; 1 · · ·n) B(~p 1)B(~p 2) · · ·B(~p n) , (2.4) where ∆(~p , ~p 1, . . . ~p n) := (2π)3δ(3)(~p − ~p 1 − · · · − ~p n). Note that the x− coordinate is common to all the fields, which is why we have restricted the transformation to the lightcone quantisation surface Σ. By inserting this expansion into (2.2) and using the requirement that the transforma- tion be canonical, Ettle and Morris succeeded in deriving a very simple expression for the coefficients Y. After translating to our conventions (see appendix A), they are given by: Y(~p ; 12 · · ·n) = ( 2ig)n−1 〈12〉〈23〉 · · · 〈n− 1, n〉 . (2.5) The first few terms in (2.4) are then: A(~p ) =B(~p ) + 2igp+ d3p1d3p2 (2π)3 δ(3)(~p − ~p 1 − ~p 2) 〈12〉 B(~p )B(~p − 2g2p+ d3p1d3p2d3p3 (2π)6 δ(3)(~p − ~p 1 − ~p 2 − ~p 3) 〈12〉〈23〉 B(~p 1)B(~p 2)B(~p 3) + · · · . (2.6) Similarly, one can write down the expansion of the negative helicity field Ā, which, as discussed above, is linear in B̄, but is an infinite series in the new field B. In [34] it was shown that the coefficients in the expansion of Ā are very closely related to those for A.2 The expansion of B̄ turns out to be simply Ā(~p ) =− (2π)3 ∆(~p , ~p , . . . , ~p (ps+) (p+)2 Y(~p ; 1 · · ·n) B(~p 1)· · ·B̄(~p s)· · ·B(~p n) (2π)3 ∆(~p , ~p 1, . . . , ~p n) (p+)2 Y(~p ; 1 · · ·n) (ps+) 2 B(~p ) · · · B̄(~p s) · · ·B(~p n) . (2.7) Thus we see that at each order in the expansion, we need to sum over all possible positions of B̄. Explicitly, the first few terms are: Ā(~p ) = B̄(~p )− d3p1d3p2 (2π)3 δ(3)(~p − ~p 1 − ~p 2) 1 〈12〉× (p1+) 2B̄(~p 1)B(~p 2) + (p2+) 2B(~p 1)B̄(~p 2) + 2g2 d3p1d3p2d3p3 (2π)6 δ(3)(~p − ~p 1 − ~p 2 − ~p 3) 1 〈12〉〈23〉× (p1+) 2B̄(~p 1)B(~p 2)B(~p 3)+(p2+) 2B(~p 1)B̄(~p 2)B(~p 3)+(p3+) 2B(~p 1)B(~p 2)B̄(~p 3) + · · · (2.8) Using the above results, it is in principle straightforward to derive the terms that arise on inserting the Mansfield transformation into the two remaining vertices of the theory. For the simplest cases, one can see explicitly that these combine to produce MHV vertices, and some arguments were also given in [33, 34] that this must be true in general. In supersymmetric theories, the MHV vertices are enough to reproduce complete scat- tering amplitudes at one loop [43]. However, as we mentioned earlier, for pure YM it is clear that the terms in the MHV Lagrangian (2.3) will not be enough to generate complete quantum amplitudes. For instance, the scattering amplitude with all gluons with positive helicity, which at one loop is finite and given by a rational term, cannot be obtained by only using MHV diagrams, for the simple reason that one cannot draw any diagram contributing to it by only resorting to MHV vertices.3 Another amplitude which cannot be derived within conventional MHV diagrams is the amplitude with only one gluon of negative helicity. Simi- larly to the all-plus amplitude, this single-minus amplitude vanishes at tree level, and at one loop is given by a finite, rational function of the spinor variables. 2This is perhaps easiest to see [42] by considering that, in the context of N = 4 SYM, A and B are part of the same lightcone superfield. 3On the other hand, it was shown in [35] that the parity conjugate all-minus amplitude is correctly generated by using MHV diagrams. The lesson we learn from this is that, in order to apply the MHV method to derive complete amplitudes in pure YM, one should look more closely at the change of variables in the full quantum theory. There are several possible subtleties one should pay careful attention to at the quantum level. First of all, it is possible that the canonical nature of the transformation is not preserved, leading to a non–trivial Jacobian which could provide the missing amplitudes. Another possible source of contributions could come from violations of the equivalence theorem. This theorem states that, although correlation functions of the new fields are in general different from those of the old fields, the scattering amplitudes are actually the same4, as long as the new fields are good interpolating fields. These issues were explored in some detail in [35] (see also [34, 42]) where it was shown, for a different (non-canonical) field redefinition, how a careful treatment of these effects can combine to reproduce some of the amplitudes that would seem to be missing at first sight. Another aim of [35] was to demonstrate how to reproduce one of the above–mentioned rational amplitudes, the one with all–minus helicities, in the MHV formalism. This amplitude is slightly less mysterious than the all–plus amplitude in the sense that one can write down the contributing diagrams using only MHV vertices; however a calculation without a suitable regulator in place would give a vanishing answer, despite the fact that this amplitude is finite. In [35], it was shown, using dimensional regularisation, that the full nonzero result arises from a slight mismatch between four– and D (= 4− 2ǫ)–dimensional momenta. It is natural therefore to expect that dimensional regularisation will be helpful also for the problem at hand, which is to recover the rational amplitudes of pure Yang–Mills after the Mansfield transformation. Decomposing the regularised lightcone Lagrangian into a pure four-dimensional part and the remaining ǫ–dependent terms, and performing the transfor- mation on the four-dimensional part only, will give rise to several new ǫ–dependent terms that can potentially give finite answers when forming loops. Although this approach shows promise, it is not the one we will make use of in the fol- lowing. Instead, motivated by the fact that the Mansfield transformation seems to be deeply rooted in four dimensions, we would like to look for a purely four–dimensional regularisation scheme. We now turn to a review of the particular scheme we will use. 2.2 A four–dimensional regulator for lightcone Yang–Mills In the above we explained why a näıve application of the Mansfield transform leads to puzzles at the quantum level, and discussed possible ways to improve the situation. The conclusion was that, since the missing amplitudes arise from subtle mismatches in regularisation, one should be careful to perform the Mansfield transform on a suitably regularised version of the lightcone Yang–Mills action. Here we will review one approach to the regularisation of lightcone Yang–Mills, which, despite several slightly unusual features, appears to be ideally 4Modulo a trivial wave-function renormalisation. suited for the problem at hand. The regularisation we propose to use is inspired by recent work of CQT [39, 40, 41] on Yang–Mills amplitudes in the lightcone worldsheet approach [44, 45]. This is an attempt to understand gauge–string duality which is similar in spirit to ’t Hooft’s original work on the planar limit of gauge theory [46], and aims at improving on early dual model techniques [47, 48]. We recall that one of the main goals in those works is to exhibit the string worldsheet as made up of very large planar diagrams (“fishnets”). In their recent work, Thorn and collaborators make this statement more precise, using techniques that were unavailable when the original ideas were put forward. It is hoped that, by understanding how to translate a generic Yang–Mills planar diagram to a configuration of fields (with suitable boundary conditions) on the lightcone worldsheet, it will eventually become possible to perform the sum of all these diagrams. This approach to gauge–string duality is thus complementary to that using the AdS/CFT correspondence. The field content and structure of the worldsheet theory dual to Yang–Mills theory is rather intricate (see e.g. [45]), but for our purposes the details are not important. What is most relevant for us is that one of the principles of this approach is that all quantities on the Yang–Mills side should have a local worldsheet description. This includes the choice of regulator that needs to be introduced when calculating loop diagrams. This requirement led Thorn [49] (see also [50, 51]) to introduce an exponential UV cutoff, which we will discuss in a short while. Since one of the goals of this programme is to translate an arbitrary planar diagram into worldsheet form (and eventually calculate it), it is an important intermediate goal to understand how to do standard Yang–Mills perturbation theory in “worldsheet–friendly” language. In [39, 40, 41] CQT do exactly that for the simplest case, that of one–loop diagrams in Yang–Mills theory, by analysing how familiar features like renormalisation are affected by the unusual regularisation procedure and other special features of the lightcone worldsheet formalism. To conclude this brief overview of the lightcone worldsheet formalism, the main point for our current purposes is that it provides motivation and justification for a slightly unusual regularisation of lightcone Yang–Mills, which we will now describe. Let us momentarily focus on the self–dual part of the lightcone Yang–Mills Lagrangian: L = L−+ + L++− = −Az̄�Az + 2ig[Az, ∂+Az̄](∂+)−1(∂z̄Az) . (2.9) This action provides one of the representations of self-dual Yang-Mills theory. After trans- forming to momentum space, we find that the only vertex in the theory is the following (suppressing the gauge index structure): A2 A1 = −2g [p1+p z̄ − p2+p1z̄] = − [12] . (2.10) As for propagators, following [40], we will use the Schwinger representation: dTe+Tp . (2.11) In (2.11) p2 is understood to be the appropriate (p2 < 0) Wick rotated version of the Minkowski space inner product. For our choice of signature, the latter is p · q = p+q− + p−q+ − p · q = p+q− + p−q+ − (pzqz̄ + pz̄qz) , (2.12) so that p2 = 2(p+p− − pzpz̄). We will also make use of the dual or “region momentum” representation, where one assigns a momentum to each region that is bounded by a line in the planar diagram. By convention, the actual momentum of the line is given by the region momentum to its right minus that on its left, as given by the direction of momentum flow5. Clearly such a pre- scription can only be straightforwardly implemented for planar diagrams, which is the case considered in [40]. This is also sufficient for our purposes, since we are calculating the lead- ing single–trace contribution to one–loop scattering amplitudes. Non–planar (multi–trace) contributions can be recovered from suitable permutations of the leading–trace ones (see e.g. [52]). To demonstrate the use of region momenta, a sample one–loop diagram is pictured in Figure 1. Figure 1: A sample one–loop diagram indicating the labelling of region momenta. The outgoing leg momenta are p1 = k1 − k4 , p2 = k2 − k1 , p3 = k3 − k2 , p4 = k4 − k3, while the loop momentum (directed as indicated) is l = q − k1. 5In [40] the flow of momentum is chosen to always match the flow of helicity, but we will not use this convention. The “worldsheet–friendly” regulator that CQT employ is simply defined as follows [49]: For a general n–loop diagram, with qi being the loop region momenta, one simply inserts an exponential cutoff factor exp(−δ q2i ) (2.13) in the loop integrand, where δ is positive and will be taken to zero at the end of the calcula- tion. This clearly regulates UV divergences (from large transverse momenta), but, as we will see, has some surprising consequences since it will lead to finite values for certain Lorentz– violating processes, which therefore have to be cancelled by the introduction of appropriate counterterms. Note that q2 = 2qzqz̄ has components only along the two transverse directions, hence it breaks explicitly even more Lorentz invariance than the lightcone usually does. This might seem rather unnatural from a field-theoretical point of view, however it is crucial in the lightcone worldsheet approach. Indeed, the lightcone time x− and x+ (or in practice its dual momentum p+) parametrise the worldsheet itself, and are regulated by discretisation; thus, they are necessarily treated very differently from the two transverse momenta qz, qz̄ which appear as dynamical worldsheet scalars. Fundamentally, this is because of the need to preserve longitudinal (x+) boost invariance (which eventually leads to conservation of discrete p+). The fact that the regulator depends on the region momenta rather than the actual ones is a consequence of asking for it to have a local description on the worldsheet. The main ingredient for what will follow later in this paper is the computation of the (++) one–loop gluon self–energy in the regularisation scheme discussed earlier. This is performed on page 10 of [40], and we will briefly outline it here. This helicity–flipping gluon self–energy, which we denote by Π++, is the only potential self–energy contribution in self–dual Yang–Mills; in full YM we would also have Π+− and, by parity invariance, Π−−. There are two contributions to this process, corresponding to the two ways to route helicity in the loop, but they can be easily shown to be equal so we will concentrate on one of them, which is pictured in Figure 2. Figure 2: Labelling of one of the selfenergy diagrams contributing to Π++. In Figure 2, p,−p are the outgoing line momenta, l is the loop line momentum, and k, k′, q are the region momenta, in terms of which the line momenta are given by p = k′ − k, l = q − k′ . (2.14) Remembering to double the result of this diagram, we find the following expression for the self–energy: Π++ =8g2N (2π)4 −(p + l)+ (p+lz̄ − l+pz̄) l2(p+ l)2 (−p+)(p+ l)+ ((−p+)(pz̄ + lz̄)− (p+ + l+)(pz̄)) (p+)2 (p+lz̄ − l+pz̄)(p+(pz̄ + lz̄)− (p+ + l+)pz̄) l2(p + l)2 (2.15) Although we are suppressing the colour structure, the factor of N is easy to see by thinking of the double–line representation of this diagram6. One of the crucial properties of (2.15) is that the factors of the loop momentum l+ coming from the vertices have cancelled out, hence there are no potential subtleties in the loop integration as l+ → 0. This means that, although for general loop calculations one would have to follow the DLCQ procedure and discretise l+ (as is done for other processes considered in [39, 40, 41]), this issue does not arise at all for this particular integral, and we are free to keep l+ continuous. To proceed, we convert momenta to region momenta, rewrite propagators in Schwinger representation, and regulate divergences using the regulator (2.13). Employing the unbroken shift symmetry in the + region momenta to further set k+ = 0, (2.15) can be recast as: Π++ = dT1dT2 (k′+) eT1(q−k) 2+T2(q−k ′)2−δq2× k′+(qz̄ − k′z̄)− (q+ − k′+)(k′z̄ − kz̄) k′+(qz̄ − kz̄)− q+(k′z̄ − kz̄) (2.16) Since q− only appears in the exponential, the q− integration will lead to a delta function containing q+, which can be easily integrated and leads to a Gaussian–type integral for qz, qz̄. Performing this integral, we obtain (setting T = T1 + T2, x = T1/(T1 + T2)) Π++ = dT δ2 [xkz̄ + (1− x)k′z̄]2 (T + δ)3 eTx(1−x)p 2− δT (xk+(1−x)k′)2 . (2.17) Notice that, had we not regularised using the δ regulator, we would have obtained zero at this stage. Instead, now we can see that there is a region of the T integration (where T ∼ δ) that can lead to a nonzero result. On performing the T and x integrations, and sending δ to zero at the end, we obtain the following finite answer: Π++ = 2 (kz̄) 2 + (k′z̄) 2 + kz̄k . (2.18) 6 For simplicity, we take the gauge group to be U(N). Notice that this nonvanishing, finite result violates Lorentz invariance, since it would imply that a single gluon can flip its helicity. Also, it explicitly depends on only the z̄ components of the region momenta. Such a term is clearly absent in the tree-level Lagrangian (unlike e.g. the Π+− contribution in full Yang–Mills theory), thus it cannot be absorbed through renormalisation – it will have to be explicitly cancelled by a counterterm. This counterterm, which will play a major rôle in the following, is defined in such a way that: + = 0 , (2.19) in other words it will cancel all insertions of Π++, diagram by diagram. Let us note here that, had we been doing dimensional regularisation, all bubble contributions would vanish on their own, so there would be no need to add any counterterms. So this effect is purely due to the “worldsheet–friendly” regulator (2.13). It is also interesting to observe that in a supersymmetric theory this bubble contribution would vanish7 so this effect is only of relevance to pure Yang–Mills theory. 2.3 The one–loop (++++) amplitude Now let us look at the all–plus four-point one–loop amplitude in this theory. It is easy to see that it will receive contributions from three types of geometries: boxes, triangles and bubbles. It is a remarkable property8 that the sum of all these geometries adds up to zero. In particular, with a suitable routing of momenta, the integrand itself is zero. Pictorially, we can state this as: + 4× + 2× + 8× = 0 . (2.20) The coefficients mean that we need to add that number of inequivalent orderings. So we see (and refer to [40] for the explicit calculation) that the sum of all the diagrams that one can construct from the single vertex in our theory, gives a vanishing answer. However, as discussed in the previous section, this is not everything: we need to also include the contribution of the counterterm that we are forced to add in order to preserve Lorentz invariance. Since this counterterm, by design, cancels all the bubble graph contributions, we are left with just the sum of the box and the four triangle diagrams. In pictures, A++++ = +4× + 2× + 8× + 2× + 8× (2.21) 7This can in fact be derived from the results of [53], where similar calculations were considered with fermions and scalars in the loop. 8This observation is attributed to Zvi Bern [40]. where A++++ is the known result [54] for the leading–trace part of the four–point all-plus amplitude: A++++(A1A2A3A4) = i [12][34] 〈12〉〈34〉 , (2.22) and the terms in the parentheses clearly cancel among themselves. This leaves the box and triangle diagrams, which are exactly those appearing in the calculation of the parity conjugate amplitude using dimensional regularisation [35], where the bubbles were zero to begin with. Following [40], we make the obvious, but important for the following, observation that one can change the position of the parentheses: A++++ = + 4× + 2× + 8× +2× +8× (2.23) where again the terms in the parentheses are zero (by (2.20)). This demonstrates that one can compute the all-plus amplitude just from a tree-level calculation with counterterm insertions (of course, these diagrams are at the same order of the coupling constant as one– loop diagrams because of the counterterm insertion). This remarkable claim is verified in [40], where CQT explicitly calculate the 10 counterterm diagrams and recover the correct result for the four-point amplitude (see pp. 22-23 of [40])9. This result, apart from being very appealing in that one does not have to perform any integrals (apart from the original integral that defined the counterterm) so that the calcula- tion reduces to tree–level combinatorics, will also turn out to be a convenient starting point for performing the Mansfield transformation. Specifically, our claim is that the whole series of all-plus amplitudes will arise just from the counterterm action. In the following we will show how this works explicitly for the four-point all-plus case, and then we will argue for the n-point case that the corresponding expression derived from the counterterm has all the correct singularities (soft and collinear), giving strong evidence that the result is true in general. 3 The all-plus amplitudes from a counterterm Having reviewed the relevant new features that arise when doing perturbation theory with the worldsheet–motivated regulator of [49], we now have all the necessary ingredients to perform the Mansfield change of variables on the regulated lightcone Lagrangian. In this section, we will carry out this procedure. We will first regulate lightcone self–dual Yang–Mills, which, as 9In practice, these authors choose to insert the self-energy result (2.18) in the tree diagrams, so what they compute is minus the all–plus amplitude. discussed, will require us to introduce an explicit counterterm in the Lagrangian. Then we will perform the Mansfield transformation on the original Lagrangian (converting it to a free theory). We will then show that, upon inserting the change of variables into the counterterm Lagrangian, we recover the all–plus amplitudes as vertices in the theory. 3.1 Mansfield transformation of LCT As we saw, the “worldsheet-friendly” regularisation requires us to add a certain counterterm to the lightcone Yang–Mills action, required in order to cancel the Lorentz-violating helicity– flipping gluon selfenergy. As mentioned previously, the calculation of the all–plus amplitude can be tackled purely within the context of self-dual Yang–Mills, which we will focus on from now on. We see that, as a result of this regularisation, the complete action at the quantum level becomes: L(r)SDYM = L+− + L++− + LCT , (3.1) where L+− + L++− is the classical Lagrangian for self-dual Yang-Mills introduced in (2.9). Although CQT do not write down a spacetime Lagrangian for LCT, it is easy to see that the following expression would have the right structure: LCT = − d3kid3kj Ai j(k i, kj)[(kiz̄) 2 + (k 2 + kiz̄k j , ki) . (3.2) This expression depends explicitly on the dual, or region, momenta. In (3.2) we have made use of the simplest way to associate region momenta to fields, which is to assign a region momentum to each index line in double–line notation [46], and thus a momentum ki, kj to each of the indices of the gauge field Ai j (now slightly extended into a dipole, as would be natural from the worldsheet perspective, where an index is associated to each boundary). Since each line has a natural orientation, the actual momentum of each line can be taken to be the difference of the index momentum of the incoming index line and the outgoing index line. So the momentum of Ai j(k i, kj) is taken to be p = kj−ki. As discussed above, this assignment can only be performed consistently for planar diagrams, which is sufficient for our purposes. Clearly, the structure of (3.2) is rather unusual. First of all, it depends only on the antiholomorphic (z̄) components of the region momenta, and so is clearly not (lightcone) covariant. Even more troubling is the fact that it does not depend only on differences of region momenta, but also on their sums. Since each field thus carries more information than just its momentum, LCT is a non–local object from a four–dimensional point of view (although, as shown in [40], it can be given a perfectly local worldsheet description). Leaving the above discussion as food for thought, we will now rewrite (3.2) in a more conventional way that is most convenient for inserting into Feynman diagrams, LCT = − d3p d3p′ δ(p+ p′) Ai j(p ′)((kiz̄) 2 + (k 2 + kiz̄k i(p) . (3.3) In this expression, which can be thought of as the zero–mode or field theory limit of (3.2), all the region momentum dependence is confined to the polynomial factor (kiz̄) 2+kiz̄k z̄. This vertex, inserted into tree diagrams, would exactly reproduce the effects of the counterterm pictured in (2.19). Although (3.3) still exhibits some of the apparently undesirable features we discussed above, the calculations in [40] demonstrate that, after summing over all possible insertions of this term, the final result is covariant and correctly reproduces the all–plus amplitudes10. Therefore, we believe that its problematic properties are really a virtue in disguise, and (as we will see explicitly) they seem to be crucial in obtaining the full series of n–point all–plus amplitudes from the Mansfield transformation of a single term. We are now ready to perform the Mansfield change of variables. In the spirit of the discussion earlier, we will perform the transformation on the classical part of the action only: L+−(A, Ā) + L++−(A, Ā) = L+−(B, B̄) (3.4) Hence the classical part of the action has been converted to a free theory. Without a regulator, this would be the whole story. However we now see that, within the particular regularisation we are working with, the full Lagrangian L(r)SDYM contains one extra, one–loop piece, given by LCT in (3.3), which is quadratic in the positive helicity fields A. To complete the Mansfield transformation, we will clearly need to expand this term in the new fields B, using the Ettle–Morris coefficients (2.4). Since LCT depends only on the holomorphic A fields, we will only need the expansion of A in terms of B given in (2.4). As a first check that LCT leads to the right kind of structure, note that since A depends only on the holomorphic B fields, all the new vertices are all–plus. Thus, the full action, when expressed in terms of the B fields, takes the schematic form: L(r)SDYM(A, Ā) = L+−(B, B̄) + L++(B) + L+++(B) + L++++(B) + · · · (3.5) In the next section we will calculate the four–point term L++++ and demonstrate that, when restricted on–shell, it reproduces the known form (2.22) for the all–plus amplitude. 3.2 The four–point case To begin with, we focus on the derivation of the four-point all-plus vertex, whose on-shell version will give us the four-point scattering amplitude. We will thus expand the old fields A in the counterterm (3.3) (or (3.2)) up to terms containing four B-fields. When inserting the Ettle–Morris coefficients into (3.3), one has to sum over all possible cyclic orderings with which this can be done. A complication is that now the counterterm 10Note that similar–looking treatments using index momenta instead of line momenta for vertices, but which in the end sum up to covariant results have appeared in the context of noncommutative geometry (see e.g. [55]). Although it is possible to write e.g. (3.2) in star–product form, at this stage it is not clear whether that is a useful reformulation. itself depends on the ordering. In other words, we need to sum over all the ways of assigning dual momenta to the indices. Schematically, the inequivalent terms that we obtain are: AA → ( B1B2)( B3B4) + ( B2B3)( B4B1) B1B2B3)B4 + ( B2B3B4)B1 + ( B3B4B1)B2 + ( B4B1B2)B3 , (3.6) where the terms on the first line arise from doing two quadratic substitutions and those on the second from doing one cubic substitution. All the other possibilities are related by cyclicity of the trace. For definiteness, let us now write down what one of these terms means explicitly:11 B1B2B3)B4 = −2g2 tr dpdp4δ(p+p4) dp1dp2dp3δ(p−p1−p2−p3) p+√ 〈12〉〈23〉× × B(p1)B(p2)B(p3) (k3z̄) 2 + (k4z̄) 2 + k4z̄k B(p4) = 2g2 dp1dp2dp3dp4δ(p1 + p2 + p3 + p4)× (k3z̄) 2 + (k4z̄) 2 + k4z̄k 〈12〉〈23〉 tr B(p1)B(p2)B(p3)B(p4) (3.7) The reason this particular combination of kz̄’s appears here is that, given the ordering we chose, after the Mansfield transformation the counterterm ends up being on leg 4, and its line bounds the regions with momenta k3 and k3. This is represented pictorially in Figure 3. Figure 3: One of the contributions to the four–point all-plus vertex. Although Figure 3 might suggest that there is a propagator between the counterterm insertion and the location of the original A, which has now split into three B’s, this is of course not the case since the whole expression is a vertex at the same point. We have drawn the diagram in this fashion to emphasise which leg the counterterm is located on after the transformation. On the other hand, this vertex is nonlocal (as discussed above, it was nonlocal even in the original variables, but this is now compounded by the Mansfield coefficients, which contain momenta in the denominator), so this notation serves as a useful reminder of that fact. 11We suppress the overall factor of −g2N/(12π2) until the end of this section. Also, the integrals are implicitly taken to be on the quantisation surface Σ. It is interesting to note that (3.7) is essentially the same expression as the sum of the two channels with the same region momentum dependence that appear in CQT’s calculation of this amplitude using tree–level diagrammatics (compare with Eq. 83 in [40]), which we illustrate in Fig. 4. Thus we have a picture where one post–Mansfield transform vertex (with Figure 4: The two diagrams with counterterm insertions on leg 4 that arise in the calculation of CQT, and, combined, add up to the contribution in Fig. 3. B’s) effectively sums two tree–level pre–transformation (with A’s) Feynman diagrams. This is a first indication that our calculation of the all–plus vertex can be mapped, practically one–to–one, to that of the all–plus amplitude on pp. 22-23 of [40]. Another type of contribution to the vertex arises when we transform both of the A’s in LCT. One of the two terms that we find is: B2B3)( B4B1) = −2g2 tr dp dp′δ(p+ p′) dp2dp3δ(p− p2 − p3) p+√ 〈23〉B(p 2)B(p3) (k1z̄) 2 + (k3z̄) 2 + k1z̄k dp4dp1δ(p′ − p4 − p1) 〈41〉B(p 4)B(p1) = −2g2 dp1 · · ·dp4δ(p1+p2+p3+p4) (p2+ + p + + p ((k1z̄) 2 + (k3z̄) 2 + k1z̄k 〈23〉〈41〉 B(p1)B(p2)B(p3)B(p4) (3.8) This contribution can also be mapped to one of the two terms with bubbles on internal lines in CQT. We can now tabulate all the terms that we obtain in this way by making the schematic form (3.6) precise. Since the delta–function and trace over B parts are the same for all these terms, in Table 1 we just list the rest of the integrand. To obtain the final form of the vertex, we are now instructed to sum over all these contributions. Thus we can write L++++(B) = 2g2 dp1dp2dp3dp4δ(p1+p2+p3+p4) V(4) tr[B(p1)B(p2)B(p3)B(p4)] (3.9) Schematic form Pictorial form Integrand B1B2B3)B4 +k3k4 〈12〉〈23〉 B2B3B4)B1 +k1k4 〈23〉〈34〉 B3B4B1)B2 +k2k1 〈34〉〈41〉 B4B1B2)B3 +k2k3 〈41〉〈12〉 B2B3)( B4B1) − +k1k3 〈23〉〈41〉 B1B2)( B3B4) − +k4k2 〈34〉〈12〉 Table 1: The various contributions to the all–plus four–point vertex. Note that we use the simplifying notation ki := k where V(4) is given by the following expression:12 V(4) = 1√ 〈12〉〈23〉〈34〉〈41〉× 3 + k 4 + k3k4)〈34〉〈41〉+ p1+ 1 + k 4 + k1k4)〈12〉〈41〉 + p2+ 2 + k 1 + k2k1)〈12〉〈23〉+ p3+ 3 + k 2 + k2k3)〈23〉〈34〉 − (p2+ + p3+)(p1+ + p4+)(k21 + k23 + k1k3)〈12〉〈34〉 − (p3+ + p4+)(p2+ + p1+)(k24 + k22 + k4k2)〈23〉〈41〉 (3.10) Comparing this to the expected answer (2.22), we see that the (quadratic) antiholomorphic momentum dependence should arise from the various kz̄ factors in (3.10). In [40], CQT start from essentially the same expression and demonstrate that it gives the correct result for the all-plus amplitude. Therefore, following practically the same steps as those authors, we can easily see that we obtain the expected answer. However, since we would like to find the full vertex V, we will need to keep off–shell information, and so we will choose a slightly different route. 12For the sake of brevity we omit a subscript z̄ in the region momenta appearing in (3.10). The main complication in bringing (3.10) into a manageable form is clearly the presence of the region momenta. We would like to disentangle their effects as cleanly as possible. Therefore, our derivation will proceed by the following steps: 1. First, we will show that (3.10) can be manipulated so that the quadratic dependence on region momenta drops out, leaving only terms linear in the region momenta. 2. Second, we will decompose the resulting expression into a part that depends on the region momenta and one that does not. The k–dependent part turns out to have a very simple form, and vanishes on–shell. 3. Finally, we will show that the k–independent part reduces to the known amplitude. For the first step, we will need the following identity, which is proved in appendix B: +〈34〉〈41〉+ p1+ +〈12〉〈41〉+ p2+ +〈12〉〈23〉+ p3+ +〈23〉〈34〉 − (p2+ + p3+)(p1+ + p4+)〈12〉〈34〉 − (p3+ + p4+)(p2+ + p1+)〈23〉〈41〉 = 0 (3.11) Also, using the shorthand notation Kij := (k 2 + (k 2 + kiz̄k z̄: we note the following very useful identity: Kij = Kik + (k z̄ − kkz̄ )(kiz̄ + k z̄ + k z̄ ) = Kik + (k z̄ − kkz̄ )lijk (3.12) where 1 ≤ k ≤ n and lijk = kiz̄+k z̄ . Noting that, for j > k, k z̄−kkz̄ = pk+1z̄ +pk+2z̄ + · · · p we can use this to rewrite all the region momentum combinations appearing in (3.10) in the following way: K34 = (K12 +K23 +K34 +K41 + (p̄3 + p̄4)(l124 + l234) + 2(p̄2 + p̄3)l134) K14 = (K12 +K23 +K34 +K41 − (p̄2 + p̄3)(l134 + l123) + 2(p̄3 + p̄4)l124) K12 = (K12 +K23 +K34 +K41 − (p̄3 + p̄4)(l124 + l234)− 2(p̄2 + p̄3)l123) K23 = (K12 +K23 +K34 +K41 + (p̄2 + p̄3)(l134 + l123)− 2(p̄3 + p̄4)l234) K13 = (K12 +K23 +K34 +K41 + (p̄3 − p̄2)l123 + (p̄1 − p̄4)l134) K24 = (K12 +K23 +K34 +K41 + (p̄4 − p̄3)l234 + (p̄2 − p̄1)l124) (3.13) where we have introduced the notation p̄i = p z̄. We have thus expressed all the quadratic region momentum dependence in terms of the common factor K12 +K23 +K34 +K41, and, given (3.11), it is clear that this contribution will vanish.13 13One could have chosen a different combination of the Kij ’s, but we find the symmetric choice in (3.13) convenient. After this step, we are left with an expression which is linear in the region momenta. We will now proceed in a similar way, and rewrite all the expressions that contain lijk in terms of a suitably chosen common factor: l124 + l234 = (k1z̄ + k z̄ + k z̄ + k (p1z̄ + p l134 + l123 = (k1z̄ + k z̄ + k z̄ + k (p2z̄ + p 2l234 = (k1z̄ + k z̄ + k z̄ + k z̄) + (2p2z̄ + p z̄ − p1z̄) 2l123 = (k1z̄ + k z̄ + k z̄ + k z̄) + (2p1z̄ + p z̄ − p4z̄) 2l134 = (k1z̄ + k z̄ + k z̄ + k z̄) + (2p3z̄ + p z̄ − p2z̄) 2l124 = (k1z̄ + k z̄ + k z̄ + k z̄) + (2p4z̄ + p z̄ − p3z̄) (3.14) In appendix B we show that the total coefficient of the common (k1z̄ + k z̄ + k z̄ + k z̄) factor is +(+(p̄3 + p̄4) + (p̄2 + p̄3))〈34〉〈41〉+ p1+ +(−(p̄2 + p̄3) + (p̄3 + p̄4))〈12〉〈41〉 + p2+ +(−(p̄3 + p̄4)− (p̄2 + p̄3))〈12〉〈23〉+ p3+ +(+(p̄2 + p̄3)− (p̄3 + p̄4))〈23〉〈34〉 − (p2+ + p3+)(p1+ + p4+)( (p̄3 − p̄2) + (p̄1 − p̄4))〈12〉〈34〉 − (p3+ + p4+)(p2+ + p1+)( (p̄4 − p̄3) + (p̄2 − p̄1))〈23〉〈41〉 = − 3 [(12) + (23) + (34) + (41)] (3.15) where (pi) 2 is the full covariant momentum squared, and (ij) = pi+p z − p z. Thus we see that the complete dependence on the region momenta can be rewritten as follows: = − 3 (12) + (23) + (34) + (41) 〈12〉〈23〉〈34〉〈41〉 . (3.16) It is rather satisfying that the region momentum dependence of the vertex takes this simple form, which clearly vanishes when the external legs are on–shell, and thus will not contribute to the all–plus amplitudes. Having completely disentangled the region momenta kz̄ from the actual momenta pz̄, we will now focus on the terms containing only the latter, which were produced during the decompositions in (3.14). After a few simple manipulations, they can be rewritten as14 V (4)p = +[(p̄1 + p̄2)(p̄1 − p̄2) + (p̄3 + p̄2)(p̄3 − p̄2)]〈34〉〈41〉 + p1+ +[(p̄2 + p̄3)(p̄2 − p̄3) + (p̄4 + p̄3)(p̄4 − p̄3)]〈41〉〈12〉 + p2+ +[(p̄3 + p̄4)(p̄3 − p̄4) + (p̄1 + p̄4)(p̄1 − p̄4)]〈12〉〈23〉 + p3+ +[(p̄4 + p̄1)(p̄4 − p̄1) + (p̄2 + p̄1)(p̄2 − p̄1)]〈23〉〈34〉 − (p2+ + p3+)(p1+ + p4+)[(p̄3 − p̄2)(p̄1 − p̄4)− (p̄1 + p̄2)2]〈12〉〈34〉 − (p3+ + p4+)(p2+ + p1+)[(p̄4 − p̄3)(p̄2 − p̄1)− (p̄2 + p̄3)2]〈23〉〈41〉 (3.17) This expression, together with (3.16) is our proposal for the off–shell four–point all–plus vertex that should be part of the MHV-rules formalism at the quantum level. It would be very interesting to elucidate its structure and bring it into a more compact form. For the moment, however, we will be content to demonstrate that (3.17) is equal on shell to the sought–for amplitude. To that end, we will follow a similar approach to CQT, and rewrite all the holomorphic spinor brackets in terms of the following three: 〈12〉〈34〉, 〈23〉〈41〉, 〈12〉〈41〉. To achieve this, we use momentum conservation and a certain cyclic identity (see appendix A) to write +〈34〉〈41〉 = p4+ p3+〈42〉 − p4+〈23〉 +〈42〉 − (p4+)2 p1+〈12〉 − p3+〈32〉 − (p4+)2〈23〉 = p4+ +〈12〉〈41〉 − p4+(p4+ + p3+)〈23〉〈41〉 . (3.18) In a similar way, we can show that +〈12〉〈23〉 = p2+ +〈12〉〈41〉 − p2+(p2+ + p3+)〈34〉〈12〉 , +〈23〉〈34〉 =− p3+(p +)〈12〉〈34〉 − p3+(p1++p2+)〈23〉〈41〉+p3+ +〈12〉〈14〉 (3.19) Collecting all the terms together, and manipulating the resulting expressions, it is straight- 14We write V (4) = +〈12〉〈23〉〈34〉〈41〉V(4). forward to show that (3.17) simplifies to just V (4)p = 〈23〉〈41〉{34}(p1+ + p2+)[(p̄1 − p̄2)− (p̄2 + p̄3)] +〈12〉〈34〉{23}(p2+ + p3+)[(p̄1 + p̄2) + (p̄1 − p̄4)] +〈12〉〈41〉 (p̄1 + p̄2)({41}+ {32})+(p̄2 + p̄3)({12}+ {43}) (3.20) where we use the notation [34] {ij} = pi+p z̄ − pj+piz̄ = (1/ +[ij]. Converting to the usual antiholomorphic bracket notation, we rewrite (3.20) as V (4)p = 〈23〉〈41〉 +[34](p + + p +)[(p̄1−p̄2)− (p̄2+p̄3)] + 〈12〉〈34〉 +[23](p + + p +)[(p̄1+p̄2) + (p̄1−p̄4)] + 〈12〉〈41〉 (p̄1 + p̄2)(p +[41] + p +[32]) + (p̄2 + p̄3)(p +[12] + p +[43]) (3.21) Note that so far this expression is completely off shell. We will now show that on shell it reduces to the known result (2.22). In doing this we will keep track of the p2 terms that appear when applying momentum conservation in the form 〈ik〉[kj] = . (3.22) These terms are collected in appendix B. We start by rewriting each of the terms in the last two lines of (3.21) as follows 〈12〉〈41〉[41] p1+ +(p̄1 + p̄2) = −〈23〉〈41〉[34] p1+ +(p̄1 + p̄2) 〈12〉〈41〉[32] p3+ +(p̄1 + p̄2) = −〈12〉[32]〈42〉p2+p3+(p̄1 + p̄2) − 〈12〉〈34〉[23] p3+ +(p̄1 + p̄2) 〈12〉〈41〉[12] p1+ +(p̄2 + p̄3) = −〈12〉〈34〉[23] p1+ +(p̄2 + p̄3) 〈12〉〈41〉[43] p3+ +(p̄2 + p̄3) = −〈41〉〈23〉[34] p3+ +(p̄2 + p̄3) − 〈41〉[43]〈42〉p4+p3+(p̄2 + p̄3) . (3.23) We also transform the 〈12〉〈34〉 term using the Schouten identity and also momentum con- servation, 〈12〉〈34〉[23] +=〈23〉〈41〉[34] ++〈14〉〈23〉[13] +−〈13〉〈42〉[23] + , (3.24) and add up all contributions to the 〈23〉〈41〉 term, which are 〈23〉〈41〉[34] 4(p2+p̄1 − p1+p̄2) + 2(p3+p̄1 − p1+p̄3) 〈23〉〈41〉[34] +[4{21}+ 2{31}] . (3.25) Converting to the spinor bracket, the first of these terms is +[12]〈23〉[34]〈41〉 , (3.26) while the remaining terms from (3.23) and (3.24) combine to give 〈14〉〈23〉[13] + − 〈13〉〈42〉[23] (p2+ + p +)[(p̄1+p̄2) + (p̄1−p̄4)] + 〈12〉[32]〈42〉p2+[p2+(p̄1 + p̄2)− p4+(p̄2 + p̄3)] =− 〈14〉[13]〈12〉p3+(p2+ + p3+)[(p̄1+p̄2) + (p̄1−p̄4)] + 〈12〉[32]〈42〉p2+[p2+(p̄1 + p̄2)− p4+(p̄2 + p̄3)] =− 〈14〉[13]〈12〉p3+(2(p2+ + p3+)p̄1 − 2p1+(p̄2 + p̄3)) = 2〈14〉[13]〈12〉p3+{41} (3.27) (where we suppress an overall 1/(4 2)) and we see that (3.27) cancels the second term in (3.25), thus showing that (3.26) is the complete on-shell answer. Reintroducing all the prefactors, we thus find that the amplitude is A(4) = − g 〈12〉〈23〉〈34〉〈41〉 × +[12]〈23〉[34]〈41〉 [12][34] 〈12〉〈34〉 . (3.28) Now note that, as discussed in appendix A, in order to convert to the usual Yang–Mills theory normalisation we need to send g → g/ 2. We conclude that A(4) gives precisely the result (2.22) for the all–plus scattering amplitude. 3.3 The general all–plus amplitude We have just given an explicit derivation of the four point all-plus amplitude, from the two-point counterterm (3.3). We will argue in the following that this two-point counterterm contains all the all-plus amplitudes. First, we can see immediately that the counterterm (3.3) has the right kind of structure. Consider the n–point all–plus amplitude [56]: A(n) = 1≤i<j<k<l≤n 〈ij〉[jk]〈kl〉[li] 〈12〉 · · · 〈n1〉 . (3.29) In terms of spinor brackets this amplitude has terms of the form 〈 〉2−n[ ]2. A quick look at the Ettle-Morris coefficients shows that, for an n–point vertex coming from LCT, they contribute exactly 2 − n powers of the spinor brackets 〈 〉. Furthermore, there are exactly two powers of [ ] coming from the counterterm Lagrangian LCT ∼ (k2z̄)A2 – one for each power of k. Thus the general structure of LCT is appropriate to reproduce (3.29). Pictorially, we can represent the general n–point amplitude, arising from the counterterm in the new variables, as in Figure 5. ki kj Figure 5: The structure of a generic term contributing to the n–point vertex. All momenta are taken to be outgoing, and all indices are modulo n. Thus we can write this n–point all–plus vertex as follows: A(n)+···+ = 1···n δ(p+ p′) 1≤i<j≤n Y(p; j + 1, . . . , i) (kiz̄) 2 + (k 2 + kiz̄k Y(p′; i+ 1, . . . , j)× × tr[BiBi+1 · · ·BjBj+1 · · ·Bi−1] 2i)n−2 1···n δ(p1+· · ·+pn) 1≤i<j≤n + + · · ·+ pi+) 〈j + 1, j + 2〉 · · · 〈i− 1, i〉× (kiz̄) 2 + (k 2 + kiz̄k ) (pi+1+ + · · ·+ pj+) pi+1+ p 〈i+ 1, i+ 2〉 · · · 〈j − 1, j〉tr[B1 · · ·Bn] . (3.30) Focusing only on the relevant part of the above expression, and ignoring all coefficients, the general structure we obtain is the following: V(n)+···+ = 〈12〉 · · · 〈n1〉× 1≤i<j≤n 〈j, j + 1〉〈i, i+ 1〉 + − ki+)2((kiz̄)2 + (k 2 + kiz̄k  (3.31) where we have extracted the denominator at the expense of introducing the two missing holomorphic factors 〈j, j + 1〉 and 〈i, i+ 1〉 in the numerator. We also made use of the fact kj − ki = pi+1 + pi+2 + · · ·+ pj = −(pj+1 + pj+2 + · · ·+ pi) , (3.32) applied to the + components, to rewrite the two p+ sums in the numerator in terms of the k’s (this gives rise to a minus which we suppress). It is easy to verify that, for n = 4, this sum reproduces the 6 contributions that appeared in the four–point case, and (as we explicitly showed above) combined to give the expected answer. Therefore, we would like to propose that the vertex (3.31) will reduce on–shell to an expression proportional to (3.29). We will not attempt to prove this statement here15, but will instead move on to study the general properties of the n-point expression (3.30). Whilst the explicit calculation for the four point case was rather involved as we saw earlier, the study of the general properties of the n–point amplitudes proves much simpler. In particular, we will show that the collinear and soft limits of the expressions proposed for the n–point case can be very easily shown to be correct. Let us start by introducing some simplifying notation. One can write the change of variables for the A field as A1 = Y12B2 +Y123B2B3 +Y1234B2B3B4 + · · · , (3.33) where Y12 = δ12, Y123 = , Y1234 = (23)(34) , (3.34) and generally Y12...n = 1+3+4+ . . . (n− 1)+ (23)(34) . . . (n− 1 n) (3.35) (for simplicity, we are dropping inconsequential constant factors in this discussion). This notation is similar to that of [34]. Integrations and the insertion of suitable delta functions are understood, and can be illustrated by comparing the short-hand expressions above with the full equations given earlier. It will prove convenient to define Kij = k i + k j + kikj, ki := k z̄. (3.36) We will use the expression Y•12...n in the following, where the dot in the first placemark in the Y means that one substitutes in that place the negative of the sum of the other momenta. Then the result which we have proved above for the four point amplitude V1234 can be expressed as V1234 =K43Y•4Y•123 +K14Y•1Y•234 +K21Y•2Y•341 +K32Y•3Y•412 +K31Y•23Y•41 +K24Y•12Y•34 , (3.37) 15It is perhaps interesting to remark that the proof would involve converting the double sum in (3.31) to the quadruple sum in (3.29)—a state of affairs which has appeared before in a rather different context [20]. or very simply V1234 = 1≤i<j≤4 KijY• j+1...iY• i+1...j . (3.38) It is clear that the general conjecture that all the n–point all plus amplitudes are generated from the two-point counterterm (3.3) translates into the proposal that the n-point all-plus amplitude V12...n is given by V12...n = 1≤i<j≤n KijY• j+1...iY• i+1...j , (3.39) Let us now show that the expression on the right-hand side of (3.39) has precisely the same soft and collinear limits as the known amplitude on the left-hand side. Collinear limits Under the collinear limit pi → zP , pi+1 → (1− z)P , P 2 → 0 , (3.40) the n-point amplitude V12...n behaves as V12...n → z(1− z) (i i+ 1) V12...i i+2...n , (3.41) where we relabel P → pi after the limit is taken (the i+ and (i i + 1) factors involve momenta rather than spinors, which is why the z-dependent factor is 1/z(1−z), rather than the conventional 1/ z(1− z)). Consider the behaviour of the right-hand side of (3.39) under the limit (3.40). The first point is that if the indices i, i + 1 lie on different Y’s, then there are no poles generated in this collinear limit. This is clear from the explicit expressions for the Y’s in (3.35). Thus we may ignore any terms of this type. It is then immediate from the explicit forms of the Y’s Y12...s → z(1− z) (i i+ 1) Y12...i i+2...s , (3.42) for any i = 2, . . . s − 1, with s ≤ n (the first index in Y never contributes in a collinear limit, as one can see from the conjecture (3.39)). Thus we see that the Y expressions have the right sort of collinear behaviour. It is straightforward to see that the K coefficients in (3.39) also get relabelled correctly in the collinear limit; they are not explicitly involved as they refer to pairs of momenta attached to different Y fields, and as we saw, these do not contribute. It is then immediate to see that the summation over the products of Y’s in (3.39) reduces correctly in the collinear limit to the required summation over products of Y’s with one fewer leg in total. Hence the proposal (3.39) for the amplitude has precisely the same collinear limits as the physical amplitude. Soft limits We also find that there is a simple derivation of the soft limits of the expression in (3.39). In the soft limit pj → 0 , (3.43) the n-point amplitude V12...n behaves as V12...n → S(j) V12...j−1 j+1...n , (3.44) where we assume cyclic ordering as usual, so that, for example, pn+1 = p1. The soft function S(j) is given in terms of the momentum brackets by S(j) = j+(j − 1 j + 1) (j − 1 j) (j j + 1) . (3.45) The Y functions have a simple behaviour under soft limits. One has immediately that in the soft limit pj → 0, Y12...s → S(j) Y12...j−1 j+1...s , (3.46) for j = 3, . . . s− 1 (with s ≤ n). For the soft limits corresponding to the case missing in the above, we need the results Y•s+1...j = Y•s+1...j−1 (j − 1)+ (j − 1 j) , Y•j...s = Y•j+1...s (j + 1)+ (j j + 1) , (3.47) which follow from the definitions of the Y’s, and (j + 1)+ (j j + 1) (j − 1)+ (j − 1 j) = j+(j − 1 j + 1) (j − 1 j) (j j + 1) = S(j) , (3.48) which follows from the cyclic identity i+(jk) + j+(ki) + k+(ij) = 0. Finally, from relabelling the K’s we have in the soft limit that Ksj → Ksj−1. Then it follows that in the soft limit Ksj Y•s+1...j Y•j+1...s +Ksj−1 Y•s+1...j−1 Y•j...s → S(j)Ksj−1 Y•s+1...j−1 Y•j+1...s , (3.49) as required. Again, it is then easy to see that the summation over the products of Y’s in (3.39) reduces correctly in the soft limit to the required summation over products of Y’s with one fewer leg in total. Hence the proposal (3.39) for the amplitude has precisely the same soft limits as the physical amplitude. 4 Discussion Whilst new, twistor-inspired methods for calculating amplitudes in gauge theory have led to much progress, the lack of a systematic action-based formulation which incorporates these new ideas has been an impediment to further developments. MHV diagrams have the two advantages of being closely allied to the twistor picture, as well as providing an explicit realisation of the dispersion and phase space integrals fundamental to unitarity- based methods. However, without an action formalism, standard MHV methods have so far been mainly restricted to massless theories at one-loop level, and to the cut-constructible parts of amplitudes. The advent of a classical MHV Lagrangian for gauge theory, derived from lightcone YM theory [32, 33, 34], provides the basis for transcending these limitations. In order for this to be realised, it is necessary to describe the quantum MHV theory. What we have done in this paper is to investigate this quantum theory. Using the regularisation methods of [39, 40, 41], we have provided arguments that the simplest one-loop counterterm in the quantum MHV theory – a two point vertex – provides an extraordinarily concise generating function for the infinite sequence of one-loop, all-plus helicity amplitudes in YM theory. We showed this by explicit calculation for the four-point case, and then proved that the soft and collinear limits of the conjectured n-point amplitude precisely matched those of the correct answer. We would like to emphasise that the simplicity of our approach — which reduced the calculations of the loop amplitudes we considered to tree–level algebraic manipulations— is largely due to the four–dimensional nature of the regularisation scheme we employed. By staying in four dimensions, we preserve the appealing features of the inherently four– dimensional field redefinition of [32, 33]. Based upon this result, it is very natural to conjecture that the full quantum YM theory is correctly described by this quantum MHV Lagrangian. The correct ingredients appear to be present. For example, in the approach of [39, 40, 41] there arise one-loop counterterms with helicities (++), (+ + −), (−−), (− − +). We studied the (++) counterterm in this paper, arguing that when expressed in the (B, B̄) variables this generates the full set of all- plus amplitudes. Transforming the (++−) counterterm to (B, B̄) variables will generate an infinite sequence of single–minus vertices. There will be other contributions to single-minus vertices from combinations of all-plus vertices and MHV vertices. It would be surprising if the combined contributions of these did not lead to the correct YM single-minus expressions. Certainly all of these have the correct powers of spinor brackets for this to be the case. Transforming the (−−) and (− − +) counterterms to (B, B̄) variables will lead to new contributions to MHV vertices16. The MHV vertices from the classical MHV Lagrangian only generate the cut-constructible parts of YM loop amplitudes, such as the one-loop MHV 16In the MHV case there are additional counterterms noted in [41] which may also need to be taken into account in future discussions. amplitude. These new contributions might be expected to lead to the missing, rational parts. This would also potentially explain why in [57] the combination of all-plus vertices with MHV tree vertices did not yield the correct single-minus amplitudes – these additional MHV contributions are missing. Further evidence for the conjecture that the quantum MHV Lagrangian is equivalent to quantum YM theory would be welcome. One could start with seeking explicit proofs of the above proposals. One can also investigate beyond massless one-loop gauge theory – an advantage of the Lagrangian approach is that the inclusion of masses, and of fermions and scalars, is in principle clear. There are other issues raised by this work. It is plausible that the potential quantum versions of the twistor space formulations of gauge theory [58, 59, 60] are most likely to be allied to the quantum theory discussed here – one simple reason for believing this is that the regularisation employed here keeps one in four dimensions. Perhaps there are simple twistor space analogues of the counterterms discussed above. Finally, although for our purposes the lightcone worldsheet approach to perturbative gauge theory provided simply the motivation for a particular choice of regularisation scheme, we believe that it would be fruitful to further explore possible connections between that framework and the twistor string programme. Addendum: We would like to thank Paul Mansfield and Tim Morris for having informed us that they have recently been pursuing research related to that presented in this paper. Their work, which is complementary to ours in that it employs dimensional regularisation, has now appeared in [61]. Acknowledgements It is a pleasure to thank Paul Heslop, Gregory Korchemsky, Paul Mansfield, Tim Morris and Adele Nasti for discussions. We would like to thank PPARC for support under the Rolling Grant PP/D507323/1 and the Special Programme Grant PP/C50426X/1. The work of GT is supported by an EPSRC Advanced Fellowship EP/C544242/1 and by an EPSRC Standard Research Grant EP/C544250/1. A Notation Lightcone conventions Here we summarise our lightcone conventions. We start off by introducing lightcone coordinates x± := x0 ± x3√ , xz := x1 + ix2√ , xz̄ := x1 − ix2√ . (A.1) We also have x+ = x−, x z = −xz̄ , and so on. The scalar product between two vectors A and B is written as A · B := A+B− + A−B+ − AzBz̄ −Az̄Bz . (A.2) We choose x− as our lightcone time coordinate, therefore the lightcone gauge used in this paper is defined by A− = 0 . (A.3) This condition can be written as η ·A = 0, where η is a constant null vector, chosen to have components η := (1/ 2, 0, 0, 1/ 2) (hence η− = 1, η+ = ηz = ηz̄ = 0). To any four-vector p we associate the bispinor paȧ defined by paȧ := p− −pz −pz̄ p+ . (A.4) We also define holomorphic and anti-holomorphic spinors as λa := , λ̃ȧ := , (A.5) from which it follows that λaλ̃ȧ := pzpz̄ −pz̄ p+ . (A.6) This is of course consistent with the on-shell condition p− = pzpz̄/p+. Furthermore, compar- ing (A.4) and (A.6) and choosing η as specified earlier, we see that a generic off-shell vector p can be decomposed as p = λλ̃ + zη , (A.7) where p−p+ − pzpz̄ 2(p · η) . (A.8) (A.7) and (A.8) are the familiar decompositions of off-shell vectors in the MHV literature [62, 17, 63, 15]. The off-shell holomorphic spinor product is defined as: 〈ij〉 = z − p , (A.9) whereas for the antiholomorphic spinors we define [ij] = z̄ − pj+piz̄ . (A.10) In these conventions, one finds 2(pi · pj) = 〈i j〉 [i j] + (pi)2 + (pj)2 , (A.11) or, in the case where pi and pj are on shell, 2(pi · pj) = 〈i j〉 [i j]. In the standard QCD literature conventions it is customary to define 2(pi · pj) = 〈i j〉 [j i]; this can be obtained by simply re-defining the inner product of two anti-holomorphic spinors, [i j], to be the negative of the right hand side of (A.10). Useful identities The form (A.9) is very convenient for deriving identities for 〈ij〉 that also involve the p+ components. For instance, one has: pi+〈jk〉+ +〈ki〉+ pk+〈ij〉 pi+(p z − pk+pjz) z − pi+pkz) pk+(p z − p = 0 . (A.12) It is also easy to see how to apply momentum conservation, take say 〈ij〉, and substitute pj = − k 6=j pk (for each component). (A.13) Then we have +〈ij〉 = pi+(− k 6=j p z) + ( k 6=j p k 6=j z − pk+piz k 6=j pk+〈ki〉 . (A.14) We have also used the momentum bracket notation from [34] (ij) = pi+p z − p z , {ij} = pi+p z̄ − pj+piz̄ . (A.15) Lightcone Yang–Mills action Here we give the form of the lightcone Yang–Mills action that we use in this paper. As discussed in more detail in [35], starting from the YM Lagrangian −(1/4) trF 2, imposing the lightcone gauge (A.3), and integrating out the A+ component which appears quadratically, the final lightcone theory contains only the two physical components Az and Az̄ [64, 65, 66], which we associate with positive and negative helicity respectively. The Lagrangian takes the simple form (2.1) LYM = L+− + L++− + L−−+ + L++−− , (A.16) L+− = −2 tr{Az̄(∂+∂− − ∂z∂z̄)Az} , L++− = 2ig tr{[Az, ∂+Az̄](∂+)−1(∂z̄Az)} , L−−+ = 2ig tr{[Az̄, ∂+Az](∂+)−1(∂zAz̄)} , L++−− = −2g2 tr{[Az̄, ∂+Az](∂+)−2[Az, ∂+Az̄]} . (A.17) Note that, in agreement with CQT, we have used the normalisation tr{T aT b} = δab. In order to convert to the usual conventions for Yang–Mills theory, we therefore need to rescale g → g/ Relation to the notation of CQT To compare our notation to that of [39, 40, 41], note that we employ outgoing momenta instead of incoming, therefore the all–plus amplitudes in these works would be all–minus from our perspective, and should thus be conjugated when comparing. Also, our time evolution coordinate is taken to be x− rather that x+, which (among other changes) implies that p+ of CQT becomes p+. Our metric is also taken to have opposite signature to that in CQT. Finally, CQT define momentum brackets K∧ij and K ij , which are just our (ij) and {ij} brackets respectively. B Details on the four–point calculation In this appendix we prove two results that were used in section 3, namely equations (3.11) and (3.15). To make the expressions more compact, instead of momentum brackets we use the following notation: fij = − . (B.1) The fij variables satisfy the simple relation: fij = fik + fkj , (B.2) while momentum conservation is applied as pi+fij = − pk+fkj . (B.3) Also, to minimise clutter, in this appendix we use the notation qi := p Proof of the quadratic identity In order to show (3.11), it is convenient to divide out by the + factor (which is there anyway in (3.10)) in order to bring it to the form q24f34f41 + q 1f12f41 + q 2f12f23 + q 3f23f34 − (q2 + q3)(q1 + q4)f12f34 − (q3 + q4)(q2 + q1)f23f41 = 0 , (B.4) Expanding out the two last terms in (B.4) as − (q1q3 + q2q4)(f12f34 + f23f41)− (q1q2 + q3q4)f12f34 − (q2q3 + q4q1)f23f41 , (B.5) we apply momentum conservation on each of the four components of the first term of (B.5), in the following way: − q1q3f12f34 = q1f12(q1f14 + q2f24) = −q21f12f41 + q1q2f12f24 , − q1q3f23f41 = q3f23(q2f42 + q3f43) = −q23f23f34 + q2q3f23f42 , − q2q4f12f34 = q4(q3f13 + q4f14)f34 = −q24f34f41 + q3q4f13f34 , − q2q4f23f41 = q2f23(q2f21 + q3f31) = −q22f12f23 + q2q3f31f23 . (B.6) Clearly these transformations have been chosen to cancel the first four terms in (B.4). Col- lecting the remaining terms, we obtain q1q2f12(f24 − f34) + q2q3f23(f42 + f31 − f41) + q3q4f34(f13 − f12)− q1q4f23f41 = q1q2f12f23 + q2q3f23f32 + q3q4f34f23 + q1q4f23f14 = f23[q2(q1f12 + q3f32) + q4(q3f34 + q1f14)] = f23[−q2(q4f42)− q4(q2f24)] (B.7) thus showing (3.11). Proof of the linear identity We will now outline the proof ot the linear (in region momenta) identity (3.15). Con- verting it to the notation used in the appendix, and performing simple manipulations, we find (suppressing the overall 3/8 factor): X = q24((p̄3 + p̄4) + (p̄2 + p̄3))f34f41 + q 1(−(p̄2 + p̄3) + (p̄3 + p̄4))f12f41 + q22(−(p̄3 + p̄4)− (p̄2 + p̄3))f12f23 + q23(+(p̄2 + p̄3)− (p̄3 + p̄4))f23f34 (q2 + q3)(q1 + q4)[(p̄3 − p̄2) + (p̄1 − p̄4)]f12f34 (q3 + q4)(q1 + q2)[(p̄4 − p̄3) + (p̄2 − p̄1)]f23f41 = (p̄3 − p̄1)(q24f34f41 − q22f12f23) + (p̄4 − p̄2)(q21f12f41 − q23f23f34) − (q2 + q3)(q1 + q4)(p̄3 + p̄1)f12f34 − (q3 + q4)(q1 + q2)(p̄2 + p̄4)f23f41 = (p̄3 − p̄1)(q24f34f41 − q22f12f23) + (p̄4 − p̄2)(q21f12f41 − q23f23f34) − (p̄1 + p̄3)q2q4(f12f34 − f23f41) + (p̄2 + p̄4)q1q3(f12f34 − f23f41) − (p̄1 + p̄3)(q1q2 + q3q4)f12f34 + (p̄1 + p̄3)(q2q3 + q4q1)f23f41 . (B.8) Similarly to the previous case, we will rewrite the second line in the final expression in such a way that we completely cancel all the terms in the first line. To do that we use −(p̄1 + p̄3)q2q4(f12f34 − f23f41) =(p̄3 − p̄1)(q22f12f23 − q24f34f41)+ + q1q2p̄ 1f12f31 − q4q1p̄1f41f13+ + q3q4p̄ 3f34f13 − q2q3p̄3f23f31 (B.9) (p̄2 + p̄4)q1q3(f12f34 − f23f41) =(p̄4 − p̄2)(q23f23f34 − q21f12f41)+ + q2q3p̄2f23f42 − q1q2p̄2f12f24+ + q4q1p̄4f41f24 − q3q4p̄4f34f42 . (B.10) What remains after substituting these is X = p̄1q1f31(q2f12 + q4f41) + q3p̄3f13(q4f34 + q2f23) + p̄2q2f42(q3f23 + q1f12) + q4p̄4f24(q1f41 + q3f34) − (p̄1 + p̄3)(q1q2 + q3q4)f12f34 + (p̄1 + p̄3)(q2q3 + q4q1)f23f41 = p̄1q1q2f12f41 + p̄3q3q4f34f23 + p̄1q4q1f41f21 + p̄3q2q3f23f43 + p̄2q2f42(q3f23 + q1f12) + q4p̄4f24(q1f41 + q3f34) − (p̄1q3q4 + p̄3q1q2)f12f34 + (p̄1q2q3 + p̄3q4q1)f23f41 . (B.11) Now we collect various terms together to rewrite X as X = p̄1q2f41(q1f12 + q3f23) + p̄3q4f23(q3f34 + q1f41) + p̄1q4f21(q1f41 + q3f34) + p̄3q2f43(q3f23 + q1f12) + p̄2q2f42(q3f23 + q1f12) + p̄4q4f24(q1f41 + q3f34) = p̄1q2f41(2q3f23 − q4f42) + p̄3q4f23(2q1f41 − q2f24) + p̄1q4f21(2q1f41 − q4f42) + p̄3q2f43(2q3f23 − q4f42) + p̄2q2f42(2q3f23 − q4f42) + p̄4q4f24(2q1f41 − q2f24) = 2[q2q3f23(p̄1f41 + p̄3f43 + p̄2f42) + q4q1f41(p̄3f23 + p̄1f21 + p̄4f24)] + (p̄1 + p̄2 + p̄3 + p̄4)q2q4f24f42 . (B.12) Clearly the term on the last line vanishes by momentum conservation. We now restore all labels to write the final result as X =2 (32)[f4(p z̄ + p z̄ + p z̄)− p1z̄f1 − p2z̄f2 − p3z̄f3]+ + 2 (14)[f2(p z̄ + p z̄ + p z̄)− p3z̄f3 − p1z̄f1 − p4z̄f4] , (B.13) where we used that q2q3f23 = p + − p3z/p3+) = p3+p2z − p2+p3z = (32) (and similarly for (14)), and where fi = p +. Using momentum conservation on both terms, we rewrite them as X = −2[(32) + (14)] p1z̄p p2z̄p p3z̄p p4z̄p . 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Litvak, On the evaluation of integrals in the light cone gauge, Nucl. Phys. B241 (1984) 463. http://xxx.lanl.gov/abs/hep-th/0410280 Introduction Background The classical MHV Lagrangian A four–dimensional regulator for lightcone Yang–Mills The one–loop (++++) amplitude The all-plus amplitudes from a counterterm Mansfield transformation of LCT The four–point case The general all–plus amplitude Discussion Notation Details on the four–point calculation

0704.0246

Fermi-liquid effects in the transresistivity in quantum Hall double layers near $\nu= 1/2 $

Fermi-liquid effects in the transresistivity in quantum Hall double layers near ν = 1/2 Natalya A. Zimbovskaya Department of Physics and Astronomy, St. Cloud State University, 720 Fourth Avenue South, St. Cloud, MN 56301, USA; Urals State Mining University, Kuibysheva Str. 30, Yekaterinburg, Russia, 620000 (Dated: November 4, 2018) Here, we present theoretical studies of the temperature and magnetic field dependences of the Coulomb drag transresistivity between two parallel layers of two dimensional electron gases in quantum Hall regime near half filling of the lowest Landau level. It is shown that Fermi-liquid interactions between the relevant quasiparticles could give a significant effect on the transresistivity, providing its independence of the interlayer spacing for spacings taking on values reported in the experiments. Obtained results agree with the experimental evidence. PACS numbers: 71.27.+a 73.43.-f During the last decade double-layer two-dimensional electron gas (2DEG) systems were of significant interest due to many remarkable phenomena they exhibit, includ- ing so called Coulomb drag. In Coulomb drag experi- ments two 2DEGs are arranged close to each other, so that they can interact via Coulomb forces. A current I is applied to one layer of the system, and the voltage VD in the other nearby layer is measured, with no current allowed to flow in that layer. The ratio −VD/I gives a transresistivity ρD which characterizes the strength of the effect. The physical interpretation of the Coulomb drag is that momentum is tranferred from the current carrying layer to the nearby one due to interlayer inter- actions [1, 2, 3]. It was shown theoretically [4, 5] and confirmed with experiments [5] that the transresistivity between two 2DEGs in quantum Hall regime at one half filling of the lowest Landau level for both layers is proportional to T 4/3 (T is the temperature of the system) which is quite different from the temperature dependence of ρD in the absence of the external magnetic field applied to 2DEGs. This temperature dependence of the drag at ν = 1/2 originates from the ballistic contribution to the transresistivity. The latter reflects the response of the two-layer system to the driving disturbance of finite wave vector q and finite frequency ω when considering scales are smaller than the mean free path l of electrons (ql ≫ 1) , and times are shorter than their scattering time τ (ωτ ≫ 1) [6]. In further experiments [7] the Coulomb drag was mea- sured between 2DEGs where the layer filling factor was varied around ν = 1/2. The transresistivity was re- ported to be enhanced quadratically with ∆ν = ν− 1/2. It was also reported that the curvature of the enhance- ment depended on temperature but it was insensitive to both sign of ∆ν and distance d between the layers. The present work is motivated with these experiments of [7]. We calculate the transresistivity between two layers of 2DEGs subject to a strong magnetic field which provides ν close to 1/2 for both layers. We start from the well-known expression [1, 3] which relates the Coulomb drag transresistivity to density- density components of the polarization in the layers Π(1)(q, ω) and Π(2)(q, ω) : 2(2π)2 (2π)2 sinh2(h̄ω/2T ) ∣U(q, ω) ImΠ(1)(q,ω)ImΠ(2)(q,ω). (1) Here, U(q, ω) is the screened interlayer Coulomb inter- action, and electron densities in the layers are supposed to be equal (n1 = n2 = n). Within the usual Composite Fermion (CF) approach [8] a single layer polarizability describes that part of the density-current electromagnetic response which is irre- ducible with respect to the Coulomb interaction. Adopt- ing for simplicity the RPA, we obtain the following ex- pression for the 2× 2 polarizability matrix: Π−1 = (K0)−1 + C−1. (2) Here, the matrix K0 gives the response of noninteract- ing CFs and C is the Chern-Simons interaction matrix. Assuming for certainty the wave vector q to lie in the ”x” direction we have: . (3) Starting from the expression (2) we arrive at the fol- lowing results for the density-density response function Π00(i)(q, ω) : Π00(i)(q, ω) = Π(i)(q, ω) 00(i) (q,ω) 8iπh̄ K001(i)(q,ω)− ∆(i)(q,ω) . (4) http://arxiv.org/abs/0704.0246v1 ∆(i)(q,ω) = K 00(i)(q,ω)K 11(i)(q,ω) + K001(i)(q,ω) Within the RPA response functions included in Eqs. (4), (5) are simply related to components of the CF conduc- tivity tensor σ̃ [8]: xx(q,ω) 00(i) (q,ω) 00(i) (q,0) σ̃(i)yy (q,ω) = − K011(i)(q,ω)−K 11(i)(q,0) σ̃(i)xy = −σ̃ K001(i)(q,ω). (6) To proceed we calculate the components of the CF conductivity at ν slightly away from 1/2. In this case CFs experience a nonzero effective magnetic field Beff = B−B1/2. We concentrate on the ballistic contribution to the transresistivity, so we need asymptotics for the rel- evant conductivity components applicable in a nonlocal (ql ≫ 1) and high frequency (ωτ ≫ 1) regime. Cor- responding expressions for σ̃ij were obtained in earlier works [8]. However, these results are not appropriate for our analysis for they do not provide a smooth passage to the Beff → 0 limit at finite q. Due to this reason we do not use them in further calculations. To get a suitable approximation for the CF conductivity we start from the standard solution of the Boltzmann transport equation for the CF distribution function. This gives us the fol- lowing results for the CF conductivity components for a single layer [9]: σ̃αβ = (2πh̄)2 dψvα(ψ) exp ′′)dψ′′ ′) exp ′′)dψ′′ (ψ′ − ψ)(1− iωτ) dψ′. (7) Here, m∗, Ω are the CF effective mass and the cyclotron frequency at the effective magnetic field Beff ; ψ is the angular coordinate of the CF cyclotron orbit. Now we carry out some formal transformations of this expression (7) following the way proposed before [9, 10]. First, we expand the CF velocity components vβ(ψ ′) in Fourier series: vkβ exp(ikψ ′). (8) Substituting this expansion (8) into (7) we obtain: (2πh̄)2 dψvα(ψ) exp(ikψ) ikΩ− iω + + iqvx(ψ) vx(ψ +Ωθ ′)− vx(ψ) dθ (9) where θ = (ψ′ − ψ)/Ω. Then we introduce a new variable η which is related to the variable θ as follows: ikΩ− iω + + iqvx(ψ) vx(ψ +Ωθ ′)− vx(ψ) dθ′, (10) and we arrive at the result: σ̃αβ = im∗e2 (2πh̄)2 vα(ψ) exp(ikψ) ω + i/τ − kΩ− qvx(ψ +Ωθ) dψ. (11) Under the conditions of interest ωτ ≫ 1, ql ≫ 1, and also assuming that the filling factor is close to ν = 1/2, so that qvF ≫ Ω (vF is the CFs Fermi velocity), the variable θ is approximately equal to ητ(1 + iql cosψ + ikΩτ − iωτ)−1. Taking this into account and expanding the last term in the denominator of (11) in powers of Ωθ we obtain: qvx(ψ +Ωθ) ≈ qvx(ψ) + ηΩqτ(1 + iql cosψ + ikΩτ − iωτ)−1 (Ωτ)2(1 + iql cosψ + ikΩτ − iωτ)−2 . (12) Substituting this asymptotic expression into (9) we can calculate first terms of the expansions of relevant compo- nents of the CF conductivity in powers of the small pa- rameter (qR)−1 where R = vF /Ω is the CF cyclotron radius. Within the ”collisionless” limit 1/τ → 0 we have: σ̃xx = −N 1− δ2 (1− δ2)5 2(qR)2 1− δ2 ; (13) σ̃yy = N 1− δ2 + iδ + 2(qR)2 (1− δ2)5 (1− δ2)3 ; (14) σ̃xy = iN 1− δ2 (1− δ2)3 . (15) Here, N = m∗/2πh̄2 is the density of states at the CF Fermi surface, and δ = ω/qvF . Using these re- sults we can easily get approximations for the functions αβ(i) (q, ω) (α, β = 0.1) and, subsequently, the de- sired density-density response function given by (4). It was shown [3] that the integral over ω in the expres- sion for ρD (1) is dominated by ω ∼ T, and the ma- jor contribution to the integral over q in this expres- sion comes from q ∼ kF (T/T0)1/3, where kF is the Fermi wave vector and the scaling temperature T0 is de- fined below. Therefore we get an estimate for δ, namely δ ∼ (T/µ)(T0/T )1/3, where µ is the chemical potential of a single 2DEG included in the bilayer. For the param- eter T0 taking on values of the order of room tempera- ture, δ is small compared to unity at low temperatures (T ∼ 1K). Here, we limit ourselves to the case of two identical layers (Π(1) = Π(2) ≡ Π) . For δ ≪ 1 we obtain the approximation: Π00(q, ω) − 8πih̄ωkF 1 + 2(kFR) (qR)−2 Here, dn/dµ is the compressibility of the ν = 1/2 state which is defined as [3]: ≡ Π00(q → 0; ω → 0) = 8πh̄2 . (17) This differs from the compressibility of the noninteracting 2DEG in the absence of an external magnetic field (the latter equals N). The difference in the compressibility values is a manifestation of the Chern-Simons interaction in strong magnetic fields. In following calculations we adopt the expression used in the work [3] for the screened interlayer potential U(q,ω), namely: U(q,ω) = Vb + Ub 1 + Π(q,ω)(Vb + Ub) Vb − Ub 1 + Π(q,ω)(Vb − Ub) where Vb(q) = 2πe 2/ǫq and Ub(q) = (2πe 2/ǫq)e−qd are Fourier components of the bare Coulomb potentials for intralayer and interlayer interactions, respectively, and ǫ is the dielectric constant. Substitung (18) into (1) and using our result (16) for Π(q, ω) we can present the transresistvity in the ”ballistic” regime as: ρD = ρD0 + δρD. (19) Here, the first term ρD0 is the transresistivity at ν = 1/2 when the effective magnetic field is zero, and the sec- ond term gives a correction arising in a nonzero effective magnetic field (away from ν = 1/2 ). As it was to be expected, our expression for ρD0 coincides with the al- ready known result [3]: ρD0 = Γ(7/3)ζ(4/3) with T0 = πe2nd/ǫ (1 + α), and 2πe2d . (21) The leading term of the correction δρD at low tempera- tures (T/T0) ≪ 1 can be writen as follows: δρD = (kFR)2 ρD0∆ν + 4a2 (∆ν)2. Here, the dimensionless positive constant a2 can be ap- proximated as: sinh2 y y4/3 cosh2 y dy. (23) We have to remark that our result (23) cannot be used in the limit T → 0. Actually, this expression pro- vides a good asymptotic form for the coefficient a2 when (TkF l/µ) 1/3 ≥ 1.5. Assuming that the mean free path is of the order of 1.0µm as in the experiments [11] on dc magnetotransport in a single modulated 2DEG at ν close to 1/2, and using the estimate of [7] for the electron density n = 1.4 × 1015m−2, we obtain that the expres- sion (23) gives good approximation for a2 when T/µ is no less than 10−2 . It follows from our results (19), (22) that transresis- tivity ρD enhances nearly quadratically with ∆ν when the filling factor deviates from ν = 1/2. The linear in ∆ν term is also present in the expression for δρD. This causes an asymmetric shape of the plot of Eq. (22) rel- ative to ∆ν = 0. However, this asymmetry is not very significant for the linear term is smaller than the last term on the right hand side of (22). This difference in magnitudes is due to different temperature dependences of the considered terms. The first term including the lin- ear in (kFR) −1 correction is proportional to (T/T0) whereas the second one is proportional to (T/T0) 2/3 and predominates at low temperatures. So, the magnetic field dependence of the transresistivity near ν = 1/2 matchs that observed in the experiments (See Fig. 1). Keeping only the greatest term in (22), the ratio ρD/ρD0 can be presented in the form: = 4β(∆ν)2 + 1. (24) FIG. 1: Scaled drag resistivity versus ∆ν at T = 0.6; lowest dashed curve is the plot of Eq. (22) at m∗ = 4mb; A0 = 15, and remaining curves present experimental data of [7]; Here, the coefficient β equals: Γ(7/3)ζ(4/3) . (25) This coefficient is proportional to the curvature of the plot of Eq. (22) assuming that the first term is neglected. The curvature reveales a strong dependence on tempera- ture whose character also agrees with experiments of [7] as it is shown in Fig. 2. A striking feature in the experimental results is that they appear to be nonsensitive to the distance between the 2DEGs. Sets of data corresponding to samples with different interlayer spacings dA = 10nm and dB = 22.5nm fall on the same curve. This concerns both mag- netic field dependence of the transresistivity and tem- perature dependence of the parameter β . Results of the present analysis provide a possible explanation for this feature. It follows from (20)–(25) that the dependence of ρD of the interlayer spacing is completely included in the characteristic temperature T0 which is defined with Eq. (21). The above quantity is nearly independent of the interlayer separation d when the parameter α takes on values larger that unity. Estimating the parameter α as it is given by Eq. (21), we obtain that the con- dition α > 1 could be satisfied for small values of the compressibility of the ν = 1/2 state. However, within the RPA the effective mass of CFs coincides with the single electron band mass mb which takes on the value mb ≈ 0, 07me for GaAs wells (me is the mass of a free FIG. 2: Temperature dependence of the coefficient β−1 for interlayer distances d = 10nm (upper curve) and d = 22.5nm (lower curve) compared to the summary of experi- mental curvature at both spacings [7] electron). Using this value to estimate the compressibil- ity as it is introduced by Eq. (17) we get α ≈ 0.44. This is too small to provide insensitivity of the coefficient β determined by Eq. (25) to the interlayer distance for in- terlayer spacings reported in the experiments [3]. The above discrepancy could be removed taking into account Fermi liquid interactions among quasiparticles (CFs). To include Fermi liquid effects into consideration we write the renormalized polarizability Π∗ in the form [8]: Π∗−1 = Π−1 + F(0) + F(1). (26) Here, Π is the polarizability of noninteracting CFs de- fined with Eq. (2), and the remaining terms present con- tributions arising due to Fermi liquid interaction in the CF system. Only contributions from the first and great- est two terms in the expansion of the Fermi liquid in- teraction function in Legendre polynomials ( f0 and f1 , respectively) are kept in Eq. (26) to avoid too lengthy calculations. Matrix elements of the 2× 2 matrices F(0) and F(1) equal: F(0) = F(1) = m∗ −mb m∗ −mb Within the Fermi liquid theory the effective mass m∗ is related to the ”bare” mass mb as follows: 2πh̄2 1 +A1 . (28) Using these expressions (26)–(28) and carrying out cal- culations within the relevant limit δ ≪ 1, we obtain that the expression for the density-density response function for a single layer keeps the form given by Eq. (16) where the compressibility dn/dµ is replaced with the quantity dn∗/dµ renormalized due to the Fermi liquid interaction: 8πh̄2 8πh̄2 . (29) For strongly correlated quasiparticles this renormaliza- tion may significantly reduce the compressibility of the CF liquid, and, consequently, increase the value of the parameter α. It is usually assumed [3, 8] that the Fermi liquid renormalization of the effective mass significantly changes its value: m∗ ∼ 5 − 10 mb. This gives for the Fermi liguid coefficient A1 values of the order of 10. Us- ing this estimate, and substituting our renormalized com- pressibility (29) into the expression (21) we arrive at the conclusion that dn∗/dµ is low enough for the condition α > 1, to be satisfied when the Fermi liquid parameter A0 ≡ f0/2πh̄ 2 takes on values of the order of 10 − 100. This conclusion does not seem an unrealistic one for it is reasonable to expect A0 to be of the order or greater than the next Fermi liquid parameter A1. We obtain a reasonably good agreement between the plot of our Eq. (22) and the experimental results, using A0 = 15 and A1 = 3 (m ∗ = 4mb). (Fig. 1). Our results for temperature dependence of β−1 also agree with the results of experiments [7]. The up- per curve in Fig.2 corresponds to the double-layer sys- tem with with smaller interlayer spacing dA = 10nm which gives T0 = 487K, and the lower curve exhibits the temperature dependence of β−1 for greater spacing dB = 22.5nm (T0 = 587K). The curves do not coincide but they are arranged rather close to each other. Finally, the results of the present analysis enable us to qualitatively describe all important features observed in experiments of [7] on the Coulomb drag slightly away from one half filling of lowest Landau levels of both in- teracting 2DEG. They also give us grounds to treat these experimental results as one more evidence of strong Fermi liquid interaction in the CF system near one half filling of the lowest Landau level. The above interaction pro- vides a significant reduction of the compressibility of the CF liquid and a consequent enhancement in the screen- ing length in single layers. Essentially, the parameter α characterizes the ratio of the Thomas–Fermi screening length in a single 2DEG at ν = 1/2 and the separation between the layers [3]. When α > 1, intralayer interac- tions predominate those between the layers which could be the reason for low sensitivity of the bilayer to changes in the interlayer spacing. It is likely that here is an expla- nation for the reported nearly independence of the drag on the interlayer separation [7]. We believe that at larger distances between the layers the dependence of the tran- sresistivity of d could be revealed in the experiments. At the same time the results of [7] give us a valuable opportunity to estimate a strength of Fermi liquid inter- actions between quasiparticles at ν = 1/2 state which is important for further studies of such systems. Acknowledgments: The author thank K.L. Haglin and G.M. Zimbovsky for help with the manuscript. [1] L. Zheng and A.H. MacDonald, Phys. Rev. B 48, 8203 (1993). [2] A. Kamenev and Y. Oreg, Phys. Rev. B 52, 7516 (1995). [3] I. Ussishkin and A. Stern, Phys. Rev. B 56, 4013 (1997). [4] S. Sakhi, Phys. Rev. B 56, 4098 (1997); [5] M.P. Lilly, J.P. Eisenstein, L.N. Pfeiffer and K.W. West, Phys. Rev. Lett. 80, 1714 (1998). [6] When the external driving disturbance applied to one of the layers is of small q,ω (ql ≪ 1, ωτ ≪ 1) the tran- sresisitivity is dominated with the diffusion contribution, and new effects could emerge (See e.g. F. von Oppen, S.H. Simon and A. Stern, Phys. Rev. Lett. 87, 106803 (2001) and references therein). [7] M.P. Lilly, J.P. Eisenstein, L.N. Pfeiffer and K.W. West, cond-mat/9909231. [8] B.I. Halperin, P.A. Lee and N. Read, Phys. Rev. B 47, 7312 (1993); S.H. Simon and B.I. Halperin, Phys. Rev. B 48, 17368 (1993). [9] N.A. Zimbovskaya and J.L. Birman, Phys. Rev. B, 60 16762 (1999). [10] N.A. Zimbovskaya, ”Local Geometry of the Fermi Surface and High-Frequency Phenomena in Metals”, Springer-Verlag, New York, 2001. [11] R.L. Willett, K.W. West, and L.N. Pfeiffer Phys. Rev. Lett. 83, 2624 (1999). http://arxiv.org/abs/cond-mat/9909231

0704.0247

Geometry of four-dimensional Killing spinors

arXiv:0704.0247v2 [hep-th] 15 May 2007 Preprint typeset in JHEP style - HYPER VERSION IFUM-890-FT UB-ECM-PF-07-05 Geometry of four-dimensional Killing spinors Sergio L. Cacciatori,ad Marco M. Caldarelli,b Dietmar Klemm,cd Diego S. Mansicd and Diederik Roeste a Dipartimento di Scienze Fisiche e Matematiche, Università dell’Insubria, Via Valleggio 11, I-22100 Como. b Departament de F́ısica Fonamental, Universitat de Barcelona, Diagonal, 647, 08028 Barcelona, Spain. c Dipartimento di Fisica dell’Università di Milano, Via Celoria 16, I-20133 Milano. d INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano. e Departament Estructura i Constituents de la Materia, Facultat de F́ısica, Universitat de Barcelona, Diagonal, 647, 08028 Barcelona, Spain. E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: The supersymmetric solutions of N = 2, D = 4 minimal ungauged and gauged supergravity are classified according to the fraction of preserved supersymmetry using spinorial geometry techniques. Subject to a reasonable assumption in the 1/2- supersymmetric time-like case of the gauged theory, we derive the complete form of all supersymmetric solutions. This includes a number of new 1/4- and 1/2-supersymmetric possibilities, like gravitational waves on bubbles of nothing in AdS4. Keywords: Superstring Vacua, Black Holes, Supergravity Models. http://arxiv.org/abs/0704.0247v2 mailto:[email protected], [email protected], [email protected], [email protected], [email protected] mailto:[email protected], [email protected], [email protected], [email protected], [email protected] http://jhep.sissa.it/stdsearch Contents 1. Introduction 2 2. G-invariant Killing spinors in 4D 4 2.1 Orbits of Dirac spinors under the gauge group 4 2.2 The ungauged theory 7 2.3 The gauged theory 8 2.4 Generalized holonomy 10 3. Null representative 1 + ae1 11 3.1 Constant Killing spinor, da = 0 12 3.2 Killing spinor with da 6= 0 15 3.3 Half-supersymmetric backgrounds 17 4. Timelike representative 1 + be2 19 4.1 Conditions from the Killing spinor equations 19 4.2 Geometry of spacetime 20 4.3 Half-supersymmetric backgrounds 25 4.4 Time-dependence of second Killing spinor 28 5. Timelike half-supersymmetric examples 33 5.1 Static Killing spinors and b = b(z) 34 5.1.1 AdS2 ×H2 space-time (α = 0) 36 5.1.2 AdS4 space-time (β 2 = 4αγ) 39 5.1.3 The Reissner-Nordström-Taub-NUT-AdS4 family 41 5.2 Harmonic b solutions 43 5.2.1 Deformations of AdS2 ×H2 44 5.2.2 Deformations of AdS4 44 5.2.3 Deformations of Reissner-Nordström-Taub-NUT-AdS4 44 5.3 Imaginary b solutions 45 5.4 Action of the PSL(2,R) group on the imaginary b solutions 49 5.5 Gravitational Chern-Simons system and G0 = ψ− = 0 solutions 50 6. Final remarks 52 A. Spinors and forms 54 – 1 – B. Spinor bilinears 56 C. The case P ′ = 0 57 D. Half-supersymmetric solutions with G0 = 0 58 1. Introduction Throughout the history of string and M-theory an important part in many develop- ments in the subject has been played by supersymmetric solutions of supergravity, i.e. by backgrounds which admit a number of Killing spinors ǫ which are parallel with respect to the supercovariant derivative1: Dµǫ = 0. Due to their ubiquitous role it has long been realised that it would be advantageous to have classifications of all super- symmetric solutions of a given theory. For purely gravitational backgrounds the supersymmetric possibilities follow from the Berger classification of the possible Riemannian holonomies [1] (see [2, 3] for an extension to the Lorentzian case). However, in the presence of additional force fields (carried by e. g. scalars, gauge potentials or a cosmological constant) it has proven very difficult to obtain knowledge of all supersymmetric possibilities. The reason for the complication in the presence of additional fields lies in the holonomy of the supercurvature Rµν = D[µDν]. For purely gravitational backgrounds the holonomy of the supercurvature is generically given by H = Spin(d − 1, 1) in d dimensions, and hence coincides with the Lorentz group. In such cases the Lorentz gauge freedom allows one to choose constant Killing spinors. Another simplification is that if there is one Killing spinor with a specific stability subgroup, i.e. it is invariant under some Lorentz subgroup, all other spinors with the same stability subgroup are Killing as well. For more general solutions including fields other than gravity, the holonomy is generically extended to a larger group H ⊃ Spin(d− 1, 1). For example, in the present paper we consider gauged minimal four-dimensional N = 2 supergravity, which has H = GL(4,C) [4]. In such cases one cannot choose constant Killing spinors nor are all spinors with the same stability subgroup automatically Killing. For these reasons the 1For the purpose of this discussion we will ignore possible additional Killing spinor equations coming from the variation of dilatinos and gauginos. – 2 – classification of the backgrounds that allow for Killing spinors is more convoluted, or richer, in such cases. For a long time the only classification available was in ungauged minimal four-dimensional N = 2 supergravity [5, 6], which has H = SL(2,H). A new impulse was given to the subject with the introduction of G-structures and the method of spinor bilinears to solve the Killing spinor equations [7]. In this approach, space-time forms are constructed as bilinears from a Killing spinor and one analyses the constraints that these forms imply for the background. Using this framework, a number of complete classifications [8–10] and many partial results (see e.g. [11–21] for an incomplete list) have been obtained. By complete we mean that the most general so- lutions for all possible fractions of supersymmetry have been obtained, while for partial classifications this is only available for some fractions. Note that the complete classi- fications mentioned above involve theories with eight supercharges and H = SL(2,H), and allow for either half- or maximally supersymmetric solutions. An approach which exploits the linearity of the Killing spinors has been proposed [22] under the name of spinorial geometry. Its basic ingredients are an explicit oscillator basis for the spinors in terms of forms and the use of the gauge symmetry to transform them to a preferred representative of their orbit. In this way one can construct a linear system for the background fields from any (set of) Killing spinor(s) [23]. This method has proven fruitful in e.g. the challenging case of IIB supergravity [24–26]. In addition, it has been adjusted to impose ’near-maximal’ supersymmetry and thus has been used to rule out certain large fractions of supersymmetry [27–30]. Finally, a complete classification for type I supergravity in ten dimensions has been obtained [32]. In the present paper we would like to address the classification of supersymmetric solutions in four-dimensional minimal N = 2 supergravity. As will also be reviewed in section 2, the ungauged case has been classified completely [5,6]. For the gauged case, the discussion of 1/4 supersymmetry splits up in a time-like and a light-like class (de- pending on the causal nature of the Killing vector associated to the Killing spinor). The time-like class is completely specified by a single complex function depending on three spatial coordinates b = b(z, w, w̄), subject to a second-order differential equation which can not be solved in general [13]. The light-like class can be given in all generality, and in addition its restriction to 1/2-BPS solutions has been derived [16]. Furthermore, there are no backgrounds with 3/4 supersymmetry [29] and AdS4 is the unique possi- bility with maximal supersymmetry. Therefore the remaining open question concerns half-supersymmetric backgrounds in the gauged theory2. In the following, we will first re-analyse the 1/4-supersymmetric backgrounds us- ing the method of spinorial geometry, and in fact find an additional possibility in the 2The addition of external matter was considered in [31]. – 3 – light-like case: a half supersymmetric bubble of nothing in AdS4 and its Petrov type II generalization, a new 1/4 BPS configuration that has the interpretation of grav- itational waves propagating on the bubble of nothing. This completes the analysis of the null class in all its generality. Then we will derive the constraints for half- supersymmetric backgrounds for the timelike class. Subject to a single assumption on the time-dependence of the second Killing spinor these will be solved in general, up to a second order ordinary differential equation. The assumption will be justified by solving the full set of conditions in a number of examples which illustrate the possible spatial dependence of b. All these cases turn out to have time-dependence of the assumed form. The different examples are: • the b = b(z) family of solutions, comprising part of the Reissner-Nordström-Taub- NUT-AdS4 backgrounds, • waves on the previous backgrounds with b = b(z, w), • solutions with b imaginary and their PSL(2,R) transformed counterparts, • solutions of the dimensionally reduced gravitational Chern-Simons model that can be embedded in the equations for a timelike Killing spinor [16]. We determine when these backgrounds preserve 1/2 supersymmetry and provide the explicit Killing spinors. Moreover, in the subcases consisting of AdS4 and AdS2×H2, the action of the isometries of these backgrounds on the Killing spinors is given explicitly. The outline of this paper is as follows. In section 2, we discuss the orbits of Killing spinors and review the known classification results in the theory at hand. In section 3, we go through the complete classification of the null class. In section 4, we discuss the constraints for 1/4 and 1/2 supersymmetry in the timelike class. We derive the time-dependence of the second Killing spinor and solve the equations for the case of linear time-dependence (G0 = 0). A number of examples of the 1/2 BPS timelike class are provided in section 5. Finally, in section 6 we present our conclusions and outlook. In appendix A we review our notation and conventions for spinors, while in appendix B the associated bilinear forms are given. Appendix C deals with the special case P ′ = 0, to be defined in section 4.4. Finally, in appendix D, we will give the details of the G0 = 0 case. 2. G-invariant Killing spinors in 4D 2.1 Orbits of Dirac spinors under the gauge group In order to obtain the possible orbits of Spin(3,1) in the space of Dirac spinors ∆c, – 4 – we first consider the most general positive chirality spinor3 a1 + be12 (a, b ∈ C) and determine its stability subgroup. This is done by solving the infinitesimal equation αcdΓcd(a1 + be12) = 0 . (2.1) First of all, notice that a1 + be12 is in the same orbit as 1, which can be seen from eγΓ13eψΓ12eδΓ13ehΓ02 1 = ei(δ+γ)eh cosψ 1 + ei(δ−γ)eh sinψ e12 . This means that we can set a = 1, b = 0 in (2.1), which implies then α02 = α13 = 0, α01 = −α12, α03 = α23. The stability subgroup of 1 is thus generated by X = Γ01 − Γ12 , Y = Γ03 + Γ23 . (2.2) One easily verifies that X2 = Y 2 = XY = 0, and thus exp(µX + νY ) = 1 + µX + νY , so that X, Y generate R2. Spinors of negative chirality are composed of odd forms, i.e. ae1 + be2. One can show in a similar way that they are in the same orbit as e1, and the stability subgroup is again R2, with the above generators X, Y . For definiteness and without loss of generality we will always assume that the first Killing spinor has a non-vanishing positive chirality component, and use (part of) the Lorentz symmetry to bring this to the form 1. Hence we can write a general spinor as 1 + ae1 + be2. Now act with the stability subgroup of 1 to bring ae1 + be2 to a special form: (1 + µX + νY )(1 + ae1 + be2) = 1 + be2 + [a + 2b(ν − iµ)]e1 . In the case b = 0 this spinor is invariant, so the representative is 1+ ae1, with isotropy group R2. If b 6= 0, one can bring the spinor to the form 1+ be2, with isotropy group I. The representatives4 together with the stability subgroups are summarized in table 1. In the ungauged theory, we therefore can have the following G-invariant Killing spinors. The R2-invariant Killing spinors are spanned by 1 and e1 and there can be up to four of these. The I-invariant Killing spinors are spanned by all four basis elements and there can be up to eight of these. In the first two case, the vector Va bilinear in the spinor ǫ is lightlike, whereas in the last case it is timelike, see table 1. The existence of a globally defined Killing spinor ǫ, with isotropy group G ∈ Spin(3,1), gives rise to a G-structure. This means that we have an R2-structure in the null case and an identity structure in the timelike case. 3Our conventions for spinors and their description in terms of forms can be found in appendix A. 4Note the difference in form compared to the Killing spinors of the corresponding theories in five and six dimensions: in six dimensions these can be chosen constant [9] while in five dimensions they are constant up to an overall function [28]. In four dimensions such a choice is generically not possible. – 5 – In U(1) gauged supergravity, the local Spin(3,1) invariance is actually enhanced to Spin(3,1) × U(1). Thus, in order to obtain the stability subgroup, one determines the Lorentz transformations that leave a spinor invariant up to an arbitrary phase factor, which can then be gauged away using the additional U(1) symmetry. For the representative 1, one gets in this way an isotropy group generated by X, Y and Γ13 obeying [Γ13, X ] = −2Y , [Γ13, Y ] = 2X , [X, Y ] = 0 , i. e. G ∼= U(1)⋉R2. For ǫ = 1 + ae1 with a 6= 0, the stability subgroup R2 is not enhanced, whereas the I of the representative 1+ be2 is promoted to U(1) generated by Γ13 = iΓ•̄•. The Lorentz transformation matrix aAB corresponding to Λ = exp(iψΓ•̄•) ∈ U(1), with ΛΓBΛ −1 = aABΓA, has nonvanishing components a+− = a−+ = 1 , a••̄ = e 2iψ , a•̄• = e −2iψ . (2.3) Finally, notice that in U(1) gauged supergravity one can choose the function a in 1+ae1 real and positive: Write a = R exp(2iδ), use eδΓ13(1 + ae1) = e iδ1 + e−iδae1 = e iδ(1 +Re1) , and gauge away the phase factor exp(iδ) using the electromagnetic U(1). ǫ G ⊂ Spin(3,1) G ⊂ Spin(3,1) × U(1) Va = D(ǫ,Γaǫ) 1 R2 U(1)⋉R2 (1, 0,−1, 0) 1 + ae1 R 2 (a ∈ R) (1 + |a|2, 0,−1− |a|2, 0) 1 + be2 I U(1) (1 + |b|2, 0,−1 + |b|2, 0) Table 1: The representatives ǫ of the orbits of Dirac spinors and their stability subgroups G under the gauge groups Spin(3,1) and Spin(3,1) × U(1) in the ungauged and gauged theories, respectively. The number of orbits is the same in both theories, the only difference lies in the stability subgroups and the fact that a is real in the gauged theory. In the last column we give the vectors constructed from the spinors. In the gauged theory the classification of G-invariant spinors is therefore slightly more complicated. There can be at most two U(1)⋉R2-invariant Killing spinors, spanned by 1. The four R2-invariant spinors are spanned by 1 and e1. Then there are the U(1)-invariant spinors, spanned by 1 and e2. Finally, for generic enough Killing spinors, one does not fall in any of the above classes and the common stability subgroup is I. Note that in the gauged theory the presence of G-invariant Killing spinors will in general not lead to a G-structure on the manifold but to stronger conditions. The – 6 – structure group is in fact reduced to the intersection of G with Spin(3,1), and hence is equal to the stability subgroup in the ungauged theory. We will now consider the possible supersymmetric solutions to the equation Dµǫ = 0 in various sectors of N = 2, D = 4 in terms of the stability subgroup G of the Killing spinors. 2.2 The ungauged theory The supercovariant derivative of ungauged minimal N = 2 supergravity in four dimen- sions reads Dµ = ∂µ + ωabµ Γab + FabΓabΓµ . (2.4) As mentioned in the introduction, a first point to notice is that there is no complex conjugation on the Killing spinor. Therefore, the number of supersymmetries that are preserved is always even: if ǫ is Killing, then so is iǫ. First consider purely gravitational solutions with F = 0. In this case the superco- variant connection truncates to the Levi-Civita connection and has Spin(3,1) holonomy. This implies the following. If ǫ is Killing, then so are5 Γ3∗ǫ and Γ012∗ǫ (where ∗ denotes complex conjugation). Together, the operations i, Γ3∗ and Γ012∗ generate four linearly independent Killing spinors from any null spinor ǫ = 1 or ǫ = 1+ae1 and eight from any time-like spinor ǫ = 1+ be2. This illustrates the general statement in the introduction: if the gauge group equals the holonomy, as in this case, then there is only one possible number of Killing spinors for every stability subgroup. Therefore there are only two classes of supersymmetric solutions, which are listed in table 2, and which consist of the gravitational wave and Minkowski space-time, respectively. G = \ N = 4 8 Table 2: Gravitational solutions with G-invariant Killing spinors in the ungauged theory. Now let us also allow for fluxes F . The supercovariant connection no longer equals the Levi-Civita connection due to the flux term. In particular, this implies that Γ012∗ no longer commutes with Dµ. However, this does still hold for the other operation: Γ3 ∗ ǫ is Killing provided ǫ is. The combined operations of i and Γ3∗ generate four linearly 5These operations anti-commute and commute with the Γ-matrices, respectively. – 7 – independent spinors from any null or time-like spinor. Thus the number of supersym- metries is always N = 4p, as illustrated in table 3. Indeed the generalised holonomy of the supercovariant connection in the ungauged case is SL(2,H) [4], consistent with the supersymmetries coming in quadruplets. G = \ N = 4 8 Table 3: General solutions with G-invariant Killing spinors in the ungauged theory. The half-supersymmetric solution have been classified by Tod [5] and consist of the plane wave and the Israel-Wilson-Perjes metric, respectively. The maximally supersym- metric solutions are AdS2 × S2 and its Penrose limits, the Hpp wave and Minkowski space-time [6]. 2.3 The gauged theory The supercovariant derivative of gauged minimalN = 2 supergravity in four dimensions reads Dµ = ∂µ + ωabµ Γab − iℓ−1Aµ + 12ℓ −1Γµ + FabΓabΓµ . (2.5) Due to the gauging the structure of Γ-matrices is richer, but there still is no complex conjugation on the Killing spinor. Therefore, the number of supersymmetries that are preserved is always even: if ǫ is Killing, then so is iǫ. Again, we first consider the purely gravitational solutions. In this case the super- covariant derivative has SO(3,2) holonomy. The operation Γ012∗ commutes with Dµ and therefore generates additional Killing spinors. Together, the operations i and Γ012∗ generate four linearly independent Killing spinors from generic null or time-like spinors. The exception is the null spinor ǫ = 1+e1, in which case ǫ and Γ012∗ are linearly depen- dent, and hence allows for two instead of four Killing spinors. The possibilities allowed for by this analysis of the supercovariant derivative can be found in table 4. However, although all these entries are allowed for by the spinor orbit structure and the crude analysis of the supercurvature above, not all of them have an actual field theoretic realisation in supergravity. In other words, there are no solutions to the Killing spinor equations for all of the above sets of Killing spinors. The lightlike cases were considered in [16]: The 1/4-BPS case is the Lobatchevski wave while imposing more supersymmetries leads to the maximally supersymmetric AdS4 solution (with – 8 – G = \ N = 2 4 6 8 U(1)⋉ R2 × × × × 2 √ ◦ × × U(1) × ◦ × × I × ◦ ◦ Table 4: Gravitational solutions with G-invariant Killing spinors in the gauged theory. Check marks indicate entries with actual solutions, while circles stand for allowed entries which are not realized. G=1). The N = 4 and G = R2 entry is thus effectively empty. In particular, this implies that imposing a single Killing spinor 1 + ae1 with a 6= 1 leads to AdS4. Also note that the N = 6 and G = 1 entry must be empty since any time-like spinor plus 1+e1 leads to maximal supersymmetry, while all other Killing spinors come in groups of four. The only remaining entries are N = 4 and G = U(1) or G = I. Using the results of [13,16], it is straightforward to show that in these purely gravitational timelike cases the geometry is given by ds2 = −z 2 + n2 (dt− 2n cosh θdφ)2 + ℓ z2 + n2 + (z2 + n2)(dθ2 + sinh2 θdφ2) , where n = ±ℓ/2. But this is simply AdS4 written as a line bundle over a three- dimensional base manifold, so both N = 4 entries are empty as well. We conclude that there are no 1/2-supersymmetric gravitational solutions in the gauged theory, only the 1/4-supersymmetric Lobatchevski waves and maximally supersymmetric AdS4. We now come to the general supersymmetric solutions in the gauged case. Due to the gauging and flux terms, neither Γ012∗ nor Γ3∗ commute with Dµ. Therefore we have the cases as listed in table 5. The supercovariant connection in the gauged case has generalized holonomy GL(4,C) [4], again consistent with the supersymmetries coming in doublets. The 1/4-BPS solutions with G = R2 and G = U(1) were derived in [13], and we will show there is no solution with G = U(1)⋉R2. In addition, it was shown in [16] that any additional supersymmetries in the null case are always timelike, i.e. end up in the N = 4 and G = 1 entry. Again, the N = 4 and G = R2 entry is empty. It would be interesting to see if there is a nice explanation for this. In addition, the maximally supersymmetric case is always AdS4. Recently, it has been shown in [29] that the N = 6 and G = 1 entry is empty as well, because imposing three complex Killing spinors implies that the spacetime is AdS4 and thus maximally supersymmetric. – 9 – G = \ N = 2 4 6 8 U(1)⋉ R2 ◦ × × × 2 √ ◦ × × Table 5: General solutions with G-invariant Killing spinors in the gauged theory. Check marks indicate entries with actual solutions, while circles stand for allowed entries which are not realized. The most general 1/2-BPS solution in the timelike case remains an open issue and will be studied in this paper. 2.4 Generalized holonomy In minimal gauged supergravity theories with eight supercharges, the generalized holon- omy group for vacua preserving N supersymmetries, whereN = 0, 2, 4, 6, 8, is GL(8−N 2 [4]. To see this, assume that there exists a Killing spinor ǫ1. By a local GL(4,C) transformation, ǫ1 can be brought to the form ǫ1 = (1, 0, 0, 0) T . This is annihilated by matrices of the form that generate the affine group A(3,C) ∼= GL(3,C)⋉C3. Now impose a second Killing spinor ǫ2 = (ǫ 2, ǫ2) T . Acting with the stability subgroup of ǫ1 yields eAǫ2 = ǫ02 + b , where bT = aTA−1(eA − 1) . We can choose A ∈ gl(3,C) such that eAǫ2 = (1, 0, 0)T , and b such that ǫ02 + bT ǫ2 = 0. This means that the stability subgroup of ǫ1 can be used to bring ǫ2 to the form ǫ2 = (0, 1, 0, 0). The subgroup of A(3,C) that stabilizes also ǫ2 consists of the matrices 1 0 b2 b3 0 1 B12 B13 0 0 B22 B23 0 0 B32 B33 ∈ GL(2,C)⋉ 2C2 . Finally, imposing a third Killing spinor yields GL(1,C) ⋉ 3C as maximal generalized holonomy group, which is however not realized in N = 2, D = 4 minimal gauged – 10 – supergravity [16, 29]. It would be interesting to better understand why such preons actually do not exist. In section 4.3, we explicitely compute the generalized holonomy group for N = 2, D = 4 minimal gauged supergravity in the case N = 2 and show that it is indeed contained in A(3,C), supporting thus the classification scheme of [4]. 3. Null representative 1 + ae1 In this section we will analyse the conditions coming from a single null Killing spinor. As we saw in section 2.1, there are two orbits of such spinors, one with representative ǫ = 1 and stability subgroup G = U(1)⋉R2 and one with ǫ = 1 + ae1 and G = R Owing to local U(1) gauge invariance, it is always possible to choose the function a real and positive, so in the following we set a = eχ, χ ∈ R. The Killing spinor equations become E •̄ − 2iF+•̄E− = 0 , E •̄ + 2iF+•̄E− = 0 , ω−• + 2iF−•E •̄ + = 0 , ω−• + −2iF−•E •̄ + = 0 , (3.1) where φ ≡ F+− + F •̄• and Ω ≡ ω+− + ω•̄•. The conditions for the special U(1)⋉R2-orbit with ǫ = 1 can be obtained as the singular limit χ → −∞ of the above equations. Note however that, in this limit, the second line implies the constraint ℓ−1−iφ = 0, while the fourth line leads to ℓ−1+iφ = 0. Clearly, for ℓ−1 6= 0 this does not allow for a solution. Hence, in the gauged theory, there are no backgrounds with U(1)⋉R2-invariant Killing spinors. The only null possibility is therefore given by the R2-invariant Killing spinor ǫ = 1 + eχe1. We will now analyse the above conditions for the generic case with χ finite. In fact, we will furthermore assume it is positive. This does not constitute any loss of generality since one can flip the sign of χ by changing chirality (a spinor 1 + eχe1 with χ negative is gauge equivalent to a spinor e1 + e χ̃1 with χ̃ = −χ positive), and hence the resulting background will not depend on this sign. From the last two equations one obtains the constraints F−• = F−•̄ = 0 , φ = − i tanhχ (3.2) – 11 – on the field strength, as well as ω−• = ω−•̄ = − 1√ 2ℓ coshχ E− (3.3) for the spin connection. (3.2) implies F+− = 0 and F •̄• = − i tanhχ. The first two equations of (3.1) yield then ω+− = 2eχH3E − − 1 coshχ ω•̄• = 2i sinhχH1E cosh 2χ coshχ A = −ℓ coshχH1E− − sinhχE3 , dχ = −2 coshχH3E− + sinhχE1 , (3.4) where E1 = (E• + E •̄)/ 2, iE3 = (E• − E •̄)/ 2, and we defined F+• + F+•̄√ = H1 , F+• − F+•̄√ = iH3 . In order to proceed, we distinguish two subcases, namely dχ = 0 and dχ 6= 0. 3.1 Constant Killing spinor, da = 0 If a and hence χ are constant, eqn. (3.4) implies χ = H3 = 0. Next we impose vanishing torsion. The torsion two-form reads T− = dE− + E1 ∧ E− , T+ = dE+ − E1 ∧ ω+1 + + ω+3 ∧ E3 , T 1 = dE1 + E− ∧ ω+1 + T 3 = dE3 + E1 ∧ E3 − ω+3 ∧ E− . (3.5) From T− = 0 one gets E−∧dE− = 0, so by Fröbenius’ theorem there exist two functions η and u such that locally E− = ηdu . Plugging this into T− = 0 yields d log η + ∧ du = 0 , – 12 – so that there exists a function ξ such that E1 = − ℓ dη + ξdu . The gauge field and its field strength can now be written as A = −ℓηH1du , F = H1dη ∧ du , and the Bianchi identity F = dA implies dH1 + H1d log η ∧ du = 0 . This means that H1η 3/2 can depend only on u, 3/2 = −ϕ where the prefactor and the derivative were chosen in order to conform with the notation of [13]. Let us define a new coordinate x = −η−1/2, so that E1 = ℓ dx+ξdu, E− = x−2du A = −xϕ′(u)du . (3.6) One can now use part of the residual gauge freedom, given by the stability subgroup 2 of the null spinor 1 + ae1, in order to simplify E 1. To this end, consider an R2 transformation with group element Λ = 1 + µX + νY , where X and Y are given in (2.2). Defining α = µ+ iν, this can also be written as Λ = 1 + αΓ+• + ᾱΓ+•̄ . (3.7) Given the ordering A,B = +,−, •, •̄, the Lorentz transformation matrix aAB corre- sponding to Λ ∈ R2 ⊆ Spin(3,1) reads aAB = 0 1 0 0 1 −4|α|2 2ᾱ 2α 0 −2ᾱ 0 1 0 −2α 1 0 . (3.8) The transformed vielbein αEA = aABE B is thus given by αE• = E• − 2αE− , αE1 = E1 − 2 (α + ᾱ)E− , αE •̄ = E •̄ − 2ᾱE− , αE3 = E3 + 2i (α− ᾱ)E− , αE− = E− , αE+ = E+ + 2ᾱE• + 2αE •̄ − 4|α|2E− . (3.9) – 13 – Choosing α + ᾱ = ξx2/ 2, we can eliminate E1u, so one can set ξ = 0 without loss of generality. Note that this still leaves a residual gauge freedom associated to the imaginary part of α, which will be used below. From dT 3 = 0 we get d(ω+3/x) ∧ du = 0, and thus there exist two functions β, β̃ such that ω+3 = −xdβ + β̃du . Plugging this into T 3 = 0 yields d(xE3 + βdu) = 0, which is solved by E3 = − ℓ dy + βdu , (3.10) where y denotes some function that we shall use as a coordinate. Using the remaining gauge freedom (3.8) with Imα = −βx2/2 2 allows to set also β = 0. The equation T 1 = 0 tells us that ω+1 + E+/ℓ = γdu for some function γ. Using this together with T+ = 0, one shows that E− ∧ E+ E− ∧ E+ which means that the surface described by E− and E+ is integrable, so that E+ = ℓ2 du+ hdV , (3.11) for some functions G, h, V . The metric becomes then ds2 = 2E−E+ + Gdu2 + 2h dudV + dx2 + dy2 . (3.12) Finally, the equation T+ = 0 implies ∂xh = ∂yh = 0 , ∂V G = ∂uh , (3.13) ∂xG , β̃ = − ∂yG . h can be eliminated by introducing a new coordinate v(u, V ) with ∂V v = h/ℓ 2 and shifting G → G + 2∂uv, which leads to ds2 = Gdu2 + 2dudv + dx2 + dy2 . (3.14) Note that, due to (3.13), G is independent of v, therefore ∂v is a Killing vector. One easily verifies that it coincides with the Killing vector constructed from the Killing spinor as − ℓ2 D(ǫ,Γµǫ). – 14 – All that remains is to impose the Maxwell and Einstein equations. One finds that the former are automatically satisfied by the gauge potential (3.6). The same holds for the Einstein equations, except for the uu-component, which gives the Siklos equation with sources ∆G − 2 ∂xG = − ϕ′(u)2 . (3.15) This family of solutions enjoys a large group of diffeomorphisms which leave the solution invariant in form but change the function G. This is the Siklos-Virasoro invariance, discussed in [16, 33]. In conclusion, the geometry of solutions admitting the constant null spinor 1 + e1 is given by the Lobachevski waves with metric (3.14) and gauge field (3.6), where G satisfies (3.15) and ϕ(u) is arbitrary. This coincides exactly with the results of [13], where it was shown moreover that there is a second covariantly constant spinor iff the wave profiles G and ϕ have the form Gα(x, y, u) = − + 2αx3 − α2ℓ2(x2 + y2) , ϕ(u) = u , (3.16) up to Siklos-Virasoro transformation, with α ∈ R constant. In this case, the solution does also belong to the timelike class [13]. While the α 6= 0 solution only has the obvious Killing vectors ∂v and ∂y, the special α = 0 case is maximally symmetric with a five-dimensional isometry group. 3.2 Killing spinor with da 6= 0 If da and hence also dχ do not vanish, one can use the R2 stability subgroup of the spinor 1 + eχe1 to eliminate the fluxes F+• and F+•̄. To see this, observe that under an R2 transformation (3.8), αF+• = F+• − 2iα tanhχ , αF •̄• = F •̄• , so by choosing α = − iℓ F+• cothχ one can achieve αF+• = 0. Note that this would not be possible if χ = 0. With this gauge fixing, one has sinhχE1 , A = − sinhχE3 , F = −1 tanhχE1 ∧ E3 . (3.17) Next we impose vanishing torsion. Using (3.17), one easily shows that T− = 0 leads to e2χ − 1 = 0 , and therefore one can introduce a function u with e2χ − 1 E− = du . (3.18) – 15 – Before we come to the other torsion components, let us consider the Bianchi identity and the Maxwell equations. The gauge field strength reads F = dχ sinh 2χ Requiring it to be equal to dA implies that A/ tanhχ is closed, so that locally tanhχdΨ . (3.19) Note that the functions χ, u and Ψ must be independent, because otherwise E1, E− and E3 would not be linearly independent. We can thus use these three functions as coordinates. Using ∗F = −1 tanhχE− ∧ E+ , the Maxwell equations d∗F = 0 imply E− ∧ E+ sinh 2χ E− ∧ E+ = 0 . By Fröbenius’ theorem and (3.18), E+ can thus be written as du+ hdV , where K̃, h and V are some functions, and we can use V as the remaining coordinate. Substituing E+ into the Maxwell equations one obtains a constraint on the function h, e2χ + 1 ∧ du ∧ dV = 0 , and hence h = h0(u, V ) e2χ + 1 In what follows, we define K = K̃/(e2χ + 1) and use ω+1 = (ω+• + ω+•̄)/ 2, ω+3 = (ω+• − ω+•̄)/ 2i. We now come to the remaining torsion components. From T 3 = 0 and T 1 = 0 one obtains respectively ω+3 = AE− , ω+1 = − E ℓ coshχ +BE− , where A and B are some functions to be determined. Finally, T+ = 0 yields ∂VK = 2∂uh0 , A = − e4χ − 1 ) sinhχ√ tanhχ ∂ΨK , B = e4χ − 1 sinhχ∂χK . – 16 – The line element is given by ds2 = 2E−E+ + = cothχ Kdu2 + 2h0dudV ℓ2dχ2 4 sinh2 χ sinhχ coshχ . (3.20) As before, one can eliminate h0 by introducing a new coordinate v(u, V ) with ∂V v = h0 and shifting K → K + 2∂uv, whereupon the metric becomes ds2 = cothχ Kdu2 + 2dudv ℓ2dχ2 4 sinh2 χ sinhχ coshχ . (3.21) Notice that, owing to (3.20), K is independent of v, therefore ∂v is a Killing vector. It coincides with the Killing vector − 2D(ǫ,Γµǫ) constructed from the Killing spinor. All that remains now is to impose Einstein’s equations. One finds that they are all satisfied except for the uu component, which yields again a Siklos-type equation for K, ∂2ΨK + 4 tanhχ∂2χK − cosh2 χ ∂χK = 0 . (3.22) In conclusion, the bosonic fields for a configuration admitting a null Killing spinor with dχ 6= 0 are given by (3.19) and (3.21), with K satisfying (3.22)6. As we will discuss in section 5.3, the K = 0 solution is of Petrov type D and represents a bubble of nothing in anti-De Sitter space-time. When K 6= 0, the metric becomes of Petrov type II and the Weyl scalar signalling the presence of gravitational radiation acquires a non-vanishing value. Hence the general solution represents a gravitational wave on a bubble of nothing. To our knowledge these solutions have not featured in the literature before. 3.3 Half-supersymmetric backgrounds In the previous subsections we have addressed the conditions for preserving one null Killing spinor of the form ǫ1 = 1 or ǫ1 = 1 + e χe1. It is natural to enquire about the possibility of these backgrounds admitting an additional Killing spinor with the same 2 stability subgroup, i.e. of the form ǫ2 = c01+ c1e1. Using the fact that ǫ1 is Killing, the second Killing spinor equation Dµǫ2 = 0 can then be rewritten as (c0 − c1)Dµ1 + ∂µc01 + ∂µc1e1 = 0 , (3.23) 6This solution escaped a majority of the present authors in [13]. The reason for this is that equ. (4.32) of [13] is not correct; it must be R+−ij = 0, which yields no information on the constant κ. Thus, in addition to the solutions with κ = 0 found in [13] (the Lobachevski waves), there are also the κ = 1 solutions, which are exactly the ones found here with dχ 6= 0. – 17 – in the U(1)⋉R2 case and (c0 − c1e−χ)Dµ1 + ∂µc01 + (∂µc1 − c1∂µχ)e1 = 0 , (3.24) in the R2 case. Furthermore, we can assume that (c0 − c1) 6= 0 and (c0 − c1e−χ) 6= 0 in the two cases, respectively, since otherwise the second Killing spinor would be linearly dependent on the first and there would not be any additional constraints. Hence the e2 and e12 components of Dµ1 have to vanish separately. In particular, this implies that ω−• = 0 (as can be seen from the third line of (3.1) in the singular limit χ → −∞). However, this is clearly incompatible with (3.3). We conclude that, in the gauged theory, there are no backgrounds with four R2-invariant Killing spinors. In other words, there are no half-supersymmetric backgrounds with an R2-structure. This is unlike the ungauged case, where the half-supersymmetric gravitational waves provide such solutions. Therefore, the only possibility to augment the supersymmetry of the null solutions above is to add a Killing spinor which breaks the R2 invariance, i.e. with a non-vanishing e2 and/or e12 component. From a linear combination of the first and second Killing spinor one can then always construct a time-like Killing spinor, and hence this brings us to the next section. For the convenience of the reader, we will already summarise how to restrict the 1/4-supersymmetric null solutions to allow for a time-like Killing spinor as well. For the case with constant null Killing spinors, dχ = 0, the restriction was al- ready discussed in [13] and is given in (3.16). For the other case, with dχ 6= 0, it is straightforward to show that the solution (3.19), (3.21) admits a second Killing spinor iff ∂χG = ∂ΨG = 0, so that G depends only on u. By a simple diffeomorphism one can then set G = 0. The general solution to the Killing spinor equations reads in this case ǫ = λ1(1 + e χe1) + e4χ − 1 (e2 + e χe1 ∧ e2) , (3.25) where λ1,2 ∈ C are constants. The invariants constructed from ǫ, as defined in appendix B, are 2 cothχ(|λ2|2dv − |λ1|2du)− sinh 2χ (λ2λ̄1 − λ̄2λ1) dΨ , B = − 2(|λ1|2du+ |λ2|2dv) + e4χ − 1 sinhχ (λ̄1λ2 + λ1λ̄2) dχ , f = i(λ1λ̄2 − λ̄1λ2) tanhχ , g = (λ̄1λ2 + λ1λ̄2) cothχ . The norm of the Killing vector V is given by V 2 = − 2 sinh 2χ (λ̄1λ2 + λ1λ̄2) 2 − 4|λ1λ2|2 tanhχ . – 18 – Since χ > 0, this is negative unless λ1 = 0 or λ2 = 0, so indeed the solution (3.19), (3.21) with G = 0 must belong also to the timelike class. It turns out that it is identical to the bubble of nothing of section 5.3 with imaginary b and L < 0. The coordinate transformation 2A2(t− Ly)− z , v = − 2A2(t− Ly)− z Ψ = −2A2t , χ = artanhX (3.26) with A8 = −1/4L brings the metric (3.21) (with G = 0) to (5.60), and the field strength of (3.19) to (5.61). Note that, in the new coordinates, the above invariants become V = ∂t as a vector, and B = dz, in agreement with section 4.2. 4. Timelike representative 1 + be2 We will now turn to the timelike case and first recover the general 1/4-BPS solutions [13]. Afterwards we will study the conditions for 1/2 supersymmetry. This will complete the classification since we already know that no 3/4-supersymmetric solutions can arise and AdS4 is the unique maximally supersymmetric possibility. 4.1 Conditions from the Killing spinor equations Acting with the supercovariant derivative (2.5) on the representative 1 + be2 yields the linear system ω•̄•+ − ω+−+ − bA+ = 0 , ω•̄•+ + ω+−+ − F •̄• + ib√ F+− = 0 , ω•−+ + i 2bF•− = ω•++ = 0 , (4.1) ω•̄•− − ω+−− − bA− + F •̄• − i√ F+− = 0 , ω•̄•− + ω+−− − A− = 0 , b ω•+− + i 2F•+ = ω•−− = 0 , (4.2) – 19 – ω•̄•• − ω+−• − bA• − i 2F •̄− = 0 , ω•̄•• + ω+−• − A• − i 2bF •̄+ = 0 , ω•−• + − ib√ F •̄• − ib√ F+− = 0 , b ω•+• + F •̄• + i√ F+− = 0 , (4.3) ∂•̄b+ ω•̄••̄ − ω+−•̄ − bA•̄ = 0 , ω•̄••̄ + ω+−•̄ − A•̄ = 0 , ω•−•̄ = b ω •̄ = 0 . (4.4) From eqns. (4.1) - (4.4) one obtains the gauge potential and the fluxes in terms of the spin connection and the function b, − ∂+b − ω•̄•+ , A− = ω••̄− , A• = (ω••̄• + ω • ) , F+− = i√ (b ω•+• − b−1ω•−• ) , F•+ = ω+−•̄ , F••̄ = i√ (b ω•+• + b −1ω•−• ) + , F•− = i ω•−+ . (4.5) Furthermore, the system (4.1) - (4.4) determines almost all components of the spin connection (with the exception of ω••̄) in terms of the function b and its spacetime derivatives, ω+−+ = , ω+−− = 0 , ω ω+•+ = ω •̄ = 0 , ω − = − , ω+•• = ω−•+ = −b ∂•̄b̄ , ω−•− = ω−••̄ = 0 , ω−•• = . (4.6) In what follows, we assume b 6= 0. One easily shows that b = 0 leads to ℓ−1 = 0, so this case appears only in ungauged supergravity. 4.2 Geometry of spacetime In order to obtain the spacetime geometry, we consider the spinor bilinears Vµ = D(ǫ,Γµǫ) , Bµ = D(ǫ,Γ5Γµǫ) , – 20 – whose nonvanishing components are 2 b̄b , V− = − 2 , B+ = 2 b̄b , B− = As V 2 = −4b̄b = −B2, V is timelike and B is spacelike. Using eqns. (4.1) - (4.4), it is straightforward to show that V is Killing and B is closed, i. e. , ∂AVB + ∂BVA − ωCB|AVC − ωCA|BVC = 0 , ∂ABB − ∂BBA − ωCB|ABC + ωCA|BBC = 0 . There exists thus a function z such that B = dz locally. Let us choose coordinates (t, z, xi) such that V = ∂t and i = 1, 2. The metric will then be independent of t. Note also that the system (4.1) - (4.4) yields ∂tb = 2 (|b|2∂− − ∂+)b = 0 , so b is time-independent as well. In terms of the vierbein EAµ the metric is given ds2 = 2E+E− + 2E•E •̄ , (4.7) where E+µ = Bµ + Vµ 2|b|2 , E−µ = Bµ − Vµ From V 2 = −4|b|2 and V = ∂t as a vector we get Vt = −4|b|2, so that V = −4|b|2(dt+σ) as a one-form, with σt = 0. Furthermore, V • = 0 yields E•t = 0, and thus E• = E•zdz + E The component E•z can be eliminated by a diffeomorphism xi = xi(x′j , z) , = −EIz , I = •, •̄ . As the matrix EIi is invertible 7, one can always solve for ∂xi/∂z. Note that the metric is invariant under t→ t+ χ(xi, z) , σ → σ − dχ , 7One has det(EIi ) = − det(EAµ ), and the latter is always nonzero. – 21 – where χ(xi, z) denotes an arbitrary function. This second gauge freedom can be used to eliminate σz. Hence, without loss of generality , we can take σ = σidx i, and the metric (4.7) becomes ds2 = −4|b|2(dt+ σidxi)2 + 4|b|2 + 2E iE •̄j dx j . (4.8) Next one has to impose vanishing torsion, ν − ∂νEAµ + ωAµBEBν − ωAνBEBµ = 0 . One finds that some of these equations are already identically satisfied, while the re- maining ones yield (using the expressions (4.6) for the spin connection) the constraints ∂zσi = − 4|b|2 (E •̄ − E•iEj•)∂j ln(b/b̄) , (4.9) ∂iσj − ∂jσi = (E•iE •̄j −E•jE •̄i ) ∂z ln(b/b̄) + , (4.10) ω••̄t = −2|b|2∂z ln(b/b̄) + − 2b̄ , (4.11) j − ∂jE•i = (E•iE •̄j −E•jE •̄i )ω••̄•̄ , (4.12) as well as ∂z + ω ∂z ln(b̄b) + E•i = 0 . (4.13) In (4.9), EiI denotes the inverse of E j . In order to obtain the above equations, one has to make use of the inverse tetrad E+ = − 2|b|2∂z , E− = 2|b|2 2 ∂z , E• = E •(∂i − σi∂t) . (4.13) can be solved to give E•i = |b|Ê i exp dz ω••̄z − , (4.14) where Ê•i is an integration constant that depends only on the coordinates x j . At this point it is convenient to use the residual U(1) gauge freedom of a combined local Lorentz and gauge transformation to eliminate ω••̄z . This is accomplished by the transformation (2.3), with dz ω••̄z . – 22 – Note that ψ is real, as it must be. Defining Φ := − 1 , (4.15) we have thus E•i = |b|Ê i expΦ . (4.16) Using (4.16) in (4.12), one gets for the only remaining unknown component ω••̄• of the spin connection ω••̄• = ω̂••̄• − Êi•∂i |b| exp(−Φ) , where ω̂••̄• denotes the spin connection following from the zweibein Ê In what follows, we shall choose the conformal gauge for the two-metric hij = ÊIiÊ j , i. e. , hij = e 2ξ[(dx1)2 + (dx2)2] . (4.17) with ξ depending only on the coordinates xi. Furthermore, we choose an orientation such that Ê•i Ê j − Ê•j Ê •̄i = −ie2ξǫij , where ǫ12 = 1. To be concrete, we shall take (ÊIi ) = The eqns. (4.9) and (4.10) then simplify to ∂zσi = − 4|b|2 ǫij∂j ln(b/b̄) , (4.18) ∂iσj − ∂jσi = − |b|2 e 2(Φ+ξ)ǫij ∂z ln(b/b̄) + . (4.19) Moreover, one has ω••̄• = −∂• ln |b|e−Φ−ξ . (4.20) In [13] it has been shown that in the case where the Killing vector constructed from the Killing spinor is timelike, the Einstein equations follow from the Killing spinor equations, so all that remains to do at this point is to impose the Bianchi identity and the Maxwell equations. Using the spin connection (4.6) and (4.11) in (4.5), the gauge – 23 – potential and the field strength become A = i(dt + σ)(b− b̄) + ℓ ǫij∂j(Φ + ξ) dx i − iℓ d ln(b/b̄) , F = i(dt + σ) ∧ d (b̄− b) + 1 4|b|2dz ∧ dx iǫij∂j(b+ b̄) 2|b|2 ∂z(b+ b̄) + e2(Φ+ξ)ǫijdx i ∧ dxj . (4.21) The Bianchi identity F = dA yields ∆(Φ + ξ) = e2(Φ+ξ) , (4.22) with ∆ = ∂i∂i denoting the flat space Laplacian in two dimensions. As for the Maxwell equations, −gFµν) = 0 , the only nontrivial information comes from the t-component, which gives 4e2(Φ+ξ) b2∂2z − b̄2∂2z + b2∆ − b̄2∆1 = 0 , (4.23) where we used eqns. (4.18) and (4.19). Let us now show that the equations (4.22) and (4.23) are actually the same as the ones in [16]. If we set F = − 1 , eφ = 2eΦ+ξ , (4.24) (4.22) yields exactly equation (2.3) of [16]. On the other hand, deriving (4.22) with respect to z and using (4.15), one obtains ∆A + e2φ 3A∂zA− 3B∂zB + A3 − 3AB2 + ∂2zA = 0 , (4.25) where A and B denote the real and imaginary part of F respectively. This can be used in (4.23) to get ∆B + e2φ ∂2zB + 3B∂zA+ 3A∂zB − B3 + 3A2B = 0 , which, together with (4.25), yields ∆F + e2φ F 3 + 3F∂zF + ∂ = 0 , (4.26) i. e. , equation (2.2) of [16]. For a complete identification of the present results with the ones in [16], one also has to set σ = ω. – 24 – In conclusion, the metric of the general 1/4-supersymmetric solution is given by ds2 = −4|b|2(dt+ σ)2 + 1 4|b|2 dz2 + 4e2(Φ+ξ)dw dw̄ , (4.27) where b and φ are determined by the system (4.22), (4.23) and w = x1 + ix2 ≡ x+ iy. The one-form σ follows then from (4.18) and (4.19), and the gauge field strength is given by (4.21). Note that (4.23) represents also the integrability condition for (4.18), (4.19). As noted in [16], this system of equations is invariant under PSL(2,R) transformations8. If we define a new coordinate z′ through the Möbius transformation αz + β γz + δ , (4.28) with α, β, γ and δ arbitrary real constants satisfying αδ − βγ = 1, then the functions b̃(z′, xi) and Φ̃(z′, xi) defined by (γz′ − α)2b − γz′ − α , e Φ̃ = (γz′ − α)2 eΦ , (4.29) solve the system in the new coordinate system (z′, xi), with the function ξ(xi) left invariant and z seen as a function of z′. This symmetry allows to generate new BPS solutions from the known ones. Note however that it is only a symmetry of the equations for 1/4 supersymmetry, and if we apply it to solutions with additional Killing spinors, it will in general not preserve them, as we shall show explicitely in some examples. 4.3 Half-supersymmetric backgrounds We now would like to investigate the possibility of adding a second Killing spinor. Since the first Killing spinor ǫ1 has stability subgroup 1, one cannot use Lorentz transforma- tions to bring the second spinor to a preferred form. Therefore we use the most general ǫ2 = c01 + c1e1 + c2e2 + c12e1 ∧ e2 . (4.30) The corresponding linear system simplifies significantly after inserting the results from ǫ1. These determine all the fluxes and the spin connection in terms of the functions b, ξ and their derivatives. First it is convenient to introduce the new basis9 b−1c2 − c0 8It might be of interest to investigate the possible relation between this ’hidden symmetry’ and the Ehlers group for solutions of four-dimensional vacuum gravity with a Killing vector. 9Note that ǫ1 = (1, 0, 0, 0) in this basis. – 25 – in which the Killing spinor equations for ǫ2 read (∂A +MA)α = 0 , (4.31) with the connection MA given by 0 −∂+ ln b̄ 0 0 0 ∂+ ln b̄ −∂• ln b ∂• ln b 0 0 b̄−b√ ∂+ ln b̄− ∂+ ln b 0 −|b|2∂•̄ ln b̄ 0 b̄−b√2ℓ − ∂+ ln(b̄b) 0 0 |b|−2∂• ln b̄ −|b|−2∂• ln b̄ 0 ∂− ln b −|b|−2∂• ln b̄ |b|−2∂• ln b̄ 0 ∂•̄ ln b b−b̄√ 2ℓ|b|2 − ∂− ln(b̄b) 0 0 0 − − ∂− ln b̄ b−b̄√2ℓ|b|2 + ∂− ln 0 −∂• ln b̄ 0 0 0 ∂• ln(b̄b) 0 0 b̄− ∂+ ln b −∂• ln |b|e−Φ−ξ b+ ∂+ ln b̄ 0 −∂• ln |b|e−Φ−ξ M•̄ = 0 0 −∂− ln b̄ ∂− ln b̄+ 0 0 ∂− ln(b̄b) + −∂− ln(b̄b)− 0 0 ∂•̄ ln e−Φ−ξ −∂•̄ ln b 0 0 −∂•̄ ln b̄ ∂•̄ ln e−Φ−ξ Let us first of all consider the simpler possibility of a second Killing spinor of the form ǫ2 = c01 + c2e2. As discussed in section 2.1, both ǫ1 and ǫ2 are invariant under the same U(1) symmetry, and hence this case constitutes the G = U(1) case with four supersymmetries. As can easily be seen from the above Killing spinor equations with α1 6= 0 and α2 = α12 = 0, this restricts the derivatives of the coefficient b to be ∂−b = − , ∂+b = − , ∂•b = ∂•̄b = 0 . (4.32) Hence this corresponds to ∂zb = −1/ℓ. As will be discussed in section 5.1, this restric- tion uniquely leads to the half-supersymmetric anti-Nariai space-time. Hence AdS2×H2 is the only possibility for backgrounds with four U(1)-invariant Killing spinors. In the more general case with α2 and α12 non-vanishing, i.e. with trivial stability subgroup, the Killing spinor equations do not so readily provide information about b – 26 – and one has to resort to their integrability conditions. The first integrability conditions for the linear system (4.31) are Nµνα ≡ (∂µMν − ∂νMµ + [Mµ,Mν ])α = 0 , (4.33) where the matrices Mµ = E µMA are given by 2(|b|2M− −M+) , Mz = 2|b|2 (M+ + |b|2M−) , Mw = σwMt + eΦ+ξM• , Mw̄ = σw̄Mt + eΦ+ξM•̄ , and we introduced the complex coordinates w = x + iy, w̄ = x − iy. For half- supersymmetric solutions, the six matrices Nµν must have rank two. (As at least one Killing spinor exists, namely ǫ1 = (1, 0, 0, 0), we already know that the Nµν can have at most rank three. Rank one is not possible, because 3/4 BPS solutions cannot exist [29]. Rank zero corresponds to the maximally supersymmetric case, which implies that the spacetime geometry is AdS4 [13].) Let us define ѵν ≡ SNµνT , 1 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 , T = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 1 The similarity transformation S corresponds to adding the first line to the second one and T adds the last column to the third one. This does not alter the rank of Nµν . One finds Ñwt = 2b∂∂z b̄+ ∂b̄ −2|b| e−Φ−ξ ∂2b̄+ 1 ∂b̄∂b̄ ∂z b̄+ ∂ ln b̄ 0 −2∂(Φ + ξ)∂b̄ 2b̄∂∂zb+ ∂b −2|b| e−Φ−ξ ∂2b+ 1 ∂ ln b 0 −2∂(Φ + ξ)∂b) 2|b|3e−Φ−ξ∂̄∂ ln b 2b̄∂∂zb 0 −2|b|3eΦ+ξb−2 2∂zb+ ∂ ln b 2|b|3e−Φ−ξ∂̄∂ ln b̄ 2b∂∂z b̄− 2ℓ∂b̄0 −2|b|3eΦ+ξ b̄−2 2∂z b̄+ ∂z b̄+ ∂z b̄+ ∂ ln b̄ – 27 – Ñw̄t = 2b∂̄∂z b̄ −2|b|e−Φ−ξ∂̄∂ ln b̄ ∂z b̄+ ∂̄ ln b̄ ℓ|b|e 2∂z b̄+ + 2b|b|b̄e 2∂z b̄+ ∂z b̄+ 2b̄∂̄∂zb −2|b|e−Φ−ξ∂̄∂ ln b ∂̄ ln b ℓ|b|e 2∂zb+ + 2b̄|b|be 2∂zb+ 2|b|b̄e−Φ−ξ ∂̄2b+ 1 ∂̄b∂̄b 2b̄∂̄∂zb+ 0 −2∂̄(Φ + ξ)∂̄b ∂̄ ln b 2|b|be−Φ−ξ ∂̄2b̄+ 1 ∂̄b̄∂̄b̄ 2b∂̄∂z b̄− 4ℓ ∂̄b̄0 −2∂̄(Φ + ξ)∂̄b̄ ∂z b̄+ ∂̄ ln b̄ where ∂ = ∂w, ∂̄ = ∂w̄. The other four integrability conditions give no additional information, because the lines of the corresponding matrices are proportional to the lines of Ñwt and Ñw̄t As the upper right 3× 3 determinant of Ñwt must vanish, we obtain ∂b = 0 or e−2(Φ+ξ)b̄∂b̄ e−2(Φ+ξ)b∂b e−2(Φ+ξ)b∂b e−2(Φ+ξ)b̄∂b̄ = 0 . (4.34) Let us assume that the expression in (4.34) does not vanish. One has then ∂b = 0 as well as ∂b̄ = 011. But then also (4.34) holds, which leads to a contradiction. Thus (4.34) must be satisfied in any case. Note that the vanishing of the first column of ѵν implies that also the first column of T−1NµνT is zero, and thus T −1NµνT ∈ a(3,C), hence the generalized holonomy in the case of one preserved complex supercharge is contained in the affine group A(3,C). This supports the classification scheme of [4]. Of course, depending on the particular solution, the generalized holonomy may also be a subgroup of A(3,C). 4.4 Time-dependence of second Killing spinor In this section we will utilize the above Killing spinor equations to derive the time- dependence of the second Killing spinor. In addition, we will show that the Killing spinor equations can be completely solved when the second Killing spinor is time- independent. Let us first simplify the Killing spinor equations (4.31). In the following we set b = reiϕ and define ψ = Φ + ξ, ψ1 = r 2α1, ψ2 = re −ψα2, ψ12 = re −ψα12 and ψ± = 10In order to show this, one has to make use of eqns. (4.22) and (4.26). 11This follows from the vanishing of the 3 × 3 determinant that is obtained from Ñwt by deleting the first column and the third line. – 28 – ψ2 ± ψ12. First of all, use the integrability conditions (4.33), that can be rewritten as ѵνT −1α = 0. Defining P = e−2ψb∂b, the second component for µ = w, ν = t gives ′ + ψ−∂P = 0 , (4.35) with ′ = ∂z . Let us assume P ′ 6= 0 (the case P ′ = 0 is considered in appendix C and will lead to the same conclusions). If we define g(t, z, w, w̄) = −ψ−/P ′, we get ψ− = −gP ′ , ψ1 = g∂P . The third component of the (w, t) integrability condition is of the form ψ1f1 + ψ2∂b+ ψ−f− = 0 , for some functions f1, f− that depend on z, w, w̄ but not on t. Using the above form of ψ1 and ψ−, this becomes f1g∂P + ψ2∂b− f−gP ′ = 0 . (4.36) Now, if g = 0, the latter equation implies ψ2∂b = 0, and hence (since ∂b 6= 0 due to P ′ 6= 0) ψ2 = 0. Furthermore, ψ1 = ψ− = 0 in this case, so there exists no other Killing spinor. Thus, g 6= 0 and we can write g = expG. Dividing (4.36) by g and deriving with respect to t yields ∂t(ψ2/g) = 0 and hence ψ2 = e Gψ02(z, w, w̄) . It is then plain that ∂tψi = ψi∂tG, i = 1, 2, 12. The Killing spinor equations are of the form ∂µψi = Mµijψj , for some time-independent matrices Mµ. Taking the derivative of this with respect to t, one gets ∂µ∂tG = 0, whence G = G0t + G̃(z, w, w̄) , with G0 ∈ C constant. We have thus ∂tψi = G0ψi and hence also ∂tαi = G0αi. Fur- thermore, the time-dependence of α0 can be easily deduced from the Killing spinor equations: if G0 does not vanish it is of the same exponential form as the other com- ponents of the second Killing spinor, i.e. ∂tα0 = G0α0, while if G0 vanishes there can be a linear part in t, i.e. ∂tα0 = c for some constant c. Hence, in terms of the basis elements, the time-dependence of the second Killing spinor takes the form12 G0 = 0 : ǫ2 = c01 + c1e1 + c2e2 + c12e1 ∧ e2 + ct(1 + be2) , G0 6= 0 : ǫ2 = eG0t(c01 + c1e1 + c2e2 + c12e1 ∧ e2) , (4.37) 12We will loosely refer to Killing spinors with G0 = 0 as time-independent, despite the possible linear time-dependence, to distinguish from the G0 6= 0 exponential time-dependence. – 29 – where c0, c1, c2, c12 are time-independent functions of the spatial coordinates, and c is a constant. This was derived assuming P ′ does not vanish, but as we show in appendix C is in fact a completely general result. Hence, adding a second Killing spinor to ǫ1 = 1 + be2, the Killing spinor equations imply that ǫ2 always has the above time- dependence. Plugging this time-dependence into the subsystem of the Killing spinor equations not containing α0 one obtains in terms of ψi ψ′1 − ψ− = 0 , (4.38) ψ′2 − ψ12 = 0 , (4.39) ψ′12 − e−2ψ ψ12 = 0 , (4.40) ψ′1 − ψ− = 0 , (4.41) ψ′2 + e −2ψ ∂̄b ψ2 = 0 , (4.42) ψ′12 − ψ12 = 0 , (4.43) ∂ψ1 − σwG0ψ1 = 0 , (4.44) ∂ψ2 − σwG0 + − 2∂ψ ψ2 = 0 , (4.45) ∂ψ12 + σwG0 + − 2∂ψ ψ12 = 0 , (4.46) ∂̄ψ1 − σw̄G0 + ψ1 + e − ψ12 = 0 ,(4.47) ∂̄ψ2 − σw̄G0 + ψ12 = 0 ,(4.48) ∂̄ψ12 − σw̄G0 + ψ12 = 0 .(4.49) ForG0 = 0, these equations simplify significantly, and allow for a complete solution. As is shown in appendix D, under the additional assumption ψ− 6= 0, ψ1 6= 0, the metric – 30 – and the field strength for half-supersymmetric solutions with G0 = 0 are given in terms of a single real function H depending only on the combination Z−w− w̄ and satisfying the second order differential equation 1 + e−2H Ḧ + Ḣ2 1− 3α e2H + 1− α2 , (4.50) where α ∈ R denotes an arbitrary constant and γ = 0, 1. The new coordinate Z is defined by Z = z for γ = 0 and Z = ℓ ln 1 + z for γ = 1. Furthermore, in the remainder of this section and in appendix D, a dot denotes a derivative with respect to Z − w − w̄. Given a solution of (4.50), one defines the functions χ, ρ by e2H + 1− α2 − Ḣ2χ2 . (4.51) Note that χ is imaginary and ρ is real. b and ψ are then given by b = eγZ/ℓρ eiϕ , e2ψ = e2(H+γZ/ℓ) , where tanϕ = so that the metric reads ds2 = −4ρ2e2γZ/ℓ(dt+ σ)2 + 1 + e2Hdwdw̄ , (4.52) where the shift vector satisfies ∂Zσw = e−γZ/ℓ , ∂σw̄ − ∂̄σw = − e−γZ/ℓ Finally, the gauge field strength is given by (4.21). Equation (4.50) is actually the Euler-Lagrange equation for the following standard action for the scalar H d (Z − w − w̄) M(H)Ḣ2 − V (H) , (4.53) where M(H) = e2H + 1 (e2H + 1− α2)3/2 , V (H) = − γ e2H + 1− 2α2 (e2H + 1− α2)1/2 . (4.54) – 31 – Thus it is possible to use the energy conservation law of that model in order to evaluate the “velocity” Ḣ in terms of H . Since dH = Ḣd (Z − w − w̄) one has M(H)Ḣ2 + V (H) = 0 , (4.55) so that there must exist a constant E such that [E − V (H)] = e2H + 1− α2 e2H + 1 e2H + 1− 2α2√ e2H + 1− α2 (4.56) The key-point is to consider now, as a new coordinate, the function H in place of w+ w̄13 and to write down the full solution, say metric plus gauge field, in terms of H . Using w = x+ iy, the general solution is given by ds2 = −4ρ2e2γZ/ℓ dt + e−γZ/ℓσ̂ydy dy2 + dZ2 + e2H dZ − dH A = ℓḢ −2iρ2χeγZ/ℓdt+ 1− e2Hχ2 − i ℓ d log , (4.57) where Ḣ is given in equation (4.56), the functions χ and ρ are defined in (4.51) and the shift vector reads σ̂y = − If γ = 1, a simple example of this set of solutions can be obtained by setting α = 0, Ḣ = 1/ℓ , b = . (4.58) As will be shown in section 5.1.2, this corresponds to the maximally supersymmetric AdS4 solution. More general γ = 1 solutions will be two-parameter deformations thereof, the parameters being α and the energy E of the associated scalar system. Setting γ = 0 the potential V (H) vanishes and the parameter E can be fixed by a simple rescaling of the coordinates. Thus we are left with a one-parameter family of solutions. Since the metric does no more depend explicitly on Z, it is useful to replace the coordinate Z instead of x by H . Defining a new coordinate r such that 13This is possible by simply requiring that Ḣ 6= 0. – 32 – r4 ≡ 16 e2H + 1− α2 and a new parameter Q = 4 α, the complete solution reads ds2 = − dt− 2ℓ r4 + ℓ2Q2 h(r)2 r4 + ℓ2Q2 − 16 dx2 + dy2 A = −Q dy − i ℓ d log , (4.59) where h(r) = r4 + ℓ2Q2 r4 + ℓ2Q2 − 16 . (4.60) The parameter Q can thus be interpreted as an electric charge. The Petrov type of the solution is D or simpler. If one sets Q = 4/ℓ the Petrov type is reduced to N , so that there is a gravitational wave. In order to complete the classification of G0 = 0 solutions, we need to study separately the cases where either ψ1 or ψ− vanishes (it can easily be seen from (4.39) and (4.40) that there is no solution if both vanish). As one can see by looking at equations (4.38) and (4.41), the condition ψ1 = 0 leads to b = b(z), which is studied in detail in section 5.1. The other possibility, ψ− = 0, is more involved, but as we show in appendix D it boils down to three different cases, that can be completely solved: the AdS2 × H2 anti-Nariai spacetime studied in section 5.1.1, the imaginary b case solved in section 5.3, and finally the half BPS solution coming from the gravitational Chern-Simons model, that we analyse in section 5.5. We would like to remark that the assumptionG0 = 0 on the overall time-dependence of the second Killing spinor seems a reasonable choice since all known 1/2-supersymmetric solutions to be studied in the next section are contained in this class, or can be brought to this class by a general coordinate transformation. Hence we expect the G0 = 0 class to form an important subclass of all 1/2-supersymmetric solutions. 5. Timelike half-supersymmetric examples The problem of finding all half BPS configurations in the timelike class involves the solution of the integrability conditions we obtained above. To obtain explicit examples of half BPS solutions, we shall restrict to some simple subclasses with particular b. This will determine the fraction of preserved supersymmetry for the solutions which are already known to be 1/4 supersymmetric, and will also lead to new solutions. – 33 – 5.1 Static Killing spinors and b = b(z) The timelike vector field V , constructed as a bilinear of the Killing spinor, is static if the associated one-form V = dt + σ satisfies the Fröbenius condition V ∧ dV = 0. Obviously, there can be static BPS solutions with V not being static itself, due to the choice of coordinates; we shall loosely refer the Killing spinors whose vector bilinear is static as static Killing spinors. The staticity condition, in turn, implies dσ = 0 and puts strong constraints on the function b. Indeed, equation (4.18) implies that the phase ϕ of b depends only on z. Then, (4.19) gives the modulus r of b in terms of its phase, sinϕ(z) lϕ′(z) . (5.1) As a consequence, r and therefore the complete complex function b, depend on the single variable z. The full solution is therefore determined by the single real function ϕ, which has to satisfy the equations for supersymmetry (together with the conformal factor ψ). However, since the equations can be exactly solved for arbitrary b(z), we will stick to this more general case and eventually comment on the static subcase. If b depends only on z, the equations of motion simplify to b2∂2z = 0 , (5.2) e−2ξ∆ξ = . (5.3) Here we have used the fact that Φ, defined in (4.15), depends only on the coordinate z. In principle there is also an integration constant K(w, w̄) with arbitrary dependence on the transverse coordinates, but since Φ appears only in the combination Φ+ ξ in all the equations, we can always absorb the (w, w̄) dependence into the conformal factor ξ. Now the left hand side of equation (5.3) depends only on the coordinates w and w̄, while the right hand side depends only on z. This equation can be therefore satisfied only if both sides are equal to some constant κ. The system of equations is then e2ξ = 0 , (5.4) e2Φ(z) = − l κ . (5.5) Note that the first one is the Liouville equation, whose solution describes the transverse two-dimensional manifold, which has therefore constant curvature κ. – 34 – Equations (5.2) and (5.5) can easily be solved [16]. Their solution is given by14 b̄ = −αz 2 + βz + γ ℓ(2αz + β) , (5.6) with α, β, γ ∈ C. Then ξ solves the Liouville equation for a constant curvature two- manifold with scalar curvature15 κ = 8(αγ̄ + ᾱγ)− 4ββ̄ . (5.7) This solution generically belongs to the supersymmetric Reissner-Nordström-Taub- NUT-AdS4 family of spacetimes. The values α = 0 and β 2 = 4αγ are special cases and will be treated separately in the following. Note that the coefficients α, β and γ are not three independent parameters, as they can be rescaled without changing the function b: the solutions depend only on their ratios. For example, if α 6= 0, one can use β/α and γ/α as independent complex parameters of the family of solutions. The solutions with static Killing spinor form a subset of this family. For (5.6) the staticity condition (5.1) yields the condition αβ̄− ᾱβ = 0. Recalling the expression for the NUT charge of these solutions, , (5.8) this charge must vanish for non-vanishing α, as one could have guessed. On the other hand, for α = 0 the solution is anti-Nariai, as we shall see below. We conclude that the most general supersymmetric configuration with static Killing vector constructed as a Killing spinor bilinear is either of the form (5.6) – i. e. in the fourth row of table 1 of [34] – with vanishing NUT charge, or it is anti-Nariai spacetime. The supersymmetric static solutions discussed so far are generically 1/4-BPS. We want to see what further condition ensures the presence of an additional Killing spinor. Inserting the staticity ansatz b = b(z) into the integrability equations and requiring these matrices to be of rank smaller or equal to two, one finds the following condition (in particular this is obtained from the vanishing of the minor of the last row of Ñwt and the first two rows of Ñw̄t) 2b′ + = 0 , (5.9) 14With this definition, the constants α, β and γ coincide with a, b and c of [16] respectively. 15This scalar curvature differs from the one given in [16] for the case in which all coefficients are real, k = 4αγ − β2 = κ/4. The factor of 4 comes from the different definition of the conformal factor of the transverse metric, our ξ is related to the old γ by ξ = γ − ln 2. – 35 – As an aside, note that we have only used the ansatz b = b(z) so far and not the staticity condition (5.1), i.e. the precise relation between r and ϕ. The static solutions are therefore in general still a subset of the solutions under consideration. Condition (5.9) calls for the following three different cases, corresponding to the vanishing of its three factors. 5.1.1 AdS2 ×H2 space-time (α = 0) Requiring the first factor of (5.9) to vanish leads to b = −z + ic with constant c, corresponding to α = 0 in (5.6). We can absorb the imaginary part of c by a shift of the coordinate z and henceforth will assume c ∈ R. In this case κ = −4 and we have a hyperbolic transverse space. As a solution of (5.4) we can take e2ξ = . (5.10) Moreover, eΦ = l|b| and σ = 0, therefore giving the metric ds2 = −4 dt2 + dx2 + dy2 . (5.11) This is the anti-Nariai AdS2 × H2 solution, with the AdS2 factor written in Poincaré coordinates for c = 0 and in global coordinates for c 6= 0. The coordinate transforma- tions between Poincaré coordinates (tP , zP ) (with c = 0) to global ones (tgl, zgl) (with c 6= 0) is given by (zgl − z2gl + ℓ 2c2 cos(4ctgl/ℓ)) , tP = − z2gl + ℓ 2c2 sin(4ctgl/ℓ) zgl − z2gl + ℓ 2c2 cos(4ctgl/ℓ) . (5.12) The electromagnetic field strength (4.21) in this case is given by F = − 1 dx ∧ dy , (5.13) i.e. only lives on the hyperbolic part and is independent of the coordinates of the AdS part of space-time. This solution preserves precisely 1/2 of the supersymmetries, as was already shown in [35]. To obtain the form of the Killing spinors admitted by this metric we first observe that the integrability conditions impose α2 = α12 = 0. Then the Killing spinor equations are easily solved, but one should treat separately the cases c = 0 and c 6= 0: – 36 – • If c = 0, then α0 = λ1 + λ2 , α1 = , (5.14) where λ1,2 ∈ C are integration constants. This yields the following Killing spinors, spanning a two-dimensional complex space, λ1 + λ2 1 + b λ1 + λ2 e2 . (5.15) Note that λ1 = 1, λ2 = 0 corresponds to the original Killing spinor. Also note that the constant G0, corresponding to the time-dependence of the second Killing spinor with λ2 6= 0, is zero. The form of the scalar invariant corresponding to the general spinor ǫ is b̃ = b |λ1|2 + |λ2|2 λ̄1λ2 + λ1λ̄2 λ̄1λ2 − λ1λ̄2 . (5.16) Here the first term is real, while the second is imaginary. Note that the latter is in fact constant. Then the Killing vector Ṽ built from ǫ will have a norm Ṽ 2 = −4|b̃|2, and will be timelike unless b̃ vanishes. This is however not possible, because both the real and imaginary parts of b̃ should vanish, but since λ1,2 do not depend on the coordinates, the real part cannot vanish. Therefore, every Killing spinor of this solution belongs to the timelike class. • If c 6= 0 we have [λ1−iλ2+(λ1+iλ2) −4ict/ℓ , α1 = − |b| (λ1+iλ2)e −4ict/ℓ , (5.17) and the most general Killing spinor is parametrized by λ̃1,2 ∈ C as follows (λ1 − iλ2)(1 + be2) + 2|b|(λ1 + iλ2)e −4ict/ℓ(1 + b∗e2) . (5.18) Note that the combination λ1− iλ2 corresponds to the first Killing spinor 1+ be2, while the orthogonal combination λ1 + iλ2 gives rise to the second Killing spinor proportional to 1 + b∗e2. Any combination with λ2 6= 0 has G0 = −4ic/ℓ. In this case, the real part of the invariant b̃ is given by Re(b̃) = |λ1|2 (−z + z2 + ℓ2c2 cos(4ct/ℓ)) + |λ2|2 (−z − z2 + ℓ2c2 cos(4ct/ℓ))+ 2 + λ2λ z2 + ℓ2c2 sin(4ct/ℓ)) , (5.19) while the imaginary part is identical to that of (5.16). – 37 – It can easily be checked that the coordinate transformation (5.12) indeed relates the complex scalar b̃, which is composed of spinor bilinears, in (5.16) and (5.19) to each other. Let’s now check how the isometries of AdS2 act on the Killing spinors. It is useful to do this by embedding AdS2 with metric ds2 = −4 dt2 + ) (5.20) into the three-dimensional flat space Xa = (U, T,X) with metric ds2 = −dU2 − dT 2 + dX2 . (5.21) Then, AdS2 is obtained as the hyperboloid defined by −U2 − T 2 +X2 = ℓ , (5.22) and its isometry group SO(2,1) will act as the three-dimensional Lorentz group on the embedding coordinates Xa (here a is a three-dimensional Lorentz index). If c = 0, the AdS2 metric (5.20) is in the Poincaré form, and can be seen to be the induced metric on the hyperboloid by parameterizing it with the coordinates (t, z) given by z = U +X , t = 2(U +X) . (5.23) Then, if one defines the 3d Lorentz vector |λ1|2 − |λ2|2 (λ∗1λ2 + λ1λ 2) ,− |λ1|2 + |λ2|2 , (5.24) one explicitly checks that the invariant b̃ can be put in the form b̃ = XaΛ ΛaΛa. (5.25) Now, the real and imaginary part of b̃ are independently manifestly invariant under the AdS2 isometries, as they should be (since they transform respectively as pseudoscalar and scalar under diffeomorphism16). If c 6= 0 we have AdS2 in global coordinates, and the embedding is modified to U = − ℓ + c2 cos , T = − ℓ + c2 sin , X = . (5.26) 16Note that Λ doesn’t depend on the sum of the phases of λ1,2; this is diffeomorphism invariant but transforms under U(1) gauge transformations. – 38 – The invariant (5.19) takes again the manifestly invariant form (5.25), as expected, and the isometries of AdS2 are realized linearly on the Killing spinors through their action on Λa. This result may be useful to study in detail quotients of AdS2 and to see whether this operation breaks some supersymmetry. 5.1.2 AdS4 space-time (β 2 = 4αγ) The following subcase corresponds to the vanishing of the second factor of the inte- grability condition (5.9). The function b is then given by b = − z + ic, which can be obtained as the special case β2 = 4αγ from (5.6). This corresponds to AdS4, the only maximally supersymmetric solution of the theory. Indeed the integrability condition matrices vanish in this case. Let’s see in detail the form of the metric arising from different values of c. As in the previous case we can take the constant c to be real. If c = 0, the metric is static, σ = 0, ξ = 0 and e2Φ = |b|4, and we obtain anti-de Sitter in Poincaré coordinates, ds2 = −z dt2 − dx2 − dy2 dz2 . (5.27) On the other hand, for c 6= 0, the metric appears in non-static coordinates, σ = − ℓdy , e2ξ = 4c2x2 , e2Φ = |b|4 , (5.28) which give ds2 = − + 4c2 dt− ℓdy 16c2x2 dx2 + dy2 + 4c2 dz2 . (5.29) The field strength (4.21) vanishes in this case. We shall now obtain the form of the Killing spinors for AdS4, and will do this in the simpler c = 0 case. The solution of the Killing spinor equations yields α0 = λ1 − λ3 , α2 = − λ2 , α12 = 1− zt λ4 , (5.30) where the coefficients λ1,...,4 span a four dimensional complex space, as expected in the case of maximal supersymmetry. In the form basis of the spinors ǫ = c01+c1e1+c2e2+ – 39 – c12e1 ∧ e2, we obtain c0 = λ1 − λ3 , c2 = − λ3 + λ4 , c12 = λ4 . (5.31) The new Killing spinors corresponding to λ2 and λ4 both have 17 G0 = 0. To study the action of the AdS4 isometries it is useful to embed the hyperboloid in a five-dimensional flat space (U, V, T,X, Y ) with metric ds2 = −dU2 + dV 2 − dT 2 + dX2 + dY 2. (5.32) Then, AdS4 is the hypersurface −U2 + V 2 − T 2 +X2 + Y 2 = −ℓ2/4 and its isometries are realized as the SO(3,2) isometries of the embedding space. The relation with the Poincaré coordinates is U − V , U − V , U − V , z = 2(U − V ) . (5.33) If we define the vectors ℓΛa = |λ1|2 − |λ2|2 + |λ3|2 − |λ4|2 |λ1|2 + |λ2|2 − |λ3|2 − |λ4|2 λ3λ̄4 + λ̄3λ4 − λ̄1λ2 − λ1λ̄2 λ2λ̄4 + λ̄2λ4 − λ̄1λ3 − λ1λ̄3 λ2λ̄4 − λ̄2λ4 + λ̄1λ3 − λ1λ̄3 , Xa = , (5.34) where the index a = 1, . . . , 5 is an SO(3,2) index raised and lowered using the metric (5.32), then a = − 1 λ3λ̄4 − λ̄3λ4 + λ̄1λ2 − λ1λ̄2 ]2 ≥ 0 , (5.35) and the invariant b̃ for the Killing spinors reads b̃ = c∗0c2 + c1c 12 = XaΛ ΛaΛa . (5.36) 17Note that this does not hold for λ3, whose time-dependence is not of the form derived in section 4.4. There is no contradiction however, since all solutions in this class have P = 0 and hence are treated separately in appendix C. It is interesting to find that nevertheless the time-dependence of many Killing spinors in this class have the canonical G0 time-dependence. – 40 – This form of b̃ is manifestly invariant under the AdS4 isometries, and shows that under Λa transforms in the fundamental representation of SO(3,2) under these transforma- tions. Note that it has precisely the same form (5.25) as in the anti-Nariai case. Again, the explicit knowledge of the AdS4 isometry group action on the Killing spinors is important to study the supersymmetry of its quotients. 5.1.3 The Reissner-Nordström-Taub-NUT-AdS4 family The last subcase corresponds to the vanishing of the third factor of the integrability condition (5.9). Note that this is precisely the expression in square brackets of equation (5.5) and the condition reads simply κ = 0. Then ξ is an harmonic function and the transverse space is flat. In particular, the solution (5.6) admits a second Killing spinor |β|2 = 2(αγ̄ + ᾱγ) . (5.37) Since α 6= 0 we can define ζ = Im(β/α) and δ = Im(γ/α). Moreover, all equations are invariant under rigid translations in the z directions, since the coordinate z never appears explicitly in them. One can use this freedom to eliminate the real part of β/α by performing the redefinition z 7→ z − 1 Re(β/α). Hence this complete family of 1/2 BPS solutions is determined by two real parameters ζ and δ, b = −1 z2 − iζz + 1 ζ2 − iδ 2z − iζ . (5.38) Then σ = −2ζ(r/ℓ)2dϑ and the resulting metric is ds2 = − z2 + ζ + (ζz + δ) z2 + ζ dt− 2ζ z2 + ζ z2 + ζ + (ζz + δ) dr2 + r2 dϑ2 , (5.39) where we used polar coordinates (r, ϑ) in the (w, w̄) plane. The charges of the solution M = −δζ , n = , P = −ζ , Q = −δ . (5.40) Essentially, the imaginary part of γ gives the electric charge and the imaginary part of β determines the NUT charge. Note that the quantization condition P = −(kℓ2+4n2)/2ℓ is also satisfied. In terms of the charges, the solution is given by b = −1 (z − in)2 + 2n2 + iℓQ 2 (z − in) . (5.41) – 41 – The subfamily of static half BPS configurations is obtained by imposing the static- ity condition ζ = 0 or equivalently vanishing NUT charge. It is parameterized by the single parameter left, δ ∈ R and the solutions are restricted to have the following charges M = 0 , n = 0 , P = 0 , Q = −δ In terms of the charges, the solution is given by b = −1 z2 + iℓQ . (5.42) The metric and electromagnetic field strength for this solution read ds2 = − dt2 + + 4ℓ2z2 dwdw̄ , (5.43) F = −Q dt ∧ dz . (5.44) This is simply the backreacted AdS4 filled with the electric field generated by an electric charge Q placed in its center ζ = 0. The solution has a singularity there. Note that this solution was already shown to be 1/2 supersymmetric in [36]. It was also shown there that the Killing spinors are preserved if one compactifies the transverse two-dimensional plane to a two-torus. We will now discuss the Killing spinors for these metrics. The integrability condi- tions impose α2 = 0 and α4 . (5.45) With these constraints, the Killing spinor equations simplify, and can be solved to give α0 = λ1 + 2iζw̄λ2 , α1 = 0 , (5.46) z2 + iζz + ζ 4z2 + ζ2 , α12 = α2 − 4z2 + ζ2 , (5.47) where λ1,2 ∈ C parameterize the two dimensional space of Killing spinors. Then the most general Killing spinor for these metrics is ǫ = (λ1 + 2iζw̄λ2) 1− ℓλ2 2z + iζ 2z − iζ e1 +b (λ1 + 2iζw̄λ2) e2 − z2 − iζz + ζ2 4z2 + ζ2 λ2 e1 ∧ e2 . (5.48) – 42 – Again the second Killing spinor has G0 = 0 time-dependence. Finally, the correspond- ing orbit of the Killing spinor is determined by the invariant b̃ = b|λ1|2 + z2 + iζz + ζ 2z − iζ + 4ζ 2bww̄ |λ2|2 + 2iζb w̄λ̄1λ2 − wλ1λ̄2 . (5.49) It is easy to show now that b̃ is non vanishing for any choice of λ1,2: indeed if b̃ = 0, we have ∂∂̄b̃ = 4ζ2b|λ2|2 = 0 and either λ2 = 0, which implies in turn λ1 = 0, or ζ = 0. In the latter case, it is very easy to see that b̃ = 0 iff ǫ = 0. Therefore, all Killing spinors of this family of metrics belong to the timelike class, and the solution is purely timelike. Summary of the b = b(z) case: 1. The only supersymmetric solutions with static Killing spinor (i.e. whose timelike Killing vector constructed as a Killing spinor bilinear is static) are AdS4, the anti-Nariai spacetime and the Reissner-Nordström-AdS4 solutions of the fourth row of table 1 of [34], i. e. solutions of the form (5.6) with vanishing NUT charge. 2. The only 1/2 BPS solutions with static Killing spinor are the anti-Nariai space- time and the solution (5.43) with field strength (5.44). 3. The most general half BPS solution with b = b(z) are the anti-Nariai spacetime and the solution (5.39) with charges (5.40) describing an electric charge in the center of AdS4. The natural way to continue this approach is to study half BPS solutions with b harmonic, and this will be the subject of the next paragraph. 5.2 Harmonic b solutions The previous class of solutions can be generalized by requiring ∆b = 0 instead of b = b(z) [16]. This implies that ∆1/b = 0 and hence (4.23) still simplifies in exactly the same way as in the b = b(z) case. Indeed, the solution is b̄ = −αz 2 + βz + γ ℓ(2αz + β) , (5.50) where now α, β and γ are no more constants but arbitrary functions of (w, w̄). It is then easy to show that the ∆b = 0 condition requires these functions to be harmonic and all (anti-)holomorphic, that is α, β and γ all depending either only on w or only on w̄, and this is the most general solution with ∆b = 0. The b = b(z) configurations – 43 – are particular cases of this larger class, and are obtained for α, β and γ constant. Note that also the ∂b = 0 and ∂b̄ = 0 subclasses fall into this family. Let’s take for definiteness α, β, γ all anti-holomorphic, then b = b(z, w). The requirement that the integrability conditions allow for an extra Killing spinor, i.e. that they are of rank ≤ 2, in this case leads to several conditions. One of these is obtained from the minor of the last three lines of Ñwt and reads 2∂z b̄+ ∂z b̄+ ∂b∂b − 2∂(Φ + ξ)∂b ∂b = 0. (5.51) This gives three different cases to be analysed, corresponding to the vanishing of the first three factors of this equation (vanishing of the fourth factor implies b = b(z) and hence brings one back to the previous section). 5.2.1 Deformations of AdS2 ×H2 The vanishing of the first factor in (5.51) implies b = −z + ic(w), where c(w) is an arbi- trary holomorphic function. These are the α(w) = 0 supersymmetric Kundt solutions of Petrov type II, describing gravitational and electro-magnetic waves propagating on anti-Nariai space-time [16]. The remaining integrability conditions however imply α1 = α2 = α12 = 0, in which case there is no second Killing spinor, or ∂c = 0. Therefore there are no new half BPS solutions with non constant c. In this class c constant is the half supersymmetric anti-Nariai spacetime and the other preserve only 1/4 of the supersymmetries. 5.2.2 Deformations of AdS4 The vanishing of the second factor in (5.51) implies b = − z + ic(w). In this case we are considering the β2 = 4αγ supersymmetric Kundt solutions, describing gravitational and electro-magnetic waves propagating on AdS4 spacetime [16]. Again the remaining integrability equations have to solutions: α1 = α2 = α12 = 0 or ∂c = 0. Hence, as in the previous case, we find that there are no harmonic deformations of AdS4 preserving half supersymmetry. 5.2.3 Deformations of Reissner-Nordström-Taub-NUT-AdS4 Not considering the previous two special cases, the general solution represents expand- ing gravitational and electro-magnetic waves propagating on a Reissner-Nordström- Taub-NUT-AdS4 spacetime [16]. When Im(β) = 0, the solution can be put in Robinson- Trautman form and is of Petrov type II. The vanishing of the third factor in (5.51) is given by ∂b∂b − 2∂(Φ + ξ)∂b = 0 . (5.52) – 44 – With b given in (5.50) this case can be solved for the derivative of Φ + ξ and implies ∂̄(Φ + ξ) = , (5.53) and therefore ∆(Φ + ξ) = 0. Then (5.3) fixes the transverse manifold to be flat and κ(w) = 8(αγ̄ + ᾱγ)− 4ββ̄ = 0. (5.54) But α,β and γ being holomorphic, this last equation can be satisfied if and only if they are constant, and we are back to the previous case, i. e. there are no new 1/2 BPS solutions. Summary of the harmonic case: There are no new half BPS solutions in the harmonic b case. The only half BPS solutions are those with b = b(z), and as soon as one deforms these solutions by adding some harmonic (w, w̄)-dependence, one breaks supersymmetry further to 1/4. 5.3 Imaginary b solutions Another subcase we want to study is b̄ = −b, i. e. b purely imaginary. For notational convenience we introduce18 b = iX , where X is real. From (4.15) one gets Φ = 0. All quantities in the Bianchi iden- tity (4.22), apart from b and hence X , are then z-independent. The only consistent possibility is to take ∂zX = 0. The remaining equations (4.22) and (4.23) read e2ξ , ∆ e2ξ = 0 . (5.55) Examples of 1/4 supersymmetric solutions of this class, i.e. with imaginary b, that were discussed in [16] are X = (x/ℓ)α with α = −2 and α = 1 . These correspond to a particular Petrov type I solution and an electrovac AdS travelling wave of Petrov type N, respectively. It was shown that the latter actually preserves a second, null Killing spinor. In this section we will derive the general condition for 1/2 supersymmetry in the case of imaginary b and will find that there is a one-parameter family of such solutions. The condition for 1/2 supersymmetry is very simple in this case. Assuming that ∂X is not equal to zero, which would clearly be incompatible with (5.55), there is 18In the following we will assume that X is positive without loss of generality. – 45 – only one differential constraint which needs to be satisfied for the existence of a second Killing spinor, i. e. for the matrices of integrability conditions to have rank 2, namely ∂2X−1 − 2∂ξ∂X−1 = 0 . (5.56) The above three differential equations can be integrated to e2ξ = −iK̄(w̄)∂X−1 , ∂X−1 = i , (5.57) where K(w) is an arbitrary holomorphic function and L is a real constant. The func- tion K(w) corresponds to the freedom to choose holomorphic coordinates on the two- dimensional space, and hence it can be gauged away. A convenient gauge choice will be K(w) = iℓ. Note that, for this choice, the imaginary part of the right hand side of the last equation vanishes, and therefore that ∂yX = 0. For L = 0, (5.57) can be integrated to give , (5.58) which is (up to a rescaling of the coordinate x) the example given above with α = 1 This was already found to be 1/2 supersymmetric in [16]. Here we find that this solution is a special case of the most general possibility. For other values of the constant L it is convenient to use X as a new coordinate instead of solving for X(x). From (4.18) and (4.19) it follows that σ can be chosen to . (5.59) Then the metric reads ds2 = −4X2 dz2 + ℓ2dX2 X2(1 + 4LX4) 1 + 4LX4 dy2 . (5.60) Finally, from (4.21) we obtain the gauge field strength F = 2dt ∧ dX . (5.61) Note that the geometry (5.60) is generically of Petrov type D, and becomes of Petrov type N for L = 0. Now let us turn our attention to the form of the second Killing spinor. First of all, the integrability conditions imply that it takes the form αT = (β1, β2, iX 3eξβ2, iX 3eξβ2) , – 46 – where β1 and β2 are arbitrary space-time dependent functions. The Killing spinor equations (4.31) yield β1 = λ1 − 12λ2b −2 , β2 = λ2b where λ1 and λ2 are integration constants. This implies that the new Killing spinor takes the form ǫ = λ1ǫ1 + λ2ǫ2, where ǫ1 = 1 + iXe2 , ǫ2 = X−2(1− iXe2) + X−4 + L (e1 − iXe1 ∧ e2) . (5.62) Note that G0 = 0 as well in this class. One interesting aspect of the second Killing spinor ǫ2 is the norm of its associated Killing vector Vµ = D(ǫ2,Γµǫ2). We find VµV µ = −4X2L2, hence the second Killing spinor is indeed null for the case L = 0, as was noticed before, while it is timelike for L 6= 0. In the latter case, to understand whether the solution belongs also to the null class of supersymmetric solutions, we have therefore to study the most general linear combination of the two Killing spinors. The Killing vector Ṽ constructed from ǫ = λ1ǫ1 + λ2ǫ2 has norm Ṽ 2 = λ̄1λ2 − λ1λ̄2 )2 − 4X2 L|λ1|2 + |λ2|2 which can vanish only if L ≤ 0. We have therefore three cases: 1. L > 0, pure timelike class, Petrov type D. 2. L = 0, belongs to both null and timelike classes, Petrov type N. This is the homogeneous half BPS pp-wave in AdS. (In the terminology of [16] it has a wave profile Gα with α = 0). 3. L < 0, belongs to both null and timelike classes, Petrov type D. Actually the solutions (5.60) with L > 0 can be cast into a simpler form. This is done by trading the coordinate y for a new variable ψ = Ly − t. For convenience, let us also introduce the Schwarzschild coordinate r and rescale z, r = − ℓ√ , ζ = Lz . (5.63) In the new coordinates, the metric and the gauge field strength read ds2 = − dt2 + dψ2 + dζ2 , F = qe dt ∧ dr , (5.64) – 47 – where we have defined qe = 2ℓ/ L. This is precisely the half BPS solution obtained in [36], the massless limit of an electrically charged toroidal black hole, which forms a naked singularity. It is also interesting to note that the charge qe diverges in the L→ 0 limit. This limit is naively singular in these coordinates, but it can be taken if we perform a Penrose limit [37, 38]. The existence of this limit explains why we obtained a one-parameter family of geometries (5.60) connecting the massless limit of toroidal black holes and a pp-wave. Indeed, define the new coordinates (X+, X−, R, Z) and the rescaled charge Qe by ψ + t = 2ǫ2X+ , ψ − t = 2X− , r = 1 , ζ = ǫZ , qe = . (5.65) Then, the singular limit ǫ→ 0 yields is a regular solution of the theory and corresponds to the half supersymmetric solution (5.60) with L = 0, ds2 = 4 dX+dX− − Q dX−2 + dR2 + dZ2 , F = Qe dX− ∧ dR . (5.66) In the procedure, we have blown up the metric in the neighborhood of a geodesic with ψ + t constant near the boundary r → ∞ of AdS. We now turn to the L < 0 case, which is both timelike and lightlike. Let us define L = −µ2. We can perform a coordinate transformation inspired from the previous one, ψ = Ly − t , r = − ℓ , ζ = z , (5.67) under which the metric and the field strength become ds2 = dt2 + − q2e −dψ2 + dζ2 , F = qe dt ∧ dr , (5.68) where we have defined qe = 2ℓ/µ. We see that this is the precisely the metric for L > 0 after the double analytic continuation t 7→ it , ψ 7→ iψ , qe 7→ −iqe . (5.69) This solution represents therefore a bubble of nothing in AdS [39–42]. Note that the metric is singular for r = ℓqe. One should compactify t, in such a way to eliminate the conical singularity on the (t, r) hypersurface. Then, if we compactify also ζ , this S1 will have a minimal radius for r = ℓqe (the boundary of the bubble of nothing) and then grow with r. Note that for r → ∞ one locally recovers AdS spacetime, and that the L = 0 solutions can again be understood as a Penrose limit of this metric. – 48 – 5.4 Action of the PSL(2,R) group on the imaginary b solutions We can now generate new supersymmetric solutions by acting with the PSL(2,R) symmetry group (4.28)-(4.29) on the known ones. It is easy to check that the AdS4 and AdS2×H2 solutions are invariant under this group (although it acts non trivially on the Killing spinors). Its action on the b = b(z) subfamily of the RNTN-AdS4 solutions was studied in [16], where it was shown that it acts non trivially on the charges, by mixing them. Here we want to apply it to the imaginary b solutions of the previous paragraph. The new solution solution of the supersymmetry equations (4.22)-(4.23) generated by the transformation (4.28)-(4.29) is b̃ = − γ 2γ2ℓXz + i , e2(Φ̃+ξ) = 1 + 4LX4 , (5.70) where, without loss of generality, we eliminated α by means of a translation of z19, and dropped the prime of the new coordinate z′. The shift function is then determined by solving equations (4.18) and (4.19), σx = 0 , σy = 1 + 4LX4 4γ2X4z2 . (5.71) Then, defining the new coordinates (T, σ, p, q) through 2ℓ2γ2 , σ = , p = − ℓ , q = 2ℓ2γ2z , (5.72) the metric reads ds2 = − Q(q) q2 + p2 P (p) q2 + p2 dq2 + q2 + p2 P (p) (q2 + p2)P (p) dσ2, (5.73) Q(q) = , P (p) = p4 + 4Lℓ2 , (5.74) and the gauge field (4.21) is F = d ℓ(q2 + p2) ∧ dT + d q2 + p2 ∧ dσ . (5.75) 19After this translation the limit γ → 0 is not anymore well-defined. To perform it, one has to substitute preliminarily z with z − α/γ everywhere. – 49 – The form of the metric suggests some connection with the Plebanski-Demianski family of solutions, and indeed these geometries are of Petrov type D for L 6= 0, and of Petrov type N for L = 0, but we were not able to find the precise relation. Note also that the parameter γ has been reabsorbed in the new variables, and we are left with a one-parameter (L) family of solutions. The left hand side of the necessary condition (4.34) for the existence of a second Killing spinor reads, for this solution, − 9iX 4 (1 + 4LX4) ℓ2 (1 + 4γ4ℓ2X2z2) γ2 (5.76) which clearly vanishes only for γ = 0, i.e. if the PSL(2,R) transformation is trivial. Therefore, the new solutions (5.73)-(5.75) preserves only 1/4 of the supersymmetries, and we explicitly see that the PSL(2,R) transformations can break any additional supersymmetry. Also note that if we perform the PSL(2,R) transformation adapting the original metric to a different Killing spinor, we could in principle end up with other supersymmetric solutions. Surprisingly, we find that the L = 0 solution can be cast in the Lobatchevski wave form, even though it only has a time-like Killing spinor. This can be seen by trading the coordinates (q, p) for (x, z) defined by , z = , (5.77) in the metric (5.73) with L = 0, which becomes ds2 = −2 dTdσ + z x2 + z2 x2 + z2 x2 + z2 dT 2 + dz2 + dx2 . (5.78) The field strength can be easily obtained from equation (5.75) but the result is not particularly enlightening and therefore we do not report it. This metric represents a 1/4 BPS Lobatchevski wave, whose Killing spinor falls in the timelike class. This does not contradict the results obtained in the null case, since the null Lobatchevski had a field strength (3.6) of the form F = φ′(T )dT ∧dz, while this solution has a much more complicated gauge field. It is however interesting to note that the solutions of the null case do not exhaust all possible supersymmetric Lobatchevski waves. 5.5 Gravitational Chern-Simons system and G0 = ψ− = 0 solutions A number of the previously studied subcases can be combined into the interesting Ansatz b = −1 αz2 + βz + γ 2αz + β − iη(w, w̄) , (5.79) – 50 – where α, β and γ are three real constants. For α = β = 0 this reduces to b imaginary, while η = 0 leads to the real subcase of b = b(z). With this assumption, the equations for a timelike Killing spinor reduce to e2ξ (k − 3η) = 0 , ∆η + e2ξ kη − η3 = 0, (5.80) where we have defined k = 4αγ − β2 and ∆ = 4∂∂̄. Interestingly, as shown in [16], this system of equations follows from the dimensionally reduced Chern-Simons action [43, 44], (2)Rη + η3 , (5.81) if we use the conformal gauge (2)gijdx idxj = e2ξ (dx2 + dy2) and η is the curl of a vector potential, (2)g ǫijη = ∂iAj−∂jAi. To obtain equations (5.80) we vary the action with respect to Ai and ξ. When varying the dimensionally reduced Chern-Simons action with respect to gij there is however an additional equation to (5.80). Using the results of Grumiller and Kummer [48], one obtains the most general solution to the dimensionally reduced Chern-Simons system [16] e2ξ = η4, (5.82) where L is an integration constant and dη = e2ξdx. Trading the coordinate x for η, we get the following configuration of the fields ds2 = − 4 P ′22 + η [dt + σ] P ′22 + η dz2 + P 22 e−2ξdη2 + e2ξdy2 A = 2 P ′22 + η [dt + σ] + Vdy − i ℓ d log , (5.83) where P2(z) = αz 2 + βz + γ, k is defined as above and the shift function reads αη2 + dy . (5.84) These solutions preserve 1/4 of the original supersymmetry. In fact, the k = 0 solutions coincide with the imaginary b ones and their PSL(2,R) transforms of sections 5.3 and 5.4. For k non-vanishing these are different solutions. As can be seen from the Poisson bracket (4.34), the only possibility to have 1/2 supersymmetry is α = 0 and hence k ≤ 0. In fact, starting from any solution with k ≤ 0, one can always obtain α = 0 by an appropriate PSL(2,R) transformation. The – 51 – non-trivial part of the PSL(2,R) symmetry is z 7→ −1/(z + δ), whose action on the parameters α, β and γ of the Ansatz (5.79) is given by α 7→ αδ2 − βδ + γ , β 7→ 2αδ − β , γ 7→ α , (5.85) which keeps k fixed. Indeed, for k ≤ 0, there is always a PSL(2,R) transformation that sets α = 0, while this is impossible for k > 0. The requirement α = 0 leads to the half-supersymmetric imaginary b solution of section 5.3 for k = 0. In the case of k negative, when α = 0 one can scale β to 1 in (5.79) without loss of generality, and γ can be put to zero by a translation in z. Hence the function b is given by b = −1 1− iη . (5.86) The metric is given in (D.32) and is generically of Petrov type D. The second Killing spinor can be found in (D.33). As shown in appendix D, the G0 = ψ− = 0 solutions are either the imaginary b ones, anti-Nariai spacetime or the above 1/2 supersymmetric solution with k = −1. We would like to mention that (5.82) is the most general solution to the dimen- sionally reduced Chern-Simons system, but not to the equations (5.80). The reason for this is the additional constraint one obtains when varying (5.81) with respect to gij. An example of this is provided by the Petrov type I solution with b = i(x/ℓ)2 in section 5.3 and its PSL(2,R) transform given in eq. (2.44) of [16]. 6. Final remarks In this paper, we applied spinorial geometry techniques to classify all supersymmetric solutions of minimal N = 2 gauged supergravity in four dimensions. In the presence of null Killing spinors, the problem can be completely solved, and all 1/4- and 1/2-supersymmetric solutions have been written down explicitly. We showed that there are no 1/4-BPS backgrounds with U(1)⋉R2-invariant Killing spinors and those with R2-invariant Killing spinors have been derived in sections 3.1 and 3.2. The backgrounds in the latter section were previously unknown and are Petrov type II configurations describing gravitational waves propagating on a bubble of nothing in AdS4. In addition, it turned out that there are no 1/2-BPS backgrounds with R invariant Killing spinors and hence any additional Killing spinor is timelike. In section 3.3 we gave the backgrounds with one null and one timelike Killing spinor. For a timelike Killing spinor we derived the conditions for the corresponding back- grounds in section 4.1 and 4.2. We worked out the first integrability conditions nec- essary for the existence of a second Killing spinor in section 4.3. We explicitly solved – 52 – these equations in a number of subcases in section 5, and thereby found several new solutions, like the bubbles of nothing in AdS4, already obtained in the null formalism, and their PSL(2,R)-transformed configurations. Furthermore, our results showed that the generalized holonomy in the case of one preserved complex supercharge is contained in A(3,C), supporting thus the classification scheme of [4]. In addition, the time-dependence of a second time-like Killing spinor was shown to be an overall exponential factor with coefficient G0 in section 4.4. In the case G0 = 0 these equations have been solved in full generality, up to a second order ordinary differential equation. We expect this class to comprise a large number of interesting 1/2-BPS solutions. Indeed, all the examples of section 5 either have vanishing G0 or can be transformed to that case by a coordinate transformation. There are several interesting points that remain to be understood. First of all, it would be desirable to get a deeper insight into the underlying geometric structure in the case of U(1) invariant spinors. In five dimensions, spacetime is a fibration over a four-dimensional Hyperkähler or Kähler base for ungauged and gauged supergravity respectively [8, 12], whereas in four-dimensional ungauged supergravity one has a fi- bration over a three-dimensional flat space [5]. This suggests that the base for D = 4 gauged supergravity might be an odd-dimensional analogue of a Kähler manifold, i. e. , a Sasaki manifold. From the equations (4.22) and (4.23) this is not obvious. Secondly, in [16], a surprising relationship between the equations (4.22), (4.23) gov- erning 1/4 BPS solutions and the gravitational Chern-Simons theory [43] was found. Why such a relationship should exist is not clear at all, and deserves further investiga- tions. The third point concerns preons, which were conjectured in [45] to be elementary constituents of other BPS states. In type II and eleven-dimensional supergravity, it was shown that imposing 31 supersymmetries implies that the solution is locally maximally supersymmetric [27,30,46]. Similar results in four- and five-dimensional gauged super- gravity were obtained in [28,29]. This implies that preonic backgrounds are necessarily quotients of maximally supersymmetric solutions. While M-theory preons cannot arise by quotients [47], it remains to be seen if 3/4 supersymmetric solutions to N = 2, D = 4 or D = 5 gauged supergravities really do not exist. The only maximally super- symmetric backgrounds in these theories are AdS4 [13] and AdS5 [12] respectively, so the putative preonic configurations must be quotients of AdS. Finally, it would be interesting to apply spinorial geometry techniques to classify all supersymmetric solutions of four-dimensional N = 2 matter-coupled gauged super- gravity. Work in this direction is in progress [49]. – 53 – Acknowledgments We are grateful to Alessio Celi, Marcello Ortaggio and Christoph Sieg for useful dis- cussions. This work was partially supported by INFN, MURST and by the European Commission program MRTN-CT-2004-005104. D.R. wishes to thank the Università di Milano for hospitality. Part of this work was completed while he was a post-doc at King’s College London, for which he would like to acknowledge the PPARC grant PPA/G/O/2002/00475. In addition, he is presently supported by the European EC- RTN project MRTN-CT-2004-005104, MCYT FPA 2004-04582-C02-01 and CIRIT GC 2005SGR-00564. A. Spinors and forms In this appendix, we summarize the essential information needed to realize the spinors of Spin(3,1) in terms of forms. For more details, we refer to [50]. Let V = R3,1 be a real vector space equipped with the Lorentzian inner product 〈·, ·〉. Introduce an orthonormal basis e1, e2, e3, e0, where e0 is along the time direction, and consider the subspace U spanned by the first two basis vectors e1, e2. The space of Dirac spinors is ∆c = Λ ∗(U⊗C), with basis 1, e1, e2, e12 = e1∧e2. The gamma matrices are represented on ∆c as Γ0η = −e2 ∧ η + e2⌋η , Γ1η = e1 ∧ η + e1⌋η , Γ2η = e2 ∧ η + e2⌋η , Γ3η = ie1 ∧ η − ie1⌋η , (A.1) where ηj1...jkej1 ∧ . . . ∧ ejk is a k-form and ei ∧ η = (k − 1)!ηij1...jk−1ej1 ∧ . . . ∧ ejk−1 . One easily checks that this representation of the gamma matrices satisfies the Clifford algebra relations {Γa,Γb} = 2ηab. The parity matrix is defined by Γ5 = iΓ0Γ1Γ2Γ3, and one finds that the even forms 1, e12 have positive chirality, Γ5η = η, while the odd forms e1, e2 have negative chirality, Γ5η = −η, so that ∆c decomposes into two complex chiral Weyl representations ∆+c = Λ even(U ⊗ C) and ∆−c = Λodd(U ⊗ C). Let us define the auxiliary inner product αiei, βjej〉 = α∗iβi (A.2) – 54 – on U ⊗ C, and then extend it to ∆c. The Spin(3,1) invariant Dirac inner product is then given by D(η, θ) = 〈Γ0η, θ〉 . (A.3) In many applications it is convenient to use a basis in which the gamma matrices act like creation and annihilation operators, given by Γ+η ≡ (Γ2 + Γ0) η = 2 e2⌋η , Γ−η ≡ (Γ2 − Γ0) η = 2 e2 ∧ η , Γ•η ≡ (Γ1 − iΓ3) η = 2 e1 ∧ η , Γ•̄η ≡ (Γ1 + iΓ3) η = 2 e1⌋η . (A.4) The Clifford algebra relations in this basis are {ΓA,ΓB} = 2ηAB, where A,B, . . . = +,−, •, •̄ and the nonvanishing components of the tangent space metric read η+− = η−+ = η••̄ = η•̄• = 1. The spinor 1 is a Clifford vacuum, Γ+1 = Γ•̄1 = 0, and the representation ∆c can be constructed by acting on 1 with the creation operators Γ+ = Γ−,Γ •̄ = Γ•, so that any spinor can be written as φā1...ākΓ ā1...āk1 , ā = +, •̄ . The action of the Gamma matrices and the Lorentz generators ΓAB is summarized in the table 6. 1 e1 e2 e1 ∧ e2 Γ+ 0 0 2e2 − 2e1 ∧ e2 0 0 2e1 0 2e1 ∧ e2 0 Γ•̄ 0 Γ+− 1 e1 −e2 −e1 ∧ e2 Γ•̄• 1 −e1 e2 −e1 ∧ e2 Γ+• 0 0 −2e1 0 Γ+•̄ 0 0 0 2 Γ−• −2e1 ∧ e2 0 0 0 Γ−•̄ 0 2e2 0 0 Table 6: The action of the Gamma matrices and the Lorentz generators ΓAB on the different basis elements. – 55 – Note that ΓA = UA aΓa, with 1 0 1 0 −1 0 1 0 0 1 0 −i 0 1 0 i ∈ U(4) , so that the new tetrad is given by EA = (U∗)AaE B. Spinor bilinears Given a Killing spinor ǫ = c01 + c1e1 + c2e2 + c12e1 ∧ e2 , (B.1) one can construct the bilinears f̃ = −iD(ǫ, ǫ) = −i (c0c∗2 − c1c∗3 − c2c∗0 + c12c∗1) , (B.2) g̃ = −iD(ǫ,Γ5ǫ) = c0c∗2 + c1c∗3 + c2c∗0 + c12c∗1 , (B.3) Ṽ = D(ǫ,Γµǫ) dx |c2|2 + |c12|2 − |c0|2 − |c1|2 |c2|2 + |c12|2 + |b|2 |c0|2 + |c1|2 (dt + σ) ψ [(c2c 1 − c0c∗12) dw + (c1c∗2 − c12c∗0) dw̄] , (B.4) B̃ = D(ǫ,Γ5Γµǫ) dx |c2|2 − |c12|2 + |c0|2 − |c1|2 |c2|2 − |c12|2 − |b|2 |c0|2 − |c1|2 (dt + σ) ψ [(c2c 1 + c0c 12) dw + (c1c 2 + c12c 0) dw̄] , (B.5) D(ǫ,Γµνǫ) dx µ ∧ dxν = − (c0c∗2 − c1c∗12 + c2c∗0 − c12c∗1) dt ∧ dz 12 + |b|2c0c∗1 dt ∧ dw − 2e 0 + 4|b|2c1c∗0 dt ∧ dw̄ 2 − c1c∗12 + c2c∗0 − c12c∗1)σw + 2|b|3 c2c 2|b|c0c dz ∧ dw 2 − c1c∗12 + c2c∗0 − c12c∗1)σw̄ + 2|b|3 c12c 2|b|c1c dz ∧ dw̄ – 56 – 12σw̄ − c12c∗0σw + |b|2 (c0c∗1σw̄ − c1c∗0σw) 4|b| (c0c 2 + c1c 12 − c2c∗0 − c12c∗1) dw ∧ dw̄ . (B.6) Given the first Killing spinor of the form ǫ1 = 1 + be2 and the second Killing spinor ǫ2 = c01 + c1e1 + c2e2 + c12e1 ∧ e2, one can also construct mixed bilinears of the type D(ǫ1,Γ···ǫ2), which verify the same differential equations as the bilinears built from the original two Killing spinors: f̂ = −i(b̄c0 − c2) , ĝ = b̄c0 + c2 , (B.7) (c2 + bc0) (dt + σ) + (c2 − bc0) dz + b̄c1 − c12 dw̄ , (B.8) (c2 − bc0) (dt+ σ) + (c2 + bc0) dz + b̄c1 + c12 dw̄ . (B.9) C. The case P ′ = 0 In section 4.3, we simplified the equations for the second Killing spinor under the assumption P ′ 6= 0, where P = e−2ψb∂b. Here we consider the case P ′ = 0. To this end, we need the following subset of the Killing spinor equations (4.31): ∂+ψ2 − ψ12 = 0 , (C.1) ∂+ψ12 − re−ψ∂•̄ ln b̄ ψ1 − ψ12 = 0 , (C.2) ∂−ψ2 + e−ψ∂•̄ ln b ψ1 − ψ2 = 0 , (C.3) ∂−ψ12 − ψ12 = 0 , (C.4) re−ψ∂• e2ψψ2 ψ1 = 0 , (C.5) re−ψ∂• e2ψψ12 ψ1 = 0 . (C.6) If P ′ = 0, (4.35) implies ψ− = 0 or ∂P = 0. Let us first assume the former, i. e. , ψ2 = ψ12. From (C.6) – (C.5) one obtains then ψ1 = 0 or = 0 . (C.7) – 57 – • If ψ1 = 0, (C.4) – (C.3) yields ψ2 = 0, and thus there exists no further Killing spinor. • If (C.7) holds, one can use (C.1) and (C.4) to show that ∂+ψ2 = ∂−ψ2 = 0, or equivalently ∂tψ2 = ψ 2 = 0. Using this in (C.2) and (C.3) and deriving with respect to t, one gets ∂̄b̄ ∂tψ1 = ∂̄b ∂tψ1 = 0. When ∂tψ1 6= 0, this means that ∂̄b = ∂b = 0, so b = b(z), which is a case analyzed in section 5.1. If instead ∂tψ1 = 0, all the ψi are independent of t, and the Killing spinor equations reduce to the system (4.38) to (4.49) with G0 = 0. In the case ∂P = 0, consider the integrability condition ′ + ψ−∂Q = 0 , (C.8) where Q = e−2ψ b̄∂b̄, following from the first line of Ñwt. As long as Q ′ 6= 0, with the same reasoning as in section 4.3, one obtains the system (4.38) to (4.49). If Q′ = 0, (C.8) implies ψ− = 0 or ∂Q = 0. The case ψ− = 0 was already considered above, so the only remaining case is P ′ = ∂P = Q′ = ∂Q = 0. For P = Q = 0 we get again b = b(z), so without loss of generality we can assume P 6= 0 or Q 6= 0. Suppose that Q = 0, P 6= 0, so b = b(w, z). Take the logarithm of e−2ψb∂b = P (w̄), derive with respect to z, use (4.15), and apply ∂̄. This leads to ∂b = 0, which is a contradiction to the assumption P 6= 0. In the same way one shows that P = 0, Q 6= 0 is not possible, so that both P and Q must be nonvanishing. Now use the third row of Ñw̄t, which leads to Q̄ψ2 = 0 and hence ψ2 = 0. Finally, the last row of Ñw̄t yields ψ− = 0, i. e. , the case already considered above. Hence, the conclusion is that in the case P ′ = 0, the second Killing spinor either has G0 time-dependence of the form (4.37), or leads to solutions with b = b(z). The latter are treated separately in section 5.1. As can be found there, all 1/2-BPS solutions with b = b(z) also have second Killing spinors with G0 time-dependence of the form (4.37). Hence this time-dependence is a completely general result20 for second Killing spinors in the time-like case. D. Half-supersymmetric solutions with G0 = 0 From the difference of equations (4.39)−(4.43) and (4.48)−(4.49) one gets ψ− = ψ−(w). Furthermore, [(4.42)−(4.40)+e−2ψ(4.47)] and (4.44) yield ψ1 = ψ1(z). Assuming ψ− 6= 20The only counterexample is the third Killing spinor of AdS4, see (5.30), but since this is maximally supersymmetric it does not contradict the result. – 58 – 0, eqns. (4.38) and (4.41) can be written in the form = 0 , = 0 , (D.1) where β = ψ1/ψ−. Deriving (D.1) with respect to w̄ gives + ∂∂̄ = 0 , + ∂∂̄ = 0 . Now use (D.1) in the difference between the first equation and the complex conjugate of the second one to get β̄β ′ − β̄ ′β = 0 . Observe that β̄β ′ − β̄ ′β = |ψ−|−2 1 − ψ̄′1ψ1 (z), so that for ψ̄1ψ 1 − ψ̄′1ψ1 6= 0 there must exist a real function B(z) and a generic function h(w, w̄) such that b = B(z)h(w, w̄) . Plugging this into (D.1), we conclude that = 0 , so that the phase of the function h is fixed, h = hR(w, w̄)e iϕ0 , with hR real. Using (4.18), the constancy of the phase of b implies that the shift vector σ does not depend on z. (4.19) gives then = 0 , or, using (4.15), cosϕ0 +B′ = 0 , and thus B′ = c , cosϕ0 = −c , where c denotes a real constant. Now we have to distinguish to cases: 1. c 6= 0: In this case b(z) = B0 − 2 cosϕ03ℓ z eiϕ0 . Plugging this into the first of eqns. (D.1) one gets = 0 , which is solved by ψ1 = ηb where η is a constant. But this yields ψ̄1ψ 1−ψ̄′1ψ1 = 0, which contradicts our assumption. – 59 – 2. c = 0: In this case b(w, w̄) = ihR(w, w̄). The combination (4.40)+(4.42)−(4.39)− (4.43) leads to ψ− = 0, which again contradicts one of our assumptions. We thus conclude that ψ̄1ψ 1− ψ̄′1ψ1 = 0, and hence ψ1 = ζ(z)eiθ0 where θ0 is a constant and ζ(z) is a real function. Sending ψi → e−iθ0ψi we can take ψ1 real and non-negative without loss of generality. Let us now consider the case where both ψ1 and ψ− are non-vanishing. This allows to introduce new coordinates Z,W, W̄ such that ψ1(z) dz , dW = ψ−(w) , dW̄ = ψ̄−(w̄) Note that one can set ψ− = 1 using the residual gauge invariance w 7→ W (w), ψ 7→ ψ̃ = ψ − 1 ln(dW/dw)− 1 ln(dW̄/dw̄) leaving invariant the metric e2ψdwdw̄. We can thus take W = w in the following. Equations (4.38) and (4.41) are then equivalent to (∂Z + ∂)ϕ = 0 , ∂Z lnψ1 − (∂Z + ∂) ln r = 0 . From the real part of the first equation we have ϕ = ϕ(Z − w − w̄) . Using ψ1 = ψ1(Z), the second equation implies (∂Z + ∂) = 0 , and hence = ρ(Z − w − w̄) . The function b must thus have the form b(Z,w + w̄) = ψ1(Z)B(Z − w − w̄) , where B(Z − w − w̄) = ρ(Z − w − w̄)eiϕ(Z−w−w̄). The difference between (4.45) and (4.46) yields (∂Z + ∂) (lnψ1 − ψ) = 0 , so that lnψ1 − ψ = −H(Z − w − w̄) with H real. This gives e2ψ = ψ1(Z) for the conformal factor. In terms of the new coordinate Z, (4.15) reads ∂Zψ + = 0 . – 60 – Using the definition of H we get Ḣ + ∂Z lnψ1 + = 0 , (D.2) where a dot denotes a derivative with respect to Z−w− w̄. We can thus conclude that ∂Z lnψ1 = γ/ℓ for some constant γ, i. e. ,ψ1(Z) = ψ γZ/ℓ. By shifting Z one can set 1 = 1. Calling χ = ψ+/ψ−, the only remaining nontrivial Killing spinor equations ∂Zχ− 2 χ+ 2iϕ̇+ = 0 , − Ḣ + γ χ− 2ie−2H ϕ̇− 1 = 0 , ∂χ + 2 χ− 2iϕ̇− 1 = 0 , ∂̄χ+ 2 χ− 2iϕ̇ = 0 , χ + 2 1 + e−2H − Ḣ + γ = 0 . Summing the first and the third equation yields χ = χ(Z −w− w̄), so that we are left χ̇− 2 χ+ 2iϕ̇+ = 0 , (D.3) − Ḣ + γ χ− 2ie−2H ϕ̇− 1 = 0 , (D.4) − χ̇+ 2 ρ̇ χ− 2iϕ̇ = 0 , (D.5) χ + 2 1 + e−2H − Ḣ + γ = 0 . (D.6) Adding (D.4) and (D.6) one gets Ḣχ + = 0 , (D.7) which means that χ is purely imaginary. From (D.2) and (D.7) we obtain then the function B, + Ḣ(1 + χ) = 0 . (D.8) – 61 – Using this, the remaining Killing spinor equations reduce further to 1 + e2H = 0 , (D.9) = 0 , (D.10) 1 + χ2 − 2 ρ̇ 1 + e−2H . (D.11) Note that (D.9) automatically implies the integrability condition for the system (4.18), (4.19), which reduces to ∂Zσw = , ∂σw̄ − ∂̄σw = − . (D.12) Thus, also equation (4.23) is satisfied, whereas (4.22) reads 1 + e−2H 2Ḧ + Ḣ2 1 + 3χ2 . (D.13) From (D.8) we obtain the phase ϕ and the modulus ρ of B, tanϕ = i − Ḣ2χ2 . Plugging this into equation (D.10) yields 2ḦḢχ 1− χ2 − Ḣ2χ̇ 1 + 3χ2 χ̇ = 0 . Using (D.13), this can be rewritten as 1− χ2 1 + e−2H = 0 , so that either Ḧ = 0 or Ḣχ (1− χ2) + 1 + e−2H χ̇ = 0. It is straightforward to show that the first case leads to AdS4, whereas the second one implies e2H + 1 1− χ2 = −α 2 , (D.14) where α is a real integration constant. Equations (D.9) and (D.11) are then identically satisfied. Solving (D.14) for χ and plugging into (D.13) yields finally the ordinary differential equation (4.50), which determines half-supersymmetric solutions with G0 = 0. Putting together all our results, we obtain (4.52) for the metric. Note that in the – 62 – case γ 6= 0 one can always set γ = 1 by rescaling the coordinates. The second Killing spinor for these backgrounds is given by αT = (α0, ρ −2e−γZ/ℓ, eH) , where α0 = − + α̂0(Z,w, w̄) , and α̂0 is a solution of the system ∂Zα̂0 = − iϕ̇ + γ ∂α̂0 = − + iϕ̇ , (D.15) ∂̄α̂0 = − σw̄ + + iϕ̇ γχ e2H ℓψ1ρ2 It is straightforward to verify that the integrability conditions for this system are al- ready implied by (D.9), (D.10) and (D.12). Consider now the case ψ− = 0. From the difference of equations (4.38) and (4.41) it follows that b′/b is real. Then (4.38) and (4.44) imply that ψ is a real function, depending only on z, ψ1 = ψ1(z). Moreover, since ψ12 = ψ2, the difference of equations (4.45) and (4.46) imply that b′/b+ 1/ℓb is imaginary. The conditions b′/b real and b′/b+ 1/ℓb imaginary can be satisfied simultaneously in three different ways: • b′/b = 0 hence b = b(w, w̄) is an imaginary function independent of z. This case is solved completely in section 5.3. • b′/b+ 1/ℓb = 0 implies b = −z/ℓ + c and corresponds to AdS2 ×H2, analyzed in section 5.1.1. It is also a subcase of the following, general case, • if we are not in one of the previous special cases, the function b must take the b = −1 1− iY (w, w̄) , (D.16) where Y (w, w̄) is some real function to be determined. We thus have to solve just for the ansatz (D.16). Equation (4.38) implies ψ′1/ψ1 = b than is solved by ψ1 = z, where we have reabsorbed the integrability constant in the – 63 – scale of z. Equation (4.39) (or equivalently (4.43)) tells us that ψ2 = ψ2(w, w̄), so that the remaining independent equations read iz2e−2ψ 1 + Y 2 − ψ2 = 0 , ∂ψ2 + ∂ 1 + Y 2 ψ2 − iY = 0 , ∂̄ψ2 + ∂̄ log 1 + Y 2 ψ2 = 0 . The first equation allows us to define a function H(w, w̄) such that eψ = zeH(w,w̄) , (D.17) while the last one implies that there must exist a holomorphic function C(w) such that 1 + Y 2 . (D.18) Thus we are left with e2HC(w) = i∂̄Y , (D.19) e2HC(w) = ie2HY 1 + Y 2 . (D.20) This set of equations automatically implies the integrability condition for the system (4.18), (4.19), which reduces to ∂zσ = i , (D.21) ∂σ̄ − ∂̄σ = iℓ2e2HY 1 + Y 2 . (D.22) Thus also (4.23), which reads ∂∂̄Y − e2HY 1 + Y 2 = 0 , (D.23) is satisfied and it turns out that also the Bianchi identity (4.22), namely ∂∂̄2H − e2H 1 + 3Y 2 = 0 , (D.24) holds. We conclude that a solution to the system (D.19), (D.20) describes a 1/2-BPS configuration of the “gravitational Chern-Simons” system discussed in [16]. If C(w) = 0 then necessarily also Y = 0 so that we are left with AdS. If C(w) 6= 0 then we can define new variables W and W̄ such that ∂W = C(w)∂ , ∂W̄ = C̄(w̄)∂̄ , (D.25) – 64 – so that we have e2HCC̄ = i∂W̄Y , e2HCC̄ = ie2HCC̄Y 1 + Y 2 As what we did in the previous case, we can set C(w) = 1 using the residual gauge invariance w 7→ W (w), ψ 7→ ψ̃ = ψ − 1 ln(dW/dw) − 1 ln(dW̄/dw̄) leaving invariant the metric e2ψdwdw̄. We can thus take W = w without loss of generality, and get e2H = i∂̄Y , (D.26) 2∂H = iY 1 + Y 2 (D.26) implies Y = Y [i(w − w̄)] and hence H = H [i(w− w̄)]. Denoting with a dot the derivative with respect to the combination i(w − w̄) we have e2H = Ẏ , (D.27) 2Ḣ = Y 1 + Y 2 . (D.28) The equations for the shift form can now be integrated, giving Ẏ d (w + w̄) (D.29) Plugging (D.27) into (D.28) leads to Ÿ = Ẏ Y (1 + Y 2) , (D.30) which, integrated once, gives Y 2 + Y 4 ≡ P (Y ) , (D.31) where L is a real constant and k = −121. We can thus use Y as a new coordinate, instead of i(w − w̄). 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0704.0248

A Rigorous Time-Domain Analysis of Full--Wave Electromagnetic Cloaking (Invisibility)

A Rigorous Time-Domain Analysis of Full–Wave Electromagnetic Cloaking (Invisibility) ∗† Ricardo Weder‡ Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas Departamento de Métodos Matemáticos y Numéricos Universidad Nacional Autónoma de México Apartado Postal 20-726, México DF 01000 [email protected] Abstract There is currently a great deal of interest in the theoretical and practical possibility of cloaking objects from the observation by electromagnetic waves. The basic idea of these invisibility devices [8, 9, 13], [18] is to use anisotropic transformation media whose permittivity and permeability ελν , µλν, are obtained from the ones, ελν0 , µ of isotropic media, by singular transformations of coordinates. In this paper we study electromagnetic cloaking in the time-domain using the for- malism of time-dependent scattering theory [23]. This formalism provides us with a rigorous method to analyze the propagation of electromagnetic wave packets with finite energy in transformation media. In particular, it allows us to settle in an unambiguous way the mathematical problems posed by the singularities of the inverse of the permittivity and the permeability of the transformation media on the boundary of the cloaked objects. Von Neumann’s theory of self-adjoint extensions of symmetric operators plays an important role on this issue. We write Maxwell’s equations in Schrödinger form with the electromagnetic propa- gator playing the role of the Hamiltonian. We prove that the electromagnetic propagator outside of the cloaked objects is essentially self-adjoint. This means that it has only one self-adjoint extension, AΩ, and that this self-adjoint extension generates the only possible unitary time evolution, with constant energy, for finite energy electromagnetic waves, propagating outside of the cloaked objects. ∗PACS classification scheme 2006: 41.20.Jb, 02.30.Tb,02.30.Zz, 02.60.Lj. †Research partially supported by CONACYT under Project P42553F. ‡Fellow Sistema Nacional de Investigadores. http://arxiv.org/abs/0704.0248v4 Moreover, AΩ is unitarily equivalent to the electromagnetic propagator in the medium ελν0 , µ 0 . Using this fact, and since the coordinate transformation is the identity outside of a ball, we prove that the scattering operator is the identity. This implies that for any incoming finite-energy electromagnetic wave packet the outgoing wave packet is precisely the same. In other words, it is not possible to detect the cloaked objects in any scattering experiment where a finite-energy wave packet is sent towards the cloaked objects, since the outgoing wave packet that is measured after interaction is the same as the incoming one. Our results give a rigorous proof that the construction of [8, 9, 13], [18] cloaks passive and active devices from observation by electromagnetic waves. Actually, the cloaking outside is independent of what is inside the cloaked objects. As is well known, self-adjoint extensions can be understood in terms of bound- ary conditions. Actually, for the electromagnetic fields in the domain of AΩ the component tangential to the exterior of the boundary of the cloaked objects of both, the electric and the magnetic field have to be zero. This boundary condi- tion is self-adjoint in our case because the permittivity and the permeability are degenerate on the boundary of the cloaked objects. Furthermore, we prove cloaking for general anisotropic materials. In particular, our results prove that it is possible to cloak objects inside general crystals. 1 Introduction There is currently a great deal of interest in the theoretical and practical possibility of cloaking objects from the observation by electromagnetic fields. The basic idea of these in- visibility devices [8, 9, 13], [18] is to use anisotropic transformation media whose permittivity and permeability, ελν , µλν , are obtained from the ones, ελν0 , µ 0 , of isotropic media, by sin- gular transformations of coordinates. The singularities lie on the boundary of the objects to be cloaked. Here the material interpretation is taken. Namely, the ελν , µλν and the ελν0 , µ represent the components in flat Cartesian space of the permittivity and the permeability of physical media with different material properties. It appears that with existing technology it is possible to construct media as described above using artificially structured metamaterials. In [8, 9] a proof of cloaking was given for the conductivity equation -i.e., in the case of zero frequency- from detection by measurement of the Dirichlet to Neumann map that relates the value of the electric potential on the boundary to its normal derivative. The papers [13] and [18] consider electromagnetic waves in the geometrical optics approximation, i.e. for large frequencies. In [24] a experimental verification of cloaking is presented and [4] and [5] give a numerical simulation. A rigorous prof of cloaking has already been given by [7] where fixed frequency waves were studied, i.e., in the frequency domain. They consider a class of finite energy solutions to Maxwell’s equations in a bounded set, O, that contains the cloaked object on its interior, and they prove cloaking, at any frequency, with respect to the measurement of the Cauchy data of these solutions on the boundary of O. We give further comments on this paper below. For other results on this problem see [25] and [15]. In [16] cloaking of elastic waves is considered, and the history of invisibility is discussed. In this paper we study electromagnetic cloaking in the time-domain using the formalism of time-dependent scattering theory [23]. This formalism provides us with a rigorous method to analyze the propagation of electromagnetic wave packets with finite energy in transformation media. In particular, it allows us to settle in an unambiguous way the mathematical problems posed by the singularities of the inverse of the permittivity and the permeability of the transformation media on the boundary of the cloaked objects. Von Neumann’s theory of self- adjoint extensions of symmetric operators plays an important role on this issue. We write Maxwell’s equations in Schrödinger form with the electromagnetic propagator playing the role of the Hamiltonian. We prove that the electromagnetic propagator outside of the cloaked objects is essentially self-adjoint. This means that it has only one self-adjoint extension, AΩ, and that this self-adjoint extension generates the only possible unitary time evolution, with constant energy, for finite energy electromagnetic waves propagating outside of the cloaked objects. Moreover, AΩ is unitarily equivalent to the electromagnetic propagator in the medium ελν0 , µ 0 . Using this fact, and since the coordinate transformation is the identity outside of a ball, we prove that the scattering operator is the identity. This implies that for any incoming finite-energy electromagnetic wave packet the outgoing wave packet is precisely the same. In other words, it is not possible to detect the cloaked objects in any scattering experiment where a finite-energy wave packet is sent towards the cloaked objects, since the outgoing wave packet that is measured after interaction is the same as the incoming one. Our results give a rigorous proof that the construction of [8, 9, 13], [18] cloaks passive and active devices from observation by electromagnetic waves. Actually, the cloaking outside is independent of what is inside the cloaked objects. As is well known, self-adjoint extensions can be understood in terms of boundary condi- tions. Actually, for the electromagnetic fields in the domain of AΩ the component tangential to the exterior of the boundary of the cloaked objects of both, the electric and the mag- netic field have to be zero. This boundary condition is self-adjoint in our case because the permittivity and the permeability are degenerate on the boundary of the cloaked objects. Furthermore, we prove cloaking for general anisotropic materials. In particular, our results prove that it is possible to cloak objects inside general crystals. Even though, as mentioned above, the cloaking is independent of the cloaked objects, and in particular, the cloaking outside is not affected by the presence of passive and/or active devices inside the cloaked objects, we discuss the dynamics of electromagnetic waves inside the cloaked objects for completeness, since it helps to understand the above mentioned independence of cloaking from the properties of the cloaked objects. We prove that every self-adjoint extension of the electromagnetic propagator in a trans- formation medium is the direct sum of the unique self-adjoint extension in the exterior of the cloaked objects, AΩ, with some self-adjoint extension of the electromagnetic propagator in the interior of the cloaked objects. Each of these self-adjoint extensions corresponds to a possible unitary time evolution for finite energy electromagnetic waves. As is well known, the fact that time evolution is unitary assures us that energy is conserved. This results implies that the electromagnetic waves inside and outside of the cloaked objects completely decouple from each other. Actually, the electromagnetic waves inside the cloaked objects are not allowed to leave them, and viceversa, the electromagnetic waves outside can not go inside. In terms of boundary conditions, this means that transmission conditions that link the electromagnetic fields inside and outside the cloaked objects are not allowed, since they do not correspond to self-adjoint extensions of the electromagnetic propagator, and then, they do not lead to a unitary dynamics that conserves energy. Furthermore, choosing a particular self-adjoint extension of the electromagnetic propagator of the cloaked objects amounts to choosing some boundary condition on the inside of the boundary of the cloaked objects. In other words, any possible unitary dynamics implies the existence of some boundary condition on the inside of the boundary of the cloaked objects. The fact that there is a large class of self-adjoint extensions -or boundary conditions- that can be taken inside the cloaked objects could be useful in order to enhance cloaking in practice, where one has to consider approximate transformation media as well as in the analysis of the stability of cloaking. Actually, we consider a slightly more general construction than the one of [8, 9, 13], [18] since we allow for a finite number of cloaked objects. In [7] a very general construction for cloaking is introduced. In the case of Maxwell’s equations all their constructions are made within the context of the permittivity and the permeability tensor densities being conformal to each other, i.e., multiples of each other by a positive scalar function. In particular, all isotropic media are included in this category. They mention that both for mathematical and practical reasons it would be very interesting to understand cloaking for general anisotropic materials in the absence of this assumption. In this paper we actually solve this problem, since we prove cloaking for all general anisotropic materials. In particular, our results prove that it is possible to cloak objects inside general crystals. Note, moreover, that [7] also considers the cases of the Helmholtz equation. We do not discuss this problems here. Furthermore, remark that the existing theorems in the uniqueness of inverse scattering do not apply under the present conditions. In [7] cloaking is proven with respect to the Cauchy data at any fixed frequency given on a surface that encloses the cloaked object. In the case where the permittivity and the permeability are bounded above and below it is well known that the Cauchy data at a fixed frequency is equivalent to the scattering matrix at the same frequency. See for example [17] and [26]. This equivalence is, however, not proven in the case where the permittivity and the permeability are degenerate on the boundary of the objects. In fact, it is perhaps even not true for general degenerate media that are not transformation media since in this case it is possible that there are finite energy electromagnetic waves that are absorbed by the boundary of the objects as t → ±∞. If this is true, the equivalence will not hold since the Cauchy data in a surface that encloses the objects will not contain information on the waves that are asymptotically absorbed by the boundary of the objects. It is a problem of independent interest to see if this actually happens or not for general degenerate permittivities and permeabilities. For an example of scattering by a bounded obstacle with a singular boundary and Neumann boundary condition, where this happens see [10]. For a similar situation in the scattering of electromagnetic waves by a Schwarzschild black-hole see [2]. Note that in our approach we directly consider the scattering operator that is measured in scattering experiments. In the analysis of Maxwell’s equations with permittivity and permeability that are inde- pendent of frequency the dispersion of the medium is not taken into account. This means that cloaking will hold for electromagnetic wave packets with a narrow enough range of frequencies, such that this assumption is valid. The paper is organized as follows. In Section 2 we prove our results in electromagnetic cloaking. In Section 3 we consider the propagation of electromagnetic waves in the interior of the cloaked objects. In Section 4 we formulate cloaking as a boundary value problem outside of the cloaked objects for the Maxwell equations at a fixed frequency, following our analysis of the self-adjoint extensions of the electromagnetic propagator. In particular, we give the appropriate boundary condition on the outside of the boundary of the cloaked objects. Finally, in Section 5 we prove cloaking of infinite cylinders. This is of interest since this is the case considered in the experimental verification in [24] and in the numerical simulations of [4] and [5]. Of course, [24] only consider a slice of the cylinder. In Sections 3 and 4 we give further comments on the results of [7]. Addendum After the previous version of this paper was posted in the arXiv we published the paper [29] where we generalized the results of this paper on spherical cloaks to the case of high- order cloaks, and where we also discussed cloaking in the frequency domain. Moreover, in our paper [30] we identified the cloaking boundary condition that has to be satisfied in the inside of the boundary of the cloaked objects, in the case where the permittivity and the permeability are bounded above and below inside the cloaked objects. 2 Electromagnetic Cloaking Let us consider Maxwell’s equations, ∇× E = − B, ∇×H = D, (2.1) ∇ ·B = 0,∇ ·D = 0, (2.2) in a domain, Ω ⊂ R3, as follows, Ω := R3 \ ∪Nj=1Kj, Kj ∩Kl = ∅, j 6= l (2.3) where Kj, j = 1, 2, · · · , N, are the objects to be cloaked. We assume that each Kj is a ball with center cj and radius aj, i.e., x ∈ R3 : |x− cj | ≤ aj , j = 1, 2, · · · , N. (2.4) The cloaked objects are denoted by K := ∪Nj=1Kj . We designate the Cartesian coordinates of x by xλ, λ = 1, 2, 3 and by Eλ, Hλ, B λ, Dλ, λ = 1, 2, 3, respectively, the components of E,H,B, and D. As usual, we denote by ελν and µλν , respectively, the permittivity and the permeability. We have that, Dλ = ελνEν , B λ = µλνHν , (2.5) where we use the standard convention of summing over repeated lower and upper indices. We consider now a transformation from Ω0 := R 3 \ {c1, c2, · · · , cN} onto Ω that was first used to obtain cloaking for the conductivity equation, i.e. at zero frequency, by [8, 9] and then by [18] for cloaking electromagnetic waves (for a related result in two dimensions using conformal mappings see [13]). For any y ∈ R3 we denote, ŷ := y/|y|. Let yλ, λ = 1, 2, 3, designate the cartesian coordinates of y ∈ Ω0. Take bj > aj, j = 1, 2, · · · , N . Then, for 0 < |y− cj| ≤ bj , we define, x = x(y) = f(y) := cj + bj − aj |y − cj |+ aj ŷ − cj. (2.6) Note that this transformation blows up the point cj onto ∂Kj and that it sends the punctu- ated ball B̃cj (bj) := {y ∈ R 3 : 0 < |y − cj | ≤ bj} onto the spherical shell, aj < |x− cj | ≤ bj . We assume that, B̃cj (bj) ∩ B̃cl(bl) = ∅, j 6= l, 1 ≤ j, l ≤ N. (2.7) For y ∈ R3\∪Nj=1B̃cj (bj) we define the transformation to be the identity, x = x(y) = f(y) := y. Our transformation is a bijection from Ω0 onto Ω. By y = y(x) := f −1(x) we designate the inverse transformation. We denote the elements of the Jacobian matrix by Aλλ′ , Aλλ′ := . (2.8) Note that the Aλλ′ ∈ C Ω0 \ ∪ j=1∂B̃cj (bj) . We designate by Aλ λ the elements of the Jacobian of the inverse bijection, y = y(x) := f−1(x), Ω \ ∪Nj=1∂B̃cj (bj) . (2.9) The papers [8, 9] and [18] considered the case where N = 1, c1 = 0. We take here the so called material interpretation and we consider our transformation as a bijection between two different spaces, Ω0 and Ω. However, our transformation can be considered, as well, as a change of coordinates in Ω0. Of course, these two point of view are mathematically equivalent. This means, in particular, that under our transformation the Maxwell equations in Ω0 and in Ω will have the same invariance that they have under change of coordinates in three-space. See, for example, [21]. Let us denote by ∆ the determinant of the Jacobian matrix (2.8). Then, bj − aj ( bj−aj |y− cj|+ aj |y − cj| , for 0 < |y − cj | ≤ bj . (2.10) This result is easily obtained rotating into a coordinate system such that, y − cj = (|y − cj|, 0, 0) [25]. For y ∈ Ω0 \ ∪ j=1B̃cj (bj),∆ ≡ 1. Let us denote by E0,H0,B0,D0, ε 0 , µ 0 , respectively, the electric and magnetic fields, the magnetic induction, the electric displacement, and the permittivity and permeability of Ω0. ε 0 , µ 0 , are positive, Hermitian matrices that are constant in Ω0. The electric field is a covariant vector that transforms as, Eλ(x) = A λ (y)E0,λ′(y). (2.11) The magnetic fieldH is a covariant pseudo-vector, but as we only consider space transfor- mations with positive determinant, it also transforms as in (2.11). The magnetic induction B and the electric displacement D are contravariant vector densities of weight one that transform as Bλ(x) = (∆(y)) Aλλ′(y)B 0 (y), (2.12) with the same transformation for D. The permittivity and permeability are contravariant tensor densities of weight one that transform as, ελν(x) = (∆(y)) Aλλ′(y)A ν′(y) ε 0 (y), (2.13) with the same transformation for µλν . The Maxwell equations (2.1, 2.2) are the same in both spaces Ω and Ω0. Let us denote by ελν , µλν, ε0λν , µ0λν , respectively, the inverses of the corresponding permittivity and permeability. They are covariant tensor densities of weight minus one that transform as, ελν(x) = ∆(y)A λ (y)A ν (y) ε0λ′ν′(y), µλν(x) = ∆(y)A λ (y)A ν (y)µ0λ′ν′(y). (2.14) Note that det ελν = ∆−1 det ελν0 , detµ λν = ∆−1 detµλν0 , (2.15) det ελν = ∆det ε0λν , detµλν = ∆detµ0λν . (2.16) We now introduce the Hilbert spaces of electric and magnetic fields with finite energy. The E0,H0,B0,D0, were defined in Ω0, but since R 3 \ Ω0 = {cj} j=1 is of measure zero, we can consider them as defined in R3, what we do below. We denote by H0E the Hilbert space of all measurable, square integrable, C 3− valued functions defined on R3 with the scalar product, 0ν dy 3. (2.17) We similarly define the Hilbert space,H0H , of all measurable, square integrable, C 3− valued functions defined on R3 with the scalar product, 0ν dy 3. (2.18) The Hilbert space of finite energy fields in R3 is the direct sum H0 := H0E ⊕H0H . (2.19) Moreover, we designate byHΩE the Hilbert space of all measurable, C 3− valued functions defined on Ω that are square integrable with the weight ελν , with the scalar product, (1),E(2) 3. (2.20) Finally, we denote by HΩH the Hilbert space of all measurable, C 3− valued functions defined on Ω that are square integrable with the weight µλν , with the scalar product, (1),H(2) 3. (2.21) The Hilbert space of finite energy fields in Ω is the direct sum HΩ := HΩE ⊕HΩH . (2.22) We now write the Maxwell’s equations (2.1) in Schrödinger form. We first consider the case of R3. We denote by ε0 and µ0, respectively, the matrices with entries ε0λν and µ0λν . Recall that (∇×E) = sλνρ ∂ Eρ where s λνρ is the permutation contravariant pseudo- density of weight −1 (see section 6 of chapter II of [21], where a different notation is used). By a0 we denote the following formal differential operator, ε0∇×H0 −µ0∇× E0 . (2.23) Here, as usual, we denote, ε0∇ × H0 := ε0λν(∇ × H0) ν , and µ0∇ × E0 = µ0λν(∇ × E0) Then, equations (2.1) are equivalent to, . (2.24) Let us denote by C10(R 3) the set of all C6−valued continuously differentiable functions on R3 that have compact support. Then, a0 with domain C 3) is a symmetric operator in H0, i.e., a0 ⊂ a 0. Moreover, it is essentially self-adjoint in H0, i.e., it has only one self-adjoint extension, that we denote by A0. Its domain is given by, D(A0) = , (2.25) ∈ D(A0), (2.26) where the derivatives are taken in distribution sense. These results follow easily from the fact that -via the Fourier transform- a0 is unitarily equivalent to multiplication by a matrix valued function that is symmetric with respect to the scalar product of H0. Moreover, it follows from explicit computation that the only eigenvalue of A0 is zero, that it has infinite multiplicity, and that, H0⊥ := (kernelA0) ∈ H0 : ελν0 E0ν = 0, µλν0 H0ν = 0 . (2.27) Furthermore, A0 has no singular-continuous spectrum and its absolutely-continuous spec- trum is R. See, for example, [27, 28]. Taking any ∈ H0⊥ ∩D(A0) (2.28) we obtain a finite energy solution to the Maxwell equations (2.1, 2.2) as follows (t) = e−itA0 . (2.29) This is the unique finite energy solution with initial value at t = 0 given by (2.28). Note that as e−itA0H0⊥ ⊂ H0⊥ equations (2.2) are satisfied for all times if they are satisfied at t = 0. Let us now consider the case of Ω. We denote by ε and µ, respectively, the matrices with entries ελν and µλν . We now define the following formal differential operator, −µ∇× E . (2.30) Equations (2.1) are equivalent to, Let us denote by C10(Ω) the set of all C 6−valued continuously differentiable functions on Ω that have compact support. Then, aΩ with domain C 0(Ω) is a symmetric operator in HΩ. To construct a unitary dynamics that preserves energy we have to analyse the self-adjoint extensions of aΩ. We denote by UE the following unitary operator from H0E onto HΩE , (UEE0)λ (x) := A λ E0λ′(y), (2.31) and by UH the unitary operator from H0H onto HΩH , (UHH0)λ (x) := A λ H0λ′(y). (2.32) Then, U := UE ⊕ UH (2.33) is a unitary operator from H0 onto HΩ. We denote by a00 the restriction of a0 to C 0(Ω0). The operator a00 is essentially self-adjoint and its only self-adjoint extension is A0. This follows from the essential self- adjointness of a0 and from the fact that any function in C 3) can be approximated in the graph norm of a0 by functions in C 0(Ω0). To prove this take any continuously differentiable real-valued function, φ, defined on R such that, φ(y) = 0, |y| ≤ 1 and φ(y) = 1, |y| ≥ 2. Then, for any ∈ C10(R we have that, φ(n|y− cj|) ∈ C10(Ω0) and moreover, s- lim φ(n|y− cj|) s- lim φ(n|y− cj|) where by s- lim we designate the strong limit in H0. As a00 is essentially self-adjoint, it follows from the invariance of Maxwell equations that aΩ is essentially self-adjoint, and that its unique self-adjoint extension, that we denote by AΩ, satisfies AΩ = U A0 U ∗. (2.34) For the proof of these facts see [29]. Hence, we have the following theorem. THEOREM 2.1. The operator aΩ is essentially self-adjoint, and its unique self-adjoint extension, AΩ, satisfies (2.34). The unitary equivalence given by (2.34) implies that AΩ has the same spectral prop- erties that A0. Namely, it has no singular-continuous spectrum, the absolutely-continuous spectrum is R and the only eigenvalue is zero and it has infinite multiplicity. Moreover, HΩ⊥ := (kernelAΩ) ∈ HΩ : ελνEν = 0, µλνHν = 0 . (2.35) Furthermore, taking any ∈ HΩ⊥ ∩D(AΩ) (2.36) we obtain a finite energy solution to the Maxwell equations (2.1, 2.2) as follows (t) = e−itAΩ . (2.37) This is the unique finite energy solution with initial value at t = 0 given by (2.36). Note that as e−itAΩHΩ⊥ ⊂ HΩ⊥ equations (2.2) are satisfied for all times if they are satisfied at t = 0. We can consider more general solutions by considering the scale of spaces associated with AΩ, but we do not go into this direction here. The facts that aΩ is essentially self-adjoint and that its unique self-adjoint extension AΩ is unitarily equivalent to the propagator A0 of the homogeneous medium are strong statements. They mean that the only possible unitary dynamics in Ω that preserves energy is given by (2.37) and that this dynamics is unitarily equivalent to the free dynamics in R3 given by (2.29). In fact, ∂Ω acts like a horizon for electromagnetic waves propagating in Ω in the sense that the dynamics is uniquely defined without any need to consider the cloaked objects K = ∪Nj=1Kj . As we will prove below this implies electromagnetic cloaking for all frequencies in the strong sense that the scattering operator is the identity. Since D(AΩ) = UD(A0), for any (E,H) T ∈ D(AΩ) there is a (E0,H0) T ∈ D(A0) such . (2.38) Then, it follows from (2.31, 2.32, 2.33) that E× n = 0,H× n = 0, in ∂K+, (2.39) where ∂K+ denotes the outside of the boundary of the cloaked objects, K, and n is the normal vector to ∂K+, if (E0,H0) are, for example, bounded near ∂K+. That is to say, for electromagnetic fields in the domain of AΩ the tangential components of both, the electric and the magnetic field vanish in the exterior of the boundary of the cloaked objects. This is a self-adjoint boundary condition because the permittivity and the permeability are degenerate on ∂K+. Let χΩ be the characteristic function of Ω, i.e., χΩ(x) = 1,x ∈ Ω, χΩ(x) = 0,x ∈ R 3 \ Ω. We define, (x) := χΩ(x) (x). (2.40) By (2.6, 2.10, 2.13), ∣ελν(x) ∣ ≤ C, ∣µλν(x) ∣ ≤ C, x ∈ Ω. Then, J is a bounded operator from H0 into HΩ. The wave operators are defined as follows, W± = s- lim eitAΩ Je−itA0P0⊥, (2.41) where P0⊥ denotes the projector onto H0⊥. Let us designate by W1,2(R3) the Sobolev space of C6 valued functions. We denote by I the identity operator on H0. Then, LEMMA 2.2. W± = UP0⊥. (2.42) Proof: Denote, W (t) := eitAΩ J e−itA0P0⊥. By (2.34), for any ϕ ∈ H0 W (t)ϕ = ψ(t) + UP0⊥ϕ, (2.43) ψ(t) := U eitA0 (U∗J − I) e−itA0P0⊥ϕ. (2.44) Let BR denote the ball of center zero and radius R in R 3. Since for |y| ≥ R, with R large enough, our transformation, x = f(y), is the identity, x = y, and in consequence, Aλλ′(y) = δ λ′ for |y| ≥ R, we have that, (U∗J − I) = (U∗J − I)χBR . (2.45) It follows that, s- lim ψ(t) = U s- lim eitA0ϑ(t) (2.46) with, ϑ(t) := (U∗J − I)χBR e−itA0P0⊥ϕ. (2.47) We have that, ‖ϑ(t)‖ e−itA0P0⊥ϕ e−itA0P0⊥ϕ e−itA0P0⊥ϕ . (2.48) Then, as (A0 + i) −1P0⊥ is bounded from H0 into W 1,2(R3) [27] [28], it follows from the Rellich local compactness theorem that (A0 + i) −1P0⊥ is a compact operator in H0. Suppose that ϕ ∈ D(A0) ∩ H0⊥. Then, s- lim e−itA0P0⊥ϕ = s- lim (A0 + i) −1P0⊥e −itA0(A0 + i)ϕ = 0, (2.49) and whence, by (2.48), s- lim ϑ(t) = 0, (2.50) and it follows that in this case, s- lim ψ(t) = 0. (2.51) By continuity this is also true for ϕ ∈ H0⊥. Then, (2.42) follows from (2.43) and (2.51). The scattering operator is defined as S := W ∗+W−. (2.52) COROLLARY 2.3. S = P0⊥. (2.53) Proof: This is immediate from (2.42) because U∗ U = I. Let us denote by S⊥ the restriction of S to H0⊥. S⊥ is the physically relevant scattering operator that acts in the Hilbert space H0⊥ of finite energy fields that satisfy equations (2.2). We designate by I⊥ the identity operator on H0⊥. We have that, COROLLARY 2.4. S⊥ = I⊥. (2.54) Proof: This follows from Corollary 2.3. The fact that S⊥ is the identity operator on H0⊥ means that there is perfect cloaking for all frequencies. Suppose that for very negative times we are given an incoming wave packet e−itA0ϕ−, with ϕ− ∈ H0⊥. Then, for large positive times the outgoing wave packet is given by e−itA0ϕ+ with ϕ+ = S⊥ϕ−. But, as S⊥ = I, we have that ϕ+ = ϕ− and then, e−itA0ϕ− = e −itA0ϕ+. Since the incoming and the outgoing wave packets are the same there is no way to detect the cloaked objects K from scattering experiments performed in Ω. In this paper we considered transformation media obtained from a singular transformation that blows up a finite number of points, by simplicity, and since this is the situation in the applications. Suppose that we have a transformation that is singular in a set of points that we call M and denote now Ω0 := R 3 \M . What we really used in the proofs is that W1,2(R3) = W 0 (Ω0) where W 0 (Ω0) denotes the completion of C 0 (Ω0) in the norm of W1,2(R3). We also assumed that ελν0 , µ 0 are constant. What was actually needed is that a0 is essentially self-adjoint. All our results hold under this more general conditions provided that in (2.41, 2.42) and (2.53) we replace P0⊥ by the projector onto the absolutely-continuous subspace of A0 and that we assume that D(A0) ∩H0ac ⊂ W 1,2(R3), where we have denoted the absolutely-continuous subspace of A0 by H0ac. Moreover, S⊥ has to be defined as the restriction of S to H0ac and in (2.54) I⊥ has to be the identity operator on H0ac. Note that under these general assumptions A0 could have non-zero eigenvalues and singular-continuous spectrum. For example, W1,2(R3) = W 0 (Ω0) if M has zero Sobolev one capacity [1, 11, 12]. Moreover, assume that the permittivity and the permeability tensor densities ελν0 , µ 0 are bounded below and above. Under this condition a0 is essentially self-adjoint. Furthermore, let us denote by Ĥ0 the Hilbert space of finite energy solutions defined as in (2.19) but with ελν0 = µ 0 = δ λµ. Let Â0, Ĥ0⊥ be, respectively, the electromagnetic propagator in Ĥ0 and the orthogonal complement of its kernel. We have that H0 and Ĥ0 are the same set of functions with equivalent norms. Furthermore, D(A0) = D(Â0), kernel Â0 = kernelA0. Moreover, (E0,H0) T ∈ H0⊥ if and only if E0 = ε0Ê0,H0 = µ0Ĥ0 for some (Ê0, Ĥ0) ∈ Ĥ0⊥. As [27, 28] D(Â0) ∩ Ĥ0⊥ ⊂ W 1,2(R3) we have that D(A0) ∩ H0⊥ ⊂ W 1,2(R3) if ε0, µ0 are bounded operators on W1,2(R3) and this is true if the derivatives ∂ µ0 are bounded operators on Ĥ0 for ρ = 1, 2, 3. Note, furthermore, that H0ac ⊂ H0⊥. 3 Electromagnetic Waves Inside the Cloaked Objects Let us now consider the propagation of electromagnetic waves in the cloaked objects. For this purpose we assume that in each Kj the permittivity and the permeability are given by ελνj , µ j , with inverses εjλν, µjλν and where εj, µj are the matrices with entries εjλν, µjλν. Furthermore, we assume that 0 < ελνj , µ j ≤ C,x ∈ Kj and that for any compact set Q con- tained in the interior of Kj there is a positive constant CQ such that det ε j > CQ, detµ CQ,x ∈ Q. In other words, we only allow for possible singularities of εj, µj on the boundary of Kj. We designate by HjE the Hilbert space of all measurable, C 3− valued functions defined on Kj that are square integrable with the weight ε j , with the scalar product, jν dx 3. (3.1) Similarly, we denote by HjH the Hilbert space of all measurable, C 3− valued functions defined on Kj that are square integrable with the weight µ j , with the scalar product, jν dx 3. (3.2) The Hilbert space of finite energy fields in Kj is the direct sum Hj := HjE ⊕HjH, (3.3) and the Hilbert space in the cloaked objects K is the direct sum, HK := ⊕ j=1Hj . The complete Hilbert space of finite energy fields including the cloaked objects is, H := HΩ ⊕HK . (3.4) We now write (2.1) as a Schrödinger equation in each Kj as before. We define the following formal differential operator, εj∇×Hj −µj∇× Ej . (3.5) Equation (2.1) in Kj is equivalent to . (3.6) Let us denote by C10(K̂j) the set of all C 6−valued continuously differentiable functions on Kj that have compact support in the interior of Kj, that we denote by K̂j := Kj \ ∂Kj . Then, aj with domain C 0(K̂j) is a symmetric operator in Hj . We denote, a := aΩ ⊕ j=1 aj , (3.7) with domain, D(a) := ⊕Nj=1 ∈ C10(Ω)⊕ j=1 C 0(K̂j) . (3.8) The operator a is symmetric in H. The possible unitary dynamics that preserve energy for the whole system including the cloaked objects K are given by the self-adjoint extensions of a. Let us denote a the closure of a, with similar notation for aΩ, aj , j = 1, · · · , N . Then, a = AΩ ⊕ j=1 aj, where we used the fact that as aΩ is essentially self-adjoint, aΩ = AΩ. The adjoint of a is given by, D(a∗) = ⊕Nj=1 ∈ H : ∈ D(AΩ), aj , (3.9) ⊕Nj=1 ⊕Nj=1 aj , (3.10) ⊕Nj=1 ∈ D(a∗). (3.11) Let us denote by KΩ± := kernel(i∓ a Ω),Kj± := kernel(i∓ a j) the deficiency subspaces of aΩ and aj, j = 1, · · · , N . Since aΩ is essentially self-adjoint KΩ± = {0}. Let K± := ⊕ j=1Kj± be the deficiency subspaces of aK := ⊕ j=1aj . Suppose that K± have the same dimension. Then, it follows from Corollary 1 in page 141 of [22] that there is a one-to-one correspondence between self-adjoint extensions of aK and unitary maps from K+ into K−. If V is such a unitary, then the corresponding self-adjoint extension AKV is given by, D(AKV ) = {ϕ+ ϕ+ + V ϕ+ : ϕ ∈ D(aK), ϕ+ ∈ K+} , AKϕ = aKϕ+ iϕ+ − iV ϕ+. Hence, since KΩ± = {0} and a = AΩ⊕aK there is a one-to-one correspondence between self- adjoint extensions of a and unitary maps, V , from K+ into K−. The self-adjoint extension AV corresponding to V is given by, AV = AΩ ⊕ AKV . Thus, we have proven the following theorem. THEOREM 3.1. Every self-adjoint extension, A, of a is the direct sum of AΩ and of some self-adjoint extension, AK of aK , i.e., A = AΩ ⊕AK . (3.12) This theorem tells us that the cloaked objects K and the exterior Ω are completely decoupled and that we are free to choose any boundary condition inside the cloaked objectsK that makes aK self-adjoint without disturbing the cloaking effect in Ω. Boundary conditions that make AK self-adjoint are well known. See for example, [19], [20], [14] and [6]. It follows from explicit computation that zero is an eigenvalue of every AK with infinite multiplicity and that, HK⊥ := (kernelAK) ∈ HK : ελνK Eν = 0, µλνK Hν = 0 , (3.13) where by ελνK (x) := ε j (x) for x ∈ Kj, and µ K (x) := µ j (x) for x ∈ Kj , j = 1, 2, · · · , N . It follows that zero is an eigenvalue of A with infinite multiplicity and that, H⊥ := (kernelA) = HΩ⊥ ⊕HK⊥. (3.14) For any ϕ = ϕΩ ⊕ ϕK ∈ H⊥ ∩D(A), e−itAϕ = e−itAΩ ϕΩ ⊕ e −itAK ϕK (3.15) is the unique solution of Maxwell’s equations (2.1, 2.2) with finite energy that is equal to ϕ at t = 0. This shows once again that the dynamics in Ω and in K are completely decoupled. If at t = 0 the electromagnetic fields are zero in Ω, they remain equal to zero for all times, and viceversa. Actually, electromagnetic waves inside the cloaked objects are not allowed to leave them, and viceversa, electromagnetic waves outside can not go inside. This implies, in particular, that the presence of active devices inside the cloaked objects has no effect on the cloaking outside. In terms of boundary conditions, this means that transmission conditions that link the electromagnetic fields inside and outside the cloaked objects are not allowed. Furthermore, choosing a particular self-adjoint extension of the electromagnetic propagator of the cloaked objects amounts to choosing some boundary condition on the inside of the boundary of the cloaked objects. In other words, any possible unitary dynamics implies the existence of some boundary condition on the inside of the boundary of the cloaked objects. The particular boundary condition that nature will take depends on the specific properties of the metamaterial used to build the transformation media as well us on the properties of the media inside the cloaked objects. Note that this does not mean that we have to put any physical surface, a lining, on the surface of the cloaked object to enforce any particular boundary condition on the inside, since as we already mentioned this plays no role in the cloaking outside. It would be, however, of theoretical interest to see what the interior boundary condition turns out to be for specific cloaked objects and metamaterials. These results apply to the exact transformation media that we consider on this paper. However, the fact that there is a large class of self-adjoint extensions -or boundary conditions- that can be taken inside the cloaked objects could be useful in order to enhance cloaking in practice, where one has to consider approximate transformation media as well as in the analysis of the stability of cloaking. The fact that for the single coating there has to be boundary conditions on the inside of ∂K has already been observed by [7]. In Definition 4.1 of [7] a definition of finite energy solutions is given. Furthermore, is proven in Theorem 6.1 that in the case of the single coating -where the permittivity and the permeability are bounded above and below inside the cloaked object- the tangential components of the electric and the magnetic field of these solutions have to vanish in the inside of the boundary of the cloaked object. Note that in this case in order to have a self-adjoint extension of the electromagnetic propagator inside the cloaked object we are only allowed to require that either the tangential component of E or the tangential component of H vanishes, but not both. These boundary conditions are called hidden boundary conditions in [7] where also the case of the Helmholtz equation is considered. In the case of Maxwell’s equations they propose two solutions to this issue. One of them is a lining, i.e., a physical material on the boundary of the cloaked object that enforces a particular boundary condition, for example, they propose a lining by a perfect electric conductor. Note that this raises now the question of what is the boundary condition between the lining and the cloaking metamaterial. In fact, we face the same problem as before, since we can always consider that the lining is part of the cloaked objects, and then, the question of what is the appropriate boundary condition remains. The second proposal of [7] is a double coating that corresponds to surrounding both the inner and the outersurface of the cloaked objects with appropriately matched metamaterials. As our permittivities and permeabilities inside K are allowed to vanish as they approach ∂K the double coating fits in our formalism. In Theorem 5.1 of [7] cloaking is proven for all frequencies and active devices, with the double coating, with respect to the Cauchy data of the finite energy solutions that they define in Definition 4.1. Remark that there is no real contradiction between our results and the ones of [7]. Our results imply that there is always a hidden boundary condition on the inside of the boundary of the cloaked objects, that is imposed upon us by the fundamental principle of the con- servation of the energy of the electromagnetic waves, that implies that time evolution has to be given by a unitary group generated by a self-adjoint extension of the electromagnetic propagator, and this amounts to a boundary condition at the inside of the boundary of the cloaked objects. Note that we do not exclude here the possibility that in some cases the electromagnetic propagator of the cloaked objects could be essentially self-adjoint, and in this situation the dynamics inside the cloaked objects will be uniquely defined. In this case the hidden boundary condition will be uniquely determined by the boundary conditions satisfied by the functions in the domain of the unique self-adjoint realization of the elec- tromagnetic propagator in the cloaked objects. Note, however, that we have proven that for exact transformation media the cloaking outside is actually independent of the cloaked objects. 4 Cloaking as a Boundary Value Problem It is a question of independent interest to consider cloaking as a boundary value problem for the Maxwell’ system at a fixed frequency ∇× E = iλB, ∇×H = −iλD, λ 6= 0, (4.1) ∇ ·B = 0,∇ ·D = 0. (4.2) As we have already shown, cloaking is independent of the cloaked object, and this means that we only have to consider these equation in Ω. The main question now is to decide what is an appropriate class of solutions with locally finite energy. Our analysis of the self-adjoint extensions of the electromagnetic propagator shows that we have to take solutions that are locally in the domain of AΩ, that is to say that they are given by (2.38) with (E0,H0) T locally in the domain of A0, i.e., (E0,H0) T are in the domain of A0 when multiplied by any function in C∞0 (R 3). It follows from (2.39) that the solutions with locally finite energy have to satisfy the boundary condition, E× n = 0,H× n = 0, in ∂K+, where ∂K+ is the outside of the boundary of the cloaked object. This is the only self- adjoint boundary condition on ∂K+. Note that we define in the same way solutions with (locally) finite energy in a bounded subset of Ω. In [7] a different definition of solutions with (locally) finite energy is given in Definition 4.1. 5 Cloaking an Infinite Cylinder We discuss now the case of an infinite cylinder. For simplicity we consider one cylinder centered at zero and with its axis the vertical line L := (0, 0, x3), x3 ∈ R. Then, x = (x1, x2, x3) ∈ R3 : |x1|2 + |x2|2 ≤ a, x3 ∈ R , Ω := R3 \K. (5.3) The set Ω0 is now given by, Ω0 = R 3 \ L. (5.4) Let us denote by x := (x,1 , x2) the vectors in the x1 − x2 plane and x̂ := x/|x|. The transformation (2.6) is replaced by x = x(y) = f(y) := x = ( b−a |y|+ a)ŷ, x3 = y3, (5.5) for 0 < |y| ≤ b and with b > a. This transformation blows up the line L onto ∂K and it sends Kb \ L onto Kb \K where Kb := y = (y1, y2, y3) ∈ R3 : |y1|2 + |y2|2 ≤ b, y3 ∈ R For y ∈ R3 \Kb we define the transformation to be the identity, x = y. The Hilbert spaces of finite energy electromagnetic fields, the unitary operator U and a0, A0, aΩ, aK , a, are defined as in Section 2. THEOREM 5.1. The operator aΩ is essentially self-adjoint, and its unique self-adjoint extension, AΩ, satisfies AΩ = U A0 U ∗. (5.6) Proof: The theorem is proven as Theorem 2.1 observing that W 1,2(R2) = W 2 \ 0) since {0} has zero Sobolev one capacity in R2 [1, 11, 12]. We now consider the wave and the scattering operators. For simplicity we assume below that ελν0 = ε̃ δ λν , µλν0 = µ̃ δ λν . The wave operators are defined as in (2.41) but now the operator J is defined as follows, (x) := χΩ(x)φ(x) where φ is continuous and it satisfies φ(x) = (|x| − a), a ≤ |x| ≤ a + δ and φ(x) = 1 for |x| ≥ a+ 2δ, for some δ > 0. LEMMA 5.2. W± = UP0⊥. (5.7) Proof: The lemma is proven as in the proof of Lemma 2.2, but in (2.45, 2.46, 2.47, 2.48) we have to replace χBR , by χCR where, CR := {y ∈ R 3 : |y| ≤ R} for R large enough. Now we can not prove (2.49, 2.50, 2.51) by compactness arguments because K is unbounded. Instead we use propagation estimates for A0. The following results are well known. See for example [3, 27, 28, 31] where the general anisotropic case is considered. For any ϕ ∈ H0⊥, e−itA0ϕ = (2π)3/2 eik·y e−iω+(k)tP+(k)ϕ̂(k) + e −iω−(k)tP−(k)ϕ̂(k) d3k (5.8) where ϕ̂ is the Fourier transform of ϕ, ω±(k) = ±|k|c with c := (ε̃µ̃) −1/2, and P±(k) are projectors on R3 that are infinitely differentiable for k ∈ R3\0. Suppose that ϕ̂ ∈ C∞0 (R and let O be a bounded open set such that O ⊂ R3 \ L and support ϕ̂ ⊂ O. Denote Ô := : k ∈ O Then by the (non) stationary phase Theorem (see the Corollary to Theorem XI.14 of [23] ), for any n = 1, 2, · · · there is a constant Cn such that e−itA0ϕ ∣ ≤ Cn (1 + |y|+ |t|) /∈ Ô. (5.9) We write, −itA0ϕ = φ1 + φ2 (5.10) φ1 := χ(±y/(ct)/∈Ô)χCRe −itA0ϕ (5.11) φ2 := χ(±y/(ct)∈Ô)χCRe −itA0ϕ. (5.12) by (5.9) s- lim φ1 = 0. (5.13) Note, furthermore, that there is an ǫ > 0 such that |k| ≥ ǫ for any k ∈ Ô. Then, for any ∈ Ô, |y| ≥ c|t|ǫ. It follows that there is a T such that φ2 = 0, for|t| ≥ T. (5.14) By (5.10, 5.13, 5.14) s- lim −itA0ϕ = 0, (5.15) and (2.50, 2.51) follow. Note that P0⊥ is not needed because ϕ ∈ H0⊥. By continuity this is true for all ϕ ∈ H0⊥. Then, (5.7) follows from (2.43, 2.51). The scattering operator is defined as in (2.52). COROLLARY 5.3. S = P0⊥. (5.16) Proof: This is immediate from (5.7) because U∗ U = I. Let us denote by S⊥ the restriction of S to H0⊥. S⊥ is the physically relevant scattering operator that acts in the Hilbert space H0⊥ of finite energy fields that satisfy equations (2.2). We designate by I⊥ the identity operator on H0⊥. We have that, COROLLARY 5.4. S⊥ = I⊥. (5.17) Proof: This follows from Corollary 5.3. Again, the fact that S⊥ is the identity operator on H0⊥ means that there is cloaking for all frequencies. In Theorem 7.1 of [7] cloaking is proven for all frequencies with respect to the Cauchy data of the finite energy solutions that they define in Definition 4.1 and furthermore, in Theorem 8.2, they prove cloaking for all frequencies with the SHS boundary condition with respect to the Cauchy data of the finite energy solutions that they define in Definition 8.1. Theorem 3.1 remains true in the case of the cylinder. The proof is the same. Furthermore, all the remarks about finite energy solutions, and cloaking and that we made in Sections 2, 3, are true in the case of a cylinder. We do not repeat them here. Moreover, equations (2.38) hold. However, since now the transformation (5.5) only acts on the plane orthogonal to the axis of the cylinder equations (2.39) has to be replaced by E× x̂ = 0, H× x̂ = 0, in ∂K+, (5.18) where E := (E1, E2),H := (H1, H2). As in Section 4 we define solutions to (4.1, 4.2) with locally finite energy as solutions that are locally in the domain of AΩ, that is to say that they are given by (2.38) with (E0,H0) T locally in the domain of A0, i.e., (E0,H0) T are in the domain of A0 when multiplied by any function in C∞0 (R 3). It follows from (5.18) that the solutions with locally finite energy have to satisfy the boundary condition, E× x̂ = 0, H× x̂ = 0, in ∂K+. (5.19) Note that (5.19) is the SHS boundary condition considered in [7]. We have proven here that (5.19) is the only self-adjoint boundary condition on ∂K+ allowed by energy conserva- tion. Acknowledgement This work was partially done while I was visiting the Institut für Theoretische Physik, Eidgenössische Techniche Höchschule Zurich. I thank professors Gian Michele Graf and Jürg Fröhlich for their kind hospitality. References [1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Second Edition, Academic Press, Amsterdam, 2003 [2] A. Bachelot, Gravitational scattering of electromagnetic fields by Schwarzschild black- holes, Ann. Inst. H. Poincaré 54 261-320 (1991). [3] R. Courant and D. Hilbert, Methoden der Mathematischen Physik II, Zweite Auflage, Springer-Verlag, Berlin, 1968. [4] W. Cai, U.K. Chettiar, A.V. Kildishev and V.M. Shalaev, Optical Cloaking with meta- materials, Nature Photonics 1 224-226 (2007). [5] S.A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith and J. Pendry, Full-wave simulation of electromagnetic cloaking structures, Phys. Rev. E 74 036621 (2006). [6] M. Sh. Birman and M.Z. Solomyak, The self-adjoint Maxwell operator in arbitrary domains, Algebra i Analiz. 1 96-110 (1989). English transl. Leningrad Math. J. 1 99- 115 (1989). [7] A. Greenleaf, Y. Kurylev, M. Lassas and G. Ulhmann, Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys. 275 749-789 (2007). [8] A. Greenleaf, M. Lassas, and G. Uhlmann, Anisotropic conductivities that cannot be detected by EIT, Physiol. Meas. 24 413-419 (2003). [9] A. Greenleaf, M. Lassas, and G. Uhlmann, On nonuniqueness for Calderón’s inverse problem, Math. Res. Let. 10 685-693 (2003). [10] R. Hempel and R. Weder, On the completeness of wave operators under loss of local compactness, Journal of Functional Analysis, 113 391-412 (1993). [11] T. Kilpeläinen, J. Kinnunen and O. Martio, Sobolev spaces with zero boundary values on metric spaces, Potential Anal. 12 233-247 (2000). [12] J. Kinnunen and O. Martio, The Sobolev capacity on metric spaces, Ann. Acad. Sci. Fenn. Math. 21 367-382 (1997). [13] U. Leonhardt, Optical conformal mapping, Science 312 1777-1780 (2006). [14] R. Leis, Initial Boundary Value Problems in Mathematical Physics, John Wiley & Sons, New York, 1986. [15] U. Leonhardt and T. G. Philbin, General relativity in electrical engineering, New J. Phys. 8 247 (2006). [16] G. W. Milton, M. Briane, and J. R. Willis, On cloaking for elasticity and physical equations with transformation invariant form, New J. Phys. 8 248 (2006). [17] A. Nachman, Reconstruction from boundary measurements, Ann. of Math. 128 71-96 (1988). [18] J. B. Pendry, D. Schurig, and D. R. Smith, Controlling electromagnetic fields, Science 312 1780-1782 (2006). [19] R. Picard, Ein Randwertproblem für die zeitunabhängigen Maxwellschen Gleichungen mit der Randbedingung n · εE = n · µH = 0 in beschränken Gebieten beliebigen Zusammenhangs Appl. Anal. 6 207-221 (1977). [20] R. Picard, On the low frequency asymptotics in electromagnetic theory, J. Reine Angew. Math. 354 50-73 (1984). [21] E.J. Post, Formal Structure of Electromagnetics General Covariance and Electromag- netics, Dover Publications, Mineola, New York, 1997. [22] M. Reed and B. Simon, Methods of Modern Mathematical Physics II Fourier Analysis and Self-Adjointness, Academic Press, New York, 1975. [23] M. Reed and B. Simon, Methods of Modern Mathematical Physics III Scattering Theory, Academic Press, New York, 1979. [24] D. Schurig, , J.J. Mock, B.J. Justice, S.A. Cummer, J.B.. Pendry, A.F. Starr and D. R. Smith, Metamaterial electromagnetic cloak at microwave frequencies Science 314 977-980 (2006). [25] D. Schurig, J.B. Pendry, and D. R. Smith, Calculation of material properties and ray tracing in transformation media, Opt. Exp. 14 9794-9804 (2006). [26] G. Uhlmann, Scattering by a metric, Chap. 6.1.5 in Encyclopedia on Scvattering, Aca- demic Press, R. Pike and P. Sabatier, eds (2002), 1668-1677. [27] R. Weder, Analyticity of the scattering matrix for wave propagation in crystals, J. Math. Pures et Appl. 64 121-148 (1985). [28] R. Weder, Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media. Applied Mathematical Sciences 87 Springer-Verlag, New York, 1991. [29] R. Weder, A rigorous analysis of high-order electromagnetic invisibility cloaks, arXiv: 0711.0507, 2007, J. Phys. A: Mathematical and Theoretical 41 065207 (2008). IOP- Select. [30] R. Weder, The boundary conditions for electromagnetic invisibility cloaks, arXiv: 0801.3611, 2008. [31] C.H. Wilcox, Asymptotic wave functions and energy distributions in strongly propaga- tive media, J. Math. Pures et Appl. 57 275-231 (1978). Introduction Electromagnetic Cloaking Electromagnetic Waves Inside the Cloaked Objects Cloaking as a Boundary Value Problem Cloaking an Infinite Cylinder

0704.0249

Non-perturbative conserving approximations and Luttinger's sum rule

Non-perturbative conserving approximations and Luttinger’s sum rule Jutta Ortloff, Matthias Balzer, Michael Potthoff Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Am Hubland, D-97074 Würzburg, Germany Weak-coupling conserving approximations can be constructed by truncations of the Luttinger- Ward functional and are well known as thermodynamically consistent approaches which respect macroscopic conservation laws as well as certain sum rules at zero temperature. These properties can also be shown for variational approximations that are generated within the framework of the self- energy-functional theory without a truncation of the diagram series. Luttinger’s sum rule represents an exception. We analyze the conditions under which the sum rule holds within a non-perturbative conserving approximation. Numerical examples are given for a simple but non-trivial dynamical two- site approximation. The validity of the sum rule for finite Hubbard clusters and the consequences for cluster extensions of the dynamical mean-field theory are discussed. PACS numbers: 71.10.-w, 71.10.Fd I. INTRODUCTION Continuous symmetries of a Hamiltonian imply the existence of conserved quantities: The conservation of total energy, momentum, angular momentum, spin and particle number is enforced by a not explicitly time- dependent Hamiltonian which is spatially homogeneous and isotropic and invariant under global SU(2) and U(1) gauge transformations. For the treatment of a macro- scopically large quantum system of interacting fermions, approximations are inevitable in general. Approxima- tions, however, may artificially break symmetries and thus lead to unphysical violations of conservations laws. Baym and Kadanoff1,2 have analyzed under which cir- cumstances an approximation for time-dependent corre- lation functions, and for one- and two-particle Green’s functions in particular, respect the mentioned macro- scopic conservation laws. They were able to give cor- responding rules for a proper construction of approxima- tions, namely criteria for selecting suitable classes of dia- grams, within diagrammatic weak-coupling perturbation theory. Weak-coupling approximations following these rules and thus respecting conservation laws are called “conserving”. Frequently cited examples for conserving approximations are the Hartree-Fock or the fluctuation- exchange approximation.1,3,4 Baym2 has condensed the method of constructing con- serving approximations into a compact form: A con- serving approximation for the one-particle Green’s func- tion G is obtained by using Dyson’s equation G = 1/(G−10 − Σ) with (the free, U = 0, Green’s function G0 and) a self-energy Σ = ΣU [G] given by a univer- sal functional. Apart from G, the universal functional ΣU must depend on the interaction parameters U only. Furthermore, the functional must satisfy a vanishing-curl condition or, alternatively, must be derivable from some (universal) functional ΦU [G] as TΣU [G] = δΦU [G]/δG (the temperature T is introduced for convenience). In short, “Φ-derivable” approximations are conserving. Φ-derivable approximations have been shown2 to ex- hibit several further advantageous properties in addition. One of these concerns the question of thermodynamical consistency. There are different ways to determine the grand potential of the system from the Green’s function which do not necessarily yield the same result when us- ing approximate quantities. On the one hand, Ω may be calculated by integration of expectation values, accessi- ble by G, with respect to certain model parameters. For example, Ω may be calculated by integration of the av- erage particle number, as obtained from the trace of G, with respect to the chemical potential µ. On the other hand, Ω can be obtained as Ω = Φ + Tr lnG − Tr(ΣG) without integration. A Φ-derivable approximation con- sistently gives the same result for Ω in both ways. At zero temperature T = 0 there is another non-trivial theorem which is satisfied by any Φ-derivable approxima- tion, namely Luttinger’s sum rule.5,6 This states that the volume in reciprocal space that is enclosed by the Fermi surface is equal to the average particle number. The orig- inal proof of the sum rule by Luttinger and Ward5 is based on the existence of Φ in the exact theory and is straightforwardly transferred to the case of a Φ-derivable approximation. This also implies that other Fermi-liquid properties, such as the linear trend of the specific heat at low T and Fermi-liquid expressions for the T = 0 charge and the spin susceptibility are respected by a Φ-derivable approximation. There is a perturbation expansion5,7 which gives the Luttinger-Ward functional ΦU [G] in terms of closed skeleton diagrams (see Fig. 1). As a manageable Φ- = + + +Φ FIG. 1: Diagrammatic representation of the Luttinger-Ward functional ΦU [G]. Double lines stand for the interacting one- particle Green’s function G, dashed lines represent the ver- tices U . http://arxiv.org/abs/0704.0249v2 derivable approximation must specify a (universal) func- tional ΦU [G] that can be evaluated in practice, one usu- ally considers truncations of the expansion and sums up a certain subclass of skeleton diagrams only. This, how- ever, means that the construction of conserving approx- imations is restricted to the weak-coupling limit. One purpose of the present paper is to show that it is possible to construct Φ-derivable approximations for lat- tice models of correlated fermions with local interactions which are non-perturbative, i.e. do not employ trunca- tions of the skeleton-diagram expansion. The idea is to employ the self-energy-functional theory (SFT).8,9,10 The SFT constructs the Luttinger-Ward functional ΦU [G], or its Legendre transform FU [Σ], in an indirect way, namely by making contact with an exactly solvable reference sys- tem. Thereby, the exact functional dependence of FU [Σ] becomes available on a certain subspace of self-energies which is spanned by the self-energies generated by the reference system. The obvious question is whether those non- perturbative Φ-derivable approximations have the same properties as the weak-coupling Φ-derivable ap- proximations suggested by Baym and Kadanoff. This requires the discussion of the following points: (i) Macroscopic conservations laws. For fermionic lat- tice models, conservation of energy, particle number and spin have to be considered. Besides the static thermody- namics, the SFT concept concentrates on the one-particle excitations. For the approximate one-particle Green’s function, however, it is actually simple to prove that the above conservation laws are respected. A short discus- sion is given in Appendix A. (ii) Thermodynamical consistency. This issue has al- ready been addressed in Ref. 11. It has been shown that the µ derivative of the (approximate) SFT grand poten- tial (including a minus sign) equals the average particle number 〈N〉 as obtained by the trace of the (approxi- mate) Green’s function. The same holds for any one- particle quantity coupling linearly via a parameter to the Hamiltonian, e.g. for the average total spin 〈S〉 coupling via a field of strength B. (iii) Luttinger sum rule. This is the main point to be discussed in the present paper. There are different open questions: First, it is straightforward to prove that weak-coupling Φ-derivable approximations respect the sum rule as one can directly take over the proof for the exact theory. For approximations constructed within the SFT, a different proof has to be given. Second, it turns out that a non-perturbative Φ-derivable approximation respects the sum rule if and only if the sum rule holds for the reference system that is used within the SFT. As the original and thereby the related reference system may be studied in the strong-coupling regime, this raises the question which reference system does respect the sum rule, i.e. which approximation is consistent with the sum rule. Third, it will be particularly interesting to study reference systems which generate dynamical impurity ap- proximations (DIA)8,9 and variational cluster approxi- mations (VCA),10,12 as these consist of a finite number of degrees of freedom. Does the Luttinger sum rule hold for finite systems? Do the DIA and the VCA respect the sum rule? What is the simplest approximation consistent with the sum rule? Note that finite reference systems consist- ing of a few sites only have been shown9,13,14,15,16,17,18,19 to generate approximations which qualitatively capture the main physics correctly. Finally, it is important to un- derstand these issues in order to understand whether and how a violation of the sum rule is possible within cluster extensions20,21,22,23 of the dynamical mean-field theory (DMFT).24,25,26,27,28 Note that the SFT comprises the DMFT and certain29 cluster extensions and that possi- ble violations of the sum rule in the two-dimensional lat- tice models have been reported,30,31,32 including a study using the dynamical cluster approximation (DCA).33 The paper is organized as follows: A brief general dis- cussion of the Luttinger sum rule is given in the next section, and a form of the sum rule specific to systems with a finite number of spatial degrees of freedom is derived. Sec. III clarifies the status of the sum rule with respect to non-perturbative approximations gener- ated within the SFT framework. The results are elu- cidated by several numerical examples obtained for the most simple but non-trivial non-perturbative conserving approximation in Sec. IV. Violations of the sum rule in fi- nite systems and their consequences are discussed in Sec. V. Finally, Sec. VI summarizes our main conclusions. II. LUTTINGER SUM RULE A system of interacting electrons on a lattice is gen- erally described by a Hamiltonian H(t,U) = H0(t) + H1(U) consisting of a one-particle part H0(t) and an interaction H1(U) with one-particle and interaction pa- rameters t and U , respectively. As a prototype, let us consider the single-band Hubbard model34,35,36 on a translationally invariant D dimensional lattice consist- ing of L sites with periodic boundary conditions. The Hamiltonian is given by: iσcjσ + niσni−σ . (1) Here, i = 1, ..., L refers to the sites, σ =↑, ↓ is the spin projection, ciσ (c iσ) annihilates (creates) an electron in the one-electron state |iσ〉, and niσ = c ciσ. Fourier transformation diagonalizes the hopping matrix t and yields the dispersion ε(k). There are L allowed k points in the first Brillouin zone. Let G = Gt,U denote the one-electron Green’s func- tion of the model H(t,U). In case of the Hubbard model, its elements are given by Gij(ω) = 〈〈ciσ ; c jσ〉〉ω . In the absence of spontaneous symmetry breaking, the Green’s function is spin-independent and diagonal in reciprocal space. It can be written as Gk(ω) = 1/(ω + µ − ε(k) − Σk(ω)) where µ is the chemical potential and Σk(ω) the self-energy. We also introduce the notation Σt,U for the self-energy, and Gt,0 = 1/(ω + µ − t) for the free (non-interacting) Green’s function which exhibits the de- pendence on the model parameters but suppresses the frequency dependence. Dyson’s equation then reads as Gt,U = 1/(G t,0 −Σt,U ). The Luttinger sum rule5,6 states that 〈N〉 = 2 Θ(Gk(0)) (2) where N = niσ is the particle-number operator, 〈N〉 its (T = 0) expectation value, and Θ the Heavy- side step function. The factor 2 accounts for the two spin directions. Since Gk(0) −1 = µ − ε(k) − Σk(0), the sum gives the number of k points enclosed by the in- teracting Fermi surface which, for L → ∞, is defined via µ−ε(k)−Σk(0) = 0. In the thermodynamic limit the sum rule therefore equates the average particle number with the Fermi-surface volume (apart from a factor (2π)D/L). Note that, as Θ(Gk(0)) = Θ(1/Gk(0)), the sum rule Eq. (2) also includes the so-called Luttinger volume37 which (for L → ∞) is enclosed by the zeros of Gk(0). The standard proof of the sum rule can be found in Ref. 5. It is based on diagrammatic perturbation theory to all orders which is used to construct the Luttinger- Ward functional ΦU [G] as the sum of renormalized closed skeleton diagrams (see Fig. 1). We emphasize that the original proof straightforwardly extends also to finite sys- tems. For L < ∞ the sum in Eq. (2) is discrete. Actually, the proof is performed for finite L first, and the thermo- dynamic limit (if desired) can be taken in the end. The limit T → 0, on the other hand, is essential and is re- sponsible for possible violations of the sum rule (see Sec. Below we need an alternative but equivalent formu- lation of the sum rule. We start from the following (Lehmann) representation for the Green’s function: Gk(ω) = αm(k) ω + µ− ωm(k) . (3) Here, ωm(k)−µ are the (real) poles and αm(k) the (real and positive) weights. For real frequencies ω, it is then easy to verify the identity: Θ(Gk(ω)) = Θ(ω+µ−ωm(k))− Θ(ω+µ−ζn(k)) where ζn(k) − µ is the n-th (real) zero of the Green’s function, i.e. Gk(ζn(k)− µ) = 0. For temperature T = 0 we have 〈N〉 = dω(−1/π)ImGk(ω + i0 +) and thus 〈N〉 = αm(k)Θ(µ−ωm(k)). Hence, the Luttinger sum rule reads: αm(k)Θ(µ− ωm(k)) Θ(µ− ωm(k))− Θ(µ− ζn(k)) This form of the sum rule is convenient for the discussion of finite systems with L < ∞. III. SELF-ENERGY-FUNCTIONAL THEORY AND LUTTINGER SUM RULE Within the self-energy-functional theory (SFT),8,9,10 the grand potential Ω is considered as a functional of the self-energy: Ωt,U [Σ] = Tr ln t,0 −Σ + FU [Σ] . (6) Here, the trace Tr of a quantity A is defined as TrA ≡ eiωn0 Ak(iωn) where iωn = i(2n+ 1)πT are the fermionic Matsubara frequencies, and the functional FU [Σ] is the Legendre transform of the Luttinger-Ward functional ΦU [G]. The self-energy functional (6) is sta- tionary at the physical self-energy, δΩt,U [Σt,U ]/δΣ = 0, and, if evaluated at the physical self-energy, yields the physical value for the grand potential: Ωt,U [Σt,U ] = Ωt,U ≡ −T ln tr exp(−β(H(t,U)−µN)) where β = 1/T . Comparing with the self-energy functional Ωt′,U [Σ] = Tr ln t′,0 −Σ + FU [Σ] (7) of a reference system with the same interaction but a modified one-particle part, i.e. with the Hamiltonian H(t′,U), the not explicitly known but only U -dependent functional FU [Σ] can be eliminated: Ωt,U [Σ] = Ωt′,U [Σ] + Tr ln t,0 −Σ − Tr ln t′,0 −Σ An approximation is constructed by searching for a sta- tionary point of the self-energy functional on the sub- space of trial self-energies spanned by varying the one- particle parameters t′: ∂Ωt,U [Σt′,U ] = 0 . (9) Inserting a trial self-energy into Eq. (8) yields Ωt,U [Σt′,U ] = Ωt′,U +Tr ln t,0 −Σt′,U − Tr lnGt′,U . The decisive point is that the r.h.s. can be evaluated ex- actly for a reference system which is exactly solvable. Apart from the free Green’s function Gt,0, it involves quantities of the reference system only. This strategy to generate approximations has several advantages: (i) Contrary to the usual conserving approxi- mations, the exact functional form of Ωt,U [Σ] is retained. Any approximation is therefore non-perturbative by con- struction. On the level of one-particle excitations, macro- scopic conservation laws are respected as shown in Ap- pendix A. (ii) With Ωt,U [Σt′,U ] evaluated at the station- ary point t′ = t′s, an approximate but explicit expression for a thermodynamical potential is provided. As all phys- ical quantities derive from this potential, the approxima- tion is thermodynamically consistent in itself (see Ref. 11 for details). (iii) As different reference systems gen- erate different approximations, the SFT provides a uni- fying framework that systematizes a class of “dynamic” approximations (see Refs. 29,38 for a discussion). In the following we discuss the question whether or not a dynamic approximation respects the Luttinger sum rule. For this purpose consider first the Tr ln(· · · ) terms in Eq. (10). These can be evaluated using the ana- lytical and causal properties of the Green’s functions as described in Ref. 9 (see Eq. (4) therein). Using −T ln(1 + exp(−ω/T )) → ωΘ(−ω) for T → 0 yields: Tr ln t,0 −Σt′,U (ωm(k)− µ)Θ(µ− ωm(k)) (ζn(k)− µ)Θ(µ− ζn(k)) . (11) Analogously, we have Tr lnGt′,U = 2 (ω′m(k)− µ)Θ(µ− ω m(k)) (ζn(k)− µ)Θ(µ− ζn(k)) . Note that the reference system is always assumed to be in the same macroscopic state as the original system, i.e. it is considered at the same temperature and, more importantly here, at the same chemical potential µ. Fur- thermore, it has been used that, by construction of the approximation, the self-energy and hence its poles at ζn(k) − µ are the same for both, the original and the reference system. This implies that the second terms on the r.h.s. of Eq. (11) and (12), respectively, cancel each other in Eq. (10). Finally, a (large but) finite system (L < ∞) and a finite reference system are considered. Hence, the set of poles of the Green’s function and of the self-energy as well as sums over k are discrete and finite. Taking the µ derivative on both sides of Eq. (10) then yields: ∂Ωt,U [Σt′,U ] ∂Ωt′,U Θ(µ− ωm(k)) Θ(µ− ω′m(k)) . (13) Here we have assumed the ground state of the refer- ence system to be non-degenerate with respect to the particle number. From the (zero-temperature) Lehmann representation39 it is then obvious that, within a sub- space of fixed particle number, the µ-dependence of the Green’s function is the same as its ω-dependence, i.e. G(ω) = G̃(ω+µ) with a µ-independent function G̃. Via the Dyson equation of the reference system, this prop- erty can also be inferred for the self-energy and, via the Dyson equation of the original system, for the (approx- imate) Green’s function of the original system. Conse- quently, the poles of (G−1 t,0 −Σt′,U ) −1 and of Gt′,U are linearly dependent on µ, i.e. ωm(k) and ω m(k) in Eqs. (11) and (12) are independent of µ. We once more exploit the fact that the self-energy of the original system is identified with the self-energy of the reference system. Using Eq. (4) one immediately arrives 〈N〉 = 〈N〉′ + 2 Θ(Gk(0))− 2 (0)) . (14) This is the final result: The Luttinger sum rule for the original system, Eq. (2), is satisfied if and only if it is satisfied for the reference system, i.e. if 〈N〉′ = (0)). A few remarks are in order. For the reference sys- tem, the status of the Luttinger sum rule is that of a general theorem (as long as the general proof is valid); 〈N〉′ and G′ (0) represent exact quantities. The above derivation shows that the theorem is “propagated” to the original system irrespective of the approximation that is constructed within the SFT. This propagation also works in the opposite direction. Namely, a possible violation of the exact sum rule for the reference system would im- ply a violation of the sum rule, expressed in terms of approximate quantities, for the original system. Eq. (14) holds for any choice of t′. Note, however, that stationarity with respect to the variational parameters t′ is essential for the thermodynamical consistency of the approximation. In particular, consistency means that the average particle number 〈N〉 = −∂Ωt,U [Σt′,U ]/∂µ on the l.h.s. can be obtained as the trace of the Green’s function. Stationarity is thus necessary to get the sum rule in the form (5). There are no problems to take the thermodynamic limit (if desired) on both sides of Eq. (14) (after divi- sion of both sides by the number of sites L). The k sums turn into integrals over the unit cell of the reciprocal lat- tice. For a D-dimensional lattice the D − 1-dimensional manifolds of k points with Gk(0) = ∞ or Gk(0) = 0 form Fermi or Luttinger surfaces, respectively. For the above derivation, translational symmetry has been assumed for both, the original as well as the refer- ence system. Nothing, however, prevents us from repeat- ing the derivation in case of systems with reduced (or completely absent) translational symmetries. One sim- ply has to re-interprete the wave vector k as an index which, combined with m, refers to the elements of the diagonalized Green’s function matrix G. The exact sum rule, Eq. (5), generalizes accordingly. The result (14) re- mains valid (with the correct interpretation of k) for an original system with reduced translational symmetries. It is also valid for the case of a translationally symmet- ric original Hamiltonian where, due to the choice of a reference system with reduced translational symmetries, the symmetries of the (approximate) Green’s function of the original system are (artificially) reduced. A typical example is the variational cluster approximation (VCA) where the reference system consists of isolated clusters of finite size. IV. TWO-SITE DYNAMICAL-IMPURITY APPROXIMATION While the Hartree-Fock approximation may be con- sidered as the most simple weak-coupling Φ-derivable approximation, the most simple non-perturbative Φ- derivable approximation is given by the dynamical- impurity approximation (DIA). This shall be demon- strated in the following for the single-band Hubbard model (1) as the original system to be investigated. The DIA is generated by a reference system consisting of a decoupled set of single-impurity Anderson models with a finite number of sites ns and is known 8 to recover the dynamical mean-field theory in the limit ns → ∞. As long as the Luttinger sum rule holds for the single- impurity reference system, the DIA must yield a one- particle Green’s function and a self-energy respecting the sum rule. The Hamiltonian of the reference system is H(t′,U) =∑L i=1 H i with H ′i = iσciσ + niσni−σ aikσ + ciσ + h.c.) . For a homogeneous phase, the variational parameters ′ = ({ε 0 , ε }) can be assumed to be indepen- dent of the site index i: ε0 ≡ ε 0 , εk ≡ ε , Vk ≡ V For the sake of simplicity, we consider the two-site DIA (ns = 2), i.e. a single bath site per correlated site only. In this case there are three independent variational pa- rameters only: the on-site energies of the correlated and of the bath site, ε0 and εc ≡ εk=2, respectively, as well as the hybridization strength V ≡ Vk=2. As the reference 0 0.2 0.4 0.6 0.8 1 filling n U=W=4 FIG. 2: Filling dependence of the variational parameters at their respective optimized values and of the chemical po- tential. Calculations for the Hubbard model with a semi- elliptical free density of states of band width W = 4 and interaction strength U = W = 4 using the two-site DIA. system consists of replicated identical impurity models which are spatially decoupled, the trial self-energy is lo- cal and site-independent, Σij(ω) = δijΣ(ω). Calculations have been performed for the Hub- bard model with a one-particle dispersion ε(k) = e−ik(Ri−Rj)tij such that the density of one- particle energies D(ε) is semi-elliptic. For |ε| ≤ W/2, D(ε) = δ(ε− ε(k)) = (W/2)2 − ε2 . (16) The free band width is set to W = 4. This serves as the energy scale. The computation of the SFT grand potential is per- formed as described in Ref. 9. Stationary points of the resulting function Ω(ε0, εc, V ) ≡ Ωt,U [Σε0,εc,V ] are ob- tained via iterated linearizations of its gradient. There is a unique non-trivial stationary point (with V 6= 0). Fig. 2 shows the variational parameters at this point as functions of the filling n. For the entire range of fillings, the ground state of the reference system lies in the in- variant subspace with Ntot = iσciσ + a iσaiσ) = 2. The parameters as well as the chemical potential are smooth functions of n. We have checked that the ther- modynamical consistency condition n = −L−1∂Ω/∂µ =∫ 0 ρ(ω)dω is satisfied within numerical accuracy. Here ρ(ω) = D(ω + µ− Σ(ω)) (17) is the interacting local density of states (DOS). At half-filling the values of the optimized on-site en- ergies are consistent with particle-hole symmetry. With ε0 − µ = −U/2 and εc − µ = 0 the reference system is in the Kondo regime with a well-formed local moment at the correlated site. The finite hybridization strength V leads, for U = W , to a finite DOS ρ(ω = 0) > 0 and thus 0 0.2 0.4 0.6 0.8 1 filling n 2S-DMFT two-site DIA FIG. 3: Quasi-particle weight z as a function of the filling within the two-site DIA (full lines) and the two-site DMFT40 (dashed lines). Calculations for U = W and U = 2W . to a metallic Fermi liquid as it is expected for the Hub- bard model within a (dynamical) mean-field description. Due to the simple structure of the self-energy generated by the two-site reference system, however, quasi-particle damping effects are missing. Decreasing the filling from n = 1 to n = 0 drives the reference system more and more out of the Kondo regime. While εc stays close to the chemical potential, the on-site energy of the correlated site ε0 crosses µ close to quar- ter filling and lies above µ eventually. Note that ε0 = 0 within the DMFT, i.e. for ns → ∞, while for finite ns there is a clear deviation from ε0 = 0 which is necessary to ensure thermodynamical consistency. For fillings very close to n = 0, the grand potential Ωt,U [Σε0,εc,V ] be- comes almost independent of Σ. This implies that it be- comes increasingly difficult to locate the stationary point with the numerical algorithm used. The slight upturn of ε0 below n = 0.01 (see Fig. 2) might be a numerical ar- tifact. It is instructive to compare the parameters with those of the two-site DMFT (2S-DMFT).40 The 2S-DMFT is a simplified version of the DMFT where a mapping onto the two-site single impurity Anderson model is achieved by means of a simplified self-consistency equation. As- suming ε0 = 0 as in the full DMFT, there are two pa- rameters left (εc and V ) which are fixed by considering the first non-trivial order in the low- and in the high- frequency expansion of the self-energy and the Green’s function in the DMFT self-consistency equation. Al- though being well motivated, this approximation is es- sentially ad hoc. One therefore has to expect that the 2S- DMFT is thermodynamically inconsistent and exhibits a violation of Luttinger’s sum rule. A comparison of the DIA for ns = 2 with the 2S-DMFT is thus ideally suited to demonstrate the advantages gained by constructing approximations within the variational framework of the First of all, there are differences in fact. At half-filling the 2S-DMFT predicts the hybridization to be somewhat larger than the two-site DIA while the value for εc is again fixed by particle-hole symmetry. Deviations grow with decreasing filling. Contrary to the two-site DIA, V monotonously increases and is larger in the entire filling range, ε0 = 0 by construction, and εc even diverges for n → 0 within the 2S-DMFT (see Ref. 40). On the other hand, the system is essentially uncorrelated in the limit n → 0. Strong differences in the parameters, which en- ter the self-energy only, therefore do not necessarily im- ply strongly different physical quantities. This is demon- strated by Fig. 3 which shows the quasi-particle weight calculated via dΣ(ω = 0) as a function of the filling. While there are obvious differ- ences when comparing the results from the two-site DIA with those of the 2S-DMFT, the qualitative trend of z is very similar in both approximations. Both approxima- tions also compare well with the full DMFT: There is a quadratic behavior of z(n) for n → 1 in the Fermi-liquid phase (U = W ) and a linear trend when approaching the Mott phase (U = 2W ). The critical interaction strength for the Mott transition is found to be Uc ≈ 1.46W for the two-site DIA and Uc = 1.5W within the 2S-DMFT. For details on the Mott transition see Refs. 9,40. In case of a local and site-independent self-energy, the Luttinger sum rule can be written in the form41 µ = µ0 +Σ(ω = 0) , (19) where µ0 is the chemical potential of the free (U = 0) system at the same particle density. Eq. (19) implies that not only the enclosed volume but also the shape of the Fermi surface remains unchanged when switching on the interaction. Using Eq. (17) this immediately implies41 ρ(0) = D(µ0) = ρ0(0) , (20) i.e., in case of a correlated metal, the value of the inter- acting local density of states at ω = 0 is independent of U and thus fixed to the value of the density of states of the non-interacting system at the same filling. The interacting and the non-interacting DOS are plot- ted in Fig. 4 for different fillings and for U = W and U = 2W . The impurity self-energy of the two-site ref- erence system is an analytical function of ω except for two first-order poles on the real axis. Via Eq. (17) this two-pole structure implies that the DOS consists of three peaks the form of which is essentially given by the non- interacting DOS. At half-filling the three peaks are eas- ily identified as the lower and the upper Hubbard band and the quasi-particle resonance as it is characteristic for a (dynamical) mean-field description.27 For U = W the resonance still has a significant weight. The weight decreases upon approaching the critical interaction, and the resonance has disappeared in the Mott insulator for -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 8 10 n=1.0 n=0.75 n=0.5 n=0.25 n=0.0 FIG. 4: Interacting local density of states ρ(ω) (solid lines) for different fillings as indicated. Calculations using the two- site DIA for U = W (left) and U = 2W (right). For n = 0.25, n = 0.5 and n = 0.75 the non-interacting DOS ρ0(ω) is shown for comparison (dashed lines). Note that ρ(0) = ρ0(0). The dotted line for U = 2W in the top panel is the DOS for n = 0.99. U = 2W . Hole doping of the Mott insulator is accom- plished by the reappearance of the resonance at ω = 0 which preempts the creation of holes in the lower Hub- bard band.42 As can be seen in the spectrum for n = 0.99 in the top panel (dotted line), the quasi-particle res- onance appears within the Mott-Hubbard gap. With decreasing filling, the upper Hubbard band gradually shifts to higher excitation energies and loses weight. This weight is transfered to the low-energy part of the spec- trum. For lower fillings where the Kondo regime has been left, one would actually expect that the quasi-particle res- onance disappears by merging with the lower Hubbard band. This, however, cannot be described with the sim- ple two-pole structure of the self-energy. One therefore should interprete the gap around ω = −1 at n = 0.25 as an artifact of the approximation. Furthermore, the widths of the Hubbard bands is considerably underes- timated as damping effects are missing completely. The filling-dependent spectral-weight transfer across the Hub- bard gap as well as the energy positions of the main peaks, however, are in overall agreement with general expectations.34,43 It is worth emphasizing that this simple two-site 0 0.2 0.4 0.6 0.8 1 -0.02 filling n Hubbard-I 2S-DMFT two-site DIA FIG. 5: Numerical results for the difference between the volume enclosed by the Fermi surface VFS and the filling n as a function of n for U = W = 4. The Luttinger sum rule (VFS− n = 0) is exactly respected by the two-site DIA. Results for the 2S-DMFT and the Hubbard-I approximation are shown for comparison. Dashed line: difference between the filling n and the average occupation of the correlated (impurity) site in the reference system at stationarity for the two-site DIA. dynamical-impurity approximation exactly fulfills the Luttinger sum rule. In Fig. 4 this can be seen by compar- ing with the DOS of the non-interacting system (dashed lines). The non-interacting DOS cuts the interacting one at ω = 0 which shows that Eq. (20) is satisfied. Note that this is trivial for n = 1 as this is already enforced by particle-hole symmetry. Off half-filling, however, the pinning of the DOS to its non-interacting value at ω = 0 is a consequence of Φ-derivability and thereby a highly non-trivial feature. In contrast, the 2S-DMFT does show a violation of Luttinger’s sum rule which, however, must be attributed to the ad hoc nature of the approximation. Fig. 5 shows 0 0.2 0.4 0.6 0.8 1 filling n 2S-DMFT two-site DIA FIG. 6: Filling dependence of the compressibility κ for U = W as obtained within the 2S-DMFT via κ = ∂n/∂µ (solid line) and via a general Fermi-liquid relation (Eq. (22), dashed line). Using the two-site DIA identical results are obtained for both cases. the difference between the volume enclosed by the Fermi surface VFS = Θ(µ−ε(k)−Σ(0)) = 2 dεD(ε+µ−Σ(0)) and the filling n as a function of the filling. As can be seen, there is an artificial violation of the sum rule for the 2S-DMFT which is of the order of a few per cent while for the Φ-derivable two-site DIA the sum rule is fully re- spected. Note that, unlike the DMFT and also unlike the simplified 2S-DMFT, the two-site DIA predicts a fill- ing which slightly differs from the average occupation of the correlated impurity site in the reference system (see dashed line in Fig. 5). For a finite number of bath sites ns this appears to be necessary to fulfill the Luttinger sum rule. The figure also shows the result obtained within the Hubbard-I approximation.34 Here a very strong (ar- tificial) violation of up to 100 % (for n close to half-filling) is obtained. This should be considered as a strong draw- back which is typical for uncontrolled mean-field approx- imations. There are more relations which, analogously to the Luttinger sum rule, can be derived by means of per- turbation theory to all orders6 in the exact theory and which are respected by weak-coupling conserving approx- imations. For example, the compressibility, defined as κ = ∂n/∂µ, can be shown to be related to the interact- ing DOS and the self-energy at the Fermi edge via κ = 2ρ(0) ∂Σ(0) . (22) Fig. 6 shows that for the 2S-DMFT it makes a difference whether κ is calculated as the µ-derivative of the filling or via Eq. (22). Again, this must be attributed to the fact that the 2S-DMFT is not a Φ-derivable approxima- tion. Contrary, the two-site DIA does respect the general Fermi-liquid property (22) and thus yields the same re- sult in both cases (see Fig. 6). V. VIOLATION OF LUTTINGER’S SUM RULE IN FINITE SYSTEMS The preceding section has demonstrated that the two- site DIA satisfies the Luttinger sum rule. According to Eq. (14), we can conclude that the Luttinger sum rule must hold for the corresponding reference system, i.e. for the two-site single-impurity Anderson model. Of course, this can be verified more directly by evaluating Eq. (5). In case of a finite system or a system with reduced trans- lational symmetries, the Green’s function is a matrix with elements Gαβ(ω) where α refers to the one-particle basis states, and the Luttinger sum rule reads: α(k)m Θ(µ−ω m ) = Θ(µ−ω(k)m )− Θ(µ− ζ(k)n ) . Here the index k labels the elements of the diagonalized Green’s function, i.e. Eq. (5) is generalized by replacing (k, σ) → k. In case of an impurity model, Eq. (23) ac- tually represents the Friedel sum rule.44,45 For the two- site single-impurity Anderson model, the different one- particle excitation energies ω m − µ, the zeros of the Green’s function ζ n − µ and the weights α m are eas- ily determined by full diagonalization. We find that Eq. (23) is satisfied in the entire parameter space (except for V = 0, see below). Note that a violation of the sum rule occurs when, as a function of a model parameter x, a zero of the Green’s function crosses ω = 0 for x = xc. At xc the number of negative zeros counted by the second term on the r.h.s. changes by one while the first term as well as the l.h.s. remain constant since (unlike a pole) a zero of the Green’s function is generically not connected with a change of the ground state (level crossing). This implies that the sum rule would be violated for x < xc or for x > xc. The case V = 0 is exceptional. Within the two-site DIA this corresponds to the Mott insulator (see Fig. 4, topmost panel for U = 2W ). For V = 0 the reference system consists of two decoupled sites, and the Green’s function becomes diagonal in the site index. There is no zero of the local Green’s function corresponding to the uncorrelated site. We can thus concentrate on the correlated site where the local Green’s function exhibits a zero at η−µ = ε0+U/2. In the sector with one electron at the correlated site (ε0 < µ < ε0 + U), the second term on the r.h.s. changes by two at µ = µc = ε0 + U/2 because of the two-fold degenerate ground state. In this case Luttinger’s sum rule in the form (23) is violated for µ < µc and for µ > µc. This “violation”, however, is a trivial one which immediately disappears if the ground- state degeneracy is lifted by applying a weak field term, for example. Fig. 7 shows a phase diagram of the single-impurity Anderson model with ns = 4 sites as obtained by full diagonalization. The diagram covers the entire range of the total particle number N = 〈c†σcσ〉 +∑ k=2〈a akσ〉 from N = 0 to N = 2ns = 8. A non- degenerate ground state is enforced by applying a small but finite magnetic field. No violation of the Luttinger sum rule is found. We have repeated the same calculation also for ns = 10 using the Lanczos technique. 46 Again, the sum rule is found to be always satisfied (We have performed calculations for different U and bath parame- ters). This might have been expected as the (ns → ∞) Anderson model can generally be classified as a (local) Fermi liquid.47 The situation is less clear in the case of correlated lat- tice models such as the Hubbard or the t-J model. For two dimensions there are several numerical studies using high-temperature expansion,30 quantum Monte-Carlo,31 extended DMFT,32,48 and dynamical cluster approxima- tion (DCA)33 which indicate a violation in the strongly correlated metallic phase close to half-filling. For studies of large clusters or studies directly working in the thermo- -1 0 1 2 3 SIAM n =4s FIG. 7: Phase diagram µ vs. ε of the single-impurity An- derson model with ns = 4 sites. Total particle numbers are indicated by Roman figures. Results have been obtained by full diagonalization for the following model parameters. One- particle energies: ε0 = 0 (correlated site), εk = ε + (k − 3) with k = 2, 3, 4 (uncorrelated bath sites). Hubbard interac- tion: U = 2ε. Hybridization strength: Vk = 0.1 for k = 2, 3, 4. To lift Kramers degeneracy in case of an odd particle number, a weak (ferromagnetic) field of strength b = 0.001 is coupled to the local spins. The dashed line marks the particle-hole symmetric case. Luttinger’s sum rule is found to be satisfied in the entire parameter space. dynamic limit, a definite conclusion on the validity of the sum rule is difficult to obtain as finite-temperature or ar- tificial broadening effects etc. must be controlled numer- ically. Contrary, full diagonalization of Hubbard clusters consisting of a few sites only can provide exact results. While their direct relevance for the thermodynamic limit is less clear, it is important to note that reference sys- tems with a finite number of sites or a finite number of correlated sites provide the basis for a number of cluster approaches within the SFT framework. Via Eq. (14) their properties are transferred to the approximate treatment of lattice models in the thermodynamic limit. The validity of Eq. (23) has been checked for Hubbard clusters of different size and in different geometries. The µ vs. U phase diagram for an L = 4-site open Hubbard chain with nearest-neighbor hopping in Fig. 8 shows a representative example. Again, a small but finite field term is added to avoid a ground-state degeneracy. As the chemical potential, for fixed U , is moved off the particle- hole symmetric point µ = U/2 and exceeds certain criti- cal values (red lines), the particle number N [as obtained from the l.h.s. of Eq. (23)] changes from N = L down to (up to) N = 0 (N = 2L). A critical µ value indicates a change of the ground state (level crossing) that is accom- panied by a change of the ground-state particle number. In the one-particle Green’s function this is characterized by a pole ω m −µ crossing ω = 0. The blue lines indicate -5 0 5 10 15 Hubbard L=4 FIG. 8: Phase diagram µ vs. U of the Hubbard model with L = 4 sites (open chain) as obtained by full diagonalization. Nearest-neighbor hopping t = −1. A weak (ferromagnetic) field of strength b = 0.01 is applied to lift Kramers degener- acy. The dashed line marks the particle-hole symmetric case. Particle numbers (l.h.s. of Eq. (23)) are indicated by Roman figures. R.h.s. of Eq. (23): Arabic figures. Luttinger’s sum rule is found to be violated for sufficiently strong U . Uc1, Uc,2: critical interactions. those chemical potentials at which a zero of the Green’s function ζ n − µ crosses ω = 0. Whenever this happens the r.h.s. of Eq. (23) changes while the l.h.s. is constant. Fig. 8 shows that this occurs several times in the N = L sector. At the particle-hole symmetric point µ = U/2 the Luttinger sum rule is obeyed while it is violated in a wide region of the parameter space corresponding to half- filling N = L. However, a critical interaction strength Uc turns out to be necessary. The value for Uc strongly varies for different cluster sizes and geometries but has always been found to be positive and finite. Note that for L = 4 the sum rule is fulfilled for any particle num- ber N 6= L. Qualitatively similar results can be found for the L = 2-site Hubbard cluster where calculations can be done even analytically. Again, a violation of the sum rule is found in the half-filled sector beyond a certain critical This has already been noticed by Rosch49 and was used in combination with a strong-coupling expansion to ar- gue that a violation of the sum rule generically occurs for a Mott insulator. Stanescu et al.50 have shown quite generally that the sum rule is fulfilled when particle-hole symmetry is present (the Luttinger surface is the same as the Fermi surface of the non-interacting system) but violated in the Mott insulator away from particle-hole symmetry. It is interesting to note that these arguments cannot be used to construct a violation of the sum rule within DMFT or for a single-impurity Anderson model: For an (almost) particle-hole symmetric case and model parameters describing a Mott insulator (within DMFT), -4 -2 0 2 4 6 8 Luttinger sum rule particle number U=16t FIG. 9: Ground-state particle number (red Roman figures, l.h.s. of Eq. (23)) and prediction by the Luttinger sum rule (blue Arabic figures, r.h.s. of Eq. (23)) as functions of the chemical potential for a L = 9-site Hubbard cluster with pe- riodic boundary conditions. Arabic numbers are only given when different from Roman ones. Calculations using the Lanczos method and a finite but small magnetic field and finite but small on-site potentials to lift ground-state degen- eracies. an odd number of sites ns (with ns → ∞) must be consid- ered and thus a magnetic field is needed to lift Kramers degeneracy. Even an infinitesimal field, however, leads (at zero temperature) to a finite and even large polar- ization corresponding a well-formed but unscreened local moment. This polarization is incomplete for any finite U as the DMFT predicts a small but finite double occu- pancy for a Mott insulator. Still there is a proximity to the fully polarized band insulator which finally results in a weakly correlated state and thus in a situation which is unlikely to show a violation of the sum rule. We have also considered Hubbard clusters with L = 9 and L = 10 sites by using the Lanczos technique.46 Calculations have been performed for different Lanczos depths lmax to ensure that the results are independent of lmax. Fig. 9 displays an example for L = 9 and a highly symmetric cluster geometry with periodic bound- ary conditions and a well-defined reciprocal space. To lift ground-state degeneracies resulting from spatial symme- tries as well as the Kramers degeneracy, small but fi- nite on-site potentials and a small magnetic-field term are included in the cluster Hamiltonian. Fig. 10 shows an example for L = 10 sites without any spatial sym- metries. Kramers degeneracy for odd N is removed by applying a small magnetic field. With the figures we com- pare the expressions on the left-hand and the right-hand side of Eq. (23). Obviously, the sum rule is respected in -4 -2 0 2 4 6 8 Luttinger sum rule particle number U=16t FIG. 10: The same as Fig. 9 but for 10 sites. most cases. Violations are seen for half-filling N = L, i.e. in the “Mott-insulating phase”, which is consistent with Ref. 49. However, the sum rule is also violated in the “metallic phase” close to half-filling, namely for N = L − 1 (Fig. 9, L = 9) and N = L − 1, L − 2 (Fig. 10, L = 10). This nicely corresponds to the generally observed trend30,31,32,33,48 for violations in the slightly doped metallic regime. We have also verified that the sum rule is restored by lowering U . Fig. 9 and 10 demonstrate that the sum rule is violated in the whole µ range corresponding to N = L − 1. This is an important point as it shows that it is irrelevant whether the T = 0 limit is approached by holding 〈N〉 fixed and adjusting µ = µ(T ) or by fixing µ and let 〈N〉 = 〈N〉(T ) be T -dependent. A violation of the sum rule is found in both cases. Kokalj and Prelovs̆ek51 have demonstrated that viola- tions of the sum rule can also be found for the t-J model on a finite number of sites. Our result provides an ex- plicit example showing that not only for t-J51 but also for Hubbard clusters a violation can be found when the chemical potential is set to µ = limT→0 µ(T ) with µ(T ) obtained for given 〈N〉 = const. Anyway, the original proof5 does not depend on this choice for µ but appears to work for any µ. The results raise the question which assumptions used in the original proof of the theorem are violated or where the proof breaks down. Note that the recently proposed alternative topological proof52 assumes a Fermi-liquid state from the very beginning and thus cannot be applied to a finite system. Using weak symmetry-breaking fields, a more or less trivial breakdown due to ground-state de- generacy has been excluded. An analysis of the ground state of the L = 2 and L = 4 Hubbard clusters which are accessible with exact (analytical or numerical) meth- ods has shown that, for model parameters where the sum rule is violated, the interacting ground state can never- theless be adiabatically connected to the non-interacting one. This excludes level crossing as a potential cause for the breakdown. While we cannot make a definite state- ment, it appears at least plausible that the violation of the sum rule results from a non-commutativity of two limiting processes, the infinite skeleton-diagram expan- sion and the limit T → 0. Using a functional-integral formalism, the Luttinger- Ward functional at finite T can also be constructed in a non-perturbative way, i.e. avoiding an infinite summation of diagrams, as has been shown recently.53 Formally, the Luttinger sum rule can be obtained by exploiting a gauge invariance of the Luttinger-Ward functional [see Ref. 53]: ∂(iωn) ΦU [G(iωn)] = 0 . (24) If at all, this invariance can only be shown for T = 0 where iωn becomes a continuous variable. Unfortunately, the non-perturbative construction of ΦU requires a T > 0 formalism. Hence, the validity of the sum rule depends on question whether the limit T → 0 commutes with the frequency differentiation. Necessary and sufficient condi- tions for this assumption are not easily worked out. An understanding of the main reason for the possible break- down of the sum rule in finite systems, very similar to the case of Mott insulators, is therefore not yet available (see also the discussion in Ref. 49). VI. CONCLUSIONS Φ-derivable approximations are conserving, thermody- namically consistent and, for T = 0, formally respect cer- tain non-trivial theorems such as the Luttinger sum rule. As the construction of the Luttinger-Ward functional Φ is by no means trivial and may conflict with the limit T → 0 or different other limiting processes, however, the validity of the sum rule may be questioned. Violations of the sum rule can be found in fact for the case of strongly correlated electron systems. For Mott insulators and fi- nite systems in particular, a breakdown is documented easily. This implies that a general approximation for the spectrum of one-particle excitations (of the one-particle Green’s function) may violate the sum rule for two pos- sible reasons, namely because (i) the sum rule is violated in the exact theory, or (ii) the approximation generates an artificial violation. Within the usual weak-coupling conserving approxima- tions, such as the fluctuation-exchange approximation, the sum rule always holds as the formal steps in the gen- eral proof of the sum rule can be carried over to the approximation – but with the important simplification of a limited class of diagrams. This also implies that weak-coupling conserving approximations, when applied beyond the weak-coupling regime, might erroneously pre- dict the sum rule to hold. The present paper has focussed on non-perturbative conserving approximations. Non-perturbative approxi- mations, constructed within the framework of the self- energy-functional theory and referring to a certain ref- erence system, are Φ-derivable and consequently respect certain macroscopic conservation laws and are thermo- dynamically consistent. Whether or not the sum rule holds within the approximate approach, however, cannot be answered generally. We found that Luttinger’s sum rule holds within an (SFT) approximation if and only if it holds exactly in the corresponding reference system. The reference system that leads to the most simple but non-trivial example for a non-perturbative conserv- ing approximation consists of a single correlated and a single bath site. For this two-site system, we have found the sum rule to be valid in the entire parameter space. Consequently, the resulting two-site dynamical- impurity approximation (DIA) – opposed to more ad hoc approaches like the two-site DMFT – fully respects the sum rule as could be demonstrated in different ways. In view of the simplicity of the approximation this is a re- markable result. Since the sum rule dictates the low- frequency behavior of the one-particle Green’s function, important mean-field concepts, such as the emergence of a quasi-particle resonance at the Fermi edge, are quali- tatively captured correctly, even away from the particle- hole symmetric case. This qualifies the two-site DIA for a quick but rough estimate of mean-field physics, including phases with spontaneously broken symmetries. Full diagonalization and the Lanczos method have been employed to show that also the single-impurity An- derson model with a finite number of ns > 2 sites respects the sum rule. Consequently, this property is transferred to an ns-site DIA. For ns → ∞ the full dynamical mean- field theory is recovered which is thereby recognized as the prototypical non-perturbative conserving approxima- tion. Clearly, in the case of the DMFT, Φ-derivability is well known27 and obvious, for example, when construct- ing the DMFT with the help of the skeleton-diagram ex- pansion. Using as a trial self-energy the self-energy of a cluster with L > 1 correlated sites, generates an approximation where short-range spatial correlations are included up to the cluster extension. These variational cluster approxi- mations provide a first step beyond the mean-field con- cept. Again, whether or not the sum rule is respected within the VCA depends on the reference system itself. For the L = 2 Hubbard cluster, analytical calculations straightforwardly show that violations of the sum rule occur at half-filling, beyond a certain critical interaction strength. In the thermodynamic limit, this would corre- spond to the Mott-insulating regime. Applying the Lanc- zos method to larger clusters, has shown, however, that a breakdown of the Luttinger sum rule is also possible for fillings off half-filling. For sufficiently strong U , the sum rule is violated in the whole N = L − 1-particle sector. This would correspond to a (strongly correlated) metallic state in the thermodynamic limit. Whether or not a VCA calculation is consistent with the sum rule, then depends on the set of cluster hopping parameters t′ which make the self-energy functional stationary. First VCA calculations54 for the D = 2 Hubbard model at low doping and using clusters with up to L = 10 sites do predict a violation in fact. It is by no means clear a priori what happens in a clus- ter approach using additional bath degrees of freedom as variational parameters, as e.g. in the cellular DMFT.21,22 The usual periodization of the self-consistent C-DMFT self-energy, however, should be avoided when testing the sum rule as this introduces an additional (though physi- cally motivated) approximation. Instead, Eq. (5) must be used with k re-interpreted as an index referring to the el- ements of the self-consistent diagonalized lattice Green’s function. Employing the dynamical cluster approximation (DCA)20 represents an alternative which directly oper- ates in reciprocal space. From a real-space perspective, the DCA is equivalent with the cellular DMFT but ap- plied to a modified model H = H(t,U) → H(t,U) with modified hopping parameters which are invariant under superlattice translations as well as under translations on the cluster.29,55 In the limit L → ∞ the replacement t → t becomes irrelevant. Analogous to the C-DMFT, the sum rule then holds within the DCA if and only if it holds for the individual cluster at self-consistently de- termined cluster parameters. Note, however, that this requires that (besides the DCA self-energy) the modified hopping t instead of the physical hopping has to be con- sidered in the computation of the volume enclosed by the Fermi (Luttinger) surface of the lattice model. This is exactly what is usually done in DCA calculations. Within this context and in view of the violations found for finite Hubbard clusters, it is possible to understand why a non-perturbative cluster approximation, like the VCA,54 or a cluster extension of the DMFT, like the DCA,33 can produce results that are inconsistent with Luttinger’s theorem. Acknowledgments We thank Robert Eder and Achim Rosch for valu- able discussions. The work is supported by the Deutsche Forschungsgemeinschaft within the Forschergruppe FOR APPENDIX A: MACROSCOPIC CONSERVATION OF ENERGY, PARTICLE NUMBER AND SPIN The one-particle Green’s function as obtained within an approximation generated by the choice of a reference system respects the macroscopic conservation laws which result from symmetries of the system with respect to con- tinuous transformation groups: Energy conservation is apparently respected as by con- struction the approximate SFT Green’s function depends on a single frequency only, i.e. is invariant under time translations. Conservation of the total particle number and spin is respected if the approximate G transforms in the same way as the exact Green’s function under global U(1) and SU(2) gauge transformations. Consider a general trans- formation of the form c†α → c with unitary S such that the interaction part H1(U) of the Hamiltonian is invariant (α refers to the states of the one-particle basis). In a diagrammatic approach, the invariance of H1(U) implies that the corresponding conservation law is respected “locally” at each vertex. Hence, for a conserving approximation in the sense of Baym and Kadanoff, the transformation behavior of the free Green’s function is then propagated by the diagram rules to the full Green’s function. Consequently, the lat- ter must transform under S in the same way as the exact G, i.e. Gαβ → Gαβ = . (A2) Consider now the case of the SFT. One has to show that the approximate Green’s function G for the trans- formed system with Hamiltonian H is given by G = † if G is the approximate Green’s function of the model H . Applying the transformation (A1) to H , one finds H = H0(t) +H1(U) → H = H0(t) +H1(U) with t = StS†. Again, S is assumed to leave the interaction part invariant. The Green’s function G of the transformed model is (approximately) constructed via from the free Green’s function of the transformed model and the SFT self-energy which is the self-energy of the reference system H ′ = H0(t s) +H1(U) at the stationary point t For the transformed problem H , the stationary point s is determined from the SFT Euler equation: )ω;αβ = 0 . As an ansatz to solve the Euler equation we take = St′1S † (A5) with t′1 to be determined. The transformation law (A2) for the exact Green’s function of the reference system = GSt′ S†,U = SGt′1,US †. This also holds for the free Green’s function. Using the Dyson equation of the reference system we can deduce Σ = SΣt′ Furthermore, for the free Green’s function of the trans- formed original model we have G t,0 = SGt,0S †. Using these results, we see that Eq. (A4) is equivalent to t,0 −Σt′1,U ∂(Σt′ ,U )ω;αβ = 0 . But this is just the Euler equation for the original model which is solved by t′1 = t s. Remembering the ansatz made, we now have for the stationary point s = St †. Inserting this into Eq. (A3) gives G = S(G−1 t,0 − Σt′s,U ) † = SGS† which is the desired re- sult. Note that one has to ensure that the stationary point for the transformed problem t s = St † lies within the space of one-particle parameters characteristic for the ref- erence system. For models with local interaction part and for local (and also global) gauge transformations, however, this is always easily satisfied. 1 G. Baym and L. P. Kadanoff, Phys. Rev. 124, 287 (1961). 2 G. Baym, Phys. Rev. 127, 1391 (1962). 3 N. E. Bickers, D. J. Scalapino, and S. R. 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0704.0250

2D-MIT as self-doping of a Wigner-Mott insulator

2D-MIT as self-doping of aWigner-Mott insulator S. Pankov ∗ V. Dobrosavljevic National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306 Abstract We consider an interaction-driven scenario for the two-dimensional metal-insulator transition in zero magnetic field (2D-MIT), based on melting the Wigner crystal through vacancy-interstitial pair formation. We show that the transition from the Wigner- Mott insulator to a heavy Fermi liquid emerges as an instability to self-doping, resembling conceptually the solid to normal liquid transition in He3. The resulting physical picture naturally explains many puzzling features of the 2D-MIT. Key words: Strong correlation; disorder; metal-insulator transition; Hubbard model PACS: 71.27.+a, 72.15.Rn, 71.30.+h A series of fascinating experiments, as first performed some ten years ago by Kravchenko and co-workers [1], have deeply changed our thinking about the two dimensional electron gas (2DEG). These and many later experiments demonstrated convincingly that the 2DEG can exhibit typ- ical metallic behavior [2] above a well defined critical den- sity nc. One of the most prominent features of this metallic phase is an unprecedented resistivity drop, which is found only in the low density regime n & nc close to the tran- sition. These findings suggested that a well-defined metal- insulator transition (MIT) may exist even in two dimen- sions, in contrast to long held-beliefs based on the theories for noninteracting disordered electrons. These experiments are typically performed at such low electron density where the relative strength of the Coulomb interaction is so large (rs & 10), that the localization pro- cesses could conceivably be suppressed by interaction ef- fects. A diffusion-mode theory describing such interaction renormalizations at weak disorder has been developed by Finkelshtein and Punnoose [3], suggesting that sufficiently strong interactions may indeed stabilize the metallic phase. However, this theory can provide guidance only within a narrowdiffusive regime restricted to very low temperatures. In contrast, the most striking experimental results have been established in a broad parameter range well outside this regime, and are most pronounced in the cleanest sam- ples [2]. In particular, the best established experimental sig- nature of the transition relies on careful effective mass mea- surements, which is found to diverges as m∗ ∝ (n− nc) ∗ Corresponding author; email: [email protected] 1 1.05 1.1 -1.002 -0.998 -0.996 Fig. 1. Evolution of the free energy profile W [δ], as the electron density is increased across the MIT (from top to bottom). The cusp corresponds to the Wigner solid. The self-doped transition (blue line) takes place before the instability emerges at half-filling (δ = 0, orange line) and the Wigner insulating state is replaced by a heavy Fermi liquid. while the Lande g∗-factor remains largely unrenormalized. Such phenomena cannot be understood within a low-energy diffusion mode theory, which relies on Anderson localiza- tion to produce an insulating phase. A fundamental question is thus posed by these experi- ments: what is the basic mechanism that drives the metal- insulator transition in these systems? Does one have to rely on disorder effects at all in the zero-th order approximation, or can one understand the most important experimental features by interaction effects alone? In this work, we con- centrate solely on the effects of strong Coulomb interactions in the clean limit, and try to establish which experimental Preprint submitted to Elsevier 31 October 2018 http://arxiv.org/abs/0704.0250v1 features can be explained by entirely ignoring the disorder. We approach the transition from the insulating side, starting with the Wigner-Mott insulator, and examine its melting by quantum fluctuations as density increases. These are believed to be dominated [4] by the vacancy- interstitial pair excitations on top of the classical triangu- lar lattice configuration. Within the Wigner crystal, the electrons are tightly bound, and the vacancy-interstitial excitations can be well represented by a mapping to a two- band Hubbard (i.e. charge-transfer) model, respectively corresponding to the lattice and the interstitial electrons. As the density increases, the charge-transfer gap eventu- ally closes, and a metal-insulator transition assumes the character of a Mott-metal-insulator transition, leading to a strongly correlated metallic state on the metallic side. Our model Hamiltonian reads: fiσ + ecc ciσ − iσciσ + c iσfiσ) + fi↓ (1) where f †, f and c†, c are creation and annihilation opera- tors for site and interstitial electrons respectively. It is as- sumed that the inter-cell hopping tij exists only between interstitial orbitals, while only the site electrons are subject to onsite Coulomb repulsion U , and the interstitial orbitals are coupled to the site orbitals via hybridization V . The local electrostatic potentials for the two bands are denoted by ef and ec. Wemodel the strong onsite repulsion by exclusion of dou- ble ocupancy, using the standard slave-boson mean-field formalism. The free energy per electron then reads: W [λ, Z, µ, δ] = ln (1 + exp (−(ε̃lk − µ)/T )) (Z − 1) + µ (2) where εlk are renormalized band energies, λ is the Lagrange multiplier enforcing the slave-boson constraint, Z is the quasiparticle weight, µ is the chemical potential. The chem- ical potential µ is an internal parameter here, arising simi- larly to λ, from constraining the electron density per unit area. The self-doping δ measures deviation in the number of electrons (per elementary cell) from the half filling. In the classical limit of low electron density δ = 0, but it be- comes δ 6= 0 at higher density, where quantum effects are important. This is reminiscent of the liquid solid transition in He3 [5]. The equations of the state are given by the sad- dle point of the free energy W [λ, Z, µ, δ]. A peculiarity of this model is that the band energies are not fixed, but are self-consistently determined through their dependence on the occupation of the site and inter- stitial orbitals. It immediately follows that the system is unstable to the self-doping at the MIT. Indeed, by care- fully accounting for the electrostatic energy balance due to such charge transfer, we find that the free energy takes a lower value (relative to the classical value at δ = 0) on one of the branches at δ 6= 0 before the transition at half filling is found (see Fig. 1). With appropriately chosen model parameters we can ex- plain the basic experimental results, which otherwise can- not be captured by the diffusion mode theory. Similarly to what is seen experimentally, we observe strong renor- malization of the effective mass near the transition m∗ ∼ (n − nc) −1. In this strongly correlated regime any extrin- sic disorder is very effectively screened by interaction ef- fects [7], providing a plausible scenario for the large resis- tivity drop [8]. The magneto-resistance data are also natu- rally explained by our two band model. The experiment [9] has shown that in high parallel magnetic fields the trans- port takes place by activated processes, with an activation gap that vanishes linearly at some density n1 > nc. In the strong field our model reduces to a trivial model of non- interacting spinless electrons. For the density n < n1 the system is a band insulator with the lower band completely filled. The bands broaden as the density increases, and the insulating gap closes linearly at some density n1 > nc. Our static lattice model does not capture the collective charge density fluctuations – the phonons of the Wigner crystal. Because the Coulomb interaction is long ranged, these collective modes are very soft, and play an important role in renormalizing the model parameters. It is this strong renormalization of the charge transfer gap ec − ef that pushes the transition to such low density [6] (rs ≫ 1)), in a fashion that is conceptually very similar to the formation of the Coulomb gap [10]. At present, these effects are incorpo- rated in through the choice of the electrostatic parameters of our model. In future work, we would like to systemati- cally incorporate these soft collective modes, the effects of which are currently included in a semi-phenomenological fashion. This program can be achieved by a variety of meth- ods, including extended dynamical mean field approaches (EDMFT), by exploiting unique properties of the Coulomb potential, and by employing the techniques recently devel- oped in the Coulomb glass context [10]. References [1] S. V. Kravchenko et al., Phys. Rev. B 51, 7038 (1995). [2] For a review, see S. V. Kravchenko and M. P. Sarachik, Rep. Prog. Phys. 67, 1 (2004). [3] A. Punnoose and A. M. Finkelstein, Phys. Rev. Lett. 88, 016802 (2002). [4] B. Tanatar and D. M. Ceperley Phys. Rev. B 39, 5005 (1989). [5] D. Vollhardt, Rev. Mod. Phys. 56, 99 (1984). [6] Z. Lenac and M. Sunjic, Phys. Rev. B 52, 11238 (1995). [7] D. Tanasković, et al., Phys. Rev. Lett. 91, 066603 (2003). [8] M. C. O. Aguiar et al., Europhys. Lett. 67, 226 (2004). [9] J. Jaroszynski et al, Phys. Rev. Lett. 92, 226403 (2004). [10] S. Pankov and V. Dobrosavljević, Phys. Rev. Lett. 94, 046402 (2005). References

0704.0251

Entanglement of Subspaces and Error Correcting Codes

Entanglement of Subspaces and Error Correcting Codes Gilad Gour1, ∗ and Nolan R. Wallach2, † 1Institute for Quantum Information Science and Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 2Department of Mathematics, University of California/San Diego, La Jolla, California 92093-0112 (Dated: November 4, 2018) We introduce the notion of entanglement of subspaces as a measure that quantify the entanglement of bipartite states in a randomly selected subspace. We discuss its properties and in particular we show that for maximally entangled subspaces it is additive. Furthermore, we show that maximally entangled subspaces can play an important role in the study of quantum error correction codes. We discuss both degenerate and non-degenerate codes and show that the subspace spanned by the logical codewords of a non-degenerate code is a k-totally (maximally) entangled subspace. As for non-degenerate codes, we provide a mathematical definition in terms of subspaces and, as an example, we analyze Shor’s nine qubits code in terms of 22 mutually orthogonal subspaces. PACS numbers: 03.67.Mn, 03.67.Hk, 03.65.Ud I. INTRODUCTION AND DEFINITIONS Bipartite entanglement has been recognized as a cru- cial resource for quantum information processing tasks such as teleportation [1] and super dense coding [2]. As a result, in the last years there has been an enormous effort to understand and study the characterization, manipu- lation and quantification of bipartite entanglement [3]. Yet, despite a great deal of progress that was achieved, the theory on mixed bipartite entanglement is incom- plete and a few central important questions such as the additivity of the entanglement of formation [4] remained open. Perhaps the richness and complexity of mixed bi- partite entanglement can be found in the fact that a finite set of measures of entanglement is insufficient to com- pletely quantify it [5]. In this paper we shed some light on mixed bipartite entanglement with the introduction of a new kind of measure of entanglement which we call entanglement of subspaces (EoS). We will see that EoS can play an important role in the study of quantum error correcting codes (QECC). It has been shown recently [6, 7] that geometry of high-dimensional vector spaces can be counterintuitive especially when subspaces with very unique properties are more common than one intuitively expects. That is, roughly speaking, if a high dimensional subspace is se- lected randomly it is quite likely to have strange proper- ties. For example, in [7] it has been demonstrated that a randomly chosen subspace of a bipartite quantum system will likely contain nothing but nearly maximally entan- gled states even if the dimension of the subspace is almost of the same order as the dimension of the original sys- tem. This kind of result has implications, in particular, to super-dense coding [8] and for quantum communication in general (see also [9] for other implications of randomly ∗Electronic address: [email protected] †Electronic address: [email protected] selected subspaces). The quantification of the entangle- ment of such subspaces is therefore very important and we start with its definition. Definition 1. Let HA and HB be finite dimensional Hilbert spaces and let WAB be a subspace of HA ⊗HB. The entanglement of WAB is defined as: ≡ min ψAB∈WAB : ‖ψAB‖ = 1 , (1) where E is the entropy of entanglement of ψAB. Note that if the subspace WAB contains a product state then E(WAB) = 0. On the other hand, if, for example, WAB is orthogonal to a subspace spaned by an unextendible product basis (UPB) [11, 12] then E(WAB) > 0. Claim: Let dA = dimHA and dB = dimHB. If E(WAB) > 0 then dimWAB ≤ (dA − 1)(dB − 1). (2) This claim follows from [10] and also related to the fact that the number of (bipartite) states in a UPB is at least dA + dB − 1 [11]. Note that for two qubits (i.e. dA = dB = 2) E(WAB) can be greater than zero only for one dimensional subspaces. We can use Eq. (1) to define another measure of en- tanglement on bipartite mixed states. Definition 2. Let ρ ∈ B HA ⊗HB be a bipartite mixed state and let SABρ be the support subspace of ρ. Then, the entanglement of the support of ρ is defined as ESupport(ρ) ≡ E(SABρ ) . It can be easily seen that this measure is not continuous and therefore can not be considered as a proper measure of entanglement. Nevertheless, this measure can serve as a mathematical tool to find lower bounds for other mea- sures of entanglement that are more difficult to calculate http://arxiv.org/abs/0704.0251v2 mailto:[email protected] mailto:[email protected] especially in higher dimensions. For example, the entan- glement of the support of ρ provides a lower bound for the entanglement of formation. It can be shown that in lower dimensions the bound is generally not tight. For example, for two qubits in a mixed state ρ, the entangle- ment of the support ESupport(ρ) = 0 (see Eq. (2)). On the other hand, in higher dimensions the bound can be very tight [6, 7]. II. ENTANGLEMENT OF SUBSPACES In this section we study some of the properties of EoS with a focus on additivity properties. The EoS provides a lower bound on the entanglement of formation and our in- terest in its additivity properties is due to one of the most important unresolved questions in quantum information, namely the additivity conjecture for the entanglement of formation. In particular, the additivity question of EoS is identical to the additivity conjecture of quantum channel output entropy [13] that has been shown to be equivalent to the additivity conjecture of entanglement of formation [4]. Thus, additivity properties of EoS can shed some light on this topic. A. Additivity properties of the entanglement of subspaces Here we consider the additivity properties of EoS. We start by showing that if UAB and V A ′B′ are two sub- spaces such that E(UAB) > 0 and/or E(V A′B′) > 0 then E(UAB ⊗ V A′B′) > 0. Consider W = Cn ⊗ Cm. Let ej, j = 1, ..., n be the standard basis of Cn. We will also use the notation fj for the standard basis of Cm. An element of a tensor product of two vector spaces, A and B will be called a product if it is of the form a⊗ b with a ∈ A and b ∈ B. Proposition 1. Let u1, ..., ud, v1, ..., vd ∈ W be such that if x = i bivi is a product then x = 0. If z = i ui ⊗ vi is a product in (Cn ⊗ Cn)⊗ (Cm ⊗ Cm) then z = 0. Proof. We write ui = j=1 ej ⊗ uij and vj = j=1 ej ⊗ vij . Assume that iui⊗vi is a product in (Cn⊗Cn)⊗(Cm⊗Cm). This means that there exists z ∈ Cn⊗Cn and w ∈ Cm⊗Cm such that i,k,l (ek⊗el)⊗(uik⊗vil) = z⊗w. If we write out z = k,lzklek⊗el with zkl ∈ C then we must have uik⊗vil = zklw for all k, l. We now write uik = ikfm and w = mfm⊗wm. The displayed formula now implies that (k.l,m fixed) umikvil = zklwm. This implies that (with k and m fixed) we have umikel⊗vil = ( zklel)⊗wm. Hence ikvi is a product. Our assumption implies that it must be 0. Hence i,k,m umikek⊗fm⊗vi = ui⊗vi. As was to be proved. Note that the proposition above states that if none of the decompositions of a bipartite mixed state, ρ, contain a product state, then also none of the decompositions of ρ ⊗ σ (σ is a bipartite mixed state) contain a prod- uct state. This property is related to the additivity con- jecture [4] for the entanglement of formation (and other measures) and one of the main questions that we will consider here is wether the EoS is additive. That is, does E(UAB ⊗ V A ′B′) = E(UAB) + E(V A ′B′) ? Clearly, if the EoS were additive then the proposition above would have been a trivial consequence of that. However, we were not able to prove the additivity of EoS (in general) although for some special cases it has been tested numerically in [14] and no counter example has been found. The proposition below provides a lower bound. Proposition 2. Let N = min{dimUAB, dimV A′B′}. E(UAB) + E(V A )− logN ≤ E UAB ⊗ V A . (3) The equation above provides a lower bound whereas the upper bound E UAB ⊗ V A′B′ ≤ E(UAB)+E(V A′B′) follows directly from the definition of EoS. Thus, for N = 1 the EoS is additive. Note also that even if N is small (e.g. N = 2), E(UAB) and E(V A′B′) can be arbitrarily large (i.e. depending on dA and dB but not on N). Proof. Let χ be a normalized vector in UAB⊗V A′B′ . We can write χ in its Schmidt decomposition as follows: i ⊗ vA where i pi = 1 (pi ≥ 0) and the uABi ’s (vA i ’s) are orthonormal. Now, from the strong subadditivity of the von-Neumann entropy we have S(ρA′) + S(ρB) ≤ S(ρAB) + S(ρAA′) , where ρA ≡ TrA′BB′χ ⊗ χ∗, ρB ≡ TrAA′B′χ ⊗ χ∗, etc. Now, note that S(ρAA′) = E(χ) and S(ρAB) = H({pi}) ≤ logN , where H({pi}) is the Shanon entropy. Furthermore, note that ρA′ = piωi and ρB = where ωi ≡ TrB′vA i ⊗ vA and σi ≡ TrAuABi ⊗ uABi Hence, since the von-Neumann entropy is concave we S(ρA′) ≥ piS(ωi) = piE(v i ) ≥ E and similarly S(ρB) ≥ E . Combining all this we ≤ logN + E(χ) , for all χ ∈ UAB ⊗ V A′B′ . This complete the proof. B. Maximally entangled subspaces As we have seen above, if N = 1 then the EoS is clearly additive. As we will see in the next subsection, it is also additive for maximally entangled subspaces: Definition 3. Let W be a subspace of HA ⊗ HB and let dA = dimHA and dB = dimHB. W is said to be a maximally entangled subspace in HA ⊗HB if E(W ) = logm , (4) where m ≡ min{dA, dB}. The term maximally entangled subspace have been used in [6, 7] for a subspace W with E(W ) slightly less than logm. In this paper, we will call such subspaces nearly maximally entangled to distinguish from (exactly) maximally entangled subspaces as defined above. In [15] it has been shown that the average entangle- ment of a pure state φ ∈ HA ⊗ HB which is chosen randomly according to the unitarily invariant measure satisfies 〈E(φ)〉 ≥ log2 dA − 2 ln 2dB where without loss of generality dA ≥ dB. Later on, in [6, 7] this result has been extended to subspaces and in particular it has been shown, somewhat surprisingly, that a randomly chosen subspace of bipartite quantum system will likely be a nearly maximally entangled sub- space. Thus, as nearly maximally entangled subspaces are quite common it is important to understand their structure. As a first step in this direction, in the following we study the structure of (exactly) maximally entangled subspaces. Let φ be a state in HA⊗HB. If e1, ..., em is an or- thonormal basis of HB we may write φi⊗ei . We define a dB× dB Hermitian matrix B = [〈φi|φj〉] (i.e. B is the reduced density matrix). Let λ1, ..., λdB be the set of eigenvalues of B counting multiplicity. Then the entanglement of φ is E(φ) = − λi log(λi) . It is easy to show that E(φ) ≤ logm and equality is attained if and only if B = 1 P with P a projection matrix onto a d dimensional subspace of CdB . Clearly this definition of entropy is independent of the choice of basis and could also be given using an orthonormal basis of HA and analyzing the corresponding dA coefficients in HB. Under the condition of equality φ is maximally entangled, and this in particular implies that if dA ≥ dB 〈φi|φj〉 = δij . Proposition 3. Assume that dA ≥ dB and set m = dB . Let UAB be a maximally entangled subspace in HA⊗HB of dimension d. If e1, ..., em is an orthonormal basis of HB then there exist U1, ..., Um subspaces of HA such that 〈Ui|Uj〉 = 0 if i 6= j, dimUj = d > 0 for all j = 1, ...,m and unitary operators Ti : C d → Ui i = 1, ...,m such that UAB = { Tiw⊗ei|w ∈ Cd}. Conversely, if U1, ..., Um are mutually orthogonal sub- spaces of HA such that dimUj = d > 0 for all j = 1, ...,m and we have unitary operators Ti : C d → Ui i = 1, ...,m such that UAB = { Tiw⊗ei|w ∈ Cd} , then UAB is maximally entangled. Proof. Let ψ1, ..., ψd be an orthonomal basis of U Then we can write ψij⊗ei with 〈ψij |ψkj〉 = 1mδik. The condition on U AB is that if a ∈ Cd is a unit vector then ajψj is maximally entangled in HA⊗HB. This implies that ajψlj ajψkj δl,k . Fix l 6= k and let p 6= q ≤ d be two integers. Let a = (a1, ..., ad) with aj = 0 for j 6= p or j 6= q. Set ap = b, aq = c and |b|2 + |c|2 = 1. Then we have 〈bψlp + cψlq|bψkp + cψkq〉 = 0. On the other hand we have 〈bψlp + cψlq|bψkp + cψkq〉 = bc 〈ψlp|ψkq〉+ cb 〈ψlq|ψkp〉 Set z = bc. We look at two cases: first z = 1 (b = c = ) and second z = i√ (b = 1√ , c = i√ ). Thus we have 〈ψlp|ψkq〉+ 〈ψlq|ψkp〉 = 0 for the first case and 〈ψlp|ψkq〉 − 〈ψlq|ψkp〉 = 0 for the second. Hence, 〈ψlp|ψkq〉 = 〈ψlq|ψkp〉 = 0. We set Ul = Span{ψlp|p = 1, ..., d}. Then 〈Ul|Uk〉 = 0 if l 6= k. We now consider what happens when l = k. We first note that taking ap = 1 and all other entries equal to 0 we have 〈ψlp|ψlp〉 = 1m . Now using b, c as above for p 6= q we have 〈ψlp|ψlp〉+ 〈ψlp|ψlq〉+ 〈ψlq|ψlp〉+ 〈ψkp|ψkp〉 = 〈ψlp|ψlp〉+ i 〈ψlp|ψlq〉 − i 〈ψlq|ψlp〉+ 〈ψkp|ψkp〉 = Hence as above we find that 〈ψlp|ψlq〉 = 0 if p 6= q. Thus√ mψl1, ..., mψld is an orthonormal basis of Ul. This implies that the spaces U1, ..., Um have the desired prop- erties. Let u1, ..., ud, be the standard orthonormal basis of Cd and define Tiuj = mψij . With this notation in place UAB has the desired form. The converse is proved by the obvious calculation. Corollary 4. If UAB is a maximally entangled subspace in HA⊗HB (dA ≥ dB), then dimUAB ≤ Furthermore, there always exists a maximally entangled subspace of dimension ⌊dA/dB⌋. Proof. Assume that dimHA = dA ≥ dimHB = dB. According to the first part of Proposition 4, if UAB is a maximally entangled subspace of dimension d then d × dB ≤ dA. On the other hand, if d ≤ ⌊dA/dB⌋ then the second half of the statement implies that there is a maximally entangled subspace of dimension d. In the following we find necessary and sufficient con- ditions for a subspace to be maximally entangled. In section III we use this to show that maximally entan- gled subspaces can play an important role in the study of error correcting codes. As above we consider the space HA⊗HB with dimHA = dA ≥ dimHB = dB and a max- imally entangled subspace UAB ⊂ HA⊗HB. We will also consider End(HB) to be a Hilbert space with inner prod- uct 〈X |Y 〉 = Tr(X†Y ) for any two operators X and Y in End(HB). Proposition 5. Let UAB ⊂ HA⊗HB be a subspace and dA ≥ dB. Then, UAB is maximally entangled if and only if the map End(HB)⊗UAB → HA⊗HB given by X⊗u 7→√ dB(I ⊗X)u is an isometry onto its image. Proof. Let d = dimUAB and let the notation be as in Proposition 3. Thus, if UAB is maximally entangled and if e1, ..., edB is an orthonormal basis of HB then an ele- ment of UAB is of the form T (w) = Ti(w) ⊗ ei , with Ti a unitary operator from C d onto a subspace Ui of HA and Ui and Uj are orthogonal for all i 6= j. We now calculate 〈(I⊗X)T (w)|(I⊗Y )T (z)〉 = 〈Ti(w)|Tj(z)〉 〈Xei|Y ej〉 . Now, since 〈Ti(w)|Tj(z)〉 = δij〈w|z〉 (see Proposition 3) we have: 〈(I ⊗X)T (w)|(I ⊗ Y )T (z)〉 = δij〈w|z〉 〈Xei|Y ej〉 = 〈w|z〉Tr(X†Y ) . That is, we proved that if UAB is maximally entangled then the map is an isometry. For the converse we note that we have an isometry of Cd onto UAB given by T (w) = Ti(w) ⊗ ei . Now, if the map defined in the proposition is an isometry 〈(I ⊗X)T (w)|(I ⊗ Y )T (z)〉 = 〈w|z〉Tr(X†Y ) . That is, 〈Ti(w)|Tj(z)〉 〈Xei|Y ej〉 = 〈w|z〉Tr(X†Y ) , for all X,Y ∈ End(HB). Hence, we must have 〈Ti(w)|Tj(z)〉 = δij〈w|z〉/ and from Proposition 3 the subspace UAB is maximally entangled. C. Additivity of maximally entangled subspaces We now discuss the additivity properties of maximally entangled subspaces. Proposition 6. Let UAB ⊂ HA⊗HB and V A′B′ ⊂ HA′⊗HB′ be maximally entangled subspaces. Then, UAB ⊗ V A = logm+ logm′ , (5) where m ≡ min{dA, dB} and m′ ≡ min{dA′ , dB′}. Remark. From the above proposition it follows that if dA ≥ dB and dA′ ≥ dB′ or dB ≥ dA and dB′ ≥ dA′ then UAB⊗V A ′B′ is maximally entangled in (HA⊗HA′)⊗(HB⊗HB′). However, if for example dA > dB and dA′ < dB′ then U AB⊗V A′B′ is NOT maximally entangled in (HA⊗HA′)⊗(HB⊗HB′) because mm′ < min{dAdA′ , dBdB′}. Proof. There are basically two possibilities (up to inter- changing factors): the first is dA ≥ dB and dA′ ≥ dB′ , and the second is dA ≥ dB and dA′ < dB′ . In the first case we have as in the statement of proposition 3 the subspaces Uj and the unitaries Tj : d → Uj such that UAB = { i Tiw⊗ei|w ∈ Cd}. We also have the orthonormal basis fi of HB , the sub- spaces Vj and the unitaries Sj : C d′ → Vj such that ′B′ = { i Siw ′⊗fi|w′ ∈ Cd ′}. Thus, as a subspace of (HA⊗HA′)⊗(HB⊗HB′), UAB⊗V A′B′ is spanned by the elements (Tiw⊗Sjw′)⊗(ei⊗fj). Thus if we identify Cd⊗Cd′ with Cdd′ then the converse assertion in proposition 3 implies that UAB⊗V A′B′ is a maximally entangled space. This implies that UAB⊗V A = log dB + log dB′ = logm+ logm We now consider the second case. For UAB we have ex- actly as above UAB = { i Tiw⊗ei|w ∈ Cd}. For V A we denote by fj an orthonormal basis of HA (not of HB′ as above). Thus, according to proposition 3 we have ′B′ = { i fi ⊗ Siw′|w′ ∈ Cd ′}. As a subspace of (HA⊗HA′)⊗(HB⊗HB′), UAB⊗V A′B′ is spanned by the elements (Tiw⊗fj)⊗(ei⊗Sjw′). We will assume first that d′ ≤ d. Let w′1, ..., w′d′ be an orthonormal basis of Cd . Thus, if φ is a state in UAB⊗V A′B′ we can write it as i,j,k (Tiwk⊗fj)⊗(ei⊗Sjw′k) , where wk are some non-normalized vectors in C d. Fur- thermore, 〈φ|φ〉 = dBdA′ ‖uk‖2 . Hence, if φ is normalized then ‖uk‖2 = dBdA′ Now, since Sjw k is an orthonormal set of vectors for all j and k (see proposition 3), the entanglement of φ as an element of (HA⊗HA′)⊗(HB⊗HB′) is given by dBdA′S(B) where B = [〈wi|wj〉]1≤i,j≤d′ , and if λ1, ..., λd′ are the eigenvalues of B then the von-Neumann entropy of B is S(B) = − λi logλi . Now B is the most general d′ × d′ self adjoint, positive semidefinate matrix with trace 1/dBdA′ . The minimum of the entropy for such matrices is log(dadB) This proves the proposition for the case d′ ≤ d. If d < d′ then we can prove the proposition by using the same argument, this time with wk an orthonormal basis of C and with B′ = w′i|w′j 1≤i,j≤d. This completes the proof. The above proposition also shows that the entangle- ment of formation is additive for bipartite states with maximally entangled support. If ρ is a mixed state in HA⊗HB then the entanglement of formation is defined in terms of the convex roof extension: EF(ρ) = min piE(φi) where the minimum taken over all decompositions piφi⊗φ∗i with φi a pure bipartite state and pi > 0 and pi = 1. Corollary 7. Let ρ and σ be mixed states in B(HA⊗HB) and B(HA′⊗HB′), respectively. If the support subspaces Sρ and Sσ are maximally entangled then EF(ρ⊗ σ) = EF(ρ) + EF(σ) . The proof of this corollary follows directly from the fact that for states with maximally entangled support EF(ρ) = E(Sρ). Note that the class of mixed states with maximally entangled support is extremely small (i.e. of measure zero). In particular, it is a much smaller class than the one found by Vidal, Dur and Cirac [16] . III. ERROR CORRECTING CODES A. Definitions We consider error correcting codes that are used to en- code l qubits in n ≥ l qubits in such a way that they can correct errors on any subset of k or fewer qubits. These codes, which we call (n, l, k) error correcting codes, can be classified into two classes (for example see [17]): de- generate and non-degenerate (or orthogonal) codes. We start with a general definition of error correcting codes that is equivalent to the definition given (for example) in [17] , but here we define the codes in terms of sub- spaces. Definition 4. Let X ∈ End(⊗kC2) and 0 ≤ i0 < i1 < ... < ik−1 ≤ n − 1. The operator Xi0i1···ik−1 on ⊗nC2, that represents the errors on the k qubits i1, ..., ik−1, is defined by Xi0...ik−1v = σ(X ⊗ I)σ−1v, where (a) σ ∈ Sn (acting on {0, 1, ..., n − 1} by permutations) is defined such that σ(j) = ij , (b) σ can act on ⊗nC2 by σ(v0 ⊗ v1 ⊗ · · · ⊗ vn−1) = vσ(0) ⊗ vσ(1) ⊗ · · · ⊗ vσ(n−1) and (c) ⊗nC2 is viewed as (⊗kC2)⊗ (⊗n−kC2) (putting together the k tensor factors that correspond to the k qubits i1, ..., ik−1 and the rest n− k tensor factors). An (n, l, k) error correcting code is defined from its following ingredients: I. An isometry T : ⊗lC2 → ⊗nC2. II. Let V0 = T (⊗lC2). There are V1, ..., Vd mutually orthogonal subspaces of ⊗nC2 that are also orthogonal to V0. III. For each Vj there is a unitary isomorphism, Uj, of Vj onto V0 with U0 = I. IV. Xi0i1···ik−1V0 ⊂ ⊕dj=0Vj . V. Let Pj be the ortogonal projection of ⊗nC2 onto Vj then if v ∈ V0 is a unit vector and Pj(Xi0i1···ik−1v) 6= 0 Pj(Xi0i1···ik−1v) ∥Pj(Xi0i1···ik−1v) equals v up to a phase. In the next subsection we study Shor’s (9, 1, 1) error correcting code and show that it satisfies this definition. However, before that, let us introduce the notion of k- totally entangled subspaces which will play an important role in our discussion of QECC. Definition 5. Let H be the space of n qubits, ⊗nC2. Corresponding to any choice of k qubits (tensor factors) we can consider H = HA ⊗HB with HA = ⊗n−kC2 and HB = ⊗kC2. For k ≤ n/2 we will say that a subspace, V , of H is k-totally entangled if it is maximally entangled relative to every decomposition of H as above. It is interesting to note that all the subspaces spanned by the logical codewords of the different non-degenerate error correcting codes given in [18, 19, 20] are 2-totally entangled subspaces. On the other hand, the subspaces spanned by the logical codewords of degenerate codes, like Shor’s 9 qubits code, are in general only partially maximally entangled subspaces (i.e. maximally entan- gled for some choices of k qubits but not for all choices). In the following subsections we will see the reason for that. B. Analysis of Shor’s 9 qubits code We start with the following notations. We set u± = (|000〉 ± |111〉) so that the two logical codewords in Shor’s 9 qubit code are v+ = u+ ⊗ u+ ⊗ u+ and v− = u−⊗u−⊗u−. The subspace spanned by these codewords is denoted by V0 = Cv+ ⊕ Cv−. We also denote u0± = (|100〉 ± |011〉)/ 2, u1± = (|010〉 ± |101〉)/ 2 and u2± = (|001〉 ± |110〉)/ Using these notations, we define 21 mutually orthogo- nal 2 dimensional subspaces orthogonal to V0: V1 = Cu− ⊗ u+ ⊗ u+ ⊕ Cu+ ⊗ u− ⊗ u−, V2 = Cu+ ⊗ u− ⊗ u+ ⊕ Cu− ⊗ u+ ⊗ u−, V3 = Cu+ ⊗ u+ ⊗ u− ⊕ Cu− ⊗ u− ⊗ u+, V4+i = Cu +⊗u+⊗u+⊕Cui−⊗u−⊗u−, for i = 0, 1, 2, V7+i = Cu+⊗ui+⊗u+⊕Cu−⊗ui−⊗u−, for i = 0, 1, 2, V10+i = Cu+⊗u+⊗ui+⊕Cu−⊗u−⊗ui−, for i = 0, 1, 2, V13+i = Cu −⊗u+⊗u+⊕Cui+⊗u−⊗u−, for i = 0, 1, 2, V16+i = Cu+⊗ui−⊗u+⊕Cu−⊗ui+⊗u−, for i = 0, 1, 2, V19+i = Cu+⊗u+⊗ui−⊕Cu−⊗u−⊗ui+, for i = 0, 1, 2. If X ∈ End(C2) (linear maps of C2 to C2) then we denote by Xi the linear map of ⊗9C2 to itself that is the tensor product of the identity of C2 in every tensor factor but the i-th and is X in the i-th factor thus X0 = X ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I, X1 = I ⊗X ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I ⊗ I Then we have (here ⌊x⌋ = max{m|m ≤ x,m ∈ Z}) XiV0 ⊂ V0 ⊕ V⌊i/3⌋+1 ⊕ Vi+4 ⊕ Vi+13, 0 ≤ i ≤ 8. We choose an observable R with R|Vi = λiI, 0 ≤ i ≤ 21 R|W = µI, where W is the orthogonal complement of ⊕21i=0Vi and λi 6= λj for i 6= j and λi 6= µ for any i. We define a unitary operator Uj : Vj → V0 as follows: we denote the Pauli matrices by , A2 = , A3 = and then define U0 = I, Ui = (A1)3i−1, for i = 1, 2, 3, Ui = (A2)i−4, for 4 ≤ i ≤ 12 and Ui = (A3)i−13, for 13 ≤ i ≤ 21. This gives an one qubit error correcting code since if v ∈ V0 is a state and if we have an error in the i-th position then we will have Xiv ∈ V0 ⊕ V⌊i/3⌋+1 ⊕ Vi+4 ⊕ Vi+13 . Thus, if we measure the observable R on Xiv then the measurement will yield one of λj with j = 0, ⌊i/3⌋+1, i+4 or i + 13 and Xiv will have collapsed up to a phase to Ujv; hence applying Uj will fix the error. Remark. Note that the subspace V0 is not 2-totally en- tangled subspace. Nevertheless, V0 has very special properties. In particular, if we group the 9 qubits as (1, 2, 3) : (4, 5, 6) : (7, 8, 9), then for any choice of 2 qubits that are not from the same group, the subspace V0 is maximally entangled with respect to the decomposition between the 2 qubits and the rest 7 qubits. If the 2 qubits are chosen from the same group then the entanglement of V0 with respect to this decomposition is 1ebit. Thus, out of the 36 different decompositions, with respect to 27 of them E(V0) = 2ebits and with respect to the other 9 decompositions E(V0) = 1ebit. C. Orthogonal codes We now consider a somewhat more intuitive class of codes known as non-degenerate codes which we also name as orthogonal codes. Definition 6. Let A0 = I, A1, A2, A3 the Pauli basis and define A j0j1···jk−1 i0i1···ik−1 to be (Aj0 ⊗Aj1 ⊗ · · · ⊗Ajk−1)i0···ik−1 , where 0 ≤ jr ≤ 3 and 0 ≤ i0 < i1 < ... < ik−1 ≤ n − 1. Let Σ be the set of distinct operators of the form A j0j1···jk−1 i0i1···ik−1 . Then an orthogonal (n, l, k) code is an (n, l, k) error correcting code such that if we label Σ as the set of d + 1 operators S0 = I, S1, ..., Sd then Vj = SjV0. Note that Σ has d+ 1 = elements. Thus, a necessary condition that there exist an (n, l, k) code is the quantum Hamming bound [17]: ≤ 2n−l. Proposition 8. A 2l dimensional subspace V of ⊗nC2 is the V0 of an (n, l, k)-orthoganal error correcting code if and only if V is 2k-totally entangled. Proof. Let V be a 2k-totally entangled subspace in H = ⊗nC2, and let X : ⊗kC2 → ⊗kC2 be a linear map on k qubits. As above, for any i0 < i1 < ... < ik−1 (1 ≤ il ≤ n) we denote by Xi0i1...ik−1 the operation X on H, when acting on the k qubits i0, i1, ...., ik−1 (the rest of the n − k qubits are left ”untouched”). Let also Z ≡ {X ∈ |TrX = 0} and for any i0 < ... < ik−1 let Ui0...ik−1 ≡ {Xi0...ik−1V |X ∈ Z}. We define the subspace W = V + i0<...<ik−1 Ui0...ik−1 . That is, W consists of all the possible states after an error on k or less qubits has been occurred. Now, let A0 = I, A1, A2, A3 be an orthonormal basis of End(C with Ai invertible (e.g. the Pauli basis of 2×2 matrices). As in Definition 6, we denote by A j0...jk−1 i0...ik−1 the operator Xi0...ik−1 that corresponds to X = Aj0 ⊗· · ·⊗Ajk−1 , and the set of all such operators we denote by Σ ≡ {Aj0...jk−1i0...ik−1 |1 ≤ il ≤ n, 0 ≤ jl ≤ 3} . Now, let v1, ..., vd be an orthonormal basis of V and define B ≡ {Svi S ∈ Σ, 1 ≤ i ≤ d}. We now argue that B is an orthonormal basis of W . Clearly, the vectors in B span W . It is therefore enough to show that the vectors in B are orthogonal. Let Svi and S′vj be two vectors in B with S = A j0...jk−1 i0...ik−1 S′ = A ...j′ ...i′ . We denote byHB the Hilbert space of the qubits i0, ..., ik−1 and i 0, ..., i k−1, and by HA the Hilbert space of the rest of the qubits. Note that HB consists of at most 2k qubits. Now, since V is 2k-totally entan- gled subspace, it is maximally entangled relative to the decomposition H = HA ⊗HB. Thus, from Proposition 5 we clearly have 〈Svi|S′vj〉 = 〈vi|vj〉Tr S̃†S̃′ = δijδSS′ , where S ≡ IA ⊗ S̃ and S′ ≡ IA ⊗ S̃′; that is, S̃ and S̃′ are the projections of S and S′ onto HB, respectively. Hence, B is an orthonormal basis of W . Since B is an orthonormal basis we can construct an observable (i.e. Hermition operator) R such that for all v ∈ V R(Sv) = λSSv with all of the λS distinct. We also define R to be zero on the orthogonal complement to W in H. Now, suppose that an element v has been changed by a k-qubit transformation yielding Xi0...ik−1v. We do a measurment of R and since the image is in W the outcome is λS for some S. After the measurment, the quantum state is Sv and so we recover v by applying S−1 (actually S if we used the Pauli basis). The converse fol- lows from the same lines in the opposite direction. This completes the proof. Note that Corollary 4 together with the proposi- tion above is consistent with the quantum Singleton bound [22], n ≥ 4k + l, which also follows trivially from the quantum Hamming bound for the case of orthogonal codes that we considered in this subsection. IV. SUMMERY AND CONCLUSIONS We introduced the notion of entanglement of subspaces as a measure that quantify the entanglement of bipartite states in a randomly selected subspace. We discussed its properties and suggested that it is additive. We were not able to prove this conjecture (which is equivalent to the additivity conjecture of the entanglement of formation) although some numerical tests [14] supports that and for maximally entangled subspaces we proved that it is ad- ditive. We then extended the definition of maximally entangled subspaces into k-totally entangled subspaces and showed that the later can play an important role in the study of quantum error correction codes. We considered both degenerate and non-degenerate codes and showed that the subspace spanned by the log- ical codewords of a non-degenerate code is a k-totally (maximally) entangled subspace. This observation, fol- lowed by an analysis of the degenerate Shor’s nine qubits code in terms of 22 mutually orthogonal subspaces, mo- tivated us to define a general (possible degenerate) error correcting code in terms of subspaces. We believe that further investigation in this direction would lead to a better understanding of degenerate quantum error cor- recting codes. Acknowledgments:— The authors would like to thank Aram Harrow and David Meyer for fruitful discussions. [1] C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres and W. K. Wootters, Phys. Rev. Lett. 70, 1895 (1993). [2] C. H. Bennett and S.J. Wiesner, Phys. Rev. Lett. 69, 2881 (1992). [3] M. B. Plenio, S. Virmani, Quant. Inf. Comp. 7, 1 (2007). [4] P. W. Shor, Commun. Math. Phys. 246, 453 (2004). [5] G. Gour, Phys. Rev. A 72, 022323 (2005). [6] P. Hayden, quant-ph/0409157 [7] P. Hayden, D. W. Leung and A. Winter, Comm. Math. Phys. 265, 95 (2006). [8] A. Abeyesinghe, P. Hayden, G. Smith and Andreas Win- ter, IEEE Trans. Inform. Theory 52, 3635 (2006). [9] P. Shor, Lecture Notes 2002, http://www.msri.org/ pub- lications/ln/msri/2002/quantumcrypto/shor/1/ [10] N. R. Wallach, ”An Unentangled Gleason’s theorem”, Quantum computation and information (Washington, DC, 2000), 291–298,Contemp. Math. 305(2002). [11] C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin and B. M. Terhal, Phys. Rev. Lett. 82, 5385 (1999). [12] D. P. DiVincenzo, T. Mor, P. W. Shor, J. A. Smolin and B. M. Terhal, Comm. Math. Phys. 238, 379 (2003). [13] A. Roy and G. Gour, in preparation. [14] A. W. Harrow, private communication. [15] D. N. Page, Phys. Rev. Lett. 71, 1291 (1993). [16] G. Vidal, W. Dur and J. I. Cirac, Phys. Rev. Lett. 89, 027901 (2002). [17] M. A. Nielsen and I. L. Chuang, ”Quantum Computa- tion and Quantum Information” (Cambridge University Press, 2000). [18] A. M. Steane, Phys. Rev. Lett. 77, 793 (1996); Proc. R. Soc. London A, 452, 2551 (1996). [19] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters, Phys. Rev. A 54, 3824 (1996). [20] R. Laflamme, C. Miquel, J. P. Paz and W. H. Zurek, Phys. Rev. Lett. 77, 198 (1996). [21] P. Shor, Phys. Rev. A 52, 2493 (1995). [22] E. Knill and R. Laflamme, Phys. Rev. A 55, 900 (1997). http://arxiv.org/abs/quant-ph/0409157 http://www.msri.org/

0704.0252

Does the present data on B_s - bar B_s mixing rule out a large enhancement in the branching ratio of B_s --> mu+ mu- ?

Does the present data on Bs − B̄s mixing rule out a large enhan ement in the bran hing ratio of Bs → µ Ashutosh Kumar Alok and S. Uma Sankar Indian Institute of Te hnology, Bombay, Mumbai-400076, India In this letter, we onsider the onstraints imposed by the re ent measurement of Bs− B̄s mixing on the new physi s ontribution to the rare de ay Bs → µ +µ−. New physi s in the form ve tor and axial-ve tor ouplings is already severely onstrained by the data on B → (K,K∗)µ+µ−. Here, we show that Bs−B̄s mixing data, together with the data on K0−K̄0 mixing and KL → µ +µ− de ay rate, strongly onstrain the s alar-pseudos alar ontribution to Bs → µ +µ−. We on lude that new physi s an at best lead to a fa tor of 2 in rease in the bran hing ratio of Bs → µ +µ− ompared to its Standard Model expe tation. The �avour hanging neutral intera tion (FCNI) b → sµ+µ− serves as an important probe to test the Standard Model (SM) and its possible extensions. This four fermion intera tion gives rise to semi-leptoni de ays B → (K,K∗)µ+µ− and also the purely leptoni de ay Bs → µ+µ−. The semi-leptoni de ays B → (K,K∗)µ+µ− have been observed experimentally [1, 2, 3℄ with bran hing ratios lose to their SM predi tions [4, 5, 6℄. At present there is only an upper limit, 1.0×10−7 at 95% C.L., on the bran hing ratio of the de ay Bs → µ +µ− [7, 8℄. The SM predi tion for this bran hing ratio is (3.2±1.5)×10−9 [9℄ or ≤ 7.7×10−9 at 3σ level. Bs → µ +µ− will be one of the important rare B de ays to be studied by the experiments at the up oming Large Hadron Collider (LHC). We expe t that the present upper limit will be redu ed signi� antly in these experiments. A non-zero value of this bran hing ratio is measurable, if it is ≥ 10−8 [10℄. In a previous publi ation [11℄, we studied the onstraints on new physi s ontribution to the bran hing ratio of Bs → µ +µ− oming from the experimentally measured values of the bran hing ratios of B → (K,K∗)µ+µ−. We found that if the new physi s intera tions are in the form of ve tor/axial-ve tor operators, then the present data on B(B → (K,K∗)µ+µ−) does not allow a large boost in B(Bs → µ +µ−). By large boost we mean an enhan ement of at least an order of magnitude in omparison to the SM predi tion. However, if the new http://arxiv.org/abs/0704.0252v1 physi s intera tions are in the form of the s alar/pseudos alar operators, then the presently measured rates of B → (K,K∗)µ+µ− do not put any useful onstraints on Bs → µ +µ− and BNP (Bs → µ +µ−) an be as high as the present experimental upper limit. Therefore we are led to the on lusion that if future experiments measure Bs → µ +µ− with a bran hing ratio greater than 10−8, then the new physi s giving rise to this de ay has to be in the form of s alar/pseudos alar intera tion. Re ently Bs− B̄s mixing has been observed experimentally [13℄, with a very small exper- imental error. In this paper, we want to see what onstraint this measurement imposes on the new physi s ontribution to the bran hing ratio of Bs → µ +µ−. In parti ular, we on- sider the question: Does it allow new physi s in the form of s alar/pseudos alar intera tion to give a large boost in BNP (Bs → µ +µ−) ? We start by onsidering the Bs → µ +µ− de ay. The e�e tive new physi s lagrangian for the quark level transition b̄ → s̄µ+µ− due to s alar/pseudos alar intera tions an arise from tree and/or ele troweak penguin and/or box diagrams. We parametrize it as b̄→s̄µ+µ− = G1 b̄(g S + g P γ5)s µ̄(g S + g P γ5)µ, (1) where G1 is a dimensional fa tor hara terizing the overall s ale of new physi s, with di- mension (mass)−2. This fa tor essentially arises due to the s alar propagator in tree or ele troweak penguin diagrams (or s alar propagators in box diagrams) whi h ouples the quark bilinear to the lepton bilinear. gsbS,P and g S,P are dimensionless numbers, hara - terizing, respe tively, b − s and µ − µ ouplings due to new physi s s alar/pseudos alar intera tions. Ele tromagneti penguins ne essarily have ve tor ouplings in the lepton bi- linear so they do not ontribute to the e�e tive lagrangian in eq. (1). The amplitude for the de ay Bs → l +l− is given by M(Bs → µ +µ−) = G1 g ∣b̄γ5s ∣Bs〉 [g S ū(pµ)v(pµ̄) + g P ū(pµ)γ5v(pµ)] . (2) The pseudos alar matrix element is, ∣b̄γ5s ∣Bs〉 = −i mb +ms , (3) where mb and ms are the masses of bottom and strange quark respe tively. The al ulation of the de ay rate gives ΓNP (Bs → µ +µ−) = (gsbP ) 2 + (g f 2BsM (mb +ms)2 . (4) We see that the de ay rate depends upon the new physi s ouplings (gsbP ) and G2 2]. To obtain information on these parameters, we look at Bs − B̄s mixing together with KL → µ +µ− de ay and K0 − K̄0 mixing. Let us onsider Bs − B̄s mixing to obtain a onstraint on (g . Repla ing leptoni bilinear by quark bilinear in eq. 1, we get ∆B = 2 Lagrangian, LSPBs−B̄s = G2 b̄(g S + g P γ5)s b̄(g S + g P γ5)s, (5) where G2 is another dimensional fa tor. As in the ase of G1, introdu ed in eq. (1), G2 also arises due to the s alar propagator (or progators in the ase of box diagrams). Therefore it also has dimension (mass)−2 and is of the same order of magnitude as G1. From eq. (5), we al ulate the mass di�eren e of the Bs mesons to be ∆mBs = G2 (g 2B̂Bs f 2BsM (mb +ms)2 . (6) Thus the e�e tive b− s pseudos alar oupling is obtained to be (gsbP ) ∆mBs(mb +ms) 2B̂Bsf M3BsG2 . (7) We now onsider the de ay KL → µ +µ−. The same new physi s leading to the e�e tive b̄ → s̄µ+µ− lagrangian in eq. (1), also leads a similar e�e tive lagrangian for s̄ → d̄µ+µ− transition. The only di�eren e will be the e�e tive s alar/pseudos alar ouplings in the quark bilinear. Thus we have, s̄→d̄µ+µ− = G1 s̄(g S + g P γ5)d µ̄(g S + g P γ5)µ. (8) The al ulation of de ay rate gives ΓNP (KL → µ +µ−) = 2(gsdP ) 2 + (g f 2KM (md +ms)2 . (9) Here extra fa tor of 2 o urs be ause the amplitudes A(K0 → µ+µ−) = A(K̄0 → µ+µ−) and KL = K0+K̄0√ . We see that G2 2 + (g 2] an be al ulated from Γ(KL → µ +µ−), on e we know the value of (gsdP ) . In order to determine the value of (gsdP ) , we onsider K0 − K̄0 mixing. The e�e tive s alar/pseudos alar new physi s lagrangian for this pro ess an be obtained from that of s̄ → d̄µ+µ− by repla ing lepton urrent by orresponding quark urrent or equaivalently from e�e tive lagrangian of eq. (5) where b − s quark bilinear is repla ed by s− d quark bilinear, K0−K̄0 = G2 s̄(g S + g P γ5)d s̄(g S + g P γ5)d. (10) From this lagrangian, we obtain the KL −KS mass di�eren e to be ∆mK = G2 (g f 2KM (ms +md)2 . (11) Thus the e�e tive s− d pseudos alar oupling is (gsdP ) 2∆mK(md +ms) B̂Kf 2 M3KG2 . (12) Substituting the above value of (gsdP ) in eq. (9), we get 2 + (g 2πG2B̂K M2K∆mK ΓNP (KL → µ +µ−). (13) Substituting the value of G2 2 + (g 2] from eq. (13) and (gsbP ) from eq. (7) in eq. (4), we get ΓNP (Bs → µ +µ−) = )2 (∆mBs ΓNP (KL → µ +µ−). (14) The bran hing ratio is given by, BNP (Bs → µ +µ−) = )2 (∆mBs τ(Bs) τ(KL) BNP (KL → µ +µ−). (15) We wish to obtain the largest possible value for B(Bs → µ +µ−). To this end, we make the liberal assumption that the experimental values of ∆mBs , ∆mK and BNP (KL → µ +µ−) are saturated by new physi s ouplings. The de ay rate for KL → µ +µ− onsists of both long distan e and short distan e ontributions. The new physi s we onsider here, ontributes only to the short distan e part of the de ay rate. In ref [14℄, an upper limit on the short distan e ontribution to B(KL → µ +µ−) is al ulated to be 2.5× 10−9. The mass di�eren e of the Bs mesons is re enly measured by the CDF ollaboration to be ∆mBs = (1.17±0.01)× 10−11GeV [13℄. The bag parameters for theK and the Bs mesons are B̂K = (0.58±0.04) and B̂Bs = (1.30±0.10) [15℄. The values of the other parameters of eq. (15) are taken from Review of Parti le Properties [16℄: ∆mK = (3.48±0.01)×10 −15GeV τ(Bs) = (1.47±0.06)×10 −12 Sec and τ(KL) = (5.11± 0.02)× 10 −8 Sec. Substituting these values in eq. (15), we get BNP (Bs → µ +µ−) = (6.30± 0.75)× 10−9, (16) where all the errors are added in quadrature. At 3σ, BSM(Bs → µ +µ−) < 7.7× 10−9 where as BNP (Bs → µ +µ−) < 8.55× 10−9. Thus we see that this upper bound is almost the same as the SM predi tion even if we maximize the new physi s ouplings by assuming that they saturate the experimental values. Therefore the present data on Bs − B̄s mixing together with data on K0 − K̄0 mixing and KL → µ +µ− de ay puts a strong onstraint on new physi s s alar/pseudos alar ouplings and doesn't allow a large boost in the bran hing ratio of Bs → µ We now assume that the new physi s involving s alar/pseudos alar ouplings a ounts for the di�eren e between the experimental values and the SM predi tions of ∆mK , ∆mBs and the short distan e ontribution to Γ(KL → µ +µ−). The SM value for Bs − B̄s is given by [17, 18℄, (∆mBs)SM = ηBMBs B̂Bsf M2WS(xt) |Vts| = (1.16± 0.32)× 10−11GeV, (17) with fBs B̂Bs = (262±35)MeV [15℄ , ηB = 0.55±0.01[18℄ and |Vts| = 0.0409±0.0009 [16℄. S(xt) with xt = m W is one of the Inami-Lim fun tions [19℄. The SM value for K 0 − K̄0 mixing is given by [20℄, (∆mK)SM = λ∗2c η1S(xc) + λ t η2S(xt) + 2λ η3S(xc, xt) , (18) where λj = V jsVjd, xj = m W . The fun tions S are given by [21, 22℄, S(xt) = 2.46 170GeV , S(xc) = xc. (19) S(xc, xt) = xc 4(1− xt) 3x2t ln xt 4(1− xt)2 . (20) Using η1 = (1.32 ± 0.32) [23℄, η2 = (0.57 ± 0.01) [18℄, η3 = (0.47 ± 0.05) [24, 25℄, B̂K = (0.58±0.04) [15℄ ; fK = (159.8±1.5)MeV , |Vcs| = 0.957±0.017±0.093, |Vcd| = 0.230±0.011, |Vts| = 0.0409± 0.0009 and |Vtd| = 0.0074± 0.0008 [16℄, we get (∆mK)SM = (1.87± 0.49)× 10 −15GeV. (21) All the masses were taken from [16℄. Considering only the short-distan e e�e ts, the SM bran hing ratio for KL → µ +µ− in next-to-next-to-leading order of QCD is (0.79± 0.12)× 10−9 [26℄. Substra ting out the SM ontribution from the experimental values of ∆mBs , ∆mK and BNP (KL → µ +µ−) , we get BNP (Bs → µ +µ−) = (∆mBs)exp − (∆mBs)SM (∆mK)exp − (∆mK)SM τ(Bs) τ(KL) Bexp(KL → µ +µ−)short − BSM(KL → µ . (22) Substituting the experimental values and the SM predi tions in the above equation, and adding all the errors in quadrature, we get BNP (Bs → µ +µ−) = (0.08± 2.54)× 10−9. (23) whi h is onsistent with zero. At 3σ, the upper limit on the new physi s ontribution is lose to SM predi tion. Thus the present data on ∆mBs along with ∆mK and BNP (KL → µ puts strong onstraints on new physi s s alar/pseudos alar ouplings and doesn't allow a large enhan ement in the bran hing ratio of BNP (Bs → µ +µ−) mu h beyond the SM predi tions. New physi s at most an ause a fa tor of two enhan ement but not an order of magnitude. Hen e the total bran hing ratio whi h is the sum of SM ontribution and new physi s ontribution will be of the order of 10−8 and hen e rea hable at LHC. Con lusions: In this letter, we onsidered the onstraints on the New Physi s ouplings of s alar/pseudos alar type in the b → s transition. It was shown previously that only su h New Physi s an give rise to an order of magnitude enhan ement of the de ay rate for Bs → µ +µ−. Using the re ent data on Bs − B̄s mixing, together with the data on K 0 − K̄0 mixing and the short distan e ontribution to KL → µ +µ−), we obtained very strong bounds on B(Bs → µ +µ−). New Physi s in the form of s alar/pseudos alar ouplings an at most in rease the B(Bs → µ +µ−) by a fa tor of 2 ompared to its Standard Model predi tion. An order magnitude enhan ement, previously allowed, is ruled out. A knowledgments We thank Prof. Rohini Godbole for posing a question whi h led to this investigation. We also thank Prof. B. Ananthanarayan for a riti al reading of the manus ript. [1℄ Babar Collaboration: B. Aubert et al., Phys. Rev. Lett. 91, 221802 (2003). [2℄ Belle Collaboration: A. Ishikawa et al., Phys Rev. Lett. 91, 261601 (2003). [3℄ Babar Collaboration: B. Aubert et al., hep-ex/0604007. [4℄ A. Ali, E. Lunghi, C. Greub and G. Hiller, Phys. Rev. D 66, 034002 (2002). http://arxiv.org/abs/hep-ex/0604007 [5℄ E. Lunghi, hep-ph/0210379. [6℄ F. Kruger and E. Lunghi, Phys. Rev. D 63, 014013 (2001). [7℄ V. M. Abazov et al., D0 Collaboration Phys. Rev. Lett. 94, 071802 (2005). [8℄ D. Tonelli for the CDF Collaboration, hep-ex/0605038. [9℄ A. J. Buras, hep-ph/0101336 and Phys. Lett. B 566 115 (2003). [10℄ R. Forty, WHEPP-8 pro eedings. [11℄ A. K. Alok and S. Uma Sankar, Phys. Lett. B 620 61 (2005) . [12℄ Belle Collaboration: A. Ishikawa et al., hep-ex/0603108. [13℄ CDF Collaboration: A. Abulen ia, et al., Phys. Rev. Lett. 97, 242003 (2006). [14℄ G. Isidori and R. Unterdorfer, JHEP 0401: 009 (2004). [15℄ S. Hashimoto Int. J. Mod. Phys. A 20 5133 (2005). [16℄ Review of Parti le Physi s: W. M. Yao et al., J. Phys. G 33 1 (2006) . [17℄ Monika Blanke et. al., JHEP 0610 003 (2006). [18℄ A. J. Buras et. al., Nu l. Phys. B347 491 (1990). [19℄ T. Inami and C. S. Lim, Prog. Theor. Phys. 65 297 (1981) [Erratum-ibid 65 1772 (1981)℄. [20℄ A. J. Buras , hep-ph/0505175. [21℄ A. J. Buras et. al., Nu l. Phys. B238 529 (1984). [22℄ A. J. Buras et. al., Nu l. Phys. B245 369 (1984). [23℄ S. Herrli h and U. Nierste, Nu l. Phys. B419 292 (1994). [24℄ S. Herrli h and U. Nierste, Phys. Rev. D 52, 6505 (1995). [25℄ S. Herrli h and U. Nierste, Nu l. Phys. B476 27 (1996). [26℄ M. Gorbahn and U. Haisah, Phys. Rev. Lett. 97, 122002 (2006). http://arxiv.org/abs/hep-ph/0210379 http://arxiv.org/abs/hep-ex/0605038 http://arxiv.org/abs/hep-ph/0101336 http://arxiv.org/abs/hep-ex/0603108 http://arxiv.org/abs/hep-ph/0505175 Acknowledgments References

0704.0253

The Spitzer c2d Survey of Large, Nearby, Interstellar Clouds VIII. Serpens Observed with MIPS

Version 0.99; 31 Jan 2007 The Spitzer c2d Survey of Large, Nearby, Interstellar Clouds VIII. Serpens Observed with MIPS Paul M. Harvey1, Luisa M. Rebull2, Tim Brooke3, William J. Spiesman1, Nicholas Chapman4, Tracy L. Huard5, Neal J. Evans II1, Lucas Cieza1, Shih-Ping Lai4, Lori E. Allen5, Lee G. Mundy4, Deborah L. Padgett2, Anneila I. Sargent3, Karl R. Stapelfeldt6 Philip C. Myers5, Ewine F. van Dishoeck7, Geoffrey A. Blake8, David W. Koerner9 ABSTRACT We present maps of 1.5 square degrees of the Serpens dark cloud at 24, 70, and 160µm observed with the Spitzer Space Telescope MIPS Camera. We describe the observations and briefly discuss the data processing carried out by the c2d team on these data. More than 2400 compact sources have been extracted at 24µm, nearly 100 at 70µm, and 4 at 160µm. We estimate completeness limits for our 24µm survey from Monte Carlo tests with artificial sources inserted into 1Astronomy Department, University of Texas at Austin, 1 University Station C1400, Austin, TX 78712-0259; [email protected], [email protected], [email protected], [email protected] 2Spitzer Science Center, MC 220-6, Pasadena, CA 91125; [email protected]; [email protected] 3Division of Physics, Mathematics, & Astronomy, MS 105-24, California Institute of Technology, Pasadena, CA 91125; [email protected]; [email protected] 4Astronomy Department, University of Maryland, College Park, MD 20742; [email protected], [email protected], [email protected] 5Smithsonian Astrophysical Observatory, 60 Garden Street, MS42, Cambridge, MA 02138; [email protected] .edu, [email protected], [email protected] 6Jet Propulsion Laboratory, MS 183-900, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109; [email protected] 7Leiden Observatory, Postbus 9513, 2300 RA Leiden, Netherlands; [email protected] 8Division of Geological and Planetary Sciences, MS 150-21, California Institute of Technology, Pasadena, CA 91125; [email protected] 9Northern Arizona University, Department of Physics and Astronomy, Box 6010, Flagstaff, AZ 86011- 6010; [email protected] http://arxiv.org/abs/0704.0253v1 – 2 – the Spitzer maps. We compare source counts, colors, and magnitudes in the Serpens cloud to two reference data sets, a 0.50 deg2 set on a low-extinction region near the dark cloud, and a 5.3 deg2 subset of the SWIRE ELAIS N1 data that was processed through our pipeline. These results show that there is an easily identifiable population of young stellar object candidates in the Serpens Cloud that is not present in either of the reference data sets. We also show a comparison of visual extinction and cool dust emission illustrating a close correlation between the two, and find that the most embedded YSO candidates are located in the areas of highest visual extinction. Subject headings: infrared: general — clouds: star forming regions 1. Introduction The Spitzer Space Telescope Legacy project “From Molecular Cores to Planet-forming Disks” includes IRAC and MIPS mapping of five large star-forming clouds (Evans et al. 2003). The Serpens cloud covers more than 10 square degrees as mapped by optical extinction (Cambrésy 1999), but for reasons of practicality the c2d project was only able to observe 1.5 deg2 with the MIPS instrument on Spitzer (further Spitzer observations of a larger area of Serpens are planned as part of an extended survey of the Gould Belt, Allen 2007, in prep.). At an assumed distance of 260 pc (Straizys, Cernis, & Bartasiute 1996), the area mapped by c2d corresponds to ∼ 4.5 × 7 pc. This paper is one in a series describing the IRAC and MIPS observations of each of the c2d clouds. Previous papers include those on IRAC observations of Serpens (Harvey et. al. 2006), Chamaeleon (Porras et al. 2007), and Perseus (Jorgensen et al. 2006), as well as MIPS observations of Chamaeleon (Young et al. 2005), Perseus (Rebull et al. 2007), Lupus (Chapman et al. 2007), and Ophiuchus (Padgett et al. 2007). Our observations of Serpens cover an area that includes the well studied “core” clus- ter region, Cluster A, together with the newly discovered Cluster B (Harvey et. al. 2006; Djupvik et al. 2006) to the south, as well as the Herbig Ae/Be star, VV Ser. Significant portions of this cloud have been studied by previous space infrared missions, including IRAS (Zhang et al. 1988; Zhang, Laureijs, & Clark 1988) and ISO (Kaas et al. 2004; Djupvik et al. 2006). The much higher sensitivity and longer wavelength capability of the Spitzer MIPS instrument, however, allows us to detect both very low luminosity infrared-excess objects and to map very cool diffuse dust emission in the region. Our results are also complementary to the 1.1mm mapping of the same region by Enoch et al. (2007). The combined results on Serpens using both the MIPS and IRAC observations are discussed in a companion paper – 3 – where we also give detailed object lists (Harvey et al. 2007). In §2 we describe details of the observations obtained from the MIPS instrument for Serpens and the data processing pipeline used to reduce the observations. In §3 we describe a number of results from our MIPS observations and correlations between them and the 2MASS catalog (Skrutskie et al. 2006). We show in §3.1 that there is an excellent correlation between the coolest dust that we can observe which emits at 160µm and the optical extinction in Serpens. We investigate the possibility of time variability at 24µm in our two-epoch data set in §3.2. In §3.3 we discuss our results statistically in terms of source counts and compare these to predictions of models of the Galaxy as well as to the counts in the reference fields. We present color-color and color-magnitude plots of the population of infrared sources in §3.4 and discuss the separation of likely cloud members from the extensive background population of stars and extragalactic objects. In the final part of §3 we briefly describe some details of individual sources of particular interest. 2. Observations and Data Reduction The MIPS observations cover an area where A > 6 in the contour maps of Cambrésy (1999). In addition, a nearby off-cloud region of 0.5 square degrees was mapped for com- parison with the cloud region. A summary of the regions observed is listed in Table 1 with the AOR (Astronomical Observation Request) number to facilitate access from the Spitzer archive. The regions covered at 24µm are outlined in Figure 1 against the 25 µm IRAS sky. The observing strategy and basic MIPS data analysis for the c2d star-forming clouds have been described in detail by Rebull et al. (2007), but we summarize here the most important details. Fast scan maps were obtained at two separate epochs with a spacing between adja- cent scan legs of 240” in each epoch. The second epoch observations were offset by 125” from the first in the cross-scan direction to fill in the 70µm sky coverage that would otherwise have been missed due to detector problems. The second epoch scan was also offset 80” from the first in the scan direction to minimize missing 160µm data. For some of the c2d clouds, these offsets together with sky rotation were sufficient to give essentially complete one-epoch coverage at 160µm, but for Serpens there were still small gaps between every two scan lines. Table 2 lists the sky coverage at each wavelength. The two observation epochs were sepa- rated in time by ∼ 6 hours to allow identification of asteroids in the images; over this time period asteroids will typically move 0.3 – 2 arcminutes. Because of Serpens’ relatively large ecliptic latitude, ∼ 24 degrees, only a very small number of asteroids were seen, all of which were removed by requiring 2-epoch detection in our final source lists. Typical integration times are 30 seconds at 24µm, 15 seconds at 70µm and 3 seconds at 160µm. Additional GTO – 4 – observations east of the region of highest emission are not included in this analysis because a different observing strategy was used. Those observations could, however, be added to ours in order to construct a somewhat larger mosaic of the region. Figure 2 shows the three individual images produced for the MIPS bands as well as a false color image of the three together. Harvey et al. (2007) show an additional image combining the 24µm data with IRAC observations as well as enlargements of the two main clusters observed. Note that unlike the IRAC instrument, the three wavelengths of MIPS all have diffraction limited spatial resolution which means the resolution varies dramatically between 24µm (∼ 6”) and 160µm (∼ 40”). Our data reduction is described in detail by Evans et al. (2007) but we summarize the important details here. In addition, previous versions of the c2d pipeline, some of which still apply to these data, have been described in more detail by Rebull et al. (2007) and Young et al. (2005). We began our data reduction with the BCD images, processed in this case by the standard SSC S13.2 pipeline. Following this the three MIPS channels underwent slightly different processing paths in our c2d reduction. The 24µm data were mosaicked with the SSC’s Mopex software (Makovoz & Marleau 2005) after processing in the c2d pipeline to reduce artifacts, e.g. “jailbars” near bright sources. Point sources were extracted with “c2dphot” (Harvey et al. in prep.), a source extractor based on “Dophot” (Schechter, Mateo, & Saha 1993), which utilizes the mosaics for source identification but the stack of individual BCD’s for each identified object to provide the photometry and position information. We have estimated our completeness limit at 24µm in a manner similar to that described for our IRAC photometry (Harvey et. al. 2006). We inserted a number of artificial sources into the 24µm mosaic at random positions over a range of brightness covering the range 2 < [24] < 12 mag. and then tested whether they were properly extracted. We also produced a mosaic with only artificial sources (no real ones) but a noise level comparable to that in the observed image, and tested the completeness of extraction from that artificial image to estimate the effects of confusion in this relatively high source density region. Figure 3 shows the results from these tests. Clearly at the fainter flux levels, the effects of high source density are important to the true completeness level in Serpens, e.g. [24] > 9.5 mag. The processing of the 70µm data followed a path similar to that at 24µm with two exceptions. At 70µm the SSC produces two sets of BCD’s, one of which is simply calibrated and another that is filtered spatially and temporally in a manner that makes point source identification easier but which does not conserve flux for brighter sources nor for diffuse emis- sion. We produced mosaics of both the unfiltered and the filtered products using Mopex on the native BCD pixel scale. Point sources were extracted using APEX (Makovoz & Marleau 2005). Source reality was checked by hand inspection and comparison with the 24µm source – 5 – list. Generally the filtered mosaics were used for point source extraction, but above F(70) ∼ 2 Jy, we used the unfiltered data. Above F(70) ∼ 23 Jy, sources begin to be saturated. At these very high flux levels we used a procedure to fit the wings of the source profile; these data have been assigned a higher uncertainty of because of the inherent uncertainties in this procedure. Complete tables of source positions and flux densities for likely cloud members in Ser- pens are given by Harvey et al. (2007) for our 3.6 – 70µm observations. At 160µm our processing was limited to producing a native pixel scale mosaic using interpolation to fill in missing pixels and point source extraction from the unfiltered mosaic. We extracted four nominal point sources in the entire mapped area. Two of these are associated with obvious multiple clumps of 24/70µm sources. The other two, SSTc2dJ1829167+0018225 (associ- ated with IRAS 18267+0016) and SSTc2dJ18293197+0118429 (associated with source 159 of Kaas et al. (2004)) are likely powered mostly by single, shorter wavelength sources. Table 3 lists the positions and flux densities of these four nominal point sources with short com- ments, since their 160µm photometry is not described in any of our other publications on Serpens. None of these is in the core area of either of the main clusters. This is because large areas in those clusters are saturated, and the close spacing of many bright sources leads to the complicated, extended structure seen in Figure 2 at 160µm, without obvious point-like sources. After extraction, the source lists were bandmerged with our IRAC source lists for Ser- pens (Harvey et. al. 2006) and the 2MASS catalog of J, H, and K photometry (Skrutskie et al. 2006) as described by Evans et al. (2007). The radius for source matching with shorter wave- length detections was 4” at 24µm and 8” at 70µm. Table 4 lists the number of sources ex- tracted at 24 and 70µm, and some examples of statistics of numbers identified with shorter wavelength sources. In addition to bandmerging, sources undergo a classification process based on the available photometry, 2MASS, IRAC, and MIPS. For the purposes of this paper the most important classification is that of “star” which implies a spectral energy distribution that is well-fit as a reddened stellar photosphere without requiring any excess infrared emission from possible circumstellar dust. The data reported here consist of a sub- set of all the sources extracted in Serpens. The entire catalog is available from the SSC website (http://ssc.spitzer.caltech.edu/legacy/all.html). For this paper we have limited our discussion to sources with a signal-to-noise ratio greater than 5 and to sources found in both epochs of observation to eliminate asteroids. These limits lead to a very high reliability for the objects reported here, probably greater than 98%. In addition to our reduction of the Serpens Cloud and off-cloud data, we have also processed a 5.3 deg2 portion of the SWIRE Spitzer Legacy data (Surace et al. 2004) from http://ssc.spitzer.caltech.edu/legacy/all.html – 6 – the ELAIS N1 field through our c2d pipeline. Since this field is almost entirely populated by Galactic stars and extragalactic objects, it provides an additional control field against which to compare our Serpens Cloud population as discussed below. Note that the SWIRE observations go approximately a factor of 4 deeper than c2d due to increased integration time. 3. Results 3.1. Extended Emission The 160µm emission traces the coolest and most extended dust seen with MIPS. Figure 4 shows an image of the 160µm emission together with contours of the optical extinction. Also shown are the locations of the two main clusters of young stellar objects in Serpens, the core Cluster A, and Cluster B (also called the G3-G6 cluster by Djupvik et al. (2006)). The optical extinction has been estimated by our fitting of the objects that were well characterized as extincted stellar photospheres. This figure shows a very close correlation between the coolest dust and the dust that is associated with optical extinction. The figure also clearly shows that the two high-stellar-density clusters, Cluster A and B, are located in areas of maximum extinction, as we discuss further in §3.5. 3.2. 24µm Time Variability Since many pre-main-sequence stars exhibit variable optical emission, we conducted a simple examination of the 24µm fluxes from the two observed epochs, similar to that in Perseus by Rebull et al. (2007) and for the IRAC data in Serpens (Harvey et al. 2007). As shown in Table 1, the time difference between the two epochs of observation was of order 4 hours. Figure 5 shows the ratio of the 24µm flux density between the two epochs for all the extracted sources whose signal-to-noise ratio was above 5 that were detected in both epochs of observation. Although there are a few outliers beyond the limits expected on the basis of the signal-to-noise ratios, these are all readily explained as due to poor photometry near the edges of the mosaic or problems due to source confusion or adjacency to bright sources. This is consistent with the findings of Rebull et al. (2007) for the Perseus 24µm sources and by Harvey et al. (2007) for the Serpens IRAC sources. Although there are undoubtedly some variable sources in these clouds, the observing techniques of the c2d program were not designed to enable reliable detection of modestly variable objects. – 7 – 3.3. Source Counts Because the Serpens star-forming cloud is so close to the Galactic plane, b ∼ 5 degrees, the vast majority of the sources detected at the shorter wavelengths are background stars in the Galaxy. At tfainter flux levels, background extragalactic objects constitute a significant population. In order to estimate the background Galactic star numbers we have used the Wainscoat et al. (1992) model provided by J. Carpenter (private communication). Figure 6 shows the predicted star counts from the model together with the observed counts at 24µm for both the Serpens Cloud and the off-cloud region. Also shown in the figure are the source counts from the c2d-processed SWIRE ELAIS N1 field which are largely extragalactic for fluxes below a few mJy. This figure shows that contamination by Galactic stars at the brighter fluxes and by extragalactic sources at the faint end is a significant problem for identifying Serpens Cloud members. To address this problem we discuss our use of several color and flux criteria in the following section. It is also apparent that there is an excess of bright (F > 300 mJy) sources relative to the expected background counts. This excess is, in fact, real and represents the bright end of the YSO candidate population discussed in the following section. 3.4. Color-Magnitude Diagrams The c2d team has discussed in a number of studies how the use of color-magnitude and color-color diagrams can separate likely young cloud members with infrared excesses from reddened stars and many background extragalactic sources (Young et al. 2005; Harvey et. al. 2006; Rebull et al. 2007; Harvey et al. 2007). Since nearly half of the area covered by our MIPS 24µm observations was not observed with IRAC (Harvey et. al. 2006), we utilize the color and magnitude criteria developed by Young et al. (2005) and refined by Rebull et al. (2007) and Chapman et al. (2007) to isolate a candidate YSO population without requiring the existence of IRAC data. The most populated diagram is naturally the color-magnitude diagram of K versus K - [24] because of the much larger number of 24µm sources than 70µm ones. Figure 7 shows the distribution of sources in this diagram for the 1453 sources with S/N above 5 at 24µm and with 2MASS K matches within 4”. This distribution is very similar to that seen in other well-populated c2d clouds such as Perseus (Rebull et al. 2007). A comparison of the SWIRE results, the Serpens off-cloud results, and the Serpens Cloud data shows: 1) objects in our “star” class fall in a relatively narrow band with blue -[24] colors (K -[24] < 1) as would be expected, and 2) the part of the diagram toward redder colors is populated by a number of sources in Serpens that are not seen in either the off-cloud region or in the SWIRE data set, except at K magnitudes fainter than K – 8 – 14. This allows us to assign a high probability that sources in the region K < 14 and K -[24] > 2 are Serpens Cloud YSO candidates with excess emission at 24µm probably due to circumstellar dust. Note that the off-cloud area does have a population of moderately reddened objects (K -[24] < 2), well-fit as stellar photospheres that are not seen in the SWIRE sample, simply because even the off-cloud area has more reddening than the high Galactic latitude ELAIS N1 region. In order to categorize our YSO candidates crudely in terms of evolutionary state, we have drawn lines in Figure 7 indicating where objects would fall based on the YSO source classification criteria of Greene et al. (1994) using the K -[24] color to measure the spectral slope. Table 5 lists the number of candidates and the number in each of the four classes. Although AGB stars with substantial mass loss also exhibit mid- infrared excesses, Harvey et. al. (2006) have argued that the number expected in this area is less than or of order a half dozen (four of which have already been confirmed spectroscopically as AGB interlopers by Merin et al. (in prep.). The positions of and photometry for the YSO candidates that are not in the area covered by IRAC are given by Harvey et al. (2007) along with those in the IRAC area. Harvey et al. (2007) discuss the comparison between YSO’s selected by the criteria used here (K and 24µm data only) and the more restrictive criteria possible with the combination of IRAC data. They basically find that we actually may have missed 8 or 9 YSO’s in the area not covered by IRAC and included a very few, 3 or 4, that may be background extragalactic sources. But the overall conclusion is that there is a good correspondence between the YSO candidates found using only MIPS and 2MASS versus those selected with a more complete data set. It is also clear that the area mapped by both IRAC and MIPS, 0.85 deg2 contains a much higher density of YSO’s, 235 or 276 deg−2 than does the area only covered by MIPS/24µm with 51 YSO’s or 54 deg−2. Even if we exclude the area of the two high density clusters, the area covered by the combined IRAC/MIPS observations has a YSO density a factor of 4 higher than the area not included in the IRAC observations. We have also plotted our photometry in two other color/magnitude spaces for compar- ison with other c2d clouds. Figure 8 shows the distribution of sources in K vs. K -[70] space. As observed by Rebull et al. (2007) in Perseus, there are a large number of likely cloud members at much brighter K magnitudes than seen for SWIRE extragalactic objects. In addition, there is a small population of faint (in K ) objects that are redder than any of the SWIRE objects in both Serpens and Perseus. The four objects redder than K -[70] = 15 are all likely to be slightly less extreme versions of the sources discussed in the next section. Two of these are located in cluster A, but tend to be around the outside of the tight cluster of very red objects. The other two are in a small grouping associated with the second of the four 160µm point sources listed in Table 3. Since all of these objects were also observed in our program with IRAC, they are also listed in the appropriate tables of Harvey et al. – 9 – (2007), and all are considered high probability YSO’s. The final color-magnitude diagram, [24] vs [24]-[70] is shown in Figure 9. Again this distribution is qualitatively similar to that in Perseus, although we find many fewer sources in the area overlapping the red edge of the extragalactic distribution than did Rebull et al. (2007) for their “rest of the cloud”. The Serpens distribution is qualitatively more similar to that for the NGC1333 portion of Perseus. Since many of the sources represented in this diagram for Serpens are located in one of the two principal clusters, A and B, in Serpens, it is perhaps not surprising that they would mimic some of the properties of similar young clusters like NGC 1333. 3.5. The Most Embedded Objects We have selected the coldest, most obscured sources from our sample by looking for objects not detected in the 2MASS survey but detected with reasonable signal-to-noise at both 24 and 70µm. There are 11 such objects in our surveyed area, and these are listed in Table 6. Interestingly all 11 are located in the heart of either Cluster A or B. Additionally, as shown in Table 6 all were detected in some or all IRAC bands. Their energy distributions are all consistent with a designation of Class I even though they are not included in Figure 7 since they were not detected in the 2MASS survey. In fact, several of these objects are strongly enough peaked in the far-infrared that they have energy distributions consistent with some nominal Class 0 sources despite the fact that all were detected with IRAC. The class status of these will be discussed further using mm data by Enoch et al. (2007, in prep.). Figure 10 shows the SED’s for the two most embedded objects from Table 6. Each of these appears to be associated with an outflow in its respective cluster, and both have very similar SED’s that differ only in their absolute flux level by a factor of ∼ 10. Table 6 shows also that the most embedded object in Cluster B (whose SED is shown in Figure 10) was not selected as a YSO by Harvey et al. (2007). The reason is that the flux at 3.6µm was too faint to meet the selection criteria of that study. The area within 15” of that source contains two other extracted compact sources in the c2d data set. The positions and photometry for all three are shown in Table 7 and an image of the area is shown in Figure 11. Although the source density is quite high, the 70µm contours shown in the figure are clearly centered on the northernmost source, “C”. Source “B” is a slightly extended source that may represent a separate exciting object or may just be the location of the most visible jet emission that has been discussed briefly by Harvey et al. (2007) in this region. Source “A” is a faint, but very red object about 6” to the west of source “C” and appears to be a point-like object in the images. – 10 – Figure 4 shows clearly that Cluster A and B are located in the highest extinction parts of the cloud. Therefore the lack of detection of the objects in Table 6 at 1 – 2.3µm may be due at least partly to the extinction of the cloud material in which they are embedded in addition to individual circumstellar material. Although the nominal extinction values in these areas range up to A ∼ 35 – 40, the fact that these values result from smoothing over 90 arcseconds of the stellar distribution means that they probably underestimate the extinction in the most extreme regions. This association of the coldest objects with the highest extinction regions is similar to the correlation seen by Enoch et al. (2007) between extinction and location of dense mm cores. 4. Summary We have described the basic observational characteristics of the c2d MIPS observations of the Serpens Cloud. In a 1.5 deg2 area we have found 250 YSO candidates on the basis of the K -[24] color. An additional 11 objects can be identified on the basis of their 24 and 70µm fluxes and lack of detection by 2MASS. All of these YSO candidates will be discussed in more detail in a companion paper (Harvey et al. 2007). All the most embedded objects are found in the central area of the two main clusters of YSO’s previously identified in Serpens. The images and source catalogs derived from these data are all available on the SSC website, http://ssc.spitzer.caltech.edu/legacy/all.html. Support for this work, part of the Spitzer Legacy Science Program, was provided by NASA through contracts 1224608, 1230782, and 1230779 issued by the Jet Propulsion Lab- oratory, California Institute of Technology, under NASA contract 1407. Astrochemistry in Leiden is supported by a NWO Spinoza grant and a NOVA grant. JKJ was supported by NASA Origins grant NAG5-13050. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by NASA and the National Science Foundation. We also acknowledge extensive use of the SIMBAD data base. http://ssc.spitzer.caltech.edu/legacy/all.html – 11 – REFERENCES Cambrésy, L. 1999 A & A, 345, 965 Chapman, N. et al. 2007, ApJ, in press. 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A. et al. 2004, The SWIRE ELAIS N1 Image Atlases and Source Catalogs, (Pasadena: Spitzer Science Center), http://ssc.spitzer.caltech.edu/legacy/ Wainscoat, R. J. et al. 1992, ApJS, 83, 111 http://ssc.spitzer.caltech.edu/legacy/ – 12 – Young, K. E. 2005, ApJ, 628, 283 Zhang, C. Y. et al. 1988, A&A, 199, 170 Zhang, C. Y., Laureijs, R. J., & Clark, F. O. 1988, A&A, 196, 236 This preprint was prepared with the AAS LATEX macros v5.2. – 13 – Table 1. Summary of Observations Region AOR Time-Date l a b a (UT) (deg) (deg) Serpens 5713408 2004-04-05 23:40 31.5 5.4 5713920 2004-04-06 04:05 31.5 5.3 5713664 2004-04-06 00:22 31.6 5.2 5714176 2004-04-06 04:48 30.6 5.1 Off Cloud 5716736 2004-04-06 01:26 35.2 4.4 5716992 2004-04-06 05:52 35.2 4.3 a l and b are listed for the center of the 24 µm AOR. – 14 – Table 2. Serpens Cloud Sky Coverage Region 24 µm 70 µm 160 µm (deg2) (deg2) (deg2) Serpens 1.81 1.57 1.49 Off-Cloud 0.47 0.36 0.41 – 15 – Table 3. 160µm Point Sources RA (J200) Dec (J200) Flux (mJy) Comment YSO# a 18 29 32.3 +01 18 56 24000 Single 24/70µm Source 104 18 29 52.9 +00 36 09 18200 Cluster of four 24µm Sources 18 29 16.7 +00 18 20 10000 Single 24/70µm Source 88 18 28 15.7 −00 03 11 6070 Cluster of four 24µm Sources aYSO number from Harvey et al. (2007). – 16 – Table 4. Serpens Cloud Detection Statistics Wavelength(s) Source Number 24µm > 3σ 2635 24µm > 5σ 1494 70µm > 3σ 97 70µm > 5σ 88 24 & 70µm > 5σ 75 24µm & 2MASS K > 5σ 1085 24µm & any IRAC 1040a 70µm & any IRAC 77 aThe greater number of matches between 24µm and K versus IRAC is due to the smaller area coverage of the IRAC data. – 17 – Table 5. Classification based on K −[24] Classification Serpens Source Counta number with K −[24]>2, K <14 250 number with K −[24]>2, K <14, and Class I K −[24] color 15 (6%) number with K −[24]>2, K <14, and “flat” K −[24] color 21 (8%) number with K −[24]>2, K <14, and Class II K −[24] color 158 (63%) number with K −[24]>2, K <14, and Class III K −[24] color 56 (22%) aSince a 2MASS detection is required to be included in these statistics, very cold or deeply embedded sources are not present in these counts, e.g. those sources in Table 6. Table 6. The Most Embedded Objects Name/Position YSO # a 3.6 µm 4.5 µm 5.8 µm 8.0 µm 24.0 µm 70.0 µm Associated Source b SSTc2dJ... (mJy) (mJy) (mJy) (mJy) (mJy) (mJy) 18285404+0029299 40 5.81±0.50 27.6± 2.3 44.8± 2.6 56.4± 3.2 918± 85 11100± 1040 D62/66 18285486+0029525 42 1.94±0.12 10.6± 0.6 20.4± 1.1 30.2± 1.6 765± 70 7250± 675 D65 18290619+0030432 67 8.05±0.41 45.0± 2.8 93.9± 4.8 129± 7 1320± 139 7240± 713 D90 18290675+0030343 68 3.27±0.21 11.7± 0.7 14.9± 0.8 20.7± 1.2 1000± 105 11400± 1180 D94 18290906+0031323 < 0.12 0.29±0.03 0.40±0.09 0.31±0.08 64.6± 6.0 6380± 611 D101 18294810+0116449 135 1.96±0.10 6.98±0.42 12.1± 0.6 16.7± 0.8 219± 21 14900± 1420 K241, SMM9 18294963+0115219 141 0.85±0.08 2.64±0.27 2.32±0.28 3.54±0.31 1180± 117 82800± 7810 K258a, SMM1 18295219+0115478 150 7.38±0.41 33.0± 2.1 41.3± 2.2 40.0± 2.6 1640± 154 15200± 1420 K270, SMM10 18295285+0114560 155 8.65±0.44 34.6± 1.8 72.0± 3.4 110± 5 1040± 96 5570± 523 K276 18295927+0114016 195 2.72±0.28 5.76±0.44 7.78±1.16 36.0± 5.4 109± 19 12200± 1160 SMM3 18295992+0113116 198 2.77±0.16 29.5± 1.5 103± 4 199± 10 2620± 249 6830± 675 K331 aIdentifying number from YSO table in Harvey et al. (2007). bReferences are: D: (Djupvik et al. 2006), K: (Kaas et al. 2004), SMM: Davis et al. (1999). Table 7. Sources Marked In Figure 11 Marker Name/Position 3.6 µm 4.5 µm 5.8 µm 8.0 µm 24.0 µm 70.0 µm SSTc2dJ... (mJy) (mJy) (mJy) (mJy) (mJy) (mJy) Aa 18290904+0031280 0.95±0.11 2.78±0.23 2.92±0.24 5.03±0.40 14.0± 1.9 · · · B 18290864+0031305 0.06±0.03 0.32±0.02 0.47±0.05 0.62±0.07 36.2± 3.4 · · · C 18290906+0031323 < 0.12 0.29±0.03 0.40±0.09 0.31±0.08 64.6± 6.0 6380± 611 aThis is YSO # 75 in Harvey et al. (2007). – 20 – Fig. 1.— IRAS 25 µm map showing the observed c2d regions in the Serpens cloud, both the star-forming region marked “SERPENS” and the low-extinction “OFF-CLOUD” area. – 21 – Fig. 2.— Registered Serpens 24µm, 70µm and 160µm images of the c2d MIPS region. The color image is a composite of all three bands, and includes only the 1.27 square degree area where data are available for each of the three bands. Colors represent red:160µm green:70 µm and blue:24µm. The black outline shows the region where 4 bands of IRAC data were observed (Harvey et. al. 2006). – 22 – Fig. 3.— Completeness test at 24µm. The upper solid line shows the measured completeness fraction for artificial sources inserted into the observed 24µm mosaic image of Serpens as a function of magnitude. The slightly higher dash-dot line shows the completeness fraction for sources inserted into an artificial image with no real sources but with a noise level equal to that in the observed data. The lower solid line (mostly equal to zero) shows the fraction of “unreliable” sources, i.e. sources extracted which were not real. – 23 – Fig. 4.— Contours of A at levels of 5,10,20,30 mag determined from 2MASS and Spitzer c2d IRAC data are overlaid on the Serpens 160 µm image. The visual extinction and 160µm emission are quite well correlated. The locations of Cluster A and B are indicated. – 24 – 0 1 2 3 4 log flux@24 Fig. 5.— A search for time variability in the Serpens 24µm data; plot of log flux ratio of epoch1 to epoch2 versus log flux density (mJy) for the combined epoch data. There is no verifiable time variable source in the cloud based on these data. – 25 – Fig. 6.— 24µm source counts in the Serpens MIPS field (dark line), and off-cloud region (dashed line). SWIRE galaxy counts (thin line) fall below the Serpens data at our flux limit of 1 mJy. The predicted source counts from the Wainscoat model at 25 µm (Wainscoat et al. 1992) are shown by the dot-dash line. – 26 – 0 2 4 6 8 10 12 14 Ks-[24] (mag) SWIRE 0 2 4 6 8 10 12 14 Ks-[24] (mag) Class III Class II flat Class I Serpens 0 2 4 6 8 10 12 14 Ks-[24] (mag) Class III Class II flat Class I Serpens OC Fig. 7.— Color-magnitude diagram for K vs. K − [24] for objects in SWIRE (left) and Serpens (center) and off-cloud region (right). The SWIRE counts are shown as a surface density with darker implying higher density. Objects in SWIRE are expected to be mostly galaxies (objects with K &14) or stellar photospheres (objects with K − [24] .1). For the Serpens and off-cloud plots, filled gray circles are objects with SEDs resembling photospheres, and plus signs are the remaining objects. An additional box around a point denotes that it was also detected at 70µm. Objects that are candidate young objects have colors unlike those objects found in SWIRE, e.g., K .14 and K − [24] &1. Dashed lines denote the divisions between Class I, flat, Class II, and Class III objects; to omit foreground and background stars, we have further imposed a K − [24] >2 requirement on our Class III objects (see text). – 27 – 0 5 10 15 20 Ks-[70] Fig. 8.— Color-magnitude diagram of K vs. K − [70] for Serpens (crosses) with data from the full SWIRE survey (grey dots) included for comparison. – 28 – Fig. 9.— Color-magnitude diagram of [24] vs. [24] − [70] for Serpens (crosses) with data from the full SWIRE survey (grey dots) included for comparison. – 29 – Fig. 10.— Spectral energy distribution for the two most embedded sources in Table 6, one in Cluster A (open squares, SSTc2dJ1829463+0115219) and one in Cluster B (open diamonds, SSTc2dJ18290906+0031323, source “C” in Table 7), both of which appear to be associated with outflows. – 30 – Fig. 11.— Three color image of the eastern end of Cluster B where the most embedded source, C, is located. This is the likely exciting source for an HH-like outflow visible in the IRAC data. The color scheme is: blue/4.5µm, green/8.0µm, and red/24µm. The contours of 70µm emission are also superimposed with levels at 40, 80, 160, 240, and 320 MJy/sr. Also shown are the positions of two other compact sources extracted from the images in this region. The letters correspond to positions/fluxes in Table 7. Introduction Observations and Data Reduction Results Extended Emission 24m Time Variability Source Counts Color-Magnitude Diagrams The Most Embedded Objects Summary

0704.0254

Unravelling the sbottom spin at the CERN LHC

Unravelling the sbottom spin at the CERN LHC Alexandre Alves1, ∗ and Oscar Éboli1, † Instituto de F́ısica, Universidade de São Paulo, São Paulo, Brazil (Dated: 02/04/07) Establishing that a signal of new physics is undoubtly supersymmetric requires not only the discovery of the supersymmetric partners but also probing their spins and couplings. We show that the sbottom spin can be probed at the CERN Large Hadron Collider using only angular correlations in the reaction pp → b̃b̃∗ → bb̄/pT , which allow us to distinguish a universal extra dimensional interpretation with a fermionic heavy bottom quark from supersymmetry with a bosonic bottom squark. We demonstrate that this channel provides a clear indication of the sbottom spin provided the sbottom production rate and branching ratio into bχ̃01 are sufficiently large to have a clear signal above Standard Model backgrounds. I. INTRODUCTION Supersymmetric models are promising candidates for physics beyond the Standard Model (SM), despite the present lack of direct experimental evidence on supersymmetry (SUSY). The CERN Large Hadron Collider (LHC) has a large reach for the discovery of SUSY [1] that largely relies on the production and decay of strongly interacting new particles i.e. squarks and gluinos [2, 3]. Establishing that a signal of new physics at the LHC is indeed supersymmetric requires not only the discovery of the new supersymmetric partners but also probing their interactions and spins [4, 5, 6, 7]. Previously, the squark spin has been studied through the long decay chain q̃ → χ̃02 → ℓ̃ → χ̃01 which is also used to measure its mass [4, 6] or in conjunction with the gluino spin analysis in the decay chain g̃ → b̃ → χ̃02 → ℓ̃ → χ̃01 [7]. In this work we probe the potential of the CERN Large Hadron Collider (LHC) for unravelling the bottom squark spin using angular correlations in the short decay chain pp → b̃b̃∗ → bb̄/pT , analogously to what has been done for the analysis of the slepton spin [4, 5]. A nice feature of this reaction is that only sbottoms and the lightest SUSY particle (LSP) takes place in it, providing a further check of sbottom spin obtained in the long decay chain studies. Unfortunately, this analysis can not be extended straightforwardly to light flavor squarks due to large QCD backgrounds. To determine the spin nature of the sbottoms at the LHC we compare the SUSY sbottom production and decay chain with another scenario where the new intermediate states have the same spin as the SM particles and leads to the same final state bb̄/pT . Such a model are Universal Extra Dimensions (UED) [8] where each SM particle has a heavy Kaluza–Klein (KK) partner which can mimic the SUSY cascade decay, provided we employ the mass spectra extracted from the decay kinematics match [9]. Here we are not focusing on UED searches but we use UED only for comparison with the SUSY predictions. There are many observables which we can use to discriminate ‘typical’ UED and SUSY models, like the production rate or the mass spectrum. Nevertheless, at the LHC we measure only production cross sections times branching ratios, and the UED as well as the SUSY mass spectra are unlikely to be what we currently consider ‘typical’. On the other hand, spin information is generally extracted from angular correlations. Therefore, we base our analysis exclusively on distributions of the outgoing SM b quarks as predicted by UED and by SUSY. We demonstrate that this final state bb̄/pT provides a clear indication of the sbottom spin provided the sbottom production rate and branching ratio into bχ̃01 are sufficiently large to have a clear signal above SM backgrounds. ∗Electronic address: [email protected] †Electronic address: [email protected] http://arxiv.org/abs/0704.0254v1 mailto:[email protected] mailto:[email protected] II. UED INTERACTIONS AND PARAMETERS We assumed one extra dimension with size R, where all SM fields propagate [8, 9], leading to a tower of discrete KK excitations for each of the SM fields (n = 0). In this scenario, the 5-dimensional wave functions for an SU(2)–doublet fermion are dL cos dR sin . (1) On the other hand, the roles of the left and right handed n-th KK excitations are reversed for SU(2) singlets. Just like in the MSSM, the spinors of the singlet (q) and doublet (Q) KK–fermion mass eigenstates can be expressed in terms of the SU(2) doublet and singlet fields ψd,s Q(n) = cosα(n)ψ d + sinα (n)ψ(n)s , q(n) = sinα(n)γ5ψ d − cosα (n)γ5ψ(n)s . (2) In general, the mixing angle α(n) is suppressed by the SM fermion mass over the KK–excitation mass plus one–loop corrections except for the top quark due to its large mass. tan 2α(n) = Q + δm The non–degenerate KK–mass terms δm(n) contain tree level and loop contributions to the KK masses, including possibly large contributions from non–universal boundary conditions. The neutral KK gauge fields will play the role of neutralinos in the alternative description of the sbottom production and decay. Just as in the SM, there is a KK–weak mixing angle which for each n rotates the interaction eigenstates into mass eigenstates γ(n)µ = cos θ µ + sin θ 3,µ , Z(n)µ = −sin θ(n)w B(n)µ + cos θ(n)w W 3,µ . (4) The n-th KK weak mixing angle is again mass suppressed tan 2θ(n)w = v2 g gY /2 )2 − (δm(n)B )2 + v2 (g2 − g2Y ) /4 where δm(n) contains tree level as well as loop corrections to the KK gauge boson masses. Generally (δm 2 ≫ v2 g2 − g2Y [9] and the lightest KK partner is the B(1), with basically no admixture from the heavy We only considered the first set of KK excitations to formulate an alternative interpretation of production and decay of sbottoms at the LHC, using the UED decay b(1) → bγ(1) to mimic a sbottom decay. The relevant interactions for this decay are Lγ1q1q = igY q̄(0)γµ Yd cosα (1)PL + Ys sinα (1)PR Q(1) − q̄(0)γµ Yd sinα (1)PL + Ys cosα (1)PR γ(1)µ . (6) In general, the KK partners of the SM particles do not have a mass spectrum similar to what we expect in SUSY, however, we imposed that the first KK excitations have the same mass as the SUSY particles. Assigning the LSP mass to the Lightest KK Particle (LKP) and the Next-LSP mass to the Next-LKP mass fixes the KK-weak mixing angle by means of Eq. (5) θ(1)w = arctan ggY v 2(m2NLSP −m2LSP ) . (7) test point particle mass (GeV) branching ratio S5 χ̃01 122. stable S5 b̃1 633. Br(b̃1 → bχ̃ 1) = 26.6% S5 b̃2 663. Br(b̃2 → bχ̃ 1) = 78.8% SPS1a χ̃01 97. stable SPS1a b̃1 517. Br(b̃1 → bχ̃ 1) = 4.4% SPS1a b̃2 547. Br(b̃2 → bχ̃ 1) = 29% L1 χ̃01 97.8 stable L1 b̃1 280. Br(b̃1 → bχ̃ 1) = 40% L1 b̃2 354. Br(b̃2 → bχ̃ 1) = 20% Table I: Masses of sbottoms, the lightest neutralino, and branching ratios of b̃1,2 → bχ̃ 1 for the test points considered in our numerical simulations. III. EVENT SIMULATION AND TEST POINTS We considered three scenarios for the new particle spectrum. Our first test point is the LHC point 5 (S5) [10] that exhibits rather heavy sbottoms with sizeable decays into bχ̃01; see Table I. Our second reference point is SPS1a [11] where the sbottom masses are close to ones in the first scenario, however the decays b̃1,2 → bχ̃01 are suppressed. The third test point exhibits a somewhat light squark spectrum which could be eventually produced at a 1 TeV International e+e− Linear Collider; we denote this point by L1. The masses for the L1 parameter point are mb̃1 = 280 GeV, mb̃2 = 354 GeV, mt̃1 = 339 GeV, mt̃2 = 371 GeV, and mχ̃01 = 97.8 GeV. For the SPS1a and S5 parameter choices the lighter of the two sbottoms is almost entirely a left state b̃1 ∼ b̃L while for L1 it is approximately a maximal mixture of left and right states b̃1 ∼ 2/2b̃L + 2/2b̃R. The salient features of these reference points is summarized in Table I. In order to correctly treat all spin correlations we performed a parton–level analysis including the UED interactions in MADGRAPH [12] and using SMADGRAPH [13] for the SUSY simulation. In our calculations we used CTEQ6L1 parton distribution functions [14] with the factorization and renormalization scales are fixed by µF = µR = (mb̃1 + mb̃2)/2 for the signal and µF = µR = mV + i pTi for the SM backgrounds, where mV is the mass of the relevant electroweak gauge boson and i pTi is the sum of the transverse momentum of additional jets. We simulated experimental resolutions by smearing the energies (but not directions) of all final state partons with a Gaussian error. We considered a jet resolution ∆E/E = 0.5/ E ⊕ 0.03 and σ/ET = 0.46 ET for the missing transverse energy where ET is the sum of the jet transverse energies. In addition to that we included a b-tagging efficiency of ε = 60% with a mistagging probability of 1/200 for light jets [15]. No K-factors were applied to signals or backgrounds since we do not expect QCD corrections to change significantly the shape of kinematical distributions [2, 3]. Nevertheless, given the increasing interest in studying spins correlations at the LHC by means of long and short decay chains, it is important to check if this is indeed the case. IV. ANALYSIS AND RESULTS At the LHC we analyzed the production of sbottom pairs followed by their decay in a b and the LSP pp→ b̃1,2b̃∗1,2 → bb̄χ̃01χ̃01 → bb̄/pT , (8) i.e. the signal is characterized by two b-tagged jets and missing transverse energy. One feature of the reaction (8) is that the s–channel subprocesses qq̄ → b̃b̃∗ present the well–known angular distribution d cos θ∗ ∝ 1− cos2 θ∗ (9) where θ∗ is the polar angle of produced scalar particles in their center-of-mass frame. However, this clean signature of the sbottom spin is contaminated by the subprocess gg → b̃b̃∗ which contains t-channel diagrams and quartic couplings; see Fig. 1 left panel. Therefore, it is easier to decipher the new state spin if we enhance the importance of the qq̄ s–channel subprocesses via a judicious choice of cuts. Notwithstanding, the cuts to isolate the signal must not introduce bias in the angular distributions used to study the sbottom spin. -1 -0.5 0 0.5 1 gg+qq no cuts cosθ* -1 -0.5 0 0.5 1 gg+qq S5 no cuts cosθ* Figure 1: Left panel: cos θ∗ distribution for the production of sbottoms coming from qq̄ and gg fusions. Right panel: same distribution as in the right panel but for the production of b(1). We used the test point S5 spectrum. On the other hand, the center–of–mass angular distribution of KK bottoms in UED produced by qq̄ fusion is d cos θ∗ ∝ 1 + E2b1 −m E2b1 +m cos2 θ∗ , (10) where Mb1 and Eb1 are the mass and energy respectively of the b (1) in the center–of–mass frame. This distribution peaks in the forward and backward directions being quite distinct of the SUSY prediction. Moreover, we must also include the gg → b(1)b̄(1) which contains t- and s-channel contributions which present a peak towards the forward and backward regions as well; see Fig. 2 right panel. At the LHC we can not reconstruct the the polar angle (θ∗) of produced particles in their center-of-mass frame due to the presence of undetected χ̃01 or γ (1). Therefore, we must use an alternative variable that retains part of the information carried by θ∗. A convenient variable to use in our analysis is [5] cos θ∗bb ≡ tanh where ∆ηbb is the rapidity separation of the b-tagged jets. Notice that ∆ηbb is invariant under boosts along the collision axis, and consequently, cos θ∗bb is invariant under boosts as well. The angle θ bb is the polar angle between each reconstructed bottom jet direction in the longitudinally boosted frame in which the rapidities of the bottoms are equal and opposite. In order to cos θ∗bb carry information of the produced particle spin, the flight directions of sbottoms and bottoms must be correlated. We depict in the left panel of Fig. 2 the cosine of the opening angle between the bottom and the sbottom flight directions in the b̃b̃∗ center–of–mass system for the S5 and L1 spectra. Clearly, the bulk of the signal is concentrated in region of small opening angles as a consequence of the large energy of the sbottoms after cuts. Fig. 3 shows that cos θ∗bb is indeed strongly correlated to the cosine of production polar angle (θ ∗) of the sbottoms and b(1)’s in their center–of–mass system. Therefore, we must expect the shape of cos θ∗bb distributions to resemble the polar angle spectra of sbottoms and KK bottoms apart from some smearing effects due to non-perfect correlations between the bottoms and sbottoms (KK bottoms) flight directions. Nevertheless, a clear separation between SUSY and UED distributions should be possible as was demonstrated in the case of smuon pair production at the LHC [5]. Taking a closer look at Fig. 3 we already realize that UED events are slightly more concentrated near cos θ∗bb = ±1 while SUSY events are homogeneously distributed along the direction cos θ∗bb = cos θb̃. In our analysis, we included the following backgrounds for the process pp→ bb̄/pT : • SM QCD and electroweak production of bb̄Z with Z → νν̄ that accounts for ≃ 91% of the total background after cuts for the S5 and SPS1a test points and ≃ 72% for the L1 scenario. 0.5 0.6 0.7 0.8 0.9 1 normalized distributions cosθsb/b Figure 2: Cosine of the opening angle between the bottom and the sbottom in center–of–mass system for two different mass spectra considered here. -1 -0.5 0 0.5 1 cosθ* -1 -0.5 0 0.5 1 cosθ* Figure 3: In the left (right) panel we display a scattered plot cos θ∗bb ⊗ cos θ ∗ for sbottoms (b(1)) production. Here we chose the parameter point S5. • Reducible electroweak and QCD backgrounds like jjZ, jjW , jτντ , bb̄j, jj, jjj, and jjb where some of the decay products evade detection and j stands for a light jet that might be mistagged as a b jet. These dangerous backgrounds are efficiently reduced by the requirement of two b-tagged jets. The second largest background after cuts is bbW where the lepton from the W boson has a large rapidity |ηℓ| > 2.5. • SUSY processes, excluding b̃b̃∗ production, that lead to the final state bb̄χ̃01χ̃01. In order to properly trigger [16] the event and tag the b-jets [15] the following acceptance cuts were imposed in all cases |ηb| < 2.5 , pbT > 100 GeV , ∆Rbb > 0.7 , /pT > 100 . (12) A potentially large background is the QCD production of dijets once we take into account that mismeasurements of the jets properties can lead to missing transverse momentum and that this process has a huge cross section. This background can be efficiently reduced by the missing transverse momentum cut Eq. (12) and by requiring that the azimuthal angle between the jets and the missing transverse momentum satisfy |∆Φ(/pT , pTj)| > 0.3 . (13) In the L1 test point simulations we applied the following cuts not only to enhance the signal and deplete the background but also to augment the importance of the qq̄ s-channel subprocess Mbb̄ > 300 GeV , Meff > 600 GeV , |∆Φ(/pT , pTjsoft)| < 2.4 , (14) where Meff is the sum of all jet and missing transverse momenta and ∆Φ(/pT , pTjsoft) the azimuthal angle between the missing transverse momentum and the softest jet. After cuts the SUSY signal cross section is 38.1 fb while the background cross section is 2.4 fb, leading to S/B ≃ 16.6. For the parameter point S5 we used the following cuts to optimize the signal Mbb̄ > 600 GeV , Meff > 1 TeV , |∆Φ(/pT , pTjsoft)| < 2.4 . (15) The SUSY signal cross section at this test point is 4.55 fb with a background of 1.24 fb and S/B ≃ 3.7 while the qq̄ fusion accounts for ≃ 40% of the signal events. On the other hand, ≃ 45% of the UED signal stems from qq̄ fusion. The cuts employed in this case were harder for two reasons: first, the bottoms are harder in this case once the sbottom is much heavier than the neutralino compared to L1 case; second, the signal cross section is significantly smaller than in the L1 case which required a deeper suppression of backgrounds to avoid a severe bias on the signal angular distributions. Finally, we imposed the following cuts for the test point SPS1a Mbb̄ > 600 GeV , Meff > 1 TeV , (16) which lead to a signal (background) cross section of 1.07 (1.46) fb and S/B ≃ 0.73. In the left panel of Fig. 4 we show the impact of cuts on the dσ/d cos θ∗bb distributions for SUSY and UED predictions assuming the S5 spectrum. We normalized the UED signal cross section to the SUSY one. As the background events populate the bins of larger | cos θ∗bb|, the selection cuts tend to suppress that region enhancing the central region for both SUSY and UED. The variable ∆Φ(/pT , pTj) is efficient in rejecting dangerous SM backgrounds like dijet production, however, it has a potential to bias the distributions as we can see in the left panel of Fig. 4. Therefore, a harder cut on this variable is not recommended in the study of spin correlations. A natural question at this point is whether we can mimic the SUSY signal by varying the UED mixing angles. According to Section II there is not very much room to modify the UED Lagrangian to bring kinematical correlations closer to the SUSY prediction. The KK weak mixing angle θ w in Eq. (5) is fixed by the KK masses, so we can not change it while keeping the masses fixed. The same limitations hold when we try to adjust the mixing between the singlet and doublet KK fermions, described by the angle α(n); see Eq. (2). In contrast to the 3rd–generation sfermion sector in the MSSM, the UED mixing angle is not a (third) free parameter, even if we move around the masses invoking boundary conditions. Notwithstanding, for illustration purpose we vary α(1) in the right panel Fig. 4 to check whether the SUSY cos θ∗bb can be reproduced by a UED decay chain with different couplings to the fermions. From Eq. (6) we see that varying α(1) effectively enhances the left or right couplings of the KK bottom decay into bottom plus LKP. This figure allows us to see that the changes in the UED parameters are not sufficient to mimic the SUSY predictions. We depict in Fig. 5 the cos θ∗bb spectrum with/without adding the background for the S5 test point, an integrated luminosity of 300 fb−1, and after applying cuts (12) and (15). To avoid using any information but the spin we assume the S5 spectrum for the UED particles and normalize their production cross section times branching fractions to the SUSY rate. From Figure 5, we can easily see that the production of fermionic strongly interacting states (UED) favor large separations (cos θ∗bb) between the b-tagged jets while the production of scalar particles (SUSY) leads to a rather constant distribution. Note that these distributions indeed resembles the distributions of the production angles in the center–of–mass system which reflects the correlation between these observables. Moreover, the UED cos θ∗bb distribution is similar to the SM background one since they take place through similar diagrams containing KK partners or SM particles with the same spin quantum numbers. For the assumed integrated luminosity it is rather easy to distinguish between the two models. These two possibilities can be disentangled, for instance, through the asymmetry: σ(| cos θ∗bb| < 0.5)− σ(| cos θ∗bb| > 0.5) σ(| cos θ∗bb| < 0.5) + σ(| cos θ∗bb| > 0.5) . (17) -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 S5 Spectrum 300 fb-1 (a) SUSY all cuts (b) SUSY basic cuts (c) UED all cuts (d) UED basic cuts -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 L = 300 fb-1 S5 Spectrum UED: α(1) = 0 UED: α(1) = π/4 UED: α(1) = π/2 cosθ* bb Figure 4: Left panel: impact of cuts on the dσ/d cos θ∗bb distributions for SUSY and UED signals. We normalized the UED signal cross sections to the SUSY ones using the S5 spectrum. Right panel: SUSY and UED dσ/d cos θ∗bb distributions without backgrounds but varying the KK bottom mixing angle α(1). The basic cuts are the acceptance defined in Eq. (12) plus Mbb̄ > 600 GeV. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Total Bckg 300 fb-1 cosθ* bb -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 total bkg others 300 fb-1 cosθ* bb Figure 5: In the left (right) panel we plot the cos θ∗bb distribution without (with) the addition of the background contribution. Here we used the parameter point S5 and assumed an integrated luminosity of 300 fb−1. The error bars indicate the expected statistical uncertainties. This asymmetry is +0.238 ± 0.023 for the UED spin assignment while the SUSY interpretation it is significantly larger +0.373± 0.022 where the quoted errors are statistical. We estimate that an integrated luminosity of 300 fb−1 is needed to reach a 5σ level signal for the S5 spectrum. The determination of the sbottom spin is much easier for the test point L1 due to the large b̃ → bχ̃01 branching ratio and production cross section. We depicted in Fig. 6 the cos θ∗bb distribution and without/with adding the SM background. In this case a mere integrated luminosity of 15 fb−1 is enough to discriminate at the 5σ level UED and SUSY. The asymmetries are given by +0.565± 0.019 for SUSY and +0.365± 0.021 for UED for this integrated luminosity. Therefore, there is a clear distinction between the two spin assignments (UED × SUSY) for both S5 and L1 mass spectra and the above integrated luminosities. The results for the reference point SPS1a are quite different from the S5 and L1 ones. Due to small b̃ → bχ̃01 branching ratio, the SM backgrounds play a major role. We can see in the left panel of Fig. 7 that the pure SUSY and UED cos θ∗bb spectrum are quite different. However, once we add the SM background, which has a shape similar to UED, it is no longer easy to separate the SUSY from UED, even for an integrated luminosity of 1 ab−1; see the -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Total Bckg 50 fb-1 cosθ* bb -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 total bkg others 50 fb-1 cosθ* bb Figure 6: Same as Fig. 5 but for the parameter point L1 and an integrated luminosity of 50 fb−1. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 naive background subtraction SPS1a 1 ab-1 cosθ* bb -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 total bkg others SPS1a 1 ab-1 cosθ* bb Figure 7: Same as Fig. 5 but for the parameter point SPS1a and an integrated luminosity of 1 ab−1. right panel of Fig. 7. For instance, the asymmetry (17) is 0.282± 0.019 for SUSY and 0.200± 0.019 for UED and this extremely large luminosity. In this case SUSY and UED can be discriminated only at at ∼ 4σ level. Therefore, it is important for this point to subtract the SM background which can be estimated, for instance, from the measurement of the bb̄µ+µ− cross section. If we neglect systematic errors associated to this extraction from data, the sbottom spin can be determined at 5σ level for an integrated luminosity of ≃ 500 fb−1. The left panel of Fig. 7 displays the distributions of SUSY and UED after a naive background subtraction where we just did not add the background to the signals and the error bars calculated as S +B. A dedicated study is necessary to determine the actual impact of the statistical and systematic errors [17]. In general, we noticed that the expected shape of distributions in cos θ∗bb are similar to the ones in smuon pair production and decay to µχ01 [5], which shows the universality and robustness of the method. Moreover, in both cases (b̃∗sbx and µ̃∗µ̃) the S5 spectrum seems to be more promising as compared to the SPS1a spectrum, for example, SUSY versus UED discrimination of smuon spin assignments is possible in the case of S5 (SPS1a) for an integrated luminosity of 200 fb−1(500 fb−1). As a final remark note that, apart from negligible effects from bb̄ initiated contributions which include interactions with gluinos and electroweak interactions with Z bosons, the production rate does not depend upon the left or right nature of the sbottom since the QCD interactions to the gluons are blind to these details. On the other hand, the sbottom–bottom–neutralino vertex is sensitive to the relative content of mass eigenstates in terms of left and right states which by its turn depend upon the mass spectrum. As we have pointed out in the Section III the lightest sbottom is almost entirely a left state in the S5 mass spectrum, while it is an equal mixture of left and right states in the L1 mass spectrum. As can be seen in the Figs. 5 and 6, it seems plausible to conclude that the distributions are not sensitive to the particular mixtures of left and right states of the sbottom mass eigenstates, however a more detailed study including more test points is necessary to confirm this indication. V. CONCLUSIONS In the near future the LHC will start its endeavor in searching for new physics signals. Once those signals have been identified as the production of new states the next logical step will be the determination of the underlying model among all candidates by measuring the properties and interactions of the new particles. The size of production cross sections and mass spectra might provide valuable hints about the underlying model, nevertheless, an undisputed discrimination will only be possible after determination of the spins of the new states. We showed that the discrimination between a SUSY spin interpretation against an UED one is possible in the case of scalar bottom (fermionic KK bottom) pair production in the reaction pp → bb̄/pT . Using the variable proposed in Ref. [5], see Eq. (11), we demonstrated that a clear determination of the spin of the decaying strongly interacting particle is possible provided the production cross sections and branching ratios into bχ̃01 are sufficiently large.Even in the worst scenario studied, the SPS1a spectrum, the determination of spins might be possible after background subtraction, however, a dedicated study of the impact of statistical and systematic errors on the spin determinations will be necessary in this test point. Acknowledgments We would like to thank Tilman Plehn for insightful comments and Fabio Maltoni and the CP3 team for their help with the implementation of UED model into Madgraph. This research was supported in part by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) and by Conselho Nacional de Desenvolvimento Cient́ıfico e Tecnológico (CNPq). [1] For reviews of SUSY, see e.g. : I. J. R. Aitchison, hep-ph/0505105; S. P. Martin, hep-ph/9709356. [2] S. Dawson, E. Eichten and C. Quigg, Phys. Rev. D 31, 1581 (1985). [3] W. Beenakker, R. Höpker, M. Spira and P. M. Zerwas, Nucl. Phys. B 492, 51 (1997); T. Plehn, arXiv:hep-ph/9809319; W. Beenakker, M. Krämer, T. Plehn, M. Spira and P. M. Zerwas, Nucl. Phys. B 515, 3 (1998); W. Beenakker, M. Klasen, M. Krämer, T. Plehn, M. Spira and P. M. Zerwas, Phys. Rev. Lett. 83, 3780 (1999); Propspino2.0 publicly available from www.ph.ed.ac.uk/~tplehn [4] A. J. Barr, Phys. Lett. B 596, 205 (2004); [5] A. J. Barr, JHEP 0602, 042 (2006). [6] J. 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B 638, 119 (2006). http://arxiv.org/abs/hep-ph/0505105 http://arxiv.org/abs/hep-ph/9709356 http://arxiv.org/abs/hep-ph/9809319 www.ph.ed.ac.uk/~tplehn http://arxiv.org/abs/hep-ph/0605067 http://arxiv.org/abs/hep-ph/0510418 http://arxiv.org/abs/hep-ph/0601186 http://arxiv.org/abs/hep-ph/0507284 http://arxiv.org/abs/hep-ph/0510204 http://arxiv.org/abs/hep-ph/0201195 Introduction UED interactions and parameters Event simulation and test points Analysis and results Conclusions Acknowledgments Bibliography

0704.0255

Modeling the three-point correlation function

Draft version October 26, 2018 Preprint typeset using LATEX style emulateapj v. 08/22/09 MODELING THE GALAXY THREE-POINT CORRELATION FUNCTION Felipe A. Maŕın , Risa H. Wechsler , Joshua A. Frieman and Robert C. Nichol Draft version October 26, 2018 ABSTRACT We present new predictions for the galaxy three-point correlation function (3PCF) using high- resolution dissipationless cosmological simulations of a flat ΛCDM Universe which resolve galaxy-size halos and subhalos. We create realistic mock galaxy catalogs by assigning luminosities and colors to dark matter halos and subhalos, and we measure the reduced 3PCF as a function of luminosity and color in both real and redshift space. As galaxy luminosity and color are varied, we find small differences in the amplitude and shape dependence of the reduced 3PCF, at a level qualitatively con- sistent with recent measurements from the SDSS and 2dFGRS. We confirm that discrepancies between previous 3PCF measurements can be explained in part by differences in binning choices. We explore the degree to which a simple local bias model can fit the simulated 3PCF. The agreement between the model predictions and galaxy 3PCF measurements lends further credence to the straightforward association of galaxies with CDM halos and subhalos. Subject headings: cosmology: large-scale structure of universe — galaxies: formation — galaxies: statistics — galaxies: halos 1. INTRODUCTION Observations of the higher-order statistics of the galaxy distribution can provide fundamental tests of the stan- dard cosmological model. For example, higher-order cor- relation functions of the mass are predicted to be zero in linear perturbation theory for Gaussian initial condi- tions, which are expected in the simplest inflation models of the early Universe (Peebles 1980, Bernardeau et al. 2002, Szapudi 2005 and references therein). In the late Universe, however, non-linear gravitational cluster- ing and biased galaxy formation lead to non-Gaussianity in the galaxy density field, resulting in non-zero con- nected N-point correlation functions (NPCFs) with N > 2. By studying higher-order galaxy statistics on large scales, we can test the nature of the initial conditions; on smaller scales, the NPCFs can constrain models of biased galaxy formation (e.g., Fry & Gaztañaga 1993; Frieman & Gaztañaga 1994) and the relationship be- tween galaxies and their host dark matter halos. The three-point correlation function (3PCF, or ζ), and its Fourier-space equivalent, the bispectrum, are the first in the hierarchy of higher-order statistics and measure the shape dependence of the number of galaxy triplets as a function of scale. The 3PCF is sensitive to, for example, the shapes of dark matter halos and the presence of filamentary structures in the large-scale structure of the Universe (Sefusatti & Scoccimarro 2005). Since the pioneering work of Peebles and col- laborators (see Peebles 1980 and references therein, 1 Department of Astronomy & Astrophysics, Kavli Institute for Cosmological Physics, The University of Chicago, Chicago, IL 60637 USA 2 Kavli Institute for Particle Astrophysics & Cosmology, Physics Department, and Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94305 3 Center for Particle Astrophysics, Fermi National Accelerator Laboratory, P.O. Box 500, Batavia, IL 60510 USA 4 Institute of Cosmology & Gravitation, University of Portsmouth, Portsmouth, PO1 2EG, UK 5 e-mail: [email protected] in particular Groth & Peebles 1977), there have been many measurements of the 3PCF and bispectrum using a variety of angular (Peebles & Groth 1975; Groth & Peebles 1977; Gaztañaga & Frieman 1994; Frieman & Gaztañaga 1999; Huterer et al. 2001; Szapudi et al. 2001, 2002; Ross et al. 2006) and red- shift (Gaztañaga & Frieman 1994; Jing & Börner 1998; Verde et al. 1998; Scoccimarro et al. 2001a; Feldman et al. 2001; Jing & Börner 2004) catalogs. There has also been considerable theoretical work to understand the 3PCF and bispectrum using non-linear perturbation theory (see Bernardeau et al. 2002 and references therein), the halo model (see Ma & Fry 2000; Scoccimarro et al. 2001b; Wang et al. 2004; Fosalba et al. 2005), and cosmological simulations (Barriga & Gaztañaga 2002; Scoccimarro et al. 1999; Gaztañaga & Scoccimarro 2005; Hou et al. 2005). In recent years, there has been renewed inter- est in the 3PCF due to the availability of large redshift surveys. These surveys now provide both the volume and the number of galaxies required to make robust measurements of the 3PCF over a range of scales. For example, recent papers by Kayo et al. (2004), Hikage et al. (2005), Nichol et al. (2006), Nishimichi et al. (2006), Kulkarni et al. (2007) have presented measurements of the 3PCF from the Sloan Digital Sky Survey (SDSS; York et al. 2000) as a function of scale, galaxy luminosity, and color. Nichol et al. (2006) also quantified the effect of large- scale structures on the shape dependence of the 3PCF. Likewise, several recent papers (Jing & Börner 2004; Baugh et al. 2004; Croton et al. 2004; Gaztañaga et al. 2005; Pan & Szapudi 2005; Croton et al. 2006) provide new measurements of the 3PCF and high-order correla- tions from the Two-degree Field Galaxy Redshift Survey (2dFGRS; Colless et al. 2001). The main results from these recent SDSS and 2dF- GRS analyses of the 3PCF are: i) on large scales, the observed 3PCF is in qualitative agreement with expec- http://arxiv.org/abs/0704.0255v2 2 Maŕın, Wechsler, Frieman and Nichol tations for the growth of structure from Gaussian initial conditions; ii) on smaller scales (i.e., in the non-linear and weakly non-linear regimes), the 3PCF measured in redshift space scales with the redshift-space two-point correlation function (2PCF, or ξ) as ζ ∼ ξ2, consis- tent with the “hierarchical clustering” ansatz (Peebles 1980); iii) the shape dependence of the reduced 3PCF depends at most weakly on galaxy luminosity; iv) the amplitude of the 3PCF is larger for elongated triangle configurations than for more symmetric triangle shapes (Gaztañaga & Scoccimarro 2005; Gaztañaga et al. 2005; Nichol et al. 2006; Kulkarni et al. 2007) — again consis- tent with expectations from non-linear clustering theory. It is important to make detailed comparisons of these 3PCF observations with theoretical predictions that in- corporate a realistic prescription for modeling galax- ies and that account for observational effects such as redshift-space distortions. Such comparisons can be car- ried out in two ways: either the observations can be corrected for the redshift-space effects and compared to theory in real space, e.g., using the projected 3PCF (Zheng 2004), which is analogous to the projected 2PCF (Zehavi et al. 2005), or one can build mock galaxy cata- logs from cosmological simulations and measure the the- oretical 3PCF directly in redshift space. Since we also have access to the real-space 3PCF from such mock cata- logs, we can investigate in detail the relationship between the real- and redshift-space correlation functions. In this paper, we pursue the second of these method- ologies, using state-of-the-art high-resolution dissipation- less dark matter cosmological simulations. These simu- lations have the spatial resolution required to identify the dark matter (DM) halos and subhalos that host indi- vidual galaxies and at the same time encompass a large enough volume to probe large-scale structure in a sta- tistically reliable way. The model we use assigns galaxy properties (luminosity, color, etc.) directly to these DM galactic halos and subhalos using simple, empirically- based assumptions about these properties. This ap- proach differs in both assumptions and the resolution required from methods which build galaxy catalogs by statistically assigning several galaxies to each (more mas- sive) halo using a Halo Occupation Distribution (HOD; e.g. Berlind & Weinberg 2002; Wang et al. 2004; see Kulkarni et al. (2007) for constraints on HOD param- eters from the SDSS Luminous Red Galaxy sample’s 3PCF) or from semi-analytic methods. The method used here was first implemented by Kravtsov et al. (2004), and has been applied successfully to different statisti- cal studies, including the 2PCF (Conroy et al. 2006), galaxy-galaxy lensing (Tasitsiomi et al. 2004), and close pair statistics (Berrier et al. 2006) among others (see also Vale & Ostriker 2004, 2006; Conroy et al. 2007). Here we extend the study of this model to the 3PCF as a function of luminosity, color, and redshift. Where pos- sible, we make direct comparisons in redshift space with recent observations. In §2, we describe the simulations used in this paper and our methods for constructing mock galaxy catalogs based on resolved DM halos. We also review the tech- niques used to estimate the NPCFs. In §3, we present measurements of the 3PCF in both real and redshift space for both the dark matter and galaxy catalogs, while in §4, we study the dependence of the model 3PCF on galaxy luminosity and color. In §5, we compare the model 3PCF with SDSS observations and discuss the ef- fects of binning. We relate the 3PCF to a simple non- linear bias model in §6. In §7, we summarize and discuss our findings. 2. METHODS 2.1. Dark matter simulations We investigate clustering statistics using cosmological N -body simulations of structure formation in the concor- dance, flat ΛCDM cosmology with ΩΛ = 0.7 = 1 − Ωm, h = 0.7, and σ8 = 0.9, where Ωm, ΩΛ are the present matter and vacuum densities in units of the critical den- sity, h is the Hubble parameter in units of 100 km s−1 Mpc−1, and σ8 specifies the present linear rms mass fluc- tuation in spheres of radius 8 h−1Mpc. The simulations used here were run using the Adaptive Refinement Tree N−body code (ART, see Kravtsov et al. 1997 for de- tails), which implements successive refinements in space and time in high-density environments. The primary simulation box we use is 120 h−1Mpc on a side (hereafter, L120); the number and mass of each dark matter particle are Np = 512 3 ≈ 1.34× 108 and mp = 1.07× 10 9h−1M⊙ respectively. This simulation has been previously used to measure several properties of dark matter halos and subhalos (e.g., Allgood et al. 2006; Wechsler et al. 2006; Conroy et al. 2006; Berrier et al. 2006). In order to in- clude more massive halos and study the effects of the size of the sample on the statistical analysis, we also use a second simulation with the same cosmological param- eters in a bigger box, with 200 h−1Mpc on a side (which was also used to measure halo shapes in Allgood et al. 2006). This box contains Np = 256 3 particles with mass mp = 3.98× 10 10 h−1M⊙, therefore it will lack low mass (and luminosity) objects that are included in the L120 From these dark matter samples, virialized concentra- tions of particles are identified as halos. In order to find these halos and their constituent subhalos (concentra- tions of virialized matter inside bigger halos), a variant of the Bound Density Maxima halo finding algorithm of Klypin et al. (1999) is used. This algorithm assigns den- sities to each particle using a smoothing kernel on the 32 nearest neighbors; centering on the highest-overdensity particle, each center is surrounded by a sphere of radius rfind = 50h −1kpc. The algorithm removes unbound par- ticles when calculating the properties of the halos. The halo catalog is complete for halos with more than 50 particles, which corresponds to a minimum halo mass of 1.6×1010 h−1M⊙ for the L120 box and 2.0×10 12 h−1M⊙ for halos in the L200 box. Henceforth, we will use the terms “distinct halo” to mean any halo that is not within the virial radius of a larger halo, “subhalo” to indicate a halo that is within the virial radius of a larger halo, and “galactic halo” to refer to the halo directly hosting a galaxy. Using this terminology, the galactic halo of a satellite galaxy is a subhalo while the galactic halo of a central or isolated galaxy will be a distinct halo. 2.2. From halos to ‘galaxies’ It is expected that galaxy properties depend in detail not only on the dark matter clustering, but also on the Modeling galaxy three-point statistics 3 gas dynamics, radiative processes, and feedback mecha- nisms that affect the baryonic components. A program to include all those physical processes in building mock galaxy catalogs from simulations would require introduc- ing a number of free parameters and assumptions which could partially obscure the relevant mechanisms that de- termine the shape and amplitude of the 3PCF. Our ap- proach is instead more empirical—to associate galaxies of given properties with simulated dark matter halos and subhalos, using halo properties that we can measure in the simulation, and to see whether this one-to-one corre- spondence predicts a galaxy 3PCF that is consistent with the observations. As described below, the assignment of galaxies to halos was designed to reproduce certain fea- tures of the observed galaxy distribution, but the 3PCF was not one of these. As a result, the 3PCF constitutes a non-trivial test of this approach. The primary galaxy samples used here are created by assigning galaxy luminosities and colors drawn from the SDSS redshift survey to dark matter halos and subhalos, using the maximum circular velocity at z = 0, Vmax, as an indicator of the halo virial mass. Vmax has been found to be a good proxy of the galaxy potential well, which is a good indicator of stellar mass (Kravtsov et al. 2004; Conroy et al. 2006). A galaxy luminosity in the r−band is assigned to each halo by matching the cumulative velocity function n(> Vmax) of all galactic halos (distinct halos and sub- halos) to the observed SDSS r-band luminosity function (Blanton et al. 2003b) at z = 0.1 (the approximate mean redshift of the main spectroscopic SDSS galaxy sample used to estimate the luminosity function). To correct to z = 0 magnitudes, we use the code kcorrect v3 2 (Blanton et al. 2003a). Since the limited size of the box gives us an upper limit on the luminosities which can be reliably studied (in the statistical sense), and at the same time we cannot sample the lowest-luminosity ob- jects due to limited spatial resolution, for the L120 box we present results for galaxies in the absolute magnitude range −19 ≥ Mhr ≥ −22, where M r ≡ Mr − 5 logh. The L200 box has a lower spatial resolution, therefore it contains only brighter objects, with Mhr ∼ − 20. In order to assign colors to the galaxies, we use the proce- dure described by Wechsler (2004) and Tasitsiomi et al. (2004). This method uses the relation between local galaxy density (defined as the distance to the tenth near- est neighbor brighter than Mhr = −19.7 and within cz = 1000 km/s) and color observed in the SDSS, using the CMU-Pitt Value Added Catalog constructed from DR1 (Abazajian et al. 2003), to assign a color to each mock galaxy. We use the distant observer approxima- tion (Bernardeau et al. 2002) to obtain positions in red- shift space. Table 1 describes the subsamples used in this study. Although the general association of galaxies with dark matter subhalos seems quite robust, the detailed asso- ciation of galaxy properties with subhalo properties is less clear. While galaxy luminosity is expected to be quite tightly connected to velocity, the circular velocity of a given subhalo decreases with time due to tidal strip- ping as it interacts with its host halo. Observable galaxy properties such as luminosity and color will likely be less affected by this process. This implies that galaxy ob- TABLE 1 Subsamples at z = 0 Box Subsample Number Density of objects [(h−1Mpc)−3] L120 All objects Mhr < −19 25371 1.5× 10 L120 −19 < Mhr < −20 15432 8.9× 10 L120 −20 < Mhr < −21 8153 4.7× 10 L120 red (g − r > 0.7) 10059 5.8× 10−3 L120 blue (g − r < 0.7) 15312 8.9× 10−3 L200 All objects Mhr < −20 43564 5.4× 10 L200 −20 < Mhr < −21 30575 3.8× 10 L200 −21 < Mhr < −22 9921 1.2× 10 L200 red (g − r > 0.7) 17278 2.2× 10−3 L200 blue (g − r < 0.7) 23748 2.9× 10−3 servables may be more strongly correlated with Vmax,acc, the maximum circular velocity of the halos at the mo- ment of accretion onto their host, than with the current maximum circular velocity, Vmax,now. This conclusion is supported by measurements of two-point statistics on both large and small scales in simulations (Conroy et al. 2006; Berrier et al. 2006). Motivated by these consider- ations, we construct galaxy catalogs using both Vmax,acc and Vmax,now. We also use galaxy catalogs at different redshifts in order to study the evolution of the 3PCF with time. All these additional catalogs use the L120 box and have the same spatial density as the sample of halos selected by Vmax,now. We note that the model for as- signing color is also not uniquely determined. It is, how- ever, sufficient to match the two-point clustering length for red and blue galaxies and several observed properties of galaxy clusters (Zehavi et al. 2005). Future measure- ments of both clustering statistics and of the properties of galaxies in groups and clusters should help to further refine these galaxy assignment models. 2.3. Measuring the 3PCF Just as the 2PCF measures the excess probability of finding two objects separated by a distance r, the 3PCF describes the probability of finding three objects in a particular triangle configuration compared to a random sample. The probability of finding three objects in three arbitrary volumes dV1, dV2, and dV3, at positions r1, r2 and r3 respectively, is given by (Peebles 1980) P = [1 + ξ(r12) + ξ(r23) + ξ(r31) + ζ(r12, r23, r31)]× n̄3dV1dV2dV3, (1) where n̄ is the mean density of the objects, rij ≡ ri − rj the distance between two objects, ξ is the 2PCF, and ζ is the 3PCF: ξ(r12)= 〈δ(r1)δ(r2)〉 (2) ζ(r12, r23, r31)= 〈δ(r1)δ(r2)δ(r3)〉; (3) here δ is the fractional overdensity in the dark matter field or in the distribution of galaxies. Since the 3PCF depends on the configuration of the three distances, it is sensitive to the 2-D shapes of the spatial structures, at large and small scales (Sefusatti & Scoccimarro 2005; Gaztañaga & Scoccimarro 2005). Motivated by the “hi- erarchical” form of the N-point functions, ζ ∝ ξ2, found 4 Maŕın, Wechsler, Frieman and Nichol by Peebles & Groth (1975), we use the reduced 3PCF Q(r, u, α) to present our results: Q(r, u,α ) = ζ(r, u, α) ξ(r12)ξ(r23) + ξ(r23)ξ(r31) + ξ(r31)ξ(r12) . (4) This quantity is useful since Q is found to be close to unity over a large range of scales even though ξ and ζ vary by orders of magnitude (Peebles 1980). To parametrize the triangles for the 3PCF measurements, r ≡ r12 sets the scale size of the triangle, while the shape param- eters are given by the ratio of two sides of the trian- gle, u = r23/r12, and the angle between the two sides of the triangle α = cos−1(r̂12 · r̂23), where r̂12, r̂23 are the unit vectors of the first two sides. Following Gaztañaga & Scoccimarro (2005), triangles where α is close to 0◦ or 180◦ are referred to as “elongated config- urations”, while those with α ∼ 50◦ − 120◦ are referred to as “rectangular configurations”, We calculate the 2PCF using the estimator of Landy & Szalay (1993), DD − 2DR+RR . (5) Here, DD is the number of data pairs normalized by ND ×ND/2, DR is the number of pairs using data and random catalogs normalized by NDNR, and RR is the number of random data pairs normalized by NR×NR/2. The 3PCF is calculated using the Szapudi & Szalay (1998) estimator: DDD − 3DDR+ 3DRR−RRR , (6) where DDD, the number of data triplets, is normalized by N3D/6, and RRR, the random data triplets, is nor- malized by N3R/6. DDR is normalized by N DNR/2, and DRR by NDN We estimate the errors using jack-knife re-sampling. From each galaxy catalog, we construct sixteen subsam- ples of L120 or L200; within each of them we remove a different region (30 × 602 (h−1Mpc)3 for the L120 box, and 50×1002 (h−1Mpc)3 for the L200 box). The variance σJK of Q is calculated as: σ2JK = N − 1 (Qi − Q̄i) 2, (7) where N = 16 is the number of subsamples, Qi is the value for the i−th subsample, and Q̄i is the mean of the Qi. We note in passing that the validity of jack-knife resampling as a method to estimate the errors has not been explicitly tested with mock catalogs for three-point statistics. Although it is beyond the scope of this paper, this may be an interesting topic of future investigation especially once the statistical power of the measurements improves. To compute the 2PCF and 3PCF, we use the NPT software developed in collaboration with the Auton Lab at Carnegie Mellon University. NPT is a fast implemen- tation of the NPCFs using multi-resolution kd-trees to compute the number of pairs and triplets in a dataset. For more details and information on the algorithm, see Moore et al. (2001), Gray et al. (2004), and Nichol et al. (2006). 3. THE 3PCF OF GALAXIES AND DARK MATTER We estimate the reduced 3PCF for the distribution of dark matter and for galaxies for different triangle con- figurations, focusing on the scale and shape dependence of the 3PCF. We also investigate its time evolution and how it depends on the selection criterion for subhalos. We study Q(r, u, α) in both real and redshift space, in order to compare our results with current observations and disentangle galaxy biasing effects from those which are consequences of redshift distortions. In order to distinguish scale and, most importantly, shape effects, and to keep the errors as small as possi- ble, we have chosen an intermediate-resolution binning scheme. For studies of equilateral triangles (u = 1 and α = π/3 rad), we use bins of size ∆ log(r) = 0.1. For measurements of the shape dependence of the 3PCF, we use triangles with four different scales r = 0.75, 1.5, 3, 6, and 9 h−1Mpc, using the shape parameters u = 2, and 15 angular bins separated by ∆α = π/15 rad; the resolution of the bins is given by ∆rij = ±0.03rij. This resolution is sufficient to see the most important features of the 3PCF even on small scales, although it is not sufficient to dis- tinguish the “finger-of-god” effect at the smallest scales in redshift space, where Q(α) varies very little except at very small or elongated angles, where it increases to many times the mean value (Gaztañaga & Scoccimarro 2005). 3.1. The 3PCF in real space Figures 1 and 2 show the 3PCF for dark matter par- ticles and for galaxies from the L120 and L200 simula- tions. Here we plot results for dark matter (thick solid line), galaxies in halos selected by Vmax,now (thin solid line), and galaxies in halos selected by Vmax,acc (long- dashed line) for z = 0 and for halos selected by Vmax,now at z = 1 (short-dashed line) and z = 2 (dotted line). Figure 1 shows the reduced 3PCF for equilateral trian- gles, Qeq(r), in real (left panel) and redshift space (right panel). In real space, the reduced 3PCF for the dark matter is only weakly scale dependent on small scales, decreases rapidly with increasing scale around r ∼ 3h−1 Mpc, and falls off more slowly on larger scales. This behavior is broadly consistent with previous N-body re- sults (e.g. Scoccimarro et al. 1998) and with expectations from leading order non-linear perturbation theory on the largest scales (shown as the thick red long-dashed curve in Figure 1 left panel), with loop-corrected perturbation theory on intermediate scales where the rms perturba- tion amplitude δ(r) is of order unity (the transition to the strongly non-linear regime; adding more orders to the calculation would increase the agreement to the N -body dark matter 3PCF amplitude), and with quasi-stable hi- erarchical clustering on the smallest scales. Tests with the L200 box indicate that the downturn in Qeq for dark matter at scales larger than r ∼ 8h−1 Mpc is likely due to finite volume effects. At scales below r ∼ 10h−1 Mpc, the dark matter Qeq is larger than that for the galaxies; this behavior is broadly expected if galaxies are more strongly clustered than (positively biased with respect to) the mass, cf. eqn.(10). Modeling galaxy three-point statistics 5 Fig. 1.— The reduced 3PCF, Qeq(r), for equilateral triangles in the L120 box. Left: Results in real space. Thick solid line (black): dark matter; thin solid line (black): galactic halos selected by Vmax,now at z = 0; short-dashed (green): galactic halos selected by Vmax,now at z = 1; dotted (red): galactic halos selected by Vmax,now at z = 2; long-dashed (blue): galactic halos selected by Vmax,acc at z = 0; thick long-dashed (red): leading-order perturbation theory, dark matter. Right: Results in redshift space. Line types correspond to the same dark matter and halo samples as in the left-hand plot. Error bars are calculated using jack-knife resampling and are only shown for one of the samples for clarity. At higher redshift, evolution is seen in Qeq(r) that is con- sistent with expectations from non-linear gravitational evolution: on the largest scales, the amplitude of Qeq is unchanging, as predicted from leading order perturbation theory, while the sharp break associated with the tran- sition to the strongly non-linear regime moves to larger scales as the density perturbation amplitude increases with time. Comparing results for subhalos selected by Vmax,now and by Vmax,acc, differences in the amplitude of Qeq ap- pear on small scales, r . 3 h−1 Mpc. In halo-model language, on these scales the 3PCF is sensitive to the internal structure of halos, i.e., to the one- and two-halo terms, while the three-halo term dominates the 3PCF on larger scales (Wang et al. 2004; Takada & Jain 2003). In Figure 2, the top panels show how the reduced 3PCF depends on triangle shape in real space. In general, the 3PCF for elongated configurations is greater than for rectangular configurations. This is a consequence of the fact that, in non-linear gravitational instability, ve- locity flows tend to occur along gradients of the density field (Bernardeau et al. 2002). The 3PCF is larger for dark matter than for galaxies for all shapes and scales, although the difference is larger for elongated configura- tions. The difference in 3PCF amplitude between rect- angular and elongated configurations is larger on large scales, in broad agreement with leading-order theoreti- cal predictions (Bernardeau et al. 2002): on large scales, the strong shape dependence is determined by pertur- bative non-linear dynamics; on smaller scales, the shape dependence is washed out since the coherence between the velocity and density fields gives way to virialized mo- tions. This scale dependence of the 3PCF shape is also reflected in the redshift evolution: in the L120 box (left panel), the galaxy 3PCFs at z = 1 and 2 (green-dashed and red-dotted curves) essentially retain the primordial shape dependence of leading-order non-linear perturba- tion theory, i.e., at those redshifts, these scales are still close to the quasi-linear regime. At r = 3 h−1Mpc, the largest evolution in Q(α) is found for elongated configu- rations. As was seen in Figure 1, the effect of changing halo se- lection from Vmax,acc to Vmax,now on the 3PCF shape ap- pears only on small scales, r . 1.5h−1 Mpc, i.e., roughly within the scale of a typical cluster-mass host halo. On the larger scales probed in the L200 box (right panel of Figure 2), the galaxy reduced 3PCF (open cir- cles) tracks the shape of the dark matter 3PCF fairly well. The difference between the galaxy and dark matter 3PCF amplitudes on these scales is reasonably well fit by a simple bias prescription: the thin blue curve is the biased 3PCF that results from fitting the galaxy 3PCF with eqn. (10); see §6. We also see that the jack-knife errors increase on the largest scales, where the effects of the finite box size start to become evident. For compar- ison, the red long-dashed curve is the 3PCF of the dark matter from leading-order non-linear perturbation the- ory (Bernardeau et al. 2002; Jing & Borner 1997). On the largest scales, it is in reasonable agreement with the measured 3PCF for the dark matter. 3.2. The 3PCF in redshift space Redshift distortions have been studied in depth (and are a useful tool to constrain cosmological parameters) for the power spectrum (e.g., Bernardeau et al. 2002; da Ângela et al. 2005; Tinker 2007) and for the bis- pectrum (Scoccimarro et al. 1999; Verde et al. 2002; Sefusatti et al. 2006). Some comparisons have been made for the 3PCF as well (Matsubara & Suto 1994; Takada & Jain 2003; Wang et al. 2004). Gaztañaga & Scoccimarro (2005) found that the redshift distortions do not have a strong dependence on the cosmological parameters. The right panel of Figure 1 and the bottom panels 6 Maŕın, Wechsler, Frieman and Nichol 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 Fig. 2.— Measurement of the reduced 3PCF as a function of triangle shape, Q(α), for different scales r, with side ratio u = r23/r12 = 2 fixed. Left: Results in the L120 box for r = 0.75, 1.5, and 3 h−1Mpc (from left to right) in real space (top panels) and redshift space (bottom panels) for dark matter and galaxies; line types correspond to the same dark matter and halo samples as in Figure 1. Right: Results in the L200 box for r = 6 (left) and 9 h−1Mpc (right) in real (top) and redshift space (bottom); Thick solid line (black): dark matter; open circles: galaxies in halos selected by Vmax,now at z = 0; thin solid line (blue): predicted galaxy 3PCF using the dark matter 3PCF Qdm and eqn. (10) with best-fit bias parameters c1, c2 obtained from fitting the galaxy 3PCF at r = 9h −1 Mpc (see §6); long dashed (red): leading order non-linear perturbation theory prediction for dark matter reduced 3PCF. Error bars calculated using jack-knife resampling method. of Figure 2 show the 3PCF in redshift space. The first feature that can be seen is a dramatic decrease in the amplitude and in the scale and shape dependence of Q compared to the real-space measurements. For exam- ple, for equilateral triangles, the redshift space Qz(r) is reduced compared to the real space Q(r) at small scales and increased with respect to the real-space re- sults on larger scales. The overall effect is that Qz(r) is nearly scale-independent, i.e., the clustering appears more hierarchical in redshift space (Suto & Matsubara 1994; Matsubara & Suto 1994; Scoccimarro et al. 1999). Moreover, in redshift space, the suppression of the galaxy 3PCF relative to that of the dark matter is apparent on all scales; it appears to be relatively independent of scale and configuration and is larger than the relative suppression in real space, consistent with earlier results (Wang et al. 2004; Gaztañaga et al. 2005). It also ap- pears that there is very little redshift evolution of the galaxy 3PCF in redshift space; the measurements at z = 1 and z = 2 are nearly indistinguishable from each other. With regard to halo selection, as in real space we find that QVmax,now < QVmax,acc, but the differences between them are smaller than in real space. Together, these results suggest that the shape and scale dependence of the reduced 3PCF in redshift space on the scales shown here are largely determined by redshift dis- tortions, with non-linear gravitational evolution playing a subdominant role. 4. OBSERVING THE 3PCF: LUMINOSITY AND COLOR DEPENDENCE To investigate the dependence of the three-point clus- tering on galaxy luminosity and color and to make direct comparisons with measurements from recent redshift sur- veys, we calculate the 3PCF for galaxies with luminos- ity and color cuts similar to those that have been ap- plied to redshift survey data samples, with luminosity and color information obtained as described in §2. The luminosities are assigned according to Vmax,now in the L120 and L200 boxes, since we hace those measurements for both boxes. As seen in the previous section, the use of Vmax,now is justified since the differences with respect to Vmax,acc are only significant on small scales (r<∼ 1.5 h−1Mpc) in real space and are almost completely dis- sapear on redshift space. We adopt the same binning scheme used in the previous section. 4.1. Luminosity dependence The left panels of Figure 3 show results for the re- duced 3PCF for equilateral configuratons in two lumi- nosity bins, in real and redshift space. For equilateral triangles, there is a small difference in 3PCF between the luminosity samples in real space: the fainter galaxies have larger Q(r) than the brighter ones, as expected in a simple linear bias model (eqn. 10). The redshift-space 3PCFs for these galaxies are roughly constant with r, Qz,gal(r) ∼ 0.7, in agreement with SDSS measurements for these configurations (see Figures 7-9 in Kayo et al. 2004). There is a very slight difference between the re- duced 3PCF amplitudes for different luminosity bins in redshift space, but it is not statistically significant for a dataset of this size. This result qualitatively agrees with the SDSS results of Kayo et al. (2004) who also found almost no luminosity dependence of the reduced 3PCF on these scales. They found a slightly higher amplitude for the reduced 3PCF for the −19 > Mhr > −20 sample compared to that of brighter galaxies, but their results for the two luminosity samples were consistent within the error bars. The brightest galaxies (dotted line) show significant fluctuations in both real and redshift space; we think this behavior is due to the small size of the box Modeling galaxy three-point statistics 7 by luminosity real space by luminosity redshift space by color real space by color redshift space Fig. 3.— The reduced 3PCF, Q(r), for equilateral triangles as a function of galaxy luminosity and color in real and redshift space. Top left : Qeq(r) in real space for galaxies divided into luminosity bins; long dash-dotted (cyan): −19 > Mhr > −20; short-dashed (green): −20 > Mhr > −21; dotted (magenta): −21 > M r > −22. The brightest sample comes from the L200 box, the other two are from the L120 box. Bottom left: Results in redshift space; line types correspond to the same galaxy samples as in the top left panel. Top right: Qeq(r) in real space for galaxies divided accord- ing to color, using the L120 box; long-dashed (red): red galaxies (g− r > 0.7); short dash-dotted (blue): blue galaxies (g− r < 0.7). Bottom right Results in redshift space; line types correspond to the same galaxy samples as in the top right panel. Error bars are calculated using jack-knife respampling. and the low density of objects. Figure 4 shows the dependence of Q(α) on galaxy lu- minosity in real (curves) and redshift space (points). We use the same ordinate scale for all the plots to empha- size where configuration effects are more important. The top panels show results for small scales, calculated with the L120 box. On these scales, the reduced redshift- space 3PCF of the brighter sample (−20 > Mhr > −21, short dashed curve for real space, filled triangles for red- shift space) is slightly lower than for the fainter sample (−19 > Mhr > −20, long dashed-dotted curve for real space, filled squares for redshift space) for all angles, consistent with the results for Qz(r) using equilateral triangles. The lower plots in Figure 4 show the lumi- nosity dependence on larger scales, measured using the L200 box; note that the luminosity bins in the lower plots are −20 > Mhr > −21 and −21 > M r > −22. The characteristic U-shape of Q(α) appears clearly in the redshift-space measurements on scales larger than r = 6 h−1Mpc. The luminosity dependence of Q on these scales appears non-existent in redshift space and only slight in real space. There are hints that the reduced 3PCF for fainter galaxies may have slightly higher am- plitude and stronger shape dependence than for brighter galaxies, but these trends are not statistically significant in the samples studied here. Rather, the strong lumi- nosity dependence observed for the 2PCF (Zehavi et al. 2005) appears to be closely matched by a correponding dependence of the 3PCF, such that the reduced 3PCF Q is roughly independent of luminosity. L120 box L120 box 0 50 100 150 L200 box 0 50 100 150 L200 box Fig. 4.— Q(α) shape dependence for galaxies divided by lumi- nosity, for r = 1.5, 3, 6, and 9 h−1Mpc, with fixed u = 2 in real (lines) and redshift space (symbols). Top two plots are re- sults for the L120 box: filled squares and long dash-dotted (cyan): −19 > Mhr > −20; filled triangles and short-dashed (green): −20 > Mhr > −21; triangles are slightly shifted to the right for clarity. Bottom plots show results for the L200 box: filled trian- gles and short-dashed (green): −20 > Mhr > −21; filled circles and dotted (magenta): −21 > Mhr > −22; circles are slightly shifted to the right for clarity. Error bars are calculated using jack-knife resampling and are shown only for one of the samples. L120 box L120 box 0 50 100 150 L200 box 0 50 100 150 L200 box Fig. 5.— Q(α) shape dependence for galaxies divided by color, in the L120 box (top) and L200 box (bottom) in real (lines) and redshift space (symbols). Open squares and long-dashed line (red): red (g − r > 0.7) galaxies. open triangles and short dash-dotted (blue): blue (g − r < 0.7) galaxies; squares are slightly shifted to the right for clarity. Error bars are calculated using jack-knife resampling and are shown only for one of the samples. 4.2. Color dependence The right panels of Figure 3 show the reduced 3PCF Q(r) for equilateral triangles separately for red and blue galaxies in real (top) and redshift space (bottom). Due to limited statistics, we use the full luminosity range in each color bin, as opposed to Kayo et al. (2004), who di- 8 Maŕın, Wechsler, Frieman and Nichol vided the color bins into luminosity subsamples as well (see their Figure 9). Since the luminosity functions for red and blue galaxies differ, our red and blue samples have different characteristic luminosities — the red sam- ple is on average brighter. In both real and redshift space, the red galaxies appear to have a slightly higher reduced 3PCF amplitude. Comparing with the left panels, this difference is in the opposite sense from that expected from the fact that the red galaxies are brighter; put an- other way, if we were to compare red and blue samples of the same luminosity, the color difference in Q would likely be larger than that seen here. On the other hand, we should not overinterpret these trends, since the differ- ences are within the statistical errors. Kayo et al. (2004) found a similarly weak dependence of Q in redshift space on color. Figure 5 shows the color dependence of Q(α) in real (curves) and redshift space (points), for the same sam- ples and configurations as in Figure 4. In real space, on scales larger than about 3 h−1 Mpc, Q for red galaxies is larger for rectangular configurations and smaller for elongated configurations than for blue galaxies. This be- havior is qualitatively consistent with a picture in which red galaxies preferentially occupy the inner regions of clusters, while blue galaxies tend to trace out more el- liptical or filamentary structures. In redshift space, the trend with color is largely washed out. These redshift space results appear more consistent with the observa- tions of the 2dFGRS (Gaztañaga et al. 2005), where the color differences for Q in redshift space are smaller than those seen in the SDSS by Kayo et al. (2004). The fact that red galaxies have a larger two-point clus- tering amplitude than blue galaxies and that the reduced 3PCF for red galaxies is also larger than for blue galax- ies (at least for rectangular configurations) suggests that red galaxies have a larger quadratic (non-linear) bias, Cf. eqn. 10. This is qualitatively consistent with the ob- served morphology-density or color-density relation ac- cording to which red galaxies are preferentially found in dense regions, since the latter contribute more strongly to the higher-order correlations. The qualitative agree- ment between the two- and three-point observations and our method for assigning galaxy colors confirms that the color of a galaxy, a consequence of many physical pro- cesses occurring inside the galaxies, depends largely on the surrounding environment. However, we note that the color assignment for these subhalos is the most un- certain part of the model; future work will be required to determine how clustering statistics will change if more sophisticated schemes are adopted. 4.3. Redshift distortions: Galaxy Type and Evolution So far, we have explored some differences between mea- surements in real and redshift space of the 3PCF for dif- ferent simulated galaxy samples. Here we investigate in more detail whether the redshift distortions of the 3PCF are universal or instead depend on the type of galaxy studied and to what extent they evolve with time. We study the behavior of the quantity ∆Q(r, u, α) ≡ Qz(r, u, α)−Qr(r, u, α), (8) where Qr, Qz represent the reduced 3PCF measured in real and redshift space, respectively. We explore ∆Q L120 box dark matter L120 box blue galaxies red galaxies L120 box L200 box Fig. 6.— ∆Q(r, u, α) ≡ Qz(r, u, α)−Qr(r, u, α) as a function of galaxy type and redshift. Top: ∆Q for dark matter at z = 0 and for galaxy samples at different redshifts in the L120 box, for equilateral triangles (top left) and as a function of angle for triangles with r = 3 h−1Mpc and u = 2 (top right). Bottom: ∆Q as a function of galaxy type for equilateral triangles in the L120 box (bottom left) and as a function of angle for r = 6 h−1Mpc and u = 2 in the L200 box (bottom right). as a function of galaxy type (luminosity and color) and epoch. Figure 6 shows ∆Q for equilateral triangles as a function of scale and also the configuration dependence for triangles with fixed r, u = 2, and different opening angles α. In general, the trends are similar to those seen above: Qz < Qr for small scales and for elongated triangle con- figurations, while the opposite behavior is seen for larger scales and for rectangular configurations. At z = 0, for equilateral triangles ∆Q(r) appears to display a roughly universal scale dependence, independent of galaxy type, with ∆Q(r) ≃ 0.67r−2 − 2.6r−1 − 0.02r + 0.88 over the range 0.5 ≤ r ≤ 5 h−1Mpc. On the other hand, for r = 6h−1 Mpc and u = 2, the shape dependence of ∆Q shows more dependence on galaxy type, with blue galaxies having larger values than red galaxies and bright galaxies smaller values than faint galaxies. Note that the redshift distortions of Q appear insensitive to whether the subhalos are identified at the present or at the time they are first accreted onto host halos. We see clear evolution of ∆Q with redshift, as expected since redshift distortion effects should become more pro- nounced as perturbations become more non-linear. For equilateral triangles, the upper left panel of Figure 6 shows that the scale r where ∆Q ∼ 0 increases with time. The shape dependence of ∆Q at fixed scale also appears to increase with time. 5. OBSERVING THE 3PCF: COMPARISON TO SDSS DATA AND BINNING EFFECTS Below, we compare our results with measurements of the 3PCF from the SDSS by Nichol et al. (2006). The SDSS sample is magnitude limited, with mr < 17.5, and has additional cuts in absolute magnitude, −19 < Mr < −22, and in redshift, 0.05 < z < 0.15. In order to com- pare with this data, we randomly resample the galaxies from the L120 simulation (in particular, the Vmax,noew Modeling galaxy three-point statistics 9 Fig. 7.— Top: Qz(r, u = 2, α) at r = 1.0, 2.0 and 4.0 h −1Mpc from SDSS observations (green points) from Nichol et al. (2006) and from our pseudo-flux-limited-sample in the L120 box (long- dashed lines) with the same binning scheme used in the mentioned paper. Bottom: Effect of binning in the 3PCF measurements. Qz(r, u = 2, α) at r = 2.0 and 4.0 h −1Mpc in the L120 box and r = 10 h−1Mpc in the L200 box for the simulations in a volume- limited sample. solid (red): results using a wide binning scheme (Nichol et al. 2006); long-dashed (black): results with narrow bin- ning. In the middle panel, we plot the results using SDSS data with wide (green points) and narrow (triangles) binning. sample at z = 0), so that they have the same absolute magnitude distribution as the SDSS flux-limited sample used in Nichol et al. (2006); we call these pseudo-flux- limited samples. This resampling technique does not properly model the distance-dependent selection func- tion of the SDSS, but it should reproduce its clustering properties on average. Moreover, since, as we have just shown, the luminosity dependence of the reduced 3PCF is weak, we expect that this procedure should be suffi- ciently accurate for our purposes. Nichol et al. (2006) measure the shape dependence of Q(α) for four different length scales, using wide bins in r, u, and α (see their paper for more details): ∆r = 1 h−1Mpc, ∆u = 1 and ∆α = 0.1 rad. In Figure 7, we compare the SDSS results (green points) to the cal- culation of Qz(r, u = 2, α) using our model redshift- space, volume-limited sample 3PCF (solid lines) and our pseudo-magnitude-limited sample (dotted line), each with the same binning as the data. We use the L120 box for small scales and the L200 box for the r = 10 h−1Mpc measurements. As the top panels of Figure 7 show, the model and data show good agreement within the model jack-knife error bars in amplitude as well as in the shape of Q(α). Over the range of scales shown here, both the SDSS data and the simulations show relatively little shape de- pendence for the reduced 3PCF amplitude. This is in contrast to the results of §4 above and to the 2dFGRS re- sults of Gaztañaga et al. (2005) on similar scales, where a significantly stronger shape dependence of Q(α) is evi- dent. As noted by Gaztañaga & Scoccimarro (2005) and Kulkarni et al. (2007), the differences can be traced to the binning scheme: the relatively wide binning scheme used here results in smearing and therefore suppression of the U-shape of Q over most scales of interest. This effect is not due primarily to the binning in α: small bins in both r and u are necessary to see the effects of shape-dependent clustering. To illustrate these effects in more detail, in the lower panels of Figure 7 we also show results for the same sam- ple but with a narrower binning scheme, using ∆r = 0.1 h−1Mpc (ten times smaller), ∆u = 0.2 (five times smaller), and ∆α = 0.05 rad (two times smaller). With the narrower bins (shown by the dashed curves in Figure 7), the shape-dependence ofQ is more pronounced for the simulation, especially on larger scales. For comparison, for r = 4 h−1Mpc (lower center panel), the solid trian- gles shows results for the SDSS flux-limited sample us- ing a similar narrow binning scheme Nichol et al. (2006), again showing good agreement between the model and the data. 6. GALAXY BIAS AND THE 3PCF As we have seen, the 3PCF predicted for galax- ies differs systematically from that expected for dark matter. These differences reflect differences in the spatial distributions of these two populations; higher- order statistics can therefore provide important con- straints upon the bias between galaxies and dark mat- ter (Fry & Gaztañaga 1993; Frieman & Gaztañaga 1994) and its dependence upon galaxy properties. On large scales, where the rms dark matter and galaxy overdensities are small compared to unity, it is com- mon to adopt a deterministic, local bias model (e.g. Fry & Gaztañaga 1993), δgal = f(δdm) = b1δdm + δ2dm + ..., (9) where δgal and δdm are the local galaxy and dark matter overdensities smoothed over some scale R. We can use the simulations above to test how well this simple bias prescription characterizes the galaxy distribution and its clustering statistics. Figure 8 shows the relation between δgal and δdm for all subhalos in the L200 box. The points show the over- densities for the galaxy and dark matter fields in ran- domly placed spheres of radius 10h−1 Mpc in both real (left panel) and redshift space (right panel). The solid black curves show the best quadratic fits of the form in eqn. (9). The quadratic local bias model appears to do a reasonable job in characterizing the mean rela- tion. Nonetheless, there is significant scatter, either due to stochastic bias or dependence of bias on other prop- erties than δdm, that is not captured by this simple bias model. The errors on these fits, also extended to samples divided by galaxy luminosity, are shown in Fig. 9 by the light solid contours. To leading order, this bias prescription leads to a relation between the galaxy and dark matter reduced 3PCF amplitudes of the form (Fry & Gaztañaga 1993; Gaztañaga & Scoccimarro 2005) Qgal = (Qdm + c2) , (10) where c1 = b1 and c2 = b2/b1. Also to leading order, at low overdensities the relation between the galaxy and dark matter 2PCF amplitudes in this model is given just 10 Maŕın, Wechsler, Frieman and Nichol Fig. 8.— Galaxy vs. dark matter overdensities for all galaxies in the L200 box measured in randomly placed spheres of radius 10h−1 Mpc in real (left panel) and redshift space (right panel). Solid black curve denotes best fit using the quadratic bias relation of eqn. 9. Long-dashed red line indicates the best linear bias fit (b2 = 0) to the 2PCF, i.e., estimating b = ξgal/ξdm. Short-dashed blue curve indicates best fit of eqn. 10 to the 3PCF of the galaxies and dark matter. by the linear bias, ξgal = b 1ξdm. We can test how well this bias prescription captures the clustering statistics by fitting these relations to the dark matter and galaxy 2- and 3-point correlation functions in the simulation in both real and redshift space and extracting the parame- ters c1 and c2. Since the relation (9) holds for the den- sity fields on some smoothing scale, we should fit the correlation functions for separations comparable to or slightly larger than this scale. For the 3PCF, we use triangles with u ≡ r23/r12 = 2, r = 9 h −1Mpc, and weight equally all configurations with 00 < α < 1800 using the L200 box. To calculate the likelihood func- tion, we use a method similar to that described in Gaztañaga & Scoccimarro (2005), which is based on an eigenmode analysis of the covariance matrix. We use the jack-knife subsamples in real and redshift space to construct covariance matrices, and we use only the dom- inant eigemodes with values > 2/N where N = 16 is the number of jack-knife subsamples; we found that adding further eigenmodes just increases artificially the signal. The best-fit parameter values from the 3PCF, substi- tuted into Eqn. 9, are shown as the short-dashed blue curves in Figure 8. Also, in the right panels of Fig- ure 2 we show the predicted galaxy 3PCF (solid blue curves) using the measured dark matter Qdm and the best fitting c1, c2 parameters in Eqn. (10) from the fit at r = 9h−1 Mpc. We see that the agreement with the mea- sured galaxy 3PCF is very good in both real and redshift space for triangles with r = 9h−1 Mpc; for configurations with r = 6h−1 Mpc, the agreement in real space is still quite good while in redshift space some deviations in the shape-dependence appear. In Figure 9 we show the fits for c1 and c2 for the dif- ferent galaxy samples in the L200 box in both real and redshift space: all galaxies (top panels), galaxies in the absolute range −20 > Mhr > −21 (middle), and those in the range −21 > Mhr > −22 (bottom). The thick oval contours indicate the 1 and 2σ confidence intervals for a ∆χ2 distribution with two free parameters, constrained using the 3PCF, via Eq. (10), and the large points in- dicate the maxiumum likelihood values from the 3PCF. The thin solid contours show the constraints on c1 and c2 using the fit of Eq. (9) to the measurements in Figure 8, and the vertical line in each panel shows the estimate of c1 from comparing the 2PCF amplitudes for galaxies and dark matter using pairs selected from the triplets used to measure the 3PCF. The best-fit values are shown in Ta- ble 2: the first two columns show the best-fit parameters 1 , c 2 using the reduced 3PCF (eq. 10), the next two columns, cδ1 and c 2, are calculated using the quadratic bias model (eq. 9) fit to the counts in cells, and in the last column c 1 is obtained from comparing the 2PCF of galaxies and dark matter assuming a linear bias model, c21 = ξgal/ξdm. In agreement with previous measurements in surveys and simulations, the bias parameters obtained from the 3PCF are degenerate, resulting in elongated contour el- lipses; this could be mitigated to some extent by using a larger variety of triangle configurations. In real space (left panels of Figure 9), we see that the three methods of extracting the bias parameters are in rough agree- ment: there is a preference for c1 ∼ 1 and negative c2 as was found for the 2dFGRS 3PCF measurements (Gaztañaga et al. 2005). Note that the 3PCF fit tends to overestimate c1 and to slightly overestimate c2 com- pared to the other two methods. In redshift space (right panels), the opposite is true: the 3PCF constraint tends to underestimate c1. The right panels of Figs. 8 and 9 show that there is a larger discrepancy between the 3PCF fits and the counts-in-cells fit to Eqn. (9) in red- shift space, suggesting that the relation (10) may not be a good representation in redshift space. While this comparison with the deterministic local bias model is suggestive, to test it more quantitatively one should use more triangle configurations and larger- volume catalogs to enable more precise calculation of the correlation matrices. 7. SUMMARY We have studied the 3PCF of dark matter and galax- ies in high-resolution dissipationless cosmological simu- lations. The galaxy model associates dark matter ha- los with galaxies by matching the halo velocity function, including subhalos, with the observed galaxy luminosity function by abundance, and has been shown previously to provide an excellent match to observed two-point statis- tics for galaxies (Conroy et al. 2006). Our primary re- sults are as follows: 1. The reduced real-space 3PCF for both galaxies and dark matter has strong dependence on scale and shape. The shape dependence of the 3PCF strengthens with increasing scale, in agreement with previous simulation results for dark matter and with expectations from non-linear perturba- tion theory. 2. On small scales, or alternatively with increasing time, the shape dependence of Q washes out as Modeling galaxy three-point statistics 11 TABLE 2 Best-fit bias parameters in the L200 box Subample c All objects r-space 1.16 +0.21 −0.15 -0.19 +0.19 −0.16 +0.05 −0.12 -0.20 +0.19 −0.22 1.06±0.09 All objects z-space 0.86 +0.11 −0.14 -0.31 +0.15 −0.11 +0.05 −0.07 -0.32 +0.12 −0.18 1.03±0.08 −20 < Mhr < −21 r-space 1.08 +0.28 −0.18 -0.20 +0.28 −0.23 +0.07 −0.16 -0.08 +0.08 −0.19 1.01±0.09 −20 < Mhr < −21 z-space 0.86 +0.21 −0.16 -0.30 +0.20 −0.15 +0.06 −0.12 -0.34 +0.22 −0.16 1.00±0.08 −21 < Mhr < −22 r-space 1.42 +0.48 −0.29 -0.15 +0.33 −0.35 +0.06 −0.14 -0.36 +0.31 −0.05 1.19±0.10 −21 < Mhr < −22 z-space 0.99 +0.15 −0.35 -0.20 +0.21 −0.30 +0.12 −0.05 -0.43 +0.33 −0.07 1.13±0.13 0.5 1 1.5 0.5 1 1.5 Fig. 9.— Constraints on the model bias parameters c1 and c2 in real (left panels) and redshift space (right panels), measured in the L200 box for all galaxies (top), galaxies with −20 > Mhr > −21 (middle), and galaxies with −21 > Mhr > −22 (bottom). Thick ellipses correspond to 1σ and 2σ constraints on parameters using the reduced 3PCF fit to equation 10; symbols denote the minimum χ2 value. Thin ellipses come from fitting the smoothed halo and dark matter overdensities (equation 9) shown in Figure 8. The vertical lines show the constraint on c1 from comparing the dark matter and galaxy 2PCF amplitudes and fitting the ratio with a linear bias model, c2 = ξgal/ξdm. virial motions within halos replace coherent infall on larger scales. 3. Redshift-space distortions attenuate the shape and scale dependence of the reduced 3PCF and weaken the evolution with redshift. 4. The reduced 3PCF shows only weak dependence on galaxy luminosity and color; put another way, the scaling between the 3PCF amplitude and the 2PCF is predicted to be nearly independent of galaxy type in this model. The trend of Q with color is somewhat stronger than with luminosity: the re- duced 3PCF is slightly enhanced for red galaxies over blue, especially for elongated configurations. 5. Our model predictions are in excellent agreement with the shape and scale dependence of the galaxy 3PCF measured in the SDSS when the same bin- ning scheme is used. Since the results are highly sensitive to the binning scheme, caution must be exercised in comparing theory and observations of the 3PCF. In combination with earlier results, this comparison indicates that a simple scheme in which galaxies and dark matter halos and subhalos are associated in a one-to-one fashion based on maxi- mum circular velocity can provide a good match to a wide range of galaxy clustering statistics. 6. The effect on the 3PCF of changing the selection of subhalos (i.e., of connecting galaxy luminosity to Vmax,acc instead of Vmax,now) is evident on small scales (r < 2 h−1Mpc) and for elongated configu- rations, but is negligible on larger scales. Future measurements of the 3PCF will help constrain dif- ferent models for the association of galaxy luminos- ity and color with subhalo properties. 7. On scales of order 10h−1 Mpc, a local, determinis- tic bias scheme is in reasonable agreement with the galaxy and dark matter distributions of the model. The bias parameters extracted fromQgal are in rea- sonable agreement with the δgal-δdm relation in real space, less so in redshift space. Nevertheless, the redshift-space constraints on the bias parameters are in agreement with the 2dFGRS measurements of the 3PCF. We are indebted to Anatoly Klypin and Brandon All- good for running and making available the simulations used in this paper, which were run on the Columbia ma- chine at NASA Ames and on the Seaborg machine at NERSC (Project PI: Joel Primack), to Andrey Kravtsov for running some of the halo catalogs used in this study, and to Charlie Conroy for providing us with measure- ments of vmax,acc. We thank Enrique Gaztañaga for enlightening comments on measuring and interpreting the 3PCF and comparing results of Q(α) with his es- timator; and Cameron McBride for discussions on the measurements of the bias parameters. We additionally thank Andrey Kravtsov, Issha Kayo and Roman Scoc- cimarro for several useful discussions. RHW was par- tially supported by NASA through Hubble Fellowship grant HST-HF-01168.01-A awarded by the Space Tele- scope Science Institute, and also recieved support from the U.S. Department of Energy under contract number DE-AC02-76SF00515. This work was supported in part by the Kavli Institute for Cosmological Physics through the grant NSF PHY-0114422, and by the U.S. Depart- ment of Energy at Fermilab and at U. Chicago. FAM thanks the Fulbright Program and CONICYT-Chile for additional support. This study also made use of the SDSS DR3 Archive, for which funding has been provided by the Alfred P. Sloan 12 Maŕın, Wechsler, Frieman and Nichol Foundation, the Participating Institutions, the National Aeronautics and Space Administration, the National Sci- ence Foundation, the U.S. Department of Energy, the Japanese Monbukagakusho, and the Max Planck Soci- ety. The SDSS Web site is http://www.sdss.org/. 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0704.0256

Multi-spectral Observations of Lunar Occultations: I. Resolving The Dust Shell Around AFGL 5440

Multi-spectral Observations of Lunar Occultations: I. Resolving The Dust Shell Around AFGL 5440 Paul M. Harvey1 and Andrew Oldag1 ABSTRACT We present observations and modeling of a lunar occultation of the dust- enshrouded carbon star AFGL 5440. The observations were made over a contin- uous range of wavelengths from 1 – 4µm with a high-speed spectrophotometer designed expressly for this purpose. We find that the occultation fringes cannot be fit by any single-size model. We use the DUSTY radiative transfer code to model a circumstellar shell and fit both the observed occultation light curves and the spectral energy distribution described in the literature. We find a strong constraint on the inner radius of the dust shell, Tmax = 950 K ± 50K, and op- tical depth at 5µm of 0.5 ± 0.1. The observations are best fit by models with a density gradient of ρ ∝ r−2 or the gradient derived by Ivezić & Elitzur for a radiatively driven hydrodynamic outflow. Our models cannot fit the observed IRAS 60µm flux without assuming a substantial abundance of graphite or by assuming a substantially higher mass-loss rate in the past. Subject headings: stars: AGB and post-AGB – techniques: high angular resolu- tion – stars: mass loss 1. Introduction Mass loss from post-main-sequence stars provides a large fraction of the heavy element abundance and solid particle content of the interstellar medium (e.g. Wallerstein & Knapp 1998; Ferrarotti & Gail 2006). The mechanism(s) of the mass loss during the AGB phase of stellar evolution are thought to involve both radiation pressure and stellar pulsation (Suh 1997; Wallerstein & Knapp 1998; Schröder et al. 1998), but most details of this process are not well understood. Many uncertainties about these processes can be clarified by studies of the spatial structure of circumstellar mass-loss shells. For example, a number of molecular 1Astronomy Department, University of Texas at Austin, 1 University Station C1400, Austin, TX 78712- 0259; [email protected], [email protected] http://arxiv.org/abs/0704.0256v1 – 2 – line studies of the extended envelopes around AGB stars have found strong evidence for periodic variations in mass-loss rates leading to the appearance of “rings” in the radial distribution of molecular emission lines, (e.g. Fong, Meixner & Shah 2003, Olofsson et al. 1996). Very deep, sensitive imaging studies have also found similar phenomena in the dust around the most nearby extreme example of these objects IRC+10216 (Mauron & Huggins 2000). In order to study the inner regions of these circumstellar shells, however, angular resolutions well under 1 arcsec are required. For example, at a distance of 1 kpc, the dust evaporation radius around a 104 L⊙ star corresponds to an angular radius of 20 milli- arcsec (mas). Speckle interferometry and more recently adaptive optics observations have enabled resolutions of order 0.1 arcsec (Hofmann et al. 2001; Biller et al. 2005), while lunar occultation observations and multi-aperture interferometry have pushed angular resolutions to the milli-arcsecond level, e.g. reviews by Quirrenbach (2004) and Monnier (2003). Until recently most interferometric observations have been made in typically one or two relatively broad bands. We present here observations of a lunar occultation of the star AFGL 5440 (aka OH 06.86-1.5, IRAS 18036-2344), made with a high-speed infrared spectropho- tometer, pMIRAS (Harvey & Wilson 2003), developed as a prototype for a more ambitious instrument now nearing completion. This star has been classified as a carbon-rich AGB star on the basis of its IRAS LRS spectrum (Zuckerman & Dyck 1986; Volk & Cohen 1989; Kwok, Volk & Bidelman 1997). Groenewegen et al. (2002) have estimated the distance to be 2.25 kpc. Near-infrared through far-infrared photometry of the source has been summa- rized by Guglielmo et al. (1993) and more recently by Guandalini et al. (2006), including a combination of ground based photometry, the IRAS values, and more recent MSX results. The reported distance and photometry imply a luminosity of 1.4×104 L⊙. In addition to the observations of AFGL 5440, we also discuss observations that we have made of two “calibration” stars in order to understand the limitations of our obser- vation/analysis process. These objects are cool stars with no detectable circumstellar dust shell based on their near-infrared and IRAS colors, IRC+00233 (M7) and HD 155292 (K2). Our observations cover the entire 1 - 4µm spectral region with a resolving power that varies from ∼ 20 at the shortest wavelengths to ∼ 100 at the long end. Our time resolution of 8 msec permits an effective angular resolution of a couple milli-arcsec, with the exact resolution being a strong function of the signal-to-noise ratio as discussed later. The broad wavelength coverage allows us to observe simultaneously the Fresnel fringe pattern of the obscured central star at the shorter wavelengths together with the circumstellar dust emis- sion from the warmest part of the dust distribution. In §2 we describe the details of the observations and instrumental parameters and the basic data reduction process. Then in §3 we describe various ways we have modelled the star+shell in order to determine the limits – 3 – on the circumstellar shell structure placed by our observations. Finally in §4 we summarize the implications of these results for the mass loss of this object. 2. Observations and Data Reduction The observations of AFGL 5440 were made during an immersion occultation event on 28 Aug 2001 at approximately 02:44:00 UT. The elevation of the Moon at the time of the event was 36◦, and the Sun was 17◦ below the horizon. The sky conditions were not completely photometric but cloud cover was minimal and intermittent. The position angle of the occultation event on the lunar limb was 54◦, and the lunar phase was 0.72. We used the pMIRAS instrument on the McDonald Observatory 2.7-m telescope. The details of the instrument have been described by Harvey & Wilson (2003), but we summarize the most important characteristics here. The instrument is essentially a long/wide slit, high-speed spectrophotometer using a NaCl prism to disperse the light from the slit, which is then imaged onto a 32×100 pixel portion of a 2562 InSb array detector. The slit width is chosen to be the minimum acceptable in order to minimize the background on the detector within the limitations imposed by the seeing conditions. For these observations the slit width was set at about 5 arcsec for the typical seeing of 1.5 arcsec. The detector is read-out every 8 msec, with photon integration occuring over essentially the full 8 msec time. Therefore, the Fresnel occultation fringe pattern is averaged over this 8 msec time (as well as over the 2.7-m telescope aperture). Because we use a refractive dispersive element, the dispersion/spectral resolution is not equal at all wavelengths; the highest dispersion is at the longest wavelengths, a feature that minimizes the background photon count at those wavelengths. On average over the 1 – 4µm waveband covered by the instrument, one pixel corresponds roughly to λ/∆λ = 100. Because of uncompensated seeing effects and the lower spectral dispersion at shorter wavelengths, the true resolving power at the shortest wavelengths is R ∼ 20. In the spatial direction the plate scale is 0.4 arcsec/pixel. The instrument is read-noise limited shortward of 2.5µm and background-limited longward of 3µm. A typical observation consists of taking 5000 frames at a time roughly centered on the occultation event. This is accomplished by using a buffer that holds the most recent 5000 frames and terminating the data acquisition a few seconds after the event is observed on a real-time display. This is an important feature since the predicted times of occultation events are often in error by as much as 10 seconds due to irregularities in the lunar limb as well as imperfect stellar astrometry. Our observations of the occultation event of the comparison star, IRC+00233, were obtained during an immersion event on 29 Jun 2001, and those of HD 155292 were obtained during an immersion event on 26 Aug 2001. – 4 – The data reduction process consists of several typical steps. Because the spectrum is being observed with much higher time resolution than the seeing timescale, the spectrum moves around by several pixels over the course of an occultation event. Therefore, to con- struct a light curve with minimal spectral blurring, the images must be shifted to correspond to the same wavelength/pixel scale. This is done with a simple cross-correlation algorithm that works well for the high S/N data discussed here. The spectrum is also not perfectly aligned with the X/Y axes of the detector, so we take out this tilt as well during the pro- cessing to simplify later steps. Because we use a “long slit”, ∼ 10 arcsec, we can use the sky measurements on either side of the stellar spectrum to provide an accurate and high-time- resolution sky subtraction. For the data discussed here we produce a weighted average in the spectral direction that is two pixels wide and sum the pixels in the spatial direction that have detectable signal. We have experimented with more elaborate photometric extraction schemes, but this technique appears to produce S/N ratios as high as any more complex algorithms. We have also experimented with various flat-fielding methods but have found only a small improvement in S/N with these methods. The end result of the data reduc- tion process is a sequence of ∼ 100 light curves for which we extract a few hundred frames centered on the occultation event. The frames that are taken well before the Fresnel fringe pattern of the occultation becomes evident are used to estimate the noise level in the data, due both to read-noise, background photon statistics, and the often non-negligible amount of seeing/scintillation noise caused by the atmosphere. Because the exact location of the spectrum on the detector varies both due to seeing and between observing runs after adjustments to the instrument, we perform the wavelength calibration by fitting the NaCl dispersion function to the observed positions of the J, H, K, and L atmospheric transmission maxima in the actual data for each occultation event. The accuracy of this calibration is probably good to ± .01 λ throughout the 1 – 4µm region that is observed. Because of variable instrumental efficiency depending on the placement of the star image on the input slit, and the common occurrence of non-photometric sky conditions during occultation events, we specifically do not attempt to derive a flux calibration for our data. We have, however, compared the relative signal from AFGL 5440 to that of relatively well characterized stars observed on the same night and conclude that the stellar magnitudes in the published literature for AFGL 5440 are consistent with values that we would have derived from our signal strengths to within ± 30%. – 5 – 3. Source Modeling The fringe pattern of a lunar occultation event is a convolution of the pattern for a point source over several parameters that all act to blur the fringes. These parameters include: the telescope aperture, the integration time of the detector, the wavelength bandwidth of the observation, and, most importantly, the source size/structure. In order to extract the source size or more detailed properties of the spatial structure, the typical procedure is to model the combination of all the above “blurring” parameters with various possible source models to find the best fit to the data (e.g. Nather & McCants 1970; Richichi et al. 1995). An additional uncertainty in the observations is the basic frequency of the fringe pattern, i.e. the speed of the lunar shadow. Although this parameter is calculated by the software that we use to predict occultation events, small uncertainties in the shape of the lunar limb (roughness due to craters, etc) can produce differences in the predicted shadow speed up to several tens of percent. Therefore, this parameter must also be fit in addition to the parameters that blur the fringes. We began our modeling process by assuming a uniform disk as the simplest possible model with which to try fitting the data for the observed objects, AFGL 5440, IRC+00233, and HD 155292. We use a simple χ2 test for the best model, allowing one or more parameters to vary during the process. Typically we first allow both the size and lunar shadow velocity to vary until we find an approximate fit to both. This fit can be done either individually at each wavelength or globally using the entire waveband. 3.1. Comparison Stars For IRC+00233 and HD 155292 we assumed that a single source size was likely to be appropriate for the entire wavelength range within our observational uncertainties, and we fit the observations globally for source size and lunar velocity. Rough estimates of the angular sizes of these two stars can be derived simply by assuming that they are blackbodies of the effective temperatures given by their spectral types. Using the empirical relation between angular size and B-K color from van Belle (1999) for IRC+00233 (B = 11.06, K = 1.95), we would expect an angular size of 3.5 mas; the star is, however, likely to be mildly variable, so the size at the time of our observations might have been different by ± 20%. For HD155292 (B = 11.0, K = 4.9) a similar calculation gives an angular size of 0.6 mas. Since typical departures from a uniform disk model are at the level of 10 – 30% (e.g. Thompson, Creech- Eakland & van Belle 2003; and Scholz 2003), the resolution required to detect them reliably for even IRC+00233 is below 1 mas. Based on tests we have done with model data, this is beyond the capabilities of our current data set which is limited by both spectral resolution – 6 – and signal-to-noise ratio to accuracies on the order of ±2 mas. The best fit size for IRC+00233 is between 4.5 and 5.0 mas. Figure 1 shows observed and model light curves for a model assuming a 4.5 mas uniform disk for a subset of the wavelengths observed. Figure 2 shows the χ2 and signal-to-noise ratio as a function of wavelength over the entire observed band for this model. Both figures show that this model provides a very good fit to the observations except for a small range of wavelengths around ∼ 1.7µm where a substantially larger size would provide a better fit. We do not have a good explanation for this discrepancy; it may be due to some systematic noise effect or to a real difference in the stellar photosphere in that region. For comparison Schmidtke et al. (1986) observed this same star in an occultation event using narrow-band filters near 1.6 and 2.2µm. They found a size at those wavelengths of ∼ 3 mas, similar to but smaller than our calculated blackbody size. For HD 155292 all uniform-disk models with a size less than about 3 mas were able to fit the data reasonably well (Figures 3 and 4). Since the blackbody size of the star is less than a milli-arcsecond, this is consistent with the expected uncertainties in our data and modeling. The signal-to-noise ratio for this star was low enough that we had no effective narrow-band information beyond 2.5µm, and at the shorter wavelengths some periodic electronic pickup had a non-negligible effect on the observed fringe patterns as well. Note that for both these comparison stars there are wavelengths with reasonable S/N as shown in Figures 1 and 3 where there is a less than adequate fit, so these are difficult to explain solely as due to telluric atmospheric absorption effects. 3.2. AFGL 5440 For AFGL 5440 a quick glance at the observed light curves (Figure 5) indicated that a single source size was unlikely to fit over the entire 1 – 4µm bandwidth (Harvey & Wilson 2003). This suggests that we are seeing the combination of the emission from the central star and a circumstellar shell of material due to mass loss from the star. To demonstrate the poor fit with a single size uniform disk, we show in Figures 5 and 6 the results of trying to fit the data with one example uniform disk, 11 mas. As can be seen in the plots of χ2 as well as the observed versus model light curves, the 11 mas disk provides a passable fit in the mid-range of wavelengths, 2.5 – 3.2µm, but produces fringes that are too sharp at the longer wavelengths and too broadened at the shortest wavelengths. This result motivated us to pursue a full radiative transfer model for the object that could be used to compare both the size constraints provided by our occultation data and the spectral energy distribution (Guandalini et al. 2006) which contains important and different information about the relative amount of dust at different temperatures. The DUSTY code – 7 – (Ivezić, Nenkova & Elitzur 1999) was originally created, in fact, for modeling the emission from AGB stars surrounded by mass-loss shells. Its output includes model source images as well as the total energy distribution, so it is ideal for our purposes. Our approach to using the code was to choose a particular combination of input parameters and then vary the dust optical depth to find the best fit to the energy distribution for those parameters. We then used the output source images that were computed as a function of wavelength to calculate expected occultation fringes for the model and compared those to the observed fringes. Since AFGL 5440 has been classified as a carbon-rich star, we assumed a carbon- rich dust composition, typically some combination of amorphous carbon, silicon carbide, and graphite as the major constituents. The other critical parameters for the models are the dust temperature at the inner radius and the radial density gradient. The outer radius of the dust shell makes essentially no difference to the observed characteristics at λ < 60µm as long as it is at least 100 times the inner radius. The goal of our modeling was to find some reasonable fit to our occultation data and the rough spectral energy distribution for the circumstellar dust shell; we have not attempted to extract details of the stellar photosphere or attempt a thorough examination of all possible dust size/composition models since our data do not bear directly on those issues for reasons of wavelength coverage and spectral resolution. In particular, we did not try to find any better than superficial agreement with the IRAS LRS data. We explored more than 200 models to understand the effect of varying the input pa- rameters on the quality of the fit. Basically all the models that provided an approximate fit to the occultation observations and the spectral energy distribution had several features in common. First, the dust temperature at the inner radius of the dust shell was of order 950K ± 50K. Models with a maximum dust temperature below 900 K did not have enough hot dust to fit the occultation fringes at the longer wavelengths, while models with hotter inner edges had even more difficulty than the best-fit model in reproducing the energy distribu- tion longward of 10µm. Secondly, the radial density gradient of models with a reasonable fit was close to r−2 (or to DUSTY’s calculation of the gradient appropriate for a radiatively driven wind which approximates an r−2 distribution for large radii). Models with a density gradient of r−1.8 came closer to producing the IRAS 60µm flux, but did not fit the shape of the 5 – 20µm energy distribution well. Models with a density gradient of r−2.2 can fit the energy distribution out to 20µm and also provide a good fit to the occultation results, but have substantially worse fits to the IRAS 60µm flux than our best fit models. The optical depth for best fit at the fiducial wavelength of 5µm was typically in the range 0.3 – 0.6. Finally, the dust composition that provided the best fit included a small amount of SiC together with comparable amounts of amorphous carbon and graphite. Other carbon-rich compositions produced reasonable fits except at the longest wavelengths or in the 11µm SiC – 8 – feature. Note that there is a range of optical properties for different forms of amorphous car- bon (Andersen, Liodl,& Hofner 1999) that we did not explore. We experimented with two grain size distributions, the MRN slope (Mathis, Rumpl & Nordsieck 1977), and another, the KMH shape (Kim, Martin & Hendry 1994), used by Ivezić & Elitzur (1996) for models of IRC+10216, that has a smoother fall-off on each end. We found that we could obtain reasonably good fits with either distribution. Figures 7 and 8 show the fits to the spectral energy distribution and to the occultation light curves for the best-fit model. Figure 9 shows the quality of the fit versus wavelength, and figure 10 illustrates the spatial profiles of this best-fit model. Figures 11 and 12 likewise show for comparison the fits for a model discussed below that does not use any graphite. Finally, figures 13 and 14 show results for a third model that fits the energy distribution but gives a poor fit to the occultation results because of a lack of enough warm dust close to the star. 4. Discussion and Summary Model dust shells have been computed for a number of carbon-rich AGB stars by various authors. Le Bertre, Gougeon, & Le Sidaner (1995) and Le Bertre (1997) found that the energy distributions for nearly two dozen carbon-rich stars could be modelled with very similar dust shell parameters. They found a common value of maximum dust temperature of order 950 K with shell density gradient of ρ ∝ r−2 for spherically symmetric shells. The best fit dust property implied a dust emissivity, ǫ ∝ λ−1.3, consistent with that expected for an amorphous carbon-rich dust composition as also found by Jura (2004). These conclusions were enhanced by their ability to fit the energy distributions over a range of phases of the observed variability of many of the stars. On the other hand, Suh (1997) suggested that a “superwind” phase of mass loss could improve model fits to the energy distributions for a number of carbon-rich AGB dust shells by enhancing the emission shortward of 30µm because of a high rate of mass loss in recent times, e.g. at small radii where most of the emission would be due to hotter dust. Ivezić & Elitzur (1996) used a self-consistent radiatively driven hydrodynamic model for the density distribution around IRC+10216 and were able to fit both the spectral energy distribution and the near-infrared angular visibility data from speckle observations. Interestingly they derived an inner dust shell temperature substantially lower than ours and most other studies, ∼ 750K. Winters et al. (1997) computed a full time- dependent model of periodic outflow from AFGL 3068 and were able to produce a good fit to the energy distribution and observed light curves. Virtually all of these modeling efforts used dust emissivities appropriate for some form of amorphous carbon (usually including SiC), but with no graphite (see also Lorenz-Martins 2001), contrary to our best model fit above. The model of Ivezić & Elitzur (1996) utilized a grain size distribution that included – 9 – substantially larger grains than most earlier models. Our lunar occultation observations add constraints to these results most directly in defining the amount of dust in the innermost region around AFGL 5440, since our longest observed wavelength is 4µm. Basically, our data imply that the density of grains emitting in the 3 – 4µm spectral region must be sufficiently large to produce the substantial fringe “blurring” observed in our light curves. Graphically, this amount of emission is illustrated in figure 10 showing the source profiles for the best-fit model. For any given assumed dust emissivity, this constraint then implies a fairly narrow range of optical depth and ratio of near-infrared to mid-infrared dust emission. Thus, we constrain the dust density gradient in the innermost part of the circumstellar shell. Finally, as mentioned above, the fact that we were unable to find a satisfactory fit to the data with any model having a maximum dust temperature lower than 900 K is a strong constraint on the location of the inner edge of the dust shell. This result follows from the fact that dust cooler than 900 K cannot emit sufficiently to produce the required amount of 3 – 4µm emission. Note that this temperature is nearly a factor of two below the expected condensation temperatures for dust around these stars (Egan & Leung 1995). The physical value of this inner radius for AFGL 5440 for the dust properties of the best-fit model is 3.7×1014 cm, or an angular radius at the assumed distance of 2.25 kpc of 11 mas. For reference the angular diameter of the central star is of order 3 mas, based on a blackbody approximation for its likely photospheric temperature of 2500K ± 300K and 2µm magnitude. Since our modeling is the first of which we are aware for this particular star over the entire infrared wavelength region, we also discuss here the constraints on the circumstellar cloud properties implied by the overall energy distribution. As mentioned above, our best-fit model utilized roughly equal amounts of amorphous carbon and graphite in addition to the small amount of SiC required to fit the 11µm feature seen in the IRAS LRS spectrum. This result is contrary to the large amount of evidence against substantial amounts of graphite in stars like AFGL 5440. The factor that drove us to include graphite was its relatively flat emissivity vs. wavelength dependence between 10 and 50µm that enabled us to fit the IRAS 60µm flux. Models with only amorphous carbon and SiC failed to fit that flux by factors of 3 or more. Figures 11 and 12 discussed above show the energy distribution and model light curves for the best-fitting model that uses only amorphous carbon and SiC, and which uses the radiatively driven hydrodynamic density gradient of Ivezic & Elitzur (1996) as computed by DUSTY. The light curves fit the observed occultation data nearly as well as those for the best-fit model. Clearly, however, the IRAS 60µm flux cannot be fit by any similar model without graphite, and substantial modifications would have to be made to the assumed density law – 10 – in the outer regions or to the dust emissivity in order to come close to fitting the 60µm flux. Explaining this problem is beyond the scope of our study, but the fact that there is abundant evidence for non-constant outflow from carbon stars (Fong, Meixner & Shah 2003; Mauron & Huggins 2000) provides a convenient (if ad hoc) explanation. If the mass-loss rate were greater in the past, then the amount of dust at radii appropriate for emission at 60µm might well be larger than a simple extrapolation from the dust responsible for the 3 – 20µm emission. This would describe the opposite situation from that proposed by (Suh 1997) who suggested higher mass-loss rates in the recent past to explain observations of a number of other similar carbon stars. The radial location of dust emitting strongly at 60µm is of order 1 to a few arcseconds from the star. For an assumed distance of 2.25 kpc and its measured outflow velocity of 22 km s−1 (Groenewegen et al. 2002), this would correspond to a time of order 500 years to a couple thousand years in the past for the proposed higher mass-loss rate. This time scale is comparable to the period of fluctuation seen by Mauron & Huggins (2000) for IRC+10216. The fact that we have constrained the absolute value and radial dependence of the dust density with our observations means that we have constrained the recent dust mass-loss rate for AFGL 5440 as well. The dust mass-loss rate implied by our derived optical depth and radial density dependence is Ṁdust = 6.5 × 10 −8 M⊙/yr for the optical constants of amorphous carbon used by DUSTY (Ivezić, Nenkova & Elitzur 1999). Groenewegen et al. (2002) computed dust and gas mass-loss rates individually for AFGL 5440 on the basis of the IRAS 60µm flux for the dust, and millimeter CO observations for the gas. They derived a gas mass-loss rate of 3.1×10−5 M⊙/yr and dust mass-loss rate of 5.1×10 −8 M⊙/yr implying a gas-to-dust mass ratio of 600. Interestingly, their dust mass-loss rate is slightly lower than the value we have derived in spite of their using the 60µm IRAS flux for normalization. This result reinforces the overall uncertainties in model assumptions and absolute dust opacties, particularly at longer infrared wavelengths. In any case, a dust mass-loss rate of order 5 – 10×10−8M⊙/yr and gas mass-loss rate a few hundred times larger are consistent with the data. In summary, our observations have separately resolved the stellar photosphere and the inner edge of the circumstellar dust shell around AFGL 5440. We have strongly constrained the inner radius of the dust shell surrounding the star as well as the near-infrared optical depth. Our constraints together with the spectral energy distribution suggest a dust density gradient consistent with that expected for radiatively driven mass loss with the exception that the far-infrared flux may imply a recent decrease in mass-loss rate from the time when the far-ir-emitting dust was ejected. – 11 – 5. Acknowledgments We thank a number of people and institutions that have supported this work since its earliest stages. M. Simon greatly encouraged our efforts and provided many useful comments over the course of this work. He also provided the basic prediction software (developed by L. Cassar) that we use for determining the times and elements of occultation events. We also acknowledge illuminating conversations with A. Richichi, S. Guilloteau, D. Evans, and R. E. Nather and very helpful suggestions from two anonymous referees. D. Wilson developed the initial versions of a number of the reduction algorithms used in this work and provided a great deal of assistance during the observations. C. Young provided help in the intricacies of DUSTY. This project has been supported by: internal McDonald Observatory funding, NASA Grant NAG5-10458, and NSF grant AST-0096626. We also acknowledge extensive use of the NASA Astrophysics Data System and SIMBAD, and P. Harvey thanks the University of Colorado’s Center for Astrophysics and Space Astronomy for graciously hosting him during a sabbatical while much of this paper was written. REFERENCES Andersen, A. C., Liodl, R. & Hofner, S. 1999, A&A, 349, 243 Biller, B. A. et al. 2005, ApJ, 620, 450 Egan, M. P. & Leung, C. M. 1995, ApJ, 444, 251 Ferrarotti, A. S. & Gail, H.-P. 2006, A&A, 447, 553 Fong, D., Meixner, M. & Shah, R. Y. 2003, ApJ, 582, L39 Groenewegen, M. A. T., Sevenster, M., Spoon, H. W. 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DUSTY Models of AFGL 5440 Model Tmax ρ ∝ r −N τ Amorph C Graphite SiC Size Dist. K @5µm % %) %) 194 950 -2.0 0.41 40 50 10 MRN1 200 950 -2.0 0.55 95 0 5 KMH2 206 850 rad-flow3 0.45 95 0 5 KMH2 Note. — 1Mathis, Rumpl & Nordsieck (1977) with amin = .005µm; amax = 0.25µm. 2Kim, Martin & Hendry (1994) with amin = .005µm; amax = 0.2µm. 3 Radiatively driven outflow computed by DUSTY as described by Ivezić & Elitzur (1996) – 14 – Fig. 1.— Plots of observed light curves (crosses) versus computed model light curves for the 4.5 mas uniform disk model of IRC+00233 at a sampling of the range of wavelengths observed between 1 and 4µm. In each panel the lower curve of triangles shows the difference of observed minus model. – 15 – Fig. 2.— Plots of the χ2 (solid) for the model fit of the 4.5 mas uniform disk to the observed light curve for IRC+00233 over the range of wavelengths observed between 1 and 4µm. The observed signal-to-noise ratio is also shown (dashed) as a function of wavelength for comparison. – 16 – Fig. 3.— Plots of observed light curves (crosses) versus computed model light curves for the 2.0 mas uniform disk model of HD155292 at a sampling of the range of wavelengths observed between 1 and 4µm. In each panel the lower curve of triangles shows the difference of observed minus model. – 17 – Fig. 4.— Plots of the χ2 (solid) for the model fit of the 2.0 mas uniform disk to the observed light curve for HD155292 over the range of wavelengths observed between 1 and 4µm. The observed signal-to-noise ratio is also shown (dashed) as a function of wavelength for comparison. – 18 – Fig. 5.— Plots of observed light curves (crosses) versus computed model light curves for the 11 mas uniform disk model for AFGL 5440 at a sampling of the range of wavelengths observed between 1 and 4µm. In each panel the lower curve of triangles shows the difference of observed minus model. – 19 – Fig. 6.— Plots of the χ2 (solid) for the model fit of the 11 mas uniform disk to the observed light curve for AFGL 5440 over the range of wavelengths observed between 1 and 4µm. The observed signal-to-noise ratio is also shown (dashed) as a function of wavelength for comparison. – 20 – Fig. 7.— Plot of the observed spectral energy distribution of AFGL 5440 versus that predicted by model 194, the best fit model. The open diamonds show the values from Guandalini et al. (2006) and Guglielmo et al. (1993). The light grey line indicates the IRAS LRS spectrum. – 21 – Fig. 8.— Plots of observed light curves (crosses) versus computed model light curves for model 194 for AFGL 5440 at a sampling of the range of wavelengths observed between 1 and 4µm. In each panel the lower curve of triangles shows the difference of observed minus model. – 22 – Fig. 9.— Plots of the χ2 (solid) for the fit of model 194 (the best fit) to the observed light curve for AFGL 5440 over the range of wavelengths observed between 1 and 4µm. The observed signal-to-noise ratio is also shown (dashed) as a function of wavelength for comparison. – 23 – Fig. 10.— Plots of the model spatial distributions for AFGL 5440 predicted by model 194, the best fit model. The curves are drawn for 1.0 and 1.2µm (solid, 1.0µm is the more extended), and 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6, 3.8, and 4.0µm, with the longest wavelengths being the most extended of these. – 24 – Fig. 11.— Plot of the observed spectral energy distribution of AFGL 5440 versus that predicted by model 200, the best fit model that does not use graphite as for fig. 7. – 25 – Fig. 12.— Plots of observed light curves (crosses) versus computed model light curves for model 200 for AFGL 5440 at a sampling of the range of wavelengths observed between 1 and 4µm. In each panel the lower curve of triangles shows the difference of observed minus model. – 26 – Fig. 13.— Plot of the observed spectral energy distribution of AFGL 5440 versus that predicted by model 206, illustrating the fit for a shell with a cooler inner dust temperature as for fig. 7. – 27 – Fig. 14.— Plots of observed light curves (crosses) versus computed model light curves for model 206 for AFGL 5440 at a sampling of the range of wavelengths observed between 1 and 4µm. In each panel the lower curve of triangles shows the difference of observed minus model. Note the larger residuals at the longer wavelengths than for the best fit model 194. Introduction Observations and Data Reduction Source Modeling Comparison Stars AFGL 5440 Discussion and Summary Acknowledgments

0704.0257

Orbifold cohomology of abelian symplectic reductions and the case of weighted projective spaces

Orbifold cohomology of abelian symplectic reductions and the case of weighted projective spaces Tara S. Holm Abstract. These notes accompany a lecture about the topology of symplectic (and other) quotients. The aim is two-fold: first to advertise the ease of computation in the symplectic category; and second to give an account of some new computations for weighted projective spaces. We start with a brief exposition of how orbifolds arise in the symplectic category, and discuss the techniques used to understand their topology. We then show how these results can be used to compute the Chen-Ruan orbifold cohomology ring of abelian symplectic reductions. We conclude by comparing the several rings associated to a weighted projective space. We make these computations directly, avoiding any mention of a stacky fan or of a labeled moment polytope. Contents 1. Symplectic manifolds and quotients 2 2. Orbifolds and their cohomology 5 3. Why the symplectic category is convenient 9 4. The case of weighted projective spaces 12 References 18 The notion of an orbifold has been present in topology since the 1950’s [S1, S2]. More recently, orbifolds have played an important role in differential and alge- braic geometry, and in mathematical physics. A fundamental theme is to compute topological invariants associated to an orbifold, with one ostensible goal to under- stand Gromov-Witten invariants for these spaces. The aim of the present article is modest: to expound how techniques from symplectic geometry may be used to un- derstand the degree-zero genus-zero Gromov-Witten invariants with three marked points, the so-called Chen-Ruan orbifold cohomology ring; and to make ex- plicit the details of these techniques in the case of weighted projective spaces. In the symplectic category, orbifolds arise as symplectic quotients. We recount the techniques from symplectic geometry that may be used to compute topolog- ical invariants of a symplectic quotient. This is based on Kirwan’s seminal work 1991 Mathematics Subject Classification. Primary 53D20; Secondary 14N35, 53D45, 57R91. Key words and phrases. Symplectic quotient, orbifold, cohomology. TSH is grateful for the support of the NSF through the grant DMS-0604807. http://arxiv.org/abs/0704.0257v1 2 TARA S. HOLM [Ki]; and for orbifold invariants, the author’s joint work with Goldin and Knutson [GHK]. The quotients we consider are by a compact connected abelian group. We employ techniques coming from algebraic topology, most notably using equivariant cohomology. For those used to working with finite groups, it is important to note that, whereas for finite groups the invariant part of a cohomology ring is identical to the equivariant cohomology, this is not the case for connected groups. The main example in this article is a weighted projective space CPn . Its definition depends on a sequence (b) = (b0, . . . , bn) of positive integers. Kawasaki showed that the ordinary cohomology groups, with integer coefficients, of the un- derlying topological space of a weighted projective space are identical to the co- homology groups of a smooth projective space [Ka], but there is a twisted ring structure. We review the details of his work. Then in Theorem 4.2, we compute the cohomology of the orbifold [CPn ], proving that (0.1) H∗([CPn(b)];Z) = H (S2n+1;Z) ∼= 〈b0 · · · bnun+1〉 Whereas Kawasaki finds a twist in the the ring structure, we find torsion in high degrees of the ring (0.1). There is a natural map from Kawasaki’s ring to this one, and we describe the map explicitly. Finally in Theorem 4.3, we compute the Chen- Ruan cohomology ring of this orbifold. We make this computation using integer coefficients, generalizing results in [J, Ma1, Ma2]. Moreover, we give explicit generators and relations, and avoid mentioning a stacky fan [BCS] or a labeled polytope [LT, GHK]. The definitions in this article make sense for arbitrary coefficient rings. Indeed, all computations in the final section use integer coefficients. Moore and Witten have suggested that the torsion in K-theory has more physical significance than torsion in cohomology [MoWi]. The author together with Goldin, Harada and Kimura, is investigating a K-theoretic version of [GHK] and of the computations herein, building on the work of Harada and Landweber [HL]. The remainder of the paper is organized as follows. In Section 1 we give a quick exposition of how orbifolds arise in the symplectic category. We then introduce several cohomology rings associated to an orbifold in Section 2. We advertise the ease of computation for these rings in Section 3. The novel results in this article are the computations in Section 4. We include detailed proofs that avoid much of the symplectic machinery used in [GHK]. Acknowledgments. Many thanks are due to Tony Bahri, Matthias Franz, Re- becca Goldin, Megumi Harada, Ralph Kaufmann, Takashi Kimura, Allen Knutson, Eugene Lerman, Reyer Sjamaar, and Alan Weinstein for many helpful conversa- tions; and to Yoshiaki Maeda and the organizers and sponsors of Poisson 2006 in Tokyo, Japan, where this work was presented. 1. Symplectic manifolds and quotients We begin with a very brief introduction to the symplectic category; a more detailed account of the subject can be found in [CdS]. A symplectic form on a manifold M is a closed non-degenerate two-form ω ∈ Ω2(M). Thus, for any tangent vectors X ,Y ∈ TpM , ωp(X ,Y) ∈ R. The key examples include the following. Example 1.1: M = S2 = CP 1 with ωp(X ,Y) equal to the signed area of the parallelogram spanned by X and Y. This is the Fubini-Study form on CP 1. COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 3 Example 1.2: M any orientable Riemann surface with ω as in Example 1.1. Note that orientability is a necessary condition on a symplectic manifold M , because the top exterior power of the symplectic form is a volume form. Example 1.3: M = R2d with ω = dxi ∧ dyi. Example 1.4: M = Oλ a coadjoint orbit of a compact connected semisimple Lie group, equipped with ω the Kostant-Kirillov-Soriau form. Example 1.3 gains particular importance because of Darboux’s Theorem 1.5. Let M be a symplectic 2d-manifold with symplectic form ω. Then for every point p ∈ M , there exists a coordinate chart U about p with coordinates x1, . . . , xd, y1, . . . , yd so that on this chart, (1.1) ω = dxi ∧ dyi. Thus, whereas Riemannian geometry uses local invariants such as curvature to distinguish metrics, symplectic forms are locally indistinguishable. The symmetries of a symplectic manifold may be encoded as a group action. Here we restrict ourselves to a compact connected abelian group T = (S1)n. An action of T on M is symplectic if it preserves ω; that is, ρ∗gω = ω, for each g ∈ T , where ρg is the diffeomorphism corresponding to the group element g. The action is Hamiltonian if in addition, for every ξ ∈ t, the vector field (1.2) Xξ = [exp(tξ)]|t=0 is a Hamiltonian vector field. That is, we require that ω(Xξ, ·) = dφ ξ is an exact one-form. Each φξ is a smooth function on M , determined up to a constant. Taking them together, we may define a moment map (1.3) Φ : M −→ t∗ p 7−→ Φ(p) : t → R ξ 7→ φξ(p) Returning to our examples, we have Hamiltonian actions in all but the second example. Example 1.6: The circle S1 acts on M = S2 = CP 1 by rotations. If we use angle and height coordinates on S2, then the vector field this action generates is tangent to the latitude lines, so in coordinates, X ξ = ∂ , and since ω = dθ ∧ dh, ω(Xξ, ·) = dh, so a moment map is the height function on S 2, as shown in Figure 1.1 below. Example 1.7: If M is a two-torus M = T 2 = S1 × S1, then S1 × S1 acts on itself by multiplication. This action is symplectic, but is not Hamiltonian. In fact, no Riemann surface with non-zero genus has a nontrivial Hamiltonian torus action. Example 1.8: The torus T d = (S1)d ⊂ Cd acts by coordinate-wise multiplication on M = R2d = Cd. This action rotates each copy of C = R2 (at unit speed), and is Hamiltonian. Identifying t∗ ∼= Rd, a moment map is (1.4) Φ(z1, . . . , zn) = (|z1| 2, . . . , |zd| up to a constant multiple. 4 TARA S. HOLM PSfrag replacements Figure 1.1. The vector field and moment map for S1 acting by rotations on S2. Example 1.9: Each coadjoint orbit M = Oλ ⊆ g ∗ may be identified as a homoge- nous space G/L, where L is a Levi subgroup of the Lie group G. Thus G and its maximal torus T act on M by left multiplication. A G-moment map is inclusion (1.5) ΦG : Oλ →֒ g and a T -moment map is the G-moment map composed with the natural projection g∗ → t∗ that is dual to the inclusion t →֒ g. In each of these examples, the image of the (torus) moment map is a convex subset of Rn. This is true more generally. Convexity Theorem 1.10 ([A],[GuSt]). If M is a compact Hamiltonian T -space, then Φ(M) is a convex polytope. It is the convex hull of Φ(MT ), the images of the T -fixed points. The convexity theorem is an example of a localization phenomenon: a global feature (the image of the moment map) that is determined by local features of the fixed points (their images under the moment map). The convexity property is a recurring theme in symplectic geometry; its many guises are illustrated in [GuSj]. The moment map is a T -invariant map: it maps entire T -orbits to the same point in t∗. Thus when α is a regular value, the level set Φ−1(α) is a T -invariant submanifold of M . Moreover, the action of T on a regular level set is locally free: it has only finite stabilizers. This follows directly from the moment map condition: at a regular value, dφξ is never zero, implying that Xξ is not zero, so there is no 1-parameter subgroup fixing points in the level set. Thus, at a regular value the symplectic reduction M//T (α) = Φ−1(α) is an orbifold. In fact, Marsden and Weinstein proved Theorem 1.11 ([MaWe]). If M is a Hamiltonian T -space and α is a regular value of the moment map Φ, then the symplectic reduction M//T (α) is a symplectic orbifold. COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 5 More generally, the symplectic reduction M//T (α) at a critical value is a symplectic stratified space [SL]. Symplectic reduction is an important technique for constructing new symplec- tic manifolds from old. From our examples, we may construct several classes of symplectic manifolds. Example 1.12: For the action of S1 on M = S2 = CP 1 by rotation, the level set of a regular value is a latitude line, which the circle rotates. The quotient is a point. Note that if S1 acts by rotation S2 at twice the usual speed, then the quotient, as an orbifold, is [pt/Z2]. Thus, every orbifold [pt/Zk] is a quotient of S by a rotation action. Example 1.13: T d acts on M = R2d = Cn by rotation of each copy of C = R2. The level set of a regular value is a copy of T d, and so again the quotient is a point (or potentially an orbipoint, for different actions of T d). However, we may also restrict our attention to subtori K ⊆ T n. The action of K is still Hamiltonian, and for certain choices of K, Cn//K(α) is a symplectic toric orbifold. Lerman and Tolman show that every effective symplectic toric orbifold may be constructed in this way [LT]. Example 1.14: For the T -action on a coadjoint orbit M = Oλ, the symplectic reduction M//T (α) is known as a weight variety. One may determine the possi- ble orbifold singularities by analyzing the combinatorics of G and its Weyl group. See, for instance, [Kn] or [GHK]. This reduced space plays an important role in representation theory. 2. Orbifolds and their cohomology We now turn to orbifolds in the topological category. In terms of local models, an orbifold is a topological space where each point has a neighborhood homeo- morphic (or diffeomorphic) to the quotient of a (fixed dimensional) vector space by a finite group. Satake introduced this notion in the 1950’s [S1, S2], originally calling the spaces V -manifolds. Thurston coined the term orbifold when he redis- covered them in the 1970’s (see [Th]) in his study of 3-manifolds. This local model, however, makes it difficult to define very basic pieces in the theory of orbifolds: overlap conditions on orbifold charts, suborbifolds, and maps between orbifolds. As is evident already in the work of Haefliger [Hæ], the proper way to think of an orbifold is as a Morita equivalence class of groupoids, one of which is a proper étale groupoid (see [Moe]); or equivalently as a smooth Deligne-Mumford stack (see [DM]). For example, using this structure, a map of orbifolds should simply be a morphism of the appropriate objects. While groupoids or stacks provide the correct mathematical framework, the technology is a bit beyond the scope of this article. Indeed, for us it is sufficient to work with the local models, largely because we restrict our attention to orbifolds that arise as global quotients. Nevertheless, we will need to distinguish between an orbifold X or [X ] and its underlying topological space (or coarse moduli space) X . In particular, when X is presented as a global quotient of a manifold M by a group G, we will use square brackets [M/G] to denote the orbifold, and M/G to denote the underlying coarse moduli. For an orbifold X, at each point x ∈ X, we have a local isotropy group Γx at x. We will be interested in almost complex orbifolds, that is orbifolds that have 6 TARA S. HOLM local models isomorphic to Cd/Γx, with Γx ⊆ U(d) a finite group acting unitarily on Cd. Our main example in this section is the orbisphere shown in the figure below. This is a symplectic toric orbifold in the sense of Tolman and Weitsman Figure 2.1. An orbisphere with two orbifold points, a Zp singu- larity at the north pole and a Zq singularity at the south pole. All other points in the space have local isotropy group the one-element group. When p and q are relatively prime, this is a weighted pro- jective space CP 1p,q. [LT]. In that context, it corresponds to the labeled polytope that is an edge, with the vertices labeled p and q. This orbifold cannot (always) be presented as a global quotient by a finite group, although it can be presented as a symplectic reduction. When p and q are relatively prime, it is the reduction of C2 by the Hamiltonian S1-action (2.1) e2πiθ · (z1, z2) = (e p·2πiθ · z1, e q·2πiθ · z2). Thus, it is a global quotient of the level set Φ−1(α) ≈ S3 by a locally free S1 action. In this case, it is also the global quotient of CP 2 by a Zp × Zq action 1. When p and q are not relatively prime, the orbisphere is not a global quotient by a finite group, but it is still a symplectic quotient of C2 by a Hamiltonian S1 × Zg action, where g = gcd(p, q) (following [LT]). Note also that an orbisphere is isomorphic to a weighted projective space CP 2p,q exactly when p and q are relatively prime. If p and q are not relatively prime, the weighted projective space is not reduced: it has a global stabilizer. On the other hand, the orbisphere described above is always reduced. Next we turn to algebraic invariants that we may attach to an orbifold X. The hope is that these invariants are computable and at the same time retain some information of the orbifold structure of X. All the invariants are isomorphic to singular cohomology if X = X is in fact a manifold. Definition 2.1. The ordinary cohomology ring of an orbifold X is the singular cohomology of the underlying topological space X, (2.2) H∗(X ;R), with coefficients in a commutative ring R. 1With an apology to number theorists, we take the topologist’s notation: Zp denotes the integers modulo p. COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 7 This ring is computable using standard techniques from algebraic topology, but it does not distinguish between the orbisphere in Figure 2.1 and a smooth sphere. For the second invariant, we restrict our attention to orbifolds X presented as the quotient of a manifold M by the locally free action of a Lie group G. It is conjectured that every orbifold can be expressed as such a global quotient, and it is known to be true for effective or reduced orbifolds, those that do not have a global finite stabilizer (see, for example, [EHKV, Theorem 2.18]). A presentation as a global quotient is desirable because then the topology or geometry of the quotient X is simply the G-equivariant topology or geometry of the manifold M . This principle motivates the following definition. Definition 2.2. Given a presentation of an orbifold X = [M/G] as a global quo- tient, the cohomology ring of the orbifold X is the equivariant cohomology (2.3) H∗(X;R) := H∗G(M ;R), with coefficients in a commutative ring R. Recall that equivariant cohomology is a generalized cohomology theory in the equivariant category. Using the Borel model, we define (2.4) H∗G(M ;R) := H ∗((M × EG)/G;R), where EG is a contractible (though infinite dimensional) space with a free G action, andG acts diagonally onM×EG. There are well-developed methods for computing equivariant cohomology, hence this invariant is still computable. Whenever X = [M/G] is a global quotient, the associated quotient map (2.5) q : M → X is a G-invariant map. This induces a continuous map (2.6) q : (M × EG)/G → X. When X = X is a manifold (i.e. G acts on M locally freely), this map is a fibration with fiber BG. The map q induces a map in cohomology, (2.7) q∗ : H∗(X ;R) −→ H∗((M × EG)/G;R) = H∗G(M ;R). This induced map is an isomorphism when 1. G acts freely on M ; 2. G is a finite group and R is a ring in which |G| is invertible; and 3. G acts locally freely on M and R is a field of characteristic 0. This last item implies that the cohomology of the orbifold differs from the coho- mology of its coarse moduli only in its torsion. Notably, when R = Z, this ring does in fact distinguish between the orbisphere in Figure 2.1 and a smooth sphere. The third invariant was introduced by Chen and Ruan [CR] to explain mathe- matically the stringy Betti numbers and stringy Hodge numbers that physi- cists have attached to orbifolds. To define this third invariant, we need to introduce the first inertia orbifold (2.8) I1(X) := (x, (g)Γx) x ∈ X and (g)Γx is a conjugacy class in Γx This is again an orbifold, and X is the suborbifold called the identity sector whose pairs consist of a point in X together with the identity element coset. The other connected components of I1(X) are called the twisted sectors. For a global 8 TARA S. HOLM quotient X = [M/G] with G abelian, we may identify I1(X) = g∈G[M g/G]. On the other hand, when X = [M/G] is global quotient with G finite, we may identify I1(X) = g∈T [M g/C(g)], where the union is over T a set of representatives of conjugacy classes in G. For the orbisphere example, the inertia orbifold is shown in Figure 2.2. PSfrag replacements · · · · · · Figure 2.2. The inertia orbifold for the the orbisphere with two orbifold points as in Figure 2.1. Each of the p − 1 points to the right of the north pole represents a [pt/Zp] and each of the q − 1 to the right of the south pole a [pt/Zq]. Definition 2.3 ([CR]). Given a presentation of an orbifold X = [M/G] as a global quotient, the Chen-Ruan orbifold cohomology of X, as a vector space, is defined to be the cohomology of the first inertia orbifold, (2.9) HCR(X;R) := H(I 1(X);R), with coefficients in a commutative ring R. The Chen-Ruan ring is endowed with a Q grading, different from the grading coming from singular cohomology. For a connected component Z of I1(X) that lies in the (g) piece of the first inertia, the grading of H(Z;R) is shifted by a rational number which is twice the age of Z. The age is determined by the weights of the action of the group element g on the normal bundle to Z inside of X. This is precisely where we need X to be (stably) almost complex. These rational shifts ensure that the ranks of the Chen-Ruan cohomology groups agree with the stringy invariants of an orbifold. The dependence of the rational shifts on the normal bundles ν(Z ⊂ X) means that the Chen-Ruan ring is not in general functorial for arbitrary morphisms of orbifolds. To define a product on the Chen-Ruan cohomology, we must define higher inertia. The nth inertia orbifold consists of tuples, a point in the orbifold and an n-tuple of conjugacy classes. Restricting to the 2nd inertia, there are natural maps (2.10) e1, e2, e3 : I 2(X) −→ I1(X) defined by e1(p, (g)Γp , (h)Γp) = (p, (g)Γp),(2.11) e2(p, (g)Γp , (h)Γp) = (p, (h)Γp), and(2.12) e3(p, (g)Γp , (h)Γp) = (p, ((gh) −1)Γp)(2.13) COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 9 Chen and Ruan define the product of two classes α, β ∈ HCR(X;R) to be be (2.14) α ⌣ β := (e3)∗(e 1α ∪ e 2β ∪ ε), where e∗1 and e 2 are pull-back maps, (e3)∗ is the push-forward, ∪ is the usual cup product, and ε is the Euler class of the obstruction bundle. This Euler class should be viewed as a quantum correction term. It ensures that the product respects the Q-grading and that the product is associative. Neither of these properties is immediately obvious, and proving the latter requires a rather substantial argument. Since we can avoid mention of the obstruction bundle in our computations, we suppress further details here, and refer the curious reader to [ℵGV, CR, GHK] for additional information. The Chen-Ruan cohomology ring is the degree 0 part of the (small) quantum cohomology ring. Hence, it is generally the most difficult of these three invariants to compute. It has been computed for orbifolds that are global quotients of a manifold by a finite group in [FG]. The definition was extended to the algebraic category in [ℵGV], and in [BCS], the ring is computed for toric Deligne-Mumford stacks with Q, R and C coefficients. In the next section, we review how to compute this ring for abelian symplectic quotients, as demonstrated in [GHK]. In the last section, we will compute each of these invariants explicitly for weighted projective spaces. We conclude this section with a table of these rings for an orbisphere that has a Z2 singularity at the north pole and is otherwise smooth. This is an example of a weighted projective space, and is denoted CP 11,2. (2.15) X = [CP 11,2] Ring Grading H∗(X ;Z) Z[x]/〈x2〉 deg(x) = 2 H∗(X;Z) Z[x]/〈2x2〉 deg(x) = 2 H ∗CR(X;Z) Z[x, u]/〈2x 2, 2xu, u2 − x〉 deg(x) = 2, deg(u) = 1 3. Why the symplectic category is convenient In the past thirty years, tremendous progress has been made in understanding the equivariant topology of Hamiltonian T -spaces and its relationship to the ordi- nary topology of their quotients. For a compact torus T = (S1)n, the classifying bundle ET is an n-fold product of infinite dimensional spheres, and the classifying space BT is an n-fold product of copies of CP∞. Thus, (3.1) H∗T (pt;Z) = H ∗(BT ;Z) = Z[x1, . . . , xn], where deg(xi) = 2. The key ingredient to understanding topology of Hamiltonian T -spaces is the moment map. Frankel [F] proved that for a Hamiltonian T -action on a Kähler manifold, each component φξ of the moment map is a Morse-Bott function on M , and generically the critical set is the fixed point set MT . In his paper [A] on the Convexity Theorem 1.10, Atiyah generalized this work to the purely symplectic setting. Building on the work of Frankel and Atiyah, Kirwan developed techniques to prove two fundamental theorems that allow us to understand the cohomology of Hamiltonian T -spaces and their quotients. The first is a version of localization: it allows us to make global computations by understanding fixed point data. While 10 TARA S. HOLM this theorem is not explicitly stated in her book [Ki], it does follow immediately from her work in Chapter 5. Injectivity Theorem 3.1 ([Ki]). Let M be a compact Hamiltonian T -space. The inclusion map MT →֒ M induces (3.2) i∗ : H∗T (M ;Q) −→ H T ;Q) an injection in equivariant cohomology. The compactness hypothesis is stronger than strictly necessary. We may re- place it with a properness condition on the moment map. The proof relies on the fact that a generic component of the moment map is an equivariantly perfect Morse-Bott function on M . The image of this injection has been computed in many examples, including toric varieties, coadjoint orbits of compact connected semisimple Lie groups, and coadjoint orbits of Kac-Moody groups. These compu- tations initially appeared in [CS, GKM] and further generalizations are described in [GoH, GuH, GuZ, HHH]. The second theorem relates the equivariant topology of a Hamiltonian T -space to the ordinary topology of its reduction. Surjectivity Theorem 3.2 ([Ki]). Let M be a compact Hamiltonian T -space, and α a regular value of the moment map. The inclusion Φ−1(α) →֒ M induces (3.3) κ : H∗T (M ;Q) −→ H −1(α);Q) a surjection in equivariant cohomology. Again for surjectivity, compactness is more than is necessary. Most importantly, this result does apply to linear actions of a torus on Cd with a proper moment map. The key idea in the proof is to use the function ||Φ − α||2 as a Morse-like function, now known as a Morse-Kirwan function. The critical sets are not non-degenerate, but one may still explicitly understand them via a local normal form. It is then possible to prove that ||Φ−α||2 is an equivariantly perfect function on M . The kernel of the map κ can be computed using methods in [Go, JK, TW]. Using the fact that at a regular value, H∗T (Φ −1(α);Q) ∼= H∗(M//T (α);Q), we have a diagram (3.4) 0 // ker(κ) // H∗T (M ;Q)) κ // // H∗(M//T (α);Q) // 0 H∗T (M T ;Q) Thus, by computing im(i∗) and ker(κ), we may derive an explicit presentation of the cohomology H∗([M//T (α)];Q) of an orbifold arising as a symplectic quotient. We now turn to a generalization of Theorem 3.2 in the context of orbifolds and the Chen-Ruan ring. CR Surjectivity Theorem 3.3 ([GHK]). Let M be a compact Hamiltonian T - space, and α a regular value of the moment map. The inclusion Φ−1(α) →֒ M induces (3.5) K : H∗T (M g;Q) −→ H∗T (Φ −1(α)g ;Q) COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 11 a surjection. Moreover, these are R×T -graded rings, K is a map of graded rings, and there is an isomorphism of graded rings (3.6) H∗T (Φ −1(α)g ;Q) ∼= H CR(M//T (α);Q). The surjectivity (3.5) is a direct consequence of the Surjectivity Theorem 3.2 applied to each space Mg (and again the compactness is not strictly necessary). The hard work is defining the grading and ring structure, and proving that K is a map of graded rings. Goldin, Knutson, and the author define a ring structure ⌣ that generalizes a definition (for G finite) of Fantechi and Göttsche [FG]. Us- ing this definition, we may deduce (3.6); on the other hand, associativity of this product is not at all obvious. Making use of the injection i∗ on each piece, there is an alternative product ⋆ on the ring g∈T H g;Q) that is much simpler to compute, and clearly associative. This alternative product has the advantage that it avoids all mention of the obstruction bundle; instead it relies only on fixed point data (i.e. the topology of the fixed point set and isotropy data for the action of the torus on the normal bundles to the fixed point components). Another key point is that although the ring on the left of (3.5) is quite large, there is a finite subgroup Γ of T , generated by all finite-order elements that stabilize some regular point in M , so that the Γ-subring (3.7) H∗T (M still surjects onto the Chen-Ruan cohomology of the reduction. Thus, while it appears that we have made the computation much more complicated, it turns that there is still an effective algorithm to complete it. For full details, please refer to [GHK]. We now return to our examples. Example 3.4: For a symplectic toric orbifold Cn//K(α), we may use the com- binatorics of its labeled polytope to establish an explicit presentation of the Chen- Ruan cohomology of these orbifolds [GHK, § 9]. In the cases where the symplectic picture is identical to the algebraic, the [GHK] results replicate those of [BCS]. Example 3.5: For the T -action on a coadjoint orbit M = Oλ, the symplectic reduction M//T (α) is a weight variety. The equivariant cohomology of M may be read directly from its moment polytope, as may the orbifold singularities of the reduction. In this case, the Theorem 3.3 yields an explicit combinatorial description of the Chen-Ruan cohomology of the weight variety. We conclude this section with a brief remark on coefficients. In both Theo- rems 3.1 and 3.2, the rational coefficients are necessary. For the Injectivity The- orem 3.1, we may prove the result over Z with the additional hypothesis that H∗(MT ;Z) contains no torsion. The Surjectivity Theorem 3.2 over Z requires much stronger hypotheses. We will see in the next section that we may compute inte- grally for weighted projective spaces, but that a simple product of two weighted projective spaces yields a counter-example to the general theorem. Tolman and Weitsman verify surjectivity over Z for a rather restrictive class of torus actions [TW]. This topic is being more closely examined for a larger collection of actions by Susan Tolman and the author [HT]. 12 TARA S. HOLM 4. The case of weighted projective spaces Let b = (b0, . . . , bn) be an (n+1)-tuple of positive integers. Consider the circle action on Cn+1 given by (4.1) t · (z0, . . . , zn) = (t b0z0, . . . , t bnzn) for each t ∈ S1. This action preserves the unit sphere S2n+1, and the weighted projective space CPn is the quotient of S2n+1 by this locally free circle action. This is a symplectic reduction because S2n+1 is (up to equivariant homeomorphism) a regular level set for a moment map Φ for the weighted S1 action on Cn+1. Thus, (4.2) CPn(b) n+1//S1. Nonetheless, we may continue our analysis without invoking the full symplectic machinery: the arguments simplify greatly in this special case. When the bi are relatively prime (that is, gcd(b0, . . . , bn) = 1), this fits into the framework of symplectic toric orbifolds discussed in Example 3.4. When the bi are not relatively prime, we let g = gcd(b0, . . . , bn), and note that there is a global Zg stabilizer. In this case, the orbifold [CP ] is not reduced. Its coarse moduli space is the same as the coarse moduli space of CPn (b/g) , where (b/g) denotes the sequence of integers (b0/g, . . . , bn/g); and as a non-reduced orbifold, it corresponds to a gerbe [CPn(b)] → [CP (b/g)]. It is important to include the case when the bi are not relatively prime, because such non-reduced weighted projective spaces may well show up as suborbifolds of a reduced weighted projective space. The cohomology of the topological space CPn . Kawasaki studied the sin- gular cohomology ring of (the coarse moduli space of) weighted projective spaces [Ka]. To present the product structure, we will need the integers (4.3) ℓk = ℓ k := lcm bi0 · · · bik gcd(bi0 , . . . , bik) 0 ≤ i0 < · · · < ik ≤ n for each 1 ≤ k ≤ n. Theorem 4.1 ([Ka]). The integral cohomology of CPn (4.4) Hi(CPn(b);Z) Z if i = 2k, 0 ≤ k ≤ n, 0 otherwise. Moreover, letting γi denote the generator of H 2i(CPn ;Z), we have (4.5) γk ∪ γm = ℓk · ℓm γm+k. Outline of the Proof. Let Gk denote the group of k th roots of unity. Then as a topological space, CPn is homeomorphic to a quotient of ordinary projective space CPn by the finite group (4.6) G(b) = Gb0 × · · · ×Gbn . COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 13 Explicitly, using standard homogeneous coordinates, the map pb : CP n −→ CPn(b)(4.7) [z0 : · · · : zn] 7−→ [z 0 : · · · : z n ](4.8) induces the homeomorphism CPn/G(b) ∼= CP . This then induces an isomor- phism in singular cohomology with rational coefficients, since over Q we have the isomorphisms (4.9) H∗(CPn;Q) ∼= H ∗(CPn;Q)G(b) ∼= H ∗(CPn/G(b);Q). Over the integers, the computation is a bit more subtle. Using twisted lens spaces, Kawasaki verifies that just as for CPn, the cohomology ring H∗(CPn is torsion-free with a copy of Z in each even degree between 0 and 2n; however the product structure is twisted by the weights bi. Moreover, there are cases when we need all n generators γ1, . . . , γn to present this ring. Kawasaki showed that the map (4.10) p∗b : H 2k(CPn(b);Z) −→ H 2k(CPn;Z) is multiplication by ℓk for all 1 ≤ k ≤ n. From this, we may deduce that (4.11) γ1 ∪ γk = ℓ1 · ℓk+1 γk+1, and hence γk ∪ γm = ℓk · ℓm γm+k. The result now follows. � It is important to note that pb is not an isomorphism of orbifolds. Indeed, the isotropy group at any point in CPn is the stabilizer group of any lift of the point in S2n+1. Thus, all isotropy groups for CPn are cyclic. On the other hand, the orbifold CPn/G(b) has points with isotropy group G(b), which may not be cyclic. Nevertheless we will make use of the map pb to understand the structure of the cohomology of the orbifold [CPn ] = [S2n+1/S1 ]. Here, the subscript (b) on S1 indicates that the circle action is weighted by the integers (b) = (b0, . . . , bn). The cohomology of the orbifold [CPn(b)]. Since the weighted projective space ] is a symplectic reduction, it is possible invoke the results from Section 3 to determine the cohomology of the orbifold H∗([CPn ];Q), and to apply results from [GHK, §9] to obtain a presentation over Z. We give a direct argument here that is similar in spirit, but that avoids much of this big machinery; we then compare this to Kawasaki’s Theorem 4.1. Theorem 4.2. The cohomology of the orbifold [CPn (4.12) H∗([CPn(b)];Z) = H (S2n+1;Z) ∼= 〈b0 · · · bnun+1〉 Moreover, the natural map (4.13) q∗(b) : H ∗(S2n+1/S1(b);Z) −→ H (S2n+1;Z) is completely determined by q∗ (γ1) = ℓ1 · u = lcm(b0, . . . , bn) · u. Proof. Consider the (weighted) circle action of S1 on Cn+1 given by (4.14) t · (z0, · · · , zn) = (t b0 · z0, · · · , t bn · zn). 14 TARA S. HOLM The unit sphere S2n+1 is invariant under this action, so we get a long exact sequence in S1-equivariant cohomology for the pair (Cn+1, S2n+1), (4.15) · · · → HiS1 (Cn+1, S2n+1;Z) → HiS1 (Cn+1;Z) → HiS1 (S2n+1;Z) → · · · . Thinking of (Cn+1, S2n+1) as a disk and sphere bundle over a point, we may use the Thom isomorphism to identify Hi (Cn+1, S2n+1;Z) ∼= H i−2(n+1) (Cn+1;Z). Under this identification, the map α is the cup product with the equivariant Euler class (4.16) eS1 (Cn+1) = bo · · · bnu Thus, the map α is injective, so the long exact sequence splits into short exact sequences (4.17) 0 → HiS1 (Cn+1, S2n+1;Z) → HiS1 (Cn+1;Z) → HiS1 (S2n+1;Z) → 0. Thus, we have a surjection (4.18) β : Z[u] ∼= H (Cn+1;Z) −→ H∗S1 (S2n+1;Z). Moreover, the exactness of (4.17) means that the kernel of β is equal to the image of α, namely all multiples of the equivariant Euler class. This establishes (4.12). Turning to (4.13), the map (4.19) q∗(b) : H ∗(S2n+1/S1(b);Z) −→ H (S2n+1;Z), is exactly the one defined in (2.7). We know that this is an isomorphism over Q, so q∗ must map γ1 to a multiple of u. Moreover, because b0 · · · bnu n+1 is zero in (S2n+1;Z), we must have that (4.20) (q∗(b)(γ1)) n+1 ∈ b0 · · · bnu To determine the image of the class γ1, we return to the map pb : CP n → CPn(b). This map lifts to maps on S2n+1 and Cn+1 given by (4.21) Πb // S2n+1 πb //? S2n+1 0 6= (z0, · · · , zn) 7−→ |zbii | zb00 , · · · , z (4.22) (0, . . . , 0) 7−→ (0, . . . , 0)(4.23) The maps in this diagram are all equivariant with respect to the standard circle action on the left-hand spaces and the (b)-weighted circle action on the right-hand COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 15 spaces. Thus, we have a diagram of maps (4.24) (Cn+1 × S1)/S1 Πb // (Cn+1 × S1 (S2n+1 × S1)/S1 πb // (S2n+1 × S1(b))/S CPn pb // CPn Applying singular cohomology H∗( ;Z) and identifying equivariant cohomology, we have a commutative diagram (4.25) Z[x] H∗ (Cn+1;Z) oo (Cn+1;Z) Z[u] 〈xn+1〉 H (S2n+1;Z) oo (S2n+1;Z) 〈b0···bnun+1〉 〈xn+1〉 H ∗(CPn;Z) oo ∗(CPn(b);Z) Z⊕ Zγ1 ⊕ · · · ⊕ Zγn Because Cn+1 equivariantly deformation retracts to a point, the map Π∗b maps the generator u to x. The commutativity of the top square then implies that π∗b (u) = x. Thus, we know that π∗b (q (b)(γ1) = q ∗(p∗b(γ1))(4.26) = q∗(ℓ1 · x), by Kawasaki’s result,(4.27) = ℓ1 · x, since q ∗ is an equality,(4.28) = π∗b (ℓ1 · u).(4.29) In low degree, π∗b is injective, so we may conclude that q (b)(γ1) = ℓ1 · u. Noting that ℓ1 = lcm(b0, . . . , bn) completes the proof. � Over the integers, this invariant does distinguish a weighted projective space from the standard one; however, it may not differentiate between two weighted projective spaces. For example, the cohomology rings of the orbifolds [CP 12,2] and [CP 14,1] are identical. They are both (4.30) 〈4u2〉 We note that these surjectivity techniques do not generally work over the inte- gers. To see this, we note that for any abelian reduction of affine space, the domain of the Kirwan map H∗T (C N ;Z) has terms only in even degrees. If we consider the simple product [CP 11,2 × CP 1,2], we may compute the cohomology of this orbifold using the above result and the Künneth formula. Since [CP 11,2] has 2-torsion in high degrees, the Tor term from the Künneth formula plays a role, yielding 2-torsion in 16 TARA S. HOLM high odd degrees in the cohomology of the orbifold [CP 11,2 × CP 1,2]. Thus, surjec- tivity must fail over the integers in this example. We note that any failure over Z must be due to problems with torsion, because surjectivity does hold over Q. The Chen-Ruan orbifold cohomology of [CPn ]. When computing the Chen- Ruan ring, it is important to recall that a weighted projective space is a circle reduc- tion. Thus, the finite group Γ for which the Γ-piece surjects onto H ∗CR([CP (b)];Z) is a cyclic group. For any vector v that is non-zero is a single coordinate, say the ith coordinate, the stabilizer of v is Zbi . Thus, the group Γ generated by all finite stabilizers is the ℓth roots of unity Zℓ ⊂ S 1, where ℓ = lcm(b0, . . . , bn). Hence, we have a surjection (4.31) Z[u, α0, α1, . . . , αℓ−1] // // H CR([CP (b)];Z). In this case, thinking of e ℓ = ζk ∈ Zℓ ⊂ S 1, αk denotes a generator for (4.32) H∗S1 ((Cn+1)ζk ;Z). To complete the computation, we must determine the orbifold product (4.33) αi ⌣ αj = αi ⋆ αj and the kernel of the orbifold Kirwan map (3.5). For any integer m ∈ Z, we let [m] denote the smallest non-negative integer congruent to m modulo ℓ. For any rational number q ∈ Q, 〈q〉f denotes its fractional part. Finally, we let (4.34) ak(m) := [bk ·m] bk ·m This is the rational number such that ζm acts on the k th coordinate by e2πiak(m). Theorem 4.3. The Chen-Ruan orbifold cohomology of CPn (4.35) H ∗CR([CP (b)];Z) Z[u, α0, α1, . . . , αℓ−1] I + J where u is a class in degree 2, (4.36) deg(αj) = 2 ak(j). Here, I is the ideal (4.37) I = αiαj − (bku) ak(i)+ak(j)−ak(i+j) α[i+j] generated by the ⋆ product structure, and J is (4.38) J = ak(j)=0 the kernel of the surjection K of the orbifold Kirwan map. Remark 4.4: The generator u is the generator of S1-equivariant cohomology and hence has degree 2. The generator αk is a placeholder for the cohomology of the ζk-sector. COHOMOLOGY OF ABELIAN SYMPLECTIC REDUCTIONS 17 Remark 4.5: Note that the generator α0 is the placeholder for the identity sector. Indeed, we always have (4.39) α0 ⋆ α0 = α0 as a consequence of the relation in (4.37) where i = j = 0, hence we may think of α0 as 1. Remark 4.6: The reader may use this theorem to check that H ∗CR( ;Z) does distinguish [CP 12,2] from [CP 4,1]. Proof. We use the ⋆ product given by Equation (2.1) in [GHK]. In the case of a weighted circle action on Cn+1, there is exactly one fixed point (the origin), any generator αi restricted to that fixed point is 1, and the equivariant Euler class for the kth coordinate is precisely bku, whence (4.40) αi ⋆ αj = (bku) ak(i)+ak(j)−ak(i+j) α[i+j]. Turning to the kernel computation, each (Cn+1)ζj has a weighted S1 action, and so we apply Theorem 4.2 to this subspace. Thus, for the ζj-sector, the kernel contribution is the equivariant Euler class of (Cn+1)ζj times the placeholder αj . We note that (Cn+1)ζj contains the kth coordinate subspace precisely when ak(j) = 0. Hence, (4.41) eS1 ((Cn+1)ζj ) = ak(j)=0 and the theorem follows. � This theorem is an immediate consequence of Theorem 4.2 and [GHK]. The importance of this description is its ease in computation, since it avoids any com- putation of a labeled moment polytope (á la [LT]) or of a stacky fan (á la [BCS]). We demonstrate this computational facility in the following concluding example. Example 4.7: Consider the weighted projective space [CP 51,2,2,3,3,3]. This is a symplectic reduction of C6, the group Γ is Z/6Z, and so the Chen-Ruan orbifold cohomology of [CP 51,2,2,3,3,3] is a quotient of (4.42) Z[u, α0, α1, α2, α3, α4, α5]. 18 TARA S. HOLM The following chart contains the data needed to compute the ideals I and J . (4.43) g ζ0 ζ1 ζ2 ζ3 ζ4 ζ5 (C6)g C6 {0} 3C(3) 2C(2) 3C(3) {0} aC(1)(g) 0 aC(2)(g) 0 aC(3)(g) 0 2 · age(g) 0 14 generator of ((C6)g;Z) α0 α1 α2 α3 α4 α5 ((C6)g) 108u6 1 27u3 4u2 27u3 1 Note that because of the multiplicities, (4.44) 2 · age(g) = 2 · a1(g) + 2a2(g) + 3a3(g) Since α0 = 1, and α1 and α5 are in the kernel ideal J , we only need to compute the products among α2, α3 and α4. For example, we may compute (4.45) α2 ⋆ α2 = (u) (3u)0+0−0 α4 = 4u All of the products contributing to I, then, are summarized in the following table. (4.46) ⋆ α2 α3 α4 α2 4u 2α4 α5 = 0 4u α3 27u 4 uα1 = 0 α4 uα2 Thus, as a ring, (4.47) H ∗CR([CP 1,2,2,3,3,3];Z) Z[u, α0, α1, α2, α3, α4, α5] I + 〈108u6, α1, 27u3α2, 4u2α3, 27u3α4, α5〉 This generalizes Jiang’s computation [J] to a computation over Z. References [ℵGV] D. Abramovich, T. Graber, and A. Vistoli, “Algebraic orbifold quantum products.” Contemp. 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[S2] I. Satake, “The Gauss-Bonnet Theorem for V-manifolds.” J. Math. Soc. Japan 9 (1957) 464–492. [SL] R. Sjamaar and E. Lerman, “Stratified symplectic spaces and reduction.” Ann. of Math. (2) 134 (1991) no. 2, 375–422. [Th] W. Thurston, Three-dimensional geometry and topology. Vol. 1. Princeton Mathemat- ical Series, #35. Princeton University Press, 1997. ISBN 0-691-08304-5. [TW] S. Tolman and J. Weitsman, “The cohomology ring of symplectic quotients.” Communi- cations in Analysis and Geometry 11 (2003) no. 4, 751–773. Preprint math.DG/9807173. Department of Mathematics, Cornell University, Ithaca, NY 14853-4201 USA E-mail address: [email protected] http://arxiv.org/abs/math/9807173 1. Symplectic manifolds and quotients 2. Orbifolds and their cohomology 3. Why the symplectic category is convenient 4. The case of weighted projective spaces References

0704.0258

Correlation functions in the Non Perturbative Renormalization Group and field expansion

Correlation functions in the Non Perturbative Renormalization Group and field expansion Diego Guerra,∗ Ramón Méndez-Galain,† and Nicolás Wschebor‡ Instituto de F́ısica, Facultad de Ingenieŕıa, Univ. de la República, J.H.y Reissig 565, 11000 Montevideo, Uruguay (Dated: November 4, 2018) Abstract The usual procedure of including a finite number of vertices in Non Perturbative Renormalization Group equations in order to obtain n-point correlation functions at finite momenta is analyzed. This is done by exploiting a general method recently introduced which includes simultaneously all vertices although approximating their momentum dependence. The study is performed using the self-energy of the tridimensional scalar model at criticality. At least in this example, low order truncations miss quantities as the critical exponent η by as much as 60%. However, if one goes to high order truncations the procedure seems to converge rapidly. PACS numbers: 03.75.Fi,05.30.Jp ∗Electronic address: [email protected] †Electronic address: [email protected] ‡Electronic address: [email protected] http://arxiv.org/abs/0704.0258v2 mailto:[email protected] mailto:[email protected] mailto:[email protected] I. INTRODUCTION In nearly all fields in physics, there are systems having a large number of strongly cor- related constituents. These cannot be treated with usual perturbative methods. Phase transitions and critical phenomena, disordered systems, strongly correlated electrons, quan- tum chromodynamics at large distances, are just a few examples which demand a general and efficient method to treat non-perturbative situations. In problems as those just quoted, the calculation of correlation functions of the configuration variables is, in general, a very complicated task. The non perturbative renormalization group (NPRG) [1, 2, 3, 4, 5] has proven to be a powerful tool to achieve this goal. It presents itself as an infinite hierarchy of flow equations relating sequentially the various n-point functions. It has been successfully applied in many different problems, either in condensed matter, particle or nuclear physics (for reviews, see e.g. [6, 7, 8]; a pedagogical introduction can be found in [9]). In most of these problems however, one is interested in observables dominated by long wavelength modes. In these cases, it is then possible to approximately close the infinite hierarchy of NPRG equations performing an expansion in the number of derivatives of the field. This approximation scheme is known as the derivative expansion (DE) [10]. The price to pay is that the n-point functions can be calculated only at small external momenta, i.e. smaller than the smallest mass on the problem (vanishing momenta in the case of critical phenomena). In many other physical problems however, this is not enough: the full knowledge of the momentum dependence of correlation functions is needed in order to calculate quantities of physical interest (e.g. to get the spectrum of excitations, the shape of a Fermi surface, the scattering matrix, etc.). There have been many attempts to solve the infinite system of flow equations at finite momenta; most of them are based on various forms of an early proposal by Weinberg [11]. Although some of these attempts [12, 13, 14, 15] introduce sophisticated ansatz for the unknown correlation functions appearing in a given flow equation, most efforts simply ignore high order vertices. In all these works, only low order vertices are taken into account: usual calculations do not even include the complete flow of the 3- and 4-point functions. Moreover, it is not possible a priori to gauge the quality of such approximations schemes. Recently, an alternative general method to get n-point functions at any finite momenta within the NPRG has been proposed [16]. It has many similarities with DE. First, it is an approximation scheme that can be systematically improved. Second, the scheme yields a closed set of flow equations including simultaneously an infinite number of vertices; one thus goes far beyond schemes including a small number of vertices, as those quoted in the previous paragraph. Moreover, it has been proven [16] that in their corresponding limits, both perturvative and DE results are recovered; this remains valid at each order of the respective expansion. Finally, in the large-N limit of O(N) models, the leading-order (LO) of the approximation scheme becomes exact for all n-point functions. (The expression “leading order” means the first step in the approximation scheme; it does not refer to an expansion in a small parameter which usually does not exist in these kind of problems.). In [17], the method has been applied, in its leading order, to the calculation of the self- energy of the scalar model, at criticality. That is, we have fine-tuned the bare mass of the model in order for the correlation length to be infinite, and then we have studied the full range of momenta, from the high momenta Gaussian regime to the low momenta scaling one. At this order of the approximation scheme the self-energy is expected to include all one loop contributions and to achieve DE at next-to-leading order (NLO) precision, in the corresponding limit [16]. The numerical solution found in [17] verifies these properties. Moreover, the function has the expected physical properties in all momenta regime. First, it presents the correct scaling behavior in the infrared limit. The model reproduces critical exponent η with a level of precision comparable to the DE at NLO. Moreover, contrarily to DE, the anomalous power-law behavior can be read directly from the momentum dependence of the 2-point function. Second, it shows the expected logarithmic shape of the perturbative regime even though the coefficient in front of the logarithm, which is a 2-loops quantity, is only reproduced with an error of 8%. In order to check the quality of the solution in the intermediate momentum region, a quantity sensitive to this crossover sector has been calculated: one gets a result almost within the error bars of both Monte-Carlo and resumed 7-loops perturbative calculations. Please observe that this quantity is extremely difficult to calculate: even these sophisticated methods give an error of the order of 10%. Another interesting similarity between DE and the method presented in [16] is that, as a price to pay in order to close the equations including an infinite number of vertices, one has to study the problem in an external constant field. Accordingly, one ends up with partial differential equations which may be difficult to solve. A useful approximation scheme, widely used in DE calculations, is to perform, on top of the expansion in derivatives of the field, an extra expansion in powers of the field (see e.g. [7]), in the spirit of Weinberg proposal. During the last 10 years, this strategy has been widely used [18, 19, 20, 21, 22, 23, 24, 25]; in many studied situations this expansion seems to converge (generally oscillating) [19, 20, 21, 23, 24, 25], while in many others it does not [22, 26]. In d = 2, the field expansion has been explored with no indication of convergence for critical exponents, even going to high orders [27]. In this work we shall explore this procedure of expansion in powers of the field, in the framework of the calculation scheme presented in [16]. More precisely, we shall make a field expansion on top of the already approximated 2-point function flow equation solved in [17]. Then we shall compare results with and without field expansion. In doing so, we have two goals. First, we shall study the apparent convergence of this procedure. This comparison is essential if one hopes to apply the scheme described in [16] to situations more complicated than that considered in [17]. For example, within DE scheme, when trying to go to higher orders or when considering more involved models, the expansion in powers of the field on top of the corresponding approximate flow equations is sometimes the only practical strategy to solve them [8, 23, 28]. The second and more important goal is the following: as we shall see in section III, truncation in powers of the field is equivalent to ignoring high order vertices in the flow equations. Thus, the comparison presented here can help to estimate the quality of the calculations made so far to get n-point functions at finite momenta neglecting high order vertices. The article is organized as follows. In the next section we describe the basics ingredients of both the NPRG and the approximation scheme introduced in [16]. We also present the results obtained in [17], when this scheme is used to find the 2-point function of the scalar model. In section III, we apply the expansion in the field at various orders and compare these results with those found in [17]. Finally, we present the conclusions of the study. II. GENERAL CONSIDERATIONS Let us consider a scalar field theory with the classical action (∂µϕ(x)) ϕ2(x) + ϕ4(x) . (1) Here, r and u are the microscopic mass and coupling, respectively. The NPRG builds a family of effective actions, Γκ[φ] (where φ(x) = 〈ϕ(x)〉J is the ex- pectation value of the field in presence of an external source J(x)), in which the magnitude of long wavelength fluctuations are controlled by an infrared regulator depending on a con- tinuous parameter κ. One can write for Γκ[φ] an exact flow equation [5, 14, 21, 29]: ∂κΓκ[φ] = (2π)d ∂κRκ(q Γ(2)κ +Rκ , (2) where Γ κ is the second functional derivative of Γκ with respect to φ(x), and Rκ denotes a family of “cut-off functions” depending on κ: Rκ(q) behaves like κ 2 when q ≪ κ and it vanishes rapidly when q ≫ κ [19, 20]. The effective action Γκ[φ] interpolates between the classical action obtained for κ = Λ (where Λ−1 is the microscopic length scale), and the full effective action obtained when κ → 0, i.e., when all fluctuations are taken into account (see e.g. [7]). By differentiating eq. (2) with respect to φ(x), and then letting the field be constant, one gets the flow equation for the n-point function Γ κ in a constant background field φ. For example, for the 2-point function one gets: κ (p;φ) = (2π)d ∂κRk(q) Gκ(q;φ)Γ κ (p, q,−p− q;φ) ×Gκ(q + p;φ)Γ κ (−p, p+ q,−q;φ)Gκ(q;φ) Gκ(q;φ)Γ κ (p,−p, q,−q;φ)Gκ(q;φ) , (3) where G−1κ (q;φ) ≡ Γ κ (q,−q;φ) +Rκ(q 2), (4) and we used the definition (2π)d δ(d) Γ(n)κ (p1, . . . , pn;φ) = ddx1 . . . ddxne δφ(x1) . . . δφ(xn) φ(x)≡φ The flow equation for a given n-point function involves the n+1 and n+2 point functions (see, e.g., eq. (3)), so that the flow equations for all correlation functions constitute an infinite hierarchy of coupled equations. In [16], a general method to solve this infinite hierarchy was proposed. It exploits the smoothness of the regularized n-point functions, and the fact that the loop momentum q in the right hand side of the flow equations (such as eq. (2) or eq. (3)) is limited to q <∼ κ due to the presence of ∂κRκ(q). The leading order of the method presented in [16] thus consists in setting Γ(n)κ (p1, p2, ..., pn−1 + q, pn − q) ∼ Γ κ (p1, p2, ..., pn−1, pn) (6) in the r.h.s. of the flow equations. After making this approximation, some momenta in some of the n-point functions vanish, and their expressions can then be obtained as derivatives of m-point functions (m < n) with respect to a constant background field. Specifically, in the flow equation for the 2-point function, eq. (3), after setting q = 0 in the vertices of the r.h.s., the 3- and 4-point functions will contain one and two vanishing momenta, respectively. These can be related to the following derivatives of the 2-point function: Γ(3)κ (p,−p, 0;φ) = κ (p,−p;φ) , Γ(4)κ (p,−p, 0, 0;φ) = κ (p,−p;φ) . (7) One then gets a closed equation for Γ κ (p;φ): 2;φ) = J d (p, κ;φ) κ (p,−p;φ) d (κ;φ) κ (p,−p;φ) , (8) where d (p; κ;φ) ≡ (2π)d κ∂κRκ(q 2)Gκ(p+ q;φ)G (n−1) κ (q;φ), (9) d (κ;φ) ≡ (2π)d κ∂κRκ(q 2)Gnκ(q;φ). (10) In fact, in order to preserve the relation Γ(2)κ (p = 0;φ) = , (11) Vκ(φ) = Γκ[φ(x) ≡ φ]/Vol being the effective potential, it is better to make the approxima- tion (6) (followed by (7)) in the flow equation for Σκ(p;φ) defined as Σκ(p;φ) = Γ κ (p;φ)− p 2 − Γ(2)κ (p = 0;φ). (12) The 2-point function is then obtained from Γ(2)(p;φ) = ∂2Vκ(φ)/∂φ 2 + p2 +Σκ(p;φ), which demands the simultaneous solution of the flow equations for Vκ(φ) and Σκ(p;φ). As shown in [17], even if the complete solution of these equations is a priori complicated, a simple, and still accurate, way of solving them consists in assuming in the various integrals G−1κ (q;φ) ≃ Zκq 2 + ∂2Vκ(φ)/∂φ 2 +Rκ(q 2), (13) where Zκ ≡ Zκ(φ = 0), with Zκ(φ) ≡ 1+∂Σκ(p;φ)/∂p 2|p=0. This approximation is consistent with an improved version of the Local Potential Approximation (LPA, the first order of the DE), which includes explicitly a field renormalization factor Zκ [7]. Doing so, the “p = 0” sector decouples from the p 6= 0 one. Here, by “p = 0” we mean the sector describing vertices and derivative of vertices at zero momenta, i.e., flow equations for Vκ and Zκ. Moreover, it is useful to use the regulator [20] 2) = Zκ(κ 2 − q2) Θ(κ2 − q2), (14) which allows the functions J d (p; κ;φ) and I d (κ;φ) to be calculated analytically. The corresponding expressions can be found in [17]. In fact, all quantities are functions of ρ ≡ φ2/2. The problem is then reduced to the solution of the three flow equations for Vκ(ρ) and Zκ(ρ), for the p = 0 sector, and for Σκ(p; ρ), in the p 6= 0 one. As only the “effective mass” m2κ(ρ) ≡ ∂2Vκ(φ) ∂Vκ(ρ) ∂2Vκ(ρ) (and its derivatives with respect to ρ) enters in the p 6= 0 sector, it is more convenient to work with the flow equation for m2κ(ρ) instead of that for Vκ(ρ) itself. The non-trivial fact is that by differentiating twice the flow equation for Vκ(ρ) w.r.t φ, one gets a closed equation for m2κ(ρ). In order to make explicit the fixed point in the κ → 0 limit, it is necessary to work with dimensionless variables: µκ(ρ̃) ≡ Z −2 m2κ(ρ) , χκ(ρ̃) ≡ Z κ Zκ(ρ) , ρ̃ ≡ K d Zκ κ 2−d ρ , (16) which, in the critical case, have a finite limit when κ → 0. Above, Kd is a constant conveniently taken as K−1d ≡ d 2 d−1 πd/2 Γ(d/2) (e.g., K3 = 1/(6π 2)). In the p 6= 0 sector, the dimensionful variable p in the self-energy flow equation makes Σκ(p; ρ̃) reach a finite value when κ → 0. As discussed in [17], the inclusion of the flow equation for the renormalization factor Zκ(ρ̃) is essential in order to preserve the correct scaling behavior of Γ (2)(p; ρ̃) in the infrared limit. Doing so, in the critical case, the function Γ(2)(p; ρ̃)/(Zκκ 2) has to reach a fixed point expression depending on ρ̃ and p/κ, when κ, p ≪ u and ρ̃ ∼ 1. Putting all together, in d = 3, the three flow equations that have to be solved are κ∂κµκ(ρ̃) = −(2 − ηκ)µκ(ρ̃) + (1 + ηκ)ρ̃µ κ(ρ̃)− µ′κ(ρ̃) + 2ρ̃µ κ(ρ̃) (1 + µκ(ρ̃))2 4ρ̃µ′κ(ρ̃) (1 + µκ(ρ̃))3 κ∂κχκ(ρ̃) = ηκχκ(ρ̃) + (1 + ηκ)ρ̃χ κ(ρ̃)− 2ρ̃ µ′2κ (ρ̃) (1+µκ(ρ̃))4 1− ηκ 8ρ̃χ′κ(ρ̃) µ′κ(ρ̃) (1+µκ(ρ̃))3 χ′κ(ρ̃)+2ρ̃χ κ(ρ̃) (1+µκ(ρ̃))2 , (18) together with χ′κ(0) χ′κ(0)/5 + (1 + µκ(0)) , κ∂κZκ = −ηκZκ , (19) for the p = 0 sector, and κ∂κΣκ(p, ρ̃) = (1 + ηκ)ρ̃Σ κ(p, ρ̃) + 2ρ̃µ′2κ (ρ̃)κ (1+µκ(ρ̃))2 fκ(p̃, ρ̃)− 2(1−ηκ/5) (1+µκ(ρ̃))2 2ρ̃fκ(p̃,ρ̃) (1+µκ(ρ̃))2 2µ′κ(ρ̃)Σ κ(p, ρ̃) + Σ′2κ (p,ρ̃) (1−ηκ/5) (1+µκ(ρ̃))2 (Σ′κ(p, ρ̃) + 2ρ̃Σ κ(p, ρ̃)) (20) for the p 6= 0 one. In these equations, the prime means ∂ρ̃ and we used the explicit expression for I 3 = 2K3κ 5−2nZ1−nκ (1−ηκ/5)/(1 + µκ(ρ̃)) n. In eq. (20), we introduced the dimensionless expression fκ defined as J 3 (p; κ; ρ) ≡ K3κ −1Z−2κ /(1 + µκ(ρ̃)) 2 × fκ(p̃; ρ̃), with p̃ ≡ p/κ. In [17], this strategy is used to get the 2-point function of the scalar model at criticality and zero external field (i.e., Σ(p = 0, ρ = 0) = 0), in d = 3. As recalled above, the function thus obtained has the correct shape, either in the scaling, perturbative and intermediate momenta regimes. III. EXPANSION IN POWERS OF THE FIELD In this section, we shall compare the solution obtained in [17] using the procedure de- scribed above, with the solution of the same three flow equations expanded in powers of ρ̃ and truncating up to a given order. Before doing so, let us first consider only the flow equation for the potential or, equivalently, that for the effective mass, i.e., eq. (17), with Zκ ≡ 1 (ηκ ≡ 0). This corresponds to the pure LPA sector and it is thus independent of the scheme presented in [16]. In d = 3, its expansion in powers of the field has been widely studied during the last ten years, using various regulators [22, 24]. Recently, another inter- esting truncation scheme has also been considered in [30] showing much better convergence properties. However, here we shall consider the simpler expansion in powers of the fields; as shall be seen bellow this is the field expansion that can be compared to usual truncation in the number of vertices. It has been shown [25] that, using the regulator we consider here (see eq. (14)), this expansion seems to converge. This result follows when expanding both around finite and zero external field, although faster in the first case. In [25] the convergence in this situation has been discussed studying the critical exponent ν. In order to strengthen this conclusion, as a first step in our study, we have analyzed the effect of the expansion on the function µκ(ρ̃): µκ(ρ̃) = µ(n)κ ρ̃ n. (21) More precisely, we shall gauge the impact of truncating this sum on the fixed point values of the coefficients µ κ , which are proportional to vertices at zero momenta and zero external field. This study is motivated by the fact that these µ κ shall appear in the Σκ(p; ρ̃) flow equation, eq. (20), when the later shall be expanded around ρ̃ = 0. Results are shown in Figure 1. The four plots present the fixed point value for the first 4 couplings, µ n = 0, · · · , 3. For each coupling, we present the result which follows by solving the complete LPA equation, eq. (17), together with the result obtained with the equation expanded in powers of ρ̃. For example, when going only up to the first order (i.e., neglecting all µ κ with n ≥ 2), the corresponding equations for µ κ and µ κ , are: κ = (ηκ − 2)µ (1− ηκ/5)µ (1 + µ κ = (2ηκ − 1)µ 6(1− ηκ/5)(µ (1 + µ . (23) which have to be solved simultaneously. (In fact, if solving just the LPA, ηκ = 0; nevertheless, we have kept ηκ in eqs. (22)-(23) for a later use of these equations). When going to the second order, eq. (23) acquires a new term and a new flow equation, that for µ κ , appears; and so on. According to Figure 1, an apparent convergence shows up. In all cases one observes 1 2 3 4 5 6 7 8 9 order -0.18 -0.15 -0.12 -0.09 4 5 6 7 8 9 -0.189 -0.1875 -0.186 4 5 6 7 8 9 0.246 0.2475 0.249 1 2 3 4 5 6 7 8 9 order 0.135 0.225 6 7 8 9 0.093 0.0945 0.096 2 3 4 5 6 7 8 9 order 0.075 3 4 5 6 7 8 9 order 0.0280 0.0315 0.0350 0.0385 FIG. 1: First four dimensionless fixed point couplings at zero momenta and zero external field: results obtained by truncating the flow equation, as a function of the order; the corresponding value for the complete equation is represented by the dotted-line. that: 1) there seems to be an oscillating convergence, 2) the value of µ(i) is found with about 1% error truncating at order i+ 3. Let us now turn to the study of the flow equation for the 2-point function coming from the scheme proposed in [16]. As the effective potential (or the effective mass), Γ κ (p; ρ) can also be expanded in powers of the external field: Γ(2)κ (p; ρ) = (2n)! Γ(2n+2)κ (p,−p, 0, 0, · · · , 0; ρ)|ρ=0 ρ n, (24) because Γ(m+2)κ (p,−p, 0, 0, · · · , 0; ρ) = κ (p;φ) 3,5×10 3,6×10 2,0×10 2,5×10 FIG. 2: Comparison of the self-energy when expanding only the flow equations for the self-energy Σκ(p; ρ̃) and its derivative Zκ(ρ̃) (strategy I): truncation is made at first (double dotted-dashed), second (dotted-dashed), third (dashed) and fourth (dotted) order; the complete solution is given by the straight line. In the figure, u = 5.9210−4Λ. and we used that, at zero field, all odd vertex functions vanish. Equation (24) makes clear the point stated above: once approximation (6) is performed, truncating the expansion in powers of the external field is equivalent to neglecting high order vertices. Moreover, eqs. (24) and (25) show that the procedure proposed in [16] indeed includes all vertices, although approximately. We have now all the ingredients to discuss the main goal of this paper: the analysis of the expansion of the three flow equations for µκ(ρ̃), Zκ(ρ̃) and Σκ(p; ρ̃), eqs. (17-20), around ρ̃ = 0. In doing so, one can write: Σκ(p, ρ̃) = Σ(n)κ (p) ρ̃ n. (26) χκ(ρ̃) = χ(n)κ ρ̃ n. (27) together with eq. (21). For example, when going to the first order, the six equations that have to be solved are: κ (p) = − (1− ηκ/5)Σ κ (p) (1 + µ κ (p) = (1 + ηκ)Σ κ (p) + (1 + µ fκ(p̃, 0)− 2(1− ηκ/5) (1 + µ 2fκ(p̃, 0) (1 + µ 2µ(1)κ Σ κ (p) + κ (p) 2(1− ηκ/5)µ κ (p) (1 + µ , (29) which correspond to the expansion of eq. (20), κ = ηκχ (1− ηκ/5)χ (1 + µ κ = (1 + 2ηκ)χ (1 + µ κ (1− ηκ/5) (1 + µ , (31) which correspond to the expansion of eq. (18), together with eqs. (22) and (23). In fact, it is possible to perform two kinds of expansion. First, in order to isolate the effect of the field expansion just in the flow equations provided by the scheme presented in [16], we shall expand only the flow equations for Σκ(p; ρ̃) and its derivative at zero momenta Zκ(ρ̃), eqs. (20) and (18), solving exactly the differential flow equation for µκ(ρ̃), eq. (17). For example, at first order, one should solve simultaneously eqs. (28-29), (30-31), and (17). This is called “strategy I”. Second, to consider all the effects, we shall make the expansion in the three flow equations. For example, at first order, one should solve simultaneously eqs. (28-29), (30-31), and (22-23). We call this “strategy II”. Notice that, as explained in [17], in order to get the correct scaling behavior it is mandatory to treat the equations for Zκ(ρ̃) and Σκ(p; ρ̃) with the same approximations; it is then not possible to solve one of them completely while expanding the other one. Figure 2 presents the self-energy one gets truncating up to fourth order, following strategy I; it is also shown the function obtained in [17] (from now on, the latter function, obtained by solving the 3 differential equations, eqs. (17), (18) and (20), shall be called the “complete solution”). Figure 3 presents the same results when following strategy II. These Figures show that, in both strategies of expansion, by truncating at first order one already gets a function with the correct shape in all momenta regimes. 3,5×10 3,6×10 1,9×10 2,0×10 FIG. 3: Comparison of the self-energy when expanding the three flow equations (strategy II): truncation is made at first (double dotted-dashed), second (dotted-dashed), third (dashed) and fourth (dotted) order; the complete solution is given by the straight line. In the figure, u = 5.9210−4Λ. In order to make a quantitative evaluation of the approximate solution obtained doing the expansion, we have calculated different numbers describing the physical properties of the self-energy. First, as can be seen in both figures above, all solutions have, in the infrared (p ≪ u), the potential behavior characterizing the scaling regime: Σ(p) + p2 ∼ p2−η, where η is the anomalous dimension. We have checked that, at each order and in both strategies, the resulting self-energy does have scaling, and we extracted the corresponding value of η. In fact, this can be done in two different ways: either using the κ–dependence of Zκ (η = − limκ→0 κ∂κ logZκ) or the p–dependence of Σ(p) stated above. We checked that those two values always coincide, within numerical uncertainties. Figure 4 presents the relative error for η, at each order, when compared with the value following from the complete solution. One observes: 1) in both strategies of expansion there is an apparent convergence, which is oscillatory; 2) the solution from strategy I reaches faster the correct result; 3) when following strategy I, already with a second order truncation the error is about 3% and it drops to less that 1% at the third order. Nevertheless, due to the mixed characteristic of 3 4 5 6 7 8 1 2 3 4 5 6 7 8 order FIG. 4: Relative error (measured in percent) for the anomalous dimension, with respect to the value coming from the complete solution, as a function of the truncation order. Full line: expanding only the flow equations for the self-energy Σκ(p; ρ) and its derivative Zκ(ρ) (strategy I); dashed line: expanding all flow equations (strategy II). strategy I, when using this strategy at high order numerical problems arise: indeed, this task demands the numerical evaluation of high order derivatives of µκ(ρ̃), to be used in the various flow equations obtained when expanding that of Σκ(p; ρ̃). If high precision in the result is required, strategy II is then numerically preferable. It is important to observe here that the procedure which can be compared to the usual truncation including a finite number of vertices is strategy II. Moreover, the inclusion of high order vertices without performing any other approximation is difficult; for example, the complete inclusion of the 6-point vertex has never been done. Accordingly, as can be seen in fig. 4, when including only up to the 4-point vertex, as it usually done, the error in η can be as large as 60%. A second number to assess the quality of the approximate solution is the critical exponent ν. In order to calculate it, we extract the renormalized dimensionful mass from m2R = κ2µκ(ρ̃ = 0) and we relate it to the microscopic one by m2R(κ = 0) ∝ (m R(κ = Λ)−m R,crit(κ = Λ)) 2ν , (32) where mR,crit is the critical renormalized mass. With the complete solution one gets ν = 3 4 5 6 7 8 1 2 3 4 5 6 7 8 order FIG. 5: Relative error (measured in percent) for the critical exponent ν, with respect to the value coming from the complete solution, as a function of the truncation order. Full line: expanding only the flow equations for the self-energy Σκ(p; ρ) and its derivative Zκ(ρ) (strategy I); dashed line: expanding all flow equations (strategy II). 0.647, to be compared with the best accepted value [31]: ν = 0.6304 ± 0.0013. Figure 5 presents the relative error of the value of ν extracted from the expansion. Once again, one observes that the convergence is much faster when following strategy I, i.e., when considering the effect of the expansion only on the self-energy equation. The large momenta regime (p ≫ u) of the self-energy can be calculated using perturbation theory, yielding the well known logarithmic shape: Σ(p) ∼ A log(p/B), where A and B are constants. For the complete solution presented in [17] one can prove analytically that A = u2/9π4, which is only 8% away from the exact result A = u2/96π2 (please observe that this coefficient is given by a 2-loop diagram for the self energy which is only approximatively included at this order). The proof of this analytical result remains valid when performing the field expansion, at any order and within both strategies. We have checked that our numerical solution always has the correct shape, with A = u2/9π4. This is due to the fact that already the first order in the expansion of Σκ(p; ρ̃) around ρ̃ = 0 contains the same 2-loop diagrams contributing to the complete solution. In order to study the quality of the self-energy in the intermediate momenta regime, we 2 3 4 1 2 3 4 order FIG. 6: Relative error (measured in percent) for ∆〈φ2〉, with respect to the value coming from the complete solution, as a function of the truncation order. Full line: expanding only the flow equations for the self-energy Σκ(p; ρ) and its derivative Zκ(ρ) (strategy I); dashed line: expanding all flow equations (strategy II). have calculated a quantity which is very sensitive to this cross-over region: ∆〈φ2〉 = (2π)3 p2 + Σ(p) . (33) (the integrand is non zero only in the region 10−3 <∼ p/u <∼ 10, see for example [15]). This quantity received recently much attention because it has been shown [32] that for a scalar model with O(N) symmetry, in d = 3 and N = 2, it determines the shift of the critical temperature of the weakly repulsive Bose gas. It has then been widely evaluated by many methods, for different values of N , in particular, for N = 1. With the numerical solution found in [17], one gets a number almost within the error bars of the best accepted results available in the literature, using lattice and 7 loops resumed perturbative calculations. Please observe that these errors are as large as 10%, which is an indication that this quantity is particularly difficult to calculate. In Figure 6 we plot the relative error in ∆〈φ2〉, at each order of the expansion, when compared with the complete solution result found in [17]. One can appreciate that 1) for both expansion strategies there is an apparent convergence, which is also oscillatory; 2) in both strategies, already with a second order truncation the error is about 1%. IV. SUMMARY AND CONCLUSIONS In this article, the inclusion of a finite number of vertices in NPRG flow equations is analyzed. An unsolved difficulty of this usual strategy (originally proposed by Weinberg) is the estimation of the error introduced at a given step. Moreover, without performing further approximations, it is very hard to reach high orders of the procedure. The study of its convergence is thus a difficult task. In the present work we analyse this problem using a different approximation scheme [16]: instead of considering a finite number of vertices, this procedure includes all of them, although approximately. Within this context, it is possible to estimate the error of the Weinberg approximation, order by order. To do so one can perform, on top of the approximation presented in [16], the usual truncation in the number of vertices. The analysis has been done in the particular case of the 2-point function of the scalar field theory in d = 3 at criticality. It has been shown [17] that, at least in this case, the procedure proposed in [16] yields very precise results. Another interesting outcome of the present work follows from the fact that, within the approximation [16], truncation in the number of vertices is equivalent to an expansion in powers of a constant external field. The latter is usually employed in the DE context in order to deal with complicated situations. The analysis of the present paper generalizes this expansion procedure when non zero external momentum are involved. The calculation of the 2-point function demands the study of both the p = 0 and the p 6= 0 sectors. While the first one is given by the well studied DE flow equations, the latter follows from the approximation scheme introduced in [16] to calculate the flow of Σκ(p; ρ). We used two different strategies to perform the field expansion, both of them around zero external field: either expanding only the flow equation for the self-energy (and its derivative) (strategy I), or both the effective potential and the self-energy (and its derivative) flow equations (strategy II). We have studied the convergence of various quantities measuring physical properties of the self-energy in all momenta regimes: the critical exponents η and ν of the infrared regime, the coefficient of the ultraviolet logarithm, and ∆〈φ2〉 which is dominated by the crossover momenta regime. As stated in section III, the strategy that can be compared to the usual truncation which includes a finite number of vertices is strategy II. For example, including completely the 4-point vertex as it is usually done (i.e., in the language of field expansion, going only up to the first order of the expansion), when describing the deep infrared regime one could make errors as big as 60% in the critical exponent η (see figure 4). If one wants results with less that 5% error for this quantity, the inclusion of up to 8-point vertices (i.e., going up to third order) is necessary. However, when going to higher orders in the field expansion, the series for all considered quantities seem to converge rapidly, within both strategies. The convergence is faster when using strategy I, i.e., when making the expansion only for the approximate flow equation resulting from the method presented in [16]. For example, using strategy I, a third order truncation introduces a relative error smaller that 1% for all studied quantities; while using strategy II, in order to reach the same error one needs 6th order for η, 4th order for ν and 2nd order for ∆〈φ2〉. Nevertheless, due to numerical difficulties, if trying to go to high order expansions, it is preferable to use strategy II, i.e., expanding also the effective potential flow equation. It is difficult to assess the generality of these results on the use of field expansion on top of the strategy proposed in [16]. Of course, there are situations where expanding in an external field is not a priori convenient. One can mention as a first example, situations where there is a physical external field (as in a broken phase or when an external source for the field is considered). A second example is two-dimensional systems where even in the DE, the field expansion does not seem to converge. 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Forgacs and H. Giacomini, J. Phys. A 40 (2007) F215 [arXiv:hep-th/0611306]. [31] R. Guida, J. Zinn-Justin, J. Phys. A31, 8103 (1998). [32] G. Baym, J.-P. Blaizot, M. Holzmann, F. Laloë, and D. Vautherin, Phys. Rev. Lett. 83, 1703 (1999). http://arxiv.org/abs/hep-th/0611306 Introduction General considerations Expansion in powers of the field Summary and Conclusions References

0704.0259

Formation and Collisional Evolution of Kuiper Belt Objects

FORMATION AND COLLISIONAL EVOLUTION OF KUIPER BELT OBJECTS Scott J. Kenyon Smithsonian Astrophysical Observatory Benjamin C. Bromley Department of Physics, University of Utah David P. O’Brien Planetary Science Institute Donald R. Davis Planetary Science Institute This chapter summarizes analytic theory and numerical calculations for the formation and collisional evolution of KBOs at 20–150 AU. We describe the main predictions of a baseline self-stirring model and show how dynamical perturbations from a stellar flyby or stirring by a giant planet modify the evolution. Although robust comparisons between observations and theory require better KBO statistics and more comprehensive calculations, the data are broadly consistent with KBO formation in a massive disk followed by substantial collisional grinding and dynamical ejection. However, there are important problems reconciling the results of coagulation and dynamical calculations. Contrasting our current understanding of the evolution of KBOs and asteroids suggests that additional observational constraints, such as the identification of more dynamical families of KBOs (like the 2003 EL61 family), would provide additional information on the relative roles of collisional grinding and dynamical ejection in the Kuiper Belt. The uncertainties also motivate calculations that combine collisional and dynamical evolution, a ‘unified’ calculation that should give us a better picture of KBO formation and evolution. 1. INTRODUCTION Every year in the Galaxy, a star is born. Most stars form in dense clusters of thousands of stars, as in the Orion Nebula Cluster (Lada and Lada 2003; Slesnick et al. 2004). Other stars form in small groups of 5–10 stars in loose asso- ciations of hundreds of stars, as in the Taurus-Auriga clouds (Gomez et al. 1993; Luhman 2006). Within these associa- tions and clusters, most newly-formed massive stars are bi- naries; lower mass stars are usually single (Lada 2006). Large, optically thick circumstellar disks surround nearly all newly-formed stars (Beckwith and Sargent 1996). The disks have sizes of ∼ 100–200 AU, masses of ∼ 0.01- 0.1 M⊙, and luminosities of ∼ 0.2–1 L⋆, where L⋆ is the luminosity of the central star. The masses and geometries of these disks are remarkably similar to the properties of the minimum mass solar nebula (MMSN), the disk required for the planets in the solar system (Weidenschilling 1977b; Hayashi 1981; Scholz et al. 2006). As stars age, they lose their disks. For solar-type stars, radiation from the opaque disk disappears in 1–10 Myr (Haisch et al. 2001). Many older stars have optically thin debris disks comparable in size to the opaque disks of younger stars but with much smaller masses, . 1M⊕, and luminosities, . 10−3 L⋆ (Chapter by Moro-Martin et al.). The lifetime of this phase is uncertain. Some 100 Myr-old stars have no obvious debris disk; a few 1–10 Gyr-old stars have massive debris disks (Greaves 2005). In the current picture, planets form during the transi- tion from an optically thick protostellar disk to an opti- cally thin debris disk. From the statistics of young stars in molecular clouds, the timescale for this transition, ∼ 105 yr, is comparable to the timescales derived for the forma- tion of planetesimals from dust grains (Weidenschilling 1977a; Youdin and Shu 2002; Dullemond and Dominik 2005) and for the formation of lunar-mass or larger planets from planetesimals (Wetherill and Stewart 1993; Weidenschilling et al. 1997a; Kokubo and Ida 2000; Nagasawa et al. 2005; Kenyon and Bromley 2006). Because the grains in debris disks have short collision lifetimes, . 1 Myr, compared to the ages of their parent stars, & 10 Myr, high velocity collisions between larger objects must main- tain the small grain population (Aumann et al. 1984; Backman and Paresce 1993). The inferred dust produc- tion rates for debris disks around 0.1–10 Gyr old stars, ∼ 1020 g yr−1, require an initial mass in 1 km ob- http://arxiv.org/abs/0704.0259v1 jects, Mi ∼ 10–100 M⊕, comparable to the amount of solids in the MMSN. Because significant long-term de- bris production also demands gravitational stirring by an ensemble of planets with radii of 500–1000 km or larger (Kenyon and Bromley 2004a; Wyatt et al. 2005), debris disks probably are newly-formed planetary sys- tems (Aumann et al. 1984; Backman and Paresce 1993; Artymowicz 1997; Kenyon and Bromley 2002b, 2004a,b). KBOs provide a crucial test of this picture. With objects ranging in size from 10–20 km to ∼ 1000 km, the size dis- tribution of KBOs yields a key comparison with theoretical calculations of planet formation (Davis and Farinella 1997; Kenyon and Luu 1998, 1999a,b). Once KBOs have sizes of 100–1000 km, collisional grinding, dynamical perturba- tions by large planets and passing stars, and self-stirring by small embedded planets produce features in the distribu- tions of sizes and dynamical elements that observations can probe in detail. Although complete calculations of KBO formation and dynamical evolution are not available, these calculations will eventually yield a better understanding of planet formation at 20–100 AU. The Kuiper belt also enables a vital link between the solar system and other planetary systems. With an outer radius of & 1000 AU (Sedna’s aphelion) and a current mass of ∼ 0.1 M⊕ (Luu and Jewitt 2002; Bernstein et al. 2004, Cahpter by Petit et al.), the Kuiper belt has prop- erties similar to those derived for the oldest debris disks (Greaves et al. 2004; Wyatt et al. 2005). Understanding planet formation in the Kuiper belt thus informs our inter- pretation of evolutionary processes in other planetary sys- tems. This paper reviews applications of coagulation theory for planet formation in the Kuiper belt. After a brief introduc- tion to the theoretical background in §2, we describe results from numerical simulations in §3, compare relevant KBO observations with the results of numerical simulations in §4, and contrast the properties of KBOs and asteroids in §5. We conclude with a short summary in §6. 2. COAGULATION THEORY Planet formation begins with dust grains suspended in a gaseous circumstellar disk. Grains evolve into planets in three steps. Collisions between grains produce larger aggregates which decouple from the gas and settle into a dense layer in the disk midplane. Continued growth of the loosely bound aggregates leads to planetesimals, gravita- tionally bound objects whose motions are relatively inde- pendent of the gas. Collisions and mergers among the en- semble of planetesimals form planets. Here, we briefly de- scribe the physics of these stages and summarize analytic results as a prelude to summaries of numerical simulations. We begin with a prescription for the mass surface density Σ of gas and dust in the disk. We use subscripts ‘d’ for the dust and ‘g’ for the gas and adopt Σd,g = Σ0d,0g 40 AU , (1) where a is the semimajor axis. In the MMSN, n = 3/2, Σ0d ≈ 0.1 g cm −2, and Σ0g ≈ 5–10 g cm (Weidenschilling 1977b; Hayashi 1981). For a disk with an outer radius of 100 AU, the MMSN has a mass of ∼ 0.03 M⊙, which is comparable to the disk masses of young stars in nearby star-forming regions (Natta et al. 2000; Scholz et al. 2006). The dusty midplane forms quickly (Weidenschilling 1977a, 1980; Dullemond and Dominik 2005). For inter- stellar grains with radii, r ∼ 0.01–0.1 µm, turbulent mixing approximately balances settling due to the vertical compo- nent of the star’s gravity. As grains collide and grow to r ∼ 0.1–1 mm, they decouple from the turbulence and settle into a thin layer in the disk midplane. The timescale for this process is ∼ 103 yr at 1 AU and ∼ 105 yr at 40 AU. The evolution of grains in the midplane is uncertain. Be- cause the gas has some pressure support, it orbits the star slightly more slowly than the Keplerian velocity. Thus, orbiting dust grains feel a headwind that drags them to- ward the central star (Adachi et al. 1976; Weidenschilling 1984; Tanaka and Ida 1999). For m-sized objects, the drag timescale at 40 AU, ∼ 105 yr, is comparable to the growth time. Thus, it is not clear whether grains can grow by di- rect accretion to km sizes before the gas drags them into the inner part of the disk. Dynamical processes provide alternatives to random ag- glomeration of grains. In ensembles of porous grains, gas flow during disruptive collisions leads to planetesimal for- mation by direct accretion (Wurm et al. 2004). Analytic es- timates and numerical simulations indicate that grains with r ∼ 1 cm are also easily trapped within vortices in the disk (e.g. de la Fuente Marcos and Barge 2001; Inaba and Barge 2006). Large enhancements in the solid-to-gas ratio within vortices allows accretion to overcome gas drag, enabling formation of km-sized planetesimals in 104–105 yr. If the dusty midplane is calm, it becomes thinner and thinner until groups of particles overcome the local Jeans criterion – where their self-gravity overcomes local orbital shear – and ‘collapse’ into larger objects on the local dy- namical timescale, ∼ 103 yr at 40 AU (Goldreich and Ward 1973; Youdin and Shu 2002; Tanga et al. 2004). This pro- cess is a promising way to form planetesimals; however, turbulence may prevent the instability (Weidenschilling 1995, 2003, 2006). Although the expected size of a col- lapsed object is the Jeans wavelength, the range of plan- etesimal sizes the instability produces is also uncertain. Once planetesimals with r ∼ 1 km form, gravity dom- inates gas dynamics. Long range gravitational interactions exchange kinetic energy (dynamical friction) and angular momentum (viscous stirring), redistributing orbital energy and angular momentum among planetesimals. For 1 km objects at 40 AU, the initial random velocities are compara- ble to their escape velocities, ∼ 1 m s−1 (Weidenschilling 1980; Goldreich et al. 2004). The gravitational binding en- ergy (for brevity, we use energy as a shorthand for specific energy), Eg ∼ 10 4 erg g−1, is then comparable to the typ- ical collision energy, Ec ∼ 10 4 erg g−1. Both energies are smaller than the disruption energy – the collision energy needed to remove half of the mass from the colliding pair of objects – which is Q∗D ∼ 10 5–107 erg g−1 for icy mate- rial (Davis et al. 1985; Benz and Asphaug 1999; Ryan et al. 1999; Michel et al. 2001; Leinhardt and Richardson 2002; Giblin et al. 2004). Thus, collisions produce mergers in- stead of debris. Initially, small planetesimals grow slowly. For a large ensemble of planetesimals, the collision rate is nσv, where n is the number of planetesimals, σ is the cross-section, and v is the relative velocity. The collision cross-section is the geometric cross-section, πr2, scaled by the gravitational focusing factor, fg, σc ∼ πr 2fg ∼ πr 2(1 + β(vesc/evK) 2) , (2) where e is the orbital eccentricity, vK is the orbital ve- locity, vesc is the escape velocity of the merged pair of planetesimals, and β ≈ 2.7 is a coefficient that ac- counts for three-dimensional orbits in a rotating disk (Greenzweig and Lissauer 1990; Spaute et al. 1991; Wetherill and Stewart 1993). Because evK ≈ vesc, gravitational focusing factors are small and growth is slow and orderly (Safronov 1969). The timescale for slow, orderly growth is ts ≈ 30 1000 km 250 yr 0.1 g cm−2 Gyr , where P is the orbital period (Safronov 1969; Lissauer 1987; Goldreich et al. 2004). As larger objects form, several processes damp parti- cle random velocities and accelerate growth. For objects with r ∼ 1–100 m, physical collisions reduce particle ran- dom velocities (Ohtsuki 1992; Kenyon and Luu 1998). For larger objects with r & 0.1 km, the smaller objects damp the orbital eccentricity of larger particles through dynam- ical friction (Wetherill and Stewart 1989; Kokubo and Ida 1995; Kenyon and Luu 1998). Viscous stirring by the large objects excites the orbits of the small objects. For planetes- imals with r ∼ 1 m to r ∼ 1 km, these processes occur on short timescales, . 106 yr at 40 AU, and roughly bal- ance when these objects have orbital eccentricity e ∼ 10−5. In the case where gas drag is negligible, Goldreich et al. (2004) derive a simple relation for the ratio of the eccentric- ities of the large (‘l’) and the small (‘s’) objects in terms of their surface densities Σl,s (see also Kokubo and Ida 2002; Rafikov 2003c,b,d,a), , (4) with γ = 1/4 to 1/2. Initially, most of the mass is in small objects. Thus Σl/Σs ≪ 1. For Σl/Σs ∼ 10 −3 to 10−2, el/es ≈ 0.1–0.25. Because esvK ≪ vl,esc gravitational focusing factors for large objects accreting small objects are large. Runaway growth begins. Runaway growth relies on positive feedback between accretion and dynamical friction. Dynamical friction pro- duces the largest fg for the largest objects, which grow faster and faster relative to the smaller objects and contain an ever growing fraction of the total mass. As they grow, these protoplanets stir the planetesimals. The orbital ve- locity dispersions of small objects gradually approach the escape velocities of the protoplanets. With esvK ∼ vl,esc, collision rates decline as runaway growth continues (eqs. (2) and (4)). The protoplanets and leftover planetesimals then enter the oligarchic phase, where the largest objects – oligarchs – grow more slowly than they did as runaway objects but still faster than the leftover planetesimals. The timescale to reach oligarchic growth is (Lissauer 1987; Goldreich et al. 2004) to ≈ 30 250 yr 0.1 g cm−2 Myr , (5) For the MMSN, to ∝ a −3. Thus, collisional damping, dynamical friction and gravitational focusing enhance the growth rate by 3 orders of magnitude compared to orderly growth. Among the oligarchs, smaller oligarchs grow the fastest. Each oligarch tries to accrete material in an annular ‘feed- ing zone’ set by balancing the gravity of neighboring oli- garchs. If an oligarch accretes all the mass in its feed- ing zone, it reaches the ‘isolation mass’ (Lissauer 1987; Kokubo and Ida 1998, 2002; Rafikov 2003a; Goldreich et al. 2004), miso ≈ 28 40 AU 0.1 g cm−2 M⊕ . (6) Each oligarch stirs up leftover planetesimals along its or- bit. Smaller oligarchs orbit in regions with smaller Σl/Σs. Thus, smaller oligarchs have larger gravitational focusing factors (eqs. (2) and (4)) and grow faster than larger oli- garchs (Kokubo and Ida 1998; Goldreich et al. 2004). As oligarchs approach miso, they stir up the velocities of the planetesimals to the disruption velocity. Instead of mergers, collisions then yield smaller planetesimals and de- bris. Continued disruptive collisions lead to a collisional cascade, where leftover planetesimals are slowly ground to dust (Dohnanyi 1969; Williams and Wetherill 1994). Ra- diation pressure from the central star ejects dust grains with r . 1–10 µm; Poynting-Robertson drag pulls larger grains into the central star (Burns et al. 1979; Artymowicz 1988; Takeuchi and Artymowicz 2001). Eventually, plan- etesimals are accreted by the oligarchs or ground to dust. To evaluate the oligarch mass required for a disruptive collision, we consider two planetesimals with equal mass mp. The center-of-mass collision energy is Qi = v i /8 , (7) where the impact velocity v2i = v 2+v2esc (Wetherill and Stewart 1993). The energy needed to remove half of the combined mass of two colliding planetesimals is Q∗D = Qb + ρQg , (8) 1 3 5 7 log Radius (cm) e = 0.001 e = 0.01 e = 0.1 Fig. 1.— Disruption energy, Q∗D, for icy objects. The solid curve plots a typical result derived from numerical simulations of collisions that include a detailed equation of state for crystalline ice (Qb = 1.6 × 10 7 erg g−1, βb = −0.42, ρ = 1.5 g cm −3, Qg = 1.5 erg cm−3, and βg = 1.25; Benz and Asphaug 1999). The other curves plot results using Qb consistent with model fits to comet breakups (βb ≈ 0; Qb ∼ 10 3 erg g−1, dashed curve; Qb ∼ 10 erg g−1, dot-dashed curve; Asphaug and Benz 1996). The dashed horizontal lines indicate the center of mass collision energy (eq. (7)) for equal mass objects with e = 0.001, 0.01, and 0.1. Col- lisions between objects with Qi ≪ Q D yield merged remnants; collisions between objects with Qi ≫ Q D produce debris. where Qbr βb is the bulk (tensile) component of the binding energy and ρQgr βg is the gravity component of the bind- ing energy (Davis et al. 1985; Housen and Holsapple 1990, 1999; Holsapple 1994; Benz and Asphaug 1999). We adopt v ≈ vesc,o, where vesc,o = (Gmo/ro) 1/2 is the escape ve- locity of an oligarch with mass mo and radius ro. We define the disruption mass md by deriving the oligarch mass where Qi ≈ Q D. For icy objects at 30 AU md ∼ 3× 10 107 erg g−1 M⊕. (9) Figure 1 illustrates the variation of Q∗D with radius for several variants of eq. (8). For icy objects, detailed nu- merical collision simulations yield Qb . 10 7 erg g−1, −0.5 . βb . 0, ρ ≈ 1–2 g cm −3, Qg ≈ 1–2 erg cm and βg ≈ 1–2 (solid line in Fig. 1, Benz and Asphaug 1999, see also Chapter by Leinhardt et al.)). Models for the breakup of comet Shoemaker-Levy 9 suggest a smaller component of the bulk strength, Qb ∼ 10 3 erg g−1 (e.g., Asphaug and Benz 1996), which yields smaller disruption energies for smaller objects (Fig. 1, dashed and dot-dashed curves). Because nearly all models for collisional disrup- tion yield similar results for objects with r & 1 km (e.g., Kenyon and Bromley 2004d), the disruption mass is fairly independent of theoretical uncertainties once planetesimals become large. For typical Q∗D ∼ 10 7–108 erg g−1 for 1–10 km objects (Fig. 1), leftover planetesimals start to disrupt when oligarchs have radii, ro ∼ 200–500 km. Once disruption commences, the final mass of an oli- garch depends on the timescale for the collisional cascade (Kenyon and Bromley 2004a,b,d; Leinhardt and Richardson 2005). If disruptive collisions produce dust grains much faster than oligarchs accrete leftover planetesimals, oli- garchs with mass mo cannot grow much larger than the disruption radius (maximum oligarch mass mo,max ≈ md). However, if oligarchs accrete grains and leftover planetes- imals effectively, oligarchs reach the isolation mass before collisions and radiation pressure remove material from the disk (eq. (6); Goldreich et al. 2004). The relative rates of accretion and disruption depend on the balance between collisional damping and gas drag – which slow the colli- sional cascade – and viscous stirring and dynamical friction – which speed up the collisional cascade. Because deriv- ing accurate rates for these processes requires numerical simulations of planetesimal accretion, we now consider simulations of planet formation in the Kuiper belt. 3. COAGULATION SIMULATIONS 3.1. Background Safronov (1969) invented the current approach to plan- etesimal accretion calculations. In his particle-in-a-box method, planetesimals are a statistical ensemble of masses with distributions of orbital eccentricities and inclinations (Greenberg et al. 1978; Wetherill and Stewart 1989, 1993; Spaute et al. 1991). This statistical approximation is es- sential: N -body codes cannot follow the n ∼ 109–1012 1 km planetesimals required to build Pluto-mass or Earth- mass planets. For large numbers of objects on fairly cir- cular orbits (e.g., n & 104, r . 1000 km, and e . 0.1), the method is also accurate. With a suitable prescription for collision outcomes, solutions to the coagulation equa- tion in the kinetic theory yield the evolution of n(m) with arbitrarily small errors (e.g., Wetherill 1990; Lee 2000; Malyshkin and Goodman 2001). In addition to modeling planet growth, the statistical approach provides a method for deriving the evolution of orbital elements for large ensembles of planetesimals. If we (i) assume the distributions of e and i for planetesimals follow a Rayleigh distribution and (ii) treat their motions as perturbations of a circular orbit, we can use the Fokker- Planck equation to solve for small changes in the orbits due to gas drag, gravitational interactions, and physical col- lisions (Hornung et al. 1985; Wetherill and Stewart 1993; Ohtsuki et al. 2002). Although the Fokker-Planck equation cannot derive accurate orbital parameters for planetesimals and oligarchs near massive planets, it yields accurate solu- tions for the ensemble average e and i when orbital reso- nances and other dynamical interactions are not important (e.g., Wetherill and Stewart 1993; Weidenschilling et al. 1997a; Ohtsuki et al. 2002). Several groups have implemented Safronov’s method for calculations relevant to the outer solar system (Greenberg et al. 1984; Stern 1995, 2005; Stern and Colwell 1997a,b; Davis and Farinella 1997; Kenyon and Luu 1998, 1999a,b; Davis et al. 1999; Kenyon and Bromley 2004a,d, 2005). These calculations -3 -2 -1 0 1 2 3 4 log Radius (km) 0 Myr 40-47 AU -3 -2 -1 0 1 2 3 4 log Radius (km) Fig. 2.— Evolution of a multiannulus coagulation model with Σ = 0.12(ai/40 AU)−3/2 g cm−2. Left: cumulative mass distribution at times indicated in the legend. Right: eccentricity distributions at t = 0 (light solid line), t = 10 Myr (filled circles), t = 100 Myr (open boxes), t = 1 Gyr (filled triangles), and t = 5 Gyr (open diamonds). As large objects grow in the disk, they stir up the leftover planetesimals to e ∼ 0.1. Disruptive collisions then deplete the population of 0.1–10 km planetesimals, which limits the growth of the largest objects. adopt a disk geometry and divide the disk into N concen- tric annuli with radial width ∆ai at distances ai from the central star. Each annulus is seeded with a set of planetes- imals with masses mij , eccentricities eij , and inclinations iij , where the index i refers to one of N annuli and the index j refers to one of M mass batches within an annulus. The mass batches have mass spacing δ ≡ mj+1/mj . In most calculations, δ ≈ 2; δ ≤ 1.4 is optimal (Ohtsuki et al. 1990; Wetherill and Stewart 1993; Kenyon and Luu 1998). Once the geometry is set, the calculations solve a set of coupled difference equations to derive the number of ob- jects nij , and the orbital parameters, eij and iij , as func- tions of time. Most studies allow fragmentation and ve- locity evolution through gas drag, collisional damping, dy- namical friction and viscous stirring. Because Q∗D sets the maximum size mc,max of objects that participate in the collisional cascade, the size distribution for objects with m < mc,max depends on the fragmentation parameters (eq. (8); Davis and Farinella 1997; Kenyon and Bromley 2004d; Pan and Sari 2005). The size and velocity distributions of the merger population with m > mc,max are established during runaway growth and the early stages of oligarchic growth. Accurate treatment of velocity evolution is impor- tant for following runaway growth and thus deriving good estimates for the growth times and the size and velocity dis- tributions of oligarchs. When a few oligarchs contain most of the mass, col- lision rates depend on the orbital dynamics of individual objects instead of ensemble averages. Safronov’s statisti- cal approach then fails (e.g., Wetherill and Stewart 1993; Weidenschilling et al. 1997b). Although N-body methods can treat the evolution of the oligarchs, they cannot follow the evolution of leftover planetesimals, where the statis- tical approach remains valid (e.g., Weidenschilling et al. 1997b). Spaute et al. (1991) solve this problem by adding a Monte Carlo treatment of binary interactions between large objects to their multiannulus coagulation code. Bromley and Kenyon (2006) describe a hybrid code, which merges a direct N-body code with a multiannulus coagula- tion code. Both codes have been applied to terrestrial planet formation, but not to the Kuiper belt. Current calculations cannot follow collisional growth ac- curately in an entire planetary system. Although the six order of magnitude change in formation timescales from ∼ 0.4 AU to 40 AU is a factor in this statement, most modern supercomputers cannot finish calculations involv- ing the entire disk with the required spatial resolution on a reasonable timescale. For the Kuiper belt, it is possi- ble to perform a suite of calculations in a disk extending from 30–150 AU following 1 m and larger planetesimals. These calculations yield robust results for the mass distri- bution as a function of space and time and provide interest- ing comparisons with observations. Although current cal- culations do not include complete dynamical interactions with the giant planets or passing stars ((see, for example, Charnoz and Morbidelli 2007), sample calculations clearly show the importance of external perturbations in treating the collisional cascade. We begin with a discussion of self-stirring calculations without interactions with the gi- ant planets or passing stars and then describe results with external perturbers. 3.2. Self-Stirring To illustrate in situ KBO formation at 40–150 AU, we consider a multiannulus calculation with an initial ensem- ble of 1 m to 1 km planetesimals in a disk with Σ0d = 0.12 g cm−2. The planetesimals have initial radii of 1 m to 1 km (with equal mass per logarithmic bin), e0 = 10 i0 = e0/2, mass density 1 ρ = 1.5 g cm−3 and fragmentation parameters Qb = 10 3 erg g−1, Qg = 1.5 erg cm −3, βb = 0, 1 Our choice of mass density is a compromise between pure ice (ρ = 1 g cm−3) and the measured density of Pluto (ρ ≈ 2 g cm−3 Null et al. and βg = 1.25 (dashed curve in Fig. 1; Kenyon and Bromley 2004d, 2005). The gas density also follows a MMSN, with Σg/Σd = 100 and a vertical scale height h = 0.1 r (Kenyon and Hartmann 1987). The gas density is Σg ∝ e−t/tg , with tg = 10 Myr. This calculation uses an updated version of the Bromley and Kenyon (2006) code that includes a Richardson extrapolation pro- cedure in the coagulation algorithm. As in the Eulerian (Kenyon and Luu 1998) and fourth order Runge-Kutta (Kenyon and Bromley 2002a,b) methods employed pre- viously, this code provides robust numerical solutions to kernels with analytic solutions (e.g., Ohtsuki et al. 1990; Wetherill 1990) without producing the wavy size distribu- tions described in other simulations with a low mass cutoff (e.g. Campo Bagatin et al. 1994). Once the evolution of large (r > 1 m) objects is complete, a separate code tracks the evolution of lower mass objects and derives the dust emission as a function of time. Figure 2 shows the evolution of the mass and eccentric- ity distributions at 40–47 AU for this calculation. During the first few Myr, the largest objects grow slowly. Dynam- ical friction damps the orbits of the largest objects; colli- sional damping and gas drag circularize the orbits of the smallest objects. This evolution erases many of the initial conditions and enhances gravitational focusing by factors of 10–1000. Runaway growth begins. A few (and some- times only one) oligarchs then grow from r ∼ 10 km to r ∼ 1000 km in ∼ 30 Myr at 40 AU and in ∼ 1 Gyr at 150 AU (see eq. (5)). Throughout runaway growth, dynam- ical friction and viscous stirring raise the random velocities of the leftover planetesimals to e ≈ 0.01–0.1 and i ≈ 2o– 4o (v ∼ 50–500 m s−1 at 40–47 AU; Figure 2; right panel). Stirring reduces gravitational focusing factors and ends run- away growth. The large oligarchs then grow slowly through accretion of leftover planetesimals. As oligarchs grow, collisions among planetesimals initi- ate the collisional cascade. Disruptive collisions dramat- ically reduce the population of 1–10 km objects, which slows the growth of oligarchs and produces a significant de- bris tail in the size distribution. In these calculations, dis- ruptive collisions remove material from the disk faster than oligarchs can accrete the debris. Thus, growth stalls and produces ∼ 10–100 objects with maximum sizes rmax ∼ 1000–3000 km at 40–50 AU (Stern and Colwell 1997a,b; Kenyon and Bromley 2004d, 2005; Stern 2005). Stochastic events lead to large dispersions in the growth time for oligarchs, to (eq. (5)). In ensembles of 25–50 simu- lations with identical starting conditions, an occasional oli- garch will grow up to a factor of two faster than its neigh- bors. This result occurs in simulations with δ = 1.4, 1.7, and 2.0, and thus seems independent of mass resolution. These events occur in ∼ 25% of the simulations and lead to factor of ∼ 2 variations in to (eq. (5)). In addition to modest-sized icy planets, oligarchic 1993). The calculations are insensitive to factor of two variations in the mass density of planetesimals. 6 7 8 9 10 log Time (yr) R > 10 km (40-47 AU) R > 100 km (40-47 AU) Dust (40-150 AU) Fig. 3.— Time evolution of the mass in KBOs and dust grains. Solid line: dust mass (r . 1 mm) at 40–150 AU. Dashed (dot- dashed) lines: total mass in small (large) KBOs at 40–47 AU. growth generates copious amounts of dust (Figure 3). When runaway growth begins, collisions produce small amounts of dust from ‘cratering’ (see, for example Greenberg et al. 1978; Wetherill and Stewart 1993; Stern and Colwell 1997a,b; Kenyon and Luu 1999a). Stirring by growing oligarchs leads to ‘catastrophic’ collisions, where colliding planetes- imals lose more than 50% of their initial mass. These dis- ruptive collisions produce a spike in the dust production rate that coincides with the formation of oligarchs with r & 200–300 km (eq. (9)). As the wave of runaway growth propagates outward, stirring produces disruptive collisions at ever larger heliocentric distances. The dust mass grows in time and peaks at ∼ 1 Gyr, when oligarchs reach their maximum mass at 150 AU. As the mass in leftover plan- etesimals declines, Poynting-Robertson drag removes dust faster than disruptive collisions produce it. The dust mass then declines with time. 3.3. External Perturbation Despite the efficiency of self-stirring models in remov- ing leftover planetesimals from the disk, other mechanisms must reduce the derived mass in KBOs to current ob- servational limits. In self-stirring calculations at 40–50 AU, the typical mass in KBOs with r ∼ 30—1000 km at 4–5 Gyr is a factor of 5–10 larger than currently ob- served (Luu and Jewitt 2002, Chapter by Petit et al.). Un- less Earth-mass or larger objects form in the Kuiper belt (Chiang et al. 2007; Levison and Morbidelli 2007), exter- nal perturbations must excite KBO orbits and enhance the collisional cascade. Two plausible sources of external perturbation can re- duce the predicted KBO mass to the desired limits. Once Neptune achieves its current mass and orbit, it stirs up the orbits of KBOs at 35–50 AU (Levison and Duncan 1990; Holman and Wisdom 1993; Duncan et al. 1995; Kuchner et al. 2002; Morbidelli et al. 2004). In ∼ 100 Myr or less, Neptune removes nearly all KBOs with a . 37–38 AU. Beyond a ∼ 38 AU, some KBOs are trapped in or- bital resonance with Neptune (Malhotra 1995, 1996); oth- ers are ejected into the scattered disk (Duncan and Levison 1997). In addition to these processes, Neptune stir- ring increases the effectiveness of the collisional cascade (Kenyon and Bromley 2004d), which removes additional mass from the population of 0.1–10 km KBOs and prevents growth of larger KBOs. Passing stars can also excite KBO orbits and en- hance the collisional cascade. Although Neptune dy- namically ejects scattered disk objects with perihelia q . 36–37 AU (Morbidelli et al. 2004), objects with q & 45–50 AU, such as Sedna and Eris, require an- other scattering source. Without evidence for massive planets at a & 50 AU (Morbidelli et al. 2002), a pass- ing star is the most likely source of the large q for these KBOs (Morbidelli and Levison 2004; Kenyon and Bromley 2004c). Adams and Laughlin (2001, see also Chapter by Dun- can et al.) examined the probability of encounters between the young Sun and other stars. Most stars form in dense clusters with estimated lifetimes of ∼ 100 Myr. To ac- count for the abundance anomalies of radionuclides in so- lar system meteorites (produced by supernovae in the clus- ter) and for the stability of Neptune’s orbit at 30 AU, the most likely solar birth cluster has ∼ 2000–4000 members, a crossing time of ∼ 1 Myr, and a relaxation time of ∼ 10 Myr. The probability of a close encounter with a distance of closest approach aclose is then ∼ 60% (aclose/160 AU) (Kenyon and Bromley 2004c). Because the dynamical interactions between KBOs in a coagulation calculation and large objects like Neptune or a passing star are complex, here we consider simple cal- culations of each process. To illustrate the evolution of KBOs after a stellar flyby, we consider a very close pass with aclose = 160 AU (Kenyon and Bromley 2004c). This co-rotating flyby produces objects with orbital parameters similar to those of Sedna and Eris. For objects in the coag- ulation calculation, the flyby produces an e distribution eKBO = 0.025(a/30 AU)4 a < a0 0.5 a > a0 with a0 ≈ 65 AU (see Ida et al. 2000; Kenyon and Bromley 2004c; Kobayashi et al. 2005). This e distribution produces a dramatic increase in the debris production rate throughout the disk, which freezes the mass distribution of the largest objects2. Thus, to produce an ensemble of KBOs with r & 300 km at 40–50 AU, the flyby must occur when the Sun has an age t⊙ & 10–20 Myr (Figure 2). For t⊙ & 100 Myr, 2 The i distribution following a flyby depends on the relative orientations of two planes, the orbital plane of KBOs and the plane of the trajectory of the passing star. Here, we assume the flyby produces no change in i, which simplifies the discussion without changing any of the results significantly. the flyby is very unlikely. As a compromise between these two estimates, we consider a flyby at t⊙ ∼ 50 Myr. 6 7 8 9 10 log Time (yr) Flyby Model 40-47 AU 47-55 AU 66-83 AU Fig. 4.— Evolution of Σ after a stellar flyby. After 50 Myr of growth, the close pass excites KBOs to large e (eq. (10)) and enhances the collisional cascade. Figure 4 shows the evolution of the KBO surface den- sity in three annuli as a function of time. At early times (t . 50 Myr), KBOs grow in the standard way. After the flyby, the disk suffers a dramatic loss of material. At 40–47 AU, the disk loses ∼ 90% (93%) of its initial mass in ∼ 1 Gyr (4.5 Gyr). At ∼ 50–80 AU, the collisional cascade re- moves ∼ 90% (97%) of the initial mass in ∼ 500 Myr (4.5 Gyr). Beyond ∼ 80 AU, KBOs contain less than 1% of the initial mass. Compared to self-stirring models, flybys that produce Sedna-like orbits are a factor of 2–3 more efficient at removing KBOs from the solar system. To investigate the impact of Neptune on the collisional cascade, we parameterize the growth of Neptune at 30 AU as a simple function of time (Kenyon and Bromley 2004d) MNep ≈ 6× 1027 e(t−tN )/t1 g t < tN 6× 1027 g + C(t− t1) tN < t < t2 1.0335× 1029 g t > t2 where CNep = 1.947 × 10 21 g yr−1, tN = 50 Myr, t1 = 3 Myr, and t2 = 100 Myr. These choices enable a model Nep- tune to reach a mass of 1 M⊕ in 50 Myr, when the largest KBOs form at 40–50 AU, and reach its current mass in 100 Myr3. As Neptune approaches its final mass, its gravity stirs up KBOs at 40–60 AU and increases their orbital eccentric- ities to e ∼ 0.1–0.2 on short timescales. In the coagula- tion model, distant planets produce negligible changes in i, so self-stirring sets i in these calculations (Weidenschilling 1989). This evolution enhances debris production by a fac- tor of 3–4, which effectively freezes the mass distribution of 100–1000 km objects at 40–50 AU. By spreading the 3 This prescription is not intended as an accurate portrayal of Neptune for- mation, but it provides a simple way to investigate how Neptune might stir the Kuiper belt once massive KBOs form. 6 7 8 9 10 log Time (yr) Neptune Stirring SS: 40-47 AU NS: 40-47 AU NS: 47-55 AU Fig. 5.— Evolution of Σ(KBO) in models with Neptune stirring. Compared to self-stirring models (SS; dashed curve), stirring by Neptune rapidly removes KBOs at 40–47 AU (NS; solid cruve) and at 47–55 AU (NS; dot-dashed curve). leftover planetesimals and the debris over a larger volume, Neptune stirring limits the growth of the oligarchs and thus reduces the total mass in KBOs. Figure 5 shows the evolution of the surface density in small and large KBOs in two annuli as a function of time. At 40–55 AU, Neptune rapidly stirs up KBOs to e ∼ 0.1 when it reaches its current mass at ∼ 100 Myr. Large collision velocities produce more debris, which is rapidly ground to dust and removed from the system by radiation pressure at early times and by Poynting-Robertson drag at later times. Compared to self-stirring models, the change in Σ is dramatic, with only ∼ 3% of the initial disk mass remaining at ∼ 4.5 Gyr. From these initial calculations, it is clear that exter- nal perturbations dramatically reduce the mass of KBOs in the disk (see also Charnoz and Morbidelli 2007). Fig- ure 6 compares the mass distributions at 40–47 AU and at 4.5 Gyr for the self-stirring model in Figure 2 (solid line) with results for the flyby (dot-dashed line) and Neptune stir- ring (dashed line). Compared to the self-stirring model, the close flyby reduces the mass in KBOs by ∼ 50%. Neptune stirring reduces the KBO mass by almost a factor of 3 rel- ative to the self-stirring model. For KBOs with r & 30–50 km, the predicted mass in KBOs with Neptune stirring is within a factor of 2–3 of the current mass in KBOs. These simple calculations for the stellar flyby and Nep- tune stirring do not include dynamical depletion. In the stel- lar flyby picture, the encounter removes nearly all KBOs beyond a truncation radius, aT ∼ 48 (aclose / 160 AU) AU (Kenyon and Bromley 2004c). Thus, a close pass with aclose ∼ 160 AU can produce the observed outer edge of the Kuiper belt at 48 AU. Although many objects with initial a > aT are ejected from the Solar System, some are placed on very elliptical, Sedna-like orbits4. In the Neptune stir- 4Levison et al. (2004) consider the impact of the flyby on the scattered disk and Oort cloud. After analyzing a suite of numerical simulations, they conclude that the flyby must occur before Neptune reaches its current orbit -3 -2 -1 0 1 2 3 4 log Radius (km) Initial state Self-stirring Flyby Neptune 40-47 AU Fig. 6.— Mass distributions for evolution with self-stirring (heavy solid line), stirring from a passing star (dot-dashed line), and stirring from Neptune at 30 AU (dashed line). After 4.5 Gyr, the mass in KBOs with r & 50 km is ∼ 5% (self-stirring), ∼ 3.5% (flyby), and ∼ 2% (Neptune stirring) of the initial mass. The number of objects with r & 1000 km is ∼ 100 (self-stirring), ∼ 1 (flyby), and ∼ 10 (Neptune stirring). The largest object has rmax ∼ 3000 km (self-stirring), rmax ∼ 500–1000 km (flyby), and rmax ∼ 1000–2000 km (Neptune stirring). ring model, dynamical interactions will eject some KBOs at 40–47 AU. If the dynamical interactions that produce the scattered disk reduce the mass in KBOs by a factor of 2 at 40–47 AU (e.g., Duncan et al. 1995; Kuchner et al. 2002), the Neptune stirring model yields a KBO mass in reason- ably good agreement with observed limits (for a different opinion, see Charnoz and Morbidelli 2007). 3.4. Nice Model Although in situ KBO models can explain the current amount of mass in large KBOs, these calculations do not address the orbits of the dynamical classes of KBOs. To ex- plain the orbital architecture of the giant planets, the ‘Nice group’ centered at Nice Observatory developed an inspired, sophisticated picture of the dynamical evolution of the gi- ant planets and a remnant planetesimal disk (Tsiganis et al. 2005; Morbidelli et al. 2005; Gomes et al. 2005, and refer- ences therein). The system begins in an approximate equi- librium, with the giant planets in a compact configuration (Jupiter at 5.45 AU, Saturn at ∼ 8.2 AU, Neptune at ∼ 11.5 AU, and Uranus at ∼ 14.2 AU) and a massive planetesimal disk at 15–30 AU. Dynamical interactions between the gi- ant planets and the planetesimals lead to an instability when Saturn crosses the 2:1 orbital resonance with Jupiter, which results in a dramatic orbital migration of the gas giants and the dynamical ejection of planetesimals into the Kuiper belt, scattered disk, and the Oort cloud. Comparisons between the end state of this evolution and the orbits of KBOs in and begins the dynamical processes that populate the Oort cloud and the scattered disk. If Neptune forms in situ in 1–10 Myr, then the flyby cannot occur after massive KBOs form. If Neptune migrates to 30 AU after mas- sive KBOs form, then a flyby can truncate the Kuiper belt without much impact on the Oort cloud or the scattered disk. 6 7 8 9 10 log Time (yr) 20-25 AU 36-44 AU 66-83 AU NICE Model Fig. 7.— Evolution of Σ in a self-stirring model at 20–100 AU. At 20–25 AU, it takes ∼ 5–10 Myr to form 1000 km objects. After ∼ 0.5–1 Gyr, there are ∼ 100 objects with r ∼ 1000–2000 km and ∼ 105 objects with r ∼ 100–200 km at 20–30 AU. As these objects grow, the collisional cascade removes ∼ 90% of the mass in remnant planetesimals. The twin vertical dashed lines bracket the time of the Late Heavy Bombardment at ∼ 300–600 Myr. the ‘hot population’ and the scattered disk are encouraging (Chapter by Morbidelli et al.). Current theory cannot completely address the likelihood of the initial state in the Nice model. Thommes et al. (1999, 2002) demonstrate that n-body simulations can produce a compact configuration of gas giants, but did not consider how fragmentation or interactions with low mass planetes- imals affect the end state. O’Brien et al. (2005) show that a disk of planetesimals has negligible collisional grinding over 600 Myr if most of the mass is in large planetesimals with r & 100 km. However, they did not address whether this state is realizable starting from an ensemble of 1 km and smaller planetesimals. In terrestrial planet simulations starting with 1–10 km planetesimals, the collisional cascade removes ∼ 25% of the initial rocky material in the disk (Wetherill and Stewart 1993; Kenyon and Bromley 2004b). Interactions between oligarchs and remnant planetesimals are also important for setting the final mass and dynamical state of the terrestrial planets (Bromley and Kenyon 2006; Kenyon and Bromley 2006). Because complete hybrid cal- culations of the giant planet region are currently compu- tationally prohibitive, it is not possible to make a reliable assessment of these issues for the formation of gas giant planets. Here, we consider the evolution of the planetesimal disk outside the compact configuration of giant planets, where standard coagulation calculations can follow the evolution of many initial states for 1–5 Gyr in a reasonable amount of computer time. Figure 7 shows the time evolution for the surface density of planetesimals in three annuli from one typical calculation at 20–25 AU (dot-dashed curve; Mi = 6 M⊕), 36–44 AU (dashed curve; Mi = 9 M⊕), and 66–83 AU (solid curve; Mi = 12 M⊕). Starting from the stan- dard surface density profile (eq. 1), planetesimals at 20–25 AU grow to 1000 km sizes in a few Myr. Once the colli- sional cascade begins, the surface density slowly declines to ∼ 10% to 20% of its initial value at the time of the Late Heavy Bombardment, when the Nice model predicts that Saturn crosses the 2:1 orbital resonance with Jupiter. These results provide a strong motivation to couple co- agulation calculations with the dynamical simulations of the Nice group (see also Charnoz and Morbidelli 2007). In the Nice model, dynamical interactions with a massive plan- etesimal disk are the ‘fuel’ for the dramatic migration of the giant planets and the dynamical ejection of material into the Kuiper belt and the scattered disk. If the mass in the plan- etesimal disk declines by ∼ 80% as the orbits of the giant planets evolve, the giant planets cannot migrate as dramati- cally as in the Gomes et al. (2005) calculations. Increasing the initial mass in the disk by a factor of 3–10 may allow coagulation and the collisional cascade to produce a debris disk capable of triggering the scattering events of the Nice model. 3.5. A Caveat on the Collisional Cascade Although many of the basic outcomes of oligarchic growth and the collisional cascade are insensitive to the ini- tial conditions and fragmentation parameters for the plan- etesimals, several uncertainties in the collisional cascade can modify the final mass in oligarchs and the distributions of r and e. Because current computers do not allow coag- ulation calculations that include the full range of sizes (1 µm to 104 km), published calculations have two pieces, a solution for large objects (e.g., Kenyon and Bromley 2004a,b) and a separate solution for smaller objects (e.g., Krivov et al. 2006). Joining these solutions assumes that (i) collision fragments continue to collide and fragment until particles are removed by radiative processes and (ii) mu- tual (destructive) collisions among the fragments are more likely than mergers with much larger oligarchs. These as- sumptions are reasonable but untested by numerical cal- culations (Kenyon and Bromley 2002a). Thus, it may be possible to halt or to slow the collisional cascade before radiation pressure rapidly remove small grains with r ≈ 1–100 µm. In current coagulation calculations, forming massive oli- garchs at 5–15 AU in a massive disk requires an ineffi- cient collisional cascade. When the cascade is efficient, the most massive oligarchs have m . 1 M⊕. Slowing the cascade allows oligarchs to accrete planetesimals more efficiently, which results in larger oligarchs that contain a larger fraction of the initial mass. If collisional damping is efficient, halting the cascade completely at sizes of ∼ 1 mm leads to rapid in situ formation of Uranus and Neptune (Goldreich et al. 2004) and early stirring of KBOs at 40 AU. There are two simple ways to slow the collisional cas- cade. In simulations where the cascade continues to small sizes, r ∼ 1–10 µm, the radial optical depth in small grains is τs ∼ 0.1–1 at 30–50 AU (Kenyon and Bromley 2004a). Lines-of-sight to the central star are not purely radial, so this optical depth reduces radiation pressure and Poynting- Robertson drag by small factors, ∼ e−0.2τs ∼ 10%–30%, and has little impact on the evolution of the cascade. With τs ∝ a −s and s ∼ 1–2, however, the optical depth may re- duce radiation forces significantly at smaller a. Slowing the collisional cascade by factors of 2–3 could allow oligarchs to accrete leftover planetesimals and smaller objects before the cascade removes them. Collisional damping and gas drag on small particles may also slow the collisional cascade. For particles with large ratios of surface area to volume, r . 0.1–10 cm, colli- sions and the gas effectively damp e and i (Adachi et al. 1976; Goldreich et al. 2004) and roughly balance dynam- ical friction and viscous stirring. Other interactions be- tween small particles and the gas – such as photophoresis (Wurm and Krauss 2006) – also damp particles randome velocities and thus might help to slow the cascade. Both collisions and interactions between the gas and the solids are more effective at large volume density, so these pro- cesses should be more important inside 30 AU than outside 30 AU. The relatively short lifetime of the gas, ∼ 3–10 Myr, also favors more rapid growth inside 30 AU. If damping maintains an equilibrium e ∼ 10−3 at a ∼ 20–30 AU, oli- garchs can grow to the sizes, r & 2000 km, required in the Nice model. Rapid growth at a ∼ 5–15 AU might produce oligarchs with the isolation mass (r ∼ 10–30 R⊕; eq. 6) and lead to the rapid formation of gas giants. Testing these mechanisms for slowing the collisional cascade requires coagulation calculations with accurate treatments of collisional damping, gas drag, and optical depth for particle radii r ∼ 1–10 µm to r ∼ 10000 km. Although these calculations require factors of 4–6 more computer time than published calculations, they are possi- ble with multiannulus coagulation codes on modern parallel computers. 3.6. Model Predictions The main predictions derived from coagulation mod- els are n(r), n(e), and n(i) as functions of a. The cu- mulative number distribution consists of three power laws (Kenyon and Bromley 2004d; Pan and Sari 2005) n(r) = −αd r ≤ r1 ni r1 ≤ r < r0 −αm r ≥ r0 The debris population at small sizes, r ≤ r1, always has αd ≈ 3.5. The merger population at large sizes, r ≥ r0, has αm ≈ 2.7–4. Because the collisional cascade robs oligarchs of material, calculations with more stirring have steeper size distributions. Thus, self-stirring calculations with Qb & 10 5 erg g−1 (Qb . 10 3 erg g−1) typically yield αm ≈ 2.7–3.3 (3.5–4). Models with a stellar flyby or stir- ring by a nearby gas giant also favor large αm. The transition radii for the power laws depend on the fragmentation parameters (see Fig. 1; see also Pan and Sari (2005)). For a typical e ∼ 0.01–0.1 in self-stirring models, r0 ≈ r1 ≈ 1 km when Qb & 10 5 erg g−1. When Qb . 10 erg g−1, r1 ≈ 0.1 km and r0 ≈ 10–20 km. Thus the cal- culations predict a robust correlation between the transition radii and the power law exponents: large r0 andαm or small r0 and αm. Because gravitational stirring rates are larger than ac- cretion rates, the predicted distributions of e and i at 4–5 Gyr depend solely on the total mass in oligarchs (see also Goldreich et al. 2004). Small objects with r . r0 con- tain a very small fraction of the mass and cannot stir them- selves. Thus e and i are independent of r (Fig. 2). The e and i for larger objects depends on the total mass in the largest objects. In self-stirring models, dynamical friction and viscous stirring between oligarchs and planetesimals (during runaway growth) and among the ensemble of oli- garchs (during oligarchic growth) set the distribution of e for large objects with r & r0. In self-stirring models, vis- cous stirring among oligarchs dominates dynamical friction between oligarchs and leftover planetesimals, which leads to a shallow relation between e and r, e ∝ r−γ with γ ≈ 3/4. In the flyby and Neptune stirring models, stirring by the external perturber dominates stirring among oligarchs. This stirring yields a very shallow relation between e and r with γ ≈ 0–0.25. Other results depend little on the initial conditions and the fragmentation parameters. In calculations with different initial mass distributions, an order of magnitude range in e0, and Qb = 10 0–107 erg g−1, βb = −0.5–0, Qg = 0.5–5 erg cm−3, and βg ≥ 1.25, rmax and the amount of mass re- moved by the collisional cascade vary by . 10% relative to the evolution of the models shown in Figures 2–7. Because collisional damping among 1 m to 1 km objects erases the initial orbital distribution, the results do not depend on e0 and i0. Damping and dynamical friction also quickly erase the initial mass distribution, which yields growth rates that are insensitive to the initial mass distribution. The insensitivity of rmax and mass removal to the frag- mentation parameters depends on the rate of collisional dis- ruption relative to the growth rate of oligarchs. Because the collisional cascade starts when mo ∼ md (eq. (9)), calculations with small Qb (Qb . 10 3 erg g−1) produce large amounts of debris before calculations with large Qb (Qb & 10 3 erg g−1). Thus, an effective collisional cascade should yield lower mass oligarchs and more mass removal when Qb is small. However, oligarchs with mo ∼ md still have fairly large gravitational focusing factors and accrete leftover planetesimals more rapidly than the cascade re- moves them. As oligarchic growth continues, gravitational focusing factors fall and collision disruptions increase. All calculations then reach a point where the collisional cascade removes leftover planetesimals more rapidly than oligarchs can accrete them. As long as most planetesimals have r ∼ 1–10 km, the timing of this epoch is more sensitive to grav- itational focusing and the growth of oligarchs than the col- lisional cascade and the fragmentation parameters. Thus, rmax and the amount of mass processed through the colli- sional cascade are relatively insensitive to the fragmentation parameters. 4. Confronting KBO collision models with KBO data Current data for KBOs provide two broad tests of coagu- lation calculations. In each dynamical class, four measured parameters test the general results of coagulation models and provide ways to discriminate among the outcomes of self-stirring and perturbed models. These parameters are • rmax, the size of the largest object, • αm, the slope of the size distribution for large KBOs with r & 10 km, • r0, the break radius, which measures the radius where the size distribution makes the transition from a merger population (r & r0) to a collisional popu- lation (r . r0) as summarized in eq. (12), and • Ml, the total mass in large KBOs. For all KBOs, measurements of the dust mass allow tests of the collisional cascade and link the Kuiper belt to observa- tions of nearby debris disks. We begin with the discussion of large KBOs and then compare the Kuiper belt with other debris disks. Table 1 summarizes the mass and size distribution pa- rameters derived from recent surveys. To construct this table, we used online data from the Minor Planet Center (http://cfa−www.harvard.edu/iau/lists/MPLists.html) for rmax (see also Levison and Stern 2001) and the results of several detailed analyses for αm, rmax, and r0 (e.g., Bernstein et al. 2004; Elliot et al. 2005; Petit et al. 2006, Chapter by Petit et al.). Because comprehensive KBO sur- veys are challenging, the entries in the Table are incomplete and sometimes uncertain. Nevertheless, these results pro- vide some constraints on the calculations. Current data provide clear evidence for physical differ- ences among the dynamical classes. For classical KBOs with a = 42–48 AU and q > 37 AU, the cold population (i . 4o) has a steep size distribution with αm ≈ 3.5–4 and rmax ∼ 300–500 km. In contrast, the hot population (i & 10o) has a shallow size distribution with αm ≈ 3 and rmax ∼ 1000 km (Levison and Stern 2001). Both popu- lations have relatively few objects with optical brightness mR ≈ 27–27.5, which implies r0 ∼ 20–40 km for reason- able albedo ∼ 0.04–0.07. The detached, resonant, and scat- tered disk populations contain large objects with rmax ∼ 1000 km. Although there are too few detached or scattered disk objects to constrain αm or r0, data for the resonant population are consistent with constraints derived for the hot classical population, αm ≈ 3 and r0 ≈ 20–40 km. The total mass in KBOs is a small fraction of the ∼ 10–30 M⊕ of solid material in a MMSN from ∼ 35–50 AU (Gladman et al. 2001; Bernstein et al. 2004; Petit et al. 2006, see also Chapter by Petit et al.). The classical and res- onant populations have Ml ≈ 0.01–0.1 M⊕ in KBOs with TABLE 1. DATA FOR KBO SIZE DISTRIBUTION KBO Class Ml (M⊕) rmax (km) r0 (km) qm cold cl 0.01–0.05 400 20–40 km & 4 hot cl 0.01–0.05 1000 20–40 km 3–3.5 detached n/a 1500 n/a n/a resonant 0.01–0.05 1000 20–40 km 3 scattered 0.1–0.3 700 n/a n/a r & 10–20 km. The scattered disk may contain more mate- rial, Ml ∼ 0.3 M⊕, but the constraints are not as robust as for the classical and resonant KBOs. These data are broadly inconsistent with the predictions of self-stirring calculations with no external perturbers. Al- though self-stirring models yield inclinations, i ≈ 2o–4o, close to those observed in the cold, classical population, the small rmax and large αm of this group suggest that an ex- ternal dynamical perturbation – such as a stellar flyby or stirring by Neptune – modified the evolution once rmax reached ∼ 300–500 km. The observed break radius, r0 ∼ 20–40 km, also agrees better with the r0 ∼ 10 km ex- pected from Neptune stirring calculations than the r0 ∼ 1 km achieved in self-stirring models (Kenyon and Bromley 2004d; Pan and Sari 2005). Although a large rmax and small αm for the resonant and hot, classical populations agree reasonably well with self-stirring models, the ob- served rmax ∼ 1000 km is much smaller than the rmax ∼ 2000–3000 km typically achieved in self-stirring calcula- tions (Figure 1). Both of these populations appear to have large r0, which is also more consistent with Neptune stir- ring models than with self-stirring models. The small Ml for all populations provide additional ev- idence against self-stirring models. In the most optimistic scenario, where KBOs are easily broken, self-stirring mod- els leave a factor of 5–10 more mass in large KBOs than currently observed at 40–48 AU. Although models with Neptune stirring leave a factor of 2–3 more mass in KBOs at 40–48 AU than is currently observed, Neptune ejects ∼ half of the KBOs at 40–48 AU into the scattered disk (e.g., Duncan et al. 1995; Kuchner et al. 2002). With an es- timated mass of 2–3 times the mass in classical and reso- nant KBOs, the scattered disk contains enough material to bridge the difference between the KBO mass derived from Neptune stirring models and the observed KBO mass. The mass in KBO dust grains provides a final piece of evidence against self-stirring models. From an analysis of data from Pioneer 10 and 11, Landgraf et al. (2002) esti- mate a dust production rate of ∼ 1015 g yr−1 in 0.01–2 mm particles at 40–50 AU. The timescale for Poynting- Robertson drag to remove these grains from the Kuiper belt is ∼ 10–100 Myr (Burns et al. 1979), which yields a mass of ∼ 1022–1024 g. Figure 8 compares this dust mass with masses derived from mid-IR and submm observations of several nearby solar-type stars (Greaves et al. 1998, 2004; Williams et al. 2004; Wyatt et al. 2005) and with predic- tions from the self-stirring, flyby, and Neptune stirring mod- els. The dust masses for nearby solar-type stars roughly fol- 5 6 7 8 9 10 log Time (yr) Self-stirring Flyby Neptune Fig. 8.— Evolution of mass in small dust grains (0.001–1 mm) for models with self-stirring (dot-dashed line), stirring from a passing star (dashed line), and stirring from Neptune at 30 AU (solid line) for Qb = 10 3 erg g−1. Calculations with smaller (large) Qb produce more (less) dust at t . 50 Myr and some- what more (less) dust at t & 100 Myr. At 1–5 Gyr, models with Neptune stirring have less dust than self-stirring or flyby models. The boxes show dust mass estimated for four nearby solar-type stars (from left to right in age: HD 107146, ǫ Eri, η Crv, and τ Cet; Greaves et al. 1998, 2004; Williams et al. 2004; Wyatt et al. 2005) and two estimates for the Kuiper belt (boxes connected by solid line Landgraf et al. 2002). low the predictions of self-stirring models and flyby models with Qb ∼ 10 3 erg g−1. The mass of dust in the Kuiper belt is 1–3 orders of magnitude smaller than predicted in self-stirring models and is closer to the predictions of the Neptune stirring models. To combine the dynamical properties of KBOs with these constraints, we rely on results from N -body simu- lations that do not include collisional processing of small objects (see Chapter by Morbidelli et al.). For simplic- ity, we consider coagulation in the context of the Nice model, which provides a solid framework for interpreting the dynamics of the gas giants and the dynamical classes of KBOs. In the Nice model, Saturn’s crossing of the 2:1 res- onance with Jupiter initiates the dynamical instability that populates the Kuiper belt. As Neptune approaches a ≈ 30 AU, it captures resonant KBOs, ejects KBOs into the scat- tered disk and the Oort cloud, and excites the hot classical KBOs. Although Neptune might reduce the number of cold, classical KBOs formed roughly in situ beyond 30 AU, the properties of these KBOs probably reflect conditions in the Kuiper Belt when the instability began. The Nice model requires several results from coagula- tion calculations. Once giant planets form at 5–15 AU, col- lisional growth must produce thousands of Pluto-mass ob- jects at 20–30 AU. Unless the planetesimal disk was mas- sive, growth of oligarchs must dominate collisional grind- ing in this region of the disk. To produce the cold classical population at ∼ 45 AU, collisions must produce 1–10 Pluto- mass objects and then efficiently remove leftover planetesi- mals. To match the data in Table 1, KBOs formed at 20–30 AU should have a shallower size distribution and a larger rmax than those at 40–50 AU. Some coagulation results are consistent with the trends required in the Nice model. In current calculations, colli- sional growth naturally yields smaller rmax and a steeper size distribution at larger a. At 40–50 AU, Neptune-stirring models produce a few Pluto-mass objects and many smaller KBOs with e ∼ 0.1 and i ≈ 2o–4o. Although collisional growth produces more Plutos at 15–30 AU than at 40–50 AU, collisional erosion removes material faster from the in- ner disk than from the outer disk (Fig. 7). Thus, collisions do not produce the thousands of Pluto-mass objects at 15– 30 AU required in the Nice model. Reconciling this aspect of the Nice model with the coag- ulation calculations requires a better understanding of the physical processes that can slow or halt the collisional cas- cade. Producing gas giants at 5–15 AU, thousands of Plutos at 20–30 AU, and a few or no Plutos at 40–50 AU implies that the outcome of coagulation changes markedly from 5 AU to 50 AU. If the collisional cascade can be halted as out- lined in section §3.5, forming 5–10 M⊕ cores at 5–15 AU is straightforward. Slowing the collisional cascade at 20–30 AU might yield a large population of Pluto mass objects at 20–30 AU. Because αm and rmax are well-correlated, bet- ter constraints on the KBO size distributions coupled with more robust coagulation calculations can test these aspects of the Nice model in more detail. To conclude this section, we consider constraints on the Kuiper belt in the more traditional migration scenario of Malhotra (1995), where Neptune forms at ∼ 20–25 AU and slowly migrates to 30 AU. To investigate the relative importance of collisional and dynamical depletion at 40– 50 AU, Charnoz and Morbidelli (2007) couple a collision code with a dynamical code and derive the expected distri- butions for size and orbital elements in the Kuiper belt, the scattered disk, and the Oort cloud. Although collisional de- pletion models can match the observations of KBOs, these models are challenged to provide enough small objects into the scattered disk and Oort cloud. Thus, the results suggest that dynamical mechanisms dominate collisions in remov- ing material from the Kuiper belt. Although Charnoz and Morbidelli (2007) argue against a dramatic change in collisional evolution from 15 AU to 40 AU, the current architecture of the solar system provides good evidence for this possibility. In the MMSN, the ratio of timescales to produce gas giant cores at 10 AU and at 25 AU is ξ = (25/10)3 ∼ 15. In the context of the Nice model, formation of Saturn and Neptune at 8–11 AU in 5– 10 Myr thus implies formation of other gas giant cores at 20–25 AU in 50–150 Myr. If these cores had formed, they would have consumed most of the icy planetesimals at 20– 30 AU, leaving little material behind to populate the outer solar system when the giant planets migrate. The appar- ent lack of gas giant core formation at 20–30 AU indicates that the collisional cascade changed dramatically from 5–15 AU (where gas giant planets formed) to 20–30 AU (where gas giant planets did not form). As outlined in §3.5, under- standing the interaction of small particles with the gas and the radiation field may provide important insights into the evolution of oligarchic growth and thus into the formation and structure of the solar system. 5. KBOs and Asteroids In many ways, the Kuiper Belt is similar to the asteroid belt. Both are populations of small bodies containing rela- tively little mass compared to the rest of the Solar System; the structure and dynamics of both populations have been influenced significantly by the giant planets; and both have been and continue to be significantly influenced by colli- sions. Due to its relative proximity to Earth, however, there are substantially more observational data available for the asteroid belt than the Kuiper Belt. While the collisional and dynamical evolution of the asteroid belt is certainly not a solved problem, the abundance of constraints has allowed for the development of reasonably consistent models. Here we briefly describe what is currently understood about the evolution of the asteroid belt, what insights that may give us with regards to the evolution of the Kuiper Belt, and what differences might exist in the evolution of the two popula- tions. It has long been recognized that the primordial as- teroid belt must have contained hundreds or thousands of times more mass than the current asteroid belt (e.g. Lecar and Franklin 1973; Safronov 1979; Weidenschilling 1977c; Wetherill 1989). Reconstructing the initial mass distribution of the Solar System from the current masses of the planets and asteroids, for example, yields a pronounced mass deficiency in the asteroid belt region relative to an oth- erwise smooth distribution for the rest of the Solar System (Weidenschilling 1977c). To accrete the asteroids on the timescales inferred from meteoritic evidence would require hundreds of times more mass than currently exists in the main belt (Wetherill 1989). In addition to its pronounced mass depletion, the aster- oid belt is also strongly dynamically excited. The mean proper eccentricity and inclination of asteroids larger than ∼50 km in diameter are 0.135 and 10.9o (from the cata- log of Knežević and Milani (2003)), which are significantly larger than can be explained by gravitational perturbations amongst the asteroids or by simple gravitational pertur- bations from the planets (Duncan 1994). The fact that the different taxonomic types of asteroids (S-type, C-type, etc.) are radially mixed somewhat throughout the main belt, rather than confined to delineated zones, indicates that there has been significant scattering in semimajor axis as well (Gradie and Tedesco 1982). Originally, a collisional origin was suggested for the mass depletion in the asteroid belt (Chapman and Davis 1975). The difficulty of collisionally disrupting the largest asteroids, coupled with the survival of the basaltic crust of the ∼500-km diameter asteroid Vesta, however, suggest that collisional grinding was not the cause of the mass deple- tion (Davis et al. 1979, 1985, 1989, 1994; Wetherill 1989; Durda and Dermott 1997; Durda et al. 1998; Bottke et al. 2005a,b; O’Brien and Greenberg 2005). In addition, col- lisional processes alone could not fully explain both the dy- namical excitation and the radial mixing observed in the as- teroid belt, although Charnoz et al. (2001) suggest that col- lisional diffusion may have contributed to its radial mixing. Several dynamical mechanisms have been proposed to explain the mass depletion, dynamical excitation and radial mixing of the asteroid belt. As the solar nebula dissipated, the changing gravitational potential acting on Jupiter, Saturn, and the asteroids would lead to changes in their precession rates and hence changes in the posi- tions of secular resonances, which could ‘sweep’ through the asteroid belt, exciting e and i, and coupled with gas drag, could lead to semi-major axis mobility and the re- moval of material from the belt (e.g., Heppenheimer 1980; Ward 1981; Lemaitre and Dubru 1991; Lecar and Franklin 1997; Nagasawa et al. 2000, 2001, 2002). It has also been suggested that sweeping secular resonances could lead to orbital excitation in the Kuiper Belt (Nagasawa and Ida 2000). However, as reviewed by Petit et al. (2002) and O’Brien et al. (2006), secular resonance sweeping is gen- erally unable to simultaneously match the observed e and i excitation in the asteroid belt, as well as its radial mix- ing and mass depletion, for reasonable parameter choices (especially in the context of the Nice Model). Another possibility is that planetary embryos were able to accrete in the asteroid belt (e.g., Wetherill 1992). The fact that Jupiter’s ∼10 Earth-mass core was able to accrete in our Solar System beyond the asteroid belt suggests that embryos were almost certainly able to accrete in the aster- oid belt, even accounting for the roughly 3-4× decrease in the mass density of solid material inside the snow line. The scattering of asteroids by those embryos, coupled with the Jovian and Saturnian resonances in the asteroid belt, has been shown to be able to reasonably reproduce the observed e and i excitation in the belt as well as its radial mixing and mass depletion (Petit et al. 2001, 2002; O’Brien et al. 2006). In the majority of simulations of this scenario by both groups, the embryos are completely cleared from the asteroid belt. Thus, the observational evidence and theoretical mod- els for the evolution of the asteroid belt strongly suggest that dynamics, rather than collisions, dominated its mass depletion. Collisions, however, have still played a key role in sculpting the asteroid belt. Many dynamical fam- ilies, clusterings in orbital element space, have been dis- covered, giving evidence for ∼20 breakups of 100-km or larger parent bodies over the history of the Solar System (Bottke et al. 2005a,b). The large 500-km diameter asteroid Vesta has a preserved basaltic crust with a single large im- pact basin (McCord et al. 1970; Thomas et al. 1997). This basin was formed by the impact of a roughly 40-km projec- tile (Marzari et al. 1996; Asphaug 1997). The size distribution of main-belt asteroids is known or reasonably constrained through observational surveys down to ∼1 km in diameter (e.g. Durda and Dermott 1997; 0.1 1 10 100 1000 Diameter (km) Asteroid and TNO Size Distributions Bernstein (2004) Fit to Data 0.1 1 10 100 1000 Diameter (km) Asteroid and TNO Size Distributions Subaru SKADS Spacewatch Cataloged Asteroids Fig. 9.— Observational estimates of the main belt and TNO size distributions. The pentagons (with dashed best-fit curve) show the total TNO population as determined from the Bernstein et al. (2004) HST survey, converted to approximate diameters assum- ing an albedo of 0.04. Points with arrows are upper limits given by non-detections. The solid line is the population of observed asteroids, and open circles are from debiased Spacewatch main- belt observations (Jedicke and Metcalfe 1998). These data, con- verted to diameters, were provided by D. Durda. The two dashed lines are extrapolations based on the Sloan Digital Sky Survey (Ivezić et al. 2001) and the Subaru Sub-km Main Belt Aster- oid Survey (Yoshida et al. 2003), and diamonds show the debi- ased population estimate from the SKADS survey (Gladman et al. 2007). Error bars are left out of this plot for clarity. Note that the TNO population is substantially more populous and massive, by roughly a factor of 1000, than the asteroid population. Jedicke and Metcalfe 1998; Ivezić et al. 2001; Yoshida et al. 2003; Gladman et al. 2007). Not surprisingly, the largest uncertainties are at the smallest sizes, where good orbits are often not available for the observed asteroids, which makes the conversion to absolute magnitude and diame- ter difficult (e.g., Ivezić et al. 2001; Yoshida et al. 2003). Recent results from the Sub-Kilometer Asteroid Diame- ter Survey (SKADS, Gladman et al. 2007), the first survey since the Palomar-Leiden Survey designed to determine or- bits as well as magnitudes of main-belt asteroids, suggest that the asteroid magnitude-frequency distribution may be well represented by a single power law in the range from H=14.0 to 18.8, which corresponds to diameters of 0.7 to 7 km for an albedo of 0.11. These observational constraints are shown in Fig. 9 alongside the determination of the TNO size distribution from Bernstein et al. (2004). While over some size ranges, the asteroid size distribu- tion can be fit by a single power law, over the full range of observed asteroid diameters from ∼1-1000 km, there are multiple bumps or kinks in the size distribution (namely around 10 and 100 km in diameter). The change in slope of the size distribution around 100 km is due primarily to the fact that asteroids larger than this are very difficult to disrupt, and hence the size distribution of bodies larger than 100 km is likely primordial. The change in slope around 10 km has a different origin—such a structure is produced as a result of a change in the strength properties of aster- oids, namely the transition from when a body’s resistance to disruption is dominated by material strength to when it is dominated by self-gravity. This transition in strength prop- erties occurs at a size much smaller than 10 km, but re- sults in a structure that propagates to larger sizes (see, e.g., Durda et al. 1998; O’Brien and Greenberg 2003). The pres- ence of this structure in the asteroid size distribution is con- sistent with the asteroids being a collisionally-relaxed pop- ulation, i.e. a population in which the size distribution has reached an approximate steady state where collisional pro- duction and collisional destruction of bodies in each size range are in balance. The collisional evolution of the asteroid belt has been studied by many authors (e.g. Davis et al. 1985; Durda 1993; Davis et al. 1994; Durda and Dermott 1997; Durda et al. 1998; Campo Bagatin et al. 1993, 1994, 2001; Marzari et al. 1999). The most recent models of collisional evolution of the asteroid belt incorporate aspects of dynamical evo- lution as well, such as the removal of bodies by reso- nances and the Yarkovsky effect, and the enhancement in collisional activity during its massive primordial phase (O’Brien and Greenberg 2005; Bottke et al. 2005a,b). In particular, Bottke et al. (2005a) explicitly incorporate the results of dynamical simulations of the excitation and clear- ing of the main belt by embedded planetary embryos per- formed by Petit et al. (2001). Such collisional/dynamical models can be constrained by a wide range of observational evidence such as the main belt size distribution, the num- ber of observed asteroid families, the existence of Vesta’s basaltic crust, and the cosmic ray exposure ages of ordi- nary chondrite meteorites, which suggest that the lifetimes of meter-scale stony bodies in the asteroid belt are on the order of 10-20 Myr (Marti and Graf 1992). One of the most significant implications of having an early massive main belt, which was noted in early colli- sional models (e.g. Chapman and Davis 1975) and recently emphasized in the case of collisional evolution plus dynam- ical depletion (e.g., Bottke et al. 2005b), is that the majority of the collisional evolution of the asteroid belt occurred dur- ing its early, massive phase, and there has been relatively lit- tle change in the main-belt size distribution since then. The current, wavy main-belt size distribution, then, is a ‘fossil’ from its first few hundred Myr of collisional and dynamical evolution. So how does the Kuiper Belt compare to the asteroid belt in terms of its collisional and dynamical evolution? Evi- dence and modeling for the asteroid belt suggest that dy- namical depletion, rather than collisional erosion, was pri- marily responsible for reducing the mass of the primordial asteroid belt to its current level. In the case of the Kuiper Belt, this is less clear. As shown in Sec. 3, collisional erosion, especially when aided by stellar perturbations or the formation of Neptune, can be very effective in remov- ing mass. At the same time, dynamical models such as the Nice Model result in the depletion of a large amount of mass through purely dynamical means and are able to match many observational constraints. Recent modeling that couples both collisional fragmentation and dynami- cal effects suggests that collisional erosion cannot play too large of a role in removing mass from the Kuiper Belt, oth- erwise the Scattered Disk and Oort Cloud would be too de- pleted to explain the observed numbers of short- and long- period comets (Charnoz and Morbidelli 2007). That model currently does not include coagulation. Further modeling work, which self-consistently integrates coagulation, colli- sional fragmentation, and dynamical effects, is necessary to fully constrain the relative contributions of collisional and dynamical depletion in the Kuiper Belt. We have noted that the asteroid belt has a collisionally- relaxed size distribution that is not well-represented by a single power law over all size ranges. Should we expect the same for the Kuiper Belt size distribution, and is there evi- dence to support this? The collision rate in the Kuiper Belt should be roughly comparable to that in the asteroid belt, with the larger number of KBOs offsetting their lower in- trinsic collision probability (Davis and Farinella 1997), and as noted earlier in this chapter, the primordial Kuiper Belt, like the asteroid belt, would have been substantially more massive than the current population. This suggests that the Kuiper Belt should have experienced a degree of col- lisional evolution roughly comparable to the asteroid belt, and thus is likely to be collisionally relaxed like the asteroid belt. Observational evidence thus far is not detailed enough to say for sure if this is the case, although recent work (Kenyon and Bromley 2004e; Pan and Sari 2005) suggests that the observational estimate of the TNO size distribution by Bernstein et al. (2004), shown in Fig. 9, is consistent with a collisionally-relaxed population. While the Kuiper Belt is likely to be collisionally re- laxed, it is unlikely to mirror the exact shape of the aster- oid belt size distribution. The shape of the size distribu- tion is determined, in part, by the strength law Q∗D, which is likely to differ somewhat between asteroids and KBOs. This is due to the difference in composition between as- teroids, which are primarily rock, and KBOs, which con- tain a substantial amount of ice, as well as the difference in collision velocity between the two populations. With a mean velocity of ∼5 km/s (Bottke et al. 1994), collisions between asteroids are well into the supersonic regime (rel- ative to the sound speed in rock). For the Kuiper belt, col- lision velocities are about a factor of 5 or more smaller (Davis and Farinella 1997), such that collisions between KBOs are close to the subsonic/supersonic transition. For impacts occurring in these different velocity regimes, and into different materials, Q∗D may differ significantly (e.g., Benz and Asphaug 1999). The difference in collision velocity can influence the size distribution in another way as well. With a mean collision velocity of ∼5 km/s, a body of a given size in the aster- oid belt can collisionally disrupt a significantly larger body. Thus, transitions in the strength properties of asteroids can lead to the formation of waves that propagate to larger sizes and manifest themselves as changes in the slope of the size distribution, as seen in Fig. 9. For the Kuiper belt, with col- lision velocities that are about a factor of 5 or more smaller than in the asteroid belt, the difference in size between a given body and the largest body it is capable of disrupting is much smaller than in the asteroid belt, and waves should therefore be much less pronounced or non-existent in the KBO size distribution (e.g., O’Brien and Greenberg 2003). There is still likely to be a change in slope at the largest sizes where the population transitions from being primordial to being collisionally relaxed, and such a change appears in the debiased observational data of Bernstein et al. (2004) (shown here in Fig. 9), although recent observations sug- gest that the change in slope may actually occur at smaller magnitudes than found in that survey (Petit et al. 2006). Is the size distribution of the Kuiper Belt likely to be a ‘fossil’ like the asteroid belt? The primordial Kuiper Belt would have been substantially more massive than the cur- rent population. Thus, regardless of whether the depletion of its mass was primarily collisional or dynamical, colli- sional evolution would have been more intense early on and the majority of the collisional evolution would have oc- curred early in its history. In either case, its current size distribution could then be considered a fossil from that early phase, although defining exactly when that early phase ends and the size distribution becomes ‘fossilized’ is not equally clear in both cases. In the case where the mass depletion of the Kuiper Belt occurs entirely through collisions, there would not necessarily be a well-defined point at which one could say that the size distribution became fossilized, as the collision rate would decay continuously with time. In the case of dynamical depletion, where the mass would be re- moved fairly rapidly as in the case of the Nice Model de- scribed in Sec. 3.4, the collision rate would experience a correspondingly rapid drop, and the size distribution could be considered essentially fossilized after the dynamical de- pletion event. As noted earlier in this section, an important observable manifestation of collisions in the asteroid belt is the for- mation of families, i.e. groupings of asteroids with similar orbits. Asteroid families are thought to be the fragments of collisionally disrupted parent bodies. These were first rec- ognized by Hirayama (1918) who found 3 families among the 790 asteroids known at that time. The number increased to 7 families by 1926 when there were 1025 known as- teroids (Hirayama 1927). Today, there are over 350,000 known asteroids while the number of asteroid families has grown to about thirty. Given that the Kuiper Belt is likely a collisionally evolved population, are there collisional families to be found among these bodies? Families are expected to be more difficult to recognize in the Kuiper Belt than in the asteroid belt. Families are identified by finding statistically significant clusters of asteroid orbit elements—mainly the semi-major axis, eccentricity and inclination. The colli- sional disruption of a parent bodies launches fragments with speeds of perhaps a few hundred meters/sec relative to the original target body. This ejection speed is small compared with the orbital speeds of asteroids, hence the or- bits of fragments differ by only small amounts from that of the original target body and, more importantly, from each other. Thus, the resulting clusters of fragments are easy to identify. However, in the Kuiper Belt, where ejection velocities are likely to be about the same but orbital speeds are much lower, collisional disruption produces a much greater dis- persion in the orbital elements of fragments. This reduces the density of the clustering of orbital elements and makes the task of distinguishing family members from the back- ground population much more difficult (Davis and Farinella 1997). To date, there are over 1000 KBOs known, many of which have poorly-determined orbits or are in resonances that would make the identification of a family difficult or impossible. Chiang et al. (2003) applied lowest-order secu- lar theory to 227 non-resonant KBOs with well-determined orbits and found no convincing evidence for a dynamical family. Recently, however, Brown et al. (2007) found evi- dence for a single family with at least 5 members associated with KBO 2003 EL61. This family was identified based on the unique spectroscopic signature of its members, and con- firmed by their clustered orbit elements. Given the small numbers involved, it cannot be said whether or not finding a single KBO family at this stage is statistically that different from the original identification of 3 asteroid families when there were only 790 known aster- oids (Hirayama 1918). However, the fact that the KBO fam- ily associated with 2003 EL61 was first discovered spectro- scopically, and its clustering in orbital elements was later confirmed, while nearly all asteroid families were discov- ered based on clusterings in orbital elements alone, suggests that even if comparable numbers of KBO families and aster- oid families do exist, the greater dispersion of KBO families in orbital element space may make them more difficult to identify unless there are spectroscopic signatures connect- ing them as well. Perhaps when the number of non-resonant KBOs with good orbits approaches 1000, more populous Kuiper Belt families will be identified, and as can be done now with the asteroid belt, these KBO families can be used as con- straints on the interior structures of their original parent bodies as well as on the collisional and dynamical history of the Kuiper Belt as a whole. 6. Concluding Remarks Starting with a swarm of 1 m to 1 km planetesimals at 20–150 AU, the growth of icy planets follows a stan- dard pattern (Stern and Colwell 1997a,b; Kenyon and Luu 1998, 1999a,b; Kenyon and Bromley 2004a,d, 2005). Col- lisional damping and dynamical friction lead to a short pe- riod of runaway growth that produces 10–100 objects with r ∼ 300–1000 km. As these objects grow, they stir the orbits of leftover planetesimals up to the disruption veloc- ity. Once disruptions begin, the collisional cascade grinds leftover planetesimals into smaller objects faster than the oligarchs can accrete them. Thus, the oligarchs always con- tain a small fraction of the initial mass in solid material. For self-stirring models, oligarchs contain ∼ 10% of the initial mass. Stellar flybys and stirring by a nearby gas gi- ant augment the collisional cascade and leave less mass in oligarchs. The two examples in §3.3 suggest that a very close flyby and stirring by Neptune leave ∼ 2% to 5% of the initial mass in oligarchs with r ∼ 100–1000 km. This evolution differs markedly from planetary growth in the inner solar system. In ∼ 0.1–1 Myr at a few AU, run- away growth produces massive oligarchs, m & 0.01M⊕, that contain most of the initial solid mass in the disk. Aside from a few giant impacts like those that might produce the Moon (Hartmann and Davis 1975; Cameron and Ward 1976), collisions remove little mass from these objects. Al- though the collisional cascade removes many leftover plan- etesimals before oligarchs can accrete them, the lost ma- terial is much less than half of the original solid mass (Wetherill and Stewart 1993; Kenyon and Bromley 2004b). For a & 40 AU, runaway growth leaves most of the mass in 0.1–10 km objects that are easily disrupted at modest colli- sion velocities. In 4.5 Gyr, the collisional cascade removes most of the initial disk mass inside 70–80 AU. Together with numerical calculations of orbital dynam- ics (Chapter by Morbidelli et al.), theory now gives us a foundation for understanding the origin and evolution of the Kuiper belt. Within a disk of planetesimals at 20– 100 AU, collisional growth naturally produces objects with r ∼ 10–2000 km and a size distribution reasonably close to that observed among KBOs. As KBOs form, migration of the giant planets scatters KBOs into several dynamical classes (Chapter by Morbidelli et al.). Once the giant plan- ets achieve their current orbits, the collisional cascade re- duces the total mass in KBOs to current levels and produces the break in the size distribution at r ∼ 20–40 km. Contin- ued dynamical scattering by the giant planets sculpts the inner Kuiper belt and maintains the scattered disk. New observations will allow us to test and to refine this theoretical picture. Aside from better measures of αm, rmax, and r0 among the dynamical classes, better limits on the total mass and the size distribution of large KBOs with a ∼ 50–100 AU should yield a clear discriminant among theoretical models. In the Nice model, the Kuiper belt was initially nearly empty outside of ∼ 50 AU. Thus, any KBOs found with a ∼ 50–100 AU should have the collisional and dynamical signatures of the scattered disk or detached population. If some KBOs formed in situ at a & 50 AU, their size distribution depends on collisional growth modi- fied by self-stirring and stirring by ∼ 30 M⊕ of large KBOs formed at 20–30 AU and scattered through the Kuiper belt by the giant planets. From the calculations of Neptune stir- ring (§3.3), stirring by scattered disk objects should yield a size distribution markedly different from the size distribu- tion of detached or scattered disk objects formed at 20–30 AU. Wide-angle surveys on 2–3 m class telescopes (e.g., Pan-Starrs) and deep probes with 8–10 m telescopes can provide this test. Information on smaller size scales – αd and r1 – place additional constraints on the bulk properties (fragmentation parameters) of KBOs and on the collisional cascade. In any of the stirring models, there is a strong correlation between r0, r1, and the fragmentation parameters. Thus, direct mea- sures of r0 and r1 provide a clear test of KBO formation calculations. At smaller sizes (r . 0.1 km), the slope of the size distribution αd clearly tests the fragmentation al- gorithm and the ability of the collisional cascade to remove KBOs with r ∼ 1–10 km. Although the recent detection of KBOs with r ≪ 1 km (Chang et al. 2006) may be an instrumental artifact (Jones et al. 2006; Chang et al. 2007), optical and X-ray occultations (e.g., TAOS) will eventually yield these tests. Finally, there is a clear need to combine coagulation and dynamical calculations to produce a ‘unified’ picture of planet formation at a & 20 AU. Charnoz and Morbidelli (2007) provide a good start in this direction. Because the collisional outcome is sensitive to internal and external dy- namics, understanding the formation of the observed n(r), n(e), and n(i) distributions in each KBO population re- quires treating collisional evolution and dynamics together. A combined approach should yield the sensitivity of αm, rmax, and r0 to the local evolution and the timing of the formation of giant planets, Neptune migration, and stellar flybys. These calculations will also test how the dynami- cal events depend on the evolution during oligarchic growth and the collisional cascade. Coupled with new observations of KBOs and of planets and debris disks in other planetary systems, these calculations should give us a better under- standing of the origin and evolution of KBOs and other ob- jects in the outer solar system. We thank S. Charnoz, S. Kortenkamp, A. Morbidelli, and an anonymous reviewer for comments that consider- ably improved the text. We acknowledge support from the NASA Astrophysics Theory Program (grant NAG5- 13278; BCB & SJK), the NASA Planetary Geology and Geophysics Program (grant NNX06AC50G; DPO), and the JPL Institutional Computing and Information Services and the NASA Directorates of Aeronautics Research, Science, Exploration Systems, and Space Operations (BCB & SJK). REFERENCES Adachi, I., Hayashi, C., and Nakazawa, K. (1976). The gas drag effect on the elliptical motion of a solid body in the primordial solar nebula. Progress of Theoretical Physics, 56, 1756–1771. Adams, F. C. and Laughlin, G. (2001). Constraints on the Birth Aggregate of the Solar System. Icarus, 150, 151– Artymowicz, P. (1988). Radiation pressure forces on parti- cles in the Beta Pictoris system. Astrophys. J. Lett., 335, L79–L82. Artymowicz, P. (1997). Beta Pictoris: an Early Solar Sys- tem? Annual Review of Earth and Planetary Sciences, 25, 175–219. Asphaug, E. (1997). Impact origin of the Vesta family. 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This 2-column preprint was prepared with the AAS LATEX macros v5.2. INTRODUCTION COAGULATION THEORY COAGULATION SIMULATIONS Background Self-Stirring External Perturbation Nice Model A Caveat on the Collisional Cascade Model Predictions Confronting KBO collision models with KBO data KBOs and Asteroids Concluding Remarks

0704.0260

On Existence of Boundary Values of Polyharmonic Functions

ON EXISTENCE OF BOUNDARY VALUES OF POLYHARMONIC FUNCTIONS 1 M.L. GORBACHUK and S.M. TORBA ABSTRACT. In trigonometric series terms all polyharmonic functions inside the unit disk are described. For such functions it is proved the existence of their boundary values on the unit circle in the space of hyperfunctions. The necessary and sufficient conditions are presented for the boundary value to belong to certain subspaces of the space of hyperfunctions. The purpose of this paper is to find necessary and sufficient conditions for a solution of the equation ∆mu = 0 inside a domain to have a limit on the boundary of the domain in various functional spaces. We consider the simplest situation where a domain is the unit disk K = {z = reit, 0 ≤ r < 1, 0 ≤ t ≤ 2π}. The case of m = 1 has been investigated during 20th century by a lot of mathematicians (we refer for details to [1 - 5]. The case of m = 2 was considered in [6]. For an arbitrary m the problem of existence of boundary values in the space L2(∂K) (∂K - is the unit circle) was discussed in [7]. 1. Denote by D = D(∂K) the set of all infinitely differentiable functions on ∂K. We say that a sequence ϕn ∈ D converges to ϕ ∈ D, n → ∞, and write ϕn → ϕ, if for every k ∈ N0 = {0, 1, 2, . . .}, the sequence ϕ n (t) converges to ϕ (k)(t) uniformly in t ∈ ∂K. Let also A = A(∂K) be the set of analytic functions on ∂K. The convergence in A is introduced in the following way: a sequence ϕn ∈ A converges to ϕ in A (ϕn → ϕ) if there exists a neighbourhood U of ∂K in which all the functions ϕn(t) converge to ϕ(t) uniformly on any compact set from For a number α > 0 we put Aα = {ϕ ∈ D ∣∣∃c > 0 ∀N0 max |ϕ(k)(t)| ≤ cαkk!}. The linear set Aα is a Banach space with respect to the norm ‖ϕ‖Aα = sup |ϕ(k)(t)| It is not hard to show that if α < α′, then Aα ⊆ Aα′ , A = ind lim and the dense continuous embeddings A ⊂ D ⊂ Lp(∂K) = Lp, 1 ≤ p < ∞, 1Mathematics Subject Classification. Primary 35J30 Key words and phrases. Polyharmonic equation, solution inside of a domain and its boundary value, hyperfunc- tion, Fourier series. Supported by CRDF and Ukr. Government (Project UM 1-2507-OD-03) http://arxiv.org/abs/0704.0260v1 hold. Let D′ and A′ are the spaces of continuous antilinear functionals on D (distributions) and A (hyperfunctions), respectively (see [8]). In the following, < F, ϕ > denotes an action of the functional F ∈ A′ (F ∈ D′) onto ϕ ∈ A (ϕ ∈ D). By convergence in A′ (in D′) we mean the weak one, that is, Fn → F (Fn → F ) if for any ϕ ∈ A (ϕ ∈ D), the number sequence < Fn, ϕ > converges to < F, ϕ >. As ek(t) = e ikt ∈ A (k ∈ Z), the Fourier coefficients ck(F ) =< F, ek > can be determined for F ∈ A′. It is known (see e.g. [9]) that ck(F )e ikt A and one can easily verify that the below assertion is valid. Proposition 1 The following equivalence relations hold: F ∈ D ⇐⇒ ∀α > 0 ∃c > 0 |ck(F )| ≤ c|k| F ∈ A ⇐⇒ ∃α > 0 ∃c > 0 |ck(F )| ≤ ce −α|k|; F ∈ D′ ⇐⇒ ∃α > 0 ∃c > 0 |ck(F )| ≤ c|k| F ∈ A′ ⇐⇒ ∀α > 0 ∃c > 0 |ck(F )| ≤ ce α|k|. Moreover, the series ck(F )e ikt converges to F in the corresponding space. The sequence {Fn}n∈N, whose elements Fn belong to one of the spaces D,A, D ′ or A′, converges to F in this space if and only if the constants c and α in the above estimates for |ck(Fn)| do not depend on n and for any k ∈ Z, ck(Fn) → ck(F ), n → ∞. 2. A function u(r, t) = u(reit) ∈ C2m(K) is called m-harmonic in K if it satisfies the equation ∆mu(r, t) = 0, 0 ≤ r < 1, t ∈ [0, 2π]. (1) Note, that no conditions on the behaviour of u(r, t) near ∂K are imposed. Theorem 1. In order that a function u(r, t) ∈ C2m(K) be m-harmonic in K, it is necessary and sufficient that the representation u(r, t) = (r2 − 1)j−1 ck(Fj)r |k|eikt, Fj ∈ A ′, (2) be admissible, where Fj are uniquely determined by u(r, t). Proof. By Proposition 1, ∀α > 0 ∃cj = cj(α) > 0 ∀k ∈ Z |ck(Fj)| ≤ cje α|k|. So the series ck(Fj)r |k|eikt converges uniformly in the disk KR = {z ∈ C : |z| ≤ R} of radius R < e−α and determines an infinitely differentiable function there. The direct check shows that the functions (r2 − 1)j−1 ck(Fj)r |k|eikt, j = 1, 2, . . . , m, satisfy (1) in KR. Since α > 0 is arbitrary, these functions are solutions of the equation (1) inside K. To prove the necessity, suppose at first m = 1. Let u(r, t) be a harmonic function in K. Then for a fixed r < 1, u(r, t) is infinitely differentiable in t, and it may be written in the form u(r, t) = ck(r)e ikt, ck(r) = u(r, t)e−ikt dt, (3) where the series and all its derivatives converge uniformly in t ∈ [0, 2π]. The coefficients ck(r) are infinitely differentiable on [0, 1) and satisfy the equation ′′(r) + rck ′(r) = k2ck(r). Hence, ck(r) = r |k|ck, ck ∈ C. It follows from the convergence of the series in (3) that ∀r < 1 r|k||ck| = e −α|k||ck| ≤ c, where α = − ln r > 0 is arbitrary. By Proposition 1, ck are the Fourier coefficients of a certain hyperfunction F1, and u(r, t) = r|k|ck(F1)e ikt, F1 ∈ A ′. (4) Thus, the representation (2) is valid when m = 1. Assume the representation (2) to be true for an (m−1)-harmonic insideK function u(r, t) (m ≥ 2), and we shall prove that such a representation holds for an m-harmonic function. If u(r, t) is anm-harmonic function, then ∆u(r, t) is an (m−1)-harmonic one. By assumption, there exist Ej ∈ A ′, j = 1, 2, . . . , m− 1, such that ∆u(r, t) = (r2 − 1)j−1 ck(Ej)r . (5) If we choose ũ ∈ C2(K) so that ∆(u(r, t)− ũ(r, t)) = 0, (6) then, because of (4), we shall have u(r, t) = ũ(r, t) + ck(F1)e , F1 ∈ A Let us find at first ũ(r, t) in the case where the equation (5) is of the form ∆u(r, t) = r|k|ck(E1)e ikt, E1 ∈ A By using the identity (r2 − 1)j r|k|ck(F )e (r2− 1)j−1(|k|+ j)+ (j− 1)(r2− 1)j−2 r|k|ck(F )e for F ∈ A′, one can verify that the function ũ2(r, t) = u2(r, t), where u2(r, t) = (r2 − 1) |k|+ 1 ck(E1)e (ũ1(r, t) ≡ 0), satisfies (6). Set ck(F2) = ck(E1) 4(|k|+1) . By Proposition 1, F2 ∈ A ′. So, in the case under consideration u(r, t) = (r2 − 1)j−1 r|k|ck(Fj)e Suppose now that we know solutions ul(r, t) of the equations ∆u(r, t) = (r2 − 1)j−1 r|k|ck(Ej)e ikt, (8) for all l ≤ s, s ≤ m− 2 is fixed. We show how to find a solution of the equation ∆u(r, t) = (r2 − 1)j−1 r|k|ck(Ej)e ikt. (9) We put us+2(r, t) = (r 2 − 1)s+1 4(s+ 1)(|k|+ s+ 1) ck(Es+1)e It follows from (7) and (8) that if u(r, t) is a solution of (9), then ∆(u− us+2)(r, t) = (r2 − 1)j−1 ck(Ej)e (r2 − 1)sr|k|ck(Es+1)e ikt − s(r2 − 1)s−1 |k|+ s+ 1 + (r2 − 1)s r|k|ck(Es+1)e ikt = (r2 − 1)j−1 r|k|ck(Ej)e ikt + (r2 − 1)s−1 ck(Es)− s · ck(Es+1) |k|+ s + 1 eikt. Taking into account that Es, Es+1 ∈ A ′, we conclude, by Proposition 1, that there exists E ′s ∈ A such that s) = ck(Es)− s · ck(Es+1) |k|+ s+ 1 whence ∆(u− us+2)(r, t) = (r2 − 1)j−1 j ∈ A where E ′j = Ej as j = 1, . . . , s− 1. By assumption, we can find ũs+1(r, t) so that ∆(u− us+2 − ũs+1)(r, t) = 0. Setting ũs+2(r, t) = us+2(r, t) + ũs+1(r, t), we arrive at the equality ∆(u− ũs+2)(r, t) = 0. It is not hard to observe that for the desired function ũ(r, t) we have the formula ũ(r, t) = ũm(r, t) = um(r, t) + um−1(r, t) + · · ·+ u2(r, t) = (r2 − 1)m−1 4(m− 1)(|k|+m− 1) ck(Em−1)e ikt + . . . +(r2 − 1)2 4 · 2(|k|+ 2) ck(E2)e ikt + (r2 − 1) 4(|k|+ 1) ck(E1)e u(r, t) = (r2 − 1)j−1 r|k|ck(Fj)e ikt, Fj ∈ A where ck(Fj) = ck(Ej−1) 4(j − 1)(|k|+ j − 1) Since for F ∈ A′ r|k|ck(F ) → ck(F ), r → 1, and |r |k|ck(F )| < |ck(F )|, we have, by Proposition 1, that (r2 − 1)j−1 r|k|ck(Fj)e ikt A F1 if j = 1 0 if j > 1, as r → 1. The elements Fj ∈ A ′ are determined uniquely by the function u(r, t) in the following F1 = lim u(r, ·), Fj+1 = lim u(r, ·)− (r2 − 1)p−1 r|k|ck(Fp)e (r2 − 1)j where the limit is taken in the space A′. This completes the proof. Because of harmonicity in K of the functions uj(r, t) = r|k|ck(Fj)e the representation (2) implies, in particular, the next assertion (cf. [4]). Corollary 1. Let u(r, t) be an m-harmonic in K function. Then it admits a representation of the form u(r, t) = (r2 − 1)j−1uj(r, t), (10) where the functions uj(r, t) are harmonic in K. When proving the theorem, it was also established the following fact. Corollary 2. If u(r, t) is an m-harmonic in K function, then there exists its radial boundary value u(1, ·) on ∂K in the space A′, that is, u(r, ·) → u(1, ·) as r → 1. 3. Let Φ be a complete linear Hausdorff space such that the continuous embeddings A ⊂ Φ ⊂ A′ hold. We say that F ∈ Φ is a boundary value on ∂K of an m-harmonic in K function u(r, t) and write F = u(1, ·) if u(r, ·) → F as r → 1. It is seen from Theorem 1 and Corollary 2 that every m-harmonic in K function has a boundary value in A′. Moreover, each element F ∈ A′ is the boundary value of a certain m- harmonic in K function. The natural question arises: under what conditions on an m-harmonic in K function u(r, t) its boundary value u(1, ·) belongs to Φ? Theorem 2. The boundary value u(1, ·) of an m-harmonic in K function u(r, t) belongs to the space Φ if and only if the set {u(r, ·)}r<1 is compact in Φ. Proof. Necessity. It is known that if r0 < 1, then u(r0, ·) ∈ A, and u(r, ·) → u(r0, ·) as 1 > r → r0. Since the embedding A ⊂ Φ is continuous, u(r, ·) → u(r0, ·) (r → r0). By assumption, u(r, ·) → u(1, ·) if r → 1. So, the set {u(r, ·)}r<1 is compact in Φ. Sufficiency. Let the set {u(r, ·)}r<1 be compact in Φ. Suppose r → 1. Then there exists a subsequence rk → 1 such that u(rk, ·) converges in Φ (rk → 1) to a certain element F ∈ Φ. Since Φ ⊂ A continuously, u(rk, ·) converges in A ′. Taking into account that u(r, ·) → u(1, ·) as r → 1, we have u(1, ·) = F ∈ Φ which completes the proof. In the partial case where Φ = L2(∂K), Theorem 2 was obtained in [7]. By using compactness criteria for sets, one can find the sufficient conditions for the boundary value of a polyharmonic function to belong to Lp(∂K), 1 ≤ p < ∞. For instance, the following assertion is valid. Corollary 3. Let u(r, t) be an m-harmonic inside the disk K function. In order that u(r, t) have a boundary value in Lp = Lp(∂K), it is necessary and sufficient that: 1) sup 0≤r<1 ‖u(r, ·)‖Lp < ∞; rei(t−τ) − u(reit) ∣∣p dt → 0 (τ → 0) uniformly in r ∈ [0, 1). Now we consider in more detail the case of L2. Let F ∈ A′ ∣∣∣∣∣ ‖F‖Bj = sup 0≤r<1 (1− r2)j r2|k||ck(F )| The set Bj with norm ‖ · ‖Bj forms a Banach space. Theorem 3. If u(r, t) is an m-harmonic in K function, then 0≤r<1 |u(r, t)|2 dt < ∞ ⇐⇒ F1 ∈ L2, Fj ∈ Bj−1 if 2 ≤ j ≤ m, where Fj are taken from representation (2). Moreover, u(r, ·) → F1 (r → 1) weakly in the space Proof. Assume that in the representation (2) F1 ∈ L2, Fj ∈ Bj−1 (j = 2, . . . , m). Then |u(r, t)|2 dt = r2|k| ∣∣ck(F1) + (r2 − 1)ck(F2) + · · ·+ (r2 − 1)m−1ck(Fm) ∣∣2 ≤ r2|k|(r2 − 1)2(j−1)|ck(Fj)| 2 ≤ c. Conversely, let sup 0≤r<1 ‖u(r, ·)‖L2 < ∞. Then, as was shown in [4, Lemma 7], each summand in (10) is bounded, too: 0≤r<1 ‖(1− r2)j−1uj(r, ·)‖L2 < ∞, j = 1, 2, . . . , m. This is equivalent to the inequality 0≤r<1 (1− r2)2(j−1) r2|k||ck(Fj)| 2 < ∞, that is, F1 ∈ L2, Fj ∈ Bj−1 (j = 2, . . . , m). It still remains to prove the weak convergence of u(r, ·) to F1 (r → 1) in L2. Since u(r, ·) as r → 1 and eikt ∈ A (k ∈ Z), we have u(r, t)eikt dt = lim r|k|[c−k(F1) + (r 2 − 1)c−k(F2) + · · ·+ c−k(Fm)] = ck(F1) =< F1, ek >= F1(t)e Thus, u(r, ·) → F1 (r → 1) weakly in L2 on a total set, and sup 0≤r<1 ‖u(r, ·)‖L2 < ∞. It follows from here that u(r, ·) → F1 (r → 1) weakly in L2. The proof is complete. Let u(r, t) be a harmonic in K function. It follows from (2) that ‖u(r, ·)‖2L2 = r2|k||ck(F1)| In view of ‖u(r, ·)‖L2 ≤ c, the well-known Fatou lemma and the Lebesgue theorem on passage to the limit yield |ck(F1)| 2 < ∞, F1 = u(1, ·) ∈ L2, ‖u(r, ·)‖L2 → ‖u(1, ·)‖L2, r → 1. Therefore the weak convergence of u(r, t) to u(1, t) implies the strong one. As was shown in [7], in the case of m = 2 the boundedness of ‖u(r, ·)‖L2 does not guarantee the convergence of u(r, ·) (r → 1) in L2. We pass now to the Sobolev spaces 2 = W 2 (∂K) = F ∈ A′ |k|2α|ck(F )| , α ∈ R. The following statement is valid. Theorem 4. The embeddings 2 ⊂ Bj ⊂ W hold. Proof. Since the function f(r) = (1 − r2)2jr2k, j, k ∈ N0, reaches its maximum at the point r2 = k , and 0≤r<1 f(r) = k + 2j )2j ( k + 2j we have 0≤r<1 (1− r2)2j r2|k||ck(F )| < cj |ck(F )| |k|2j that is, W 2 ⊂ Bj . Suppose now F ∈ Bj. Then, substituting z := r ∃c > 0 (1− z)2j z|k||ck(F )| 2 < c. Multiplying this inequality by (1− z)−α and then integrating along [0, 1), we obtain |ck(F )| (1− z)2j−αz|k| dz < ∞. (11) If we put δ = 2j − α+ 1, we get for n ∈ N (1− z)2j−αzn dz = δ(δ + 1) . . . (δ + n) Since = 1 + the relation an = O is fulfilled. Taking in (11) α = 1− ε, ε ∈ (0, 1), we conclude that |ck(F )| |k|2j+ε that is, F ∈ W 2 , which completes the proof. The next theorem is devoted to the question on the existence of boundary values in the space D′ of distributions. Theorem 5. In order that an m-harmonic in K function u(r, t) admit a representation of the form (2) with Fj ∈ D ′ (j = 1, . . . , m), it is necessary and sufficient that ∃α ≥ 0 ∃c > 0 sup t∈[0,2π] |u(reit)| ≤ c(1− r)−α. (12) Proof. Let the inequality (12) hold. Then for p ∈ N, p > α, the function v(r, t) = (1−r2)pu(r, t) is (m+ p)-harmonic in K, and it is not difficult to verify that 0≤r<1 ‖v(r, ·)‖L2 ≤ 2πc. By Theorems 3,4, the function v(r, t) may be represented in the form (2) where Fj ∈ W −(j+p) Since D′ = W−α2 , we have Fj ∈ D The necessity of condition (12) for m = 1 was proved in [5]. Namely, it was shown there that for a harmonic function of the form u(r, t) = r|k|ck(F )e ikt, f ∈ D′ there exists α ≥ 0 such that |u(r, t)| ≤ c(1− r)−α. If we take α = max αj , where αj corresponds to Fj from (2), we obtain the estimate (12) for an m-harmonic function (m is arbitrary). Corollary 4. An m-harmonic in K function u(r, t) has a boundary value in D′ if and only if it satisfies (12). For a number β > 1 we put G{β} = G{β}(∂K) = {ϕ ∈ D ∣∣∃α > 0 ∃c > 0 ∀k ∈ N0 max |ϕ(k)(t)| ≤ cαkkkβ}. (13) The linear space G{β} is endowed with the inductive limit topology of the Banach spaces G{β,α} of functions ϕ ∈ D satisfying (13) with a fixed constant α. The norm in G{β,α} is defined as ‖ϕ‖G{β,α} = sup |ϕ(k)(t)| αkkkβ It is evident, that A ⊂ G{β} ⊂ D ⊂ L2 ⊂ D ′ ⊂ G′{β} ⊂ A where G′{β} denotes the dual of G{β}. Theorem 6. An m-harmonic in K function u(r, t) admits a representation of the form (2) with Fj ∈ G {β} (j = 1, . . . , m) if and only if ∀α > 0 ∃c = c(α) > 0 sup 0≤r<1 |u(r, t)| ≤ ceα(1−r) , q = β − 1 . (14) The proof follows the scheme like that in Theorem 5 if to take into account that F ∈ G′{β} ⇐⇒ ∀α > 0 |ck(F )| < ce −α|k|1/β , and the series ck(F )e ikt converges to F in G′{β}-topology. Corollary 5. In order that an m-harmonic in K function u(r, t) have a boundary value in the space G′{β}, it is necessary and sufficiant that the condition (14) be satisfied. References [1] Privalov I.I. Boundary Properties of Single-Valued Analytic Functions, Gostekhizdat, Moskva-Leningrad, 1950 (in Russian). [2] Koosis P. Introduction to Hp Spaces, Cambridge University Press, London-New York-New Rochelle-Melburnr-Sydney, 1980. [3] Gorbachuk V.I., Knyazyuk A.V. Boundary values of solutions of operator differential equa- tions, Uspekhi Mat. Nauk 44 (1989), no. 3, 55-91. [4] Komatsu H. Ultradistributions and Hyperfunctions, Lecture Notes Math. 287 (1973), 180- [5] Gorbachuk V.I. On solutions of an operator differential equation with singularilies, Bound- ary Value Problems for Differential Equations, Sborn. Nauchn. Trudov, Inst. Matem. Ukrain. AN, 1992, 8-36. [6] Gorbachuk M.L., Denche M. Representation and boundary values of biharmonic functions, Uspekhi Mat. Nauk 46 (1991), no. 6, p. 202. [7] Mikhailov V.P. On the existence of boundary values of solutions of a polyharmonic equation on the boundary of a domain, Mat. Sborn. 187 (1996), no. 11, 89-115. [8] Berezansky Yu.M., Sheftel Z.G., Us G.F. Functional Analysis. Vol. 1,2, Birkhauser, Basel- Boston-Berlin, 1966. [9] Gorbachuk V.I. On Fourier series of periodic ultradistributions, Ukrain. Mat. Zh. 34 (1982), no. 2, 144-150. Institute of Mathematics National Academy of Sciences of Ukraine 3 Tereshchenkivs’ka Kyiv 01601, Ukraine E-mail: [email protected], [email protected]

0704.0261

Constraints on the Self-Interaction Cross-Section of Dark Matter from Numerical Simulations of the Merging Galaxy Cluster 1E 0657-5

Constraints on the Self-Interaction Cross-Section of Dark Matter from Numerical Simulations of the Merging Galaxy Cluster 1E 0657-56 Scott W. Randall1, Maxim Markevitch1,2, Douglas Clowe3,4, Anthony H. Gonzalez5, and Marusa Bradač6 ABSTRACT We compare recent results from X-ray, strong lensing, weak lensing, and optical observations with numerical simulations of the merging galaxy cluster 1E 0657-56. X-ray observations reveal a bullet-like subcluster with a prominent bow shock, which gives an estimate for the merger velocity of 4700 km s−1, while lensing results show that the positions of the total mass peaks are consistent with the centroids of the collisionless galaxies (and inconsistent with the X-ray brightness peaks). Previous studies, based on older observational datasets, have placed upper limits on the self-interaction cross-section of dark matter per unit mass, σ/m, using simplified analytic techniques. In this work, we take advantage of new, higher-quality observational datasets by running full N-body simulations of 1E 0657-56 that include the effects of self-interacting dark matter, and com- paring the results with observations. Furthermore, the recent data allow for a new independent method of constraining σ/m, based on the non-observation of an offset between the bullet subcluster mass peak and galaxy centroid. This new method places an upper limit (68% confidence) of σ/m < 1.25 cm2 g−1. If we make the assumption that the subcluster and the main cluster had equal mass-to-light ratios prior to the merger, we derive our most stringent constraint of σ/m < 0.7 cm2 g−1, which comes from the consistency of the subcluster’s ob- served mass-to-light ratio with the main cluster’s, and with the universal cluster 1Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138, USA 2Space Research Institute, Russian Academy of Science, Profsoyuznaya 84/32, Moscow 117997, Russia 3Steward Observatory, University of Arizona, 933 N. Cherry Ave., Tuscon, AZ 85721, USA 4Department of Physics and Astronomy, Ohio University, Clippinger Lab 251B, Athens, OH 45701, USA 5Department of Astronomy, University of Florida, 211 Bryant Space Science Center, Gainesville, FL 32611, USA 6Kavli Institute for Particle Astrophysics and Cosmology, P.O. Box 20450, MS-29, Stanford, CA 94309, http://arxiv.org/abs/0704.0261v1 – 2 – value, ruling out the possibility of a large fraction of dark matter particles being scattered away due to collisions. Our limit is a slight improvement over the pre- vious result from analytic estimates, and rules out most of the 0.5 − 5 cm2 g−1 range invoked to explain inconsistencies between the standard collisionless cold dark matter model and observations. Subject headings: dark matter — clusters: individual (1E0657-56) — methods: numerical — large scale structure of universe 1. Introduction The nature of dark matter, which accounts for the majority of the mass in the Universe, is one of the major outstanding problems of modern astrophysics. Although it is often assumed that dark matter is collisionless, there is no a priori reason to believe that this is the case, and it has been noted by other authors that a non-zero self-interaction cross-section can have important astrophysical implications (e.g., Spergel & Steinhardt 2000). In particular, self-interacting dark matter (SIDM) has been invoked to alleviate some apparent problems with the standard cold dark matter (CDM) model, such as the non-observation of cuspy mass profiles in galaxies (e.g., Moore 1994; Flores & Primack 1994; cf. Navarro et al. 1997; Moore et al. 1999b) and the overprediction of the number of small sub-halos within larger systems (e.g., Klypin et al. 1999; Moore et al. 1999a). Previous simulations and theoretical studies suggest that a self-interaction cross-section per unit mass of σ/m ∼ 0.5− 5 cm2 g−1 is needed to explain the observed mass profiles of galaxies (e.g., Davé et al. 2001; Ahn & Shapiro 2003, though see also Ahn & Shapiro 2005). Earlier studies have found stringent upper limits on σ/m, inconsistent with the above range (e.g., Yoshida et al. 2000a; Hennawi & Ostriker 2002; Miralda-Escudé 2002, though see also Sand et al. 2002). However, in general these studies require non-trivial assumptions or statistical samples of clusters and full cosmological simulations. Furlanetto & Loeb (2002) pointed out that if one observes an offset between the gas and dark matter in a merging cluster, arising because of the ram pressure acting on the gas but not the dark matter, it can be used to constrain the collisional nature of dark matter. Markevitch et al. (2002, hereafter M02) found just such a cluster, 1E 0657-56, which in the Chandra image shows a bullet-like subcluster exiting the core of the main cluster, with prominent bow shock and cold front features, and a uniquely simple merger geometry (Markevitch et al. 2002, hereafter M02). This gas bullet lags behind the subcluster galaxies, which led M02 to suggest that this cluster could be used to determine whether or not dark matter is collisional. If dark matter were collisionless, one would expect the subcluster dark matter – 3 – halo to be coincident with the collisionless subcluster galaxies. A map of the dark matter distribution was subsequently derived by Clowe et al. (2004) using weak lensing observations, which showed that the subcluster dark matter clump lay ahead of the gas bullet, close to the centroid of the subcluster galaxies (see also Clowe et al. 2006a, hereafter C06). The X-ray image of 1E 0657-56 is shown in Fig. 1 with the most recently derived weak lensing mass contours of C06 overlain. The weak lensing contours are shown instead of the strong lensing contours since they are better for showing the overall structure, as they are derived from a wider field of view. The positions of the total mass peaks from strong and weak lensing analyses are consistent with one another, and the general structures are similar. The more massive main cluster is on the left and the high-velocity merging bullet subcluster is on the right. The main and subcluster mass peaks are clearly visible in the mass map, as is the offset between the gas bullet and the corresponding dark matter (DM) peak. C06 argued that this offset is direct evidence for the existence of dark matter. The weak lensing mass map of Clowe et al. (2004) was used by Markevitch et al. (2004, hereafter M04), in conjunction with the X-ray and optical observations available at the time, to analytically estimate upper limits on the self-interaction cross-section per unit mass of DM, σ/m, using three independent methods. These methods were based on the observed offset between the gas bullet and the DM subclump, the high merger velocity of the subcluster, and the survival of the DM subclump (more precisely, the subcluster’s M/L ratio being equal to that observed in other clusters and in the main cluster). M04 assume a King mass profile, based on the original weak lensing mass map, and that the subcluster has passed only once through the main cluster, close to the main cluster core, as indicated by the X-ray image. Their most stringent limit comes from the observed survival of the DM subclump, from which they infer that σ/m < 1 cm2 g−1. Although the analytic estimates performed by M04 provide useful upper limits on σ/m, several conservative simplifying assumptions were necessary. For instance, the effects of dynamical friction as the subcluster disturbs the main cluster mass distribution were ignored, as was the possibility of multiple scatterings per particle. Although these effects are relatively small, their inclusion may lead to tighter constraints. Furthermore, the analytic estimates cannot address any structure that may be found in a high-resolution mass map (e.g., tails in the DM distribution, similar to the gas tails seen in the bullet, due to collisional stripping of DM, as described by M04). This argues for full N-body simulations that would include the effects of SIDM with varying cross-sections. Additionally, new data (both X-ray and lensing) have become available for 1E 0657-56. Analysis of data from 450 ks of total exposure with Chandra gives a more accurate shock Mach number of M = 3.0 ± 0.4 (all uncertainties 68%), which corresponds to a shock (and – 4 – bullet) velocity of 4700±630 km s−1 (Markevitch 2005). Recent weak and strong lensing analyses of a much larger optical dataset, which includes HST observations, give a higher quality mass map and a more accurate determination of the subcluster dark matter and galaxy centroids (Bradač et al. 2006, hereafter B06; C06). In particular, the accuracy of the total mass and galaxy centroids is now sufficient for an additional method of constraining σ/m. In this paper, we will concentrate on the most sensitive method from M04, which is based on the observed mass-to-light (M/L) ratios, and on this new test. The best way to interpret the new high-quality data is through comparisons with detailed numerical simula- tions of the merger which allow for SIDM with varying cross-sections. We present results from such simulations and give constraints on the self-interaction cross-section of dark matter particles. We assume Ω0 = 0.3, ΩΛ = 0.7, and H0 = 70 km s −1 Mpc−1, for which 1′′ = 4.42 kpc at the cluster redshift of z = 0.296. 2. The Simulations 2.1. Simulation Code and Parameters All simulations were performed using a modified version of the publicly available TreeSPH code GADGET2 (Springel 2005). To model the self-interaction of the DM particles, we adopted a Monte Carlo method used previously by other authors (e.g., Burkert 2000; Yoshida et al. 2000b). At each simulation time step, the scattering probability for the ith particle is given by Pi = ρiσvrel∆t, (1) where ρi is the local density, vrel is the relative velocity between the ith particle and its nearest neighbor, and ∆t is the time step size. The local density is determined using GADGET2’s smoothed particle hydrodynamic (SPH) capabilities. Collisions are assumed to be elastic and scattering isotropic in the center-of-mass frame. In order for this relation to be valid, ∆t must be chosen such that Pi ≪ 1. We ran a series of merger simulations with σ/m varying between 0 and 1.25 cm2 g−1. Each simulation run included 106 DM particles (gas was not included in the simulations, see discussion in § 4.5). Additionally, we performed a convergence test run with 107 DM particles and σ/m ≈ 1 cm2 g−1, which agreed well with the lower resolution run for all tests we performed. We interpret this agreement as indicating that the effects of individual self-interacting DM particles are well modeled by the large computation particles used in the simulations, and that the results we present here are not seriously affected by numerical resolution effects. The ratio of DM particles in the main cluster and subcluster was set equal – 5 – to the initial total mass ratio of the clusters, which is known analytically from the King models used to build the clusters. In this work, we apply a new method for constraining σ/m based on the absence of an offset between the subcluster total mass and galaxy centroids. For this, we added another family of particles to the simulations to represent the collisionless galaxies. We choose 105 “normal” galaxy particles for the main cluster and 2.5×104 for the subcluster throughout. The ratio of the number of normal galaxy particles was estimated based on galaxy counts given by Barrena et al. (2002). The galaxies were initially distributed like the DM in each run. The mass in galaxies was assumed to be roughly 5% of the total mass for each cluster, which, combined with the number of galaxy particles, gives a low mass per galaxy (2.15×108 M⊙). Using a large number of light-weight galaxies was chosen over using a more realistic mass per galaxy so that accurate galaxy centroids could be determined. Test simulations run with the more realistic average mass per galaxy of 1011 M⊙, also determined from results given by Barrena et al. (2002), showed similar results to those given below in §3, though, as expected, with a larger scatter. Two cD galaxies, one at the center of each cluster, were also included in the simulations (though we note that three cD galaxies are observed, two associated with the main cluster and one with the subcluster). Their inclusion leads to conservative estimates of the effects of DM self-interaction, since a lower central DM density is required to reproduce the observed total mass profile (and the scattering probability depends on the local DM density via Eqn. 1). The cD galaxies were each given a mass of 1013M⊙. The gravitational softening length was chosen to be 2 kpc throughout, which is on the order of the mean inter-particle separation in the densest region in each simulation (i.e., at core passage). The softening length for the cD galaxy particles was set to 60 kpc throughout. This large softening length was chosen since on the scale of the simulations cD galaxies are significantly extended objects and treating them as concentrated point-like masses would be unrealistic. Since the lensing observations do not give an accurate mass profile for the subcluster, King models with density profiles of ρ(r) = ρ0(1 + r 2/r2c ) −3/2, where ρ0 is the central density and rc is the core radius, were conservatively chosen for the mass profiles of each cluster. Such a choice gives conservative limits for the effects of self-interacting DM, since King models do not have strongly concentrated “cuspy” cores, as compared to NFW (Navarro et al. 1995) and Hernquist (1990) profiles. Thus the central density is lower and the total number of DM particle collisions is conservatively reduced. Further discussion of the impact of the bullet mass profile on our results can be found in §4.2. As suggested by the X-ray morphology (M04), all simulated mergers were head-on collisions with zero impact parameter and an initial separation of 4 Mpc. The effects of a possible non-zero impact parameter are discussed in §4.1. – 6 – 2.2. Initial Conditions For each run, the initial conditions were chosen such that the projected mass profiles of the main cluster and the bullet subcluster, after core passage and at the observed separation (720 kpc), roughly matched those from the most recent combined strong and weak lensing results derived by B06, which are given in the last row of Table 2. A relatively small contribution from the observed distribution of gas mass was subtracted from the B06 total mass, so that the resulting values could be directly compared with the simulations. The gas masses were computed from the X-ray observations. The details of the derivation of the gas mass map will be given in a future paper. A summary of the parameters for each simulation run is given in Table 1, which gives the initial central density and core radius for the main cluster (ρc,1, rc,1) and the bullet subcluster (ρc,2, rc,2), and σ/m for each run. The mass profiles were truncated at 20 rc. As pointed out by C06, weak lensing is expected to underestimate the mass of the lens by 10-20% in the dense central regions. Furthermore, weak lensing can underestimate masses due to mass-sheet degeneracy, where the mass map is affected by the non-detection of mass at the edges of the field of view. The effect can be seen by comparing the total (i.e., without the gas mass subtracted) weak lensing mass estimates of C06 to the mass profiles derived by B06, which combine strong and weak lensing observations. We chose to match the projected mass profiles in the simulations to those given by B06, since strong lensing is expected to give better results in the central core regions. We are most interested in these regions since most of the particle scattering depth accumulates near the center. We explore the effects of decreasing the total mass of the system on our results in § 4.3. 2.3. Matching Simulations to Observations Columns 4 & 5 of Table 2 give the total projected mass within 150 kpc of the mass peak for the main cluster (M1(r < 150kpc)) and the subcluster (M2(r < 150kpc)) at the observed separation for each simulation run and from the lensing observations. Some trial-and-error was necessary to determine what initial conditions gave the desired projected mass profiles at the desired separation. In general, larger σ/m values required a more concentrated subcluster initially. This effect occurs because the self-interaction of the DM particles causes them to be scattered away, particularly in high density regions. Consequently, the subcluster mass profile is spread out during core passage. For the purposes of comparing the simulations with observations, we take the simulation snapshot where the offset between the clusters is closest to the observed separation. The time resolution of the snapshots was small enough to match this value to within a few kpc (< 1′′), which is within the observational error. We – 7 – require the simulated mass profiles at this moment to be consistent with the strong lensing mass map to within 10% in the inner regions (r < 500 kpc). 2.4. Stability of Simulated Halos It is of interest to evaluate the stability of our simulated clusters, particularly in the pres- ence of SIDM. In the case of non-self-interacting DM, the phase-space distribution function can be computed and used to generate a gravitationally stable King model density profile. However, in the case of SIDM, particle collisions will tend to transfer kinetic energy from one region of the cluster to another, consequently altering the density profile (see, e.g., Burkert 2000). In section § 3.2, we will draw conclusions based on the fraction of particles scattered away from the core of the subcluster due to the merger event. It is therefore necessary to determine what fraction of particles might flow from the central region of the bullet due to the instability resulting from SIDM collisions. To this end, we ran simulations of the subclus- ter DM halo, allowing it to evolve in isolation over the timescale of the merger simulations (about 1 Gyr). We used the same cluster parameters as the subcluster that has the highest central density of all the clusters in Table 1. The results are shown in Figure 3, which gives the initial density profile (solid line) and the density profile after 1 Gyr for σ/m = 0 cm2 g−1 (dotted line) and σ/m = 0.7 cm2 g−1 (dashed line). The density in the inner regions is marginally enhanced in the case of SIDM. This result is similar to the core-collapse phase seen by Burkert (2000), where weak interactions between the kinematically hot core and the cooler outer regions result in an outward transport of kinetic energy (though this effect is expected to be somewhat curtailed here due to the near isothermality of the King profile at small radii). For the purposes of the test described in § 3.2, we are only concerned with the total mass within projected radii. This quantity is plotted for each run in Figure 4. The above effect of SIDM on the projected mass profile is negligible, particularly for projected radii x ≥ 150 kpc, which is the minimum radius considered for the test described in § 3.2. Thus, if we find that a large fraction of SIDM particles scattered outside this radius, it can be assumed to be caused by the merger event as opposed to any halo instability. Further- more, the collisionless galaxies are expected to adjust to any change in the overall potential (which is dominated by the DM), thereby acting to further stabilize the mass-to-light ratio. Indeed, in these isolated subcluster runs, the mass-to-light ratios within a projected radius of 150 kpc from the cluster centers stay within 2% of their initial values, regardless of the DM self-interaction. – 8 – 3. Results 3.1. Galaxy – Dark Matter Centroid Offset For non-self-interacting DM, the centroids of the subclump DM and galaxy distributions are expected to be coincident throughout the simulation, since gravity is the only operating force. However, when σ/m > 0, the subcluster DM halo experiences a drag force as it passes through the main cluster, and subsequently lags the collisionless galaxies, just as the fluid- like subcluster gas core is observed to lag the DM halo (see Fig. 1). We ran simulations with a range of values for σ/m and calculated the centroids for each particle type by taking the average projected position of the particles in some large region, centering on this position with a smaller region, and repeating with smaller and smaller regions (down to a region with a radius of 200 kpc). Column 6 of Table 2 gives ∆x, the offset between the subcluster galaxy and DM centroids, for each run, for the moment when the subcluster is close to the observed separation of 720 kpc from the main cluster. The dependence of ∆x on σ/m is also plotted in Figure 5 (solid line). Results from the run with σ/m = 0 indicate that the offsets from the simulations are accurate to about ±2 kpc (0.5′′). It is clear from Table 2 that the centroid offset is a strong function of σ/m. An X-ray image close-up of the bullet region with error contours for the subcluster total mass and galaxy centroids overlain is shown in Figure 2. Details of the derivation of the total mass centroids are given in C06. The centroid of the galaxy distribution was calculated from the ACS photometry, using all galaxies for which the F814W-F606W color is within 0.15 mag of the red sequence. We used an Epanechnikov kernel with h = 30′′ (Merritt & Tremblay, 1994; Gonzalez et al. 2002) to determine the centroid, and a bootstrap technique to quantify the uncertainty. The centroid of the subcluster galaxies is found to be 5.7′′±6.6′′ (25 ± 29 kpc) west of the corresponding weak lensing mass peak. Given the observational errors on the centroid positions (roughly 5′′, or 22 kpc, on the subcluster mass peak and galaxy centroid), the absence of a larger offset means that σ/m < 1.25 cm2 g−1. We note that, although this upper limit is greater than the best constraint of σ/m < 1 cm2 g−1 found by M04, it is more robust, since it does not rely on the assumption that the subcluster and the main cluster had equal M/L ratios prior to the merger, as is the case with the limit from M04 (see § 3.2). This distinction is relevant since, although there is evidence for a universal M/L ratio for clusters, the level scatter for individual clusters is not negligible (see Dahle 2000). – 9 – 3.2. Subcluster M/L Ratio In a merger scenario, SIDM is expected to give a lower M/L ratio for the subcluster that has just passed through a dense core as compared to collisionless DM. This is because during the merger, DM particles are scattered away due to collisions, while the collisionless galaxies are relatively unaffected. To estimate the change in the M/L ratio in the simulations due to the merger, we simply take the ratio of the total mass to galaxy mass within 150 kpc (projected) of the bullet DM centroid and compare the values at the start of the simulations and at the observed separation. The results are tabulated in Column 7 of Table 2, which gives f , the fractional decrease in the bullet M/L ratio within 150 kpc, and also plotted in Figure 5 (dashed line). We note that for σ/m ≈ 1 cm2 g−1, the subcluster loses about 38% of its mass within 150 kpc, which is in agreement with a conservative estimate of 20 - 30% given by M04. As expected, the numerical results yield somewhat tighter constraints on σ/m as compared to the analytic estimates when using the same method and observational constraints. Using the latest lensing mass map from B06, we rederived M/L ratios for each of the two subcluster within a projected 150 kpc of the total mass peaks (for previous results see Clowe et al., 2004). For the subcluster, the mass contribution from the outskirts of the main cluster has been approximately subtracted, whereas for the main cluster, the total mass is used, since the contribution from the subcluster is negligible. The projected mass contribution from the main cluster to the subcluster is estimated by taking the average mass in an annulus at the distance of the subcluster (excluding the region of the subcluster itself). This gives a conservative estimate for the upper limit on σ/m, since scattering due to putative DM collisions is expected to result in an anomalously low M/L value for the subcluster as compared to the main cluster, and by reducing the observed mass of the subcluster we minimize the effect of the collisions that we want to constrain. We find M/LB = 471± 28, 422± 25 and M/LI = 179± 11, 214± 13 for the subcluster and the main cluster, respectively (for a discussion of the errors on the mass measurements, see B06). The ratios agree with one another to within about the 68% confidence intervals. From the I band data, we find that the ratio of M/L ratios of the subcluster and main cluster is 0.84± 0.07. We conservatively choose to use the I band data only, since we want to put a firm lower limit on this ratio, andM/LB is larger for the subcluster than for the main cluster. Assuming each cluster started out with similar M/L values, which appears to be a reasonable assumption for clusters in general (e.g., Mellier 1999; Dahle 2000), we conclude that the subcluster could not have lost more than ∼ 23% of its initial mass. A comparison with the results from simulations plotted in Figure 5 (dashed line) shows that this implies σ/m . 0.6 cm2 g−1, which is a slight improvement over the previous best limit of σ/m . 1 cm2 g−1 from the conservative estimates of M04. – 10 – 3.3. Structure in Subcluster Dark Matter Distribution M04 suggested that scattered DM particles, which would account for about 1/5 of the total subcluster mass, might form tail features in the DM distribution, similar to the tails seen in the X-ray image of the gas bullet (see Figure 1). The simulations allow us to determine whether the non-observation of such tails in the mass map could be used to constrain σ/m. We find that, rather than forming a tail, the scattered particles are mostly deposited in the core of the main cluster, and do not form any features at a level that is interesting for constraining σ/m. 4. Discussion 4.1. Non-zero Impact Parameter As M04 argue, the morphology of the X-ray image, in particular, the symmetry of the North-South X-ray bar (most likely an oblate spheroid viewed edge-on) between the main cluster and subcluster mass peaks around the axis of symmetry set by the shape of the X-ray bullet (which gives its present velocity direction), combined with the line-of-sight velocity and X-ray derived Mach number, indicate a merger axis that is ∼ 10◦ from the plane of the sky, and that the cluster cores must have passed close to one another, certainly within the ∼ 200 kpc core radius of the main cluster. In all simulations previously discussed, it was assumed that the bullet subcluster passed directly through the center of the main cluster core, i.e., that the impact parameter of the merger, b, is zero. For b > 0, we expect that the effects of self-interacting DM will be reduced, since the density is at a maximum when the core centers pass directly through one another, and the scattering probability is proportional to the density (Eqn. 1). To test the strength of this effect, we re-ran the simulation R4 (see Table 1) with an impact parameter of b = 200 kpc. Aside from the impact parameter, the initial mass and velocity distributions were identical to those for run R4, so that the relative effects of b > 0 could be investigated (specifically, no adjustments were made to the initial conditions to more closely match the current observed mass profiles). The resulting projected total mass profiles for the subcluster within 150 kpc and 250 kpc at the observed separation agreed with those from the b = 0 run to within 4%. For the main cluster, the match was better than 1%. The resulting offset between the galaxy and DM centroids during the post core passage phase was systematically smaller than the offset seen in the b = 0 run. At the observed separation, the difference in the centroid offsets, as compared to the b = 0 run, was about 4 kpc, which is on the order of both the observational error and the accuracy of our numerical – 11 – technique. The fractional change in the M/L ratio of the subcluster was similarly affected; for the b = 0 case, the M/L ratio within 150 kpc drops by about 27%, whereas for the run with b = 200 kpc, it drops by 22%. Assuming, as we did in § 3.2, that the subcluster could not have lost more than ∼ 23% of its initial mass, we find the constraint that σ/m < 0.7 cm2 g−1. We therefore conclude that, although a non-zero impact parameter reduces the effects of self- interacting DM as expected, the level of the effect is relatively small. This is likely due to the assumed King mass profile of the main cluster. The radial density gradient is relatively small within the core radius of the main cluster (which in this case is 151 kpc), so it is not surprising that the effects of self-interacting DM are not significantly reduced by increasing the impact parameter, so long as it is comparable to the core radius of the main cluster. Naturally, the effects of a non-zero b would be increased if the main cluster had a strongly peaked mass profile, though the current lensing data suggest otherwise. We conclude that any value of impact parameter that is consistent with the observations will only slightly alter our results. 4.2. Alternative Bullet Mass Profiles As noted in §2, the choice of a King mass profile for the subcluster is expected to give conservative estimates on the effects of collisional DM, based on the M/L ratio, since the central density is low as compared to models with cuspy cores such as NFW and Hernquist models. However, in the case of the galaxy/DM centroid offset test, one might argue that since the subcluster galaxies are more tightly bound in the center for more highly concen- trated mass profiles, it will be more difficult to displace them, which could lead to a smaller offset between the centroids despite the increased action of DM collisions. We therefore ran a test simulation, using a King model for the main cluster and a Hernquist model for the subcluster, with σ/m = 0.72 cm2 g−1. The Hernquist profile is given by ρ(r) = (r + a)3 , (2) whereM is the total cluster mass and a is the scale length (Hernquist 1990). As before, initial parameters were chosen such that the bullet mass profile roughly matches the observed profile at the current separation (we used M = 3.13× 1014 M⊙, a = 100 kpc). This is expected to be the most conservative model combination for this test, since the main cluster King model minimizes the effects of DM self-interaction while the subcluster Hernquist model maximizes the binding energy of the subcluster galaxies. The results show that, when comparing to run R4 in Table 2, the galaxy/DM centroid offset was only slightly less than that found with the King model subcluster, on the order of the accuracy of the simulation offset values (less than – 12 – 1 kpc). The change in the subcluster’s M/L was similarly only weakly affected (f = 0.27 for the King model bullet subcluster, whereas for the Hernquist model we find f = 0.31, consistent with the King profile being the conservative case). The agreement is likely due to the fact that the centroids become offset from one another after core passage, and it is during core passage that the central density peak of the bullet is mostly “smoothed away” due to DM collisions (recall that DM scattering is more frequent in high density regions, so high density structures are more efficiently destroyed by DM self-interactions). Although strongly peaked density profiles have been found to be unstable to SIDM (e.g., Burkert 2000; Yoshida et al. 2000b), in our simulation a significant change in density only occurred at small radii, such that the total projected mass of the subcluster within 50 kpc remained stable up until the merger event. Therefore the subcluster mass distribution remained significantly more peaked than a King profile cluster with the same projected mass within 150 kpc. We conclude that our results are only weakly dependent on the mass profile chosen for the bullet, so long as we require that the observed mass profile is reproduced. For the initially more centrally concentrated profiles, the effects of the increased binding energy in the core are balanced by the increased scattering frequency in this region. 4.3. Mass Profile Dependence As mentioned in § 2, weak lensing is expected to underestimate the mass of the lens by 10-20%. There are two separate effects that contribute to this underestimation. First, near the core of a cluster there is a large region without weak lensing galaxies, and this region is effectively smoothed over when computing the mass map. Additionally, galaxies near the regions where strong lensing dominates are measured in the weak lensing approximation, which also leads to an underestimate of the mass in the core. Second, the total cluster mass can be underestimated due to mass-sheet degeneracy, where the mass-map is affected by the non-detection of mass at the edge of the field of view. Although projection of foreground and/or background structures unassociated with the clusters will artificially increase the mass, it is highly unlikely that such projected structures significantly contributed to the de- tected lensing signal (C06). Results from strong lensing, which is not susceptible to the same systematic underestimation as weak lensing, do indeed give systematically higher projected masses for this system, by about a factor of 2 within the inner few hundred kpc, which is the region we are most interested in for this analysis (compare B06 and C06; see C06 for further discussion of this discrepancy). Though we chose to use the mass estimates from strong lensing, since it should give a more reliable estimate of the projected mass near the cluster cores, it is interesting to explore the dependence of our results on the lensing mass estimates. To this end, we conducted a simulation run similar to run R4, but with the initial cluster – 13 – central densities chosen such that the projected mass profiles at the observed separation were about 2 times lower than the masses derived from strong lensing observations, roughly in agreement with the weak lensing results given by C06 (the initial core radii were the same as for run R4). Since the scattering probability depends on the density, we expect these less massive halos to be more weakly affected by SIDM. Results obtained from the simulations with the lower mass normalization show that the effects of SIDM are diminished, as expected for a linear dependence of the scattering probabilities on the projected mass. In run R4, the M/L ratio dropped by 27%, whereas for the run with 1/2 the total mass it dropped by 14% (roughly a factor of 2 less). Similarly, the galaxy/DM centroid offset was 11.1 kpc, again, about a factor of 2 down from the 24.1 kpc offset seen in run R4. If we assume that this factor of two effect can be applied to all of values given in columns 6 & 7 of Table 2 and consider the more sensitive M/L test, we find that requiring f . 0.20 would correspond to σ . 1.25 cm2 g−1. This is done as a test of the method only, since these low halo mass values are not realistic, as they are insufficient to produce the system of strong arcs observed in the HST images (C06). 4.4. Low Merger Velocity All of the simulations discussed so far have assumed a merger velocity that is consistent with that derived from X-ray observations (Markevitch, 2005), which give a Mach number for the shock front of M = 3.0±0.4, and it is assumed that the subcluster has the same velocity as the shock (though see Springel & Farrar, 2007). Since the subcluster could have slowed down, or the shock front accelerated, it is interesting to ask what effect a lower velocity would have on the inferred upper limit on σ/m, particularly since the observed velocity is larger than would be expected from free fall of the subcluster onto the main cluster (Farrar & Rosen 2007). In order to test the dependence of our upper limit on merger velocity, we ran a simulation with σ/m = 0.72 cm2 g−1 such that the relative velocity of the cluster DM halos at observed separation was 1.5 times lower, about 3100 km s−1 (M ≈ 2). This is close to the expected free-fall velocity of the subcluster, and to the relative velocity of 2860 km s−1 found by Springel & Farrar (2007) from hydrodynamical simulations of this system. The results showed little difference from the higher velocity run (compare to run R4 in Table 2): ∆x was 30.2 kpc (vs. 24.1 kpc) and f was 0.25 (vs. 0.27). We therefore conclude that our results are relatively insensitive to merger velocities that are not in large disagreement with the observations. A weak dependence of the M/L ratio on the subcluster velocity v is easy to understand: the particle scatters out of the subcluster as long as v/2 is much greater than the escape velocity from the subcluster, which it is by a large margin (M04). – 14 – 4.5. Effects of Diffuse Gas As mentioned in § 2, the intracluster gas observed in the X-ray band was not included in the simulations (doing so would greatly increase the computing time and the complexity involved with matching the observations in detail). The only way for the gas to affect the results is via gravitational interaction (we ignore the possibility of non-gravitational baryon- DM interactions, the cross-section of which has been shown to be extremely small, e.g., Chen et al. 2002). In general, the gas is expected to contribute about 10% of the total mass of the system, a figure which appears to be consistent with the lensing and X-ray observations (B06). One might worry that, when matching the observed mass profiles, some “extra” DM is needed to account for the missing gas. As mentioned in § 2, gas masses have been subtracted from the lensing masses using a detailed model of the gas distribution derived from fitting the X-ray observations. In terms of the test involving the decrease in the subcluster M/L ratio (see § 3.2), we needn’t worry about the subcluster gas for the simple reason that the gas bullet is far from the subcluster mass peak (roughly 23′′, or 102 kpc). Therefore, the gas in the region of the bullet mass peak is not centrally concentrated and will not significantly add to the binding energy of potentially scattered DM particles. For the galaxy and total mass centroid offset test (see § 3.1), the exclusion of the gas is expected to give a conservative result: the gas bullet and bar feature seen in Figure 1 will act to decelerate the subcluster DM halo and galaxies. However, if, as is the case with SIDM, the DM halo starts to lag behind the galaxies and gets closer to the gas cores than the main concentration of the galaxies, it will experience a larger deceleration, thereby increasing the offset between the two. Due to the relatively low mass of the gas components, and the large distance between the gas peaks and the subcluster DM halo and galaxies (as compared to the offset of the latter), the strength of this effect will be quite small. We therefore conclude that including gas in the simulations would not significantly affect our results. 5. Summary We have combined results from new X-ray, optical and lensing observations and our N-body simulations of the merging galaxy cluster 1E 0657-56 in order to derive an upper limit on the self-interaction cross-section of dark matter particles, σ/m. We give constraints on σ/m based on two independent methods: from the lack of offset between the total mass peak and galaxy centroid of the subcluster that would arise during the merger due to drag on the subcluster halo from DM particle collisions, and from the lack of a decreased mass- to-light ratio of the subcluster due to scattering of DM particles. From the former, we find σ/m < 1.25 cm2 g−1, and from the latter, σ/m < 0.7 cm2 g−1, which includes the – 15 – uncertainty in the impact parameter of the merger (upper limits are from 68% confidence intervals). Our best constraint is a modest improvement of the previous best constraint from conservative analytic estimates of σ/m < 1 cm2 g−1 (M04). Furthermore, our limit of σ/m < 1.25 cm2 g−1 is more robust than the best analytic limit, since this method does not depend on the assumption that the subcluster and main cluster M/L ratios were equal prior to the merger. Previous studies have found that σ/m ∼ 0.5−5 cm2 g−1 is needed produce the observational effects that self-interacting dark matter has been invoked to explain (e.g., non- peaked galaxy mass profiles and the underabundance of small halos within larger systems). Our results rule out almost this full range of values, at least under the assumption that σ is velocity-independent. We would like to thank Volker Springel, Naoki Yoshida, Yago Ascasibar, and Alexey Vikhlinin for useful discussions and for providing access to various private codes. Simulations were performed on a Beowulf cluster at the ITC in the Harvard-Smithsonian Center for Astrophysics. Support for this work was partially provided for by the NASA Chandra grants G04-5152X and TM6-7010X, and NASA contract NAS8-39073. REFERENCES Ahn, K., & Shapiro, P.R. 2003, J. Korean Astronomical Soc., 36, 89 Ahn, K., & Shapiro, P.R. 2005, MNRAS, 363, 1092 Barrena, R., Biviano, A., Ramella, M., Falco, E. E., & Seitz, S. 2002, A&A, 386, 816 Bradač, M., Clowe, D., Gonzalez, A. H., Marshall, P., Forman, W., Jones, C., Markevitch, M., Randall, S., Schrabback, T., & Zaritsky, D. 2006, ApJ, 652, 937 (B06) Burkert, A. 2000, ApJ, 534, L143 Chen, X., Hannestad, S., & Scherrer, R. J. 2002, Phys. Rev. D65, 123515 Clowe, D., Gonzalez, A. H., & Markevitch M. 2004, ApJ, 604, 596 Clowe, D., Bradač, M., Gonzalez, A. 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N., & Steinhardt, P. J. 2000, Phys. Rev. Lett., 84, 3760 Springel, V., & Farrar, G. 2007 (in press, astro-ph/0703232) Springel, V. 2005, MNRAS, 364, 1105 Yoshida, N., Springel, V., White, S. D. M., Tormen, G. 2000a, 535, L103 Yoshida, N., Springel, V., White, S. D. M., Tormen, G. 2000b, 544, L87 This preprint was prepared with the AAS LATEX macros v5.2. http://arxiv.org/abs/astro-ph/0511345 http://arxiv.org/abs/astro-ph/0703232 – 17 – Table 1. Initial Simulation Parameters Run Name NDM σ/m ρc,1 rc,1 ρc,2 rc,2 (cm2 g−1) (106 M⊙ kpc −3) (kpc) (106 M⊙ kpc −3) (kpc) R1 106 0 3.27 213 4.59 149 R2 106 0.24 3.27 213 4.59 149 R3 106 0.48 4.42 183 6.57 129 R4 106 0.72 7.03 151 11.75 108 R5 106 0.96 6.26 167 9.76 124 R6 106 1.25 6.26 167 9.76 124 – 18 – Table 2. Conditions at Observed Separation Run Name NDM σ/m M1(r < 150kpc) M2(r < 150kpc) ∆ x (cm2 g−1) (1013 M⊙) (10 13 M⊙) (kpc) R1 106 0 12.0 11.1 1.8 0.0 R2 106 0.24 11.5 10.4 5.4 0.08 R3 106 0.48 11.8 10.4 15.0 0.16 R4 106 0.72 12.6 11.0 24.1 0.27 R5 106 0.96 12.4 10.9 37.9 0.32 R6 106 1.25 11.4 9.8 53.9 0.38 Obs. 11.9±1.6 10.6±0.4 25± 29 0.16± 0.07 a∆ x is the offset between the subcluster total mass and galaxy centroids. bf is the fractional decrease in the mass-to-light ratio of the subcluster within 150 kpc. – 19 – Fig. 1.— X-ray image with weak lensing mass contours overlain. The gas bullet lags the subcluster DM halo. The current separation of the subcluster and main cluster mass peaks is 720 kpc. – 20 – Fig. 2.— Close up of the subcluster bullet region, with the DM (blue) and galaxy (red) centroid error contours overlain. The contours show the 68.3% and 99.7% error regions. The left panel shows the X-ray Chandra image, while the right shows the optical HST image. – 21 – Fig. 3.— Density profile of an isolated King model cluster at t = 0 (solid line), and after evolving for 1 Gyr with σ/m = 0 (dotted line) and σ/m = 0.7 cm2 g−1 (dashed line). – 22 – Fig. 4.— Total mass within projected radius x for the cluster plotted in Figure 3. Line-type indications are the same as in Figure 3. – 23 – Fig. 5.— The dependence of the subcluster galaxy and total mass centroid offset (∆ x, solid line) and the fractional change in the subcluster M/L ratio (f , dashed line) on σ/m. Based on the values given in Table 2. Introduction The Simulations Simulation Code and Parameters Initial Conditions Matching Simulations to Observations Stability of Simulated Halos Results Galaxy – Dark Matter Centroid Offset Subcluster M/L Ratio Structure in Subcluster Dark Matter Distribution Discussion Non-zero Impact Parameter Alternative Bullet Mass Profiles Mass Profile Dependence Low Merger Velocity Effects of Diffuse Gas Summary

0704.0262

Stringy Instantons at Orbifold Singularities

arXiv:0704.0262v3 [hep-th] 18 Jun 2007 Preprint typeset in JHEP style. - HYPER VERSION SISSA-16/2007/EP Stringy Instantons at Orbifold Singularities Riccardo Argurio1, Matteo Bertolini2, Gabriele Ferretti3, Alberto Lerda4 and Christoffer Petersson3 1Physique Théorique et Mathématique and International Solvay Institutes Université Libre de Bruxelles, CP 231, 1050 Bruxelles, Belgium 2SISSA/ISAS and INFN - Sezione di Trieste Via Beirut 2; I-34014 Trieste, Italy 3Department of Fundamental Physics Chalmers University of Technology, 412 96 Göteborg, Sweden 4Dipartimento di Scienze e Tecnologie Avanzate Università del Piemonte Orientale, I-15100 Alessandria, Italy Istituto Nazionale di Fisica Nucleare - sezione di Torino, Italy [email protected], [email protected], [email protected], [email protected], [email protected] Abstract: We study the effects produced by D-brane instantons on the holomorphic quantities of a D-brane gauge theory at an orbifold singularity. These effects are not limited to reproducing the well known contributions of the gauge theory instantons but also generate extra terms in the superpotential or the prepotential. On these brane instantons there are some neutral fermionic zero-modes in addition to the ones expected from broken supertranslations. They are crucial in correctly reproducing effects which are dual to gauge theory instantons, but they may make some other interesting contributions vanish. We analyze how orientifold projections can remove these zero-modes and thus allow for new superpotential terms. These terms contribute to the dynamics of the effective gauge theory, for instance in the stabilization of runaway directions. Keywords: Instantons, D-branes. http://arxiv.org/abs/0704.0262v3 mailto:[email protected], [email protected], [email protected], [email protected], [email protected] http://jhep.sissa.it/stdsearch?keywords=Instantons+D-branes Contents 1. Introduction 1 2. Preliminaries 3 3. The N = 1 Z2 × Z2 orbifold 7 3.1 Instanton sector 10 3.2 Recovery of the ADS superpotential 11 3.3 Absence of exotic contributions 14 3.4 Study of the back-reaction 16 4. The N = 1 Z2 × Z2 orientifold 17 4.1 Instanton sector 19 5. An N = 2 example: the Z3 orientifold 21 5.1 Instanton sector 24 6. Conclusions 25 1. Introduction It has long been realized that instantons in string theory are often in close correspon- dence with instantons in gauge theories [1, 2, 3, 4, 5, 6]. Recently it was found that in some situations stringy instantons can dynamically generate some terms which from a low-energy effective point of view enter as ordinary external couplings in the superpo- tential of gauge theories living on space-filling branes [7, 8, 9, 10, 11, 12, 13, 14]. By instantons in string theory we generally mean instantons which are geometrically real- ized as Euclidean extended objects wrapped on some non-trivial cycles of the geometry. Thus, in a sense, a stringy instanton has a “life of its own”, not requiring an underlying gauge theory. This opens up the possibility of having contributions originating from instantons that do not admit a standard gauge theory realization. We shall refer to these instantons as exotic. There has been some debate in the recent literature about the instances where such exotic instantons can actually contribute to the gauge theory superpotential in a non-trivial manner. In this work we will contribute to such a debate by considering backgrounds where a simple CFT description is possible, such as orbifolds or orientifolds thereof. We present various simple examples of what we believe to be a rather generic situation. Namely, the presence of extra zero-modes for these instantons, in addition to those required by the counting of broken symmetries, makes some of their contributions vanish. Such extra zero-modes should not come as a surprise, since a D-brane instanton in a CY manifold breaks a total of four out of eight supercharges, i.e. it has two extra fermionic zero-modes from the point of view of holomorphic N = 1 gauge theory quantities. We give some arguments as to why the backreaction of the space-filling branes on the geometry might not help in lifting these extra zero-modes. We further argue that only more radical changes of the background, such as the introduction of fluxes, deformations of the CY geometry or the introduction of orientifold planes, can remove these zero-modes. When this happens, exotic instantons do contribute to the gauge theory superpotential and may provide qualitative changes in the low energy effective dynamics, as for instance the stabilization of otherwise runaway directions. We will be interested in Euclidean D-branes in type II theories. We will work with IIB fractional branes at orbifold and orientifold singularities rather than type IIA wrapped branes. The motivation for this choice of setting is two-fold. First, recent advances in the gauge/gravity correspondence require the study of exotic instantons, whose effects tend to stabilize the gauge theory rather than unstabilize it [15, 16, 9, 17], and the gauge/gravity correspondence is more naturally defined in the context of IIB theory. Second, similar effects are used in string phenomenology to try to understand possible mechanisms for neutrino masses [7, 8, 13]. This latest activity is mainly done in the type IIA scenario, but we find it easier to address some subtle issues in the IIB orbifold case. While working in an exact string background, our considerations will nonetheless be only local, i.e. we will not be concerned with global issues such as tadpole cancellation that arise in proper compactifications. This is perfectly acceptable in the context of the gauge/gravity correspondence where the internal manifold is non-compact but, even for string phenomenology, the results we obtain stand (locally) when properly embedded in a consistent compactification. The paper is organized as follows: In section 2 we set up the notation and discuss some preliminary material. In section 3 we discuss our first case, namely the N = 1 Z2 × Z2 orbifold. After briefly recovering the usual instanton generated corrections to the superpotential we discuss the possible presence of additional exotic contributions and find that they are not present because of the additional zero-modes. We conclude by giving a CFT argument on why such zero-modes are not expected to be lifted even by taking into account the backreaction of the D-branes, unless one is willing to move out the orbifold point in the CY moduli space. Sections 4 and 5 present two separate instances where exotic contributions are present after having removed the extra zero-modes by orientifolding. The first is an N = 1 orientifold, the second is an N = 2 orientifold, displaying corrections to the superpotential and the prepotential, respectively. We end with some conclusions and a discussion of further developments. 2. Preliminaries In this section we briefly review the generic setup in the well understoodN = 4 situation in order to introduce the notation for the various fields and moduli and their couplings. The more interesting theories we will consider next will be suitable projections of the N = 4 theory. In fact, the exotic cases can all be reduced to orbifolds/orientifolds of this master case once the appropriate projections on the Chan-Paton factors are performed. Since we are interested in instanton physics (for comprehensive reviews see [18] and the recent [19]) we will take the ten dimensional metric to be Euclidean. We consider a system where both D3-branes and D(−1)-branes (D-instantons) are present. To be definite, we take N D3’s and k D-instantons 1. Quite generically we can distinguish three separate open string sectors: • The gauge sector, made of those open strings with both ends on a D3-brane. We assume the brane world-volumes are lying along the first four coordinates xµ and are orthogonal to the last six xa. The massless fields in this sector form an N = 4 SYM multiplet [22]. We denote the bosonic components by Aµ and X a. Written in N = 1 language this multiplet is formed by a gauge superfield whose field strength is denoted by Wα and three chiral superfields Φ 1,2,3. With a slight abuse of notation, the bosonic components of the chiral superfields will also be denoted by Φ, i.e. Φ1 = X4+ iX5 and so on. In N = 2 language we have instead a gauge superfield A and a hypermultiplet H , all in the adjoint representation. The low energy action of these fields is a four dimensional N = 4 gauge theory. All these fields are N ×N matrices for a gauge group SU(N). 1These D3/D(−1) brane systems (and their orbifold projections) are very useful and efficient in studying instanton effects from a stringy perspective even in the presence of non-trivial closed string backgrounds, both of NS-NS type [20] and of R-R type [21]. • The neutral sector, which comprises the zero-modes of strings with both ends on the D-instantons. It is usually referred to as the neutral sector because these modes do not transform under the gauge group. The zero-modes are easily ob- tained by dimensionally reducing the maximally supersymmetric gauge theory to zero dimensions. We will use an ADHM [23] inspired notation [5, 6]. We denote the bosonic fields as aµ and χ a, where the distinction between the two is made by the presence of the D3-branes. The fermionic zero-modes are denoted by MαA and λα̇A, where α and α̇ denote the (positive and negative) four dimensional chi- ralities and A is an SU(4) (fundamental or anti-fundamental) index denoting the chirality in the transverse six dimensions. The ten dimensional chirality of both fields is taken to be negative. In Euclidean space M and λ must be treated as independent. When needed, we will also introduce the triplet of auxiliary fields Dc, directly analogous to the four dimensional D, that can be used to express the various interactions in an easier form as we will see momentarily. All these fields are k × k matrices where k is the instanton number. • The charged sector, comprising the zero-modes of strings stretching between a D3-brane and a D-instanton. For each pair of such branes we have two conjugate sectors distinguished by the orientation of the string. In the NS sector, where the world-sheet fermions have opposite modding as the bosons, we obtain a bosonic spinor ωα̇ in the first four directions where the GSO projection picks out the neg- ative chirality. In the conjugate sector, we will get an independent bosonic spinor ω̄α̇ of the same chirality. Similarly, in the R sector, after the GSO projection we obtain a pair of independent fermions (one for each conjugate sector) both in the fundamental of SU(4) which we denote by µA and µ̄A. These fields are rectangular matrices N × k and k ×N . The couplings of the fields in the gauge sector give rise to a four dimensional gauge theory. The instanton corrections to such a theory are obtained by constructing the Lagrangian describing the interaction of the gauge sector with the charged sector zero- modes while performing the integral over all zero-modes, both charged and neutral. A crucial point to notice and which will be important later is that while the neutral modes do not transform under the gauge group, their presence affects the integral because of their coupling to the charged sector. The part of the interaction involving only the instanton moduli is well known from the ADHM construction and it is essentially the reduction of the interacting gauge Lagrangian for these modes in a specific limit where the Yukawa terms for λ and the quadratic term for D are scaled out (see [18, 6] for details). The final form of this part of the interaction is: S1 = tr − [aµ, χ + χaω̄α̇ω α̇χa + (Σ̄a)ABµ̄ AµBχa − (Σ̄a)ABM αA[χa,M µ̄Aωα̇ + ω̄α̇µ A + σ βα̇[M βA, aµ] λα̇A − iD ω̄α̇(τ c) α̇ωβ̇ + iη̄ µν [a µ, aν ] (2.1) where the sum over colors and instanton indices is understood. τ denotes the usual Pauli matrices, η̄ (and η) the ’t Hooft symbols and Σ̄ (and Σ) are used to construct the six-dimensional gamma-matrices Σ̄a 0 . (2.2) The above interactions can all be understood in terms of string diagrams on a disk with open string vertex operators inserted at the boundary in the α′ → 0 limit. The interaction of the charged sector with the scalars of the gauge sector can be worked out in a similar way and yields S2 = tr ω̄α̇X (Σ̄a)ABµ̄ . (2.3) Let us rewrite the above action in a way which will be more illuminating in the following sections. Since we will be mainly focusing on situations where we have N = 1 super- symmetry, it is useful to write explicitly all indices in SU(4) notation, and then break them into SU(3) representations. We thus write the six scalars Xa as the antisymmetric representation of SU(4) as follows XAB = −XBA ≡ (Σ̄ a)ABXa . (2.4) The action S2 then reads S2 = tr ǫABCDω̄α̇XABXCDω µ̄AXABµ . (2.5) Splitting now the indices A into i = 1 . . . 3 and 4, we can identify Φ i ≡ Xi4 in the 3̄ of SU(3) and Φi ≡ 1 ǫijkXjk in the 3 of SU(3). Thus we can rewrite the action (2.5) as S2 = tr ωα̇ + ǫijkµ̄ iΦjµk . (2.6) In the above form, it is clear which zero-modes couple to the holomorphic superfields and which others couple to the anti-holomorphic ones. This distinction will play an important role later. The main object of our investigation is the integral of e−S1−S2 over all moduli Z = C d{a, χ,M, λ,D, ω, ω̄, µ, µ̄} e−S1−S2 , (2.7) where we have lumped all field independent normalization constants (including the instanton classical action and the appropriate powers of α′ required by dimensional analysis) into an overall coefficient C. There are, of course, other interactions involving the fermions and the gauge bosons but, as far as the determination of the holomorphic quantities are concerned, they can be obtained from the previous ones and supersym- metry arguments. For example, a term in the superpotential is written as the integral over chiral superspace dx4dθ2 of a holomorphic function of the chiral superfields, but such a function is completely specified by its value for bosonic arguments at θ = 0. Thus, if we can “factor out” a term dx4dθ2 from the moduli integral (2.7), whatever is left will define the complex function to be used in the superpotential and similarly for the prepotential in the N = 2 case if we succeed in factoring out an integral over N = 2 chiral superspace dx4dθ4. The coordinates x and θ must of course come from the (super)translations bro- ken by the instanton and they will be associated to the center of mass motion of the D-instanton, namely, xµ = tr aµ and θαA = trMαA for some values of A.2 One must pay attention however to the presence of possible additional neutral zero-modes coming either from the traceless parts of the above moduli or from the fields λ and χ. These modes must also be integrated over in (2.7) and their effects, as we shall see, can be quite dramatic. In particular, the presence of λ in some instances is crucial for the implementation of the usual ADHM fermionic constraints whereas in other circum- stances it makes the whole contribution to the superpotential vanish. These extra λ zero-modes are ubiquitous in orbifold theories and generically make it difficult to obtain exotic instanton corrections for these models. As we shall see, they can however be easily projected out by an orientifold construction making the derivation of such terms possible. In the full expression for the instanton corrections there will also be a field-inde- pendent normalization factor coming from the one-loop string diagrams and giving for instance the proper gYM dependence in the case of the usual instanton corrections. In this paper we will only focus on the integral over the zero-modes, which gives the proper field-dependence, referring the reader to [10, 11] for a discussion of these other issues. 2Obviously, for the case of an anti-instanton, the roles of M and λ are reversed. 3. The N = 1 Z2 × Z2 orbifold In order to present a concrete example of the above discussion, let us study a simple C3/Z2 × Z2 orbifold singularity. The resulting N = 1 theory is a non-chiral four-node quiver gauge theory with matter in the bi-fundamental. Non-chirality implies that the four gauge group ranks can be chosen independently [24]. This corresponds to being able to find a basis of three independent fractional branes in the geometry (for a review on fractional branes on orbifolds see e.g. [25]). The field content can be conveniently summarized in a quiver diagram, see Fig. 1, which, together with the cubic superpotential W = Φ12Φ23Φ31 − Φ13Φ32Φ21 + Φ13Φ34Φ41 − Φ14Φ43Φ31 +Φ14Φ42Φ21 − Φ12Φ24Φ41 + Φ24Φ43Φ32 − Φ23Φ34Φ42 , (3.1) uniquely specifies the theory. SU(N ) SU(N ) SU(N )SU(N ) Figure 1: Quiver diagram for the Z2 × Z2 orbifold theory. Round circles correspond to SU(Nℓ) gauge factors while the lines connecting quiver nodes represent the bi-fundamental chiral superfields Φℓm. A stack of N regular D3-branes amounts to having one and the same rank assign- ment on the quiver. The gauge group is then SU(N)4 and the theory is anN = 1 SCFT. Fractional branes correspond instead to different (but anomlay free) rank assignments. Quite generically, fractional branes can be divided into three different classes, depend- ing on the IR dynamics they trigger [26]. The non-chiral nature and the particularly symmetric structure of the orbifold under consideration allows one to easily construct any such instance of fractional brane class. If we turn on a single node, we are left with a pure SU(N) SYM gauge theory, with no matter fields and no superpotential. This theory is believed to confine. The geometric dual effect is that the corresponding fractional brane leads to a geometric transition where the branes disappear leaving behind a deformed geometry. Indeed, there is one such deformation in the above singularity. Turning on two nodes leads already to more varied phenomena. There are now two bi-fundamental superfields, but still no tree level superpotential. Thus, the system is just like two coupled massless SQCD theories or, by a slightly asymmetric point of view, massless SQCD with a gauged diagonal flavor group. The low-energy behavior depends on the relative ranks of the two nodes. If the ranks are different, the node with the highest rank is in a situation where it has less flavors than colors. Then an Affleck-Dine-Seiberg (ADS) superpotential [27, 28] should be dynamically generated, leading eventually to a runaway behavior. This set up of fractional branes is sometimes referred to as supersymmetry breaking fractional branes [29, 26, 30]. If the ranks are the same we are in a situation similar to Nf = Nc SQCD for both nodes. Hence we expect to have a moduli space of SUSY vacua, which gets deformed, but not lifted, at the quantum level. This moduli space is roughly identified in the geometry with the fact that the relevant fractional branes are interpreted as D5-branes wrapped on the 2-cycle of a singularity which is locally C× (C2/Z2). Such a fractional brane can move in the C direction. This is what is called an N = 2 fractional brane since, at least geometrically, it resembles very much the situation of fractional branes at N = 2 singularities. In what follows we use the two-node example as a simple setting in which we can analyze the subtleties involved in the integration over the neutral modes. For the gauge theory instanton case it is known that there are extra neutral fermionic zero-modes in addition to those required to generate the superpotential. Their integration allows to recover the fermionic ADHM constraints on the moduli space of the usual field theory instantons. For such instantons, we will be able to obtain the ADS superpotential and corresponding runaway behavior in the familiar context with Nc and Nf fractional branes at the respective nodes, for Nf = Nc−1. On the other hand, we will argue that the presence of such extra zero-modes rules out the possibility of having exotic instanton effects, such as terms involving baryonic operators in the Nf = Nc case. It was the desire to study such possible contributions that constituted the original motivation for this investigation. We will first show that such effects are absent for this theory as it stands, and we will later discuss when and how this problem can be cured.3 3In a situation where the CFT description is less under control than in the setting discussed in the present paper, it has been argued in [17] that such baryonic couplings do arise in the context of fractional branes on orbifolds of the conifold, possibly at the expense of introducing O-planes. Also in a IIA set up similar to the ones of [7, 8, 10, 11, 13] it seems reasonable that one can wrap an ED2-brane along an O6-plane and produce such couplings on other intersecting D6-branes. Our orbifold theory can be easily obtained as an orbifold projection of N = 4 SYM. The orbifolding procedure and the derivation of the superpotential (3.1) are by now standard. We briefly recall the main points in order to fix the notation and because some of the details will be useful later in describing the instantons in such a set up. The group Z2 × Z2 has four elements: the identity e, the generators of the two Z2 that we denote with g1 and g2 and their product, denoted by g3 = g1g2. If we introduce complex coordinates (z1, z2, z3) ∈ C z1 = x4 + ix5 , z2 = x6 + ix7 , z3 = x8 + ix9 (3.2) the action of the orbifold group can be defined as in Table 1. z1 z2 z3 e z1 z2 z3 1 −z2 −z3 g2 −z 1 z2 −z3 g3 −z 1 −z2 z3 Table 1: The action of the orbifold generators. Let γ(g) be the regular representation of the orbifold group on the Chan-Paton factors. If the orbifold is abelian, as always in the cases we shall be interested in, we can always diagonalize all matrices γ(g). We will assume that the two generators have the following matrix representation γ(g1) = σ3 ⊗ 1 = 1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −1 , γ(g2) = 1⊗ σ3 = 1 0 0 0 0 −1 0 0 0 0 1 0 0 0 0 −1 (3.3) where the 1’s denote Nℓ ×Nℓ unit matrices (ℓ = 1, ..., 4). Then, the orbifold projection amounts to enforcing the conditions Aµ = γ(g)Aµγ(g) −1 , Φi = ±γ(g)Φiγ(g)−1 (3.4) where the sign ± must be chosen according to the action of the orbifold generators g that can be read off from Table 1. With the choice (3.3), the vector superfields are block diagonal matrices of different size (N1, N2, N3, N4), one for each node of the quiver, while the three chiral superfields Φi have the following form [24] 0 × 0 0 × 0 0 0 0 0 0 × 0 0 × 0 , Φ2 = 0 0 × 0 0 0 0 × × 0 0 0 0 × 0 0 , Φ3 = 0 0 0 × 0 0 × 0 0 × 0 0 × 0 0 0 , (3.5) where the crosses represent the non-zero entries Φℓm appearing in the superpotential (3.1). 3.1 Instanton sector Now consider D-instantons in the above set up. Such instantons preserve half of the 4 supercharges preserved by the system of D3-branes plus orbifold. In this respect recall that the fractional branes preserve exactly the same supercharges as the regular branes.4 Using the N = 4 construction of the previous section and the structure of the orbifold presented in eq. (3.5), we now proceed in describing the zero-modes for such instantons. The neutral sector is very similar to the gauge sector. Indeed, in the (−1) su- perghost picture, the vertex operators for such strings will be exactly the same, except for the eip·X factor which is absent for the instanton. The Chan-Paton structure will also be the same, so that the same pattern of fractional D-instantons will arise as for the fractional D3-branes. In particular, the only regular D-instanton (which could be thought of as deriving from the one of N = 4 SYM) is the one with rank (instanton number) one at every node. All other situations can be thought of as fractional D- instantons, which can be interpreted as Euclidean D1-branes wrapped on the two-cycles at the singularity, ED1 for short. Generically, we can then characterize an instanton configuration in our orbifold by (k1, k2, k3, k4). Following the notation introduced in section 2, the bosonic modes will comprise a 4×4 block diagonal matrix aµ, and six more matrix fields χ1, . . . χ6, that can be paired into three complex matrix fields χ1+iχ2, χ3+iχ4, χ5+iχ6, having the same structure as (3.5) but now where each block entry is a kℓ×km matrix. On the fermionic zero-modes MαA and λα̇A (also matrices) the orbifold projection enforces the conditions MαA = R(g)AB γ(g)M αBγ(g)−1 , λα̇A = γ(g)λα̇Bγ(g) −1R(g)BA (3.6) 4There is another Euclidean brane which preserves two supercharges, namely the Euclidean (anti) D3-branes orthogonal to the 4 dimensions of space-time. We will be considering here only the D- instantons, leaving the complete analysis of the other effects to future work. In this context, note that the extended brane instantons would have an infinite action (and thus a vanishing contribution) in the strict non-compact set up we are using here. where R(g) is the orbifold action of Table 1 in the spinor representation which can be chosen as R(g1) = −Γ 6789 , R(g2) = −Γ 4589 . (3.7) It is easy to find an explicit representation of the Dirac matrices such thatMαA and λα̇A for A = 1, 2, 3 also have the structure of (3.5) while for A = 4 they are block diagonal. Equivalently, one could write the spinor indices in the internal space in terms of the three SO(2) charges associated to the embedding SO(2) × SO(2) × SO(2) ⊂ SO(6) ≃ SU(4) Mα−++ =Mα1 , Mα+−+ =Mα2 , Mα++− =Mα3 , Mα−−− =Mα4 , λα̇+−− = λα̇1 , λα̇−+− = λα̇2 , λα̇−−+ = λα̇3 , λα̇+++ = λα̇4 . (3.8) The most notable difference between the neutral sector and the gauge theory on the D3-branes is that, whereas in the four-dimensional theory the U(1) gauge factors are rendered massive by a generalization of the Green-Schwarz mechanism and do not appear in the low energy action, for the instanton they are in fact present and enter crucially into the dynamics. Let us finally turn to the charged sector, describing strings going from the instan- tons to the D3-branes. The analysis of the spectrum and the action of the orbifold group on the Chan-Paton factors show, in particular, that the bosonic zero-modes are diagonal in the gauge factors. There are four block diagonal matrices of bosonic zero- modes ωα̇, ω̄α̇ with entries Nℓ×kℓ and kℓ×Nℓ respectively and eight fermionic matrices µA, µ̄A with entries Nℓ × km and km × Nℓ, that again display the same structure as above – same as (3.5) for A = 1, 2, 3 and diagonal for A = 4. 3.2 Recovery of the ADS superpotential The measure on the moduli space of the instantons and the ADHM constraints are simply obtained by inserting the above expressions into the moduli integral (2.7). If one chooses some of the Nℓ or kℓ to vanish one can deduce immediately from the structure of the projection which modes will survive and which will not. As a consistency check, one can try to reproduce the ADS correction to the super- potential [27, 28] for the theory with two nodes. Take fractional branes corresponding to a rank assignment (Nc, Nf , 0, 0), and consider the effect of a ED1 corresponding to instanton numbers (1, 0, 0, 0). The only chiral fields present are the two components of Φ1 connecting the first and second node 0 Q 0 0 Q̃ 0 0 0 0 0 0 0 0 0 0 0 . (3.9) Since the instanton is sitting only at one node, all off diagonal neutral modes are absent, as they connect instantons at two distinct nodes. Thus, the only massless modes present in the neutral sector are four bosons xµ, denoting the upper-left component of aµ, two fermions θα denoting the upper-left component of Mα4 and two more fermions λα̇ denoting the upper-left component of λα̇4. We have identified the non zero entries of aµ and Mα4 with the super-coordinates xµ and θα since they precisely correspond to the Goldstone modes of the super-translation symmetries broken by the instanton and do not appear in S1+ S2 (cfr. (2.1) and (2.3)). Their integration produces the integral over space-time and half of Grassmann space which precedes the superpotential term to which the instanton contributes. On the contrary, λα̇ appears in S1 and when it is integrated it yields the fermionic ADHM constraint. In the charged sector, we have bosonic zero-modes ωuα̇ and ω̄α̇u, with u an index in the fundamental or anti-fundamental of SU(Nc). In addition, there are fermionic zero-modes µu and µ̄u with indices in SU(Nc), together with additional fermionic zero- modes µ′f and µ̄′f where the index f is now in the fundamental or anti-fundamental of SU(Nf ). 5 Note that the µ zero-modes carry an SU(4) index 4 (being on the diagonal) while the µ′ zero-modes carry an SU(4) index 1, since they are of the same form as Φ1. All this can be conveniently summarized in a generalized quiver diagram as rep- resented in Fig. 2, which accounts for both the brane configuration and the instanton zero-modes. For a single instanton, the action (2.1) greatly simplifies since many fields are vanishing as well as all commutators and one gets S1 = i (µ̄uω α̇ + ω̄α̇uµ u) λα̇ − iDcω̄α̇u(τ . (3.10) Similarly, the coupling of the charged modes to the chiral superfield can be expressed by writing eq. (2.3) as ω̄α̇u v + Q̃ ωα̇v − µ̄uQ̃ µ̄′fQ u . (3.11) Note that it is the anti-holomorphic superfields that enter in the couplings with the fermionic zero-modes, as is clear by comparing with (2.6). The above action is exactly the same which appears in the ADHM construction as reviewed in [18]. 5Recall that the bosonic zero-modes are diagonal in the gauge factors; therefore there are no ω and ω̄α̇f zero-modes. SU(N ) SU(N )f Figure 2: Quiver diagram describing an ordinary instanton in a SU(Nc) × SU(Nf ) theory. Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1 is wrapped on the same cycle as the color branes. All zero-modes are included except the θ’s and the xµ’s, which only contribute to the measure for the integral over chiral superspace. We are now ready to perform the integral (2.7) over all the existing zero-modes. Writing dx4dθ2W , (3.12) we see that the instanton induced superpotential is W = C d{λ,D, ω, ω̄, µ, µ̄} e−S1−S2 . (3.13) The integrals over D and λ enforce the bosonic and fermionic ADHM constraints, respectively. Thus W = C d{ω, ω̄, µ, µ̄} δ(µ̄uω α̇ + ω̄α̇uµ u) δ(ω̄α̇u(τ ) e−S2 . (3.14) We essentially arrive at the point of having to evaluate an integral over a set of zero- modes which is exactly the same as the one discussed in detail in the literature, e.g. [18]. We thus quickly go to the result referring the reader to the above review for further details. First of all, it is easy to see that, due to the presence of extra µ modes in the integrand from the fermionic delta function, only when Nf = Nc − 1 we obtain a non-vanishing result. After having integrated over the µ and µ′, we are left with a (constrained) gaussian integration that can be performed e.g. by going to a region of the moduli space where the chiral fields are diagonal, up to a row/column of ze- roes. Furthermore, the D-terms in the gauge sector constrain the quark superfields to obey QQ† = Q̃†Q̃, so that the bosonic integration brings the square of a simple determinant in the denominator. The last fermionic integration conspires to cancel the anti-holomorphic contributions and gives WADS = Λ2Nc+1 det(Q̃Q) , (3.15) which is just the expected ADS superpotential for Nf = Nc − 1, the only case where such non-perturbative contribution is generated by a genuine one-instanton effect and not by gaugino condensation. In (3.15) Λ is the SQCD strong coupling scale that is reconstructed by the combination of e−8π 2/g2 coming from the instanton action with various dimensional factors coming from the normalization of the instanton measure [18]. 3.3 Absence of exotic contributions Until now, we have reproduced from stringy considerations the effect that is supposed to be generated also by instantons in the gauge theory. Considering a slightly different set up, we would like to study the possibility of generating other terms. Let us consider a system with rank assignment (Nc, Nf , 0, 0), as before, but frac- tional instanton numbers (0, 0, 1, 0). In other words, we study the effect of a single fractional instanton sitting on an unoccupied node of the gauge theory. The quiver diagram, with the relevant zero-modes structure, is given in Fig. 3. The neutral zero-modes of the instanton sector are the same as before. This is because the quantization of this sector does not know the whereabouts of the D3- branes and thus all nodes are equivalent, in this respect. In the mixed sector, we have no bosonic zero-modes now, since the ω and ω̄ are diagonal. Note that, although we always have four mixed (ND) boundary conditions, due to the quiver structure induced by the orbifold, here we effectively realize the same situation one has when there are eight ND directions, namely that the bosonic sector of the charged moduli is empty. On the other hand, there are fermionic zero-modes µu, µ̄u, µ ′f and µ̄′f , as in the previous case. Note that despite having the same name, these zero-modes correspond actually to different Chan-Paton matrix elements with respect to the previous ones, the difference being in the instanton index that is not written explicitly. In particular we can think of µ and µ′ as carrying an SU(4) index 2 and 3 respectively. Because of the absence of bosonic charged modes, the action (2.1) is identically SU(N )fSU(N ) Figure 3: Quiver diagram describing an exotic instanton in a SU(Nc) × SU(Nf ) theory. Gauge theory nodes are represented by round circles, instanton nodes by squares. The ED1 is wrapped on a different cycle with respect to both sets of quiver branes. zero and the action (2.3) contains only the last term: S1 = 0 µ̄′fQ̃ u. (3.16) Note that in this case it is the holomorphic superfields which appear above, as is clear from (2.6) and from noticing that the diagonal fermionic zero-mode µ4 is not present. We are thus led to consider W = C d{λ,D, µ, µ̄} e−S2 . (3.17) One notices right away that the integral over the charged modes is non vanishing (only) for the case Nf = Nc and gives a tantalizing contribution proportional to BB̃, where B = detQ and B̃ = det Q̃ are the baryon fields of the theory. However, we must carefully analyze the integration over the remaining zero-modes of the neutral sector. Now neither D nor λ appear in the integrand. The integral over D does not raise any concern: it is, after all, an auxiliary field and its disappearance from the integrand is due to the peculiarities of the ADHM limit. Before taking this limit, D appeared quadratically in the action and could be integrated out, leaving an overall normalization constant. The integral over λ is another issue. In this case, λ is absent from the integrand even before taking the ADHM limit and its integration multiplies the above result by zero, making the overall contribution of such instantons to the superpotential vanishing. Of course, the presence of such extra zero-modes should not come as a surprise since they correspond to the two extra broken supersymmetries of an instanton on a CY. Therefore we see that the neutral zero-modes contribution, in the exotic instanton case, plays a dramatic role and conspires to make everything vanishing (as opposite to the ADS case analyzed before). A natural question is to see whether these zero- modes get lifted by some effect we have not taken into account, yet. For one thing, supersymmetry arguments would make one think that taking into account the back- reaction of the D3-branes might change things. However, in the following subsection we show that this seems not to be the case. 3.4 Study of the back-reaction Let us stick to the case Nf = Nc, which is the only one where the integral (3.17) might give a non-vanishing contribution. In this case the fractional brane system is nothing but a stack of (Nc) N = 2 fractional branes. These branes couple to only one of the 3 closed string twisted sectors [24]. More specifically, they source the metric hµν , the R-R four-form potential Cµνρσ and two twisted scalars b and c from the NS-NS and R-R sector respectively. This means that the disk one-point function of their vertex operators [31, 32] is non vanishing when the disk boundary is attached to such D3-branes. (Indeed in this way or, equivalently, by using the boundary-state formalism [33, 34], one can derive the profile for these fields.) If the back-reaction of these fields on the instanton lifted the extra zero-modes λ’s, this should be visible when computing the one point function of the corresponding closed string vertex operators on a disk with insertions on this boundary of the vertex operators for such moduli. To see whether such coupling is there, we first need to write down the vertex operators for the λ’s in the (±1/2) superghost pictures. The vertex in the (−1/2) picture is found e.g. in [6] and reads λ (z) = λα̇AS α̇(z)SA(z)e−φ(z)/2 , (3.18) where Sα̇(z) and SA(z) are the spin-fields in the first four and last six directions re- spectively. For our argument we need to focus on the SA(z) dependence. Since the modulus that survives the orbifold projection is, with our conventions, λα̇4 = λα̇+++, we write the corresponding spin-field as S+++(z) = eiH1(z)/2eiH2(z)/2eiH3(z)/2, (3.19) where Hi(z) is the free boson used to bosonize the fermionic sector in the i-th complex direction: ψi(z) = eiHi(z). The vertex operator in the +1/2 picture can be obtained by applying the picture-changing operator to (3.18) λ (z) = [QBRST, ξV λ (z)] . (3.20) The crucial part in QBRST is [31] QBRST = ψµ∂Xµ + ψ̄i∂Z i + ψi∂Z̄ i + . . . (3.21) Because of the nature of the supercurrent, we see that (3.21) flips at most one sign in (3.19), hence the product V λ will always carry an unbalanced charge in some of the three internal SO(2) groups. On the other hand, the vertex operators for the fields sourced by the fractional D3’s cannot compensate such an unbalance. Hence, their correlation function on the D-instanton with the insertion of V λ carries a charge unbalance and therefore vanishes. Therefore, at least within the above perturbative approach, the neutral zero-modes seem not to get lifted by the back-reaction of the D3-branes. One might consider some additional ingredients which could provide the lifting. A natural guess would be moving in the CY moduli space or adding suitable background fluxes [35, 36]. There are indeed non-vanishing background fields at the orbifold point, i.e. the b fields of the twisted sectors which the N = 2 fractional branes do not couple to. These fields, however, being not associated to geometric deformations of the internal space should be described by a CFT vertex operator uncharged under the SO(2)’s, simply because of Lorentz invariance in the internal space. Therefore, the only way to get an effective mass term for the zero-modes λ would be to move out of the orbifold point in the CY moduli space. Indeed, the other moduli of the NS-NS twisted sector, being associated to geometric blow-ups of the singularity, are charged under (some of) the internal SO(2)’s and can have a non vanishing coupling with the λ’s. More generically, complicated closed string background fluxes might be suitable. This is an interesting option which however we do not pursue here, since we want to stick to situations where a CFT description is available. A more radical thing to do is to remove the zero-modes from the very start, for instance by means of an orientifold projection [37, 38]. This is the option we are going to consider in the remainder of this work. 4. The N = 1 Z2 × Z2 orientifold In this section we supplement our orbifold background by an O3 orientifold and show that in this case exotic instanton contributions do arise and provide new terms in the superpotential. We refer to e.g. [39, 40, 41] for a comprehensive discussion of N = 1 and N = 2 orientifolds. The first ingredient we need is the action of the O3-plane on the various fields. Denote by Ω the generator of the orientifold. The action of Ω on the vertex operators for the various fields (ignoring for the time being the Chan-Paton factors) is well known. The vertex operators for the bosonic fields on the D3-brane contain, in the 0 picture, the following terms: Aµ ∼ ∂τx µ and Φi ∼ ∂σz̄ i. They both change sign under Ω, the first because of the derivative ∂τ and the second because the orientifold action for the O3-plane is always accompanied by a simultaneous reflection of all the transverse coordinates zi. The action of the orientifold on the Chan-Paton factors is realized by means of a matrix γ(Ω) which in presence of an orbifold must satisfy the following consistency condition [39] γ(g)γ(Ω)γ(g)T = + γ(Ω) (4.1) for all orbifold generators g. This amounts to require that the orientifold projection commutes with the orbifold projection. The matrix γ(Ω) can be either symmetric or anti-symmetric. We choose to perform an anti-symmetric orientifold projection on the D3 branes and denote the corresponding matrix by γ−(Ω). This requires having an even number Nℓ of D3 branes on each node of the quiver so that we can write γ−(Ω) = ǫ1 0 0 0 0 ǫ2 0 0 0 0 ǫ3 0 0 0 0 ǫ4 (4.2) where the ǫℓ’s are Nℓ × Nℓ antisymmetric matrices obeying ǫ ℓ = −1. Using (3.3) and (4.2) it is straightforward to verify that the consistency condition (4.1) is verified. The field content of the stacks of fractional D3-branes in this orientifold model is obtained by supplementing the orbifold conditions (3.4) with the orientifold ones Aµ = −γ−(Ω)A µγ−(Ω) −1 , Φl = −γ−(Ω)Φ lTγ−(Ω) −1. (4.3) This implies that Aµ = diag (A µ) with A µ = ǫℓA µ ǫℓ. Thus, the result- ing gauge theory is a USp(N1) × USp(N2) × USp(N3) × USp(N4) theory. The chiral superfields, which after the orbifold have the structure (3.5), are such that the Φℓm component joining the nodes ℓ and m of the quiver, must obey the orientifold condi- tion Φℓm = ǫℓΦ mℓǫm. In the following, we will take N3 = N4 = 0 so that we are left with only two gauge groups and no tree level superpotential. 4.1 Instanton sector Let us now consider the instanton sector, starting by analyzing the zero-mode content in the neutral sector. There are two basic changes to the previous story. The first is that the vertex operator for aµ is now proportional to ∂σx µ, not to ∂τx µ and it remains invariant under Ω (the vertex operator for χa still changes sign). The second is that the crucial consistency condition discussed in [38] requires that we now represent the action of Ω on the Chan-Paton factors of the neutral modes by a symmetric matrix which can be taken to be γ+(Ω) = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , (4.4) where the 1’s are kℓ × kℓ unit matrices. The matrix aµ will be 4 × 4 block diagonal, e.g. aµ = diag (a µ), but now a µ = a µ . The most generic situation is to have a configuration with instanton numbers (k1, k2, k3, k4). By considering a configuration with k3 = 1 and k1 = k2 = k4 = 0, we can project out all bosonic zero-modes except for the four components a3µ that we denote by xµ. The scalars χ 4 . . . χ9 are off-diagonal and we shall not consider them further. The nice surprise comes when considering the orientifold action on the fermionic neutral zero-modes MαA and λα̇A. The orbifold part of the group acts on the spinor indices as in (3.7), while the orientifold projection acts as the reflection in the transverse space, namely R(Ω) = −iΓ456789 (4.5) Putting together the orbifold projections (3.6) with the orientifold ones MαA = RAB(Ω)γ+(Ω)(M αB)Tγ+(Ω) −1 , λα̇A = γ+(Ω)(λα̇B) Tγ+(Ω) −1RBA(Ω) (4.6) we can find the spectrum of surviving fermionic zero-modes. Using (4.4) and (4.5), it is easy to see that (4.6) implies MαA = (MαA)T , λα̇A = −(λα̇A) T . (4.7) Thus, for the simple case where k3 = 1 and k1 = k2 = k4 = 0, all λ’s are projected out and only two chiral M zero-modes remain: Mα−−−, to be identified with the N = 1 chiral superspace coordinates θα. Also the charged zero-modes are easy to discuss in this simple scenario. There are no bosonic modes since the D-instanton and the D3-branes sit at different nodes while the bosonic modes are necessarily diagonal. Most of the fermionic zero-modes µA and µ̄A are also projected out by the orbifold condition µA = R(g)ABγ(g)µ Bγ(g)−1 , µ̄A = R(g)ABγ(g)µ̄ Bγ(g)−1 . (4.8) Finally, the orientifold condition relates this time the fields in the conjugate sectors, allowing one to express µ̄ as a linear combination of the µ µ̄A = R(Ω)ABγ+(Ω)(µ B)Tγ−(Ω) −1 . (4.9) The only charged modes surviving these projections can be expressed, in block 4 × 4 notation, as 0 0 µ13 0 0 0 0 0 0 0 0 0 0 0 0 0 , µ̄2 = 0 0 0 0 0 0 0 0 µ̄31 0 0 0 0 0 0 0 0 0 0 0 0 0 µ23 0 0 0 0 0 0 0 0 0 , µ̄3 = 0 0 0 0 0 0 0 0 0 µ̄32 0 0 0 0 0 0 , (4.10) where the entries, to be thought of as column/row vectors in the fundamental/anti- fundamental of SU(Nℓ) depending on their position, are such that µ̄31 = −µ 13ǫ1 and µ̄32 = −µ 23ǫ2. Thus, in the case where we have fractional D3 branes (N1, N2, 0, 0) and an exotic instanton (0, 0, 1, 0), the only surviving chiral field is Φ12 ≡ ǫ1Φ 21ǫ2, the orientifold projection eliminates the offending λ’s and we are left with just the neutral zero-modes xµ and θ α and the charged ones µ13 and µ23. This is summarized in the generalized quiver of Fig. 4. In this case the instanton partition function is dx4dθ2 W (4.11) where the superpotential W is W = C dµ e−S1−S2 = C dµ13dµ23 e iµT13ǫ1Φ12µ23 . (4.12) This integral clearly vanishes unless N1 = N2, in which case we have W ∝ det(Φ12) (4.13) USP(N )USP(N )1 2 Φ Φ1221 2313µ Figure 4: The generalized Z2 ×Z2 orientifold quiver and the exotic instanton contribution. We thus see that exotic instanton corrections are possible in this simple model.6 It is interesting to note that the above correction is present in the same case (N1 = N2 ≡ N) where the usual ADS superpotential for USp(N) is generated [42] WADS = Λ2N+3 det(Φ12) (4.14) and its presence stabilizes the runaway behavior and gives a theory with a non-trivial moduli space of supersymmetric vacua given by det(Φ12) = const. Of course, the ADS superpotential for this case can also be constructed along the same lines as section 3.2, see e.g. [18]. In fact, this derivation is somewhat simpler than the one for the SU(N) gauge group since there are no ADHM constraints at all in the one instanton case. We think the above situation is not specific to the background we have been consid- ering, but is in fact quite generic. As soon as the λ zero-modes are consistently lifted, we expect the exotic instantons to contribute new superpotential terms. As a further example, in the next section we will consider a N = 2 model, where exotic instantons will turn out to contribute to the prepotential. 5. An N = 2 example: the Z3 orientifold Let us now consider the quiver gauge theory obtained by placing an orientifold O3-plane at a C × C2/Z3 orbifold singularity. In what follows we will use N = 1 superspace notation. We first briefly repeat the steps that led to the constructions of such a quiver 6The gauge invariant quantity above can be rewritten as the Pfaffian of a suitably defined mesonic matrix. theory in the seminal paper [39]. Define ξ = e2πi/3 and let the generator of the orbifold group act on the first two complex coordinates as 0 ξ−1 , (5.1) while leaving the third one invariant. This preserves N = 2 SUSY. The action of the generator g on the Chan-Paton factors is given by the matrix γ(g) = 1 0 0 0 ξ 0 0 0 ξ2  . (5.2) The N = 2 theory obtained this way, summarized in Fig. 5, is a three node quiver gauge theory with gauge groups SU(N1)× SU(N2)× SU(N3), supplemented by a cubic superpotential which is nothing but the orbifold projection of the N = 4 superpotential (its precise form is not relevant for the present purposes). SU(N ) SU(N ) SU(N )1 Figure 5: The Z3 (un-orientifolded) theory. The lines with both ends on a single node represent adjoint chiral multiplets which, together with the vector multiplets at each node constitute the N = 2 vector multiplets. Similarly, lines between nodes represent chiral mul- tiplets which pair up into hyper-multiplets, in N = 2 language. As for the action of Ω on the Chan-Paton factors, we choose again to perform the symplectic projection on the D3-branes. To do so, we must take N1 to be even and N2 = N3, so that we can write γ−(Ω) = ǫ 0 0 0 0 1 0 −1 0  , (5.3) where ǫ is a N1×N1 antisymmetric matrix obeying ǫ 2 = −1 and the 1’s denote N2×N2 identity matrices. The matrices γ(g) and γ−(Ω) satisfy the usual consistency condi- tion [38, 39] as in (4.1). The field content on the fractional D3-branes at the singularity will be given by implementing the conditions Aµ = γ(g)Aµγ(g) −1 , Φi = ξ−iγ(g)Φiγ(g)−1 , Aµ = −γ−(Ω)A µγ−(Ω) −1 , Φi = −γ−(Ω)Φ iTγ−(Ω) −1 . (5.4) The orbifold part of these conditions forces Aµ and Φ 3 to be 3 × 3 block diagonal matrices, e.g. Aµ = diag (A µ), while the orientifold imposes that A µ = ǫA and A2µ = −A µ . The resulting gauge theory is thus a USp(N1)× SU(N2) theory. It is convenient, however, to still denote A2µ and A µ diagramatically as belonging to different nodes with the understanding that these should be identified in the above sense. The projection on the chiral fields can be done similarly and we obtain, denoting by Φℓm the non-zero entries of the fields Φ 1 and Φ2 (only one can be non-zero for each pair ℓm) Φ12 = −ǫΦ 31, Φ13 = +ǫΦ 21, Φ23 = Φ 23, Φ32 = Φ 32 . (5.5) The field content is summarized in Table 2. USp(N1) SU(N2) Φ12 � � Φ21 � � Φ13 � � Φ31 � � Φ23 · �� Φ32 · �� Table 2: Chiral fields making up the quiver gauge theory. The theory we want to focus on in the following has rank assignment (N1, N2) = (0, N). This yields an N = 2 SU(N) gauge theory with an hyper-multiplet in the sym- metric/(conjugate)symmetric representation. We denote the N = 2 vector multiplet by A whose field content in the block 3× 3 notation is thus 0 0 0 0 A 0 0 0 −AT  . (5.6) In what follows we will be interested in studying corrections to the prepotential F coming from exotic instantons associated to the first node (the one that is not populated by D3-branes). Let us then analyze the structure of the stringy instanton sector of the present model, first. 5.1 Instanton sector The most generic situation is to have a configuration with instanton numbers (k1, k2) (later we will be mainly concerned with a configuration with instanton numbers (1, 0)). Let us start analyzing the zero-modes content in neutral sector. The story is pretty similar to the one discussed in the previous section. The vertex operator for aµ is proportional to ∂σx µ and so it remains invariant under Ω. The action on the Chan- Paton factors of these D-instantons must now be represented by a symmetric matrix which we take to be γ+(Ω) = 1′ 0 0 0 0 1 0 1 0  (5.7) where 1′ is a k1 × k1 unit matrix and the 1’s are k2 × k2 unit matrices. Because of the different orientifold projection, the matrices of bosonic zero-modes behave slightly differently. The matrices aµ, χ 8 and χ9 will still be 3×3 block diagonal, e.g. aµ = diag (a µ), but now a µ = a µ and a µ = a µ whereas the same relations for χ8 and χ9 will have an additional minus sign. The remaining fields χ4...7 are off diagonal and we shall not consider them further since we will consider only the case of one type of instanton. By considering a configuration with k1 = 1 and k2 = 0, we can project out all bosonic zero-modes except for the four components a1µ that we denote by xµ. Let us now consider the orientifold action on the fermionic neutral zero-modesMαA and λα̇A. The orbifold part of the group acts on the internal spinor indices as a rotation R(g) = e Γ45e− Γ67 , (5.8) while the orientifold acts through the matrix R(Ω) given in (4.5). The orbifold and orientifold projections thus require MαA = R(g)ABγ(g)M αBγ(g)−1 , λα̇A = γ(g)λα̇Bγ(g) −1R(g)BA , (5.9) MαA = R(Ω)ABγ+(Ω)(M αB)Tγ+(Ω) −1 , λα̇A = γ+(Ω)(λα̇B) Tγ+(Ω) −1R(Ω)BA . Using the explicit expressions for the various matrices, we see that, for the simple case where k1 = 1 and k2 = 0, all λ’s are projected out and only four chiral M zero- modes remain: Mα−−− and Mα++− to be identified with the N = 2 chiral superspace coordinates θ1α and θ α. Hence, also in this case the orientifold projection has cured the problem encountered in section 3 (albeit in a N = 2 context now) and we can rest assured that the integration over the charged modes will yield a contribution to the prepotential. Let us now move to the charged zero-modes sector. Just as in the previous model, there are no bosonic modes since the D-instanton and the D3-branes sit at different nodes while the bosonic modes are necessarily diagonal. Most of the fermionic zero- modes µA and µ̄A are projected out by the orbifold condition which is formally the same as in (4.8), while the orientifold condition relates the fields in the conjugate sectors, giving µ̄ as a linear combination of the µ’s according to µ̄A = R(Ω)ABγ+(Ω)(µ B)Tγ−(Ω) −1 . (5.10) To summarize, the only charged modes surviving the projection can be expressed, in block 3× 3 notation as 0 0 0 0 0 0 µ 0 0  , µ̄1 = 0 µT 0 0 0 0 0 0 0 0 0 0 µ′ 0 0 0 0 0  , µ̄2 = 0 0 −µ′T 0 0 0 0 0 0  (5.11) where the entries are to be thought of as column/row vectors in the fundamental/anti- fundamental of SU(N) depending on their position. As anticipated, the configuration we want to consider is a (0, N) fractional D3- branes system together with an exotic (1, 0) instanton. The quiver structure, including the relevant moduli, is depicted in Fig. 6. It is now easy to see that inserting the expressions (5.6) and (5.11) into Eqs. (2.1), (2.3) and (2.7) we finally obtain dx4dθ4F with F = C dµdµ′ eiµ TAµ′ ∝ detA . (5.12) It would be interesting to study the potential implications of this result in the gauge theory. There are many other simple models that could be analyzed along these lines. 6. Conclusions In this paper we have presented some simple examples of what seem to be rather generic phenomena in the context of string instanton physics. We paid particular attention to Φ Φ3223 U(N) U(N) Figure 6: The extended Z3 orientifold theory with (0, N) fractional D3-branes and (1, 0) instanton number. The upper node (which would represent the USp(N1) gauge group and disappears when we set N1 = 0 as in the case under consideration) is where the instanton sits. The lower nodes denote only one gauge group. The charged fermionic zero-modes follow Eq. (5.11). For simplicity we have not drawn the lines denoting the adjoint. the study of the fermionic zero-modes and their effects on the holomorphic quantities of the theory. We have seen both examples where the instanton contributions vanish due to the presence of extra zero-modes and where they do not. In the second case, as explicitly shown in a N = 1 example, exotic instantons can have a stabilizing effect on the theory. Although we have only considered some simple examples, we would like to stress that these results are quite generic and can be carried over to all orbifold gauge theories. A future direction would be to try to be more systematic and analyze the various possibilities encountered in more complex N = 2 and N = 1 models. In a similar spirit, one should analyze the multi-instanton contributions as well, since the total correction to the holomorphic quantities will be the sum of all such terms. The study of the zero-modes is expected to be even more relevant in this case as it will probably make many contributions vanish. With an eye to string phenomenology, one should also incorporate these models into globally consistent compactifications and study the effects of these terms there. Lastly, it would be interesting to study the dynamical implications of some of the terms generated. We briefly touched upon this at the end of section 4 when we men- tioned the stabilizing effect of the exotic instanton on the USp(N) theory. Although from the strict field theory point of view these terms are thought of as ordinary polyno- mial terms in the holomorphic quantities,7 they are “special” when seen from the point of view of string theory and they might therefore induce a particular type of dynamics. Acknowledgements We would like to thank many people for discussions and email exchanges at various stages of this work that helped us sharpen the focus of the presentation: M. Bianchi, M. Billò, P. Di Vecchia, S. Franco, M. Frau, F. Fucito, S. Kachru, R. Marotta, L. Mar- tucci, F. Morales, B. E. W. Nilsson, D. Persson, I. Pesando, D. Robles-Llana, R. Russo, A. Tanzini, A. Tomasiello, A. Uranga, T. Weigand and N. Wyllard. R.A., M.B. and A.L. are partially supported by the European Commission FP6 Programme MRTN-CT-2004-005104, in which R.A is associated to V.U. Brussel, M.B. to University of Padova and A.L. to University of Torino. R.A. is a Research Associate of the Fonds National de la Recherche Scientifique (Belgium). The research of R.A. is also supported by IISN - Belgium (convention 4.4505.86) and by the “Interuniver- sity Attraction Poles Programme –Belgian Science Policy”. M.B. is also supported by Italian MIUR under contract PRIN-2005023102 and by a MIUR fellowship within the program “Rientro dei Cervelli”. The research of G.F. is supported by the Swedish Research Council (Vetenskapsr̊adet) contracts 622-2003-1124 and 621-2002-3884. 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0704.0263

Turbulent Diffusion of Lines and Circulations

7 Turbulent Diffusion of Lines and Circulations Gregory L. Eyink Department of Applied Mathematics & Statistics The Johns Hopkins University 3400 N. Charles Street Baltimore, MD 21218 Tel: 410-516-7201, Fax: 410-516-7459 e-mail: [email protected] Abstract We study material lines and passive vectors in a model of turbulent flow at infinite- Reynolds number, the Kraichnan-Kazantsev ensemble of velocities that are white-noise in time and rough (Hölder continuous) in space. It is argued that the phenomenon of “spon- taneous stochasticity” generalizes to material lines and that conservation of circulations generalizes to a “martingale property” of the stochastic process of lines. PACS: 47.27.Jv, 52.65.Kj, 02.50.Fz, 05.45.Df keywords: turbulence, material lines, circulations, Kraichnan model, dynamo, fractals http://arxiv.org/abs/0704.0263v1 The evolution of material lines and surfaces passively carried by turbulent flow has long been a subject of interest [1]. This is motivated in part by questions surrounding dynamically relevant objects, such as vortex lines [2, 3, 4] and magnetic field-lines [5, 6], which have been argued to behave similar to material lines at high Reynolds numbers. However, in that limit, the turbulent velocity field is no longer differentiable in space but only Hölder continuous [7, 8, 9]. Observations from experiments and simulations suggest that material objects advected by such a rough velocity become fractal, with a Hausdorff dimension strictly greater than their topological dimension [10, 11, 12, 13, 14, 15]. This poses a difficulty to the view that vortex lines behave as material lines—a consequence of the Kelvin-Helmholtz theorem [16, 17]—since circulations are a priori not defined for non-rectifiable loops. It has recently been argued that the Kelvin theorem in fact breaks down in turbulent flows, in the sense that the circulation is not strictly conserved for every loop [18, 19]. Similar breakdown of Alfvén’s theorem on magnetic-flux conservation [20] is expected in plasma turbulence at high magnetic Reynolds numbers [21]. These questions have been sharpened by recent work on the Kraichnan model of advection by a Gaussian random velocity field that is delta-correlated in time [22]. A novel phenomenon has been discovered there called spontaneous stochasticity: Lagrangian particle trajectories for a non-Lipschitz advecting velocity are non-unique and split to form a random process in path- space for a fixed velocity realization [23, 24, 25, 26, 27, 28, 29]. This phenomenon raises many fundamental questions, including whether material objects such as lines and surfaces can even exist in the limit of infinite Reynolds number. It is the purpose of this Letter to outline a new approach to the evolution of such geometric objects in the Kraichnan model. We focus on material lines and passive vectors, which are dual objects in the same sense as material particles and passive scalar fields [30]. In particular, we shall sketch the proof of a “martingale property” that has previously been proposed [18] as a generalization of the conservation of circulations for a rough velocity field. We consider stochastic flows [31] on a d-dimensional manifold M driven by Brownian vector fields that are not Lipschitz regular in space. To simplify the presentation, we use Euclidean space M = Rd or the torus M = Td to illustrate the main ideas. More precisely, u(x, t) is a Gaussian random vector field, with mean u(x, t) and fluctuating part ũ(x, t) with covariance 〈ũi(x, t)ũj(x ′, t′)〉 = Dij(x,x ′; t)δ(t − t′). (1) for x,x′ ∈ M. We are mainly interested in the case that u(x, t) ≡ 0 and u(x, t) ≡ ũ(x, t) is a homogeneous random field, with D(x,x′; t) = D(x− x′, t). The quantity ∆(x,x′; t) = tr[D(x,x; t) +D(x′,x′; t)− 2D(x,x′; t)] (2) is 〈‖u(x) − u(x′)‖2〉, the mean of the Euclidean norm squared, for a random field u(x) with covariance D(x,x′; t). The case of greatest interest to us has ∆(x,x′; t) ∝ ‖x − x′‖2α with 0 < α < 1. In that case, u(x) is Hölder continuous with exponent α at every point in space. We consider oriented lines (1-cells) given parametrically as continuous, one-to-one maps C : [0, 1] → M. A material line satisfies (d/dt)C(σ, t) = u(C(σ, t), ◦ t). (3) for σ ∈ [0, 1] and t ∈ R. The circle “◦” means that we interpret equation (3) in the Stratonovich sense. The (forward) Ito equation (d/dt)C(σ, t) = u(C(σ, t), t) equivalent to (3) has the mean changed to u∗i (x, t) = ui(x, t) + (1/2)(∂/∂x k)Dik(x,x ′; t)| ′=x ([31], section 3.4.) If M = R d and if u(x, t) is a homogeneous random field, then the Ito and Stratonovich interpretations of equation (3) are equivalent. Now let P [C, t] denote the conditional probability distribution of lines for a fixed velocity realization u. This distribution satisfies a stochastic Liouville equation: (d/dt)P [C, t] = − δCi(σ) (ui(C(σ), ◦ t)Pu[C, t]) . (4) Equation (4) is a direct consequence of equation (3) and must also be interpreted in the Stratonovich sense. It is formally equivalent to the Ito equation: (d/dt)P [C, t] = − δCi(σ) ([u∗i (C(σ), t) + ũi(C(σ), t)]Pu[C, t]) δCi(σ)δCj(σ′) Dij(C(σ), C(σ ′); t)P [C, t] . (5) Averaging equation (5) over the Gaussian ensemble of velocities ũ yields a functional Fokker- Planck equation for distributions in the space of free lines C on the manifold M : (d/dt)P [C, t] = − δCi(σ) ([u∗i (C(σ), t)P [C, t]) δCi(σ)δCj(σ Dij(C(σ), C(σ ′); t)P [C, t] . (6) The first term on the righthand side represents a drift with the mean velocity u∗ and the second term represents a diffusion arising from the velocity covariance D. Similar diffusions on the path- and loop-spaces of a manifold M have been much studied, motivated in part by questions from quantum field theory [32, 33]. The above considerations are rigorously justifiable for the case of a Lipschitz velocity with α = 1 but are only formal when α < 1. A more careful (and also more physically realistic) approach in the latter case is to replace the advecting velocity u with a “coarse-grained” or smoothed velocity uλ = ϕλ ∗ u, by convolution with a smooth filter kernel ϕλ(r) = λ −dϕ(r/λ). The length-scale λ can be interpreted as a mathematical representation of the viscous cutoff in a true turbulent velocity field [25, 26]. The exact solution of the Liouville equation (4) for such a smoothed velocity is (dC, t|C0, t0) = δ(C − ξ (C0))dC, (7) with initial condition P (dC, t0|C0, t0) = δ(C − C0)dC. Here ξ : M → M is the stochastic flow of diffeomorphisms generated by the smoothed velocity-field uλ ([31], section 4.6). Despite (7), a nontrivial diffusion process in line-space can be obtained if the limit λ → 0 is taken appropriately. Consider a “nice” distribution Gρ(dC) which is supported on lines entirely contained in the ball B(0, ρ) of radius ρ at the origin 0 and take the weak limit Gρ(dC (dC, t|C0 + C 0, t0)Ψ(C) = (dC, t|C0, t0)Ψ(C) (8) for bounded, continuous functionals Ψ(C) and t > t0. In the “weakly compressible regime” [24, 25, 28]— and, in particular, for a divergence-free velocity field satisfying ∇·u = 0— this limit should yield a non-degenerate diffusion. 1 This is a generalization of the phenomenon of spontaneous stochasticity to the turbulent advection of lines, with initial line C0 at time t0 splitting into a random ensemble of lines C at time t. As for the case of smooth advection, an unconditional diffusion satisfying equation (6) may be obtained by averaging over the velocity u. The instantaneous realizations C of this diffusion process should be fractal objects when the advecting velocity is Hölder continuous with exponent 0 < α < 1 and rigorous estimates of their Hausdorff dimensions would be of much interest. These questions may also be addressed numerically using Lagrangian Monte Carlo techniques [34, 35, 36]. In such a study, the material line C(t) would be represented by a discrete approximation CN (t) constructed from N +1 Lagrangian particles xa(t), a = 0, ..., N : CN (σ, t) = (1− θN (σ))xaN (σ)(t) + θN (σ)xaN (σ)+1(t). (11) Here aN (σ) = [Nσ] with [x] the greatest integer less than or equal to x (modulo N for loops) and θN (σ) = (Nσ) where (x) = x− [x] is the fractional part of x. Thus, (11) corresponds to a piecewise-linear curve with linear segments connecting the successive Lagrangian particles. So long as δN (t) = maxa |xa(t)−xa+1(t)| . λ, the discrete approximation CN (t) represents well the material line C(t) and the approximation becomes better as N → ∞ and δN (t) ≪ λ. However, the same is not true in the opposite limit, where λ ≪ δN (t). The phenomenon of “spontaneous stochasticity” for a rough velocity field makes it very doubtful that material lines even exist if the limit λ → 0 is taken before evolving in time and an initial line then presumably “explodes” 1We note that the same diffusion process should also be obtained by using Duhamel’s formula to solve equation (5) as the Ito integral Pu[C, t] = S 0 (t, t0)P [C, t0]− (t, t δCi(σ) [eui(C(σ), t )]Pu[C, t , (9) where S∗0 (t, t ′) = Texp ∗(t′) (t) = − δCi(σ) i (C(σ), t)·) + δCi(σ)δCj(σ′) Dij(C(σ), C(σ ); t)· is the Fokker-Planck operator of the (mean) diffusion in line-space. Equation (9) can be solved iteratively to generate a representation Pu[C, t] = S (t, t0)P [C, t0] as a Wiener chaos expansion in white-noise eu; cf. [28, 29]. into a disconnected cloud of particles at any time t > 0. Thus, the velocity smoothing in (8) appears to be necessary to define appropriately a line-diffusion for a rough (Hölder) velocity. Alternatively, a stochastic regularization might be employed that adds a white-noise κdW (t) to the evolution equation of Lagrangian particles [30]. We now turn to the dual problem of a passive vector (1-form) A advected by the velocity u = u+ ũ: ∂tA(x, t) + (u(x, ◦t)·∇)A(x, t) + (∇u(x, ◦t))A(x, t) = 0, (12) This stochastic equation is interpreted again in the Stratonovich sense. Equation (12) for d = 3 is equivalent by vector calculus identities to ∂tA+∇(u·A)− u×(∇×A) = 0. The latter has the form of Ohm’s law, E+ u×B = ηJ, (13) in the ideal limit of zero resistivity (η = 0) for an electric field E = −∂tA − ∇(u·A) and magnetic field B = ∇×A given by a vector potential A, with the electric current J = ∇×B. Taking the curl of (13) yields an induction equation ∂tB = ∇×(u×B)+η△B for the magnetic field. With this interpretation, the passive vector equation was introduced by Kazantsev [37] as a soluble model of the kinematic dynamo. (See also [38, 39, 40].) The “circulation” (or “holonomy”) of A along C is defined in Lagrangian form as ΦL(C, t) = A(t)·dx, (14) where C(t) is the material line advected by u(t) which started as line C at the initial time t = t0. Conservation of “circulation”, (d/dt)ΦL(C, t) = 0, follows formally from (12) for any space dimension d ≥ 1. It is rigorously true for the case of a smooth advecting velocity with α = 1 ([31], section 4.9). If C is a closed loop (1-cycle), then the line-integral (14) represents gauge-invariant magnetic flux and the conservation law corresponds to Alfvén’s theorem [20]. We now consider the generalization of this result for α < 1. For this purpose, it is useful to reformulate the passive vector equation (12) as a passive scalar in line-space: ∂tΦE(C, t) + dσ [ui(C(σ), t) + ũi(C(σ), ◦t)] δCi(σ) ΦE(C, t) = 0. (15) In this equation, ΦE(C, t) = A(t)·dx is the Eulerian circulation of A along a fixed (non- advected) line C. An exactly analogous reformulation of the incompressible Euler equation (as an active scalar in loop-space) was advanced some time ago by Migdal [41]. Note that conservation of circulations is just the formal solution of (15) by the method of characteristics. We shall take the equation (15) as our primitive formulation of the passive vector; one of the immediate advantages is that we can avoid (for the moment) the question how to define line-integrals over fractal lines. We then convert (15) to Ito formulation: ∂tΦE(C, t) = − dσ [u∗i (C(σ), t) + ũi(C(σ), t)] δCi(σ) ΦE(C, t) dσ′ Dij(C(σ), C(σ ′), t) δCi(σ)δCj(σ′) ΦE(C, t) (16) This stochastic equation is solved by the method of LeJan and Raimond [28, 29] (cf. footnote #1), writing it as a (backward) Ito integral and iterating to obtain ΦE(C, t) = Su(t, t ′)ΦE(C, t where the Markov operator semi-group S (t, t′) is defined by a Wiener chaos expansion. More intuitively, this solution is expressed as ΦE(C, t) = (dC ′, t′|C, t)ΦE(C ′, t′), t′ < t, (17) in terms of the turbulent diffusion of lines (backward in time). We see that the circulations are not conserved, except on average. This is precisely the “martingale property” that was conjectured (for solutions of incompressible Euler equations) in [18]. Note that this property imposes an irreversible arrow of time, since Eulerian circulations are given as averages over their past values, not future ones. This “generalized Alfvén theorem” should be related to dynamo action in the Kazantsev model [37, 38, 39, 40]. In the physical context of the dynamo, there is an additional resistive term η dxj∂i δσij (x) ΦE(C, t) on the righthand side of (15), where δ/δσij(x) is the “area derivative” in the loop calculus of Migdal [41]. This term breaks time-reversal symmetry and should select the backward-martingale solution (17) in the ideal limit η → 0. Of course, it cannot be ruled out a priori that the η → 0 limit of the resistive regularization and the λ → 0 limit for regularized velocity, as in (8), shall yield distinct weak solutions of the loop-equation (15), as occurs in the intermediate compressibility regime of the passive scalar problem [24, 25, 26, 28]. These results can be generalized to turbulent diffusion processes of higher-dimensional ma- terial objects, k-dimensional oriented submanifolds of M or k-cells Ck(t). The dual object is the passive k-form ωk, which satisfies (in Stratonovich sense) k + L ωk = 0 (18) with L the Lie-derivative along the vector field u ([31], section 4.9). This equation is formally equivalent to conservation of the integral invariants I(Ck, t) = Ck(t) ωk(t) (19) for any k-cell Ck(t) comoving with u [42]. Then k = 0 is the passive scalar, k = 1 the passive vector and k = d the passive density [24]. A theory similar to that developed here for k = 1 applies for any integer k. A unified approach to all these results is to consider directly the turbulent diffusion of the Lagrangian flow maps ξt,t , which satisfy the stochastic equation (d/dt)ξt,t (a) = u(ξt,t (a), ◦t), ξt ′,t′(a) = a. (20) In this framework one can derive for the distribution P [ξ, t] on maps exact analogues of the Liouville equation, in Stratonovich form (4) or Ito form (5). It is natural to formulate the problem as an infinite-dimensional diffusion in the Hilbert space H = L2(M,Rd) . It is known for the cases M = Rd or Td that the semigroup S(M) of Borel volume-preserving maps is a closed subset of this Hilbert space, and that the group G(M) of volume-preserving diffeomor- phisms is dense in S(M) for the L2-topology [43]. This construction is a close analogue of the “generalized Euler flows” of Brenier, but for the Cauchy initial-value problem. To summarize: We have outlined an approach to the study of material lines in a model of turbulent flow at infinite Reynolds number and to the dual problem of a passive vector in the same flow. The main conclusions are (1) that a non-degenerate diffusion should exist for material lines, generalizing the phenomenon of “spontaneous stochasticity” of material points, and (2) that the Kelvin/Alfvén theorem on conservation of circulations should generalize to a “martingale property”. Although the approach sketched here depends heavily on the white- noise character of the velocity field in time, we expect that similar results hold for more realistic velocity ensembles with the crucial property that realizations are rough (Hölder continuous) in space. See [18, 21] for related rigorous results on the solutions of incompressible fluid equations. The two properties discussed in the context of this model problem should be an essential feature of real fluid turbulence in the high Reynolds number limit. Acknowledgements. We thank S. Chen, M. Chertkov, L. Chevillard, R. Ecke, C. Meneveau, K. R. Sreenivasan and E. T. Vishniac for useful conversations. This work was supported by the NSF grant # ASE-0428325 at the Johns Hopkins University and by the Center for Nonlinear Studies at Los Alamos National Laboratory, where the research was begun. References [1] G. K. Batchelor, “The effect of homogeneous turbulence on material lines and surfaces,” Proc. Roy. Soc. Lond. 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(North-Holland, Amsterdam, 2003), pp.55–86. http://arxiv.org/abs/hep-th/9303130 http://arxiv.org/abs/hep-th/9306152 http://arxiv.org/abs/hep-th/9310088

0704.0264

Gluon Radiation of an Expanding Color Skyrmion in the Quark-Gluon Plasma

arXiv:0704.0264v1 [hep-ph] 2 Apr 2007 CCNY-HEP-07/x March 2007 Gluon Radiation of an Expanding Color Skyrmion in the Quark-Gluon Plasma Jian Dai1 Physics Department City College of the CUNY New York, NY 10031 Abstract The density of states and energy spectrum of the gluon radiation are calculated for the color current of an expanding hydrodynamic skyrmion in the quark gluon plasma with a semiclassical method. Results are compared with those in literatures. E-mail: [email protected] http://arxiv.org/abs/0704.0264v1 1 Introduction In this letter, we address the issue of gluon radiation during the hydrodynamic stage in the evolution of the deconfined hot QCD matter or quark gluon plasma (QGP) [1] (for review see for example [2]). The medium induced gluon radiation has been thoroughly explored in the context of final state partonic energy loss or “jet quenching” [3]. The spatially extended nuclear matter affects the processes of fragmentation and hadronization of the hard partons produced in the relativistic heavy ion collisions. Essentially all high p⊥ hadronic observables are affected at collider energies and the degree of the medium modification can give a characterization of the hot QCD matter in the deconfined phase. In principle, the medium induced radiation effect emerges from thermal QCD per se. However, in practice, different approximation schemes are applied giving consistent results [4, 5]. On the other hand, gluon radiation has also been considered in the context of gluon density saturation in the initial stage, where a strongly interacting gluonic atmosphere is crucial for the rapid local thermalization for the deconfined QCD matter [6]. The time evolution of the RHIC “fireball” can influence the observable particle production spectra. Given a strong initial interaction, the resulting state of matter is usually modeled as a relativistic fluid undergoing a hydrodynamic flow. Generalized fluid mechanics that charac- terizes the long-distance physics of the transport of color charges has been developed for this purpose [7] (for review see [8]). Recently, we discovered a type of single skyrmion solutions in color fluid [9]. Moreover, we found an interesting case in which the time-dependent skyrmion expands in time, which is in accordance with the expanding nature of the fireball generated in RHIC experiments [10]. The pattern of gluon radiation pertaining to the color current of these non-static configurations is an important character of this color skyrmion. So in this letter we calculate this radiation spectrum in a semiclassical approach. The main results from our calculation are the following. There is a fast fall-off in the UV side of the spectrum but a smooth peak dominates the intermediate energy. And in IR, a long tail is the characteristic feature. The organization of this paper is the following. In Sect. 2, after a brief review of the nonabelian fluid mechanics, we calculate the nonabelian current corresponding to the soliton solution. In Sect. 3, semiclassical gluon radiation is calculated. In Sect. 4, comparison of the radiation spectrum in our hydrodynamic approach and in other approaches is carried out. 2 Color current of an expanding soliton Given the thermalization of hot QCDmatter above the deconfinement transition temperature, the transport of the color charges in the volume of the nuclear size can be modeled by a nonlinear sigma model in a first-order formalism L = jµωµ − F (n)− geffJaµAaµ. (1) This nonlinear sigma model describes an ideal fluid system. The configuration of this fluid is described by a group element field U , which shows up in the velocity field ωµ ωµ = − Tr(σ3U †∂µU). (2) Conjugate to the velocity is the abelian charge current jµ. It is easy to see that the first term in the lagrangian density (1) gives rise to the canonical structure of the fluid system. The fact that we will consider only one abelian charge current means that U takes value in an SU(2) group. The information about the equation of state (EOS) of the fluid is contained in the second term, which is essentially the free energy density of the fluid. In fact, energy and pressure densities are given by the ideal fluid formula ǫ = F, p = nF ′ − F. (3) Here n is the invariant length of jµ, n2 = jµjµ. The third term is the gauge coupling of the fluid with an external gluon field Aaµ with an effective coupling geff . J aµ is the nonabelian charge current which is related to the abelian current by the Eckart factorization Jaµ = Qajµ where Qa is the nonabelian charge density of the fluid configuration Tr(σ3U †σaU). (4) For SU(2) group, a = 1, 2, 3. When the temperature is relatively high, we approximate the EOS by ǫ = 3p (5) which is known in relativistic fluid mechanics to describe radiation. As a result, the free energy density can be obtained by integrating Eq. (3), n4/3 (6) where β is a dimensionless constant of integration. In this case, and without an external gluon field, the fluid system in (1) possesses a class of expanding soliton solutions which can be studied via variational and collective coordinate methods [10]. U = U , R(t) ≈ R0( + 1)4/3θ(t) (7) where R0 and τ are the spacial and temporal characterizations of the variational soliton and θ(t) the usual step function in time direction. Physically, it is certainly very interesting to understand the origin of these two scales from a fundamental level. The approximation in (7) is valid provided τ ≪ R0. This condition enables us to define a small parameter . (8) For our purpose, we calculate the nonabelian current in (1) corresponding to the soliton solution in Eq. (7). To do so, the hedgehog ansatz is specified for the solution (7) U = cosφ+ iσ · x̂ sinφ (9) where x̂ is the unit vector and φ is given by the stereographic map sinφ = 1 + s2 , cosφ = ±1− s 1 + s2 . (10) We write s as the dimensionless coordinate x/R(t). The sign in the expression of cosφ signifies a topological charge which is the skyrmion number. The negative sign gives the skyrmion number +1 or a skyrmion and the positive sign the skyrmion number is −1 or an anti-skyrmion. We will take the positive sign in the following. By expressing the abelian current jµ in terms of the velocity ωµ through the equation of motion, we derive the following expression for the nonabelian current d3xJaµ = (1 + s2)6 · (ŝ23s2Ṙ2 − 1) · δa3 (1− 6s2 + s4) + 4ǫa3bŝbs(1− s2) + 8ŝ3ŝas2  −ŝ3s(1 + s2)Ṙ 2ŝ1ŝ3s 2 − 2ŝ2s 2ŝ2ŝ3s 2 + 2ŝ1s 2ŝ23s 2 − s2 + 1  .(11) The current in (11) has a natural form of a multipole expansion due to the skyrmion orienta- tion in the color space. In this letter we only consider the effect of the lowest mode and the effects of higher polarization will be considered elsewhere. The spherically symmetric part in the current is contained only in the third component d3xJa3 = −δa3 )3 d3s (1 + s2)6 P6(s) (12) where P6(s) = 1− 7s2 + 7s4 − s6. 3 Semiclassical gluon radiation Now we consider the interaction between the expanding color skyrmion and the hard partons. Since the transfer momentum between hard partons is in high order to that between hard parton and soliton, we expect a hierarchy between the partonic coupling gYM and the effective coupling geff . Accordingly, gluon self-interaction in terms like F aµν can be omitted so we can work with a free parton picture. Then the gauge coupling in (1) becomes the coupling between a classical current and a free quantum field for gluon. In this approximation, the lowest order semiclassical amplitude is given by iM = geff 〈1| d4xJaµÂaµ|0〉. (13) |0〉 and |1〉 are gluonic Fock vacuum and one-gluon state. The gluon factor in (13) is given by the wave function 〈1|Âaµ(x)|0〉 = ϕaεµ eik·x√ where the color and helicity parts ϕ, ε will be summed over eventually. Putting the current in, we have iM = A(k) dteiωt (1 + s2)6 e−iR(t)k·sP6(s) (15) where A(k) = −(2/β)3geffϕ3ε3/ 2ω. The spatial Fourier transformation can be completed analytically iM = B(k) dteiωt−R(t)kQ4(R(t)k) (16) where B(k) = π2A(k)/120 and Q4(x) = 5x 2 − 5x3 + x4. To go further, we need to specify R(t) in this equation to the form given in (7). This gives iM = B(k)e−iωτ η dteiηt−t 4/3) (17) where η = ωτ/(kR0) 3/4. With onshell condition ω = k, η = λκ1/4 where κ is defined to be R0k. Accordingly, )(geff ϕ3ε3e )( iM̃λ(κ) where iM̃λ(κ) = dteiλκ 1/4t−t4/3Q4(t 4/3) (19) The radiation spectrum is given by dE = kdN . E(k) is the total energy radiated over the entire time of expansion as a function of k. The number distribution is |M|2d3k (20) where the summation is over colors and helicities of the gluon. In a spherically symmetric setting, dN = ndk where n is the density of states n = 4πk2 |M|2. (21) By straightforward calculation, n = αR0λ 2κ−1/2|M̃λ(κ)|2, (22) = αλ2κ1/2|M̃λ(κ)|2. (23) where α ≡ (2π5/225)(g2eff /β6). The numerical results for λ = 1/15, 2/15, 1/5 are given in Fig. 1. 0.2 0.5 1 2 5 10 ��������� 0.2 0.5 1 2 5 10 ���������� Figure 1: Density of states and energy spectrum for λ = 1/5 (Black), 2/15 (Deep Gray) and 1/15 (Light Gray). 4 Comparison and discussion Understanding the pattern of gluon radiation in relativistic heavy ion collision processes is important for making an accurate determination of the physical mechanisms from the measurement of its decay products. In [6], the authors extracted the asymptotic behavior of the number density in small k is of the 1/k form. In our case, the asymptotic of the number density in small k is ∼ 1/ k. (See Fig. 2.) The difference comes from the fact that the medium size is taken to be infinitely large in [6] while in our case the medium size is characterized by the soliton size R0. So the IR behavior in our case is softer. For the case of jet quenching, the radiation energy lost is due to scattering off the hard quarks. A popular approach is to model the medium as a collection of colored static scattering 0.02 0.05 0.1 0.2 0.5 1 0.115 0.125 0.135 �!!!! ������������� Figure 2: n/(1/ κ) in small k for λ = .2 centers [11]. This approach can be extended to the expanding medium [4] though the gluon radiation by the expanding medium itself is not included. In fact, the medium induced gluon radiation is characterized by the frequency q̂L2 (24) where q̂ is the quenching parameter, estimated to be .04 ∼ .16GeV 2/fm, and L is the in- medium path length of a hard parton [12]. In general ωC is significantly larger than the characteristic momentum in our case 1/R0. So there is a hierarchy between the medium induced gluon radiation spectrum and the gluon radiation spectrum by the medium. Our hydrodynamical approach opens up another interesting possibility to address the eccentricity of the elliptic flow either intrinsically by considering the nonabelian color current or exogenously by considering the gluon radiation patterns. This will be the topic of the follow-up to this work. Acknowledgment. This work was supported by a CUNY Collaborative Research In- centive grant. The author has greatly benefited from the mentoring by V. P. Nair. References [1] PHENIX Collaboration, K. Adcox, et al, Nucl. Phys. A757 (2005) 184-283, nucl- ex/0410003; I. Arsene et al. BRAHMS collaboration, Nucl. Phys. A757 (2005) 1-27, nucl-ex/0410020; B. B. Back et al (PHOBOS), Nucl. Phys. A757 (2005) 28-101, nucl- ex/0410022; STAR Collaboration: J. Adams, et al, Nucl. Phys. A757 (2005) 102-183, nucl-ex/0501009. [2] Berndt Muller, James L. Nagle, nucl-th/0602029. [3] Alexander Kovner, Urs A. Wiedemann, “Gluon Radiation and Parton Energy Loss”, in Quark Gluon Plasma 3 Editors: R. C. Hwa and X. Wang World Scientific Singapore, hep-ph/0304151. [4] Carlos A. Salgado, Urs Achim Wiedemann, Phys. Rev. D68 (2003) 014008, hep- ph/0302184. [5] Urs A. Wiedemann, Nucl. Phys. B588 (2000) 303, hep-ph/0005129. [6] Yuri V. Kovchegov, Dirk H. Rischke, Phys. Rev. C56 (1997) 1084, hep-ph/9704201. [7] R. Jackiw, V.P. Nair, So-Young Pi, Phys. Rev. D62 (2000) 085018, hep-th/0004084; B. Bistrovic, R. Jackiw, H. Li, V.P. Nair, S.-Y. Pi, Phys. Rev. D67 (2003) 025013, hep-th/0210143. [8] R. Jackiw, V.P. Nair, S.-Y. Pi, A.P. Polychronakos, J. Phys.A. Math. Gen. 37 (2004) R327. [9] Jian Dai, V.P. Nair, Phys. Rev. D74 (2006) 085014, hep-ph/0605090. [10] Jian Dai, “Stability and Evolution of Color Skyrmions in the Quark-Gluon Plasma”, hep-ph/0612260. [11] M. Gyulassy, X. Wang, Nucl. Phys. B420 (1994) 583. [12] Miklos Gyulassy, Ivan Vitev, Xin-Nian Wang, Ben-Wei Zhang, “Jet Quenching and Radiative Energy Loss in Dense Nuclear Matter”, in Quark Gluon Plasma 3 Editors: R. C. Hwa and X. Wang World Scientific Singapore, nucl-th/0302077.

0704.0265

REM near-IR and optical multiband observations of PKS2155-304 in 2005

Astronomy & Astrophysics manuscript no. PKS2155˙astroph˙erratum c© ESO 2018 October 25, 2018 REM near-IR and optical multiband observations of PKS 2155-304 in 2005⋆ A. Dolcini1, F. Farfanelli2, S. Ciprini2, A. Treves1, S. Covino3, G. Tosti2, E. Pian4, B. Sbarufatti1, E. Molinari3, G. Chincarini3 ,5, F. M. Zerbi3, G. Malaspina3, P. Conconi3, L. Nicastro6, E. Palazzi6, V. Testa7, F. Vitali7, L. A. Antonelli7, J. Danziger4, G. Tagliaferri3, E. Meurs8, S. Vergani8, A. Fernandez-Soto9 , E. Distefano10, G. Cutispoto10, and F. D’Alessio7 1 Università degli Studi dell’Insubria, Dipartimento di Fisica e Matematica, via Valleggio 11, 22100 Como, Italy 2 Dipartimento di Fisica e Osservatorio Astronomico, Università di Perugia, Via. A. Pascoli, 06123 Perugia, Italy 3 INAF, Osservatorio Astronomico di Brera, via E. Bianchi 46, 23807 Merate (LC), Italy 4 INAF, Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34143 Trieste, Italy 5 Università degli Studi di Milano-Bicocca, Dipartimento di Fisica, Piazza delle Scienze, 3, 20126 Milan, Italy 6 INAF/IASF Bologna, via Gobetti 101, 40129 Bologna, Italy 7 INAF, Osservatorio Astronomico di Roma, via Frascati 33, 00040 Monteporzio Catone, Italy 8 Dunsink Observatory, Castleknock, Dublin 15, Ireland 9 Observatori Astronomic, Universitat de Valencia, Aptdo. Correos 22085, Valencia 46071, Spain 10 INAF, Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123 Catania, Italy Received 29 September 2006 / Accepted 26 March 2007 Abstract Context. Spectral variability is the main tool for constraining emission models of BL Lac objects. Aims. By means of systematic observations of the BL Lac prototype PKS 2155-304 in the infrared-optical band, we explore variability on the scales of months, days and hours. Methods. We made our observations with the robotic 60 cm telescope REM located at La Silla, Chile. VRIJHK filters were used. Results. PKS 2155-304 was observed from May to December 2005. The wavelength interval explored, the total number of photometric points and the short integration time render our photometry substantially superior to previous ones for this source. On the basis of the intensity and colour we distinguish three different states of the source, each of duration of months, which include all those described in the literature. In particular, we report the highest state ever detected in the H band. The source varied by a factor of 4 in this band, much more than in the V band (a factor ≈ 2). The source softened with increasing intensity, contrary to the general pattern observed in the UV-X-ray bands. On five nights of November we had nearly continuous monitoring for 2-3 hours. A variability episode with a time scale of τ ≈24 h is well documented, a much more rapid flare with τ=1-2 h, is also apparent, but is supported by relatively few points. Conclusions. The overall spectral energy distribution of PKS 2155-304 is commonly described by a synchrotron-self-Compton model. The optical infrared emission is however in excess of the expectation of the model, in its original formulation. This can be explained by a variation of the frequency of the synchrotron peak, which is not unprecedented in BL Lacs. Key words. galaxies: active - galaxies: BL Lacertae objects: PKS 2155-304 1. Introduction PKS 2155-304 (z=0.116, Falomo et al. 1991) is a prototype of high frequency peaked BL Lac objects. It has been observed in the entire electromagnetic spectrum, from radio to TeV gamma-rays. It was the target of several multifrequency campaigns, the main scope of which was to study the variability of the spectral energy distribution (SED), in order to constrain emission models. In particular we refer to the 1991 and 1994 campaigns involving IUE, ROSAT, ASCA, EUVE and ground based telescopes (see Edelson et al. 1995, Urry et al. 1997, and references therein). There were noticeable differences in source behaviour between these two epochs. While in 1991 the multiwavelength variability was almost achromatic, and the X-ray variation led that in the UV by a couple of hours, in 1994 the variability was more pronounced in X-rays than in UV-optical, with a lag of the latter ⋆ This paper is the corrected version astroph 0704.0265 published in A&A 469 503. It contains the material in the ”Errata Corrrige”, in press in A&A. http://arxiv.org/abs/0704.0265v2 2 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 Period of observation Nights of observation Number of photometric points Total exposure time May 6 129 14520 s September 8 159 18080 s October 3 102 11590 s November 21 1581 173540 s December 6 64 7030 s Table 1. Outline of observations accomplished in 2005. by two days. The general pattern was that of a hardening of the spectrum with increasing intensity. More recently Zhang et al. (2006b) studied a large set of data covering the period 2000-2005 obtained with the XMM-Newton satellite, which allowed a direct comparison of the X-ray and UV-optical band, the latter deriving from the Optical Monitor on board the satellite. The complexity of the variability pattern is confirmed. Some episodes of achromatic variation were detected, but a general tendency of increasing variability amplitude with increasing frequency, and spectral hardening with increasing intensity was found. Optical photometry has been performed by several groups in several occasions (see e.g. Miller et al. (1983), Smith et al. (1992), Xie et al. (1996), Paltani et al. (1997), Pesce et al. (1997), Fan & Lin (2000), Tommasi et al. (2001) and references therein). All this material is rather fragmented, consisting of few hours of observations during few nights. The difficulty of a systematic observing campaign covering many nights is partly overcome by the possibility of observing using remotely guided or robotic telescopes. The REM telescope, originally designed for a prompt detection of gamma ray bursts (see Molinari et al. (2006)), is particularly apt for photometric studies of BL Lacs (see also the previous results for PKS 0537-441 by Dolcini et al. 2005, and for 3C 454.3 by Fuhrmann et al. 2006) and, being located at La Silla (Chile), it is ideally fit to study PKS 2155-304. We report on extensive and intensive photometric campaign performed in 2005 in the V, R, I, J, H, K bands. For the total number of photometric points, for the time resolution (minutes) and spectral range this campaign seems to supersede all the IR-optical photometric material presented thus far. 2. REM, Photometric procedure, data analysis 2.1. REM The Rapid Eye Mount (REM) Telescope is a 60 cm fully robotic instrument. It has two cameras fed at the same time by a dichroic filter that allows the telescope to observe in the NIR (z’, J, H, K) as well as optical (I, R, V). Further information on the REM project may be found in Zerbi et al. (2001), Chincarini et al. (2003) and Covino et al. (2004). 2.2. Observations and data analysis REM observed the PKS 2155-304 field during May, September, October, November and December 2005 in VRIH bands. Only during three nights in September the telescope observed also in J and K filters. To allow intranight and short time-scale variability monitoring, very intensive observations (2-3 h, quasi-continuously) were made during five of the nights in November. An outline of the observations is reported in Table 1, while the complete log is only available in Table A.1 (see Appendix A): we report for each photometric point the band, the epoch, the integration time, the intensity and its uncertainty. Typical integration times are ≤100 s and statistical uncertainties are always ≤ 10% and ≤ 3% in the highest state (November 2005, see following). Reduction of the REM NIR and optical frames followed standard procedures. Photometric analysis of the frames was done using the GAIA1 and DAOPHOT packages (Stetson 1986). Relative calibration was obtained by calculating magnitude shifts relative to three bright isolated stars in the field, indicated by A, B, C in Fig. 1 (image taken from ESO Digitized Sky Survey2). The NIR frames were calibrated using the magnitudes of the A, B and C stars as reported in the 2MASS catalogue3. For the optical, we exposed on 2006 June 29 the standard field G156-31 (Landolt, 1992), and immediately after this the PKS 2155-304 field. We calculated the zero points which were then used to calibrate all of our data. The observed magnitudes in the REM filters for the reference objects A, B, and C are reported in Table 2. We have monitored the relative intensities of the A, B, C reference stars during the entire observation period, and we have detected no indication of variability within 0.1 mag (error on the average ≤0.01 mag). 1 http://star-www.dur.ac.uk/ pdraper/gaia/gaia.html 2 http://archive.eso.org/dss/dss 3 http://irsa.ipac.caltech.edu A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 3 Figure 1. PKS2155-304 field (DSS-1 survey). Letters indicate stars used for calibration. A B C RA 21:58:46.505 21:58:43.807 21:58:42.337 DEC -30:17:51.29 -30:17:15.71 -30:10:27.41 K 11.171±0.024 12.475±0.030 12.648±0.024 H 11.182±0.027 12.556±0.026 12.769±0.027 J 11.510±0.027 12.838±0.026 13.091±0.029 I 12.184±0.005 13.421±0.009 13.216±0.006 R 12.981±0.004 13.434±0.006 13.671±0.010 V 13.179±0.005 13.822±0.009 13.899±0.013 Table 2. Coordinates, IR and optical magnitudes for the reference stars. Note that we found significant deviations from the optical calibrations provided by the finding charts for AGN of the Heidelberg University4 (Hamuy & Maza, 1989). In particular the star C is also used as a calibrator by these authors and our optical zeropoint differs by about 0.3 mag from theirs. Relative and absolute calibration errors have been added in quadrature to the photometric error derived from the procedure. 3. Results 3.1. Long term variability In this section we report the results of the long term photometric analysis. The light curves in the H, R, I, V filters are given in Fig. 2. The intensity is normalized with respect to the average over the entire observation period. These averages are given in Table 3. It is immediately apparent that the total variability range is very different in the various filters, being a factor ≈ 4 in H and a factor ≈ 2 in V (see Table 3) . The shapes of the light curves are similar in the various filters. A flare-like structure is apparent in all filters at t ≈ 680 (first days of November). The ratio between the V- and H-band fluxes, designated as V/H, is reported in Fig. 3. In order not to introduce spurious effects due to small time scale variability, the V/H ratio has been computed for pairs of V and H measurements spaced apart in time by no more than 10 minutes. It seems that there are two main colour states: the source softens rather abruptly, in response to the November flare. On the basis of the light curve and the colour curve we divide the observations in three epochs: 1 500-525, 2 640-660, 3 670-725, expressed in MJD5. 4 http://www.lsw.uni-heidelberg.de/projects/extragalactic/charts/2155-304.html 5 For the Modified Julian Date we use the convention MJD=JD-2,453,000.5 4 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 500 550 600 650 700 500 550 600 650 700 500 550 600 650 700 H filter I filter R filter V filter 500 550 600 650 700 Figure 2. Normalized light curves of PKS 2155-304. Flux is reported in arbitrary unit (a. u.). In each boxes a typical error bar is plotted. Filter H I R V Average 114.9±3.3 34.45±6.5 30.89±5.13 30.70±5.05 Max value 156.5 46.4 38.3 37.4 Min value 36.5 19.1 16.2 16.2 Average ep.1 39.3±1.4 21.4±1.5 18.7±1.3 18.1±0.7 Average ep.2 65.9±5.2 28.4±3.3 27.2±2.5 20.3±3.4 Average ep.3 122.9±6.1 38.8±1.9 34.1±1.5 33.5±1.7 Table 3. Average intensities for all epochs and all filters. All data are in mJy units. Epoch 1 corresponds to May 2005 observa- tions, epoch 2 to September-October 2005 observations and epoch 3 to November-December 2005 observations. 3.2. Short time-scale variability We report in Fig. 4 the light curves for five nights in November 2005, when the observations were more intensive. All the nights belong to epoch 3, corresponding to the high state of the source. The mean intensity and the 1-sigma values for each night are given in Table 4. A χ2 analysis indicates that in each night the significance of variability is very high, but for the nights of Nov 4 and Nov 18 for the H band and Nov 19 for the V band. In the box of Nov 4 - V band we also report the photometry of a comparison star which illustrates directly the significance of the source variability. Though the shapes of intensity curves are different (see Fig. 4), there is a rather regular colour-intensity dependence (see Fig. 5) indicating harder states for higher intensities. We adopt the usual definition of time scale variability τ = 11+z d f/dt . Following Montagni et al. (2006), a variability time scale is taken as reliable if the light curve can be approximated with a linear dependence, and it contains at least 10 points. In particular this gives a time scale of ≈ 24 h for the November 4 night (Fig. 4, V band - Nov 4 box). The simultaneous H light curve does not show any regular variability. We note that on November 8 in the H curve there is a flare-like event. If one connects 4 points A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 5 Time (MJD) 500 525 550 575 600 625 650 675 700 725 Figure 3. V/H flux ratio evolution during 2005. Error bars are comparable with symbol size. Figure 4. Light curves in the H and V filters for five nigths in November 2005, when the observations were more intensive. Dates of observations are reported in each box. The solid line in V band - 4 Nov box results from a linear regression analysis. The solid line in H band - 8 Nov box connects the four points of the flare-like structure. In each box it is given a typical error bar. In V band - 4 Nov box the light curve of one comparison star is also plotted, with a fixed enhancement of 9 mJy. 6 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 Figure 5. V/H flux ratio versus intensity for the five more intensively observed nights of epoch 3. In each box a typical error bar is plotted. Night 4/11 8/11 18/11 19/11 20/11 Average H 119.3±1.7 119.3±3.0 120.4±2.1 124.5±2.8 130.8±3.0 Average I 39.1±0.6 36.4±0.5 38.7±0.6 38.7±0.6 38.3±0.7 Average R 38.5±0.8 36.40±.8 37.3±0.5 38.0±1.6 37.1±0.5 Average V 33.2±1.1 33.0±1.1 33.4±1.1 33.5±1.1 34.7±0.1 Table 4. Average intensities and 1-sigma values for all filters for all five nights with more intensive observations in November 2005. All values are in mJy units. as suggested in Fig. 4 H band - Nov 8 box, the time scale variability is as short as 1.5 h. Unfortunately the V light curve is too sparse to confirm the presence of the flare also in this band. 3.3. The NIR-Optical spectral energy distribution We had six filter coverage (K,H,J,I,R,V) during three nights of Sept. 2005 (epoch 2) and representative SEDs for these nights are reported in Fig. 6. The delays between exposures in the different filters are less than 20 minutes. Reddening corrections are less than 6% in V and have been neglected. A fit with a single power law yields α ≈0.9 and it is clearly not good. The main deviation derives from the J filter, exceeding substantially our photometric precision of about 10%. An improvement in the fit is obtained by using a broken power law with spectral indices α ≈0.4 for the IR data and α ≈0.9 for the optical data. For comparison, we report in Fig. 7 the SED of June 29, 2006, exposure used for calibration purpose: its profile is rather similar to that of Sept. 2005. At the other epochs the SED consists of 4 points (H, I, R, V), and in Figs. 8 and 10 we give representative examples of SEDs acquired on epoch 1 and 3. The time differences between observations at various filters are less than 20 minutes. A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 7 Frequency (Hz) 19/9 data 21/9 data 26/9 data Average data Figure 6. September 2005 spectra for observations including the K and J filters. The spectral fit on average data with a single power law yields a spectral index α=0.91±0.07. Frequency (Hz) Figure 7. 29 June 2006 spectrum. The spectral fit with a single power law yields a spectral index α=0.90±0.16. In Fig. 8, which refers to a low state, we report also the estimated contribution of the host galaxy, which was calculated adopting the H magnitude of the galaxy measured by Kotilainen et al. (1998) and the Mannucci at al. (2001) template spectrum for giant ellipticals. It is apparent that the contribution of the galaxy never exceeds 20% of the BL Lac signal. At the other epochs the contribution from galaxy is negligible and it is not relevant for explaining the excess in J with respect a single power law noted above. The epoch 2 photometry (Fig. 9) is compared with spectrophotometry obtained with the ESO 3.6m telescope by R. Falomo6 on July 25, 2001 (Sbarufatti et al. (2006)). The source was found in a similar, but somewhat lower brightness state and some deviations from a power law are apparent. The HRIV points at epoch 3 (Fig. 10) are roughly fitted by a single power law of α ≈1.3. In any case the comparison of the SEDs at the three epochs clearly indicate a softening with increasing intensity. 6 spectrum available at the ZBLLAC online library, http://www.oapd.inaf.it/zbllac http://www.oapd.inaf.it/zbllac 8 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 Figure 8. 13 May 2005 spectrum - epoch 1. We report also the spectrum of the host galaxy (see text). The spectral fit with a single power law yields a spectral index α=0.77±0.16. Figure 9. 19 September 2005 spectrum - epoch 2. For comparison we report the ESO 3.6m telescope spectrophotometry which correspond to a slightly lower state of the source. The spectral fit with a single power law yields a spectral index α=0.88±0.05. 4. Discussion A collection of near-IR/optical SEDs of PKS2155-304 obtained by various authors at different epochs is presented in Fig. 11 and in Table 5. Our data encompass all those reported in the literature. In the historical observations of PKS2155-304 the delays between exposures at different filters are typically of the order of hours, instead of about 10 minutes as in our data set. Comparing literature data with our data it is apparent that the maximum we observed on 20 November 2005 in the H filter light curve is the highest state ever reported in this band. Note that the V state was comparable with states reported in the literature, likely because the coverage of the source in the optical band is less sparse than that in the NIR. A most noticeable result of our photometry is the discovery of long term H-band variability, the amplitude of which is much larger than that in the optical. A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 9 Frequency (Hz) Figure 10. 4 November 2005 spectrum - epoch 3. The spectral fit with a single power law yields a spectral index α=1.32±0.25. Error bars are comparable with symbols size. Data set α V (mJy) This work (13/5/2005) 0.77±0.16 16.485±0.263 This work (19/9/2005) 0.88±0.05 24.370±0.238 This work (4/11/2005) 1.32±0.24 35.278±0.498 Bertone at al. (2000) 0.42±0.26 26.20±0.58 Pesce et al. (1997) 0.62±0.30 24.50±0.67 Zhang & Xie (1996) 0.62±0.16 22.90±0.63 Bersanelli et al. (1992) 0.61±0.38 51.88±1.56 (J band) Treves et al. (1989) (1/12/1983) 0.51±0.31 19.80±0.36 Treves et al. (1989) (11/11/1984) 0.51±0.41 26.20±0.48 Miller & McAlister (1983) 0.62±0.56 17.8 Table 5. Spectral index values and V values for all spectra plotted in Fig. 11. α vs V plot is reported in Fig. 12. In Fig. 12 we plot the spectral index vs the V magnitude, as reported in table 5. There is no apparent correlation. It is noticeable however that the highest state in all bands (our observation of Nov 2005) corresponds to a rather soft spectral shape. This contrasts with the usual source behaviour of hardening with increasing intensity, as found in the UV-X-ray band (see Introduction). It contrasts also with the short time scale variability, as reported in section 3.2. There is a general consensus that the blazar SED can be explained by the superposition of a synchrotron component, and an inverse Compton one due either to scattering off the synchrotron photons (synchrotron-self Compton, SSC), or to external photons like those of the broad line region or of a thermal disk (e.g. Tavecchio et al. 1998, Katarzynski et al. 2005). This results in a typical two-maxima shape of the blazar SED. In Fig. 13 we report examples of the SED modeling proposed for PKS 2155- 304, on the basis of data taken in 1997. The models are detailed in Chiappetti et al. (1999). The object is a typical HBL, with the synchrotron peak in the soft X-rays. A well known critical point of this model, is that the source size is essentially constrained by variability, and variability itself requires that the SED is constructed using simultaneous observations in all bands. A further step of the modelling consists in identifying the physical origin of the relativistic jet and of its variability, see e.g. Katarzynski & Ghisellini (2006). With this premise it is obvious that the optical-IR photometric study, non simultaneous with that in other regions of the SED, has only a limited relevance in clarifying the overall picture. However we would like to make some remarks. If the SSC models reported in Fig. 13 truly represent the behaviour of the SED in 1997, as suggested by the good match with the X-ray and TeV energy data, and if our 2005 optical-IR spectra are also due to the SSC mechanism, then the latter represent a different condition in the jet and point to different critical parameters within the SSC scenario. While the IR-optical spectrum in May 2005 (triangles) has the same shape as predicted in 1997, but different normalization, the November 2005 IR-optical spectrum is different in both 10 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 This work (13/5/2005) This work (19/9/2005) This work (4/11/2005) Bertone et al. 2000 (24/5/1996) Pesce et al. 1997 (19/5/1994) Zhang and Xie 1996 (5/7/1994) Bersanelli et al. 1992 (17/1/1987) Treves et al. 1989 (1/12/1983) Treves et al. 1989 (11/11/1984) Miller and McAlister 1983 (19/11/1981) Frequency (Hz) Figure 11. Different spectra of PKS2155-304 from observations at other epochs reported in the literature. Symbols correspond to following works: filled circles: this work (13/5/2005 data), filled squares: this work (19/9/2005 data), filled up triangles: this work (4/11/2005 data), open diamonds: Bertone et al. (2000; 24/5/1996 data), open circles: Pesce et al. (1997; 19/5/1994 data, the Hamuy & Maza (1989) calibration is used), open up triangles: Zhang and Xie (1996; 5/7/1994 data), open squares: Bersanelli et al. (1992; 17/1/1987 data), open crosses: Treves et al. (1989; 1/12/1983 data), open stars: Treves et al. (1989; 11/11/1984 data), asterisks: Miller and McAlister (1983; 19/11/1981 data). Spectral index values and V magnitudes for all data sets are reported in Table 5. shape and normalization. The May 2005 observation suggests that the synchrotron peak may be located at a frequency similar to the one observed in 1997 (approximately between extreme UV and soft X-rays), the total energy being somewhat higher (about a factor 2, see Figure 13) than observed in 1997. The slope of the November 2005 spectrum suggests instead a much lower synchrotron peak energy, around the IR-optical domain or even redward, i.e. about 2-3 orders of magnitude lower than observed in 1997 and inferred in May 2005. While a variation of the synchrotron peak energy of this amplitude and on this time scale (the September 2005 slope is intermediate between those of May and November 2005, suggesting a monotonic change) it is not unprecedented in blazars (Mkn501 exhibited a similar variation in a much more rapid time scale, Pian et al. 1998), this would be the first observation of this kind in PKS 2155-304. Therefore, our interpretation is only tentative, although supported by the large observed IR variability. Alternatively, in order to explain the optical-IR flux excess we observe in 2005 with respect to the SSC prediction based on the earlier multiwavelength data (Fig. 13), one could invoke a thermal component, possibly from hot dust associated with the “dusty torus” surrounding the central region of the active nucleus, as suggested in the cases of other blazars with excess in the optical-infrared band (De Diego et al. 1997, for blazar 3C 66A; Pian et al. 1999 for 3C 279; Pian et al. 2002, 2006, for blazar PKS 0537-441). However, this seems somewhat less likely, because high emission states, as observed by us, are expected to be dominated by non-thermal beamed relativistic radiation. The continuation of this and other similar optical-IR studies, which have been proven to be promising but do not provide enough information for a physical interpretation of the data, requires that the observations are extended to other wavelengths. Simultaneous observations over a large wavelength range is the only tool to provide the necessary information for a physical interpretationof the observed variability of blazars. REM monitorings of the kind reported here could be an effective trigger to A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 11 V (mJy) 15 17.5 20 22.5 25 27.5 30 32.5 35 Figure 12. α vs V plot for data reported in Fig. 11. Symbols are the same as used in Fig. 11. X-ray satellites, and programs along these lines are foreseen with SWIFT. Cross correlation procedures, which up to now have been limited mainly to the X-ray band (Zhang et al. 2005, 2006a, 2006b, Sembay et al. 2002, Edelson et al. 1995), would be extended to a much larger portion of the SED. Appendix A: Table of observations Filter Epoch (MJD) Integr time s Intensity mJy Sigma K 633.4872 120 73.993 3.640 K 633.4966 120 73.345 3.640 K 633.5012 120 73.382 2.938 K 633.5094 120 76.417 1.043 K 633.5254 120 72.309 2.698 K 635.4835 120 70.598 2.579 K 635.6612 120 66.373 2.810 K 635.6670 120 67.794 2.548 K 635.6736 120 67.857 2.096 K 635.6794 120 69.182 2.758 K 635.6832 120 66.987 2.263 K 639.8339 120 67.982 0.689 K 639.8355 120 67.345 0.702 H 503.7708 120 38.943 3.048 H 503.7723 120 39.412 2.977 H 503.7741 120 38.907 2.903 H 503.8028 120 40.628 3.180 H 503.8043 120 43.135 7.497 H 507.7738 120 38.373 0.314 H 507.7753 120 38.302 0.314 H 507.7839 120 38.692 0.317 H 507.7854 120 38.586 0.316 12 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 H 507.7873 120 38.835 0.601 H 507.7881 120 38.871 0.318 H 508.7487 120 37.881 1.999 H 508.7501 120 37.224 2.168 H 508.7516 120 37.771 2.127 H 508.7531 120 38.444 1.924 H 508.7549 120 37.847 1.146 H 508.7565 120 37.396 1.786 H 510.6465 120 39.339 2.005 H 510.7174 120 39.850 1.559 H 510.7192 120 39.557 1.584 H 510.7208 120 39.594 1.513 H 510.7229 120 40.274 1.466 H 510.7244 120 40.071 1.240 H 510.7379 120 39.484 1.258 H 510.7394 120 39.740 1.555 H 510.7413 120 39.960 1.382 H 510.7427 120 39.412 1.399 H 510.7448 120 40.219 2.635 H 510.7463 120 40.256 1.246 H 514.7073 120 38.799 1.271 H 514.7089 120 38.056 1.247 H 514.7107 120 38.515 1.297 H 514.7122 120 38.267 1.741 H 514.7143 120 37.881 1.379 H 514.7156 120 38.887 1.415 H 514.7068 120 37.881 2.448 H 514.7710 120 37.812 1.342 H 514.7734 120 37.259 1.288 H 514.7749 120 36.478 1.162 H 515.6989 120 40.929 2.272 H 515.7003 120 40.108 2.117 H 515.7022 120 40.966 2.833 H 515.7037 120 41.155 2.210 H 515.7085 120 40.703 3.926 H 515.7086 120 41.422 2.149 H 515.7091 120 41.042 2.316 H 515.7106 120 41.117 2.320 H 515.7124 120 41.498 2.266 H 515.7139 120 40.816 2.117 H 635.3400 120 61.436 2.148 H 635.3489 120 63.801 0.405 H 635.3552 120 63.566 0.794 H 635.3648 120 62.349 0.855 H 637.5426 120 57.494 0.952 H 637.5481 120 57.334 1.721 H 637.5502 120 57.230 1.591 H 637.5619 120 55.928 1.162 H 637.5779 120 55.671 1.427 H 642.6039 120 59.597 1.427 H 642.6054 120 59.597 1.394 H 643.4546 120 61.267 0.902 H 643.4561 120 61.948 0.509 H 643.4581 120 62.119 1.031 H 643.4596 120 62.925 1.627 H 643.4607 120 61.663 0.741 H 643.4631 120 62.607 0.992 H 645.5581 120 64.095 1.014 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 13 H 645.5622 120 65.528 1.049 H 645.5641 120 64.036 1.104 H 645.5655 120 63.860 2.105 H 646.4435 120 60.873 2.048 H 646.4464 120 60.817 2.036 H 646.4490 120 60.482 2.044 H 646.4510 120 60.705 3.685 H 646.4521 120 62.291 2.194 H 646.4536 120 61.834 3.389 H 650.5509 120 64.213 2.708 H 650.5524 120 66.134 2.500 H 650.5537 120 64.036 3.012 H 650.5552 120 62.810 5.245 H 650.5571 120 65.047 3.372 H 650.5586 120 64.629 2.247 H 650.5606 120 63.449 2.176 H 650.5621 120 65.770 2.425 H 650.5641 120 66.256 2.454 H 650.5709 120 64.808 4.160 H 650.5743 120 64.692 2.418 H 655.3672 120 73.253 2.759 H 655.3688 120 72.048 1.800 H 655.3708 120 71.322 1.508 H 655.3716 120 71.916 1.493 H 655.3741 120 71.718 1.505 H 655.3756 120 73.312 1.762 H 655.3772 120 71.454 1.996 H 655.3787 120 71.257 1.886 H 655.5449 120 71.652 1.500 H 655.5464 120 71.652 1.695 H 655.5481 120 70.474 1.475 H 655.5496 120 70.799 1.482 H 655.5515 120 71.191 1.490 H 655.5530 120 71.718 1.501 H 655.5548 120 72.115 1.641 H 655.5563 120 71.718 1.566 H 655.5581 120 72.381 2.305 H 655.5596 120 72.448 1.516 H 655.5615 120 71.536 1.562 H 655.5630 120 71.060 1.552 H 655.5648 120 71.652 1.565 H 655.5663 120 71.454 1.561 H 655.5681 120 71.119 1.620 H 655.5696 120 70.799 2.577 H 665.3660 120 118.912 3.138 H 678.3451 120 121.458 1.503 H 678.3466 120 116.957 1.524 H 678.3480 120 117.173 1.628 H 678.3501 120 121.458 1.687 H 678.3516 120 120.455 1.491 H 678.3531 120 121.570 1.689 H 678.3551 120 117.497 1.821 H 678.3566 120 120.123 3.080 H 678.3581 120 119.902 1.562 H 678.3604 120 121.123 2.447 H 678.3619 120 117.822 1.971 H 678.3634 120 119.131 1.601 H 678.3656 120 120.012 1.932 14 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 H 678.3671 120 119.571 1.661 H 678.3685 120 119.022 2.584 H 678.3707 120 119.681 1.927 H 678.3723 120 117.065 1.885 H 678.3738 120 117.931 1.460 H 678.3759 120 118.366 1.500 H 678.3774 120 118.584 1.593 H 678.3788 120 118.366 1.767 H 678.3811 120 118.803 1.506 H 678.3826 120 118.803 1.470 H 678.3840 120 118.803 1.470 H 678.3862 120 117.822 1.493 H 678.3877 120 118.693 2.064 H 678.3892 120 117.931 1.760 H 678.3911 120 117.931 1.638 H 678.3926 120 119.022 1.653 H 678.3942 120 119.902 1.790 H 678.3963 120 118.475 1.466 H 678.3978 120 118.039 1.461 H 678.3993 120 118.257 1.765 H 678.4014 120 118.039 1.461 H 678.4028 120 117.605 1.968 H 678.4043 120 118.912 1.472 H 678.4064 120 120.123 1.728 H 678.4078 120 121.123 1.499 H 678.4093 120 119.792 2.164 H 678.4114 120 118.584 2.225 H 678.4128 120 117.281 1.528 H 678.4143 120 117.281 2.457 H 678.4164 120 117.497 1.754 H 678.4178 120 115.140 1.500 H 678.4193 120 119.022 1.776 H 678.4214 120 117.822 1.535 H 678.4218 120 118.257 1.643 H 678.4244 120 115.352 1.503 H 678.4265 120 118.912 1.711 H 678.4280 120 116.098 1.437 H 678.4294 120 120.900 1.496 H 678.4346 120 119.131 1.601 H 678.4362 120 120.677 2.019 H 678.4376 120 120.677 1.494 H 678.4398 120 120.789 1.495 H 678.4412 120 120.566 3.387 H 678.4427 120 121.123 2.359 H 678.4447 120 122.019 1.891 H 678.4462 120 120.789 1.495 H 678.4476 120 120.344 2.092 H 678.4497 120 120.900 2.811 H 678.4512 120 121.794 2.372 H 678.4526 120 121.794 2.372 H 678.4546 120 120.012 2.605 H 678.4562 120 120.566 3.189 H 678.4576 120 121.906 1.693 H 678.4597 120 121.458 1.882 H 678.4612 120 122.582 2.214 H 678.4626 120 121.794 1.887 H 682.3480 120 118.693 1.260 H 682.3495 120 117.497 1.152 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 15 H 682.3510 120 116.634 3.660 H 682.3532 120 119.681 1.270 H 682.3547 120 119.681 1.270 H 682.3562 120 119.131 3.121 H 682.3583 120 119.351 1.170 H 682.3597 120 117.822 1.155 H 682.3618 120 118.693 1.163 H 682.3633 120 118.257 1.851 H 682.3647 120 118.257 1.159 H 682.3669 120 118.366 2.313 H 682.3684 120 117.173 1.192 H 682.3700 120 116.420 1.575 H 682.3721 120 116.420 1.184 H 682.3735 120 118.803 1.689 H 682.3771 120 116.527 3.454 H 682.3771 120 117.389 1.246 H 682.3785 120 117.389 1.246 H 682.3801 120 117.497 1.928 H 682.3821 120 119.681 1.329 H 682.3836 120 120.900 1.556 H 682.3735 120 120.677 1.183 H 682.3872 120 118.148 1.254 H 682.3887 120 119.902 1.175 H 682.3903 120 117.713 1.757 H 682.3917 120 118.693 1.260 H 682.3938 120 117.065 1.747 H 682.3953 120 119.131 2.328 H 682.3968 120 118.148 2.121 H 682.3989 120 116.849 2.006 H 682.4005 120 116.634 2.662 H 682.4019 120 119.131 1.693 H 682.4041 120 121.011 1.284 H 682.4055 120 118.257 2.216 H 682.4070 120 119.351 3.641 H 682.4156 120 120.234 1.547 H 682.4171 120 122.695 2.299 H 682.4186 120 125.900 3.406 H 682.4222 120 132.198 2.373 H 682.4257 120 119.571 3.235 H 682.4271 120 118.693 5.296 H 682.4305 120 120.677 4.421 H 682.4324 120 119.571 3.235 H 682.4338 120 121.011 1.284 H 682.4359 120 117.931 1.310 H 682.4374 120 118.803 1.860 H 682.4388 120 121.682 1.418 H 682.4398 120 123.488 5.620 H 682.4412 120 123.148 1.666 H 682.4427 120 120.012 4.503 H 682.4449 120 119.571 1.464 H 682.4464 120 113.143 1.319 H 682.4479 120 114.189 2.702 H 682.4500 120 120.344 1.549 H 682.4506 120 122.582 1.830 H 682.4543 120 118.039 2.596 H 682.4558 120 126.714 1.551 H 682.4573 120 123.375 7.051 H 683.3333 120 114.611 0.668 16 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 H 683.3348 120 112.312 1.052 H 683.3363 120 114.717 0.985 H 683.3385 120 112.002 0.653 H 683.3401 120 112.105 0.653 H 683.3415 120 113.560 0.662 H 683.3435 120 113.770 1.442 H 683.3451 120 113.665 0.662 H 683.3466 120 115.885 1.568 H 683.3486 120 113.770 0.809 H 683.3501 120 114.505 0.737 H 683.3516 120 113.665 0.808 H 683.3535 120 113.143 0.804 H 683.3551 120 117.605 0.685 H 683.3566 120 114.400 1.258 H 683.3585 120 115.140 0.671 H 683.3600 120 116.849 0.681 H 683.3616 120 117.065 1.005 H 683.3636 120 112.519 0.656 H 683.3651 120 115.246 0.819 H 683.3666 120 112.830 0.802 H 683.3687 120 115.034 0.818 H 683.3702 120 114.928 0.740 H 683.3717 120 113.979 1.739 H 683.3739 120 114.190 0.812 H 683.3754 120 112.934 0.727 H 683.3769 120 115.565 2.991 H 683.3795 120 112.519 1.054 H 683.3810 120 118.584 0.691 H 683.3826 120 114.400 0.736 H 683.3845 120 112.416 2.109 H 683.3860 120 115.459 0.991 H 683.3875 120 114.084 0.811 H 683.3889 120 115.034 0.740 H 683.3904 120 113.665 0.890 H 683.3920 120 116.527 0.750 H 684.3339 120 120.234 1.488 H 684.3353 120 119.681 1.662 H 684.3368 120 121.123 1.743 H 684.3389 120 119.571 1.558 H 684.3403 120 119.792 1.561 H 684.3441 120 119.681 1.662 H 684.3456 120 122.695 2.570 H 684.3470 120 123.148 1.983 H 684.3484 120 123.148 1.983 H 684.3505 120 119.792 1.929 H 684.3519 120 121.682 1.585 H 684.3541 120 120.900 1.624 H 684.3556 120 122.356 1.700 H 684.3571 120 122.356 1.514 H 684.3591 120 120.123 1.487 H 684.3606 120 120.900 1.532 H 684.3621 120 121.682 2.641 H 684.3644 120 120.789 1.945 H 684.3660 120 121.011 1.577 H 684.3674 120 121.682 3.418 H 684.3696 120 116.205 1.514 H 684.3711 120 122.244 1.698 H 684.3726 120 122.131 2.291 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 17 H 684.3748 120 122.356 1.514 H 684.3762 120 120.234 3.279 H 684.3777 120 123.034 1.653 H 684.3798 120 122.808 2.055 H 684.3813 120 123.034 1.523 H 684.3828 120 124.059 1.536 H 684.3848 120 122.808 1.767 H 684.3863 120 121.906 2.646 H 684.3878 120 124.516 2.896 H 684.3898 120 122.582 1.517 H 684.3912 120 122.019 1.639 H 684.3927 120 122.469 1.596 H 684.3948 120 124.861 1.627 H 684.3962 120 123.602 1.778 H 684.3977 120 121.123 1.808 H 685.3274 120 125.091 1.945 H 685.3289 120 121.346 1.851 H 685.3303 120 127.769 2.978 H 685.3324 120 120.123 1.803 H 685.3409 120 119.792 3.144 H 685.3424 120 116.957 2.644 H 685.3439 120 117.389 2.202 H 685.3459 120 113.770 2.201 H 685.3474 120 117.389 2.202 H 685.3489 120 114.084 2.500 H 685.3510 120 114.506 2.358 H 685.3525 120 118.803 1.888 H 685.3540 120 115.352 2.038 H 685.3561 120 115.992 1.741 H 685.3575 120 116.313 1.746 H 685.3590 120 114.295 1.715 H 685.3610 120 115.459 1.795 H 685.3625 120 117.605 2.206 H 685.3639 120 118.257 3.015 H 685.3660 120 116.527 1.778 H 685.3676 120 119.022 2.451 H 685.3690 120 115.885 2.386 H 685.3711 120 116.420 2.397 H 685.3726 120 115.671 1.838 H 685.3740 120 115.992 1.741 H 685.3761 120 116.527 1.947 H 685.3776 120 119.461 1.793 H 685.3790 120 115.246 1.979 H 685.3810 120 117.822 1.768 H 685.3825 120 114.928 2.223 H 685.3840 120 117.497 1.763 H 689.3300 120 121.682 1.856 H 689.3315 120 120.900 1.844 H 689.3330 120 119.792 1.827 H 689.3351 120 120.344 1.913 H 689.3365 120 116.205 1.995 H 689.3380 120 119.792 1.827 H 689.3401 120 118.039 1.972 H 689.3423 120 120.455 2.723 H 689.3438 120 119.461 1.858 H 689.3458 120 121.794 2.035 H 689.3473 120 119.902 2.003 H 689.3487 120 121.011 2.138 18 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 H 689.3508 120 119.681 1.861 H 689.3523 120 117.822 1.797 H 689.3537 120 117.931 1.799 H 689.3557 120 123.034 2.055 H 689.3572 120 118.912 4.965 H 689.3586 120 118.584 2.095 H 690.3843 120 135.402 1.676 H 690.3857 120 130.866 8.745 H 690.3892 120 156.468 6.088 H 690.3907 120 133.791 6.165 H 690.3922 120 132.442 1.543 H 690.3943 120 132.809 1.209 H 690.3958 120 131.712 1.072 H 690.3972 120 131.470 1.196 H 690.3994 120 134.408 1.094 H 690.4009 120 136.278 1.109 H 690.4023 120 133.668 3.077 H 690.4044 120 133.545 1.146 H 690.4059 120 136.530 1.111 H 690.4073 120 136.153 1.891 H 690.4094 120 136.781 1.499 H 690.4109 120 132.686 1.207 H 690.4124 120 133.422 1.145 H 690.4144 120 134.532 1.224 H 690.4158 120 135.278 1.393 H 690.4172 120 134.161 1.221 H 690.4193 120 135.902 2.436 H 690.4208 120 134.904 1.158 H 690.4222 120 137.666 2.020 H 690.4244 120 138.047 2.360 H 690.4259 120 133.791 1.217 H 690.4274 120 137.160 1.905 H 690.4297 120 136.404 1.171 H 690.4309 120 135.153 0.992 H 690.4324 120 138.812 1.521 H 691.3710 120 131.955 1.019 H 691.3724 120 123.148 2.555 H 691.3739 120 133.668 2.891 H 691.3760 120 135.777 2.111 H 691.3774 120 150.531 4.325 H 691.3789 120 134.285 1.639 H 691.3812 120 133.176 2.647 H 691.3827 120 135.527 1.334 H 691.3842 120 132.809 2.524 H 691.3864 120 142.176 1.193 H 691.3878 120 137.033 1.349 H 691.3893 120 134.408 1.751 H 691.3913 120 140.355 5.166 H 691.3928 120 135.527 1.766 H 691.3943 120 141.914 1.849 H 691.3964 120 135.153 1.761 H 691.3977 120 136.781 0.972 H 691.3999 120 126.948 0.980 H 692.3336 120 119.022 2.366 H 692.3351 120 118.366 2.780 H 692.3366 120 118.803 2.361 H 692.3387 120 122.469 3.639 H 692.3402 120 128.952 2.681 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 19 H 692.3417 120 118.257 2.777 H 692.3439 120 120.455 2.343 H 692.3453 120 118.584 3.864 H 692.3468 120 118.803 2.311 H 692.3489 120 120.012 2.641 H 692.3503 120 121.011 3.349 H 692.3518 120 118.693 2.726 H 692.3539 120 116.634 3.385 H 692.3553 120 119.792 2.751 H 692.3569 120 119.681 2.690 H 692.3590 120 117.714 3.835 H 692.3605 120 119.241 2.370 H 692.3619 120 118.584 2.916 H 692.3717 120 124.631 2.448 H 692.3729 120 121.346 3.439 H 692.3743 120 118.912 2.517 H 692.3759 120 119.131 2.400 H 692.3778 120 119.681 2.943 H 692.3793 120 119.681 3.392 H 692.3807 120 117.065 2.327 H 692.3828 120 120.566 2.507 H 692.3843 120 117.714 2.371 H 692.3857 120 119.022 2.366 H 692.3878 120 117.822 2.315 H 692.3907 120 120.234 2.459 H 692.3922 120 120.677 2.348 H 692.3937 120 120.900 2.839 H 692.3959 120 118.257 3.352 H 692.3973 120 121.235 2.847 H 692.3988 120 120.789 2.433 H 692.4003 120 122.582 2.595 H 692.4017 120 121.458 2.525 H 692.4032 120 119.131 2.340 H 692.4053 120 122.582 2.385 H 692.4067 120 120.900 2.403 H 692.4082 120 119.681 2.811 H 692.4103 120 122.356 2.432 H 692.4118 120 119.461 3.467 H 692.4133 120 121.235 2.668 H 692.4155 120 121.346 2.384 H 692.4169 120 123.716 2.407 H 692.4184 120 118.148 2.298 H 692.4500 120 118.803 2.790 H 692.4514 120 123.148 2.419 H 692.4551 120 120.789 2.715 H 692.4565 120 119.461 2.374 H 692.4580 120 116.527 2.423 H 692.4602 120 122.469 2.876 H 692.4616 120 121.011 2.377 H 693.3339 120 124.631 1.014 H 693.3353 120 124.516 1.364 H 693.3368 120 123.261 1.618 H 693.3389 120 123.602 1.440 H 693.3403 120 124.516 1.451 H 693.3417 120 121.794 1.692 H 693.3438 120 121.011 1.410 H 693.3453 120 120.123 1.093 H 693.3467 120 121.570 0.989 20 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 H 693.3489 120 124.173 1.130 H 693.3503 120 122.695 1.117 H 693.3519 120 121.123 1.102 H 693.3541 120 119.351 1.025 H 693.3556 120 122.131 1.111 H 693.3571 120 123.261 1.122 H 693.3593 120 121.458 1.503 H 693.3607 120 129.189 2.209 H 693.3622 120 122.695 1.263 H 693.3643 120 123.488 1.195 H 693.3658 120 122.244 1.112 H 693.3673 120 120.455 1.096 H 693.3694 120 122.131 1.792 H 693.3708 120 120.234 1.032 H 693.3723 120 122.131 0.994 H 693.3745 120 123.261 1.712 H 693.3760 120 121.570 1.176 H 693.3774 120 121.682 1.882 H 693.3795 120 121.906 1.984 H 693.3798 120 123.261 4.246 H 693.3813 120 121.458 2.901 H 693.3834 120 125.091 1.288 H 693.3849 120 120.566 1.035 H 693.3878 120 124.516 1.133 H 693.3892 120 125.784 2.151 H 693.3907 120 122.808 1.520 H 693.3927 120 125.322 1.373 H 693.3942 120 124.861 1.368 H 693.3959 120 125.206 1.139 H 693.3977 120 126.831 2.486 H 693.3992 120 126.132 1.220 H 693.4006 120 130.625 3.120 H 693.4022 120 127.182 3.482 H 693.4037 120 125.669 2.463 H 693.4052 120 124.746 1.071 H 693.4073 120 124.287 2.861 H 693.4088 120 125.553 1.143 H 693.4091 120 125.669 4.777 H 693.4126 120 125.437 3.655 H 693.4141 120 126.598 3.466 H 693.4155 120 127.652 1.873 H 693.4176 120 124.631 1.636 H 693.4191 120 126.016 1.654 H 693.4205 120 124.631 1.928 H 693.4226 120 125.322 2.040 H 693.4241 120 124.746 2.340 H 693.4256 120 125.322 4.429 H 693.4276 120 126.248 5.251 H 693.4291 120 128.714 4.663 H 693.4305 120 128.359 6.955 H 693.4327 120 127.887 7.968 H 693.4342 120 129.070 2.860 H 693.4357 120 132.077 6.561 H 693.4377 120 126.016 6.373 H 693.4392 120 129.786 6.214 H 693.4407 120 126.948 7.566 H 693.4421 120 126.714 6.067 H 693.4436 120 130.986 8.754 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 21 H 693.4451 120 129.427 5.964 H 693.4471 120 124.746 9.127 H 693.4487 120 122.582 1.115 H 693.4501 120 124.287 1.067 H 693.4522 120 121.458 1.782 H 693.4537 120 120.455 1.165 H 693.4551 120 123.148 1.616 H 694.3585 120 126.598 1.567 H 694.3599 120 130.866 1.659 H 694.3615 120 125.784 1.557 H 694.3604 120 123.944 1.921 H 694.3649 120 126.248 1.563 H 694.3665 120 127.299 1.900 H 694.3685 120 123.944 1.921 H 694.3700 120 128.714 1.677 H 694.3715 120 128.241 1.587 H 694.3735 120 127.417 1.902 H 694.3750 120 129.666 5.188 H 694.3765 120 128.241 1.845 H 694.3785 120 129.905 2.530 H 694.3800 120 130.505 2.183 H 694.3815 120 129.382 1.639 H 694.3834 120 130.025 3.869 H 694.3849 120 129.308 4.726 H 694.3864 120 129.905 3.331 H 694.3884 120 129.308 3.316 H 694.3899 120 128.477 2.888 H 694.3913 120 127.769 1.581 H 694.3934 120 127.534 1.579 H 694.3948 120 130.625 3.140 H 694.3964 120 130.145 2.925 H 694.3983 120 130.265 1.697 H 694.3998 120 130.025 2.439 H 694.4013 120 130.986 1.707 H 694.4034 120 129.308 4.726 H 694.4049 120 131.349 2.654 H 694.4064 120 131.228 1.663 H 694.4078 120 131.955 2.764 H 694.4093 120 130.866 1.659 H 694.4108 120 126.714 1.606 H 694.4129 120 131.470 2.286 H 694.4144 120 128.833 1.633 H 694.4159 120 128.714 1.677 H 694.4175 120 131.228 2.113 H 694.4189 120 130.625 1.656 H 694.4204 120 132.809 1.683 H 694.4219 120 131.834 1.671 H 694.4233 120 132.931 1.984 H 694.4248 120 136.781 1.734 H 694.4270 120 136.153 2.554 H 694.4291 120 138.174 4.931 H 694.4307 120 133.914 3.762 H 694.4321 120 131.349 1.626 H 694.4341 120 134.532 1.705 H 694.4355 120 130.745 2.187 H 694.4370 120 132.564 1.727 H 694.4385 120 132.320 2.130 H 694.4399 120 132.564 2.878 22 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 H 694.4414 120 130.625 1.879 H 694.4435 120 130.265 1.809 H 694.4450 120 132.931 1.685 H 694.4465 120 132.564 3.083 H 694.4479 120 132.320 3.393 H 694.4495 120 133.176 2.498 H 694.4509 120 137.286 1.789 H 694.4529 120 133.299 1.650 H 694.4544 120 137.920 2.686 H 694.4559 120 135.527 4.146 H 695.3516 120 126.481 0.737 H 695.3531 120 128.241 1.476 H 695.3546 120 128.005 0.939 H 695.3566 120 126.365 1.029 H 695.3580 120 126.481 0.737 H 695.3596 120 124.631 0.642 H 695.3616 120 124.631 2.313 H 695.3630 120 126.132 0.735 H 695.3645 120 127.887 1.694 H 695.3667 120 129.786 0.952 H 695.3682 120 125.553 0.824 H 695.3696 120 126.714 0.738 H 695.3711 120 124.746 0.819 H 695.3726 120 128.952 1.156 H 695.3740 120 130.265 1.277 H 695.3765 120 129.427 0.754 H 695.3779 120 126.714 1.242 H 695.3794 120 129.666 1.493 H 695.3815 120 124.631 0.642 H 695.3829 120 124.402 1.115 H 695.3844 120 125.669 1.775 H 695.3857 120 125.437 1.229 H 695.3879 120 132.564 1.990 H 695.3894 120 128.359 0.842 H 695.3915 120 128.596 1.817 H 695.3929 120 126.481 0.737 H 695.3944 120 123.602 0.811 H 695.3965 120 125.322 0.645 H 695.3979 120 129.189 1.939 H 695.3994 120 130.745 0.673 H 695.4015 120 124.746 0.915 H 695.4030 120 128.596 1.260 H 695.4044 120 128.005 1.696 H 695.4065 120 123.716 0.637 H 695.4080 120 129.905 1.835 H 695.4095 120 126.365 0.927 H 696.3481 120 129.070 1.375 H 696.3496 120 126.016 1.437 H 695.1936 120 129.547 1.477 H 696.3533 120 129.189 1.748 H 696.3547 120 131.955 1.505 H 696.3562 120 132.442 1.510 H 696.3583 120 128.833 1.417 H 696.3597 120 130.625 1.436 H 696.3612 120 132.564 1.877 H 696.3633 120 131.470 1.861 H 696.3647 120 133.791 1.587 H 696.3662 120 129.308 1.831 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 23 H 696.3677 120 128.596 1.820 H 696.3698 120 126.598 1.444 H 696.3713 120 127.652 1.404 H 696.3727 120 128.714 1.665 H 696.3749 120 131.470 1.499 H 696.3763 120 128.477 1.738 H 696.3778 120 130.745 1.438 H 696.3815 120 130.986 2.217 H 696.3829 120 130.745 1.691 H 696.3844 120 131.228 2.319 H 696.3865 120 129.905 2.013 H 696.3879 120 133.054 2.351 H 696.3894 120 130.986 1.554 H 696.3915 120 130.505 1.765 H 696.3930 120 130.265 1.612 H 696.3945 120 134.038 1.897 H 696.3966 120 132.199 2.142 H 696.3981 120 132.564 1.641 H 696.3995 120 131.107 1.556 H 696.4016 120 130.986 2.030 H 696.4031 120 130.986 3.365 H 696.4053 120 131.712 7.665 H 696.4073 120 131.712 1.502 H 696.4088 120 131.107 2.935 H 699.3295 120 124.976 1.120 H 699.3309 120 127.887 1.259 H 699.3325 120 124.402 1.225 H 699.3345 120 124.058 1.785 H 699.3360 120 121.011 1.256 H 699.3375 120 122.356 1.761 H 699.3396 120 123.716 1.159 H 699.3411 120 125.437 1.175 H 699.3426 120 123.944 2.071 H 699.3454 120 123.944 3.210 H 699.3469 120 123.944 1.358 H 699.3484 120 123.944 1.161 H 699.3505 120 120.123 2.904 H 699.3521 120 123.716 1.433 H 699.3535 120 123.148 1.426 H 699.3555 120 122.131 1.850 H 699.3569 120 122.244 1.145 H 699.3584 120 123.944 2.474 H 699.3605 120 124.631 3.879 H 699.3619 120 125.091 1.707 H 699.3634 120 125.437 1.536 H 699.3654 120 122.921 1.862 H 699.3669 120 123.148 2.872 H 699.3683 120 121.458 1.089 H 699.4052 120 126.831 1.137 H 699.3718 120 121.123 1.192 H 699.3734 120 123.602 1.217 H 699.3754 120 127.299 1.474 H 699.3768 120 127.652 2.654 H 699.3784 120 130.265 1.282 H 699.3821 120 126.481 1.465 H 699.3835 120 128.833 1.207 H 699.3850 120 126.948 3.288 H 699.3871 120 126.831 1.248 24 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 H 699.3885 120 129.666 3.247 H 699.3900 120 130.866 3.389 H 699.3922 120 126.831 3.176 H 699.3936 120 125.437 1.997 H 699.3950 120 125.091 1.371 H 699.3971 120 123.716 1.356 H 699.3986 120 122.356 1.146 H 699.4000 120 123.830 1.110 H 699.4023 120 126.948 1.250 H 699.4038 120 122.469 1.147 H 699.4052 120 124.402 1.225 H 699.4073 120 126.481 5.734 H 699.4088 120 124.287 1.978 H 699.4103 120 125.322 1.534 H 699.4123 120 124.402 1.441 H 699.4138 120 122.921 1.769 H 699.4153 120 125.784 1.378 H 699.4173 120 125.437 2.196 H 699.4188 120 124.516 1.524 H 699.4202 120 123.034 1.425 H 699.4224 120 123.944 1.220 H 699.4238 120 122.808 1.274 H 699.4253 120 121.906 1.492 H 700.3828 120 124.059 1.010 H 700.3843 120 123.944 1.128 H 700.3857 120 125.553 1.215 H 700.3879 120 124.402 1.068 H 700.3893 120 122.131 1.111 H 700.3908 120 129.427 1.053 H 700.3929 120 126.715 1.088 H 700.3943 120 126.831 1.306 H 700.3958 120 122.356 1.113 H 700.3978 120 127.534 1.579 H 700.3994 120 127.065 1.156 H 700.4009 120 125.206 1.643 H 700.4030 120 125.553 1.376 H 700.4044 120 128.952 1.413 H 700.4059 120 124.516 1.926 H 700.4079 120 125.669 1.144 H 700.4091 120 126.365 1.385 H 700.4108 120 125.553 1.078 H 700.4129 120 126.249 1.221 H 700.4143 120 125.206 1.019 H 700.4158 120 124.976 1.287 H 700.4178 120 124.631 1.206 H 700.4193 120 126.365 2.371 H 700.4208 120 124.402 1.633 H 700.4244 120 127.652 1.096 H 700.4259 120 127.065 1.392 H 700.4274 120 126.132 1.027 H 700.4288 120 126.249 1.149 H 700.4291 120 125.206 2.141 H 700.4305 120 130.745 2.344 H 700.4326 120 127.417 3.488 H 700.4341 120 126.831 2.273 H 700.4355 120 123.944 1.627 H 700.4376 120 124.861 3.200 H 700.4391 120 124.516 1.827 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 25 H 700.4406 120 127.887 1.401 H 700.4420 120 126.715 1.859 H 700.4435 120 123.602 1.530 H 700.4449 120 127.769 1.489 H 700.4464 120 124.173 1.130 H 700.4479 120 123.602 1.354 H 700.4499 120 122.695 1.430 H 700.4514 120 123.375 1.059 H 700.4529 120 125.553 1.744 H 700.4549 120 124.516 1.069 H 700.4448 120 123.830 1.357 H 700.4463 120 123.375 1.352 H 700.4598 120 123.716 1.062 H 700.4614 120 126.948 1.228 H 700.4628 120 120.012 1.236 H 700.4642 120 122.469 1.261 H 701.3506 120 117.281 1.487 H 701.3521 120 117.389 1.752 H 701.3535 120 120.012 1.521 H 701.3558 120 122.244 2.293 H 701.3572 120 117.497 1.632 H 701.3588 120 116.312 1.674 H 701.3608 120 117.605 1.692 H 701.3622 120 118.912 1.652 H 701.3637 120 116.098 1.733 H 701.3658 120 113.143 1.434 H 701.3672 120 113.770 1.978 H 701.3688 120 114.611 1.917 H 701.3708 120 113.247 1.823 H 701.3724 120 112.209 1.422 H 701.3738 120 110.364 1.777 H 701.3759 120 112.002 1.874 H 701.3774 120 116.420 1.476 H 701.3789 120 115.671 1.664 H 701.3810 120 116.205 1.734 H 701.3824 120 117.065 1.747 H 701.3839 120 119.131 1.601 H 701.3862 120 118.475 1.768 H 701.3876 120 115.671 1.664 H 701.3890 120 111.078 1.931 H 701.3911 120 116.527 1.677 H 701.3926 120 117.281 1.687 H 701.3940 120 115.778 1.728 H 701.3962 120 115.671 1.507 H 701.3977 120 118.803 2.065 H 701.3996 120 117.931 1.697 H 701.4011 120 117.281 1.687 H 701.4026 120 118.693 1.546 H 701.4045 120 119.131 1.714 H 701.4060 120 117.497 1.579 H 701.4075 120 119.351 1.555 H 701.4095 120 118.693 1.772 H 701.4110 120 119.131 1.714 H 701.4125 120 116.742 1.445 H 701.4147 120 115.991 1.511 H 701.4162 120 116.957 2.033 H 701.4177 120 116.742 1.809 H 701.4199 120 116.420 1.517 26 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 H 701.4214 120 117.713 1.694 H 701.4229 120 114.189 1.447 H 702.3953 120 124.173 1.680 H 702.3967 120 123.602 1.814 H 702.3982 120 123.716 2.229 H 702.4003 120 125.784 1.743 H 702.4017 120 125.437 1.739 H 702.4032 120 121.235 1.640 H 702.4053 120 125.091 1.782 H 702.4067 120 124.746 2.679 H 702.4082 120 123.602 1.814 H 702.4103 120 122.695 1.701 H 702.4117 120 125.900 1.703 H 702.4132 120 125.437 2.181 H 702.4152 120 124.631 1.651 H 702.4174 120 122.356 1.621 H 702.4194 120 124.059 2.157 H 702.4214 120 122.582 1.799 H 702.4224 120 124.631 1.686 H 702.4244 120 124.976 1.732 H 702.4259 120 122.244 2.051 H 702.4274 120 125.437 1.662 H 702.4294 120 125.437 1.900 H 702.4309 120 123.148 1.928 H 702.4324 120 124.402 2.087 H 702.4344 120 123.716 1.674 H 702.4359 120 125.206 1.735 H 702.4374 120 124.631 1.888 H 702.4393 120 124.631 1.727 H 702.4408 120 123.944 1.819 H 702.4423 120 123.830 1.875 H 702.4443 120 123.375 1.669 H 702.4458 120 124.861 2.095 H 702.4473 120 123.148 2.066 H 702.4493 120 126.249 1.750 H 702.4508 120 123.602 1.672 H 702.4523 120 124.173 1.680 H 702.4537 120 122.808 1.802 H 703.4019 120 116.957 1.096 H 703.4033 120 114.295 1.478 H 703.4048 120 115.885 1.086 H 703.4062 120 113.979 1.068 H 703.4077 120 113.770 1.066 H 703.4092 120 118.039 1.106 H 703.4113 120 115.671 1.200 H 703.4128 120 112.727 1.056 H 703.4142 120 113.979 1.068 H 703.4163 120 112.416 1.166 H 703.4177 120 116.849 1.681 H 703.4192 120 118.257 1.448 H 703.4213 120 115.034 1.078 H 703.4228 120 112.209 1.104 H 703.4242 120 116.098 1.205 H 703.4263 120 117.497 1.288 H 703.4277 120 115.034 1.078 H 703.4298 120 116.312 1.145 H 703.4313 120 116.312 1.090 H 703.4340 120 113.874 1.067 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 27 H 703.4354 120 118.475 1.450 H 703.4370 120 115.992 2.706 H 703.4390 120 115.459 1.136 H 703.4405 120 118.257 1.370 H 703.4420 120 112.727 1.056 H 668.7222 120 114.190 1.070 H 703.4456 120 114.928 1.077 H 703.4470 120 116.527 1.427 H 703.4485 120 114.084 1.184 H 703.5240 120 114.822 1.852 H 703.5255 120 118.475 2.026 H 703.5270 120 116.205 2.329 H 703.5284 120 116.098 1.872 H 703.5300 120 117.173 2.051 H 703.5314 120 115.459 2.072 H 703.5334 120 118.475 1.910 H 703.5349 120 116.420 2.205 H 703.5364 120 115.459 1.862 H 703.5385 120 116.205 1.874 H 703.5399 120 115.991 1.870 H 703.5414 120 117.713 2.501 H 703.5428 120 114.611 1.960 H 703.5443 120 115.885 1.901 H 703.5458 120 118.366 2.072 H 703.5479 120 116.098 1.942 H 703.5493 120 114.822 2.010 H 703.5508 120 116.742 2.789 H 703.5529 120 116.849 1.884 H 703.5543 120 117.713 1.898 H 703.5558 120 118.584 1.984 H 703.5583 120 114.084 1.951 H 703.5598 120 116.849 2.097 H 703.5613 120 115.778 2.192 H 703.5634 120 113.979 2.158 H 703.5648 120 117.389 2.223 H 703.5663 120 117.605 2.167 H 703.5684 120 116.634 2.628 H 703.5698 120 116.312 1.989 H 703.5713 120 117.281 1.891 H 709.5557 120 114.505 2.709 H 709.5571 120 113.665 2.793 H 710.5570 120 118.148 0.925 H 710.5585 120 118.912 1.503 H 710.5613 120 118.366 1.315 H 710.5627 120 119.241 1.000 H 710.5643 120 118.039 1.876 H 710.5663 120 116.742 2.758 H 710.5673 120 117.930 1.973 H 710.5693 120 118.693 0.929 H 720.5547 120 118.584 2.774 H 720.5562 120 116.312 3.596 H 720.5577 120 120.344 3.037 H 720.5592 120 119.791 2.861 H 720.5607 120 123.716 3.172 H 720.5621 120 121.458 2.842 H 720.5636 120 121.794 4.078 H 720.5651 120 115.459 3.119 H 720.5666 120 119.791 3.423 28 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 H 721.5507 120 131.955 3.061 H 721.5521 120 125.206 3.027 H 721.5535 120 125.322 3.448 H 721.5557 120 131.591 3.109 H 721.5571 120 132.442 3.342 H 721.5585 120 130.385 3.655 H 721.5607 120 127.299 3.041 H 721.5621 120 128.005 2.970 H 721.5636 120 129.547 3.132 H 722.5481 120 123.261 8.973 H 722.5496 120 117.605 11.772 H 722.5510 120 127.065 9.366 J 633.4851 120 60.078 5.054 J 633.4935 120 59.637 5.078 J 633.5005 120 57.374 4.908 J 633.5074 120 61.026 5.165 J 633.5208 120 59.637 5.047 J 635.5070 120 57.163 1.229 J 635.6336 120 56.483 1.373 J 635.6411 120 56.691 0.912 J 635.6470 120 58.495 1.927 J 635.6531 120 56.223 0.904 J 635.6575 120 58.065 1.328 J 635.6643 120 56.743 1.339 J 635.6752 120 57.163 1.519 J 639.6642 120 55.657 1.692 J 639.6657 120 55.172 1.728 I 504.27 60 23.344 0.637 I 504.27 60 22.294 0.406 I 504.27 60 23.130 0.210 I 508.27 60 21.095 0.384 I 508.27 60 21.487 0.391 I 508.28 60 21.487 0.391 I 509.29 60 19.239 0.175 I 509.29 60 19.063 0.173 I 509.29 60 19.063 0.173 I 511.25 60 20.332 0.185 I 511.26 60 20.332 0.185 I 511.26 60 20.710 0.188 I 511.28 60 20.520 0.187 I 511.28 60 20.332 0.185 I 511.28 60 20.520 0.187 I 516.25 60 23.560 0.214 I 516.25 60 23.344 0.212 I 516.25 60 23.344 0.425 I 516.25 60 23.778 0.216 I 516.25 60 23.998 0.437 I 516.31 60 21.095 0.384 I 516.31 60 20.902 0.380 I 516.31 60 20.710 0.377 I 517.24 60 21.290 0.194 I 517.24 60 21.095 0.192 I 517.24 60 21.095 0.192 I 619.21 30 23.344 0.425 I 619.21 30 23.130 0.421 I 619.21 30 23.344 0.425 I 619.21 30 23.344 0.425 I 619.22 30 23.778 0.433 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 29 I 619.22 30 23.344 0.425 I 619.22 30 23.560 0.429 I 619.22 30 23.344 0.425 I 619.23 30 25.362 0.231 I 619.23 30 25.362 0.231 I 619.24 30 25.362 0.231 I 619.24 30 25.362 0.231 I 619.24 30 25.129 0.229 I 619.24 30 25.362 0.231 I 619.25 30 25.129 0.229 I 619.25 30 25.362 0.231 I 619.25 30 25.362 0.231 I 619.26 30 25.362 0.231 I 631.13 30 24.220 0.441 I 631.13 30 23.778 0.433 I 631.13 30 23.998 0.437 I 631.13 30 25.362 0.462 I 632.98 30 30.774 0.280 I 632.98 30 30.774 0.280 I 632.99 30 31.346 0.285 I 632.99 30 31.929 0.291 I 632.99 30 33.127 1.206 I 632.99 30 33.434 0.304 I 633 30 30.492 0.555 I 633 30 30.492 0.555 I 633 30 29.661 0.540 I 633 30 29.661 0.540 I 633.01 30 28.852 0.263 I 633.01 30 28.852 0.525 I 633.01 30 28.066 0.255 I 633.01 30 28.066 0.511 I 633.02 30 27.554 0.501 I 633.02 30 27.051 0.492 I 633.03 30 27.554 0.501 I 633.03 30 27.301 0.248 I 634.98 30 26.314 0.479 I 634.99 30 25.362 0.462 I 634.99 30 25.129 0.457 I 634.99 30 25.129 0.457 I 635.1 30 31.929 0.581 I 635.1 30 33.743 0.614 I 635.12 30 25.597 0.932 I 635.12 30 25.833 0.940 I 635.13 30 25.597 0.699 I 635.13 30 26.314 0.718 I 635.14 30 28.326 0.516 I 635.15 30 28.066 0.511 I 635.15 30 28.326 0.516 I 635.16 30 28.588 0.520 I 635.16 30 29.119 0.530 I 635.17 30 29.119 0.795 I 635.17 30 28.852 0.525 I 635.17 30 28.852 0.525 I 635.17 30 28.066 0.511 I 635.18 30 29.119 0.795 I 635.18 30 29.389 0.535 I 635.18 30 29.119 0.530 I 635.19 30 28.852 2.100 30 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 I 640.22 120 29.119 0.265 I 640.22 120 29.119 0.265 I 641.11 120 29.389 0.535 I 641.11 120 28.852 0.525 I 643.21 120 30.774 0.280 I 643.22 120 30.774 0.280 I 644.1 120 31.346 0.285 I 644.1 120 31.346 0.285 I 648.2 120 32.224 0.586 I 648.21 120 32.522 0.592 I 648.22 120 32.224 0.586 I 655.03 120 32.224 1.759 I 655.03 120 32.224 0.586 I 655.2 120 31.929 0.581 I 655.03 120 28.066 0.255 I 655.03 120 27.809 0.253 I 655.2 120 34.370 0.313 I 655.2 120 34.370 0.313 I 655.21 120 35.009 0.319 I 655.21 120 35.009 0.319 I 655.22 120 34.688 0.316 I 655.22 120 35.009 0.319 I 678.03 120 39.827 0.626 I 678.03 120 39.608 0.639 I 678.04 120 39.974 0.645 I 678.04 120 39.718 0.713 I 678.06 120 38.885 0.618 I 678.06 120 39.608 0.794 I 678.07 120 39.173 0.615 I 678.07 120 39.901 0.655 I 678.09 120 39.245 0.657 I 678.09 120 39.245 0.624 I 678.1 120 39.644 0.639 I 678.12 120 38.387 0.610 I 678.12 120 38.529 0.612 I 678.13 120 38.422 0.630 I 678.13 120 38.493 0.612 I 679.08 120 37.756 1.039 I 679.09 120 38.778 0.848 I 680.01 120 39.974 0.635 I 680.02 120 38.000 0.665 I 680.03 120 37.410 0.671 I 680.03 120 37.375 0.639 I 680.04 120 36.896 0.595 I 680.05 120 37.617 0.629 I 680.06 120 37.169 0.599 I 680.06 120 38.035 0.613 I 680.08 120 37.410 0.655 I 680.08 120 37.272 0.815 I 680.14 120 35.726 0.568 I 680.14 120 35.924 0.564 I 681.1 120 38.458 0.643 I 681.1 120 38.210 0.607 I 681.11 120 38.529 0.612 I 681.12 120 36.457 0.579 I 681.13 120 39.245 0.937 I 681.13 120 39.901 0.716 I 682.03 120 38.000 0.604 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 31 I 682.03 120 37.860 0.602 I 682.04 120 37.169 0.591 I 682.05 120 36.896 0.586 I 682.06 120 36.491 0.580 I 682.06 120 36.357 0.596 I 682.07 120 36.457 0.672 I 682.08 120 36.794 0.616 I 682.1 120 36.457 0.731 I 682.1 120 36.189 0.605 I 682.11 120 36.223 0.747 I 682.11 120 36.693 0.851 I 682.13 120 35.660 0.610 I 682.13 120 36.090 0.767 I 683.02 120 38.210 0.788 I 683.02 120 37.169 0.599 I 683.02 120 36.223 0.584 I 683.03 120 36.457 0.588 I 683.03 120 35.594 0.574 I 683.05 120 35.431 0.593 I 683.05 120 35.792 0.577 I 683.06 120 35.924 0.579 I 683.06 120 38.849 0.680 I 684.02 120 38.885 0.802 I 684.02 120 39.390 0.626 I 684.04 120 39.608 0.771 I 684.04 120 38.529 0.605 I 684.05 120 37.479 0.604 I 684.05 120 39.101 0.631 I 684.07 120 39.029 0.719 I 684.07 120 36.189 0.633 I 689.02 120 36.761 0.603 I 689.02 120 36.896 0.662 I 689.03 120 35.957 0.645 I 689.04 120 41.132 0.663 I 690.08 120 40.718 0.647 I 690.08 120 42.285 0.759 I 690.09 120 42.052 0.704 I 690.11 120 42.480 0.851 I 690.11 120 41.589 0.696 I 690.11 120 42.794 1.022 I 691.06 120 41.704 0.655 I 691.06 120 42.013 0.718 I 691.08 120 42.207 0.951 I 691.08 120 38.387 0.619 I 692.02 120 38.105 0.684 I 692.02 120 39.426 0.660 I 692.04 120 39.101 0.669 I 692.04 120 37.825 0.601 I 692.06 120 38.210 0.639 I 692.08 120 38.778 0.616 I 692.08 120 38.885 0.611 I 692.14 120 39.827 0.734 I 693.02 120 39.608 0.629 I 693.03 120 39.499 0.709 I 693.04 120 38.849 0.680 I 693.04 120 38.458 0.643 I 693.05 120 38.316 0.609 I 693.06 120 39.426 0.747 32 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 I 693.07 120 39.029 0.700 I 693.08 120 38.635 0.634 I 693.09 120 37.548 0.775 I 693.09 120 37.965 0.603 I 693.1 120 37.686 0.592 I 693.11 120 37.756 0.600 I 693.12 120 37.930 0.596 I 693.12 120 38.387 0.610 I 693.14 120 37.513 0.596 I 694.05 120 38.210 0.600 I 694.05 120 38.493 0.612 I 694.08 120 38.493 0.612 I 694.08 120 39.462 0.627 I 694.09 120 39.974 0.635 I 694.1 120 40.011 0.636 I 694.11 120 39.499 0.769 I 694.11 120 40.868 0.649 I 694.11 120 40.568 0.679 I 694.13 120 39.499 0.675 I 694.13 120 39.390 0.689 I 694.14 120 39.353 0.635 I 694.14 120 39.101 0.614 I 695.05 120 40.680 0.639 I 695.05 120 39.938 0.635 I 695.06 120 38.921 0.628 I 695.07 120 38.671 1.275 I 695.08 120 38.635 0.614 I 695.08 120 38.458 0.611 I 695.09 120 39.137 0.685 I 695.1 120 38.422 0.604 I 696.05 120 39.101 0.621 I 696.05 120 39.245 0.704 I 696.06 120 40.531 0.693 I 696.06 120 40.196 0.639 I 696.08 120 40.643 0.646 I 696.08 120 40.755 0.794 I 696.09 120 40.943 0.660 I 699.03 120 41.475 0.785 I 699.03 120 41.436 0.651 I 699.04 120 40.906 0.650 I 699.04 120 41.436 0.693 I 699.04 120 41.551 0.787 I 699.06 120 42.052 0.796 I 699.06 120 41.132 0.675 I 699.08 120 41.360 0.667 I 699.08 120 41.666 0.729 I 699.09 120 40.755 0.648 I 699.1 120 41.589 0.653 I 699.11 120 40.943 0.651 I 699.11 120 41.936 0.666 I 700.04 120 41.284 0.723 I 700.04 120 41.170 0.849 I 700.05 120 46.388 0.107 I 700.05 120 40.793 0.714 I 700.05 120 40.793 0.641 I 700.08 120 41.246 0.648 I 700.08 120 40.943 0.651 I 700.09 120 40.755 0.713 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 33 I 700.1 120 40.270 0.742 I 700.11 120 40.755 0.713 I 700.11 120 40.680 0.647 I 701.03 120 40.943 0.735 I 701.03 120 37.895 0.648 I 701.04 120 37.721 0.695 I 701.05 120 38.387 0.610 I 701.06 120 38.387 0.619 I 701.06 120 38.778 0.874 I 701.07 120 39.137 0.702 I 701.07 120 37.479 0.615 I 701.07 120 37.895 0.648 I 701.09 120 37.101 0.590 I 701.09 120 37.238 0.592 I 702.04 120 39.499 0.691 I 702.04 120 40.307 0.763 I 702.05 120 40.382 0.634 I 702.05 120 40.270 0.633 I 702.07 120 41.094 0.653 I 702.07 120 40.568 0.645 I 702.08 120 41.132 0.654 I 702.08 120 41.513 0.669 I 703.05 120 36.592 0.657 I 703.06 120 37.341 0.593 I 703.06 120 37.375 0.613 I 703.08 120 36.964 0.647 I 703.08 120 37.341 0.638 I 704.04 120 36.390 0.622 I 704.04 120 36.056 0.581 I 704.06 120 36.998 0.597 I 704.06 120 36.625 0.575 I 704.08 120 36.896 0.586 I 704.08 120 37.721 0.599 I 711.07 120 36.524 0.574 I 711.08 120 36.090 0.592 I 715.04 120 37.032 0.633 I 715.04 120 36.794 0.616 I 715.05 120 36.964 0.596 I 715.05 120 36.727 0.643 I 721.06 120 34.212 0.585 I 721.06 120 35.236 0.560 I 722.06 120 37.965 0.783 I 722.06 120 38.281 0.705 R 504.27 60 20.601 0.375 R 504.28 60 19.856 0.361 R 504.28 60 20.791 0.757 R 508.28 60 17.942 0.163 R 508.28 60 17.778 0.162 R 508.29 60 17.778 0.162 R 509.29 60 17.778 0.162 R 509.29 60 17.615 0.160 R 509.29 60 17.778 0.162 R 510.3 60 16.214 0.295 R 510.3 60 16.364 0.149 R 510.3 60 16.364 0.149 R 511.26 60 18.962 0.173 R 511.26 60 18.788 0.171 R 511.26 60 19.137 0.174 34 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 R 511.28 60 18.962 0.173 R 511.28 60 18.788 0.171 R 511.28 60 18.788 0.171 R 516.25 60 21.178 0.193 R 516.25 60 18.445 0.168 R 516.25 60 18.616 0.169 R 517.24 60 19.674 0.179 R 517.24 60 19.314 0.176 R 517.24 60 19.493 0.177 R 517.25 60 19.493 0.177 R 517.25 60 19.493 0.177 R 517.25 60 19.493 0.177 R 619.2 30 26.175 0.476 R 619.21 30 26.175 0.476 R 619.21 30 26.175 0.476 R 619.21 30 26.175 0.476 R 619.22 30 23.653 0.215 R 619.22 30 23.653 0.215 R 619.23 30 23.872 0.217 R 619.23 30 23.872 0.217 R 619.24 30 23.653 0.215 R 619.24 30 23.872 0.217 R 619.24 30 23.653 0.215 R 619.24 30 23.872 0.217 R 619.25 30 24.093 0.219 R 619.25 30 24.093 0.219 R 619.25 30 24.093 0.219 R 619.25 30 24.093 0.219 R 631.13 30 26.661 0.485 R 631.13 30 26.908 0.490 R 632.98 30 28.700 0.261 R 632.98 30 28.700 0.261 R 632.99 30 28.966 0.264 R 632.99 30 28.700 0.261 R 632.99 30 28.966 0.264 R 632.99 30 28.966 0.264 R 632.99 30 29.234 0.266 R 632.99 30 29.234 0.266 R 633 30 28.700 0.261 R 633 30 28.700 0.261 R 633.01 30 27.918 0.254 R 633.01 30 28.176 0.256 R 633.01 30 27.662 0.252 R 633.01 30 27.408 0.249 R 633.02 30 26.661 0.243 R 633.02 30 26.661 0.243 R 633.02 30 26.417 0.240 R 633.02 30 26.417 0.240 R 634.98 30 23.653 0.215 R 634.98 30 23.008 0.209 R 634.98 30 23.872 0.217 R 634.98 30 23.872 0.217 R 635.1 30 27.408 0.249 R 635.1 30 27.408 0.249 R 635.13 30 28.176 0.769 R 635.13 30 28.966 0.527 R 635.13 30 28.437 1.035 R 635.13 30 28.437 0.776 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 35 R 635.14 30 29.777 0.813 R 635.14 30 28.700 0.784 R 635.15 30 28.176 0.769 R 635.15 30 28.700 0.784 R 635.15 30 26.175 0.238 R 635.15 30 25.697 0.468 R 635.16 30 25.935 0.236 R 635.16 30 25.935 0.236 R 635.17 30 24.093 0.658 R 635.17 30 25.935 0.236 R 635.17 30 25.935 0.236 R 635.18 30 26.417 0.240 R 635.18 30 26.661 0.485 R 635.18 30 25.697 0.468 R 635.18 30 24.997 0.455 R 635.19 30 25.935 0.472 R 635.19 30 26.417 0.481 R 640.22 120 28.966 0.264 R 640.22 120 27.408 0.249 R 641.11 120 31.760 0.578 R 641.11 120 31.760 0.578 R 644.1 120 27.408 0.499 R 644.1 120 27.662 0.503 R 648.21 120 28.437 0.259 R 648.21 120 28.966 0.264 R 648.22 120 28.437 0.518 R 648.22 120 30.053 1.641 R 652.05 120 31.760 0.289 R 655.02 120 26.417 0.240 R 655.02 120 26.417 0.240 R 655.03 120 25.461 0.232 R 655.03 120 25.461 0.232 R 655.2 120 32.351 0.294 R 655.2 120 32.351 0.294 R 655.21 120 32.351 0.294 R 655.21 120 32.351 0.294 R 665.04 30 31.469 0.573 R 665.04 30 32.054 0.583 R 678.02 30 34.824 0.634 R 678.02 120 35.480 0.915 R 678.02 120 35.980 0.927 R 678.04 120 35.980 0.957 R 678.04 120 35.409 0.914 R 678.05 120 35.303 0.925 R 678.05 120 35.409 0.922 R 678.07 120 35.303 0.919 R 678.07 120 35.551 0.920 R 678.08 120 34.846 0.989 R 678.08 120 34.846 0.908 R 678.1 120 34.846 0.904 R 678.1 120 34.637 0.909 R 678.11 120 34.222 0.893 R 678.11 120 34.119 0.891 R 678.13 120 33.406 0.869 R 678.13 120 33.676 0.876 R 679.08 120 34.187 0.958 R 679.08 120 33.880 0.891 R 680.01 120 33.507 1.134 36 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 R 680.01 120 33.507 1.248 R 680.02 120 33.138 1.062 R 680.03 120 32.707 0.857 R 680.04 120 32.972 0.869 R 680.04 120 33.238 0.933 R 680.05 120 33.005 1.185 R 680.06 120 33.005 0.970 R 680.07 120 32.707 1.318 R 680.07 120 32.905 1.159 R 680.13 120 32.707 0.896 R 680.13 120 31.731 0.862 R 681.09 120 31.699 0.845 R 681.1 120 34.360 0.892 R 681.11 120 34.498 0.900 R 681.11 120 34.291 0.960 R 681.12 120 34.429 0.929 R 681.13 120 34.671 0.917 R 682.02 120 35.021 0.934 R 682.03 120 34.846 0.939 R 682.04 120 34.846 1.176 R 682.04 120 33.744 1.023 R 682.05 120 33.272 1.046 R 682.06 120 32.806 1.156 R 682.07 120 32.905 0.874 R 682.07 120 33.812 0.889 R 682.09 120 33.744 1.120 R 682.09 120 31.539 0.997 R 682.11 120 31.476 0.847 R 682.11 120 32.021 0.890 R 682.12 120 31.317 0.862 R 682.12 120 31.190 0.832 R 683.01 120 31.444 0.832 R 683.01 120 31.190 0.858 R 683.03 120 31.635 0.831 R 683.03 120 31.476 0.828 R 683.04 120 31.731 0.853 R 683.04 120 32.086 0.842 R 683.06 120 31.731 0.834 R 683.06 120 30.782 0.816 R 684.01 120 31.001 0.821 R 684.01 120 33.339 0.936 R 684.02 120 33.305 0.884 R 684.02 120 34.085 0.896 R 684.03 120 33.541 0.877 R 684.03 120 34.050 0.919 R 684.05 120 34.602 0.997 R 684.05 120 34.256 0.915 R 684.06 120 34.050 0.919 R 684.06 120 33.812 0.879 R 689.01 120 33.744 0.882 R 689.02 120 31.892 0.837 R 689.03 120 32.183 0.844 R 689.03 120 32.054 0.870 R 690.07 120 32.444 0.851 R 690.08 120 36.161 0.953 R 690.09 120 35.056 0.935 R 690.09 120 35.909 0.934 R 690.1 120 36.125 0.934 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 37 R 690.11 120 36.269 0.998 R 691.06 120 36.415 0.946 R 691.06 120 36.891 1.216 R 691.07 120 38.277 1.190 R 691.07 120 37.410 1.027 R 692.02 120 37.112 1.137 R 692.02 120 33.914 0.886 R 692.03 120 34.256 0.915 R 692.03 120 33.272 0.883 R 692.05 120 33.406 1.032 R 692.06 120 33.138 0.867 R 692.07 120 33.272 0.871 R 692.08 120 33.071 0.866 R 692.09 120 33.948 0.887 R 692.09 120 33.575 0.878 R 692.13 120 33.507 0.916 R 692.13 120 33.914 0.927 R 693.02 120 34.567 0.902 R 693.02 120 34.222 0.893 R 693.03 120 33.710 0.881 R 693.04 120 33.238 0.870 R 693.05 120 33.812 0.883 R 693.05 120 33.642 0.879 R 693.06 120 33.105 0.867 R 693.07 120 33.205 0.869 R 693.07 120 33.205 0.869 R 693.07 120 32.905 0.862 R 693.08 120 33.005 0.864 R 693.09 120 32.608 0.855 R 693.1 120 32.938 0.863 R 693.1 120 32.872 0.861 R 693.11 120 32.839 0.860 R 693.12 120 32.740 0.858 R 694.05 120 35.729 0.929 R 694.05 120 35.409 0.922 R 694.06 120 35.587 0.926 R 694.06 120 35.622 0.927 R 694.07 120 34.916 0.910 R 694.07 120 34.811 0.907 R 694.09 120 35.232 0.917 R 694.09 120 35.551 0.931 R 694.1 120 35.515 0.924 R 695.04 120 34.637 0.903 R 695.05 120 34.741 0.906 R 695.06 120 33.880 0.881 R 695.06 120 34.085 0.890 R 695.07 120 33.744 0.888 R 695.08 120 33.778 0.883 R 695.09 120 34.187 0.892 R 695.09 120 33.914 0.886 R 696.04 120 35.338 0.974 R 696.04 120 34.986 0.907 R 696.06 120 34.532 0.901 R 696.06 120 34.567 0.902 R 696.07 120 35.056 0.913 R 696.08 120 35.126 0.915 R 696.09 120 36.016 0.931 R 696.09 120 36.089 1.021 38 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 R 699.02 120 37.186 1.180 R 699.02 120 36.927 1.217 R 699.04 120 35.444 1.217 R 699.04 120 36.744 1.146 R 699.05 120 36.089 0.933 R 699.05 120 36.488 0.991 R 699.07 120 35.837 0.932 R 699.08 120 35.658 0.923 R 699.09 120 36.233 0.941 R 699.09 120 35.622 1.040 R 699.1 120 36.089 0.933 R 699.11 120 36.415 0.989 R 700.03 120 36.161 0.940 R 700.03 120 35.837 0.932 R 700.05 120 35.409 0.922 R 700.05 120 35.480 0.923 R 700.06 120 35.338 0.920 R 700.06 120 35.303 0.919 R 700.07 120 34.846 0.908 R 700.08 120 35.021 0.912 R 700.09 120 34.222 0.893 R 700.09 120 35.091 0.914 R 700.1 120 35.056 0.909 R 700.11 120 35.303 0.919 R 701.02 120 33.440 0.875 R 701.03 120 33.473 0.871 R 701.04 120 34.085 0.890 R 701.04 120 34.016 0.997 R 701.05 120 33.676 0.932 R 701.06 120 33.812 1.008 R 701.07 120 33.778 1.024 R 701.07 120 32.972 0.914 R 701.08 120 33.473 0.871 R 701.08 120 33.105 1.143 R 702.03 120 36.306 1.076 R 702.04 120 36.488 0.947 R 702.05 120 35.622 0.927 R 702.05 120 35.694 0.928 R 702.06 120 36.016 0.936 R 702.06 120 36.125 0.934 R 702.08 120 36.634 0.965 R 702.08 120 36.670 0.958 R 703.04 120 32.313 0.910 R 703.04 120 32.411 0.855 R 703.05 120 32.608 0.917 R 703.06 120 32.641 0.895 R 703.07 120 32.510 0.852 R 703.08 120 32.575 0.854 R 704.04 120 32.021 0.879 R 704.04 120 32.248 0.846 R 704.05 120 31.795 0.831 R 704.05 120 32.313 0.886 R 704.07 120 32.740 0.878 R 704.07 120 32.248 0.866 R 711.07 120 32.641 0.976 R 711.07 120 32.248 1.138 R 715.04 120 32.806 1.178 R 715.03 120 32.608 0.959 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 39 R 715.05 120 32.674 1.048 R 715.05 120 32.707 1.030 R 721.06 120 31.412 0.943 R 721.06 120 31.763 1.513 R 722.05 120 33.071 1.682 R 722.07 120 34.986 0.912 R 723.05 120 35.021 1.626 V 504.29 60 18.243 0.332 V 504.29 60 18.928 0.172 V 504.29 60 18.076 0.164 V 504.3 60 18.243 0.166 V 504.3 60 18.243 0.166 V 504.3 60 18.076 0.164 V 508.29 60 17.910 0.163 V 508.29 60 17.910 0.163 V 508.29 60 17.910 0.163 V 509.29 60 17.583 0.160 V 509.29 60 17.422 0.159 V 509.29 60 17.583 0.160 V 510.3 60 16.185 0.295 V 510.3 60 16.185 0.147 V 510.3 60 16.185 0.147 V 511.26 60 18.412 0.168 V 511.26 60 18.412 0.168 V 511.26 60 18.412 0.168 V 511.28 60 18.412 0.168 V 511.28 60 18.582 0.169 V 511.28 60 18.412 0.168 V 516.31 60 18.412 0.168 V 516.31 60 18.754 0.171 V 516.31 60 18.076 0.164 V 517.24 60 18.582 0.169 V 517.24 60 18.582 0.169 V 517.24 60 18.582 0.169 V 517.25 60 18.412 0.168 V 517.25 60 18.412 0.168 V 517.25 60 18.412 0.168 V 619.2 30 25.183 0.458 V 619.2 30 24.952 0.454 V 619.21 30 25.651 0.467 V 619.21 30 24.952 0.454 V 619.22 30 23.179 0.211 V 619.22 30 23.179 0.211 V 619.22 30 23.394 0.213 V 619.22 30 23.179 0.211 V 619.23 30 23.179 0.211 V 619.23 30 23.394 0.213 V 619.23 30 23.394 0.213 V 619.23 30 23.394 0.213 V 619.24 30 23.610 0.215 V 619.24 30 23.394 0.213 V 619.25 30 23.610 0.215 V 619.25 30 23.610 0.215 V 619.26 30 23.610 0.215 V 619.26 30 23.610 0.215 V 632.98 30 28.386 0.258 V 632.98 30 28.648 0.261 V 632.99 30 29.724 0.270 40 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 V 632.99 30 28.648 0.261 V 633 30 28.386 0.258 V 633 30 28.386 0.258 V 633.01 30 27.359 0.249 V 633.01 30 27.108 0.247 V 633.01 30 26.860 0.244 V 633.01 30 26.613 0.242 V 633.01 30 26.128 0.238 V 633.02 30 26.128 0.238 V 634.98 30 22.341 0.203 V 634.98 30 22.136 0.201 V 635.1 30 26.128 0.476 V 635.1 30 25.651 0.467 V 635.12 30 28.386 0.775 V 635.13 30 29.181 0.797 V 635.13 30 28.648 1.043 V 635.13 30 28.386 1.808 V 635.14 30 29.181 0.797 V 635.14 30 26.613 0.242 V 635.14 30 26.860 0.244 V 635.14 30 27.108 0.247 V 635.15 30 26.860 0.489 V 635.15 30 26.860 0.489 V 635.16 30 26.860 0.489 V 635.16 30 26.369 0.480 V 635.16 30 26.369 0.480 V 635.16 30 26.128 0.476 V 635.17 30 25.651 0.467 V 635.17 30 26.369 0.480 V 635.18 30 26.128 0.238 V 635.18 30 26.369 0.480 V 635.19 30 26.860 0.733 V 635.19 30 26.369 0.720 V 640.21 120 27.868 0.254 V 640.21 120 27.612 0.251 V 641.1 120 32.292 0.294 V 641.11 120 31.996 0.291 V 643.21 120 28.386 0.258 V 643.21 120 28.386 0.258 V 644.1 120 29.181 0.266 V 644.1 120 28.648 0.261 V 648.2 120 28.386 0.258 V 648.2 120 28.914 0.263 V 648.21 120 28.126 0.256 V 648.21 120 28.126 0.256 V 648.22 120 30.839 1.123 V 648.22 120 29.724 0.541 V 651.08 120 33.504 0.915 V 651.08 120 32.591 0.890 V 651.2 120 32.591 0.297 V 651.2 120 33.814 0.308 V 651.21 120 32.591 0.297 V 651.22 120 32.591 0.297 V 651.23 120 32.591 0.297 V 651.23 120 32.591 0.297 V 651.24 120 32.591 0.297 V 651.24 120 32.591 0.297 V 652.07 120 34.762 0.316 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 41 V 652.08 120 34.443 0.313 V 652.2 120 31.124 0.283 V 652.2 120 30.839 0.281 V 652.21 120 31.412 0.286 V 652.21 120 31.124 0.283 V 652.23 120 31.412 0.286 V 652.23 120 31.412 0.286 V 652.24 120 31.124 0.283 V 652.24 120 31.124 0.283 V 655.02 120 25.183 0.229 V 655.02 120 25.183 0.229 V 655.2 120 32.591 0.297 V 655.2 120 32.591 0.297 V 655.21 120 32.292 0.294 V 655.21 120 32.591 0.297 V 655.22 120 32.591 0.297 V 655.22 120 32.591 0.297 V 664.2 30 34.127 0.621 V 664.2 30 34.443 0.627 V 664.21 30 35.083 0.639 V 664.21 30 27.359 0.249 V 678.02 120 35.278 0.498 V 678.02 120 35.083 0.505 V 678.03 120 34.096 0.482 V 678.03 120 34.443 0.506 V 678.05 120 33.783 0.477 V 678.05 120 33.320 0.643 V 678.06 120 33.258 0.757 V 678.06 120 33.258 0.733 V 678.08 120 32.712 0.697 V 678.08 120 32.832 0.846 V 678.09 120 33.106 0.487 V 678.09 120 31.996 1.190 V 678.11 120 32.442 1.289 V 678.11 120 32.174 1.444 V 678.12 120 31.820 0.724 V 678.12 120 31.762 1.101 V 679.07 120 33.075 0.486 V 679.08 120 33.106 0.777 V 679.09 120 30.109 0.944 V 679.1 120 29.316 0.755 V 679.13 120 32.893 0.847 V 679.13 120 31.441 0.568 V 679.14 120 31.239 1.869 V 679.15 120 31.412 0.627 V 680 120 33.075 0.486 V 680.01 120 32.501 0.490 V 680.02 120 32.382 0.625 V 680.02 120 32.772 0.482 V 680.03 120 32.144 0.803 V 680.04 120 32.382 0.646 V 680.05 120 32.263 0.856 V 680.05 120 32.742 0.537 V 680.07 120 32.115 0.472 V 680.07 120 30.726 1.273 V 680.13 120 33.014 0.485 V 680.13 120 32.832 0.483 V 681.09 120 36.805 0.785 42 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 V 681.09 120 37.282 0.821 V 681.1 120 36.300 0.852 V 681.11 120 35.967 0.650 V 681.12 120 37.111 0.985 V 681.12 120 36.635 0.807 V 682.02 120 33.474 0.518 V 682.02 120 34.096 0.491 V 682.03 120 32.292 0.456 V 682.04 120 32.292 0.475 V 682.05 120 31.908 0.451 V 682.05 120 31.732 0.491 V 682.06 120 32.471 0.861 V 682.07 120 31.908 0.494 V 682.09 120 33.474 1.049 V 682.09 120 35.408 1.139 V 682.1 120 33.846 0.770 V 682.1 120 34.127 0.728 V 682.12 120 32.412 0.810 V 682.12 120 32.681 1.051 V 683.01 120 32.352 0.784 V 683.01 120 32.115 0.497 V 683.02 120 31.762 0.723 V 683.04 120 31.499 0.860 V 683.04 120 31.211 0.564 V 683.05 120 30.641 0.450 V 683.05 120 31.355 0.461 V 684.01 120 33.350 0.735 V 684.01 120 33.474 0.473 V 684.03 120 34.666 0.920 V 684.03 120 32.681 0.919 V 684.04 120 34.507 0.997 V 684.04 120 32.561 0.915 V 684.06 120 32.651 0.816 V 684.06 120 31.996 0.874 V 689.01 120 31.849 0.507 V 689.01 120 32.233 0.486 V 689.02 120 31.996 0.470 V 689.03 120 32.352 0.476 V 690.06 120 36.434 0.536 V 690.07 120 36.872 0.555 V 690.07 120 35.769 0.526 V 690.07 120 36.266 0.533 V 690.08 120 35.473 0.549 V 690.09 120 36.000 1.070 V 690.1 120 35.670 0.891 V 690.1 120 35.375 0.911 V 691.05 120 37.351 0.578 V 691.05 120 36.974 0.557 V 691.07 120 37.145 0.560 V 691.07 120 36.838 0.542 V 692.01 120 32.233 0.486 V 692.01 120 33.014 0.526 V 692.03 120 33.412 0.481 V 692.03 120 33.412 0.481 V 692.04 120 33.908 0.556 V 692.05 120 32.412 0.466 V 692.05 120 32.471 0.467 V 692.06 120 34.570 0.508 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 43 V 692.07 120 34.634 0.498 V 692.07 120 33.566 0.483 V 692.08 120 34.285 0.493 V 692.09 120 34.570 0.508 V 692.13 120 31.879 0.846 V 692.13 120 31.557 0.837 V 693.01 120 34.411 0.602 V 693.03 120 34.826 0.695 V 693.03 120 33.628 0.484 V 693.04 120 34.002 0.541 V 693.05 120 33.939 0.488 V 693.06 120 33.443 0.481 V 693.06 120 33.628 0.484 V 693.06 120 33.136 0.477 V 693.08 120 33.535 0.505 V 693.08 120 32.953 0.484 V 693.09 120 32.832 0.593 V 693.1 120 33.075 0.498 V 693.11 120 33.228 0.600 V 693.11 120 32.712 0.894 V 693.12 120 33.197 0.488 V 693.13 120 33.535 0.493 V 694.04 120 34.890 0.513 V 694.04 120 34.762 0.511 V 694.06 120 35.278 0.546 V 694.06 120 34.666 0.510 V 694.07 120 35.343 0.520 V 694.07 120 35.703 0.525 V 694.08 120 35.083 0.516 V 694.08 120 34.987 0.514 V 694.1 120 34.954 0.982 V 694.1 120 35.703 0.525 V 694.11 120 34.222 0.515 V 694.12 120 34.285 0.504 V 694.12 120 34.159 0.515 V 694.13 120 34.443 0.506 V 694.13 120 31.732 0.448 V 695.04 120 34.285 0.504 V 695.04 120 34.507 0.507 V 695.05 120 34.033 0.500 V 695.06 120 34.064 0.501 V 695.07 120 33.939 0.511 V 695.07 120 34.096 0.527 V 695.08 120 33.721 0.496 V 695.09 120 33.721 0.496 V 696.04 120 34.602 0.509 V 696.04 120 34.762 0.511 V 696.05 120 34.826 0.512 V 696.05 120 34.316 0.505 V 696.07 120 34.507 0.520 V 696.07 120 34.666 0.715 V 696.08 120 35.019 0.515 V 696.09 120 35.703 0.689 V 699.02 120 35.245 0.518 V 699.02 120 35.506 0.914 V 699.03 120 35.310 0.519 V 699.03 120 34.411 0.506 V 699.05 120 34.348 0.505 44 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 V 699.05 120 34.538 0.508 V 699.06 120 35.083 0.516 V 699.06 120 34.666 0.510 V 699.07 120 33.814 0.497 V 699.07 120 35.148 0.517 V 699.08 120 34.602 0.509 V 699.09 120 35.148 0.517 V 699.1 120 35.604 0.523 V 699.1 120 35.441 0.521 V 700.03 120 35.868 0.527 V 700.03 120 35.441 0.521 V 700.04 120 35.525 0.530 V 700.04 120 34.922 0.513 V 700.06 120 34.634 0.522 V 700.06 120 35.019 0.527 V 700.07 120 34.380 0.505 V 700.07 120 34.634 0.669 V 700.08 120 34.253 0.504 V 700.09 120 34.890 0.610 V 700.1 120 34.858 0.845 V 700.1 120 33.908 0.511 V 701.02 120 32.203 0.513 V 701.02 120 32.501 0.490 V 701.03 120 32.292 0.486 V 701.04 120 31.791 0.479 V 701.05 120 32.501 0.490 V 701.05 120 32.712 0.972 V 701.06 120 31.326 0.585 V 701.08 120 32.115 0.852 V 701.08 120 32.893 0.484 V 702.03 120 35.051 0.515 V 702.03 120 35.834 0.527 V 702.04 120 35.769 0.526 V 702.04 120 35.148 0.517 V 702.06 120 35.736 0.525 V 702.06 120 35.901 0.716 V 702.07 120 35.571 0.809 V 702.07 120 35.245 0.531 V 703.03 120 31.211 0.459 V 703.04 120 31.703 0.916 V 703.05 120 30.613 0.740 V 703.05 120 31.470 0.739 V 703.06 120 30.924 0.541 V 703.07 120 31.879 0.595 V 703.07 120 31.791 0.747 V 703.07 120 31.067 0.468 V 704.03 120 31.239 0.459 V 704.03 120 31.067 0.457 V 704.05 120 31.239 0.459 V 704.05 120 31.239 0.666 V 704.06 120 31.441 0.474 V 704.06 120 31.470 0.445 V 704.07 120 31.616 0.590 V 704.07 120 31.268 0.498 V 710.06 120 29.669 0.447 V 710.07 120 29.154 0.601 V 711.07 120 31.586 0.765 V 711.07 120 31.674 0.941 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 45 V 715.03 120 31.703 0.592 V 715.03 120 32.442 0.990 V 715.04 120 31.820 0.771 V 715.04 120 31.967 0.638 V 721.05 120 29.806 0.678 V 721.05 120 30.248 0.710 V 722.05 120 32.591 0.695 V 722.05 120 32.681 0.590 V 723.05 120 32.115 0.754 V 723.05 120 31.820 0.724 Table A.1: Log of observations. Epoch of observations is reported in JD- 2453000.5 unit. References Bersanelli, M., Bouchet, P., Falomo, R, & Tanzi, E. G., 1992, AJ, 104, 28 Bertone, E., Tagliaferri, G., Ghisellini, G., et al. 2000, A&A, 356, 1 Cardelli, J. A., Clayton, G. C. & Mathis, J. S. 1989, ApJ, 345, 245 Chincarini, G., Zerbi, F., Antonelli, A., et al. 2003, The Messenger, 113, 40 Covino, S., Stefanon, M., Fernandez-Soto, A., et al. 2004, SPIE, 5492, 1613 Chiappetti, L., Maraschi, L., Tavecchio, F., et al. 1999, ApJ, 521, 552 De Diego, J. A., Kidger, M. R., Gonzáles-Pérez, J. N., and Letho, H. 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R., McAlister, 1983, ApJ, 272, 26 Molinari et al., 2006, [arXiv:astro-ph/0612607] Montagni, F., Maselli, A., Massaro, E., et al. 2006, A&A, 451, 435 Paltani, S., Courvoisier, T. J.-L., Blecha, A., & Bratschi, P. 1997, A&A, 327, 539 Pesce, J. E., Urry, C. M., Maraschi, L., et al. 1997, ApJ, 486, 770 Pian, E., Vacanti, G., Tagliaferri, G., et al. 1998, ApJ, 492, L17 Pian, E., Urry, C. M., Maraschi, L., et al. 1999, ApJ, 521, 112-120 Pian, E., Falomo, R., Hartman, R.C., et al. 2002, ApJ, 392, 407-415 Pian, E., Romano, P., Treves, A., et al. 2006, in preparation Sbarufatti, B., Falomo, R., Treves, A. & Kotilainen, J. 2006, A&A, 457, 35 Sembay, S., Edelson, R., Markowitz, A., et al. 2002, ApJ, 574, 634 Smith, P. S., Hall, P. B., Allen, R. G., & Sitko, M. L. 1992, ApJ, 400, 115 Stetson, P. B., 1986, PASP, 99, 191 Tavecchio, F., Maraschi, L. and Ghisellini, G., 1998, ApJ, 509, 608 Tanihata, C., Kataoka, J., Takahashi, T., & Madejski, G. M., 2004, ApJ, 601, 759 Tommasi, L., Diaz, R., Palazzi, E., et al. 2001, ApJ Supplement Series, 132, 73 Treves, A., Morini, M., Chiappetti, L., et al. 1989, ApJ, 341, 733 Urry, C. M., Maraschi, L., Edelson, R., et al. 1997, ApJ, 486, 799 Xie, G. Z., Zhang, Y. H., Li, K. H., et al. 1996, AJ, 111, 3 Zerbi F. M., Chincarini, G., Ghisellini, G., et al. 2001, AN, 322, 275 Zhang, Y. H., Xie, G. Z. 1996, A&A Supplement Series, 116, 289 Zhang, Y. H., Treves, A., Celotti, A., Qin, Y. P., & Bai, J. M., 2005, ApJ, 629, 686 http://arxiv.org/abs/astro-ph/0610801 http://arxiv.org/abs/astro-ph/0612607 46 A. Dolcini et al.: REM monitoring of PKS2155-304 during 2005 Figure 13. SED of PKS 2155-304 in two states, adapted from Chiappetti et al. (1999) (see the paper for details). Data from this work are also plotted. Filled triangles correspond to epoch 1 (13/5/2005 data), while filled hexagons belong to epoch 3 data (20/11/2005). Optical, UV and REM data are dereddened using E(B-V)=0.026 and parameters given by Cardelli et al. (1989). Zhang, Y. H., Treves, A., Maraschi, L., Bai, J. M., & Liu, F. K. 2006a, ApJ, 637, 699 Zhang, Y. H., Bai, J. M., Zhang, S. N., et al., 2006b, ApJ, 651, 682 Introduction REM, Photometric procedure, data analysis Observations and data analysis Results Long term variability Short time-scale variability The NIR-Optical spectral energy distribution Discussion Table of observations

0704.0266

Supernova Polarization and the Type IIn Classification

Supernova Polarization and the Type IIn Classification Jennifer L. Hoffman Department of Astronomy, UC Berkeley, 601 Campbell Hall, Berkeley, CA 94720-3411 Abstract. While the members of the Type IIn category of supernovae are united by the presence of strong multicomponent Balmer emission lines in their spectra, they are quite heterogeneous with respect to other properties such as Balmer line profiles, light curves, strength of radio emission, and intrinsic brightness. We are now beginning to see variety among SNe IIn in their polarimetric characteristics as well, some but not all of which may be due to inclination angle effects. The increasing number of known “hybrid” SNe with IIn-like emission lines suggests that circumstellar material may be more common around all types of SNe than previously thought. Investigation of the correlations between spectropolarimetric signatures and other IIn attributes will help us address the question of classification of “interacting SNe” and the possibility of distinguishing different groups within the diverse IIn subclass. Keywords: supernovae, type IIn, supernova classification PACS: 97.60.Bw Type IIn (“narrow-line”) supernovae (SNe IIn) are Type II events whose primary distinction is the existence of strong narrow hydrogen Balmer emission lines in the spectra [1, 2]. These lines indicate the presence of circumstellar material ejected by the progenitor star and excited by the UV and X-ray photons from the supernova explosion. The IIn subclass includes 2–5% of all Type II supernovae [3]. Turatto [4] categorized the SNe IIn as a special class of core-collapse supernovae related to the SNe Ib/c and hypernovae. However, in recent years several new discoveries have blurred the boundaries between the IIn category and other types of supernovae. It should be noted that since the label of IIn is assigned based on spectral characteristics, while that of II-L or II-P is based on features of the light curve, these three categories are not mutually exclusive, and some objects may be given more than one classification in the literature. But observations of “hybrid” supernovae such as SN 2002ic [5, 6] and SN 2005gj [7], which showed IIn-like Hα lines superposed on type Ia-like spectra, and of “chameleon” supernovae such as SN 2001em [8, 9], which evolved from a Type Ic to a Type IIn over the span of a few years, suggest that not only SNe II but potentially all supernovae may show signatures of interaction with circumstellar material. Consequently, Turatto’s revised classification scheme [10] includes a broad category of “interacting SNe” that spans all SN types and includes the SNe IIn. The IIn category has always been heterogeneous, as noted by Filippenko [2], whose Figure 14 presented a non-coeval collection of SN IIn spectra. Figure 1 shows that even when compared at similar ages, SNe IIn have quite diverse spectral characteristics. The primary feature of SNe IIn spectra, the strong Hα emission line, also varies substantially in strength and profile between objects of comparable age (Figure 2) and even for a given object over time. In particular, the very narrow (FWHM < 200 km/s) component of Hα http://arxiv.org/abs/0704.0266v2 FIGURE 1: Deredshifted optical spectra of four Type IIn super- novae at similar epochs, showing the diversity of spectral character- istics within the subtype. All dates are post-discovery. Detailed analy- ses of SN 1997eg and SN 2000P have been conducted by [11] and [12], respectively; the other data are presented courtesy of A. Fil- ippenko (priv. comm.). For the unpublished spectra, redshift cor- rections are based on information in the Asiago Supernova Catalog [13]. Hydrogen Balmer lines are labeled. does not necessarily exist at all times in the evolution of a SN IIn. In SN 1997eg (Figure 3; [11]), this narrow Hα component disappeared after 100 days post-discovery, was replaced by a small symmetric absorption feature, and then reappeared around day 400. The Hα line in SN 1998S ([12]; their Figure 7) also lost its narrow emission component early on, but was very asymmetric and changed much more dramatically over time. Because of interaction with circumstellar material, the light curves of SNe IIn often decline quite slowly in comparison with other core-collapse objects, but not all members of the subclass show this slow decline [2]. The canonical IIn SN 1988Z decreased in brightness by only 5 magnitudes in 1000 days [14], but others decline more quickly (e.g., SN 1998S; [15]). SN 1994W faded by ∼6 magnitudes in B in only 100 days [16]. Circumstellar interaction can also cause some SNe IIn to become strong radio and X-ray emitters; SN 1988Z and SN 1986J are among the brightest radio supernovae ever observed [17]. Nearly all the SNe IIn detected in X-rays have also been strong radio sources [18, 19]. However, not all SNe IIn become radio-loud [18], and there is considerable heterogeneity even among the “radio-quiet” subset [2]. The overlap between the IIn, II-P, and II-L categories makes it difficult to identify trends in radio brightness with core-collapse subtype. FIGURE 2. As in Fig. 1, but for the Hα region of the spectrum. FIGURE 3. Time evolution of the multi-component Hα profile of SN 1997eg [11]. Spectra are normal- ized to day 16 at v =−10,000 km s−1. FIGURE 4. Polarized flux spectra (percent polarization multiplied by flux) for SN 1998S [12] and SN 1997eg [11], binned to 10 Angstroms in each case. Dashed lines show total flux spectra for comparison, scaled by 0.037 and 0.036, respectively. Polarization data are corrected for estimated interstellar polariza- tion (ISP) effects; these estimates are somewhat uncertain in each case, but a simple correction for ISP could not account for all the differences between these two polarized flux spectra. Finally, SNe IIn show considerable variety in their spectropolarimetric characteristics (Figure 4). Supernovae of all types are known to be polarized due to intrinsic asphericity of the ejecta. The circumstellar material surrounding SNe IIn can produce its own po- larization signature in addition to that arising from the ejecta. Some of the polarization variations between SNe IIn are likely due to inclination; that is, they may have similar geometrical distributions of circumstellar material, but have different polarization sig- natures due to different viewing angles. To study this effect, I am constructing a grid of Monte Carlo radiative transfer models of the Hα line polarization produced by circum- stellar matter distributions with various geometries and seen at various viewing angles [20]. Preliminary results suggest that differences in viewing angle may account for some of the polarimetric variety in SNe IIn. However, such geometric effects cannot explain all of the IIn diversity; for example, the transformation of SN 2001em from a Ic into a IIn [8, 9] is very unlikely to be due to a sudden change in inclination! These examples show that the IIn classification is currently something of a “catchall” for a number of related but distinct objects, all with close connections to Turatto’s [10] new category of “interacting SNe”. Now that circumstellar interaction has been recognized to occur for a broad range of supernova types, and now that high-quality time-dependent data are more readily available, we are in a good position to reconsider the categorization of SNe IIn and other interacting supernovae. An initiative underway within the Berkeley supernova group will quantify the spectral and spectropolarimetric characteristics of SNe IIn and search for correlations between these features and other properties such as light curve shape and radio/X-ray behavior. If found, such correlations will illuminate the relationships between diverse members of the IIn category and perhaps ultimately argue for the category’s subdivision. In the future, full understanding and correct classification of these important objects will depend critically on obtaining a broad range of observational information, including multiwavelength and polarimetric data, with as much time coverage as possible. In addition, supernova researchers should expand their collaborations with the evolved star community. If the circumstellar media around SNe IIn and related objects are indeed created by winds and outflows from massive stars, we can use nearby evolved star nebulae as analogous cases to the circumstellar envelopes of supernovae. Such an effort could greatly improve our understanding of supernova progenitors and help shed light on the physical causes of the diversity we observe among interacting supernovae. ACKNOWLEDGMENTS This research was funded by an NSF Astronomy & Astrophysics Postdoctoral Fellow- ship, AST-0302123; by NSF grant AST-0607485 to A. V. Filippenko; and by the Na- tional Energy Research Scientific Computing Center, US DOE Contract #DE-AC03- 76SF00098. I thank Alex Filippenko at UC Berkeley and Peter Nugent at the Lawrence Berkeley Laboratory for their support and invaluable contributions. REFERENCES 1. E. M. Schlegel, MNRAS 244, 269 (1990). 2. A. V. Filippenko, ARAA 35, 309 (1997). 3. E. Cappellaro, M. Turatto, D. Y. Tsvetkov, et al., A&A 322, 431 (1997). 4. M. Turatto, S. Benetti, and E. Cappellaro, “Variety in Supernovae,” in From Twilight to Highlight: The Physics of Supernovae, Springer, Berlin, 2003, p. 200. 5. J. Deng, K. S. Kawabata, Y. Ohyama, et al., ApJ 605, L37 (2004). 6. M. Hamuy, M. M. Phillips, N. B. Suntzeff, et al., Nature 424, 651 (2003). 7. G. Aldering, P. Antilogus, S. Bailey, et al., ApJ 650, 510 (2006). 8. A. M. Soderberg, A. Gal-Yam, and S. R. Kulkarni, GCN 2586, 1 (2004). 9. N. N. Chugai, and R. A. Chevalier, ApJ 641, 1051–1059 (2006). 10. M. Turatto, “Classification of Supernovae,” in This Volume, AIP, New York, 2007. 11. J. L. Hoffman, D. C. Leonard, R. Chornock, et al. (2007), in prep. 12. D. C. Leonard, A. V. Filippenko, A. J. Barth, et al., ApJ 536, 239 (2000). 13. R. Barbon, V. Buondí, E. Cappellaro, et al., A&AS 139, 531 (1999). 14. M. Turatto, E. Cappellaro, I. J. Danziger, et al., MNRAS 262, 128 (1993). 15. A. Fassia, W. P. S. Meikle, W. D. Vacca, et al., MNRAS 318, 1093 (2000). 16. R. J. Cumming, and P. Lundqvist, “Supernova Progenitor Constraints from Circumstellar Interaction: Type II,” in Advances in Stellar Evolution, Cambridge University Press, Cambridge, 1997, p. 297. 17. R. A. Chevalier, C. Fransson, and T. K. Nymark, ApJ 641, 1029 (2006). 18. S. D. van Dyk, M. Hamuy, and A. V. Filippenko, AJ 111, 2017 (1996). 19. D. Pooley, W. H. G. Lewin, D. W. Fox, et al., ApJ 572, 932 (2002). 20. J. L. Hoffman, “Polarized Line Profiles as Diagnostics of Circumstellar Geometry in Type IIn Supernovae,” in Circumstellar Media and Late Stages of Massive Stellar Evolution, Rev. Mex. AA Ser. Conf., 2007, in press.

0704.0267

Near-Infrared Spectra of the Black Hole X-Ray Binary, A0620-00

Near-Infrared Spectra of the Black Hole X-ray Binary A0620-00 Cynthia S. Froning1 [email protected] Center for Astrophysics and Space Astronomy, University of Colorado, 593 UCB, Boulder, CO 80309-0593 Edward L. Robinson [email protected] Department of Astronomy, University of Texas at Austin, Austin, TX 78712 Martin A. Bitner [email protected] Department of Astronomy, University of Texas at Austin, Austin, TX 78712 ABSTRACT We present broadband NIR spectra of A0620-00 obtained with SpeX on the IRTF. The spectrum is characterized by a blue continuum on which are super- imposed broad emission lines of H I and He II and a host of narrower absorption lines of neutral metals and molecules. Spectral type standard star spectra scaled to the dereddened spectrum of A0620-00 in K exceed the A0620-00 spectrum in J and H for all stars of spectral type K7V or earlier, demonstrating that the donor star, unless later than K7V, cannot be the sole NIR flux source in A0620-00. In addition, the atomic absorption lines in the K3V spectrum are too weak with respect to those of A0620-00 even at 100% donor star contribution, restricting the spectral type of the donor star in A0620-00 to later than K3V. Comparison of the A0620-00 spectrum to scaled K star spectra indicates that the CO ab- sorption features are significantly weaker in A0620-00 than in field dwarf stars. Fits of scaled model spectra of a Roche lobe-filling donor star to the spectrum of A0620-00 show that the best match to the CO absorption lines is obtained when the C abundance is reduced to [C/H] = -1.5. The donor star contribution in the H waveband is determined to be 82 ± 2%. Combined with previous published results from Froning & Robinson (2001) and Marsh et al. (1994), this gives a precise mass for the black hole in A0620-00 of M1 = 9.7± 0.6 M⊙. http://arxiv.org/abs/0704.0267v1 – 2 – Subject headings: binaries: close — infrared: stars — stars: individual (A0620– 00) — stars: variables: other 1. Introduction Among X-ray binary systems (XRBs), 18 or more have been identified as containing probable black hole (BH) accretors (McClintock & Remillard 2005). The BH masses mea- sured to date appear to fall into a limited range. From a Bayesian analysis of the observa- tional parameters of several low-mass XRBs, Bailyn et al. (1998) concluded that 6 of the 7 systems measured had BH masses clustered around 7 M⊙and that the overall population was heavily biased away from low BH masses (3 – 5 M⊙). Since that study, at least one system, J0422+32, has been found to have a measured mass of about 4 M⊙(Gelino & Harrison 2003), but the overall trend toward higher BH masses persists. This result is in conflict with theoretical evolutionary models of the formation of BH XRBs, which predict a continuous distribution of BH masses from neutron star masses up to 10–15 M⊙ (Fryer & Kalogera 2001). Confirmation or elimination of the low-mass BH “gap” would provide important constraints on the role of massive star evolution, supernova ener- getics, and subsequent binary evolution in BH formation (Fryer & Kalogera 2001). Further analysis of the mass distribution of BHs in compact binary systems is currently hampered by the generally poor precision of existing mass estimates (e.g., see Table 4.2 in McClintock & Remillard 2005). In this manuscript, we address this limitation by analyzing near-infrared (NIR) spectra to obtain a precise BH mass for one XRB, A0620-00. A0620-00 was discovered in 1975 when it erupted in an X-ray nova (Elvis et al. 1975). After its return to quiescence, A0620-00 was revealed as an interacting binary system with a K star donating mass via an accretion disk to a compact object (Oke 1977; McClintock et al. 1983). Later, the orbital period of the binary and the radial velocity amplitude of the donor star were measured and yielded a mass function, f(M) = 3.17 M⊙, which established A0620- 00 as a likely BH XRB (McClintock & Remillard 1986). Further observations established the binary mass ratio and determined the masses of the stars to within one unknown, the binary orbital inclination: M1 = (3.09±0.09) sin 3 i M⊙ and M2 = (0.21±0.09) sin 3 i M⊙, where M1 and M2 are the BH and donor star masses, respectively (Marsh, Robinson & Wood 1994; 1Visiting Astronomer at the Infrared Telescope Facility, which is operated by the University of Hawaii under Cooperative Agreement no. NCC 5-538 with the National Aeronautics and Space Administration, Science Mission Directorate, Planetary Astronomy Program. – 3 – Orosz et al. 1994). Several groups have determined values for the inclination, with the num- bers ranging from 38◦ ≤ i ≤ 75◦. As a result, estimates of the BH mass in A0620-00 vary from 3.3 to 13.6 M⊙(Haswell et al. 1993; Shahbaz, Naylor & Charles 1994; Froning & Robinson 2001; Gelino et al. 2001). The broad range of derived inclinations and BH masses for A0620-00 result from long- term variability in the system and from uncertain determinations of the amount of veiling, or dilution, by sources other than the donor star. The inclination is determined by modeling the amplitude of the ellipsoidal variations in the donor star light curve, so both of these effects will alter the derived BH mass. In particular, an additional source will dilute the amplitude of the ellipsoidal variation, leading to an underestimate of the inclination and a corresponding overestimate of the BH mass if not taken into account. The best way to determine the true donor star contribution in A0620-00 is to model its spectrum, particularly in the NIR, where the late type donor star is expected to domi- nate. Shahbaz, Bandyopadhyay & Charles (1999) modeled a low S/N, K-band spectrum of A0620-00, from which they concluded that the accretion disk contributes at most 27% of the continuum in the NIR. They fit only the 12CO bandhead at 2.29 µm, however, which is sen- sitive to both temperature and luminosity of the donor star and may be prone to metallicity effects in compact binary systems (Froning & Robinson 2001). Harrison et al. (2007) also recently published a K-band spectrum of A0620-00 in which they confirmed that the 12CO absorption lines are anomalously weak. What is needed to settle the debate over the con- tribution of the accretion disk to the NIR spectrum of A0620-00 are higher S/N, broadband spectra. To this end, we have obtained and present here 0.8 – 2.4 µm spectroscopy of A0620- 00 obtained with SpeX at the NASA InfraRed Telescope Facility (IRTF). This manuscript is organized as follows: § 2 summarizes the data reduction and calibration steps; § 3 presents the data analysis and modeling of the donor star spectrum; and § 4 gives discussion and conclusions. 2. Observations and Data Reduction We observed A0620-00 on 2004 January 8 – 10 using SpeX on the IRTF (Rayner et al. 2003). The weather was clear with good seeing conditions (. 0.7′′) throughout the run. We observed A0620-00 using the ShortXD mode, which has a cross-dispersed echelle configu- ration and covers 0.8 – 2.5 µm simultaneously in 6 orders. All observations were obtained through the 0.′′5 slit, resulting in a spectral resolution in the center of each order of R = 1200 (250 km s−1). A nearby A0V star was observed hourly to sample the atmospheric absorption spectrum. We also observed several spectral type calibration stars in the same configuration – 4 – used for A0620-00 (supplemented with similar data from a previous SpeX observing run). The observations are summarized in Table 1. All data reduction, calibration, and spectral extraction steps were performed using Spextool, an IDL-based package developed by the IRTF (Cushing, Vacca, & Rayner 2004; Vacca, Cushing, & Rayner 2003). The calibration processing steps included flat-fielding, sky subtraction, optimal spectral extraction, and wavelength calibration. Each A0620-00 exposure was extracted individually to preserve the full 300 sec time resolution. The spectra were corrected for telluric absorption and flux-calibrated using the A0V stellar spectra as described by Vacca, Cushing, & Rayner (2003). Finally, the orders were merged for each exposure to create single spectra covering the full wavelength range. The S/N per resolution element was ∼4 in the individual spectra. The slit was kept aligned to the parallactic angle during data acquisition, so the relative fluxes along the full 0.8 – 2.4 µm range are accurate to ≤ 2% (where the upper limit is the uncertainty at 0.9 µm when guiding at 2.4 µm; Cushing et al. 2004). In addition, stable observing conditions resulted in absolute flux calibrations of comparable quality. We have not quantified this number as our analysis does not depend on the absolute flux of the spectrum, but we note for completeness that a rough comparison of our mean in-band colors to those of Froning & Robinson (2001) and Gelino et al. (2001) yielded JHK colors within 0.1 mag of their results, well within the level of variability observed in A0620-00 over long time periods. For the time-averaged spectrum of A0620-00, we shifted each 300 sec spectrum to the rest frame of the donor star before median combining, using the orbital ephemeris of McClintock & Remillard (1986) and the donor star radial velocity amplitude fromMarsh, Robinson & Wood (1994). The error bars were determined by calculating the median absolute deviation of each pixel and then propagated through the smoothing and de-reddening steps. We did not cor- rect for the effects of orbital smearing within an exposure time, but we note that this is a negligible effect (≤30 km s−1) at our 250 km s−1 spectral resolution. The time-averaged spectrum is shown in Figure 1. The spectrum has been boxcar-smoothed by 3 pixels, equiva- lent to one resolution element. Based on the scatter around linear fits to (relatively) line-free spectral regions, we find that the S/N in the time-averaged spectrum is ∼55 in H and K and ∼45 in J (> 1µm, ∼30 at shorter wavelengths). Also shown in Figure 1 is the dereddened spectrum of A0620-00, calculated assuming a reddening along the line of sight of E(B–V) = 0.39 (Wu et al. 1976). Figures 2 – 4 show ex- panded views of the J,H, and K bands of the spectrum, with prominent spectral absorption and emission features labeled. Line identifications were made using multiple sources, in- cluding Wallace et al. (2000), Meyer et al. (1998), Kleinmann & Hall (1986), Harrison et al. – 5 – (2004), and the Atomic Line List2. 3. Analysis The broadband NIR spectrum of A0620-00 shown in Figures 1 – 4 is characterized by a blue continuum on which are superposed broad (full width at zero intensities ≥4000 km s−1) emission lines of H I and He II and narrow (full width at half minima ≃ 250 – 400 km s−1) absorption lines of neutral metals, including transitions of Na I, Mg I, Al I, Si I, K I, Ca I, Ti I, and Fe I. The emission lines are believed to originate in the accretion disk, while the absorption features originate in the photosphere of the donor star. The absorption spectrum is similar to that of a K star, with previous estimates of the spectral type ranging from K3V to K7V (Oke 1977; González Hernández et al. 2004). In Figure 5, we show the dereddened spectrum of A0620-00 compared to that of a field K5V star. The K star has been normalized to A0620-00 near 2.29 µm, just blueward of the 12CO (2,0) bandhead. The comparison immediately shows that a K5V (or hotter) star cannot be the only flux source in the NIR. If the K5 star is scaled to the flux of A0620-00 in the K band, it exceeds the dereddened flux by >20% at bluer wavelengths. Adopting the K4V and K3V spectral types of Gelino et al. (2001) and González Hernández et al. (2004) results in an even larger disparity between the expected and observed J- and H-band fluxes. Even a spectral type as late at K7V exceeds the observed flux in A0620-00 by up to 10% over most of the J and H bands when normalized to 100% contribution in K. Modest increases in the assumed reddening along the line of sight do not reconcile the spectrum of A0620-00 with that of a K5V star. Adopting a reddening of E(B–V) = 0.45 brings the 0.8 µm fluxes into agreement, but the template spectrum is still brighter than the spectrum of A0620-00 at longer wavelengths, including most of the J and H bands. Because the relative reddening values between H and K are small, the reddening must be increased to E(B–V) > 1.0 to bring the spectrum of the normalized K5V template below the spectrum of A0620-00 at all NIR wavelengths. It is extremely difficult to reconcile a reddening this high with the observed depth of the interstellar absorption feature at 2200 Å in the spectrum of A0620-00 (Wu et al. 1976). As a result, the fundamental conclusion remains: if the donor star is the sole source of NIR emission in A0620-00, its spectral type must be later than that of a K7V. Otherwise, some level of dilution must be present. The absorption spectrum of A0620-00 resembles that of the K5V template, but there 2http://www.pa.uky.edu/p̃eter/atomic/ – 6 – is at least one important difference between them: the CO molecular absorption features in A0620-00 are significantly weaker relative to the metal lines than in the template spectrum. This difference can affect determinations of the donor star contribution to the NIR spectrum. For example, the dilution analysis performed by Shahbaz, Bandyopadhyay & Charles (1999) on the K-band spectrum of A0620-00 is unlikely to be a valid determination of the contribu- tion of the donor star to the NIR spectrum, since they relied entirely on the relative strength of the 12CO 2.29 µm feature. The weakness of the CO lines also suggests that other anomalies may exist in the spectrum, necessitating that its decomposition be undertaken over a wide wavelength range and using multiple line species and features. Accordingly, we compare the line equivalent widths and line ratios in A0620-00 with those of field star populations and model the spectrum using both spectral type standards and synthetic spectra. 3.1. Classification Based on Spectral Indices Studies using equivalent widths (EWs) and EW ratios to determine the spectral type of a star or stellar population have been pursued by several groups (e.g., Origlia, Moorwood, & Oliva 1993; Ali et al. 1995; Förster Schreiber 2000; and sources therein). Of particular interest to us is the work of Förster Schreiber (2000; hereafter FS), who examined H and K-band absorption lines to find temperature and luminosity indices and indices sensitive to dilution of the stellar spectrum by other sources. FS was primarily interested in spectral trends in giant and supergiant stars as the dominant stellar source in extragalactic NIR spectra, but his analysis includes some dwarf stars as well. For comparison, we calculated the EWs of several prominent stellar absorption lines in the time-averaged spectrum of A0620-00 to compare to the spectral indices in FS. Table 2 gives the measured EWs. The lines were chosen to correspond to those in Table 3 of FS and were calculated using the same continuum normalization and integration limits. Where applicable, we also applied the EW correction for spectral resolution from Equations 2 – 4 of FS. The error bars on the EWs are the standard deviation of the mean for several measurements of each line with variable estimates of the continuum placement. We first compared our EWs to the spectral indices given in Figure 5 of FS, which presents temperature and luminosity indicators for stars with solar or near-solar abundances. With the exception of Si I λ1.59 µm and Mg I λ2.28 µm, all of the lines under analysis show a strong trend of increasing EWs with decreasing stellar temperature. The EWs in A0620-00 are on the low side of the distributions for these lines, indicating stellar temperatures of ≥5000 K. – 7 – We also compared our EWs with those presented in Ali et al. (1995), who concentrated on temperature indices for dwarf stars. Note that the EWs in Table 2 used to compare to the Ali et al. (1995) indices are larger than those used for the FS indices because Ali et al. used a wider wavelength interval for their measurements, which we mirrored. Using their EW-temperature relationships, we obtain a temperature of 4600±300 K from Ca I and 5000±450 K from Na I. Therefore, if uncorrected for dilution, EWs in A0620-00 point to a donor star of roughly type K3V star or earlier. However, stars of spectral type K7V or earlier exceed the observed spectrum of A0620-00 in J and H when zero dilution is assumed in K. Therefore, we conclude that a diluting continuum source must be present in the NIR spectrum of A0620-00. The amount of dilution of the stellar spectrum by another source is determined in FS by comparing the line ratios of adjacent atomic and molecular features (their Table 8). Unfortunately, their line ratios use the H and K band CO molecular absorption features, which we have already seen are not normal in A0620-00. Indeed, a comparison of the 12CO 1.62 µm and 2.29 µm features in A0620-00 gives a result so disproportionately strong in the 1.62 µm line that the ratio doesn’t even appear on the FS spectral index plot. Similarly, a comparison of the 2.29 µm feature to the nearby Na I and Ca I EWs indicates that the CO feature is weaker in A0620-00 than in any of the dwarf stars analyzed by FS. These results indicate that the CO features cannot be used to estimate the non-stellar dilution component A0620-00. 3.2. Fitting Spectral Type Standard Stars to the Spectrum of A0620-00 In an effort to quantify the contribution of the donor star to the NIR spectrum of A0620- 00 and its dilution by other sources, we compared its spectrum to those of K3V, K5V, and K7V spectral type standard stars. The standard stars (listed in Table 1) were observed with the same instrument configuration and calibrated using the same procedures as for the A0620-00 observations. Before comparing the A0620-00 and template spectra, both were boxcar-smoothed over 3 pixels (one resolution element). The spectrum of A0620-00 was also dereddened with E(B–V) = 0.39 (using the reddening curve of Cardelli, Clayton, & Mathis (1989) and the standard star spectra were convolved with a Gaussian of 83 km s−1 FWHM to mimic the rotational broadening of the donor star in A0620-00 (Marsh, Robinson & Wood 1994). Note, however, that the rotational broadening is smaller than the 250 km s−1 resolu- tion of the spectra and has a minimal effect on the results. We wrote an IDL program to fit scaled template spectra to the spectrum of A0620-00 using the following steps. First, we selected a small wavelength range (typically, 0.1 µm or – 8 – smaller) and fit a spline function to the continuum. The continuum points were selected by eye. After normalizing both the spectrum of A0620-00 and that of the template star and placing both spectra on a common, linear dispersion, we multiplied the template spectrum by a fraction, f , which represents the donor star contribution to the spectrum of A0620-00, and subtracted the scaled donor star spectrum from that of A0620-00. We varied f from 0 to 1 in increments of 0.01 to find the fraction that minimized the rms of the residual in each waveband. Finally, we repeated this analysis over several spectral lines and groups of lines over the full NIR spectral range. The fit regions we examined are given in Table 3. The resulting best values of f and the rms for each fit region and template spectrum is given in Table 4. There are few absorption lines in the J band that are both relatively strong and un- contaminated by emission lines in the A0620-00 spectrum, so our fits were restricted to the portion of the long J band between Pγ and Pβ (1.10 – 1.26 µm). This region contains a blend of singly-ionized atomic species, including transitions of Mg I, Fe I, and Si I. The best fit fractions range from 0.78 to 1.0. There is a disparity between the strongest line in this range, Mg I λ1.183 µm and the other lines in this band: the Mg I line is best fit at f ∼0.9, but the other lines are weaker in the template than in A0620-00 even at f = 1. The situation is less ambiguous in the H band. The best fit to the full H band spectrum using the K5V standard star is shown in Figure 6. The fit has f = 0.76 and an rms of 0.016. Although the χ2 statistics are relatively poor (χ2ν = 13.5), there is a good qualitative correspondence between the morphologies of the observed and template spectra. Similar results are obtained with the K3V and K7V templates. Many of the stronger transitions (predominately Mg I and Si I lines) are too weak in the f = 0.76 template, however. If we restrict the fits to narrower regions around these lines, we generally obtain higher f and better fits (e.g., χ2ν = 5.2 for the 1.48 – 1.52 µm region fit by the K5V template). The large χ2ν values for our fits indicate that our error bars are undersized relative to the true uncertainty in the fits. This is unsurprising, given the systematic uncertainties that affect modeling of NIR spectra in faint compact binaries, including the influence of sky background and telluric absorption correction, uncertain placement of the continuum level where no true continuum exists, and complex line blending wherein small temperature and/or abundance variations between the template and target stars can affect fit results. As a result, we have chosen to determine the mean value and uncertainty in the H-band donor star contribution by using an average of fits to multiple lines and multiple templates over the full waveband, rather than thorough the use of χ2 statistics, as we believe the scatter between line fits provides a more rigorous sampling of uncertainties, particularly systematics. We determined the best representative value for the H band donor star contribution by – 9 – averaging the best-fit f values for the three narrowband H fit regions (1.48 – 1.52 µm, 1.56– 1.61 µm, and 1.70–1.72 µm) and the K5V and K7V templates. Averaging these values for the K5V and K7V template stars gives a donor star fraction f = 0.82±0.02. In K, we fit three regions: the spectrum shortward of He I λ2.058 µm, which is dominated by Ca I absorption lines; the region between He I and Bγ, which includes transitions of Mg I, Al I, and Si I; and the region longward of Bγ, which contains a rich blend of features, including transitions of Na I, Ca I, Fe I, Ti I, Mg I, and CO. Note that we do not have any fit results for the K3V template to the K-band spectrum, because the absorption lines in the K3V template are too weak relative to those of A0620-00, even at f=1. We have already shown from the comparison of the SEDs of A0620-00 and a K3V star that a star this hot cannot be the sole emission source (i.e., have f = 1) in A0620-00 in K without exceeding the observed spectrum at shorter wavelengths. Now we see that a K3V star also cannot be reconciled to the spectrum of A0620-00 by decreasing its fractional contribution, because its absorption lines are already too weak at f = 1 to match those observed in A0620-00. As a result, we restrict our fits in K to K5V and K7V templates. The Ca I lines from 1.90 – 2.02 µm could not be simultaneously fit by a single value for f . The fit values given in Table 4 appear to the eye to be overdiluted as a result of spurious features (residuals of the telluric absorption correction) driving the fits. The strongest lines — 1.978 and 1.987 µm — are fit by f ∼ 0.5, but at this value the other lines in this region are too weak. This spectral region may be contaminated by Bδ λ1.945 µm emission from the accretion disk. Problems also beset the spectral fits in the 2.07 – 2.15 µm region. The K5V and K7V stellar spectra have anomalous emission bumps at 2.14 µm that cause visibly overdiluted fits over the full wavelength range. If the fits are restricted to 2.07 – 2.13 µm, best fits are obtained for f ∼ 0.65, while fits to the strongest feature alone, Al I 2.117 µm, gives f = 0.75 for both the template fits. The final fit region was the long-K portion of the spectrum, 2.18 – 2.42 µm, which includes numerous atomic species transitions and the CO bandheads. Figure 7 shows the best fits for the K5V template over the full fit region and when the fit is restricted to λ < 2.28 µm, excluding the region dominated by the CO lines. When long K is fit in its entirety, the fits are driven to large dilutions of the donor star to match the weak CO absorption in the A0620-00 spectrum. At these low donor fractions (f = 0.45 for the K5V template and f = 0.37 for the K7V template), the atomic lines are too weak in the template spectrum relative to those in A0620-00. If the fit is restricted to 2.18 – 2.28 µm, the donor fractions rise to f = 0.81 (K5V) and f = 0.76 (K7V). At these values, the atomic absorption spectrum of A0620-00 is well fit although there remain discrepencies between template and target spectra, most notably in the red components of Na I λ2.209 and λ2.339 µm and in – 10 – the Fe I lines from 2.226 – 2.247 µm, all of which are too weak in the template relative to A0620-00. A single discrepancy in Fe I may explain these deviations, as there are Fe I transitions coincident with both of the ”Na I” lines (see Figure 6 of FS for an illustration of the complex line blending in this spectral region). 3.3. Fitting Model Spectra to the Spectrum of A0620-00 In addition to modeling A0620-00 with standard star spectra, we used the LinBrod program to generate synthetic spectra for a Roche lobe-filling star in a compact binary with the geometry of A0620-00 (Bitner & Robinson 2006). We adopted the following pa- rameter values for the models: Porb = 0.323 d, q = 0.067, i = 41 ◦, and K2 = 433 km s (McClintock & Remillard 1986; Marsh, Robinson & Wood 1994; Gelino et al. 2001). We cre- ated phase-resolved spectra for donor star temperatures of T = 4000, 4250, 4500, 4750, and 5000 K. Finally, we also created models at each temperature in which the carbon abundance in the star was reduced to [C/H] = -0.5, -1.0, -1.5, and -2.0. To compare to the observed spectrum of A0620-00, we averaged the model spectra over the binary orbit after removing the donor star orbital motion and smoothed the spectra to the observed spectral resolution. Because the SED and spectral type standard star fits to the A0620-00 spectrum point to a donor star spectral type later than K3V, or T < 5000 K, we concentrated on the T = 4000, 4250, and 4500 K models. In an effort to characterize the carbon depletion required to match the observed CO line depths, we also focused on the long-K portion of the spectrum (2.18 – 2.42 µm) . Table 5 gives fit results for the LinBrod model fits. We first fit the solar abundance models and then repeated the fits for the carbon-depleted spectra. Figure 8 shows the normalized long K spectrum of A0620-00 with the solar abundance, T = 4000 K model and with the T = 4000 K, [C/H] = -1.5 spectrum. The model spectra has been scaled by f = 0.77, the best fit donor fraction for the 2.18 – 2.28 µm region. The [C/H] = -1.5 models provided the best fit to the spectrum of A0620-00 for all three of the donor temperatures examined (χ2ν = 4.0 for the T=4000 K, [C/H] = -1.5 model fit to 2.28 – 2.39 µm, versus χ2ν = 5.9 and χ ν = 5.2 for the [C/H] = -1.0 and -2.0 models, respectively). The -0.5 and -1.0 models had CO lines that were still too strong relative to those of A0620-00, while the CO features were virtually absent in the -2.0 models and too weak to match the observed spectrum. – 11 – 4. Discussion and Conclusions 4.1. The Donor Star in A0620–00 4.1.1. The Donor Star Spectral Type and Fractional Contribution to the NIR Spectrum Our analysis of the NIR spectrum of A0620-00 has demonstrated three principal results: 1) the donor star is not the only NIR flux source, with 18±2% of the H-band flux originating in another component of the binary; 2) the donor star must be later than a K3V spectral type; and 3) the CO absorption lines are anomalously weak, requiring a carbon abundance of [C/H] = -1.5 in the donor star to match the observed line depths. A comparison of the broadband NIR SED of A0620-00 with those of spectral type standard star spectra shows that the donor star cannot be the sole source of NIR flux. If the standard star spectra are normalized to the dereddened spectrum of A0620-00 in the K-band, they exceed the observed flux in the J and H wavebands. This result is true for all standard star spectra earlier than M0V. The discrepancy cannot be reconciled by changes in the differential reddening correction because the relative reddening correction between K and J and H is too small. Additionally, the K-band absorption lines in the K3V standard star are too weak to match the line depths seen in the A0620-00 spectrum even at a 100% donor star contribution. Decreasing the donor star contribution to f < 1 will only make the template lines weaker, so a K3V spectral type for the donor star is ruled out. The most precise measurement of the donor star temperature is by González Hernández et al. (2004), who fit synthetic spectra created from model atmospheres to the visible spectrum of A0620-00. They found that T2 = 4900 ± 100 K, which corresponds to a K2/K3V spectral type. This result is not in agreement with our requirement that the donor star be later than K3V. There are reasons to believe that the González Hernández et al. (2004) results may be unreliable, however. To determine the stellar parameters they used 24 Fe I lines, to which they fit models with five free parameters: stellar temperature, gravity, and metallicity, as well as a normalization and slope to represent dilution from the accretion disk. The use of Fe I lines alone to constrain all stellar parameters (plus the disk contribution) is uncommon practice for stellar modeling, which typically relies on independent determinations of the stellar temperature, as well as on both Fe I and Fe II transitions. González Hernández et al. (2004) also determined that the abundances of the elements they fit were slightly above solar values. The adoption of a cooler donor star temperature, as required by our results, will reduce their derived abundances. We conclude that the spectral type of the donor star is most likely between K5V to K7V, but do not attempt to constrain the spectral type more precisely. The rms values – 12 – for the dilution fits to various regions of the A0620-00 spectrum are comparable for both spectral types and we cannot rely on the CO features to create more precise temperature indices. We therefore averaged the dilution values from both spectral type fits for the three narrow regions in the H-band to derive our H-band donor star fraction: f = 0.82 ± 0.02, or an 82% donor star contribution in H. The donor fraction in long K (> 2.2µm) is 81% for a K5V template or 76% for the K7V. Figure 9 shows the spectrum of A0620-00 with a K5V template star scaled to 82% of the H-band flux and the A0620-00 spectrum after the contribution of the donor star has been subtracted. While the spectrum of A0620-00 is dominated by the K type donor, there is a significant second component consisting of a blue continuum and strong H I and He II line emission. Our results of a K5V to K7V donor star spectral type and 18% – 24% veiling in H and K do not agree with those of Gelino et al. (2001), who found that the NIR SED of A0620-00 matched that of a K4V (T = 4600 K) star with no dilution. As pointed out by Hynes et al. (2005), however, there is a degeneracy between the spectral type of the star and the amount and distribution of a veiling spectrum in an XRB: a cooler donor star plus a diluting component can result in the same SED as a hotter donor star with no contamination. Hynes et al. showed that even a 100 K overestimate of the temperature of the donor star could result in a factor of two underestimate of the veiling. Because our data show, both in the line EWs and the NIR SED, that a K type donor star cannot be the only NIR flux source, we conclude that Gelino et al. (2001) overestimated the temperature of the donor star and consequently underestimated the spectral dilution in the NIR. Our results also disagree with the recent work of Harrison et al. (2007), who argue by analogy with the IR spectrum of the cataclysmic variable SS Cygni that A0620-00 has <4% dilution in the K-band. Their argument can be summarized as follows: a) A0620-00 and SS Cyg have similar K-band spectral slopes, binary properties, and quiescent mass accretion rates; b) SS Cyg also has a mid-IR excess; c) the NIR and MIR SEDs of SS Cyg can be well fit by a K4V stellar spectrum plus free-free emission; d) because A0620-00 and SS Cyg are similar, application of the star plus wind model can be extended from the latter to the former to estimate a ∼4% limit on the contamination level of the NIR spectrum of A0620-00. We have several concerns about this line of reasoning. First, as already discussed above, the similarity of a SED to that of a field star does not preclude contaminating emission from the disk. The data shown in Harrison et al. reinforce this: the JHK colors of SS Cyg are consistent with that of an undiluted K4V star even as the ellipsoidal light curves show clear evidence of contamination. In fact, Harrison el al. point out that the K-band spectrum of A0620-00 has a different slope from those of the field stars, which provides fairly clear evidence of contamination. Second, it is not known whether free-free emission is the correct – 13 – explanation for the excess MIR flux observed in SS Cyg. Another possibility discussed by Dubus et al. (2004), who originally published the IR SED shown by Harrison et al., is that circumbinary disk emission is responsible. Indeed, Muno & Mauerhan (2006) have found a MIR component in A0620-00, which they fit with a ≃600 K blackbody component, consistent with emission from circumbinary dust rather than free-free emission in a wind. The cool blackbody component found by Muno & Mauerhan (2006) does not affect the NIR spectrum of A0620-00, which suggests that the attempt to use MIR data from SS Cyg to estimate the NIR contamination in A0620-00 is a red herring. Finally, we reiterate what our simultaneous, JHK spectra of A0620-00 indicate directly about the donor star contamination in the system: based on the shape and fluxes of the JHK continuum spectrum, the absolute EWs of the atomic absorption lines, and dilution analyses using both field stars and a Roche-lobe filling model spectrum, the donor star must be significantly diluted (∼20%), even in the K-band. 4.1.2. Weak CO Features in the Donor Star Spectrum Although the atomic absorption line spectrum in A0620-00 is consistent with a K5V – K7V spectral type, the molecular 12CO lines are significantly weaker in A0620-00 than in a field star, corresponding to CO line strengths normally seen in early G stars. Using the LinBrod model spectra of a Roche lobe-filling star, we found that reducing the C abundance to [C/H] = -1.5 results in a better match to the CO lines in the spectrum than the [C/H] = 0, -1.0, and -2.0 models when the donor star fraction is fit to f = 0.77. The weakness of the 12CO lines in A0620-00 has already been noted by Harrison et al. (2007), who determine that the 12C abundance must be decreased 50% to match the depth of the 12CO bandheads. The 50% reduction in 12C abundance ([12C/H] = -0.3) found by Harrison et al. is significantly smaller than the 97% drop we claim. Again, however, we have several concerns about the method by which Harrison et al. obtained their results. First, they started with the line list included in the spectral synthesis program SPECTRUM (Gray & Corbally 1994), but when they were unable to match the spectra of standard spec- tral type field stars using this line list (and Kurucz atmospheres) due to the presence of strong absorption features in the model but not the observed spectra, they abandoned it and constructed one consisting only of Na I, Mg I, and CO transitions. When they found that the lines in their new models were too weak to match those of field stars at the correct temperature, they globally increased the log(gf) values of every line in their new line list to match up with observations. They then applied this revised model to the spectrum of A0620-00, adjusting the C isotopic abundances until the best fit by eye was achieved. We are not confident in the quantitative reliability of spectral model fits in which spectral – 14 – lines have been dropped and oscillator strengths altered in order to achieve even a rough fit to a K5V field star. We note in contrast that our LinBrod models give fits to the spectrum of A0620-00 of comparable quality of those using template field stars with no deletions or alterations to the spectral synthesis line lists required. Our other concern with the Harrison et al. fits is the placement of the continuum and evaluation of the best model fit. They determined the best-fit model by eye. To our eyes, however, the fits they show in their Figure 3 do not properly take into account the actual data quality: their pseudo-continuum levels appear too high and their CO line minima are too low in their preferred model. A rough comparison of their observed spectrum to ours shows that the 12CO normalized line depths are similar in both data sets, suggesting that a statistical model fit to their spectrum would result in a 12C abundance similar to the one we obtain. Finally, Harrison et al. contend that in addition to a low 12C abundance in A0620-00, the 13C abundance is enhanced, such that 13C/12C = 1. They base their identification of 13CO on the presence of a feature coinciding with the 13CO(3,1) 2.374 µm bandhead. However, they also note that the 13CO(2,0) 2.345 µm is not seen in their spectrum, despite being the stronger feature of the two. Given the poor atmospheric conditions at the time their data was taken and the increasing amount of telluric H20 absorption at these wavelengths, we do not believe that their data provide a clear detection of 13CO and certainly not of 13C at equal abundance with 12C. In our spectrum of A0620-00, there is a feature at the location of the 13CO(2,0) 2.345 µm bandhead, but no feature coinciding with 13CO(3,1) 2.374 µm. We have marked the locations of the first two 13CO bandheads in our Figure 4. Dhillon et al. (2002) point out, however, that a Ti I feature is coincident with the 13CO(2,0) 2.345 µm bandhead and is a much more likely explanation of the feature we see. As a result, we do not believe that an unambiguous detection of 13CO can be made in our spectrum and we find no evidence of any enhancement of this species in A0620-00. Anomalously weak CO absorption features have been seen in other compact binary sys- tems. The NIR spectra of several cataclysmic variables show weak or absent 12CO absorption (Harrison et al. 2000, 2004). In the dwarf nova U Gem, the NIR CO absorption lines are sig- nificantly weaker than in the M3V standard star spectrum that provides a good match to the atomic lines. Model fits to the FUV spectrum of the metal-enriched white dwarf in U Gem show that the C abundance on the WD surface is [C/H] = -1.0, while the N abundance is highly super-solar, [N/H] = 0.7 (Sion et al. 1998; Long & Gilliland 1999; Froning et al. 2001). Similarly, in the UV spectrum of the BH XRB XTE J1118+480 in outburst, typically strong emission lines of C IV and O V are undetectable, while the N V λ1240 Å appears enhanced (Haswell et al. 2002). The UV line ratios are inconsistent with photoionization models, leading Haswell et al. to conclude that the emission spectrum is indicative of the accretion of C-depleted material from the donor star. As a result, A0620-00 can be added – 15 – to the ever-increasing list of cataclysmic variables and XRBs that show depleted C (and, in some cases, enhanced N), pointing to a common history of nuclear processing of C to N in compact binary systems. 4.2. The Mass of the Black Hole in A0620–00 Based on previous work, the mass of the BH in A0620-00 is known to the value of one unknown, the binary inclination: M1 = (3.09± 0.09) sin −3 i (Marsh, Robinson & Wood 1994). The orbital inclination can be obtained by modeling the light curve of the donor star. The donor star fills its Roche lobe and is distorted in shape, which leads to a double-humped ellipsoidal variation in the light curve, the amplitude of which is dependent on inclination. If a source besides the donor star contributes to the light curve, the amplitude of the ellipsoidal variation will be diluted, leading to an underestimate of the inclination and a corresponding overestimate of the BH mass if the contaminating source is not taken into account. The most precise inclination results to date were reported by Gelino et al. (2001), who modeled JHK light curves of A0620-00. Based on good agreement between the target SED and that of a dereddened K4 star, they concluded that the donor star is the only NIR continuum source in A0620-00. Using this assumption, they modeled the ellipsoidal light curve and found i = 40.◦75± 3◦ and M1 = 11.0± 1.9 M⊙. As discussed above, however, the NIR SED alone is insufficient to resolve the degeneracy between donor star temperature and veiling by another flux source in the system. Our results indicate that the donor star cannot be the only NIR flux source in A0620-00 and that consequently, the Gelino et al. results overestimate the BH mass in A0620-00. We previously modeled the H-band light curve in A0620-00 with a donor star plus accre- tion disk model and determined 38◦ ≤ i ≤ 75◦, or 3.3 ≤ M1 ≤ 13.6M⊙ (Froning & Robinson 2001). The broad range of values was caused by a degeneracy between the inclination and the fractional contribution of the accretion disk to the H-band light. Table 5 of Froning & Robinson (2001) gives the inclination in A0620-00 as a function of the fractional contribution of diluting sources in the H-band. In § 3.2 of this paper, we determined that the donor star contributes 82±2% of the H-band flux in A0620-00. This result, a diluting fraction of 18±2%, combined with Table 5 in Froning & Robinson (2001) gives a binary inclination for A0620-00 of i = 43± 1◦. Based on this inclination, we obtain the mass of the BH accretor in A0620-00: M1 = 9.7±0.6M⊙. This result is comparable to previous estimates of the BH mass in the literature. Shahbaz, Naylor & Charles (1994) found a BH mass of 10 M⊙, while Gelino et al. (2001) – 16 – found M1 = 11.0±1.9 M⊙. The inclination we derive is within the error interval of Gelino et al. The difference in BH masses between the two results comes from the slightly higher inclination we adopt as a result of our determination that the donor star cannot be the sole NIR emission source. The error bar on our derived BH mass represents the propagated statistical errors in the result. A potential source of systematic uncertainty is our assumption that the atomic absorption spectrum of A0620-00 can be modeled by template spectra with solar abundances. The fact that the C abundance in A0620-00 has to be decreased significantly to match the CO lines suggests the need for caution in this regard. However, the relative line ratios of the atomic transitions largely agree with each other in the derived fractional donor star contributions. These lines are also strong transitions that lie on the flat portion of the curve of growth and will not be sensitive to small abundance variations. Finally, we note that while González Hernández et al. (2004) derived slightly super-solar abundances for the metal lines in A0620-00, they used a stellar temperature that is too hot to be consistent with the NIR SED. The adoption of a cooler temperature for the donor star will cause the metal line abundances required to fit the optical spectrum to decrease. Another potential source of systematic error is the large time interval (8 years) between acquisition of the NIR light curve and the spectra. Our analysis assumed that the donor star diluting fraction found by analyzing the spectra applies equally well to the light curve data. This assumption is valid, we believe, because while A0620-00 is variable, its NIR colors typi- cally don’t vary by more than 0.2 mag, and six observations of the H-band light curve spaced over days to years had mean colors that agreed to within 0.04 mag (Froning & Robinson 2001; Gelino et al. 2001). Our rough estimate of the absolute calibration of our time-averaged spectrum was also consistent with the previous measurements. We can estimate the uncer- tainty in the non-donor star contribution by assuming that it could vary by ±4%, consistent with the previous measurements. This would cause the donor star fraction to range from 79 ≤ f ≤ 86, which results in i = 43± 1◦, consistent with our statistical uncertainties. We thank Nathaniel Cunningham for assistance in observing A0620-00, and the staff at the IRTF for their support. We also thank Chris Sneden and Niall Gaffney for their help in calculating the LinBrod donor star spectra and for useful discussions. Facilities: IRTF(SpeX) – 17 – REFERENCES Ali, B., Carr, J. S., Depoy, D. L., Frogel, J. A., & Sellgren, K. 1995, AJ, 110, 2415 Bailyn, C. D., Jain, R. K., Coppi, P., & Orosz, J. A. 1998, ApJ, 499, 367 Bitner, M. A. & Robinson, E. L. 2006, AJ, 131, 1712 Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 422, 158 Cushing, M. C., Vacca, W. D., & Rayner, J. T. 2004, PASP, 116, 362 Dhillon, V. S., Littlefair, S. P., Marsh, T. R., Sarna, M. 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SpeX Observations Object Date (UT) Instrument Texp (min) Φ A0620-00 2004 Jan 8 SpeX 290 0.59 – 1.47 A0620-00 2004 Jan 9 SpeX 280 0.69 – 1.49 A0620-00 2004 Jan 10 SpeX 250 0.92 – 1.61 HD42606 (K2.5 III) 2004 Jan 18 SpeX 1.2 · · · HD 3765 (K2 V) 2004 Jan 18 SpeX 2 · · · HD16160 (K3 V) 2004 Jan 19 SpeX 1.8 · · · 61 Cyg A (K5 V) 2000 Sept 15 SpeX 0.5 · · · 61 Cyg B (K7 V) 2000 Sept 15 SpeX 0.5 · · · aOrbital phase coverage of A0620–00 for each night’s observations, based on the ephemeris of McClintock & Remillard (1986). – 22 – Table 2. Equivalent Widths of Selected Absorption Lines Feature EWa (Å) EWb (Å) Si I 1.5892 1.52±0.04 · · · 12CO (6,3) 1.6187 0.86±0.07 · · · Na I 2.2076 0.94±0.06 2.18±0.04 Fe I 2.2263 0.36±0.05 · · · Fe I 2.2387 0.40± · · · Ca I 2.2636 1.06±0.06 3.21±0.03 Mg I 2.2814 0.43±0.04 0.68±0.06 12CO (2,0) 2.2935 0.38±0.06 2.26±0.08 12CO (3,1) 2.3227 0.22±0.02 · · · 13CO (2,0) 2.3448 0.64±0.04 · · · aEWs calculated using the integration lim- its of Förster Schreiber 2000. bEWs calculated using the integration lim- its of Ali et al. (1995). – 23 – Table 3. Wavelength Ranges for Spectral Fits Waveband λ Range (mum) Description J 1.10 – 1.26 Long J 1.175 – 1.215 Blend incl. Mg I, Fe I, Si I H 1.4 – 1.8 Full H band 1.48 – 1.52 Mg1 lines. 1.56 – 1.61 Blend incl. Mg I, Si I 1.70 – 1.72 Mg1 K 1.90 – 2.02 Several Ca I lines. 2.07 – 2.15 Short K incl. Mg I, Si I, Al I. 2.18 – 2.42 Long K incl. CO bandhead. 2.18 – 2.28 No CO, lines incl. Na I, Fe I, Ca I, Mg I. – 24 – Table 4. Template Star Fits to A0620–00 Spectrum Template Wavelength Range Template Fraction rms Spectral Type (µm) (f) K3V 1.10–1.26 0.78 0.012 1.175 – 1.215 0.82 0.010 1.4–1.8 0.76 0.016 1.48 – 1.52 0.71 0.011 1.56 – 1.61 0.81 0.011 1.70 – 1.72 0.91 0.009 K5V 1.10 – 1.26 0.90 0.013 1.175 – 1.215 1.0 0.010 1.4 – 1.8 0.78 0.016 1.48 – 1.52 0.82 0.010 1.56 – 1.61 0.88 0.015 1.70 – 1.72 0.78 0.010 1.90 – 2.02 0.37 0.02 2.07 – 2.15 0.52 0.01 2.18 – 2.42 0.45 0.012 2.18 – 2.28 0.81 0.009 K7V 1.10 – 1.26 0.87 0.013 1.175 – 1.215 0.99 0.011 1.4 – 1.8 0.76 0.017 1.48 – 1.52 0.82 0.010 1.56 – 1.61 0.85 0.015 1.70 – 1.72 0.77 0.009 1.90 – 2.02 0.39a 0.021 2.07 – 2.15 · · · b · · · 2.18 – 2.42 0.37 0.014 2.18 – 2.28 0.76 0.009 – 25 – aFits too diluted due to noise in spectra. bFits compromised by spurious feature in template. – 26 – Table 5. LinBrod Fits to A0620–00 Spectrum Model Temperature Model C Abundance Wavelength Range Template Fraction rms (K) (log[C/H]/[C/H]⊙) (µm) (f) 4000 0.0 2.18 – 2.28 0.72 0.011 4250 0.0 2.18 – 2.28 0.76 0.011 4500 0.0 2.18 – 2.28 0.99 0.011 Full Long K Region Including CO Lines 4000 0.0 2.18 – 2.38 0.14 0.015 4000 -0.5 2.18 – 2.38 0.28 0.014 4000 -1.0 2.18 – 2.38 0.54 0.012 4000 -1.5 2.18 – 2.38 0.78 0.011 4000 -2.0 2.18 – 2.38 0.74 0.012 4250 0.0 2.18 – 2.38 0.14 0.015 4250 -0.5 2.18 – 2.38 0.28 0.014 4250 -1.0 2.18 – 2.38 0.56 0.013 4250 -1.5 2.18 – 2.38 0.82 0.012 4250 -2.0 2.18 – 2.38 0.81 0.012 4500 0.0 2.18 – 2.38 0.17 0.015 4500 -0.5 2.18 – 2.38 0.35 0.014 4500 -1.0 2.18 – 2.38 0.71 0.013 4500 -1.5 2.18 – 2.38 0.96 0.012 4500 -2.0 2.18 – 2.38 0.93 0.013 Longward of the 2.29 µm Bandhead Onlya 4000 -0.5 2.28 – 2.38 0.77 0.025 4000 -1.0 2.28 – 2.38 0.77 0.014 4000 -1.5 2.28 – 2.38 0.77 0.012 4000 -2.0 2.28 – 2.38 0.77 0.013 4250 -0.5 2.28 – 2.38 0.77 0.024 – 27 – Table 5—Continued Model Temperature Model C Abundance Wavelength Range Template Fraction rms (K) (log[C/H]/[C/H]⊙) (µm) (f) 4250 -1.0 2.28 – 2.38 0.77 0.014 4250 -1.5 2.28 – 2.38 0.77 0.012 4250 -2.0 2.28 – 2.38 0.77 0.013 4500 -0.5 2.28 – 2.38 0.77 0.020 4500 -1.0 2.28 – 2.38 0.77 0.014 4500 -1.5 2.28 – 2.38 0.77 0.013 4500 -2.0 2.28 – 2.38 0.77 0.014 aDonor star fraction fixed in these models to the best-fit value to the nearby atomic lines from the template star fits. – 28 – Fig. 1.— The NIR spectrum of A0620–00, obtained in 2004 January. The solid line shows the time-averaged spectrum of A0620. Individual exposures were shifted to remove the orbital motion of the donor star before averaging. The dotted line shows the spectrum after dereddening, assuming E(B–V) = 0.39 (Wu et al. 1976). Fig. 2.— The spectrum of A0620-00 in J. Prominent spectral features are labeled. An error bar representative of the statistical uncertainty per resolution element is plotted on the far right of the plot. Also shown at the bottom of the figure is the spectrum of HD45137, the A0V star used for telluric correction of the A0620–00 spectra. The spectrum retains the throughput profile of each spectral order but is not shown with absolute flux calibration. The H I lines intrinsic to the A0V spectrum have been fitted and removed using the xtellcor program developed by the IRTF. Fig. 3.— The spectrum of A0620-00 in H. Prominent spectral features are labeled. A representative error bar for a resolution element is plotted on the far right. The telluric spectrum is also shown at the bottom of the figure. Fig. 4.— The spectrum of A0620-00 in K. Prominent spectral features are labeled. A representative error bar for a resolution element is plotted on the far right. The telluric spectrum is also shown. Fig. 5.— Shown in black is the dereddened spectrum of A0620-00. Shown in gray is the spectrum of 61 Cyg A, a K5V spectral type star. The spectrum of 61 Cyg A has been normalized to the flux of A0620 just blueward of the 12CO 2.29 µm bandhead. Fig. 6.— The normalized H-band spectrum of A0620–00 with a scaled spectral type standard star fit. The standard star, shown in red, is 61 Cyg A, a K5V star. It has been scaled by f = 0.76. Fig. 7.— The normalized K-band spectrum of A0620–00 with a scaled spectral type standard star fit. The template star is 61 Cyg A, a K5V star. The solid red spectrum shows the template scaled by f = 0.45, the best fit over the full 2.18 – 2.42 µm range. The dashed red spectrum shows the template scaled by f = 0.81, the best fit to the 2.18 – 2.28 µm region. Fig. 8.— The normalized K-band spectrum of A0620–00 with scaled LinBrod T = 4000 K model spectra fits. The solid red line shows the LinBrod model with [C/H] = -1.5, while the dashed red line shows the solar abundance model. To avoid confusion, the latter is shown only for λ > 2.288 µm. The models are scaled by f = 0.77. Fig. 9.— The top panel shows the dereddened spectrum of A0620-00 in black and the spectrum of 61 Cyg A, a K5V star, in gray. The spectrum of the K5V template has been – 29 – scaled to 82% of the A0620-00 flux at the center of the H band, 1.6 µm. The lower panel shows the NIR spectrum of the accretion disk in A0620-00, created by subtracting the template spectrum from that of A0620-00. – 30 – f1.eps – 31 – f2.eps – 32 – f3.eps – 33 – f4.eps – 34 – f5.eps – 35 – f6.eps – 36 – f7.eps – 37 – f8.eps – 38 – f9.eps Introduction Observations and Data Reduction Analysis Classification Based on Spectral Indices Fitting Spectral Type Standard Stars to the Spectrum of A0620-00 Fitting Model Spectra to the Spectrum of A0620-00 Discussion and Conclusions The Donor Star in A0620–00 The Donor Star Spectral Type and Fractional Contribution to the NIR Spectrum Weak CO Features in the Donor Star Spectrum The Mass of the Black Hole in A0620–00

0704.0268

Automated Generation of Layout and Control for Quantum Circuits

Automated Generation of Layout and Control for Quantum Circuits Mark Whitney, Nemanja Isailovic, Yatish Patel and John Kubiatowicz University of California, Berkeley {whitney, nemanja, yatish, kubitron}@eecs.berkeley.edu To appear in the ACM International Conference on Computing Frontiers, 2007 Abstract We present a computer-aided design flow for quantum circuits, complete with automatic layout and control logic extraction. To motivate automated layout for quantum circuits, we investigate grid-based layouts and show a performance variance of four times as we vary grid structure and initial qubit placement. We then propose two polynomial-time design heuristics: a greedy algorithm suitable for small, congestion- free quantum circuits and a dataflow-based analy- sis approach to placement and routing with implicit initial placement of qubits. Finally, we show that our dataflow-based heuristic generates better layouts than the state-of-the-art automated grid-based lay- out and scheduling mechanism in terms of latency and potential pipelinability, but at the cost of some area. 1 Introduction Quantum computing offers us the opportunity to solve certain problems thought to be intractable on a classical machine. For example, the follow- ing classically hard problems benefit from quan- tum algorithms: factorization [19], unsorted database search [6], and simulation of quantummechanical sys- tems [26]. In addition to significant work on quantum al- gorithms and underlying physics, there have been several studies exploring architectural trade-offs for quantum computers. Most such research [3, 16] has focused on simulating quantum algorithms on a fixed layout rather than on techniques for quantum circuit synthesis and layout generation. These studies tend Classical Control: HDL Format (plus annotations for scheduling) Quantum Layout (including initial qubit placement) Basic Blocks Custom Modules Quantum Circuit Specification CAD Flow for Quantum Circuits New Custom Module Figure 1: The goal of our CAD flow is to automate the laying out of a quantum circuit to generate a physical layout, an intelligent initial placement of qubits, the associated classical control logic and an- notations to help the online scheduler better use the layout optimizations as they were intended. This flow may then be used recursively to design larger blocks using previously created modules. to use hand-generated and hand-optimized layouts on which efficient scheduling is then performed. While this approach is quite informative in a new field, it quickly becomes intractable as the size of the circuit grows. Our goal is to automate most of the tasks involved in generating a physical layout and its associated con- trol logic from a high-level quantum circuit specifica- tion (Figure 1). Our computer-aided design (CAD) flow should process a quantum circuit specification and produce the following: • a physical layout in the desired technology • an intelligent initial qubit placement in the lay- • classical control circuitry specified in some hard- ware description language (HDL), which may then be run through a classical CAD flow http://arxiv.org/abs/0704.0268v1 • a set of annotations or “hints” for the online scheduler, allowing a tighter coupling of layout optimizations to actual runtime operation Much like a classical CAD flow, this quantum CAD flow is intended to be used hierarchically. We begin with a set of technology-specific basic blocks (some ion trap technology examples are given in Section 2). We then lay out some simple quantum circuits with the CAD flow, thus creating custom modules. The CAD flow may then be used recursively to create ever larger designs. This approach allows us to develop, evaluate and reuse design heuristics and avoids both the uncertainty and time-intensive nature of hand- generated layouts. 1.1 Motivation for a Quantum CAD Quantum circuits that are large enough to be “inter- esting” require the orchestration of hundreds of thou- sands of physical components. In approaching such problems, it is important to build upon prior work in classical CAD flows. Although the specifics of quan- tum technologies (such as are discussed in Section 2) are different from classical CMOS technologies, prior work in CAD research can give us insight into how to approach the automated layout of quantum gates and channels. Further, quantum circuits exhibit some interesting properties that lend themselves to automatic synthe- sis and computer-aided design techniques: Quantum ECC Quantum data is extremely frag- ile and consequently must remain encoded at all times – while being stored, moved, and com- puted upon. The encoded version of a circuit is often two or three orders of magnitude larger than the unencoded version. Further, the ap- propriate level of encoding may need to be se- lected as part of the layout process in order to achieve an appropriate “threshold” of error-free execution. Rather than burdening the designer with the complexities of adding fault-tolerance to a circuit, computer-aided synthesis, design and verification can perform such tasks automat- ically. Ancillae Quantum computations use many helper qubits known as ancillae. Ancillae consist of bits that are constructed, utilized and recycled as part of a computation. Sometimes, ancillae are explicit in a designer’s view of the circuit. Often, however, they should be added automat- ically in the process of circuit synthesis, such as during the construction of fault-tolerant circuits from high-level circuit descriptions. An auto- matic design flow can insert appropriate circuits to generate and recycle ancillae without involv- ing the designer. Teleportation Quantum circuits present two pos- sibilities for data transport: ballistic movement and teleportation. Ballistic movement is rela- tively simple over short distances in technologies such as ion traps (Section 2). Teleportation is an alternative that utilizes a higher-overhead distri- bution network of entangled quantum bits to dis- tribute information with lower error over longer distances [9]. The choice to employ teleportation is ideally done after an initial layout has deter- mined long communication paths. Consequently, it is a natural target for a computer-aided design flow. 1.2 Contributions In this paper, we make the following contributions: • We propose a CAD flow for automated design of quantum circuits and detail the necessary com- ponents of the flow. • We describe a technique for automatic synthe- sis of the classical control circuitry for a given layout. • We show that different grid-based architectures, which have been the focus of most prior work in this field, exhibit vastly varying performance for the same circuit. • We present heuristics for the placement and routing of quantum circuits in ion trap technol- • We lay out some quantum error correction cir- cuits and evaluate the effectiveness of the heuris- tics in terms of circuit area and latency. Dead End Gate Straight Channel Three-Way Intersection Four-Way Intersection P0P0P0P0 P1P1P1 Straight Channel Gate Figure 2: Example library of basic macroblocks. Each macroblock has a specific number of ports (shown as P0-P3) along with a set of electrodes used for ion movement and trapping. Some macroblocks contain a trap region where gates may be performed (black square). 1.3 Paper Organization The rest of this paper is organized as follows. We introduce our chosen technology in Section 2, fol- lowed by an overview of prior work in the field in Section 3. In Section 4, we detail our proposed CAD flow and our evaluation metrics. In Section 5, we de- scribe the control circuitry interface and scheduling protocol that we use in the following sections. Sec- tion 6 contains a study of grid-based layouts, which have been the basis of most prior work on this sub- ject. In Section 7, we present a greedy approach to laying out quantum circuits, followed in Section 8 by a much more scalable dataflow analysis-based ap- proach to layout. Section 9 contains our experimental results for all three approaches to layout generation, and we conclude in Section 10. 2 Ion Traps For our initial study, we choose trapped ions [4, 17] as our substrate technology. Trapped ions have shown good potential for scalability [10]. In this technol- ogy, a physical qubit is an ion, and a gate is a loca- tion where a trapped ion may be operated upon by a modulated laser. The ion is both trapped and ballistically moved by applying pulse sequences to discrete electrodes which line the edges of ion traps. Figure 3a shows an experimentally-demonstrated layout for a three-way intersection [7]. A qubit may be held in place at any trap region, or it may be ballistically moved between them using the gray electrodes lining the paths. Rather than using ion traps as basic blocks, we de- fine a library of macroblocks consisting of multiple traps for two reasons. First, macroblocks abstract out some of the low-level details, insulating our anal- yses from variations in the technology implementa- tions of ion traps. Details such as which ion species is used, specific electrode sizing and geometry (clearly variable in the layout in Figure 3a) and exact voltage levels necessary for trapping and movement are all encapsulated within the macroblock. Second, ballis- tic movement along a channel requires carefully timed application of pulse sequences to electrodes in non- adjacent traps. By defining basic blocks consisting of a few ion traps, we gain the benefit that crossing an interface between basic blocks requires communica- tion only between the two blocks involved. We use the library of macroblocks shown in Fig- ure 2, each of which consists of a 3x3 grid of trap re- gions and electrodes, with ports to allow qubit move- ment between macroblocks. The black squares are gate locations, which may not be performed at inter- sections or turns in ion trap technology. Each of these macroblocks may be rotated in a layout. This library is by no means exhaustive, however it does provide the major pieces necessary to construct many physi- cal circuits. The macroblocks we present are abstrac- tions of experimentally-demonstrated ion trap tech- nology [7, 18]. In Figure 3, we show how one can map a demonstrated layout (Figure 3a) to our macroblock abstractions (Figure 3b). We model this layout as a set of StraightChannel and ThreeWayIntersection macroblocks. Above the ion trap plane is an array of MEMS mirrors which routes laser pulses to the gate locations in order to apply quantum gates [11], as shown in Figure 3c. Some key differences between this quantum circuit technology and classical CMOS are as follows: • “Wires” in ion traps consist of rectangular chan- nels, lined with electrodes, with atomic ions sus- pended above the channel regions and moved ballistically [13]. Ballistic movement of qubits requires synchronized application of voltages on channel electrodes to move data around. Thus each wire requires movement control circuitry to handle any qubit communication. • A by-product of the synchronous nature of the qubit wire channels is that these circuits can be used in a synchronous manner with no ad- ditional overhead. This enables some convenient pipelining options which will be discussed in Sec- tion 8.1. Figure 3: a) Experimentally demonstrated physical layout of a T-junction (three-way intersection). b) Abstraction of the circuit in (a), built using the StraightChannel and ThreeWayIntersection macroblocks shown in Figure 2. c) The ion traps are laid out on a plane, above which is an array of MEMS mirrors used to route and split the laser beams that apply quantum gates. • Each gate location will likely have the ability to perform any operation available in ion trap technology. This enables the reuse gate locations within a quantum circuit. • Scalable ion trap systems will almost certainly be two-dimensional due to the difficulty of fab- ricating and controlling ion traps in a third di- mension [8]. This means that all ion crossings must be intersections. • Any routing channel may be shared by multi- ple ions as long as control circuits prevent multi- ion occupancy. Consequently, our circuit model resembles a general network, although schedul- ing the movement in a general networking model adds substantial complexity to our circuit. • Movement latency of ions is not only dependent on Manhattan distance but also on the geometry of the wire channel. Experimentally, it has been shown that a right angle turn takes substantially longer than a straight channel over the same dis- tance [18, 7]. 3 Related Work Prior research has laid the groundwork for our quan- tum circuit CAD flow. Svore et al [22, 23] proposed a design flow capable of pushing a quantum program down to physical operations. Their work outlined various file formats and provided initial implementa- tions of some of the necessary tools. Similarly, Balen- siefer et al [2, 3] proposed a design flow and compi- lation techniques to address fault-tolerance and pro- vided some tools to evaluate simple layouts. While our CAD flow builds upon some of these ideas, we concentrate on automatic layout generation and con- trol circuitry extraction. Additionally, initial hand-optimized layouts have been proposed in the literature. Metodi et al [15] pro- posed a uniform Quantum Logic Array architecture, which was later extended and improved in [24]. Their work concentrated on architectural research and did not delve into details of physical layout or scheduling. Finally, Metodi et al [16] created a tool to automati- cally generate a physical operations schedule given a quantum circuit and a fixed grid-based layout struc- ture. We extend and improve upon their work by adding new scheduling heuristics capable of running on grid-based and non-grid-based layouts. Maslov et al [14] have recently proposed heuristics for the mapping of quantum circuits onto molecules used in liquid state NMR quantum computing tech- nology. Their algorithm starts with a molecule to be used for computation, modeled as a weighted graph with edges representing atomic couplings within the molecule. The dataflow graph of the circuit is mapped onto the molecule graph with an effort to minimize overall circuit runtime. Our techniques fo- cus on circuit placement and routing in an ion trap technology and do not use a predefined physical sub- strate topology as in the NMR case. A new ion trap geometry is instead generated by our toolset for each circuit. 4 Quantum CAD Flow The ultimate goal of a quantum CAD flow is identical to that of a standard classical CAD flow: to automate High-Level Description Synthesizer Tech-Independent Netlist Tech Mapping (Tech-Specific Gates, Encoding, Fault Tolerance) Tech Parameter File (Basic Blocks) Custom Modules Tech-Dependent Netlist Interconnect (if necessary) Placement and Routing Classical Control Synthesis for Qubit Movement Geometry-Aware Netlist Figure 4: An overview of our CAD flow for quantum circuits. Ovals represent files; rectangles represent tools. The gray area highlights the portions on which we focus in this paper. the synthesis and laying out of a circuit. For a quan- tum CAD flow, the output circuit consists of both the quantum portion and the associated classical control logic. The quantum CAD flow we present elaborates on the design flows described in prior works [3, 22, 23]. Unlike prior work, our CAD flow addresses the need to integrate automatic generation of classical control into the flow. Figure 4 shows an overview of our CAD toolset. Rectangles are tools, while ovals represent intermediate file formats. Our toolset is built to be as similar to classical CAD flows as possible, while still accounting for the differences between classical and quantum computing described in Section 1.1. At the top, we begin with a high-level description of the desired quantum circuit. At present this spec- ification consists of a sequence of quantum assembly language (QASM [3]) instructions implementing the desired circuit, since this is a convenient format al- ready being used by various third-party tools. We are currently investigating extension of this high-level de- scription to other formats, such as schematic entry, mathematical formulae or a more general high-level language. The synthesizer parses the QASM file and gener- ates a technology-independent netlist stored in XML format. From this point onward (downward in the figure), all file formats are XML. Additionally, infor- mation may be modified or added but generally not removed. As we move down the flow, we add more and more low level details, but we also keep high- level information such as encoded qubit groupings, nested layout modules, distinction between ancillae and data, etc. This allows low-level tools to make more intelligent decisions concerning qubit placement and channel needs based on high-level circuit struc- ture. It likewise allows logical level modification at the lowest levels without having to attempt to deduce qubit groupings. A technology parameter file specifies the complete set of basic blocks available for the layout (see exam- ples in Figure 2), as well as design rules for connect- ing them. A basic block specification contains the following: • the geometry of the block in enough detail to allow fabrication • control logic for each operation possible within the block (including both movement and gates) • control logic for handling each operation possible at each interface The most basic function of the technology map- ping tool is to take a technology-independent netlist and map it onto allowed basic blocks to create the technology-dependent netlist. This may be more or less complicated depending upon the complexity of the basic blocks. In addition, it may need to trans- late to technology-specific gates (in case the QASM file uses gates not available in this technology), en- code the qubits used in the circuit (perhaps also auto- matically adding the ancilla and operation sequences necessary for error correction) and add fault tolerance to the final physical circuit. In the initial technology-dependent netlist, all qubits are physical qubits, meaning that encoding levels have been set (though they may still be modi- fied later). At this point, any technology-specific op- timizations may optionally be applied to the physical circuit encapsulated in this netlist. Additionally, if the circuit is complex enough to warrant the inclusion of a teleportation-based interconnection network [9], it is added to the netlist here using the higher level qubit grouping information in the netlist. Once the designer is happy with the netlist, a place- ment and routing tool lays out the netlist and adds any further channels needed for communication. This geometry-aware netlist may be iterated upon as nec- essary to refine the layout. Once the layout is final- ized, the classical control synthesis tool combines the control logic of the various components of the design, integrates interface control mechanisms to function properly and generates the unified control structure for the entire layout. Our control synthesis tool gen- erates a Verilog file, which may then be run through a classical CAD flow for implementation. The layout specification along with the con- trol logic file together comprise the geometry-aware netlist, which is the end result for the quantum cir- cuit initially specified in the high-level description. In order to allow hierarchical design of larger quan- tum circuits, we may now add this geometry-aware netlist to our set of custom modules. Future technol- ogy mappings may use both the basic blocks speci- fied in the technology parameter file and any custom modules we create (or acquire). The gray area in Figure 4 identifies the portions we shall be focusing on for the rest of this paper. We currently process the high-level description (a QASM file) directly into a technology-dependent netlist for ion traps using the macroblocks shown in Figure 2. Thus we perform a tech mapping, but no automatic encoding, interconnect or addition of gates for fault tolerance. In this paper, we focus on laying out low- level circuits, such as those for encoded ancilla gen- eration and error correction. The classical control synthesis box of the CAD flow is discussed in Sec- tion 5, while placement and routing are analyzed and compared in Sections 6, 7, 8 and 9. We use two main metrics to evaluate the perfor- mance of our CAD flow: area and latency. For area, we consider the bounding box around the layout, so irregularly-shaped layouts are penalized (since they have wasted space). To determine latency of circuit execution, we use the scheduling heuristic described in Section 5.2 and extended in Section 8.3. A third metric of interest is fault-tolerance. For small layouts and circuits, we can use third-party tools to deter- mine whether a given layout and schedule is fault- tolerant [5], but we do not currently use the fault- tolerance metric in our iterative design flow. We use area and latency because, to a first approximation, lower area and lower latency are likely to decrease de- coherence. Previous algorithms to accurately deter- mine the error tolerance of a quantum circuit have in- volved very computationally-intensive analyses that would be inappropriate for circuits with more than a few dozen gates [1]. However, we are looking into ways to incorporate fault tolerance as a metric. 5 Control The classical control system is responsible for exe- cuting the quantum circuit, including deciding where and when gate operations occur and tracking and managing every qubit in the system. It is composed of the following major components: instruction issue logic, gate control logic and macroblock control logic. Instruction issue logic handles all instruction schedul- ing and determines qubit movement paths. Gate con- trol logic oversees laser resource arbitration, deciding which requested gate operations may occur at any given time. The macroblock control logic, which con- sists of an individual logic block for each macroblock in the system, handles all the internals of the mac- roblock, including details of gate operation for each gate possible within the macroblock, qubit movement within the macroblock and qubit movement into and out of the ports. 5.1 Control Interfaces The first step in the control flow involves process- ing the quantum circuit’s high-level description (the QASM file). The instruction issue logic accepts this stream of instructions as input and creates a series of qubit control messages. Using these qubit control messages, macroblock control logic blocks can deter- mine where to move qubits and when to execute a gate operation. Qubit control messages are simple bit streams composed of a qubit ID, along with a se- quence of commands, as shown in Figure 5. When a qubit needs to perform an action, the instruction issue logic sends to it an appropriate control message which travels with the qubit as it traverses the lay- out. Once a macroblock receives a qubit and its corre- sponding control message, it uses the first command in the sequence to determine the operation it must perform. The macroblock then removes the com- Figure 5: Example of how a qubit control message is constructed to move a qubit through a series of mac- roblocks. The qubit enters M0 and travels through M1 and M2, arriving at M3 where it is instructed to perform a CNOT. mand bits used and passes on the remaining control message to the next macroblock into which the qubit travels. In this manner, the instruction issue logic can create a multi-command qubit control message that specifies the path a qubit will traverse through consecutive macroblocks, along with where gate op- erations take place. The instruction issue logic only has to transmit this control message to the source macroblock, relying on the inter-macroblock commu- nication interface to handle the rest. Communication between the instruction issue logic and the macroblocks takes place using a shared con- trol message bus in order to minimize the number of wire connections required by the instruction issue logic. Each macroblock listens to the control message bus for messages addressed to it and only processes messages with a destination ID that match the mac- roblock’s ID. A macroblock is only responsible for monitoring the control message bus if it contains a qubit that has no remaining command bits. This con- dition generally occurs after a gate operation, when the instruction issue logic is deciding what action the qubit should take next. Once the instruction issue logic sends a new control message for the qubit, the macroblock resumes operation. Macroblocks communicate with each other via con- trol signals associated with each quantum port in the macroblock. Each port has signals to control qubit movement into the macroblock and signals to control movement out of the macroblock via that port. These signals are connected to the corresponding signals of the neighboring macroblocks. The macroblocks as- sert a request signal to a destination macroblock when a qubit command indicates the qubit should cross into the next macroblock. If an available signal response is received, the qubit, along with its control message, can move across into the neighbor- ing macroblock; if not, the qubit must wait until the available signal is present. The macroblock interface enables the instruction issue logic to schedule qubit movement as a path through a sequence of macroblocks, without concern- ing itself with the low level details of qubit move- ment. This modular system allows macroblocks to be replaced with any other macroblock that imple- ments the defined interface, without modifying the instruction issue logic. Additionally, macroblocks have an interface to the laser control logic. Whenever a macroblock is in- structed to perform a gate operation, it must request a laser resource through the laser control logic. The laser controller is responsible for aggregating requests from all the macroblocks in the system, and decid- ing when and where to send laser pulses. The laser controller also attempts to parallelize as many oper- ations as possible. Once the laser pulses have com- pleted, the laser controller notifies the macroblocks, indicating that the gate operation is complete. 5.2 Instruction Scheduling The instruction issue logic is responsible for deter- mining the runtime execution order of the instruc- tions in the quantum circuit, which involves both preprocessing and online scheduling. The instruc- tion sequence is first preprocessed to assign priori- ties that will help during scheduling. The sequence is traversed from end to beginning, scheduling instruc- tions as late as dependencies allow, using realistic gate latencies but ignoring movement. Essentially, each instruction is labeled with the length of its crit- ical path to the end of the program. This is similar to the method used in [16], but we use critical path with gate times rather than the size of the dependent subtree. The instruction preprocessing generates an opti- mal schedule assuming infinite gates and zero move- ment cost. However, we wish to evaluate a layout with more realistic characteristics. Our scheduler is designed to schedule on an arbitrary graph, but the layouts provided to it by the place and route tool are in fact planar layouts using only right angles. In ad- dition, the scheduler requires that the qubit initial positions be provided as well. Our scheduler implements a greedy scheduling technique. It keeps the set of instructions which have had all their dependencies fulfilled (and thus are ready to be executed). It attempts to schedule them in priority order. So the highest priority ready instruction (according to critical path) is attempted first and is thus more likely to get access to the re- sources it needs. These contested resources include both gates and channels/intersections. Once all pos- sible instructions have been scheduled, time advances until one or more resources is freed and more instruc- tions may be scheduled. This scheduling and stalling cycle continues until the full sequence has been exe- cuted or until deadlock occurs, in which case it is de- tected and the highest priority unscheduled instruc- tion at the time of deadlock is reported. Since we are interested in evaluating layouts rather than in designing an efficient online scheduler, we use very thorough searches over the graph in both gate assignment and pathfinding. This causes the sched- uler to take longer but takes much of the uncertainty concerning schedule quality out of our tests. In addi- tion, the scheduler reports stalling information which may be used for iterating upon the layout. 5.3 Control Extraction Armed with well defined component interfaces and a method to execute the quantum instructions, all that remains to create the control system for a given quan- tum circuit is putting the pieces together. The quan- tum datapath is composed of an arbitrary number of macroblocks pulled from the component library. Each macroblock in our component library has asso- ciated with it classical control logic. The control logic handles all the internals of the macroblock including details of ion movement, ion trapping and gate oper- ation. In our library, the macroblock control logic is specified using behavioral Verilog modules. When the layout stage of the CAD flow creates a physical layout of macroblocks, we extract the cor- responding control logic blocks and assemble them together in a top-level Verilog module for the full control system, stitching together all necessary mac- roblock interfaces. This module instantiates all the appropriate macroblock control modules, along with the instruction issue logic and laser controller unit. Combined, these modules are assembled into a sin- Figure 6: QPOS grid structure constructed by tiling the highlighted 2× 2 macroblock cell. gle Verilog module which implements the full classi- cal control system for the quantum circuit and which may be input to a classical CAD flow for synthesis. 6 Grid-based Layouts We begin our exploration of placement and routing heuristics by considering grid-based layouts. A ma- jority of the work done in the field has concentrated on these types of layouts. In all of these works, a lay- out is constructed by first designing a primitive cell and then tiling this cell into a larger physical layout. For example, the authors of [15, 16] manually design a single cell, and for any given quantum circuit, they use that cell to construct an appropriately sized lay- out. In [23], the authors automate the generation of an H-Tree based layout constructed from a single cell pattern. Similarly, [3] uses a cell such as in [23] but also provides some tools to evaluate the performance of a circuit when the number of communication chan- nels and gate locations within the cell is varied. We use a combination of these methods to implement a tool that automatically creates a grid-based physical layout for a given quantum circuit. The grid-based physical layouts generated by our tools are constructed by first creating a primitive cell out of the macroblocks mentioned in Section 2 and then tiling the cell to fill up the desired area. For example, Figure 6 shows how a 2 × 2 sized cell can be tiled to create the layout used in [16] (referred to henceforth as the QPOS grid). These types of simple structures are easy to automatically generate given only the number of qubits and gate operations in the quantum circuit. Furthermore, grid-based structures are very appealing to consider because, apart from 1000 2000 3000 4000 5000 6000 7000 8000 9000 0 100 200 300 400 500 600 700 800 900 Grid Structure [[23,1,7]] Golay Encode Grid Search (3x2 cell) Min - Mean - Max (b)(a) Figure 7: Variations in runtime of various grid-based physical layouts for [[23, 1, 7]] Golay encode circuit. For each grid structure the minimum, mean, and maximum time are plotted. Figure 8: Comparison of the best 3×2 cell for two dif- ferent circuits. (a) The best cell for the [[23, 1, 7]] Go- lay encode circuit. (b) The best cell for the [[7, 1, 3]] L1 correct circuit. selecting the number of cells in the layout and the initial qubit placement, no other customization is re- quired in order to map a quantum circuit onto the layout. The regular pattern also makes it easy to de- termine how qubits move through the system, as sim- ple schemes such as dimension-ordered routing can be used. The approach we use to generate the grid-based layout for a given quantum circuit is as follows: 1. Given the cell size, create a valid cell structure out of macroblocks. 2. Create a layout by tiling the cell to fill up the desired area. 3. Assign initial qubit locations. 4. Simulate the quantum circuit on the layout to determine the execution time. The first step finds a valid cell structure. A cell is valid if all the macroblocks that open to the perime- ter of the cell have an open macroblock to connect to when the cell is tiled. Also, a cell cannot have an iso- lated macroblock within it that is unreachable. Once we tile this valid cell to create a larger layout, we must decide on how to assign initial qubit locations. The two methods we utilize are: a systematic left to right, one qubit per cell approach, and a randomized place- ment. The systematic placement allows us to fairly compare different layouts. However, since the initial placement of the qubits can affect the performance of the circuit, the tool also tries a number of random placements in an effort to determine if the systematic placement unfairly handicapped the circuit. This layout generation and evaluation procedure is iterated upon until all valid cell configurations of the given size are searched. We then repeat this process for different cell sizes. The cell structure that results in the minimum simulated time for the circuit is used to create the final layout. As an example, Figure 7 shows the results of searching for the best layout composed of 3× 2 sized cells targeting the [[23, 1, 7]] Golay encode circuit [21], one of our benchmarks shown in Table 1. More than 900 valid cell configurations were tested. For each cell configuration, we try multiple initial qubit place- ments (as mentioned earlier) resulting in a range of runtimes for each cell configuration. Differences in the runtime of the circuit are not limited to just vari- ations on the cell configuration but are in fact also highly dependent on the initial qubit placement. Figure 8 shows the best cell structure found by conducting a search of all 2× 2, 2× 3, and 3× 2 sized cells for two different circuits. The main result of this search is that the best cell structure used to create the grid-based layout is dependent on what circuit will be run upon it. By varying the location of gates and communication channels, we tailor the structure of the layout to match the circuit requirements. While this type of exhaustive search of physical layouts is capable of finding an optimal layout for a quantum circuit, it suffers from a number of draw- backs. Namely, as the size of the cell increases, the number of possible cell configurations grows exponen- tially. Searching for a good layout for anything but the smallest cell sizes is not a realistic option. Fur- thermore, while small circuits may be able to take advantage of primitive cell based grids, larger cir- cuits will require a less homogeneous layout. One approach to doing this is to construct a large layout out of smaller grid-based pieces, all with different cell configurations. While this approach is interesting, we feel a more promising approach is one that resembles a classical CAD flow, where information extracted from the circuit is used to construct the layout. 7 Greedy Place and Route One problem we observed in the regular grid layout design was that the high amount of channel conges- tion due to limited bandwidth causes densely-packed (occupied) gates. Additionally, we found that a num- ber of gate locations and channels in many of the grids were not even used by the scheduler to perform the circuit. We present a new heuristic that attempts to solve some of these problems. The heuristic is a simple greedy algorithm that starts with only as many gate locations as qubits (because we assume that qubits only rest in storage/gate locations) and no channels connecting the gates. It iterates with the circuit scheduler, moving and connecting gate locations un- til the qubits can communicate sufficiently to perform the specified circuit. The current layout is fed into the circuit scheduler which tries to schedule until it finds qubits in gate locations that cannot communi- cate to perform a gate. The place and router then connects the problematic gate locations and tries scheduling on the layout again. The iteration fin- ishes once the circuit can be successfully completed. Our algorithm bears some similarity to the iterative procedure in adaptive cluster growth placement [12] in classical CAD. Gate locations are placed from the center outward as the circuit grows to fit a rectilinear boundary. The placer can move gate locations that have to be connected if they are not already connected to something else. The router connects gate locations by making a direct path in the x and y directions between them and placing a new channel, shifting existing channels out of the way. Since channels are allowed to overlap, intersections are inserted where the new channels cut across existing ones. This technique has the advantage that, since the circuit scheduler prioritizes gates based on gate delay critical path, potentially critical gates are mapped to gate locations and connected early in the process. Thus critical gates tend to be initially placed close together to shorten the circuit critical path. Ad- ditionally, gate locations that need to communicate can be connected directly instead of using a general shared grid channel network, where congestion can occur and cause qubits to be routed along unneces- sarily long paths. A disadvantage of this heuristic is that gate place- ment is done to optimize critical path, not to min- imize channel intersections. This means that the layout could end up having many 4-way channel in- tersections and turns, both of which have more de- lay than 2-way straight channels. Additionally, even though critical gates are mapped and placed near each other, the channel routing algorithm tends to spread these gate locations apart as more channels cut through the center of the circuit. We discuss our experimental evaluation of this heuristic in Section 9. 8 Dataflow-Based Layouts As described in Section 6, a systematic row by row initial placement for qubits allows us to make some- what accurate comparisons between different grid- based layouts, while a random initial qubit placement allows us to test a single grid’s dependence on qubit starting positions. However, in laying out a quantum circuit, we would like to have a more intelligent and natural means of determining initial qubit placement. For this, we turn to the dataflow graph representation of the circuit. 8.1 Dataflow Graph Analysis Figure 9a shows a QASM instruction sequence con- sisting of Hadamard gates (H) and controlled bit- flips (CX) operating on qubits Q0, Q1, Q2 and Q3, with each instruction labeled by a letter. Figure 9b shows the equivalent sequence of operations in stan- dard quantum circuit format. Either of these may A) H Q0 B) H Q1 C) H Q2 D) H Q3 E) CX Q0,Q1 F) CX Q2,Q3 G) CX Q1,Q2 H) CX Q2,Q3 I) CX Q0,Q2 (b) (c)(a) Figure 9: a) A QASM instruction sequence. b) A quantum circuit equivalent to the instruction sequence in (a). c) A dataflow graph equivalent to the instruction sequence in (a). Each node represents an instruction, as labeled in (a). Each arc represents a qubit dependency. A B C D A B C D A B C D NG4NG3NG2 (b) (c)(a) NG4NG3NG2 NG4NG3NG2 Figure 10: a) Each node (instruction) is initialized in its own node group (NG, outlined by the dotted lines), which corresponds to a physical gate location in a layout. Once placed, we extract physical distances between the nodes (the edge labels). b) We find the longest edge weight on the longest critical path (the length 5 edge on the path C-F-G-H-I; solid bold arrows) and merge its two node groups to eliminate that latency. c) We recompute the critical path (A-E-I; dashed bold arrows) and merge its node groups, and so on. be translated into the dataflow graph shown in Fig- ure 9c, where each node represents a QASM instruc- tion (as labeled in Figure 9a) and each arc represents a qubit dependency. With this dataflow graph, we may perform some analyses to help us place and route a layout for our quantum circuit. The general idea is that we shall create node groups in the dataflow graph which correspond to distinct gate locations that may then be placed and routed on a layout. All instructions within a single node group are guaranteed to be executed at a single gate location, as elaborated upon in Section 8.3. To be- gin with, we create a node group for each instruction, giving us a dataflow group graph, as shown in Fig- ure 10a. If we lay out this group graph with a distinct designated gate for each instruction (using heuristics discussed in Section 8.2), we get a layout in which the starting location of each qubit is specified implic- itly by its first gate location, so no additional initial placement heuristic is needed. From this layout we can extract movement latency between nodes and label the edges with weights (as in Figure 10a). We now find the longest critical path by qubit. The critical path A-E-I of qubit Q0 has length 14 (the dashed bold arrows), while the critical path C-F-G-H-I of qubit Q2 has length 15 (the solid bold arrows). We select the longest edge on the longest critical path, which is the edge G-H with weight 5. We merge these two node groups to eliminate this la- tency, in effect specifying that these two instructions should occur at the same gate location (Figure 10b). We then update the layout and recompute distances. Assuming we merged these two node groups to the location of H (NG8), then the weight of edge F-G changes to 1 (to match the weight of edge F-H) and the weight of edge E-G probably changes to 6 (former E-G plus former G-H), but the exact change really depends on layout decisions. The new critical path is now A-E-I, so if we do this again, we merge node groups NG5 and NG9 to eliminate the edge of weight 8, and we get the group graph in Figure 10c. In merging nodes, there is the possibility that two qubit starting locations get merged, complicating the assignment of initial placement. For this reason, we add a dummy input node for each qubit before its first instruction. The merging heuristic doesn’t allow more than one input node in any single node group, so we maintain the benefit of having an intelligent initial qubit placement without extra work. There is an important trade-off to consider when taking this merging approach. A tiled grid layout provides plenty of gate location reuse but is un- likely to provide any pipelinability without great ef- fort. A layout of the group graph in Figure 10a (with each instruction assigned to a distinct gate location) provides no gate location reuse at all but high potential pipelinability. This raises the ques- tion of whether we wish to minimize area and time (for critical data qubits), maximize throughput of a pipeline (for ancilla generation), or compromise at some middle ground where small sets of nearby nodes are merged in order to exploit locality while still re- taining some pipelinability. We intend to further ex- plore this topic in the future. 8.2 Placement and Routing Taking the group graph from the dataflow analysis heuristic, the placement algorithm takes advantage of the fanout-limited gate output imposed by the No- Cloning Theorem [25] to lay out the dataflow-ordered gate locations in a roughly rectangular block. We adopt a gate array-style design, where gate locations are laid out in columns according to the graph, with space left between each pair of columns for necessary channels. This can lead to wasted space due to a linear layout of uneven column sizes, so we may also perform a folding operation, wherein a short column may be folded in (joined) with the previous column, thus filling out the rectangular bounding box of the layout as much as possible and decreasing area. The columns are then sorted to position gate locations that need to be connected roughly horizontal to one another. This further minimizes channel distance be- tween connected gate locations and reduces the num- ber of high-latency turns. Once gate locations are placed, we use a grid-based model in which we first route local wire channels be- tween gate locations that are in adjacent or the same columns. These channels tend to be only a few mac- roblocks long each. A separate global channel is then inserted between each pair of rows and between each Technology- Dependent Netlist Dataflow Analysis & Gate Combination Sorted Dataflow Placement Local/Global Channel Router Geometry-Aware Netlist Gate Scheduler/ Simulator Placement and Routing Figure 11: The placement and routing portion of our CAD flow (shown in Figure 4) takes a technology- dependent netlist and translates it into a geometry- aware netlist through an iterative process involving dataflow analysis and placement and routing tech- niques. pair of columns of gate locations. These global chan- nels stretch the full length of the layout. There are no real routing constraints in our simple model since channels are allowed to overlap and turn into 3- or 4-way intersections. We depend on the dataflow col- umn sorting in the placement phase to reduce the number of intersections and shared local channels. While local channels could technically be used for global routing and vice versa, we’ve found that this division in routing tends to divide the traffic and sep- arate local from long-distance congestion. With these basic placement and routing schemes, we may now iterate upon the layout, as shown in Fig- ure 11. The technology-dependent netlist is trans- lated into a dataflow group graph with a separate gate location for each instruction (Figure 10a). This group graph is then placed, routed and scheduled to get latency and identify the runtime critical path (as opposed to the critical path in the group graph, which fails to take congestion into account). The longest latency move on the runtime critical path (between two node groups) is merged into one node group, thus eliminating the move since a node group represents a single gate location. This new group graph is then placed, routed and scheduled again to find the next pair of node groups to merge. Once this process has iterated enough times, we reach a point where congestion at some heavily merged node group is actually hurting the latency with each further merge. We alleviate this conges- tion by adding storage nodes (essentially gate loca- tions that don’t actually perform gates) near the con- gested node group. This increases the area slightly but maintains the locality exploited by the merging heuristic. If congestion persists, we halt the algo- rithm, back up a few merging steps and output the geometry-aware netlist. 8.3 Annotated Scheduling The scheduling heuristic described in Section 5.2 schedules an arbitrary QASM instruction sequence on an arbitrary layout. However, once we have assigned instructions in a dataflow graph to node groups (as described in Section 8.1), we wish those instructions to be executed at their proper location on any lay- out placed and routed from the group graph. To this end, we annotate each instruction in the instruction sequence with the name of the gate location where it must be executed. Additionally, since we have the gate locations in advance, we can incorporate move- ment in the back-prioritization of the instruction se- quence. Thus, the priority assigned to each qubit now incorporates both gate latencies and movement through an uncongested layout, which gives us a bet- ter approximation of each qubit’s critical path. We use this extended scheduler in our dataflow-based ex- periments presented in Section 9. 9 Results We now present our simulation results for the heuris- tics described in earlier sections. 9.1 Benchmarks Relatively high error rates of operations in a quantum computer necessitate heavy encodings of qubits. As such, we focus on encoding circuits (useful for both data and ancillae) and error correction circuits to ex- periment with circuit layout techniques. We lay out a number of error correction and encoding circuits to Qubit Gate Circuit name count count [[7, 1, 3]] L1 encode [20] 7 21 [[23, 1, 7]] L1 encode [21] 23 116 [[7, 1, 3]] L1 correction [1] 21 136 [[7, 1, 3]] L2 encode [20] 49 245 Table 1: List of our QECC benchmarks, with quan- tum gate count and number of qubits processed in the circuit. evaluate the effectiveness of the heuristics used in our CAD flow in terms of circuit area and latency, as de- termined by our scheduler. Our circuit benchmarks are shown in Table 1. We use two level 1 (L1) encod- ing circuits, a level 2 (L2) recursive encoding circuit and a fault-tolerant level 1 correction circuit. The idea of the encoding circuits is that they will provide a constant stream of encoded ancillae to in- teract with encoded data qubit blocks. Thus, for these circuits, throughput is a more important mea- sure than latency, implying that they would benefit greatly from pipelining. Nonetheless, a high latency circuit could introduce non-trivial error due to in- creased qubit idle time. On the other hand, correc- tion circuits are much more latency dependent, since they are on the critical path for the processing of data qubit blocks. 9.2 Evaluation We have evaluated a variety of layout design heuris- tics on the four benchmarks shown in Table 1. The results are in Table 2. “QPOS Grid” refers to the best scheduled layout from the literature [16] (see Section 6). “Optimal Grid” refers to the best grid with an area matching the QPOS Grid used that was found by the exhaustive search described in Section 6. “Greedy” refers to the heuristic de- scribed in Section 7. “DF” refers to the dataflow- based approach from Section 8. “Non-folded” means the dataflow graph is laid out with varying column widths; “folded” means the layout has been made more rectangular by stacking columns. The num- ber of global channels is between each pair of rows and columns of gate locations. “Critical combining” refers to our dataflow group graph merging heuristic. The exhaustive search over grids yields the best latency for all benchmarks, which is not surprising. Circuit Heuristic Latency (µs) Area [[7, 1, 3]] L1 encode QPOS Grid 548.0 49 Optimal Grid 509.0 49 Greedy channel and gate location placement 648.0 36 Non-folded DF, 2 global channels, critical combining 768.2 231 Folded DF, 1 global channels, critical combining 795.4 126 Folded DF, 2 global channels, critical combining 712.4 182 [[23, 1, 7]] Golay encode QPOS Grid 2268.0 575 Optimal Grid 1801.0 575 Greedy channel and gate location placement 2457.0 168 Non-folded DF, 2 global channels, critical combining 2169.2 3880 Folded DF, 1 global channels, critical combining 2264.0 713 Folded DF, 2 global channels, critical combining 2248.2 1394 [[7, 1, 3]] L1 correction QPOS Grid 1300.0 1271 Optimal Grid 771.0 1271 Greedy channel and gate location placement 1932.0 756 Non-folded DF, 2 global channels, critical combining 999.8 2378 Folded DF, 1 global channels, critical combining 1501.2 690 Folded DF, 2 global channels, critical combining 1121.2 1496 [[7, 1, 3]] L2 encode QPOS Grid 2411.0 1365 Optimal Grid 1367.0 1365 Greedy channel and gate location placement 4791.0 936 Non-folded DF, 2 global channels, critical combining 1582.4 4087 Folded DF, 1 global channels, critical combining 1828.6 1617 Folded DF, 2 global channels, critical combining 1944.8 3381 Table 2: Latency results for a variety of ECC circuits with different placement and routing heuristics. This kind of search becomes intractable quickly as circuit size grows, and additionally, it is based on the unproven assumption that a regular layout pattern is the best approach. We include this data point as something to keep in mind as a target latency. Among the polynomial-time heuristics, we first note that no single heuristic is optimal for all four benchmarks and that, in fact, no single heuristic op- timizes both latency and area for any single circuit. Dataflow-based place and route techniques in general produce the lowest latency circuits. We find that the optimal global channel count per column (1 or 2) de- pends on the circuit being laid out. This is an artifact of the lack of maturity in our routing methodology. We intend to explore more adaptive routing optimiza- tion in our ongoing work. The dataflow approach and the QPOS Grid tend to trade off between latency and area. However, we expect that the dataflow approach will show greater potential for pipelining, thus allowing us to target cir- cuits such as an encoded ancilla generation factory, for which throughput is of greater importance than latency. We also observe that non-folded dataflow layouts are likely to have even greater pipelinability than folded ones, but at the likely cost of greater area. Although, we should note that the area estimates for the non-folded DF-based layouts are in fact overes- timates due to our use of a liberal bounding box for these calculations. We find that the greedy heuristic tends to find the best design area-wise, but the latency penalty increases with circuit complexity. This is expected, as greedy is unable to handle congestion problems, so it works best for small circuits where congestion is not an issue. It is for the opposite reason that the DF heuristics fail on the [[7, 1, 3]] L1 encode. They insert too much complexity into an otherwise simple problem. 10 Conclusion We presented a computer-aided design flow for the layout, scheduling and control of ion trap-based quantum circuits. We focused on physical quantum circuits, that is, ones for which all ancillae, encod- ings and interconnect are explicitly specified. We explored several mechanisms for generating optimal layouts and schedules for our benchmark circuits. Prior work has tended to assume a specific regular grid structure and to schedule operations within this structure. We investigated a variety of grid structures and showed a performance variance of a factor of four as we varied grid structure and initial qubit place- ment. Since exhaustive search is clearly impractical for large circuits, we also explored two polynomial- time heuristics for automated layout design. Our greedy algorithm produces good results for very sim- ple circuits, but quickly begins to be suboptimal as circuit size grows. For larger circuits, we investigated a dataflow-based analysis of the quantum circuit to assist a place and route mechanism which leverages from classical algorithms. We found that our our dataflow approach generally offers the best latency, often at the cost of area. However, we expect that a layout based on the dataflow graph analysis also of- fers better potential for pipelining than a grid-based approach, and we intend to investigate this further in the future. References [1] P. Aliferis, D. Gottesman, and J. Preskill. Quantum accuracy threshold for concatenated distance-3 codes. Arxiv preprint quant- ph/0504218, 2005. [2] S. Balensiefer, L. Kreger-Stickles, and M. Oskin. QUALE: quantum architecture layout evaluator. Proceedings of SPIE, 5815:103, 2005. [3] S. Balensiefer, L. Kregor-Stickles, and M. Oskin. An evaluation framework and instruction set ar- chitecture for ion-trap based quantum micro- architectures. Proc. 32nd Annual International Symposium on Computer Architecture, 2005. [4] J. I. Cirac and P. Zoller. Quantum computations with cold trapped ions. Phys. Rev. Lett, 74:4091– 4094, 1995. [5] A. Cross. qasm-tools. http://www.media.mit.edu/quanta/quanta- web/projects/qasm-tools/, 2006. [6] L. Grover. Symposium on Theory of Computing (STOC 1996), pages 212–219. [7] W. Hensinger, S. Olmschenk, D. Stick, D. Hucul, M. Yeo, M. Acton, L. Deslauriers, C. Monroe, and J. Rabchuk. T-junction ion trap array for two-dimensional ion shuttling, storage, and ma- nipulation. Applied Physics Letters, 88(3):34101, 2006. [8] D. Hucul, M. Yeo, W. K. Hensinger, J. Rabchuk, S. Olmschenk, and C. Monroe. On the transport of atomic ions in linear and multidimensional ion trap arrays. quant-ph/0702175, 2007. [9] N. Isailovic, Y. Patel, M. Whitney, and J. Ku- biatowicz. Interconnection Networks for Scal- able Quantum Computers. Proceedings of the 33rd International Symposium on Computer Ar- chitecture (ISCA), 2006. [10] D. Kielpinski, C. Monroe, and D.J. Wineland. Architecture for a large-scale ion-trap quantum computer. Nature, 417:709–711, 2002. [11] J. Kim, S. Pau, Z. Ma, H. McLellan, J. Gages, A. Kornblit, and R. Slusher. System design for large-scale ion trap quantum information pro- cessor. Quantum Information and Computation, 5(7):515–537, 2005. [12] CM Kyung, JM Widder, and DA Mlynski. Adaptive cluster growth (ACG); a new algo- rithm for circuit packingin rectilinear region. De- sign Automation Conference, 1990. EDAC. Pro- ceedings of the European, pages 191–195, 1990. [13] M.J. Madsen, W.K. Hensinger, D. Stick, J.A. Rabchuk, and C. Monroe. Planar ion trap ge- ometry for microfabrication. Applied Physics B: Lasers and Optics, 78:639 – 651, 2004. [14] D. Maslov, S. M. Falconer, and M. Mosca. Quantum circuit placement: Optimizing qubit- to-qubit interactions through mapping quan- tum circutis into a physical experiment. quant- ph/0703256, 2007. [15] T. Metodi, D. Thaker, A. Cross, F. Chong, and I. Chuang. A Quantum Logic Array Microar- chitecture: Scalable Quantum Data Movement and Computation. Proceedings of the 38th Inter- national Symposium on Microarchitecture (MI- CRO), 2005. [16] T.S. Metodi, D.D. Thaker, A.W. Cross, F.T. Chong, and I.L. Chuang. Scheduling physical operations in a quantum information processor. Proceedings of SPIE, 6244:62440T, 2006. [17] C. Monroe, D. M. Meekhof, B. E. King, W. M. Itano, and D. J. Wineland. Demonstration of a universal quantum logic gate. Phys. Rev. Lett., 75:4714–4717, 1995. [18] C. Pearson, D. Leibrandt, W. Bakr, W. Mallard, K. Brown, and I. Chuang. Experimental investi- gation of planar ion traps. Phys. Rev. A, 73(3), 2006. [19] P.W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. 35’th Ann. Symp. on Foun- dations of Comp. Science (FOCS), pages 124– 134, 1994. [20] A. M. Steane. Simple quantum error correcting codes. Phys. Rev. A, 54:4741–4751, 1996. [21] A.M. Steane. Overhead and noise threshold of fault-tolerant quantum error correction. Phys. Rev. A, 68(4):42322, 2003. [22] K. Svore, A. Aho, A. Cross, I. Chuang, and I. Markov. A Layered Software Architecture for Quantum Computing Design Tools. Computer, 39(1):74–83, 2006. [23] K. Svore, A. Cross, A. Aho, I. Chuang, and I. Markov. Toward a software architecture for quantum computing design tools. Proceedings of the 2nd International Workshop on Quantum Programming Languages (QPL), pages 145–162, 2004. [24] D.D. Thaker, T.S. Metodi, A.W. Cross, I.L. Chuang, and F.T. Chong. Quantum Memory Hierarchies: Efficient Designs to Match Avail- able Parallelism in Quantum Computing. Pro- ceedings of the 33rd International Symposium on Computer Architecture (ISCA), 2006. [25] W. Wootters and W. Zurek. A single quantum cannot be cloned. Nature, 299:802–803, 1982. [26] C. Zalka. Simulating quantum systems on a quantum computer. Proceedings: Math- ematical, Physical and Engineering Sciences, 454(1969):313–322, 1998. Introduction Motivation for a Quantum CAD Flow Contributions Paper Organization Ion Traps Related Work Quantum CAD Flow Control Control Interfaces Instruction Scheduling Control Extraction Grid-based Layouts Greedy Place and Route Dataflow-Based Layouts Dataflow Graph Analysis Placement and Routing Annotated Scheduling Results Benchmarks Evaluation Conclusion

0704.0269

Modeling the Spectral Energy Distribution and Variability of 3C 66A during the WEBT campaign of 2003 -- 2004

Submitted to The Astrophysical Journal Modeling the Spectral Energy Distribution and Variability of 3C 66A during the WEBT campaign of 2003 – 2004 M. Joshi1 and M. Böttcher1 ABSTRACT The BL Lac object 3C 66A was observed in an extensive multiwavelength monitoring campaign from July 2003 till April 2004. The spectral energy dis- tribution (SED) was measured over the entire electromagnetic spectrum, with flux measurements from radio to X-ray frequencies and upper limits in the very high energy (VHE) γ-ray regime. Here, we use a time-dependent leptonic jet model to reproduce the SED and optical spectral variability observed during our multiwavelength campaign. Our model simulations could successfully reproduce the observed SED and optical light curves and predict an intrinsic cutoff value for the VHE γ-ray emission at ∼ 4 GeV. The effect of the optical depth due to the intergalactic infrared background radiation (IIBR) on the peak of the high- energy component of 3C 66A was found to be negligible. Also, the presence of a broad line region (BLR) in the case of 3C 66A may play an important role in the emission of γ-ray photons when the emission region is very close to the cen- tral engine, but further out, the production mechanism of hard X-ray and γ-ray photons becomes rapidly dominated by synchrotron self-Compton emission. We further discuss the possibility of an observable X-ray spectral variability pattern. The simulated results do not predict observable hysteresis patterns in the optical or soft X-ray regimes for major flares on multi-day time scales. Subject headings: galaxies: active — BL Lacertae objects: individual (3C 66A) — gamma-rays: theory — radiation mechanisms: non-thermal 1Astrophysical Institute, Department of Physics and Astronomy, Ohio University, Athens, OH 45701, USA http://arxiv.org/abs/0704.0269v1 – 2 – 1. Introduction Blazars are the most extreme class of Active Galactic Nuclei (AGN) exhibiting rapid variability at all wavelengths and a high degree of linear polarization in the optical. They have been observed at all wavelengths, from radio through VHE γ-rays and are characterized by non-thermal continuum spectra and radio jets with individual components often exhibit- ing apparent superluminal motion. This class of AGNs is comprised of BL Lac objects and flat-spectrum radio quasars (FSRQs), which are distinguished primarily on the basis of the absence or presence of broad emission lines in their optical spectra. The broadband spectra of blazars are associated with non-thermal emission and exhibit two broad spectral components. The low energy component is due to synchrotron emis- sion from non-thermal electrons in a relativistic jet whereas the high energy component is attributed either to the Compton upscattering of low energy radiation by the synchrotron emitting electrons (for a recent review see, e.g., Böttcher (2006)) or the hadronic processes initiated by relativistic protons co-accelerated with the electrons (Mücke & Protheroe 2001; Mücke et al. 2003) . Blazars are often known to exhibit variability at all wavelengths, vary- ing on time scales from months, to a few days, to even less than an hour in some cases. The radio emission of blazars shows variability on a time scale of weeks to months whereas the optical emission for some blazars might vary on a time scale of around one and a half hours. At X-ray energies, some HBLs exhibit characteristic loop features when the pho- ton energy spectral index, α, is plotted against the X-ray flux. These plots are known as hardness-intensity diagrams (HIDs) and the loop structures are called spectral hysteresis. This spectral hysteresis can be interpreted as the signature of synchrotron radiation, due to the gradual injection and/or acceleration of ultrarelativistic electrons in the emitting region and their subsequent radiative cooling (Kirk et al. 1998; Georganopoulos & Marscher 1998; Kataoka et al. 2000; Kusunose et al. 2000; Li & Kusunose 2000; Böttcher & Chiang 2002). 3C 66A is classified as a low-frequency peaked (or radio selected) BL Lac object (LBL). The peak of the low-frequency component of LBLs generally lie in the IR or optical regime, whereas the high-energy component peak is located at several GeV, and the γ-ray output is typically comparable to or slightly higher than the spectral output of the synchrotron component. The redshift of 3C 66A has a relatively uncertain determination of z = 0.444 (Bramel et al. 2005). It has exhibited rapid microvariability at optical and near infrared in the past and has been suggested as a promising candidate for detection by the new generation of atmospheric Čerenkov telescope facilities like H.E.S.S., MAGIC, or VERITAS (Costamante & Ghisellini 2002). This object has been studied in radio, IR, optical, X-rays and γ-rays in the past. Its low-frequency component is known to peak in the IR - UV regime whereas the high-frequency component generally peaks at multi MeV - GeV energies. The – 3 – multiwavelength SED and correlated broadband spectral variability behaviour of 3C 66A have been very poorly understood. For this reason, Böttcher et al. (2005) organized an intensive multiwavelength campaign to observe this object from July 2003 through April 2004, with the core campaign period being Sept. - Dec. 2003. As described in Böttcher et al. (2005), the object exhibited several outbursts in the optical. The variation was on the order of ∆m ∼ 0.3-0.5 over a timescale of several days. The minimum variability timsecale of 2 hr provided an estimate for the size of the emitting region to be on the order of 1015 cm. The optical flares suggested the presence of an optical spectral hysteresis pattern with the B - R hardness peaking several days before the R- and B- band flux peaked. The RXTE PCA data indicated a transition between the synchrotron and the high-energy component at photon energies of & 10 keV. The broadband SED of 3C 66A suggested that the synchrotron component peaked in the optical. In the VHE γ-ray regime, STACEE provided an upper limit at Eγ & 150 GeV whereas an upper limit at Eγ > 390 GeV resulted from simultaneous Whipple observations. In this paper, we use a leptonic jet model to reproduce the broadband SED and the observed optical spectral variability patterns of 3C 66A and make predictions regarding observable X-ray spectral variability patterns and γ-ray emission. In §2, we describe the time-dependent leptonic jet model used to reproduce the observed SED and optical spectral variability patterns of 3C 66A. The parameters used to simulate the observed results are described in §3. The modeling results and VHE γ-ray predictions are discussed in §4. We summarize in §5. Throughout this paper, we refer to α as the energy spectral index, Fν [Jy] ∝ ν −α. A cosmology with Ωm = 0.3, ΩΛ = 0.7, and H0 = 70 km s −1 Mpc−1 is used. In this cosmology, and using the redshift of z = 0.444, the luminosity distance of 3C 66A is dL = 2.46 Gpc. 2. Model Description The SEDs and optical variability patterns of 3C 66A were modeled using a one-zone homogeneous leptonic jet model. The model assumes injection of a population of ultrarel- ativistic non-thermal electrons and positrons into a spherical emitting volume (the “blob”) of comoving radius Rb at a time-dependent rate. Since the positrons lose equal amount of energy as the electrons via the same radiative loss mechanisms so we do not distinguish between them throughout the paper. The injected electron population is described by a single power law distribution with a particle spectral index q, comoving injection density Qinje (γ; t) (cm −3s−1) and low- and high-energy cutoffs γ1 and γ2, respectively, such that – 4 – Qinje (γ) = Q 0 (t)γ −q for γ1 ≤ γ ≤ γ2, where Q 0 (t) is the injection function and is given by, 0 (t) = Linj(t) if q 6= 2 Linj(t) mec2ln(γ2/γ1) if q = 2 where Linj specifies the power of the injected pair population and V b is the blob volume in the comoving frame. The randomly oriented magnetic field B has uniform strength throughout the blob and is determined by an equipartition parameter eB ≡ uB/ue (in the comoving frame), where uB is the magnetic field energy density and ue is the electron energy density. We keep eB constant so that the magnetic field value changes according to the evolving electron energy density value as determined by equation 2. The initial injection of the electron population into the blob takes place at a height z0 above the plane of the central accretion disk. The emitting region travels relativistically with a speed v/c = βΓ = (1 − 1/Γ 2)1/2 along the jet. The jet is directed at an angle θobs with respect to the line of sight. The Doppler boosting of the emission region with respect to the observer’s frame is determined by the Doppler factor δ = [Γ(1− βΓ cos θobs)] −1, where Γ is the bulk Lorentz factor. As the emission region propagates in the jet, the electron population inside the blob continuously loses its energy due to synchrotron emission, Compton upscattering of syn- chrotron photons (SSC) and/or Compton upscattering of external photons (EC). The seed photons for the EC process include the UV soft X-ray emission from the disk entering the jet either directly (Dermer et al. 1992; Dermer & Schlickeiser 1993) or after getting reprocessed in the BLR or other circumnuclear material (Sikora et al., 1994; Dermer et al. 1997). The time-dependent evolution of the electron and photon population inside the emission region is governed, respectively, by, ∂ne(γ, t) ne(γ, t) +Qe(γ, t)− ne(γ, t) te,esc ∂nph(ǫ, t) = ṅph,em(ǫ, t)− ṅph,abs(ǫ, t)− nph(ǫ, t) tph,esc Here, (dγ/dt)loss is the radiative energy loss rate, due to synchrotron, SSC and/or EC emission, for the electrons. Qe(γ, t) is the sum of external injection and intrinsic γ − γ pair – 5 – production rate and te,esc is the electron escape time scale. ṅph,em(ǫ, t) and ṅph,abs(ǫ, t) are the photon emission and absorption rates corresponding to the electrons’ radiative losses and, tph,esc = (3/4)Rb/c is the photon escape timescale. The time-dependent evolution of the electron and photon population inside the blob is followed and radiative energy loss rates as well as photon emissivities are calculated using the time-dependent radiation transfer code of Böttcher & Chiang (2002). The model only follows the evolution of the emission region out to sub-pc scales and as a result only the early phase of γ-ray production can be simulated. Since the radiative cooling is strongly dominant over adiabatic cooling during this phase and the emission region is highly optically thick out to GHz radio frequencies, the simulated radio flux is well below the actual radio data. We do not simulate the phase of the jet components in which they are expected to gradually become transparent to radio frequencies as that would require the introduction of several additional, poorly constrained parameters. 3. Model Parameters The model independent parameters that were estimated using the SED and optical intraday variability measurements (see Böttcher et al. 2005) were used to develop an initial set of input parameters: δ ≈ 15 R ≈ 3.3× 1015 cm B ≈ 2.9 ǫ γ1 ≈ 3.1× 10 γ2 ≈ 1.5× 10 p ≈ 4 (4) Here p is the equilibrium spectral index that determines the optical synchrotron spec- trum and p = q+1 for strongly cooled electrons. The initial set of parameters was modified to reproduce the quiescent as well as the flaring state of 3C 66A. Approximately 350 sim- ulations were carried out to study the effects of variations of various parameters, such as γ1, γ2, q, B and Γ, on the resulting broadband spectra and light curves. The set of model parameters that provided a satisfactory fit to the quiescent state of 3C 66A involved a value of the Doppler factor, δ = Γ = 24 and a viewing angle of θobs = 2.4 o. These parameters were chosen on the basis of VLBA observations that provided the limits on the superluminal motion and indicated bending of the jet towards the line of sight thus resulting in a smaller – 6 – viewing angle and a higher Doppler boosting of the emission region as compared to the values inferred from the superluminal measurements on larger scales (Jorstad et al. 2005; Böttcher et al. 2005). The fitting of the SED both in the quiescent as well as flaring state of 3C 66A was carried out such that the simulated quiescent state does not overpredict the X-ray photon flux as X-ray photons are expected to be dominated by the flaring episodes. On the other hand, the flaring state was simulated such that the resulting time-averaged spectrum passes through the observed time-averaged optical as well as X-ray data points. This was achieved by varying individual parameters, such as, γ1, γ2 and q between the values for quiescent and flaring states with time profiles as discussed in the next section. A value of γ1 = 2.1× 10 3, γ2 = 4.5× 10 4 and q = 2.4 provided a satisfactory fit to the flaring state. Also, during our multiwavelength campaign of 2003 - 2004, flux upper limits at multi-GeV - TeV energies could be obtained and as a result we could get upper limits on the respective parameters governing the EC component. The various model parameters used to simulate the two states of 3C 66A are listed in Table 1. Figures 1 and 2 respectively show the reproduction of the SED of 3C 66A, for both the quiescent and flaring state observed during the campaign period. The quiescent state is a reproduction of the state observed around 1st October 2003 whereas the flaring state is the reproduction of a generic 10 day flaring period corresponding to the timescale of several of the major outbursts that were observed during the campaign. The simulated time-averaged spectrum of 3C 66A in the flaring state is shown in Figure 3. The simulations, corresponding to fits 1 and 2 of Table 1, were carried out for a pure SSC emission process by artificially setting LD = 0, where LD is the bolometric disk luminosity. Fit 3 of Table 1 refers to an EC+SSC case with LD = 1.0× 10 45 ergs s−1 and is shown in Figure 9. The value of LD was chosen such that it is more than the value of the jet luminosity used in the simulations and at the same time does not produce a blue bump in the simulated SED. In order to assess the possible effect of EC emission in 3C66A, an upper limit to the optical depth of the BLR was first determined using XSTAR, which returns the ionization balance and temperature, opacity, and emitted line (Hα, Hβ) and continuum fluxes. The BLR was modeled as a spherical shell with rBLR,in = 0.045 pc and rBLR,out = 0.050 pc, where rBLR,in and rBLR,out stand for the inner and outer radii of the broad line region. A Thomson optical depth of 0.3 for the BLR was chosen as a reasonable upper limit such that the line emission is weak enough or absent to be consistent with the observed featureless continuum. – 7 – ν [Hz] Oct. 1 Nov. 1 Nov. 11 Dec. 28 STACEE Whipple (99 % UL)RXTE 2003 XMM−Newton ROSAT EXOSAT EGRET BeppoSAX 1999/2001 EINSTEIN 1979/1980 STACEE (99 % UL, Γ = 2.5) (99 % UL, Γ = 3.0) STACEE (99 % UL) Fig. 1.— Reproduction of the quiescent state of 3C 66A observed around October 1st 2003. The simulation of this state was carried out using parameters that do not overpredict the X- ray photon flux. The black colored solid line indicates the instantaneous spectrum generated by the simulation after the system (blob + injected electron population) attains equilibrium. The low-energy component peaks in the optical at νsyn ≈ 4.8 × 10 14 Hz whereas the high- energy SSC component peaks in the MeV regime at νSSC ≈ 1.6× 10 21 Hz. The synchrotron cooling timescale in the observer’s frame is ≈ 1.2 hours, which is on the order of observed minimum optical variability timescale of 2 hours. The diamond shaped STACEE upper limit is a new addition and is provided by Lindner (2006). All data that are indicated by dotted curves are archival data and are shown for comparison. The historical average of the 5 EGRET pointings is also included to provide a guideline for our simulated VHE emission. – 8 – ν [Hz] Oct. 1 Nov. 1 Nov. 11 Dec. 28 STACEE Whipple (99 % UL) RXTE 2003 XMM−Newton ROSAT EXOSAT EGRET BeppoSAX 1999/2001 EINSTEIN 1979/1980 STACEE (99 % UL, Γ = 2.5) (99 % UL, Γ = 3.0) STACEE (99 % UL) Fig. 2.— Simulation of the flaring state for a generic 10 day flare corresponding to the timescale of several major outbursts that were observed in the optical regime during our campaign. The various curves show the instantaneous spectral energy distribution of 3C 66A at several different times in the observer’s frame: black (red in the online version) dotted line (∼ 5th hour), gray (green) dashed line (∼ 8th hour), black (blue) dot-dashed line (∼ 14th hour), gray (yellow) long-dashed line (∼ 20th hour), long-dashed black line (∼ 8th day, highest state attained by the system during flaring), gray solid line (∼ 9th day), dotted black (violet) line (∼ 16th day), gray (cyan) colored solid line (∼ 18th day), dashed black (magenta) colored line (∼ 20th day) and black (red) solid line (∼ 22nd day, equilibrium state reached by the system after the flaring episode is over). The synchrotron component of the flaring state peaks at νsyn ≈ 1.1× 10 15 Hz and the SSC component peaks at νSSC ≈ 2.7× 10 22 Hz. The SSC component of this state cuts off at νSSC,cutoff ≈ 2.3 × 10 24 Hz. The synchrotron cooling timescale in the optical regime is ≈ 37 minutes for the flaring state. – 9 – 4. Results and Discussion As can be seen in Figure 3, the time-averaged simulated spectrum passes through the time-averaged optical data points whereas the high energy end of the synchrotron compo- nent passes through the time averaged X-ray data indicating the dominance of synchrotron emission in the production of such photons in case of flaring. For X-ray photons with energy beyond 10-12 keV, the data is less reliable due to low count rates and possible source con- fusion with 3C 66B. The spectral upturn at ≥ 7 keV occurs due to the presence of the SSC component in the simulation. The presence of this component cannot be suppressed because in order to suppress it the population of seed photons would have to be diluted, which can be done by increasing the size of the emission region. But the size of the emission region cannot be increased any further due to the strict constraint on the maximum size of the blob that comes from the observed minimum variability timescale in the optical region, which is 2 hrs. Hence, the emission region size cannot exceed 3.6 × 1015 (D/24) cm. Thus, our model suggests that the harder X-ray photons come from the SSC and not the synchrotron mechanism with the expected spectral hardening taking place at ∼ 7 keV. The high energy component, due to the SSC emission, for the time-averaged spectrum (see Figure 3) cuts off at ∼ 1.0× 1024 Hz or 4 GeV. From the simulated level of VHE emission we predict that the object is well within the observational range of MAGIC, VERITAS and, especially, GLAST (see Figure 3) whose sensitivity limit is 50 times lower than that of EGRET at 100 MeV and even more at higher energies and its two year limit for source detection in an all-sky survey is 1.6 × 10−9 photons cm−2 s−1 (at energies > 100 MeV). Thus it will be possible to extract the spectral and variability information for this object at such high energies in future observations. Flaring above the quiescent state of 3C 66A was reproduced using a flaring profile for the electron injection power (Linj(t)) that was Gaussian in time (see Figure 4): Linj(t) = L inj(t) + (Lflinj − L (z−rc)2 ] (5) Here, qu and fl stand for the quiescent and flaring state respectively, z determines the position of the emission region in the jet at time t, rc indicates the position of the center of the simulated flare and σ stands for the Gaussian width of the flare. The rest of the parameters such as γ1 and γ2 and q were also changed accordingly. In order to simulate the observed optical flare, the system was first allowed to come to an equilibrium and after the equilibrium was set up the flare was introduced with a Gaussian width, σ corresponding to 14 days in the observer’s frame. Although the flare was introduced – 10 – ν [Hz] VERITAS STACEE Whipple (99 % UL) RXTE 2003 XMM−Newton ROSAT EXOSAT EGRET BeppoSAX 1999/2001 EINSTEIN 1979/1980 STACEE (99 % UL, Γ = 2.5) (99 % UL, Γ = 3.0) GLAST MAGIC (Large ZA) STACEE (99 % UL) Fig. 3.— Time-averaged spectral energy distribution of 3C 66A for a period of 23 days around a flare as shown in Figure 2. The filled black (colored in the online version) circles are the time-averaged optical and IR data points for the entire campaign period and the “RXTE 2003” denotes the time-averaged X-ray data points. The dot-dashed black line is the contribution from the synchrotron component only whereas the long-dashed black line indicates the contribution of the SSC component only. The time-averaged synchrotron component peaks at νsyn ≈ 7.2× 10 14 Hz whereas the time-averaged SSC component peaks at νSSC ≈ 5.3 × 10 21 Hz. The synchrotron component cuts off near 7 keV whereas the SSC component cuts off at ∼ 4 GeV. The black colored dashed line indicates the attenuation due to the optical depth at VHE energies. The γγ absorption effect becomes significant at ∼ 200 GeV. The black (green, maroon and magenta) lines indicate the sensitivity limits for an observation time of 50 hours for MAGIC, VERITAS and MAGIC (Large Zenith Angle) and for GLAST for an observation time of 1 month. – 11 – in order to simulate the observed major optical outbursts lasting for 10 days, the choice of 14 days for the Gaussian width was made such that the width of the simulated flare matches that of the observed flare, rc was adjusted such that the centre of the simulated flare aligns with that of the observed one and the value of Linj was varied such that the peak of the simulated flare matches that of the observed one. The observed lightcurves did not agree well with a flaring profile that was top-hat or triangular in time as can be seen in the figure. The presence of a flare that is Gaussian in time might represent an initial injection of particles into the emission region at the base of the jet. The particles slowly get accelerated as a shock wave ploughs through the region and finally dies out in time. Crucial information on the dominant acceleration mechanism comes from the change in the shape of the particle injection spectral index with time, which might also indicate a possible change in the B-field orientation. According to the current understanding of acceleration mechanisms, parallel shocks generally produce electron spectra of Qe(γ) ∝ γ −q with 2.2 . q . 2.3 (Achterberg et al. 2001; Galant et al. 1999), whereas oblique shocks produce much softer injection spectral indices. On the other hand, 2nd order Fermi acceleration behind the shock front might give rise to a harder injection index of the order of q ∼ 1 or beyond (Virtanen & Vainio 2005). In order to reproduce the flaring state, the simulation first starts out in the quiescent state with quiescent state parameters and then the value of these parameters is changed to the flaring state parameters as the flaring is introduced in the simulation. Since, the value of q, in our simulations, changes from 3.1 (quiescent state) to 2.4 (flaring state) it might indicate a possible change in the orientation of the B-field from oblique to parallel during the flaring episode or an interplay between the 1st and 2nd order Fermi acceleration thereby making the particle spectra harder. The contribution from such acceleration mechanisms and the shear acceleration (Rieger & Duffy 2004) might play an important role in accelerating the particles to higher energies. The simulated optical variability in the R band (0.55 mag) matches the observed value (0.3-0.5 mag) for a 10 day period outburst. The predicted variability in B is more than that of R by ∼ 0.15 mag as also observed, which indicates that the spectrum is becoming harder (see Figure 5) with the spectral upturn occuring at B-R ≈ 0.72 mag as shown in Figure 6. Figure 6 is a hardness intensity graph that shows that the object follows a positive correlation of becoming harder in B-R while getting brighter in both the bands during the 10-day flare simulated in Figure 2. This agrees well with the observed optical variability pattern. In this study, we are not addressing the variability that was observed on intraday timescales as that analysis would open up an even larger parameter space, which cannot be reasonably well constrained without any variability information in the X-ray regime. The flare declines faster as compared to the time taken by the flare to rise. This might – 12 – 2940 2945 2950 JD - 2450000 Fig. 4.— The simulated lightcurves for various flaring profiles that have been superimposed on the observed R-band lightcurve (see Figure 7 of Böttcher et al. (2005)) for an outburst on ∼ November 1st 2003. The solid black line denotes a flaring profile that is Gaussian in time as used for the flare in Figure 2, the dash-dotted black line is a trianglular flaring profile whereas the dashed black line is a flaring profile that is top-hat in time. As can be seen, the Gaussian flaring profile closely matches the width as well as the profile of the observed flare. – 13 – 0.0e+00 2.0e+05 4.0e+05 6.0e+05 8.0e+05 1.0e+06 1.2e+06 time (sec) 2e+12 3e+12 4e+12 5e+12 0.0e+00 5.0e+11 1.0e+12 1.5e+12 1 keV 10 keV 15 keV 3 keV 0e+00 2e+12 4e+12 6e+12 1 MeV 100 MeV 10 GeV ∆R ~ 0.55 mag Fig. 5.— Simulated lightcurves for the optical, X-rays and γ-ray energy regimes shown in the three panels respectively. The simulated variability in the R band is ≈ 0.55 mag as indicated by the arrows. The B band, denoted by the black dotted line exhibits a higher variability of ≈ 0.7 mag, in the simulation, than that in the R band, which is consistent with our observations. The simulated lightcurve at 1 keV is indicated by a black dashed curve and exhibits an amplitude variation of ≈ 1.4 × 1012 Jy Hz. The 3, 10 and 15 keV lightcurves, denoted by the black solid line, black long-dashed line and the black dot-dashed curve, respectively, on the other hand do not exhibit much variability. In the VHE regime, the 1 MeV lightcurve is denoted by a black solid line. The 100 MeV lightcurve is indicated by a black long-dashed curve and the simulated variability amplitude in this energy regime is on the order of 1012 Jy Hz. The black dot-dashed line indicates the lightcurve at 10 GeV. – 14 – 13.6013.7013.8013.9014.0014.10 R magnitude 0.640 0.660 0.680 0.700 0.720 0.740 0.760 0.780 Fig. 6.— The simulated hardness-intensity diagram indicates a positive correlation between R- and B-band for an outburst lasting for ∼ 10 days. The object becomes brighter in R and harder in B-R as shown by the arrows. The spectral upturn takes place at B-R ≈ 0.72 mag where the flux in B equals that in R (corresponding to αBR = 0). – 15 – indicate that the particles’ synchrotron cooling timescale is less than or equal to the light crossing time. τ obscool,sy ≈ 2.8× 10 −1/2 ( 15 s (6) We can calculate the observed synchrotron cooling timescale, τ obscool,syn in the optical regime from equation 6 (Böttcher et al. 2005) using δ = 24, B = 2.4 G and ν15 = 0.48 for the quiescent state and B = 2.8 G and ν15 = 1.1 for the flaring state (see Figures 1 and 2), where ν15 is the characteristic synchrotron frequency in units of 10 15 Hz. This yields a value of τ obscool,sy ∼ 1.2 hours for the quiescent state whereas for the flaring state it reduces to 37 minutes. The observed minimum variability timescale of ∼ 2 hours might therefore correspond to the observed dynamical timescale, where τ obsdyn ≈ 1 + z . (7) This implies that it takes time to build up the electron population in the emission region through flaring but once built up the electrons lose their energy efficiently to produce synchrotron photons. This can be used to constrain the value of the magnetic field in the jet, which has been allowed to evolve in time keeping eB = 1 and has an average value of 2.4 Gauss in the simulated quiescent state and 2.8 Gauss in the simulated flaring period. The crossover of X-ray lightcurves, in our simulations, is a result of the dominance of the SSC component in hard X-rays (see Figure 5). The lightcurve of soft X-ray photons of energy 1 keV exhibits a greater variability of ∼ 1.4 × 1012 Jy Hz in its flux as compared to their optical counterpart. This is expected because the soft X-ray photons, during the flaring episode, are produced from synchrotron emission of electrons that are accelerated to very high energies and as a result have a very short cooling timescale and thus greater variability. In case of hard X-rays no significant variability is predicted. This is because such photons are produced from Compton upscattering of synchrotron photons off the low-energy electrons and as a result the cooling timescale is much longer as compared to the cooling timescale of their soft X-ray and optical counterparts. Hence, the variability information gets washed out. The predicted X-ray spectral variability pattern of large variability in the low X-ray energy band and negligible variability in the high X-ray energy band is similar to what has also been observed in BL Lacertae on several occasions (see for e.g., Ravasio et al. 2003, 2002). As can be seen in Figure 7, spectral hysteresis patterns are not predicted for optical as – 16 – well as soft X-ray photons. This is expected because the cooling timescale of their parent electron population is so short that what is observed is the average effect of this cooling over the dynamical timescale and hence any hysteresis pattern gets smeared out. On the other hand, one expects to see these patterns at higher energies because as explained earlier, this photon population comes from Compton upscattering off low-energy electrons, which have a longer cooling timescale and as a result the photon population gradually builds up over time and then dies away giving rise to a hysteresis pattern (see Figure 8). The slight spectral softening at 10 keV seen in its hysteresis pattern (see Figure 7) for higher values of νFν indicates a small synchrotron contribution near the peak of the flare. The simulated instantaneous SED, for a pure SSC model, shows a definite presence of γ-ray emission in 3C66A, in the quiescent as well as the flaring state (see Figure 1 and 2). The intrinsic cutoff of VHE emission in the flaring state, according to the simulations, for the time averaged spectrum is ∼ 1.0 × 1024 Hz or 4 GeV. In our simulations, the emission of VHE γ-ray photons is produced by the SSC mechanism in the quiescent as well as the flaring state. Figure 5 shows the simulated lightcurves for VHE photons and as can be seen the νFν value changes by ∼ 4.17×10 12 Jy Hz at 100 MeV. The variability in VHE photons is expected as they are the result of Compton upscattering off the higher energy electrons and due to this the hysteresis pattern is not seen at such high energies as the cooling timescale of such high energy electrons is very short (see Figure 8). From Figure 9, it can be seen that the high-energy component of 3C 66A, in the flaring state, could start out with a dominant contribution of the EC emission, shown by the red solid line. But as the blob travels further away and passes the outer edge of the broad line region, the EC contribution becomes less significant and the SSC emission takes over. This is indicated by the black long-dashed line in the figure. We might actually find that this maximum contribution would be just enough to explain the historical EGRET flux and that there could be GeV flaring due to early external Comptonization. The effect of an optical depth due to the IIRB on the spectra of 3C 66A was also evaluated and was found to be insignificant in the energy range we are interested in as shown in Figure 3. The optical depth due to the IIRB was determined using the analytic expression given in Stecker et al. (2006). The γ − γ absorption till ∼ 100 GeV is negligible and becomes slightly observable at ∼ 200 GeV as the optical depth takes a value of, τγγ ≈ 2.9. Hence, the SSC emission cutoff value at ∼ 4 GeV is intrinsic. – 17 – 0.61 0.71 0.81 0.91 νFν [Jy Hz] 0.09 0.29 0.49 0.69 0.89 0.55 0.65 0.75 0.85 0.95 1 keV 10 keV Fig. 7.— Simulated spectral hysteresis pattern in the R-band, 1 keV and 10 keV energy regimes, shown in the three panels respectively. As can be seen, the hysteresis pattern starts to show up in the 10 keV energy regime. – 18 – 0.45 0.55 0.65 0.75 0.85 0.95 νFν [Jy Hz] 0.15 0.35 0.55 0.75 0.95 0.02 0.22 0.42 0.62 0.82 α 10 GeV 1 MeV 100 MeV Fig. 8.— Simulated hysteresis pattern for 1 MeV, 100 MeV and 10 GeV energy regimes, shown in the three panels repectively. The hysteresis pattern is prominent for the 1 MeV energy regime but starts to become absent at higher energies. – 19 – ν [Hz] Oct. 1 Nov. 1 Nov. 11 Dec. 28 STACEE Whipple (99 % UL)RXTE 2003 XMM−Newton ROSAT EXOSAT EGRET BeppoSAX 1999/2001 EINSTEIN 1979/1980 STACEE (99 % UL, Γ = 2.5) (99 % UL, Γ = 3.0) STACEE (99 % UL) Fig. 9.— Simulation of the effect of the BLR on the instantaneous spectral energy distri- bution of 3C 66A for the first 3 days of a simulation similar to Figure 2. The curves in the figure denote the instantaneous spectra obtained from the simulation. The gray (red in the online version) solid line denotes one of the initial instantaneous spectrum at the beginning of the simulation whereas the black long-dashed line indicates the last spectrum obtained from the simulation. – 20 – 5. Summary An extensive analysis of the data of 3C 66A, obtained from the multiwavelength mon- itoring campaign on 3C 66A from July 2003 to April 2004, was carried out using a time- dependent leptonic jet model. The analysis was targeted towards understanding the dom- inant radiation mechanism in the production of the high-energy component of the SED of 3C 66A in the quiescent as well as the flaring state. Our simulations yielded predictions regarding the observable variability patterns in the X-ray as well as the VHE energy regimes where such patterns could not be detected during the campaign. The object was well sam- pled in the optical, especially in the R-band, during the campaign. It had exhibited several major outbursts (∼10 days) in this regime with a varibility of ∆m ∼ 0.3-0.5. The X-ray data covered the 3-10 keV range with the onset of the high-energy component expected at ≥ 10 keV photon energies. Only upper limits in the VHE regime had been obtained. The simulations from our model could successfully reproduce the observed SED as well as the optical spectral variability patterns. The model suggests the dominance of the SSC mechanism in the production of hard X-ray as well as VHE photons. On the other hand, soft X-ray photons exhibit spectral softening during flaring indicating the onset of the synchrotron component in this energy range. According to the simulated time-averaged spectrum, the synchrotron component is expected to cut off near 7 keV whereas the SSC component cuts off at ∼ 4 GeV. A flaring profile that was Gaussian in time could successfully reproduce the observed flaring profile for a timescale of ∼ 10 days. The simulated varibility in R (∆m ∼ 0.55) agreed well with the observed variability. According to the simulations, the object flares up in R and B simultaneoulsy with τ obscool,syn (37 minutes) being less than or equal to the light crossing time (2 hours) during flaring. No significant variability is predicted in the hard X-ray regime. This is due to the production of such photons from Compton upscattering off low-energy electrons with cooling timescales much longer than the light crossing time, 3Rb/4c. On the other hand, the simulated lightcurves of VHE γ-ray photons exhibit significant variability as such photons are produced from the Compton upscattering off higher energy electrons with shorter cooling timescales than the light crossing time. The effect of the optical depth due to γ − γ absorption by the IIBR on the SED of 3C 66A was also evaluated. The simulations do not predict a significant effect on the SED due to the optical depth. The SSC emission cutoff predicted to be at ∼ 4 GeV can be taken as the intrinsic SSC emission cutoff value for this object. We predict the object to be well within the observational range of MAGIC, VERITAS and GLAST. Finally, the EC emission for this object was also calculated and it appears that the EC emission could be dominant in the high-energy component initially, but as the emission region travels further – 21 – away from the BLR, the EC contribution becomes less significant and the SSC emission takes over. It is highly probable that this maximum contribution of the EC component might explain the historical EGRET flux and that there could be GeV flaring due to early external Comptonization. This work was partially supported through NRL BAA 76-03-01, contract no. N00173- 05-P-2004. REFERENCES Achterberg, A., et al., 2001, MNRAS, 328, 393 Böttcher, M., 2006, in proc. “The Multi-Messenger Approach to High-Energy Gamma-Ray Sources”, Barcelona, Spain, 2006, Astroph. & Space Sci., in press Böttcher, M., et al., 2005, ApJ, 631, 169 Böttcher, M., & Chiang, J., 2002, ApJ, 581, 127 Böttcher, M., Mause, H., & Schlickeiser, R., 1997, A&A, 324, 395 Bramel, D. A., et al., 2005, ApJ, 629, 108 Costamante, L., & Ghisellini, G., 2002, A&A, 384, 56 Dermer, C. D., Sturner, S. J., & Schlickeiser, R., 1997, ApJS, 109, 103 Dermer, C. D., & Schlickeiser, R., 1993, ApJ, 416, 458 Dermer, C. D., Schlickeiser, R., & Mastichiadis, A., 1992, A&A, 256, L27 Gallant, Y. A., et al., 1999, A&AS, 138, 549 Georganopoulos, M., & Marscher,A. P., 1998, ApJ, 506, 621 Jorstad, S. G., et al., 2005, AJ, 130, 1418 Kataoka, J., et al., 2000, ApJ, 528, 243 Kirk, J. G., Rieger, F. M., & Mastichiadis, A., 1998, A&A, 333, 452 Kusunose, M., Takahara, F., & Li, H., 2000, ApJ, 536, 299 – 22 – Li, H., & Kusunose, M., 2000, ApJ, 536, 729 Lindner, T., PhD thesis, McGill University, 2006 Mücke, A., & Protheroe, R. J., 2001, Astropart. Phys., 15, 121 Mücke, A., et al., 2003, Astropart. Phys., 18, 593 Ravasio, M., et al., 2003, A&A, 408, 479 Ravasio, M., et al., 2002, A&A, 383, 763 Rieger, F. M., & Duffy, P., 2004, ApJ, 617, 155 Sikora, M., Begelman, M. C., & Rees, M. J., 1994, ApJ, 421, 153 Stecker, F. W., Malkan, M. A., Scully, S. T., 2006, in press Virtanen, J. J. P., & Vainio, R., 2005, ApJ, 621, 313 This preprint was prepared with the AAS LATEX macros v5.2. – 23 – Table 1. Model Parameters used to reproduce the quiescent and flaring state of 3C 66A as shown in Figures 1 and 2, respectively. Note: Linj is the luminosity with which electron population is injected into the blob. γ1,2 are the low- and high-energy cutoffs of electron injection spectrum and q is the particle spectral index. Profile stands for the flare profile used to reproduce the optical variability pattern, eB is the equipartition parameter and magnetic field B is the equipartition value. Γ is the bulk Lorentz factor, Rb is the comoving radius of the blob, θobs is the viewing angle and τT,BLR is the radial Thomson depth of the Fit Linj [10 41 ergs/s] γ1 [10 3] γ2 [10 4] q Profile eB B [G] Γ Rb [10 15 cm] θobs [deg] τT,BLR 1 2.7 1.8 3.0 3.1 ——– 1 2.4 24 3.59 2.4 0 2 8.0 2.1 4.5 2.4 Gaussian 1 2.8 24 3.59 2.4 0 3 8.0 2.1 4.5 2.4 Gaussian 1 2.8 24 3.59 2.4 0.3 Introduction Model Description Model Parameters Results and Discussion Summary

0704.0270

The HARPS search for southern extra-solar planets. X. A m sin i = 11 Mearth planet around the nearby spotted M dwarf GJ 674

Astronomy & Astrophysics manuscript no. Gl674 c© ESO 2018 October 30, 2018 The HARPS search for southern extra-solar planets? X. A m sin i = 11 M⊕ planet around the nearby spotted M dwarf GJ 674 X. Bonfils1, M. Mayor2, X. Delfosse3, T. Forveille3, M. Gillon2, C. Perrier3, S. Udry2, F. Bouchy4, C. Lovis2, F. Pepe2, D. Queloz2, N. C. Santos1,2,5, and J.-L. Bertaux6 1 Centro de Astronomia e Astrofı́sica da Universidade de Lisboa, Observatório Astronómico de Lisboa, Tapada da Ajuda, 1349-018 Lisboa, Portugal e-mail: [email protected] 2 Observatoire de Genève, 51 ch. des Maillettes, CH-1290 Sauverny, Switzerland 3 Laboratoire d’Astrophysique, Observatoire de Grenoble, BP 53, F-38041 Grenoble, Cedex 9, France 4 Institut d’Astrophysique de Paris, CNRS, Université Pierre et Marie Curie, 98bis Bd Arago, 75014 Paris, France 5 Centro de Geofisica de Évora, Rua Romão Ramalho 59, 7002-554 Évora, Portugal 6 Service d’Aéronomie du CNRS, BP 3, 91371 Verrières-le-Buisson, France Received January 09, 2007; accepted xxxx xx, 2007 ABSTRACT Context. How planet properties depend on stellar mass is a key diagnostic of planetary formation mechanisms. Aims. This motivates planet searches around stars which are significantly more massive or less massive than the Sun, and in particular our radial velocity search for planets around very-low mass stars. Methods. As part of that program, we obtained measurements of GJ 674, an M2.5 dwarf at d=4.5 pc, which have a dispersion much in excess of their internal errors. An intensive observing campaign demonstrates that the excess dispersion is due to two superimposed coherent signals, with periods of 4.69 and 35 days. Results. These data are well described by a 2-planet Keplerian model where each planet has a ∼11 M⊕ minimum mass. A careful analysis of the (low level) magnetic activity of GJ 674 however demonstrates that the 35-day period coincides with the stellar rotation period. This signal therefore originates in a spot inhomogeneity modulated by stellar rotation. The 4.69-day signal on the other hand is caused by a bona-fide planet, GJ 674b. Conclusions. Its detection adds to the growing number of Neptune-mass planets around M-dwarfs, and reinforces the emerging conclusion that this mass domain is much more populated than the jovian mass range. We discuss the metallicity distributions of M dwarf with and without planets and find a low 11% probability that they are drawn from the same parent distribution. Moreover, we find tentative evidence that the host star metallicity correlates with the total mass of their planetary system. Key words. stars: individual: GJ 674 – stars: planetary systems – stars: late-type – technique: radial-velocity 1. Introduction M dwarfs, the most common stars in our Galaxy, were added to the target lists of planet-search programs soon after the first exo- planet discoveries. Compared to Sun-like stars, they suffer from some drawbacks: they are faint and photon noise therefore of- ten limits measurements of their radial velocity, and many are at least moderately active and thus prone to so-called “radial- velocity jitter” (Saar & Donahue 1997). On the other hand, the smaller masses of M dwarfs result in a higher wobble ampli- tude for a given planetary mass, and their p-mode oscillations have both smaller amplitudes and shorter periods than those of solar type stars. These oscillations therefore average out much faster. As a result, the detection of an Earth-like planet in the – closer – habitable zone of an M dwarf is actually within reach of today’s best spectrographes. Perhaps most importantly, how- ever, M dwarfs represent unique targets to probe the dependance on stellar mass of planetary formation, thanks to the wide mass range (0.1 to 0.6 M�) spanned by that spectral class alone. Send offprint requests to: X. Bonfils ? Based on observations made with the HARPS instrument on the ESO 3.6 m telescope under the GTO program ID 072.C-0488 at Cerro La Silla (Chile). The first planet found to orbit an M dwarf, GJ 876b (Delfosse et al. 1998; Marcy et al. 1998), was only the 9th exoplanet discovered around a main sequence star. Besides showing that Jupiter-mass planets can form at all around very-low-mass stars, its discovery suggested that they might be common, since it was found amongst the few dozen M dwarfs that were observed at that time. Against these early expectations, no other M dwarf was reported to host a planet until 2004, though a second planet (GJ 876c, mp sin i = 0.56 MJup – Marcy et al. 2001) was soon found around GJ 876 itself. In 2004, the continuous improvement of the radial-velocity techniques resulted in the quasi-simultaneous discovery of three Neptune-mass planets, around µ Ara (mp sin i = 14 M⊕ – Santos et al. 2004), ρ Cnc (mp sin i = 14 M⊕ – McArthur et al. 2004) and GJ 436 (mp sin i = 23 M⊕ – Butler et al. 2004; Maness et al. 2006). Of those three, GJ 436b, orbits an M dwarf, and put that spectral class back on the discovery forefront. It was soon fol- lowed by another two, a single planet around GJ 581 (mp sin i = 17 M⊕ – Bonfils et al. 2005b) and a very light (mp = 7.5 M⊕) third planet in the GJ 876 system (Rivera et al. 2005). As a re- sult, planets around M dwarfs today represent a substantial frac- tion (30%) of all known planets with m sin i <∼ 30 M⊕. 2 X. Bonfils et al.: An 11 M⊕ planet around the nearby M dwarf GJ 674 Even with GJ 849b (mp sin i = 0.82 MJup – Butler et al. 2006) now completing the inventory of M-dwarf planets found with radial-velocity techniques, the upper-range of planet masses re- mains scarcely populated. This contrasts both with the (still very incompletely known) Neptune-mass planets orbiting M dwarfs and with the jovian planets around Sun-like stars. At larger sep- arations, microlensing surveys similarly probe the frequency of planets as a function of their mass. That technique has detected four putative planets that likely orbit M dwarfs: OGLE235- MOA53b (mp ∼ 1.5 − 2.5 MJup – Bond et al. 2004), OGLE-05- 071Lb (mp = 0.9 MJup – Udalski et al. 2005), OGLE-05-390Lb (mp = 0.017 MJup – Beaulieu et al. 2006) and OGLE-05-169Lb (mp = 0.04 MJup – Gould et al. 2006). Two of these four plan- ets have likely masses below 0.1 MJup . Given the detection bias of that technique towards massive companions, this again sug- gests that Neptune-mass planets are much more common than Jupiter-mass ones around very-low-mass stars. Here we report the discovery of a 11 M⊕ planet orbiting GJ 674 every 4.69 days. GJ 674b has the 5th lowest mass of the known planets, and coincidentally is also the 5th planetary system centered on a M dwarf. Its detection adds to the small in- ventory of both very-low mass planets and planets around very- low mass stars. After reviewing the properties of the GJ 674 star (§2), we briefly present our radial velocity measurements (§3) and their Keplerian analysis (§4). A careful analysis of the mag- netic activity of GJ 674 (§5) assigns one of the two periodicities to rotational modulation of a stellar spot signal, and the other one to a bona fide planet. We conclude with a brief discussion of the properties of the detected planet. 2. The properties of GJ 674 GJ 674 (HIP 85523, LHS 449) is a M2.5 dwarf (Hawley et al. 1997) in the Altar constellation. At 4.5 pc (π = 220.43±1.63 mas – ESA 1997), it is the 37th closest stellar system, the 54th clos- est star (taking stellar multiplicity into account)1, and only the 2nd closest known planetary system (after � Eridani, and slightly closer than GJ 876). Its photometry (V = 9.382 ± 0.012; K = 4.855 ± 0.018 – Turon et al. 1993; Cutri et al. 2003) and parallax imply absolute magnitudes of MV = 11.09 ± 0.04 and MK = 6.57 ± 0.04. GJ 674’s J − K color (= 0.86 – Cutri et al. 2003) and the Leggett et al. (2001) colour-bolometric relation result in a K-band bolo- metric correction of BCK = 2.67, and in a 0.016 L� luminosity. The K-band mass-luminosity relation of Delfosse et al. (2000) gives a 0.35 M� mass and the Bonfils et al. (2005a) photometric calibration of the metallicity results in [Fe/H] = −0.28 ± 0.2. The moderate X-ray luminosity (Lx/Lbol ' 5.10−5 – Hünsch et al. 1999) and Ca ii H & K emission depict a modestly active M dwarf (Fig. 1). Its UVW galactic velocities place GJ 674 be- tween the young and old disk populations (Leggett 1992), sug- gesting an age of ∼ 108−9yr. Last but not least, since we are concerned with radial veloc- ities, the high proper motion of GJ 674 (1.05 arcsec yr−1 – ESA 1997) changes the orientation of its velocity vector along the line-of-sight (e.g. Kürster et al. 2003) to result in an apparent secular acceleration of 0.115 m s−1 yr−1. At our current precision this acceleration will not be detectable before another decade. 1 on Mar. 1st 2007 (http://www.chara.gsu.edu/RECONS/TOP100.htm) 3962.5 3965 3967.5 3970 3972.5 3975 Wavelength [Å]Wavelength [Å] Fig. 1. Emission reversal in the Ca ii H line of GJ 674 (top) and GJ 581 (bottom). Within our sample GJ 581 has one of the weakest Ca ii emission and illustrates a very quiet M dwarf. GJ 674 has much stronger emission and is moderately active. Table 1. Observed and inferred stellar parameters for GJ 674 Parameter GJ 674 Spectral Type M2.5 V 9.382 ± 0.012 π [mas] 220.43 ± 1.63 Distance [pc] 4.54 ± 0.03 MV 11.09 ± 0.04 K 4.855 ± 0.018 MK 6.57 ± 0.04 L? [L�] 0.016 Lx/Lbol 5.10−5 v sin i [km s−1] . 1 dvr/dt [m s−1yr−1] 0.115 [Fe/H] −0.28 M? [ M�] 0.35 age [Gyr] 0.1-1 Teff [K] 3500-3700 3. Radial-velocity data We observed GJ 674 with the HARPS echelle spectrograph (Mayor et al. 2003) mounted on the ESO 3.6-m telescope at La Silla Observatory (Chile). After demonstrating impressive planet finding capabilities right after its commissioning (Pepe et al. 2004), this spectrograph now defines the state of the art in radial- velocity measurements, delivering a significantly better preci- sion than its ambitious 1 m s−1 specification. As one recent pub- lished example, Lovis et al. (2006a) obtained a 0.64 m s−1 disper- sion for the residuals of their orbital solution of the 3 Neptune- mass planets of HD 69830. We observed GJ 674 without interlaced Thorium-Argon light to obtain cleaner spectra for spectroscopic analysis, at some small cost in the ultimate Doppler precision. Since June 2004 we have gathered 32 exposures of 900 s each with a median S/N ratio of ∼ 90. Their Doppler information content, evalu- ated according to the prescriptions of Bouchy et al. (2001), is mostly below 1 m s−1. Our internal errors additionally include, in quadrature sum, an “instrumental” uncertainty of 0.5 m s−1 for the nightly drift of the spectrograph (since we do not use the ThAr lamp to monitor it) and the measurements uncertainty of the daily wavelength zero point calibration. We did benefit of the recent improvements of the HARPS wavelength calibration, which is now stable to 0.1 m s−1 (Lovis et al. 2006b). A constant radial velocity gives a very large reduced chi- square (χ̄2 = 132) for the time series, which reflects a disper- X. Bonfils et al.: An 11 M⊕ planet around the nearby M dwarf GJ 674 3 σ = 7.35 m s−1 χ̄2ν = 132 53000 53250 53500 53750 54000 Julian date −2,400,000 [day]Julian date −2,400,000 [day] 1 10 100 1000 Period [day]Period [day] Fig. 2. Upper panel: Radial-velocity measurements of GJ 674 as a func- tion of time. The high dispersion (σ = 7.35 m s−1) and chi-square value (χ̄2 = 132) betray a (coherent or incoherent) signal in the data. Bottom panel: the Lomb-Scargle periodogram of the velocities has prominent power excess around P = 4.69 days (downward arrow), which indi- cates that much of the excess dispersion reflects a coherent signal with a period close to that value. The second highest peak, at 1.27 day, is a one-day alias of the 4.69 days period (1.27 ' 1 + 1/4.69). sion (∼ 7.4 m s−1) well above our internal errors (Fig. 2). This prompted a search for an orbital (§4) and/or magnetic activity (§5) signal. 4. Orbital analysis A Lomb-Scargle periodogram (Press et al. 1992) of the velocity measurements shows a narrow peak around 4.69-day (Fig. 2). Adjustment of a single Keplerian orbit demonstrates that it is best described by a m2 sin i =12.7 M⊕ planet (0.040 MJup) re- volving around GJ 674 every P2 = 4.6940 ± 0.0005 days in a slightly eccentric orbit (e2 = 0.10 ± 0.02). The residuals around this low-amplitude orbit (K1 = 9.8±0.2 m s−1) have a dispersion of 3.27 m s−1 (Fig. 3), still well above our measurement errors, and the reduced chi-square per degree of freedom is χ̄2 = 30.6. A periodogram of the residuals indicates that much of this ex- cess dispersion stems from a broad power peak centered around 35 days, prompting us to perform a 2-planet fit. We searched for 2-planet Keplerian solutions with Stakanof (Tamuz, in prep.), a program which uses genetic algorithms to efficiently explore the large parameter space of multi-planet models. Stakanof quickly converged to a 2-planet solution that describes our measurements much better than the single planet fit (σ = 0.82 m s−1, χ̄2 = 2.57 per degree of freedom – Fig. 4). The orbital parameters of the 4.69-day planet change little from the 1-planet fit, except for the eccentricity which increases to e2 = 0.20±0.02. Its mass is revised down to M2 sin i = 11.09 M⊕, and the period hardly changes, P2 = 4.6938 ± 0.0007 day. The second planet would have a P3 = 34.8467 ± 0.0324 day pe- riod, an e3 = 0.20 ± 0.05 eccentricity and a minimum mass of m3 sin i = 12.58 M⊕. Such periods would correspond to semi- major axes of 0.04 and 0.15 AU. Those are sufficiently disjoint σ = 3.27 m s−1 χ2 = 30.6 0 0.25 0.5 0.75 1 53000 53250 53500 53750 54000 Julian date −2,400,000 [day]Julian date −2,400,000 [day] 1 10 100 1000 Period [day]Period [day] Fig. 3. Upper panel: Radial velocities of GJ 674 (red filled circles) phase-folded to the 4.6940 days period of the best 1-planet fit (curve). The dispersion around the fit (σ = 3.27 m s−1) and its reduced chi- square (χ̄2 = 30.6 per degree of freedom) indicate that a single planet does not describe the data very well. Middle panel: Radial-velocity residuals of the 1-planet fit Bottom panel: The Lomb-Scargle peri- odogram of the residuals shows a broad peak centered around 35 days. that mutual interactions can be neglected over observable time scales, and that the system would be stable over longer time scales. The low dispersion around the solution and the lack of any significant peak in the Lomb-Scargle periodogram of its residu- als shows that our current radial-velocity measurements contain no evidence for an additional component. 5. Activity analysis Apparent Doppler shifts unfortunately do not always originate in the gravitational pull of a companion: in a rotating star, stellar surface inhomogeneities such as plages and spots can break the exact balance between light emitted in the red-shifted and blue- shifted halves of the star. Observationally, these inhomogeneities translate into flux variations as well as into changes of both the shape and the centroid of spectral lines (Saar & Donahue 1997; Queloz et al. 2001). Spots typically also impact spectral indices, whether designed to probe the chromosphere (to which photo- spheric spots have strong magnetic connections), or the photo- sphere (because spots have cooler spectra). Of the two candidate periods, the 4.69-day one is unlikely to reflect stellar rotation. We measure from our GJ 674 spectra a rotational velocity of v sin i . 1 km s−1, which would need a rather unprobable stellar 4 X. Bonfils et al.: An 11 M⊕ planet around the nearby M dwarf GJ 674 σ = 0.82 m s−1 χ2 = 2.57 0 0.25 0.5 0.75 1 53000 53250 53500 53750 54000 Julian date −2,400,000 [day]Julian date −2,400,000 [day] 1 10 100 1000 Period [day]Period [day] Fig. 4. Top two panels: Radial velocity measurements phased to each of the two periods, after subtraction of the other component of our best 2- planet Keplerian model. Third panel: Residuals of the best 2-planet fit as a function of time (O−C, Observed minus Computed). Bottom panel: Lomb-Scargle periodogram of these residuals. inclination (i . 15◦) to match such a short period. The moderate activity level of GJ 674 on the other hand leaves the nature of the second signal a priori uncertain, and the very small rotation velocity removes much of the power of the usual bisector test (Appendix A). We therefore investigated its magnetic activity through photometric observations (§5.1) and detailed examina- tion of the chromospheric features in the clean HARPS spectra (§5.2). 5.1. Photometric variability We obtained photometric measurements with the CCD cam- era of the Euler Telescope (La Silla) during 21 nights between September 2nd and October 19th 2006. GJ 674 was observed through a VG filter which, amongst the available filters, opti- ∆F = −0.013 sin(2π/35.68 + 54259.93) 53980 53990 54000 54010 54020 54030 Julian date −2,400,000 [day]Julian date −2,400,000 [day] 1 10 100 1000 Period [day]Period [day] Fig. 5. Upper panel: Differential photometry of GJ 674 as a function of time. The star clearly varies with a 1.3% amplitude. Bottom panel: The periodogram of the GJ 674 photometry exhibits significant power excess peaked at 35 days (small black arrow). mizes the flux ratio between GJ 674 and its two brightest refer- ence stars. This relatively blue filters also happens to have good sensitivity to spots on cool stars such as GJ 674. To minimize at- mospheric scintillation noise we took advantage of the low stel- lar density to defocus the images to FWHM ∼ 11′′, so that we could use longer exposure times. The increased read-out and sky background noises from the larger synthetic aperture which we then had to use remain negligible compared to both stellar pho- ton noise and scintillation. We gathered 14 to 75 images per night with a median expo- sure time of 20 seconds. We used the Sept. 24th data, which have the longest nightly time base, to tune the parameters of the Iraf Daophot package and optimize the set of reference stars (HD 157931, CD 4611534 and 7 anonymous fainter stars) to min- imize the dispersion in the GJ 674 photometry for that night. These parameters were then fixed for the analysis of the full data set. The nightly light curves for GJ 674 were normalized by that of the sum of the references, clipped at 3-σ to remove a small number of outliers, and averaged to one measurement per night to examine the long term photometric variability of GJ 674. GJ 674 clearly varies with a ∼1.3% amplitude, and a (quasi-)period close to 35 days (Fig. 5). To verify that this vari- ability does not actually originate in one of the reference stars, we repeated the analysis alternately using as reference star HD 157931 alone and the average of the 8 other references. Both light curves are very similar to Fig. 5. The photometric observations are consistent with the signal of a single spot, within the limitations of their incomplete phase coverage: the variations are approximately sinusoidal, and their ∼0.2-0.3 radian phase shift from the corresponding radial veloc- ity signal closely matches the difference expected for a spot. The spot would cover 2.6% of the stellar surface if completely dark, corresponding to a ∼ 0.16 R? radius for a circular spot. X. Bonfils et al.: An 11 M⊕ planet around the nearby M dwarf GJ 674 5 Hα Ca II H+K−5 0 1 2 3 4 5 IndexIndex Ca II H+K 0 0.2 0.4 0.6 0.8 1 Ca II H+K 1 10 100 1000 Period [day]Period [day] Fig. 6. Upper panel: Differential radial velocity of GJ 674, corrected for the signature of the 4.69 days planet in our 2-planet Keplerian fit, as a function of the Hα (red filled circles) and Ca ii H&K (green filled squares) spectral indices defined in the text. Bottom right panels: the Ca ii H+K and Hα indexes phased to the longer period of the 2-planet Keplerian model. Bottom left panels: Power Density spectra of the spectroscopic indexes. A clear power excess peaks at 34.8 days (vertical dashed lines). 5.2. Variability of the spectroscopic indices The emission reversal in the core of the Ca iiH&K resonant lines results from non-radiative heating of the chromosphere, which is closely coupled to spots and plages through magnetic con- nections between the photosphere and chromosphere. The Hα line is similarly sensitive to chromospheric activity. We mea- sured these chromospheric spectral features results in the clean HARPS spectra used to measure the radial velocities, and exam- ine their variability. Like the well known Mt. Wilson S index (Baliunas et al. 1995), our Ca ii H+K index is defined as: Index = H + K B + V . (1) with H and K sampling the two lines of the Ca ii doublet, and B and V the continuum on both sides of the doublet. Our H and K intervals are 31 km s−1 wide and centered on 3933.664 and 3968.47 Å, while B and V are respectively integrated over [3952.6, 3956 Å] and [3974.8, 3976 Å]. This H+K index varies with a clear period of ∼34.8 days (Fig. 6). Within the combined errors this is consistent with both the photometric period and the longer radial velocity period. The phasing of the chromospheric index and the photometry is such that lower photometric flux matches higher Ca ii emission, as ex- pected if active chromospheric regions hover over photospheric spots. A plot of the (apparent) radial-velocity as a function of the H+K spectral index similarly shows the characteristic loop pat- tern expected for a spot. The radial velocity effect of a spot can- cels out when it crosses the sub-observer meridian, which oc- curs twice during a rotation period: once on the hemisphere fac- ing the observer, and once on the opposite hemisphere. During the front-facing crossing the spot has maximal projected area, hence maximal chromospheric emission, while it has a minimal projected area (and is possibly hidden, depending on its latitude and the stellar inclination) during the back-facing crossing. As a result, both extrema of the chromospheric index correspond to radial-velocity zero-crossings. At intermediate phases the spot produces intermediate chromospheric emission levels, and it in- duces positive (respectively negative) radial-velocity shifts when the masked area is on the rotationally blue- (respectively red- shifted) half of the star. The net result in a plot of chromospheric emission as a function of radial velocity is a closed loop. Chromospheric filling-in of photospheric Hα absorption has similarly been found a powerful activity diagnostic for M dwarfs. Kürster et al. (2003) found that in Barnard’s star it corre- lates linearly with the radial-velocity variations, and interpreted that finding as evidence that active plage regions inhibit the con- vective velocity field. The variation pattern in GJ 674 definitely differs from a linear correlation between Hα and the radial- velocity residuals, and needs a different explanation. 6 X. Bonfils et al.: An 11 M⊕ planet around the nearby M dwarf GJ 674 Similarly to Kürster et al. (2003) we define our Hα index as: Index = F1 + F2 . (2) with FHα sampling the Hα line, and F1 and F2 the contin- uum on both sides of the line. Our FHα interval is 31 km s−1 wide and centered on 6562.808 Å, while F1 and F2 are respec- tively integrated over [6545.495, 6556.245 Å] and [6575.934, 6584.684 Å]. The Hα index behaves similarly to the Ca ii H+K index. The chromospheric indices vary by factors of ∼2 and ∼1.3 (for our specific choices of continuum windows), and are thus much more contrasted than the photometry. They do not how- ever vary as smoothly with phase as the photometry, perhaps due to (micro-)flares. This somewhat reduces their value as diag- nostics of spot-induced radial velocity variations, but these mea- surements on the other hand require no new observation. They undoubtedly reinforce the spot interpretation here, and they will be extremely useful in cases where photometry cannot be imme- diately obtained. 5.3. Planets vs. activity In §4 we showed that our 32 radial-velocity measurements of GJ 674 are well described by two Keplerian signals, as illustrated by the low reduced chi-square of that model. The above analy- sis (§5) however demonstrates that the rotation period of GJ 674 coincides with the longer of the two Keplerian periods. Both the stellar flux and the Ca iiH+K emission vary with that period, im- plying that the surface of GJ 674 has a magnetic spot. This spot must induce radial-velocity changes, with the observed phase relative to the photometric signal. As a consequence, some, and probably all, of the 35-day radial-velocity signal must originate in the spot. Planet-induced activity through magnetic coupling (e.g. Shkolnik et al. 2005) would in principle be an alternative explanation of the correlation, but here it is not a very attrac- tive one: the inner planet is at least as massive as the hypothet- ical 35-day planet, and would, at least naively, be expected to have stronger interactions with the magnetosphere of GJ 674. The 4.69-day period however is only seen in the radial velocity signal, and it has no photometric or chromospheric counterpart. 6. Discussion 6.1. Characteristics of GJ 674b Perhaps the most important result of the above analysis is that the ∼4.69-day planet of GJ 674 is robust: variability identifies the stellar rotation period as ∼35 days, and the 4.69-day period therefore cannot reflect rotation modulation. The short period signal, in spite of its larger amplitude, also has no counterpart in either photometry or chromospheric emission, further excluding a signal caused by magnetic activity. The 1-planet fit, which effectively treats the activity sig- nal as white noise, results in a minimum mass for GJ 674b of m2 sin i = 12.7 M⊕. The 2-planet fit by contrast filters out this signal. That filtering obviously uses a physical model which is not completely appropriate, but that remains preferable to handling a (partly) coherent signal as white noise. We there- fore adopt the corresponding estimate of the minimum mass, m2 sin i = 11.09 M⊕. At 0.039 AU from its parent star, the temperature of GJ 674 b is ∼450 K. Planets above a few Earth masses planets can, but need not, accrete a large gas fraction, leaving its composition – mostly gaseous or mostly rocky – unclear. The orbital eccentric- ity might shed light on the structure of GJ 674b, if confirmed by additional measurements: rocky and gaseous planets have rather different dissipation properties, and significant eccentric- ity at the short period of GJ 674 b needs a high Q factor, unless it is pumped by an additional planet at a longer period (e.g. Adams & Laughlin 2006). For now, the stellar activity leaves the statis- tical significance of the eccentricity slightly uncertain, and we therefore prefer to stay clear from overinterpreting it. 6.2. Properties of M-dwarf planets One important motivation in searching for planets around M dwarfs is to investigate whether the planet-metallicity correla- tion found for Jupiter-mass planets around solar-type stars ex- tends to very-low-mass stars. Our photometric calibration of M dwarfs metallicity (Bonfils et al. 2005b) gives respective metal- licities of [Fe/H]=−0.03, −0.25, +0.14, +0.03, and −0.28 for GJ 436, GJ 581, GJ 849, GJ 876 and GJ 674. M dwarfs with known planets therefore have an average metallicity of −0.078 and a median of −0.03. By comparison, the 44 M dwarfs of the Bonfils et al. (2005b) volume limited sample which are not cur- rently known to host a planet have average and median metal- licities of −0.181 and −0.160. M dwarfs with planets therefore appear slightly more metal-rich than M dwarfs without planets. A Kolmogorov-Smirnov test (Press et al. 1992) of the two sam- ples gives an 11% probability that they are drawn from the same distribution. The significance of the discrepancy is therefore still modest, limited by small-number statistics. One can additionally note that the two stars which host giant planets, GJ 876 and GJ 849, occupy the metal-rich tail of the M dwarf metallicity distribution, with GJ 849 almost as metal-rich as the most metal-rich star of the comparison sample. The next most metal-rich of the M dwarfs with planets, GJ 436, has an ad- ditional long-period companion (P>6 yr) which might well be a giant planet (Maness et al. 2006) and would then strengthen that trend. If confirmed by additional data, this would validate the theoretical predictions (Ida & Lin 2004; Benz et al. 2006) that only Jovian-mass planets are more likely to form around metal- rich stars. Current observations are consistent with this predic- tion, but not yet very conclusively so (Udry et al. 2006). Much recent theoretical work has gone into examining how planet formation depends on stellar mass. Within the “core ac- cretion” paradigm, Laughlin et al. (2004) and Ida & Lin (2005) predict that giant planet formation is inhibited around very- low-mass stars, while Neptune-mass planets should inversely be common. Within the same paradigm, but assuming that M dwarfs have denser protoplanetary disks, Kornet et al. (2006) predict instead that Jupiter-mass planets become more frequent in inverse proportion to the stellar mass. Finally, Boss (2006) ex- amines how planet formation depends on stellar mass for plan- ets formed by disk instability, and concludes that frequency of Jupiter-mass planet is independent of stellar mass, as long as disks are massive enough to become unstable. To date, none of the ∼300 M dwarfs scrutinized for plan- ets by the various radial-velocity searches (Bonfils et al. 2006; Endl et al. 2006; Butler et al. 2006) has been found to host a hot Jupiter. Conversely, GJ 674b is already the 4th hot Neptune. Though that cannot be established quantitatively yet, these sur- veys are likely to be almost complete for hot Jupiters, which are easily detected. Hot Neptune detection, on the other hand, is definitely highly incomplete. Setting aside this incompleteness for now, simple binomial statistics shows that the probability of X. Bonfils et al.: An 11 M⊕ planet around the nearby M dwarf GJ 674 7 −0.75 −0.5 −0.25 0 0.25 [Fe/H][Fe/H] Fig. 7. Upper panel: Metallicity distributions of 44 M dwarfs without known planets (gray shading) and of the 5 M dwarfs known to host planets (black shading). Bottom panel: Corresponding cumulative dis- tributions (solid and dashed lines, respectively). Table 2. Keplerian parameterization for GJ 674b and GJ 674’s spot. Parameter GJ 674b Spot P [days] 4.6938 ± 0.007 34.8467 ± 0.0324 T [JD] 2453780.085 ± 0.078 2453767.13 ± 0.92 e 0.20 ± 0.02 0.20 ± 0.05 ω [deg] 143 ± 6 113 ± 9 K [m s−1] 8.70 ± 0.19 5.06 ± 0.19 a1 sin i [AU] 3.68 10−6 1.59 10−5 f (m) [M�] 3.0 10−13 4.4 10−13 m2 sin i [MEarth] 11.09 12.58 a [AU] 0.039 0.147 finding no and 4 detections in 300 draws of the same function is only 3%. There is a thus 97% probability that hot Neptunes are more frequent than hot Jupiter around M dwarfs. Accounting for this detection bias in more realistic simulations (Bonfils et al. in prep.) obviously increases the significance of the difference. Planet statistics around M dwarfs therefore favor the theoreti- cal models which, at short periods, predict more Neptune-mass planets than Jupiter-mass planets. Acknowledgements. We are grateful to the anonymous referee for construc- tive comments. XB and NCS acknowledge support from the Fundação para a Ciência e a Tecnologia (Portugal) in the form of fellowships (references SFRH/BPD/21710/2005 and SFRH/BPD/8116/2002) and a grant (reference POCI/CTE-AST/56453/2004). The photometric monitoring has been performed on the EULER 1.2 meter telescope at La Silla Observatory. We are grateful to the SNF (Switzerland) for its continuous support. This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France. References Adams, F. C. & Laughlin, G. 2006, ApJ, 649, 1004 Baliunas, S. L., Donahue, R. A., Soon, W. H., et al. 1995, ApJ, 438, 269 Beaulieu, J.-P., Bennett, D. P., Fouqué, P., et al. 2006, Nature, 439, 437 Benz, W., Mordasini, C., Alibert, Y., & Naef, D. 2006, in Tenth Anniversary of 51 Peg-b: Status of and prospects for hot Jupiter studies, ed. L. Arnold, F. Bouchy, & C. Moutou, 24–34 Bond, I. 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The bisector signal therefore decreases faster with decreasing rotational velocities than the radial-velocity signal, and disappears faster in measure- ment noise. For GJ 674 we measure a very low rotation velocity (v sin i < 1km s−1). It is therefore unsurprising that the cor- 8 X. Bonfils et al.: An 11 M⊕ planet around the nearby M dwarf GJ 674 −7.5 −5 −2.5 0 2.5 5 7.5 RV [m/s]RV [m/s] 1 10 100 1000 Period [day]Period [day] Fig. A.1. Bisector analysis for GJ 674 measurements relation between the bisector span and radial velocity is weak (Fig. A.1) and not statistically significant.

0704.0271

An individual based model with global competition interaction: fluctuations effects in pattern formation

arXiv:0704.0271v2 [cond-mat.stat-mech] 27 Feb 2008 An individual based model with global competition interaction: fluctuation effects in pattern formation E. Brigatti ⋆±, V. Schwämmle† and Minos A. Neto⋆ †Centro Brasileiro de Pesquisas F́ısicas, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil ⋆Instituto de F́ısica, Universidade Federal Fluminense, Campus da Praia Vermelha, 24210-340, Niterói, RJ, Brazil ±e-mail address: [email protected] (October 26, 2018) We present some numerical results obtained from a simple individual based model that describes clustering of organisms caused by competition. Our aim is to show that, even when a deterministic description developed for continuum models predicts no pattern formation, an individual based model displays well defined patterns, as a consequence of fluctuation effects caused by the discrete nature of the interacting agents. 87.17.Aa,87.23.Kg, 87.23.-n, 05.10.Ln I. INTRODUCTION Birth and death processes are two of the most relevant characteristics of the dynamics of biological populations and can be responsible for the emergence of stable spa- tial patterns [1]. In fact, the intrinsic asymmetry in the nature of birth and death processes can enhance small initial differences in the spatial population density and lead to the formation of structures [2–4]. These clusters are resistant to some levels of diffusion and emerge as soon as the birth of new individuals outcompetes their movements. For this reason, simplified models combin- ing birth and death processes with Brownian movement are able to describe aggregation of individuals [5]. Another central ingredient, present in ecological sys- tems, that can cause the generation of spatial structures, is the competition for resources [6–8]. Different individ- uals struggle for nutrients with a competition strength directly dependent on the individuals’ spatial density within the competing range. Reproduction and/or death rates depending on the number of individuals in the sur- roundings can stand for this kind of interaction. This fea- ture has attracted the interest of experts from a variety of fields, ranging from pure mathematics [9] and non-linear physics [10–14,27] to population biology [16,17] and the- oretical studies in evolutionary theory [20–25]. In ad- dition, similar behaviors can be found in physical sys- tems, such as, for example, in mode interaction in crys- tallization fronts [18] and in spin-wave patterns [19]. It is remarkable that the structured state generated by this kind of frequency-dependent interaction exists only for some specific form of the interaction [24] and is reached through a transition in the parameter space. This transi- tion, (segregation transition [10]), drives the steady state of the system from a spatially homogeneous distribution to one marked by some clearly distinguishable inhomo- geneities. All these models, characterized by diffusion effects and an implementation of frequency dependent birth and death processes, permit multiple interpretations. In a common interpretation the system space directly represents the physical space where the organisms live and the diffusion represents their spatial movement. Competition between individuals corresponds to a mech- anism of growth control caused by limited common re- sources. In this case, pattern formation can reproduce the evolution of bacterial colonies [6], plankton concen- tration [5], development of vegetation [8] or spatial dis- tribution of predators [7]. On the other hand, a different interpretation enables us to describe the speciation process: the generation of two different species starting from one single continuous population of interbreeding organisms. To be specific, we can describe the speciation process by representing all the phenotypic characteristics that determine the bi- ological success of an individual by a number, the strat- egy value, that labels each individual. By reproduction, that includes a mutation process, an offspring inherits a strategy that slightly differs from that of its parent. In order to model natural selection, a frequency-dependent mechanism that mimics competition, completes the in- gredients necessary for the emergence of population clus- tering. In this scenario, the generation of a new cluster is interpreted, in a broad sense, as a speciation event. Now, if the model space represents the mentioned strat- egy space and the diffusion models the mutation process during reproduction, we can identify the mechanism of growth control with natural selection and the branching events with the speciation process. This different inter- pretation justifies the analogy between a model that de- scribes the speciation process and the ones that describe spatial pattern formation in the evolution of bacterial colonies, vegetation or predation. We must remember that, since this model does not in- clude sexual reproduction, we are describing trait diver- gence in an asexual population, rather than speciation. Anyway, apart from effects strictly related to sexual re- production, the dynamics characterized by the individu- http://arxiv.org/abs/0704.0271v2 als’ diffusion from regions of low viability in favor of more viable ones, is the essential core of these two phenomena. A detailed and motivated discussion of these processes can be found in the following references [20–25]. We can describe such processes starting from an indi- vidual based model, that yields information on the be- havior of a finite system (finite population) and accounts for fluctuation effects caused by the discrete nature of the interacting agents. Another approach, that neglects fluc- tuations, describes individuals just with the use of a field that represents the population density at each position in space over time. This method, usually denominated con- tinuous mean-field description [15,27], becomes exact in the infinite-size limit if fluctuations are small compared to averages [4]. Note that we are not making a mean- field approximation in the nature of the interaction (see, for instance, ref. [26]), that, in our model, is local and different for each individual. Choosing this second strategy, the generalization of a well-investigated equation (Fisher-Kolmogoroff- Petrovsky-Piscounoff [28]) is quite common at present. In addition to a diffusion process with coefficient D and a population growth mechanism (rate a), this equation incorporates a growth-limiting process controlled by the parameter b [11,12,14]: ∂ρ(x, t) = D∇2ρ(x, t) + aρ(x, t) − (1.1) b ρ(x, t) ρ(y, t)F (x, y)dy , where ρ is the density of individuals at position x and time t. Competition is obtained by varying the death probability for each individual, and is controlled through the influence function F (x, y). Let us focus on the shape of the influence function: it can range from a simple box- like function to a globally uniform interaction. However, the Gaussian function should be considered particularly relevant. If, for instance, we need to represent the ac- tivity of a sedentary animal the interaction represented in the influence function should take into account the in- dividual’s daily excursion around the fixed breeding site that can be represented by Brownian motion and, for this reason, by a Gaussian distribution. In the same way, if we want to represent the habitat degeneration induced by the growth of a colony of plants [29] we can think that the colony is originated by a single individual that disperses its seeds in a way also well described by Brownian motion. More generally, for a biological interaction that does not stop at some defined length (presence of a cutoff), and that is nonlocal and controlled by a purely stochastic process, the Gaussian function should be the most natu- ral choice. On the other hand, this choice is a source of complications. Deterministic descriptions, in the case of a Gaussian influence function, predict no pattern forma- tion [12,30]. However, such descriptions do not take into account fluctuations arising from the discrete character of individuals. The importance of these fluctuations has been recently pointed out in a quite paradigmatic exam- ple, where random walking organisms that reproduce and die at a constant rate spontaneously aggregate [2–5,31]. The deterministic approximation is not able to show this behavior, incapable of capturing the essential asym- metry between birth, a multiplicative process that incre- ments the density in the regions adjacent to the parent, and death events, that occur anywhere. Even when the patterns can be obtained within the deterministic description, a recent work outlines the importance of fluctuations by showing their impact on affecting transi- tion points and amplitudes [14,32,33]. In this work we present some numerical results ob- tained by means of a simple individual based model. Our aim is to show the appearance of a segregation transition in a model where the deterministic instability, produced by the non-local interaction, is not sufficient for gener- ating inhomogeneities [12], but the superimposed micro- scopic stochastic fluctuations permit the emergence of patterns. Moreover, we compare the model implemen- tation in the strategy space with the implementation in the physical space. In the first, used to characterize the speciation process, diffusion corresponds to a mutation phenomenon, operating just one time in each individ- ual’s life. In the second, directly related to the reaction- diffusion equation (eq. (1.1)), diffusion describes a typical Brownian motion. The paper is organized as follows. The next section describes the model used in our simulations. Section III shows, for specific values of the parameters, the emer- gence of spatially inhomogeneous steady states. In sec- tion IV we prove that these patterns are not caused by some finite size effect. Section V is devoted to illustrate the segregation transition and general conditions that al- low spatial segregation of arbitrary wavelengths. In sec- tion VI we describe, in the light of the existing literature, the cluster size dependence on diffusion rate and popu- lation size and give some hints related to the behavior of fluctuations. Conclusion are reported in section VII. II. THE MODEL The simulations start with an initial population of N0 individuals located along a ring of length L, i.e. we take periodic boundary conditions. At each time step, our model is controlled by the following microscopic rules: 1) Each individual, characterized by its position x, dies with probability P , P = K · exp(− (x− yj)2 ) , (2.1) whereN(τ) is the total number of individuals at the ac- tual step τ and yj is the respective individuals’ positions. The distance between two individuals is obtained by tak- ing the shorter distance on the ring. The strength of competition declines with increasing distance according to a Gaussian function with deviation C. The parameter K depicts the carrying capacity. 2) If the individual survives this death selection, it reproduces. The newborn, starting from the parent’s lo- cation, moves in a random direction a distance obtained from a Gaussian distribution of standard deviation σ. This change represents the effect of mutations in the offspring phenotype. As soon as all the individuals have passed the death selection and eventually reproduced, the next time step begins. This model implementation is analogous to the diffusion process described by eq. (1.1). To establish a more direct comparison between that mean field description and the individual based simula- tion, we have also implemented the model with an exact microscopic representation of the diffusion term. In this case, at any given time step, we perform first a loop over all particles where individuals move some distance, in a random direction, chosen from a Gaussian distribution of standard deviation σ. At the end of this loop, a second one starts, where: 1) Each individual with strategy x dies with a proba- bility obtained from eq. (2.1). 2) If the individual survives the death selection pro- cess, it reproduces and the newborn maintains the same location of the parent. If, in eq. (1.1), we measure time in units of the simula- tion time step, the coefficient D is related to our simula- tion parameter through D ∝ σ2. The influence function is given by a Gaussian of standard deviation C and the effective growth rate is 1− P , with P given by eq. (2.1). This non constant growth rate can be represented by eq. (1.1) for a = 1 and the frequency dependent part included in the integral term (see ref. [14]). Essentially, the only difference between these two ver- sions of our model is that, in the first one, individuals move only at birth, while in the second version they can move at every time step throughout their life. Since the death probability, at equilibrium, is approximately 1/2, usually, in the second version, an individual will move between one and two times during its entire life. For this reason, there should be no relevant differences in the qualitative behavior of the two model implementations. As shown by the measures reported in section IV, the only significant effect is the appearance of slightly wider distributions. III. MODULATION In the following we present some typical examples of steady states generated by the dynamics of the model that clearly show the emergence of patterns for some spe- cific values of the parameters. For a global competition that results to be extremely long-ranged (large C values, in relation to the values of parameters σ and K), the steady state is characterized by a spatially homogeneous occupancy. If the σ value is sufficiently large or/and the K value sufficiently small, totally homogeneous distributions are obtained (Figure 1), otherwise the solution is smooth but with the pop- ulation concentrated in one region of the ring, with its width controlled by σ. As stated above, a simple heuristic analysis of eq. (1.1) in the Fourier space shows that there is a necessary con- dition for the emergence of inhomogeneity: the Fourier transform of the influence function must have negative values and large enough magnitude [12,13]. A Gaus- sian in an infinite domain has a positive counterpart in Fourier space and so does not match such requirements. In contrast with these results, the fluctuations present in our individual based model arrive to excite one specific mode and modulations of this wavelength appear. The tuning of the parameters allows modulations of arbitrary wavelengths (Figure 1). When C is decreased, the competition between modes becomes stronger and no single mode dominates. In this situation, some small regions of the ring are occupied forcing all the remaining areas, up to some range, to be nearly empty. The landscape results populated by several living colonies divided by dead regions. There is almost no competition between individuals of different colonies and the space separating them can be identified with an effective interaction length. This steady state (spiky state [10]) corresponds to a sequence of isolated colonies (spikes) and seen in the Fourier space, many active wavelengths contribute to it (Figure 2). Finally, for extremely short-ranged competition, in re- lation to the σ value, no collective cooperation between different excited modes emerges and a noisy spatially homogeneous distribution appears (Figure 2). We describe these paradigmatic steady states of the system by characterizing the related spatial structure with the help of a structure function S(q) [14] defined as follows: S(q) = exp[i2πq · xj(τ)] (3.1) where the sum is performed over all individuals j with their positions determined by xj(τ). Note, that, for con- venience, the structure function is calculated only over the closed interval [0, L]. The function is averaged over some time interval T in order to avoid noisy data. S(0) corresponds to the square of the mean number of individ- uals in the system. The maxima of S(q) identify the rel- evant periodicity present in the steady state. We will see that the position of the global maximum (qM ) provides an appropriate order parameter for the identification of the segregation transition. In our study we explored two different initial condi- tions. In the first (local i.c.), the colony is located in a finite and compact region of the space. In the second (global i.c.), the individuals are spread all over the space. The final distribution is independent of this choice and, generally, local initial conditions make the system reach the steady state more slowly. For this reason, if not dif- ferently specified, our results are obtained from global initial conditions. IV. FINITE SIZE EFFECTS Our analysis starts by exploring the model dependence on the space size L. The reason for such interest is given by the necessity of testing if the pattern formation is not merely a product of some finite size effect. This is im- portant, in the light of what was reported by Fuentes et al. in ref. [12]. In their work, a numerical solution of eq. (1.1) with a Gaussian influence function, showed a segregation transition. But such a transition was just the effect of the finite domain size that acted like a cut-off for the Gaussian. Evidence of this interpretation came from the observation that the amplitude of the pattern depended on the ratio of the standard deviation of the in- fluence function to the domain size - the critical values of the standard deviation corresponding to the segregation transition depended linearly on the domain size - and the same patterns appeared for a modified Gaussian, which vanishes abruptly beyond a cutoff. The study of our individual based model gave differ- ent results. By running some simulations with exactly the same parameters but changing the ring extension, we were able to show that the system is not influenced by the domain size. If we choose data from spiky steady states, that permit clear quantitative measures, it is possible to remark that the general morphology of the patterns does not change increasing the L value. In fact, both the pop- ulation density and the mean number of peaks per space interval remains constant. Moreover, in order to provide a more precise test of possible small variations in the distribution, we measured the cluster size. This quan- tity was calculated by evaluating the standard deviation (< x2 >i −< x >i2)1/2 of the position of the i individu- als confined in each peak, then averaged over the different peaks present at step τ and, finally, averaged over many time steps after the system has reached the steady state. Varying the system size caused no changes in the clus- ters size (see Figure 3). From this result, we concluded that the general aspect of the steady state does not change with L. In particular, in contrast with what hap- pens when the mean field equation is solved numerically, the patterns do not depend on the ratio C . For example, for C = 0.2 and L = 50 we obtained a spiky steady state, for C = 0.004 and L = 1 (same ratio) we obtained an homogeneous steady state. Taking into account these re- sults, from now on, all our simulations are implemented on a ring of size 1. We have just shown how the average of the population size < N > scales with L in the steady state. Now, we will give, through a simple heuristic argument, an esti- mation of < N > as a function of the parameters K and C, that will be useful also in the following of our analysis. We can assume that, locally and in the steady state, the number of deaths must be, on average, compensated by the number of newborns, in order to comprise a sta- ble population. For this reason, the death probability P must equal 1/2. Assuming that the number of neighbors that compete with a single individual are the ones living up to a distance C and that, in these surroundings, the average density N/L can be considered to be uniform, P reduces to K ·2C ·N/L. Thus, N ∝ L/(CK). Looking at Figure 4 we can see that this crude evaluation, that ne- glects diffusion, inhomogeneity and reduces the influence function to a box, describes well the general behavior of the data obtained from our simulations. V. SEGREGATION TRANSITION In the previous paragraphs we supported the fact that steady states, depending on the parameter values, can assume inhomogeneous spatial distributions. Now, we will try to describe the transition towards these states (segregation transition). The structure function intro- duced in eq. (3.1) provides a proper order parameter to describe this transition. Different regions in the pa- rameter space, coinciding with different steady states, correspond to different positions of the global maximum (obviously we are not taking into account S(q) at q = 0) of the structure function. The transition from a homo- geneous to an inhomogeneous distribution (see Figure 2) matches the jump of the position of the global maxi- mum (qM ) to a clear integer value, corresponding to the number of clusters present in the space. For this rea- son we can characterize the transition by looking at the shape assumed by S(q), or looking at the value of qM . If the space is homogeneously occupied, the structure function does not present an integer maximum. On the contrary, the maximum is located at qM ≃ 1.4. This value corresponds to an uniform distribution of individ- uals in the interval [0,1], approximated by the expression exp(i2πq ·x)dx . The segregation transition is char- acterized by the passage of qM from 1.4 to an integer value as soon as a modulation becomes dominant. In Figure 5 we show qM as a function of C, varying K and σ. First of all, from the analysis of these data, we can observe that the number of clusters scales as C−1 (or, equivalently, the periodicity of the inhomogeneous phase has wave lengths proportional to C). Moreover, a criti- cal value of C exists for which the transition takes place. This Ccritic grows with 1/K and with σ. An analysis of the available data suggests the possible dependence: Ccritic ∝ σ2/3K−1/3 . Finally, for larger values of the parameter C, in this range of the parameters σ and K, the distributions are characterized by just one peak. For any value of the competition strength, as can be seen in Figure 6, there exists a critical value σcritic, de- pendent on C and K, above which no spatial structures emerge. Another measurement, that permits us to state this relation in a different and clearer way, is presented in the next paragraph. VI. CLUSTER SIZE AND FLUCTUATIONS In the following we describe, in the light of the existing literature, the cluster size dependence on diffusion rate and population size for the two different implementations of the model. We start by analyzing the typical size S of the clus- ters that appear in the spiky phase. The data exposed in Figure 7 show a dependence of the cluster size on the dif- fusion coefficient: S ∝ σ, equivalent to S ∝ D. These results are in accordance with the data presented in ref. [33] obtained from an individual based model. In addi- tion, this work pointed out how this behavior deviates from the conclusions obtained from the deterministic ap- proximation, where the cluster size was only weakly de- pendent on the diffusion coefficient; another fact support- ing the relevance of fluctuation effects in these systems. We can easily interpret the dependence of the clus- ter size on the diffusion coefficient by assuming that the individuals confined in a cluster diffuse a distance pro- portional to DJ where J is the number of jumps the individual performs in its life. In the case of the first implementation of the model, where the diffusion is due to the mutation process, J is obviously 1. Similar results are obtained with the second implementation (see Fig- ure 7), apart from a slightly wider cluster size (in this case, individuals move, on average, more than just once during their lifetime). Even so, the data show the same dependence on the diffusion coefficient. Finally, we present S as a function of K: S ∝ The reasons for this behavior are already explained in ref. [34]: the cluster size is not controlled just by the single individual’s number of jumps; in fact the diffusive process continues with its descendants. For this reason, it is proportional to the mean lifetime of a family, esti- mated to be proportional toK−1 (see ref. [34] for details). We introduce a new quantity that is useful for describ- ing the existence of a critical diffusion value, giving an estimation of its dependence on others parameters and a confirmation of previous results. This quantity, which we call mobility M(τ), estimates the mean mobility of individuals. At a given time step τ , we choose an indi- vidual i. Then we look for the closest agent, among all the population, at time step τ − 1. We identify as di the distance between these two individuals. Averaging over the entire population N(τ) we obtain: M(τ) = di. (6.1) The values assumed by M on varying the parameters σ and C are shown in Figure 8. It is easy to distinguish two clearly different behaviors. If the system is organized in a spiky state (when σ ≤ σcritic), M(τ) ∝ σ. M is another way of measuring the mean distance that an individual moves during its lifetime inside the region defined by the cluster. For this reason, this measure is coherent with the data obtained from the direct evaluation of the standard deviation of the clusters. In contrast, when the system is organized in the homogeneous phase (when σ > σcritic) M becomes independent of σ and is proportional to the inverse of the occupation density M(τ) ∝ KC. The val- ues of the mobility obtained from simulations with dif- ferent values of C and K can be easily collapsed into one function (see the inset of Figure 8). The collapse is performed using the scaling σ → σ/C K. This indicates that the characteristic value of the crossover, σcritic, that separates the two different behaviors of M(τ), scales as: σcritic ∝ C K. (6.2) We conclude our study with some measurements try- ing to catch some properties of the system fluctuations. First, we estimated the fluctuations of the total popula- tion, averaging over different simulations. The variance turned out to be constant throughout the time evolution and of the order of the square root of the total population. The mechanism of auto-regulation of the population di- mension does not allow the growth of big differences in the total number of individuals. For this reason, we focused our attention on the spa- tial distribution of the population and tried to measure some properties of these fluctuations. We studied the variation of the local number of individuals in the same simulation, for different times. We analyzed the evo- lution of the system starting from local initial condi- tions, with the population concentrated in the interval [0.49, 0.51]. In this situation, the system evolves in time with a small cluster fluctuating around the initial space interval. This situation changes when a branching event occurs that generates two well-defined clusters. Our in- terest is in showing the behavior of local space fluctua- tions and capturing possible variations in correspondence with the branching event. First of all, we looked at the mean value of the spatial local fluctuations Fs(τ), defined as Fs(τ) = ( f2j ) 1/2, where fj is the occupancy varia- tion of the bin j from time step τ − 1 to time step τ . We performed the average over all the b bins the ring was divided in and obtained Fs(τ) = N(τ), with no rel- evant variations throughout the time evolution, even in the time interval corresponding to the branching event. More interesting is the shape of the frequency distribu- tion of the size of fj . In fact, a simple Gaussian does not fit this distribution, that presents extended tails (see Figure 9). Throughout the system time evolution, the shape of the normalized distribution is conserved. For global initial conditions the same frequency distribution, with extended tails, is recovered at the steady state. It is identical to the one obtained with local initial condi- tions and measured at the steady state. We think that the deviation of the distribution from a Gaussian can be considered as a hint that fluctuations play a relevant role in the dynamics of the systems. VII. CONCLUSIONS We presented some results regarding clustering of or- ganisms caused by a frequency-dependent interaction that represents competition. We showed how this way of modeling competition can be used not only to de- scribe spatial phenomena in population biology, but also, through a more abstract interpretation, to test ideas of evolutionary theory (for example, studying the speciation process). From this unifying perspective, our study, obtained from an extensive collection of data coming from sim- ulations of an individual based model with global com- petition, pointed out the relevance of fluctuation effects in pattern formation. For the influence function adopted, the mean field description predicts the absence of spatial structures. On the contrary, fluctuations are able to ex- cite the emergence of well defined patterns, which can not be generated from a deterministic instability. Furthermore, we discussed other fundamental proper- ties of our model in the light of the existing literature, unfolding a comparison with other models that describe spatial segregation originated by some deterministic in- stability. We showed that the observed patterns are not due to a finite size effect, we characterized the behavior of the segregation transition in various regions of the pa- rameter space and we studied the existence of a critical diffusion value. We analyzed the dependence of the clus- ter size on the diffusion coefficient and pointed out some characteristics of the fluctuations of the system. ACKNOWLEDGMENTS We are grateful to J.S. Sá Martins for a critical reading of the manuscript and thank the Brazilian agency CNPq for financial support. [1] S. A. Levin and L. A. Segel, SIAM Rev. 27, 45 (1985); R. Durrett and S. A. Levin, Philos. Trans. R. Soc. London, Ser. B 343, 329 (1994). [2] Y.-C. Zhang, M. Serva and M. Polikarpov, J. Stat. Phys. 58, 849 (1990). [3] N. M. Shnerb, Y. Louzoun, E. Bettelheim and S. Solomon, Proc. Natl. Acad. Sci. U.S.A. 97, 10322 (2000). [4] B. Houchmandzadeh, Phys. Rev. E 66, 052902 (2002). [5] W. R. Young, A. J. Roberts and G. Stuhne, Nature 412, 328 (2001). [6] E. Ben-Jacob, I. Cohen, and H. Levine, Adv. Phys. 49, 395 (2000); J.Wakita, K. Komatsu, A. Nakahara, T. Mat- suyama, and M. Matsushita, J. Phys. Soc. Jpn. 63, 1205 (1994). [7] K. S. McCann, J. B. Rasmussen, and J. Umbanhowar, Ecol. Lett. 8, 513 (2005). [8] J. B. Wilson and A. D. Q. Agnew, Adv. Ecol. Res. 23, 263 (1992); R. Lefever and O. Lejeune, Bull. Math. Biol. 59, 263 (1997); N. M. Shnerb, P. Sarah, H. Lavee, and S. 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Hernandez-Garcia and C. Lopez, J. Phys.: Condens. Matter 17, S4263 (2005). FIG. 1. Homogeneous steady state distribution ( top, C = 4.0, 1/K = 80000, σ = 0.01) and modulated steady state distribution (bottom, C = 0.9, 1/K = 18000, σ = 0.01). The insets show the structure functions, S(q), of the correspond- ing simulations. We show the distributions at time step 1000, whereas the structure functions are averaged over 500 time steps. FIG. 2. Spiky (top, C = 0.059, 1/K = 500, σ = 0.001) and homogeneous (bottom, C = 0.005, 1/K = 500, σ = 0.001) steady states. The insets show the structure functions S(q). We show the distributions at time step 2000, whereas the structure functions are averaged over 1000 time steps. The transition between these two states, in this typical range of parameters, has been extensively studied. 0 10 20 30 40 50 0.005 FIG. 3. Dependence of the cluster size on the ring size L. We present data from the model with mutation (trian- gles) and from the one that implements diffusion (circles), C = 0.2, K = 0.0029, σ = 0.001. The average is carried out over all the clusters present at a given time step and over different time steps. FIG. 4. The number of individuals N present in the steady state is proportional to (CK)−1. This result is in accordance with the one obtained for a box-type influence function of length C (see ref. [32]). We present data for different simu- lations with 1/K ∈ [50, 500] and σ ∈ [0.0001, 0.01]. This last parameter does not influence the final number of individuals. The dashed line has slope -1. FIG. 5. Segregation transition at Ccritical. Upper figure: variation in dependence of K, where 1/K = 50, 150, 400, 500 and σ = 0.001. Lower figure: variation in dependence of σ, where σ = 0.0005, 0.001, 0.002, 0.005, 0.01 and 1/K = 200. FIG. 6. As shown in these figures, a critical σ value exists, above which no spatial structures emerge. Upper figure: vari- ation in dependence on C; we set 1/K = 100. Lower figure: variation in dependence on K; we set C = 0.01. FIG. 7. Top: Cluster size as a function of σ; 1/K = 300. The solid line has slope 1. Bottom: cluster size as a function of 1/K; σ = 0.0001. The solid line has slope 1/2. Trian- gles represent data from the simulations where diffusion is implemented through mutations, circles for the direct imple- mentation of the diffusive process; we set C = 0.09. FIG. 8. Mobility dependence on the diffusion parameter σ for different C values; K = 0.01. In the inset, the data collapse for arbitrary values of the parameters C and K. FIG. 9. Spatial local fluctuation distribution at the steady state: deviation from a Gaussian (C = 0.1, K = 0.00005, σ = 0.0017). Data are averaged over 5 time steps. The con- tinuous line is the best fit Gaussian.

0704.0272

A Comparison between Anomalous 6-cm H$_2$CO Absorption and CO(1-0) Emission in the L1204/S140

A comparison between anomalous 6-cm H2CO absorption and CO(1-0) emission in the L1204/S140 region Mónica Ivette Rodŕıguez1,2 Tommy Wiklind1,3 Ronald J. Allen1 Vladimir Escalante2 Laurent Loinard2 1Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA monica, rjallen, [email protected] 2Centro de Radiostronomı́a y Astrof́ısica, Universidad Nacional Autónoma de México, Apartado Postal 72 - 3, C.P. 58091, Morelia, Michoacán, México m.rodriguez, l.loinard, [email protected] 3Affiliated with the Space Sciences Department of the European Space Agency ABSTRACT We report observations of the dust cloud L1204 with the Onsala 25-m tele- scope in the 6 cm (111-110) transition of H2CO. The observed region includes the S140 Hα arc. This spectral line is seen here in absorption against the cosmic microwave background, indicating the presence of widespread warm molecular gas at intermediate densities. Overall, the distributions of H2CO and CO (taken from the literature) are fairly similar, though significant differences exist at small scales. Most notably, while the CO peak is nearly coincident with the S140 Hα arc, the maximum H2CO absorption is clearly separated from it by a full 10 ′ beam (∼ 3 pc). We argue that these differences result from differing abundances and ex- citation requirements. The CO(1-0) line is more optically thick and more biased towards warm gas than the H2CO 6 cm line. On the other hand, formaldehyde is more easily photodissociated and is, therefore, a poorer tracer of the molecular gas located immediately behind Photon Dominated Regions. Subject headings: ISM: clouds — ISM: molecules – radio lines: ISM – stars: formation – galaxies: ISM http://arxiv.org/abs/0704.0272v1 – 2 – 1. Introduction Since H2 –the most abundant molecule in space– lacks a permanent dipole moment, its rotational transitions are prohibited. Although the quadrupolar transitions exist, they are of little use for the syudy of the bulk of molecular gas in the ISM because they require high temperature to be excited. Instead, the structure and properties of cold molecular clouds in the interstellar medium are usually studied using low-energy rotational transitions of simple non-symmetric polar molecules. For practical reasons, the first rotational transition (J = 1→0) of carbon monoxide (CO), at 115.27 GHz has been the most popular choice. This transition, however, has long been known to be nearly always optically thick, so – for a given filling factor– its intensity is expected to increase monotonically with the kinetic temperature of the emitting gas. Clearly, this could have adverse effects on efforts to establish the distribution of molecular gas in the ISM from CO observations alone, because very low temperature gas might go unnoticed in sensitivity-limited CO observations while warm regions (& 20 K) will stand out even if they are not those with the highest molecular content. The sources where these effects might be most noticeable are those with large temperature gradients; for instance in molecular clouds located in the immediate vicinity of hot stars. While all molecular emission tracers share this temperature dependance to some degree, absorption lines can be detected even in very cold gas, provided sufficiently bright background continuum sources are available. The scarcity of such sources at the wavelength of the common molecular tracers, however, has limited the usefulness of absorption measurements in the study of specific Galactic molecular clouds (e.g Evans et al. 1980). The 6-cm (110-111) transition of ortho formaldehyde (H2CO) offers an interesting alternative. Owing to collisions with neutral particles that selectively overpopulate the lower energy level, the excitation temperature of the 110 → 111 transition lies below 2.7 K (Townes & Cheung 1969). This allows the transition to be observed in absorption against the cosmic microwave background (CMB) (Snyder et al. 1969), and makes it a potentially powerful tracer of molecular gas in any direction of the sky. The excitation requirements are such the the 6 cm H2CO line is a good indicator of the presence of cool to warm molecular gas (T & 10 K) at intermediate densities (103 cm−3 ≤ n ≤ 105cm−3). Unfortunately, the absorption line is weak, so very large amounts of telescope time are required to map large areas of the sky. Recently, Rodŕıguez et al. (2006) conducted a blind search for H2CO absorption and compared CO emission and H2CO absorption profiles towards the Galactic anticenter. They found a rough, large-scale correlation between these two tracers, and concluded that both lines preferentially trace warm and dense molecular gas. Here, we will examine this relation between CO and H2CO at a somewhat smaller scale using observations of the well-known, nearby star-forming region Sharpless 140 (S140 –Sharpless 1959) associated with the dark – 3 – dust cloud Lynds 1204 (L1204 –Lynds 1962). L1204 is centered at l = 107◦. 47, b = +4◦. 82 and covers an area of 2.5 square degrees (Lynds 1962). At its southwest edge lies S140, a prominent compact arc-shaped H ii region with an angular size of ∼ 2′ × 6′. The ionization of S140 is maintained by the nearby B0V star HD211880 (Blair et al. 1978). The distance of S140/L1204 deduced from the brightness of the exciting star is 910 pc (Crampton & Fisher 1974). S140 has been the subject of many observational studies, that have usually focused on the Photon Dominated Region (PDR) on the edge of L1204, and on the embedded infrared sources located right behind it (e.g., Preibisch et al. 2001; Hayashi & Murata 1992; Preibisch & Smith 2002; Bally et al. 2002). Remarkably, while the dust cloud is seen as an extended dark feature covering more than two square degrees, the CO emission peaks immediately behind the Hα arc (Heyer et al. 1996; Evans et al. 1987; Blair et al. 1978), while only relatively faint emission extends deep within the dust cloud (Helfer & Blitz 1997). The 6-cm line of H2CO was detected in absorption against the CMB in L1204 near S140 by Blair et al. (1978) with the NRAO 43 m telescope, and unexpectedly by Evans et al. (1987) during VLA observations of the bright condensation just northwest of S140. However neither of those studies provided a extensive mapping of the H2CO CMB absorption in L1204, and the exact extension of the gas traced by H2CO remains unclear. In this article, we will present such a extensive mapping of the 6 cm CMB absorption of H2CO over most of the large dust complex L1204, and compare our results with existing CO observations taken from the literature. 2. Data The H2CO observations were obtained during two sessions (January and September- October 2004, respectively) with the 25.6-m telescope of the Onsala Space Observatory (OSO) in Sweden. At 6 cm, the angular resolution is 10′, and our pointing precision was always better than 20′′. Frequency-switching, with a frequency throw of 0.4 MHz was used, and both polarizations of the incoming signal were recorded simultaneously in two indepen- dent units of the autocorrelation spectrometer. Each of these units provided 800 2 kHz-wide channels. At the observed frequency of 4829.660 MHz, this setup provided a total bandwidth of 99 km s−1 and a (Hanning-smoothed) velocity resolution of 8 kHz ≡ 0.49 km s−1. The spectrometer was centered at the systemic velocity of S140, VLSR = −8.0 km s−1. Daily observations of the supernova remnant Cas A were used to check the overall performance of the system. The system temperature during our observations varied from 33 to 36 K. In order to map the entire region behind S140, we observed 72 positions on a regular square grid with a 10′ spacing, centered at l = 107◦. 0, b = +5◦. 3; the resulting map uniformly covers a 1◦. 0 × 1◦. 8 rectangular region (Fig. 1) . The off-line data reduction was done – 4 – with the CLASS program of the GILDAS software package (Guilloteau & Forveille 1989), and involved only the subtraction of (flat) baselines from individual integrations and the averaging of all spectra taken at the same pointing position. The total integration time for each of these positions was about 10 hours, yielding a typical final noise level of 3 mK (T∗A). The distribution of radio continuum sources in the region of L1204 has been studied in detail by Allen Machalek & Jia (in preparation), using data from the Canadian Galactic Plane Survey. Fairly bright continuum emission is associated with the Hα arc and the embedded massive protostars located behind it –but, as we will see momentarily no formaldehyde was detected from either of these regions. In addition, a number of extragalactic background sources as well as diffuse emission associated with the dust cloud L1204 itself contribute to the overall radio continuum. The typical brightness temperature average over the Onsala beam at 6 cm, however, is only about 0.2 K, except towards the Hα arc and the embedded massive protostars (where again, no absorption was detected). Since the brightness temperature is so small, any H2CO absorption profiles features detected must be absorption of the cosmic microwave background radiation at 2.7 K. In the analysis of our new observations, we will also make use of 12CO(1-0) observations of L1204/S140 kindly provided by Dr. Tamara Helfer, and published in Heyer et al. (1996), and Helfer & Blitz (1997). These data were obtained with the 14-m telescope of the Five College Radio Astronomy Observatory (FCRAO) in Amherst (MA), and have an intrinsic angular resolution of 45′′. For comparison with our formaldehyde data, we have smoothed the CO(1-0) observations to 10′, and resampled them on our observing grid. 3. Results Formaldehyde absorption was detected in at least 16 of our 72 observed positions (see Fig. 2). The maximum absorption is located 10′ arcmin behind the S140 H ii region at a LSR velocity of −8.0 km s−1, similar to that of the CO emission detected in that area. Table 1. Source positions. Source Position (l, b) Size (∆α,∆δ) α(J2000.0), δ(J2000.0) Reference L1204 107◦. 37, +4◦. 87 1◦. 0 × 2◦. 5 22h26m41s.+63◦15′36′′. Lynds (1962) S140(Hα) 106 ◦. 8, +5◦. 3 2′ × 6′ 22h19m23s.+63◦18′16′′. Sharpless (1959) Our survey 107◦. 0, +5◦. 3 1◦. 0 × 1◦. 8 22h20m52s.+63◦24′49′′. This paper – 5 – Fig. 1.— This figure shows a sketch of the L1204/S140 region, with our observed positions shown as ”+” symbols. The correspondence between Galactic coordinates (used throughout the paper) and equatorial coordinates (that have usually been preferred for observations of S140) is indicated. The circle represents the 2◦. 5 size of the dust cloud L1204 as reported by Lynds (1962). The central positions of L1204 and S140 are shown as ”” symbols. The contour represents the lowest value of the H2CO absorption. A second spatio-kinematical structure is detected towards the north-east (here, and in the rest of the paper, north and all other directions refer to Galactic coordinates), at VLSR ∼ −11 km s−1. Both components are presumably associated with L1204, and have clear CO counterparts (Fig. 2 – Blair et al. 1978; Evans et al. 1987; Sugitani & Fukui 1987; Park & Minh 1995). There is also an isolated absorption feature towards the southeast, at VLSR ∼ −2.5 km s−1. Given its low LSR velocity, this feature is likely unrelated to L1204, and is probably a local cloud along the line of sight. Thus, while Sugitani & Fukui (1987) identified three molecular components associated with L1204 in their 13CO observations, we only find two in our formaldehyde data. We do find evidence, however, for a systematic velocity gradient across the cloud. Park & Minh (1995) argued that this complex overall spatio- kinematical morphology was created when S140 and L1204 were swept up by an expanding shell associated with the Cepheus bubble. Our data do not illuminate this assertion any further, and a more thorough study is necessary to understand the detailed structure of this region. – 6 – Fig. 2.— (a) Mosaic of H2CO CMB absorption spectra observed in the L1204/S140 region. The (0,0) position corresponds to l = 107◦. 0, b = 5◦. 3, about 12′ east of S140, and about 40′ north-west of the nominal center of L1204 (Lynds 1962). (b) Corresponding CO(1-0) observations smoothed to 10′ (see text). Note that the CO emission is located a full 10′ west of the maximum H2CO absorption. 4. Comparison with other molecular tracers The S140/L1204 region has been observed in many different molecular tracers (e.g. Tafalla et al. 1993, Zhou et al. 1993, Park & Minh 1995), but most of these observations have focused either on the S140 PDR or on the embedded infrared sources located just behind S140, while only a few observations covered the entire dust cloud. Indeed, the first CO observations of S140 (Blair et al. 1978) only covered a limited part of the region. To our knowledge, the only existing large-scale CO map of L1204 is that obtained in the 90s with the FCRAO telescope (see §2) and published by Heyer et al. (1996) and Helfer & Blitz (1997)1. As mentioned earlier, we will use a smoothed version of that dataset here in order to compare with our formaldehyde observations. In general, the CO emission and H2CO absorption morphologies in this region are quite similar (Fig. 3). This was already noticed by Blair et al. (1978) in their 6′ observations. It is also in good agreement with the results obtained towards the Galactic anticenter by 1The region lies on the edge of, and is only partly covered by, the CfA Galactic plane survey of Dame et al. (2001). – 7 – Fig. 3.— All panels show a grey-scale version of the DSS-red image of the region around l = 107◦. 0, b = 5◦. 4. The Hα arc of S140 is clearly visible on these (red) images near l = 106◦. 8, b = 5◦. 3. In the three left panels (a1–a3), CO contours taken from the smoothed CO data of Heyer et al. (1996) are overlaid on top of the DSS image, whereas in the three right panels (b1–b3), our H2CO contours are overlaid. The contours in the top two panels (a1–b1) include the entire velocity range associated with L1204 (from -12 to -5 K km s−1), whereas in the middle two (a2–b2) and bottom two (a3–b3) panels, the contours correspond only to the -11 K km s−1 and the -8 K km s−1 components, respectively. The asterisks correspond to the position of the peak H2CO absorption for each component. – 8 – Rodŕıguez et al. (2006), and towards the Orion molecular complex by Cohen et al. (1983). There are, however, several noteworthy differences between the CO emission and H2CO absorption in S140. The first difference is the fact that the CO peak and the H2CO absorption maximum are not located at the same position. The CO integrated intensity map (Fig. 3, see also Fig. 2) shows that the maximum CO emission occurs just behind the S140 Hα arc at the western edge of L1204, while only comparatively fainter emission extends to greater longitudes. The maximum H2CO absorption, however, is located about 10 ′ eastward of the CO peak. A second notable difference between the CO and H2CO profile is the existence of a CO emission ”tail” in the south/southeast part of the main cloud with little or no H2CO counterpart (Figs. 2 and 3). -0.35 -0.25 -0.15 -0.05 0 10 20 30 40 50 I (12CO) K km/sec Intensity ratio S140 + L1204 L1204 Galactic Anticenter Fig. 4.— Correlation between the H2CO absorption line intensity and the 12CO(1-0) emission line intensity at corresponding points in L1204. The squares correspond to data from Table 2, the open circles correspond to H2CO upper limits and the triangles correspond to the Galactic anticenter data previously published in Rodŕıguez et al. (2006). The horizontal and vertical lines show the “best case” detection limit. The dashed line is the least-squares fit for the L1204/S140 region data, and the dash-dotted line is the least-squares fit for the Galactic anticenter data from Rodŕıguez et al. (2006). Note that the fits do not pass through the (0,0) point, suggesting that the relation is not linear at low intensity values. In order to study the relation between H2CO absorption and CO(1-0) emission in a more quantitative way, we have computed the moments of the profiles shown in Fig. 2. The results are listed in Table 2 (Appendix B). When two velocity components are visible at a given pointing, the moments for each were computed separately. Intensities above 3σ are shown as squares in Fig. 4, and were used to make least-square fits (see below). The two – 9 – spatio-kinematical components that we identified in our formaldehyde dataset behave quite similarly with respect to the CO-H2CO relation, and are plotted together in Fig. 4. The best least-squares fit to a straight line for the entire L1204 dataset yields: I(H2CO) = (4.4± 1.1)× 10−3 I(CO) + (34± 16)× 10−3 K km s−1. (1) We shall see momentarily that the CO emission near S140 may be particularly bright because of local heating. Ignoring the pointings very near S140, however, yields a fairly similar relation between CO and H2CO: I(H2CO) = (3.8± 0.9)× 10−3 I(CO) + (31± 9)× 10−3 K km s−1. (2) The small difference between these relations presumably reflects the differing excitation requirements for the two lines. The relation between CO and H2CO given by Eqs. 1 and 2 for the L1204 region is almost identical to that found towards the Galactic anticenter by Rodŕıguez et al. (2006): I(H2CO) = (4.1± 0.5)× 10−3 I(CO) + (34± 13)× 10−3 K km s−1. (3) It is important to note, however, that, in spite of the agreement between the fits to the Galactic anticenter and S140 data, there is very significant scatter in the CO-H2CO relation, some points lying nearly 10σ away from the linear relation. This situation was already noticed by Rodŕıguez et al. (2006) in their study of the Galactic anticenter. This lack of a detailed correspondence between CO emission and H2CO absorption presumably reflects differences in the excitation conditions of the two tracers, as we will now discuss in the next section. 5. Discussion The comparison between H2CO absorption and CO(1-0) emission profiles in the Galac- tic Anticenter and in the L1204/S140 region has led us to three important observational conclusions: 1. Qualitatively, the morphology of CO and H2CO are quite similar, and quantitatively, the line integrated intensities correlate quite well with one another. – 10 – 2. The scatter in the CO-H2CO relation is, however, significantly larger than the obser- vational errors. 3. In the specific case of S140, the CO emission peak is offset by about 3 pc from the locus of the deepest formaldehyde absorption, and there is a region south of the main cloud where significant CO emission is detected with little or no H2CO counterpart. From the general large-scale correspondance between the CO(1-0) and H2CO 6 cm integrated maps (Fig. 3), and from the fair correlation between their line intensities (Fig. 4), we conclude that the physical conditions needed for the excitation of both lines are quite similar. The calculations presented in the Appendix A, indeed show that both lines preferentially trace warm gas at intermediate densities (103.6< n < 105 for H2CO; n > 10 for CO). In this scheme, the offset between the CO(1-0) peak and the H2CO maximum absorption may seem puzzling. Note that a similar trend is seen at higher resolution: while the CO peak in the full-resolution CO map published by Heyer et al. (1996) and Helfer & Blitz (1997) is at l = 106o.8, b = +5o.3, the H2CO absorption feature seen in the high- resolution VLA images published by Evans et al. (1987) is centered around l = 106o.9, b = +5o.3, again a few arcminutes to the east. We suggest that a combination of two effects may explain this puzzling result. First, it can be seen from the excitation analysis presented in Appendix A that the H2CO absorption strength ”saturates” for T & 30 K, whereas the temperature of the CO emission continues to rise at higher kinetic temperature (Fig. 5). For example, while the H2CO line strength increases by only about 30% when the kinetic temperature goes from 20 to 40 K, the CO(1-0) line intensity increases by more than a factor of two. According to Park & Minh (1995), the CO brightness temperature is about 40 K at the peak and 20 K for the rest of the cloud. Thus, the strong CO peak behind S140 may well be largely due to enhanced kinetic temperatures related to local heating (by the external star providing the ionization of S140, and/or by the infrared sources embedded in the cloud). As one progresses into the cloud, the local heating diminishes, and the CO line intensity fades. Also, it should be pointed out that the formaldehyde calculations presented in the Appendix show that the 6-cm line should be seen in emission rather than absorption when the density exceeds 105 cm−3. The fact that this is not the case near the CO peak (neither in our low-resolution data, nor in the high-resolution VLA data presented by Evans 1978) suggests that the gas density there is lower than 105 cm−3. Other effects that could explain the offset between the CO and the H2CO peaks are the lower dissociation energy, and the lower abundance (and, therefore, lower self-shielding) of formaldehyde compared to CO (see Appendix A.3). In a photo-dissociated region, these effects should combine to create a stratified distribution where CO survives nearer the source of the UV photons than H2CO. This stratification, combined with the heating of the CO, – 11 – would naturally lead to the offset between H2CO and CO seen in the present data. Finally, the origin of the other main difference between CO and H2CO in S140, namely the existence of CO emission at the south of L1204 with no or little formaldehyde counterpart, is likely related to another aspect of the excitation differences between the 6 cm line of formaldehyde and the 1-0 transition of carbon monoxide. Fig. 6 of the Appendix A.2 shows that the density detection limit for H2CO line is ∼10 times larger than the density limit for the CO(1-0) line. We therefore suggest that the gas traced by the CO emission to the south of L1204 is of relatively very low density. It is interesting to note, indeed, that classical high-density molecular tracers (e.g. CS or NH3) have only been detected around the CO peak behind S140, and not in the southern region of the cloud. Thus, we conclude that the CO(1-0) and H2CO 6 cm lines both tend to preferentially trace warm gas at intermediate densities. There are, however, significant differences related either to differing excitation requirements or to differing abundances. These differences can easily explain the large scatter in the CO–H2CO relation. 6. Conclusions The main conclusions of this work are the following: 1. We have mapped a large region (70′ × 110′) around L1204/S140 in the 6 cm line of formaldehyde, observing a total of 72 regularly-spaced positions every 10′ on a regular grid. The center of our map was at l = 107◦. 0, b = +5◦. 3, and formaldehyde was detected against the cosmic microwave background in at least 16 of our 72 positions (Fig. 2). 2. The formaldehyde emission can be separated in three spatio-kinematical components (Fig. 3): two (at VLSR ∼ –11 km s−1 in the northeast part of the cloud, and at VLSR ∼ –8 km s−1 just behind S140) are clearly associated with L1204, whereas the other (an isolated component at VLSR ∼ –2.5 km s−1 towards the southeast) is most likely a local foreground cloud unrelated to S140/L1204. 3. Both qualitatively and quantitatively, the CO(1-0) emission and the formaldehyde 6 cm absorption lines correlate fairly well. An excitation analysis shows that both preferentially trace warm gas at intermediate densities. 4. There are, however, notable differences between the CO and H2CO lines, that can be traced to differing excitation requirements and abundances. Those differences are most likely the origin of the large scatter in the CO-H2CO intensity correlation. – 12 – We thank Professor Roy Booth, director (retired) of the radio observatory at Onsala, for generous allocations of telescope time and for his warm hospitality during our several visits to the observatory. We are also grateful to the observatory technical and administrative staff for their capable assistance with our observing program. We thank Tamara Helfer for supplying us with the CO data cube of S140. We acknowledge the financial support of the Dirección General de Asuntos del Personal Académico (DGAPA), Universidad Nacional Autónoma de México (UNAM) and Consejo Nacional de Ciencia y Tecnoloǵıa (CONACyT), in México, and the Director’s Discretionary Research Fund at the Space Telescope Science Institute. The Digitized Sky Surveys were produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions. A. Model calculations A.1. Collisional pumping Observations of H2CO in dark clouds show that the anomalous absorption of the 6 and 2 cm lines is due to collisions with H2 that selectively overpopulate the lower levels of the lines (Evans et al. 1975). We calculated the non–LTE equilibrium populations of the first 40 levels of ortho–H2CO assuming excitation by the 2.7 K background and collisions with H2. Green (1991) has calculated excitation rates of these levels for collisions with He taking advantage of the spherical symmetry of the He potential for kinetic temperatures T = 10 to 300 K. According to Green (1991), excitation rates by H2 collisions could be 2.2 times higher than those by He because of the smaller reduced mass and differences in the interaction potentials. We will show the results of the calculations under the assumption that the H2– H2CO collisional rates are the same as the He–H2CO rates. Probabilities for the radiative transitions were taken from Jaruschewski et al. (1986). The optical depths of the transitions involved in the pumping mechanism are generally larger than 1 at high densities and a radiative transfer calculation is required. Two limiting approximations in the radiation transport are often considered in molecular clouds: the large velocity gradient (LVG) model and the microturbulent model (Leung & Liszt 1976). The LVG model assumes that the line profile is dominated by systematic motion of the gas while the microturbulent model assumes that the turbulent velocity is much larger than any systematic motion. S 140 is likely to have several velocity components, but the existence of large systematic motions in molecular clouds and the validity of the LVG model has not – 13 – been well established in other molecular clouds (e.g. Evans et al. (1975), Zuckerman & Evans (1974), Zhou et al. (1990)). We therefore use the microturbulent model, and for simplicity we will use the escape probability formalism to account for photon trapping in a turbulent medium in plane-parallel slab of mean total optical depth τt that is perpendicular to the line of sight. In this case the A–value of a transition in the equations of statistical equilibrium is multiplied by a “loss probability” P (τ, τt) that depends on the mean optical depth τ in the slab. There are many different ways to define P (τ, τt), which can differ by several orders of magnitude for large optical depths. We will use the form suggested by Hummer & Storey (1992) for a uniform medium with no continuum absorption: P (τ, τt) = [K2(τ) +K2(τt − τ)] , (A1) where K2(τ) = dx φ(x)E2[τφ(x)] , (A2) and E2 is the second exponential integral function. The function K2(τ) can be calculated from fits by Hummer (1981) for a normalized Doppler profile, φ(x) = exp(−x2)/ Equation (A1) can be viewed as the single flight escape probability through either side of the slab averaged over the line profile. The probability that a photon of an isotropic background reaches optical depth τ in this slab is also given by equation (A1), and the blackbody continuum is thus attenuated by a factor P (τ, τt). The mean optical thickness is given by nu/gu nl/gl dz , (A3) where σlu is the absorption cross section of the transition and ∆νD is the Doppler width. We assumed a Doppler width of 2 km s−1 (FWHM = 2 ln 2∆νD = 3.3 km s −1). The level population and its statistical weight are given by n and g respectively with subindex u for the upper and l for the lower level. The emergent line brightness temperature with subtracted background T0 is found by direct integration of the source function throughout the slab as ∆Tb = Tb − T0 = T0[ exp(−T )− 1] exp(−T + τ) nl/gl nu/gu dτ , (A4) where – 14 – 1− nu/gu nl/gl dz , (A5) is the total mean optical depth of a slab of thickness L and T = τtφ(0) is the line–center total optical depth. Equation A4 gives the correct asymptotic limits: Tb → T0 when the density goes to 0 and Tb → T for high densities. The population densities and the optical depths as a function of position in the slab in equations (A1) and (A3) were calculated iteratively. A convergence of 10−3 K in Tb was achieved after a few iterations for T ≤ 40 K and H2 densities n(H2) < 10 6 cm−3. For higher densities and temperatures the procedure becomes unstable. Fig. 5 shows the calculated ∆Tb for a 1 pc–thick slab as a function of the H2 density and a constant H2CO abundance of 2 × 10−9 with respect to H2 (Hasegawa et al. (1992), Leung et al. (1984)). The assumed thickness of the slab has an important effect in the anomalous absorption as shown in Fig. 5. As the thickness of the slab decreases, the effectiveness of the pumping mechanism that cools the line decreases. (a) (b) (c) Fig. 5.— (a) Brightness temperature minus background continuum vs. density for different kinetic temperatures T of a 1–pc thick slab. (b) Contour intervals are 0.2 K from ∆Tb = −1 to 1 K. (c) Brightness temperature minus background continuum vs. density for different slab thicknesses at T = 40 K. Garrison et al. (1975) identified some transitions, like 111 → 312, that produce selection effects in the excitation of some levels that cool the H2CO doublets. We have tested our model for possible variations of the collision rates. An overall increase of collisional rates by a factor of 2.2 decreases ∆Tb in figure 5 by 0.3 to 0.5 K for T ≥ 20 K and n(H2) > 1.6 × 104 cm−3. For lower T and n(H2) there is very little variation in the predicted Tb. – 15 – A.2. A PDR model for CO The UV field has a higher influence on the CO brightness temperature than on the H2CO brightness temperature. Both molecules are quickly photodissociated near the edge of clouds, but the larger abundance of the CO molecule makes its chemistry and interaction with radiation more complex. In order to take into account the variation of CO abundance along the line of sight, we used the Meudon PDR code to calculate the CO brightness temperature of a plane–parallel slab irradiated by a UV field (Le Bourlot et al. 1993). A detailed description of a revised version of the code is given by Le Petit et al. (2006). The sharp separation between the molecular, atomic and ionized emissions suggests that the L1204/S140 interface is a PDR viewed nearly edge–on (Hayashi & Murata 1992) irradiated by HD 211880. The angle of incidence of the star’s radiation on the PDR boundary is an unknown parameter but appears to be more-or-less perpendicular. Furthermore, the infrared embedded sources are probably young stars that may also enhance the radiation field (Evans et al. 1989). We ran the code with its parameters set to represent a plane–parallel slab irradiated from one side by a UV field with an enhancement factor χ = 200 with respect to the Draine (1978) average interstellar radiation field. In a PDR, the gas is heated by the photoelec- tric emission from grains and PAH’s, H2 formation in grains, UV pumping in the Lyman and Werner bands, gas–grain collisions, photoionization, and photodissociation. As the UV radiation is absorbed deeper into the cloud, other processes like cosmic rays and chemical reaction energies become important. Cooling is produced by fine–structure and molecular line emission. Shocks and turbulence can keep a PDR away from isobaric equilibrium. How- ever in our model the temperature and density of the PDR were kept constant in order to compare the results with our H2CO model in Fig. 6. We used the chemical network given for S140 by the Meudon group at its Internet site2, which does not include H2CO. Thus the CO and H2CO calculations represent different models, and Fig. 6 is given only as indication of the local conditions that produce the emission and absorption for each molecule. A.3. Photodissociation of H2CO The H2CO molecule is quickly photodissociated into CO and H2 or H in the average UV interstellar radiation field with a rate of 1.0 × 10−9 exp(−1.7AV ) s−1 (van Dishoeck 1988). Keene et al. (1985) estimated that far–ultraviolet radiation (FUV) from the star HD211880 2http://aristote.obspm.fr/MIS/pdr/exe.html – 16 – CO H CO Fig. 6.— Brightness temperature minus background continuum vs. density for different kinetic temperatures T for both the 12CO(1-0) line and the 6 cm H2CO line (Note that ∆Tb(k) is plotted ”negative” compared to Figure 5). will have an enhancement factor of G0 = 150 with respect to the average interstellar field (Habing 1968) at the ionization front, although Spaans et al. (1997) found that a more intense radiation field may be needed to explain the H2 rotational emission. H2CO has a dissociation energy of 3.61 ± 0.03 eV (Suto Wang & Lee 1986), and a photodissociation rate of 1.0 × 10−9 sec−1 in the interstellar field (van Dishoeck 1988) while for CO the values are 11.2 eV and 2.0 × 10−10 sec−1 respectively. Thus it is possible that the H2CO will be selectively photodissociated near the S140 ionization front and the bright PDR region, where the CO emission peaks. Detailed PDR model calculations by Li et al. (2002) show that at 20′ from the ionization front, where we observe the H2CO maximum, G0 ≤ 20 and AV ∼ 15. We added 62 reactions involving H2CO and H2CO + taken from the UMIST data base (Woodall et al., 2007) 3 to the chemical network of the Meudon group mentioned above and ran a PDR model with G0 = 200, T = 40K and constant density of 10 3 cm−3. We found that H2CO has significant abundance only at depths of Av > 7 while CO becomes important at Av > 4, which shows that photodestruction could explain the offset between the CO and the H2CO peaks. 3http://www.udfa.net/ – 17 – B. Profile moments in detail – 18 – Table 2. Profile moments for each position of Fig. 2. The upper limits correspond to 3σ. The symbol “. . . ” indicates no data were available, while “nf” indicates that data were available but no reliable fit could be made. Values marked with “yes” in column 6 are plotted as squares in Fig. 4, and were used for the least square fits. Offset (l,b) 1000× I(H2CO) 〈V 〉 I(CO) 〈V 〉 Included in arcmin K km s−1 km s−1 K km s−1 km s−1 fits 40, -70 < -24.3 nf . . . . . . 30, -70 < -18.9 nf . . . . . . 20, -70 < -25.8 nf . . . . . . 10, -70 < -23.4 nf 4.1 ± 0.7 -7.6 ± 1.3 0, -70 -42.1 ± 8.4 -7.5 ± 1.5 3.5 ± 0.6 -8.7 ± 1.5 yes -10, -70 -30.3 ± 7.1 -6.2 ± 1.5 8.6 ± 0.8 -8.3 ± 0.8 yes -20, -70 < -21.6 nf < 2.1 nf 40, -60 < -22.5 nf . . . . . . 30, -60 < -24.0 nf . . . . . . 20, -60 < -26.7 nf 5.9 ± 0.8 -7.0 ± 1.1 10, -60 < -23.7 nf 5.1 ± 0.6 -8.3 ± 1.0 0, -60 < -19.2 nf < 1.8 nf -10, -60 -27.9± 7.1 -6.7 ± 0.5 < 2.1 nf -20, -60 < -28.8 nf < 1.5 nf 40, -50 < -25.5 nf . . . . . . 30, -50 -109.3 ± 8.9 -2.6 ± 0.2 14.8 ± 1.1 -1.8 ± 0.2 yes 20, -50 < -25.2 nf 10.5 ± 0.8 -5.0 ± 0.4 10, -50 < -21.6 nf 8.2 ± 0.6 -7.0 ± 0.5 0, -50 < -18.9 nf < 2.4 nf -10, -50 < -16.5 nf < 2.4 nf -20, -50 < -24.6 nf < 4.8 nf 40, -40 < -23.1 nf . . . . . . 30, -40 < -22.8 nf 3.1 ± 0.6 -6.2 ± 1.2 20, -40 < -22.2 nf 7.2 ± 0.8 -7.2 ± 0.9 10, -40 < -23.1 nf 4.5 ± 0.5 -8.8 ± 1.1 0, -40 < -22.5 nf < 2.7 nf -10, -40 < -26.1 nf < 2.7 nf -20, -40 < -18.6 nf < 3.0 nf 40, -30 < -22.5 nf 3.4 ± 0.5 -9.0 ± 1.5 30, -30 < -22.8 nf 4.0 ± 0.7 -8.9 ± 1.5 – 19 – Table 2—Continued Offset (l,b) 1000× I(H2CO) 〈V 〉 I(CO) 〈V 〉 Included in arcmin K km s−1 km s−1 K km s−1 km s−1 fits 20, -30 < -25.5 nf < 1.8 nf 10, -30 < -30.9 nf 5.2 ± 0.6 -8.6 ± 1.0 0, -30 < -20.1 nf < 4.2 nf -10, -30 < -26.7 nf < 1.8 nf -20, -30 < -22.2 nf < 2.1 nf 40, -20 . . . . . . < 2.7 nf 30, -20 < -30.9 nf 3.5 ± 0.7 -9.4 ± 1.9 20, -20 < -24.0 nf < 2.7 nf 10, -20 -31.6 ± 6.0 -8.2 ± 1.6 8.2 ± 1.1 -8.9 ± 1.2 yes 0, -20 < -29.1 nf 8.5 ± 1.4 -9.5 ± 1.7 -10, -20 < -19.5 nf 6.8 ± 0.3 -8.7 ± 0.9 -20, -20 < -22.2 nf < 1.5 nf 40, -10 . . . . . . < 1.8 nf 30, -10 < -35.7 nf < 2.4 nf 20, -10 < -16.8 nf < 3.0 nf 10, -10 -43.1 ± 8.2 -8.1 ± 1.6 4.7 ± 1.1 -10.3 ± 2.8 yes 0, -10 -76.2 ± 13.8 -9.5 ± 1.8 11.7 ± 0.8 -9.0 ± 0.6 yes -10, -10 -82.7 ± 7.7 -8.1 ± 0.8 18.6 ± 0.6 -8.2 ± 0.3 yes -20, -10 < -24.3 nf < 2.7 nf 40, 0 < -33.0 nf 2.8 ± 0.5 -11.6 ± 2.3 30, 0 < -28.8 nf < 3.3 nf 20, 0 -63.8 ± 7.8 -7.8 ± 0.9 3.8 ± 1.1 -8.9 ± 3.0 yes 10, 0 -93.7 ± 11.3 -8.7 ± 1.1 9.4 ± 0.7 -9.9 ± 0.8 yes 0, 0 -272.6 ± 9.8 -7.8 ± 0.3 21.6 ± 0.7 -8.0 ± 0.3 yes -10, 0 -159.8 ± 8.6 -8.0 ± 0.4 46.5 ± 1.6 -7.8 ± 0.3 yes -20, 0 < -24.0 nf 10.2 ± 0.7 -7.7 ± 0.6 40,+10 . . . . . . < 1.7 nf 30, +10 < -28.8 nf < 0.9 nf 20, +10 -52.0 ± 8.2 -8.7 ± 1.4 5.3 ± 0.4 -9.1 ± 0.7 yes 10, +10 -114.1 ± 9.7 -6.2 ± 0.6 11.1 ± 1.3 -7.1 ± 0.9 yes – 20 – Table 2—Continued Offset (l,b) 1000× I(H2CO) 〈V 〉 I(CO) 〈V 〉 Included in arcmin K km s−1 km s−1 K km s−1 km s−1 fits -70.3 ± 7.7 -10.3 ± 1.2 13.5 ± 1.1 -10.6 ± 1.0 yes 0, +10 -247.0 ± 10.1 -7.6 ± 0.3 22.8 ± 1.6 -8.5 ± 0.6 yes -10, +10 -118.9 ± 9.2 -8.1 ± 0.6 22.3 ± 2.0 -8.6 ± 0.8 yes -20, +10 < -19.2 nf < 0.9 nf 40, +20 . . . . . . < 1.2 nf 30, +20 < -25.2 nf 3.1 ± 0.4 -8.3 ± 1.7 20, +20 -42.3 ± 9.3 -7.1 ± 1.6 6.7 ± 0.7 -9.9 ± 1.0 yes -82.0 ± 7.4 -11.3 ± 1.1 nf nf 10, +20 -49.8 ± 6.9 -6.9 ± 1.0 7.0 ± 1.2 -8.0 ± 1.4 yes -63.1 ± 5.5 -10.8 ± 1.0 8.3 ± 0.9 -11.0 ± 1.4 yes 0, +20 -68.3 ± 9.7 -8.9 ± 1.3 6.1 ± 0.7 -10.8 ± 1.3 yes -10, +20 -56.4 ± 6.8 -8.6 ± 1.1 5.5 ± 0.5 -10.8 ± 1.1 yes -20, +20 < -29.7 nf < 2.7 nf 40, +30 . . . . . . < 1.6 nf 30, +30 < -27.9 nf < 1.6 nf 20, +30 < -19.5 nf 3.4 ± 0.8 -11.4 ± 2.7 10, +30 -54.1 ± 8.8 -9.3 ± 1.6 5.6 ± 0.7 -10.3 ± 1.2 yes 0, +30 < -18.0 nf 5.0 ± 0.6 -11.1 ± 1.3 -10, +30 < -27.0 nf 2.0 ± 0.4 -10.2 ± 2.2 -20, +30 < -23.4 nf < 3.0 nf – 21 – REFERENCES Bally, J., Reipurth, B., Walawender, J., & Armond, T., 2002, AJ, 124, 2152 Blair, G. 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0704.0273

Dimers on surface graphs and spin structures. II

DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II DAVID CIMASONI AND NICOLAI RESHETIKHIN Abstract. In a previous paper [3], we showed how certain orientations of the edges of a graph Γ embedded in a closed oriented surface Σ can be understood as discrete spin structures on Σ. We then used this correspondence to give a geometric proof of the Pfaffian formula for the partition function of the dimer model on Γ. In the present article, we generalize these results to the case of compact oriented surfaces with boundary. We also show how the operations of cutting and gluing act on discrete spin structures and how they change the partition function. These operations allow to reformulate the dimer model as a quantum field theory on surface graphs. Contents Introduction 2 Acknowledgements 3 1. The dimer model on graphs with boundary 3 1.1. Dimers on graphs with boundary 3 1.2. Dimers on surface graphs with boundary 4 2. Kasteleyn orientations on surface graphs with boundary 5 2.1. Kasteleyn orientations 5 2.2. Discrete spin structures 5 2.3. The Pfaffian formula for the partition function 6 3. Cutting and gluing 8 3.1. Cutting and gluing graphs with boundary 8 3.2. Cutting and gluing surface graphs with boundary 9 3.3. Cutting and gluing discrete spin structures 10 3.4. Cutting Pfaffians 12 4. Quantum field theory for dimers 13 4.1. Quantum field theory on graphs 13 4.2. Quantum field theory for dimers on graphs 13 4.3. The dimer model as the theory of free Fermions 14 5. Dimers on bipartite graphs and height functions 16 5.1. Composition cycles on bipartite graphs 16 5.2. Height functions for planar bipartite graphs 16 5.3. Height functions for bipartite surface graphs 19 5.4. The dimer quantum field theory on bipartite surface graphs 22 References 23 Date: October 22, 2018. 1991 Mathematics Subject Classification. Primary: 82B20; Secondary: 57R15. http://arxiv.org/abs/0704.0273v1 2 DAVID CIMASONI AND NICOLAI RESHETIKHIN Introduction A dimer configuration on a graph Γ is a choice of a family of edges of Γ, called dimers, such that each vertex of Γ is adjacent to exactly one dimer. Assigning weights to the edges of Γ allows to define a probability measure on the set of dimer configurations. The study of this measure is called the dimer model on Γ. Dimer models on graphs have a long history in statistical mechanics [6, 12], but also show interesting aspects involving combinatorics, probability theory [10, 4], real algebraic geometry [9, 8], etc... A remarkable fact about dimer models was discovered by P.W. Kasteleyn in the 60’s: the partition function of the dimer model can be written as a linear combination of 22g Pfaffians of N ×N matrices, where N is the number of vertices in the graph and g the genus of a closed oriented surface Σ where the graph can be embedded. The matrices are signed-adjacency matrices, the sign being determined by an orientation of the edges of Γ called a Kasteleyn orientation. If the graph is embedded in a surface of genus g, there are exactly 22g equivalence classes of Kasteleyn orientations, defining the 22g matrices. This Pfaffian formula for the partition function was proved by Kasteleyn in [6] for the cases g = 0, 1, and only stated for the general case [7]. A combinatorial proof of this fact and the exact description of coefficients for all oriented surfaces first appeared much later [11, 14]. The number of equivalence classes of Kasteleyn orientations on a graph Γ em- bedded in Σ is also equal to the number of equivalence classes of spin structures on Σ. An explicit construction relating a spin structure on a surface with a Kasteleyn orientation on a graph with dimer configuration was suggested in [10]. In [3], we in- vestigated further the relation between Kasteleyn orientations and spin structures. This allows to understand Kasteleyn orientations on a graph embedded in Σ as discrete spin structures on Σ. We also used this relation to give a geometric proof of the Pfaffian formula for closed surfaces. Our final formula can be expressed as follows: given a graph Γ embedded in a closed oriented surface Σ of genus g, the partition function of the dimer model on Γ is given by Z(Γ) = ξ∈S(Σ) Arf(ξ)Pf(Aξ(Γ)), where S(Σ) denotes the set of equivalence classes of spin structures on Σ, Arf(ξ) = ±1 is the Arf invariant of the spin structure ξ, and Aξ(Γ) is the matrix given by the Kasteleyn orientation corresponding to ξ. The first part of the present paper is devoted to the extension of the results obtained in [3] to dimer models on graphs embedded in surfaces with boundary (Sections 1 and 2). We then show how the operations of cutting and gluing act on discrete spin structures and how they change the partition function (Section 3). These operations define the structure of a functorial quantum field theory in the spirit of [2, 13], as detailed in Section 4. We then give two equivalent reformulations of the dimer quantum field theory: the “Fermionic” version, which describes the partition function of the dimer model as a Grassman integral, and the “Bosonic” version, the equivalent description of dimer models on bipartite surface graphs in terms of height functions. This special case of bipartite graphs is the subject of Section 5. DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 3 Throughout this paper, Σ is a compact surface, possibly disconnected and possi- bly with boundary, endowed with the counter-clockwise orientation. All results can be extended to the case of non-orientable surfaces, which will be done in a separate publication. We refer to [14] for a combinatorial treatment of dimer models on non-orientable surface graphs. Acknowledgements. We are grateful to J. Andersen, M. Baillif, P. Teichner and A. Vershik for inspiring discussions. We also thankfully acknowledge the hospitality of the Department of Mathematics at the University of Aarhus. The work of D.C. was supported by the Swiss National Science Foundation. This work of N.R. was partially supported by the NSF grant DMS–0307599, by the CRDF grant RUM1– 2622, by the Humboldt foundation and by the Niels Bohr research grant. 1. The dimer model on graphs with boundary 1.1. Dimers on graphs with boundary. In this paper, a graph with boundary is a finite graph Γ together with a set ∂Γ of one valent vertices called boundary vertices . A dimer configuration D on a graph with boundary (Γ, ∂Γ) is a choice of edges of Γ, called dimers , such that each vertex that is not a boundary vertex is adjacent to exactly one dimer. Note that some of the boundary vertices may be adjacent to a dimer of D, and some may not. We shall denote by ∂D this partition of boundary vertices into matched and non-matched. Such a partition will be called a boundary condition for dimer configurations on Γ. A weight system on Γ is a positive real valued function w on the set of edges of Γ. It defines edge weights on the set D(Γ, ∂Γ) of dimer configurations on (Γ, ∂Γ) w(D) = w(e), where the product is taken over all edges occupied by dimers of D. Fix a boundary condition ∂D0. Then, the Gibbs measure for the dimer model on (Γ, ∂Γ) with weight system w and boundary condition ∂D0 is given by Prob(D | ∂D0) = Z(Γ;w | ∂D0) where Z(Γ;w | ∂D0) = D:∂D=∂D0 w(D), the sum being on all D ∈ D(Γ, ∂Γ) such that ∂D = ∂D0. Let V (Γ) denote the set of vertices of Γ. The group G(Γ) = {s : V (Γ) → R>0} acts on the set of weight systems on Γ as follows: (sw)(e) = s(e+)w(e)s(e−), where e+ and e− are the two vertices adjacent to the edge e. Note that (sw)(D) = v s(v)w(D) and Z(Γ; sw | ∂D0) = v s(v)Z(Γ;w | ∂D0), both products being on the set of vertices of Γ matched by D0. Therefore, the Gibbs measure is invariant under the action of the group G(Γ). Note that the dimer model on (Γ, ∂Γ) with boundary condition ∂D0 is equivalent to the dimer model on the graph obtained from Γ by removing all edges adjacent to non-matched boundary vertices. 4 DAVID CIMASONI AND NICOLAI RESHETIKHIN Given two dimer configurations D and D′ on a graph with boundary (Γ, ∂Γ), let us define the (D,D′)-composition cycles as the connected components of the symmetric difference C(D,D′) = (D ∪D′)\(D ∩D′). If ∂D = ∂D′, then C(D,D′) is a 1-cycle in Γ with Z2-coefficients. In general, it is only a 1-cycle (rel ∂Γ). 1.2. Dimers on surface graphs with boundary. Let Σ be an oriented compact surface, not necessarily connected, with boundary ∂Σ. A surface graph with bound- ary Γ ⊂ Σ is a graph with boundary (Γ, ∂Γ) embedded in Σ, so that Γ ∩ ∂Σ = ∂Γ and the complement of Γ \ ∂Γ in Σ \ ∂Σ consists of open 2-cells. These conditions imply that the graph Γ := Γ ∪ ∂Σ is the 1-skeleton of a cellular decomposition of Note that any graph with boundary can be realized as a surface graph with boundary. One way is to embed the graph in a closed surface of minimal genus, and then to remove one small open disc from this surface near each boundary vertex of the graph. A dimer configuration on a surface graph with boundary Γ ⊂ Σ is simply a dimer configuration on the underlying graph with boundary (Γ, ∂Γ). Given two dimer configurations D and D′ on a surface graph Γ ⊂ Σ, let ∆(D,D′) denote the homology class of C(D,D′) in H1(Σ, ∂Σ;Z2). We shall say that two dimer configurations D and D′ are equivalent if ∆(D,D′) = 0 ∈ H1(Σ, ∂Σ;Z2). Note that given any three dimer configurations D,D′, and D′′ on Γ ⊂ Σ, we have the identity (1) ∆(D,D′) + ∆(D′, D′′) = ∆(D,D′′) in H1(Σ, ∂Σ;Z2). Fix a homology class β ∈ H1(Σ, ∂Σ;Z2), a dimer configuration D1 ∈ D(Γ, ∂Γ) and a boundary condition ∂D0. The associated partial partition function is defined Zβ,D1(Γ;w | ∂D0) = D:∂D=∂D0 ∆(D,D1)=β w(D), where the sum is taken over all D ∈ D(Γ, ∂Γ) such that ∂D = ∂D0 and ∆(D,D1) = The equality (1) implies that Zβ,D1(Γ;w | ∂D0) = Zβ+∆(D0,D1),D0(Γ;w | ∂D0). Furthermore, the relative homology class β′ = β + ∆(D0, D1) lies in the image of the canonical homomorphism j : H1(Σ,Z2) → H1(Σ, ∂Σ;Z2). Hence, Zβ′,D0(Γ;w | ∂D0) = α:j(α)=β′ Zα(Γ, w | ∂D0), where the sum is taken over all α ∈ H1(Σ,Z2) such that j(α) = β ′, and Zα(Γ;w | ∂D0) = D:∂D=∂D0 ∆(D,D0)=α w(D). Therefore the computation of the partition function Zβ,D1(Γ;w | ∂D0) boils down to the computation of Zα(Γ;w | ∂D0) with α ∈ H1(Σ;Z2). We shall give a Pfaffian formula for this latter partition function in the next section (see Theorem 2.4). DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 5 2. Kasteleyn orientations on surface graphs with boundary 2.1. Kasteleyn orientations. Let K be an orientation of the edges of a graph Γ, and let C be an oriented closed curve in Γ. We shall denote by nK(C) the number of times that, traveling once along C following its orientation, one runs along an edge in the direction opposite to the one given by K. A Kasteleyn orientation on a surface graph with boundary Γ ⊂ Σ is an orien- tation K of the edges of Γ = Γ ∪ ∂Σ which satisfies the following condition: for each face f of Σ, nK(∂f) is odd. Here ∂f is oriented as the boundary of f , which inherits the orientation of Σ. Using the proof of [3, Theorem 3.1], one easily checks that if ∂Σ is non-empty, then there always exists a Kasteleyn orientation on Γ ⊂ Σ. More precisely, we have the following: Proposition 2.1. Let Γ ⊂ Σ be a connected surface graph, possibly with boundary, and let C1, . . . , Cµ be the boundary components of Σ with the induced orientation. Finally, let n1, . . . , nµ be 0’s and 1’s. Then, there exists a Kasteleyn orientation on Γ ⊂ Σ such that 1 + nK(−Ci) ≡ ni (mod 2) for all i if and only if n1 + · · ·+ nµ ≡ V (mod 2), where V is the number of vertices of Γ. Proof. First, let us assume that there is a Kasteleyn orientation K on Γ ⊂ Σ such that 1 + nK(−Ci) ≡ ni for all i. Let Σ ′ be the closed surface obtained from Σ by pasting a 2-disc Di along each boundary component Ci. Let Γ ′ ⊂ Σ′ be the surface graph obtained from Γ as follows: for each i such that ni = 1, add one vertex in the interior of Di and one edge (arbitrarily oriented) between this vertex and a vertex of Ci. The result is a Kasteleyn orientation on Γ ′ ⊂ Σ′, with Σ′ closed. By [3, Theorem 3.1], the number V ′ of vertices of Γ′ is even. Hence, 0 ≡ V ′ ≡ V + n1 + · · ·+ nµ (mod 2). Conversely, assume Γ ⊂ Σ is a surface graph with n1+ · · ·+nµ ≡ V (mod 2). Paste 2-discs along the boundary components of Σ as before. This gives a surface graph Γ′ ⊂ Σ′ with Σ′ closed and V ′ even. By [3, Theorem 3.1], there exists a Kasteleyn orientation K ′ on Γ′ ⊂ Σ′. It restricts to a Kasteleyn orientation K on Γ ⊂ Σ with 1 + nK(−Ci) ≡ ni for all i. � Recall that two Kasteleyn orientations are called equivalent if one can be ob- tained from the other by a sequence of moves reversing orientations of all edges adjacent to a vertex. The proof of [3, Theorem 3.2] goes through verbatim: if non-empty, the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ is an affine H1(Σ;Z2)-space. In particular, there are exactly 2 b1(Σ) equivalence classes of Kasteleyn orientations on Γ ⊂ Σ. 2.2. Discrete spin structures. As in the closed case, any dimer configuration D on a graph Γ allows to identify equivalence classes of Kasteleyn orientations on Γ ⊂ Σ with spin structures on Σ. Indeed, [3, Theorem 4.1] generalizes as follows. Given an oriented simple closed curve C in Γ, let ℓD(C) denote the number of vertices v in C whose adjacent dimer of D sticks out to the left of C in Σ. Also, let V∂D(C) be the number of boundary vertices v in C not matched by D, and such that the interior of Σ lies to the right of C at v. 6 DAVID CIMASONI AND NICOLAI RESHETIKHIN Theorem 2.2. Fix a dimer configuration D on a surface graph with boundary Γ ⊂ Σ. Given a class α ∈ H1(Σ;Z2), represent it by oriented simple closed curves C1, . . . , Cm in Γ. If K is a Kasteleyn orientation on Γ ⊂ Σ, then the function qKD : H1(Σ;Z2) → Z2 given by qKD (α) = Ci · Cj + (1 + nK(Ci) + ℓD(Ci) + V∂D(Ci)) (mod 2) is a well-defined quadratic form on H1(Σ;Z2). Proof. Fix a dimer configuration D on (Γ, ∂Γ) and a Kasteleyn orientation K on Γ ⊂ Σ. Let Σ′ be the surface (homeomorphic to Σ) obtained from Σ by adding a small closed collar to its boundary. For every vertex v of ∂Γ that is not matched by a dimer of D, add a vertex v′ near v in the interior of the collar and an edge between v and v′. Let us denote by Γ′ the resulting graph in Σ′. Putting a dimer on each of these additional edges, and orienting them arbitrarily, we obtain a perfect matching D′ and an orientation K ′ on Γ′. Although Γ′ ⊂ Σ′ is not strictly speaking a surface graph, all the methods of [3, Section 4] apply. Indeed, Kuperberg’s vector field defined near Γ′ clearly extends continuously to the collar. As in the closed case, it also extends to the faces with even index singularities. Using the perfect matching D′ on Γ′, we obtain a vector field f(K ′, D′) with even index singularities, which determines a spin structure ξf(K′,D′) on Σ ′. Johnson’s theorem [5] holds for surfaces with boundary, so this spin structure defines a quadratic form q on H1(Σ ′;Z2) = H1(Σ;Z2). If C is a simple close curve in Γ ′ ⊂ Σ′, then q([C]) + 1 = nK (C) + ℓD′(C) as in the closed case. The proof is completed using the equalities nK (C) = nK(C) and ℓD′(C) = ℓD(C) + V∂D(C). � Since Johnson’s theorem holds true for surfaces with boundary and [3, Proposi- tion 4.2] easily extends, we have the following corollary. Corollary 2.3. Let Γ ⊂ Σ be a surface graph, non-necessarily connected, and pos- sibly with boundary. Any dimer configuration D on Γ ⊂ Σ induces an isomorphism of affine H1(Σ;Z2)-spaces ψD : K(Γ ⊂ Σ) −→ S(Σ) from the set of equivalence classes of Kasteleyn orientations on Γ ⊂ Σ onto the set of spin structures on Σ. Furthermore, ψD − ψD′ is equal to the Poincaré dual of ∆(D,D′). In particular, ψD = ψD′ if and only if D and D ′ are equivalent dimer configurations. � 2.3. The Pfaffian formula for the partition function. Let Γ be a graph, not necessarily connected, and possibly with boundary, endowed with a weight system w. Realize Γ as a surface graph Γ ⊂ Σ, and fix a Kasteleyn orientation K on it. The Kasteleyn coefficient associated to an ordered pair (v, v′) of distinct vertices of Γ is the number aKvv′ = εKvv′(e)w(e), where the sum is on all edges e in Γ between the vertices v and v′, and εKvv′(e) = 1 if e is oriented by K from v to v′; −1 otherwise. DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 7 One also sets aKvv = 0. Let us fix a boundary condition ∂D0 and enumerate the matched vertices of Γ by 1, 2, . . . , 2n. Then, the corresponding coefficients form a 2n× 2n skew-symmetric matrix AK(Γ;w | ∂D0) = A K called the Kasteleyn matrix . Let D be a dimer configuration on (Γ, ∂Γ) with ∂D = ∂D0, given by edges e1, . . . , en matching vertices iℓ and jℓ for ℓ = 1, . . . , n. Let σ be the permutation (1, . . . , 2n) 7→ (i1, j1, . . . , in, jn), and set εK(D) = (−1)σ εKiℓjℓ(eℓ), where (−1)σ denotes the sign of σ. Note that εK(D) does not depend on the choice of σ, but only on the dimer configuration D. Finally, recall that the Arf invariant of a (possibly degenerate) quadratic form q on H := H1(Σ;Z2) is defined by Arf(q) = (−1)q(α). If there is a component γ of ∂Σ such that q(γ) 6= 0, then one easily checks that Arf(q) = 0. On the other hand, if q(γ) = 0 for all boundary components γ of Σ, then Arf(q) takes the values +1 or −1. Theorem 2.4. Let Γ ⊂ Σ be a surface graph, not necessarily connected, and possi- bly with boundary. Let b1(Σ) denote the dimension of H1(Σ;Z2), and let g denote the genus of Σ. Then, Zα(Γ;w | ∂D0) = 2b1(Σ) (−1)q εK(D0)Pf(A for any α ∈ H1(Σ;Z2), and Z(Γ;w | ∂D0) = Arf(qKD0)ε K(D0)Pf(A where both sums are over the 2b1(Σ) equivalence classes of Kasteleyn orientations on Γ ⊂ Σ. Furthermore, Arf(qKD0)ε K(D0) does not depend on D0. Proof. First note that if the theorem holds for two surface graphs, then it holds for their disjoint union. Therefore, it may be assumed that Σ is connected. The first formula follows from Theorem 2.2: the proof of Theorem 4 and the first half of the proof of Theorem 5 of [3] generalize verbatim to the case with (possible) boundary. The second formula can be obtained from the first one by summing over all α ∈ H1(Σ;Z2). However, this requires some cumbersome computations, so let us give another proof of this equality. As mentioned in Section 1, the dimer model on (Γ, ∂Γ) with boundary condition ∂D0 is equivalent to the dimer model on the graph Γ′ = Γ′(∂D0) obtained from Γ by removing all edges adjacent to non- matched boundary vertices. Let w′ denote the restriction of w to Γ′. If Γ ⊂ Σ is a surface graph with boundary, then Γ′ ⊂ Σ′ is a surface graph, where Σ′ is the closed oriented surface obtained from Σ by gluing discs along all boundary components. By [3, Theorem 5.3], Z(Γ;w | ∂D0) = Z(Γ ′;w′) = Arf(qK (D0)Pf(A K′(Γ′;w′)), 8 DAVID CIMASONI AND NICOLAI RESHETIKHIN the sum being on all equivalence classes of Kasteleyn orientations on Γ′ ⊂ Σ′. Such a Kasteleyn orientation K ′ extends uniquely to a Kasteleyn orientation K on Γ ⊂ Σ such that qKD0(γ) = 0 for all boundary component γ of Σ. Furthermore, (D0) = ε K(D0) and A K′(Γ′;w′) = AK(Γ;w | ∂D0). Since Arf(q ) = 0 for all other Kasteleyn orientations, the theorem follows. � 3. Cutting and gluing 3.1. Cutting and gluing graphs with boundary. Let (Γ, ∂Γ) be a graph with boundary, and let us fix an edge e of Γ. Let (Γ{e}, ∂Γ{e}) denote the graph with boundary obtained from (Γ, ∂Γ) as follows: cut the edge e in two, and set ∂Γ{e} = ∂Γ ∪ {v′, v′′}, where v′ and v′′ are the new one valent vertices. Iterating this procedure for some set of edges E leads to a graph with boundary (ΓE, ∂ΓE), which is said to be obtained by cutting (Γ, ∂Γ) along E. Note that a dimer configuration D ∈ D(Γ, ∂Γ) induces an obvious dimer config- uration DE ∈ D(ΓE, ∂ΓE): cut in two the dimers of D that belong to E. A weight system w on Γ induces a family of weight systems (wt )t on ΓE indexed by t : E → R>0, as follows: if e is an edge of Γ which does not belong to E, set (e) = w(e); if e ∈ E is cut into two edges e′, e′′ of ΓE, set w (e′) = t(e)w(e)1/2 and (e′′) = t(e)−1w(e)1/2. Note that this family of weight systems is an orbit under the action of the subgroup of G(ΓE) consisting of elements s such that s(v) = 1 for all v ∈ V (Γ) and s(v′) = s(v′′) whenever v′, v′′ ∈ ∂ΓE come from the same edge of Let us now formulate how the cutting affects the partition function. The proof is straightforward. Proposition 3.1. Fix a boundary condition ∂D0 on (Γ, ∂Γ) and a set E of edges of Γ. Then, given any parameter t : E → R>0, Z(Γ;w | ∂D0) = Z(ΓE;w | ∂DI0), where the sum is taken over all subsets I of E and ∂DI0 is the boundary condition on (ΓE, ∂ΓE) induced by ∂D0 and I: a vertex of ∂ΓE is matched in ∂D 0 if and only if it is matched in ∂D0 or it comes from an edge in I. � The operation opposite to cutting is called gluing: pick a pair of boundary vertices of Γ, and glue the adjacent edges e′, e′′ along these vertices into a single edge e. In order for the result to be a graph, it should be assumed that e′ and e′′ are different edges of Γ. We shall denote by (Γϕ, ∂Γϕ) the graph obtained by gluing (Γ, ∂Γ) according to a pairing ϕ of several vertices of ∂Γ. Note that a dimer configuration D ∈ D(Γ, ∂Γ) induces a dimer configuration Dϕ ∈ D(Γϕ, ∂Γϕ) if and only if the boundary condition ∂D on ∂Γ is compatible with ϕ, i.e: ϕ relates matched vertices with matched vertices. Obviously, a dimer configuration DE is compatible with the pairing ϕ which glues back the edges of E, and (DE)ϕ = D on ((ΓE)ϕ, (∂ΓE)ϕ) = (Γ, ∂Γ). An edge weight system w on Γ induces an edge weight system wϕ on Γϕ as follows: wϕ(e) = w(e) if e is an edge of Γ; w(e′)w(e′′) if e is obtained by gluing the edges e′ and e′′ of Γ. DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 9 If E is a set of edges of Γ and ϕ is the pairing which glues back these edges, then )ϕ = w for any t : E → R>0. The effect of gluing on the partition function is best understood in the language of quantum field theory. We therefore postpone its study to Section 4. 3.2. Cutting and gluing surface graphs with boundary. Let Γ ⊂ Σ be a surface graph with boundary. Let C be a simple curve in Σ which is “in general position” with respect to Γ, in the following sense: (i) it is disjoint from the set of vertices of Γ; (ii) it intersects the edges of Γ transversally; (iii) its intersection with any given face of Σ is connected. Let ΣC be the surface with boundary obtained by cutting Σ open along C. Also, let ΓC := ΓE(C) be the graph with boundary obtained by cutting (Γ, ∂Γ) along the set E(C) of edges of Γ which intersect C, as illustrated in Figure 1. Γ ⊂ Σ ΓC ⊂ ΣC Figure 1. Cutting a surface graph Γ ⊂ Σ along a curve C. Obviously, ΓC ⊂ ΣC is a surface graph with boundary. We will say that it is obtained by cutting Γ ⊂ Σ along C. Abusing notation, we shall write wtC for the weight system wt E(C) on ΓC . A class β ∈ H1(Σ, ∂Σ;Z2) induces βC ∈ H1(ΣC , ∂ΣC ;Z2) via H1(Σ, ∂Σ;Z2) → H1(Σ, ∂Σ ∪N(C);Z2) ≃ H1(ΣC , ∂ΣC ;Z2). Here N(C) denotes a neighborhood of C in Σ, the first homomorphism is induced by inclusion, and the second one is the excision isomorphism. Note that given any two dimers configurations D and D′ on Γ ⊂ Σ, ∆(DC , D C) = ∆(D,D ′)C in H1(ΣC , ∂ΣC ;Z2). This easily leads to the following refinement of Proposition 3.1. Proposition 3.2. Fix β ∈ H1(Σ, ∂Σ;Z2), D ′ ∈ D(Γ, ∂Γ), and a boundary condi- tion ∂D0 on (Γ, ∂Γ). Then, given any parameter t : E(C) → R>0, Zβ,D′(Γ;w | ∂D0) = I⊂E(C) ZβC ,D′C (ΓC ;w C | ∂D where the sum is taken over all subsets I of E(C) and ∂DI0 is the boundary condition on (ΓC , ∂ΓC) induced by ∂D0 and I. � Let us now define the operation opposite to cutting a surface graph with bound- ary. Pick two closed connected subsets M1,M2 of ∂Σ, which are not points, and satisfy the following properties: (i) M1 ∩M2 ⊂ ∂M1 ∪ ∂M2 and ∂M1 ∪ ∂M2 is disjoint from ∂Γ; 10 DAVID CIMASONI AND NICOLAI RESHETIKHIN (ii) the intersection of each given face of Σ with M1 ∪M2 is connected; (iii) there exists an orientation-reversing homeomorphism ϕ : M1 → M2 which induces a bijection M1 ∩ ∂Γ → M2 ∩ ∂Γ such that for all v in M1 ∩ ∂Γ, v and ϕ(v) are not adjacent to the same edge of Γ. Let Γϕ ⊂ Σϕ be obtained from the surface graph Γ ⊂ Σ by identifying M1 and M2 via ϕ and removing the corresponding vertices of Γ. This is illustrated in Figure 2. By the conditions above, the pair Γϕ ⊂ Σϕ remains a surface graph. It is said to be obtained by gluing Γ ⊂ Σ along ϕ. Γ ⊂ Σ Γϕ ⊂ Σϕ −→ M2 Figure 2. Gluing a surface graph Γ ⊂ Σ along ϕ : M1 → M2. Note that any surface graph ΓC ⊂ ΣC obtained by cutting Γ ⊂ Σ along some curve C in general position with respect to Γ satisfies the conditions listed above. Furthermore, (ΓC)ϕ ⊂ (ΣC)ϕ = Γ ⊂ Σ, where ϕ is the obvious homeomorphism identifying the two closed subsets of ∂ΣC coming from C. Conversely, if C denotes the curve in Σϕ given by the identification ofM1 andM2 via ϕ, then it is in general position with respect to Γϕ, and (Γϕ)C ⊂ (Σϕ)C . 3.3. Cutting and gluing discrete spin structures. Let Γ ⊂ Σ be a surface graph with boundary, and let C be a simple curve in Σ in general position with respect to Γ. As noted above, any dimer configurationD on (Γ, ∂Γ) induces a dimer configuration DC on (ΓC , ∂ΓC). If two dimer configurations D,D ′ ∈ D(Γ, ∂Γ) are equivalent, then DC , D C ∈ D(ΓC , ∂ΓC) are equivalent as well: ∆(DC , D C) = ∆(D,D ′)C = 0 ∈ H1(ΣC , ∂ΣC ;Z2). A Kasteleyn orientation K on Γ ⊂ Σ induces a Kasteleyn orientation KC on ΓC ⊂ ΣC as follows. Let KC be equal to K on all edges of ΓC coming from edges of Γ. For all the new edges of ΓC , there is a unique orientation which satisfies the Kasteleyn condition, since each face of Σ is crossed at most once by C. One easily checks that if K and K ′ are equivalent Kasteleyn orientations, then KC and K are also equivalent. Hence, there is a well-defined operation of cutting discrete spin structures on a surface with boundary. This is not a surprise. Indeed, the inclusion ΣC ⊂ ΣC ∪ N(C) = Σ induces a homomorphism i∗ : H1(ΣC ;Z2) → H1(Σ;Z2). The assignment q 7→ qC = q ◦ i∗ defines a map from the quadratic forms on H1(Σ;Z2) to the quadratic forms on H1(ΣC ;Z2), which is affine over the restriction homomorphism i ∗ : H1(Σ;Z2) → H1(ΣC ;Z2). By Johnson’s theorem, it induces an affine map between the sets of DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 11 spin structures S(Σ) → S(ΣC). By Corollary 2.3, there is a unique map K(Γ ⊂ Σ) → K(ΓC ⊂ ΣC) which makes the following diagram commute: (2) K(Γ ⊂ Σ) ∼= ψD // K(ΓC ⊂ ΣC) ∼= ψDC S(Σ) // S(ΣC). This map is nothing but [K] 7→ [KC ]. Now, let K be a Kasteleyn orientation on a surface graph Γ ⊂ Σ, and let ϕ : M1 →M2 be an orientation-reversing homeomorphism between two closed con- nected subsets in ∂Σ, as described above. We shall say that a Kasteleyn orientation K on Γ ⊂ Σ is compatible with ϕ if the following conditions hold: (i) whenever two edges e′, e′′ of Γ are glued into a single edge e of Γϕ, the orientation K agrees on e′ and e′′, giving an orientation Kϕ on e; (ii) the induced orientation Kϕ is a Kasteleyn orientation on Γϕ ⊂ Σϕ. The Kasteleyn orientationKϕ on Γϕ ⊂ Σϕ is said to be obtained by gluing K along Given any Kasteleyn orientation K on Γ ⊂ Σ, the induced orientation KC on ΓC ⊂ ΣC is compatible with the map ϕ such that (ΣC)ϕ = Σ; furthermore, (KC)ϕ is equal to K. Conversely, if K is a Kasteleyn orientation on Γ ⊂ Σ which is compatible with ϕ, and C denotes the curve in Σϕ given by the identification of M1 and M2 via ϕ, then (Kϕ)C is equal to K. With these notations, any dimer configuration D on Γ which is compatible with ϕ satisfies (Dϕ)C = D. Therefore, diagram (2) gives K(Γϕ ⊂ Σϕ) ∼= ψDϕ // K(Γ ⊂ Σ) ∼= ψD S(Σϕ) // S(Σ), where both horizontal maps are affine over i∗ : H1(Σϕ;Z2) → H 1(Σ;Z2). Under- standing the gluing of Kasteleyn orientations (up to equivalence) now amounts to understanding the restriction homomorphism i∗. Using the exact sequence of the pair (Σϕ,Σ), one easily checks the following results: – The restriction homomorphism i∗ is injective, unlessM1 andM2 are disjoint and belong to the same connected component of Σ. In this case, the kernel of i∗ has dimension 1. – The homomorphism i∗ is onto unless M1 ∪ M2 is a 1-cycle and the cor- responding connected component of Σϕ is not closed. In this case, the cokernel of i∗ has dimension 1. This leads to the four following cases. Fix a Kasteleyn orientation K on Γ ⊂ Σ. (1) If i∗ is an isomorphism, then there exist a Kasteleyn orientation K ′ equiv- alent to K which is compatible with ϕ. Furthermore, the assignment [K] 7→ [K ′ϕ] gives a well-defined map between K(Γ ⊂ Σ) and K(Γϕ ⊂ Σϕ). (2) If i∗ is onto but not injective, then there exist K ′,K ′′ ∼ K which are compatible with ϕ, inducing two distinct well-defined maps [K] 7→ [K ′ϕ] and [K] 7→ [K ′′ϕ] between K(Γ ⊂ Σ) and K(Γϕ ⊂ Σϕ). 12 DAVID CIMASONI AND NICOLAI RESHETIKHIN (3) If i∗ is injective but not onto, then M1 ∪M2 is a 1-cycle, oriented as part of the boundary of Σ. There exist K ′ ∼ K which is compatible with ϕ if and only if the following condition holds: nK(M1) + n K(M2) ≡ 0 (mod 2) if M1 and M2 are disjoint; nK(M1 ∪M2) ≡ 1 (mod 2) otherwise. (Note that this condition only depends on the equivalence class of K.) In this case, it induces a well-defined class [K ′ϕ] in K(Γϕ ⊂ Σϕ). (4) Finally, assume i∗ is neither onto nor injective. If K satisfies the condition above, then there exist K ′,K ′′ ∼ K which are compatible with ϕ, inducing two well-defined maps [K] 7→ [K ′ϕ] and [K] 7→ [K ϕ]. On the other hand, if K does not satisfy the condition above, then it does not contain any representative which is compatible with ϕ. 3.4. Cutting Pfaffians. Let us conclude this section with one last observation. Let Γ ⊂ Σ be a surface graph with boundary, and let C be a simple curve in Σ. The equality Z(Γ;w | ∂D0) = I⊂E(C) Z(ΓC ;w C | ∂D of Proposition 3.1 can be understood as the Taylor series expansion of the function Z(Γ;w | ∂D0) in the variables (w(e))e∈E(C). Clearly, if E(C) = {ei1 , . . . , eik}, then w(eiℓ) ∂kZ(Γ;w | ∂D0) ∂w(ei1) · · · ∂w(eik) (0) = Z(ΓC ;w C | ∂D By Theorem 2.4, the partition function Z(Γ;w | ∂D0) can be expressed as a lin- ear combination of Pfaffians of matrices AK(Γ;w | ∂D0) depending on Kasteleyn orientations K of Γ ⊂ Σ such that qKD0(γ) = 0 for all boundary component γ of Σ. Recall that any such orientation K extends to a Kasteleyn orientation KC on ΓC ⊂ ΣC . Furthermore, all equivalence classes of Kasteleyn orientations such that (D0)C (γ) = 0 for all boundary component γ of ΣC are obtained in this way. (This follows from the fact that the map [K] 7→ [KC ] is affine over the restriction homo- morphism.) Finally, the partition function Z(ΓC ;w C | ∂D0) can also be expressed as a linear combination of Pfaffians of matrices AKC (ΓC ;w C | ∂D0) via Theorem Gathering all these equations, we obtain a relation between the Pfaffian of the matrix AK(Γ;w | ∂D0) and the Pfaffian of the matrix A KC (ΓC ;w C | ∂D0). This relation turns out to be exactly the equation below, a well-known property of Pfaf- fians. Proposition 3.3. Let A = (aij) be a skew-symmetric matrix of size 2n. Given an ordered subset I of the ordered set α = (1, . . . , 2n), let AI denote the matrix obtained from A by removing the ith row and the ith column for all i ∈ I. Then, for any ordered set of indices I = (i1, j1, . . . , ik, jk), ∂kPf(A) ∂ai1j1 · · · ∂aikjk = (−1)σ(I)Pf(AI), where (−1)σ(I) denote the signature of the permutation which sends α to the ordered set I(α\I). � DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 13 4. Quantum field theory for dimers 4.1. Quantum field theory on graphs. Let (Γ, ∂Γ) be a graph with boundary, and let us assume that each vertex v in ∂Γ is oriented, that is, endowed with some sign εv. In the spirit of the Atiyah-Segal axioms for a (0 + 1)-topological quantum field theory [2, 13], let us define a quantum field theory on graphs as the following assignment: (1) Fix a finite dimensional complex vector space V . (2) To the oriented boundary ∂Γ, assign the vector space Z(∂Γ) = εv=+1 εv=−1 where V ∗ denotes the vector space dual to V . (3) To a finite graph Γ with oriented boundary ∂Γ and weight system w, assign some vector Z(Γ;w) ∈ Z(∂Γ), with Z(∅;w) = 1 ∈ C = Z(∅). Note that any orientation preserving bijection f : ∂Γ → ∂Γ′ induces an isomor- phism Z(f) : Z(∂Γ) → Z(∂Γ′) given by permutation of the factors. This assign- ment is functorial: if g : ∂Γ′ → ∂Γ′′ is another orientation preserving bijection, then Z(g ◦ f) = Z(g) ◦Z(f). Finally, if f : ∂Γ → ∂Γ′ extends to a homeomorphism F : Γ → Γ′, then Z(f) maps Z(Γ) to Z(Γ′). Note also that Z(−∂Γ) = Z(∂Γ)∗, and that Z(∂Γ ⊔ ∂Γ′) = Z(∂Γ)⊗ Z(∂Γ′). The main point is that we require the following gluing axiom. Let Γ be a graph with oriented boundary ∂Γ, such that there exists two disjoint subsets X1, X2 of ∂Γ and an orientation reversing bijection ϕ : X1 → X2 (i.e. εϕ(v) = −εv for all v ∈ X1). Obviously, ϕ induces a linear isomorphism Z(ϕ) : Z(X1) → Z(X2) ∗. Let Γϕ denote the graph with boundary ∂Γϕ = ∂Γ \ (X1 ∪X2) obtained by gluing Γ according to ϕ, and let wϕ be the corresponding weight system on Γϕ (recall Section 3.1). Let Bϕ denote the composition Z(∂Γ) = Z(∂Γϕ)⊗ Z(X1)⊗ Z(X2) → Z(∂Γϕ)⊗ Z(X2) ∗ ⊗ Z(X2) → Z(∂Γϕ), where the first homomorphism is given by id⊗Z(ϕ)⊗ id, and the second is induced by the natural pairing Z(X2) ∗ ⊗ Z(X2) → C. We require that Bϕ(Z(Γ;w)) = Z(Γϕ;wϕ). Remark. In the same spirit, one can define a quantum field theory on surface graphs . Here, the vector Z(Γ ⊂ Σ;w) ∈ Z(∂Γ) might depend on the realization of Γ as a surface graph Γ ⊂ Σ, and the gluing axiom concerns gluing of surface graphs, as defined in Section 3.2. 4.2. Quantum field theory for dimers on graphs. Let us now explain how the dimer model on weighted graphs with boundary defines a quantum field theory. As vector space V , choose the 2-dimensional complex vector space with fixed basis a0, a1. Let α0, α1 denote the dual basis in V ∗. To a finite graph Γ with oriented boundary ∂Γ and weight system w, assign Z(Γ;w) = Z(Γ;w | ∂D) a(∂D) ∈ Z(∂Γ), where the sum is on all possible boundary conditions ∂D on ∂Γ, and a(∂D) = εv=+1 aiv(∂D) ⊗ εv=−1 αiv(∂D) ∈ Z(∂Γ). 14 DAVID CIMASONI AND NICOLAI RESHETIKHIN Here, iv(∂D) = 1 if the vertex v is matched by ∂D, and iv(∂D) = 0 otherwise. Let us check the gluing axiom. First note that Bϕ(a(∂D)) = 0 unless ∂D is compatible with ϕ (i.e: unless ϕ(v) is matched in ∂D if and only if v is matched in ∂D). In such a case, B(a(∂D)) = a(∂D|∂Γϕ), where ∂D|∂Γϕ denotes the restriction of the boundary condition ∂D to ∂Γϕ ⊂ ∂Γ. All the possible boundary conditions ∂Dϕ on ∂Γϕ are given by such restrictions. Therefore, Bϕ(Z(Γ;w)) = ∂D⊃∂Dϕ Z(Γ;w | ∂D) a(∂Dϕ), the interior sum being on all boundary conditions ∂D on ∂Γ that are compatible with ϕ, and such that ∂D|∂Γϕ = ∂Dϕ. By definition, ∂D⊃∂Dϕ Z(Γ;w | ∂D) = D:∂D⊃∂Dϕ w(D) = Z(Γϕ;wϕ | ∂Dϕ). Therefore, the gluing axiom is satisfied. 4.3. The dimer model as the theory of free Fermions. Let W be an n- dimensional vector space. The choice of an ordered basis in W induces an isomor- phism between its exterior algebra W = ⊕nk=0 W and the algebra generated by elements φ1, . . . , φn with defining relations φiφj = −φjφi. This space is known as the Grassman algebra generated by φ1, . . . , φn. The choice of an ordered basis in W also defines a basis in the top exterior power of W . The integral over the Grassman algebra of W of an element a ∈ W is the coordinate of a in the top exterior power of W with respect to this basis. It is denoted by a dφ. There is a scalar product on the Grassman algebra generated by φ1, . . . , φn; it is given by the Grassman integral (3) < F,G >= F (φ)G(ψ)dφdψ. Note that the monomial basis is orthonormal with respect to this scalar product. One easily shows (see e.g. the Appendix to [3]) that the Pfaffian of a skew symmetric matrix A = (aij) can be written as Pf(A) = i,j=1 φiaijφj Let us now use this to reformulate the quantum field theory of dimers in terms of Grassman integrals. Let Γ ⊂ Σ be a (possibly disconnected) surface graph, possibly with boundary. Let us fix a numbering of the vertices of Γ, a boundary condition ∂D0 on ∂Γ and a Kasteleyn orientation K on Γ ⊂ Σ. Let a ij be the Kasteleyn coefficient associated to K and the vertices i, j of Γ (recall Section 2). By Theorem 2.4 and the identity above, Z(Γ;w | ∂D0) = Arf(qKD0)ε K(D0) i,j∈V (D0) dφ∂D0 , where the sum is over all 2b1(Σ) equivalence classes of Kasteleyn orientations on Γ ⊂ Σ, V (D0) denotes the set of vertices of Γ that are matched by D0, and dφ∂D0 = DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 15 ∧i∈V (D0)dφi. This leads to the formula Z(Γ;w | ∂D0) = i,j∈V (D0) DK∂D0φ, where DK∂D0φ = Arf(q )εK(D0) d∂D0φ. Let us point out that this measure does not depend on the choice of D0, but only on the induced boundary condition ∂D0. Now, the numbering of the vertices of Γ gives a numbering of the vertices of ∂Γ. This induces a linear isomorphism between Z(∂Γ) and the Grassman algebra generated by (φi)i∈∂Γ. The image of the partition function under this isomorphism is the following element of the Grassman algebra of boundary vertices: Z(Γ;w) = Z(Γ;w | ∂D0) i∈V (∂D0) (∂Γ), where V (∂D0) = V (D0) ∩ ∂Γ. This leads to Z(Γ;w) = i,j∈V (D0) DK∂D0φ i∈V (∂D0) i,j∈V (Γ) where DKφ = Arf(qKD0)ε K(D0) ∧i/∈∂Γ dφi. This measure depends only on K, but not on D0. We can now formulate the dimer model as the theory of free (Gaussian) Fermions: (1) To the boundary of Γ ⊂ Σ, we assign (∂Γ), the Grassman algebra gener- ated by the ordered set ∂Γ; (2) To a surface graph Γ ⊂ Σ with ordered set of vertices V (Γ) and weight system w, we assign the element Z(Γ ⊂ Σ;w) of (∂Γ) given by Z(Γ ⊂ Σ;w) = i,j∈V (Γ) where the sum is over all 2b1(Σ) equivalence classes of Kasteleyn orientations on Γ ⊂ Σ, and DKφ = Arf(qKD0)ε K(D0) ∧i/∈∂Γ dφi. The gluing axiom now takes the following form. Let Γϕ ⊂ Σϕ denote the surface graph with boundary obtained by gluing Γ ⊂ Σ along some orientation-reversing homeomorphism ϕ : M1 → M2 (see Section 3.2). Recall that ϕ induces a bijection between the two disjoint sets X1 = ∂Γ ∩M1 and X2 = ∂Γ ∩M2. Therefore, it induces an isomorphism Z(ϕ) : (X1) → (X2). Consider the map Bϕ given by the composition (∂Γ) = (∂Γϕ)⊗ (X1)⊗ (X2) → (∂Γϕ)⊗ (X2) → (∂Γϕ). Here, the first homomorphism is given by id⊗ (h ◦Z(ϕ))⊗ id, where h : (X2) → ∗ is the isomorphism induced by the scalar product (3). Then, we require Bϕ(Z(Γ ⊂ Σ;w)) = Z(Γϕ ⊂ Σϕ;wϕ). We already know that this equality holds. Indeed, Z(Γ ⊂ Σ;w) just depends on (Γ, w), and the formula above is nothing but the gluing axiom for Z(Γ;w) translated in the formalism of Grassman algebras. However, it can also be proved from scratch using the results of Section 3.3 together with well-known properties of Pfaffians. 16 DAVID CIMASONI AND NICOLAI RESHETIKHIN 5. Dimers on bipartite graphs and height functions 5.1. Composition cycles on bipartite graphs. Recall that a bipartite structure on a graph Γ is a partition of its set of vertices into two groups, say blacks and whites, such that no edge of Γ joins two vertices of the same group. Equivalently, a bipartite structure can be regarded as a 0-chain v black v white v ∈ C0(Γ;Z). A bipartite structure induces an orientation on the edges of Γ, called the bipartite orientation: simply orient all the edges from the white vertices to the black ones. Using this orientation, a dimer configuration D ∈ D(Γ, ∂Γ) can now be regarded as a 1-chain with Z-coefficients e ∈ C1(Γ;Z) such that ∂D = β in C0(Γ, ∂Γ;Z) = C0(Γ;Z)/C0(∂Γ;Z). Therefore, given two dimer configurations D,D′ on Γ, their difference D −D′ is a 1-cycle (rel ∂Γ) with Z-coefficients, denoted by C(D,D′). Its connected components are called (D,D′)- composition cycles . In short, a bipartite structure on a graph allows to orient the composition cycles. 5.2. Height functions for planar bipartite graphs. Let us now assume that the bipartite graph Γ is planar without boundary, i.e. that it can be realized as a surface graph Γ ⊂ S2. Let X denote the induced cellular decomposition of the 2-sphere, which we endow with the counter-clockwise orientation. Since H1(X ;Z) = H1(S 2;Z) = 0, the 1-cycle C(D,D′) is a 1-boundary, so there exists σD,D′ ∈ C2(X ;Z) such that ∂σD,D′ = C(D,D ′). Let hD,D′ ∈ C 2(X ;Z) be given by the equality σD,D′ = f∈F (X) hD,D′(f) f ∈ C2(X ;Z), where the sum is over all faces of X . The cellular 2-cochain hD,D′ is called a height function associated to D,D′. Since H2(X ;Z) = H2(S 2;Z) = Z, the 2-chain σD,D′ is uniquely defined by D,D′ up to a constant, and the same holds for hD,D′ . Hence, one can normalize all height functions by setting hD,D′(f0) = 0 for some fixed face f0. This is illustrated in Figure 3. Alternatively, hD,D′ can be defined as the only h ∈ C 2(X ;Z) such that h(f0) = 0 and h increases by 1 when a (D,D′)-composition cycle is crossed in the positive direction (left to right as we cross). It follows that for any height function h and any two 2-cells f1 and f2, |h(f1)− h(f2)| ≤ d(f1, f2), where d(f1, f2) is the distance between f1 and f2 in the dual graph, i.e. the minimal number of edges crossed by a path connecting an point inside f1 with a point inside f2. This can be regarded as a Lipschitz property of height functions. Note also that for any three dimer configurations D, D′ and D′′ on Γ, the following cocycle equality holds: hD,D′ + hD′,D′′ = hD,D′′ . DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 17 Figure 3. An example of a bipartite planar graph with two dimer configurations D (solid) and D′ (traced lines). The corresponding height function hD,D′ (where f0 is the outer face) and (D,D composition cycles are pictured on the right hand side. The Lipschitz condition stated above leads to the following definition. Given a fixed 2-cell f0 of the cellular decomposition X induced by Γ ⊂ S 2, set H(X, f0) = {h ∈ C 2(X ;Z) |h(f0) = 0 and |h(f1)− h(f2)| ≤ d(f1, f2) ∀f1, f2}. Given h ∈ H(X, f0), let C(h) denote the oriented closed curves formed by the set of oriented edges e of Γ such that h increases its value by 1 when crossing e in the positive direction. (In other words, C(h) = ∂σ, where σ ∈ C2(X ;Z) is dual to h ∈ C2(X ;Z).) Obviously, there is a well-defined map D(Γ)×D(Γ) → H(X, f0), (D,D ′) 7→ hD,D′ with C(hD,D′) = C(D,D ′). However, this map is neither injective nor surjective in general. Indeed, the number of preimages of a given h is equal to the number of dimer configurations on the graph obtained from Γ by removing the star of C(h). Depending on Γ ⊂ S2, this number can be zero, or arbitrarily large. To obtain a bijection, we proceed as follows. Fix a dimer configuration D0 on Γ. Let C(D0) denote the set of all C ⊂ Γ consisting of disjoint oriented simple 1-cycles, such that the following condition holds: for all e ∈ D0, either e is contained in C or e is disjoint from C. Finally, set HD0 (X, f0) = {h ∈ H(X, f0) |C(h) ∈ C(D0)}. Proposition 5.1. Given any h ∈ HD0(X, f0), there is unique dimer configuration D ∈ D(Γ) such that hD,D0 = h. Furthermore, given any two dimer configurations D0, D1 on Γ, we have a canonical bijection HD0(X, f0) → HD1(X, f0) given by h 7→ h+ hD0,D1 . Proof. One easily checks that the assignment D 7→ C(D,D0) defines a bijection D(Γ) → C(D0). Furthermore, there is an obvious bijection HD0(X, f0) → C(D0) given by h 7→ C(h). This induces a bijection D(Γ) → HD0(X, f0) and proves the first part of the proposition. The second part follows from the first one via the cocycle identity hD,D0 + hD0,D1 = hD,D1 . � 18 DAVID CIMASONI AND NICOLAI RESHETIKHIN Let us now consider an edge weight system w on the bipartite planar graph Γ. Recall that the Gibbs measure of D ∈ D(Γ) is given by Prob(D) = Z(Γ;w) where w(D) = e∈D w(D) and Z(Γ;w) = D∈D(Γ)w(D). Let us now fix a dimer configurationD0 and a face f0 ofX , and use the bijectionD(Γ) → HD0(X, f0) given by D 7→ hD,D0 to translate this measure into a probability measure on HD0(X, f0). To do so, we shall need the following notations: given an oriented edge e of Γ, wβ(e) = w(e) if the orientation on e agrees with the bipartite orientation; w(e)−1 otherwise. This defines a group homomorphism wβ : C1(X ;Z) → R>0. Finally, given any f ∈ F (X), set qf = wβ(∂f), where ∂f is oriented as the boundary of the counter-clockwise oriented face f . This number qf is called the volume weight of the face f . Proposition 5.2. The Gibbs measure on D(Γ) given by the edge weight system w translates into the following probability measure on HD0(X, f0): ProbD0(h) = ZD0,f0(X, q) where q(h) = f∈F (X) f and ZD0,f0(X ; q) = h∈HD0(X,f0) q(h). Furthermore, this measure is independant of the choice of f0. Finally, the bijection HD0(X, f0) → HD1(X, f0) given by h 7→ h+hD0,D1 is invariant with respect to the measures ProbD0 and ProbD1 . Proof. For any D ∈ D(Γ), we have w(D)w(D0) w(e)−1 = wβ(C(D,D0)) = wβ(∂σD,D0) = wβ f∈F (X) hD,D0(f)∂f f∈F (X) wβ(∂f) hD,D0(f) = f∈F (X) hD,D0 (f) f = q(hD,D0). The proposition follows easily from this equality. � Let V (Γ) (resp. E(Γ)) denote the set of vertices (resp. of edges) of Γ. Recall that the group G(Γ) = {s : V (Γ) → R>0} acts on the set of weight systems on Γ by (sw)(e) = s(e+)w(e)s(e−), where e+ and e− are the two vertices adjacent to the edge e. As observed in Section 1.1, the Gibbs measure on D(Γ) is invariant under the action of the group G(Γ). Note also that this action is free unless Γ is bipartite. In this later case, the 1-parameter family of elements sλ ∈ G(Γ) given by sλ(v) = λ if v is black and DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 19 sλ(v) = λ −1 if v is white act as the identity on the set of weight systems. Hence, if Γ is bipartite, the number of “essential” parameters is equal to |E(Γ)|− |V (Γ)|+1. If this bipartite graph is planar, then |E(Γ)| − |V (Γ)|+ 1 = |F (X)| − χ(S2) + 1 = |F (X)| − 1. The |F (X)| volume weights qf are invariant with respect to the action of G(Γ). They can be normalized in such a way that f∈F (X) qf = 1, giving exactly |F (X)| − 1 parameters. Thus, in the height function formulation of the Gibbs measure, only essential parameters appear. 5.3. Height functions for bipartite surface graphs. Let us now address the general case of a bipartite surface graph Γ ⊂ Σ, possibly disconnected, and possibly with boundary ∂Γ ⊂ ∂Σ. Fix a family γ = {γi} i=1 of oriented simple curves in Γ representing a basis in H1(Σ, ∂Σ;Z). Note that such a family of curves exists since Γ is the 1-squeletton of a cellular decomposition X of Σ. Given any D,D′ ∈ D(Γ, ∂Γ), the homology class of C(D,D′) = D −D′ can be written in a unique way [C(D,D′)] = D,D′(i)[γi] ∈ H1(Σ, ∂Σ;Z), with a D,D′(i) ∈ Z. Hence, C(D,D i=1 a D,D′(i)γi is a 1-boundary (rel ∂X), that is, there exists σ D,D′ ∈ C2(X, ∂X ;Z) = C2(X ;Z) such that (4) C(D,D′)− ∂σ D,D′ − D,D′(i)γi ∈ C1(∂X ;Z). The 2-cochain h D,D′ ∈ C 2(X ;Z) dual to σ D,D′ is called a height function associated to D,D′ with respect to γ. Since Z2(X, ∂X ;Z) = H2(X, ∂X ;Z) = H2(Σ, ∂Σ;Z) ∼= H0(Σ;Z), the 2-chain σ D,D′ is uniquely determined byD,D ′ and γ up to an element of H0(Σ;Z), and the same holds for h D,D′ . In other words, the set of height functions associated toD,D′ with respect to γ is an affineH0(Σ;Z)-space: it admits a freely transitive action of the abelian group H0(Σ;Z). One can normalize the height functions by choosing some family F0 of faces of X , one for each connected component of X , and by setting h D,D′(f0) = 0 for all f0 ∈ F0. Given h ∈ C2(X ;Z), set C(h) = ∂σ ∈ C1(X ;Z), where σ ∈ C2(X ;Z) is dual to h ∈ C2(X ;Z). Given a fixed D0 ∈ D(Γ, ∂Γ), let C(D0) denote the set of all C ⊂ Γ consisting of disjoint oriented 1-cycles (rel ∂Γ) such that the following condition holds: for all e ∈ D0, either e is contained in C or e is disjoint from C. Finally, let H (X,F0) denote the set of pairs (h, a) ∈ C 2(X ;Z) × Zb1 which satisfy the following properties: – h(f0) = 0 for all f0 in F0; – there exists C ∈ C(D0) such that C − C(h)− i=1 a(i)γi ∈ C1(∂X ;Z). We obtain the following generalization of Proposition 5.1. The proof is left to the reader. Proposition 5.3. Given any (h, a) ∈ H (X,F0), there is a unique dimer config- uration D ∈ D(Γ, ∂Γ) such that h = h and a = a. Furthermore, given any 20 DAVID CIMASONI AND NICOLAI RESHETIKHIN two dimer configuration D0, D1 ∈ D(Γ, ∂Γ), there is a canonical bijection (X,F0) → H (X,F0) given by (h, a) 7→ (h+ h D0,D1 , a+ a D0,D1 Recall that the boundary conditions on dimer configurations induce a partition D(Γ, ∂Γ) = D(Γ, ∂Γ | ∂D′0), where D(Γ, ∂Γ | ∂D′0) = {D ∈ D(Γ, ∂Γ) | ∂D = ∂D 0}. This partition translates into a partition of H (X,F0) via the bijection D(Γ, ∂Γ) → H (X,F0) given by D 7→ (h ). Indeed, let F∂(X) denote the set of boundary faces of X , that is, the set of faces of X that are adjacent to ∂Σ. The choice of a boundary condition ∂D′0 (together with F0) determines h (f) for all D such that ∂D = ∂D′0 and all f ∈ F∂(X). The actual possible values of h on the boundary faces depend on γ, D0 and F0; they can be determined explicitely. We shall denote by ∂h such a value of a height function on boundary faces, and call it a boundary condition for height functions. In short, we obtain a partition (X,F0) = (X,F0 | ∂h indexed by all possible boundary conditions on height functions h . Each bound- ary condition on dimer configurations corresponds to one boundary condition on height functions via D 7→ h Let us now consider an edge weight system w on the bipartite graph Γ, and a fixed boundary condition ∂D′0. Recall that the Gibbs measure for the dimer model on (Γ, ∂Γ) with weight system w and boundary condition ∂D′0 is given by Prob(D | ∂D′0) = Z(Γ;w | ∂D′0) where Z(Γ;w | ∂D′0) = D∈D(Γ,∂Γ | ∂D′0) w(D). Let us realize Γ as a surface graph Γ ⊂ Σ, fix a dimer configuration D0 ∈ D(Γ, ∂Γ), a family γ = {γi} of oriented simple curves in Γ representing a basis in H1(Σ, ∂Σ;Z), and a collection F0 of faces of the induced cellular decomposition X of Σ, one face for each connected component of X . We can use the bijection D(Γ, ∂Γ | ∂D′0) → H (X,F0 | ∂h 0) given by D 7→ (h ) to translate the Gibbs measure into a probability measure on H (X,F0 | ∂h To do so, let us first extend the weight system w to all edges of X by setting w(e) = 1 for all boundary edges of X . As in the planar case, define wβ : C1(X ;Z) → R>0 as the group homomorphism such that, for any oriented edge e of X , wβ(e) = w(e) if the orientation on e agrees with the bipartite orientation; w(e)−1 otherwise. DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 21 Note that this makes sense even for boundary edges where there is no bipartite orientation, as w(e) = 1 for such edges. Consider the parameters qf = wβ(∂f) for all f ∈ F (X) \ F∂(X); qi = wβ(γi) for all 1 ≤ i ≤ b1. We obtain the following generalization of Proposition 5.2: Proposition 5.4. Given an element (h, a) ∈ H (X,F0), set q(h, a) = f∈F (X)\F∂(X) 1≤i≤b1 Then, the Gibbs measure for the dimer model on (Γ, ∂Γ) with weight system w and boundary condition ∂D′0 translates into the following probability measure on (X,F0 | ∂h ProbD0(h, a | ∂h q(h, a) D0,F0 (X ; q | ∂h′0) where D0,F0 (X ; q | ∂h′0) = (h,a)∈H (X,F0 | ∂h q(h, a). Furthermore, the measure is independant of the choice of F0. Finally, the bijection (X,F0 | ∂h 0) → H (X,F0 | ∂h 1) given by (h, a) 7→ (h+ h D0,D1 , a+ a D0,D1 invariant with respect to the measures ProbD0 and ProbD1 . Proof. For any D ∈ D(Γ, ∂Γ | ∂D′0), equation (4) leads to C(D,D0)− ∂σ i=1 a (i)γi Computing the first term, we get wβ(C(D,D0)) = w(e)−1 = w(D)w(D0) As for the second one, wβ(∂σ ) = wβ f∈F (X) hD,D0(f)∂f f∈F (X) hD,D0 (f) Since wβ(γi) = qi, these equations lead to w(D) = w(D0) f∈F (X) 1≤i≤b1 i = λ · q(h where λ = w(D0) f∈F∂(X) f depends only on D0 and ∂D 0. The proposi- tion follows easily from this equality. � Let us count the number of essential parameters in the dimer model on (Γ, ∂Γ) with some boundary condition partitioning ∂Γ into (∂Γ)nm ⊔ (∂Γ)m, matched and non-matched vertices. We have |E(Γ)|− |(∂Γ)nm| edge weights, with an action of a (|V (Γ)|−|(∂Γ)nm|)-parameter group. Since Γ is bipartite, there is a b0(Γ)-parameter 22 DAVID CIMASONI AND NICOLAI RESHETIKHIN subgroup acting as the identity. Therefore, the number of essential parameters is equal to |E(Γ)| − |V (Γ)|+ b0(Γ) = |E(X)| − |∂Γ| − |V (X)|+ b0(X) = |F (X)| − |∂Γ| − χ(X) + b0(X) = |F (X) \ F∂(X)|+ b1(Σ)− b2(Σ). The numbers |F (X) \ F∂(X)| and b1(Σ) correspond to the parameters qf and qi. Furthermore, the parameters qf can be normalized by f qf = 1, the product being on all faces of a given closed component of Σ. Therefore, we obtain exactly the right number of parameters in this height function formulation of the dimer model. Remark. Note that all the results of the first part of the present section can be adapted to the general case of a non-necessarily bipartite surface graph: one simply needs to work with Z2-coefficients. However, the height function formulation of the dimer model using volume weights does require a bipartite structure. It is unknown whether a reformulation of the dimer model with the right number of parameters is possible in the general case. 5.4. The dimer quantum field theory on bipartite surface graphs. Let us now use these results to reformulate the dimer quantum field theory on bipartite graphs. Let Γ ⊂ Σ be a bipartite surface graph, and let X denote the induced cellular decomposition of Σ. Fix a dimer configuration D0 ∈ D(Γ, ∂Γ), a family γ = {γi} of oriented simple curves in Γ representing a basis in H1(Σ, ∂Σ;Z), and a choice F0 of one face in each connected component of X . (1) To ∂X , assign Z(∂X) = f∈F∂(X) where W is the complex vector space with basis {αn}n∈Z, and F∂(X) de- notes the set of faces of X adjacent to the boundary. (2) To X with weight system q = {qf}f∈F (X) ∪ {qi}1≤i≤b1(Σ), assign D0,F0 (X ; q) = D0,F0 (X ; q | ∂h)α(∂h) ∈ Z(∂X), where D0,F0 (X ; q | ∂h) = (h,a)∈H (X,F0 | ∂h) f∈F (X)\F∂(X) 1≤i≤b1(Σ) and α(∂h) = f∈F∂(X) f αh(f). Recall the notation a(∂D) ∈ Z(∂Γ) of Section 4.2. The bijection D(Γ, ∂Γ) → (X,F0) induces an inclusion j : Z(∂Γ) →֒ Z(∂X) such that j(a(∂D)) = f∈F∂(X) DIMERS ON SURFACE GRAPHS AND SPIN STRUCTURES. II 23 Therefore, using the proof of Proposition 5.4, j(Z(Γ;w)) = Z(Γ;w | ∂D) j(a(∂D)) D0,F0 (X ; q | ∂h)w(D0) f∈F∂(X) f∈F∂(X) αh(f) = w(D0)Z D0,F0 (X ; q), where the weight system q is obtained from w by qf = wβ(∂f) and qi = wβ(γi). In this setting, the gluing axiom makes sense only when the data β, D0 and γ are compatible with the gluing map ϕ. In such a case case, it holds by the equality above and the results of Section 4.2. The equivalence between the quantum field theories formulated in Section 4.3 and in the present section should be regarded as a discrete version of the boson- fermion correspondence on compact Riemann surfaces (see [1]). References 1. L. Álvarez-Gaumé, J.-B. Bost, G. Moore, P. Nelson and C. Vafa, Bosonization on higher genus Riemann surfaces, Comm. Math. Phys. 112, (1987) 503–552. 2. M. Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68, (1988) 175–186. 3. D. Cimasoni and N. Reshetikhin, Dimers on surface graphs and spin structures. I, to appear in Comm. Math. Phys. 4. H. Cohn, R. Kenyon and J. Propp, A variational principle for domino tilings. J. Amer. Math. Soc. 14, (2001) 297–346. 5. D. Johnson, Spin structures and quadratic forms on surfaces. J. London Math. Soc. (2) 22, (1980) 365–373. 6. W. Kasteleyn, Dimer statistics and phase transitions. J. Mathematical Phys. 4, (1963) 287– 7. W. Kasteleyn, Graph Theory and Theoretical Physics (Academic Press, London 1967) pp. 43–110. 8. R. Kenyon and A. Okounkov, Planar dimers and Harnack curves. Duke Math. J. 131, (2006) 499–524. 9. R. Kenyon, A. Okounkov and S. Sheffield, Dimers and amoebae. Ann. of Math. (2) 163, (2006) 1019–1056. 10. G. Kuperberg, An exploration of the permanent-determinant method. Electron. J. Combin. 5, (1998) Research Paper 46, 34 pp. (electronic). 11. A. Galluccio and M. Loebl, On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electron. J. Combin. 6, (1999) Research Paper 6, 18 pp. (electronic). 12. B. McCoy and T.T. Wu, The two-dimensional Ising model (Harvard University Press, Cam- bridge Massachusetts 1973). 13. G. Segal, The definition of conformal field theory, Differential geometrical methods in theo- retical physics (Como, 1987), 165–171, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 250, Kluwer Acad. Publ., Dordrecht, 1988. 14. G. Tesler, Matchings in graphs on non-orientable surfaces, J. Combin. Theory Ser. B 78 (2000), no. 2, 198–231. Department of Mathematics, UC Berkeley, 970 Evans Hall, Berkeley, CA 94720, USA E-mail address: [email protected] E-mail address: [email protected] Introduction Acknowledgements 1. The dimer model on graphs with boundary 1.1. Dimers on graphs with boundary 1.2. Dimers on surface graphs with boundary 2. Kasteleyn orientations on surface graphs with boundary 2.1. Kasteleyn orientations 2.2. Discrete spin structures 2.3. The Pfaffian formula for the partition function 3. Cutting and gluing 3.1. Cutting and gluing graphs with boundary 3.2. Cutting and gluing surface graphs with boundary 3.3. Cutting and gluing discrete spin structures 3.4. Cutting Pfaffians 4. Quantum field theory for dimers 4.1. Quantum field theory on graphs 4.2. Quantum field theory for dimers on graphs 4.3. The dimer model as the theory of free Fermions 5. Dimers on bipartite graphs and height functions 5.1. Composition cycles on bipartite graphs 5.2. Height functions for planar bipartite graphs 5.3. Height functions for bipartite surface graphs 5.4. The dimer quantum field theory on bipartite surface graphs References

0704.0274

New version announcement for TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95

New version announcement for TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95 G.M. von Hippel 1 Department of Physics, University of Regina, Regina, Saskatchewan, S4S 0A2, Canada Abstract We present a new version of TaylUR, a Fortran 95 module to automatically compute the numerical values of a complex-valued function’s derivatives with respect to sev- eral variables up to an arbitrary order in each variable, but excluding mixed deriva- tives. The new version fixes a potentially serious bug in the code for exponential- related functions that could corrupt the imaginary parts of derivatives, as well as being compatible with a wider range of compilers. Key words: automatic differentiation, higher derivatives, Fortran 95 PACS: 02.60.Jh, 02.30.Mv 1991 MSC: 41-04, 41A58, 65D25 NEW VERSION PROGRAM SUMMARY Manuscript Title: New version announcement for TaylUR, an arbitrary-order diag- onal automatic differentiation package for Fortran 95 Authors: G.M. von Hippel Program Title: TaylUR Journal Reference: Catalogue identifier: Licensing provisions: none Programming language: Fortran 95 Computer: Any computer with a conforming Fortran 95 compiler Operating system: Any system with a conforming Fortran 95 compiler Keywords: automatic differentiation, higher derivatives, Fortran 95 PACS: 02.60.Jh, 02.30.Mv Email address: [email protected] (G.M. von Hippel). URL: http://uregina.ca/~vonhippg/ (G.M. von Hippel). 1 Corresponding author Preprint submitted to Elsevier Science 4 November 2018 http://arxiv.org/abs/0704.0274v1 Classification: 4.12 Other Numerical Methods, 4.14 Utility Catalogue identifier of previous version: ADXR v1 0 Journal reference of previous version: Comput. Phys. Commun. 174 (2006) 569-576 Does the new version supersede the previous version?: yes Nature of problem: Problems that require potentially high orders of derivatives with respect to some variables or derivatives of complex-valued functions, such as e.g. expansions of Feyn- man diagrams in particle masses in perturbative Quantum Field Theory. Solution method: Arithmetic operators and Fortran intrinsics are overloaded to act correctly on ob- jects of a defined type taylor, which encodes a function along with its first few derivatives with respect to the user-defined independent variables. Derivatives of products and composite functions are computed using Leibniz’s rule and Fàa di Bruno’s formula. Reasons for the new version: The previous version [1] contained a potentially serious bug in the functions over- loading the exponential-related intrinsics (EXP, LOG, SIN, COS, TAN, SINH, COSH, TANH), which could corrupt the imaginary parts of derivatives. It also contained some features which caused it to crash when compiled with certain compilers (no- tably the NAG and Lahey/Fujitsu compilers). Summary of revisions: The bug in the exponential-related intrinsics has been corrected. A number of ad- ditional changes have been made to the code to enable better compatibility with a greater range of compilers, including the NAG and Lahey/Fujitsu compilers. Users of some of these compilers may have to define useintrinsic as a preprocessor sym- bol when compiling TaylUR. Restrictions: Memory and CPU time constraints may restrict the number of variables and Taylor expansion order that can be achieved. Loss of numerical accuracy due to cancella- tion may become an issue at very high orders. Unusual features: No mixed higher-order derivatives are computed. The complex conjugation opera- tion assumes all independent variables to be real. Running time: The running time of TaylUR operations depends linearly on the number of vari- ables. Its dependence on the Taylor expansion order varies from linear (for linear operations) through quadratic (for multiplication) to exponential (for elementary function calls). References: [1] G. M. von Hippel, TaylUR, an arbitrary-order diagonal automatic differentiation package for Fortran 95, Comput. Phys. Commun. 174 (2006) 569-576.

0704.0275

Mapping radii of metric spaces

Mapping radii of metric spaces∗ George M. Bergman November 4, 2018 Dedicated to the memory of David Gale Abstract It is known that every closed curve of length ≤ 4 in Rn (n > 0) can be surrounded by a sphere of radius 1, and that this is the best bound. Letting S denote the circle of circumference 4, with the arc-length metric, we here express this fact by saying that the mapping radius of S in Rn is 1. Tools are developed for estimating the mapping radius of a metric space X in a metric space Y. In particular, it is shown that for X a bounded metric space, the supremum of the mapping radii of X in all convex subsets of normed metric spaces is equal to the infimum of the sup norms of all convex linear combinations of the functions d(x,−) : X → R (x ∈ X). Several explicit mapping radii are calculated, and open questions noted. 1 The definition, and three examples. Definition 1. We will denote by Metr the category whose objects are metric spaces, and whose morphisms are nonexpansive maps. That is, for metric spaces X and Y we let (1) Metr(X,Y ) = {f : X → Y | (∀x0, x1 ∈ X) d(f(x0), f(x1)) ≤ d(x0, x1)}. Throughout this note, a map of metric spaces will mean a morphism in Metr. Given a nonempty subset A of a metric space Y, we define its radius by (2) radY (A) = infy∈Y supa∈A d(a, y), a nonnegative real number or +∞. For metric spaces X and Y, we define the mapping radius of X in Y (3) map-rad(X,Y ) = supf∈Metr(X,Y ) radY (f(X)) = supf∈Metr(X,Y ) infy∈Y supx∈X d(f(x), y). If X is a metric space and Y a class of metric spaces, we likewise define (4) map-rad(X,Y) = supY ∈Y map-rad(X,Y ) = supY ∈Y, f∈Metr(X,Y ) infy∈Y supx∈X d(f(x), y). (The term “mapping radius” occurs occasionally in complex analysis with an unrelated meaning [15, Def. 7.11].) All vector spaces in this note will be over the field of real numbers unless the contrary is stated. The result stated in the first sentence of the abstract has been discovered many times [5], [6], [18], [19], [25]. (Usually, the length of the closed curve is given as 1 and the radius of the sphere as 1/4, but the scaled-up version will be more convenient here.) Let us obtain it in somewhat greater generality. ∗2000 Mathematics Subject Classifications. Primary: 54E40. Secondary: 46B20, 46E15, 52A40. Keywords: nonexpansive map between metric spaces, maximum radius of image, convex subset of a normed vector space. Any updates, errata, related references etc., learned of after publication will be noted at http://math.berkeley.edu/∼ gbergman/papers/ . http://arxiv.org/abs/0704.0275v2 http://math.berkeley.edu/\protect Lemma 2. Let S denote the circle of circumference 4, with the arc-length metric. Then for any nonzero normed vector space V, we have map-rad(S, V ) = 1. Proof. In V, any 1-dimensional subspace U is isometric to the real line R, and we can map S into R, and hence into U, by “folding it flat”, getting for image an interval of length 2. Since this interval has points at distance 2 apart, its radius in V cannot be less than 1, so map-rad(S, V ) ≥ 1. For the reverse inequality, consider any map f : S → V. We wish to find a point y ∈ V having distance ≤ 1 from every point of f(S). Let p and q be any two antipodal points of S, and let y = (f(p) + f(q))/2. Every point x ∈ S lies on a length-2 arc between p and q in S, hence d(p, x) + d(q, x) = 2, hence d(f(p), f(x))+d(f(q), f(x)) ≤ 2, i.e., (d(f(p), f(x))+d(f(q), f(x)))/2 ≤ 1, so d((f(p)+f(q))/2, f(x)) ≤ 1, as claimed. Let us make explicit the argument used at the very last step above. It is the c1 = c2 = 1/2 case of (5) If c1, . . . , cn are nonnegative real numbers summing to 1, and v1, . . . , vn are elements of a normed vector space V, then for all w ∈ V, d( ci vi, w) ≤ ci d(vi, w). This can be seen by writing the left-hand side as ||( civi) − w|| = || ci (vi − w)|| ≤ ||ci (vi − w)|| = ci d(vi, w). Consider next the union X of two circles S0 and S1, each of circumference 4, intersecting in a pair of points antipodal in each (e.g., take for S0 and S1 any two distinct great circles on a sphere of circumfer- ence 4), again with the arc-length metric. We can show that this X also has mapping radius ≤ 1 in V by the same argument as before, except that where we previously used an arbitrary pair of antipodal points, we are now forced to use precisely the pair at which our circles intersect. We are not so restricted in the example showing that radius 1 can actually be achieved – we can stretch one circle taut between any two antipodal points, and for most choices of those points, we have a great deal of freedom as to what to do with the other circle. In any case, we have Lemma 3. Let X be the union of two circles S0 and S1, each of circumference 4, intersecting in a pair of points antipodal in each, with the arc-length metric. Then for any nonzero normed vector space V, we have map-rad(X,V ) = 1. We could apply the same method to any number of circles joined at a common pair of antipodal points; but let us move in a different direction. Again picturing S0 and S1 as great circles on a sphere of circumference 4 in Euclidean 3-space, assume they meet at right angles, and call their points of intersection the north and south poles. Let us bring in a third circle, S2, the equator, and let X = S0 ∪ S1 ∪ S2, again with the arc length metric. We no longer have a pair of antipodal points belonging to all three circles; rather, we have three pairs of points, S1 ∩ S2 = {p0, q0}, S2 ∩ S0 = {p1, q1}, and S0 ∩ S1 = {p2, q2}. Now given a normed vector space V and a map f : X → V in Metr, suppose we let (6) y = (f(p0) + f(q0) + f(p1) + f(q1) + f(p2) + f(q2))/6. What can we conclude about d(y, f(x)) for x ∈ X ? Say x ∈ S2. Since both {p0, q0}, and {p1, q1} are pairs of antipodal points of S2, we have d(p0, x) + d(q0, x) = d(p1, x) + d(q1, x) = 2. The same will not be true of d(p2, x) and d(q2, x). To determine how large these can get, let us take x ∈ S2 as far as possible (under our arclength metric) from the intersections of S2 with our two circles through the poles p2 and q2. This happens when x is at the midpoint of any of the quadrants into which p0, q0, p1 and q1 divide S2; in this situation, d(p2, x) = d(q2, x) = 3/2. (Each quadrant has arc-length 1, and one has to go a quadrant and a half to get from p2 or q2 to x.) We see, in fact, that for any x ∈ S2 we have d(p2, x) = d(q2, x) ≤ 3/2, hence d(p2, x) + d(q2, x) ≤ 3. Now applying any map f : X → V, and invoking (5) with all ci = 1/6, we see that for y as in (6) we have d(y, f(x)) ≤ (2 + 2 + 3)/6 = 7/6. We have proved this for x ∈ S2; by symmetry, it is also true for x lying on S0 or S1. This allows us to conclude, not that map-rad(X,V ) = 1 as in the preceding two cases, but (7) map-rad(X,V ) ≤ 7/6. And in fact, there do exist maps f : X → V with radV (f(X)) > 1. To describe such a map, note that X can be identified with the 1-skeleton of a regular octahedron of edge 1. In the next few paragraphs, let us put aside our picture of X in terms of great circles on a sphere, and replace it with this (straight-edged) octahedral skeleton. If we look at our octahedron in Euclidean 3-space from a direction perpendicular to one of its faces, we see that face and the opposite one as overlapping, oppositely oriented equilateral triangles, with vertices joined by the remaining 6 edges, which look like a regular hexagon. Now suppose we regard these two opposite triangular faces as made of stiff wire, and the other 6 edges as made of string. Then if we bring the planes of the two wire triangles closer to one another, the string edges will loosen. Suppose, however, that we rotate the top triangle clockwise as they approach one another, so that three of those strings are kept taut, while the other three become still looser. When the planes of our wire triangles meet, those wire triangles will coincide, and the three taut string edges will fall together with the three edges of that triangle, while the three loose ones become loops, hanging from the three vertices. Let us lock the two wire triangles together, and pull the three loops taut, radially away from the center of symmetry of the triangle. What we then have is the image of a certain map f in Metr from our octahedral skeleton X into a plane, which we can identify with R2. We see that radR2(f(X)) will be the distance from the center of symmetry of our figure to each of the three points to which the drawn-out loops are stretched; i.e., the sum of the distance from the center of symmetry to each vertex of the triangle, and the length of the stretched loop attached thereto. The former distance is two thirds of the altitude of the triangle, (2/3)( 3/2) = 1/ and the latter length is 1/2 (since the loop doubles back), so (8) radR2(f(X)) = 1/ 3 + 1/2. Hence, map-rad(X,R2) ≥ 1/ 3 + 1/2 > 1. This shows that our three-circle space does indeed behave differently from the preceding one- and two- circle examples. However 1/ 3 + 1/2 ≈ 1.0773, which falls well short of the upper bound 7/6 ≈ 1.1667 of (7). We can overcome this deficiency by using a different norm on R2. Let V be R2 with the norm whose unit disc is the region enclosed by a regular hexagon H of unit side. Note that the 6 sides of H are parallel to the 6 radii joining 0 to the vertices of H, hence these sides have length 1 in the new metric, just as in the Euclidean metric, and indeed, any line segment in one of those directions will have the same length in both metrics. Now let us map X, still pictured as the 1-skeleton of a regular octahedron of side 1 in Euclidean 3-space, into V so that, as before, two opposite triangles are embedded isometrically (now under the metric of V ), and made to fall together with each other and with three of the other edges, while the remaining three edges form loops that are stretched radially outward as far as they will go. Let us moreover take the sides of our image-triangle to be parallel to three sides of H. The map X → R2 that does this is almost the same one as before. The 9 edges that end up parallel to edges of H are mapped exactly as before, since distances in those directions are the same in the two metrics. The three folded loops end up set-theoretically smaller than before, since the new metric is greater in their direction than is the Euclidean metric, and they go out a distance 1/2 in the new metric before turning back; but they still contribute the value 1/2 to the calculation of the radius of our image of X. The significant change in that calculation concerns the distance from the center of our triangle to its three vertices. Looking at our triangle as a translate of one of the 6 equilateral triangles into which H is decomposed by its radii, we see that the altitude of that triangle is equal to its side in this metric (since the midpoint of a side of H has the same distance, 1, from the origin as a vertex of H does). Hence the distance from the center to a vertex is 2/3. Adding to this the distance 1/2 from that vertex to the end of the loop attached to it, we get 2/3 + 1/2 = 7/6. Assuming that the center of our triangle is indeed the minimizing point defining the radius (i.e., is a value of y that yields the infimum (2); we will verify this in Lemma 5), this achieves the upper bound (7). Summarizing, and making a few supplementary observations, we have Lemma 4. Let X be the 1-skeleton of a regular octahedron of side 1, under the arc-length metric. Then for any nonzero normed vector space V, (9) 1 ≤ map-rad(X,V ) ≤ 7/6. The exact value of map-rad(X,V ) is 1 if V is 1-dimensional, is ≥ 1/ 3 + 1/2 if V is Rn (n ≥ 2) under the Euclidean norm, and is 7/6 if V is R2 under the norm having for unit circle a regular hexagon. Proof. The lower bound 1 in (9) is gotten as in the last full sentence before Lemma 3, by regarding X as S0 ∪ S1 ∪ S2, straightening out one of these circles to cover a segment of length 2 in a 1-dimensional subspace of V, and letting the other two circles collapse into that line in any way. (Or for a construction that relies less on geometric intuition, pick any p ∈ S0, map X into R by the function d(p,−), note that this map sends p and the point antipodal to p on S0 to 0 and 2, respectively, and embed R in V.) As before, such an image of X has points 2 units apart, and so has radius ≥ 1 in V by the triangle inequality. The upper bound 7/6 was obtained in (7). To see that when V itself is 1-dimensional, the value 1 is not exceeded, note that the distance between any two points of X is ≤ 2. Hence the image of X under any map into such a V is a segment of length ≤ 2, hence of radius ≤ 1. The lower bounds 1/ 3 + 1/2 and 7/6 for V = R2 with the two indicated norms were obtained above by explicit mappings. Let us now justify the assumption we made just before the statement of the above lemma, about the center from which we computed the radius. Lemma 5. Let V be a normed vector space, A a nonempty subset of V, and G a finite group of isometries of V which preserve A. Then (10) radV (A) = infy∈V G supa∈A d(a, y), where V G is the fixed-point set of G. In particular, if V G is a singleton {v0}, then (11) radV (A) = supa∈A d(a, v0). Proof. Given v ∈ V, let (12) y = |G|−1 and note that this point lies in V G, and that for any a ∈ A, (13) d(a, y) = d(a, |G|−1 gv) ≤ |G|−1d(a, gv) = |G|−1 d(g−1a, v) ≤ |G|−1 supb∈A d(b, v) = supb∈A d(b, v). Here the first inequality holds by (5) and the second by considering b = g−1a. Hence for y ∈ V G defined by (12), supa∈A d(a, y) ≤ supa∈A d(a, v), from which (10) follows. The final assertion is a special case. For V = R2 with a regular hexagon as unit circle, the group G generated by a rotation by 2π/3 about any point is an isometry of V, and if we take that point to be the center of symmetry of the set f(X) we were looking at above, G preserves f(X) and has that center of symmetry as unique fixed point; so the above lemma justifies our description of the radius of f(X) in terms of distance from that point. In the earlier computation using the Euclidean metric on R2, we “saw” that the radius was measured from the center of symmetry; this is now likewise justified by Lemma 5. Lemma 4 leaves open Question 6. For V = R2 under the Euclidean norm, and X the 1-skeleton of a regular octahedron of side 1, where does map-rad(X,V ) lie within [1/ 3 + 1/2, 7/6] ? For V = Rn, again with the Euclidean norm, but n > 2, is the answer the same? Having whetted our appetite with this example, let us prove some general results. 2 General properties of mapping radii. Lemma 7. Let X, X ′, Y, Y ′ be nonempty metric spaces, Y and Y′ classes of such metric spaces, and V and V ′ normed vector spaces. (i) If there exists a surjective map h : X → X ′ (or more generally, a map X → X ′ with dense image) in Metr, then map-rad(X ′, Y ) ≤ map-rad(X,Y ). (ii) If Y ′ ⊆ Y, then for any nonempty subset A of Y ′ we have radY ′(A) ≥ radY (A). Here equality will hold if Y ′ is a retract of Y ; i.e., if the inclusion of Y ′ in Y has a left inverse in Metr. Hence if Y ′ is a retract of Y, then map-rad(X,Y ′) ≤ map-rad(X,Y ). In particular, this is true if Y is a normed vector space (or more generally, a convex subset of such a space) and Y ′ the fixed subspace (respectively, subset) of a finite group G of affine isometries of Y. (iii) If Y′ ⊆ Y, then map-rad(X,Y′) ≤ map-rad(X,Y). (iv) In contrast to (i) and (ii), for X ′ ⊆ X, either of the numbers map-rad(X ′, Y ) and map-rad(X,Y ) can be greater than the other, and if Y ′ ⊆ Y, or if Y ′ is a surjective image of Y in Metr, either of the numbers map-rad(X,Y ′) and map-rad(X,Y ) can be greater than the other. Proof. (i) Suppose h : X → X ′ has dense image. Then for any f : X ′ → Y, fh(X) is dense in f(X ′), hence radY (fh(X)) = radY (f(X ′)), so the terms of the supremum defining map-rad(X,Y ) include all the terms of the supremum defining map-rad(X ′, Y ), from which the asserted inequality follows. (ii) The terms of the infimum defining radY (A) include the terms of the infimum defining radY ′(A), giving the first inequality. If there exists a retraction e of Y onto Y ′, then for every y ∈ Y and a ∈ A we have d(a, e(y)) ≤ d(a, y), since e is nonexpansive and fixes points of A. Hence supa∈A d(a, e(y)) ≤ supa∈A d(a, y), and taking the infimum of this over y ∈ Y, we get radY ′(A) ≤ radY (A). This and the previous inequality give the asserted equality. Since Metr(X,Y ′) ⊆Metr(X,Y ), we also get map-rad(X,Y ′) ≤ map-rad(X,Y ), as claimed. If Y is a convex subset of a normed vector space, and Y ′ the fixed set of a finite group G as in the final assertion, note that the function e(v) = |G|−1 gv used in the proof of Lemma 5 is nonexpansive: d(e(v), e(w)) = ||e(v)− e(w)|| = ||e(v − w)|| ≤ ||v − w|| = d(v, w), and is a retraction of Y onto Y G. (iii) This is again a case of suprema of a smaller and a larger set of real numbers. (iv) The assertion for X ′ ⊆ X can be seen from the following mapping radii, where subsets of R are given the induced metric: map-rad({0}, {0, 2}) = 0, map-rad({0, 2}, {0, 2}) = 2, map-rad({0, 1, 2}, {0, 2}) = 0. The assertion for Y ′ ⊆ Y is shown by the observations map-rad({0, 2}, {0}) = 0, map-rad({0, 2}, {0, 2}) = 2, map-rad({0, 2}, {0, 1, 2}) = 1. Finally, to get the case where Y ′ is a surjective image of Y, note that we have surjections {0, 3} → {0, 2} → {0, 1} in Metr, and that map-rad({0, 2}, {0, 3}) = 0, map-rad({0, 2}, {0, 2}) = 2, map-rad({0, 2}, {0, 1}) = 1. (With a bit more work, one can construct sets X, Y0, Y1, Y2 ⊆ R such that each Yi+1 is both a subset and a surjective image of Yi, and such that map-rad(X,Y0) < map-rad(X,Y1) > map-rad(X,Y2).) To state consequences of the above results, let us fix some notation. Definition 8. For n ≥ 0, n-dimensional Euclidean space, i.e., Rn with the Euclidean norm, will be denoted n. The class of all Euclidean spaces, {En | n ≥ 0}, will be denoted Euc. The class of all normed vector spaces, regarded as metric spaces, will be denoted NmV. The class of all convex subsets of normed vector spaces, regarded as metric spaces, will be denoted Conv. The diameter of a metric space X will be defined by diam(X) = supx,y∈X d(x, y). Corollary 9. If X is a nonempty metric space, then (14) map-rad(X,E1) ≤ map-rad(X,E2) ≤ . . . ≤ map-rad(X,En) ≤ . . . , with supremum map-rad(X,Euc). Further, (15) diam(X)/2 = map-rad(X, E1) ≤ map-rad(X, Euc) ≤ map-rad(X, NmV) ≤ map-rad(X, Conv) ≤ map-rad(X, Metr) = radX(X) ≤ diam(X). Proof. Since En is the fixed subspace of a reflection of En+1, the final assertion of Lemma 7(ii) gives (14). (We could have put “0 = map-rad(X, E0) ≤” at the left end of (14); but this would complicate some references we will want to make to (14) later.) By definition, map-rad(X,Euc) is the supremum of these values. To see the initial equality of (15), note on the one hand that under any nonexpansive map f : X → E1, the images of any two points of X are ≤ diam(X) apart, hence f(X) must lie in an interval of length ≤ diam(X), and any interval in E1 has radius half its length, so map-rad(X, E1) ≤ diam(X)/2. On the other hand, for x, y ∈ X, the function d(x,−) : X → E1 is nonexpansive, and the images of x and y under this map are d(x, y) apart, whence the radius of f(X) is at least half this value. Taking the supremum over all x and y, we get map-rad(X, E1) ≥ diam(X)/2. The next four steps, inequalities among mapping radii, are instances of Lemma 7(iii). In the equality following these, the direction “≤” simply says that nonexpansive maps are radius-nonincreasing, while “≥” holds because one of the maps in the supremum defining map-rad(X, Metr) is the identity map of X. The final inequality is immediate. We note in passing some cases where these mapping radii are easy to evaluate. Corollary 10. If a metric space X satisfies radX(X) = diam(X)/2, then all terms of (15) through radX(X) are equal (and hence also equal to all terms of (14)). In particular, this is true whenever (i) X is a finite tree, with edges of arbitrary positive lengths, under the arc-length metric, or (ii) X has an isometry ρ with a fixed point 0 such that for every x ∈ X, d(x, ρ(x)) = 2 d(x, 0). Proof. The first sentence is clear from (15). To get the two classes of examples, it suffices to show in each case that radX(X) ≤ diam(X)/2, since (15) gives the reverse inequality. In case (i), X is compact, so we may choose x, y ∈ X with d(x, y) = diam(X). The unique non-self- intersecting path between x and y is isometric to a closed interval, and so has a midpoint p, satisfying d(x, p) = d(y, p) = diam(X)/2; it now suffices to show that d(z, p) ≤ diam(X)/2 for all z ∈ X. Consider the unique non-self-intersecting path from p to z. Because X is a tree, that path out of p cannot have nontrivial intersection with both the path from p to x and the path from p to y; assume it meets the latter only in p. Then the unique non-self-intersecting path from z to y is the union of the path from z to p and the path from p to y, and we know that it has length ≤ diam(X), so subtracting off diam(X)/2, the length of the path from p to y, we conclude that the length of the path from z to p is ≤ diam(X)/2, as required. In case (ii), we have radX(X) ≤ supx∈X d(x, 0) = supx∈X d(x, ρ(x))/2 ≤ diam(X)/2. Examples falling under case (ii) above include all centrally symmetric subsets of normed vector spaces containing 0, under the induced metric, and a hemisphere under the geodesic metric. A less trivial result, now. Recall that in proving the upper bounds on the mapping radii of Lemmas 2, 3 and 4, we in effect chose formal weighted combinations of points of X, and used these to specify convex linear combinations of points of f(X) ⊆ V. We abstract this technique below. In the statement of the theorem, as a convenient way to express formal weighted combinations of points of X, we use probability measures on X with finite support. (Recall that a probability measure on X is a nonnegative-valued measure µ such that µ(X) = 1, and that µ is said to have support in a set X0 if it is zero on every subset of X −X0. Apologies for the double use of “d” below, for the distance function of the metric space and the “d” of integration.) Theorem 11. Let X be a nonempty metric space. Then (16) map-rad(X,Conv) = infµ supx∈X d(x, z) dµ(z), where the infimum is over all probability measures µ on X with finite support. Proof. We first prove “≤”, imitating the argument of Lemmas 2, 3 and 4. We must show, for any nonexpan- sive map f : X → C, where C is a convex subset of a normed vector-space V, and any probability measure µ on X with finite support, that (17) radC(f(X)) ≤ supx∈X d(x, z) dµ(z). For any point x of X, let µx denote the probability measure on X with singleton support {x}. Since the µ of (17) is a probability measure with finite support, it has the form c1µx1 + · · · + cnµxn , where x1, . . . , xn are points of X, and c1, . . . , cn are nonnegative real numbers summing to 1. The point ci f(xi) lies in C, so by definition of the radius, the left-hand side of (17) is ≤ supx∈X d(y, f(x)) = supx∈X d( ci f(xi), f(x)), which by (5) is ≤ supx∈X ci d(f(xi), f(x)), which, because f is nonex- pansive, is ≤ supx∈X ci d(x, xi). The sum in this expression is the integral in (17), giving the desired inequality. In proving the direction “≥” in (16), we may assume the metric space X is bounded, since otherwise it has infinite diameter, in which case (15) tells us that the left hand side of (16) is infinite. Assuming boundedness, we shall display a particular embedding e of X in a convex subset C of a normed vector space U, such that radC(e(X)) is greater than or equal to the right-hand side of (16). Let U be the space of all continuous bounded real-valued functions on X, under the sup norm, let e : X → U take each x ∈ X to the function d(x,−) (this e is easily seen to be nonexpansive) and let C be the convex hull of e(X). Now for x ∈ X, its image e(x) = d(x,−) can be written y 7→ d(y, z) dµx(z). Hence an arbitrary u ∈ C, i.e., a convex linear combination of these functions, will have the same form, but with µx replaced by a convex linear combination µ of the measures µx, i.e., a general probability measure µ on X with finite support. For such a function u, and any x ∈ X, the distance d(e(x), u) in C is the sup norm of u− e(x), which is at least the value of u− e(x) at x ∈ X, which is u(x)− 0 = d(x, z) dµ(z). The radius of e(X) in C is thus at least the infimum over all µ of the supremum over all x of this integral, which is the right-hand side of (16). Recall that when we obtained our bound (7) on the mapping radius of the 1-skeleton of an octahedron, analogy and good luck led us to the formal linear combination of points of X used in (6) (in effect, a probability measure µ), which turned out to give the optimal bound. In general we ask Question 12. Let X be a finite graph with edges of possibly unequal lengths, under the arc-length metric. Must there be a probability measure µ on X with finite support that realizes the infimum of (16)? Is there an algorithm for finding such a µ if it exists, or if not, for evaluating (16)? We cannot expect in general that a measure of the desired sort will have support in the set of vertices of the graph X, as happened in Lemma 4. E.g., if X is isometric to a circle with arc-length metric, one can show that a measure µ realizes the infimum of (16) if and only if it gives equal weight to p and q whenever p and q are antipodal points; so if X is, say, an equilateral polygon with an odd number of vertices, µ cannot be concentrated in the vertices. A class of examples generalizing our octahedral skeleton, which it would be of interest to examine, are the 1-skeleta of cross polytopes [7]. A situation simpler than that of Question 12 is that of a finite metric space X. Here the determination of the right-hand side of (16) is a problem in linear programming; whether it has an elegant solution I don’t know. The determination of map-rad(X,En) for such a space X is, similarly, in principle, a problem in calculus. In Corollary 10, we saw that the mapping radius is easy to compute for a space that has “a robust center”. Using the preceding theorem, let us show the same for a space with a pair of “robust antipodes”. Corollary 13. Suppose the metric space X has a pair of points p and q such that (18) (∀ r ∈ X) d(p, r) + d(r, q) = d(p, q). Then letting D = d(p, q), we have diam(X) = D, and map-rad(X,Conv) = D/2. Thus, the terms of (15) through map-rad(X,Conv) are all equal to D/2. In particular, this is true if X is the 1-skeleton of a regular tetrahedron or of a parallelopiped (in particular, of a cube), with the arc-length metric, or is the 0-skeleton of any of the regular polyhedra other than the tetrahedron, with metric induced by the arc-length metric on the 1-skeleton of that polyhedron. The property (18) is, of course, inherited by any subspace of X containing p and q. Proof. For any two points r, r′ ∈ X, we have 2 d(r, r′) ≤ (d(r, p)+d(p, r′))+(d(r, q)+d(q, r′)) = (d(p, r)+d(r, q))+(d(p, r′)+d(r′, q)) = 2D, so d(r, r′) ≤ D, whence diam(X) = D. Now let µ be the probability measure giving weight 1/2 to each of p and q. For this µ, the integral on the right-hand side of (16) has value D/2 for all x, hence the supremum of that integral over x is D/2, hence (16) shows that map-rad(X,Conv) ≤ D/2. Comparing with the first term of (15), we see that all the the terms of (15) through map-rad(X,Conv) (though not, as before, through map-rad(X,Metr)) are equal. For X the 1-skeleton of a regular tetrahedron, we get (18) on taking for p and q the midpoints of two opposite edges. For X the 1-skeleton of a parallelopiped, we can use any two antipodal points (not necessarily vertices. In picturing this case, it may help to note that X is isometric to the 1-skeleton of a rectangular parallelopiped.) In the 0-skeleton cases, we use any pair of opposite vertices. In each case, the verification of (18) is not hard. The final sentence is clear. So, for instance, for the 1-skeleta of the tetrahedron and cube of edge 1, the 6-tuples of terms of (15) (not distinguishing terms shown connected by equals-signs) are (1, 1, 1, 1, 3/2, 2) and (3/2, 3/2, 3/2, 3/2, 3, 3) respectively. (The reason the last two numbers are equal for the cube, but distinct for the tetrahedron, is that for the cube, the function x 7→ supy d(x, y) is 3 for all x, while for the tetrahedron, it ranges from a maximum value 2 at the midpoints of the edges to a minimum value 3/2 at the vertices. In neither of these cases is the maximum twice the minimum, so neither of them falls under Corollary 10.) Let us note a curious feature of the construction used in Theorem 11: it has what at first looks like a universal property (part (i) of the next result) but turns out not to be (part (ii)). Corollary 14 (to proof of Theorem 11). Let X be a bounded metric space, let U be the space of continuous bounded real-valued functions on X under the sup norm (cf. second half of the proof of Theorem 11), and let e : X → U be the map taking each x ∈ X to the function d(x,−). Now let f : X → V be any map (in Metr) from X into a normed vector space V. Then (i) For every family of points x1, . . . , xn ∈ X, every family c1, . . . , cn of nonnegative real numbers summing to 1, and every x ∈ X, one has (19) d(f(x), ci f(xi)) ≤ d(e(x), ci e(xi)). However, (ii) Given points x1, . . . , xn ∈ X, and two families of nonnegative real numbers b1, . . . , bn and c1, . . . , cn, each summing to 1, it is not necessarily true that (20) d( bi f(xi), ci f(xi)) ≤ d( bi e(xi), ci e(xi)). Thus, the convex hull of e(X) need not admit a map (in Metr) to the convex hull of f(X) making a commuting triangle with e and f. Proof. (i) may be seen by combining the calculations of the last sentence of the proof of the “≤” direction of Theorem 11, which shows that d(f(x), ci f(xi)) ≤ ci d(x, xi), and the end of the proof of the “≥” direction, which, by evaluating e(xi) and e(x) as elements of the function-space U at the point x, shows that d(e(x), ci e(xi)) ≥ ci d(x, xi). To get (ii), let X again be a circle of circumference 4 with arc-length metric, and let x0, x1, x2, x3 ∈ X be four points equally spaced around it. Note that for any u ∈ X, we have d(x0, u) + d(x2, u) = 2 = d(x1, u)+d(x3, u). Hence if we choose the bi and ci so that the right-hand-side of (20) is d((e(x0)+e(x2))/2, (e(x1)+e(x3))/2), we see that this value is 0. On the other hand, if we map X into E 1 by f(x) = 1 − max(d(x0, x), 1), then of the f(xi), only f(x0) is nonzero, so the left-hand side is not 0, so (20) fails. There are, in fact, a different normed vector space U and mapping e : X → U for which the universal property of (20) does hold [23, Theorem 2.2.4]; we examine this construction in an appendix, §6. 3 Some explicit mapping radii. A classical result of H. E.W. Jung is, in effect, an evaluation of the mapping radius in En of a very simple metric space. Theorem 15 (after Jung [17]). Let D∞ denote an infinite metric space in which the distances between distinct points are all 1. (The cardinality does not matter as long as it is infinite.) Then the values of map-rad(D∞, E n) for n = 0, 1, 2, . . . are, respectively, (21) 0 < 1/2 < 1/ 3/8 < . . . < n/(2(n+ 1)) < . . . . Hence, map-rad(D∞, Euc) = 1/ Likewise, for any positive integer m, if we let Dm be an m-element metric space with all pairwise distances 1, then for every n ≥ 0, (22) map-rad(Dm, E r/(2(r + 1)) , where r = min(m−1, n). Hence, map-rad(Dm, Euc) = (m− 1)/(2m) . Summary of proof. The main result of [17] is that every subset of En of diameter ≤ 1 has radius ≤ n/(2(n+ 1)) . This gives map-rad(D∞, E n/(2(n+ 1)) . On the other hand, the n+ 1 vertices of the n-simplex of edge 1 in En form a subset of radius exactly n/(2(n+ 1)) , and clearly D∞ can be mapped onto that set, establishing equality. Taking the limit of this increasing sequence as n → ∞, one gets map-rad(D∞, Euc) = 1/ Clearly, the hypothesis m > n works as well as m =∞ in concluding as above that map-rad(Dm, En) = n/(2(n+ 1)) . For m ≤ n, on the other hand, any image of Dm in En lies in an affine subspace that can be identified with Em−1, so in that case we get map-rad(Dm, E n) = map-rad(Dm, E m−1) = (m− 1)/(2m) . Combining these results, we get (22) and the final conclusion. The inequalities (21) show that each step of (14) can be strict. What about the steps of (15)? If we identify terms connected by equal-signs, then (15) lists six possibly distinct values, connected by five ≤-signs. Three of these ≤-signs are shown strict by the 3-point metric space D3 of the above theorem, for which, I claim, the 6-tuple of values is (1/2, 1/ 3 , 2/3, 2/3, 1, 1). The first of these values, and the last two, are clear, and the second comes from the above theorem (line after (22)). To evaluate the remaining two values, map-rad(D3, NmV) and map-rad(D3,Conv), consider the embedding e : D3 → U as in the last paragraph of the proof of Theorem 11. The space U used there can in this case be described as R3 under the sup norm; let C be the convex hull in U of (23) e(D3) = {(0, 1, 1), (1, 0, 1), (1, 1, 0)}. Then Lemma 7(ii) (in particular, the final sentence) tells us that radC(e(D3)) is the common distance of the three points of e(D3) from the unique point of C invariant under cyclic permutation of the coordinates, namely (2/3, 2/3, 2/3). This common distance is 2/3 (since each member of e(D3) has a zero coordinate), so radC(e(D3)) = 2/3, and by Theorem 11, this is map-rad(D3,Conv). Since map-rad(D3, NmV) ≤ map-rad(D3, Conv), to show that map-rad(D3, NmV) is also 2/3 it will suffice to obtain a nonexpansive map f of D3 into a vector space V such that radV (f(D3)) = 2/3. This may be done by using the same mapping as above, but translated by (−2/3,−2/3,−2/3), so that the affine span of its image becomes a vector subspace of R3, which, with its induced norm, we take as our V. The preceding argument now gives radV (f(D3)) = 2/3. For a space showing strict inequality at the final step of (15), radX(X) ≤ diam(X), one can use any nontrivial instance of Corollary 10; for instance, the unit interval [0, 1], for which that corollary shows that the 6-tuple in question is (1/2, 1/2, 1/2, 1/2, 1/2, 1). This leaves the step (24) map-rad(X,NmV) ≤ map-rad(X,Conv). I thought at first that equality had to hold here: that for a C a convex subset of a normed vector space V and any A ⊆ C (in particular, the image of any map of a metric space into C), one had radV (A) = radC(A). However, this is not so: consider the untranslated case (23) of the above D3 example, and note that the point (1/2, 1/2, 1/2) ∈ U has distance 1/2 from each point of (23); so radU (e(D3)) ≤ 1/2 < 2/3 = radC(e(D3)). Nonetheless we have seen that for X = D3, equality holds in (24). Here, however, is an example (which it took attempts spread over many months to find) for which that inequality is strict. Consider the graph with 7 vertices, x, y0, y1, y2, z0, z1, z2, and 9 edges: a length-1 edge from x to each of the yi, and a length-2 edge from yi to zj whenever i 6= j; and let X be the vertex-set of this graph, with arc-length metric. Thus, for all i 6= j we have (25) d(x, yi) = 1, d(x, zi) = 3, d(yi, yj) = 2, d(yi, zj) = 2, d(yi, zi) = 4, d(zi, zj) = 4. Let us first find map-rad(X,Conv), using Theorem 11. We must maximize the infimum (16) over the convex linear combinations of µx, . . . , µz2 . By Lemma 5, it suffices to maximize that expression over points invariant under permutations of the subscripts; i.e., over convex linear combinations of (26) µx, µy = (µy1 + µy2 + µy3)/3, µz = (µz1 + µz2 + µz3)/3. We find that (27) µx(x) = 0, µx(yi) = 1, µx(zi) = 3, µy(x) = 1, µy(yi) = 4/3, µy(zi) = 8/3, µz(x) = 3, µz(yi) = 8/3, µz(zi) = 8/3. Any convex linear combination of these three functions has value ≥ 8/3 at each zi; so every value of the supremum in (16) is at least 8/3. Moreover, taking µ = µy (or more generally, µ = (1− t)µy + tµz for any t ∈ [0, 5/6]), we see that this value 8/3 is attained; so (28) map-rad(X,Conv) = 8/3. The idea of our verification that map-rad(X,NmV) is strictly smaller than (28) will be to use the non- convex affine combination (3µy − µx)/2 of the functions (27), so as to reduce somewhat the highest values of µy, those at the zi, without bringing the values at other points up by too much. But since we don’t have the analog of Theorem 11 for non-convex combinations (and indeed, that analog is not true in general – if it were, then 2µy − µx would lead to a still better result, but it does not), we must calculate by hand rather than calling on such a theorem. So suppose f is a nonexpansive map of X into a normed vector space V, and let (29) p = (f(y0) + f(y1) + f(y2)− f(x))/2. We need to bound the distances between p and the points of f(X). In view of the symmetry of (29), it will suffice to bound the distances to f(x), f(y0) and f(z0). We calculate (30) d(p, f(x)) = || (f(y0) + f(y1) + f(y2)− f(x)− 2f(x))/2 || ≤ (||f(y0)− f(x)||+ ||f(y1)− f(x)|| + ||f(y2)− f(x)||)/2 ≤ (1 + 1 + 1)/2 = 3/2. d(p, f(y0)) = || (f(y0) + f(y1) + f(y2)− f(x)− 2f(y0))/2 || ≤ (||f(y1)− f(y0)||+ ||f(y2)− f(x)||)/2 ≤ (2 + 1)/2 = 3/2. d(p, f(z0)) = || (f(y0) + f(y1) + f(y2)− f(x)− 2f(z0))/2 || ≤ (||f(y0)− f(x)||+ ||f(y1)− f(z0)||+ ||f(y2)− f(z0)||)/2 ≤ (1 + 2 + 2)/2 = 5/2. Taking the maximum of these values, we get (31) map-rad(X,NmV) ≤ 5/2 < 8/3 = map-rad(X,Conv), a strict inequality, as claimed. The above observations suggest the question: Which normed vector spaces V have the property that the radius of every subset X of V is the same whether evaluated in V, or in an arbitrary convex subset of V containing X ? This is examined in an appendix, §7. The example of (23) showed that the radius of a subset of a normed vector space could change when one passed to a larger normed vector space. Let us note a curious consequence. Lemma 16. Let U be R3 under the sup norm, and U0 ⊆ U be {(a, b, c) ∈ U | a + b + c = 0}. Then there is no isometric reflection U → U having U0 as its fixed subspace. In fact, no finite group of affine isometries of any normed vector space W containing U has U0 as its fixed subspace. Proof. Let W be any normed vector space containing U, and let f : D3 → U0 be given by f(x) = e(x) − (2/3, 2/3, 2/3), for e as in the paragraph containing (23). The first sentence of Lemma 7(ii) gives radW (f(D3)) ≤ radU (f(D3)), which we saw is < radU0(f(D3)). On the other hand, if W had a finite group G of affine isometries with fixed subspace U0, then Lemma 5 would give radW (f(D3)) = radU0(f(D3)). Returning to (14) and (15), let us for simplicity reduce the number of independent values by “normalizing” to the case diam(X) = 2, and ask for more detailed information than those inequalities. Question 17. Let X run over all metric spaces of diameter 2. What can one say about the geometry of the resulting sets of sequences (32) {(map-rad(X,E1), map-rad(X,E2), . . . , map-rad(X,En), . . . )} ⊆ RN, (33) {(map-rad(X, Euc), map-rad(X, NmV), map-rad(X, Conv), map-rad(X, Metr))} ⊆ R4 ? Can one describe them exactly? Are they convex; or do they become convex on replacing the entries by their logarithms, or under some other natural change of coordinates? If two successive terms of a member of (32) are equal, is the sequence constant from that point on? Another family of questions, suggested by Theorem 15, is Question 18. For n ≥ 2, what can one say about the set of nonnegative real numbers that can be written map-rad(X,En) for finite metric spaces X in which all distances are integers? Are all such real numbers “constructible”, i.e., obtainable from rational numbers by a finite sequence of square roots and ring operations? Is this set well-ordered for each n ? (It has a smallest element 0, and a next-to-smallest element 1/2.) Does this set change if “finite metric spaces X . . . ” is weakened to “bounded metric spaces X . . . ”? For m < n, can one assert any inclusion between the sets of mapping radii into Em, and into En ? Are there values that occur as map-rad(X,Euc) for some X, but not as map-rad(X ′,En) for any X ′ and n ? (E.g., can 1/ 2 be written in the latter form?) We end this section with an observation made in [12] for the spaces En, which in fact holds for closed convex subsets of arbitrary finite-dimensional normed spaces. Lemma 19 (cf. [12, Proposition 29, p.14, and second paragraph of p.46]). If C is a closed convex subset of a normed vector space V of finite dimension n, and A a subset of C with > n elements, then radC(A) = supA0 radC(A0), where A0 runs over the n+1-element subsets of A. Proof. “≥” is clear; so it suffices to show that if for some real number r, each A0 is contained in a closed ball of radius r centered at a point of C, then so is A. Now for each a ∈ A, the set of v ∈ C such that a lies in the closed ball in C of radius r about v is the closed ball in C of radius r about a, hence a compact convex subset of V. To say that a set A0 is contained in some closed ball of radius r centered at a point of C is to say that the intersection of these sets, as a runs over A0, is nonempty. By Helly’s Theorem ([14], [8]), if a family of compact convex subsets of Rn has the property that every system of n+1 members of this family has nonempty intersection, then so does the whole family; which in this case means that all of A is contained in a ball of the indicated sort. The above lemma does not imply the corresponding statement for mapping radii. For example, let X = {x, y0, y1, y2}, where x has distance 1/2 from each of the yi, and these have distance 1 from each other. The maximum of the mapping radii in E2 of 3-element subsets of X is map-rad({y0, y1, y2},E2) = map-rad(D3,E 2) = 1/ 3. But map-rad(X,E2) ≤ radX(X) = 1/2. On the other hand, for this example, map-rad(X,E2) can be described as the infimum over p ∈ X of the supremum of map-rad(X0,E 2) over all 3-element subsets X0 of X containing p. So we ask Question 20. Does there exist, for every positive integer n, a positive integer N and a formula which for every metric space X of ≥ N elements, and every normed vector space V of dimension n, expresses map-rad(X,V ), using the operations of suprema and infima, in terms of the numbers map-rad(X0, V ), for N -element subsets X0 ⊆ X ? 4 Realizability of mapping radii. For a subset A of a metric space Y, let us say that radY (A) is realized if the infimum in the definition (2) of that expression is attained, that is, if there exists y ∈ Y such that A is contained in the closed ball of radius radY (A) about y. Likewise, for metric spaces X and Y, let us say that map-rad(X,Y ) is realized if the supremum in the definition of that expression is attained; that is, if there exists an f : X → Y such that radY (f(X)) = map-rad(X,Y ). (This does not presume that radY (f(X)) is realized.) Lemma 21. Let X and Y be nonempty metric spaces. (i) If Y is compact, then for any subset A ⊆ Y, radY (A) is realized. (ii) If X and Y are both compact, then map-rad(X,Y ) is realized. However (iii) For X compact and Y bounded and complete, or for X bounded and complete and Y compact, map-rad(X,Y ) may fail to be realized. Proof. (i) follows from the fact that for bounded A, supa∈A d(a, y) is a continuous function of y, hence assumes a minimum on Y. To get (ii), we note that Metr(X,Y ) is a closed subset of the function space Y X , which is compact because Y is, so Metr(X,Y ) is compact in the function topology. We would like to say that the real-valued map on this space given by f 7→ radY (f(X)) is continuous, and hence assumes a maximum. For general X, this continuity does not hold, as will follow from the second statement of (iii); but I claim that it holds if X is compact. For given f ∈Metr(X,Y ) and ε > 0, compactness allows us to cover X by finitely many open balls of radius ε/3, say centered at x1, . . . , xn. Consider the neighborhood of f in Metr(X,Y ) given U = {g ∈Metr(X,Y ) | d(f(xi), g(xi)) < ε/3 (i = 1, . . . , n)}. Taking any x ∈ X and y ∈ Y, note that there exists i such that d(xi, x) < ε/3; hence for g ∈ U, |d(f(x), y) − d(g(x), y)| ≤ d(f(x), g(x)) ≤ d(f(x), f(xi)) + d(f(xi), g(xi)) + d(g(xi), g(x)) ≤ ε/3 + ε/3 + ε/3 = ε. Thus, the two functions associating to every y ∈ Y the numbers supx∈X d(f(x), y) and supx∈X d(g(x), y) differ everywhere by ≤ ε, whence the infima of these functions, radY (f(X)) and radY (g(X)) differ by ≤ ε, giving continuity of f 7→ radY (f(X)), which, as noted above, yields (ii). (iii) For an example with X but not Y compact, let X = D2, i.e., a space consisting of two points at distance 1 apart, and let Y = {y2, y3, . . . , yn, . . . } ∪ {z}, with d(ym, yn) = 1 (m 6= n) and d(yn, z) = 1 − 1/n. Note that the radius in Y of a point-pair {z, yn} or {ym, yn} with m < n is 1 − 1/n. Now Metr(X,Y ) consists of all set-maps X → Y, and it follows from the above calculation that map-rad(X,Y ) = supn(1− 1/n) = 1, but that this value is not achieved. (If we had not specified that Y should be complete, we could have used the simpler example, X = D2, Y = [0, 1).) For an example with Y but not X compact, let X = {x2, x3, . . . , xn, . . . }, with d(x2n, x2n+1) = 2−1/n, and all other pairs of distinct points having distance 1; and let Y = [0, 2] ⊆ E1. Note that if a map X → Y is to have radius > 1/2, it must send some pair of points to values differing by > 1, and by our metric on X, these two points must have the forms x2n, x2n+1. Since all other points have distance 1 from these two, the images of all other points must fall within the interval between their images. Hence the image of our map falls within an interval of length ≤ 2− 1/n for some positive n, i.e., of length < 2, and hence of radius < 1. But such images can have radii arbitrarily close to 1, again giving a mapping radius that is not realized. Corollary 22. Suppose Y is a metric space in which every closed bounded subset is compact. Then (i) For every bounded nonempty subset A ⊆ Y, radY (A) is realized. (ii) If the isometry group of Y is transitive, or more generally, if Y has a bounded subset which meets every orbit of that group, then for every compact nonempty metric space X, map-rad(X,Y ) is realized. Proof. (i) Let radY (A) = r, choose any a0 ∈ A, and let Y ′ be the closed ball of any radius r′ > diam(A) ≥ r about a0 in Y. By assumption Y ′ is compact. We see that A ⊆ Y ′, and that every point y ∈ Y with supa∈A d(a, y) ≤ r′ lies in Y ′. Since infy∈Y supa∈A d(a, y) = r, the space Y contains points y for which supa∈A d(a, y) comes arbitrarily close to r; hence it will contain points for which that value is arbitrarily close to r and is ≤ r′. Points with this latter property lie in Y ′, whence radY ′(A) is also equal to r, and applying part (i) of the preceding lemma with Y ′ for Y gives the desired conclusion. (ii) Suppose every orbit of the isometry group of Y meets the closed ball of radius c about y0 ∈ Y, and let x0 be any point of X. Then every f : X → Y may be adjusted by an isometry of Y (which will preserve the radius of f(X)) so that we get d(f(x0), y0) ≤ c, and after this adjustment, f(X) will lie in the closed ball of radius c + diam(X) about y0. Letting Y ′ denote the closed ball of any radius r′ > c + diam(X) about y0, we see as in the proof of (i) that the radii of these image sets f(X) in Y ′ will equal their radii within Y, and applying part (ii) of the preceding lemma with Y ′ for Y, we get the desired conclusion. 5 Related literature (and one more question). Lemma 2 above, determining the mapping radius of a circle in a normed vector space V, occurs frequently in the literature (with E3 or En for V ) as an offshoot of the proof of Fenchel’s Theorem, the statement that the total curvature of a closed curve C in E3 is at least 2π, with equality only when C is planar and convex [9, Satz I]. To prove that theorem, Fenchel noted that this total curvature is the length of the curve in the unit sphere S2 traced by the unit tangent vector to C, and that that curve cannot lie wholly in an open hemisphere of S2 (nor in a closed hemisphere unless C is planar). He completed the proof by showing [9, Satz I′ ] that a closed curve of length < 2π (respectively, equal to 2π) in S2 must lie in an open hemisphere (respectively, must either lie in an open hemisphere or be a union of two great semicircles). In our language, this says that a circle of circumference < 2π, made a metric space using arc-length, has mapping radius < π/2 in S2 and (along with some additional information) that the circle of arc-length exactly 2π has mapping radius π/2. Subsequent authors [5], [6, Lemma on p.30], [16], [19], [21], [22] gave simpler proofs of Fenchel’s Satz I′ (similar to our proof of Lemma 2), and/or generalized that result from S2 to Sn, and/or obtained the more precise result that the mapping radius of a circle of length L ≤ 2π in Sn is L/4, and/or noted that the same method also gives the analogous result with En, or indeed any of a large class of geometric structures, in place of Sn. The last-mentioned generalizations were based on the observation that the concept of the midpoint of a pair of points can be defined, and behaves nicely, in many geometric contexts. I do not know whether more general convex linear combinations, such as we used in (5) and in the proof of Theorem 11, can be defined outside the context of vector spaces so as to behave nicely; hence the emphasis in this note on vector spaces and their convex subsets. A.Weinstein (personal communication) suggests that an approach to “averaging” of points introduced by Cartan and developed further by Weinstein in [24] might serve this function. J.Lott (personal communication) points similarly to the concepts of Hadamard space [2] and Busemann convex space [4]. The results on closed curves of length L in the unit sphere cited above all take L ≤ 2π. If we write S1L for a circle of circumference L with arc-length metric, and Sn for the unit n-sphere (of circumference 2π) with geodesic distance as metric, it is clear that the result map-rad(S1L, S n) = L/4 cannot be expected to hold when L > 2π; but it would be interesting to investigate how that mapping radius does behave as a function of L. For all L, map-rad(S1L, S n) < π, since a curve of fixed length cannot come arbitrarily close to every point of Sn, and if it misses the open disk of geodesic radius r about a point p, then it is contained in the closed disc of geodesic radius π − r about the antipodal point. Many of the papers referred to above consider arcs as well as closed curves; i.e., also study map-rad([0, L], Sn), and prove that for L ≤ π, this equals L/2. Again, the case of larger L would be of interest. So we Question 23. For fixed n > 1, how does map-rad(S1L, S n) behave for L > 2π, and how does map-rad([0, L], Sn) behave for L > π, as functions of L ? For instance, are these two functions piecewise analytic? It seems likely that there will be ranges of values of L in which different configurations of a closed curve or arc give maximum radius, and that the value of this radius will be an analytic function of L within each such range. (I conjecture that for all L between 2π and a value somewhat greater than 3π, map-rad(S1L, S will be realized by a “3-peaked crown”, consisting of 6 arcs of great circles, with midpoints equally spaced along a common equator. For map-rad(S1L, S n) with n > 2, I have no guesses.) Many papers in this area also consider the smallest “box” – in various senses – into which one can fit all curves, or closed curves, of unit length [5] [13] [20], or all point-sets of unit diameter [10]. These do not translate into statements about our concept of mapping radius for two reasons. First, they deal with arc length in the Euclidean metric, but with “boxes” which, though they could in many cases be considered closed balls in another metric, are not balls in the Euclidean metric; and our formalism of mapping radius does not look at more than one metric on Y at a time. Second, they generally allow rotations as well as translations in fitting the box around the curve, while in looking at radii we only have one closed ball of each radius centered at a given point. The intuitive interest of Question 23 above arises in part from a special property of the sphere: that a large open or closed ball, i.e., one that falls just short of covering Sn, has for complement a small closed or open ball. For spaces Y not having this property, the most natural analogs of those questions might be the corresponding questions about “mapping co-radii”, given by the definitions (34) coradY (A) = supy∈Y infa∈A d(a, y) (A ⊆ Y ), (35) map-corad(X,Y ) = inff∈Metr(X,Y ) coradY (f(X)) = inff∈Metr(X,Y ) supy∈Y infx∈X d(f(x), y) (cf. (2) and (3)). So, for instance, one might ask about the values of map-corad([0, L], B2) for B2 the closed unit disc in R2, as a function of L. (I’m not sure that “co-radius” is a good choice of term: one could argue that that term would more ap- propriately apply either to radY (Y−A), or to what in the notation of (34) would be written coradY (Y −A). So the above names are just suggestions, which others may choose to revise.) 6 Appendix: The Arens-Eells space of X. At the end of §2, I mentioned that every metric space X admits an embedding in a normed vector space U having the universal property that Corollary 14(ii) showed that the embedding we were considering there did not have. The construction in question was introduced by Arens and Eells [1], and its universal property noted by Weaver [23, Theorem 2.2.4], who calls it the Arens-Eells space of X. Weaver is there most interested in this space as a pre-dual to the Banach space of Lipschitz functions on X. I will sketch below a motivation for the same object in terms of the universal property. My description will also make a couple of technical choices different from those of [1] and [23]. Essentially the same construction arises in mathematical economics, in the study of the “transportation problem” [11], cf. [23, §2.3]. What to us will be the norm of an element of the Arens-Eells space appears there as the minimum cost of transporting goods from a given set of sources to a given set of markets. To lead up to the construction, let a metric space X be given, consider any map (as always, nonexpansive) f of X into a normed vector space V, and let us ask, as a sample question: If we know the distances among four points x1, x2, x3, x4 ∈ X, what can we say about ||f(x1) + f(x2)− f(x3)− f(x4)|| ? Clearly, this will be bounded above by ||f(x1)−f(x3)|| + ||f(x2)−f(x4)|| ≤ d(x1, x3) + d(x2, x4). The other way of pairing terms of opposite sign similarly gives the bound d(x1, x4) + d(x2, x3). Hence (36) ||f(x1) + f(x2)− f(x3)− f(x4)|| ≤ min(d(x1, x3) + d(x2, x4), d(x1, x4) + d(x2, x3)). For a similar, but slightly less straightforward case, suppose we want to bound ||3f(x1) + f(x2) − 2f(x3) − 2f(x4)||. We cannot, as before, pair off terms whose coefficients in this expression happen to be the same except for sign. There are, however, ways of breaking up that expression as a linear combination of differences; and a little experimentation shows that all ways of doing so are convex combinations of two extreme decompositions. These two cases lead to the bound (37) ||3f(x1) + f(x2)− 2f(x3)− 2f(x4)|| ≤ min(2d(x1, x3)+d(x1, x4)+d(x2, x4), 2d(x1, x4)+d(x1, x3)+d(x2, x3)). We will not stop here to prove that (36) and (37) are best bounds. Let us simply observe that these considerations suggest that the norm of such a linear combination of images of points of X under a universal map e : X → U should be given by an infimum of linear combinations of the numbers d(x, y) (x, y ∈ X) with nonnegative real coefficients, the infimum being taken over all such linear expressions which, when each d(x, y) is replaced by e(x)− e(y), give the required element. An obvious problem is that the only elements we get in this way are those in which the sum of the coefficients of the members of e(X) is 0. This difficulty is intrinsic in the situation: There will not in fact exist a nonexpansive map of X into a normed vector space having the standard sort of universal mapping property with respect to such maps, because, though the condition of nonexpansivity bounds the distances among images of points of X, it does not bound the distances between such images and 0; so universality would force the images of points of X to have infinite norm. What we can get, rather, is a set-map e of X into a vector space U, and a norm on the subspace U0 of linear combinations of images of points of X with coefficients summing to 0, such that for all x, y ∈ X, ||e(x) − e(y)|| ≤ d(x, y), and which has the universal property that given any nonexpansive map f of X into a normed vector space V, there exists a unique vector-space homomorphism g : U → V which satisfies f = ge, and is nonexpansive on U0. Observe that the norm on U0 induces a metric on each coset of that subspace; in particular, on the coset U1 of elements in which the sum of all coefficients is 1, which is the affine span of the image of X. The map of X into that coset is nonexpansive, and the asserted universal property of e is easily seen to yield (20), the property that the construction of §2 failed to have. Weaver’s answer to the same distance-to-0 problem is to use metric spaces with basepoint, and basepoint- respecting maps, the basepoint of a vector space being 0. This has the advantage of giving a universal property in the conventional sense, with both U and V in the category of normed vector spaces. However, it requires one to make a possibly unnatural choice of basepoint in X ; changes in that choice induce isometries on the universal space, which, though affine, are not linear. The approach I actually find most natural is to regard what I have called U1 as a “normed affine space”, that is, a set with a simply transitive group of “translation” maps by elements of a normed vector space, and to note that U1 has a genuine universal property in the category of normed affine spaces. However, the development of that concept would be an excessive excursion for this appendix. Still another approach would be to work with “normed” vector spaces where the norm is allowed to take on the value +∞. In any case, it is straightforward to verify that the Arens-Eells space of X as described in [23] and my U0 are isometrically isomorphic, so below I will quote results of Weaver’s, tacitly restated for my version of the construction. The details, now: let U be the vector space of all real-valued (i.e., not necessarily nonnegative) measures µ on X with finite support, and, as before, for each x ∈ X let µx be the probability measure with support {x}. Thus, {µx | x ∈ X} is a basis of U. Let U0 ⊆ U denote the subspace of measures µ satisfying µ(X) = 0. Let W similarly denote the space of all real-valued measures on X × X with finite support; for each (x, y) ∈ X ×X, let νx,y be the probability measure with support {(x, y)}, and let W+ ⊆ W be the cone of nonnegative linear combinations of the νx,y, i.e., the nonnegative-valued measures on X ×X. Finally, let D : W → U be the linear map defined by the condition (38) D(νx,y) = µx − µy for x, y ∈ X, which clearly has image U0. We now define the norm of any µ ∈ U0 by (39) ||µ|| = inf ν∈W+, D(ν)=µ (x,y)∈X×X d(x, y) dν . It is easy to verify that this indeed gives a norm with the desired universal property. The one verification that is not immediately obvious is that it is a norm rather than a pseudonorm; i.e., that it is nonzero for nonzero µ ∈ U0. To get this, one first proves the desired universal property in the wider context of pseudonormed vector spaces, then notes that given any nonzero µ = ai µxi ∈ U0 (I finite, all ai nonzero), one can find a nonexpansive map f : X → R which is zero at all but one of the xi, say xi0 , from which it follows by the universal property that ||µ|| ≥ | ai f(xi)| = |ai0 f(xi0)| > 0. Weaver [23, Theorem 2.3.7(b)] shows that the infimum in (39) is always attained, and in fact, by a ν whose “support” in X (the set of points which appear as x or y in terms νx,y having nonzero coefficient in the expression for ν) coincides with the support of µ (the set of x such that µx appears with nonzero coefficient in the expression for µ). Our next proposition strengthens this result a bit. For brevity, we will call on Weaver’s result in the proof, but I will sketch afterward how the argument can be made self-contained. We will use the following notation and terminology. Given ν = bj νxj ,yj ∈ W (where J is a finite set, the pairs (xj , yj) for j ∈ J are distinct, and all bj 6= 0), let Γ(ν) be the directed graph having for vertices all points of X, and for directed edges the finitely many pairs (xj , yj) (j ∈ J). Let us define the positive support of a directed graph Γ as the set of vertices which are initial points of its edges, and its negative support as the set of vertices which are terminal points. For ν ∈ W, we will call the positive and negative supports of Γ(ν) the positive and negative supports of ν. On the other hand, for µ = ai µxi ∈ U0, let us define its positive support to be {xi | ai > 0}, and its negative support to be {xi | ai < 0}. These are clearly disjoint. Note that when ν ∈W+, the positive support of D(ν) is contained in the positive support of ν, and contains all elements thereof that are not also in the negative support of ν, and that the negative support of D(ν) has the dual properties. When we speak of a cycle in a directed graph, we shall mean a cycle in the corresponding undirected graph; we shall also understand that in a cycle no vertex is traversed more than once. Note that a cycle of length 1 in Γ(ν) can only arise when a term νx,x has nonzero coefficient in ν, while a cycle of length 2, i.e., the presence of two edges between x and y, can only occur if νx,y and νy,x both have nonzero coefficients. But a cycle of length n > 2 involving a given sequence of vertices may arise in any of 2n ways, depending on the orientations of the edges. We now prove Proposition 24 (cf. [11, Theorem 3.3, p.84]). Let µ ∈ U0. Then the infimum of (39) is attained by an element ν ∈W+ (not necessarily unique) whose positive and negative supports coincide respectively with the positive and negative supports of µ, and whose graph Γ(ν) has no cycles. Proof. As mentioned, Weaver proves the existence of a ν ∈ W+ with D(ν) = µ which achieves the infi- mum (39) and has the same support as µ. Let ν be chosen, first, to have these properties; second, among such elements, to minimize the total number of edges in Γ(ν), and finally, to minimize the sum of the coefficients of all the νx,y in its expression. This last condition is achievable because the set of elements of W+ which are linear combinations of a given finite family of the νx,y, and for which the coefficients of these elements are all ≤ some constant, is compact; so after finding some ν with D(ν) = µ which achieves the minimum (39), has the same support as µ, and minimizes the number of edges in Γ(ν), we may restrict our search for elements also minimizing the coefficient-sum to the compact set of elements having these properties and having every coefficient less than or equal to the coefficient-sum of the element we have found. Suppose, now, that Γ(ν) has a cycle. Thus, we may choose distinct vertices p1, . . . , pk, and for each j ∈ {1, . . . , k}, a term νpj ,pj+1 or νpj+1,pj occurring with positive coefficient in ν, where the subscripts j are taken modulo k. (If k > 2 and both νpj ,pj+1 and νpj+1,pj occur in ν, we choose one of these arbitrarily. If k = 2, we make sure that the terms we choose for j = 1, 2 are distinct, one being νp1,p2 and the other νp2,p1 .) For each j ∈ {1, . . . , k} let us now define ν′pj ,pj+1 to be νpj ,pj+1 if that is the jth term in the list we have chosen, or −νpj+1,pj if the jth term in that list is νpj+1,pj , and let ν′ = ν′pj ,pj+1 ∈ W. In general, ν′ /∈ W+, but for all λ ∈ R near enough to 0, we have ν + λν′ ∈ W+, since the relevant coefficients in ν are strictly positive. Note that for each j, D(ν′pj ,pj+1) = µj − µj+1, hence D(ν ′) = 0, hence D(ν + λν′) = D(ν) = µ. Clearly, (x,y)∈X×X d(x, y) d(ν+λν′) is an affine function of λ. Hence it must be constant, otherwise, using small λ of appropriate sign, we would get a contradiction to the assumption that ν achieves the minimum of (39); so all the elements ν+λν′ achieve this same minimum. Now some choice of λ will cause λν′ to exactly cancel the smallest among the coefficients of terms νpj ,pj+1 or νpj+1,pj in our cycle in Γ(ν). Thus, ν + λν′ contradicts the minimality assumption on the number of edges in Γ(ν). This contradiction shows that Γ(ν) has no cycles. Next, let us compare the positive and negative supports of ν with those of µ. We have chosen ν so that its support, namely the union of its positive and negative supports, coincides with the support of µ; and since µ = D(ν), the positive and negative supports of ν will each contain the corresponding support of µ. So if these inclusions are not both equalities, we must have a vertex p which is both in the positive and the negative support of ν; i.e., such that there is an edge (q, p) of Γ(ν) leading into p, and an edge (p, r) leading out of it. Let ν′ = νq,r − νq,p − νp,r. Like the element denoted by that symbol in the preceding argument, this satisfies D(ν′) = 0. Let us again form ν + λ ν′, this time choosing the value λ > 0 which leads to the cancellation of the smaller of the coefficients of νq,p and νp,r in ν, or of both if these coefficients are equal. Since this does not reverse the sign of either of these coefficients, ν + λ ν′ still belongs to W+. Note that (x,y)∈X×X d(x, y) d(ν+λν′) ≤ (x,y)∈X×X d(x, y) dν, since by the triangle inequality, d(q, r) ≤ d(q, p) + d(p, r); so the property of minimizing the latter integral among elements of W+ mapped to µ by D has not been lost. Also, Γ(ν + λν ′) has dropped at least one edge that belonged to Γ(ν), since at least one coefficient was canceled, and has gained at most one edge, namely (q, r) (if that was not previously present); so the total number of edges has not increased. Finally, when we look at the sum of all the coefficients, we see that the coefficient of νq,r has increased by λ, while those of νq,p and νp,r have both decreased by λ, so there has been a net change of −λ < 0. Thus, we have a contradiction to our choice of ν as minimizing that sum. This completes the proof of the main assertion of the proposition. Let us verify, finally, the parenthetical comment that the ν of the proposition may not be unique. Let X be a 4-point space {x1, x2, x3, x4} where the distance between every pair of distinct points is 1, and let µ = µx1 + µx2 − µx3 − µx4 . It is not hard to check that in this case, the only elements ν that can possibly satisfy the conditions of the proposition are νx1,x3 + νx2,x4 and νx1,x4 + νx2,x3 . Since these give the same value for the integral of (39), d(x1, x3) + d(x2, x4) = 2 = d(x1, x4) + d(x2, x3), each satisfies our conditions. (Of course, for most choices of metric on this set X, one of these two values is smaller than the other, and we then get a unique ν satisfying the conditions of the proposition.) To get a self-contained version of the above proof which includes the existence result we cited from [23], one may start by looking at any finite subset X0 of X containing the support of µ, verify by compactness as above that the infimum of (39) over elements ν with support contained in X0 is achieved, then note that any element in the support of ν but not in the support of µ must belong to both the positive and negative supports of ν, a situation excluded by the proof. Letting X0 then run over all finite subsets of X containing the support of µ, one sees that the infimum of (39) exists, and is simply the infimum with ν restricted to have support in the support of µ. We remark that the final condition of the above proposition, that Γ(ν) have no cycles, is not entailed by the other conditions. E.g., returning to X = {x1, x2, x3, x4} with all distances 1, we see that every convex linear combination ν of the two elements that we found, νx1,x3 + νx2,x4 and νx1,x4 + νx2,x3 , still minimizes (39), and still has support X ; but if ν is a proper convex combination of those two elements, then Γ(ν) = Γ(νx1,x3+ νx2,x4) ∪ Γ(νx1,x4+ νx2,x3), which contains (indeed, is) a cycle. Let us now show, however, that when, as in the statement of the proposition, Γ(ν) is cycle-free, it uniquely determines ν. Thus, the calculation of the norm (39) reduces in principle to checking finitely many ν. Lemma 25. Let µ ∈ U0, and let Γ be a directed graph with vertex-set X and without cycles. Then there is at most one ν ∈W (and so, a fortiori, at most one ν ∈ W+) such that D(ν) = µ and Γ(ν) ⊆ Γ. To characterize this element ν, consider any edge (x, y) in Γ. Let Γx (containing x) and Γy (containing y) be the connected components into which the connected component of Γ containing (x, y) separates when that edge is removed. Then the coefficient in ν of νx,y is the common value of dµ and − dµ, i.e., is both the sum of the coefficients of µz over z in Γx, and the negative of the corresponding sum over Γy. Proof. We will prove the assertion of the second paragraph, from which that of the first clearly follows. Writing µ = D(ν), the contributions to the expression dµ from any term νp,q such that both p and q lie in Γx clearly cancel, while terms such that neither p nor q lies in Γx contribute nothing. This leaves the νx,y term, which contributes precisely its coefficient, leading to the first description of that coefficient. Likewise, this term contributes the negative of its coefficient to dµ, yielding the second description. Corollary 26. Suppose µ ∈ U0 is integer-valued, and let ν be an element of W+ with the properties that D(ν) = µ, that ν has the same positive and negative supports as µ, and that Γ(ν) has no cycles. Then for every x ∈ X such that the coefficient of µx in µ is ±1, the vertex x is a leaf of Γ(ν). Hence, if µ has the property that the coefficient of every µx is ±1, then ν is induced, in the obvious way, by a bijection between the positive support of µ and the negative support of µ. Thus, in that case, letting n be the common cardinality of these supports, there are exactly n! such ν ∈ W+. Proof. Consider any x such that µx has coefficient +1 in µ. Then x is in the positive support of Γ(ν), but not in the negative support. The latter condition says that ν involves no terms νy,x (y ∈ X), so +1 is the sum of the coefficients in ν of the terms νx,y (y ∈ X − {x}). Since ν ∈ W+, these coefficients are nonnegative, and by Lemma 25 they are integers, so as they sum to 1, only one of them can be nonzero, making x a leaf. The same argument, mutatis mutandis, gives the case where the coefficient of µx is −1. The assertion of the final paragraph clearly follows, since a directed graph in which every vertex is a leaf corresponds to a bijection between “source” and “sink” vertices. As sample applications, recall the two computations at the beginning of this section, with which we motivated the construction of our universal embedding e : X → U0. In our present notation, what we were doing was evaluating the norms in U0 of elements of the two forms µx1 + µx2 − µx3 − µx4 and 3µx1 + µx2 − 2µx3 − 2µx4. In the first case, the last paragraph of the above corollary leads to just two graphs, and hence two values of ν one of which must achieve the infimum (39), namely νx1,x3 + νx2,x4 and νx1,x4 + νx2,x3 , showing that if f is our universal map e, equality holds in (36), and for general f, (36) is the best bound. (This also establishes the example that we said was “not hard to check” in the next-to-last paragraph of the proof of Proposition 24.) In the case µ = 3µx1 + µx2 − 2µx3 − 2µx4, Corollary 26 says that x2 is a leaf of Γ(ν). As it lies in the positive support of µ, the vertex it is attached to must lie in the negative support, i.e., must be either x3 or x4. In the former case, subtracting νx2,x3 from ν will give an element ν ′ ∈ W+ which is sent by D to 3µx1 − µx3 − 2µx4 . Since this ν′ has only one element, x1, in its positive support, its graph is uniquely determined, giving ν′ = νx1,x3 +2νx1,x4 , hence ν = νx2,x3 + νx1,x3 +2νx1,x4 . The case where x2 is attached to x4 similarly gives ν = νx2,x4 + 2νx1,x3 + νx1,x4 , and these together show that (37) is a best bound. I referred earlier to the mathematical economist’s “transportation problem”. There, our d(x, y) corre- sponds to the cost of transporting a unit quantity of goods from location x to location y; so our definition of ||µ|| describes the minimum cost of transporting goods produced and consumed at locations and in quantities specified by µ. Incidentally, the first assertion of Corollary 26 does not remain true if we weaken “the coefficient of µx in µ is ±1” to “the coefficient of µx has least absolute value among the nonzero coefficients occurring in µ.” For instance, suppose µ has the form 3µx1 − 4µx2 + 2µx3 − 4µx4 + 3µx5 . Then one of the elements of W+ satisfying the conditions of Corollary 26 is ν = 3νx1,x2 + νx3,x2 + νx3,x4 + 3νx5,x4 . Here Γ(ν) has the form x1 → x2 ← x3 → x4 ← x5, so x3, despite having smallest coefficient in µ, is not a leaf. Proposition 24 sheds some light on our earlier “partial universality” result, Corollary 14(i). Given any convex linear combination of points of our universal image of X, aiµxi (ai > 0, ai = 1), and any point x ∈ X (which for simplicity we will assume is not one of the xi, though the argument can be adjusted to the case where it is), the difference µx − aiµxi is an element of U0 with positive support {x} and negative support {xi | i ∈ I}. For this situation, the conditions of Proposition 24 clearly lead to a unique Γ(ν), to wit, the tree whose edges are all pairs (x, xi) (i ∈ I), and hence to the unique choice ν = ai νx,xi . Thus, the right-hand side of (39) comes to ai d(x, xi), which is equal to the right-hand side of (19). In contrast, when one considers the difference between two general convex linear combinations of elements µx, as in Corollary 14(ii), there may be many directed graphs satisfying the conditions of Proposition 24, so the norm of that difference doesn’t have a simple expression. The universality of the Arens-Eells space U yields a formula for map-rad(X,NmV) analogous to our description (16) of map-rad(X,Conv); namely, (40) map-rad(X,NmV) = infµ∈U1 supx∈X infν∈W+, D(ν)=µ−µx y,z∈X d(y, z) dν(y, z). But this is cumbersome to use. E.g., the reader might try working through a verification, for the space described by (25), of the statement that the µ implicit in (29), (µy0 + µy1 + µy2 − µx)/2, does indeed lead to the infimum of (40), showing that map-rad(X,NmV) = 5/2, and not a smaller value. Given an element µ ∈ U0, it would be interesting to look for bounds on the number of distinct graphs Γ(ν) corresponding to elements ν ∈W+ as in the first sentence of Corollary 26. (This is simply a function of the coefficients occurring in µ, as a family of positive real numbers with multiplicities.) To start with, one might look for bounds in terms of the cardinalities of the positive and negative supports of µ. Weaver [23] also gets a description of the universal nonexpanding map of X into a normed complex vector space, paralleling the description for the real case, but he notes [23, p.43, next-to-last paragraph of §2.2] that in the complex case it is no longer true that the infimum corresponding to (39) is always attained by a ν having the same support as µ. (In the complex version of (39), by the way, one must replace dν by |dν|, instead of restricting ν to a “positive cone” as above, since there is no natural analog of that cone. Weaver takes this approach for both the real and complex cases; my use of W+ for the real case is one of the different technical choices that I have made.) For instance, if X = {x, y0, y1, y2} with d(x, yi) = 1 and d(yi, yj) = 2 (i 6= j), and if µ = µy0 + ωµy1 + ω2µy2 , where ω is a primitive cube root of unity, then the minimizing ν is νy0,x + ωνy1,x + ω 2νy2,x, which makes that integral 3, while the best ν having support in the support of µ, {y0, y1, y2}, is 13 (1− ω)νy0,y1 + (ω − ω2)νy1,y2 + 13 (ω 2 − 1)νy2,y0 , of which each term contributes 1 3 · 2 to that integral, giving a total of 3 · 2 = 12 > 3. If in this space we replace the point x by a sequence of points x1, x2, . . . , such that d(xm, yi) = 1 + 1/m and d(xm, xn) = |1/m− 1/n|, the above µ still has ||µ|| = 3, but the infimum defining that norm is not achieved. 7 Appendix: Translating convex sets to 0. In §3, we saw that the radius of a subset X of a normed vector space V could be larger when measured within a convex subset C of V than within the whole space V. If we regard this as a pathology, we would like to know in which V it does not occur. We shall obtain partial results below, which, we will see, make it likely that for n > 2, the only norms on Rn for which it does not happen are those giving a structure isomorphic to En. Observe that the radius of X, whether within V or within a convex subset C, is determined by the set of closed balls containing X, and that these are all convex; hence that radius is a function of the convex hull of X. So our question reduces to the case where X is convex. Moreover, if X shows the above behavior with respect to one convex subset C of V, it will show it with respect to any smaller convex subset in which it lies; these two observations reduce our question to the case where C = X. This reduction is the equivalence of conditions (41) and (42) of the next lemma. Condition (43) then reformulates the problem. (Note that in (43) and similar statements throughout this section, an expression such as “C − v” will denote the translate of the set C by the vector −v, in contrast to notations such as X−{x} for set-theoretic difference, used occasionally in earlier sections.) Lemma 27. If V is a locally compact normed vector space, with closed unit ball B, then the following conditions are equivalent: (41) For every nonempty subset X of V, and convex subset C of V containing X, one has radV (X) = radC(X). (42) For every nonempty convex subset C of V, one has radV (C) = radC(C). (43) Every nonempty closed convex subset C of B has a translate C − v which contains 0 and is again contained in B. Proof. We have noted the equivalence of (41) and (42); let us prove (42) equivalent to (43). (43)⇒ (42): Dilating by arbitrary constants, we see that if (43) holds for B, then it holds for rB for all positive real numbers r. Moreover, the statement that C − v contains 0 and is contained in rB is equivalent to saying that v ∈ C and that v+ rB contains C; i.e., that C is contained in the ball of radius r about v ∈ C. Thus (43) says that if a closed convex set C is contained in some closed ball about some point of V (taken without loss of generality to be 0), then it is contained in a ball of the same radius about one of its own points. This yields the case of (42) where C is closed. The facts that the convex hull of a finite subset of V is compact, hence closed, and that the radius of an arbitrary set is the supremum of the radii of its finite subsets, allow us to deduce the general case of (42) from the case of closed C. (42)⇒ (43): If C is a closed convex subset of V contained in B, then radV (C) ≤ 1, so by (42), radC(C) ≤ 1. Moreover, compactness of C implies that the set of radii of closed balls containing C and centered at points v ∈ C achieves this minimum radC(C) ≤ 1, so that C is contained in a translate v+B (v ∈ C), i.e., C − v ⊆ B. Now (43) is a statement purely about the convex set B in the topological vector space V, so our question becomes that of which subsets B of a topological vector space V satisfy it. (In the statement of the lemma, the topology and the set B both arise from the normed structure on V ; but that relation is not needed by the statement of (43) alone.) Here are some pieces of language, one ad hoc, the rest more or less familiar, that we will use in examining this question. Definition 28. If C ⊆ B are convex subsets of a real vector space V, with 0 ∈ B, we shall call C parkable in B if there exists v ∈ C such that C − v ⊆ B. When clear from context, “in B” may be omitted. If V is a real topological vector space, we will call sets of the form {x ∈ V | L(x) = a}, where L is a nonzero continuous linear functional on V and a ∈ R, hyperplanes, while sets of the form {x ∈ V | L(x) ≥ a} will be called closed half-spaces. A subset S of a vector space V will be called centrally symmetric if S = −S. A center of symmetry of a subset S of V will mean a point v ∈ V such that S − v is centrally symmetric; equivalently, such that S = 2v − S. (Note that a center of symmetry of a nonempty convex set belongs to that set.) Lemma 29. Let B be a compact convex subset of Rn containing 0. Then the following conditions are equivalent: (44) The intersection of B with every hyperplane A that meets B is parkable. (45) The intersection of B with every closed half-space H that meets B is parkable. (46) Every nonempty closed convex subset C of B is parkable (= (43) above). Proof. (46)⇒ (44) is clear; we will show (44)⇒ (45)⇒ (46). (44)⇒ (45): Let H be a closed half-space in Rn, bounded by a hyperplane A, and meeting B. If H∩B contains 0 it is trivially parkable, so assume the contrary. Thus B meets both H and its complement, hence it meets their common boundary A, so by (44) there exists v ∈ A ∩ B such that (A ∩B) − v ⊆ B. I claim that (H ∩ B) − v is also contained in B. Indeed, let p ∈ H ∩ B; we wish to show p − v ∈ B. Intersecting our sets with the subspace of V spanned by p and v, and taking appropriate coordinates, we may assume that n = 2, that A is the line {(x, y) | y = 1} ⊆ R2, and that v is the point (0, 1). H will be the closed half-plane {(x, y) | y ≥ 1}, so we can write p = (xp, yp) with yp ≥ 1. In this situation, A ∩ B will be a line segment (possibly degenerate) extending from a point (s, 1) to a point (t, 1) (s ≤ t). Since (A ∩ B) − v ⊆ B, B also contains the segment from (s, 0) to (t, 0). Note that if xp were > t, then the point where the line segment from p = (xp, yp) ∈ B to (t, 0) ∈ B meets A would have x-coordinate > t, contradicting the assumption that A ∩ B terminates on the right at (t, 1); so xp ≤ t. Similarly, xp ≥ s. Thus, xp ∈ [s, t], so (xp, 0) ∈ B. Hence p − v = (xp, yp−1) lies on the line segment connecting p = (xp, yp) ∈ B with (xp, 0) ∈ B, hence lies in B, as claimed. (45)⇒ (46): Suppose C is a nonempty closed convex subset of B which is not parkable. By compactness of B, among the translates of C contained in B there is (at least) one that minimizes its distance to 0 in the Euclidean norm on Rn; let us assume C itself has this property. Let p be the point of C nearest to 0 in that norm, and let A be the hyperplane passing through p and perpendicular (again in the Euclidean norm) to p regarded as a vector. Then C will lie wholly in the half-space H bounded by A and not containing 0. (For if we had q ∈ C not lying in H, then points close to p on the line segment from p to q would be nearer to 0 than p is.) Assuming (45), H ∩B is parkable; say v ∈ H ∩B with (H ∩B)− v ⊆ B. Since v ∈ H, if we write v as the sum ap+ q of a scalar multiple of p and a vector q perpendicular to p, the coefficient a will be ≥ 1, and so in particular, positive. It follows that for sufficiently small positive c, the point p − cv will be closer to 0 than p is; moreover, if we take such a c that is ≤ 1, (H ∩ B) − cv will still be contained in B, since H ∩B and (H ∩B)− v are. Hence C − cv is contained in B, and has a point p− cv which is closer to 0 than p is, contradicting our minimality assumption on C and p. (In the above result, we could have replaced Rn by any real Hilbert space.) Clearly, the closed Euclidean unit ball in Rn satisfies (44), and hence (45) and (46); hence since those conditions are preserved by invertible linear transformations, so does the closed region enclosed by any ellipsoid centered at 0. On the other hand, our example in the paragraph containing (23), of a normed vector space in which (41) failed, had for its unit ball B a 3-cube centered at 0, showing that our properties fail for that B. To see geometrically the failure of (44) for that B, choose a vertex of that cube and pass a plane A through the three vertices adjacent thereto; it is not hard to see that A ∩ B is not parkable. One can similarly show that none of the regular polyhedra centered at 0 satisfy (44), nor a circular cylinder centered at 0, nor the solid obtained by attaching a hemisphere to the top and bottom of such a cylinder. In fact, for n > 2, I know of no compact convex subset of Rn with nonempty interior that does satisfy that condition, other than the regions enclosed by ellipsoids centered at 0. The situation is different for n = 2, as shown by point (d) of the next result. Lemma 30. Suppose B is a centrally symmetric convex subset of Rn. Then (a) Any nonempty convex subset C ⊆ B that has a center of symmetry is parkable in B. Hence, assuming in the remaining points that B is also compact, we have (b) If the intersection of B with every hyperplane A that meets B has a center of symmetry, then B satisfies the equivalent conditions (44)-(46). In particular, (c) If B is the closed region enclosed by an ellipsoid in Rn, then B satisfies (44)-(46), and (d) If n = 2, then without further restrictions, B satisfies (44)-(46). Proof. Let C be as in (a), with center of symmetry z ∈ C. Then for every x ∈ C, 2z − x ∈ C ⊆ B, so by central symmetry of B, we have x− 2z ∈ B. Averaging x and x− 2z, we get x− z ∈ B. Thus C− z ⊆ B, so C is parkable. It follows that any B as in (b) satisfies (44), hence by Lemma 29, all of (44)-(46). In the situation of (c), the intersection of B with a hyperplane A, if nonempty, is either a point or the region enclosed by an ellipsoid in A, hence has a center of symmetry, while in (d) the intersection of B with every line in R2 that meets B is a point or a closed line segment, hence has a center of symmetry; so in each case, (b) gives the asserted conclusion. Question 31. Let n ≥ 3, and suppose B is a compact convex subset of Rn having nonempty interior and containing 0. Of the implications (i)⇒ (ii)⇒ (iii), which we have noted hold among the conditions listed below, is either or both reversible? (i) B is an ellipsoid centered at 0. (ii) B is centrally symmetric, and for every hyperplane A meeting B, A ∩B has a center of symmetry. (iii) Every closed convex subset of B is parkable in B. Branko Grünbaum has pointed out to me a similarity between this question and the result of W.Blaschke [3, pp.157–159] that if E is a smooth compact convex surface in R3 with everywhere nonzero Gaussian curvature, such that when E is illuminated by parallel rays from any direction, the boundary curve of the bright side lies in a plane, then E is an ellipsoid. I believe that methods similar to Blaschke’s may indeed show that both implications of Question 31 are reversible. To see why, suppose B is a compact convex subset of R3 with nonempty interior containing 0, which satisfies (iii) above, and whose boundary E is (as in Blaschke’s result) a smooth surface with everywhere nonzero Gaussian curvature. Let A be any plane through 0, and A′ the plane gotten by shifting A a small distance. Now the vectors that can possibly park A′ ∩ B are constrained by the directions of the tangent planes to E at the points of A′ ∩ E (which are well-defined because E is assumed smooth), and if we take A′ sufficiently close to A, these tangent planes become close to the corresponding tangent planes at the points of A ∩ E. Applying the above observations to planes A′ on both sides of A, one can deduce that all the tangent planes to E along A∩E must contain vectors in some common direction (I am grateful to Bjorn Poonen for this precise formulation of a rough idea I showed him); in other words, that A ∩ E is the boundary of the bright side when E is illuminated by parallel rays from that direction. By definition, A∩E lies in the plane A; so we have the situation that Blaschke considered, except that we have started with planarity and concluded that the curve is a boundary of illumination, rather than vice versa. That last difference is probably not too hard to overcome. More serious is the smoothness assumption on E, used in both the above discussion and Blaschke’s argument. Finally, can the result be pushed from n = 3 to arbitrary n ≥ 3 ? I leave it to those more skilled than I in the subject to see whether these ideas can indeed be turned into a proof that (iii)⇒(i) in Question 31. A related argument which can be extracted from a step in Blaschke’s development shows that a compact convex subset B of R2 containing 0 and satisfying (46), whose boundary is a smooth curve containing no line segments, must be centrally symmetric. Again, one would hope to remove the conditions on the boundary. One can ask about a converse to another of our observations: Question 32. Suppose C is a compact convex subset of Rn (n > 2) such that for every centrally symmetric compact convex subset B of Rn containing a translate C′ of C, the set C′ is parkable in B. Must C have a center of symmetry? Here the behavior of a given C can change depending on whether the dimension of the ambient vector space is 2 or – as in the above question – larger: a triangle C has the above property in R2 by Lemma 30(d), but not in R3, as we saw in the example where B was a cube. Returning to the “pathology” which motivated the considerations of this section, one important case is where the radius of a subset X of a normed vector space V decreases when V is embedded in a larger normed vector space W. The next lemma determines how far down the radius of a given X can go. Lemma 33. Let V be a normed vector space, and X a bounded subset of V. Then (47) infW⊇V radW (X) = radV {(x− y)/2 | x, y ∈ X}, where W ranges over all normed vector spaces containing V. This infimum is realized by a W in which V has codimension 1. Proof. First consider any normed vector space W containing V, and suppose X is contained in the closed ball of radius r about w ∈W. That ball has w as a center of symmetry, so it also contains {2w−y | y ∈ X}, hence taking midpoints of segments connecting that set to points x ∈ X, it contains {w+ (x− y)/2 | x, y ∈ X}. Translating by −w, we see that the ball of radius r about 0 contains {(x− y)/2 | x, y ∈ X}, so r is at least the right-hand side of (47). This gives the inequality “≥ ” in (47); it remains to construct a W for which radW (X) equals that right-hand side. Before doing this, note that (47) holds trivially if X is empty or a singleton; so assuming it is neither, let us re-scale and assume without loss of generality that the right-hand side of (47) equals 1. Since the set whose radius is taken there is centrally symmetric, that set is contained in the closed unit ball BV of V. (Cf. the proof of Lemma 30(a), which works not just for Rn, but for any normed vector space with B its closed unit ball; or the proof of Lemma 5, applied to the 2-element group generated by x 7→ −x.) Now let W = V ⊕ R, let us identify V with V × {0} ⊆ W, and let us take for the closed unit ball BW of W the closure of the convex hull of (48) {(x, 1) | x ∈ X} ∪ BV ∪ {(−x,−1) | x ∈ X}. (We understand “closure” to mean “with respect to the product topology”, since we don’t have a norm until we have made the above definition.) It is easy to see that any point in the convex hull of (48) whose second coordinate is 0 is a convex linear combination of a point of BV and a member of the set on the right-hand side of (47); but by assumption that set is contained in BV ; so in fact, BW ∩ V = BV , so the norm of W indeed extends that of V. But BW contains the translate {(x, 1) | x ∈ X} of X, hence X is contained in the closed ball of radius 1 about (0,−1), hence has radius ≤ 1 in W. Even the case V = En is not immune to this phenomenon, since even in that case, the overspace W of the above construction is generally not Euclidean. For instance, if we take for X an equilateral triangle in 2 centered at the origin, it is not hard to see that {(x− y)/2 | x, y ∈ X} is a hexagon whose vertices are the midpoints of the edges a regular hexagon with the same circumcircle as X ; so the radius of X decreases in W by the ratio of the inradius to the circumradius of a regular hexagon, in other words, by This will not, of course, happen for a centrally symmetric X (cf. Lemma 30 or (47)). Other cases for which it cannot happen depend on the metric structure: if X is a right or obtuse triangle in E2, or more generally, any bounded set containing a diameter of a closed ball in which it lies, its radius clearly cannot go down under extension of the ambient normed vector space (cf. Corollary 10). 8 Acknowledgements. In addition to persons acknowledged above, I am indebted to W.Kahan for showing me an exercise he had given his Putnam-preparation class, of proving Lemma 2 in E3, and for subsequently pointing out that my solution to that exercise worked in any normed vector space; to Nik Weaver for information about his results in [23], and to David Gale for pointing out the connection between the construction of §6 and results in mathematical economics. References [1] Richard F. Arens and James Eells, Jr., On embedding uniform and topological spaces, Pacific J. Math. 6 (1956) 397–403. MR 18, 406e. [2] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Math- ematischen Wissenschaften, v.319. Springer, 1999. MR 2000k:53038. [3] Wilhelm Blaschke, Kreis und Kugel, Leipzig, 1916, reprinted by Chelsea Publishing Company, New York, 1949. MR 17, 887b. [4] Herbert Busemann, Spaces with non-positive curvature, Acta Math. 80 (1948) 259–310. MR 10,623g. [5] G. D. Chakerian and M. S. Klamkin, Minimal covers for closed curves, Math. Mag., 46 (1973) 55–61. MR 47#2496. [6] S. S. Chern, Curves and surfaces in Euclidean space, Studies in Global Geometry and Analysis, pp.16-56, Math. Assoc. Amer., Studies in Math., v.4, 1967. MR 35#3610. [7] H. S. M. Coxeter, Regular Polytopes, Methuen & Co.; Pitman, 1948; 1949. MR 10,261e, and for subsequent editions, MR 27#1856 and MR 51#6554. [8] Ludwig Danzer, Branko Grünbaum and Victor Klee, Helly’s theorem and its relatives, Proc. Sympos. Pure Math., Vol. VII, pp. 101–180, Amer. Math. Soc., Providence, R.I., 1963. MR 28#524. [9] Werner Fenchel, Über Krümmung und Windung geschlossener Raumkurven, Math. Ann., 101 (1929) 238–252. [10] David Gale, On inscribing n-dimensional sets in a regular n-simplex, Proc. Amer. Math. Soc., 4 (1953) 222–225. MR 14,787b. [11] David Gale, The theory of linear economic models, McGraw-Hill, 1960; University of Chicago Press, 1989. MR 22#6599. [12] Hugo Hadwiger and Hans Debrunner, Combinatorial geometry in the plane, translated by Victor Klee, with a new chapter and other additional material supplied by the translator. Holt, Rinehart and Win- ston, New York 1964 vii + 113 pp. MR 29#1577. [13] J. H̊astad, S. Linusson and J. Wästlund A smaller sleeping bag for a baby snake, Discrete Comput. Geom., 26 (2001) 173–181. MR 2002b:52012. [14] E. Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten, Jahresber. Deutsch. Math.- Verein., 32 (1923) 175–176. [15] Einar Hille, Analytic function theory, v. II, Ginn and Co., Boston, 1962. MR 34#1490. [16] R. A. Horn, On Fenchel’s Theorem, Amer. Math. Monthly, 78 (1971) 380-381. MR 44#2142. [17] Heinrich W. E. Jung, Über die kleinste Kugel, die eine räumliche Figur einschliesst, J. Reine Angew. Math., 123 (1901) 241–257. [18] J. C. C. Nitsche, The smallest sphere containing a rectifiable curve, Amer. Math. Monthly, 78 (1971) 881–882. MR 45#480. [19] H. Rutishauser and H. Samelson, Sur le rayon d’une sphére dont la surface contient une courbe fermée, C. R. Acad. Sci. Paris, 227 (1948) 755–757. MR 10, 321c; correction, MR 10, p.856. [20] Jonathan Schaer and John E. Wetzel, Boxes for curves of constant length, Israel J. Math., 12 (1972) 257–265. MR 47#5726. [21] B. Segre, Sui circoli geodetici di una superficie a curvatura totale constante, che contengono nell’interno una linea assegnata, Boll. Un. Mat. Ital., 13 (1934) 279–283. Zbl 10, 271. [22] Philip C. Tonne, A simple closed curve on a hemisphere, Houston J. Math., 10 (1984) 585. MR 86b:53004. [23] Nik Weaver, Lipschitz algebras, World Scientific Publishing, River Edge, NJ, 1999, xiv+223 pp., ISBN: 981-02-3873-8. MR 2002g:46002. [24] Alan Weinstein, Almost invariant submanifolds for compact group actions, J. Eur. Math. Soc. 2 (2000) 53–86. MR 2002d:53076. [25] John E. Wetzel, Covering balls for curves of constant length, Enseignement Math., (2) 17 (1971) 275– 277. MR 48#12315. George M. Bergman Department of Mathematics University of California Berkeley, CA 94720-3840 [email protected] The definition, and three examples. General properties of mapping radii. Some explicit mapping radii. Realizability of mapping radii. Related literature (and one more question). Appendix: The Arens-Eells space of X. Appendix: Translating convex sets to 0. Acknowledgements.

0704.0276

A New Model For The Loop-I (The North Polar Spur) Region

A New Model For The Loop-I (The North Polar Spur) Region M. Wolleben1,2 ABSTRACT The North Polar Spur (NPS) is the brightest filament of Loop I, a large circular feature in the radio continuum sky. In this paper, a model consisting of two synchrotron emitting shells is presented that reproduces large-scale structures revealed by recent polarization surveys. The polarized emission of the NPS is reproduced by one of these shells. The other shell, which passes close to the Sun, gives rise to polarized emission towards the Galactic poles. It is proposed that X-ray emission seen towards the NPS is produced by interaction of the two shells. Two OB-associations coincide with the centers of the shells. A formation scenario of the Loop I region is suggested. Subject headings: ISM: magnetic fields — ISM: structure — polarization — solar neighborhood 1. Introduction Total intensity surveys reveal a number of radio spurs that can be joined into small circles on the sky, so-called radio loops. One of these is Loop I, which has an intriguing filament called the North-Polar Spur (NPS). It has been concluded by several authors (e.g. Berkhuijsen et al. 1971; Heiles 1979; Salter 1983, and references therein) that the radio loops are correlated with expanding gas and dust shells, energized by supernovae or stellar winds. Accordingly, the Loop I superbubble was attributed to stellar winds from the SCO-CEN OB association and supernova activity in the same vicinity, with the NPS being the brightest segment of a supernova remnant (SNR). Its magnetic field (B-field) has been modelled by Spoelstra (1972) and Heiles (1998) based on radio and optical polarization data, suggesting that the local B-field is deformed by an expanding shell. According to a model by Weaver 1Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4. 2National Research Council Canada, Herzberg Institute of Astrophysics, Dominion Radio Astrophysical Observatory, P.O. Box 248, Penticton, BC, V2A 6J9, Canada; [email protected] http://arxiv.org/abs/0704.0276v1 – 2 – (1979), an SNR, produced by a member of the SCO-CEN association, has expanded inside the Loop I bubble. Its shell is just beginning to encounter and interact with the surface of the surrounding H I shell (GS 331+14-15). There has been a debate over the expansion velocity (vexp) of GS 331+14-15: Weaver gives vexp ≈ 2 km s −1, based on data from the northern hemisphere, while Heiles (1984) and Sofue et al. (1974) find a higher vexp of 19 and 25 km s−1. In a picture by Bochkarev (1987) the H I shell has already passed the Sun. The relatively low expansion velocity of the neutral gas surrounding Loop I implies an age of some 106 years and hence suggests the NPS to be part of an old SNR. At this age, however, an SNR is well beyond its radiative phase and its optical and radio emission become invisible. On the other hand, X-ray and radio emission have been observed towards the NPS, indicating an age at least one order of magnitude less. Based on X-ray data, Borken & Iwan (1977) suggest that Loop I is an old but recently reheated SNR. This was refined by Egger (1995) who argue that Loop I is caused by a shock from a recent supernova, heating the inner wall of the SCO-CEN supershell, and giving rise to the prominent X-ray feature of the Recently, Wolleben et al. (2006) published the new DRAO Low-Resolution Polarization Survey of the northern sky at 1.4 GHz1, which is part of the Canadian Galactic Plane Survey (Taylor et al. 2003). Making use of these new data as well as the WMAP polarization data at 23 GHz (Page et al. 2006), a new and detailed study of the Loop I region can be performed. In this paper I describe a model consisting of two interweaving magnetic shells, which explains some newly detected polarization and depolarization structures, correctly predicts the shape of the NPS, as well as providing a clue to the origin of X-ray emission observed towards the NPS. 2. Large Polarized Structures In The DRAO Survey The polarization data used in this paper are shown in Figures 1 and 2 The NPS has long been known to be one of the most intensely polarized features of the sky. Its filaments cover large parts of the northern Galactic hemisphere at latitudes of b & 30◦, with an abrupt drop of polarized emission below 30◦ latitude at 1.4 GHz. Surprisingly, structures of the NPS are very evident in polarized intensity but its filamentary structure is not detectable in polarization angle. In the following, previously unknown polarization features detected in 1See also Wolleben (2005). The data are publicly available and can be downloaded at: http://www.drao.nrc.ca/26msurvey, http://www.mpifr-bonn.mpg.de/div/konti/26msurvey, or from CDS, Strasbourg (Wolleben et al. 2005) http://www.drao.nrc.ca/26msurvey http://www.mpifr-bonn.mpg.de/div/konti/26msurvey – 3 – the DRAO polarization survey, which are of relevance for this study, are described: 2.1. High Latitude Polarized Emission (HLPE) In Fig. 1 (top), towards the Galactic poles (b = ±90◦), a systematic pattern of polarized emission and polarization angle is visible (HLPE, hereafter). At 408 MHz, the all-sky map of Haslam et al. (1982) shows a weak counterpart of the HLPE in total intensity (Stokes-I) with excess emission at the Galactic poles of ∼ 4 K (northern pole) and ∼ 5 K (southern pole). The Stockert survey at 1420 MHz (Reich & Reich 1986) also shows weak Stokes-I emission at the poles of ∼ 60 mK (northern pole) and ∼ 200 mK (southern pole). These temperatures imply a spectral index of −3.4 and −2.6 for the northern and southern HLPE, respectively, and thereby suggest that the HLPE is synchrotron emission. What could cause enhanced synchrotron emission around the Galactic poles – the HLPE? First, the HLPE could be (polarized) synchrotron emission from the local spiral arm, which would produce a similar pattern of polarization angles towards the Galactic poles. Second, the HLPE could be caused by the Galactic halo. However, neither can explain the distinct boundaries of the HLPE, which are observed in the second and third Galactic quadrant (cf. thin white lines in Fig. 1 top). In fact, synchrotron emission from the halo or local spiral arm should be observable at all latitudes where the local Galactic B-field is perpendicular to the line-of-sight, and not just at the poles. 2.2. New Radio Loop A previously unknown filament of polarized emission in the southern Galactic hemi- sphere (hereafter referred to as “New Loop”) is revealed by the DRAO polarization survey. The New Loop has an excess polarized intensity of ∼ 170 mK at 1.4 GHz relative to its surroundings. The elongated shape of the New Loop can be fitted by a small circle. From a fit by eye, the center of this circle is at l = 345◦, b = 0◦, with a radius of 65◦ (dashed line in Fig. 1 top). Faint counterparts can be found in total intensity at 408 MHz (∼ 5 K) and 1420 MHz (∼ 250 mK), which gives a spectral index of −2.4. This filament can also be seen in the WMAP polarization data at 23 GHz. – 4 – 2.3. Depolarization Band A minimum of polarized emission from the first Galactic quadrant from ∼ 0◦ to ∼ 70◦ at 1.4 GHz, within a band confined by remarkably sharp boundaries at b ≈ ±30◦ (cf. Fig 1 top), is seen with great clarity for the first time. The percentage polarization within this area is around 3%, compared to 20 - 30% outside the band. Obviously, strong depolarization within the Galactic plane must “destroy” the polarized emission from the first Galactic quadrant. The Depolarization Band is prominent in the DRAO data at 1.4 GHz but not seen in the WMAP data at 23 GHz, which ephasises the assumption that it is caused by depolarization due to Faraday rotation. The length of the line-of-sight makes differential Faraday rotation a likely depolarization mechanism2. However, the absence of polarized emission from the NPS below b = 30◦, in the DRAO data at 1.4 GHz as well as in the WMAP data at 23 GHz, cannot be attributed solely to the Depolarization band. It must reflect an inherent property of the emission region. This is explained by the model presented in this paper. 3. The Model The model consists of two synchrotron emitting shells: S1 and S2. The shells are spherical with constant shell thickness. No emission is produced outside the shells. The Sun resides within S1 between its inner and outer surface (see Fig. 3 and 4). Theoretical models of magnetic fields in supershells (e.g. Ferriere et al. 1991; Tomisaka 1992) predict that the ambient B-field is pushed by the shock wave and compressed within the expanding shell. The model described here resembles these theoretical B-fields in the simplest way: Magnetic field lines run from the polar caps (the magnetic poles) along longitudes of the shell. The appearance of the B-field of the shell, that is, its B⊥ and B‖ components, depends on the vantage point. In case of a large nearby shell the projection can result in a complicated B-field pattern on the sky. Each shell is described by 8 parameters (Tab. 1): the center coordinates l, b, d; the inner (rin) and outer (rout) radius of the shell; the angle between the B-field and the line- of-sight to the Galactic center (Bθ) and to Galactic north (Bφ); and two scaling factors describing the intrinsic brightness of each shell. For simplicity, it is assumed that there is no Faraday rotation within the shells. The model described here is an attempt to reproduce the polarized emission – the emission tracing the regular B-field – because this model does 2Differential Faraday rotation occurs if synchrotron emitting and Faraday rotating regions are mixed. In this case, polarization vectors with different orientation may superimpose and cancel each other (see also Burn 1966). – 5 – not take irregular or turbulent B-field components into account. Therefore, the model must be fitted to polarization maps, as is done here. Moreover, polarization maps at frequencies around 1 GHz or less are believed to show rather local emission due to depolarization, which helps avoiding confusion with unrelated background emis- sion. The model uses different field orientations in the two shells. Where the shells overlap in space this is obviously not correct. But the complex shape of the edge of the Local Bubble (cf. Fig 3) suggests a very complex evolution, probably the outcome of many earlier stellar winds and supernova events. Any previous large-scale magnetic field has probably been tangled by this complex evolution. Presumably the Local Bubble is not a unique circumstance, so the implication is that most regions in the Galaxy are not characterized by a large-scale quasi- uniform component. The two-field model proposed is a simple approximation to a complex reality, and is unlikely to be correct over the full extent of the shells. However, the good fit to the data demonstrates that it is an adequate model for most of the Loop I region. The model was used to calculate the integrated Stokes U and Q values for each line- of-sight through the shell complex. Polarized intensity and polarization angle were derived from these integrated values, thereby accounting for depolarization due to superposition of differently oriented polarization vectors along the line-of-sight (see Appendix for a more detailed description). In the northern hemisphere, DRAO data were preferred over WMAP data because of the better sensitivity. Some regions in the DRAO maps were masked out however: the “Fan-Region” (see Fig. 1), the two H II-regions S 27 (l = 4◦, b = 22◦) and S 7 (l = 350◦, b = 22◦), and a ±3◦ strip along the Galactic plane. In the southern hemisphere, 23 GHz WMAP data (scaled to 1.4 GHz using a spectral index of −3) were preferred over the 1.4 GHz polarization survey of Testori et al. (2004) because the western caps lie at b ≈ 0◦ where depolarization at 1.4 GHz is likely to be strong. In the WMAP data the Galactic plane was masked out because a simple “scaling up” of WMAP data from 23 GHz to 1.4 GHz, without taking depolarization effects into account, would result in wrong polarized intensities in regions where Faraday rotation along the line-of-sight is high. The southern Galactic pole in the WMAP was masked out because here sensitivity is too low to permit accurate fitting. Fitting was done by computer, applying an algorithm that searched for the minimum of the square-root of the sum of the differences between modelled and observed polarized intensities. The algorithm randomly modified model parameters until a good fit was achieved. Different initial start values for the fit were chosen to evaluate the uniqueness of the best-fit. In order to determine the confidence ranges of the best-fit parameters, models with slightly different sets of parameters were tried and visually inspected. – 6 – 4. Discussion Figures 2 and 5 show polarized intensity and polarization angle maps of the best-fit model. At high Galactic latitudes the HLPE is correctly reproduced by S1. Only S1 produces HLPE because the Sun is located inside this shell, leading to local emission around the Sun. S2 produces the polarized emission of the NPS. At intermediate latitudes, where the line-of-sight through the B⊥ component of S1 is longest, the New Loop is seen. At low latitudes (|b| . 30◦) synchrotron emission from the two shells is reduced because the path lengths through the shells are short and B⊥ is small. Although the model was only fitted to polarized intensities, the polarization angles at intermediate and high Galactic latitudes are remarkably well reproduced. The predicted western cap region does not fit the observations as well as the eastern cap region. However, the western caps are likely to be inside the Local Bubble, while the eastern caps may have expanded into a denser medium. Hence, the “real” shells are not likely to be of perfectly spherical shape. However, a comprehensive model which includes the full complexity of the ISM around Loop I is beyond the scope if this paper, and is probably impossible for want of adequate data. Portions of the two proposed synchrotron shells currently overlap in space, which, pro- jected onto the sky, results in a ring-like region in the northern Galactic hemisphere (see Fig. 5 bottom). This region roughly agrees with the location of 1.5 keV X-ray emission from the NPS. It is therefore suggested that the X-ray emission is produced by recent interaction of the two shells within this region. 5. Possible Formation History The SCO-CEN association can be divided into three subgroups: Lower Centaurus Crux (LCC), Upper Centaurus Lupus (UCL), and Upper Scorpius (US). Máız-Apellániz (2001) calculated the positions of the center of each subgroup in the past, taking the effects of solar motion, Galactic rotation, and motions in the z-direction into account. Accordingly, the stars whose paths come closest to the center of S1 are those of the LCC3, which crossed this point 6± 2 Myr ago at a distance of 70 pc from the Sun. Closest to the center of S2 is the US group at its current location (l = 350◦, b = 20◦, 145 pc away from the Sun). Stellar activity within the LCC subgroup started 11 to 12 Myr ago, and 5 to 6 Myr ago within the 3Note that the scattering of members of these subgroups across the sky means that the UCL is almost as good a candidate. – 7 – US (de Geus et al. 1989). The picture that emerges from these data can be summarized as follows. The first supernovae are expected to take place about 3-5 Myr after formation of an association. Thus, about 7 to 8 Myr ago, supernova explosions began to occur within the LCC and, subsequently, delivered enough energy to inflate a bubble around LCC – the S1-shell. The US, instead, has only recently reached its evolutionary time scale and has just begun inflating the Loop I bubble (S2). The shock front of Loop I (S2) hit the LCC bubble (S1) just recently (104 years ago or less), giving rise to the X-ray emission observed toward the NPS. In this picture S1 is about 6 Myr old, based on the positional coincidence with the LCC at this time. S2 is 1-2 Myr old, based on the the age of the US. This suggests that S1 is more evolved than S2, which may explain why almost no observational tracers of S1 can be found in total intensity surveys. S1 seems to be an almost dissolved shell, whose last observable remnants are the HLPE and the New Radio Loop. The scenario proposed here has compelling similarities to the models of H I shells in this region by Weaver (1979) and Bochkarev (1987), mentioned earlier in the Introduction. The ages derived above and best-fit radii of the model give expansion velocities of 8 and 32 km s−1 for S1 and S2, respectively, using the standard model for the kinematic age of stellar wind bubbles with tkin = 0.6R/vexp (Weaver et al. 1977). Taking into account the uncertainties in this estimate, these velocities agree with the velocities of 2 km s−1 (Weaver 1979) and 19-25 km s−1 (Heiles 1984; Sofue et al. 1974) found for H I gas in this region. This may imply that the H I distribution in this region is the result of two superimposed H I shells that are expanding with different velocities. 6. Conclusions Based on an analysis of the new DRAO polarization survey and scaled WMAP data, a model consisting of two synchrotron emitting shells is developed that reproduces large-scale structures in the polarized sky. One of these shells, S1, has reached the Sun and gives rise to polarized synchrotron emission at high Galactic latitudes, the HLPE described in this paper. Where the path length through S1 is longest and its B-field is perpendicular to the line-of-sight over its whole length in the emitting region, a new radio loop is detected in polarized intensity. A younger shell, S2, correctly reproduces emission from the NPS. A scenario is proposed in which S2 has recently started to interact with S1, possibly giving rise to the observed X-ray emission. The model also predicts correctly the low polarization from S1 and S2 within the Depolarization Band. The picture of the local ISM suggested in – 8 – this paper is in agreement with previous studies outlined in the Introduction, although the two-shell geometry of the model makes a comparison with previous models difficult. The author would like to thank T. L. Landecker, E. M. Berkhuijsen, R. Kothes, A. D. Gray, and P. Vaudrevange for comments on the manuscript. The Dominion Radio Astrophys- ical Observatory is a National Facility operated by the National Research Council Canada. The Canadian Galactic Plane Survey is a Canadian project with international partners, and is supported by the Natural Sciences and Engineering Research Council (NSERC). I acknowl- edge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. A. Magnetic Field Used In The Model In this model an approximation for the magnetic field ~Bi inside an expanding shell is used. Its strength and direction depend on the direction (θ, ϕ) of the ambient magnetic field B̃ as well as the inner and outer radii of the shell (rin, rout). Looking at Fig. 6, it is easy to see that the magnitude of ~Bi is maximal / zero where the expansion is perpendicular / parallel to B̃. For |~x| > rout and |~x| < rin the magnetic field strength ~Bi is assumed to be zero. The direction of ~Bi is given by B̂i = |~ρi| , (A1) where ~ρi = b̂ − (b̂ · x̂)x̂ with b̂ the unit vector of the ambient field B̃ and x̂ the unit vector pointing towards the i-th line-of-sight element. The field strength of ~Bi can be evaluated to ni = (b̂ · x̂) 2, (A2) giving rise to the Stokes parameters ui ∝ ni B (γ+1)/2 ⊥ sin 2 arctan qi ∝ ni B (γ+1)/2 ⊥ cos 2 arctan for the i-th line-of-sight. Here B⊥ = B2l +B b and Bl, Bb are the projections of ~Bi onto the line-of-sight. An energy spectral index of the synchrotron radiation of γ ≈ 2.8 is taken. Assuming no intrinsic Faraday rotation, the polarized intensity and the polarized position angle is finally found U2 +Q2, PA = 1 arctan – 9 – where U = ui and Q = REFERENCES Berkhuijsen, E. M., Haslam, C. G. T., & Salter, C. J. 1971, A&A, 14, 252 Bochkarev, N. G. 1987, Ap&SS, 138, 229 Borken, R. J., & Iwan, D.-A. C. 1977, ApJ, 218, 511 Burn, B. 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R., et al. 2003, AJ, 125, 3145 http://arxiv.org/abs/astro-ph/0603450 – 10 – Testori, J. C., Reich, P., & Reich, W. 2004, in The Magnetized Interstellar Medium, ed. B. Uyaniker, W. Reich, & R. Wielebinski, 57 Tomisaka, K. 1992, PASJ, 44, 177 Weaver, R., McCray, R., Castor, J., Shapiro, P., & Moore, R. 1977, ApJ, 218, 377 Weaver, H. 1979, IAU Symp. 84: The Large-Scale Characteristics of the Galaxy, 84, 295 Wolleben, M. 2005, Ph.D. Thesis, Wolleben, M., Landecker, T. L., Reich, W., & Wielebinski, R. 2005, VizieR Online Data Catalog, 344, 80411 Wolleben, M., Landecker, T. L., Reich, W., & Wielebinski, R. 2006, A&A, 448, 411 This preprint was prepared with the AAS LATEX macros v5.2. – 11 – Table 1. Best-fit parameters of the model. l b d rin rout Bθ Bφ Shell (deg) (deg) (pc) (pc) (pc) (deg) (deg) S1 346 3 78 72 91 71 −72 ±5 ±5 ±10 ±10 ±10 ±30 ±30 S2 347 37 95 63 87 25 25 ±15 ±15 ±10 ±15 ±10 ±30 ±30 – 12 – Fig. 1.— Map of polarized intensity in units of mK (top) and polarization angle (bottom) in Galactic coordinates, taken from the DRAO polarization survey at 1.4 GHz (northern hemi- sphere) and the WMAP polarization data smoothed to 4◦ resolution (southern hemisphere). The data are shown in rectangular projection to expose structures around the Galactic poles. The polarized intensity map exhibits three extended features: the NPS (from l = 310◦ to 50◦ and b = 30◦ to 80◦), the New Radio Loop (centered at l = 40◦, b = −55◦, about 40◦ in diameter), and the so-called “Fan-Region” (centered at l = 140◦, b = 5◦, about 60◦ in diameter). The solid lines mark the observable extent of the HLPE in the second and third Galactic quadrant. The dashed line shows a circle fitted through the New Loop and one of the two extended “arms” of the NPS (at about b = 70◦). Black contours in the bottom panel show 1.5 keV X-ray emission associated with the NPS at levels of 200 and 350×10−6counts/s (taken from Snowden et al. 1995). – 13 – Fig. 2.— Observed (left) and modelled (right) maps of the northern (top) and southern (bottom) Galactic poles with polarization vectors overlayed. The contours show polarized intensity from the DRAO and WMAP data from 50 to 400 mK, in steps of 50 mk. – 14 – Fig. 3.— The sketch displays cuts through the model viewed looking down on the Galactic plane towards negative latitudes (top) and the vertical plane through the Sun and perpen- dicular to the line-of-sight looking towards l = 90◦ (bottom). The two shells are indicated by solid (S1) and dashed (S2) lines. The Sun is indicated by the circled dot. Thin lines indicate the B-field orientation of each shell (the B-field component parallel to the image plane). The stippled region shows the Na I distribution around the Local Bubble (taken from Sfeir et al. 1999). Filled triangles and squares show the centers of the LCC and US subgroup today, 5 Myr ago, and 10 Myr ago (only for LCC) (taken from Máız-Apellániz 2001). – 15 – Fig. 4.— An “engineering drawing” to help the reader depict the 3-dimensional structure of the two proposed shells. The solid line indicates S1, and the dashed line indicates S2.In contrast to Fig. 3 these drawings show projections of the two shells rather than cuts through the Galaxy. The Sun is located at the origin of the coordinate system. The figure shows the shells as seen from the Galactic anti-center towards the Galactic center (top left), seen sideways (top right), and seen from above (bottom). – 16 – Fig. 5.— Modelled maps of polarized intensity in units of mK (top) and polarization angle (bottom). The dashed line from Fig. 1 is repeated in the top panel. The black contour in the bottom panel indicates the region where the S1 and S2 shells overlap in space. The model was merely fitted to polarized intensity which means that “Fig. 5 top” (model) was fitted to “Fig. 1 top” (survey). Nevertheless, the predicted polarization angles (“Fig. 5 bottom”) resemble the observed pattern (“Fig. 1 bottom”) remarkably well. The Fan-Region is not part of the model described in this paper. – 17 – Fig. 6.— Geometry of the spherical shell. rin and rout are the inner and outer radii of the shell, respectively; B̃ is the background magnetic field and b̂ is its unit vector; ~Bi is the magnetic field inside the shell along the i-th line-of-sight element; and x̂ is the unit vector pointing towards the i-th line-of-sight element. Region I and II indicate regions with strong and weak B-fields, respectively. Introduction Large Polarized Structures In The DRAO Survey High Latitude Polarized Emission (HLPE) New Radio Loop Depolarization Band The Model Discussion Possible Formation History Conclusions Magnetic Field Used In The Model

0704.0277

Leray numbers of projections and a topological Helly type theorem

arXiv:0704.0277v1 [math.CO] 2 Apr 2007 7 Leray Numbers of Projections and a Topological Helly Type Theorem Gil Kalai∗ Roy Meshulam† November 4, 2018 Abstract Let X be a simplicial complex on the vertex set V . The rational Leray number L(X) of X is the minimal d such that H̃i(Y ;Q) = 0 for all induced subcomplexes Y ⊂ X and i ≥ d. Suppose V = i=1 Vi is a partition of V such that the induced sub- complexes X[Vi] are all 0-dimensional. Let π denote the projection of X into the (m − 1)-simplex on the vertex set {1, . . . ,m} given by π(v) = i if v ∈ Vi. Let r = max{|π −1(π(x))| : x ∈ |X|}. It is shown L(π(X)) ≤ rL(X) + r − 1 . One consequence is a topological extension of a Helly type result of Amenta. Let F be a family of compact sets in Rd such that for any F ′ ⊂ F , the intersection F ′ is either empty or contractible. It is shown that if G is a family of sets such that for any finite G′ ⊂ G, the intersection G′ is a union of at most r disjoint sets in F , then the Helly number of G is at most r(d+ 1). Math Subject Classification. 55U10, 52A35 Keywords and Phrases. Helly’s Theorem, d-Leray Complexes ∗Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel, and Departments of Computer Science and Mathematics, Yale University. e-mail: [email protected] . Research supported by ISF, BSF and NSF grants. †Department of Mathematics, Technion, Haifa 32000, Israel. e-mail: meshu- [email protected] . Research supported by the Israel Science Foundation. http://arxiv.org/abs/0704.0277v1 1 Introduction Let F be a family of sets. The Helly number h(F) of F is the minimal positive integer h such that if a finite subfamily K ⊂ F satisfies K′ 6= ∅ for all K′ ⊂ K of cardinality ≤ h, then K 6= ∅. Helly’s classical theorem (1913, see e.g. [3]) asserts that the Helly number of the family of convex sets in Rd is d+ 1. Helly’s theorem and its numerous extensions are of central importance in discrete and computational geometry (see [3, 10]). It is of considerable interest to understand the role of convexity in these results, and to find suitable topological extensions. Indeed, it is often the case that topological methods provide a deeper understanding of the underlying combinatorics behind Helly type theorems. Helly himself realized in 1930 (see [3]) that in his theorem, convex sets can be replaced by topological cells if you impose the additional requirement that all non-empty intersections of these cells are again topological cells. Helly’s topological version of his theorem also follows from the later nerve theorems of Borsuk, Leray and others (see below). The following result was conjectured by Grünbaum and Motzkin [8] , and proved by Amenta [1]. A family of sets G is an (F , r)-family if for any finite G ′ ⊂ G, the intersection G ′ is a union of at most r disjoint sets from F . Theorem 1.1 (Amenta). Let F be the family of compact convex sets in Rd. Then for any (F , r)-family G h(G) ≤ r(d+ 1) . The main motivation for the present paper was to find a topological ex- tension of Amenta’s Theorem. Let X be a simplicial complex on the vertex set V . The induced sub- complex on a subset of vertices S ⊂ V is X [S] = {σ ∈ X : σ ⊂ S}. The link of a subset A ⊂ V is lk(X,A) = {τ ∈ X : τ ∪ A ∈ X, τ ∩ A = ∅ } . The geometric realization of X is denoted by |X|. We identify X and |X| when no confusion can arise. All homology groups considered below are with rational coefficients, i.e. Hi(X) = Hi(X ;Q) and H̃i(X) = H̃i(X ;Q). The rational Leray number L(X) ofX is the minimal d such that H̃i(Y ) = 0 for all induced subcomplexes Y ⊂ X and i ≥ d. The Leray number can be regarded as a simple topologically based “complexity measure” of X . Note that L(X) = 0 iff X is a simplex, and L(X) ≤ 1 iff X is the clique complex of a chordal graph (see [9]). It is well-known (see e.g. [7]) that L(X) ≤ d iff H̃i(lk(X, σ)) = 0 for all σ ∈ X and i ≥ d. Leray numbers have also significance in commutative algebra, since L(X) is equal to the Castelnuovo- Mumford regularity of the Stanley-Reisner ring of X over Q (see [7]). From now on we assume that V1, . . . , Vm are finite disjoint 0-dimensional complexes, and denote their join by V1 ∗ · · · ∗ Vm. Let ∆m−1 be the simplex on the vertex set [m] = {1, . . . , m}, and let π denote the simplicial projection from V1 ∗ · · · ∗ Vm onto ∆m−1 given by π(v) = i if v ∈ Vi. For a subcomplex X ⊂ V1 ∗ · · ·∗Vm, let r(X, π) = max{|π −1(π(x))| : x ∈ |X|}. Our main result is the following Theorem 1.2. Let Y = π(X) and r = r(X, π). Then L(Y ) ≤ rL(X) + r − 1 . (1) Example: For r ≥ 1, d ≥ 2 let m = rd, and consider a partition [m] = k=1Ak with |Ak| = d. For i ∈ [m] let Vi = {i} × [r]. Denote by ∆(A) the simplex on vertex set A, with boundary ∂∆(A) ≃ S |A|−2. For k, j ∈ [r] let Akj = Ak × {j}, and let Xk = ∆(A1k) ∗ · · · ∗∆(Ak−1,k) ∗ ∂∆(Akk) ∗∆(Ak+1,k) ∗ · · · ∗∆(Ark) . Let X = k=1Xk. Then L(X) = d − 1, and the projection π : X → ∆m−1 satisfies r(X, π) = r. Since π(X) = ∂∆m−1, it follows that L(π(X)) = m−1. Hence equality is attained in (1). As mentioned earlier, Theorem 1.2 is motivated by an application in com- binatorial geometry. The nerve N(F) of a family of sets F , is the simplicial complex whose vertex set is F and whose simplices are all F ′ ⊂ F such that F ′ 6= ∅. It is easy to see that h(F) ≤ 1 + L(N(F)). (2) A finite family F of compact sets in some topological space is a good cover if for any F ′ ⊂ F , the intersection F ′ is either empty or contractible. If F is a good cover in Rd, then by the Nerve Lemma (see e.g. [2]) L(N(F)) ≤ d, hence follows the Topological Helly’s Theorem: h(F) ≤ d + 1. Theorem 1.2 implies a similar topological generalization of Amenta’s theorem. Theorem 1.3. Let F is a good cover in Rd. Then for any (F , r)-family G h(G) ≤ r(d+ 1) . The proof of Theorem 1.2 combines a vanishing theorem for the multiple point sets of a projection, with an application of the image computing spec- tral sequence due to Goryunov and Mond [5]. In Section 2 we describe the Goryunov-Mond result. In Section 3 we prove our main result, Proposition 3.1, which is then used to deduce Theorem 1.2. The proof of Theorem 1.3 is given in Section 4. 2 The Image Computing Spectral Sequence For X ⊂ V1 ∗ · · · ∗ Vm and k ≥ 1 define the multiple point set Mk by Mk = {(x1, . . . , xk) ∈ |X| k : π(x1) = · · · = π(xk)} . Let W be a Q-vector space with an action of the symmetric group Sk. Denote Alt = 1 sign(σ)σ ∈ Q[Sk]. Then AltW = {Altw : w ∈ W} = {w ∈ W : σw = sign(σ)w for all σ ∈ Sk} . (3) The natural action of Sk on Mk induces an action on the rational chain complex C∗(Mk) and on the rational homology H∗(Mk). The idempotence of Alt implies that Alt H∗(Mk) ∼= H∗(AltC(Mk)) . (4) The following result is due to Goryunov and Mond [5] (see also [4] and [6]). Theorem 2.1 (Goryunov and Mond). Let Y = π(X) and r = r(X, π). Then there exists a homology spectral sequence {Erp,q} converging to H∗(Y ) with E1p,q = AltHq(Mp+1) 0 ≤ p ≤ r − 1, 0 ≤ q 0 otherwise Remark: The E1 terms in the original formulation of Theorem 2.1 in [5], are given by E1p,q = AltHq(D p+1) where k = closure{(x1, . . . , xk) ∈ |X| k : π(x1) = · · · = π(xk), xi 6= xj for i 6= j} . The isomorphism AltHq(D p+1) ∼= AltHq(Mp+1) which implies (5), is proved in Theorem 3.4 in [6]. Indeed, as noted there, the inclusionDp+1 → Mp+1 induces an isomorphism Alt Cq(D p+1) ∼= AltCq(Mp+1) already at the alternating chains level. 3 Homology of the Multiple Point Set In this section we study the homology of a generalization of the multiple point set. For subcomplexes X1, . . . , Xk ⊂ V1 ∗ · · · ∗ Vm, let M(X1, . . . , Xk) = {(x1, . . . , xk) ∈ |X1| × · · · × |Xk| : π(x1) = · · · = π(xk)} . In particular, if X1 = · · · = Xk = X then M(X1, . . . , Xk) = Mk. We identify the generalized multiple point set M(X1, . . . , Xk) with the simplicial complex whose p-dimensional simplices are {wi0 , . . . , wip}, where 1 ≤ i0 < · · · < ip ≤ m, wij = (vij ,1, . . . , vij ,k) ∈ V and {vi0,r, . . . , vip,r} ∈ Xr for all 1 ≤ r ≤ k. The main ingredient in the proof of Theorem 1.2 is the following Proposition 3.1. H̃j M(X1, . . . , Xk) = 0 for j ≥ i=1 L(Xi). The proof of Proposition 3.1 depends on a spectral sequence argument given below. We first recall some definitions. Let K be a simplicial complex. The subdivision sd(K) is the order complex of the set of the non-empty simplices of K ordered by inclusion. For σ ∈ K let DK(σ) denote the order complex of the interval [σ, ·] = {τ ∈ K : τ ⊃ σ}. Let DK(σ) denote the order complex of the interval (σ, ·] = {τ ∈ K : τ % σ}. Note that DK(σ) is isomorphic to sd(lk(K, σ)) via the simplicial map τ → τ − σ. Since DK(σ) is contractible, it follows that Hi DK(σ), DK(σ) ∼= H̃i−1 lk(K, σ) for all i ≥ 0. For σ ∈ V1 ∗ · · · ∗ Vm, let σ̃ = i∈π(σ) Vi. Note that if σ2 ∈ X2, . . . , σk ∈ Xk then there is an isomorphism M(X1, σ2, . . . , σk) ∼= X1[∩ i=2σ̃i] . (6) For 0 ≤ p ≤ n = i=2 dimXi let S ′p = {(σ2, . . . , σk) ∈ X2 × · · · ×Xk : dim σi ≥ n− p} and let Sp = S p − S p−1. For σ = (σ2, . . . , σk) ∈ S p let Aσ = M(X1, σ2, . . . , σk)×DX2(σ2)× · · · ×DXk(σk) , Bσ = M(X1, σ2, . . . , σk)× DX2(σ2)× · · · × DXj (σj)× · · · ×DXk(σk) Proposition 3.2. There exists a homology spectral sequence {Erp,q} converg- ing to H∗ M(X1, . . . , Xk) such that E1p,q = i1,...,ik≥0 i1+···+ik=p+q i=2σ̃i] H̃ij−1 lk(Xj, σj) for 0 ≤ p ≤ n , 0 ≤ q , and E1p,q = 0 otherwise. Proof: For 0 ≤ p ≤ n let σ∈S′p Aσ ⊂ M(X1, . . . , Xk)× sd(X2)× · · · × sd(Xk). Write K = Kn, and consider the projection on the first coordinate θ : K → M(X1, . . . , Xk). Let (x1, . . . , xk) ∈ M(X1, . . . , Xk), and let σi denote the minimal simplex in Xi that contains xi. Then the fiber (x1, . . . , xk) = {(x1, . . . , xk)} ×DX2(σ2)× · · · ×DXk(σk) is a cone, hence K is homotopy equivalent to M(X1, . . . , Xk). The filtration ∅ ⊂ K0 ⊂ · · · ⊂ Kn = K gives rise to a homology spectral sequence {E converging to H∗(K) ∼= H∗(M(X1, . . . , Xm)). The E p,q terms are computed as follows. First note that Kp−1 = Bσ . (8) Secondly, Aσ −Bσ ∩ Aσ′ = ∅ for σ 6= σ ′ ∈ Sp. Hence H∗(Aσ, Bσ) . (9) Applying excision, (8),(9), and the Künneth formula we obtain: E1p,q = Hp+q(Kp, Kp−1) ∼= Hp+q Hp+q(Aσ, Bσ) ∼= i1,...,ik≥0 i1+···+ik=p+q M(X1, σ2, . . . , σk) DXj (σj), DXj(σj) i1,...,ik≥0 i1+···+ik=p+q i=2σ̃i] H̃ij−1 lk(Xj , σj) Proof of Proposition 3.1: If L(Xj) = 0 for all 1 ≤ j ≤ k, then all the Xj’s are simplicies, say Xj = σj . It follows that M(X1, . . . , Xk) is isomorphic to the simplex j=1 π(σj) and thus has vanishing reduced homology in all nonnegative dimensions. Suppose then that m = j=1 L(Xj) > 0. Without loss of generality we may assume that L(X1) > 0. Let i1, . . . , ik ≥ 0 such that j=1 ij ≥ m. Then either i1 ≥ L(X1) and then Hi1 i=2σ̃i] = 0, or there exists a 2 ≤ j ≤ k such that ij − 1 ≥ L(Xj) and then H̃ij−1 lk(Xj, σj) By (7) it follows that E1p,q = 0 if p + q ≥ m, hence H̃j M(X1, . . . , Xk) for all j ≥ m. Remark: If all the Vj’s are singletons then M(X1, . . . , Xk) is isomorphic to j=1Xj . Hence Proposition 3.1 implies the following result of [7]. Corollary 3.3 ([7]). If X1, . . . , Xk are simplicial complexes on the same vertex set, then Xj) ≤ L(Xj) . Proof of Theorem 1.2: Let Y = π(X) and r = r(X, π). Assuming as we may that L(X) > 0, we have to show that Hm(Y ) = 0 for m ≥ rL(X)+r−1. By Theorem 2.1 it suffices to show that Alt Hq(Mp+1) = 0 for all pairs (p, q) such that p ≤ r − 1 and p + q ≥ rL(X) + r − 1. Indeed, p ≤ r − 1 implies that q ≥ rL(X) ≥ (p + 1)L(X), thus Hq(Mp+1) = 0 by Proposition 3.1. 4 A Topological Amenta Theorem Proof of Theorem 1.3: Suppose G = {G1, . . . , Gm} is an (F , r)-family. Write Gi = j=1 Fij , where ri ≤ r and Fij Fij′ = ∅ for 1 ≤ j 6= j ′ ≤ ri. Let Vi = {Fi1, . . . , Firi} and consider the nerve X = N({Fij : 1 ≤ i ≤ m, 1 ≤ j ≤ ri}) ⊂ V1 ∗ · · · ∗ Vm . Let ∆m−1 be the simplex on the vertex set {G1, . . . , Gm} and let π denote the projection of V1 ∗ · · · ∗Vm into ∆m−1 given by π(Fij) = Gi. Then π(X) = N(G). Let y ∈ |N(G)| and let σ = {Gi : i ∈ I} be the minimal simplex in N(G) such that y ∈ |σ|. Then |π−1(y)| = |{ (ji : i ∈ I) : Fiji 6= ∅ }| . (10) On the other hand (ji:i∈I) Fiji (11) and the union on the right is a disjoint union. The assumption that G is an (F , r) family, together with (10) and (11), imply that |π−1(y)| ≤ r for all y ∈ |N(G)|. Since F is a good cover in Rd, the Leray number of the nerve satisfies L(X) = L(N(F)) ≤ d. Therefore by (2) and Theorem 1.2 h(G) ≤ 1 + L(N(G)) = 1 + L(π(X)) ≤ 1 + rL(X) + r − 1 ≤ r(d+ 1) . References [1] N. Amenta, A short proof of an interesting Helly-type theorem. Discrete Comput. Geom. 15(1996), 423–427. [2] A. Björner, Topological methods, in Handbook of Combinatorics (R. Graham, M. Grötschel, and L. Lovász, Eds.) , 1819–1872, North- Holland, Amsterdam, 1995. [3] J. Eckhoff, Helly, Radon and Carathéodory type theorems, in Handbook of Convex Geometry (P.M. Gruber and J.M. Wills Eds.), North–Holland, Amsterdam, 1993. [4] V. Goryunov, Semi-simplicial resolutions and homology of images and discriminants of mappings. Proc. London Math. Soc. 70(1995), 363–385. [5] V. Goryunov, D. Mond, Vanishing cohomology of singularities of map- pings. Compositio Math. 89(1993), 45–80. [6] K. Houston, An introduction to the image computing spectral sequence. Singularity theory (Liverpool, 1996), 305–324, London Math. Soc. Lec- ture Note Ser., 263, Cambridge Univ. Press, Cambridge, 1999. [7] G. Kalai, R. Meshulam, Intersections of Leray complexes and regu- larity of monomial ideals, Journal of Combinatorial Theory Ser. A., 113(2006), 1586–1592. [8] B. Grünbaum, T. Motzkin, On components in some families of sets, Proc. Amer. Math. Soc. 12(1961), 607–613. [9] G. Wegner, d-Collapsing and nerves of families of convex sets, Arch. Math. (Basel) 26(1975) 317–321. [10] R. Živaljević, Topological methods, in Handbook of discrete and compu- tational geometry, (J. Goodman and J. O’Rourke Eds.), 209–224, CRC Press, Boca Raton, 1997.

0704.0278

q-Deformed spin foam models of quantum gravity

q-deformed spin foam models of quantum gravity Igor Khavkine1 and J. Daniel Christensen2 Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada Department of Mathematics, University of Western Ontario, London, Ontario, Canada E-mail: [email protected] and [email protected] November 4, 2018 Abstract We numerically study Barrett-Crane models of Riemannian quantum grav- ity. We have extended the existing numerical techniques to handle q-deformed models and arbitrary space-time triangulations. We present and interpret expec- tation values of a few selected observables for each model, including a spin-spin correlation function which gives insight into the behaviour of the models. We find the surprising result that, as the deformation parameter q goes to 1 through roots of unity, the limit is discontinuous. PACS numbers: 04.60.Pp 1 Introduction Spin foam models were first introduced as a space-time alternative to the spin network description of states in loop quantum gravity [3]. The most studied spin foam models are due to Barrett and Crane [8,9]. A spin foam is a discretization of space-time where the fundamental degrees of freedom are the areas labelling its 2-dimensional faces. An important goal in the investigation of spin foam models is to obtain predictions that can be compared to the large scale, classical, or semiclassical behavior of gravity. This work continues the numerical investigation of the physical properties of spin foam models of Riemannian quantum gravity begun in [5–7, 13]. In this paper, we extend the computations to the q-deformed Barrett-Crane model and to larger space-time triangulations. The main applications of q-deformation are two-fold. On the one hand, it can act as a regulator for divergent models, as is apparent in the link between the Ponzano- Regge [27] and Turaev-Viro [31] models. On the other hand, Smolin [30] has argued that q-deformation is necessary to account for a positive cosmological constant. Both of these aspects are explored in more detail in Section 2.2. A surprising result of our work is evidence that the limit, as the cosmological constant is taken to zero through positive values, is discontinuous. Large triangulations are necessary to approximate semiclassical space-times. The possibility of obtaining numerical results from larger triangulations takes us one step closer to that goal and increases the number of facets from which the physical proper- ties of a spin foam model may be examined. As an example, we are able to study how the spin-spin correlation varies with the distance between faces in the triangulation. http://arxiv.org/abs/0704.0278v1 This paper is structured as follows. We begin in Section 2 by reviewing the basics of q-deformation and discussing in detail its aforementioned applications. Section 3 reviews the details of the Barrett-Crane model, summarizes the necessary changes for its q-deformation, and defines several observables associated to spin foams. In Section 4, we review the existing numerical simulation techniques and how they need to be generalized to handle q-deformation and larger triangulations. Section 5 presents the results of our numerical simulations. In Section 6, we give our conclusions and list some avenues for future research. The Appendix briefly summarizes our notational conventions and useful formulas. 2 Deformation of su(2) In this section, we describe the q-deformation of the Lie algebra su(2) into the algebra suq(2) (also denoted Uq(su(2))), the representations of suq(2), and the applications of q-deformation. The deformations of spin(4) are then obtained through the isomor- phism spin(4) ∼= su(2)⊕ su(2). The following is part of the general subject of quantum groups [21]. Here we shall concentrate solely on the su(2) and spin(4) cases. 2.1 The algebra suq(2) and its representations The Lie algebra su(2) is generated by the well known Pauli matrices σi, which obey the commutation relations [σ+, σ−] = 4σ3, [σ3, σ+] = 2σ+, [σ3, σ−] = −2σ−, (1) where σ± = σ1 ± iσ2. The universal enveloping algebra of su(2) is the associative algebra generated by σ± and σ3 subject to the above identities, with the Lie bracket being interpreted as [A,B] = AB −BA. The q-deformed algebra suq(2) is constructed by replacing σ3 with another gen- erator. Formally, it is thought of as Σ = q σ3 , where q ∈ C with the exceptions q 6= 0, 1,−1. The Lie bracket relations are replaced by the identities [σ+, σ−] = 4 Σ2 − Σ−2 q − q−1 , Σσ+ = qσ+Σ, Σσ− = −qσ−Σ. (2) We can rewrite q = 1+ 2ε and think of ε as a small complex number. Then, formally at leading order in ε, the substitution Σ = q σ3 = 1 + εσ3 + O(ε 2) reduces the deformed identities (2) to the standard Lie algebra relations (1). The associative algebra generated by σ± and σ3 subject to the deformed identities (2) is the algebra suq(2). For generic q, that is, when q is not a root of unity, the finite-dimensional irreducible representations of suq(2) are classified by a half-integer, j = 0, 1/2, 1, 3/2, . . . , referred to as the spin, in direct analogy with the representations of su(2) and the theory of angular momentum. The dimension of the representation j is 2j + 1. When q = exp(iπ/r) is a 2rth root of unity (ROU), where r > 2 is an integer called the ROU parameter, the representations j are still defined, but become reducible for j > (r − 2)/2. They decompose into a sum of representations with spin at most (r − 2)/2 and so-called trace 0 ones, whose nature will be explained below. For the purposes of this paper we are concerned only with intertwiners between representations of suq(2), i.e., linear maps commuting with the action of the algebra, and their (quantum) traces1. Any such intertwiner can be constructed from a small set of generators and elemen- tary operations on them. These constructions, as well as traces, can be represented graphically. Such graphs are called (abstract) spin networks. Their calculus is well developed and is described in [18], whose conventions we follow throughout the paper with one exception: we use spins (half-integers) instead of twice-spins (integers). A brief review of our notation and conventions can be found in the Appendix. Trace 0 representations of suq(2) are so called because the trace of an intertwiner from such a representation to itself is always zero. Thus, they can be freely discarded, as they do not contribute to the evaluation of q-deformed spin networks. 2.2 Applications of q-deformation Deformation, especially with q = exp(iπ/r) a 2rth primitive ROU, is important for spin foam models for at least two reasons. Replacing q = 1 by some ROU can act as a regulator for a model whose partition function and observable values are otherwise divergent. Also, suq(2) spin networks 2 naturally appear when considering a positive cosmological constant in loop quantum gravity. The original Ponzano-Regge model [27] attempts to express the path integral for 3-dimensional Riemannian general relativity as a sum over labelled triangulations of a 3-manifold. The edges of the triangulation are labelled by discrete lengths, identified with spin labels of irreducible SU(2) representations. Each tetrahedron contributes a 6j-symbol factor to the summand, normalized to ensure invariance of the overall sum under change of triangulation. Unfortunately, the Ponzano-Regge model turned out to be divergent. Motivated by the construction of 3-manifold invariants, Turaev and Viro were able to regularize the Ponzano-Regge model [1, 31] by replacing the SU(2) 6j-symbols with their q-deformed analogs at a ROU q. The key feature of the regu- larization is the truncation of the summation to only the irreducible representations of suq(2) of non-zero trace, which leaves only a finite number of terms in the model’s partition function. A version of the Barrett-Cranemodel, derived from a group field theory by De Pietri, Freidel, Krasnov and Rovelli [16] (DFKR for short), was also found to be divergent. A q-deformed version of the same model at a ROU q is similarly regularized (see Sec- tion 3.2). Some numerical results for the regularized version of this model are given in Section 5.2. The argument linking q-deformation to the presence of a positive cosmological constant is due to Smolin [29] and is given in more refined form in [30]. It is briefly summarized as follows. Loop quantum gravity begins by writing the degrees of free- dom of general relativity in terms of an SU(2) connection on a spatial slice and the slice’s extrinsic curvature. A state in the Schrödinger picture, a wave function on the space of connections, can be constructed by integrating the Chern-Simons 3-form over the spatial slice. This state, known as the Kodama state, simultaneously satisfies all the canonical constraints of the theory and semiclassically approximates de Sit- 1When q = 1, this notion of trace reduces up to sign to the usual trace of a linear map, but is slightly different otherwise, cf. [10, Chapter 4]. 2These are graphs embedded in a 3-manifold, labelled by representations of suq(2). They are similar to but distinct from the abstract spin networks referred to above. See [4] for the distinction. ter spacetime, which is a solution of the vacuum Einstein equations with a positive cosmological constant. The requirement that the Kodama state also be invariant un- der large gauge transformations implies discretization of the cosmological constant, Λ ∼ 1/r, with r a positive integer. The coefficients of the Kodama state in the spin network basis are obtained by evaluating the labelled graph, associated to a basis state, as an abstract suq(2) spin network. Here the deformation parameter q is a ROU, q = exp(iπ/r), where the ROU parameter r is identified with the discretization parameter of the cosmological constant. Given the heuristic link [4] between spin networks of loop quantum gravity and spin foams, it is natural to q-deform a spin foam model as an attempt to account for a positive cosmological constant. With this aim, Noui and Roche [23] have given a q-deformed version of the Lorentzian Barrett-Crane model. The possibility of q- deformation has been with the Riemannian Barrett-Crane model since its inception [8] and all the necessary ingredients have been present in the literature for some time. In the next section these details are collected in a form ready for numerical investigation. 3 Deformation of the Barrett-Crane model Consider a triangulated 4-manifold. Let ∆n denote the set of n-dimensional simplices of the triangulation. The dual 2-skeleton is formed by associating a dual vertex, edge and polygonal face to each 4-simplex, tetrahedron, and triangle of the triangulation, respectively. A spin foam is an assignment of labels, usually called spins, to the dual faces of the dual 2-skeleton. Each dual edge has 4 spins incident on it, while each dual vertex has 10. A spin foam model assigns amplitudes AF , AE and AV , that depend on all the incident spins, to each dual face, edge and vertex, respectively. The amplitude Z(F ) assigned to a spin foam F is the product of the amplitudes for individual cells of the 2-complex, while the total amplitude Ztot assigned to a triangulation is obtained by summing over all spin foams based on the triangulation: Z(F ) = AF (f) AE(e) AV (v), Ztot = Z(F ). (3) Some models, such as those based on group field theory [16,17,24], also include a sum over triangulations in the definition of the total partition function. 3.1 Review of the undeformed model The Riemannian Barrett-Crane model was first proposed in [8]. Its relation to the Crane-Yetter [15] spin foam model is analogous to the relation of the Plebanski [26] formulation of general relativity (GR) to 4-dimensional BF theory with Spin(4) as the structure group. Both BF theory and the Crane-Yetter model are topological and the latter is considered a quantization of the former [2]. In the Plebanski formulation, GR is a constrained version of BF theory. Similarly, the Barrett-Crane model restricts the spin labels summed over in the Crane-Yetter model. With this restriction, Barrett and Crane hoped to produce a discrete model of quantum (Riemannian) GR. 3.1.1 Dual vertex amplitude All amplitudes are defined in terms of spin(4) spin networks. However, given the isomorphism spin(4) ∼= su(2)⊕ su(2), all irreducible representations of spin(4) can be written as tensor products of irreducible representations of su(2). The Barrett-Crane model specifically limits itself to balanced representations, which are of the form j⊗ j, where j is the irreducible representation of su(2) of spin j. Since the tensor product corresponds to a juxtaposition of edges in a spin network, any spin(4) spin network may be written as an su(2) spin network where an edge labelled j ⊗ j is replaced by two parallel edges, each labelled j. To avoid redundancy of notation, we use a single j instead of j ⊗ j to label spin(4) spin network edges. We then distinguish them from su(2) networks by placing a bold dot at every vertex. The Barrett-Crane vertex is an intertwiner between four balanced representations: e . (4) The graphs on the right hand side of the definition are su(2) spin networks and the sum runs over all admissible labels e. The graphical notation and the conditions for admissibility are defined in the Appendix. The above expression defines the Barrett-Crane vertex in a way that breaks ro- tational symmetry. However, it can be shown that the vertex is in fact rotation- ally symmetric. Up to normalization, this property makes the Barrett-Crane vertex unique [28]. The above formula defines a vertical splitting of the vertex. A ninety degree rotation will define an analogous horizontal splitting. Both possibilities are important in the derivation of the algorithm presented in Section 4.1. Given a 4-simplex v of a triangulation, the corresponding vertex of the dual 2- complex is assigned the amplitude AV (v) = j1,0j1,1 j1,4j1,2 j2,1 j2,4 j2,2 j2,3 . (5) This spin network is called the 10j-symbol. The 4-simplex v is bounded by five tetra- hedra, which correspond to the vertices of the 10j graph. The four edges incident on a vertex correspond to the four faces of the corresponding tetrahedron; the spin labels are assigned accordingly. The edge joining two vertices corresponds to the face shared by corresponding tetrahedra. Evaluation of the 10j-symbol is discussed in Section 4.1. While the crossing structure depicted above is immaterial in the undeformed case, it is essential at nontrivial values of q. It is given here for reference. 3.1.2 Dual edge and face amplitudes The original paper of Barrett and Crane did not specify dual edge and face amplitudes. Three different dual edge and face amplitude assignments were considered in a previous paper [7]. We concentrate on the same possibilities. For the Perez-Rovelli model [25], we have AF (f) = j , AE(e) = j1 j2 j3 j4 . (6) For the DFKR model [16], we have AF (f) = j , AE(e) = . (7) For the Baez-Christensen model [7], we have AF (f) = 1, AE(e) = . (8) The bubble diagram, when translated into su(2) spin networks, corresponds to two bubbles (see Appendix) . (9) and evaluates to (2j + 1)2. The so-called eye diagram simply counts the dimension of the space of 4-valent intertwiners, which is also the number of admissible e-edges summed over in Equa- tion (4). In symmetric form, it is given by 1 + min{2j, s− 2J} if positive and s is integral, 0 otherwise, where s = k jk, j = mink jk, and J = maxk jk. 3.2 The q-deformed model Thanks to graphical notation, the q-deformation of the spin foam amplitudes described above is straightforward, with only a few subtleties. The main distinction is that q-deformed graphs are actually ribbon (framed) graphs with braiding. Thus, any undeformed spin network has to be supplemented with information about twists and crossings before evaluation. In [32], Yetter generalized the Barrett-Crane 4-vertex for a q-deformed version of spin(4). Since spin(4) ∼= su(2) ⊕ su(2), there is a two parameter family of possible deformations of the Lie algebra, spinq,q′ (4) ∼= suq(2) ⊕ suq′(2). Yetter singles out the one parameter family q′ = q−1, restricted to balanced representations, since it preserves the invariance of the Barrett-Crane vertex under rotations. This family also has especially simple curl and twist identities: , (11) where the left factor of j ⊗ j corresponds to suq(2) and the right one to suq−1(2), and the 3-vertex is the obvious juxtaposition of two suq(2) and suq−1(2) 3-vertices. Once this deformation is adopted, the ribbon structure can be ignored [32], so one only needs to specify the crossing structure for a given spin(4) spin network to obtain a well-defined q-evaluation. There are three basic graphs needed to define the Barrett-Crane simplex ampli- tudes: the bubble, the eye, and the 10j-symbol. The evaluation of the bubble graph, Equation (9), is [2j + 1]2, where the quantum integer [2j + 1] is defined in the Ap- pendix. Remarkably, the value of the eye diagram turns out not to depend on q and its value is still given by Equation (10). The only exception is when q is a ROU with parameter r. Then, the dimension of the space of 4-valent intertwiners changes to 1 + min{2j, s− 2J} r − 1−max{2J, s− 2j} if positive and s is integral, 0 otherwise, where again s = k jk, j = mink jk, and J = maxk jk. The 10j-symbol is the only network with a non-planar graph. Originally, it was defined in terms of the 15j-symbol from the Crane-Yetter model. This 15j-symbol was defined with q-deformation in mind, so its crossing and ribbon structure was fully specified [14, Section 3]. Adapted to the 10j-graph, it can be summarized as follows: Consider a 4-simplex. The dual 1-skeleton of the boundary has five dual vertices and ten dual edges, and is the complete graph K5 on these five dual vertices. If we remove one of the (non-dual) vertices from the boundary of the 4-simplex, what remains is homeomorphic to R3. For any such homeomorphism, the embedding of K5 into R can be projected onto a 2-dimensional plane. The crossing structure of the 10j graph is defined by such a projection. It is illustrated in Equation (5). Although, with crossings, the 10j graph is no longer manifestly invariant under permutations of its vertices, it can be shown to be so. 3.3 Observables The definition of observables in a spin foam model of quantum gravity is still open to interpretation (see Section 6 of [7] for a brief discussion). For a fixed spin foam, the half-integer spin labels of its faces are the fundamental variables of the model. Practically speaking, any observable of a spin foam model should be an expectation value of some function O(F ) of the spin labels of a spin foam F , averaged over all spin foams with amplitudes specified by Equation (3): 〈O〉 = O(F )Z(F ) . (13) In this paper we choose to concentrate on a few observables representative of the kind of quantities computable in a spin foam model. As before, fix a triangulation of a 4-manifold, let ∆2 represent the set of its faces and let j : ∆2 → {0, 1/2, 1, . . .} be the spin labelling. We define: J(F ) = ⌊j(f)⌉ , (14) (δJ)2(F ) = (⌊j(f)⌉ − 〈J〉) , (15) A(F ) = ⌊j(f)⌉ ⌊j(f) + 1⌉, (16) Cd(F ) = f,f ′∈∆2 dist(f,f ′)=d ⌊j(f)⌉ ⌊j(f ′)⌉ − 〈J〉 〈(δJ)2〉 . (17) where ⌊n⌉ denotes a quantum half-integer (see Appendix), | · | denotes cardinality, dist(f, f ′) denotes the distance between faces, and Nd is a normalization factor (see below for the definition of distance and Nd). These observables represent average spin per face, variance of spin per face, average area per face, and spin-spin correlation as a function of d. The choice of observables given above is somewhat arbitrary. For instance, there are several subtly distinct choices for the expression for (δJ)2. Fortunately, they all yield expectation values that are nearly identical. The expression given above has the technical advantage of falling into the class of so-called single spin observables. These are observables whose expectation value can be directly obtained from the knowledge of probability with which spin j occurs on any face of a spin foam. All of J , (δJ)2, and A are single spin observables, while Cd is not. Note that on a fixed triangulation with no other background geometry, there is no physical notion of distance. We can, instead, define a combinatorial analog. For any two faces f and f ′ of a given triangulation, let dist(f, f ′) be the smallest number of face-sharing tetrahedra that connect f to f ′. Given the discrete structure of our spacetime model, it is conceivable that this combinatorial distance, multiplied by a fundamental unit of length, approximates some notion of distance derived from the dynamical geometry of the spin foam model. The correlation function Cd may be thought of as analogous to a normalized 2- point function of quantum field theory. The d-degree of face f is the number of faces f ′ such that dist(f, f ′) = d. If the d-degree of every face is the same, the normalization factor Nd can be taken to be the number of terms in the sum (17), that is, the number of face pairs separated by distance d. This choice ensures the inequality |Cd| ≤ 1. If not all faces have the same d-degree, then the normalization factor has to be modified Nd = |∆2|Dd, (18) where Dd is the maximum d-degree of a face, which reduces to the simpler definition in the case of uniform d-degree. The choice of the q-dependent expression ⌊j⌉, instead of simply using the half- integer j, is motivated in Section 5.1. For some q, the argument of the square root in A(F ) may be negative or even complex. In that case, a branch choice will have to be made. Luckily, if q = 1, q is a ROU, or q is real, the expression under the square root is always non-negative. 4 Numerical simulation The key development that made possible numerical simulation of variations of the (undeformed) Barrett-Crane model [6,7] is the development by Christensen and Egan of a fast algorithm for evaluating 10j-symbols [13]. In this section, we show how this algorithm generalizes to the q-deformed case and discuss numerical evaluation of observables for the previously described spin foam models. 4.1 The q-deformation of the fast 10j algorithm The derivation of the Christensen-Egan algorithm given in [13] is contingent on the possibility of splitting the Barrett-Crane 4-vertex as in Equation (4) and on the re- coupling identity, Equation (43) of the Appendix. Both identities still hold in the q-deformed case. The validity of the 4-vertex splitting was proved by Yetter [32] and the recoupling identity is a standard part of suq(2) representation theory. The only remaining detail of the algorithm’s generalization is the crossing structure of the 10j graph, which was established in Section 3.2. However, its only consequence is an extra factor from the twist implicit in the bubble diagram of Section 4 of [13], cf. Equation (50) of the Appendix. We will not reproduce the derivation of the algo- rithm here. However, the way in which the twist arises is schematically illustrated in Figure 1. Note that the triviality of the twist for Yetter’s balanced representations, Equation (11), does not apply here since the twist occurs separately in distinct suq(2) networks. The algorithm itself can be summarized in the following form: {10j} = (−)2S m1,m2 φ tr[M4M3M2M1M0]. (19) The 10j-symbol depends on the ten spins ji,k, (i = 1, 2, k = 0, . . . , 4) specified in Equation (5). The overall prefactor depends on the total spin S = i,k ji,k and the per-term prefactor is φ = (−)m1−m2 [2m1 + 1][2m2 + 1]q m1(m1+1)−m2(m2+1). (20) (a) (b) (c) (d) Figure 1: In reference to [13], (a) corresponds to Equation (1), (b) corresponds to Equation (2), while (c) and (d) correspond to the “ladder” and “bubble” diagrams of Section 4, respectively. The illustrated twist introduces the explicitly q-dependent factor into Equation (20). The exponents of (−) and q are always integers. The Mk are matrices (not all of the same size) of dimensions compatible with the five-fold product and trace. Their matrix elements are [2lk + 1](T1) θ(j2,k−1, lk+1, j1,k) θ(j2,k+1, lk+1, j1,k+1) , (21) lk j2,k mi lk+1 j2,k−1 j1,k θ(j2,k, lk+1,mi) . (22) The quantum integers [n], as well as the theta θ(a, b, c) and tetrahedral Tet[· · ·] suq(2) spin networks are defined in the Appendix. The quantities lk and mi are spin labels (half-integers). They are constrained by admissibility conditions (parity conditions and triangle inequalities). The parity of each index is determined by the conditions lk ≡ j1,k + j2,k ≡ j1,k−1 + j2,k−2, (23) mi ≡ lk + j2,k−1, (24) for i = 1, 2 and k = 0, . . . , 4, where ≡ denotes equivalence mod 1 and the second subscript of j is taken mod 5. Summation bounds are determined by the triangle in- equalities, which must be checked for each trivalent vertex introduced in the derivation of the algorithm. They boil down to lb3(j1,k, j2,k, j2,k−1) ≤ mi ≤ j1,k + j2,k + j2,k−1, (25) |j1,k−1 − j2,k−2| ≤ lk ≤ j1,k−1 + j2,k−2, (26) |j1,k − j2,k| ≤ lk ≤ j1,k + j2,k, (27) |mi − j2,k−1| ≤ lk ≤ mi + j2,k−1, (28) for i = 1, 2 and k = 0, . . . , 4, where we have used the notation lb3(a, b, c) = 2max{a, b, c} − (a+ b+ c). (29) When q = exp(iπ/r) is a ROU, extra inequalities must be taken into account to exclude summation over reducible representations. These are mi ≥ j1,k + j2,k + j2,k−1 − (r − 2), (30) mi ≤ ub3(j1,k, j2,k, j2,k−1) + (r − 2), (31) lk ≤ (r − 2)− (j1,k + j2,k), (32) lk ≤ (r − 2)− (j1,k−1 + j2,k−2), (33) lk ≤ (r − 2)− (m+ j2,k−1), (34) where now ub3(a, b, c) = 2min{a, b, c} − (a+ b+ c). If any of the parity constraints or inequalities cannot be satisfied, the 10j-symbol evaluates to zero. This algorithm has been implemented and tested in the q = 1 and ROU cases, for both j and r up to several hundreds. Unfortunately, for generic q, when Q = max{|q|, |q|−1} > 1, the quantum integers grow exponentially as |[n]| ∼ Qn. Such a rapid growth makes the sums involved in this algorithm numerically unstable. It is still possible to use this algorithm with Q close to 1 or symbolically, using rational functions of q instead of limited precision floating point numbers. Symbolic computa- tion is, however, significantly slower (by up to a factor of 106) than its floating point counterpart. The software library spinnet which implements these and other spin network evaluations is available from the authors and will be described in a future publication. 4.2 Positivity and statistical methods The sums involved in evaluating expectation values of observables, as in Equation (13), are very high-dimensional. For instance, a minimal triangulation of the 4-sphere (seen as the boundary of a 5-simplex) contains 20 faces. Hence, any brute force evaluation of an expectation value, even on such a small lattice, involves a sum over the 20- dimensional space of half-integer spin labels. Fortunately, in the undeformed case, the total amplitude Z(F ) for a closed spin foam is never negative3 [5]. The proof for the q = 1 case generalizes to the ROU case. One need only realize two facts. The first is that, in the ROU case, quantum integers are non-negative. The second is that, for q a ROU, an suq−1(2) spin network evaluates to the complex conjugate of the corresponding suq(2) spin network. The 3We expect the same thing to hold in Lorentzian signature [5, 12]. disjoint union of any two such spin networks evaluates to their product, the absolute value squared of either of them, and hence is non-negative. Then, the same positivity result follows as from Equation (1) of [5]. This positivity allows us to treat Z(F ) as a statistical distribution and use Monte Carlo methods to extract expectation values with much greater efficiency than brute force summation. The main tool for evaluating expectation values is the Metropolis algorithm [20,22]. The algorithm consists of a walk on the space of spin labellings. Each step is randomly picked from a set of elementary moves and is either accepted or rejected based on the relative amplitudes of spin foam configurations before and after the move. An expectation value is extracted as the average of the observable over the configurations constituting the walk. Elementary moves for spin foam simulations are discussed in the next section. A Metropolis-like algorithm is possible even if individual spin foam amplitudes Z(F ) are negative or even complex. However, if the total partition function Ztot sums to zero, then the expectation values in Equation (13) become ill defined. Moreover, in numerical simulations, if Ztot is even close to zero, expectation value estimates may exhibit great loss of precision and slow convergence. In the path-integral Monte Carlo literature, this situation is known as the sign problem [11]. Still, the sign problem need not occur or, depending on the severity of the problem, there may be ways of effectively dealing with it. Independent Metropolis runs can be thought of as providing independent estimates of a given expectation value. Thus, the error in the computed value of an observable can be estimated through the standard deviation of the results of many independent simulation runs [19]. 4.3 Elementary moves for spin foams The choice of elementary moves for spin foam simulations must satisfy several criteria. Theoretically, the most important one is ergodicity. That is, any spin foam must be able to transform into any other one through a sequence of elementary moves which avoid configurations with zero amplitude. Practically, it is important that these moves usually preserve admissibility. A spin foam F is called admissible if the associated amplitude Z(F ) is non-zero. If, starting with an admissible spin foam, most elementary moves produce an inadmissible spin foam, the simulation will spend a lot of time rejecting such moves without any practical benefit. As before, consider a fixed triangulation of a compact 4-manifold. The parity conditions (23) imposed on the ji,k, j1,k + j2,k ≡ j1,k−1 + j2,k−2, 0 ≤ k ≤ 4, when taken together with the total spin foam amplitude (3), provide strong constraints on admissible spin foams. One can show that a move that changes spin labels by ±1/2 (mod 1) on each face of a closed surface in the dual 2-skeleton preserves the parity constraint. We take as the elementary moves the moves that change the spin labels by ±1/2 on the boundaries of the dual 3-cells of the dual 3-complex; the dual 3-cells correspond to the edges of the triangulation. If the manifold has non-trivial mod 2 homology in dimension 2, additional moves would be necessary, but for the examples we consider the moves above suffice. From a practical point of view, extra moves might improve the simulation’s equilibration time. For instance, in the ROU case, parity preserving moves that change the spins from 0 to (r − 2)/2 or (r − 3)/2 were introduced, since spins close to either admissible extreme may have large amplitudes. This property of the Perez-Rovelli and Baez-Christensen models is illustrated in the following section. Unfortunately, the inequalities constraining spin labels do not have a similar geo- metric interpretation and cannot be used to easily restrict the set of elementary moves in advance. 5 Results Using methods described in the previous section, we ran simulations of the three vari- ations of the Barrett-Crane model described in Section 3 and obtained expectation values for observables listed in Section 3.3. While previous work [7] performed simula- tions only on the minimal triangulation of the 4-sphere, which we will refer to simply as the minimal triangulation, we have extended the same techniques to arbitrary tri- angulations of closed manifolds. 5.1 Discontinuity of the r → ∞ limit The most striking result we can report is a discontinuity in the transition to the limit r → ∞, where r, a positive integer, is the ROU parameter with q = exp(iπ/r). As r → ∞, the deformation parameter q tends to its classical value 1. If we interpret the cosmological constant as inversely proportional to r, Λ ∼ 1/r, this limit also corresponds to Λ → 0, through positive values. For a fixed spin foam, the amplitudes and observables we study tend continuously to their undeformed values as r → ∞. However, we find that observable expectation values do not tend to their undeformed values in the same limit, that is, 〈O〉r 9 〈O〉q=1 as r → ∞. The discontinuity is most simply illustrated with the single spin distribution, that is the probability of finding spin j at any spin foam face. This probability can be estimated from the histogram of all spin labels that have occurred during a Monte Carlo simulation. The points in Figure 2(a) show the single spin distributions for the Baez-Christensen model with r = 50 and q = 1. The curves show the corresponding single bubble amplitude. It is the amplitude Z(Fj) of a spin foam Fj with all spin labels zero, except for the boundary of an elementary dual 3-cell, whose faces are all labelled with spin j. The amplitudes and distributions are normalized as probabil- ity distributions so their sums over j yield 1. The similarity between the points and the continuous curves is consistent with the hypothesis that spin foams with isolated bubbles dominate the partition function sum. The behavior of the single spin dis- tribution for the Perez-Rovelli model is very similar, except that its peaks are much more pronounced. Note that the undeformed single spin distribution has a single peak at j = 0, while the r = 50 case has two peaks, one at j = 0 and the other at j = (r − 2)/2, the largest non-trace 0 irreducible representation. The bimodal nature of the single spin distribution has an important impact on the large r behavior of observable expectation values, as is most easily seen with single spin observables (Section 3.3). For instance, if we consider the average, j̄, of the half-integers j, the large j peak would dominate the expectation value and 〈j̄〉 would diverge linearly in r, as r → ∞. On the other hand, since J is the average of the quantum half-integers ⌊j⌉, 〈J〉 at least approaches a constant in the same limit. This is illustrated in Figure 2(b). Figure 2: (a) Single spin distribution and single bubble amplitude for the Baez- Christensen model. The distribution was obtained from 109 steps of Metropolis sim- ulation on a triangulation with 202 faces (cf. Section 5.3). (b) Some single spin ob- servables as functions of j, with r = 50. However, as shown in Figure 3, this limit is not the same as the undeformed expectation value. At the same time, as can be seen from the plot of the Perez- Rovelli average area in the same figure, there are some observables whose large r limits are at least very close to the undeformed values. The area observable summand ⌊j⌉ ⌊j + 1⌉ is exactly zero at both j = 0 and j = (r − 2)/2, while the spin observable summand Jj = ⌊j⌉ is zero at j = 0 but still positive at j = (r − 2)/2, Figure 2(b). The large j peak of the Perez-Rovelli model is very narrow and thus the expectation value of a single spin observable is strongly influenced by its value at j = (r − 2)/2. The data for larger triangulations is qualitatively similar. 5.2 Regularization of the DFKR model As expected, the ROU deformation of the DFKR model yields a finite partition func- tion and finite expectation values. For instance, its single spin distribution for r = 40 is illustrated in Figure 4. The divergence of the amplitude for large spins in the unde- formed, q = 1, case makes numerical simulation impossible without an artificial spin cutoff. Thus, we do not have an undeformed analog of the single spin distribution. For the minimal triangulation, the ROU spin distribution deviates slightly from the single bubble amplitude close to the boundaries of admissible j. For the larger trian- gulation, the deviation is much more pronounced and is not restricted to the edges. This suggests that there are other significant contributions to the partition function besides single bubble spin foams. Note the large weight associated with spins around j = r/4. Around this value of j, both the area Aj = ⌊j⌉ ⌊j + 1⌉ and the spin Jj = ⌊j⌉ attain their maximal values and are proportional to r. Thus, it is natural to expect their expectation values to grow linearly in r, which is consistent with the divergent nature of the undeformed DFKR model. This is precisely the behavior shown in Figure 5. On the minimal triangulation, the best linear fits for the average spin expectation value and for the Figure 3: Observables for the Baez-Christensen (BCh) and Perez-Rovelli (PR) models as functions of the ROU parameter r. For large r, observables do not in general tend to their undeformed, q = 1, values; arrows show the deviation. Some observables were scaled to fit on the graph. Data is from Metropolis simulations on the minimal triangulation. square root of the average spin variance are 〈J〉r = 0.146 r − 0.064, (35) (δJ)2 = 0.014 r + 0.187. (36) For larger triangulations, the dependence of these observables is also approximately linear in r, with only slight variation in the effective slope. 5.3 Spin-spin correlation The ability to work with larger lattices allows us to explore a broader range of observ- ables. One of them is the spin-spin correlation function Cd defined in Section 3.3. In general 〈C0〉 = 1 and 〈Cd〉 → 0 for large d. The decay of the correlation shows how quickly the spin labels on different spin foam faces become independent. A positive value of 〈Cd〉 indicates that, on average, any two faces distance d apart both have spins above (or both below) the mean 〈J〉. On the other hand, a negative value of 〈Cd〉 indicates that, on average, any two faces distance d apart have one spin above and one below the mean 〈J〉. A small triangulation limits the maximum distance between faces. For example, the minimal triangulation has maximum distance d = 3. Larger triangulations of the 4-sphere were obtained by refining the minimal one by applying Pachner moves randomly and uniformly over the whole triangulation. We restricted the Pachner moves to those that did not decrease the number of simplices. Figure 4: Single spin distributions and single bubble amplitudes for the DFKR model. The distributions were obtained from 109 steps of Metropolis simulation on the mini- mal triangulation and on a triangulation with 202 faces (cf. Section 5.3). The largest triangulation we have used has maximum distance d = 6. Its corre- lations for different models are shown in Figure 6 along with those from the minimal triangulation. Correlation functions for different values of ROU parameter r (including the q = 1 case) and other triangulations are qualitatively similar. Notice the small negative dip for small values of d for the Perez-Rovelli and Baez- Christensen models. As discussed in previous sections, the partition functions of these models are dominated by spin foams with isolated bubbles. The correlation data is consistent with this hypothesis. The values of the spins assigned to faces of the bubble will be strongly correlated, while the values of the spins on two faces, one of which lies on the bubble and the other does not, should be strongly anti-correlated. Since a given face usually has fewer nearest neighbors that lie on the same bubble than that do not, on average, the short distance correlation is expected to be negative. At slightly larger distances, the correlation function turns positive again. This indicates that on a larger triangulations, spin foams with several isolated bubbles contribute strongly to the partition function. Although, with so few data points, it is difficult to extrapolate the behavior of the correlation function to larger triangulations and distances, its features are qualitatively similar to that of a condensed fluid, where the density-density correlation function exhibits oscillations on the scale of the molecular dimensions. Note that the behavior of the DFKR correlation function is significantly different from the other two. This is also consistent with the already observed fact that its partition function has strong contributions from other than single or isolated bubble spin foams. Figure 5: Observables for the DFKR model: area 〈A〉, average spin 〈J〉, spin standard deviation 〈(δJ)2〉. Metropolis simulation, minimal triangulation. Error bars are smaller than the data points. 6 Conclusion We have numerically investigated the behavior of physical observables for the Perez- Rovelli, DFKR, and Baez-Christensen versions of the Barrett-Crane spin foam model. Each version assigns different dual edge and face amplitudes to a spin foam, and these choices greatly affect the behavior of the resulting model. The behavior of the models was also greatly affected by q-deformation. The limiting behavior of observables was found to be discontinuous in the limit of large ROU parameter r, i.e., q = exp(iπ/r) close to its undeformed value of 1. This result is at odds with the physical interpretation of the relation Λ ∼ 1/r between the cosmological constant Λ and the ROU parameter. Finally, the behavior of the exam- ined physical observables, especially of the spin-spin correlation function, indicates the dominance of isolated bubble spin foams in the Perez-Rovelli and Baez-Christensen partition functions, while less so for the the DFKR one. Some questions raised by these results deserve attention. For instance, it is not known whether the same q → 1 limit behavior will be observed when q is taken through non-ROU values. While calculations with max{|q|, |q|−1} > 1 are numerically unstable, they should still be possible for |q| ∼ 1. Another important project is to perform a more extensive study of the effects of triangulation size in order to better understand the semi-classical limit. Finally, all of this work should also be carried out for the Lorentzian models, which are physically much more interesting but computationally much more difficult. These and other questions will be the subject of future investigations. Figure 6: Spin-spin correlation functions for the Baez-Christensen (BCh), Perez- Rovelli (PR) and DFKR models, on the minimal triangulation (6 vertices, 15 edges, 20 faces, 15 tetrahedra, and 6 4-simplices) as well as a larger triangulation (23 vertices, 103 edges, 202 faces, 200 tetrahedra, and 80 4-simplices). ROU parameter r = 10. Acknowledgements The authors would like to thank Wade Cherrington for helpful discussions. The first author was supported by NSERC and FQRNT postgraduate scholarships and the second author by an NSERC grant. Computational resources for this project were provided by SHARCNET. A Spin network notation and conventions Quantum integers are a q-deformation of integers. For an integer n, the corresponding quantum integer is denoted by [n] and is given by [n] = qn − q−n q − q−1 . (37) In the limit q → 1, we recover the regular integers, [n] → n. Note that [n] is invariant under the transformation q 7→ q−1. When q = exp(iπ/r) is a root of unity (ROU), for some integer r > 1, an equivalent definition is [n] = sin(nπ/r) sin(π/r) . (38) This expression is non-negative in the range 0 ≤ n ≤ r. Quantum factorials are defined as [n]! = [1][2] · · · [n]. (39) In many cases, q-deformed spin network evaluations can be obtained from their unde- formed counterparts by simply replacing factorials with quantum factorials. For con- venience, when dealing with half-integral spins, we also define quantum half-integers ⌊j⌉ = when j is a half-integer. Abstract suq(2) spin networks can be approached from two different directions. They can represent contractions and compositions of suq(2)-invariant tensors and in- tertwiners [10]. At the same time, they can represent traces of tangles evaluated according to the rules of the Kauffman bracket [18]. Either way, the computations turn out to be the same. We present here formulas for the evaluation of a few spin networks of interest. The single bubble network evaluates to what is sometimes called the superdimension of the spin-j representation: j = (−)2j [2j + 1]. (41) (As in the rest of the paper, the spin labels are half-integers.) Up to a constant, there is a unique 3-valent vertex (corresponding to the Clebsch- Gordan intertwiner) whose normalization is fixed up to sign by the value of the θ- network : θ(a, b, c) = (−)s[s+ 1]![s− 2a]![s− 2b]![s− 2c]! [2a]![2b]![2c]! , (42) where s = a+ b + c. The θ-network is non-vanishing, together with the three-vertex itself, if and only if s is an integer and the triangle inequalities are satisfied: a ≤ b+ c, b ≤ c+ a, and c ≤ a+ b. In addition, when q is a ROU, one extra inequality must be satisfied: s ≤ r − 2. The triple (a, b, c) of spin labels is called admissible if θ(a, b, c) is non-zero. The recoupling identity gives the transformation between different bases for the linear space of 4-valent tangles (or intertwiners): (−)2e[2e+ 1]Tet a b e c d f θ(a, d, e) θ(c, b, e) e , (43) where the sum is over all admissible labels e and the value of the tetrahedral network a b e c d f m≤S≤M (−)S [S + 1]! i[S − ai]! j [bj − S]! , (44) where [bj − ai]! E ! = [2A]![2B]![2C]![2D]![2E]![2F ]! (45) a1 = (a+ d+ e) b1 = (b+ d+ e+ f) (46) a2 = (b + c+ e) b2 = (a+ c+ e+ f) (47) a3 = (a+ b+ f) b3 = (a+ b+ c+ d) (48) a4 = (c+ d+ f) m = max{ai} M = min{bj}. (49) Due to parity constraints, the ai, bj, m, M , and S are all integers. Since the three-vertex is unique up to scale, its composition with with a braiding applied to two incoming legs yields a multiplicative factor: = (−)a+b−cqa(a+1)+b(b+1)−c(c+1) . (50) Note that the above braiding factor is not invariant under the transformation q 7→ q−1, while the bubble, tetrahedral and θ-networks are all invariant under this transforma- tion, by virtue of their expressions in terms of quantum integers. References [1] Archer F and Williams R M 1991 Physics Letters B 273 438–44 [2] Baez J C 1996 Letters in Mathematical Physics 38 129–43 (Preprint arXiv:q-alg/9507006) [3] Baez J C 1998 Classical and Quantum Gravity 15 1827–58 (Preprint arXiv:gr-qc/9709052) [4] Baez J C 2000 Lecture Notes in Physics 543 25–94 (Preprint arXiv:gr-qc/9904025) [5] Baez J C and Christensen J D 2002 Classical and Quantum Gravity 19 2291–306 (Preprint arXiv:grqc/0110044) [6] Baez J C, Christensen J D, and Egan G 2002 Classical and Quantum Gravity 19 6489–513 (Preprint arXiv:gr-qc/0208010) [7] Baez J C, Christensen J D, Halford T R, and Tsang D C 2002 Classical and Quantum Gravity 19 4627–48 (Preprint arXiv:gr-qc/0202017) [8] Barrett J W and Crane L 1998 Journal of Mathematical Physics 39 3296–302 (Preprint arXiv:gr- qc/9709028) [9] Barrett J W and Crane L 2000 Classical and Quantum Gravity 17 3101–18 (Preprint arXiv:gr- qc/9904025) [10] Carter J S, Flath D E, and Saito M 1995 The Classical and Quantum 6j-Symbols (Princeton, New Jersey: Princeton University Press) [11] Ceperley D and Alder B 1986 Science 231 555–60 [12] Cherrington J W and Christensen J D 2006 Classical and Quantum Gravity 23 721–36 (Preprint arXiv:gr-qc/0509080) [13] Christensen J D and Egan G 2002 Classical and Quantum Gravity 19 1184–93 (Preprint arXiv:gr-qc/0110045) [14] Crane L, Kauffman L H, and Yetter D N 1997 Journal of Knot Theory and Its Ramifications 6 177–234 (Preprint arXiv:hep-th/9409167) [15] Crane L and Yetter D N 1993 A categorical construction of 4D topological quantum field theories Quantum Topology ed L H Kauffman and R A Baadhio (Singapore: World Scientific Press) pp 120–30 [16] De Pietri R, Freidel L, Krasnov K, and Rovelli C 2000 Nuclear Physics B 574 785–806 (Preprint arXiv:hep-th/9907154) [17] Freidel L 2005 International Journal of Theoretical Physics 44 1769–83 (Preprint arXiv:hep- th/0505016) [18] Kauffman L H and Lins S L 1994 Temperley-Lieb Recoupling Theory and Invariants of 3- Manifolds (Princeton, New Jersey: Princeton University Press) [19] Kikuchi M and Ito N 1993 Journal of the Physical Society of Japan 62 3052 [20] Landau D P and Binder K 2005 A Guide to Monte Carlo Simulations in Statistical Physics 2nd ed (Cambridge: Cambridge University Press) [21] Majid S 2000 Foundations of Quantum Group Theory (Cambridge: Cambridge University Press) [22] Metropolis N, Rosenbluth A W, Rosenbluth M N, Teller A H, and Teller E 1953 Journal of Chemical Physics 21 1087–92 [23] Noui K and Roche P 2003 Classical and Quantum Gravity 20 3175–214 (Preprint arXiv:gr- qc/0211109) [24] Perez A 2003 Classical and Quantum Gravity 20 R043–104 (Preprint arXiv:gr-qc/0301113) [25] Perez A and Rovelli C 2001 Nuclear Physics B 599 255–82 (Preprint arXiv:gr-qc/0006107) [26] Plebański J F 1977 Journal of Mathematical Physics 18 2511–20 [27] Ponzano G and Regge T 1968 Semiclassical limit of Racah coefficients Spectroscopic and Group Theoretical Methods in Physics ed F Bloch et al (Amsterdam: North-Holland) pp 1–98 [28] Reisenberger M P 1999 Journal of Mathematical Physics 40 2046–54 (Preprint arXiv:gr- qc/9809067) [29] Smolin L 1995 Journal of Mathematical Physics 36 6417–55 (Preprint arXiv:gr-qc/9505028) [30] Smolin L 2002 Quantum gravity with a positive cosmological constant Preprint arXiv:hep- th/0209079 [31] Turaev V G and Viro O Y 1992 Topology 31 865–902 [32] Yetter D N 1999 Journal of Knot Theory and Its Ramifications 8 815–29 (Preprint arXiv:math.QA/9801131) Introduction Deformation of su(2) The algebra suq(2) and its representations Applications of q-deformation Deformation of the Barrett-Crane model Review of the undeformed model Dual vertex amplitude Dual edge and face amplitudes The q-deformed model Observables Numerical simulation The q-deformation of the fast 10j algorithm Positivity and statistical methods Elementary moves for spin foams Results Discontinuity of the r limit Regularization of the DFKR model Spin-spin correlation Conclusion Spin network notation and conventions

0704.0279

HI velocity dispersion in NGC 1058

H I velocity dispersion in NGC 1058 A. O. Petric Astronomy Department, Columbia University, New York, NY USA [email protected] M. P. Rupen National Radio Astronomy Observatory, P.O. Box O, Socorro, NM, 87801, USA [email protected] ABSTRACT We present excellent resolution and high sensitivity Very Large Array (VLA) observations of the 21cm H I line emission from the face-on galaxy NGC 1058, providing the first reliable study of the H I profile shapes throughout the entire disk of an external galaxy. Our observations show an intriguing picture of the interstellar medium; throughout this galaxy velocity– dispersions range between 4 to 15 km s−1 but are not correlated with star formation, stars or the gaseous spiral arms. The velocity dispersions decrease with radius, but this global trend has a large scatter as there are several isolated, resolved regions of high dispersion. The decline of star light with radius is much steeper than that of the velocity dispersions or that of the energy in the gas motions. Subject headings: galaxies: kinematics and dynamics — individual:NGC 1058; galaxies:ISM 1. Introduction Observations of H I velocities perpendicular to the disk (vz) are necessary for studies of both the interstellar medium (ISM) (McKee & Ostriker 1977, Kulkarni & Heiles 1988, Braun 1992, 1997) and disk dynamics (Oort 1932; Rupen 1987; Lockman & Gehman 1991; Merriefield 1993; Malhotra 1994, 1995; Olling 1995) because they set a direct upper limit on the thermal and kinetic temperature of the gas. Hence the H I velocities perpendicular to the disk are an important dynamical tracer and as such can be used to constrain, both the gas mass distribution in the plane and its vertical structure (i.e., density as a function of height-z above the plane)(van der Kruit & Shostak 1982,1984, Lockman & Gehman 1991, Malhotra 1995). http://arxiv.org/abs/0704.0279v1 – 2 – The behavior of the velocity dispersions as a function of galactic radius is important for determinations of the shape of dark matter halos. To date even the most sophisticated methods (e.g. Olling 1995, 1996) assume either a constant or an azimuthally symmetric velocity dispersion. Our measurements can therefore be used with studies of edge-on systems to determine radial variations in the mass to light (M/L) ratio. Processes associated with star formation,such as stellar winds and multiple supernova explosions are thought to put energy into the ISM in the form of mechanical energy, starlight (which leads to photoelectric emission from dust grains), and cosmic rays. The velocity dispersion in the z direction is intimately connected to the forces holding the gas against gravitational instabilities and hence to star-formation in the disk (e.g. Mac Low & Klessen 2004, Li, Mac Low & Klessen 2005). Measuring the degree of correlation between the locations of star-forming regions and those of high dispersion is a good method to investigate the relation between star-related energy sources and H I bulk motions. The face-on spiral galaxy NGC 1058 (e.g. Eskridge et al. 2002) is ideal for studies of H I vz dispersions. Its low inclination (4 ◦–11◦) (Lewis 1987, van der Kruit & Shostak 1984) means that the gradient in rotational velocity is small across the beam and therefore it does not significantly corrupt measurements of velocities perpendicular to the disk. Single dish studies of NGC 1058 (Allen & Shostak 1979, Lewis 1975, Lewis 1984) lack the resolution to trace the dispersion across the disk but through modeling of the rotational component these authors estimate it to range between 7 and 9 km/sec. In a series of papers (1982-1984) van der Kruit & Shostak analyze H I emission profiles in a number of face-on galaxies and determine that the velocity dispersions in NGC 1058 range only between 7 to 8 km/sec at all radii with very little variation. Dickey, Hanson, & Helou (1990) find the that velocity dispersion in NGC 1058 decreases with optical surface brightness but that in the extended gas disk, beyond the Holmberg radius the velocity dispersion is 5.7 km/sec everywhere, that is no variations with spiral phase or H I surface density are found. All previous determinations of the H I velocity dispersion in NGC 1058 have been hindered by low spatial (e.g., Lewis 1984) and/or spectral (e.g., van der Kruit & Shostak 1984) resolution as well as by relatively poor sensitivity, requiring smoothing over large sections of the galactic disk, or missing up to 40% of the total flux (Dickey, Hanson, & Helou 1990). These trade-offs have led to significantly different conclusions about the H I velocity dispersion. Our sensitive observations at high spatial and velocity resolution as well as recovery of the entire single dish flux, allowed us to accurately measure the profile widths even in the outskirts of the H I disk, to resolve the arm from the interarm regions, and to analyze in detail the H I profile shapes, not just their breadths throughout the H I disk. – 3 – 2. Observations and Data Reductions The 21cm line of neutral hydrogen in NGC 1058 was observed with the VLA in the C and CS1configurations. The C configuration data was taken on 14 and 15 June 1993 for a total time on-source of 12.23 hours. The D configuration observation were performed on 7 and 8 November 1993 for a total time on source of 2.67 hours, and CS configuration data was collected on January 3, 1995 for a total time on-source of 5.42 hours. Rupen (1997,1998) gives a detailed account of the UV coverage in each of the configurations, compares the merits of each configuration, and discusses the benefits of combining them. Both the C and CS configurations have a maximum baseline of 3.6 km, while the D configuration has a maximum baseline 1 km). The minimum baseline, which determines the size of the most extended feature which can be observed by the VLA, is 35 m. All observations were taken in dual polarization mode and Hanning smoothing was applied on- line; resulting in 127 independent spectral channels with a velocity width of 2.58 km/sec. We followed the normal AIPS calibration procedures and used the same flux (3C48) and phase (0234+285) calibrators throughout. The continuum emission was approximated as a linear fit to visibilities in 20 line-free channels on each side of the signal, and this fit was then subtracted from the uv-data in all the channels. Rupen (1999), gives a detailed description of the bandpass calibration and continuum subtraction. The data cube presented here was deconvolved using the CLEAN algorithm as imple- mented by AIPS task IMAGR.2, iterated until the residuals were nearly zero and the flux density in the CLEAN model was stable. The cube was tapered to a resolution of 30′′× 29′′, or 1.3 × 1.3 kpc at a distance of 10 Mpc (Ferguson et al. 1998). Rupen (1997), presents a detailed comparison of several cleaning algorithms and motivates the use of the CLEAN algorithm for this data. A more general discussion of CLEAN as implemented in AIPS is given in chapter 5, of the AIPS cookbook as well as in Cornwell, Braun, & Briggs 1999. A more specific examination of deconvolution algorithms as applied on our NGC 1058 data is presented in Rupen (1997,1999). The RMS noise level in the line channels of the cube was 0.5 and mJy/beam, corre- sponding to a column density of 1.6× 1018 cm−2 per channel. The H I integrated line profile 1The CS (shortened C) configuration moves two antennas from intermediate stations in the standard C configuration to the center of the array. The resulting short spacings significantly increase the sensitivity of the array to extended structure, while maintaining the same spatial resolution (Rupen 1997). 2A description of IMAGR can be found in the AIPS cookbook available online at http://www.aoc.nrao.edu/aips/cook.html. http://www.aoc.nrao.edu/aips/cook.html – 4 – agrees with those obtained in single dish studies (Allen & Shostak 1979) after both the single dish and the VLA data are corrected for primary beam response. Figure 1 presents the frames of the 30′′ data cube. Each image (traditionally called a channel map) in this figure represents the 21 cm line emission at a certain velocity, the abscisa and ordinate axis are the RA and Dec coordinates. The lack of artifacts in these images (such as a negative bowl around the galaxy) also suggests that the images have been correctly deconvolved. 3. Results and Analysis The general properties of NGC 1058 are presented in Table 1. Figure 2 shows intensity weighted mean velocity contours atop the H I intensity map and presents the H I spiral structure of NGC 1058. Figure 2 also illustrates the superb sensitivity and resolution of these studies, which allowed us to measure the H I emission at distances of approximatively 10 kpc from the center of the disk and to differentiate between the arms and the inter-arms. We characterize the widths of the H I profiles by the relative dispersions σv of the best Gaussian fit 3; the fits were done using a least squares minimization algorithm. Figures 3 and 4, show the observed profiles for a few pixels throughout NGC 1058 from the 45′′ and the 30′′ data sets respectively. The single Gaussians which best approximate the shapes of these profiles, as well as their residuals are also shown. Note, that each of the profiles is representative of the remainder of the spectra, it is not a best find or the result of averaging over large areas of the disk or velocity space. While the residual patterns suggest that a single Gaussian is not a good functional description of the H I profiles in NGC 1058, the FWHM and σv derived from the single Gaussian fits track well the intrinsic width of the H I spectrum. This proportionality allows us to describe the widths of the profile in terms of the results of our least squares fitting. The general characteristics of the velocity dispersion will be discussed in terms of the 45′′and 30′′ cubes. 3 The FWHM is often use to characterize the H I line widths. This tradition is based upon the fact that H I profiles are modeled by one or multiple Gaussians, where the flux as a function of velocity v is given by f(v) = (v − v0) and v0 is the velocity at associated with the peak flux. – 5 – 4. General Characteristics of the Velocity Dispersion Figure 5 presents the distribution of velocity dispersions across the disk of NGC 1058. Unlike previous observers of NGC 1058, we find a wide range of dispersions from 4 to 14 km sec−1 in addition to a few extremely narrow profiles with σv ∼ 3.5 km sec−1. These narraow profiles are found in regions of relatively low column density at radii greater than 300′′ or 13 kpc. There are three regions of high dispersion which stand out in Figure 5: one in the center (labeled C and ∼ 4.5 kpc across) and two others symmetric about the center in the North-West (N ∼ 3 kpc) and South-East (S ∼ 3 × 5 kpc) of the center. We find no obvious correlation between high H I velocity dispersion and stars or star formation tracers such as Hα (Figure 10), radio contiuum, SNe except in the central region C. The most probable explanation for the observed highest dispersions outside the central region (i.e. in N and S) are small scale (≤ 0.7 kpc) bulk motions (see section 7 below). In the southern, part the disk could be warping (van der Kruit & Shostak 1984, Shen & Sellwood 2006) leading to the observed broad profiles. Figure 6 shows that H I profiles from N and S are also assymmetric. However, a similar explanation for region N would suggest rather impressive small-scale structure in the warp as it would requiere the inclination to change ∼ 3 degrees over a region smaller than 0.7 kpc in diameter, if we assume from Tully Fisher an intrinsic rotation velocity of 150 km s−1 . A more exciting alternative explanation for the bulk motions observed in N is that they are caused by the infall of gas left over from galaxy formation. However, this is somewhat difficult to reconcile with the relatively low column density in these regions. Two global trends are evident from the derived dispersions: a radial fall-off, shown in Figure 7, and a predominance of the broadest profiles in the inter-arm regions of the galaxy (Figure 9). Ferguson et al. (1998) used deep Hα observations to reveal the presence of H II regions in the central 6 kpc of NGC 1058. There are several knots of high dispersion (12 to 13.5 km/sec) in region C, with most of the profiles measuring between 7.5 and 11 km/sec. However, none of the star formation sites outside the central ∼2 kpc discovered in that study seem to affect the width of the profiles. Also, regions N and S are located in the inter-arm regions and are not associated with sufficiently strong star formation to be detected in the Ferguson et al. (1998) study. Therefore we find that the dispersions do not correlate with star formation as shown from the overlay of Feruson et al (1998)’s Hα map atop contours of velocity dispersion Figure 10. Figure 8 shows the kinetic energy in the gas associated with motions perpendicular to the disk. Because only a qualitative behaviour was of interest here, the kinetic energy in vertical motions at a certain pixel location was roughly approximated as the product of total intensity times the square of the velocity dispersion. Approximated as such, the kinetic – 6 – energy in vertical motions does not follow the decline in star light which drops with radius as ∼ exp . Figure 9 shows that, the broadest profiles seem to be found in relatively low column density areas between the spiral arms (as traced by H I). As such we do not find a correlation between the velocity dispersion of stars in the disk or the H I column density. The dissimilarity between stars, star-formation, H I intensity, and the kinetic energy in the gas implies that processes other than those directly associated with stars put energy into the ISM. Sellwood & Balbus (1999) suggested that magnetic fields with strengths of a few micro-gauss in these extended disks allow energy to be extracted from galactic differential rotation through MHD-driven turbulence. While that mechanism predicted a uniform dis- persion outside of the optical disk, in an attempt to explain lower quality data on NGC 1058, a similar mechanism has the potential of explaining the level and behaviour of the velocity dispersions as a function of radius (Sellwood, private communication). The Sellwood & Bal- bus (1999) paper generated significant work on numerical models that predict the occurance of the magnetorotational instability in galactic disks (e.g. Dziourkevitch, Elstner, & Rudiger 2004, Pionteck & Ostriker 2004). 5. Profile Shapes Any model that would explain how energy is put into the ISM must account for the shape of the profiles in NGC 1058. A single Gaussian least-squares fitting routine was run on the 30′′ and 45′′ data sets. In both cases, we found that while the signal to noise for most profiles was excellent the chisq per degree of freedom was larger than a few, the residuals also suggested that the wings were broader than those of a Gaussian. To understand whether how the H I line shapes varied throughout the galaxy, the profiles were normalized by flux, aligned so that their peaks were at the same central velocity. These were plotted in units of FWHM (2.354σv), using the parameters from the single Gaussian fits to control the scaling. This was done to reveal only the difference in the line shapes and not other differences such as the width of peak intensity. While stacking up the 45′′ profiles it became clear that almost all profiles appeared to have the same shape. Figure 13 suggests that despite it being non-Gaussian, the shapes of the line profiles are identical throughout most of the galaxy when scaled by σv and their peak flux and aligned so that their peaks occur at the same velocity. For the 45′′ data, median shapes from profiles within different width and peak intensity ranges were compared and found to be identical within the error-bars. The method used to derive such median line shapes is fairly straightforward. After the pixel selection (by FWHM, – 7 – location in the galaxy, etc.) the profiles corresponding to every pixel were normalized in intensity dividing by the peak flux. A grid of 63 channels for the 45′′ data and 108 for the 30′′ data was set up to replace the velocity axis from units of km/sec to units of FWHM. For example suppose that the velocity corresponding to the peak of a certain profile (in the 45′′ cube) is Vcen km s −1 and that its FWHM is FW km s−1. Only the channels between Vcen − 3×FW and Vcen + 3×FW were used in deriving the median shape. The normalized fluxes corresponding to these channels were then resampled onto a grid where each bin (i.e.; channel) is 6FW divided by the number of channels. Each bin therefore contains a certain number distribution of normalized fluxes; these fluxes were then sorted and the middle value is taken as the median. Median profiles were also derived and compared from various areas throughout the disk and the line shapes appeared similar everyhwere except in N and S, where the profiles where more asymmetric as previously discussed. For brevity we present just two of these tests in Figure 14. The same median comparison tests were done on the 30′′ data. At the 30′′ resolution median profiles derived for certain ranges of peak flux and from various areas throughout the galaxy were also identical. However, the median profiles derived for various ranges of FWHM appeared to vary in the shape of their wings, perhaps because of the lower signal to noise in this data set, and to the smaller number of broad (σv ≥ 10km s−1) than that of narrow lines. Throughout our analysis we assumed that the noise characteristic in each profile is random and that the rms noise is the same regardless of the strengh of the signal. This assumption need not be true as deconvolution algorithms seem to produce noise that is proportional in a non-linear fashion with signal (Rupen 97). However different noise for different flux levels will hardly lead to a universal, non-Gaussian line shape. A double Gaussian (a narrow and a broad component) as shown in Figure 15 is a good fit to the median profile derived from the 45′′ data. 6. Kinetic Energy Distribution The uniformity of the profile shape in NGC 1058 suggests that on scales of 2.5 kpc, the neutral gas is being stirred into the same distribution of energy per unit mass and that this distribution is different than that for other galaxies (e.g. the Milky Way). Figure 16 shows the normalized kinetic energy (KE) distribution for the Milky Way and for NGC 1058. This comparison is only qualitative. The term “normalized” in the case of the Milky Way refers to the fact that the KE distribution was obtained from a model (i.e. double Gaussian fit) of the H I emission at the North Galactic Pole; this model was presented in Kulkarni – 8 – & Fich (1985), hereafter KF85, and it only includes “normal” emission, i.e. it does not include emission from the H I falling into the disk. These authors corrected for the infalling emission by assuming that the huge bump on one side of the profile represented infalling gas. To remove the bump they reflected the profile about the velocity corresponding to the peak flux and obtained the profile shown in Figure 16. The units of the KF plot are Kelvin2 km2 s−2. The units for the NGC 1058 are arbitrary, and the term “normalized” in this case means that instead of flux or temperature, we use flux divided by peak flux, and bin numbers instead of velocities. To compare those qualitatively we aligned the KF profile with the NGC 1058 median profile from the entire 45′′ data set. The aligning was done by fitting a single Gaussian to the KF85 profile and to the NGC 1058 median profile. We require and that the limits of the KF85 and the NGC 1058 profiles span an equal number of FWHM. The striking feature in the Galactic energy distribution, also noted by Kulkarni & Fich (1985) is the almost constant kinetic energy for about 50 km/sec. In contrast, NGC 1058’s KE curve is more centrally peaked. Presumably the KE distribution is set both by the galactic potential as well as explosive ISM events (such as SNe, star formation, and infalling gas). It is not perfectly clear how these factors have shaped the energy distribution as a function of velocity of either the Milky Way or NGC 1058. function of velocity. 7. Beam Smearing and Bulk Motions An accurate study of the profile shapes throughout the galaxy require us to understand the effect of beam smearing on our measurements. Consider a round spiral disk with gas moving in circular orbits at a velocity vcirc; attaching polar coordinates to this disk (rd, θ) and letting the angle between the normal to the plane of the galaxy and line-of sight be referred to as the inclination angle (i) the observed radial velocity on a set of sky-coordinates (x, y) will be vz(x, y) = vz(rd, θ)cos(i) + vcirc(rd) sin(i)cos(θ) + vred where vred is the velocity of the galaxy for which with respect to the observer. Obviouosly a smaller i is (a more face-on) makes it easier to measure the true vz distribution. A gradient in the vcirc(rd)sin(i)cos(θ) across the resolution element (beam) will increase the width of the profile and confuse the measurements of the velocities perpendicular to the disk. This problem is known as beam smearing. Two tests were performed using our highest resolution (15′′) data to quantify the effect of beam smearing on our measurements of σv. First, we determined the maximum in-plane velocity difference within a beam which would contribute to the width of the line profile at a certain position in the galaxy (i.e. at a pixel). This was done by finding the maximum – 9 – difference (hereafter Mdiff) between the central velocity vcen (i.e. the velocity associated with the peak flux as derived from the single Gaussian fit to the 15′′ data) of the H I pixel and the central velocities of all the pixels within a square with 32′′ sides centered on that pixel. Figure 11 shows the map of these maximum difference. This method is based on the assumption that differences in vcen are due to gas motions in the plane of the galaxy. This test shows that σv is correlated with Mdiff . The correlation between σv and Mdiff suggests the exitence of bulk motions on scales smaller and equal to those probed by our highest resolution data 15′′ (0.7 kpc). Finding Mdiff across NGC 1058’s disk gives an upper limit to the broadening of the H I profiles. To better understand the effect of beam smearing on our observations we constructed a simple model of how H I in NGC 1058 would appear if it was an infinitely cold disk; we then convolved this model with a 30′′ beam and ran our Gaussian least squares fitting routine on the resulting H I profiles. The widths of these final model profiles were significantly smaller than those measured in NGC 1058 (Figure 12) suggesting that beam smearing does not have a significant impact on our observations. 8. Summary and Conclusions Excellent resolution and high sensitivity H I observations of NGC 1058 show an intrigu- ing picture of the interstellar medium throughout this galaxy: the velocity dispersion ranges from 4 to 14 km/sec but is not correlated with star formation or the spiral arms, which is another major ISM regulator. Global trends such as a radial fall-off must be explained in the context of significant local effects; most notable among these are isolated, resolved regions of high velocity dispersions as well as significant scatter in the dispersion at a given radius. In summary unlike some previous studies, we find that the dispersion is not constant and it does not simply decline with radius. we also find that there is no tight correlation between the width of the profiles and the spiral arms. The most probable source for the highest dispersions observed outside the central regions are small scale (≤0.7 kpc) bulk motions. The energy sources supporting such motions are not entirely clear. The disk is warped in the southern part (van der Kruit & Shostak 1984, Shen & Sellwood 2006), leading to the observed broad profiles: however, a similar explanation for region N would suggest a rather impressive small-scale structure in the warp, as it would require the inclination to change by ∼ 3 degrees over a size smaller than 0.7 kpc. There is no obvious correlation with stars or star formation tracers such as Hα, radio continuum, SNe except in region C; nor is it clear what role, if any, is played by spiral arms – 10 – in driving the observed small scale bulk motions. Some of the measured velocity dispersions are higher than the 10 km s−1 canonical sound speed in the ISM, but since we cannot easily measure directly the pressure and 3-dimensional density structure of the gas, we cannot determine the exact sound speed to know if we are indeed seeing supersonic motions. The shapes of the H I profiles in NGC 1058 are non-Gaussian and hence cannot be explained as emission from single temperature gas. Therefore, it is not clear whether these narrow profiles are evidence of a lower thermal balance point between heating and cooling mechanisms in NGC 1058’s outskirts as compared to the rest of the galaxy. A double Gaussian description of the H I profile is far from a complete surprise. The surprise is the constancy between the broad and narrow components throughout NGC 1058’s H I disk. In previous studies (e.g. Mebold 1972, Young & Lo 1996) it was found that some of the H I profiles were well described by double Gaussians, and these associated the narrow Gaussians with the CNM and the broad with the WNM. Young & Lo (1996) found that the narrow component existed only in regions of high H I column density, next to areas with active star formation. It is unlikely that the universal profile in NGC 1058 can be explained as a combination of cold and warm medium for the narrow and broad component respectively, because it seems difficult to have the same ratio of warm to cold gas in regions associated with stars and star fromation and at radii three times the optical R25. Also, high resolution observations in other galaxies (Braun 1998) showed that the CNM dissappears at the edges of the optical disk. Further quests on the observational front such as (at what resolution will this universality break down, is this spatial scale particular to NGC 1058, can we see the same shape and/or its universality in other systems), as well as theoretical efforts to model mechanisms of injecting energy into the ISM, and determine how that energy dissipates throughout a fractal ISM are necessary to understand the full significance of the universal profile in our 45′′ data cube. A.P. would like to thank Jacqueline van Gorkom for invaluable help in designing the experiment, as well as during during the analysis process and in editing this document. A.P. would also like to thank Liese van Zee, Mordecai Mac-Low and Jennifer Donovan for their helpfull suggestions and discussions. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.. REFERENCES Allen, R. J., and Shostak, G. S. 1979, Astron. Astrophys. Suppl. 35, 163 – 11 – Braun, R. and Walterbros R. A. M. , 1992 ApJ, 386, 120 Braun, R., 1997 ApJ484, 637 Braun, R., astro-ph/9804320 Interstellar Turbulence, Proceedings of the 2nd Guillermo Haro Conference. Edited by Jose Franco and Alberto Carraminana. Cambridge University Press, 1999., p.12 Dickey, J.M. and Lockman, F.J., 1990, Annu. Rev. Astron. Astrophys. 28, 215 Dickey, J.M., Mebold, U., Stanimirovic, S., Staveley-Smith L., 2000, ApJ, 536. 756D Dziourkevitch, N., Elstner, D., & Rudiger, G., 2004, A& A, 423, L29 Eskridge, P. B., Frogel, J. A., Pogge, R. W., Quillen, A. C., et al. 2002, ApJS, 143, 73 Ferguson, A., Wyse, R. F. G., Gallagher, J.S., Hunter, D.A., 1998 ApJ506, 19 Ferguson, A., Gallagher, J. S., Wyse, R. F. G., AJ, 116, 673 Kulkarni, S.R. and Fich, M., 1985 ApJ289, 792 Kulkarni, S.R. and Heiles, C., 1988 in Galactic and Extragalactic Radio Astronomy, ed. G.L. Verschuur& K. I. Kellermann (New York: Springer Verlag), 95 Lewis, B.M., 1984 ApJ285, 453 Lewis, B.M., 1987 ApJS63, 515 Lewis, B.M., 1987 Obs. 107, 201L Li, Y., Mac Low, M.M., & Klessen, R.S., 2005, ApJ, 620, L19 Lockman, F. J., 1984, ApJ, 283, 90 Lockman, F. J. and Gehman, C. S., 1991, ApJ, 382, 182 Mac Low, M.M., & Klessen, R. S., 2004, Rev. Mod. Phys. 76, 125 Malhotra, S., 1994, ApJ, 433, 687 Malhotra, S., 1995 ApJ, 448, 132 McKee, C.F.,Ostriker, J.P. 1977, ApJ, 218, 148 Mebold, U., 1972, A& A, 19, 13 http://arxiv.org/abs/astro-ph/9804320 – 12 – Merrifield, M. R., 1993, MNRAS, 261, 233 Olling, R., 1995, PhD Thesis Olling, R., 1996, AJ 112, 457 Oort, J. H., 1932, Bull. Astron. Inst. Netherlands, 6, 349 Piontek, R. A., & Ostriker, E. C., 2004, ApJ 601, 905 van der Kruit, P.C., Shostak, G.S., 1982 A&A, 115, 293S Rupen, M. P., 1987 PhD thesis Rupen, M. P., 1997, VLA Scientific Memorandum, No. 172: A Test of the CS (Shortened C) Configuration available at http://www.vla.nrao.edu/memos/sci Rupen, M. P., 1998, VLA Scientific Memorandum, No. 172: A Test of the CS (Shortened C) Configuration available at http://www.vla.nrao.edu/memos/sci/175/cstest2/ Rupen, M.P., 1999, Spectral Line Observing II: Calibration and Analysis in ASP Conf. Ser.,180, Synthesis Imaging in Radio Astronomy II ed. by Taylor, G.B., Carilli, C.L., & Perley, R.A. pg. 229 van der Kruit, P.C., Shostak, G.S., 1984 A&A, 134, 258V Shen, J. & Sellwood, J. A., MNRAS, 2006 370, 2 Sellwood, J. A. & Balbus, S.A., ApJ 1999, 511, 660 Shostak, G.S., van der Kruit, P.C., 1984 A&A, 132, 20S Young, L.M., Lo, K.Y. 1996 ApJ, 462, 203 Young, L.M., Lo, K.Y. 1997 ApJ, 476, 127 Young, L.M., Lo, K.Y. 1997 ApJ, 490, 710 This preprint was prepared with the AAS LATEX macros v5.2. http://www.vla.nrao.edu/memos/sci http://www.vla.nrao.edu/memos/sci/175/cstest2/ – 13 – Table 1. General Properties NGC 1058 R.A.(B1950) 02 40 23.2 Dec (B1950) +37 07 48.0 Morphological type Sc Vsys [km/sec] 518 LB [L⊙] 1.5× 109 MHI [M⊙] 2.3× 109 SFR [M⊙ yr −1]a 3.5× 10−2 D25×d25 [arcmin]b 3.0× 2.8 Distance[Mpc]c 10 Physical equivalent of 1′′ 48.5 pc Inclinationd 4–11◦ Environmente member of the NGC 1023 Group aSFR stands for Star Formation Rate, it was calculated from Hα fluxes by Ferguson, Gallagher & Wyse (1998) bNASA/IPAC extragalactic database (NED) cFerguson, Gallager, & Wyse (1998) dvan der Kruit & Shostak (1984) e Lewis (1975) – 14 – Fig. 1.— Sample channel maps for NGC 1058–Each square image represents the H I 21 cm line emission within a velocity range of 2.58 km s−1 where the central velocity of that range is given on the upper left corner of each image. The x and y axis of every channel map gives the Right Ascension (RA) and Declination (Dec) coordinates,and are identical to these in Figure 2. Such sample channel maps can be assembled together in the same way the frames of a movie are put together to make what is refered to as a data cube. – 15 – Fig. 2.— Intensity weighted mean velocity contours atop H I intensity map for NGC 1058 Intensity weighted mean velocity contours atop H I intensity map (grey) for NGC 1058—This figure was made using the 15′′ data cube with a sensitivity of 0.5 mJy/beam corresponding to a column density of 1.6 × 1018 cm−2. The physical resolution of this image is 0.7 kpc and the velocity contours range between 500 and 558 km/sec in 2 km/sec increments. The H I disk in NGC 1058 extends to a diameter of more than 20 kpc. – 16 – Fig. 3.— Four sample of H I profiles (crosses), the Gaussian fit (solid line) and the residual pattern (red) with σv from the Gaussian fit: 12.7 km/sec (upper left, pixel from a region of high dispersion), 5.95 km/sec (upper right, pixel from a region of low dispersion), 7.6 km/sec (lower left pixel from an interarm region), 9.3 km/sec (lower right, pixel from an arm region). The x axis is in km/sec and the y axis represents H I intensity in mJy/beam. The shown residuals indicate that a single Gaussian function does not adequately describe the line shapes. However the width of the Gaussian does track the breadth of the H I profile. This figure is based on the 45′′ data cube and the single Gaussian fits done for that cube. – 17 – Fig. 4.— Three samples of H I profiles (solid line), the Gaussian fit (squares) and the residual pattern (dashed) with σv from the Gaussian fit: 3.8, 7.6, and 13.2 respectively. Figure based on the 30′′ data cube and the single Gaussian fits done for that cube. The x axis is in km/sec and the y axis represents H I intensity in mJy/beam. – 18 – Fig. 5.— Distribution of dispersions throughout NGC 1058; the regions of highest dispersion are labeled N, C, and S. The x and y axis are the RA and Dec in B1950 coordinates. This figure is based on the results of the single Gaussian fit performed on the 30′′ data. The contours are in km/sec and start in steps of 0.5 km/sec. Black is used for dispersions between 5.5 to 7, cyan for 7.5 to 9, green for 9.5 to 11, red for 11.5 to 13, and magenta for 13.5 to 15 km/sec. – 19 – Fig. 6.— Four sample profiles in the regions with highest asymmetries (of order few percent); The x axis of each of the four plots represents velocity and is in units of km s−1 and the y axis represents H I intensity in mJy/beam, where the beam refers to the point spread function of the observations. The upper left is from region N, upper right from C, lower left from S, and lower right from a region West of S. Regions N,C, and S are shown and labeled in Figure 3. – 20 – Fig. 7.— Radial dependece of NGC 1058’s σvs—The x axis (in arcseconds) gives the radius while the y axis (in km/sec) gives the σv as derived from single Gaussian least squares fits to the 30′′ data cube. The filled circles represent points with error bars,less than 12.5% of σv, the empty circles – points with error bars between 12.5% and 25% and the dots– points with errors greater than 25%. Despite a few high σv regions (N,S in Figure 5, the radial falloff is evident. – 21 – Fig. 8.— The energy in the neutral gas was roughly approximated as the product of the total H I intensity and the square of the velocity dispersion. Both the green and the black lines represent azimuthal averages of concentric rings around the center of NGC 1058. The black error bars show the rms in each of these rings. The red line shows the exponential fit to the stellar data. The energy in the gas falls off with radius much slower than the stellar luminosity suggesting that processes other than those associated with star input energy are responsible for heating the gas at large radiae. – 22 – Fig. 9.— σv contours atop of an H I total intensity map—The contours range from 4 to 14 km/sec in steps of 0.5 km/sec and are based on the single Gaussian fit to the 30′′, NGC 1058 data cube. Note that regions N and S of high σv are located in the inter-arms. – 23 – Fig. 10.— Hα greyscale from Ferguson, Gallager, & Wyse 1998, atop dispersion contours as in Figure 5. – 24 – Fig. 11.— Contours of maximum potential beam smearing in km s−1. Black is used for dispersions maximum beam smearing effect of 1, 2, and 3 km/sec, cyan for 4,5,6, green for 7,8,9, red for 10, 11, and 12, and magenta for values of 30 km/sec and above. 5.5 to 7, cyan for 7.5 to 9, green for 9.5 to 11, red for 11.5 to 13, and magenta for 13.5 to 15 km/sec. The magenta contours are regions where the H I emssion was very faint or non-existent. As such the Gaussian fitting routine employed produced spurious results. The square shape of some of the contours is an artifact of the method employed in determining the maximum beam smearing. This figure is based on the 15′′ data cube. Note that the highest velocity gradients are found in regions N and S and south-west of C. Regions N, S, and C are shown and labeled in Figure 5. – 25 – Fig. 12.— Effect of beam smearing on σv measurements at 30 ′′ resolution. The x axis in in km/sec and the y axis is in mJy/beam. The connected squares represent the observed profile. The narrow profiles were obtained by modeling the velocity profiles associated with an infinitely cold disk and then convolving that model with a 30′′ beam, and running the Gaussian least squares fitting routine on the convolved cube. – 26 – Fig. 13.— Median Profile (red) atop all H I profiles from NGC 1058 45′′ data cube. The x axis is flux/peak flux and the y axis is velocity minus the central velocity and divided by the FWHM. – 27 – Fig. 14.— Median profiles derived for certain FWHM ranges (left pannel);and for peak flux ranges (right pannel) from the 45′′ NGC 1058 data set. The x axis is not in [km/sec] but represents the grid (bin number) on which the profiles were set. Please refer to text for a detailed explanation. The y axis is the flux divided by peak flux. In the left pannel black is used for median profiles with widths (FWHM) between 14 to 18 km/sec, cyan for 18 to 22 km/sec, green 22 and red for 26 to 30 km/sec. In the right pannel a solid line is used for profiles with peak fluxes between 10 and 25 mJy/beam, a dotted line is used for profiles with peaks between 25 and 40 mJy/beam, short dash for 40 to 55, and long dash for 55 to 70 mJy/beam. – 28 – Fig. 15.— Double Gaussian Fit to the Universal Profile — The x axis is in bins as described in the text and the y axis is the flux normalized to the peak intensity. The two Gaussian components used to fit the Median profile and their sum are shown in dotted lines. The residuals are shown in red. Error bars based on the rms in each bin are also given. The ratio between the areas of the broad and narrow components is 1.35 while that between their FWHM is 2.09. – 29 – Fig. 16.— Normalized Kinetic Energy distributions for the Milky Way (top) and NGC 1058 (bottom) and corresponding H I profiles. The top figure was obtained from a double Gaussian decomposition of the North Galactic Pole H I emission, from Kulkarni & Fich (1985). For the top figure, the x axis represents velocity in units of km s−1 and the y axis is the normalized energy in units of Kelvin km2 s−2. The inset upper right figure shows the H I profile from which the normalized energy curve for the Milky Way’s North Galactic Pole emission was estimated. The bottom figure shows the qualitative behaviour of the kinetic energy distribution with velocity in NGC 1058. Here the x axis represents a bin number (ref.to text) while the y axis represents the qualitative behaviour of the kinetic energy distribution in NGC 1058. This figure suggests that the kinetic energy in the Galactic North Galactic Pole emission is more evenly distributed in velocity than that in NGC 1058. Introduction Observations and Data Reductions Results and Analysis General Characteristics of the Velocity Dispersion Profile Shapes Kinetic Energy Distribution Beam Smearing and Bulk Motions Summary and Conclusions

0704.0280

Common Envelope Evolution Redux

Common Envelope Evolution Redux Ronald F. Webbink Department of Astronomy, University of Illinois, 1002 W. Green St., Urbana, IL 61801, USA Summary. Common envelopes form in dynamical time scale mass exchange, when the envelope of a donor star engulfs a much denser companion, and the core of the donor plus the dense companion star spiral inward through this dissipative en- velope. As conceived by Paczynski and Ostriker, this process must be responsible for the creation of short-period binaries with degenerate components, and, indeed, it has proven capable of accounting for short-period binaries containing one white dwarf component. However, attempts to reconstruct the evolutionary histories of close double white dwarfs have proven more problematic, and point to the need for enhanced systemic mass loss, either during the close of the first, slow episode of mass transfer that produced the first white dwarf, or during the detached phase preceding the final, common envelope episode. The survival of long-period inter- acting binaries with massive white dwarfs, such as the recurrent novae T CrB and RS Oph, also presents interpretative difficulties for simple energetic treatments of common envelope evolution. Their existence implies that major terms are missing from usual formulations of the energy budget for common envelope evolution. The most plausible missing energy term is the energy released by recombination in the common envelope, and, indeed, a simple reformulation the energy budget explicitly including recombination resolves this issue. 1 Introduction From the realization [26, 27, et seq.] that all cataclysmic variables (CVs) are interacting binary stars, their existence posed a dilemma for theories of binary evolution. The notion that close binary stars might evolve in ways fundamentally different from isolated stars was rooted in the famous ‘Algol paradox’ (that the cooler, lobe-filling subgiant or giant components among these well-known eclipsing binaries are less massive, but more highly evolved, than their hotter main-sequence companions). The resolution of that para- dox invoked large-scale mass transfer reversing the initial mass ratios of these binaries [34]. Indeed, model calculations assuming conservation of total mass and orbital angular momenum are qualitatively consistent with the main fea- tures of Algol-type binaries. Even if quantitative consistency between models http://arxiv.org/abs/0704.0280v1 2 Ronald F. Webbink and observational data generally requires some losses of mass and angular momentum among Algol binaries (e.g., [10, 21, 8]), the degree of those losses is typically modest, and the remnant binary is expected to adhere closely to an equilibrium core mass-radius relation for low-mass giant stars (see, e.g., the pioneering study of AS Eri by Refsdal, Roth & Weigert [46]). Those rem- nant binaries are typically of long orbital period (days to weeks) in compari- son with CVs, and furthermore typically contain helium white dwarfs of low mass, especially in the short-period limit. In contrast, CVs evidently contain relatively massive white dwarfs, in binary systems of much shorter orbital periods (hours), that is, with much smaller total energies and orbital angular momenta. In an influential analysis of the Hyades eclipsing red dwarf/white dwarf binary BD +16◦ 516 (= V471 Tau), Vauclair [57] derived a total system mass less than the turnoff mass of the Hyades, and noted that the cooling age of the white dwarf component was much smaller than the age of the cluster. He speculated that V471 Tau in its present state was the recent product of the ejection of a planetary nebula by the white dwarf. Paczynski [41] realized that, immediately prior to that event, the white dwarf progenitor must have been an asymptotic giant branch star of radius ∼600 R⊙, far exceeding its current binary separation ∼3R⊙. He proposed that the dissipation of orbital energy provided the means both for planetary nebula ejection and for the severe orbital contraction between initial and final states, a process he labeled ‘common envelope evolution’ (not to be confused with the common envelopes of contact binary stars). Discovery soon followed of the first ‘smoking gun’, the short-period eclipsing nucleus of the planetary nebula Abell 63 [3]. Over the succeeding three decades, there have been a number of attempts to build detailed physical models of common envelope evolution (see [55] for a review). These efforts have grown significantly in sophistication, but this phenomenon presents a daunting numerical challenge, as common envelope evolution is inherently three-dimensional, and the range of spatial and tem- poral scales needed to represent a common envelope binary late in its inspiral can both easily exceed factors of 103. Determining the efficiency with which orbital energy is utilized in envelope ejection requires such a code to conserve energy over a similarly large number of dynamical time scales. Theoretical models of common envelope evolution are not yet capable of predicting the observable properties of objects in the process of inspiral. If envelope ejection is to be efficient, then the bulk of dissipated orbital energy must be deposited in the common envelope on a time scale short compared with the thermal time scale of the envelope, else that energy be lost to ra- diation. The duration of the common envelope phase thus probably does not exceed ∼103 years. However, general considerations of the high initial orbital angular momenta of systems such as the progenitor of V471 Tau, and the fact that most of the orbital energy is released the envelope only very late in the inspiral have led to a consensus view [60, 32, 33, 65, 50] that the planetary nebulae they eject should be bipolar in structure, with dense equatorial rings Common Envelope Evolution Redux 3 absorbing most of the initial angular momentum of the binary, and higher- velocity polar jets powered by the late release of orbital energy. Indeed, this appears to be a signature morphology of planetary nebulae with binary nuclei (e.g., [2]), although it may not be unique to binary nuclei. 2 The Energetics of Common Envelope Evolution Notwithstanding the difficulties in modeling common envelope evolution in detail, it is possible to calculate with some confidence the initial total energy and angular momentum of a binary at the onset of mass transfer, and the corresponding orbital energy and angular momentum of any putative remnant of common envelope evolution. Consider an initial binary of component masses M1 and M2, with orbital semimajor axis Ai. Its initial total orbital energy is Eorb,i = − GM1M2 . (1) Let star 1 be the star that initiates interaction upon filling its Roche lobe. If M1c is its core mass, and M1e = M1 − M1c its envelope mass, then we can write the initial total energy of that envelope as Ee = − GM1M1e λR1,L , (2) where R1,L is the Roche lobe radius of star 1 at the onset of mass transfer (the orbit presumed circularized prior to this phase), and λ is a dimensionless pa- rameter dependent on the detailed structure of the envelope, but presumably of order unity. For very simplified models of red giants – condensed poly- tropes [40, 14, 16] – λ is a function only of me ≡ Me/M = 1 − Mc/M , the ratio of envelope mass to total mass for the donor, and is well-approximated λ−1 ≈ 3.000− 3.816me + 1.041m e + 0.067m e + 0.136m e , (3) to within a relative error < 10−3. For the final orbital energy of the binary we have Eorb,f = − GM1cM2 , (4) where Af is of course the final orbital separation. If a fraction αCE of the difference in orbital energy is consumed in unbinding the common envelope, αCE ≡ orb − E , (5) 4 Ronald F. Webbink αCEλr1,L M1 −M1c , (6) where r1,L ≡ R1,L/Ai is the dimensionless Roche lobe radius of the donor at the start of mass transfer. In the classical Roche approximation, r1,L is a function only of the mass ratio, q ≡ M1/M2 [7]: r1,L ≈ 0.49q2/3 0.6q2/3 + ln(1 + q1/3) . (7) Typically, the second term in brackets in (6) dominates the first term. As formulated above, our treatment of the outcome of common envelope evolution neglects any sources or sinks of energy beyond gravitational terms and the thermal energy content of the initial envelope (incorporated in the parameter λ). The justification for this assumption is again that common envelope evolution must be rapid compared to the thermal time scale of the envelope. This implies that radiative losses (or nuclear energy gains – see below) are small. They, as well as terminal kinetic energy of the ejecta, are presumably reflected in ejection efficiencies αCE < 1. We neglect also the rotational energy of the common envelope (invariably small in magnitude compared to its gravitational binding energy), and treat the core of the donor star and the companion star as inert masses, which neither gain nor lose mass or energy during the course of common envelope evolution. One might imagine it possible that net accretion of mass by the companion during inspiral might compromise this picture. However, the common envelope is typically vastly less dense than the companion star. and may be heated to roughly virial temperature on infall. A huge entropy barrier arises at the interface between the initial photosphere of the companion and the common envelope in which it is now embedded, with a difference in entropy per particle of order (µmH/k)∆s ≈ 4–6. The rapid rise in temperature and decrease in density through the interface effectively insulates the accreting companion thermally, and strongly limits the fraction of the very rarified common envelope it can retain upon exit from that phase [62, 15]. Common envelope evolution entails systemic angular momentum losses as well as systemic mass and energy losses. Writing the orbital angular momen- tum of the binary, GM21M 2A(1 − e M1 +M2 , (8) in terms of the total orbital energy, E = −GM1M2/2A, we find immediately that the ratio of final to initial orbital angular momentum is )3/2 ( M1c +M2 M1 +M2 )−1/2 ( )1/2 ( 1− e2f 1− e2i . (9) Since M1c < M1 and we expect the initial orbital eccentricity to be small (ei ≈ 0), it follows that any final energy state lower than the initial state (|Ef | > Common Envelope Evolution Redux 5 |Ei|) requires the loss of angular momentum. The reverse is not necessarily true, so it is the energy budget that most strongly constrains possible outcomes of common envelope evolution. 3 Does Common Envelope Evolution Work? As an example of common envelope energetics, let us revisit the pre-CV V471 Tau, applying the simple treatment outlined above. It is a member of the Hyades, an intermediate-age metal-rich open cluster (t = 650 Myr, [Fe/H] = +0.14) with turnoff mass MTO = 2.60 ± 0.06M⊙ [28]. The cooling age of the white dwarf is much smaller than the age of the cluster (tcool,WD = 10 yr [39] – but see the discussion there of the paradoxical fact that this most massive of Hyades white dwarfs is also the youngest). Allowing for the possi- bility of significant mass loss in a stellar wind prior to the common envelope phase, we may take MTO for an upper limit to the initial mass M1 of the white dwarf component. The current masses for the white dwarf and its dK2 companion, as determined by O’Brien et al. [39] are MWD = 0.84± 0.05M⊙, MK = 0.93±0.07M⊙, with orbital separation A = 3.30±0.08R⊙. A 2.60M⊙ star of Hyades metallicity with a 0.84M⊙ core lies on the thermally-pulsing asymptotic giant branch, with radius (maximum in the thermal pulse cycle) which we estimate at Ri = 680R⊙ = R1,L, making Ai = 1450R⊙. With this combination of physical parameters, we derive an estimate of αCEλ = 0.057 for V471 Tau. Equation (3) then implies αCE = 0.054. This estimate of course ignores any mass loss prior to common envelope evolution (which would drive αCE to lower values), or orbital evolution since common envelope evolution (which would drive αCE to higher values). In any event, the status of V471 Tau would appear to demand only a very small efficiency of envelope ejection.1 The fact that V471 Tau is a double-lined eclipsing member of a well- studied cluster provides an exceptionally complete set of constraints on its prior evolution. In all other cases of short-period binaries with degenerate or compact components, available data are inadequate to fix simultaneously both the initial mass of the compact component and the initial binary separation, for example. To validate the energetic arguments outlined above, one must resort to consistency tests, whether demonstrating the existence of physically- plausible initial conditions that could produce some individual system, or else 1 The anomalously small value of αCE deduced for V471 Tau may be connected to its puzzlingly high white dwarf mass and luminosity: O’Brien et al. [39] suggest that it began as a heirarchical triple star, in which a short-period inner binary evolved into contact, merged (as a blue straggler), and later engulfed its lower- mass companion in a common envelope. An overmassive donor at the onset of common envelope evolution would then have a more massive core than produced by its contemporaries among primordially single stars, and it would fill its Roche lobe with a more massive envelope at somewhat shorter orbital period, factors all consistent with a larger value of αCE having led to V471 Tau as now observed. 6 Ronald F. Webbink following a plausible distribution of primordial binaries wholesale through the energetics of common envelope evolution and showing that, after application of appropriate observational selection effects, the post-common-envelope pop- ulation is statistically consistent with the observed statistics of the selected binary type. In the cases of interacting binaries, such as CVs, one should allow further for post-common-envelope evolution. Nevertheless, within these limi- tations, binary population synthesis models show broad consistency between the outcomes of common envelope evolution and the statistical properties of CVs and pre-CVs [4, 24, 44, 17, 63], as well as with most super-soft X-ray sources [5], for assumed common envelope ejection efficiencies typically of or- der αCE ≈ 0.3–0.5. A useful tool in reconstructing the evolutionary history of a binary, used implicitly above in analyzing V471 Tau, is the mass-radius diagram spanned by single stars of the same composition as the binary. Figure 1 illustrates such a diagram for solar-composition stars from 0.08 M⊙ to 50 M⊙. In it are plotted various critical radii marking, as a function of mass, the transition from one evolutionary phase to the next.2 Since the Roche lobe of a binary component represents a dynamical limit to its size, its orbital period fixes the mean density at which that star fills its Roche lobe, logPorb(d) ≈ log(RL/R⊙)− log(M/M⊙)− 0.455 , (10) to within a very weak function of the binary mass ratio. The mass and radius of any point in Fig. 1 therefore fixes the orbital period at which such a star would fill its Roche lobe, just as the orbital period of a binary fixes the evolutionary state at which such a star initiates mass transfer. 2 Not all evolutionary phases are represented here. In a binary, a donor initiates mass transfer when it first fills its Roche lobe; if it would have done so at a prior stage of evolution, then its present evolutionary state is ‘shadowed’, in the sense that it only occurs by virtue of the binary not having filled its lobe previously. Thus, for example, low- and intermediate-mass stars cannot in general initiate mass transfer during core helium burning, because they would have filled their Roche lobes on the initial ascent of the giant branch. Fig. 1 (facing page). The mass-radius diagram for stars of solar metallicity, con- structed from the parametric models of stellar evolution by Hurley, Pols, & Tout [19] and models of thermally-pulsing asymptotic giant branch stars by Wagenhuber & Weiss [58]. Also plotted in the locus of asymptotic giant branch stars at the onset of the superwind, after Willson [64]; beyond this radius, systemic mass loss drives orbital expansion faster than nuclear evolution drives stellar expansion, and a bi- nary will no longer be able to initiate tidal mass transfer. The unlabeled dotted line terminating at the junction between lines labeled ‘helium core flash’ and ‘core helium ignition’ marks the division between those helium cores (at lower masses) which evolve to degeneracy if stripped of their envelope, and those (at higher masses) which ignite helium non-degenerately and become helium stars. Common Envelope Evolution Redux 7 8 Ronald F. Webbink In Fig. 2, the corresponding core masses of low- and intermediate-mass stars are plotted in the mass-radius diagram. For a binary which is the imme- diate product of common envelope evolution, the mass of the most recently formed white dwarf (presumably the spectroscopic primary) equals the core mass of the progenitor donor star. That donor (presuming it to be of solar metallicity) must be located somewhere along the corresponding core mass sequence in Fig. 2, with the radius at any point along that sequence corre- sponding to the Roche lobe radius at the onset of the mass transfer, and the mass a that point corresponding to the initial total mass of the donor. Thus, if the mass of the most recently-formed white dwarf is known, it is possible to identify a single-parameter (e.g., initial mass or initial radius of the donor) family of possible common-envelope progenitors. Using a mapping procedure similar to this, Nelemans & Tout [35] recently explored possible progenitors for detached close binaries with white dwarf components. Broadly speaking, they found solutions using (6) for almost all systems containing only one white dwarf component. Only three putative post-common-envelope systems failed to yield physically-plausible values of αCEλ: AY Cet (G5 III + DA, Porb = 56.80 d [49]), Sanders 1040 (in M67: G4 III + DA, Porb = 42.83 d [56]), and HD 185510 (=V1379 Aql: gK0 + sdB, Porb = 20.66 d [22]). The first two of these systems are non-eclipsing, but photometric masses for their white dwarf components are extremely low (estimated at ∼0.25M⊙ and 0.22M⊙, respectively), with Roche lobe radii consistent with the limiting radii of very low-mass giants as they leave the giant branch (cf. Fig. 2, above). They are thus almost certainly post-Algol binaries, and not post-common-envelope binaries. HD 185510 is an eclipsing binary; a spectroscopic orbit exists only for the gK0 component [9]. The mass (0.304 ± 0.015M⊙) and radius (0.052 ± 0.010R⊙) of the sdB component, deduced from model atmosphere fitting of IUE spectra combined with solution of the eclipse light curve, place it on a low-mass white dwarf cooling curve, rather than among helium-burning subdwarfs [22]. Indeed, from fitting very detailed evolutionary models to this system, Nelson & Eggleton [38] found a Fig. 2 (facing page). The mass-radius diagram for low- and intermediate-mass stars, as in Fig. 1, but with loci of constant core mass added. The solid lines added correspond to core masses interior to the hydrogen-burning shell, dashed lines to those interior to the helium-burning shell. Solid lines intersecting the base of the giant branch (dash-dotted curve) correspond to helium core masses of to 0.15, 0.25, 0.35, 0.5, 0.7, 1.0, 1.4, and 2.0 M⊙; those between helium ignition and the initial thermal pulse to 0.7, 1.0, 1.4, and 2.0 M⊙, and those beyond the initial thermal pulse to 0.7, 1.0, and 1.4 M⊙. Dashed lines between helium ignition and initial thermal pulse correspond to carbon-oxygen core masses of 0.35, 0.5, 0.7, 1.0, and 1.4 M⊙. Beyond the initial thermal pulse, helium and carbon-oxygen core masses converge, with the second dredge-up phase reducing helium core masses above ∼0.8 M⊙ to the carbon-oxygen core. Common Envelope Evolution Redux 9 10 Ronald F. Webbink post-Algol solution they deemed acceptable. It thus appears that these three problematic binaries are products of quasi-conservative mass transfer, and not common envelope evolution. The close double white dwarfs present a more difficult conundrum, how- ever. Nelemans et al. [36, 37, 35] found it impossible using the energetic argu- ments (6) outlined above to account for the existence of a most known close double white dwarfs. Mass estimates can be derived for spectroscopically de- tectable components of these systems from their surface gravities and effective temperatures (determined from Balmer line fitting). The deduced masses are weakly dependent on the white dwarf composition, and may be of relatively modest accuracy, but they are independent of the uncertainties in orbital inclination afflicting orbital solutions. These mass estimates place the great majority of detectable components in close double white dwarf binaries below ∼0.46M⊙, the upper mass limit for pure helium white dwarfs (e.g., [51]). They are therefore pure helium white dwarfs, or perhaps hybrid white dwarfs (low- mass carbon-oxygen cores with thick helium envelopes). While reconstructions of their evolutionary history yield physically-reasonable solutions for the final common envelope phase, with values for 0 < αCEλ < 1, the preceding phase of mass transfer, which gave rise to the first white dwarf, is more problematic. If it also proceeded through common envelope evolution, the deduced values of αCEλ ≤ −4 for that phase are unphysical. Nelemans & Tout [35] interpreted this paradox as evidence that descriptions of common envelope evolution in terms of orbital energetics, as described above, are fundamentally flawed. 4 An Alternative Approach to Common Envelope Evolution? Nelemans et al. [36] proposed instead parameterizing common envelope evo- lution in terms of γ, the ratio of the fraction of angular momentum lost to the fraction of mass lost: Ji − Jf M1 −M1,c M1 +M2 . (11) Both initial and final orbits are assumed circular, so the ratio of final to initial orbital separations becomes M1c +M2 M1 +M2 M1 −M1c M1 +M2 . (12) Among possible solutions leading to known close double white dwarfs, Nele- mans & Tout [35] find values 1 < γ . 4 required for the second (final) common envelope phase, and 0.5 . γ < 3 for the first (putative) common envelope phase. They note that values in the range 1.5 < γ < 1.7 can be found among possible solutions for all common envelope phases in their sample, not only Common Envelope Evolution Redux 11 those leading to known double white dwarfs, but those leading to known pre- CV and sdB binaries as well. The significance of this finding is itself open to debate. At one extreme, it would seem implausible for any mechanism to remove less angular momentum per unit mass than the orbital angular momentum per unit mass of either component in its orbit (so-called Jeans-mode mass loss). At the other extreme, a firm upper limit to γ is set by vanishing final orbital angular momentum, Jf . If M1c and M2 can be regarded as fixed, the corresponding limits on γ are M1 +M2 M1 −M1c > γ > M1 +M2 M1 −M1c M1 +M2 M1c +M2 . (13) In a fairly typical example, M1c = M2 = M1, γ is inevitably tightly con- strainted for any conceivable outcome: 5 > γ > 5 . The ratio of final to initial orbital separation, Af/Ai, is extremely sensitive to γ near the upper limit of its range. It is therefore not surprising to find empirical estimates of γ clus- tering as they do – their values merely affirm the fact that Af must typically be much smaller than Ai. The unphysically large or, more commonly, negative values of αCEλ noted above for the first mass transfer phase in the production of close white dwarf binaries [35] implies that the orbital energies of these binaries have increased through this phase (or, at any rate, decreased by significantly less than the nominal binding energies of their common envelopes). Such an increase in orbital energy is a hallmark of slow, quasi-conservative mass transfer, on a thermal or nuclear time scale. Thermal time scale mass transfer is driven by relaxation of the donor star toward thermal evolution; the re-expansion of the donor following mass ratio reversal is powered by the (nuclear) energy outflow from the core of the star. Likewise, the bulk expansion of the donor star in nuclear time scale mass transfer draws energy from nuclear sources in that star. It appears, therefore, that the first phase of mass transfer among known close double white dwarfs cannot have been a common envelope phase, but must instead have been a quasi-conservative phase, notwithstanding the difficulties that conclusion presents, as we shall now see. The dilemma that the close double white dwarfs present is illustrated in Figs. 3 and 4. Figure 3 shows the distribution of immediate remnants of mass transfer among solar-metallicity binaries of low and intermediate mass, for a relatively moderate initial mass ratio. Conservation of total mass and orbital angular momentum have been assumed. The remnants of the intial primary include both degenerate helium white dwarfs, and nondegenerate he- lium stars which have lost nearly all of their hydrogen envelopes. The helium white dwarfs lie almost entirely along the left-hand boundary, the line labeled ‘envelope exhaustion’ in Fig. 1. (The extent of this sequence is more apparent in the distribution of remnant secondaries.) Their progenitors have enough angular momentum to accommodate core growth in the terminal phases of mass transfer. In the calculation shown, the least massive cores grow from 12 Ronald F. Webbink Common Envelope Evolution Redux 13 0.11M⊙ to ∼0.18M⊙ by the completion of mass transfer. In contrast, virtu- ally all binaries leaving nondegenerate helium star remnants have too little angular momentum to recover thermal equilibrium before they have lost their hydrogen envelopes; for them, there is no slow nuclear time scale phase of mass transfer, and core growth during mass transfer is negligible. The lowest- mass helium star remnants have nuclear burning lifetimes comparable to their hydrogen-rich binary companions, now grown through mass accretion. Those more massive than ∼0.8M⊙ develop very extended envelopes during shell he- lium burning, and will undergo a second phase of mass transfer from primary to secondary, not reflected here; such massive white dwarfs are absent in the Nelemans & Tout [35] sample, and so are omitted here. In Fig. 4, the remnants of the first phase of conservative mass transfer illustrated in Fig. 3 are followed through the second phase of mass transfer, using (6). Because the remnants of the first phase have second-phase donors much more massive their companions, and nearly all have deep convective en- velopes, they are unstable to dynamical time scale mass transfer, and undergo common envelope evolution. The systems labeled ‘Without Wind Mass Loss’ have been calculated assuming that no orbital evolution or mass loss occurs between the end of the first phase of mass transfer and common envelope evolution. It is assumed furthermore that αCE = 1, in principle marking the most efficient envelope ejection energetically possible. Binary orbital periods of 0.1, 1.0, and 10 days (assuming equal component masses) are indicated for reference. Observed close double white dwarfs, as summarized by Nelemans & Tout [35], have a median orbital period of 1.4 d, and mass (spectroscopic pri- mary) 0.39M⊙. Among double-lines systems, nearly equal white dwarf masses are strongly favored, with the median q = 1.00. Clearly, most observed dou- ble white dwarfs are too long in orbital period (have too much total energy and angular momentum) to have evolved in the manner assumed here. Fur- thermore, the computed binary mass ratios are typically more extreme than observed, with the second-formed core typically 1.3-2.5 times as massive as the first. The problem is that, while remnant white dwarfs or low-mass he- Fig. 3 (facing page). Products of mass- and angular momentum-conservative mass transfer for a typical initial mass ratio. The radii indicated refer to Roche lobe radii at the onset or termination of mass transfer, as appropriate. To avoid common envelope evolution, the donor stars (the region outlined in bold toward the lower right in the diagram) must have radiative envelopes, and so arise between the terminal main sequence and base of the giant branch. Their mass transfer remnants are outlined in bold at the center-left of the diagram, with the remnant accretors at upper right. The regions mapped are truncated in each case at a lower initial donor mass of 1.0 M⊙ and upper initial donor core mass of 0.7 M⊙. Lines of constant initial core mass (with values as in Fig. 2) are indicated for the initial and remnant primaries. Lines of constant remnant primary mass are indicated for the remnant secondaries. 14 Ronald F. Webbink Common Envelope Evolution Redux 15 lium stars with suitable masses can be produced in the first, conservative mass transfer phase, the remnant companions have envelope masses too large, and too tightly bound, to survive the second (common envelope) phase of inter- action at orbital separations and periods as large as observed. Evidently, the progenitors of these double white dwarfs have lost a significant fraction of their initial mass, while gaining in orbital energy, prior to the final common envelope phase. These requirements can be fulfilled by a stellar wind, pro- vided that the process is slow enough that energy losses in the wind can be continuously replenished from nuclear energy sources. The requisite mass loss and energy gain are possible with stellar wind mass loss during the non-interactive phase between conservative and common envelope evolution, or with stellar winds in nuclear time-scale mass transfer or the terminal (recovery) phase of thermal time-scale mass transfer. Systemic mass loss during or following conservative mass transfer will (in the absence of angular momentum losses) shift the remnant regions to the left and upward in Fig. 3 (subject to the limit posed by envelope exhaustion), while systemic angular momentum losses shift them downwards. More extreme initial mass ratios shift them downwards to the left. The net effect of wind mass loss is illustrated by the regions labeled ‘With Wind Mass Loss’ in Fig. 4. For simplicity, it is assumed here that half the remnant mass of the original secondary was lost in a stellar wind prior to the common envelope phase. Mass loss on this scale not only significantly re- duces the mass of the second-formed white dwarf relative to the first, but the concomitant orbital expansion produces wider remnant double white dwarfs, bringing this snapshot model into good accord with the general properties of real systems. Losses of this magnitude might be unprecedented among single stars prior to their terminal superwind phase, but they have been a persis- tent feature of evolutionary studies of Algol-type binaries [10, and references Fig. 4 (facing page). Remnants of the second, common envelope, phase of mass transfer of the systems shown in Fig. 3. Masses refer to the final remnants of the original secondaries, and radii to their Roche lobe radii. Two groups of remnants are shown. Those at lower right labeled ‘Without Wind Mass Loss’ follow directly from the distributions of remnant primaries and secondaries shown in Fig. 3. Because the remnant secondaries straddle the helium ignition line in Fig. 3, across which core masses are discontinuous (see Fig. 2), the distribution of their post-common- envelopes remnants is fragmented, some appearing as degenerate helium white dwarf remnants (lower left), some as helium main sequence star remnants (lower center), and the remainder as shell-burning helium star remnants (upper center). These latter two groups overlap in the mass-radius diagram. The remnant distributions labeled ‘With Wind Mass Loss’ assume that the remnant secondaries of conservative mass transfer lose half their mass in a stellar wind prior to common envelope evolution. They too are fragmented, into degenerate helium white dwarf remnants (lower left) and shell-burning helium stars (upper right). Within each group of remnants, lines of constant remnant primary mass are shown, as in Fig. 3. 16 Ronald F. Webbink therein] and, indeed, of earlier studies of close double white dwarf forma- tion [11]. In the present context, their existence appears inescapable, if not understood. 5 Long-Period Post-Common-Envelope Binaries and the Missing Energy Problem If the properties of short-period binaries with compact components can be reconciled with the outcomes of common envelope evolution as expected from simple energetics arguments, a challenge to this picture still comes from the survival of symbiotic stars and recurrent novae at orbital separations too large to have escaped tidal mass transfer earlier in their evolution. Notwithstanding this author’s earlier hypothesis that the outbursting component in the recur- rent nova T CrB (and its sister system RS Oph) might be a nondegenerate star undergoing rapid accretion [59, 29, 61], it is now clear that the hot components in both of these systems must indeed be hot, degenerate dwarfs [48, 6, 1, 18]. Furthermore, the short outburst recurrence times of these two binaries de- mand that the degenerate dwarfs in each must have masses very close to the Chandrasekhar limit. The complexion of the problem posed by these systems can be illustrated by a closer examination of T CrB itself. Its orbital period (P = 227.53 d) and spectroscopic mass function (f(m) = 0.299M⊙) are well-established from the orbit of the donor M3 III star [23]. The emission-line orbit for the white dwarf [25] now appears very doubtful [18], but the system shows very strong ellipsoidal variation (e.g., [1]), suggesting that the system is near a grazing eclipse. Following Hric, et al. [18], I adopt MWD = 1.38 M⊙ amd MM3 = 1.2 M⊙. The Roche lobe radius of the white dwarf is then RL,WD = 84 R⊙, nearly an order of magnitude larger than can be accom- modated from the energetics arguments presented above, even for αCE = 1, assuming solar metallicity for the system (see Fig. 5). A similar discrepancy occurs for RS Oph. It is evident that these long-period binaries are able to tap some en- ergy source not reflected in the energy budget in (6). One possibility, dis- cussed repeatedly in studies of planetary nebula ejection ([30, 42], more re- cently [58, 12, 13]) is that the recombination energy of the envelope comes Fig. 5 (facing page). Post-common-envelope masses and Roche lobe radii for bina- ries consisting of a white dwarf or helium star plus a 1.2M⊙ companion, computed with αCE = 1. Remnant systems inhabit the regions outlined in bold, and spanned vertically by lines of constant white dwarf/helium star mass of 0.25, 0.35, 0.5, 0.7, 1.0, 1.4, and 2.0M⊙. Other initial sequences, encoded as in Fig. 2, have been mapped through common envelope evolution. The location in this diagram of the white dwarf in the recurrent nova T CrB is also indicated. Common Envelope Evolution Redux 17 18 Ronald F. Webbink into play. For solar composition material (and complete ionization), that re- combination energy amounts to 15.4 eVamu−1, or 1.49 × 1013 erg g−1. For tightly-bound envelopes on the initial giant branch of the donor, this term is of little consequence; but near the tip of the low-mass giant branch, and on the upper aymptotic giant branch of intermediate-mass stars, it can be- come comparable with, or even exceed, the gravitational potential energy of the envelope. In the model calculations of thermally-pulsing asymptotic giant branch stars by Wagenhuber & Weiss [58], the threshold for spontaneous ejec- tion by envelope recombination occurs consistently when the stellar surface gravity at the peak thermal pulse luminosity falls to log gHRI = −1.118± 0.042 . (14) This threshold marks the presumed upper limit to the radii of lower-mass asymptotic giant branch stars in Figs. 1 et seq. in the present paper. In fact, the total energies of the envelopes of these stars formally becomes decidedly positive even before the onset of the superwind phase, also shown in these figures. Whether single stars successfully tap this ionization energy in ejecting planetary nebulae is still debated, but the circumstances of mass transfer in binary systems would seem to provide a favorable environment for doing so. In the envelopes of extended giants and asymptotic giant branch stars, pho- tospheric electron densities and opacities are dominated by heavy elements; the middle of the hydrogen ionization zone is buried at optical depths of order τ ∼ 105. Adiabatic expansion of the envelope of the donor into the Roche lobe of its companion can therefore trigger recombination even as the recombina- tion radiation is itself trapped and reprocessed within the flow, much as the same process occurs in rising convective cells. Other possible energy terms exist that have been neglected in the energet- ics arguments above: rotational energy, tidal contributions, coulomb energy, magnetic fields, etc. But Virial arguments preclude most of these terms from amounting to more than a minor fraction of the internal energy content of the common envelope at the onset of mass transfer, when the energy budget is established. The only plausible energy source of significance is the input from nuclear reactions. In order for that input to be of consequence, it must of course occur on a time scale short compared with the thermal time scale of the common envelope. Taam [53] explored the possibility that shell burning in an asymptotic giant branch core could be stimulated by mixing induced dynamically in the common envelope (see also [54, 52]). Nothing came of this hypothesis: mixing of fresh material into a burning shell required tak- ing low-density, high-entropy material from the common envelope and mixing it downward many pressure scale heights through a strongly stable entropy gradient to the high-density, low-entropy burning region. In the face of strong buoyancy forces, dynamical penetration is limited to scales of order a pressure scale height. Common Envelope Evolution Redux 19 6 Common Envelope Evolution with Recombination The notion that recombination energy my be of importance specifically to common envelope evolution is not new. It has been included, at least para- metrically in earlier studies, for example, by Han et al. [13], who introduced a second α-parameter, αth, characterizing the fraction of the initial thermal energy content of the common envelope available for its ejection. The initial energy kinetic/thermal content of the envelope is constrained by the Virial Theorem, however, and it is not clear that there is a compelling reason for treating it differently from, say, the orbital energy input from the inspiraling cores. We choose below to formulate common envelope evolution in terms of a single efficiency parameter, labeled here βCE to avoid confusion with αCE as defined above. By combining the standard stellar structure equations for hydrostatic equi- librium and mass conservation, we can obtain an expression for the gravita- tional potential energy, Ωe, of the common envelope: Ωe ≡ − dM = 3 PV P dV , (15) where subscripts c refer to the core-envelope boundary, and ∗ to the stellar surface. This is, of course, the familiar Virial Theorem applied to a stellar interior. It is convenient to split the pressure in this integral into non-relativistic (particle), Pg, and relativistic (photon), Pr, parts. The envelopes of giants un- dergoing common envelope evolution are sufficiently cool and non-degenerate to make the classical ideal gas approximation an excellent one for the particle gas. One can then write P = Pg + Pr = ur , (16) where ug and ur are kinetic energy densities of particle and radiation gases, respectively. The total internal energy density of the gas is u = ug + ur + uint , (17) where the term uint now appearing represents non-kinetic contributions to the total energy density of the gas, principally the dissociation and ionization energies plus internal excitation energies of bound atoms and molecules. The overwhelmingly dominant terms in uint are the ionization energies: uint ≈ ρχeff . Integrating over the stellar envelope, we obtain for the total energy Ee of the envelope: Ee = Ωe + Ue 3 PU | − 2Ug − Ur + (Ug + Ur + Uint) = −4πR3cPc − Ug + Uint , (18) 20 Ronald F. Webbink where we explicitly take P∗ → 0. In fact, experience shows that, for red-giant like structures, Rc is so small that the first right-hand term in the last equality can generally be neglected. In that case, we get the familiar Virial result, but with the addition of a term involving the ionization/excitation/dissociation energy available in the gas, Uint ≈ Meχeff , which becomes important for dif- fuse, loosely-bound envelopes. In the context of common envelope evolution, it is of course the dissipated orbital energy, E orb−E orb, that must unbind the envelope. However, the inclu- sion of Uint in Ee now opens the possibility that the common envelope began with positive total energy; that is, in the usual αCE-prescription, it is possible for λ−1 to be zero or even negative, which has the undesirable consequence that αCE need not lie in the interval 0 ≤ αCE ≤ 1 for all physically-possible outcomes. However, the gravitational potential energy of the envelope, Ωe, is negative-definite, and by comparing it with all available energy sources (or- bital energy released plus internal energy of the envelope), we can define an ejection efficiency βCE that has the desired property, 0 ≤ βCE ≤ 1: βCE ≡ orb − E orb)− Ue 4πR3cPc + 2Ug + Ur orb − E orb + Ug + Ur + Uint . (19) By analogy to the form factor λ in the conventional αCE formalism above, we can define separate form factors λΩ for the gravitational potential energy and λP for the gas plus radiation contributions to the (kinetic) internal energy of the envelope: and Ug + Ur = , (20) In contrast, the recombination energy available can be written simply in terms of an average ionization energy per unit mass, Uint = Meχeff . (21) The ratio of final to initial orbital separation then becomes 1 + 2 βCEλΩr1,L λPr1,L χeffAi M1 −M1c In the limit that radiation pressure Pr, ionization energy (Uint), and the boundary term (4πR3cPc) are all negligible, then 2λΩ → λP → λ and βCE → 2αCE/(1 + αCE). Fig. 6 (facing page). Post-common-envelope masses and Roche lobe radii as in Fig. 5, but with recombination energy included, computed from (22) with βCE = 1, with the approximation 2λΩ = λP = λ from (3). At small separations, the differences are inconsequential, but substantially larger final separations are allowed when Af & 10R⊙ (RL & 3R⊙). Common Envelope Evolution Redux 21 22 Ronald F. Webbink The ability to tap the recombination energy of the envelope has a profound effect on the the final states of the longest-period intermediate-mass binaries, those that enter common envelope evolution with relatively massive, degener- ate carbon-oxygen (or oxygen-neon-magnesium) cores. As is evident in Fig. 6, possible final states span a much broader range of final orbital separations. Indeed, for the widest progenitor systems, the (positive) total energy of the common envelope can exceed the (negative) orbital energy of the binary, mak- ing arbitrarily large final semimajor axes energetically possible.3 The inclusion of recombination energy brings both T CrB and RS Oph within energetically accessible post-common-envelope states. It suffices as well to account for the exceptionally long-period close double white binary PG 1115+166, as sug- gested by Maxted, et al. [31]. 7 Conclusions Re-examination of global constraints on common envelope evolution leads to the following conclusions: Both energy and angular momentum conservation pose strict limits on the outcome of common envelope evolution. Of these two constraints, however, energy conservation is much the more demanding. The recent study of close double white dwarf formation by Nelemans & Tout [35] shows clearly that their progenitors can have lost little orbital en- ergy through their first episodes of mass transfer. Since common envelope ejection must be rapid if it is to be efficient, its energy budget is essentially fixed at its onset by available thermal and gravitational terms. The preser- vation of orbital energy through that first phase of mass transfer therefore indicates that the observed close double white dwarfs escaped common en- velope formation in that first mass transfer phase. They evidently evolved through quasi-conservative mass transfer. However, strictly mass- and angular momentum-conservative mass transfer leaves remnant accretors that are too massive and compact to account for any but the shortest-period close double white dwarfs. Significant mass loss and the input of orbital energy prior to the onset of the second (common envelope) phase of mass transfer are required. The requisite energy source must be of nuclear evolution, which is capable of driving orbital expansion and stellar wind losses during the slower (thermal recovery or nuclear time scale) phases of quasi-conservative mass transfer, or during the interval between first and second episodes of mass transfer. Details of this process remain obscure, however. 3 The final orbit remains constrained by the finite initial orbital angular momentum of the binary. Final semimajor axes much in excess of the initial semimajor axis may be energetically allowed, but the finite angular momentum available means that they cannot be circular – see (9) – an effect which has been neglected in Fig. 6. Common Envelope Evolution Redux 23 Long-period cataclysmic variables such as T CrB and RS Oph pose a more extreme test of common envelope energetics. With their massive white dwarfs, the evident remnants of much more massive initial primaries, they are nevertheless too low in total systemic mass to be plausible products of quasi-conservative mass transfer, but too short in orbital period to have es- caped tidal mass transfer altogether. They must be products of common en- velope evolution, but to have survived at their large separations, they demand the existence of a latent energy reservoir in addition to orbital energy to as- sist in envelope ejection. It appears that these binaries efficiently tap ioniza- tion/recombination energy in ejecting their common envelopes. That reservoir is demonstrably adequate to account for the survival of these binaries. Its in- clusion requires only a simple revision to the parameterization of common envelope ejection efficiency. Acknowledgement. This work owes its existence to both the encouragement and the patience of Gene Milone, to whom I am most grateful. 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0704.0281

The Source of Turbulence in Astrophysical Disks: An Ill-posed Problem.

The Source of Turbulence in Astrophysical Disks: An Ill-posed Problem. Denis Richard, NASA Ames Research Center UCSC-Ames Planet and Star Formation Meeting March 29, 2007 Astrophysical Disks Disks are ubiquitous in Astrophysics : * Planetary disks * Circumstellar Disks (around young stars) * Binary systems * Active Galactic Nuclei (around black holes) Therefore, understanding disks is fundamental to understand planetary and stellar formation and evolution. Wide range of sizes : From Saturn's rings ( ~ 107 km) to AGN disks (~ parsec = 3.1 10 13 km) Wide variety of complex physical processes, one of them being the transport of angular momentum. Astrophysical Disks (Artist view) Astrophysical Disks (HST) Turbulence in Disks * Accretion Disk : gas and “dust” falling inward, toward the central object. * Thin disk = H/R << 1 → Keplerian rotation : Ω ~ r -3/2 (no radial or vertical velocity in first approximation) * To maintain stationary rotation, angular momentum needs to be transported outward. * Need for an adequate transport mechanism. * Molecular viscosity is too small. * Turbulence : instability mechanism ? Transport properties ? * Early models : Shakura & Sunyaev 1973 : * Turbulence most likely generated by differential rotation (shear flow). * Ad hoc model for transport : turbulent viscosity based on smallest constraints : = α . Cs . H A Short History of Turbulence Models for Accretion Disks * 1973 – 1991 : Shakura & Sunyaev era : - Source of turbulence unknown (shear flow assumption ?) - Work on turbulence, in order to improve transport model. - meanwhile, transport model : ν = α . Cs . H * 1991 : Magneto Rotational Instability rediscovered Weak magnetic field coupled to shear flow gives rise to a linear instability. (Chandrasekhar 1960, Balbus & Hawley 1991) * 1991 – present : MRI era : - Source of turbulence : MRI - Turbulent model : ν = α . Cs . H Turbulent Viscosity Model What is it ? * Analytically : A description of turbulent transport as a diffusive mechanism. * For numerical simulation : a type of basic subgrid model. Where does it come from ? * Built ad hoc (alpha-viscosity) : relevant length scale x relevant velocity. * Measured experimentally (lab or numerical) : Reynold stress. When should it be used ? * When studying anything BUT turbulence as a fundamental physical process. * When simulations do not have adequate resolution to describe the whole range of scales of the flow, in which case the viscosity model has to be chosen as to describe only the subgrid scales. From Instabilities to Turbulence Two types of Instabilities : * Linear : flow unstable to infinetesimal perturbations (super-critical transitions) ex : thermal convection, Rayleigh/centrifugal instability in rotating flows. * Non-Linear : flow unstable to finite amplitudes (sub-critical transitions) ex : plan shear flow, Differential rotation (?) Analytical : Linear can be well treated (transition) / no general model for Non-Linear. Numerical : Linear : generally low Reynolds, large scale flows = “lower” resolution Non-linear : generally high Reynolds, small scale flows = “high” resolution Laboratory : In theory can study both types equally well. In an Astrophysical context : Both types are equally difficult to study, because what ultimately matters is the turbulent state, which will generally be at very high Reynolds, thus difficult to describe. Thus, while MRI has been around for more than 15 years in the disk community, there is no associated description for turbulent transport. An instability easy to describe does not mean that the induced turbulence is equally easy to quantify. Polemic ? What Polemic ? * Little to no doubt that MRI is at work in most accretion disks. * But : Is MRI the only turbulent mechanism relevant to Astrophysical disks ? “Schools of thoughts” : MRI school * MRI is necessary to power accretion disks (Only MRI can provide adequate angular momentum transport.) * MRI is sufficient to power all accretion disks (giving birth to such things as “Dead zone models”) Instability X school * Instability X is also relevant to Astrophysical Disks dynamics. (Differential Rotation, Plane Shear, Strato-rotational, Baroclinic,...) No-school school * Just tell me how turbulence acts in my disk, so that I can react, coagulate, form a planet, evolve, etc... Arguments against Differential Rotation Analytical : radial and azimuthal fluctuations can not grow at the same time. (Balbus, Hawley & Stone, 1996) Problem : this set of equations is linear, thus irrelevant to non-linear instabilities (~ Rayleigh criterion) Numerical : No-show in numerical simulations (Balbus, Hawley & Stone, 1996) Problem : Reynolds numbers are “low” (a 1024 3 grid can simulate a maximum Re of order 10,000) Experimental : H.Ji, M.Burin, E.Schartman & J.Goodman, 2006 (accompanied by comments by S.Balbus) Can Differential Rotation lead to Turbulence in Disks ? * High Reynolds Shear flow : Astrophysical Disks Re ~ 106 to 1020 * Early (1930's) laboratory experiments : Couette-Taylor flow unstable at high Re. (Richard & Zahn, 1999) Data from Wendt (1933) and Taylor (1936) Can Differential Rotation lead to Turbulence in Disks ? * New experimental setup (Richard, 2001). Conclusion : In a laboratory experiment (a not-so-close analog to a disk), differential rotation can give rise to turbulence despite published arguments. Differential Rotation may lead to turbulence in Keplerian disks. Princeton Experiment (H.Ji, M.Burin, E.Schartman & J.Goodman, 2006) (Comments requested by meeting organizers...) * Couette-Taylor flow. * No flow-visualization, due to technical difficulties (private communication, M.Burin). Turbulent flows can not be positively differenciated from laminar flows. (As stated in the paper itself.) This is why Ji et al never claim that the flows are stable. They merely discuss fluctuation levels. * Boundary conditions : * Stubby aspect ratio with independently rotating rings to compensate and avoid large scale circulation. * clever setup but : - experimental calibration does not agree with numerical simulations of this setup (see Burin et al, 2006). - calibration done “blind” (no visualization) and only close by the Rayleigh boundary where the flow could be turbulent (priv. comm. M.Burin). Should have been done in a regime where there are no doubts that the flow is laminar. Could have been returning the flow to a laminar regime after a sub- critical transition. Princeton Experiment (H.Ji, M.Burin, E.Schartman & J.Goodman, 2006) (Comments requested by meeting organizers...) * Main argument for Astrophysical flows : * measured β < 6.2 10-6 * then compares α and β parameters numerically, by deriving a formulation of the α viscosity for the laboratory setup. Concludes that α and β should have a similar value, thus that α ~ β < 6.2 10-6 is too small (α ~ 10 -3). * BUT : this formulation for the α viscosity has meaning only in a thin keplerian disk where Cs = Ω.H. It makes absolutely no sense for this experimental setup. * What makes sense is to compare α and β value in an Astrophysical context. For α.Ω.H2 = β.Ω.R2, then β / α = (H/R)2, therefore for α ~ 10 -3, and 0.001< H < 0.1, β ~ 10 -9 - 10 -5, still provides adequate transport of angular momentum. Conclusion * The issue of differential rotation is still very much open. * The debate about the origin of turbulence in disks should be a search to characterize the turbulent state of disks and to model transport properties. Today, it sometimes seems to be more an effort to eliminate non-fashionable instabilities. * Considering the complexity of disks, it would be naive to think that one process only participate in angular momentum process. * Discouraging work on various instabilities is damaging for disk understanding. Not only angular momentum is transported. Chemistry, planet formation, etc. are affected by turbulence that may not ultimately be relevant for angular momentum transport. * Should be acknowledged : - The limitations of our tools : numerical, analytical, and experimental. Being able to describe a process better/more easily does not make it more relevant or important. - The lack of observational constraints on disks physics. Real object Analytical model LES / DNS models Laboratory model ...but quite possibly...Hopefully... ...while most often presented as :Tools

0704.0282

On Punctured Pragmatic Space-Time Codes in Block Fading Channel

On Punctured Pragmatic Space-Time Codes in Block Fading Channel Samuele Bandi†, Luca Stabellini, Andrea Conti and Velio Tralli †Corresponding author. Authors are with ENDIF, University of Ferrara, via Saragat 1, 44100 Ferrara, Italy (e-mail: [email protected],[email protected],[email protected],[email protected]) Abstract—This paper considers the use of punctured con- volutional codes to obtain pragmatic space-time trellis codes over block-fading channel. We show that good performance can be achieved even when puncturation is adopted and that we can still employ the same Viterbi decoder of the convolutional mother code by using approximated metrics without increasing the complexity of the decoding operations. I. INTRODUCTION A relevant result obtained in wireless communications is that the use of Space-Time Codes (STC) with both mul- tiple transmitting and receiving antennas can be exploited to mitigate the effect of fading without sacrificing spectral efficiency. Several research activities in the last decade has been devoted to find STC able to improve the performance in terms of achieved diversity and coding gain [1]. Since the introduction of STC, the problem of determining the best STC has been always a difficult task, especially for a large number of transmitting antennas. In [2] a new approach to space-time coding called Pragmatic space-time coding (PSTC) has been devised. The pragmatic approach consists in using common convolutional codes as STC over a MIMO channel; it has also been shown that PSTC could be used with both QPSK and BPSK to achieve maxi- mum diversity and good performance. The use of convolu- tional codes as STC allows decoding by using the same Viterbi decoder with only a proper change in metrics evaluation. However, in the case of high-rate codes, R = k/n, the number of operations that the decoder has to perform grows exponentially with k, and the number of paths to be stored increases rapidly. Puncturation was initially introduced for convolutional codes to avoid the complexity issue of the Viterbi decoder in case of high rate codes. Puncturation consists in deleting one or more bits of a codeword; in this way rate p/(p+m) punctured code can be obtained by periodically puncturing a low rate 1/n convolutional code, i.e. by erasing m bits for each period of length p. In this paper, following the same aforementioned pragmatic approach to STC, we will introduced puncturation, using therefore punctured convolutional codes as ST codes. We will call this family of codes punctured pragmatic space-time codes (P2-STC). The use of puncturation in conjunction with STC has also been investigated in [3], where the pragmatic approach was not adopted and only the case of two transmitting antennas was addressed by designing the codes to preserve full diversity gain only for short error sequences. II. PRAGMATIC SPACE-TIME CODES: In this work we consider P2-STC in a block fading channel (BFC)(see [4] [5]) where fading coefficients for each couple of transmitting and receiving antennas are constants in blocks of B bits and independent block by block. L different blocks are experimented by a codeword. In this situation, being N the number of transmitting antennas and M the number of receiving antennas, we can generalize the Singleton bound for BFC to obtain the following diversity bound div ≤ 1 + bLN(1−R)c . (1) where div is the maximum achievable diversity degree per receiving antenna. This means that, by puncturing a rate 1/2 convolutional code in quasi-static fading channel (L = 1) with N = 2,M = 1, we cannot achieve div = 2 since R > 1/2. For a rate 1/3 convolutional code with N = 3, M = 1 and L = 1, we cannot achieve div = 3 if R > 1/3. Our results prove however, that we can still implement puncturation and preserve div = 2 if 1/3 < R < 2/3. To describe a system with P2-STC we refer to the block scheme in Fig.1. We denote by C(t) = [c(t)1 c 2 ...c T a super- symbol, i.e. a vector of symbols simultaneously transmitted by the N transmitting antennas at the time t. A frame is composed by a sequence of F super-symbols (every Ts seconds, N symbols are sent in parallel on the N transmitting antennas). If we indicate the complex envelope of the signal received by antenna s at time t with r(t)s and with α i,j the fading coefficient between antenna s and i, at time t, we have: r(t)s = Es + η s (2) Fig. 1. Block scheme for pragmatic space-time codes (transmitter side), 4−states, 1bps/Hz, BPSK modulation, n = 2 antennas, (5, 7)8 generators, Decoder stands for Viterbi decoder with proper branch-metrics. To perform maximum likelihood decoding, the Viterbi algo- rithm should estimate the input bit sequence b̂ in this way: b̂ = arg min |r(t)s − i (b) Es|2 (3) where c(t)i (b) are the transmitted symbols corresponding to the path in the trellis labeled by the bit sequence b, and ∆M (t) = |r(t)s − Es|2 (4) are the metric increments. The fundamental problem is, how to compute these metric increments when we adopt puncturation. Consider for example a rate 1/2 code used with the punctura- tion matrix 1 0 1 1 1 1 0 1 with two transmitting antennas and BPSK modulation. Fig. 2. Relationship between received symbols and coded bits for a P2-STC obtained from a R=1/2 convolutional code with N=2, M=1 Fig. 2 shows the relationship between the received symbols r(t) and the coded bits. We can see that the second and third transition of the trellis diagram are labeled with only one coded bit, i.e., symbol r(2) carries coded bits from two different transitions and therefore we must use the same received symbol r(2) to compute two transition metrics on the trellis. Since the coded bit c(2)1 cannot be recovered from the trellis diagram at instant t = 2 we can use it only for the third transition metric. To implement ML decoding we should join two transitions into one such that no symbol contains bits from two transitions anymore, as already pointed out in [6]. This however, would mean that we could not use a conventional Viterbi decoder of the rate 1/2 code. To exploit the same Viterbi decoder we must actually choose an approximation of ML decoding that uses approximated metrics. In [6] a very similar problem arise in the context of trellis coded modulation. In a similar way we suggest to split the second transition metric in two components that can be used for two different transitions on the trellis diagram. In this way, at the time t = 2, ∆M (t) is approximated with ∆M̃ (t) as follows: ∆M̃ (t) = 1/2 min |rs − [α 1,s(2b1 − 1)− α 2,s(2b2 − 1)] +1/2 min |rs − [α 1,s(2b1 − 1)− α 2,s(2b2 − 1)] This metric increment can be easily split on the two transitions at time t = 2, which we name as left and right transitions supporting transmission at time t. To improve decoding, we also propose another approxi- mated metric whose computation requires an interaction with the Viterbi algorithm: in other words, with this second ap- proach the left part of the metrics at time t = 2 can be computed by using the outcomes of the Viterbi algorithm after having processed left transitions at time t. If the metric com- putation runs in parallel with the Viterbi algorithm, when we know the survivor path at each state of the trellis between left and right transitions, we can compute the metric increments on the right transition using the information on the survivor path at each departing state σ. The second approximated metric can be written as ∆M̃ (t) = (1− β) min |rs − [α 1,s(2b1 − 1)− α 2,s(2b2 − 1)] |rs − [α 1,s(2b1(σ)− 1)− α 2,s(2b2 − 1)] where an additional parameter β is included for further opti- mization. We have compared these two metrics and we had confir- mation that the second one performs better, since it exploits the additional information provided by the Viterbi algorithm. In the next section, we will address the way of generalizing this method of evaluating decoding metrics for different codes with BPSK modulation and different puncturation matrices. III. DECODING METRIC COMPUTATION When the transmitted super-symbol carries coded bits from two different transitions due to puncturing, the approximated metric increment that exploits the information on the survivor path has proved to give the best performance. However, up to now, it can only be computed for a puncturation pattern similar to that given in (5). The next step is to generalize the metric computation for the generic puncturation matrix and for a generic number of transmitting antennas. We will do this in two steps: first we will find a generalization for the case with two transmitting antennas (N = 2) and a generic puncturation matrix and then we will remove this hypothesis and consider the case with N > 2. We first considered a two columns puncturation matrix as follows: 1 0 1 1 1 1 1 1 1 1 0 1 In this case we have three symbols (δ = 3) carrying coded bits belonging to different transitions of the trellis diagram. As the decoder computes three metrics using the three received symbols, we must split these three metrics into four trellis transition. The relationship between the received symbols r(t) and the coded bits is shown in Fig.3. Since r(2) carries bits from the second and third transitions on the trellis we will use it to compute the metric increments for the left and right transitions at time t = 2. Since also r(3) carries bits from the third transition we will use both r(2) and r(3) to compute the metric for the third transition (which is also the left transition at time t = 3). On the left metric computed at time t = 3 the symbol c(3)1 is unknown; therefore this left metric is evaluated with the value of c(3)1 Fig. 3. Relationship between received symbols and coded bits of P2-STC obtained from a R=1/2 convolutional code with N=2, M=1 and puncturation matrix as in (9). that minimizes it. On the right metric computed at time t = 2 the information from the Viterbi algorithm can be exploited to derive the value of c(2)2 for the computation of the metric. Each metric increment coming from received symbols carrying bits belonging to two trellis transitions can be split into two components, a left metric ∆m̃(t)L and a right metric ∆m̃ the former will be approximately computed by considering the coded bits of the right transition at time t as unknown, whereas the latter will be computed by considering the coded bits on the left transition at time t as those of the survivor path at the trellis state σ(t). That is, for t = 2, 3, 4: ∆M̃ (t) = ∆m̃(t)L + ∆m̃ R (9) where ∆m̃(t)L = (1− β)minb2 |r(t)s − α 1,s(2b 1 − 1) − α(t)2,s(2b2 − 1) Es|2 (10) ∆m̃(t)R = β |r(t)s − α 1,s(2b (t))− 1) − α(t)2,s(2b 2 − 1) Es|2 (11) In the decoding algorithm for this example, the second transition of the trellis will be labelled with ∆m̃(2)L , the third transition with ∆m̃(2)R +∆m̃ L , the fourth transition with ∆m̃(3)R +∆m̃ L and the last transition with ∆m̃ We have investigated the optimum value of β for different values of δ and for a rate 1/2 code with puncturation matrices with 2 zeros (as in eq. (9)). We found that the optimum value of β is a decreasing function of δ. The reason of this behaviour is that the information held by the survivor path at each state exploited in the right metric will be more and more unreliable as the δ increases. For δ = 1 this value approaches 0.5. The generalization of this approach to the N > 2 case can be easily done if we assume that the puncturation pattern does not allow the transmission of super-symbols carrying coded bits from more than two trellis transition. Also in this case each approximated metric increment can be split into a left and right components. These two component may be related to more than one coded bits and the value of β should be suitably chosed and optimized for each puncturation pattern, for each value of N and even for each time t. However, we can give the following rules of thumb to compute the parameter β in the approximated metric increment at time t: β − 1 = nL , β = nR where nL is the number of not erased coded bits in the label of the left transition at time t and nR is the number of not erased coded bits in the label of the rigth transition at time t. For example, if we refer to matrices in (20), we suggest using β = 1/3 in the approximated metric at time t = 2. A. A simplified metric We propose here another possible approach for the general- ization of the metric computation, which is sligthly simplified with respect to the one already presented. We name this approximated metric as Type-2 metric to distinguish it from the former metric named Type-1 metric. With this second approach, if we refer to the example in Fig.3 as before, we will exploit the information of the survivor path at each state only in the last transition with an erased bit, i.e the right-metric increment is used only at the fifth transition. As we want our decoder to be an approximation of an ML decoder, we have to weight each component of each metric increment with an appropriate parameter: the first δ components (left metrics) and the last component have to be multiplied by parameters whose sum must equal to δ (equal to 3 in the example). Instead of using δ + 1 different parameters we choose to use only two different weights: the first one, ωa, is used for the left metrics, whereas the second one, ωb, is used for the last right metric. Once again we include a parameter β to balance the two different weights. ωa = (1− β) δ + β(1− δ) ωb = β δ + β(1− δ) With this choice of ωa and ωb, if β = 0 then ωa = 1 and ωb = 0, i.e. we do not trust the right metric using survivor paths information. If β = 1 then ωa = 0 and ωb = 1, i.e., we only exploit the last right metric If β = 0.5 then ωa = ωb, i.e. we weight the metrics uniformly. For our decoding rule to be an approximation of the ML decoder we require that: δωa + ωb = δ (15) which is clearly satisfied by our construction of parameters ωa and ωb. If we refer to the same example as before, the expression for the approximated metric increments at time t = 2, 3, 4 becomes: ∑ t=2,3,4 ∆M̃ (t) = t=2,3,4 ∆m̃(t)L + ∆m̃ R (16) where ∆m̃(t)L = ωamin |rs − α 1,s(2b 1 − 1) − α(t)2,s(2b2 − 1) Es|2 (17) ∆m̃(t)R = ωb |rs − α 1,s(2b (t))− 1) − α(t)2,s(2b 2 − 1) Es|2 (18) With this approach we have split δ metric increments corresponding to the δ symbols received at t = 2, 3, 4 into δ+ 1 metric increments. As in the previous case the optimum values of β is a decreasing function of δ since the state is more and more untrustworthy as δ increases. A reasonable value of β is between 0.5 and 1. In rate-adaptive applications this allows to change simple parameters for metric computation in the Viterbi decoder to decode very different puncturation matrices, keeping the overall trellis structure of the decoder unchanged. IV. SEARCH FOR GOOD PUNCTURATION MATRICES We address now the issue of finding good puncturation matrices that allow to design the wanted rate R for the P2- STC starting from a convolutional mother code of rate 1/N for BPSK modulation (this is the case addressed in this paper: we can easily extend the approach for any pragmatic STC, i.e. as an example by considering a mother code with rate 2/(2N) or 1/(2N) for QPSK modulation). As a first step, we investigate the behavior of simple punc- turation patterns of N zeros to be used as building blocks for good puncturation matrices with N rows (N is also the number of transmitting antennas). We conjecture that the position of a basic pattern inside a puncturation matrix do not affect the performance (or its effect is negligible). A simple observation will guide our search for good pat- terns: if the number of super-symbols which carry coded bits of different transitions of the trellis diagram increases, the performance of the decoder get worse because of the increasing for the number of approximated metric increments. On the other hand, if we limit this number to zero by considering a puncturing pattern of N zeros that erases all the bits in a single transition (in this case the entire super-symbol is erased), the performance may be degradated because all the erased bits are concentrated in a single position. We consider basic patterns obtained by placing N zeros on one or more subsequent columns (in the latter case δ is the number of columns). The simulations have proved that putting all zeros in the same column is not the best solution. We made simulations comparing the performance obtained by each different pattern when parameter β is fixed to the optimum value for each case. For N = 2 we obtained as best puncturing patterns the following: ... 1 0 ... ... 0 1 ... ... 0 1 ... ... 1 0 ... For N = 3 all the best patterns have one zero on the first column and two on the second column, e.g.  ... 1 0 ...... 1 0 ... ... 0 1 ...  P =  ... 1 0 ...... 0 1 ... ... 1 0 ...  (20) As a second step in our search, we try to use this basic patterns to build up more complex puncturation matrices that allow to reach higher code rates. In the search for good puncturation matrices we put the zeros in a way to satisfy the rate-compatibility rule. In table IV we show a family of rate- compatible puncturation matrices for the N = 2 case with a puncturation matrix with p = 10. In table II, instead, we addressed the case with three transmitting antennas (N = 3) and a puncturation matrix with p = 10. The final rate R of the code will be p/[(m− p)N ], where m is the number of basic puncturing patterns of N bits in the matrix. TABLE I rate rate-compatible R=5/9 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 R=5/8 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 R=5/7 1 1 0 1 0 1 0 1 1 1 1 0 1 0 1 0 1 1 1 1 R=5/6 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 1 TABLE II rate rate-compatible R=10/27 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 R=10/24 1 0 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 0 1 0 1 1 1 1 1 1 1 R=10/21 1 0 1 0 1 0 1 1 1 1 1 0 1 0 1 0 1 1 1 1 0 1 0 1 0 1 1 1 1 1 R=10/18 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 1 1 R=10/15 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 5 10 15 20 25 !!"#type-II !!$#type-II !!%#type-II !!&#type-II !!"#type-I !!&#type-I Fig. 4. Effects of different basic puncturation patterns of 2 zeros on the performance of the P2-STC scheme with N = 2. The mother code is(133,171) R=1/2, with a puncturing period p = 10 V. NUMERICAL RESULTS The Fig.4, shows the sensitivity of the performance of a P2-STC with the same rate to different puncturation patterns of 2 zeros and to the two types of approximated metrics discussed before. We can easily note that Type-I metric is better than Type-II metric as expected, and in general, as Fig. 5. Performance of P2-STC obtained from a rate 1/3 mother code (133,145,175) and L=1,N=3,M=1. a)R=1/3, b)R=10/27, c)R=10/24, d)R=10/21, e)R=10/18 Fig. 6. Performance over a BFC of a R = 5/8 P2-STC obtained from rate 1/2 convolutional code (133,171) with N=2,M=1. a)L=10, b)L=8, c)L=6, d)L=4, e)L=2 already commented in previous sections, the performance degrades as δ (distance between the 2 zeros) increases. The best case is δ = 1, even though for Type-II metric performance changes sligthly. In Fig.5 we show that it is possible to achieve the maximum diversity allowed on a quasi static channel with a P2-STC with 3 transmitting antennas: for 1/3 < R < 2/3, the maximum diversity degree is 2 and can be achieved with all the codes with rate larger than 10/21. It is evident that there is a threashold effect around rate 0.5: if we puncture too much, we lose 1 degree of diversity. In Fig.6 we show the performance over a BFC of a rate 5/8 P2-STC obtained from the well-known rate 1/2 mother code (133,171) with a puncturing pattern like the one in (8) for varying fading levels per codeword. This code, which can not achieve a diversity degree larger than 1 on a quasi-static channel, is able to capture a significantly large diversity degree according to the limits given by (1). VI. CONCLUSIONS In this paper, we have shown that it is possible to use punctured convolutional codes to build pragmatic space-codes over a block-fading channel. We have also shown that by an appropriate choice of the metric increments a single Viterbi algorithm on the same trellis diagram can still be used for decoding different rate-compatible codes. We have proposed two different approximated metrics and explained how to construct rate-compatible puncturation matrices that give good performances. In the results we have shown that with this scheme we can easily obtain codes with rates higher than 1/N which can approach or achieve the maximum allowed diversity degree on both quasi-static channel and block-fading channel. VII. ACKNOWLEDGMENTS Authors would like to acknowledge Prof. M. Chiani for his helpful discussions. This work has been done within the European Network of Excellence in Wireless Communications (NEWCOM). REFERENCES [1] Vahid Tarokh, Nambi Seshadri, A. R. Calderbank “Space-Time Codes for high Data Rate Wireless Communication: Performance Criterion and Code Construction” IEEE Transactions on information theory Volume: 44 no. 2, 2001 Page(s): 744-65 March 1998 [2] Chiani, M.; Conti, A.; Tralli, V.; “A pragmatic approach to space- time coding” Communications, 2001. ICC 2001. IEEE International Conference on, Volume: 9, 2001 Page(s): 2794-2799 vol.9 [3] Chan-Soo Hwang; Seung Hoon Nam; Jaehak Chung; Byungiang Jeong; “Design of punctured space-time trellis codes” Personal Indoor and Mobile Radio Communications IEEE Proceedings on, vol.2, pp. 1698- 1702, Sept. 2003. [4] Chiani, M. “Error probability for block codes over channel with block interference,”IEEE Transactions on information theory vol.44, Issue 7, pp. 2998-3008, November 1998. [5] McEliece, R.J ; Stark, W.E. “Channel with block interference” IEEE Transactions on information theory vol.IT-30, pp. 44-53, Jan. 1984. [6] Woerz, T;Schweikrt, R. “Performance of Punctured Pragmatic Codes,” Global Telecommunications Conference, ’95., IEEE vol.1, pp. 13-17, November 1995. [7] E. Malkamaki, H. Leib, “Coded diversity on Block-Fading Channels” IEEE Trans. on Information Theory on, Volume: 45, March 1999 Introduction Pragmatic space-time codes: Decoding metric computation A simplified metric Search for good puncturation matrices Numerical results Conclusions Acknowledgments References

0704.0283

On the Markov trace for Temperley--Lieb algebras of type $E_n$

arXiv:0704.0283v1 [math.QA] 2 Apr 2007 ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En R.M. Green Department of Mathematics University of Colorado Campus Box 395 Boulder, CO 80309-0395 E-mail: [email protected] Abstract. We show that there is a unique Markov trace on the tower of Temperley– Lieb type quotients of Hecke algebras of Coxeter type En (for all n ≥ 6). We explain in detail how this trace may be computed easily using tom Dieck’s calculus of diagrams. As applications, we show how to use the trace to show that the diagram representation is faithful, and to compute leading coefficients of certain Kazhdan– Lusztig polynomials. 1. Introduction In the paper [17], Jones introduced a certain Markov trace on the tower of Hecke algebras H(An−1) associated to the Coxeter groups Sn = W (An−1), which are the symmetric groups. When Jones’ trace is restricted to one of the algebras H = H(An−1), it is degenerate, but its radical is an ideal, J , of H and so we obtain a generically nondegenerate trace on the algebra H/J , which is the Temperley–Lieb algebra TLn occurring in statistical mechanics [25] (the trace is the matrix trace of a transfer matrix algebra). In [19], Kazhdan and Lusztig introduced a remarkable polynomial Px,w(q) for any elements x, w in a Coxeter group W . These polynomials have important ap- plications in representation theory. Although the polynomials have an elementary 1991 Mathematics Subject Classification. 20C08, 20F55, 57M15. Typeset by AMS-TEX http://arxiv.org/abs/0704.0283v1 2 R.M. GREEN definition, the only obvious way to compute them is using a rather complicated recurrence relation. One of the main obstructions to computing the polynomials efficiently is a fast way to compute the integer µ(x, w), which is the coefficient of q(ℓ(w)−ℓ(x)−1)/2 in Px,w(q). In [12], the author showed how Jones’ trace can be used to compute the leading coefficients µ(x, w) ∈ Z in the case where x and w are fully commutative elements of W (in the sense of [24]). In this paper, we will investigate the analogous phenomenon in Coxeter type En. This includes Coxeter groups of types A and D as special cases. The algebras TLn may be defined in terms of generators and relations in a way that generalizes readily to Coxeter systems of other types. These generalized Temperley–Lieb algebras have been studied for Coxeter type En by a number of people [2, 3, 7]. Although the Coxeter groups of type En are infinite for n > 8, the Hecke algebra quotient TL(En) in this case is still finite dimensional. In [2], tom Dieck constructed a diagrammatic representation of TL(En), although the question of whether this is a realisation—a faithful representation—is not tackled. In §9, we will prove Theorem 1.1. The diagrammatic representation of TL(En) given in [2] is injec- tive. The closing remarks of [2] state without proof that this representation can be used to define a Markov trace on the tower of algebras TL(En). In Theorem 8.11, we will prove this claim and furthermore we will show that there is a unique such Markov trace. Although this is similar to what happens in type A, the analogous claim for Coxeter type D is false. This trace is also remarkable for other reasons: after suitable rescaling, it is a tabular trace in the sense of [10], and a generalized Jones trace in the sense of [12]. The fact that the trace is tabular implies that it is (generically) nondegenerate on the algebras TL(En). The fact that we have a generalized Jones trace will lead to the following theorem (proved in §9) where the monomial basis elements bw are ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 3 defined in §3. Theorem 1.2. Let {bw : w ∈ Wc} be the monomial basis of TL(En) indexed by the fully commutative Coxeter group elements, and let tr be the unique Markov trace on the tower of algebras TL(En). If x, y ∈ Wc, then the coefficient of v−1 in tr(bxby−1) (after expansion as a power series) is µ̃(x, y), where µ̃(x, y) = µ(x, y) if x ≤ y, µ(y, x) if x 6≤ y, and µ(a, b) is the integer defined in [19]. We will also show in §9 how µ̃(x, y) may be evaluated non-recursively using the diagram calculus. 2. Traces and Markov traces By a trace on an R-algebra A, we mean an R-linear map t : A −→ R such that t(ab) = t(ba) for all a, b ∈ A. The radical of the trace is the set of all a ∈ A such that t(ab) = 0 for all b ∈ A. The radical is always an ideal of A, and if it is trivial, the trace is said to be nondegenerate. In any case, if I is the radical of t, then t induces a nondegenerate trace on the quotient algebra R/I. The set of traces on an R-algebra A has a natural R-module structure. In the special case where ρ is a representation of an R-algebra A, then the matrix trace associated to ρ is a trace in the above sense, which means that, if A is semisimple, the Grothendieck group of A gives a Z-lattice in the space of traces, generated by the traces of the simple modules. We will be particularly concerned with algebras where the base ring R is obtained by extending scalars from the ring of Laurent polynomials A = Z[v, v−1] to some ring F ⊗ A. This has the effect of specializing the parameter v to an invertible element of F . In this situation, a trace is called generically nondegenerate if it is nondegenerate as a trace over A, and if it also remains nondegenerate as a trace over F ⊗A for all but finitely many specializations of v. 4 R.M. GREEN Suppose now that R is an integral domain and {An : n ≥ N} is a family of unital R-algebras such that An is a subalgebra of An+1 for all n ≥ N . Let A∞ be the associated direct limit. Suppose also that there is a set of elements {gn : n ∈ N} such that gn+1 ∈ An+1\An for all n and such that {gn : n ≤ M} is an algebra generating set for AM . Following [5, §4], we may now introduce the notion of Markov trace. Definition 2.1. Maintain the above notation, and let F be a field containing R. A Markov trace on A∞ with parameter z ∈ F is an F -linear map τ : A∞ −→ F satisfying the following conditions: (i) τ(1) = 1; (ii) τ(hbn+1) = zτ(h) for n ≥ N and h ∈ An; (iii) τ(hh′) = τ(h′h) for all h, h′ ∈ A∞. Jones [17] proved that there is a unique Markov trace with parameter z on the tower of Hecke algebras of type An, and that the only one of these traces that passes to the Temperley–Lieb quotient is the one with parameter z = (v + v−1)−1. This is an important observation in the construction of the Jones polynomial, be- cause conditions (ii) and (iii) for the trace are what is needed to ensure that the polynomial is invariant under the two types of “Markov move”. Some other notable work on Markov traces includes that of Geck and Lam- bropoulou [4], who classified the Markov traces in Coxeter types B and D, using a suitable extension of the above definition. Lambropoulou [20] extended this work (in type B) to generalized and cyclotomic Hecke algebras of type B. For the purposes of studying Temperley–Lieb type quotients of Hecke algebras, a better definition of Markov traces seems to be one that appears in work of Seifert [22] and recent work of Gomi [6, Definition 3.7]. In this case, one retains conditions (i) and (iii) of Definition 2.1 and replaces condition (ii) by the requirement that τ(aTs) = zsT (a) whenever we have a ∈ H(WI) for some parabolic subgroup WI corresponding to ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 5 I ⊆ S\{s}. (In other words, we require condition (ii) to hold for all generators of An+1, not just one particular generator.) Here, zs is an indeterminate depending on the conjugacy class of s in W . In this paper, we will restrict our attention to the tower of algebras TL(En), and in this case, the above definitions happen to agree; however, they do not agree in the corresponding question for type Dn. In the latter case, it can be shown that the Seifert–Gomi formulation produces a unique Markov trace, and Definition 2.1 does not. 3. The algebras TL(En) Let X = X(En) be a Coxeter graph of type En, where n ≥ 6. Following [3], we label the vertices of X by 0, 1, . . . , n− 1 in such a way that 1, 2, 3, . . . , n− 1 lie in a straight line, and such that 3 is the unique vertex of degree 3, which is adjacent to 2, 4 and 0. Figure 1 shows the case n = 6. Figure 1. Coxeter graph of type E6 4 5 3 2 1 Let W (En) be the associated Coxeter group with distinguished set of generating involutions S(En) = {si : i is a vertex of X(En)}. In other words, W = W (En) is given by the presentation W = 〈S(En) | (st)m(s,t) = 1 for m(s, t) < ∞〉, where m(s, s) = 1, m(s, t) = 2 if s and t are not adjacent in X , and m(s, t) = 3 if s and t are adjacent in X . The elements of S = S(En) are distinct as group elements, and m(s, t) is the order of st. Denote by Hq = Hq(En) the Hecke algebra 6 R.M. GREEN associated to W . This is a Z[q, q−1]-algebra with a basis consisting of (invertible) elements Tw, with w ranging over W , satisfying TsTw = Tsw if ℓ(sw) > ℓ(w), qTsw + (q − 1)Tw if ℓ(sw) < ℓ(w), where ℓ is the length function on the Coxeter group W , w ∈ W , and s ∈ S. If n > 8, the group W is infinite and Hq has infinite rank as an A-algebra. For the applications we have in mind, it is convenient to extend the scalars of Hq to produce an A-algebra H, where A = Z[v, v−1] and v2 = q, and to define a scaled version of the T -basis, {T̃w : w ∈ W}, where T̃w := v−ℓ(w)Tw. We will write A+ and A− for Z[v] and Z[v−1], respectively. A product w1w2 · · ·wn of elements wi ∈ W is called reduced if ℓ(w1w2 · · ·wn) = i ℓ(wi). We reserve the terminology reduced expression for reduced products w1w2 · · ·wn in which every wi ∈ S. We write L(w) = {s ∈ S : ℓ(sw) < ℓ(w)} R(w) = {s ∈ S : ℓ(ws) < ℓ(w)}. The set L(w) (respectively, R(w)) is called the left (respectively, right) descent set of w. Call an element w ∈ W complex if it can be written as a reduced product x1wss′x2, where x1, x2 ∈ W and wss′ is the longest element of some rank 2 parabolic subgroup 〈s, s′〉 such that s and s′ correspond to adjacent vertices in the Coxeter graph En. Denote by Wc(En) the set of all elements of W that are not complex. The elements of Wc = Wc(En) are the fully commutative elements of [24]; they are characterized by the property that any two of their reduced expressions may be obtained from each other by repeated commutation of adjacent generators. Let J(En) be the two-sided ideal of H generated by the elements T1 + Ts + Tt + Tst + Tts + Tsts, ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 7 where (s, t) runs over all pairs of elements of S for which m(s, t) = 3. Follow- ing Graham [7, Definition 6.1], we define the generalized Temperley–Lieb algebra TL(En) to be the quotient A-algebra H(En)/J(En). We denote the corresponding epimorphism of algebras by θ : H(En) −→ TL(En). Let tw (respectively, t̃w) de- note the image in TL(En) of the basis element Tw (respectively, T̃w) of H. If s ∈ S, we define bs ∈ TL(En) by bs = v−11 + t̃s. A more convenient description of TL(En) for the purposes of this paper is by generators and relations (as in [3, §2.2]). Since the Laurent polynomial v + v−1 occurs frequently, we denote it by δ. Proposition 3.1. As a unital A-algebra, TL(En) is given by generators {bs : s ∈ S} and relations b2s = δbs, bsbt = btbs if m(s, t) = 2, bsbtbs = bs if m(s, t) = 3. The following basis theorem will be used freely in the sequel. Theorem 3.2 [3, 7]. (i) The set {t̃w : w ∈ Wc} is a free A-basis for TL(En). (ii) If w ∈ Wc and w = si1si2 · · · sir is reduced, then the element bw = bsi · · · bsir is a well-defined element of TL(En). (iii) The set {bw : w ∈ Wc} is a free A-basis for TL(En). Proof. Part (i) is due to Graham [7, Theorem 6.2]. Parts (ii) and (iii) are stated by Fan in [3, §2.2], and more details may be found in [13, Proposition 2.4]. � Definition 3.3 [3, §2.3]. Let P = P (n) denote the set of subsets of the Coxeter graph En that consist of non-adjacent vertices. We allow P to include the empty set, 8 R.M. GREEN ∅. For any A ∈ P , let i(A) be the product of the elements of S(En) corresponding to the vertices in A (with i(∅) = 1); note that the order of the product is immaterial since the vertices in A correspond to commuting generators. Let A,B ∈ P . We say that A and B are neighbours if and only if 1 + #(A ∩ B) = #A = #B, and the two vertices in (A∪B)\(A∩B) are adjacent in En. Define an equivalence relation on P by taking the reflexive and transitive closure of the relation A ∼ B if A and B are neighbours. Let P̄ denote the set P/ ∼ . Example 3.4. In type E7, let A = {0, 2, 4, 6} and B = {0, 1, 4, 6}. In this case, i(A) = b0b2b4b6 and i(B) = b0b1b4b6, A and B are neighbours, and the equivalence class of A is precisely {A,B}. Definition 3.5 [3, §6.3]. Let n ≥ 6. If n is odd, we define P ′ = P ′(n) to be the subset of P (n) consisting of the sets {(n− 1)− 2j : 0 ≤ j ≤ N} : 0 ≤ N ≤ n− 1 together with the set {n− 1, n− 3, n− 5, . . . , 4} ∪ {0} and the empty set. If n is even, we define P ′ = P ′(n) be the subset of P (n) consisting of the sets {(n− 1)− 2j : 0 ≤ j ≤ N} : 0 ≤ N ≤ n− 2 together with the empty set. Example 3.6. In type E6, we have P ′ = {{5}, {5, 3}, {5, 3, 1}, ∅} . In type E7, we have P ′ = {{6}, {6, 4}, {6, 4, 2}, {6, 4, 2, 0}, {6, 4, 0}, ∅} . The importance of the set P ′ comes from the following Proposition 3.7 (Fan, [3, Lemma 8.1.2]). The set P ′ constitutes a complete set of equivalence class representatives for P with respect to ∼. � ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 9 4. Cells and the a-function In §4, we recall the definitions of the a-function and cells arising from the mono- mial basis. Most of this material comes from the papers [3] and [10], or is implicit in them. Definition 4.1 [3, Definition 2.3.1]. The a-function a : Wc −→ Z≥0 is defined by a(w) := max {#A : w = xi(A)y is reduced} for w ∈ Wc. Proposition 4.2. Let w ∈ Wc and let f ∈ A. Define the degree, deg f , of f to be the largest integer n such that vn occurs with nonzero coefficient in f , with the convention that deg 0 = −∞. Denote the structure constants with respect to the monomial basis by gx,y,z ∈ A, namely bxby = gx,y,zbz. (i) The structure constant gx,y,z is either zero or a nonnegative power of δ, and, given x and y, we have gx,y,z 6= 0 for a unique z. (ii) If s ∈ S and gs,y,z 6∈ Z, then gs,y,z = δ, ℓ(sy) < ℓ(y) and y = z. Similarly, if gx,s,z 6∈ Z, then gx,s,z = δ, ℓ(xs) < ℓ(x) and x = z. (iii) We have a(w) = maxx,y∈Wc deg gx,y,w. (iv) We have a(w) = maxx,y∈Wc deg gw,x,y. Proof. Parts (i) and (ii) are well known and follow easily from [3, Proposition 5.4.1]. Part (iii) is proved in [10, Proposition 4.2.3] using the results of [3]. The proof of [3, Theorem 5.5.1] shows that deg gw,x,y ≤ min(a(w), a(x)), which means that x,y∈Wc deg gw,x,y ≤ a(w). 10 R.M. GREEN Conversely, [3, Lemma 5.2.6] shows that bwbw−1 = (v + v −1)a(w)bd for some d ∈ Wc, so taking x = w−1 and y = d, we find that x,y∈Wc deg gw,x,y ≥ a(w), which completes the proof of (iv). � Definition 4.3 [3, Definition 4.1]. For any w,w′ ∈ Wc, we say that w′ ≤L w if there exists bx such that gx,w,w′ 6= 0, where g is as in Proposition 4.2. For any w,w′ ∈ Wc, we say that w′ ≤R w if there exists bx such that gw,x,w′ 6= 0. For any w,w′ ∈ Wc, we say that w′ ≤LR w if there exist bx and by such that bxbwby = cbw′ for some c 6= 0. We write w ∼L w′ to mean that both w′ ≤L w and w ≤L w′. Similarly, we define w ∼R w′ and w ∼LR w′. The relation ∼L (respectively, ∼R, ∼LR) is an equivalence relation, and the corresponding equivalence classes of Wc are called the left (respectively, right, two- sided) cells. It is clear from the definitions and the fact that the identity element is a monomial basis element that two-sided cells are unions of left cells, and also unions of right cells. Proposition 4.4. (i) Let w ∈ Wc. If we have w = xi(A)y reduced for some A such that #A = a(w), then i(A) ∼LR w and w ∼R xi(A). (ii) The a-function is constant on left, right, and two-sided cells. (iii) If w,w′ ∈ Wc are such that w′ ≤R w and w′ 6∼R w, then a(w′) > a(w). An analogous statement holds for left cells and two-sided cells. (iv) The right cell containing i(A) is precisely the set {w ∈ Wc : w = i(A)x reduced, a(w) = #A}. ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 11 (v) A left cell and a right cell contained in the same two-sided cell intersect in a unique element. Proof. Statement (i) is proved during the argument establishing [3, Theorem 4.5.1.]. The fact that the a-function is constant on two-sided cells is implicit in the proof of [3, Theorem 4.5.1]. Since two-sided cells are unions of left (or right) cells, part (ii) follows. Suppose now that w,w′ ∈ Wc are such that w′ ≤R w and w′ 6∼R w. An inductive argument using the definition of ≤R reduces the problem to the case where there is some s ∈ S such that bwbs is a multiple of bw′ , so let us assume that this is the situation. By [3, Corollary 4.2.2], the assumption that w′ ≤R w implies that a(w′) ≥ a(w). The statement follows unless a(w′) = a(w), so suppose we are in this case. Let us write w = xi(A)y as in statement (i). Now [3, Lemma 4.2.5], applied to the element xi(A) and the sequence of generators corresponding to ys, shows that we have w′ = xi(A)y′ reduced. By part (i), we find that w′ ∼R xi(A), and thus that w′ ∼R w, a contradiction. The statement for left cells follows by a symmetrical argument, and the statement for two-sided cells follows from the previous claims and the fact that if w′ ≤LR w, then there is a chain w′ = w1, w2, w3, . . . , wk = w where, for each 1 ≤ i < k, we have either wi ≤L wi+1 or wi ≤R wi+1. This completes the proof of (iii). Part (iv) is [3, Proposition 4.4.3]. Part (v) is well known and follows from the proof of [3, Theorem 6.1.2]. � Remark 4.5. For finite and affine Weyl groups, the a-function defined above is known by [23, Theorem 3.1] to be the restriction of Lusztig’s more general a- function [21] restricted to the subset Wc. Although it is not true that each of the monomial cells studied above is a cell 12 R.M. GREEN in the sense of Kazhdan–Lusztig [19], it can be shown fairly easily that each left (respectively, right, two-sided) monomial cell is a subset of some left (respectively, right, two-sided) Kazhdan–Lusztig cell. 5. Traces on the algebras TL(En) In §5, we will extend scalars and deal with a K-form of TL(En), where K is a field containing A and a square root of δ. (The existence of δ is needed for compatibility with [3], but can ultimately be removed; see Remark 6.4.) We write TLK(En) := K ⊗A TL(En). We aim to classify the traces, τ : TLK(En) −→ K, that is, linear functions τ with the property that τ(ab) = τ(ba) for all a, b ∈ TLK(En). It is clear that the set of all traces on TLK(En) is a K-vector space (dependent in principle on K and δ). The main result of §5 is that there is a basis for this vector space in natural bijection with the set P ′ of §3. The next result shows how τ naturally induces a function P/ ∼ −→ K. Lemma 5.1. Maintain the notation of Definition 3.3. Suppose A,B ∈ P are such that A ∼ B, and let τ : TLK(En) −→ K be a trace. Then τ(i(A)) = τ(i(B)). Proof. The proof immediately reduces to the case where A and B are neighbours. Let s (respectively, t) be the element of S corresponding to the unique element of A\B (respectively, B\A). It is immediate from the definitions that i(A) = bsi(A ∩B) = i(A ∩B)bs and i(B) = bti(A ∩B) = i(A ∩B)bt. We then have τ(i(A)) = τ(bsi(A ∩B)) = τ(bsbtbsi(A ∩B)) = τ(btbsi(A ∩B)bs) = τ(btbsbsi(A ∩B)) = δτ(btbsi(A ∩B)) = τ(btbtbsi(A ∩B)) = τ(btbsi(A ∩B)bt) = τ(btbsbti(A ∩B)) = τ(bti(A ∩B)) = τ(i(B)), as required. � ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 13 Lemma 5.2. Any trace τ : TLK(En) −→ K is determined by its values on the set {i(A) : A ∈ P}. Proof. Suppose the values of τ(i(A)) are known for each A ∈ P . We will show how to compute the value of τ(bw), where w ∈ Wc is arbitrary. Let us write w = xi(A)y reduced as in Proposition 4.4 (i). Using a reverse induction, we will assume that the values of τ(bw′) for a(w ′) > a(w) = #A, if such w′ exist, have been determined. By the defining relations of TL(En), we have bi(A)bi(A) = δ #Abi(A), and so we have τ(bw) = τ(bxbi(A)by) = δ−#Aτ(bxbi(A)bi(A)by) = δ−#Aτ(bi(A)bybxbi(A)). Now i(A)y and xi(A) lie in Wc because w does, and Proposition 4.4 (i) and (ii) shows that a(w) = a(xi(A)). By Proposition 4.2 (i), we have bi(A)bybxbi(A) = bi(A)ybxi(A) = δ for some z ∈ Wc, and it is clear from the definitions that z ≤L xi(A). By Proposi- tion 4.4 (ii) and (iii), we see that a(z) ≥ a(xi(A)) = a(w) = #A. If a(z) > #A then our inductive hypothesis determines the value of τ(δcbz), which in turn determines the value of τ(bw). We may therefore assume that a(z) = #A. To complete the proof, it is enough to show that z = i(A), because the value of τ(bz) will then have been determined by our assumptions. Let s ∈ A. Since bsbi(A) = δbi(A) by the defining relations, the definition of bz shows that bsbz = δbz. By Proposition 4.2 (ii), this means that ℓ(sz) < ℓ(z), and it follows that A ⊆ L(z). Because A is a set of commuting generators, standard 14 R.M. GREEN properties of Coxeter groups show that we can write z = i(A)z′ reduced. Applying Proposition 4.4 (iv) to the fact that a(z) = #A shows that z ∼R i(A). A symmet- rical argument then shows that we have z ∼L i(A). By Proposition 4.4 (v), this can only happen if z = i(A). � Theorem 5.3. For each Ā ∈ P̄ (as in Definition 3.3), there is a unique trace τĀ : TLK(En) −→ K such that for each B ∈ P we have τĀ(i(B)) = 1 if B ∈ Ā, 0 otherwise. The set {τĀ : Ā ∈ P̄} is a K-basis for the set of all traces τ : TLK(En) −→ K. Proof. It is clear from the definition of trace that the traces from TLK(En) to K form a K-vector space. Lemmas 5.1 and 5.2 show that this space has dimension at most the size of P̄ . Fan [3, Theorem 5.6.1] shows that TLK(En) is semisimple and that is then a direct sum of |P̄ | matrix rings. This proves that the dimension of the space of traces is at least the size of P̄ , and thus that the space has the claimed dimension. A dimension count, together with another application of lemmas 5.1 and 5.2, then shows that there are unique traces τĀ with the properties claimed, and that they form a basis. � We now come to the central definition of the paper. Definition 5.4. The trace tr : TLK(En) −→ K is defined by Ā∈P̄ δ−#AτĀ, where τĀ is as in Theorem 5.3. ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 15 Corollary 5.5. Any trace τ : TLK(En) −→ K satisfies τ(bw) = τ(bw−1) for all w ∈ Wc. Proof. It follows from Proposition 3.1 that there is a unique A-linear antiautomor- phism ∗ : TL(En) −→ TL(En) fixing the generators bs. We may extend this to a K-linear antiautomorphism ∗ : TLK(En) −→ TLK(En). If a ∈ TLK(En), let us write a∗ for ∗(a). Note that if A ∈ P , then i(A) is invariant under ∗, because i(A) is a product of commuting generators bs. Given a trace τ : TLK(En) −→ K, the K-linear map τ ′ : TLK(En) −→ K defined by τ ′(a) = τ(a∗) is also a trace. Since τ and τ ′ agree on all elements i(A) for A ∈ P , Lemma 5.2 shows that τ = τ ′, and the assertion follows. � Remark 5.6. The trace tr will turn out to induce the Markov trace of the title. Note that the definition makes sense because Ā, B̄ ∈ P̄ implies #A = #B. Traces on Hecke algebras of finite Coxeter groups are known have a property similar to that given in Corollary 5.5; see [5, Corollary 8.2.6] for more details. 6. Cellular structure and the a-funtion In §6, we explain how the trace tr is particularly compatible with the structure of TL(En) as a cellular algebra, in the sense of [8]. We will not recall the complete definition of a cellular algebra here, but we summarize below the properties of the cellular structure that are important for our purposes. Definition 6.1. Let Λ be the set of two-sided cells for TL(En), equipped with the partial order induced by ≤LR. For each λ ∈ Λ, let M(λ) be an indexing set for the left cells contained in λ; note that the inversion map on the Coxeter group W induces a bijection between the set of left cells in λ and the set of right cells in λ (see the remarks at the end of [3, §4.4]). Proposition 6.2. Maintain the above notation. (i) Let T, U ∈ M(λ) for some fixed λ ∈ Λ. Then T ∩ U contains a unique element, w, and we define CT,U = bw. 16 R.M. GREEN (ii) The A-algebra anti-automorphism ∗ : TL(En) −→ TL(En) defined by ∗(bw) = bw−1 satisfies ∗(CT,U ) = CU,T . In particular, we have w2 = 1 if and only if bw = CT,T for some T . (iii) Suppose that CP,Q and CR,S are arbitrary monomial basis elements, and define CT,U by the condition CP,QCR,S = δ aCT,U (which makes sense by Proposition 4.2 (i)). If P,Q,R, S, T and U all belong to the same two-sided cell, then P = T and S = U ; if, furthermore, we have Q = R, then a = a(CT,U ). If it is not the case that P = T , S = U and Q = R, then we have a < a(CT,U ). Proof. Parts (i) and (ii), which are originally due to Graham [7], are proved in [10, Proposition 4.2.1]. Part (iii) is proved in [10, propositions 4.2.1 and 4.2.3] using the results of [3]. � Proposition 6.3. For all w ∈ Wc, we have tr(bw) = δa, where a = −a(w) if w2 = 1, and a < −a(w) otherwise. Proof. Let λ be the two-sided cell containing w. We will prove the statement by induction on the partial order on two-sided cells given in Definition 6.1. Writing w = CT,U for T, U ∈ M(λ), as in Proposition 6.2 (i), and applying Proposition 6.2 (ii), we see that the condition w2 = 1 is equivalent to T = U . By Proposition 4.4, there exists a product of a(w) commuting generators, i(A), in λ. Define V ∈ M(λ) by the condition CV,V = bi(A). Since tr is a trace, Proposition 6.2 (iii) shows that tr(CT,U ) = δ −a(w)tr(CT,V CV,U ) = δ −a(w)tr(CV,UCT,V ). By Proposition 4.2 (i), we have CV,UCT,V = δ bCX,Y for some b ≥ 0 and some basis element CX,Y . There are now two cases to consider. ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 17 The first possibility is that CX,Y comes from the two-sided cell λ. (If T = U , this case must occur by Proposition 6.2 (iii).) In this case, we have X = Y = V , and thus CX,Y = bi(A). Proposition 6.2 (iii) then shows that b = a(w) if T = U , and b < a(w) otherwise. Since we have tr(bi(A)) = δ −a(w) by definition of tr, we have tr(CT,U ) = δ −a(w)+b−a(w), and the result follows. The other possibility is that CX,Y comes from a two-sided cell λ ′ with λ′ < λ, and T 6= U . In this case, Proposition 4.4 (iii) shows that a(CX,Y ) > a(w). By the inductive hypothesis, we know that tr(CX,Y ) = δ a′ , where a′ ≤ −a(CX,Y ) < −a(w). This means that tr(CT,U ) = δ−a(w)+b+a . By propositions 4.2 (iii) and 4.4 (ii), we have b ≤ a(w), and thus tr(CT,U ) = δa for a < −a(w), as required. � Remark 6.4. The above proposition shows that we do not actually need δ ∈ k to define tr. From now on, we need only assume that K is a field containing A. Proposition 6.5. If K is the field of fractions of the power series ring Z[[v−1]], then tr is a nondegenerate trace on TLK(En), and tr(CP,QCR,S)− δQRδPS ∈ v−1Q[[v−1]], where δQR and δPS are the Kronecker delta. Proof. An element x of K is uniquely representable in the form where λi ∈ Q for all i. If x 6= 0, we define deg x to be the largest integer j such that λj 6= 0. If x, y 6= 0 then deg(xy) = deg x + deg y, so the facts that deg δ = 1 and deg 1 = 0 imply that deg δa = −a. The second assertion follows from the fact that deg δa = −a combined with Proposition 4.4 (ii), Proposition 6.2 (iii) and Proposition 6.3. We will now show that for any nonzero a ∈ TLK(En), we have tr(aa∗) 6= 0, from which the assertion follows. We have λwbw, 18 R.M. GREEN and by clearing denominators (thus multiplying a by a nonzero scalar), we may assume that we have λw ∈ A for all w ∈ Wc. Choose w′ with λw′ 6= 0 and N(w′) := deg λw′ maximal, and let cw′ be the (integer) coefficient of v N(w′) in λw′ . Setting aw′ = v −N(w′)λw′bw′ , we then have tr(aw′a w′) = c 2 mod v−1Q[[v−1]]. If λw′′ 6= 0 but deg λw′′ is not maximal, we may again define aw′′ = v−N(w ′′)λw′′bw′′ , but then tr(aw′′a w′′) ∈ v−1Q[[v−1]]. Since the integers c2 are strictly positive, it follows that tr((v−N(w ′)a)(v−N(w ′)a)∗) 6∈ v−1Q[[v−1]], which completes the proof. � Proposition 6.6. Let K be the field of fractions of the power series ring Z[[v−1]], and let K ′ be the subfield of K consisting of the field of fractions of Z[[v−2]]. (i) The field TLK(En) has a unique structure as a Z2-graded algebra over K which vn has degree n mod 2 and K ′ is precisely the set of elements of degree 0 mod 2. (ii) The algebra TLK(En) has a unique structure as a Z2-graded algebra over K in which vn has degree n mod 2 and the generators bs have degree 1 mod 2. We denote the even subalgebra consisting of elements of degree 0 mod 2 by TLK′(En). (iii) Let τ : TLK(En) −→ K be any trace. Then there are unique K ′-linear maps τ(0), τ(1) : TLK′(En) −→ K ′ such that τ(0) + vτ(1) is the restriction of τ to TLK′(En), and furthermore, τ(0) and τ(1) are themselves traces. Proof. Recall from the proof of Proposition 6.5 that K = Q((v−1)) = Q[v][[v−1]], so that each element x ∈ K has a unique expression of the form ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 19 where qi ∈ Q and N ∈ Z depends on x. Similar reasoning shows that the subfield K ′ of K then consists precisely of those elements for which qi = 0 whenever i is odd. Part (i) is a consequence of this construction. The assertion of (ii) is immediate from the observation that the defining relations of Proposition 3.1 respect the given grading. Let π : K −→ K ′ be the map where q′i = qi if i is even, 0 otherwise. Our description ofK ′ shows that π is a K ′-linear map. Denoting the restriction of τ to TLK′(En) by τ ′, it follows that π ◦τ ′ is a trace on TLK′(En). Since τ(0) = π ◦τ ′, the maps τ(0), vτ(1) = τ ′ − τ(0) and τ(1) are also traces, completing the proof of (iii). � Note that any trace from TLK′(En) to K ′ extends uniquely to a trace from TLK(En) to K by tensoring by K ⊗K′ −. Lemma 6.7. The trace tr : TLK(En) −→ K arises from a trace tr′ : TLK′(En) −→ K ′ by extension of scalars. Proof. We use the notation of §5. Note that if A ∈ P , then i(A) is an element of TLK(En) of degree #A mod 2. We also have tr(i(A)) = δ #A, which is an element of K of degree #A mod 2. Recall that TLK′(En) is a K ′-subalgebra of TLK(En) and note that if y, z are homogeneous elements of TLK(En), then yz and zy have the same degree. The argument of Lemma 5.2 now shows that if x is an element of TLK′(En), we have a relation tr(x) = tr Ā∈P̄ λĀi(A) 20 R.M. GREEN where for each Ā ∈ P̄ , we have λĀ(i(A)) ∈ TLK′(En). By the first paragraph of the proof, λĀ must be homogeneous of degree #A mod 2, and tr(λĀi(A)) ∈ K ′. The proof is completed by the observation that any x ∈ TLK(En) is uniquely expressible as x(0) + vx(1) for x(0), x(1) ∈ TLK′(En) (compare with Proposition 6.6 (iii)). � Corollary 6.8. If w ∈ Wc and tr(bw) = δa as in Proposition 6.3, then a ≡ λ(w) mod 2. Proof. By Lemma 6.7, we have deg tr(bw) = ℓ(w) mod 2, so the assertion follows from the fact that deg δ = 1. � §7. tom Dieck’s diagram calculus In [2], tom Dieck introduced a diagram calculus for the algebras TL(En). To give a rigorous definition of tom Dieck’s diagram calculus, as we do here, we first need to recall the graphical definition of the Temperley–Lieb algebra. We start by recalling Jones’ formalism of k-boxes [18], following the approach of Martin and the author in [15]. For further details and references, the reader is referred to [11, Definition 7.1. Let k be a nonnegative integer. The standard k-box, Bk, is the set {(x, y) ∈ R2 : 0 ≤ x ≤ k + 1, 0 ≤ y ≤ 1}, together with the 2k marked points 1 = (1, 1), 2 = (2, 1), 3 = (3, 1), . . . , k = (k, 1), k + 1 = (k, 0), k + 2 = (k − 1, 0), . . . , 2k = (1, 0). Definition 7.2. Let X and Y be embeddings of some topological spaces (such as lines) into the standard k-box. Multiplication of such embeddings to obtain a new embedding in the standard k-box shall, where appropriate, be defined via the following procedure on k-boxes. The product XY is the embedding obtained by placing X on top of Y (that is, X is first shifted in the plane by (0, 1) relative to Y , so that marked point (i, 0) in X coincides with (i, 1) in Y ), rescaling vertically by a ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 21 scalar factor of 1/2 and applying the appropriate translation to recover a standard k-box. Definition 7.3. Let k be a nonnegative integer. Consider the set of smooth em- beddings of a single curve (which we usually call an “edge”) in the standard k-box, such that the curve is either closed (isotopic to a circle) or its endpoints coincide with two marked points of the box, with the curve meeting the boundary of the box only at such points, and there transversely. By a smooth diffeomorphism of this curve we mean a smooth diffeomorphism of the copy of R2 in which it is embedded, that fixes the boundary, and in particular the marked points, of the k-box, and takes the curve to another such smooth embedding. (Thus, the orbit of smooth diffeomorphisms of one embedding contains all embeddings with the same endpoints.) A concrete Brauer diagram is a set of such embedded curves with the property that every marked point coincides with an endpoint of precisely one curve. (In examples we can represent this set by drawing all the curves on one copy of the k-box. Examples can always be chosen in which no ambiguity arises thereby.) Two such concrete diagrams are said to be equivalent if one may be taken into the other by applying smooth diffeomorphisms to the individual curve embeddings within it. There is an obvious map from the set of concrete diagrams to the set of pair partitions of the 2k marked points. It will be evident that the image under this map is an invariant of concrete diagram equivalence. The set Bk(∅) is the set of equivalence classes of concrete diagrams. Such a class (or any representative) is called a Brauer diagram. Let D1, D2 be concrete diagrams. Since the k-box multiplication defined above internalises marked points in coincident pairs, corresponding curve endpoints in D1D2 may also be internalised seamlessly. Each chain of curves concatenated in this way may thus be put in natural correspondence with a single curve. Thus the multiplication gives rise to a closed associative binary operation on the set of 22 R.M. GREEN concrete diagrams. It will be evident that this passes to a well defined multiplication on Bk(∅). Let R be a commutative ring with 1. The elements of Bn(∅) form the basis elements of an R-algebra PBn (∅) with this multiplication. A curve in a diagram that is not a closed loop is called propagating if its endpoints have different y-values, and non-propagating otherwise. (Some authors use the terms “through strings” and “arcs” respectively for curves of these types.) Note that in a Brauer diagram drawn on a single copy of the k-box it is not gen- erally possible to keep the embedded curves disjoint. Let Tk(∅) ⊂ Bk(∅) denote the subset of diagrams having representative elements in which the curves are disjoint. Representatives of this kind are called Temperley–Lieb diagrams. It will be evident that PBn (∅) has a subalgebra with basis the subset Tk(∅). (That is to say, the disjointness property is preserved under multiplication.) We denote this subalgebra Pn(∅) Because of the disjointness property there is, for each element of Tk(∅), a unique assignment of orientation to its curves that satisfies the following two conditions. (i) A curve meeting the r-th marked point of the standard k-box, where r is odd, must exit the box at that point. (ii) Each connected component of the complement of the union of the curves in the standard k-box may be oriented in such a way that the orientation of a curve coincides with the orientation induced as part of the boundary of the connected component. Note that the orientations match up automatically in composition. If D1 and D2 are equivalent concrete Temperley–Lieb diagrams, the diffeomorphisms that give rise to the equivalence set up a bijection between the connected components of D1 and those of D2. ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 23 Figure 2. A pillar diagram corresponding to an element of T8(∅) 1 2 3 4 5 6 7 8 910111213141516 Definition 7.4. A pillar diagram consists of a pair (D, f), where D ∈ Tk(∅) is a Temperley–Lieb diagram and f is a function from the connected components of D to Z≥0, such that any component with anticlockwise orientation is mapped to zero. On the diagram D, we indicate the values of f on the clockwise connected com- ponents either by writing in the appropriate integer, or by inserting k disjoint discs (the “pillars” of [2]). The set of pillar diagrams arising from the set Tk(∅) will be denoted Tk(•). Example 7.5. Let k = 8. A pillar diagram corresponding to an element of Tk(•) is shown in Figure 2. Note that there are 10 connected components, precisely 7 of which inherit a clockwise orientation. The values of f on these 7 components are 3, 2, 2, 1, 0, 0, 0. We define an algebra Pn(•), analogous to Pn(∅), with the set Tk(•) as a basis. The multiplication is k-box multiplication with the added convention that function values on the connected components are additive. (This is natural if one represents the function values with pillars as in Figure 2.) For our purposes, we need to apply an equivalence relation on the concrete diagrams of Tk(•). Locally, this is given by the relation shown in Figure 3. 24 R.M. GREEN Figure 3. A topological reduction rule In the notation where clockwise regions are labelled by nonnegative integers, the relation of Figure 3 is that shown in Figure 4. Figure 4. Alternative notation for the topological reduction If the regions labelled k and l are connected to each other, Figure 3 shows that we have k = l > 1 and p = k − 1. On the other hand, if the regions labelled k and l are genuinely distinct, that is, the arcs shown on the left hand side of figure 3 are not sections of some longer arc, then we have p = k+ l− 1 ≥ 1. In the latter case, it is not possible for any regions labelled by the integer zero to be created or destroyed by the topological reduction. Note that the other partial regions shown in figures 2 and 3 have anticlockwise orientation, and as such they are labelled by the integer 0. Definition 7.6. If L is a closed loop in a concrete diagram of Tk(•), we define m(L) to be the integer label of the region immediately interior to L; in particular, we have m(L) = 0 if L has anticlockwise orientation. Let R be a commutative ring with 1. The R-algebra PEn (•) is the quotient of the R-algebra Pn(•) obtained by applying the following three relations: (i) for each closed loop L whose immediate interior is labelled 1 and whose imme- diate exterior is necessarily labelled 0, relabel the immediate interior of L by 0 ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 25 and remove L; (ii) for each closed loop L whose immediate interior is labelled 0 and whose imme- diate exterior is labelled k, relabel the immediate interior of L by k, remove L and multiply by δ; (iii) for each region R labelled by k ≥ 2 (whether or not R is a closed loop), decrease the label of R by 1 and multiply by δ. A basis for PEn (•) may be obtained by using the notion of “reduced” diagrams given in [2, §2] and Bergman’s diamond lemma [1]. However, we do not pursue this because we do not need it for our purposes. Definition 7.7. Suppose n > 1 and 1 ≤ k < n. The diagram Enk of PEn (•) is the one where each point i is connected by a propagating edge to point 2n+1− i, unless i ∈ {k, k+1, 2n−k, 2n+1−k}. Points k and k + 1 are connected by an edge, as are points 2n − k and 2n + 1 − k. All regions are labelled by 0. The diagram Bnk of PEn (•) is the one where each point i is connected by a propagating edge to point 2n+ 1− i, and all regions are labelled by 0, except the rectangular region bounded by k, k+1, 2n− k and 2n+1− k, which is labelled by Proposition 7.8. There is a unique homomorphism ρ : TL(En) −→ PEn (•) of unital A-algebras sending b0 to Bn3 and bs to Ens for i ∈ {1, 2, . . . , n − 1}, where the numbering of generators is as in §3. Proof. This is a routine (but important) exercise using the presentation of Propo- sition 3.1, and is essentially the same as the proof of [2, Theorem 2.5]. � We shall see later that ρ is in fact a faithful representation. We will not determine the image of ρ, but this can be done by an inductive combinatorial argument similar to those in [9, §5]. 26 R.M. GREEN §8. Existence and uniqueness of the Markov trace There is a well-known embedding ιn : TL(En) −→ TL(En+1) sending bs to bs for each generator of TL(En) (see [3, §6.3]). This means that the tower of algebras TL(En), equipped with the generators bs, fits into the framework of Markov traces defined in §2. We recall the definition in order to fix some notation. Definition 8.1. Let K be a field containing A. A Markov trace on TLK(E∞) with parameter z ∈ K is a K-linear map τ : TLK(E∞) −→ K satisfying the following conditions: (i) τ(1) = 1; (ii) τ(hbn) = zτ(h) for n ≥ 6 and h ∈ TLK(En); (iii) τ(hh′) = τ(h′h) for all n ≥ 6 and h, h′ ∈ TLK(En). Remark 8.2. Note that in condition (ii), bn is the unique generator in TL(En+1) that does not lie in TL(En). As mentioned in [3, §2.2], the algebras TL(En) are quotients of the Hecke algebras of the Coxeter groups W (En), and bs = q −1/2(Ts+ 1), where the Ts are the usual generators for the Hecke algebra as given in [16, §7]. This means that the Markov trace can also be regarded as a trace on a tower of Hecke algebras. Proposition 8.3. If τ is a Markov trace on TLK(E∞), then the parameter z must be equal to δ−1, and τ is unique. Restricted to TL(En), such a Markov trace must agree with the trace tr. Proof. Let n ≥ 6. Part (ii) of Definition 8.1 shows that τ(bn−1bn) = zτ(bn−1). On the other hand, the defining relations and part (iii) of the definition show that τ(bn−1bn) = δ −1τ(bn−1(bn−1bn)) = δ −1τ(bn−1bnbn−1) = δ −1τ(bn−1), proving the assertion about the parameter. To prove the other assertions, it suffices to show that, regarding TLK(En) as a subalgebra of TLK(E∞), we have τ(i(A)) = δ −#A for A ∈ P = P (n). Choose ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 27 such an A. It follows from Definition 3.3 that for sufficiently large N ≥ n, and identifying A in the obvious way with an element of P (N), we can find B ∈ P (N) with A ∼ B and B∩{b0, b1, . . . , b5} = ∅. The first assertion together with repeated applications of part (ii) of Definition 8.1 (and one application of part (i)) now show that τ(i(B)) = δ−#B = δ−#A, and Lemma 5.1 completes the proof. � To prove that the Markov trace on TLK(E∞) exists, we make use of the diagram calculus, as hinted in [2, §6]. Definition 8.4. Let k be a nonnegative integer. The standard k-cone is obtained from the standard k-box by identifying each pair of points {(x, 0), (x, 1)} for each 0 ≤ x ≤ k+1, and identifying all the points in the set {(k+1, y) : 0 ≤ y ≤ 1}. The standard k-cone is homeomorphic to a closed disc. Let D be a diagram in PEk (•). The trace diagram, D, of D is obtained by identifying the boundary points of the k-box bounding D to form the standard k-cone. Figure 5. The trace diagram of the pillar diagram in Figure 2 Example 8.5. The trace diagram D corresponding to the diagram D of Figure 2 is shown in Figure 5. Notice that the outer part of the trace diagram (regarded as a disc) will always have an anticlockwise orientation and thus be labelled by 0. Consequently, any 28 R.M. GREEN regions in the trace diagram not labelled by zero must be bounded by at least one closed loop. (It is possible for the closed loops to be nested.) Definition 8.6. Let g : Z≥0 −→ Z≥0 be given by g(c) = 1 if c = 0, c− 1 if c ≥ 1. If D is a trace diagram for TL(En), we define the content, c(D), of D to be the integer g(f(L)), where the sum is over all the connected components L of D that are interior to at least one closed loop, and where f(L) is the integer assigned to L as in Definition Example 8.7. The content of the trace diagram in Figure 5 is g(2) + g(3) + g(3) = 5. Lemma 8.8. The content of a trace diagram D is invariant under the topological reduction rule shown in Figure 3. Proof. Consider the application of the topological reduction rule to a diagram that looks locally like the situation in Figure 6. Figure 6. Labelling of points involved in the topological relation C D As in the discussion following Figure 4, there are two cases to consider, according as the two pillar regions are connected or not in D. ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 29 There are four cases to consider, according as there is an oriented curve in D from point A to point C, and (independently) according as there is an oriented curve in D from point D to point B. Suppose first that there is no oriented curve in D from point A to point C, and also that there is no oriented curve in D from point D to point B. In this case, the two pillar regions are genuinely distinct, and applying the topological relation does not produce any new closed loops. We are then in the case p = k + l − 1 ≥ 1 of Figure 4, so the summands (k − 1) and (l − 1) appearing in Definition 8.6 are replaced by a single ((k + l − 1)− 1), leaving the content unchanged. We next deal with the case where there is an oriented curve from point A to point C, but no oriented curve from point D to point B. In this case, the two pillar regions are connected to each other, and the application of the topological rule produces a new closed loop (labelled zero) from the curve originally connecting point A to point C. We are now in the case k = l > 1 of Figure 4. This will change one of the summands (k− 1) of Definition 8.6 to (k− 2), and a new summand of 1 will be produced, corresponding to the new closed loop. The content thus remains unchanged. Consideration of the case where there is an oriented curve from point D to point B, but not from point A to point C, proceeds in exactly the same way. The last case, in which both oriented curves exist, also works similarly, except that the oriented curves shown in Figure 6 are already part of a closed loop. Application of the topological relation splits this closed loop into two closed loops, again producing an extra summand of 1 and changing a summand (k − 1) to (k − 2), leaving the content unchanged. � Lemma 8.9. There is a well-defined K-linear map τ•n : PEn (•) −→ K such that for each pillar diagram D, τ•n(D) = δ c(D). If x, y ∈ PEn (•), we have τ•n(xy) = τ n(yx). 30 R.M. GREEN Proof. For the first assertion, we need to check relations (a)–(c) of Definition 7.6. Relation (iii) holds by Lemma 8.8. In relation (i), we have D = D1, where D1 is the result of removing a loop labelled 1 from D. Since c(D) = c(D1), we have τ n(D) = τ n(D1). In relation (ii), we have D = δD2, where D2 is the result of removing a loop labelled 0 from D. Since c(D) = c(D2) + 1, we have τ n(D) = τ n(D2). By linearity, we only need check the second assertion in the case where x and y are pillar diagrams, and this is immediate from the construction of trace diagrams from pillar diagrams. � It is not hard to see that there is an algebra embedding ι•n : PEn (•) −→ PEn+1(•) analogous to the map ιn. Given a pillar diagram D of PEn (•), ι•(D) is the diagram obtained by adding a vertical line on the right of the diagram. Lemma 8.10. Let D be a pillar diagram of PEn (•). (i) We have τ•n+1(ι n(D)) = δτ n(D). (ii) Let En+1n be as in Definition 7.7. Then we have τ n(D) = τ n+1(ιn(D)En). Proof. Part (i) follows from the observation that the trace diagram ι•n(D) differs from the trace diagram D only in having a single extra closed loop, labelled 0. A short calculation involving diagrams shows that the trace diagrams D and ιn(D)En are equivalent, from which part (ii) follows. � Theorem 8.11. Let τn : TLK(En) −→ K be the trace defined by τn(x) = δ −nτ•n(ρ(x)). The family of traces {τn : n ≥ 6} is compatible with the direct limit of algebras TLK(En) and gives the unique Markov trace on TLK(E∞). Furthermore, the Markov trace agrees with the traces tr of Definition 5.4. Proof. The maps τn are traces by Proposition 7.8 and Lemma 8.9. They are compatible with the direct limit by Lemma 8.10 (i). Since τ•n(1) = δ n, we have ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 31 τn(1) = 1. Condition (ii) of Definition 8.1 follows from part (ii) of Lemma 8.10. Uniqueness of the Markov trace, and agreement with the traces tr, is given by Proposition 8.3. � 9. Proofs and applications Proof of Theorem 1.1. We need to show that the homomorphism ρ of Proposition 7.8 is injective, and there is no loss in passing to the field of fractions K of Z[[v−1]]. In this case, Proposition 6.5 and Theorem 8.11 show that the unique Markov trace on TLK(En), which can be defined on Im(ρ), is nondegenerate on TLK(En). The conclusion follows. � Proposition 9.1. The linear map (1 + v−2)nτn = v −nτ•n ◦ ρ restricted to TL(En) takes values in A. It is a tabular trace in the sense of [10], and a positive generalized Jones trace in the sense of [12]. Proof. The first assertion comes from the fact that τ•n evaluated on a diagram (such as an element of the form ρ(bw) for w ∈ Wc) yields a nonnegative integer power of To check that (1+v−2)nτn is a tabular trace, we need to check that axiom (A5) of [10, Definition 1.3.4] is satisfied. We have just shown that (1+v−2)nτn takes values in A, and it is clear from Theorem 8.11 that (1+ v−2)nτn is a trace. We have seen in Corollary 5.5 and Proposition 6.2 (ii) that (1 + v−2)nτn(x) = (1 + v −2)nτn(x for all x ∈ TL(En). All that remains to check is that τ(va(CS,T )CS,T ) = δS,T mod v −1A−. This follows from propositions 6.2 (ii) and 6.3 once we observe that we have (1 + v−2)n = 1 mod v−2Q[[v−1]], 32 R.M. GREEN regarded as power series in Q[v][[v−1]]. To show that (1+ v−2)nτn is a generalized Jones trace (see [12, Definition 2.9]), two further conditions must be checked. One of these is precisely that established by Lemma 6.7; the other is that, for x, y ∈ Wc, we should have (1 + v−2)nτn(cxcy−1) = 1 mod v−1A− if x = y, 0 mod v−1A− otherwise, where {cw : w ∈ Wc} is the canonical basis of TL(En) defined by J. Losonczy and the author in [14]. By [14, Theorem 3.6], this is nothing other than the basis {bw : w ∈ Wc} in this case. The corresponding property for tr (instead of (1+v−2)nτn) follows from Proposition 6.5, and the assertion for (1+v −2)nτn follows from the fact that (1 + v−2)n = 1 mod v−2A−. A generalized Jones trace is positive if it sends canonical basis elements to el- ements of N[v, v−1]. This holds for (1 + v−2)nτn by Proposition 6.3: in this case, (1 + v−2)nτn(bw) = δ b for some b ≥ 0, so that (1 + v−2)n ∈ N[v, v−1]. � Remark 9.2. Proposition 9.1 corrects the proof of [10, Theorem 4.3.5], where the proof that the tabular trace takes the same values on x and x∗ contains a gap. Proof of Theorem 1.2. By [12, Theorem 7.10], the conclusion of Theorem 1.2 holds for a generalized Jones trace if the underlying Coxeter group has “Property F” and a bipartite Coxeter graph. Clearly the graphs En are bipartite, because they contain no circuits. Property F holds by [12, Remark 3.5]; see [13, Lemma 5.6] for a fuller explanation. To complete the proof, we simply have to transfer the result from (1+v−2)nτn to the Markov trace, which follows from the fact that (1+v−2)n = 1 mod v−2A−. � The next result is an easier to use version of Theorem 1.2. Corollary 9.3. Let x, y ∈ Wc(En). Then we have µ̃(x, y) = 1 if τ•n ◦ ρ(bxby−1) = δn−1, 0 otherwise. ON THE MARKOV TRACE FOR TEMPERLEY–LIEB ALGEBRAS OF TYPE En 33 Proof. This follows from Theorem 1.2 together with the observation that bxby−1 = δbbw for some b ≥ 0 and w ∈ Wc, and the fact that τ•n sends diagrams to positive powers of δ. � Remark 9.4. It follows from [13, Theorem 4.6 (iv)] and [14, Theorem 3.6] that the monomial basis element bx is the projection of the Kazhdan–Lusztig basis element C′x ∈ H(En). Regarding tr and τ•n ◦ ρ as traces on the Hecke algebra, Theorem 1.2 and Corollary 9.3 can be used to evaluate the trace on products of certain Kazhdan–Lusztig basis elements, without evaluating the product (which would be difficult). Another noteworthy property of these results is that they give non- recursive formulae for certain of the integers µ(x, y). Remark 9.5. In [7, §9], Graham showed that if x, w ∈ Wc for TL(En) then µ(x, y) ∈ {0, 1}, and also produced a nonrecursive method of finding all the x with µ(x, y) = 1 for a fixed y. (In [7], x and y are said to be “close” if µ̃(x, y) = 1.) However, unlike the results above, this does not give an efficient way to compute µ(x, y) when both of x and y are specified. Corollary 9.3 can therefore be regarded as a quick way to tell if two elements are close or not. Remark 9.6. It is possible to modify Theorem 1.2 and Corollary 9.3 so that they provide a nonrecursive way to test whether two diagrams represent the same algebra element. However, we do not pursue this here for reasons of space. Example 9.7. Consider the Coxeter system of type En with n = 6, and generators s0, . . . , s5 as numbered in Figure 1. Define y = s1s2s4s0s5 and w = s1s2s3s4s0s3s5s2s4s1s3s2s0s3s4s5; these are both reduced expressions for fully commutative elements. The diagrams ρ(by) and ρ(bw) are shown in figures 7 and 8 respectively. To evaluate τ n(bybw−1), we invert the diagram for bw, compose it with by and identify boundary points to produce a trace diagram. The trace diagram so obtained is shown in Figure 9 (up 34 R.M. GREEN to equivalence), and by inspection, it has content 1 + 1 + 1 + (3− 1) = 5 = n − 1. It follows from Corollary 9.3 that µ(y, w) = 1. Figure 7. The diagram ρ(by) of Example 9.7 REFERENCES 35 Figure 8. The diagram ρ(bw) of Example 9.7 Figure 9. The trace diagram corresponding to τ•6 ◦ ρ(bybw−1) of Example 9.7 Acknowledgement I am grateful to P.P. Martin for helpful comments on an early version of this paper. References 36 REFERENCES [1] G.M. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978), 178–218. [2] T. tom Dieck, Bridges with pillars: a graphical calculus of knot algebra, Topology Appl. 78 (1997), 21–38. [3] C.K. Fan, Structure of a Hecke algebra quotient, J. Amer. Math. Soc. 10 (1997), 139–167. [4] M. Geck and S. Lambropoulou, Markov traces and knot invariants related to Iwahori–Hecke algebras of type B, J. Reine Angew. Math. 482 (1997), 191–213. [5] M. Geck and G. Pfeiffer, Characters of finite Coxeter groups and Iwahori–Hecke algebras, Oxford University Press, Oxford, 2000. [6] Y. Gomi, The Markov traces and the Fourier transforms, J. Algebra 303 (2006), 566–591. [7] J.J. Graham, Modular representations of Hecke algebras and related algebras, Ph.D. thesis, University of Sydney, 1995. [8] J.J. Graham and G.I. Lehrer, Cellular algebras, Invent. Math. 123 (1996), 1–34. [9] R.M. Green, Generalized Temperley–Lieb algebras and decorated tangles, J. Knot Th. Ram. 7 (1998), 155–171. [10] R.M. Green, Tabular algebras and their asymptotic versions, J. Algebra 252 (2002), 27–64. [11] R.M. Green, On planar algebras arising from hypergroups, J. Algebra 263 (2003), 126–150. [12] R.M. Green, Generalized Jones traces and Kazhdan–Lusztig bases, J. Pure Appl. Alg. (to appear; math.QA/0509362). [13] R.M. Green, Star reducible Coxeter groups, Glasgow Math. J. 48 (2006), 583–609. [14] R.M. Green and J. Losonczy, Canonical bases for Hecke algebra quotients, Math. Res. Lett. 6 (1999), 213–222. [15] R.M. Green and P.P. Martin, Constructing cell data for diagram algebras, J. Pure Appl. Alg. (in press; math.RA/0503751). [16] J.E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cam- bridge, 1990. [17] V.F.R. Jones, Hecke algebra representations of braid groups and link polynomials, Ann. of Math. (2) 126 (1987), 335–388. [18] V.F.R. Jones, Planar Algebras, I (preprint). [19] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184. [20] S. Lambropoulou, Knot theory related to generalized and cyclotomic Hecke algebras of type B, J. Knot Th. Ram. 8 (1999), 621–658. [21] G. Lusztig, Cells in affine Weyl groups, Algebraic groups and related topics, Adv. Studies Pure Math 6, North-Holland and Kinokuniya, Tokyo and Amsterdam, 1985, pp. 255–287. [22] B.G. Seifert, The spherical trace on inductive limits of Hecke algebras of type A, B, C, D and factors, Quart J. Math. 41 (1990), 109–126. [23] J.Y. Shi, Fully commutative elements and Kazhdan–Lusztig cells in the finite and affine Coxeter groups, II, Proc. Amer. Math. Soc. 133 (2005), 2525–2531. [24] J.R. Stembridge, On the fully commutative elements of Coxeter groups, J. Algebraic Combin. 5 (1996), 353–385. [25] H.N.V. Temperley and E.H. Lieb, Relations between percolation and colouring problems and other graph theoretical problems associated with regular planar lattices: some exact results for the percolation problem, Proc. Roy. Soc. London Ser. A 322 (1971), 251–280. http://arxiv.org/abs/math/0509362 http://arxiv.org/abs/math/0503751

0704.0284

Second Order Perturbative Calculation of Quasinormal Modes of Schwarzschild Black Holes

hep-th Second Order Perturbative Calculation of Quasinormal Modes of Schwarzschild Black Holes Hsien-chung Kao Department of Mathematics, University of Durham, Durham, DH1 3LE, UK. 1 Department of Physics, National Taiwan Normal University, Taipei, Taiwan 116. [email protected] Abstract We analytically calculate to second order the correction to the asymptotic form of quasinormal frequencies of four dimensional Schwarzschild black holes based on the monodromy analysis pro- posed by Motl and Neitzke. Our results are in good agreement with those obtained from numerical calculation. 1on leave from National Taiwan Normal University. http://arxiv.org/abs/0704.0284v1 1 Introduction Quasinormal modes (QNMs) were originally observed in considering the scattering or emission of gravitational waves by Schwarzschild black holes [1]. It was found that a characteristic damped oscillation, which only depends on the black hole mass, dominated the time evolution in a certain period of time. Since then QNMs have been investigated extensively both analytically and numerically. For a general review and classification, see Refs. [2, 3]. From numerical studies, an asymptotic formula for quasinormal frequencies of Schwarzschild black holes was obtained [4]: 2GMωn ≈ 0.0874247+ i+O[n−1/2]. (1) The real part in the above formula was later postulated to be 1 ln 3 [5] based on a discrete area spectrum of quantum black holes proposed in Ref. [6]. This was confirmed later by Motl and Neitzke [7]. The recent surge of interest in the QNMs derived from its possible application in determining the Immirzi parameter in loop quantum gravity[8]. The numerical value ln 3 in the real part of the asymptotic quasinormal frequencies in Schwarzschild black holes was at first taken as a hint that the relevant gauge group in loop quantum gravity is SO(3) instead of the commonly believed SU(2). However, as shown in Ref. [7], the value ln 3 is not universal and one should take the argument with a grain of salt. Another interesting application of QNMs was pointed out by Horowitz and Hubeny in their study of a scalar field in the background of a Schwarzschild anti-de Sitter black hole [9]. According to AdS/CFT correspondence, a large black hole in AdS spacetime corresponds to a thermal state in CFT [10]. They argued the decay of the scalar field corresponds to the decay of a perturbation of this state. In the BTZ black hole, a one-to-one correspondence was found between the QNMs in the bulk and the poles of the retarded correlation function in the dual conformal field theory on the boundary [11]. The idea of dS/CFT correspondence has also been proposed and formulated [15]. Since there is a cosmological horizon in de Sitter spacetime, QNMs may also be defined in principle. Similar studies of QNMs have also been carried out in de Sitter spacetime trying to lent support for such correspondence [16]. However, the situation there is more subtle and it seems QNMs only exist in odd dimensions [3]. Therefore, it is not clear whether such correspondence makes sense in even dimensions, and further study is necessary. 2 Perturbative calculation of the asymptotic form of quasi- normal frequencies In Ref. [12], the author calculated the first order correction to the asymptotic form of quasinormal frequencies of a Schwarzschild black hole using a WKB analysis. The result was extended to include the scalar field case using the monodromy analysis developed by Motl and Neitzke [13]. The agreement with numerical results is excellent. We will begin with a brief review of their method which made systematic expansion more accessible. In a background spacetime described by a metric gµν , a massless scalar Φ satisfies the following Klein-Gordon equation: −g∂νΦ = 0. (2) For four dimensional Schwarzschild black holes, the metric is given by ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ2, with f(r) = (1− r0 ) and r0 = 2GM. Let Φ(r, t,Ω) = r φ(r)Ylm(Ω) e iωt. (3) φ(r) now satisfies the following equation: − f(r) d + V (r)φ = ω2φ, (4) V (r) = (1 − r0 l(l + 1) By a simple modification in the potential V (r) [2], V (r) = (1− r0 l(l + 1) (1− j2)r0 , (5) the previous equation can also describes linearized perturbation of the metric or an electromagnetic test fields. Here, j = 0, 1, 2 which is the spin of the relevant field. They can also be classified as the tensor, vector, and scalar types of perturbation to the background Schwarzschild metric using the master equations derived by Ishibashi and Kodama [14]. Introducing the tortoise coordinate: x(r) = r + r0 ln(r/r0 − 1), one obtain a Schrodinger-like equation + V [r(x)] φ = ω2φ. (6) Because of our convention in eq (3), QNMs are defined through the following out-going wave boundary condition: φ(x) ∼ eiωx as x → −∞ (horizon), e−iωx as x → ∞ (spatial infinity), assuming Reω > 0. Define f(x) = eiωx φ ∼ e2iωx as x → −∞, 1 as x → ∞. According to Ref. [7], the boundary condition at the horizon translates to the monodromy of f(x) around it M(r0) = e4πωr0 . (9) The same monodromy can also be accounted for by those around r = 0 and r = ∞, and it has been shown that only the former one is non-trivial. To find the monodromy around r = 0, one need to introduce the complex coordinate variable z = ω(x− iπr0) = ω[r + r0 ln(1− r/r0)], (10) which is vanishing at the black hole singularity r = 0. In the limit |r/r0| ≪ 1, the potential can be expanded as a series in z/(ωr0): V (z) = −ω 2(1− j2) 3l(l+ 1) + 1− j2 2(−ωr0)1/2z3/2 − 3l(l+ 1) + 1− j 2(−ωr0)3/2z1/2 + . . . . (11) Note that the third term in the above expression is of order (−ωr0)−3/2 and would not contribute until we consider third order perturbation. To second order in perturbation theory, the wavefunction can be expanded as φ = φ(0) + φ(1) + φ(2) +O(ω−3/2). (12) The zeroth, first and second order equations are given by dφ(0) 1− j2 φ(0) = 0; (13) dφ(1) 1− j2 φ(1) = −ωr0 δV (z)φ(0); (14) dφ(2) 1− j2 φ(2) = −ωr0 δV (z)φ(1), (15) respectively. Here, δV (z) = 3l(l+ 1) + 1− j2 2(−ωr0)1/2z3/2 . (16) Define φ (z) to be the two linearly independent solutions to the zeroth order equation (z) = J±j/2(z). (17) In the asymptotic region z ≫ 1 (z) ≈ cos[z − π(1 ± j)/4]. (18) It has been shown by Musiri and Siopsis that φ can be expressed in terms of φ + (z) = Cφ + (z) dz1 δV (z1)φ (z1)φ + (z1)− Cφ dz1 δV (z1)φ + (z1)φ + (z1); (19) (z) = Cφ + (z) dz1 δV (z1)φ (z1)φ (z1)− Cφ(0)− (z) dz1 δV (z1)φ + (z1)φ (z1). (20) where C = −ωr0/ sin(πj/2) [13]. Similarly, φ(2)± can in turn be expressed in terms of φ + (z) = Cφ + (z) dz2 δV (z2)φ (z2)φ + (z2)− Cφ dz2 δV (z2)φ + (z2)φ + (z2); (21) (z) = Cφ + (z) dz2 δV (z2)φ (z2)φ (z2)− Cφ(0)− (z) dz2 δV (z2)φ + (z2)φ (z2). (22) In the limit, z → ∞, (z) = c−± φ + (z)− c+± φ (z); (23) (z) = d−± φ + (z)− d+± φ (z). (24) Here, c±± = C dz1 δV (z1)φ (z1)φ (z1); (25) d±± = C dz1δV (z2) δV (z1)φ + (z2)φ (z1)− φ(0)− (z2)φ + (z1) (z1). (26) Notice that φ defined in eq (17) are in fact linearly dependent to each other when j is an even integer. As a result, each of these coefficients is divergent by itself in these cases. It is reassuring to see that all the divergent pieces cancel among themselves so that physically interested quantities do have a smooth limit when j is an even integer. In zeroth order, the combination φ(0)(z) = φ + (z)− e−iπ(j/2)φ (z) ∼ e−iz (27) in the asymptotic region z ≫ 1. This can be extended to second order φ(z) = + (z) + + (z) + + (z) − e−iπ(j/2) 1− ξ√ (z) + (z) + , (28) by introducing two parameters ξ and ζ. Naturally, they are determined by the condition that the coefficient of the eiz term is vanishing when z → ∞: ξ = ξ+ + ξ−; (29) ζ = −ξξ− + d++ eiπj/2 − d+− + d−− e−iπj/2 − d−+, (30) where ξ+ = c++ e iπj/2 − c+−, ξ− = c−− e−iπj/2 − c−+. (31) Substitute the above result back to eq (28), we have φ(z) = i eiπ(1−j)/4 sin(πj/2) e−iz 1− ξ−√ ξ(ξm + c+−)− d−− e−iπj/2 + d−+ , (32) where the identity c−+ = c+− has been used to simplify the expression. When going around the black hole singularity by 3π, φ and φ both pick up an extra phase: ( e3iπz) = e3iπ(2±j)/2φ (−z); (33) ( e3iπz) = e3iπ(3±j)/2φ (−z). (34) Consequently, φ( e3iπz) = e3iπ(1+j)/2 + (−z)− i + (−z)− + (−z) − e−iπ(j/2) 1− ξ√ e3iπ(1−j)/2 (−z)− i 1√ (−z)− 1 . (35) To second order, φ( e3iπz) = −i eiπ(1−j)/4 sin(3πj/2) e−iz (1 + ie3iπj)ξ+ + (1 + i)ξ−√ −ωr0(−1 + ei3πj) −(1 + i)ξ ξ− + [(1 + ei3πj)(d++ eiπj/2 − d−+) + 2d−− e−iπj/2 − 2d+−] −ωr0(−1 + ei3πj) + . . . , (36) where the term eiz is not relevant for our calculation and has been neglected. Taking the ratio between the coefficients of the term e−iz in eqs (36) and (32), we obtain the monodromy to second order: M(r0) = −[1 + 2 cos(jπ)] ∆2c +∆2d . (37) Here, (1 + ie3iπj)ξ+ + (i+ e 3iπj)ξ− (−1 + ei3πj) ; (38) ∆2c = −(1− i)ξ+ ξ− − ξc+−; (39) ∆2d = (1 + ei3πj)(d++ e iπj/2 + d−− e −iπj/2)− 2d+− − 2d−+ ei3πj (−1 + ei3πj) . (40) The terms ∆2c and ∆2d depend on coefficients cµν and dµν , respectively. Although our expression for ∆1 here is different from that in Ref. [13] by a phase factor, our final result is identical to their. Making use of the formula I1(µ, ν) ≡ dz z−1/2Jµ(z)Jν(z) = π/2Γ(1+2µ+2ν Γ(3−2µ−2ν )Γ(3+2µ−2ν )Γ(3−2µ+2ν , (41) one can obtain explicitly c++ = 3l2 + 3l+ 1− j2 ) Γ(1−2j ) Γ(1+2j ) sin[ π(1−2j) 2 sin( j π ; (42) c−− = 3l2 + 3l+ 1− j2 ) Γ(1−2j ) Γ(1+2j ) sin[ π(1+2j) 2 sin( j π ; (43) c+− = 3l2 + 3l+ 1− j2 ) Γ(1−2j ) Γ(1+2j ) sin[ π(1−2j) ] sin[ π(1+2j) 2 sin( j π . (44) Note that c−− = −c++(j → −j); c+− = −c−+(j → −j). (45) These relation are also obeyed by dµν ’s, which can be used to reduce our work. With the above results, we are ready to find ∆1 and ∆2c in eq (40): ∆1 = − i(3l2 + 3l + 1− j2) Γ2(1 ) cos( ) cos(jπ) 2π3/2[1 + 2 cos(jπ)] ; (46) ∆2c = − (3l2 + 3l + 1− j2)2 Γ4(1 ) Γ2(1−2j ) Γ2(1+2j ) cos(jπ) 1152π3 . (47) The double integral I2(µ2, ν2;µ1, ν1) ≡ dz1 z 1 Jµ2(z2)Jν2(z2)Jµ1(z1)Jν1 (z1) (48) can be expressed in terms of the generalized hypergeometric functions, but the general formula is quite complicated and not particularly illuminating. Therefore, we will just give the final result explicitly for the coefficients d++ and d+−: d++ = − 3l2 + 3l+ 1− j2 cot( jπ ) 5G4( , 1+j , 1−j , 1, 2+j , 2−j 576 sin2( j π 3l2 + 3l+ 1− j2 cot( j π ) cot(j π) Γ(1+2j ) Γ(1+2j ) Γ2(1+j 288 sin( jπ 1 + 2j 1 + j 1 + j 1 + 2j 2 + j 2 + j 5 + 2j , 1 + j; 1) 3l2 + 3l+ 1− j2 ) Γ2(1−2j ) Γ2(1+2j ) sin2[ π(1−2j) ] sin[ π(1+2j) 2π3 sin2( j π ; (49) d+− = − 3l2 + 3l+ 1− j2 ) 5G4( , 1+j , 1−j , 1, 2+j , 2−j 1152 sin3( j π 3l2 + 3l+ 1− j2 cot2( j π ) cot(j π) Γ(1−2j ) Γ(1−2j ) Γ2(1−j 1− 2j 1− 2j 5− 2j , 1− j; 1) 3l2 + 3l+ 1− j2 ) Γ2(1−2j ) Γ2(1+2j ) sin2[ π(1−2j) ] sin2[ π(1+2j) 2304 π3 sin2( j π . (50) Here, we have used the regularized generalized hypergeometric function 5G4(a1, a2, a3, a4, a5; b1, b2, b3, b4; z) so that the pole structure of each term in these expressions are more explicit. It is related to the usual generalized hypergeometric function by 5G4(a1, a2, a3, a4, a5; b1, b2, b3, b4; z) = 5F4(a1, a2, a3, a4, a5; b1, b2, b3, b4; z) Γ(b1)Γ(b2)Γ(b3)Γ(b4) . (51) The other two coefficients can be obtained by relations analogous to those in eq (45) d−− = −d++(j → −j); d−+ = −d+−(j → −j). (52) On the face of it, each of the dµν ’s has a third order pole coming from terms involving the generalized hypergeometric function when j is an even integer. On closer look, we see there are some cancelation among the divergences and in the end all they have are just simple poles in such limit similar to the cµν ’s. Another possible divergence arises in d−− when j = 1, which will again be canceled when we calculate the monodromy. It is now straightforward to obtain ∆2d by making use of the following two identities −4π7/2 cos2( ) cot(jπ)Γ( 1 + 2j 1 + 2j ) Γ2( 1 + j 1 + 2j 1 + j 1 + j 1 + 2j 2 + j 2 + j 5 + 2j , 1 + j; 1) +4π7/2 cos2( ) cot(jπ)Γ( 1 − 2j 1− 2j ) Γ2( 1− 2j 1− 2j 5− 2j , 1− j; 1) −Γ4(1 ) Γ2( 1 − 2j ) Γ2( 1 + 2j ) sin[ π(1− 2j) ] sin[ π(1 + 2j) ] = 0; (53) −4π7/2 cos(jπ 1 + 2j 1 + 2j ) Γ2( 1 + j 1 + 2j 1 + j 1 + j 1 + 2j 2 + j 2 + j 5 + 2j , 1 + j; 1) −4π7/2 cos( 1− 2j 1 − 2j ) Γ2( 1 − j 1− 2j 1− 2j 5− 2j , 1− j; 1); +8π5 Γ( ) 5G4( 1 + j 2 + j −Γ4(1 ) Γ2( 1 − 2j ) Γ2( 1 + 2j ) cos( )[1− cos(jπ)] = 0. (54) Eventually, we achieve the following nice result ∆2d = (3l2 + 3l+ 1− j2)2 Γ4(1 ) Γ2(1−2j ) Γ2(1+2j ) cos(jπ) 1152π3[1 + 2 cos(jπ)] , (55) where all divergences have been canceled out. Together with the result from eq (47), the asymptotic form of quasinormal frequencies of a four dimensional Schwarzschild black hole is found to be 4πωr0 = (2n+ 1)πi+ ln[1 + 2 cos(jπ)] i(3l2 + 3l + 1− j2) Γ2(1 ) Γ(1−2j ) Γ(1+2j ) cos( jπ ) cos(jπ) 2π3/2[1 + 2 cos(jπ)] (3l2 + 3l+ 1− j2)2 Γ4(1 ) Γ2(1−2j ) Γ2(1+2j ) cos2(jπ) 576π3[1 + 2 cos(jπ)]2(−ωr0) +O[(−ωr0)−3/2]. (56) The physically interested cases are ≈ (2n+ 1)πi+ ln 3 + 1− i√ (l2 + l − 1)Γ4(1/4) 2π3/2 (l2 + l − 1)2Γ8(1/4) 2592π3 , for j = 2; (57) ≈ (2n+ 1)πi+ ln 3 + 1− i√ (l2 + l + 1/3)Γ4(1/4) 2π3/2 (l2 + l+ 1/3)2Γ8(1/4) 288π3 , for j = 0; (58) ≈ 2nπi+ i2π(l 2 + l)2 , for j = 1. (59) A few comments are in order. First, all the second order corrections are purely imaginary. In particular, when j = 2 (gravitational perturbation) the numerical coefficients of the i/n term (after divided by 4π) are 0.739, 3.58, 49.7 for l = 2, 3, 6, respectively. They are in good agreement with the known numerical studies [4]. As for the real part, our result predicts vanishing correction. For j = 2, this is again consistent with the numerical results in Ref. [4] for l = 2, 3. For l = 6 the numerical result is 0.263, which seems to be contradictory to ours. However, the numerical value for l = 6 has opposite sign relative to those of l = 2, 3. This is peculiar, since in all other cases a given type of corrections are always of the same sign irrespective of the specific value of angular momentum. Therefore, we believe more study is needed to clarify whether there is really a discrepancy. As for the j = 1 case, the numerical study in Ref. [17] suggests the leading correction is of the form b . However, this does not necessarily mean the two results are inconsistent. In fact, one can only extract the behavior of the leading correction to the real part from their Fig. 2 and further numerical study is needed to confirm or refute our prediction. 3 Conclusion In sum, we have calculated to second order the correction to the asymptotic form of quasinormal frequencies for Schwarzschild black holes in four dimensions. Most of our results are consistent with the numerical ones when available. In cases where there seem to be contradiction, we think further numerical studies are needed to clarify the situation. It would also be helpful if more detailed numerical studies can be carried out for the j = 0 case so that more thorough comparisons are possible. It would be interesting to generalize the method to other spacetime backgrounds [18]. Extension to higher order is also desirable. It might enable us to find a quantitative prediction for the ”algebraically special” frequencies in Schwarzschild black holes, where the quasinormal frequency is purely imaginary and it increases with the fourth power of l [19, 4]. Acknowledgment The author thanks Chong-Sun Chu for helpful discussions. The work is supported in part by the National Science Council and the National Center for Theoretical Sciences, Taiwan. References [1] C. V. Vishveshwara, ”Scattering of gravitation radiation by a Schwarzschild black-hole,” Nature 227, 936 (1970); W. H. Press, ”Long Wave Trains of Gravitational Waves from a Vibrating black hole,” Astrophys. J. Lett. 170, L105 (1971). [2] H. P. Nollert, ”Quasinormal modes: the characteristic ’sound’ of black holes and neutron stars,” Class Quantum Grav. 16, R159 (1999); K. D. Kokkotas and B. G. Schmidt, ”Quasi-normal modes of stars and black holes,” Living Rev. Relativ. 2, 2 (1999) [gr-qc/9909058]. http://arxiv.org/abs/gr-qc/9909058 [3] J. 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Shu http://arxiv.org/abs/gr-qc/0303095 http://arxiv.org/abs/hep-th/0308168 http://arxiv.org/abs/hep-th/0305147 http://arxiv.org/abs/hep-th/0305185 http://arxiv.org/abs/hep-th/0308128 http://arxiv.org/abs/hep-th/0106109 http://arxiv.org/abs/hep-th/0106113 http://arxiv.org/abs/hep-th/0106247 http://arxiv.org/abs/hep-th/0204030 http://arxiv.org/abs/hep-th/0208065 http://arxiv.org/abs/hep-th/0304231 http://arxiv.org/abs/gr-qc/0311064 http://arxiv.org/abs/gr-qc/0309112 http://arxiv.org/abs/hep-th/0511113 and Y.-G. Shen, ” it Perturbative Calculation of Quasinormal Modes of d-Dimensional Black Holes,” JHEP 0608, 087 (2006) [hep-th/0605128]. [19] S. Chandrasekhar, ”On Algebraically Special Perturbations of Black Holes,” Proc. R. Soc. London A392, 1 (1984). http://arxiv.org/abs/hep-th/0605128 Introduction Perturbative calculation of the asymptotic form of quasinormal frequencies Conclusion

0704.0285

Epitaxial graphene

Epitaxial graphene Walt A. de Heer a,∗ Claire Berger a,b Xiaosong Wu a Phillip N. First a Edward H. Conrad a Xuebin Li a Tianbo Li a Michael Sprinkle a Joanna Hass a Marcin L. Sadowski b Marek Potemski b Gerard Martinez b aSchool of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USA bGrenoble High Magnetic Field Laboratory, CNRS, Grenoble, France Abstract Graphene multilayers are grown epitaxially on single crystal silicon carbide. This system is composed of several graphene layers of which the first layer is electron doped due to the built-in electric field and the other layers are essentially undoped. Unlike graphite the charge carriers show Dirac particle properties (i.e. an anoma- lous Berry’s phase, weak anti-localization and square root field dependence of the Landau level energies). Epitaxial graphene shows quasi-ballistic transport and long coherence lengths; properties which may persists above cryogenic temperatures. Paradoxically, in contrast to exfoliated graphene, the quantum Hall effect is not ob- served in high mobility epitaxial graphene. It appears that the effect is suppressed due to absence of localized states in the bulk of the material.Epitaxial graphene can be patterned using standard lithography methods and characterized using a wide array of techniques. These favorable features indicate that interconnected room temperature ballistic devices may be feasible for low dissipation high-speed nano- electronics. Key words: PACS: ∗ Corresponding author. Email address: [email protected] (Walt A. de Heer). Preprint submitted to Solid State Communications 12 February 2013 http://arxiv.org/abs/0704.0285v1 1 Introduction Carbon nanotubes are prototypical of quasi-one dimensional graphene nanos- tructures. The approximate electronic structure of a carbon nanotube with diameter D is understood starting from the graphene dispersion relation, i.e. the Dirac cone E = ~v0|k|, and quantizing the angular momentum about the axis so that En = ~v0 k2z + k n, where kn = (n + a)π/R, where a is 0 or 1/2 depending on whether the nanotube is metallic or semiconducting. A metal- lic nanotube has two dispersionless bands that cross the Fermi level while a semiconducting nanotube has a band-gap Eg = γ0a0/R ∼ 0.4eV·nm/R. This property, that graphene nanostructures can be metallic or semiconducting de- pending on their shape carries over to nanopatterned graphene ribbons as shown below. High purity multiwalled carbon nanotubes (as well as single walled nanotubes) were found to be room-temperature ballistic conductors [1]. This property re- quires (at least) that electrons traverse the length of the nanotube without scattering. This discovery coincided with predictions of the effect by Ando [2,3], and by Todorov and White [4] who demonstrated that the chiral nature of the charge carriers in nanotubes inhibits backscattering [in all graphene structures (including graphene), chirality results from the equivalence of the A and B sub-lattices]. Ando first recognized the formal analogy between neu- trino wave functions and those that describe electrons near the Fermi level in nanotubes (and in graphene). Neutrinos are massless fermions that are de- scribed by the Weyl’s equation (or massless Dirac equation) [3]. The quantum number associated with chirality is the pseudospin which, like spin, can have two values. Unlike spin, the pseudospin is coupled to the momentum. In order to backscatter an electron, the scattering potential must reverse both the mo- mentum and the pseudospin. Interactions that act equivalently on A and B atoms (like long-range potentials) conserve pseudospin and cannot backscatter charge carriers. Ballistic conduction is only one of the favorable electronic properties of carbon nanotubes. Others are the extremely weak electron-phonon coupling [1,5], the excellent FET characteristics [6], and the robustness of the material itself. All of these properties indicate that nanotubes could be used for nanoelectronics. Unfortunately, incorporation of nanotubes in large-scale integrated electronic architectures proves to be so daunting that it may never be realized. Harness- ing these properties requires graphitic materials that are related to carbon nanotubes, but which are more manageable. Precisely these theoretical considerations led us in 2001 to speculate that 2D graphene could serve these purposes. We initiated experiments on epitaxially grown graphene on single crystal silicon carbide. Much of the earlier efforts fo- cused on producing and characterizing the epitaxial graphene material. While we have achieved some success, much work remains. To fully exploit the prop- erties of nanopatterned epitaxial graphene, one must control the graphene material, its structure, and the chemistry and morphology of defined edges. These are the challenges for graphene-based nanoelectronics. The most im- portant feature of 2D epitaxial graphene is that interconnected structures can in principle be patterned on the scale of an entire wafer. If, like carbon nanotubes, the carriers remain ballistic, it will lead to a fascinating world of coherent carbon-based electronics. The discovery of the intriguing properties of deposited exfoliated graphene has recently caused overwhelming excitement in the 2D electron gas community [7,8,9,10]. This very fascinating material clearly demonstrates the chiral na- ture of the charge carriers, as it manifests in several properties, of which the anomalous phase in the quantum Hall effect is the most striking. The spon- taneous rippling caused by the Mermin-Wagner transition [11,12,13] and the absence of the weak anti-localization, possibly due to the gauge field at the ripples [14], as well as the recently discovered high-field splitting of the Landau levels [15] are all very important effects that still require full explanation. The possibility that epitaxial graphene may serve as a platform for carbon- based nanoelectronics has further greatly amplified the interest in this field, especially in the electronics community. However, epitaxial graphene and de- posited exfoliated graphene are very different materials. Epitaxial graphene is generally multi-layered whereas exfoliated graphene has only one layer. There- fore, epitaxial graphene, is a much more complex material; in fact it represents a class of materials. It may seem that epitaxial graphene is simply ultrathin graphite, but this is emphatically not so. Experimentally, the charge carri- ers in epitaxial graphene are found to be chiral and the band structure is clearly related to the Dirac cone [16,17,18,19,20,21]. To lowest order, epitaxial graphene appears to consist of stacked, non-interacting graphene sheets, the first of which is highly charged and the others carry much lower charge. In con- trast to deposited exfoliated graphene, anomalous phase-transition-like state- changes are often observed in transport measurements of epitaxial graphene, that are probably related to weak interlayer interactions. These first measurements suggest that, like most layered quasi-2D conducting materials, epitaxial graphene is poised to present a host of interesting new phenomena. A snapshot of the emerging science and technology of epitaxial graphene is given here. 0 1 2 3 4 5 6 Number of graphene layers 100 200 300 400 500 Energy(eV) (a) Si:C=0.75 (b) Si:C=1.59 (c) Si:C=0.14 Fig. 1. Model of Si:C Auger peak intensity ratio versus number of graphene layers for SiC(0001) substrates. Solid line: Model with interface layer of C adatoms at 1/3 their bilayer density. Dotted line: Model with interface layer of Si adatoms at 1/3 their bilayer density. Dashed line: Model with bulk-terminated SiC(0001). Inset shows Auger spectra obtained after (a) ex-situ H2 etching (no UHV preparation), (b) UHV anneal at 1150◦C (LEED 3 pattern), (c) UHV anneal at 1350◦C (LEED 6 3 × 6 3 pattern). 2 Epitaxial graphene formation and characterization It is well known that ultrathin graphitic films grow on hexagonal silicon carbide crystals [22,23,24,25,26]. Specifically they grow on the 0001 (silicon- terminated) and 0001 (carbon-terminated) faces of 4H- and 6H-SiC when crys- tals are heated to about 1300◦C in ultra-high vacuum (UHV). It is also possi- ble to grow these films at more moderate vacuum conditions using ovens with controlled background gas. The epitaxial growth is established by examining, for example, the LEED patterns after various growth times (see e.g., Fig. 3). Growth on the Si face is slow and terminates after relatively short times at high temperatures. The growth on the carbon face apparently does not self-limit so that relatively thick layers (∼ 4 up to 100 layers) can be achieved. For thin layers, we can estimate the graphene thickness by modeling mea- sured Auger-electron intensities [16] or photoelectron intensities [17]. Fig. 1 shows model results for the Si:C Auger intensity ratio for graphene grown on SiC(0001) substrates, with three different assumptions for the interface layer between bulk SiC and the graphene layers (see caption). The Auger model, valid for both 4H and 6H polytypes, includes the relative sensitivity factors for Si and C [27], attenuation of the 3keV incident electrons and of the Auger electrons exiting from successively deeper layers [28,29,30], and the electron collection angle (42◦). Thicker multilayer graphene can be measured via con- ventional ellipsometry. Scanning tunneling microscopy images of monolayer graphene on the surfaces of 4H- and 6H-SiC(0001) (Si-face) show large flat regions with a characteristic Fig. 2. STM topographs (0.8 V sample bias, 100 pA) of nominally 1 ML epitaxial graphene on SiC(0001). Top: Image showing large flat regions of 6 3 × 6 3 re- construction and regions where the reconstruction has not fully formed. Next-layer islands are also seen. Bottom: A region of 6 3 reconstruction, imaged through the overlying graphene layer. hexagonal corrugation of ∼ 0.3 Å on a 1.9-nm period (Fig. 2). Small-scale images resolve the graphene atomic lattice throughout [16,23], but with a factor 10×-20× smaller amplitude. Imaging for the monolayer is apparently dominated by interface states of an underlying reconstruction of the SiC. In conjunction with the graphene overlayer is a 6 3 × 6 3R30◦ reconstruction with respect to the bulk-terminated SiC surface. The detailed reconstruction of this surface is still a matter of debate [31]. Successive graphene layers show much less influence of the interface states [16], but the 1.9-nm corrugation period (6 × 6 with respect to the SiC bulk-terminated surface) is still visible in both STM and LEED for the thickest Si-face films we have prepared [5-6 monolayers (ML)]. To date, most transport measurements have been done on multilayer graphene grown on the carbon face [SiC(0001) substrates]. This material is grown in an RF-induction furnace at pressures of ∼ 10−5 Torr. Because the initial film- growth is very rapid, it is rare to obtain films thin enough for direct STM and LEED studies of those layers near the SiC interface. As a consequence of charge transfer from the SiC, these layers are the most important for electrical transport. Surface x-ray scattering has proved to be a useful tool for extracting quantitative information about the C-face-grown material. Figure 3 shows LEED patterns from two graphene films grown on 4H-SiC(0001) substrates. According to the Auger ratios, these were nominally (a) 3 ML graphene, and (b) 4 ML graphene. The LEED pattern in Fig. 3(a) shows rel- = 0.022Å-1 = 0.003Å-1 (Å-1) 2.90 2.92 2.94 2.96 2.98 3.00 Si-Face C-Face Graphite (011) rotated 32.2o = 0.005Å-1 φφφφ (degrees) -3 -2 -1 0 1 2 3 -2.2o 2.2o φφφφ (degrees) -33 -32 -31 -30 -29 -28 -27 (011) Graphite Graphite [01 ] (c) (d) (a) (b)Graphite [1100] [1000] Fig. 3. LEED and x-ray diffraction from multilayer graphene grown on 4H-SiC(0001) substrates. (a) LEED pattern (71 eV) for ∼ 3 ML graphene, (b) LEED pattern (103 eV) for ∼ 4 ML graphene (unlabeled sets of 6-fold spots in (a) and (b) are from a√ 3R30◦ SiC interface reconstruction). (c) Radial x-ray scans through (top) the (10ℓ) graphite rod, and (bottom) across the diffuse arcs seen in (b). (d) Azimuthal x-ray scans across (top) the graphite (10ℓ) rod and (bottom) the diffuse rods seen in (b). atively good registry to the SiC substrate (with the unit cell rotated by 30◦, as for Si-face material), whereas the film in Fig. 3(b) shows some rotational disorder. The evidence suggests that epitaxial growth does occur at the inter- face, but that succeeding graphene sheets do not have strong rotational order. Interestingly, the diffuse rings in Fig. 3(b) are clearly centered around a mini- mum in intensity on the SiC azimuth, indicating some preferential alignment, as discussed below. While there is azimuthal disorder in the film, the long range vertical order of the film is much larger than is observed for Si-face grown films [32]. This is demonstrated in Fig. 3(c) that shows radial x-ray diffraction scans through both the graphite (10ℓ) graphite rod (φ = −30.0◦ in the [1100] SiC direction) and through the diffuse rings (φ = 2.2◦ in the SiC [1000] direction). The x- ray profiles for both the (10ℓ = 1.5) and diffuse rods on the C-face graphene are nearly 10 times narrower than those for Si-face films. The profile widths are inversely related to the size of order graphene domains; L = 2π/∆qr. For Si-face films the order graphene regions are ∼ 290Å while for the C-face films the domains are ∼ 2100Å. The domain size estimated this way is most likely a lower limit on the actual size of a graphene sheet. A continuous graphene sheet (typically 3000Å terrace width) folded over a SiC step would break the scattered x-ray coherence from the two regions, but may have a much smaller influence on the electronic structure. Note that even the diffuse rings have domain sizes of ∼ 1200Å. In fact, the rotationally disordered graphene has a structure. Fig. 3(d) shows x- ray azimuthal scans through both the graphite (10ℓ) and diffuse graphite rods. The diffuse rings are in fact peaked at ±2.2◦ relative to the SiC azimuth. This angle is not arbitrary. It corresponds to a structure were two vertically stacked graphene sheets are commensurate if rotated with respect to one another by cos 11/13 = 32.204◦ [33]. Both 30◦ and ±2.204◦ rotated graphene are also nearly commensurate with the SiC 6 3 × 6 3 R30 seen in Si-face grown graphene [see Fig. 3(a)] It therefore seems that during graphitization large graphene sheets are free to rotate with respect to each other and lock in, on average, to these preferred orientations on the SiC C-face. In addition to the difference in long range and orientational order of films grown on the two polar faces of SiC, the vertical roughness of the multilayer graphene is very different. X-ray diffraction reveals that the rms roughness of the C-face multilayer films is less than 0.05 Å over the 2 µm coherence length of the beam [34]. On the Si-face the roughness is much larger (∼ 0.2Å [35]), presumably as a consequence of the 6 × 6 corrugation (see Fig. 2). Finally, x-ray reflectivity experiments show two other important features of multilayer graphene grown on the C-face of SiC. First, the first layer of graphene sits 1.62Å above the last SiC layer [34,36]. This bond length is nearly equal to the bond length of diamond (1.54Å) and suggest that the substrate bond to the first graphene layer is much stronger than a van der Waals in- teraction. In fact ab intio calculations find and very similar bond distance [36]. These calculations show that the first graphene layer is in fact insulat- ing. Only the formation of the second graphene layer gives rise to an electron dispersion curve showing a Dirac cone. Thus the first graphene layer can be interpreted as a “buffer” layer between the substrate and an isolated layer with the electronic properties of an isolated graphene sheet. The second important result from the x-ray reflectivity is that the graphene interlayer spacing is significantly larger than bulk graphite [34]. The measured value is 3.368Å which is between the value of bulk graphite and turbostratic graphite. This larger spacing suggest a significant density of stacking faults. This is not too surprising given the rotational disorder in the C-face films. For a random stacking fault model the layer spacing can be used to estimate the stacking fault density to be one every other layer [34,37]. This type of density suggest that the AB stacking order, that would destroy the graphene electronic character, is nearly lost in these films and may significantly impact the transport properties of these films. Fig. 4. Infrared transmission spectroscopy of epitaxial graphene with about 10 layers revealing Landau level structure. (a) Infrared transmission spectrum at B = 0.4 T and T = 1.9 K, showing a series of absorption peaks. (Inset) The absorption maxima positions as a function of field showing the B dependence that is characteristic for a chiral “massless” Dirac particle. (b) Schematic diagram of the Landau levels En(B) in which the only parameter is v0 that is found to be 10 8 cm/s. The arrows indicate the observed transitions. EF is determined from the lowest field for which the n = 0 to n = 1 transition is observed. 3 Landau level spectroscopy of epitaxial graphene Dirac particle properties of the charge carries in epitaxial graphene multilayers have been beautifully demonstrated in Landau Level spectroscopy by Sadowski et al. [20]. (See Sadowski et. al. in this issue for a summary and update). We summarize some of the results here. In these measurements, an epitaxial graphene sample is illuminated by infrared light in a magnetic field at low temperatures. The absorption is measured as a function of photon energy at various magnetic field strengths. An example of such a spectrum is shown in Fig. 4. The various absorption lines are identified as transitions between various Landau levels. The transitions energy are found to accurately follow En = v0 2ne~B. The exact B dependence is the hallmark of a ”massless” Dirac particle (more precisely, of a linear density of states); massive particles have a linear B dependence. Moreover, a gap at the tip of the Dirac cone also distorts the B behavior. The Fermi velocity is determined from the dispersion of the transitions with magnetic field to be v0 = 1.03 × 108 cm/s, which is close to its value for exfoliated graphene. The n = 0 to n = 1 transition is observed only for B ≥ 0.16 T, which indicates that the n = 1 level is just depopulated at that field. Hence, -15 meV < EF < 15 meV and n ≈ 1.5×1010 /cm2 and the Fermi wavelength is ≈ 300 nm. It is further found that the intensity of the signal scales with the thickness of the film. These experiments demonstrate that epitaxial graphene consists of stacked graphene layers, whose electronic band structure is characterized by a Dirac cone with chiral charge carriers. Remarkably, there is no evidence for a gap nor for a SiC with scratches Flatten SiC by H2 etching Graphitization Deposit contacts E-beam lithography DevelopO2 plasma Lift off HSQ Bare SiC HSQ Exposed Graphene Metal contact Resist spin- coating bonding Fig. 5. Patterning epitaxial graphene deviation of the linear density of states: undistorted Dirac cone properties are directly observed as close as 20 meV to the Dirac point in the n = 0 − n = 1 transitions. Epitaxial graphene is clearly not graphite, which has a different spectrum and an entirely different electronic structure (see Sadowski et al. in this issue). This difference reflects that epitaxial graphene does not have the Bernal stacking that would lift the pseudospin degeneracy [34]. Hence epitaxial graphene is a form of multilayered graphene that is structurally and electronically distinct from graphite. These experiments probe the low charge density bulk of the epitaxial graphene layer. Below we discuss the highly charged interface layer. 4 Patterning epitaxial graphene Epitaxial graphene samples are patterned using a variety of microelectronics patterning methods. Features down to several tens of nanometers are produced by standard e-beam lithography methods. The method is outlined in Fig. 5. 5 Transport in 2D epitaxial graphene The first published transport measurements on epitaxial graphene were made on a Hall bar patterned on a graphene film with about 3 layers on the sili- con face of 4H-SiC [16]. The mobility of the sample was relatively low (1100 cm2/V·s) nevertheless the Shubnikov-de Haas oscillations are clearly distin- guished (see Fig. 6) [38]. Resistance maxima in graphene are expected at fields Bn when the Fermi energy intercept the Landau levels, i.e. for EF = 2ne~Bn, where v0 ≈ 108 cm/s is the Fermi velocity, hence Bn = (EF /v0)2/2ne~ = B1/n. For normal electrons maxima are found when EF = (n + 1/2)eBn~/m, Fig. 6. 2D transport measured in a 400 µm by 600 µm Hall bar on 3 layer epitaxial graphene on the Si face. Mobility µ = 1200 cm2/V·s, coherence length lφ = 300 nm. (a) Magnetoresistance at T =0.3, 2 and 4 K showing well developed SdH peaks, indicated with their Landau indices n; the Hall resistance at 0.3 K (dashed line), shows a weak feature at the expected Hall plateau position. The amplitude of the weak localization peak at B = 0 corresponds to 1G0. (b) Landau plot; the linear extrapolation passes through the origin demonstrating the anomalous Berry’s phase characteristic of graphene. (c) The Lifshitz-Kosevich analysis of the n = 2 and n = 3 peaks which correspond to graphene with a Fermi velocity vF = 7.2 × 105 cm/s. hence Bn = EF m/(n + 1/2)~e. Therefore the Landau plot (a plot of n ver- sus 1/Bn) of a Dirac particle intercepts the origin whereas the Landau plot of a normal electron intercepts the y axis at n = 1/2. The intercept should occur at 0 when the Berry’s phase is anomalous. This shows that the Landau plot provides a ready method to identify a Dirac particle when the quantum Hall measurements are not feasible. The Landau plot (Fig. 7) for data on a sample similar to the one of [16] passes through the origin indicating that the Berry’s phase is anomalous. The Hall coefficient at 0.3 K is found to be 330 Ω/T corresponding to a charge density of 2 × 1012 electrons/cm2. (Note that for a Dirac particle it should be 6500/B1 = 450 Ω/T.) From v0 = 10 8 cm/s we further find that EF ≈ 1680 K. The large charge density is caused by the built-in electric field at the SiC-graphene interface, which dopes the interfa- cial graphene layer. This layer carries most of the current (and causes the SdH oscillations). The charge density of the top layers is more than 2 order of mag- nitude smaller (see above) and they are expected to be much more resistive. The temperature dependence of the SdH peak amplitudes is determined by the Landau level spacing En+1(B)−En(B) and given by the Lifshitz-Kosevich equation: An(T ) ∼ u/ sinh(u) where u = 2πk2BT/∆E(B) [39]. From this fit we find that at B = 7 T, (E3(B) − E2(B))/kB = 250 K (compared with 340 K predicted for graphene at this carrier density) and that the Dirac point is about 1290 K below EF . This sample shows ample evidences that the carriers in the high-charge-density layer, like those in the low-density layers, are Dirac electrons. However the quantum Hall effect is not observed. Instead, only weak undulations are seen in the Hall resistance. It was assumed that higher mobility samples would enhance the QHE and subsequent work progressed in that direction. Note also the intense weak localization peak near B = 0 indicative of significant point- defect scattering. Due to the high current density, the interface graphene layer dominates the transport, although the other layers are expected to contribute, and more so in 2D structures than in quasi 1D structures (see below). Graphene grown on the Si face typically has low electron mobilities. The very thin films are relatively unprotected from even slight residual oxidizing gases that damage the graphene [32]. Work is still progressing to improve Si face graphene films. On the other hand, graphene grown on the C face has much higher mobili- ties [18]. The films are also considerably thicker so that the high-density layer at the interface is more protected [34]. Fig. 7 shows the MR measurements of a Hall bar (100 µm ×1000 µm) at several temperatures [21]. The SdH oscillations are barely discernable, which is generally the case for our high mobility 2D samples. The reason for this is not likely due to sample inho- mogeneity. The Landau plot of the oscillations reveals the anomalous Berry’s phase, characteristic of Dirac electrons. Furthermore the charge density is 3.8× 1012 electrons/cm2. The charge density from the Hall effect is 4.6× 1012 electrons/cm2. The Lishitz-Kosevich analysis of the peak heights agrees with the expected Landau level spacing for a Dirac particle. A striking feature of this sample is that the weak localization peak is very weak, ∼ 0.07G0 (compared with the sample in Fig. 6) which indicates that point defect density in this sample is low and these defects are possibly lo- calized entirely at the patterned edges of the Hall bar. On the other hand, a marked temperature dependent depression of the conductance at low fields is observed. This feature suggests weak anti-localization that is expected when Dirac electrons are scattered by long-range potentials [2,3]. These could be due to the localized counterions in the SiC substrate. In fact the amplitude, field and temperature dependence of this feature match predictions of the weak anti-localization very well [40]. Another typical feature is the large positive magnetoresistance and a kink in the Hall resistance at low fields. These features (as well as the small discrep- ancy in the charge density) could be due to the other layers of density n . 1010 −5 0 5 B (T) −0.02 0 0.02 −0.02 0 0.02 B (T) 3 4 5 6 7 8 9 B (T) 0 0.1 0.2 0.3 0.4 1/B (T−1) 0 2 4 6 8 B (T) (a) (b) (d) (e) Fig. 7. 2D transport in a 100 µm × 1000 µm Hall bar on a ∼10 layer eptitaxial graphene film on the C face. a) Resistance as a function of the magnetic field. Inset, dash-dot lines, low field MR at various temperatures (1.4, 4.2, 7, 10, 15, 20, 30, 50 K). (b) Low field MR after subtracting 50 K data as a background. dash-dot lines, experimental data, which show suppressed weak localization peak around zero. The positive MR above 0.02 T reveal the weak anti-localization effect. Solid lines, fits to the theory by McCann et al.. (c) High field MR after subtracting a parabolic background at several temperatures(4, 7, 15, 30 K). Well defined SdH oscillations can be seen down to 2.5 T. (d) Landau plot for SdH oscillations, which intercept y axis at zero. (e) Landau level spacing obtained by Lifshitz-Kosevich analysis. Squares: experiment. Solid line, theoretical prediction for ∆E assuming vF = 0.82 × 108 cm/s, dash-dot line: vF = 108 cm/s. /cm2 [20], although no SdH features can be attributed to them. It should be noted that the critical field Bc for which extreme quantum limit is reached (where EF coincides with the n = 0 Landau level, i.e. at about 30 meV above the Dirac point) is also very low: Bc ≤ 160 mT (see Fig. 4). The Hall resistance is featureless (except for extremely weak ripples) and shows no evidence for quantum Hall plateaus, as is for a typical high mobility 2D samples. The transport properties of a narrower ribbon are shown in Fig. 8. It is at once clear that the SdH oscillations are much more pronounced. The Landau plot corresponds quite well with the expectations for a Dirac particle with a velocity 0.7×108 cm/s. This ribbon shows evidence for weak anti-localization. A more pronounced weak localization peak compared with Fig. 7 is observed. 3 4 5 6 7 8 9 B (T) 0 0.1 0.2 0.3 (T−1) 0 5 10 B (T) (b) (c) Fig. 8. Intermediate width Hall bar: 1 µm × 5 µm. The zero field resistance is 502 Ω. (a) High field MR after subtracting a smooth background at several tempera- tures(4, 10, 20, 30, 50, 70 K). (b) Landau plot. B1 = 53T, intercept 0.13±0.02 (c) Square: Landau level spacing ∆E obtained by fitting the temperature dependence of SdH amplitudes to LK equation. Solid line, theoretical prediction for ∆E assuming vF = 0.7 × 108 cm/s, dash-dot line: vF = 108 cm/s However the Hall resistance, which is quite similar to that in Fig. 6, shows no evidence for the QHE. 6 Transport in quasi-1D epitaxial graphene Quantum confinement effects manifest in narrow ribbons. As for 2D Hall bars, this interface graphene layer is charged with about 4 × 1012 electrons/cm2 which corresponds to a Fermi wavelength of about 20 nm. Since the Fermi wavelength of the low-density layers is about 400 nm, consequently for ribbons that are narrower than 500 nm, these layers contribute little to the transport. For very narrow ribbons (≤ 100 nm) with rough edges, the low-density layers are expected to be insulating, since there are no propagating modes (channels). Figure 9 shows the Hall resistance and the magnetoresistance of a narrow ribbon (see Ref. [18] for details). The Landau levels for a graphene ribbon are approximately given by En(B, W ) ≈ [En(W )4 + En(B)4]1/4 (1) where EB(n) = 2neBv20~ and EW (n) = nπ~v0/W [41]. Confinement effects become apparent for low fields, approximately when the cyclotron diameter becomes greater than the ribbon width. Confinement will then cause deviation Fig. 9. Narrow Hall bar 500 nm × 6 µm. The zero field resistance is 1125 Ω. (a) Magnetoresistance oscillations for temperatures ranging from 4-58 K after subtrac- tion of a smooth background. (b) Landau plot of the magnetoresistance peaks. The deviation for large from linearity is due to quantum confinement. (c) The energy gap between the Fermi level and the lowest unoccupied Landau level is found from the Lifshitz-Kosevich analysis (inset) of the peaks and increases linearly with field for large fields and saturates for low fields. The saturation confirms quantum con- finement. from the linearity in the Landau plot as seen in Fig. 9. The Lifshitz-Kosevich analysis confirms the confinement. For high magnetic fields the energy sepa- ration between the Landau levels increases with increasing field as expected, while for low field the energy separation saturates and is determined by the quantum confinement. Note that this analysis does not require a determination of the locations of the magnetoresistance peaks (Ref. [18]). The mobilities of the graphene ribbons appears to increase with decreasing width, Fig. 11. This effect may be related to the reduced back-scattering with decreasing number of conducting channels. On the other hand, back-scattering at the ribbon edges should become relatively more important with decreasing width. The amplitudes of the SdH oscillations are much more pronounced for narrow ribbons than for high mobility 2D Hall bars. A relatively large fraction of the high-mobility narrower Hall bar samples do not exhibit SdH oscillations at all, as seen in Fig. 10. Occasionally rather complex magnetoresistance structures that in many cases appear not to be random but exhibit features that are approximately linear in field (like in the Aharonov-Bohm effect). Several of these systems are found to be coherent and ballistic. In one case the resistance of a 0.5× 5 µm Hall bar abruptly and reversibly drops by an order of magnitude at T = 200 K to below 10 Ω/sq. It appears that scattering at the edges is specular without any back-scattering. 0 2 4 6 8 B (T) Fig. 10. Magnetoresistance of a 0.2 µm × 1 µm ribbon. The experiment were done at 4, 8, 12, 30, 45, 60, 90 K, from top to bottom. The resistance has been shifted for clarity, except for 4 K. The amplitude of the weak localization peak at zero field is about 1G0. 10 -1 100 101 102 103 Width (micron) mobility @4K with Ns=3.4 1012/cm2 10 -2 10-1 100 101 102 Width ( µm ) T=4 K T=250K Width (µm) Width (µm) Fig. 11. The width dependence of mobility. The effects point to a correlated electronic system (Levy, Berger, de Heer et al., to be published). 7 Structure dependent properties and the absence of the quantum Hall effect A key focus of epitaxial graphene research is to develop a new graphene- based electronics material with shape tunable properties. The intrinsic width dependent bandgap of graphene ribbons has been born out experimentally in back-gated deposited exfoliated graphene ribbons [42]. We have not yet demonstrated the effect in epitaxial graphene, primarily due to problems in gating the material, which we hope to solve soon. Currently we have reasonable statistics that appear to suggest that the mobil- ities of the ribbons actually increase with decreasing ribbon width (Fig. 11). This intriguing property could be due to the fact that the system becomes more one-dimensional with decreasing width and thereby that backscattering is inhibited. On the other hand, the decreasing width also implies that the edges (which are presumed to be rough) become more important and enhance the scattering. Apparently that effect is not dominant. It is remarkable that the SdH oscillations are extremely weak except for very low mobility samples, that are known to be quite defective (as in Fig. 6). In fact the SdH oscillations are almost imperceptible in the 2D sample (the amplitudes are only 0.001 of the mean resistance) even though they are well resolved up to the 15th Landau level. The weak localization peak is weak (∼ 0.07G0) and evidence is seen for weak anti-localization. In contrast, the oscillation of the 2nd Landau level in the low mobility sample 2D is large (0.3 of the mean resistance); this sample may exhibit the quantum Hall effect at high fields. Furthermore, the weak localization peak is intense (∼ 1G0) In the intermediate regime, the 1 µm width ribbon exhibits well resolved SdH peaks (0.016 of the mean resistance) while the weak localization peak is 0.52G0, weak anti-localization is also present. Narrow ribbons exhibit more intense weak-localization peaks, well-resolved SdH oscillations, quantum confinement peaks, and high mobilities but no ev- idence for the quantum Hall effect. It may be assumed that the QHE in the high-density layer is shorted out by the low density layers, however this is not bourn out in simulations. For example, it is not possible to ”convert” the oscillations of Fig. 9 to those of Fig. 7 by adding the conductivity of many graphene layers to the former. Note that the relative SdH oscillation ampli- tudes in Fig. 7 are 16 times smaller than in Fig. 8, while they are more than 20 times smaller in Fig. 6, while the square resistances of all three are within a factor of 3 from each other. 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0704.0286

Mathematics of thermoacoustic tomography

7 Mathematics of thermoacoustic tomography Peter Kuchment∗and Leonid Kunyansky† November 2, 2018 Abstract The paper presents a survey of mathematical problems, techniques, and challenges arising in the Thermoacoustic (also called Photoacous- tic or Optoacoustic) Tomography. 1 Introduction Computerized tomography has had a huge impact on medical diagnostics. Numerous methods of tomographic medical imaging have been developed and are being developed (e.g., the “standard” X-ray, single-photon emission, positron emission, ultrasound, magnetic resonance, electrical impedance, op- tical) [55, 59, 75, 76, 77]. The designers of these modalities strive to in- crease the image resolution and contrast, and at the same time to reduce the costs and negative health effects of these techniques. However, these goals are usually rather contradictory. For instance, some cheap and safe methods with good contrast (like optical or electrical impedance tomogra- phy) suffer from low resolution, while some high resolution methods (such as ultrasound imaging) often do not provide good contrast. Recently re- searchers have been developing novel hybrid methods that combine different physical types of signals, in hope to alleviate the deficiencies of each of the types, while taking advantage of their strengths. The most successful exam- ple of such a combination is the Thermoacoustic Tomography (TAT) ∗Mathematics Department, Texas A& M University, College Station, TX 77843-3368, USA. [email protected] †Mathematics Department, University of Arizona, Tucson, AZ 77843-3368, USA. [email protected] http://arxiv.org/abs/0704.0286v2 1[62]. Albeit not being a common feature in clinics yet, TAT scanners are actively researched, developed and already manufactured, for instance by OptoSonics, Inc. (http://www.optosonics.com/), founded by the pioneer of TAT R. Kruger. After a substantial effort, major breakthroughs have been achieved in the last couple of years in the mathematical modeling of TAT. The aim of this article is to survey this recent progress and to describe the relevant models, mathematical problems, and reconstruction procedures arising in TAT, and to provide references to numerous research publications on this topic. The main thrust of this text is toward mathematical methods; consid- erations of the text length, as well as authors’ background do not let us discuss in any detail industrial and physical set-ups and parameters of the TAT technique, and limitations of the corresponding mathematical models. Fortunately, the excellent recent surveys by M. Xu and L.-H. V. Wang [117] and by A. A. Oraevsky and A. A. Karabutov [87, 88] accomplish all of these tasks, and thus the reader is advised to consult with them for all such is- sues (see also the recent textbook [113]). On the other hand, in spite of the significant recent progress in mathematics of TAT, there is no compre- hensive survey text addressing in details the relevant mathematical issues, although the surveys [88, 117] do mention some mathematical reconstruction techniques. The structure of the paper is a follows: Section 2 contains a brief descrip- tion of the TAT procedure. The next Section 3 provides the mathematical formulation of the TAT problem. In general, it is formulated as an inverse problem for the wave equation. However, in the case of the constant sound speed, it can be also described in terms of a spherical mean operator (a spherical analog of the Radon transform). The section also contains the list of natural questions to be addressed concerning this model. These issues are addressed then one by one in the following sections. In particular, Section 4 discusses uniqueness of reconstruction, i.e. the question of whether the data collected in TAT is sufficient for recovery of the information of interest. Albeit, for all practical purposes this issue is resolved in Corollary 2, we pro- vide an additional discussion of unresolved uniqueness problems, which are probably of more academic interest. Section 5 addresses inversion formulas and algorithms. In Section 6 effects of having only partial data are discussed. 1TAT is also called Photoacoustic (PAT) or Optoacoustic (OAT) Tomography and is sometimes abbreviated as TCT, which stands for Thermoacoustic Computed Tomography http://www.optosonics.com/ Section 7 contains results concerning the so called range conditions, i.e. the conditions that all ideal data must satisfy. Section 8 provides additional remarks and discussions of the issues raised in the previous sections. The paper ends with an Acknowledgments section and bibliography. Concerning the latter, we need to mention that the engineering and biomedical literature on TAT is rather vast and no attempt has been made in this text to create a comprehensive bibliography of the topic from the engineering prospective. The references in [87, 88, 109, 112, 117] to a large extent fill this gap. The authors, however, have tried to present a sufficiently complete review of the existing literature on mathematics of TAT. 2 Thermoacoustic tomography In TAT, a short duration EM pulse is sent through a biological object (e.g., woman’s breast in mammography) with the aim of triggering a thermoa- coustic response in the tissue. As it is explained in [117], the radiofrequency (RF) and the visible light frequency ranges are currently considered to be the most suitable for this purpose. Since mathematics works exactly the same way in both of these frequency ranges, we will not make such distinction and will be talking about just “an EM pulse”. E.g., in Figure 1 a microwave pulse is assumed. In most cases the pulse is spatially wide, so that the Figure 1: The TAT procedure. whole object is more or less uniformly irradiated. Some part of EM energy is absorbed throughout the object. The amount of energy absorbed at a location strongly depends on local biological properties of the cells. Oxygen saturation, concentration of hemoglobin, density of the microvascular net- work (angiogenesis), ionic conductivity, and water content are among the parameters that influence the absorption strongly [117]. Thus, if the energy absorption distribution function f(x) were known, it would provide a great diagnostic tool. For instance, it could be useful for detecting cancerous cells that absorb several times more energy in the RF range than the healthy ones [62, 88, 115, 117]. However, as an imaging tool neither RF waves, nor visual light alone would provide acceptable resolution. In the RF case, this is due to the long wave length. One can use shorter microwaves, but this will be at the expense of the penetration depth. In the optical region, the problem is with the multiple scattering of light. So, a different mechanism, the so called Photoacoustic Effect [46, 107, 113, 117], is used to image f(x). Namely, the EM energy absorption results in thermoelastic expansion and thus in a pressure wave p(x, t) (an ultrasound signal) that can be measured by transducers placed around the object. Now one can attempt to recover the function f(x) (the image) from the measured data p(x, t). Such a measuring scheme, utilizing two types of waves, brings about the high resolution of the ultrasound diagnostics and the high contrast of EM waves. It overcomes the adverse effect of the low contrast of ultrasound with respect to soft tissue. In fact, such a low contrast is a good thing here, allowing one to assume in the first approximation that the sound speed is constant. This often used approximation is not always appropriate, but it is the most studied case at the moment. Later on in this text we will describe some initial considerations of the variable sound speed case, following [4, 57]. For this TAT method (and in particular, for the mathematical model described below) to work, several conditions must be met. For instance, the time duration of the EM pulse must be shorter than the time it takes the sound wave to traverse the smallest feature that needs to be reconstructed. The ultrasound detector must be able to resolve the time scale of the duration of the EM pulse. On the other hand, the transducer must be also able to detect much lower frequencies. Thus, one needs to have extra-wide-band transducers, and these are currently available. One can find the technical discussion of all these issues, for instance, in [88, 117]. In this text we will assume that all these conditions are met and thus the mathematical models described are applicable. In the next section we present a mathematical description of the relation between f(x) and p(x, t) (similar mathematical problems arise in sonar [73] and radar [81] imaging, as well as in geophysics [27]). 3 Mathematical model of TAT: wave equation and the spherical mean trans- 3.1 The wave equation model We assume that the ultrasound speed at location x is equal to c(x). Then, modulo some constant coefficients that we will assume all to be equal to 1, the pressure wave p(x, t) satisfies the following problem for the standard wave equation [28, 107, 115]: ptt = c 2(x)∆xp, t ≥ 0, x ∈ R p(x, 0) = f(x), pt(x, 0) = 0 The goal is to find, using the data measured by transducers, the initial value f(x) at t = 0 of the solution p(x, t). In order to formalize what data is in fact measured, one needs to specify what kind of transducers is used, as well as the geometry of the measurement. By the geometry of the measurement we mean the distribution of locations of transducers used to collect the data. We briefly describe here the commonly considered measurement proce- dure, which uses point detectors. Line and planar detectors have also been suggested (see Section 8.1.1). It is too early to judge which one of them will become most successful, but the one using point transducers has been more thoroughly studied mathematically and experimentally, and thus will be mostly addressed in this article. In this case, the transducers are assumed to be point-like, i.e. of sufficiently small dimension. A transducer at time t measures the average pressure over its surface at this time, which for the small size of the transducer can be assumed to be just the value of p(y, t) at the location y of the transducer. Dimension count shows immediately that in order to have enough data for reconstruction of the function f(x), one needs to collect data from the transducers’ locations y running over a surface S in 3. Thus, the data at the experimentalist’s disposal is the function g(y, t) that coincides with the restriction of p(x, t) to the set of points y ∈ S. Taking into account that the measurements produce the values g(y, t) of the pressure p(x, t) of (1) on S × R+, the set of equations (1) extends to become ptt = c 2(x)∆xp, t ≥ 0, x ∈ R p(x, 0) = f(x), pt(x, 0) = 0 p(y, t) = g(y, t), y ∈ S × R+ Figure 2: An illustration to (2). The problem now reduces to finding the initial value f(x) in (2) from the knowledge of the lateral data g(x, t) (see Figure 3.1). A person familiar with PDEs might suspect first that there is something wrong with this problem, since we seem to have insufficient data for the recovery of the solution of the wave equation in a cylinder from the lateral values alone. This, however, is an illusion, since in fact there is a significant additional restriction: the solution holds in the whole space, not just inside the cylinder S × R+. We will see soon that in most cases, the data is sufficient for recovery of f(x). 3.2 Spherical mean model We now introduce an alternative formulation of the problem that works in the constant speed case only. We will assume that the units are chosen in such a way that c(x) = 1. The known Poisson-Kirchhoff formula [25, Ch. VI, Section 13.2, Formula (15)] for the solution of (1) gives p(x, t) = c (t(Rf)(x, t)) , (3) where (Rf)(x, r) = |y|=1 f(x+ ry)dA(y) (4) is the spherical mean operator applied to the function f(x), and dA is the normalized area element on the unit sphere in R3. Hence, knowledge of the function g(x, t) for x ∈ S and all t ≥ 0 essentially means knowledge of the spherical mean Rf(x, t) at all points (x, t) ∈ S × R+. One thus is lead to studying the spherical mean operator R : f → Rf and in particular its restriction RS to the points x ∈ S only (these are the points where we place transducers): RSf(x, t) = |y|=1 f(x+ ty)dA(y), x ∈ S, t ≥ 0. (5) This explains why, in many works on TAT, the spherical mean operator has been the model of choice. Albeit the (unrestricted) spherical mean operator has been studied rather intensively and for a long time (e.g., [17, 25, 58]), its version RS with the centers restricted to a subset S appears to have been studied since early 1990s only [1]-[14], [16, 26, 30, 31, 33, 32, 34, 35, 36, 39, 41, 63, 64, 68, 69, 71, 72, 73, 77, 80, 82, 83, 89, 90, 91, 95, 96, 97, 104, 121] and offers quite a few new and often hard questions. In what follows, we will alternate between these two (PDE and integral geometry) interpretations of the TAT model, since each of them has its own advantages. 3.3 Main mathematical problems of TAT We now formulate the typical list of problems one would like to address in order to implement the TAT reconstruction. 1. For which sets S ∈ R3 is the data collected by transducers placed along S sufficient for unique reconstruction of f? In terms of the spheri- cal mean operator, the question is whether RS has zero kernel on an appropriate class of functions, say continuous with compact supports. 2. If the data collected from S is sufficient, what are inversion formulas and algorithms? 3. How stable is the inversion? 4. What happens if the data is “incomplete”? 5. What is the space of all possible “ideal” data g(t, y) collected on a sur- face S? Mathematically (and in the constant sound speed case) it is the question of describing the range of the operator RS in appropriate function spaces. This question might seem to be unusual (for instance, to people used to partial differential equations), but in tomography im- portance of knowing the range of Radon type transforms is well known. Such information is used to improve inversion algorithms, complete in- complete data, discover and compensate for certain data errors, etc. (e.g., [30, 38, 39, 40, 53, 54, 55, 76, 77, 90]). 4 Uniqueness of reconstruction Many of the problems of interest to TAT can be formulated in any dimension d, albeit the practical dimensions are only d = 3 and d = 2. We will consider an arbitrary dimension d whenever we see this suitable. Let S ⊂ Rd be the set of locations of transducers and f be a compactly supported function (one can show that for purposes of uniqueness of recon- struction problem, one can always assume that f is smooth [7]). Does the absence of the signal on the transducers, i.e. g(t, y) = 0 for all t and y in S, imply that f = 0? If the answer is a “yes,” we call S - a uniqueness set, otherwise a non-uniqueness set. In other words, in terms of TAT, the uniqueness sets are those that distributing transducers along them provides enough data for unique reconstruction of the function f(x). In terms of the wave equation, uniqueness sets are the sets of complete observability, i.e. such that observing the motion on this set only, one gets enough information to reconstruct the whole oscillation. In terms of the spherical mean operator, the question is of whether the equality RSf = 0 implies that f = 0. We will address this problem for the constant sound speed case first. 4.1 Constant speed case As it has been discussed, the dimension count makes it clear that S must be (d − 1)-dimensional, i.e. a surface in 3D or a curve in 2D. We will also see that most of such surfaces are “good”, i.e. are uniqueness ones (or, in other words, provide enough information for reconstruction). Thus, we should rather discuss the problem of describing the “bad”, non-uniqueness sets. The following simple statement is very important and not immediately obvious. Lemma 1 [7, 71, 72, 121] Any non-uniqueness set S is a set of zeros of a (non-trivial) harmonic polynomial. In particular, 1. If there is no non-zero polynomial vanishing on S, then S is a unique- ness set. 2. If there is no non-zero harmonic function vanishing on S, then S is a uniqueness set. The proof of this lemma is very simple. It works under the assumption of exponential decay of the function f(x), not necessarily of compactness of its support. It also introduces some polynomials that play significant role in the whole analysis of the spherical mean operator RS. Let k ≥ 0 be an integer. Consider the convolution Qk(x) = |x| 2k ∗ f(x) = |x− y|2kf(y)dy. (6) This is clearly a polynomial of degree at most 2k. Rewriting the integral in polar coordinates centered at x and using radiality of |x − y|, one sees that Qk(x) is determined if we know the values Rf(x, t) of the spherical mean of f centered at x: Qk(x) = cd t2k+d−1Rf(x, t)dt. In particular, If RSf ≡ 0, then each polynomial Qk vanishes on S. Another observation that is easy to justify is that if the function f is exponentially decaying (e.g., is compactly supported), then if all polynomials Qk vanish identically, the function itself must be equal to zero. (This is not necessarily true anymore if f and its derivatives decay only faster than any power, rather than exponentially.) Thus, we conclude that if f is not identically equal to zero, then there is at least one non-zero polynomial Qk. Since, as we discussed, equality RSf = 0 implies that Qk|S = 0, we conclude that S must be algebraic. Now notice the following simple to verify equality (with a non-zero con- stant ck): ∆Qk = ckQk−1, (7) where ∆ is the Laplace operator. This implies that the lowest k non-zero polynomial Qk is harmonic. Since Qk|S = 0, this proves the lemma. Consider now the case when S is a closed (hyper-)surface (i.e., the bound- ary of a bounded domain). Since, as it is well known, there is no non-zero harmonic function in the domain that would vanish at the boundary (the spectrum of the Dirichlet Laplace operator is strictly positive), we conclude that such S is a uniqueness set for harmonic polynomials. Thus, we get the following important Corollary 2 [7, 63] Any closed surface is uniqueness set for the spherical mean Radon transform. An older alternative proof of this corollary provides an additional insight into the problem. We thus sketch it here. Let us assume for simplicity that the dimension d ≥ 3 is odd (even dimensions require a little bit more work). Suppose that the closed surface S remains stationary (nodal) for the oscillation described by (1). Since the oscillation is unconstrained and the initial perturbation is compactly supported, after a finite time, the interior of S will become stationary. On the other hand, we can think that S is fixed (since it is not moving anyway). Then, the energy inside S must stay constant. This is the contradiction that proves the statement of Corollary 2. We will see in the next Section that the same method works in some cases of variable sound speed, providing the needed uniqueness of reconstruction result. This corollary resolves the uniqueness problems for most practically used geometries. It fails, however, if f does not decay sufficiently fast (see [3], where it is shown in which Lp(Rd) classes of functions f(x) closed surfaces remain uniqueness sets). It also provides uniqueness for some “limited data” problems. For in- stance, if S is an open (even tiny) piece of an analytic closed surface Σ, it suffices. Indeed, if it did not, then it would be a part of an algebraic non- uniqueness surface. Uniqueness of analytic continuation would show then that the whole Σ is a non-uniqueness set, which we know to be incorrect. This result, however, does not say that it would be practical to reconstruct using observations from a tiny S. We will see later that this would not lead to a satisfactory reconstructions, due to instabilities. A geometry sometimes used is the planar one, i.e. detectors are placed along a plane S (line in the 2D). In this case, there is no uniqueness of reconstruction when the sound speed is constant. Indeed, if f(x) is odd with respect to S, then clearly all measured data g(t, y) will vanish. However, it is well known [25, 58] that functions even with respect to S can be recovered. What saves the day in TAT is that the object to be imaged is located on one side of S. Then, extending f(x) as an even function with respect to S, one can still recover it from the data. Although, for all practical purposes the uniqueness of reconstruction prob- lem is essentially resolved by the Corollary 2, the complete understanding of uniqueness problem has not been achieved yet. Thus, we include below some known theoretical results and open problems. 4.1.1 Non-uniqueness sets in R2. In this Section, we follow the results and exposition of [7, 71, 72] in discussing uniqueness sets in 2D. What are simple examples of non-uniqueness sets? As we have already mentioned, any line S (or a hyperplane in higher dimensions) is a non-uniqueness set, since any function f odd with respect to S will clearly produce no signal: RSf = 0. Analogously, consider a Coxeter system ΣN of N lines passing through a point and forming equal angles (see Fig. 3). Figure 3: Coxeter cross ΣN . Choosing the intersection point as the pole and expanding functions into Fourier series with respect to the polar angle, it is easy to discover existence of an infinite dimensional space of functions that are odd with respect to each of the N lines. Thus, such a cross ΣN is also a non-uniqueness set. Less obviously, one can use the infinite dimensional freedom just mentioned to add any finite set Φ of points still preserving non-uniqueness. The following major and very non-trivial result was conjectured in [71, 72] and proven in [7]. It shows that there are no other bad sets S besides the ones we have just discovered: Theorem 3 A set S ⊂ R2 is a non-uniqueness set for the spherical mean transform in the space of compactly supported functions, if and only if S ⊂ ωΣN ∪ Φ, where ΣN is a Coxeter system of lines, ω is a rigid motion of the plane, and Φ is a finite set. A sketch of a rather intricate proof of this result is provided in Section 4.1.2 Higher dimensions Here we present a believable conjecture of how the result should look like in higher dimensions. Conjecture 4 [7]A set S ⊂ Rd is a non-uniqueness set if and only if S ⊂ ωΣ ∪ Φ, where Σ is the surface of zeros of a homogeneous harmonic polyno- mial, ω is a rigid motion of Rd, and Φ is an algebraic surface of dimension at most d− 2. Figure 4: A picture of a 3-dimensional non-uniqueness set. The progress towards proving this conjecture has been slow, albeit some partial cases have been treated ([1]-[12]). E.g., in some cases one can prove that S is a ruled surface (i.e., consists of lines), but proving that these lines (rules) pass through a common point remains a challenge. It is known, though, that both the zero sets of homogeneous harmonic polynomials and algebraic subsets of dimension at most d − 2 are non-uniqueness sets [2, 7], and thus one should avoid using them as placements of transducers for TAT. 4.1.3 Relations to other areas of analysis The problem of injectivity of RS has relations to a wide variety of areas of analysis (see [1, 7] for many examples). In particular, the following interpre- tation is important: Theorem 5 [7, 63] The following statements are equivalent: 1. S ⊂ Rd is a non-uniqueness set for the spherical mean operator. 2. S is a nodal set for the wave equation, i.e. there exists a non-zero compactly supported f such that the solution of the wave propagation problem  = ∆u, u(x, 0) = 0, ut(x, 0) = f(x) vanishes on S for any moment of time. 3. S is a nodal set for the heat equation, i.e. there exists a non-zero compactly supported f such that the solution of the problem = ∆u, u(x, 0) = f(x) vanishes on S for any moment of time. The interpretation in terms of the wave equation provides important PDE tools and insights, which have lead to a recent progress [33, 12] (albeit it has not lead yet to a complete alternative proof of Theorem 3). The rough idea, originally introduced in [33], is that if S is a nodal set, then it might be considered as the fixed boundary. In this case, the signals must go around S. However, in fact, there is no obstacle, so signals can propagate along straight lines. Thus, in order to avoid discrepancies in arrival times, S must be very special. One can find details in [33] and in [12]. 4.2 Uniqueness in the case of a variable sound speed It is shown in [35, Theorem 4] that uniqueness of reconstruction also holds in the case of a smoothly varying (strictly positive) sound speed, if the source function f(x) is completely surrounded by the observation surface S (in other words, if there is no US signal coming from outside of S). The proof uses the celebrated unique continuation result by D. Tataru [108]. One can also establish uniqueness of reconstruction in the case of the source not necessarily completely surrounded by S. However, here we need to impose an additional non-trapping condition on the sound speed. We assume that the sound speed is strictly positive c(x) > c > 0 and such that c(x)− 1 has compact support, i.e. c(x) = 1 for large x. Consider the Hamiltonian system in R2nx,ξ with the Hamiltonian H = c2(x) |ξ|2:  x′t = = c2(x)ξ ξ′t = − ∇ (c2(x)) |ξ|2 x|t=0 = x0, ξ|t=0 = ξ0. The solutions of this system are called bicharacteristics and their projections into Rnx are rays. We will assume that the following non-trapping condition holds: all rays (with ξ0 6= 0) tend to infinity when t→ ∞. Theorem 6 [4] Under the non-trapping conditions formulated above, com- pactly supported function f(x) is uniquely determined by the data g measured on S for all times. (No assumption of f being supported inside S is imposed.) One should mention that ray trapping can occur for some sound speed profiles. For instance, if c(x) = |x| for some range r1 < |x| < r2, then there are rays trapped in this spherical shell. We are not sure what happens in this case to the uniqueness of reconstruction statement of Theorem 6 and inversion formula of Theorem 7. 5 Reconstruction: formulas and examples Here we will address the procedures of actual reconstruction of the source f(x) from the data g(t, y) measured by transducers. 5.1 Constant sound speed We assume here that the sound speed is constant and normalized to be equal to 1. 5.1.1 Inversion formulas Before we move to our case of interest, which is spheres centered on a closed surface S surrounding the object to be imaged, we briefly refer to related but somewhat different work. Namely, the problem of recovering functions from integrals over spheres centered on a (hyper)plane S has attracted a lot of attention over the years. Albeit, as it has been mentioned before, there is no uniqueness in this case (functions odd with respect to S are annihilated), even functions can be recovered. Thus also functions supported on one side of the plane can be as well, by means of their even extension. Many explicit inversion formulas and procedures have been obtained for this situation [16, 26, 31, 39, 41, 60, 77, 80, 89, 90, 101, 102, 103]. We will not provide any details here, since this acquisition geometry is not very useful. In particular, this is due to “invisibility” of some parts of the interfaces, see Section 6, which arises from truncating the plane. The same problem is encountered with some other unbounded acquisition surfaces, such as a surface of an “infinitely” long cylinder. Thus, it is more practical to place transducers along a closed surface surrounding the object. The simplest surface of this type is a sphere. 5.1.2 Fourier expansion methods Let us assume that S is the unit sphere in Rn. We would like to reconstruct a function f(x) supported inside S from the known values of its spherical integrals g(y, r) with the centers on S: g(y, r) = f(y + rω)rn−1dω, y ∈ S. The first inversion procedures for the case of spherical acquisition were de- scribed in [82] in 2D and in [83] in 3D. These solutions were obtained by harmonic decomposition of the measured data and the sought function, and by equating coefficients of the corresponding Fourier series. In particular, the 2-D algorithm of [82] is based on the Fourier decompo- sition of f and g in angular variables: f(x) = fk(ρ)e ikϕ, x = (ρ cos(ϕ), ρ sin(ϕ)) (9) g(y(θ), r) = gm(r)e ikθ, y = (R cos(θ), R sin(θ)). Following [82] we consider the Hankel transform ĝm,J(λ) of the Fourier coef- ficients gm(r) (divided by 2πr) ĝm,J(λ) = gm(r)J0(λr)dr = H0 gm(r) . (10) To simplify the presentation we introduce the convolution GJ(λ, y) of the sought function with the Bessel function J0(λ|x− y|). GJ(λ, y) = f(x)J0(λ|x− y|)dx, (11) One can notice that ĝm,J(λ) are the Fourier coefficients of GJ(λ, y) in θ: ĝm,J(λ) = GJ(λ, y)e −imθdθ. (12) Now coefficients fm(ρ) can be recovered from gm(r) by application of the addition theorem for the Bessel function J0(λ|x− y|): J0(λ|x− y|) = Jm(λ|x|)Jm(λ|y|)e −im(ϕ−θ). (13) Indeed, by substituting equations (9) and (13) into (11), and (11) into (12) one obtains ĝm,J(λ) = 2πJm(λ|R|) fm(ρ)Jm(λρ)ρdρ = Hm(fm(ρ)), where Hm is the m-th order Hankel transform. Since the latter transform is self-invertible, the coefficients fm(ρ) can be recovered by the following formula fm(ρ) = Hm ĝm,J(λ) Jm(λ|R|) Jm(λ|R|) gm(r) , (14) which is the main result of [82]. Function f(x) can now be reconstructed by summing series (9). Note that the above method requires a division of the Hankel transform of the measured data by Bessel functions Jm that have infinitely many zeros. Theoretically, there is no problem; the Hankel transform H0 gm(r) has to have zeros that would cancel those in the denominator. However, since the measured data always contain some error, the exact cancelation is not likely to happen, and one needs a sophisticated regularization scheme to keep the total error bounded. This problem can be avoided by replacing in (10) Bessel function J0 by Hankel function H ĝm,H(λ) = gm(r)H 0 (λr)dr. The addition theorem for H 0 (λ|x− y|) takes form 0 (λ|x− y|) = Jm(λ|x|)H m (λ|y|)e −im(ϕ−θ), and by proceeding as before one can obtain the following formula for fm(ρ): fm(ρ) = Hm ĝm,H(λ) m (λ|R|) m (λ|R|) gm(r)H 0 (λr)dr Unlike Jm, Hankel functions H m (t) do not have zeros for all real values of t and therefore problems with division by zeros do not arise in this amended version of the method [82]. This derivation can be repeated in 3-D, with the exponentials eikθ replaced by the spherical harmonics, and with cylindrical Bessel functions replaced by their spherical counterparts. By doing this one will arrive at the Fourier series method of [83]. Our use of Hankel function H 0 above is similar to the way the authors of [83] utilized spherical Hankel function h 0 to avoid the divisions by zero. 5.1.3 Filtered backprojection methods The favorite way of inverting Radon transform for tomography purposes is by using filtered backprojection type formulas, which involve filtration in Fourier domain followed (or preceded) by a backprojection. In the case of the set of spheres centered on a closed surface (e.g., sphere) S, one expects such a formula to involve a filtration with respect to the radius variable and then some integration over the set of spheres passing through the point of interest. For quite a while, no such type formula had been discovered. This did not prevent practitioners from reconstructions, since good approximate inversion formulas (parametrices) could be developed, followed by an iterative improvement of the reconstruction, see e.g. reconstruction procedures in [114, 115, 118, 119, 120], and especially [96, 97]. The first set of exact inversion formulas of the filtered backprojection type was discovered in [33]. These formulas were obtained only in odd dimensions. Several different variations of such formulas (different in terms of exact order of the filtration and backprojection steps) were developed. Let us denote by g(p, r) = r2RSf the spherical integral, rather than the average, of f . Then various versions of the 3D inversion formulas that reconstruct a function f(x) supported inside S from its the spherical mean data RSf , read: f(x) = − 1 g(y, |y − x|)dA(y), f(x) = − 1 g(y, t) ) ∣∣∣∣∣ t=|y−x| dA(y), f(x) = − 1 g(y,t) )) ∣∣∣∣∣ t=|y−x| dA(y). Recently, analogous formulas were obtained for even dimensions in [32]. De- noting by g, as before the spherical integrals (rather than averages) of f , the formulas of [32] in 2D look as follows: f(x) = g(y, t) log(t2 − |x− y|2) dt dl(y), (16) f(x) = g(y, t) log(t2 − |x− y|2) dt dl(y), (17) A different set of explicit inversion formulas that work in arbitrary dimensions was presented in [69]. f(x) = 4(2π)n−1 n(y)h(y, |x− y|)dA(y). (18) h(y, t) = Y (λt) J(λt′)g(y, t′)dt′ −J(λt) Y (λt′)g(y, t′)dt′ λ2n−3dλ, (19) J(t) = Jn/2−1(t) tn/2−1 , Y (t) = Yn/2−1(t) tn/2−1 Jn/2−1(t) and Yn/2−1(t) are respectively the Bessel and Neumann functions of order n/2− 1, and n(y) is the vector of exterior normal to ∂B. In 2-D equations (18), (19) can be simplified to yield the following recon- struction formula f(x) = − g(y, t′) |x− y|2 − t′2  dl(y). A similar simplification is also possible in 3D resulting in the formula f(x) = g(y, t) ) ∣∣∣∣∣ t=|y−x| dA(y). (20) Equation (20) is equivalent to one of the formulas derived in [116] for the 3D case. It is interesting to notice that the “universal” formula of [116] holds for all geometries when the backprojection type formulas are known: spherical, cylindrical, and planar. It is not very likely that such explicit formulas would be available for any closed surfaces S different from spheres (see a related discussion in [15, 27]). 5.1.4 Series solutions for arbitrary geometries Although, as we have just mentioned, we do not expect such explicit formulas to be derived for non-spherical closed surfaces S, there is, however, a different approach [70] that theoretically works for any closed S and that is practically useful in some non-spherical geometries. Let λ2m and um(x) be the eigenvalues and normalized eigenfunctions of the Dirichlet Laplacian −∆ on the interior Ω of a closed surface S: ∆um(x) + λ mum(x) = 0, x ∈ Ω, Ω ⊆ R n, (21) um(x) = 0, x ∈ S, ||um|| |um(x)| 2dx = 1. As before, we would like to reconstruct a compactly supported function f(x) from the known values of its spherical integrals g(y, r) with the centers on S: g(y, r) = f(y + rω)rn−1dω, y ∈ S. We notice that um(x) is the solution of the Dirichlet problem for the Helmholtz equation with zero boundary conditions and the wave number λm, and thus it admits the Helmholtz representation um(x) = Φλm(|x− y|) um(y)ds(y) x ∈ Ω, (22) where Φλm(|x− y|) is a free-space rotationally invariant Green’s function of the Helmholtz equation (21). The eigenfunctions {um(x)} 0 form an orthonormal basis in L2(Ω). There- fore, f(x) can be represented by the series f(x) = αmum(x) (23) um(x)f(x)dx. Since f(x) is C10 , series (23) converges pointwise. A reconstruction formula of αm, and thus of f(x), will result if we substitute representation (22) into (23) and interchange the order of integrations. Indeed, after a brief calculation we will get um(x)f(x)dx = I(y, λm) um(y)dA(x), (24) where I(y, λ) = Φλ(|x− y|)f(x)dx. (25) Certainly, the need to know the spectrum and eigenfunctions of the Dirichlet Laplacian imposes a severe constraint on the surface S. However, there are simple cases when the eigenfunctions are well known, and fast sum- mation formulas for the corresponding series are available. Such is the case of a cubic measuring surface S (see [70]); the eigenfunctions um are products of sine functions um(x) = πm1x1 πm2x2 πm3x3 , (26) where m = (m1, m2, m3), m1, m2, m3 ∈ N, and the eigenvalues are easily found as well λm = π 2|m|2/R2. (27) Sum (23) is just a regular 3-D Fourier sine series easily computable by ap- plication of the Fast Sine Fourier transform algorithm. The algorithmic trick that allows one to calculate fast the coefficients αm consists in com- puting first integrals (25) on a uniform mesh in λ. This is easily done by a one-dimensional Fast Cosine Fourier transform algorithm, with Φλ(t) = cos(λt)/t. The normal derivatives of um(x) are also products of sine func- tions, this time two-dimensional ones. This, in turn, permits rapid evaluation of integrals I(y, λ) ∂ um(y)dA(x) for each mesh value of λ, and for each one of the six faces ∂Ωi, i = 1, ..., 6 of the cube. Finally, the computation of αm using equation (24) reduces to the interpolation in the spectral param- eter λ, since the integrals in the right hand side of this equation have been computed for the mesh values of this parameter (not for λm). Due to oscil- latory nature of the integrals (25) a low order interpolation here would lead to inaccurate reconstructions. Luckily, however, these integrals are analytic functions of parameter λ (due to the finite support of g). Hence, high order polynomial interpolation is applicable, and numerics yields very good results. The algorithm we just described requires O(m3 logm) floating point op- erations if the reconstruction is to be performed on an m×m×m Cartesian grid, from comparably discretized data measured on a cubic surface. In prac- tical terms, it yields reconstructions in the matter of several seconds on grids with total number of nods exceeding a million [70]. 5.1.5 Time reversal (backpropagation) methods In the constant speed case, the following approach is possible in 3D: due to the validity of the Huygens’ principle (i.e., the signal escapes from any bounded domain in finite time), the pressure p(t, x) inside S will become equal to zero for any time T larger than the time required to cross the domain (i.e., time that it takes the sound to move along the diameter of S, which for c = 1 equals the diameter). Thus, one can impose the zero conditions on p(t, x) for t = T and solve the wave equation (2) back in time, using the measured data g as the boundary values. The solution of this well posed problem at t = 0 gives the desired source function f(x). Such methods have been successfully implemented [22]. Although in 2D or in presence of sound speed variations, Huygens’ princi- ple does not hold anymore, and thus the signal theoretically will stay forever, one can find good approximate solutions using a similar approach [4, 18], see discussions of the variable speed case below. 5.1.6 Examples of reconstructions and additional remarks about the inversion formulas • It is well known that different analytic inversion formulas in tomogra- phy can behave differently in numerical implementation (e.g., in terms of their stability), However, numerical implementation seems to show that the analytic (backprojection type) formulas (15)-(20), in spite of some of them being not equivalent, work equally well. See, for example the results of an analytic formula reconstruction in 3D shown in Fig. • It is worth noting that although formulas (15)-(16) and (18)-(20) will yield identical results when applied to functions that can be represented as the spherical mean Radon transform of a function supported inside S, they are in general not equivalent when applied to functions with larger supports. Simple examples (e.g., of f being the characteristic Figure 5: A mathematical phantom in 3D (left) and its reconstruction using an analytic inversion formula. set of a large ball containing S) show that these two types of formulas provide different reconstructions. • An interesting observation is that backprojection formulas (15)-(20) do not reconstruct the function f correctly inside the surface S, if f has support reaching outside S. For instance, applying the reconstruction formulas to the function RS(χ|x|≤3) leads to an incorrect reconstruction of the value of f = χ|x|≤3 inside S = {|x| ≤ 1}. (Here by χV we denote the characteristic function of the set V , i.e. it takes the value 1 in V and zero outside. So, χ|x|≤3 is the characteristic function of the ball of radius 3 centered at the origin.) An another example: if one adds to the phantom shown in Fig. 5 two balls to the right of the surrounding sphere S, this leads to strong artifacts, as seen on Fig. 6. What is the reason for such a distortion? If one does not know in ad- vance that f has support inside S, the backprojection formulas shown before use insufficient information to recover a function with a larger support, and thus uniqueness of reconstruction is lost. Then the for- mulas misinterpret the data, wrongly assuming that they came form a function supported inside S and thus reconstructing the function in- correctly. Notice that the series reconstruction of the preceding Section is free of such problem. E.g., the reconstruction shown in Fig. 7 confirms this. Figure 6: A perturbed reconstruction, due to presence of two additional balls outside S (not shown on the picture). Figure 7: In the phantom shown on the left, most disks are located outside the square acquisition surface S indicated by the dotted line. This does not perturb the reconstruction inside S (right). 5.2 Reconstruction in the variable speed case We will assume here that the sound speed c(x) is smooth, positive, constant for large x, and non-trapping. Although most analytic techniques we de- scribed above do not work in the variable speed case, some formulas can be derived and algorithms can be designed. This work is in a beginning stage and the results described below most surely can and will be improved. 5.2.1 “Analytic” inversions Let us denote by Ω the interior of the observation surface S, i.e. the area where the object to be imaged is located. Consider in Ω the operator A = −c2(x)∆ with zero Dirichlet conditions on the boundary S = ∂Ω. This operator is self-adjoint, if considered in the weighted space L2(Ω; c−2(x)). We also denote by E the operator of harmonic extension, which trans- forms a function φ on S to a harmonic function on Ω which coincides with φ on S. The following result provides a formula for reconstructing f from the data Theorem 7 [4] The function f(x) in (2) can be reconstructed in Ω as fol- lows: f(x) = (Eg|t=0)− 2 sin (τA 2 )E(gtt)(x, τ)dτ. (28) The validity of this result hinges upon decay estimates for the solution (so called local energy decay [29, 110, 111]), which hold under the non-trapping condition. These estimates guarantee a qualified decay of the solution p(t, x) inside any bounded region, e.g. in Ω, when time t increases. In odd dimen- sions decay is exponential, but only polynomial in even dimensions. The decay can be used instead of Huygens’ principle to solve the wave equation backwards, starting at the infinite time. This leads to the formula (28). Due to functions of the operator A being involved, it is not that clear how explicit this formula can be made. For instance, it would be interesting to see whether one can derive from (28) a backprojection inversion formula for the case of a constant sound speed and S being a sphere (we have already seen that such formulas are known). 5.2.2 Backpropagation The exponential decay at large values of time can be used as follows: for a sufficiently large T , one can assume that the solution is practically zero at t = T . Thus, imposing zero initial conditions at t = T and solving in reverse time direction, one arrives at t = 0 to an approximation of f(x) [18]. 5.2.3 Eigenfunction expansions One natural way to try to use the formula (28) is to use eigenfunction expan- sion of the operator A in Ω (assuming that such expansion is known). This immediately leads to the following result: Theorem 8 Under the same conditions on the sound speed as before, func- tion f(x) can be reconstructed inside Ω from the data g in (2), as the following L2(B)-convergent series: f(x) = fkψk(x), (29) where the Fourier coefficients fk can be recovered using one of the following formulas: fk = λ k gk(0)− λ sin (λkt)g k(t)dt, fk = λ k gk(0) + λ cos (λkt)g k(t)dt, or fk = −λ sin (λkt)gk(t)dt = −λ sin (λkt)g(x, t) (x)dxdt, gk(t) = g(x, t) (x)dx. Here ν denotes the external normal to S. One notices that this is a generalization to the variable sound speed case of the expansion method of [70], discussed in Section 5.1.4. An interesting feature is that, unlike in [70], we do not need to know the whole space Green’s function for A (which is certainly not known). It is not clear yet how feasible numerical implementation of this approach could be. 6 Partial data. “Visible” and “invisible” sin- gularities Uniqueness of reconstruction does not imply practical recoverability, since the reconstruction procedure might be severely unstable. This is well known to be the case, for instance, in incomplete data situations in X-ray tomography, and even for complete data problems in some imaging modalities, such as the electrical impedance tomography [64, 68, 76, 77]. In order to describe the results below, we need to explain the notion of the wave front set WF (f) of a function f(x). This set carries detailed information on singularities of f(x). It consists of pairs (x, ξ) of a point x in space and a wave vector (Fourier domain variable) ξ 6= 0. It is easier to say what it means that a point (x0, ξ0) is not in the wave front set WF (f). This means that one can smoothly cut-off f to zero at a small distance from x0 in such a way that the Fourier transform φ̂f(ξ) of the resulting function φ(x)f(x) decays faster than any power of ξ in directions that are close to the direction of ξ0. We remind the reader that if this Fourier transform decays that way in all directions, then f(x) is smooth near the point x0. So, the wave front set contains pairs (x0, ξ0) such that f is not smooth near x0, and ξo indicates why it is not: the Fourier transform does not decay well in this direction. For instance, if f(x) consists of two smooth pieces joined non- smoothly across a smooth interface Σ, then WF (f) contains pairs (x, ξ) such that x is in Σ and ξ is normal to Σ at x. One can find simple introduction to the notions of microlocal analysis, such as the wave front set, for instance in [106]. Analysis done in [99] for the constant speed case (equivalently, for the spherical mean transform RS), showed which parts of the wave front (and thus singularities) of a function f can be recovered from its partial X-ray data. An analog of this result also holds for the spherical mean transform RS [73] (see also [120] for a practical discussion). We formulate it below in an imprecise form (see [73] for precise formulation). Theorem 9 [73] A wavefront set point (x, ξ) of f is “stably recoverable” from RSf if and only if there is a circle (sphere in higher dimensions) centered on S, passing through x, and normal to ξ at this point. As we have already mentioned, this result does not exactly hold the way it is formulated and needs to include some precise conditions (see [73, Theorem 3]). The statement is, for instance, correct if S is a smooth hypersurface and the support of f lies on one side of the tangent plane to S at the center of the sphere mentioned in the theorem. Talking about jump singularities only (i.e., interfaces between smooth regions inside the object to be imaged), this result says that in order for a piece of the interface to be stably recoverable (dubbed “visible”), one should have for each point of this interface, a sphere centered at S and tangent to the interface at this point. Otherwise, the interface will be blurred away (even if there is a uniqueness of reconstruction theorem). The reason is that if all spheres of integration are transversal to the interface, the integration smoothes off the singularity, and therefore its recovery becomes highly unsta- ble (numerically, one has to deal with inversion of a matrix with exponentially fast decaying singular values). The Figure 8 below shows an example of an incomplete data reconstruction from spherical mean data. One sees clearly the effect of disappearance of the parts of the boundaries that are not touched tangentially by circles centered at transducers’ locations. Figure 8: Effect of incomplete data: the phantom (left) and its incomplete data reconstruction. The transducers were located along a 180o circular arc (the left half of a large circle surrounding the squares). 7 Range conditions As it has already been mentioned, the space of functions g(t, y) that could arise as exact data measured by transducers (i.e., the range of the data), is very small (of infinite codimension in the spaces of all functions of t > 0, y ∈ S). Knowing this space (range) is useful for many theoretical and practical purposes (reconstruction algorithms, error corrections, incomplete data completion, etc.), and thus has attracted a lot of attention (e.g., [30, 38, 39, 40, 53, 54, 64, 66, 67, 68, 74, 76, 77, 78, 90, 100]. For instance, for the standard Radon transform f(x) → g(s, ω) = x·ω=s f(x)dx, |ω| = 1, the range conditions on g(s, ω) are: 1. evenness: g(−s,−ω) = g(s, ω) 2. moment conditions: for any integer k ≥ 0, the kth moment Gk(ω) = skg(ω, s)ds extends from the unit circle of vectors ω to a homogeneous polynomial of degree k in ω. The evenness condition is obviously necessary and is kind of “trivial”. It seems that the only non-trivial conditions are the moment ones. However, here the standard Radon transform misleads us, as it often happens. In fact, for more general transforms of Radon type it is often easy (or easier) to find analogs of the moment conditions, while analogs of the evenness conditions are often elusive (see [64, 66, 67, 76, 77, 84] devoted to the case of SPECT (single photon emission tomography)). The same happens in TAT. Let us deal first with the case of a constant sound speed, when one can think of the spherical mean transform RS instead of the wave equation model. An analog of the moment conditions was already present implicitly (without saying that these were range conditions) in [71, 72, 7] and explicitly formu- lated as such in [95]. Indeed, our discussion in Section 4 of the polynomials Qk provides the following conditions of the moment type: Moment conditions [7, 71, 72, 95] on data g(p, r) = RSf(p, r) look as follows: for any integer k ≥ 0, the moment Mk(ω) = r2k+d−1g(p, r)dr can be extended from S to a (non-homogeneous) polynomial Qk(x) of degree at most 2k. These conditions, however, are incomplete, and in fact infinitely many others, which play the role of an analog of evenness, need to be added. Complete range descriptions for RS when S is a circle in 2D were discov- ered in [13] and then in odd dimensions in [34]. They were then extended to any dimension and interpreted in several different ways in [6]. These conditions happen to be intimately related to PDEs and spectral theory. Figure 9: In order to describe these conditions, we need to introduce some notations. Let B be the unit ball in Rd, S - the unit sphere, and C - the cylinder B×[0, 2] (see Fig. 9). We introduce the spherical mean operator RS as before: RSf(x, t) = |y|=1 f(x+ ty)dA(y), x ∈ S. Several different range descriptions for RS were provided in [6], out of which we only show a few: Theorem 10 [6] The following three statements are equivalent: 1. The function g ∈ C∞0 (S × [0, 2]) is representable as RSf for some f ∈ C∞0 (B). (In other words, g represents an ideal (free of errors) set of TAT data.) 2. (a) The moment conditions are satisfied. (b) Let −λ2 be any eigenvalue of the Laplace operator in B with zero Dirichlet conditions and ψλ be the corresponding eigenfunction. Then the following orthogonality condition is satisfied: S×[0,2] g(x, t)∂νψλ(x)jn/2−1(λt)t n−1dxdt = 0. (31) Here jp(z) = cp Jp(z) is the so called spherical Bessel function. 3. (a) The moment conditions are satisfied. (b) Let ĝ(x, λ) = g(x, t)jn/2−1(λt)t n−1dt. Then, for any m ∈ Z, the mth spherical harmonic term ĝm(x, λ) of ĝ(x, λ) vanishes at all zeros λ 6= 0 of Bessel function Jm+n/2−1(λ). Remark 11 [6] 1. In odd dimensions, moment conditions are not necessary, and thus con- ditions 2(b) or 3(b) suffice. (A similar earlier result was established for a related transform in [34].) 2. The range conditions (2) of the previous Theorem are also necessary when S is the boundary of any bounded domain, not necessarily a sphere. 3. An analog of these conditions can be derived for a variable sound speed (without non-trapping conditions imposed). 8 Concluding remarks 8.1 Variations of the TAT procedure 8.1.1 Planar and linear transducers Assuming that transducers are point-like, is clearly an approximation, and in fact a transducer measures the average pressure over its area. It has been rightfully claimed that the point approximation for transducers should lead to some blurring in the reconstructions. This, as well as intricacies of reconstructions from the data obtained by point transducers, triggered recent proposals for different types of transducers (see [20, 21], [47]-[52], [92, 93]). In these papers, it was suggested to use either planar, or line detectors. In the first case [47], the detectors are assumed to be large and planar, ideally assumed to be approximations of infinite planes that are placed tan- gentially to a sphere containing the object. Thus, the data one collects is the integrals of the pressure over these planes, for all values of t > 0. If one takes the standard 3D Radon transform of the pressure p(x, t) with respect to x: P (x, t) 7→ q(s, t, ω) = x·ω=s p(x, t)dA(x), where dA is the surface measure and ω is a unit vector in R3, this is well known to reduce the 3D Laplace operator ∆x to the second derivative ∂ 2/∂s2 [30, 38, 39, 40, 53, 54], and thus the 3D wave equation to the string vibra- tion problem. The measured data provide the boundary conditions for this problem. The initial conditions in (1) mean evenness with respect to time, and thus the standard d’Alambert formula leads to the immediate realiza- tion that the measured data is just the 3D Radon transform of f(x). Thus, the reconstruction boils down to the well known inversion formulas for the Radon transform. Another proposal ([20, 21], [49]-[52], [92, 93]) is to use line detectors that provide line integrals of the pressure p(x, t). Such detectors can be implemented optically, using either Fabry-Perot [20], or Mach-Zehnder [93] interferometers. Suppose that the object is surrounded by a surface that is rotation in- variant with respect to the z-axis. It is suggested to place the line detectors perpendicular to the z-axis and tangential to the surface. The same consid- eration as above then shows that after the 2D Radon (or X-ray, which in 2D is the same) transform in each plane orthogonal to z-axis, the 3D wave equation converts into the 2D one for the Radon data. The measurements provide the boundary data. Thus, the reconstruction boils down to solving a 2D problem similar to the one in the case of point detectors, and then inverting the 2D Radon transform. Due to the recent nature of these two projects, it appears to be too early to judge which one will be superior in the end. For instance, it is not clear beforehand, whether the approximation of infinite size (length, area) of the linear or planar detectors works better than the zero dimension approxima- tion for point detectors. Further developments will resolve these questions. 8.1.2 Direct imaging techniques Some direct imaging techniques have been suggested, which might not require mathematical reconstructions. See, for instance, [79] about an acoustic lens system. 8.1.3 Using contrast agents Contrast agents to improve TAT imaging have been developed (e.g., [24]). 8.1.4 Passive thermoacoustic imaging The TAT model we have considered can be called “active thermoacoustic to- mography,” due to the set-up when the practitioner creates the signal. There has been some recent development of the “passive thermoacoustic tomogra- phy,” where the thermoacoustic signal is used to image the temperature sources present inside the body. One can find a survey of this area in [94]. 8.2 Uniqueness 8.2.1 Sketch of the proof of Theorem 10 We provide here a brief outline of the rather technical proof of Theorem 10. Suppose that f is compactly supported, not identically zero, and such that RSf = 0. Our previous considerations show that one can assume that S is an algebraic curve (not a straight line) that is contained in the set of zeros of a non-trivial harmonic polynomial. Now one touches the boundary of the support of f from outside by a circle centered on S. Then microlocal analysis of the operator RS (which happens to be an analytic Fourier Integral Operator, FIO [19, 42, 43, 44, 45, 65, 98]) shows that, due to the equality RSf = 0, at the tangency point the vector co-normal to the sphere should not belong to the analytic wave front of f (microlocal regularity of solutions of RSf = 0). This, for instance, can be also extracted from the results of [105]. On the other hand, a theorem by Hörmander and Kashiwara [56, Theorem 8.5.6] shows that this vector must be in the analytic wave front set, since f = 0 on one side of the sphere (a microlocal version of uniqueness of analytic continuation). This way, one gets a contradiction. Unfortunately, the life is not so easy, and the proof sketched above does not go through smoothly, due to possible cancelation of wavefronts at different tangency points. Then one has to involve the geometry of zeros of harmonic polynomials [37] to exclude the possibility of such a cancelation. Thus, the proof uses microlocal analysis and geometry of zeros of har- monic polynomials. Both these tools have their limitations. For instance, the microlocal approach (at least, in the form it is used in [7]) does not al- low considerations of non-compactly supported functions. Thus, the validity of the Theorem for arbitrarily fast decaying, but not compactly supported, functions is still not established, albeit it most certainly holds. On the other hand, the geometric part does not work that well in dimensions higher than two. Development of new approaches is apparently needed in order to over- come these hurdles. A much simpler PDE approach has emerged recently [33] (see also [12] and the next Section), albeit its achievements have been limited so far. 8.2.2 Some open problems concerning uniqueness As it has already been mentioned, one can consider the practical problems about uniqueness resolved. However, the mathematical understanding of the uniqueness problem for the restricted spherical mean operators RS is still unsatisfactory. Here are some questions that still await their resolution: 1. Describe uniqueness sets in dimensions larger than 2 (prove the Con- jecture 4). Recent limited progress, as well as variations on this theme can be found in [1]-[12]. 2. Prove Theorem 3 without using microlocal and harmonic polynomial tools. 3. Prove Theorem 3 on uniqueness sets S under the condition of suffi- ciently fast decay (rather than compactness of support) of the function. Very little is known for the case of functions without compact support. The main known result is of [3], which describes for which values of 1 ≤ p ≤ ∞ the result of Corollary 2 still holds: Theorem 12 [3] Let S be the boundary of a bounded domain in Rd and f ∈ Lp(Rd) such that RSf ≡ 0. If p ≤ 2d/(d−1), then f ≡ 0 (and thus S is injectivity set for this space). This fails for any p > 2d/(d− 1). 8.3 Inversion Albeit closed form (backprojection type) inversion formulas are available now for the cases of S being a plane (and object on one side from it), cylinder, and a sphere, there is still some mystery surrounding this issue. 1. Can one write a backprojection type inversion formula in the case of the constant sound speed for a closed surface S which is not a sphere? We suspect that the answer to this question is negative (see also related discussion in [15, 27]). 2. The inversion formulas for S being a sphere assume that the object to be imaged is inside S. One can check on simplest examples that if the support of function f(x) reaches outside S, the inversion formulas do not reconstruct the function correctly even inside of S. See [5] for a discussion. 3. The I. Gelfand’s school of integral geometry has developed a marvelous machinery of the so called κ operator, which provides a general ap- proach to inversion and range descriptions for transforms of Radon type [38, 39]. In particular, it has been applied to the case of integration of various collections (“complexes”) of spheres in [39, 41]. This consider- ation seems to suggest that one should not expect explicit closed form inversion formulas for RS when S is a sphere. We, however, know that such formulas have been discovered recently [33, 69]. This apparent controversy has not been resolved. 4. Can one derive any more explicit analytic formulas from (28)? 5. Can the series expansion formulas of Theorem 8 be efficiently imple- mented? One can also mention that in some works [15, 23] it is suggested to use in the TAT problem not only the values of the pressure measured by transducers on the observation surface S, but its normal derivative to S as well. If one knows both, then taking Fourier transform in the time variable and using the whole space Green’s function for the Helmholtz equation leads immediately to a reconstruction formula for the solution (which seems to be much simpler than what is proposed in [23]). The problem is that this normal derivative is not measured by TAT devices. Under some circumstances (e.g., when there are no sources of ultrasound outside S), one can prove the theoretical pos- sibility of recovering the missing normal derivative. This, however, does not seem to us to be a plausible procedure. In rare cases (planar, cylindrical, or spherical surface S), when involvement of the normal derivative can be eliminated (e.g., [15, 27]), this might lead to feasible inversion algorithms, but in these cases, as explained before in this text, explicit and nicely im- plementable analytic inversion formulas are available. So, jury is still out on this issue as well. 8.4 Stability Stability of inversion when S is a sphere surrounding the support of f(x) is the same as for the standard Radon transform, as the results of [91] and sec- ond statement of Theorem 11 show. However, if the support reaches outside, albeit Corollary 2 still guarantees uniqueness of reconstruction, stability (at least for the parts outside S) is gone. Indeed, Theorem 9 shows that some parts of singularities of f outside S will not be stably “visible.” 8.5 Range As Theorem 9 states, the range conditions 2 and 3 of Theorem 10 are neces- sary also for non-spherical closed surfaces S and for functions with support outside S. They, however, are not expected to be sufficient, since Theorem 9 indicates that one might expect non-closed ranges in some cases. The same applies for non-constant sound speed case. Acknowledgments The work of the first author was partially supported by the NSF DMS grants 0604778 and 0648786. The second author was partially supported by the DOE grant DE-FG02-03ER25577 and NSF DMS grant 0312292. Part of this work was completed when the first author was at the Isaac Newton Institute for Mathematical Sciences. The authors express their gratitude to the NSF, DOE and INI for this support. The authors thank M. Agranovsky for extremely useful discussions, M. Anastasio, G. Beylkin and M. 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Introduction Thermoacoustic tomography Mathematical model of TAT: wave equation and the spherical mean transform The wave equation model Spherical mean model Main mathematical problems of TAT Uniqueness of reconstruction Constant speed case Non-uniqueness sets in R2. Higher dimensions Relations to other areas of analysis Uniqueness in the case of a variable sound speed Reconstruction: formulas and examples Constant sound speed Inversion formulas Fourier expansion methods Filtered backprojection methods Series solutions for arbitrary geometries Time reversal (backpropagation) methods Examples of reconstructions and additional remarks about the inversion formulas Reconstruction in the variable speed case ``Analytic'' inversions Backpropagation Eigenfunction expansions Partial data. ``Visible'' and ``invisible'' singularities Range conditions Concluding remarks Variations of the TAT procedure Planar and linear transducers Direct imaging techniques Using contrast agents Passive thermoacoustic imaging Uniqueness Sketch of the proof of Theorem 10 Some open problems concerning uniqueness Inversion Stability Range

0704.0287

Search for Very High Energy Emission from Gamma-Ray Bursts using Milagro

Search for Very High Energy Emission from Gamma-Ray Bursts using Milagro P. M. Saz Parkinson for the Milagro Collaboration1 Santa Cruz Institute for Particle Physics, University of California, Santa Cruz, CA 95064 Abstract. Gamma-Ray Bursts (GRBs) have been detected at GeV energies by EGRET and models predict emission at > 100 GeV. Milagro is a wide field (2 sr) high duty cycle (> 90%) ground based water Cherenkov detector that records extensive air showers in the energy range 100 GeV to 100 TeV. We have searched for very high energy emission from a sample of 106 gamma-ray bursts (GRB) detected since the beginning of 2000 by BATSE, BeppoSax, HETE-2, INTEGRAL, Swift or the IPN. No evidence for emission from any of the bursts has been found and we present upper limits from these bursts. Keywords: gamma-ray sources; gamma-ray bursts; astronomical observations: gamma-ray PACS: 98.70.Rz,95.85.Pw Some of the most important contributions to understanding gamma-ray bursts (GRBs) have come from observations of afterglows over a wide spectral range [1]. Many GRB models predict very high energy (VHE, > 100 GeV) emission from GRBs at a level comparable to that at MeV energies (e.g. [2, 3]). EGRET detected several GRBs at energies above 100 MeV, indicating that the spectrum of GRBs extends at least out to 1 GeV, with no evidence for a spectral cut-off [4]. A second component was also found in one burst which extended up to at least 200 MeV and had a much slower temporal decay than the main burst [5]. At very high energies, there has been no conclusive emission detected for any single GRB, though a search for counterparts to 54 BATSE bursts with Milagrito, a prototype of Milagro, found evidence for emission from one burst, with an after trials significance slightly greater than 3σ [6]. At these high energies, gamma rays are attenuated by the redshift-dependent extra-galactic background light (EBL) [7], making the detection of VHE emission from GRBs very challenging. A search for an excess of events above those due to the background was carried out for each of the 106 satellite- detected GRBs in our sample (see Table 1). These represent all the GRBs known to have occurred within the field of view of Milagro during its first seven years of operations (2000-2006)2. Milagro detected no significant emission from any of these bursts, and fluence upper limits are given in Table 1. We acknowledge Scott Delay and Michael Schneider for their dedicated efforts in the construction and maintenance of the Milagro experiment. This work has been supported by the National Science Foundation (under grants PHY-0245234, -0302000, -0400424, -0504201, -0601080, and ATM-0002744) the US Department of Energy (Office of High-Energy Physics and Office of Nuclear Physics), Los Alamos National Laboratory, the University of California, and the Institute of Geophysics and Planetary Physics. REFERENCES 1. van Paradijs, J., Kouveliotou, C. & Wijers, R. A. M. J. 2000, Annual Review of Astronomy and Astrophysics 38, 379 2. Dermer, C. D., Chiang, J., & Mitman, K. E. 2000, ApJ 537, 785 3. Zhang, B. & Mészáros, P. 2001, ApJ 559, 110 4. Dingus, B. L. 2001, in Aharonian, F. A. and Volk, H. J. (eds), High Energy Gamma Ray Astronomy, AIP Conf. Proc., 558, 383 5. Gonzalez, M. M. et al. 2003, Nature 424, 749 6. Atkins, R. et al. 2000, ApJL 533, L119 7. Primack, J. R. et al. 2005, in Aharonian, F., Volk, H., and Horns, D. (eds) Gamma 2004 Heidelberg, AIP Conf. Proc., 745, 23-33 1 A. Abdo, B. T. Allen, D. Berley, E. Blaufuss, S. Casanova, B. L. Dingus, R. W. Ellsworth, M. M. Gonzalez, J. A. Goodman, E. Hays, C. M. Hoffman, B. E. Kolterman, C. P. Lansdell, J. T. Linnemann, J. E. McEnery, A. I. Mincer, P. Nemethy, D. Noyes, J. M. Ryan, F. W. Samuelson, P. M. Saz Parkinson, A. Shoup, G. Sinnis, A. J. Smith, G. W. Sullivan, V. Vasileiou, G. P. Walker, D. A. Williams, X. W. Xu and G. B. Yodh 2 GRB 060218, due to its long duration of more than 2000 s moved out of Milagro’s field of view after the start of the burst. The limit presented here is for the initial 10 s hard spike reported by the instrument team. http://arxiv.org/abs/0704.0287v1 TABLE 1. GRBs in the Milagro field of view (2000-2006). Column 1 is the GRB name. A superscript refers to the number of IPN error regions in the Milagro field of view. A superscript of one implies only one of two error regions fell in the Milagro field of view, while a two implies that both did, and they are listed one after the other. Column 2 gives the duration of the burst (in seconds), column 3 the zenith angle (in degrees), column 4 the measured redshift, column 5 the satellite(s) detecting the GRB, and column 6 gives the Milagro 99% confidence upper limit on the 0.2–20 TeV fluence in erg cm−2. Numbers in bold (also labelled with a #) take into account absorption by the EBL (using the Primack 05 model) for a redshift given in column 4. Those with three dots imply the redshifts are so high that all the emission is expected to be absorbed. GRB Dur. θ z Inst. 99% UL 000113 370 21 ... BATSE 5.5e-6 0001311 12 41 ... IPN 6.5e-7 000205 23 25 ... BSAX 6.9e-7 000206 10 39 ... BSAX 9.3e-7 000212 8 2.2 ... BATSE 1.1e-6 000220 2.4 49 ... BATSE 1.1e-5 000226 10 32 ... BATSE 3.4e-6 000226b1 94.5 32 ... IPN 7.8e-7 000301C 14 38 2.03 BATSE ... 000302 120 32 ... BATSE 6.8e-6 000314 12.8 45 ... BSAX 3.6e-5 000317 550 6.4 ... BATSE 7.9e-6 000330 0.2∗ 30 ... BATSE 1.0e-6 000331 55 38 ... BATSE 1.2e-5 000402 120 48 ... BSAX 4.5e-5 000408 2.5 31 ... BATSE 1.0e-6 000424 5 36 ... BATSE 7.6e-7 000508 30 34 ... BATSE 3.7e-6 0006071 0.12 42 ... IPN 4.6e-7 000615 10 39 ... BSAX 1.6e-6 000630 20 32 ... IPN 2.2e-6 0007072 18 43 ... IPN 1.9e-6 0007072 18 41 ... IPN 1.0e-6 000727 10 41 ... IPN 2.6e-6 000730 7 19 ... IPN 4.2e-7 0008211 8 27 ... IPN 6.9e-7 0008301 8 35 ... IPN 9.1e-7 000926 25 16 2.04 IPN ... 001017 10 42 ... IPN 2.2e-6 001018 31 32 ... IPN 2.1e-6 001019 10 20 ... IPN 1.1e-6 001105 30 8.5 ... IPN 1.4e-6 001204 0.44 48 ... BSAX 1.2e-5 010104 2 20 ... IPN 4.0e-7 010220 150 27 ... BSAX 2.1e-6 010613 152 25 ... IPN 2.9e-6 010706 48 37 ... IPN 2.6e-6 010903 41 49 ... IPN 2.9e-5 010921 24.6 10 0.45 HETE 2.9e-5# 011130 83.2 34 ... HETE 3.4e-6 011212 84.4 33 ... HETE 6.7e-6 020311 11.5 27 ... IPN 1.7e-7 0204292 16 39 ... IPN 4.6e-7 0204292 16 30 ... IPN 3.0e-7 020625b 125 38 ... HETE 5.7e-6 020702 26 34 ... IPN 1.4e-6 0209081 17 19 ... IPN 7.3e-7 020914 9 5.7 ... IPN 4.2e-7 021104 19.7 13 ... HETE 7.5e-7 021112 7.1 34 ... HETE 9.4e-7 021113 20 18 ... HETE 6.4e-7 021211 6 35 1.01 HETE ... 030413 15 27 ... IPN 1.0e-6 030823 56 33 ... HETE 2.8e-6 GRB Dur. θ z Inst. 99% UL 031026 0.24 45 ... IPN 1.1e-6 031220 23.7 43 ... HETE 4.0e-6 040506 175 49 ... IPN 6.0e-6 040924 0.6 43 0.859 HETE 1.4e-3# 041211 30.2 43 ... HETE 4.8e-6 041219a 520 27 ... INTGR. 5.8e-6 050124 4 23 ... Swift 3.0e-7 050213 17 23 ... IPN 1.3e-6 050319 15 45 3.24 Swift ... 050402 8 40 ... Swift 2.1e-6 050412 26 37 ... Swift 1.7e-6 050502 20 43 3.793 INTGR. ... 050504 80 28 ... INTGR. 1.3e-6 050505 60 29 4.3 Swift ... 050509b 0.128 10 0.226? Swift 1.1e-6# 050522 15 23 ... INTGR. 5.1e-7 050607 26.5 29 ... Swift 8.9e-7 050703 26 26 ... IPN 1.2e-6 050712 35 39 ... Swift 2.5e-6 050713b 30 44 ... Swift 4.0e-6 050715 52 37 ... Swift 1.7e-6 050716 69 30 ... Swift 1.6e-6 050820 20 22 2.612 Swift ... 051103 0.17 50 0.001? IPN 4.2e-6# 051109 36 9.7 2.346 Swift ... 051111 20 44 1.55 Swift ... 051211b 80 33 ... INTGR. 2.6e-6 051221 1.4 42 0.55 Swift 9.8e-4# 051221b 61 26 ... Swift 1.8e-6 060102 20 40 ... Swift 2.0e-6 060109 10 22 ... Swift 4.1e-7 060110 15 43 ... Swift 3.0e-6 060111b 59 37 ... Swift 2.3e-6 060114 100 41 ... INTGR. 5.1e-6 060204b 134 31 ... Swift 2.7e-6 060210 5 43 3.91 Swift ... 060218 10 44.6 0.03 Swift 3.8e-5# 060306 30 46 ... Swift 7.2e-6 060312 30 44 ... Swift 3.3e-6 060313 0.7 47 ... Swift 2.7e-6 060403 25 28 ... Swift 1.0e-6 060427b 0.22 16 ... IPN 2.1e-7 060428b 58 27 ... Swift 1.1e-6 060507 185 47 ... Swift 1.8e-5 060510b 330 43 4.9 Swift ... 060515 52 42 ... Swift 9.6e-6 060712 26 35 ... Swift 3.8e-6 060814 146 23 ... Swift 2.5e-6 060904A 80 14 ... Swift 2.4e-6 060906 43.6 29 3.685 Swift ... 061002 20 45 ... Swift 4.0e-6 061126 191 28 ... Swift 4.3e-6 061210 0.8 23 0.41? Swift 6.1e-6# 061222a 115 30 ... Swift 5.6e-6

0704.0288

Specific heat and bimodality in canonical and grand canonical versions of the thermodynamic model

Specific heat and bimodality in canonical and grand canonical versions of the thermodynamic model G. Chaudhuri ∗† and S. Das Gupta‡ Physics Department, McGill University, Montréal, Canada H3A 2T8 (Dated: October 22, 2018) Abstract We address two issues in the thermodynamic model for nuclear disassembly. Surprisingly large differences in results for specific heat were seen in predictions from the canonical and grand canon- ical ensembles when the nuclear system passes from liquid-gas co-existence to the pure gas phase. We are able to pinpoint and understand the reasons for such and other discrepancies when they appear. There is a subtle but important difference in the physics addressed in the two models. In particular if we reformulate the parameters in the canonical model to better approximate the physics addressed in the grand canonical model, calculations for observables converge. Next we turn to the issue of bimodality in the probability distribution of the largest fragment in both canonical and grand canonical ensembles. We demonstrate that this distribution is very closely related to average multiplicities. The relationship of the bimodal distribution to phase transition is discussed. PACS numbers: 25.70Mn, 25.70Pq ∗ On leave from Variable Energy Cyclotron Center, 1/AF Bidhan Nagar, Kolkata 700064, India †Electronic address: [email protected] ‡Electronic address: [email protected] http://arxiv.org/abs/0704.0288v3 mailto:[email protected] mailto:[email protected] I. INTRODUCTION In models of statistical disassembly of a nuclear system formed by the collision of two heavy ions at intermediate energy one assumes that because of multiple nucleon-nucleon collisions a statistical equilibrium is reached. The temperature rises. The system expands from normal density and composites are formed on the way to disassembly. As the system reaches between three to six times the normal volume, the interactions between composites become unimportant (except for the long range Coulomb interaction) and one can do a statistical equilibrium calculation to obtain the yields of composites at a volume called the freeze-out volume. The partitioning into available channels can be solved in the canonical ensemble where the number of particles in the nuclear system is finite (as it would be in experiments). In some experiments, the number of particles can fluctuate around a mean value. In such a case a sum of several canonical calculations could be appropriate. Even when the number of particles is fixed one can hope to replace a canonical model calculation by a grand canonical model calculation where the particle number fluctuates but the average number can be constrained to a given value. The case we will look at corresponds to this situation. Usually the grand canonical model is more easily solved. Hence it is more commonly used although in the case of nuclear physics (where particle numbers are typically ≈ 200 or less) the use of the canonical ensemble would be more appropriate. Apart from ease of calculation, there is another reason why the grand canonical model is very useful. Known properties of nuclear interactions predict that if nuclear systems were arbitrarily large (consider a fictitious system where the Coulomb interaction is switched off) disassembly of nuclear systems would show features of liquid-gas phase transition [1]. Since the grand canonical ensemble is expected to become accurate for large systems, this would seem to be a suitable framework to describe bulk properties. The canonical model, built with a constant particle number in mind, can not be pushed to arbitrarily large particle number although it can be implemented for fairly big systems containing thousands of particles. One can then extraprolate from finite particle number systems to the infinite particle number case. When this was done, a huge difference showed up between grand canonical and canonical results for cv (specific heat per particle at constant volume). In the grand canonical model cv was merely discontinuous at phase transition [2] but in the canonical model cv would go to infinity as the system particle number approached infinity [3]. This discrepancy was examined in detail for a system with 2000 particles [4]. The analysis showed that in the co-existence region, even when the average number of particles is 2000, fluctuations in the number of particles is huge if one uses the grand canonical ensemble. The cv with a fixed number of particles is sharply peaked at a temperature T where T is a function of the number of particles. Because in the grand canonical ensmble the particle number fluctuations are significant when the average number is 2000, the resulting cv is smeared out. What this analysis did not answer are two significant questions: (a) under what conditions are the grand canonical results valid and (b) under what conditions can the canonical and grand canonical results agree? We answer both these questions in this work. The apparent paradox is explained. We also address the issue of bimodality in the probability distribution of the largest fragment as a function of the mass number of the largest fragment for a finite system when the thermodynamic model is used. Clearly the canonical model is appropriate here but we also study bimodality in the grand canonical ensemble, where the average value of the number of particles is constrained to a typical value expected in heavy ion collisions. In the thermodynamic model it is easy to devise systems which may (as in the nuclear matter case) or may not (i.e., by switching off surface tension term in binding energy formula) have a phase transition (some other models may not have this versatility). Thus in this model the connection of bimodality to phase transition can be directly established. It is interesting to note (as described in detail later) that both canonical model and the approximation of a finite system with the grand canonical model show features of bimodality but quantitatively the results are quite different. The plan of the paper is as follows. In section II we set up the the formulae for the grand canonical ensemble. In section III the methodology of the canonical ensemble is presented. We point out that, although not obvious, the two ensembles actually addressed different physics causing the difference in results for specific heat. In section IV we reformulate the canonical model to better approximate the grand canonical model. We show that results for cv progressively converge. We then turn to the question of bimodality in the probability distribution of the largest fragment as a function of the fragment mass number. Formulae for the probability distribution are given in section V. In section VI we present results for this distribution and discuss the bimodality in both the canonical and grand canonical ensembles. The connection between the probability distribution of the largest fragment and average multiplicity is established in section VII. Summary is presented in section VIII. As in [1, 2, 4] we use one kind of particle and no Coulomb interaction. This is adequate for the purpose of this study and offers considerable numerical simplifications. Numerous applications of the canonical [5] and the grand canonical models [6] with two kinds of particles exist where fits experimental data are the main issues. II. FORMULAE IN THE GRAND CANONICAL MODEL If we have na particles of type a, nb particles of type b, nc particles of type c etc. all enclosed in a volume V and interactions between particles can be neglected, the grand partition function for this case can be written as Zgr = i=a,b,c... (1 + eβµiωi + e .......) = i=a,b,c,.. exp(eβµiωi) (1) Here the µi is the chemical potential and ωi the canonical partition function of one particle of type i. The average number of particles of type i is given by ∂(lnZgr)/∂(βµi) : ni = e βµiωi (2) It is possible that one of the species can be built from two other species. In reverse, a heavier species can also break up into two lighter species. If α number of particles of type a can combine with β number of particles of type b to produce γ number of particles of type c, then chemical equilibrium implies [7] that the chemical potentials of a, b and c are related by αµa + βµb = γµc. In our model we have N nucleons in a volume V (which is significantly larger than the normal nuclear volume) but these nucleons can be singles or form bound dimers, trimers etc. Chemical equilibrium implies that a composite with k bound nucleons has a chemical poten- tial kµ where µ is the chemical potential of the monomer (nucleon). Thus our ensemble has monomers, dimers, trimers etc. upto some species with kmax bound nucleons. In the actual world of nuclear physics kmax terminates around 250 because of Coulomb interaction but in the model pursued here we may terminate it arbitrarily at 1 (monomers only), 2(monomers and dimers), 3 or any large kmax. It was demonstrated in [8] that liquid-gas type phase transition occurs for large kmax > 2000. The total number of nucleons will be denoted by N . Of course, the grand canonical ensemble works best when N is very large, ideally infinite. We now look into ωi, the partition function of one composite of i nucleons.This factors into two parts, a traditional translation energy part and an intrinsic part: ωi = zi(tran)zi(int) where zi(tran) = exp(−βp2/2mi)d (2πmiT ) 3/2 (3) The intrinsic part zi(int) of course contains the key to phase transition. If we regard each composite to exist only in a ground state with energy e i , then zi(int) = exp(−βe i ). We use e i = −iW +σi 2/3 where nuclear physics sets W=16 MeV and σ = 18 MeV. This simple model itself will lead to the main results of this paper. Because of the surface term, energy per particle drops as i grows. Let us denote by F the free energy of the N nucleons where N is the total number of nucleons; E be the energy and S, the entropy: F = E−TS. At finite temperature F will go to its minimum value. The key issue is how the system of N nucleons breaks up into clusters of different sizes as the temperature changes. At low temperature E and hence F minimises by forming very large clusters (liquid). But as the temperature increases S will increase by forming larger number of clusters thus breaking up the big clusters. Gaseous phase will appear. How exactly this will happen requires calculation and these show that the system goes through a first order liquid-gas phase transition [2, 8]. As is the common practice, we used here a slightly more sophisticated model for zi(int). We make the surface tension temperature dependent in conformity with usual parametrisation [9]; σ(T ) = σ0[(T c − T 2)/(T 2c + T 2)]5/4. Here σ0 =18 MeV and Tc=18 MeV. At T = Tc surface tension vanishes and we have a fluid only. For us this is unimportant as our focus will be the temperature range 3 to 8 MeV. Also in zi we include not only the ground state but also the excited states of the composite in the Fermi-gas approximation [1, 9]. The expression for zi(int) is now complete and easily tractable. Let us now summarise the relevant equations. For k = 1 (the nucleon which has no excited states) (2πmT )3/2 exp(µ/T ) (4) and for 1 < k ≤ kmax (2πmT )3/2k3/2 exp[(µk +Wk + kT 2/ǫ0 − σ(T )k 2/3)/T ] (5) Here nk is the average number of composites with k nucleons. In the rest of the paper, for brevity, we will omit the qualifier “average”. A useful quantity is the multiplicity defined as kmax∑ nk (6) The number of nucleons bound in a composite with k nucleons is knk and obviously N = ∑kmax k=1 knk. The pressure is given by kmax∑ T (7) This follows from the identity pV = T lnZgr. Quantities like N, V, nk are all extensive variables. These equations can all be cast in terms of intensive variables like N/V = ρ, nk/N etc so that we can assume both N and V approach very large values and fluctuations in the number of particles can be ignored. Thus for a given temperature and density we solve for µ using (2πmT )3/2 (exp(µ/T ) + kmax∑ k5/2 exp[(µk +Wk + kT 2/ǫ0 − σ(T )k 2/3)/T ]) (8) The sum rule N = ∑kmax k=1 knk changes to 1 = knk/N . The energy per particle is given by kmax∑ Ek (9) where Ek = T for k=1 and for 1 < k ≤ kmax T + k(−W + ) + σ(T )k2/3 − T [∂σ(T )/∂T ]k2/3 (10) The term T [∂σ(T )/∂T ]k2/3 arises from the temperature dependence of the surface tension. The effect of this term is small. Eqs. (9) and (10) follow from the identity E = µN− ∂ lnZgr. From what we have described so far it would appear that V in eqs.(3) to (9) is the freeze-out volume V , the volume to which the system has expanded. Actually if the freeze- out volume is V then in these equations we use Ṽ which is close to V but less. The reason for this is the following. To a good approximation a composite of k nucleons is an incompressible sphere with volume k/ρ0 where the value of ρ0 is ≃ 0.16 fm −3. The volume available for translational motion (eq.(3)) is then Ṽ = V − Vexcluded where we approximate Vexcluded ≃ N/ρ0 = V0 the normal volume of a nucleus with N nucleons. Similar corrections are implicit in Van der Waals equation of state. This is meant to take care of hard sphere interactions between different particles. This answer is approximate. The correct answer is multiplicity dependent. The approximation of non-interacting composites in a volume gets to be worse as the volume decreases. We restrict our calculation to volumes V greater than 2V0. This is how the calculations reported proceed. We choose a value of V0/V = ρ/ρ0 from which V0/Ṽ = ρ̃/ρ0 = ρ/(ρ0 − ρ) is deduced. This value of ρ̃ is used in eq.(8) to calculate µ and all other quantities. We plot results as function of ρ/ρ0. If we plotted them as function of ρ̃/ρ0 the plot would shift to the right. III. THE CANONICAL MODEL SOLUTION The statistical equilibrium model as described above can be solved for a given fixed number of particles when the number of particles N is finite. No spread in the number of particles, which is inherent in the grand canonical ensemble, needs to be made. Extensive use of the canonical model has been made to fit experimental data [5] so just an outline will be presented for completeness. Among other applications, the canonical model can be used to study finite particle number effects on phase transition characteristics. Consider again N identical particles in an enclosure V and temperature T . These N nucleons will combine into monomers, dimers, trimers etc. The partition function of the system in the canonical ensemble can be written as Here ωi is the one particle partition function of a composite which has i nucleons. We already encountered ωi in section II: ωi = zi(tran)zi(int) with zi(tran) and zi(int) given in detail. Other forms for ωi can be used in the method outlined here. The summation in eq.(11) is over all partitions which satisfy N = ini. The summation is non-trivial as the number of partitions which satisfy the sum is enormous. We can define a given allowed partition to be a channel. The probablity of the occurrence of a given channel P (~n) ≡ P (n1, n2, n3....) is P (~n) = ∏ (ωi)ni . (12) The average number of composites of i nucleons is easily seen from the above equation to ni = ωi Since ini = N , one readily arrives at a recursion relation [10] kωkQN−k (14) For one kind of particle, QN above is easily evaluated on a computer for N as large as 3000 in matter of seconds. It is this recursion relation that makes the computation so easy in the model. Of course, once one has the partition function all relevant thermodynamic quantities can be computed. For example, eq. (7) still gives the expression for pressure although one could correct for the center of mass motion by reducing the multiplicity by 1: p = T (M − 1)/Ṽ . The chemical potential can be calculated from µ = F (N) − F (N − 1) where the free energy is F (N) = −T ln QN which is readily available from the calculation. IV. GENERATING GRAND CANONICAL RESULTS FROM THE CANONICAL ENSEMBLE We first consider pressure (eq.(7)) in the grand canonical ensemble. The V in eq.(7) cancels out the V in eqs. (4)and (5) and thus pressure is given in terms of intensive variables directly. We may assume that this is truly the pressure in infinite systems (V and nk arbitrarily large in which case fluctuations in the grand canonical ensemble can be ignored). However the grand canonical answer does depend upon the value of kmax. In Fig.1 we have used kmax=2000, a value large enough so that liquid-gas transition type features emerge (the flatness of pressure against ρ). For kmax significantly lower, the flatness disappears (see details in [8]). In Fig.1 we also show several canonical model results all with the same kmax=2000 but different values of N . For N=2000, the canonical results are quite different from the grand canonical results except for very low densities. In particular a region of mechanical instability is seen which can give rise to a region of negative cp, the specific heat per particle at constant pressure (see [5] for detailed discussion). In the same figure, we have also shown prssures in the canonical model when N=100,000 and 500,000. We see that the pressure approaches the grand canonical value as N increases (the periodicity obvious in the curve for N= 100,000 arises from the fact the largest composite has k =2000 but we will not get into a detailed analysis here). The conclusion here is that the grand canonical value of pressure in Fig.1 refers to a system which has N = ∞ where the largest cluster has k = 2000. This is quite different from the usual canonical model result which would have N = kmax = 2000. To address the physics of the grand canonical model we keep kmax still at 2000 but need to keep on increasing the value of N . Then the canonical results converge towards the grand canonical values. For this given problem we have approached the grand canonical result from a canonical ensemble. One can consider the reverse problem: getting the canonical result starting from the grand canonical model. It is of course obvious that the correspondence would be exact provided one uses the appropriate V in the grand canonical ensemble and then projects from it the part which has an exact N . This is because the grand canonical ensemble is a particular weighted sum of canonical ensembles with different N ’s. Let us now turn to Fig.2 which deals with cv, the specific heat per particle at constant volume. We again keep kmax = 2000. One finds that the cv in a canonical calculation for N = kmax = 2000 produces a very sharp peak. The grand canonical expression for energy per particle (eq.(9)) is an intensive quanity and so is its derivative cv. We expect this grand canonical result for cv is valid for N very large. Comparison shows that the grand canonical result differs drastically from the canonical N=2000 result in a very narrow window when the system passes from the co-existence to a pure gas phase. Can we make the results converge by successively increasing the value of N in the canonical model? The answer is “yes” as Fig.2 demonstrates. We see that the canonical result with kmax=2000 approaches the grand canonical result with kmax=2000 as the number of particles N in the canonical calculations is progressively increased beyond N=2000. To summarise: the grand canonical model is applicable when N >> kmax. The limit N >> kmax; kmax → ∞ is robust (as shown in [8]) and produces a first order phase transition. This model is distinct from the canonical model N = kmax; kmax → ∞. There is no scaling in this latter model:N = kmax, both N and kmax very large is not equivalent within a factor of scaling to a system with 2N = 2kmax. We are unable to provide a robust limit for the canonical model of N = kmax;N → ∞. Fig. 3 shows the progression of the E/N and pressure curves as N = kmax increases from 2000 to 50,000. V. BIMODALITY: THE BASIC FORMULAE IN GRAND CANONICAL AND CANONICAL ENSEMBLE In event by event analysis in experiments, one can in principle ascertain the largest mass (or the largest charge) emerging in each event from multifragmentation. The probability distribution of this largest mass can be plotted as a function of the value of mass of this largest fragment. It is shown that a bimodality in this distribution at a certain temperature is a signature of a first order phase transition: that is, if the system were infinitely large it would have a first order phase transition [12, 13]. Thus from a finite system one can have a signal for phase transition. We will now see how the probability distribution of the largest fragment as a function of the mass of the largest fragment can be computed in the thermodynamic model in the two ensembles. First the grand canonical ensemble. The grand canonical ensemble works best for a large system and we have already seen in the previous section that application of this model to multifragmentation of finite nuclei can lead to serious errors in some temperature (equivalently energy) window. Nonetheless, let us proceed to see how results can be derived. We fix a value for ρ/ρ0 (in figs. (4) and (5) we have kept this at 0.25) and choose the appropriate value of the volume so that the average number N = ∑kmax 1 knk is 150, the system whose results we show. The heaviest composite allowed in the model kmax is also 150. From eqs. (1) and (2) one can derive that the probability that a particular composite with k nucleons does not occur at all is and the probability that it occurs at least once or more is enk−1 Note that our nk here is the same as nk’s of eqs. (4) and (5), the average values in the grand canonical ensemble. The probability that k is the highest mass fragment in an event is then given by (k < kmax) Pm(k) = enk − 1 e−(nk+1+nk+2+.....nkmax) (15) From the above eq. one readily derives Pm(k + 1) Pm(k) enk+1 − 1 enk − 1 enk = enk+1 − 1 1− e−nk If nk+1 > nk then Pm(k + 1) > Pm(k). If further both nk+1 and nk are small compared to 1 Pm(k + 1) Pm(k) Let us turn to the calculation of the probability disribution of the largest fragment as a function of the mass of the largest fragment in the canonical model. A detailed formulation when two kinds of particles are present was given in a recent paper [11] but for competeness, we review the development.There is an enormous number of channels in Eq.(11). Different channels will have different values for the largest fragment. For example there is a term in the sum of Eq.(11). In this channel all the fragments and hence also the largest fragment has mass 1. The probability of this channel occurring is (from Eq.(12)) Pm(1) = The full partition function can be written as QN = Q(ω1, ω2, ω3, .......ωkmax). If we construct a QN where we set all ω’s except ω1 to be zero then this QN(ω1, 0, 0, 0..............) = and this has the largest mass 1. Consider now constructing a QN with only two ω’s : QN(ω1, ω2, 0, 0, 0, ......). This will have the largest mass sometimes 1 (as is still there) and sometimes 2 (as, for example, in the term (N−6)! ). It then follows that Pm(k) = QN (ω1, ω2, ...ωk, 0, 0, 0..)−QN (ω1, ω2, ...ωk−1, 0, 0, 0...) In the above the first term in the numerator takes care of the occurrence of all partitions where the largest fragment is between 1 and k and the second term takes care of all the partitions where the largest fragment is between 1 and k−1. The difference, divided by QN is the desired answer. Since one has the general formula for the probability Pm(k), one can compute the average value of the mass of the largest fragment as well as the root mean square deviation. In fact, these have been measured in some experiments [14] and have recently been calculated [11]. But we will not need this for this paper. VI. REPRESENTATIVE RESULTS The probability distribution Pm(k) of the largest fragment as a function of k where k is the largest fragment in an event is shown in Fig.(4) where the freeze-out density ρ/ρ0 is 0.25 and the dissociating system has N=150 (for the grand canonical ensemble the average value is 150). The canonical and grand canonical results are quite different but both display bimodality (there are two maxima with similar heights), the grand canonical at temperature ≈ 5.9 MeV and the canonical at temperature ≈ 6.2 MeV. In Fig.(5) we have compared the nk’s of the two models. Near the end value 150 the differences are very substantial at all temparatures. At lower values of k they agree very well at T = 6.8 MeV, quite well at T=6.2 MeV but gets worse at lower temperatures becoming quite different at T=5.0 MeV. These differences have been noted and discussed before [15]. VII. CONNECTION BETWEEN PROBABILITY DISTRIBUTION OF THE LARGEST FRAGMENT AND AVERAGE MULTIPLICITY The very first experiments in heavy ion collisions measured nk, the average multiplicity against k. One of the earliest postulates was the following. At low energy nk first falls with k but after reaching a minimum rises again. This is the so-called “U” shape. This shape at lower temperature is an indication that the system will undergo a liquid-gas type phase transition. As the energy of collision increases, the height of the maximum on the heavier side will decrease, will then disappear (this marks the phase transition temperature). At higher energy, nk decreases monotonically with k. This is discussed in many places including [1, 3]. Basically then one looks at the behaviour of nk as a function of k and energy as one signature of phase transition. Since bimodality in the probability distribution is also a signature of phase transition, we hope to get a connection between Pm(k) and nk. For bimodality one requires that after the minimum following the first maximum, Pm(k) will rise again witk k. Similarly in conjectures involving the multiplicity, nk, after reaching a minimum must rise again with k. These two features are intimately related. In the grand canonical model this is very simple to prove. Equation (16) shows that if nk+1 > nk then Pm(k+1) > Pm(k) and bimodality can happen. The reverse is not true; nk+1 < nk does not imply that Pm(k + 1) is less than Pm(k). There is similar connection in the canonical model. Here it can be proven that on the heavier side k > N/2, a rise of nk with k guarantees that Pm(k) will rise with k. In fact it is even more direct than that. For k > N/2, we have an equality; Pm(k) = nk. This can be proven from eq. (18) but there is a an easier proof. We can rewrite Pm(k) as a sum of terms: Pm(k) = P m(k) + P m(k) + P m(k) + ........ (19) where in each of the terms in the right hand side k is the highest mass that occurs but in P 1m(k) the composite k occurs only once, in P m(k) it occurs twice, in P m(k) it occurs three times and so on. Specifically, P 1m(k) = ; N − k = i=1 ini P 2m(k) = ; N − 2k = i ini It is clear what the structures for higher terms in the series will be. It is then also obvious nk = P m(k) + 2P m(k) + 3P m(k) + .......+O.C. (20) In the above, O.C. stands for other channels where mass k occurs but it is not the highest mass in the channel. If k > N/2 then only P 1m(k) exists. Thus for k > N/2 we have Pm(k) = nk. If nk rises with k in this region then so does Pm(k). For k ≤ N/2, the relationship is nk ≥ Pm(k), with nk usually significantly larger than Pm(k). For bimodality to exist we need to have nk rising with k in some region k > N/2. The relation Pm(k) = nk for k > N/2 is not limited to the thermodynamic model only. It is true in any number conserving model. VIII. SUMMARY This paper had two goals. One, to resolve and understand the difference between grand canonical and canonical values of specific heat and pressure in thermodynamic models as applied to heavy ion collisions. This issue we believe is resolved. Second, to understand the link between bimodality in the distribution of the heaviest fragment and the average multiplicity of fragments (which has also been linked with aspects of phase transition). We think we have gained an understanding. In a later publication we expect to show more results for bimodal distributions for realistic cases with two kinds of particles and the Coulomb interactions included. Calculations for a particular case have already appeared [11]. IX. ACKNOWLEDGEMENT This work is supported by the Natural Sciences and Engineering Research Council of Canada. [1] S. Das Gupta, A. Z. Mekjian and M. B. Tsang, Advances in Nuclear Physics, vol. 26, 91, (2001). [2] K. A. Bugaev, M. I. Gorenstein, I. N. Mishustin and W. Greiner, Phys. Rev. C62, 044320 (2000). [3] S. Das Gupta and A. Z. Mekjian, Phys. Rev. C57, 1361 (1998). [4] C. B. Das, S. Das Gupta and A. Z. Mekjian, Phys. Rev. C68 , 031601(R) (2003) [5] C. B. Das, S. Das Gupta, W. G. Lynch, A. Z. Mekjian and M. B. Tsang, Phys. Rep. 406, 1 (2005) [6] S. Das Gupta and A. Z. Mekjian, Phys. Rep. 72, 131 (1981) [7] F. Reif, Fundamentals of statistical and thermal physics,(McGraw Hill, New York, 1965) Chap- ter 8. [8] G. Chaudhuri, S. Das Gupta and M. Sutton, Phys. Rev. B74, 174106 (2006). [9] J. P. Bondorf, A. S. Botvina, A. S. Iljinov, I. N. Mishustin and K. Sneppen, Phys. Rep. 257, 133 (1995). [10] K. C. Chase, and A. Z. Mekjian, Phys. Rev C52, R2339, (1995) [11] G Chaudhuri and S. Das Gupta, Phys. Rev. C75, 034603 (2007) [12] F. Gulminelli and Ph. Chomaz, Phys. Rev. C 71, 054607 (2005). [13] M. Pichon et al., Nucl. Phys. A779, 267 (2006). [14] J. B. Elliott et al., Phys Rev C67,024609 (2003) [15] C. B. Das, S. Das Gupta, and B. K. Jennings, Phys. Rev. C70,044611(2004) 0 0.1 0.2 0.3 0.4 0.5 ρ /ρ0 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.011 T= 6 MeV k max=2000 FIG. 1: Pressures calculated in the canonical model compared with pressure calculated in the grand canonical model. For all of these the largest composite allowed has 2000 nucleons (kmax = 2000) and the temperature is 6 MeV. The grand canonical calculation is the solid curve. The canonical calculations are done with N=2000 (dash-dot), N=100,000 (dots) and N=500,000(dash). As N increases, agreement with the grand canonical result becomes better and better. 6 6.5 7 7.5 8 Temperature (MeV) 6 6.5 7 7.5 8 Temperature (MeV) ρ/ ρ0 = 0.25 max =2000 ρ/ρ0=0.25 k max=2000 FIG. 2: Specific heat at constant volume in grand canonical and canonical models. Two different scales are needed to highlight differences in values. Again the canonical calculations are done with N=2000 (dash-dot), N=100,000 (dots) and N=500,000 (dash). The difference between the grand canonical result (solid) and the N=2,000 calculation is huge around 7 MeV (upper panel) but for N=100,000 and N=500,000 the results are so close to the grand canonical values that they are nearly indistinguishable in the scale of the upper panel. In the lower panel results for N=100,000 and N=500,000 are compared with grand canonical values. Even in this expanded scale the N=500,000 canonical results are indistinguishable in the curve from the grand canonical results. 0 0.1 0.2 0.3 0.4 0.5 0.005 0.0075 6.5 7 7.5 8 Temperature (MeV) T = 6 MeV ρ/ρ0 =0.25 FIG. 3: Pressure and energy per particle for the canonical model of N = kmax for N=2000 (solid), N=10,000 (dash) and N=50,000 (dot). 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 T= 5.0 MeV T=6.2 MeV T=6.8 MeV T= 5.9 MeV FIG. 4: Probability that the largest cluster has k nucleons plotted as a function of k in the grand canonical(solid) and the canonical model(dot). Here N=150 in the canonical model and in the grand canonical model the average value is set at N=150. The value of kmax is also 150. The density is fixed at ρ/ρ0 = 0.25. In the grand canonical model bimodality is seen at about 5.9 MeV and in the canonical model this appears at about 6.2 MeV. 1 10 100 1 10 100 1 10 100 1 10 100 T= 5.0 MeV T=6.2 MeV T=6.8 MeV T= 5.9 MeV FIG. 5: For the same cases as above, the average multiplicity of each composite plotted as a function of mass number k. Note that the grand canonical results (solid) approximate the canonical results (dots) quite well at the highest temperature (except at very high mass numbers) but the agreement worsens as the temperature is lowered. Introduction Formulae in the grand canonical model The canonical model solution Generating grand canonical results from the canonical ensemble Bimodality: the basic formulae in grand canonical and canonical ensemble Representative Results Connection between probability distribution of the largest fragment and average multiplicity Summary Acknowledgement References

0704.0289

Vortex proliferation in the Berezinskii-Kosterlitz-Thouless regime on a two-dimensional lattice of Bose-Einstein condensates

Vortex proliferation in the Berezinskii-Kosterlitz-Thouless regime on a two-dimensional lattice of Bose-Einstein condensates V. Schweikhard, S. Tung, and E. A. Cornell[*] JILA, National Institute of Standards and Technology and University of Colorado, and Department of Physics, University of Colorado, Boulder, Colorado 80309-0440 (Dated: October 30, 2018) We observe the proliferation of vortices in the Berezinskii-Kosterlitz-Thouless regime on a two- dimensional array of Josephson-coupled Bose-Einstein condensates. As long as the Josephson (tun- neling) energy J exceeds the thermal energy T , the array is vortex-free. With decreasing J/T , vortices appear in the system in ever greater numbers. We confirm thermal activation as the vortex formation mechanism and obtain information on the size of bound vortex pairs as J/T is varied. PACS numbers: 03.75.Lm, 03.75.Gg, 74.50.+r, 74.81.Fa, 67.90.+z One of the defining characteristics of superfluids is long-range phase coherence [1], which may be destroyed by quantum fluctuations, as in the Mott-insulator tran- sition [2, 3], or thermal fluctuations, e.g. in one- dimensional Bose gases [4, 5] and in a double-well system [6]. In two dimensions (2D), Berezinskii [7], Kosterlitz and Thouless [8] (BKT) developed an elegant description of thermal phase fluctuations based on the unbinding of vortex-antivortex pairs, i.e. pairs of vortices of opposite circulation. The BKT picture applies to a wide variety of 2D systems, among them Josephson junction arrays (JJA), i.e. arrays of superfluids in which phase coherence is mediated via a tunnel coupling J between adjacent sites. Placing an isolated (free) vortex into a JJA is ther- modynamically favored if its free energy F = E−TS ≤ 0. In an array of period d the vortex energy diverges with ar- ray size R as E ≈ J log(R/d) [9], but may be offset by an entropy gain S ≈ log(R/d) due to the available ≈ R2/d2 sites. This leads to a critical condition (J/T )crit ≈ 1 in- dependent of system size, below which free vortices will proliferate. In contrast, tightly bound vortex-antivortex pairs are less energetically costly and show up even above (J/T )crit. The overall vortex density is thus expected to grow smoothly with decreasing J/T in the BKT crossover regime. Transport measurements, both in continuous superflu- ids [10, 11] and superconducting JJA [12] have confirmed the predictions of BKT, without however directly observ- ing its microscopic mechanism, vortex-antivortex unbind- ing. A recent beautiful experiment [13] in a continuous 2D Bose gas measured the phase-phase decay function through the BKT cross-over, and saw evidence for ther- mal vortex formation. For related theoretical studies see e.g. [14]. In this work we present more detailed vortex- formation data, collected in a 2D array of BECs with ex- perimentally controllable Joephson couplings. The sys- tem was studied theoretically in [15]. Our experiment starts with production of a partially Bose-condensed sample of 87Rb atoms in a harmonic, ax- ially symmetric magnetic trap with oscillation frequen- cies {ωx,y, ωz} = 2π{6.95, 15.0}Hz. The number of con- densed atoms is kept fixed around 6 × 105 as the tem- perature is varied. We then transform this system into 4.7µm . . .. . . .. ... .. Nwell FIG. 1: Experimental system. (a) 2D optical lattice intensity profile. A lattice of Josephson-coupled BECs is created in the white-shaded area. The central box marks the basic building block of our system, the double-well potential shown in (b). The barrier height VOL and the number of condensed atoms per well, Nwell, control the Josephson coupling J , which acts to lock the relative phase ∆φ. A cloud of uncondensed atoms at temperature T induces thermal fluctuations and phase de- fects in the array when J < T . (c) Experimental sequence: A BEC (i) is loaded into the optical lattice over 10 s, suppress- ing J to values around T. We allow 2 s for thermalization. To probe the system, we ramp off the lattice on a faster timescale tr [23] and take images of the recombined condensate. When J is reduced below T (ii)-(vii), vortices (dark spots) appear as remnants of the thermal fluctuations in the array. a Josephson junction array, as illustrated in Fig. 1. In a 10 s linear ramp, we raise the intensity of a 2D hexag- onal optical lattice [16] of period d = 4.7µm in the x-y plane. The resulting potential barriers of height VOL be- tween adjacent sites [Fig.1(b)] rise above the condensate’s chemical potential around VOL ≈ 250− 300Hz, splitting it into an array of condensates which now communicate http://arxiv.org/abs/0704.0289v3 only through tunneling. This procedure is adiabatic even with respect to the longest-wavelength phonon modes of the array [18, 19] over the full range of VOL in our exper- iments. Each of the ≈ 190 occupied sites (15 sites across the BEC diameter 2×RTF ≈ 68µm [20]) now contains a macroscopic BEC, withNwell ≈ 7000 condensed atoms in each of the central wells at a temperature T that can be adjusted between 30−70nK. By varying VOL in a range between 500Hz and 2 kHz we tune J between 1.5µK and 5nK, whereas the “charging” energy Ec, defined in [1], is on the order of a few pK, much smaller than both J and T . In this regime, thermal fluctuations of the rela- tive phases ∆φTh ≈ T/J are expected, while quantum fluctuations ∆φQ ≈ (Ec/4J)1/4 are negligible [1]. The suppression of the Josephson coupling greatly sup- presses the energy cost of phase fluctuations in the x-y plane, between condensates, J [1− cos(∆φ)], compared to the cost of axial (z) phase fluctuations inside the con- densates [21]. As a result, axial phase fluctuations re- main relatively small, and each condensate can be ap- proximated as a single-phase object [22]. After allowing 2 s for thermalization, we initiate our probe sequence. We first take a nondestructive thermom- etry image in the x-z plane, from which the temperature T and, from the axial condensate size Rz, the number of condensed particles per well, Nwell, is obtained (see below). To observe the phase fluctuations we then turn down the optical lattice on a time-scale tr [23], which is fast enough to trap phase winding defects, but slow enough to allow neighboring condensates to merge, pro- vided their phase difference is small. Phase fluctuations are thus converted to vortices in the reconnected conden- sate, as has been observed in the experiments of Scherer et al. [24]. We then expand the condensate by a factor of 6 and take a destructive image in the x-y plane. Figure 1(c) illustrates our observations: (ii)-(vii) is a sequence of images at successively smaller J/T (measured in the center of the array [25]). Vortices, with their cores visible as dark “spots” in (iii)-(vii), occur in the BEC center around J/T = 1. Vortices at the BEC edge ap- pear earlier, as here the magnetic trap potential adds to the tunnel barrier, suppressing the local J/T below the quoted value. That the observed “spots” are indeed circulation-carrying vortices and antivortices is inferred from their slow ≈ 100ms decay after the optical lattice ramp-down, presumably dominated by vortex-antivortex annihilation. From extensive experiments on vortices in our system we know that circulation-free “holes” fill so quickly due to positive mean field pressure, that they do not survive the pre-imaging expansion. Vortices with identical circulation would decay by dissipative motion to the BEC edge, in our trap over & 10 s. To investigate the thermal nature of phase fluctua- tions, we study vortex activation while varying J at different temperatures. For a quantitative study, accu- rate parameter estimates are required. The Josephson- coupling energy J is obtained from 3D numerical sim- ulations of the Gross-Pitaevskii equation (GPE) for 0.1 1 10 0.1 1 10 100 cold: 30<T<40nK hot: 55<T<70nK 10 100 1000 J [nK] cold: 30<T<40nK hot: 55<T<70nK 10 100 1000 J [nK] (b) (c) FIG. 2: Quantitative study of vortices. The areal density of vortices is quantified by the plotted D defined in the text. D is extracted only from the central 11% of the condensate region [circle in inset (a)] to minimize effects of spatial inhomogene- ity. (b) D vs J for two datasets with distinct “cold” and “hot” temperatures. Each point represents one experimental cycle. The increase in D with decreasing J . 100nK signals the spontaneous appearance of vortices, while the “background” D . 0.01 for J & 200nK is not associated with vortices. Vortices clearly proliferate at larger J for the “hot” data, indicating thermal activation as the underlying mechanism. The large scatter in D at low J is due to shot noise on the small average number of vortices in the central condensate region. (c) same data as in (b), but averaged within bins of size ∆[log(J)] = 0.15. Error bars of D are standard errors. (d) same data as (b), but plotted vs J/T . “Cold” and “hot” datasets almost overlap on what appears to be a universal vortex activation curve, as confirmed by averaging [inset (e)], clearly revealing the underlying competition of J and T . the central double-well system [6, 26] [Fig.1(b)], self- consistently including mean-field interactions of both condensed and uncondensed atoms [27]. A use- ful approximation for J in our experiments is [25]: J(VOL, Nwell, T ) ≈ Nwell × 0.315nK exp[Nwell/3950 − VOL/244Hz](1+0.59T/100nK). The finite-T correction to J arises from both the lifting-up of the BEC’s chem- ical potential and the axial compression by the thermal cloud’s repulsive mean field, but does not take into ac- count the effects of phase fluctuations on J (in condensed- matter language, we calculate the bare J). Nwell is de- termined by comparison of the experimentally measured Rz, to Rz(VOL, Nwell, T ) obtained from GPE simula- tions. Both experimental and simulated Rz are obtained from a fit to the distribution of condensed and uncon- densed atoms, to a Thomas-Fermi profile plus mean-field- modified Bose function [27]. In determination of all J values, there is an overall systematic multiplicative un- certainty ∆J/J = × 1.6, dominated by uncertainties in the optical lattice modulation contrast, the absolute in- tensity calibration, and magnification in the image used to determine Nwell. In comparing J for “hot” and “cold” clouds (see Fig. 2) there is a relative systematic error of 15% associated with image fitting and theory uncertain- ties in the thermal-cloud mean-field correction to J . The qualitative results of our work are consistent whether we use an automated vortex-counting routine or count vortices by hand, but the former shows signs of saturation error at high vortex density, and the latter is vulnerable to subjective bias. As a robust vortex-density surrogate we therefore use the “roughness” D of the con- densate image caused by the vortex cores. Precisely, we define D as the normalized variance of the measured col- umn density profile from a fit to a smooth finite-T Bose profile [27], with a small constant offset subtracted to account e.g. for imaging noise. To limit spatial inhomo- geneity in J , caused by spatially varying condensate den- sity and optical lattice intensity, to < 10%, D is extracted only from the central 11% of the condensate area which contains 20 lattice sites [Fig. 2(a)]. Comparison to au- tomated vortex-counts shows that D is roughly linear in the observed number of vortices, irrespective of the sign of their circulation, with a sensitivity of ≈ 0.01/vortex. Figure 2 shows results of our quantitative study. In Fig. 2(b), we plot D vs J for two datasets with dis- tinct temperatures. At large J & 200nK a background D . 0.01 is observed, that is not associated with vortices, but due to residual density ripples remaining after the optical lattice ramp-down. Vortex proliferation, signaled by a rise of D above ≈ 0.01, occurs around J ≈ 100nK for “hot” BECs and at a distinctly lower J ≈ 50nK for “cold” BECs [confirmed by the averaged data shown in Fig. 2(c)], indicating thermal activation as the vortex formation mechanism. Plotting the same data vs J/T in Fig. 2(d) shows collapse onto a universal vortex acti- vation curve, providing strong evidence for thermal ac- tivation. A slight residual difference becomes visible in the averaged “cold” vs “hot” data [Fig.2(e)], perhaps be- cause of systematic differences in our determination of J at different temperatures. The vortex density D by itself provides no distinction between bound vortex-antivortex pairs and free vortices. In the following we exploit the flexibility of optical po- tentials to distinguish free or loosely bound vortices from tightly bound vortex-antivortex pairs. A “slow” optical lattice ramp-down allows time for tightly bound pairs to annihilate before they can be imaged. By slowing down the ramp-down duration τ [inset of Fig. 3 (a)], 0.1 1 10 ramp-down timescale τ τ = 6 ms τ = 36 ms 0 20 40 60 smallest surviving pairs ramp-down timescale τ [ms] (J/T<<1) (J/T>>1) 0 20 40 60 ramp-down timescale τ [ms] 1.3kHz (c) I II & III larger pairs & free vortices FIG. 3: (a) Vortex density D probed at different optical lat- tice ramp-down timescales τ . A slow ramp provides time for tightly bound vortex-antivortex pairs to annihilate, allow- ing selective counting of loosely bound or free vortices only, whereas a fast ramp probes both free and tightly bound vor- tices. A fit to the vortex activation curve determines its mid- point (J/T )50%, its 27% − 73% width ∆(J/T )27−73, and the limiting values D< (D>) well below (above) (J/T )50%. (b) A downshift in (J/T )50% is seen for slow ramps, consistent with the occurrence of loosely bound or free vortices at lower J/T only. (c) Mapping between ramp-down timescale τ and estimated size of the smallest pairs surviving the ramp (up- per axis). The difference D< − D> measures the number of vortices surviving the ramp (right axis). Comparison to sim- ulated vortex distributions yields a size estimate of the small- est surviving pairs (upper axis). Inset: smallest possible pair sizes in a hexagonal array, I: d/ 3, II: d, III: 2d/ we therefore selectively probe vortices at increasing spa- tial scales. This represents an attempt to approach the “true” BKT vortex unbinding crossover that is comple- mentary to transport measurements employed so success- fully in superconductive and liquid Helium systems. Figure 3(a) shows vortex activation curves, probed with two different ramp-down times. Two points are worth noticing: First, a slow ramp compared to a fast one shows a reduction of the vortex density D< in arrays with fully randomized phases at low J/T . The difference directly shows the fraction of tightly bound pairs that an- nihilate on the long ramp. Second, a slower ramp shows vortex activation at lower (J/T )50%, confirming that free or very loosely bound vortices occur only at higher T (lower J). Specifically, the data clearly show a range around J/T ≈ 1.4 where only tightly bound pairs exist. Figure 3(b) quantitatively shows the shift of (J/T )50% from 1.4 to 1.0 with slower ramp time. We can make a crude mapping of the experimental ramp-down time- scale to theoretically more accessible vortex-antivortex pair sizes as follows: In Fig. 3(c), we see the decrease of the saturated (low-J/T ) vortex density D< with increas- ing ramp timescale τ . The right axis shows the inferred number of vortices that survived the ramp. We compare this number of surviving vortices to simulations [28] of a 20-site hexagonal array with random phases. In these simulations we find, on average, a total of 10 vortices, 6 of which occur in nearest-neighbor vortex-antivortex pairs [configuration I in Fig. 3(c)], 1.7 (0.4) occur in configuration II (III) respectively, and 1.9 occur in larger pairs or as free vortices. Experimentally≈ 11 vortices are observed for the fastest ramps, in good agreement with the expected total number of vortices. For just some- what slower ramps of τ ≈ 5ms, only 3 vortices survive, consistent with only vortices in configuration II & III or larger remaining (indicated in Fig. 3, top axis) [29]. For τ & 30ms ramps less than 2 vortices remain, according to our simulations spaced by more than 2d/ 3. Thus we infer that ramps of τ ≈ 30ms or longer allow time for bound pairs of spacing . 2d/ 3 to decay before we ob- serve them. The downward shift of (J/T )50% in Fig. 3(b) thus tells us that loosely bound pairs of size larger than 3, or indeed free vortices, do not appear in quan- tity until J/T ≤ 1.0, whereas more tightly bound vortex pairs appear in large number already for J/T ≤ 1.4. A further interesting observation concerns the width of the vortex activation curve. The relative width, de- termined from fits to data such as the ones shown in Fig. 3(a), is ∆(J/T )27−73/(J/T )50% ≈ 0.3, independent of ramp-down duration. This width is neither as broad as in a double-well system [6, 30], where the coherence factor rises over a range ∆(J/T )27−73/(J/T )50% ≈ 1.4, nor as broad as expected from our simulations [28] of an array of uncoupled phases, each fluctuating inde- pendently with ∆φRMS = T/J , for which we find ∆(J/T )27−73/(J/T )50% ≈ 0.85. Presumably collective effects in the highly multiply connected lattice narrow the curve. On the other hand, the width is 3 times larger than the limit due to spatial inhomogeneity in J , suggest- ing contributions to the width due to finite-size effects or perhaps revealing the intrinsically smooth behavior of vortex activation in the BKT regime. In conclusion, we have probed vortex proliferation in the BKT regime on a 2D lattice of Josephson-coupled BECs. Allowing variable time for vortex-antivortex pair annihilation before probing the system provides a time- to-length mapping, which reveals information on the size of pairs with varying J/T . We acknowledge illuminating conversations with Leo Radzihovsky and Victor Gurarie. This work was funded by NSF and NIST. [*] Quantum Physics Division, National Institute of Stan- dards and Technology. [1] A. Leggett, Rev. Mod. Phys. 73, 307 (2001). [2] M. Greiner et al., Nature 415, 39 (2002). [3] D. Jaksch et al., Phys. Rev. Lett. 81, 003108 (1998). [4] S. Richard et al., Phys. Rev. Lett. 91, 010405 (2003); [5] D. Hellweg et al., Phys. Rev. Lett. 91, 010406 (2003). [6] R. Gati et al., Phys. Rev. Lett. 96, 130404 (2006). [7] V. Berezinskii, Sov. Phys.-JETP 32, 493 (1971); 34, 610 (1972). [8] J. Kosterlitz, D. Thouless, J. Phys. C 6, 1181 (1973). [9] M. Tinkham, Introduction to Superconductivity, McGraw-Hill, Inc., New York (1996). [10] G. Agnolet et al., Phys. Rev. B 39, 8934 (1989). [11] A. Safonov et al., Phys. Rev. Lett. 81, 4545 (1998) [12] D. Resnick et al., Phys. Rev. Lett. 47, 1542 (1981). [13] Z. Hadzibabic et al., Nature 441, 1118 (2006). [14] T. Simula and P. Blakie, Phys. Rev. Lett. 96, 020404 (2006). [15] A. Trombettoni et al., New J. Phys. 7, 57 (2005). [16] Three circularly polarized laser beams (λ = 810.1nm) in- tersect in a tripodlike configuration, with θ = 6.6◦ angles to the z-axis. Calculation of the optical dipole potential [17] includes counterrotating terms and interaction with both the D1 and D2 lines, as well as the ‘fictitious mag- netic field’ due to the circular polarization. The tilted bias field of the TOP trap makes P ≈ 0.5[17]. [17] R. Grimm et al., Adv. At. Mol. Opt. Phys. 42, 95 (2000). [18] J. Javanainen, Phys. Rev. A 60, 4902 (1999). [19] K. Burnett et al., J. Phys. B 35, 1671 (2002). [20] To avoid radial flows during VOL ramp-up, RTF is kept constant by balancing the lattice-enhanced mean field pressure with radial confinement due to the optical lattice envelope, by the choice of a 67µm 1/e2 intensity waist. Axial (z) confinement is due to the magnetic trap alone. [21] D. Petrov et al., Phys. Rev. Lett. 87, 050404 (2001). [22] In the axial condensate region between z = −Rz/3,+Rz/3, where according to our 3D GPE simulations 85% of the tunnel current is localized and hence the relative phase is measured, axial phase fluctuations [21] vary between ≈ 600mrad (“cold” data in Fig. 2) and 800mrad (“hot”) in the regime J/T ≈ 1. [23] Within a dataset, the ramp-down rate is kept fixed, tr = τ × VOL/1.3 kHz, τ = 18ms if not otherwise indicated. [24] D. Scherer et al., Phys. Rev. Lett. 98, 110402 (2007). [25] J is averaged over junctions within the central 11% of the array area, from which all quantitative experimental results are extracted. [26] D. Ananikian and T. Bergeman, Phys. Rev. A 73, 013604 (2006). [27] M. Naraschewski and D. Stamper-Kurn, Phys. Rev. A 58, 2423 (1998). [28] Following [24], in simulations we count a vortex if all three phase differences in an elemental triangle of junc- tions are ∈ (0, π), or if all are ∈ (−π, 0). [29] The very short annihilation time of configuration I pairs [ τ < 5ms in Fig. 3(c) ] is not unexpected: their spacing 3 = 2.8µm is comparable to the diameter of a vortex core in the bulk condensate (after VOL rampdown). [30] L. Pitaevskii and S. Stringari, Phys. Rev. Lett. 87, 180402 (2001).

0704.0290

The core binary fractions of star clusters from realistic simulations

To appear in the Astrophysical Journal The core binary fractions of star clusters from realistic simulations Jarrod R. Hurley Centre for Astrophysics and Supercomputing, Swinburne University of Technology, P.O. Box 218, VIC 3122, Australia [email protected] Sverre J. Aarseth Institute of Astronomy, Madingley Road, Cambridge CB3 0HA, UK [email protected] Michael M. Shara Department of Astrophysics, American Museum of Natural History, Central Park West at 79th Street, New York, NY 10024 [email protected] ABSTRACT We investigate the evolution of binary fractions in star clusters using N -body models of up to 100 000 stars. Primordial binary frequencies in these models range from 5% to 50%. Simulations are performed with the NBODY4 code and include a full mass spectrum of stars, stellar evolution, binary evolution and the tidal field of the Galaxy. We find that the overall binary fraction of a cluster almost always remains close to the primordial value, except at late times when a cluster is near dis- solution. A critical exception occurs in the central regions where we observe a marked increase in binary fraction with time – a simulation starting with 100 000 stars and 5% binaries reached a core binary frequency as high as 40% at the end of the core-collapse phase (occurring at 16Gyr with ∼ 20 000 stars remaining). Binaries are destroyed in the core by a variety of processes as a cluster evolves, but the combination of mass-segregation and creation of new binaries in exchange interactions produces the observed increase in relative number. We also find that http://arxiv.org/abs/0704.0290v1 – 2 – binaries are cycled into and out of cluster cores in a manner that is analogous to convection in stars. For models of 100 000 stars we show that the evolution of the core-radius up to the end of the initial phase of core-collapse is not affected by the exact value of the primordial binary frequency (for frequencies of 10% or less). We discuss the ramifications of our results for the likely primordial binary content of globular clusters. Subject headings: binaries: close, general — globular clusters: general— meth- ods: N-body simulations — open clusters and associations: general — stellar dynamics 1. Introduction The binary content of a globular cluster is important in determining the frequency and nature of cluster stellar exotica, as well as the dynamical evolution of the cluster. It has long been recognized that binary formation is inevitable in a self-gravitating system1. Indeed, the presence of binaries as a central energy source is vital to avoid complete core-collapse (Goodman & Hut 1989). However, only more recently has it been realised that globular clusters must also have formed with a sizeable binary population (see Hut et al. 1992 for an early review). That globular clusters harbour a mixture of dynamically formed and primordial binaries can be used to understand observations of their stellar content, such as the diverse blue straggler population in 47 Tucanae (Mapelli et al. 2004). Knowledge of the likely primordial binary fraction of globular clusters is essential as input to models of globular cluster evolution. It also provides a constraint on the cluster formation process. Considering that the presence of binaries in the cluster core has a pro- nounced effect on the core properties and cluster evolution (Hut 1996), knowledge of the central binary frequency is also important. Indications are that this is relatively small – of the order of 20% (e.g. Bellazzini et al. 2002) or less (e.g. Cool & Bolton 2002) – when compared to the frequencies of binaries observed in the solar neighbourhood (Duquennoy & Mayor 1991) and open clusters such as M67 (Fan et al. 1996) which are of the order of 50%. It would be particularly useful to take measurements of the current binary fraction in globular clusters – whether that be in the core or outer regions – and extrapolate backwards to gain a reliable determination of the primordial binary content. However, processes involved 1The ten-body gravitational calculations of von Hoerner (1960) are the earliest N -body calculations published. They were continued until the first binary formed, at which point the calculations were halted. – 3 – in the intervening cluster evolution make this difficult. For example, binaries can be formed and destroyed in a variety of interactions between cluster members (Hurley & Shara 2002). Binaries will on average be more massive than single stars and thus are affected differently by mass segregation. Also, the escape rates of single stars and binaries will differ. Finally, the internal evolution of the components of binaries can also lead to binaries’ destruction. Current simulation techniques have been designed to model these (and other) processes (Aarseth 2003) and have reached the level of sophistication required to produce realistic clus- ter models. In this way the link between primordial and current cluster binary populations can be investigated directly (e.g. Hurley et al. 2005; Ivanova et al. 2005). Aarseth (1996) conducted an N -body simulation starting with 10 000 stars and a 5% binary frequency where notably the stars were drawn from a realistic initial mass function (IMF), the cluster was subject to the tidal field of the Galaxy, and both stellar and binary evolution were modelled. This model cluster had a half-life of about 2Gyr at which point the core binary frequency had risen to 20% primarily owing to mass-segregation. Thus binaries were not preferentially depleted. In this case it was not necessary to include a large initial binary fraction in order to halt core-collapse and yield a significant observed abundance in the central regions. The earlier work of McMillan & Hut (1994) reported N -body simulations of 2 000 stars or less and binary frequencies in the range of 5-20%. They included the Galactic tidal field but only considered point-mass dynamics. McMillan & Hut (1994) showed that there is a criti- cal primordial binary frequency of 10-15% below which the binaries are destroyed before the cluster dissolves owing to the tidal field. Furthermore, they found that above this critical value there exists a minimum possible binary mass fraction for the cluster – this result could be used with observations of present-day binary frequency to place limits on the primordial frequency. We note that the McMillan & Hut (1994) simulations were restricted to equal- mass stars, and the binaries were a factor of two heavier then single stars – this could give misleading results when applied to real clusters2. These N -body simulations were definitely in the open cluster regime. Dynamical pro- cesses which destroy (and equally may create) cluster binaries are density dependent. In addition, the central stellar density of a cluster is a function of the number, N , of cluster members. Thus, it is not clear that these prior results apply to globular cluster conditions. More recently Ivanova et al. (2005) have conducted Monte Carlo simulations of clusters 2 Binaries would naturally be twice as massive as single stars on average if binaries form by random pairings independent of the stellar IMF. In general correlated masses are assumed (e.g. Kroupa 1995) although the exact situation is unclear – the recent survey of stars in the solar neighborhood and in young open clusters compiled by Halbwachs et al. (2003) shows a distribution of mass-ratios, q, with a broad peak between 0.2− 0.7 but also a sharp peak for q > 0.8. – 4 – with up to 5 × 105 members and core number densities ranging from 103 − 106 stars pc−3. They show that an initial binary frequency of 100% is required to produce a current core binary frequency of 10% for a globular cluster such as 47 Tucanae. Depletion of binaries in the cluster core is found to be the result of stellar evolution processes as well as three- and four-body dynamical interactions. It is our intention in this paper to test these claims by utilising direct N -body simulations of star clusters with up to N = 100 000 members initially. One aspect that will affect the evolution of the cluster binary population is the orbital parameters of the primordial binaries – in particular the initial ratio of hard to soft binaries. The boundary between these two regimes is determined by the mean kinetic energy of the cluster stars (with binaries represented by their center-of-mass motion) where hard binaries have a binding energy in excess of 2/3 of the mean kinetic energy (Hut et al. 1992). We note that a useful estimate for the boundary in terms of the binary orbital separation is given by twice the cluster half-mass radius divided by N . In three-body single–binary star interactions hard binaries tend to harden and provide kinetic heating for the cluster (Heggie 1975; Hut 1983). Soft binaries are less strongly bound (and thus on average are wider) and are efficiently destroyed in three- and four-body encounters. As noted by Hut et al. (1992) it is for this reason that soft binaries are not generally included in cluster models. A common misconception is that the omission of soft binaries is to aid the speed of simulation; however it is binaries near the hard/soft boundary that provide the main threat to efficient simulation (Aarseth 2003). The omission is more a realisation that soft binaries have little impact on the cluster dynamics or exotic star formation and so the focus is on the more meaningful binaries, so to speak. Neglecting soft binaries has the capacity to alter binary fractions in the halo of a model cluster as binary encounters tend to occur in or near the cluster core. For this reason we will attempt to account for any omitted soft binary populations when making binary fraction comparisons. Our simulation method and initial conditions are detailed in Sections 2 and 3. Results are given in Section 4 followed by discussion in Section 5. We briefly summarize our results in Section 6. 2. Models All simulations utilized in this work were performed using the NBODY4 code (Aarseth 1999) on GRAPE-6 boards (Makino 2002) located at the American Museum of Natural History. NBODY4 uses the 4th-order Hermite integration scheme and an individual timestep algorithm to follow the orbits of cluster members and invokes regularization schemes to deal – 5 – with the internal evolution of small-N subsystems (see Aarseth 2003 for details). Stellar and binary evolution of the cluster stars are performed in concert with the dynamical integration as described in Hurley et al. (2001). The results of four extensive simulations (detailed below) form the dataset for this paper. We will make use of data from two simulations that have previously been reported in the literature – a simulation starting with 95 000 single stars and 5 000 binaries (Shara & Hurley 2006) and a simulation starting with 12 000 single stars and 12 000 binaries (Hurley et al. 2005). The former contained 100 000 members at birth, if we count each binary as one object, and thus had a primordial binary frequency of 5%. We will refer to this as the K100-5 simulation. After about nine Gyr of evolution the cluster membership was reduced by half and at an age of 15 − 16Gyr the model cluster had reached the end of the main core-collapse phase (associated with a minimum in core-radius, after which the size of the core stabilizes, in relative terms). Figure 1a shows the behaviour of the core radius as the K100-5 model evolves. Also shown is the 10% Lagrangian radius – the radius which encloses the inner 10% of the cluster by mass. From Figure 1a we see that initially the inner regions of the cluster expand owing to stellar evolution mass-loss before two-body effects take over and drive a prolonged period of contraction. When the cluster is about 12 half-mass relaxation times old (as denoted across the top of Fig. 1a) the core radius reaches a minimum of 0.17 pc and the main core-collapse phase is halted. The 10% Lagrangian radius at this point is 0.94 pc. The core density of the model begins at 102 stars pc−3 and increases to a maximum of 104 stars pc−3 just before termination of the model at 20Gyr. The core radius in Figure 1 is actually the density radius commonly used in N -body simulations (Casertano & Hut 1985). It is calculated from the density weighted average of the distance of each star from the density centre (Aarseth 2003). This definition, in combination with the effects of three-body interactions and the movement of binaries across the core boundary, allows for the small-scale fluctuations in core radius observed in Figure 1. Such fluctuations could be smoothed out (see Heggie, Hut & Trenti 2006, for example) but we have chosen not to do this. This N -body core-radius is distinct from observational determinations of core-radius calculated, for example, from the surface brightness profile (SBP) of a cluster. As discussed by Wilkinson et al. (2003) there is no general relation between the two quantities but usually the N -body value is the lesser of the two. This is supported by an in-depth analysis of the core-radius evolution of the K100-5 simulation that will be presented in an upcoming paper (Hurley, 2007, in preparation). Preliminary results show that the the core-radius obtained from the two-dimensional projected SBP of the K100-5 model agrees well with the N -body core radius for the first 7Gyr of evolution but is about twice as large by the time the model reaches 16Gyr of age. Thus the binary fraction within the 10% Lagrangian radius may often be a better number to compare with – 6 – central binary fractions quoted for real clusters and we will give both this and the core binary fraction in our results. The second model had a primordial binary frequency of 50% and was tailored to in- vestigate the evolution and stellar populations of the old open cluster M67. It had 24 000 members at birth and we will refer to this as the K24-50 simulation. It had a half-life of about 2Gyr and after 4Gyr of evolution only 2 000 stars and binaries remained. The core density was about 102 stars pc−3 on average, reaching a maximum of 350 stars pc−3 at 3 480Myr with a corresponding core radius of 0.3 pc. Figure 1b shows the evolution of the core and 10% Lagrangian radii for the K24-50 simulation. To investigate the evolution of binary fractions across a range of star cluster models we will also make use of two simulations that have yet to be published. These are a simulation that started with 90 000 single stars and 10 000 binaries (K100-10) and a simulation that started with 40 000 single stars and 10 000 binaries (K50-20). In Table 1 we summarize the properties of the four simulations. For each model the initial setup is as follows. Masses for the single stars are drawn from the IMF of Kroupa, Tout & Gilmore (1993) between the mass limits of 0.1 and 50M⊙. Each binary mass is chosen from the IMF of Kroupa, Tout & Gilmore (1991), as this had not been corrected for the effect of binaries, and the component masses are set by choosing a mass-ratio from a uniform distribution. We assume that all stars are on the zero-age main sequence (ZAMS) when the simulation begins and that any residual gas from the star formation process has been removed. We use a Plummer density profile (Aarseth, Hénon & Wielen 1974) and assume the stars and binaries are in virial equilibrium when assigning the initial positions and velocities. There is no primordial segregation by mass, binary properties, or any other discriminating factor in these models. Each cluster is subject to a standard Galactic tidal field – a circular orbit in the Solar neighborhood. Stars are removed from the simulation when their distance from the density centre exceeds twice that of the tidal radius of the cluster. The metallicity of the stars in the two simulations starting with 100 000 stars (K100-5 and K100-10) was set to be Z = 0.001 while both the K24-50 and K50-20 simulations were assigned solar metallicity (Z = 0.02). 3. Binary Period Distributions The orbital separations of the 5 000 primordial binaries in the K100-5 simulation (Shara & Hurley 2006) were drawn from the log-normal distribution suggested by Eggleton, Fitchett & Tout (1989) with a peak at 30 au. This distribution is based on the properties of doubly- – 7 – bright visual binaries in the Bright Star Catalogue (Hoffleit 1983) and is in agreement with the survey data of Duquennoy & Mayor (1991) for binaries in the solar neighbourhood – although the latter observations do not rule out a flat distribution. Orbital eccentricities of the primordial binaries were assumed to follow a thermal distribution (Heggie 1975). In the K100-5 model the initial separation distribution was capped at 100 au. With a half-mass radius of 6.7 pc for the initial model the hard/soft binary boundary is at about 30 au. Thus the maximum of 100 au excludes only the softest binaries from the distribution. Binaries with an initial pericentre distance less than five times the radius of the primary star were rejected in the setup of the model – for binaries closer than this it is assumed that interaction during the formation process and on the Hayashi track would lead to collision. Rather than enact such a collision we simply choose another set of binary parameters from the distributions. In this way the intended primordial binary fraction is preserved. The resulting period distribution of the K100-5 model is shown in Figure 2a. We see that the distribution is peaked at 105 d and does not extend beyond 106 d. The K100-10 simulation had the same binary setup as that of the K100-5 model. The K50-20 simulation also used the same Eggleton, Fitchett & Tout (1989) distribution of orbital separations but with a cap at 50 au. Primordial binaries in the M67 (or K24-50) simulation of Hurley et al. (2005) have orbital separations drawn from a flat distribution of log a (Abt 1983). An upper cutoff of 50 au was applied so that soft binaries were not included in the model – with a half- mass radius of 3.9 pc the hard/soft binary limit for the starting model was about 40 au. For this model very close primordial orbits were also rejected. The corresponding period distribution for the primordial binaries in the K24-50 simulation is shown in Figure 2b. We note that a goal of the K24-50 simulation was to reproduce the relatively large number of blue stragglers observed in M67. For this purpose an Eggleton, Fitchett & Tout (1989) separation distribution was ruled out as it did not lead to enough blue straggler production from Case A mass transfer in close binaries. Uncorrelated masses of the component stars in binaries were also ruled out for the same reason (see Hurley et al. 2005 for details). During this work we will be making comparisons to the Monte Carlo models presented by Ivanova et al. (2005). In their study binary periods were chosen from a uniform distribution in logP between the limits of 0.1 and 107 d. Thus they assumed a wider distribution of primordial binaries. If, for example, the Eggleton, Fitchett & Tout (1989) distribution used in the K100-5 simulation was extended to include all periods up to 107 d, rather than being curtailed at 100 au, the 5 000 binaries that make up the distribution shown in Figure 2a would represent about 5/6 of the full population. So effectively there would be 1 000 soft binaries that have been neglected and the true primordial frequency would be 6%. One could then assume that these soft binaries were broken-up at the very start of the simulation – although this may not be true for soft binaries residing in the less-dense outer regions of – 8 – the cluster. However, we note that there is no evidence that binary periods in star clusters should extend as far as 107 d (Meylan & Heggie 1997). In terms of hard binaries one could argue, for the sake of semantics, that in comparison to a population drawn from a uniform distribution of periods extending from 0.1 d (without restriction) our initial distributions are under-sampling the contribution of hard binaries. A key point here is that short-period binaries were not excluded from the primordial popula- tions of our simulations by some ad-hoc process. Instead the distribution of orbital periods is dictated by using distributions borne from observations in combination with accounting for pre-main sequence (MS) evolution – before contracting along the Hayashi track the stellar radius of a pre-MS star can be a factor of five or more greater than on the ZAMS (Siess, Dufour & Forestini 2000) and birth periods must allow for this (Kroupa 1995). Pre-MS evo- lution was not considered by Ivanova et al. (2005) although they did reject systems where one or both stars would initially fill their Roche-lobes at pericentre – this was also assumed in our models. 4. Results In Figure 3 we show the evolution of the core binary fraction for the four N -body simulations introduced above. Also shown is the binary fraction within the 10% Lagrangian radius and the overall binary fraction of the model clusters. Except at late times in the K24-50 model, when the cluster has lost more than 90% of its original mass and is nearing dissolution, we see that in each case the cluster binary fraction remains close to the primordial value. Focussing on the K100-5 simulation, Figure 4 shows the fractions of single stars and binaries (compared to their respective initial number) in the cluster. Following on from Figure 3a the fractions are similar at all times as expected. However, Figure 4 also shows the fractions of single stars and binaries that have escaped the cluster and we see that from about 2Gyr onwards the fractional escape rate of single stars is greater than that of the binaries. At the end of the simulation (20Gyr) the difference is 34%. This is offset somewhat by evolution processes (stellar and binary) that destroy binaries (see the dotted line in Figure 4). These processes include binaries becoming unbound due to supernova mass-loss and/or kicks (only relevant for the first 100Myr of evolution) and mass transfer-induced mergers in close binaries. The remaining difference is balanced by the destruction of binaries in dynamical encounters and this becomes more important as the cluster evolves. We note that even though the cluster binary fraction is relatively static as the cluster evolves the characteristics of the binary population change markedly over time with hard binaries favoured at late times. – 9 – Evident from Figure 3 is an overall trend for the core binary fraction to increase with time, irrespective of simulation type. For the core binary population of the K100-5 model we see that this rises from an initial 5% to as high as 40% around the time that the core-collapse phase is halted. After this time the core binary fraction becomes quite noisy owing to the small size of the core (see Figure 1) and the small numbers of binaries and stars in the core. However, the value always remains greater than the initial value. We see also from Figure 3a that the binary frequency within the inner 10% Lagrangian radius rises to a maximum of 16% just prior to the end of the core-collapse phase. It is important to note at this point that we are working with radii derived from spherical data whereas observational determinations of binary fractions are based on two-dimensional projected data. With our models it is possible to test the effect of this discrepancy on our findings. If we calculate the 10% Lagrangian radius for model K100-5 from a two-dimensional projection we find that the radius is reduced by about 20-40% across the evolution (the choice of projection axis does not affect this result). This is consistent with the expectation suggested by Fleck et al. (2006). A similar relationship is reported by Baumgardt, Makino & Hut (2005) in that the half-light radius (calculated from projected data) is approximately half the size of the half-mass radius (calculated from spherical data). However, the binary fraction within the projected 10% Lagrangian radius of our K100-5 model is almost indistinguishable from that of the result shown in Figure 3a (the dotted curve). We now aim to understand the processes underlying the evolution of the core binary fraction of star clusters, focussing again on the K100-5 simulation. Figure 5 shows the number of single stars and binaries in the core, relative to their total number in the cluster, as the cluster evolves. For the first 10Gyr of evolution the ratio of binaries in the core to binaries in the cluster is fairly static – roughly 1 in 10 binaries is in the core. However, the ratio of single stars found in the core is decreasing sharply over the same timeframe and thus single stars are being lost from the core at a greater rate than from the cluster in general (comparing Figs. 4 and 5). From 10Gyr onwards the ratio of binaries in the core also decreases. This corresponds to a period of increasing core density: prior to 10Gyr the core density of stars hovers around the 102 stars pc−3 mark but from 10− 15Gyr it increases by an order of magnitude. The binary fraction continues to rise in the core over this period indicating that single stars continue to be lost from the core at a greater rate than binaries. We note that mass loss from stellar evolution is reduced considerably at this stage compared to earlier in the cluster lifetime when more massive stars were present. Figure 6 confirms that the number of binaries in the core is decreasing with time even though the binary fraction, fb,c, is increasing. We also see from this figure that at least half of the binaries in the core at any time were not present in the core the last time the population – 10 – was sampled (this is done at intervals of 80Myr). So the core binary population is by no means static as many binaries are being created/destroyed, or moving in and out of the core, on the 80Myr timescale. It is important to note for comparison that the relaxation time in the core is approximately 200Myr initially and decreases to about 50Myr at late times. Individual binaries in cluster cores are both promiscuous and mobile – transient residents. In Figure 7 we examine the fraction of core binaries that were created in exchange interactions. These are short-lived 3- and 4-body gravitational encounters where a star is exchanged into an existing binary displacing one of the members of that binary (Heggie 1975). Thus it is a process by which primordial binaries can be destroyed and replaced by new dynamical, or exchange, binaries. We see from Figure 7a that these non-primordial binaries come to dominate the core population towards the end of the core-collapse phase in the K100-5 simulation. Figure 7a also shows that the double-degenerate binary content increases steadily in the core with time and comprises about 30% of the core binaries subse- quent to the completion of the core-collapse phase. In Figure 7b we see that the exchange binary content in the core of the K100-10 model does not reach the heights of the K100-5 model. Presumably this is a consequence of the lower core-density of the K100-10 model. The fraction of double-degenerate binaries is similar – any decrease in double-degenerate pro- duction via dynamical means in the K100-10 model is compensated by the increased number of primordial binaries. The fraction of exchange binaries in the core of the K24-50 simulation (Figure 7d is comparatively low whereas the K50-20 simulation (Figure 7c exhibits a much larger fraction. Clearly there is a positive correlation between core density and the fraction of exchange binaries in the core. Figure 8a shows the number of binaries created and destroyed in exchange interactions occurring in the core in intervals of 80Myr. Also shown is the number of core binaries destroyed by all processes (exchanges, orbital perturbations, supernovae, mergers) in each interval. The key point to note here is that on average exchange interactions are creating as many binaries as they are destroying. For the entire cluster there were 1 024 binaries destroyed in exchange interactions during the simulation and 933 binaries created. Figure 8b looks at the movement of binaries in and out of the core as the cluster evolves. Across each 80Myr interval it shows the fraction of core binaries that move out of the core during the interval and the fraction of binaries that have moved into the core during the interval. We see that the inwards and outwards fluxes are equal. Also shown is the fraction of binaries entering the core that have previously been in the core – most binaries that leave the core eventually revisit it. We see a pattern where binaries move outwards across the core boundary owing to recoil velocities from gravitational encounters, or as a result of the shrinking core. The core binary population is then replenished by binaries sinking inwards – 11 – owing to mass-segregation effects. In the discussion below we will refer to this pattern as binary convection. We note that binaries on radial orbits with a moderate to high eccentricity will also make an apparent contribution to this process. An analysis of binary disruption for the K100-5 simulation is given in Figure 9 in terms of cumulative events. Exchange interactions and orbital perturbations from nearby stars are by far the dominant causes of binary disruption and these are shown in the top panel. We see that perturbation events are more likely at early times in the evolution but, as soft binaries disappear and the binary population becomes skewed towards hard binaries, exchange events eventually overtake perturbations as the major cause of disruption. However there is an important distinction to make between these two types of event. Exchange interactions are counted as a disruption event in Figure 9a even if the event also leads to the creation of a new binary and as we have seen in Figure 8a this is more than likely. On the other hand if a binary is broken up owing to an orbital perturbation (also known as a fly-by) there is no possibility of a replacement binary being created in the event. The lower panel of Figure 9 shows the number of binaries that were ejected from the core and escaped the cluster. There is a sharp correlation between the incidence of escape and the increase in core density after 10Gyr. Even so, the total number of binaries lost owing to this process remains an order of magnitude less than either perturbation or exchange disruption. There is an initial burst of stellar/binary evolution induced mergers in short- period primordial binaries followed by a gradual depletion of binaries owing to this process and collisions in highly eccentric binaries. The cluster had a total of 287 binaries that experienced either a merger or an internal collision and 67 of these events occurred in the core. We also see from Figure 9b that supernova events do not make a meaningful contribution to depletion of the core binary population. Figure 10 repeats Figure 9 for the K24-50 simulation. In this simulation mergers and collisions are the most likely cause of core binary loss. This is linked to the increased primordial binary fraction and decreased core density, compared to the K100-5 simulation. For similar reasons exchange disruption is more likely than perturbed disruption over the course of the evolution. In fact in this simulation even the loss of binaries from the core as a result of escape is greater than that from perturbed break-up. A key distinction between the K24-50 and K100-5 simulations is that in the K24-50 case the ratio of binary destruction to creation in exchanges is 3:1 whereas it was close to 1:1 for the K100-5 simulation. The effect of a substantial primordial binary population on the evolution of open clusters has been documented in the past (McMillan, Hut & Makino 1990, for example and see Meylan & Heggie 1997 for a review). The main results are that in comparison to simulations without primordial binaries the core-collapse phase of evolution is less dramatic and the – 12 – cluster lifetime is reduced. Little has been done on this subject for globular clusters to date primarily because direct simulations have not been possible. However, our simulations starting with 100 000 stars can start to shed some light on the expected behaviour. We see from Table 1 that increasing the primordial binary frequency from 5% (K100-5 simulation) to 10% (K100-10) does not reduce the cluster half-life significantly. In contrast the K24-50 simulation with 50% binaries has a half-life of 2 060Myr while a comparable simulation of 30 000 single stars with no primordial binaries has a half-life of 3 600Myr. As noted in Hurley & Shara (2003) the presence of a large number of primordial binaries in an open cluster leads to an enhanced rate of escaping stars via recoil velocity kicks obtained in 3-body interactions. In comparison, the K100-5 and K100-10 clusters have deeper potential wells and also the change in binary fraction between the two models is much less than for the open cluster example. So a sharp change in the escape rate is not to be expected. Figure 11 shows that the core radius evolution of the K100-5 and K100-10 simulations is similar up to 15Gyr (when the K100-10 simulation was stopped). We note however that the core density of the K100-10 model at this time is only half that of the K100-5 model. So the presence of additional primordial binaries has reduced the number density of stars in the core. Also in Figure 11 we compare the core radius evolution of a 100 000 star simulation with no primordial binaries (a K100-0 model). Here we see that the core radius evolution is slightly more irregular but overall the evolution is once again similar up to 15Gyr. After core-collapse has been halted the situation is different as the single star model experiences a fluctuating, and generally increasing, core-radius while the core-radius of the K100-5 model remains approximately constant (see Figure 1a). The K100-0 model has a greater core density than the K100-5 model at the end of the main core-collapse phase. The fluctuating core-radius of the K100-0 model in the post-core-collapse phase is indicative of the core bounce and subsequent oscillations expected for such a model – these phenomena are more pronounced for models without primordial binaries (see the related discussion in Heggie & Hut 2003 and Heggie, Trenti & Hut 2006). In Figure 12 we investigate the radial distribution of the K100-5 binary population at times of 6, 12 and 18Gyr, i.e. before, during and after the deep core-collapse phase. We see that outside of the half-mass radius the binary fraction is effectively constant with radius and changes little with time. The binary population in this region is also dominated by primordial binaries – exchange binaries are unlikely to be found outside of the half-mass radius. Within the half-mass radius the binary fraction rises sharply towards the centre of the cluster and binaries become more centrally concentrated as the cluster evolves. Note that the inner radial bin corresponds to the inner 5% Lagrangian radius so the core is not resolved in Figure 12. – 13 – 5. Discussion Our N -body results clearly show that the core binary fraction of an evolved star cluster is expected to be greater than the primordial binary fraction. We see this behaviour in each of the models presented and at all times in the evolution. The most striking case is our main model (K100-5) which started with 95 000 single stars and 5 000 binaries and experienced a factor of eight increase in the core binary fraction after 16Gyr of evolution (coinciding with the end of the main core-collapse phase). At face value our results appear to be in clear contradiction to the Monte-Carlo results recently presented by Ivanova et al. (2005). Their main reference model has a primordial binary fraction fb,0 = 1.0 and a stellar density of nc = 10 5 stars pc−3. Processes such as exchange interactions, orbital perturbations, binary evolution and mass-segregation are included and the model is reduced to fb,c = 0.095 at 14Gyr. Ivanova et al. (2005) then repeat the simulation with fb,0 = 0.5 and end up with fb,c = 0.07 so, as they note, the relationship between primordial binary fraction and final core binary fraction is not linear. Coming at this from the other direction our models show a possible saturation effect as the primordial binary fraction increases. Looking back at Figures 3a and 3b we see that the core binary fraction of the K100-5 model at the 14Gyr mark is 0.2 (up from fb,0 = 0.05) while it is 0.3 (up from fb,0 = 0.1) for the K100-10 model. So the simulation with the lower primordial binary fraction has experienced the greater relative increase in core binary content. Our K24-50 model, which started with fb,0 = 0.5, has fb,c = 0.8 at a similar dynamical age so the relative increase is less again. These results raise the possibility that decreasing fb,0 below 0.5 in the Monte Carlo models may lead to conditions where fb,c can increase. It is interesting to note that the idealized models presented recently by Heggie, Trenti & Hut (2006) showed saturation effects in the core for initial binary frequencies greater than 10% and also recorded an increase in the core binary fraction with time. Ivanova et al. (2005) also performed a model to compare with 47 Tucanae. This was similar to their main reference model although slightly more dense and with an increased velocity dispersion. The result for fb,0 = 1.0 was fb,c = 0.07 – this lead to the conclusion that the primordial binary frequencies of globular clusters such as 47 Tucanae must have been close to 100% to explain current observations. However, Ivanova et al. (2005) also ran the same simulation with fb,0 = 0.75, 0.5 and 0.25 and reported little or no variation in the final core binary fraction. It would seem safe to assume that repeating the simulation with fb,0 = 0.1 may give the same result or even an increase in binary fraction. This would act to remove any obvious discrepancies between the N -body and Monte Carlo results. We would certainly be interested in seeing the results of a Monte Carlo simulation conducted with fb,0 = 0.1 and a similar setup of the primordial binary population as used in this work – 14 – – much easier than repeating a large N -body simulation with 100% binaries. A major distinction between our N -body models and the Monte Carlo simulations mentioned above is that the stellar density is at least an order of magnitude greater in the latter. Fortunately, Ivanova et al. (2005) performed a simulation with nc = 10 3 stars pc−3 which facilitates a more direct comparison with our K50-20 model which had a similar core density throughout the evolution. The K50-20 model experienced an almost factor of two increase in core binary fraction as it evolved from 0 − 8Gyr. The comparable Monte Carlo model showed a reduction in core binary fraction of more than a factor of two over the same period. So there is an obvious deviation in behaviour. Of course there is a large difference in the primordial binary fractions (0.2 compared to 1.0). The effect of this will be discussed further below. However, we note at this stage that the initial hard binary fraction in the Monte Carlo model was ∼ 30% (Ivanova, private communication) and this rose to 37% – so the hard binary fraction increases and subsequently the models do show agreement at some level. Another consideration is the velocity dispersion which is generally around 3−4 km s−1 for our models and was set to 10 km s−1 for most of the Monte Carlo models. However, Ivanova et al. (2005) did perform two models (D4 and M12) similar in all respects except that σ = 10 km s−1 in one and 4.5 km s−1 in the other. There was no significant difference in the final core binary fractions of these models. In Section 3 we discussed that in the setup of our models we might be neglecting a fraction of soft binaries from the true primordial population. This results from imposing a maximum initial orbital separation and at most would cause the binary fraction to be underestimated by a few per cent. Thus we are confident that our choice of initial parameters for the binary populations in our models is not affecting the result that the core binary fraction increases as a cluster evolves. We also note that differences in the setup of primordial binaries between our simulations and those of Ivanova et al. (2005) make it difficult to directly compare quoted binary frequencies. For example, by not accounting for pre-MS stellar radii as we do, Ivanova et al. (2005) have a greater relative number of close binaries in their primordial populations. Such an excess would result in a greater number of evolution- induced binary mergers. If we were to adopt the period distribution and methods used by Ivanova et al. (2005) we would need to choose ∼ 11 000 binaries in order to recover the 5 000 in our K100-5 model at birth. This gives an effective primordial binary frequency of 11%, for the sake of comparison. The effective primordial binary frequency for the K24-50 simulation would be 80%. Adopting these values, in the worst case scenario, would still not lead us to conclude that the core binary fraction of an evolved cluster is decreased from the primordial value. The comparable rates of binary disruption and creation owing to exchanges in our – 15 – K100-5 simulation indicates that 3-body interactions dominated over 4-body interactions. This is because the most likely outcome of a binary-binary encounter is a binary and two single stars. So a binary is lost from the overall count. This is not the case for binary- single encounters where the most likely outcome is a binary and a single star, although the pairing of stars in the binary and/or the orbital parameters may have changed. By contrast, exchange interactions in the K24-50 simulation produced a binary disruption rate much higher than the binary creation rate. Here we had a much higher proportion of primordial binaries and thus binary-binary encounters were more likely. Thus, in terms of exchange interactions, increasing the primordial binary fraction can lead to a greater rate of binary destruction. This would certainly be expected to be true of models with comparable stellar densities. However, a competing effect comes from the fact that the central density is less for simulations with higher primordial binary fractions. We certainly see this when comparing our K100-5 and K100-10 models. The setup of these models was identical in all respects except for the change in primordial binary frequency from 5% to 10%. The models have similar half-lives and we showed that the core radius evolution is also similar. So at any particular time in the evolution they are at a comparable dynamical age. But there is one clear difference – the model with twice as many primordial binaries has a central stellar density that is a factor of two less. This translates to a lower incidence of close stellar encounters and as we saw from Figure 7 a greatly reduced fraction of exchange binaries in the core. Previous simulations, albeit with small-N , have indicated that the effects of primordial binaries saturate at some level (Wilkinson et al. 2003) so this is not necessarily a trend that we expect will continue as the primordial binary fraction is increased towards unity. However it is certainly significant for clusters with frequencies of 10% or less. Another point to note is that in a 3-body exchange, not only is a binary not lost, but also a more massive single star is swapped for a less massive one, increasing the likelihood that the single star will be lost from the core via mass-segregation. So the exchange process has indirectly increased the core binary fraction. The process of binary convection that became evident from Figure 8b also is related to mass-segregation and acts to keep the core binary fraction healthy. Both single stars and binaries in the core are subject to velocity kicks from gravitational encounters. These kicks can remove an object from the core and even from the cluster entirely. For binaries this is less likely to occur primarily because they are on average more massive than single stars. Also, the average stellar mass decreases radially outwards in an evolved cluster. So if a core binary suddenly finds itself outside of the core it can be expected to be one of the more massive objects in its new local environment and thus to quickly sink back towards the core. We note that we found the movement of binaries inwards and outwards across the core boundary, as exhibited by Figures 6 and 8, to be quite striking. Our K100-5 N -body simulation creates a realistic model of a moderate-size globular – 16 – cluster. It combines stellar and binary evolution with a self-consistent treatment of the cluster dynamics. It includes primordial binaries and accounts for the tidal field of the Galaxy. Thus it provides us with a solid picture of how such a cluster evolves. Single stars escape from the cluster at a greater rate than binaries do – single stars are less massive on average so they are more likely to be tidally stripped after segregating to the outer regions of the cluster and also more likely to be ejected from the cluster in gravitational encounters. However, binaries are also lost from the cluster population owing to supernova disruption, evolution-induced mergers and dynamical encounters. These effects balance and the ratio of single stars to binaries is similar at all times in the evolution. As the cluster evolves binaries sink towards the centre and the binary fraction increases in the central regions. The core radius decreases as core-collapse proceeds and dynamical encounters become more prevalent. These encounters not only break-up binaries but also create new binaries. The cluster evolves to a state where primordial binaries dominate the binary population in the outer regions and non-primordial binaries dominate towards the centre. In the centre of the cluster soft binaries are broken-up as a result of orbital perturbations from gravitational encounters. Binaries become involved in exchange interactions, primarily three-body, but these tend to create as many binaries as they destroy. Hard binaries are lost when the components merge as a result of close binary evolution or a collision at periastron. These are ongoing processes as the cluster evolves. At an age of 10Gyr the rate of exchange interactions is greater than that of perturbed break-ups and mergers. However, perturbed break-ups are the dominant cause of binary loss. This is compared to the Monte Carlo model of Ivanova et al. (2005) which found that evolutionary mergers were the dominant event at the same age. We also find that after 10Gyr, as the core density increases, that binaries can be kicked out of the cluster directly from the core. Partly as a result of the combination of these processes the number of binaries in the core decreases as the cluster evolves. Also to blame is the movement of binaries outwards across the core boundary owing to the decreasing size of the core and recoil velocities invoked in gravitational encounters. However, the movement of single stars outwards across the core boundary is greater and the net effect is an increase in the core binary fraction. This is also helped by binary convection where binaries that were previously resident in the core are cycled back in. Noting that the typical membership of Galactic globular clusters exceeds 300 000 stars (Gnedin & Ostriker 1997, for example) we must ask the question – to what extent can we expect this behaviour to extend to globular clusters in general? We can start with the ejection rate, tej, of stars from an isolated cluster calculated by Hénon (1969) which gives tej ∝ ln (0.4N) trh (Binney & Tremaine 1987). Here trh is the half-mass relaxation timescale and we can relate this to behaviour near the core of a cluster if we assume that core-mass scales with total mass and that radii do not vary appreciably with cluster mass. – 17 – This indicates that the relative rate of outward binary ejection and inward mass-segregation (which occurs on a relaxation timescale) is only weakly dependent on the cluster mass. If we look in detail at the local relaxation timescale this scales as ρ ln (0.4N) (Davies, Piotto & De Angeli 2004, as derived from Binney & Tremaine 1987) where σ is the velocity dispersion of the cluster stars and ρ is the mass density. We can take σ ∝ M/rh ∝ c and ρ ∝ Mc, using the above assumptions, to show that tr ∝ M c / ln (0.4N). Here Mc is the cluster core-mass, M is the total cluster mass and rh is the half-mass radius. The timescale for a typical binary in the core of a globular cluster to have a close encounter with another star scales as tenc ∝ (Davies, Piotto & De Angeli 2004) where n is the number density and n ∼ ρ if the average stellar mass is of the order of M⊙, as it is in an evolved cluster core. This gives us tenc ∝ c . To escape the core a binary must acquire a boost in energy of the order of GMc/2 rc (where G is the Gravitational constant). So, assuming that the average energy imparted in an encounter does not vary strongly with mass, we have tej ∝ M c . This rather simplified analysis returns Hénon’s result and shows that asM (or N) increases there will be relatively less binary convection as both the ejection and relaxation timescales increase. However, the effect on the observed core binary fraction can be expected to be minimal. We cannot definitively use our results to make predictions regarding globular clusters such as 47 Tucanae because the central density in these clusters is at least an order of magnitude higher than that reached by our models. However, we note that our model with the highest core density showed the greatest increase in core binary fraction. Furthermore, we have considered a range of cluster types. It does not appear, from our simulations, that an initial binary fraction anywhere near as high as 100% is required to give a core population of 20% or less at later times. We also note that proper-motion cleaned colour-magnitude diagrams recently presented for NGC6397 (Richer et al. 2006) and M4 (Richer et al. 2004) show a distinct lack of binaries in regions outside of the cluster centre – this cannot be reconciled with a large primordial binary population. 6. Summary We have presented a range of simulations typical of rich open clusters and moderate-size globular clusters. In each case we find that the fraction of binaries in the core of a cluster – 18 – does not decrease as the cluster evolves. In fact the overriding trend is for an increase in core binary fraction from the primordial value. Thus we do not agree with Ivanova et al. (2005) that the binary fraction in the core will be depleted in time. We also do not agree that models of globular cluster evolution need necessarily include large populations of primordial binaries. Our simulations have shown that the binary population in the core of a cluster is con- tinually being replenished by stars from outside the core, many of which were previously in the core. This is a process we have termed binary convection. We also find that the binary content of an evolved star cluster is dominated by exchange binaries provided that the stellar density is relatively high. This is true of our moderate-size globular cluster models and we expect it to be true in more massive clusters. We also show that increasing the primordial binary fraction does not necessarily lead to an increase in the final binary fraction – in fact it gives more scope for binary depletion. A key and paradoxical result is that a final binary fraction that can be achieved by choosing a higher primordial binary fraction may also be replicated by choosing an initially lower binary fraction. We find that the overall binary fraction of a cluster does not vary appreciably from the primordial value as a cluster evolves. This is a result of binary destruction being balanced by a greater rate of escape of single stars compared to binaries. We also find that the primordial binary frequency of a cluster is well preserved outside of the cluster half-mass radius. Therefore, observations of the current binary fraction in these regions is a good indicator of the primordial binary fraction while determination of the core binary fraction provides an upper limit. We acknowledge the generous support of the Cordelia Corporation and that of Edward Norton which has enabled AMNH to purchase GRAPE-6 boards and supporting hardware. We thank the anonymous referee for extremely helpful comments and especially for alerting us to the scaling considerations. – 19 – REFERENCES Aarseth, S., Hénon, M., & Wielen, R. 1974, A&A, 37, 183 Aarseth, S. J. 1996, in Proc. 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T., Sills, A., & Beccari, G. 2004, ApJ, 605, L29 McMillan, S., & Hut, P. 1994, ApJ, 427, 793 McMillan, S., Hut, P., & Makino, J. 1990, ApJ, 362, 522 Meylan, G. & Heggie, D.C. 1997, A&ARv, 8, 1 Richer, H. B., Fahlman, G. G., Brewer, J., Davis, S., Kalirai, J., Stetson, P.B., Hansen, B. M. S., Rich, R. M., Ibata, R. A., Gibson, B. K., & Shara, M. M. 2004, AJ, 127, 2771 – 21 – Richer, H. B., et al. 2006, Science, 313, 936 Shara, M. M., & Hurley, J. R. 2006, ApJ, 646, 464 Siess, L., Dufour, E., & Forestini, M. 2000, A&A, 358, 593 von Hoerner, S. 1960, Z. Astrophys., 50, 184 Wilkinson, M. I., Hurley, J. R., Mackey, A. D., Gilmore, G. F., & Tout, C. A. 2003, MNRAS, 343, 1025 This preprint was prepared with the AAS LATEX macros v5.2. – 22 – Fig. 1.— Evolution of the core radius (solid line) and the radius containing the inner 10% of the cluster mass (dotted line) for: a) the K100-5 simulation; and b) the K24-50 simulation. The numbers across the top show the number of half-mass relaxation times that have elapsed. Note that Ns,0 and Nb,0 refer to the number of single stars and binaries, respectively, in the starting model. – 23 – Fig. 2.— Period distribution of the primordial binary populations in: a) the K100-5 sim- ulation (starting with 5 000 binaries); and b) the K24-50 simulation (starting with 12 000 binaries). – 24 – Fig. 3.— Evolution of the binary fraction in the core (solid line), within the 10% Lagrangian radius (dotted line), and for the entire cluster (dashed line). Results are shown for the: a) K100-5; b) K100-10; c) K50-20; and d) K24-50 simulations (see Table 1 for details). – 25 – Fig. 4.— Fraction of single stars (solid line) and binaries (dashed line) remaining in the cluster as a function of time (lines decreasing from top-left). Each population is scaled by the initial number of that population. Also shown are the fractions of single stars and binaries that have escaped from the cluster (lines increasing from bottom-left). The dotted line is the combined fraction of binaries lost to escape and binary/stellar evolution processes. Results are for the K100-5 simulation. – 26 – Fig. 5.— Number of single stars in the core as a fraction of the number of single stars in the cluster (solid line) and number of binaries in the core as a fraction of the number of binaries in the cluster (dashed line). Results are for the K100-5 simulation. – 27 – Fig. 6.— Number of core binaries as a function of time (solid line). Also shown at each time is the number of binaries that have remained in the core from the previous sampling (dashed line). Results are for the K100-5 simulation and the data are sampled every 80Myr. – 28 – Fig. 7.— Fraction of binaries in the core that were created in an exchange interaction (solid line) and fraction of core binaries that contain two degenerate stars (dotted line). Results are shown for the: a) K100-5; b) K100-10; c) K50-20; and d K24-50 simulations (as described in Table 1). – 29 – Fig. 8.— Statistics regarding core binaries across intervals of 80Myr as the K100-5 model cluster evolves. Shown are: a) number of binaries destroyed in an exchange interaction occurring in the core (solid line), number of binaries created in exchange interactions in the core (dashed line) and the number of binaries destroyed by any means (dotted line); and b) fraction of binaries that have moved out of the core but remained in the cluster (solid line: as a fraction of the number of binaries in the core at the start of the interval), number of binaries that have moved into the core (dashed line: as a fraction of the number of binaries in the core at the end of the interval) and the fraction of binaries entering the core that have previously resided in the core (dotted line). Note that the data have been moderately smoothed – over a width of three bins (or 240Myr). Further smoothing would hide the naturally irregular behaviour of the binary destruction/creation processes. – 30 – Fig. 9.— Cumulative numbers of events that lead to the destruction of binaries in the core. Shown are: a) binaries broken-up in exchange encounters (solid line) and binaries broken-up owing to orbital perturbations (dotted line); b) binaries that were ejected from the core and escaped from the cluster (dashed line), binaries broken-up as a result of supernovae explosions (dotted line) and binaries in which the stars merged (solid line) – this includes stellar evolution induced mergers and collisions at periastron in highly eccentric binaries. Results are for the K100-5 simulation. – 31 – Fig. 10.— As for Figure 9 but for the K24-50 simulation. – 32 – Fig. 11.— Comparison of core-radius evolution for models starting with 100 000 stars. The K100-5 simulation is taken as a reference model and shown are differences between the core radius of this model and models starting with 0% (dotted line) and 10% primordial binaries (K100-10: dashed line). The difference is scaled by the core radius of the K100-5 model. Note that for each simulation the core radius used is the average core radius in a 250Myr interval. – 33 – Fig. 12.— Binary data as a function of radial position for the K100-5 model. Shown at times of 6Gyr (solid line), 12Gyr (dashed line) and 18Gyr (dotted line) are: a) distribu- tion of binary fraction; b) fraction of binaries that are primordial. At each time there are twenty radial bins each containing the same mass, i.e. corresponding to Lagrangian radii incremented by 5%. Thus the core is not resolved. – 34 – Table 1. Details of the four N -body simulations utilised in this work. Columns 1 and 2 show the number of single stars and binaries in the starting model. The distribution used to select the orbital separations of the primordial binaries is given in column 3 and this is followed by the maximum applied to the distribution (in au). Column 5 lists the primordial binary fraction and in column 6 we show the typical stellar density in the core for the simulation (stars/pc3). The half-life of the simulation (time in Myr for Ns +Nb to drop to half the initial value) is given in column 7 and finally an identifying label is supplied for each simulation. Ns,0 Nb,0 ψ(a) amax fb,0 nc t1/2 label 95000 5000 EFT30 100 0.05 102 − 104 8920 K100-5 90000 10000 EFT30 100 0.10 100− 500 8850 K100-10 40000 10000 EFT30 50 0.20 103 5560 K50-20 12000 12000 log a 50 0.50 100− 350 2060 K24-50 Introduction Models Binary Period Distributions Results Discussion Summary

0704.0291

Approaching the Heisenberg limit in an atom laser

Approaching the Heisenberg limit in an atom laser M. Jeppesen,1 J. Dugué,1, 2 G. R. Dennis,1 M. T. Johnsson,1 C. Figl,1 N. P. Robins,1 and J. D. Close1 Australian Research Council Centre Of Excellence for Quantum-Atom Optics, Department of Physics, The Australian National University, Canberra, ACT 0200, Australia Laboratoire Kastler-Brossel, 24 rue Lhomond, 75231 Paris Cedex 05, France We present experimental and theoretical results showing the improved beam quality and reduced divergence of an atom laser produced by an optical Raman transition, compared to one produced by an RF transition. We show that Raman outcoupling can eliminate the diverging lens effect that the condensate has on the outcoupled atoms. This substantially improves the beam quality of the atom laser, and the improvement may be greater than a factor of ten for experiments with tight trapping potentials. We show that Raman outcoupling can produce atom lasers whose quality is only limited by the wavefunction shape of the condensate that produces them, typically a factor of 1.3 above the Heisenberg limit. PACS numbers: 03.75.Pp,03.75.Mn Experiments in ultracold dilute atomic gases have had an enormous impact on physics. The realization of Bose- Einstein condensates (BECs), degenerate Fermi gases, BEC-BCS crossover systems, and many others have re- sulted in many fundamental insights and a wealth of new results in both experiment and theory. One exciting sys- tem to emerge from this research is the atom laser, a highly coherent, directional beam of degenerate atoms, controllably released from a BEC [1, 2, 3, 4, 5, 6, 7, 8]. The atom lasers demonstrated so far have produced beams many orders of magnitude brighter than is pos- sible with thermal atomic beams [9]. Atom laser beams show great promise for studies of fundamental physics and in high precision measure- ments [10]. In the future, it will be possible to produce quadrature squeezing in atoms lasers, to use atom lasers to produce correlations and entanglement between mas- sive particles [11], as well as high precision interferome- ters both on earth and in space [12]. For all these it will be crucial to develop atom lasers with output modes that as clean as possible in amplitude and phase, to allow sta- ble modematching, just as it was crucial for optical lasers. The beam quality factor M2, introduced for atom lasers by J.-F. Riou et al. [13, 14], is a measure of how far the beam deviates from the Heisenberg limit, and is defined ∆x∆px, (1) where ∆x is the beam width, measured at the waist, and ∆px is the transverse momentum spread. An ideal (Gaussian) beam would therefore have M2 = 1 along both its principal transverse axes. A number of exper- imental works have shown that the beam quality of an atom laser is strongly affected by the interaction of the outcoupled atoms with the BEC from which it is pro- duced [13, 15, 16, 17, 18]. As the atoms fall through the condensate, the repulsive interaction acts as a diverging lens to the outcoupled atoms. This leads to a divergence in the atom laser beam and (because the BEC is a non- ideal lens) a poor quality transverse beam profile. Such behavior may cause problems in mode matching the atom laser beam to another atom laser, a cavity or to a waveg- FIG. 1: (color online). Top: Sequence of atom laser beams showing the improved beam profile of a Raman atom laser. The atom laser beams were produced using RF (a) and Ra- man (b and c) transitions. The angle between the Raman beams (see Fig. 2 (a)) was θ = 30◦ in (b) and θ = 140◦ in (c), corresponding to a kick of 0.5h̄k (0.3 cm/s) and 1.9h̄k (1.1 cm/s) respectively. The outcoupling rate differs be- tween each atom laser. Below: Comparison of experimental (dashed) and theoretical (solid) beam profiles 500 µm below the BEC. The height of each theoretical curve has been scaled to match experimental data. http://arxiv.org/abs/0704.0291v2 uide. Experiments on atom lasers in waveguides have produced beams with improved spatial profile [7]. How- ever, precision measurements with atom interferometry are likely to require propagation in free space, to avoid introducing noise from the fluctuations in the waveguide itself [12]. In a recent Letter [13], it was shown that the quality of a free space atom laser is improved by outcoupling from the base of the condensate. Our scheme, however, en- ables the production of a high quality atom laser while outcoupling from the center of the condensate. This is desirable for a number of reasons: First, because the clas- sical noise level is determined by the outcoupling Rabi frequency, then outcoupling from the center, where the density is greatest, gives the highest possible output flux for a given classical noise level [19]. Second, outcoupling from the center allows the longest operating time (for a quasicontinuous atom laser) since the condensate can be drained completely. Third, outcoupling from the center minimizes the sensitivity of the output coupling to con- densate excitations or external fluctuations. In a recent Letter [9], we have demonstrated a contin- uously outcoupled atom laser where the output coupler is a coherent multi-photon (Raman) transition [6, 20]. In this scheme, the atoms receive a momentum kick from the absorption and emission of photons. They leave the con- densate more quickly, so that adverse effects due to the mean-field repulsion from the condensate are reduced. In this Letter, we report measurements of a substantial improvement in the beam quality M2 using this outcou- pling. In Fig. 1, we show absorption images of atom laser beams outcoupled from the center of a BEC with (a) negligible momentum kick, (b) a kick of 0.3 cm/s, and 1.1 cm/s (c). As the kick increases, the divergence is reduced and the beam profile improved. In our experiment, we create 87Rb BECs of 5× 105 atoms in the |F = 1,mF = −1〉 state via stan- dard runaway evaporation of laser cooled atoms. We use a highly stable, water cooled QUIC magnetic trap (axial frequency ωy = 2π × 12 Hz and radial frequency ωρ = 2π × 128 Hz, with a bias field of B0 = 2 G). We control drifts in the magnetic bias by using high stability power supplies and water cooling. This stability allows us to precisely and repeatably address the condensate. We produce the atom laser by transferring the atoms to the untrapped |F = 1,mF = 0〉 state and letting them fall under gravity. To outcouple atoms with negligible momentum kick we induce spin flips via an RF field of a frequency corresponding to the Zeeman shift in the center of the condensate. Alternatively, we induce the spin flips via an optical Raman transition. The setup is shown in Fig. 2 (a). Two optical Raman beams, sep- arated by an angle θ, propagate in the plane of grav- ity and the magnetic trap bias field. The momentum transfer to the atoms through absorption and emission of the photons is 2h̄k sin(θ/2), with k the wave number of FIG. 2: (color online) (a) Experimental schematic (not to scale) showing the BEC, Raman lasers, and trapping coils. (b) Cross section along the two strong axes of the magnetic trap, showing the BEC, outcoupling surface, and atom laser trajectories. Note that the field of view in (b) is rotated 90◦ with respect to (a). the laser beams. The Raman laser beams are produced from one 700 mW diode laser. We can turn the laser power on or off in less than 200 ns using a fast switching AOM in a double pass configuration. After the switch- ing AOM, the light is split and sent through two separate AOMs, again each in a double pass configuration. The frequency difference between the AOMs corresponds to the Zeeman plus kinetic energy difference between the initial and final states of the two-photon Raman transi- tion. We stabilize the frequency difference by running the 80 MHz function generators driving the AOMs from a single oscillator. The beams are then coupled via single mode, polarization maintaining optical fibers directly to the BEC through a collimating lens and waveplate, pro- viding a maximum intensity of 2500 mW/cm2 per beam at the BEC. The polarization of the beams is optimized to achieve maximum outcoupling with a downward kick and corresponds to π polarization for the upper beam and σ+ for the lower beam. The outcoupling resonance is set to the center of the BEC for both RF and Raman outcoupling, as shown in Fig. 2 (b). This point is found by performing spec- troscopy on the BEC using 100 ms of weak output cou- pling at varying RF frequencies, and measuring the num- ber of atoms remaining in the condensate after the output coupling time [3]. A typical calibration curve is shown in Fig. 3 (a), in this case for RF outcoupling. We operate both RF and Raman output couplers at the point of max- imum outcoupling rate. We further check this frequency by ensuring that a continuous beam can still be produced when the initial condensate is very small, which can only happen when outcoupling from the center. We observe the system using standard absorption imaging along the y (weak trapping) direction, on the F = 2 → F ′ = 3 transition, with a 200 µs pulse of repumping light (F = 1 → F ′ = 2) 1 ms prior to imag- ing. From these images we are able to extract the rms width of the atom laser as a function of fall distance (see FIG. 3: (a) Output coupling spectroscopy showing the oper- ating point at the center of the BEC, solid curve to guide the eye. (b) The rms beam width for RF and Raman atom lasers. The dots represent experimental measurements and the solid curves our theoretical predictions. Fig. 3 (b)), which we use to calculateM2 (details below). To model the system, we use a two-step method fol- lowing [13]. Inside the condensate, we use the WKB approximation, by integrating the phase along the clas- sical trajectories of atoms moving in the Thomas-Fermi potential of the condensate (an inverted paraboloid) [16]. After this, we propagate the atom laser wavefunction us- ing a Kirchoff-Fresnel diffraction integral over the surface of the condensate: ψ(r) = dS′ · [G∇′ ψ − ψ∇′G], (2) where G = G(r, r′) is the Green’s function for the Hamil- tonian in the gravitational potential V (r) = −mgz [21]. Therefore, the model includes only interactions between condensate atoms and beam atoms; interactions between atoms within the beam are ignored. The integral in Eq. (2) is formally a two dimensional surface integral over the whole condensate. However for simplicity, fol- lowing [13], we neglect divergence in the weak trapping axis and only consider cross sections in the plane of the strong trapping axes, and so the integral becomes one dimensional. A 3D wavefunction is built up by calculat- ing the atom laser in a series of planes along the weak trapping axis. We ignore the effects of the magnetic field on the atom laser. The atom laser state |F = 1,mF = 0〉 is unaffected to first order by the magnetic field, but is weakly anti- trapped due to the second order Zeeman effect, with an effective trapping frequency of ω2nd = 2π × 2.6 Hz. The transverse position of an atom in such a potential is x(t) = x0 cosh(ω2ndt) ≈ x0(1 + ω 2/2). (3) For the 1 mm (14 ms) propagation we consider here the transverse position is affected by less than 3%. We also ignore the AC Stark effect of the Raman beams on the atom laser, because the intensity of the beams does not change significantly over the 1 mm propagation. We have checked the validity of this model against a solution of the full 3D Gross-Pitaesvskii (GP) equa- tion, including beam-beam interactions. To find the atom laser wavefunction at large distances below the con- densate (up to 1 mm), we transfer the GP model to a freely falling frame once the atom laser wavefunction has reached steady state. The details of the calculation will be the basis of a future publication. The two models give good agreement. Calculating the quality factor M2 of the atom laser directly from Eq. (1) requires measurement of the beam width at the waist ∆x0. Because the BEC acts as a diverging lens on the atom laser, the beam waist is virtual and located above the BEC, and so it is not possible to measure the beam quality M2 using Eq. 1 only. For our simulations, M2 is calculated equivalently from the wavefunction ψ(x, y, z) at some height z below the BEC in which the atom laser has reached the paraxial regime: (M2/2)2 = (∆x(z))2(∆kx(z)) 2 − C(z)2, (4) where ∆x(z) is beam width and C(z) is the curvature- beam width product [22]: C(z) = dx. (5) In practice it is difficult to measure the wavefunction phase, and hence C(z). However the beam width, in the paraxial regime, obeys: ∆(x(t))2 = (∆x0) 2 + (∆vx) 2(t− tw) 2, (6) where tw is the time when the beam is at its waist, and ∆x0 is the beam waist. In principle M 2 may be deter- mined simply from measurements of the beam width at different heights. In our experiment, we can only measure the beam width in the far field, at distances greater than 300 µm below the condensate (observation at distances less than 300 µm are prevented by the condensate expan- sion after trap switchoff.) In the far field the second term of Eq. 6 dominates, and so only the velocity spread can be measured. Therefore we calculate ∆x0 and tw from the model, tw = mC(z)/(h̄∆k x), with tw negative since the waist is virtual and located above the BEC. We then fit to the experimental data to find ∆vx. In Fig. 4, we present the theoretical and experimental results. We find that as the kick increases, the beam qual- ity is improved and the divergence is reduced. For our parameters, we find that for an RF atom laserM2 = 2.2, and for a Raman atom laser M2 = 1.4 with the maxi- mum two photon kick. As the kick increases,M2 contin- ues to improve, and approaches but does not reach the Heisenberg limit of one. It asymptotes to a limit slightly above that, which for our parameters is equal to 1.3. In this regime of large kick, the interaction of the outcou- pled atoms with the condensate becomes negligible, and FIG. 4: (a) Calculated quality factor M2 of an atom laser. The dots are the experimental measurements, and the solid line our theoretical predictions. (b) M2 as a function of trap- ping frequency for an RF atom laser (dashed line), a kick of 0.5h̄k (0.3 cm/s) (dotted line), and 2h̄k (1.1 cm/s) (solid line). The condensate number was N = 5 × 105 atoms, and the aspect ratio ωρ/ωy was 10. the transverse atom laser wavefunction is approximately the free space evolution of the condensate wavefunction (along the outcoupling surface). It is therefore limited by the non-ideal (non-Gaussian) condensate wavefunc- tion itself. We calculate the product ∆x∆px for the con- densate wavefunction (taken through the central horizon- tal plane of the condensate) to be 1.3. We have therefore improved the beam quality M2 by 50 percent, down to a factor of 1.4 above the Heisenberg limit. In addition, our simulations show that (using the same maximum two photon kick) it is possible to reach the condensate limit even for much tighter trapping potentials. In Fig 4 (b), we show the results of simulations for increasing trap fre- quencies, up to ω = 2π × 300 Hz. As the trap frequency increases, theM2 worsens, up toM2 = 14 for RF outcou- pling from a 2π×300 Hz trap. For the maximum Raman two photon kick, the increase is only to M2 = 1.7 for the same 2π × 300 Hz trap. Only for traps of less than 2π×50 Hz is the beam quality of an RF atom laser within 5 percent of that of a Raman atom laser. With higher order Raman transitions [23], it will be possible to reach the condensate limit even for exper- iments with traps of several kilohertz. It will also be possible to reach the Heisenberg limit by completely re- moving the atomic interaction, for example by using a Feschbach resonance. Using Raman lasers phase locked to the 6.8 GHz hyperfine splitting will prevent populat- ing the anti-trapped state, and produce a truly two state atom laser [18, 24]. Such lasers, combined with the high quality transverse mode of Raman atom lasers, could be used in a continuous version of the atomic Mach-Zehnder Bragg interferometer [25], and in the development of atomic local oscillators. We thank Ruth Mills for useful discussions. CF ac- knowledges funding from the Alexander von Humboldt foundation. This work was financially supported by the Australian Research Council Centre of Excellence pro- gram. Numerical simulations were done at the APAC National Supercomputing Facility. ∗ Electronic address: [email protected]; URL: http://www.acqao.org [1] H. M. Wiseman, Phys. Rev. A 56, 2068 (1997). [2] M.-O. Mewes, M. R. Andrews, D. M. Kurn, D. S. Durfee, C. G. Townsend, and W. Ketterle, Phys. Rev. Lett. 78, 582 (1997). [3] I. Bloch, T. W. Hänsch, and T. Esslinger, Phys. Rev. Lett. 82, 3008 (1999). [4] F. Gerbier, P. Bouyer, and A. Aspect, Phys. Rev. Lett. 86 (2001). [5] G. Cennini, G. Ritt, C. Geckeler, and M. Weitz, Phys. Rev. Lett. 91, 240408 (2003). [6] E. W. Hagley, L. Deng, M. Kozuma, J. Wren, K. Helmer- son, S. L. Rolston, and W. D. Phillips, Science 283, 1706 (1999). [7] W. Guerin, J.-F. Riou, J. P. Gaebler, V. Josse, P. Bouyer, and A. Aspect, Phys. Rev. Lett. 97, 200402 (2006). [8] A. Öttl, S. Ritter, and T. Köhl, M. Esslinger, Phys. Rev. Lett. 95, 090404 (2005). [9] N. P. Robins, C. Figl, S. A. Haine, A. K. Morrison, M. Jeppesen, J. J. Hope, and J. D. Close, Phys. Rev. Lett. 96, 140403 (2006). [10] M. A. Kasevich, Science 298, 1363 (2002). [11] S. A. Haine, M. K. Olsen, and J. J. Hope, Phys. Rev. Lett. 96, 133601 (2006). [12] Y. Le Coq, J. A. Retter, S. Richard, A. Aspect, and P. Bouyer, App. Phys. B 84, 627 (2006). [13] J.-F. Riou, W. Guerin, Y. L. Coq, M. Fauquembergue, V. Josse, P. Bouyer, and A. Aspect, Phys. Rev. Lett. 96, 070404 (2006). [14] A. E. Siegman, IEEE. J. Quantum Electron. 27, 1146 (1991). [15] M. Köhl, T. Busch, K. Mølmer, T. W. Hänsch, and T. Esslinger, Phys. Rev. A 72, 063618 (2005). [16] T. Busch, M. Köhl, T. Esslinger, and K. Mølmer, Phys. Rev. A 65, 043615 (2002). [17] Y. L. Coq, J. H. Thywissen, S. A. Rangwala, F. Gerbier, S. Richard, G. Delannoy, P. Bouyer, and A. Aspect, Phys. Rev. Lett. 87, 170403 (2001). [18] A. Öttl, S. Ritter, M. Kohl, and T. Esslinger, Rev. of Sci. Instrum. 77, 063118 (2006). [19] N. P. Robins, A. K. Morrison, J. J. Hope, and J. D. Close, Phys. Rev. A 72, 031606 (2005). [20] J. Ruostekoski, T. Gasenzer, and D. Hutchinson, Phys. Rev. A 68, 011604 (2003). [21] C. J. Bordé, C. R. Acad. Sci. Paris 4, 509 (2001). [22] J.-F. Riou, Ph.D. thesis, Institut D’Optique (2006). [23] M. Kozuma, L. Deng, E. W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. L. Rolston, and W. D. Phillips, Phys. Rev. Lett. 82, 871 (1999). [24] J. Dugué, N. P. Robins, C. Figl, M. Jeppesen, P. Sum- mers, M. T. Johnsson, J. J. Hope, and J. D. Close, Phys. Rev. A 75, 053602 (2007). [25] Y. Torii, Y. Suzuki, M. Kozuma, T. Sugiura, T. Kuga, L. Deng, and E. W. Hagley, Phys. Rev. A 61, 041602 (2000). mailto:[email protected] http://www.acqao.org

0704.0292

A practical Seedless Infrared-Safe Cone jet algorithm

incl-spect-pythia-lhc.eps A practical Seedless Infrared-Safe Cone jet algorithm Gavin P. Salam and Grégory Soyez∗† LPTHE, Université Pierre et Marie Curie – Paris 6, Université Denis Diderot – Paris 7, CNRS UMR 7589, 75252 Paris cedex 05, France. arXiv:0704.0292 [hep-ph] April 2007 Abstract Current cone jet algorithms, widely used at hadron colliders, take event particles as seeds in an iterative search for stable cones. A longstanding infrared (IR) unsafety issue in such algorithms is often assumed to be solvable by adding extra ‘midpoint’ seeds, but actually is just postponed to one order higher in the coupling. A proper solution is to switch to an exact seedless cone algorithm, one that provably identifies all stable cones. The only existing approach takes N2N time to find jets among N particles, making it unusable at hadron level. This can be reduced to N2 lnN time, leading to code (SISCone) whose speed is similar to that of public midpoint implementations. Monte Carlo tests provide a strong cross-check of an analytical proof of the IR safety of the new algorithm, and the absence of any ‘Rsep’ issue implies a good practical correspondence between parton and hadron levels. Relative to a midpoint cone, the use of an IR safe seedless algorithm leads to modest changes for inclusive jet spectra, mostly through reduced sensitivity to the underlying event, and significant changes for some multi-jet observables. SISCone, the C++ implementation of the algorithm, is available at http://projects.hepforge.org/siscone/ (standalone), http://www.lpthe.jussieu.fr/~salam/fastjet/ (FastJet plugin). ∗On leave from the PTF group of the University of Liège. †Current address: Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA http://arxiv.org/abs/0704.0292v2 http://arxiv.org/abs/0704.0292 http://projects.hepforge.org/siscone/ http://www.lpthe.jussieu.fr/~salam/fastjet/ Contents 1 Introduction 2 2 Overview of the cone jet-finding algorithm 5 3 IR unsafety in the midpoint algorithm 6 4 An exact seedless cone jet definition 8 4.1 One-dimensional example . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.2 The two-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2.1 General approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 4.2.2 Specific computational strategies . . . . . . . . . . . . . . . . . . . 11 4.3 The split–merge part of the cone algorithm . . . . . . . . . . . . . . . . . . 14 5 Tests and comparisons 16 5.1 Measures of IR (un)safety . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5.2 Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.3 Rsep: an inexistent problem . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.4 Physics impact of seedless v. midpoint cone . . . . . . . . . . . . . . . . . 23 5.4.1 Inclusive jet spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.4.2 Jet masses in 3-jet events . . . . . . . . . . . . . . . . . . . . . . . 26 6 Conclusions 30 A Further computational details 32 A.1 Cone multiplicities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 A.2 Computational complexity of the split–merge step . . . . . . . . . . . . . . 32 B Proof of IR safety of the SISCone algorithm 33 B.1 General aspects of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 34 B.2 Split–merge ordering variable . . . . . . . . . . . . . . . . . . . . . . . . . 37 1 Introduction Two broad classes of jet definition are generally advocated [1] for hadron colliders. One option is to use sequential recombination jet algorithms, such as the kt [2] and Cam- bridge/Aachen algorithms [3], which introduce a distance measure between particles, and repeatedly recombine the closest pair of particles until some stopping criterion is reached. While experimentally these are starting to be investigated [4, 5], the bulk of measurements are currently carried out with the other class of jet definition, cone jet algorithms (see e.g. [6]). In general there are indications [7] that it may be advantageous to use both sequential recombination and cone jet algorithms because of complementary sensitivities to different classes of non-perturbative corrections. Cone jet algorithms are inspired by the idea [8] of defining a jet as an angular cone around some direction of dominant energy flow. To find these directions of dominant energy flow, cone algorithms usually take some (or all) of the event particles as ‘seeds’, i.e. trial cone directions. Then for each seed they establish the list of particles in the trial cone, evaluate the sum of their 4-momenta, and use the resulting 4-momentum as a new trial direction for the cone. This procedure is iterated until the cone direction no longer changes, i.e. until one has a “stable cone”. Stable cones have the property that the cone axis a (a four-vector) coincides with the (four-vector) axis defined by the total momentum of the particles contained in the cone, D (pin cone, a) = 0 , with pin cone = pi Θ(R−D(pi, a)) , (1) where D(p, a) is some measure of angular distance between the four-momentum p and the cone axis a, and R is the given opening (half)-angle of the cone, also referred to as the cone radius. Typically one defines D2(p, a) = (yp− ya)2+(φp−φa)2, where yp, ya and φp, φa are respectively the rapidity and azimuth of p and a. Two types of problem arise when using seeds as starting points of an iterative search for stable cones. On one hand, if one only uses particles above some momentum threshold as seeds, then the procedure is collinear unsafe. Alternatively if any particle can act as a seed then one needs to be sure that the addition of an infinitely soft particle cannot lead to a new (hard) stable cone being found, otherwise the procedure is infrared (IR) unsafe. The second of these problems came to fore in the 1990’s [9], when it was realised that there can be stable cones that have two hard particles on opposing edges of the cone and no particles in the middle, e.g. for configurations such as pt1 > pt2; R < D(p1, p2) < (1 + pt2/pt1)R. (2) In traditional iterative cone algorithms, p1 and p2 each act as seeds and two stable cones are found, one centred on p1, the other centred on p2. The third stable cone, centred between p1 and p2 (and containing them both) is not found. If, however, a soft particle is added between the two hard particles, it too acts as a seed and the third stable cone is then found. The set of stable cones (and final jets) is thus different with and without the soft particle and there is a resulting non-cancellation of divergent real soft production and corresponding virtual contributions, i.e. the algorithm is infrared unsafe. Infrared unsafety is a serious issue, not just because it makes it impossible to carry out meaningful (finite) perturbative calculations, but also because it breaks the whole relation between the (Born or low-order) partonic structure of the event and the jets that one observes, and it is precisely this relation that a jet algorithm is supposed to codify: it makes no sense for the structure of multi-hundred GeV jets to change radically just because hadronisation, the underlying event or pileup threw a 1 GeV particle in between them. A workaround for the above IR unsafety problem was proposed in [9]: after finding the stable cones that come from the true seed particles, add artificial “midpoint” seeds between pairs of stable cones and search for new stable cones that arise from the midpoint seeds. For configurations with two hard particles, the midpoint fix resolved the IR unsafety issue. It was thus adopted as a recommendation [6] for Run II of the Tevatron and is now coming into use experimentally [10, 11]. Recently, it was observed [1] that in certain triangular three-point configurations there are stable cones that are not identified even by the midpoint procedure. While these can be identified by extended midpoint procedures (e.g. midpoints between triplets of particles) [12, 13], in this article (section 3) we show that there exist yet other 3-particle configurations for which even this fix does not find all stable cones. Given this history of infrared safety problems being fixed and new ones being found, it seems to us that iterative1 cone algorithms should be abandoned. Instead we believe that cone jet algorithms should solve the mathematical problem of demonstrably finding all stable cones, i.e. all solutions to eq. (1). This kind of jet algorithm is referred to as an exact seedless cone jet algorithm [6] and has been advocated before in [16]. With an exact seedless algorithm, the addition of one or more soft particles cannot lead to new hard stable cones being found, because all hard stable cones have already been (provably) found. Therefore the algorithm is infrared safe at all orders. Two proposals exist for approximate implementations of the seedless jet algorithm [6, 17]. They both rely on the event being represented in terms of calorimeter towers, which is far from ideal when considering parton or hadron-level events. Ref. [6] also pro- posed a procedure for an exact seedless jet algorithm, intended for fixed-order calculations, and implemented for example in the MCFM and NLOJet fixed order (NLO) codes [18, 19].2 This method takes a time O to find jets among N particles. While perfectly ad- equate for fixed order calculations (N ≤ 4), a recommendation to extend the use of such seedless cone implementations more generally would have little chance of being adopted experimentally: the time to find jets in a single (quiet!) event containing 100 particles would approach 1017 years. Given the crucial importance of infrared safety in allowing one to compare theoretical predictions and experimental measurements, and the need for the same algorithm to be used in both, there is a strong motivation for finding a more efficient way of implementing the seedless cone algorithm. Section 4 will show how this can be done, first in the context of a simple one-dimensional example (sec. 4.1), then generalising it to two dimensions (y, φ, sec. 4.2) with an approach that can be made to run in polynomial (N2 lnN) time. As 1A more appropriate name might be the doubly iterative cone algorithm, since as well as iterating the cones, the cone algorithm’s definition has itself seen several iterations since its original introduction by UA1 in 1983 [14], and even since the Snowmass accord [15], the first attempt to formulate a standard, infrared and collinear-safe cone-jet definition, over 15 years ago. 2Section 3.4.2 of [6] is the source of some confusion regarding nomenclature, because after discussing both the midpoint and seedless algorithms, it proceeds to show some fixed-order results calculated with the seedless algorithm, but labelled as midpoint. Though both algorithms are IR safe up to the order that was shown, they would not have given identical results. in recent work on speeding up the kt jet-algorithm [20], the key insights will be obtained by considering the geometrical aspects of the problem. Section 4.3 will discuss aspects of the split–merge procedure. In section 5 we will study a range of physics and practical properties of the seedless algorithm. Given that the split–merge stage is complex and so yet another potential source of infrared unsafety, we will use Monte Carlo techniques to provide independent evidence for the safety of the algorithm, supplementing a proof given in appendix B. We will examine the speed of our coding of the algorithm and see that it is as fast as publicly available midpoint codes. We will also study the question of the relation between the low- order perturbative characteristics of the algorithm, and its all-order behaviour, notably as concerns the ‘Rsep’ issue [21, 1]. Finally we highlight physics contexts where we see similarities and differences between our seedless algorithm and the midpoint algorithm. For inclusive quantities, such as the inclusive jet spectrum, perturbative differences are of the order of a few percent, increasing to 10% at hadron level owing to reduced sensitivity to the underlying event in the seedless algorithm. For exclusive quantities we see differences of the order of 10− 50%, for example for mass spectra in multi-jet events. 2 Overview of the cone jet-finding algorithm Algorithm 1 A full specification of a modern cone algorithm, governed by four param- eters: the cone radius R, the overlap parameter f , the number of passes Npass and a minimum transverse momentum in the split–merge step, pt,min. Throughout, particles are to be combined by summing their 4-momenta and distances are to be calculated using the longitudinally invariant ∆y and ∆φ distance measures (where y is the rapidity). 1: Put the set of current particles equal to the set of all particles in the event. 2: repeat 3: Find all stable cones of radius R (see Eq. (1)) for the current set of particles, e.g. using algorithm 2, section 4.2.2. 4: For each stable cone, create a protojet from the current particles contained in the cone, and add it to the list of protojets. 5: Remove all particles that are in stable cones from the list of current particles. 6: until No new stable cones are found, or one has gone around the loop Npass times. 7: Run a Tevatron Run-II type split–merge procedure [6], algorithm 3 (section 4.3), on the full list of protojets, with overlap parameter f and transverse momentum threshold pt,min. Before entering into technical considerations, we outline the structure of a modern cone jet definition as algorithm 1, largely based on the Tevatron Run-II specification [6]. It is governed by four parameters. The cone radius R and overlap parameter f are standard and appeared in previous cone algorithms. The Npass variable is new and embodies the suggestion in [1] that one should rerun the stable cone search to eliminate dark towers [21], particle pt [GeV] y φ 1 400 0 0 2 110 0.9R 0 3 90 2.3R 0 4 1.1 1.5R 0 Table 1: Particles 1–3 represent a hard configuration. The jets from this hard configuration are modified in the midpoint cone algorithm when one adds the soft particle 4. i.e. particles that do not appear in any stable cones (and therefore never appear in jets) during a first pass of the algorithm, even though they can correspond to significant energy deposits. A sensible default is Npass = ∞ since, as formulated, the procedure will in any case stop once further passes find no further stable cones. The pt,min threshold for the split–merge step is also an addition relative to the Run II procedure, inspired by [12, 7]. It is discussed in section 4.3 together with the rest of the split–merge procedure and may be set to zero to recover the original Run II type behaviour, a sensible default. The main development of this paper is the specification of how to efficiently carry out step 3 of algorithm 1. In section 3 we will show that the midpoint approximation for finding stable cones fails to find them all, leading to infrared unsafety problems. Section 4 will provide a practical solution. Code corresponding to this algorithm is available publicly under the name of ‘Seedless Infrared Safe Cone’ (SISCone). 3 IR unsafety in the midpoint algorithm Until now, the exact exhaustive identification of all stable cones was considered to be too computationally complex to be feasible for realistic particle multiplicities. Instead, the Tevatron experiments streamline the search for stable cones with the so-called ’midpoint algorithm’ [9]. Given a seed, the latter calculates the total momentum of the particles contained within a cone centred on the seed, uses the direction of this momentum as a new seed and iterates until the resulting cone is stable. The initial set of seeds is that of all particles whose transverse momentum is above a seed threshold s (one may take s = 0 to obtain a collinear-safe algorithm). Then, one adds a new set of seeds given by all midpoints between pairs of stable cones separated by less than 2R and repeats the iterations from these midpoint seeds. The problem with the midpoint cone algorithm can be seen from the configurations of table 1, represented also in fig. 1. Using particles 1 − 3, there exist three stable cones. In a pt-scheme recombination procedure (a pt weighted averaging of y and φ) they are at y ≃ {0.194R, 1.53R, 2.3R}.3 Note however that starting from particles 1, 2, 3 as seeds, one only iterates to the stable cones at y ≃ 0.194R and y = 2.3R. Using the midpoint between 3In a more standard E-scheme (four-momentum) recombination procedure the exact numbers depend slightly on R, but the conclusions are unchanged. p t/GeV p t/GeV (a) (b) y0 1 2 3−1 y0 1 2 3−1 Figure 1: Configuration illustrating one of the IR unsafety problems of the midpoint jet algorithm (R = 1); (a) the stable cones (ellipses) found in the midpoint algorithm; (b) with the addition of an arbitrarily soft seed particle (red wavy line) an extra stable cone is found. these two stable cones, at y ≃ 1.247R, one iterates back to the stable cone at y ≃ 0.194R, therefore the stable cone at y = 1.53R is never found. The result is that particles 1 and 2 are in one jet, and particle 3 in another, fig.1a. If additionally a soft particle (4) is present to act as a seed near y = 1.53R, fig.1b, then the stable cone there is found from the iterative procedure. In this case we have three overlapping stable cones, with hard-particle content 1 + 2, 2 + 3 and 3. What happens next depends on the precise splitting and merging procedure that is adopted. Using that of [6] then for f < 0.55 the jets are merged into a single large jet 1 + 2+ 3, otherwise they are split into 1 and 2 + 3. Either way the jets are different from those obtained without the extra soft seed particle, meaning that the procedure is infrared unsafe. In contrast, a seedless approach would have found the three stable cones independently of the presence of the soft particle and so would have given identical sets of jets. The infrared divergence arises for configurations with 3 hard particles in a common neighbourhood plus one soft one (and a further hard electroweak boson or QCD parton to balance momentum). Quantities where it will be seen include the NLO contribution to the heavy-jet mass in W/Z+2-jet (or 3-jet) events, the NNLO contribution to the W/Z+2-jet cross section or the 3-jet cross section, or alternatively at NNNLO in the inclusive jet cross section. The problem might therefore initially seem remote, since the theoretical state of the art is far from calculations of any of these quantities. However one should recall that infrared safety at all orders is a prerequisite if the perturbation series is to make sense at all. If one takes the specific example of the Z+2-jet cross section (measured in [10]) then the NNLO divergent piece would be regulated physically by confinement at the non-perturbative scale ΛQCD, and would give a contribution of order s ln pt/ΛQCD. Since αs(pt) ln pt/ΛQCD ∼ 1, this divergent NNLO contribution will be of the same order as the NLO piece αEWα s. Therefore the NLO calculation has little formal meaning for the midpoint algorithm, since contributions involving yet higher powers of αs Observable 1st miss cones at Last meaningful order Inclusive jet cross section NNLO NLO W/Z/H + 1 jet cross section NNLO NLO 3 jet cross section NLO LO W/Z/H + 2 jet cross section NLO LO jet masses in 3 jets, W/Z/H + 2 jets LO none Table 2: Summary of the order (α4s or α sαEW ) at which stable cones are missed in various processes with a midpoint algorithm, and the corresponding last order that can be mean- ingfully calculated. Infrared unsafety first becomes visible one order beyond that at which one misses stable cones. will be parametrically as large as the NLO term.4 The situation for a range of processes is summarised in table 2. 4 An exact seedless cone jet definition One way in which one could imagine trying to ‘patch’ the seed-based iterative cone jet- algorithm to address the above problem would be to use midpoints between all pairs of particles as seeds, as well as midpoints between the initial set of stable cones.5 However it seems unlikely that this would resolve the fundamental problem of being sure that one will systematically find all solutions of eq. (1) for any ensemble of particles. Instead it is more appropriate to examine exhaustive, non-iterative approaches to the problem, i.e. an exact seedless cone jet algorithm, one that provably finds all stable cones, as advocated already some time ago in [16]. For very low multiplicities N , one approach is that suggested in section 3.3.3 of [6] and used in the MCFM [18] and NLOJet [19] next-to-leading order codes. One first identifies all possible subsets of the N particles in the event. For each subset S, one then determines the rapidity (yS) and azimuth (φS) of the total momentum of the subset, pS = i∈S pi and then checks whether a cone centred on yS , φS contains all particles in S but no other particles. If this is the case then S corresponds to a stable cone. This procedure guarantees that all solutions to eq. (1) will be found. In the above procedure there are ∼ 2N distinct subsets of particles and establishing whether a given subset corresponds to a stable cone takes time O (N). Therefore the time to identify all stable cones is O . For the values of N (≤ 4) relevant in fixed- order calculations, N2N time is manageable, however as soon as one wishes to consider 4As concerns the measurement [10], the discussion is complicated by the confusion surrounding the nomenclature of the seedless and midpoint algorithms — while it seems that the measurement was carried out with a true midpoint algorithm, the calculation probably used the ‘midpoint’ as defined in section 3.4.2 of [6] (cf. footnote 2), which is actually the seedless algorithm, i.e. the measurements and theoretical predictions are based on different algorithms. 5This option was actually mentioned in [6] but rejected at the time as impractical. etc... Figure 2: Representation of points on a line and the places where a sliding segment has a change in its set of enclosed points. parton-shower or hadron-level events, with dozens or hundreds of particles, N2N time is prohibitive. A solution can only be considered realistic if it is polynomial in N , preferably with not too high a power of N . As mentioned in the introduction, approximate procedures for implementing seedless cone jet algorithms have been proposed in the past [6, 17]. These rely on considering the momentum flow into discrete calorimeter towers rather than considering particles. As such they are not entirely suitable for examining the full range event levels, which go from fixed- order (few partons), via parton shower level (many partons) and hadron-level, to detector level which has both tracking and calorimetry information. 4.1 One-dimensional example To understand how one might construct an efficient exact seedless cone jet algorithm, it is helpful to first examine a one-dimensional analogue of the problem. The aim is to identify all solutions to eq. (1), but just for (weighted) points on a line. The equivalent of a cone of radius R is a segment of length 2R. Rather than immediately looking for stable segments one instead looks for all distinct ways in which the segment can enclose a subset of the points on the line. Then for each separate enclosure one calculates its centroid C (weighted with the pt of the particles) and verifies whether the segment centred on C encloses the same set of points as the original enclosure. If it does then C is the centre of a stable segment. A simple way of finding all distinct segment-enclosures is illustrated in fig.2. First one sorts the points into order on the line. One then places the segment far to the left and slides it so that it goes infinitesimally beyond the leftmost point. This is a first enclosure. Then one slides the segment again until its right edge encounters a new point or the left edge encounters a contained point. Each time either edge encounters a point, the point-content of the segment changes and one has a new distinct enclosure. Establishing the stability of each enclosure is trivial, since one knows how far the segment can move in each direction without changing its point content — so if the centroid is such that the segment remains within these limits, the enclosure corresponds to a stable segment. The computational complexity of the above procedure, N lnN , is dominated by the need to sort the points initially: there are O (N) distinct enclosures and, given the sorted list, finding the next point that will enter or leave an edge costsO (1) time, as does updating the weighted centroid (assuming rounding errors can be neglected), so that the time not associated with the sorting step is O (N). (a) (b) (c) (d) Figure 3: (a) Some initial circular enclosure; (b) moving the circle in a random direction until some enclosed or external point touches the edge of the circle; (c) pivoting the circle around the edge point until a second point touches the edge; (d) all circles defined by pairs of edge points leading to the same circular enclosure. 4.2 The two-dimensional case 4.2.1 General approach The solution to the full problem can be seen as a 2-dimensional generalisation of the above procedure.6 The key idea is again that of trying to identify all distinct circular enclosures, which we also call distinct cones (by ‘distinct’ we mean having a different point content), and testing the stability of each one. In the one-dimensional example there was a single degree of freedom in specifying the position of the segment and all distinct segment enclosures could be obtained by considering all segments with an extremity defined by a point in the set. In 2 dimensions there are two degrees of freedom in specifying the position of a circle, and as we shall see, the solution to finding all distinct circular enclosures will be to examine all circles whose circumference lies on a pair of points from the set. To see in detail how one reaches this conclusion, it is useful to examine fig. 3. Box (a) shows a circle enclosing two points, the (red) crosses. Suppose, in analogy with fig. 2 that one wishes to slide the circle until its point content changes. One might choose a direction at random and after moving a certain distance, the circle’s edge will hit some point in the plane, box (b), signalling that the point content is about to change. In the 1-dimensional case a single point, together with a binary orientation (taking it to be the left or right-hand point) were sufficient to characterise the segment enclosure. However in the 2-dimensional case one may orient the circle in an infinite number of ways. We can therefore pivot the circle around the boundary point. As one does this, at some point a second point will then touch the boundary of the circle, box (c). The importance of fig. 3 is that it illustrates that for each and every enclosure, one can always move the corresponding circle (without changing the enclosure contents) into a position where two points lie on its boundary.7 Conversely, if one considers each circle 6We illustrate the planar problem rather than the cylindrical one since for R < π/2 the latter is a trivial generalisation of the former. 7There are two minor exceptions to this: (a) for any point separated from all others by more than 2R, the circle containing it can never have more than that one point on its edge — any such point forms a stable cone of its own; (b) there may be configurations where three or more points lie on the same circle whose boundary is defined by a pair of points in the set, and considers all four permutations of the edge points being contained or not in the enclosure, then one will have identified all distinct circular enclosures. Note that one given enclosure can be defined by several distinct pairs of particles, which means that when considering the enclosures defined by all pairs of particles, we are likely to find each enclosure more than once, cf. fig. 3d. A specific implementation of the above approach to finding the stable cones is given as algorithm 2 below. It runs in expected time O (Nn ln n) where N is the total number of particles and n is the typical number of particles in a circle of radius R.8 The time is dominated by a step that establishes a traversal order for the O (Nn) distinct circular enclosures, much as the one-dimensional (N lnN) example was dominated by the step that ordered the O (N) distinct segment enclosures.9 Some aspects of algorithm 2 are rather technical and are explained in the subsubsection that follows. A reader interested principally in the physics of the algorithm may prefer to skip it on a first reading. 4.2.2 Specific computational strategies A key input in evaluating the computational complexity of various algorithms is the knowl- edge of the number of distinct circular enclosures (or ‘distinct cones’) and the number of stable cones. These are both estimated in appendix A.1, and are respectively O (Nn) and (expected) O (N). Before giving the 2-dimensional analogue of the 1-d algorithm of section 4.1 we examine a simple ‘brute force’ approach for finding all stable cones. One takes all ∼ Nn pairs of points within 2R of each other and for each pair identifies the contents of the circle and establishes whether it corresponds to a stable cone, at a cost of O (N) each time, leading to an overall N2n total cost. This is to be compared to a standard midpoint cone algorithm, whose most expensive step will be the iteration of the expected O (Nn) midpoint seeds, for a total cost also of N2n, assuming the average number of iterations from any given seed to be O (1).10 One can reduce the computational complexity by using some of the ideas from the 1-d example, notably the introduction of an ordering for the boundary points of circles, and the use of the boundary points as sentinels for instability. Specifically, three elements will be required: i) one needs a way of labelling distinct cones that allows one to test whether two cones are the same at a cost of O (1); of radius R (i.e. are cocircular) — given a circle defined by a pair of them, the question of which of the others is in the circle becomes ambiguous and one should explicitly consider all possible combinations of inclusion/exclusion; a specific case of this is when there are collinear momenta (coincident points), which can however be dealt more simply by immediately merging them. 8Given a detector that extends to rapidities y < ymax, n/N ∼ πR2/(4πymax), which is considerably smaller than 1 — this motivates us to distinguish n from N . 9For comparison we note that the complexity of public midpoint algorithm implementations scales as 10In both cases one can reduce this to Nn2 by tiling the plane into squares of edge-length R and restricting the search for the circle contents to tiles in the vicinity of the circle centre. Algorithm 2 Procedure for establishing the list of all stable cones (protojets). For sim- plicity, parts related to the special case of multiple cocircular points (see footnote 7) are not shown. They are a straightforward generalisation of steps 6 to 13. 1: For any group of collinear particles, merge them into a single particle. 2: for particle i = 1 . . . N do 3: Find all particles j within a distance 2R of i. If there are no such particles, i forms a stable cone of its own. 4: Otherwise for each j identify the two circles for which i and j lie on the circumference. For each circle, compute the angle of its centre C relative to i, ζ = arctan ∆φiC 5: Sort the circles found in steps 3 and 4 into increasing angle ζ . 6: Take the first circle in this order, and call it the current circle. Calculate the total momentum and checkxor for the cones that it defines. Consider all 4 permutations of edge points being included or excluded. Call these the “current cones”. 7: repeat 8: for each of the 4 current cones do 9: If this cone has not yet been found, add it to the list of distinct cones. 10: If this cone has not yet been labelled as unstable, establish if the in/out status of the edge particles (with respect to the cone momentum axis) is the same as when defining the cone; if it is not, label the cone as unstable. 11: end for 12: Move to the next circle in order. It differs from the previous one either by a particle entering the circle, or one leaving the circle. Calculate the momentum for the new circle and corresponding new current cones by adding (or removing) the momentum of the particle that has entered (left); the checkxor can be updated by XORing with the label of that particle. 13: until all circles considered. 14: end for 15: for each of the cones not labelled as unstable do 16: Explicitly check its stability, and if it is stable, add it to the list of stable cones (protojets). 17: end for ii) one needs a way of ordering one’s examination of cones so that one can construct the cones incrementally, so as not to pay the (at least, see below) O ( n) construction price anew for each cone; iii) one needs a way limiting the number of cones for which we carry out a full stability test (which also costs at least To label cones efficiently, we assign a random q-bit integer tag to each particle. Then we define a tag for combinations of particles by taking the logical exclusive-or of all the tags of the individual particles (this is easily constructed incrementally and is sometimes referred to as a checkxor). Then two cones can be compared by examining their tags, rather than by comparing their full list of particles. With such a procedure, there is a risk of two non-identical cones ending up with identical tags (‘colliding’), which strictly speaking will make our procedure only ‘almost exact’. The probability p of a collision occurring is roughly the square of the number of enclosures divided by the number of distinct tags. Since we have O (Nn) enclosures, this gives p ∼ N2n2/2q. By taking q sufficiently large (in a test implementation we have used q = 96) and using a random number generator that guarantees that all bits are decorrelated [22], one can ensure a negligible collision probability.11 Given the ability to efficiently give a distinct label to distinct cones, one can address points ii) and iii) mentioned above by following algorithm 2. Point (ii) is dealt with by steps 2–6, 12 and 13: for each particle i, one establishes a traversal order for the circles having i on their edge — the traversal order is such that as one works through the circles, the circle content changes only by one particle at a time, making it easy to update the momentum and checkxor for the circle.12 One maintains a record of all distinct cones in the form of a hash (as a hash function one simply takes log2Nn bits of the tag), so that it only takes O (1) time to check whether a cone has been found previously. Rather than explicitly checking the stability of each distinct cone, the algorithm exam- ines whether the multiple edge points that define the cone are appropriately included/excluded in the circle around the cone’s momentum axis, step 10. All but a tiny fraction of unstable cones fail this test, so that at the end of step 14 one has a list (of size O (N)) of candidate stable cones — at that point one can carry out a full stability test for each of them. This therefore deals with point (iii) mentioned above. The dominant part of algorithm 2 is the ordering of the circles, step 5, which takes n lnn time and must be repeated N times. Therefore the overall cost is Nn ln n. As well as computing time, a significant issue is the memory use, because one must maintain a list of all distinct cones, of which there are O (Nn). One notes however that standard 11A more refined analysis shows that we need only worry about collisions between the tags of stable cones and other (stable or unstable) cones — since there are O (N) stable cones, the actual collision probability is more likely to be O /2q. In practice for N ∼ 104 and n ∼ 103 (a very highly populated event) and using q = 96, this gives p ∼ 10−18. In principle to guarantee an infinitesimal collision probability regardless of N, q should scale as lnN , however N will in any case be limited by memory use (which scales as Nn) so a fixed q is not unreasonable. 12Rounding errors can affect the accuracy of the momentum calculated this way; the impact of this can be minimised by occasionally recomputing the momentum of the circle from scratch. implementations of the split–merge step of the cone algorithm also require O (Nn) storage, albeit with a smaller coefficient. It is worth highlighting also an alternative approach, which though slower, O Nn3/2 has lower memory consumption and also avoids the small risk inexactness from the check- xor. It is similar to the brute-force approach, but uses 2-dimensional computational ge- ometry tree structures, such as quad-trees [23] or k-d trees [24]. These involve successive sub-divisions of the plane (in quadrants, or pairs of rectangles), similarly to what is done in 1-dimensional binary trees. They make it possible to check the stability of a given circle n time (the time is mostly taken by identifying tree cells near the edge of the circle, of which there are O ( n)), giving an overall cost of Nn3/2. The memory use of this form of approach is O (N n), simply the space needed to store the stable-cone contents.13 4.3 The split–merge part of the cone algorithm The split–merge part of our cone algorithm is basically that adopted for Run-II of the Tevatron [6]. It is shown in detail as algorithm 3. Since it does not depend on the procedure used to find stable cones, it may largely be kept as is. We do however include the following small modifications: 1. The run II proposal used Et throughout the split–merge procedure. This is not invariant under longitudinal boosts. We replace it with p̃t, a scalar sum of the transverse momenta of the constituents of the protojet. This ensures that the results are both boost-invariant and infrared safe. We note that choosing instead pt (a seemingly natural choice, made for example in the code of [19, 13]) would have led to IR unsafety in purely hadronic events — the question of the variable to be used for the ordering is actually a rather delicate one, and we discuss it in more detail in appendix B.2. 2. We introduce a threshold pt,min below which protojets are discarded (step 2 of algo- rithm 3). This parameter is motivated by the discussion in [6] concerning problems associated with an ‘excess’ of stable cones in seedless algorithms, notably in events with significant pileup. It provides an infrared and collinear safe way of removing the resulting large number of low pt stable cones. By setting it to zero one recovers a be- haviour identical to that of the Run-II algorithm (modulo the replacement Et → p̃t, above), and we believe that in practice zero is actually a sensible default value. We note that a similar parameter is present in PxCone [12, 7]. 13Though here we are mainly interested in exact approaches, one may also examine the question of the speed of the approximate seedless approach of Volobouev [17]. This approach represents the event on a grid and essentially calculates the stability of a cone at each point of the grid using a fast-Fourier transformation (FFT). In principle, for this procedure to be as good as the exact one, the grid should be fine enough to resolve each distinct cone, which implies that it should have O (Nn) points; therefore the FFT will require O (Nn lnNn) time, which is similar in magnitude to the time that is needed by the exact algorithm. An open question remains that of whether a coarser grid might nevertheless be ‘good enough’ for many practical applications. Algorithm 3 The disambiguated, scalar p̃t based formulation of a Tevatron Run-II type split–merge procedure [6], with overlap threshold parameter f and transverse momentum threshold pt,min. To ensure boost invariance and IR safety, for the ordering variable and the overlap measure, it uses of p̃t,jet = i∈jet |pt,i|, i.e. a scalar sum of the particle transverse momenta (as in a ‘pt’ recombination scheme). 1: repeat 2: Remove all protojets with pt < pt,min. 3: Identify the protojet (i) with the highest p̃t. 4: Among the remaining protojets identify the one (j) with highest p̃t that shares particles (overlaps) with i. 5: if there is such an overlapping jet then 6: Determine the total p̃t,shared = k∈i&j |pt,k| of the particles shared between i and 7: if p̃t,shared < fp̃t,j then 8: Each particle that is shared between the two protojets is assigned to the one to whose axis it is closest. The protojet momenta are then recalculated. 9: else 10: Merge the two protojets into a single new protojet (added to the list of protojets, while the two original ones are removed). 11: end if 12: If steps 7–11 produced a protojet that coincides with an existing one, maintain the new protojet as distinct from the existing copy(ies). 13: else 14: Add i to the list of final jets, and remove it from the list of protojets. 15: end if 16: until no protojets are left. 3. After steps 7–11, the same protojet may appear more than once in the list of protojets. For example a protojet may come once from a single original stable cone, and a second time from the splitting of another original stable cone. The original statement of the split–merge procedure [6] did not address this issue, and there is a resulting ambiguity in how to proceed. One option (as is done for example in the seedless cone code of [19]) is to retain only a single copy of any such identical protojets. This however introduces a new source of infrared unsafety: an added soft particle might appear in one copy of the protojet and not the other and the two protojets would then no longer be identical and would not be reduced to a single protojet. This could (and does occasionally, as evidenced in section 5.1) alter the subsequent split–merge sequence. If one instead maintains multiple identical protojets as distinct entities (as is done in the codes of [13, 18]), then the addition of a soft particle does not alter the number of hard protojet entries in the protojet list and the split–merge part of the algorithm remains infrared safe. We therefore choose this second option, and make it explicit as step 12 of algorithm 3. The split–merge procedure is guaranteed to terminate because the number of overlapping pairs of protojets is reduced each time an iteration of the loop finds an overlap. A proof of the infrared safety of this (and the other) parts of our formulation of the cone algorithm is given in appendix B. The computational complexity (O (N2)) of the split–merge procedure is generally smaller than that of the stable-cone search, and so we relegate its discussion to appendix A.2. Finally, before closing this section, let us return briefly to the top-level of the cone formulation, algorithm 1 and the question of the loop over multiple passes. This loop contains just the stable-cone search, and one might wonder why the split–merge step has not also been included in the loop. First consider pt,min = 0: protojets found in different passes cannot overlap, and the split–merge procedure is such that if a particle is in a protojet then it will always end up in a jet. Therefore it is immaterial whether the split– merge step is kept inside or outside the loop. The advantage of keeping it outside the loop is that one may rerun the algorithm with multiple overlap values f simply by repeating the split–merge step, without repeating the search for stable cones. For pt,min 6= 0 the positioning of the split–merge step with respect to the Npass loop would affect the outcome of the algorithm if all particles not found in first-pass jets were to be inserted into the second pass stable-cone search. Our specific formulation constitutes a design choice, which allows one to rerun with different values of f and pt,min without repeating the stable-cone search. 5 Tests and comparisons 5.1 Measures of IR (un)safety In section 4 we presented a procedure for finding stable cones that is explicitly IR safe. In appendix B we provide a proof of the IR safety of the rest of the algorithm. The latter is rather technical and not short, and while we have every reason to believe it to be correct, we feel that there is value in supplementing it with complementary evidence for the IR safety of the algorithm. As a byproduct, we will obtain a measure of the IR unsafety of various commonly used formulations of the cone algorithm. To verify the IR safety of the seedless cone algorithm, we opt for a numerical Monte Carlo approach, in analogy with that used in [25] to test the more involved recursive infrared and collinear safety (a prerequisite for certain kinds of resummation). The test proceeds as follows. One generates a ‘hard’ event consisting of some number of randomly distributed momenta of the order of some hard scale pt,H , and runs the jet algorithm on the hard event. One then generates some soft momenta at a scale pt,S ≪ pt,H , adds them to the hard event (randomly permuting the order of the momenta) and reruns the jet algorithm. One verifies that the hard jets obtained with and without the soft event are identical. If they are not, the jet algorithm is IR unsafe. For a given hard event one repeats the test with many different add-on soft events so as to be reasonably sure of identifying most hard events that are IR unsafe. One then repeats the whole procedure for many hard events. Algorithm Type IR unsafe Code JetClu Seeded, no midpoints 2h+1s [9] [13] SearchCone Seeded, search cone [21], midpoints 2h+1s [1] [13] MidPoint Seeded, midpoints (2-way) 3h+1s [1] [13] MidPoint-3 Seeded, midpoints (2-way, 3-way) 3h+1s [13] PxCone Seeded, midpoints (n-way), non-standard SM 3h+1s [12] Seedless [SM-pt] Seedless, SM uses pt 4h+1s a [here] Seedless [SM-MIP] Seedless, SM merges identical protojets 4h+1sb [here] Seedless [SISCone] Seedless, SM of algorithm 3 no [here] aFailures on 4h+1s arise only for R > π/4; for smaller R, failures arise only for higher multiplicities bFailures for 4h+1s are extremely rare, but become more common for 5h+1s and beyond Table 3: Summary of the various cone jet algorithms and the code used for tests here; SM stands for “split–merge”; Nh+Ms indicates that infrared unsafety is revealed with configurations consisting of N hard particles and M soft ones, not counting an additional hard, potentially non-QCD, particle to conserve momentum. All codes have been used in the form of plugins to FastJet (v2.1) [20]. The hard events are produced as follows: we choose a linearly distributed random number of momenta (between 2 and 10) and for each one generate a random pt (linearly distributed, 2−24pt,H ≤ pt ≤ pt,H , with pt,H = 1000GeV), a random rapidity (linearly distributed in −1.5 < y < 1.5) and a random φ. For each hard event we also choose random parameters for the jet algorithm, so as to cover the jet-algorithm parameter space (0.3<R<1.57, 0.25<f <0.95, linearly distributed, the upper limit on R being motivated by the requirement that R < π/2; the pt,min on protojets is set to 0 and the number of passes is set to 1). For each add-on soft event we generate between 1 and 5 soft momenta, distributed as the hard ones, but with the soft scale pt,S = 10 −100GeV replacing pt,H . We note that the hard events generated as above do not conserve momentum — they are analogous to events with a missing energy component or with identified photons or leptons that are not given as inputs to the jet clustering. For the safety studies on the full SISCone algorithm, we therefore also generate a set of hard events which do have momentum conservation, analogous to purely hadronic events. To validate our approach to testing IR safety, we apply it to a range of cone jet algo- rithms, listed in table 3, including the many variants that are IR unsafe. In PxCone the cut on protojets is set to 1GeV and in the SearchCone algorithm the search cone radius is set to R/2. The fraction of hard events failing the safety test is shown in fig. 4 for each of the jet algorithms.14 All jet algorithms that are known to be IR unsafe do indeed fail the tests. 14The results are based on 80 trial soft add-on events for each hard event and should differ by no more than a few percent (relative) from a full determination of the IR safety for each hard event (which would be obtained in the limit of an infinite number of trial soft add-on events for each hard event). For SISCone we only use 20 soft add-on events, so as to make it possible to probe a larger number of hard configurations. 10-5 10-4 10-3 10-2 10-1 1 Fraction of hard events failing IR safety test JetClu SearchCone PxCone MidPoint Midpoint-3 Seedless [SM-pt] Seedless [SM-MIP] Seedless (SISCone) 50.1% 48.2% 16.4% 15.6% 0.17% < 10-9 Figure 4: Failure rates for the IR safety tests. The algorithms are as detailed in table 3. Seeded algorithms have been used with a zero seed threshold. The events used do not conserve momentum (i.e. have a missing energy component), except for the seedless SM-pt case (where all events conserve momentum, to highlight the issue that arises in that case) and for SISCone (where we use a mix of momentum conserving and non-conserving events so as to fully test the algorithm). Further details are given in the text One should be aware that the absolute failure rates depend to some extent on the way we generated the hard events, and so are to be interpreted with caution. Having said that, our hard events have a complexity similar to the Born-level (lowest-order parton-level) of events that will be studied at LHC, for example in the various decay channels of tt̄H production, and so both the order of magnitudes of the failure rates and their relative sizes should be meaningful. Algorithms that fail on ‘2h+1s’ events have larger failure rates than those that fail on ‘3h+1s’ events, as would be expected — they are ‘more’ infrared unsafe. One notes the significant failure rates for the midpoint algorithms, ∼ 16%, and the fact that adding 3-way midpoints (i.e. between triplets of stable cones) has almost no effect on the failure rate, indicating that triangular configurations identified as IR unsafe in [1] are much less important than others such as that discussed in section 3. PxCone’s smaller failure rate seems to be due not to its multi-way midpoints, but rather to its specific split–merge procedure which leads to fewer final jets (so that one is less sensitive to missing stable cones). Seedless algorithms with problematic split–merge procedures lead to small failure rates (restricting one’s attention to small values of R, these values are further reduced). One might be tempted to argue that such small rates of IR safety failure are unlikely to have a physical impact and can therefore be ignored. However there is always a risk of some specific study being unusually sensitive to these configurations, and in any case our aim here is to provide an algorithm whose IR safety is exact, not just approximate. Finally, with a ‘good’ split–merge procedure, that given as algorithm 3, none of the over 5 × 109 hard events tested (a mix both with and without momentum conservation) failed the IR safety test. For completeness, we have carried out limited tests also for Npass = ∞ and with a pt,min on protojets of 100GeV, and have additionally performed tests with a larger range of rapidities (|y| < 3), collinearly-split momenta, cocircular configurations, three scales instead of two scales and again found no failures. These tests together with the proof given in appendix B give us a good degree of confidence that the algorithm truly is infrared safe, hence justifying its name. 5.2 Speed As can be gathered from the discussion in [6], reasonable speed is an essential requirement if a new variant of cone jet algorithm is to be adopted. To determine the speed of various cone jet algorithms, we use the same set of events taken for testing the FastJet formulation of the kt jet algorithm in [20] — these consist of a single Pythia [26] dijet event (with pt,jets ≃ 50GeV) to which we add varying numbers of simulated minimum bias events so as to vary the multiplicity N . Thus the event structure should mimic that of LHC events with pileup. Figure 5 shows the time needed to find jets in one event as a function of N . Among the seeded jet algorithms we consider only codes that include midpoint seeds. For the (CDF) midpoint code [13], written in C++, there is an option of using only particles above a threshold s as seeds and we consider both the common (though collinear unsafe) choice s = 1GeV and the (collinear safe but IR unsafe) s = 0GeV. The PxCone code [12], written in Fortran 77, has no seed threshold. Our seedless code, SISCone, is comparable in speed to the fastest of the seeded codes, the CDF midpoint code with a seed threshold s = 1GeV, and is considerably faster than the codes without a seed threshold (not to mention existing exact seedless codes which take ∼ 1 s to find jets among 20 particles and scale as N2N). Its run time also increases more slowly with N than that of the seeded codes, roughly in agreement with the expectation of SISCone going as Nn ln n (with a large coefficient) while the others go as N2n. The midpoint code with s = 1GeV has a more complex N -dependence presumably because we have run the timing on a single set of momenta, and the proportionality between the number of seeds and N fluctuates and depends on the event structure. For comparison purposes we have also included the timings for the FastJet (v2) kt imple- mentation, which for these values ofN uses a strategy that involves a combination ofN lnN and Nn dependencies. Timings for the FastJet implementation of the Aachen/Cambridge algorithm are similar to those for the kt algorithm. 0.001 0.01 100 1000 10000 CDF midpoint (s=0 GeV) CDF midpoint (s=1 GeV) PxCone SISCone kt (fastjet) Figure 5: Time to cluster N particles, as a function of N , for various algorithms, with R = 0.7 and f = 0.5, on a 3.4GHz Pentium R© IV processor. For the CDF midpoint algorithm, s is the threshold transverse momentum above which particles are used as seeds. 5.3 Rsep: an inexistent problem Suppose we have two partons separated by ∆R and with transverse momenta pt1 and pt2 (pt1 > pt2). Both partons end up in the same jet if the cone containing both is stable, i.e. < 1 + z , z = , (3) where the result is exact for small R or with pt-scheme recombination. Equivalently one can write the probability for two partons to be clustered into a single jet as P2→1(∆R, z) = Θ 1 + z − . (4) The limit on ∆R/R ranges from 1 for z = 0 to 2 for z = 1. This z-dependent limit is the main low-order perturbative difference between the cone algorithm and inclusive versions of sequential recombination ones like the kt or Cambridge/Aachen algorithms, since the latter merge two partons into a single jet for ∆R/R < 1, independently of their energies. Rsep = 1.3? TWO JETS 0 0.5 1 1.5 2 2.5 ∆R / R ONE JET NP: 2 jets? PT: 1 jet? Figure 6: Schematic representation of the phase space region in which two partons will end up in a single cone jet versus two jets, at the 2-parton level (PT) and, according to the Rsep statement, after showering and hadronisation (NP). A statement regularly made about cone algorithms (see for example [21, 1, 27]) is that parton showering and hadronisation reduce the stability of the cone containing the ‘original’ two partons, leading to a modified ‘practical’ condition for two partons to end up in a single jet, < min (Rsep , 1 + z) , (5) or equivalently, P2→1(∆R, z) = Θ 1 + z − ∆R Rsep − , (6) with Rsep ≃ 1.3 [28, 29].15 This situation is often represented as in figure 6, which depicts the ∆R, z plane, and shows the regions in which two partons are merged into one jet or resolved as two jets. The boundary ∆R = 1+z corresponds to eq. (3), while the alternative boundary at ∆R = Rsep is eq. (5). So large a difference between the low-order partonic expectation and hadron-level results would be quite a worrying feature for a jet algorithm — after all, the main purpose of a jet algorithm is to give as close a relation as possible between the first couple of orders of perturbation theory and hadron level.16 The evidence for the existence of eq. (6) with Rsep = 1.3 seems largely to be based [28, 29] on merging two events (satisfying some cut on the jet pt’s), running the jet-algorithm on the merged event, and examining at what distance particles from the two events end up in the same jet. This approach indicated that particles were indeed less likely to end 15The name Rsep was originally introduced [30] in the context of NLO calculations of hadron-collider jet-spectra, but with a different meaning — there it was intended as a free parameter to model the lack of knowledge about the details of the definition of the cone jet algorithm used experimentally. This is rather different from the current use as a parameter intended to model our inability to directly calculate the impact of higher-order and non-perturbative dynamics of QCD in cone algorithms. 16The apparent lack of correspondence is considered sufficiently severe that in some publications (e.g. [11]) the NLO calculation is modified by hand to compensate for this. 0 0.5 1 1.5 2 2.5 ∆R / Rcone Prob. 2 kt subjets → 1 cone jet = 1; Rcone = 0.4 a) parton level 0 0.5 1 1.5 2 2.5 ∆R / Rcone b) hadron level Figure 7: The probability P2→1(∆R, z) for two kt-algorithm subjets to correspond to a single cone jet, as a function of pt1/pt2 and ∆R for the two kt subjets. Events have been generated with Herwig [31] (hadron-level includes the underlying event) and the results are based on studying all kt jets with pt > 50GeV and |y| <1. Further details are to be found in the text. up in the same jet if they were more than 1.3R apart, however the result is an average over a range of z values making it hard to see whether eq. (6) is truly representative of the underlying physics.17 To address the question in more depth we adopt the following strategy. Rather than combining different events, we use one event at a time, but with two different jet algorithms. On one hand we run SISCone with a fairly small value of R, Rcone = 0.4. Simultaneously we run inclusive kt jet-clustering [2] on the event, using a relatively large R (Rkt = 1.0), and identify any hard kt-jets. For each hard kt jet we undo its last clustering step so as to obtain two subjets, S1 and S2 — these are taken to be the analogues of the two partons. We then examine whether there is a cone jet that contains more than half of the pt of each of S1 and S2. If there is, the conclusion is that the two kt subjets have ended up (dominantly) in a single cone jet. The procedure is repeated for many events, and one then examines the probability, P2→1(∆R, z), of the two kt subjets being identified with a single cone jet, as a function of the distance ∆R between the two subjets, S1 and S2, and the ratio z of their pt’s. The 17A preliminary version of [27] showed more differential results; these, however, seem not to be in the definitive version. results are shown in fig. 7 both at parton-shower level and at hadron level, as simulated with Herwig [31]. The middle contour corresponds to a probability of 1/2. At parton- shower level this contour coincides remarkably well with the boundary defined by eq. (3), up to ∆R/R = 1.7. It is definitely not compatible with eq. (5) with Rsep = 1.3. Beyond ∆R/R = 1.7 the contour bends a little and one might consider interpreting this as an Rsep ≃ 1.8.18 However, in that region the transition between P = 1 and P = 0 is broad, and to within the width of the transition, there remains good agreement with eq. (3) — it seems more natural therefore to interpret the small deviation from eq. (3) as a Sudakov- shoulder type structure [32], which broadens and shifts the Θ-function of eq. (4), as would happen with almost any discontinuity in a leading-order QCD distribution. Once one includes hadronisation effects in the study, fig. 7b, one finds that the transition region broadens further, as is to be expected. Now the P = 1/2 contour shifts away slightly from the 1 + z result at small z as well. However, once again this shift is modest, and of similar size as the breadth of the transition region. To verify the robustness of the above results we have examined other related indicators. One of them is the probability, P2→2 of finding two cone jets, each containing more than half of the transverse momentum of just one of the kt subjets. At two-parton level, one expects P1→2 + P2→2 = 1. Deviation from this would indicate that our procedure for matching cone jets to kt jets is misbehaving. We find that the relation holds to within around 15% over most of the region, deviating by at most ∼ 25% in a small corner of phase space ∆R/R ≃ 1.5, z ≃ 0.2. Another test is to examine the fraction F2 of the softer S2’s transverse momentum that is found in the cone that overlaps dominantly with S1. At two- parton level this should be equal to P2→1, but this would not be the case after showering if there were underlying problems with our matching procedure. We find however that F2 does agree well with P2→1. These, together with yet further tests, lead to us to believe that conclusions drawn from fig. 7 are robust. Finally, while these results have been obtained within a Monte Carlo simulation, Herwig, a similar study could equally be well carried experimentally on real events. So, in contrast to statements that are often made about the cone jet algorithm, the perturbative picture of when two partons will recombine, given by eq. (4), seems to be a relatively good indicator of what happens even after perturbative radiation and hadronisa- tion. In particular the evidence that we have presented strongly disfavours the Rsep-based modification, eq. (6). This is a welcome finding, and should help provide a firmer basis for cone-based phenomenology. 5.4 Physics impact of seedless v. midpoint cone In this section, we discuss the impact on physical measurement of switching from a mid- point type algorithm to a seedless IR-safe one such as SISCone. We study two physical observables, the inclusive jet spectrum and the jet mass spectrum in 3-jet events. The 18Such a value has been mentioned to us independently by M. Wobisch in the context of unpublished studies of jet shapes for the SearchCone algorithm [21]. spectra have been obtained by generating events with a Monte-Carlo either at fixed order in perturbation theory (NLOJet [19]) or with parton showering and hadronisation (Pythia [26]), and by performing the jet analysis on each event using three different algorithms (each with R = 0.7 and f = 0.5, and additionally in the case of SISCone, Npass = 1 and pt,min = 0): 1. SISCone: the seedless, IR-safe definition described in algorithms 1–3; 2. midpoint(0): the midpoint algorithm using all particles as seeds; 3. midpoint(1): the midpoint algorithm using as seeds all particles above a threshold of 1 GeV. We have used a version of the CDF implementation of the midpoint algorithm modified to have the split–merge step based on p̃t rather than pt (so that it corresponds to algorithm 4.3 with pt,min = 0). The motivation for this is that we are mainly interested in the physics impact of having midpoint versus all stable cones, and the comparison is simplest if the subsequent split–merge procedure is identical in both cases.19 We shall first present the results obtained for the inclusive jet spectrum and then discuss the jet mass spectrum in 3-jet events. Most studies carried out in this section have used kinematics corresponding to the Tevatron Run II, i.e. a centre-of-mass energy s = 1.96 TeV, and usually, for simplicity we have chosen not to impose any cuts in rapidity. 5.4.1 Inclusive jet spectrum As discussed in section 3, the differences between the midpoint algorithm and SISCone are expected to start when we have 3 particles in a common neighbourhood plus one to balance momentum. For pure QCD processes this corresponds to 2 → 4 diagrams, O (α4s). This is NNLO for the inclusive spectrum. Though a NNLO calculation of the inclusive spectrum is beyond today’s technology (for recent progress, see [33]), we can easily calculate the O (α4s) difference between midpoint and SISCone, using just tree-level 2 → 4 diagrams, since the difference between the algorithms is zero at orders α2s and α s, i.e. we can neglect two-loop 2 → 2 diagrams and one-loop 2 → 3 diagrams. The significance of the difference can be understood by comparing to the leading order spectrum, which is identical for the two algorithms. Figure 8 shows the resulting spectra: the upper plot gives the leading order inclusive spectrum together with the difference between SISCone and midpoint(0) at O (α4s). The lower plot shows the relative difference. One sees that the use of the IR-safe seedless cone algorithm introduces modest corrections, of order 1-2%, in the inclusive jet spectrum. This order of magnitude is roughly what one would expect, since the differences only appear at 19We could also have compared SISCone with a midpoint algorithm using pt in the split–merge (a common default); the figures we show below would have stayed unchanged at the 1% level for the inclusive spectrum, while for the jet masses the effects range between a few percent at moderate masses and 10−20% in the high-mass tail. 20 40 60 80 100 120 140 160 180 200 inclusive pT spectrum (all y) SISCone (Born level, 0(αs |midpoint(0) -- SISCone| 0(αs NLOJet R=0.7, f=0.5 20 40 60 80 100 120 140 160 180 200 pT (GeV) -0.02 -0.01 20 40 60 80 100 120 140 160 180 200 pT (GeV) -0.02 -0.01 Figure 8: (a) Inclusive jet spectrum: the upper curve gives the leading-order (O (α2s)) spectrum, while the lower (blue) curve gives the difference between the SISCone and mid- point(0) algorithm, obtained from the O (α4s) tree-level amplitude; (b) the relative differ- ence. -0.04 -0.02 50 100 150 200 pt [GeV] pp− √s = 1.96 TeV R=0.7, f=0.5, |y|<0.7Pythia 6.4 (a) hadron-level (with UE) hadron-level (no UE) parton-level 50 100 150 200 pt [GeV] pp √s = 14 TeV R=0.7, f=0.5, |y|<0.7Pythia 6.4 (b) hadron-level (with UE) hadron-level (no UE) parton-level Figure 9: Relative difference between the inclusive jet spectra for midpoint(1) and SIS- Cone, obtained from Pythia at parton level, hadron level without underlying event (UE) contributions, and hadron level with UE. Shown (a) for Tevatron collisions and (b) for LHC collisions. relative order α2s. As we will see below, larger differences will appear when one examines more exclusive quantities. In addition, we have used Herwig and Pythia to investigate the differences between midpoint(1) and SISCone with parton showering. Both generators give similar results, and we show the results just of Pythia, fig. 9a. The difference at parton level is very similar to what was observed at fixed order. At hadron level without underlying event (UE) corrections, the difference remains at the level of 1−2% (though it changes sign); once one includes the underlying event contributions, the difference increases noticeably at lower pt — this is because the midpoint(1) algorithm receives somewhat larger UE corrections than SISCone. Since the underlying event is one of the things that is likely to change from Tevatron to LHC, in figure 9b we show similar curves for LHC kinematics. At parton level and at hadron level without the underlying event, the results are essentially the same as for the Tevatron. With the underlying event included, the impact of the missing stable cones in the midpoint algorithm reaches of the order of 10 to 15%, and thus starts to become quite a significant effect. With Herwig, we find that the impact is little smaller because its underlying event is smaller than Pythia’s at the LHC. 5.4.2 Jet masses in 3-jet events As well as the inclusive jet pT spectrum, we can also study more exclusive quantities. One example is the jet-mass spectrum in multi-jet events. Jet-masses are potentially of interest for QCD studies, particle mass measurements [34] and new physics searches, where they could be used to identify highly boosted W/Z/H bosons or top quarks produced in the decays of new heavy particles [35]. The simplest multi-jet events in which to study jet masses are 3-jet events. There, the 0 10 20 30 40 50 60 70 80 90 100 M (GeV) Mass spectrum of jet 2 midpoint(0) -- SISCone SISCone NLOJet R=0.7, f=0.5 0 10 20 30 40 50 60 70 80 90 100 M (GeV) 2 Mass spectrum of jet 2 midpoint(0) -- SISCone SISCone NLOJet R=0.7, f=0.5 ∆ R23 < 1.4 Figure 10: Mass spectrum of the second hardest jet as obtained with the different cone algorithms on tree-level 4-particle events (generated with NLOJet): the plots shows the relative difference between the midpoint and SISCone results. In the upper plot we consider all three-jet events satisfying the transverse-momentum cuts, while in the lower plot (note scale) we consider only those in which second and third jet are separated by ∆R23 < 2R. 0 10 20 30 40 50 (a) SISCone midpoint(0) midpoint(1) 0 10 20 30 40 50 60 70 80 0.01 Pythia 6.4 R=0.7, f=0.5 SISCone midpoint(0) midpoint(1) -0.75 -0.25 0.25 0 10 20 30 40 50 60 70 80 M (GeV) (c) midpoint(0) 0 10 20 30 40 50 60 70 80 -0.75 -0.25 0.25 M (GeV) (d) midpoint(1) Figure 11: Mass spectrum of the third hardest jet obtained from the different cone algo- rithms run on three-jet Pythia events. The top-left (top-right) plot shows the spectrum in linear (logarithmic) scale and the bottom plots show the relative difference between each midpoint algorithm and SISCone. See the text for the details of the event selection. masses of all the jets vanish at the 3-particle level. The first order at which the jet masses become non-zero is O (α4s) and this is also the order at which differences appear between the midpoint and seedless cone algorithms. Therefore, as in section 5.4.1, we generate 2 → 4 tree-level events, but now keep only those with exactly 3 jets with pT ≥ 20 GeV in the final state. We further impose that the hardest jet should have a pT of at least 120 GeV and the second hardest jet a pT of at least 60 GeV. With these cuts we can compute the jet-mass spectrum for each of the three jets and for the three different algorithms. In the upper plot of Figure 10, we show the relative difference “(midpoint(0) - SIS- Cone)/SISCone” for the mass spectrum of the second hardest jet. In the lower plot we show the same quantity for events in which we have placed an additional requirement that the y−φ distance between the second and third jets be less than 2R (such distance cuts are often used when trying to reconstruct chains of particle decays). The midpoint algorithm’s omission of certain stable cones leads to an overestimate of the mass spectrum by up to ∼ 10% without a distance cut (much smaller differences are observed for the first and third jet) and of over 40% with a distance cut. The problem is enhanced by the presence of the distance cut because many more of the selected events then have three particles in a com- mon neighbourhood, and this is precisely the situation in which the midpoint algorithm misses stable cones (cf. section 3). We emphasise also that the NLO calculation of these mass spectra would be impossible with a midpoint algorithm, because the 10− 40% tree-level differences would be converted into an infrared divergent NLO contribution. A general comment is that the problems seen here for the midpoint algorithm without a distance cut are of the same general order of magnitude as the 16% failure rate in the IR safety tests of section 5.1, suggesting that the absolute failure rates given there are a good indicator of the degree of seriousness of issues that can arise in generic studies with the infrared unsafe algorithms. In addition to this fixed-order parton-level analysis, we have studied the jet masses in 3-jet events at hadron level (i.e. after parton showering and hadronisation) using events generated with Pythia. At hadron level many more seeds are present, due to the large particle multiplicity. One might therefore expect the midpoint algorithm to become a good approximation to the seedless one. For the mass of the second hardest jet, i.e. the quantity we studied at fixed order in figure 10, we find that the midpoint and seedless algorithms do give rather similar results at hadron level. In other words differences that we see in a leading order calculation are not propagated through to the full hadron level result. This is a serious practical issue for the midpoint algorithm, because a jet algorithm’s principal role is to provide a good mapping between low-order parton level and hadron level. Nevertheless, despite the many seeds that are present at hadron level, we find that there are still some observables for which the midpoint algorithm’s lack of stable cones does have a large impact even at hadron level. This is the case that the mass distribution of the third hardest jet, shown in figure 11 (obtained without a distance cut) on both linear and logarithmic scales so as to help visualise the various regions of the distribution. Moderate differences are present in the peak region, but in the tail of the distribution they become large, up to 50%. They are greater for midpoint(1) than for midpoint(0), because the seed threshold causes fewer stable cones to be found with the midpoint(1) algorithm. These results have been checked using the Herwig Monte-Carlo. We have observed similar differences at parton-shower level, at the hadron level and at the hadron level including underlying event, both in the peak of the distribution and in the tail. We note that hadronisation corrections are substantial in the tail of the distribution, both for the midpoint and SISCone algorithms. The above results confirm what one might naturally have expected: while very inclusive quantities may not be overly sensitive to the deficiencies of one’s jet algorithm, as one extends one’s investigations to more exclusive quantities, those deficiencies begin to have a much larger impact. 6 Conclusions Given the widespread use of cone jet algorithms at the Tevatron and their foreseen contin- ued use at LHC, it is crucial that they be defined in an infrared safe way. This is necessary in general so as to ensure that low-order parton-level considerations about cone jet-finding hold also for the fully showered, hadronised jets that are observed in practice. It is also a prerequisite if measurements are to be meaningfully compared to fixed order (LO, NLO, NNLO) predictions. The midpoint iterative cone algorithm currently in use is infrared unsafe, as can be seen by examining the sets of stable cones that are found for simple three-parton configurations. This may seem surprising given that the midpoint algorithm was specifically designed to avoid an earlier infrared safety problem — however the midpoint infrared problem appears at one order higher in the coupling, and this is presumably why it was not identified in the original analyses. The tests shown in section 5.1 suggest that the midpoint-cone infrared safety problems, while smaller than without the midpoint, are actually quite significant (∼ 15%). We therefore advocate that where a cone jet algorithm is used, it be a seedless variant. For such a proposal to be realistic it is crucial that the seedless variant be practical. The approaches adopted in fixed order codes take O time and are clearly not suitable in general. Here we have shown that it is possible to carry out exact seedless jet-finding in ex- pectedO Nn3/2 time withO Nn1/2 storage, or almost exactly20 in expected O (Nn lnn) time with O (Nn) storage (we recall that N is the total number of particles, n the typical number of particles in a jet). The second of these approaches has been implemented in a C++ code named SISCone, available also as a plugin for the FastJet package. For N ∼ 1000 it is comparable in speed to the existing CDF midpoint code with 1GeV seeds. While this is considerably slower than the N lnN and related FastJet strategies [20] for the kt and Cambridge/Aachen jet algorithms, it remains within the limits of usability and provides for the first time a cone algorithm that is demonstrably infrared and collinear safe at all orders, and suitable for use at parton level, hadron level and detector level. 20with a failure probability that can be made arbitrarily small and that we choose to be . 10−18. As well as being infrared safe, a jet algorithm must provide a faithful mapping between expectations based on low-order perturbative considerations, and observations at hadron level. There has been considerable discussion of worrisome possible violations of such a correspondence for cone algorithms, the “Rsep” issue. For SISCone we find however that the correspondence holds well. An obvious final question is that of the impact on physics results of switching from the midpoint to the seedless cone. For inclusive quantities, one expects the seedless cone jet algorithm to give results quite similar to those of the midpoint cone, because the IR unsafety of the midpoint algorithm only appears at relatively higher orders. This is borne out in our fixed order and parton-shower studies of the inclusive jet spectrum where we see differences between the midpoint and SISCone algorithms of about a couple of percent. At moderate pt at hadron level, the differences can increase to 5− 10%, because SISCone has a lower sensitivity to the underlying event, a welcome ‘fringe-benefit’ of the seedless algorithm. For less inclusive quantities, for example the distribution of jet masses in multi-jet events, differences can be significant. We find that for 3-jet events, the absence of some stable cones (i.e. infrared unsafety) in the midpoint algorithm leads to differences compared to SISCone at the∼ 10% level at leading order (α4s) in a large part of the jet-mass spectrum. Greater effects still, up to 50%, are seen with specific cuts at fixed order, and in the tails of the jet-mass spectra for parton-shower events. Thus, even if the infrared safety issues of the midpoint algorithm appear to be at the limit of today’s accuracy when examining inclusive quantities, for measurements of even moderate precision in multi-jet configurations (of increasing interest at Tevatron and omnipresent at LHC), the use of a properly defined cone algorithm such as SISCone is likely to be of prime importance. Acknowledgements We are grateful to Markus Wobisch for many instructive discussions about cone algorithms, Steve Ellis and Joey Huston for exchanges about their IR safety and Rsep, Matteo Cacciari for helpful suggestions on the SISCone code and Giulia Zanderighi for highlighting the question of collinear safety. We thank them all, as well as George Sterman, for useful comments and suggestions on the manuscript. We also gratefully acknowledge Mathieu Rubin for a careful reading of an early version of the manuscript, Andrea Banfi for pointing out a relevant reference and Torbjörn Sjöstrand for assistance with Pythia. The infrared unsafe configuration shown here was discovered subsequent to discussions with Mrinal Dasgupta on non-perturbative properties of cone jet algorithms. This work has been supported in part by grant ANR-05-JCJC-0046-01 from the French Agence Nationale de la Recherche. G.S. is funded by the National Funds for Scientific Research (Belgium). Finally, we thank the Galileo Galilei Institute for Theoretical Physics for hospitality and the INFN for partial support during the completion of this work. A Further computational details A.1 Cone multiplicities In evaluating the computational complexity of (computational) algorithms for various stages of the cone jet algorithm it is necessary to know the numbers of distinct cones and of stable cones. Such information also constitutes basic knowledge about cone jet definitions, which may for example be of relevance in understanding their sensitivity to pileup, i.e. multiple pp interactions in the same bunch crossing. Since large multiplicities will be due to pileup, let us consider a simple model for the event structure which mimics pileup, namely a set of momenta distributed randomly in y and φ and all with similar pt’s (or alternatively with random pt’s in some limited range). Given that the particles will be spread out over a region in y, φ that is considerably larger than the cone area, in addition to N , the total number of particles, it is useful to introduce also n, the number of points likely to be contained in a region of area πR2. The first question to investigate is that of the number of distinct cones. The number of pairs of points that has to be investigated is O (Nn). However some of these pairs of points will lead to identical cones. It is natural to ask whether, despite this, the number of distinct cones is still O (Nn). To answer this question, one may examine how far one can displace a cone in any given direction before its point content changes. The area swept when moving a cone a distance δR is 4RδR, and the average number of points intersected is 4ρR δR where ρ = O (n/R2) is the density of points (per unit area). Therefore the distance moved before the cone edge is likely to touch a point is δR = (4ρR)−1 = O (R/n). Correspondingly the area in which one can move the centre of cone without changing the cone’s contents is π(δR)2 = O (R2/n2). Given that the total area is O (R2N/n) we have that the number of distinct cones is O (Nn), the same magnitude as the number of relevant point pairs. Let us now consider the number of stable cones. If we take a cone at random and sum its momenta then the resulting momentum axis will differ from the original cone axis by an amount typically of order R/ n (since the standard deviation of y and φ for set of points in the cone is O (R)). The probability of the difference being . R/n in both the y and φ directions (i.e. the probability that the new axis contains the same set of particles) is ∼ (R/n)2/(R/ n)2 ∼ 1/n. Therefore the number of stable cones is O (N). This assumes a random distribution of particles. There may exist special classes of configurations for which the number of stable cones is greater than O (N). Therefore timing results that are sensitive to the number of stable cones are to be understood as “expected” results rather than rigorous upper bounds. A.2 Computational complexity of the split–merge step To study the computational complexity of the split–merge step, we work with the expec- tation that there are O (N) initial protojets (as discussed above) and that there will be roughly N/n ≪ N final jets (since there are O (n) particles per jet). It is reasonable to assume that there will be roughly equal numbers of merging and splitting operations. Split- ting leaves the number of protojets unchanged, while merging reduces it by 1. Therefore there will be O (N) split–merge steps before we reach the final list of jets. There are three kinds of tasks in the split–merge procedure. Firstly one has to maintain a list of jets ordered in p̃t, both for finding the one with highest p̃t and for searching through the remaining jets (in order of decreasing p̃t) to find an overlapping one. Maintaining the jets in order is easily accomplished with a balanced tree (for example a priority_queue or multiset in C++), at a cost of N lnN for the initial construction and lnN per update, i.e. a total of N lnN , which is small compared to the remaining steps. In examining the complexity of finding the hardest overlapping jet one needs to know the cost of comparing two jets for overlap as well as the typical number of times this will have to be done. A naive comparison of two jets takes time n. Using a 2d tree structure such as a quadtree or k-d tree (as suggested also by Volobouev [17]), this can be reduced n. The number of jets to be compared before an overlap is found will depend on the event structure — if one assumes that jet positions are decorrelated with their p̃t’s, then O (N/n) comparisons will have to be made each time around the loop. The total cost of this will therefore be N2/ n (N2) with (without) a 2d tree. Finally each merging/splitting procedure will take n (n) time with (without) a tree, so the total time spent merging and splitting will be O (N n) (or O (Nn) without a tree). The dominant step is the search for overlapping jets, which will have a total cost of n (with a sizable coefficient), or N2 without any 2d tree structures. Since in practice N2 is smaller than the Nn lnn needed to find the stable cones, here the introduction of a tree structure gives little overall advantage. A final comment concerns memory usage: when not using any tree structures, the list of protojets and their contents requires O (Nn) space, which is the same order of magnitude as the storage needed for identifying the set of stable cones in the first place. With a tree structure this can be reduced to O (N B Proof of IR safety of the SISCone algorithm In this appendix, we shall explicitly prove that SISCone, algorithms 1–3, is infrared safe. This means that if we run SISCone first with a set of hard particles, then with the same set of hard particles together with additional soft particles, then: (a) all jets found in the event without soft particles will be found also in the event with the soft particles; (b) any extra jets found in the event with soft particles will themselves be soft, i.e. they will not contain any of the hard particles. If either of these conditions fails in a finite region of phasespace for the hard particles, then the cancellation between (soft) real and virtual diagrams will be broken at some order of perturbation theory, leading to divergent jet cross sections. We will first discuss the proof using a simplifying assumption: two protojets with distinct hard particle content have distinct values for the split–merge ordering variable, p̃t. We shall then discuss subtleties associated with various ordering variables, and explain why p̃t is a valid choice. B.1 General aspects of the proof By soft particles, we understand particles whose momenta are negligible compared to the hard ones. Specifically, for any set of hard particles {p1, . . . , pn} and any set of soft ones {p̄1, . . . , p̄m}, we consider a limit in which all soft momenta are scaled to zero, so that they do not affect any momentum sums, {p̄j}→0 pi. (7) In what follows, the limit of the momenta of the soft particles being taken to zero will be implicit. Let us now compare two different runs of the cone algorithm: in the first one, referred to as the “hard event”, we compute the jets starting with a list of hard particles {p1, . . . , pN}, and, in the second one, referred to as the “hard+soft event”, we compute the jets with the same set of hard particles plus additional soft particles {p̄1, . . . , p̄M}. As mentioned above, the IR safety of the SISCone algorithm amounts to the statements (a) that for every jet in the hard event there is a corresponding jet in the hard+soft event with identical hard particle content (plus possible extra soft particles) and (b) that there are no hard jets in the hard+soft event that do not correspond to a jet in the hard event. To prove this, we shall proceed in two steps: first, we shall show that the determination of stable cones is IR safe, then that the split–merge procedure is also IR safe. The IR safety of the stable-cone determination is a direct consequence of the fact that: • each cone initially built from the hard particles only was determined by two particles in algorithm 2. This cone is thus still present when adding soft particles and, because of eq. (7), is still stable. Hence, all stable cones from the hard event are also present after inclusion of soft particles, the only difference being that they also contain extra soft particles which do not modify their momentum. • no new stable cone containing hard particles can appear. Indeed, if a new stable cone appeared, Snew with content {pα1 , . . . , pαn, p̄ᾱ1 , . . . , p̄ᾱm}, then the fact that its momentum pαi + p̄ᾱj corresponds to a stable cone, implies, by eq. (7), that the cone with just the hard momenta pαi is also stable. However as shown in section 4.2 all stable cones in the hard event have already been identified, therefore this cone cannot be new. From these two points, one can deduce that after the determination of the stable cones we end up with two different kinds of stable cones: firstly, there are those that are the same as in the hard event but with possible additional soft particles; and secondly there are stable cones that contain only soft particles. So, the ‘hard content’ of the stable cones has not been changed upon addition of soft particles and algorithm 2 is IR safe. The main idea behind the proof of the IR safety of the split–merge process, algorithm 3, is to show by induction that the hard content of the protojets evolves in the same way for the hard and hard+soft event. Since the hard content is the same at the beginning of the process, it will remain so all along the split–merge process which is what we want to prove. There is however a slight complication here: when running algorithm 3 over one itera- tion of the loop in the hard event, we sometimes have to consider more than one iteration of the loop in the hard+soft event. As we shall shortly see, in that case, only the last of these iterations modifies the hard content of the jets and it does so in the same way as in the hard event step. So, let us now follow the steps of algorithm 3 in parallel for the hard and hard+soft event, and show that they are equivalent as concerns the hard particles. In the following analysis, item numbers coincide with the corresponding step numbers in algorithm 3. 2: If pt,min is non-zero, all purely soft protojets will be removed from the hard+soft event and by eq. (7) the same set of hard protojets will be removed in the hard and hard+soft event. Thus the correspondence between the hard protojets in the two events will persist independently of pt,min. 3: In general, protojets with identical hard content will have nearly identical p̃t values, whereas protojets with different hard-particle content will have substantially different p̃t values. 21 Therefore the addition of soft particles will not destroy the p̃t ordering and the protojet with the largest p̃t in the hard event, i will have the same hard content as the one in the hard+soft event (let us call it i′). 4: The selection of the highest-p̃t protojet j (j ′ in the hard+soft case) that overlaps with i (i′) can differ in the hard and hard+soft events, and we need to consider separately the cases where this does not, or does happen. The first case, C1, is that i′ and j′ overlap in their hard content — because of the common p̃t ordering, j ′ must then have the same hard content as j. The second case, C2, is that i′ and j′ only overlap through their soft particles, so j′ cannot be the ‘same’ jet as j (since j by definition overlaps with i through hard particles). By following the remaining part of the loop, we shall show that in the first case all modifications of the hard content are the same in the hard and hard+soft events, while, for the second case, the iteration of the loop in the hard+soft event does not modify any hard content of the protojets. In this second case, we then proceed to the next iteration of the loop in the hard+soft event but stay at the same one for the hard event. C1: The two protojets i′ and j′ overlap in their hard content 6,7: We need to compute the fraction of p̃t shared by the two protojets. Since the hard contents of i (j) and i′ (j′) are identical, the fraction of overlap, given by the hard content only, will be the same in the hard and hard+soft events. Hence, the decision to split or merge the protojets will be identical. 21As mentioned already, this point is more delicate than it might seem at first sight. We come back to it in the second part of this appendix. 8: Since the centres of both protojets are the same in the hard and hard+soft events, the decision to attribute a hard particle to one protojet or the other will be the same in both events. Hence splitting will reorganise hard particles in the same way for the hard+soft event as for the hard one. 10: In both the hard and the hard+soft events, the merging of the two protojets will result in a single protojet with the same hard content. C2: The two protojets i′ and j′ overlap through soft particles only 6,7: Since the fraction of p̃t shared by the protojets will be 0 in the limit eq. (7), the two protojets will be split. 8: In the splitting, only shared particles, i.e. soft particles, will be reassigned to the first or second protojet. The hard content is therefore left untouched, as is the p̃t ordering of the protojets. 11: At the end of the splitting/merging of the overlapping protojets, we have to consider the two possible overlap cases separately: in the first case, the hard contents of the protojets are modified in the same way for the hard and hard+soft event. This case is thus IR safe. In the second case, the iteration of the loop in the hard+soft event does not correspond to any iteration of the loop in the hard event. However the hard content of the protojets in the hard+soft event is not modified and the p̃t ordering of the jets remains identical; at the next iteration of the hard+soft loop, the new j′ may once again have just soft overlap with i′ and the loop will thus continue iterating, splitting the soft parts of the jets, but leaving the hard content of the jets unchanged. This will continue until j′ corresponds to the j of the hard event, i.e. we encounter case 1.22 Therefore even though we may have gone around the loop more times in the hard+soft event, we do always reach a stage where the split–merge operation in the hard+soft event coincides with that in the hard event, and so this part of the procedure is infrared safe. 5,14: Up to possible intermediate loops involving case 2 above, when the protojet i has no overlapping protojets in the hard event, the corresponding i′ in the hard+soft event has no overlaps either. Final jets will thus be added one by one with the same hard content in the hard and hard+soft events. This completes the proof that the SISCone algorithm is IR safe, modulo subtleties related to the ordering variable, as discussed below. Regarding the ‘merge identical protojets’ (MIP) procedure: 22Note that the second case can only happen a finite number of times between two occurrences of the first case: as the p̃t ordering is not modified during the second case, each time around the loop the overlap will involve a j′ with a lower p̃t than in the previous iteration, until one reaches the j ′ that corresponds to j. 12: In algorithm 3, we do not automatically merge protojets appearing with the same content during the split–merge process. This is IR safe. If instead we allow for two identical protojets to be automatically merged, then when two protojets have the same hard content but differ as a result of their soft content, they are automatically merged in the hard event but not in the hard+soft event. This in turn leads to IR unsafety of the final jets. A final comment concerns collinear safety and cocircular points. When defining a candidate cone from a pair of points, if additional points lie on the edge of the cone, then there is an ambiguity as to whether they will be included in the cone. From the geometrical point of view, this special case of cocircular points (on a circle of radius R) can be treated by considering all permutations of the the cocircular points being included or excluded from the circle contents. SISCone contains code to deal with this general issue. The case of identically collinear particles, though a specific example of cocircularity, also adds the problem that a circle cannot properly be defined from two identical points. For explicit collinear safety we thus simply merge any collinear particles into a single particle, step 1 of algorithm 2. Given the resulting collinear-safe set of protojets, the split–merge steps preserve collinear safety, since particles at identical y−φ coordinates are treated identically. B.2 Split–merge ordering variable Suppose we use some generic variable v (which may be pt, Et, mt, p̃t, etc.) to decide the order in which we select protojets for the split–merge process. A crucial assumption in the proof of IR safety is that two jets with different hard content will also have substantially different values for v, i.e. the ordering of the v’s will not be changed by soft modifications. If this is not the case then the choice of the hard protojets that enter a given split–merge loop iteration can be modified by soft momenta, with a high likelihood that the final jets will also be modified. At first sight one might think that whatever variable is used, it will have different values for distinct hard protojets. However, momentum conservation and coincident masses of identical particles can introduce relations between the kinematic characteristics of distinct protojets. Some care is therefore needed so as to ensure that these relations do not lead to degeneracies in the ordering, with consequent ambiguities and infrared unsafety for the final jets. In particular: • Two protojets can have equal and opposite transverse momenta if between them they contain all particles in the event (and the event has no missing energy or ‘ignored’ particles such as isolated leptons). It is probably fair to assume that no two protojets will have identical longitudinal components, since in pp collisions the hard partonic reaction does not occur in the pp centre of mass frame. • Two protojets will have identical masses if they each stem exclusively from the same kind of massive particle. The two massive particles may be undecayed (e.g. fully reconstructed b-hadrons) or decayed (top, W , Z, H , or some non-standard new particle), or even one decayed and the other not (some hypothetical particle with a long lifetime).23 In the second case we can assume that two identical decayed particles have different decay planes, because there is a vanishing phase space for them to have identical decay planes. Note that in a simple two-parton event almost any choice of variable will lead to a degen- eracy (no sensible invariant will distinguish the two particles), however this specific case is not problematic because for R < π/2 neither of the two partons can be in a protojet that overlaps with anything else. From the point of view of IR safety, it is only for ‘fat’ (non-collimated) hard protojets that we need worry about the problem of degeneracies in the split–merge ordering, because only then will there be overlaps whose resolution is ambiguous in the presence of degeneracies. Let us now consider what occurs with various possible choices for the split–merge variable. pt: This choice, adopted in certain codes [13, 19], can be seen to have a problem for events with momentum conservation in the hadronic part, because if two non-overlapping protojets contain, between them, all the hard particles then they will have identical pt’s. If they each overlap with a common third protojet, the resulting split–merge sequence will be ambiguous. Table 4 provides an example of such an event. The simplest occurrences of this problem (4h+ 1s) apply only to R > π/4 (four particles must form at least 3 fat protojets). The problem arises also for smaller R values, but only at higher multiplicities. mt: A workaround for the event of table 4 is to use the transverse mass, mt = p2t +m In pure QCD, with all particles stable, this is a good variable, because even if two fat protojets have identical pt’s through momentum conservation, the fact that they are ‘fat’ implies that they will be massive (over and above intrinsic particle masses), and the phase space for them to have identical masses vanishes, thus killing any IR divergences. However, for events with two identical decaying particles, two fat protojets resulting from the particle decays can have identical pt’s (by momentum conservation) and identical masses (because the decaying particles were identical). This could happen for example in the fully hadronic decay channel for tt̄ events. Thus, this choice is not advisable in a general purpose algorithm. Et: The variable used in the original run II proposal was Et [6]. It has the drawback that it is not longitudinally boost invariant: at central rapidity it is equal to mt, while at high rapidities it tends to pt. Because the phase space for two protojets to have identical rapidities vanishes (recall that we do not fix the partonic centre-of-mass), two protojets with identical pt’s and masses will have different Et’s, because the 23Strictly speaking, for all scenarios of decayed heavy particles, the finite width Γ of the particle ensures that the two jets actually have slightly different masses, breaking any degeneracies. In practice however, ΓW,Z,t ∼ 1GeV and (for a light Higgs) ΓH ≪ ΛQCD, whereas for the width to save us from the dangers of degeneracies we would need Γ ≫ ΛQCD. event 1 n px py pz 0 86.01 66 0 1 64 -66 0 2 -77 -70 0 3 -73 70 0 4 -0.01 0 2 event 2 n px py pz 0 85.99 66 0 1 64 -66 0 2 -77 -70 0 3 -73 70 0 4 0.01 0 2 Table 4: Illustration of two events that conserve transverse momentum and differ only through a soft particle, but lead to different hard jets with a split–merge procedures that uses pt as the ordering variable and for measuring overlap. All the particles are to be taken massless. For R = 0.9 and f = 0.7 each event has stable cones consisting of {01}, {23} and {12}, as well as all single particles. The slight difference in momenta between the two events, to balance the soft particle, causes the {01} ({23}) protojet to have the largest pt in the first (second) event, it splits with {12} (merges with {12}), leading after further split–merge steps to two hard jets, {01} and {23} (one hard ‘monster’ jet, {0123}). degree of ‘interpolation’ between between pt and mt will be different. This resolves the degeneracy and should cure the resulting IR safety issue, albeit at the expense of introducing boost-dependence. p̃t: The scalar sum of transverse momenta of the protojet constituents, p̃t, has the prop- erty that it is equal to mt if all particles in the protojet have identical rapidities, while it is equal to pt (i.e. the vector sum) if all particles have identical azimuths. For a decayed massive particle, it essentially interpolates between pt and mt accord- ing to the orientation of the decay plane. The phase space for all particles to have identical azimuths vanishes, as does the phase space for the decay products of two heavy particles to have identically oriented decay planes. Therefore this choice re- solves any degeneracies, as is needed for infrared safety. Another advantage of p̃t is that adding a particle to a protojet always increases its p̃t (this is not the case for pt or Et), ensuring that the degree of overlap between a pair of jets is always bounded by 1. Since it is also boost invariant, it is the choice that we recommend and that we adopt as our default.24 Note that the above considerations hold for any split–merge procedure that relies on order- ing the jets according to a single-jet variable. One might also consider ordering according to variables determined from pairs of protojets: e.g. first split-merge the pair of protojets with the largest (or alternatively smallest) overlap, recalculate all overlaps, and then repeat until there are no further overlaps. However this specific example would also be dangerous, 24One might worry about the naturalness of a variable that depends on the decay plane of heavy particles — however, any unnaturalness is present anyway in the split–merge procedure since if two particles decay purely in the transverse plane then there is a likelihood of having overlapping protojets, whereas if they decay in longitudinally oriented decay planes they will not overlap. since the particles that are common to protojets a and b (say) could also be the particles that are common between a and c, once again leading to an ambiguous split–merge se- quence. One protojet-pair ordering variable that might be free of this problem is the y−φ distance between the protojets, however we have not investigated it in detail. A final comment concerns the impact of the split–merge procedure on non-global [36] resummations for jets [37], in which one is interested in determining which of a set of ordered soft particles are in a given hard jet. A soft and collinear splitting inside the jet can modify the p̃t (or Et or mt) of the jet by an amount of the same order of magnitude as a soft, large-angle emission near the edge of the jet. In events with two back-to-back narrow jets, for which there is a near degeneracy between the p̃t’s of the two hard jets, this can affect which of the two hard protojets split–merges first with an overlapping soft protojet, leading to ambiguities in the assignment of the soft particles to the two hard jets. This interaction between collinear and soft modes is somewhat reminiscent of that in [38], though the origin and structure are kinematical in our case. 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Seymour, JHEP 0608 (2006) 059 [hep-ph/0604094]. http://arxiv.org/abs/hep-ph/0611148 http://arxiv.org/abs/hep-ph/9208249 http://arxiv.org/abs/hep-ph/0011363 http://arxiv.org/abs/hep-ph/9710333 http://arxiv.org/abs/hep-ph/0612257 http://arxiv.org/abs/hep-ph/0703207 http://arxiv.org/abs/hep-ph/0701150 http://arxiv.org/abs/hep-ph/0701166 http://arxiv.org/abs/hep-ph/0701247 http://arxiv.org/abs/hep-ph/0702037 http://arxiv.org/abs/hep-ph/0702150 http://arxiv.org/abs/hep-ph/0104277 http://arxiv.org/abs/hep-ph/0203009 http://arxiv.org/abs/hep-ph/0206076 http://arxiv.org/abs/hep-ph/0211426 http://arxiv.org/abs/hep-ph/0508159 http://arxiv.org/abs/hep-ph/0610242 http://arxiv.org/abs/hep-ph/0604094 Introduction Overview of the cone jet-finding algorithm IR unsafety in the midpoint algorithm An exact seedless cone jet definition One-dimensional example The two-dimensional case General approach Specific computational strategies The split–merge part of the cone algorithm Tests and comparisons Measures of IR (un)safety Speed Rsep: an inexistent problem Physics impact of seedless v. midpoint cone Inclusive jet spectrum Jet masses in 3-jet events Conclusions Further computational details Cone multiplicities Computational complexity of the split–merge step Proof of IR safety of the SISCone algorithm General aspects of the proof Split–merge ordering variable

0704.0293

Isospin breaking in the yield of heavy meson pairs in e+e- annihilation near threshold

William I. Fine Theoretical Physics Institute University of Minnesota FTPI-MINN-07/08 UMN-TH-2541/07 LPT-Orsay/07-19 March 2007 Isospin breaking in the yield of heavy meson pairs in e+e− annihilation near threshold S. Dubynskiya, A. Le Yaouancb, L. Oliverb, J.-C. Raynalb and M.B. Voloshinc a School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA b Laboratoire de Physique Théorique1, Université de Paris XI, Bâtiment 210, 91405 Orsay Cedex, France c William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA and Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia Abstract We revisit the problem of interplay between the strong and the Coulomb interac- tion in the charged-to-neutral yield ratio for BB̄ and DD̄ pairs near their respective thresholds in e+e− annihilation. We consider here a realistic situation with a resonant interaction in the isospin I = 0 channel and a nonresonant strong scattering amplitude in the I = 1 state. We find that the yield ratio has a smooth behavior depending on the scattering phase in the I = 1 channel. The same approach is also applicable to the KK̄ production at the φ(1020) resonance, where the Coulomb effect in the charged-to- neutral yield ratio is generally sensitive to the scattering phases in both the isoscalar and the isovector channels. Furthermore, we apply the same approach to the treatment of the effect of the isotopic mass difference between the charged and neutral mesons and argue that the strong-scattering effects generally result in a modification to the pure kinematical effect of this mass difference. 1Unité Mixte de Recherche UMR 8627 - CNRS http://arxiv.org/abs/0704.0293v1 1 Introduction The JPC = 1−− resonances near the new flavor thresholds: Υ(4S), ψ(3770), and φ(1020) are the well known sources in e+e− experiments of pairs of the new-flavor mesons: respectively BB̄, DD̄, and KK̄. A number of experimental approaches depends on the knowledge of the relative yield of pairs of charged and neutral mesons: Rc/n = σ(e+e− → P+P−) σ(e+e− → P 0P̄ 0) , (1) where P stands for the pseudoscalar meson, i.e. B, D, or K, and dedicated measurements of such ratio have been done at the Υ(4S) resonance [1] at ψ(3770) [2] and at φ(1020) [3]. The values of the ratio Rc/n at all three discussed resonances are close to one due to these resonances being isotopic scalars, and it is the deviation of the discussed ratio from one that presents phenomenological interest. This deviation is generally contributed by the following factors: the isospin violation due to the Coulomb interaction between the charged mesons and due to the isotopic mass difference between charged and neutral mesons, and, in the case of the KK̄ production at the φ(1020) resonance, a non-negligible nonresonant isovector production amplitude. The latter effect can be studied and described as the “tail of the ρ resonance”, while the isospin breaking due to the mass difference is usually accounted for as a kinematical effect in the P wave production cross section factor p3, where p is the the c.m. momentum of each of the mesons. The Coulomb effect has attracted a considerable theoretical attention. The expression for this effect in the ratio Rc/n in the limit, where the resonance and the charged mesons are considered as point-like particles [4] has the simple textbook form: δRc/n = , (2) with α being the QED constant and v the velocity of each of the (charged) mesons in the c.m. frame. However for the production of the real-life mesons the analysis is complicated by the charge form factors of the mesons[5], by the form factor in the vertex of interaction of the resonance with the meson pair [5, 6] and generally by the strong interaction between the mesons [7, 8, 9]. In particular, it has been argued [8, 9] that the modification of the Coulomb effect by the strong (resonant) interaction between the mesons is quite significant. The previously considered picture of the strong interaction was however somewhat unrealistic. Namely, it has been assumed [8, 9] that the wave function in the I = 1 state of the meson pair is vanishing at short but finite distances, which would correspond to a singular behavior of the strong interaction at finite distances. In this paper we derive the formulas for the Coulomb effect in the ratio Rc/n under the standard assumption about the strong scattering amplitude in the channels with I = 0 and I = 1. We find that in the case of the Υ(4S) and ψ(3770) resonances, where the heavy meson pairs are produced by the isotopically singlet electromagnetic current of the corresponding heavy quark, the strong-interaction effect in the Coulomb correction depends on the scattering phase δ1 in the I = 1 channel and is a smooth function of the energy across the resonance, while in the case of the Kaon production at and near the φ(1020) there is also a smooth dependence on the nonresonant part of the strong scattering phase δ0 in the isoscalar channel inasmuch as there is a contribution of the isovector production amplitude at these energies. In either case we find that the behavior of the Coulomb effect is smooth on the scale of the resonance width, unlike the behavior previously found [8, 9] under less realistic assumptions. We further notice that essentially the same calculation can be applied to considering the effect on the ratio Rc/n of the isotopic mass difference ∆m between the charged and neutral mesons, at least in the first order in ∆m, by considering the mass difference as a perturbation by a (constant) potential. In this way we find that the result coincides with the linear in ∆m term in the ratio of the kinematical factors p3 only in the limit of vanishing strong scattering phase. Once the latter phase is taken into account, there arises a correction whose relative contribution is determined by the parameter (p a) with a being the characteristic range of the strong interaction. We therefore conclude that the conventionally used p3 approximation for this effect may be somewhat applicable to the KK̄ production at the φ(1020) resonance, where p ≈ 120MeV, but becomes quite questionable for the DD̄ production at the ψ(3770), where p ≈ 280MeV. The strong-scattering phase in the P -wave state of mesons produced in e+e− annihilation near the threshold is proportional to p3. We therefore expect the discussed effects of the strong interaction in the ratio Rc/n to exhibit a measurable variation with energy. A mea- surement of this variation can thus provide an information on the strong scattering phases, which is not readily available by other means. The material in the paper is organized as follows. In Sec. 2 we consider the production of meson pairs by an isosinglet source and derive the formula for the correction to Rc/n due to a generic isospin-violating interaction potential V (r) viewed as a perturbation. In Sec. 3 we generalize this treatment to the situation where the source is a coherent mixture of I = 0 and I = 1. The specific expressions corresponding to the Coulomb interaction and the isotopic mass difference are considered in Sec. 4. Sec. 5 contains phenomenological estimates of the constraints on the parameters of the strong interaction between heavy mesons based on the currently available data [1, 2] for BB̄ and DD̄ production. Finally, in Sec. 6 we summarize our results. 2 General formulas for an isoscalar source We start with considering the behavior of the scattering wave functions of a meson-antimeson pair in the limit of exact isotopic symmetry, i.e. neglecting any Coulomb effects and the isotopic mass difference. We adopt the standard picture (see e.g. in the textbook [10]), where the strong interaction is confined within the range of distances r < a, so that beyond that range, at r > a the motion of the mesons is free. The two relevant independent solutions to the Schrödinger equation at r > a for the radial wave function in the P wave are the free outgoing wave f(pr) = eipr (3) and its complex-conjugate, f ∗(pr), describing the incoming wave. A general wave function of a pair of neutral mesons, φn(r) as well as of a pair of charged mesons, φc(r), in this region is a linear superposition of these two solutions. In the region of strong interaction, i.e. at r < a, the isotopic symmetry selects as independent channels the states with definite isospin, I = 0 and I = 1, corresponding to the wave functions φ0 = φc + φn and φ1 = φc − φn. The detailed behavior of the I = 0 and I = 1 wave functions inside the strong interaction region is not important for the present treatment, and the important point is that the non-singular at r = 0 ‘inner’ wave functions match at r = a particular linear superpositions of the incoming and outgoing waves (which superpositions in fact correspond to standing waves): χ0(r) = e iδ0 f(pr) + e−iδ0 f ∗(pr) , χ1(r) = e iδ1 f(pr) + e−iδ1 f ∗(pr) , (4) where δ0 and δ1 are the strong scattering phases in respectively the isoscalar and isovector states. Consider now the production of meson pairs by a source localized inside the region of strong interaction, i.e. r < a, such as e.g. the electromagnetic current. The wave function of the produced meson pairs at r ≤ a is then determined by both the source and the strong interaction, and the relevant solution to the Schrödinger equation is chosen by the requirement that asymptotically at large distances, r → ∞, only an outgoing wave is present. Let us first consider the simple case where the relevant electromagnetic current is a pure isotopic singlet, which is the case for DD̄ and BB̄ pair production. Then in the limit of exact isotopic symmetry the outgoing waves for the ‘n’ and the ‘c’ channels have exactly the same amplitude, which for our present purpose can be chosen as one: φ(0)c (r) = f(pr) and φ n (r) = f(pr) at r → ∞ , (5) where the superscript (0) stands for the approximation of exact isotopic symmetry. It can be noted that the approximation of the free motion beyond the region of the strong interaction in fact makes the expressions in Eq.(5) applicable at all r > a, i.e. all the way down to the matching point r = a. It is helpful to notice for a later discussion that at the matching point the I = 1 wave function is vanishing while the I = 0 function φ 0 contains only the outgoing wave. When continued into the strong interaction region, i.e. at r < a, the function φ0 evolves into the solution determined by the strong interaction and the source. The isospin-violating effects of the Coulomb interaction and of the mass difference ∆m between the charged and neutral mesons can be generally described as being due to a presence of an extra potential V (r) in the ‘c’ channel beyond the region of the strong interaction: V = −α/r for the Coulomb interaction effect and a constant potential V = 2∆m describing the mass difference. In other words the wave function φn of the ‘n’ channel is still determined at r > a by the radial Schrödinger equation for free P -wave motion2, while the equation for the ‘c’ channel function φc reads as + p2 −mV (r)− 2 φc(r) = 0 . (6) It is assumed throughout the present consideration that the isospin-breaking potential exists only at distances beyond the range of the strong interaction, i.e. that V (r) has support only at r > a. The justification for such treatment is that in the region of the strong force 2Clearly, in the considered here first order in the isospin violation only the difference of the interaction between the two channels is important, thus any such difference can be relegated to one channel, while keeping the other one unperturbed. Also, any effect of the mass difference in the kinetic term p2/m is of order v2/c2 as compared to the discussed here effect of ∆m in the overal energy difference between the two channels, and is totally neglected in our treatment. small isospin-violating effects are compared to the energy of the strong interaction, so that the contribution of any such effects arising at r < a is very small, while in the region r > a the relative contribution of the potential V (r) is determined by its ratio to the kinetic energy of the mesons, which is small near the threshold. It should be emphasized that although the interaction at distances r > a is present only in the ‘c’ channel, the wave functions in both channels are modified in comparison with those in Eq.(5), as a result of the coupling between channels imposed by the boundary conditions at r = a. According to the setting of the problem of production of the meson pairs by a localized source, the appropriate modified functions are those containing at r → ∞ only the outgoing waves φc → (1 + x) f(pr), φn → (1 + y) f(pr) , (7) where the (complex) coefficients x and y arise due to the potential V , and are proportional to V in the considered here first order of perturbation theory. These coefficients determine the ratio of the production amplitudes: Ac/An = 1+ x− y, and the discussed here modification of the yield ratio: Rc/n = 1 + 2Rex− 2Re y . (8) The modified wave function in both channels is subject to two conditions: i: The channel with neutral mesons has only an outgoing wave at all r > a. In other words, the expression for φn(r) in Eq.(7) is valid at all r down to r = a; ii: The wave function of the channel with isospin I = 1 at r ≤ a should be proportional to the standing-wave solution matching the function χ1 in Eq.(4), since there is no source for the I = 1 state of the meson pairs. These two conditions are sufficient to fully determine the modified functions at r > a and thus to find the coefficients x and y. The first order in V (r) perturbation of the wave function in the channel with charged mesons is found in the standard way, using the P wave Green’s function G+(r, r ′) satisfying the equation + p2 − G+(r, r ′) = δ(r − r′) , (9) and the condition that G+(r, r ′) contains only an outgoing wave when either of its arguments goes to infinity. The Green’s function is constructed from two solutions of the homogeneous equation, i.e. from the functions f(pr) and f ∗(pr), as G+(r, r 2 i p [f(pr) f ∗(pr′) θ(r − r′) + f(pr′) f ∗(pr) θ(r′ − r)] , (10) where θ is the standard unit step function. The perturbation δφc is then found as δφc(r) = m G+(r, r ′) V (r′) f(pr′) dr′ . (11) One readily finds from this explicit form of the solution that δφc contains only the outgoing wave at asymptotic distances r → ∞: δφc |r→∞ = − f(pr) V (r′) |f(pr′)|2 dr′ , (12) so that the coefficient x is purely imaginary: x = − i V (r′) |f(pr′)|2 dr′ (13) and gives no contribution to the ratio of the production rates Rc/n described by Eq.(8)3. Consider now the matching of the wave functions at r = a. In this region of r one has r < r′ in the integral in Eq.(11) so that the correction in the ‘c’ channel has only an incoming wave: δφc(r) |r→a = η f ∗(pr) (14) η = − V (r′) [f(pr′)] dr′ . (15) The wave functions φ0 = φc + φn and φ1 = φc − φn corresponding to the states with isospin I = 0 and I = 1 are then found as φ0 |r→a = 2f(pr) + y f(pr) + η f ∗(pr) and φ1 |r→a = η f ∗(pr)− y f(pr) . (16) One can now apply the condition ii to determine the coefficient y. Indeed, the condition for the wave function φ1 at r → a to be proportional to f ∗(pr) + e2iδ1 f(pr) requires y to be given by y = −η e2iδ1 . (17) Upon substitution in Eq.(8) this yields Rc/n = 1 + e2iδ1 e2ipr V (r) dr  . (18) 3It can be noticed that the integral in Eq.(13) is divergent, which corresponds to the infrared-divergent behavior of the perturbation for the phase of the wave function, logarithmic for the Coulomb interaction and linear for a constant potential. This slight technical difficulty can be readily resolved, for our present purposes, by introducing an infrared regularizing factor exp(−λ r) in the potential and setting λ → 0 in the end result. 3 Mixed isoscalar and isovector source The formula (18) gives the general expression for the isospin-breaking effect in the considered yield ratio for the case where the mesons are produced by an isoscalar source. The presented consideration can also be extended to a situation where the source is a general coherent mix- ture of an isoscalar and isovector. The specific isotopic composition of the source determines the ratio of the coefficients of the amplitudes of the running outgoing waves in the I = 1 and I = 0 channels at the matching point r = a, which ratio we denote as A1/A0, thus defining A1 and A0 as the production amplitudes in the respective channels (in the limit of exact isotopic symmetry). In this situation the generalization of the expressions in Eq.(5) for radial wave functions in the ‘outer’ region r > a in the zeroth order in the isospin violation can be written as φ(0)c (r) = (A0 + A1) f(pr) and φ n (r) = (A0 − A1) f(pr) . (19) The isospin violation in the asymptotic form of these wave functions at r → ∞ can then be parametrized, similarly to Eq.(7), by complex coefficients x and y as φc → (A0 + A1) (1 + x) f(pr), φn → (A0 − A1) (1 + y) f(pr) , (20) so that the yield ratio is found from Rc/n = A0 + A1 A0 − A1 (1 + 2Rex− 2Re y ) . (21) The coefficient x, similarly to the previous discussion and the equation (13), is purely imaginary and in fact does not contribute in Eq.(21), while the coefficient y is found from the appropriately modified conditions on the wave functions. Namely, the previously discussed condition i remains applicable, so that the asymptotic expression in Eq.(20) for the ‘n’ channel function remains valid in the entire ‘outer’ region r > a down to the matching point r = a. In order to allow for the isovector component of the source the condition ii has to be modified as will be described few lines below. The perturbation by the potential V (r) of the ‘c’ channel wave function at the matching point r = a is readily found, similarly to Eq.(14), as δφc(r) | r→a = η (A0 + A1) f ∗(pr) (22) with η given by Eq.(15). One can now write the expressions for the resulting ‘outer’ wave functions in the isotopic channels at the matching point: φ0(r) | r→a = 2A0 f(pr) + η (A0 + A1) f ∗(pr) + y (A0 −A1)f(pr) = 2A0 + y (A0 −A1)− η (A0 + A1) e2iδ0 f(pr) + η (A0 + A1) e iδ0 χ0(r) (23) φ1(r) | r→a = 2A1 f(pr) + η (A0 + A1) f ∗(pr)− y (A0 −A1)f(pr) = 2A1 − y (A0 −A1)− η (A0 + A1) e2iδ1 f(pr) + η (A0 + A1) e iδ1 χ1(r) , (24) with χ0 and χ1 being the standing wave functions from Eq.(4) in the corresponding isotopic channels, which when evolved in the region of strong interaction contain no singularity at r = 0. The remaining parts in the latter expressions for the functions φ0 and φ1 describe the proper running outgoing waves. These parts, when continued down in r into the strong interaction region evolve to match the source at r < a. The ratio of the amplitudes of the isovector and the isoscalar running waves is determined by the isotopic composition of the source, and by the isotopically symmetric propagation through the strong-interaction region. Thus the ratio of the amplitudes of these waves at r = a does not depend on the isospin-breaking effects at r > a and should be equal to A1/A0. Applying this condition to the isotopic wave functions given by the expressions (23) and (24), one finds the equation for the coefficient y: 2A1 − y (A0 − A1)− η (A0 + A1) e2iδ1 2A0 + y (A0 − A1)− η (A0 + A1) e2iδ0 . (25) This equation in fact replaces in this more general situation the previously discussed condi- tion ii, which condition and the ensuing result in Eq.(17) are readily recovered in the limit A1/A0 = 0 from Eq.(25). Considering that both y and η are of the first order in the potential V , it is sufficient to use the linear expansion of the equation (25) in y and η, finding in this way the solution for y in the form y = −ηA0 e 2iδ1 − A1 e2iδ0 A0 − A1 , (26) and thus arriving at the final formula for the relative yield: Rc/n = A0 + A1 A0 −A1 2iδ1 − A1 e2iδ0 A0 − A1 e2ipr V (r) dr . (27) Given that A0 = |A0| eiδ0 and A1 = |A1| eiδ1, the amplitude-dependent factor in this formula can also be written in terms of the real ratio ρ = |A1/A0| as 2iδ1 −A1 e2iδ0 A0 −A1 = e2iδ1 1− ρ ei(δ0−δ1) 1− ρ e−i(δ0−δ1) . (28) 4 The Coulomb and the mass-difference effects The general formulas in Eq.(18) and (27) can now be applied to a discussion of the specific isospin-breaking effects in the e+e− production of meson pairs at and near the threshold resonances. We start with considering the effect of the Coulomb interaction. In a detailed treatment of this correction one should include the realistic form factors of the mesons, which cut off at short distances the difference in the electromagnetic interactions between the charged and neutral mesons. In the present discussion we replace for simplicity the gradual cutoff of the Coulomb interaction by an abrupt cutoff at an effective range r = ac, where generally ac ≥ a 4. The master integral with the Coulomb potential V (r) = −α/r in the equations (18) and (27) then takes the form e2ipr V (r) dr = cos 2pac 2(pac)2 sin 2pac − Ci(2pac) − cos 2pac sin 2pac 2(pac)2 − Si(2pac) 2 (pac)2 − ln(2 pac) + 1− γE (pac) , (29) where the integral sine and cosine are defined in the standard way: Si(z) = sin t and Ci(z) = − cos t and γE = 0.577 . . . is the Euler’s constant. The latter line in Eq.(29) shows few first terms of the expansion of the integral in the parameter (pac). This expansion illustrates the behavior of the correction toward the threshold. For the purpose of this illustration one can consider first the simpler expression in Eq.(18). The imaginary part, which determines the discussed Coulomb effect in Rc/n in the limit where there is no strong scattering, δ1 → 0, is not singular 4As previously mentioned, any extension of the isospin-breaking potential inside the strong interaction region can result only in very small corrections. at pac → 0, and the textbook formula (2) is recovered in this limit. The real part of the integral in Eq.(29) is singular at small pac, but it multiplies in Eq.(18) the factor sin δ1. The P -wave scattering phase in its turn is proportional at small momenta to p3: δ1 ∼ (pa)3, so that the overall contribution of the real part of the integral is not singular at the threshold either. Considering a more general expression for the Coulomb effect for the case of an isotopically mixed source, following from the equation (27), one can readily arrive at the same conclusion that the singular in (pac) real part of the integral (29) does not lead to an actual singularity, since it only enters the ratio Rc/n multiplied by a combination of the phases δ0 and δ1 (cf. Eq.(28)), each vanishing as p 3 toward the threshold. As previously mentioned, the effect of the isotopic mass difference corresponds to that of a constant potential V = 2∆m extending from the range of the strong interaction r = a to infinity. The master integral with such potential has the form e2ipr V (r) dr = 2 cos 2 pa + sin 2 pa+ i 2 sin 2 pa − cos 2 pa − 2 pa+ 3 i+O (pa)2 . (30) In the limit of vanishing strong scattering phases the mass correction to Rc/n is determined by only the imaginary part of the integral, which in the limit of small pa thus yields Rc/n = 1− 3∆m = 1− 3∆m , (31) where E is the total kinetic energy of the meson pair, and the found expression coincides with the linear in ∆m term in the expansion of the usually assumed ratio of the kinematical factors (p+/p0) 3. Clearly, in the more realistic case of presence of the strong scattering the real part of the integral in Eq.(30) also contributes and the simple kinematical approximation is generally invalidated. 5 Phenomenological estimates In this section we discuss application of our formulas to interpreting the data on the charged to neutral meson yield ratio Rc/n at the near-threshold resonances Υ(4S), ψ(3770) and φ(1020). The purpose of this discussion is to illustrate the effect of the strong scatering on the isospin breaking corrections, and we use here the simplified picture of a abrupt cutoff of the Coulomb interaction and of the isotopic mass difference effects. Such simplification generally can be used as long as the parameter (pa) is not large. A detailed analysis should likely involve a model of a gradual cutoff, since the details of the transition become important at lager momenta. 5.1 Υ(4S) The simplest case for the study of the isospin breaking corrections in the relative production of heavy mesons is offered by the BB̄ pair production near and at the Υ(4S) resonance. Indeed, this process only is due to the purely isosinglet electromagnetic current of the b quarks, and the isotopic mass difference between the B mesons is very small: ∆mB = −0.33±0.28MeV [11], so that any deviation of the ratio Rc/n from one is essentially entirely due to the Coulomb interaction. On the other hand, the parameter α/v for the Coulomb effect in this case is the largest due to small velocity of the B mesons: at the energy of the Υ(4S) peak vB/c ≈ 0.06. In particular, the numerical value in the expression (2) is 0.19. The experimental data [1] however indicate a significantly smaller deviation of Rc/n from one. The BaBar data with the smallest errors give Rc/n = 1.006±0.036±0.031. Such behavior is likely a result of a combined effect of the meson and production vertex form factors [5, 6] and of the discussed here modification of the Coulomb correction by the strong scattering phase. These effects can in principle be separated and studied quantitatively by measuring the energy dependence of the ratio Rc/n near the Υ(4S) resonance. With the presently available data we can only use a simplified parametrization of the form factor effects by introducing an abrupt cutoff for the Coulomb interaction at r = ac ≥ a and thereby estimate the likely regions in the (ac, δ1) plane. Such estimate from the equations (18) and (29) is shown in Fig.1 as a one-sigma area, corresponding to the BaBar data with the statistical and systematic errors added in quadrature: Rc/n = 1.006± .048. Clearly, more precise data from dedicated measurements of the ratio Rc/n are needed for a better understanding of the parameters of strong interaction between the B mesons. 0.2 0.4 0.6 0.8 1 1.2 Figure 1: The one sigma area (shaded) in the (ac, δ1) plane corresponding to the BaBar data on the B+B−/B0B̄0 yield ratio at the Υ(4S) resonance. 5.2 ψ(3770) The largest isospin-breaking effect in the DD̄ production at the ψ(3770) is that due to the mass difference between the charged and the neutral D mesons: ∆mD = 4.78±0.10MeV [11]. The most precise measurements of this process have been done [2] at the energy 3773MeV. At this energy the momentum of each charged D meson is p+ = 254MeV and that for a neutral D meson is p0 = 287MeV. Thus the ratio of the kinematical factors (p+/p0) 3 ≈ 0.69 is significantly less than one. The Coulomb effect is somewhat smaller. Indeed, the velocity of a charged meson at this energy is v+/c = 0.135 and the expression (2) gives numerically 0.085. One can notice that if the kinematical and the Coulomb factors are combined in a straightforward way to estimate Rc/n = (p+/p0) 3 [1 + πα/(2v+)] ≈ 0.75, this would be in a very good agreement with the experimental number [2]: Rc/n = 0.776 ± 0.024+0.014 −0.006. Thus it is quite likely that at this particular energy there is a considerable cancelation between the strong-interaction effects in the yield ratio, and such cancelation by itself imposes constraints on the parameters of strong interaction between the D mesons, which constraints is interesting to analyze. An analysis of the strong-interaction effects along the lines discussed in the present paper generally runs into two difficulties. One is that our approach is accurate only in the linear in ∆m approximation, while the actual effect of the isotopic mass difference between the D mesons is not very small. However, numerically, the first term in the expansion of the kinematical factor (Eq.(31)) gives 0.67, which is quite close to the mentioned above value 0.69, and it looks like the linear term gives a reasonable approximation. The other point is that the cutoff parameter ac for the Coulomb interaction at short distances does not necessarily coincide with the range parameter a used for the short-distance cutoff of the effect of the mass difference. However, as previously noted, the Coulomb effect is somewhat small at the energy of the ψ(3770) resonance, and for the purpose of preliminary estimates we set ac = a in our numerical analysis. In order to allow for possible errors introduced by our approximations in comparing with the data, we linearly add a theoretical uncertainty of 0.03 units to the combined in quadrature statistical and experimental errors. Proceeding in this way we find that the only region in the (a, δ1) plane at a < 2 fm consistent with the CLEO-c data at one sigma level is the one shown in Fig.2. 0.2 0.4 0.6 0.8 1 Figure 2: The area (shaded) in the (a, δ1) plane corresponding to the CLEO-c data on the D+D−/D0D̄0 yield ratio at the ψ(3770) resonance. The uncertainty shown includes a one sigma experimental error with our estimate of the theoretical uncertainty added linearly. It is interesting to compare the plots in the Figures 1 and 2. In the heavy quark limit applied to both b and c quarks the strong interaction between the heavy mesons should be the same, corresponding to the same range parameters a and ac. The scattering phase δ1 for these two systems is generally different due to different masses. However, provided there are no isovector ‘molecular’ bound states, the sign of the phase should be the same, with the absolute value of the phase for heavier B mesons being larger than for the D mesons. The comparison with the data for the D mesons favors small values of the range parameter, as indicated by Fig.2. If one also assumes that ac ≈ a for the B mesons, the short range of ac, according to Fig.1, is compatible with the B mesons data at a negative scattering phase δ1, which sign of δ1 is also in agreement with the D meson data. A negative sign of δ1 corresponds to a repulsion, which for the I = 1 state of heavy meson pairs can be expected on general grounds [12]. 5.3 φ(1020) We believe that the production of KK̄ pairs in e+e− annihilation at and near the φ(1020) resonance merits a separate analysis along the lines discussed in the present paper and using detailed data similar to those in Ref.[3]. As is known, this production receives a small but measurable nonresonant contribution from the isovector part of the electromagnetic current of the u and d quarks, which corresponds to an isotopically mixed source. Furthermore, it has been pointed out [13] that a detailed theoretical analysis of the K+K−/K0K̄0 yield ratio at the φ(1020) resonance produces a result which possibly is at a meaningful variance with the data. At present we limit ourselves to noticing that the formula in Eq.(27), applicable in this situation, describes a smooth behavior of the considered isospin breaking effects across the resonance in the I = 0 channel. Indeed, the I = 0 scattering phase at energy E near the resonance energy E0 is given by the Breit-Wigner formula e2iδ0 = ∆− i γ ∆+ i γ e2iδ̃0 , (32) where ∆ = E −E0, δ̃0 is the nonresonant scattering phase in the isoscalar channel, and γ is the width parameter. Both δ̃0 and γ are smooth functions of the energy proportional to p small momentum, and γ(E0) determines the resonance width Γ as γ = Γ/2. The ratio of the isovector and isoscalar production amplitudes can then be parametrized near the resonance ∆+ i γ ei (δ1−δ̃0) , (33) where µ is a parameter with dimension of energy: µ ∼ mφ − mρ. The amplitude ratio entering the correction factor in Eq.(27) can then be written in the form 2iδ1 − A1 e2iδ0 A0 − A1 = e2iδ1 µ− (∆− i γ) e−i(δ1−δ̃0) µ− (∆ + i γ) e+i(δ1−δ̃0) , (34) which manifestly shows that this ratio is a pure phase factor of a complex quantity slowly varying across the φ(1020) resonance. 6 Summary We have considered the effects of the isospin breaking by the Coulomb interaction and by the isotopic mass difference in the relative yield Rc/n of pairs of charged and neutral mesons near threshold by a compact source, such as in the production of heavy mesons in e+e− annihilation. These effects are modified by the strong interaction scattering phases. The general formula for a situation where the source is an arbitrary coherent mixture of an isoscalar and isovector is given by Eq.(27). In particular, for a purely isoscalar source, which is the case for the e+e− annihilation into DD̄ and BB̄ pairs the strong-interaction effect is determined by the scattering phase δ1 in the I = 1 channel (Eq.(18)). As a practical matter we find that under the standard assumptions about the strong scattering amplitudes in the near-threshold resonance region the ratio Rc/n has a smooth behavior with energy showing no abnormal rapid variation on the scale of the resonance width. The energy dependence of this ratio is rather determined by the non-resonant scattering scattering phase(s). In the P -wave the phase δ1 is proportional to p 3, so that a measurement of the behavior ratio Rc/n with energy can provide information on this phase, which is not readily accessible by other means. The behavior of the ratio Rc/n at larger energies away from the threshold also depends on the details of the onset of the strong interaction between the heavy mesons at short distances and on the behavior of their electromagnetic form factors, and a study of this behavior can provide an insight into these properties of the heavy-light hadrons. Acknowledgements The work of MBV is supported, in part, by the DOE grant DE-FG02-94ER40823. References [1] J.P. Alexander et al. [CLEO Collaboration], Phys.Rev.Lett. 86, 2737 (2001); S.B. Athar et al. [CLEO Collaboration], Phys.Rev. D 66, 052003 (2002); B. Aubert et al. [BABAR Collaboration], Phys.Rev. D 65, 032001 (2002); B. Aubert et al. [BABAR Collaboration], Phys.Rev. D 69, 071101 (2004); N.C. Hastings et al. [Belle Collaboration], Phys.Rev. D 67, 052004 (2003). [2] Q. He et al. [CLEO Collaboration], Phys.Rev.Lett. 95, 121801 (2005) [Erratum-ibid. 96, 199903 (2006)] [3] M.N. Achasov et al. [SND Collaboration], Phys.Rev. D 63, 072002 (2001). [4] D. Atwood and W.J. Marciano, Phys.Rev. D 41, 1736 (1990). [5] G.P. Lepage, Phys.Rev. D 42, 3251 (1990). [6] N. Byers and E. Eichten, Phys.Rev. D 42, 3885 (1990). [7] R. Kaiser, A.V. Manohar, and T. Mehen, Report hep-ph/0208194, Aug. 2002 (unpub- lished) [8] M.B. Voloshin, Mod.Phys.Lett. A 18, 1783 (2003). [9] M.B. Voloshin, Phys.Atom.Nucl. 68, 771 (2005) [Yad.Fiz. 68, 804 (2005)]. [10] L.D. Landau and E.M. Lifshits, Quantum Mechanics (Non-relativistic Theory), Third Edition, Pergamon, Oxford, 1977. [11] W.M. Yao et al. [Particle Data Group], J.Phys. G 33, 1 (2006). [12] M.B. Voloshin and L.B. Okun, JETP Lett. 23, 333 (1976). [13] A. Bramon, R. Escribano, J.L. Lucio M. and G. Pancheri, Phys.Lett. B 486, 406 (2000) http://arxiv.org/abs/hep-ph/0208194 Introduction General formulas for an isoscalar source Mixed isoscalar and isovector source The Coulomb and the mass-difference effects Phenomenological estimates (4S) (3770) (1020) Summary

0704.0294

QED x QCD Resummation and Shower/ME Matching for LHC Physics

QED⊗QCD Resummation and Shower/ME Matching for LHC Physics∗ B.F.L. Ward and S.A. Yost Department of Physics, Baylor University, Waco, TX, USA BU-HEPP-07/02 November 4, 2018 Abstract We present the theory of QED⊗QCD resummation and its inter- play with shower/matrix element matching in precision LHC physics scenarios. We illustrate the theory using single heavy gauge boson production at hadron colliders. PACS numbers: 12.38.Cy, 12.15.Lk, 11.25.Db ∗Presented by S.A.Y. at the Cracow Epiphany Conference on Precision Physics and Monte Carlos for LHC, 4 – 6 January, 2007. http://arxiv.org/abs/0704.0294v1 1 Introduction In the imminent LHC environment, where one expects to have an experi- mental luminosity precision tag at the level of 2%, [1] the requirement for the theoretical precision tag on the corresponding luminosity processes, such as single W,Z production with the subsequent decay into light lepton pairs, should be at the 0.67% level in order not to compromise, unnecessarily, the over-all precision of the respective LHC luminosity determinations. This dic- tates that multiple gluon and photon radiative effects must be controlled at the stated precision. The theory of QED ⊗ QCD exponentiation [2] allows for the simultaneous resummation of multiple gluon and multiple photon ra- diative effects in LHC physics processes, to be realized ultimately by MC methods on an event-by-event basis in the presence of parton showers, in a framework which allows us to systematically improve the accuracy of the calculations without double-counting of effects, in principle to all orders in both αs and α. Such a theoretical framework opens the way to the desired theoretical precision tag on the LHC luminosity processes. Our starting point for the new QED⊗QCD resummation theory [2] is the QCD resummation theory presented in Ref. [3]. This resummation is an exact rearrangement of the QCD perturbative series based on the N = 1 term in the exponent in the formal proof of exponentiation in non-Abelian gauge theories in the eikonal approximation, as given in Ref. [4]. This exponential is augmented with a sum of residuals which take into account the remaining contributions to the perturbative series exactly to all orders in αs. therefore have an exact result whereas the resummation theory in Ref. [4] and those in Refs. [5–7] are approximate. Recently, an alternative resummation theory, the soft-collinear effective theory(SCET) [8], has been developed to treat double resummation of soft and collinear effects. Since we have an exact re-arrangement of the perturbative series, we could introduce the results from Refs. [5–8] into our representation as well. Such introductions will appear elsewhere. The need for the extension of the QCD resummation theory to QED⊗ QCD resummation was already suggested by the results in Refs. [9–14], where 1If desired, our overall expoential factor can be made to include all of the terms in the exponent in Ref. [4], in principle. it was shown that in the evolution of the structure functions the inclusion of the QED contributions leads to effects at the level of ∼ 0.3%, already almost half of the error budget discussed above. We will find similar size effects from the threshold region of heavy gauge boson production. All of these must be taken into account if one wants ∼ 1.0% for the theoretical precision tag. The discussion is organized as follows. In Section 2, we review the exten- sion of the YFS theory to an exact resummation theory for QCD. Section 3 presents the further extension to QED⊗QCD. Section 4 contains the ap- plication to heavy gauge boson production with the attendant discussion of shower/ME matching. Section 5 contains some concluding remarks. 2 Extension of YFS Theory to QCD We consider a parton-level single heavy boson production process such as q + q̄′ → V + n(g) + X → ℓ̄ℓ′ + n(g) + X , where V = W±, Z, and ℓ = e, µ, ℓ′ = νe, νµ(e, µ) respectively for V = W +(Z), and ℓ = νe, νµ, ℓ ′ = e, µ respectively for V = W−. It has been established [3] that the cross section may be expressed as dσ̂exp = dσ̂n = eSUMIR(QCD) (2π)4 eiy·(p1+p2−q1−q2− kj)+DQCD × ˜̄βn(k1, . . . , kn) where gluon residuals ˜̄βn(k1, . . . , kn), defined by Ref. [3], are free of all in- frared divergences to all orders in αs(Q). The functions SUMIR(QCD) and DQCD, together with the basic infrared functions B QCD, B̃ QCD, and S̃ QCD are specified in Ref. [3]. We call attention to the essential compensation between the left over genuine non-Abelian IR virtual and real singularities between the phase space integrals dPh β̄n and dPh β̄n+1 that really allows us to isolate ˜̄βj and distinguishes QCD from QED, where no such compensation occurs. The result in (1) has been realized by Monte Carlo methods [3]. See also Refs. [15–17] for exact O(α2s) and Refs. [18–20] for exact O(α) results on the heavy gauge boson production processes which we discuss here. Apparently, we can not emphasize too much the exactness of (1). Some confusion seems to exist because it does not show explicitly an ordered ex- ponential operator for an appropriate ordering prescription, path-ordered, time-ordered, etc. The essential point is that, in (1), we have evaluated the matrix elements of these operators and written the result in terms of the over-all exponent shown therein and the residuals ˜̄βj. This allows us to maintain exactness to all orders in αs. 3 QED⊗QCD Resummation Theory The new QED⊗QCD theory is obtained by simultaneously exponentiating the large IR terms in QCD and the exact IR divergent terms in QED, so that we arrive at the new result dσ̂exp = e SUMIR(QCED) n,m=0 d3kj1 d3k′j2 (2π)4 eiy·(p1+q1−p2−q2− k′j2 )+DQCED × ˜̄βn,m(k1, . . . , kn; k 1, . . . , k where the new YFS [21,22] residuals, ˜̄βn,m(k1, . . . , kn;k 1, . . . , k m), with n hard gluons and m hard photons, defined in Ref. [2], represent the successive application of the YFS expansion first for QCD and subsequently for QED. The functions SUMIR(QCED), DQCED are determined from their QCD analogs SUMIR(QCD), DQCD via the substitutions BnlsQCD → B QCD + B QED ≡ B QCED, B̃nlsQCD → B̃ QCD + B̃ QED ≡ B̃ QCED, (3) S̃nlsQCD → S̃ QCD + S̃ QED ≡ S̃ everywhere in expressions for the latter functions given in Ref. [3]. We stress that if desired the exponent corresponding the N th Gatherall exponent for N > 1 can be systematically included in the QCD exponents SUMIR(QCD), DQCD if desired, with a corresponding change in the respective residuals ˜̄βn,m(k1, . . . , kn; k 1, . . . , k m). The residuals ˜̄βn,m(k1, . . . , kn; k 1, . . . , k m) are free of all infrared singularities, and the result in (2) is a representation that is exact and that can therefore be used to make contact with parton shower MC’s without double counting or the unnecessary averaging of effects such as the gluon azimuthal angular distribution relative to its parent’s momentum direction. In the respective infrared algebra (QCED) in (2), the average Bjorken x values xavg(QED) ∼= γ(QED)/(1 + γ(QED)), xavg(QCD) ∼= γ(QCD)/(1 + γ(QCD)), where γ(A) = 2αACA (Ls − 1), A = QED, QCD, with CA = Q f , CF , respec- tively, for A = QED, QCD and the big log Ls, imply that QCD dominant corrections happen an order of magnitude earlier than those for QED. This means that the leading ˜̄β0,0-level gives already a good estimate of the size of the interplay between the higher order QED and QCD effects which we will use to illustrate (2) here. 4 QED⊗ QCD Threshold Corrections and Shower/ME Matching at the LHC The cross section for the processes pp → V +n(γ)+m(g)+X → ℓ̄ℓ′+n′(γ)+ m(g) +X , where V, ℓ, ℓ′ are the vector-boson / lepton combinations defined in Section 3, may be constructed from the parton-level cross section via the usual formula (we use the standard notation here [2]) dσexp = dxidxjFi(xi)Fj(xj)dσ̂exp(xixjs), (4) In this section, we will use the result in (2) here with semi-analytical methods and structure functions from Ref. [23] to examine the size of QED⊗QCD threshold corrections. A Monte Carlo realization will appear elsewhere [24]. First, we wish to make contact with the existing literature and stan- dard practice for QCD parton showers as realized by HERWIG [25] and/or PYTHIA [26]. Eventually, we will also make contact with the new parton distribution function evolution MC algorithm in Ref. [27]. We intend to combine our exact YFS-style resummation calculus with HERWIG and/or PYTHIA by using the latter to generate a parton shower starting from the initial (x1, x2) point at factorization scale µ, after this point is provided by the {Fi}. This combination of theoretical constructs can be systematically improved with exact fully exclusive results order-by-order in αs, where cur- rently the state of the art in such a calculation is the work in Ref. [28] which accomplishes the combination of an exact O(αs) correction with HERWIG, where the gluon azimuthal angle is averaged in the combination. The issue of this being an exact rearrangement of the QCD and QED perturbative series requires some comment. Unlike the threshold resumma- tion techniques in Refs. [5–7], we have a resummation which is valid over the entire phase space. Thus, it is readily applicable to an exact treatment of the respective phase space in its implementation via MC methods. We may illustrate how the combination with PYTHIA/HERWIG may proceed as follows. We note that, for example, if we use a quark mass mq as our collinear limit regulator, DGLAP [29] evolution of the structure functions allows us to factorize all the terms that involve powers of the big log Lc = lnµ 2/m2q −1 in such a way that the evolved structure function contains the effects of summing the leading big logs L = lnµ2/µ20 where the evolution involves initial data at the scale µ0. This gives us a result independent of mq for mq ↓ 0. In the DGLAP theory, the factorization scale µ represents the largest pT of the gluon emission included in the structure function. In practice, when we use these structure functions with an exact result for the residuals in (2), it means that we must in the residuals omit the contributions from gluon radiation at scales below µ. This can be shown to amount in most cases to replacing Ls = ln ŝ/m q − 1 → Lnls = ln ŝ/µ 2 but in any case it is immediate how to limit the pT in the gluon emission that we do not double count effects. In other words, we apply the standard QCD factorization of mass singularities to the cross section in (2) in the standard way. We may do it with either the mass regulator for the collinear singularities or with dimensional regularization of such singularities. The final result should be independent of this regulator and this is something that we may use as a cross-check on the results. This would in practice mean the following: We first make an event with the formula in (4) which would produce an initial beam state at (x1, x2) for the two hard interacting partons at the factorization scale µ from the struc- ture functions {Fj} and a corresponding final state X from the exponenti- ated cross section in dσ̂exp(xixjs), where we stress that the latter has had all collinear singularities factorized so that it is much more convergent then its analog in LEP physics for the electroweak theory for example. The standard Les Houches procedure [30] of showering this event (x1, x2, X) would then be used, employing backward evolution of the initial partons. If we restrict the pT as we have indicated above, there would be no double counting of effects. Let us call this pT matching of the shower from the backward evolution and the matrix elements in the QCED exponentiated cross section. It is possible, however, to be more accurate in the use of the exact result in (2). Just as the residuals ˜̄βn,m(k1, . . . , kn; k 1, . . . , k m) are computed order by order in perturbation theory from the corresponding exact perturbative results by expanding the exponents in (2) and comparing the appropriate corresponding coefficients of the respective powers of αnαms , so too can the shower formula which is used to generate the backward evolution be expanded so that the product of the shower formula’s perturbative expansion, the per- turbative expansion of the exponents in (2), and the perturbative expansions of the residuals can be written as an over-all expansion in powers of αnαms and required to match the respective calculated exact result for given order. In this way, new shower subtracted residuals, { βn,m(k1, . . . , kn; k 1, . . . , k are calculated that can be used for the entire gluon pT phase space with an accuracy of the cross section that should in principle be improved compared with the first procedure for shower matching presented above. Both ap- proaches are under investigation, where we note that the shower subtracted 2 Here, we refer to both on-shell and off-shell emitted gluons. residuals have been realized for the exact O(α) luminosity Bhabha process at DAPHNE energies by the authors in Ref. [31]. Returning to the general discussion, we compute, with and without QED, the ratio rexp = σexp/σBorn, where we do not use the narrow resonance ap- proximation, for we wish to set a paradigm for precision heavy vector boson studies. The formula which we use for σBorn is obtained from that in (4) by substituting dσ̂Born for dσ̂exp therein, where dσ̂Born is the respective parton- level Born cross section. Specifically, we have from (1) the ˜̄β0,0-level result σ̂exp(x1x2s) = ∫ vmax dv γQCED v γQCED−1FYFS(γQCED) eδYFS σ̂Born((1− v)x1x2s) where we intend the well-known results for the respective parton-level Born cross sections and the value of vmax implied by the experimental cuts under study. What is new here is the value for the QED⊗QCD exponent γQCED = + 2CF Lnls (6) where Lnls = ln x1x2s/µ 2 when µ is the factorization scale. The functions FYFS(γQCED) and δYFS(γQCED) are well-known [22] as well: FY FS(γQCED) = e−γQCEDγE Γ(1 + γQCED) δY FS(γQCED) = γQCED + 2ζ(2)− where ζ(2) is Riemann’s zeta function of argument 2, i.e., π2/6, and γE is Euler’s constant, i.e., 0.5772. . . . Using these formulas in (4) allows us to get the results rexp = 1.1901 , QCED ≡ QCD+QED, LHC 1.1872 , QCD, LHC 1.1911 , QCED ≡ QCD+QED, Tevatron 1.1879 , QCD, Tevatron. We see that QED is at the level of .3% at both LHC and FNAL. This is stable under scale variations [2]. We agree with the results in Refs. [15,16,18–20] on both of the respective sizes of the QED and QCD effects. Furthermore, the QED effect is similar in size to structure function results found in Refs. [9–13]. 5 Conclusions We have shown that YFS theory (EEX and CEEX), when extended to non- Abelian gauge theory, allows simultaneous exponentiation of QED and QCD, QED⊗QCD exponentiation. For QED⊗QCD we find that full MC event generator realization is possible in a way that combines our calculus with HERWIG and PYTHIA in principle. Semi-analytical results for QED (and QCD) threshold effects agree with literature on Z production. As QED is at the .3% level, it is needed for LHC theory predictions at . 1%. The cor- responding analysis of the W production is in progress. We have illustrated a firm theoretical basis for the realization of the complete O(α2s , ααs, α 2) re- sults needed for the FNAL/LHC/RHIC/ILC physics and all of the latter are in progress. Acknowledgments Work partly supported by US DOE grant DE-FG02-05ER41399 and by NATO grant PST.CLG.980342. S.A.Y. thanks the organizers of the 2007 Cracow Epiphany Conference for hospitality. B.F.L.W. thanks Prof. W. 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D65 (2002) 073007; and ref- erences therein [20] Z.A. Zykunov, Eur. Phys. J.C3 (2001) 9; and references therein. [21] D.R. Yennie, S.C. Frautschi, and H. Suura,Ann. Phys. 13 (1961) 379; see also K.T. Mahanthappa, Phys. Rev. 126 (1962) 329, for a related analysis. [22] See also S. Jadach et al., Comput. Phys. Commun. 102 (1997) 229; S. Jadach, M. Skrzypek and B.F.L. Ward, Phys. Rev. D55 (1997) 1206; S. Jadach, B.F.L. Ward and Z. Wa̧s, Phys. Rev. D63 (2001) 113009; S. Jadach, B.F.L. Ward and Z. Wa̧s, Comp. Phys. Commun. 130 (2000) 260; S. Jadach et al., ibid. 140 (2001) 432, 475. [23] A.D. Martin et al., Phys. Rev. D51 (1995) 4756. [24] S. Jadach et al., to appear. [25] G. Corcella et al., hep-ph/0210213 and references therein. [26] T. Sjostrand et al., hep-ph/0308153. http://arxiv.org/abs/hep-ph/0210213 http://arxiv.org/abs/hep-ph/0308153 [27] S. Jadach and M. Skrzypek, Acta Phys. Pol. B35 (2004) 735; hep-ph/0504263, 0504205. [28] S. Frixione and B. Webber, J. High Energy Phys. 0206 (2002) 029; S. Frixione, P. Nason and B. Webber, ibid. 0308 (2003) 007; and references therein. [29] G. Altarelli and G. Parisi, Nucl. Phys. B126 (1977) 298; Yu. L. Dok- shitzer, Sov. Phys. JETP 46 (1977) 641; L.N. Lipatov, Yad. Fiz. 20 (1974) 181; V. Gribov and L. Lipatov, Sov. J. Nucl. Phys. 15 (1972) 675, 938. [30] E. Boos et al., hep-ph/0109068. [31] G. Balossini et al., Nucl. Phys. Proc. Suppl. 162, 59 (2006); in Proc. ICHEP06, in press, and references therein. http://arxiv.org/abs/hep-ph/0504263 http://arxiv.org/abs/hep-ph/0109068 Introduction Extension of YFS Theory to QCD QEDQCD Resummation Theory QED QCD Threshold Corrections and Shower/ME Matching at the LHC Conclusions

0704.0295

On the number of topological types occurring in a parametrized family of arrangements

ON THE NUMBER OF TOPOLOGICAL TYPES OCCURRING IN A PARAMETRIZED FAMILY OF ARRANGEMENTS SAUGATA BASU Abstract. Let S(R) be an o-minimal structure over R, T ⊂ Rk1+k2+ℓ a closed definable set, and π1 : R k1+k2+ℓ → Rk1+k2 , π2 : R k1+k2+ℓ → Rℓ, π3 : R k1+k2 → Rk2 the projection maps as depicted below. k1+k2+ℓ R k1+k2 For any collection A = {A1, . . . , An} of subsets of R k1+k2 , and z ∈ Rk2 , let Az denote the collection of subsets of R {A1,z, . . . , An,z} where Ai,z = Ai ∩ π 3 (z), 1 ≤ i ≤ n. We prove that there exists a constant C = C(T ) > 0 such that for any family A = {A1, . . . , An} of definable sets, where each Ai = π1(T ∩ π 2 (yi)), for some yi ∈ R ℓ, the number of distinct stable homotopy types amongst the arrangements Az, z ∈ R k2 is bounded by C · n(k1+1)k2 while the number of distinct homotopy types is bounded by C · n(k1+3)k2 . This generalizes to the o-minimal setting, bounds of the same type proved in [5] for semi-algebraic and semi-Pfaffian families. One technical tool used in the proof of the above results is a pair of topological comparison theorems reminiscent of Helly’s theorem in convexity theory and these might be of independent interest in the quantitative study of arrangements. 1. Introduction The study of arrangements is a very important subject in discrete and computa- tional geometry, where one studies arrangements of n subsets of Rk (often referred to as objects of the arrangements) for fixed k and large values of n (see [1] for a survey of the known results from this area). The precise nature of the objects in an arrangements will be discussed in more details below. Common examples consist of arrangements of hyperplanes, balls or simplices in Rk. More generally one con- siders arrangements of objects of “bounded description complexity”. This means that each set in the arrangement is defined by a first order formula in the language of ordered fields involving at most a constant number of polynomials whose degrees are also bounded by a constant (see [12]). Key words and phrases. Combinatorial Complexity, O-minimal Structures, Homotopy Types, Arrangements. The author was supported in part by NSF grant CCF-0634907. 2000 MATHEMATICS SUBJECT CLASSIFICATION 14P10, 14P25 http://arxiv.org/abs/0704.0295v3 2 SAUGATA BASU In this paper we consider parametrized families of arrangements. The question we will be interested in most, is the number of “topologically” distinct arrange- ments which can occur in such a family (precise definition of the topological type of an arrangement is given later (see Definition 1.6)). Parametrized arrangements occur quite frequently in practice. For instance, take any arrangement A in Rk1+k2 and let π : Rk1+k2 → Rk2 be the projection on the last k2 co-ordinates. Then for each z ∈ Rk2 , the intersection of the arrangement A with the fiber π−1(z), is an ar- rangement Az in R k1 and the family of the arrangements {Az}z∈Rk2 is an example of a parametrized family of arrangements. Even though the number of arrange- ments in the family {Az}z∈Rk2 is infinite, it follows from Hardt’s triviality theorem generalized to o-minimal structures (see Theorem 4.2 below) that the number of “topological types” occurring amongst them is finite and can be effectively bounded in terms of the n, k1, k2 up to multiplication by a constant that depends only on the particular family from which the objects of the arrangements are drawn. If by topological type we mean homeomorphism type, then the best known upper bound on the number of types occurring is doubly exponential in k1, k2. However, if we consider the weaker notion of homotopy type, then we obtain a singly ex- ponential bound. We conjecture that a singly exponential bound also holds for homeomorphism types as well. We now make precise the class of arrangements that we consider and also the notion of topological type of an arrangement. 1.1. Combinatorial Complexity in O-minimal Geometry. In order to put the study of the combinatorial complexity of arrangements in a more natural mathe- matical context, as well as to elucidate the proofs of the main results in the area, a new framework was introduced in [2] which is a significant generalization of the settings mentioned above. We recall here the basic definitions of this framework from [2], referring the reader to the same paper for further details and examples. We first recall an important model theoretic notion – that of o-minimality – which plays a crucial role in this generalization. 1.1.1. O-minimal Structures. O-minimal structures were invented and first studied by Pillay and Steinhorn in the pioneering papers [13, 14]. Later the theory was further developed through contributions of other researchers, most notably van den Dries, Wilkie, Rolin, Speissegger amongst others [20, 21, 22, 25, 26, 15]. We particularly recommend the book by van den Dries [19] and the notes by Coste [6] for an easy introduction to the topic as well as the proofs of the basic results that we use in this paper. Definition 1.1 (o-minimal structure). An o-minimal structure over a real closed field R is a sequence S(R) = (Sn)n∈N, where each Sn is a collection of subsets of R (called the definable sets in the structure) satisfying the following axioms (following the exposition in [6]). (1) All algebraic subsets of Rn are in Sn. (2) The class Sn is closed under complementation and finite unions and inter- sections. (3) If A ∈ Sm and B ∈ Sn then A×B ∈ Sm+n. (4) If π : Rn+1 → Rn is the projection map on the first n co-ordinates and A ∈ Sn+1, then π(A) ∈ Sn. (5) The elements of S1 are precisely finite unions of points and intervals. TOPOLOGICAL TYPES OF PARAMETRIZED ARRANGEMENTS 3 The class of semi-algebraic sets is one obvious example of such a structure, but in fact there are much richer classes of sets which have been proved to be o-minimal (see [6, 19]). 1.1.2. Admissible Sets. We now recall from [2] the definition of the class of sets that will play the role of sets with bounded description complexity mentioned above. Definition 1.2 (admissible sets). Let S(R) be an o-minimal structure over R and let T ⊂ Rk+ℓ be a fixed definable set. Let π1 : R k+ℓ → Rk (respectively π2 : R k+ℓ → Rℓ) be the projections onto the first k (respectively last ℓ) co-ordinates. T ⊂ Rk+ℓ We will call a subset S of Rk to be a (T, π1, π2)-set if S = Ty = π1(π 2 (y) ∩ T ) for some y ∈ Rℓ. If T is some fixed definable set, we call a family of (T, π1, π2)-sets to be a (T, π1, π2)-family. We wil also refer to a finite (T, π1, π2)-family as an arrangement of (T, π1, π2)-sets. 1.2. Stable Homotopy Equivalence. For any finite CW-complex X we denote by SX the suspension of X and for n ≥ 0, we denote by SnX the n-fold iterated suspension S ◦ S ◦ · · · ◦ S ︸ ︷︷ ︸ n times Note that if i : X →֒ Y is an inclusion map, then there is an obvious induced inclusion map Sni : SnX →֒ SnY between the n-fold iterated suspensions of X and Recall from [17] that for two finite CW-complexes X and Y , an element of (1.1) {X ;Y } = lim [SiX,SiY ] is called an S-map (or map in the suspension category). An S-map f ∈ {X ;Y } is represented by the homotopy class of a map f : SNX → SNY for some N ≥ 0. Definition 1.3 (stable homotopy equivalence). An S-map f ∈ {X ;Y } is an S- equivalence (also called a stable homotopy equivalence) if it admits an inverse f−1 ∈ {Y ;X}. In this case we say that X and Y are stable homotopy equivalent. If f ∈ {X ;Y } is an S-map, then f induces a homomorphism f∗ : H∗(X,Z) → H∗(Y,Z) between the homology groups of X and Y . The following theorem characterizes stable homotopy equivalence in terms of homology. Theorem 1.4. [8, pp. 604] Let X and Y be two finite CW-complexes. Then X and Y are stable homotopy equivalent if and only if there exists an S-map f ∈ {X ;Y } which induces isomorphisms f∗ : H∗(X,Z) → H∗(Y,Z). 4 SAUGATA BASU 1.3. Diagrams and Co-limits. The arrangements that we consider are all finitely triangulable. In other words, the union of objects of an arrangement is homeomor- phic to a finite simplicial complex, and each individual object in the arrangement will correspond to a sub-complex of this simplicial complex. It will be more con- venient to work in the category of finite regular cell complexes, instead of just simplicial complexes. Let A = {A1, . . . , An}, where each Ai is a sub-complex of a finite regular cell complex. We will denote by [n] the set {1, . . . , n} and for I ⊂ [n] we will denote by AI (respectively AI) the regular cell complexes Ai (respectively Ai). Notice that if J ⊂ I ⊂ [n], then AJ ⊂ AI , AI ⊂ AJ . We will call the collection of sets {|AI |}I⊂[n] together with the inclusion maps iI,J : |AI | →֒ |AJ |, J ⊂ I, the diagram of A. Notice that (even though we do not use this fact), |A[n]| is the co-limit of the diagram of A. For I ⊂ [n] we will denote by A[I] the sub-arrangement {Ai | i ∈ I}. 1.4. Diagram Preserving Maps. Now let A = {A1, . . . , An}, B = {B1, . . . , Bn} where each Ai, Bj is a sub-complex of a finite regular cell complex for 1 ≤ i, j ≤ n. Definition 1.5 (diagram preserving maps). We call a map f : |A[n]| → |B[n]| to be diagram preserving if f(|AI |) ⊂ |BI | for every I ⊂ [n]. (Notice that the above property is equivalent to f(|Ai|) ⊂ |Bi| for every i ∈ [n] but the previous property will be more convenient for us later when we extend the definition of diagram preserving maps to homotopy co-limits (see Definition 3.3).) We say that two maps f, g : |A[n]| → |B[n]| are diagram homotopic if there exists a homotopy h : |A[n]| × [0, 1] → |B[n]|, such that h(·, 0) = f, h(·, 1) = g and h(·, t) is diagram preserving for each t ∈ [0, 1]. More generally, we call a map f : SN |A[n]| → SN |B[n]| to be diagram preserving if f(SN |AI |) ⊂ S N |BI | for every I ⊂ [n]. We say that two maps f, g : S N |A[n]| → SN |B[n]| are diagram homotopic if there exists a homotopy h : SN |A[n]| × [0, 1] → SN |B[n]| such that h(·, 0) = f, h(·, 1) = g and h(·, t) is diagram preserving for each t ∈ [0, 1]. We say that f : |A[n]| → |B[n]| is a diagram preserving homeomorphism if there exists a diagram preserving inverse map g : |B[n]| → |A[n]| such that the induced maps g ◦ f : |A[n]| → |A[n]| and f ◦ g : |B[n]| → |B[n]| are Id|A[n]| and Id|B[n]|, respectively. We say that f : |A[n]| → |B[n]| is a diagram preserving homotopy equivalence if there exists a diagram preserving inverse map g : |B[n]| → |A[n]| such that the induced maps g◦f : |A[n]| → |A[n]| and f ◦g : |B[n]| → |B[n]| are diagram homotopic to Id|A[n]| and Id|B[n]|, respectively. We say that an S-map f ∈ {|A[n]|; |B[n]|} is a diagram preserving stable homo- topy equivalence if it is represented by a diagram preserving map f̃ : SN |A[n]| → SN |B[n]| such that there exists a diagram preserving inverse map g̃ : SN |B[n]| → SN |A[n]| TOPOLOGICAL TYPES OF PARAMETRIZED ARRANGEMENTS 5 for which the induced maps g̃ ◦ f̃ : SN |A[n]| → SN |A[n]|, f̃ ◦ g̃ : SN |B[n]| → SN |B[n]| are diagram homotopic to IdSN |A[n]| and IdSN |B[n]|, respectively. Translating these topological definitions into the language of arrangements, we say that: Definition 1.6 (topological type of an arrangement). Two arrangements A,B are homeomorphic (respectively homotopy equivalent, stable homotopy equivalent) if there exists a diagram preserving homeomorphism (respectively homotopy equiva- lence, stable homotopy equivalence) between them. Remark 1.7. Note that, since two definable sets might be stable homotopy equiv- alent, without being homotopy equivalent (see [18, pp. 462]), and also homotopy equivalent without being homeomorphic, the notions of homeomorphism type, ho- motopy type and stable homotopy type are each strictly weaker than the previous The main results of this paper can now be stated. 1.5. Main Results. Let S(R) be an o-minimal structure over R, T ⊂ Rk1+k2+ℓ a closed and bounded definable set, and let π1 : R k1+k2+ℓ → Rk1+k2 (respectively, π2 : R k1+k2+ℓ → Rℓ, π3 : R k1+k2 → Rk2) denote the projections onto the first k1 + k2 (respectively, the last ℓ, the last k2) co-ordinates. For any collection A = {A1, . . . , An} of (T, π1, π2)-sets, and z ∈ R k2 , we will denote by Az the collection of sets, {A1,z, . . . , An,z}, where Ai,z = Ai ∩ π 3 (z), 1 ≤ i ≤ n. A fundamental theorem in o-minimal geometry is Hardt’s trivialization theorem (Theorem 4.2 below) which says that there exists a definable partition of Rk2 into a finite number of definable sets {Ti}i∈I such that for each i ∈ I, all fibers Az with z ∈ Ti are definably homeomorphic. A very natural question is to ask for an upper bound on the size of this partition (which will also give an upper bound on the number of homeomorphism types amongst the arrangements Az, z ∈ R Hardt’s theorem is a corollary of the existence of cylindrical cell decompositions of definable sets proved in [11] (see also [19, 6]). When A is a (T, π1, π2)-family for some fixed definable set T ⊂ Rk1+k2+ℓ, with π1 : R k1+k2+ℓ → Rk1+k2 , π2 : k1+k2+ℓ → Rℓ, π2 : R k1+k2 → Rk2 the usual projections, and #A = n, the quantitative definable cylindrical cell decomposition theorem in [2] gives a doubly exponential (in k1k2) upper bound on the cardinality of I and hence on the number of homeomorphism types amongst the arrangements Az, z ∈ R k2 . A tighter (say singly exponential) bound on the number of homeomorphism types of the fibers would be very interesting but is unknown at present. Note that we cannot hope for a bound which is better than singly exponential because the lower bounds on the number of topological types proved in [5] also applies in our situation. In this paper we give tighter (singly exponential) upper bounds on the number of homotopy types occurring amongst the fibers Az, z ∈ R k2 . We prove the following theorems. The first theorem gives a bound on the number of stable homotopy types of the arrangements Az, z ∈ R k2 , while the second theorem gives a slightly worse bound for homotopy types. 6 SAUGATA BASU Theorem 1.8. There exists a constant C = C(T ) > 0 such that for any collection A = {A1, . . . , An} of (T, π1, π2)-sets the number of distinct stable homotopy types amongst the arrangements Az, z ∈ R k2 is bounded by C · n(k1+1)k2 . If we replace stable homotopy type by homotopy type, we obtain a slightly weaker bound. Theorem 1.9. There exists a constant C = C(T ) > 0 such that for any collection A = {A1, . . . , An} of (T, π1, π2)-sets the number of distinct homotopy types occuring amongst the arrangements Az, z ∈ R k2 is bounded by C · n(k1+3)k2 . 2. Background In this section we describe some prior work in the area of bounding the number of homotopy types of fibers of a definable map and their connections with the results presented in this paper. We begin with a definition. Definition 2.1 (A-sets). Let A = {A1, . . . , An}, such that each Ai ⊂ R k is a (T, π1, π2)-set. For I ⊂ {1, . . . , n}, we let A(I) denote the set (2.1) i∈I⊂[n] j∈[n]\I (Rk \Aj) and we will call such a set to be a basic A-set. We will denote by C(A) the set of non-empty connected components of all basic A-sets. We will call definable subsets S ⊂ Rk defined by a Boolean formula whose atoms are of the form, x ∈ Ai, 1 ≤ i ≤ n, an A-set. An A-set is thus a union of basic A-sets. If T is closed, and the Boolean formula defining S has no negations, then S is closed by definition (since each Ai is closed) and we call such a set an A-closed Moreover, if V is any closed definable subset ofRk, and S is anA-set (respectively A-closed set), then we will call S∩V to be an (A, V )-set (respectively (A, V )-closed set). 2.1. Bounds on the Betti numbers of Admissible Sets. The problem of bounding the Betti numbers of A-sets is investigated in [2], where several results known in the semi-algebraic and semi-Pfaffian case are extended to this general setting. In particular, we will need the following theorem proved there. Theorem 2.2. [2] Let S(R) be an o-minimal structure over R and let T ⊂ Rk+ℓ be a closed definable set. Then, there exists a constant C = C(T ) > 0 depending only on T such that for any arrangement A = {A1, . . . , An} of (T, π1, π2)-sets of k the following holds. For every i, 0 ≤ i ≤ k, (2.2) D∈C(A) bi(D) ≤ C · n TOPOLOGICAL TYPES OF PARAMETRIZED ARRANGEMENTS 7 Remark 2.3. The main intuition behind the bound in Theorem 2.2 (as well as similar results in the semi-algebraic and semi-Pfaffian settings) is that the homotopy type (or at least the Betti numbers) of a definable set in Rk defined in terms of n sets belonging to some fixed definable family, depend only on the interaction of these sets at most k + 1 at a time. This is reminiscent of Helly’s theorem in convexity theory (see [7]) but in a homotopical setting. This observation is also used to give an efficient algorithm for computing the Betti numbers of arrangements (see [3, Section 8]). However, the proof of Theorem 2.2 in [2] (as well as the proofs of similar results in the semi-algebraic [4] and semi-Pfaffian settings [10]) depends on an argument involving the Mayer-Vietoris sequence for homology, and does not require more detailed information about homotopy types. In Section 3 below, we make the above intuition mathematically precise. We prove two theorems (Theorems 3.6 and 3.7 below) and these auxiliary re- sults are the keys to proving the main results of this paper (Theorems 1.8 and 1.9). Moreover, these auxiliary results could also be of independent interest in the quantitative study of arrangements. 2.2. Homotopy types of the fibers of a semi-algebraic map. Theorem 2.2 gives tight bounds on the topological complexity of an A-set in terms of the cardi- nality of A, assuming that the sets in A belong to some fixed definable family. A problem closely related to the problem we consider in this paper is to bound the number of topological types of the fibers of a projection restricted to an arbitrary A-set. More precisely, let S ⊂ Rk1+k2 be a set definable in an o-minimal structure over the reals (see [19]) and let π : Rk1+k2 → Rk2 denote the projection map on the last k2 co-ordinates. We consider the fibers, Sz = π −1(z) ∩ S for different z in k2 . Hardt’s trivialization theorem, (Theorem 4.2 below) shows that there exists a definable partition of Rk2 into a finite number of definable sets {Ti}i∈I such that for each i ∈ I and any point zi ∈ Ti, π −1(Ti) ∩ S is definably homeomorphic to Szi × Ti by a fiber preserving homeomorphism. In particular, for each i ∈ I, all fibers Sz with z ∈ Ti are definably homeomorphic. In case S is an A-set, with A a (T, π1, π2)-family for some fixed definable set T ⊂ Rk1+k2+ℓ, with π1 : R k1+k2+ℓ → Rk1+k2 , π2 : R k1+k2+ℓ → Rℓ, π2 : R k1+k2 → k2 , the usual projections, and #A = n, the quantitative definable cylindrical cell decomposition theorem in [2] gives a doubly exponential (in k1k2) upper bound on the cardinality of I and hence on the number of homeomorphism types of the fibers of the map π3|S . A tighter (say singly exponential) bound on the number of homeomorphism types of the fibers would be very interesting but is unknown at present. Recently, the problem of obtaining a tight bound on the number of topological types of the fibers of a definable map for semi-algebraic and semi-Pfaffian sets was considered in [5], and it was shown that the number of distinct homotopy types of the fibers of such a map can be bounded (in terms of the format of the formula defining the set) by a function singly exponential in k1k2. In particular, the combinatorial part of the bound is also singly exponential. A more precise statement in the case of semi-algebraic sets is the following theorem which appears in [5]. Theorem 2.4. [5] Let P ⊂ R[X1, . . . , Xk1 , Y1, . . . , Yk2 ], with deg(P ) ≤ d for each P ∈ P and cardinality #P = n. Then, for any fixed P-semi-algebraic set S the 8 SAUGATA BASU number of different homotopy types of fibers π−1(y) ∩ S for various y ∈ π(S) is bounded by (2k1nk2d) O(k1k2). Remark 2.5. The proof of Theorem 2.4 however has the drawback that it relies on techniques involving perturbations of the original polynomials in order to put them in general position, as well as Thom’s Isotopy Theorem, and as such does not extend easily to the o-minimal setting. The main results of this paper (see Theorem 1.8 and Theorem 1.9) extend the combinatorial part of Theorem 2.4 to the more general o-minimal category. Remark 2.6. Even though the formulation of Theorem 2.4 seems a little different from the main theorems of this paper (Theorems 1.8 and 1.9), they are in fact closely related. In fact, as a consequence of Theorem 1.9 we obtain bounds on the number of homotopy types of the fibers of S for any fixed A-set S, analogous to the one in Theorem 2.4. More precisely we have: Theorem 2.7. Let S(R) be an o-minimal structure over R, and T ⊂ Rk1+k2+ℓ a closed and bounded definable set, and π1 : R k1+k2+ℓ → Rk1+k2 , π2 : R k1+k2+ℓ → ℓ, and π3 : R k1+k2 → Rk2 the projection maps. Then, there exists a constant C = C(T ) > 0, such that for any collection A = {A1, . . . , An} of (T, π1, π2)-sets, for any fixed A-set S the number of distinct homotopy types of fibers π−13 (z)∩S for various z ∈ π3(S) is bounded by C · n(k1+3)k2 . A similar result with a bound of C · n(k1+1)k2 holds for stable homotopy types as well. 3. A Topological Comparison Theorem As noted previously, the main underlying idea behind our proof of Theorem 1.8 is that the homotopy type of an A-set in Rk depends only on the interaction of sets in A at most (k + 1) at a time. In this section we make this idea precise. We show that in case A = {A1, . . . , An}, with each Ai a definable, closed and bounded subset of Rk, the homotopy type of any A-closed set is determined by a certain sub-complex of the homotopy co-limit of the diagram of A. The crucial fact here is that this sub-complex depends only on the intersections of the sets in A at most k + 1 at a time. In order to avoid technical difficulties, we restrict ourselves to the category of finite, regular cell complexes (see [24] for the definition of a regular cell complex). The setting of finite, regular cell complexes suffices for us, since it is well known that closed and bounded definable sets in any o-minimal structure are finitely tri- angulable, and hence, are homeomorphic to regular cell complexes. 3.1. Topological Preliminaries. Let A = {A1, . . . , An}, where each Ai is a sub- complex of a finite regular cell complex. We now define the homotopy co-limit of the diagram of A. TOPOLOGICAL TYPES OF PARAMETRIZED ARRANGEMENTS 9 3.1.1. Homotopy Co-limits. Let ∆[n] denote the standard simplex of dimension n−1 with vertices in [n] (and by |∆[n]| the corresponding closed geometric simplex). For I ⊂ [n], we denote by ∆I the (#I − 1)-dimensional face of ∆[n] corresponding to I. The homotopy co-limit, hocolim(A), is a CW-complex defined as follows. Definition 3.1 (homotopy co-limit). hocolim(A) = I⊂[n] ∆I ×AI/ ∼ where the equivalence relation ∼ is defined as follows. For I ⊂ J ⊂ [n], let sI,J : |∆I | →֒ |∆J | denote the inclusion map of the face |∆I | in |∆J |, and let iI,J : |AJ | →֒ |AI | denote the inclusion map of |AJ | in |AI |. Given (s,x) ∈ |∆I |×|AI | and (t,y) ∈ |∆J |×|AJ | with I ⊂ J , then (s,x) ∼ (t,y) if and only if t = sI,J(s) and x = iI,J(y). Note that there exist two natural maps fA : |hocolim(A)| → |A [n]|, gA : |hocolim(A)| → |∆[n]| defined by (3.1) fA(s,x) = s, (3.2) gA(s,x) = x. where (s,x) ∈ |∆Ic | × c, c is a cell in A [n] and Ic = {i ∈ [n] | c ∈ Ai}. Notice that we have |hocolim(A)| = I⊂[n] |∆I | × |AI | ⊂ I⊂[n] |∆I | × A Definition 3.2 (truncated homotopy co-limits). For any m, 0 ≤ m ≤ n, we will denote by hocolimm(A) the sub-complex of hocolim(A) defined by (3.3) hocolimm(A) = g A (skm(∆[n])). Definition 3.3 (diagram preserving maps between homotopy co-limits). Replacing in Definition 1.5, |A[n]| and |B[n]|, by |hocolim(A)| and |hocolim(B)| respectively, as well as |AI | and |BI | by f A (|AI |) and f B (|BI |) respectively, we get definitions of diagram preserving homotopy equivalences and stable homotopy equivalences between |hocolim(A)| and |hocolim(B)|, and more generally for anym ≥ 0, between |hocolimm(A)| and |hocolimm(B)|. Definition 3.4. We say thatA ≈m B if there exists a diagram preserving homotopy equivalence φ : |hocolimm(A)| → |hocolimm(B)|. We say that A ∼m B, if there exists a diagram preserving stable homotopy equivalence φ ∈ {hocolimm(A); hocolimm(B)}, represented by φ̃ : SN |hocolimm(A)| → S N |hocolimm(B)|, for some N > 0. 10 SAUGATA BASU Remark 3.5. Note that in the above definition the map φ need not be induced by a diagram preserving map φ : A[n] → B[n] (respectively, φ̃ : SN |hocolimm(A)| → SN |hocolimm(B)|). Indeed if it was the case then the proofs of Theorems 3.6 and 3.7 below would be simplified considerably. The two following theorems are the crucial topological ingredients in the proofs of our main results. Theorem 3.6. Let A = {A1, . . . , An},B = {B1, . . . , Bn} be two families of sub- complexes of a finite regular cell complex, such that: (1) Hi(|A [n]|,Z),Hi(|B [n]|,Z) = 0, for all i ≥ k, and (2) A ∼k B. Then, A and B are stable homotopy equivalent. Theorem 3.7. Let A = {A1, . . . , An},B = {B1, . . . , Bn} be two families of sub- complexes of a finite regular cell complex, such that: (1) dim(Ai), dim(Bi) ≤ k, for 1 ≤ i ≤ n, and (2) A ≈k+2 B. Then, A and B are homotopy equivalent. We now state two corollaries of Theorems 3.6 and 3.7 which might be of interest. Given a Boolean formula θ(T1, . . . , Tn) containing no negations and a family of sub-complexes A = {A1, . . . , An} of a finite regular cell complex, we will denote by Aθ the sub-complex defined by the formula, θA, which is obtained from θ by replacing in θ the atom Ti by Ai for each i ∈ [n], and replacing each ∧ (respectively ∨) by ∩ (respectively ∪). Corollary 3.8. Let A = {A1, . . . , An},B = {B1, . . . , Bn} be two families of sub- complexes of a finite regular cell complex, satisfying the same conditions as in The- orem 3.6. Let θ(T1, . . . , Tn) be a Boolean formula without negations. Then, |Aθ| and |Bθ| are stable homotopy equivalent. Corollary 3.9. Let A = {A1, . . . , An},B = {B1, . . . , Bn} be two families of sub- complexes of a finite regular cell complex, satisfying the same conditions as in The- orem 3.7. Let θ(T1, . . . , Tn) be a Boolean formula without negations. Then, |Aθ| and |Bθ| are homotopy equivalent. 3.2. Proofs of Theorems 3.6 and 3.7. Let A and B as in Theorem 3.6. We need a preliminary lemma. Lemma 3.10. |A[n]| is diagram preserving homotopy equivalent to |hocolim(A)|. Proof. Consider the map fA : |hocolim(A)| → |A defined in (3.1). Clearly, if x ∈ c, f−1A (c) = |∆Ic |. Now applying Smale’s version of the Vietoris- Begle Theorem [16] we obtain that fA is a homotopy equivalence. Clearly, fA is diagram preserving. Moreover, (see for instance the proof of Theorem 6 in [16]) there exists an cellular inverse map hA : |A [n]| → |hocolim(A)| such that fA ◦ hA is diagram preserving, and is a homotopy inverse of fA. � TOPOLOGICAL TYPES OF PARAMETRIZED ARRANGEMENTS 11 We can now prove Theorems 3.6 and 3.7. Proof of Theorem 3.6. Let hA : |A [n]| → |hocolim(A)| be a diagram preserving homotopy equivalence known to exist by Lemma 3.10. Since hA is cellular, and dim |A[n]| ≤ k, its image is contained in hocolimk(A) since by definition (Eqn. (3.3)) skk(hocolim(A)) ⊂ hocolimk(A). We will denote by hA,B : S N |hocolimk(A)| → S N |hocolimk(B)| a map represent- ing a diagram preserving stable homotopy equivalence known to exist by hypothesis (which we assume to be cellular). Let iB,k : S N |hocolimk(B)| →֒ S N |hocolim(B)| denote the inclusion map. The map iB,k induces isomorphisms (iB,k)∗ : Hj(hocolimk(B),Z) → Hj(hocolim(B),Z) for 0 ≤ j ≤ k − 1. Consequently, the map fB ◦ iB,k induces isomorphisms (fB ◦ iB,k)∗ : Hj(hocolimk(B),Z) → Hj(B [n],Z) for 0 ≤ j ≤ k − 1. Composing the maps, SNhA, hAB, iB,k,S NfB we obtain that the map, NfB ◦ iB,k ◦ hAB ◦ S NhA : S N |A[n]| → SN |B[n]| induces isomorphisms (SNfB ◦ iB,k ◦ hA,B,k ◦ S NhA)∗ : Hj(|A [n]|,Z) → Hj(|B [n]|,Z) for all j ≥ 0. Moreover, the map SNfB ◦ iB,k ◦ hAB ◦ S NhA is diagram preserving since each constituent of the composition is diagram preserving. It now follows from Theorem 1.4 that the S-map represented by φ = SNfB ◦ iB,k ◦ hAB ◦ S NhA : S N |A[n]| → SN |B[n]|, is a diagram preserving stable homotopy equivalence. � Before proving Theorem 3.7 we first need to recall a few basic facts from homo- topy theory. Definition 3.11 (k-equivalence). A map f : X → Y between two regular cell complex is called a k-equivalence if the induced homomorphism f∗ : πi(X) → πi(Y ) is an isomorphism for all 0 ≤ i < k, and an epimorphism for i = k, and we say that X is k-equivalent to Y . (Note that k-equivalence is not an equivalence relation). We also need the following well-known fact from algebraic topology. Proposition 3.12. Let X,Y be finite regular cell complexes with dim(X) < k, dim(Y ) ≤ k, and f : X → Y a k-equivalence. Then, f is a homotopy equivalence between X and Proof. See [23, pp. 69]. � 12 SAUGATA BASU Proof of Theorem 3.7. The proof is along the same lines as that of the proof of Theorem 3.6. Let hA : |A [n]| → |hocolim(A)| be a diagram preserving homotopy equivalence known to exist by Lemma 3.10. By the same argument as before, its image is contained in |hocolimk+2(A)|. We will denote by hA,B : |hocolimk+2(A)| → |hocolimk+2(B)| a diagram preserv- ing homotopy equivalence known to exist by hypothesis. Let iB,k+2 : |hocolimk+2(B)| →֒ |hocolim(B)| denote the inclusion map. The map iB,k+2 induces isomorphisms (iB,k+2)∗ : πj(hocolimk+2(B)) → πj(hocolim(B)) for 0 ≤ j ≤ k+1. This is a consequence of the exactness of the homotopy sequence of the pair (hocolim(B), hocolimk+2(B)) (see [18]). Consequently, the map fB ◦ iB,k induces isomorphisms (gB ◦ iB,k)∗ : πj(hocolimk+2(B)) → πj(B for 0 ≤ j ≤ k + 1. Composing the maps, hA, hAB, iB,k+2, fB we obtain that the map fB ◦ iB,k ◦ hAB ◦ hA : |A [n]| → |B[n]| induces isomorphisms (fB ◦ iB,k ◦ hA,B,k ◦ hA)∗ : πj(A [n]) → πj(B for 0 ≤ j ≤ k + 1. Moreover, the map fB◦iB,k◦hAB◦hA is diagram preserving since each constituent of the composition is diagram preserving. It now follows from Proposition 3.12 that the map φ = fB ◦ iB,k ◦ hAB ◦ hA : |A [n]| → |B[n]| is a diagram preserving homotopy equivalence. � Proof of Corollary 3.8. First note that since the formula θ does not contain nega- tions, writing θ as a disjunction of conjunctions, there exists Σ ⊂ 2[n] such that AI (respectively, Bθ = BI). Let A ′ = {AI | I ∈ Σ} (respectively, B′ = {BI | I ∈ Σ}). It follows from the hypothesis that A′ ∼k B Now apply Theorem 3.6. � Proof of Corollary 3.9. The proof is similar to that of Corollary 3.8 using Theorem 3.7 in place of Theorem 3.6 and is omitted. � 4. Proofs of the Main Theorems 4.1. Summary of the main ideas. We first summarize the main ideas under- lying the proof of Theorem 1.8. The proof of Theorem 1.9 is similar and differs only in technical details. Let A = {A1, . . . ,An} be a (T, π1, π2)- arrangement in k1+k2 . Using Proposition 4.7, we obtain a definable partition, {Cα}α∈I (say) of k2 , into connected locally closed definable sets Cα ⊂ R k2 , with the property that as z varies over Cα, we get for each I ⊂ [n] with #I ≤ k1 + 1 isomorphic (and continuously varying) triangulations of the sub-arrangement A[I]. Moreover, these triangulations are downward compatible in the sense that the restriction to A[J ] TOPOLOGICAL TYPES OF PARAMETRIZED ARRANGEMENTS 13 of the triangulation of A[I], refines that of A[J ] for each J ⊂ I (cf. Proposition 4.7 below). These facts allow us to prove that for any z1, z2 ∈ Cα the truncated homotopy co-limits |hocolimk1(Az1)| and |hocolimk1(Az2)| are homotopy equiva- lent by a diagram preserving homotopy equivalence. More precisely, we first prove that the thickened homotopy co-limits |hocolim+k1(Az1 , ε̄)| and |hocolim (Az2 , ε̄)| are homeomorphic, and then use Proposition 4.8 to deduce that |hocolimk1(Az1)| and |hocolimk1(Az2)| are homotopy equivalent. Theorem 3.6 then implies that Az1 is stable homotopy equivalent to Az2 by a diagram preserving stable homotopy equivalence. It remains to bound the number of elements in the partition {Cα}α∈I . We use Theorem 2.2 to obtain a bound of C · n(k1+1)k2 on this number, where C is a constant which depends only on T . In order to prove Theorem 1.8 we recall a few results from o-minimal geometry. We first note an elementary property of families of admissible sets (see [2] for a proof). Observation 4.1. Suppose that T1, . . . , Tm ⊂ R k+ℓ are definable sets, π1 : R k+ℓ → k and π2 : R k+ℓ → Rℓ the two projections. Then, there exists a definable sub- set T ′ ⊂ Rk+ℓ+m depending only on T1, . . . , Tm, such that for any collection of (Ti, π1, π2) families Ai, 1 ≤ i ≤ m, the union Ai is a (T ′, π′1, π 2)-family, where π′1 : R k+m+ℓ → Rk and π′2 : R k+ℓ+m → Rℓ+m are the projections onto the first k, and the last ℓ+m co-ordinates respectively. 4.2. Hardt’s Triviality for Definable Sets. One important technical tool will be the following o-minimal version of Hardt’s triviality theorem. Let X ⊂ Rk×Rℓ and A ⊂ Rk be definable subsets of Rk×Rℓ and Rℓ respectively, and let π : X → Rℓ denote the projection map on the last ℓ co-ordinates. We say that X is definably trivial over A if there exists a definable set F and a definable homeomorphism h : F ×A → X ∩ π−1(A), such that the following diagram commutes. F ×A X ∩ π−1(A) In the diagram above π2 : F ×A → A is the projection onto the second factor. We call h a definable trivialization of X over A. If Y is a definable subset of X , we say that the trivialization h is compatible with Y if there is a definable subset G of F such that h(G×A) = Y ∩ π−1(A). Clearly, the restriction of h to G×A is a trivialization of Y over A. Theorem 4.2 (Hardt’s theorem for definable families). Let X ⊂ Rk × Rℓ be a definable set and let Y1, . . . , Ym be definable subsets of X. Then, there exists a finite partition of Rℓ into definable sets C1, . . . , CN such that X is definably trivial over each Ci, and moreover the trivializations over each Ci are compatible with Y1, . . . , Ym. 14 SAUGATA BASU Remark 4.3. We first remark that it is straightforward to derive from the proof of Theorem 4.2 that the definable sets C1, . . . , CN can be chosen to be locally closed, and can be expressed as, C1 = R ℓ \ B1, C2 = B1 \ B2, . . . , CN = BN−1 \ BN for closed definable sets B1, . . . , BN . Clearly, the closed definable sets B1, . . . , BN , determine the sets Ci of the partition. Remark 4.4. Note also that it follows from Theorem 4.2, that there are only a finite number of topological types amongst the fibers of any definable map f : X → Y between definable sets X and Y . This remark would be used a number of times later in the paper. Since in what follows we will need to consider many different projections, we adopt the following convention. Notation 4.5. Given m and p, p ≤ m, we will denote by π≤pm : R m → Rp (respectively π>pm : R m → Rm−p) the projection onto the first p (respectively the last m− p) coordinates. 4.3. Definable Triangulations. A triangulation of a closed and bounded defin- able set S is a simplicial complex ∆ together with a definable homeomorphism from |∆| to S. Given such a triangulation we will often identify the simplices in ∆ with their images in S under the given homeomorphism. We call a triangulation h1 : |∆1| → S of a definable set S, to be a refinement of a triangulation h2 : |∆2| → S if for every simplex σ1 ∈ ∆1, there exists a simplex σ2 ∈ ∆2 such that h1(|σ1|) ⊂ h2(|σ2|). Let S1 ⊂ S2 be two closed and bounded definable subsets of R k. We say that a definable triangulation h : |∆| → S2 of S2, respects S1 if for every simplex σ ∈ ∆, h(σ) ∩ S1 = h(σ) or ∅. In this case, h −1(S1) is identified with a sub-complex of ∆ and h|h−1(S1) : h −1(S1) → S1 is a definable triangulation of S1. We will refer to this sub-complex by ∆|S1 . We introduce the following notational conventions in order to simplify arguments used later in the paper. Notation 4.6. If T ⊂ Rk1+k2+ℓ be any definable subset of Rk1+k2+ℓ, for each m ≥ 0, and (z,y0, . . . ,ym) ∈ R k2+(m+1)ℓ, we will denote by Tz,y0,...,ym ⊂ R k1 the definable 1≤i≤m {x ∈ Rk1 | (x, z) ∈ Tyi}. For {j0, . . . , jm′} ⊂ [m], we will denote by πm,j0,...,jm′ : R (m+1)ℓ → R(m ′+1)ℓ the projection map on the appropriate blocks of co-ordinates. It is well known that compact definable sets are triangulable and moreover the usual proof of this fact (see for instance [6]) can be easily extended to produce a definable triangulation in a parametrized way. We will actually need a family of such triangulations satisfying certain compatibility conditions mentioned before. The following proposition states the existence of such families. We omit the proof of the proposition since it is a technical but straightforward extension of the proof of existence of triangulations for definable sets. Proposition 4.7 (existence of m-adaptive triangulations). Let T ⊂ Rk1+k2+ℓ be a closed and bounded definable subset of Rk1+k2+ℓ and let m ≥ 0. For each 0 ≤ p ≤ m, there exists TOPOLOGICAL TYPES OF PARAMETRIZED ARRANGEMENTS 15 (1) a definable partition {Cp,α}α∈Ip of R k2+(p+1)ℓ, into locally closed sets, de- termined by a sequence of definable closed sets, {Bp,α}α∈Ip (see Remark 4.3 above), and (2) for each α ∈ Ip, a definable continuous map, hp,α : |∆p,α| × Cp,α → (z,y0,...,yp)∈Cp,α Tz,y0,...,yp where ∆p,α is a simplicial complex, and such that for each (z,y0, . . . ,yp) ∈ Cp,α, the restriction of hp,α to |∆p,α| × (z,y0, . . . ,yp) is a definable trian- gulation hp,α : |∆p,α| × (z,y0, . . . ,yp) → Tz,y0,...,yp of the definable set Tz,y0,...,yp respecting the subsets, Tz,y0 , . . . , Tz,yp, and (3) for each subset {j0, . . . , jp′} ⊂ [p], (Idk2 , πp,j0,...,jp′ )(Cp,α) ⊂ Cp′,β for some β ∈ Ip′ , and for each (z,y0, . . . ,yp) ∈ Cp,α, the definable triangulation of Tz,yj0 ,...,yjp′ induced by the triangulation hp,α : |∆p,α| × (z,y0, . . . ,yp) → Tz,y0,...,yp is a refinement of the definable triangulation, hp′,β : |∆p′,β| × (z,yj0 , . . . ,yjp′ ) → Tz,yj0 ,...,yjp′ (We will call the family {hp,α}0≤p≤m,α∈Ip an m-adaptive family of triangulations of T .) We will also need the following technical result. Proposition 4.8. Let Ct ⊂ R k, t ≥ 0 be a definable family of closed and bounded sets, and let C ⊂ Rk+1 be the definable set Ct × {t}. If for every 0 ≤ t < t Ct ⊂ Ct′ , and C0 = π k+1(C ∩ (π −1(0)), then there exists t0 > 0 such that, C0 has the same homotopy type as Ct for every t with 0 ≤ t ≤ t0. Proof. The proof given in [4] (see Lemma 16.17) for the semi-algebraic case can be easily adapted to the o-minimal setting using Hardt’s triviality for definable families instead of for semi-algebraic ones. � We now introduce another notational convention. Notation 4.9. Let F(x) be a predicate defined over R+ and y ∈ R+. The notation ∀(0 < x ≪ y) F(x) stands for the statement ∃z ∈ (0, y) ∀x ∈ R+ (if x < z, then F(x)), and can be read “for all positive x sufficiently smaller than y, F(x) is true”. More generally, Notation 4.10. For ε̄ = (ε0, . . . , εn) and a predicate F(ε̄) over R + we say “for all sufficiently small ε̄, F(ε̄) is true” if ∀(0 < ε0 ≪ 1)∀(0 < ε1 ≪ ε0) · · · ∀(0 < εn ≪ εn−1)F(ε̄). 16 SAUGATA BASU 4.4. Infinitesimal Thickenings of the Faces of a Simplex. We will need the following construction. Let ε̄ = (ε0, . . . , εn) ∈ R + , with 0 ≤ εn < · · · < ε0 < 1. Later we will require ε̄ to be sufficiently small (see Notation 4.10). For a face ∆J ∈ ∆[n], we denote by CJ(ε̄) the subset of |∆J | defined by CJ (ε̄) = {x ∈ |∆J | | dist(x, |∆I |) ≥ ε#I−1 for all I ⊂ J}. Note that, |∆[n]| = I⊂[n] CI(ε̄). Figure 1. The complex ∆[n]. I ⊂ J ⊂ K = [n] CI(ε̄) CI(ε̄) ∩ CJ(ε̄) ∩ CK(ε̄) CI(ε̄) ∩ CJ(ε̄) CJ(ε̄) CJ(ε̄) ∩ CK(ε̄) CK(ε̄) Figure 2. The corresponding complex C(∆[n]) with I ⊂ J ⊂ K = [n]. Also, observe that for sufficiently small ε̄ > 0, the various CJ (ε̄)’s are all home- omorphic to closed balls, and moreover all non-empty intersections between them TOPOLOGICAL TYPES OF PARAMETRIZED ARRANGEMENTS 17 also have the same property. Thus, the cells CJ (ε̄)’s together with the non-empty intersections between them form a regular cell complex, C(∆[n], ε̄), whose underly- ing topological space is |∆[n]| (see Figures 1 and 2). Definition 4.11. We will denote by C(skm(∆[n]), ε̄) the sub-complex of C(∆[n], ε̄) consisting of the cells CI(ε̄)’s together with the non-empty intersections between them where |I| ≤ m+ 1. We now use thickened simplices defined above to define a thickened version of the homotopy co-limit of an arrangement A. 4.5. Thickened Homotopy Co-limits. Given an m-adaptive family of triangu- lations of T (cf. Proposition 4.7), {hp,α}0≤p≤m,α∈Ip and z ∈ R k2 , we define a cell complex, hocolim+m(Az) (best thought of as an infinitesimally thickened version of hocolimm(Az)), whose associated topological space is homotopy equivalent to |hocolimm(Az)|. Definition 4.12 (the cell complex hocolim m(Az)). Let Cm denote the cell complex C(skm(∆[n]), ε̄) defined previously (cf. Definition 4.11). Let C be a cell of Cm. Then, C ⊂ |∆I | for a unique simplex ∆I with I = {i0, . . . , im′} ⊂ [n], m ′ ≤ m, and (following notation introduced before in Definition 4.11) C = CI1(ε̄) ∩ · · · ∩ CIp(ε̄), with I1 ⊂ I2 ⊂ · · · ⊂ Ip ⊂ I and p ≤ m We denote by K(C, ε̄) the cell complex consisting of the cells C × hm′,α(|σ|, z,yi0 , . . . ,yim′ ) with α ∈ Im′ , (z,yi0 , . . . ,yim′ ) ∈ Cα,m′ , σ ∈ ∆m′,α, and hm′,α(|σ|, z,yi0 , . . . ,yim′ ) ⊂ Az,I . We denote (4.1) hocolim+m(Az, ε̄) = K(C). The compatibility properties (properties (2) and (3) in Proposition 4.7) of the m- adaptive family of triangulations of T , {hp,α}0≤p≤m,α∈Ip , ensure that hocolim m(Az, ε̄) defined above is a regular cell complex. Notice that, since the map fA defined in Eqn. 3.1 extends to |hocolim+m(Az, ε̄), the notion of diagram preserving maps ex- tend to |hocolim+m(Az, ε̄) as well. We now prove: Lemma 4.13. Let z ∈ Rℓ and m ≥ 0. Then, for all sufficiently small ε̄ > 0, |hocolim+m(Az, ε̄)| is homotopy equivalent to |hocolimm(Az)| by a diagram preserv- ing homotopy equivalence. Proof. Let N = |hocolim+m(Az, ε̄)|. First replace εm by a variable t in the definition ofN to obtain a closed and bounded definable set, Nmt , and observe thatN t ⊂ N for all 0 < t < t′ ≪ 1. Now apply Proposition 4.8 to obtain that N is homotopy equivalent to Nm0 . Now, replace εm−1 by t in the definition of N 0 to obtain N t , and applying Proposition 4.8 obtain that Nm0 is homotopy equivalent to N 0 . Continuing in this way we finally obtain that, N is homotopy equivalent to N00 = |hocolimm(Az)|. 18 SAUGATA BASU Moreover, the diagram preserving property is clearly preserved at each step of the proof. � Proof of Theorem 1.8. Recall that for m ≥ 0, and (z,y0, . . . ,ym) ∈ R k2+(m+1)ℓ, we denote by Tz,y0,...,ym the definable set Tz,yi ⊂ R Now apply Proposition 4.7 to the set T with m = k1 to obtain an k1-adaptive family of triangulations {hp,α}1≤p≤k1,α∈Ip . We now fix {y1, . . . ,yn} ⊂ R ℓ and let A = {A1, . . . , An} with Ai = Tyi ⊂ k1+k2 . For each z ∈ Rk2 , we will denote by Az = {A1,z, . . . , An,z} where Ai,z = {x ∈ Rk1 | (x, z) ∈ Ai}. For α ∈ Ik1 , and 1 ≤ i0 < · · · < ik1 ≤ n, we will denote by Bk1,α,i0,...,ik1 ⊂ R the definable closed set Bk1,α,i0,...,ik1 = {z ∈ R ℓ | (z,y0, . . . ,yk1) ∈ Bk1,α}. α∈Ik1 {Bk1,α,i0,...,ik1 | 1 ≤ i0 < i1 < · · · < ik1 ≤ n}, and let C ∈ C(B). Theorem 1.8 will follow from the following two lemmas. Lemma 4.14. For any z1, z2 ∈ C, Az1 is stable homotopy equivalent to Az2 . Proof. Clearly, by Theorem 3.6 it suffices to prove that |hocolimk1(Az1 )| is diagram preserving homotopy equivalent to |hocolimk1(Az2 )|. The compatibility properties of the triangulations ensure that that the complex |hocolim+k1(Az1 , ε̄) is isomorphic to |hocolim (Az2 , ε̄) and hence |hocolim (Az1 , ε̄)| is homeomorphic to |hocolim+k1(Az1 , ε̄)|. Using Lemma 4.13 we get a diagram preserving homotopy equivalence φ : |hocolimk1(Az1 )| → |hocolimk1(Az2 )|. It now follows from Theorem 3.6 that the arrangements Az1 and Az2 are stable homotopy equivalent. � Lemma 4.15. There exists a constant C(T ) such that the cardinality of C(B) is bounded by C · n(k1+1)k2 . Proof. Notice that eachBk1,α, α ∈ Ik1 is a definable subset ofR k2+(k1+1)ℓ depending only on T . Also, the cardinality of the index set Ik1 is determined by T . Hence, the set B consists of definable sets, each one of them is a (Bk1,α, π k2+(k1+1)ℓ , π>k2 k2+(k1+1)ℓ for some α ∈ Ik1 . Using Observation 4.1, we have that B is a (B, π 2)-set for some B determined only by T . Now apply Theorem 2.2. � The theorem now follows from Lemmas 4.14 and 4.15 proved above. � Proof of Theorem 1.9. The proof is similar to that of Theorem 1.8 given above, except we use Theorem 3.7 instead of Theorem 3.6, and this accounts for the slight worsening of the exponent in the bound. � TOPOLOGICAL TYPES OF PARAMETRIZED ARRANGEMENTS 19 Proof of Theorem 2.7. Using a construction due to Gabrielov and Vorobjov [9] (see also [2]) it is possible to replace any given A-set by a closed bounded A′-set (where A′ is a new family of definable closely related to A with #A′ = 2k(#A)), such that the new set has the same homotopy type as the original one. Using this construction one can directly deduce Theorem 2.7 from Theorem 1.9. We omit the details. � References 1. Pankaj K. Agarwal and Micha Sharir, Arrangements and their applications, Handbook of computational geometry (J. Urrutia J.R. Sack, ed.), North-Holland, Amsterdam, 2000, pp. 49– 119. MR 1746675 2. S. Basu, Combinatorial complexity in o-minimal geometry, preprint at arXiv:math.CO/0612050, 2006, An extended abstract appears in the Proceedings of the ACM Symposium on the Theory of Computing, 2007. 3. , Algorithmic semi-algebraic geometry and topology – recent progress and open prob- lems, Surveys on Discrete and Computational Geometry: Twenty Years Later, Contemporary Mathematics, vol. 453, American Mathematical Society, 2008, pp. 139–212. 4. S. Basu, R. Pollack, and M.-F. Roy, Algorithms in real algebraic geometry, Algorithms and Computation in Mathematics, vol. 10, Springer-Verlag, Berlin, 2006 (second edition). MR 1998147 (2004g:14064) 5. S. Basu and N. Vorobjov, On the number of homotopy types of fibers of a definable map, Journal of the London Mathematical Society (2007), 757–776. 6. M. Coste, An introduction to o-minimal geometry, Istituti Editoriali e Poligrafici Internazion- ali, Pisa, 2000, Dip. Mat. Univ. Pisa, Dottorato di Ricerca in Matematica. 7. L. Danzer, B. Grunbaum, and V. Klee, Helly’s theorem and its relatives, Convexity, Pro- ceedings of Symposia In Pure Mathematics, vol. VII, American Mathematical Society, 1963, pp. 101–180. 8. Jean Dieudonné, A history of algebraic and differential topology. 1900–1960, Birkhäuser Boston Inc., Boston, MA, 1989. MR 995842 (90g:01029) 9. A. Gabrielov and N. Vorobjov, Approximation of definable sets by compact families and upper bounds on homotopy and homology, preprint at arXiv:math.AG/0710.3028v1, 2007. 10. D. Grigoriev and N. Vorobjov, Solving systems of polynomial inequalities in subexponential time, Journal of Symbolic Computation 5 (1988), 37–64. 11. J.F. Knight, A. Pillay, and C. Steinhorn, Definable sets in ordered structures. II., Trans. Amer. Math. Soc. 295 (1986), no. 2, 593–605. MR 0833698 (88b:03050b) 12. J. Matousek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, 2002. 13. A. Pillay and C. Steinhorn, Definable sets in ordered structures. I., Trans. Amer. Math. Soc. 295 (1986), no. 2, 565–592. MR 0833697 (88b:03050a) 14. , Definable sets in ordered structures. III., Trans. Amer. Math. Soc. 309 (1988), no. 2, 469–576. MR 0943306 (89i:03059) 15. J.-P. Rolin, P. Speissegger, and A. J. Wilkie, Quasianalytic Denjoy-Carleman classes and o-minimality, J. Amer. Math. Soc. 16 (2003), no. 4, 751–777 (electronic). MR 1992825 (2004g:14065) 16. S. Smale, A Vietoris mapping theorem for homotopy, Proc. Amer. Math. Soc. 8:3 (1957), 604–610. 17. E. H. Spanier and J. H. C. Whitehead, Duality in relative homotopy theory, Ann. of Math. (2) 67 (1958), 203–238. MR 0105105 (21 #3850) 18. E.H. Spanier, Algebraic topology, McGraw-Hill, New York, 1966. 19. L. van den Dries, Tame topology and o-minimal structures., LMS Lecture Notes, vol. 248, Cambridge University Press, 1998. 20. Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. MR 1404337 (97i:32008) 21. Lou van den Dries and Patrick Speissegger, The real field with convergent generalized power series, Trans. Amer. Math. Soc. 350 (1998), no. 11, 4377–4421. MR 1458313 (99a:03036) 22. , The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3) 81 (2000), no. 3, 513–565. MR 1781147 (2002k:03057) 20 SAUGATA BASU 23. O. Ya. Viro and D. B. Fuchs, Homology and cohomology, Topology. II, Encyclopaedia Math. Sci., vol. 24, Springer, Berlin, 2004, Translated from the Russian by C. J. Shaddock, pp. 95– 196. MR 2054457 24. George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York, 1978. MR 516508 (80b:55001) 25. A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), no. 4, 1051–1094. MR 1398816 (98j:03052) 26. , A theorem of the complement and some new o-minimal structures, Selecta Math. (N.S.) 5 (1999), no. 4, 397–421. MR 1740677 (2001c:03071) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A. E-mail address: [email protected] 1. Introduction 1.1. Combinatorial Complexity in O-minimal Geometry 1.2. Stable Homotopy Equivalence 1.3. Diagrams and Co-limits 1.4. Diagram Preserving Maps 1.5. Main Results 2. Background 2.1. Bounds on the Betti numbers of Admissible Sets 2.2. Homotopy types of the fibers of a semi-algebraic map 3. A Topological Comparison Theorem 3.1. Topological Preliminaries 3.2. Proofs of Theorems ?? and ?? 4. Proofs of the Main Theorems 4.1. Summary of the main ideas 4.2. Hardt's Triviality for Definable Sets 4.3. Definable Triangulations 4.4. Infinitesimal Thickenings of the Faces of a Simplex 4.5. Thickened Homotopy Co-limits References

0704.0296

Generalized Twistor Transform And Dualities, With A New Description of Particles With Spin, Beyond Free and Massless

USC-07/HEP-B3 hep-th/0704.0296 Generalized Twistor Transform And Dualities With A New Description of Particles With Spin Beyond Free and Massless1 Itzhak Bars and Bora Orcal Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089-0484, USA. Abstract A generalized twistor transform for spinning particles in 3+1 dimensions is constructed that beautifully unifies many types of spinning systems by mapping them to the same twistor , thus predicting an infinite set of duality relations among spinning systems with different Hamiltonians. Usual 1T-physics is not equipped to explain the duality relationships and unification between these systems. We use 2T-physics in 4+2 dimensions to uncover new properties of twistors, and expect that our approach will prove to be useful for practi- cal applications as well as for a deeper understanding of fundamental physics. Unexpected structures for a new description of spinning particles emerge. A unifying symmetry SU(2, 3) that includes conformal symmetry SU(2, 2) =SO(4, 2) in the massless case, turns out to be a fundamental property underlying the dualities of a large set of spinning systems, including those that occur in high spin theories. This may lead to new forms of string theory back- grounds as well as to new methods for studying various corners of M theory. In this paper we present the main concepts, and in a companion paper we give other details [1]. I. SPINNING PARTICLES IN 3+1 - BEYOND FREE AND MASSLESS The Penrose twistor transform [2]-[5] brings to the foreground the conformal symmetry SO(4, 2) in the dynamics of massless relativistic particles of any spin in 3 + 1 dimensions. The transform relates the phase space and spin degrees of freedom xµ, pµ, s µν to a twistor and reformulates the dynamics in terms of twistors instead of phase space. The 1 This work was partially supported by the US Department of Energy, grant number DE-FG03-84ER40168. http://arxiv.org/abs/0704.0296v2 twistor ZA is made up of a pair of SL(2, C) spinors µ α̇, λα, α, α̇ = 1, 2, and is regarded as the 4 components A = 1, 2, 3, 4 of the Weyl spinor of SO(4, 2) =SU(2, 2). The well known twistor transform for a spinning massless particle is [5] µα̇ = −i (x̄+ iȳ)α̇β λβ, λαλ̄β̇ = pαβ̇, (1.1) where (x̄+ iȳ) (xµ + iyµ) (σ̄µ) , and pαβ̇ = pµ (σµ)αβ̇ , while σµ = (1, ~σ) , σ̄µ = (−1, ~σ) are Pauli matrices. xµ + iyµ is a complexification of spacetime [2]. The helicity h of the particle is determined by p · y = h. The spin tensor is given by sµν = εµνρσyρpσ, and it leads to 1 sµνsµν = h 2. The Pauli-Lubanski vector is proportional to the momentum εµνρσs νρpσ = (y · p) pµ−p2yµ = hpµ, appropriate for a massless particle of helicity h. The reformulation of the dynamics in terms of twistors is manifestly SU(2, 2) covariant. It was believed that twistors and the SO(4, 2)=SU(2, 2) symmetry, interpreted as conformal symmetry, govern the dynamics of massless particles only, since the momentum pµ of the form pαβ̇ = λαλ̄β̇ automatically satisfies p µpµ = 0. However, recent work has shown that the same twistor ZA = that describes massless spinless particles (h = 0) also describes an assortment of other spinless particle dynamical systems [6][7]. These include massive and interacting particles. The mechanism that avoids pµpµ = 0 [6][7] is explained following Eq.(6.9) below. The list of systems includes the following examples worked out explicitly in previous publications and in unpublished notes. The massless relativistic particle in d = 4 flat Minkowski space. The massive relativistic particle in d = 4 flat Minkowski space. The nonrelativistic free massive particle in 3 space dimensions. The nonrelativistic hydrogen atom (i.e. 1/r potential) in 3 space dimensions. The harmonic oscillator in 2 space dimensions, with its mass ⇔ an extra dimension. The particle on AdS4, or on dS4. The particle on AdS3×S1 or on R× S3. The particle on AdS2×S2. The particle on the Robertson-Walker spacetime. The particle on any maximally symmetric space of positive or negative curvature. The particle on any of the above spaces modified by any conformal factor. A related family of other particle systems, including some black hole backgrounds. In this paper we will discuss these for the case of d = 4 with spin (h 6= 0). It must be emphasized that while the phase spaces (and therefore dynamics, Hamiltonian, etc.) in these systems are different, the twistors µα̇, λα are the same. For example, the massive particle phase space (xµ, pµ)massive and the one for the massless particle (x µ, pµ)massless are not the same (xµ, pµ) , rather they can be obtained from one another by a non-linear transformation for any value of the mass parameter m [6], and similarly, for all the other spaces mentioned above. However, under such “duality” transformations from one system to another, the twistors for all the cases are the same up to an overall phase transformation µα̇, λα massive µα̇, λα massless = · · · = µα̇, λα . (1.2) This unification also shows that all of these systems share the same SO(4, 2)=SU(2, 2) global symmetry of the twistors. This SU(2, 2) is interpreted as conformal symmetry for the massless particle phase space, but has other meanings as a hidden symmetry of all the other systems in their own phase spaces. Furthermore, in the quantum physical Hilbert space, the symmetry is realized in the same unitary representation of SU(2, 2) , with the same Casimir eigenvalues (see (7.16,7.17) below), for all the systems listed above. The underlying reason for such fantastic looking properties cannot be found in one-time physics (1T-physics) in 3+1 dimensions, but is explained in two-time physics (2T-physics) [8] as being due to a local Sp(2, R) symmetry. The Sp(2, R) symmetry which acts in phase space makes position and momentum indistinguishable at any instant and requires one extra space and one extra time dimensions to implement it, thus showing that the unification relies on an underlying spacetime in 4+2 dimensions. It was realized sometime ago that in 2T- physics twistors emerge as a gauge choice [9], while the other systems are also gauge choices of the same theory in 4+2 dimensions. The 4+2 phase space can be gauge fixed to many 3+1 phase spaces that are distinguishable from the point of view of 1T-physics, without any Kaluza-Klein remnants, and this accounts for the different Hamiltonians that have a duality relationship with one another. We will take advantage of the properties of 2T-physics to build the general twistor transform that relates these systems including spin. Given that the field theoretic formulation of 2T-physics in 4+2 dimensions yields the Standard Model of Particles and Forces in 3+1 dimensions as a gauge choice [10], including spacetime supersymmetry [11], and given that twistors have simplified QCD computations [12][13], we expect that our twistor methods will find useful applications. II. TWISTOR LAGRANGIAN The Penrose twistor description of massless spinning particles requires that the pairs µα̇, iλ̄α̇ or their complex conjugates (λα, iµ̄ α) be canonical conjugates and satisfy the he- licity constraint given by Z̄AZA = λ̄α̇µ α̇ + µ̄αλα = 2h. (2.1) Indeed, Eq.(1.1) satisfies this property provided y · p = h. Here we have defined the 4̄ of SU(2, 2) as the contravariant twistor Z̄A ≡ Z†η2,2 λ̄α̇ µ̄ , η2,2 = = SU (2, 2) metric. (2.2) The canonical structure, along with the constraint Z̄AZA = 2h follows from the following worldline action for twistors − 2hà , DτZ − iÃZA. (2.3) In the case of h = 0 it was shown that this action emerges as a gauge choice of a more general action in 2T-physics [6][7]. Later in the paper, in Eq.(4.1) we give the h 6= 0 2T-physics action from which (2.3) is derived as a gauge choice. The derivative part of this action gives the canonical structure S0 = dτiZ̄A λ̄α̇∂τµ α̇ + µ̄α∂τλα that requires µα̇, iλ̄α̇ or their complex conjugates (λα, iµ̄ α) to be canonical conjugates. The 1-form Ãdτ is a U(1) gauge field on the worldline, DτZ A is the U(1) gauge covariant derivative that satisfies δε for δεà = ∂ε/∂τ and δεZ A = iεZA. The term 2hà is gauge invariant since it transforms as a total derivative under the infinitesimal gauge transformation. 2hà was introduced in [6][7] as being an integral part of the twistor formulation of the spinning particle action. Our aim is to show that this action describes not only massless spinning particles, but also all of the other particle systems listed above with spin. This will be done by constructing the twistor transform from ZA to the phase space and spin degrees of freedom of these systems, and claiming the unification of dynamics via the generalized twistor transform. This generalizes the work of [6][7] which was done for the h = 0 case of the action in (2.3). We will use 2T-physics as a tool to construct the general twistor transform, so this unification is equivalent to the unification achieved in 2T-physics. III. MASSLESS PARTICLE WITH ANY SPIN IN 3+1 DIMENSIONS In our quest for the general twistor transform with spin, we first discuss an alternative to the well known twistor transform of Eq.(1.1). Instead of the yµ (τ) that appears in the complexified spacetime xµ + iyµ we introduce an SL(2, C) bosonic2 spinor vα̇ (τ) and its complex conjugate v̄α (τ) , and write the general vector yµ in the matrix form as yα̇β = hvα̇v̄β + ωpα̇β , where ω (τ) is an arbitrary gauge freedom that drops out. Then the helicity condition y · p = h takes the form v̄pv = 1. Furthermore, we can write λα = pαβ̇vβ̇ since this automatically satisfies λαλ̄β̇ = pαβ̇ when p 2 = (v̄pv − 1) = 0 are true. With this choice of variables, the Penrose transform of Eq.(1.1) takes the new form λα = (pv)α , µ α̇ = [(−ix̄p+ h) v]α̇ , p2 = (v̄pv − 1) = 0, (3.1) where the last equation is a set of constraints on the degrees of freedom xµ, pµ, v α̇, v̄α. If we insert the twistor transform (3.1) into the action (2.3), the twistor action turns into the action for the phase space and spin degrees of freedom xµ, pµ, v α̇, v̄α ẋµpµ − p2 + ih (v̄p)Dτv −Dτv (pv) − 2hà . (3.2) where Dτv = v̇ − iÃv is the U(1) gauge covariant derivative and we have included the Lagrange multiplier e to impose p2 = 0 when we don’t refer to twistors. The equation of motion for à imposes the second constraint v̄pv−1 = 0 that implies U(1) gauge invariance3. From the global Lorentz symmetry of (3.2), the Lorentz generator is computed via Noether’s theorem Jµν = xµpν − xνpµ + sµν , with sµν = i hv̄ (pσ̄µν + σµνp) v. The helic- 2 This is similar to the fermionic case in [5]. The bosonic spinor v can describe any spin h. 3 If this action is taken without the U(1) constraint à = 0 , then the excitations in the v sector describe an infinite tower of massless states with all helicities from zero to infinity (here we rescale 2hv → v) Sall spins = ẋµpµ − v̄pv̇ − v̇pv (3.3) The spectrum coincides with the spectrum of the infinite slope limit of string theory with all helicities v̄pv. This action has a hidden SU(2, 3) symmetry that includes SU(2, 2) conformal symmetry. This is explained in the rest of the paper by the fact that this action is a gauge fixed version of a 2T-physics master action (4.1,5.4) in 4+2 dimensions with manifest SU(2, 3) symmetry. A related approach has been pursued also in [15]-[18] in 3+1 dimensions in the context of only massless particles. Along with the manifestly SU(2, 3) symmetric 2T-physics actions, we are proposing here a unified 2T-physics setting for discussing high spin theories [14] including all the dual versions of the high spin theories related to the spinning physical systems listed in section (I). ity is determined by computing the Pauli-Lubanski vector W µ = 1 εµνλσsνλpσ = (hv̄pv) p The helicity operator hv̄pv reduces to the constant h in the U(1) gauge invariant sector. The action (3.2) gives a description of a massless particle with any helicity h in terms of the SL(2, C) bosonic spinors v, v̄. We note its similarity to the standard superparticle action [20][21] written in the first order formalism. The difference with the superparticle is that the fermionic spacetime spinor θα̇ of the superparticle is replaced with the bosonic spacetime spinor vα̇, and the gauge field à imposes the U(1) gauge symmetry constraint v̄pv − 1 = 0 that restricts the system to a single, but arbitrary helicity state given by h. Just like the superparticle case, our action has a local kappa symmetry with a bosonic local spinor parameter κα (τ), namely α̇ = p̄α̇βκβ, δκxµ = ((δκv̄)σµv − v̄σµ (δκv)) , (3.4) µ = 0, δκe = −ih κ̄ (Dτv)− , δκà = 0. (3.5) These kappa transformations mix the phase space degrees of freedom (x, p) with the spin degrees of freedom v, v̄. The transformations δκxµ, δκe are non-linear. Let us count physical degrees of freedom. By using the kappa and the τ -reparametrization symmetries one can choose the lightcone gauge. From phase space xµ, pµ there remains 3 positions and 3 momentum degrees of freedom. One of the two complex components of vα̇ is set to zero by using the kappa symmetry, so vα̇ = . The phase of the remaining component is eliminated by choosing the U(1) gauge, and finally its magnitude is fixed by solving the constraint v̄pv − 1 = 0 to obtain vα̇ = (p+)−1/2 . Therefore, there are no independent physical degrees of freedom in v. The remaining degrees of freedom for the particle of any spin are just the three positions and momenta, and the constant h that appears in sµν . This is as it should be, as seen also by counting the physical degrees of freedom from the twistor point of view. When we consider the other systems listed in the first section, we should expect that they too are described by the same number of degrees of freedom since they will be obtained from the same twistor, although they obey different dynamics (different Hamiltonians) in their respective phase spaces. The lightcone quantization of the the massless particle systems described by the actions (3.2,3.3) is performed after identifying the physical degrees of freedom as discussed above. The lightcone quantum spectrum and wavefunction are the expected ones for spinning mass- less particles, and agree with their covariant quantization given in [15]-[19]. IV. 2T-PHYSICS WITH SP(2, R) , SU(2, 3) AND KAPPA SYMMETRIES The similarity of (3.2) to the action of the superparticle provides the hint for how to lift it to the 2T-physics formalism, as was done for the superparticle [22][9] and the twistor superstring [23][24]. This requires lifting 3+1 phase space (xµ, pµ) to 4+2 phase space XM , PM and lifting the SL(2, C) spinors v, v̄ to the SU(2, 2) spinors VA, V̄ A. The larger set of degrees of freedom XM , PM , VA, V̄ A that are covariant under the global symmetry SU(2, 2) =SO(4, 2) , include gauge degrees of freedom, and are subject to gauge symmetries and constraints that follow from them as described below. The point is that the SU(2, 2) invariant constraints on XM , PM , VA, V̄ A have a wider set of solutions than just the 3+1 system of Eq.(3.2) we started from. This is because 3+1 di- mensional spin & phase space has many different embeddings in 4+2 dimensions, and those are distinguishable from the point of view of 1T-physics because target space “time” and corresponding “Hamiltonian” are different in different embeddings, thus producing the dif- ferent dynamical systems listed in section (I). The various 1T-physics solutions are reached by simply making gauge choices. One of the gauge choices for the action we give below in Eq.(4.1) is the twistor action of Eq.(2.3). Another gauge choice is the 4+2 spin & phase space action in terms of the lifted spin & phase space XM , PM , VA, V̄ A as given in Eq.(5.4). The latter can be further gauge fixed to produce all of the systems listed in section (I) including the action (3.2) for the massless spinning particle with any spin. All solutions still remember that there is a hidden global symmetry SU(2, 2) =SO(4, 2) , so all systems listed in section (I) are realizations of the same unitary representation of SU(2, 2) whose Casimir eigenvalues will be given below. For the 4 + 2 version of the superparticle [22] that is similar to the action in (5.4), this program was taken to a higher level in [9] by embedding the fermionic supercoordinates in the coset of the supergroup SU(2, 2|1) /SU(2, 2)×U(1). We will follow the same route here, and embed the bosonic SU(2, 2) spinors VA, V̄ A in the left coset SU(2, 3) /SU(2, 2)×U(1) . This coset will be regarded as the gauging of the group SU(2, 3) under the subgroup [SU (2, 2)× U (1)] from the left side. Thus the most powerful version of the action that reveals the global and gauge symmetries is obtained when it is organized in terms of the XMi (τ), g (τ) and à (τ) degrees of freedom described as 4+2 phase space XM (τ) PM (τ) ≡ XMi (τ) , i = 1, 2, doublets of Sp (2, R) gauge symmetry, group element g (τ) ⊂ SU (2, 3) subject to [SU (2, 2)× U(1) ]L × U(1)L+R gauge symmetry. We should mention that the h = 0 version of this theory, and the corresponding twistor prop- erty, was discussed in [6], by taking g (τ) ⊂SU(2, 2) and dropping all of the U(1)’s. So, the generalized theory that includes spin has the new features that involves SU(2, 2) →SU(2, 3) and the U(1) structures. The action has the following form XNj ηMN + Tr (iDτg) g − 2hà , (4.1) where εij = is the antisymmetric Sp(2, R) metric, and DτX i = ∂τX the Sp(2, R) gauge covariant derivative, with the 3 gauge potentials Aij = εikA For SU(2, 3) the group element is pseudo-unitary, g−1 = (η2,3) g † (η2,3) , where η2,3 is the SU(2, 3) metric η2,3 = . The covariant derivative Dτg is given by Dτg = ∂τg − ià [q, g] , q = 14×4 0  (4.2) where the generator of U(1)L+R is proportional to the 5×5 traceless matrix q ∈ u(1) ∈ su(2, 3)L+R . The last term of the action −2hÃ, which is also the last term of the action (2.3), is invariant under the U(1)L+R since it transforms to a total derivative. Finally, the 4× 4 traceless matrix (L) BA ∈su(2, 2) ∈su(2, 3) that appears on the left side of g (or right side of g−1) is (L) BA ≡ LMN , LMN = εijXMi X j = X MPN −XNPM . (4.3) where ΓMN = ΓM Γ̄N − ΓN Γ̄M are the 4×4 gamma-matrix representation of the 15 gen- erators of SU(2, 2). A detailed description of these gamma matrices is given in [11]. The symmetries of actions of this type for any group or supergroup g were discussed in [9][23][24][7]. The only modification of that discussion here is due to the inclusion of the U(1) gauge field Ã. In the absence of the à coupling the global symmetry is given by the transformation of g (τ) from the right side g (τ) → g (τ) gR where gR ⊂SU(2, 3)R. However, in our case, the presence of the coupling with the U(1)L+R charge q breaks the global symmetry down to the (SU(2, 2)×U(1))R subgroup that acts on the right side of g. So the global symmetry is given by global: g (τ) → g (τ) hR, hR ∈ [SU(2, 2)× U(1)]R ⊂ SU(2, 3)R. (4.4) Using Noether’s theorem we deduce the conserved global charges as the [SU(2, 2)×U(1)]R components of the the following SU(2, 3)R Lie algebra valued matrix J(2,3) J(2,3) = g , J2,3 = η2,3 (J2,3) (η2,3) , (4.5) The traceless 4× 4 matrix (J ) BA = ΓMNJMN is the conserved SU(2, 2) =SO(4, 2) charge and J0 is the conserved U(1) charge. Namely, by using the equations of motion one can verify ∂τ (J ) BA = 0 and ∂τJ0 = 0. The spinor charges jA, j̄A are not conserved4 due to the coupling of Ã. As we will find out later in Eq.(6.8), jA is proportional to the twistor J0ZA, (4.6) up to an irrelevant gauge transformation. It is important to note that J and J0 are invariant on shell under the gauge symmetries discussed below. Therefore they generate physical symmetries [SU(2, 2)×U(1)]R under which all gauge invariant physical states are classified. The local symmetries of this action are summarized as Sp (2, R)× SU (2, 2) 3 kappa kappa U (1) (4.7) The Sp(2, R) is manifest in (4.1). The rest corresponds to making local SU(2, 3) transfor- mations on g (τ) from the left side g (τ) → gL (τ) g (τ) , as well as transforming XMi = XM , PM as vectors with the local subgroup SU(2, 2)L =SO(4, 2) , and A ij under the kappa. The 3/4 kappa symmetry which is harder to see will be discussed in more detail below. These symmetries coincide with those given in previous discussions in [9][23][24][7] despite the presence of Ã. The reason is that the U(1)L+R covariant derivative Dτg in Eq.(4.2) can be replaced by a purely U(1)R covariant derivative Dτg = ∂τg + igqà because the difference drops out in the trace in the action (4.1). Hence the symmetries on left side of g (τ) → gL (τ) g (τ) remain the same despite the coupling of Ã. We outline the roles of each of these local symmetries. The Sp(2, R) gauge symmetry can reduce XM , PM to any of the phase spaces in 3+1 dimensions listed in section (I). This 4 In the high spin version of (4.1) with à = 0, the global symmetry is SU(2, 3)R and jA, j̄ A are conserved. is the same as the h = 0 case discussed in [6]. The [SU(2, 2)×U(1)]L gauge symmetry can reduce g (τ) ⊂SU(2, 3) to the coset g → t (V ) ∈SU(2, 3) /[SU(2, 2)×U(1)]L parameterized by the SU(2, 2)×U(1) spinors VA, V̄ as shown in Eq.(5.3). The remaining 3/4 kappa symmetry, whose action is shown in Eq.(5.15), can remove up to 3 out of the 4 parameters in the VA. The U(1)L+R symmetry can eliminate the phase of the remaining component in V . Finally the constraint due to the equation of motion of à fixes the magnitude of V . In terms of counting, there remains only 3 position and 3 momentum physical degrees of freedom, plus the constant h, in agreement with the counting of physical degrees of freedom of the twistors. It is possible to gauge fix the symmetries (4.7) partially to exhibit some intermediate covariant forms. For example, to reach the SL(2, C) covariant massless particle described by the action (3.2) from the 2T-physics action above, we take the massless particle gauge by using two out of the three Sp(2, R) gauge parameters to rotate the M = +′ doublet to the (τ) = , and solving explicitly two of the Sp(2, R) constraints X2 = X ·P = 0 XM = ( xµ (τ)), PM = ( x · p , pµ (τ)). (4.8) This is the same as the h = 0 massless case in [6]. There is a tau reparametrization gauge symmetry as a remnant of Sp(2, R) . Next, the [SU(2, 2)×U(1)]L gauge symmetry reduces g (τ) → t (V ) written in terms of VA, V̄ as given in Eq.(5.3), and the 3/4 kappa symmetry reduces the SU(2, 2) spinor VA → to the two components SL(2, C) doublet vα̇, with a leftover kappa symmetry as discussed in Eqs.(3.4-3.5). The gauge fixed form of g is then g = exp 2hvα̇ 0 0 0 2hv̄α 0 1 hvα̇v̄β 2hvα̇ 0 1 0 2hv̄α 1 ∈ SU (2, 3) . (4.9) The inverse g−1 = (η2,3) g † (η2,3) is given by replacing v, v̄ by (−v) , (−v̄) . Inserting the gauge fixed forms of X,P, g (4.8,4.9) into the action (4.1) reduces it to the massless spin- ning particle action (3.2). Furthermore, inserting these X,P, g into the expression for the current in (4.5) gives the conserved SU(2, 2) charges J (see Eqs.(5.9,5.20)) which have the significance of the hidden conformal symmetry of the gauge fixed action (3.2). This hidden symmetry is far from obvious in the form (3.2), but it is straightforward to derive from the 2T-physics action as we have just outlined. Partial or full gauge fixings of (4.1) similar to (4.8,4.9) produce the actions, the hidden SU(2, 2) symmetry, and the twistor transforms with spin of all the systems listed in section (I). These were discussed for h = 0 in [6], and we have now shown how they generalize to any spin h 6= 0, with further details below. It is revealing, for example, to realize that the massive spinning particle has a hidden SU(2, 2) “mass-deformed conformal symmetry”, including spin, not known before, and that its action can be reached by gauge fixing the action (4.1), or by a twistor transform from (2.3). The same remarks applied to all the other systems listed in section (I) are equally revealing. For more information see our related paper Through the gauge (4.8,4.9), the twistor transform (3.1), and the massless particle action (3.2), we have constructed a bridge between the manifestly SU(2, 2) invariant twistor action (2.3) for any spin and the 2T-physics action (4.1) for any spin. This bridge will be made much more transparent in the following sections by building the general twistor transform. V. 2T-PHYSICS ACTION WITH XM , PM , VA, V̄ A IN 4+2 DIMENSIONS We have hinted above that there is an intimate relation between the 2T-physics action (4.1) and the twistor action (2.3). In fact the twistor action is just a gauged fixed version of the more general 2T-physics action (4.1). Using the local SU(2, 2) =SO(4, 2) and local Sp(2, R) symmetries of the general action (4.1) we can rotateXM (τ) , PM (τ) to the following form that also solves the Sp(2, R) constraints Xi ·Xj = X2 = P 2 = X · P = 0 [6][7] XM = ( 0), PM = ( 0). (5.1) This completely eliminates all phase space degrees of freedom. We are left with the gauge fixed action Sh = (Dτg) g − 2hà , where (iL) → 1 ′−′, and ′−′ = 1. Due to the many zero entries in the 4×4 matrix Γ−′− [6], only one column from g in the form and one row from g−1 in the form Z̄A,−Z̄5 can contribute in the trace, and therefore the action becomes Sh = iZ̄AŻA − iZ̄5Ż5 + à Z̄5Z5 − 2h Here Z̄5Ż5 drops out as a total derivative since the magnitude of the complex number Z5 is a constant Z̄5Z5 = 2h. Furthermore, we must take into account Z̄ AZA − Z̄5Z5 = 0 which is an off-diagonal entry in the matrix equation g−1g = 1. Then we see that the 2T-physics action (4.1) reduces to the twistor action (2.3) with the gauge choice (5.1)5. Next let us gauge fix the 2T-physics action (4.1) to a manifestly SU(2, 2) =SO(4, 2) invariant version in flat 4+2 dimensions, in terms of the phase space & spin degrees of freedom XM , PM , VA, V̄ A. For this we use the [SU(2, 2)×U(1)]left symmetry to gauge fix g gauge fix: g → t (V ) ∈ SU(2, 3) [SU(2, 2)× U(1)]left (5.2) The coset element t (V ) is parameterized by the SU(2, 2) spinor V and its conjugate V̄ = V †η2,2 and given by the 5×5 SU(2, 3) matrix6 t (V ) = 1− 2hV V̄ )−1/2 1− 2hV̄ V )−1/2 2hV̄ 1  . (5.3) The factor 2h is inserted for a convenient normalization of V. Note that the first matrix commutes with the second one, so it can be written in either order. The inverse of the group element is t−1 (V ) = (η2,3) t † (η2,3) = t (−V ) , as can be checked explicitly t (V ) t (−V ) = 1. Inserting this gauge in (4.1) the action becomes Ẋ · P − AijXi ·Xj − ΩMNLMN − 2hà V̄ LV 1− 2hV̄ V (5.4) XNj ηMN − 2hà V̄ LV 1− 2hV̄ V (5.5) where i = ∂τX j − ΩMNXiN (5.6) is a covariant derivative for local Sp(2, R) as well as local SU(2, 2) =SO(4, 2) but with a composite SO(4, 2) connection ΩMN (V (τ)) given conveniently in the following forms ΩMNΓMN = (i∂τ t) t SU(2,2) ΩMNLMN = −Tr (i∂τ t) t . (5.7) Thus, Ω is the SU(2, 2) projection of the SU(2, 3) Cartan connection and given explicitly as ΩMNΓMN = 2h V̇ − V V̄ V̇ V̄ − V V̇ − V̄ V̇ V 1− 2hV̄ V 1− 2hV̄ V ) + h V̄ V̇ − V̇ V 1− 2hV̄ V ) (5.8) 5 In the high spin version of (4.1) without à (see footnote (3)), we replace Z5 = e Z̄AZA and after dropping a total derivative, the twistor equivalent becomes Sall spins = iZ̄AŻA + Z̄Zφ̇ . For a more covariant version that displays the SU(2, 3) global symmetry, we introduce a new U(1) gauge field for the overall phase of and write Sall spins = iZ̄AŻA − iZ̄5Ż5 + B̃ Z̄AZA − Z̄5Z5 6 Arbitrary fractional powers of the matrix 1− 2hV V̄ are easily computed by expanding in a series and then resuming to obtain 1− 2hV V̄ = 1+ V V̄ 1− 2hV̄ V )γ − 1 /V̄ V. The action (5.4,5.5) is manifestly invariant under global SU(2, 2) =SO(4, 2) rotations, and under local U (1) phase transformations applied on VA, V̄ A. The conserved global symmetry currents J and J0 can be derived either directly from (5.4) by using Noether’s theorem, or by inserting the gauge fixed form of g → t (V ) into Eq.(4.5)7 J(2,3) = t−1 1− 2hV V̄ 1− 2hV V̄ J0, J0 = 2hV̄ LV 1− 2hV̄ V (5.9) 1− 2hV V̄ LV 1√ 1− 2hV̄ V (5.10) According to the equation of motion for à that follows from the action (5.4) we must have the following constraint (this means U(1) gauge invariant physical sector) V̄ LV 1− 2hV̄ V = 1. (5.11) Therefore, in the physical sector the conserved [SU(2, 2)×U(1)]right charges take the form physical sector: J0 = 2h, J = 1− 2hV V̄ 1− 2hV V̄ . (5.12) Let us now explain the local kappa symmetry of the action (5.4,5.5). The action (5.4) is still invariant under the bosonic local 3/4 kappa symmetry inherited from the action (4.1). The kappa transformations of g (τ) in the general action (5.4) correspond to local coset elements exp ∈SU(2, 3)left/[SU(2, 2)×U(1)]left with a special form of the spinor KA KA = Xi · Γκi (τ) , (5.13) with κiA (τ) two arbitrary local spinors8. Now that g has been gauge fixed g → t (V ), the kappa transformation must be taken as the naive kappa transformation on g followed by a [SU(2, 2)×U(1)]left gauge transformation which restores the gauge fixed form of t (V ) t (V ) → t (V ′) = Tr (ω) t (V ) (5.14) The SU(2, 2) part of the restoring gauge transformation must also be applied on XM , PM . Performing these steps we find the infinitesimal version of this transformation [22] δκV = 1− 2hV V̄ 1− 2hV̄ V , δκX i = ω MNXiN , δκA ij = see below, (5.15) 7 In the high spin version (à = 0) the conserved charges include jA as part of SU(2, 3)R global symmetry. It is then also convenient to rescale 2hV → V in Eqs.(5.3-5.10) to eliminate an irrelevant constant. 8 In this special form only 3 out of the 4 components of KA are effectively independent gauge parameters. This can be seen easily in the special frame for XM , PM given in Eq.(5.1). where ωMN (K, V ) has the same form as ΩMN in Eq.(5.8) but with V̇ replaced by the δκV given above. The covariant derivative D̂τX i in Eq.(5.6) is covariant under the local SU(2, 2) transformation with parameter ωMN (K, V ) (this is best seen from the projected Cartan connection form Ω = [(i∂τ t) t −1]SU(2,2)). Therefore, the kappa transformations (5.15) inserted in (5.5) give δκSh = Xi ·Xj + iT r (Dτ t) t . (5.16) In computing the second term the derivative terms that contain ∂τK have dropped out in the trace. Using Eq.(5.13) we see that LK = 1 εliXMl X XLj ΓMNΓLκ j (5.17) εliXMl X j (ΓMNL + ηNLΓM − ηMLΓN )κl (5.18) εliXi ·Xj Xl · Γκj (5.19) The completely antisymmetric XMi X l ΓMNL term in the second line vanishes since i, j, l can only take two values. The crucial observation is that the remaining term in LK is proportional to the dot products Xi ·Xj. Therefore the second term in (5.16) is cancelled by the first term by choosing the appropriate δκA ij in Eq.(5.16), thus establishing the kappa symmetry. The local kappa transformations (5.15) are also a symmetry of the global SU(2, 3)R charges δκJ = δκJ0 = δκjA = 0 provided the constraints Xi · Xj = 0 are used. Hence these charges are kappa invariant in the physical sector. We have established the global SO(4, 2) and local Sp(2, R)× (3/4 kappa)×U(1) symme- tries of the phase space action (5.4) in 4+2 dimensions. From it we can derive all of the phase space actions of the systems listed in section (I) by making various gauge choices for the local Sp(2, R)× (3/4 kappa)×U(1) symmetries. This was demonstrated for the spinless case h = 0 in [6]. The gauge choices for XM , PM discussed in [6] now need to be supplemented with gauge choices for VA, V̄ A by using the kappa×U(1) local symmetries. Here we demonstrate the gauge fixing described above for the massless particle of any spin h. The kappa symmetry effectively has 3 complex gauge parameters as explained in footnote (8). If the kappa gauge is fixed by using two of its parameters we reach the following forms , V̄ A → (0 v̄α) , V̄ V → 0, 1− 2hV V̄ )−1/2 → . (5.20) By inserting this gauge fixed form of V, and the gauge fixed form of X,P given in Eq.(4.8), into the action (5.4) we immediately recover the SL(2, C) covariant action of Eq.(3.2). The U(1) gauge symmetry is intact. The kappa symmetry of the action of Eq.(3.2) discussed in Eqs.(3.4,3.5) is the residual 1/4 kappa symmetry of the more general action ((5.4). For other examples of gauge fixing that generates some of the systems in the list of section (I) see our related paper [1]. VI. GENERAL TWISTOR TRANSFORM (CLASSICAL) The various formulations of spinning particles described above all contain gauge degrees of freedom of various kinds. However, they all have the global symmetry SU(2, 2)=SO(4, 2) whose conserved charges J BA are gauge invariant in all the formulations. The most sym- metric 2T-physics version gave the J BA as embedded in SU(2, 3)R in the SU(2, 2) projected form in Eq.(4.5) SU(2,2) . (6.1) Since this is gauge invariant, when gauge fixed, it must agree with the Noether charges computed in any version of the theory. So we can equate the general phase space version of Eq.(5.9) with the twistor version that follows from the Noether currents of (2.3) as follows J = Z(h)Z̄(h) − 1 Z(h)Z̄(h) 1− 2hV V̄ 1− 2hV V̄ J0 (6.2) The trace corresponds to the U(1) charge J0 = Tr Z(h)Z̄(h) J0 = Z (h)Z̄(h) = 1− 2hV V̄ 1− 2hV V̄ . (6.3) In the case of h = 0 this becomes Z(0)Z̄(0) = L. (6.4) Therefore the equality (6.3) is solved up to an irrelevant phase by Z(h) = 1− 2hV V̄ Z(0). (6.5) By inserting (6.4) into the constraint (5.11) we learn a new form of the constraints V̄ Z(0) = 1− 2hV̄ V , V̄ Z(h) = 1. (6.6) In turn, this implies Z(0) = 1− 2hV̄ V (6.7) which is consistent9 with Z(0)Z̄(0) = L , and its vanishing trace Z̄(0)Z(0) = 0 since LL = 0 (due to X2 = P 2 = X · P = 0). Putting it all together we then have Z(h) = 1− 2hV V̄ 1− 2hV̄ V V. (6.8) We note that this Z(h) is proportional to the non-conserved coset part of the SU(2, 3) charges J2,3, that is jA = (h) given in Eqs.(4.5,4.6) or (5.10), when g and L are replaced by their gauge fixed forms, and use the constraint10 J0 = 2h. The key for the general twistor transform for any spin is Eq.(6.5), or equivalently (6.8). The general twistor transform between Z(0) and XM , PM which satisfies Z(0)Z̄(0) = L is already given in [6] as Z(0) = = −i X , λ(0)α λ̄ X+P µ −XµP+ (σµ)αβ̇ . (6.9) Note that (X+P µ −XµP+) is compatible with the requirement that any SL(2, C) vector constructed as λ must be lightlike. This property is satisfied thanks to the Sp(2, R) constraints X2 = P 2 = X · P = 0 in 4+2 dimensions, thus allowing a particle of any mass in the 3 + 1 subspace (since P µPµ is not restricted to be lightlike). Besides satisfying Z(0)Z̄(0) = L, this Z(0) also satisfies Z̄(0)Z(0) = 0, as well as the canonical properties of twistors. Namely, Z(0) has the property [6] dτ Z̄(0)∂τZ (0) = dτ ẊMPM . (6.10) From here, by gauge fixing the Sp(2, R) gauge symmetry, we obtain the twistor transforms for all the systems listed in section (I) for h = 0 directly from Eq.(6.9), as demonstrated in [6]. All of that is now generalized at once to any spin h through Eq.(6.5). Hence (6.5) together with (6.9) tell us how to construct explicitly the general twistor Z A in terms 9 To see this, we note that Eqs.(6.4,6.6) lead to LV V̄ L 1−2hV̄ V = Z(0)Z̄(0)V V̄ Z(0)Z̄(0) 1−2hV̄ V = Z (0)Z̄(0) = L. 10 For the high spin version (à = 0) we don’t use the constraint. Instead, we use Z(h) = 1√ 1−2hV V̄ Z(0) only in its form (6.5), and note that, after using Eq.(6.4), the jA in Eq.(5.10) takes the form jA = 1−2hV̄ V , and it is possible to rescale h away everywhere 2hV → V. of spin & phase space degrees of freedom XM , PM , VA, V̄ A. Then the Sp(2, R) and kappa gauge symmetries that act on XM , PM , VA, V̄ A can be gauge fixed for any spin h, to give the specific twistor transform for any of the systems under consideration. We have already seen in Eq.(6.2) that the twistor transform (6.5) relates the conserved SU(2, 2) charges in twistor and phase space versions. Let us now verify that (6.5) provides the transformation between the twistor action (2.3) and the spin & phase space action (5.4). We compute the canonical structure as follows dτ Z̄(h)∂τZ (h) = dτ Z̄(0) 1− 2hV V̄ 1− 2hV V̄ (6.11) Z̄(0) 1√ 1−2hV V̄ 1−2hV V̄ +Z̄(0) 1 1−2hV V̄ ∂τZ (6.12) Ẋ · P + Tr (i∂τ t) t (6.13) The last form is the canonical structure of spin & phase space as given in (5.4). To prove this result we used Eq.(6.10), footnote (6), and the other properties of Z(0) including Eqs.(6.4- 6.7), as well as the constraints X2 = P 2 = X · P = 0, and dropped some total derivatives. This proves that the canonical properties of Z(h) determine the canonical properties of spin & phase space degrees of freedom and vice versa. Then, including the terms that impose the constraints, the twistor action (2.3) and the phase space action (5.4) are equivalent. Of course, this is expected since they are both gauge fixed versions of the master action (4.1), but is useful to establish it also directly via the general twistor transform given in Eq.(6.5). VII. QUANTUM MASTER EQUATION, SPECTRUM, AND DUALITIES In this section we derive the quantum algebra of the gauge invariant observables J BA and J0 which are the conserved charges of [SU(2, 2)×U(1)]R. Since these are gauge invariant symmetry currents they govern the system in any of its gauge fixed versions, including in any of its versions listed in section (I). From the quantum algebra we deduce the constraints among the physical observables J BA ,J0 and quantize the theory covariantly. Among other things, we compute the Casimir eigenvalues of the unitary irreducible representation of SU(2, 2) which classifies the physical states in any of the gauge fixed version of the theory (with the different 1T-physics interpretations listed in section (I)). The simplest way to quantize the theory is to use the twistor variables, and from them compute the gauge invariant properties that apply in any gauge fixed version. We will apply the covariant quantization approach, which means that the constraint due to the U(1) gauge symmetry will be applied on states. Since the quantum variables will generally not satisfy the constraints, we will call the quantum twistors in this section ZA, Z̄ A to distinguish them from the classical Z A , Z̄ (h)A of the previous sections that were constrained at the classical level. So the formalism in this section can also be applied to the high spin theories (discussed in several footnotes up to this point in the paper) by ignoring the constraint on the states. According to the twistor action (2.3) ZA and iZ̄ A (or equivalently λα and iµ̄ α) are canon- ical conjugates. Therefore the quantum rules (equivalent to spin & phase space quantum rules) are ZA, Z̄ = δ BA . (7.1) These quantum rules, as well as the action, are manifestly invariant under SU(2, 2) . In covariant SU(2, 2) quantization the Hilbert space contains states which do not obey the U(1) constraint on the twistors. At the classical level the constraint was J0 = Z̄Z = 2h, but in covariant quantization this is obeyed only by the U(1) gauge invariant subspace of the Hilbert space which we call the physical states. The quantum version of the constraint requires Ĵ0 as a Hermitian operator applied on states (we write it as Ĵ0 to distinguish it from the classical version) Ĵ0 = A + Z̄AZA , Ĵ0|phys〉 = 2h|phys〉. (7.2) The operator Ĵ0 has non-trivial commutation relations with ZA, Z̄ A which follow from the basic commutation rules above Ĵ0, ZA = −ZA, Ĵ0, Z̄ = Z̄ A. (7.3) By rearranging the orders of the quantum operators ZAZ̄ A = Z̄AZA+4 we can extract from (7.2) the following relations Z̄Z = Ĵ0 − 2, T r = Ĵ0 + 2. (7.4) Furthermore, by using Noether’s theorem for the twistor action (2.3) we can derive the 15 generators of SU(2, 2) in terms of the twistors and write them as a traceless 4 × 4 matrix J BA at the quantum level as follows J BA = ZAZ̄B − δ BA = ZZ̄ − Ĵ0 + 2 . (7.5) In this expression the order of the quantum operators matters and gives rise to the shift J0 → Ĵ0 + 2 in contrast to the corresponding classical expression. The commutation rules among the generators J BA and the ZA, Z̄A are computed from the basic commutators (7.1), J BA , ZC = −δ BC ZA + J BA , Z̄D = δ DA Z̄ B − 1 Z̄D δ BA (7.6) J BA ,J DC = δ DA J BC − δ BC J DA , Ĵ0,J BA = 0. (7.7) We see from these that the gauge invariant observables J BA satisfy the SU(2, 2) Lie algebra, while the ZA, Z̄ A transform like the quartets 4, 4̄ of SU(2, 2) . Note that the operator Ĵ0 commutes with the generators J BA , therefore J BA is U(1) gauge invariant, and furthermore Ĵ0 must be a function of the Casimir operators of SU(2, 2) . When Ĵ0 takes the value 2h on physical states, then the Casimir operators also will have eigenvalues on physical states which determine the SU(2, 2) representation in the physical sector. From the quantum rules (7.3), it is evident that the U(1) generator Ĵ0 can only have integer eigenvalues since it acts like a number of operator. More directly, through Eq.(7.4) it is related to the number operator Z̄Z. Therefore the theory is consistent at the quantum level (7.2) provided 2h is an integer. Let us now compute the square of the matrix J BA . By using the form (7.5) we have (JJ ) = ZZ̄ − Ĵ0+2 ZZ̄ − Ĵ0+2 = ZZ̄ZZ̄ − 2 Ĵ0+2 ZZ̄ + Ĵ0+2 where we have used Ĵ0, ZAZ̄ = 0. Now we elaborate ZZ̄ZZ̄ Ĵ0 − 2 Z̄B = Ĵ0 − 1 B where we first used (7.4) and then (7.3). Finally we note from (7.5) that ZAZ̄ B = J BA + Ĵ0+24 δ Putting these observations together we can rewrite the right hand side of (JJ ) in terms of J and Ĵ0 as follows11 (JJ ) = Ĵ20 − 4 . (7.8) 11 A similar structure at the classical level can be easily computed by squaring the expression for J in Eq.(6.2) and applying the classical constraint J0 = Z̄ AZA = 2h. This yields the classical version J CA J BC = J BA + 316J A = hJ BA + 34h 2δ BA , which is different than the quantum equation (7.8). Thus, the quadratic Casimir at the classical level is computed as C2 = J20 = 3h 2 which is different than the quantum value in (7.16). This equation is a constraint satisfied by the global [SU(2, 2)×U(1)]R charges J BA , Ĵ0 which are gauge invariant physical observables. It is a correct equation for all the states in the theory, including those that do not satisfy the U(1) constraint (7.2). We call this the quantum master equation because it will determine completely all the SU(2, 2) properties of the physical states for all the systems listed in section (I) for any spin. By multiplying the master equation with J and using (7.8) again we can compute JJJ . Using this process repeatedly we find all the powers of the matrix J (J )n = αnJ + βn, (7.9) where αn(Ĵ0) = Ĵ0 − 1 Ĵ0 − 2 Ĵ0 + 2 , (7.10) βn(Ĵ0) = Ĵ20 − 4 αn−1(Ĵ0). (7.11) Remarkably, these formulae apply to all powers, including negative powers of the matrix J . Using this result, any function of the matrix J constructed as a Taylor series takes the form f (J ) = α (7.12) where Ĵ0 − 1 Ĵ0 − 2 Ĵ0 + 2 , (7.13) Ĵ0 − 1 Ĵ0 + 2 Ĵ0 − 2 Ĵ0 − 2 Ĵ0 + 2  . (7.14) We can compute all the Casimir operators by taking the trace of J n in Eq.(7.9), so we find12 Cn(Ĵ0) ≡ Tr (J )n = 4βn(Ĵ0) = Ĵ20 − 4 αn−1(Ĵ0). (7.15) In particular the quadratic, cubic and quartic Casimir operators of SU(2, 2) =SO(6, 2) are computed at the quantum level as C2(Ĵ0) = Ĵ20 − 4 , C3(Ĵ0) = Ĵ20 − 4 Ĵ0 − 4 , (7.16) C4(Ĵ0) = Ĵ20 − 4 7Ĵ20 − 32Ĵ0 + 52 . (7.17) 12 Other definitions of Cncould differ from ours by normalization or linear combinations of the Tr (J n). The eigenvalue of the operator Ĵ0 on physical states Ĵ0|phys〉 = 2h|phys〉 completely fixes the unitary SU(2, 2) representation that classifies the physical states, since the most general representation of SO(4, 2) is labeled by the three independent eigenvalues of C2, C3 and C4. Obviously, this result is a special representation of SU(2, 2) since all the Casimir eigenvalues are determined in terms of a single half integer number h. Therefore we conclude that all of the systems listed in section (I) share the very same unitary representation of SU(2, 2) with the same Casimir eigenvalues given above. In particular, for spinless particles Ĵ0 → h = 0 we obtain C2 = −3, C3 = 6, C4 = −394 , which is the unitary singleton representation of SO(4, 2) =SU(2, 2). This is in agreement with previous covariant quantization of the spinless particle in any dimension directly in phase space in d + 2 dimensions, which gave for the SO(d, 2) Casimir the eigenvalue as MN → 1 − d2/4 on physical states that satisfy X2 = P 2 = X · P = 0 [8]. So, for d = 4 we get C2 = −3 in agreement with the quantum twistor computation above. Note that the classical computation either in phase space or twistor space would give the wrong answer C2 = 0 when orders of canonical conjugates are ignored and constraints used classically. Of course, having the same SU(2, 2) Casimir eigenvalue is one of the infinite number of duality relations among these systems that follow from the more general twistor transform or the master 2T-physics theory (4.1). All dualities of these systems amount to all quantum functions of the gauge invariants J BA that take the same gauge invariant values in any of the physical Hilbert spaces of the systems listed in section (I). All the physical information on the relations among the physical observables is already captured by the quantum master equation (7.8), so it is sufficient to concentrate on it. The predicted duality, including these relations, can be tested at the quantum level by computing and verifying the equality of an infinite number of matrix elements of the master equation between the dually related quantum states for the systems listed in section (I). In the case of the Casimir operators Cn the details of the individual states within a representation is not relevant, so that computation whose result is given above is among the simplest computations that can be performed on the systems listed in section (I) to test our duality predictions. This test was performed successfully for h = 0 at the quantum level for some of these systems directly in their own phase spaces [26], verifying for example, that the free massless particle, the hydrogen atom, the harmonic oscillator, the particle on AdS spaces, all have the same Casimir eigenvalues C2 = −3, C3 = 6, C4 = −394 at the quantum level. Much more elaborate tests of the dualities can be performed both at the classical and quantum levels by computing any function of the gauge invariant J BA and checking that it has the same value when computed in terms of the spin & phase space of any of the sys- tems listed in section (I). At the quantum level all of these systems have the same Casimir eigenvalues of the Cn for a given h. So their spectra must correspond to the same unitary irreducible representation of SU(2, 2) as seen above. But the rest of the labels of the repre- sentation correspond to simultaneously commuting operators that include the Hamiltonian. The Hamiltonian of each system is some operator constructed from the observables J BA , and so are the other simultaneously diagonalizable observables. Therefore, the different systems are related to one another by unitary transformations that sends one Hamiltonian to another, but staying within the same representation. These unitary transformations are the quantum versions of the gauge transformations of Eq.(4.7), and so they are the duality transformations at the quantum level. In particular the twistor transform applied to any of the systems is one of those duality transformations. By applying the twistor transforms we can map the Hilbert space of one system to another, and then compute any function of the gauge invariant J BA between dually related states of different systems. The prediction is that all such computations within different systems must give the same result. Given that J BA is expressed in terms of rather different phase space and spin degrees of freedom in each dynamical system with a different Hamiltonian, this predicted duality is remarkable. 1T-physics simply is not equipped to explain why or for which systems there are such dualities, although it can be used to check it. The origin as well as the proof of the duality is the unification of the systems in the form of the 2T-physics master action of Eq.(4.1) in 4+2 dimensions. The existence of the dualities, which can laboriously be checked using 1T-physics, is the evidence that the underlying spacetime is more beneficially understood as being a spacetime in 4+2 dimensions. VIII. QUANTUM TWISTOR TRANSFORM We have established a master equation for physical observables J at the quantum level. Now, we also want to establish the twistor transform at the quantum level expressed as much as possible in terms of the gauge invariant physical quantum observables J . To this end we write the master equation (7.8) in the form J − 3 Ĵ0 − 2 J + 1 Ĵ0 + 2 = 0. (8.1) Recall the quantum equation (7.5) J + Ĵ0+2 = ZZ̄, so the equation above is equivalent to J − 3 Ĵ0 − 2 Z = 0. (8.2) This is a 4× 4 matrix eigenvalue equation with operator entries. The general solution is Ĵ0 + 2 V̂ (8.3) where V̂A is any spinor up to a normalization. This is verified by using the master equation (8.1) which gives J − 3 Ĵ0 − 2 J − 3 Ĵ0 − 2 J + 1 Ĵ0 + 2 V̂ = 0. Not- ing that the solution (8.3) has the same form as the classical version of the twistor transform in Eq.(6.8), except for the quantum shift J0 → Ĵ0 + 2, we conclude that the V̂A introduced above is the quantum version of the VA discussed earlier (up to a possible renormalization as belonging to the coset SU(2, 3) /[SU(2, 2)×U(1)]. Now V̂A is a quantum operator whose commutation rules must be compatible with those of ZA, Z̄ A, Ĵ0 and J BA . Its commutation rules with J BA , Ĵ0 are straightforward and fixed uniquely by the SU(2, 2)×U(1) covariance Ĵ0, V̂A = −V̂A, Ĵ0, V̂ , (8.4) J BA , V̂C = −δ BC V̂A + V̂C δ J BA , V̂ = δ DA V̂ δ BA . (8.5) Other quantum properties of V̂A follow from imposing the quantum property Z̄Z = Ĵ0 − 2 in (7.4). Inserting Z of the form (8.3), using the master equation, and observing the commutation rules (8.4), we obtain J + Ĵ0 + 2 V̂ = 1. (8.6) 13 The quantum version of V̂ is valid in the whole Hilbert space, not only in the subspace that satisfies the U(1) constraint Ĵ0 → 2h. In particular, in the high spin version, already at the classical level we must take V̂ = V ( J0) and then rescale it V 2h → V as described in previous footnotes. So in the full quantum Hilbert space we must take V̂ = 2hV (Ĵ0 + γ) −1/2 (or the rescaled version V 2h → V ) with the possibly quantum shifted operator (Ĵ0 + γ) −1/2. This is related to (5.11) if we take (5.9) into account by including the quantum shift J0 → Ĵ0 + 2. Considering (8.3) this equation may also be written as V̂ Z = Z̄V̂ = 1. (8.7) Next we impose ZA, Z̄ = δ BA to deduce the quantum rules for [V̂A, V̂ ]. After some algebra we learn that the most general form compatible with ZA, Z̄ = δ BA is V̂A, V̂ = − V̂ V̂ Ĵ0 − 1 δ BA + M(J − 3 Ĵ0 − 2 ) + (J − 3 Ĵ0 − 2 , (8.8) where M BA is some complex matrix and M̄ = (η2,2)M † (η2,2) . The matrix M BA could not be determined uniquely because of the 3/4 kappa gauge freedom in the choice of V̂A itself. A maximally gauge fixed version of V̂A corresponds to eliminating 3 of its components V̂2,3,4 = 0 by using the 3/4 kappa symmetry, leaving only A ≡ V̂1 6= 0. Then we find V̂ 1,2,4 and V̄ 3 = A†. Let us analyze the quantum properties of this gauge in the context of the formalism above. From Eq.(8.6) we determine A = (J 13 ) e−iφ, where φ is a phase, and then from Eq.(8.3) we find ZA. J 1A + Ĵ0 + 2 )−1/2 e−iφ, Z̄A = eiφ )−1/2 J A3 + Ĵ0 + 2 (8.9) We see that, except for the overall phase, ZA is completely determined in terms of the gauge invariant J BA . We use a set of gamma matrices ΓM given in ([6],[11]) to write J BA = 14iJ MN (ΓMN) A as an explicit matrix so that ZA can be written in terms of the 15 SO(4, 2) =SU(2, 2) generators JMN . We find J12 + 1 J+− + 1 ′−′ + Ĵ0+2 (J+1 + iJ+2) ′1 + iJ+ e−iφ√ , (8.10) and Z̄A = Z†η2,2 . The orders of the operators here are important. The basisM = ±′,±, i with i = 1, 2 corresponds to using the lightcone combinations X± ′ ±X1′ , X± = (X0 ±X1). From our setup above, the ZA, Z̄ A in (8.10) are guaranteed to satisfy the twistor commuta- tion rules ZA, Z̄ = δ BA provided we insure that the V̂A, V̂ have the quantum properties given in Eqs.(8.4,8.5,8.8). These are satisfied provided we take the following non-trivial commutation rules for φ φ, Ĵ0 = i, [φ, J12] = Ĵ0, e = ±e±iφ, [J12, e±iφ] = ±1 e±iφ (8.11) while all other commutators between φ and JMN vanish. Then (8.8) becomes [V̂A, V̂ ] = 0, so M BA vanishes in this gauge. Indeed one can check directly that only by using the Lie algebra for the JMN , Ĵ0 and the commutation rules for φ in (8.11), we obtain ZA, Z̄ = δ BA , which a remarkable form of the twistor transform at the quantum level. The expression (8.10) for the twistor is not SU(2, 2) covariant. Of course, this is because we chose a non-covariant gauge for V̂A. However, the global symmetry SU(2, 2) is still intact since the correct commutation rules between the twistors and JMN or the J BA as given in (7.6,7.7) are built in, and are automatically satisfied. Therefore, despite the lack of manifest covariance, the expression for ZA in (8.10) transforms covariantly as the spinor of SU(2, 2) . It is now evident that one has many choices of gauges for V̂A. Once a gauge is picked the procedure outlined above will automatically produce the quantum twistor transform in that gauge, and it will have the correct commutation rules and SU(2, 2) properties at the quantum level. For example, in the SL(2, C) covariant gauge of Eq.(5.20), the quantum twistor transform in terms of JMN is µα̇ = Jµν (σ̄ vβ̇ + ′−′vα̇, λα = ′µ (σµ)αβ̇ v β̇. (8.12) with the constraint v̄σµvJ +′µ = 1. (8.13) This gauge for V̂M covers several of the systems listed in section (I). The spinless case was discussed at the classical level in ([6]). The quantum properties of this gauge are discussed in more detail in ([1]). The result for ZA in (8.10) is a quantum twistor transform that relies only on the gauge invariants J BA or equivalently JMN . It generalizes a similar result in [6] that was given at the classical level. In the present case it is quantum and with spin. All the information on spin is included in the generators JMN = LMN +SMN . There are other ways of describing spinning particles. For example, one can start with a 2T-physics action that uses fermions ψM (τ) [27] instead of our bosonic variables VA (τ) . Since we only use the gauge invariant J MN , our quantum twistor transform (8.3) applies to all such descriptions of spinning particles, with an appropriate relation between V̂ and the new spin degrees of freedom. In particular in the gauge fixed form of V̂ that yields (8.10) there is no need to seek a relation between V̂ and the other spin degrees of freedom. Therefore, in the form (8.10), if the JMN are produced with the correct quantum algebra SU(2, 2) =SO(4, 2) in any theory, (for example bosonic spinors, or fermions ψM , or the list of systems in section (I), or any other) then our formula (8.3) gives the twistor transform for the corresponding degrees of freedom of that theory. Those degrees of freedom appear as the building blocks of JMN . So, the machinery proposed in this section contains some very powerful tools. IX. THE UNIFYING SU(2, 3) LIE ALGEBRA The 2T-physics action (4.1) offered the group SU(2, 3) as the most symmetric unifying property of the spinning particles for all the systems listed in section (I), including twistors. Here we discuss how this fundamental underlying structure governs and simplifies the quan- tum theory. We examine the SU(2, 3) charges J BA , Ĵ0, jA, j̄A given in (4.5,5.9,5.10). Since these are gauge invariant under all the gauge symmetries (4.7) they are physical quantities that should have the properties of the Lie algebra14 of SU(2, 3) in all the systems listed in section (I). Using covariant quantization we construct the quantum version of all these charges in terms of twistors. By using the general quantum twistor transform of the previous section, these charges can also be written in terms of the quantized spin and phase space degrees of freedom of any of the relevant systems. The twistor expressions for Ĵ0,J BA are already given in Eqs.(7.2,7.5) Ĵ0 = A + Z̄AZA , J BA = ZAZ̄B − Ĵ0 + 2 δ BA . (9.1) We have seen that at the classical level (jA)classical = J0ZA and now we must figure out 14 Even when jA is not a conserved charge when the U(1) constraint is imposed, its commutation rules are still the same in the covariant quantization approach, independently than the constraint. the quantum version jA = Ĵ0 + αZA that gives the correct SU(2, 3) closure property jA, j̄ = J BA + A . (9.2) The coefficient 5 is determined by consistency with the Jacobi identity jA, j̄ j̄B, jC [jC , jA, ] , j̄ = 0, and the requirement that the commutators of jA with J BA , Ĵ0 be just like those of ZA given in Eqs.(7.6,7.7), as part of the SU(2, 3) Lie algebra. So we carry out the computation in Eq.(9.2) as follows jA, j̄ Ĵ0 + αZAZ̄ Ĵ0 + α− Z̄B Ĵ0 + α Ĵ0 + αZA (9.3) Ĵ0 + α Ĵ0 + α− 1 Z̄BZA (9.4) Ĵ0 + α− 1 ZA, Z̄ + ZAZ̄ B (9.5) = δ BA Ĵ0 + α− 1 + Ĵ0 + 2 + J BA (9.6) To get (9.4) we have used the properties ZAf Ĵ0 + 1 ZA and Z̄ Ĵ0 − 1 Z̄B for any function f . These follow from the commutator Ĵ0, ZA = −ZA written in the form ZAĴ0 = Ĵ0 + 1 ZA which is used repeatedly, and similarly for Z̄ B. To get (9.6) we have used ZA, Z̄ = δ BA and then used the definitions (9.1). By comparing (9.6) and (9.2) we fix α = 1/2. Hence the correct quantum version of jA is Ĵ0 + ZA = ZA Ĵ0 − . (9.7) The second form is obtained by using ZAf Ĵ0 + 1 Note the following properties of the jA, j̄ j̄AjA = Ĵ0 − Ĵ0 − Ĵ0 − Ĵ0 − 2 (9.8) Ĵ0 + Ĵ0 + Ĵ0 + Ĵ0 + 2 (9.9) which will be used below. With the above arguments we have now constructed the quantum version of the SU(2, 3) charges written as a 5× 5 traceless matrix Ĵ2,3 = quantum (9.10) B − 1 Ĵ0 + Ĵ0 +  , (9.11) with Ĵ0,J given in Eq.(9.1). At the classical level, the square of the matrix J2,3 vanishes since L2 = 0 as follows (J2,3) classical = g−1 g = 0. (9.12) At the quantum level we find the following non-zero result which is SU(2, 3) covariant Ĵ2,3 ZZ̄ − 1 Ĵ0 + Ĵ0 + (9.13) Ĵ2,3 − 1. (9.14) By repeatedly using the same equation we can compute all powers Ĵ2,3 , and by taking traces we obtain the Casimir eigenvalues of the SU(2, 3) representation. For example the quadratic Casimir is Ĵ2,3 = −5. (9.15) Written out in terms of the charges, Eq.(9.14) becomes − 1. (9.16) Collecting terms in each block we obtain the following relations among the gauge invariant charges J , Ĵ0, j, j̄ − jj̄ + 5 + 1 = 0, (9.17) j − jĴ0 + j = 0, (9.18) −j̄j + Ĵ0 + 1 = 0. (9.19) Combined with the information in Eq.(9.9) the first equation is equivalent to the master quantum equation (7.8). After using jĴ0 = Ĵ0j+ j, the second equation is equivalent to the eigenvalue equation (8.2) whose solution is the quantum twistor transform (8.3). The third equation is identical to (9.8). Hence the SU(2, 3) quantum property Ĵ2,3 Ĵ2,3 − 1, or equivalently Ĵ2,3 + 2 Ĵ2,3 + = 0, governs the quantum dynamics of all the sytems listed in sec- tion (I) and captures all of the physical information, twistor transform, and dualities as a property of a fixed SU(2, 3) representation whose generators satisfy the given constraint. This is a remarkable simple unifying description of a diverse set of spinning systems, that shows the existence of the sophisticated higher structure SU(2, 3) for which there was no clue whatsoever from the point of view of 1T-physics. X. FUTURE DIRECTIONS One can consider several paths that generalizes our discussion, including the following. • It is straightforward to generalize our theory by replacing SU(2, 3) with the super- group SU(2, (2 + n) |N) . This generalizes the spinor VA to V aA where a labels the fundamental representation of the supergroup SU(n|N) . The case of N = 0 and n = 1 is what we discussed in this paper. The case of n = 0 and any N relates to the superparticle with N supersymmetries (and all its duals) discussed in [22] and in [6][7]. The massless particle gauge is investigated in [17], but the other cases listed in section (I) remain so far unexplored. The general model has global symmetry SU(2, 2)×SU(n|N)×U(1) ⊂ [SU(2, (2 + n) |N)]R if a U(1) gauging is included, or the full global symmetry [SU(2, (2 + n) |N)]R in its high spin version. It also has local gauge symmetries that include bosonic & fermionic kappa symmetries embedded in [SU(2, (2 + n) |N)]L as well as the basic Sp(2, R) gauge symmetry. The gauge sym- metries insure that the theory has no negative norm states. In the massless particle gauge, this model corresponds to supersymmetrizing spinning particles rather than supersymmetrizing the zero spin particle. The usual R-symmetry group in SUSY is replaced here by SU(n|N)×U(1) . For all these cases with non-zero n,N , the 2T- physics and twistor formalisms unify a large class of new 1T-physics systems and establishes dualities among them. • One can generalize our discussion in 4+2 dimensions, including the previous paragraph, to higher dimensions. The starting point in 4+2 dimensions was SU(2, 2) =SO(4, 2) embedded in g =SU(2, 3) . For higher dimensions we start from SO(d, 2) and seek a group or supergroup that contains SO(d, 2) in the spinor representation. For example for 6+2 dimensions, the starting point is the 8×8 spinor version of SO(8∗) =SO(6, 2) embedded in g =SO(9∗) =SO(6, 3) or g =SO(10∗) =SO(6, 4) . The spinor variables in 6+2 dimensions VA will then be the spinor of SO(8 ∗) =SO(6, 2) parametrizing the coset SO(9∗) /SO(8∗) (real spinor) or SO(10∗) /SO(8∗)×SO(2) (complex spinor). This can be supersymmetrized. The pure superparticle version of this program for various dimensions is discussed in [6][7], where all the relevant supergroups are classified. That discussion can now be taken further by including bosonic variables embedded in a supergroup as just outlined in the previous item. As explained before [6][7], it must be mentioned that when d + 2 exceeds 6 + 2 it seems that we need to include also brane degrees of freedom in addition to particle degrees of freedom. Also, even in lower dimensions, if the group element g belongs to a group larger than the minimal one [6][7], extra degrees of freedom will appear. • The methods in this paper overlap with those in [28] where a similar master quantum equation technique for the supergroup SU(2, 2|4) was used to describe the spectrum of type-IIB supergravity compactified on AdS5×S5. So our methods have a direct bearing onM theory. In the case of [28] the matrix insertion in the 2T-physics action was generalized to L(4,2) L(6,0) to describe a theory in 10+2 dimensions. This approach to higher dimensions can avoid the brane degrees of freedom and concentrate only on the particle limit. Similar generalizations can be used with our present better develped methods and richer set of groups mentioned above to explore various corners of M theory. • One of the projects in 2T-physics is to take advantage of its flexible gauge fixing mechanisms in the context of 2T-physics field theory. Applying this concept to the 2T-physics version of the Standard Model [10] will generate duals to the Standard Model in 3+1 dimensions. The study of the duals could provide some non-perturbative or other physical information on the usual Standard Model. This program is about to be launched in the near future [29]. Applying the twistor techniques developed here to 2T-physics field theory should shed light on how to connect the Standard Model with a twistor version. This could lead to further insight and to new computational techniques for the types of twistor computations that proved to be useful in QCD [12][13]. • Our new models and methods can also be applied to the study of high spin theories by generalizing the techniques in [14] which are closely related to 2T-physics. The high spin version of our model has been discussed in many of the footnotes, and can be supersymmetrized and written in higher dimensions as outlined above in this section. The new ingredient from the 2T point of view is the bosonic spinor VA and the higher symmetry, such as SU(2, 3) and its generalizations in higher dimensions or with supersymmetry. The massless particle gauge of our theory in 3+1 dimensions coincides with the high spin studies in [15]-[18]. Our theory of course applies broadly to all the spinning systems that emerge in the other gauges, not only to massless particles. The last three sections on the quantum theory discussed in this paper would apply also in the high spin version of our theory. The more direct 4+2 higher dimensional quantization of high spin theories including the spinor VA (or its generalizations V is obtained from our SU(2, 3) quantum formalism in the last section. • One can consider applying the bosonic spinor that worked well in the particle case to strings and branes. This may provide new string backgrounds with spin degrees of freedom other than the familiar Neveu-Schwarz or Green-Schwarz formulations that involve fermions. More details and applications of our theory will be presented in a companion paper [1]. We gratefully acknowledge discussions with S-H. Chen, Y-C. Kuo, and G. Quelin. [1] I. Bars and B. Orcal, in preparation. [2] R. Penrose, “Twistor Algebra,” J. Math. 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0704.0297

Remnant evolution after a carbon-oxygen white dwarf merger

Mon. Not. R. Astron. Soc. 000, 1–?? (2007) Printed 20 August 2019 (MN LATEX style file v2.2) Remnant evolution after a carbon-oxygen white dwarf merger S.-C. Yoon1,2⋆, Ph. Podsiadlowski3 and S. Rosswog4 1Astronomical Institute ”Anton Pannekoek”, University of Amsterdam, Kruislaan 403, 1098 SJ, Amsterdam, The Netherlands 2Department of Astronomy & Astrophysics, University of California, Santa Cruz, CA95064, USA 3Department of Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK 4School of Engineering and Science, Jacobs University Bremen†, Campus Ring1, Bremen 28759, Germany Accepted/ Received ABSTRACT We systematically explore the evolution of the merger of two carbon-oxygen (CO) white dwarfs. The dynamical evolution of a 0.9 M⊙ + 0.6 M⊙ CO white dwarf merger is followed by a three-dimensional SPH simulation. The calculation uses a state-of-the art equation of state that is coupled to an efficient nuclear reaction network that accurately approximates all stages from helium burning up to nuclear statistical equilibrium. We use an elaborate pre- scription in which artificial viscosity is essentially absent, unless a shock is detected, and a much larger number of SPH particles than earlier calculations. Based on this simulation, we suggest that the central region of the merger remnant can, once it has reached quasi-static equilibrium, be approximated as a differentially rotating CO star, which consists of a slowly rotating cold core and a rapidly rotating hot envelope surrounded by a centrifugally supported disc. We construct a model of the CO remnant that mimics the results of the SPH simulation using a one-dimensional hydrodynamic stellar evolution code and then follow its secular evo- lution, where we include the effects of rotation on the stellar structure and the transport of angular momentum. The influence of the Keplerian disc is implicitly treated by considering mass accretion from the disc onto the hot envelope. The stellar evolution models indicate that the growth of the cold core is controlled by neutrino cooling at the interface between the core and the hot envelope, and that carbon ignition in the envelope can be avoided despite high effective accretion rates. This result suggests that the assumption of forced accretion of cold matter that was adopted in previous studies of the evolution of double CO white dwarf merger remnants may not be appropriate. Specifically we find that off-center carbon ignition, which would eventually lead to the collapse of the remnant to a neutron star, can be avoided if the following conditions are satisfied: (1) when the merger remnant reaches quasi-static equilib- rium, the local maximum temperature at the interface between the core and the envelope must be lower than the critical limit for carbon-ignition. (2) Angular-momentum loss from the cen- tral merger remnant should not occur on a time scale shorter than the local neutrino cooling time scale at the interface. (3) The mass-accretion rate from the centrifugally supported disc must be sufficiently low (Ṁ . 5 × 10−6...10−5 M⊙ yr −1). Our results imply that at least some products of double CO white dwarfs merger may be considered good candidates for the progenitors of Type Ia supernovae. In this case, the characteristic time delay between the initial dynamical merger and the eventual explosion would be ∼ 105 yr. Key words: Stars: evolution – Stars: white dwarf – Stars: accretion – Supernovae: general – 1 INTRODUCTION The coalescence of two carbon-oxygen (CO) white dwarfs with a combined mass in excess of the Chandrasekhar limit has long been considered a promising path towards a Type Ia supernova (SN Ia; Iben & Tutukov 1984; Webbink 1984). Indeed, in the last few ⋆ E-mail: [email protected] (SCY); [email protected] (PhP); [email protected] (SR) † formerly International University Bremen years, a few massive double CO white dwarf systems have been found that have periods short enough for them to merge within a Hubble time (e.g. Napiwotzki et al. 2002, 2004). This double- degenerate (DD) scenario can also easily explain the lack of hy- drogen and helium lines in most SN Ia spectra and the occur- rence of SNe Ia both in old and young star-forming systems (e.g. Branch et al. 1995). Theoretically, the final fate of double CO white dwarf merg- ers has been much debated. Previous studies assumed that the dy- namical disruption of the Roche-lobe filling secondary should lead c© 2007 RAS http://arxiv.org/abs/0704.0297v2 2 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog Figure 1. Schematic illustration of the configuration of the remnant of a double CO white dwarf merger once quasi-static equilibrium has been es- tablished. to the formation of a thick disc around the primary white dwarf (Tutukov & Yungelson 1979; Mochkovitch & Livio 1989, 1990). Therefore, accretion of CO-rich matter from the thick disc onto the central cold white dwarf has been studied for investigating the evolution of such mergers by many authors (Nomoto & Iben 1985; Saio & Nomoto 1985, 1998, 2004; Piersanti et al. 2003a,b). As ac- cretion rates from the thick disc should be close to the Edding- ton limit (Ṁ ≈ 10−5 M⊙yr −1), most of those studies concluded that carbon ignition in the envelope of the accreting white dwarf is an inevitable consequence of such rapid accretion of CO-rich matter. Once carbon ignites off-center, the burning flame propa- gates inwards on a relatively short time scale (∼ 5000 yr), and the CO white dwarf is transformed into an ONeMg white dwarf (Saio & Nomoto 1985, 1998). When the mass of the ONeMg white dwarf approaches the Chandrasekhar limit, electron capture onto Ne and Mg is expected to lead to the gravitational collapse of the white dwarf to a neutron star (Nomoto & Kondo 1991; see Dessart et al. 2006 and Kitaura, Janka & Hillebrandt 2006 for re- cent studies of such collapse). However, the evolution of the remnants of double CO white dwarf mergers is not yet well understood. For instance, it has been debated whether the accretion rate decreases when the ac- creting white dwarf reaches critical rotation (Piersanti et al. 2003a; Saio & Nomoto 2004). More importantly, the canonical descrip- tion of the merger remnant as a primary white dwarf + thick disc system is clearly an oversimplification. In previous three- dimensional smoothed particle hydrodynamics (SPH) simula- tions (Benz et al. 1990; Segretain, Chabrier & Mochkovitch 1997; Guerrero et al. 2004; see also Sect. 2), a large fraction of the dis- rupted secondary and the outermost layers of the primary form an extended hot envelope around the cold core containing most of the primary mass. The rest of the secondary mass becomes a centrifu- gally supported disc in the outermost layers of the merger rem- nant. Interestingly, the merger remnant reaches a state of quasi- static equilibrium within a few minutes from the onset of the dy- namical disruption of the secondary. As the structure of the cold core plus the hot envelope appears to have a fairly spheroidal shape (see below) rather than the toroidal shape obtained with a zero- temperature equation of state (Mochkovitch & Livio 1989, 1990), the merger remnant may be better described as a differentially ro- tating single CO star consisting of a slowly rotating cold core and a rapidly rotating hot extended envelope surrounded by a Keple- rian disc, as illustrated in Fig. 1, than the previously adopted pri- mary white dwarf + thick disc system. The further evolution of the merger must therefore be determined by the thermal cooling of the hot envelope and the redistribution of the angular momentum in- side the central remnant, and accretion of matter onto the envelope from the Keplerian disc. With this new approach to the problem in mind, we here re- visit both the dynamical and the secular evolution of double CO white dwarf mergers. In the following section (Sect. 2), we present the numerical results of an SPH simulation of the dynamical evolu- tion of the coalescence of a 0.9 M⊙ WD and a 0.6 M⊙ CO white dwarf up to the stage of quasi-hydrostatic equilibrium, and we care- fully investigate the structure of the merger remnant. In Sect. 3, we construct models of the central remnant in quasi-static equilibrium state (primary + hot extended envelope) which mimic the SPH re- sult and calculate the thermal evolution of the merger remnant using a hydrodynamic stellar evolution code. In particular, the conditions for avoiding off-center carbon ignition are systematically explored. In Sect. 4, we conclude this work by discussing uncertainties in our assumptions, the implications for Type Ia supernovae and future work. 2 DYNAMICAL EVOLUTION OF THE MERGER Before discussing the subsequent thermal evolution after the co- alescence of a double CO white dwarf coalescence binary, we investigate the configuration of the remnant in quasi-static equi- librium in some detail. For this purpose, we have carried out a SPH simulation of the dynamical process of the coalescence of two CO white dwarfs of 0.9 M⊙ and 0.6 M⊙, respec- tively. Our simulation uses a 3D smoothed particle hydrodynam- ics (SPH) code that is an offspring of a code developed to simu- late neutron star mergers (Rosswog et al. 2000; Rosswog & Davies 2002; Rosswog & Liebendörfer 2003). It uses an artificial viscos- ity scheme with time-dependent parameters (Morris & Monaghan 1997). In the absence of shocks, the viscosity parameters have a very low value (α = 0.05 and β = 0.1; most SPH implemen- tations use values of α = 1...1.5 and β = 2...3); if a shock is detected, a source term (Rosswog et al. 2000) guarantees that the parameters rise to values that are able to resolve the shock prop- erly without spurious post-shock oscillations. To suppress artificial viscosity forces in pure shear flows, we additionally apply a switch originally suggested by Balsara (1995). To account for the energetic feedback onto the fluid from nuclear transmutations, we use a minimal nuclear reaction net- work developed by Hix et al. (1998). It couples a conventional α- network stretching from He to Si with a quasi-equilibrium-reduced α-network. Although a set of only seven nuclear species is used, this network reproduces the energy generation of all burning stages from He-burning to NSE very accurately. For details and tests we refer to Hix et al. (1998). We use the HELMHOLTZ equation of state (EOS), developed by the Center for Astrophysical Thermonu- clear Flashes at the University of Chicago. This EOS allows to freely specify the chemical composition of the gas and can be cou- pled to nuclear reaction networks. The electron/positron equation of state has been calculated without approximations, i.e. it makes no assumptions about the degree of degeneracy or relativity; the exact expressions are integrated numerically to machine precision. The nuclei in the gas are treated as a Maxwell-Boltzmann gas, the photons as blackbody radiation. The EOS is used in tabular form with densities ranging from 10−10 6 ρYe 6 10 11 g cm−3 and temperatures from 104 to 1011 K. A sophisticated, biquin- tic Hermite polynomial interpolation is used to enforce thermody- namic consistency (i.e. the Maxwell-relations) at interpolated val- ues (Timmes & Swesty 2000). We use a MacCormack predictor-corrector method (e.g. Lomax et al. 2001) with individual particle time steps to evolve the fluid. With our standard parameters for the tree-opening cri- c© 2007 RAS, MNRAS 000, 1–?? Remnant evolution after a carbon-oxygen white dwarf merger 3 Figure 2. Dynamical evolution of the coalescence of a 0.6 M⊙ + 0.9 M⊙ CO white dwarf binary. The panels in the left column show the density in the orbital plane, the panels in the right column the temperature in units of 106 K. Lengths are in code units (= 109 cm). c© 2007 RAS, MNRAS 000, 1–?? 4 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog Figure 3. Dynamical evolution of the coalescence of a 0.6 M⊙ + 0.9 M⊙ CO white dwarf binary. Continued from Fig. 2. c© 2007 RAS, MNRAS 000, 1–?? Remnant evolution after a carbon-oxygen white dwarf merger 5 Figure 4. The evolution of the local peak of temperature during the merger of two CO white dwarfs of 0.6M⊙ and 0.9M⊙, respectively, as a function of time after the onset of the simulation. terion and the integration, this time marching implementation con- serves the total energy to better than 4 × 10−3 and the total an- gular momentum to better than 2 × 10−4. Note that this could, in principle, be improved even further by taking into account the so-called “grad-h”-terms (Springel & Hernquist 2002; Monaghan 2002; Price 2004) and extra-terms arising from adapting gravita- tional smoothing terms (Price & Monaghan 2006). To avoid numerical artifacts, we only use equal mass SPH particles. For the initial conditions, we therefore stretch a uniform particle distribution according to a function that has been derived from solving the 1D stellar structure equations. This technique is described in detail in Rosswog, Ramirez-Ruiz & Hix (2007). This particle setup is then further relaxed with an additional damping force (e.g. Rosswog, Speith & Wynn 2004) so that the particles can settle into their true equilibrium configuration. The calculations are performed with 2×105 SPH particles, a much larger particle num- ber than could be afforded by previous calculations, and run up to a much longer evolutionary time (5 minutes) than previous calcu- lations (see Table 1). Figs. 2 and 3 show the dynamical evolution of the merging process of the double white dwarf system considered in this study. The panel in the left columns show the densities and the panel in the right columns the temperatures (in units of 106 K) in the orbital plane. The secondary is completely disrupted within 1.7 minutes, and mass accretion onto the primary induces local heating near the surface of the primary. Fig. 4 shows the evolution of the maximum temperature as a function of time. The peak in temperature reaches 1.7×109 K at t ≃ 1.0 min, where t = 0.0 marks the moment when the simulation starts. Carbon ignites when T & 109 K, but nuclear burning is quenched soon due to the local expansion of the hottest region, as is also observed in the simulations of Guerrero et al. (2004). The total amount of energy released due to nuclear burn- ing is about 1045 erg. Segretain, Chabrier & Mochkovitch (1997) considered the same initial white dwarf masses as in the present study. But they adopted the original artificial viscosity prescription of Monaghan & Varnas (1988), which is known to introduce spuri- ous forces in shear flows, and they did not include nuclear burning (Table 1). By the end of their calculation (t = 1.56 min), Tmax reached 8×108 K, while in our simulation, it decreases to 8×108 K only when t ∼ 1.7 min. Interestingly, Tmax decreases further af- Figure 5. Top: Density contour of the merger remnant in the x− z plane at t = 5.3 min. Here one code unit corresponds to 109 cm. Middle: Thermo- dynamic structure of the merger remnant at t = 5.31 min: shown are the temperature and the density as a function of distance from the centre, along the positive x- and z-axis, as indicated. Bottom: Angular velocity in units of the local Keplerian value at t = 5.31 min, along the positive/negative x- and y-axis of the merger remnant. terwards in our calculation, as shown in Fig. 4, and reaches a steady value at Tmax ≃ 5.6 × 10 8 K when t & 2.5min. In the other cal- culations by Benz et al. (1990) and Guerrero et al. (2004), the dy- namical evolution of the merger was not followed for more than 2 minutes either, and we cannot directly compare our results to theirs. However, we suspect that Tmax would also decrease further in the systems they considered if they had continued their calculations for a longer evolutionary time. It should also be noted that energy dis- sipation by artificial viscosity might lead to overheating, and that c© 2007 RAS, MNRAS 000, 1–?? 6 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog thermal diffusion – which may play an important role in the outer- most layers – is not considered in the present study. It is thus likely that Tmax in the quasi-static equilibrium state may be even lower in reality than in our simulation. Fig. 5 shows the structure of the merger remnant at quasi-static equilibrium. The central region with R . 109 cm (Mr . 1.1 M⊙) has a fairly spheroidal shape, and a centrifugally supported disc appears at R & 109 cm where the angular velocity is close to the Keplerian value. The fraction of the secondary mass contained in the Keplerian disc is larger in our simulation (about 67%) than in Segretain, Chabrier & Mochkovitch (1997) (about 41 %). The innermost core (R . 3 × 108 cm; Mr . 0.6 M⊙) is essen- tially isothermal, and the temperature has its peak value (T ≃ 5.6 × 108 K) at R ≃ 5 × 108 cm and Mr ≃ 0.85 M⊙. The disc material extends over 4× 109 cm along the z-axis as the tem- perature is still high; if thermal diffusion were included, the disc would become much thinner on a short time scale of a few hours. Therefore, our simulation confirms the remnant structure at quasi- static equilibrium that is illustrated in Fig. 1. In the next section, we investigate the secular evolution of the merger from such a quasi- static equilibrium state. 3 SECULAR EVOLUTION OF THE MERGER REMNANT 3.1 Physical assumptions and methods Our SPH simulation shows that the remnant of the merger of two CO white dwarfs (0.9 M⊙ + 0.6 M⊙) in the state of quasi-static equilibrium has the following features (see Fig. 6): (i) The core is cold and nearly isothermal. (ii) The local peak of temperature (Tp) is located at a mass co- ordinate slightly less than the primary mass. (iii) A steep gradient in temperature appears at the interface be- tween the core and the local peak of temperature. (iv) The interface is rather widely extended into the primary (∆Minterface ≈ 33 % of the primary mass), and the mass of the quasi-isothermal cold core (Mcore) is about 77 % of the primary mass. (v) The mass of the outer envelope above the local peak of tem- perature contains about 33 % of the mass of the secondary, and the rest of the secondary forms a Keplerian disc. Let us define Tp as the local peak of temperature at quasi-static equilibrium, MCM as the mass of the central remnant (cold core + hot envelope), and Mp as the location of Tp in the mass coordi- nate (i.e., Mp = Mcore + ∆Minterface; see Fig. 6). To construct models of the central remnant, we use a one-dimensional hydrody- namic stellar evolution code which incorporates the effects of ro- tation on the stellar structure, transport of angular momentum due to the shear instability, Eddington-Sweet circulation, and the Gol- dreich, Schubert and Fricke instability, and dissipation of rotational energy due to shear motions. The effects of magnetic fields are ne- glected (see Sec. 4). More details about the code are described in (Yoon & Langer 2004; hereafter YL04) and references therein. In order to mimic the temperature profile of the central rem- nant as obtained from the SPH simulation, we artificially deposit energy in the envelope, using the following prescription for a white dwarf with M = MCR: e(Mr) = A(T (Mr)− T (Mr)) [erg g ], (1) where Figure 6. Initial model of the central remnant for sequences Sa1 – Sa11. The top and middle panels show temperature as a function of the mass co- ordinate and radius, respectively. The solid curve in the bottom panel gives the angular-velocity profile as a function of radius. The dashed curve de- notes the angular velocity in units of the local Keplerian value. T ′(Mr) = 3 · 107 K+ (7 · 107 K− 3 · 107 K) Mcore if Mr < Mcore, Tp − (Tp − 7 · 10 Mp−Mr Mp−Mcore if Mcore 6 Mr 6 Mp, C − (C − Tp) log[ρ(Mr)/ρs] log[ρ(Mp)/ρs] if Mr > Mp. In this way, the temperature profile in the central remnant model follows T ′(Mr). Here, A and C are constants. We use A = 105 erg g−1 s−1 K−1 and C = 2× 108 K in most cases. A rotational profile is imposed as c© 2007 RAS, MNRAS 000, 1–?? Remnant evolution after a carbon-oxygen white dwarf merger 7 Table 1. Comparison of SPH simulations of double CO white dwarf mergers. The columns list: M1 and M2: the masses of the primary and the secondary, respectively; NoP: the total number of particles used; νsph: the type of artificial viscosity employed, ‘std.’ refers to Monaghan & Varnas (1988); Network: type of nuclear network employed; tsim: evolutionary time that has elapsed by the end of the calculation; Tmax: maximum temperature obtained during the simulation; and Tp: the local peak of temperature at the end of the calculation Ref.∗ M1 M2 NoP νsph Network tsim Tmax TP 1 1.2 M⊙ 0.9 M⊙ ∼ 7× 103 std. None 51 sec. ? ∼ 109 K 2 0.8 M⊙ 0.6 M⊙ ∼ 4× 104 std. + Balsara-switch alpha network 50 sec. 1.4× 109 K ? 2 1.0 M⊙ 0.6 M⊙ ∼ 4× 104 std. + Balsara-switch alpha network 65 sec. 1.6× 109 K ? 2 1.0 M⊙ 0.8 M⊙ ∼ 4× 104 std. + Balsara-switch alpha network 65 sec. 2.0× 109 K ? 3 0.9 M⊙ 0.6 M⊙ ∼ 6× 104 std None 1.56 min. ? ∼ 7× 108 K 4 0.9 M⊙ 0.6 M⊙ 2× 105 see Rosswog et al. (2000) QSE-alpha network 5.3 min. 1.7× 109 K 5.6× 108 K ∗1: Benz et al. (1990), 2: Guerrero et al. (2004), 3: Segretain, Chabrier & Mochkovitch (1997); 4: Present Study Table 2. Merger remnant model sequences. Each column lists the following: No.: sequence label, MCR: mass of the central remnant, Mcore: mass of the quasi-isothermal core, Mp: location of the local peak of temperature in the mass coordinate, Tp: the local peak of temperature, ρp: density at Mr = Mp, τJ: adopted time scale for angular momentum loss according to Eq. (4), Ṁacc: adopted mass accretion rate from the Keplerian disc, C-ig: off-center ignition of carbon, MWD,ig: total mass of the central remnant when off-center carbon ignition occurs, Mr,ig: location of off-center carbon ignition in the mass coordinate. No. MCR Mcore Mp Tp ρp τJ Ṁacc C-ig MWD,ig Mr,ig [M⊙] [M⊙] [M⊙] [108 K] [106 g/cm3] [yr] 10−6 M⊙/yr [M⊙] [M⊙] Sa1 1.11 0.6 0.84 5.6 0.8 ∞ 0.0 No - - Sa2 1.11 0.6 0.84 5.6 0.8 102 0.0 Yes 1.11 0.80 Sa3 1.11 0.6 0.84 5.6 0.8 103 0.0 Yes 1.11 0.80 Sa4 1.11 0.6 0.84 5.6 0.8 104 0.0 Yes 1.11 0.85 Sa5 1.11 0.6 0.84 5.6 0.8 105 0.0 No - - Sa6 1.11 0.6 0.84 5.6 0.8 105 10.0 Yes 1.34 1.09 Sa7 1.11 0.6 0.84 5.6 0.8 105 5.0 Yes 1.34 1.20 Sa8 1.11 0.6 0.84 5.6 0.8 105 2.0 No - - Sa9 1.11 0.6 0.84 5.6 0.8 105 1.0 No - - Sa10 1.11 0.6 0.84 5.6 0.8 5 · 105 5.0 No - - Sa11 1.11 0.6 0.84 5.6 0.8 5 · 105 1.0 No - - Aa1 1.25 0.6 0.93 5.0 2.3 ∞ 0.0 No - - Aa2 1.25 0.6 0.93 5.0 2.3 102 0.0 Yes 1.250 0.90 Aa3 1.25 0.6 0.93 5.0 2.3 103 0.0 Yes 1.250 0.92 Aa4 1.25 0.6 0.93 5.0 2.3 104 0.0 Yes 1.250 1.12 Aa5 1.25 0.6 0.92 5.0 2.3 105 0.0 No - - Aa6 1.25 0.6 0.92 5.0 2.3 106 0.0 No - - Aa7 1.25 0.6 0.92 5.0 2.3 105 10.0 Yes 1.360 1.20 Aa8 1.25 0.6 0.92 5.0 2.3 105 5.0 No - - Aa9 1.25 0.6 0.92 5.0 2.3 105 1.0 No - - Aa10 1.25 0.6 0.92 5.0 2.3 106 10.0 Yes 1.382 1.22 Ab1 1.25 0.7 0.92 5.0 3.1 ∞ 0.0 No - - Ab2 1.25 0.7 0.92 5.0 3.1 103 0.0 Yes 1.250 0.97 Ab3 1.25 0.7 0.92 5.0 3.1 104 0.0 No - - Ab4 1.25 0.7 0.92 5.0 3.1 105 0.0 No - - Ab5 1.25 0.7 0.92 5.0 3.1 105 10.0 Yes 1.344 1.21 Ab6 1.25 0.7 0.92 5.0 3.1 105 5.0 No - - Ac1 1.25 0.5 0.88 5.9 1.6 ∞ 0.0 Yes 1.250 0.84 Ad1 1.25 0.6 0.92 6.0 1.8 ∞ 0.0 Yes 1.250 0.90 Ad2 1.25 0.6 0.92 6.0 1.8 106 5.0 Yes 1.252 0.90 Ae1 1.25 0.6 0.90 6.8 1.5 ∞ 0.0 Yes 1.250 0.87 Ba1 1.363 0.82 0.95 5.0 12.2 ∞ 0.0 No - - Ba2 1.363 0.82 0.95 5.0 12.2 102 0.0 Yes 1.363 0.95 Ba3 1.363 0.82 0.95 5.0 12.2 103 0.0 Yes 1.363 1.12 Ba4 1.363 0.82 0.95 5.0 12.2 104 0.0 No - - Ba5 1.363 0.82 0.95 5.0 12.2 105 0.0 No - - Ba6 1.363 0.82 0.95 5.0 12.2 105 10.0 Yes 1.398 1.34 Ba7 1.363 0.82 0.95 5.0 12.2 105 5.0 Yes 1.483 1.43 Ba8 1.363 0.82 0.95 5.0 12.2 105 1.0 No - - Ta1 1.25 0.60 0.86 5.0 28.8 ∞ 0.0 No - - c© 2007 RAS, MNRAS 000, 1–?? 8 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog Figure 7. (a) Evolution of a non-rotating white dwarf accreting with a constant accretion rate of Ṁ = 10−5 M⊙ yr−1 with an initial mass of 0.9 M⊙ (Seq. N0.9) in the density – temperature plane. (b) The local effective accretion rate (Ṁeff,r := 4πr 2ρv) as a function of density in Seq. N0.9, at different evolutionary epochs as indicated by the labels. (c) – (f) The rates of energy loss/production due to neutrino (ǫν ) cooling, compressional heating (ǫcomp), nuclear energy generation (ǫnuc) and thermal diffusion (ǫth) at different evolutionary epochs. Note that here ǫν , ǫcomp and ǫnuc represent the values which are used in the evolutionary calculations, while ǫth is an order-of-magnitude estimate according to Eq. (7). Ω(Mr) = ΩO, if Mr < Mcore, ωO + (1− ωO) Mr−Mcore MCR−Mcore if Mr > Mcore, where ΩO = 0.2 GMCR/R3 and ωO = ΩO/ GMcore/r core. As shown in Fig. 6, this simple assumption gives a rotational ve- locity profile that is morphologically similar to that found in the SPH simulation: a steep gradient at the interface between the core and the envelope, and a local peak in the envelope. Within our 1-D approximation of the effects of rotation, the exact shape of the ro- tational velocity profile does not affect the main conclusions of the present work for the following reasons. Firstly, the velocity gradient at the interface is adjusted to the threshold value for the dynamical shear instability on a very short time scale (see below, and discus- sions in YL04). Secondly, our 1-D approximation underestimates the effect of the centrifugal force on the stellar structure in layers which rotate more rapidly than about 60 % critical (YL04), and un- certainties due to this limit are much greater than due to the shape of Ω(r) in the outer layers of the envelope. Possible uncertainties due to this limitation are critically discussed in Sect. 4. The central remnant may lose angular momentum by out- ward angular momentum transport into the Keplerian disc (Popham & Narayan 1991; Paczyński 1991) and/or by the gravita- tional wave radiation, e.g., due to the r-mode instability (Andersson 1998; Friedman & Morsink 1998). Our code cannot properly de- scribe any of these effects, and here we consider them simply by assuming a constant time scale for the angular momentum loss (τJ; see Knaap 2004; cf. Piersanti et al. 2003a), such that the specific c© 2007 RAS, MNRAS 000, 1–?? Remnant evolution after a carbon-oxygen white dwarf merger 9 Table 3. Accreting white dwarf model sequences with a constant accretion rate (Ṁ = 10−5 M⊙/yr). The columns list: No: sequence label, Minit: initial mass, logLinit/L⊙ : initial luminosity, MWD,ig: the total mass of the white dwarf when carbon ignites off-center, and Mr,ig: location of car- bon ignition in the mass coordinate. Sequences with ‘N’ are for non-rotating models, and ‘R’ for rotating models. No Minit logLinit/L⊙ MWD,ig Mr,ig N0.7 0.7 −2.118 0.999 0.793 N0.8 0.8 −2.128 1.010 0.862 N0.9 0.9 −2.188 1.039 0.939 N1.0 1.0 −2.137 1.087 1.024 N1.1 1.1 −2.170 1.150 1.114 N1.2 1.2 −2.119 1.225 1.207 R0.8 0.8 −2.114 1.297 1.038 R0.9 0.9 −2.119 1.249 1.050 R1.0 1.0 −2.082 1.207 1.069 R1.1 1.1 −2.050 1.205 1.127 angular momentum of each mass shell decreases over a time step ∆t by an amount ∆ji = ji [1− exp(−∆t/τJ)] . (4) Mass accretion from the Keplerian disc is also considered in some model sequences, with different values for the accretion rate (Ṁacc). The angular-momentum accretion is treated in the same way as in YL04: the accreted matter is assumed to carry angular momentum at a value close to the Keplerian value if the surface velocity of the central remnant is below critical, while no angular- momentum accretion is allowed otherwise. Model sequences with different sets of MCR, Mcore, Mp, τJ, Ṁ , and Tp are calculated, as summarized in Table 2. The initial model in Seqs S is intended to reproduce the result of our SPH sim- ulation, where MCR = 1.10 M⊙ and Mp ≈ 0.84 M⊙ are adopted. We also assume MCR = 1.25 M⊙ and Mp ≈ 0.9 M⊙ in Seq. A, and MCR = 1.364 M⊙ and Mp = 0.95 M⊙ in Seq. B, to simulate mergers of 0.9 − 1.0 M⊙ + 0.7 − 1.0 M⊙ white dwarf binaries. At a given MCR, different sets of Mcore, Mp, and Tp are marked in the sequence label by minor characters (a, b, c, d, e), while dif- ferent sets of τJ and Ṁacc are indicated by Arabian numbers. For instance, sequences Sa1 – Sa11 have the same initial merger model, but different values for τJ and Ṁacc. Rotation is neglected in a test sequence Ta1 (i.e., the models are non-rotating). The temperature and angular-velocity profiles in the initial central remnant model of Seqs Sa1 - Sa11 are shown in Fig. 6. The temperature (a few to sev- eral 108 K) and the size (∼ 109 cm) of the envelope appear to be comparable to those obtained from the SPH simulation (see Fig. 5). For comparison, we also ran model sequences for classi- cal cold-matter accretion with a constant accretion rate of Ṁ = 10−5 M⊙/yr, for both non-rotating and rotating cases, as summa- rized in Table 3. 3.2 Results 3.2.1 Classical models of cold-matter accretion Before discussing the central remnant models, let us first investi- gate the evolution of classical cold-matter accreting white dwarf models in detail. In these models, the accreted matter is assumed to have the same entropy as the surface value of the accreting white dwarf. As shown in previous studies (e.g. Nomoto & Iben 1985), the thermal evolution of rapidly accreting white dwarfs is deter- Figure 8. Local effective mass accretion rate (Ṁeff,r ≡ 4πr 2ρv) as a function of density in the models of sequence R0.9, at different evolutionary epochs. mined by the interplay of compressional heating and thermal dif- fusion. Fig. 7 shows an example of the evolution of such accret- ing white dwarf models for an initial WD mass of 0.9 M⊙ and a constant accretion rate of Ṁacc = 10 −5 M⊙ yr −1 (Seq. N0.9; Table 3). Fig. 7a shows that the temperature increases continuously in the envelope (ρ . 106 g cm−3), and finally carbon burning be- comes significant at ρ ≃ 5.6 × 105 g cm−3 and T ≃ 6× 108 K when t ≃ 1.3 × 104 yr. In Figs. 7c –f, the rates of compressional heating (ǫcomp), neutrino cooling (ǫν ), nuclear energy generation (ǫnuc) and thermal diffusion (ǫth) are shown. In our stellar evolu- tion code, the compressional heating rate is calculated according ǫcomp = (Kippenhahn & Weigert 1990). Neutrino cooling rates are obtained following Itoh et al. (1996). While ǫcomp, ǫν , and ǫnuc in the figures correspond to the values that are used for the evolutionary calcula- tions, the thermal diffusion rate (ǫth) – which is only calculated implicitly in the code – can only be estimated to within an order- of-magnitude from ǫth ≈ TCP/τth . (6) Here CP denotes the specific heat at constant pressure, and τth the local thermal diffusion time scale defined as τth ≡ H P/K , (7) where HP is the pressure scale height, and K [(4acT 3)/(3CPκρ 2)] is the thermal diffusivity. It is clear from Fig. 7 that the local peak of temperature is located where the compressional heating rate begins to dominate over the thermal diffusion rate (ρ ≈ 105 g cm−3), as expected. The neutrino cooling rate also increases as the temperature in the envelope becomes higher, but nuclear energy generation becomes significant before neutrino cooling dominates the thermal evolution, inducing a carbon-burning flash around ρ ≃ 5.6 · 105 g cm−3. As Table 3 shows, and consistent with the findings of Nomoto & Iben (1985), such off-center carbon flashes occur re- gardless of the initial mass of the white dwarf, if Ṁacc ≈ 10−5 M⊙ yr −1. The results with models including rotation show that carbon ignition may be delayed if the effect of rotation is in- cluded (Table 3; see also Piersanti et al. 2003a and Saio & Nomoto 2004). The reason is that the local effective mass accretion rate c© 2007 RAS, MNRAS 000, 1–?? 10 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog Figure 9. (a) Evolution of the central remnant in Seq. Sa1 in the log ρ − T plane. The dotted curve gives the critical temperature where the nuclear energy generation rate due to carbon burning equals the energy loss rate due to neutrino cooling. (b) The local effective accretion rate (Ṁeff,r ≡ 4πr 2ρvr) as a function of density in the merger remnant model of Seq. Sa1, at different evolutionary epochs as indicated by the labels. (c) – (f) The rates of energy loss/gain due to neutrino (ǫν ) cooling, compressional heating (ǫcomp), nuclear energy generation (ǫcomp) and thermal diffusion (ǫth) as a function of density in the central remnant models of Seq. Sa1 at different evolutionary epochs. Note that here ǫν , ǫcomp and ǫnuc represent the values which are used in the evolutionary calculations, while ǫth is an order-of-magnitude estimate according to Eq. (7). (Ṁeff,r ≡ 4πr 2ρv) inside the white dwarf at a given mass is lower because of the centrifugal force. For instance, in Seq. N0.9, we have Ṁeff,r ≈ 10 −5 M⊙ yr −1 at around ρ = 5 × 105 g cm−3 when t ≃ 104 yr (Fig. 7b), but Ṁeff,r is lowered by a factor of two in the corresponding rotating model at a similar epoch (i.e., Ṁeff,r ≈ 5 × 10 −6 M⊙ yr −1), as revealed in Fig. 8. However, carbon ignition occurs well before the white dwarf reaches the Chandrasekhar limit, in all model sequences considered. Thus, ro- tation by itself cannot change the conclusion of the previous work that the coalescence of double CO white dwarfs should lead to accretion-induced collapse rather than a thermonuclear explosion, unless the accretion rate is significantly lowered, as was also shown by Piersanti et al. (2003a) and Saio & Nomoto (2004). 3.2.2 Sequences without angular-momentum loss and mass accretion Having understood the physics of the thermal evolution of CO white dwarfs which accrete cold matter with a rate close to the Ed- dington limit, we now investigate the evolution of the central rem- nant model consisting of a cold core and a hot envelope as described in Sect. 3.1. First, we examine the results of the model sequences where both angular-momentum loss and mass accretion from the Keplerian disc are neglected (i.e., τJ = ∞ and Ṁ = 0; Seqs Sa1, Aa1, Ab1, Ac1, Ad1, Ae1, Ba1, & Ta1). Fig. 9a illustrates the evolution of the central remnant for MCR = 1.10 M⊙ in Seq. Sa1 in the density – temperature plane. Note that the local peak of temperature at t = 0.0 (Tp = 5.6× 108 K) is significantly below the critical temperature for car- c© 2007 RAS, MNRAS 000, 1–?? Remnant evolution after a carbon-oxygen white dwarf merger 11 bon ignition (TC−ig; dotted curve in Fig 9a). It is shown in Fig. 9b that the local effective accretion rate (Ṁeff,r) remains relatively high (5×10−6−10−5 M⊙ yr −1) around ρ = 106 g cm−3, where the local peak of temperature is located, for about 5000 yrs. Despite such high effective accretion rates, the temperature peak continu- ously decreases, although the inner core becomes somewhat hotter due to compression, and the central remnant finally becomes a cold white dwarf. A few remarkable differences compared to the stan- dard accreting white dwarf models are found in this regard. Firstly, since the envelope is very hot, neutrino cooling – in particular by photoneutrinos – is significant from the beginning, and even domi- nant over the thermal diffusion at the interface between the core and the envelope as shown in Fig. 9c. In cold-matter accreting white dwarfs, neutrino cooling becomes important only after a significant amount of mass has been accreted (Fig. 7). Secondly, the compres- sional heating rate is slightly lower than the neutrino cooling rate around the local peak of temperature. As the contraction of the cen- tral remnant is mainly determined by the thermal evolution of the envelope, the local accretion rate is in fact controlled by the cool- ing process. This explains why we have ǫcomp ≈ ǫν around the local peak of temperature for the initial ∼ 104 yrs, and why the local peak of temperature continuously decreases despite the rela- tively high effective accretion rate. This conclusion is the same for all other sequences with a TP that is significantly lower than TC−ig (Seq. Aa1, Ab1, & Ba1) including the non-rotating case (Seq. Ta1; Table 2). We find that, in Seq. Sa1, the differentially rotating layers at the interface between the core and the envelope in the initial central remnant model are stable against the dynamical shear instability (DSI). They are, however, unstable to the DSI in other sequences, where the interface is more degenerate (see YL04 for discussions on the DSI). Consequently, in Seq. Aa1 for example, the rate of rotational energy dissipation (ǫrot) appears to be very high initially (Fig. 10). The differentially rotating layers are rapidly smeared out by the dynamical shear instability (see the discussion in Sect. 2 in YL04), and ǫrot falls below the thermal diffusion and/or neutrino cooling rate only within 20 yrs. Hence we conclude that the rota- tional energy dissipation does not play an important role for the long-term evolution of the central remnant. Fig. 11 shows how the evolution of the central remnant changes if the local peak of temperature in the initial model (Tp) is close to or above the critical limit for carbon burning (TC−ig), with Seqs Ac1 and Ae1 as examples. In contrast to Seq. Sa1 or Aa1, car- bon burning dominates the evolution very soon in both sequences, and the temperature increases rapidly. Although the further evolu- tion has not been followed in the present study, it is most likely that the carbon-burning flame propagates inward such that the central remnant is converted into an ONeMg white dwarf within several thousand years as shown by Saio & Nomoto (1998). As summarized in Table 2, all other sequences follow the same evolutionary pattern: off-center carbon ignition is avoided in Seqs Ab1, Ba1, and Ta1 where Tp is significantly below TC−ig, while carbon ignites off-center in the other sequences where Tp & TC−ig. It is thus remarkable that the thermal evolution of the central rem- nant is sensitively determined by the local peak of temperature in the quasi-static equilibrium state. In conclusion, in the absence of angular-momentum loss and mass accretion from the Keplerian disc, the thermal evolution of the central remnant is roughly controlled by neutrino cooling at the interface between the core and the envelope, and off-center carbon burning may be avoided as long as Tp < TC−ig, while it seems inevitable if Tp & TC−ig. Figure 10. Upper panel: The angular velocity relative to the local maximum as a function of the mass coordinate in the central remnant models of Seq. Aa1 at different evolutionary epochs. Lower panel: the rate of rotational energy dissipation (ǫrot; see YL04) as a function of the mass coordinate in the corresponding models shown in the upper panel. 3.3 Effect of angular momentum loss In Seqs Sa2 – Sa5, the central remnant has the same initial condi- tions as in Seq. Sa1, angular momentum loss from the white dwarf with different time scales τJ is considered according to Eq. (4). Note that off-center carbon ignition occurs in Seqs Sa2, Sa3 & Sa4, where τJ . 10 4 yr, while it is avoided in Seq. Sa5 where τJ = 10 5 yr. These results indicate that off-center carbon igni- tion should be induced if the angular-momentum loss occurs too rapidly for neutrino cooling or thermal diffusion to control the ef- fective mass accretion. For instance, Fig. 12 shows that in Seq. Sa4, where τJ = 10 4 yr, the effective mass accretion rate reaches a few 10−5 M⊙ yr −1 at the interface between the core and the envelope (ρ ≈ 106 g cm−3), and the compressional heating rate exceeds the neutrino cooling rate. It is shown that the critical angular-momentum-loss time scale, τJ, for off-center carbon ignition (τJ,crit) is smaller for Seqs Ab and Ba than for Seqs Sa and Aa: τJ,crit ≈ 10 3 for Seqs Ab and Ba, and τJ,crit ≈ 10 4 for Seqs Sa and Aa. This is due to the different local thermodynamic properties at the interface between the core and the envelope in different central remnant models. As shown in Fig. 13, higher density and/or temperature at the interface result in a shorter neutrino cooling time, making it possible to avoid local heating for a smaller τJ. In other words, τJ,crit roughly cor- responds to the time scale for neutrino cooling at the local peak of temperature (τν,p). From this experiment, we conclude that, in the absence of c© 2007 RAS, MNRAS 000, 1–?? 12 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog Figure 11. Evolution of the central remnant in Seqs Ac1 (upper panel) and Ae1 (lower panel), in the log ρ−T plane. The dotted curve gives the critical temperature where nuclear energy generation rate due to carbon burning equals to energy loss rate due to neutrino cooling. mass accretion from the Keplerian disc, carbon ignition may be avoided in the central remnant, if Tmax,init < TC−ig, and if τJ > τν,p. 3.4 Mass accretion from the Keplerian disc In reality, mass accretion from the Keplerian disc onto the central remnant is expected. The accretion rate is determined by the vis- cosity of the disc, which is not well known. However, we expect the accretion rate from a Keplerian disc may be significantly lower than from a pressure-supported thick disc that was assumed in pre- vious studies. Our results, as summarized in Table 2, indicate that even with mass accretion, the central remnant with Tp < TC−ig can avoid off-center carbon ignition if the accretion rate is sufficiently low (i.e., Ṁ < 5× 10−6...10−5 M⊙ yr −1), and if τJ > τν,p (see Table 2). The thermal history of the central remnant in those sequences where carbon ignites off-center is similar to that of the white dwarf in classical accretion model sequences. However, as the central remnant has a rapidly rotating hot envelope, carbon ignition is sig- nificantly delayed compared to the case of classical accretion. In Seq. N1.2, where Minit = 1.2 M⊙ and Ṁacc = 10 −5 M⊙ yr carbon ignites only when about 0.025 M⊙ is accreted, while in Seq. Aa7 more than 0.15 M⊙ have to be accreted to induce carbon ignition at the same accretion rate, despite its higher initial mass. On the other hand, the comparison of Seq. Aa7 with Seq. Aa10 indicates that off-center carbon ignition is delayed if the central remnant keeps more angular momentum. The critical accretion rate for inducing off-center carbon ignition is thus difficult to precisely determine, as our 1-D models significantly underestimate the effect of the centrifugal force, especially in the envelope where carbon ignites. In addition, the physics of angular momentum loss/gain is not well understood yet, as discussed in Yoon & Langer (2005). Note that MWD,ig in Seqs Aa10, Ba6 and Ba7 is already very close, or even above the Chandrasekhar limit. However, the cen- tral density in those models is still smaller by an order of mag- nitude than the critical limit for carbon ignition due to the effect of rotation. As the carbon-burning flame will propagate inwards within several thousand years (Saio & Nomoto 1998), only about ∼ 0.05 M⊙ may be further accreted by the time the burning flame reaches the center, and the central density may not become high enough to induce a thermonuclear explosion before the whole cen- tral remnant is converted into an ONeMg white dwarf. (Super-) Chandrasekhar mass ONeMg white dwarfs produced in this way will eventually collapse to a neutron star (see Yoon & Langer 2005; Dessart et al. 2006). On the other hand, the white dwarf continuously grows to/above the Chandrasekhar limit (≈ 1.4 M⊙) without suffering carbon ignition (neither at the center nor off-center) in Seqs Sa8, Sa9, Sa10, Sa11, Aa8, Aa9, Ab6, and Ba8. The outcome in these cases is thus the formation of a (super-) Chandrasekhar mass CO white dwarf, which will eventually explode as a Type Ia super- nova. The mass of the exploding white dwarf should depend on the amount of angular momentum (Yoon & Langer 2005) and can- not exceed the mass budget of merging white dwarfs. Fig. 14 shows the evolutionary paths of the central remnant for Seqs Sa8, Sa9 & Sa11 as examples in the mass – angular momentum plane. Note that the central remnant initially has a large amount of angular momen- tum (J = 1.11×1050 erg s), such that without loss/gain of angular momentum, it should accrete matter until it reaches M ≃ 1.68 M⊙ where it explodes in a SN Ia explosion. In Seqs Sa8 and Sa9, the ac- cretion time scale (τacc) is longer than the angular momentum loss time scale, and the total angular momentum of the white dwarf con- tinuously decreases while the total mass increases. Consequently, carbon ignites at the center when the white dwarf grows to 1.50M⊙ and 1.42 M⊙ for Seqs Sa8 and Sa9, respectively. In Seq. Sa11, on the other hand, both mass and angular momentum of the central remnant continuously increase, given that τacc . τJ, and a SN Ia explosion is expected only when M ≃ 1.70 M⊙. Note that this is even larger than the mass budget of the binary system consid- ered for this sequence (i.e, 0.9 M⊙+0.6 M⊙). In nature, the white dwarf must stop growing in mass when M = 1.5 M⊙, and a SN Ia explosion will be induced only when a sufficient amount of angular momentum has been removed, e.g. via gravitational wave radiation, as illustrated by the path Sa11-B in Fig. 14. 4 CONCLUSION AND DISCUSSION We have explored the dynamical and secular evolution of the merger of double CO white dwarf binaries whose total mass ex- ceeds the Chandrasekhar limit. Based on our new SPH simula- tion of the coalescence of two CO white dwarfs of 0.9 M⊙ and 0.6 M⊙, we suggest that the immediate post-merger remnant is best described as a differentially rotating CO star consisting of a slowly rotating cold core and a rapidly rotating hot envelope that is surrounded by a Keplerian disc rather than as “cold white dwarf + thick disc” system, as in previous investigations. The evolution of such a CO star is determined by the thermal evolution of the en- velope, and the growth of the core is controlled by the cooling due c© 2007 RAS, MNRAS 000, 1–?? Remnant evolution after a carbon-oxygen white dwarf merger 13 Figure 12. Same as in Fig. 9, but for Seq. Sa4. Figure 13. Contour lines of the neutrino cooling time scale (τν ≡ TCp/ǫν ) in the log ρ − T plane. The level at each line gives log τν in units of years. The dashed line denotes the critical temperature for carbon ignition. The local peak of temperature and the corresponding density in the initial model of Seqs Sa, Aa, Ab, Ac, Ae, and Ba are marked by the filled symbols as indicated by the labels. to neutrino emission and thermal diffusion, which is fundmentally different from the assumption of “forced accretion of cold matter”. Our 1-D stellar evolution models of the central remnant, i.e. the cold core and the hot envelope, which include the effects of rotation, indicate that there are three necessary conditions for the merger remnant to avoid off-center carbon ignition such that a SN Ia may be produced: (i) The local peak of temperature of the merger remnant at the interface between the core and the envelope must be lower than the critical temperature for carbon ignition (Tp < TC−ig). (ii) The time scale for angular-momentum loss from the central remnant by must be larger than the neutrino cooling time scale at the interface (τJ > τν,P). (iii) Mass accretion from the Keplerian disc onto the cen- tral remnant must be sufficiently slow (Ṁacc . 5 × 10−6...10−5 M⊙ yr Our new SPH simulation confirms that at least the first condition (Tp < TC−ig) should be fulfilled in the CO white dwarf binary considered. As emphasized in Sect. 3.1, our 1-D models significantly un- derestimate the effect of the centrifugal force on the stellar structure in the rapidly rotating outermost layers. However, since thermal diffusion always dominates over both neutrino cooling and com- pressional heating in the outer envelope (ρ . 105...106 g cm−3) above the interface, as shown in Figs. 7 and 12, the detailed struc- ture of the rapidly rotating outermost layers above the interface may not significantly affect our results on the thermal evolution of the merger remnant, as long as the angular momentum of the envelope is not lost faster than the local neutrino cooling time scale at the interface. On the other hand, mass accretion from the Keplerian disc should occur preferentially along the equatorial plane of the envelope. As shown in the SPH simulation, the envelope is more extended along the equatorial plane, where most angular momen- tum is deposited, than along the polar axis, and the resultant com- pressional heating must be much weakened, compared to the case of our 1-D models. The enhanced role of rotation must thus help c© 2007 RAS, MNRAS 000, 1–?? 14 S.-C. Yoon, Ph. Podsiadlowski & S. Rosswog to increase the critical mass accretion rate for inducing off-center carbon ignition, in favor of producing a Type Ia supernova. We have concluded that the loss of angular momentum on a short time scale (τJ . τν,p ≈ 10 4...105 yr) may induce off-center carbon ignition even when Tmax,init < TC−ig. Rapidly rotating compact stars may experience loss of angular momentum by gravi- tational wave radiation, due to either the bar-mode instability or the r-mode instability. The onset of the dynamical or secular bar-mode instability requires a very high ratio of the rotational energy to the gravitational energy: Erot/Egrav & 0.2 for the dynamical bar- node instability, and Erot/Egrav & 0.14 for the secular bar-mode instability (e.g. Shapiro & Teukolsky 1983). As both our 1-D mod- els and SPH simulation give a value of Erot/Egrav that is much lower (about 0.06 – 0.07) than those critical limits, the bar-mode in- stability may not be relevant. The r-mode instability may operate, in principle, even with such a low Erot/Egrav (Andersson 1998; Friedman & Morsink 1998). However, we estimate that the growth time of the r-mode instability (τr), using our central remnant mod- els and following Lindblom (1999), is & 106 yr, which is much longer than the local neutrino cooling time scale (τν,p ≈ 10 4 yr). Alternatively, angular momentum might be transported from the accreting star into the Keplerian disc when the accretor reaches critical rotation. Calculations by Saio & Nomoto (2004) indicate, however, that the decrease of the total angular momentum due to such an effect is not significant in accreting white dwarfs. In con- clusion, neither gravitational wave radiation nor outward angular- momentum transport is likely to lead to a rapid loss of angular momentum from the central remnant such that τJ < τν,p, unless magnetic torques are important. The central remnant may be enforced to rotate rigidly on a short time scale in the presence of strong magnetic torques (cf. Spruit 2002). The central remnant in both our SPH simulation and 1-D models has Jtot > 10 50 erg s, which is significantly higher than the maximum limit a rigidly rotating white dwarf can retain, as shown in Fig. 14. This means that if magnetic torques led to rigid rotation, a large amount of angular momentum should be trans- ported into the Keplerian disc (Case a in Fig. 14), or mass shed- ding of super-critically spun-up layers should occur from the cen- tral remnant (Case b in Fig. 14). In Case a, the local density around the interface should increase by several factors by the time when the central remnant reaches rigid rotation as implied by Fig. 15. Off-center carbon ignition might be inevitable in this case due to a resultant high effective accretion rate, if the time for angular mo- mentum redistribution were shorter than the local cooling time due to neutrino losses. In Case b, on the other hand, the local density at the interface might not increase if mass shedding from the cen- tral remnant occurred at a sufficiently high rate. Therefore, the role of magnetic fields in the merger evolution remains uncertain at the current stage and is a challenging subject for future work. The coalescence of more massive double CO white dwarf bi- naries is likely to result in a higher maximum temperature due to the enhanced role of gravity. Consequently, given the important role of the maximum temperature in the merger remnant for its final fate, less massive binary CO white dwarfs may be favored for the pro- duction of SNe Ia from such a channel. We note that there are number of potentially important factors that have not been included in either the present study or previous simulations. These include the following points: (i) The previous and present simulations assumed that white dwarfs are cold prior to the merging process. However, Iben, Tutukov & Fedorova (1998) point out that tidal interactions Figure 14. Evolution of the central remnant in the mass – angular momen- tum plane. The thick solid curve shows the angular momentum of a rigidly rotating white dwarf with critical rotation at the surface as a function of the white dwarf mass. The thick dashed curve and the thick dot-dashed curve give the critical angular momentum for a differentially rotating CO white dwarf to reach carbon ignition at the center (ρc = 2 × 109 g cm−3), and electron-capture induced collapse (ρc = 10×1010 g cm−3), respectively, according to Yoon & Langer (2005). A SN Ia explosion is expected in the hatched region. The filled circle denotes the initial model of the central rem- nant in Seq. Sa. The evolution of the central remnant in Seqs Sa 8, Sa9 and Sa11 is shown by the thin dotted curves, as indicated. The thin solid curves denote possible evolutionary paths of the central remnant with strong mag- netic torques that may enforce rigid rotation, with loss of angular momen- tum but without mass shedding (Case ’a’), and with both loss of angular momentum and mass shedding (Case ’b’). See the text for more details. Figure 15. The density profile in the initial model of the central remnant in Seqs Sa (solid curve), and in a corresponding hot (Tc = 108 K) white dwarf model that rotates rigidly at critical rotation at the surface (dashed curve). might heat up the white dwarfs as the orbit shrinks, which could weaken the gravitational potential of the primary. Furthermore, as the temperature of white dwarfs is a function of their age, younger progenitors should have more extended envelopes, which may re- sult in a lower Tp. (ii) A thin hydrogen/helium envelope must be present initially in both the primary and the secondary. As hydrogen or helium should ignite at a much lower temperature than carbon, the influence of the release of nuclear energy during the merger process may be even more important than shown in the existing SPH simulations, c© 2007 RAS, MNRAS 000, 1–?? Remnant evolution after a carbon-oxygen white dwarf merger 15 which is likely to lower Tp. Furthermore, neutrino losses, which were neglected in the present study, would also tend to reduce Tp. (iii) At a given total mass (Mtot = Mprimary + Msecondary), different mass ratios of the white dwarf components (q ≡ Msecondary/Mprimary) must result in different merger structures. (iv) A lower q at a given Mtot may not only lead to a stronger gravitational potential of the primary, but also to a lower mass accretion rate during the dynamical mass transfer (Guerrero et al. 2004). As the former and the latter will tend to increase and de- crease Tp, respectively, quantitative studies are necessary to predict how Tp will change with q. Finally, another important ingredient that needs to be consid- ered is thermal diffusion during the dynamical evolution. As shown above, the mass-accretion rate from the Keplerian disc onto the en- velope of the central remnant is one of the most important factors that critically determine the final fate of double CO white dwarf mergers. The accretion rates depend on the structure of the Kep- lerian disc at thermal equilibrium, which can be only understood by including thermal diffusion in future simulations. But here we emphasize again that the accretion rates from a centrifugally sup- ported Keplerian disc should be significantly lower than those from a pressure-supported thick disc that was previously assumed, which opens the possibility for at least some double CO white dwarf mergers to produce SNe Ia. ACKNOWLEDGMENTS We are grateful to Norbert Langer and Ken’ichi Nomoto for many useful suggestions and comments. SCY is supported by the VENI grant (639.041.406) of the Netherlands Organization for Scientific Research (NWO). The computations have been performed on the JUMP supercomputer at the Höchstleistungsrechenzentrum Jülich. 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0704.0298

Direct Theorems in the Theory of Approximation of the Banach Space Vectors by Entire Vectors of Exponential Type

DIRECT THEOREMS IN THE THEORY OF APPROXIMATION OF BANACH SPACE VECTORS BY EXPONENTIAL TYPE ENTIRE VECTORS YA. GRUSHKA AND S. TORBA Abstra t. For an arbitrary operator A on a Bana h spa e Xwhi h is the generator of C0�group with ertain growth ondition at in�nity, the dire t theorems on onne - tion between the smoothness degree of a ve tor x ∈ X with respe t to the operator A, the rate of onvergen e to zero of the best approximation of x by exponential type entire ve tors for the operator A, and the k-module of ontinuity are established. The results allow to obtain Ja kson-type inequalities in a number of lassi spa es of periodi fun tions and weighted Lp spa es. 1. Introdu tion The dire t and inverse theorems establishing a relationship between the smoothness degree of a fun tion with respe t to the di�erentiation operator and the rate of onver- gen e to zero of its best approximation by trigonometri polynomials are well known in the theory of approximation of periodi fun tions. Ja kson's inequality is one among su h results. N. P. Kuptsov proposed a generalized notion of the module of ontinuity, expanded onto C0-groups in a Bana h spa e [1℄. Using this notion, N. P. Kuptsov [1℄ and A. P. Terekhin [2℄ proved the generalized Ja kson's inequalities for the ases of a bounded group and s-regular group. Remind that the group {U(t)}t∈R is alled s-regular if the resolvent of its generator A satis�es the ondition ∃θ ∈ R : ‖Rλ(eiθAs)‖ ≤ CImλ . G. V. Radzievsky studied the dire t and inverse theorems [3, 4℄, using the notion of K- fun tional instead of module of ontinuity, but it should be noted that the K-fun tional has two-sided estimates with regard to module of ontinuity at least for bounded C0- groups. In the papers [5, 6℄ and [7℄ the authors investigated the ase of a group of unitary operators in a Hilbert spa e and established Ja kson-type inequalities in Hilbert spa es and their rigs. These inequalities are used to estimate the rate of onvergen e to zero of the best approximation of both �nite and in�nite smoothness ve tors for the operator A by exponential type entire ve tors. We onsider the C0-groups, generated by the so- alled non-quasianalyti operators [8℄, i.e. the groups satisfying (1.1) ln ‖U(t)‖ 1 + t2 dt < ∞. As was shown in [5℄, the set of exponential type entire ve tors for the non-quasianalyti operator A is dense in X, so the problem of approximation by exponential type entire Re eived by the editors 04/04/2007. 2000 Mathemati s Subje t Classi� ation. Primary 41A25, 41A17, 41A65. Key words and phrases. Dire t and inverse theorems, modulo of ontinuity, Bana h spa e, entire ve tors of exponential type. This work was partially supported by the Ukrainian State Foundation for Fundamental Resear h (proje t N14.1/003). http://arxiv.org/abs/0704.0298v3 2 YA. GRUSHKA AND S. TORBA ve tors is orre t. On the other hand, it was shown in [9℄ that ondition (1.1) is lose to the ne essary one, so in the ase when (1.1) doesn't hold, the lass of entire ve tors isn't ne essary dense in X, and the orresponding approximation problem loses its meaning. The purpose of this work is to obtain Ja kson-type inequalities in the ase where a ve tor of a Bana h spa e is approximated by exponential type entire ve tors for a non- quasianalyti operator, and, in parti ular, Ja kson-type inequalities in various lassi al fun tion spa es. 2. Preliminaries Let A be a losed linear operator with dense domain of de�nition D(A) in the Bana h spa e (X, ‖·‖) over the �eld of omplex numbers. Let C∞(A) denotes the set of all in�nitely di�erentiable ve tors of the operator A, i.e. C∞(A) = D(An), N0 = N ∪ {0}. For a number α > 0 we set α(A) = x ∈ C∞(A) | ∃c = c(x) > 0 ∀k ∈ N0 ∥∥Akx ∥∥ ≤ cαk The set E α(A) is a Bana h spa e with respe t to the norm Eα(A) = sup ‖Anx‖ Then E(A) = α>0 E α(A) is a linear lo ally onvex spa e with respe t to the topology of the indu tive limit of the Bana h spa es E α(A): E(A) = lim ind α(A). Elements of the spa e E(A) are alled exponential type entire ve tors of the operator A. The type σ(x,A) of a ve tor x ∈ E(A) is de�ned as the number σ(x,A) = inf {α > 0 : x ∈ Eα(A)} = lim sup ‖Anx‖ Example 2.1. Let X is one of the Lp(2π) (1 ≤ p < ∞) spa es of integrable in p-th degree over [0, 2π], 2π-periodi al fun tions or the spa e C(2π) of ontinuous 2π-periodi al fun tions (the norm in X is de�ned in a standard way), and let A is the di�erentiation operator in the spa e X (D(A) = {x ∈ X∩AC(R) : x′ ∈ X}; (Ax)(t) = dx , where AC(R) denotes the spa e of absolutely ontinuous fun tions over R). It an be proved that in su h ase the spa e E(A) oin ides with the spa e of all trigonometri polynomials, and for y ∈ E(A) σ(y,A) = deg(y), where deg(y) is the degree of the trigonometri polynomial In what follows, we always assume that the operator A is the generator of the group of linear ontinuous operators {U(t) : t ∈ R} of lass C0 on X. We re all that belonging of the group to the C0 lass means that for every x ∈ X the ve tor-fun tion U(t)x is ontinuous on R with respe t to the norm of the spa e X. For t ∈ R+, we set MU (t) := sup τ∈R, |τ |≤t ‖U(τ)‖ . The estimation ‖U(t)‖ ≤ Meωt for some M,ω ∈ R implies MU (t) < ∞ (∀t ∈ R+). It is easy to see that the fun tion MU (·) has the following properties: 1) MU (t) ≥ 1, t ∈ R+; 2) MU (·) is monotoni ally non-de reasing on R+; 3) MU (t1 + t2) ≤ MU (t1)MU (t2), t1, t2 ∈ R+. DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 3 A ording to [1℄, for x ∈ X, t ∈ R+ and k ∈ N we set ωk(t, x, A) = sup 0≤τ≤t ∥∥∆kτx ∥∥ , where(2.1) ∆kh = (U(h)− I)k = (−1)k−j U(jh), k ∈ N0, h ∈ R (∆0h ≡ 1).(2.2) Moreover, let (2.3) ω̃k(t, x, A) = sup |τ |≤t ∥∥∆kτx Remark 2.1. It is easy to see that in the ase of the bounded group {U(t)} (‖U(t)‖ ≤ M, t ∈ R) the quantities ωk(t, x, A) and ω̃k(t, x, A) are equivalent within onstant fa tor (ωk(t, x, A) ≤ ω̃k(t, x, A) ≤ M ωk(t, x, A)), and in the ase of isometri group (‖U(t)‖ ≡ 1, t ∈ R) these quantities oin ide. It is immediate from the de�nition of ω̃k(t, x, A) that for k ∈ N: 1) ω̃k(0, x, A) = 0; 2) for �xed x the fun tion ω̃k(t, x, A) is non-de reasing and is ontinuous by the variable t on R+; 3) ω̃k(nt, x,A) ≤ 1 + (n− 1)MU ((n− 1)t) ω̃k(t, x, A) (n ∈ N, t > 0); 4) ω̃k(µt, x,A) ≤ 1 + µMU (µt) ω̃k(t, x, A) (µ, t > 0); 5) for �xed t ∈ R+ the fun tion ω̃k(t, x, A) is ontinuous in x. For arbitrary x ∈ X we set, a ording to [7, 6℄, Er(x,A) = inf y∈E(A) :σ(y,A)≤r ‖x− y‖ , r > 0, i.e. Er(x,A) is the best approximation of the element x by exponential type entire ve tors y of the operator A for whi h σ(y,A) ≤ r. For �xed x Er(x,A) does not in rease and Er(x,A) → 0, r → ∞ for every x ∈ X if and only if the set E(A) of exponential type entire ve tors is dense in X. Parti ularly, as indi ated above, the set E(A) is dense in X if the group {U(t) : t ∈ R} belongs to non-quasianalyti lass. 3. Abstra t Ja kson's inequality in a Bana h spa e Theorem 3.1. Suppose that {U(t) : t ∈ R} satis�es ondition (1.1). Then ∀k ∈ N there exists a onstant mk = mk(A) > 0, su h that ∀x ∈ X the following inequality holds: (3.1) Er(x,A) ≤ mk · ω̃k , x, A , r ≥ 1. Remark 3.1. If, additionally, the group {U(t)} is bounded (MU (t) ≤ M̃ < ∞, t ∈ R), then the assumption r ≥ 1 an be hanged to r > 0. Integral kernels, onstru ted in [10℄, will be used in the proving of the theorem. More- over, we need additional properties of these kernels, la king in [10℄. The following lemma shows how these kernels are onstru ted and ontinues the investigation of their proper- ties. In what follows we denote as Q the lass of fun tions α : R 7→ R, satisfying the following onditions: I) α(·) is measurable and bounded on any segment [−T, T ] ⊂ R. II) α(t) > 0, t ∈ R. III) α(t1 + t2) ≤ α(t1)α(t2), t1, t2 ∈ R. |ln(α(t))| dt < ∞. 4 YA. GRUSHKA AND S. TORBA Lemma 3.1. Let α ∈ Q. Then there exists su h entire fun tion Kα : C 7→ C that 1) Kα(t) ≥ 0, t ∈ R; −∞ Kα(t) dt = 1; 3) ∀r > 0 ∃cr = cr(α) > 0 ∀z ∈ C |Kα(rz)| ≤ cr e r|Im z| α(|z|) Proof. Without lost of generality we may assume that the fun tion α(t) satis�es addi- tional onditions: V) α(t) ≥ 1, t ∈ R; 1 VI) α(t) is even on R and is monotoni ally in reasing on R+; ∥∥α−1 L1(R) −∞ |α −1(t)|dt < ∞. It is easy to verify that assumptions V),VII) and ondition that the fun tion α(t) is even in VI) don't on�ne the general ase if one examined the fun tion α1(t) = α̃(t)α̃(−t), where α̃(t) = (1 + α(t))(1 + t2). In [11, theorems 1 and 2℄ it has been proved that the monotony ondition on α(t) in VI) doesn't on�ne the general ase too. It follows from VII) that (3.2) α(t) → ∞, t → ∞. Let β(t) = lnα(t), t ∈ R. Conditions III)-VII) and (3.2) lead to on lusion that β(t) > 0, β(−t) = β(t), β(t) → ∞, t → ∞; β(t1 + t2) ≤ β(t1) + β(t2), t1, t2 ∈ R(3.3) dt < ∞(3.4) Be ause of (3.3) there exists limit limt→∞ . And, by virtue of (3.4): (3.5) lim Also, using (3.4) it is easy to he k that (3.6) moreover, all terms of the series (3.6) are positive. From the onvergen e of series (3.6) follows the existen e of su h sequen e {Qn}∞n=1 ⊂ R that Qn > 1, Qn → ∞, n → ∞ (3.7) Qk = S < ∞. We set ak := β(k)Qk , k ∈ N. The de�nition of ak and (3.7) result in equality (3.8) ak = 1. We onstru t the sequen e of fun tions, whi h, obviously, are entire for every n ∈ N: fn(z) := Pk(z), where Pk(z) = sin akz , z ∈ C, n ∈ N. As shown in [8℄, for non-quasianalyti groups the ondition ‖U(t)‖ ≥ 1 always holds, therefore in this paper the ondition V) automati ally takes pla e. DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 5 Similarly to the proof of the Denjoy-Carleman theorem [12, p.378℄ it an be on luded that the sequen e of (entire) fun tions fn(z) onverges uniformly to the fun tion f(z) = sin akz , z ∈ C in every disk {z ∈ C | |z| ≤ R}. Thus, by Weierstrass theorem, the fun tion f(z) is entire. Using the inequality | sin z| ≤ min(1, |z|)e|Imz|, z ∈ C and taking (3.8) into a ount, when z ∈ C and r > 0, we re eive |f(rz)| = sin akrz akr|z| akr|z| akr|Im z| = er|Im z| akr|z| ≤ er|Im z| akr|z| for every N ∈ N. Using the inequality min(1, a) ·min(1, b) ≤ min(1, ab), we get: (3.9) |f(rz)| ≤ er|Im z|min2 akr|z| = er|Im z|min2 2N(∏N β(k)Qk (r|z|)N = er|Im z|min2 2NN ! · · · β(N) )N |z|NQ1 · · ·QN Be ause of the ondition Qn → ∞, n → ∞ there exists su h number n(r) ∈ N that: (3.10) ∀n > n(r) Qn ≥ It follows from (3.5) that there is T0 ∈ (0,∞) su h that: (3.11) ∀ t > T0 In [10℄ the following statement was proved: (3.12) ∀ t1, t2 ∈ R+ t1 ≤ t2 ⇒ β(t1) β(t2) Let z ∈ C and |z| ≥ max β[−1](n(r)), T0 , where β[−1] is the inverse fun tion of the fun tion β on [0,∞) (the inverse fun tion β[−1] exists due to monotony of β on [0,∞)). We substitute as N in (3.9) N := [β(|z|)], where [·] denotes the integer part of a number. Then for k ∈ {1, . . . , N}, in a ordan e with (3.11) and (3.12), we obtain k ≤ N ≤ β(|z|) ≤ |z| and (3.13) β(|z|) Using (3.9),(3.10),(3.13), we �nd |f(rz)| ≤ er|Im z| 2NN ! β(|z|) )N ( r )N |z|NQ1 · · ·QN ≤ er|Imz| 2NN ! )N ( r )N |z|NQ1 · · ·QN = er|Im z| 22NN ! Q1 · · ·QN 6 YA. GRUSHKA AND S. TORBA ≤ er|Im z| Q1 · · ·QN = er|Im z| Q1 · · ·QN Sin e Qn ≥ 1, the last inequality leads to (3.14) |f(rz)| ≤ er|Im z| )N−n(r) = er|Im z| )2n(r) e−[β(|z|)] ≤ ≤ er|Im z| )2n(r) e−(β(|z|)−1) = C(1)r er|Im z| α(|z|) where C r = e )2n(r) . When z ∈ C and |z| < max β[−1](n(r)), T0 , using (3.9), we get (3.15) |f(rz)| ≤ er|Im z| = er|Im z|α(|z|) α(|z|) ≤ er|Im z| C α(|z|) where C r = α(max(β [−1](n(r)), T0)). It follows from (3.14), (3.15) that (3.16) |f(rz)| ≤ er|Imz| α(|z|) , z ∈ C, where C(0)r = max(C(1)r , C(2)r ). Inequality (3.16) and Condition VII) imply that ‖f‖L1(R) < ∞. Thus it is enough to set Kα(z) := 1‖f‖ L1(R) f(z), z ∈ C and use (3.16) to �nish the proof. � Let α ∈ Q, and Kα : C 7→ C is the fun tion onstru ted by the fun tion α in lemma 3.1. We set Kα,r(z) := rKα(rz), z ∈ C, r ∈ (0,∞). The lemma 3.1 ensures us that the fun tion Kα,r has the following properties: 1) Kα,r(t) ≥ 0, t ∈ R; −∞ Kα,r(t) dt = 1; 3) ∀z ∈ C |Kα,r(z)| ≤ rcr e r|Im z| α(|z|) ; r > 0. Lemma 3.2. ∀r ∈ (0,∞) there exists onstant c̃r = c̃r(α) > 0, su h that ∀n ∈ N the following inequality holds: |K(n)α,r(t)| ≤ c̃r α(|t|) rn, t ∈ R Proof. In what follows in this proof we assume t ∈ R, r ∈ (0,∞), n ∈ N. Let γn,r(t) := ζ ∈ C : |ζ − t| = n Using Cau hy's integral theorem and Stirling's approximation for n!, we get |K(n)α,r(t)| ≤ γn,r(t) |Kα,r(ξ)| |ξ − t|n+1 |dξ| = n! γn,r(t) |Kα,r(ξ)||dξ| ≤ (!)rn+1√ γn,r(t) |Kα,r(ξ)||dξ|, where c(!) = sup < e1/12. DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 7 Using property 3) of the fun tion Kα,r, the ondition t ∈ R and onditions III), VI) of the fun tion α, one an �nd from the last inequality |K(n)α,r(t)| ≤ c(!)rn+1√ e−nrcr γn,r(t) er|Im ξ| α(|ξ|) |dξ| = c(!)rn+1√ α(|t|) γn,r(t) er|Im (ξ−t)|α(|(t− ξ) + ξ|) α(|ξ|) |dξ| ≤ (!)rn+1√ α(|t|) γn,r(t) er|Im (ξ−t)|α(|t− ξ|)|dξ| ≤ (!)rn+1√ α(|t|) γn,r(t) |dξ| = c̃r α(|t|) where c̃r = c (!)rcr . � Remark 3.2. If the fun tion α(t) satis�es the onditions of lemma 3.1, but, moreover, has the polynomial order of growth at in�nity, i.e. ∃m ∈ N0, ∃M > 0: (3.17) α(t) ≤ M(1 + |t|)2m, t ∈ R, another integral kernel may be used: K̃α(z) = sin z , Km = sin x In mu h the same way to the proving of the lemmas 3.1 and 3.2 one an show that ∣∣K̃α(rz) ∣∣ ≤ C̃r er|Im z| α(|z|) , where C̃r = ∣∣K̃(n)α,r(t) ∣∣ ≤ c̃r 2πnα(n α(|t|) rn, where c̃r = c (!)rC̃r , that is to say, de�ned in su h a way integral kernel satis�es lemmas 3.1 and 3.2. Proof of theorem 3.1. Let the group {U(t) : t ∈ R} satis�es (1.1). Then it follows from [11, theorems 1 and 2℄ that (3.18) ln (MU (|t|)) 1 + t2 dt < ∞. We �x arbitrary k ∈ N and set α(t) := MU (|t|) (1 + |t|)k+2, t ∈ R. The fun tion α is, obviously, even on R. Condition (3.18) and the properties of the fun tion MU (·) imply α ∈ Q, and, moreover, (3.19) (1 + |t|)MU (|t|) (1 + |t|)2 Using lemma 3.1 (or remark 3.2 if α(t) ≤ M(1+ |t|)m) for the fun tion α(t), we onstru t the family of kernels Kα,r. In what follows, we assume x ∈ X, r ∈ (0,∞) and n ∈ {1, . . . , k}. We de�ne xr,n := Kα,r(t)U(nt)x dt. 8 YA. GRUSHKA AND S. TORBA Let ν ∈ N0. Let's prove that xr,n ∈ C∞(A) = ν∈N0 D(A ν ) and (3.20) Aνxr,n = (−1)ν K(ν)α,r(t)U(nt)x dt. It follows from the property 3) of the fun tion Kα,r and from lemma 3.2 that there exists su h onstant C̃(ν, r) > 0 that K(ν)α,r(t) ≤ eC(ν,r) , t ∈ R. Thus, using (3.19), we get (3.21) ∥∥∥K(ν)α,r(t)U(nt)x ∥∥∥ dt ≤ C̃(ν, r) ‖U(t)‖n ‖x‖ dt ≤ ≤ C̃(ν, r) ‖x‖ MU (|t|)k dt ≤ 2C̃(ν, r) ‖x‖ < ∞. Therefore the integral α,r(t)U(nt)x dt onverges. We de�ne x(ν)r,n = (−1)ν K(ν)α,r(t)U(nt)x dt. Then, using losedness of the operator A and integration by parts, one an �nd for x ∈ D(A) that x(ν)r,n ∈ D(A) and (3.22) Ax(ν)r,n = (−1)ν K(ν)α,r(t)U(nt)Axdt = (−1)ν K(ν)α,r(t)(U(nt)x)′dt = = − (−1) K(ν+1)α,r (t)U(nt)x dt = x(ν+1)r,n . Let x is an arbitrary element of the spa e X. Then there exists the sequen e {xm}∞m=1 ⊂ D(A) su h that ‖xm − x‖ → 0, m → ∞. Consequently, using inequality (3.21) and relation (3.22), one an get ∥∥∥(xm)(ν)r,n − x(ν)r,n ∥∥∥ ≤ ∥∥∥K(ν)α,r(t)U(nt)(xm − x) ∥∥∥ dt ≤ 2C̃(ν, r) ‖xm − x‖ → 0; ∥∥∥A(xm)(ν)r,n − x(ν+1)r,n ∥∥∥ = ∥∥∥(xm)(ν+1)r,n − x(ν+1)r,n ∥∥∥→ 0, m → ∞. Hen e, taking into a ount losedness of the operator A, we have: (3.23) x(ν)r,n ∈ D(A), Ax(ν)r,n = x(ν+1)r,n . One an get (3.20) from (3.23) by indu tion. Using relation (3.20) and lemma 3.2, one an �nd: (3.24) ‖Aνxr,n‖ ≤ ∣∣∣K(ν)α,r(t) ∣∣∣ ‖U(nt)‖ dt ≤ ≤ ‖x‖ 2πν α α(|t|) rν ‖U(t)‖n dt ≤ ≤ c̃r ‖x‖ 2πν α )(∫ ∞ ‖U(t)‖n where, a ordingly to (3.19) and due to n ≤ k, ‖U(t)‖n ‖U(t)‖k dt ≤ 2 < ∞. Sin e β(t) = ln(α(t)), t ∈ R, as was mentioned in the proof of lemma 3.1, limτ→∞ β(τ)τ = 0 ( f. (3.5)). Thus ))1/ν = lim β( νr )) = e ·0 = 1. DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 9 Therefore from relation (3.24) one an get: lim sup ‖Aνxr,n‖ )1/ν ≤ r The last inequality brings us to the on lusion that (3.25) xr,n ∈ E(A) and σ(xr,n, A) ≤ For arbitrary x ∈ X we de�ne (3.26) x̃r,k := Kα,r(t)(x + (−1)k−1(U(t)− I)kx)dt = Kα,r(t) (−1)n+1 U(nt)x dt (the absolute onvergen e by the norm of X of the integral in the right part of (3.26) follows from inequality (3.21), so the de�nition of the ve tor x̃r,k is orre t). Using de�nition (3.26) one an get: x̃r,k = (−1)n+1 Kα,r(t)U(nt)xdt = (−1)n+1 xr,n. Therefore, a ordingly to (3.25), x̃r,k ∈ E(A) and σ(x̃r,k, A) ≤ r. Hen e for an arbitrary x ∈ X we have: Er(x,A) = inf y∈E(A) :σ(y,A)≤r ‖x− y‖ ≤ ‖x− x̃r,k‖ Using (3.26), the property 2) of the kernel Kα,r and (2.3), the last inequality implies: Er(x,A) ≤ Kα,r(t)xdt− Kα,r(t) x+ (−1)k−1(U(t)− I)kx ∥∥∥∥ ≤ Kα,r(t) ∥∥(U(t)− I)kx ∥∥ dt ≤ Kα,r(t)ω̃k(|t|, x, A) dt. So, in a ordan e with the property 4) of the fun tion ω̃k(|t|, x, A), (3.27) Er(x,A) ≤ Kα,r(t)ω̃k |rt|1 , x, A ≤ ω̃k , x, A 1 + |rt|MU (|t|) )kKα,r(t)dt. Taking into a ount properties of the fun tion MU (·), the de�nition of Kα,r, lemma 3.1 and equality (3.19), one an �nd for r ≥ 1: 1 + |rt|MU (|t|) )kKα,r(t)dt ≤ 1 + |rt|MU (rt) rKα(rt)dt ≤ (1 + τ)MU (τ) )kKα(τ)dτ ≤ c1 (1 + |τ |)MU (|τ |) dτ = 2c1 < ∞. In a ordan e with (3.27), inequality (3.1) holds for all r ∈ [1,∞) with a onstant mk = 2c1. It should be noted that onstant mk, indeed, depends on k, be ause due to 3.1, the onstant c1 = c1(α) depends on the fun tion α(t) = (MU (|t|))k(1 + |t|)k+2. 10 YA. GRUSHKA AND S. TORBA Moreover, let the group {U(t)} is bounded (MU (t) ≤ M̃, t ∈ R, M̃ ≥ 1). Taking into a ount properties of the fun tion MU (·), the de�nition of Kα,r, lemma 3.1 and equality (3.19), one an �nd for r ∈ (0,∞) (1 + |rt|MU (|t|))kKα,r(t)dt ≤ (1 + |rt|M̃MU (rt))krKα(rt)dt ≤ ≤ M̃k ((1 + τ)MU (τ)) kKα(τ)dτ ≤ 2M̃kc1 < ∞, whi h proves remark 3.1 with the onstant mk = 2M̃ kc1. � Theorem 3.1 allows us to prove the analogue of the lassi Ja kson's inequality for m times di�erentiable fun tions: Corollary 3.1. Let x ∈ D(Am), m ∈ N0. Then ∀k ∈ N0 (3.28) Er(x,A) ≤ mk+m , Amx,A , r ≥ 1, where the onstants mn (n ∈ N) are the same as in theorem 3.1. Proof. Let x ∈ D(Am) and r ≥ 1. By theorem 3.1, Er(x,A) ≤ mk+m · ω̃k+m , x, A Let t ∈ R, 0 ≤ |t| ≤ 1 . Then, using properties of the groups of the C0 lass and properties of the fun tion MU (t), one an get: ∥∥(U(t)− I)k+mx ∥∥(U(t)− I)m(U(t)− I)kx · · · ‖U(ξ1 + · · ·+ ξm)‖ ∥∥(U(t)− I)kAmx ∥∥ dξ1 . . . dξm ≤ ≤ MU (m|t|) ∥∥(U(t)− I)kAmx ∥∥ tm ≤ , Amx,A This implies ω̃k+m , x, A = sup|t|≤ 1 ∥∥(U(t)− I)k+mx ∥∥ ≤ MU ( , Amx,A , whi h proves inequality (3.28). � By setting in orollary 3.1 k = 0 and taking into a ount that ω̃0 (·, Amx,A) ≡ ‖Amx‖, one an on lude the following inequality: Corollary 3.2. Let x ∈ D(Am), m ∈ N0. Then (3.29) Er(x,A) ≤ MU (1/r) )m‖Amx‖ r ≥ 1, where the onstants mn (n ∈ N) are the same as in theorem 3.1. 4. The examples of appli ation of the abstra t Ja kson's inequality in parti ular spa es Lets onsider several examples of appli ation of theorem 3.1 in parti ular spa es. DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 11 4.1. Ja kson's inequalities in Lp(2π) and C(2π). Example 4.1. Let the spa e X and the operator A are the same as in the example 2.1. Then for x ∈ X the quantity Er(x,A) is the value of the best approximation of fun tion x by trigonometri polynomials whose degree does not ex eed r with respe t to the norm in X. It is generally known that di�erential operator A is a generator of (isometri ) group of shifts in the spa e X: (U(t)x)(ξ) = x(t+ ξ), x ∈ X; t, ξ ∈ R ‖U(t)‖ ≡ 1, t ∈ R,(4.1) where ‖U(·)‖ = ‖U(·)‖L(X) is the norm of the operator U(t) in the spa e L(X) of linear ontinuous operators over X. It follows from (4.1) that ω̃k(t, x, A) = ωk(t, x, A) = sup 0≤h≤t (−1)k−j x(·+ jh) , t ∈ R+, x ∈ X. I.e., in that ase, ω̃k(t, x, A) oin ides with lassi modulus of ontinuity of k-th degree in the spa e X. Thus, from theorem 3.1 and orollary 3.1 one an on lude all lassi Ja kson-type inequalities in the spa es C(2π) and Lp(2π), 1 ≤ p < ∞. 4.2. Ja kson's inequalities of the approximation by exponential type entire fun tions in the spa e Lp(R, µ p). We onsider the real-valued fun tion µ(t) satisfying the following onditions: 1) µ(t) ≥ 1, t ∈ R; 2) µ(t) is even, monotoni ally non-de reasing when t > 0; 3) µ(t) satis�es naturally o urring in many appli ations ondition µ(t+ s) ≤ µ(t) · µ(s), s, t ∈ R. lnµ(t) dt < ∞, or alternatively, instead of 4), the equivalent ondition holds: lnµ(k) Lets onsider several important lasses of fun tions satisfying onditions 1)�4). 1. Constant fun tion µ(t) ≡ 1, t ∈ R. 2. Fun tions with polynomial order of growth at in�nity. It is easy to he k that for su h fun tions following estimate holds: ∃k ∈ N, ∃M ≥ 1 µ(t) ≤ M(1 + |t|)k, t ∈ R. 3. Fun tions of the form µ(t) = e|t| , 0 < β < 1, t ∈ R. 4. µ(t) represented as a power series for t > 0. I.e., µ(t) = where {mn}n∈N is the sequen e of positive real numbers satisfying two onditions: • m0 = 1, m2n ≤ mn−1 ·mn+1, n ∈ N; • ∀k, l ∈ N (k+l)! The fun tion µ(t), de�ned above, obviously satis�es onditions 1) and 2). The ondition ∀k, l ∈ N (k+l)! implies (4.2) tksn−kn! k!(n− k)!mn tksn−k mkmn−k 12 YA. GRUSHKA AND S. TORBA and it is easy to see that ondition 3) follows from inequality (4.2). The Denjoy - Carleman theorem [12, p.376℄ asserts that the following onditions are equivalent: a) µ(t) satis�es ondition 4); 5. µ(t) as a module of an entire fun tion with zeroes on the imaginary axis. We onsider ω(t) = C , t ∈ R, where C ≥ 1, 0 < t1 ≤ t2 ≤ . . . , < ∞. We set µ(t) := |ω(t)|. Then µ(t) satis�es onditions 1) � 3), and, as shown in [8℄, µ(t) satis�es ondition 4) also. Lets pro eed to the des ription of the spa es Lp(R, µ p). Let the fun tion µ(t) satis�es onditions 1) � 4). One an onsider the spa e Lp(R, µ p) of the fun tions x(s), s ∈ R, integrable in p-th degree with the weight µp: Lp(R,µp) |x(s)|pµp(s) ds. Lp(R, µ p) is the Bana h spa e. We onsider the di�erential operator A (D(A) = {x ∈ Lp(R, µ p) ∩ AC(R) : x′ ∈ Lp(R, µp)}, (Ax)(t) = dxdt ). As in example 4.1, the operator A generates the group of shifts {U(t)}t∈R in the spa e Lp(R, µp). But in ontrast to example 4.1, this group isn't bounded. Indeed, lets onsider x(s) = 1, s ∈ [0, 1], 0, s 6∈ [0, 1]. Obviously, x(s) ∈ Lp(R, µp), but for t > 1 ‖U(t)x‖p = |x(t+ s)|pµp(s) ds = µp(s) ds ≥ µp(t− 1) → ∞, t → ∞. On the other hand, be ause of the property 3), ‖U(t)x‖p = |x(t+s)|pµp(s) ds ≤ µp(−t) |x(t+s)|pµp(t+s) ds = µ(−t) )p‖x‖p, so ‖U(t)‖Lp(R,µp) ≤ µ(−t) = µ(|t|), t ∈ R. 2 By the same way as in the example 4.1, modules of ontinuity ωk and ω̃k oin ides with lassi ones, but in ontrast to the example 4.1, they don't equal mutually. The spa e E(A) onsists of fast de res ent at the in�nity entire fun tions. The examples of su h fun tions have been given in [8℄. By applying theorem 3.1 one an get Corollary 4.1. ∀k ∈ N there exists onstant mk(p, µ) > 0 su h that ∀f ∈ Lp(R, µp) Er(f) ≤ mk · ω̃k , x, A , r ≥ 1. If µ(t) is ontinuous and µ(0) = 1, it is possible to show in a similar manner that ‖U(t)‖Lp(R,µp) = µ(|t|). DIRECT THEOREMS IN THE THEORY OF APPROXIMATION ... 13 Referen es 1. N. P. Kup ov, Dire t and inverse theorems of approximation theory and semigroups of operators (Russian), Uspekhi Mat.Nauk. 23 (1968), No. 4, 118-178. 2. A. P. Terehin, A bounded group of operators, and best approximation (Russian), Di�eren ial'nye Uravneniya i Vy� isl.Mat., Vyp.2 (1975), 3-28. 3. G. V. Radzievskii, On best approximations and the rate of onvergen e of expansions in root ve tors of an operator, Ukrainian Math.J. 49 (1997), No. 6, 844-864 (1998). 4. G. V. Radzievskii, Dire t and inverse theorems in problems of approximation by ve tors of �nite degree, Sb.Math. 189 (1998), No.3�4, 561-601. 5. M. L. Gorba huk and V. I. Gorba huk, On the approximation of smooth ve tors of a losed operator by entire ve tors of exponential type, Ukrainian Math.J. 47 (1995), No. 5, 713-726 (1996). 6. M. L. Gorba huk and V. I. Gorba huk Operator approa h to approximation problems, St.Petersburg Math.J. 9 (1998), No. 6, 1097-1110. 7. M. L. Gorba huk, Ya. I. Grushka and S. M. Torba, Dire t and inverse theorems in the theory of approximations by the Ritz method, Ukrainian Math.J. 57 (2005), No. 5, 751-764 (arXiv:0709.4243 [math.FA℄). 8. Ju. I. Ljubi� and V. I. Ma aev, Operators with separable spe trum (Russian), Mat.Sb.(N.S.) 56 (98) (1962), No. 4, 433-468. 9. M. L. Gorba huk, On analyti solutions of operator-di�erential equations, Ukrainian Math.J. 52 (2000), No. 5, 680-693 (2001). 10. V. A. Mar� enko, On some questions of the approximation of ontinuous fun tions on the whole real axis (Russian), Zap. Mat. Otd. Fiz-Mat. Fak. KHGU i KHMO 22 (1951), No. 4, 115-125. 11. O. I. Inozem ev and V. A. Mar� enko, On majorants of genus zero (Russian), Uspekhi Mat.Nauk (N.S.) 11 (1956), 173-178. 12. Walter Rudin, Real and Complex Analysis, M Grow-Hill, New York, 1970. E-mail address: sergiy.torba�gmail. om, grushka�imath.kiev.ua Institute of Mathemati s of the National A ademy of S ien es of Ukraine, Teresh henkovskaya 3, 01601 Kiev (Ukraine) http://arxiv.org/abs/0709.4243 1. Introduction 2. Preliminaries 3. Abstract Jackson's inequality in a Banach space 4. The examples of application of the abstract Jackson's inequality in particular spaces 4.1. Jackson's inequalities in Lp(2) and C(2) 4.2. Jackson's inequalities of the approximation by exponential type entire functions in the space Lp(R, p) References

0704.0299

Parametrized Post-Newtonian Expansion of Chern-Simons Gravity

Parametrized Post-Newtonian Expansion of Chern-Simons Gravity Stephon Alexander1 and Nicolás Yunes1 Center for Gravitational Wave Physics, Institute for Gravitational Physics and Geometry and Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA (Dated: November 4, 2018) We investigate the weak-field, post-Newtonian expansion to the solution of the field equations in Chern-Simons gravity with a perfect fluid source. In particular, we study the mapping of this solution to the parameterized post-Newtonian formalism to 1 PN order in the metric. We find that the PPN parameters of Chern-Simons gravity are identical to those of general relativity, with the exception of the inclusion of a new term that is proportional to the Chern-Simons coupling parameter and the curl of the PPN vector potentials. We also find that the new term is naturally enhanced by the non-linearity of spacetime and we provide a physical interpretation for it. By mapping this correction to the gravito-electro-magnetic framework, we study the corrections that this new term introduces to the acceleration of point particles and the frame-dragging effect in gyroscopic precession. We find that the Chern-Simons correction to these classical predictions could be used by current and future experiments to place bounds on intrinsic parameters of Chern-Simons gravity and, thus, string theory. PACS numbers: 11.25.Wx, 95.55.Ym, 04.60.-m, 04.80.Cc I. INTRODUCTION Tests of alternative theories of gravity that modify gen- eral relativity (GR) at a fundamental level are essential to the advancement of physics. One formalism that has had incredible success in this task is the parameterized post- Newtonian (PPN) framework [1, 2, 3, 4, 5, 6]. In this formalism, the metric of the alternative theory is solved for in the weak-field limit and its deviations from GR are expressed in terms of PPN parameters. Once a metric has been obtained, one can calculate predictions of the alternative theory, such as light deflection and the perihe- lion shift of Mercury, which shall depend on these PPN parameters. Therefore, experimental measurements of such physical effects directly lead to constraints on the parameters of the alternative theory. This framework, together with the relevant experiments, have already been successfully employed to constrain scalar-tensor the- ories (Brans-Dicke, Bekenstein) [7], vector-tensor the- ories (Will-Nordtvedt [8], Hellings-Nordtvedt [9]), bi- metric theories (Rosen [10, 11]) and stratified theories (Ni [12]) (see [13] for definitions and an updated review.) Only recently has this framework been used to study quantum gravitational and string-theoretical inspired ideas. On the string theoretical side, Kalyana [14] investi- gated the PPN parameters associated with the graviton- dilaton system in low-energy string theory. More re- cently, Ivashchuk, et. al. [15] studied PPN parameters in the context of general black holes and p-brane spher- ically symmetric solutions, while Bezerra, et. al. [16] considered domain wall spacetimes for low energy effec- tive string theories and derived the corresponding PPN parameters for the metric of a wall. On the quan- tum gravitational side, Gleiser and Kozameh [17] and more recently Fan, et. al. [18] studied the possibility of testing gravitational birefringence induced by quan- tum gravity, which was proposed by Amelino-Camelia, el. al. [19] and Gambini and Pullin [20]. Other non- PPN proposals have been also put forth to test quan- tum gravity, for example through gravitational waves [21, 22, 23, 24, 25, 26, 27, 28], but we shall not discuss those tests here. Chern-Simons (CS) gravity [29, 30] is one such ex- tension of GR, where the gravitational action is mod- ified by the addition of a parity-violating term. This extension is promising because it is required by all 4- dimensional compactifications of string theory [31] for mathematical consistency because it cancels the Green- Schwarz anomaly [32]. CS gravity, however, is not unique to string theory and in fact has its roots in the standard model, where it arises as a gravitational anomaly pro- vided that there are more flavours of left handed leptons than right handed ones. Moreover the CS extension to GR can arise via the embedding of the three dimensional Chern-Simons topological current into a 4D space-time manifold, decsribed by Jackiw and Pi [30] Chern-Simons gravity has been recently studied in the cosmological context. In particular, this framework was used to shed light on the anisotropies of the cos- mic microwave background (CMB) [33, 34, 35] and the leptogenesis problem [34, 36, 37]. Parity violation has also been shown to produce birefringent gravitational waves [28, 29], where different polarizations modes ac- quire varying amplitudes. These modes obey different propagation equations because the imaginary sector of the classical dispersion relation is CS corrected. Different from [20], in CS birefringence the velocity of the gravita- tional wave remains that of light. In this paper we study CS gravity in the PPN frame- work, extending the analysis of [38] and providing some missing details. In particular, we shall consider the ef- fect of the CS correction to the gravitational field of, for instance, a pulsar, a binary system or a star in the weak-field limit. These corrections are obtained by solv- http://arxiv.org/abs/0704.0299v1 ing the modified field equations in the weak-field limit for post-Newtonian (PN) sources, defined as those that are weakly-gravitating and slowly-moving [39]. Such an ex- pansion requires the calculation of the Ricci and Cotton tensors to second order in the metric perturbation. We then find that CS gravity leads to the same gravitational field as that of classical GR and, thus, the same PPN parameters, except for the inclusion of a new term in the vectorial sector of the metric, namely 0i = 2ḟ (∇× V )i , (1) where ḟ acts as a coupling parameter of CS theory and Vi is a PPN potential. We also show that this solution can be alternatively obtained by finding a formal solu- tion to the modified field equations and performing a PN expansion, as is done in PN theory. The full solution is further shown to satisfy the additional CS constraint, which leads to equations of motion given only by the di- vergence of the stress-energy tensor. The CS correction to the metric found here leads to an interesting interpretation of CS gravity and forces us to consider a new type of coupling. The interpretation consists of thinking of the field that sources the CS cor- rection as a fluid that permeates all of spacetime. Then the CS correction in the metric is due to the “dragging” of such a fluid by the motion of the source. Until now, cou- plings of the CS correction to the angular momentum of the source had been neglected by the string theory com- munity. Similarly, curl-type terms had also been consid- ered unnecessary in the traditional PPN framework, since previous alternative gravity theories had not required it. As we shall show, in CS gravity and thus in string the- ory, such a coupling is naturally occurring. Therefore, a proper PPN mapping requires the introduction of a new curl-type term with a corresponding new PPN parameter of the type of Eq. (1). A modification to the gravitational field leads natu- rally to corrections of the standard predictions of GR. In order to illustrate such a correction, we consider the CS term in the gravito-electro-magnetic analogy [40, 41], where we find that the CS correction accounts for a mod- ification of gravitomagnetism. Furthermore, we calculate the modification to the acceleration of point particles and the frame dragging effect in the precession of gyroscopes. We find that these corrections are given by δai = − c2 r2 δΩi = − ḟ c3 r2 ni − v , (2) where m and v are the mass and velocities of the source, while r is the distance to the source and ni = xi/r is a unit vector, with · and × the flat-space scalar and cross products. Both corrections are found to be naturally en- hanced in regions of high spacetime curvature. We then conclude that experiments that measure the gravitomag- netic sector of the metric either in the weak-field (such as Gravity Probe B [42]) and particularly in the non-linear regime, will lead to a direct constraint on the CS cou- pling parameter ḟ . In this paper we develop the details of how to calculate these corrections, while the specifics of how to actually impose a constraint, which depend on the experimental setup, are beyond the scope of this paper. The remainder of this paper deals with the details of the calculations discussed in the previous paragraphs. We have divided the paper as follows: Sec. II describes the basics of the PPN framework; Sec. III discusses CS modified gravity, the modified field equations and com- putes a formal solution; Sec. IV expands the field equa- tions to second order in the metric perturbation; Sec. V iteratively solves the field equations in the PN approx- imation and finds the PPN parameters of CS gravity; Sec. VI discusses the correction to the acceleration of point particles and the frame dragging effect; Sec. VII concludes and points to future research. The conventions that we use throughout this work are the following: Greek letters represent spacetime indices, while Latin letters stand for spatial indices only; semi- colons stand for covariant derivatives, while colons stand for partial derivatives; overhead dots stand for deriva- tives with respects to time. We denote uncontrolled re- mainders with the symbol O(A), which stands for terms of order A. We also use the Einstein summation con- vention unless otherwise specified. Finally, we use ge- ometrized units, where G = c = 1, and the metric signa- ture (−,+,+,+). II. THE ABC OF PPN In this section we summarize the basics of the PPN framework, following [6]. This framework was first developed by Eddington, Robertson and Schiff [1, 6], but it came to maturity through the seminal papers of Nordtvedt and Will [2, 3, 4, 5]. In this section, we de- scribe the latter formulation, since it is the most widely used in experimental tests of gravitational theories. The goal of the PPN formalism is to allow for com- parisons of different metric theories of gravity with each other and with experiment. Such comparisons become manageable through a slow-motion, weak-field expansion of the metric and the equations of motion, the so-called PN expansion. When such an expansion is carried out to sufficiently high but finite order, the resultant solu- tion is an accurate approximation to the exact solution in most of the spacetime. This approximation, however, does break down for systems that are not slowly-moving, such as merging binary systems, or weakly gravitating, such as near the apparent horizons of black hole binaries. Nonetheless, as far as solar system tests are concerned, the PN expansion is not only valid but also highly accu- rate. The PPN framework employs an order counting- scheme that is similar to that used in multiple-scale anal- ysis [43, 44, 45, 46]. The symbol O(A) stands for terms of order ǫA, where ǫ ≪ 1 is a PN expansion parameter. For convenience, it is customary to associate this parameter with the orbital velocity of the system v/c = O(1), which embodies the slow-motion approximation. By the Virial theorem, this velocity is related to the Newtonian poten- tial U via U ∼ v2, which then implies that U = O(2) and embodies the weak-gravity approximation. These expan- sions can be thought of as two independent series: one in inverse powers of the speed of light c and the other in positive powers of Newton’s gravitational constant. Other quantities, such as matter densities and deriva- tives, can and should also be classified within this order- counting scheme. Matter density ρ, pressure p and spe- cific energy density Π, however, are slightly more com- plicated to classify because they are not dimensionless. Dimensionlessness can be obtained by comparing the pressure and the energy density to the matter density, which we assume is the largest component of the stress- energy tensor, namely p/ρ ∼ Π/ρ = O(2). Derivatives can also be classified in this fashion, where we find that ∂t/∂x = O(1). Such a relation can be derived by noting that ∂t ∼ vi∇i, which comes from the Euler equations of hydrodynamics to Newtonian order. With such an order-counting scheme developed, it is instructive to study the action of a single neutral particle. The Lagrangian of this system is given by L = (gµνu −g00 − 2g0ivi − gijvivj where uµ = dxµ/dt = (1, vi) is the 4-velocity of the parti- cle and vi is its 3-velocity. From Eq. (3), note that knowl- edge of L to O(A) implies knowledge of g00 to O(A), g0i to O(A − 1) and gij to O(A − 2). Therefore, since the Lagrangian is already known to O(2) (the Newtonian so- lution), the first PN correction to the equations of motion requires g00 to O(4), g0i to O(3) and gij to O(2). Such order counting is the reason for calculating different sec- tors of the metric perturbation to different PN orders. A PPN analysis is usually performed in a particular background, which defines a particular coordinate sys- tem, and in an specific gauge, called the standard PPN gauge. The background is usually taken to be Minkowski because for solar system experiments deviations due to cosmological effects are negligible and can, in principle, be treated as adiabatic corrections. Moreover, one usu- ally chooses a standard PPN frame, whose outer regions are at rest with respect to the rest frame of the universe. Such a frame, for example, forces the spatial sector of the metric to be diagonal and isotropic [6]. The gauge em- ployed is very similar to the PN expansion of the Lorentz gauge of linearized gravitational wave theory. The differ- ences between the standard PPN and Lorentz gauge are of O(3) and they allow for the presence of certain PPN potentials in the vectorial sector of the metric perturba- tion. The last ingredient in the PPN recipe is the choice of a stress-energy tensor. The standard choice is that of a perfect fluid, given by T µν = (ρ+ ρΠ+ p)uµuν + pgµν . (4) Such a stress-energy density suffices to obtain the PN expansion of the gravitational field outside a fluid body, like the Sun, or of compact binary system. One can show that the internal structure of the fluid bodies can be ne- glected to 1 PN order by the effacement principle [39] in GR. Such effacement principle might actually not hold in modified field theories, but we shall study this subject elsewhere [47]. With all these machinery, on can write down a super- metric [6], namely g00 = −1 + 2U − 2βU2 − 2ξΦW + (2γ + 2 + α3 + ζ1 − 2ξ)Φ1 + 2 (3γ − 2β + 1 + ζ2 + ξ)Φ2 + 2 (1 + ζ3)Φ3 + 2 (3γ + 3ζ4 − 2ξ)Φ4 − (ζ1 − 2ξ)A, g0i = − (4γ + 3 + α1 − α2 + ζ1 − 2ξ)Vi (1 + α2 − ζ1 + 2ξ)Wi, gij = (1 + 2γU) δij , (5) where δij is the Kronecker delta and where the PPN potentials (U,ΦW ,Φ1,Φ2,Φ3,Φ4,A, Vi,Wi) are defined in Appendix A. Equation (5) describes a super-metric theory of gravity, because it reduces to different met- ric theories, such as GR or other alternative theo- ries [6], through the appropriate choice of PPN param- eters (γ, β, ξ, α1, α2, α3, ζ1, ζ2, ζ3, ζ4). One could obtain a more general form of the PPN metric by performing a post-Galilean transformation on Eq. (5), but such a procedure shall not be necessary in this paper. The super-metric of Eq. (5) is parameterized in terms of a specific number of PPN potentials, where one usually employs certain criteria to narrow the space of possible potentials to consider. Some of these restriction include the following: the potentials tend to zero as an inverse power of the distance to the source; the origin of the co- ordinate system is chosen to coincide with the source, such that the metric does not contain constant terms; and the metric perturbations h00, h0i and hij transform as a scalar, vector and tensor. The above restrictions are reasonable, but, in general, an additional subjective condition is usually imposed that is based purely on sim- plicity: the metric perturbations are not generated by gradients or curls of velocity vectors or other generalized vector functions. As of yet, no reason had arisen for re- laxing such a condition, but as we shall see in this paper, such terms are indeed needed for CS modified theories. What is the physical meaning of all these parame- ters? One can understand what these parameters mean by calculating the generalized geodesic equations of mo- tion and conservation laws [6]. For example, the param- eter γ measures how much space-curvature is produced by a unit rest mass, while the parameter β determines how much “non-linearity” is there in the superposition law of gravity. Similarly, the parameter ξ determines whether there are preferred-location effects, while αi rep- resent preferred-frame effects. Finally, the parameters ζi measure the amount of violation of conservation of to- tal momentum. In terms of conservation laws, one can interpret these parameters as measuring whether a the- ory is fully conservative, with linear and angular momen- tum conserved (ζi and αi vanish), semi-conservative, with linear momentum conserved (ζi and α3 vanish), or non- conservative, where only the energy is conserved through lowest Newtonian order. One can verify that in GR, γ = β = 1 and all other parameters vanish, which implies that there are no preferred-location or frame effects and that the theory is fully conservative. A PPN analysis of an alternative theory of gravity then reduces to mapping its solutions to Eq. (5) and then de- termining the PPN parameters in terms of intrinsic pa- rameters of the theory. The procedure is simply as fol- lows: expand the modified field equations in the metric perturbation and in the PN approximation; iteratively solve for the metric perturbation to O(4) in h00, to O(3) in h0i and to O(2) in hij ; compare the solution to the PPN metric of Eq. (5) and read off the PPN parameters of the alternative theory. We shall employ this procedure in Sec. V to obtain the PPN parameters of CS gravity. III. CS GRAVITY IN A NUTSHELL In this section, we describe the basics of CS gravity, following mainly [29, 30]. In the standard CS formalism, GR is modified by adding a new term to the gravitational action. This term is given by [30] SCS = d4xf (⋆R R) , (6) where mpl is the Planck mass, f is a prescribed external quantity with units of squared mass (or squared length in geometrized units), R is the Ricci scalar and the star stands the dual operation, such that R⋆R = Rαβγδǫ αβµνRγδµν , (7) with ǫµνδγ the totally-antisymmetric Levi-Civita tensor and Rµνδγ the Riemann tensor. Such a correction to the gravitational action is inter- esting because of the unavoidable parity violation that is introduced. Such parity violation is inspired from CP violation in the standard model, where such corrections act as anomaly-canceling terms. A similar scenario oc- curs in string theory, where the Green-Schwarz anomaly is canceled by precisely such a CS correction [32], al- though CS gravity is not exclusively tied to string the- ory. Parity violation in CS gravity inexorably leads to birefringence in gravitational propagation, where here we mean that different polarization modes obey differ- ent propagation equations but travel at the same speed, that of light [29, 30, 36, 47]. If CS gravity were to lead to polarization modes that travel at different speeds, then one could use recently proposed experiments [17] to test this effect, but such is not the case in CS grav- ity. Birefringent gravitational waves, and thus CS grav- ity, have been proposed as possible explanations to the cosmic-microwave-background (CMB) anisotropies [36], as well as the baryogenesis problem during the inflation- ary epoch [33]. The magnitude of the CS correction is controlled by the externally-prescribed quantity f , which depends on the specific theory under consideration. When we consider CS gravity as an effective quantum theory, then the cor- rection is suppressed by some mass scale M , which could be the electro-weak scale or some other scale, since it is unconstrained. In the context of string theory, the quan- tity f has been calculated only in conservative scenar- ios, where it was found to be suppressed by the Planck mass. In other scenarios, however, enhancements have been proposed, such as in cosmologies where the string coupling vanishes at late times [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58], or where the field that generates f cou- ples to spacetime regions with large curvature [59, 60] or stress-energy density [28, 47]. For simplicity, we here as- sume that this quantity is spatially homogeneous and its magnitude is small but non-negligible, so that we work to first order in the string-theoretical correction. There- fore, we treat ḟ as an independent perturbation parame- ter, [70] unrelated to ǫ, the PN perturbation parameter. The field equations of CS modified gravity can be ob- tained by varying the action with respect to the metric. Doing so, one obtains Gµν + Cµν = 8πTµν , (8) where Gµν is the Einstein tensor, Tµν is a stress-energy tensor and Cµν is the Cotton tensor. The latter tensor is defined via Cµν = − (µDαRν)β + (Dσf,τ ) ⋆Rτ (µ where parenthesis stand for symmetrization, g is the de- terminant of the metric, Da stands for covariant differ- entiation and colon subscripts stand for partial differen- tiation. Formally, the introduction of such a modification to the field equations leads to a new constraint, which is compensated by the introduction of the new scalar field degree of freedom f . This constraint originates by re- quiring that the divergence of the field equations vanish, namely DµCµν = Dνf ( ⋆RR) = 0, (10) where the divergence of the Einstein tensor vanished by the Bianchi identities. If this constraint is satisfied, then the equations of motion for the stress-energy DµT µν are unaffected by CS gravity. A common source of confusion is that Eq. (10) is sometimes interpreted as requiring that R⋆R also vanish, which would then force the correction to the action to vanish. However, this is not the case be- cause, in general, f is an exact form (d2f = 0) and, thus, Eq. (10) only implies an additional constraint that forces all solutions to the field equations to have a vanishing The previous success of CS gravity in proposing plau- sible explanations to important cosmological problems prompts us to consider this extension of GR in the weak- field regime. For this purpose, it is convenient to rewrite the field equations in trace-reversed form, since this form is most amenable to a PN expansion. Doing so, we find, Rµν + Cµν = 8π Tµν − , (11) where the trace of the Cotton tensor vanishes identically and T = gµνT µν is the four dimensional trace of the stress-energy tensor. To linear order, the Ricci and Cot- ton tensors are given by [30] Rµν = − �hµν +O(h)2, Cµν = − ǫ̃0αβ(µ�ηhν)β,α +O(h)2, (12) where ǫ̃αβγδ is the Levi-Civita symbol, with conven- tion ǫ̃0123 = +1, and �η = −∂2t + ηij∂i∂j is the flat space D’Alambertian, with ηµν the Minkowski metric. In Eq. (12), we have employed the Lorentz gauge condition α = h,µ/2, where h = g µνhµν is the four dimensional trace of the metric perturbation. The Cotton tensor changes the characteristic behav- ior of the Einstein equations by forcing them to become third order instead of second order. Third-order par- tial differential equations are common in boundary layer theory [43]. However, in CS gravity, the third-order con- tributions are multiplied by a factor of f and we shall treat this function as a small independent expansion pa- rameter. Therefore, the change in characteristics in the modified field equations can also be treated perturba- tively, which is justified because eventhough ḟ might be enhanced by standard model currents, extra dimensions or a vanishing string coupling, it must still carry some type of mass suppression. The trace-reversed form of the field equations is useful because it allows us to immediately find a formal solution. Inverting the D’Alambertian operator we obtain Hµν = −16π �−1η Tµν − +O(h)2, (13) where we have defined an effective metric perturbation Hµν ≡ hµν + ḟ ǫ̃0αβ(µhν)β,α. (14) Note that this formal solution is identical to the formal PN solution to the field equations in the limit ḟ → 0. Also note that the second term in Eq. (14) is in essence a curl operator acting on the metric. This antisymmetric operator naturally forces the trace of the CS correction to vanish, as well as the 00 component and the symmetric spatial part. From the formal solution to the modified field equa- tions, we immediately identify the only two possible non- zero CS contributions : a coupling to the vector compo- nent of the metric h0i; and coupling to the transverse- traceless part of the spatial metric hTTij . The latter has already been studied in the gravitational wave con- text [29, 30, 47] and it vanishes identically if we require the spatial sector of the metric perturbation to be confor- mally flat. The former coupling is a new curl-type contri- bution to the metric perturbation that, to our knowledge, had so far been neglected both by the string theory and PPN communities. In fact, as we shall see in later sec- tions, terms of this type will force us to introduce a new PPN parameter that is proportional to the curl of certain PPN potentials. Let us conclude this section by pushing the formal so- lution to the modified field equations further to obtain a formal solution in terms of the actual metric perturba- tion hµν . Combining Eqs. (13) and (14) we arrive at the differential equation hµν+ḟ ǫ̃ (µhν)β,α = −16π �−1η Tµν − +O(h)2. Since we are searching for perturbations about the gen- eral relativistic solution, we shall make the ansatz hµν = h µν + ḟζµν +O(h)2, (16) where h µν is the solution predicted by general relativity h(GR)µν ≡ −16π �−1η Tµν − , (17) and where ζµν is an unknown function we are solving for. Inserting this ansatz into Eq. (15) we obtain ζµν+ḟ ǫ̃ (µζν)β,α = 16πǫ̃ (µ∂α� Tν)β − gν)βT We shall neglect the second term on the left-hand side be- cause it would produce a second order correction. Such conclusion was also reached when studying parity viola- tion in GR to explain certain features of the CMB [35]. We thus obtain the formal solution ζµν = 16πǫ̃ (µ∂α� Tν)β − gν)βT and the actual metric perturbation to linear order be- comes hµν = −16π �−1η Tµν − + 16πḟ ǫ̃kℓi�−1η δi(µTν)ℓ,k − δi(µην)ℓT,k +O(h)2, where we have used some properties of the Levi-Civita symbol to simplify this expression. The procedure pre- sented here is general enough that it can be directly ap- plied to study CS gravity in the PPN framework, as well as possibly find PN solutions to CS gravity. IV. PN EXPANSION OF CS GRAVITY In this section, we perform a PN expansion of the field equations and obtain a solution in the form of a PN se- ries. This solution then allows us to read off the PPN parameters by comparing it to the standard PPN super- metric [Eq. (5)]. In this section we shall follow closely the methods of [6] and [61] and indices shall be manipulated with the Minkowski metric, unless otherwise specified. Let us begin by expanding the field equations to second order in the metric perturbation. Doing so we find that the Ricci and Cotton tensors are given to second order Rµν = − �ηhµν − 2hσ(µ,ν)σ + h,µν 2hρ(µ,ν)λ − hµν,ρλ − hρλ,µν hρλ,µhρλ,ν + h ν,λ (21) − hρµ,λhρν,λ + h,λ − 2hλρ,ρ hµν,λ − 2hλ(µ,ν) +O(h)3, Cµν = − ǫ̃0αβ(µ �ηhν)β,α − hσβ,αν)σ ǫ̃0αβ(µ �ηhν)β,α − hσβ,αν)σ 2hν)(λ,α) − hλα,ν) β − 2hσ(λ,β)σ + h,βλ − 2Q̂Rν)β,α ǫ̃σαβ(µ 2h0(σ,τ) − hστ,0 hτ [β,α]ν) − hν)[β,α]τ hµλǫ̃ 0αβ(λ �ηhν)β,α − hσβ,αν)σ ǫ̃0αβ(µ β,α − hσβ,ασλ) hνλ +O(h)3. (22) where index contraction is carried out with the Minkowski metric and where we have not assumed any gauge condition. The operator Q̂(·) takes the quadratic part of its operand [of O(h)2] and it is explained in more detail in Appendix B, where the derivation of the expan- sion of the Cotton tensor is presented in more detail. In this derivation, we have used the definition of the Levi- Civita tensor ǫαβγδ = (−g)1/2ǫ̃αβγδ = ǫ̃αβγδ +O(h)2, (23) ǫαβγδ = −(−g)−1/2ǫ̃αβγδ = − ǫ̃αβγδ +O(h)2. Note that the PN expanded version of the linearized Ricci tensor of Eq. (21) agrees with previous results [6]. Also note that if the Lorentz condition is enforced, several terms in both expressions vanish identically and the Cot- ton tensor to first order reduces to Eq. (12), which agrees with previous results [30]. Let us now specialize the analysis to the standard PPN gauge. For this purpose, we shall impose the following gauge conditions k − 1 h,j = O(4), k − 1 hkk,0 = O(5), (24) where hkk is the spatial trace of the metric perturbation. Note that the first equation is the PN expansion of one of the Lorentz gauge conditions, while the second equation is not. This is the reason why the previous equations where not expanded in the Lorentz gauge. Nonetheless, such a gauge condition does not uniquely fix the coordi- nate system, since we can still perform an infinitesimal gauge transformation that leaves the modified field equa- tions invariant. One can show that the Lorentz and PPN gauge are related to each other by such a gauge transfor- mation. In the PPN gauge, then, the Ricci tensor takes the usual form R00 = − ∇2h00 − h00,ih00, hijh00,ij +O(6), R0i = − ∇2h0i − h00,0i +O(5), Rij = − ∇2hij +O(4), (25) which agrees with previous results [6], while the Cotton tensor reduces to C00 = O(6), C0i = − ḟ ǫ̃0kli∇2h0l,k +O(5), Cij = − ḟ ǫ̃0kl(i∇2hj)l,k +O(4), (26) where ∇ = ηij∂i∂j is the Laplacian of flat space [see Ap- pendix B for the derivation of Eq. (26).] Note again the explicit appearance of two coupling terms of the Cotton tensor to the metric perturbation: one to the transverse- traceless part of the spatial metric and the other to the vector metric perturbation. The PN expansions of the linearized Ricci and Cotton tensor then allow us to solve the modified field equations in the PPN framework. V. PPN SOLUTION OF CS GRAVITY In this section we shall proceed to systematically solve the modified field equation following the standard PPN iterative procedure [6]. We shall begin with the 00 and ij components of the metric to O(2), and then proceed with the 0i components to O(3) and the 00 component to O(4). Once all these components have been solved for in terms of PPN potentials, we shall be able to read off the PPN parameters adequate to CS gravity. A. h00 and hij to O(2) Let us begin with the modified field equations for the scalar sector of the metric perturbation. These equations are given to O(2) by ∇2h00 = −8πρ, (27) because T = −ρ. Eq. (27) is the Poisson equation, whose solution in terms of PPN potentials is h00 = 2U +O(4). (28) Let us now proceed with the solution to the field equa- tion for the spatial sector of the metric perturbation. This equation to O(2) is given by ∇2hij + ḟ ǫ̃0kl(i∇2hj)l,k = −8πρδij , (29) where note that this is the first appearance of a Cotton tensor contribution. Since the Levi-Civita symbol is a constant and ḟ is only time-dependent, we can factor out the Laplacian and rewrite this equation in terms of the effective metric Hij as ∇2Hij = −8πρδij , (30) where, as defined in Sec. III, Hij = hij + ḟ ǫ̃0kl(ihj)l,k. (31) The solution of Eq. (30) can be immediately found in terms of PPN potentials as Hij = 2Uδij +O(4), (32) which is nothing but Eq. (13). Recall, however, that in Sec. III we explicitly used the Lorentz gauge to simplify the field equations, whereas here we are using the PPN gauge. The reason why the solutions are the same is that the PPN and Lorentz gauge are indistinguishable to this order. Once the effective metric has been solved for, we can obtain the actual metric perturbation following the pro- cedure described in Sec. III. Combining Eq. (31) with Eq. (32), we arrive at the following differential equation hij + ḟ ǫ̃ (ihj)l,k = 2Uδij. (33) We look for solutions whose zeroth-order result is that predicted by GR and the CS term is a perturbative cor- rection, namely hij = 2Uδij + ḟζij , (34) where ζ is assumed to be of O(ḟ)0. Inserting this ansatz into Eq. (33) we arrive at ζij + ḟ ǫ̃ (iζj)l,k = 0, (35) where the contraction of the Levi-Civita symbol and the Kronecker delta vanished. As in Sec. III, note that the second term on the left hand side is a second order cor- rection and can thus be neglected to discover that ζij vanishes to this order. The spatial metric perturbation to O(2) is then simply given by the GR prediction without any CS correction, namely hij = 2Uδij +O(4). (36) Physically, the reason why the spatial metric is unaf- fected by the CS correction is related to the use of a perfect fluid stress-energy tensor, which, together with the PPN gauge condition, forces the metric to be spa- tially conformally flat. In fact, if the spatial metric were not flat, then the spatial sector of the metric perturba- tion would be corrected by the CS term. Such would be the case if we had pursued a solution to 2 PN or- der, where the Landau-Lifshitz pseudo-tensor sources a non-conformal correction to the spatial metric [39], or if we had searched for gravitational wave solutions, whose stress-energy tensor vanishes [29, 36]. In fact, one can check that, in such a scenario, Eq. (30) reduces to that found by [29, 30, 36, 47] as ρ → 0. We have then found that the weak-field expansion of the gravitational field outside a fluid body, like the Sun or a compact binary, is unaffected by the CS correction to O(2). B. h0i to O(3) Let us now look for solutions to the field equations for the vector sector of the metric perturbation. The field equations to O(3) become ∇2h0i + h00,0i + ḟ ǫ̃0kli∇2h0l,k = 16πρvi, (37) where we have used that T 0i = −T0i. Using the lower order solutions and the effective metric, as in Sec. III, we obtain ∇2H0i + U,0i = 16πρvi, (38) where the vectorial sector of the effective metric is H0i = h0i + ḟ ǫ̃0klih0l,k. (39) We recognize Eq. (38) as the standard GR field equation to O(3), except that the dependent function is the effec- tive metric instead of the metric perturbation. We can thus solve this equation in terms of PPN potentials to obtain H0i = − Wi, (40) where we have used that the superpotential X satisfies X,0j = Vj − Wj (see Appendix A for the definitions.) Combining Eq. (39) with Eq. (40) we arrive at a differ- ential equation for the metric perturbation, namely h0i + ḟ ǫ̃0klih0l,k = − Wi. (41) Once more, let us look for solutions that are perturbation about the GR prediction, namely h0i = − Wi + ḟζi, (42) where we again assume that ζi is of O(ḟ)0. The field equation becomes ḟ (∇× ζ)i = (∇× V )i + (∇×W )i where (∇×A)i = ǫijk∂jAk is the standard curl operator of flat space. As in Sec. III, note once more that the second term on the left-hand side is again a second order correction and we shall thus neglect it. Also note that the curl of the Vi potential happens to be equal to the curl of the Wi potential. The solution for the vectorial sector of the actual gravitational field then simplifies to h0i = − Wi + 2ḟ (∇× V )i +O(5). (44) We have arrived at the first contribution of CS mod- ified gravity to the metric for a perfect fluid source. Chern-Simons gravity was previously seen to couple to the transverse-traceless sector of the metric perturbation for gravitational wave solutions [29, 30, 36, 47]. The CS correction is also believed to couple to Noether vector currents, such as neutron currents, which partially fueled the idea that this correction could be enhanced. However, to our knowledge, this correction was never thought to couple to vector metric perturbations. From the analy- sis presented here, we see that in fact CS gravity does couple to such terms, even if the matter source is neu- trally charged. The only requirement for such couplings is that the source is not static, ie. that the object is ei- ther moving or spinning relative to the PPN rest frame so that the PPN vector potential does not vanish. The latter is suppressed by a relative O(1) because in the far field the velocity of a compact object produces a term of O(3) in Vi, while the spin produces a term of O(4). In a later section, we shall discuss some of the physical and observational implications of such a modification to the metric. C. h00 to O(4) A full analysis of the PPN structure of a modified the- ory of gravity requires that we solve for the 00 component of the metric perturbation to O(4). The field equations to this order are ∇2h00 − h00,ih00,i + hijh00,ij = 4πρ [1 v2 − U + 1 , (45) where the CS correction does not contribute at this order (see Appendix B.) Note that the h0i sector of the metric perturbation to O(3) does not feed back into the field equations at this order either. The terms that do come into play are the h00 and hij sectors of the metric, which are not modified to lowest order by the CS correction. The field equation, thus, reduce to the standard one of GR, whose solution in terms of PPN potentials is h00 = 2U − 2U2 + 4Φ1 + 4Φ2 +2Φ3 +6Φ4 +O(6). (46) We have thus solved for all components of the metric per- turbation to 1 PN order beyond the Newtonian answer, namely g00 to O(4), g0i to O(3) and gij to O(2). D. PPN Parameters for CS Gravity We now have all the necessary ingredients to read off the PPN parameters of CS modified gravity. Let us begin by writing the full metric with the solutions found in the previous subsections: g00 = −1 + 2U − 2U2 + 4Φ1 + 4Φ2 + 2Φ3 + 6Φ4 +O(6), g0i = − Wi + 2ḟ (∇× V )i +O(5), gij = (1 + 2U) δij +O(4). (47) One can verify that this metric is indeed a solution of Eqs. (27), (29), (37) and (45) to the appropriate PN or- der and to first order in the CS coupling parameter. Also note that the solution of Eq. (47) automatically satis- fies the constraint ⋆RR = 0 to linear order because the contraction of the Levi-Civita symbol with two partial derivatives vanishes. Such a solution is then allowed in CS gravity, just as other classical solutions are [62], and the equations of motion for the fluid can be obtained di- rectly from the covariant derivative of the stress-energy tensor. We can now read off the PPN parameters of the CS modified theory by comparing Eq. (5) to Eq. (47). A vi- sual inspection reveals that the CS solution is identical to the classical GR one, which implies that γ = β = 1, ζ = 0 and α1 = α2 = α3 = ξ1 = ξ2 = ξ3 = ξ4 = 0 and there are no preferred frame effects. However, Eq. (5) contains an extra term that cannot be modeled by the standard PPN metric of Eq. (5), namely the curl contri- bution to g0i. We then see that the PPN metric must be enhanced by the addition of a curl-type term to the 0i components of the metric, namely g0i ≡ − (4γ + 3 + α1 − α2 + ζ1 − 2ξ)Vi (1 + α2 − ζ1 + 2ξ)Wi + χ (r∇× V )i , (48) where χ is a new PPN parameter and where we have multiplied the curl operator by the radial distance to the source, r, in order to make χ a proper dimension- less parameter. Note that there is no need to introduce any additional PPN parameters because the curl of Wi equals the curl of Vi. In fact, we could have equally pa- rameterized the new contribution to the PPN metric in terms of the curl of Wi, but we chose not to because Vi appears more frequently in PN theory. For the case of CS modified gravity, the new χ parameter is simply χ = 2 , (49) which is dimensionless since ḟ has units of length. If an experiment could measure or place bounds on the value of χ, then ḟ could also be bounded, thus placing a con- straint on the CS coupling parameter. VI. ASTROPHYSICAL IMPLICATIONS In this section we shall propose a physical interpreta- tion to the CS modification to the PPN metric and we shall calculate some GR predictions that are modified by this correction. This section, however, is by no means a complete study of all the possible consequences of the CS correction, which is beyond the scope of this paper. Let us begin by considering a system of A nearly spher- ical bodies, for which the gravitational vector potentials are simply [6] V i = viA + , (50) W i = (vA · nA)niA + where mA is the mass of the Ath body, rA is the field point distance to the Ath body, niA = x A/rA is a unit vector pointing to the Ath body, vA is the velocity of the Ath body and J iA is the spin-angular momentum of the Ath body. For example, the spin angular momentum for a Kerr spacetime is given by J i = m2ai, where a is the dimensionless Kerr spin parameter. Note that if A = 2 then the system being modeled could be a binary of spin- ning compact objects, while if A = 1 it could represent the field of the sun or that of a rapidly spinning neutron star or pulsar. In obtaining Eq. (50), we have implicitly assumed a point-particle approximation, which in classical GR is justified by the effacement principle. This principle pos- tulates that the internal structure of bodies contributes to the solution of the field equations to higher PN order. One can verify that this is indeed the case in classical GR, where internal structure contributions appear at 5 PN order. In CS gravity, however, it is a priori unclear whether an analogous effacement principle holds because the CS term is expected to couple with matter current via standard model-like interactions. If such is the case, it is possible that a “mountain” on the surface of a neutron star [63] or an r-mode instability [64, 65, 66] enhances the CS contribution. In this paper, however, we shall neglect these interactions, and relegate such possibilities to future work [47]. With such a vector potential, we can calculate the CS correction to the metric. For this purpose, we define the correction δg0i ≡ g0i − g(GR)0i , where g 0i is the GR prediction without CS gravity. We then find that the CS corrections is given by δg0i = 2 (vA × nA)i − (JA · nA) where the · operator is the flat-space inner product and where we have used the identities ǫ̃ijk ǫ̃klm = δilδjm − δimδjl and ǫ̃ilk ǫ̃jlm = 2δij . Note that the first term of Eq. (51) is of O(3), while the second and third terms are of O(4) as previously anticipated. Also note that ḟ couples both to the spin and orbital angular momentum. Therefore, whether the system under consideration is the Solar system (vi of the Sun is zero while J i is small), the Crab pulsar (vi is again zero but J i is large) or a binary system of compact objects (neither vi nor J i vanish), there will in general be a non-vanishing coupling between the CS correction and the vector potential of the system. From the above analysis, it is also clear that the CS correction increases with the non-linearity of the space- time. In other words, the CS term is larger not only for systems with large velocities and spins, but also in re- gions near the source. For a binary system, this fact implies that the CS correction is naturally enhanced in the last stages of inspiral and during merger. Note that this enhancement is different from all previous en- hancements proposed, since it does not require the pres- ence of charge [28, 47], a fifth dimension with warped compactifications [59, 60], or a vanishing string cou- pling [48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. Un- fortunately, the end of the inspiral stage coincides with the edge of the PN region of validity and, thus, a com- plete analysis of such a natural enhancement will have to be carried out through numerical simulations. In the presence of a source with the vector potentials of Eq. (50), we can write the vectorial sector of the metric perturbation in a suggestive way, namely g0i = viA − vA − v(eff)A (eff) A · nA − 2 (eff) where we have defined an effective velocity and angular momentum vector via viA(eff) = v A − 6ḟ J iA(eff) = J A − ḟmAviA, (53) or in terms of the Newtonian orbital angular momentum = rA × pA and linear momentum piA(N) = mAv LiA(eff) = L A(N) − 6ḟ (nA × JA) J iA(eff) = J A − ḟpiA. (54) From this analysis, it is clear that the CS corrections seems to couple to both a quantity that resembles the or- bital and the spin angular momentum vector. Note that when the spin angular momentum vanishes the vectorial metric perturbation is identical to that of a spinning mov- ing fluid, but where the spin is induced by the coupling of the orbital angular momentum to the CS term. The presence of an effective CS spin angular momen- tum in non-spinning sources leads to an interesting phys- ical interpretation. Let us model the field that sources ḟ as a fluid that permeates all of spacetime. This field could be, for example, a model-independent axion, in- spired by the quantity introduced in the standard model to resolve the strong CP problem [67]. In this scenario, then the fluid is naturally “dragged” by the motion of any source and the CS modification to the metric is nothing but such dragging. This analogy is inspired by the er- gosphere of the Kerr solution, where inertial frames are dragged with the rotation of the black hole. In fact, one could push this analogy further and try to construct the shear and bulk viscosity of such a fluid, but we shall not attempt this here. Of course, this interpretation is to be understood only qualitatively, since its purpose is only to allow the reader to picture the CS modification to the metric in physical terms. An alternative interpretation can be given to the CS modification in terms of the gravito-electro-magnetic (GED) analogy [40, 41], which shall allow us to eas- ily construct the predictions of the modified theory. In this analogy, one realizes that the PN solution to the linearized field equations can be written in terms of a potential and vector potential, namely ds2 = − (1− 2Φ)dt2 − 4 (A · dx) dt+ (1 + 2Φ) δijdxidxj , where Φ reduces to the Newtonian potential U in the Newtonian limit [41] and Ai is a vector potential related to the metric via Ai = −g0i/4. One can then construct GED fields in analogy to Maxwell’s electromagnetic the- ory via Ei = − (∇Φ)i − ∂t Bi = (∇×A)i , (56) which in terms of the vectorial sector of the metric per- turbation becomes Ei = − (∇Φ)i + 1 Bi = −1 (∇× g)i , (57) where we have defined the vector gi = g0i. The geodesic equations for a test particle then reduce to the Lorentz force law, namely F i = −mEi − 2m (v ×B)i . (58) We can now work out the effect of the CS correction on the GED fields and equations of motion. First note that the CS correction only affects g. We can then write the CS modification to the Lorentz force law by defining δai = ai − ai , where ai is the acceleration vector predicted by GR, to obtain, δai = δġi + (v × δΩ)i , (59) where we have defined the angular velocity δΩi = (∇× δg)i . (60) The time derivative of the vector gi is of O(5) and can thus be neglected, but the angular velocity cannot and it is given by δΩi = − 3 (vA · nA)niA − viA , (61) which is clearly of O(3). Note that although the first term between square brackets cancels for circular orbits because niA is perpendicular to v A to Newtonian order, the second term does not. The angular velocity adds a correction to the acceleration of O(4), namely δai = −3 (vA · nA) (vA × nA)i , (62) which for a system in circular orbit vanishes to Newto- nian order. One could use this formalism to find the perturbations in the motion of moving objects by inte- grating Eq. (62) twice. However, for systems in a circular orbit, such as the Earth-Moon system or compact bina- ries, this correction vanishes to leading order. Therefore, lunar ranging experiments [68] might not be able to con- straint ḟ . Another correction to the predictions of GR is that of the precession of gyroscopes by the so-called Lense- Thirring or frame-dragging effect. In this process, the spin angular momentum of a source twists spacetime in such a way that gyroscopes are dragged with it. The precession angular velocity depends on the vector sector of the metric perturbation via Eq. (61). Thus, the full Lense-Thirring term in the precession angular velocity of precessing gyroscopes is ΩiLT = − J iA(eff) − 3n JA(eff) · nA . (63) Note that this angular velocity is identical to the GR prediction, except for the replacement J iA → J iA(eff). In CS modified gravity, then, the Lense-Thirring effect is not only produced by the spin angular momentum of the gyroscope but also by the orbital angular momentum that couples to the CS correction. Therefore, if an ex- periment were to measure the precession of gyroscopes by the curvature of spacetime (see, for example, Gravity Probe B [42]) one could constraint ḟ and thus some in- trinsic parameters of string theory. Note, however, that the CS correction depends on the velocity of the bodies with respect to the inertial PPN rest-frame. In order to relate these predictions to the quantities that are actually measured in the experiment, one would have to transform to the experiment’s frame, or perhaps to a basis aligned with the direction of distant stars [6]. Are there other experiments that could be performed to measure such a deviation from GR? Any experiment that samples the vectorial sector of the metric would in effect be measuring such a deviation. In this paper, we have only discussed modifications to the frame-dragging effect and the acceleration of bodies through the GED analogy, but this need not be the only corrections to classical GR predictions. In fact, any predictions that de- pends on g0i indirectly, for example via Christoffel sym- bols, will probably also be modified unless the corrections is fortuitously canceled. In this paper, we have laid the theoretical foundations of the weak-field correction to the metric due to CS gravity and studied some possible cor- rections to classical predictions. A detailed study of other corrections is beyond the scope of this paper. VII. CONCLUSION We have studied the weak-field expansion of the solu- tion to the CS modified field equations in the presence of a perfect fluid PN source in the point particle limit. Such an expansion required that we linearize the Ricci and Cotton tensor to second order in the metric pertur- bation without any gauge assumption. An iterative PPN formalism was then employed to solve for the metric per- turbation in this modified theory of gravity. We have found that CS gravity possesses the same PPN parame- ters as those of GR, but it also requires the introduction of a new term and PPN parameter that is proportional to the curl of the PPN vector potentials. Such a term is enhanced in non-linear scenarios without requiring the presence of standard model currents, large extra dimen- sions or a vanishing string coupling. We have proposed an interpretation for the new term in the metric produced by CS gravity and studied some of the possible consequences it might have on GR predic- tions. The interpretation consists of picturing the field that sources the CS term as a fluid that permeates all of spacetime. In this scenario, the CS term is nothing but the “dragging” of the fluid by the motion of the source. Irrespective of the validity of such an interpretation, the inclusion of a new term to the weak-field expansion of the metric naturally leads to corrections to the standard GR predictions. We have studied the acceleration of point particles and the Lense-Thirring contribution to the precession of gyroscopes. We have found that both corrections are proportional to the CS coupling parame- ter and, therefore, experimental measurement of these ef- fects might be used to constraint CS and, possibly, string theory. Future work could concentrate on studying further the non-linear enhancement of the CS correction and the modifications to the predictions of GR. The PPN analysis performed here breaks down very close to the source due to the use of a point particle approximation in the stress energy tensor. One possible research route could consists of studying the CS correction in a perturbed Kerr back- ground [69]. Another possible route could be to analyze other predictions of the theory, such as the perihelion shift of Mercury or the Nordtvedt effect. Furthermore, in light of the imminent highly-accurate measurement of the Lense-Thirring effect by Gravity Probe B, it might be useful to revisit this correction in a frame better-adapted to the experimental setup. Finally, the CS modification to the weak-field metric might lead to non-conservative effects and the breaking of the effacement principle [47], which could be studied through the evaluation of the gravitational pseudo stress-energy tensor. Ultimately, it will be experiments that will determine the viability of CS modified gravity and string theory. Acknowledgments The authors acknowledge the support of the Center for Gravitational Wave Physics funded by the National Science Foundation under Cooperative Agreement PHY- 01-14375, and support from NSF grants PHY-05-55-628. We would also like to thank Cliff Will for encouraging one of us to study the PPN formalism and Pablo Laguna for suggesting one of us to look into the PPN expansion of CS gravity. We would also like to thank R. Jackiw, R. Wagoner and Ben Owen for enlightening discussions and comments. APPENDIX A: PPN POTENTIALS In this appendix, we present explicit expressions for the PPN potentials used to parameterize the metric in Eq. (5). These potentials are the following: |x− x′| d3x′, ρ′v′i |x− x′| d3x′, ρ′v′j(x− x′)j(x− x′)i |x− x′|3 d3x′, ρ′ρ′′ (x− x′)i |x− x′|3 (x′ − x′′)i |x− x′′| − (x− x′′)i |x′ − x′′| d3x′d3x′′, ρ′v′2 |x− x′| d3x′, Φ2 ≡ ρ′U ′ |x− x′| d3x′, |x− x′| d3x′, Φ4 ≡ |x− x′| d3x′, v′i (x− x′) |x− x′| d ρ′|x− x′|d3x′. (A1) These potentials satisfy the following relations ∇2U = −4πρ, ∇2Vi = −4πρvi, ∇2Φ1 = −4πρv2, ∇2Φ2 = −4πρU, ∇2Φ3 = −4πρΠ, ∇2Φ4 = −4πp, ∇2X = −2U (A2) The potential X is sometimes referred to as the super- potential because it acts as a potential for the Newtonian potential. APPENDIX B: LINEARIZATION OF THE COTTON TENSOR In this appendix, we present some more details on the derivation of the linearized Cotton tensor to second order. We begin with the definition of the Cotton tensor [30] in terms of the symmetrization operator, namely Cµν = − 1√ (Dσf) ǫ σαβ(µDαR β + (Dστf) Rτ(µ|σ|ν) Using the symmetries of the Levi-Civita and Riemann tensor, as well as the fact that f depends only on time, we can simplify the Cotton tensor to Cµν = (−g)−1ḟ ǫ̃0αβ(µRν)β,α + ǫ̃ 0αβ(µΓ Γ0στ ǫ̃ σαβ(µRν)τ αβ . (B2) Noting that the determinant of the metric is simply g = −1 + h, so that (−g)−1 = 1 + h, we can identify four terms in the Cotton tensor A = ḟ ǫ̃ 0αβ(µ L̂Rν)β,α B = ḟ ǫ̃ 0αβ(µhρρ L̂Rν)β,α C = ḟ ǫ̃ 0αβ(µ L̂Rλβ ǫ̃σαβ(µ L̂Γ0στ L̂Rν)ταβ E = ḟ ǫ̃ 0αβ(µ Q̂Rν)β,α , (B3) where the L̂ operator stands for the linear part of its operand, while the Q̂ operator isolates the quadratic part of its operand. For example, if we act L̂ and Q̂ on (1+h)n, where n is some integer, we obtain L̂(1 + h)n = nh, Q̂(1 + h)n n(n− 1) h2.(B4) Let us now compute each of these terms separately. The first four terms are given by A = − ǫ̃0αβ(µ β,α − hσβ,νασ B = − hǫ̃0αβ(µ β,α − hσβ,νασ C = − ǫ̃0αβ(µ hν)λ,α + h α,λ − hλα,ν) β − hσλ,βσ − hσβ,λσ + h,λβ ǫ̃σαβ(µ 2h0(σ,τ) − hστ,0 hτ [β,α] ν − hν [β,α]τ The last term of the Cotton tensor is simply the deriva- tive of the Ricci tensor which we already calculated to second order in Eq. (21). In order to avoid notation clut- ter, we shall not present it again here, but instead we combine all the Cotton tensor pieces to obtain Cµν = − ḟ ǫ̃0αβ(µ β,α − hσβ,ασν) ǫ̃0αβ(µ β,α − hσβ,ασν) 2hν)(λ,α) − hλα,ν) β − 2hσ(λ,β)σ + h,βλ − 2Q̂Rν)β,α ǫ̃σαβ(µ 2h0(σ,τ) − hστ,0 hτ [β,α] ν) − hν)[β,α]τ +O(h)3 where its covariant form is Cµν = − ǫ̃0αβ(µ �ηhν)β,α − hσβ,αν)σ ǫ̃0αβ(µ �ηhν)β,α − hσβ,αν)σ 2hν)(λ,α) − hλα,ν) β − 2hσ(λ,β)σ + h,βλ − 2Q̂Rν)β,α + hνλ β,α − hσβ,ασλ) ǫ̃σαβ(µ 2h0(σ,τ) − hστ,0 hτ [β,α]ν) − hν)[β,α]τ hµλǫ̃ 0αβ(λ �ηhν)β,α − hσβ,αν)σ +O(h)3. (B7) For the PPN mapping of CS modified gravity, only the 00 component of the metric is needed to second order, which implies we only need C00 to O(h)2. This compo- nent is given by C00 = ǫ̃ijk0 2h0(i,ℓ) − hiℓ,0 hℓ[k,j]0 − h0[k,j]ℓ h0ℓǫ̃ 0jk(ℓ �ηh0k,j − hik,j0i +O(h)3, (B8) where in fact the last term vanishes due to the PPN gauge condition. Note that this term is automatically of O(6), which is well beyond the required order we need in h00. 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0704.0300

Scaling of Resistance and Electron Mean Free Path of Single-Walled Carbon Nanotubes

Scaling of Resistance and Electron Mean Free Path of Single-Walled Carbon Nanotubes Meninder Purewal1, Byung Hee Hong2, Anirudhh Ravi2, Bhupesh Chandra3, James Hone3, and Philip Kim2 Department of Applied Physics, Columbia University, New York, New York 10027 Department of Physics, Columbia University, New York, New York 10027 and Department of Mechanical Engineering, Columbia University, New York, New York 10027 We present an experimental investigation on the scaling of resistance in individual single walled carbon nanotube devices with channel lengths that vary four orders of magnitude on the same sample. The electron mean free path is obtained from the linear scaling of resistance with length at various temperatures. The low temperature mean free path is determined by impurity scattering, while at high temperature the mean free path decreases with increasing temperature, indicating that it is limited by electron-phonon scattering. An unusually long mean free path at room temperature has been experimentally confirmed. Exponentially increasing resistance with length at extremely long length scales suggests anomalous localization effects. Single walled carbon nanotubes (SWNTs) are 1D con- ductors that exhibit a rich variety of low dimensional charge transport phenomena [1], including ballistic con- duction [2, 3, 4, 5, 6], localization [7] and 1D variable range hopping [8]. The electron mean free path, Lm, is one of the important length scales that characterize the different 1D transport regimes. One method of determin- ing Lm in SWNTs is to measure ballistic conduction for a given device channel length. However, this method yields a lower bound of Lm, and works only at low temperature [2, 3, 4, 5] or at higher temperature for small length scales (<60 nm) [6]. Another approach to obtain Lm at room temperature is to employ scanning probe microscopy to measure the linear scaling of the channel resistance [9], or use non-invasive multi-terminal measurements [10]. Due to the experimental limitations of these approaches, the characterization of Lm for the same SWNTs over a range of temperatures is yet to be realized. Recent advances in the growth of extremely long SWNTs (>1 mm) [11] now allow for an intensive study on their intrinsic properties. In this letter, we present experimental measurements on the scaling behavior of resistance in individual, millimeter long SWNTs for the temperature range of 1.6 - 300 K. From the linear scaling of resistance, the temperature dependent electron mean free path is calculated for each temperature. Beyond the linear scaling regime, we observe that the resistance in- creases exponentially with length, indicating localization behavior. Macroscopically long and straight individual SWNTs were grown on a degenerately doped Si/SiO2 substrate (tox = 500 nm) using the chemical vapor deposition method described in Ref.[11]. This was followed by the fabrication of multiple Pd electrodes with various sepa- rations (200 nm- 400 µm)(Fig. 1(a)). Pd electrodes were chosen to create highly transparent SWNT-electrode con- tacts [4]. The diameters of the SWNTs were measured by atomic force microscope (AFM). We chose SWNTs with diameter d less than 2.5 nm to exclude any possi- bility of including multiwalled nanotubes (MWNT) in -15 0 15 -30 -15 0 500 m500 m Vg (V) Vg (V) 0.8 m 0.8 m 1.5 m 1.5 m FIG. 1: (a)Optical image showing typical SWNT devices with multiple Pd electrodes. (Inset) Scanning electron microscope image of an isolated SWNT contacted with these electrodes. Room temperature ISD(Vg) of selected channel lengths for (b) metallic SWNT (M1) and (c) semiconducting SWNT (SC3) with VSD = 6.4 and 2.7 mV, respectively. this study. In addition, we confirmed that the high bias saturation current is < 30 µA for all SWNTs stud- ied [12], assuring that the samples consisted of single tubes rather than small bundles or MWNTs. The sub- strate was used as a gate electrode to tune the chemical potential of the sample by the application of a gate volt- age Vg. A small dc source-drain bias voltage (< 10 mV), VSD, was applied between pairs of consecutive electrodes, and the two-terminal linear response conductance was determined from the measured source-drain current ISD. Fig. 1(b-c) shows the measured ISD as a function of VSD for selected channel length sections on two repre- sentative SWNTs. All curves exhibit a ‘gap’ like fea- ture - a range of Vg where ISD is suppressed. On the same SWNT, every device (pair of consecutive elec- http://arxiv.org/abs/0704.0300v2 trodes) shows a similar ISD(Vg) up to a length-dependent multiplicative factor, once we align the centers of the gap region for each curve. The similarity of the ISD(Vg) behavior in different sections for each SWNT sample indicates that the corresponding ‘gap’ features are de- rived from the intrinsic electronic structure of the SWNT rather than the effects of random local variation. We use the qualitatively different ISD(Vg) behaviors of different SWNTs to categorize them as metallic (M-NT) or semiconducting nanotubes (S-NT). Typical S-NTs (Fig. 1(c)) exhibit an off current region ISD < 10 when the Fermi energy EF lies n the energy gap [13, 14]. On the other hand, a weaker suppression of ISD(Vg) is observed in the ‘small gap’ region in M-NTs (Fig. 1(b)). The ‘small gap’ in M-NTs has been attributed to the curvature-induced energy gap Eg <100 meV [15], which is distinguished from the S-NT energy gap, which scales with diameter as Eg ∼ 1/d (nm) [1]. Among the 11 SWNTs we studied in this letter, we found 4 M-NTs and 7 S-NTs. Each of these SWNTs exhibit a gap cen- tered at Vg > 0, indicating their p-doped nature. At large negative gate voltage (Vg < −20 V), EF lies well outside of the gap region and ISD(Vg) saturates to I whose value depends only on the applied VSD and chan- nel length L of the SWNT section. The two-terminal resistance of the SWNT section is then obtained from R(L) = VSD/I SD. We note that four-terminal resistance measurements are possible for each section by utilizing the available multiple electrode configuration. However, in our experiment, the four terminal measurements yield essentially similar results to the two terminalR(L), which prevents separation of the ‘contact’ resistance contribu- tion from R(L). Such inseparable contact resistance be- tween SWNT-metal electrodes was reported to be caused by the invasiveness of metal contacts [16]. We designed many pairs of electrodes with different L on each SWNT so that the scaling of R(L) can be studied for a specific sample at a given temperature T . Fig. 2(a) show R(L) of a representative SWNT measured in the temperature range of 1.6 - 300 K and with an L range of 200 nm - 50 µm. In these ranges, R(L) increases linearly and appears to converge to a finite value for small L (in- set to Fig. 2(a)). We found that this scaling behavior can be described well by a simple linear dependence with an offset: R(L) = ρL +Rc, where ρ and Rc are interpreted as the 1D resistivity and contact resistance, respectively. The solid lines in Fig. 2(a) are the two parameter line fits of the data points at a given T value. From these fits, Rc(T ) and ρ(T ) are obtained as shown in Fig. 2(b) and Fig. 2(c), respectively. For this sample, Rc remains fairly constant at ∼8 kΩ and ρ(T ) exhibits typical metal- lic behavior, i.e. it decreases with T and saturates to a value ρsat at low temperatures. Similar scaling behavior of R(L) is observed in other SWNTs, from which both Rc and ρ(T ) are extracted within the linear scaling regime. Table I summarizes d, Rc, and ρsat for the 4 M-NTs 200 K 300 K 110 K 1.65 K L ( m) T (K) T (K) 0 100 200 300 100 200 300 (b) (c) 0 20 40 L ( m) FIG. 2: (a) (Inset) R(L) for sample M1 at select temperatures ranging from 1.6 - 300 K. (Main) A log-log plot highlights the behaviors at different lengths scaling 3 orders of magnitude. From the linear fits (solid lines) of these data points, we ob- tain the 1D resistivity (b) and the contact resistance (c) at different temperatures. The dashed line in (c) represents RQ. and 7 S-NTs considered in this study. To understand the scaling of R(L) in Fig. 2, we begin with the two-terminal Landauer-Buttiker formula applied to SWNTs [9]. If we consider 4 low-energy channels in the SWNT, 2 each for spin and band degeneracy, then the scaling of resistance is given by R(L) = (h/4e2)(L/Lm + 1) + Rnc, where e and h are electron charge and Plank constant and Lm and Rnc are the electron mean free path and the non- transparent contact resistance, respectively. Note that we separate out the contribution of Rnc from the total contact resistance Rc, so that the contact resistance be- comes the quantum resistance RQ = h/4e 2 when the con- tacts become fully transparent. From the experimentally obtained ρ(T ) and Rc, we can deduce Lm = RQ/ρ(T ) and Rnc = Rc − RQ for each of our SWNT samples. In particular, we note that Rnc <∼ RQ for the majority of our samples, suggesting that the barrier at the contacts is very thin and adds only a negligible contribution when L becomes substantially large. We now discuss the temperature dependent behavior of the mean free path. Fig. 3 is the central result of this letter, showing Lm(T ) of the SWNTs listed in Ta- ble I. Overall, Lm(T ) exhibits different behaviors in two regimes separated by Tcr: (i) the high tempera- ture regime (T > Tcr) where Lm ∼ T −1 (dashed line in Fig. 3), which indicates that inelastic scattering be- TABLE I: Device characteristics for SWNTs used in this study. The character M (SC) is designated for metallic (semiconduct- ing) SWNTs. M1 M2 M3 M4 SC1 SC2 SC3 SC4 SC5 SC6 SC7 d(nm) 2.0 ± .2 1.3 ± .4 1.7 ± .6 1.6 ± .4 1.6 ± .4 1.8 ± .6 1.9 ± .4 2.1 ± .2 2.2 ± .2 2.0 ± .6 2.2 ± .2 Rc(kΩ) 7.9 ± .8 11.5 ± 2.9 8.3 ± 2.5 12.0 ± 4.4 10.2 ± 4.5 14.9 ± 5.7 10.4 ± .9 7.0 ± 2.3 25.4 ± 4.2 6.9 ± 40 21.8 ± 14 ρsat(kΩ/µm) 0.76 ± .02 0.87 ± .02 0.93 ± .01 6.5 ± .08 2.95 ± .05 3.61 ± .05 4.64 ± .01 5.91 ± .12 8.13 ± .31 14.1 ± .19 16.3 ± .13 (µm) 8.56 ± .23 7.65 ± .17 7.07 ± .08 1.00 ± .01 2.24 ± .04 1.83 ± .03 1.40 ± .01 1.10 ± .02 0.80 ± .03 0.47 ± .01 0.40 ± .01 1001 10 T (K) FIG. 3: (color online) The electron mean free path for the samples listed in Table I at different temperatures. Most metallic SWNTs (open circles) saturate at higher values than that of semiconductors (closed circles). The dashed line rep- resents T−1 dependence. The insets show scanning gate microscopy images taken on devices SC2 (upper) and SC7 (lower). Lighter color corresponds less current in the SWNT. The defects in the SWNT are highlighted by the bright region (suppressed current) on the SWNT. The scale bar is 500nm. tween electrons and acoustic phonons is dominant [9, 17] regardless of chirality [18]; and (ii) the low temperature regime (T < Tcr) where Lm saturates to the the tube specific Lsatm . In this low temperature limit, the phonons freeze out and Lsatm is determined by the temperature in- dependent elastic scattering with impurities. We believe the widely spread Lsatm values (0.4-10 µm) in (ii) are a result of each SWNT sample having a static disorder of different strengths and densities. We employ scanning gate microscopy (SGM) [19] to image this static disor- der. Indeed, the SGM images on S-NTs (insets to Fig. 3) reveal that the SWNT with a shorter Lsatm shows more defects. Note also that we have experimentally confirmed that Lm is generally much higher for M-NTs than that of S-NTs. This is an indication that the scattering of electrons is strongly suppressed in M-NTs, as predicted by Ando et al. [20] and McEuen et al. [21]. In M-NTs we have experimentally shown that the ballistic electron 0 100 200 1.65 K 110 K L (µm) T (K) 0 100 200 L (µm) 101 102 0 100 200 300 400 1.65 K 110 K L (µm) 0.5 M T (K) 10050 L (µm) 100 10210110-1 FIG. 4: R(L) in the non-linear regimes for samples (a) SC6 (Lsatm ≈ 460 nm) and (b)M3 (L m ≈ 7 µm). Note that the data is magnified in (a) for clarity. The dashed line is an ex- tension of the linear regime and the solid line is a fit for all data. Rdev shows the absolute value of the difference between the actual device resistance and the corresponding linear resis- tance at 110 K (lower inset a) and 1.65 K (lower inset b).The non-linearity increases with decreasing temperature, which is reflected in the value of Lc(upper insets). conduction is possible for channel lengths up to 8 µm at low temperature and 0.8 µm even at room temperature. Finally, we turn our attention to the non-linear scaling of R(L). Fig. 4 presents R(L) beyond the linear scal- ing regime of a representative S-NT and M-NT. At ex- tremely long length scales and low temperatures, R(L) deviates from the linear dependence extended from the linear regime (dashed lines in main figure and see also Rdev = R(L) − Rc − RQL/Lm in lower insets). Since R(L << Lsatm ) ∼ RQ for all temperatures, we empha- size here that this non-linear behavior in R(L) is solely due to increasing electron scattering in the bulk part of the SWNTs rather than an increasing barrier between the SWNT and electrodes. In order to experimentally determine the critical length scale Lc beyond which the non-linear behaviors is dominant, we use a phenomeno- logical equation: R(L) = Rc + RQ(L/Lm + e L/Lc) to fit the data (solid curves in Fig. 4). While Lc shows a strong sample dependent behavior, generally we found Lc >> Lm in all temperature ranges, with the tempera- ture dependence exhibiting a trend of increasing Lc with increasing T (upper insets to Fig. 4). This observed be- havior of Lc(T ) excludes the quantum interference re- lated to strong localization effects such as Anderson Lo- calization [7] from the possible scenarios. In particular, in the high temperature regime (T > Tcr), the phase coher- ence length Lφ is limited by the phase-breaking electron- phonon scattering, and thus Lφ ∼ Lm << Lc, inviting further study to elucidate the observed localization be- havior beyond the strong localization limit [22, 23]. In conclusion, we determine the length dependent re- sistance for SWNTs with channel lengths ranged 200 nm - 400 µm. From the scaling behavior we evaluate the electron mean free path and localization length of the SWNT for a range of temperatures. While the low tem- perature mean free path is determined by the impurity scattering, an unusually long mean free path is demon- strated at room temperature, even with the dominant electron-phonon scattering. We thank I. Aleiner, B. Altshuler, and P. Jarillo- Herrero for helpful discussions. This work is supported by the NSF NIRT(ECS 0507111), CAREER (DMR- 0349232), NSEC (CHE-0117752), and the New York State Office of Science, Technology, and Academic Re- search (NYSTAR). [1] R. Saito, G. Dresselhaus, and M.S. Dresselhaus, Physical Properties of Carbon Nanotubes (Imperial College Press, London 1998). [2] J. Kong, E. Yenilmez, T.W. Tombler, W. Kim, H. Dai, R.B. Laughlin, L. Liu, C.S. Jayanthi, and S.Y. Wu, Phys. Rev. Lett. 87, 106801 (2001). [3] W. Liang, M. Bockrath, D. Bozovic, J.H. Hafner, M. Tinkham, and H. Park, Nature 411, 665 (2001). [4] D. Mann, A. Javey, J. Kong, Q. Wang, and H. Dai, Nano Lett. 3, 1541 (2003). [5] A. Javey, J. Guo, Q. Wang, M. Lundstrom and H. 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0704.0302

Spline Single-Index Prediction Model

SPLINE SINGLE-INDEX PREDICTION MODEL Li Wang and Lijian Yang University of Georgia and Michigan State University Abstract: For the past two decades, single-index model, a special case of projection pursuit regres- sion, has proven to be an efficient way of coping with the high dimensional problem in nonparamet- ric regression. In this paper, based on weakly dependent sample, we investigate the single-index prediction (SIP) model which is robust against deviation from the single-index model. The single- index is identified by the best approximation to the multivariate prediction function of the response variable, regardless of whether the prediction function is a genuine single-index function. A poly- nomial spline estimator is proposed for the single-index prediction coefficients, and is shown to be root-n consistent and asymptotically normal. An iterative optimization routine is used which is sufficiently fast for the user to analyze large data of high dimension within seconds. Simulation experiments have provided strong evidence that corroborates with the asymptotic theory. Appli- cation of the proposed procedure to the rive flow data of Iceland has yielded superior out-of-sample rolling forecasts. Key words and phrases: B-spline, geometric mixing, knots, nonparametric regression, root-n rate, strong consistency. 1. Introduction XTi , Yi = {Xi,1, ...,Xi,d, Yi}ni=1 be a length n realization of a (d+ 1)-dimensional strictly stationary process following the heteroscedastic model Yi = m (Xi) + σ (Xi) εi,m (Xi) = E (Yi|Xi) , (1.1) in which E (εi |Xi ) = 0, E ε2i |Xi = 1, 1 ≤ i ≤ n. The d-variate functions m, σ are the unknown mean and standard deviation of the response Yi conditional on the predictor vector Xi, often estimated nonparametrically. In what follows, we let XT , Y, ε have the stationary distribution of XTi , Yi, εi . When the dimension of X is high, one unavoidable issue is the “curse of dimensionality”, which refers to the poor convergence rate of nonparametric esti- mation of general multivariate function. Much effort has been devoted to the circumventing of this difficulty. In the words of Xia, Tong, Li and Zhu (2002), there are essentially two approaches: function approximation and dimension reduction. A favorite function approxima- tion technique is the generalized additive model advocated by Hastie and Tibshirani (1990), Address for correspondence: Lijian Yang, Department of Statistics and Probability, Michigan State Univer- sity, East Lansing, MI 48824, USA. E-mail: [email protected] http://arxiv.org/abs/0704.0302v2 2 LI WANG AND LIJIAN YANG see also, for example, Mammen, Linton and Nielsen (1999), Huang and Yang (2004), Xue and Yang (2006 a, b), Wang and Yang (2007). An attractive dimension reduction method is the single-index model, similar to the first step of projection pursuit regression, see Friedman and Stuetzle (1981), Hall (1989), Huber (1985), Chen (1991). The basic appeal of single-index model is its simplicity: the d-variate function m (x) = m (x1, ..., xd) is expressed as a univariate function of xT θ0 = p=1 xpθ0,p. Over the last two decades, many authors had devised various intelligent estimators of the single-index coefficient vector θ0 = (θ0,1, ..., θ0,d) , for instance, Powell, Stock and Stoker (1989), Härdle and Stoker (1989), Ichimura (1993), Klein and Spady (1993), Härdle, Hall and Ichimura (1993), Horowitz and Härdle (1996), Carroll, Fan, Gijbels and Wand (1997), Xia and Li (1999), Hristache, Juditski and Spokoiny (2001). More recently, Xia, Tong, Li and Zhu (2002) proposed the minimum average variance estimation (MAVE) for several index vectors. All the aforementioned methods assume that the d-variate regression function m (x) is exactly a univariate function of some xT θ0 and obtain a root-n consistent estimator of θ0. If this model is misspecified (m is not a genuine single-index function), however, a goodness-of-fit test then becomes necessary and the estimation of θ0 must be redefined, see Xia, Li, Tong and Zhang (2004). In this paper, instead of presuming that underlying true function m is a single-index function, we estimate a univariate function g that optimally approximates the multivariate function m in the sense of g (ν) = E m (X)|XT θ0 = ν , (1.2) where the unknown parameter θ0 is called the SIP coefficient, used for simple interpretation once estimated; XT θ0 is the latent SIP variable; and g is a smooth but unknown function used for further data summary, called the link prediction function. Our method therefore is clearly interpretable regardless of the goodness-of-fit of the single-index model, making it much more relevant in applications. We propose estimators of θ0 and g based on weakly dependent sample, which includes many existing nonparametric time series models, that are (i) computationally expedient and (ii) theoretically reliable. Estimation of both θ0 and g has been done via the kernel smoothing techniques in existing literature, while we use polynomial spline smoothing. The greatest advantages of spline smoothing, as pointed out in Huang and Yang (2004), Xue and Yang (2006 b) are its simplicity and fast computation. Our proposed procedure involves two stages: estimation of θ0 by some n-consistent θ̂, minimizing an empirical version of the mean squared error, R(θ) = E{Y − E(Y |XT θ)}2; spline smoothing of Y on XT θ̂ to obtain a cubic spline estimator ĝ of g. The best single-index approximation to m(x) is then m̂(x) = ĝ xT θ̂ Under geometrically strong mixing condition, strong consistency and n-rate asymptotic SINGLE-INDEX PREDICTION MODEL 3 normality of the estimator θ̂ of the SIP coefficient θ0 in (1.2) are obtained. Proposition 2.2 is the key in understanding the efficiency of the proposed estimator. It shows that the derivatives of the risk function up to order 2 are uniformly almost surely approximated by their empirical versions. Practical performance of the SIP estimators is examined via Monte Carlo examples. The estimator of the SIP coefficient performs very well for data of both moderate and high dimension d, of sample size n from small to large, see Tables 1 and 2, Figures 1 and 2. By taking advantages of the spline smoothing and the iterative optimization routines, one reduces the computation burden immensely for massive data sets. Table 2 reports the computing time of one simulation example on an ordinary PC, which shows that for massive data sets, the SIP method is much faster than the MAVE method. For instance, the SIP estimation of a 200-dimensional θ0 from a data of size 1000 takes on average mere 2.84 seconds, while the MAVE method needs to spend 2432.56 seconds on average to obtain a comparable estimates. Hence on account of criteria (i) and (ii), our method is indeed appealing. Applying the proposed SIP procedure to the rive flow data of Iceland, we have obtained superior forecasts, based on a 9-dimensional index selected by BIC, see Figure 5. The rest of the paper is organized as follows. Section 2 gives details of the model spec- ification, proposed methods of estimation and main results. Section 3 describes the actual procedure to implement the estimation method. Section 4 reports our findings in an extensive simulation study. The proposed SIP model and the estimation procedure are applied in Section 5 to the rive flow data of Iceland. Most of the technical proofs are contained in the Appendix. 2. The Method and Main Results 2.1. Identifiability and definition of the index coefficient It is obvious that without constraints, the SIP coefficient vector θ0 = (θ0,1, ..., θ0,d) identified only up to a constant factor. Typically, one requires that ‖θ0‖ = 1 which entails that at least one of the coordinates θ0,1, ..., θ0,d is nonzero. One could assume without loss of generality that θ0,d > 0, and the candidate θ0 would then belong to the upper unit hemisphere Sd−1+ = (θ1, ..., θd) | p=1 θ p = 1, θd > 0 For a fixed θ = (θ1, ..., θd) , denote Xθ = X T θ, Xθ,i = X i θ, 1 ≤ i ≤ n. Let mθ (Xθ) = E (Y |Xθ) = E {m (X) |Xθ} . (2.1) Define the risk function of θ as R (θ) = E {Y −mθ (Xθ)}2 = E {m (X)−mθ (Xθ)}2 + Eσ2 (X) , (2.2) 4 LI WANG AND LIJIAN YANG which is uniquely minimized at θ0 ∈ Sd−1+ , i.e. θ0 = arg min θ∈Sd−1+ R (θ) . Remark 2.1. Note that Sd−1+ is not a compact set, so we introduce a cap shape subset of Sd−1+ Sd−1c = (θ1, ..., θd) | θ2p = 1, θd ≥ 1− c2 , c ∈ (0, 1) Clearly, for an appropriate choice of c, θ0 ∈ Sd−1c , which we assume in the rest of the paper. Denote θ−d = (θ1, ..., θd−1) , since for fixed θ ∈ Sd−1+ , the risk function R (θ) depends only on the first d− 1 values in θ, so R (θ) is a function of θ−d R∗ (θ−d) = R θ1, θ2, ..., θd−1, 1− ‖θ−d‖22 with well-defined score and Hessian matrices S∗ (θ−d) = R∗ (θ−d) , H ∗ (θ−d) = ∂θ−d∂θ R∗ (θ−d) . (2.3) Assumption A1: The Hessian matrix H∗ (θ0,−d) is positive definite and the risk function R is locally convex at θ0,−d, i.e., for any ε > 0, there exists δ > 0 such that R ∗ (θ−d)−R∗ (θ0,−d) < δ implies ‖θ−d − θ0,−d‖2 < ε. 2.2. Variable transformation Throughout this paper, we denote by Bda = x ∈ Rd |‖x‖ ≤ a the d-dimensional ball with radius a and center 0 and ∣the kth order partial derivatives of m are continuous on B the space of k-th order smooth functions. Assumption A2: The density function of X, f (x) ∈ C(4) , and there are constants 0 < cf ≤ Cf such that cf/Vold ≤ f (x) ≤ Cf/Vold , x ∈ Bda f (x) ≡ 0, x /∈ Bda For a fixed θ, define the transformed variables of the SIP variable Xθ Uθ = Fd (Xθ) , Uθ,i = Fd (Xθ,i) , 1 ≤ i ≤ n, (2.4) SINGLE-INDEX PREDICTION MODEL 5 in which Fd is the a rescaled centered Beta {(d+ 1) /2, (d+ 1) /2} cumulative distribution function, i.e. Fd (ν) = ∫ ν/a Γ (d+ 1) Γ {(d+ 1) /2}2 2d 1− t2 )(d−1)/2 dt, ν ∈ [−a, a] . (2.5) Remark 2.2. For any fixed θ, the transformed variable Uθ in (2.4) has a quasi-uniform [0, 1] distribution. Let fθ (u) be the probability density function of Uθ, then for any u ∈ [0, 1] fθ (u) = d (v) fXθ (v) , v = F d (u) , in which fXθ (v) = lim△ν→0 P (ν ≤ Xθ ≤ ν +△ν). Noting that xθ is exactly the projection of x on θ, let Dν = {x|ν ≤ xθ ≤ ν +△ν} ∩Bda, then one has P (ν ≤ Xθ ≤ ν +△ν) = P (X ∈ Dν) = f (x) dx. According to Assumption A2 cfVold(Dν) Vold (B ≤ P (ν ≤ Xθ ≤ ν +△ν) ≤ CfVold(Dν) Vold (B On the other hand Vold(Dν) = Vold−1(Jν)△ν + o (△ν) , where Jν = {x|xθ = v} ∩Bda. Note that the volume of Bda is πd/2ad/Γ (d/2 + 1) and Vold−1 (Jν) = π(d−1)/2 a2 − ν2 )(d−1)/2 Γ {(d+ 1)/2} , Vold−1(Jν) Vold (B Γ (d+ 1) }(d−1)/2 Therefore 0 < cf ≤ fθ (u) ≤ Cf <∞, for any fixed θ and u ∈ [0, 1]. In terms of the transformed SIP variable Uθ in (2.4), we can rewrite the regression function mθ in (2.1) for fixed θ γθ (Uθ) = E {m (X) |Uθ} = E {m (X) |Xθ} = mθ (Xθ) , (2.6) then the risk function R (θ) in (2.2) can be expressed as R (θ) = E {Y − γθ (Uθ)}2 = E {m (X)− γθ (Uθ)}2 + Eσ2 (X) . (2.7) 2.3. Estimation Method 6 LI WANG AND LIJIAN YANG Estimation of both θ0 and g requires a degree of statistical smoothing, and all estimation here is carried out via cubic spline. In the following, we define the estimator θ̂ of θ0 and the estimator ĝ of g. To introduce the space of splines, we pre-select an integer n1/6 ≪ N = Nn ≪ n1/5 (log n)−2/5, see Assumption A6 below. Divide [0, 1] into (N + 1) subintervals Jj = [tj, tj+1), j = 0, ..., N − 1, JN = [tN , 1], where T := {tj}Nj=1 is a sequence of equally-spaced points, called interior knots, given as t1−k = ... = t−1 = t0 = 0 < t1 < ... < tN < 1 = tN+1 = ... = tN+k, in which tj = jh, j = 0, 1, ..., N +1, h = 1/ (N + 1) is the distance between neighboring knots. The j-th B-spline of order k for the knot sequence T denoted by Bj,k is recursively defined by de Boor (2001). Denote by Γ(k−2) = Γ(k−2) [0, 1] the space of all C(k−2) [0, 1] functions that are polynomials of degree k−1 on each interval. For fixed θ, the cubic spline estimator γ̂θ of γθ and the related estimator m̂θ of mθ are defined as γ̂θ (·) = arg min γ(·)∈Γ(2)[0,1] {Yi − γ (Uθ,i)}2 , m̂θ (ν) = γ̂θ {Fd (ν)} . (2.8) Define the empirical risk function of θ R̂ (θ) = n−1 {Yi − γ̂θ (Uθ,i)}2 = n−1 {Yi − m̂θ (Xθ,i)}2 , (2.9) then the spline estimator of the SIP coefficient θ0 is defined as θ̂ = arg min θ∈Sd−1c R̂ (θ) , and the cubic spline estimator of g is m̂θ with θ replaced by θ̂, i.e. ĝ (ν) = arg min γ(·)∈Γ(2)[0,1] Yi − γ {Fd (ν)} . (2.10) 2.4. Asymptotic results Before giving the main theorems, we state some other assumptions. Assumption A3: The regression function m ∈ C(4) for some a > 0. Assumption A4: The noise ε satisfies E (ε |X) = 0, E ε2 |X = 1 and there exists a positive constant M such that sup |ε|3 |X = x < M . The standard deviation function σ (x) is continuous on Bda, 0 < cσ ≤ inf x∈Bda σ (x) ≤ sup x∈Bda σ (x) ≤ Cσ <∞. SINGLE-INDEX PREDICTION MODEL 7 Assumption A5: There exist positive constants K0 and λ0 such that α (n) ≤ K0e−λ0n holds for all n, with the α-mixing coefficient for XTi , εi defined as α (k) = sup B∈σ{Zs,s≤t},C∈σ{Zs,s≥t+k} |P (B ∩ C)− P (B)P (C)| , k ≥ 1. Assumption A6: The number of interior knots N satisfies: n1/6 ≪ N ≪ n1/5 (log n)−2/5. Remark 2.3. Assumptions A3 and A4 are typical in the nonparametric smoothing literature, see for instance, Härdle (1990), Fan and Gijbels (1996), Xia, Tong Li and Zhu (2002). By the result of Pham (1986), a geometrically ergodic time series is a strongly mixing sequence. Therefore, Assumption A5 is suitable for (1.1) as a time series model under aforementioned assumptions. We now state our main results in the next two theorems. Theorem 1. Under Assumptions A1-A6, one has θ̂−d−→ θ0,−d, a.s.. (2.11) Proof. Denote by (Ω,F ,P) the probability space on which all XTi , Yi are defined. By Proposition 2.2, given at the end of this section ‖θ−d‖2≤ R̂∗ (θ−d)−R∗ (θ−d) −→ 0, a.s.. (2.12) So for any δ > 0 and ω ∈ Ω, there exists an integer n0 (ω), such that when n > n0 (ω), R̂∗ (θ0,−d, ω) − R∗ (θ0,−d) < δ/2. Note that θ̂−d = θ̂−d (ω) is the minimizer of R̂∗ (θ−d, ω), so R̂∗ θ̂−d (ω) , ω − R∗ (θ0,−d) < δ/2. Using (2.12), there exists n1 (ω), such that when n > n1 (ω), R θ̂−d (ω) , ω − R̂∗ θ̂−d (ω) , ω < δ/2. Thus, when n > max (n0 (ω) , n1 (ω)), θ̂−d (ω) , ω −R∗ (θ0,−d) < δ/2 + R̂∗ θ̂−d (ω) , ω −R∗ (θ0,−d) < δ/2 + δ/2 = δ. According to Assumption A1, R∗ is locally convex at θ0,−d, so for any ε > 0 and any ω, if θ̂−d (ω) , ω −R∗ (θ0,−d) < δ, then ∥θ̂−d (ω)−θ0,−d ∥ < ε for n large enough , which implies the strong consistency. Theorem 2. Under Assumptions A1-A6, one has θ̂−d−θ0,−d d−→ N {0,Σ (θ0)} , where Σ (θ0) = {H∗ (θ0,−d)}−1Ψ(θ0) {H∗ (θ0,−d)}−1, H∗ (θ0,−d) = {lpq}d−1p,q=1 and Ψ(θ0) = {ψpq}d−1p,q=1 with lp,q = −2E [{γ̇pγ̇q + γθ0 γ̈p,q} (Uθ0)] + 2θ0,qθ E [{γ̇pγ̇d (Uθ0) + γθ0 γ̈p,d} (Uθ0)] +2θ−30,dE [(γθ0 γ̇d) (Uθ0)] θ20,d + θ I{p=q} + θ0,pθ0,qI{p 6=q} +2θ0,pθ 0,dE [{γ̇pγ̇q + γθ0 γ̈p,q} (Uθ0)]− 2θ0,pθ0,qθ γ̇2d + γθ0 γ̈d,d (Uθ0) 8 LI WANG AND LIJIAN YANG ψpq = 4E γ̇p − θ0,pθ−10,dγ̇d γ̇q − θ0,qθ−10,dγ̇d (Uθ0) {γθ0 (Uθ0)− Y } in which γ̇p and γ̈p,q are the values of ∂θp∂θq γθ taking at θ = θ0, for any p, q = 1, 2, ..., d−1 and γθ is given in (2.6). Remark 2.4. Consider the Generalized Linear Model (GLM): Y = g XT θ0 +σ (X) ε, where g is a known link function. Let θ̃ be the nonlinear least squared estimator of θ0 in GLM. Theorem 2 shows that under the assumptions A1-A6, the asymptotic distribution of the θ̂−d is the same as that of θ̃. This implies that our proposed SIP estimator θ̂−d is as efficient as if the true link function g is known. The next two propositions play an important role in our proof of the main results. Propo- sition 2.1 establishes the uniform convergence rate of the derivatives of γ̂θ up to order 2 to those of γθ in θ. Proposition 2.2 shows that the derivatives of the risk function up to order 2 are uniformly almost surely approximated by their empirical versions. Proposition 2.1. Under Assumptions A2-A6, with probability 1 θ∈Sd−1c u∈[0,1] |γ̂θ (u)− γθ (u)| = O log n+ h4 , (2.13) 1≤p≤d θ∈Sd−1c 1≤i≤n {γ̂θ (Uθ,i)− γθ (Uθ,i)} log n√ , (2.14) 1≤p,q≤d θ∈Sd−1c 1≤i≤n ∂θp∂θq {γ̂θ (Uθ,i)− γθ (Uθ,i)} log n√ . (2.15) Proposition 2.2. Under Assumptions A2-A6, one has for k = 0, 1, 2 ‖θ−d‖≤ ∂kθ−d R̂∗ (θ−d)−R∗ (θ−d) = o(1), a.s.. Proofs of Theorem 2, Propositions 2.1 and 2.2 are given in Appendix. 3. Implementation In this section, we will describe the actual procedure to implement the estimation of θ0 and g. We first introduce some new notation. For fixed θ, write the B-spline matrix as Bθ = {Bj,4 (Uθ,i)}n, Ni=1,j=−3 and Pθ = Bθ θ (3.1) as the projection matrix onto the cubic spline space Γ n,θ. For any p = 1, ..., d, denote Ḃp = Bθ, Ṗp = SINGLE-INDEX PREDICTION MODEL 9 as the first order partial derivatives of Bθ and Pθ with respect to θ. Let Ŝ∗(θ−d) be the score vector of R̂ ∗ (θ−d), i.e. Ŝ∗(θ−d) = R̂∗ (θ−d) . (3.2) The next lemma provides the exact forms of Ŝ∗(θ−d). Lemma 3.1. For the score vector of R̂∗ (θ−d) defined in (3.2), one has Ŝ∗ (θ−d) = −n−1 ṖpY − θpθ−1d Y , (3.3) where for any p = 1, 2, ..., d ṖpY = 2Y T (I−Pθ) Ḃp θ Y, (3.4) where Ḃp = {Bj,3 (Uθ,i)−Bj+1,3 (Uθ,i)} Ḟd (Xθ,i)h−1Xi,p }n, N i=1,j=−3 Ḟd (x) = Γ (d+ 1) aΓ {(d+ 1) /2}2 2d I (|x| ≤ a) . Proof. For any p = 1, 2, ..., d, the derivatives of B-splines in de Boor (2001) implies Ḃp = Bj,4 (Uθ,i) }n, N i=1,j=−3 Bj,4 (Uθ,i) }n, N i=1,j=−3 Bj,3 (Uθ,i) tj+3 − tj Bj+1,3 (Uθ,i) tj+4 − tj+1 Ḟd (Xθ,i)Xi,p }n, N i=1,j=−3 {Bj,3 (Uθ,i)−Bj+1,3 (Uθ,i)} Ḟd (Xθ,i) h−1Xi,p }n, N i=1,j=−3 Next, note that Ṗp = Ḃp θ +Bθ = Ḃp θ +Bθ θ +Bθ Since BTθ Bθ BTθ Bθ BTθ Bθ θ Bθ + )−1 ∂ BTθ Bθ and ∂ BTθ Bθ = ḂTp Bθ +B θ Ḃp, thus )−1 ( p Bθ +B θ Ḃp 10 LI WANG AND LIJIAN YANG Hence Ṗp = (I−Pθ) Ḃp θ +Bθ p (I−Pθ) . Thus, (3.4) follows immediately. In practice, the estimation is implemented via the following procedure. Step 1. Standardize the predictor vectors {Xi}ni=1 and for each fixed θ ∈ Sd−1c obtain the CDF transformed variables {Uθ,i}ni=1 of the SIP variable {Xθ,i} through formula (2.5), where the radius a is taken to be the 95% percentile of {‖Xi‖}ni=1. Step 2. Compute quadratic and cubic B-spline basis at each value Uθ,i, where the number of interior knots N is N = min n1/5.5 , (3.5) Step 3. Find the estimator θ̂ of θ0 by minimizing R̂ ∗ through the port optimization routine with (0, 0, ..., 1) as the initial value and the empirical score vector Ŝ∗ in (3.3). If d < n, one can take the simple LSE (without the intercept) for data {Yi,Xi}ni=1 with its last coordinate set positive. Step 4. Obtain the spline estimator ĝ of g by plugging θ̂ obtained in Step 3 into (2.10). Remark 3.1. In (3.5), c1 and c2 are positive integers and [ν] denotes the integer part of ν. The choice of the tuning parameter c1 makes little difference for a large sample and according to our asymptotic theory there is no optimal way to set these constants. We recommend using c1 = 1 to save computing for massive data sets. The first term ensures Assumption A6. The addition constrain c2 can be taken from 5 to 10 for smooth monotonic or smooth unimodel regression and c2 > 10 if has many local minima and maxima, which is very unlikely in application. 4. Simulations In this section, we carry out two simulations to illustrate the finite-sample behavior of our SIP estimation method. The number of interior knots N is computed according to (3.5) with c1 = 1, c2 = 5. All of our codes have been written in R. Example 1. Consider the model in Xia, Li, Tong and Zhang (2004) Y = m (X) + σ0ε, σ0 = 0.3, 0.5, ε i.i.d∼ N(0, 1) where X = (X1,X2) T ∼N(0, I2), truncated by [−2.5, 2.5]2 and m (x) = x1 + x2 + 4exp − (x1 + x2)2 x21 + x . (4.1) If δ = 0, then the underlying true functionm is a single-index function, i.e., m (X) = 2XT θ0+ XT θ0 , where θT0 = (1, 1) / 2. While δ 6= 0, then m is not a genuine single-index SINGLE-INDEX PREDICTION MODEL 11 function. An impression of the bivariate function m for δ = 0 and δ = 1 can be gained in Figure 1 (a) and (b), respectively. Table 1: Report of Example 1 (Values out/in parentheses: δ = 0/δ = 1) σ0 n θ0 BIAS SD MSE Average MSE 5e− 04 0.00825 7e− 05 (−0.00236) (0.02093) (0.00044) 7e− 05 −6e− 04 0.00826 7e− 05 (0.00043) (0.00174) (0.02083) (0.00043) −0.00124 0.00383 2e− 05 (−0.00129) (0.01172) (0.00014) 2e− 05 −0.00124 0.00383 2e− 05 (0.00014) (0.00110) (0.01160) (0.00013) 0.00121 0.01346 0.00018 (−0.00137) (0.02257) (0.00051) 0.00018 −0.00147 0.01349 0.00018 (0.00051) (0.00062) (0.02309) (0.00052) −0.00204 0.00639 4e− 05 (−0.00229) (0.01205) (0.00015) 4e− 05 0.00197 0.00637 4e− 05 (0.00015) (0.00208) (0.01190) (0.00014) For δ = 0, 1, we draw 100 random realizations of each sample size n = 50, 100, 300 respec- tively. To demonstrate how close our SIP estimator is to the true index parameter θ0, Table 1 lists the sample mean (MEAN), bias (BIAS), standard deviation (SD), the mean squared error (MSE) of the estimates of θ0 and the average MSE of both directions. From this table, we find that the SIP estimators are very accurate for both cases δ = 0 and δ = 1, which shows that our proposed method is robust against the deviation from single-index model. As we expected, when the sample size increases, the SIP coefficient is more accurately estimated. Moreover, for n = 100, 300, the total average is inversely proportional to n. Example 2. Consider the heteroscedastic regression model (1.1) with m (X) = sin , σ (X) = σ0 5− exp 5 + exp ) , (4.2) in which Xi = {Xi,1, ...,Xi,d}T and εi, i = 1, ..., n, are i.i.d∼ N (0, 1), σ0 = 0.2. In our simulation, the true parameter θT0 = (1, 1, 0, ..., 0, 1)/ 3 for different sample size n and dimension d. The 12 LI WANG AND LIJIAN YANG superior performance of SIP estimators is borne out in comparison with MAVE of Xia, Tong, Li and Zhu (2002). We also investigate the behavior of SIP estimators in the previously unemployed cases that sample size n is smaller than or equal to d, for instance, n = 100, d = 100, 200 and n = 200, d = 200, 400. The average MSEs of the d dimensions are listed in Table 2, from which we see that the performance of the SIP estimators are quite reasonable and in most of the scenarios n ≤ d, the SIP estimators still work astonishingly well where the MAVEs become unreliable. For n = 100, d = 10, 50, 100, 200, the estimates of the link prediction function g from model (4.2) are plotted in Figure 2, which is rather satisfactory even when dimension d exceeds the sample size n. Theorem 1 indicates that θ̂−d is strongly consistent of θ0,−d. To see the convergence, we run 100 replications and in each replication, the value of ‖θ̂ − θ0‖/ d is computed. Figure 3 plots the kernel density estimations of the 100 ‖θ̂ − θ0‖ in Example 2, in which dimension d = 10, 50, 100, 200. There are four types of line characteristics which correspond to the two sample sizes, the dotted-dashed line (n = 100), dotted line (n = 200), dashed line (500) and solid line (n = 1000). As sample sizes increasing, the squared errors are becoming closer to 0, with narrower spread out, confirmative to the conclusions of Theorem 1. Lastly, we report the average computing time of Example 2 to generate one sample of size n and perform the SIP or MAVE procedure done on the same ordinary Pentium IV PC in Table 2. From Table 2, one sees that our proposed SIP estimator is much faster than the MAVE. The computing time for MAVE is extremely sensitive to sample size as we expected. For very large d, MAVE becomes unstable to the point of the breaking down in four cases. 5. An application In this section we demonstrate the proposed SIP model through the river flow data of Jökulsá Eystri River of Iceland, from January 1, 1972 to December 31, 1974. There are 1096 observations, see Tong (1990). The response variables are the daily river flow (Yt), measured in meter cubed per second of Jökulsá Eystri River. The exogenous variables are temperature (Xt) in degrees Celsius and daily precipitation (Zt) in millimeters collected at the meteorological station at Hveravellir. This data set was analyzed earlier through threshold autoregressive (TAR) models by Tong, Thanoon and Gudmundsson (1985), Tong (1990), and nonlinear additive autoregressive (NAARX) models by Chen and Tsay (1993). Figure 4 shows the plots of the three time series, from which some nonlinear and non-stationary features of the river flow series are evident. To make these series stationary, we remove the trend by a simple quadratic spline regression and these trends (dashed lines) are shown in Figure 4. By an abuse of notation, we shall continue to use Xt, Yt, Zt to denote the detrended series. SINGLE-INDEX PREDICTION MODEL 13 In the analysis, we pre-select all the lagged values in the last 7 days (1 week), i.e., the predictor pool is {Yt−1, ..., Yt−7,Xt,Xt−1, ...,Xt−7, Zt, Zt−1, ..., Zt−7, }. Using BIC similar to Huang and Yang (2004) for our proposed spline SIP model with 3 interior knots, the following 9 explanatory variables are selected from the above set {Yt−1, ..., Yt−4,Xt,Xt−1,Xt−2, Zt, Zt−1}. Based on this selection, we fit the SIP model again and obtain the estimate of the SIP coefficient θ̂ = {−0.877, 0.382,−0.208, 0.125,−0.046,−0.034, 0.004,−0.126, 0.079}T . Figure 5 (a) and (b) display the fitted river flow series and the residuals against time. Next we examine the forecasting performance of the SIP method. We start with estimating the SIP estimator using only observations of the first two years, then we perform the out-of- sample rolling forecast of the entire third year. The observed values of the exogenous variables are used in the forecast. Figure 5 (c) shows this SIP out-of-sample rolling forecasts. For the purpose of comparison, we also try the MAVE method, in which the same predictor vector is selected by using BIC. The mean squared prediction error is 60.52 for the SIP model, 61.25 for MAVE, 65.62 for NAARX, 66.67 for TAR and 81.99 for the linear regression model, see Chen and Tsay (1993). Among the above five models, the SIP model produces the best forecasts. 6. Conclusion In this paper we propose a robust SIP model for stochastic regression under weak depen- dence regardless if the underlying function is exactly a single-index function. The proposed spline estimator of the index coefficient possesses not only the usual strong consistency and n-rate asymptotically normal distribution, but also is as efficient as if the true link function g is known. By taking advantage of the spline smoothing method and the iterative method, the proposed procedure is much faster than the MAVE method. This procedure is especially powerful for large sample size n and high dimension d and unlike the MAVE method, the performance of the SIP remains satisfying in the case d > n. Acknowledgment This work is part of the first author’s dissertation under the supervision of the second author, and has been supported in part by NSF award DMS 0405330. Appendix A.1. Preliminaries In this section, we introduce some properties of the B-spline. Lemma A.1. There exist constants c > 0 such that for j=−k+1 αj,kBj,k up to order k = 4 ch1/r ‖α‖r ≤ j=−k+1 αj,kBj,k 3r−1h )1/r ‖α‖r , 1 ≤ r ≤ ∞ ch1/r ‖α‖r ≤ j=−k+1 αj,kBj,k ≤ (3h)1/r ‖α‖r , 0 < r < 1 14 LI WANG AND LIJIAN YANG where α := (α−1,2, α0,2, ..., αN,2, ..., αN,4). In particular, under Assumption A2, for any fixed θ ch1/2 ‖α‖2 ≤ j=−k+1 αj,kBj,k ≤ Ch1/2 ‖α‖2 . Proof. It follows from the B-spline property on page 96 of de Boor (2001), j=−k+1Bj,k ≡ 3 on [0, 1]. So the right inequality follows immediate for r = ∞. When 1 ≤ r < ∞, we use Hölder’s inequality to find j=−k+1 αj,kBj,k j=−k+1 |αj,k|r Bj,k j=−k+1 1−1/r = 31−1/r j=−k+1 |αj,k|r Bj,k Since all the knots are equally spaced, −∞Bj,k (u) du ≤ h, the right inequality follows from j=−k+1 αj,kBj,k (u) du ≤ 3r−1h ‖α‖rr . When r < 1, we have j=−k+1 αj,kBj,k j=−k+1 |αj,k|r Brj,k. Since j,k (u) du ≤ tj+k − tj = kh and j=−k+1 αj,kBj,k (u) du ≤ ‖α‖rr Brj,k (u) du ≤ 3h ‖α‖ the right inequality follows in this case as well. For the left inequalities, we derive from Theorem 5.4.2, DeVore and Lorentz (1993) |αj,k| ≤ C1h−1/r ∫ tj+1 j=−k+1 αj,kBj,k (u) for any 0 < r ≤ ∞, so |αj,k|r ≤ Cr1h−1 ∫ tj+1 j=−k+1 αj,kBj,k (u) SINGLE-INDEX PREDICTION MODEL 15 Since each u ∈ [0, 1] appears in at most k intervals (tj,tj+k), adding up these inequalities, we obtain that ‖α‖rr ≤ C1h ∫ tj+k j=−k+1 αj,kBj,k (u) du ≤ 3Ch−1 j=−k+1 αj,kBj,k The left inequality follows. For any functions φ and ϕ, define the empirical inner product and the empirical norm as 〈φ,ϕ〉θ = φ (u)ϕ (u) fθ (u) du, ‖φ‖22,n,θ = n φ2 (Uθ,i) . In addition, if functions φ,ϕ are L2 [0, 1]-integrable, define the theoretical inner product and its corresponding theoretical L2 norm as ‖φ‖22,θ = φ2 (u) fθ (u) du, 〈φ,ϕ〉n,θ = n φ (Uθ,i)ϕ (Uθ,i) . Lemma A.2. Under Assumptions A2, A5 and A6, with probability 1, θ∈Sd−1c k,k′=2,3,4 1≤j,j′≤N Bj,k, Bj′,k′ Bj,k, Bj′,k′ log n Proof. We only prove the case k = k′ = 4, all other cases are similar. Let ζθ,j,j′,i = Bj,4 (Uθ,i)Bj′,4 (Uθ,i)− EBj,4 (Uθ,i)Bj′,4 (Uθ,i) , with the second moment Eζ2θ,j,j′,i = E B2j,4 (Uθ,i)B j′,4 (Uθ,i) EBj,4 (Uθ,i)Bj′,4 (Uθ,i) where EBj,4 (Uθ,i)Bj′,4 (Uθ,i) }2 ∼ N−2, E B2j,4 (Uθ,i)B j′,4 (Uθ,i) ∼ N−1 by Assumption A2. Hence, Eζ2θ,j,j′,i ∼ N−1. The k-th moment is given by ∣ζθ,j,j′,i ∣Bj,4 (Uθ,i)Bj′,4 (Uθ,i)− EBj,4 (Uθ,i)Bj′,4 (Uθ,i) ≤ 2k−1 ∣Bj,4 (Uθ,i)Bj′,4 (Uθ,i) ∣EBj,4 (Uθ,i)Bj′,4 (Uθ,i) where ∣EBj,4 (Uθ,i)Bj′,4 (Uθ,i) k ∼ N−k, E ∣EBj,4 (Uθ,i)Bj′,4 (Uθ,i) k ∼ N−1. Thus, there exists a constant C > 0 such that E ∣ζθ,j,j′,i k ≤ C2k−1k!Eζ2j,j′,i. So the Cramér’s condition is satisfied with Cramér’s constant c∗. By the Bernstein’s inequality (see Bosq (1998), Theorem 1.4, page 31), we have for k = 3 ζθ,j,j′,i ≤ a1 exp 25m22 + 5c + a2 (k)α q + 1 ])6/7 16 LI WANG AND LIJIAN YANG where δn = δ log n√ , a1 = 2 δ2 (nN) log2 n 25m22 + 5c , m22 ∼ N−1, a2 (3) = 11n , m3 = max 1≤i≤n ∥ζθ,j,j′,i ≤ cN1/3. Observe that 5cδn = o(1) by Assumption A6, then by taking q such that ≥ c0 log n, q ≥ c1n/ log n for some constants c0, c1, one has a1 = O(n/q) = O (log n), a2 (3) = o Assumption A6 again. Assumption A5 yields that q + 1 ])6/7 K0 exp q + 1 ])}6/7 ≤ Cn−6λ0c0/7. Thus, for fixed θ ∈ Sd−1c , when n large enough ζθ,j,j′,i ≤ c log n exp −c2δ2 log n + Cn2−6λ0c0/7. (A.1) We divide each range of θp, p = 1, 2, ..., d − 1, into n6/(d−1) equally spaced intervals with disjoint endpoints −1 = θp,0 < θp,1 < ... < θp,Mn = 1, for p = 1, ..., d − 1. Projecting these small cylinders onto Sd−1c , the radius of each patch Λr, r = 1, ...,Mn is bounded by cM Denote the projection of the Mn points as θr = θr,−d, 1− ‖θr,−d‖22 , r = 0, 1, ...,Mn. Employing the discretization method, sup θ∈Sd−1c 1≤j,j′≤N ∣ζθ,j,j′,i ∣ is bounded by 0≤r≤Mn 1≤j,j′≤N ∣ζθr,j,j′,i ∣+ sup 0≤r≤Mn 1≤j,j′≤N ∣ζθ,j,j′,i − ζθr,j,j′,i ∣ . (A.2) By (A.1) and Assumption A6, there exists large enough value δ > 0 such that ζθr,j,j′,i ≤ n−10, which implies that 1≤j,j′≤N ζθr ,j,j′,i N2Mnn −10 ≤ C n−3 <∞. Thus, Borel-Cantelli Lemma entails that 0≤r≤Mn 1≤j,j′≤N ζθr ,j,j′,i log n√ , a.s.. (A.3) SINGLE-INDEX PREDICTION MODEL 17 Employing Lipschitz continuity of the cubic B-spline, one has with probability 1 0≤r≤Mn 1≤j,j′≤N ζθ,j,j′,i − ζθr,j,j′,i M−1n h −6) . (A.4) Therefore Assumption A2, (A.2), (A.3) and (A.4) lead to the desired result. Denote by Γ = Γ(0)∪Γ(1)∪Γ(2) the space of all linear, quadratic and cubic spline functions on [0, 1]. We establish the uniform rate at which the empirical inner product approximates the theoretical inner product for all B-splines Bj,k with k = 2, 3, 4. Lemma A.3. Under Assumptions A2, A5 and A6, one has An = sup θ∈Sd−1c γ1,γ2∈Γ 〈γ1, γ2〉n,θ − 〈γ1, γ2〉θ ‖γ1‖2,θ ‖γ2‖2,θ log n , a.s.. (A.5) Proof. Denote without loss of generality, j=−k+1 αjkBj,k, γ2 = j=−k+1 βjkBj,k, for any two 3 (N + 3)-vectors α =(α−1,2, α0,2, ..., αN,2, ..., αN,4) , β =(β−1,2, β0,2, ..., βN,2, ..., βN,4) . Then for fixed θ 〈γ1, γ2〉n,θ = j=−k+1 αj,kBj,k (Uθ,i) j=−k+1 βj,kBj,k (Uθ,i) j=−k+1 j′=−k+1 αj,kβj′,k′ Bj,k, Bj′,k′ ‖γ1‖22,θ = j=−k+1 j′=−k+1 αj,kαj′,k′ Bj,k, Bj′ ,k′ ‖γ2‖22,θ = j=−k+1 j′=−k+1 βj,kβj′,k′ Bj,k, Bj′ ,k′ According to Lemma A.1, one has for any θ ∈ Sd−1c , c1h ‖α‖22 ≤ ‖γ1‖ 2,θ ≤ c2h ‖α‖ 2 , c1h ‖β‖ 2 ≤ ‖γ2‖ 2,θ ≤ c2h ‖β‖ c1h ‖α‖2 ‖β‖2 ≤ ‖γ1‖2,θ ‖γ2‖2,θ ≤ c2h ‖α‖2 ‖β‖2 . 18 LI WANG AND LIJIAN YANG Hence An = sup θ∈Sd−1c γ1∈γ,γ2∈Γ 〈γ1, γ2〉n,θ − 〈γ1, γ2〉θ ‖γ1‖2,θ ‖γ2‖2,θ ‖α‖∞ ‖β‖∞ c1h ‖α‖2 ‖β‖2 × sup θ∈Sd−1c k,k′=2,3,4 1≤j,j′≤N Bj,k, Bj′ ,k′ Bj,k, Bj′ ,k′ An ≤ c0h−1 sup θ∈Sd−1c k,k′=2,3,4 1≤j,j′≤N Bj,k, Bj′ ,k′ Bj,k, Bj′ ,k′ which, together with Lemma A.2, imply (A.5). A.2. Proof of Proposition 2.1 For any fixed θ, we write the response YT = (Y1, ..., Yn) as the sum of a signal vector γθ, a parametric noise vector Eθ and a systematic noise vector E, i.e., Y = γθ +Eθ +E, in which the vectors γTθ = {γθ (Uθ,1) , ..., γθ (Uθ,n)}, ET = {σ (X1) ε1, ..., σ (Xn) εn} and ETθ = {m (X1)− γθ (Uθ,1) , ...,m (Xn)− γθ (Uθ,n)}. Remark A.1. If m is a genuine single-index function, then Eθ0 ≡ 0, thus the proposed SIP model is exactly the single-index model. Let Γ n, θ be the cubic spline space spanned by {Bj,4 (Uθ,i)} , −3 ≤ j ≤ N for fixed θ. Projecting Y onto Γ n, θ yields that γ̂θ = {γ̂θ (Uθ,1) , ..., γ̂θ (Uθ,n)}T = ProjΓ(2) γθ + ProjΓ(2) Eθ + ProjΓ(2) where γ̂θ is given in (2.8). We break the cubic spline estimation error γ̂θ (uθ) − γθ (uθ) into a bias term γ̃θ (uθ)− γθ (uθ) and two noise terms ε̃θ (uθ) and ε̂θ (uθ) γ̂θ (uθ)− γθ (uθ) = {γ̃θ (uθ)− γθ (uθ)}+ ε̃θ (uθ) + ε̂θ (uθ) , (A.6) where γ̃θ (u) = {Bj,4 (u)}T−3≤j≤N V 〈γθ, Bj,4〉n,θ , (A.7) ε̃θ (u) = {Bj,4 (u)}T−3≤j≤N V 〈Eθ, Bj,4〉n,θ , (A.8) ε̂θ (u) = {Bj,4 (u)}T−3≤j≤N V 〈E, Bj,4〉n,θ . (A.9) SINGLE-INDEX PREDICTION MODEL 19 In the above, we denote by Vn,θ the empirical inner product matrix of the cubic B-spline basis and similarly, the theoretical inner product matrix as Vθ Vn,θ = θ Bθ = Bj′,4, Bj,4 j,j′=−3 ,Vθ = Bj′,4, Bj,4 j,j′=−3 . (A.10) In Lemma A.5, we provide the uniform upper bound of ∞. Before that, we first describe a special case of Theorem 13.4.3 in DeVore and Lorentz (1993). Lemma A.4. If a bi-infinite matrix with bandwidth r has a bounded inverse A−1 on l2 and κ = κ (A) := ‖A‖2 is the condition number of A, then ∞ ≤ 2c0 (1− ν) , with c0 = ν −2r ∥ , ν = κ2 − 1 )1/4r ( κ2 + 1 )−1/4r Lemma A.5. Under Assumptions A2, A5 and A6, there exist constants 0 < cV < CV such that cVN −1 ‖w‖22 ≤ wTVθw ≤ CVN−1 ‖w‖ 2 and −1 ‖w‖22≤ w Vn,θw ≤ CVN−1 ‖w‖22 , a.s., (A.11) with matrices Vθ and Vn,θ defined in (A.10). In addition, there exists a constant C > 0 such θ∈Sd−1c ≤ CN, a.s., sup θ∈Sd−1c ∞ ≤ CN. (A.12) Proof. First we compute the lower and upper bounds for the eigenvalues of Vn,θ. Let w be any (N + 4)-vector and denote γw (u) = j=−3wjBj,4 (u), then Bθw = {γw (Uθ,1) , ..., γw (Uθ,n)} and the definition of An in (A.5) from Lemma A.3 entails that ‖γw‖22,θ (1−An) ≤ w Vn,θw = ‖γw‖22,n,θ ≤ ‖γw‖ 2,θ (1 +An) . (A.13) Using Theorem 5.4.2 of DeVore and Lorentz (1993) and Assumption A2, one obtains that ‖w‖22 ≤ ‖γw‖ 2,θ = w Vθw = wjBj,4 ‖w‖22 , (A.14) which, together with (A.13), yield −1 ‖w‖22 (1−An) ≤ w Vn,θw ≤ CfCN−1 ‖w‖22 (1 +An) . (A.15) Now the order of An in (A.5), together with (A.14) and (A.15) implies (A.11), in which cV = cfC,CV = CfC. Next, denote by λmax (Vn,θ) and λmin (Vn,θ) the maximum and minimum eigenvalue of Vn,θ, simple algebra and (A.11) entail that −1 ≥ ‖Vn,θ‖2 = λmax (Vn,θ) , = λ−1min (Vn,θ) ≤ c V N, a.s., 20 LI WANG AND LIJIAN YANG κ := ‖Vn,θ‖2 = λmax (Vn,θ)λ min (Vn,θ) ≤ CV c V <∞, a.s.. Meanwhile, let wj = the (N + 4)-vector with all zeros except the j-th element being 1, j = −3, ..., N . Then clearly j Vn,θwj = B2j,4 (Uθ,i) = ‖Bj,4‖ , ‖wj‖2 = 1,−3 ≤ j ≤ N and in particular 0 Vn,θw0 ≤ λmax (Vn,θ) ‖w0‖2 = λmax (Vn,θ) , −3Vn,θw−3 ≥ λmin (Vn,θ) ‖w−3‖2 = λmin (Vn,θ) . This, together with (A.5) yields that κ = λmax (Vn,θ)λ min (Vn,θ) ≥ wT0 Vn,θw0 wT−3Vn,θw−3 ‖B0,4‖2n,θ ‖B−3,4‖2n,θ ‖B0,4‖2θ ‖B−3,4‖2θ 1 +An which leads to κ ≥ C > 1, a.s. because the definition of B-spline and Assumption A2 ensure that ‖B0,4‖2θ ≥ C0 ‖B−3,4‖ for some constant C0 > 1. Next applying Lemma A.4 with κ2 − 1 )1/16 ( κ2 + 1 )−1/16 and c0 = ν , one gets ≤ 2ν−8N (1− ν)−1 = CN, a.s.. Hence part one of (A.12) follows. Part two of (A.12) is proved in the same fashion. In the following, we denote by QT (m) the 4-th order quasi-interpolant of m corresponding to the knots T , see equation (4.12), page 146 of DeVore and Lorentz (1993). According to Theorem 7.7.4, DeVore and Lorentz (1993), the following lemma holds. Lemma A.6. There exists a constant C > 0, such that for 0 ≤ k ≤ 2 and γ ∈ C(4) [0, 1] (γ −QT (γ))(k) h4−k, Lemma A.7. Under Assumptions A2, A3, A5 and A6, there exists an absolute constant C > 0, such that for function γ̃θ (u) in (A.7) θ∈Sd−1c (γ̃θ − γθ) h4−k, a.s., 0 ≤ k ≤ 2, (A.16) Proof. According to Theorem A.1 of Huang (2003), there exists an absolute constant C > 0, such that θ∈Sd−1c ‖γ̃θ − γθ‖∞ ≤ C sup θ∈Sd−1c γ∈Γ(2) ‖γ − γθ‖∞ ≤ C h4, a.s., (A.17) SINGLE-INDEX PREDICTION MODEL 21 which proves (A.16) for the case k = 0. Applying Lemma A.6, one has for 0 ≤ k ≤ 2 θ∈Sd−1c {QT (γθ)− γθ} ≤ C sup θ∈Sd−1c h4−k ≤ C h4−k, (A.18) As a consequence of (A.17) and (A.18) for the case k = 0, one has θ∈Sd−1c ‖QT (γθ)− γ̃θ‖∞ ≤ C h4, a.s., which, according to the differentiation of B-spline given in de Boor (2001), entails that θ∈Sd−1c {QT (γθ)− γ̃θ} h4−k, a.s., 0 ≤ k ≤ 2. (A.19) Combining (A.18) and (A.19) proves (A.16) for k = 1, 2. Lemma A.8. Under Assumptions A1, A2, A4 and A5, there exists an absolute constant C > 0, such that 1≤p≤d θ∈Sd−1c {γ̃θ (Uθ,i)− γθ (Uθ,i)}ni=1 h3, a.s., (A.20) 1≤p,q≤d θ∈Sd−1c ∂θp∂θq {γ̃θ (Uθ,i)− γθ (Uθ,i)}ni=1 h2, a.s.. (A.21) Proof. According to the definition of γ̃θ in (A.7), and the fact that QT (γθ) is a cubic spline on the knots T {{QT (γθ)− γ̃θ} (Uθ,i)}ni=1 = Pθ {{QT (γθ)− γθ} (Uθ,i)} which entails that {{QT (γθ)− γ̃θ} (Uθ,i)}ni=1 = Pθ {{QT (γθ)− γθ} (Uθ,i)}ni=1 = Ṗp {{QT (γθ)− γθ} (Uθ,i)}ni=1 +Pθ {{QT (γθ)− γθ} (Uθ,i)}ni=1 . Since {{QT (γθ)− γθ} (Uθ,i)}ni=1 = (Uθ,i) {QT (γθ)− γθ} (Uθ,i)Xip applying (A.19) to the decomposition above produces (A.20). The proof of (A.21) is similar. 22 LI WANG AND LIJIAN YANG Lemma A.9. Under Assumptions A2, A5 and A6, there exists a constant C > 0 such that θ∈Sd−1c ∥n−1BTθ ∞ ≤ Ch, a.s., sup 1≤p≤d θ∈Sd−1c n−1ḂTp ≤ C, a.s., (A.22) θ∈Sd−1c ‖Pθ‖∞ ≤ C, a.s., sup 1≤p≤d θ∈Sd−1c ≤ Ch−1, a.s.. (A.23) Proof. To prove (A.22), observe that for any vector a ∈ Rn, with probability 1 ∥n−1BTθ a ∞ ≤ ‖a‖∞ max−3≤j≤N Bj,4 (Uθ,i) ≤ Ch ‖a‖∞ , n−1ḂTp a ≤ ‖a‖∞ max−3≤j≤N {(Bj,3 −Bj+1,3) (Uθ,i)} Ḟd (Xθ,i)Xi,p ≤ C ‖a‖∞ . To prove (A.23), one only needs to use (A.12), (A.22) and (3.1). Lemma A.10. Under Assumptions A2 and A4-A6, one has with probability 1 θ∈Sd−1c BTθ E = max −3≤j≤N Bj,4 (Uθ,i)σ (Xi) εi log n√ , (A.24) 1≤p≤d θ∈Sd−1c BTθ E = sup 1≤p≤d θ∈Sd−1c ḂTpE log n√ . (A.25) Similarly, under Assumptions A2, A4-A6, with probability 1 θ∈Sd−1c BTθ Eθ = sup θ∈Sd−1c −3≤j≤N Bj,4 (Uθ,i) {m (Xi)− γθ (Uθ,i)} log n√ (A.26) 1≤p≤d θ∈Sd−1c BTθ Eθ log n√ , a.s.. (A.27) Proof. We decompose the noise variable εi into a truncated part and a tail part εi = ε i,1 + εDni,2 +m i , where Dn = n η (1/3 < η < 2/5), εDni,1 = εiI {|εi| > Dn}, εDni,2 = εiI {|εi| ≤ Dn} −m i = E [εiI {|εi| ≤ Dn} |Xi] . It is straightforward to verify that the mean of the truncated part is uniformly bounded by D−2n , so the boundedness of B spline basis and of the function σ 2 entail that θ∈Sd−1c Bj,4 (Uθ,i) σ (Xi)m n−2/3 SINGLE-INDEX PREDICTION MODEL 23 The tail part vanishes almost surely P {|εn| > Dn} ≤ D−3n <∞. Borel-Cantelli Lemma implies that Bj,4 (Uθ,i) σ (Xi) ε , for any k > 0. For the truncated part, using Bernstein’s inequality and discretization as in Lemma A.2 θ∈Sd−1c 1≤j≤N Bj,4 (Uθ,i) σ (Xi) ε log n/ , a.s.. Therefore (A.24) is established as with probability 1 θ∈Sd−1c n−2/3 log n/ log n/ The proofs of (A.25), (A.26) are similar as E {m (Xi)− γθ (Uθ,i) |Uθ,i } ≡ 0, but no truncation is needed for (A.26) as sup θ∈Sd−1c 1≤i≤n |m (Xi)− γθ (Uθ,i)| ≤ C <∞. Meanwhile, to prove (A.27), we note that for any p = 1, ..., d [Bj,4 (Uθ,i) {m (Xi)− γθ (Uθ,i)}] According to (2.6), one has γθ (Uθ) ≡ E {m (X) |Uθ}, hence E [Bj,4 (Uθ) {m (X)− γθ (Uθ)}] ≡ 0,−3 ≤ j ≤ N, θ ∈ Sd−1c . Applying Assumptions A2 and A3, one can differentiate through the expectation, thus [Bj,4 (Uθ) {m (X)− γθ (Uθ)}] ≡ 0, 1 ≤ p ≤ d,−3 ≤ j ≤ N, θ ∈ Sd−1c , which allows one to apply the Bernstein’s inequality to obtain that with probability 1 [Bj,4 (Uθ,i) {m (Xi)− γθ (Uθ,i)}] (nh)−1/2 log n which is (A.27). Lemma A.11. Under Assumptions A2 and A4-A6, for ε̂θ (u) in (A.9), one has θ∈Sd−1c u∈[0,1] |ε̂θ (u)| = O log n , a.s.. (A.28) 24 LI WANG AND LIJIAN YANG Proof. Denote â ≡ (â−3, · · · , âN )T = BTθ Bθ BTθ E = V n−1BTθ E , then ε̂θ (u) = j=−3 âjBj,4 (u), so the order of ε̂θ (u) is related to that of â. In fact, by Theorem 5.4.2 in DeVore and Lorentz (1993) θ∈Sd−1c u∈[0,1] |ε̂θ (u)| ≤ sup θ∈Sd−1c ‖â‖∞ = θ∈Sd−1c n−1BTθ E ≤ CN sup θ∈Sd−1c ∥n−1BTθ E ∞ , a.s., where the last inequality follows from (A.12) of Lemma A.5. Applying (A.24) of Lemma A.10, we have established (A.28). Lemma A.12. Under Assumptions A2 and A4-A6, for ε̃θ (u) in (A.8), one has θ∈Sd−1c u∈[0,1] |ε̃θ (u)| = O log n , a.s.. (A.29) The proof is similar to Lemma A.11, thus omitted. The next result evaluates the uniform size of the noise derivatives. Lemma A.13. Under Assumptions A2-A6, one has with probability 1 1≤p≤d θ∈Sd−1c 1≤i≤n ε̂θ (Uθ,i) (nh3)−1/2 log n , (A.30) 1≤p≤d θ∈Sd−1c 1≤i≤n ε̃θ (Uθ,i) (nh3)−1/2 log n , (A.31) 1≤p,q≤d θ∈Sd−1c 1≤i≤n ∂θp∂θq ε̂θ (Uθ,i) (nh5)−1/2 log n , (A.32) 1≤p,q≤d θ∈Sd−1c 1≤i≤n ∂θp∂θq ε̃θ (Uθ,i) (nh5)−1/2 log n . (A.33) Proof. Note that ε̂θ (Uθ,i) = (I−Pθ) Ḃp θ E+Bθ p (I−Pθ)E. Applying (A.24) and (A.25) of Lemma A.10, (A.12) of Lemma A.5, (A.22) and (A.23) of Lemma A.9, one derives (A.30). To prove (A.31), note that ε̃θ (Uθ,i) {PθEθ} = ṖpEθ +Pθ Eθ = T1 + T2, (A.34) in which (I−Pθ) Ḃp −Bθ (I−Pθ) Ḃp −Bθ BTθ Bθ ḂTpBθ BTθ Bθ BTθ Eθ SINGLE-INDEX PREDICTION MODEL 25 T2 = Bθ BTθ Bθ BTθ Eθ By (A.24), (A.12), (A.22) and (A.23), one derives θ∈Sd−1c ‖T1‖∞ = O n−1/2N3/2 log n , a.s., (A.35) while (A.27) of Lemma A.10, (A.12) of Lemma A.5 θ∈Sd−1c ‖T2‖∞ = N ×O n−1/2h−1/2 log n n−1/2h−3/2 log n , a.s.. (A.36) Now, putting together (A.34), (A.35) and (A.36), we have established (A.31). The proof for (A.32) and (A.33) are similar. Proof of Proposition 2.1. According to the decomposition (A.6) |γ̂θ (u)− γθ (u)| = |{γ̃θ (u)− γθ (u)}+ ε̃θ (u) + ε̂θ (u)| . Then (2.13) follows directly from (A.16) of Lemma A.7, (A.28) of Lemma A.11 and (A.29) of Lemma A.12. Again by definitions (A.8) and (A.9), we write {(γ̂θ − γθ) (Uθ,i)} = (γ̃θ − γθ) (Uθ,i) + γ̃θ (Uθ,i) + ε̂θ (Uθ,i) . It is clear from (A.20), (A.30) and (A.31) that with probability 1 1≤p≤d θ∈Sd−1c 1≤i≤n (γ̃θ − γθ) (Uθ,i) 1≤p≤d θ∈Sd−1c 1≤i≤n ε̃θ (Uθ,i) ε̂θ (Uθ,i) )−1/2 log n Putting together all the above yields (2.14). The proof of (2.15) is similar. A.3. Proof of Proposition 2.2 Lemma A.14. Under Assumptions A2-A6, one has θ∈Sd−1c R̂ (θ)−R (θ) = o(1), a.s.. Proof. For the empirical risk function R̂ (θ) in (2.9), one has R̂ (θ) = n−1 {γ̂θ (Uθ,i)−m (Xi)− σ (Xi) εi}2 26 LI WANG AND LIJIAN YANG = n−1 {γ̂θ (Uθ,i)− γθ (Uθ,i) + γθ (Uθ,i)−m (Xi)− σ (Xi) εi}2 , hence R̂ (θ) = n−1 {γ̂θ (Uθ,i)− γθ (Uθ,i)}2 + n−1 σ2 (Xi) ε +2n−1 {γ̂θ (Uθ,i)− γθ (Uθ,i)} {γθ (Uθ,i)−m (Xi)− σ (Xi) εi} {γθ (Uθ,i)−m (Xi)}2 + 2n−1 {γθ (Uθ,i)−m (Xi)}σ (Xi) εi, where γ̂θ (x) is defined in (2.8). Using the expression of R (θ) in (2.7), one has θ∈Sd−1c ∣R̂ (θ)−R (θ) ∣ ≤ I1 + I2 + I3 + I4, I1 = sup θ∈Sd−1c {γ̂θ (Uθ,i)− γθ (Uθ,i)}2 I2 = sup θ∈Sd−1c {γ̂θ (Uθ,i)− γθ (Uθ,i)} {γθ (Uθ,i)−m (Xi)− σ (Xi) εi} I3 = sup θ∈Sd−1c {γθ (Uθ,i)−m (Xi)}2 − E {γθ (Uθ)−m (X)}2 I4 = sup θ∈Sd−1c σ2 (Xi) ε i − Eσ2 (X) {γθ (Uθ,i)−m (Xi)}σ (Xi) εi Bernstein inequality and strong law of large number for α mixing sequence imply that I3 + I4 = o(1), a.s.. (A.37) Now (2.13) of Proposition 2.1 provides that θ∈Sd−1c u∈[0,1] |γ̂θ (u)− γθ (u)| = O n−1/2h−1/2 log n+ h4 , a.s., which entail that I1 = O n−1/2h−1/2 log n , a.s., (A.38) I2 ≤ O (nh)−1/2 log n+ h4 × sup θ∈Sd−1c |γθ (Uθ,i)−m (Xi)− σ (Xi) εi| . Hence I2 ≤ O n−1/2h−1/2 log n+ h4 , a.s.. (A.39) The lemma now follows from (A.37), (A.38) and (A.39) and Assumption A6. SINGLE-INDEX PREDICTION MODEL 27 Lemma A.15. Under Assumptions A2 - A6, one has θ∈Sd−1c 1≤p≤d R̂ (θ)−R (θ) − n−1 ξθ,i,p n−1/2 , a.s., (A.40) in which ξθ,i,p = 2 {γθ (Uθ,i)− Yi} γθ (Uθ,i)− R (θ) , E (ξθ,i,p) = 0. (A.41) Furthermore for k = 1, 2 θ∈Sd−1c R̂ (θ)−R (θ) n−1/2h−1/2−k log n+ h4−k , a.s.. (A.42) Proof. Note that for any p = 1, 2, ..., d R̂ (θ) = n−1 {γ̂θ (Uθ,i)− Yi} γ̂θ (Uθ,i) , R (θ) = E {γθ (Uθ)−m (X)} γθ (Uθ) {γθ (Uθ)−m (X)− σ (X) ε} γθ (Uθ) Thus E (ξθ,i,p) = 2E {γθ (Uθ,i)− Yi} ∂∂θpγθ (Uθ,i) R (θ) = 0 and R̂ (θ)−R (θ) = (2n) ξθ,i,p + J1,θ,p + J2,θ,p + J3,θ,p, (A.43) J1,θ,p = n {γ̂θ (Uθ,i)− γθ (Uθ,i)} (γ̂θ − γθ) (Uθ,i) , J2,θ,p = n {γθ (Uθ,i)−m (Xi)− σ (Xi) εi} (γ̂θ − γθ) (Uθ,i) , J3,θ,p = n {γ̂θ (Uθ,i)− γθ (Uθ,i)} γθ (Uθ,i) . Bernstein inequality implies that θ∈Sd−1c 1≤p≤d ξθ,i,p n−1/2 log n , a.s.. (A.44) 28 LI WANG AND LIJIAN YANG Meanwhile, applying (2.13) and (2.14) of Proposition 2.1, one obtains that θ∈Sd−1c 1≤p≤d |J1,θ,p| = O log n+ h4 )−1/2 log n+ h3 n−1h−2 log2 n+ h7 , a.s.. (A.45) Note that J2,θ,p = n {γθ (Uθ,i)−m (Xi)− σ (Xi) εi} (γ̃θ − γθ) (Uθ,i) −n−1 (E+Eθ)T {Pθ (E+Eθ)} . Applying (2.13), one gets θ∈Sd−1c 1≤p≤d J2,θ,p + n −1 (E+Eθ) {Pθ (E+Eθ)} , a.s., while (A.24), (A.26) and (A.12) entail that with probability 1 θ∈Sd−1c 1≤p≤d n−1 (E+Eθ) {Pθ (E+Eθ)} log n ×N ×N ×O log n n−1N log2 n θ∈Sd−1c 1≤p≤d |J2,θ,p| = O h3 + n−1N log2 n , a.s.. (A.46) Lastly J3,θ,p − n−1 (γ̃θ − γθ) γθ (Uθ,i) = n −1 (E+Eθ) BTθ Bθ By applying (A.24), (A.26), and (A.12), it is clear that with probability 1 θ∈Sd−1c 1≤p≤d n−1BTθ E+n BTθ Bθ log n ×N ×O h+ (nN) log n n−1 log2 n+ (nN)−1/2 log n while by applying (A.16) of Lemma A.7, one has θ∈Sd−1c 1≤p≤d (γ̃θ − γθ) γθ (Uθ,i) , a.s., SINGLE-INDEX PREDICTION MODEL 29 together, the above entail that θ∈Sd−1c 1≤p≤d |J3,θ,p| = O h4 + n−1 log2 n+ (nN)−1/2 log n , a.s.. (A.47) Therefore, (A.43), (A.45), (A.46), (A.47) and Assumption A6 lead to (A.40), which, together with (A.44), establish (A.42) for k = 1. Note that the second order derivative of R̂ (θ) and R (θ) with respect to θp, θq are {γ̂θ (Uθ,i)− Yi} ∂θp∂θq γ̂θ (Uθ,i) + γ̂θ (Uθ,i) γ̂θ (Uθ,i) E {γθ (Uθ)−m (X)} ∂θp∂θq γθ (Uθ) + E γθ (Uθ) γθ (Uθ) The proof of (A.42) for k = 2 follows from (2.13), (2.14) and (2.15). Proof of Proposition 2.2. The result follows from Lemma A.14, Lemma A.15, equations (A.50) and (A.51). A.4. Proof of the Theorem 2 Let Ŝ∗p (θ−d) be the p-th element of Ŝ ∗ (θ−d) and for γθ in (2.6), denote ηi,p := 2 γ̇p − θ0,pθ−10,dγ̇d (Uθ0,i) {γθ0 (Uθ0,i)− Yi} , (A.48) where γ̇p is value of γθ taking at θ = θ0, for any p, q = 1, 2, ..., d − 1. Lemma A.16. Under Assumptions A2-A6, one has 1≤p≤d−1 Ŝ∗p (θ0,−d)− n−1 n−1/2 , a.s.. (A.49) Proof. For any p = 1, ..., d − 1 Ŝ∗p (θ−d)− S∗p (θ−d) = − θpθ−1d R̂ (θ)−R (θ) Therefore, according to (A.40), (A.41) and (A.48) ηi,p = n ξθ0,i,p − θ0,pθ ξθ0,i,d, E (ηi,p) = 0, 1≤p≤d−1 Ŝ∗p (θ0,−d)− S∗p (θ0,−d)− n−1 n−1/2 , a.s.. Since S∗ (θ−d) attains its minimum at θ0,−d, for p = 1, ..., d − 1 S∗p (θ0,−d) ≡ − θpθ−1d R (θ) which yields (A.49). 30 LI WANG AND LIJIAN YANG Lemma A.17. The (p, q)-th entry of the Hessian matrix H∗ (θ0,−d) equals lp,q given in Theo- rem 2. Proof. It is easy to show that for any p, q = 1, 2, ..., d, R (θ) = E {m (X)− γθ (Uθ)}2 = −2E γθ (Uθ) γθ (Uθ) ∂θp∂θq R (θ) = −2E γθ (Uθ) γθ (Uθ) + γθ (Uθ) ∂θp∂θq γθ (Uθ) Note that R∗ (θ−d) = R (θ)− R (θ) , (A.50) ∂θp∂θq R∗ (θ−d) = ∂θp∂θq R (θ)− ∂θp∂θd R (θ)− ∂θd∂θq R (θ) 1− ‖θ−d‖22 R (θ) + ∂θd∂θd R (θ) . (A.51) R∗ (θ−d) = −2E γθ (Uθ) γθ (Uθ) + 2θ−1 γθ (Uθ) γθ (Uθ) ∂θp∂θq R∗ (θ−d) = −2E γθ (Uθ) γθ (Uθ) + γθ (Uθ) ∂θp∂θq γθ (Uθ) +2θqθ γθ (Uθ) γθ (Uθ) + γθ (Uθ) ∂θp∂θd γθ (Uθ) 1− ‖θ−d‖22 γθ (Uθ) γθ (Uθ) +2θpθ γθ (Uθ) γθ (Uθ) + γθ (Uθ) ∂θp∂θq γθ (Uθ) −2θpθqθ−2d E γθ (Uθ) + γθ (Uθ) ∂θd∂θd γθ (Uθ) Therefore we obtained the desired result. Proof of Theorem 2. For any p = 1, 2, ..., d − 1, let fp (t) = Ŝ tθ̂−d + (1− t) θ0,−d , t ∈ [0, 1], fp (t) = tθ̂−d+(1− t) θ0,−d θ̂q − θ0,q SINGLE-INDEX PREDICTION MODEL 31 Note that Ŝ∗ (θ−d) attains its minimum at θ̂−d, i.e., Ŝ ≡ 0. Thus, for any p = 1, 2, ..., d− 1, tp ∈ [0, 1], one has −Ŝ∗p (θ0,−d) = Ŝ∗p − Ŝ∗p (θ0,−d) = fp (1)− fp (0) ∂θqθp tpθ̂−d + (1− tp) θ0,−d q=1,...,d−1 θ̂−d−θ0,−d −Ŝ∗ (θ0,−d) = ∂θq∂θp tpθ̂−d + (1− tp) θ0,−d p,q=1,...,d−1 θ̂−d − θ0,−d Now (2.11) of Theorem 1 and Proposition 2.2 with k = 2 imply that uniformly in p, q = 1, 2, ..., d − 1 ∂θq∂θp tpθ̂−d+(1− tp) θ0,−d −→ lq,p, a.s., (A.52) where lp,q is given in Theorem 2. Noting that θ̂−d−θ0,−d is represented as ∂θq∂θp tpθ̂−d + (1− tp) θ0,−d p,q=1,...,d−1 nŜ∗ (θ0,−d) , where Ŝ∗ (θ0,−d) = Ŝ∗p (θ0,−d) and according to (A.48) and Lemma A.16 Ŝ∗p (θ0,−d) = n ηp,i + o n−1/2 , a.s., E (ηp,i) = 0. Let Ψ (θ0) = (ψpq) p,q=1 be the covariance matrix of Ŝ∗p (θ0,−d) with ψpq given in Theorem 2. 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Table 2: Report of Example 2 Sample Size n Dimension d Average MSE Time MAVE SIP MAVE SIP 4 0.00020 0.00018 1.91 0.19 10 0.00031 0.00043 2.17 0.10 30 0.00106 0.00285 2.77 0.13 50 0.00031 0.00043 3.29 0.10 100 0.00681 0.00620 5.94 0.31 200 0.00529 0.00407 27.90 0.49 4 0.00008 0.00008 3.28 0.09 10 0.00012 0.00017 3.93 0.13 30 0.00017 0.00058 5.41 0.15 50 0.00032 0.00127 8.48 0.16 100 — 0.00395 — 0.44 200 — 0.00324 — 0.73 4 0.00004 0.00003 5.32 0.17 10 0.00005 0.00007 7.49 0.24 30 0.00006 0.00017 10.08 0.26 50 0.00007 0.00030 15.42 0.24 100 0.00015 0.00061 40.81 0.54 200 — 0.00197 — 1.44 4 0.00002 0.00001 14.44 0.76 10 0.00002 0.00003 24.54 0.79 30 0.00002 0.00008 32.51 0.83 50 0.00002 0.00010 52.93 0.89 100 0.00003 0.00012 143.07 0.99 200 0.00004 0.00020 386.80 1.96 400 — 0.00054 — 4.98 4 0.00001 0.00001 33.57 1.95 10 0.00001 0.00001 62.54 3.64 30 0.00001 0.00002 92.41 1.95 50 0.00001 0.00003 155.38 2.72 100 0.00001 0.00005 275.73 1.81 200 0.00008 0.00006 2432.56 2.84 400 — 0.00010 — 9.35 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 −1 0 1 2 −1 0 1 2 Figure 1: Example 1. (a) and (b) Plots of the actual surface m in model (4.1) with re- spect to δ = 0, 1; (c) and (d) Plots of various univariate functions with respect to δ = 0, 1: XTi θ̂, Yi , 1 ≤ i ≤ 50 (dots); the univariate function g (solid line); the estimated function of g by plugging in the true index coefficient θ0 (dotted line); the estimated function of g by plugging in the estimated index coefficient (dashed line) θ̂ = (0.69016, 0.72365)T for δ = 0 and (0.72186, 0.69204)T for δ = 1. −2 −1 0 1 2 n= 100 , d= 10 −2 −1 0 1 2 n= 100 , d= 50 −2 −1 0 1 2 n= 100 , d= 100 −3 −2 −1 0 1 2 3 n= 100 , d= 200 Figure 2: Example 2. Plots of the spline estimator of g with the estimated index parameter θ̂ (dotted curve), cubic spline estimator of g with the true index parameter θ0 (dashed curves), the true function m (x) in (4.2) (solid curve), and the data scatter plots (dots). 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Density Estimation, d=10 n=100 n=200 n=500 n=1000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Density Estimation, d=50 n=100 n=200 n=500 n=1000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Density Estimation, d=100 n=100 n=200 n=500 n=1000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Density Estimation, d=200 n=100 n=200 n=500 n=1000 Figure 3: Example 2. Kernel density estimators of the 100 ‖θ̂ − θ0‖/ 0 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 Figure 4: Time plots of the daily Jökulsá Eystri River data (a) river flow Yt (solid line) with its trend (dashed line) (b) temperature Xt (solid line) with its trend (dashed line) (c) precipitation Zt (solid line) with its trend (dashed line). +++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++ +++++++++++++++++++++++++++++++++++++ ++++++ +++++ +++++ ++++++++++ ++++++++++ +++++++ +++++++ ++++++++++++ +++++ ++++++++++++ +++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++ ++++++++++++++++++ +++++ +++++ ++++++ +++++++ +++++ +++++ +++++ +++++ ++++++++ +++++ ++++++++++++++ +++++++ +++++++++++++++++++++++++ ++++++++++++++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++ ++++++++++ +++++++++++++++++++++ ++++++ ++++++ ++++++ ++++++++++ +++++++++ +++++ ++++++++++++++++++++++++++++++ +++++++ ++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 0 200 400 600 800 1000 0 200 400 600 800 1000 800 900 1000 1100 ++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++ ++++++++++ +++++++++++++++++++++ ++++++ ++++++ ++++++ ++++++++++ +++++++++ +++++ ++++++++++++++++++++++++++++++ +++++++ ++++++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++ Figure 5: (a) The scatter plot of the river flow (“+”) and the fitted plot of the river flow (line) and (b) Residuals of the fitted SIP model (c) Out-of-sample rolling forecasts (line) of the river flow for the entire third year (“+”) based on the first two years’ river flow.

0704.0303

Measurement of the Aerosol Phase Function at the Pierre Auger Observatory

Measurement of the Aerosol Phase Function at the Pierre Auger Observatory S.Y. BenZvi a, B.M. Connolly a, J.A.J. Matthews b, M. Prouza a, E.F. Visbal a,c, and S. Westerhoff a aColumbia University, Department of Physics and Nevis Laboratories, 538 West 120 th Street, New York, NY 10027, USA bUniversity of New Mexico, Department of Physics and Astronomy, Albuquerque, NM 87131, USA cCarnegie Mellon University, Department of Physics, Pittsburgh, PA 15213, USA Abstract Air fluorescence detectors measure the energy of ultra-high energy cosmic rays by collecting fluorescence light emitted from nitrogen molecules along the extensive air shower cascade. To ensure a reliable energy determination, the light signal needs to be corrected for atmospheric effects, which not only attenuate the signal, but also produce a non-negligible background component due to scattered Cherenkov light and multiple-scattered light. The correction requires regular measurements of the aerosol attenuation length and the aerosol phase function, defined as the probability of light scattered in a given direction. At the Pierre Auger Observatory in Malargüe, Argentina, the phase function is measured on an hourly basis using two Aerosol Phase Function (APF) light sources. These sources direct a UV light beam across the field of view of the fluorescence detectors; the phase function can be extracted from the image of the shots in the fluorescence detector cameras. This paper describes the design, current status, standard operation procedure, and performance of the APF system at the Pierre Auger Observatory. Key words: Ultra-high energy cosmic rays; air fluorescence detectors; atmospheric monitoring; aerosol phase function PACS: 42.68.-w, 42.68.Jg, 92.60.Mt, 92.60.Sz, 96.50.sd 1 Introduction The Pierre Auger Observatory in Malargüe, Argentina, is designed to study the origin of ultra-high energy cosmic rays with energies above 1018 eV. While Preprint submitted to Astroparticle Physics 30 October 2018 http://arxiv.org/abs/0704.0303v2 still under construction, scientific data taking began in 2004, and first results have been published [1,2,3]. The Pierre Auger Observatory is a hybrid detector that combines two tech- niques traditionally used to measure cosmic ray air showers: surface particle detection and air fluorescence detection. Both detector types measure the cos- mic ray primary indirectly, using the Earth’s atmosphere as part of the de- tector medium. When the primary particle enters the atmosphere, it interacts with air molecules, initiating a cascade of secondary particles, the so-called ex- tensive air shower. Surface detectors in the form of ground arrays sample the shower front as it impacts the ground, whereas air fluorescence detectors make use of the fact that the particles in the air shower excite nitrogen molecules in the air, causing UV fluorescence. Using photomultiplier cameras to record air shower UV emission, we can observe showers as they develop through the atmosphere and obtain a nearly calorimetric estimate of the shower energy. Upon completion, the surface detector (SD) array of the Pierre Auger Ob- servatory will comprise 1600 water Cherenkov detector tanks, deployed in a hexagonal grid over an area of 3000 km2, and four fluorescence detector (FD) stations overlooking the SD from the periphery. An advantage of combining both detector types at the same site is the possibility to cross-calibrate. Based on the subset of events seen with both detectors, the nearly calorimetric in- formation of the FD provides the energy calibration of the SD. For the calibration to be meaningful, the properties of the calorimeter, i.e. the atmosphere, must be well-known. At the Pierre Auger Observatory, this is achieved by an extensive program to monitor the atmosphere within the overall FD aperture and measure atmospheric attenuation and scattering properties in the 300 to 400 nm wavelength band recorded by the FDs [4,5,6]. Two primary forms of atmospheric light scattering need to be considered: molecular, or Rayleigh, scattering, mainly due to nitrogen and oxygen molecules; and aerosol scattering due to airborne particulates. The angular distribution of scattered light in both types of scattering may be described by a phase function P (θ), defined as the probability per unit solid angle of scattering through an angle θ. Rayleigh scattering allows for an analytical treatment, and assuming isotropic scattering, the Rayleigh phase function has the well known 1 + cos2 θ angular dependence. Matters are more complicated for aerosols, because the scatter- ing cross section depends on the size distribution and shape of the scatter- ers. Forward scattering typically dominates in this case, but the fraction of forward-scattered light varies strongly with aerosol type. Moreover, a rigorous analytical treatment is not possible, though the literature gives various ap- proximations. For example, if one assumes spherical particles with a known or estimated size distribution, then aerosol scattering can be described analyti- cally using Mie theory [7]. In practice, however, aerosols vary a great deal in size and shape, and the aerosol content of the atmosphere changes on short time scales as wind lifts up dust, weather fronts pass through, or rain removes dust from the atmosphere. The FD reconstruction of the primary cosmic ray particle energy must account not only for light that is “lost” between the shower and the camera due to scattering, but also for direct and indirect Cherenkov light contributing to the FD signal. The amount of Cherenkov light seen by the FDs depends on the viewing angle, i.e. the angle between the shower axis and the FD line of sight, and can be calculated once the geometry of the air shower is determined. At small viewing angles, direct Cherenkov light dominates, while at viewing angles greater than ∼ 20◦, the FDs detect mainly “indirect” Cherenkov light scattered into the FD field of view. To calculate this scattered component, the aerosol phase function needs to be known. Finally, a small multiple scattering component also adds to the contamination of the fluorescence light and must be removed [8]. The Aerosol Phase Function (APF) light sources [9,10], in conjunction with the fluorescence detectors at the Pierre Auger Observatory, are designed to measure the aerosol phase function on an hourly basis during FD data tak- ing. The APF light sources direct a near-horizontal pulsed light beam across the field of view of a nearby FD. The aerosol phase function can then be re- constructed from the intensity of the light observed by the FD cameras as a function of scattering angle. Since the FD telescopes cover about 180◦ in az- imuth, the aerosol phase function is measured over a wide range of scattering angles. Currently, APF light sources are installed and operating at two of the FDs. With their ability to measure the angular distribution of the scattered light, the APF light sources are meant to complement other atmospheric monitoring tools at the Auger site which measure the optical depth, and therefore the amount of attenuation due to aerosols. This paper describes the design and performance of the APF light sources. It is structured as follows. Section 2 gives a description of the APF facilities. Section 3 describes how the aerosol phase function is determined from the APF data. In Section 4, we show first results for data taken between June and December 2006. Section 5 summarizes the paper. Fig. 1. Schematic layout of the Pierre Auger Observatory. The shaded area indicates the shape and size of the surface detector area. The fluorescence detectors are placed at the periphery of the surface detector array. The field of view of the 6 bays of each fluorescence detector (FD) is indicated by the lines. From the Central Laser Facility (CLF) [6] in the center of the surface detector array, a pulsed UV laser beam is directed into the sky, providing another test beam which can be observed by the FDs. 2 APF Light Sources 2.1 Detector Buildings, Optics, and Electronics The Auger FD comprises four detector stations (see Fig. 1). At present, the sites at Los Leones, Coihueco, and Los Morados are completed and fully opera- tional, while the fourth site at Loma Amarilla is under construction. APF light sources are operating at the Coihueco and Los Morados FD sites. Both were built by the University of New Mexico group [10]. Fig. 2 shows a photograph of the APF container building at Los Morados. Each APF building contains sources which operate at different wavelengths in the region of interest between 300 nm and 400 nm. During the initial studies described in this paper, only one light source with a Johnson U-band filter of central wavelength 350 nm was used. However, in the near future, we plan to operate the light sources at several wavelengths to study the wavelength dependence of the phase function over the full range of the FD sensitivity. The light beam is provided by a broad-band Xenon flash lamp source from Fig. 2. Photos of the enclosure (left) and the light source (right) at the Los Morados APF facility. In the photograph on the left, the Los Morados FD can be seen on the horizon (to the left of the container). Perkin Elmer Optoelectronics (model LS-1130-4 FlashPac with FX-1160 flash lamp). The Xenon flash lamps were chosen because of their excellent stability in intensity and pulse shape. A Johnson/Cousins (Bessel) U-band filter from Omega Optical Inc. (part number XBSSL/U/50R) selects a central wavelength of ∼ 350 nm, FWHM 60 nm) from the broad flash lamp spectrum. The beam is focused using a 20.3 cm diameter UV enhanced aluminum spherical mirror (speed f/3) from Edmund Scientific Co. (part number R43-589). All optical components are assembled on a commercial optical plate. We use Thor optical table parts, assembled from Nomex Epoxy/Fiberglass 1.91 cm panels from TEKLAM (part number N507EC). The Xenon lamps rest inside refurbished 6.1 m shipping containers, and the light is sent through a 0.749 cm thick acrylite UV transmitting window (Cyro Industries acrylite OP-4 UVT acrylic). Each light source provides a nearly horizontal beam of divergence ≤ 10 mrad pulsed across the field of view of the nearby fluorescence detector. Computer control occurs from the correspond- ing FD building. A serial radio link (YDI Wireless, model 651-900001-001 (TranzPoint ESC-II Kit)) connects the computer to a commercial ADC/relay system (model ADC-16F 16 channel 8 bit ADC and RH-8L 8-relay card from Electronic Energy Control Inc.) at the light source. Once during each hour of FD data taking, the ADC/relay system enables a 1 Hz GPS pulser (CNS Systems Inc., model CNSC01 with TAC32 software) and a 12 V to 24 V inverter to power the Xenon flash lamps. Each lamp fires a set of 5 shots, pulsed at 2 second intervals. The APF events are flagged by the FD data acquisition system and the corresponding FD data are stored on disk in especially designated APF data files. When the light sources are not operating, only the radio link and the ADC board are powered. The total current draw is therefore only ∼ 0.2 A at 12 V, and the whole system can be powered by batteries recharged during the day with 12 V solar panels (two Siemens SP75 75 W solar modules with Trace C35 controller). 2.2 APF Signals in the Fluorescence Detectors The light beam produced by the APF sources is observed by the cameras of the corresponding FD site. The FD detectors of the Pierre Auger Observatory are described in detail elsewhere [11]. Here, we only give a short summary of the main characteristics relevant for the analysis of APF shots. Each Auger FD site contains six bays, and each bay encloses a UV telescope composed of a spherical light-collecting mirror, a photomultiplier camera at the focal surface, and a UV transmitting filter in the aperture. The mirrors have a radius of curvature of 3.4 m and an area of about 3.5 × 3.5 m2. The camera consists of 440 photomultipliers with a hexagonal bialkaline photo- cathode, arranged in a 20× 22 array. Each camera has a field of view of 30.0◦ in azimuth and 28.6◦ in elevation, covering an elevation angle range from 1.6◦ to 30.2◦ above horizon. To reduce optical aberrations, including coma, the FD telescopes use Schmidt optics with a circular diaphragm of diameter 2.2 m placed at the center of curvature of the mirror, and a refractive corrector ring at the telescope aperture. Fig. 3 shows an APF shot as seen by the Coihueco FD. Five out of the 6 bays of the Coihueco FD site observe light from the Coihueco APF facility. In this figure, the light travels from right to left. Fig. 4 shows the relative positions of the APF source and the FD at the Coihueco site. The geometry is in part dictated by the local topography, and consequently is slightly different for the Los Morados site. Fig. 3. The APF pulse as seen by the Coihueco FD. The light travels from right to left, and each PMT Cluster observes 30◦ in azimuth. Note that the projection of the approximately horizontal APF beam onto the spherical FD surface results in a curved track. Fig. 4. Scheme of the location of the Coihueco APF light source relative to the Coihueco FD. Located at the center is the Coihueco FD with its field of view in- dicated. The value of α is 26◦ and β is 38◦, measured from the North. The shot direction γ is about 24◦. 3 Determination of the Aerosol Phase Function The signal from the APF light source observed by the ith pixel of a fluorescence detector can be expressed as Si = I0 · Ti · ·∆zi ·∆Ωi · ǫi . (1) In this equation, I0 is the light source intensity; Ti is the transmission factor e−ri/Λtot which accounts for light attenuation from the beam to the pixel; ri is the distance from the beam to the detector; Λtot, Λm, and Λa are the total, molecular, and aerosol extinction length, respectively; and σ−1m dσm/dΩ and σ−1a dσa/dΩ are the normalized differential molecular and aerosol scattering cross sections, respectively, which are identical to the phase functions Pm(θ) and Pa(θ). The integral of Pm(θ) and Pa(θ) over all solid angles is equal to 1. Finally, ∆zi, ∆Ωi, and ǫi are the track length, detector solid angle, and the efficiency for the ith pixel of the detector. The data come in the form of total PMT signal per pixel from a particular shot. Those data are binned as a function of azimuth and averaged between the five shots taken within 10 seconds. In this analysis, 5◦ bins are used, although the fit is relatively insensitive to the number of bins. Each FD pixel is hexagonally shaped, so for those lying at the boundary of two azimuth bins, the fractional area of the hexagon in each bin is used to properly distribute the signal. The signal in each pixel is divided by ∆zi, 1/r i and ǫi to correct for the geometry of the beam and pixel calibration. Note that in the roughly cylindrical geometry of the FD-APF beam, the ∆zi and 1/r i corrections almost completely cancel Typical values for the aerosol extinction length in dry atmospheres are be- tween 10 km and 20 km, reaching 40 km for very clear conditions. Since the perpendicular distance from the beam to the FD is only on the order of a few hundred meters, it is reasonable to assume full atmospheric transmission (Ti = 1) over the length of the beam. In reality, this assumption does not hold well for the most distant beam points, so these points are not used in the present study. In the near future, measurements of the extinction length from the Auger lidar stations [5] will be used to improve the APF analysis. In another approximation, we assume that the extinction lengths are identi- cal for each pixel for single measurements and do not require an index i. In principle, the extinction length depends on the number density of scatterers and is therefore a function of the density (temperature, pressure) of the air. Given corrections for geometry, attenuation, and pixel efficiency, Eq. 1 reduces Si = C · , (2) where C is a constant whose value is unimportant because arbitrary units are sufficient in determining the phase function. From the theory of Rayleigh scattering it is known that the Rayleigh phase function is Pm(θ) = 16 π(1 + 2 γ) (1 + 3 γ) + (1− γ) cos2 θ where γ accounts for the effect of molecular anisotropy on Rayleigh scattering. For isotropic scattering, γ = 0, this reduces to the familiar Pm(θ) = (1 + cos2 θ) . (4) The effect of the anisotropy is small and wavelength-dependent. Bucholtz [12] Fig. 5. Schematic of track seen by ith pixel. estimates γ ≃ 0.015 at 360 nm and concludes that the correction leads to a ∼ 3% systematic increase in the Rayleigh scattering cross section, and a fractional change ≤ 1.5% from the approximate (1 + cos2 θ). In our analysis, only the shape of the function is relevant, and we use Eq. 4 as an approximation of Eq. 3. The aerosol phase function is often parameterized by the Henyey-Greenstein function [13]: Pa(θ) = 1− g2 (1 + g2 − 2gµ)3/2 , (5) where µ = cos θ and g is an asymmetry parameter equal to the mean cosine of the scattering angle: g = 〈cos θ〉. The parameter g is a measure of how much light is scattered in the forward direction; a greater g means more light is forward-scattered. Values for g range from g = 1 (total forward scattering) to g = −1 (total backward scattering), with g = 0 indicating isotropic scattering. The Henyey-Greenstein function works well for pure forward scattering, but it cannot describe realistic aerosol conditions, which typically give rise to non- negligible backscattering. Following [14,15], we modify Eq. 5 so that Pa(θ) = 1− g2 (1 + g2 − 2gµ)3/2 3µ2 − 1 2(1 + g2)3/2 . (6) The new term in this expression is proportional to the second Legendre poly- nomial, and it is introduced to describe the extra backscattering component. The value f is a fit parameter used to tune the relative strength of forward to backward scattering. The binned APF signal observed in the FD is therefore subjected to a 4- parameter fit: Si = A · (1 + µ i ) +B · (1− g (1 + g2 − 2gµi)3/2 3µ2i − 1 2(1 + g2)3/2 , (7) where A, B, g and f are the fit parameters. In principle, the parameters A and B, which describe the relative amount of Rayleigh and Mie scattering, can be determined from measurements of the extinction lengths Λm and Λa and assumptions about the particle albedo, i.e. the ratio of light scattered by the aerosol particle in all directions to the amount of incoming light. The albedo is close to one if the particle is mostly reflective. Since local information on the extinction lengths was not available for this analysis, we use A and B as additional fit parameters. We find that the distinct shapes of the two phase functions does allow a determination of A and B from the data themselves. At Coihueco, the APF signal is seen in 5 out of the 6 mirrors, so the track is visible over ∼ 150◦ in azimuth. At the boundary between each mirror there is some overlap in the fields of view of pixels. This overlap produces a double counting of signal resulting in the value of bins at boundaries being too large. These bins are simply ignored in the fit. The values of the other bins and their errors are obtained from the mean and standard deviation of the five APF shots in each shot sequence. On clear nights with few or no aerosols, the fit to Eq. 7 returns unphysical val- ues for the parameters B, f , and g. In those cases, we re-fit the data to a pure Rayleigh function by setting B, f , and g equal to zero. Two examples of fits, one for a night with aerosol content, and one for a night with pure Rayleigh scattering, are shown in Fig. 6. The aerosol, molecular, and total phase func- tions are shown. The aerosol phase function is obtained by subtracting the molecular component determined by the fit. We fit the data only over a subrange of the available scattering angles, from θmin ≃ 32.5 ◦ to θmax ≃ 147.3 ◦. As Fig. 6 indicates, the data deviates from the theoretical prediction for scattering angles below θmin and above θmax. At smaller and larger angles, several effects corrupt the signal and make it unusable for the fit to the phase function. Due to the local geometry at the Coihueco site (see Fig. 4), the APF shot is not visible for θ < 24◦, and below 30◦, the signal is incomplete because the beam is still partially beneath the detector field of view. At large scattering angles, the beam is at a rapidly in- creasing distance to the corresponding FD bay, and attenuation of light from the beam to the detector becomes important. As mentioned earlier, because local measurements of the optical depth are not yet available, we simply as- sume T = 1. As measurements of T become available, the attenuation of light scattered at large angles can be used to correct the data. ]° [θscattering angle 0 20 40 60 80 100 120 140 160 180 A 46.2± 8602 B 1814± 2.696e+05 g 0.0036± 0.6804 f 0.0147± 0.4977 Aerosol Phase Function Total phase function Rayleigh phase function Mie phase function ]° [θscattering angle 0 20 40 60 80 100 120 140 160 180 10000 15000 20000 25000 30000 35000 40000 45000 50000 A 19± 1.011e+04 Aerosol Phase Function Total phase function Rayleigh phase function Mie phase function Fig. 6. Two examples for APF data fits on different days. In the upper plot (June 28, 2006, 5:12 am local time) aerosols are visible. Data are fit to the function given in Eq. 7. The phase function in the lower plot (July 2, 2006, 3:12 am local time) is consistent with pure Rayleigh scattering. Data are fit to Eq. 7, with B = 0, f = 0, and g = 0. Error bars for both plots are the standard deviation of the 5 APF events. In order to apply geometrical corrections when binning the data, the angle at which the APF light source shoots (γ in Fig. 4) with respect to the FD and the elevation angle of the shot direction needs to be known. We determined these values from the data themselves. The elevation angle was determined from a reconstruction of APF shots with the FD offline reconstruction [16], and γ was determined from the analysis of APF shots on nights where aerosol scattering was negligible. The data from these nights were fit to the Rayleigh component of the phase function, with the position of the minimum (nominally at 90◦ scattering angle) as a free parameter. The fit value of this angle was then used to deduce the direction which the APF light source shoots relative to the FD (∼ 24◦ at Coihueco). 4 First Results We have applied the analysis described in Section 3 to data recorded between June and December 2006 at the Coihueco site. Since the APF light sources operate during all nights of FD operation, this data set includes all moonless nights, with the exception of nights with rain or strong winds when the FDs remain closed. Fig. 7 shows the distribution of the asymmetry parameter g (left) and the backscatter parameters f (right). For most nights with aerosol contamination, the value of g at the experiment site in Malargüe is ∼ 0.6, with an average of 0.59 and a standard deviation of 0.07 for the data period analyzed here. Values of g = 0 indicate hours where the measured phase function can be described by pure Rayleigh scattering, so the aerosol phase function is effectively negligible. Fig. 7 also shows the asymmetry parameter as a function of time for the analyzed period. With the limited amount of data taken so far, no conclusions concerning seasonal variations can be drawn. The asymmetry parameter appears to be stable during the observed time period. With more data becoming available over the next few years, we plan to monitor the month-to-month variation in g and analyze possible correlations with other weather measurements. One of the main tasks of the APF, in addition to providing the in situ aerosol phase function for every hour of FD data taking, is the identification of “clear” nights with small aerosol contamination. These nights play an important role in the calibration of other atmospheric monitoring devices such as the Central Laser Facility (CLF) [6]. On clear nights, the measured phase function can be described by pure Rayleigh scattering (measurements where this is the case appear as g = 0 in Fig. 7). To confirm the reliability of the fit where both the normalization of the Mie and the Rayleigh contribution are fit parameters, Fig. 8 shows the Rayleigh normalization factor A for the same data set. One might expect the molecular asymmetry parameter g 0 0.2 0.4 0.6 0.8 1 backscatter parameter f -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Jun 2006 Jul 2006 Aug 2006 Sep 2006 Oct 2006 Nov 2006 Dec 2006 Fig. 7. Top: Distribution of the asymmetry parameter g (top left) and the backscatter parameter f (top right) for all measurements performed between June and December 2006. Values of g = 0 (and f = 0) indicate that the phase function can be described with pure Rayleigh scattering. Bottom: Asymmetry parameter g as a function of time. contribution to be rather stable, and in fact this parameter does not change much with time. It is instructive to compare the average asymmetry parameter obtained from the APF with model expectations and measurements at comparable locations. Typically, measurements are performed at optical wavelengths and cannot be directly compared to measurements at UV wavelengths. However, a compi- lation at different wavelengths from 450 nm to 700 nm [17] shows that the wavelength dependence of g is small; values at 450 nm are a few percent larger than at 550 nm. Rayleigh normalization 4000 6000 8000 10000 12000 14000 16000 Fig. 8. Distribution of the Rayleigh normalization parameter A for all measurements performed between June and December 2006. To first order, g = 0.7 is often used as a generic value for g in radiative transfer models. A smaller value for g is expected at dry locations. A parameterization of aerosol optical properties by d’Almeida et al. [18] suggests values for g between 0.64 and 0.83 at 550 nm depending on aerosol type and season, with higher averages for high relative humidity. The Pierre Auger Observatory is located east of the Andes in the Pampa Amarilla, an arid high plateau at 1420 m a.s.l., so values around 0.6 are within expectations. For comparison, recent measurements carried out in the South- ern Great Planes of the US [19] yield values for g at 550 nm of 0.60± 0.03 for dry conditions and 0.65± 0.05 for ambient conditions. The aerosol phase function most commonly used in fluorescence detector data analysis, both for the High Resolution Fly’s Eye (HiRes) Experiment [20], which operated in Utah between 1997 and 2006, and the Pierre Auger FD de- tectors, is the function obtained from a desert aerosol simulation by Longtin [21]. Longtin’s desert model is based on Mie scattering theory and assumes that the desert atmosphere has three major components: carbonaceous particles, water- soluble particles, and sand. For each aerosol component, the model assumes a characteristic log normal size distribution and refractive index. Longtin per- formed his calculations for several wavelengths and wind speeds; those made at 550 nm with a wind speed of 10 m/s most closely match the 300 nm to 400 nm nitrogen fluorescence band observed by the FDs and have therefore been traditionally used in air fluorescence data analysis. ]° [θscattering angle 0 20 40 60 80 100 120 140 160 180 Aerosol Phase Function = 550 nm, wind = 10 m/s)λLongtin ( This work: f = 0.5, g = 0.7 This work: f = 0.4, g = 0.6 Fig. 9. Comparison of the Longtin aerosol phase function (desert atmosphere simu- lated with a wind speed of 10 m/s) with the default phase function used in the Auger atmospheric database (f = 0.5, g = 0.7) and the typical phase function measured by the APF (f = 0.4, g = 0.6). Fig. 9 compares the Longtin aerosol phase function at 550 nm to the modi- fied Henyey-Greenstein function of Eq. 6 with two sets of f and g: f = 0.5 and g = 0.7, the default values used by the Auger atmospheric database; and f = 0.4 and g = 0.6, the values determined in this study to be more typical of the detector location. The comparison shows that, on average, the differ- ence between the Longtin function and the measured phase function is small for those scattering angles relevant in fluorescence measurements — ∼ 30◦ to 150◦. Only at the largest scattering angles above 160◦ do the phase func- tions differ notably. This region is outside the current range of validity of our measurement. Our primary interest in aerosol scattering is its effect on the air shower recon- struction, most notably the determination of the shower energy. However, it is not straightforward to estimate the extent to which the use of measured rather than averaged values of f and g changes the energy reconstruction, as this depends strongly on other atmospheric parameters, for example the aerosol optical depth. Rather than singling out the phase function measurement, we need to study the effect of the combined measurement of all atmospheric pa- rameters, a task which is beyond the scope of this paper. We can, however, get an estimate of the relevance of the phase function mea- surement by studying its effect on the energies of events that are of particu- lar importance for the energy calibration of the detector, the “golden hybrid hist1 Mean 0.6996 RMS 0.7913 ) / Eapf - Estd(E -10 -8 -6 -4 -2 0 2 4 6 8 10 210 hist1 Mean 0.6996 RMS 0.7913 f > 0, g > 0 Mean 0.71 RMS 0.79 f = 0, g = 0 Mean 1.87 RMS 2.34 Weather Conditions f > 0, g > 0 f = 0, g = 0 Fig. 10. Differences in the energies of golden hybrid events reconstructed with de- fault phase function values (Estd) and those reconstructed using phase function fit parameters determined from APF measurements (Eapf). The red (bold) histogram represents data taken during nights with measurable aerosols; the blue (light) his- togram depicts events observed on purely molecular nights. events.” These are events observed by one or more fluorescence detectors and three or more surface array tanks. For “golden hybrid events” observed by the Coihueco FD site between June and December 2006, we performed the recon- struction twice: first, using the default parameters f = 0.5 and g = 0.7 to estimate aerosol scattering; and second, using the fit parameters determined from APF measurements. In both cases atmospheric extinction was simu- lated using an average aerosol profile model representative of the Malargüe site [22,23]. Fig. 10 depicts the relative differences in energies caused by reconstructing showers with the default phase function and the measured phase function. The red (bold) histogram represents data taken during nights with aerosol contamination (f > 0, g > 0) while the blue (light) histogram represents data taken during nights where according to the APF analysis scattering is purely molecular. The correction is typically of order one percent. However, on those nights when aerosol loading is extremely low, so that atmospheric scattering may be characterized as purely molecular, the use of the default scattering parameters causes larger errors in the shower reconstruction. Under such conditions, the total phase function lacks the strong forward-scattering component typical of aerosols. During these periods, incorrectly accounting for aerosol scattering starts to impact the energy calibration of the detector. A correct determination of the phase function on a regular basis is therefore an important part of the atmospheric monitoring efforts at the site. 5 Conclusions and Outlook As part of the atmospheric monitoring program at the Pierre Auger Obser- vatory, the aerosol phase function at 350 nm is routinely measured at two of the four FD sites. A first analysis of data taken from June to December 2006 shows that values of g = 〈cos θ〉 ≃ 0.6 for the mean cosine of the scattering angle θ are typical for aerosols at the site of the experiment. Over the next several years, the APF light sources will produce a data set of unprecedented size of the scattering properties of aerosols. This data set will enable us to carefully study any seasonal change in the aerosol content. The APF light sources and the other atmospheric monitoring instruments at the Auger site will accumulate one of the largest sets of continuous measurements in the 300 nm to 400 nm range ever recorded for a single location. The APF light sources are currently operating at a wavelength of 350 nm only. In the near future, we will add regular measurements at 330 nm and 390 nm to study the dependence of the phase function on the wavelength of the scattered light. Acknowledgements We are grateful to the following agencies and organizations for financial sup- port: The APF light sources were built by a grant from the Department of Energy (DOE) Office of Science (USA) (DE-FG03-92ER40732). Parts of the APF analysis were performed during the 2006 REU (Research Experience for Undergraduates) program at Columbia University’s Nevis Laboratories which is supported by the National Science Foundation (USA) under contract num- ber NSF-PHY-0452277. References [1] P. Mantsch (for the Pierre Auger Collaboration), The Pierre Auger Observatory Status and Progress, in Proc. 29th Int. Cosmic Ray Conference, Pune, India (2005) (astro-ph/0604114); www.auger.org. [2] J. Abraham et al. (Pierre Auger Collaboration), Astroparticle Phys. 27 (2007) 155. [3] J. Abraham et al. (Pierre Auger Collaboration), Astroparticle Phys., in press (2007) (astro-ph/0607382). http://arxiv.org/abs/astro-ph/0604114 http://arxiv.org/abs/astro-ph/0607382 [4] M. Mostafá (for the Pierre Auger Collaboration), Atmospheric Monitoring for the Pierre Auger Fluorescence Detector, in Proc. 28th Int. Cosmic Ray Conference, Tsukuba, Japan (2003), 2 (HE1.3), 465. [5] S.Y. BenZvi et al., Nucl. Instr. Meth. A, in press (2007) (astro-ph/0609063). [6] B. Fick et al., Journal of Instrumentation 1 (2006) P11003. [7] G. Mie, Ann. Phys. 25 (1908) 377. [8] M.D. Roberts, J. Phys. G: Nucl. Part. Phys 31 (2005) 1291. [9] J.A.J. Matthews, R. Clay, for the Pierre Auger Collaboration, Atmospheric Monitoring for the Auger Fluorescence Detector, in Proc. 27th Int. Cosmic Ray Conference, Hamburg, Germany (2001), 2 (HE 1.8), 745. [10] J.A.J. Matthews et al., APF Light Sources for the Auger Southern Observatory, in Proc. 28th Int. Cosmic Ray Conference, Tsukuba, Japan (2003), 2 (HE 1.5), [11] The Pierre Auger Collaboration, Performance of the Fluorescence Detectors of the Pierre Auger Observatory, in Proc. 29th Int. Cosmic Ray Conference, Pune, India (2005) (astro-ph/0508389). [12] A. Bucholtz, Appl. Opt. 34 (1995) 2765. [13] L. Henyey and J. Greenstein, Astrophys. Journal 93 (1941) 70. [14] E.S. Fishburne, M.E. Neer, and G. Sandri, Voice Communication via Scattered Ultraviolet Radiation, Report 274, Vol. 1, Aeronautical Research Associates of Princeton, Princeton, NJ (1976). [15] F. Riewe and A.E.S. Green, Applied Optics 17 (1978) 1923. [16] S. Argirò et al., submitted to Computer Physics Communications (2007). [17] M. Fiebig and J.A. Ogren, J. Geophys. Res. 111 (2006) D21204. [18] G.A. d’Almeida, P. Koepke, and E.P. Shettle, Atmospheric Aerosols: Global Climatology and Radiative Characteristics, A. Deepak Publishing, Hampton, Virginia (1991). [19] E. Andrews et al., J. Geophys. Res. 111 (2006) D05S04. [20] G.B. Thomson et al. (for the HiRes Collaboration), Nucl. Phys. Proc. Suppl. 136 (2004) 28; www.cosmic-ray.org. [21] D.R. Longtin, A Wind Dependent Desert Aerosol Model: Radiative Properties, Air Force Geophysics Laboratories, AFL-TR-88-0112, 1988. [22] S.Y. BenZvi et al. (for the Pierre Auger Collaboration), Measurement of Aerosols at the Pierre Auger Observatory, in Proc. 30th Int. Cosmic Ray Conference, Mérida, México (2007). http://arxiv.org/abs/astro-ph/0609063 http://arxiv.org/abs/astro-ph/0508389 [23] M. Prouza (for the Pierre Auger Collaboration), Systematic Study of Atmosphere-Induced Influences and Uncertainties on Shower Reconstruction at the Pierre Auger Observatory, in Proc. 30th Int. Cosmic Ray Conference, Mérida, México (2007). Introduction APF Light Sources Detector Buildings, Optics, and Electronics APF Signals in the Fluorescence Detectors Determination of the Aerosol Phase Function First Results Conclusions and Outlook Acknowledgements References

0704.0304

The World as Evolving Information

The World as Evolving Information Carlos Gershenson1,2,3 1 Computer Sciences Department, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas Universidad Nacional Autónoma de México1 Ciudad Universitaria, A.P. 20-726, 01000 México D.F. México [email protected] http://turing.iimas.unam.mx/∼cgg 2 New England Complex Systems Institute 238 Main Street Suite 319 Cambridge, MA 02142, USA 3Centrum Leo Apostel, Vrije Universiteit Brussel Krijgskundestraat 33 B-1160 Brussel, Belgium This paper discusses the benefits of describing the world as information, especially in the study of the evolution of life and cognition. Traditional studies encounter prob- lems because it is difficult to describe life and cognition in terms of matter and energy, since their laws are valid only at the physical scale. However, if matter and energy, as well as life and cognition, are described in terms of information, evolution can be described consistently as information becoming more complex. The paper presents eight tentative laws of information, valid at multiple scales, which are generalizations of Darwinian, cybernetic, thermodynamic, psychological, philosophical, and complexity principles. These are further used to discuss the no- tions of life, cognition and their evolution. 1Current affiliation. A considerable part of this work was developed while at other institu- tions. http://arxiv.org/abs/0704.0304v3 http://turing.iimas.unam.mx/~cgg 1 Introduction Throughout history we have used concepts from our current technology as metaphors to describe our world. Examples of this are the description of the body as a factory during the Industrial Age, and the description of the brain as a computer during the Information Age. These metaphors are useful because they extend the knowledge acquired by the scientific and technological developments to other areas, illuminating them from a novel perspective. For example, it is common to extend the particle metaphor used in physics to other domains, such as crowd dynamics [27]. Even when people are not particles and have very com- plicated behaviour, for the purposes of crowd dynamics they can be effectively described as particles, with the benefit that there is an established mathemati- cal framework suitable for this description. Another example can be seen with cybernetics [4, 28], where the system metaphor is used: everything is seen as a system with inputs, outputs, and a control that regulates the internal variables of the system under the influence of perturbations from its environment. Yet another example can be seen with the computational metaphor [60], where the universe can be modelled with simple discrete computational machines, such as cellular automata or Turing machines. Having in mind that we are using metaphors, this paper proposes to extend the concept of information to describe the world: from elementary particles to galaxies, with everything in between, particularly life and cognition. There is no suggestion on the nature of reality as information [58]. This work only explores the advantages of describing the world as information. In other words, there are no ontological claims, only epistemological. In the next section, the motivation of the paper is presented, followed by a section describing the notion of information to be used throughout the paper. In Section 4, eight tentative laws of information are put forward. These are applied to the notions of life (Section 5) and cognition (Section 6). The paper closes presenting future work and conclusions. 2 Why Information? There is a great interest in the relationship between energy, matter, and infor- mation [32, 54, 43]. One of the main reasons for this arises because this rela- tionship plays a central role in the definition of life: Hopfield [30] suggests that the difference between biological and physical systems is given by the meaning- ful information content of the former ones. Not that information is not present in physical systems, but—as Roederer puts it—information is passive in physics and active in biology [49]. However, it becomes complicated to describe how this information came to be in terms of the physical laws of matter and energy. In this paper the inverse approach is proposed: let us describe matter and energy in terms of information. If atoms, molecules and cells are described as information, there is no need of a qualitative shift (from non-living to living matter) while describing the origin and evolution of life: this is translated into a quantitative shift (from less complex to more complex information). There is a similar problem when we study the origin and evolution of cog- nition [20]: it is not easy to describe cognitive systems in terms of matter and energy. The drawback with the physics-based approach to the studies of life and cognition is that it requires a new category, that in the best situations can be referred to as “emergent”. Emergence is a useful concept, but it this case it is not explanatory. Moreover, it stealthily introduces a dualist view of the world: if we cannot relate properly matter and energy with life and cognition, we are forced to see these as separate categories. Once this breach is made, there is no clear way of studying or understanding how systems with life and cognition evolved from those without it. If we see matter and energy as particular, simple cases of information, the dualist trap is avoided by following a continuum in the evolution of the universe. Physical laws are suitable for describing phenomena at the physical scale. The tentative laws of information presented below aim at being suitable for describing phenomena at any scale. Certainly, there are other approaches to describe phenomena at multiple scales, such as general systems theory and dynamical systems theory. These approaches are not exclusive, since one can use several of them, including information, to describe different aspects of the same phenomena. Another benefit of using information as a basic descriptor for our world is that the concept is well studied and formal methods have already been developed [14, 46], as well as its philosophical implications have been discussed [19]. Thus, there is no need to develop a new formalism, since information theory is well established. I borrow this formalism and interpret it in a new way. Finally, information can be used to describe other formalisms: not only par- ticles and waves, but also systems, networks, agents, automata, and computers can be seen as information. In other words, it can contain other descriptions of the world, potentially exploiting their own formalisms. Information is an inclusive formalism. 3 What Is Information? Extending the notion of Umwelt [57], the following notion of information can be given: Notion 1 Information is anything that an agent can sense, perceive, or observe. This notion is in accordance with Shannon’s [52], where information is seen as a just-so arrangement, a defined structure, as opposed to randomness [12, 13], and it can be measured in bits. This notion can be applied to everything that surrounds us, including matter and energy, since we can perceive it—because it has a defined structure—and we are agents, according to the following notion: Notion 2 An agent is a description of an entity that acts on its environment [22, p. 39]. Noticing that agents (and their environments) are also information (as they can be perceived by other agents, especially us, who are the ones who describe them as agents), an agent can be a human, a cell, a molecule, a computer program, a society, an electron, a city, a market, an institution, an atom, or a star. Each of these can be described (by us) as acting in their environment, simply because they interact with it. However, not all information is an agent, e.g. temperature, color, velocity, hunger, profit. Notion 3 The environment of an agent consists of all the information interact- ing with it. Information will be relative to the agent perceiving it2. Information can exist in theory “out there”, independently of an agent, but for practical purposes, it can be only spoken about once an agent—not necessarily a human—perceives / interacts with it. The meaning of the information will be given by the use the agent perceiving it makes of it [59], i.e. how the agent responds to it [7]. Thus, Notion 1 is a pragmatic one. Note that perceived information is different from the meaning that an agent gives to it. Meaning is an active product of the interaction between information and the agent perceiving it [13, 44]. Like this, an electron can be seen as an agent, which perceives other electrons as information. The same description can be used for molecules, cells, and animals. We can distinguish: First order information is that which is perceived directly by an agent. For example, the information received by a molecule about another molecule Second order information is that which is perceived by an agent about in- formation perceived by another agent. For example, the information per- ceived by a human observer about a molecule receiving information about another molecule. Most of the scientific descriptions about the world are second order informa- tion, as we perceive how agents perceive and produce information. The present approach also introduces naturally the role of the observer in science, since ev- erything is “observing” the (limited, first order) information it interacts with from its own perspective. Humans would be second-level observers, observing the information observed by information. Everything we can speak about is observed, and all agents are observers. Information is not necessarily conserved, i.e. it can be created, destroyed, or transformed. These can take place only through interaction. Computation can be seen as the change in information, be it creation, destruction, or transfor- mation. Matter and energy can be seen as particular types of information that cannot be created or destroyed, only transformed, along with the well-known properties that characterize them. 2Shannon’s information [52] deals only with the technical aspect of the transmission of information and not with its meaning, i.e. it neglects the semantic aspect of communication. The amount of information required to describe a process, system, object, or agent determines its complexity [46]. According to our current knowledge, during the evolution of our universe there has been a shift from simple informa- tion towards more complex information [2] (the information of an atom is less complex than that of a molecule, than that of a cell, than that of a multicellular organism, etc.). This “arrow of complexity”[11] in evolution can guide us to explore general laws of information. 4 Tentative Laws of Information Seeing the world as information allows us to describe general laws that can be applied to everything we can perceive. Extending Darwin’s theory [15], the present framework can be used to reframe “universal Darwinism” [17], which explores the idea of evolution beyond biological systems. In this work, the laws that describe the general behaviour of information as it evolves are introduced. These laws are only tentative, in the sense that they are only presented with arguments in favour of them, but they still need to be thoroughly tested. 4.1 Law of Information Transformation Since information is relative to the agents perceiving it, information will poten- tially be transformed as different agents perceive it. Another way of stating this law is the following: information will potentially be transformed by interacting with other information. This law is a generalization of the Darwinian principle of random variation, and ensures novelty of information in the world. Even when there might be static information, different agents can perceive it differ- ently and interact with it, potentially transforming it. Through evolution, the transformation of information generates a variety or diversity that can be used by agents for novel purposes. Since information is not a conserved quantity, it can increase (created), de- crease (destroyed), or be maintained as it is transformed. As an example, RNA polymerase (RNAP) can make errors while copying DNA onto RNA strands. This slight random variation can lead to changes in the proteins for which the RNA strands serve as templates. Some of these changes will lead to novel proteins that might improve or worsen the function of the original proteins. The transformation of information can be classified as follows: Dynamic. Information changes itself. This could be considered as “objective, internal” change. Static. The agent perceiving the information changes, but the information itself does not change. There is a dynamic change but in the agent. This could be considered as “subjective, internal” change. Active. An agent changes information in its environment. This could be con- sidered as an “objective, external” change. Stigmergic. An agent makes an active change of information, which changes the perception of that information by another agent. This could be con- sidered as “subjective, external” or “intersubjective” change. 4.2 Law of Information Propagation Information propagates as fast as possible. Certainly, only some information manages to propagate. In other words, we can assume that different informa- tion has a different “ability” to propagate, also depending on its environment. The “fitter” information, i.e. that which manages to persist and propagate faster and more effectively, will prevail over other information. This law generalizes the Darwinian principle of natural selection, the maximum entropy production principle [37] (entropy can also be described as information), and Kauffman’s tentative fourth law of thermodynamics3. It is interesting that this law contains the second law of thermodynamics, as atoms interact, propagating informa- tion homogeneously. It also describes living organisms, where genetic informa- tion is propagated across generations. And it also describes cultural evolution, where information is propagated among individuals. Life is “far from thermo- dynamic equilibrium” because it constrains [32] the (more simple) information propagation at the thermodynamic scale, i.e. the increase of entropy, exploiting structures to propagate (or maintain) the (more complex) information at the biological scale. In relation with the law of information transformation, as information re- quires agents to perceive it, information will be potentially transformed. This source of novelty will allow for the “blind” exploration of better ways of propa- gating information, according to the agents perceiving it and their environments. Extending the previous example, if errors in transcription made by RNAP are beneficial for its propagation (which entails the propagation of the cell producing RNAP), cells with such novel proteins will have better chances of survival than their “cousins” without transcription errors. The propagation of information can be classified as follows: Autonomous. Information propagates by itself. Strictly speaking, this is not possible, since at least some information is determined by the environment. However, if more information is produced by itself than by its environment, we can call this autonomous propagation (See Section 5). Symbiotic. Different information cooperates, helping to propagate each other. Parasitic. Information exploits other information for its own propagation. Altruistic. Information promotes the propagation of other information at the cost of its own propagation. 3“The workspace of the biosphere expands, on average, as fast as it can in this coconstruct- ing biosphere” [32, p. 209] 4.3 Law of Requisite Complexity Taking into account the law of information transformation, transformed infor- mation can increase, decrease, or maintain its previous complexity, i.e. amount [46]. However, more complex information will require more complex agents to perceive, act on, and propagate it. This law generalizes the cybernetic law of requisite variety [4]. Note that simple agents can perceive and interact with part of complex information, but they cannot (by themselves) propagate it. An agent cannot perceive (and thus contain) information more complex than itself. For simple agents, information that is complex for us will be simple as well. As stated above, different agents can perceive the same information in different ways, giving it different meanings. The so called “arrow of complexity” in evolution [11] can be explained with this law. If we start with simple information, its transformation will produce by simple drift [39, 41] increases in the complexity of information, without any goal or purpose. This occurs simply because there is an open niche for information to become more complex as it varies. But this also promotes agents to become more complex to exploit novel (complex) information and propagate it. Evolution does not need to favour complexity in any way: information just propagates to every possible niche as fast as possible, and it seems that there is often an “adjacent possible” [32] niche of greater complexity. For example, it can be said that a protein (as an agent) perceives some information via its binding sites, as it recognizes molecules that “fit” a site. More complex molecules will certainly need more complex binding sites. Whether complex molecules are better or worse is a different matter: some will be better, some will be worse. But for those which are better, the complexity of the proteins must match the complexity of the molecules perceived. If the binding site perceives only a part of the molecule, then this might be confused with other molecules which share the perceived part. Following the law of information transformation, there will be a variety of complexities of information. The law of requisite complexity just states that the increase in complexity of information is determined by the ability of agents to perceive, act on, and propagate more complex information. Since more complex information will be able to produce more variety, the speed of the complexity increase will escalate together with the complexity of the information. 4.4 Law of Information Criticality Transforming and propagating information will tend to a critical balance be- tween its stability and its variability. Propagating information maintains itself as much as possible, but transforming information varies it as much as possi- ble. This struggle leads to a critical balance analogous to the “edge of chaos” [36, 31], self-organized criticality [8, 1], and the “complexity from noise” princi- ple [6]. The homeostasis of living systems can also be seen as the self-regulation of information criticality. This law can generalize Kauffman’s four candidate laws for the coconstruc- tion of a biosphere [32, Ch. 8]. Their relationship with this framework demands further discussion, which is out of the scope of this paper. A well known example can be seen with cellular automata [36] and random Boolean networks [31, 21, 23]: stable (ordered) dynamics limit considerably or do not allow change of states so information cannot propagate, while variable (chaotic) dynamics change the states too much, losing information. Following the law of information propagation, information will tend to a critical state between stability and variability to maximize its propagation: if it is too stable, it will not propagate, and if it is too variable, it will be transformed. In other words, “critical” information will be able to propagate better than stable or variable one, i.e. as fast as possible (cf. law of information propagation). 4.5 Law of Information Organization Information produces constraints that regulate information production. These constraints can be seen as organization [32]. In other words, evolving information will be organized (by transformation and propagation) to regulate information production. According to the law of information criticality, this organization will lie at a critical area between stability and variability. And following the law of information propagation, the organization of information will enable it to propagate as fast as possible. This law can also be seen as information having a certain control over its environment, since the organization of information will help it withstand pertur- bations. It has been shown [33, 47, 34] that using this idea as a fitness function can lead to the evolution of robust and adaptive agents, namely maximizing the mutual information between sensors and environment. A clear example of information producing its own organization can be seen with living systems, which are discussed in Section 5. 4.6 Law of Information Self-organization Information tends to its preferred, most probable state. This is actually a tautol- ogy, since observers determine probabilities after observing tendencies of infor- mation dynamics. Still, this tautology can be useful to describe and understand phenomena. This law lies at the heart of probability theory and dynamical sys- tems theory [5]. The dynamics of a system tend to a subset of its state space, i.e. attractors, depending on its history. This simple fact reduces the possibility space of information, i.e. a system will tend towards a small subset of all pos- sible states. If we describe attractors as “organized”, then we can describe the dynamics of information in terms of self-organization [25]. Pattern formation can be described as information self-organizing, and re- lated to the law of information propagation. Information will self-organize in “fit” patterns that are the most probable (defined a posteriori). Understanding different ways in which self-organization is achieved by trans- forming information can help us understand better natural phenomena [24] and design artificial systems [22]. For example, random Boolean networks can be said to self-organize towards their attractors [23]. 4.7 Law of Information Potentiality An agent can give different potential meanings to information. This implies that the same information can have different meanings. Moreover, meaning— while being information—can be independent of the information carrying it, i.e. depend only on the agent observing it. Thus, different information can have the same potential meaning. The precise meaning of information will be given by an agent observing it within a specific context. The potentiality of information allows the effective communication between agents. Different information has to be able to acquire the same meaning (homonymy), while the same information has to be able to acquire different meanings (polysemy) [44]. The relationship between the laws of information and communication is clear, but beyond the scope of this paper. The law of information potentiality is related to a passive information trans- formation, i.e. a change in the agent observing information. In spite of information potentiality, not all meanings will be suitable for all information. In other words, pure subjectivism cannot dictate meanings of information. By the law of information propagation, some meanings will be more suitable than others and will propagate. The suitability of meanings will be determined by their use and context [59]. However, there is always a certain freedom to subjectively transform information. For example, a photon can be observed as a particle, as a wave, or as a particle-wave. The suitability of each given meaning is determined by the context in which the photon is described/observed. 4.8 Law of Information Perception The meaning of information is unique for an agent perceiving it in unique, always changing open contexts. If meaning of information is determined by the use an agent makes of it, which is embedded in an open environment, we can go to such a level of detail that the meaning will be unique. Certainly, agents make generalizations and abstractions of perceptions in order to be able to respond to novel information. Still, the precise situation and context will never be repeated. This makes perceived information unique. The implication of this is that the response to any given information might be “unexpected”, i.e. novelty can arise. Moreover, the meaning of information can be to a certain extent arbitrary. This is related with the law of information transformation, as the uniqueness of meaning allows the same information perceived differently by the same or different agents to be statically transformed. This law is a generalization of the first law of human perception: “whatever is perceived can be perceived only from a uniquely situated place in the overall structure of points of view” [29, p. xxiv] (cited in [44, p. 250]). We can describe agents perceiving information as filtering it. An advantage of humans and other agents is that we can choose which filter to use to perceive. The suggestion is not that “unpleasant” information should be solipsistically ignored, but that information can be potentially actively transformed. For example, T lymphocytes in an immune system can perceive foreign agents and attack them. Even when the response will be similar for similar foreign agents, each perception will be unique, a situation that always leaves space for novelty. Scales of perception Different information is perceived at different scales of observation [9]. As the scale tends to zero, then the information tends to infinite. For lower scales, more information and details are perceived. The uniqueness of information perception dominates at these very low (spatial and temporal) scales. However, as gener- alizations are made, information is “compressed”, i.e. only relevant aspects of information are perceived4. At higher scales, more abstractions and general- izations are made, i.e. less information is perceived. When the scale tends to infinite, the information tends to zero. In other words, no information is needed to describe all of the universe, because all the information is already there. This most abstract understanding of the world is in line with the “highest view” of Vajrayana Buddhism [45]. Implications at this level of description cannot be right or wrong, because there is no context. Everything is contained, but no information is needed to describe it, since it is already there. This “maximum” understanding is also described as vacuity, which leads to bliss [45, p. 42]. Following the law of information criticality, agents will tend to a balance where the perceived information is minimal but maximally predictive [51] (at a particular scale): few information is cheaper, but more information in general entails a more precise predictability. The law of requisite complexity applies at particular scales, since a change of scale will imply a change of complexity of information [9]. 5 On the Notion of Life There is no agreed notion of life, which reflects the difficulty of defining the concept. Still, many researchers have put forward properties that characterize important aspects of life. Autopoiesis is perhaps the most salient one, which notes that living systems are self-producing [55, 38]. Still, it has been argued that autopoiesis is a necessary but not sufficient property for life [50]. The relevance of autonomy [10, 42, 35] and individuality [40, 35] for life have also been highlighted . These approaches are not unproblematic, since no living system is completely autonomous. This follows from the fact that all living systems are open. For 4The relevance is determined by the context, i.e. different aspects will be relevant for different contexts. example, we have some degree of autonomy, but we are still dependent on food, water, oxygen, sunlight, bacteria living in our gut, etc. This does not mean that we should abandon the notion of autonomy in life. However, we need to abandon the sharp distinction between life and non-life [11, 35], as different degrees of autonomy escalate gradually, from the systems we considered as non-living to the ones we consider as living. In other words, life has to be a fuzzy concept. Under the present framework, living and non-living systems are information. Rather than a yes/no definition, we can speak about a “life ratio”: Notion 4 The ratio of living information is the information produced by itself over the information produced by its environment. Being more specific—since all systems also receive information—a system with a high life ratio produces more (first order) information about itself than the one it receives from its environment. Following the law of information orga- nization, this also implies that living information produces more of its own con- straints (organization) to regulate itself than the ones produced by its environ- ment, and thus it has a greater autonomy. All information will have constraints from other (environmental) information, but we can measure (as second-order information) the proportion of internal over external constraints to obtain the life ratio. If this is greater than one, then the information regulates by itself more than the proportion that is regulated by external information. In the opposite case, the life ratio would be less than one. Following the law of information propagation, evolution will tend to informa- tion with higher life ratios, simply because this can propagate better, as it has more “control” and autonomy over its environment. When information depends more on its environment for its propagation, it has a higher probability of being transformed as it interacts with its environment. Note that the life ratio depends on spatial and temporal scales at which information is perceived. For example, for some microorganisms observed at a scale of years , the life ratio would be less than one, but if observed at a scale of seconds, the life ration would be greater than one. Certainly, some artificial systems would be considered as living under this notion. However, we can make a distinction between living systems embodied in or composed by biological cells [16], i.e. life as we know it, and the rest, i.e. life as it could be. The latter ones are precisely those explored by artificial life. 6 On the Notion of Cognition Cognition is certainly related with life [53]. The term has taken different mean- ings in different contexts, but all of them can be generalized into a common notion [20]. Cognition comes from the Latin cognoscere, which means “get to know”. Like this, Notion 5 A system is cognitive if it knows something [20, p.135]. From Notion 2, all agents are cognitive, since they “know” how to act on their environment, giving (first order) meaning to their environmental informa- tion. Thus, there is no boundary between non-cognitive and cognitive systems. Throughout evolution, however, there has been a gradual increase in the com- plexity of cognition [20]. This is because all agents can be described as possessing some form of cognition, i.e. “knowledge” about the (first-order) information they perceive5. Following the law of requisite complexity, evolution leads to more complex agents, to be able to cope with the complexity of their environment. This is precisely what triggers the (second-order) increase in the complexity of cognition we observe. Certainly, there are different types of cognition6. We can say that a rock “knows” about gravity because it perceives its information, which has an effect on it, but it cannot react to this information. Throughout evolution, infor- mation capable of maintaining its integrity has prevailed over that which was not. Robust information is that which can resist perturbations to maintain its integrity. The ability to react to face perturbations to maintain information makes information adaptive, increasing its probability of maintenance. When this reaction is made before it occurs, the information is anticipative7. As in- formation becomes more complex (even if only by information transformation), the mechanisms for maintaining this information also become more complex, as stated by the law of requisite complexity. This has led gradually to the advanced cognition that animals and machines posses. 7 Future Work The ideas presented here still need to be explored and elaborated further. One way of doing this would be with a simulation-based method. Being inspired by ǫ-machines [51, 26], one could start with “simple” agents that are able to per- ceive and produce information, but cannot control their own production. These would be let to evolve, measuring if complexity increases as they evolve. The hypothesis is that complexity would increase (under which conditions still re- mains to be seen), to a point where “ǫ-agents” will be able to produce themselves depending more on their own information than that of the environment. This would be similar to the evolution in Tierra [48] or Avida [3] systems, only that self-replication would not be inbuilt. The tentative laws of information presented in Section 4 would be better defined if such a system was studied. One important aspect that remains to be studied is the representation of 5One could argue that, since agency (and thus cognition) is already assumed in all agents, this approach is not explanatory. But I am not trying to explain the “origins” of agency, since I assume it to be there from the start. I believe that we can only study the evolution and complexification of agency and cognition, not their “origins”. 6For example, human, animal, plant, bacterial, immune, biological, adaptive, systemic, and artificial [20]. 7For a more detailed treatment on robustness, adaptation, and anticipation, see [22] thermodynamics in terms of information. This is because the ability to per- form thermodynamic work is a characteristic property of biological systems [32]. This work can be used to generate the organization necessary to sustain life (cf. law of information organization). It is difficult to describe life in terms of thermodynamics, since it entails new characteristic properties not present in thermodynamic systems. But if we see the latter ones as information, it will be easier to describe how life—also described as information—evolves from them, as information propagates itself at different scales. A potential application of this framework would be in economy, considering capital, goods, and resources as information (a non-conserved quantity) [18]. A similar benefit (of non-conservation) could be given in game theory: if the payoff of games is given in terms of information (not necessarily conserved), non- zero sum games could be easier to grasp than if the payoff is given in material (conserved) goods. It becomes clear that information (object), the agent perceiving it (subject) and the meaning-making or transformation of information (action) are deeply interrelated. They are part of the same totality, since one cannot exist without the others. This is also in line with Buddhist philosophy. The implications of an informational description of the world for philosophy have also to be addressed, since some schools have focussed on partial aspects of the object-subject-action trichotomy. Another potential application of the laws of information would be in ethics, where value can be described accordingly to the present framework. 8 Conclusions This paper introduced general ideas that require further development, extension and grounding in particular disciplines. Still, a first step is always necessary, and hopefully feedback from the community will guide the following steps of this line of research. Different metaphors for describing the world can be seen as different lan- guages: they can refer to the same objects without changing them. And each can be more suitable for a particular context. For example, English has several advantages for fast learning, German for philosophy, Spanish for narrative, and Russian for poetry. In other words, there is no “best” language outside a par- ticular context. In a similar way, I am not suggesting that describing the world as information is more suitable than physics to describe physical phenomena, or better than chemistry to describe chemical phenomena. It would be redun- dant to describe particles as information if we are studying only particles. The suggested approach is meant only for the cases when the physical approach is not sufficient, i.e. across scales, constituting an alternative worth exploring to describe evolution. It seems easier to describe matter and energy in terms of information than vice versa. Moreover, information could be used as a common language across scientific disciplines [56]. Acknowledgements I should like to thank Irun Cohen, Inman Harvey, Francis Heylighen, David Krakauer, Antonio del Rı́o, Marko Rodriguez, David Rosenblueth, Stanley Salthe, Mikhail Prokopenko, Clément Vidal, and Héctor Zenil for their useful comments and suggestions. 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[60] Wolfram, Stephen, A New Kind of Sciene, Wolfram Media (2002). 1 The World as Evolving Information 1 Introduction 2 Why Information? 3 What Is Information? 4 Tentative Laws of Information 4.1 Law of Information Transformation 4.2 Law of Information Propagation 4.3 Law of Requisite Complexity 4.4 Law of Information Criticality 4.5 Law of Information Organization 4.6 Law of Information Self-organization 4.7 Law of Information Potentiality 4.8 Law of Information Perception 5 On the Notion of Life 6 On the Notion of Cognition 7 Future Work 8 Conclusions

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Polymerization Force Driven Buckling of Microtubule Bundles Determines the Wavelength of Patterns Formed in Tubulin Solutions

Polymerization Force Driven Buckling of Microtubule Bundles Determines the Wavelength of Patterns Formed in Tubulin Solutions Yongxing Guo, Yifeng Liu, Jay X. Tang, and James M. Valles, Jr. Physics Department, Brown University, Providence, RI 02912 (Dated: June 8, 2021) We present a model for the spontaneous formation of a striated pattern in polymerizing microtubule solutions. It describes the buckling of a single microtubule (MT) bundle within an elastic network formed by other similarly aligned and buckling bundles and unaligned MTs. Phase contrast and polarization microscopy studies of the temporal evolution of the pattern imply that the polymerization of MTs within the bundles creates the driving compressional force. Using the measured rate of buckling, the established MT force-velocity curve and the pattern wavelength, we obtain reasonable estimates for the MT bundle bending rigidity and the elastic constant of the network. The analysis implies that the bundles buckle as solid rods. Microtubules (MTs), a major component of the eukary- otic cytoskeleton [1], can form various structures and pat- terns. For example, in vivo, MTs organize into the spindles and asters essential for mitosis [2] and the parallel arrays and stripes necessary for directing early processes in embryogene- sis [3, 4]. Many in vitro studies of MT organization have been performed in order to elucidate the mechanisms underlying the formation of these structures [5, 6, 7, 8]. Of particular relevance here are the striped birefringent patterns [Fig. 1(a)], which spontaneously form from polymerizing a purified tubu- lin solution without motor proteins or MT associated proteins. Hitt et al. attributed these patterns to the formation of ne- matic liquid crystalline domains [5]. Tabony et al., on the other hand, proposed that a reaction-diffusion based mecha- nism drives the formation of MT stripes [6]. Our recent in- vestigations imply a starkly different scenario in which the local MT alignment into wave-like structures occurs through a collective process of MT bundling and buckling [9]. MTs that are aligned by a static magnetic field [10, 11] or convec- tive flow [9] during the initial stage of polymerization sponta- neously form bundles in tubulin solutions with concentrations of a few mg/ml. These bundles elongate and buckle in co- ordination with neighboring bundles into a wave-like shape. The nesting of the buckled bundles can quantitatively account for the MT density and orientation variations leading to the striped birefringent pattern [9]. We proposed that a compres- sional force is generated by MT polymerization occurring uni- formly along the bundle contours. The buckling wavelength is controlled by the bending rigidity of the bundles and the elasticity of the background network of MTs. This interesting initial assessment calls for further investigation of the micro- scopic picture of the bundle elongation, the MT buckling force and the buckling mode selection mechanism. Here we present a mechanical model for the process in addition to new experimental data on the time evolution of the bundle contour length and solution birefringence that pro- vide direct support for the validity of the model. The model considers the instability of a single MT bundle under a com- pressional force, embedded in an elastic network formed by both bundled and dispersed MTs. Time lapse phase contrast and quantitative polarized light microscopy imply that MT polymerization within the bundles provides the compressional FIG. 1: Image of a MT birefringent pattern and a sketch of the me- chanical buckling model. (a) Striped birefringent pattern [23]. The image was taken between crossed polarizers with the polarization di- rections at 45◦ with respect to the x axis. (b) Schematic drawing of buckled MT bundles surrounded by an elastic MT network (gray background). The white dashed line depicts the central bundle before buckling. The white sinusoidal curve depicts the elongated bundle after buckling and the gray sinusoidal curves represent the neighbor- ing MT bundles. ξ (x) is the transverse displacement for the central bundle. force. Specifically, they reveal that the bundles elongate uni- formly along their contours while maintaining constant radii consistent with growth through the elongation of the individ- ual MTs comprising them. We make predictions for the char- acteristic buckling wavelength using the bundle bending rigid- ity and the critical buckling force estimated from the measured MT force-velocity curve. The measured wavelength of about 600 µm implies that the bundles bend as solid rods. We envision initially the microtubule solution to consist of an array of straight and parallel bundles aligned along the x axis and embedded in a network composed of dispersed MTs as in Fig. 1(b). All of the bundles experience a similar com- pressional force that grows to a critical value, causing them to buckle. To describe the buckling, we consider a single bun- dle in the center of the sample and characterize its interaction with the network using a single elastic constant, α , such that αξ (x) is the elastic restoring force exerted by the network on the bundle per unit length. Treating the bundle as a rod with a bending rigidity, K, under a uniform compressional force, F , the force balance in the y direction at the onset of the buckling is given by [12, 13, 14] ξ (x) ∂ξ (x) ]+αξ (x) = 0 (1) Performing a standard normal mode stability analysis of Eq. (1) using ξ (x) ∝ eikx yields a relation between the angu- lar wavenumber, k, and the compressional force, F = α/k2 + Kk2, which suggests a minimum or critical compressional force Fc for a buckling solution. The critical compressional force is Fc = 2 Kα , and the characteristic wavelength is λc = 2π/k = π 8K/Fc = 2π K/α (2) The resultant characteristic wavelength [Eq. (2)] agrees with the prediction for λc based on energy minimization [9]. This model predicts buckling in a higher mode than the fun- damental one as in classic Euler buckling. In agreement with experiments, this model implies that the orientation of MT bundles in a striped sample varies continu- ously in space [9]. In contrast, previous models had suggested that discrete and alternate angular orientations of the MTs formed the striated patterns [15]. In addition, the weak depen- dence of the buckling wavelength on the mechanical parame- ters is consistent with the small variations in both the observed buckling wavelength across a single macroscopic sample and the patterns formed under different conditions (for example, samples with different tubulin concentrations and samples in containers with different size.). Time lapse phase contrast microscopy reveals that the MT bundles elongate uniformly along their contour during buck- ling, which is consistent with polymerization occurring uni- formly along the bundles. The elongation is illustrated in the phase images Fig. 2(a) and Fig. 2(b), showing a fixed region taken 12 and 100 minutes after polymerization initiation, re- spectively. The three white curves in each image are computer generated traces of bundle contours that extend between se- lected fiducial marks. The fiducial marks are visible as dark spots in the images. To generate the white curves, we pre- sumed that the bundles followed the striations in the images and traced the stripes between the fiducial marks, whose po- sitions were tracked using the MetaMorph imaging software (Universal Imaging, West Chester, PA). Specifically, we de- termined the local striation orientation at each pixel by calcu- lating a Fast Fourier Transform (FFT) of the area around the pixel, shown, for example, in Fig. 2(c). The FFT appeared as an elongated spot oriented perpendicular to the striation di- rection [Fig. 2(d)]. The radially integrated FFT intensity has a peak at a specific azimuthal angle [Fig. 2(d)] that is perpen- dicular to the striation orientation. In this way, the lengths of three segments along a MT bundle were recorded every 30 seconds and plotted in Fig. 2(f), (g) and (h). The normal- ized lengths of these three segments grew at nearly the same, constant rate, shown in Fig. 2(i), implying that the MT bun- dles elongate uniformly along their contour instead of growing FIG. 2: Illustration and measurements of the uniform elongation of MT bundles [23]. (a,b) Phase contrast images of a sample region, show progression of the pattern over one hour. MT bundles are dis- cerned by the thin striations. The image contrast is enhanced for bet- ter visualization. Segments 1 through 3 are adjacent pieces of a con- tour followed by bundles. The segment ends are defined by fiducial marks. (c) Magnified view of the region denoted by the white box in (b), showing an encircled fiducial mark. (d) Fast Fourier Transform (FFT) of (c). (e) The radially averaged FFT intensity plotted versus the azimuthal angle θ and fit using a Gaussian function. The local bundle orientation is orthogonal to the angle at which the Gaussian fit peaks. (f-h) Length of segments 1 (f), 2 (g) and 3 (h) as a func- tion of time. (i) Lengths of the three segments as functions of time, normalized to their lengths at 46 minutes. solely at their ends. It further suggests that the bundles elon- gate through polymerization of their constituent MTs, which start and end at random places along a bundle. The uni- form growth of all MTs within the bundle justifies a uniform elongation rate and the use of a uniform compressional force throughout the bundle in the mechanical model, giving rise to the sinusoidal ξ (x) over the entire pattern. Additional quantitative information about the microscopic picture of the buckling is gained through time-lapse bire- fringence measurements. PolScope (CRI, Cambridge, MA) images, taken sequentially at a fixed sample region [16], yielded the time evolution at each pixel of both the retar- dance (∆ ≡ bire f ringence× h, where h is the sample thick- ness) and the slow axis direction (ϕ(x), orientation of MT bundles) [17]. Two representative PolScope images of a sin- gle region taken at different stages of self-organization are shown in Fig. 3(a) and 3(b). The slow axis variation, ϕ(x), along the white lines in Fig. 3(a) and 3(b) can be fit to ϕ(x) = atan[A 2π cos( 2π (x + x0))], indicating that the bundle follows ξ (x) = Asin( 2π (x+ x0)) with a single wavelength λ , buckling amplitude A, and offset x0 [Fig. 3(c)]. The resultant wavelength, λ ≈ 600 µm, is plotted in Fig. 3(d). The nor- FIG. 3: Time evolution of a MT pattern obtained by measuring the retardance and slow axis of the sample using a PolScope imaging system [16]. (a,b) Retardance images of a sample region at 12 and 100 min of self-organization, respectively. The gray bar shows the retardance magnitude scale and the white pins provide the slow axis orientation. The straight white lines represent the slow axis line scan position. (c) Slow axis line scan (black) and the fitted slow axis ori- entation ϕ(x) = atan[A 2π cos( 2π (x+x0))] (gray) at 100 min. (d) The dominant buckling wavelength λ , obtained from the fitted shapes of the bundle at individual time points. (e) The length evolution of the fitted bundle contour. L0 = 1544 µm is the initial unbuckled length of the bundle. The segment before the arrow designates a latent period prior to the onset of the buckling. (f) The magnitude of the retardance averaged over the white lines as shown in (a,b) versus the normalized length L/L0. malized contour length calculated from the fits, L(t)/L0, grew nearly linearly with time at a normalized rate of L̇(t)/L0≈ 1 % per min [Fig. 3(e)]. Simultaneously, the retardance magnitude averaged over the white line in Fig. 3(a) increased roughly in proportion to L(t)/L0 [Fig. 3(f)]. Based on the nesting model we proposed earlier and assuming that neighboring MT bundles do not coalesce, the average retardance goes as ∆(t) ∼ δ × n(t)L(t)/L0 [9, 17], where n(t) is the number of MTs in the cross section of a bundle and δ is the retardance of a single MT. Therefore, the linear relation between ∆(t) and L(t)/L0 implies that n(t) remains constant throughout buck- ling. Thus, the elongation of MT bundles occurs through the polymerization of MTs within the bundles and does not in- volve the incorporation of new MTs to existing bundles. With the above observations and model, we can quanti- tatively characterize the elastic properties of the bundle (K) and network (α). We begin with the implications of the mea- sured wavelength λ . In order to predict λ from the mechani- cal buckling model, we need to estimate K and F [Eq. (2)]. Two limits exist for K. If tight packing (solid model) of the MTs inside the bundle is assumed, then Ksolid = n2KMT, where KMT ≈ 3.4× 10−23 N ·m2 is the bending rigidity of a single MT [18, 19]. If MTs slide freely inside the bun- dle, then Kslip = nKMT. We employ the measured force- velocity relation, f (v) = C1 ln[C2/(v + C3)] (C1 = 1.89 pN, C2 = 1.13 µm/min and C3 = −0.08 µm/min [18]), for a sin- gle MT and presume F = n f (v), where v is the average elongation rate of individual MT inside the bundle. Writ- ing the average length of MTs inside the bundle as lMT, the elongation rate of a single MT is then approximately v(lMT) = lMT × L̇(t)/L0. Using the models for K, F and Eq. (2), we derive predictions of λ for both the solid model, λsolid = π 8nKMT/ f (v(lMT)), and the slip model, λslip = 8KMT/ f (v(lMT)). Each depends on lMT and n. Using n = 280 [9], we plot the wavelength over a reasonable range of individual MT lengths ([1]) in Fig. 4. The solid model for K appears much more reasonable than the slip model. The fact that K depends quadratically on n in our system suggests that MTs are fully coupled (acting like a solid material) inside the bundle, similar to the behavior of F-actin bundles held to- gether through depletion forces [20]. The bundling of initially aligned MTs can be attributed to the depletion force induced by unpolymerized tubulin dimers, oligomers and even short MTs [9]. The conclusion that the bundles bend as solid rods appar- ently conflicts with the picture of elongation, that involves the growth and relative sliding of individual MTs within the bun- dles. We speculate that the explanation involves two distinct time scales: the time for a MT to come to mechanical equi- librium with its neighbors following the insertion of a tubulin dimer to its end, τmech, and the average interval between in- sertions, τdimer. In the limit τmech < τdimer, strong coupling be- tween the MTs in the bundle can occur leading to the solid rod result. The opposite limit intuitively leads to weak coupling between the MTs within a bundle. We estimate τdimer ≈ 0.1s from our data, which seems quite long compared to the times characterizing the relative motion of neighboring MTs on the molecular length scales relevant to τmech. The exact molecu- lar picture, which goes beyond the scope of our model, needs further study. Using the solid model for K, we can calculate the remain- ing model parameter, α , from Eq. (2): α = Kslip(2π/λexpt)4 ≈ 0.032Pa. This value is remarkably small compared to that estimated for a single MT buckling inside a cell (α∗ ≈ 2700Pa [12]). We identify two contributors to the difference between α and α∗. In general, α ∼ G, where G is the elas- tic shear modulus of the surrounding network. G ∼ 1Pa in our system [21], while G∗ ∼ 1000Pa for the surrounding cy- toskeleton network inside the cell [22]. The other contributor is the coordination of the buckling of the MT bundles, which reduces the distortion of the surrounding network, and thus weakens the effective restoring force and α (analysis in prepa- FIG. 4: Theoretically calculated wavelength (λ ) as a function of the average length of MTs (lMT) inside the bundle at the onset of buck- ling. In the solid model λsolid = π 8nKMT/ f (v(lMT)), and in the slip model λslip = π 8KMT/ f (v(lMT))). λexpt is the experimentally observed buckling wavelength (dashed line). ration). In summary, using microscopic studies of the temporal evo- lution of the striated MT patterns, we show that the polymer- ization of MTs within the bundles causes uniform elongation. This in turn creates the driving compressional force which ultimately causes the MT bundles to buckle. It is this coor- dinated buckling that produces the striped birefringent pat- tern. The proposed mechanical buckling model adequately describes the buckling process. It predicts a critical buckling force and a characteristic wavelength, which depend on the elasticity of the surrounding network and the bending rigid- ity of the MT bundles. Combing the bending rigidity of MT bundles and the established MT force-velocity curve with the mechanical model, we obtain a reasonable estimate for the elastic constant of the network and find that MTs inside the bundle are fully coupled. We thank Allan Bower for help in understanding the elas- tic constant α and thank L. Mahadevan and Thomas R. Powers for valuable discussions. This work was supported by NASA (NNA04CC57G, NAG3-2882) and NSF (DMR 0405156, DMR 0605797). [1] A. Desai and T. J. Mitchison, Annu. Rev. Cell Dev. Biol. 13, 83 (1997). [2] D. Bray, Cell Movement: From Molecules to Motility (Garland, New York, 2001). [3] R. P. Elinson and B. Rowning, Dev. Biol. 128, 185 (1988). [4] G. Callaini, Development 107, 35 (1989). [5] A. L. Hitt, A. R. Cross, and R. C. Williams, J. Biol. Chem. 265, 1639 (1990). [6] J. Tabony, Science 264, 245 (1994). [7] F. J. Nédélec, T. Surrey, A. C. Maggs, and S. Leibler, Nature 389, 305 (1997). [8] C. E. Walczak, I. Vernos, T. J. Mitchison, E. Karsenti, and R. Heald, Current Biology 8, 903 (1998). [9] Y. Liu, Y. Guo, J. M. Valles, and J. Tang, Proc. Natl. Acad. Sci. U.S.A. 103, 10654 (2006). [10] W. Bras, G. P. Diakun, J. F. Dı́az, G. Maret, H. Kramer, J. Bor- das, and F. J. Medrano, Biophys. J. 74 (1998). [11] N. Glade and J. Tabony, Biophys. Chem. 115, 29 (2005). [12] C. P. Brangwynne, F. C. MacKintosh, S. Kumar, N. A. Geisse, J. Talbot, L. Mahadevan, K. K. Parker, D. E. Ingber, and D. A. Weitz, J. Cell Biol. 173, 733 (2006). [13] J. R. Gladden, N. Z. Handzy, A. Belmonte, and E. Villermaux, Phys. Rev. Lett. 94, 035503 (2005). [14] L. D. Landau and E. M. Lifshitz, Theory of Elasticity (Oxford, New York, 1986), 3rd ed. [15] J. Tabony and N. Glade, Langmuir 18, 7196 (2002); C. Papa- seit, L. Vuillard, and J. Tabony, Biophys. Chem. 79, 33 (1999); J. Tuszynski, M. V. Sataric, and et al., Physics Letters A 340, 175 (2005). [16] The sample was polymerized from a 5 mg/ml tubulin solution (2 mM GTP, 3.5% in molar ratio of Oregon Green conjugated taxol to tubulin dimers, 100 mM pipes, 1 mM EGTA, 2 mM MgSO4, PH 6.9) in a 40×8×0.4mm3 glass cuvette which was exposed to 9 T vertical static magnetic field for 5 minutes at 37◦C (the magnetic field direction is along the long axis of the cuvette). The cuvette was then laid flat on the microscope stage and a coherently buckled area was chosen for observation and measurement at 30◦C. [17] R. Oldenbourg, E. D. Salmon, and P. T. Tran, Biophys. J. 74, 645 (1998). [18] M. Dogterom and B. Yurke, Science 278, 856 (1997). [19] J. A. Tuszyński, T. Luchko, S. Portet, and J. M. Dixon, Eur. Phys. J. E 17, 29 (2005). [20] M. M. A. E. Claessens, M. Bathe, E. Frey, and A. R. Bausch, Nature Materials 5, 748 (2006). [21] M. Sato, W. H. Schwartz, S. C. Selden, and T. D. Pollard, J. Cell Biol. 106, 1205 (1988). [22] R. E. Mahaffy, C. K. Shih, F. C. MacKintosh, and J. Käs, Phys. Rev. Lett. 85, 880 (2000). [23] The sample was polymerized from 8 mg/ml tubulin solution (same buffer condition as in [16]) in a 40×10×1mm3 quartz cuvette and was subjected to convective flow (induced by asym- metrical thermal contacts, with the left and bottom surfaces in contact with a 37◦C waterbath-warmed aluminum holder and other sides exposed to 30◦C ambient) for the first 9 minutes. The cuvette was then laid flat on the microscope stage for ob- servation and measurement at 30◦C. References

0704.0306

Neutron Inelastic Scattering Processes as Background for Double-Beta Decay Experiments

Neutron Inelastic Scattering Processes as Background for Double-Beta Decay Experiments D.-M. Mei,1, 2, ∗ S.R. Elliott,1 A. Hime,1 V. Gehman,1, 3 and K. Kazkaz3, † Los Alamos National Laboratory, Los Alamos, NM 87545 The Department of Earth Science and Physics, University of South Dakota, Vermillion, South Dakota 57069 Center for Experimental Nuclear Physics and Astrophysics, and Department of Physics, University of Washington, Seattle, WA 98195 (Dated: October 29, 2018) We investigate several Pb(n, n′γ) and Ge(n, n′γ) reactions. We measure γ-ray production from Pb(n, n′γ) reactions that can be a significant background for double-beta decay experiments which use lead as a massive inner shield. Particularly worrisome for Ge-based double-beta decay experi- ments are the 2041-keV and 3062-keV γ rays produced via Pb(n, n′γ). The former is very close to the 76Ge double-beta decay endpoint energy and the latter has a double escape peak energy near the endpoint. We discuss the implications of these γ rays on past and future double-beta decay experiments and estimate the cross section to excite the level that produces the 3062-keV γ ray. Excitation γ-ray lines from Ge(n, n′γ) reactions are also observed. We consider the contribution of such backgrounds and their impact on the sensitivity of next-generation searches for neutrinoless double-beta decay using enriched germanium detectors. PACS numbers: 23.40.-s, 25.40.Fq, 29.40.Wk I. INTRODUCTION Neutrinoless double-beta decay plays a key role in understanding the neutrino’s absolute mass scale and particle-antiparticle nature [1, 2, 3, 4]. If this nuclear de- cay process exists, one would observe a mono-energetic line originating from a material containing an isotope subject to this decay mode. One such isotope that may undergo this decay is 76Ge. Germanium-diode detectors fabricated from material enriched in 76Ge have estab- lished the best half-life limits and the most restrictive constraints on the effective Majorana mass for the neu- trino [5, 6]. One analysis [7] of the data in Ref. [6] claims evidence for the decay with a half-life of 1.2 × 1025 y. Planned Ge-based double beta decay experiments [8, 9] will test this claim. Eventually, these future experiments target a sensitivity of > 1027 y or ∼ 1 event/ton-year to explore mass values near that indicated by the atmo- spheric neutrino oscillation results. The key to these experiments lies in the ability to re- duce intrinsic radioactive background to unprecedented levels and to adequately shield the detectors from ex- ternal sources of radioactivity. Previous experiments’ limiting backgrounds have been trace levels of natural decay chain isotopes within the detector and shielding components. The γ-ray emissions from these isotopes can deposit energy in the Ge detectors producing a con- tinuum, which may overwhelm the potential neutrino- ∗Permanent Address: The Department of Earth Science and Physics, University of South Dakota, Vermillion, South Dakota 57069 †Permanent Address: Lawrence Livermore National Laboratory, Livermore CA 94550 less double-beta-decay signal peak at 2039 keV. Great progress has been made identifying the location and ori- gin of this contamination, and future efforts will substan- tially reduce this contribution to the background. The background level goal of 1 event/ton-year, however, is an ambitious factor of ≈ 400 improvement over the currently best achieved background level [6]. If the efforts to re- duce the natural decay chain isotopes are successful, pre- viously unimportant components of the background must be understood and eliminated. The potential for neutron reactions to be one of these background components is the focus of this paper. The work of Mei and Hime[10] recognized that (n, n′γ) reactions will become important for ton-scale double-beta decay experiments. Specifically, we have studied neutron reactions in Pb and Ge, materi- als that play important roles in the Majorana [8] design. But since lead is used by numerous low-background ex- periments, the results will have wider utility. This paper presents measurements and simulations of Pb(n, n′γ) and Ge(n, n′γ) reactions and estimates the re- sulting background for Ge-detector based, double-beta decay experiments for a given neutron flux. With these results, we then use the neutron flux, energy spectrum, angular distribution, multiplicity and lateral distribu- tions determined in [10] to estimate the background in Ge detectors situated in underground laboratories. In Section II we describe the experiments, the data, and the simulations. In Sections III and IV we describe the analy- sis of these data. Section IV also discusses the important Pb(n, n′γ) production of γ rays at 2041 and 3062 keV. The former is dangerously near the 2039-keV Q-value for zero-neutrino double-beta decay in 76Ge and the latter can produce a double-escape peak line at 2040 keV. These dangerous processes for Ge-based double-beta decay ex- periments are discussed for the first time in this work. Section V determines an overall background model for http://arxiv.org/abs/0704.0306v4 our detector and the implications of this model for fu- ture experimental designs. It also considers the relevant merits of Cu versus Pb as shielding materials, and the use of depth to mitigate these backgrounds is discussed. We also consider the possibility that the double-escape peak of the 3062-keV γ ray could contribute to the signal claimed in Ref. [7]. Finally, we summarize our conclu- sions in Section VI. II. THE MEASUREMENTS We collected five data sets to explore the implications of (n, n′γ) for double-beta decay experiments. All mea- surements were done in our basement laboratory at Los Alamos National Laboratory. The laboratory building is at an atmospheric depth of 792 g/cm2 and provides about 1 mwe concrete (77 g/cm2) overburden against cosmic ray muons. Three data sets were taken with a CLOVER detec- tor [11]. This detector is a set of 4 n-type, segmented germanium detectors. The four crystals have a total nat- ural germanium mass of 3 kg and each crystal is seg- mented in half. The CLOVER detector and its opera- tion in our laboratory were described in Ref. [12]. The remaining two measurements were done with a PopTop detector [13] set up in coincidence with a NaI detector. The PopTop is a 71.8-mm long by 64-mm diameter p- type Ge detector. Taking into account the central bore, the detector is 215 cm3 or 1.14 kg. The NaI crystal is 15.25-cm long by 15.25-cm diameter and is directly con- nected to a photo-tube. All data were read out using a pair of X Ray Instrumentation Associates (XIA) [14] Digital Gamma Finder Four Channel (DGF4C) CAMAC modules. The CAMAC crate is connected to the PCI bus of a Dell Optiplex computer running Windows 2000. The system was controlled using the standard software sup- plied by XIA. This data acquisition software runs in the IGOR Pro environment [15] and produces binary data files that were read in and analyzed using the ROOT framework[16]. The data sets include: 1. A background run with the CLOVER 2. A Th-wire source run with the CLOVER 3. An AmBe source run with the CLOVER using two different geometries of moderator 4. An AmBe source run with the PopTop surrounded by lead 5. An AmBe source run with the PopTop surrounded by copper In this section we describe the experiments and the data collected. A. The Experimental Configurations The CLOVER was surrounded by 10 cm of lead shield- ing to reduce the signal from ambient radioactivity. Un- derneath and above the lead was 5 cm of 30%-loaded borated polyethylene to reduce thermal neutrons. The background run done in this configuration lasted 27.13 live-days. The configuration for the Th-source run was similar, but with some lead removed to expose the detec- tor to the source. The Th source run had a live time of 1337 seconds. The setup was modified somewhat from this background-run configuration for the measurements with the AmBe source. Fig. 1 shows the configuration for one of the AmBe measurements. For these data, the CLOVER was shielded on four sides with 10 cm of lead. The AmBe source, 30 mCi of 241Am with a calibrated neutron yield of ≈ 63,000 Hz (±0.7%), was on one side of the CLOVER with 5 cm of lead and a layer of pure polyethylene moderator (either 10 or 15 cm thick) be- tween the source and detector. The data acquisition sys- tem is inactive during data transfer. Only the AmBe runs had a large enough event rate for the dead time to be appreciable. A 6.13-h live-time data run (57% live) was taken with 15 cm of moderator, and another 3.57-h live-time data (38% live) run was taken with 10 cm mod- erator (pictured). For the analysis presented below, the data from these two configurations were combined, and thus the AmBe-CLOVER data set contains 9.7 h of live time. The observed energy spectrum extended from ≈ 10-3100 keV for these data sets. During the analysis of the AmBe data, we observed a weak line at 3062 keV. This energy corresponds to a γ-ray transition in 207Pb, and we therefore hypothesized that it was generated via Pb(n, n′γ). The double-escape- peak (DEP) energy (2040 keV) associated with this γ ray is very dangerous for 76Ge neutrinoless double-beta de- cay experiments because it falls so close to the transition energy (2039 keV). Furthermore because the DEP is a single-site energy deposition, it cannot be distinguished from double-beta decay through event topology. This is in contrast to a full-energy γ-ray peak, which tends to consist of several interactions and therefore is a multiple- site deposition. (See [12] for a discussion of the use of event topology to reduce background in Ge detectors.) The final two measurements were intended to study this 3062-keV line in the spectrum and demonstrate its origin. In both cases a PopTop Ge detector faced a 15.25 cm by 15.25 cm NaI detector for coincidence data. By se- quentially placing a Pb and then a Cu absorber between an AmBe source and a PopTop Ge detector, we tested the hypothesis that the line was due to neutron interac- tions in Pb. By looking for an coincident energy deposit in the NaI detector, we could be assured the Ge detector signal originated from a neutron interaction in the sam- ple. An energy deposit threshold in the NaI of greater than 200 keV was required for a coincidence. The Pop- Top was placed 27.3 cm from the NaI detector with the source placed 20.3 cm (7 cm) from the Ge (NaI) detector. For the lead study, 5 cm of lead was placed directly be- tween the Ge detector and the source. Additional lead, in the form of 5-cm-thick bricks was positioned around the 4 sides of the Ge detector to reduce room background. For the copper study, a 0.5-cm thick Cu tube was placed around the PopTop and a 5-cm Cu block was placed be- tween the PopTop and the source. For this final run, all the lead was removed. For both of these sets of data, the observed spectra extended from ≈ 125 keV to ≈ 9 MeV. For the PopTop data, the Pb and Cu runs were of 19.12 h and 17.76 h live-time, respectively. FIG. 1: The CLOVER detector as configured for the AmBe source run. The setup at the time of this photograph used 4′′ of polyethylene. One wall of the lead shield was removed only to clarify the relationship between the AmBe source, moderator, and the CLOVER. B. The Data sets The crystals were individually calibrated, and the re- sulting spectra summed together to form a single his- togram. The peaks within each of the 3 CLOVER data sets were identified and their intensities determined. If an event had 2 crystals that responded in coincidence, the histogram would have two entries. Therefore the spectra we analyzed and simulated included all single-crystal en- ergy deposits. By not eliminating events that registered signals in more than 1 of the CLOVER Ge detectors, we maximized the event rate. The peak strengths were estimated by fitting a Gaussian shape to peaks and a flat background to the spectrum in the region near the peak. For the nuclear recoil lines, the peak shape was assumed to be a triangle and not Gaussian. In Table I the uncertainties derive from this fit. A summary of the peak strengths is given in Table I and the spectra them- selves are shown in Fig. 2. The data sets were chosen to help decouple line blendings. Because the rates in all peaks and continua are much higher for the source- induced data than for the background, features in those spectra are due to the sources and other contributions can be safely ignored. For example, the 2614.5-keV line can arise from either the decay of 208Tl or 208Pb(n, n′γ). When exposed to a Th source, Tl decay dominates the spectrum, whereas when exposed to an AmBe source, (n, n′γ) dominates. Hence by normalizing the rate in this line to the rate in a pure neutron-induced transition (e.g. the 596-keV 74Ge(n,n’γ)), we can determine the relative contribution of the two processes to the back- ground spectrum. In fact, in the background data, both processes contribute to this line. Some comments on our choices for line identification are in order. For an isotope such as 72Ge where a neutron capture leads to a stable nucleus, almost all (n,γ) lines could also be interpreted as (n, n′γ) lines in the result- ing nucleus; in this case 73Ge. For isotopes within the detector however, such as the 53.5-keV 72Ge(n,γ) tran- sition, the competing 73Ge(n, n′γ) line would be a sum of this γ-ray energy and the recoil nucleus energy. At these low energies where the recoil is a fair fraction of the γ-ray energy, the (n, n′γ) would simply contribute to the continuum and not be observed as a line. For the high energy cases, the blend of a mono-energetic γ-ray line and a (n, n′γ) process might be present. For the calibration runs, our threshold was approxi- mately 70 keV. Also note that we used a thorium wire as a calibration source. Since the wire is pure natural Th, we observe the Th X rays in that data. In contrast, the background run shows lines from the thorium chain as a contaminant, therefore those lines are absent. In all spectra, there are a few lines we have not iden- tified. TABLE I: A summary of the observed lines in the various spectra taken with the CLOVER detector. Blank entries indicate that no significant peak feature above the continuum was found. Single and double escape peaks are labeled by SEP and DEP respectively. Line assignments for which we are unsure are indicated by a question mark. Line energies are taken from the Table of Isotopes [17]. Energy Process Count Rates (keV) backgrnd Thorium CLOVER (per hr) (Hz) AmBe (Hz) 23.4 70Ge(n,γ) 1.017(5) 46.5 210Pb 112.75(42) 72Ge(n,γ) 61.01(31) 2.079(8) 63.2 234Th 93.02(38) 67.7 230Th 24.05(19) 0.591(4) 68.8 72Ge(n,γ) 0.440(4) 72.80 Pb x-ray 10.5(1) 0.506(4) 74.97 Pb x-ray 663.5(1.0) 34.1(2) 1.703(7) 76.7 Unidentified 29.0(2) 228Th Pb x-ray Pb x-ray 115.28(42) 10.9(1) 0.930(5) Pb x-ray 55.65(29) 16.8(1) 0.262(3) Continued TABLE I – continued Energy Process Count Rates (keV) backgrnd Thorium CLOVER (per hr) (Hz) AmBe (Hz) 89.9 Th x-ray 32.4(2) 92.7 234Th 171.38(51) 0.060(1) 93.4 Th x-ray 47.5(2) 96.0 115In(n,γ) ? 0.166(2) 99.5 228Ac 13.71(15) 3.0(1) 105.3 Unidentified 8.99(12) 104.8 105.6 Th x-ray 20.6(1) 108.7 Th x-ray 7.6(1) 109.9 19F(n, n′γ) 43.00(26) 0.506(4) 129.1 228Ac 12.91(14) 3.2(1) 139.7 74Ge(n,γ) 47.20(27) 2.339(8) 143.9 230Th 20.03(18) 154.0 228Ac 7.69(11) 1.5(1) 159.7 77mGe 0.114(2) 162.4 115In(n,γ) 10.71(13) 1.073(6) 174.9 70Ge(n,γ) 7.45(11) 0.763(5) 186.1 186.2 226Ra 115In(n,γ) 114.60(42) 0.323(3) 197.1 198.4 19F(n, n′γ) 71Ge sum 81.04(35) 2.328(8) 199.2 228Ac 0.66(2) 202.6 115In(n,γ) 0.061(1) 209.5 228Ac 19.38(17) 10.9(1) 215.5 228Th 2.43(6) 0.92(3) 238.6 212Pb 295.77(67) 139.9(1) 0.105(2) 242.0 214Pb 57.49(30) 9.2(1) 247.1 70Ge(n,γ) 0.070(1) 253.7 74Ge(n,γ) 2.76(7) 0.410(3) 270.2 228Ac 21.39(18) 9.1(1) 273.0 115In(n,γ) 0.055(1) 277.4 208Tl 208Pb(n, n′γ) 12.98(14) 5.4(1) 0.086(2) 284.6 Unidentified 2.79(7) 288.1 212Bi ? 1.03(3) 295.2 214Pb 58.02(30) 297.2 298.7 72Ge(n,γ) 115In(n,γ) 0.068(1) 300.1 212Pb 18.59(17) 9.7(1) 306.2 70Ge(n,γ) 1.12(4) 0.046(1) 321.4 228Ac 0.67(2) 326.0 70,72Ge(n, n′γ) 0.487(4) 328.3 228Ac 10.02(12) 8.2(1) 332.9 228Ac 1.14(3) 335.5 115In(n,γ) 0.028(1) 338.7 228Ac 45.13(26) 31.5(2) 351.9 214Pb 95.11(38) 354.1 Unidentified 0.043(1) 385.1 115In(n,γ) 0.048(1) 391.3 70Ge(n,γ) 0.053(1) 409.8 228Ac 4.05(8) 4.3(1) 416.9 116mIn 2.21(6) 0.359(3) 438.9 Unidentified 2.54(6) 445.2 74Ge(n,γ) 0.037(1) 452.3 212Bi? 0.82(2) 463.3 228Ac 10.96(13) 9.2(1) 474.0 72Ge(n,γ) ? 2.54(6) Continued TABLE I – continued Energy Process Count Rates (keV) backgrnd Thorium CLOVER (per hr) (Hz) AmBe (Hz) 478.6 228Ac 0.39(2) 470-485 10B(n,α) 7Li∗(γ)7Li Doppler Broadened signf. 492.9 73Ge(n,γ) 0.123(2) 499.9 70Ge(n,γ) 0.453(4) 503.9 228Ac 0.34(2) 509.3 510.7 510.7 510.9 228Ac 208Tl 208Pb(n, n′γ) Annih.γ 171.93(51) 16.8(1) 3.409(10) 516.2 35Cl(n,γ) 0.160(2) 537.5 206Pb(n, n′γ) 5.12(9) 0.158(2) 562.9 563.0 228Ac 76Ge(n, n′γ) 12.83(14) 1.52(3) 0.244(3) 569.7 207Pb(n, n′γ) 14.17(15) 0.422(4) 572.3 228Ac 0.53(2) 574.7 74Ge(n, γ) 0.091(2) 583.1 208Tl 208Pb(n, n′γ) 71.64(33) 49.6(2) 0.256(3) 595.9 74Ge(n, n′γ) 73Ge(n,γ) 59.90(30) 1.869(7) 608.3 73Ge(n,γ) 0.333(3) 609.2 214Bi 60.11(30) 629.6 72Ge(n, n′γ) 0.078(2) 648.2 115In(n,γ) 0.025(1) 657.2 206Pb(n, n′γ) 0.047( 1) 662.0 137Cs 9.04(12) 663.8 206Pb(n, n′γ) 0.069(1) 669.0 70Ge(n, n′γ) 0.030(1) 692.4 72Ge(n, n′e−) 87.70(37) 2.406(8) 701.0 74Ge(n, n′γ) 0.082(2) 708.2 70Ge(n,γ) 0.176(2) 727.3 212Bi 15.72(16) 11.7(1) 747.7 70Ge(n,γ) 0.047(1) 755.3 228Ac 2.19(6) 1.52(3) 763.1 208Tl 0.85(3) 763.1 208Pb(n, n′γ)? 0.032(1) 766.6 768.4 224mPa 214Bi 4.85(9) 771.8 228Ac 2.02(6) 2.11(4) 782.0 228Ac 0.67(2) 785.5 212Bi 4.19(8) 1.57(3) 786.3 786.8 35Cl(n,γ) 208Pb(n, n′γ) 0.041(1) 788.4 788.7 35Cl(n,γ) 70Ge(n,γ) 0.064(1) 795.0 228Ac 7.92(11) 6.1(1) 798.0 208Pb(n, n′γ) 0.023(1) 803.1 206Pb(n, n′γ) 20.90(18) 0.850(5) 806.2 214Bi 3.03(7) 808.2 70Ge(n,γ) 0.048(1) 818.6 116mIn 0.064(1) 824.9 1.03(4) 830.4 228Ac 0.70(2) 834.1 72Ge(n, n′γ) 45.15(26) 0.290(3) Continued TABLE I – continued Energy Process Count Rates (keV) backgrnd Thorium CLOVER (per hr) (Hz) AmBe (Hz) 835.6 228Ac 2.30(4) 840.4 228Ac 1.19(3) 843.8 27Al(n, n′γ) 4.48(8) 0.112(2) 846.9 76Ge(n, n′γ) 0.062(1) 860.4 860.4 208Tl 208Pb(n, n′γ) 8.64(12) 6.0(1) 0.090(2) 865.0 Unidentified 0.094(2) 867.9 73Ge(n,γ) 4.25(8) 0.466(4) 881.0 206Pb(n, n′γ) 2.50(6) 0.151(2) 892.9 212Bi 0.42(2) 894.3 72Ge(n, n′γ) 0.029(1) 897.8 207Pb(n, n′γ) 6.28(10) 0.199(2) 904.1 228Ac 0.94(3) 911.2 228Ac 48.86(27) 37.1(2) 934.1 214Bi 1.99(6) 958.4 228Ac 0.37(2) 960.9 74Ge(n, n′γ) 0.095(2) 964.4 228Ac 11.22(13) 6.2(1) 968.8 228Ac 26.83(20) 21.6(1) 981.0 206,8Pb(n, n′γ) 0.035(1) 988.4 228Ac 1.95(5) 0.20(1) 993.7 995.1 74Ge(n, n′γ) 206Pb(n, n′γ) 0.026(1) 999.5 74Ge(n, n′γ) 0.034(1) 1001.5 224mPa 8.03(11) 1004.5 228Ac 0.17(1) 1014.5 27Al(n, n′γ) 7.46(11) 0.173(2) 1033.1 228Ac 0.20(1) 1040.1 1040.8 70Ge(n, n′γ) 16.89(16) 0.210(2) 1063.7 207Pb(n, n′γ) 8.47(11) 0.145(2) 1065.0 228Ac 0.47(2) 1078.8 212Bi 1.11(4) 0.62(2) 1093.9 208Tl sum 511+583 0.90(3) 1095.8 1096.9 207Pb(n, n′γ) 70Ge(n,γ) 116mIn 7.81(11) 0.464(4) 1101.3 74Ge(n, n′γ) 0.123(2) 1105.6 74Ge(n, n′γ) 0.019(1) 1110.4 228Ac sum 0.52(2) 1120.6 214Bi 11.70(13) 1122.5 228Ac sum 0.26(1) 208Pb(n, n′γ) 72Ge(n, n′γ) 0.022(1) 1131.6 73Ge(n,γ)? 1.34(5) 0.034(1) 1139.4 70Ge(n,γ) 1.34(5) 0.077(2) 1153.5 228Ac 0.15(1) 1155.2 214Bi 1.08(4) 1164.9 1166.0 35Cl(n, n′γ) 72Ge(n, n′γ) 0.49(3) 0.072(1) 1173.5 60Co 13.67(14) 1201.2 p(n,γ)d DEP 0.415(3) 1204.2 73Ge(n,γ) 74Ge(n, n′γ) 9.39(12) 0.163(2) 1226.7 74Ge(n, n′γ) 0.017(1) 1238.4 214Bi 5.22(9) 1246.9 228Ac 0.58(2) Continued TABLE I – continued Energy Process Count Rates (keV) backgrnd Thorium CLOVER (per hr) (Hz) AmBe (Hz) 1261.0 74Ge(n, n′γ) 0.019(1) 1281.0 214Bi 0.74(3) 1286-7 228Ac Blend 0.14(1) 1293.5 116mIn 4.09(8) 0.462(4) 1298.8 70Ge(n,γ) 0.087(2) 1332.5 74Ge(n, n′γ) 0.018(1) 1332.5 60Co 12.27(14) 1344.5 1345.9 1347.7 74Ge(n,γ) 206Pb(n, n′γ) 70Ge(n,γ) 0.52(3) 0.017(1) 1374.2 228Ac sum 964 + 409 911 + 463 0.28(1) 1378.0 214Bi 3.24(7) 1378.8 70Ge(n,γ) 0.065(1) 1393.8 206Pb(n, n′γ) 0.016(1) 1401.5 214Bi 0.58(3) 1408.6 214Bi 1.84(5) 1413.6 73Ge(n, n′γ) 0.018(1) 1431.1 228Ac 0.15(1) 1433.5 206Pb(n, n′γ) 0.020(1) 1436.9 208Pb(n, n′γ) 0.017(1) 1459.2 228Ac 0.80(2) 1461.0 40K 30.18(22) 0.066(1) 1463.9 72Ge(n, n′γ) 0.114(2) 1466.8 206Pb(n, n′γ) 0.032(1) 1471.6 73Ge(n,γ) 0.047(1) 1489.2 74Ge(n, n′γ) 0.025(1) 1496.2 228Ac 0.73(3) 0.85(3) 1501.7 228Ac 0.42(2) 1508.9 116mIn 0.069(1) 1508.9 214Bi 1.95(3) 1512.7 212Bi 0.38(2) 1538 214Bi 0.54(3) 1557.1 228Ac 0.14(1) 1580.8 228Ac 0.68(3) 0.55(2) 1588.3 228Ac 6.06(10) 2.94(5) 1592.5 1592.5 1593.0 208Tl DEP 208Pb DEP 207Pb(n, n′γ) 7.24(11) 2.10(4) 0.107(2) 1599.3 214Bi 0.38(2) 1601.1 1602.0 35Cl(n,γ) 74Ge(n, n′γ) 0.018(1) 1614.9 208Pb(n, n′γ)? 0.020(1) 1620.5 212Bi 1.81(5) 1.32(3) 1625.0 228Ac 0.34(2) 1630.7 228Ac 1.82(5) 1.46(3) 1631.5 1632.0 74Ge(n, n′γ) 70Ge(n,γ) 0.021(1) 1634.0 76Ge(n,γ) 0.016(1) 1638.3 228Ac 0.43(3) 0.41(2) 1640.4 208Pb(n, n′γ) 74,76Ge(n, n′γ) 0.026(1) 1661.3 214Bi 0.36(2) 1666.3 228Ac 0.17(1) 1699.5 206Pb(n, n′γ)? 0.021(1) 1704.5 206Pb(n, n′γ) 0.72(3) 0.041(1) Continued TABLE I – continued Energy Process Count Rates (keV) backgrnd Thorium CLOVER (per hr) (Hz) AmBe (Hz) 1710.9 72Ge(n, n′γ) 0.327(3) 1712.2 p(n,γ)d SEP 0.156(2) 1725.7 207Pb(n, n′γ) 0.029(1) 1729.6 214Bi 3.59(7) 1764.7 214Bi 11.68(13) 1779.0 27Al(n,γ) 28Al ⇒28 Si 2.13(6) 0.127(2) 1806.0 212Bi 0.11(1) 1844.5 206Pb(n, n′γ) 0.044(1) 1846.9 214Bi 2.26(6) 1940.4 74Ge(n, n′γ) 0.027(1) 1951.1 35Cl(n,γ) 0.032(1) 1959.3 35Cl(n,γ) 0.023(1) 2092.1 2092.7 206Pb(n, n′γ) 207Pb(n, n′γ) 0.039(1) 2103.8 208Tl SEP 208Pb SEP 5.21(9) 2.25(4) 0.090(2) 2112.1 116mIn 0.061(1) 2118.5 214Bi 0.64(3) 2204.0 214Bi 3.73(8) 2223.3 p(n,γ)d 5.813(13) 2390.5 116mIn 0.015(1) 2448.5 214Bi 0.51(3) 2614.5 208Tl 208Pb(n, n′γ) 39.39(25) 16.3(1) 0.729(5) 2650.3 206Pb(n, n′γ)? 0.011(1) 2686 sum 208Tl? 0.12(1) 2892 sum 208Tl 0.08(1) 3061.9 207Pb(n, n′γ) 0.010(1) C. Neutron Spectra Simulation Fast neutrons (from 100 MeV to 1 GeV or more) tend to produce additional neutrons through nuclear reactions as they traverse high-Z material. In particular the flux of neutrons will increase several-fold while the average neutron energy decreases through these processes. As a result, fast neutrons will penetrate deep into a shield pro- ducing additional neutrons at lower energies. These low energy neutrons (∼< 20 MeV) give rise to a substantial γ-ray flux because (n,n γ) cross sections are large near 10 MeV, but become small at higher energies. Hence it is these secondary lower-energy neutrons that inter- act with the shield and detector materials to produce γ rays, which can give rise to background in double-beta decay experiments. To understand the process by which high energy neutrons influence the low-energy neutron flux and, in turn, the observed γ-ray flux, we simulated neutrons impinging on an outer shield and tracked how their spectrum changed as the particles traversed the shield. We also simulated the production of neutron- induced γ rays and how the Ge detector responded to Energy (keV) 0 50 100 150 200 250 300 350 AmBe Spectrum Background Spectrum Energy (keV) 400 500 600 700 800 900 AmBe Spectrum Background Spectrum Energy (keV) 1000 1200 1400 1600 1800 2000 AmBe Spectrum Background Spectrum Energy (keV) 2000 2200 2400 2600 2800 3000 3200 AmBe Spectrum Background Spectrum FIG. 2: The AmBe and background spectra taken with the CLOVER. them. Specifically, we performed simulations of several geometries including: 1. A simulation of the cosmic-ray produced neutrons with energy up to 1 GeV at our lab in Los Alamos and their propagation through a 10-cm Pb shield. The response of the CLOVER detector to γ rays produced by neutron interactions in the shield was simulated. This simulation, compared to our back- ground data, tests the precision to which we can model neutron production, scattering with sec- ondary neutron production and (n,n’γ) interac- tions. 2. A simulation of the neutron flux induced on the CLOVER from the AmBe source (neutron energy ≤11.2 MeV) passing through 15 cm of polyethylene before impinging on the 10-cm Pb shield. Since the neutron flux is of low energy, this simulation tests the precision to which we model (n,n’γ) interac- tions. 3. A simulation of the neutron flux with neutron en- ergies up to a few GeV expected at 3200 mwe deep due to cosmic-ray µ interactions in the surround- ing rock and 30-cm lead shield and the resulting response of the CLOVER to the flux of γ rays aris- ing from this flux. This simulation permits us to estimate rates in detectors situated in underground laboratories. The first two of these simulations are to verify the code’s predictive power. The third is to aid in understanding the utility of depth to avoid neutron-induced backgrounds. The simulation package GEANT3-GCALOR [18, 19] is described in detail in Ref. [10]. In general, (n, n′) reactions leave the target nucleus in a highly excited state which subsequently decays via a γ-ray cascade to the ground state. In the simulation, inelastic scattering cross sections for excitation to a given level depends on the properties of the ground and excited states. These cross sections were calculated using in-house-written code based on Hauser-Feshbach [20] theory modified by Moldauer [21]. The validation of the Hauser-Feshbach theory has been the subject of several studies [22, 23, 24]. The simulated γ-ray flux arises from the relaxation of the initial excited-state distribution, which includes a large number of levels (60 states for 208Pb(n, n′γ) reactions, for example). The nuclear levels and their decay chan- nels were provided by the ENSDF[25] database through the GEANT package. Note however, that the simula- tion did not predict every possible transition. In particu- lar the important 2041-keV and 3062-keV emissions from Pb were not part of this simulation. This situation arises because the simulation packages only have (n, n′) cross sections for the lowest lying excited states for most nu- clei. It is set to zero for most other levels. The details of this simulation are described in detail in Ref. [10]. Here we study the effectiveness of the simulation to predict spectra resulting from (n,n’γ). The simulation was done by generating neutrons with the appropriate energy spectrum outside the lead shield and propagating them through the shield including sec- ondary interactions that may add to the neutron flux and alter the energy spectrum. Fig. 3 shows a comparison between the data and the simulations for the CLOVER background run and AmBe run. Note only neutrons as primary particles were simulated for this comparison and the dominant difference between the two spectra is due to the room’s natural radioactivity and non-neutron µ-induced processes. Here we excluded those processes from the simulation to emphasize the spectral shape, in- cluding lines, that are a direct result of neutron inter- actions. The similarity of the spectra in Fig. 3 indicates that the measured background spectrum is dominated by neutron-induced reactions. The uncertainty in the simulation is calculated by com- paring the well known peaks in Table II which shows a comparison of the simulation to the line production for both background and AmBe runs. The measured neutron produced lines are within about 5% of the pre- dicted values from simulation, as is the continuum rate in the AmBe data. Therefore the (n,n’γ) rates are well- simulated for nuclear states with well-defined cross sec- tions. (The continuum for the background data includes processes that were not simulated and hence is not a good measure of the uncertainty.) Because the neutron flux estimates come from these line strengths (See Sec- tion III), the uncertainty in the flux cancels in these es- timates. The uncertainty in the measured neutron flux and spectrum underground(≈35%) constrains the preci- sion to which such simulations can be verified and is well described in Ref. [10]. This 35% uncertainty due to the flux is much larger than the uncertainty for the γ-ray line production described above. Therefore, a total un- certainty of 35% is used for all predictions of line rates underground throughout this paper. III. THE NEUTRON FLUX In this section, we use the data to determine the neu- tron fluxes we observed during our various experimental configurations. We then compare our measured cosmic- ray induced flux with that predicted from past measure- ments and our simulation. A. Ge(n, n′γ) Analysis Spectral lines that indicate neutron interactions in nat- ural Ge detectors have been studied previously. See Ref- erences [26, 27, 28, 29], for example. In particular, the sawtooth-shaped peaks due to 72,74Ge(n, n′) at 693 keV and 596 keV respectively are clear indications of neu- trons and have been used to deduce neutron fluxes [30]. Operating Ge detectors in a low-background configura- tion, these lines can be used to help interpret the back- ground components. Recent double-beta decay experi- ments [5, 6] have constructed their detectors from Ge en- riched in isotope 76. Although an appreciable amount of 74Ge remained (14%), 70,72,73Ge are depleted. For such Energy (keV) 500 1000 1500 2000 2500 3000 Am-Be, MC Am-Be, Data Background, Data Background, MC Am-Be, MC Am-Be, Data Background, Data Background, MC Energy (keV) 500 550 600 650 700 750 800 Am-Be, MC Am-Be, Data Background, Data Background, MC Am-Be, MC Am-Be, Data Background, Data Background, MC FIG. 3: Comparison of the measured and simulated AmBe spectra for the CLOVER detector surrounded by 10 cm lead and 15 cm of moderator. The upper plot shows the energy range between 10 - 3100 keV. The lower plot shows the range 470 - 830 KeV where the most significant (n, n′γ) lines can be seen. The simulated AmBe neutron spectrum was nor- malized to the AmBe source strength for 6.13-h live-time. The measured total neutron flux in the background spectrum (see Section IIIC) was used to normalize the simulated back- ground spectrum. Note, only neutrons as primary particles were simulated for this comparison and the difference between the spectra is due to the room’s natural radioactivity and non- neutron, µ-induced processes. detectors, only lines originating in isotopes 74 and 76 are useful for neutron interaction analysis. As these ex- periments reach for lower background, neutron-induced backgrounds become a greater concern and the diagnos- tic tools more important. Neutrons from (α,n) and fission reactions have an en- ergy spectrum with an average energy similar to the AmBe spectrum used in this study. Furthermore, the average energy of the AmBe neutrons is similar to that of the neutrons within the hadronic cosmic-ray flux im- pinging on our surface laboratory although the latter ex- tend to much higher energies. Therefore the Ge-detector signatures indicating the presence of neutrons described above will be similar to those arising from neutrons origi- nating from the rock walls of an underground laboratory. However, low-background experiments that use Ge de- tectors are typically deep underground and are shielded TABLE II: A comparison of the simulated to measured rates (Background: per hour and AmBe: Hz) for several lines pro- duced by neutron interactions. The 2041-keV and 3062-keV lines are not included in the simulation. Process γ-ray Background-CLOVER Energy Simulation Measurement 74Ge(n, n′γ) 596 keV 56.21 59.90(30) 74Ge(n, n′γ) 254 keV 2.63 2.76(7) 76Ge(n, n′γ) 2023 keV 3.2×10−7 below sensitivity 206Pb(n, n′γ) 537 keV 4.82 5.12(9) 207Pb(n, n′γ) 898 keV 6.21 6.28(10) 206Pb(n, n′γ) 1706 keV 0.69 0.72(3) 206Pb(n, n′γ) 2041 keV none not seen Continuum region 2000-2100 keV 110.2 187.35(19) 207Pb(n, n′γ) 3062 keV none not seen Process γ-ray AmBe-CLOVER Energy Simulation Measurement 74Ge(n, n′γ) 596 keV 1.8 1.87 74Ge(n, n′γ) 254 keV 0.36 0.41 76Ge(n, n′γ) 2023 keV 8.5×10−4 below sensitivity 206Pb(n, n′γ) 537 keV 0.15 0.16 207Pb(n, n′γ) 898 keV 0.14 0.20 206Pb(n, n′γ) 1706 keV 0.04 0.04 206Pb(n, n′γ) 2041 keV none not seen Continuum region 2000-2100 keV 7.01 7.33 207Pb(n, n′γ) 3062 keV none 0.01 from environmental radioactivity by a thick shield. This shield, typically made of Pb, is then usually surrounded by a neutron moderator. This configuration is effective at greatly reducing the neutron flux originating from (α,n) and fission reactions in the cavity walls of the under- ground laboratory. In contrast, although neutrons origi- nating from µ interactions underground are much rarer, they have much higher energy. Therefore these µ-induced neutrons can penetrate the shield more readily and be- come a major fraction of the neutrons impinging on the detector. B. The AmBe Neutron Flux The estimate of the flux of neutrons with energies greater than 692 keV is given by [30, 31, 32] Φn = k , (1) where I is the counts s−1 under the asymmetric 692-keV peak, V is the volume of the detector in cm3 (566 cm3) and k is a parameter found by Ref. [30] to be 900 ± 150 cm. For the 15-cm moderator data, this formula predicts a neutron flux of 2.3/(cm2 s) whereas our simulation, using the known flux of the source, predicts 1.8/(cm2 s). This difference (20-30%) is somewhat greater than the 17% uncertainty claimed by Ref. [30]. The geometry for our measurement was complicated and perhaps this added complexity of neutron propagation contributes to the difference. For the uncertainty associated with the flux of neutrons produced from cosmic ray µ, we use the 35% value as it is much larger than the value associated with Eqn. 1. For the Am-Be neutron source, the rate in the 692-keV peak is 2.406 ± 0.008 Hz. This results in Φamben = 3.8 ± 1.1 /(cm2 s). This rate is an average over the two moderator configurations. The neutron flux during the 10-cm moderator run is estimated to be about a factor 2.3 larger than for the 15-cm moderator run. For the PopTop-AmBe run on Pb for the raw data (in coinci- dence with the NaI detector), the effective flux was 8.6 ± 2.6/(cm2 s) (0.26 ± 0.08 /(cm2 s)). C. Cosmic-ray Induced Neutron Fluxes In the background spectrum the rate in the 692-keV peak is 87.7 ± 0.4/hr. Using Eqn. (1) with I = (2.44 ± 0.06)×10−2 Hz for the background spectrum, one obtains a fast neutron flux of Φbackn = (3.9 ± 1.2)×10 −2 /(cm2 s) at the detector in our surface laboratory. Ref. [30] provides a similar formula to estimate the thermal neutron flux, which is accurate to approximately 30%. Using the intensity of the 139.68-keV γ-ray line of 75mGe: 980I139.68 139.68 + 1.6)V , (2) 139.68 ≃ 1− 1− e−V V 1/3 where I = 47.2 ± 0.3 /h = 0.013 Hz is the event rate in the peak of 139.68-keV line and V is the volume of the detector in cm3. Using V = 566 cm3 we obtain Φbackth = (9.1 ± 2.7)×10−3 /(cm2 s). We also measure the thermal neutron flux for the Am-Be neutron source, Φambeth = 1.6 ± 0.5 /(cm2 s). Thus the total neutron flux incident on the Ge detector measured for the background run is approximately Φbacktot = Φbackn + Φ th = (4.8 ± 0.7) × 10 −2 /(cm2 s). D. Neutron Flux as a Function of Depth In our basement laboratory, there are 3 primary sources of environmental neutrons. The largest contri- bution comes from the hadronic cosmic ray flux. The next largest arises from µ interactions in the 77 g/cm2 thick overhead concrete layer in the building. Finally there is the negligible contribution from (α,n) and fission neutrons from natural radioactivity in the room. The at- mospheric depth at the altitude of our laboratory is 792 g/cm2. Including the concrete, the depth is 869 g/cm2. Using the analysis of Ziegler [33, 34], the flux at our lab due to the hadronic flux can be estimated to be 3.0 times larger than that at sea level. The flux at sea level has been measured to be 1.22 × 10−2 /(cm2 s) [35] result- ing in a flux in our laboratory of 3.7 × 10−2 /(cm2 s). To estimate the additional neutron flux originating from µ interactions in the concrete above our laboratory, we rely on our simulations of neutron generation and prop- agation. The simulation predicts 1.4 × 10−2 /(cm2 s) (3.3 × 10−2 /(cm2 s)) for the muon-induced (hadronic) neutron flux inside the lead shield for a total simulated neutron flux of 4.7 × 10−2 /(cm2 s) in acceptable agree- ment with our measurement of (4.8 ± 2.2) × 10−2 /(cm2 s) = (1.51 ± 0.69) × 106 ± /(cm2 y). The success of this simulation lends credence to the neutron flux estimate in the following sections. The neutron flux onto the detector will be increased due to the neutron interactions with shield materials and neutron back-scattering from the cavity walls. For exam- ple, our simulations show that the fast neutron flux will increase by a factor of ≈ 10 by traversing a 30-cm lead layer. Also, neutrons will backscatter from the cavity walls and reflect back toward the experimental appara- tus, effectively increasing the impinging neutron flux by a factor of 2-3 depending on the specific geometries of the detector and experimental hall. Therefore, it is im- portant to account for these effects when estimating the neutron flux at the detector. Muon-induced neutron production in different shield- ing materials and in the detector itself was also stud- ied in Ref. [10]. For example, with 30 cm of lead sur- rounding a CLOVER-style detector at 3200 mwe, the to- tal muon-induced neutron flux impinging on the detector was calculated to be (8.6 ± 4.0) × 10−8 /(cm2 s) = 2.7 ± 1.2/(cm2 y). Some of the interactions resulting from these neutrons would be eliminated by a µ veto. Assum- ing a veto efficiency of 90% for muons traversing this lead shield, the effective neutron flux is estimated to be (2.0 ± 0.9) × 10−8 /(cm2 s) = 0.63 ± 0.29 /(cm2 y). The energy spectrum of neutrons at the lead/detector bound- ary at 3200 mwe is shown in Fig. 4 and has an average value of 45 MeV. The average energy of the µ-induced neutrons is 100- 200 MeV and much higher than that of (α,n) neutrons (≈ 5 MeV). The simulated flux of the µ-induced neutrons ((2 ± 0.9) × 10−8 /cm2 s = 0.63 ± 0.29/cm2 y) inside the detector shield at a depth of 3200 mwe is a factor 2.4 greater than the simulated (α,n) flux surviving the shield ((0.85 ± 0.39)× 10−8 /cm2 s = 0.27 ± 0.12/cm2 y). The average energy of these (α,n) originating neutrons is 3-5 MeV at the detector surface. With this estimate of the neutron flux at depth and with our measurements of the Pb and Ge neutron- induced detector response, we can proceed to estimate these processes in underground Ge-detector experiments. There are effects in addition to the incident flux, however, that must be taken into account when extrapolating our surface laboratory results to different geometries and lo- Energy (MeV) 10 1 10 : Lead/Detector boundary! ,n): Lead/Detector boundary!( FIG. 4: The effective neutron flux onto the simulated detector described in the text at a depth of 3200 mwe. Shown are the neutron flux from two sources: (1) the effective neutron flux induced by muons that transverse the surrounding rock and shielding materials assuming a 90% muon-veto efficiency and (2) the neutron flux from (α,n) reactions in the rock. cations. 1. As the thickness of the Pb shield increases, addi- tional secondary neutrons will be generated. Our simulation predicts that a factor kshield = 2.16 more neutrons will be produced by a 30-cm thick shield as compared to a 10-cm thick shield. 2. As the energy of the neutrons increases, the number of multiply scattered neutrons increases and there- fore the number of interactions that might produce a γ ray increases. For the average energy of neu- trons at our surface laboratory (at 3200 mwe), the average scattering length is λL=7.1 cm (λUG = 12.5 3. Also as the energy increases, the number of states that can be excited in the target nucleus increases. In the shield at our surface laboratory (at 3200 m.w.e), the average neutron energy is 6.5 MeV (45 MeV). All of these factors can be incorporated into a scaling formula derived from our simulation. The rate (RUGROI) of background near the region of interest (ROI) in an underground laboratory can be related to that measured in our surface laboratory (RLROI) as ROI = (1 + )kshield(( EUGn − Ex ELn − Ex )0.8) ROI , where ΦLn (Φ n ) is the neutron flux in our surface labo- ratory (at 3200 mwe), En is the neutron energy and Ex is the excitation energy for a typical level. This formula reproduces our simulated results well and the uncertainty of its use is dominated by the precision of simulation. Us- ing the 2.6-MeV level in 208Pb as an example, RUGROI ∼ 1.7 × 10−5RLROI . Fig. 5 shows a comparison between the Monte Carlo simulation and the scaling formula Eqn. 4 for several lines. -ray Energy (keV)γ 500 1000 1500 2000 2500 -610× Monte Carlo simulation Scaling formula FIG. 5: The comparison between the Monte Carlo simulation of a detector as described in the text and the scaling formula for several excitation lines. The 35% uncertainties shown in this figure arise from the cross section uncertainty and the statistical uncertainty of determining the peak counts in the simulated spectra. The latter dominates. IV. ANALYSIS A. Pb(n, n′γ) analysis If fast neutrons are present, then one will also see γ-ray lines from Pb(n, n′) interactions. In very low background configurations, γ rays from neutron-induced excitations in 208Pb and 207Pb can be masked by or confused for de- cays of 208Tl and 207Bi respectively. Therefore it is the stronger transitions in 206Pb (537.5, 1704.5 keV) that are most useful for determining if these processes are taking place. In 207Pb the relative strength of the 898-keV tran- sition, compared to the 570- and 1064-keV transitions, is much stronger when it originates from 207Pb(n,n’γ) as opposed to 207Bi β decay to 207Pb. Therefore this line can also be used as a tell-tale signature of neutron inter- actions. Our data show indications of 206,207,208Pb(n, n′γ). As noted earlier, the 2614-keV γ ray from 208Pb can orig- inate from 208Tl decay or from 208Pb(n,n’γ). The 692- keV peak arises only from neutron interactions on 72Ge. Since the Pb shielding was similar in both the background and AmBe runs, we can compare the ratio of the rate in the 2614-keV peak to that in the 72Ge(n, n′γ) 692-keV peak in the two data sets to deduce the fraction of the 2614-keV in the background run that can be attributed to neutron interactions. This ratio in the Am-Be spec- trum is 0.30 and that in the background spectrum is 0.45, and therefore, we conclude that ≈ 67% of the strength in the background run is due to neutron reactions and the remainder is due to 208Tl decay. Clearly, in our sur- face laboratory, environmental neutrons are a significant contributor to the observed signal. B. The Special Cases of the 2023-keV, 2041-keV and 3062-keV γ rays The 2023-keV level in 76Ge can be excited by neutrons. The simulation predicts that this line is too weak to be seen in the CLOVER AmBe data, but the CLOVER is built of natural Ge. In the enriched detectors planned by future double-beta decay experiments, the fraction of isotope 76 is much larger and this line would be enhanced. Still our simulation (Table V) predicts it would be a very small peak. The 3714-keV level in 206Pb can emit a 2041-keV γ ray. We only observed a candidate γ-ray peak in the coincidence data (Ge detector event in coincidence with a 4.4-MeV γ ray in the NaI detector) with the AmBe source radiating the Pb shield around the PopTop detec- tor. The magnitude of this peak if it exists is small and not convincing. We use this data set to place a limit on the production rate of this line as it results in the most conservative limit. In the AmBe-irradiated CLOVER and the non- coincidence PopTop spectra, we observed a 3062-keV γ ray that we assign to a transition from the 3633-keV level in 207Pb. (See Fig. 6.) This line is only present when Pb surrounds the detector: It is absent when Cu forms the shield. The statistical sensitivity was too weak in the PopTop coincidence spectrum to observe this weak line. In the CLOVER AmBe spectrum, the rate of this line is 5.3 × 10−3 that of 596-keV 74Ge peak rate and in the raw AmBe PopTop with Pb spectrum the ratio is 4.5 × 10−3. From these data we can estimate an approx- imate rate that these dangerous backgrounds would be produced for a given neutron flux. In our surface labora- tory, the CLOVER background rate for the 74Ge 596-keV peak was 59.9 events/hr. This leads to a predicted rate of 0.3 events/hr in the 3062-keV peak. Note that our data indicate that any peak at 3062 keV is statistically weak (≤ 0.2 events/hr) but reasonably consistent with this pre- diction. Note, in the AmBe-CLOVER runs, polyethylene blocks were used to increase the flux of thermal neutrons. It appears that these blocks contain some Cl and there- fore we see indications of Cl(n,γ) lines. Even though 35Cl has a neutron capture line at 3062-keV, we do not assign the observed line in our data to that process. Because the 35Cl(n,γ) line at 2863 keV is not observed and because the line at 1959 keV is weak, we conclude that assigning this line to Cl would be inconsistent with the predicted line ratios for neutron capture. However, a concern re- garding our assignment of the 3062-keV line to 207Pb is that one also expects a 2737-keV emission from the same 3633-keV level. This companion γ ray is not observed in our data and we plan future measurements dedicated to measuring the neutron-induced relative intensities of these two lines. If we assume that the entire rate (0.01 Hz) of the 3062-keV line is due to (n, n′γ), we can make a crude estimate of the cross section by scaling to the rate in the 2614-keV line. The cross section for the 2614-keV (n,n’γ) is 2.1 b ± 10% [23]. Using this cross section, the relative rates in the two peaks, the different isotopic ra- tios of 208Pb and 207Pb, and the different γ-ray detection efficiencies, the average cross section for 207Pb(4.5-MeV n, n′ 3062-keV γ ray) is estimated to be 75 mb. The uncertainty is estimated to be about 20% or ∼ 15 mb. Energy (keV) 3000 3020 3040 3060 3080 3100 AmBe activation, 6.1 h with 6" poly, 3.6 h with 4" poly FIG. 6: The energy spectrum near 3000 keV showing the 207Pb(n, n′γ) 3062-keV γ-ray line in the AmBe spectrum with the CLOVER. From measurements with the CLOVER and a 56Co source, which has γ-ray energies near 3100 keV, we expect 0.13 DEP events per full-energy γ-ray event. Therefore, in our surface lab, we expect 0.03 events/hr in the DEP at 2039 keV due to 207Pb(n, n′γ). This is well below our continuum rate of 2.5 events/keV-hr or 10/hr in an energy window corresponding to a 4-keV wide peak. Since our simulation does not predict all these lines, we summarize the measured rates normalized to the neutron flux in Table III to provide simple scaling to different ex- perimental configurations. The uncertainties in Table III are estimates based on a minor contribution of the sta- tistical uncertainty in the peak strengths and a major contribution resulting from the ≈ 35% uncertainty in the neutron flux determination as described in Section III B. Because the uncertainty is mostly systematic, there is a good possibility that the total uncertainties for each in- dividual measurement are correlated. Therefore, to esti- mate the average values in this Table, we took a straight average of the individual values and then assigned an un- certainty equal to the largest fractional value. This pro- cedure, although not rigorous, is more conservative than a weighted average. In addition, some peaks were not observed in all spectra. The upper limits on the strength of these peaks were estimated from the rates of weak- est peaks observed near the associated energy region in the spectrum. Such peaks are considered to represent the level of sensitivity of our peak detection procedures. The 2041-keV line is a special case. We quote an up- per limit based on the only spectrum that indicated a possible peak. These measurements were done for a CLOVER de- tector inside a 10-cm Pb shield. The relative energy- dependent efficiency (ǫrel) for a full-energy peak in the CLOVER can be approximated by, TABLE III: The raw count rates for select processes normalized to neutron flux of 1/(cm2y). When extrapolating to neutron energies distant from that near the measurements, the uncertainty (35%) associated with the extrapolation must be included. See text for a discussion of the uncertainty estimates in this table especially with respect to the average. Process Rate Rate Rate Rate AmBe-CLOVER AmBe-PopTop Background Average ( events 206Pb(n, n′537− keV γ) 13.9±4.9 20.5±7.2 12.2±4.3 15.5±5.4 74Ge(n, n′596− keV γ) 164±57 unresolveda 142±50 153±54 207Pb(n, n′898− keV γ) 17.5±6.1 21.2±7.4 14.9±5.2 17.9±6.3 206Pb(n, n′1705− keV γ) 3.6±1.3 3.2±1.1 1.7±0.6 2.8±1.0 206Pb(n, n′2041− keV γ) <6.3 < 0.5 <1.7 < 5.0b 207Pb(n, n′3062− keV γ) 0.88±0.31 0.6±0.21 <1.0 0.7±0.2 ( events (keV ty) Continuum Rate from Pb,Ge(n, n′γ) 2.6±0.9 2.0±0.7 2.5±0.9 2.4±0.8 aIn the PopTop data, the 596-keV line was not resolved from nearby lines. bThe 2041 line was not observed in any of our spectra, however, a weak peak-like feature was present in the AmBe-PopTop coinci- dence data. We used the upper limit for the rate in that peak as the ”average” as we considered this to be most conservative. ǫrel = 0.15 + 0.93e −(Eγ−148) 766 , (5) where Eγ is the γ-ray energy in keV. This expression is normalized to 1.0 at 209 keV and is estimated to have an accuracy of about 20% near 200 keV improving to about 10% at 2600 keV. The quoted relative efficiency for each of the 4 individual CLOVER detectors is 26% at 1.33 MeV as quoted by the manufacturer. Table III does not incorporate this efficiency correction, therefore the table presents the measured count rates with a minimum of assumptions. The thickness of Pb is large compared to the mean free path of the γ rays of interest, therefore, the scaling should hold for other thick-shield configurations. Even so, the rates will be geometry-dependent so these results can only be considered guides when applied to other experimental designs. The rate of these excitations also depends on neutron energy. For the background run (AmBe run) the average neutron energy is ≈ 6.5 MeV (≈ 5.5 MeV). Our simulations predict that the rate of these excitations scales as energy to the 0.81 power. V. DISCUSSION A. A Model of the CLOVER Background We can use these experimental results to create a back- ground model for our surface lab and deduce the contri- bution to the continuum near 2039 keV due to (n,n’γ) re- actions. We then use simulation of high-energy neutron production and propagation to extrapolate this model to better understand experiments done at depth. The measured rate for the continuum near 2039 keV was 14.8 events/(keV kg d). For the Th-wire data, this continuum rate was 0.10 events/(keV s) (2900 events/(keV kg d)) and for the AmBe data it was 0.09 events/(keV s) (2600 events/(keV kg d)). To determine the neutron-induced continuum rates in the AmBe data, however, we have to correct for the contribution from the tail of two high- energy γ rays that are not part of the neutron-induced spectrum in the background. These are the γ rays from the 2223-keV p(n, γ)d and the 4.4-MeV γ rays originat- ing from the (α,n) reaction of the AmBe source itself. Although only ≈ 10% of these AmBe γ rays penetrate the 5-cm Pb shield, there is still a significant flux. A simple simulation of the detector response to 2223- keV γ rays can easily determine ratio of the rate in the 2039-keV region to that in the full-energy peak. Simula- tion indicates that this ratio is 5.2 × 10−2 /keV. Since the full-energy peak count rate is 5.813 Hz, we find this contribution to the continuum is 0.03 events/(keV s). For the 4.4-MeV γ ray from the source itself, simulation must determine an absolute rate in the continuum because the high-energy threshold prevented the observation of the full-energy peak or its escape peaks. The simulation pre- dicts 0.03 events/(keV s). Subtracting these two contri- butions from the continuum rate for the AmBe source near 2039 keV results in a final value of 0.03 events/(keV s) or 860 events/(keV kg d). Our background measurements were done without a cosmic ray anti-coincidence system. From auxiliary mea- surements with a scintillator in coincidence with the CLOVER and a similar shielding geometry, we measured the rate of µ passing through the detector. In the contin- uum near the 2039-keV region, the rate is 5.4 events/(keV kg d). From the Th-wire source data, we measure the ratio TABLE IV: A summary of the count rate in the CLOVER background data in the energy region near 2039 keV based on the model deduced for the surface lab described in the text. The precision of the neutron-induced and muon-induced spectra simulations (Section IIC and Ref [10]) is estimated to be about 35%. We take this to be a conservative estimate for the uncertainties associated with this Table. Process CLOVER Event Rate Surface Lab events/(keV kg d) neutron-induced 8.3±2.9 208Tl Compton scattering 0.7±0.3 high energy µ continuum 5.4±1.9 Total from model 14.4±5.0 Measured Rate 14.8±0.2 of the rate in the continuum near 2039 keV to that in the 2614 keV (16.3 Hz) peak to be 6 × 10−3/keV. Of the 2614-keV peak rate in the background data, ≈ 33% is due to 208Tl decay. Scaling from the 2614-keV peak in the background data, the count rate near the 2039-keV region due to the Compton tail of the 208Tl 2614-keV peak is 0.7 events/(keV kg d). The remainder of the 2614-keV peak is due to neutron- induced processes. The contribution due to neutrons can be estimated from the AmBe data. For the AmBe data, the ratio of the rate in the continuum near the 2039- keV region (0.03 events(keV s)) to that for the 596-keV 74Ge(n, n′γ) peak (1.87 Hz) is 1.6 × 10−2/keV. Scaling from the 74Ge peak rate in the background data (59.9/h) indicates a rate of 7.8 events/(keV kg d) in the contin- uum near 2039 keV. That is, 53% of the events in that region are due to neutrons. One can do a similar scal- ing from the 692-keV 72Ge rates. Here the ratio is 1.3 × 10−2/keV and the continuum rate is 8.8 events/(keV kg d). We use the average of the Ge values as our estimate (8.3 events/(keV kg d) = 3030 events/(keV kg y)) for the neutron induced contribution to the continuum rate. Table IV summarizes the deduced contributions to the spectrum in the 2039-keV region in the CLOVER back- ground spectrum and the following section discusses how these data are used along with simulation to estimate rates in experimental apparatus underground. B. Solving the Problem with Overburden The primary purpose of this study is to better under- stand the impact of neutrons on the background for fu- ture double-beta decay experiments. In this subsection, we use neutron fluxes from our simulations of the surface laboratory, measurements with the AmBe source, and simulations of the neutron flux in an underground lab- oratory to estimate the contribution of neutron-induced backgrounds underground. In the following subsection, we examine data from previous underground experi- ments. The simulation of neutron processes in the 10-cm Pb shield and Ge comprising the CLOVER detector at the altitude of our laboratory predicts about 1594±558 events/(keV kg y) between 2000 and 2100 keV due to lead excitation and about 1337±468 events/(keV kg y) in this energy region due to germanium excitation. Our mea- sured value for the neutron-induced events is 3030±1061 events/(keV kg y) to be compared with this predicted value of 2931±1026 events/(keV kg y). The simulation of the CLOVER within a 30-cm lead shield at a depth of 3200 mwe, predicts about 0.019±0.007 events/(keV kg y) contributed from lead excitation and about 0.016±0.006 events/(keV kg y) contributed from germanium excitation for a total of 0.035±0.012 events/(keV kg y). One can also just scale our surface-laboratory measurement of the neutron- induced rate near 2039 keV by the factor derived from Eqn. 4 above. This results in 0.05±0.02 events/(keV kg y). For a detector like the CLOVER, analysis based on pulse shape discrimination (PSD), and the response of individual segments or crystals can help reduce back- ground based on its multiple-site energy deposit nature. These backgrounds can then be distinguished from the single-site energy deposit character of double-beta decay. We have measured the background reduction factor via these techniques to be ≈ 5.9 for the CLOVER [12]. Reference [10] provides a quick reference formula to estimate the neutron flux as a function of depth. The µ flux and its associated activity is reduced by ≈ 10 for each 1500 m.w.e of added depth. Future double-beta decay experiments hope to reach backgrounds near 0.25 events/(keV t y). Our estimate of the rate at 3200 mwe is 35-50 events/(keV t y), which is a factor of 150 above the goal. Hence, greater depths would be desirable. C. Discussion of Previous Underground Experiments Previous Ge-based double-beta decay experiments con- ducted deep underground [5, 6] set the standard for low levels of background. The future proposals however [8, 9] hope to build experiments with much lower backgrounds. In this subsection, we estimate how large the neutron contribution was to the previous efforts and future de- signs. Using the scaling summarized in Eqn. 4, we can compare the expectations of our simulated underground apparatus with previously published results. Table V shows this comparison. This table also presents a sum- mary of how the rates would be affected by a change in depth only. The IGEX collaboration [5] has not pub- lished its data in sufficient detail to do a similar com- parison. Other underground Ge detector experiments do not have the required sensitivity. The Heidelberg-Moscow experiment [6] is a critical case study for such backgrounds and it was operated at TABLE V: A summary of the key count rates arising from neutron interactions in the CLOVER background data in the energy region near 2039 keV as predicted by our analysis for three representative depths. The shield thickness is taken to be 30 cm and a veto system with an assumed efficiency of 90% is included. Except for the 2023-keV line, we used the scaling of Eqn. 4 to scale our CLOVER background measurements to the 3200 mwe depth and then used the muon fluxes at WIPP, Gran Sasso, and SNOLAB[10] to scale to the other depths. The Ge rates are also scaled for an enriched detector (86% isotope 76, 14% isotope 74). The scalings require the results from the simulations. The uncertainty is dominated by the simulated flux uncertainty and is estimated to be 35%. Since we did not observe the 2023-keV line, we used simulation to predict the rate. We used the measured upper limit for the 2041-keV line. For comparison, the results of Ref. [37] is shown. Reference [37] claims a result for zero-neutrino double-beta decay in an experiment performed at 3200 mwe. We entered the claimed event rate for that process in the same row as the 2041-keV line for comparison. The rate limits for the other lines assigned to Ref.[37] result from our estimates based on the figures in their papers and does not come directly from their papers. Process 1600 mwe 3200 mwe 6000 mwe Ref. [37] 74Ge 596 keV 19400±6800/(t y) 1130±400/(t y) 15±5/(t y) <800/(t y) 76Ge 2023 keV 30±10/(t y) 2±1/(t y) 0.02±0.01/(t y) 300/(t y) 206Pb 537 keV 4400±1500/(t y) 250±88/(t y) 3.4±1.2/(t y) 207Pb 898 keV 5300±1900/(t y) 310±110/(t y) 4.2±1.5/(t y) 206Pb 1705 keV 610±210/(t y) 36±13/(t y) 0.5±0.2/(t y) 206Pb 2041 keV <1300±450/(t y) <74±26/(t y) <1.0±0.3/(t y) 400/(t y) 207Pb 3062 keV 145±51/(t y) 8.4±2.9/(t y) 0.1±0.03/(t y) <71/(t y) continuum 880±310/(keV t y) 50±17/(keV t y) 0.7±0.24/(keV t y) 110/(keV t y) a depth of 3200 mwe. One is clearly led to consider if the 3062-keV γ ray can explain the signal reported in Ref. [36, 37]. Figure 36 in Ref. [37], shows that no more than a few counts can be assigned to a 3062-keV γ ray. If the 23 counts assigned to double-beta decay were actually a DEP from this γ ray, the one would expect 175 counts or so in the 3062-keV peak. Therefore, it is difficult to explain the claimed peak by this mechanism. It is also clear that the predicted rate of the 2041-keV γ ray is too low to explain their data. The data from Ref. [37] show lines at 570 and 1064 keV and the authors assigned these lines to 207Bi present in the Cu. However, the spectra dis- played in Fig. 13 of that paper shows that only detectors surrounded by Pb indicate the 570-keV line. Since there is no evidence for the 898-keV line in the data, we agree with the 207Bi assignment, however, we hypothesize that it must reside in the Pb and not the Cu. Perhaps this contamination is cosmogenically produced in Pb when it resides on the surface and not as a result of bomb testing as hypothesized by the authors[37]. Reference [37] also observed lines at 2011, 2017, 2022, and 2053 keV. These lines had rates of approximately 500/(t y), 500/(t y), 300/(t y) and 380/(t y) respectively. The line at 2022 keV is near a line we predict at 2023 keV. Reference [37] attributes these lines to weak transitions in 214Bi. From our analysis it is indicated that a negligi- ble fraction of the peak at 2023 keV is neutron-induced. However, since the predicted strength of the tell-tale lines that would indicate a presence of neutron interactions is just below the sensitivity of that experiment, this conclu- sion is not without uncertainty. It has been pointed out that the strength in the 2022-keV line is too strong with respect to the 214Bi branching ratios even when sum- ming uncertainties are taken into account [38, 39]. The analysis in Ref. [38], however, was based on a incorrectly normalized Fig. 1 in Ref [6]. A recent analysis [39] tak- ing this into account still points to an inconsistency in the line strengths. This discrepancy could be resolved if one attributes a significant fraction of that peak to neu- tron interactions on 76Ge. Such an attribution is not supported by our simulations. Reference [36] simulates the background in the Heidelberg-Moscow experiment resulting in a predicted signal of 646 ± 93 counts in the region between 2000 and 2100 keV during an exposure of 49.6 kg-y. This is a count rate of 130/(keV t y) to be compared with the quoted measured value for the data period simulated of 160/(keV t y). Their estimate indicates that only 0.2/(keV t y) are due to neutrons and they argue that µ-generated neu- trons are a negligible contribution. Our estimates indi- cate that neutrons are a more significant contribution and that the µ contribution is significant. We are aware of no direct neutron flux measurements for neutrons above 25 MeV. The flux of neutrons with energy greater than 25 MeV is estimated in Ref. [36] to be 10−11/(cm2 s) and they considered these neutrons to produce a negligible contribution to the background. In contrast, the simula- tion in Ref. [10] gives 56 × 10−11/(cm2 s) at 3200 m.w.e for the neutrons with energy greater than 25 MeV. We use this higher flux value and as a result, our estimate of the background rate near 2 MeV of 50/(keV t y) is comparable to the excess (30/(keV t y)) of the measured rate in comparison to the simulated rate in Ref. [36]. D. Is Copper an Alternative to Lead? One has to consider the existence of a DEP line at the double-beta decay endpoint a serious design considera- tion for Ge-detector experiments. From the above anal- ysis, the dangerous lines at 2041 and 3062 keV due to Pb(n, n′γ) are not significant contributors to the spec- trum of Ref. [37]. However, as future efforts reduce the natural activity irradiating the detectors, these Pb- neutron interactions will become important. One solu- tion could be the use of Cu as a shield instead of Pb. Copper is rather expensive and building the entire shield of this material is probably not necessary. A thick inner liner of Cu might suffice, but if a peak is observed and Pb is present near the detector, arguments based on the spectrum near 3062 keV will be critical. Although the problematic lines we observed in the Pb data were absent in our Cu data, the shields were too dissimilar to make a quantitative comparison regarding the effectiveness of reducing the continuum background. Furthermore, our experience with the lead and the sim- ulation of (n, n′γ) spectra reduces confidence in the con- clusion regarding the Cu in the absence of such data. We are preparing better experimental studies to address this question. VI. CONCLUSION As double-beta decay experiments become more sen- sitive, the potential background must be constrained to ever-lower levels. Much progress has been made in reduc- ing naturally-occurring radioactive isotopes from mate- rials from which the detector is constructed. As these isotopes that have traditionally limited the experimen- tal sensitivity are eliminated, rarer processes will become the dominant contributors. Here we have considered neutron-induced processes and have quantified them. Re- actions involving neutrons can result in a wide variety of contributions to the background. That is, no single com- ponent is likely to dominate. Therefore, tell-tale signa- tures for neutrons are needed and were identified in this work. In addition to the general continuum background that neutrons might produce, two specific dangerous Pb(n, n′γ) lines were identified. These two backgrounds can be significantly reduced using depth and/or an inner layer of Cu within the shield. In particular, the 3062- keV transition in 207Pb has a double escape peak at the endpoint energy for double-beta decay in 76Ge. A com- parison of past double-beta decay data indicates the rate of this transition is too small to explain a claim of double- beta decay. VII. ACKNOWLEDGMENT We thank R.L. Brodzinski for discussions regarding the historical production of 207Bi and its possible presence in the environment. We thank John Wilkerson and Jason Detwiler for useful suggestions and discussion. Finally, we also thank Alan Poon and Werner Tornow for use- ful discussions and a careful reading of the manuscript. This work was supported in part by Laboratory Directed Research and Development at Los Alamos National Lab- oratory. [1] Steve R. Elliott and Petr Vogel, Annu. Rev. Nucl. Part. Sci. 115 (2002) [2] Steven R. Elliott and Jonathan Engel, J. Phys. G: Nucl. Part. Phys. 30, R183 (2004). [3] F.T Avignone III, G.S. King III and Yuri Zdesenko, New Journal of Physics (in press 2004). [4] A.S. Barabash, Physics of Atomic Nuclei, 67, No. 3, 438 (2004). [5] C.E. Aalseth, et al., Phys. Rev. C v59, 2108 (1999). [6] H.V. Klapdor-Kleingrothaus, et al., Eur. Phys. J A12, 147 (2001); [7] H.V. Klapdor-Kleingrothaus, et al., Nucl. Instr. and Meth. A522, 371(2004). 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0704.0307

Periodic accretion from a circumbinary disk in the young binary UZ Tau E

To appear in the Astronomical Journal, July 2007 Periodic accretion from a circumbinary disk in the young binary UZ Tau E Eric L. N. Jensen1, Saurav Dhital1,2, Keivan G. Stassun2, Jenny Patience3, William Herbst4, Frederick M. Walter5, Michal Simon5, Gibor Basri6 ABSTRACT Close pre–main-sequence binary stars are expected to clear central holes in their protoplanetary disks, but the extent to which material can flow from the circumbinary disk across the gap onto the individual circumstellar disks has been unclear. In binaries with eccentric orbits, periodic perturbation of the outer disk is predicted to induce mass flow across the gap, resulting in accretion that varies with the binary period. This accretion may manifest itself observationally as periodic changes in luminosity. Here we present a search for such periodic accretion in the pre–main-sequence spectroscopic binary UZ Tau E. We present BV RI photometry spanning three years; we find that the brightness of UZ Tau E is clearly periodic, with a best-fit period of 19.16± 0.04 days. This is consistent with the spectroscopic binary period of 19.13 days, refined here from analysis of new and existing radial velocity data. The brightness of UZ Tau E shows significant random variability, but the overall periodic pattern is a broad peak in enhanced brightness, spanning more than half the binary orbital period. The variability of the Hα line is not as clearly periodic, but given the sparseness of the data, some periodic component is not ruled out. The photometric variations are in good agreement with predictions from simulations of binaries with orbital 1Swarthmore College Department of Physics and Astronomy, 500 College Ave., Swarthmore PA 19081; [email protected]. 2Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235 3Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, CA 91125 4Astronomy Department, Wesleyan University, Middletown, CT 06459 5Department of Physics and Astronomy, State University of New York, Stony Brook, NY 11794-3800 6Astronomy Department, University of California, 521 Campbell Hall, Berkeley, CA 94720 http://arxiv.org/abs/0704.0307v2 – 2 – parameters similar to those of UZ Tau E, suggesting that periodic accretion does occur from circumbinary disks, replenishing the inner circumstellar disks and possibly extending the timescale over which they might form planets. Subject headings: accretion disks — circumstellar matter — stars: binaries: spec- troscopic — stars: individual (UZ Tau E) — stars: pre-main sequence — plane- tary systems: protoplanetary disks 1. Introduction It is now well-established that most stars are members of binary systems at birth, and that many of these stars are surrounded by disks similar to those found around young single stars (see, e.g., the recent review by Monin et al. 2006). Thus, understanding the origin of binaries is vital to understanding the star formation process. The predominance of binaries also means that, based on number of systems alone, most potential sites of planet formation lie in multiple systems. However, interactions between stars and disks in binary systems can alter disk structure (Beckwith et al. 1990; Jensen et al. 1994; Osterloh & Beckwith 1995; Jensen et al. 1996a,b; Jensen & Mathieu 1997), resulting in a more complicated environment for planet formation. Nonetheless, the discovery of planets in relatively close binary systems (Eggenberger et al. 2004) shows that binary systems are viable sites of planet formation. Understanding the extent to which binaries modify the structure of their surrounding disks is important for understanding the possible diversity of planetary system environments. In addition, though the mass ratios are different, the interactions between stellar binary companions and disks involve the same physics as those between planetary companions and disks (e.g., D’Angelo et al. 2006) but are more easily observable. Thus, an understanding of binary-disk interactions may help us understand planet formation around single stars as well. A binary star system may have up to three disks: two circumstellar disks, one each around the primary and secondary; and a circumbinary disk outside the binary orbit. Both analytic calculations and numerical simulations show that the region between these disks is not stable for orbiting disk material. However, the question of how easily material can flow from the outer, circumbinary disk across the gap to the circumstellar disks has not been clear, either from an observational or a theoretical standpoint. The spectral energy distributions of some young binaries show the clear signature of a cleared central region, while other, apparently similar systems do not (Jensen & Mathieu 1997). Theoretical analyses by Lin & Papaloizou (1993) and Artymowicz & Lubow (1994) suggested that material in the region around the binary orbit is cleared, creating a quasi-equilibrium structure with three – 3 – distinct disks and a cleared region between them. If the gap between disks is impermeable, the disks evolve independently of each other. Since the presence of a binary companion may increase the rate of accretion (Clarke 1992; Ostriker et al. 1992) and the binary orbit presents a constraint on the size of each circumstellar disk, the circumstellar disks would be exhausted much more quickly than disks around single stars. However, smoothed particle hydrodynamic simulations by Artymowicz & Lubow (1996) (hereafter AL96; see also Günther & Kley 2002) predicted that material may indeed flow from the circumbinary disk to the circumstellar environment, with the accretion rate varying with the phase of the binary orbit. If such periodic accretion occurs in young binaries, it may be detectable observationally by a periodic brightening of the system as the material flowing from the circumbinary disk shocks when it collides with the circumstellar disk(s) or accretes onto the stellar surface(s). Observations of the T Tauri spectroscopic binary DQ Tau by Mathieu et al. (1997) showed such brightening, occurring at the binary orbital period. In addition, DQ Tau shows periodic variations in spectral veiling and emission line intensities with orbital phase (Basri et al. 1997), providing strong support for the broad picture of mass flow across gaps suggested by AL96. However, subsequent searches in other young, short-period binary systems surrounded by disks have yielded mixed results. Alencar et al. (2003) did not find periodic photometric variations in AK Sco, but they did find that the blue wing of the Hα line, and both the blue and red wings of the Hβ line, vary with the binary orbital period, as does the total Hα equivalent width. V4046 Sgr has shown periodic photometric variations at the binary orbital period (Quast et al. 2000; Mekkaden 2000), though an earlier study did not find such variations (Byrne 1986). Like AK Sco, however, V4046 Sgr does show variations in the equivalent width and shape of Balmer lines as a function of orbital phase (Stempels & Gahm 2004). No dedicated photometric monitoring of UZ Tau E has been reported in the literature to date, but in previous spectroscopic observations, neither Hα equivalent width nor spectral veiling has shown any obvious dependence on binary orbital phase (Mart́ın et al. 2005). Motivated by previous observational work and a desire to understand accretion in binary systems, we have undertaken a photometric monitoring campaign for the pre–main-sequence (PMS) spectroscopic binary UZ Tau E. UZ Tau, in the Taurus-Auriga star-forming region, was first discovered to be variable by Bohlin during a bright outburst in 1921 (Bailey 1921; Bohlin 1923) and identified as a ∼ 3.′′7 binary by Joy & van Biesbroeck (1944), one of the first pre–main-sequence binaries to be identified. Subsequently, both components of the binary were found to be binary themselves, making this a quadruple system. Simon et al. (1992) and Ghez et al. (1993) identified UZ Tau W as a binary system, and Mathieu et al. (1996) identified UZ Tau E to be a single-lined spectroscopic binary with a 19.1-day period. UZ Tau W, a 0.′′34 binary (47.6 AU assuming a distance of 140 pc to Taurus-Auriga; Kenyon et al. 1994), is separated from UZ Tau E by 3.′′78 (530 AU; Simon et al. 1995a). Prato et al. (2002) – 4 – detected absorption lines of the secondary star in the near infrared spectrum of UZ Tau E, measuring the mass ratio M2/M1 = 0.28 ± 0.01. Mart́ın et al. (2005) presented additional radial velocity data for UZ Tau E; they found a binary orbital period of 18.979 days and an eccentricity of 0.14. UZ Tau E shows strong Hα emission, indicative of ongoing accretion, and strong infrared and millimeter excess emission from circumstellar and circumbinary disks; the circumbinary disk has been resolved at λ = 1.3 mm (Jensen et al. 1996a) and 2.6 mm (Dutrey et al. 1996) with a size and mass comparable to disks around single stars, showing that the close spectroscopic companion has not significantly decreased the disk mass, in contrast to the 50-AU pair in UZ Tau W, where the presence of a companion at a separation comparable to typical disk sizes has greatly reduced the presence of circumstellar material. In this paper, we present our new photometric observations, as well as a re-determination of the binary orbital parameters from new and existing radial velocity observations. We then examine periodicities in the data, showing that the photometric data vary at the binary orbital period. Finally, we interpret the results in the context of the model of pulsed accretion in binary systems. 2. Observations 2.1. Photometry In order to search for periodic photometric variations, we have obtained new photometry of UZ Tau E. Our photometric observations were made with the 0.6-m Perkin Telescope at the Van Vleck Observatory (VVO) at Wesleyan University, and with ANDICAM on the 1.3-m SMARTS Telescope at CTIO. See Table 1 for details of the observations. The data were reduced using standard techniques. Since UZ Tau E and UZ Tau W are separated by 3.′′78, we sought to minimize con- tamination of the UZ Tau E photometry by light from UZ Tau W by rejecting images with FWHM greater than 7 pixels, and by using a relatively small (3-pixel-radius) photometric aperture. Light curves of UZ Tau E and W show no correlation, indicating that the UZ Tau E photometry is uncontaminated. We performed differential photometry on UZ Tau E using USNO-B1.0 1158-0057597 as a comparison star. The UZ Tau field is relatively sparse, and in some of the SMARTS images, this was the only star that was sufficiently bright to serve as a comparison star, so we used it as the sole comparison in all of our photometry. The star was verified to be non-variable at the few percent level, showing a standard deviation of 0.02 magnitudes by comparison with several other stars of similar brightness using the wider-field VVO images. – 5 – The USNO-B1.0 Catalog (Monet et al. 2003) gives magnitudes for this star from the Second Palomar Sky Survey of B = 16.39, R = 13.75, and I = 12.8. Although these filters are not identical to those used in our CCD observations, we adopted these values for the magnitude of the reference star to set the zero point of our light curves. In addition, we adopted V = 14.99 by noting that the R − I color for the comparison star suggests a spectral type of ∼ M0, and adopting a corresponding B − V color. Adopting these values allows us to determine approximate colors for UZ Tau E, though we caution that the absolute scaling of both the individual magnitudes and the colors is uncertain. The differential photometry and the color changes, which form the basis of our analysis, are both unaffected by this systematic uncertainty in the zero point. 2.2. Spectroscopy We did not acquire new spectra of UZ Tau E solely for this program, but we did make new measurements of a number of spectra taken during the course of other programs. Some of these spectra were kindly supplied by Marcos Huerta. These are echelle spectra from McDonald observatory spanning 14 nights in January 2002, with R = 46, 000 and wavelength coverage of 5460–6760 Å. The observations and data reduction are described in detail in Huerta et al. (2005). Additional echelle spectra of UZ Tau E were taken at Keck (R = 31, 000) with the setup described in Basri & Reiners (2006) and at Lick (R = 48, 000) with the setup described in Alencar & Basri (2000). We used the spectra from Huerta to measure radial velocities of UZ Tau E. Spectra of the weak-lined T Tauri star V819 Tau were used as a radial velocity standard. By cross- correlating the UZ Tau E spectra against the V819 Tau spectra, we measured heliocentric radial velocities of UZ Tau E, assuming vhelio = 14.4±1.5 km s −1 for V819 Tau (Walter et al. 1988). The resulting velocities are given in Table 2. Radial velocities were measured using several different echelle orders with strong absorption lines; the quoted uncertainties reflect the dispersion in these different measurements, as well as the uncertainty of V819 Tau’s radial velocity. In addition, we also measured the equivalent width of the Hα line in both sets of spectra in order to track changes in accretion rate over time. Equivalent widths are given in Table 2, with an estimated uncertainty of 10%. – 6 – 3. Binary orbital parameters In order to assess whether or not any periodic photometric variations detected in UZ Tau E are synchronized with the binary orbit, we need to have an accurate knowledge of the orbital parameters. These have been determined previously by Mathieu et al. (1996), Prato et al. (2002), and Mart́ın et al. (2005), but the number of radial velocity points avail- able is still relatively small, especially for the secondary, leaving open the prospect of further improvements to the orbital parameters. To that end, we have re-analyzed the spectroscopic orbit using data published in Prato et al. (2002) and Mart́ın et al. (2005) as well as our new radial velocity measurements (Section 2.2). We fit the radial velocity data using the Binary Star Combined Solution software (Gudehus 2001), the ORBIT code (Forveille et al. 1999), and our own custom-written IDL code; all gave the same solution. The best-fit phased radial velocity curve is shown in Figure 1 and the orbital parameters are given in Table 3. The best-fit period of 19.131±0.003 days is inconsistent with the value of 18.979±0.007 days found by Mart́ın et al. (2005). Examination of the power spectrum of the velocity data used by Mart́ın et al. shows that the 18.979-day period appears to be an alias of the true period, caused by beating of the 19.131-day period with two six-year gaps in the radial velocity data; there is a corresponding alias at 19.3 days. Re-fitting only the data used by Mart́ın et al., we find that the two periods both correspond to local minima in χ2 space, with reduced χ2 = 8.8 for P = 19.131 days and reduced χ2 = 10.7 for P = 18.979 days. When the new radial velocity data are added, the fit for P = 19.131 days improves to reduced χ2 = 8.1 while that for P = 18.979 days worsens to χ2 = 11.7, as expected if 19.131 days is the correct period. We note that we have not added any additional radial velocity measurements of the secondary, and thus the mass ratio remains more uncertain than the other orbital elements, resting on the six secondary radial velocities presented by Prato et al. (2002). UZ Tau E is one of only a handful of pre–main-sequence systems with measured stellar masses (see Mathieu et al. 2006 for a recent review). Because the total system mass has been measured (Simon et al. 2000), the spectroscopic orbital parameter M sin3 i can be used to determined the orbital inclination. This can then be compared with the observed inclination of the circumbinary disk. While this was done by Simon et al. (2000) and Prato et al. (2002), we revisit this issue here using our newly-determined orbital parameters for UZ Tau E. Combined with M = 1.31 ± 0.08 M⊙ (Simon et al. 2000), our orbital parameters give sin iorbit = 0.81± 0.05, or iorbit = 54 ◦ ± 5◦. This is in excellent agreement with the disk inclinations of 54◦±3◦ and 56◦±2◦ measured from interferometric images of the λ = 1.3 mm – 7 – continuum emission and the CO line emission, respectively (Simon et al. 2000). Thus, the binary orbit and the circumbinary disk are coplanar. Since the disk inclination is measured at scales of ∼ 100 AU and the binary orbit is only a few tenths of an AU, this co-planarity apparently extends over the entire disk. Though there are several theoretical studies of how tilted circumstellar disks interact with a binary system (e.g., Papaloizou & Terquem 1995; Larwood et al. 1996; Bate et al. 2000; Lubow & Ogilvie 2000), we know of no studies of the timescale for alignment of a circumbinary disk if it is initially tilted relative to the binary orbit. Studies of the effects of a planetary-mass companion on a tilted external disk (Lubow & Ogilvie 2001) suggest that such disk tilts do decay over time, however. The example of UZ Tau E shows that co-planarity over the entire disk exists already by an age of a few Myr, suggesting that circumbinary disks either form already aligned with the orbit, or come into alignment very quickly. This is similar to the result of Jensen et al. (2004), who found that circumstellar disks in young binaries tend to be aligned with each other, and thus presumably with the binary orbit. 4. Periodic variations 4.1. Photometry In order to determine whether the system is varying in phase with the binary orbit, or in any other systematic way, we have searched the photometric data for periodic signals. We begin by searching for evidence of periodicity, without pre-supposing a particular period. A Lomb-Scargle periodogram (Scargle 1982) of the I-band data (the band with the largest number of points and best time coverage; Table 1) is shown in Figure 2. There is a strong peak at a period of 19.20± 0.03 days. The false-alarm probability (FAP) of this peak is less than 0.001 according to the formulation of Horne & Baliunas (1986). While this FAP calculation is strictly applicable only to evenly-spaced data, a Monte Carlo bootstrapping method (e.g., Stassun et al. 1999) confirms that this period is statistically significant at better than 99.9% confidence. The period uncertainty reported above, which follows from the formulation of Kovacs (1981), is probably underestimated as it assumes that the underlying signal is well described by a single sinusoid. Though it is slightly off the main peak of the power spectrum, there is significant power at the binary period of 19.131 days. Periodograms of the B, V , and R data are similar (Figure 3), showing peaks near the binary period, but with broader peaks and higher false- alarm probabilities, perhaps due to the more limited time coverage of the data in those bands. – 8 – To refine the period and to better estimate its uncertainty we next performed a phase dispersion minimization (PDM) analysis (Stellingwerf 1978), which is particularly well-suited to periodic variability that is highly non-sinusoidal and/or to data with large intrinsic scatter, both of which apply to the photometry of UZ Tau E. The PDM search of the I-band data yields a best period of P = 19.17±0.05 d, where the uncertainty was determined empirically from the 1/e folding scale of the PDM merit function. The same analysis on the V -band data gives P = 19.15± 0.04 d. Schwarzenberg-Czerny (1989) argues that a related test, the one-way analysis of vari- ance, is the most powerful statistic of this kind for detection of periodic signals. Applying that test to our data yields P = 19.16± 0.03 d for the I-band data, and P = 19.15± 0.05 d for the V -band data. Following Schwarzenberg-Czerny (1989), the period uncertainty was determined using a “post-mortem” analysis that measures the 1-σ confidence interval of the primary periodogram peak, defined by its width at the mean noise power level of the peri- odogram in the vicinity of this peak. As above, a Monte Carlo permutation analysis of the light curves confirms that this period is statistically significant at better than 99.9% confi- dence. Combining these estimates, our best-fit photometric period is P = 19.16 ± 0.04 d, consistent with the binary orbital period. Figure 4 shows I-band light curves for all three observing seasons, folded at the binary orbital period of 19.131 days. As suggested by the periodogram analysis, all show indications of periodic behavior, with a broad minimum near orbital phase 0.5. The data from the 2004– 2005 season show the smoothest variability, but this is also the season with the smallest number of data points. Clearly there is significant random variability as well, with scatter of roughly 0.6 magnitudes at all orbital phases. Figure 5 shows folded B, V , R, and I lightcurves from the 2003–2004 and 2005–2006 seasons. Broadly speaking, the BV R data show the same behavior as the I-band light curves, with large-amplitude variability that appears to have both periodic and random components. We note that the R band includes the Hα line, which may complicate the interpretation of the light curve. All four bands show a gradual increase in brightness over the three years of our ob- servations, with the mean magnitude changing by 0.3 mags at I band from 2003–2004 to 2005–2006. To separate long-term variations from the shorter-term variations of interest here, a linear trend (with a slope of roughly 0.15 mag / yr at I band) has been fit to each band. The folded light curves using these de-trended data, and combining all three observing seasons, are shown in Figure 6. In addition to our photometric data, previous data on UZ Tau have shown some evidence – 9 – of both long-term trends and periodic variations. Bohlin (1923), in one of the first papers to mention UZ Tau, reported on a major flare and then a four-magnitude overall decline in brightness from 1921–1923. He also noted that there was a short-period variation with a period of 10–20 days, which encompasses the period of the variations reported here. Bohlin’s measurements are for the entire UZ Tau system, but later examination of Bohlin’s plates by Herbig (1977) showed that it was UZ Tau E that brightened dramatically in 1921. Variations in color of the system can also give clues about the cause of the variability. Figure 7 shows the V − I color as a function of I magnitude and orbital phase. The system shows a behavior commonly seen in T Tauri stars, appearing redder when fainter and bluer when brighter (Herbst et al. 1994). This behavior is consistent either with periodic changes in extinction (causing both dimming and reddening when the extinction is higher) or in accretion (adding additional blue light when the accretion rate is higher). 4.2. Spectroscopy If the periodic photometric variations are due to changes in accretion rate, one might expect accompanying variations in Hα emission or spectral veiling, common tracers of accre- tion. Mart́ın et al. (2005) searched for both of these in UZ Tau E and did not find evidence of either. With our new spectra, we can revisit the question of Hα variability. Figure 8 shows the equivalent width of the Hα emission line as a function of binary orbital phase. The variations do not appear to be strongly correlated with orbital phase. There is some evidence for lower Hα equivalent widths around phase 0.4–0.8, as seen in the photometric data, but the data are relatively sparse in that phase range as well. While a lack of periodic variability would be at odds with the photometric data, we note that periodicity may not be as obvious in the spectroscopic data, since the two datasets differ in two important respects. First, the Hα data have much sparser sampling; they span a total of eight years (1994–2002), but with only a handful of points during a given year. Second, they do not overlap at all with the photometric data. Given that the photometric data show both long-term trends and short-term scatter in addition to the periodic variations, and that T Tauri stars in general are known to show significant random variability at Hα, it may be difficult to separate random and periodic variations (if any) without a dedicated monitoring campaign, preferably one that includes simultaneous photometric and spectroscopic mea- surements. We conclude that while periodic spectroscopic variations similar to those seen in the photometry are not definitively present in these spectroscopic data, neither are they ruled out. – 10 – 5. Discussion The photometric data (and possibly the spectroscopic data) show periodic variability at the binary orbital period, suggesting that there is a link between the variability and in- teractions of the binary with its circumstellar and/or circumbinary material. In this section, we first argue that the periodic variations are unlikely to be due to stellar rotation, and then we examine how well the observed behavior matches what is expected from the pulsed accretion model of Artymowicz & Lubow (1996). Finally, we examine the available data for other spectroscopic binaries to assess whether or not there is evidence for periodic accretion as a general phenomenon. 5.1. Could the variations be due to rotation? Periodic variability is not uncommon in photometric studies of PMS stars. Indeed, dedi- cated monitoring surveys of rich star-forming regions (e.g., Mandel & Herbst 1991; Attridge & Herbst 1992; Choi & Herbst 1996; Stassun et al. 1999; Rebull 2001; Herbst et al. 2002a) have now discovered hundreds of PMS stars exhibiting periodic variability, the result of surface- brightness inhomogeneities (i.e. starspots) that rotate in and out of view with the stellar rotation period. The periodic variability observed in UZ Tau E is very unlikely to be the result of such rotationally modulated spot signals, for several reasons. First, the rotation periods of low-mass PMS stars are nearly always shorter than about 12 days, while the pe- riod of the variations reported here is 19 days. Among 150 low-mass PMS stars in the Orion Nebula Cluster, only two stars (∼ 1%) have Prot > 15 d (Herbst & Mundt 2005). Second, rotationally modulated spot signals are typically sinusoidal, and stable over many cycles or, in some cases, many years. In contrast, the periodic signal we have found in UZ Tau E is decidedly non-sinusoidal, with considerable scatter; the “bright” state has a duty cycle of ∼ 60%. Thus, it is either intrinsically non-sinusoidal, or shows substantial phase shift- ing from one cycle to the next; neither of these is consistent with rotationally-modulated variability. The rotation period distributions discussed above are presumably dominated by single stars or member of wide binaries, while tidal interactions between the stars in a close binary system can synchronize the orbital and rotational periods. However, in the case of eccentric systems like UZ Tau E, pseudo-synchronization (in which the stellar angular velocity is synchronized with the orbital angular velocity at periastron) occurs instead, since the tidal interactions are strongest around periastron (Hut 1981). The predicted pseudo-synchronous rotation period for UZ Tau E, using the weak friction formulation of Hut (1981) and the orbital parameters in Table 3, is 11.4 ± 1.2 d, inconsistent with the observed variability – 11 – period. We can also estimate the rotation period of the UZ Tau E primary directly if three quantities are known: the inclination irot of the star’s rotation axis, the star’s projected rotational velocity v sin irot, and the stellar radius. Of these, the inclination is typically impossible to measure, except under special circumstances. As noted in Section 3, the dynamical mass measurement of UZ Tau E allows us to determine the binary orbital inclination iorbit. Based on studies of other binary systems, it is reasonable to assume that this inclination is the same as that of the stellar rotation axis, irot. The most detailed study comparing the orientations of these axes in binary systems is that of Hale (1994). Considering spectral types of F5–K5, he finds that binaries with separations less than 30–40 AU tend to exhibit co-planarity between rotational equators and orbital planes, while wider binaries have random orientations. Using a similar method, Weis (1974) found a tendency for the stellar rotational equators to align with the binary orbit among primaries in F star binaries. Interestingly, Weis (1974) did not find a tendency toward co-planarity between rotational and orbital planes among A stars, suggesting that caution is necessary when comparing stars of different masses. Similarly, Guthrie (1985) found no correlation between orbital inclination and v sin i among 23 A2–A9 binaries with semi-major axes of 10–70 AU. The low mass and short period of UZ Tau E suggest, however, that the conclusions of Hale (1994) are most applicable here. Prato et al. (2002) find L = 0.63+0.19 −0.17 L⊙ and Teff = 3700 ± 150 K for the primary in UZ Tau E. Combining these values yields R = 1.9 ± 0.2 R⊙. Hartmann & Stauffer (1989) find v sin i = 15.9±4.0 km s−1 for UZ Tau E using optical spectra, consistent with the value v sin i = 16 ± 2 km s−1, which we measure from our new spectra and adopt here. Since absorption lines of the secondary of UZ Tau E have only been seen in near-infrared spectra and are not evident in any of our optical spectra, we take this to be the projected rotation velocity of the primary. Combining these measurements with sin iorbit = 0.81±0.05 (Section 3), and assuming iorbit = irot, we find Prot = 4.9± 0.8 d. If iorbit 6= irot, we find Prot ≤ 6± 1 d since sin i ≤ 1. Thus, uncertainty on the inclination cannot reconcile the photometric period with the inferred rotation period. The most uncertain remaining quantity is v sin i, but since Hartmann & Stauffer (1989) measured v sin i from 11 different spectra of UZ Tau E, with self-consistent results from two different parts of the spectrum (including spectra near λ = 5200 Å) and consistency with our new v sin i measurement, it is unlikely that line broadening from photospheric lines of the faint, red secondary could lead to an overestimate of v sin i by a factor of four. Similarly, given the uncertainties on L and Teff , it is difficult to see how the radius could be underestimated by a factor of four. Thus, we conclude that the observed periodic variations are unlikely to – 12 – be due to stellar rotation. 5.2. Evidence for pulsed accretion We have shown above that UZ Tau E exhibits periodic photometric variations that have the same period as the binary orbit, and that these variations are unlikely to be caused by stellar rotation. Here, we examine the predictions made by the pulsed accretion model of AL96 and compare them to our observations. 5.2.1. What are the predictions? Broadly speaking, Artymowicz & Lubow (1996) predict that a binary with an eccentric orbit and a circumbinary disk will have an accretion flow from the circumbinary disk, and thus onto the circumstellar disks or stellar surfaces, that varies periodically at the binary orbital period. The exact behavior of the accretion rate with orbital phase depends on the binary orbital parameters. AL96 show the results of two simulations, one for mass ratio M2/M1 = 0.43 and eccentricity e = 0.1, and another for M2/M1 = 0.79 and e = 0.5. The former shows accretion that varies relatively smoothly over the orbital period, while the latter is strongly peaked at periastron. As noted by AL96, the exact timing of the accretion variability depends on the orbital parameters, most strongly on e. Some previous observational studies of T Tauri spectroscopic binaries have focused specifically on looking for enhanced accretion near periastron; however, we note here that the actual prediction of the model is more general than that, and that the peak accretion rate need not come near periastron. 5.2.2. How well do the data match the predictions? First, we note that our observations match the general predictions of AL96 quite well, in that there are indeed periodic photometric variations at the binary orbital period, which are readily interpretable as a variable accretion rate. The comparison with the spectroscopic data is more ambiguous; if more intensive monitoring of the Hα line in UZ Tau E were to show that there are no orbit-modulated Hα variations, it would present a problem for the model. For a more specific comparison with our data, Figure 9 shows the variations of accretion – 13 – with orbital phase predicted by AL96 for a binary with M2/M1 = 0.43, e = 0.1. UZ Tau E has a more extreme mass ratio (M2/M1 = 0.30) and larger eccentricity (e = 0.33) than this, but these parameters are closer to those of UZ Tau E than those of the other simulation in AL96. AL96 do note that the timing of the maxima of the accretion depend largely on e rather than M2/M1. Since e for UZ Tau E is intermediate between the two models calculated by AL96, we might then expect the maximum accretion to come between the phase of ∼ 0.75 they calculate for the low-e case and the phase of ∼ 1 for the high-e case. For comparison with our data, we have taken the logarithm of the variations of accretion rate predicted by AL96 to shift them onto a “magnitude-like” scale, and added an arbitrary offset and scale factor to match the mean of the data and amplitude of the variations. The phase of the minimum predicted by this simulation does not match our data well; when the model is given a shift of +0.2 in orbital phase, there is better agreement between the model predictions and the data. This scaling and shifting to match the data is obviously ad hoc, but it allows us to compare the phase width of the observed variations, which appear to match the predictions relatively well. In addition, this shifted position of the maximum is indeed between the two cases calculated by AL96, as expected if eccentricity is the dominant factor in determining the timing of maximum accretion. 5.3. Evidence for periodic accretion in other T Tauri binaries The discussion and data above show that looking for evidence of periodic accretion can be complicated, with other sources of variability perhaps being important and masking the effect in small datasets, and with the exact behavior expected to be a function of the specific binary orbital parameters. That said, is evidence for pulsed accretion seen in other young binary systems? In Table 4 we present characteristics of young binaries with periods of less than one year and evidence of circumbinary material, in order of increasing eccentricity. Below, we examine the observational data for some of these systems, attempting to relate them to what we see in UZ Tau and exploring similarities and differences. Unfortunately, the small number of systems and their somewhat heterogeneous properties means that it is difficult to generalize, so we offer these comments in the spirit of attempting to pull together the existing data, rather than arguing one way or the other for the validity of the AL96 model for the sample as a whole. – 14 – 5.3.1. DQ Tau DQTau was the first system to be scrutinized for evidence of pulsed accretion. Mathieu et al. (1997) showed that the photometric variations are modulated at the binary orbital period, and Basri et al. (1997) showed that the Hα line and spectral veiling are as well. Fortuitously, the mass ratio and eccentricity of DQ Tau are quite similar to those of the high-eccentricity case modeled by AL96, allowing for specific comparison with the theory. The timing and phase width of the photometric and spectroscopic variations match the predictions well, being sharply peaked near periastron. However, the DQ Tau observations did show consid- erable orbit-to-orbit variation, with the periastron brightening being seen roughly 65% of the time. This is reminiscent of the large scatter that we see in the UZ Tau light curves; clearly the periodic accretion process is not exactly repeatable from orbit to the next, nor is it the only source of variability. 5.3.2. AK Sco The orbital eccentricity and binary mass ratio of AK Sco are quite similar to those of DQ Tau, and indeed, simulations by Günther & Kley (2002) for a binary with AK Sco’s orbital parameters predict pulsed accretion. Thus, it comes as some surprise that the system does not show periodic photometric variability, despite extensive monitoring (Alencar et al. 2003). The overall variability is large (up to 1.5 mags in y), but apparently random. There are periodic variations in the Balmer lines, but they are not sharply peaked around periastron. Examining Table 4, we note two properties of AK Sco that are quite different from those of DQ Tau or UZ Tau. First, AK Sco is considerably hotter and more luminous. Thus, accretion variations of the same luminosity as those occurring in UZ Tau and DQ Tau would result in substantially smaller magnitude changes, which could be swamped by the large random variability. Second, AK Sco has considerably lower millimeter flux than either of the other two systems. If the systems are fit with similar disk models (in which the disk is assumed to be optically thin at millimeter wavelengths in its outer regions), AK Sco’s disk mass is an order of magnitude smaller than that of DQ Tau or UZ Tau E (Jensen et al. 1996a; Jensen & Mathieu 1997; Mathieu et al. 1997). Alencar et al. (2003) fit AK Sco with an optically-thick disk model that has a comparable mass to the disk models fit to DQ Tau and UZ Tau E. However, such disk models have not been fit to DQ Tau or UZ Tau E, and would presumably result in even larger disk masses for those systems. In a direct comparison of λ = 1.1 mm flux, DQ Tau and UZ Tau E are 3 and 5 times brighter than AK Sco at roughly the same distance, presumably reflecting larger disk masses. It is possible that a somewhat lower-mass disk has different dynamics, and that the accretion flow in the AK Sco – 15 – system is fundamentally different than that in the other systems with more massive disks. 5.3.3. GW Ori This system has a near-circular orbit, and thus would not be expected to show pulsed accretion under the model set forth by AL96. However, Stempels & Gahm (2004) quote Artymowicz, private communication, as saying that pulsed accretion is possible for systems with circular orbits as well, and indeed D’Angelo et al. (2006) show that this occurs for giant planets embedded in disks. Thus, pulsed accretion appears to be possible for at least some circular-orbit systems and thus may be for GW Ori as well, though the larger gap cleared by a stellar companion (Artymowicz & Lubow 1994) will clear some of the disk resonances that might contribute to disk eccentricity growth in a system with a planetary-mass companion. Like AK Sco, GW Ori is very luminous, and shows significant random variability, though no obvious periodic variability. It does have a much more massive disk than AK Sco, however, and indeed than any of the systems considered here. Because of its much larger semimajor axis, and to some extent its circular orbit, GW Ori is much more likely to have significant circumstellar disks, as the stars do not approach each other very closely at periastron. Thus, material flowing from the circumbinary disk may merge with the circumstellar disks and then accrete more gradually onto the stars, rather than falling directly on (or near) the stellar surfaces as is expected to happen in the shorter-period systems. If the infalling material does not shock strongly as it merges with the circumstellar disk, and if any density enhancements are smoothed out somewhat by the time the material reaches the stellar surface, then any photometric signature of the periodic infall would be weakened. We note that UZ Tau E likely has circumstellar disks as well (Jensen et al. 1996a), so a similar effect could be at work in reducing the amplitude of the periodic variability relative to the stochastic variability. 5.3.4. V4046 Sgr Like GW Ori, V4046 Sgr has a nearly circular orbit. However, V4046 Sgr has shown periodic photometric variations at the binary orbital period (Quast et al. 2000; Mekkaden 2000). These variations persist over several years and are relatively sinusoidal (Walter, unpublished data, 2003–2005). Unlike the other binaries discussed here, in this case stellar rotation is a plausible explanation of the observed variations. It is common for stellar rotational periods to become synchronized with the binary orbital period, particularly for short-period binaries like V4046 Sgr. Given the short period (resulting in stronger tidal – 16 – interactions and a shorter synchronization time scale) and the somewhat older age of this system (∼ 10 Myr), synchronization is plausible, and indeed is supported by detailed analysis of the system (Stempels & Gahm 2004). However, rotation does not explain the periodic Balmer line variations observed, which Stempels & Gahm (2004) attribute to accumulations of gas co-rotating with the binary orbit. 5.3.5. ROXs 42 and ROXs 43B These two spectroscopic binaries are both weak-lined T Tauri stars (Bouvier & Appenzeller 1992; Walter et al. 1994), indicating less-active accretion than some of the other systems dis- cussed here. Neither has been detected at millimeter wavelengths, yielding only an upper limit on the disk masses (Skinner et al. 1991; Jensen et al. 1996b). Both systems show mid-infrared excesses, indicating the presence of circumbinary material, and a lack of near- infrared excess, which can be modeled as a cleared central region in the disk (Jensen & Mathieu 1997). The fact that both are higher-order multiple systems complicates matters; ROXs 42 (NTTS 162814−2427) is a triple system with a separation of 0.′′15 (Lee 1992; Ghez et al. 1993), while ROXs 43B (NTTS 162819−2423S) has a wide companion at 4.′′8 which is itself a close binary system (Walter et al. 1994; Simon et al. 1995b). Since the evidence for the pres- ence of a substantial disk rests on the low-spatial-resolution IRAS detections, it is possible that the excess is associated with the wider companions rather than arising from circumbi- nary disks around the spectroscopic binaries. In any case, the lack of millimeter detections indicates that there is less disk mass in these two systems than in the others discussed here. Neither system has been intensively monitored over timespans that would be necessary to detect periodic photometric variations at the relatively long orbital periods. ROXs 42 shows evidence for some semi-regular variations over roughly 1.5 orbital periods (Zakirov et al. 1993), while the combined light of the ROX 43 system shows only a 0.1-magnitude varia- tion, with evidence of a 1.5-day or 3-day periodicity, presumably attributable to rotation of one or more of the stars (Shevchenko & Herbst 1998). 5.3.6. KH 15D The unusual pre–main-sequence system KH 15D (V582 Mon) is a spectroscopic bi- nary that undergoes deep (∆I ∼ 3.5 mag) eclipses, thought to arise due to occultation from a circumbinary disk (Hamilton et al. 2001; Herbst et al. 2002b; Hamilton et al. 2005; Winn et al. 2006 and references therein). While the system has an eccentricity and mass ratio that would suggest that pulsed accretion might be present, the photometric variations – 17 – at the binary orbital period are dominated by the deep eclipses. Furthermore, the depth and detailed shape of these eclipses are evolving with time (Winn et al. 2003; Johnson & Winn 2004; Maffei et al. 2005; Johnson et al. 2005; Winn et al. 2006), making it very difficult to determine whether there might currently be an additional, smaller-amplitude component with the same period that is related to accretion rather than occultation. Winn et al. (2003) showed that the current deep eclipses did not occur during the first half of the twentieth century, raising the possibility of searching for evidence of accretion-related variability at earlier epochs. Their limit of one mag on the variability during that time does not preclude accretion-related variations like those seen in UZ Tau E. The ∼ 0.9-mag periodic varia- tions seen from the 1960’s through the 1980’s (Johnson & Winn 2004; Maffei et al. 2005; Johnson et al. 2005), however, are relatively smooth and are well-fit by the eclipse model (Winn et al. 2006), placing a limit on how much any accretion-related component was con- tributing to the variability during that time. Since the inferred mass ratio and eccentricity for KH 15D are similar to those of DQ Tau (Table 4), we might expect accretion-related vari- ability to be strongly peaked around periastron, which is also when the current deep eclipses occur. This might help explain several anomalously bright points seen during eclipses in the late 1990’s that are not well-fit by the model of Winn et al. (2006). The precessing circumbinary disk occultation model of Winn et al. (2006) is quite suc- cessful in reproducing the shape and ongoing evolution of the light curve, and we do not suggest that accretion explains most of the photometric variations. We note, however, the possibility that such an additional component might be sporadically present (with the same period) and that, if it is, this could complicate the modeling of the historical evolution of the light curve, especially during earlier, more-sparsely-sampled epochs. 6. Conclusions We have shown that the pre–main-sequence binary UZ Tau E shows clear photometric variability at the binary orbital period of 19.13 days. This variability is consistent with a model in which material in the circumbinary disk is periodically perturbed by the binary in its eccentric orbit and falls from the outer disk, across the cleared central gap and onto the stars or their circumstellar disks. There is significant scatter in the light curves, indicating that this “pulsed accretion” may not occur during every binary orbit. Hα equivalent widths show some suggestion of periodic variability, but it is not definitive. The apparently intermittent behavior of the accretion, and the presence of other, random sources of variability, suggest that searches for this sort of accretion signature require well- sampled datasets with long time baselines in order to detect any periodic component. In – 18 – particular, simultaneous photometric and spectroscopic monitoring of UZ Tau E in the future will help determine whether the Hα variations show a periodic component, as the photometric variations do. The good overall agreement between theory and observations suggests that resonant interactions between stars (and, by extension, planets) and disks are indeed important in determining disk structure and dynamics, while the random component of the observed behavior shows that there is still work to be done in understanding the full complexity of these interactions. We gratefully acknowledge the support of the National Science Foundation through grant AST-0307830. We thank the referee, Steve Lubow, for useful comments that improved this paper. We are grateful to Michael Meyer, David Cohen, and Larry Marschall for useful discussions; to Marcos Huerta and Pat Hartigan for use of their spectra of UZ Tau E; to Matthew Richardson for assistance with data reduction; to Peter Collings for translating early papers on UZ Tau from German to English; and to Thierry Forveille for use of his ORBIT code. 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A., & Grankin, K. N. 1993, Informational Bulletin on Variable Stars, 3898, 1 This preprint was prepared with the AAS LATEX macros v5.2. – 23 – Table 1. Observations of UZ Tau E Telescope Exp. time (s) Filter(s) Season # of nights Timespan in days SMARTS (1.3m) 5 BV RI 2003–2004 63 170 2005–2006 6 42 30 V RI 2005–2006 9 33 VVO (0.6m) 60 I 2004–2005 16 126 2005–2006 20 128 – 24 – Table 2. Radial Velocities and Hα EW Julian Date vhelio (km s −1) Hα EW (Å)a 2450416.82 · · · 88.4 2450783.93 · · · 42 2450783.94 · · · 45.1 2450784.96 · · · 38 2450785.11 · · · 57 2450835.65 · · · 54: 2451060.99 · · · 45.6 2451061.00 · · · 39.8 2451077.14 · · · 63.8 2451120.92 · · · 101 2451137.94 · · · 51.3 2451138.88 · · · 62.6 2451162.86 · · · 69.3 2451163.82 · · · 57.8 2451164.79 · · · 57.5 2451165.78 · · · 58.3 2451166.79 · · · 61.7 2451169.84 · · · 48.9 2451504.73 · · · 35 2451507.68 · · · 25: 2451508.67 · · · 49.1 2451509.71 · · · 57 2451510.66 · · · 71 2451517.98 · · · 75 2451523.68 · · · 42.7 2451524.74 · · · 49.3 2451525.76 · · · 40.8 2451527.68 · · · 46.6 2451528.68 · · · 58.9 2451529.66 · · · 61.8 2451530.60 · · · 69.1 – 25 – Table 2—Continued Julian Date vhelio (km s −1) Hα EW (Å)a 2452280.60 −4.3 ± 2.1 44 2452281.74 −3.1 ± 1.5 51 2452282.76 −5.8 ± 1.8 81 2452283.66 −4.9 ± 4.6 78 2452284.71 · · · b 77 2452286.66 6.2 ± 2.6 90 2452287.65 17.1 ± 1.7 87 2452288.68 27.2 ± 2.1 59 2452289.69 29.6 ± 3.0 65 2452290.72 37.8 ± 7.0 67 2452291.67 28.5 ± 2.1 58 2452292.64 29.1 ± 3.2 50 2452293.67 22.0 ± 3.6 45 2452579.11 · · · 50 aPositive values denote emission. bThe spectrum on this date was too noisy to allow measurement of an accurate radial velocity. – 26 – Table 3. Binary orbital parameters for UZ Tau E Period (days) 19.131± 0.003 e 0.33± 0.04 JD of periastron 2451328.3± 0.5 ω 239◦ ± 9◦ a sin i (AU) 0.124± 0.003 γ (km s−1) 13.9± 0.7 K1 (km s −1) 17.3± 1.4 K2 (km s −1) 57.4± 4.7 M sin3 i (M⊙) 0.69± 0.13 M2/M1 0.30± 0.03 Table 4. CTTS spectroscopic binaries Binary Period e M2/M1 Spectral L Disk Mass a Photometric Balmer line References System (days) Type (L⊙) (M⊙) periodicity? periodicity? V4046 Sgr 2.421 ≤ 0.01 0.94 K5 0.82 0.0085 Yes (∆B≈0.1) Yes 1, 2, 3 GW Ori 241.9 0.04 SB1 G0 26 0.3 ? (∆V≈0.7) ? 4, 5 UZ Tau E 19.131 0.33 0.30 M1 0.91 0.063 Yes (∆I≈0.8) Maybe 6, 7, 8 ROXs 43B 89.1 0.41 SB1 G0 0.4 < 0.00037 ? (∆V=0.1) ? 1, 9, 10, 11 AK Sco 13.609 0.47 0.99 F5 8.40 0.002 No (∆y≈1.5) Yes 1, 12, 13 ROXs 42 35.95 0.48 0.92 K4 0.4 < 0.00025 ? (∆V=0.4) ? 1, 10, 11, 14, 15 DQ Tau 15.804 0.56 0.97 K7–M1 0.95 0.020 Yes (∆V≈0.5) Yes 16, 17 KH 15D 48.38 0.57–0.65 0.83b K7c 0.4c · · · Eclipse (∆I≈3.5) ? 18, 19, 20 aDisk mass estimates were made assuming that the disk is at least partially optically thin at mm wavelengths. An optically- thick model for AK Sco (Alencar et al. 2003) yields a disk mass of 0.02 M⊙. bDerived from the stellar luminosity ratio that best fits the eclipse data (Winn et al. 2006). cProperties of the secondary star, since the primary is never visible. References. — 1. Jensen & Mathieu (1997). 2. Quast et al. (2000). 3. Mekkaden (2000). 4. Mathieu et al. (1991). 5. Mathieu et al. (1995). 6. This work. 7. Prato et al. (2002). 8. Jensen et al. (1996a). 9. Shevchenko & Herbst (1998). 10. Manset & Bastien (2003). 11. Bouvier & Appenzeller (1992). 12. Alencar et al. (2003). 13. Manset et al. (2005). 14. Lee (1992) 15. Walter et al. (1994). 16. Mathieu et al. (1997). 17. Basri et al. (1997). 18. Hamilton et al. (2001). 19. Hamilton et al. (2005). 20. Winn et al. (2006). – 28 – Fig. 1.— The best-fit spectroscopic orbit for UZ Tau E. Crosses show velocities of the pri- mary; those enclosed in boxes (in red in the on-line edition) show new radial velocity mea- surements presented here. Open diamonds show velocities of the secondary from Prato et al. (2002). – 29 – Fig. 2.— The Lomb-Scargle periodogram for the I-band data. The periodogram peaks at a period of 19.20 days, with a false-alarm probability of 0.001. The dashed line shows the binary orbital period of 19.131 days. The smaller peaks visible flanking the main peak (lower panel) are near the alias periods expected for beat periods between one-year and two-year periods (caused by the seasonal gaps in the data) and the binary period. – 30 – Fig. 3.— The Lomb-Scargle periodograms for the B, V , R, and I-band data. All show significant power near the binary orbital period. – 31 – Fig. 4.— The I-band magnitude for UZ Tau E folded at the binary orbital period of 19.131 days and plotted against the binary orbital phase. Top to bottom, data from 2003–2004, 2004–2005, and 2005–2006. – 32 – Fig. 5.— The BV RI magnitudes for UZ Tau E folded at the binary orbital period and plotted against the binary orbital phase. Left, 2003–2004; right, 2005-2006. The open circles in the lower right plot show the VVO I-band data, which do not have corresponding B, V, and R data. – 33 – Fig. 6.— The BV RI magnitudes for UZ Tau E folded at the binary orbital period and plotted against the binary orbital phase, after removing a long-term linear trend from each band. – 34 – Fig. 7.— V − I color vs. I magnitude and vs. orbital phase for 2003–2004 (closed circles) and 2005–2006 (open circles). The system is redder when fainter and bluer when brighter, the expected behavior either for changes in extinction or for brightening due to increased accretion. – 35 – Fig. 8.— Top: Equivalent width of the Hα line as a function of binary orbital phase. Squares are our new measurements; triangles are from Mart́ın et al. (2005). Bottom: For comparison, the phased I-band data. There is some suggestion of reduced Hα equivalent width at phases of 0.4–0.8 as seen in the photometric data, but the data are too sparse there for there to be clear evidence for periodic variability of the Hα emission. – 36 – Fig. 9.— Left: The theoretical predictions of Artymowicz & Lubow (1996) for the depen- dence of accretion rate on binary orbital phase in a binary with mass ratio M2/M1 = 0.43, e = 0.1. The top curve shows the total accretion, while the lower curves show accretion onto the secondary (higher dark curve) and primary (lower dark curve). Right: The same total accretion curve, but placed onto a logarithmic scale and shifted vertically for comparison with the phased I-band data. The model here has been given an ad hoc shift of +0.2 in phase, roughly what is expected given the binary eccentricity (see Section 5.2.2). Introduction Observations Photometry Spectroscopy Binary orbital parameters Periodic variations Photometry Spectroscopy Discussion Could the variations be due to rotation? Evidence for pulsed accretion What are the predictions? How well do the data match the predictions? Evidence for periodic accretion in other T Tauri binaries DQ Tau AK Sco GW Ori V4046 Sgr ROXs 42 and ROXs 43B KH 15D Conclusions

0704.0308

Effect of node deleting on network structure

Effect of node deleting on network structure Ke Deng,∗ Heping Zhao, and Dejun Li Department of Physics, Jishou University, Jishou, Hunan 416000, People’s Republic of China The ever-increasing knowledge to the structure of various real-world networks has uncovered their complex multi-mechanism-governed evolution processes. Therefore, a better understanding to the structure and evolution of these networked complex systems requires us to describe such processes in more detailed and realistic manner. In this paper, we introduce a new type of network growth rule which comprises of adding and deleting of nodes, and propose an evolving network model to investigate the effect of node deleting on network structure. It is found that, with the introduction of node deleting, network structure is significantly transformed. In particular, degree distribution of the network undergoes a transition from scale-free to exponential forms as the intensity of node deleting increases. At the same time, nontrivial disassortative degree correlation develops spontaneously as a natural result of network evolution in the model. We also demonstrate that node deleting introduced in the model does not destroy the connectedness of a growing network so long as the increasing rate of edges is not excessively small. In addition, it is found that node deleting will weaken but not eliminate the small-world effect of a growing network, and generally it will decrease the clustering coefficient in a network. I. INTRODUCTION Network structure is of great importance in the topological characterization of complex systems in reality. Actually, these networked complex systems have been found to share some common structural characteristics, such as the small- world properties, the power-law degree distribution, the degree correlation, and so on [1, 2, 3]. In the theoretical description of these findings, the Watts-Strogatz (WS) model [4] provides a simple way to generate networks with the small-world properties. Barabási and Albert (BA) [5], with a somewhat different aim, proposed an evolving network model to explain the origin of power-law degree distribution. In this model, by considering two fundamental mechanisms: growth and preferential attachment (PA), power-law degree distribution emerges naturally from network evolution. Based on the framework of BA model, many other mechanisms were introduced into network evolution to reproduce some more complex observed network structures [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17], such as the degree distribution of broad scale and single scale [6], as well as the degree correlation [17]. These further studies show that real networked systems may undergo a very complex evolution process governed by multiple mechanisms on which the occurrence of network structures depends. Therefore, to get a better understanding of the structure and evolution of complex networks, describing such processes in more detailed and realistic manner is necessary. In the BA’s framework, the growing nature of real-world networks is captured by a BA-type growth rule. According to this rule, one node is added into the network at each time step, intending to mimic the growing process of real systems. This rule gives an explicit description to the real-network’ growing process which, however, can in fact be much more complex. One fact is that in many real growing networks, there are constant adding of new elements, but accompanied by permanent removal of old elements (deletion of nodes) [18, 19, 20, 21, 22, 23]. Take the food webs for a example: there are both additions and losses of nodes (species) at ecological and evolutionary time scales by means of immigration, emigration, speciation, and extinction [18]. Likewise, for Internet and the World Wide Web (WWW), node-deleting is reported experimentally in spit of their rapid expansion of size [19, 20, 21, 22, 23]. In the Internet’s Autonomous Systems (ASs) map case, a node is an AS and a link is a relationship between two ASs. An AS adding means a new Internet Service Provider (ISP) or a large institution with multiple stub networks joins the Internet. An AS deleting happens due to the permanent shutdown of the corresponding AS as it is, for example, out of business. Investigations of the evolution of real Internet maps from 1997 to 2000 verified such network mechanism [19, 20, 21]. The same is for the evolution of WWW, in which the deletions of invalid web pages are also frequently discovered [22, 23]. In most cases, the deletion of a node is also accompanied with the removal of all edges once attached to it. These facts justify the investigation of node-deletion’s influence on network structure. In this paper, we introduce a new type of network growth rule which comprises of adding and deleting of nodes, and propose an evolving network model to investigate the effect of node deleting on the network structure. Before now several authors have proposed some models on node removal in networks, such as AJB networks in which a portion nodes are simultaneously removed ∗Electronic address: [email protected] http://arxiv.org/abs/0704.0308v1 mailto:[email protected] from the network [24], and also the decaying [25] and mortal [26] networks, which concerns networks’ scaling property and critical behavior respectively. Sarshar et al [27] investigated the ad hoc network with node removal, focusing on the compensatory process to preserve true scale-free state. They are different from present work, in which node deleting is treated as an ubiquitous mechanism accompanied with the evolution of real-world networks. This paper is organized as follows. In Section II, an evolving network model taking account of the effect of node deleting is introduced which reduces to a generalized BA model when the effect of node deleting vanishes. Then the effect of node deleting on network structure are investigated in five aspects: degree distribution (Section III), degree correlation (Section IV), size of giant component (Section V), average distance between nodes (Section VI) and clustering (Section VII). Finally, Section VIII presents a brief summary. II. THE MODEL We consider the following model. In the initial state, the network has m0 isolated nodes. At each time step, either a new node is added into the network with probability Pa or a randomly chosen old node is deleted from the network with probability Pd = 1 − Pa, where Pa is an adjustable parameter. When a new node is added to the network, it connects to m (m 6 m0) existing node in the network according to the preferential probability introduced in the BA model [5], which reads kα + 1 β(kβ + 1) where kα is the degree of node α. When an old node is deleted from the network, edges once attached to it are removed as well. In the model, Pa is varied in the range of 0.5 < Pa ≤ 1, since in the case of Pa 6 0.5 the network can not grow. In order to give a chance for isolated nodes to receive a new edge, we choose preferential probability Πα proportional to kα + 1 [7]. Note that when Pa = 1, our model reduces to a generalized BA model [28]. To get a general knowledge to the effect of node deleting on network structure, firstly, a simple analysis to the surviving probability D(i, t) is helpful. Here, D(i, t) is defined as the probability that a node is added into the network at time step i, and this node (the ith node) has not been deleted until time step t, where t > i. Supposing that a node-adding event happens at time step i , and the probability that the i′th node has not been deleted until time step t is denoted as D′(i′, t). Then, due to the independence of events happened at each time step, it is easy to verify that D′(i′, t + 1) = D′(i′, t)[1 − (1 − Pa)/N(t)] with D′(i′, i′) = 1, where N(t) = (2Pa − 1)t is the number of nodes in the network at moment t (in the limit of large t). In the continuous limit, we obtain ∂D′(i′, t) = − (1− Pa) (2Pa − 1)t D′(i′, t), (2) which yields D′(i′, t) = )−(1−Pa)/(2Pa−1) . (3) Thus to get the D(i, t) we should multiply D′(i′, t) with Pa, i.e. D(i, t) = Pa )−(1−Pa)/(2Pa−1) . (4) One can easily find that D(i, t) decreases rapidly as t increases and/or as i decreases provided 0.5 < Pa < 1. It is well known that highly connected nodes, or hubs, play very important roles in the structural and functional properties of growing networks [1, 2, 3]. The formation of hubs needs a long time to gain a large number of connections. As a consequence, according to Eq. (4), a large portion of potential hubs are deleted during the network evolution. Thus it can be expected that the introduction of node deleting has nontrivial effects on network structure. In the following we show how network structure can be effected by the node deleting introduced in present model. III. DEGREE DISTRIBUTION The degree distribution p(k), which gives the probability that a node in the network possesses k edges, is a very important quantity to characterize network structure. In fact, p(k) has been suggested to be used as the first criteria to classify real-world networks [6]. Therefore it is necessary to investigate the effect of node deleting on the degree distribution of networks firstly. Now we adopt the continuous approach [29] to give a qualitative analysis of p(k) for our model with slight node deletion (i.e., when Pd is very small). Supposing that there is a node added into the network at time step i′, and this node is still in the network at time t, let k(i′, t) be the degree of the i′th node at time t, where t > i′. Then the increasing rate of k(i′, t) is ∂k(i′, t) = Pam k(i′, t) + 1 − (1− Pa) k(i′, t) , (5) where S(t) = D′(i′, t)[k(i′, t) + 1] (6) and the ′ denotes the sum of all i′ during the time step between 0 and t. It is easy to verify that the first term in Eq. (5) is the increasing number of links of the i′th node due to the preferential attachment made by the newly added node. The second term in Eq. (5) accounts for the losing of a link of the i′th node during the process of node deletion, which happened with the probability k(i′, t)/N(t). Firstly we solve for the S(t) and get S(t) = (2Pa − 1) (2Pam+ 1) t (7) (see the Appendix for details). Inserting Eq. (7) back into Eq. (5), one gets ∂k(i′, t) Ak(i′, t) +B , (8) where 2P 2am− Pam+ Pa − 1 (2Pa − 1)(2Pam+ 1) (2Pa − 1)(2Pam+ 1) . (10) When Ak +B > 0, the solution of Eq. (8) is k(i′, t) = (Am+B) . (11) Now, to get the probability p(k, t) that a randomly selected node at time t will have degree k, we need to calculate the expected number of nodes Nk(t) with degree k at time t. Then the p(k, t) can be obtained from p(k, t) = Nk(t)/N(t), where N(t) is the total number of nodes at time t. Let Ik(t) represent the set of all possible nodes with degree k at time t, then one gets p(k, t) = Nk(t) i∈Ik(t) D(i, t). (12) In the continuous-time approach, the number of nodes in Ik(t) is the number of i’s for which k 6 k(i, t) 6 k + 1, and it is approximated to |∂k(i, t)/∂i|−1i=ik , where ik is the solution of the equation k(i, t) = k. To proceed with our analysis, now we make the approximation that all nodes in Ik(t) have the same surviving probability D(ik, t) [44]. Under this mean-field approximation, Eq. (12) can be written as p(k, t) = D(ik, t) ∂k(i, t) . (13) From Eq. (11), we obtain Ak +B t. (14) 1 2 3 4 5 6 7 8 9 10 FIG. 1: Pmina [defined in Eq. (20)] as a function of m. ∂k(i, t) = (Am+B) t (Ak +B) −(A+1)/A . (15) Inserting Eq. (14) back into Eq. (4) we get D(ik, t) = Pa Ak +B )(A−B)/A Inserting Eqs. (15) and (16) into Eq. (13), and noting that N(t) = (2Pa − 1)t, we get p(k, t) = 2Pa − 1 (Am+B) (B−A+1)/A (Ak +B) −(B+1)/A , (17) which is a generalized power-law form with the exponent B + 1 = 2 + Pam+ 1 2P 2am− Pam+ Pa − 1 . (18) We point out again that equation (11) is only valid when Ak +B > 0, which translates into A > 0, i.e. 2P 2am− Pam+ Pa − 1 > 0. (19) Considering that Pa > 0.5, Eq. (19) is satisfied when Pa > P (m− 1) + m2 + 6m+ 1 . (20) In Fig. 1, we plot Pmina as a function of m. One can see from Fig. 1 that the curve divides our model into two regimes. (i) Pa > P a : in this case Ak + B > 0 and equation (11) is valid. Thus, the degree distribution of the network p(k) exhibits a generalized power-law form. (ii) Pa > P a : In this case Ak + B > 0 can not be always satisfied and equation (11) is not valid. Therefore, our continuous approach fails to predict the behavior of p(k), and we will investigate it with numerical simulations. The Pmina (m), as one can find from Fig. 1, decreases with the increase of In the power-law regime [Pa > P a (m)], the behavior of p(k) is predicted by Eqs. (17) and (18), which are obtained using a mean-field approximation [Eq. (13)]. One can easily verify that such approximation is only exact when Pa = 1, in which case Eq. (18) turns into γ = 3 + 1/m, in good agreement with the results obtained from generalized BA 100 101 102 103 1.00 0.95 0.90 power-law fit pa=1.0 pa=0.95 pa=0.9 pa=0.8 pa=0.7 pa=0.6 pa=0.55 pa=0.51 exponential fit FIG. 2: Cumulative degree distribution P (k) for networks with system size N = 100000 and different values of Pa, in logarithmic scales. The dash line is power-law fit for Pa = 1. The solid line is the exponential fit for Pa = 0.51. In the simulation, we set m0 = m = 5 and each distribution is based on 10 independent realizations. Inset plots the power-law exponential γ as a function of Pa. The continuous curve is according to the analytic result of Eq. (18), and circles to the simulation results. model studied in Ref [28]. If Pmina (m) < Pa < 1, Eqs. (17) and (18) still give qualitative predictions for the model: with slight node deletion, p(k) of the network is still power-law, and the exponential γ increases with the decrease of Pa (inset of Fig. 2). In remaining regime [Pa < P a (m)], the limiting case is Pa → 0.5, in which the growth of network is suppressed (a very slowly growing one). Similar non-growing networks have been studied, for example, for the Model B in Ref[30], and the degree distribution has the exponential form. Here we conjecture that, in this regime, p(k) of our model crossovers to an exponential form, which is verified by the numerical simulation results below. Now we verify the above analysis with numerical simulations. In Fig. 2, we give the cumulative degree distributions P (k) [3] of the networks with different Pa. As Pa gradually decreases from 1 to 0.5, Fig. 2 shows an interesting transition process which can be roughly divided into three stages. (1) 0.9 6 Pa 6 1: In this stage, the model works in the power-law regime and the power-law exponent γ increases as Pa decreases. Inset of Fig. 2 gives the comparison between the value of γ predicted by Eq. (18) and the one obtained from numerical simulations. One sees that the theory and the simulation results are in perfect agreement for Pa = 1. As Pa decreases, however, the agreement is only qualitative and the deviation between theory and simulation becomes more and more obvious. As we have mentioned above, such increasing deviation is due to the mean-field approximation used in the analysis. These results tell us that slight node deletion does not cause deviation of the network from scale-free state, but only increases its power-law exponent. Such robustness of power-low p(k) revealed here gives an explanation to the ubiquity of scale-free networks in reality. It should be noted that a very similar robustness has also been found in the study of network resilience, where simultaneously deleting of a portion of nodes was taken into account in static scale-free networks [24]. (2) 0.5 < Pa 6 0.6: In this stage, the model works in the regime of Pa < P a (m). As one sees from Fig. 2, P (k) of the network behaviors exponentially. This result indicates that with manifest node deletion, the network will deviate from scale-free state and become exponential. (3) 0.6 < Pa < 0.9: In this stage, a crossover of the model from the power-law regime to the exponential regime is found, in which the P (k) is no longer pure scale-free but truncated by an exponential tail. As one can see, the truncation in P (k) increases as Pa decreases. Besides the power-law degree distribution, it is now known that p(k) in real world may deviate from a pure power- law form [18, 31, 32, 33, 34]. According to the extent of deviation, p(k) of real systems has been classified into three groups [6]: scale-free (pure power-law), broad scale (power-law with a truncation), and single scale (exponential). Many mechanisms, such as aging [6, 8, 9], cost [6], and information filtering [10], have been introduced into network growth to explain these distributions. Here, the results of Fig. 2 indicate that a modified version of growth rule can lead to all the three kinds of p(k) in reality, and it provides another explanation for the origin of the diversity of degree distribution in real-world: such diversity may be a natural result of network growth. 0.0 2.0x104 4.0x104 6.0x104 8.0x104 1.0x105 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 pa=1 pa=0.9 pa=0.8 pa=0.7 pa=0.6 pa=0.55 FIG. 3: Assortativity coefficient r plotted with network size N , for different Pa in the model. In the simulation, m0 = m = 5. Result of each curve is based on 10 independent realizations. IV. DEGREE CORRELATION It has been recently realized that, besides the degree distribution, structure of real networks are also characterized by degree correlations [19, 35, 36, 37, 38]. This translates into the fact that degrees at the end of any given edge in real networks are not usually independent, but are correlated with one another, either positively or negatively. A network in which the degrees of adjacent nodes are positively (negatively) correlated is said to show assortative (disassortative) mixing by degree. An interesting observation emerging from the comparing of real networks of different types is that most social networks appear to be assortatively mixed, whereas most technological and biological networks appear to be disassortative. The level of degree correlation can be quantified by the assortativity coefficient r lying in the range −1 6 r 6 1, which can be written as i jiki − (ji + ki) (j2i + k (ji + ki) for practical evaluation on an observed network, where ji, ki are the degrees of the vertices at the ends of the ith edge, with i = 1, . . . ,M [35]. This formula gives r > 0(r < 0) when the corresponding network is positively (negatively) correlated, and r = 0 when there is no correlation [45]. Recently, Maslov et al [39] and Park et al [40] have proposed a possible explanation for the origin of such correlation. They show for a network the restriction that there is at most one edge between any pair of nodes induces negative degree correlations. This restriction seems to be an universal mechanism (indeed, there is no double edges in most real networks), therefore, the authors of Ref. [40] conjecture that disassortativity by degree is the normal state of affairs for a network. Although only a part of the measured correlation can be explained in the way of Ref. [40], this universal mechanism does give a promising explanation for the origin of degree correlation observed in real networks of various types. It will be of great interest to discuss the effect of node deleting on degree correlation. In Fig. 3, we give the assortativity coefficient r as a function of network size N , for different Pa in our model, for m = 5. As one sees from Fig. 3, for each value of Pa, after a transitory period with finite-size effect, each r of networks tends to reach a steady value. When Pa = 1, r → 0 as N becomes large. This result indicates that networks in the BA model are uncorrelated, in agreement with results obtained in previous studies [35, 38]. When Pa < 1, nontrivial negative degree correlations spontaneously develop as networks evolve. One can see from Fig. 3 that the steady value of r in the model decreases with the decreasing Pa. In particular, when Pa 6 0.6, the value of r is about −0.1. These results indicate that node deleting leads to disassortative mixing by degree in evolving networks. To make such relation more clear, in Fig. 4, we plot r of networks in our model as a function of Pa, for different m. As the Fig. 3 indicates, when the network size is larger than 40000, the assortativity coefficient r is nearly stable. So all results in Fig. 4 are obtained from networks with N = 40000. Fig. 4 gives us the same relation between r and Pa shown in Fig. 3. What is more, it tells us that for a given Pa, r will increase with the increasing m. The increment gets its maximum between m = 1 1.0 0.9 0.8 0.7 0.6 0.5 -0.20 -0.16 -0.12 -0.08 -0.04 m=14 m=15 FIG. 4: Assortativity coefficient r as a function of Pa, for different m in the model. In the simulation, N = 40000. Result of each curve is based on 10 independent realizations. 0.0 2.0x104 4.0x104 6.0x104 8.0x104 1.0x105 -0.06 pa=1 pa=0.9 pa=0.8 pa=0.7 pa=0.6 pa=0.55 FIG. 5: Assortativity coefficient r plotted with network size N , for different Pa in the randomly growing network model. In the simulation, m0 = m = 5 and each curve is based on 10 independent realizations. and other values. We point out that this is because when m = 1, the network has been broke up into small separate components (see the following section). We can also find from Fig. 4 that the gap between different curves decreases with the increasing m and the curves tend to merge at large m. Now we give some explanations to the above observations. In the BA model, the network being uncorrelated is the result of a competition between two factors: the growth and the preferential attachment (PA). On the one hand, networks with pure growth is positively correlated. This is because the older nodes, also tending to be higher degree ones, have a higher probability of being connected to one another, since they coexisted earlier. In Fig. 5, we compute the assortativity coefficient r of a randomly growing network, which grows by the growth rule of BA-type, while the newly added nodes connect to randomly chosen existing ones. As one can see from Fig. 5 that pure growth leads to positive r. On the other hand, the introduction of PA makes the connection between nodes tend to be negatively correlated, since newly added nodes (usually low degree ones) prefer to connect to highly connected ones. Then degree correlation characteristic of the BA model is determined by this two factors. In Fig. 6, we plot the average degree of the nearest neighbor < k >nn as a function of k in the BA model. It is found that nodes with large k show no obvious biases in their connections. But there is a short disassortative mixing region when k is relatively small (also reported in Ref. [41], see Fig.1a therein). Such phenomenon can be explained by the effect of these two factor: 10 20 30 40 50 60 70 80 90 100 FIG. 6: Average degree of the nearest neighbor as a function of k for the BA model. In the simulation, N = 10000 and m = m0 = 5. Result of each curve is based on 1000 independent realizations. 1.0 0.9 0.8 0.7 0.6 0.5 1.0 0.9 0.8 0.7 0.6 0.5 FIG. 7: The relative size of the largest component S as a function of Pa for m = 2, 3, 4, 5. Inset gives the same curve for m = 1. In the simulations, N = 100000. All results are based on 10 independent realizations. Growth together with PA makes nodes with large k equally connect to both large and small degree nodes, and the latter makes nodes with small degree be disassortatively connected. Now, we introduce node-deletion. According to Eq. (4), depression of the growth of large-degree nodes also decreases the connections between them, therefore makes the correlation negative. We also investigate the effect of node deleting on the r of the randomly growing network, and obtained similar results. As one sees from Fig. 5, depression of connections between higher degree nodes causes the network less positively correlated, and with stronger node-deletion, negatively correlated. Finally, with regard to the effect of m in this relation (Fig. 4), larger m means more edges are established according to the PA probability Eq. (1). We conjecture that the orderliness of newly added nodes connecting to large degree nodes will be weakened by the increasing randomness as m becomes larger, thus leading to a less negative correlation. Such randomness can not always increase and, as we see from Fig. 4, for large m, e.g., m ≥ 14, the curves tend to merge together. V. SIZE OF GIANT COMPONENT In a network, a set of connected nodes forms a component. If the relative size of the largest component S in a network approaches a nonzero value when the network is grown to infinite size, this component is called the giant component of the network [1, 2, 3]. In most previously studied growing models [1, 2, 3], due to the BA-type growth rule they adopted, there is only one huge component in the network, i.e., S ≡ 1. In this extreme case the network gains a perfect connectedness. The opposite case of S = 1 is the extreme of S = 0, in which case the network, made up of small components, exhibits no connectedness. Experiments indicate that some real networks seem to lie in somewhere between these two extreme: they contain a giant component as well as many separate components [2, 3, 42, 43]. For example, According to Ref.[42], in May of 1999, the entire WWW, containing 203 × 106 pages, consisted of a giant component of 186× 106 pages and the disconnected components (DC) of about 17× 106 pages. In general, the introduction of node deletion in our model will cause the emergence of separate components even isolated nodes in the network. What we interest here is the connectedness of the network. In Fig. 7 we plot the relative size of the largest component S in the model, as a function of Pa, for m = 2, 3, 4, 5, where m is the number of edges generated with the adding of a new node. One sees from Fig. 7 that for any 0.5 < Pa ≤ 1, a giant component can be observed in the model if m > 1. In addition, for the same Pa, S increase as the increase of m. While when m = 1, the network is found to be broke up into separate components if Pa < 1. For example, when Pa = 0.9, S of the network with N = 100000 rapidly drops to 0.034. Inset of Fig. 7 gives the S Vs Pa curve for m = 1. These results indicate that node deleting does not destroy the connectedness of a growing network so long as the increasing rate of edges is not excessively small. VI. AVERAGE DISTANCE BETWEEN NODES Now we study the effect of node deletion on networks’ average distance L between nodes. Here the distance between any two nodes is defined as the number of edges along the shortest path connecting them. It has been revealed that, despite their often large size, most real networks present a relatively short L, showing the so-called small-world effect [1, 2, 3, 4]. Such an effect has a more precise meaning: networks are said to show the small-world effect if the value of L scales logarithmically or slower with network size for fixed mean degree. This logarithmic scaling can be proved for a variety of network models [1, 2, 3]. As we have demonstrated in Section V, node deleting does not destroy the connectedness of the network in our model for any m > 1, since there is always a giant component exists. Here in our simulation, we calculate L of the giant component of the network in our model using the burning algorithm [3]. In Fig. 8, we plot L as a function of network size N , for different Pa in our model. As one can see from the figure, for any 0.5 < Pa ≤ 1, a logarithmic scaling L ∼ lnN is obtained, while the proportional coefficient increases with the decrease of Pa. Furthermore, for a given N , L increases with the decrease of Pa. These results tell us that node deleting will weaken but not eliminate the small-world effect of a growing network. VII. CLUSTERING Finally, we investigate the effect of node deletion on network’s cluster coefficient C, which is defined as the average probability that two nodes connected to a same other node are also connected. For a selected node i with degree ki in the network, if there are Ei edges among its ki nearest neighbors, the cluster coefficient Ci of node i is defined as ki (ki + 1) . (22) Then the clustering coefficient of the whole network is the average of all individual Ci. In Fig. 9, we plot C of the giant component in the network as a function of network size N , for different Pa. As one sees from Fig. 9, for each Pa, the clustering coefficient C of our model decreases with the network size, following approximately a power law form. Such size-dependent property of C is shared by many growing network model [1, 2, 3]. Moreover, as Fig. 9 shows, for the same network-size N , C decreases as Pa decreases. The results of Fig. 9 indicate that node deleting weakens network’s clustering. VIII. CONCLUSION In summary, we have introduced a new type of network growth rule which comprises of adding and deleting of nodes, and proposed an evolving network model to investigate effects of node deleting on network structure. It has been 102 103 104 105 Pa=1 Pa=0.9 Pa=0.8 Pa=0.7 Pa=0.6 Pa=0.55 Pa=0.51 FIG. 8: Average distance L of the giant component in the network as a function of network size N , for different Pa in the model. The chose of some parameters: m0 = m = 5. These curves are results of 10 independent realizations. 102 103 104 105 Pa=1 Pa=0.9 Pa=0.8 Pa=0.7 Pa=0.6 Pa=0.55 FIG. 9: Cluster coefficient C of the giant component in the network as a function of network size N , for different Pa. In the simulation we set m0 = m = 5. These curves are results of 10 independent realizations. found that, with the introduction of node deleting, network structure was significantly transformed. In particular, degree distribution of the network undergoes a transition from scale-free to exponential forms as the intensity of node deleting increased. At the same time, nontrivial disassortative degree correlation spontaneously develops as a natural result of network evolution in the model. We also have demonstrated that node deleting introduced in our model does not destroy the connectedness of a growing network so long as the increasing rate of edge is not excessively small. In addition, it has been observed that node deleting will weaken but not eliminate the small-world effect of a growing network. Finally, we have found that generally node deleting will decrease the clustering coefficient in a network. These nontrivial effects justify further studies of the effect of node deleting on network function [3], which include topics such as percolation, information and disease transportation, error and attack tolerance, and so on. Acknowledgments The authors thank Doc. Ke Hu for useful discussions. This work is supported by the National Natural Science Foundation of China, Grant No. 10647132, and Natural Science Foundation of Hunan Province, China, Grant No. 00JJY6008. APPENDIX: THE CALCULATION OF S(T ) To get S(t), we multiply both sides of Eq. (5) by D′(i′, t) and sum up all i′ between 0 and t: ∂k(i′, t) D′(i′, t) = Pa(m− 1)− 1− Pa (2Pa − 1)t S(t) + 1. (A.1) To get the above equation we have used the definition of S(t) [Eq. (6)] and the following equation: D′(i′, t) = D(i, t)di. (A.2) The left-hand side of Eq. (A.1) can be simplified as: ∂ {[k(i′, t) + 1]D′(i′, t)} [k(i′, t) + 1] ∂D′(i′, t) [k(i′, t) + 1]D′(i′, t) − [k(t, t) + 1]D(t, t) [k(i′, t) + 1]D′(i′, t) Pa − 1 (2Pa − 1)t Substituting the above expression in Eq. (A.1), and noting that k(t, t) = m and D(t, t) = Pa, we get ∂S(t) 2(Pa − 1) (2Pa − 1)t S(t) + 2Pam+ 1. The solution to the above equation is S(t) = (2Pa − 1) (2Pam+ 1) t. [1] R. Albert and A.-L. Barabási, Rev. Mod. 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Zaliznyak, e-print cond-mat/0205379. [40] J. Park and M. E. J. Newman, Phys. Rev. E 68, 026112 (2003). [41] Huang Zhuang-Xiong, Wang Xin-Ran and Zhu Han, Chinese Physics 13, 273 (2004). [42] A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, and J. Wiener, in Proceedings of the 9th WWW Conference (Elsevier, Amsterdam, 2000), p. 309. [43] S. N. Dorogovtsev, J. F. F. Mendes and A. N. Samukhin, Phys. Rev. E 64, 025101(R) (2001). [44] It seems that this is not a very good approximation, since investigations indicate that values of (∂D(i, t)/∂i) |i=ik are large and increase rapidly with the decrease of Pa. Thus the analysis here is a qualitative one and only suit for the condition of slight node deletion in the model. [45] Another way to represent degree correlation is to calculate the mean degree of the nearest neighbors of a vertex as a function of the degree k of that vertex. Although such way is explicit to characterize degree correlation for highly heterogeneously organized networks, for less heterogeneous networks (this is the case in the proposed model when the intensity of node deleting increases, see Fig. 2), it may be very nosy and difficult to interpret. So here we adopt the assortativity coefficient r to characterize degree correlation in the model. http://arxiv.org/abs/cond-mat/0205379 INTRODUCTION THE MODEL DEGREE DISTRIBUTION DEGREE CORRELATION SIZE OF GIANT COMPONENT AVERAGE DISTANCE BETWEEN NODES CLUSTERING CONCLUSION Acknowledgments THE CALCULATION OF S(T) References

0704.0309

The Complexity of HCP in Digraps with Degree Bound Two

The Complexity of HCP in Digraps with Degree Bound Two Guohun Zhu Guilin University of Electronic Technology, No.1 Jinji Road,Guilin, Guangxi, 541004,P.R.China [email protected] Abstract. The Hamiltonian cycle problem (HCP) in digraphs D with degree bound two is solved by two mappings in this paper. The first bijection is between an incidence matrix Cnm of simple digraph and an incidence matrix F of balanced bipartite undirected graph G; The sec- ond mapping is from a perfect matching of G to a cycle of D. It proves that the complexity of HCP in D is polynomial, and finding a second non-isomorphism Hamiltonian cycle from a given Hamiltonian digraph with degree bound two is also polynomial. Lastly it deduces P = NP base on the results. 1 Introduction It is well known that the Hamiltonian cycle problem(HCP) is one of the standard NP-complete problem [1]. As for digraphs, even when the digraphs on this case: planar digraphs with indegree 1 or 2 and outdegree 2 or 1 respectively, it is still NP − Complete which is proved by J.Plesńık [2]. Let us named a simple strong connected digraphs with at most indegree 1 or 2 and outdegree 2 or 1 as Γ digraphs. This paper solves the HCP of Γ digraphs with following main results. Theorem 1. Given an incidence matrix Cnm of Γ digraph, building a mapping:F = , then F is a incidence matrix of undirected balanced bipartite graph G(X,Y ;E), which obeys the following properties: c1. |X | = n,|Y | = n,|E| = m ∀xi ∈ X ∧ 1 ≤ d(xi) ≤ 2 ∀yi ∈ Y ∧ 1 ≤ d(yi) ≤ 2 c3. G has at most n components which is length of 4. Let us named the undirected balanced bipartite graph G(X,Y : E) of Γ digraph as projector graph. http://arxiv.org/abs/0704.0309v3 Theorem 2. Let G be the projector graph of a Γ graph D(V,A), determining a Hamiltonian cycle in Γ digraph is equivalent to find a perfect match M in G and r(C′) = n − 1, where C′ is the incidence matrix of D′(V, L) ⊆ D and L = {ai|ai ∈ D ∧ ei ∈ M}. Let the each component of G corresponding to a boolean variable, a mono- tonic function f(M) is build to represents the number of component in D. Based on this function, the maximum number of non-isomorphism perfect matching is linear, thus complexity of Γ digraphs has a answer. Theorem 3. Given the incidence matrix Cnm of a Γ digraph , the complexity of finding a Hamiltonian cycle existing or not is O(n4) The concepts of cycle and rank of graph are given in section 2. Then theorems 1,2,3 are proved in sections 3,4,5 respectively. The last section discusses the P versus NP in more detail. 2 Definition and properties Throughout this paper we consider the finite simple (un)directed graph D = (V,A) (G(V,E), respectively), i.e. the graph has no multi-arcs and no self loops. Let n and m denote the number of vertices V and arcs A (edges E, respectively), respectively. As conventional, let |S| denote the number of a set S. The set of vertices V and set of arcs of A of a digraph D(V,A) are denoted by V = {vi|1 ≤ i ≤ n} and A = {aj |(1 ≤ j ≤ m) ∧ aj =< vi, vk >, (vi 6= vk ∈ V )} respectively, where < vi, vk > is a arc from vi to vk. Let the out degree of vertex vi denoted by d+(vi), which has the in degree by denoted as d −(vi) and has the degree d(vi) which equals d +(vi) + d −(vi). Let the N +(vi) = {vj | < vi, vj >∈ A}, and N−(vi) = {vj | < vj , vi >∈ A}. Let us define a forward relation ⊲⊳ between two arcs as following, ai ⊲⊳ aj = vk iff ai =< vi, vk > ∧aj =< vk, vj >. It is obvious that |ai ⊲⊳ ai| = 0 . A cycle L is a set of arcs (a1, a2, . . . , al) in a digraph D, which obeys two conditions: c1. ∀ai ∈ L, ∃aj , ak ∈ L \ {ai}, ai ⊲⊳ aj 6= aj ⊲⊳ ak ∈ V c2. | ai 6=aj∈L ai ⊲⊳ aj | = |L| If a cycle L obeys the following conditions, it is a simple cycle. c3. ∀L′ ⊂ L, L′ does not satisfy both conditions c1 and c2. A Hamiltonian cycle L is also a simple cycle of length n = |V | ≥ 2 in digraph. As for simplify, this paper given a sufficient condition of Hamiltonian cycle in digraph. Lemma 1. If a digraph D(V,A) include a sub graph D′(V, L) with following two properties, the D is a Hamiltonian graph. c1. ∀vi ∈ D′ → d+(vi) = 1 ∧ d−(vi) = 1, c2. |L| = |V | ≥ 2 and D′ is a strong connected digraph. A graph that has at least one Hamiltonian cycle is called a Hamiltonian graph. A graph G=(V ;E) is bipartite if the vertex set V can be partitioned into two sets X and Y (the bipartition) such that ∃ei ∈ E, xj ∈ X, ∀xk ∈ X \ {xj}, (ei ⊲⊳ xj 6= ∅ → ei ⊲⊳ xk = ∅) (ei, Y , respectively). if |X | = |Y |, We call that G is a balanced bipartite graph. A matching M ⊆ E is a collection of edges such that every vertex of V is incident to at most one edge of M , a matching of balanced bipartite graph is perfect if |M | = |X |. Hopcroft and Karp shows that constructs a perfect matching of bipartite in O((m + n) n) [3]. The matching of bipartite has a relation with neighborhood of X . Theorem 4. [4] A bipartite graph G = (X,Y ;E) has a matching from X into Y if and only if |N(S)| ≥ S, for any S ⊆ X. Lemma 2. A even length of simple cycle consist of two disjoin perfect matching. Two matrices representation for graphs are defined as follows. Definition 1. [5] The incidence matrix C of undirected graph G is a two di- mensional n×m table, each row represents one vertex, each column represents one edge, the cij in C are given by cij = 1, if vi ∈ ej; 0, otherwise. It is obvious that every column of an incidence matrix has exactly two 1 entries. Definition 2. [5] The incidence matrix C of directed graph D is a two dimen- sional n×m table, each row represents one vertex, each column represents one arc the cij in C are given by cij = 1, if < vi, vi >⊲⊳ aj = vi; −1, if aj ⊲⊳< vi, vi >= vi; 0, otherwise. It is obvious to obtain a corollary of the incidence matrix as following. Corollary 1. Each column of an incidence matrix of digraph has exactly one 1 and one −1 entries. Theorem 5. [5] The C is the incidence matrix of a directed graph with k com- ponents the rank of C is given by r(C) = n− k (3) In order to convince to describe the graph D properties, in this paper, we denotes the r(D) = r(C). 3 Divided incidence matrix and Projector incidence matrix Firstly, let us divided the matrix of C into two groups. C+ = {cij |cij ≥ 0 otherwise 0 } (4) C− = {cij |cij ≤ 0 otherwise 0 } (5) It is obvious that the matrix of C+ represents the forward arc of a digraph and C− matrix represents the backward arc respectively. A corollary is deduced as following. Corollary 2. A digraph D = (V,A) is strong connected if and only if the rank of divided incidence matrix satisfies r(C+) = r(C−) = |V |. Secondly, let us combined the the C+ and C− as following matrix. In more additional, let F represents as an incidence matrix of undirected graph G(X,Y ;E). The F is named as projector incidence matrix of C and G is named as projector graph , where X represents the vertices V + of D, Y represents the vertices of V − respectively. In another words we build a mapping F : D → G and denotes it as G = F (D). So the F (D) has 2n vertices and m edges if D has n vertices and m arcs. We also build up a reverse mapping: F−1 : G → D When G is a projector graph. To simplify, we also denotes the arcs ai = F −1(ei), v i = F −1(xi) and v i = F −1(yi). 3.1 Proof of Theorem 1 Firstly, let us prove the theorem 1. Proof. c1. Since Γ digraph is strong connected, then each vertices of Γ digraph has at least one forward arcs, each row of C+ has at least one 1 entries, and the U represents the C+ , so |U | = n the same principle of C−, each row of C− has at least one −1 entries, and the V represents the C− , so |V | = n Since the columns of F equal to the columns of C, |E| = m c2. Since the degree of each vi of Γ digraph is 1 ≤ d+(vi) ≤ 2, ∀ui ∈ U ∧ 1 ≤ d(ui) ≤ 2 Since the degree of each vi of Γ digraph is 1 ≤ d−(vi) ≤ 2, ∀vi ∈ V ∧ 1 ≤ d(vi) ≤ 2 c3. Let us prove by contradiction, suppose there are k > n components with length of 4 in G. Since D is strong connected, according to the corollary 2, r(F ) = 3n − q ≥ r(C+) = n, where q ≥ k is number of components (in- cluding k components with length of 4). Thus q ≤ n , then there are only x components without length 4, where x is x = q − k < n Suppose the remind x components with length of t (at least t vertices con- nected by some edges), then 4k + xt = 3n . So tx = 3n − 4k < n . According to the equation 7, the t < 2. It is contradict that the D is strong connected. 3.2 The cycle in digraph corresponding matching in projector graph Secondly, let us given the properties after mapping Hamiltonian cycle L of D into the sub graph M of projector graph G. Lemma 3. If a Hamiltonian cycle L of D mapping into a forest M of projector graph G, the forest M consist of |L| number of trees which has only two node and one edge, and M has a unique perfect matching. Proof. Let the Γ digraph D(V,A) has a sub digraph D′(V, L) which exists one Hamiltonian cycle and |L| = n, the incidence matrix C of L could be permutation as follows. 1 0 0 . . . 0 −1 −1 1 0 . . . 0 0 0 −1 1 . . . 0 0 0 0 −1 . . . 0 0 0 0 0 . . . 0 0 0 0 0 . . . −1 1 . (8) It is obvious that each row of F has only one 1 entry and each column of F has two 1 entries. According to theorem 1, F represents a balanced bipartite graph G(X,Y ;E) that each vertex has one edge connected, and each edge ei connect on vertex xi ∈ X , another in Y , in another words, ∃ei ∈ E xj ∈ X ,∀xk ∈ X \ {xj}, ei ⊲⊳ xj 6= ∅ → ei ⊲⊳ xk = ∅(ei, Y ,respectively). According the matching definition, M is a matching, since |E| = |L|, E is a perfect matching. and pair of vertices between X and Y only has one edge, so M is a forest, and each tree has only two node with one edge. 4 Proof of Theorem 2 Proof. ⇒ Let the Γ digraph D(V,A) has a sub digraph D′(V, L) which is a Hamiltonian cycle and |L| = n, let matrix C′ represents the incidence matrix of D′, so r(C′) = n − 1; According to lemma 3, the projector graph F (D′) has a perfect matching, thus F (D) also has a perfect matching. ⇐ Let G(X,Y ;E) be a projector graph of the Γ graphD(V,A),M is a perfect matching in G. Let D′(V, L) be a sub graph of D(V,A) and L = {ai|ai ∈ D∧ei ∈ M}. Since r(L) = n− 1, D′(V, L) is a strong connected digraph. it deduces that ∀vi ∈ D′,d+(vi) ≥ 1 ∧ d−(vi) ≥ 1. Suppose ∃vi ∈ D′, d+(vi) > 1 (d−(vi) > 1 respectively), Since |M | = n, it deduces that i=1 d(vi) > 2n+ 1, which imply that |L| > n. this is contradiction with L = {ai|ai ∈ D ∧ ei ∈ M} and |M | = n. So ∀vi ∈ D′, d+(vi) = d−(vi) = 1, According the lemma 1, D′ has a Hamiltonian cycle. 5 Number of perfect matching in projector graph Let us considering the number of perfect matching in G . Firstly, let us consid- ering a example as shown in figure 1. Figure 1. Original Digraph D Then the projector graph is shown in figure 2. Figure 2. Projector graph G . . . Given a perfect matching M , each component(cycle) in G has two partition edges belong to M . Let us code component Gi which |Gi| > 2 and matching M to a binary variable. 1, if Gi ∩M = {ej, ek, . . .}; 0, if Gi ∩M = {el, eq, . . .}. Now there are two cases for the number of perfect matching. Label edge. In that cases, the Code(M1) = {0, 0, 1} is different with Code(M2) = {0, 1, 0}. If there are k number of components(cycles), then there are 2k perfect match- Unlabel edge. In that cases, the Code(M1) = {0, 0, 1} is isomorphic to Code(M2) = {0, 1, 0}. The same principle that Code(M3) = {0, 1, 1} is isomorphic to Code(M4) = {1, 1, 0} but is not isomorphic to Code(M1). Then let us summary the maximal number of perfect matching in these two cases. Lemma 4. The maximal number of labeled perfect matching in a projector graph G is 2 4 , but the maximal number of unlabeled perfect matching in a projector graph G is n Proof. According to the theorem 1, there at most n components with a compo- nents which is length of k = 4. When k=2, there are only one perfect matching in G; When k = 4, there are n components which is C4, and so on when k = 6, there are n components which is C6, etc, so on. According to the lemma 2, each simple cycle has divided the perfect matching into two class. So maximal number perfect matching in the non isomorphism cycle which is 2 4 . Since in unlabeled cases, every C4 cycle is isomorphism, the maximal number of perfect matching is 2 ∗ n Review the example 1 again, it is easy find that follow proposition. Proposition 1. Given two perfect matching M1 and M2 in projector graph G, if code(M1) = code(M2), then the r(F−1(M1)) = r(F−1(M2)). 5.1 Proof of Theorem 3 Now let us proof the theorem 3. Proof. Let G be a project balanced bipartition of D. According theorem 1, the Γ graph is equivalent to find a perfect match M in a project G. According to the lemma 4, the maximal number non isomorphism perfect matching in G is only n. Thus it is only need exactly enumerate all of non isomorphism perfect match- ing M , then obtain the value = r(F−1(M)),if value = n − 1, then the ei ∈ M is also ei ∈ C, where C ⊂ D is a Hamiltonian cycle. Since the complexity of rank of matrix is O(n3), finding a simple cycle in a component with degree 2 is O(n2), and obtaining a perfect matching of a bipartite graph is O((m+ n) n) < O(n2) [3]. Then all exactly algorithms need to calculate the n time o(n3). Thus the complexity is O(n4). Since the non isomorphism perfect matching comes from the coding of edges in the component of G, it is not easy implementation. Let us give two recursive equation to obtain a perfect matching M from G. Suppose there are k component G1, G2, . . .Gk in G where Gi is a component with degree 2 and |Ei| ≥ 3. M ′ = M(t)⊗Gt, Gt is a cycle ; M(t), otherwise. M(t+ 1) = M ′, if r(F−1(M ′)) > r(F−1(M(t))) ; M(t), otherwise. where t ≤ k − 1, when t = 0, M(0) is the initial perfect matching from G. When r(F−1(M(t))) = n− 1, According the theorem 1, the A = F−1(M(t)) is a Hamiltonian cycle solution. If all of r(F−1(M(t))) < n − 1, then there has no Hamiltonian cycle in D. Since the non isomorphism perfect matching M in G is poset, the function r(F−1(M)) in G is monotonic, so this approach is exactly approach. Let us give a example to illustrate the approach in detail. Example 1. Considering the digraph D in figure 1, then the projector graph G in figure 2. Let M(0) = {e1, e8, e22, e9, e10, e3, e20, e11, e19, e5, e18, e6, e17, e7, e15, e16}. Thus the r(F−1(M(0)) = n−3. LetM ′ = r(F−1(M(0)⊗G3),then r(F−1(M ′) = n−4, thus M(1) = M(0) and then turn to G2,G1. At last it obtain the solution. Considering the equation 11, let it substituted by following equations when r(M ′) = n− 1 and t < k − 1. M(t+ 1) = M ′ if r(F−1(M ′)) ≥ r(F−1(M(t))) (12) It is obvious that all non-isomorphism Hamiltonian cycle could obtain by the repeat check the equation 12 and the equation r(M ′) = n− 1. In conversely, if a Hamiltonian cycle of Γ digraphs is given, it represents a perfect matching M in its projector graph G. Thus the equation 12 and Theo- rem 3 follows a corollary. Corollary 3. Given a Hamiltonian Γ digraph, the complexity of determining another non-isomorphism Hamiltonian cycle is polynomial time. 5.2 The HCP in digraph with bound two Let us extend the Theorem 3 to digraphs with d+(v) ≤ 2 and d−(v) ≤ 2 in this section. Theorem 6. The complexity of finding a Hamiltonian cycle existing or not in digraphs with degree d+(v) ≤ 2 and d−(v) ≤ 2 is polynomial time. Proof. Suppose a digraph D(V,A) having a vertex vi is shown as figure 3, which is d(vi) = 2 ∧ d−(vi) = 2 Figure 3. A vertex with degree than 2 ❍❍❍❍❍❍❥ ♠ ❍❍❍❍❍❍❥a3 Let us spilt this vertex to two vertices that one of vertex has degree with in degree 2 or out degree 1 , another vertex has degree with in degree 1 or out degree 2 as shown in figure 4. Then the D is derived to a new Γ graph S. Figrue 4 A vertex in D is mapping to a vertex in Γ digraph It is obvious that each vertex in the Γ graph S has increase 1 vertices and 1 arcs of D. Suppose the worst cases is each vertex in D has in degree 2 and out degree 2, the total vertices in S has 2n vertices. According to the theorem 3, obtain a Hailtonian cycle L′ in S is no more then O(n4), then the D will has a Hamiltonian cycle L′ = L ∩ A. 6 Discussion P versus NP The P versus NP is a famous open problem in computer science and math- ematics, which means to determine whether very language accepted by some nondeterministic algorithm in polynomial time is also accepted by some deter- ministic algorithm in polynomial time [6]. Cook give a proposition for the P versus NP . Proposition 2. If L is NP-complete and L ∈ P , then P = NP . According above proposition and the result above section, P versus NP problem has a answer. Theorem 7. P = NP Proof. As the result of [2], the complexity of HCP in digraph with bound two is NP − complete. According the theorem 6, the complexity of HCP in digraph with bound two is also P , thus according to proposition 2, P = NP . In fact, the [2] proves that 3SAT �p HCP of Γ digraph, since 3SAT is a NPC problem, which also implies that P = NP . 7 Conclusion According to the theorem 6, the complexity of determining a Hamiltonian cycle existence or not in digraph with bound degree two is in polynomial time. And according to the theorem 7, P versus NP problem has closed, P = NP . Acknowledgements The author would like to thank Prof. Kaoru Hirota for valuable suggestions, thank Prof. Jørgen Bang-Jensen who called mine attention to the paper [2], and thank Andrea Moro for useful discussions. References 1. Papadimitriou, C. H. Computational complexity , in Lawler, E. L., J. K. Lenstra, A. H. G. Rinnooy Kan, and D. B. Shmoys, eds., The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, Chichester, UK. (1985), 37–85 2. J.Plesńık,The NP-Completeness of the Hamiltonian Cycle Problem in Planar di- graphs with degree bound two, Journal Information Processing Letters, Vol.8(1978), 199–201 3. J.E. Hopcroft and R.M. Karp , An n5/2 Algorithm for Maximum Matchings in Bipartite Graphs . SIAM J. Comput. Vol.2, (1973), 225–231 4. P. Hall, On representative of subsets, J. London Math. Soc. 10, (1935), 26–30 5. Pearl, M, Matrix Theory and Finite Mathematics,McGraw-Hill, New York,(1973), 332–404. 6. Stephen Cook. The P Versus NP Problem ,”http://citeseer.ist.psu.edu/302888.html” ,2000. The Complexity of HCP in Digraps with Degree Bound Two Guohun Zhu Introduction Definition and properties Divided incidence matrix and Projector incidence matrix Proof of Theorem ?? The cycle in digraph corresponding matching in projector graph Proof of Theorem ?? Number of perfect matching in projector graph Proof of Theorem ?? The HCP in digraph with bound two Discussion P versus NP Conclusion

0704.0310

VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz

Astronomy & Astrophysics manuscript no. ex0069 November 4, 2018 (DOI: will be inserted by hand later) VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz X. Liu1, L. Cui1,2, W. -F. Luo1,2, W. -Z. Shi1,2, and H. -G. Song1 1 National Astronomical Observatories/Urumqi Observatory, CAS, 40-5 South Beijing Road, Urumqi 830011, China 2 Graduate University of the Chinese Academy of Sciences, Beijing 100049, China Received / Accepted Abstract. Aims and Methods: We present the results of VLBI observations of nineteen GHz-Peaked-Spectrum (GPS) radio sources at 1.6 GHz. Of them, 15 sources are selected from the Parkes Half Jansky (PHJ) sample (Snellen 2002), 4 others are from our previous observation list. We aimed at imaging the structure of GPS sources, searching for Compact Symmetric Objects (CSOs) and studying the absorption for the convex radio spectra of GPS sources. Results: We obtained total intensity 1.6 GHz VLBI images of 17 sources for the first time. Of them, 80% show mini-double-lobe radio structure, indicating that they are CSOs or candidates, and their host AGNs could be edge-on to us. This result suggests that there is a high incidence of mini double-lobe sources (or CSOs) in the PHJ sample. The sources J0323+0534, J1135−0021, J1352+0232, J2058+0540, J2123−0112 and J2325−0344 with measured redshift, showing double-lobe structure with sizes of < 1 kpc, are classified as CSOs. Three sources J1057+0012, J1600−0037 and J1753+2750 are considered as core-jet sources according to their morphologies and flux variability. Key words. galaxies: nuclei – quasars: general – radio continuum: galaxies 1. Introduction GHz-Peaked-Spectrum (GPS) radio sources are powerful (P1.4 GHz ≥ 10 25 W Hz−1), compact (≤ 1 kpc), and have con- vex radio spectra, and they make up a significant fraction (≈ 10%) of the bright radio source sample, see O’Dea (1998) for a review. In general, the presence of large scale emission asso- ciated with GPS galaxies is rare, about a few percent in a GPS sample (Stanghellini et al. 2005). Most GPS sources appear to be truly compact and isolated. Their small size is most likely due to their youth (< 104 years) according to a spectral aging analysis (Murgia 2003). A couple of GPS sources are certainly young radio sources whose kinematic age from lobe proper motions has been measured and these sources are also identified as Compact Symmetric Objects (CSOs). There is compelling evidence in favour of the youth scenario of GPS sources and CSOs, see e.g. Owsianik & Conway (1998), Tschager et al. (2000), Polatidis & Conway (2003), and Orienti et al. (2007). The GPS sources and CSOs are the key objects to study the early evolution of power- ful radio-loud AGN. A unification scenario assumes that GPS sources evolve into Compact Steep Spectrum sources (1-15 kpc), which in turn, evolve into classical extended radio sources (> 15 kpc), i.e. FR I/II radio sources (Fanti et al. 1995, Snellen et al. 2000, de Vries et al. 2007). Send offprint requests to: X. Liu: [email protected] GPS galaxies are dominated by lobe/jet emission on both sides of the central engine, and are thought to be relatively free of beaming effects. The GPS galaxies show very low polariza- tion (about less than 0.5% at 5 GHz, Dallacasa 2004, Xiang et al. 2006). The low integrated polarization could be due to large Faraday depths around the radio source, which would de- polarize the radio emission, implying that their host-AGNs are probably edge-on to us. Since GPS sources live in the narrow line region of AGN, it is likely that their low frequency radio emission will be ab- sorbed due to either synchrotron self-absorption or free-free absorption, giving rise to a peaked radio spectrum. Therefore, GPS sources are also suitable for studying radio absorption and scattering in AGNs. We have carried out EVN (European VLBI Network) ob- servations of 19 GPS sources, 15 of them are from the Parkes Half Jansky (PHJ) sample (Snellen et al. 2002) with declina- tion > −5◦ and not observed with VLBI before. Four sources are from our previous observation list which we have observed with the EVN at 2.3/8.4 GHz and/or 5 GHz (see Xiang et al. 2005, 2006). We aimed at imaging the GPS sources at 1.6 GHz, in order to confirm whether the GPS sources are double-lobe sources, and to find CSO candidates. For the sources with ob- servations at 2.3, 5.0 and 8.4 GHz, the 1.6 GHz images will further provide information on their source structure and inten- http://arxiv.org/abs/0704.0310v1 2 X. Liu et al.: VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz sity at lower frequency, for further spectral study of the GPS sources in the future. 2. Observations and data reduction The observations were carried out on 3 March 2006 at 1.65 GHz using the MK5 recording system with a band- width of 32 MHz and sample rate of 256 Mbps in dual circular polarization. The EVN antennae in this experiment were Effelsberg, Westerbork, Jodrell, Medicina, Noto, Onsala, Torun, Hartebeesthoek, Urumqi and Shanghai. Snapshot obser- vations of 19 sources (Table 1) in a total of 24 hours were made. OQ208 and DA193 were observed as calibrators. The data cor- relation was completed at JIVE. The total flux densities of the sources were also measured at 5 GHz with Urumqi 25m telescope in order to find any flux variability. The values are listed in Table 2. The Astronomical Image Processing System (AIPS) has been used for editing, a-priori calibration, fringe-fitting, self- calibration, imaging and model fitting of the data. 3. Results and comments on individual sources We list the basic information of the sources in Table 1, and the parameters derived from the VLBI images in Table 3. We com- ment on the results of each source and give a short discussion. We use S ∝ ν−α to define the spectral index. Optical informa- tion and redshifts of the GPS sources in the PHJ sample are given by de Vries et al. (2007), as listed in Table 1. 3.1. J0210+0419 (PKS B0208+040) The 1.6 GHz VLBI image (Fig. 1) is the first VLBI image of the source. It shows a double-lobe structure, and is most likely a CSO. Optical observations did not result in an identification with a lower limit of mR > 24.1, but it is identified with a mag- nitude of Ks=18.3 (de Vries et al. 2007). 3.2. J0323+0534 (4C+05.14) The 1.6 GHz VLBI image (Fig. 2) is the first VLBI image of the source, and it exhibits a strong diffuse component and a weak extended component in the south. Both are likely lobe emis- sion. About 38% total flux density (estimated from Table 1) is resolved out in the VLBI image, due to the diffuse components. For its size of 490 pc, the source can be a CSO. 3.3. J0433−0229 (4C−02.17) The 1.6 GHz VLBI image (Fig. 3) is the first VLBI image of the source, and the main component is diffuse and extended in the north-south direction, and a possible weak component in the south. About 18% total flux density (estimated from Table 1) is resolved out in the VLBI image. Either a core-jet or a CSO classification is possible for the source. 3.4. J0913+1454 (PKS B0910+151) The 1.6 GHz VLBI image (Fig. 4) is the first VLBI image of the source. It shows double structure and both components are further resolved. There is probably a hot-spot imbedded in the bright one. We consider it as a CSO candidate. 3.5. J1057+0012 (PKS B1054+004) The 1.6 GHz VLBI image (Fig. 5) is the first VLBI image of the source. There is a bright compact component followed by a secondary component and a series of possible weak compo- nents in the east, indicating this is a core-jet source. A flux vari- ability of (−11.4 ± 3.6)% over 15 years at 5 GHz, as reported in Table 2, is consistent with the core-jet classification. 3.6. J1109+1043 (PKS B1107+109) The 1.6 GHz VLBI image (Fig. 6) is the first VLBI image of the source. It is a double structure, and can be a CSO candi- date. The total flux density (1270 mJy estimated from Table 1) is completely restored in the VLBI image (1370 mJy, increased by 8%). There is also an indication of total flux increasing (4.9 ± 12.4)% at 5 GHz in Table 2 but with a large error. 3.7. J1135−0021 (4C−00.45) The 1.6 GHz VLBI image (Fig. 7) is the first VLBI image of the source. It shows a double-lobe structure, and with a size of 720 pc, we classify the source as a CSO. 3.8. J1203+0414 (PKS B1200+045) The 1.6 GHz VLBI image (Fig. 8) is the first VLBI image of the source. The triple structure may consist of a core and two sided emission, or a one sided core-jet source. The quasar as newly identified by de Vries et al (2007), is possibly a core-jet one, but still we keep the source as a CSO candidate. 3.9. J1352+0232 (PKS B1349+027) The 1.6 GHz VLBI image (Fig. 9) is the first VLBI image of the source. It shows a double-lobe like structure, for its size of 918 pc we consider it as a CSO. 3.10. J1352+1107 (4C+11.46) The 1.6 GHz VLBI image (Fig. 10) is the first VLBI image of the source. It appears to have a compact double structure or a core-jet alike, and seems diffuse emission around the source. About 30% total flux density (estimated from Table 1) is re- solved out in the VLBI image. Either a core-jet or a compact double classification is possible. 3.11. J1600−0037 (PKS B1557−004) The 1.6 GHz VLBI image (Fig. 11) is the first VLBI image of the source, and it has an overall double structure, the eastern X. Liu et al.: VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz 3 Table 1. The GPS sources. Columns (1),(2) source names; (3) optical identification (G: galaxy, QSO: quasar, EF: empty field); (4) optical magnitude; (5) redshift (de Vries et al. 2007, those with * are a photometric estimated by Tinti et al. 2005); (6) linear scale factor pc/mas [H0 = 71kms −1Mpc−1 and q0 = 0.5 have been assumed]; (7) maximum angular size from the observation; (8) maximum linear size; (9) 1.4 GHz flux density from the NVSS; (10) 2.7 GHz flux density from Snellen sample and the NED; (11) low frequency spectral index; (12) higher frequency spectral index ( computed from columns 9 and 10); (13) turnover frequency; (14) peak flux density; (15) references for the spectral information, 1 Snellen et al. 2002, 2 de Vries et al. 1997, 3 Stanghellini et al. 1998, where S ∝ ν−α. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 S ource id mR z pc/mas θ L S 1.4 S 2.7 αl αh νm S m re f mas pc mJy Jy GHz Jy J0210+0419 B0208+040 G 18.3Ks 1.5* 6.1 90 948 0.56 0.80 0.4 1.3 1 J0323+0534 4C+05.14 G 19.2 0.1785 2.7 180 490 2793 1.60 0.85 0.4 7.1 1 J0433−0229 4C−02.17 G 19.1 0.530 5.1 80 408 1462 1.04 0.52 0.4 3.0 1 J0913+1454 B0910+151 G 22.9 0.47* 4.9 80 881 0.54 0.75 0.6 1.1 1 J1057+0012 B1054+004 G 22.3 0.65* 5.5 80? 898 0.58 0.67 0.4 1.6 1 J1109+1043 B1107+109 G 22.6 0.55* 5.2 60 1481 0.80 0.94 0.5 2.4 1 J1135−0021 4C−00.45 G 21.9 0.975 6.0 120 720 1268 0.76 0.78 0.4 2.9 1 J1203+0414 B1200+045 QSO 18.8 1.221 6.1 75 458 1146 0.85 0.45 0.4 1.4 1 J1352+0232 B1349+027 G 20.0 0.607 5.4 170 918 1145 0.78 0.58 0.4 2.0 1 J1352+1107 4C+11.46 G 21.0 0.891 5.9 50 295 1538 0.78 1.03 0.4 3.6 1 J1600−0037 B1557−004 G 50 1168 0.54 1.17 1.0 1.2 1 J1648+0242 4C+02.43 G 22.1 0.824 5.8 0.61 0.4 3.4 1 J2058+0540 4C+05.78 G 23.4 1.381 6.1 160 970 1213 0.65 0.95 0.4 3.1 1 J2123−0112 B2121−014 G 23.3 1.158 6.1 80 488 1087 0.64 -0.56 0.75 0.5 1.8 2 J2325−0344 B2322−040 G 23.5 1.509 6.0 75 450 1224 0.91 -0.42 0.75 1.4 1.3 2 J0917+1113 B0914+114 EF 190 800 0.31 -0.1 1.6 0.3 2.3 3 J1753+2750 B1751+278 G 21.7 0.86* 5.9 50 625 0.46 -0.27 0.57 1.4 0.6 2 J1826+2708 B1824+271 G 22.9 45 332 0.23 -0.39 0.75 1.0 0.4 2 J2325+7917 B2323+790 G 19.5V 32 1136 -0.3 0.75 1.4 1.2 2 component has some extension in the west-east direction. A flux variability of (13± 6.9)% at 5 GHz in Table 2 may suggest this is a core-jet source. 3.12. J1648+0242 (4C+02.43) The GPS source is not detected with VLBI. It is an NVSS double-lobe source, and totally resolved out in the VLBI ob- servation. 3.13. J2058+0540 (4C+05.78) The 1.6 GHz VLBI image (Fig. 12) is the first VLBI image of the source. It shows a double-lobe source, and for the size of 970 pc, we suggest this is a CSO. 3.14. PKS B2121−014 The 1.6 GHz VLBI image (Fig. 13) shows a double-lobe struc- ture, it is similar to that at 2.3 and 5 GHz (Xiang et al. 2005, 2006), except that a weak jet-like emission ‘B’, which appears at 2.3 and 5 GHz, is missing, probably due to absorption at the lower frequency 1.6 GHz. The source is a CSO for the source size of 488 pc. 3.15. PKS B2322−040 The 1.6 GHz VLBI image (Fig. 14) exposes a central emission region between the two lobes ‘A’ and ‘B’, which is probably a core embedded in the central region. The ‘core’ emission is not detected at higher frequencies (Xiang et al. 2005, 2006), but it emerges at 1.6 GHz near the peak frequency (1.4 GHz) of the GPS source. There is a flux increase of (4.0 ± 1.9)% over 15 years at 5 GHz (Table 2), may suggest that the core is currently active. The source can be a CSO for its size of 450 pc. 3.16. PKS B0914+114 The 1.6 GHz VLBI image (Fig. 15) exhibits a core ‘A’, jet fea- ture ‘B’ and two lobes ‘C’, ‘E’. The western one ‘E’ emerges at this frequency. Labiano et al. (2007) have identified an empty field (> 25 mR) at the FIRST position of the source, and con- cluded that the previously identified nearby disk galaxy (a red- shift of 0.178) is not the host of this radio source 0914+114. For the typical compact symmetric structure, we consider the source is a CSO. 4 X. Liu et al.: VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz Table 2. Source flux and possible variability, columns 2-4 are flux densities at 5.0 GHz (PKS90), 4.85 GHz (Gregory & Condon 1991, and Griffith et al. 1995) and 4.85 GHz flux measured with the Urumqi 25m telescope on 2007/1/24 (J1648+0242, J2058+0540, 1824+271 and 2121−014 were not well measured due to source confusion or weak); column 5 is a flux variability computed from columns 3,4. S ource S 5.0 S 4.85 S 4.85Ur δS 4.85 mJy mJy mJy % J0210+0419 300 298±19 302 ± 10 1.3 ±3.1 J0323+0534 830 819±44 868 ± 9 6.0±4.6 J0433−0229 640 640±35 637 ± 14 -0.5±3.3 J0913+1454 300 315±43 297 ± 8 -5.7±10.3 J1057+0012 370 396±23 351 ± 6 -11.4±3.6 J1109+1043 400 408±56 428 ± 8 4.9±12.4 J1135−0021 440 446±25 427 ± 8 -4.3±3.6 J1203+0414 520 640±35 611 ± 7 -4.5 ±4.1 J1352+0232 470 469 ± 7 J1352+1107 410 447±62 418 ± 5 -6.5±11.9 J1600−0037 180 187±14 212 ± 3 13.4±6.9 J1648+0242 260 337±20 J2058+0540 340 356±21 2121−014 320 345±21 2322−040 500 524±29 545 ± 20 4.0 ±1.9 0914+114 140 134±19 140 ± 1 4.5 ±14.1 1751+278 298±39 292 ± 6 -2.0±10.8 1824+271 122±17 2323+790 491 ± 7 OQ208 2421±217 2514 ± 12 3.8±8.8 3.17. 1751+278 (MG2 J175301+2750) The 1.6 GHz structure (Fig. 16) is similar to what we got be- fore at 1.6 GHz (Xiang et al. 2002), and confirms that there is jet-like emission ‘C’ and ‘D’ associated with the southern component ‘B’, indicating this is a core-jet source. 3.18. B2 1824+271 The 1.6 GHz VLBI image (Fig. 17) exposes a symmetric dou- ble structure and jet-like emission associated with the two lobes, confirming this is a CSO as we have suggested (Xiang et al. 2006). 3.19. [WB92] 2323+790 The 1.6 GHz image (Fig. 18) shows a central component ‘A’ and a weak one ‘B+C’ in the north-west, and the components ‘A’ and ‘B+C’ show steep spectra between 1.6 GHz and 5 GHz (Xiang et al. 2006). The source can be a CSO candidate. 4. Discussion In the sample (Table 1), J1648+0242 is an NVSS double source and is not detected in this VLBI observation; all others are point-like in the NVSS images, indicating that GPS sources are compact. Except four sources (J1057+0012, J1352+1107, J1600−0037 and 1751+278), 14 out of 18 sources exhibit dou- ble or triple VLBI structure and can be CSOs or CSO can- didates though some of them have no measured redshift. The sources with redshift show double or triple structure with sizes < 1 kpc, suggesting these GPS sources are certainly compact and likely CSOs. The mini double-lobe sources or CSOs could be more sta- ble in flux density than other type of compact sources. We have measured the flux densities for the sources (Table 2) at 4.85 GHz and compared with the values observed 15 years ago, we found that 12 among 14 GPS sources are likely stable in flux (1σ level), two sources (J1057+0012 and J1600−0037) show about 10% variability in 3σ and 2σ level respectively. The flux variability on J1057+0012 and J1600−0037 is consistent with their core-jet classification. ‘Core-jet’ sources are defined to show a one-sided jet, and the jet is often closely pointing to us (from a pole-on AGN). It is hard to estimate the real source size due to Doppler boosting, hence the ‘core-jet’ sources might not be young radio sources even if they appear to be compact in some cases. In addition, some sources are resolved out in our VLBI image by more than 10% of total flux estimated from Table 1, probably due to diffuse emission associated with lobes and tail/jet emission. They are J0210+0419 (-14%), J0323+0534 (-38%), J0433−0229 (-18%), J1352+0232 (- 15%), J1135+1107 (-31%), J2058+0540 (-12%), 2322−040 (- 15%), and J1648+0242 is completely resolved out. The VLBI flux densities of the other nine sources at 1.6 GHz are consis- tent with the estimated total flux densities within an error of X. Liu et al.: VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz 5 Table 3. The component parameters of the VLBI images at 1.6 GHz. The columns give: (1) source name and possible classifica- tion (CSOc: CSO candidate, cj: core-jet); (2) total cleaned flux density of image at 1.6 GHz; (3) component identification labled to Xiang et al. 2002, 2005, 2006; (4),(5) peak and integral intensity of a fitted Gaussian component at 1.6 GHz in the AIPS task JMFIT; (6),(7) major/minor axes and position angle of component at 1.6 GHz; (8),(9) distance and position angle relative to the first component; (10) brightness temperature of component. 1 2 3 4 5 6 7 8 9 10 Name S vlbi Comp S p S int θ1 × θ2 PA d PA T b class mJy mJy mJy mas×mas ◦ mas ◦ 108K◦ J0210+0419 715 A 290 375 5.7 × 2.9 169 0 11.9 CSOc B 118 205 10 × 3.6 175 68.2 ± 0.1 -153.2 ± 0.1 2.2 J0323+0534 1497 A 536 1270 32.6 × 11.1 70 0 0.5 CSO B 117 548 57.5 × 21.7 18 122.7 ± 2.9 -167.2 ± 0.4 0.1 J0433−0229 1095 A 407 1045 14.5 × 4.7 171 0 2.4 CSOc/cj B 49 111 9.9 × 5.5 88 53.0 ± 0.3 164.4 ± 0.2 0.3 J0913+1454 796 A 211 501 8.2 × 4.9 65 0 2.1 CSOc B 55 157 8.9 × 6.2 80 56.8 ± 0.1 73.3 ± 0.1 0.4 J1057+0012 810 A 381 550 3.6 × 2.7 6.6 0 17.5 cj B 37 58 6.2 × 2.1 9.7 9.7 ± 0.2 114.8 ± 0.9 1.3 J1109+1043 1370 A 420 984 5.7 × 4.7 110 0 6.6 CSOc B 143 311 5.4 × 4.2 91 46.3 ± 0.1 104.4 ± 0.1 2.6 J1135−0021 1025 A 279 454 6.6 × 2.7 142 0 4.9 CSO B 143 271 8.6 × 3.0 153 85.8 ± 0.1 164.4 ± 0.1 1.7 J1203+0414 1029 A 571 850 4.7 × 2.9 107 0 25.1 CSOc B 51 81 5.0 × 3.8 78 18.1 ± 0.2 103.0 ± 0.4 1.6 C 31 35 6 × 6 0 58.2 ± 0.2 104.2 ± 0.2 1.9 J1352+0232 885 A 173 480 7.4 × 4.5 53 0 2.1 CSO B 30 120 8.9 × 6.3 110 165.7 ± 0.3 -111.9 ± 0.1 0.2 J1352+1107 896 A 198 395 8.1 × 5.4 11.7 0 2.0 CSOc/cj B 106 202 6.8 × 6.3 10 3.5 ± 0.1 53.1 ± 0.1 1.1 J1600−0037 936 A 394 607 5.4 × 4.1 157 0 cj B 125 255 7.9 × 5.0 86 24.7 ± 0.1 83.5 ± 0.1 J2058+0540 914 A 356 513 7.6 × 3.7 163 0 8.2 CSO B 182 403 12.6 × 4.5 128 127.2 ± 0.1 172.1 ± 0.1 2.3 2121−014 976 A 363 594 5.1 × 3.9 123 0 10.4 CSO C 191 415 7.4 × 5.0 126 59.4 ± 0.1 85.4 ± 0.1 2.9 2322−040 965 A 229 509 14.6 × 3.3 161 0 3.6 CSO B 74 160 11.7 × 5.3 162 40.6 ± 0.2 171.3 ± 0.1 0.8 C 65 196 19.3 × 4.7 1 22.0 ± 0.4 171.1 ± 0.2 0.5 0914+114 578 A 37 50 4.9 × 2.7 16 0 CSO B 29 65 9.1 × 4.3 66 45.6 ± 0.1 80.1 ± 0.1 C 242 360 4.2 × 3.9 90 84.2 ± 0.1 81.6 ± 0.1 E 28 40 4.4 × 3.4 63 86.1 ± 0.1 -95.9 ± 0.1 1751+278 596 A 400 522 4.8 × 2.9 71 0 14.5 cj B 26 37 8 × 3 15 20.5 ± 0.1 -128.6 ± 0.2 0.5 C 12.4 21 7 × 5 59 26.6 ± 0.3 -119.8 ± 0.3 0.2 D 8 20 19.5 × 3.6 176 41.4 ± 0.4 -104.4 ± 0.8 0.1 1824+271 296 A 145 174 3.9 × 2.1 164 0 CSO B 42 69 6.1 × 4.3 137 21.8 ± 0.1 -83.4 ± 0.1 2323+790 900 A 439 621 5.8 × 2.1 159 0 CSOc B+C 107 155 7.9 × 3.1 118 19.2 ± 0.1 -71.1 ± 0.1 10% the estimated amplitude uncertainty of the EVN observa- tions. 5. Summary and conclusion 1. We obtained total intensity 1.6 GHz VLBI images of 17 GPS sources for the first time. The majority (80%) show mini-double-lobe radio structure, indicating that they are CSOs or candidates, and their host AGNs could be edge- on to us. This result suggests that there is a high incidence of mini double-lobe sources and CSOs in the GPS source sample. 6 X. Liu et al.: VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz 2. The sources J0323+0534, J1135−0021, J1352+0232, J2058+0540, 2121−014 and 2322−040 with measured red- shift, are double-lobed with sizes of < 1 kpc, and are clas- sified as CSOs. 3. Three sources (J1057+0012, J1600−0037 and 1751+278) are classified as core-jet sources according to their mor- phologies and flux variability. 4. The 1.6 GHz images of the sources 0914+114, 1824+271, 2121−014 and 2322−040, for which we had observations at 2.3, 5.0 and 8.4 GHz, have provided information on their source structure and spectra at the lower frequency, permit- ting further spectral study in the future. Acknowledgements. We thank the referee Alvaro Labiano, and Nathan de Vries for comments. The European VLBI Network is a joint facility of European, Chinese, South African and other radio as- tronomy institutes funded by their national research councils. This re- search has made use of the NASA/IPAC Extragalatic Database (NED) which is operated by the Jet Propulsion Laboratory, Caltech, under contract with NASA. This work was partly supported by the Natural Science Foundation of China (NSFC). References de Vries N., Snellen I. A. G., Schilizzi R. T., Lehnert M. D., Bremer M. N., 2007, A&A 464, 879 de Vries W. H., Barthel P. D., O’Dea C. P., 1997, A&A 321, 105 Dallacasa D., 2004, in Proceedings of the 7th EVN Symposium, eds: Bachiller R., Colomer F., et al. Fanti, C., Fanti, R., Dallacasa, D., Schilizzi, R. T., Spencer, R. E., Stanghellini, C. 1995, A&A 302, 317 Gregory P. C., Condon J. J., 1991, ApJS 75, 1011 Griffith M. R., Wright A. E., Burke B. F., Ekers R. D., 1995, ApJS 97, Labiano A, Barthel P. D., O’Dea C. P., de Vries W. H., Perez I., Baum S. A., 2007, A&A 463, 97L Murgia M., 2003, PASA 20, 19 O’Dea C. P. 1998, PASP 110, 493 Orienti M., Dallacasa D., Stanghellini C., 2007, A&A 461, 923 Polatidis A. G., Conway J. E., 2003, PASA 20, 69 Owsianik I., Conway J. E., 1998, A&A 337, 69 Snellen I. A. G., Lehnert M. D., Bremer M. N., Schilizzi R. T., 2002, MNRAS 337, 981 Snellen I. A. G., Schilizzi R. T., Miley G. K., de Bruyn A. G., Bremer M. N., Röttgering H. J. A., 2000, MNRAS 319, 445 Tschager W., Schilizzi R. T., Röttgering H. J. A., Snellen I. A. G., Miley G. K., 2000, A&A 360, 887 Stanghellini C., O’Dea C. P., Dallacasa D., Cassaro P., Baum S. A., Fanti R., Fanti C., 2005, A&A 443, 891 Stanghellini C., O’Dea C. P., Dallacasa D., Baum S. A., Fanti R., Fanti C., 1998, A&AS 131, 303 Tinti S.,Dallacasa D., de Zotti G., Celotti A., Stanghellini C., 2005, A&A 432, 31 Xiang L., Stanghellini C., Dallacasa D., Haiyan Z., 2002, A&A 385, Xiang L., Dallacasa D., Cassaro P., Jiang D., Reynolds C., 2005, A&A 434, 123 Xiang L., Reynolds C., Strom R. G., Dallacasa D., 2006, A&A 454, X. Liu et al.: VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz 7 J0210+0419 0 100 200 MilliARC SEC 60 40 20 0 -20 -40 -60 -80 -100 Fig. 1. J0210+0419 at 1.65 GHz, the restoring beam is 10.2 × 7.5 mas with PA −75.5◦, the peak is 293 mJy/beam, the con- tours are 12 mJy/beam times levels -1, 1, 2, 4, 8, 16, 32, 64, 100, 200, 400, 800, and the same levels are used in the follow- ing images, the grey scale unit is mJy. J0323+0534 MilliARC SEC 150 100 50 0 -50 -100 -150 Fig. 2. J0323+0534 at 1.65 GHz, the restoring beam is 21.3 × 17.7 mas with PA −20.2◦, the peak is 586 mJy/beam, the first contour is 50 mJy/beam. J0433-0229 0 100 200 300 400 MilliARC SEC 100 80 60 40 20 0 -20 -40 -60 -80 Fig. 3. J0433−0229 at 1.65 GHz, the restoring beam is 8.0×6.3 mas with PA −1.7◦, the peak is 435 mJy/beam, the first contour is 15 mJy/beam. J0913+1454 0 50 100 150 200 MilliARC SEC 100 80 60 40 20 0 -20 -40 Fig. 4. J0913+1454 at 1.65 GHz, the restoring beam is 6.9×4.6 mas with PA 24.4◦, the peak is 217 mJy/beam, the first contour is 6 mJy/beam. 8 X. Liu et al.: VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz J1057+0012 0 100 200 300 MilliARC SEC 100 80 60 40 20 0 -20 -40 Fig. 5. J1057+0012 at 1.65 GHz, the restoring beam is 7.3×3.3 mas with PA 13.4◦, the peak is 390 mJy/beam, the first contour is 6 mJy/beam. J1109+1043 0 100 200 300 400 MilliARC SEC 100 80 60 40 20 0 -20 -40 -60 Fig. 6. J1109+1043 at 1.65 GHz, the restoring beam is 7.9×3.3 mas with PA 16.5◦, the peak is 434 mJy/beam, the first contour is 10 mJy/beam. J1135-0021 0 100 200 MilliARC SEC 100 80 60 40 20 0 -20 -40 -60 -80 Fig. 7. J1135−0021 at 1.65 GHz, the restoring beam is 7.8×5.3 mas with PA 20.4◦, the peak is 281 mJy/beam, the first contour is 8 mJy/beam. J1203+0414 0 200 400 MilliARC SEC 100 80 60 40 20 0 -20 -40 -60 Fig. 8. J1203+0414 at 1.65 GHz, the restoring beam is 7.0×4.9 mas with PA 15.2◦, the peak is 575 mJy/beam, the first contour is 10 mJy/beam. X. Liu et al.: VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz 9 J1352+0232 MilliARC SEC 50 0 -50 -100 -150 -200 Fig. 9. J1352+0232 at 1.65 GHz, the restoring beam is 6.7×3.2 mas with PA 9.9◦, the peak is 183 mJy/beam, the first contour is 8 mJy/beam. J1352+1107 MilliARC SEC 100 50 0 -50 -100 Fig. 10. J1352+1107 at 1.65 GHz, the restoring beam is 14.0× 12.5 mas with PA −75.3◦, the peak is 379 mJy/beam, the first contour is 20 mJy/beam. J1600-0037 0 100 200 300 400 MilliARC SEC 80 60 40 20 0 -20 -40 -60 Fig. 11. J1600−0037 at 1.65 GHz, the restoring beam is 7.4 × 5.5 mas with PA −39.8◦, the peak is 400 mJy/beam, the first contour is 8 mJy/beam. J2058+0540 MilliARC SEC 150 100 50 0 -50 -100 -150 Fig. 12. J2058+0540 at 1.65 GHz, the restoring beam is 10.6× 6.5 mas with PA 0.7◦, the peak is 362 mJy/beam, the first con- tour is 10 mJy/beam. 10 X. Liu et al.: VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz B2121-014 0 100 200 300 MilliARC SEC 100 80 60 40 20 0 -20 -40 -60 Fig. 13. 2121−014 at 1.65 GHz, the restoring beam is 5.9× 5.5 mas with PA −1.4◦, the peak is 370 mJy/beam, the first contour is 15 mJy/beam. 2322-040 MilliARC SEC 80 60 40 20 0 -20 -40 -60 -80 Fig. 14. 2322−040 at 1.65 GHz, the restoring beam is 8.6× 6.8 mas with PA −2.1◦, the peak is 233 mJy/beam, the first contour is 10 mJy/beam. 0914+114 MilliARC SEC 50 0 -50 -100 -150 -200 Fig. 15. 0914+114 at 1.65 GHz, the restoring beam is 8.3× 4.7 mas with PA 16.8◦, the peak is 246 mJy/beam, the first contour is 1 mJy/beam. MilliARC SEC 80 60 40 20 0 -20 -40 -60 -80 1751+278 Fig. 16. 1751+278 at 1.65 GHz, the restoring beam is 10.7×6.0 mas with PA 7.0◦, the peak is 406 mJy/beam, the first contour is 3 mJy/beam. X. Liu et al.: VLBI observations of nineteen GHz-Peaked-Spectrum radio sources at 1.6 GHz 11 1824+271 MilliARC SEC 60 40 20 0 -20 -40 -60 -80 Fig. 17. 1824+271 at 1.65 GHz, the restoring beam is 9.0× 5.1 mas with PA 8.1◦, the peak is 146 mJy/beam, the first contour is 1 mJy/beam. 2323+790 MilliARC SEC 60 40 20 0 -20 -40 -60 -80 Fig. 18. 2323+790 at 1.65 GHz, the restoring beam is 11.6×5.4 mas with PA −82◦, the peak is 438 mJy/beam, the first contour is 10 mJy/beam. Introduction Observations and data reduction Results and comments on individual sources J0210+0419 (PKS B0208+040) J0323+0534 (4C+05.14) J0433-0229 (4C-02.17) J0913+1454 (PKS B0910+151) J1057+0012 (PKS B1054+004) J1109+1043 (PKS B1107+109) J1135-0021 (4C-00.45) J1203+0414 (PKS B1200+045) J1352+0232 (PKS B1349+027) J1352+1107 (4C+11.46) J1600-0037 (PKS B1557-004) J1648+0242 (4C+02.43) J2058+0540 (4C+05.78) PKS B2121-014 PKS B2322-040 PKS B0914+114 1751+278 (MG2 J175301+2750) B2 1824+271 [WB92] 2323+790 Discussion Summary and conclusion

0704.0311

Moment switching in nanotube magnetic force probes

7 Moment switching in nanotube magnetic force probes John R Kirtley1,2,3, Zhifeng Deng4, Lan Luan4, Erhan Yenilmez1, Hongjie Dai5, and Kathryn A Moler1,4 1 Department of Applied Physics and Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305 USA E-mail: [email protected] 2 IBM Watson Research Center, Route 134 Yorktown Heights, NY 10598 USA 3 Faculty of Science and Technology and MESA+ Institute for Nanotechnology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands 4 Department of Physics and Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305 USA 5 Department of Chemistry, Stanford University, Stanford, California 94305 USA Abstract. A recent advance in improving the spatial resolution of magnetic force microscopy (MFM) uses as sensor tips carbon nanotubes grown at the apex of conventional silicon cantilever pyramids and coated with a thin ferromagnetic layer [1]. Magnetic images of high density vertically recorded media using these tips exhibit a doubling of the spatial frequency under some conditions [1]. Here we demonstrate that this spatial frequency doubling is due to the switching of the moment direction of the nanotube tip. This results in a signal which is proportional to the absolute value of the signal normally observed in MFM. Our modeling indicates that a significant fraction of the tip volume is involved in the observed switching, and that it should be possible to image very high bit densities with nanotube magnetic force sensors. PACS numbers: 75.75.+a,78.67.Ch Spatial period doubling has been observed for several carbon nanotube tips with different track widths and bit densities. The MFM images reported here were made using the nanotube tip shown in the inset of Figure 1. It was approximately 250nm long, with a ferromagnetic coating to a total tip diameter of 16nm. The means of producing metal-coated carbon nanotube tips on AFM cantilevers and the techniques for imaging magnetic media using these tips have been described previously [1, 2]. Briefly, carbon nanotubes were grown using wafer-scale chemical vapor deposition at the apexes of the pyramids of commercial silicon tips intended for tapping mode atomic force microscopy. For the present measurements the nanotubes were shortened to a length of about 250 nm using an electrical cutting method [3], aligned approximately perpendicular to the cantilever using a focused ion beam [2], and then coated to a total thickness of about 16 nm with a Ti/Co/Ti trilayer by e-beam evaporation from a direction parallel to the http://arxiv.org/abs/0704.0311v1 Moment switching in nanotube magnetic force probes 2 nanotube long axis (see the inset in Figure 1). The magnetic imaging was done using the Tapping/LiftTM mode of a Digital Instruments Nanoscope III SPM at room temperature in air. In this mode, the topography of the sample is first determined for each line with a scan at low tip-sample spacing z0, then the tip is retracted a specified distance and a second line scan is made while recording the deviation of the phase angle δ of the cantilever response with the cantilever driven slightly below its resonance frequency. δ is proportional to dFz/dz0, the derivative of the force on the cantilever with respect to The images presented here were made on a vertically polarized magnetic medium. A collection of phase shift images at 300 kilo-flux changes per inch (kfci) and different tip-sample spacings (z0’s) are shown in Figure 2(a), and at constant lift height (15nm) and different bit densities in Figure 2(b). Cross-sections of the data through the center of the tracks are displayed in Figure 2(c,d). Modeling of this data as described below is shown in Figure 2(e,f). At high values of z0 and high bit densities the phase images have the same period as written, but at low z0’s and low bit densities anomalies in the images gradually develop into sharp features with double the original periodicity. These anomalies are due to switching of the orientation of the magnetic moment of the tip. This can be demonstrated most convincingly by inspecting the images and cross-sections of Figure 2(b,d). In this case a background from the sections of the image without written bits has been subtracted out, and it can be seen that at low bit densities the phase shift always stays below the average background level, keeping the tip-sample force attractive by switching of the tip moment direction just as the force derivative crosses zero. This conclusion is supported by detailed modeling: We assume that the magnetic medium is composed of slabs of length s in the x direction, height h in the z direction, and width W in the y direction (Figure 1(a)). The slabs have a uniform magnetization with moment direction alternating between parallel and anti-parallel to the z-axis direction. The tip is assumed to have a square cross-section with width w and length L, with the tip end a distance z0 from the upper surface of the medium. The magnetic fields above the sample (displayed as field lines in Figure 1(b)) were calculated both analytically and numerically. In our numerical modeling the vector between the individual medium dipole moments [xm, ym, zm] and a position [x, y, z] is ~r = (x− xm)x̂+ (y − ym)ŷ + (z − zm)ẑ. Then the z and x-component of the field from the individual dipoles is given by Bz(~r) = 3(z − zm)2 Bx(~r) = 3µ0mm (x− xm)(z − zm), (1) with r = |~r|. By is given by setting x → y and xm → ym in the second equation. If we assume that the tip can also be represented by a sum of point dipoles ~mt = mtn̂, where n̂ is a unit vector in the tip moment direction, the force gradient on the tip is Moment switching in nanotube magnetic force probes 3 Medium z0+L1 Figure 1. (a) Tip and sample geometry in model. The inset is a scanning electron microscope image of the cobalt-coated nanotube tip. (b) Calculated field lines for a 300 kfci track above the center of the domains in the magnetic medium, using the parameters described in the text. The arrows indicate calculated moment orientations for a series of tip-sample spacings of 15, 22.5, 28, 33, 39, and 45 nm using the model and parameters described in the text. At small tip-sample spacings the tip moment flips to be always anti-aligned with the moment directly below the tip. At large tip-sample spacings the tip moments are only slightly perturbed by the medium fields. given by mt cos(φ) d2Bz(x, y, z) tip,medium 3µ0mmmt cos(φ) − 30(z − zm) 35(z − zm)4 where rtm is the distance between the individual dipole elements in the tip and medium, and φ is the angle between the tip moment direction and the z-axis. Moment switching in nanotube magnetic force probes 4 Figure 2. (a) Collection of phase shift images for a 300 kfci track of alternating vertically recorded media moments at different lift heights. Each line of the image has an offset such that the average phase shift value for that line is zero. (b) Image of a section of vertically recorded media with a series of tracks with different bit densities, at a lift height of 15 nm. A background has been subtracted from this image such that the regions between the tracks have zero averaged phase shifts. (c) Cross-sections through the centers of the tracks in (a). (d) Cross-sections through the centers of the tracks in (b). The zero phase shift levels for each cross-section are indicated by dashed lines. (e) Modeling of (c) as described in the text. The lines in (c) and (e) have been offset vertically for clarity. (f) Modeling of the tracks in (d) as described in the text, with the zero phase shift levels indicated by dashed lines. Moment switching in nanotube magnetic force probes 5 For positions near the center of the tracks we obtained analytical expressions for the magnetic fields by assuming that the magnetically oriented slabs have infinite width W in the y direction. If we take the boundary condition Hz(~r)|z=0+ −Hz(~r)|z=0− = Hz(~r)|z=−h− −Hz(~r)|z=−h+ = σ(~r), (3) where σ(~r) is the surface magnetic charge, the magnetic field above the sample can be written as [4] Hz(~r, z) = d2~kA −i~k·~r, (4) where H,z = e −kz(1− e−kh) d2~r σ(~r)ei ~k·~r. (5) Taking the surface magnetic charge σ(~r) to be uniform in y, σ(~r) = M 2ns < x < (2n+ 1)s = −M (2n− 1)s < x < 2ns, (6) We find tan−1 (2n+ 1)s+ x + tan−1 (2n− 1)s+ x − 2 tan−1 z 2ns+ x − tan−1 z + h (2n+ 1)s+ x − tan−1 z + h (2n− 1)s+ x + 2 tan−1 z + h 2ns+ x Since Hz = dΦ/dz, Φ a scalar potential, the x-component of the field can be written as Hx = dΦ/dx, which leads to Hx = − ((2n+ 1)s+ x)2 + log ((2n− 1)s+ x)2 −2 log (2ns+ x)2 − log (z + h)2 ((2n+ 1)s+ x)2 − log (z + h)2 ((2n− 1)s+ x)2 + 2 log (z + h)2 (2ns+ x)2 Hy is zero by symmetry. Figure 3 displays the calculated fields in the z and x directions (a,c), the tip moment orientation angle φ (b), and the switching fields < Hz >c and < Hx >c (c) for the best fit to the data as described below. It is interesting to note that the discontinuous switches of φ are controlled predominantly by the size of the x component of the field. This can be understood by examining the trajectory in field that the tip takes in moving from one domain to the next. The ovals in Figure 3c are the calculated trajectories of < Hz > vs. < Hx > for the tip heights listed in the caption. The “asteroid” is the critical field calculated for the energy functional form of Eq. 1 of the main text and has the form first predicted by Stoner and Wohlfarth[5]. The tip moment direction is predicted to switch when the field trajectories cross the critical field asteroid (solid dots in Figure 3c). This happens for relatively large values of | < Hx > | and small values of | < Hz > |. When the field trajectory crosses the Stoner-Wohlfarth Moment switching in nanotube magnetic force probes 6 0 1 2 3 4 15 nm 45 nm 0 1 2 3 4 15 nm 30 nm 33 nm 45 nm –2 –1 0 1 2 <Hx>Mt/K1 Figure 3. (color online) (a) Calculated magnetic fields, averaged over the tip volume and multiplied by the tip saturation magnetization Mt divided by the anisotropy parameter K1, above a vertically recorded magnetic medium with infinite extent in the y-direction, 12 nm thick in the z-direction, with magnetization oriented in the z-direction and alternating in the x-direction with period 2s = 185 nm, for spacings between the bottom of the tip and the medium of z0 = 15, 22.5, 26, 28, 30, 33, 39, and 45 nm. The tip has an assumed square cross-section 16nm on a side, and the averaging is over a tip length L1 = 48nm. The calculated magnetic fields < Hz > normal to the magnetic medium are offset by 3 units. The best fit values MtMm/K1 = 17.5, σw/µ0L1K1 = 0 (see main text Figure 3) were used for these calculations. (b) Calculated variation of the tip moment orientation angle φ relative to the z axis. (c) The ovals show the calculated trajectories of < Hz > Mt/K1 vs < Hx > Mt/K1 for the various tip heights. The “asteroid” plots the values of the critical fields. The intersections of these two sets of curves, indicated by solid symbols, are the fields at which switching occurs in the simulations. Moment switching in nanotube magnetic force probes 7 asteroid at large values of | < Hz > | the tip has already switched to the low energy configuration. For numerical work the medium was assumed to be composed of a collection of individual dipole moments ~m = mmẑ, where mm = ±Mmv, with Mm the saturation magnetization and v the volume of the individual medium elements. In what follows we take both the tip and medium volume elements to be cubes 4 nm on a side. This results in agreement to within a few percent between our numerical work and analytical expressions for the medium magnetic fields. Halving the size of the volume elements (to cubes 2 nm on a side) changes the calculated force derivative curve for z0 = 30 nm (see Figure 2c) by about 2%. To model the dynamics of the tip flip process, we conceptually divide the tip into two domains, one with length L1 close to the medium, the other with length L − L1 further away. Each has sufficiently strong exchange fields that the entire volume within each domain has the same moment orientation[6]. The section of the tip furthest from the medium is assumed to have its moment parallel to the z axis; that closest to the medium has its moment at an angle φ relative to the z-axis (Figure 1). Then the energy of the tip in an external magnetic field can be written as: E(φ) = µ0w 2L1K1 µ0L1K1 (1− cos φ) + sin2 φ (< Hz > cos φ+ < Hx > sin φ) where we have taken the simplest non-trivial forms for the domain wall energy (first term) and the anisotropy energy (second term)[5, 7]. The third term in Eq. 9 is the energy of the dipole moments of the tip in the external magnetic field. Here µ0K1 is the anisotropy energy density, σw is the domain wall energy per unit area, Mt is the tip saturation magnetization, and < Hz > and < Hx > are the magnetic fields in the z and x directions respectively averaged over the tip volume from z = z0 to z = z0 + L1. To simulate the magnetic force images, the tip moment is at first assumed to be parallel to the z-axis. The tip is moved to a new position, the local fields are calculated and averaged over the tip volume, φ is moved to the new local minimum in energy (Eq. 9), the force gradient is calculated, and the process is repeated. This modeling results in the cross-sections displayed in Figure 2(c,f), which reproduce the absence of tip switching at high bit densities and high lift heights, and the presence of tip switching at low bit densities and low lift heights. When tip switching occurs, the modeling also reproduces the fact that the tip-sample force gradient always stays negative, with the tip moment reversing as the z-component of the field crosses zero. The quantitative interpretation of MFM images in the presence of tip switching is straightforward once it is recognized that the phase shift δ is proportional to the negative of the absolute value of the tip sample force gradient (-|dFz/dz0|). The experimental phase shift oscillation amplitudes decrease much more rapidly than the modeling for bit densities above about 500 kfci (Figure 2f). We believe that this is because the as written bits do not have as abrupt moment orientation reversals as our idealized model. Our modeling indicates Moment switching in nanotube magnetic force probes 8 that nanotube tips with the geometry of Figure 1 could be used to image bits with sharp moment direction transitions with densities above 2000 kfci (13 nm/flux reversal). Figure 4 compares the maximum minus the minimum value for δ along a cross- section through the center of the bits at a bit density of 300 kfci as a function of tip height z0. The modeling results in this Figure are labeled by the length L1 of tip that is allowed to reorient its magnetic moment. There are three parameters in this analysis - a global multiplicative factor, MtMm/K1, and the reduced domain wall energy σw/µ0L1K1. Figure 4b plots the best fit values for MtMm/K1 and (∆δexperimental−∆δmodel)2/(N−1) (N the number of data points). In all cases the best fit value for σw/µ0L1K1 is 0, and the χ 2 value at σw/µ0L1K1 = 1 is approximately double that when σw/µ0L1K1 = 0. Increasing the domain wall energy requires larger switching fields: the best fit value at L1= 64 nm for MtMm/K1 increases from 20.8 to 28.6 when σw/µ0L1K1 increases from 0 to 1. The domain wall energy for Co is reported to be σw = 25 ± 3J/m2[8]. This leads to σw/µ0L1K1 = 0.85, using L1 = 48nm and K1=0.25 M t (neglecting crystalline anisotropy), with Mt = 1.4× 106A/m, so that our fits are consistent with the calculated wall energy, if one allows for a doubling of the best χ2 value. The tip end may be magnetically poorly coupled to the rest of the tip because of an inhomogeneity or grain boundary: it appears (Figure 1 inset) to have granularity on the scale of a few tens of nm and a kink about 50 nm from its end. We have observed qualitatively similar spatial frequency doubling using several nanotube tips, particularly in the smallest diameter tips, where inhomogeneities and weak magnetic coupling are fundamentally more difficult to avoid. The shift ∆ω in the resonance frequency ω0 of the cantilever due to a force gradient between the tip and sample dFz/dz0 is given by ∆(ω)/ω0 = −(dFz/dz0)/2k, where k is the spring constant of the cantilever. The phase shift δ of the cantilever response is then given by tan δ = /ω − ω/ω′ , (10) where Q is the quality factor, ω is the driving frequency and ω is the perturbed resonance frequency of the cantilever. At z0 = 15nm our model predicts an excursion of ∆(dF/dz)/µ0MtMm ≈ 2 × 10−9m. Using a driving frequency at optimal sensitivity ω = ω (1 − 1/ 8Q), k = 2.8N/m, Q=350, ω0 = 168kHz, and estimating Mt = 1.4 × 106A/m [9] and Mm = 2 × 105A/m [10], this corresponds to an excursion in the phase shift ∆δ = 0.7o, in reasonable agreement with the experimental value of ∆δ = 0.4o given the uncertainties in the values for the saturation magnetizations. Using Mt| < Hx >c |/K1 <∼ 2 [5], the best fit value MtMm/K1 ∼ 17 (Figure 4) implies a critical field of approximately 2.4×104 A/m, much smaller than the saturation magnetization of cobalt of 1.4×106 A/m, but comparable to a switching field of 3.2×104 A/m reported for 30 nm thick, 0.34 µm wide, 2.04 µm long ellipsoidal amorphous cobalt nanodots [11]. This reduction in switching field could result from competition between the crystalline and shape anisotropies [9] if, for example, the uniaxial crystalline anisotropy favors moment alignment along the tip radial direction, while the shape Moment switching in nanotube magnetic force probes 9 10 20 30 40 50 z0 (nm) Experiment L1= 0 L1= 4 nm L1= 16 nm L1= 32 nm L1= 48 nm L1= 256 nm 0 100 200 300 L1 (nm) 0 100 200 300 0.0000 0.0025 0.0050 0.0075 0.0100 Figure 4. (color online) (a) Full-scale variation in phase angle along cross-sections through the centers of the recorded tracks in Figure 2b. The + symbols represent experiment. The other symbols represent modeling as described in the text. The modeling curves are labeled by the length L1 of tip that switches magnetic moment orientation. (b) Best fit values for MtMm/K1, and χ (∆δexp −∆δmod)2/(N − 1) as a function of the tip switching length L1, with σw/µ0L1K1 = 0. anisotropy favors the axial direction. The tip material could also have a complicated structure incorporating grain and domain boundaries, reducing the anisotropy energy. The highly non-uniform fields in our case could also play a role in reducing the switching field. Although our analysis has been presented using a specific model for the tip dynamics, the conclusion that the tip moment flips at relatively small fields can be presented simply: The magnetic field at the tip must be less than the magnetic field at the medium surface, which is given by the saturation magnetization of the medium. The fields required to flip the tip are expected, using for example the Stoner-Wohlfarth model [5], to be about the saturation magnetization of the tip, which is for epitaxial cobalt much larger than the saturation magnetization of the medium. The switching fields of our nanotube, just as for amorphous nanodots, are much smaller than those of epitaxial Co nanodots [7, 9], which are comparable to the saturation magnetization Moment switching in nanotube magnetic force probes 10 of cobalt. It might be possible to avoid switching and the attendant spatial frequency doubling by developing processes to epitaxially coat the nanotube or by using single- crystal nanorods. However, such tips would also generate larger local magnetic fields at the sample, increasing the possibility of changing the magnetic state of the sample, particularly for the smallest samples. Our results show that reliable information on the moment orientations of the media can be inferred even in the presence of tip switching if it is realised that the MFM signal is proportional to the absolute magnitude of the tip-sample force gradient. Acknowledgments We would like to thank Dennis Adderton of First Nano and Dr. Steve Minne of Veeco Instruments for the AFM probes, and Dr. David Guarisco of Maxtor Corporation for the recorded disks. This work was supported by the Center for Probing the Nanoscale (CPN), an NSF NSEC, NSF Grant No. PHY-0425897, by NSF Grant No. DMR 0103548, by the Dutch Foundation for Research on Matter (FOM), the Netherlands Organization for Scientific Research (NWO), and the Dutch STW NanoNed program. References [1] Deng Z, Yenilmez E, Leu J, Hoffman J E, Straver E W J, Dai H, and Moler K A 2004 Appl. Phys. Lett. 85 6263 [2] Deng Z, Yenilmez E, Reilein A, Leu J, Dai H and Moler K A 2006 Appl. Phys. Lett. 88 023119 [3] Yenilmez E, Wang Q, Chen R J, Wang D W, and Dai H J 2002 Appl. Phys. Lett. 80 2225 [4] Steifel B, “Magnetic Force Microscopy at Low Temperatures and in Ultra High Vacuum - Application on High Temperature Superconductors”, Inauguraldissertation, University of Basel, 1998. [5] Stoner E C and Wohlfarth E P 1948 Phil. Trans. Royal Soc. London A 240 599 [6] Kittel C, 1946 Physical Review 70 965 [7] Bonet E, Wernsdorfer W, Barbara B, Benoit A, Mailly D and Thiaville A, 1999 Phys. Rev. Lett. 83 4188 [8] Hehn M, Padovani S, Ounadjela K and Bucher J P, 1996 Phys. Rev. B 54 3428 [9] Otani Y, Kohda T, Novosad V, Fukamichi K, Yuasa S and Katayama T, 2000 J. Appl. Phys. 87 [10] Ross C A 2001 Annu. Rev. Mater. Res. 203 [11] Johnson J A, Grimsditch M, Metlushko V, Vavassori P, Illic B, Neuzil P and Kumar R 2000 Appl. Phys. Lett. 77 4410

0704.0312

Power Spectra to 1% Accuracy between Dynamical Dark Energy Cosmologies

Mon. Not. R. Astron. Soc. 000, 1–9 (2007) Printed 31 October 2018 (MN LATEX style file v2.2) Power Spectra to 1% Accuracy between Dynamical Dark Energy Cosmologies⋆ Matthew J. Francis1†, Geraint F. Lewis1 and Eric V. Linder2 1 School of Physics, University of Sydney, NSW 2006, Australia 2 University of California, Berkeley Lab, Berkeley, CA 94720, USA ABSTRACT For dynamical dark energy cosmologies we carry out a series of N-body gravitational simulations, achieving percent level accuracy in the relative mass power spectra at any redshift. Such accuracy in the power spectrum is necessary for next generation cosmological mass probes. Our matching procedure reproduces the CMB distance to last scattering and delivers subpercent level power spectra at z = 0 and z ≈ 3. We discuss the physical implications for probing dark energy with surveys of large scale structure. Key words: methods:N-body simulations — methods: numerical — dark matter — dark energy — large-scale structure of Universe 1 INTRODUCTION The mass power spectrum plays a central role in understand- ing the large scale structure of the Universe and the cosmic expansion affecting the development of structure. Cosmolog- ical probes of the growth history or the expansion history such as weak gravitational lensing, baryon acoustic oscilla- tions, or galaxy/cluster abundances depend crucially on ac- curate knowledge of the mass power spectrum for physical interpretation of the data from large scale surveys. Most commonly the mass power spectrum is approxi- mated by the ‘Halofit’ form of Smith et al (2003), and this is used to determine cosmological parameters from the data (or estimate future precision of parameter extraction). However, the Smith et al (2003) formula is calibrated only on ΛCDM models, and for these has a precision of ∼10%. This will be insufficient for future large structure surveys that aim to ex- plore the acceleration of the cosmic expansion and the prop- erties of dark energy responsible for it. Huterer & Takada (2005) estimate that for weak gravitational lensing surveys, for example, 1% accuracy in knowledge of the mass power spectrum will be required. N-body simulations provide a well-tested technique for calculating the dark matter power spectrum at the percent level (Heitmann et al 2005). While this treats purely gravi- tational forces, leaving out baryonic effects including heat- ing and cooling, this should be a sufficient approximation for wavemodes k < 3hMpc−1 (Jing et al. 2006; Zhan & Knox ⋆ Research undertaken as part of the Commonwealth Cosmology Initiative (CCI: www.thecci.org), an international collaboration supported by the Australian Research Council † Email: [email protected] 2004; White 2004), or the scales larger than galaxies, that are of most relevance for large surveys. However, carrying out simulations for every possible cosmological model is ob- viously impractical. If one could devise a mapping procedure that matched models with the same key physical quantities, ideally to a single class of cosmologies like ΛCDM, then this would greatly aid the study of the cosmological information carried by the distribution and growth of large scale struc- ture. In Section 2 we outline the approach to such a mapping procedure and compare to previous work. Section 3 describes the details of the simulations performed and tests carried out. The qualitative physical consequences of the mapping are interpreted in Section 4. We present the computational results in Section 5 and identify several interesting features in the wavemode and redshift dependence of the power spec- trum. Physical interpretation of these results are discussed in Section 6, with conclusions and future directions sum- marised in Section 7. 2 DARK ENERGY AND COSMIC STRUCTURE While the influence of dark energy on the linear growth fac- tor of matter density perturbations can be calculated simply (see below), the full, nonlinear mass power spectrum requires N-body simulations. Such simulation studies of the non- linear power spectrum that exist for dark energy other than a cosmological constant have tended to be for a constant dark energy pressure to density ratio, or equation of state ra- tio w (Ma et al. 1999; White & Vale 2004; Linder & White 2005; McDonald et al 2006). c© 2007 RAS http://arxiv.org/abs/0704.0312v1 2 Francis, Lewis & Linder A few simulation studies have considered the effects of dynamical dark energy on the power spectrum, either through a parameterized time (or scale factor a) dependent form w(a) or a specific scalar field potential (Klypin et al 2003; Macciò et al 2004; Ma 2006). Efforts to calibrate non-linear power spectrum fitting formulas include early work by Ma et al. (1999), which was much improved upon with the advances in computing power by McDonald et al (2006). Both of these studies looked at modifying existing fits in the case of constant w cosmologies. Linder & White (2005) (LW) investigated the effects of w on the non-linear power spectrum, searching for key physical quantities, and discovered a simple matching prescription for calculating the non-linear mass power spectrum to within one to two percent. This work extends the study of the dark energy ef- fects on the full, non-linear mass power spectrum to models with dynamical dark energy, utilizing the model indepen- dent, physically motivated (Linder 2003) evolving equation of state w(a) = w0+wa(1−a). At the same time, we employ the approach of seeking central physical matching quanti- ties that incorporate CMB data through agreeing on the distance to the last scattering surface. In studying the non-linear power spectrum of mass fluc- tuations, a natural place to start is with the linear power spectrum. The effects of dark energy on the linear mass power spectrum can be calculated through the relation P (k, a) = D2(a) D2(ai) P (k, ai) (1) (see e.g. Coles & Lucchin (2002)) with the growth factor D(a) given by the formula (e.g. Hu (2002); Linder & Jenkins (2003)) 1 +X(a) 1 +X(a) = 0 (2) with derivatives with respect to scale factor a, and where X(a) is the ratio of the matter to dark energy densities, given by X(a) = Ωm d ln a′w(a′), with Ωm the di- mensionless present matter density. The non-linear power spectrum cannot be written in terms of a simple differential equation and requires the use of large volume, high resolution, N-body simulations. These are computationally expensive and therefore accurate semi- analytic fitting formulas derived from simulation results are a valuable tool. The most widely adopted current for- mula, sometimes called Halofit, was presented in Smith et al (2003). This formula is motivated by the halo model of struc- ture growth with free parameters in the function set by fit- ting to a large suite of simulations. All these simulations, however, were of cosmological constant, w = −1, cosmolo- gies. McDonald et al (2006) produced a fitting formula as a multipolynomial series for constant w models, intended to be used to modify the Smith et al result. This modification was estimated to be accurate to within a few percent in the range of cosmologies encompassed by the simulation grid. Taking a different approach, LW demonstrated that when the linear growth factors between different w =const models were matched at a high redshift point as well as at z = 0, by compensating with other cosmological parameters, the non-linear power spectrum from N-body simulations also matched to much better than a percent at those red- shifts, as well as matching to one to two percent at any inter- mediate redshift. Additionally, LW also found that the dis- tance to the surface of last scattering, dlss, closely matched when their growth matching criteria was implemented, pre- serving CMB constraints. With this formalism the power spectrum for a dark energy model can be matched to, say, a ΛCDM cosmology. Hence one can either use an appropri- ately matched Halofit result or carry out a vastly reduced suite of only ΛCDM simulations to achieve the desired ac- curacy on the mass power spectrum. This article concentrates on developing accurate match- ing of the non-linear mass power spectrum for dynamical dark energy models. We employ a somewhat different match- ing procedure from LW, explicitly matching the distance to CMB last scattering dlss and the mass fluctuation amplitude σ8 at the present and studying the effect on the growth. In this respect, our approach is essentially the converse of the LW approach. The geometric factor of the distance to CMB last scattering suffices to incorporate substantially the CMB constraints on the dark energy parameters. Since dark en- ergy had a negligible density in the early universe (except in special, early dark energy models, e.g. see Wetterich (2004); Doran & Robbers (2006); Linder (2006)), the physical size and nature of features in the CMB at the surface of last scat- tering is largely insensitive to the properties of dark energy. However, the angular size of such features is set through the angular diameter distance, which does depend on the properties of dark energy, since it relates to the expansion history of the universe a(t). Therefore, dark energy mod- els giving the same distance to the last scattering surface are largely degenerate with respect to the CMB (some dif- ferences relating to secondary anisotropies such as the ISW effect remain, see Hu & Dodelson (2002)). For a given dy- namic dark energy model (w0, wa), there is a corresponding constant equation of state weff , say, that gives the same dlss as the dynamical model, holding all other cosmological pa- rameters (such as the physical matter density Ωmh 2) fixed. This article examines the relation between the non-linear mass power spectra of the dynamical and the weff models. Once a tight correspondence is established, one can then either employ a constant w fitting formula such as from McDonald et al (2006), carry out only a suite of constant w simulations, or adjust the other cosmological parameters such that one chooses weff = −1 and requires only ΛCDM simulations. We discuss these alternatives further in §7. 3 SIMULATION DETAILS The simulations were performed using the GADGET-2 N- body code (Springel 2005), modified to incorporate the back- ground evolution a(t) appropriate for dynamical dark energy cosmologies with w(a) = w0 + (1 − a)wa. Fiducial simula- tion runs use 2563 dark matter particles in a 256 h−1Mpc periodic box with a 5123 force grid; the initial redshift was z = 24 and the force softening was set to a constant co- moving length of 60h−1Kpc. In order to check numerical convergence, runs were also performed with combinations of box size and particle number a factor of 2 greater and smaller than the fiducial. In addition, runs checking conver- gence were performed for numerical parameters including the start time, softening length, PM grid spacing, time and c© 2007 RAS, MNRAS 000, 1–9 Power Spectra to 1% Accuracy 3 force accuracy and tree update frequency. The ratio of power between the different dark energy models were largely insen- sitive to these parameters, changing by a small fraction of a percent out to k < 3hMpc−1. The linear matter power spectra used to cre- ate the initial conditions were calculated using CAMB (Lewis, Challinor & Lasenby 2000). Initial conditions were generated from the power spectrum using part of the GRAFIC program within the COSMICS package (Ma & Bertschinger 1995). For each set of distance-matched runs, the same in- put power spectrum, generated by CAMB using the w = constant model, was used for each model. In order to match the amplitude of linear growth today (identical σ8 at z = 0) for simulations of different cosmologies, the initial density and velocity perturbations of the particles were scaled in the Zel’dovich approximation using the linear growth fac- tor ratio D(astart)/D(a = 1) for the different models. This ansatz for initial conditions is robust as long as the dark energy does not change the shape of the linear power spec- trum at zstart, i.e. the dark energy plays little role in the very early universe. We have verified this to high accuracy using a version of CAMB modified for (w0, wa) models with- out dark energy perturbations. Note that in the presence of dark energy perturbations, the initial power spectrum over our range of k = 0.1 − 3hMpc−1 is affected by less than 1% for constant w models that are not distance matched. We expect that distance-matched (w0, wa) models with per- turbations will show less effect but future work will address this. The calculation of the power spectrum in simulations outputs used the ‘chaining the power’ method described in Smith et al (2003) utilising the cloud in a cell assignment scheme. No correction was made for shot noise, as the quan- tity of interest was the ratio of the power between different models. See McDonald et al (2006) for an extended discus- sion of the usefulness of taking power spectrum ratios to eliminate many numerical errors in this type of study. See also White (2005). All simulations in this paper used the best fit cosmo- logical parameters from Spergel et al (2006) of Ωm = 0.234, h = 0.74, Ωb = 0.0407 and σ8 = 0.76 in a flat ΛCDM universe. For each set of simulations, a constant equation of state weff is selected and several values for the parameters w0 and wa that maintained the same dlss were calculated. One consequence of this methodology is that these w(a) models cross the value w = −1 at some point in cosmic history. De- bate exists surrounding the physical validity of crossing be- tween the phantom regime, defined as w < −1, and w > −1. This will eventually be settled by a microphysical theory for dark energy, rather than merely a phenomenological de- scription. With this in mind we do not consider the issue of phantom energy and phantom crossing further. We select three values, weff = −0.9,−1,−1.1, as the foundations for our comparison of w(a) cosmologies. This range is in accord with constraints on constant w from current cosmological data sets (Spergel et al 2006; Seljak, Slosar, & McDonald 2006) and provides a reasonable variety for testing the matching procedure. For each con- stant w model, simulations were carried out for four more w(a) models with matching distance to the LSS. The dark energy models used are summarized in Table 1. weff = −0.9 weff = −1.0 weff = −1.1 w0 wa w0 wa w0 wa -1.1 0.620 -1.2 0.663 -1.3 0.707 -1.0 0.319 -1.1 0.341 -1.2 0.363 -0.8 -0.336 -0.9 -0.359 -1.0 -0.381 -0.7 -0.686 -0.8 -0.732 -0.9 -0.778 Table 1. Distance Matched Models. Simulations were carried out for five models (including w = weff) for each of three values of weff , where all five models in a column had identical distances to CMB last scattering. Figure 1. Dark energy equation of state vs. a for the models matched to the CMB last scattering surface distance for w = −1. 4 THE CONSEQUENCES OF DISTANCE MATCHING If the simple distance matching procedure outlined in this paper is to succeed in producing a good match in the mat- ter power spectrum, it might be expected to keep a range of physical conditions similar through cosmic history. This is indeed what is found. In particular, a variety of physical quantities exhibit pivot or crossover points, indicating not only near equality at that epoch, but a tendency toward agreement of quantities integrated over cosmic history. For instance, Figure 1 plots w(a) for the four models with match- ing distance to the w = −1 model; there is a clear epoch at a ≈ 0.7 where the values all cross w = −1. The linear growth factors of the various distance matched models also closely track each other. While the lin- ear growth is matched at a = 1 by construction, there is an additional epoch at high redshift where the linear growth of all associated models closely matches. For the weff = −1 set of models shown in Figure 2, the matching point is a = 0.24 or z = 3.12. The other two sets of models match at a similar value and show a similar trend. This behaviour illustrates the converse of what was found in LW, where dlss was found to closely match when the linear growth was matched at some high redshift point. The crossover is important, since rather than all models diverging from the z = 0 match, c© 2007 RAS, MNRAS 000, 1–9 4 Francis, Lewis & Linder Figure 2. Ratio of the linear growth factor D̄(a) ≡ D(a/D(1) relative to the central w = −1 model for the w(a) models with matched dlss. Figure 3. Ratio of ΩDE(a) relative to the central w = −1 model for the w(a) models with matched dlss. the curves track each other relatively closely. Since the non- linear power spectrum is intimately tied to the linear be- havior, this provides hope that a mapping of the full power spectrum between models can be realized. The growth crossover behaviour results from a change in sign of the relative rate of growth at a ≈ 0.5. In the linear regime the growth of fluctuations can be seen as a balance between the mutual gravitational attraction of the overden- sity, which is amplified by higher mean matter density, and the expansion rate of the universe characterised by H(a), which acts like a frictional term, suppressing growth the higher the expansion rate. In this picture, greater relative matter domination at a particular epoch will produce a more rapid growth rate at that time compared to a less matter dominated model. With this in mind it is worth comparing the matter domination history of our suite of cosmologies. Figure 3 shows the relative dark energy density Ωde(a) for the w(a) models matched to w = −1. Comparing figures 2 and 3 we see that where the rela- tive dark energy density is higher, the relative rate of linear growth is lower. For instance, the long dashed model has a Figure 4. Ratio of the non-linear mass power spectrum at z=0 relative to the w = −1 model for models with matched dlss. more negative slope in the region a < 0.5 in figure 2 than the other models, corresponding to a higher relative dark energy density in this region as shown in figure 3. The magnitude of growth is greater initially in the long dashed model in order to achieve the match at z = 0, however the growth initially grows more slowly in comparison to the other models. The change in sign of the relative growth rates at a ≈ 0.5 in fig- ure 2 corresponds to the crossover point in relative matter domination in Figure 3. The energy density, like the linear growth, exhibits a striking crossover point, again keeping physical conditions similar between models throughout cos- mic history. In a chain of related conditions, the crossover point of the equation of state w(a) (see Figure 1) causes the con- vergence of the dark energy density Ωde(a) (see Figure 3, and the crossover of Ωde(a) leads to the convergence of the growthD(a) (see Figure 2), which then creates a crossover in the growth at higher redshift. This in turn will keep the non- linear power spectrum closely matched between the models over the entire range z = 0− 3. 5 RESULTS The ratio of power measured in the simulation outputs at z = 0 relative to the central weff model for each of the sets of simulations is given in Figures 4, 5 and 6. The most outstanding result is the excellent agreement between the mapped power spectra, at the 0.1% level for k < 1hMpc−1 and . 1% for k < 3hMpc−1 (. 0.5% at the higher k for the less rapidly varying dark energy models). These figures show very similar trends, regardless of the fiducial weff model chosen. Since the trends are similar, the remaining figures will show the results for the w = −1 central models only, for brevity. The simulations shown in these figures started with an identical realisation of initial conditions, albeit scaled with respect to the linear growth factor to produce matched linear growth at z = 0. As the deviations between models are small, it is important to take care to ensure that any features are real and not the results of a spurious numerical effect. c© 2007 RAS, MNRAS 000, 1–9 Power Spectra to 1% Accuracy 5 Figure 5. As Fig. 4, for the central model w = −0.9. Figure 6. As Fig. 4, for the central model w = −1.1. The convergence tests described in Section 3 addressed the effects of numerical parameters 1. There are also two other potential sources of error, the limited volume of the simulation box and the error sam- pling error in calculating the power spectrum of the simula- tion snapshots. Care must be take that these effects are not causing spurious results. Figure 7 shows the results for the w = −1 central model simulations with rms sampling errors from the FFT power spectrum calculation plotted. For clar- ity these have been omitted from the other plots, however the errors are similar in all cases. From this plot the deviation between models is roughly 1 Changing particle resolution did cause a slight systematic shift in features seen in the power spectrum ratios. The onset of the dispersion between the models seen in figures 4-6 at k ∼ 1hMpc−1 shifted to lower k with reduced particle resolution and higher k with an increase. This shift was of order 0.1 in log k for factors of two differences in particle resolution. We cannot fully account for this numerical effect, however the difference in power at a given k-mode due to the shift is at most ∼ 0.1% and since subpercent effects are beyond the ability of N-body simulations to probe ac- curately, we do not believe this effect is of significant consequence. Figure 7. As Fig. 4, with rms sampling errors included, shown by thin lines of the same line style as each model. Figure 8. The effect on the power spectrum due to different real- isations. Displayed are the power in three realisations of a single cosmological model, w = −1, as a ratio to the power in a fourth realisation. The rms errors for each power spectrum calculation are also shown. As expected the finite volume error decreases as k increases due to the greater number of modes present at higher wavenumbers. a factor of 1−2 that of the rms error. In order to verify that the effects seen are genuinely due to the difference in dark energy models, another three sets of simulations with the same parameters but different realisations were performed. The scatter in the calculated power spectrum due to dif- ferent realisations is shown in Figure 8. The scatter in this figure clearly demonstrates the inability to accurately deter- mine the absolute power with the box sizes and number of realisations used in this study due to finite volume errors. From the figures shown it is clear that the difference be- tween realisations for a single cosmological model is greater than the difference within a single realisation for the differ- ent models. This makes accurate modeling of the absolute value of the power spectrum an extremely challenging task, requiring larger boxes, many more realisations and highly detailed consideration of sources of numerical error. Instead, we are interested in the effect of dark energy models relative c© 2007 RAS, MNRAS 000, 1–9 6 Francis, Lewis & Linder Figure 9. The effect of different realisations on the ratio of a dynamical dark energy power spectrum to the associated w = const model. Shown are 4 realisations of the ratio between the strongly time varying (w0, wa) = (−1.2, 0.663) model and the w = −1 model at z = 1, plotted as a ratio of the main realisation used in the paper. This is a typical example of the magnitude of the variation due to different realisations. to one another and therefore what is important is how much the ratios between models (such as shown in Figures 4-6) are affected by different realisations. Fortunately the effect of different realisations is vanishingly small as seen in Fig- ure 9, which shows a typical example of the variation in power ratios across the four realisations used. From Figure 9 we can be confident that the computed non-linear power spectra ratios are not visibly affected by spurious finite volume errors or effects due to the FFT cal- culation of the power spectrum. For cosmological structure probes, we are interested not just in how well we can predict the power spectrum at z = 0 but across all redshifts. In the simulations performed, data was output at a number of times. From Figure 2, two epochs are of particular interest. The first is the crossover in linear growth at a = 0.24 and the second is at a = 0.5 where the linear growth is most varied between models. The latter cor- responds to z = 1, which is extremely relevant to a number of forthcoming cosmological surveys. Hence an accurate es- timation of power here, provided by the distance matching scheme, is of great importance for understanding possible constraints on dark energy cosmologies. The ratio of power measured in the simulations boxes at these epochs are shown in Figures 10 and 11. We can do even better, however, by realizing that much of the difference in power, particularly at a ≈ 0.5, can be accounted for by the difference in linear power. Scaling this out via Equation 1, the results are as shown in Figures 12 and 13. From these figures we can see that the combined dis- tance and growth matching procedure is generally accurate to better than 1%. The greatest deviation found in all sim- ulation outputs is 2% for k ≈ 3 hMpc−1 at a = 0.5. Since the results shown thus far display a good match for distance matched models, it is worth considering how much improvement this matching achieves compared to ar- bitrary dark energy models that are not distance matched. Figure 10. As Fig. 4, but at a = 0.24 where from Fig. 2 the linear growths closely match (slightly lower than for w = −1). Figure 11. As Fig. 4, but at a = 0.5 where from Fig. 2 the linear growths are most divergent. Much of the difference in power comes from this difference in linear power. Figure 12. As Fig. 10, but with the linear growth difference scaled out. Note the reduced y-axis scale relative to Fig. 10. c© 2007 RAS, MNRAS 000, 1–9 Power Spectra to 1% Accuracy 7 Figure 13. As Fig. 11, but with the linear growth difference scaled out. Note the reduced y-axis scale relative to Fig. 11. Figure 14. The ratio of the non-linear power in several non- distance matched cosmologies to a ΛCDM model at z = 1. The amplitude of linear growth at z = 0 is the same for all models as in previous figures. Note that the divergence between models increases with k but that the divergence begins at a lower k and is significantly greater than with the distance matched models shown previously In other words, how much of a role do the values of the dark energy parameters play in structure formation, if all other parameters are kept fixed? Figure 14 shows the ra- tio of power at z = 1 between a ΛCDM model and several non-distance matched models with the same linear growth amplitude at z = 0. The divergence between these mod- els is significantly greater than what is seen in the distance matched models (and of course they will disagree on the CMB), illustrating the improvement achieved by this sim- ple scheme. 6 EVOLVING DARK ENERGY AND STRUCTURE GROWTH The matching prescription used in this article produces a mapping between the matter power spectra of dark energy cosmologies accurate to . 1% over a wide range of wave- modes and a wide range of cosmic history. This agreement 1) indicates that simple physical quantities determine the nonlinear power spectrum over this range, leading to the prospect of understanding structure formation on a funda- mental level even in dynamical dark energy cosmologies, 2) points the way to significant advances in computational effi- ciency by reducing the dimension of the grid of simulations necessary to produce accurate estimations of power spectra required for interpretation of cosmological probes such as weak gravitational lensing, baryon acoustic oscillation, and other large scale structure surveys, and 3) identifies a degen- eracy that makes it difficult to distinguish between models lying on a particular subsurface of the cosmological param- eter space. To try to ameliorate the degeneracy, we note that an evolving equation of state does imprint a small but system- atic effect on the non-linear matter power spectrum. The general trend shown by Figures 4, 5 and 6 is that dark en- ergy with a less negative value today but more negative value at high redshift (i.e. negative wa) gives greater non-linear power at k & 1hMpc−1 than its dlss-matched weff model. Similarly, more negative equations of state today with posi- tive wa possess less power in the same range. This deviation is however relatively small, remaining less than 2% out to k = 3hMpc−1. Even so, this partial degeneracy is not too worrying since it can readily be broken by other cosmolog- ical dependencies (e.g. the geometric distance dependence entering with the mass power spectrum into the weak lens- ing shear power spectrum or the baryon acoustic scale) or by complementary cosmological probes. Thus the model map- ping technique does not appear to have any real drawbacks to detract from its physical and computational advantages. Elaborating on the physical import of the mapping, a striking feature is the marked difference at z = 0 between k < 1hMpc−1 where the power spectra between models match extremely closely and k > 1hMpc−1 where they di- verge. This seems to suggest a transition between the lin- ear region at low k where by design the power should be matched, and the fully non-linear region at high k where differences in cosmic evolution have imprinted a different signature on the growth of structure on small scales, per- haps reflecting the effect of dark energy on conditions when structure formed at high redshift. Those models that show a greater non-linear growth are those that in the early universe had a greater contribution of the matter density relative to the dark energy density; these correspond to the models with today w0 > weff and in the recent universe had lower matter density relative to dark energy density. This suggests that non-linear growth is more sensitive to conditions (including the effects of dark energy) in the early, matter dominated universe than it is to conditions in the later, accelerated era of dark energy domination. Carrying this forward, one conjecture is that the “tran- sition” in the behavior of the z = 0 power spectrum at k = 1 hMpc−1 might be related to early, rather than z = 0, non-linear effects. While the non-linear scale at z = 0 should be near k . 0.2, the power spectrum remains well matched here, possibly because the non-linear growth was already matched as a result of the model mapping, i.e. “pinned down” by the agreement at a = 0.24. So the greatest dif- c© 2007 RAS, MNRAS 000, 1–9 8 Francis, Lewis & Linder ference in non-linear growth, arising from times earlier than the a = 0.24 matching, might appear at the non-linear scale associated with a < 0.24, or k & 1hMpc−1, rather than the z = 0 non-linear scale. In any case, the accurate ap- proximation of the power spectrum utilizing the matching prescription indicates that reasonably simple physics lies be- hind even the non-linear mass power spectrum. 7 CONCLUSION The mass power spectrum lies at the foundation of many cos- mological observables, such as the weak lensing shear statis- tics of galaxies, the large scale structure clustering distri- bution (including baryon acoustic oscillations), and cluster abundances. To utilize any of these cosmological measure- ments as next generation probes of large scale structure, cosmology, or dark energy requires clear and accurate un- derstanding of the mass power spectrum over the range of models under consideration, e.g. dynamical dark energy not just ΛCDM. The main results of this article (see Figs. 4,12 and 13) demonstrate that non-linear mass power spectra of dynam- ical dark energy models with smooth equation of state evo- lution can be determined to percent accuracy by calculating the power for the constant equation of state cosmology that gives a matching distance to the CMB last scattering sur- face. By varying other parameters as well, such as the matter density Ωm and h keeping Ωmh 2 constant (see LW), one can envision mapping a wide variety of dark energy models to ΛCDM models, resulting in significant gains in computa- tional efficiency. This can also alleviate concerns regarding phantom crossing. Finding the distance matched models as described in this paper is a trivial task numerically requiring the inte- gration of a differential equation and a one dimensional pa- rameter search. This simple procedure however provides a mapping that is accurate to a percent. Of course the ac- curacy of the resultant power spectrum estimation is ulti- mately only as good as the accuracy in the model being mapped to. For that model we can then utilise fitting for- mulas, such as Halofit, for a rough, ∼ 10% accuracy or more generally perform N-body simulations with the desired pa- rameters. However, the distance matching scheme in this case allows a much reduced grid of simulations to be carried out while still maintaining a high degree of accuracy. The physics behind the matching prescription involves a chain of consequences from the crossover in the behavior of one variable (e.g. equation of state) to the convergence in another, then leading to matching in the large scale struc- ture growth. Further physics, not yet fully elucidated, points to the full nonlinear power spectrum being dependent not on the linear growth, but the linear growth history , where the conditions (e.g. matter density or growth factor) at one epoch directly manifest in the nonlinear behavior at a later epoch. Future work should pursue this further, as well as inves- tigating the prospects for discerning the signatures of more complicated equations of state or perturbations. While the prescription here for mapping the mass power spectrum to percent accuracy between cosmologies is useful in itself and for computational gains, the most exciting prospects are for improved analytic fitting formulas and deeper physical un- derstanding. ACKNOWLEDGMENTS MF acknowledges the support of a Science Faculty UPA, thanks Chris Power and Jeremy Bailin for helpful discus- sions, advice and pieces of code, and thanks LBNL and SNAP for hospitality and support during much of the writ- ing of this article. We thank Martin White for pointing out the nice method of getting high resolution FFT’s without needing large arrays and for other useful discussions. This work has been supported by the Australian Research Coun- cil under grant DP 0665574 and in part by the Director, Office of Science, US Department of Energy under grant DE-AC02-05CH11231. REFERENCES “Cosmology: The origin and evolution of cosmic structure” Coles P. & Lucchin, F., Wiley & Sons (Textbook) Hawkins, E. et al, 2003, MNRAS, 346, 78 Dolag, K. et al, 2004, A&A, 416, 853 Doran, M. & Robbers, G., 2006, JCAP, 0606, 026 Heitmann, K., Ricker, P., Warren, M., & Habib, S., 2005, ApJS, 160 28 Hu, W. 2002, PhRvD, 66, 083515 Hu W. & Dodelson S., 2002, ARA&A, 40, 171 Huterer, D., & Takada, M. 2005, Astroparticle Physics, 23, Jing, Y. P., Zhang, P., Lin, W. 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B, 594, 17 White, M. 2004, Astroparticle Physics, 22, 211 White, M. 2005, Astroparticle Physics, 24, 334 White, M., & Vale, C. 2004, Astroparticle Physics, 22, 19 c© 2007 RAS, MNRAS 000, 1–9 http://arxiv.org/abs/astro-ph/0610213 http://arxiv.org/abs/astro-ph/0603449 Power Spectra to 1% Accuracy 9 Zhan, H., & Knox, L. 2004, ApJL, 616, L75 This paper has been typeset from a TEX/ LATEX file prepared by the author. c© 2007 RAS, MNRAS 000, 1–9 Introduction Dark Energy and Cosmic Structure Simulation details The Consequences of Distance Matching Results Evolving Dark Energy and Structure Growth Conclusion