Wrapping Interactions and the Genus Expansion of the 2-Point Function of Composite Operators

Nuclear Physics B 723 (2005) 3–32

Wrapping interactions and the genus expansion of the 2-point function of composite operators

Christoph Sieg a, Alessandro Torrielli b

a Max-Planck-Institut für Physik, Föhringer Ring 6, D-80805 München, Germany b Humboldt-Universität zu Berlin, Institut für Physik, Newtonstraße 15, D-12489 Berlin, Germany Received 26 May 2005; accepted 7 June 2005 Available online 1 July 2005

Abstract We perform a systematic analysis of wrapping interactions for a general class of theories with color degrees of freedom, including N = 4 SYM. Wrapping interactions arise in the genus expansion of the 2-point function of composite operators as ﬁnite size effects that start to appear at a certain order in the coupling constant at which the range of the interaction is equal to the length of the operators. We analyze in detail the relevant genus expansions, and introduce a strategy to single out the wrapping contributions, based on adding spectator ﬁelds. We use a toy model to demonstrate our procedure, performing all computations explicitly. Although completely general, our treatment should be particularly useful for applications to the recent problem of wrapping contributions in some checks of the AdS/CFT correspondence.  2005 Elsevier B.V. All rights reserved.

1. Introduction

5 The AdS/CFT correspondence [1] claims that type IIB string theory on an AdS5 ×S background with Ramond–Ramond (RR) ﬂux is dual to N = 4 super-Yang–Mills (SYM) theory with gauge group SU(N) which is invariant under the superconformal symmetry 5 group SU(2, 2|4). The common curvature radius R of AdS5 and of S in units of the string

E-mail addresses: [email protected] (C. Sieg), [email protected] (A. Torrielli).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.06.011 4 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 √ length α is related to the ’t Hooft coupling λ via R2 √ = λ, λ = g2N, g2 = 4πg , (1.1) α s where g and gs are the SYM and string coupling constants, respectively. One important 5 motivation for this conjecture is that the isometry group SO(2, 4) × SO(6) of AdS5 ×S can be identified with the bosonic subgroup SO(2, 4)×SU(4) of the superconformal group. A general proof of the conjecture is still out of reach, mainly due to the fact that a quantization of string theories in backgrounds with RR flux is difficult. Furthermore, the duality relates the strong coupling regime of one theory to the weak coupling regime of the other theory, preventing one from directly using perturbation theory to compute results in both theories that can be compared in a common regime of the parameters. However, it is possible to probe the correspondence in different limits. In one of these limits one considers classical solutions on the string side [2,3] (see also the review [4] and references therein). In this case quantitative results can be computed in the string theory as well as in its dual gauge theory. In the regime of large quantum numbers of the string (here collectively denoted by√ the angular momentum J of a rotating string solution) and in the large tension limit ( λ 1) the classical energy has a regular expansion in the modified ’t Hooft coupling λ = λ whichisheldfixed[5]. The analysis can be extended J 2 by including quantum fluctuations around the classical string solution [3,5]. The classical string sector shows integrability, e.g., rotating (rigid) string solutions have been shown to be described by the classical integrable Neumann system [6]. See also [7] for a generalization and [8–12] for related work on integrability in this context. The AdS/CFT correspondence predicts a matching of the classical energy of closed strings with the eigenvalues of the anomalous dimension matrix that describes the mixing of composite operators of N = 4 SYM under renormalization. These operators contain a single trace over the gauge group and can be regarded as discretized closed strings. In particular, the limit of interest requires that the operators contain a huge number of fields, implying that their mixing matrix, that has to be diagonalized, contains a huge number of entries. The mixing matrix itself can be obtained from the 2-point functions of the composite operators. Alternatively, the mixing problem can be recast in terms of an eigenvalue problem for the dilatation operator of N = 4SYM[13] (see also the reviews [14,15] and references therein). Furthermore, in the planar limit, the mixing problem has been refor- mulated in terms of integrable spin chains [16]. There, the composite single-trace operators are regarded as cyclic spin chains. Each fundamental field ‘flavor’ within the trace is interpreted as a spin projection eigenvalue at the corresponding site of the chain. The dilatation operator itself becomes the Hamiltonian of the chain. Formulated in these terms, the Bethe ansatz [17] (see [18] for a review) provides a tool for finding the energy eigenvalues for the spin chain states and thus the eigenvalues of the anomalous dimension matrix. Remarkable agreement of the classical string energies with the anomalous dimensions is found up to two loops. However, a mismatch was observed at three loops [19]. Another mismatch has been observed in the BMN limit [20] of the AdS/CFT correspondence. This limit corresponds to an expansion around a pointlike classical string in the center of AdS5 that moves along a great circle in S5. In this case, moving with the pointlike string with 5 the velocity of light, the AdS5 ×S background of the AdS/CFT correspondence is trans- C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 5 formed via the Penrose–Güven limit [21,22] to the 10-dimensional plane wave background [23,24]. The first order quantum fluctuations around the pointlike solutions are then described by strings in this plane wave background [3]. The corrections to the energy caused by higher order quantum fluctuations, i.e., curvature corrections to the plane wave background [25], show a mismatch with the corresponding anomalous dimensions starting at three loops [26–28]. There have been first indications for a mismatch between the quantum fluctuations at one loop order of extended string solutions, rotating in S5 [5] and in both, 5 AdS5 and S [29], with the respective gauge theory results [30,31]. However, agreement at this order was found later by including an overlooked contribution to the Bethe ansatz [32].1 A possible explanation for the mismatch, that does not insist on the unsatisfactory possibility of a breakdown of the AdS/CFT correspondence and of integrability in the planar limit, might be a non-commutativity of the limits taken on both sides [33,34]. On the string side, J →∞is taken first, and then the expression for the energy is expanded in powers of λ. However, on the gauge side the anomalous dimensions are computed as a perturbation expansion in λ that is valid for sufficiently large but finite J , and then the limit J →∞is taken, keeping λ fixed. Thereby, one has ignored finite size effects, that become relevant whenever the spin chain length is less than the range of the interaction. If this is the case, there are contributions in which the interaction wraps around the state [33,34]. They are denoted as wrapping interactions. First quantitative results including these interactions have been obtained in the context of a computer-based study of the plane wave matrix model at four loop order [35]. The purpose of this paper is to present a systematic and general study of wrapping interactions, not relying on the issues of a concrete theory and on a given perturbative order in the coupling constant. Instead, it contains statements valid in a class of theories which includes N = 4 SYM. Our general results are then checked at low and fixed loop order with the help of a simple example, that is then refined in subsequent steps. In this way we hope to have provided an initial step to understand the role of the wrapping interactions in the concrete challenge of explaining the above mentioned mismatch between semiclassical string energies and anomalous dimensions. The paper is organized as follows. In Section 2 we introduce and discuss the 2-point function of composite operators, being generated by connecting the legs of the operators with a Green function of the theory. We then work out the differences of the genus expansion of such a Green function and of the obtained 2-point function. In Section 3 we define and classify wrapping interactions as particular contributions to the Green functions that contribute at lower genus to the 2-point function. We work out some general statements for wrapping interactions by adding spectator fields to the composite operators. A special case of particular interest in the context of the AdS/CFT correspondence is the case of planar contributions to the 2-point function. There, we find that the planar wrapping diagrams originate from genus one contribution to the Green functions. We identify a unique structure for the spectator fields of planar wrapping diagrams.

1 We would like to thank I.Y. Park for bringing this fact to our attention. 6 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32

In Section 4 we present the expressions for the effective vertices and show their spectator structures. In Section 5 we apply our results to a toy model, given by scalar φ4 theory with a massless colored field of a single flavor. The toy model is then extended to multiple flavors and interactions with different flavor fluxes. At the end we arrive at the 4-scalar interaction of N = 4SYM. In Section 6 we discuss a possibility to choose the spectator fields in such a way that wrapping interactions can be automatically projected out in computer-based symbolic computations. In Appendix A we review and summarize some counting rules for Feynman diagrams used in the main text. We then apply them to extract some issues of (planar) wrapping diagrams. In Appendix B we collect some useful formulae for SU(N) and U(N) which enter our calculations.

2. The 2-point function of composite operators

2.1. Building blocks

Before we can define wrapping interactions we should fix our notations and conventions when we deal with the 2-point functions of composite operators. Oe = = By R, e 1, 2 we denote two local operators that consist of r 1,...,R elementary fields φnr where each nr denotes one field ‘flavor’ of the theory. Of particular interest are single-trace operators of the form 1 OR = √ tr(φn ···φn ), (2.1) ( N)R−2 1 R where the trace runs over the gauge group, i.e., the elementary fields φnr are decomposed = a a a × as φnr φnr T , and T are N N representation matrices of the Lie algebra of the gauge group. Later we will refer to the chosen normalization factor. For the sake of simplicity, from now on we will use the abbreviation a1 a2 an a1a2 ···an = tr T T ···T , (2.2) for the traces. We represent OR by a rectangular box with R elementary external legs attached to one of its sides. Counting of the lines starts with that line which with view into the outgoing direction has no direct neighbour on its right-hand side, and it ends with the line that has no left-hand neighbour, see Fig. 1(a). By V2R we denote the Green functions of the theory with 2R elementary external legs. R of them will be regarded as ingoing and outgoing, respectively. We represent V2R as a rectangular box where R elementary field lines enter from the left-hand side and the R remaining lines from the right-hand side. Counting of the lines is clockwise and starts from the lower left corner, see Fig. 1(b). Oe The 2-point function of two composite operators R, is defined as the correlation func- O1 O2 tion in which the R legs of the operator R [ R] are entirely contracted with the R ingoing C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 7

Fig. 1. The composite operators OR (a) and the Green function V2R (b) are the building blocks of the 2-point O1 O2 function R,V2R, R . O1 O2 [outgoing] fields of V2R. We denote such a 2-point function by R,V2R, R . When the Oe two R and V2R are combined within a 2-point function, the elementary fields within the Oe O1 traces of R are counted with increasing labels from left to right, starting with R from O2 + 1toR, and continuing with R using the labels R 1to2R. The 2-point function is obtained by contracting the 2R fields of the operators with the fields of V2R that carry the same label. This procedure guarantees that there are no crossings between the connecting lines if the lines leave the operator on the left- (right-)hand side from its right- (left-)hand side.

2.2. Coupling and genus expansion of V2R

In perturbation theory, the Green function V2R can be regarded as a series in powers K of the coupling constant g. Each fixed power gK contains a number of elementary Feynman K diagrams. Denoting a particular diagram by D2R, one can write = K V2R D2R. (2.3) K D K K In this expansion, any diagram D2R is of the order g . Furthermore, one can perform −2 K a genus expansion in powers of N [36]. Each Feynman diagram D2R is itself a sum K(h) of diagrams D2R which differ in the genus h. One defines the genus h of a diagram as the minimal genus of all compact Riemannian surfaces on which the diagram can be drawn in double-line notation without any crossings of lines. On the compactified Riemann surface, the 2R external lines end at a common vertex (at the point representing ∞) with the reversed ordering, i.e., they are attached counter-clockwise, starting with the lowest label. This configuration is shown in Fig. 2. Its genus can be obtained by Euler’s relation for the Euler character χ

χ = 2 − 2h = V − P + I, (2.4) where V is the number of vertices, P is the number of propagators, and I is the number of index loops. 8 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32

Fig. 2. On a compact Riemann surface, the external lines of the Green function V2R are connected to the vertex at ∞. These connections separate the Riemann surface into two parts. Lines that cross these connections require adding of the wrapping handle, depicted in gray.

The expansion of the Green function V2R then reads = (h) = K(h) V2R V2R D2R , (2.5) h h K D (h) 2−2h where each contribution√ V2R with ﬁxed genus h is of order N , and each diagram K(h) K 2−2h D2R is of the order ( λ) N . The vertex at ∞ carries a normalization factor N 1−R, which can be understood as follows. The vertex at ∞ can be regarded as an effective vertex for a planar connected (C = 1) tree-level (L = 0) diagram with E = 2R external legs. According√ to (A.3) it can be built out of 2R − 2 3-vertices. Hence, it should depend on ( λ)2R−2. The normalization factor is obtained by providing the required N dependence to transform the YM coupling g into λ, and then setting λ = 1 for the effective vertex.

2.3. Coupling and genus expansion of the 2-point function O1 O2 Similarly to V2R, one can expand the 2-point function R,V2R, R in powers of g −2 K(h) and N . A single diagram D2R generates a single diagram of the 2-point function if all 2 (R!) permutations within the two bundles of R elementary legs are contained within V2R. K(h) This means each permutation of the external legs is regarded as a different diagram D2R . Then, one only has to consider a unique way to contract the 2R legs of the two operators Oe K(h) R with the external legs of D2R . Consider the genus expansion of the 2-point function O1 ,V , O2 .By O1 ,DK(h), R 2R R R 2R O2 (H ) K(h) R we denote the genus H contribution of a particular diagram D2R to the 2-point K(h) function. The genus h of a particular diagram D2R does not uniquely determine the genus H . This effect has two reasons. First of all, drawing a particular diagram of the 2-point function, one has drawn only one representative of an R2-dimensional equivalence class of diagrams. It contains all diagrams C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 9

Fig. 3. The cyclic permutations of the external legs have been moved to the vertex at ∞. The resolution of the cyclic permutation requires adding one handle (a). An exception is the case (b) in which the crossing leg contains a non-planar self energy correction. In this case the genus does not change. Two cyclic permutations in the same (c) or in opposite directions (d) can be resolved with a single handle. that differ only by cyclic permutations within each of the two pairs of legs that connect the K(h) Oe diagram D2R to the two operators 2R. The equivalence relation is based on the fact that the operators (2.1) are invariant under cyclic permutations of their elementary fields. K(h) K(h) Two different representations contain two different diagrams D2R and D2R that can be mapped to each other by acting with cyclic permutations on the ingoing and/or outgoing legs. All these diagrams are again members of an equivalence class, for which elements one has h = h in general. Let us denote by h the minimal genus of all the diagrams within one equivalence class. A cyclic permutation applied to either the incoming or outgoing legs increases the genus h by at most one. This can be seen by resolving the crossing at the vertex at ∞,seeFig. 3(a). It requires adding at most one handle. In case that the crossing line contains a non-planar self energy contribution, no further handle has to be added. A further cyclic permutation acting on the remaining R legs does not change the genus further if the cyclic permutation applied to the other R legs has already increased the genus. This is because the same handle can be used to resolve the crossing, see Fig. 3(c) and (d). However, in case that no handle had to be added for the first set of R legs (like in case of a crossing line with a non-planar self energy contribution), at most one handle has to be added for the cyclic permutation applied to the second set of R fields. 2 K(h ) = Under the R diagrams D2R there is at least one with minimal genus h h.The remaining ones have genus h h h + 1 because the same handle can be used to resolve the two crossings caused by both cyclic permutations. This effect means that in O1 K(h) O2 (H ) = − = R,D2R , R the genus H can have values H h 1orH h. O1 O2 The second reason for the genus expansion of V2R and of R,V2R, R to differ from each other is caused by the fact that the 2-point function does not have external lines which, when it is drawn on a compact Riemann surface, have to be connected to an additional vertex at ∞. The bundles of ingoing and outgoing lines, interacting with each other in V2R and in the vertex at ∞, form a closed contour which divides the Riemann surface into two parts, see Fig. 2. That means, there has to be one handle for all the field lines crossing this contour. We will call this handle the wrapping handle. While being needed in certain K(h) O1 K(h) O2 diagrams D2R , it can be removed in R,D2R , R . Due to the separation of the two Oe operators R their contraction with the external lines of V2R no longer forms a closed 10 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32

Oe Fig. 4. On a compact Riemann surface, the two operators R in the 2-point function have a ﬁnite separation. This allows one to occupy the wrapping path, depicted in gray.

Fig. 5. Decomposition of the 2-point function O1 ,V , O2 into interaction part V2R without any permutations R 2R R SR of the external legs and the permutations SR without cyclic permutations C . V contains only one diagram CR R 2R K(h) D2R of each equivalence class with minimal genus h. The cyclic permutations are taken into account by using Oe Oe all cyclic permutations of the operators R, denoted by CR( R). contour, that divides the Riemann surface into two parts. There is a direct connection, from now on called the wrapping path, that replaces the wrapping handle (compare Figs. 2 and 4). The genus H of O1 ,DK(h), O2 hence obeys h − 1 H h. Combining the R 2R R O1 K(h) O2 (H ) above described effects, one expects that the diagrams R,D2R , R obtained from K(h) − a genus h diagram D2R in general have h 2 H h. Since here we are interested only in the change of the genus caused by removing the wrapping handle, we will disentangle both effect. From each equivalence class of dia- K(h) grams we will only use one representative D2R with the minimal genus h to build the corresponding diagram of the 2-point function. The additional R2 − 1 diagrams generated Oe by cyclic permutations are then taken care of by the two operators R. That means, to each of the operators we associate R − 1 copies that differ by cyclic permutations. They are combined in R2 possible ways to obtain all contributions to the 2-point function that C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 11

K(h ) + K(h) contain the diagrams D2R with h h h 1 in the equivalence class of D2R .The distribution of the permutations of the external legs of V2R is depicted in Fig. 5. K(h) K(h) After that we are left with two types of genus h diagrams D2R . One type of D2R O1 K(h) O2 contributes to genus h diagrams of the 2-point function R,D2R , R , the other type contributes to genus h − 1 diagrams, and its elements are called wrapping diagrams. In the next section we will analyze these contributions in detail.

3. Wrapping diagrams

3.1. Deﬁnition of wrapping and non-wrapping diagrams

− K(h) We deﬁne the genus h 1 wrapping diagrams Dw,2R as the genus h contributions to = − O1 O2 V2R that lead to a genus H h 1 contribution to the 2-point function R,V2R, R . K(h) Correspondingly, the non-wrapping diagrams Dnw,2R are the remaining genus h diagrams = O1 O2 that generate genus H h contributions to R,V2R, R . One has the relations (h) = (h) + (h) V2R Vnw,2R Vw,2R, (3.1) O1 (h) O2 = O1 (h) O2 (h) + O1 (h) O2 (h−1) R,V R , R R,V , R, R R,V , R, R , 2 nw 2 w 2 O1 O2 (H ) = O1 (H ) O2 + O1 (H +1) O2 R,V2R, R R,Vnw,2R, R R,Vw,2R , R . (3.2) (h) O1 O2 (H ) The translation of the genus expansion between V2R and R,V2R, R is visualized in Fig. 6.

3.2. Spectator ﬁelds

Here we develop a useful tool to distinguish between planar wrapping and non-wrapping diagrams based on adding spectator ﬁelds. They come in pairs of two ﬁelds ψ, ψˆ that

Fig. 6. The translation of the genus expansion between V (h) and O1 ,V , O2 (H ). The non-wrapping (nw) 2R R 2R R (h) = O1 O2 (H ) genus h contributions Vnw,2R to the Green function become genus H h contributions R,V2R, R to (h) = − the 2-point function. The wrapping (w) contributions Vw,2R become genus H h 1 contributions to the 2-point function. 12 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32

O1 O2 are inserted into the trace of respectively 2R and 2R, defined in (2.1), at any position ˆ between two φnr . The contraction of a pair of spectator fields, denoted by ψψ, generates a further connection line within the diagram, not interacting with any other lines. Pairs of spectator fields must be added such that their connection lines do not cross each other, and such that not two pairs are equivalent. Two pairs are equivalent if both, the two ψ and the ˆ ψ are direct neighbours within the two traces, not being separated by at least one φnr .The spectator pairs can be used to test some aspects of the topology of a diagram by observing how the genus H of the 2-point function behaves when their connection lines are added. In contrast to the wrapping diagrams, for non-wrapping ones there exists at least one representation in which the wrapping path introduced in Section 2.3 is unoccupied. Hence, following this empty path, it is possible to add a pair of spectator fields such that their connection does not cross any of the other lines. Due to the fact that the lines along the wrapping path differ between two representations of the same diagram, it is not sufficient for a diagram to be wrapping that its wrapping path is occupied by at least one field line. Likewise, it is not necessary for a diagram to be non-wrapping that its wrapping path is unoccupied. Instead, to distinguish between a wrapping and a non-wrapping diagram, one has to ensure that none or respectively at least one equivalent diagram with an unoccupied wrapping path exists. Consider a candidate for a genus h − 1 wrapping diagram. In a given representation of such a diagram one can always free the wrapping path by removing some lines. No further lines that have no influence on the occupation of the wrapping path are removed. Oe In a second step one removes the two operators 2R. In this way one arrives at a non- K(h) K(h) wrapping diagram Dnw,2R that is a truncated version of the original diagram D2R , and that obeys K K, h h − 1. If the original diagram DK(h) was wrapping, then the 2R K (h ) = truncated diagram Dnw,2R is necessarily connected (C 1 in the notation introduced in K(h) Appendix A,see(A.1)). This statement is only true if one forbids that D2R contains connected pieces that are entirely attached to only one of the two sets of R external legs. We show that if C>1 under the above assumptions, the original diagram was non-wrapping. If C>1, one has two possibilities to add two distinct spectator lines without introducing new handles or crossings, that separate the two connected pieces. One of them obstructs the wrapping path. It is possible to add these spectator lines such that they connect the operators with each other. This is possible since we demand that each connected piece was connected to both operators. Restoring the wrapping lines would force one to remove one of the spectator lines, since otherwise it had to be crossed, but the other spectator line can be kept, and hence the diagram would be non-wrapping for C>1. In Appendix A we show that for a theory with vertices with k + 2 legs that are of order gk in the YM coupling constant, planar wrapping interactions of order K in the YM coupling constant exist for K 2R. As long as K 2(2R − 1) they can be constructed from non-wrapping diagrams by adding a single line that runs along the wrapping path. This observation might be useful in explicit computer based calculations like [35], putting the analysis at higher order (K 2(2R − 1)) on an equal footing with the critical case (K = 2R). C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 13

3.3. The encoding of the topology of planar wrapping diagrams by the spectator ﬁelds

Restricting to genus h 1 contributions to the Green function V2R, one finds from the considerations in Section 3.2 that a necessary and sufficient condition for a diagram to be wrapping (non-wrapping) is that there exists no (at least one) possibility to add a pair of spectator fields ψψˆ to the diagram such that the crossings of their connection line with the other field lines do not require adding one handle. We should remark that in this statement the restriction to h 1 is essential. For h 2 one can find counter-examples to this statement, in which the handles allow one to add spectator lines to wrapping diagrams without changing their genera. We have depicted some examples in Fig. 7. Instead of adding one pair of spectator fields, one can directly put R pairs of spectator Oe fields into the R positions between the φnr of each of the two operators 2R. No crossings between spectator connections are allowed. Keeping fixed the position of the ψ, there are ˆ R possible cyclic permutations for the ψ. This means that to each of the R2 contributions O1 O2 Oe to R,V2R, R that come from the cyclic permutations of the operators 2R in (2.1), O2 one has to associate R variants by cyclic permutations only of the spectators within 2R, keeping fixed the fields φnr in its trace. With the help of these R variants of the spectator structure, one can now distinguish between wrapping and non-wrapping diagrams. In the case of a wrapping diagram, all pairwise connections between the spectators in each of the R variants crosses at least one other field line. In case of a non-wrapping diagram, there is at least one variant in which one spectator line obstructs the unoccupied wrapping path such that only up to R − 1 spectator lines cross other lines. O1 O2 (0) In the case of planar contributions R,V2R, R to the 2-point function, each connection of a pair of spectators that crosses other lines increases the genus H by one. It is important to stress that only in the planar case one necessarily has to add a handle for each spectator line that crosses other lines. For higher genus diagrams this correspondence breaks down as is shown by some examples in Fig. 7. One reason for this is that one handle can be used not only to cross another line, but even to resolve an additional crossing between lines that run along this handle. For H 1 there seems to be no universal way to identify the connections that are responsible for the genus reduction. K(1) The result from the above considerations is that a diagram Dw,2R is a planar wrapping diagram if the genus of the corresponding diagram of the 2-point function O1 K(1) O2 = 2R,Dw,2R, 2R with R spectator pairs is always increased by R , i.e., to H R.Here O1 O2 we have introduced the modified 2-point function 2R,V2R, 2R , where the difference Oe in the number of elementary fields at both 2R and V2R is given by the number of spectator fields. In case of genus H 1, this statement is sufficient for a diagram to be wrapping, but it is not necessary, as argued above. The classification via spectators is now used to project out the wrapping contributions. = = (1) For this purpose, consider the connected (C 1) genus h 1 contributions V2R to V2R. They generate planar wrapping contributions and genus H = 1 non-wrapping contributions O1 O2 to the 2-point function R,V2R, R . We now add R pairs of spectators in R different O1 O2 ways as described above and obtain the modified 2-point function 2R,V2R, 2R .We (1) focus on one of its contributions generated by V2R .IfallR cyclic permutations of the R 14 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32

Fig. 7. Some examples of higher genus diagrams with R = 2 (a)–(c) and R = 3 (d). Their spectator connections are depicted in gray. A genus H = 1 non-wrapping diagram (a). A genus H = 1 wrapping diagram (b) where both spectator connections increase the genus H by one to H = 3. A genus H = 1 wrapping diagram (c) where only the upper spectator connection increases the genus H by one to H = 2. No further handle has to be added for the downer spectator line to resolve its crossings, since this is done by the handle that resolves the crossing between the two ﬁeld lines. A genus H = 2 wrapping diagram (d), in which a non-planar self energy contribution of an internal line is responsible for the genus reduction. The two handles for the other ﬁeld lines resolve the crossings of the line that runs along the wrapping path.

ˆ O2 = spectator fields added to one of the operators, lets say ψ in 2R, lead to genus H R contributions, the contribution is a wrapping one. Applying this procedure in case of non- wrapping diagrams, one finds contributions of genus H = R − 1,...,R+ 1. H = R − 1: One spectator line obstructs the unoccupied wrapping path. Another spectator line does not cross the other field lines because of the single handle in the h = 1 contributions. C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 15

H = R: Either one spectator line obstructs the unoccupied wrapping path and all other spectator lines cross other field lines, or the wrapping path remains unobstructed and one spectator line does not cross the other field lines because of the single handle. H = R + 1: No spectator line obstructs the wrapping path and all other spectator lines cross other field lines. One thus has the relation 1 O1 (1) O2 = O1 (1) O2 (R+δ) + O1 (1) O2 (R) 2R,V2R , 2R 2R,Vnw,2R, 2R 2R,Vw,2R, 2R (3.3) δ=−1 2 for the C = 1 contributions to V2R. The above equation is now evaluated as follows: one Oe first adds the R spectator pairs to the operators 2R, only ensuring that their connections never cross each other. Then, for a given contribution to V2R one performs all contractions of the φnr , using the rules (B.5). In this way, one obtains an expression in which the R spectator pairs are distributed among a number of 1 T 2R traces. These distributions encode a classification of the ··· diagrams. To see this, we interpret each trace ψ1 ψst over st spectator fields as a vertex with st directed legs. The direction distinguishes legs associated to ψ from legs which represent ψˆ . Each pairwise contraction is regarded as a propagator connecting an outgoing with an ingoing leg of these vertices. As usual, these diagrams can be classified by the number of vertices, in this case given by T , the number of connected pieces C, the number of loops L. The number of propagators is fixed to P = R and the number of external legs is E = 0. In this picture, the removal of a spectator pair corresponds to a removal of a propagator. The execution of a contraction between pairs either becomes the fusion of two vertices or the fission of a single vertex into two, depending on whether the propagator contracts two legs at distinct vertices or at only one vertex. The diagrams that are generated from wrapping contributions are now identified as follows: each removal of a pair ψ, ψˆ must decrease the genus H by one and hence lead to an additional factor N 2. Thus, each removal of a spectator pair must increase the trace number T by one, compared to the case where this pair is kept and finally contracted. Each trace at the very end contributes a factor (1) = N. Another N-dependent factor comes from changing the normalization of the operators (2.1) that is required when the number of fields is changed. Furthermore, the above given issues of the spectator diagrams should remain true if all cyclic permutations of the ψˆ are taken into account. Hence, for the spectator diagrams of planar wrapping contributions a necessary condition is that no contractions must occur within a single trace. Otherwise, the removal of this pair of spectators would not change the genus. Instead of avoiding the fusion of two traces the fission of that trace would be inhibited. Since such a configuration must not occur in any of the cyclic permutations, the spectators within one trace must only be of one type, either ψ or ψˆ . Furthermore, the increasing of the trace number by removing one spectator pair must be independent of the contraction or removal of other spectators. This forbids that the connections form any closed loops since, after contracting some of them, one always would end up in a contraction of one spectator pair within a single trace. The spectator diagrams of wrapping

2 For C 2, the sum includes δ −2 and the wrapping contribution is absent. 16 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 contributions are therefore always tree-level and consist of 1 C R separate pieces. Any contraction of a pair ψ, ψˆ is an application of the fusion rule (B.5) such that, after having contracted all spectator pairs, C separate traces (1) remain, leading to a factor N C . On the other hand, removing all pairs ψ, ψˆ , one is left with T traces (1), contributing N T . A factor N R has to be considered for the change in the normalization of the operators. According to the last equation in (A.2), in the special case of tree-level diagrams one has the relation R = T − C, and hence concludes: Each tree-level spectator diagram that is built out of C connected pieces contributes a factor N C if the spectators are kept and it contributes a factor N 2R+C if the spectators are removed. In our case C has to be maximal, i.e., C = R. One can find several arguments for this statement. First of all, removing all the spectator pairs, one obtains a planar diagram, and 1 planar diagrams contribute to the leading power in the N expansion. This means, if C were not maximal for the planar wrapping diagrams, their contribution would be subdominant. K(h) Secondly,√ one can regard a diagram D2R as an effective vertex which is proportional to ( λ)K . The dependence on N one should associate with such a vertex can be interpreted as the N power which has to be absorbed into an effective ’t Hooft coupling that√ is later set to 1. For a planar vertex with 2R external legs one has to absorb a factor ( N)2R−2, i.e., the N dependence is given by its reciprocal value N 1−R. When this effective vertex contains h handles, and it is contracted with a planar effective vertex in a planar way, the number of traces is reduced by h. This means, compared to the case of a planar diagram which contains two planar vertices, the power in N found from the traces is reduced by h. Therefore, to remove a factor N 2 for each handle in the final result, one has to change the N dependent normalization of an effective vertex of genus h to N 1−R−h. That means, in case of planar wrapping contributions which have h = 1, including the traces over all Oe spectator pairs, and the normalization of the two operators 2R taken from (2.1), the total N dependence is N 2+C−3R. This factor should be equal to N 2−2H with H = R (see (3.3)), such that one deduces C = R.3 From the above considerations it follows that one can uniquely identify the wrapping diagrams that contribute to the planar 2-point function by finding the spectator structure ψ ψ ··· ψ ψˆ ··· ψˆ ψˆ = ψ ψˆ ψ ψˆ ··· ψ ψˆ . (3.4)

The issue that all traces contain only one spectator field enables one in principle to di- (1) rectly project out the planar wrapping contributions generated by V2R . One has to find an appropriate matrix for ψ and ψˆ such that traces which include more than one spectator field vanish. Such traces are always included in the remaining genus h = 1 non-wrapping contributions. This issue is discussed in Section 6.

3 This argument is only true if the contractions of the φnr never generate powers in N. According to (B.3), this would happen if two contractible ﬁelds φnr become direct neighbours within one trace without further ﬁelds. That this cannot happen here is guaranteed by the spectators that separate all φnr from each other. C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 17

4. Planar contributions to the 2-point function

4.1. Non-interacting case O1 O2 The simplest contribution to the 2-point function R,V2R, R is given by a pairwise planar (h = 0) connection of all the legs of the two operators without any interactions. The 0(0) corresponding contribution to V2R is given by the diagram D2R which is a direct product of free propagators that connect the ﬁelds with label r and 2R − r + 1, r = 1,...,R.The color part of this contribution, in case of a U(N) gauge group, is given by 1 0(0) 2 2 1 2 2 2 O ,D , O → R a ···a a + ···a = R N(1) = R N , (4.1) R 2R R N R−2 1 R R 1 2R where the factor R2 stems from the R2 pairs of cyclic permutations of the ﬁelds in each of O1 the two operators R. The second equality in (4.1) follows after applying the fusion rules for the traces (B.5) and the identity (B.3) with a0 = 0. The N dependent prefactor is the square of the normalization of the operators in (2.1), and it is chosen such that the above result is of the order N 2. A genus H contribution to O1 O2 2−2H R,V2R, R will then be of the order N .

4.2. Generic planar connected contributions to V2R

(0) Consider the planar connected contributions to V2R denoted as V2R . Their general form is shown in Fig. 8. From the fact that in the planar case lines must not cross in the corresponding ribbon graphs, it is obvious that they can always be represented as an effective vertex of the form √ 1 V (0) = f (0)( λ) a ···a , (4.2) 2R 2R N R−1 2R 1 √ (0) where f2R ( λ) captures the individual properties of the diagrams, i.e., their coupling dependence and the contributions from the loop integrals. The N dependence has been ﬁxed such that any planar diagram including this vertex is of the order N 2.

Fig. 8. A generic planar connected contribution to V2R. The picture shows that a planar contribution contracts the two index lines of one external line with one of the index lines of the two neighboured external lines. This corresponds to a contraction of the indices of the representation matrices. The gray-ﬁlled structure in the middle represents any planar diagram. It has no inﬂuence on the index lines that belong to the external legs. Thus all planar contributions consists of a single trace over the representation matrices. 18 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32

Including R spectator pairs ψψˆ , the generic planar contribution (4.2) to the 2-point function is given by O1 ,V(0), O2 2R 2R 2R ˆ ˆ ˆ ∝ a1ψa2 ···ψaRψ aR ···a1a2R ···aR+1 ψaR+1ψ ···a2R−1ψa2R

+ cycl. perm. ψˆ = ψ ··· ψ ψψˆ ψˆ ··· ψˆ + cycl. perm. ψ,ˆ (4.3) where in the ﬁrst equality we have used the invariance of the trace under cyclic permutations for a reordering of (4.2). The contribution given explicitly in the second equality of (4.3) is the one that identiﬁes the diagram as non-wrapping, since the removal of the spectator pair in the single trace does not increase the power in N and hence does not decrease the genus of the diagram. The connection line of this spectator pair obstructs the wrapping path. The other contributions to (4.3) are of the form ψ ··· ψ ψψˆ ψˆ ··· ψˆ . (4.4)

In contrast to ﬁrst contribution in the second line of (4.3), the removal of any spectator pair increases the power in N by two and hence reduces the genus of the corresponding diagram by one. In these contributions, no spectator line obstructs the wrapping path.

4.3. Generic planar wrapping contributions to V2R

(1) Consider the planar wrapping contributions to V2R denoted by Vw,2R. In a genus ex- O1 O2 = pansion of the 2-point function R ,V2R, R they contribute at planar level if R R, but all their contributions become genus H = 1forR >R. The most general planar wrapping contribution is obtained by taking the most general planar diagram and then filling the wrapping path with the most general planar structure. One then arrives at the ribbon graph shown in Fig. 9. From the restriction that lines must not cross in this graph, one can read off the general effective vertex for a generic planar wrapping diagram. It is given by √ (1) (1) 1 V = f ( λ) a ···a a ···a + , (4.5) w,2R w,2R N R R 1 2R R 1 √ (1) where fw,2R( λ) captures the individual properties of the wrapping diagrams, i.e., their coupling dependence and the contributions from the loop integrals. The N dependence has Oe been fixed such that any diagram with two operators 2R , R >Rincluding this vertex is of the order N 0. The vertex (4.5) is obviously invariant under separate cyclic permutations of the ingoing and outgoing legs. This is even obvious in Fig. 9, since these cyclic permutations do not change the structure represented by the gray-filled region. The symmetry found for wrapping diagrams is consistent with the observation, that the wrapping handle C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 19

Fig. 9. A generic planar wrapping contribution to V2R. The occupation of the wrapping path separates the ingoing and outgoing fields from each other and breaks the single trace of ordinary planar diagrams in Fig. 8 into two traces. The gray-filled structure in the middle represents any planar diagram. It has no influence on the lines that belong to the external legs. Thus all planar wrapping contributions consists of a product of two separate traces over the representation matrices for the ingoing and outgoing legs.

is sufﬁcient to resolve all those crossings within the ingoing and outgoing external legs of K(1) a given diagram D2R that can be traced back to cyclic permutations. This can be easily understood, looking at Fig. 3, and adding the wrapping handle that connects the regions above and below the vertex at ∞. ˆ O1 (1) O2 Including R spectator pairs ψψ, the 2-point function 2R,Vw,2R, 2R of the generic planar wrapping contribution (4.5) is given by

O1 (1) O2 2R,Vw,2R, 2R ˆ ˆ ˆ ∝ R a1ψa2 ···ψaRψ aR ···a1 a2R ···aR+1 ψaR+1ψ ···a2R−1ψa2R

= R ψ ··· ψ ψ ψˆ ψˆ ··· ψˆ , (4.6)

where the factor R in the first equality is generated by the cyclic permutations of the ψˆ that in this case do not lead to different contributions. The final result of (3.4) is the one that identifies the diagram as planar wrapping, since each removal of a spectator pair increases the number of traces and hence the power in N by one. With the additional factor of N from the change in the normalization this leads to a factor N 2, showing the decreasing of the genus H of the diagram by one. 20 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32

4.4. Generic planar wrapping contributions to V2R including ﬂavor

We extend the effective vertex (4.5) of planar wrapping interactions by considering theories which include flavor degrees of freedom such that the flavor structure is described by Kronecker δs. In the subsector of composite operators which contain no trace terms in their flavor indices4 one can write down a rather simple expression for this vertex. It is given by (1) 1 = ··· ··· + Vw,2R R aR a1 a2R aR 1 N √ f (1) ( λ) × π,2R δm1 δm2 ···δmR , (4.7) R mω(π(2R)) mω(π(2R−1)) mω(π(R+1)) π∈SR/CR ω∈CR √ (1) where fπ,2R( λ) captures the individual properties of the diagrams, and mr is the corresponding flavor index carried by the leg with color index ar . Some remarks should be made. First of all, the above expression is valid to arbitrary high order in λ. Secondly, the vertex is chosen such that its flavor dependence is invariant under separate cyclic permutations applied to the two pairs of R ingoing and outgoing legs. Since the color dependence respects the same symmetry, it is a symmetry of the complete vertex. The choice of this effective vertex enables one to reduce the number of coefficient functions from R! in the original vertex to (R − 1)!. This reduction should be useful in practical calculations. Instead of explicitly computing the coefficient functions by summing up the diagrams of the perturbation expansion, one could try to determine the coefficient functions by fitting them to available data. In this case, the chosen effective vertex should of course contain the minimum number of coefficient functions. The choice of (4.7) would be a first step. As a second step, one should try to use additional symmetries, unitarity, non- renormalization theorems, etc., to further reduce the number of independent coefficient functions.

5. The toy model

Let us consider the following toy-example of a single scalar ﬁeld in the adjoint representation of U(N), with a standard kinetic term and an interaction Lagrangian: g2 S [φ]= tr(φφφφ). (5.1) int 4! From here one can derive the interaction vertex as g2 a1a2a3a4 V = a1a2a3a4 + a1a2a4a3 + a1a3a2a4 3! + a2a1a3a4 + a1a4a2a3 + a1a4a3a2 , (5.2)

4 This is true for instance in the SU(2) subsector of N = 4 SYM, consisting of two scalars X = √1 (φ + iφ ), 2 1 2 Y = √1 (φ + iφ ). 2 3 4 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 21

Fig. 10. The three possible types of connected (C = 1) Feynman diagrams occurring at the perturbative order g4 in V4. where we have used the abbreviation (2.2) for the traces. Due to the cyclic invariance of the trace, only six inequivalent permutations out of 4!=24 possible permutations remain. We compute the ∝ g4 contribution to the 2-point function of the composite operator O = 2 = 2 tr(φ ) in perturbation theory. As derived in Appendix A, this order (K 4) is precisely the critical case for this operator (R = 2), where wrapping issues appear ﬁrst. This example reproduces, in a fairly simpliﬁed version, one of the situations one encounters in the more complicated case of N = 4SYM. In a theory with vertices only of type (5.1) at order g4, there are only three possible types of connected (C = 1) Feynman diagrams contributing to V4. We have depicted them in Fig. 10. Combining the six terms of the two 4-vertices (5.2) together, and considering all possible permutations of the external legs, one gets (4!)2 diagrams for the t- and the u-channel (Fig. 10(a)) and for the s-channel (Fig. 10(b)). The self energy contribution Fig. 10(c) leads to 4(4!)2 diagrams. All terms can be grouped together into 17 different color structures that enter the full vertex obeying crossing symmetry. The result reads

4 2g V a1a2a3a4 = N (α + α + 8γ ) a a a a + a a a a 4,sym. ! 2 t s 4 1 2 3 4 3 2 1 (3 ) + (α + α + 8γ ) a a a a + a a a a t u 4 1 3 2 4 2 3 1 + (αu + αs + 8γ ) a4a3a1a2 + a4a2a1a3 + (α + α + α + 3γ ) a a a a + a a a a t u s 1 2 3 4 1 3 2 4 + a a a a + a a a a 1 2 4 3 2 1 4 3 + a a a a + a a a a 1 3 4 2 3 1 4 2 + a2a3a4 a1 + a3a2a4 a1 + (4α + α + α ) a a a a t u s 1 4 2 3 + (α + 4α + α ) a a a a t u s 1 3 2 4 + (αt + αu + 4αs) a1a2 a3a4 , (5.3) where αt , αu, αs , γ describe the spacetime factors, including integrals over the positions of the two vertices and depending on the coordinates x1, x2 and y1, y2 of the ingoing and 22 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 outgoing fields, respectively. To obtain (5.3) we have repeatedly used the U(N)fusion and fission rules (B.5) and the trace of the identity (1) = N. To match our definition of the Green function, when used as a building block of the 2-point function, which is given in Section 2.3 (see also Fig. 5), we divide out the cyclic permutations acting separately on the incoming and outgoing legs. In this case these are simply the exchanges a1 ↔ a2 and a3 ↔ a4, under which the 17 color terms in (5.3) boil down to 6 distinct structures. We obtain the building block 2g4 V a1a2a3a4 = N (α + β + 8γ) a a a a + (α + 4γ) a a a a 4 ! 2 4 3 2 1 4 1 3 2 (3 ) + (2α + β + 3γ) a a a a + a a a a 1 2 3 4 1 3 4 2 1 + (5α + β) a a a a + (α + 2β) a a a a . (5.4) 2 1 4 2 3 1 2 3 4

Furthermore, we have identiﬁed x1 = x2 = x and y1 = y2 = y such that αt ,αu → α, αs → β and γ → γ . The coefﬁcients α, β, γ contain the corresponding 3-loop integrals. With the coordinate expression for the propagator d − ( 2 1) 1 Ixy = (5.5) d d −1 4π 2 ((x − y)2 − i ) 2 they read in d = 4 − 2ε dimensions

= d d 2 = d d 2 2 2 = α d u d vIxuIxvIuvIuy Ivy,β d u d vIxuIuvIvy,γ0. (5.6) In the last equality the vanishing of all tadpole-type graphs in dimensional regularization has been used. Moreover, they would not in any case affect the analysis of planar wrapping diagrams, since it is straightforwardly seen that they never lead to planar wrapping contributions. Explicit results for α and β can be obtained by the Gegenbauer x-space technique (see (2.20) in [37]), and for β more easily by the methods based on uniqueness [38]. O = 2 The 2-point function of the operator 2 tr(φ ) is obtained from (5.4) by contracting a1 a3 the indices a1, a2 and a3, a4 with tensors proportional to δa2 and δa4 , respectively. This is 2 a b obvious after rewriting tr(φ ) as φaφb tr(T T ) and using (B.1). The color part reduces to either N 4 or N 2 if the corresponding diagram is of genus H = 0orH = 1, respectively. This is in accord with the genus expansion of the 2-point function in Section 2.3. The planar contributions to the 2-point function are identified as the ones that contain the color structures a4a3a2a1 and a1a2 a3a4 . By drawing the diagrams, one can check that the structure with a single trace is associated to the non-wrapping diagrams, while the double trace contribution takes care of the planar wrapping diagrams. This coincides with the generic results (4.2) and (4.5). The identification of planar wrapping diagrams is simple in this special case, where the number of diagrams is small enough to draw all of them. We can therefore explicitly check our procedure based on spectator insertions as introduced in Section 3.2, adding two pairs of spectator fields such that the modified 2-point function becomes a1a2a3a4 ˆ ˆ + Oe + ˆ a1ψa2ψ V4 ψa3ψa4 cycl. perm. 2 cycl. perm. ψ. (5.7) C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 23

Oe Here, cyclic permutations within 2 corresponds to the 4 possible combinations obtained ↔ ↔ Oe by the changes a1 a2, a3 a4 within the two operators 2. The terms in (5.4) lead to the distinct spectator structures ˆ ˆ ˆ ˆ a4a3a2a1 → ψ ψψ ψ + ψ ψψ ψ , ˆ ˆ ˆ ˆ a4a1a3a2 → ψψψψ + ψψψψ , ˆ ˆ ˆ ˆ a1a2a3 a4 → ψ ψψψ + ψ ψψψ , ˆ ˆ ˆ ˆ a1a3a4 a2 → ψψψ ψ + ψψψ ψ , ˆ ˆ ˆ ˆ a1a4 a2a3 → ψψ ψψ + ψψ ψψ , ˆ ˆ a1a2 a3a4 → 2 ψ ψ ψ ψ . (5.8)

As can be easily seen, the only case in which the traces completely factorize into four single pieces is the wrapping case, in accord with our general arguments in Section 3.3. The toy example we presented up to here is interesting, but of course it does not include the full N = 4 complexity. In order to get closer to that case, in a first step we add flavor degrees of freedom to the interaction Lagrangian (5.1) in a simple way, such that the number of terms in the symmetrized interaction vertex (5.2) is not enlarged. In a second step, we then consider the 4-scalar commutator interaction of the full N = 4 theory, which symmetrized interaction vertex contains several terms with different flavor dependence.

5.1. 4-vertex with cyclic symmetric ﬂavor ﬂux

In our first refinement we take as interaction term g2 S [φ]= tr(φ φ φ φ ), (5.9) int 4! m n m n where we have introduced flavor indices m, n = 1,...,Nf. The two flavor lines that enter and leave the vertex at its four legs cross each other as depicted in Fig. 11(a).

Fig. 11. The ﬂow of a ﬂavor line through a vertex of the types tr(φmφnφmφn) (a) and tr(φmφmφnφn) (b). In the second case, the two permutations have to be taken into account in the symmetrized vertex. 24 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32

Fig. 12. Contributions to planar wrapping interactions in the presence of ﬂavor. The structure in Fig. 10(a) leads to contributions with a closed ﬂavor loop as shown in (a). The structure in Fig. 10(b) leads to contributions m1 m2 depending on δm3 δm4 as shown in (b).

Since the structure with crossing flavor lines is invariant under cyclic permutations, the symmetrized vertex consists of six inequivalent terms as in (5.2), but with an additional simple factor for the flavor dependence. One gets g2 a1a2a3a4 = m1 m2 + m1 m2 Vm m m m a1a2a3a4 δm δm a1a2a4a3 δm δm 1 2 3 4 3! 3 4 4 3 + a a a a δm1 δm3 + a a a a δm1 δm2 1 3 2 4 m2 m4 2 1 3 4 m4 m3 + a a a a δm1 δm3 + a a a a δm1 δm2 , (5.10) 1 4 2 3 m2 m4 1 4 3 2 m3 m4 where mr is the corresponding flavor index carried by the leg with color index ar , arranged as in Fig. 10. 1 The genus expansion is an expansion in powers of N , the inverse of the number of colors. Adding flavor degrees of freedom does not influence this expansion, it simply extends the result of a single field by multiplying each term by a flavor dependent factor. We con- O = sider the 2-point function of the operator 2 tr(φmφm) which now include a summation over all flavors. The planar wrapping diagrams turn out to be given by g4 a a a a αN δm1 δm2 + 2βδm1 δm2 . (5.11) (3!)2 1 2 3 4 f m4 m3 m3 m4 This result can be brought into the symmetrized form (4.7), where the single coefficient is proportional to αNf + 2β. In the corresponding ribbon graph Fig. 9 the color structure a1a2 a3a4 has a simple interpretation as the breaking of the single surrounding index line by the structure along the wrapping path. Adding flavor lines to the corresponding diagrams, one finds an interpretation even for the flavor structure in (5.11). A vertex with the flavor flow as given in Fig. 11(a) yields the diagram depicted in Fig. 12(a). A factor Nf can be traced back to a closed flavor loop which is present in diagrams of the type shown in Fig. 10(a). The re- m1 m2 maining flavor lines can be associated to the factor δm4 δm3 . The other contributions are based on the type of diagrams shown in Fig. 10(b). They do not contain closed flavor loops m1 m2 and can be associated to the flavor structure δm3 δm4 ,seeFig. 12(b). C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 25

5.2. 4-vertex with commutators

Now we get more into contact with N = 4 SYM adopting the proper commutator interaction given by

1 g2 g2 g2 S [φ]= tr[φ ,φ ][φ ,φ ]= tr(φ φ φ φ ) − tr(φ φ φ φ ), (5.12) int 2 4! m n m n 4! m n m n 4! m m n n 1 where we have introduced the factor 2 for convenience. The interaction in (5.12) appears as the combination of the previous one (5.9) with another term in which the flavor lines ‘repulse’ each other, as depicted in Fig. 11(b). The vertex with repulsing flavor lines is not invariant under all cyclic permutations, but only under those where the legs are shifted by an even number. So one has to keep track of two contributions, differing by an odd number of shifts. The explicit form of the vertex turns out to be 2g2 V a1a2a3a4 = a a a a 2δm1 δm2 − δm1 δm3 − δm1 δm2 m1m2m3m4 ! 1 2 3 4 m3 m4 m2 m4 m4 m3 4 + a a a a 2δm1 δm2 − δm1 δm2 − δm1 δm3 1 2 4 3 m4 m3 m3 m4 m2 m4 + a a a a 2δm1 δm3 − δm1 δm2 − δm1 δm2 1 3 2 4 m2 m4 m3 m4 m4 m3 + a a a a 2δm1 δm2 − δm1 δm2 − δm1 δm3 2 1 3 4 m4 m3 m3 m4 m2 m4 + a a a a 2δm1 δm3 − δm1 δm2 − δm1 δm2 1 4 2 3 m2 m4 m3 m4 m4 m3 + a a a a 2δm1 δm2 − δm1 δm3 − δm1 δm2 . (5.13) 1 4 3 2 m3 m4 m2 m4 m4 m3 The identification of the planar wrapping diagrams proceeds as in Section 5.1.Aftera tedious but straightforward computation of the flavor factor one ends up with the result

g4 a a a a ! 2 1 2 3 4 2(3 ) × (α − 2β + βN )δm1 δm3 + (α + β)δm1 δm2 + 2α(N − 2)δm1 δm2 . (5.14) f m2 m4 m3 m4 f m4 m3 Its flavor dependence can again be reconstructed by taking into account all the possible flavor flows generated by combining the vertices in Fig. 11. This result can be brought into the symmetrized form (4.7), where the single coefficient is proportional to 2αNf − 3α + β.

6. A candidate for ψ

In this section we want to show how one can project out the planar wrapping contribu- (1) tions from all contributions to the 2-point function that contain V2R , by choosing suitable matrices for the spectator pairs ψ, ψˆ . The planar wrapping contributions have the prop- (1) erty that they are the only contributions to V2R that generate the spectator structure (3.4) where the trace of all spectator fields is taken separately. If one could find a non-traceless matrix with the property that its positive powers p are traceless up to a sufficiently high 26 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 order p0,2 p p0, then one could replace all spectator fields by this matrix. The non- (1) wrapping contributions to V2R would then vanish automatically when used to build the Oe 2-point function of these modified operators 2R. ˆ ˆ 0 a In order to find such a matrix, we parameterize ψ, ψ as ψ = ψ = ψ0T + ψaT , a = 1,...,N2 − 1 with respect to the decomposition of U(N) into a U(1) part and an SU(N) part. The coefficients ψ0 and ψa are complex. A non-traceless ψ requires that 2 ψ0 = 0. The condition that tr(ψ ) = 0 amounts to the ‘light-cone’ condition a + 2 = ψaψ ψ0 0, (6.1) wherewehaveused(B.1) and (B.2). The vanishing of the traces that contain higher powers of ψ is greatly simplified by the recursive use of the relations found up to the step before. For example, evaluation of tr(ψ3) and use of (6.1) gives a b c 2 3 ψaψbψc tr T T T − √ ψ = 0. (6.2) N 0 After a little thought, one realizes that the general form of this equation at order n is c a1 an n n ψa ···ψa tr T ···T + √ ψ = 0, (6.3) 1 n ( N)n−2 0 where the cn are determined by recursion as − n n 1 n c = 1 − − c (6.4) n 2 m m m=3 for n 3 (for n = 3 the sum has to be considered empty). This recursion is solved by

n cn = (−1) (n − 1), (6.5) as can be easily veriﬁed using properties of binomial sums. The constraint on ψ is given by (6.1) combined with (−1)n a1 an n ψa ···ψa tr T ···T + √ (n − 1)ψ = 0 (6.6) 1 n ( N)n−2 0 for n = 3,...,n0 up to a desired order, which coincides with the highest possible power of ψ within a single trace, obviously given by n0 = 2R. The choice ψ = √1 normalizes the trace (ψ) = 1, such that the additional matrices 0 N within the traces do not change the prefactor for the planar wrapping interactions. The conditions (6.1) and (6.6) then read 1 (−1)n ψ ψa =− ,ψ···ψ tr T a1 ···T an + (n − 1) = 0. (6.7) a N a1 an N n−1 The important question to answer is whether this set of conditions has a solution in gen- ··· eral. This involves determining the components of the tensors ka1 an = tr(T (a1 ···T an)) appearing in (6.6), where the parentheses indicate complete symmetrization with unit weight (i.e., summing over all permutations, and dividing by n!). These objects are a fam- ily of symmetric invariant tensors of SU(N) [39]. Not all of them are independent. For C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 27

··· n N + 1 ka1 an can be expressed in terms of the lower order ones. Instead of working with these objects, we have tried to test whether it is possible to ﬁnd any matrix M with a non-vanishing trace, such that M2, M3 and M4 have vanishing trace. We were able to ﬁnd ¯ a diagonal matrix of rank 12 with pairs of complex conjugated eigenvalues λi , λi of unit length given by √ √ 1 + i 3 −1 + i 3 λ =−1,λ= ,λ= . (6.8) 1,2 3 2 4,5,6 2

7. Conclusions

In this paper we have studied the genus expansion of the 2-point function of single- trace operators containing R elementary fields, and of one of its building blocks which is the Green function with 2R external legs. We have shown that the two genus expansions do not coincide. The contributions to the Green function of a fixed genus h lead to genus h − 2 H h contributions to the 2-point function. We have discussed the two origins for this effect. Firstly, a reduction of the genus is caused by the invariance of the trace of the composite operators under cyclic permutations. Secondly, a reduction of the genus is caused by the fact that certain crossings between field lines in some contributions to the Green function can be resolved when it is used for the construction of the 2-point function. In these wrapping contributions some field lines occupy a special (wrapping) path, present in case of the 2-point function, thereby avoiding some crossings that would require adding a handle. These wrapping contributions play a role whenever the range of an interaction becomes larger than the length of the object it acts on. In the context of the AdS/CFT correspondence it is believed that they can explain an observed mismatch between the energies of classical strings and the anomalous dimensions eigenvalues of the corresponding operators starting at three loops. To avoid the additional genus changes caused by the cyclic symmetry of the local operators, we have introduced equivalence classes for the diagrams of the Green function. Diagrams that differ only by cyclic permutations within the ingoing and outgoing legs are identified, and for the construction of the 2-point function only one diagram with minimal genus h is taken from each equivalence class. As a result of the analysis we have found that all genus H contributions to the 2-point function of composite single-trace operators are obtained from these genus h = H and h = H + 1 contributions to the Green function. In our opinion this observation should be of particular importance. At a given sufficiently high order in the coupling constant, one can divide the contributions to the 2-point function and to the Green function into two classes. One class contains those contributions which are universal, i.e., they have to be used in the construction of the 2-point function of operators with a larger number of legs. The second class contains those contributions which are non-universal. Its elements change whenever the number of legs of the operators is changed. However, our above given result shows that even these non-universal contributions are contained within a universal quantity, the Green function. Thus, the non- universality of the elements within the second class can be understood as a non-universality of the projection operation used to obtain these elements. A better understanding of the 28 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 projection operation might therefore provide a way to obtain the wrapping contributions without the need to implement a selection process on each diagram separately. As a first step into this direction, we have worked out a technique, that allows one to identify the wrapping contributions in the planar (H = 0) case, which is of particular interest in the context of the AdS/CFT correspondence. The technique is based on adding pairs of spectator fields and their pairwise connections to the diagrams of the 2-point function. The planar wrapping contributions are then identified as the only contributions in which, after contracting all the other fields, the trace over each matrix-valued spectator field is taken separately. A particular choice for the components of the spectator fields should therefore project out the planar wrapping contributions automatically. We have determined the equations which such a matrix must fulfill and we have presented a simple example to demonstrate that the problem itself should have a solution. Clearly, our analysis is only a first step to a complete understanding of the wrapping interactions. As a next step one could try to restrict the freedom in our proposed general vertex that should describe wrapping interactions in the traceless SU(2) subsector. It would be interesting to discuss whether such an interaction term can be compatible with integrability.

Acknowledgements

We would like to thank B. Eden, J. Plefka and E. Sokatchev for deep and enlightening discussions. The work of C.S. is supported by DFG (Deutsche Forschungsgemeinschaft) within ‘Graduiertenkolleg 271’ and within project ER 301/1-4. The work of A.T. is supported by DFG within the ‘Schwerpunktprogramm Stringtheorie 1096’.

Appendix A. Counting rules for Feynman diagrams

We use the quantities

K = order of the diagram gK , E = number of external lines, P = number of propagators, k = k Vi number of vertices of order g with i legs, V = number of vertices, L = number of loops, C = number of connected pieces, (A.1) that classify a generic Feynman diagram. The following relations hold = k = k = k − K kVi ,V Vi ,E iVi 2P, i,k i,k i,k P = V + L − C. (A.2) C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 29

The equations for E can be cast into the form = − k − − E (i 2)Vi 2(L C). (A.3) i,k k It simplifies if the Vi obey k = k Vi Vk+2δi,k+2, (A.4) i.e., from the coupling constant g of the three point vertex all coupling constants of the higher point vertices are uniquely determined by adding one power in g for each additional leg. For example, in YM theories one only has the cubic and quartic vertices, and therefore 1 2 respectively V3 and V4 are different from zero. With (A.4) one finds that (A.2) simplifies to = k = k = − − K kVk+2,V Vk+2,EK 2(L C), k k P = V + L − C. (A.5) O1 O2 Consider now the special case of a 2-point function R,V2R, R of two composite op- Oe erators R, where all vertices in V2R are of the form (A.4). One can regard the operators as two vertices that do not contribute to the order in the coupling constant and that have R 0 = = O1 legs, i.e., VR 2. Furthermore, one has E 0 since all legs of R are contracted with the external legs of V2R. From the equation for E in (A.2) one then finds in this case K = 2(L − C − R + 2). (A.6) One can now determine the minimum order K for a planar wrapping diagram to appear. For this purpose, one has to estimate the minimum number of L. The simplest non-interacting case in Section 4.1 with K = 0, C = 1 inserted into (A.6) gives L = R − 1. In this case one can add R spectators to the diagram without making it non-planar by crossing any other lines. Therefore, each of the R − 1 loops has to be divided into at least 2-loops by a line that crosses a spectator line. Furthermore, one has to add at least one line that runs along the wrapping path, increasing the loop number by at least one. The number of loops in a wrapping diagram thus fulfills L 2R − 1. Clearly C = 1, and (A.6) then leads to a lower bound for K that becomes

K 2R. (A.7) The structure of a generic planar wrapping diagram is arbitrarily complicated. In particular, the interaction part that fills the wrapping path can itself be an arbitrary planar diagram, see Fig. 9. However, up to a certain order K in the coupling constant, all wrapping diagrams can be obtained from planar non-wrapping connected diagrams of maximum interaction length by adding a single line along the wrapping path. For a given wrapping diagram one can use the cyclic symmetry to minimize the number of lines that run along the wrapping path. One only has to guarantee that the order K in the coupling constant is sufficiently low such that at least one spectator line can be added that only crosses a single line of the planar diagram. This means all the other spectator lines in this case are allowed to cross at most two other lines. If one of them crosses more 30 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 than two other lines, one automatically finds a diagram at the same order where none of the spectator lines crosses only a single line. This is due to the fact that one could replace the third crossed line by a line that generates a second crossing of the spectator line that previously crossed only one other line. Since each of the lines that connect neighboured fields of the operators in a given diagram effectively contributes with a factor g2, the order K for which all wrapping diagrams can be obtained by adding a single line to the planar non-wrapping connected diagrams of maximum interaction length has to fulfill K 2 2(R − 1) + 1 = 2(2R − 1). (A.8)

Appendix B. Rules for U(N) and SU(N)

a i 2 Let (T ) j , a = a0,...,N − 1, i, j = 1,...,N be the N × N representation matrices for the Lie algebras of U(N) (with a0 = 0) and SU(N) (with a0 = 1). They are chosen such that they fulﬁll T a,T b = if abcT c, tr T aT b = δab, (B.1) where the trace tr is taken over the fundamental indices i, j, respectively. In particular, the U(1) generator is given by 1 T 0 = √ 1. (B.2) N The representation matrices then fulﬁll the relation

2− N1 a T a i T a k = δiδk − 0 δi δk, (B.3) j l l j N j l a=a0 where as above a0 = 0 and a0 = 1forU(N) and SU(N), respectively. With the help of the above relation it is straightforward to obtain the fusion and ﬁssion rules for the traces a tr T aA tr T aB = tr(AB) − 0 tr(A) tr(B), N a tr T aAT aB = tr(A) tr(B) − 0 tr(AB). (B.4) N In the main text we use an abbreviated notation where the traces are simply represented by parentheses and indices are no longer written as superscripts. The above rules then read a (aA)(aB) = (AB) − 0 (A)(B), N a (aA aB) = (A)(B) − 0 (AB). (B.5) N

References

[1] J.M. Maldacena, The large N limit of superconformal ﬁeld theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231–252, hep-th/9711200. C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32 31

[2] S.S. Gubser, I.R. Klebanov, A.M. Polyakov, A semi-classical limit of the gauge/string correspondence, Nucl. Phys. B 636 (2002) 99–114, hep-th/0204051. 5 [3] S. Frolov, A.A. Tseytlin, Semiclassical quantization of rotating superstring in AdS5 ×S , JHEP 0206 (2002) 007, hep-th/0204226. [4] A.A. Tseytlin, Spinning strings and AdS/CFT duality, hep-th/0311139. [5] S.A. Frolov, I.Y. Park, A.A. Tseytlin, On one-loop correction to energy of spinning strings in S5,Phys.Rev. D 71 (2005) 026006, hep-th/0408187. 5 [6] G. Arutyunov, S. Frolov, J. Russo, A.A. Tseytlin, Spinning strings in AdS5 ×S and integrable systems, Nucl. Phys. B 671 (2003) 3–50, hep-th/0307191. 5 [7] G. Arutyunov, J. Russo, A.A. Tseytlin, Spinning strings in AdS5 ×S : New integrable system relations, Phys. Rev. D 69 (2004) 086009, hep-th/0311004. 5 [8] L.F. Alday, G. Arutyunov, A.A. Tseytlin, On integrability of classical superstrings in AdS5 ×S ,hep- th/0502240. 5 [9] I. Swanson, On the integrability of string theory in AdS5 ×S , hep-th/0405172. [10] I. Swanson, Quantum string integrability and AdS/CFT, Nucl. Phys. B 709 (2005) 443–464, hep-th/0410282. [11] V.A. Kazakov, A. Marshakov, J.A. Minahan, K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP 0405 (2004) 024, hep-th/0402207. 5 [12] G. Arutyunov, S. Frolov, Integrable Hamiltonian for classical strings on AdS5 ×S , JHEP 0502 (2005) 059, hep-th/0411089. [13] N. Beisert, C. Kristjansen, M. Staudacher, The dilatation operator of N = 4 super-Yang–Mills theory, Nucl. Phys. B 664 (2003) 131–184, hep-th/0303060. [14] J.C. Plefka, Lectures on the plane-wave string/gauge theory duality, Fortschr. Phys. 52 (2004) 264–301, hep-th/0307101. [15] N. Beisert, The dilatation operator of N = 4 super-Yang–Mills theory and integrability, Phys. Rep. 405 (2005) 1–202, hep-th/0407277. [16] J.A. Minahan, K. Zarembo, The Bethe-ansatz for N = 4 super-Yang–Mills, JHEP 0303 (2003) 013, hep- th/0212208. [17] H. Bethe, On the theory of metals, 1: Eigenvalues and eigenfunctions for the linear atomic chain, Z. Phys. 71 (1931) 205–226. [18] L.D. Faddeev, How algebraic Bethe ansatz works for integrable model, hep-th/9605187. [19] N. Beisert, S. Frolov, M. Staudacher, A.A. Tseytlin, Precision spectroscopy of AdS/CFT, JHEP 0310 (2003) 037, hep-th/0308117. [20] D. Berenstein, J.M. Maldacena, H. Nastase, Strings in ﬂat space and pp waves from N = 4 super-Yang– Mills, JHEP 0204 (2002) 013, hep-th/0202021. [21] R. Penrose, A remarkable property of plane waves in general relativity, Rev. Mod. Phys. 37 (1965) 215–220. [22] R. Güven, Plane wave limits and T-duality, Phys. Lett. B 482 (2000) 255–263, hep-th/0005061. [23] M. Blau, J. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, A new maximally supersymmetric background of IIB superstring theory, JHEP 0201 (2002) 047, hep-th/0110242. [24] M. Blau, J. Figueroa-O’Farrill, G. Papadopoulos, Penrose limits, supergravity and brane dynamics, Class. Quantum Grav. 19 (2002) 4753, hep-th/0202111. 5 [25] J. Callan, G. Curtis, et al., Quantizing string theory in AdS5 ×S : Beyond the pp-wave, Nucl. Phys. B 673 (2003) 3–40, hep-th/0307032. [26] J. Callan, G. Curtis, T. McLoughlin, I. Swanson, Holography beyond the Penrose limit, Nucl. Phys. B 694 (2004) 115–169, hep-th/0404007. [27] J. Callan, G. Curtis, T. McLoughlin, I. Swanson, Higher impurity AdS/CFT correspondence in the near- BMN limit, Nucl. Phys. B 700 (2004) 271–312, hep-th/0405153. [28] T. McLoughlin, I. Swanson, N-impurity superstring spectra near the pp-wave limit, Nucl. Phys. B 702 (2004) 86–108, hep-th/0407240. 5 [29] I.Y. Park, A. Tirziu, A.A. Tseytlin, Spinning strings in AdS5 ×S : One-loop correction to energy in SL(2) sector, JHEP 0503 (2005) 013, hep-th/0501203. [30] M. Lubcke, K. Zarembo, Finite-size corrections to anomalous dimensions in N = 4 SYM theory, JHEP 0405 (2004) 049, hep-th/0405055. [31] V.A. Kazakov, K. Zarembo, Classical/quantum integrability in non-compact sector of AdS/CFT, JHEP 0410 (2004) 060, hep-th/0410105. 32 C. Sieg, A. Torrielli / Nuclear Physics B 723 (2005) 3–32

[32] N. Beisert, A.A. Tseytlin, K. Zarembo, Matching quantum strings to quantum spins: One-loop vs. ﬁnite-size corrections, Nucl. Phys. B 715 (2005) 190–210, hep-th/0502173. [33] D. Serban, M. Staudacher, Planar N = 4 gauge theory and the Inozemtsev long range spin chain, JHEP 0406 (2004) 001, hep-th/0401057. [34] N. Beisert, V. Dippel, M. Staudacher, A novel long range spin chain and planar N = 4 super-Yang–Mills, JHEP 0407 (2004) 075, hep-th/0405001. [35] T. Fischbacher, T. Klose, J. Plefka, Planar plane-wave matrix theory at the four loop order: Integrability without BMN scaling, JHEP 0502 (2005) 039, hep-th/0412331. [36] G. ’t Hooft, A planar diagram theory for the strong interactions, Nucl. Phys. B 72 (1974) 461. [37] K.G. Chetyrkin, A.L. Kataev, F.V. Tkachov, New approach to evaluation of multiloop Feynman integrals: The Gegenbauer polynomial x space technique, Nucl. Phys. B 174 (1980) 345–377. [38] D.I. Kazakov, A.V. Kotikov, The method of uniqueness: Multiloop calculations in QCD, Theor. Math. Phys. 73 (1988) 1264. [39] J.A. de Azcárraga, A.J. Macfarlane, A.J. Mountain, J.C. Pérez Bueno, Invariant tensors for simple groups, Nucl. Phys. B 510 (1998) 657–687, physics/9706006. Nuclear Physics B 723 (2005) 33–52

Partial breaking of N = 2 supersymmetry and of gauge symmetry in the U(N) gauge model

K. Fujiwara a,H.Itoyamaa, M. Sakaguchi b

a Department of Mathematics and Physics, Graduate School of Science, Osaka City University, 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan b Osaka City University Advanced Mathematical Institute (OCAMI), 3-3-138, Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan Received 22 March 2005; received in revised form 15 June 2005; accepted 20 June 2005 Available online 12 July 2005

Abstract We explore vacua of the U(N) gauge model with N = 2 supersymmetry recently constructed in hep-th/0409060. In addition to the vacuum previously found with unbroken U(N) gauge symmetry in which N = 2 supersymmetry is partially broken to N = 1, we ﬁnd cases in which the gauge n N = symmetry is broken to a product gauge group i=1 U(Ni).The 1 vacua are selected by the requirement of a positive deﬁnite Kähler metric. We obtain the masses of the supermultiplets appearing on the N = 1 vacua.  2005 Elsevier B.V. All rights reserved.

1. Introduction

This is the sequel to our previous papers [1,2] and we intend to investigate further properties of our model arising from an interplay between N = 2 supersymmetry and non-Abelian gauge symmetry. In Ref. [1], we have successfully constructed the N = 2su- persymmetric U(N) gauge model in four spacetime dimensions, generalizing the Abelian self-interacting model given some time ago in Ref. [3]. The gauging of U(N) isometry

E-mail addresses: [email protected] (K. Fujiwara), [email protected] (H. Itoyama), [email protected] (M. Sakaguchi).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.06.023 34 K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 associated with the special Kähler geometry, and the discrete R symmetry are the primary ingredients of our construction. The model describes spontaneous breaking of N = 2su- persymmetry to N = 1. The second supersymmetry realized on the broken phase acts as an approximate fermionic U(1) shift symmetry. This, combined with the notion of prepotential as an input function, tells that the model should be interpreted as a low energy effective action designed to apply microscopic calculations, invoking spectral Riemann surfaces [4,5], matrix models (see [6] for a recent review) and/or string theory [7],tovari- ous physical processes. Although we will not investigate them in this paper, it is interesting to try to find the origin of and the role played by the three parameters e, m and ξ of our model, along the developments beginning with the works of [8–10]. Connection to various compactification schemes of strings, branes and M-theory [11] is anticipated and some work along this direction has already appeared [12]. Partial spontaneous breaking of extended supersymmetry appears not possible from the consideration of the algebra among the supercharges. The basic mechanism enabling the partial breaking is a modification of the local version of the extended supersymmetry algebra by an additional spacetime independent term which forms a matrix with respect to extended indices [13,14]: ¯ j Sm = n j m + m j Qα˙ , αi(x) 2 σ αα˙ δi Tn (x) σ αα˙ Ci . (1.1) Note that this last term is not a vacuum expectation value but simply follows from the algebra of the extended supercurrents and from the fact that the triplet of the auxiliary fields Da is complex undergoing the algebraic constraints of [15]. Our model predicts j j Ci =+4mξ(τ 1)i . (1.2) Separately, we find that the scalar potential takes a vacuum expectation value V = ∓ 2mξ = 2|mξ|. (1.3) (See (4.17).) Once these are established, it is easy to see that partial breaking of extended supersymmetry is a reality: after a ninety degree rotation, the vacuum annihilates half of the supercharges while the remaining half takes non-vanishing and in fact infinite (∼|mξ| d4x) matrix elements. The thrust of the present paper is to explore vacua of the model in which the U(N)gauge symmetry is spontaneously broken to various product gauge groups and to compute the mass spectrum on the N = 1 vacua as a function of input data, the prepotential derivatives, e, m and ξ. In the next section, we review the N = 2 supersymmetric U(N) gauge model. Analysis of the vacua is given in Section 3 and we determine the vacuum expectation value of the auxiliary fields Da. We also collect some properties of the derivatives of the prepotential. In Section 4, we exhibit the Nambu–Goldstone fermion of the model. The N = 1 vacua are selected by the requirement of the positivity of the Kähler metric. In Section 5, we show that the vacua permit various breaking patterns of the U(N) gauge N = group into product gauge groups i=1 U(Ni). Also, the “triplet–doublet splitting” at N 5 is discussed. Finally in Section 6, we compute the masses of the bosons and the fermions of the model, and obtain three types of N = 1 (on-shell) supermultiplets. In Appendix A, we collect some formulas associated with the standard basis of the u(N) Lie algebra. K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 35

It will be appropriate to give a brief sketch of our result here, introducing the notation to label the generators of u(N) Lie algebra by indices. A set of u(N) generators is ﬁrst labelled by indices a,b,... = 0, 1,...,N2 − 1. Here 0 refers to the overall U(1) generator. In a basis in which the decoupling of U(1) is manifest, we label the generators belonging to the maximal Cartan subalgebra by i, j, k, ... while the generators belonging to the roots (the non-Cartan generators) are labelled by r, s, .... In our analysis of vacua, we employ the standard basis of the u(N) Lie algebra. See Appendix A for this basis. The diagonal generators in this basis are labelled by i, j, k, ... and are referred to as those in the eigenvalue basis. The non-Cartan generators associated with unbroken gauge symmetry are labelled by r, s, ... while the remaining non-Cartan generators representing broken gauge symmetry are labelled by µ, ν, .... Our physics output, mass spectrum of our model only distinguishes the broken generators from the unbroken ones. We therefore introduce α, β, γ , ... as a union of i, j, k, ... and r, s, ... in order to label the entire unbroken generators. Our ﬁnal formula will be expressible in terms of α, β, ...and µ, ν, ...only.

2. Review of the U(N) gauge model

The N = 2 U(N) gauge model constructed in [1] is composed of a set of N = 1 chi- a a ral multiplets Φ = Φ ta and a set of N = 1 vector multiplets V = V ta, where N × N = 2 − [ ]= c Hermitian matrices ta, (a 0,...,N 1), generate u(N), ta,tb ifabtc. These super- a a a a a a a a fields, Φ and V , contain component fields (A ,ψ ,F ) and (vm,λ ,D ), respectively. This model is described by an analytic function F(Φ).1 The kinetic term of Φ is given a ∗a = i aF ∗ − ∗aF by the Kähler potential K(Φ ,Φ ) 2 (Φ a Φ a) of the special Kähler geometry 2 2 ¯ a ∗a a ∗a as LK = d θd θK(Φ ,Φ ). The Kähler metric gab∗ ≡ ∂a∂b∗ K(A ,A ) = Im Fab admits isometry U(N).TheU(N) gauging is accomplished [16,17] by adding to LK LΓ =− b ∗c d which is specified by the Killing potential Pa igabfcdA A . The kinetic term of V is L =−i 2 F WaWb + Wa a given as W2 4 d θ ab c.c., where is the gauge field strength of V . L = 2 + = 0 + F This model contains the superpotential term√W d θW c.c. with W eA m 0 as 0 well as the Fayet–Iliopoulos D-term LD = 2ξD associated with the overall U(1) [18]. Owing to the non-trivial prepotential F, overall U(1) sector communicates with the non- Abelian sector in the spontaneous supersymmetry breaking [1]. Gathering these together, the total Lagrangian of the N = 2 U(N) model is given as

L = LK + LΓ + LW2 + LW + LD. (2.1) By eliminating the auxiliary ﬁelds by their equations of motion √ a = ˆ a − 1 ab + 0 D D g Pb 2 2ξδb , √ 2 2 ∗ Dˆ a ≡− gab F ψd λc + F ψ¯ d λ¯ c , (2.2) 4 bcd bcd

1 Fa ≡ ∂aF and Fab ≡ ∂a∂bF. 36 K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52

a ˆ a ab∗ ∗ F = F − g ∂b∗ W , i ∗ ∗ Fˆ a ≡ gab F λ¯ cλ¯ d − F ψcψd , (2.3) 4 bcd bcd the Lagrangian L (2.1) takes the following form: L = L + L + L + L + L kin pot Pauli mass fermi4 , (2.4) with

∗ 1 1 L =− ∗ D aDm b − a bmn − F mnpq a b kin gab mA A gabvmnv Re( ab) vmnvpq 4 8 1 a m ¯ b 1 a m ¯ b + − Fabλ σ Dmλ − Fabψ σ Dmψ + c.c. , (2.5) 2 2 1 ∗ ∗ L =− ab + 2 0 0 − ab ∗ pot g PaPb ξ δaδb g ∂aW∂b W , (2.6) 8 i cd∗ ∗ a b a b Lmass = − g Fabc∂d∗ W ψ ψ + λ λ + c.c. 4 √ 1 ∗ ∗ + √ ∗ c − ∗ c − 0 cdF a b + gac k gbc ka 2ξδc g abd ψ λ c.c. , (2.7) 2 2 b D a ≡ a − 1 a b c where we have defined the covariant derivative by mΨ ∂mΨ 2 fbcvmΨ for a ∈{ a a a} a ≡ a − a − 1 a b c Ψ A ,ψ ,λ , and vmn ∂mvn ∂nvm 2 fbcvmvn. The holomorphic Killing vec- b b bc∗ tors ka = ka ∂b are generated by the Killing potential Pa as ka =−ig ∂c∗ Pa and ∗b = b∗c L L ka ig ∂cPa. Here, we have omitted Pauli and fermi4 as they are irrelevant for our purposes in this paper. In [1], Lagrangians, L and L, are shown to be invariant up to surface terms under the R-action λa ψa R : → ,ξ→−ξ, (2.8) ψa −λa and for L in addition ∗ ∗ ∗ ∗ a + ac ∗ → b + db F g ∂c W F g ∂d W, R : 1 1 (2.9) Dc + gcdP →− Dc + gcdP . 2 d 2 d This property guarantees N = 2 supersymmetry of the model as follows. By construction, the action, S ≡ L or L , is invariant under N = 1 supersymmetry, δ1S = 0. Operating −1 −1 the R-action on this equation, one finds 0 = Rδ1SR = Rδ1R S. This implies that −1 S is invariant under the second supersymmetry δ2 ≡ Rδ1R as well, in addition to the first supersymmetry δ1.Wehavealsogivenin[1] (see Appendix A) a proof of N = 2 supersymmetry of our action, using the canonical transformation acting only on the fields without invoking ξ →−ξ.TheN = 2 supersymmetry transformations are obtained by covariantizing the N = 1 transformations with respect to R. Using the doublet of fermions a ¯ a λ Ia λ Ja λ a ≡ , λIa ≡ IJλ a, λ¯ ≡ , λ¯ a ≡ λ¯ , (2.10) I ψa J ψ¯ a I IJ K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 37 and the doublet of supersymmetry transformation parameters ¯ ≡ η1 I ≡ IJ ¯ J ≡ η1 ¯ ≡ ¯ I ηI , η ηJ , η , ηJ JIη , (2.11) η2 η¯2

IJ 12 21 where is given by = 21 = 1 and = 12 =−1, the N = 2 transformations are written as √ a = Ja δA 2ηJ λ , (2.12) a = ¯ Ja − a ¯ J δv iηJ σmλ iλJ σmη , (2.13) m √ a = mn a + m ¯ D a δλJ σ ηJ vmn 2i σ ηJ mA 1 ∗ + i τ · Da K η − η f a A bAc. (2.14) J K 2 J bc Here, Da represent the three-vectors √ ∗ ∗ ∗ Da = Dˆ a − 2gab ∂ ∗ EA 0 + MF , (2.15) √ b 0 ˆ √2ImF a Dˆ a = − 2ReFˆ a , (2.16) Dˆ a 0 0 E = −e , M = −m , (2.17) ξ 0 and τ are the Pauli matrices. Let us also note that √ 0 a a Im D = δ 0 − 2m 1 . (2.18) 0 This simply follows from (3.4) and also a consequence from the superspace constraints derived in [15]. The construction of the extended supercurrents is somewhat involved as is fully discussed in [1] (see [19,20] for this). Nonetheless it is easy to extract the piece contributing j to Ci in the algebra (1.1). It comes from the structure of ∗ ∗ ∗ τ · D bτ · Da = D b · Da1 + i D b × Da · τ. (2.19)

The second term is non-vanishing only for complex Da and this is the piece responsible j for Ci . After using (2.15) and (2.18), we derive (1.2).

3. Analysis of vacua

Let us examine the scalar potential of the model, V =−Lpot, 38 K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 1 ∗ V = ab + 2 0 0 + ∗ g PaPb ξ δaδb ∂aW∂b W 8 1 ∗ = gab P P + ∂ EA0 + MF · ∂ ∗ EA0 + MF . (3.1) 8 a b a 0 b 0 For our present purpose, it is more useful to convert this expression into

1 1 ∗ V = g P bP c + gbcD˜ · D˜ , (3.2) 8 bc 2 b c where a = ab =− a ∗c d P g Pb ifcdA A , (3.3) and √ √ 0 ∗ ∗ ∗ ˜ ≡ ˜ a =− ∗ E 0 + MF = ∗ Db gbaD 2∂b A 0 2 ∂b W , (3.4) 0 −ξδb as well as (2.18). We derive from (3.2)–(3.4), √ ∂V 1 1 i ∗ 2 = g ∂ P bP c + ∂ g P bP c + F D˜ b · D˜ c − F Mδb · D˜ c ∂Aa 4 bc a 8 a bc 4 abc 2 abc 0 1 1 i = g ∂ P bP c + ∂ g P bP c + F D˜ b · D˜ c. (3.5) 4 bc a 8 a bc 4 abc This is the expression which is valid in any vacuum and generalizes the one found in [1] (Eq. (4.17)) for the unbroken vacuum. In order to examine the general vacua, let indices i(r)label the (non) Cartan generators of u(N) respectively (see Appendix A). We are interested in the vacuum at which Ar =0. The vacuum condition (3.5) reduces [2] to i 0 = F Db · Dc , (3.6) 4 abc a=− a ∗i j = a because P ifij A A 0. Here and henceforth, we drop the tilde on D noting that D˜ a=Da as the vacuum expectation value. It is more convenient to work on another set of bases (the eigenvalue bases) in which the Cartan subalgebra of u(N) is spanned by k k i (ti)j = δi δj (i = 1 ∼ N, j,k = 1 ∼ N). (3.7)

These ti correspond to Hi in Appendix A. Let us introduce a matrix which transforms the standard bases labelled by i, j, k of the Cartan generators into the eigenvalue bases labelled by i, j, k as

j ti = Oi tj , (3.8) j ti = Oi tj . (3.9)

j k k j k k This matrix satisﬁes Oi Oj = δi and Oi Oj = δi . We normalize the standard u(N) = 1 Cartan generators ti as tr(titj ) 2 δij , which implies that the overall u(1) generator is K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 39

√1 t = 1N×N . Let us derive useful relations which will be exploited in what follows. 0 2N √ Summing up Eq. (3.8) with respect to i, we find 2Nt = O it , and thus O i = √ 0 j j i j j 2Nδ i . Taking the trace of (3.8), we obtain 0 2 O 0 = . (3.10) i N N 0 = j = Taking the trace of (3.9), we obtain 2 δi j Oi . It follows from (3.9) with i 0 that j 1 O0 = √ . (3.11) 2N ± =± ± On the other hand, non-Cartan generators tr are Ei j Ej i in Appendix A. Thus Φ can = i + r = i + 1 i j + + i j − be expanded as Φ i Φ ti r Φ tr i Φ ti 2 i,j (i =j)(Φ+ Ei j Φ− Ei j ) i j ji with Φ± =±Φ± .InAppendix A, we explain our notation in some detail. Let us collect some properties of Fab and Fabc for the following F: k C F = tr Φ. (3.12) ! =0 We note that the matrix Φ is complex and normal, and the eigenvalues λi are in general i complex. Letting Φ=λ ti , we find that the non-vanishing Fab are C − F = λi 2, (3.13) i i ( − 2)!  i −1− j −1 2F  C (λ ) (λ ) λi = λj , ∂ 2(−1)! i − j if F± ± ≡ = λ λ (3.14) i j, i j i j i j  − C i 2 i = j ∂Φ± ∂Φ± 2(−2)! λ if λ λ . See Appendix A for details. These imply that gab is diagonal: gi j ∝δi j , grs∝δrs ir and gir=g =0. In addition, we have F+i j,+i j =F−i j,−i j ,i.e.g+i j,+i j = i j i j g−i j,−i j . In particular we note that, for directions Φ± with λ = λ , F±i j,±i j = 1 F = 1 F 2 i i ,i.e. g±i j,±i j 2 gi i .For abc , the non-vanishing components are C − F = λi 3, (3.15) i i i ( − 3)!  i −1− j −1  C δ ∂ + δ ∂ (λ ) (λ ) if λi = λj , 2(−1)! i k ∂λi j k j i − j F ± ± = ∂λ λ λ (3.16) k, i j, i j  − C i 3 i = j 2(−3)! δi k λ if λ λ . Obviously ∂iFabc...=∂iFabc.... Finally, we note here remarkable relations  i −2− j −2 Fi i −Fj j  √ C (λ ) (λ ) = √ if λi = λj , 2 2N(−2)! λi −λj 2 2N(λi −λj ) F0,±i j,±i j = (3.17)  √1 i j Fi i i if λ = λ , 2 2N 40 K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 which play a key role in the analysis of the mass spectrum. Let us return to the vacuum√ condition (3.6). This condition is automatically satisfied for = r =− rs E 0 + MF ∗ = a r because D 2 g ( δs 0s) 0. Noting that the only non-vanishing second and third derivatives for the Cartan directions are the diagonal ones, namely, Fi i and Fi i i , the vacuum condition (3.6) reduces to j j Fj j j D · D = 0 with j not summed, 1 j N. (3.18)

The points speciﬁed by Fj j j =0 correspond to unstable vacua because ∂j ∂j ∗ V=0. At the stable vacua, we obtain Dj · Dj =0, or equivalently

Dj · Dj =0, (3.19) where     √ 0 √ 0 = j =  ∂ W ∗  =  eO 0 + mO iF ∗  Di Oi Dj 2 ∂λ∗i 2 i 0 i i . (3.20) −ξ ∂ (A∗0) −ξO 0 ∂λ∗i i We have determined the vacuum expectation values of the following quantities: O 0 F ∗ =− j ∓ =− ∓ m j j j (e iξ) 2(e iξ), (3.21) O0 ξ gj j = ∓ 2 , (3.22) m ξ 0 Dj = 2√ ±i ≡ d(±), (3.23) N −1 0 m ± Dj = √ −i ≡ d( ). (3.24) N ±1 We use · · · for vacuum expectation values which are determined as the solutions to (3.18). Note that the sign factor ± can be chosen freely in each j.LetM+ be the subset of j = 1,...,N in which the + sign is chosen in (3.24) and let M− be the one in which the − sign is chosen. We will see shortly that this determines the number of spontaneously broken supersymmetries. As one sees from (2.5), Im Fab measures the inverse of the 1 F squared coupling constant 2 , while Re ab is the θ-angle of QCD. gYM

4. Partially broken supersymmetry and NG fermions

Let us examine the supersymmetry transformation of fermions (2.14), which reduces at the vacuum to a = · a J δλI i τ D I ηJ . (4.1) As Dr = 0, the fermion λr on the vacuum is invariant under the supersymmetry I r = δλI 0. (4.2) K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 41

i The Nambu–Goldstone fermion signaling supersymmetry breaking is contained in λI .For i × · i δλI ,the2 2matrixτ D is easily diagonalized as i i λ ± ψ 1 i i δ √ =∓√ D ∓ iD (η1 ∓ η2). (4.3) 2 2 2 3 j If either M− or M+ is empty, D (j = 1,...,N) we have obtained are all identical. j (+) When M− is empty, the matrix τ · D = τ · d is of rank 1, which signals the partial supersymmetry breaking: λi + ψi 2 δ √ = im (η − η ), (4.4) N 1 2 2 λi − ψi δ √ = 0. (4.5) 2 i i λ √+ψ i In the original Cartan basis, this means that δ( ) = 2imδ (η1 − η2) wherewehave √ 2 0 i i √1 i i used the fact Oj = 2Nδ . As we will show in Section 6, the fermion (λ + ψ ) j 0 2 are massless while √1 (λi −ψi) are massive. Thus, N = 2 supersymmetry is spontaneously 2 broken to N = 1 and we obtain the Nambu–Goldstone fermion √1 (λ0 + ψ0) associated 2 with the overall U(1) part. The same reasoning holds when M+ is empty. On the other hand, if neither M+ nor M− is empty, we have two independent rank one matrices + − τ · d( ) and τ · d( ), and thus N = 2 supersymmetry is spontaneously broken to N = 0. Which part of the Cartan subalgebra of u(N) contains two Nambu–Goldstone fermions depends upon the type of grouping of 1 ∼ N into M+ and M−.Leti = (i ,i ) with i ∈ M+ and i ∈ M−, then

λi + ψi 2 λi + ψi δ √ = im (η1 − η2), δ √ = 0, (4.6) 2 N 2 λi − ψi 2 λi − ψi δ √ =−im (η1 + η2), δ √ = 0. (4.7) 2 N 2 As is obvious from the similar analysis given in Section 6, the fermions, √1 (λi + ψi ) 2 and √1 (λi − ψi ), are massless and contain two Nambu–Goldstone fermions of N = 2 2 supersymmetry broken to N = 0. We comment on the vacuum value of the scalar potential V.FortheN = 1 vacua, we have

V = ∓ 2mξ, (4.8) where the ∓ signs correspond respectively to the cases ∀i ∈ M±.FortheN = 0 vacua, on the other hand, we have 1 V = − 2mξ ord(M+) − ord(M−) , (4.9) N 42 K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 where ord(M±) is the number of elements of M±. We now impose the positivity criterion for the Kähler metric to select the physical vacua. For the N = 1 vacua, this criterion requires ξ g = ∓ 2 > 0. (4.10) i i m ξ ≶ Depending upon m 0, we must choose either one of the two possibilities discussed above. For the N = 0 vacua, however, the Kähler metric cannot be positive deﬁnite, caus- ing unconventional signs for the kinetic term. The N = 0 vacua are regarded as unphysical. In the subsequent sections, we will analyze the N = 1 vacua. Here, we summarize some of the properties in the original bases. i i j i j Da = δa Di = δa Oi Dj = δa Oi d(±) j i 0 N 0 = δ δ d± = δ D . (4.11) a i 2 a 0 0 Recalling ∂aW = eδa + mFa0,wesee

0 0 Fa0 = δa F00 , ga0 = δa g00 . (4.12) Hence ∗ ∗ 0 = D · D = 2 (∂ W)2 + ξ 2 = 2 e + m F 2 + 2ξ 2, (4.13) 0 0 0 00 e ξ F = − ± i , (4.14) 00 m m e Re F00 = − , (4.15) m ξ ξ g00 = ∓ = . (4.16) m m After exhausting all possibilities, we conclude that partial spontaneous supersymmetry breaking takes place in the overall U(1) sector. As for V , we obtain

V = ∓ 2mξ = 2|mξ|. (4.17)

5. Gauge symmetry breaking

Following the analysis of the vacua of our model, we turn to spontaneous breaking of gauge symmetry. Let us recall that with the generic prepotential (3.12), the vacuum condition is given by (3.21):

Fi i + 2ζ = 0, (5.1) K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 43 with k=deg F C − F = l λi 2 i i ( − 2)! 1 1 = C + C λi + C λi 2 + C λi 3 +···. (5.2) 2 3 4 2! 5 3! Here, we have introduced a complex parameter e ξ ζ ≡ ± i . (5.3) m m Eq. (5.1) is an algebraic equation for λi with degree deg F − 2 = k − 2 and provides k − 2 complex roots denoted by λ(,±), = 1 ∼ k − 2. Thus each λi is determined to be one of these k − 2 complex roots. As is well known, this deﬁnes a grouping of N eigenvalues into k − 2 sets and hence determines a breaking pattern of U(N) gauge symmetry into a k−2 k−2 = product gauge group i=1 U(Ni) with i=1 Ni N:   λ(1,±)  .   ..     λ(1,±)     λ(2,±)     ..   .  A =  ±  .  λ(2, )     ..   .   (k−2,±)   λ   .  .. λ(k−2,±) (5.4) α ∈{ |[ ] = } In fact, as we will see in the next section, vm are massless if tα ta ta, A 0 µ ∈{ |[ ] = } α while vm are massive if tµ ta ta, A 0 . The massless vm contain gauge ﬁelds of unbroken i U(Ni) and the superpartner of the Nambu–Goldstone fermion lies in the overall U(1) part. We introduce du ≡ dim U(Ni). (5.5) i Let us, as a warm up, work out the case N = 2 and the case N = 3.

(1) N = 2: there are two distinct cases for A (±) (±) λ 0 λ 0 ± ± (i) , (ii) with λ( ) = λ ( ). (5.6) 0 λ(±) 0 λ (±) The diagonal entries of (5.6) are chosen from λ(,±), = 1 ∼ k − 2. In respective cases, N = 2 supersymmetry and U(2) gauge symmetry are broken respectively to (i) N = 1, U(2) unbroken, (ii) N = 1, U(1) × U(1). 44 K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52

(2) N = 3: λ(±) 00 λ(±) 00 ± ± (i) 0 λ(±) 0 , (ii) 0 λ(±) 0 with λ( ) = λ ( ), 00λ(±) 00λ (±) λ(±) 00 ± ± ± ± (iii) 0 λ (±) 0 with λ( ) = λ ( ) = λ ( ) = λ( ), 00λ,(±) and the unbroken symmetries are respectively (i) N = 1, U(3) unbroken, (ii) N = 1, U(2) × U(1), (iii) N = 1, U(1) × U(1) × U(1).

This pattern of symmetry breaking persists at general N.Letm± ≡ ord(M±), namely, the number of elements of M±. Because either m+ = 0orm− = 0, the unbroken symmetries are N = 1, as is established in the last section, and the product gauge groups n − i=1 U(Ni) with n k 2,N. Let us ﬁnally discuss the condition under which zero eigenvalues of A emerge. This condition is simply

C2 ± 2ζ = 0, (5.7) as is read off from Eqs. (5.1) and (5.2). The complex coefficients C are to be determined from underlying microscopic theory and Eq. (5.7) tells that we can always finetune the single complex parameter ζ to obtain the zero eigenvalues. When Fi i is an even function of λi , the condition means that Eq. (5.1) has a double root. As a prototypical example, let Fi i be even and k = 6. The roots are − C4 ± C4 2 − · C6 + 2 ( 2 ) 4 4! (C2 2ζ) λi =± . (5.8) · C6 2 4! When Eq. (5.7) is satisfied, one of the two pairs of complex roots coalesces and indeed develops into zero consisting of a double root. This could be exploited to realize the “triplet–doublet splitting” at N = 5 in the context of SU(5) N = 1 SGUT. In the vacuum in which U(5) is broken to U(3) × U(2) (or SU(5) to SU(3) × SU(2) × U(1),the standard model gauge group), we are able to obtain   λ 0    λ  =   A  λ  . (5.9) 0 00 In this case, the degeneration of eigenvalues is favored.

6. Mass spectrum

We examine the mass spectrum for the N = 1 vacua for which ∀i ∈ M±. K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 45

6.1. Fermion mass spectrum

In this subsection, we compute the fermion masses. We examine the fermion mass term (2.7) for ψa and λa i ∗ ∗ − gcd F ∂ ∗ W ψaψb + λaλb 4 abc d √ 1 ∗ ∗ + √ ∗ c − ∗ c − 0 cdF a b gac k gbc ka 2ξδc g abd ψ λ . (6.1) 2 2 b The ﬁrst term in (6.1) becomes i ∗ i ∗ − gklF ∂ ∗ W ψiψj + λiλj − gij F ∂ ∗ W ψr ψs + λr λs , (6.2) 4 ij k l 4 rsi j ∗ ∗ ∗ ∗ = i ∗ F = because ∂a W δa∗ ∂i W and ij r 0. The second term in (6.1) reduces to 1 0k i j 1 t ∗i s r r s 1 0i r s − ξg Fkij ψ λ + √ gstf A ψ λ − ψ λ − ξg Firs ψ λ , (6.3) 2 2 2 ri 2 ∗ ∗ ∗ ∗ ∗ c a b = ∗ c i a r = t i s r c i = as gac kb ψ λ gac friA ψ λ gstfriA ψ λ .Wehaveusedfri A t c ∗i friδt A in the last equality. We thus conclude that the mass terms of the fermions with i index are decoupled from those of the fermions with r index. i From (6.2) and (6.3), the mass term for the λI in the eigenvalue basis is written as 1 iI J i 2 λ (Mi i)I λJ with ∗ −iξO0 ∂ ∗ W J =−i i iF i i (Mi i)I g i i i ∗ − ∗ + 0 2 ∂i W iξOi 1 i = √ Fi i i τ · D . (6.4) 2 2 It is easy to show that the mass term becomes of the form √i i + i F i + i 2 D2 iD3 i i i λ ψ 8 2 i + √i i − i F i − i 2 D2 iD3 i i i λ ψ . (6.5) 8 2 i

Noting that for the N = 1 vacua, ∀i ∈ M±, m Di ∓ iDi = 2i √ , Di ± iDi = 0, (6.6) 2 3 N 2 3 we ﬁnd that fermions √1 (λi ± ψi) are massless while fermions √1 (λi ∓ ψi) are massive 2 2 i i i i with mass |m g F0i i | after using (3.11).Hereg comes from the normalization of the kinetic term. 46 K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52

r On the other hand, it follows from (6.2) and (6.3) that the mass term of λI is written as 1 rI J s 2 λ (Mrs)I λJ with ∓ + J = mrs mrs mrs (Mrs)I − ± + , (6.7) mrs mrs mrs =−1 F = √1 t ∗i − t ∗i mrs m 0rs ,mrs gtr f λ gts f λ . (6.8) 2 2 2 si ri It is convenient to express the index r as a union of the two indices r and µ such that tr ∈{tr |[tr , A ] = 0} and tµ ∈{tr |[tr , A ] = 0}. Since [tµ, A ] belongs to {tµ} and gst is diagonal, mrs is non-vanishing only for mµν . On the other hand, mrs is non- vanishing only for mrs ,as F0µν = 0. This last equality is proven from (3.17) and (5.1) by

Fi i − Fj j −2ζ − (−2ζ) F0,±i j,±i j = √ = √ = 0. (6.9) 2 2N(λi − λj ) 2 2N(λi − λj ) r µ r Thus, we find that the λI mass term decouples from the λI mass term. The λI mass term can be easily diagonalized as was done for λi , and we find that √1 (λr ± ψr ) are massless I 2 and √1 (λr ∓ ψr ) are massive with mass |m gr r F | . Combining the result on 2 0r r i r = ∪ ∈{ |[ ] = } λI with that on λI , and letting α i r namely tα ta ta, A 0 , we find that √1 α α √1 α α αα (λ ± ψ ) are massless while (λ ∓ ψ ) are massive with mass |m g F0αα | . 2 2 µ i j i = j Finally, we examine the mass of λI . Denoting E± with λ λ by tµ± for short, we =− [ ]∝ find that mµν is non-vanishing only for mµ+µ− mµ−µ+ , since A ,tµ± tµ∓ . (See µ± Appendix A.) In these indices, the λI mass term is written as 1 µ+I 2mµ+µ− 0 J µ− λ λJ . (6.10) 2 02mµ+µ− I The summation is implied only for either one of the two indices µ+ or µ−, and it is over µ | νν |=| µ−µ− |= half of the broken generators. This implies that λI has mass g mµν g mµ+µ− √1 ν ∗i µ− ∗i µ+ ∗i |f λ | where we have used the facts gµ+µ+ = gµ−µ− and f λ =−f λ . 2 µi µ+i µ−i

6.2. Boson mass spectrum

In order to obtain the boson masses of our model, we evaluate the second variation of the scalar potential on the vacuum δδV . Recall that

V = V1 + V2, (6.11) V = 1 a b a =− a ∗c d 1 gabP P ,P ifcdA A , (6.12) 8 ∗ V = gab ξ 2δ 0δ 0 + ∂ W∂ W 2 a b a b 2 2 00 2 b0 2 bc = ξ + e g + m g00 + 2me(Re F0b)g + m (Re F0b)g Re F0c. (6.13) K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 47

a a a j Let us ﬁrst compute δδV1 .From P = 0 and A = δ j A , we obtain a a µ˜ δP = δ ˜ δP , (6.14) µ ˜ ˜ ∗ δP µ = δP µ µ −−−−−−−− → −−−→ µ˜ ∗j µ˜ j δA ∗ µ˜ t µ = (−i)f λ ,(+i)f λ ≡ f⊥λ · δA . (6.15) jµ jµ δA∗µ µ

µ˜ Here fjµ is the structure constant of the u(N) Lie algebra which is read off from Appen- dix A. For a given µ specified by a pair of indices (k,l),1 k l N, µ˜ is uniquely determined and vice versa, and the summation over j reduces to that on j = k and j = l. We obtain 1 µ˜ µ˜ δδV = δP g ˜ ˜ δP . (6.16) 1 4 µµ µ˜ 2 The summation over µ˜ is for N − du directions of the broken generators, where du is defined in (5.5). 2 In order to separate the N − du Nambu–Goldstone zero modes (one mode for each µ), we introduce the following projector of a 2 × 2 matrix as a diad for each µ: −−−−−−−→−−−−−−−− → µ˜ 1 µ˜ ∗ µ˜ t P µ ≡ −−−−−−−→ (f⊥λ) µ f⊥λ µ , (6.17) µ˜ 2 (f⊥λ) µ where −−−−−−−→ −−−−−−−− → −−−−−−−→ µ˜ 2 ∗ µ˜ t µ˜ (f⊥λ) µ ≡ f⊥λ µ · (f⊥λ) µ = µ˜ j 2 2 fjµλ . (6.18) We express δP µ˜ as −−−−−−−− → −−−→ µ˜ ∗ µ˜ t µ˜ µ δP = f⊥λ µ · P µδA . (6.19) −−−→ µ˜ µ The mode orthogonal to (P µδA ) is the one belonging to the zero eigenvalue for each µ in Eq. (6.16) and is absorbed into the longitudinal components of the corresponding gauge fields. It is given by −−−→ µ˜ µ 12 − P µ δA , (6.20) with ˜ µ j ˜ 1 fjµλ ˜ ∗ ˜ 1 − Pµ = f µ λ j ,fµ λj . (6.21) 2 µ −−−−−−→ µ˜ ∗j jµ jµ µ˜ 2 fjµλ (f⊥λ)µ Substituting (6.19) into (6.16), we obtain −−−→ −−−→ 1 µ˜ j 2 µ˜ µ ∗t µ˜ µ δδV = g ˜ ˜ f λ P δA · P δA . (6.22) 1 4 µµ jµ µ µ µ˜ 48 K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52

Let us turn our attention to δδV2 . We ﬁnd 2 2 −1 −1 00 2 δV2 =− ξ + e g δgg + m (δg)00 b0 −1 −1 b0 + 2me(δ Re F0b)g − 2me(Re F0b) g δgg 2 bc 2 −1 −1 bc + 2m (δ Re F0b)g Re F0c − m (Re F0b) g δgg (Re F0c). (6.23)

It is easy to check δV2 = 0, as it should be, from the properties (4.14)–(4.16) listed in the end of Section 4, It is straightforward to carry out one more variation, which we will not spell out here. Again from (4.14)–(4.16), we obtain ∗ δδV = 2m2 δF gcd δF 2 0c 0d ∗ ∗ = 2m2δA b F gcd F δAa 0bc 0ad = 2 ∗α F ∗ αα F α 2m δA 0αα g 0αα δA . (6.24) In deriving the last line, we have used Eq. (6.9) derived in the preceding subsection. Combining the two calculations (6.22) and (6.24), we can read off the mass formula of the scalar bosons from δδV = δδV1 + δδV2 . The scalar masses arising from αα the directions of the unbroken generators are given by |m g F0αα | . Here, we have interpreted gαα as a factor due to the normalization of the kinetic term. The scalar masses ˜ arising from the directions of the broken generators are given by √1 |f µ λj |, which is 2 jµ normalized as we have just discussed. Finally we read off the mass of the massive gauge bosons from −Lkin , 1 a b c 1 a mb ∗c −L = g f v A f v A kin aa 2 bc m 2 b c 1 ˜ = f µ λj 2v µvmµ. (6.25) 4 jµ m µ ˜ This implies that the mass is given by √1 |f µ λj |. 2 jµ We summarize the spectrum of the bosons and the fermions in Table 1. We ﬁnd the following supermultiplets. First, we consider the massless particles associated with √1 (λα ± ψα) and vα . These are labelled as A in the table. These form massless 2 m N = 1 vector multiplets of spin (1/2, 1), in which the Nambu–Goldstone vector multiplet K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 49

Fig. 1. is contained in the overall U(1) part. Secondly, massive particles labelled as B, which are √1 α α α αα associated with (λ ∓ψ ) and A , have masses given by |m g F0αα | . These form 2 massive N = 1 chiral multiplets of spin (0, 1/2). Finally, we consider massive particles la- µ µ belled as C. The zero modes of A are absorbed into vm as the longitudinal modes to form massive vector ﬁelds. These form two massive multiplets of spin (0, 1/2, 1) with the massive modes of Aµ and λµ. The masses of these supermultiplets are given by √1 |f ν λi|. I 2 µi In Fig. 1, the masses of these three types of N = 1 supermultiplets are schematically drawn.

Acknowledgements

The authors thank Koichi Murakami and Yukinori Yasui for useful discussions. This work is supported in part by the Grant-in-Aid for Scientiﬁc Research (16540262) from the Ministry of Education, Science and Culture, Japan. Support from the 21 century COE program “Constitution of wide-angle mathematical basis focused on knots” is gratefully appreciated. The preliminary version of this work was presented at the international workshop “Frontier of Quantum Physics” in the Yukawa Institute for Theoretical Physics, Kyoto University (17–19 February 2005). We wish to acknowledge the participants for stimulat- ing discussions.

Appendix A

Let Ei j , i,j = 1,...,N, be the fundamental matrix, which has 1 at the (i,j)- component and 0 otherwise. Cartan generators of u(N) are Hi ≡ Ei i , which are denoted as ti in Section 3. Non-Cartan generators can be written as

+ + 1 − − i E = E ≡ (E + E ), E =−E ≡− (E − E ), i = j, (A.1) i j j i 2 i j j i i j j i 2 i j j i ± 2 = 1 which are normalized as tr(Ei j ) 2 . Commutation relations are ± =± ∓ + ∓ Hi,Ej k iδi j Ei k iδi kEi j , (A.2) ± ± =± − + − + − ± − Ei j ,Ek l 2iδj kEi l 2iδi kEj l 2iδj lEi k 2iδi lEj k, (A.3) 50 K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 + − =− + + − + + Ei j ,Ek l 2iδj k Ei l δi lHi 2iδi k Ej l δj lHj + + + + + + 2iδj l Ei k δi kHi 2iδi l Ej k δj kHj . (A.4)

i By introducing the vacuum expectation value Φ=λ ti , it follows that ± = ± i =∓ j − k ∓ Ej k, A Ej k,λ Hi i λ λ Ej k. (A.5)

± j = k j = k In the text, Ej k are denoted as tr when λ λ , while as tµ when λ λ . Explicitly, a Φ = Φ ta is expanded as follows:

i r Φ = Φ ti + Φ tr , i i r r µ Φ ti = Φ ti,Φtr = Φ t + Φ tµ, r 1 i j + 1 i j − r = + Φ tr Φ+ Ei j Φ− Ei j , 2 2 i j λ =λ µ = 1 i j + + 1 i j − Φ tµ Φ+ Ei j Φ− Ei j . (A.6) 2 2 λi =λj F F = C F Let be ! tr Φ , then ab are evaluated as

−2 C − − F = tr H AlH A 2 l i j ( − 1)! i j l=0 − 2 C l j −2−l = tr(H H H H ) λi λ ( − 1)! i i j j l=0 C − = δ λi 2, (A.7) ( − 2)! i j −2 2F ∂ C ± ± i l j −2−l F± ± = = tr E H E H λ λ i j, i j i j i j ( − 1)! i j i i j j ∂Φ± ∂Φ± =  l 0  i −1− j −1 C (λ ) (λ ) (λi = λj :broken) = 2(−1)! λi −λj  (A.8) C i −2 i = j 2(−2)! (λ ) λ λ : unbroken) and the others vanish. We have used (A.5) and

± ± 1 E E = (δ δ ± δ δ )(H + H ) + non-Cartan generators, (A.9) i j k l 4 i k j l i l j k i j ± ∓ i E E =± (δ δ ∓ δ δ )(H − H ) + non-Cartan generators (A.10) i j k l 4 i k j l i l j k i j in the calculation. K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52 51

References

[1] K. Fujiwara, H. Itoyama, M. Sakaguchi, Supersymmetric U(N) gauge model and partial breaking of N = 2 supersymmetry, Prog. Theor. Phys. 133 (2005) 429, hep-th/0409060. [2] K. Fujiwara, H. Itoyama, M. Sakaguchi, U(N) gauge model and partial breaking of N = 2 supersymmetry, hep-th/0410132. [3] I. Antoniadis, H. Partouche, T.R. Taylor, Spontaneous breaking of N = 2 global supersymmetry, Phys. Lett. B 372 (1996) 83, hep-th/9512006. [4] N. Seiberg, E. Witten, Electric–magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang–Mills theory, Nucl. Phys. B 426 (1994) 19, hep-th/9407087; N. Seiberg, E. Witten, Nucl. Phys. B 430 (1994) 485, Erratum. [5] A. Gorsky, I. Krichever, A. Marshakov, A. Mironov, A. Morozov, Integrability and Seiberg–Witten exact solution, Phys. Lett. B 355 (1995) 466, hep-th/9505035; E.J. Martinec, N.P. Warner, Integrable systems and supersymmetric gauge theory, Nucl. Phys. B 459 (1996) 97, hep-th/9509161; T. Nakatsu, K. Takasaki, Whitham–Toda hierarchy and N = 2 supersymmetric Yang–Mills theory, Mod. Phys. Lett. A 11 (1996) 157, hep-th/9509162; R. Donagi, E. Witten, Supersymmetric Yang–Mills theory and integrable systems, Nucl. Phys. B 460 (1996) 299, hep-th/9510101; T. Eguchi, S.K. Yang, Prepotentials of N = 2 supersymmetric gauge theories and soliton equations, Mod. Phys. Lett. A 11 (1996) 131, hep-th/9510183; H. Itoyama, A. Morozov, Integrability and Seiberg–Witten theory: curves and periods, Nucl. Phys. B 477 (1996) 855, hep-th/9511126; H. Itoyama, A. Morozov, Prepotential and the Seiberg–Witten theory, Nucl. Phys. B 491 (1997) 529, hep- th/9512161; H. Itoyama, A. Morozov, Integrability and Seiberg–Witten theory, hep-th/9601168; M. Matone, Instantons and recursion relations in N = 2 SUSY gauge theory, Phys. Lett. B 357 (1995) 342, hep-th/9506102; G. Bonelli, M. Matone, M. Tonin, Solving N = 2 SYM by reflection symmetry of quantum vacua, Phys. Rev. D 55 (1997) 6466, hep-th/9610026. [6] A. Morozov, Challenges of matrix models, hep-th/0502010. [7] T.R. Taylor, C. Vafa, RR flux on Calabi–Yau and partial supersymmetry breaking, Phys. Lett. B 474 (2000) 130, hep-th/9912152; P. Mayr, On supersymmetry breaking in string theory and its realization in brane worlds, Nucl. Phys. B 593 (2001) 99, hep-th/0003198; C. Vafa, Superstrings and topological strings at large N, J. Math. Phys. 42 (2001) 2798, hep-th/0008142; F. Cachazo, K.A. Intriligator, C. Vafa, A large N duality via a geometric transition, Nucl. Phys. B 603 (2001) 3, hep-th/0103067; F. Cachazo, S. Katz, C. Vafa, Geometric transitions and N = 1 quiver theories, hep-th/0108120; F. Cachazo, B. Fiol, K.A. Intriligator, S. Katz, C. Vafa, A geometric unification of dualities, Nucl. Phys. B 628 (2002) 3, hep-th/0110028; J. Louis, A. Micu, Type II theories compactified on Calabi–Yau threefolds in the presence of background fluxes, Nucl. Phys. B 635 (2002) 395, hep-th/0202168; F. Cachazo, C. Vafa, N = 1andN = 2 geometry from fluxes, hep-th/0206017. [8] R. Dijkgraaf, C. Vafa, Matrix models, topological strings, and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3, hep-th/0206255; R. Dijkgraaf, C. Vafa, On geometry and matrix models, Nucl. Phys. B 644 (2002) 21, hep-th/0207106; R. Dijkgraaf, C. Vafa, A perturbative window into non-perturbative physics, hep-th/0208048. [9] L. Chekhov, A. Mironov, Matrix models vs. Seiberg–Witten/Whitham theories, Phys. Lett. B 552 (2003) 293, hep-th/0209085; H. Itoyama, A. Morozov, The Dijkgraaf–Vafa prepotential in the context of general Seiberg–Witten theory, Nucl. Phys. B 657 (2003) 53, hep-th/0211245; H. Itoyama, A. Morozov, Experiments with the WDVV equations for the gluino-condensate prepotential: The cubic (two-cut) case, Phys. Lett. B 555 (2003) 287, hep-th/0211259; 52 K. Fujiwara et al. / Nuclear Physics B 723 (2005) 33–52

H. Itoyama, A. Morozov, Calculating gluino condensate prepotential, Prog. Theor. Phys. 109 (2003) 433, hep-th/0212032; M. Matone, Seiberg–Witten duality in Dijkgraaf–Vafa theory, Nucl. Phys. B 656 (2003) 78, hep-th/0212253; L. Chekhov, A. Marshakov, A. Mironov, D. Vasiliev, DV and WDVV, Phys. Lett. B 562 (2003) 323, hep- th/0301071; A. Dymarsky, V. Pestun, On the property of Cachazo–Intriligator–Vafa prepotential at the extremum of the superpotential, Phys. Rev. D 67 (2003) 125001, hep-th/0301135; H. Itoyama, A. Morozov, Gluino-condensate (CIV-DV) prepotential from its Whitham-time derivatives, Int. J. Mod. Phys. A 18 (2003) 5889, hep-th/0301136; H. Itoyama, H. Kanno, Supereigenvalue model and Dijkgraaf–Vafa proposal, Phys. Lett. B 573 (2003) 227, hep-th/0304184; S. Aoyama, T. Masuda, The Whitham deformation of the Dijkgraaf–Vafa theory, JHEP 0403 (2004) 072, hep-th/0309232; H. Itoyama, H. Kanno, Whitham prepotential and superpotential, Nucl. Phys. B 686 (2004) 155, hep- th/0312306. [10] F. Cachazo, M.R. Douglas, N. Seiberg, E. Witten, Chiral rings and anomalies in supersymmetric gauge theory, JHEP 0212 (2002) 071, hep-th/0211170. [11] E. Witten, String theory dynamics in various dimensions, Nucl. Phys. B 443 (1995) 85, hep-th/9503124. [12] P. Kaste, H. Partouche, On the equivalence of N = 1 brane worlds and geometric singularities with flux, JHEP 0411 (2004) 033, hep-th/0409303. [13] J.T. Lopuszanski, The spontaneously broken supersymmetry in quantum field theory, Rep. Math. Phys. 13 (1978) 37. [14] S. Ferrara, L. Girardello, M. Porrati, Spontaneous breaking of N = 2toN = 1 in rigid and local supersymmetric theories, Phys. Lett. B 376 (1996) 275, hep-th/9512180; H. Partouche, B. Pioline, Partial spontaneous breaking of global supersymmetry, Nucl. Phys. B (Proc. Suppl.) 56 (1997) 322, hep-th/9702115. [15] R. Grimm, M. Sohnius, J. Wess, Extended supersymmetry and gauge theories, Nucl. Phys. B 133 (1978) 275; M.F. Sohnius, Bianchi identities for supersymmetric gauge theories, Nucl. Phys. B 136 (1978) 461; M. de Roo, J.W. van Holten, B. de Wit, A. Van Proeyen, Chiral superfields in N = 2 supergravity, Nucl. Phys. B 173 (1980) 175. [16] J. Bagger, E. Witten, The gauge invariant supersymmetric nonlinear sigma model, Phys. Lett. B 118 (1982) 103. [17] C.M. Hull, A. Karlhede, U. Lindstrom, M. Rocek, Nonlinear sigma models and their gauging in and out of superspace, Nucl. Phys. B 266 (1986) 1. [18] P. Fayet, J. Iliopoulos, Spontaneously broken supergauge symmetries and goldstone spinors, Phys. Lett. B 51 (1974) 461; P. Fayet, Fermi–Bose hypersymmetry, Nucl. Phys. B 113 (1976) 135. [19] S. Ferrara, B. Zumino, Transformation properties of the supercurrent, Nucl. Phys. B 87 (1975) 207. [20] H. Itoyama, M. Koike, H. Takashino, N = 2 supermultiplet of currents and anomalous transformations in supersymmetric gauge theory, Mod. Phys. Lett. A 13 (1998) 1063, hep-th/9610228. Nuclear Physics B 723 (2005) 53–76

Uniﬁcation without supersymmetry: Neutrino mass, proton decay and light leptoquarks

Ilja Dorsner a, Pavel Fileviez Pérez b

a The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 31014 Trieste, Italy b Pontiﬁcia Universidad Católica de Chile, Facultad de Física, Casilla 306, Santiago 22, Chile Received 11 May 2005; accepted 10 June 2005 Available online 5 July 2005

Abstract We investigate predictions of a minimal realistic non-supersymmetric SU(5) grand uniﬁed theory. To accomplish uniﬁcation and generate neutrino mass we introduce one extra Higgs representation— a 15 of SU(5)—to the particle content of the minimal Georgi–Glashow scenario. Generic prediction of this setup is a set of rather light scalar leptoquarks. In the case of the most natural implementation of the type II see-saw mechanism their mass is in the phenomenologically interesting region (O(102–103) GeV). As such, our scenario has a potential to be tested at the next generation of collider experiments, particularly at the Large Hadron Collider (LHC) at CERN. The presence of the 15 generates additional contributions to proton decay which, for light scalar leptoquarks, can be more important than the usual gauge d = 6 ones. We exhaustively study both and show that the scenario is not excluded by current experimental bounds on nucleon lifetimes.  2005 Elsevier B.V. All rights reserved.

1. Introduction

The most predictive grand uniﬁed theory (GUT) based on an SU(5) gauge symmetry is a minimal non-supersymmetric model of Georgi and Glashow [1] (GG). However, the failure to accommodate experimentally observed fermion masses and mixing and to unify

E-mail addresses: [email protected] (I. Dorsner), fileviez@higgs.fis.puc.cl, fi[email protected] (P. Fileviez Pérez).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.06.016 54 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 electroweak and strong forces decisively rule it out. Nevertheless, the main features of the underlying theory, e.g., partial matter unification and one-step symmetry breaking, are so appealing that there has been a number of proposals to enlarge its structure by adding more representations to have the theory in agreement with experimental data. Since the number of possible extensions is large it is important to answer the following question. What is the minimal number of extra particles that renders an SU(5) gauge theory realistic? Such an extension of the GG model with the smallest possible number of extra particles that furnish full SU(5) representation(s) has all prerequisites to be the most predictive one. Thus, if there is a definite answer to the first question it is important to ask the second one: what are the possible experimental signatures and associated uncertainties of such a minimal setup? If uncertainties of the minimal extension are significant the same is even more true of a more complicated structure unless additional assumptions are imposed. We address both questions in great detail and present truly minimal, i.e., minimal in terms of number of fields, realistic non-supersymmetric SU(5) scenario. As we demonstrate later, unification, in the minimal scenario, points towards existence of light scalar leptoquarks which could generate very rich phenomenological signatures. These are looked for in the direct search experiments as well as in the experiments looking for rare processes. Their presence generates novel proton decay contributions which can be very important. Moreover, since in our scheme their coupling to matter is through the Majorana neutrino Yukawas, their observation might even allow measurements of and/or provide constraints on neutrino Yukawa coupling entries. This makes our scenario extremely attractive. At the same time, rather low scale of vector leptoquarks that is inherent in non-supersymmetric theories exposes our scenario to the tests via nucleon lifetime measurements. We investigate all relevant experimental signatures of the scenario, including its status with respect to the present bounds on nucleon lifetime. In the next section we define our framework. In Section 3 we discuss how it is possible to get gauge couplings unification in agreement with low energy data. Then, in Section 4 we discuss possible experimental signatures of the minimal scenario. We conclude in Section 5. Appendix A contains relevant details and notation of the minimal non-supersymmetric realistic SU(5) we refer to throughout the manuscript. The origin of theoretical bounds on the familiar gauge d = 6 proton decay operators is critically analyzed in Appendix B. Appendix C contains details on the two-loop running of the gauge couplings that is presented for completeness of our work.

2. A minimal realistic non-supersymmetric SU(5) scenario

In order to motivate the minimal SU(5) grand unifying theory (GUT), where we de- ﬁne such a theory to be the one with the smallest possible particle content that renders it realistic, we ﬁrst revisit the GG model [1] and discuss its shortcomings. Only then do we present the minimal realistic scenario and investigate its experimental signatures and related uncertainties. The GG model fails from the phenomenological point of view for a number of reasons:

(1) It does not incorporate massive neutrinos; I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 55

(2) It yields charged fermion mass ratios in gross violation of experimentally observed values; (3) It cannot account for the gauge coupling unification. (For one of the first rulings on (non-)unification see for example Ref. [2].)

The first flaw is easy to fix; one either introduces three right-handed neutrinos—singlets of the Standard Model (SM)—to use a type I see-saw mechanism [3] to generate their mass or adds a Higgs field—a 15 of SU(5)—to generate neutrino mass through the so- called type II see-saw [4,5]. One might also use the combination of the two. The fourth option to use the Planck mass suppressed higher-dimensional operators [6,7] does not look promising since it generates too small scale for neutrino mass to explain the atmospheric and solar neutrino data. Nevertheless, it might still play an important role [8]. We focus our attention on the second option, i.e., addition of 15, in view of the fact that the right-handed neutrinos, being singlets of the SM, do not contribute to the running. Hence, their mass scale cannot be sufficiently well determined or constrained unless additional assumptions are introduced. The second flaw can be fixed by either introducing the higher-dimensional operators in the Yukawa sector [9] or resorting to a more complicated Higgs sector à la Georgi and Jarl- skog [10]. The former approach introduces a lot more parameters into the model (see for example [11]) but unlike in the neutrino case these operators might have just a right strength to modify “bad” mass predictions for the charged fermions. In order to keep as minimal as possible the number of particles we opt for the scenario with the non-renormalizable terms. The third flaw requires presence of additional non-trivial split representations besides those of the GG SU(5) model. It can thus be fixed in conjunction with the first and second one. For example, introduction of an extra 45 of Higgs to fix mass ratios of charged fermions à la Georgi and Jarlskog [10] allows one to achieve unification and, at the same time, raise the scale relevant for proton decay. (For the studies on the influence of an extra 45 on the running and other predictions see for example [12,13].) We will see that the addition of one extra 15 of Higgs plays a crucial role in achieving the unification in our case. So, what we have in mind as the minimal realistic SU(5) model is the GG model supplemented by the 15 of Higgs to generate neutrino mass and which incorporates non- renormalizable effects to fix the Yukawa sector of charged fermions. We analyze exclusively non-supersymmetric scenario for the following three reasons: first, this guarantees the minimality of the number of fields; second, there are no problems with the d = 4 and d = 5 proton decay operators; third, since the grand unifying scale is lower than in supersymmetric scenario the setup could possibly be verified or excluded in the next generation of proton decay experiments.

3. Uniﬁcation of gauge couplings

The main prediction, besides the proton decay, of any GUT is the uniﬁcation of the strong and electroweak forces. We thus show that it is possible to achieve gauge coupling uniﬁcation in a consistent way in our scenario. 56 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76

At the one-loop level the running of gauge couplings is given by −1 = −1 + 1 MGUT αi M αGUT bi ln , (1) Z 2π MZ where i = 1, 2, 3forU(1), SU(2), and SU(3), respectively. bi are the appropriate one-loop = 2 coefficients [14] and αGUT gGUT/(4π) represents the gauge coupling at the unifying scale MGUT. The SM coefficients for the case of n light Higgs doublet fields are 40 n 20 n b = + ,b=− + ,b=−7. (2) 1 10 10 2 6 6 3 Even though the SM coefficients do not generate unification in both the n = 1 and n = 2 case for any value of αGUT and MGUT that is not an issue since the SM does not predict the gauge coupling unification in the first place. On the other hand, a GUT, which does predict one, automatically introduces a number of additional particles with respect to the SM case that, if light enough, can change the outcome of the SM running. This change is easily incorporated if one replaces bi in Eq. (1) with the effective one-loop coefficients Bi defined by ln M /M B = b + b r ,r= GUT I , (3) i i iI I I ln M /M I GUT Z where biI are the one-loop coefficients of any additional particle I of mass MI (MZ MI MGUT). Basically, given a particle content of the GUT and Eqs. (1) and (3) we can investigate if the unification is possible. Following Giveon et al. [13],Eqs.(1) can be further rewritten in a more suitable form in terms of differences in the effective coefficients Bij (= Bi − Bj ) and low energy observables. They find two relations that hold at MZ: B 5 sin2 θ − α /α 23 = w em s , 2 (4a) B12 8 3/8 − sin θw M 16π 3/8 − sin2 θ ln GUT = w . (4b) MZ 5αem B12 2 Adopting the following experimental values at MZ in the MS scheme [15]:sin θw = ± −1 = ± = ± 0.23120 0.00015, αem 127.906 0.019 and αs 0.1187 0.002, we obtain B 23 = 0.719 ± 0.005, (5a) B12 M 184.9 ± 0.2 ln GUT = . (5b) MZ B12 Last two equations allow us to constrain the mass spectrum of additional particles that leads to an exact unification at MGUT. (In what follows we consistently use central values presented in Eqs. (5) unless specified otherwise. The inclusion of the two-loop effects and threshold corrections is addressed in detail in Appendix C.) The fact that the SM with one (two) Higgs doublet(s) cannot yield unification is now more transparent in light of Eq. (5a). Namely, the resulting SM ratio is simply too small I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 57

Table 1 Contributions to the Bij coefﬁcients. The mass of the SM Higgs doublet is taken to be at MZ

Higgsless SM ΨD ΨT VΣ8 Σ3 Φa Φb Φc 11 1 − 1 − 7 − 1 1 2 1 − 5 B23 3 6 6 rΨT 2 rV 2 rΣ8 3 rΣ3 3 rΦa 6 rΦb 6 rΦc 22 − 1 1 − − 1 − 1 − 7 8 B12 3 15 15 rΨT 7rV 0 3 rΣ3 15 rΦa 15 rΦb 15 rΦc

(B23/B12 = 0.53 for n = 1) to satisfy equality in Eq. (5a). What is needed is one or more particles that are relatively light and with suitable bi coefficients that can increase the value of the B23/B12 ratio. The most efficient enhancement is realized by a field that increases B23 and decreases B12 simultaneously. For example, light Higgs doublet is such a field (see the ΨD (⊂ 5H ) coefficients in Table 1) and it takes at least eight of them, at the one- loop level, to bring B23/B12 in accord with experiments. Other fields that could generate the same type of improvement in our scenario are light Σ3 (⊂ 24H ), Φa (⊂ 15H ) and Φb (⊂ 15H ). Bij coefficients of all the particles in our scenario are presented in Table 1 and the relevant notation is set in Appendix A. The improvement can also be due to the field that lowers B12 only or lowers B12 at sufficiently faster rate than B23. Looking at Table 1 we see that the superheavy gauge fields V comprising X and Y gauge bosons and their conjugate partners can accomplish the latter. (Note that the gauge contribution improves unification at the one-loop level only if n = 0 and the improvement grows with the increase of n [13,16]. This is because the B23/B12 ratio of the higgsless SM coefficients is the same as for the corresponding ratio of V coefficients.) But, their contribution to running has to be subdominant; otherwise one runs into conflict with the experimental data on nucleon lifetimes. All in all, the fields capable of improving unification in our minimal SU(5) grand unified scenario are ΨD, Σ3, Φa, Φb and V . Again, we refer reader to Appendix A for our notation. We treat their masses as free parameters and investigate the possibility for consistent scenario with the exact one-loop unification. Since all other fields in the Higgs sector, i.e., ΨT , Σ8 and Φc, simply worsen unification we simply assume they live at or above the grandunifying scale. In order to present consistent analysis we now discuss the constraints coming from proton decay on Bij coefficients. These enter via Eq. (5b) and assumption that MV = MGUT. As we show these constraints are rather weak if the gauge d = 6 contributions are dominant as is usually assumed in non-supersymmetric GUTs [17]. For example, if we use the latest bounds on nucleon decay lifetimes we obtain, in the context of an SU(5) non-supersymmetric GUT, in the case of maximal (minimal) suppression in the Yukawa 13 15 sector [19] MV > 2.5 × 10 GeV (MV > 1.5 × 10 GeV). (The minimal suppression = T = T case corresponds to the GG scenario with YU YU and YD YE , where YU , YD and YE are the Yukawa matrices of charged fermions. Non-renormalizable contributions violate both of those relations. The same is also true for the running in the Yukawa sector from the GUT scale where those relations hold to the scale relevant for the Yukawa couplings entering nucleon decay. On the other hand, maximal suppression corresponds to a case with particular relation between unitary matrices responsible for bi-unitary transformations in the Yukawa sector [19] that define physical basis for quarks and leptons.) In both cases we −1 = → 0 + use αGUT 35 and the best limit on partial lifetime which is established for p π e 58 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 decay channel (τ>5.0 × 1033 years). This gives conservative bound for the suppressed case since it is always possible to rotate away proton decay contributions for individual channels [19]. The uncertainty in extracting the limits on MV from experimental data is easy to under- 4 stand. Namely, even though the nucleon lifetime is proportional to MV , which would make extraction rather accurate and precise, the lifetime is also proportional to the fourth power of a term which is basically a sum of entries of unitary matrices which are a priori unknown unless the Yukawa sector of the GUT theory is specified and which, in magnitude (see Eqs. (B.2)), can basically vary from Vub to 1 [18,19]. (For full discussion see also Appendix B.) If we now adopt the MGUT ≡ MV assumption and use Eq. (5b) the above limits translate into B12 < 7.0(B12 < 6.1) for the suppressed (unsuppressed) case. So, all SU(5) GUTs with B12 > 7.0 are excluded by the usual gauge d = 6 contributions to proton decay. The theories with 6.1

• The (Σ3,Φa) case with n = 2 exactly mimics the n = 8 case in terms of quantum numbers. (Recall that it takes at least eight light Higgs doublets on top of the higssless SM content to unify the couplings. Associated corrections to the higssless SM coefficients are = 6 + 2 =−6 − 2 B23 6 r 6 and B12 15 r 15 , where r, as defined in Eq. (3), is very close to one and we take two of the doublets to be at MZ.) The unification scale is rather low and I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 59

Table 2 B23 and B12 corrections due to the degenerate pairs of ﬁelds and the associated scales for the n = 2 case

(Σ3,Φa)(Σ3,Φb)(Φa,Φb) 6 + 2 3 + 2 5 + 2 B23 6 r 6 6 r 6 6 r 6 − 6 − 2 − 12 − 2 − 8 − 2 B12 15 r 15 15 r 15 15 r 15 13 13 MGUT 5 × 10 GeV – 9 × 10 GeV Mr 1 TeV – 200 GeV

very close to the experimentally set limit for maximally suppressed case. Lightness of Φa goes against the idea behind the type II see-saw if one assumes that the parameter c3 (see Eqs. (A.5) and (A.11)) is at the GUT scale, but at this point the scenario is not ruled out experimentally. • The (Φa,Φb) case has a slightly higher unification scale than the (Σ3,Φa) case. This time both Φa and Φb have mass in phenomenologically interesting region. Lightness of Φa again requires large suppression in the Yukawa sector for neutrinos to generate correct mass scale via type II see-saw. However, such a suppression would be beneficial in suppressing novel contributions to proton decay due to the mixing between Φb and ΨT . • The (Σ3,Φb) case is the most promising. Even though it fails to unify at the one- loop level its correction to B12 is the largest of all three cases. As such, it represents the best possible candidate to maximize MV . Moreover, the Φa contribution to the running to produce unification for light Σ3 and Φb is small which implies that its mass could be in the range that is optimal for the type II see-saw for the most “natural” value of c3 coefficient.

Again, the (Σ3,Φb) case not only maximizes MV but also places MΦa at the right scale to explain neutrino masses.

The three special cases discussed above all demonstrate that large MV scale prefers Φb light regardless of the relevant scale of other particles since it is Φb coefficients that decrease B12 the most. These conclusions persist in more detailed one- and two-loop studies. Is there a way to tell between the three limiting cases we just discussed? The (Σ3,Φa) case can be tested and excluded by slight improvement in the nucleon lifetime data; other low energy signatures depend on how light Φa is. One could also test and distinguish between the (Σ3,Φb) and (Φa,Φb) cases since both favor light Φb leptoquarks that can be detected by LHC. If and when these are detected the two cases could be distinguished by the scalar leptoquark contributions to the rare processes. In the (Φa,Φb) case the suppression in the neutrino Yukawa sector would selectively erase some of these contribution while in the (Σ3,Φb) case all these contributions would be sizable. Expected improvements in the table-top experiments would then be sufficient to tell the two. Since we have MGUT and masses of V , Σ3, Φa and Φb as free parameters and only two equations—Eqs. (5a) and (5b)—we present four special cases based on certain simplifying assumptions in Figs. 1–3 and discuss each case in turn. All examples we present generate consistent unification in agreement with low energy data. (Note that the change in the −1 parameters also affects the value of αGUT. We do not present that change explicitly, which, for the range of values we use, vary from 36 to 40. In our plots we also allow MV to be at 60 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76

= Fig. 1. Plot of lines of constant value of MV (solid lines) and MΦa (dashed lines) in the MGUT–(MΣ3 MΦb ) plane. The area to the left of a steep solid line denotes the region where MV does not contribute to the running, i.e., MV MGUT. To generate the plot we consider exact one-loop unification and use central values for the gauge couplings as given in the text. This is a scenario with one light Higgs doublet (n = 1). most factor of three or four lower than the GUT scale. Once we switch to two-loop analysis with threshold corrections accounted for we appropriately set MV = MGUT.) In Fig. 1 we present the n = 1 case when the pair (Σ3,Φb) is taken to be degenerate with 2 the mass close to electroweak scale (∼ O(10 ) GeV). The parameter c3 has to be between 103 and 107 GeV to explain the neutrino mass through the type II see-saw if the Yukawa coupling for neutrinos is of order one. On the other hand, the gauge boson mass varies only × 14 slightly for a given range of MΦb and MΣ3 around 2.5 10 GeV. Clearly, unification itself allows MΦb and MΣ3 to be much heavier then 1 TeV on account of decrease of MΦa but in that case MV would be getting lighter. This, on the other hand would require additional conspiracy in the Yukawa sector in order to sufficiently suppress proton decay to avoid the experimental limit. The two light Higgs doublet case is presented in Fig. 2. This case is well motivated on the baryogenesis grounds. Namely, the interaction of the 15 of Higgs explicitly breaks B − L symmetry (see Appendix A). This opens a door for possible explanation of the baryon asymmetry in the Universe within our framework. However, since the successful generation of baryon asymmetry requires at least two higgses in the fundamental representation we study a consistent unification picture for that case. (See references for the baryogenesis mechanisms in the context of SU(5) model with two higgses in the fundamental representation [20–23].) The n = 2 case has higher scale of superheavy gauge I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 61

= Fig. 2. Plot of lines of constant value of MV (solid lines) and MΦa (dashed lines) in the MGUT–(MΣ3 MΦb ) plane. The area to the left of a steep solid line denotes the region where MV does not contribute to the running, i.e., MV MGUT. This is the n = 2 scenario.

= Fig. 3. Plot of lines of constant value of MV (solid lines) and MΦa MΣ3 in GeV units (dashed lines) in the MGUT–(MΦb ) plane. The scenario with one (two) Higgs doublet(s) corresponds to long (short) dashed lines. 62 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 bosons compared to the n = 1 case (this does not hold at the two-loop level though) and the mass of the field Φa is in the region relevant for the type II see-saw. Hence, if the mass of scalar leptoquarks is in phenomenologically interesting region (∼ O(102) GeV) we can explain neutrino masses naturally. Again, as in the n = 1 case, the gauge boson mass varies very slightly, this time around 3 × 1014 GeV. Given the last two examples we can again conclude that the exact unification in this minimal realistic scenario points towards light scalar leptoquarks. In order to understand better these results we show two more examples in Fig. 3.This timewesetΣ3 = Φa for simplicity and present both n = 1 and n = 2 cases. This scenario is disfavored by the fact that MV tends to be “small” but cannot be excluded on experimental grounds. It is evident from the plot that MV does not depend on a number of light Higgs doublets and MGUT. The reason for that is very simple and is valid only at the one-loop level. Namely, the ratio B23/B12 is the same for ΨD coefficients (=−5/2) as for the sum of corresponding Σ3 and Φa coefficients provided these are degenerate. Thus, any change in the number of light doublets in Eq. (5a) is simply compensated by the change in degenerate mass of Σ3 and Φa fields for a fixed value of Φb mass. This trend can be clearly seen in Fig. 3. The mass of Φa is generally rather low to generate neutrino mass of correct magnitude unless Yukawa couplings of neutrinos are extremely small. In all our examples, Σ3 is allowed to be much lighter than Σ8. But, this seems in conflict with the tree-level analysis of the Σ potential that is invariant under Σ →−Σ transforma- = tion which yields a well-known relation MΣ3 4MΣ8 [17]. This apparent mismatch has a simple remedy. In order to generate sufficiently large corrections to charged fermion masses via higher- dimensional operators in the Yukawa sector we need terms linear in Σ/MPl. If that is the case it is no longer possible to require that the Lagrangian is invariant under transformation Σ →−Σ. It is then necessary to include a cubic term into the Σ potential besides the usual quadratic and quartic ones. But, the potential with the cubic term (TrΣ3) violates = the validity of MΣ3 4MΣ8 relation [24–26] and allows a possibility where Σ3 is light while Σ8 is superheavy. We analyze this situation in Appendix A in some detail. Note that we do not require nor insist on the lightness of Σ3 though. In Appendix C we present 7 the two-loop analysis of the scenario where Φa is relatively light (∼ 10 GeV) and Σ3 is at the GUT scale. Our intention is solely to demonstrate that there are more possibilities available unless additional assumptions, such as Σ →−Σ transformation, are imposed on the SU(5) theory. Note, however, that the maximization of MV always requires Φb to be very light (∼ 102 GeV). In Fig. 4 we show an example where it is possible to achieve = = unification at the two-loop level (see details in Appendix C)forn 1, MΦb 250 GeV, = MΦa 1.54 TeV and the field Σ3 is at the GUT scale. What about the possible mass spectrum of Φa, Φb and Φc? The relevant potential is in Appendix A. Clearly, there are more parameters than mass eigenvalues. The tree-level analysis revels that it is possible to obtain any possible arrangement including, for example, Φb Φa <Φc. This sort of split is quite similar to the split behind the well-known doublet–triplet problem. Our framework yields rather low mass for vector leptoquarks that varies within very narrow range around 3 × 1014 GeV for a most plausible scenarios. This makes the framework testable through nucleon decay measurements. (More precisely, large portion of the I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 63

Fig. 4. Unification of the gauge couplings at the two-loop level for central values of low-energy observables [15]. The SM case with n = 1 is presented by dashed lines. Solid lines correspond to the n = 1 scenario with Φb and = = Φa below the GUT scale. Vertical lines mark the relevant scales: MZ, MΦb 250 GeV, MΦa 1.54 TeV and 14 MGUT = 0.96 × 10 GeV. parameter space of the setup has already been excluded by existing measurements on nucleon lifetime.) It is easy to understand this generic and robust prediction. We have seen 14 14 that MGUT can be at most 3.2 × 10 GeV (4.4 × 10 GeV) if we exclude the V contribution from the running for the n = 1(n = 2) case. If we now start lowering MV below MGUT we lower B12 as well. This, on the other hand, starts to increase MGUT but still keeps MV at almost the same value. Basically, the “decoupling” of MV and MGUT takes place, where the mass of X and Y gauge bosons remains in vicinity of the value before the “decoupling” while MGUT rapidly approaches the Planck scale. Another way to say this is that the Bij coefficients of superheavy gauge fields are very large compared to all other relevant coefficients (see Table 1). Thus, any small change in MV corresponds to a large change in other running parameters. Let us finally investigate possible experimental signatures coming from our consistent minimal realistic SU(5) model.

4. Experimental signatures

Our framework has potential to be tested through the detection—direct or indirect—of light leptoquarks and/or observation of proton decay. Let us investigate each of these tests in turn.

4.1. Light leptoquarks

To get consistent uniﬁcation in agreement with low energy data, neutrino mass and proton decay in our minimal framework we generate very light leptoquarks Φb. The lighter the Φb is the heavier the V becomes. Thus, in the most optimistic scenario MΦb is close to the present experimental limit ∼ O(102) GeV. In what follows we specify all relevant properties of Φb and existing constraints on its couplings and mass. 64 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76

¯T ¯ The 5a ChabΦ5b coupling yields the following interactions: CT = CT 1 − 2 d ChΦbl d Ch φb e φb ν , (6) 1 2 − where the leptoquarks φb and φb have electric charges 2/3 and 1/3, respectively, and symmetric matrix h coincides with the Yukawa coupling matrix of Majorana neutrinos (≡ Yν ) if we neglect the Planck suppressed operators. (See the last line in Eq. (A.4).) The above leptoquark interactions in the physical basis read as CT T ∗ ∗ M ∗ diag M † ∗ 1 d C D E K V Yν V K φ e, (7) C PMNS PMNS b CT T ∗ ∗ M ∗ diag 2 d C DC E K VPMNS Yν φb ν, (8) C where DC and E are the matrices which act on d quarks and e, respectively, to bring them into physical basis. (See Appendix B for exact convention.) K is a matrix containing M three CP violating phases, and VPMNS is the leptonic mixing in the Majorana case. (In the = T = GG SU(5) where YE YD one has DC E. However, that is not the case in a realistic model for fermion masses.) There are many studies about the contributions of scalar and vector leptoquarks in different processes [27,28]. For a model independent constraints on leptoquarks from rare processes see for example [28,29]. The most stringent bound on the scalar leptoquark coupling to matter comes from the limits on µ–e conversion on nuclei [30]. 2 The bound we present should be multiplied by (MΦb /100 GeV) . In our case it reads T † † ∗ −6 T × (DC hE)11(E h DC)21 < 10 . The bounds for all other elements of (DC hE)ij † † ∗ T ∗ † ∗ (E h DC)kl and (DC hN)ij (N h DC)kl are weaker. The currents bounds on leptoquarks production are set by Tevatron, LEP and HERA [31]. Tevatron experiments have set limits on scalars leptoquarks with couplings to eq of MLQ > 242 GeV. The LEP and HERA experiments have set limits which are model dependent. The search for these novel particles will be continued soon at the CERN LHC. Preliminary studies by the LHC experiments ATLAS [32] and CMS [33] indicate that clear signals can be established for masses up to about MLQ ≈ 1.3 TeV. For several studies about production of scalar leptoquarks at the LHC, see Ref. [34]. Thus, it could be possible to test our scenario at the next generation of colliders, particularly in the Large Hadron Collider (LHC) at CERN, through the production of light leptoquarks. Therefore even without the proton decay experiments we could have tests of this non-supersymmetric GUT scenario.

4.2. Proton decay

Proton decay is the most generic prediction coming from matter unification; therefore, it is the most promising test for any grand unified theory. (For new experimental bounds see [15,35].) In our minimal and consistent scenario the relevant scale for gauge bosons is around 3 × 1014 GeV regardless of how high the GUT scale goes in order to get consistent unification in agreement with low energy data. Careful study within the two-loop context with the inclusion of threshold effects revels that the highest possible value of MV in the n = 2(n = 1) case is 3.19 × 1014 GeV (3.28 × 1014 GeV) for central values of coupling constants [15] while the 1σ departure allows for the maximum value of 3.35 × 1014 GeV in the n = 2 case, for example. I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 65

Fig. 5. Contributions to the decay of the proton induced by the 15 of Higgs.

There are several contributions to the decay of the proton in our minimal scenario. We have the usual Higgs and gauge d = 6 operators but there are also new contributions due to the mixing between ΨT and Φb, with Φb being extremely light in our case. These contributions are very important. Using the relevant triplet interactions: ab + ab ∗ + C ab C ∗ + C ab C qaCA qbΨT qaCC lbΨT ua CD db ΨT ua CB eb ΨT (9) (for the expressions of A, B, C, and D matrices see for example [11]) and the interaction ¯T ¯ − term 5a ChabΦ5b, it is easy to write down the contributions for the B L non-conserving + + + 0 0 0 0 0 decays p → (K ,π ,ρ )νi , and n → (π ,ρ ,η ,w ,K )νi . We present the relevant diagram in Fig. 5 [36]. Notice that in this scenario we have the usual B − L conserving decays, i.e., the decays into a meson and antileptons, and the B − L non-conserving decays mentioned above. Since Φb has to be light, the B − L violating decays are very important. The rate for the decays into neutrinos is given by

3 2 − 2 2 2 0 2 + + (mp mK ) 2 c3 H Γ p → K ν = Γ p → K νi = A 32πm3 f 2 L M4 M4 i=1 P π ΨT Φb 4m2 D2 ˜ C C C 2 p × βC νi,s,d +˜αC νi,s ,d 9m2 B + 2 ˜ C C C 2 mp(D 3F) + βC νi,d,s +˜αC νi,d ,s 1 + , (10) 3mB C = T + T C C = † † ∗ where C(νi,dα,dβ ) (U (A A )D)1α and C(νi,dα ,dβ ) (DCD UC)α1. (See Appendix B for notation.) As you can appreciate from the above expressions, the predictions coming from these contributions are quite model dependent. Using mp = 938.3MeV,D = 0.81, F = 0.44, ˜ 3 mB = 1150 MeV, fπ = 139 MeV, AL = 1.43, and α˜ = β = α = 0.003 GeV we get 4 4 − − M M 1.95 × 10 64 GeV 6 ΨT Φb c2 3 C C C 2 C C C 2 > 0.19 C νi,s,d + C νi,s ,d + 2.49 C νi,d,s + C νi,d ,s . (11) = = 14 = 3 Let us see an example, using the values MΨT c3 10 GeV and MΦb 10 GeV, the left-hand side of the above equation is equal to 1.95 × 10−24; therefore, the sum of the C −12 6 coefficients has to be basically 10 . In the case that coefficient c3 is smaller (∼ 10 GeV), 66 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 a possibility that is not excluded, the sum of C coefficients would be around 10−4, which is their “natural” value. Moreover, the scenario would then prefer Φa at the same scale 6 (∼ 10 GeV) if Yν is taken to be proportional to Ye. Also we can suppress the relevant =− = contributions in different ways. For example, we could choose A ij A ji and Dij 0 except for i = j = 3, or set to zero these coefficients in specific models for fermion masses. In any case, if the gauge d = 6 contributions are the dominant ones for proton decay, we can get the following bounds for the proton lifetime (see Appendix B) allowing the full freedom in the Yukawa sector: 31 38 1 × 10 <τp < 2 × 10 years. (12) −1 = = × 14 Here we use αGUT 39 and MGUT 3.2 10 GeV. (See Fig. 6 for example.) Having in mind the experimental limit of 5.0 × 1033 years [15] we see that a significant portion of available parameter space has already been excluded. We hope that in the next generation of proton decay experiments this scenario would be constrained even further. How does this scenario compare to other possible extensions of the GG model? We mention only few listing them by increasing order in particle number.

• The most obvious extension is to add one more fundamental representation in the Higgs sector to the n = 2 scenario we analyze the most at the one-loop level. This addition would not raise but actually lower MV since the scalar leptoquarks which influence B12 the most would get slightly heavier than in the n = 2 case. In certain way, this actually makes the n = 1 scenario with the 15 of Higgs very unique. It is the minimal extension of the GG model with the highest available scale for MV . We focus our attention on the n = 2 case on the grounds of baryogenesis. In the same manner, the n = 3 scenario would be well motivated by the possibility of addressing the issue of the SM model CP violation [37] (see also [38] and references therein). • Very interesting possibility would be the n = 1 case with two 10s of Higgs. Such a scenario could have a very high GUT scale and still very promising phenomenological consequences due to light leptoquarks. For example, successful two-loop unification with = × 11 light Φbs (there are two now) at 250 GeV requires MΣ3 2.1 10 GeV and the GUT 15 scale at 1.0 × 10 GeV for central values of αi satMZ. This model would also require at least right-handed neutrinos and non-renormalizable operators to be completely realistic. • Another possibility would be the n = 1 case with one 10 and one 15 of Higgs. Such a scenario would have a same maximal value for the GUT scale as the two 10 case above. To be completely realistic it would require non-renormalizable operators. • The next scenario is the one proposed by Murayama and Yanagida [16] (MY). They shown that addition of two 10s of Higgs to the GG model and with n = 2 it is possible to achieve unification for extremely light scalar leptoquarks in the 10s. This allows MY to forward a “desert” hypothesis, within which the particles are either light (∼ O(102) GeV) or heavy (∼ O(1014) GeV). (Note that the scalar leptoquarks with exactly the same quantum numbers as Φb (⊂ 15) reside in the 10 as well. To see that one can use 5⊗5 = 10⊕15. Note that the couplings of Φb in the 10 (15)tothe5s are antisymmetric (symmetric).) Their model is ruled out by direct searches for scalar leptoquarks due to the extreme lightness of Φb’s. However, if one allows for splitting between Σ3 and Σ8 one can raise unification scale sufficiently to avoid proton decay bounds and resurrect their model although one I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 67 would have to abandon the “desert” hypothesis MY forwarded. The additional contribu- =−14 − 5 − 2 tion to the higgsless SM coefficient in this case is basically B12 15 rΦb 15 rΣ3 15 = which is much more than any of the cases considered in Table 2. For example, for MΦb = 10 = × 14 300 GeV we find, at the one-loop level, MΣ3 10 GeV and MGUT 9 10 GeV. Note that the presence of the 10 of Higgs yields the same type of coupling as we specify in Eq. (6) except that h, in this case, would be antisymmetric. This, however, would not a priori prevent proton decay. If there are two or more higgses in the fundamental representation in the model there exist the proton decay process schematically represented in Fig. 4 unless additional symmetry is introduced. Finally, this model would also require at least right-handed neutrinos and non-renormalizable operators to be completely realistic. • The Georgi–Jarlskog model [10] with an extra 45 of Higgs to fix fermion masses represents natural extension of the GG model. However, even though unification takes place [12,13] and the extension has good motivation the predictivity of the model is lost unless additional assumptions are introduced. More minimal extension than that would be, for example, an extra 24 of Higgs to the scenario we consider.

5. Summary

We have investigated the possibility to get a consistent uniﬁcation picture in agreement with low energy data, neutrino mass and proton decay in the context of the minimal realistic non-supersymmetric SU(5) scenario. This scenario is the Georgi–Glashow model extended by an extra 15 of Higgs. As generic predictions from the running of the gauge couplings we have that a set of scalar leptoquarks is light, with their mass, in the most optimistic case, being around O(102–103) GeV. This makes possible the tests of this scenario at the next generation of collider experiments, particularly in the Large Hadron Collider (LHC) at CERN. In the “least” optimistic scenario the mass of the scalar leptoquarks would be around 105 GeV. The proton decay issue has been studied in detail, showing that it is possible to satisfy all experimental bounds with very small, at 2% level, suppression in Yukawa sector. Rather low scale of vector leptoquarks (∼ 3 × 1014 GeV) already allows for signiﬁcant exclusion of available parameter space of our scenario. Further reduction is expected in near future with new limits on proton decay lifetime. We have particularly studied the case with two higgses in the fundamental representation since in this case it could be possible to explain the baryon asymmetry of the Universe. We have also compared this scenario with other, well motivated, extensions of the GG model. There are uncertainties related to predictions of the proposed scenario but, in view of the fact that it truly represents the minimal realistic extension of the GG model, the same is even more true of all other extensions.

Acknowledgements

We would like to thank Stephen M. Barr, Borut Bajc, M.A. Diaz, Goran Senjanovic´ and Qaisar Shaﬁ for useful discussions and comments. I.D. would also like to thank Trudy and Arthur on their support. P.F.P. thanks the Max Planck Institute for Physics (Werner- 68 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76

Heisenberg-Institute) for warm hospitality. I.D. thanks the Bartol Research Institute for hospitality. The work of P.F.P. has been supported by CONICYT/FONDECYT under contract No. 3050068.

Appendix A. Particle content and relevant interactions

In this appendix we define a minimal realistic non-supersymmetric SU(5) model. By minimal we refer to the minimal number of physical fields such a model requires. In the GG ¯ C C C model the matter is unified in two representations: 5a = (d ,l)a, and 10a = (u ,q,e )a, where a = 1, 2, 3 is a generation index. The Higgs sector comprises 5H = Ψ = (ΨD,ΨT ) = = and 24H Σ (Σ8,Σ3,Σ(3,2),Σ(3¯,2),Σ24). However, this model does not achieve unification and fails to correctly accommodate fermion masses; therefore, it is ruled out. In the introduction we discuss how it is possible to solve all phenomenological problems of the GG model introducing a minimal set of fields. Namely, it is sufficient to introduce an extra representation: 15H = Φ = (Φa,Φb,Φc). The SM decomposition of the Higgs sector is given by

24H = Σ = (8, 1, 0) + (1, 3, 0) + (3, 2, −5/6) + (3, 2, 5/6) + (1, 1, 0), (A.1)

15H = Φ = (1, 3, 1) + (3, 2, 1/6) + (6, 1, −2/3), (A.2)

5H = Ψ = (1, 2, 1/2) + (3, 1, −1/3). (A.3)

The relevant Yukawa potential, up to order 1/MPl,is m m = ij kl m + ij kl Σn n + ij kn l Σn VYukawa ij klm 10a fab10b Ψ 10a f1ab10b Ψ 10a f2ab10b Ψ MPl MPl Σi Σk + ∗ ij ¯ + ∗ j jk ¯ + ∗ ij j ¯ Ψi 10a gab5bj Ψi 10a g1ab5bk Ψi 10a g2ab 5bk MPl MPl ¯ i ¯ j ij ¯ ¯ (5aiΨ )h1ab(5bj Ψ ) + Φ 5aihab5bj + , (A.4) MPl where i, j, k, l, m represent SU(5) indices. Impact of the non-renormalizable operators on the fermion masses is discussed in [9]. Notice that we could replace MPl by a scale Λ, where MGUT <Λ

+ ∗ i ∗ jk ∗ i j k + ∗ ij ∗ k b3Ψi Ψ ΦjkΦ b4Ψi Σ j Σ kΨ b5Ψi Φ ΦjkΨ + ij ∗ k l + ∗ j kl i b6Φ ΦjkΣ lΣ i b7Φij Σ kΦ Σ l. (A.5) We assume no additional global symmetries. It is easy to generalize this potential to describe the case of two or more higgses in the fundamental representation. (We do not include non-renormalizable terms in the Higgs potential since the split between Σ3 and Σ8 masses that we frequently use in running can already be achieved at the renormalizable level.) The condition that the symmetry breaking to the SM is a local minimum of the Higgs potential for Σ (the ﬁrst line in Eq. (A.5))is[26]   15 − 4 2  32 (γ 15 ), γ > 15 , β> − 1 ,γ= 2 , (A.6)  16 15  − 1 2 120γ , 0 <γ < 15 , where dimensionless variables are deﬁned as 2 µ bΣ a 7 β = Σ ,γ= Σ + . 2 cΣ bΣ 15 √ The vacuum expectation value of Σ is Σ =λ/ 30 diag(2, 2, 2, −3, −3), where [26] c β 1/2 1 1/2 1 c β 1/2 λ = Σ + + = Σ h(βγ ). 1 1/2 (A.7) bΣ γ 120βγ (120βγ) bΣ γ Finally, the mass of X and Y gauge bosons is given by 5 M = g λ (A.8) V 12 GUT and 1/2 2 = 1 + √5 γ 1 2 MΣ bλ , 8 3 30 β h(βγ ) 4 5 γ 1/2 1 M2 = − √ bλ2, Σ3 3 30 β h(βγ ) 1 M2 = 1 − 2bγ λ2. Σ24 1 + (1 + 120βγ)1/2 → Here we note the following. In the limit that cΣ 0 we√ obtain the well know results: M2 = 4M2 [17]. However, in the limit where λ → cΣ 30 we obtain Σ3 Σ8 bΣ 8 b M2 M2 → 5/3b λ2 Σ V , Σ8 Σ π αGUT M2 → 0, Σ3 2 M2 → γ − 2bλ2. Σ24 5 70 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76

Clearly, it is technically possible to achieve a large split between MΣ3 and MΣ24 although this is highly unnatural. The relevant interactions for the see-saw mechanism are the following:

V =−M2 Φ†Φ − Y lT CΦ l + c Ψ T Φ†Ψ + . ., see-saw φa Tr a a ν a 3 D a D h c (A.9) where + 0 − √δ δ ν = 2 = T = 0 + Φa + ,l ,ΨD H ,H . (A.10) − √δ δ++ e 2 In this case the neutrino mass is given by H 0 2 Mν ≈ Yνc3 . (A.11) M2 φa This is the so-called type II see-saw [4,5]. Notice that from this equation in order to satisfy the neutrino mass experimental constraint M2 /c has to be around 1013−14 GeV. φa 3

Appendix B. Proton lifetime bounds

The dominant contribution towards nucleon decay in non-supersymmetric GUT usually comes from the gauge d = 6 proton decay operators. Even though these operators carry certain model dependence we have recently shown that unlike in the case of d = 5 operators we can still establish very firm absolute bounds on their strength that are equally valid for all unifying gauge groups [19]. We investigate the impact these bounds have on the non- supersymmetric grand unified model building in what follows. The upper bound for the total nucleon lifetime [19] in grandunifying theories, in the Majorana neutrino case, reads M4 τ . × 41 V , p 1 1 10 years 2 (B.1) αGUT 16 where MV —the mass of the superheavy gauge bosons—is given in units of 10 GeV. We stress that there exist no upper bound for partial lifetimes since we can always set to zero the decay rate for a given channel. In order to fully understand the implications of the experimental results we now specify the theoretical lower bounds on the nucleon decay in GUTs. These are applicable in the case of non-supersymmetric models if the gauge d = 6 contributions are the dominant as well as in the case of supersymmetric models where the d = 4 and d = 5 operators are either forbidden or highly suppressed. The relevant coefficients for the proton decay amplitudes in the physical basis of matter fields are [42] C 2 11 αβ 1β † α1 c e ,dβ = k V V + (V1VUD) V2V , (B.2a) α 1 1 2 UD C = 2 11 βα + 2 † β1 † 1α c eα,dβ k1V1 V3 k2 V4VUD V1VUDV4 V3 , (B.2b) I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 71 C = 2 1α βl + 2 βα † 1l c νl,dα,dβ k1(V1VUD) (V3VEN) k2V4 V1VUDV4 V3VEN , α = 1orβ = 1, (B.2c) C C = 2 † β1 † lα + βα † † l1 c νl ,dα,dβ k2 V4VUD UENV2 V4 UENV2VUD , α = 1orβ = 1, (B.2d) where α, β = 1, 2. The physical origin of the relevant terms is as follows. The terms pro- = −1 portional to k1 ( gGUTMV ) are associated with the mediation of the superheavy gauge fields V = (X, Y ) = (3, 2, 5/6), where the X and Y fields have electric charges 4/3 and 1/3, respectively. This is the case of theories based on the SU(5) gauge group. On the other hand, an exchange of V = (X,Y) = (3, 2, −1/6) bosons yields the terms proportional to = −1 k2 ( gGUTMV ).InSO(10) theories all these superheavy fields are present. = † = † = † = † The relevant mixing matrices are V1 UCU, V2 ECD, V3 DCE, V4 DCD, † † C† C † VUD = U D, VEN = E N, UEN = E N , and VUD = U D = K1VCKMK2, where K1 and K2 are diagonal matrices containing three and two phases, respectively. The leptonic = D = M mixing VEN K3VPMNSK4 in case of Dirac neutrino, or VEN K3VPMNS in the Majo- D M rana case. VPMNS and VPMNS are the leptonic mixing matrices at low energy in the Dirac and Majorana case, respectively. Our convention for the diagonalization of the up, down T = diag T = diag and charged lepton Yukawa matrices is specified by UC YU U YU , DC YDD YD , T = diag and EC YEE YE . To establish the lower bound on the nucleon lifetime we first specify the maximum value for all the coefficients listed above for SO(10) theory only. The SU(5) case is well known and can be reproduced by setting k2 = 0 in the expressions below. The upper bounds are C 2 c eα ,dβ 2k1, (B.3) SO(10) C 2 + 2 c eα,dβ SO(10) k1 k2, (B.4) 3 C ∗ C 4 βδ + 4 + 2 2 c νl,dα,dβ SO(10)c νl,dγ ,dδ SO(10) k1δ k2 2k1k2, (B.5) l=1 3 C C ∗ C C 4 + αδ c νl ,dα,dβ SO(10)c νl ,dγ ,dδ SO(10) k2 3 δ , (B.6) l=1 which translates into the following bounds on the amplitudes in the case the neutrinos are Majorana particles: + m Γ p → π ν¯ p A2 |α|2( + D + F)2 k2 + k2 2, 2 L 1 1 2 (B.7) 8πfπ 2 − 2 2 2 2 + (mp m ) 2mpD mp Γ p → K ν¯ K A2 |α|2 + 1 + (D + 3F) 8πf 2m3 L 3m 3m π p B B m D m × 2 + 2 2 + 4 p + p + 2 2 + 4 k1 k2 1 (D 3F) 2k1k2 k2 , 3mB 3mB (B.8) 72 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76

Table 3 Lower bounds for partial proton lifetime in years for the Majorana neutrino case 4 2 16 in units of MV /αGUT, where the mass of gauge bosons is taken to be 10 GeV SU(5) SO(10) Channel τp τp + p → π ν¯ 7.3 × 1033 1.8 × 1033 + p → K ν¯ 17.4 × 1033 4.8 × 1033 → 0 + × 33 × 33 p π eβ 3.0 10 1.8 10 → 0 + × 33 × 33 p K eβ 8.5 10 5.3 10

+ m Γ p → π 0e p A2 |α|2( + D + F)2 k4 + k2k2 + k4 , β 2 L 1 5 1 2 1 2 2 (B.9) 16πfπ 2 − 2 2 2 + (m m ) m Γ p → K0e p K A2 |α|2 + p (D − F) k2 + k2k2 + k4 . β 2 3 L 1 5 1 2 1 2 2 8πfπ mp mB (B.10) Using these expressions it is easy to extract lower bounds on lifetimes. In Table 3 we list all lower bounds for the proton lifetime in SU(5) and SO(10) models. Again, we use mp = 938.3MeV,D = 0.81, F = 0.44, mB = 1150 MeV, fπ = 139 MeV, AL = 1.43, and the most conservative value α = 0.003 GeV3. In the case of SO(10) models we set k1 = k2 for simplicity. Note that lower bounds are well deﬁned for the partial lifetimes while upper bound is meaningful for the total lifetime only. Finally, we can establish the theoretical bounds for the lifetime of the proton in any given GUT. In what follows we use

M4 M4 34 V <τ < 41 V . 10 years 2 p 10 years 2 (B.11) αGUT αGUT These bounds are useful since we can say something more speciﬁc about the allowed values of MV or αGUT or both. For example, if we take αGUT we can put lower limit on the value of MV using experimental data on nucleon lifetime. Also, given the value of MV and αGUT we can set the limits on the proton lifetime range within the given scenario.

Appendix C. The two-loop running

We present the details on the two-loop running. In order to maximize MV one needs extremely light Φb. In any given mass splitting scheme we thus set the mass of Φb at 250 GeV which is just above the present experimental limit of 242 GeV. Then, Σ3 or Φa or both are allowed to vary between MΦb and MGUT in order to yield unification and all other fields except for the SM ones are taken to be at the GUT scale. The identification MV ≡ MGUT is justified through the inclusion of boundary conditions at the GUT scale [43] − − λ α 1 = α 1 − i , {λ ,λ ,λ }={5, 3, 2}, (C.1) i GUT GUT 12π 1 2 3 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 73 and the relevant two-loop equations for the running of the gauge couplings take the well- known form 3 dαi(µ) bi 2 1 2 µ = α (µ) + bij α (µ)αj (µ). (C.2) dµ 2π i 8π 2 i j=1 bi and bij coefficients for the SM case for arbitrary n are well-documented (see [44] for general formula). In addition to those we have:       1 0 3  30   5  bΣ3 =  1  ,bΦb =  1  ,bΦa =  2  , (C.3) i 3 i 2 i 3 0 1 0 3     1 3 8 000  150 10 15  Σ3 =  28  Φb =  1 13  bij 0 3 0 ,bij 10 2 8 , 000 1 3 22   15 3 108 72 0  25 5  bΦa =  24 56  , (C.4) ij 5 3 0 000 which should be added to the SM ones at the appropriate particle mass scale. The two-loop Yukawa coupling contribution to the running of the gauge couplings (with the corresponding one-loop running of Yukawas) is not included in order to make meaningful comparison between n = 1 and n = 2 cases. (For the n = 2 case one vacuum expectation value of the light Higgs doublets is arbitrary and needs to be specified in order to extract fermion Yukawas for the running, i.e., tan β ambiguity. There is no such ambiguity present in the n = 1 case since the only low-energy vacuum expectation value is accurately determined by electroweak precision measurements.) The outcome of the exact numerical unification is presented in Figs. 6–8. The GUT scale as well as appropriate intermediate scales are indicated on the plots. For example, in the n = 2 scenario with light Σ3 in Fig. 6 the GUT scale is close to the one-loop results (see Fig. 1 in particular) and comes out to be 3.19 × 1014 GeV for central values of coupling constants [15].The1σ departure allows for the maximum value 14 of 3.35 × 10 GeV in that case. Note that the n = 1 case with light Σ3 presented in Fig. 7 yields somewhat higher GUT scale. The reason behind this trend is simple: the Φb contribution to B12 is seven times that of the Higgs doublet but, at the same time, they contribute the same to B23. Thus, Φb is more efficient in simultaneously improving the running and raising the GUT scale than any extra Higgs doublets. One might object that lightness of Σ3, which requires miraculous fine-tuning, makes the scenario with an extra 15 rather unattractive. Again, we do not insist on Σ3 being at intermediate scale. Note that the unification with almost the same GUT scale as in the intermediate Σ3 case is achieved at the two-loop level in the scenario where Φa is at intermediate and Σ3 is at the GUT scale. See Figs. 4 and 8 for the n = 1 and n = 2 cases, respectively. As discussed in the text, intermediate scale for Φa would require either small 74 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76

Fig. 6. Uniﬁcation of the gauge couplings at the two-loop level for central values of low-energy observables [15]. The SM case with n = 2 is presented by dashed lines. Solid lines correspond to the n = 2 scenario with Φb and = = × 4 Σ3 below the GUT scale. Vertical lines mark the relevant scales: MZ, MΦb 250 GeV, MΣ3 4.95 10 GeV 14 and MGUT = 3.19 × 10 GeV.

Fig. 7. Uniﬁcation of the gauge couplings at the two-loop level for central values of low-energy observables [15]. TheSMcasewithn = 1 is presented by dashed lines. Solid lines correspond to the n = 1 scenario with Φb, = = Σ3 and Φa below the GUT scale. Vertical lines mark the relevant scales: MZ, MΦb MΣ3 250 GeV and 14 MGUT = 3.28 × 10 GeV.

Yukawas for Majorana neutrinos or small c3. In the latter case, the novel contributions towards proton decay would be automatically suppressed. The former case could be probed if and when the leptoquarks are detected since some of the rare processes involving neutrinos would be significantly suppressed compared with the charged lepton ones. We note that in the scenario with intermediate Φa the GUT scale grows with the number of light Higgs doublets in contrast to the case when Σ3 is at the intermediate scale. Again, the reason is that the B12 coefficient of Φa is the same as the appropriate coefficient of the Higgs doublet but its contribution to B23 is four times bigger. Thus, the very efficiency of Φa in improving unification makes its impact on MGUT rather small. I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76 75

Fig. 8. Uniﬁcation of the gauge couplings at the two-loop level for central values of low-energy observables [15]. The SM case with n = 2 is presented by dashed lines. Solid lines correspond to the n = 2 scenario with Φb and = = × 6 Φa below the GUT scale. Vertical lines mark the relevant scales: MZ, MΦb 250 GeV, MΦa 4.41 10 GeV 14 and MGUT = 1.17 × 10 GeV.

References

[1] H. Georgi, S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [2] W.J. Marciano, BNL-34599 Presented at Miniconf. on Low-Energy Tests of Conservation Laws in Particle Physics, Blacksburg, VA, 11–14 September 1983. [3] P. Minkowski, Phys. Lett. B 67 (1977) 421; T. Yanagida, in: A. Sawada, A. Sugamoto (Eds.), Proceedings of the Workshop on Uniﬁed Theories and Baryon Number in the Universe, KEK Report No. 79-18, Tsukuba, 1979; S. Glashow, in: M. Lévy, et al. (Eds.), Quarks and Leptons, Cargèse, 1979, Plenum, New York, 1980; M. Gell-Mann, P. Ramond, R. Slansky, in: P. Van Niewenhuizen, D. Freeman (Eds.), Proceedings of the Supergravity Stony Brook Workshop, New York, 1979, North-Holland, Amsterdam, 1979; R. Mohapatra, G. Senjanovic,´ Phys. Rev. Lett. 44 (1980) 912. [4] G. Lazarides, Q. Shaﬁ, C. Wetterich, Nucl. Phys. B 181 (1981) 287. [5] R.N. Mohapatra, G. Senjanovic,´ Phys. Rev. D 23 (1981) 165. [6] S. Weinberg, Phys. Rev. Lett. 43 (1979) 1566. [7] R. Barbieri, J.R. Ellis, M.K. Gaillard, Phys. Lett. B 90 (1980) 249. [8] E.K. Akhmedov, Z.G. Berezhiani, G. Senjanovic,´ Phys. Rev. Lett. 69 (1992) 3013, hep-ph/9205230. [9] J.R. Ellis, M.K. Gaillard, Phys. Lett. B 88 (1979) 315. [10] H. Georgi, C. Jarlskog, Phys. Lett. B 86 (1979) 297. [11] P. Nath, Phys. Rev. Lett. 76 (1996) 2218; P. Nath, Phys. Lett. B 381 (1996) 147; B. Bajc, P. Fileviez Pérez, G. Senjanovic,´ hep-ph/0210374. [12] K.S. Babu, E. Ma, Phys. Lett. B 144 (1984) 381. [13] A. Giveon, L.J. Hall, U. Sarid, Phys. Lett. B 271 (1991) 138. [14] T.P. Cheng, E. Eichten, L.F. Li, Phys. Rev. D 9 (1974) 2259. [15] Particle Data Group, S. Eidelman, et al., Phys. Lett. B 592 (2004) 1. [16] H. Murayama, T. Yanagida, Mod. Phys. Lett. A 7 (1992) 147. [17] A.J. Buras, J.R. Ellis, M.K. Gaillard, D.V. Nanopoulos, Nucl. Phys. B 135 (1978) 66. [18] S. Nandi, A. Stern, E.C.G. Sudarshan, Phys. Lett. B 113 (1982) 165. [19] I. Dorsner, P. Fileviez Pérez, hep-ph/0410198. [20] S.M. Barr, G. Segre, H.A. Weldon, Phys. Rev. D 20 (1979) 2494. [21] D.V. Nanopoulos, S. Weinberg, Phys. Rev. D 20 (1979) 2484. [22] A. Yildiz, P. Cox, Phys. Rev. D 21 (1980) 906. 76 I. Dorsner, P. Fileviez Pérez / Nuclear Physics B 723 (2005) 53Ð76

[23] M. Fukugita, T. Yanagida, Phys. Rev. Lett. 89 (2002) 131602, hep-ph/0203194. [24] L.F. Li, Phys. Rev. D 9 (1974) 1723. [25] A.H. Guth, S.H.H. Tye, Phys. Rev. Lett. 44 (1980) 631; A.H. Guth, S.H.H. Tye, Phys. Rev. Lett. 44 (1980) 963, Erratum. [26] A.H. Guth, E.J. Weinberg, Phys. Rev. D 23 (1981) 876. [27] W. Buchmuller, R. Ruckl, D. Wyler, Phys. Lett. B 191 (1987) 442; W. Buchmuller, R. Ruckl, D. Wyler, Phys. Lett. B 448 (1999) 320, Erratum. [28] S. Davidson, D.C. Bailey, B.A. Campbell, Z. Phys. C 61 (1994) 613, hep-ph/9309310. [29] M. Herz, hep-ph/0301079. [30] SINDRUM II Collaboration, C. Dohmen, et al., Phys. Lett. B 317 (1993) 631. [31] CDF Collaboration, S.M. Wang, hep-ex/0405075; ALEPH Collaboration, R. Barate, et al., Eur. Phys. J. C 12 (2000) 183, hep-ex/9904011; DELPHI Collaboration, P. Abreu, et al., Phys. Lett. B 446 (1999) 62, hep-ex/9903072; L3 Collaboration, M. Acciarri, et al., Phys. Lett. B 489 (2000) 81, hep-ex/0005028; OPAL Collaboration, G. Abbiendi, et al., Phys. Lett. B 526 (2002) 233, hep-ex/0112024; OPAL Collaboration, G. Abbiendi, et al., Eur. Phys. J. C 31 (2003) 281, hep-ex/0305053; ZEUS Collaboration, S. Chekanov, et al., Phys. Rev. D 68 (2003) 052004, hep-ex/0304008. [32] V.A. Mitsou, N.C. Benekos, I. Panagoulias, T.D. Papadopoulou, hep-ph/0411189. [33] S. Abdullin, F. Charles, Phys. Lett. B 464 (1999) 223, hep-ph/9905396. [34] M. Kramer, T. Plehn, M. Spira, P.M. Zerwas, hep-ph/0411038; A. Belyaev, C. Leroy, R. Mehdiyev, A. Pukhov, hep-ph/0502067. [35] Super-Kamiokande Collaboration, K. Kobayashi, et al., hep-ex/0502026. [36] R.N. Mohapatra, Uniﬁcation and Supersymmetry: The Frontiers of Quark–Lepton Physics, Springer-Verlag, Berlin, 1986. [37] G.C. Branco, Phys. Rev. D 23 (1981) 2702. [38] G.C. Branco, L. Lavoura, J.P. Silva, CP Violation, Clarendon, Oxford, 1999. [39] C. Panagiotakopoulos, Q. Shaﬁ, Phys. Rev. Lett. 52 (1984) 2336. [40] H. Ruegg, Phys. Rev. D 22 (1980) 2040. [41] F. Buccella, G.B. Gelmini, A. Masiero, M. Roncadelli, Nucl. Phys. B 231 (1984) 493. [42] P. Fileviez Pérez, Phys. Lett. B 595 (2004) 476, hep-ph/0403286. [43] L.J. Hall, Nucl. Phys. B 178 (1981) 75. [44] D.R.T. Jones, Phys. Rev. D 25 (1982) 581. Nuclear Physics B 723 (2005) 77–90

Phase diagram of QCD with four quark ﬂavors at ﬁnite temperature and baryon density

Vicente Azcoiti a, Giuseppe Di Carlo b, Angelo Galante b,c, Victor Laliena a

a Departamento de Física Teórica, Universidad de Zaragoza, Pedro Cerbuna 12, E-50009 Zaragoza, Spain b INFN, Laboratori Nazionali del Gran Sasso, 67010 Assergi (L’Aquila), Italy c Dipartimento di Fisica dell’Università di L’Aquila, 67100 L’Aquila, Italy Received 11 March 2005; received in revised form 31 May 2005; accepted 23 June 2005 Available online 13 July 2005

Abstract We analyze the phase diagram of QCD with four staggered flavors in the (µ, T ) plane using a method recently proposed by us. We explore the region T 0.7TC and µ 1.4TC,whereTC is the transition temperature at zero baryon density, and find a first order transition line. Our results are quantitatively compatible with those obtained with the imaginary chemical potential approach and the double reweighting method, in the region where these approaches are reliable, T 0.9TC and µ TC. But, in addition, our method allows us to extend the transition line to lower temperatures and higher chemical potentials.  2005 Elsevier B.V. All rights reserved.

1. Introduction

The study of QCD at ﬁnite baryon density is one elusive problem of utmost importance for the understanding of strong interactions. Many features of the behavior of hadronic and quark matter at high baryon density have been suggested mainly from the analysis of effective theories. On this basis, a rich structure of phases and phase transitions is expected,

E-mail addresses: [email protected] (V. Azcoiti), [email protected] (G. Di Carlo), [email protected] (A. Galante), [email protected] (V. Laliena).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.06.026 78 V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90 and the dependence of the phase structure on the quark masses has been conjectured [1]. Unfortunately, the sign problem prevents the direct use of Monte Carlo simulations at finite baryon density. However, in the last years methods to get the transition line at high temperature and low baryon density, overcoming the sign problem, have been proposed. One method uses double reweighting (in the two parameters, β and µa) of the configurations generated at the transition temperature at zero chemical potential, following the transition line in the (µa, β) plane [2]. The rational of this method is that it is expected that the overlap between the reweighted ensemble and the ensemble at a given point of the transition line will be improved respect to the Glasgow method [3], since the original ensemble is itself a mixture of configurations corresponding to the confined and deconfined phase. This approach has been used in a first attempt to locate the expected critical endpoint of QCD with 2 + 1 flavors by using only first principles [4]. Another method ex- ploits the well-known fact that there is no sign problem if the chemical potential is purely imaginary [5]. Then, it was realized in [6] that it is possible to determine, by means of numerical simulations, a pseudo-transition line at imaginary chemical potential and extend it analytically to real chemical potential. This method has been used to study the phase diagram at small chemical potential around the zero density transition point in QCD with two and three degenerate quark flavors in [6] and with four flavors in [7]. Another proposal that is being employed to extract information about the phase transition in the region of small chemical potential is to compute the expectation values of the derivatives respect to µa at µa = 0, in order to reconstruct several terms of the Taylor series [8,9]. Recently we devised another method which can be regarded as a generalization of the imaginary chemical potential approach. The former has several advantages over the later, for it seems that it can be used to determine the transition line at lower temperatures and higher densities [10]. Especially interesting is the fact that it may be used to locate the critical endpoint expected in two flavor QCD [11]. In this paper we report the results of the phase transition line of lattice QCD with four degenerate flavors of staggered quarks obtained with this new method. At zero density a very clear first order transition separates the low and high temperature phases, at a transition temperature, TC, that, for small quark masses, ranges from 100 to 170 MeV [12]. It is expected that the transition continues along a line in the (µ, T ) plane, with the transition temperature lowering as µ increases. This is what has been obtained using the double reweighting [2] and the imaginary chemical potential [7] approaches. We also find a first order transition line starting at µ = 0 and T = TC and continuing at lower temperatures as µ increases. Our results are quantitatively compatible with those obtained in [7] with the imaginary chemical potential approach and with the results of the double reweighting method [2], in the region where these approaches are reliable (see Sections 3 and 4 for a discussion on this point). But, in addition, our method allows us to extend the transition line to lower temperatures and higher chemical potentials. The remaining of the paper is organized as follows: in next section we explain our approach. In section three we describe the numerical results and in section four we present our conclusions. V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90 79

2. Review of the method

The numerical method used in this work is based on the deﬁnition of a generalized QCD action which depends on two free parameters (x, y). This generalized action suffers also from the sign problem for real values of y but not for imaginary values of it. Simulations will be then performed at imaginary values of y and at the end analytical extensions will be needed. The main advantage of this approach when compared with the imaginary chemical potential method is that we can explore the phase transition line at imaginary values of y at any given physical temperature, i.e., we are not forced, as in the case of imaginary chemical potential, to perform simulations at so high temperatures that the system is in the quark– gluon plasma phase for any real value of µa. In this section we shall describe this method. The interested reader can ﬁnd more details of it in [10]. The lattice action for QCD with staggered fermions and chemical potential µ is

3 1 ¯ † S = S + ψ η (n) U ψ + − U ψ − PG 2 n i n,i n i n−i,i n i n i=1 1 ¯ µa −µa † ¯ + ψ η (n) e U ψ + − e U ψ − + ma ψ ψ , (1) 2 n 0 n,0 n 0 n−0,0 n 0 n n n n where SPG is the standard Wilson action for the gluonic ﬁelds, which contains β, the inverse gauge coupling, as a parameter, ηi(n) and η0(n) are the Kogut–Susskind phases, m the fermion mass, and a the lattice spacing. Let us now deﬁne the following generalized action

3 ¯ 1 ¯ † S = S + ma ψ ψ + ψ η (n) U ψ + − U ψ − PG n n 2 n i n,i n i n−i,i n i n n i=1 + Sτ (x, y), (2) with 1 † S (x, y) = x ψ¯ η (n) U ψ + − U ψ − τ 2 n 0 n,0 n 0 n−0,0 n 0 n 1 ¯ † + y ψ η (n) U ψ + + U ψ − , (3) 2 n 0 n,0 n 0 n−0,0 n 0 n where x and y are two independent parameters. The QCD action is recovered by setting x = cosh(µa) and y = sinh(µa). Monte Carlo simulations of the model (2) for real values of x,y are not feasible since we meet the sign problem. However if y is a pure imaginary number, y = iy¯, where y¯ is real, the sign problem disappears since the fermionic matrix is the sum of a constant diagonal matrix plus an anti-Hermitean matrix which anticommutes with the staggered version of the γ5 Dirac matrix, and numerical simulations become feasible. Imaginary chemical potential is a particular case of this, obtained by setting x = cos(µa) and y¯ = sin(µa). 80 V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90

Fig. 1. Minimal phase diagram conjectured for the generalized QCD. The solid line is line of phase transitions. The discontinuous line is the physical line, x2 − y2 = 1.

The expected phase diagram for this model in the (x, y) plane is shown in Fig. 1 [10]. The solid line is a presumed line of phase transitions and the discontinuous line is the physical line, x2 −y2 = 1, along which one recovers standard QCD at finite baryon density. The intersection of the solid line with the discontinuous one will therefore give us the transition chemical potential of QCD at a given temperature. A change in the physical temperature can be simulated by changing β keeping fixed Lt or vice versa. In both cases the solid line in Fig. 1 will move and the intersection point which gives the transition chemical potential will change with the physical temperature. For small β the transition line crosses the y = 0 axis at x>1 and, therefore, intersects the physical line also at x>1, producing a physical phase transition at µa > 0. By increasing β and keeping fixed the temporal lattice extent Lt , the transition point on the y = 0 axis moves toward x = 1 and eventually crosses it. Clearly, the value of β at which the transition line intersects the physical line at x = 1 and y = 0 is the zero density transition point. For larger β the transition line and the physical line do not intersect and the system is in the deconfined phase whatever the chemical potential. From an analysis of the symmetries of the action (2) it is not difficult to realize that the partition function depends on x and y only through the combinations u = x2 − y2, v = (x + y)3Lt + (x − y)3Lt . (4) 2 3L 2 For imaginary values of y (y = iy¯)wehaveu = ρ and v = 2ρ t cos(3Lt η), where ρ = x2 +¯y2 and tan η =¯y/x, and, therefore, the free energy will be a periodic function of η with period 2π/3Lt . In particular if the phase transition line of Fig. 1 continues to imaginary values of y, the expected phase diagram in the (x, y)¯ plane is displayed in Fig. 2, where we have incorporated the property of periodicity. In Fig. 2 we have also included the line ρ = 1 (dotted) which is the locus of the points accessible to numerical simulations of QCD at imaginary chemical potential. One can see now how this approach has more potentialities V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90 81

Fig. 2. Conjectured phase diagram in the (x, y)¯ plane. We have incorporated the periodicity. The dashed line contains the points corresponding to imaginary chemical potential. than the imaginary chemical potential approach. Indeed by increasing the inverse gauge coupling β, the phase transition line of Fig. 2 moves toward the origin of coordinates. In some interval (βm,βM ) the transition line intersects the ρ = 1 line and then a phase transition will appear at imaginary chemical potential. In such a situation, the physical temperature is so high that the system is in an unconﬁned phase for any real value of the chemical potential. The advantage of our approach is that in our simulations ρ is not enforced to be one. The variables u and v have the interesting property that are real for both y real and pure imaginary, so that they map the two planes (x, y) and (x, y)¯ onto a single plane (u, v). The line v = 2u3Lt /2 separates the regions corresponding to each plane: the region above it in Fig. 3 corresponds to real y and is not accessible to numerical simulations, while the region below it corresponds to imaginary y and can be explored by means of simulations. Thephysicallineisu = 1 with v 2. The imaginary chemical potential points are mapped onto the line u = 1, −2 v 2. The analytical extension from imaginary y to the physical real y becomes in the (u, v) plane the extrapolation from the region accessible to numerical simulations to u = 1, v 2. In order to get the transition chemical potential we must determine the coordinates of the intersection point between the solid line and the physical line in Fig. 1.Fromthe symmetries of the partition function we know that the phase transition line is an even function of x and y. Therefore, we can write the following equation for the transition line at ﬁxed β in the (x, y) plane1 2 2 4 6 x = 1 + a0(β) + a2(β)y + a4(β)y + O y . (5)

By ﬁxing the lattice temporal extent Lt and the gauge coupling β one ﬁxes the physical temperature T . The intersection point of the transition line with the physical line 2 = + 2 + 4 + 6 yc a0(β) a2(β)yc a4(β)yc O yc , (6)

1 Here and in the following we do not write explicitly the dependence of the coefﬁcients ai , β0, bi , etc., on the temporal lattice extent, Lt , and the quark masses, ma. 82 V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90

Fig. 3. Phase diagram in the (u, v) plane. The solid line, v = 2u3Lt /2 corresponds to y = 0, and separates the region where numerical simulations are feasible (below the line) from the region where the sign problem prevents numerical simulations (above the line). The discontinuous line is a hypothetical phase transition line. The line u = 1 is also displayed. For v 2 it is the physical line while for −2 v 2 it corresponds to imaginary chemical potential. will give us the transition chemical potential, yc = sinh(µca), at this temperature. The strategy for the determination of the transition chemical potential is then the following. From numerical simulations at imaginary values of y, y = iy¯, near the phase transition 1/2 point ((1 + a0) , 0) one can locate several phase transition points in the (x, y)¯ plane (see Fig. 2). By fitting these points with Eq. (5) with the + sign of the coefficient proportional to y2 replaced by −, we can numerically measure the first coefficients. Ignoring the quartic term, the transition chemical potential, µca, will then be given by 1/2 −1 a0 µca =±sinh . (7) 1 − a2 An alternative procedure, which is indeed the one employed in this work, is to project of the phase diagram onto the (y, β) plane. In practice, we fix x = x0 > 1, Lt and the lattice quark masses2 and perform simulations for different values of y¯ and β. In this way we can easily find an accurate estimate of the transition point at a given y¯ by interpolation in β via reweighting. The qualitative phase diagram in the (y, β) plane is displayed in Fig. 4.The solid lines are the physical lines = =± 2 − y yph x0 1, (8) and the discontinuous line is a line of phase transitions. Using again the symmetries of the partition function we can write for this line the following equation 2 4 β = β0(x0) + b2(x0)y + O y . (9)

2 Notice that in the imaginary chemical potential approach x0 is enforced to be less than one. V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90 83

Fig. 4. Conjectured phase diagram in the (y, β) plane, with x = x0 > 1 ﬁxed. The discontinuous line is a line of = =± 2 − phase transitions, whereas the solid lines are the physical lines, y yph x0 1.

As in the case discussed previously, one can measure β0 and b2 from simulations at y = 0 and at imaginary y = iy¯ (keeping x0 fixed). Then, we have to find the intersection point between the physical and phase transition lines of Fig. 4. = 2 +¯2 Equivalently, one may use the plane (u, β), where u x0 y > 1. From simulations = ¯ 2 at fixed x x0 and y 0 we can get a phase transition line β(u) for u x0 > 1. Then one can extrapolate this line to u = 1, thus obtaining the physical transition coupling at −1 chemical potential µa = cosh (x0).

3. Numerical results

We performed simulations of QCD with four degenerate flavors of staggered quarks using the hybrid Monte Carlo algorithm on lattices of sizes 63 × 4 and 83 × 4. Our method is computationally very expensive and it is rather difficult to go to larger lattices. The quark mass in lattice units was fixed to ma = 0.05. On each lattice, we repeated the simulations for several values of x, y¯, and β. We measured the plaquette, the chiral condensate, and the Polyakov loop after each molecular dynamics trajectory of unit time. For each simulation we accumulate between thirty and forty thousand measurements. Most of the simulations have been performed on the Linux clusters of LNGS-INFN, and some of them in the Linux cluster of Departamento de Física Teórica of Universidad de Zaragoza. At zero chemical potential (x = 1 and y¯ = 0) there is a very clear signal of a first order phase transition controlled by β, at which the plaquette, the chiral condensate, and the Polyakov loop vary abruptly and show a clear two state structure. This first order transition persists for x>1 and y¯ 0, with the transition coupling depending on x and y¯. To determine the transition lines numerically, we proceeded as follows. For each pair (x, y)¯ we estimate the transition coupling by reweighting à la Ferrenberg–Swendsen (in the parameter β) the configurations corresponding to the value of β at which a clear two state signal appeared. We used the maximum of the plaquette susceptibility as the criterion to define the transition coupling on the finite lattice, since it gives the best signal. 84 V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90

Table 1 Parameters of the χ2 ﬁts of the transition lines in the (y,β)¯ (or (u, β)) plane for different values of x, extracted from simulations on the L = 6andL = 8 lattices 2 Lx β0 βph b2 χ /ndof ndof 61.02 5.0359(2) 5.008(1) 0.68(2) 0.41 61.045 5.028(1) 4.962(3) 0.72(3) 1.63 1 81.0201 5.0367(5) 5.0089(10) 0.684(15) 0.82 4 81.0314 5.0332(3) 4.9869(12) 0.726(13) 1.11 5 81.0453 5.0292(5) 4.9636(21) 0.708(21) 0.97 5 81.0811 5.0179(4) 4.8943(20) 0.732(10) 0.36 7 81.1276 5.0047(4) 4.8017(28) 0.747(9) 0.97 6

In this way, for each value of x we have a transition line in the plane (y,β)¯ .Weﬁtthis line with a second order polynomial: 2 β(y)¯ = β0(x) + b2(x)y¯ . (10) The analytic continuation of this functions to real y¯ =−iy is trivial: β(y) = β (x) − b (x)y2. (11) 0 2 √ 2 The physical value of y is yph = x − 1. Hence, the physical transition coupling is 2 β(yph) = β0(x) − b2(x)(x − 1). We can also use the variable u = x2 +¯y2, and then the transition line in the plane (u, β) is ﬁt by the linear function

β(u) = βph(x) + b2(x)(u − 1). (12) Notice that the fits in the (y,β)¯ and (u, β) are identical since both the data and the fit functions are related by the same transformation. In the (u, β) plane the analytic continuation to y¯ =−iyph corresponds to the extrapolation to u = 1. Hence, the physical transition coupling for each value of x is directly given by the parameter βph. Table 1 collects the parameters of the best χ2 fits of the transition lines given by Eqs. (10) or (12) extracted from simulations on the 63 × 4 and 83 × 4. The transition lines at fixed x are displayed in Figs. 5 and 6. The left panels represent the (y,β)¯ plane, for the open symbols and the dashed line, and the (y, β) plane, for the solid and dotted lines, and the filled circle. The right panels represent the (u, β) plane. In all plots the open symbols are the numerical estimates of the phase transition points obtained form the simulations on the 83 × 4 lattice. On the left panels, the dashed line is the best χ2 fit to a function of the form (10) and the solid line is its analytical√ continuation to y = iy¯. The dotted line is the 2 physical line, given by y = yph = x − 1, and the filled circle is the physical transition point. On the right panels, the solid line is the fit of the transition points to a function of the form (12) and the dashed line is the boundary of the region where numerical simulations without sign problem can be performed (i.e., the line u = x2). The filled circle is the physical transition point obtained extrapolating the transition line to the physical region, u = 1. In the transition lines displayed in Fig. 6 the points at the largest values of y¯ (or u) deviate from the smooth growing of the points at smaller y¯ (or u), and they are not included V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90 85

Fig. 5. Phase diagrams at ﬁxed x in the (y,β)¯ and (y, β) planes (left panels) and in the (u, β) plane (right panels).

on the fits. They are a reflection of the 2π/3Lt periodicity that we would see if we look at the phase diagram onto the constant β plane instead of onto the plane of constant x (see Section 2). The data of Fig. 6 (especially those for x = 1.0811, for which many points of the phase transition line were determined) strongly suggest that there is a cusp separating the β increasing curve from the β decreasing one. This means that the 2π/3Lt periodicity is realized by the periodic replication of a nonperiodic analytic function, producing a cusp at the points where a replica ends and a new replica starts. This cusp is a nonanalyticity of the phase transition line which has the same origin as the Roberge–Weiss transition [13] found in QCD at µ =±iπ/3Lt in the high temperature region [14]. Indeed, as it was conjectured in [15], this singular behavior realized as a cusp is to be expected in a wide variety of models characterized by a quantized charge coupled to a phase. In our case the quantized charge is the baryonic charge and the phase φ = arctan(y/x)¯ . Under this conditions, one must exclude the points to the right of the cusp from the fits since they belong to a different analytic function. Notice also that the presence of the nonanalytic cusp does not necessarily imply that the convergence radius of the Taylor series around y = 0(u = 1) is limited by this singularity. Ref. [15] contains a simple illustrative example on that in what we called Gaussian model. The transition lines displayed in Figs. 5 and 6 are very smooth (they are essentially straight lines in the (u, β) plane until the cusp) and suggests a linear fit in the (u, β) plane. 86 V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90

Fig. 6. Phase diagrams at ﬁxed x in the (y,β)¯ and (y, β) planes (left panels) and in the (u, β) plane (right panels).

In fact the data reported in Table 1 imply actually a very high confidence level for the fits and to add higher order corrections, as for instance a quartic term in Eq. (10), seems meaningless. In other words, if higher order corrections were relevant at real y, it would be very hard to measure them from the data produced at imaginary y. This makes really difficult any serious analysis of systematic errors possibly induced by the fit ansatz. Fig. 7 displays the phase diagram in the plane (µa, β), together with the results obtained by Fodor and Katz with double reweighting [2], and by D’Elia and Lombardo with imaginary chemical potential simulations [7]. There is good agreement with the results of V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90 87

Fig. 7. Phase diagram in the (µa, β) plane. The solid line is the analytical continuation of the imaginary chemical potential pseudotransition line, and the dashed lines mark the error band. The authors of Ref. [7] give credit to this analytical continuation up to µa ≈ 0.3.

Fodor and Katz until µa ≈ 0.3, especially if we take into account that different methods to locate the pseudotransition coupling at finite volume are used. The agreement with D’Elia and Lombardo is very satisfactory in the whole range of µa explored. The main difference, which is already seen at µa = 0, should be attributed to a volume effect, since these authors used a 163 × 4 lattice. D’Elia and Lombardo give credit to their results for µa 0.3, which is the interval where they found agreement with Fodor and Katz. We have seen that indeed the results (ours and theirs) are reliable at least until µa = 0.5, which is the maximum chemical potential at which we performed simulations. The disagreement with Fodor and Katz for µa 0.3 is likely due to the poor overlap of the ensemble of reweighted configurations with the ensemble of typical configurations at µa 0.3. Fig. 8 shows the phase diagram in the plane (µ, T ) in physical units, with the scale set by the transition temperature at µ = 0, TC. The relative lattice spacings have been determined by means of the two loop beta function. For comparison, we also plot the result of D’Elia and Lombardo. Our results can be fit with a power function of the form T 2 p = 1 − c(µ/TC) . (13) TC 88 V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90

Fig. 8. Phase diagram in physical units, with the zero density transition temperature, TC, setting the scale.

The best χ2 ﬁt gives c ≈ 0.446(9) and an exponent, p ≈ 0.173(7). One may be curious about the extrapolation of this line to zero temperature. If this were done, we would ﬁnd a zero temperature transition at chemical potential µC ≈ 1.5TC, which, in terms of the nucleon mass [16], mN,isµC ≈ mN/5.

4. Conclusions

Using a recently proposed method to determine the phase transition lines at ﬁnite baryon density, we have obtained the phase transition line of QCD with four degenerate quark ﬂa- vors from the zero density high temperature transition, at TC,downtoT ≈ 0.7TC and µ ≈ 1.4TC. Our results are in reasonably good agreement with those of D’Elia and Lom- bardo, obtained from simulations at imaginary chemical potential. These authors give credit to their results for T 0.9TC, since this is the region where their results showed reasonably small statistical and systematic errors and in addition agree with those obtained by Fodor and Katz with the double reweighting method. Our results agree reasonably well with the central value reported by D’Elia and Lombardo in the whole region T 0.7TC and µ 1.4TC. V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90 89

We believe we can explore much lower temperatures with our method. Since it is based on analytical continuation/extrapolation there are uncontrolled systematic errors that grow in decreasing the temperature. It is therefore very difficult to estimate the minimum temperature at which our method will give a reliable prediction of the location of the phase transition point. In [10] we verified that in the three-dimensional Gross–Neveu model at large N this minimum temperature is amazingly low. The study of four flavor QCD thermodynamics at lower temperatures is very interesting, and we left it for future work. Since our method is computationally very expensive, we decided to concentrate the present effort in the more interesting cases of two and two plus one flavor QCD.

Acknowledgements

This work received ﬁnancial support from CICyT (Spain), project FPA2003-02948, from Ministerio de Ciencia y Tecnología (Spain), project BFM2003-08532-C03-01/FISI, and from an INFN–CICyT collaboration. The authors thank the Consorzio Ricerca Gran Sasso that has provided most of the computer resources needed for this work. V.L. is a Ramón y Cajal fellow.

References

[1] J.B. Kogut, M.A. Stephanov, The Phases of Quantum Chromodynamics, Cambridge Univ. Press, Cam- bridge, 2004; J.B. Kogut, Nucl. Phys. B (Proc. Suppl.) 119 (2003) 210–221. [2] Z. Fodor, S.D. Katz, Phys. Lett. B 534 (2002) 87. [3] I.M. Barbour, S.E. Morrison, E.G. Klepﬁsh, J.B. Kogut, M.P. Lombardo, Nucl. Phys. B (Proc. Suppl.) 60 (1998) 220. [4] Z. Fodor, S.D. Katz, JHEP 0203 (2002) 014; Z. Fodor, S.D. Katz, JHEP 0404 (2004) 050; Z. Fodor, S.D. Katz, K.K. Szabo, JHEP 0405 (2004) 046. [5] E. Dagotto, A. Moreo, R.L. Sugar, D. Toussaint, Phys. Rev. B 41 (1990) 811; A. Hasenfratz, D. Toussaint, Nucl. Phys. B 371 (1992) 539. [6] Ph. de Forcrand, O. Philipsen, Nucl. Phys. B 642 (2002) 290; Ph. de Forcrand, O. Philipsen, Nucl. Phys. B 673 (2003) 170; Ph. de Forcrand, O. Philipsen, Prog. Theor. Phys. Suppl. 153 (2004) 127–138. [7] M. D’Elia, M.P. Lombardo, Phys. Rev. D 67 (2003) 014505; M. D’Elia, M.P. Lombardo, Phys. Rev. D 70 (2004) 074509; M.P. Lombardo, Prog. Theor. Phys. Suppl. 153 (2004) 26–39. [8] C.R. Allton, S. Ejiri, S.J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, C. Schmidt, L. Scorzato, Phys. Rev. D 66 (2002) 074507; C.R. Allton, S. Ejiri, S.J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, C. Schmidt, Phys. Rev. D 68 (2003) 014507; C.R. Allton, M. Döring, S. Ejiri, S.J. Hands, O. Kaczmarek, F. Karsch, E. Laermann, K. Redlich, hep- lat/0501030. [9] R.V. Gavai, S. Gupta, R. Roy, Prog. Theor. Phys. Suppl. 153 (2004) 270; R.V. Gavai, S. Gupta, Phys. Rev. D 68 (2003) 34506, hep-lat/0412035. [10] V. Azcoiti, G. Di Carlo, A. Galante, V. Laliena, JHEP 0412 (2004) 010. 90 V. Azcoiti et al. / Nuclear Physics B 723 (2005) 77–90

[11] V. Azcoiti, G. Di Carlo, A. Galante, V. Laliena, in preparation. [12] F.R. Brown, F.P. Butler, H. Chen, N.H. Christ, Z. Dong, W. Schaffer, L.I. Unger, A. Vaccarino, Phys. Lett. B 251 (1990) 181. [13] A. Roberge, N. Weiss, Nucl. Phys. B 275 (1986) 734. [14] A. Hart, M. Laine, O. Philipsen, Phys. Lett. B 505 (2001) 141. [15] V. Azcoiti, A. Galante, V. Laliena, Prog. Theor. Phys. 109 (2003) 843. [16] K.D. Born, E. Laermann, N. Pirch, T.F. Walsh, P.M. Zerwas, Phys. Rev. D 40 (1989) 1653. Nuclear Physics B 723 (2005) 91–116

Two-loop QCD corrections to the heavy quark form factors: Anomaly contributions

W. Bernreuther a, R. Bonciani b, T. Gehrmann c, R. Heinesch a, T. Leineweber a, E. Remiddi d

a Institut für Theoretische Physik, RWTH Aachen, D-52056 Aachen, Germany b Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg, D-79104 Freiburg, Germany c Institut für Theoretische Physik, Universität Zürich, CH-8057 Zürich, Switzerland d Dipartimento di Fisica dell’Università di Bologna, and INFN, Sezione di Bologna, I-40126 Bologna, Italy Received 25 April 2005; received in revised form 1 June 2005; accepted 23 June 2005 Available online 18 July 2005

Abstract 2 We present closed analytic expressions for the order αs triangle diagram contributions to the matrix elements of the singlet and non-singlet axial vector currents between the vacuum and a quark– antiquark state. We have calculated these vertex functions for arbitrary momentum transfer and for four different sets of internal and external quark masses. We show that both the singlet and non- singlet vertex functions satisfy the correct chiral Ward identities. Using the exact expressions for the ﬁnite axial vector form factors, we check the quality and the convergence of expansions at production threshold and for asymptotic energies.  2005 Elsevier B.V. All rights reserved.

PACS: 11.15.Bt; 11.30.Rd; 12.38.Bx; 14.65.Fy; 14.65.Ha; 13.88.+e

Keywords: Chiral anomaly; Multi-loop calculations; Vertex diagrams; Heavy quarks; Asymptotic expansion

E-mail addresses: [email protected] (W. Bernreuther), [email protected] (R. Bonciani), [email protected] (T. Gehrmann), [email protected] (R. Heinesch), [email protected] (T. Leineweber), [email protected] (E. Remiddi).

1. Introduction

This paper is the third of a series that deals with the computation of the electromag- netic and neutral current form factors of heavy quarks Q at the two-loop level in QCD [1,2]. Knowledge of these form factors puts forward the aim of a completely differential description of the electroproduction of heavy quarks, e+e− → γ ∗, Z∗ → QQX¯ , to order 2 αs in the QCD coupling. This, in turn, will allow precise calculations or predictions of many observables, including forward–backward asymmetries, which are relevant for the (re)assessment of existing data on b quark production at the Z resonance [3], or which will become important for the production of b quarks and t quarks at a future linear collider [4]. A detailed list of references can be found in [1,2]. 2 In [1,2] we presented analytic results to order αs of the heavy quark vector and axial vector vertex form factors for arbitrary momentum transfer, up to anomalous contributions. Here we close this gap and analyze the triangle diagram contributions to the axial vector vertex which exhibit the Adler–Bell–Jackiw (ABJ) anomaly [5,6]. We use dimensional regularization and the prescription of [7,8] for implementing the Dirac matrix γ5 in D = 4, which is an adaptation of the method of [9]. In Section 2 we set up the notation and discuss chiral Ward identities which must be satisfied by the appropriately renormalized axial vector current of a heavy quark, and the flavor-singlet and non-singlet axial vector currents, 2 respectively. In Section 3 we give closed analytic expressions to order αs for the anomalous (pure singlet) contributions to the axial vector form factors for four different sets of internal and external quark masses. This is adequate for a number of applications, including the leptoproduction of b quarks at the Z resonance and of t quarks. First we present our results for spacelike squared momentum transfer s; then the form factors are analytically continued into the timelike region, and we determine also their threshold and asymptotic expansions. We check that the form factors satisfy the chiral Ward identities discussed in Section 2. In the context of this check we have calculated also the triangle diagram contributions to the pseudoscalar vertex function and the vertex function of the gluonic operator GG˜ .In[12] the triangle contributions to the bb¯ vertex function of the non-singlet current ¯ ¯ tγµγ5t − bγµγ5b were computed for massless b quarks, using the method of dispersion relations. Our chirality-conserving form factors for this specific mass combination confirm that result. In Section 4, we expand our exact expressions for the finite axial vector form factors at the production threshold and at asymptotic energies. Detailed comparison of these expansions with the full result allows us to assess the convergence properties of these widely used expansion techniques [13,14]. We conclude in Section 5.

2. Chiral Ward identities

We consider QCD with Nf − 1 massless and one massive quark. This includes the case of Nf = 6 with all quarks but the top quark taken to be massless. We analyze the following external currents: W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 93

¯ ¯ • The currents J5µ(x) = Q(x)γµγ5Q(x) and J5 = Q(x)γ5Q(x), where Q denotes the field associated with the massive quark. These currents are called axial vector and pseudoscalar current in what follows. • S = ¯ The flavor-singlet axial vector current J5µ(x) q(x)γµγ5q(x), where the sum extends over all quark flavors including Q. • NS = 3 ¯ The flavor non-singlet neutral axial vector current J5µ i=1 ψiγµγ5τ3ψi , where ψi = (u, d)i is the ith generation quark isodoublet and τ3 is the 3rd Pauli matrix. This current couples to the Z boson. The corresponding pseudoscalar current is denoted NS by J5 .

NS The current J5µ is conserved in the limit of vanishing quark masses, while the currents S J5µ and J5µ are anomalous at the quantum level [5,6].

2.1. Axial vector and pseudoscalar vertex functions

In the following we study the matrix elements of the axial vector and pseudoscalar ¯ 2 vertex functions between the vacuum and an on-shell qq pair to order αs in the QCD coupling, where q denotes one of the Nf quark ﬂavors. The bare 1-particle irreducible (1PI) vertex functions involving J5µ and J5 are denoted by Γq,µ and Γq , respectively. The kinematics is J(p)→ q(p1) +¯q(p2), with p = p1 + p2:

Γq,µ(s) = γµγ5 + Λq,µ(s), (1)

Γq (s) = γ5 + Λq (s). (2)

Following [5] it is useful to distinguish between two types of contributions to Λq,µ and Λq and, likewise, for the other vertex functions considered below: those where the axial vector (pseudoscalar) vertex γµγ5 (γ5) is attached to the external quark line q are called type A. For the axial vector and pseudoscalar currents defined above these are non-zero only for q = Q. Type B contributions are called those where γµγ5, respectively γ5, is attached to 2 a quark loop. To order αs these are the triangle diagrams Fig. 1(a) for the heavy quark S S NS NS currents J5,µ and J5, and the diagrams Fig. 1(a), (b) for J5,µ, J5 , J5,µ, and J5 .For massless quarks in the triangle Fig. 1(b) only Fig. 1(a) contributes to the matrix elements S NS of J5 and J5 . The anomalous Ward identities, which will be discussed in Section 2.3, involve also the truncated matrix element of the gluonic operator GG˜ between the vacuum and an on-shell qq¯ pair. Here ˜ a,µν a,αβ G(x)G(x) ≡ µναβ G (x)G (x), (3) with Ga,µν being the gluon field strength tensor. The operator (3) induces, to zeroth order µ α in the QCD coupling, the momentum-space vertex 8k1 k2 µναβ with the gluon momentum k1 (k2) flowing into (out of) the vertex. The truncated one-loop qq¯ vertex function Fq which will be needed in Section 2.3 is shown in Fig. 2. We use dimensional regularization in D = 4 − 2 space–time dimensions. For the treatment of γ5 we use a pragmatic approach: the calculation of the type A contributions is made using a γ5 that anticommutes with γµ in D dimensions. This approach works for 94 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116

Fig. 1. Triangle diagram (type B) contributions to the various axial vector and pseudoscalar vertex functions considered in this paper. The double and single straight line triangles in (a) and (b) denote massive and massless quarks, respectively. The external q, q¯ quarks are either massive or massless. The dashed line denotes either an axial vector or a pseudoscalar vertex. The crossed diagrams are not drawn.

Fig. 2. One-loop anomalous vertex function Fq . open fermion lines, but is known to be algebraically inconsistent, in general, for closed fermion loops with an odd number of γ5 matrices. The appropriate implementation of γ5 for type B diagrams is the prescription of ’t Hooft and Veltman [9,10] or its adaption by [7,8] where1 i γ = γ µ1 γ µ2 γ µ3 γ µ4 , (4) 5 4! µ1µ2µ3µ4 i γ γ = γ µ1 γ µ2 γ µ3 (5) µ 5 3! µµ1µ2µ3 is used within dimensional regularization. In these relations, the Lorentz indices are taken to be D-dimensional. The product of two -tensors is thus expressed as a determinant over metric tensors in D dimensions. We employ these relations in this paper. In the pragmatic approach chiral invariance is respected, to second order in the QCD coupling, also by the unrenormalized type A contributions. For the above vertex functions the following well-known chiral Ward identity can be derived in straightforward fashion:

µ (A) = (A) − − − p ΛQ,µ 2m0,QΛQ ΣQ(p1)γ5 γ5ΣQ( p2), (6)

1 0123 We use the convention 0123 =− =+1. W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 95 where m0,Q and ΣQ denote the bare mass and self-energy of Q. Because the derivation of (6) was based on an anticommuting γ5 we have

−γ5ΣQ(−p2) = ΣQ(p2)γ5. (7)

2.2. Renormalization of the currents

We renormalize the ultraviolet (UV) divergences by defining αs in the MS scheme, while the renormalized quark mass and the quark wavefunction renormalization constant Z2 are defined in the on-shell scheme. That is, the renormalized quark self-energy is zero = OS OS on-shell. For the on-shell mass mQ of the quark Q we have m0 Zm mQ, where Zm denotes the mass renormalization constant in the on-shell scheme. We described this hybrid renormalization procedure in detail in [1,2]. The renormalization of the (non-)singlet axial vector and pseudoscalar currents for the prescriptions (4), (5) in dimensional regularization was clarified in [8]. In that work also the various current renormalization constants were computed for massless quarks within the MS scheme, to three loops for the singlet axial vector and the pseudoscalar currents, and the respective finite counterterms were determined which are necessary for obtaining the correct chiral Ward identities. (Ref. [11] confirmed several of the three-loop results of [8].) Because the UV divergences caused by the currents defined by (4), (5) can be removed by a mass-independent scheme like the MS scheme, the finite renormalization constants of [8] apply also to the massive case. Our calculations presented in Section 3 yield the triangle diagram contributions to various two-loop renormalization constants of the axial vector and pseudoscalar currents for massive quarks. Comparing these results with those of [8] (see below) serves as a useful cross-check of our computations. Eq. (6) implies that, within the pragmatic approach and to second order in the QCD cou- ¯ pling, neither the axial vector current J5µ nor the operator m0,QQγ5Q do get renormalized with respect to type A contributions. Thus the renormalized second-order axial vector and pseudoscalar vertex functions of type A are given by (A),R = OS (A) ΓQ,µ Z2 ΓQ,µ, (8) (A),R = OS (A) = OS OS (A) mQΓQ m0,QZ2 ΓQ mQZm Z2 ΓQ . (9) Nevertheless, the axial vector current must be renormalized because of the ABJ anomaly which is present in the triangle diagrams. When using an anticommuting γ5 for computing the type A and (4), (5) for the type B contributions the correctly renormalized current to 2 order αs turns out to be R = finite J5µ ZJ ZJ J5µ, (10) i.e., the second-order triangle diagram contributions to the axial vector vertex function are renormalized according to (B),R = (B) + + ΛQ,µ ΛQ,µ (δJ δ5)γµγ5, (11) where

δJ ≡ ZJ − 1, 96 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116

≡ ﬁnite − δ5 ZJ 1. (12)

The renormalization constant ZJ removes the UV divergences present in the diagrams Fig. 1(a), (b). From their computation—see the next section—we find that this constant, which we define in the MS scheme, is given by 3C T α 2 Z = 1 + F R s C2() + O α3 , (13) J 2 2π s = 2 − = = + where CF (Nc 1)/2Nc, TR 1/2, C() (4π) (1 ), and Nc denotes the number of colors. This result is in agreement with [8]. The use of (4) and (5) requires, in addition, a finite finite renormalization ZJ in order to obtain the correct anomalous Ward identity for the axial vector current [5]. That is to say, for the renormalized (singlet) axial vector currents the non-renormalization [15] of the ABJ anomaly is to be maintained. For the axial vector current this reads α ∂µJ = 2im (Qγ¯ Q) + s T (GG)˜ . (14) 5µ R Q 5 R 4π R R From our calculation of the respective vertex functions involving massive and/or massless finite quarks, presented in Section 3, we find that ZJ is given by 3C T α 2 Zfinite = 1 + F R s + O α3 . (15) J 4 2π s Again, we can compare with the analogous result obtained in [8] for massless QCD, and we find agreement. For a pseudoscalar vertex the triangle diagrams Fig. 1(a) are UV and infrared finite. 2 This implies that, to order αs , (B),R = (B) ΛQ ΛQ , (16) ¯ ¯ and m0,QQγ5Q = mQ(Qγ5Q)R within the pragmatic approach. We emphasize that no ¯ finite renormalization of the pseudoscalar operator Qγ5Q is necessary in order that the Ward identities (24) and (25) below are satisfied. ¯ Next we consider the renormalization of the truncated QQ vertex function αsFQ(s) of ˜ µ S the operator αsGG. This operator mixes with ∂ J5µ under renormalization [16]: ˜ = ˜ + µ S (GG)R ZGG˜ GG ZGJ ∂ J5µ, (17) S where J5µ is the flavor-singlet current. The renormalization constants which we need here were given, e.g., in [8].IntheMS prescription we have = + O ZGG˜ 1 (αs), (18) 6C α Z = F s C() + O α2 . (19) GJ 2π s R Eqs. (17) and (19) imply that the renormalized one-loop vertex function FQ is given by R = + µ FQ FQ iZGJ p γµγ5. (20) W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 97

We recall that Eq. (5) is to be used for γµγ5. NS NS = Finally we consider the non-singlet currents J5µ and J5 , for definiteness in Nf 6 flavor QCD with five massless quarks and the massive quark Q being the top quark. The µ NS = ¯ divergence ∂ J5µ 2im0,t tγ5t is non-anomalous, and within our prescription for imple- NS ¯ menting γ5 the operators J5µ and m0,t tγ5t do not get renormalized. Let us consider the NS NS ¯ 1PI vertex functions involving J5µ and J5 and an external on-shell tt pair. Using again the on-shell scheme for defining the wavefunction and mass of the top quark, the respective renormalized vertex functions, which contain both type A and type B contributions, are NS,R = OS NS Γt,µ Z2 Γt,µ , (21) NS,R = OS OS NS Γt Zm Z2 Γt . (22)

2.3. Renormalized Ward identities

Let us now discuss the chiral Ward identities for the various renormalized vertex functions introduced in Section 2.2. First to those of the axial vector and pseudoscalar currents. For the renormalization scheme speciﬁed above one derives from Eq. (6) that the type A contributions to the on-shell second-order renormalized vertex functions (8), (9) must satisfy the Ward identity µ (A),R = (A),R p ΓQ,µ 2mQΓQ . (23)

(A),R 2 In [2] we computed the vertex function ΓQ,µ to order αs using an anticommuting γ5.We (A),R have checked by explicit computation of the pseudoscalar vertex function ΓQ ,alsoto second order in the QCD coupling, that our results given in [2] satisfy Eq. (23). For the second-order renormalized type B contributions the anomalous Ward identity α pµΛ(B),R = 2m Λ(B),R − i s T F R (24) Q,µ Q Q 4π R Q must hold. This equation follows from Eqs. (14) and (17). With Eq. (24) we can immediately write down the Ward identity for the triangle diagram ¯ S contributions to the QQ vertex function of the ﬂavor-singlet current J5,µ. It reads

N f α pµ Λ(B),R(s, m ,m ) = 2m Λ(B),R(s, m ,m ) − i s N T F R(s, m ), Q,µ j Q Q Q Q Q 4π f R Q Q j=1 (25) where mj denotes the mass of the quark in the triangle loop. All triangle diagrams Fig. 1(a), (b) contribute to the left side of (25), while for the pseudoscalar vertex the Dirac trace over a triangle is non-zero only for the massive quark. R R Next we consider the vertex functions Λq,µ and Λq which correspond to the matrix elements of J5µ,R and J5,R between the vacuum and an on-shell qq¯ pair with q = Q.To second-order in the QCD coupling the only contributions that are non-zero are those of ¯ R = 2 type B. For a massless on-shell qq pair we have Λq 0 to order αs due to the chiral invariance of the quark–gluon vertices. The same statement holds for the vertex function 98 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116

R ˜ ¯ Fq of (GG)R. Thus for external massless on-shell qq quarks the corresponding second- order renormalized axial vector vertex function must fulfill µ R = p Λq,µ 0. (26) The renormalization prescription is that of Eqs. (8) and (11). µ NS = Finally to the non-singlet vertex functions. Because the operator equation ∂ J5µ ¯ 2m0,t tγ5t does not get renormalized within the pragmatic approach, the renormalized on- shell vertex functions (21) and (22) must satisfy the canonical chiral Ward identity µ NS,R = NS,R p Γt,µ 2mt Γt . (27) The type A and type B contributions to the vertex functions fulfill Eq. (27) separately. NS NS The type B contributions to Γt,µ and Γt are shown in Fig. 1(a), (b). Because we take massless u, d, c, s, and b quarks, only the t, b quark pair survives the cancellation of the NS = = weak isodoublets in the triangle loops. In fact Γt Γt because we have put mb 0. If one takes a massless on-shell qq¯ pair instead of tt¯, the corresponding non-singlet axial vector vertex function must vanish upon contraction with pµ: µ NS,R = p Γq,µ 0. (28) In the next section we show by explicit computation that Eqs. (24)–(28) are satisfied by the respective type B contributions.

3. Results for the triangle diagrams

We compute now the contributions Fig. 1(a), (b) to the axial vector and pseudoscalar 2 vertex functions to order αs and the one-loop anomalous vertex function Fq . The on-shell vertex functions are sandwiched between Dirac spinors ¯ (B) u(p1)Λq,µv(p2), (29) ¯ (B) u(p1)Λq v(p2), (30)

u(p¯ 1)Fq v(p2), (31) where color indices are suppressed.

3.1. Calculation

Due to parity and CP invariance of QCD the following decompositions hold in four dimensions:

(B) = + 1 Λq,µ γµγ5G1(s, mi,mq ) pµγ5G2(s, mi,mq ), (32) 2mq (B) = Λq A(s, mi,mq )γ5, (33)

Fq = 2imq f(s,mq )γ5. (34) W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 99

The form factors G1, G2, A, and f are dimensionless. As before the mass of the external quark is denoted by mq , while mi is the mass of the quark circulating in the triangle loop. Because of helicity conservation the form factors G2 = A = Fq = 0 for massless external quarks. Moreover, A = 0 when the quarks in the triangle of Fig. 1(b) are massless. Thus A = 0 only if both the internal and external quarks are massive. We compute the renormalized type B two-loop vertex functions Fig. 1(a), (b) for the mass combinations (mi,mq ) = (m, m), (0,m), (m, 0), and (0, 0). These cases, which reduce the calculations to a single scale problem, are adequate for a number of applications, for instance for determining the triangle diagram contributions to Z → qq(q¯ = t) and to ¯ the tt leptoproduction amplitude. In view of mZ mb and of the large t − b mass splitting it is an excellent numerical approximation to put in these triangle contributions the masses of all internal and external quarks but the top quark to zero. Furthermore, we determine f(s,m). Let us first consider the axial vector vertex. Its (B),R renormalization is specified in Eq. (11). The functions Λq,µ are also infrared (IR) finite for D → 4 for any of the above mass combinations. Thus the renormalized form factors Gj,R are obtained from these functions by appropriate projections, which could be carried out, in principle, in four dimensions using (32). We consider the spinor traces [2] (B) P1,R = Tr γµγ5(/p1 + mq ) Λ + (δJ + δ5)γµγ5 (/p2 − mq ) , (35) q,µ P = + (B) + + − 2,R Tr γ5pµ(/p1 mq ) Λq,µ (δJ δ5)γµγ5 (/p2 mq ) . (36) (B),R Rather than calculating Λq,µ first and then performing in (35), (36) the index contractions and traces in four dimensions, it is of course technically much simpler to compute first the individual, UV-divergent pieces appearing on the right-hand sides of (35) and (36) in D dimensions. Therefore γµγ5 and γ5 appearing in the above traces have to be treated according to (4), (5),eveniftwoγ5 appear in the same Dirac trace. Because P1,R, P2,R are finite quantities, the relations between them and the form factors can be evaluated in four dimensions. We obtain sP ,R + 2mq P ,R G (s, m ,m ) = 1 2 , (37) 1,R i q 2 − 4s(4mq s) and, if mq = 0, P + 2 − P mq s 1,R (6mq s) 2,R G (s, m ,m ) =−m . (38) 2,R i q q 2 2 − s (4mq s) The form factor associated with the triangle diagram contributions to the pseudoscalar vertex function is both UV- and IR-finite and is given by 1 A (s,m,m)=− Tr γ (/p + m)Λ(B)(/p − m) . (39) R 2s 5 1 Q 2 Eq. (4) is to be used in the computation of both the triangle loop Fig. 1(a) and of the trace (39). The anomalous vertex function Fq is non-zero only for a massive quark. The renormalized form factor, which is IR-finite, is given by i f (s, m) = Tr γ (/p + m) F + iZ pµγ γ (/p − m) . (40) R 4ms 5 1 Q GJ µ 5 2 100 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116

The above form factors are computed with the technique that was applied in [1,2] to the two-loop vector and type A axial vector form factors. (Cf. these papers for detailed list of references.) As in [1,2] we use the Feynman gauge. The results below and in [1,2] were obtained by three independent calculations within our group. Performing the D-dimensional γ algebra using [17] we obtain the form factors expressed in terms of a set of different scalar integrals. These integrals are reduced [18] to a small set of master integrals. These integrals were evaluated with the method of differential equations [19–22] in [23–25].(Part of the results of [23–25] were conﬁrmed by [26].) The master integrals and thus the above form factors are expressed in terms of 1-dimensional harmonic polylogarithms H(a; x) up to weight four [27,28] which are functions of the dimensionless variable x deﬁned by Eq. (41). Similar functions were discussed before in [29,30].

3.2. Results for spacelike s<0

We give our results ﬁrst in the kinematical region in which s is spacelike. Here the form factors are real. The form factors involving an internal and/or external mass m are also expressed in terms of the variable √ √ −s + 4m2 − −s x = √ √ , (41) −s + 4m2 + −s with 0

− 16ζ(2)2x2 − 80ζ(3)x2 + 90x2 − 40ζ(2)x2 − 50ζ(2)x + 80ζ(3)x − 45x − 45 + 10ζ(2) / (x − 1)2(x + 1)3 , (43) where µ is the mass scale associated with the MS renormalization of J5,µ and ζ(n)denotes the Riemann zeta function. The chirality-ﬂipping form factors do not get renormalized. For the completely massive case we ﬁnd G (s,m,m)= 32 2x3 − x2 + 8x + 3 x3H(0, 0, 0; x)/ (x + 1)3(x − 1)5 2 − 256x3H(0, 0, −1, 0; x)/ (x − 1)3(x + 1)3 + 64x2 H(0, 1, 0; x) − 2H(1, 0, 0; x) − 2H(0, −1, 0; x) / (x − 1)2(x + 1)2 − 8x2 H(0, 0, 0, 0; x) + 6H(0, 0, 0, 1; x) / (x − 1)3(x + 1) + 32x2 H(1, 0, 0, 0; x) − H(0, 0, 1, 0; x) / (x − 1)(x + 1)3 + 32 x2 − 6x + 1 x2H(0, 1, 0, 0; x)/ (x − 1)3(x + 1)3 + 4 3x6 + 8ζ(2)x5 − 12ζ(3)x5 − 2x5 + 8ζ(2)x4 − 3x4 + 32ζ(3)x4 − 40ζ(3)x3 + 4x3 + 64ζ(2)x3 − 3x2 + 32ζ(3)x2 + 8ζ(2)x2 − 12ζ(3)x + 8ζ(2)x − 2x + 3 xH(0; x) / (x + 1)3(x − 1)5 + 4 13x5 − 13x4 − 4ζ(2)x4 − 30x3 − 20ζ(2)x3 + 20ζ(2)x2 − 2x2 + 4ζ(2)x + x − 1 xH(0, 0; x)/ (x − 1)4(x + 1)3 + 16 x2 + 2ζ(2)x + 2x + 1 xH(1, 0; x)/ (x − 1)(x + 1)3 + 8x H(0, 1; x) − 4H(−1, 0; x) / (x − 1)(x + 1) 8 + 15ζ(2)x5 − 5x5 − 8ζ(2)2x4 − 55ζ(2)x4 + 40ζ(3)x4 5 − 5x4 + 10x3 − 100ζ(2)x3 − 40ζ(3)x3 − 60ζ(2)x2 + 10x2 − 40ζ(3)x2 + 8ζ(2)2x − 5x − 35ζ(2)x + 40ζ(3)x − 5ζ(2) − 5 x/ (x − 1)4(x + 1)3 . (44) For the case of massless internal quarks we obtain: G (s, 0,m)= 3ln m2/µ2 + 8x2H(0, 1, 0; x)/ (x + 1)3(x − 1) 1 + 4x2H(0, 0, 0; x)/ (x + 1)3(x − 1) + 8x2H(0, 0, 1; x)/ (x + 1)3(x − 1) + 16x2H(0, 1, 1; x)/ (x + 1)3(x − 1) − 2 x2 + 3x − 2 xH(0, 0; x)/(x + 1)3 − 4 x2 + 3x − 2 xH(0, 1; x)/(x + 1)3 − 2 3x2 − 2x + 3 H(1; x)/(x + 1)2 102 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 − 2 x2 + 1 H(1, 0; x)/(x + 1)2 + −3x4 + 2x3 + 8ζ(2)x2 − 2x + 3 H(0; x)/ (x + 1)3(x − 1) − 4 x2 + 1 H(1, 1; x)/(x + 1)2 − 9x4 + 6ζ(2)x4 + 18x3 + 16ζ(2)x3 + 8ζ(3)x2 − 40ζ(2)x2 − 18x + 16ζ(2)x − 9 + 2ζ(2) (x + 1)3(x − 1) , (45) and G (s, 0,m)= 32x3H(0, 1, 0; x)/ (x − 1)3(x + 1)3 2 + 16x3H(0, 0, 0; x)/ (x − 1)3(x + 1)3 + 32x3H(0, 0, 1; x)/ (x − 1)3(x + 1)3 + 64x3H(0, 1, 1; x)/ (x − 1)3(x + 1)3 − 4 x3 + 5x2 − 3x + 1 xH(0, 0; x)/ (x + 1)3(x − 1)2 − 8 3x2 − 2x + 3 xH(1; x)/ (x + 1)2(x − 1)2 − 8 x3 + 5x2 − 3x + 1 xH(0, 1; x)/ (x + 1)3(x − 1)2 + 4 −3x4 + 2x3 + 8ζ(2)x2 − 2x + 3 xH(0; x) / (x − 1)3(x + 1)3 − 8 x2 + 1 xH(1, 0; x)/ (x + 1)2(x − 1)2 − 16 x2 + 1 xH(1, 1; x)/ (x + 1)2(x − 1)2 − 8 x4 + ζ(2)x4 + 2x3 + 8ζ(2)x3 − 16ζ(2)x2 + 4ζ(3)x2 + 8ζ(2)x − 2x − ζ(2) − 1 x/ (x − 1)3(x + 1)3 . (46) For the case of massless external quarks we get: 2 2 2 4 G1(s, m, 0) = 3ln −s/µ + 2 5x − 8x + 7 x H(0, 0; x)/(x − 1) − 8x2H(0, 1, 0; x)/(x − 1)4 − 4x2H(0, 0, 0; x)/(x − 1)4 − 2 −3x3 + 4ζ(2)x2 + 9x2 − 9x − 4ζ(2)x + 3 + 4ζ(2) xH(0; x)/(x − 1)4 − 16xH(1, 0, 0; x)/(x − 1)2 − 16xH(0, −1, 0; x)/(x − 1)2 + 2(x + 1) x2 + 1 H(1, 0; x)/(x − 1)3 + 6H(1; x) − 8(x + 1)H (−1, 0; x)/(x − 1) − 2ζ(2)x4 + 9x4 + 8ζ(3)x3 − 8ζ(2)x3 − 36x3 + 54x2 − 36x + 8ζ(2)x + 8ζ(3)x + 9 − 2ζ(2) /(x − 1)4, (47) and 2 G1(s, 0, 0) = 3ln −s/µ − 9 + 2ζ(2). (48) W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 103

Moreover, we have

G2(s, m, 0) = G2(s, 0, 0) = 0. (49) As already mentioned above the form factor associated with the pseudoscalar vertex function is non-zero only for the completely massive case. It is given by α 2 A (s,m,m)= s C T −16x2H(0, 0, 0; x)/ (x − 1)3(x + 1) R 2π F R + 4 x + ζ(2)x − ζ(2) + 1 xH(0, 0; x)/ (x + 1)(x − 1)2 + 4 3ζ(3)x2 − 4ζ(2)x − 6ζ(3)x + 3ζ(3) xH(0; x) / (x − 1)3(x + 1) + 12xH(0, 0, 0, 1; x)/ (x + 1)(x − 1) − 8ζ(2)xH (1, 0; x)/ (x + 1)(x − 1) + 2xH(0, 0, 0, 0; x)/ (x + 1)(x − 1) + 8xH(0, 0, 1, 0; x)/ (x + 1)(x − 1) − 8xH(0, 1, 0, 0; x)/ (x + 1)(x − 1) − 8xH(1, 0, 0, 0; x)/ (x + 1)(x − 1) + 8/5 2ζ(2)x + 5x − 2ζ(2) + 5 ζ(2)x/ (x + 1)(x − 1)2 . (50) Finally the form factor associated with the anomalous vertex function Fig. 2 is α f (s, m) = s 2C 3ln m2/µ2 − (x − 1)H (0, 0; x)/(x + 1) R 2π F − 2(x − 1)H (0, 1; x)/(x + 1) − 7x + 4ζ(2)x + 7 − 4ζ(2) /(x + 1) . (51)

3.3. Analytical continuation above threshold

We perform the analytical continuation of the form factors to the physical region above threshold, s>4m2, −1 0 for massless external quarks, with the usual prescription s → s + iδ. For the cases involving masses the variable x becomes a phase factor if 0 4m2 we deﬁne √ √ s − s − 4m2 y = √ √ , (52) s + s − 4m2 and the continuation in x is performed by the replacement

x →−y + iδ. (53) The real and imaginary parts of the form factors are deﬁned through the relations

Gj,R(s + iδ) = Re Gj,R(s) + iπ Im Gj,R(s), (54) 104 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 and likewise for AR and fR, where also a factor π is taken out of the respective imaginary part. In the completely massive case we get for s>4m2: Re G (s,m,m)=−8 2y3 + 3y2 + 4y − 1 y2H(0, 0, 0; y)/ (y − 1)3(y + 1)3 1 + 32y2H(−1, 0, 0, 0; y)/ (y + 1)(y − 1)3 + 64y2H(0, −1, 0, 0; y)/ (y + 1)(y − 1)3 − 192ζ(2)y2H(0, −1; y)/ (y + 1)(y − 1)3 − 32y2H(0, 0, −1, 0; y)/ (y + 1)(y − 1)3 + 64y2H(0, 0, 1, 0; y)/ (y + 1)(y − 1)3 − 32yH(0, 1, 0; y)/(y − 1)2 + 16yH(0, −1, 0; y)/(y − 1)2 + 4 3y3 − y2 − 3y − 4ζ(2)y + 1 yH(0, 0; y) / (y + 1)(y − 1)3 − 32yH(−1, 0, 0; y)/(y − 1)2 + 96ζ(2)yH (−1; y)/(y − 1)2 + 3y6 + 2y5 + 40ζ(2)y5 + 32ζ(3)y4 + 64ζ(2)y4 − 3y4 + 64ζ(2)y3 + 64ζ(3)y3 − 4y3 + 32ζ(3)y2 − 32ζ(2)y2 − 3y2 + 2y − 8ζ(2)y + 3 H(0; y)/ (y − 1)3(y + 1)3 + 8(y + 1)H (1, 0; y)/(y − 1) − 4 y4 − 2y2 + 16ζ(2)y2 + 1 H(−1, 0; y)/ (y + 1)(y − 1)3 − 1/5 45y5 + 170ζ(2)y5 + 80ζ(3)y4 + 70ζ(2)y4 − 45y4 − 90y3 − 256ζ(2)2y3 − 200ζ(2)y3 + 80ζ(3)y3 − 160ζ(2)y2 − 256ζ(2)2y2 + 90y2 − 80ζ(3)y2 − 80ζ(3)y + 45y + 110ζ(2)y − 45 + 10ζ(2) / (y + 1)2(y − 1)3 + 3ln m2/µ2 , (55) and Im G (s,m,m)=−8 2y3 + 3y2 + 4y − 1 y2H(0, 0; y)/ (y − 1)3(y + 1)3 1 + 32y2H(−1, 0, 0; y)/ (y + 1)(y − 1)3 + 64y2H(0, −1, 0; y)/ (y + 1)(y − 1)3 − 32y2H(0, 0, −1; y)/ (y + 1)(y − 1)3 + 64y2H(0, 0, 1; y)/ (y + 1)(y − 1)3 + 4 3y3 − y2 − 4ζ(2)y − 3y + 1 yH(0; y)/ (y + 1)(y − 1)3 − 32yH(−1, 0; y)/(y − 1)2 + 16yH(0, −1; y)/(y − 1)2 − 32yH(0, 1; y)/(y − 1)2 − 4(y + 1)H (−1; y)/(y − 1) + 8(y + 1)H (1; y)/(y − 1) W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 105 + 3y4 − 4y3 + 8ζ(2)y3 + 2y2 + 32ζ(3)y2 − 4y − 8ζ(2)y + 3 / (y + 1)(y − 1)3 . (56) For the chirality-ﬂipping form factor we have Re G (s,m,m)= 32 2y3 + y2 + 8y − 3 y3H(0, 0, 0; y)/ (y − 1)3(y + 1)5 2 − 256y3H(0, 0, 1, 0; y)/ (y + 1)3(y − 1)3 − 384ζ(2)y2H(−1; y)/ (y + 1)2(y − 1)2 − 8y2H(0, 0, 0, 0; y)/ (y + 1)3(y − 1) − 32 y2 + 6y + 1 y2H(0, −1, 0, 0; y)/ (y + 1)3(y − 1)3 + 128y2H(−1, 0, 0; y)/ (y + 1)2(y − 1)2 − 32y2H(−1, 0, 0, 0; y)/ (y + 1)(y − 1)3 − 64y2H(0, −1, 0; y)/ (y + 1)2(y − 1)2 + 128y2H(0, 1, 0; y)/ (y + 1)2(y − 1)2 + 32y2H(0, 0, −1, 0; y)/ (y + 1)(y − 1)3 + 48y2H(0, 0, 0, −1; y)/ (y + 1)3(y − 1) − 4 13y5 − 2ζ(2)y4 + 13y4 − 30y3 − 14ζ(2)y3 − 14ζ(2)y2 + 2y2 − 2ζ(2)y + y + 1 yH(0, 0; y)/ (y + 1)4(y − 1)3 + 8 y4 + 12ζ(2)y3 − 2y2 + 72ζ(2)y2 + 12ζ(2)y + 1 × yH(0, −1; y)/ (y + 1)3(y − 1)3 + 16 y2 + 4ζ(2)y − 2y + 1 yH(−1, 0; y)/ (y + 1)(y − 1)3 − 4 3y6 + 12ζ(3)y5 + 40ζ(2)y5 + 2y5 + 32ζ(2)y4 + 32ζ(3)y4 − 3y4 + 40ζ(3)y3 − 4y3 + 128ζ(2)y3 + 32ζ(3)y2 − 3y2 − 64ζ(2)y2 + 2y + 12ζ(3)y − 8ζ(2)y + 3 yH(0; y)/ (y − 1)3(y + 1)5 − 32yH(1, 0; y)/ (y + 1)(y − 1) + 4/5 10y4 + 165ζ(2)y4 + 80ζ(3)y3 + 29ζ(2)2y3 − 80ζ(2)y3 − 20y3 − 314ζ(2)2y2 − 170ζ(2)y2 + 29ζ(2)2y + 80ζ(2)y + 20y − 80ζ(3)y − 10 + 5ζ(2) y/ (y + 1)3(y − 1)3 , (57) and Im G (s,m,m)=−256y3H(0, 0, 1; y)/ (y + 1)3(y − 1)3 2 + 32 2y3 + y2 + 8y − 3 y3H(0, 0; y)/ (y − 1)3(y + 1)5 + 128y2H(0, 1; y)/ (y + 1)2(y − 1)2 106 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 + 32y2H(0, 0, −1; y)/ (y + 1)(y − 1)3 − 64y2H(0, −1; y)/ (y + 1)2(y − 1)2 − 32 y2 + 6y + 1 y2H(0, −1, 0; y)/ (y + 1)3(y − 1)3 + 128y2H(−1, 0; y)/ (y + 1)2(y − 1)2 − 8y2H(0, 0, 0; y)/ (y + 1)3(y − 1) − 32y2H(−1, 0, 0; y)/ (y + 1)(y − 1)3 − 32yH(1; y)/ (y + 1)(y − 1) + 16yH(−1; y)/ (y + 1)(y − 1) − 4 13y5 + 13y4 + 2ζ(2)y4 − 30y3 − 18ζ(2)y3 + 2y2 − 18ζ(2)y2 + y + 2ζ(2)y + 1 yH(0; y)/ (y + 1)4(y − 1)3 − 4 3y4 + 12ζ(3)y3 − 4y3 + 8ζ(2)y3 + 2y2 + 8ζ(3)y2 − 4y + 12ζ(3)y − 8ζ(2)y + 3 y/ (y + 1)3(y − 1)3 . (58) We obtain for the non-zero pseudoscalar and anomaly form factors: 2 αs CF TRy Re AR(s,m,m)= 2π (1 − y)(1 + y)3

× 4 3ζ(3) + 3ζ(3)y2 + 6yζ(3) − 8yζ(2) H(0; y) − 2(1 + y) 2 − 2y + yζ(2) + ζ(2) H(0, 0; y) + 16yH(0, 0, 0; y) − 12(1 + y)2H(0, 0, 0, −1; y) + 2(1 + y)2H(0, 0, 0, 0; y) − 8(1 + y)2H(0, 0, −1, 0; y) − 24ζ(2)(1 + y)2H(0, −1; y) + 8(1 + y)2H(0, −1, 0, 0; y) − 16ζ(2)(1 + y)2H(−1, 0; y) + 8(1 + y)2H(−1, 0, 0, 0; y)

58 29 29 − yζ2(2) − 4y2ζ(2) − ζ 2(2) + 4ζ(2) − ζ 2(2)y2 , 5 5 5 (59) 2 αs CF TRy Im AR(s,m,m)= 2π (1 − y)(1 + y)3 × 2(1 + y) yζ(2) + 2y − 2 + ζ(2) H(0; y) + 16yH(0, 0; y) + 2(1 + y)2H(0, 0, 0; y) − 8(1 + y)2H(0, 0, −1; y) + 8(1 + y)2H(0, −1, 0; y) + 8(1 + y)2H(−1, 0, 0; y) + 24yζ(3) + 12ζ(3)y2 + 12ζ(3) , (60) W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 107 α 2C Re f (s, m) = s F (1 + y)H(0, 0; y) − 2(1 + y)H(0, −1; y) R 2π 1 − y + ζ(2)(1 + y) + 3(1 − y)ln m2/µ2 − 7(1 − y) , (61) α 2C Im f (s, m) = s F (1 + y)H(0; y). (62) R 2π 1 − y In the case of massless internal quarks we get for s>4m2: Re G (s, 0,m)=−8y2H(0, 0, −1; y)/ (y − 1)3(y + 1) 1 + 4y2H(0, 0, 0; y)/ (y − 1)3(y + 1) − 8y2H(0, −1, 0; y)/ (y − 1)3(y + 1) + 16y2H(0, −1, −1; y)/ (y − 1)3(y + 1) + 4 y2 − 3y − 2 yH(0, −1; y)/(y − 1)3 − 2 y2 − 3y − 2 yH(0, 0; y)/(y − 1)3 − 3y4 + 2y3 + 4ζ(2)y2 − 2y − 3 H(0; y)/ (y − 1)3(y + 1) + 2 3y2 + 2y + 3 H(−1; y)/(y − 1)2 − 4 y2 + 1 H(−1, −1; y)/(y − 1)2 + 2 y2 + 1 H(−1, 0; y)/(y − 1)2 − 9y4 − 18y3 − 4ζ(2)y3 − 10ζ(2)y2 + 8ζ(3)y2 + 18y − 4ζ(2)y + 2ζ(2) − 9 / (y − 1)3(y + 1) + 3ln m2/µ2 , (63) Im G (s, 0,m)=−8y2H(0, −1; y)/ (y − 1)3(y + 1) 1 + 4y2H(0, 0; y)/ (y − 1)3(y + 1) − 2 y2 − 3y − 2 yH(0; y)/(y − 1)3 + 2 y2 + 1 H(−1; y)/(y − 1)2 − 3y4 + 2y3 − 4ζ(2)y2 − 2y − 3 / (y − 1)3(y + 1) , (64) Re G (s, 0,m)= 32y3H(0, −1, 0; y)/ (y + 1)3(y − 1)3 2 + 32y3H(0, 0, −1; y)/ (y + 1)3(y − 1)3 − 64y3H(0, −1, −1; y)/ (y + 1)3(y − 1)3 − 16y3H(0, 0, 0; y)/ (y + 1)3(y − 1)3 + 4 3y4 + 2y3 + 4ζ(2)y2 − 2y − 3 yH(0; y) / (y + 1)3(y − 1)3 − 8 3y2 + 2y + 3 yH(−1; y)/ (y − 1)2(y + 1)2 − 8 y3 − 5y2 − 3y − 1 yH(0, −1; y)/ (y − 1)3(y + 1)2 108 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 − 8 y2 + 1 yH(−1, 0; y)/ (y − 1)2(y + 1)2 + 4 y3 − 5y2 − 3y − 1 yH(0, 0; y)/ (y − 1)3(y + 1)2 + 16 y2 + 1 yH(−1, −1; y)/ (y − 1)2(y + 1)2 − 4 ζ(2)y4 − 2y4 + 4y3 + 4ζ(2)y3 + 8ζ(2)y2 − 8ζ(3)y2 − 4y + 4ζ(2)y + 2 − ζ(2) y/ (y + 1)3(y − 1)3 , (65) and Im G (s, 0,m)= 32y3H(0, −1; y)/ (y + 1)3(y − 1)3 2 − 16y3H(0, 0; y)/ (y + 1)3(y − 1)3 + 4 y3 − 5y2 − 3y − 1 yH(0; y)/ (y − 1)3(y + 1)2 − 8 y2 + 1 yH(−1; y)/ (y − 1)2(y + 1)2 + 4 3y4 + 2y3 − 4ζ(2)y2 − 2y − 3 y/ (y + 1)3(y − 1)3 . (66) Now to the case of massive internal and massless external quarks. An on-shell two-gluon intermediate state in Fig. 1(a) could lead for s>0 to an imaginary part of G2 only (for an off-shell Z boson), as a consequence of the Landau–Yang theorem [31]. But this form factor is zero for massless on-shell q,q¯. The form factor G1 has an absorptive part only if 2 2 s>4m . In the region 0 4m2 we ﬁnd 2 2 2 4 Re G1(s, m, 0) = 3ln s/µ + 2 5y + 8y + 7 y H(0, 0; y)/(y + 1) − 4y2H(0, 0, 0; y)/(y + 1)4 + 8y2H(0, −1, 0; y)/(y + 1)4 + 2 3y3 + 9y2 + 4ζ(2)y2 + 10ζ(2)y + 9y + 4ζ(2) + 3 × yH(0; y)/(y + 1)4 − 16yH(−1, 0, 0; y)/(y + 1)2 − 16yH(0, 1, 0; y)/(y + 1)2 − 6 y2 − 8ζ(2)y + 2y + 1 H(−1; y)/(y + 1)2 + 8(y − 1)H (1, 0; y)/(y + 1) − 2(y − 1) y2 + 1 H(−1, 0; y)/(y + 1)3 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 109 − 9y4 + 32ζ(2)y4 + 36y3 + 56ζ(2)y3 − 8ζ(3)y3 + 42ζ(2)y2 + 54y2 + 36y − 8ζ(2)y − 8ζ(3)y + 9 − 2ζ(2) /(y + 1)4, (68) and Im G (s, m, 0) = 8y2H(0, −1; y)/(y + 1)4 − 4y2H(0, 0; y)/(y + 1)4 1 + 2 5y2 + 8y + 7 y2H(0; y)/(y + 1)4 − 16yH(0, 1; y)/(y + 1)2 − 16yH(−1, 0; y)/(y + 1)2 + 8(y − 1)H (1; y)/(y + 1) − 2(y − 1) y2 + 1 H(−1; y)/(y + 1)3 + 2 3y3 + 9y2 + 4ζ(2)y2 + 9y + 6ζ(2)y + 3 + 4ζ(2) y/(y + 1)4 − 3. (69) For the completely massless case we have for s>0: 2 Re G1(s, 0, 0) = 3ln s/µ − 9 + 2ζ(2), (70) and

Im G1(s, 0, 0) =−3. (71)

3.4. Check of the chiral Ward identities

The anomalous Ward identity (24) reads in terms of the form factors

s αs G ,R(s,m,m)+ G ,R(s,m,m)= AR(s,m,m)+ TRfR(s, m), (72) 1 4m2 2 4π and Eq. (25) yields

s αs G ,R(s, 0,m)+ G ,R(s, 0,m)= TRfR(s, m). (73) 1 4m2 2 4π The results of Section 3.3 satisfy this equation, both for spacelike and timelike s.Eqs.(26) and (28) are trivially satisﬁed because G1,Ru(p¯ 1)/pγ5v(p2) = 0 for massless external quarks. For the type B contributions the non-singlet Ward identity (27) has the form s G ,R + G ,R = AR(s,m,m), (74) 1 4m2 2 where

Gj,R = Gj,R(s,m,m)− Gj,R(s, 0,m)= Gj (s,m,m)− Gj (s, 0,m). (75) As Eqs. (72) and (73) are satisfied by our form factors, Eq. (74) is of course fulfilled, too. A specific combination of the above vertex functions, namely G1(s, m, 0)− G1(s, 0, 0), which is relevant for Z → massless bb¯, was computed in [12] using Cutkosky rules and dispersion relations. Applying the notation of that paper, one obtains the relation 110 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116

G1(s, m, 0) − G1(s, 0, 0) =−I2 where the function I2 is given in [12] for s<0 and 0

4. Threshold and asymptotic expansions

Next we expand the axial vector and pseudoscalar form factors for a massive external ¯ ∼ 2 = − 4m2 QQ pair near threshold (s 4m ), in powers of β 1 s . This is useful for applications to, e.g., e+e− → tt¯ near threshold. Up to and including terms of order (β)2—respectively of order (β)3 if the leading term vanishes—the real and imaginary parts are: 19 23 2 2 Re G1(s,m,m)=− ζ(2) − + 16ζ(2) ln(2) + 3ln m /µ 3 3 11 64 8 + β2 ζ(2) − ζ(2) ln(2) + , 15 5 3 32 16 44 Im G (s,m,m)= β3 ln(β) + ln(2) − , 1 9 3 9 21 34 2 Re G2(s,m,m)=− ζ(3) − 4ζ(2) ln(2) + ζ(2) + + 4ln(2) − 12ζ(2)β 2 3 3 226 35 4 44 + β2 −6 − ζ(2) + ζ(3) + ln(2) + ζ(2) ln(2) , 15 2 3 5 2 Im G (s,m,m)=−2 + 3ζ(2) + 4β 1 − ln(2) + β2 − − 5ζ(2) , 2 3 4 8 2 23 2 2 Re G1(s, 0,m)= ζ(2) − ln (2) + 8ln(2) − + 3ln m /µ 3 3 3 8 38 4 + β2 ln2(2) + − ζ(2) , 15 15 15 8 8 Im G (s, 0,m)=−4 + ln(2) − β2 ln(2), 1 3 15 4 8 2 2 Re G2(s, 0,m)=− ζ(2) + ln (2) − 4ln(2) + 3 3 3 38 16 16 8 + β2 − − ln2(2) + ln(2) + ζ(2) , 15 5 3 5 8 8 16 Im G (s, 0,m)= 2 − ln(2) + β2 − + ln(2) . (76) 2 3 3 5 2 The form factor G1(s, m, 0) is given, for r = s/m → 0, approximately by 3 10 329 G (s, m, 0) = 3ln m2/µ2 + − r − r2, (77) 1 2 27 10800 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 111 while for s ∼ 4m2 its imaginary part has the small β expansion

16 64 Im G (s, m, 0) = Θ s − 4m2 β3 ln(β) + 8ln(2) − . (78) 1 3 9 For brevity we give for the pseudoscalar form factor only the terms of order (β)0: α 2 21 Re A (s,m,m)= s C T 5ζ(2) − ζ(3) + 12ζ(2) ln(2) , R 2π F R 2 α 2 Im A (s,m,m)= s 3ζ(2)C T . (79) R 2π F R Finally we study the behaviour of the form factors involving the massive quark for large squared momentum transfer s m2. We obtain, up to and including terms of order r−2, with L = ln(r): 2 Re G1(s,m,m)= 3ln s/µ − 9 + 2ζ(2) 2 + −L2 + 3 − 4ζ(2) L − 1 + 12ζ(2) − 8ζ(3) r 1 + −2 + 24L + 140ζ(2) + 32Lζ(3) − 32Lζ(2) − 18L2 r2

4L3 256 + − ζ 2(2) + 8ζ(2)L2 + 8(−2L + 2)ζ(2) − 64ζ(3) , 3 5 2 Im G (s,m,m)=−3 + 2L − 3 + 4ζ(2) 1 r 1 + −24 + 36L − 4L2 − 32ζ(3) + 32ζ(2) − 16Lζ(2) , r2 2 2 Re G2(s,m,m)= L − 6L + 4 − 2ζ(2) r 1 + −32L + 4L2 − 72ζ(2) − 48Lζ(3) + 64ζ(3) r2

116 1 − ζ 2(2) + L4 − 4ζ(2)L2 + 32Lζ(2) , 5 3

4 1 4 Im G (s,m,m)= (3 − L) + 32 − 8L + 48ζ(3) − 8Lζ(2) − L3 − 32ζ(2) , 2 r r2 3 2 2 2 Re G1(s, 0,m)= 3ln s/µ − 9 + 2ζ(2) + −L + 3L − 1 r 1 5 2 + + 33L − 10ζ(2) + L3 + 8ζ(3) − 4Lζ(2) − 13L2 , r2 2 3 2 1 Im G (s, 0,m)=−3 + (2L − 3) + −33 + 26L − 2L2 − 4ζ(2) , 1 r r2 2 1 Re G (s, 0,m)= L2 − 6L + 4 − 2ζ(2) + −32L + 8ζ(2) + 12L2 , 2 r r2 112 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116

4 1 Im G (s, 0,m)= (3 − L) + (32 − 24L), 2 r r2 2 8 Re G1(s, m, 0) =−9 + 3ln s/µ + 2ζ(2) + ζ(3) − ζ(2)L r 1 27 2 + − 6ζ(2) + L3 + 3L − L2 − 16ζ(3) r2 2 3

+ 12Lζ(2) + 8(2 − 2L)ζ(2) ,

8ζ(2) 1 Im G (s, m, 0) =−3 + + −3 + 2L − 4ζ(2) − 2L2 . (80) 1 r r2 The leading asymptotic terms of the pseudoscalar form factor are α 2 1 1 29 Re A (s,m,m)= s C T L4 + 4ζ(2) − ζ 2(2) R 2π F R r 12 5

− 2L2 − ζ(2)L2 − 12Lζ(3) , α 2 1 1 Im A (s,m,m)= s C T 4L + 12ζ(3) − 2Lζ(2) − L3 . (81) R 2π F R r 3

Fig. 3. The real and imaginary parts of CF TRG1,2(s,m,m) as functions of the rescaled c.m. energy and their threshold and asymptotic expansions. W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 113

Fig. 4. The real and imaginary parts of CF TRG1,2(s, 0,m) as functions of the rescaled c.m. energy and their threshold and asymptotic expansions.

The limit r →∞corresponds to the massless limit m → 0. Therefore the chirality-ﬂipping 0 form factors G2 and AR are, as expected, of order 1/r, and the terms of order r in the above expansion of the chirality-conserving form factors G1 are equal to Re G1(s, 0, 0) and Im G1(s, 0, 0), respectively. G In Figs. 3–6 we have plotted the real and imaginary√ parts of the form factors CF TR j as functions of the rescaled center-of-mass energy s/(2m),forµ = m and Nc = 3. Also shown are the values of the asymptotic expansions of these form factors given in Eq. (80), and of the threshold expansions, Eqs. (77) and (78). As far as the threshold expansions are concerned, these ﬁgures show that both the next-to-leading terms in the expansions and restriction to a relatively small region above threshold are required in order to get a satisfactory approximation to the exact result. An exception is the expansion of Re G1(s, m, 0) around√s ∼ 0 where the terms up to second-order approximate this form factor very well up to s ∼ 1.5m. The asymptotic expansions (80), on the other hand, provide a very good approximation to the form factors over a wide range of center-of-mass energies.

5. Summary

In this paper we calculated the triangle diagram contributions to the form factors of a heavy-quark axial vector current, of the ﬂavor-singlet and non-singlet axial vector cur- 114 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116

Fig. 5. The real and imaginary parts of CF TRG1(s, m, 0) as functions of the rescaled c.m. energy and their threshold and asymptotic expansions.

Fig. 6. The real and imaginary parts of CF TRG1(s, 0, 0) as functions of the rescaled c.m. energy. rents, and of the corresponding pseudoscalar currents to second-order in the QCD coupling, for four different combinations of internal and external masses. Moreover, we have given the threshold and asymptotic expansions of these form factors to a degree of approximation which should prove useful for applications. We compared our exact results with these expansions at threshold and for large energies. For the threshold expansions we found that they approximate the exact results only over a very restricted energy range above threshold, and that at least the second-order terms in the expansion are required for a reliable description. The large energy expansions appear to converge very well, and do in general yield a good description even for energies well below the asymptotic regime. In this calculation we have used the implementation of the γ5 matrix according to [7,8] in dimensional regularization. This method is algebraically consistent and convenient for computations, but breaks chiral invariance in D = 4. To restore chiral invariance in this scheme, finite counterterms are required. These were determined in [8] for massless quarks to three-loop order. In this work, we calculated, for the case of massive quarks, the two-loop finite counterterms required by chiral invariance in this scheme. As expected, all counterterms agree with the massless results of [8], illustrating their purely ultraviolet origin. Using W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116 115 these counterterms, we have shown that the appropriately renormalized vertex functions, which are infrared finite, satisfy the correct chiral Ward identities. However, it is a separate issue to apply the method of [8] to Green functions (with massless and/or massive quarks) that are infrared-divergent—using dimensional regularization as well—and ask whether these Green functions satisfy the correct chiral Ward identities. We shall report on this issue and on applications of our results in this paper and in [1,2] to the electroproduction and, in particular, to the forward–backward asymmetry of heavy quarks in a future publication.

Acknowledgements

This work was supported by Deutsche Forschungsgemeinschaft (DFG), SFB/TR9, by DFG-Graduiertenkolleg RWTH Aachen, and by the Swiss National Science Foundation (SNF) under contract 200021-101874.

References

[1] W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia, E. Remiddi, Nucl. Phys. B 706 (2005) 245, hep-ph/0406046. [2] W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia, E. Remiddi, Nucl. Phys. B 712 (2005) 229, hep-ph/0412259. [3] LEP Collaboration, A combination of preliminary electroweak measurements and constraints on the standard model, hep-ex/0412015. [4] ECFA/DESY LC Physics Working Group Collaboration, J.A. Aguilar-Saavedra, et al., TESLA technical + − design report part III: physics at an e e linear collider, DESY-report 2001-011, hep-ph/0106315. [5] S.L. Adler, Phys. Rev. 177 (1969) 2426. [6] J.S. Bell, R. Jackiw, Nuovo Cimento A 60 (1969) 47. [7] D.A. Akyeampong, R. Delbourgo, Nuovo Cimento A 17 (1973) 578. [8] S.A. Larin, Phys. Lett. B 303 (1993) 113, hep-ph/9302240. [9] G. ’t Hooft, M. Veltman, Nucl. Phys. B 44 (1972) 189. [10] P. Breitenlohner, D. Maison, Commun. Math. Phys. 52 (1977) 39. [11] J.A. Gracey, Phys. Lett. B 488 (2000) 175, hep-ph/0007171. [12] B.A. Kniehl, J.H. Kühn, Nucl. Phys. B 329 (1990) 547. [13] R. Harlander, M. Steinhauser, Prog. Part. Nucl. Phys. 43 (1999) 167, hep-ph/9812357. [14] V.A. Smirnov, Applied Asymptotic Expansions in Momenta and Masses, Springer-Verlag, Heidelberg, 2002. [15] S.L. Adler, W.A. Bardeen, Phys. Rev. 182 (1969) 1517. [16] D. Espriu, R. Tarrach, Z. Phys. C 16 (1982) 77. [17] J.A.M. Vermaseren, Symbolic Manipulation with FORM, Version 2, CAN, Amsterdam, 1991; J.A.M. Vermaseren, New features of FORM, math-ph/0010025. [18] S. Laporta, E. Remiddi, Phys. Lett. B 379 (1996) 283, hep-ph/9602417; S. Laporta, Int. J. Mod. Phys. A 15 (2000) 5087, hep-ph/0102033. [19] T. Gehrmann, E. Remiddi, Nucl. Phys. B 580 (2000) 485, hep-ph/9912329. [20] A.V. Kotikov, Phys. Lett. B 254 (1991) 158. [21] E. Remiddi, Nuovo Cimento A 110 (1997) 1435, hep-th/9711188. [22] M. Caffo, H. Czy˙z, S. Laporta, E. Remiddi, Acta Phys. Pol. B 29 (1998) 2627, hep-th/9807119; M. Caffo, H. Czy˙z, S. Laporta, E. Remiddi, Nuovo Cimento A 111 (1998) 365, hep-th/9805118. [23] R. Bonciani, P. Mastrolia, E. Remiddi, Nucl. Phys. B 661 (2003) 289, hep-ph/0301170; R. Bonciani, P. Mastrolia, E. Remiddi, Nucl. Phys. B 702 (2004) 359, Erratum. [24] R. Bonciani, P. Mastrolia, E. Remiddi, Nucl. Phys. B 676 (2004) 399, hep-ph/0307295. 116 W. Bernreuther et al. / Nuclear Physics B 723 (2005) 91–116

[25] R. Bonciani, P. Mastrolia, E. Remiddi, Nucl. Phys. B 690 (2004) 138, hep-ph/0311145. [26] M. Czakon, J. Gluza, T. Riemann, Nucl. Phys. B (Proc. Suppl.) 135 (2004) 83, hep-ph/0406203; M. Czakon, J. Gluza, T. Riemann, Phys. Rev. D 71 (2005) 073009, hep-ph/0412164. [27] E. Remiddi, J.A.M. Vermaseren, Int. J. Mod. Phys. A 15 (2000) 725, hep-ph/9905237. [28] T. Gehrmann, E. Remiddi, Comput. Phys. Commun. 141 (2001) 296, hep-ph/0107173. [29] A.B. Goncharov, Math. Res. Lett. 5 (1998) 497. [30] D.J. Broadhurst, Eur. Phys. J. C 8 (1999) 311, hep-th/9803091. [31] L.D. Landau, Dokl. Akad. Nauk USSR 60 (1948) 207; C.N. Yang, Phys. Rev. 77 (1950) 242. [32] L. Lewin, Polylogarithms and Associated Functions, North-Holland, Amsterdam, 1984. Nuclear Physics B 723 (2005) 117–131

Five-brane calibrations and fuzzy funnels

David S. Berman a, Neil B. Copland b

a Queen Mary College, University of London, Department of Physics, Mile End Road, London E1 4NS, England, UK b Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, England, UK Received 22 April 2005; received in revised form 2 June 2005; accepted 10 June 2005 Available online 5 July 2005

Abstract We present a generalisation of the Basu–Harvey equation that describes membranes ending on intersecting ﬁve-brane conﬁgurations corresponding to various calibrated geometries.  2005 Elsevier B.V. All rights reserved.

1. Introduction

There are many different but physically equivalent descriptions of how a D1 brane may end on a D3 brane. From the point of view of the D3 brane the configuration is described by a monopole on its world volume. From the point of view of the D1 brane the configuration is described by the D1 opening up into a D3 brane where the extra two dimensions form a fuzzy two sphere whose radius diverges at the origin of the three-brane. These different view points are the stringy realisation of the Nahm transformation. The BPS equation obeyed by the D1 brane is Nahm’s equation. These differing perspectives on the D1, D3 system have been explored in a variety of papers [1–6] where the fluctuation properties, the profile and the coupling to RR fields are examined and shown to match where the different approximations schemes are both valid. The M-theory equivalent of this system is that of the membrane ending on the M five- brane. There are well-known problems though of describing the theory of coincident branes

E-mail addresses: [email protected] (D.S. Berman), [email protected] (N.B. Copland).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.06.015 118 D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 in M-theory, both for the membrane and the five-brane. It is believed that matrix valued fields will not be the appropriate degrees of freedom as they are for D-branes since entropy considerations imply a different number of degrees of freedom than one would expect from matrix valued fields. What is known is the five-brane equivalent of the BIon solution. This is the self-dual string solution of Howe, Lambert and West [7] whose properties have recently been investigated in [8]. The absence of a non-Abelian membrane theory however meant that the equivalent of Nahm’s equation for the self-dual string was missing. Recently, Basu and Harvey [9], made an ansatz for such an equation. Their goal being to produce a similar fuzzy funnel description of the membrane opening up into the five-brane. From their generalised Nahm equation they went on to infer (essentially through an inverse Bogomol’nyi argument) a non-Abelian membrane action with sextic interaction. Again, the profile and the fluctuations were shown to agree with the self-dual string description. Given the somewhat ad-hoc way in which the Basu–Harvey equation has been determined it would be good to see if one could find other tests for its validity. The goal of this paper will be to show that the Basu–Harvey equation with a natural generalisation can describe not only the membrane ending on a five-brane but also the membrane ending on various intersecting five-brane configurations. These five-brane configurations will correspond to calibrated cycles. Essentially this is the M-theory generalisation of [10] where the Nahm equation was generalised to describe the D1 ending on the configurations of D3 branes that correspond to three-branes wrapped on calibrated cycles. We proceed by first describing Nahm’s equation and the generalisation that leads to the description of a D1 ending on intersecting D3-brane configurations. As an aside we also demonstrate that the solutions previously derived using the linearised approximation in [10] are actually solutions of the full non-linear theory. This is undoubtedly a consequence of the BPS nature of these solutions. Then we discuss the M-theory generalisation of Nahm’s equation as introduced by Basu and Harvey. Finally we describe a generalisation of the Basu–Harvey equation and its solutions that correspond to membranes ending on five-branes wrapped on calibrated cycles.

2. Nahm type equations

The Nahm equation [11] is given by ∂Φi i =± Φj ,Φk . (1) ∂x9 2 ij k 1 This equation is derived in string theory by simply examining the 2 BPS equation for the D1 brane [12,13]. Its solutions correspond to D1 branes opening up into a fuzzy funnel to form a D3 brane. Note, the boundary conditions are taken such that Φi(x9) is defined over the semi-infinite line as opposed to a finite interval which is the usual case. This corresponds to infinite mass monopoles which have the interpretation of infinite D1 strings ending on the brane. Explicitly, the solutions are: 1 Φi =± αi. (2) 2(σ − σ0) D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 119

Where αi obey the su(2) algebra, [αi,αj ]=2iij k αk. The sign choice is related to whether the solution is BPS or anti-BPS, in what follows a particular sign will be chosen though one should keep in mind that one can choose the opposite sign and the resulting solutions will simply be the anti-BPS equivalent. Nahm’s equation was generalised in [10] by considering 1 not just the 2 BPS equation of the D1 brane but instead by looking at the BPS equation that arises from preserving a lower number of supersymmetries. This produced the following generalised Nahm equation: ∂Φi i =± c Φj ,Φk , (3) ∂x9 2 ij k where cij k is some totally anti-symmetric constant tensor with i, j, k = 1,...,d. Along with this Nahm type equation there is also a set of algebraic equations that arise from the BPS conditions. This generalised Nahm equation along with the algebraic equations together imply the equations of motion. (We will not give the algebraic equations here since they are dependent on the details of the preserved supersymmetry, see [10] for details.) Spinors that obey the supersymmetry projection conditions may be used to write down i j k a calibration form, c = cij k dx ∧ dx ∧ dx using cij k =¯Γij k . The components of this form cij k are then what appear in the modified Nahm equation given above. (It should be stated that this modified Nahm equation with a specific c has appeared in the pre D-brane literature, for example [14].) The solutions to this equation then correspond to D1 branes ending on a three-brane that wraps the calibrated cycle or equivalently a set of intersecting three-branes. The description of the relation between calibrated cycles and intersecting brane configurations in string theory was described in [15,16]. The analysis performed in [10] examined the linearised D1-brane action. Here we examine the full non-Abelian Born–Infeld action for the D1, given by =− 2 − + 2 i −1 j ij S T1 d σ STr det ηab λ ∂aΦ Qij ∂bΦ det Q , (4) ij = ij + [ i j ] = 2 where Q δ iλ Φ ,Φ and λ 2πls . STr denotes the symmetrised trace prescription [17]. In 2 dimensions the gauge field carries no propagating degrees of freedom and may be completely gauged away, which is why only partial derivatives appear in the above action. (It is known that there is some possible ambiguity in the non-Abelian Born–Infeld theory since the derivative approximation is not valid in a non-Abelian theory yet this action has been shown to possess many of the right properties, see for example [3,4].) We now wish to write down the energy so as to obtain a Bogomol’nyi style argument. If we restrict ourselves to a static solution with three non-zero scalars, which only depend on Φ9 we can expand the determinant to give an expression for the energy 1 1 2 E = T dσ STr I + λ2∂Φi∂Φi − λ2 Φi,Φj 2 − λ2 ∂Φi Φj ,Φk , 1 2 2 ij k (5) the terms under the square root can then be rewritten using the Nahm equation as a perfect square so that 1 1 E = T dσ STr I + λ2 ∂Φi∂Φi − Φi,Φj 2 , (6) 1 2 2 120 D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 and we can clearly see that energy reduces to the linear form and a solution to the Nahm equation solves the full non-linear theory. It was from this linearised starting point that [10] proceeded by using the Bogomol’nyi trick to write the energy (minus the constant piece) as 1 1 2 E = T dσ STr λ2 ∂Φi − c Φj ,Φk + T , (7) 1 2 2 ij k where T is a topological piece. In order to be able to write the energy in this form one requires that 1 c c Tr Φj ,Φk Φl,Φm = Tr Φi,Φj Φi,Φj . (8) 2 ij k ilm One can then show that the above equation (8) holds along with the equations of motion derived from the action (4) if the modified Nahm equation is obeyed along with the algebraic conditions that follow from the vanishing of the supersymmetry variation. The modified Nahm equation then manifestly appears as the Bogomol’nyi equation derived by minimising (7). The first configuration considered in [10] is with two intersecting three-branes. This configuration requires (3) to be satisfied with c123 = c145 = 1 as well as the associated algebraic equations that follow from the supersymmetry. Expanding the energy for five non-zero scalars we have 2 E = T1 d σ 1 1 2 × STr I + λ2 ∂Φi ∂Φi − Φi ,Φj 2 + λ6 ∂Φi Φj ,Φk Φl,Φm 2, 2 2 ij klm (9) where we have used the modified Nahm equation and associated algebraic conditions to write the first square in that form. Thus if the epsilon term vanishes for a solution to the linear equations of motion, it is also solution to the full non-linear equations of motion. Indeed one can check that for the solutions given in [10] that this is the case and so their solutions are again solutions of the full Born–Infeld theory. It is interesting to note however that there do exist solutions to (3) which do not have the form described in [10] where this second term in the energy does not vanish. These solutions correspond to the case where the calibration is deformed away from the flat intersection [18,19]. The non-linearity of the brane action then plays a key role. In what follows we will restrict ourselves to the case of flat intersecting branes so that the linear equations will be enough. It would be interesting for future work to examine the case where the calibration is deformed away from the simple intersection, from the membrane perspective. In summary, the non-linear action is such that solutions that saturate the bound of the linear theory are also solutions of the non-linear theory. In the case of less than half supersymmetry there are algebraic equations in addition to the Nahm type equation. These algebraic equations are actually necessary to derive a Bogomol’nyi bound. These are the guiding properties that we will use to extend the Basu–Harvey equation and the associated membrane action to describe more complicated five-brane configurations. D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 121

3. A Nahm equation for M-theory

The following M-theory version of the Nahm equation was proposed in [9], to describe membranes ending on a ﬁve-brane, ∂Xi M3 1 + 11 G ,Xj ,Xk,Xl = 0, (10) ∂s 64π ij kl 4! 5 where the quantum Nambu 4-bracket is deﬁned by

[ ]= A1,A2,A3,A4 sgn(σ )Aσ1 Aσ2 Aσ3 Aσ4 (11) permutations σ and G5 is a difference of projection operators defined in Appendix A. The solution to this equation as given in [9] is √ i 2π 1 Xi(s) = √ Gi, (12) 3/2 s M11 where the set of matrices {Gi} are in a particular representation of Spin(4) (see Appendix A). Thus, the solution is again that of a fuzzy funnel but this time there is a fuzzy three-sphere whose radius diverges to form the five-brane. Just as the Nahm equation was generalised to describe D1 branes ending on a three- brane that wraps a calibrated cycle so we wish to modify the Basu–Harvey equation to describe membranes ending on a five-brane that wraps some calibrated cycle. The ability of the Basu–Harvey equations to be modified in a natural way so as to allow these more general configurations will be taken as supporting evidence in favour of the validity of their equation in describing membranes. The natural generalisation of the Basu–Harvey equation which we propose is i 3 ∂X M 1 ∗ + 11 g H ,Xj ,Xk,Xl = 0, (13) ∂s 64π ij kl 4! where gij kl is an anti-symmetric constant tensor that is associated to the components of the relevant calibration form that represents an intersecting five-brane configuration. H ∗ has ∗ 2 properties analogous to those of G5, namely (H ) = 1 and for the solutions we consider {H ∗,Xi}=0. In [9] an expression for the membrane energy in such a configuration was also postu- lated, 2 1 E = T d2σ Tr Xi + G ,Xj ,Xk,Xl 2 ij kl 4! 5 2 1/2 1 1 + 1 − Xi , G ,Xj ,Xk,Xl . (14) 2 ij kl 4! 5

Now the G5’s drop out of this expression, and using the Basu–Harvey equation it can be rewritten as 1 1 2 1/2 E = T d2σ Tr 1 + ∂ Xi 2 − Xi,Xj ,Xk 2 , (15) 2 2 a 2.3! 122 D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 where [Xi,Xj ,Xk] is a Nambu 3-bracket containing all permutations of the three entries with signs. So for three active scalars if the Basu–Harvey equation is obeyed the action proposed in [9] is equivalent to its linearised version 1 1 S =−T d3σ Tr 1 + ∂ Xi 2 − Xj ,Xk,Xl 2 . (16) 2 2 a 2.3! We propose that when more scalars are activated we can use our modified Basu–Harvey equation (13) to rewrite the action as the linear piece squared plus other squared terms. When these other terms are zero, the linearised action is the full action. The energy is then given by T 1 E = 2 d2σ Tr Xi Xi − Xj ,Xk,Xl Xj ,Xk,Xl , (17) 2 3! i where we have subtracted the constant piece and assumed that the X depend only on σ2 (= X10 in this gauge). Following the usual Bogomol’nyi construction we rewrite this as 2 T 1 ∗ E = 2 d2σ Tr Xi + g H ,Xj ,Xk,Xl + T , (18) 2 ij kl 4! where T is a topological piece given by 3 i 2 M11 ∂X 1 ∗ j k l T =−T2 d σ Tr gij kl H ,X ,X ,X (19) 64π ∂σ2 4! (with factors restored). This reproduces the correct energy density for the five-brane on which the membranes end in the case where gij kl = ij kl . Now, to rewrite in the energy in this way we have imposed that 1 ∗ ∗ g g H ,Xj ,Xk,Xl H ,Xp,Xq ,Xr ! ij kl ipqr Tr 3 ∗ ∗ = Tr H ,Xi,Xj ,Xk H ,Xi,Xj ,Xk , (20) which using {H ∗,Xi}=0 and (H ∗)2 = 1 is equivalent to the following algebraic constraint 1 g g Tr Xj ,Xk,Xl Xp,Xq ,Xr = Tr Xi,Xj ,Xk Xi,Xj ,Xk . (21) 3! ij kl ipqr 2 5 Note, this is satisfied for the case gij kl = ij kl when the only scalars activated are X to X . The Bogomol’nyi equation found from minimising the energy given in Eq. (18) is then our modified Basu–Harvey equation (13). We now show explicitly that the equation of motion following from the action (16) when combined with the modified Basu–Harvey equation imply the constraint (21). The equation of motion is given by

1 Xi =− Xj ,Xk, Xi,Xj ,Xk , (22) 2 where the three bracket A,B,C is the sum of the six permutations of the three entries, but with the sign of the permutation determined only by the order of the ﬁrst two entries, i.e. D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 123

ABC, ACB and CAB are the positive permutations. By using the Bogomol’nyi equation (13) twice on the left-hand side we get: 1 g g Xk,Xl, Xp,Xq ,Xr =− Xj ,Xk, Xi,Xj ,Xk . (23) 3! ij kl jpqr After multiplying by Xi and taking the trace, we recover the above constraint equation, (21). Thus in summary, the solutions of the generalised Basu–Harvey equation (13) that obey the constraint equation (23) are solutions to the equations of motion of the proposed membrane action (16).

4. Supersymmetry

In the D-brane case, both the Nahm like equation and the additional algebraic relations could be derived from imposing that the necessary supersymmetry variation vanished. The approach we have described above is equivalent but less efficient. The Bogomol’nyi argument effectively implies the Nahm type equation and the necessary algebraic relations. We would like to encode this information by imposing by fiat a supersymmetry transformation whose vanishing will imply the above equations. As to whether this can be made more concrete by constructing a supersymmetric membrane action with this supersymmetry variation we leave as an interesting open question. The advantage of this imposed supersymmetry variation is that it will allow us to relate the solutions of the membrane equations to intersecting five-branes that preserve various fractions of supersymmetry. The obvious suggested susy variation is 1 1 ∗ δλ = ∂ XiΓ µi − H ,Xi,Xj ,Xk Γ ij k . (24) 2 µ 2.4! Now, we use the modified Basu–Harvey equation in the first term and rearrange so that the requirement that the supersymmetry variation vanishes becomes that i j k ij k ij kl# X ,X ,X Γ 1 − gij kl Γ = 0, (25) i

gij kl =0

Here we sum over triplets i, j, k, such that gij kl = 0 for all l. Using the projectors allows us to express these as a set of conditions on the 3-brackets alone.

5. Five-brane conﬁgurations

We will now describe the specific equations that emerge which correspond to the various possible intersecting five-brane configurations. The five-branes must always have at least one spatial direction in common corresponding to the direction in which the membrane intersects the five-branes. These configurations of five-branes can also be thought of as a single five-brane stretched over a calibrated manifold [20]. These five-brane intersections can be found in [15,16,21]. We list the conditions following from the modified Basu–Harvey equation, those following from the supersymmetry conditions (27) (with ν the fraction of preserved supersymmetry) and then discuss any remaining conditions required to satisfy the constraint (23). In string case, [10] only the supersymmetry conditions and the Jacobi identity were required to satisfy the equivalent constraint.

5.1. Conﬁguration 1

The first configuration corresponding to the single five-brane [9] is M5: 12345 M2: 1# g2345 = 1,ν= 1/2, (28) ∗ ∗ X2 =−H X3,X4,X5 ,X3 = H X4,X5,X2 , ∗ ∗ X4 =−H X5,X2,X3 ,X5 = H X2,X3,X4 .

5.2. Conﬁguration 2

For the next case we consider two ﬁve-branes intersecting on a three-brane corresponding to an SU(2) Kähler calibration of a two surface embedded in four dimensions. The activated scalars are X2 to X7. M5: 12345 M5: 123 67 M2: 1# g2345 = g2367 = 1,ν= 1/4, (29) D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 125 ∗ ∗ X2 =−H X3,X4,X5 − H X3,X6,X7 , ∗ ∗ X3 = H X4,X5,X2 + H X6,X7,X2 , ∗ ∗ X4 =−H X5,X2,X3 ,X5 = H X2,X3,X4 , ∗ ∗ X6 =−H X7,X2,X3 ,X7 = H X2,X3,X6 , X2,X4,X6 = X2,X5,X7 , X2,X5,X6 =− X2,X4,X7 , X3,X4,X6 = X3,X5,X7 , X3,X5,X6 =− X3,X4,X7 , X4,X5,X6 = X4,X5,X7 = X4,X6,X7 = X5,X6,X7 = 0. In order to satisfy the constraint we need the Xi ’s to satisfy the following equations:

Choose m ∈{2, 3}, i,j,k,l∈{4, 5, 6, 7} i m m j k ij k X ,X , X ,X ,X = 0 (no sum over m), i j m k l ij kl X ,X , X ,X ,X = 0. (30) In the string theory case there were no additional equations, as apart from the Nahm like equations and algebraic conditions on the brackets all that was needed to solve the constraint was the Jacobi identity. If Xm anti-commutes with Xi , Xj , Xk then the ﬁrst equation reduces to the Jacobi identity. Similarly if Xm anti-commutes with Xi , Xj , Xk, Xl the second equation reduces to

i j k l ij kl X X X X = 0. (31)

5.3. Conﬁguration 3

Three ﬁve-branes can intersect on a three-brane corresponding to an SU(3) Kähler calibration of a two surface embedded in six dimensions. The active scalars are X2 to X9. M5: 12345 M5: 123 67 M5: 123 89 M2: 1# g2345 = g2367 = g2389 = 1,ν= 1/8, (32) ∗ ∗ ∗ X2 =−H X3,X4,X5 − H X3,X6,X7 − H X3,X8,X9 , ∗ ∗ ∗ X3 = H X4,X5,X2 + H X6,X7,X2 + H X8,X9,X2 , ∗ ∗ X4 =−H X5,X2,X3 ,X5 = H X2,X3,X4 , ∗ ∗ X6 =−H X7,X2,X3 ,X7 = H X2,X3,X6 , ∗ ∗ X8 =−H X9,X2,X3 ,X9 = H X2,X3,X8 , X2,X4,X6 = X2,X5,X7 , X2,X5,X6 =− X2,X4,X7 , X2,X4,X8 = X2,X5,X9 , X2,X5,X8 =− X2,X4,X9 , 126 D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 X2,X6,X8 = X2,X7,X9 , X2,X7,X8 =− X2,X6,X9 , X3,X4,X6 = X3,X5,X7 , X3,X5,X6 =− X3,X4,X7 , X3,X4,X8 = X3,X5,X9 , X3,X5,X8 =− X3,X4,X9 , X3,X6,X8 = X3,X7,X9 , X3,X7,X8 =− X3,X6,X9 , X4,X5,X6 + X6,X8,X9 = 0, X4,X5,X7 + X7,X8,X9 = 0, X4,X5,X8 + X6,X7,X8 = 0, X4,X5,X9 + X6,X7,X9 = 0, X4,X6,X7 + X4,X8,X9 = 0, X5,X6,X7 + X5,X8,X9 = 0, X4,X6,X8 = X4,X7,X9 + X5,X6,X9 + X5,X7,X8 , X5,X7,X9 = X5,X6,X8 + X4,X7,X8 + X4,X6,X9 . In order to satisfy the constraint again we have additional algebraic constraints for certain Xi ’s, for this we deﬁne “pairs” as {2, 3}, {4, 5}, {6, 7} and {8, 9}

Choose m ∈{2, 3}, i,j,k,l∈{4, 5, 6, 7, 8, 9} such that {i, j}, {k,l} are pairs i m m j k ij k X ,X , X ,X ,X = 0 (no sum over m), i j m k l ij kl X ,X , X ,X ,X = 0. (33)

5.4. Conﬁguration 4

The next conﬁguration has 3 ﬁve-branes intersecting over a string which corresponds to an SU(3) Kähler calibration of a four surface in six dimensions. There are only 6 activated scalars. M5: 12345 M5: 123 67 M5: 14567 M2: 1# g2345 = g2367 = g4567 = 1,ν= 1/8, (34) ∗ ∗ X2 =−H X3,X4,X5 − H X3,X6,X7 , ∗ ∗ X3 = H X2,X4,X5 + H X2,X6,X7 , ∗ ∗ X4 =−H X2,X3,X5 − H X5,X6,X7 , ∗ ∗ X5 = H X2,X3,X4 + H X4,X6,X7 , ∗ ∗ X6 =−H X2,X3,X7 − H X4,X5,X7 , ∗ ∗ X7 = H X2,X3,X6 + H X4,X5,X6 , X2,X4,X6 = X2,X5,X7 + X3,X5,X6 + X3,X4,X7 , X3,X5,X7 = X3,X4,X6 + X2,X5,X6 + X2,X4,X7 . D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 127

In order to satisfy the constraint again we have to satisfy additional algebraic constraints for certain Xi ’s, for this we deﬁne “pairs” as {2, 3}, {4, 5} and {6, 7}

Choose i, j, k, l, m ∈{2, 3, 4, 5, 6, 7} such that {i, j}, {k,l} are pairs i m m j k ij k X ,X , X ,X ,X = 0 (no sum over m), i j m k l ij kl X ,X , X ,X ,X = 0. (35)

5.5. Conﬁguration 5

In the next configuration we are forced by supersymmetry to have an additional anti- brane. Even though there are only three independent projectors this configuration has three five-branes and an anti-five-brane intersecting over a membrane. This corresponds to the SU(3) special Lagrangian calibration of a three surface embedded in six dimensions.

M5: 12345 M5: 12 4 6 8 M5¯ : 123 67 M5: 12 5 78 M2: 1# g2345 = g2468 =−g2367 = g2578 = 1,ν= 1/8, (36) ∗ ∗ X2 =−H X3,X4,X5 − H X4,X6,X8 ∗ ∗ + H X3,X6,X7 − H X5,X7,X8 , ∗ ∗ X3 = H X2,X4,X5 − H X2,X6,X7 , ∗ ∗ X4 =−H X2,X3,X5 − H X2,X6,X8 , ∗ ∗ X5 = H X2,X3,X4 + H X2,X7,X8 , ∗ ∗ X6 = H X2,X3,X7 − H X2,X4,X8 , ∗ ∗ X7 =−H X2,X3,X6 − H X2,X5,X8 , ∗ ∗ X8 = H X2,X4,X6 + H X2,X5,X7 , X3,X4,X7 = X3,X5,X6 , X4,X3,X8 = X4,X5,X6 , X5,X3,X8 = X5,X7,X4 , X6,X4,X7 = X6,X8,X3 , X7,X3,X8 = X7,X6,X5 , X8,X4,X7 = X8,X6,X5 , X2,X3,X8 + X2,X4,X7 + X2,X6,X5 = 0, X6,X7,X8 + X4,X5,X8 + X3,X4,X6 + X3,X5,X7 = 0. In order to satisfy the constraint once again we have similar additional algebraic constraints for certain Xi ’s, for this we deﬁne “pairs” as {3, 8}, {4, 7} and {5, 6} 128 D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131

Choose m ∈{2,...,8}, i,j,k,l,∈{3, 4, 5, 6, 7, 8} such that {i, j}, {k,l} are pairs Xi,Xm, Xm,Xj ,Xk = 0 (no sum over m), ij k i j m k l ij kl X ,X , X ,X ,X = 0. (37)

6. Solutions

The Basu–Harvey equation is solved by √ i 2π 2n + 6 1/2 1 Xi(s) = √ Gi, (38) 3/2 2n + 1 s M11 where n labels the specific representation of Spin(4) (see Appendix A). We can solve the cases of intersecting five-branes analogously to the intersecting three-branes of [10] by effectively using multiple copies of this solution. The first multi-five-brane case (29) is solved by setting √ i 2π 2n + 6 1/2 1 Xi(s) = √ H i, (39) 3/2 2n + 1 s M11 where the H i are given by the block-diagonal 2N × 2N matrices H 2 = diag G1,G1 , H 3 = diag G2,G2 , H 4 = diag G3, 0 , H 5 = diag G4, 0 , H 6 = diag 0,G3 , H 7 = diag 0,G4 , ∗ H = diag G5,G5 , (40) which are such that n + 3 1 ∗ H i + g H ,Hj ,Hk,Hl = 0. (41) 8(2n + 1) ij kl 4! This makes sure that the conditions following from the generalised Basu–Harvey equation vanish. The remaining conditions in (29), that is those following from the supersymmetry transformation, are satisfied trivially as the three brackets involved all vanish for this solution. The first additional algebraic equation of (30) is satisfied for the solution as the indices must be chosen such that for each diagonal block at least one of the Xi ’s appearing in the bracket that has a zero there, thus the term with each permutation vanishes independently. Again the second additional algebraic equation is trivially satisfied as there are no non-zero products of 5 different Xi ’s. D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 129

The more complicated cases follow easily: conﬁguration 3 is given by the block- diagonal 3N × 3N matrices H 2 = diag G1,G1,G1 , H 3 = diag G2,G2,G2 , H 4 = diag G3, 0, 0 , H 5 = diag G4, 0, 0 , H 6 = diag 0,G3, 0 , H 7 = diag 0,G4, 0 , H 8 = diag 0, 0,G3 , H 9 = diag 0, 0,G4 , ∗ H = diag G5,G5,G5 , (42) and conﬁguration 4 by H 2 = diag G1,G1, 0 , H 3 = diag G2,G2, 0 , H 4 = diag G3, 0,G1 , H 5 = diag G4, 0,G2 , H 6 = diag 0,G3,G3 , H 7 = diag 0,G4,G4 , ∗ H = diag G5,G5,G5 . (43)

Conﬁguration 5 is H 2 = diag G1,G1,G1,G1 , H 3 = diag G2, 0,G2, 0 , H 4 = diag G3,G2, 0, 0 , H 5 = diag G4, 0, 0,G2 , H 6 = diag 0,G3,G4, 0 , H 7 = diag 0, 0,G3,G3 , H 8 = diag 0,G4, 0,G4 , ∗ H = diag G5,G5,G5,G5 . (44) 130 D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131

Acknowledgements

We wish to thank Anirban Basu, Jeff Harvey, Sanjaye Ramgoolam and David Tong for discussions. D.S.B. is supported by EPSRC grant GR/R75373/02 and would like to thank DAMTP and Clare Hall College Cambridge for continued support. N.B.C. is supported by a PPARC studentship. This work was in part supported by the EC Marie Curie Research Training Network, MRTN-CT-2004-512194.

Appendix A. The fuzzy 3-sphere

To make this self-contained we include a brief description of the fuzzy 3-sphere construction following that in [9]. The approach was developed in [22–25], with an interesting string interpretation further developed in [26]. Unlike even fuzzy spheres we must use reducible representations of spin(4) = SU(2) × R+ R− n+1 n−1 n−1 n+1 SU(2). and are the ( 4 , 4 ) and ( 4 , 4 ) representations, respectively, where n is an odd integer. The dimension of R = R+ ⊕ R− is given by N = (n + 1)(n + 3)/2. The coordinates on the fuzzy S3 are the N × N matrices Gi (i = 1 to 4). These matrices are deﬁned by n n i i i G = PR+ ρs Γ P− PR− + PR− ρs Γ P+ PR+ , (A.1) s=1 s=1 where n i = i ⊗···⊗ +···+ ⊗···⊗ i ρs Γ Γ 1 1 Γ sym, (A.2) s=1 where sym stands for the completely symmetrised n-fold tensor product representation of 1 spin(4).HereP± = (1±Γ5), and PR+ , PR− are projection operators onto the irreducible + 2 − representations R , R respectively of spin(4). The matrix G5 which is important to the construction is given by

G5 = PR+ − PR− . (A.3) i (Some intuition can be gained from the fact that for n = 1, the matrices G and G5 become i Γ and Γ5, respectively.) i R i = i + i i = 1 ± i The G are elements of End( ). We can write G G+ G− with G± 2 (1 G5)G i and then G± act as homomorphisms from R∓ to R±. From the above deﬁnitions, after much manipulation [9] we can obtain the equation n + 3 Gi + G Gj GkGl = 0. (A.4) 8(2n + 1) ij kl 5 Thus the Basu–Harvey equation (10) is solved by (12) in the large N limit. However a solution can be found for any n by taking √ i 2π 2n + 6 1/2 1 Xi(s) = √ Gi. (A.5) 3/2 2n + 1 s M11 D.S. Berman, N.B. Copland / Nuclear Physics B 723 (2005) 117–131 131

References

[1] C.G. Callan, J.M. Maldacena, Brane dynamics from the Born–Infeld action, Nucl. Phys. B 513 (1998) 198, hep-th/9708147. [2] G.W. Gibbons, Born–Infeld particles and Dirichlet p-branes, Nucl. Phys. B 514 (1998) 603, hep-th/9709027. [3] D. Brecher, BPS states of the non-Abelian Born–Infeld action, Phys. Lett. B 442 (1998) 117, hep-th/ 9804180. [4] R.C. Myers, Dielectric-branes, JHEP 9912 (1999) 022, hep-th/9910053. [5] P. Cook, R. de Mello Koch, J. Murugan, Non-Abelian BIonic brane intersections, Phys. Rev. D 68 (2003) 126007, hep-th/0306250. [6] N.R. Constable, R.C. Myers, O. Tafjord, The noncommutative bion core, Phys. Rev. D 61 (2000) 106009, hep-th/9911136. [7] P.S. Howe, N.D. Lambert, P.C. West, The self-dual string soliton, Nucl. Phys. B 515 (1998) 203, hep- th/9709014. [8] D.S. Berman, Aspects of M-5 brane world volume dynamics, Phys. Lett. B 572 (2003) 101, hep-th/0307040. [9] A. Basu, J.A. Harvey, The M2–M5 brane system and a generalized Nahm’s equation, hep-th/0412310. [10] N.R. Constable, N.D. Lambert, Calibrations, monopoles and fuzzy funnels, Phys. Rev. D 66 (2002) 065016, hep-th/0206243. [11] W. Nahm, A simple formalism for the BPS monopole, Phys. Lett. B 90 (1980) 413. [12] D.E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys. B 503 (1997) 220, hep-th/ 9608163. [13] D. Tsimpis, Nahm equations and boundary conditions, Phys. Lett. B 433 (1998) 287, hep-th/9804081. [14] E. Corrigan, C. Devchand, D.B. Fairlie, J. Nuyts, First order equations for gauge ﬁelds in spaces of dimension greater than four, Nucl. Phys. B 214 (1983) 452. [15] G.W. Gibbons, G. Papadopoulos, Calibrations and intersecting branes, Commun. Math. Phys. 202 (1999) 593, hep-th/9803163. [16] J.P. Gauntlett, N.D. Lambert, P.C. West, Branes and calibrated geometries, Commun. Math. Phys. 202 (1999) 571, hep-th/9803216. [17] A.A. Tseytlin, On non-Abelian generalisation of the Born–Infeld action in string theory, Nucl. Phys. B 501 (1997) 41, hep-th/9701125. [18] N.D. Lambert, Moduli and brane intersections, Phys. Rev. D 67 (2003) 026006, hep-th/0209141. [19] J. Erdmenger, Z. Guralnik, R. Helling, I. Kirsch, A world-volume perspective on the recombination of intersecting branes, JHEP 0404 (2004) 064, hep-th/0309043. [20] R. Harvey, H.B. Lawson, Calibrated geometries, Acta Math. 148 (1982) 47. [21] B.S. Acharya, J.M. Figueroa-O’Farrill, B. Spence, Branes at angles and calibrated geometry, JHEP 9804 (1998) 012, hep-th/9803260. [22] Z. Guralnik, S. Ramgoolam, On the polarization of unstable D0-branes into non-commutative odd spheres, JHEP 0102 (2001) 032, hep-th/0101001. [23] S. Ramgoolam, On spherical harmonics for fuzzy spheres in diverse dimensions, Nucl. Phys. B 610 (2001) 461, hep-th/0105006. [24] S. Ramgoolam, Higher-dimensional geometries related to fuzzy odd-dimensional spheres, JHEP 0210 (2002) 064, hep-th/0207111. [25] M.M. Sheikh-Jabbari, Tiny graviton matrix theory: DLCQ of IIB plane-wave string theory, a conjecture, JHEP 0409 (2004) 017, hep-th/0406214. [26] M.M. Sheikh-Jabbari, M. Torabian, Classiﬁcation of all 1/2 BPS solutions of the tiny graviton matrix theory, hep-th/0501001. Nuclear Physics B 723 (2005) 132–162

Open string integrability and AdS/CFT

Tristan McLoughlin, Ian Swanson

California Institute of Technology, Pasadena, CA 91125, USA Received 9 May 2005; accepted 10 June 2005 Available online 5 July 2005

Abstract 5 We present a set of long-range Bethe ansatz equations for open quantum strings on AdS5 × S and demonstrate that they diagonalize bosonic su(2) and sl(2) sectors of the theory in the near-pp- wave limit. Results are compared with energy spectra obtained by direct quantization of the open = 2 string theory, and we find agreement in this limit to all orders in the ’t Hooft coupling λ gYMNc. We also propose long-range Bethe ansätze for su(2) and sl(2) sectors of the dual N = 2 defect conformal field theory. In accordance with previous investigations, we find exact agreement between string theory and gauge theory at one- and two-loop order in λ, but a general disagreement at higher loops. It has been conjectured that the sudden mismatch at three-loop order may be due to long-range interaction terms that are lost in the weak coupling expansion of the gauge theory dilatation operator. These terms are thought to include interactions that wrap around gauge-traced operators, or around the closed-string worldsheet. We comment on the role these interactions play in both open and closed sectors of string states and operators near the pp-wave/BMN limit of the AdS/CFT correspondence.  2005 Elsevier B.V. All rights reserved.

1. Introduction

In 2002 Berenstein, Maldacena and Nastase demonstrated that the energy spectrum of type IIB string theory on a pp-wave background geometry can be matched via the AdS/CFT correspondence to the anomalous dimensions of a special class of planar, large R-charge operators in N = 4 super-Yang–Mills (SYM) theory [1–3]. This observation sparked a number of detailed tests of AdS/CFT which go beyond the supergravity limit of the string

E-mail address: [email protected] (I. Swanson).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.06.014 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 133 theory. Many of these investigations have probed simplifying limits of Maldacena’s origi- 5 nal proposal equating IIB superstring theory on AdS5 × S with N = 4 SYM theory in four flat spacetime dimensions [4]. It has been realized in the course of these studies that both the string and gauge theory sides of this duality harbor integrable structures associated with quantum spin chains. The discovery of such structures is tantalizing, as it indicates progress toward finding exact solutions within certain sectors of the correspondence. A more immediate benefit is that the arsenal of techniques associated with the Bethe ansatz methodology has been useful for simplifying the computation of both string energies and gauge theory operator dimensions. Building on previous investigations (see, e.g., [5–16]), we study the integrability of open string states and explore the matchup with corresponding operator dimensions in a dual defect conformal field theory. This allows us to speculate on the role of a particular subset of non-perturbative interactions that have been conjectured to exist in the planar (large-Nc) limit of N = 4 SYM theory. In this paper we will focus on a branch of work involving the so-called near-pp-wave limit of the string theory [17–23].Itwasshownin[24–26] that the ten-dimensional pp-wave geometry is a consistent supergravity background that can be realized as a large- 5 radius Penrose limit of AdS5 ×S . Finite-radius corrections to this limit induce interactions on the string worldsheet that lift the highly degenerate energy spectrum on the pp-wave. These near-pp-wave perturbations were computed for the full supersymmetric string theory in [18,19], where the complete energy spectrum of two-impurity string states (characterized as having two mode excitations on the worldsheet) was compared in this limit to a corresponding set of operator dimensions computed in a perturbative regime of the gauge theory. The string and gauge theory were found to exhibit a remarkably intricate agreement = 2 at one- and two-loop order in the ’t Hooft coupling λ gYMNc, but suffer from a general mismatch at three-loop order and beyond. This pattern of agreement at low orders in perturbation theory followed by higher-order disagreement has been shown to repeat itself for the larger and more detailed spectrum of three-impurity string states in [20], and for the complete set of N-impurity spectra in [22]. The sudden three-loop disagreement has also appeared in related studies of the AdS/CFT correspondence involving the comparison of semiclassical extended string configurations in anti-de Sitter backgrounds with dual gauge theory operators [27–30]. The matching procedure between string and gauge theory in this setting is presented schematically in Fig. 1. The prevailing explanation for the higher-loop mismatch is that the comparison of string energies with operator dimensions derived in the perturbative λ expansion suffers from an order-of-limits problem [31–33]. In the upper left-hand re- 5 gion we begin with the conjectured equivalence of IIB superstring theory on AdS5 × S and N = 4 SYM theory. On the string theory side of the duality one first takes a large- 5 radius Penrose limit by boosting string states in AdS5 × S to large angular momentum J along an equatorial geodesic in the S5 subspace. This large-J limit is combined with the 2 ’t Hooft large-Nc limit, which is taken such that the ratio Nc/J remains fixed and finite. For comparison with perturbative gauge theory, the large-J Penrose limit is followed by = 2 2 an expansion in small λ gYMNc/J (the so-called modified ’t Hooft coupling), leading to the lower right-hand region in Fig. 1. On the gauge theory side, operator dimensions are first derived perturbatively in the small-λ expansion. The string angular momentum maps to the scalar component R of the SU(4)R-charge, so comparison with string theory 134 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

5 J 1 near-pp-wave IIB on AdS5×S , N =4 SYM string theory λ1 λ 1

perturbative gauge theory three-loop mismatch R1

Fig. 1. The ordering of limits in the comparison of near-pp-wave string theory with perturbative gauge theory.

(a) (b)

(c)

= 2 Fig. 2. Feynman diagrams at (a) one-, (b) two- and (c) n-loop order in λ gYMNc.

2 requires a large-R expansion, keeping Nc/R fixed. This leads, via the left-hand route in Fig. 1, to the matchup with string energies in the lower right-hand corner. The outstanding question is whether the large-J (or large-R) and small-λ (or small-λ) limits are commutative, and whether non-commutativity can account for the mismatch at three-loop order in λ. In the spin chain picture of the gauge theory [34], single-trace operators are identified with cyclic lattice configurations of interacting spins. The excitation of magnon states on the spin lattice corresponds to the insertion of R-charge impurities in the trace; this defines a basis on which the dilatation operator, or spin-chain Hamiltonian, can be diagonalized. At one-loop order in λ the dilatation operator can be shown to mix only neighboring spins on the lattice (i.e., dilatations of single-trace operators can only mix fields that are adjacent in the trace), and the corresponding statement at nth order is that the n-loop Hamiltonian can mix fields on the lattice that are separated by at most n lattice sites (see Fig. 2). Within certain sectors of N = 4 SYM theory, one strategy for extracting operator dimensions at higher loop order in λ has been to derive the appropriate n-loop spin chain Hamiltonian by enumerating all possible interaction terms of maximum length n and fix the coefficients of these terms using symmetry constraints (or otherwise) handed down from the gauge theory [35–38]. One way in which the limits in Fig. 1 could be non-commutative in this context is that this procedure may neglect the contribution of interactions on the spin lattice that have a range greater than the total length of the lattice itself [31]. The effect of these interactions would be apparent in any calculation that attempts to apply asymptotic Bethe equations (derived for long chains) to spin chains of finite length. This is precisely the situation encountered when comparing finite R-charge corrections near the BMN limit to near-pp-wave corrections to the string energy spectrum. The suggestion is that if one were T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 135

Table 1 Extended dimensions of D3- and D5-branes in the ten-dimensional background 0123456789 D3 ××× × D5 ×××××× able to eliminate such ultra long-range effects, complete agreement with corresponding string predictions would be obtained. We attempt to shed light on the specific contribution of wrapping1 interactions by studying the matchup of open string states with corresponding gauge theory operators. We formulate a prescription for determining long-range, open-chain Bethe ansätze based on the scattering matrices that appear in the analogous closed-chain equations. On the string side we test our formulas by computing the full, all-loop energy spectrum of open string states in the near-pp-wave limit and comparing with results obtained by direct quantization and diagonalization of the string lightcone Hamiltonian. The string states of interest arise 5 2 from the IIB theory on AdS5 × S with a D5-brane wrapping an AdS4 × S subspace [8, 39]. The spacetime dimensions occupied by both the D3- and D5-branes are indicated in 5 Table 1. Alternatively, one may study an AdS5 × S /Z2 space, which is the near-horizon geometry of a large number of D3-branes at an orientifold 7-plane in type I theory [5]. (The CFT dual is an Sp(N) gauge theory.) For the purposes of this paper, however, we find it sufficient to focus on the D3–D5 system, whose dual description is an N = 2 defect conformal field theory. We apply our Bethe ansatz on the CFT side of the duality by using the closed (or protected2) su(2) sector of N = 4 SYM theory to derive an open-chain, long-range Bethe ansatz for an su(2) sector of operators in the N = 2 defect theory. Let us note here that the mapping of gauge theory physics to the dynamics of integrable open spin chains has been studied in various different settings in the literature. The authors of [6], for example, studied the Sp(N) superconformal theory containing one hypermultiplet in the antisymmetric representation and four in the fundamental. This theory had previously been studied in the BMN limit in [5], and its mixing matrix was shown in [6] to map to the Hamiltonian of an integrable open spin chain. Similarly, DeWolfe and Mann [7] considered a general defect conformal field theory [39] (derived as a perturbation to N = 4 SYM theory) and demonstrated that integrable structures again emerge within certain sectors. This system, which we focus on in this paper, was also examined in [8], where comparisons were carried out at one-loop order between the full pp-wave limit of the string theory and the full BMN limit of the N = 2 gauge theory (see also reference [39]). We also note that analyses of semiclassical open string states and corresponding gauge theory operator dimensions were presented in [9–11]. The reader is referred to [12–16,40] for further related investigations.

1 To be certain, we use the term ‘wrapping’ here to specify interactions which literally wrap around the lattice. 2 To avoid confusion with open and closed strings, we will refer to the closed, non-mixing sectors of the string and gauge theory as protected sectors. 136 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

In Section 2 we establish notation, review the derivation of the bosonic interacting string Hamiltonian in the near-pp-wave limit and compute energy spectra of open string states. This is carried out for a protected su(2) sector of bosonic symmetric-traceless open strings in the S5 subspace with Dirichlet boundary conditions, and for an analogous sl(2) sector in the AdS5 subspace with Neumann boundary conditions. For pedagogical reasons, we treat separately the distinct cases of states with completely unequal mode excitations and those with mode numbers that are allowed to overlap (although we carry this out only for the su(2) sector). In Section 3 we formulate a long-range quantum Bethe ansatz for open strings based on long-range scattering matrices developed for corresponding sectors 5 of closed string states in the pure AdS5 × S background. We solve our resulting Bethe equations in the near-pp-wave limit for general open string states in both the Dirichlet su(2) and Neumann sl(2) sectors, and we compare our results with corresponding energy spectra computed directly from the string theory in Section 2. In Section 4 we apply our long-range Bethe ansatz to the su(2) and sl(2) sectors of the dual N = 2 defect conformal ﬁeld theory. The operators of interest are not gauge-traced, and the open-chain Bethe ansätze should be free of any interference from wrapping interactions. Upon solving the Bethe equations in the near-BMN limit, we obtain agreement between operator dimensions in these sectors and the corresponding su(2) and sl(2) string energy predictions at one and two-loop order in λ. Once again, however, this agreement breaks down for both sectors at three-loop order and beyond. We conclude in the ﬁnal section with a summary and discussion of these results.

2. Open string energy spectra near the pp-wave limit

5 The metric of AdS5 × S can be written in global coordinates as ds2 = Rˆ2 − cosh2 ρdt2 + dρ2 + sinh2 ρdΩ2 + cos2 θdφ2 + dθ2 3 + 2 ˜ 2 sin θdΩ3 , (2.1) 5 ˆ where the common scale radius of both the AdS5 and S subspaces is denoted by R, t is the 2 ˜ 2 global time direction, and dΩ3 and dΩ3 indicate separate three-spheres. As in previous studies, we invoke the following reparameterizations

1 + z2/4 1 − y2/4 cosh ρ = , cos θ = , (2.2) 1 − z2/4 1 + y2/4 and work with the metric 1 2 1 2 1 + z2 1 − y2 dz dz dy dy ds2 = Rˆ2 − 4 dt2 + 4 dφ2 + k k + k k . − 1 2 + 1 2 − 1 2 2 + 1 2 2 1 4 z 1 4 y (1 4 z ) (1 4 y ) (2.3) This version of the spacetime metric is useful when working with fermions (see [19] for details), though we will restrict ourselves to the bosonic sector of the string theory in the present study. The Penrose limit is reached by boosting string states to lightlike momentum T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 137

J along an equatorial geodesic in the S5 subspace. Under the rescaling prescriptions − + + x zk yk t → x ,φ→ x + ,zk → ,yk → , (2.4) Rˆ2 Rˆ Rˆ this limit is reached by taking Rˆ →∞.TheS5 angular momentum J is thereby related to the scale radius Rˆ by ˆ2 p−R = J, (2.5) and the lightcone momenta are given by i J −p+ = ∆ − J, −p− = i∂x− = ∂φ =− . (2.6) Rˆ2 Rˆ2 The transverse Cartesian coordinates zk and yk span an SO(4) × SO(4) subspace, with zk 5 lying in AdS5 (k ∈ 1,...,4) and yk in the S subspace (k ∈ 5,...,8). In the Penrose limit ˆ p− isheldfixedasJ and R become infinite. We also work in the planar limit, where the 2 number of colors (or number of D3 branes) Nc becomes large, but the quantity Nc/J is held fixed. The lightcone momentum p− is equated via the AdS/CFT dictionary with 1 J − = √ = p , (2.7) λ 2 gYMNc where, as noted in the introduction, λ is known as the modified ’t Hooft coupling, and J is equated on the CFT side with the scalar R-charge R. For reasons described in [18,19], the lightcone coordinates in Eq. (2.4) admit many simplifications, the most important of which is the elimination of normal-ordering contributions to the lightcone Hamiltonian (at least to the order of interest). In terms of these coordinates, we obtain the following large-Rˆ expansion of the metric: + − + ds2 = 2 dx dx + dz2 + dy2 − z2 + y2 dx 2 1 − + 1 + − + −2y2 dx dx + y4 − z4 dx 2 + dx 2 ˆ2 R 2 1 1 + z2 dz2 − y2 dy2 + O 1/Rˆ2 , (2.8) 2 2 where the coordinates xA span an SO(8) subspace, with A ∈ 1,...,8. The pp-wave metric emerges at leading order, and the string theory on that background is free [41,42].The background curvature correction appearing at O(1/Rˆ2) induces a set of interaction corrections to the free string spectrum in the Penrose limit. As noted in the introduction, these corrections lift the degeneracy of the free theory on the pp-wave, allowing for much more detailed comparisons of string energy spectra with gauge theory anomalous dimensions. In the Penrose limit, the lightcone Green–Schwarz action for open superstrings takes the form 2π 1 S = dτ dσ (L + L ), (2.9) pp 2π B F 0 138 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 where 1 L = x˙A 2 − x A 2 − xA 2 , (2.10) B 2 i L = iψ†ψ˙ + ψ†Πψ + ψψ + ψ†ψ † . (2.11) F 2 (We have set α = 1.) The fields ψα are eight-component complex spinors formed from two Majorana–Weyl SO(9, 1) spinors of equal chirality, and Π is defined in terms of eight- dimensional SO(8) Dirac gamma matrices by Π ≡ γ 1γ¯ 2γ 3γ¯ 4 (see [19] for further details). A A A A The shorthand notation x˙ and x denotes the worldsheet derivatives ∂τ x and ∂σ x , respectively. The pp-wave lightcone Hamiltonian is easily derived from Eqs. (2.10) and (2.11): p− 1 i i H = xA 2 + (p )2 + x A 2 + iρΠψ + ψψ − ρρ , pp A 2 (2.12) 2 2p− 2 2p−

A where the fields pA and ρα are conjugate to x and ψα, respectively. Including the first finite-radius curvature correction to the spacetime metric in Eq. (2.8) gives rise to an interacting Hamiltonian appearing at O(1/Rˆ2) in the large-radius expansion. This Hamiltonian was computed and analyzed extensively in [18–20,22]. The bosonic sector, labeled by HBB, is quartic in fields and mixes purely bosonic string states:

= 1 1 − 2 2 + 2 + 2 + 2 2 + 2 + 2 + p− A 2 2 HBB y pz z 2y z py y 2z x ˆ2 4p− 8 R 1 1 − (p )2 2 + (p )2 x A 2 + x A 2 2 + x Ap 2 . 3 A 2 A 3 A (2.13) 8p− 2p−

We have used the shorthand symbols pz and py to denote bosonic momentum ﬁelds in the 5 AdS5 and S subspaces, respectively. Since we will not be dealing with fermions, the string A† states of interest are formed by acting with bosonic creation operators an on a vacuum state |J carrying J units of angular momentum in the S5 subspace:

|N; J ≡aA1†aA2† ···aAN † |J . n1 n2 nN N Such states are generically referred to as N-impurity bosonic string states. In [18–20] it was demonstrated that there are certain special sectors of the full interacting Hamiltonian that completely decouple from the theory. Among these are two 5 bosonic sectors consisting of pure-boson states restricted to either the AdS5 or S subspaces and projected onto symmetric-traceless irreps of spacetime SO(4), plus one sector comprised of purely fermionic excitations symmetrized in spinor indices (symmetrized spinors survive in this setting because they come with additional mode-number labels). For low impurity number, these sectors are conveniently labeled according to their trans- × formation properties under the residual SO(4)AdS SO(4)S5 symmetry that survives in the Penrose (or near-Penrose) limit. Employing the SU(2)2 ×SU(2)2 notation used extensively in [18–20], the two-impurity bosonic sectors transform as (3, 3; 1, 1) and (1, 1; 3, 3), while T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 139

Table 2 × Neumann and Dirichlet directions on the SO(4)AdS SO(4)S5 transverse subspace +–12345678 + − j j j j j j j j x x zN zN zN zD yN yD yD yD the two-impurity fermionic sectors are labeled by (3, 1; 3, 1) and (1, 3; 1, 3). (The three- impurity version of this irrep decomposition is given in [20].) On the gauge theory side the symmetric-traceless AdS5 (3, 3; 1, 1) bosons map to a closed sector of the dilatation generator invariant under an sl(2) subalgebra of the full superconformal algebra psu(2, 2|4). The protected sector of (1, 1; 3, 3) bosonic S5 string states maps to an su(2) sector, and the (3, 1; 3, 1) + (1, 3; 1, 3) fermions correspond to an su(1|1) sector. We will use these algebraic labels when we refer to these decoupled subsectors in both the string theory and the gauge theory. Since we are interested in general multi-impurity states within these protected subsectors, we will restrict ourselves in the sl(2) sector to open strings with Neumann boundary conditions, and the su(2) sector will be comprised of open strings with Dirichlet boundary conditions. The available bosonic worldsheet fields are indicated in Table 2, where, j for example, z indicates fields excited in the AdS5 subspace with Neumann boundary N j 5 conditions, and yD are S fluctuations with Dirichlet boundary conditions.

2.1. Dirichlet SO(4)S5 (su(2)) sector

Since the goal is to compute O(1/J ) interaction corrections to the free energy spectrum in the pp-wave limit, we find it convenient to isolate such corrections according to the energy expansion N { } = + 2 + { } + 2 E nj ,N,J 1 nj λ /4 δE nj ,N,J O 1/J . (2.14) j=1 The near-pp-wave energy spectra we are interested in are thus collected into the O(1/J ) correction δE({nj },N,J), which generically depends on the set of mode numbers {nj } carried by the corresponding (unperturbed) energy eigenstate, the total number of worldsheet mode excitations N, and the string S5 angular momentum J . We will focus first on a set of N-impurity bosonic open string states in the protected su(2) sector. We form symmetric-traceless (in SO(4)S5 indices) states by combining excitations in the Dirichlet (y6,y7,y8)S5 directions: (j † j † j )† | ; = 1 2 ··· N | N J D an1 an2 anN J , (2.15) where the S5 labels j indicate the (y6,y7,y8) directions and contributions to the traced state are understood to be absent. The Fourier expansion of bosonic fluctuations in the Dirichlet directions is given by [43] ∞ i nσ − A = A iωnτ − A† iωnτ xD(τ, σ ) √ sin an e an e . (2.16) ωn 2 n=1 140 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

The equations of motion in the pp-wave limit A A 2 A x¨ − x + p−x = 0 (2.17) are therefore satisﬁed by the dispersion relation 2 2 ωn = p− + (n/2) , (2.18)

A A† where the mode index n is integer-valued and the oscillators an and an satisfy the usual [ A B†]= A,B commutation relations: an ,am δn,mδ . We are interested in computing diagonal matrix elements of HBB between physical string states which, in ﬁrst-order perturbation theory, will involve equal numbers of creation and annihilation operators (i.e., we need not be concerned with sectors of HBB that mix states with different numbers of impurities). We may therefore restrict to terms appearing in HBB with two creation and two annihilation operators. Following the approach in [22], we simplify the projection onto symmetric-traceless S5 string states by forming the following bosonic oscillators: ≡ √1 6 + 7 ¯ ≡ √1 6 − 7 an an ian , an an ian . (2.19) 2 2 Generic matrix elements of the form J |a a ···a (H )a† a† ···a† |J (2.20) n1 n2 nN BB n1 n2 nN 6 7 therefore select out excitations in the (yD,yD) plane and make the symmetric-traceless projection manifest. Restricting to the Dirichlet directions, we are also free to project onto 6 8 7 8 5 the (yD,yD) and (yD,yD) planes. We exclude the yN direction because we do not want to mix Neumann and Dirichlet boundary conditions (although, in general, such states are 5 5 certainly allowed), and states in the S subspace built strictly from yN excitations would be projected out by the traceless condition. As described in [22], the oscillator expansion of HBB admits two generic structures: one characterized by a contraction of SO(4) indices attached to pairs of creation or annihilation operators †A †A B B an am al ap , and one containing pairs of contracted creation and annihilation operators †A †B A B an al amap .

(The indices {n, l, m, p} are mode numbers appearing in the mode expansion of HBB.) Expanding in the ﬁelds deﬁned in Eq. (2.19), we obtain †A †A B B = † ¯† +¯† † ¯ +¯ an am al ap (6,7) anam anam (al ap al ap), (2.21) †A †B A B =¯† ¯† ¯ ¯ + † † an al amap (6,7) anal amap anal amap. (2.22) Only the second term in Eq. (2.22) will contribute to the matrix elements in (2.20).Wemay therefore simplify the calculation of the energy spectrum in this sector of the theory by † † projecting the Hamiltonian onto terms containing only the oscillator structure anal amap. T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 141

To begin we will compute energy eigenvalues for the simplest eigenstates in which all mode numbers of the symmetric-traceless state a† a† ···a† |J n1 n2 nN are taken to be unequal (n1 = n2 =··· = nN ). Between these states, the interaction Hamil- tonian will admit matrix elements whose structure is deﬁned by J |a a ···a a†a†a a a† a† ···a† |J n1 n2 NB n l m p n1 n2 NB 1 N = (δ δ δ δ + δ δ δ δ 2 nj ,n nk,l nj ,m nk,p nj ,n nk,l nk,m nj ,p j,k=1 j =k + + δnj ,lδnk,nδnj ,mδnk,p δnj ,lδnk,nδnk,mδnj ,p). (2.23) The appropriate energy eigenvalue in this sector can thus be computed by attaching coef- † † ﬁcients of anal amap in HBB to the matrix element structure in Eq. (2.23).TheO(1/J ) energy correction for this protected su(2) sector of symmetric-traceless S5 open string states is thereby found, for completely unequal mode indices, to be N n2 + n2λω2 { } =− 1 k j nk δES5 ni ,N,J . (2.24) 8J ωn ωn j,k=1 j k j =k By expanding in small λ { } δES5 ni ,N,J

N 1 1 1 = − n2 + n2 λ + n4 + n4 (λ )2 J 8 j k 64 j k j,k=1 = j k 1 − 3n6 + n4n2 + n2n4 + 3n6 (λ )3 + O (λ )4 , (2.25) 1024 j j k j k k it is easy to see that the spectrum for these states exhibits the expected property that at mth order in the expansion the energy scales with 2m factors of the mode numbers {ni}. We can extend this result to the most general set of states in this sector, formed by a set † of an creation oscillators grouped into M subsets, with all mode indices equal within these subsets: (a† )N1 (a† )N2 (a† )NM √n1 √n2 ··· √nM |J . N1! N2! NM !

The jth subset contains Nj oscillators with mode index nj , and the total impurity number is N, such that M Nj = N. (2.26) j=1 142 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

† † The matrix element of the oscillator structure anal amap between these states is N N † N † N (a ) n1 (a ) nM (a ) n1 (a ) nM J | n1 ··· nM a†a†a a n1 ··· nM |J ! ! n l m p ! ! Nn1 NnM Nn1 NnM M = − Nnj (Nnj 1)δn,nj δl,nj δm,nj δp,nj j=1 1 M + N N (δ δ δ δ + δ δ δ δ 2 nj nk n,nk l,nj m,nk p,nj n,nj l,nk m,nk p,nj j,k=1 j =k + + δn,nk δl,nj δm,nj δp,nk δn,nj δl,nk δm,nj δp,nk ). (2.27) The energy shift at O(1/J ) for completely general N-impurity open string states in the Dirichlet su(2) sector of the theory is therefore given by δE 5 {ni}, {Ni},M,J S M 2 + 2 M 2 + 2 2 1 n (6 n λ ) n n λ ωn =− N (N − ) j j + N N k j k . j j 1 2 j k (2.28) 8J 4ω ωn ωn j=1 nj j,k=1 j k j =k We see that the structure obtained for states with completely inequivalent mode indices in Eq. (2.24) appears in the second term above. This term can be thought of as the contribution to the energy from scattering among excitations with differing mode numbers; the ﬁrst term represents scattering within subsets of equal mode numbers. For eventual comparison with corresponding quantities in the dual gauge theory, we expand Eq. (2.28) in small λ: { } { } δES5 ni , Nni ,M,J M 1 3 1 1 = N (N − 1) − n2λ + n4(λ )2 − n6(λ )3 J k k 16 k 64 k 256 k k=1 M 1 1 1 + N N − n2 + n2 λ + n4 + n4 (λ )2 J j k 8 j k 64 j k k,j=1 = j k 1 − 3n6 + n4n2 + n2n4 + 3n6 (λ )3 + O (λ )4 . (2.29) 1024 j j k j k k Within each of the M subsectors of overlapping mode numbers (labeled by the indices j and k) we again observe the desired result that the contribution to the energy scales at mth 2m order in λ in proportion to nk .

2.2. Neumann SO(4)AdS (sl(2)) sector

From Table 2 we see that we can repeat the above calculation for the protected sl(2) sector of symmetric-traceless bosonic open string states excited in the AdS5 subspace. T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 143

Analogous to Eq. (2.19) above, we can project onto this sector by forming the oscillators ≡ √1 1 + 2 ¯ ≡ √1 1 − 2 an an ian , an an ian , (2.30) 2 2 and restricting to unperturbed string states of the form a† a† ···a† |J . n1 n2 nN 1 2 2 3 1 3 Here we are free to project onto the (zN ,zN ), (zN ,zN ) or (zN ,zN ) planes, and we choose 4 to exclude zD to avoid mixing boundary conditions. The states of interest carry Neumann boundary conditions, and the mode expansion is given by ∞ i nσ − A = A iωnτ − A† iωnτ xN (τ, σ ) √ cos an e an e . (2.31) ωn 2 n=0

(The equations of motion again require that ωn satisfy the dispersion relation given in Eq. (2.18).) The computation of the energy shift at O(1/J ) for the sl(2) sector of open string states follows in complete analogy with the su(2) sector described above. We form general matrix elements of the form given in Eq. (2.27) and attach coefficients of the Hamiltonian propor- † † tional to the oscillator structure anal amap, the only difference being that this sector of the interacting Hamiltonian is defined in terms of the worldsheet excitations in Eq. (2.31). Leaving out the computational details, we obtain the following energy shift for completely general open string states in the sl(2) sector of the theory: δE {n }, {N },M,J AdS i i 1 M n2(2 + n2λ ) M n2n2λ =− N (N − ) k k + N N j k . k k 1 2 j k (2.32) 32J ω ωn ωn k=1 nk j,k=1 j k j =k For comparison with the gauge theory, we expand in λ: δEAdS {ni}, {Ni},M,J M 1 1 1 1 = N (1 − N ) n2λ + n4(λ )2 − n6(λ )3 J k k 16 k 64 k 256 k k=1 M 1 1 1 + N N − n2n2(λ )2 + n2n2 n2 + n2 (λ )3 J j k 32 j k 256 j k j k j,k=1 j =k + O (λ )4 . (2.33) The result again breaks up into contributions from interactions among worldsheet excitations with both equal and inequivalent mode numbers. There are also no contributions from zero-mode fluctuations: this is an important consistency check, since such corrections are prohibited in general [18,19]. One may also expand in small λ to see that the scaling with mode numbers follows the same pattern demonstrated in Eqs. (2.24) and (2.28). 144 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

3. Open string Bethe equations

There has recently been a great deal of work exploring the existence and role of inte- 5 grable structures in both type IIB string theory on AdS5 × S and N = 4 SYM theory. It was originally noticed, first for the bosonic theory [44] and later for the full Green–Schwarz 5 action [45], that the classical coset sigma model of the string on AdS5 × S possesses an infinite set of mutually commuting conserved charges. In [46], Arutyunov and Staudacher used a set of Bäcklund transformations to construct a generating function for these charges and matched their results to a corresponding generating function computed in the dual gauge theory. In [47], the Riemann–Hilbert problem for the classical finite-gap solutions of the sigma model, restricted to an S2 ⊂ S5 subspace, was shown to be equivalent to the classical limit of the corresponding gauge theory Bethe equations. This result was extended to include the full sigma model in [48–51]. This was accompanied by [52], in which Berkovits used a pure-spinor formalism to argue that the tower of commuting conserved charges persists in the quantum theory (see also [23] for an exploration of these charges in the quantum theory near the BMN limit). Finally, we note that in [53] Arutyunov and Frolov were able to construct a Lax representation of the classical bosonic string Hamil- tonian in a specific gauge; this result was extended by Alday, Arutyunov and Tseytlin to the gauge-fixed physical superstring in [54]. (The approach to studying higher conserved charges using monodromy matrices was related to earlier investigations using Bäcklund transformations in [55].) Finally, we note that studies of integrability for semiclassical rotating strings were extended beyond the planar limit in [56,57]. On the gauge theory side, the presence of integrable structures was originally pointed out by Minahan and Zarembo [34], who showed that the action of the SYM dilatation operator on single-trace operators in a protected SO(6)-invariant sector of the theory can be mapped to that of an integrable Hamiltonian acting on a closed Heisenberg spin chain. The eigenvalues of the Hamiltonian, which are identified with corresponding operator dimensions, may thereby be obtained by solving a system of Bethe equations. Since we aim to formulate and solve a set of Bethe equations for the open string states examined in Sec- tion 2, we will briefly review how this procedure is applied to the su(2) sector of N = 4 SYM theory at one-loop order in λ. (We will return to the higher-loop treatment of the gauge theory in Section 4.) In the planar limit, single-trace operators in the su(2) sector are built from complex scalars, typically denoted by Z and φ.TheZ fields are charged under the U(1)R compo- ∼ nent of the SO(6) = U(1)R × SO(4) decomposition of the full SU(4)R-symmetry group, while the φ fields carry zero R-charge and act as impurity insertions. We obtain a basis of length-L operators by forming field monomials that carry N impurities and total R-charge equal to R = L − N:

− − − − − − − tr φN ZL N , tr φN 1ZφZL N 1 , tr φN 2Zφ2ZL N 1 ,....(3.1)

Unlike the so(6) sector originally studied by Minahan and Zarembo, which is protected from mixing at one-loop order in λ,thesu(2) sector is protected at all orders in perturbation T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 145

3 (2n) theory. There is a separate spin chain Hamiltonian Hsu(2) appropriate for each order in the λ expansion: = + 2n (2n) Hsu(2) N g Hsu(2), (3.2) n where g2 ≡ λ/8π 2. The loop expansion determines the generic form of the nth-order (2n) Hamiltonian, since Hsu(2) contains interactions that exchange ﬁelds separated by at most (2) n lattice sites. At one-loop order, Hsu(2) is given explicitly by N (2) = − Hsu(2) 2 (1 Pk,k+1), (3.3) k=1 where Pk,k+1 exchanges ﬁelds on the kth and (k + 1)th lattice sites. (The two- and three- loop extensions are given in [35–37].) The eigenvalues of this Hamiltonian can be obtained by solving the following Bethe equation N ipkL e = S(pk,pj ), (3.4) j=1 j =k where the closed-chain su(2) scattering matrix S(pk,pj ) is given by

uk − uj + i S(pk,pj ) = . (3.5) uk − uj − i The product on the right-hand side of Eq. (3.4) runs over all impurity excitations, excluding the kth impurity, and the Bethe roots uk encode the pseudomomenta pk of lattice excitations via the relation 1 p u = cot k . (3.6) k 2 2 The form of (3.4) can be understood intuitively: the exponent on the left-hand-side represents the phase acquired by a pseudoparticle excitation as it is transported once around the lattice, and the equation states that this phase must be equal to that which is acquired by scattering the excitation off of every other impurity as is passes around the chain. The S matrix interpretation of such equations was recently studied in detail in [59], where it was emphasized that the form of scattering matrices is tightly restricted by the constraints of integrability (scattering terms must be factorizable into pieces that encode at most two- body interactions; for related results relying on a virial expansion of the multi-loop spin chain Hamiltonians in N = 4 SYM theory, see [60]). For gauge-traced operators, which map to spin chain systems with periodic boundary conditions (i.e., closed spin chains),

3 In other words, the operators in Eq. (3.1) will not mix with other operators in the theory, and the anomalous dimension matrix can be diagonalized on this basis (at leading order in the large-Nc expansion). We note, however, that a recent study by Minahan [58] indicates that mixing might occur in the su(2) sector in the strong coupling regime. 146 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

Eq. (3.4) is supplemented by the condition N u + i/2 1 = k , (3.7) uk − i/2 k=1 which enforces momentum conservation on the lattice and corresponds to the usual level 5 matching condition in the dual sector of closed IIB string states in AdS5 × S . (In other { } words, the set of mode numbers nk of physical states on the lattice are required by = Eq. (3.7) to satisfy k nk 0.) The Hamiltonian of this system appears as one in an infinite tower of conserved charges in the theory which, as noted above, is typical of integrable systems. The entire infinite set of charges is diagonalized by the Bethe ansatz in Eqs. (3.4), (3.7), and the eigenvalues associated with each charge can be expressed in terms of the Bethe roots uk. Operator anomalous dimensions at O(λ) in this sector of the gauge theory are thus identified with the energy eigenvalues of the su(2) one-loop spin chain:

λ N 1 Esu(2) {ni} = . (3.8) 8π 2 u2 + 1/4 k=1 k The task of computing anomalous dimensions in this sector of N = 4 SYM theory is thereby reduced to that of solving Eqs. (3.4), (3.7) for the Bethe roots uk. This application of the Bethe ansatz was extended to the full theory at one-loop order by Beisert and Staudacher in [61], where the authors formulated a set of Bethe equations for the full set of psu(2, 2|4) ﬁelds in the theory. This methodology was extended beyond one-loop order in λ in [33], where Serban and Staudacher used an Inozemtsev spin chain [62,63] to capture higher-loop effects. (Such systems are referred to as long-range spin chains, because the Hamiltonian can mix spins that are not on neighboring lattice sites.) The energies (anomalous dimensions) admitted by the Inozemtsev system, however, do not conform to the requirements of BMN scaling at all orders in perturbation theory (i.e., at O(λn), anomalous dimensions should scale at ﬁnite R-charge as 1/R2n or, in the language of the string angular momentum, as 1/J 2n). Beisert, Dippel and Staudacher were subsequently able to formulate a long-range Bethe ansatz that met the constraints of BMN scaling and correctly reproduced su(2) anomalous dimensions to three-loop order in λ [31]. Higher-loop physics in the su(2) sector of the gauge theory is captured by the following generalized S matrix:

φ(pk) − φ(pj ) + i Ssu(2)(pk,pj ) = , (3.9) φ(pk) − φ(pj ) − i where the functions φ(pk) are given by 1 p p φ(p ) ≡ cot k 1 + 8g2 sin2 k . (3.10) k 2 2 2 (We note, however, that this scattering matrix is derived asymptotically and, strictly speak- ing, is only valid for long spin chains.) Multi-loop operator dimensions are identified with T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 147 the energies N 2 Esu(2) {ni},N,R = L + g q2(pk), (3.11) k=1 where the functions qr (pk) are defined as p r−1 pk + 2 2 k − 1 2sin( (r − 1)) 1 8g sin ( 2 ) 1 q (p ) ≡ 2 . (3.12) r k r−1 − pk g r 1 2g sin( 2 ) As described in the introduction, these formulas lead to a general disagreement with string theory at three-loop order in λ and beyond. The anomalous dimensions implied by the above scattering matrix disagree in this respect for all physical N-impurity states with 5 the corresponding near-pp-wave energy spectrum of the closed string theory on AdS5 × S [18–20,22]. In a remarkable paper by Arutyunov, Frolov and Staudacher [64], the authors discovered a modification to the long-range gauge theory S matrix in Eq. (3.9) that yields predictions consistent with the near-pp-wave energy spectrum of the string theory, and exhibits the correct λ1/4 scaling behavior at strong coupling [3]. They accomplished this by including an additional phase contribution Φ(pk,pj ) to the su(2) scattering matrix in Eq. (3.9). The resulting quantum string scattering matrix SIIB takes the form S5 IIB = SS5 (pk,pj ) Ssu(2)(pk,pj )Φ(pk,pj ), (3.13) where Φ(pk,pj ) is given by ∞ + g2 r 2 Φ(p ,p ) ≡ exp 2i q + (p )q + (p ) − q + (p )q + (p ) . k j 2 r 2 k r 3 j r 3 k r 2 j r=0 (3.14) It was demonstrated in [22] that the energies implied by the corresponding quantum string Bethe equations perfectly match energy spectra derived directly from the string theory for general closed string states in the su(2) sector (in the near-pp-wave limit). Since the energy spectra contain rather specialized structures (as we have already seen in the case of the open string states in Section 2), this is a non-trivial result: it stands as evidence that the quantum string theory is integrable, at least in this limit. In what follows we apply similar methods to the su(2) and sl(2) sectors of the bosonic open string states studied above. Inspired by the Bethe ansatz methodology presented at one-loop order for the matching of open string energies with the dimensions of corresponding gauge theory operators in [5–8], we derive long-range Bethe equations that diagonalize the interacting Hamiltonian HBB on completely general open string states.

3.1. Dirichlet SO(4)S5 (su(2)) ansatz

On the open spin chain, pseudoparticle excitations (or magnons) pass from one end of the chain, scatter off of the boundary and return to the starting point. Each transition across the length of the chain contributes a factor of exp(ipL) to the phase (the effect on the phase should be additive, so the momentum is understood to be oriented in the positive 148 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 direction after it scatters from the boundary). The total phase acquired in this process must be equal to that generated by scattering from each impurity on the chain as it travels to the boundary and back (after scattering from the boundary, the relative momenta of the other impurities on the chain acquire an overall sign change). We use this simple picture to propose a long-range Bethe ansatz for the su(2) Dirichlet sector of bosonic open string states4: N 2ipkL = IIB IIB − e SS5 (pk,pj )SS5 (pk, pj ) j=1 j =k N = Ssu(2)(pk,pj )Ssu(2)(pk, −pj )Φ(pk,pj )Φ(pk, −pj ). (3.15) j=1 j =k

The long-range scattering matrices Ssu(2) are deﬁned in Eq. (3.9) above, and the phase corrections Φ(pk,pj ) for the string are given in Eq. (3.14). We might have included some prefactor dependent on pk to capture phase contributions due to scattering from the boundary itself. We will see, however, that in this sector such contributions turn out to be trivial (this situation changes when we examine the sl(2) sector below). We also note that for open spin chains there is no momentum condition that enforces level matching among the mode numbers (just as there is no level matching condition for the open string states studied in Section 2). To simplify the presentation, and to review a general strategy for solving Eq. (3.15) [34, 64], we will ﬁrst compute energy spectra in the near-pp-wave limit for length-L states of the open chain whose impurities do not form bound states. These are the bosonic, symmetric- traceless open string states given in Eq. (2.15), with (n1 = n2 =··· = nN ). We must solve the Bethe ansatz in Eq. (3.15) order by order in the large-J expansion and extract contributions to the energy spectrum at O(1/J ). In terms of the string angular momentum J and the total impurity number N, the lattice length L for the corresponding integrable spin chain is

L = J − 1 + N. (3.16) When we analyze the dual gauge theory in Section 4 below we will see how this equation is determined. For the moment, however, we will proceed without further motivation. A useful technique, as demonstrated in [34,64], for example, is to expand the excitation momenta according to

πn p(1) p(2) p = k + k + k + O 1/J 5/2 , (3.17) k J J 3/2 J 2 (1) (2) and solve Eq. (3.15) for pk , pk , etc. The leading contribution carries the integer mode number nk of the pseudomomentum pk. (To compute energy eigenvalues to O(1/J ),we

4 This ansatz is motivated by gauge theory proposals at one-loop order in λ [6,7]. We note, however, that the speciﬁc equation given in [7] appears with a sign discrepancy. T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 149

2 will not need to find contributions to pk beyond O(1/J ).) We first note that the coeffi- (1) cient pk is non-zero only in the presence of bound momentum states, where some set of pk share the same mode number nk. In the absence of bound states, the contribution to Eq. (3.15) from the phase factors Φ(pk,pj ) is

Φ(pk,pj )Φ(pk, −pj ) 2inkπ 2 2 2 = 1 − n (ω − p−) − n (ω − p−) + O 1/J . (3.18) 2 − 2 k nj j nk Jp−(nk nj ) The Bethe equation in (3.15) is then satisﬁed by N (2) =− − − nkπ 2 − + 2 + pk nkπ(N 1) nk(ωnj p−) nj (ωnk p−) . (3.19) p−(n2 − n2) j=1 k j j =k From Eq. (3.11), we therefore ﬁnd the following contribution to the energy spectrum at O(1/J ): N (2) N n2 + n2λω2 1 nkpk 1 k j nk δE {ni},N,J = = . (3.20) J 4πp−ωn J ωn ωn k=1 k j,k=1 j k j =k This matches the corresponding energy correction for string states with no overlapping mode numbers computed directly from the string lightcone Hamiltonian in Eq. (2.24). Solving the Bethe equation in the presence of bound states is slightly more complicated. To align with conventions in the literature we will adopt a notation similar to that presented in [64], wherein the complete set of N mode numbers associated with each impurity excitation is divided into M subsets of equal mode numbers labeled by the index k: { } { } { } n1,n1,...,n1, n2,n2,...,n2,..., nM ,nM,...,nM . (3.21) k=1 k=2 k=M

The kth subset contains Nk total numbers nk, which are individually labeled by an index mk ∈ 1,...,Nk. The Bethe ansatz therefore takes the general form

Nk = IIB IIB − exp(2ipk,mk L) SS5 (pk,mk ,pk,lk )SS5 (pk,mk , pk,lk ) lk=1 lk =mk M Nj × IIB IIB − SS5 (pk,mk ,pj,mj )SS5 (pk,mk , pj,mj ), (3.22) j=1 mj =1 j =k where the momenta now carry the labels k and mk, and are expanded in large J according to (1) (2) πn p p p = k + k,mk + k,mk + O 1/J 5/2 . (3.23) k,mk J J 3/2 J 2 150 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

Within each bound state, the contribution from factors of the phase correction Φ(pk,mk , pk,lk ) is − Φ(pk,mk ,pk,lk )Φ(pk,mk , pk,lk ) = − inkπ 2 + 2 − + 2 1 8p− nk 8p−ωnk O 1/J , (3.24) 4p−ωnk J and the coefﬁcient p(1) is now nonzero in the presence of bound states: k,mk

N n2π 2ω k 1 (1) =− k nk pk,m . (3.25) k p− p(1) − p(1) lk=1 k,mk k,lk lk =mk As for the closed-chain case in [64], we have found a generalized Stieltjes problem [65] which is solved by setting

n2π 2ω (1) 2 =− k nk 2 p uN ,m , (3.26) k,mk 2p− k k where uN ,m are the roots of the Hermite polynomials QN (u) satisfying Q (u) − k k k uQ (u) + NkQ(u) = 0. At the next subleading order we ﬁnd that p(2) is given by k,mk Nk (2) 1 (2) (2) ωnk p = (1 − N)πnk − p − p + nkπ − 2 k,mk 2 k,mk k,lk p− lk=1 lk =mk M n πN − k j 2 − + 2 + nk(ωnj p−) nj (ωnk p−) . (3.27) p−(n2 − n2) j=1 k j j =k We can simplify matters by noting that when we expand the energy formula in Eq. (3.11) to the order of interest, the correction at O(1/J ) involves a sum over the index mk:

M 2Nkωnk E {nk} = J + p− k=1 M Nk 1 1 2 (1) 2 2 (2) + p− p + 2πn ω p . (3.28) 2 3 k,mk k nk k,mk p−J 8π ωn k=1 mk=1 k

Under this sum the roots uNk,mk of the Hermite polynomials QNk (u) satisfy

Nk u2 = N (N − 1). (3.29) Nk,mk k k mk=1 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 151

Furthermore, contributions to the right-hand side of Eq. (3.27) involving p(2) and p(2) k,mk k,lk will cancel: N k n k p(2) k,mk 4πωnk mk=1 2 N n N N ωn ω = k N ( − N ) + j k n4 j − n4 nk . k 1 k 2 2 k j (3.30) 8p− 8p−(n − n ) ωn ωn j=1 j k k j j =k We therefore arrive at a ﬁnal expression for the correction to the energy spectrum at O(1/J ) in the near-pp-wave limit δE 5 {ni}, {Ni},M,J S M 2 2 M 2 + 2 2 1 n (6 + n λ ) n n λ ωn =− N (N − ) k k + N N k j k . k k 1 2 j k (3.31) 8J 4ω ωn ωn k=1 nk j,k=1 j k j =k This matches the general su(2) string theory prediction in Eq. (2.28).

3.2. Neumann SO(4)AdS (sl(2)) ansatz

We can extend this analysis to the protected sl(2) sector of bosonic open string states. The open-string Bethe equation we propose is based on the long-range quantum string ansatz presented in [59] for the corresponding sector of sl(2) closed-string states:

N ipkL = IIB e SAdS(pk,pj ), (3.32) j=1 j =k

IIB where the scattering matrix SAdS is given by φ(p ) − φ(p ) − i IIB ≡ k j SAdS(pk,pj ) Ψ(pk,ppj ), (3.33) φ(pk) − φ(pj ) + i and the phase Ψ(pk,ppj ) is deﬁned as Ψ(p ,p ) k j ∞ + g2 r 1 ≡ exp −2i q + (p )q + (p ) − q + (p )q + (p ) . (3.34) 2 r 1 k r 2 j r 2 k r 1 j r=0 This ansatz was derived in [59] using general considerations of integrability and of the structure of the near-pp-wave energy spectrum computed from the string theory [22].The lattice length L obeys

L = J − 1. (3.35) 152 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

This formula is again based on gauge theory considerations, and will be discussed in Sec- tion 4. Using this scattering matrix, we propose the following general Bethe ansatz (allowing for conﬂuent mode numbers) for the sl(2) sector of open string states ﬂuctuating in the Neumann directions of the AdS5 subspace

Nk − = 2ipk,mk IIB IIB − exp(2ipk,mk L) e SAdS(pk,mk ,pk,lk )SAdS(pk,mk , pk,lk ) lk=1 lk =mk M Nj × IIB IIB − SAdS(pk,mk ,pj,mj )SAdS(pk,mk , pj,mj ). (3.36) j=1 mj =1 j =k −2ip The factor e k,mk on the right-hand side is a phase contribution interpreted as being associated with scattering from the boundary. Note also that in the small-λ expansion contributions to the Bethe equation from Ψ(pk,pj ) enter at a lower order than those for Φ(pk,pj ) in the su(2) case considered above. Expanding the pseudoparticle momenta according to Eq. (3.23), we find the following equation at first subleading order: N n2π 2ω k 1 (1) = k nk pk,m . (3.37) k p− p(1) − p(1) lk=1 k,mk k,lk lk =mk This differs by an overall sign from the corresponding su(2) equation computed in (3.25) above. At next order in the 1/J expansion we find N k M F (n ,n ) (2) = 1 (2) − (2) − nkπ + 1 j k pk,m pk,l pk,m ωnk Nj , (3.38) k 2 k k p− F2(nj ,nk) lk=1 j=1 lk =mk j =k where, for convenience, we have defined √ 2 2 2 2 F1(nj ,nk) ≡−n n π 8 + n + n λ − λ 8 + nk(nk − nj )λ ωn j k j k √ j + ω 8 + n (n − n )λ − 8 λω , (3.39) nk j j k √ nj √ ≡ 2 − 2 − + + + − F2(nj ,nk) nj nk 4 nj nkλ 4 λ (ωnk ωnj λ ωnk ωnj ) . (3.40) Using the general expansion of the energy in Eq. (3.28) above, we arrive at the following formula for the O(1/J ) energy shift in the near-pp-wave limit: δE {n }, {N },M,J AdS i i 1 M n2(2 + n2λ ) M n2n2λ =− N (N − ) k k + N N j k . k k 1 2 j k (3.41) 32J ω ωn ωn k=1 nk j,k=1 j k j =k This is precisely the energy expression computed for the sl(2) sector of the string theory in Eq. (2.32). T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 153

4. SYM long-range Bethe equations

As noted in Section 3, a long-range Bethe ansatz for the protected sector of single-trace su(2) operators in N = 4 SYM theory was formulated in [31], and an analogous sl(2) ansatz was given more recently in [59]. Building on our successful derivation of the open- string su(2) Bethe equations in (3.15), (3.22) based on corresponding equations formulated for closed string states, we can immediately write down long-range, open-chain Bethe ansatz equations for su(2) and sl(2) sectors of the N = 2 defect conformal ﬁeld theory dual to the corresponding open string states described in Section 2. The theory we are interested in describes a stack of Nc coincident D3-branes with one D5-brane extended along three of the worldvolume dimensions of the D3 stack (see Table 1). This theory, studied in [7, 8,39], contains a three-dimensional N = 4 SU(Nc) hypermultiplet in addition to the bulk four-dimensional hypermultiplet. The D5-brane defect preserves an SO(3, 2) subgroup of the conformal group and eight of the supersymmetries, though it breaks the R-symmetry from SU(4) to SU(2)H × SU(2)V . We can decompose the bulk N = 4 vector multiplet into a three-dimensional N = 4 vector multiplet and a three-dimensional N = 4 adjoint hypermultiplet. The vector multiplet contains the following bosonic ﬁelds:

1 2 3 I Aµ,X,X,X,D3X , with µ ∈ 0, 1, 2 and I ∈ 4, 5, 6. The hypermultiplet contains the component of the gauge 4 5 6 A field normal to the defect, as well as the scalars X , X , X and D3X , with A ∈ 1, 2, 3. m The three-dimensional N = 4 SU(Nc) hypermultiplet contains a set of complex scalars q (with m ∈ 1, 2) which couple canonically to the gauge fields. On the string side we take the Penrose limit by boosting in the direction dual to the (1, 2) plane. The charge J 3 SU(2)H is identified with the string angular momentum J (which we will refer to simply as the R-charge), and the fields of interest are thus charged under SU(2)H according to

X1,X2,X3 (J = 1), X4,X5,X6 (J = 0), qm, q˜m (J = 1/2). The Bethe ground state is taken to be

q¯1Z ···Zq2, with Z ≡ X1 + iX2. (The indices on the ﬁelds qm are lowered by Chan-Paton factors; see [7] for further details.) R-charge impurities in the su(2) sector are denoted by W ≡ X4 + iX5, and we form a basis of length-L (i.e., operator monomials containing a total of L ﬁelds), N-impurity operators given by

N L−N−2 N−1 L−N−3 q¯1W Z q2, q¯1W ZWZ q2, N−2 2 L−N−4 q¯1W ZW Z q2, .... (4.1) This basis corresponds to the su(2) sector of open Dirichlet string states in the S5 subspace. With the R-charge assignments given above, we now have the relation given in Eq. (3.16):

Lsu(2) = J − 1 + N. (4.2) 154 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

The sector of sl(2) open string states with Neumann boundary conditions is dual to operators with derivative insertions (with D ≡ D1 + iD2): N L−N−2 N−1 L−N−3 q¯1D Z q2, q¯1D ZDZ q2, N−2 2 L−N−4 q¯1D ZD Z q2,... (4.3) so that the lattice length of the corresponding sl(2) open spin chain is

Lsl(2) = J − 1, (4.4) as given in Eq. (3.35). By mapping the dilatation generator to the Hamiltonian of an integrable open spin chain, Bethe ansatz equations for similar sectors of operators were derived at one-loop order in λ in [6,7].

4.1. Long-range su(2) ansatz

We will focus ﬁrst on the closed-chain, long-range scattering matrix for the su(2) spin chain, derived in [31] and given above in Eq. (3.9). In terms of this S matrix, we propose the following long-range, open-chain ansatz for this sector: N 2ipkL e = Ssu(2)(pk,pj )Ssu(2)(pk, −pj ). (4.5) j =k As discussed in Section 3, we could include a phase contribution due to scattering at the boundaries. Given what we have learned from the corresponding su(2) string Bethe equations, however, we expect that such contributions are trivial. The full Bethe equation takes the form

Nk = − exp(2iLpk,mk ) Ssu(2)(pk,mk ,pk,lk )Ssu(2)(pk,mk , pk,lk ) lk=1 lk =mk M Nj × − Ssu(2)(pk,mk ,pj,mj )Ssu(2)(pk,mk , pj,mj ). (4.6) j=1 mj =1 j =k Expanding pseudoparticle momenta in the usual fashion

(1) (2) n π p p p = k + k,mk + k,mk + O 1/J 3/2 , (4.7) k,mk J J 3/2 J we find that Eq. (4.6) is satisfied to first subleading order within the kth subset of overlapping mode numbers for

N n2π 2ω k 1 (1) =− k nk pk,m . (4.8) k p− p(1) − p(1) lk=1 k,mk k,lk lk =mk T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 155

As in the case of the string Bethe ansatz above, this Stieltjes problem is solved by setting π 2n2ω (1) 2 =− k nk 2 p uN ,m , (4.9) k,mk 2p− k k which is identical to the string theory solution in Eq. (3.26). At the next subleading order we find Nk 1 p−πn p(2) = (1 − N)πn − p(2) − p(2) − k k,mk k k,mk k,lk 2 ωnk lk=1 lk =mk M 2 2πnknj ωnk + Nj . (4.10) p−(n2 − n2) j=1 j k j =k The contribution from p(2) to the O(1/J ) energy correction again involves a sum over k,mk the index m , so that contributions to δE from terms proportional to p(2) − p(2) will k k,mk k,lk cancel. Proceeding in the same fashion as before, we find a final expression for the energy spectrum at O(1/J ) in the near-BMN limit: δEsu(2) {ni}, {Ni},M,J M 2 M 2 1 nk 1 nk = N (N − )(p− − ω ) − N N . 2 k k 1 4 nk j k (4.11) p−J 16ω p−J 4ωn k=1 nk k,j=1 k j =k The energy formula again breaks neatly into contributions associated with scattering within and between bound momentum states of the spin chain. Expanding in small λ, we obtain δEsu(2) {ni}, {Ni},M,J M 1 3 1 1 = N (N − 1) − n2λ + n4(λ )2 − n6(λ )3 J k k 16 k 64 k 512 k k=1 M 1 1 1 + N N − n2 + n2 λ + n4 + n4 (λ )2 J j k 8 j k 64 j k k,j=1 = j k 3 − n6 + n6 (λ )3 + O (λ )4 . (4.12) 1024 j k Once again, we find perfect agreement with the corresponding su(2) string theory predictions in Eq. (2.29) at one- and two-loop order, but a disagreement at three loops and beyond!

4.2. Long-range sl(2) ansatz

One might guess that the correct closed-chain N = 4 sl(2) scattering matrix can be obtained from Eq. (3.33) simply by eliminating the phase contribution from Ψ(pk,pj ). 156 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

From the structure of the phase factor Ψ(pk,pj ), we can immediately see that this would lead to a general mismatch at two-loop order and beyond. It turns out that this simple hypothesis is incorrect. The correct asymptotic gauge theory scattering matrix was derived in [59]

φ(p ) − φ(p ) − i k j 2iθ(pk,pj ) Ssl(2)(pk,pj ) ≡ e . (4.13) φ(pk) − φ(pj ) + i

The phase shift θ(pk,pj ) was originally formulated in the computation of a long-range scattering matrix for the fermionic su(1|1) sector of N = 4 SYM via Staudacher’s method of the perturbative asymptotic Bethe ansatz (see [59] for further details)

iθ(pk,pj ) Ssu(1|1)(pk,pj ) =−e . (4.14) The sl(2)Smatrix in Eq. (4.13) was then derived from this su(1|1) scattering matrix and the corresponding matrix in the su(2) sector (3.9) using the following remarkable relationship:

= −1 Ssl(2) Ssu(1|1)Ssu(2)Ssu(1|1). (4.15) According to [59], this equation appears to hold for both the string theory and gauge theory. The phase θ(pk,pj ) was calculated perturbatively in small λ in [59], yielding: p p θ(p ,p ) = 2g2 sin2 k sin p − 2g4 sin4 k sin 2p k j 2 j 2 j p pj + 8g4 sin p sin2 k sin2 − (p p ). (4.16) k 2 2 k j

The expansion to this order (O(λ2)) makes the resulting scattering matrix (4.13) accurate up to and including three-loop order in λ; a higher-order expansion would require speciﬁc knowledge of certain four-loop vertices in the N = 4 gauge theory.5 In terms of the sl(2)Smatrix in Eq. (4.13), we arrive at the following long-range, open-chain Bethe ansatz for the sl(2) sector (4.3) of the defect conformal ﬁeld theory:

N −2ipk exp(2ipkL) = e Ssl(2)(pk,pj )Ssl(2)(pk, −pj ). (4.17) j=1 j =k

− The additional phase shift e 2ipk includes the effects of boundary scattering, analogous to the corresponding open string ansatz in Eq. (3.36). We can solve this equation in the near-BMN limit using the methods described in detail above. Omitting the computational details, we ﬁnd the following general energy shift (for all possible mode-index distributions) in the near-BMN limit

5 We thank Matthias Staudacher for clariﬁcation on this point. T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 157 δEsl(2) {ni}, {Ni},M,J M 1 1 1 3 = N (1 − N ) n2λ + n4(λ )2 − n6(λ )3 J k k 16 k 64 k 512 k k=1 M 1 1 5 + N N − n2n2(λ )2 + n2n2 n2 + n2 (λ )3 J j k 32 j k 1024 j k j k j,k=1 j =k + O (λ )4 . (4.18) Comparing with the corresponding expansion in the near-pp-wave limit of the open-string sl(2) Neumann sector in Eq. (2.33), we again ﬁnd precise agreement at one- and two-loop order in λ, followed by a mismatch at three loops and beyond.

5. Summary and discussion

We have formulated a set of long-range Bethe ansätze for open string states that arise 5 6 from standard deformations of IIB string theory on AdS5 × S . In a bosonic symmetric- traceless S5 sector we studied open string states with Dirichlet boundary conditions, labeled by an su(2) subalgebra of psu(2, 2|4). The open-string Bethe ansatz is formulated in this sector from the closed-string scattering matrix SIIB(p ,p ) in Eq. (3.13) according to S5 k j ipkL = IIB closed su(2) strings: e SS5 (pk,pj ), (5.1) = j k 2ipkL = IIB IIB − open su(2) strings: e SS5 (pk,pj )SS5 (pk, pj ). (5.2) j =k In this sector we found that there is no phase contribution from boundary scattering. The corresponding prescription in the sector of sl(2) symmetric-traceless open string states ﬂuctuating in AdS5 with Neumann boundary conditions was found to be ipkL = IIB closed sl(2) strings: e SAdS(pk,pj ), (5.3) = j k − 2ipkL = 2ipk IIB IIB − open sl(2) strings: e e SAdS(pk,pj )SAdS(pk, pj ). (5.4) j =k IIB The closed-string scattering matrix SAdS(pk,pj ) appears in Eq. (3.33). In this sector the open-string Bethe equation receives a contribution that may be interpreted as being due to − boundary scattering; this appears as the e 2ipk prefactor on the right-hand side of Eq. (5.4). In both sectors we computed the near-pp-wave energy spectra of general N-impurity states directly from the string theory. The Bethe ansätze presented here reproduce these results precisely, which is consistent with the expectations of integrability.

6 Let us note that our methods may also be useful in more general deformations of the theory, some examples of which were considered recently in [66]. 158 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

Fig. 3. Wrapping interaction diagram.

When the above prescription is applied to the corresponding sectors of operators in the dual N = 2 defect conformal field theory, we obtain ipkL closed su(2) SYM spin chains: e = Ssu(2)(pk,pj ), (5.5) = j k 2ipkL open su(2) SYM spin chains: e = Ssu(2)(pk,pj )Ssu(2)(pk, −pj ), j =k (5.6) in the su(2) sector, and ipkL closed sl(2) SYM spin chains: e = Ssl(2)(pk,pj ), (5.7) = j k 2ipkL −2ipk open sl(2) SYM spin chains: e = e Ssl(2)(pk,pj )Ssl(2)(pk, −pj ), j =k (5.8) in the sl(2) sector. We find that the operator anomalous dimensions (or energy spectra of the corresponding spin chains) in both the N = 4 (5.5), (5.7) and N = 2 (5.6), (5.8) gauge theories match the energy spectra of general su(2) and sl(2) physical string states to one- and two-loop order in λ, but disagree at three-loop order and higher. In [31], Beisert, Dippel and Staudacher suggested a generic type of term that could lead to order-of-limits issues in the comparison of string and gauge theory in this setting:7 λL . (c + λ)L Terms of this form will lead to different contributions depending on the sequence of limits taken in Fig. 1. While the range of interactions associated with such terms would typically be greater than the length of the chain, they would not necessarily have to wrap around the spin chain: they need only involve a number of vertices that exceeds the total number of lattice sites on the chain. For closed chains, this could obviously include wrapping interactions, pictured schematically in Fig. 3. For both open and closed chains, however, these interactions might also be of the form shown in Fig. 4, which does not require periodicity on the lattice. We note that, by definition, both types of diagram are absent in the limit of infinite chain length. (Other examples include interactions that are defined to stretch between the boundaries of an open chain, or contain a large number of vertices associated with other intermediate states.)

7 Apart from the fact that such terms might explain the mismatch with string theory beyond two loops, there is no other reason to suspect that the gauge theory produces such functions. See [31] for details. T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 159

Fig. 4. A long-range, non-wrapping diagram.

In the su(2) sector, the difference between the closed-chain scattering matrices for the string and gauge theory is collected into the phase Φ(pk,pj ): IIB = SS5 (pk,pj ) Ssu(2)(pk,pj )Φ(pk,pj ). (5.9) If the general disagreement between string and gauge theory is due to wrapping interactions alone, we should expect that the contribution from this phase correction to the open-string Bethe ansatz

Φ(pk,pj )Φ(pk, −pj ) (5.10) will drop out. While some terms from Φ(pk,pj ) do cancel in Eq. (5.6), there is still a non- zero contribution from these phases to the open string energy spectrum (see Eqs. (3.18) and (3.24)). One conjecture is that this observation in fact isolates stringy contributions to the Bethe equations from each type of interaction pictured in Figs. 3 and 4. Contributions from long-range wrapping (Fig. 3) and non-wrapping (Fig. 4) interactions might then be encoded by those terms in Φ(pk,pj ) that do or do not cancel in the phase correction (5.10) to the open-string Bethe equations, respectively. Another interpretation of our results is that the prescription for extending the closed- string, long-range Bethe ansatz to open strings (5.2), (5.4) cannot be carried over analogously to the dual gauge theory (5.6), (5.8). It is a logical possibility that the discrepancy between the closed-string and closed-chain energy spectra is due solely to wrapping interactions, and that in moving to the open-chain formulation of the gauge theory we have eliminated all order-of-limits issues, but at the same time introduced some other set of unwanted contributions. While this seems unlikely, a direct higher-loop gauge theory calculation might settle any uncertainty.

Acknowledgements

We would like to thank John Schwarz and Curt Callan for their ongoing support and encouragement. We also thank Niklas Beisert, Andrei Mikhailov and Matthias Staudacher for explanations and useful discussions. This work was supported by US Department of Energy grant DE-FG03-92-ER40701.

References

[1] D. Berenstein, J.M. Maldacena, H. Nastase, Strings in ﬂat space and pp waves from N = 4 super-Yang– Mills, JHEP 0204 (2002) 013, hep-th/0202021. 160 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

[2] E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253–291, hep-th/9802150. [3] S.S. Gubser, I.R. Klebanov, A.M. Polyakov, Gauge theory correlators from non-critical string theory, Phys. Lett. B 428 (1998) 105–114, hep-th/9802109. [4] J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231–252, hep-th/9711200. [5] D. Berenstein, E. Gava, J.M. Maldacena, K.S. Narain, H. Nastase, Open strings on plane waves and their Yang–Mills duals, hep-th/0203249. [6] B. Chen, X.J. Wang, Y.S. Wu, Integrable open spin chain in super-Yang–Mills and the plane-wave/SYM duality, JHEP 0402 (2004) 029, hep-th/0401016. [7] O. DeWolfe, N. Mann, Integrable open spin chains in defect conformal field theory, JHEP 0404 (2004) 035, hep-th/0401041. [8] P. Lee, J.-W. Park, Open strings in pp-wave background from defect conformal field theory, Phys. Rev. D 67 (2003) 026002, hep-th/0203257. [9] J. Stefanski, Open spinning strings, JHEP 0403 (2004) 057, hep-th/0312091. [10] Y. Susaki, Y. Takayama, K. Yoshida, Open semiclassical strings and long defect operators in AdS/CFT correspondence, hep-th/0410139. [11] B. Chen, X.-J. Wang, Y.-S. Wu, Open spin chain and open spinning string, Phys. Lett. B 591 (2004) 170–180, hep-th/0403004. [12] V. Balasubramanian, M.-X. Huang, T.S. Levi, A. Naqvi, Open strings from N = 4 super-Yang–Mills, JHEP 0208 (2002) 037, hep-th/0204196. [13] H. Takayanagi, T. Takayanagi, Open strings in exactly solvable model of curved space-time and pp-wave limit, JHEP 0205 (2002) 012, hep-th/0204234. [14] Y. Imamura, Open string: BMN operator correspondence in the weak coupling regime, Prog. Theor. Phys. 108 (2003) 1077–1097, hep-th/0208079. [15] K. Skenderis, M. Taylor, Open strings in the plane wave background. I: Quantization and symmetries, Nucl. Phys. B 665 (2003) 3–48, hep-th/0211011. [16] K. Skenderis, M. Taylor, Open strings in the plane wave background. II: Superalgebras and spectra, JHEP 0307 (2003) 006, hep-th/0212184. [17] A. Parnachev, A.V. Ryzhov, Strings in the near plane wave background and AdS/CFT, JHEP 0210 (2002) 066, hep-th/0208010. [18] C.G. Callan Jr., H.K. Lee, T. McLoughlin, J.H. Schwarz, I. Swanson, X. Wu, Quantizing string theory in 5 AdS5 × S : Beyond the pp-wave, Nucl. Phys. B 673 (2003) 3–40, hep-th/0307032. [19] C.G. Callan Jr., T. McLoughlin, I. Swanson, Holography beyond the Penrose limit, Nucl. Phys. B 694 (2004) 115–169, hep-th/0404007. [20] C.G. Callan Jr., T. McLoughlin, I. Swanson, Higher impurity AdS/CFT correspondence in the near-BMN limit, Nucl. Phys. B 700 (2004) 271–312, hep-th/0405153. 5 [21] I. Swanson, On the integrability of string theory in AdS5 × S , hep-th/0405172. [22] T. McLoughlin, I. Swanson, N-impurity superstring spectra near the pp-wave limit, Nucl. Phys. B 702 (2004) 86–108, hep-th/0407240. [23] I. Swanson, Quantum string integrability and AdS/CFT, Nucl. Phys. B 709 (2005) 443–464, hep-th/0410282. [24] M. Blau, J. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, A new maximally supersymmetric background of IIB superstring theory, JHEP 0201 (2002) 047, hep-th/0110242. [25] M. Blau, J. Figueroa-O’Farrill, C. Hull, G. Papadopoulos, Penrose limits and maximal supersymmetry, Class. Quantum Grav. 19 (2002) L87–L95, hep-th/0201081. [26] M. Blau, J. Figueroa-O’Farrill, G. Papadopoulos, Penrose limits, supergravity and brane dynamics, Class. Quantum Grav. 19 (2002) 4753, hep-th/0202111. [27] N. Beisert, S. Frolov, M. Staudacher, A.A. Tseytlin, Precision spectroscopy of AdS/CFT, JHEP 0310 (2003) 037, hep-th/0308117. [28] A.A. Tseytlin, Spinning strings and AdS/CFT duality, hep-th/0311139. [29] G. Arutyunov, M. Staudacher, Two-loop commuting charges and the string/gauge duality, hep-th/0403077. [30] J.A. Minahan, Higher loops beyond the SU(2) sector, JHEP 0410 (2004) 053, hep-th/0405243. [31] N. Beisert, V. Dippel, M. Staudacher, A novel long range spin chain and planar N = 4 super-Yang–Mills, JHEP 0407 (2004) 075, hep-th/0405001. T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162 161

[32] I.R. Klebanov, M. Spradlin, A. Volovich, New effects in gauge theory from pp-wave superstrings, Phys. Lett. B 548 (2002) 111–118, hep-th/0206221. [33] D. Serban, M. Staudacher, Planar N = 4 gauge theory and the Inozemtsev long range spin chain, JHEP 0406 (2004) 001, hep-th/0401057. [34] J.A. Minahan, K. Zarembo, The Bethe-ansatz for N = 4 super-Yang–Mills, JHEP 0303 (2003) 013, hep- th/0212208. [35] N. Beisert, C. Kristjansen, M. Staudacher, The dilatation operator of N = 4 super-Yang–Mills theory, Nucl. Phys. B 664 (2003) 131–184, hep-th/0303060. [36] N. Beisert, The su(2|3) dynamic spin chain, Nucl. Phys. B 682 (2004) 487–520, hep-th/0310252. [37] N. Beisert, The dilatation operator of N = 4 super-Yang–Mills theory and integrability, Phys. Rep. 405 (2005) 1–202, hep-th/0407277. [38] N. Beisert, Higher-loop integrability in N = 4 gauge theory, C. R. Physique 5 (2004) 1039–1048, hep- th/0409147. [39] O. DeWolfe, D.Z. Freedman, H. Ooguri, Holography and defect conformal ﬁeld theories, Phys. Rev. D 66 (2002) 025009, hep-th/0111135. [40] S.G. Naculich, H.J. Schnitzer, N. Wyllard, pp-wave limits and orientifolds, Nucl. Phys. B 650 (2003) 43–74, hep-th/0206094. [41] R.R. Metsaev, Type IIB Green–Schwarz superstring in plane wave Ramond–Ramond background, Nucl. Phys. B 625 (2002) 70–96, hep-th/0112044. [42] R.R. Metsaev, A.A. Tseytlin, Exactly solvable model of superstring in plane wave Ramond–Ramond background, Phys. Rev. D 65 (2002) 126004, hep-th/0202109. [43] A. Dabholkar, S. Parvizi, Dp-branes in pp-wave background, Nucl. Phys. B 641 (2002) 223–234, hep- th/0203231. [44] G. Mandal, N.V. Suryanarayana, S.R. Wadia, Aspects of semiclassical strings in AdS5, Phys. Lett. B 543 (2002) 81–88, hep-th/0206103. 5 [45] I. Bena, J. Polchinski, R. Roiban, Hidden symmetries of the AdS5 × S superstring, Phys. Rev. D 69 (2004) 046002, hep-th/0305116. [46] G. Arutyunov, M. Staudacher, Matching higher conserved charges for strings and spins, JHEP 0403 (2004) 004, hep-th/0310182. [47] V.A. Kazakov, A. Marshakov, J.A. Minahan, K. Zarembo, Classical/quantum integrability in AdS/CFT, JHEP 0405 (2004) 024, hep-th/0402207. [48] V.A. Kazakov, K. Zarembo, Classical/quantum integrability in non-compact sector of AdS/CFT, JHEP 0410 (2004) 060, hep-th/0410105. [49] N. Beisert, V.A. Kazakov, K. Sakai, Algebraic curve for the SO(6) sector of AdS/CFT, hep-th/0410253. [50] S. Schafer-Nameki, The algebraic curve of 1-loop planar N = 4 SYM, hep-th/0412254. 5 [51] N. Beisert, V.A. Kazakov, K. Sakai, K. Zarembo, The algebraic curve of classical superstrings on AdS5 ×S , hep-th/0502226. 5 [52] N. Berkovits, Quantum consistency of the superstring in AdS5 × S background, hep-th/0411170. 5 [53] G. Arutyunov, S. Frolov, Integrable Hamiltonian for classical strings on AdS5 × S , JHEP 0502 (2005) 059, hep-th/0411089. 5 [54] L.F. Alday, G. Arutyunov, A.A. Tseytlin, On integrability of classical superstrings in AdS5 × S ,hep- th/0502240. 5 [55] G. Arutyunov, M. Zamaklar, Linking Bäcklund and monodromy charges for strings on AdS5 × S ,hep- th/0504144. [56] K. Peeters, J. Plefka, M. Zamaklar, Splitting spinning strings in AdS/CFT, JHEP 0411 (2004) 054, hep- th/0410275. [57] K. Peeters, J. Plefka, M. Zamaklar, Splitting strings and chains, Fortschr. Phys. 53 (2005) 640–646, hep- th/0501165. [58] J.A. Minahan, The SU(2) sector in AdS/CFT, hep-th/0503143. [59] M. Staudacher, The factorized S-matrix of CFT/AdS, hep-th/0412188. [60] C.G. Callan Jr., J. Heckman, T. McLoughlin, I. Swanson, Lattice super-Yang–Mills: A virial approach to operator dimensions, Nucl. Phys. B 701 (2004) 180–206, hep-th/0407096. [61] N. Beisert, M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439–463, hep-th/0307042. 162 T. McLoughlin, I. Swanson / Nuclear Physics B 723 (2005) 132–162

[62] V.I. Inozemtsev, On the connection between the one-dimensional s = 1/2 Heisenberg chain and Haldane Shastry model, J. Stat. Phys. 59 (1990) 1143, JINR-E5-89-490. [63] V.I. Inozemtsev, Integrable Heisenberg–van Vleck chains with variable range exchange, Phys. Part. Nucl. 34 (2003) 166–193, hep-th/0201001. [64] G. Arutyunov, S. Frolov, M. Staudacher, Bethe ansatz for quantum strings, JHEP 0410 (2004) 016, hep- th/0406256. [65] B.S. Shastry, A. Dhar, Solution of a generalized Stieltjes problem, cond-mat/0101464. [66] S.A. Frolov, R. Roiban, A.A. Tseytlin, Gauge–string duality for superconformal deformations of N = 4 super-Yang–Mills theory, hep-th/0503192. Nuclear Physics B 723 (2005) 163–197

Finite volume gauge theory partition functions in three dimensions

Richard J. Szabo

Department of Mathematics, School of Mathematical and Computer Sciences, Heriot-Watt University, Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, UK Received 18 May 2005; received in revised form 21 June 2005; accepted 27 June 2005 Available online 20 July 2005

Abstract We determine the fermion mass dependence of Euclidean finite volume partition functions for three-dimensional QCD in the -regime directly from the effective field theory of the pseudo- Goldstone modes by using zero-dimensional non-linear σ -models. New results are given for an arbitrary number of flavours in all three cases of complex, pseudo-real and real fermions, extending some previous considerations based on random matrix theory. They are used to describe the microscopic spectral correlation functions and smallest eigenvalue distributions of the QCD3 Dirac operator, as well as the corresponding massive spectral sum rules.  2005 Elsevier B.V. All rights reserved.

1. Introduction and summary

The -regime of quantum chromodynamics in four spacetime dimensions (QCD4) [1,2] provides one of the few examples wherein exact non-perturbative results can be derived from QCD. It corresponds to the infrared sector of the Euclidean Dirac operator spectrum which is related to the mechanism of chiral symmetry breaking. The -expansion of chiral perturbation theory in this case, which describes the dynamics of the Goldstone modes [3], is taken with respect to the small quantity = L−1 where L is the linear size of the given spacetime volume. If L is much larger than the QCD scale, then spontaneously

E-mail addresses: [email protected], [email protected] (R.J. Szabo).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.06.028 164 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 broken chiral symmetry dictates the interactions and the leading contributions come from the zero-momentum pseudo-Goldstone fields. This regime represents a domain wherein a detailed understanding can be achieved of the lowest-lying Dirac operator eigenvalues, which dominate the physical low-energy observables, and in which numerical computations in lattice QCD are possible [4]. Most of the recent understanding of the -regime of QCD comes from its relation to universal random matrix theory results [5,6]. In this paper we will study the infrared behaviour of the Dirac operator spectrum in three-dimensional QCD (QCD3). In this case the analog of chiral symmetry breaking is played by the spontaneous breaking of flavour symmetry [7]. The finite volume gauge theory is connected to three universality classes in random matrix theory, the unitary ensemble (fundamental fermions with Nc 3 colours) [8,9], the orthogonal ensemble (fundamental fermions with colour group SU(2)) [10] and the symplectic ensemble (adjoint fermions with colour group SU(Nc)) [11]. The random matrix theory representation of the effective QCD3 partition function in the -regime has lead to a number of conjectures regarding the derivation of the spectral properties of the three-dimensional Dirac operator from joint eigenvalue probability distributions in the pertinent matrix model [8,12]. The patterns of flavour symmetry breaking in all three instances have been identified [8,10,11] and supported by Monte Carlo simulations [13], and the microscopic spectral correlation functions have been calculated [6,14–18]. To derive non-perturbative analytic results for the smallest eigenvalues of the Dirac operator it is necessary to have a complete proof that the universal random matrix theory results coincide with the low-energy limit of the effective field theory. From the field theoretic point of view [2] these observables should be computable entirely within the framework of finite volume partition functions which are generating functionals for the order parameter for flavour symmetry breaking [15].ForQCD3 this problem has only been thoroughly investigated for the unitary ensemble by using the supersymmetric formulation of partially quenched effective Lagrangians [19], the replica limit [20], and the relationship [21] between finite volume partition functions and the τ -function of an underlying integrable KP hierarchy [22,23]. The goal of this paper is to derive finite volume partition functions in all three cases mentioned above and to use them to explore spectral character- istics of the QCD3 Dirac operator directly in the low-energy effective field theory. In all three ensembles the effective field theory is described by a non-linear σ -model in a maximally symmetric compact space [8,10,11]. These target spaces fall into the classical Cartan classification of symmetric spaces, and confirm part of the Zirnbauer classification of random matrix universality classes in terms of Riemannian symmetric superspaces [24]. This enables us to exploit the well-known geometrical properties of these spaces [25,26] (see [27] for a recent review in the context of random matrix theory) and hence explore the differences between the three classes of fermions in QCD3. For example, we will find that the finite volume partition functions for adjoint fermions are drastically different in functional form from those for complex and pseudo-real fermions, and this difference can be understood from the form of the Riemannian metrics on the corresponding symmetric spaces. A similar geometric feature also exhibits the difference between even and odd numbers of quark flavour components. In the latter cases the theory is plagued by a severe sign problem which is rather delicate from the effective field theory point of view [8].We derive new results in all three instances for the finite volume partition functions with odd R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 165 numbers of flavours. In the case of the unitary ensemble, we use them to derive a new simplified expression for the microscopic spectral correlators, extending the known effective field theory results for the even-flavoured case [19]. We also study some of the spectral sum rules which constrain the eigenvalues of the Dirac operator [2,8,28], and briefly describe its smallest eigenvalue distribution functions [29]. On a technical level, we present a complete and thorough analysis only for the case of complex fermions. In the cases of pseudo-real and real fermions, we complete the derivations only in the cases of completely degenerate mass eigenvalues, as there are some algebraic obstructions which prevent us from obtaining closed analytical expressions for the finite volume partition functions for arbitrary configurations of quark masses. Never- theless, we present a very general algebraic description of the differences and similarities between the three universality classes, and also of the distinction between the even and odd flavoured cases, based on a remnant discrete subgroup of the flavour symmetry acting on the mass parameters of the gauge theory. This residual symmetry is described in general in Section 2, where we also review various aspects of the non-linear σ -models and describe the geometry of their symmetric target spaces in a general setting that applies also to other symmetric space σ -models such as those encountered in QCD4. Sections 3–5 then use this general formalism to analyse in detail the three classes of fermions in turn. In addition to obtaining new results in the odd-flavoured cases, the main new predictions of the effective field theory approach concern the (equal mass) partition functions for adjoint fermions, which have a far more intricate functional structure than the other ensembles and which involve functions usually encountered in QCD4 [2,30], not in QCD3.

2. Finite volume partition functions

The Euclidean Dirac operator governing the minimal coupling of quarks in the QCD3 action is given by µ iD/ = iσ (∂µ + iAµ), (2.1) where σ µ, µ = 1, 2, 3 are the Pauli spin matrices 01 0 −i 10 σ 1 = ,σ2 = ,σ3 = , (2.2) 10 i0 0 −1 and Aµ are gauge fields of the local colour group SU(Nc), Nc 2. When multi-colour fermion fields are coupled to this operator, the three-dimensional field theory also possesses a global flavour symmetry described by rotations of the quark field components via elements of some Lie group G. If one introduces quark masses in equal and opposite pairs, then the fermion determinant is positive definite and the quantum field theory is anomaly free [8]. The addition of such mass terms preserves three-dimensional spacetime parity [16] but breaks the flavour symmetry, and thereby permits one to analyse the possibility of spontaneous flavour symmetry breaking in QCD3. In this section we will construct the general low-energy effective field theory of the associated pseudo-Goldstone bosons in the ergodic -regime wherein exact, non-perturbative information about the spectrum of the QCD3 Dirac operator (2.1) can be obtained. 166 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

2.1. Non-linear σ -models of QCD3

We are interested in the mechanism of flavour symmetry breaking in three-dimensional QCD, which we consider defined at fixed ultraviolet cutoff Λ, while the quantum field theory in the infrared is regulated by a three-volume V . The effective field theory for the low-momentum modes of the Nambu–Goldstone bosons can be derived by using the three-dimensional version of chiral perturbation theory [3]. It is consistent with the flavour symmetry of the original quantum field theory represented through transformations of the quark fields by the Lie group G. In the ergodic regime, the effective Lagrangian in Euclid- ean space is given by 2 f ∗ Σ ∗ L = π Tr ∂ U∂µU − Tr M U + U + O M2 , (2.3) eff 4 µ 2 where fπ is the decay constant of the pseudo-Goldstone modes, Σ is the infinite-volume quark–antiquark condensate which is the order parameter for flavour symmetry breaking, and M is the Nf × Nf mass matrix induced by integrating out the Nf 1 flavours of fermion fields in the QCD3 partition function. When the number of fermion flavours Nf = 2n is even, so that n is the number of four-component spinors, we assume that the eigenvalues of M occur in parity-conjugate pairs (mi, −mi), mi 0, i = 1,...,n. When Nf = 2n + 1 is odd, there is a single unpaired quark of mass m0 0 which breaks parity symmetry explicitly when m0 > 0 and radiatively when m0 = 0 [31]. The higher order terms in (2.3) represent contributions involving gluons and confined quarks. The symbol Tr, in the representations that we shall use in this paper, will always correspond to an ordinary Nf ×Nf matrix trace. The Nambu–Goldstone field U lives in the appropriate vacuum manifold for the given symmetry breaking. We will use the Dyson index β = 1, 2, 4, which is borrowed from random matrix theory terminology [32]. It labels the anti-unitary symmetries of the QCD3 Dirac operator which in turn depends on the types of fermions that are present in the original field theory [9]. The Goldstone manifold is a coset space

Gβ (Nf ) = G/HΓ , (2.4) where HΓ is the stability subgroup of the original flavour symmetry group G which leaves fixed an Nf × Nf parity matrix Γ representing an involutive automorphism of G that reflects the parity symmetry of the original field theory. The fields in (2.3) may then be parametrized as U = UΓUt (2.5) with U ∈ Gβ (Nf ).In(2.5) the superscript t denotes the appropriate conjugation involution on the Lie group G, given by U t = U † for β = 2 and U t = U for β = 1, 4. Given the non-linear σ -model (2.3),wewouldnowliketostudytheregionwherethe zero-momentum mode of U dominates. This is the -regime in which the linear dimension of the system is much smaller than the Compton wavelength of the Nambu–Goldstone bosons. The spacetime integration over the effective Lagrangian (2.3) then produces an overall volume factor V , and the partition function of the non-linear σ -model simplifies to a finite-dimensional coset integral. The region of interest is obtained by simultaneously R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 167

→∞ M → taking the thermodynamic limit V and the chiral limit 0Nf of massless quarks, with the rescaled masses M = VΣM finite. The effective field theory of the Goldstone bosons is then the correct description of the dynamics below the scale set by the mass gap. Because of the tuning of the quark masses to zero, the higher-order corrections to this effective theory are exponentially small in the mass gap and come from integrating out the heavy states. The quantum field theory is thus adequately described by the finite volume partition function [8,10,11] (Nf ) = t Zβ (M) dνβ (U) exp Re Tr MUΓU

Gβ (Nf ) 1 q + 1 − (−1) Nf (−1)ND/ exp Re Tr −MUΓUt , (2.6) 2 = where qNf is the even/odd congruence class of the flavour number Nf given by q2n 0, q2n+1 = 1, and ND/ is the number of Dirac operator eigenvalues (defined, for example, in a lattice regularization or in random matrix theory). The second line of (2.6) only contributes when there is an odd number of fermion flavours. It arises from the fact that, generally, Nf ND/ the QCD3 partition function changes by a phase (−1) under a parity transformation of the quark masses. When Nf = 2n + 1 is odd, the fermion determinant is not positive definite and its sign determines two gauge inequivalent vacuum states which both have to be included in the low-energy limit of the QCD3 partition function in the -regime [8]. We will always ignore irrelevant numerical constants (independent of M) in the evaluation of (2.6), as they will not contribute to any physical quantities and can be simply absorbed into the normalization of the partition function. The measure dνβ (U) is inherited from the invariant Haar measure for integration over the Lie group G. It will be constructed in the next subsection. The low-energy effective partition function of the zero-dimensional σ -model (2.6) contains sufficient information to completely constrain the low-energy Dirac operator spectrum. For instance, by matching exact results from random matrix theory with the low-energy effective field theory (2.6), it is possible to express the microscopic limit of the spectral k-point correlation functions k (Nf ) ; M = − + M β (λ1,...,λk ) Tr δ(λl iD/ i ) (2.7) l=1 β of the QCD3 Dirac operator (2.1) in terms of a ratio of finite volume partition functions, namely (2.6) and one which involves βk additional species of valence quarks of imaginary masses iλl [15,22].In(2.7) the average is taken over SU(Nc) gauge field configurations weighted by the three-dimensional Yang–Mills action. Both (2.6) and the microscopic limit (Nf ) ± = ρ2 of (2.7) depend only on the eigenvalues µi , i 1,...,n(and µ0 when Nf is odd) of the scaled mass matrix M. 168 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

For β = 2, 4 one explicitly has

k n (N ) q β f ; = (k) + Nf 2 + 2 2 − 2 ρβ (ζ1,...,ζk M) Cβ (iζl µ0) ζl µi ζm ζp l=1 i=1 m

k (Nf ) (0,...,0; M) 1 k (Nf ) ; = β = lim ρβ (ζ1,...,ζk M) k . (2.9) ζ1,...,ζk→∞ (V Σ) π For β = 1, 2, 4, the ﬁnite volume partition functions may be used to compute the normal- (Nf ) ; [− ]⊂R ized hole probability Eβ (ζ M), i.e., the probability that the interval ζ,ζ is free of Dirac operator eigenvalues. One has the formula [29] (N ) Z f ( M2 + ζ 2) (Nf ) − 1 ζ 2 β E (ζ ; M)= e 4 (2.10) β (Nf ) Zβ (M) from which the smallest eigenvalue distribution is given by

(Nf ) ; (N ) ∂Eβ (ζ M) P f (ζ ; M)=− . (2.11) β ∂ζ

Furthermore, by matching (2.6) with the formal expansion of the full QCD3 partition function n (N ) p q p Z f M = 2 + 2 β + Nf β β ( ) det D/ mi det(D/ im0) (2.12) i=1 β with p1 = p2 = 1 and p4 = 1/2, one can express the mass-dependent susceptibilities in terms of derivatives of the ﬁnite volume partition functions with respect to one or more quark mass eigenvalues [28]. This leads to a set of massive spectral sum rules once (2.6) is known explicitly. The simplest such sum rule expresses the mass-dependent condensate for the ﬂavour symmetry breaking in the case of equal quark masses as

(Nf ) 1 1 ∂ ln Z (µ,...,µ) = β . 2 2 (2.13) ζ + µ 2Nf µ ∂µ ζ>0 β R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 169

In the massless case µ → 0, the left-hand side of (2.13) can be computed directly from the coset integral (2.6) [8–11] and is given as a function of the Dyson index β by − 1 Nf qNf = , (2.14) 2 2(Nf −qN ) ζ β − f + ζ>0 2(Nf 1) β 1 for even values of ND/ . The formula (2.14) generalizes the one given in [10,11,17,33] to include the case of odd Nf .Forβ = 2 it was ﬁrst obtained in [8]. Note that for an odd number of Dirac operator eigenvalues, the ﬁnite volume partition function (2.6) is not positive and vanishes when the unpaired fermion is massless, i.e., µ0 = 0 [16]. It is therefore not suitable to describe physical quantities such as chiral spectral sum rules, while it can appear radiatively [15].

2.2. Parametrization of the Goldstone manifold

We will now describe how to evaluate the non-linear σ -model partition functions (2.6). For this, we shall first review some elementary Lie theory that will be of use to us and which will fix some notation. We assume that the quark flavour symmetry is represented by a compact, connected, simple Lie group G, which we take to be realized by Nf × Nf invertible matrices U.Lett be a Cartan subalgebra of the Lie algebra g of G, so that T = exp(it) is a maximal torus of G. Any element X ∈ g can be taken to lie in the Cartan subalgebra t without loss of generality, since this can always be achieved via rotations by elements of G if necessary. After performing such a rotation in the given Cartan decomposition of G, the Jacobian for the change of Haar integration measure is given by a Weyl determinant ∆G(X) of G which is determined through the product ∆G(X) = (α, X), (2.15) α>0 where (α, X) are the positive roots of G evaluated on the Cartan element X.Itsmost useful form is obtained by choosing an orthonormal basis ei , i = 1,...,dim T , of vectors in weight space, ei ·ej = δij . Then the root vectors α can be identified with dual elements dimT ∨ α = αiei, (2.16) i=1 while dimT X = xiHi, (2.17) i=1 where Hi are the generators of t in the matrix basis corresponding to the orthonormal ∨ = weights. By identifying (2.17) with the weight vector x i xiei , the pairing in (2.15) may then be written as the inner product dimT ∨ ∨ (α, X) = α · x = αixi. (2.18) i=1 170 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

Table 1 The ﬂavour symmetry groups G of rank r = dim T 2 and their corresponding Dyson index β. The vectors ei , i = 1,...,r, form an orthonormal basis of weight space, while Eµ,ν is a corresponding orthonormal basis of matrix units (Eµ,ν)λρ = δµλδνρ . The Dynkin diagrams all have r nodes. The symmetric group Sr permutes the eigenvalues xi of a given Cartan element X ∈ t, while the cyclic group Z2 reﬂects them

G Positive weights Hi Dynkin diagram Weyl group U(r) ei −ej Ei,i ... Sr β = 21 i

With U = exp(iX), the suitably normalized invariant Haar measure on the Lie group G is given by

dimT 2 [dU]= dxi sinh (α, ln U). (2.19) i=1 α>0 Its inﬁnitesimal form induces a measure on the Lie algebra g as

dimT 2 [dX]= dxi ∆G(X) . (2.20) i=1 For the Lie groups that we shall deal with in this paper, the Cartan subalgebra t is represented in the basis (2.17) by Nf × Nf diagonal matrices. The residual ﬂavour symmetry after such a transformation is given by the discrete Weyl group WG of G which consists of the inequivalent transformations on t → t given by the adjoint actions

− X → Xw = wXw 1,w∈ G. (2.21)

Geometrically, WG is the group of automorphisms of the root lattice of G corresponding to Weyl reﬂections. To each w ∈ WG we may associate a sign factor sgn(w) =±1 which is the even/odd Z2-parity of the Weyl element according to its action (2.21) on the diagonal elements of the Cartan matrices. For the partition functions that will be of interest to us in this paper, the relevant group theoretical data are summarized in Table 1. We are now ready to construct the measure dνβ (U) for integration over the Goldstone manifold (2.4). For this, we shall introduce polar coordinates on the coset space Gβ (Nf ) [25], which in each case will be an irreducible, maximally symmetric compact space falling into the Cartan classiﬁcation. Let = ⊕ ⊥ g hΓ hΓ (2.22) R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 171 be a Cartan decomposition of the simple algebra g with respect to the stability subgroup = ⊥ = HΓ exp(ihΓ ), with hΓ g hΓ the orthogonal complement of hΓ in g with respect to the Cartan–Killing form. The condition for (2.4) to be a symmetric space is then [ ]⊂ ⊥ ⊂ ⊥ ⊥ ⊥ ⊂ hΓ , hΓ hΓ , hΓ , hΓ hΓ , hΓ , hΓ hΓ . (2.23)

⊥ N Let tΓ be a maximal Abelian subalgebra in the subspace hΓ , and let tΓ be the normal- C ⊂ N izer subgroup of tΓ in HΓ with tΓ tΓ its centralizer subgroup. As HΓ is a compact symmetric subgroup of the ﬂavour symmetry group G, every coset element U ∈ Gβ (Nf ) can be written by means of the adjoint representation of the stationary subgroup HΓ as [25]

− U = VRV 1, (2.24)

∈ C ∈ = where V HΓ / tΓ while R TΓ exp(itΓ ) up to transformations by the adjoint actions N C of elements of the factor group tΓ / tΓ , which can be identified with the Weyl group of the restricted root system on the symmetric space Gβ (Nf ). This new root lattice differs from that inherited from the original group G if the Cartan subalgebra t is a subset of ⊂ ⊥ hΓ , and it is defined by completing the maximal Abelian subalgebra tΓ hΓ with the generators in hΓ that commute with tΓ to give a new representation of t that lies partly in ⊥ ∈ G hΓ . The matrix R in (2.24) is the radial coordinate of the point U β (Nf ), while V is its angular coordinate. This defines a foliation of the Goldstone manifold (2.4) by conjugacy classes under the adjoint action of the stability group HΓ of the symmetric space. The decomposition (2.24) means that every matrix U ∈ Gβ (Nf ) can be diagonalized by a similarity transformation in the subgroup HΓ , and the radial coordinates {ri} are exactly the set of eigenvalues of U. Using the Haar measure (2.19) on the Lie group G,the corresponding Jacobian for the change of integration measure by (2.24) can be computed by standard techniques and one finds [26,34]

dimTΓ mα dνβ (U) =[dV ] dri sinh(αΓ , ln R) Γ , (2.25) i=1 αΓ >0 = ∈[ π ] = where ln R i i riHi with ri 0, 2 for i 1,...,dim TΓ . The second product in G (2.25) goes over positive roots of the restricted root lattice on β (Nf ), and mαΓ is the multiplicity of the root αΓ in the given Cartan decomposition, i.e., the dimension of the subspace of raising operators corresponding to the root αΓ in the algebra hΓ [26].One can further exploit the ambiguity in the choice of radial coordinates R labelled by the re- N C stricted Weyl group tΓ / tΓ to alternatively parametrize the radial decomposition (2.24) in a way more tailored to the evaluation of (2.6). The particular details of this parametrization will depend in general on the random matrix ensemble in which we are working. As N C we will see, the radial ﬂavour symmetry group tΓ / tΓ of the Goldstone manifold naturally distinguishes the cases β = 1, 2 from β = 4, and in all cases provides a nice algebraic distinction between the cases of an even or odd number of ﬂavours Nf . 172 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

2.3. The Harish-Chandra formula

The computation of (2.6) can be compared with the Harish-Chandra formula which computes the group integral [35] − 1 [dU] exp Tr XUYU 1 = sgn(w) exp Tr XY w , (2.26) ∆G(X)∆G(Y ) ∈ G w WG for X, Y ∈ t. It expresses the group integral as a finite sum over terms, one for each element of the Weyl group, and it is intimately related to the Fourier transform of the Weyl character formula for the flavour symmetry group G. Its main characteristic is that the integration in (2.26) may be rewritten as an integral over a coadjoint orbit, which is a symmetric space to which the Duistermaat–Heckman localization formula may be applied [36]. The Weyl group can be described as the factor group WG = NG/T, where NG is the normalizer subgroup of fixed points of the left action of T on the orbit G/T . In other words, the right- hand side of (2.26) is the stationary phase approximation to the group integral, and the Harish-Chandra formula states that in this case the approximation is exact. For this to be true it is important that the matrices X and Y live in the Lie algebra g of G.

3. Fundamental fermions

The simplest instance is when the quarks are described by complex fundamental Dirac fermion fields with Nc 3 colours. Although these models have been the most extensively studied ones thus far, it will be instructive for later comparisons and simply for completeness to rederive the finite volume partition functions in this case using the formalism of the previous section. In matrix model language it corresponds to the unitary ensemble β = 2. The global flavour symmetry group is then U(Nf ), while the mass and parity matrices are given in a suitable basis of the vector space CNf by   µ1   σ 3 ⊗ .. ,N= 2n,  . f  µn  M = µ0    µ1      3 . ,N= 2n + 1,  σ ⊗ ..  f µn % 3 &σ ⊗ 1n,N' f = 2n, Γ = 1 (3.1) 3 ,Nf = 2n + 1. σ ⊗1n The mass matrix explicitly breaks the flavour symmetry, and its pairing into quark masses of opposite sign preserves three-dimensional spacetime parity. The Goldstone manifolds are [8]

G2(2n) = U(2n)/U(n) × U(n), (3.2) R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 173

G2(2n + 1) = U(2n + 1)/U(n + 1) × U(n), (3.3) 2 of real dimensions dim G2(2n) = 2n and dim G2(2n + 1) = 2n(n + 1).

3.1. Partition functions

3.1.1. Nf = 1 The case of a single massive quark corresponds to formally setting n = 0in(3.3).In this case there is no ﬂavour symmetry breaking, the Goldstone manifold is a point, and there are no massless propagating degrees of freedom. The ﬁnite volume partition function is then trivially obtained from (2.6) in the form ( cosh µ, N even, Z(1)(µ) = D/ (3.4) 2 sinh µ, ND/ odd.

3.1.2. Nf = 2n The restricted root lattice of the symmetric space (3.2) is the root lattice Cn of the = = = symplectic group Sp(2n) (see Table 1), with multiplicities mei ±ej 2, m2ei 1fori, j 1,...,n, i

2sin(ri − rj ) sin(ri + rj ) = cos 2ri − cos 2rj (3.6) and deﬁning λi = cos 2ri ∈[−1, 1], we may write (3.5) in the simpler form [ ] n dV 2 dν2(U) = dλi ∆(λ1,...,λn) . (3.7) n2 2 i=1

For any X ∈ u(n) represented as a diagonal n × n matrix with entries xi ∈ R,wehave introduced its Vandermonde determinant which is the Weyl determinant (2.15), (2.18) for the unitary group G = U(n) given by ∆U(n)(X) = (xi − xj ) = ∆(x1,...,xn). (3.8) i

n 1 (2n) 1 λ (µ +µ ) Z (µ ,...,µ ) = (wˆ +wˆ −) λ i wˆ +(i) wˆ −(i) 2 1 n 2 sgn d i e ∆(µ1,...,µn) ˆ ∈ = w± Sn i 1−1 (3.15) involving the unfolded masses µi , i = 1,...,n. Since the integration measure and domain in (3.15) are invariant under permutations of the λi ’s, we may reduce the double sum over −1 the Weyl group to a single sum over the relative permutation wˆ =ˆw+wˆ − .Theλi integrals are elementary and expressed in terms of the function

1 sinh x 1 K(x) = = dλ exλ, (3.16) x 2 −1 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 175 which for purely imaginary argument defines the spectral sine-kernel of the unitary ensemble [32]. The sum over wˆ ∈ Sn defines a determinant, and in this way one may finally arrive at the simple n × n determinant formula

det[K(µ + µ )] = Z(2n)(µ ,...,µ ) = i j i,j 1,...,n . 2 1 n 2 (3.17) ∆(µ1,...,µn) This is precisely the form of the ﬁnite volume partition function that was obtained in [19] from considerations based on partially-quenched effective ﬁeld theories.

3.1.3. Nf = 2n + 1 The restricted root lattice of the symmetric space (3.3) is the non-reduced irreducible root lattice BCn given by the union of the root systems Bn and Cn of the groups SO(2n+1) = = = and Sp(2n) (see Table 1), with multiplicities mei ±ej 2, m2ei 1 as above, and mei 2 [25,27]. In this case the measure (2.25) thus reads

n 2 2 2 dν2(U) =[dV ] dri sin 2ri sin ri sin (ri − rj ) sin (ri + rj ) i=1 i

2 = 1 − where we have used the double angle identity sin ri 2 (1 cos 2ri). With the definition (3.8), the radial coordinate matrix 1 R = (3.19) exp(iσ 1 ⊗ ρ) satisfies the requisite condition (3.11). The angular matrices V ∈ U(n + 1) × U(n) are decomposed as in (3.12) with V+ ∈ U(n+ 1) and V− ∈ U(n), and the non-linear σ -model action is now modified to      µ0 1  µ   λ   †  1   1  −1 Re Tr MUΓU = Tr . V+ . V+  .. .. µn λn      µ1 λ1  .   .  −1 + Tr .. V− .. V− . (3.20) µn λn

The Harish-Chandra formula applied to the integrations over V+ ∈ U(n+ 1) and V− ∈ U(n) then yields 176 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

(2n+1) Z (µ0,µ1,...,µn) 2 n − −1 i=1(µi µ0) µ ˆ = (wˆ +) (wˆ −) w+(0) 2 sgn sgn e ∆(µ1,...,µn) wˆ +∈Sn+1 wˆ −∈Sn n 1 ( + µ →−µ × dλ eλi (µwˆ +(i) µwˆ −(i)) + (−1)ND/ 0 0 . (3.21) i µ →−µ = i i i 1 −1

Note how the extra factors of (1 − λi) in (3.18) have nicely cancelled against the (n + 1) × (n + 1) Vandermonde determinant ∆(1,λ1,...,λn) coming from the V+ integral. In the sum over wˆ − ∈ Sn, we may regard Sn as the subgroup of the permutation group Sn+1 acting only on the indices i = 1,...,nin {0, 1,...,n}. Then as before we can truncate the −1 double Weyl group sum in (3.21) to a single sum over wˆ =ˆw+wˆ − ∈ Sn+1 by extending wˆ − to all of Sn+1 through the deﬁnition wˆ −(0) = 0. After performing the λi integrals, the ﬁnite volume partition function is thus given by the simple (n + 1) × (n + 1) determinant formula

(2n+1) Z (µ0,µ1,...,µn) 2 n −1 µ µ = (µi − µ0) e 0 e j = i 1 det 2 + + ∆(µ1,...,µn) K(µ0 µi) K(µi µj ) = i,j 1,...,n − − + e µ0 e µj + − n ND/ ( 1) det + + . (3.22) K(µ0 µi) K(µi µj ) i,j=1,...,n

This is a new simplified expression for the finite volume partition function of QCD3 with an odd number of fundamental fermions. It can be readily checked to coincide with previous expressions (given by (2n + 1) × (2n + 1) determinant formulas [15,20]) for low numbers of flavours.

3.2. Correlation functions

For β = 2, the k-level correlation functions can be expressed as a ratio of partition functions with Nf and Nf + 2k flavours as in (2.8).Byusing(3.17) one finds the scaled correlators for an even number of quark flavours given by

ρ(2n)(ζ ,...,ζ ; µ ,...,µ ) 2 1 k 1 n K(µ +µ ) K(µ +iζ ) det i j i l k K(µ −iζ ) K(iζ −iζ ) i,j=1,...,n 1 j l l l = = l,l 1,...,k , (3.23) π det[K(µi + µj )]i,j=1,...,n (k) = 1 k where the matching condition (2.9) in this case fixes C2 ( 2π ) . This again agrees with the field theoretic results of [19]. In the form (3.17), the expression (2.8), originally derived in the context of random matrix theory [15], is thereby naturally tied to the field theory (2n) derivation of the spectral functions ρ2 . R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 177

For an odd number of flavours one finds (2n+1) ; ρ2 (ζ1,...,ζk µ0,µ1,...,µn) ( eµ0 eµj eiζl →− + + µ0 µ0 + + + k n ND/ det K(µ0 µi ) K(µi µj ) K(µi iζl ) + (−1) µi →−µi i,j=1,...,n →− K(µ − iζl) K(µj − iζl) K(iζl − iζ ) = ζl ζl = 0 l l,l 1,...,k ( . eµ0 eµj n+N µ →−µ (2π)k det + (−1) D/ 0 0 K(µ + µ ) K(µ + µ ) µ →−µ 0 i i j i,j=1,...,n i i (3.24) This is a new general expression for the spectral k-point functions in the case of an odd number of quarks at β = 2. It coincides with the known expressions from random matrix theory [16] and the replica method approach to QCD3 [21]. In particular, the formula (3.24) shows explicitly that the correlators are not even functions of the scaled Dirac operator eigenvalues ζl for M = 02n+1 [21]. This expression also covers the case of a single flavour Nf = 1(n = 0), which can be obtained by omitting the corresponding n × n blocks in the determinants above.

3.3. Sum rules

We may work out the massive spectral sum rules (2.13) by taking the equal mass limits in (3.17) and (3.22). This yields indeterminate forms which can be computed by regulating the mass matrices in any way that removes the n-fold and (n + 1)-fold eigenvalue degen- eracies, and then taking the degenerate limits using l’Hôpital’s rule. However, this is rather cumbersome to do in practise and it is much simpler to go back and work directly with the radial coset representation of the ﬁnite volume partition functions.

3.3.1. Nf = 2n From (3.14) we see that in the case of equal quark masses µi = µ ∀i = 1,...,n,the σ -model action is independent of the angular degrees of freedom V± ∈ U(n) and the low- energy dynamics is mediated entirely by the n radial Goldstone boson degrees of freedom. The partition function (2.6) is then simply given by

n 1 (2n) = 2µλi 2 Z2 (µ,...,µ) dλi e ∆(λ1,...,λn) . (3.25) = i 1−1 We now use the identity [32] = j−1 ∆(λ1,...,λn) det λi i,j=1,...,n (3.26) and expand the resulting two determinants in (3.25) into sums over permutations wˆ ± ∈ Sn. As in Section 3.1, the partition function depends only on the relative permutation wˆ = −1 wˆ +wˆ − ∈ Sn, and the λi integrals are now expressed in terms of derivatives of the spectral kernel deﬁned by (3.16) as 1 dmK(x) 1 K(m)(x) = = dλλmexλ. (3.27) dxm 2 −1 178 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

In this way we arrive at the n × n determinant formula (2n) = (i+j−2) Z2 (µ,...,µ) det K (2µ) i,j=1,...,n. (3.28) We may present the partition function (3.28) in a somewhat more explicit form by ex- pressing the generic derivatives (3.27) in terms of the sine-kernel K(x) itself. Starting from the integral representation (3.27) we write 1 1 1 − K(m)(x) = dλ exλ 1 − m dssm 1 2 −1 λ 1 m − − = K(x) − dssm 1 exs − e x . (3.29) 2x −1 This yields a simple linear inhomogeneous recursion relation for the sequence of functions K(m)(x), m 0in(3.27) given by m − 1 cosh x K(m)(x) + K(m 1)(x) = K(x) + 1 − (−1)m − K(x) . (3.30) x 2 x It is straightforward to iterate the recurrence (3.30) in m and express the solution in terms of K(x) and other elementary functions of x, and in this way the equal mass partition function (3.28) can be written as (2n) = + − Z2 (µ,...,µ) det Ki j 2(2µ) i,j=1,...,n, (3.31) where + 1 m m! 1 k 1 (x) = − Km − ! 2 = (m k) x k0 + × 1 − (−1)m k (sinh x − cosh x) − 2sinhx . (3.32) From (3.31) one may now proceed to derive the massive spectral sum rules (2.13).For the first few even numbers of quark flavours we find 1 2µ coth 2µ − 1 = , 2 2 2 ζ + µ = 4µ ζ>0 2 Nf 2 2 2 1 1 2sinh 2µ − µ sinh 4µ − 4µ = , ζ 2 + µ2 4µ2 2 − 2 2 Nf =4 4µ sinh 2µ ζ>0 1 2 2 ζ + µ = ζ>0 2 Nf 6 1 2µ(16µ4 + 60µ2 + 3) cosh 2µ − 28µ2(4µ2 + 3) sinh 2µ − 6µ cosh3 2µ + 9sinh3 2µ = . 12µ2 4µ2(4µ2 + 3) sinh 2µ − 16µ3 cosh 2µ − sinh3 2µ (3.33) These all agree with (2.14) in the limit µ → 0. The first two sum rules in (3.33) were also obtained in [14]. R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 179

3.3.2. Nf = 2n + 1 From (3.18) and (3.20) it follows that in the equal mass limit µ0 = µi = µ ∀i = 1,...,n the ﬁnite volume partition function (2.6) is given by (2n+1) Z2 (µ,...,µ) n 1 µ 2µλi 2 ND/ = e dλi (1 − λi)e ∆(λ1,...,λn) + (−1) {µ →−µ}. (3.34) = i 1−1 Proceeding exactly as above, and using the fact that the functions (3.32) obey the reﬂection m property Km(−x) = (−1) Km(x), we can write (3.34) explicitly as the n × n determinant formula (2n+1) = µ − Z (µ,...,µ) e det Ki+j−2(2µ) Ki+j−1(2µ) = 2 i,j 1,...,n N −µ + − D/ + − + + − ( 1) e det Ki j 2(2µ) Ki j 1(2µ) i,j=1,...,n, (3.35) with the determinants omitted for the n = 0 case as before. For an even number ND/ of Dirac operator eigenvalues, from (3.4) and (3.35) we obtain the massive spectral sum rules 1 tanh µ = , 2 2 ζ + µ = 2µ ζ>0 2 Nf 1 1 1 2µ2 coth µ − µ(1 + 3 cosh 2µ) − 2sinh2µ = , 2 2 2 (3.36) ζ + µ = 6µ 2µ + sinh 2µ ζ>0 2 Nf 3 which again agree with (2.14) in the limit µ → 0.

4. Pseudo-real fermions

The next simplest case is the parity-invariant Dirac operator iD/ acting on Nc = 2 colours of Dirac quarks in the fundamental representation of the SU(Nc) gauge group. In contrast to the previous case, the Dirac operator now possesses a special anti-unitary symmetry, [C, iD/ ]=0, generated by an operator C with C2 = 1 [9,10]. The operator C can be expressed as a combination of the usual charge conjugation C = iσ 2 with C(σµ)∗C−1 = σ µ for µ = 1, 2, 3, an infinitesimal rotation by the matrix iσ 2 in colour space, and complex conjugation. In a basis of spinors |ψk wherein C|ψk=|ψk, the matrix elements ψk|iD/ |ψl are all real [10]. This basis is obtained by augmenting the flavour components of the fermion fields by the two-colour group, thereby enhancing the original U(Nf ) flavour symmetry to a higher pseudo-real global flavour symmetry group Sp(2Nf ).Inma- trix model language this corresponds to the orthogonal ensemble β = 1. The group USp(2n) acts on the vector space C2n preserving its canonical inner product 2n 2n as well as a symplectic structure Jn : R → R , i.e., a real antisymmetric 2n × 2n matrix J2 =− C2n with n 12n. In a suitable basis of , it consists of unitary matrices V satisfying 2 V JnV = Jn, Jn = iσ ⊗ 1n. (4.1) 180 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

With respect to the same basis, the mass and parity matrices are given by  µ1   3 ⊗ 2 ⊗ . =  σ iσ .. ,Nf 2n,   µn  = 2 M  µ0iσ     µ1     = +   3 2 .  ,Nf 2n 1,  σ ⊗iσ ⊗ ..  % µn σ 3 ⊗ J ,N= 2n, = & n ' f Γ iσ 2 = + (4.2) 3 ,Nf 2n 1. σ ⊗Jn The antisymmetric forms of these matrices follow from the Grassmann nature of the quark spinor ﬁelds ψk described above. Again the mass matrices break the USp(4n) and USp(4n + 2) ﬂavour symmetries, and the corresponding Goldstone manifolds are

G1(2n) = USp(4n)/USp(2n) × USp(2n), (4.3)

G1(2n + 1) = USp(4n + 2)/USp(2n + 2) × USp(2n), (4.4) 2 of real dimensions dim G1(2n) = 4n and dim G1(2n + 1) = 4n(n + 1). As expected, the number of Goldstone bosons here is exactly twice that of the previous section.

4.1. Partition functions

4.1.1. Nf = 2n The restricted root lattice of the symmetric space (4.3) is again the root space Cn of = = Sp(2n), but now with multiplicities mei ±ej 4, m2ei 3 [25,27]. The corresponding radial integration measure (2.25) on G1(2n) is thus n 3 4 4 dν1(U) =[dV ] dri sin 2ri sin (ri − rj ) sin (ri + rj ) i=1 i

ξ = σ 2 ⊗ ρ (4.7) commutes with the symplectic structure as

ξJn = Jnξ. (4.8) R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 181

It follows that R = R−1 = R†, and the radial matrix (4.6) anticommutes with the parity matrix Γ as in (3.11). For the angular degrees of freedom V ∈ USp(2n) × USp(2n) we again take the decomposition (3.12) with V± ∈ USp(2n), so that from (4.1) we have VΓV = Γ. (4.9) 2 Using these relations, along with the identity cos(σ ⊗η) = 12 ⊗cos η for any η ∈ u(n), we can now write the action of the non-linear σ -model (2.6) similarly to (3.14) as Re Tr MUΓU 1 − − = Tr ΓMV R2 + R 2 V 1 2 −1 = Tr ΓMV(14 ⊗ cos 2ρ)V        µ1 λ1   .    .  −1 = Tr 12 ⊗ .. V+ 12 ⊗ .. V+ µ λ   n    n   µ1 λ1   .    .  −1 + Tr 12 ⊗ .. V− 12 ⊗ .. V− . µn λn (4.10) The ﬁnite volume partition function (2.6) thereby becomes

(2n) Z1 (µ1,...,µn) 1 n = − 2 4 (n) ; 2 dλi 1 λi ∆(λ1,...,λn) Ω1 (λ1,...,λn µ1,...,µn) , (4.11) = i 1 −1 where (n) ; Ω1 (λ1,...,λn µ1,...,µn) %    µ1   .  = [dV ] exp Tr 12 ⊗ .. Sp(2n) µn     λ1   .  −1 ×V 12 ⊗ .. V . (4.12) λn It is not at all clear at this stage how to proceed with the calculation. The Lie algebra sp(2n) consists of 2n×2n matrices X obeying the equation XJn +JnX = 0. The diagonal source matrices in (4.12) are not elements of this Lie algebra, nor are the original mass and parity matrices in (4.2) elements of sp(4n). Thus the Harish-Chandra formula (2.26) cannot be immediately applied to the angular integral (4.12). For the same reason this integral cannot be evaluated by using character expansion techniques for the symplectic group Sp(2n) [38]. It would be interesting to determine the corresponding analytical continuation of the 182 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

Harish-Chandra formula to this case, in analogy with that done in the unitary case G = U(n)[39] which is based on the fact that the representation theories of the Lie groups U(n) C (1) ; and GL(n, ) are essentially the same. Note that the integral Ω1 (λ µ) can be computed by means of the Harish-Chandra formula for SU(2) =∼ Sp(2). In [18] the finite volume partition function in this case is evaluated by applying a formalism based on skew-orthogonal polynomials in the orthogonal ensemble of random matrix theory, resulting in the Pfaffian expression Pfaff[K (µ˜ −˜µ )] = Z(2n)(µ ,...,µ ) = a b a,b 1,...,2n , (4.13) 1 1 n n 2 2 2 i=1 µi∆(µ1,...,µn) where µ˜ 2i−1 =−µi , µ˜ 2i = µi for i = 1,...,n. Using the permutation expansion of the Pfaffian, the integral representation (3.27) and by comparing (4.13) with (4.11), it is possible to write a conjectural formula for the integral (4.12). There are several qualitative features of the expression (4.13) that make its derivation directly in field theory appear feasible. First of all, the denominator of (4.13) can be expressed in terms of the square of the Weyl determinant ∆Sp(2n)(X) of an element X ∈ sp(2n) with eigenvalues µ1,...,µn (see (2.15) and Table 1), as appears in the expression (2.26). Furthermore, the Weyl group WSp(2n) is composed of permutations of eigenvalues xi → xπ(i), i = 1,...,n, π ∈ Sn, along with reflections xi →−xi . After the appropriate radial integration, the sum over WSp(2n) can thereby produce the required Pfaffian in the numerator of the expression (4.13), which in the random matrix theory calculation arises naturally from quaternion determinants generated in the skew-orthogonal polynomial formalism. In the next subsection shall explicitly work out the σ -model partition functions in the case of equal quark masses and thus provide further evidence that the direct field theory calculation indeed does reproduce this Pfaffian expression.

4.1.2. Nf = 2n + 1 The restricted root lattice of the symmetric space (4.4) is again the root space BCn, with = = = multiplicities mei ±ej 4, m2ei 3 as above, and in addition mei 4 [25,27]. The coset space measure (2.25) in this case is thus given by n 3 4 4 4 dν1(U) =[dV ] dri sin 2ri sin ri sin (ri − rj ) sin (ri + rj ) i=1 i

(2n+1) Z1 (µ0,µ1,...,µn) 1 n = − 2 − 2 4 dλi 1 λi (1 λi) ∆(λ1,...,λn) = i 1 −1 (n+1) (n) × Ω (1,λ1,...,λn; µ0,µ1,...,µn)Ω (λ1,...,λn; µ1,...,µn) 1 ( 1 µ →−µ + (−1)ND/ 0 0 . (4.16) µi →−µi

= (1) ; + For n 0 this expression is understood as being equal to Ω1 (1 µ0) − ND/ (1) ;− ( 1) Ω1 (1 µ0).In[18] an expression for the ﬁnite volume partition function in this case is derived from random matrix theory in the case of a vanishing unpaired quark mass µ0 = 0 (so that the matrix model measure is positive and the partition function vanishes when ND/ is odd). In the next subsection we derive a formula for the equal mass limit of (4.16) without this restriction.

4.2. Sum rules

4.2.1. Nf = 2n As in the previous section, to work out the massive spectral sum rules (2.13) in the even-ﬂavoured case at β = 1 it is convenient to work directly with the partition function (4.11) for equal quark masses µi = µ ∀i = 1,...,n, which is determined entirely by the n radial Goldstone boson degrees of freedom and is simply given by

1 n (2n) = − 2 4µλi 4 Z1 (µ,...,µ) dλi 1 λi e ∆(λ1,...,λn) . (4.17) = i 1 −1 Because of the original fermion determinant required in (2.12) and the doubling of the flavour symmetry in this case, we can anticipate the appearance of a Pfaffian expression for the finite volume partition function, in contrast to the determinant formula of the previous section. For this, we write the power of the Vandermonde determinant in (4.17) as the determinant of a 2n × 2n matrix given by [32,40]

j−1 4 λi ∆(λ1,...,λn) = det − . (4.18) (j − 1)λj 2 i=1,...,n i j=1,...,2n

Expanding the determinant into a sum over permutations wˆ ∈ S2n, and using the permutation symmetry of the integration measure and domain in (4.17) to appropriately rearrange rows, we may bring the equal mass partition function into the form 184 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

1 n (2n) = ˆ − 2 4µλi Z1 (µ,...,µ) sgn(w) dλi 1 λi e ˆ ∈ = w S2n i 1 −1 × ˆ − w(ˆ 2i)+ˆw(2i−1)−3 w(2i) 1 λi . (4.19) > ˆ Let S2n denote the subgroup of S2n consisting of “increasing” permutations w, i.e., those obeying w(ˆ 2i) > w(ˆ 2i − 1) ∀i = 1,...,n. Then the expression (4.19) can be written as

1 n (2n) = ˆ − 2 4µλi Z1 (µ,...,µ) sgn(w) dλi 1 λi e ˆ ∈ > i= w S2n 1 −1 × ˆ −ˆ − w(ˆ 2i)+ˆw(2i−1)−3 w(2i) w(2i 1) λi . (4.20) The expression (4.20) now has the standard form of a Pfafﬁan [32] and we may thereby write it as (2n) = Z1 (µ,...,µ) Pfaff A2n(4µ), (4.21) where A2n(x) is the 2n × 2n antisymmetric matrix with elements 1 2 i+j−3 xλ A2n(x)ij = (i − j) dλ 1 − λ λ e − 1 + − + − = 2(i − j) K(i j 3)(x) − K(i j 1)(x) . (4.22) These matrix elements can be alternatively expressed as

+ − 1 s d i j 3 A (x) = 2(i − j) dss dλ exλ 2n ij dx −1 −1 + − d i j 3 K (x) = (i − j) . (4.23) dx x The substitution of (4.23) into (4.21) can be straightforwardly shown to coincide with the equal mass limit of the finite volume partition function (4.13) obtained from the orthogonal ensemble of random matrix theory. Proceeding as in Section 3.3 to rewrite derivatives of the spectral sine-kernel in terms of the functions (3.32), the equal mass partition function in the case of an even number of pseudo-real quarks acquires the Pfaffian form (2n) = − + − − + − Z1 (µ,...,µ) Pfaff (i j) Ki j 3(4µ) Ki j 1(4µ) i,j=1,...,2n. (4.24) It can be computed explicitly by using the Laplace expansion of the Pfaffian [32] 2n i+j Pfaff A = (−1) (A)ij Pfaff Aji,ji (4.25) j=1 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 185 for any fixed i = 1,...,2n, where the Pfaffian minor Aji,ji is obtained from the antisymmetric 2n × 2n matrix A by deleting both its ith and jth rows and columns. The massive spectral sum rules can thus be extracted from (3.32) and (4.24) by the substitution Nf → 2Nf in (2.13) to account for the symplectic doubling of fermion degrees of freedom in this case, and for the first few even numbers of quark flavours we find 1 1 3 = − , ζ 2 + µ2 4µ coth 4µ − 1 16µ2 1 Nf =2 ζ>0 1 2 2 ζ + µ = ζ>0 1 Nf 4 1 36µ(4µ2 + 5) sinh 8µ − 9(64µ2 + 5) sinh2 4µ − 16µ2(256µ4 + 216µ2 + 27) = . 16µ2 9(8µ2 + 1) sinh2 4µ − 36µ sinh 8µ + 128µ6(16µ2 + 9) (4.26) These results agree with (2.14) in the limit µ → 0 of massless quarks.

4.2.2. Nf = 2n + 1 The equal mass limit µi = µ ∀i = 0, 1,...,n of the partition function (4.16) is given by

(2n+1) Z1 (µ,...,µ) 1 n = 2µ − 2 − 2 4µλi 4 + − ND/ { →− } e dλi 1 λi (1 λi) e ∆(λ1,...,λn) ( 1) µ µ . = i 1 −1 (4.27) The calculation is identical to that of the even-flavoured case above. The only difference is that the antisymmetric matrix (4.22) now contains an additional measure factor (1 − λ)2, which when expanded leads to the equal mass finite volume partition function in the case of an odd number of symplectic fermions in the Pfaffian form + Z(2n 1)(µ,...,µ) 1 2µ = e Pfaff (i − j) K + − (4µ) − 2K + − (4µ) i j 3 i j 2 + − 2Ki+j (4µ) Ki+j+1(4µ) = i,j 1,...,2n N −2µ + (−1) D/ e Pfaff (i − j) K + − (4µ) + 2K + − (4µ) i j 3 i j 2 − + − + + 2Ki j (4µ) Ki j 1(4µ) i,j=1,...,2n. (4.28)

This expression also covers the case Nf = 1 of a single fermion of mass µ, obtained by dropping the Pfafﬁans in (4.28), which for an even number ND/ of Dirac operator eigenvalues leads to the massive spectral sum rule 1 tanh 2µ = . 2 2 (4.29) ζ + µ = 4µ ζ>0 1 Nf 1 186 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

Sum rules for higher numbers of flavours are also straightforward to work out, but they become increasingly complicated and will not be displayed here. In each case one finds that the massless spectral sum rule (2.14) for β = 1 is indeed obeyed. Eqs. (2.10), (2.11), (4.24) and (4.28) suffice to express to smallest eigenvalue distribution of the QCD3 Dirac operator for SU(2) colour group in a closed form for an arbitrary number of fermion flavours in the case of equal quark masses.

5. Adjoint fermions

Finally, we consider the case of quark spinors which transform in the adjoint representation of the SU(Nc) colour group for Nc 2, which is real. As in the previous section, the adjoint representation of the Euclidean Dirac operator iD/ uniquely defines an anti- 2 unitary operator C0 with [C0, iD/ ]=0, but now (C0) =−1 [9,11]. The operator C0 is given as a combination of charge conjugation C = iσ 2 and complex conjugation. In this case the Dirac operator can be expressed in terms of real quaternions [11], and the condition C0|ψk=|ψk defines a basis |ψk of Majorana spinors of the second kind. The generic unitary U(Nf ) flavour symmetry thereby truncates to its real orthogonal subgroup O(Nf ). In matrix model language this corresponds to the symplectic ensemble β = 4. The mass and parity matrices are given exactly as in (3.1), and so the Goldstone manifolds are

G4(2n) = O(2n)/O(n) × O(n), (5.1)

G4(2n + 1) = O(2n + 1)/O(n + 1) × O(n) (5.2) 2 of real dimensions dim G4(2n) = n and dim G4(2n + 1) = n(n + 1). As expected, the number of Goldstone bosons here is exactly half that of Section 3.

5.1. Partition functions

5.1.1. Nf = 2 The “degenerate” case Nf = 2 corresponding to two-flavour component adjoint fermions is very special because it represents flavour symmetry breaking from O(2) to a finite subgroup Z2 × Z2. The coset space (5.1) in this case is the orbifold

G4(2) = O(2)/Z2 × Z2. (5.3)

The quotient by one of the Z2 factors in (5.3) can be used to eliminate one of the connected components of the O(2) group, thereby realizing the orbifold explicitly as a one- dimensional manifold diffeomorphic to an open interval ∼ G4(2) = SO(2)/Z2 = (0,π). (5.4) After flavour symmetry breaking, there is thus a single massless degree of freedom described in the mesoscopic scaling regime by a real scalar field θ ∈ (0,π). To evaluate the corresponding finite volume partition function (2.6), we parametrize the elements U ∈ SO(2) appearing in (5.4) as R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 187 cos θ − sin θ U = (5.5) sin θ cos θ and work out the effective σ -model action to be Re Tr MUΓU = 2µ cos 2θ. (5.6) With the normalized Haar measure [dU]=dθ/π on SO(2), the partition function (2.6) in the two flavour case is thereby found to be

(2) = Z4 (µ) I0(2µ), (5.7) where generally

π dφ + I (x) = ex cos φ imφ (5.8) m 2π −π is the modiﬁed Bessel function of the ﬁrst kind of integer order m.

5.1.2. Nf = 2n Let us now turn to the case of generic even integer values Nf = 2n with n>1. The restricted root lattice of the symmetric space (5.1) is the root space Dn of the special orthog- = = onal group SO(2n) (see Table 1), with root multiplicities mei ±ej 1fori, j 1,...,n, i

G4(2n) = SO(2n)/SO(n) × SO(n) × Z2. (5.10) In this case, it is more convenient to parametrize the radial directions of this orbifold in terms of the angle variables θi = 2ri ∈[0,π], and using (3.6) we may write (5.9) as n [dV ] dν4(U) = dθi ∆(cos θ1,...,cos θn) . (5.11) 1 n(n+1) 2 2 i=1 To ensure positivity of this measure when the absolute value signs are dropped, we will restrict the integration region for the θi ’s to the domain Dθ ={ | ··· } n (θ1,...,θn) 0 θ1 θ2 θn π . (5.12) This can always be achieved without complications for the integration of symmetric functions. 188 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

The particular details of the radial decomposition here are identical to those of Sec- tion 3.1, and we thereby arrive at the non-linear σ -model action      µ1 cos θ1  .   .  −1 Re Tr MUΓU = Tr .. V+ .. V+ µ cos θ  n   n   µ1 cos θ1  .   .  −1 + Tr .. V− .. V− , µn cos θn (5.13) where now V± ∈ O(n). The ﬁnite volume partition function is thus given by (2n) Z4 (µ1,...,µn) n = (n) ; 2 dθi∆(cos θ1,...,cos θn)Ω4 (cos θ1,...,cos θn µ1,...,µn) , (5.14) = Dθ i 1 n where (n) ; Ω4 (x1,...,xn µ1,...,µn) %      µ1 x1  .   .  −1 = [dV ] exp Tr .. V .. V . (5.15) SO(n) µn xn As in the previous section, the integral (5.15) cannot be evaluated using the Harish-Chandra formula (2.26) because the Lie algebra so(n) consists of antisymmetric n×n matrices. The ﬁrst two members of this sequence of angular integrals are easily evaluated explicitly to be

Ω(1)(x; µ) = eµx, 4 (2) 1 (µ +µ )(x +x ) 1 Ω (x ,x ; µ ,µ ) = 2e 2 1 2 1 2 I (µ − µ )(x − x ) . (5.16) 4 1 2 1 2 0 2 1 2 1 2 For n = 3 the integral (5.15) can be computed by means of the Harish-Chandra formula for SU(2) =∼ SO(3). In [18] the finite volume partition function in this instance is computed using a skew- orthogonal polynomial formalism in the symplectic ensemble of random matrix theory. For n = 2 with 1 it is necessary to assume that the mass eigenvalues µi appear in degenerate pairs, i.e., µ2a−1 = µ2a for each a = 1,...,, in order to ensure positivity of the matrix model measure. In this case a Pfaffian expression for the partition function is obtained in terms of integrals of the spectral sine kernel (3.16) (rather than derivatives as in the previous section), or equivalently in terms of exponential integral functions Ei(x). In general, this restriction does not aid the field theory calculation, which should thereby capture the case of generic mass eigenvalue configurations. In the next subsection we will evaluate (5.14) in the equal mass limit and find a striking contrast between the field theory and random matrix theory expressions. As in (5.7), the field theory calculation will naturally involve modified Bessel functions of the first kind (along with sine kernel derivatives), R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 189 which is reminiscent of results for QCD in four spacetime dimensions [2,30]. This may be a consequence of the known relations between the QCD3 and QCD4 partition functions in the -regime [20,22]. Comparison of our expressions with those of [18] imply that some non-trivial simplifying identities among these Bessel functions ought to hold, but we have not been able to find any. For n = 2 + 1, in addition to pairwise degeneracy, it is necessary to assume in the random matrix theory approach that the single unpaired quark mass vanishes, again for positivity reasons. In the next subsection we derive the equal mass limit of (5.14) without this restriction. From the field theory perspective, we shall also encounter a distinction between the cases Nf = 4 and Nf = 4 + 2, and this distinction is very natural. From Table 1 we see that the Weyl group sum for the group SO(2 + 1) is the same as that of the symplectic group Sp(2). In contrast, the Weyl group of SO(2) contains the usual −1 permutation part S and the factor (Z2) is the normal subgroup given by the possible assignments of an even number of sign flips to the eigenvalues. These facts, along with observations similar to those spelled out in the previous section, support the precise matching between the field theory and random matrix theory results in these instances.

5.1.3. Nf = 2n + 1 The restricted root lattice of the Goldstone manifold (5.2) is the root lattice Bn of the + = = special orthogonal group SO(2n 1) (see Table 1), with multiplicities mei ±ej mei 1 for i, j = 1,...,n, i

5.2. Sum rules

To compute the massive spectral sum rules (2.13), we proceed as previously to compute directly the ﬁnite volume partition functions (2.6) in the limit of equal quark masses for the various congruence classes modulo 4 of the ﬂavour number Nf .

5.2.1. Nf = 2 For Nf = 2, we substitute (5.7) into (2.13) to compute 1 1 I (2µ) = 1 . 2 2 (5.20) ζ + µ = 2µ I0(2µ) ζ>0 4 Nf 2

Once again this agrees with (2.14) in the limit µ → 0.

5.2.2. Nf = 4 For Nf = 2n with n>1 the partition function (5.14) for equal quark masses is given by

n (2n) = 2µ cos θi Z4 (µ,...,µ) dθi e ∆(cos θ1,...,cos θn). (5.21) = Dθ i 1 n

Again, since the functional integration in the original QCD3 partition function is over real Majorana fermion fields, thereby producing a square-root fermion determinant in (2.12), we can anticipate a Pfaffian expression for (5.21). With this in mind, we will proceed to evaluate the finite volume partition function using the method of integration over alternate variables [32,40], whose details again depend crucially on the even/odd parity of the flavour rank n. For n = 2 with 1 we can insert the determinant representation (3.26) to write

(4) Z4 (µ,...,µ)

θ a 2 2µ cos θ 2µ cos θ − j−1 = 2a − 2a 1 dθ2a e dθ2a 1 e det cos θi i,j=1,...,2, (5.22) a=1 Dθ θ2a−2 2 where we have deﬁned θ0 = 0. We formally insert the integrals into the determinant in (5.22) by using its multilinearity to write R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 191

(4) Z4 (µ,...,µ) j−1 cos θ2b 2µ cos θ2a = dθ2a e det . (5.23) θ2b 2µ cos θ j−1 b=1,..., = θ − dθ e cos θ = Dθ a 1 2b 2 j 1,...,2 2 Since the integrand of (5.23) is a symmetric function of the integration variables, we may extend the lower limit of integration of each variable inside the determinant down to 0. This symmetry also allows us to extend the range of each of the remaining integration variables to [0,π]. After rearranging rows in the determinant and expanding it as a sum over permutations, we arrive at the expression π ˆ − (4) = ˆ 2µ cos θ2a w(2a) 1 Z4 (µ,...,µ) sgn(w) dθ2a e cos θ2a ˆ ∈ = w S2 a 1 0

θ2a ˆ − − × dθ e2µ cos θ cosw(2a 1) 1 θ (5.24) 0 > ⊂ which may be written as a sum over the subgroup S2 S2 of increasing permutations given by (4) Z4 (µ,...,µ) π θ2a 2µ cos θ2a 2µ cos θ = sgn(w)ˆ dθ2a e dθ e wˆ ∈S> a=1 2 0 0 w(ˆ 2a)−1 w(ˆ 2a−1)−1 w(ˆ 2a−1)−1 w(ˆ 2a)−1 × cos θ2a cos θ − cos θ2a cos θ . (5.25) As before, the expression (5.25) now has the standard form of a Pfaffian and we may write it as (4) = Z4 (µ,...,µ) Pfaff B2(2µ) (5.26) where B2(x) is the 2 × 2 antisymmetric matrix with elements π π x cos θ i−1 x cos θ j−1 B2(x)ij = dθ e cos θ dθ sgn(θ − θ ) e cos θ . (5.27) 0 0 The integrals in (5.27) can be evaluated explicitly by using the Fourier series representation of the sign function ∞ 4 sin(2k + 1)φ sgn(φ) = , (5.28) π 2k + 1 k=0 for φ ∈[−π,π], and defining the sequence of functions π x cos θ+(2k+1)iθ I2k+1(x) = dθ e , (5.29) 0 192 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 for k ∈ N0. We may then express the matrix elements (5.27) as series in derivatives of the functions (5.29) given by ∞ (i−1) (j−1) ∗ (j−1) (i−1) ∗ i I + (x)I + (x) − I + (x)I + (x) B (x) = 2k 1 2k 1 2k 1 2k 1 . (5.30) 2 ij 2π 2k + 1 k=0 Let us now explicitly evaluate the derivatives of the functions (5.29). The real parts can be written in terms of the modified Bessel functions (5.8) as (i−1) = (i−1) Re I2k+1 (x) πI2k+1 (x). (5.31) For the imaginary parts, we use the expansion + sin(2k + 1)θ = Im(cos θ + isinθ)2k 1 k l 2k + 1 l − = sin θ (−1)a cos2k 2a θ (5.32) 2l + 1 a l=0 a=0 along with the change of integration variable λ = cos θ ∈[−1, 1] in (5.29) to evaluate them in terms of the sine kernel derivatives (3.27) as k l (i−1) 2k + 1 l − + − Im I (x) = 2 (−1)a K(2k 2a i 1)(x). (5.33) 2k+1 2l + 1 a l=0 a=0 The sums over l and a in (5.33) can be interchanged, after which the sum over p = l − a for fixed a can be evaluated explicitly by using the combinatorial identity k−a 2k + 1 p + a − 2k − a = 4k a (5.34) 2p + 2a + 1 a a p=0 that is straightforwardly proven by induction. Substituting in the explicit functional forms (3.32) for the sine kernel derivatives, and plugging (5.31) and (5.33) into (5.30) then enables us to write the equal mass partition function (5.26) explicitly as Z(4)(µ,...,µ) 4 ∞ (i−1) (j−1) I + (2µ)I2k;j−1(2µ) − I + (2µ)I2k;i−1(2µ) = Pfaff 2k 1 2k 1 , 2k + 1 k=0 i,j=1,...,2 (5.35) where k+i−b + 1 k 2 (−1)a2k−2a(k − 2a + i)! k − a 1 b 1 ; (x) = − Ik i − − + ! 2 = = (k 2a b i) a x b0 a 0 × 1 − (−1)b (sinh x − cosh x) − 2sinhx . (5.36) From (5.35) one may now extract the massive spectral sum rules (2.13) by using the derivative identities dIm(x) m dIk;i(x) = I + (x) + I (x), = I ; + (x). (5.37) dx m 1 x m dx k i 1 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 193

For instance, for four ﬂavours of adjoint fermions one ﬁnds 1 2 2 ζ + µ = ζ>0 4 Nf 4

4k2−1 4k+3 I2k+1(2µ)(Ik;2(2µ)− Ik;0(2µ))−Ik;0(2µ)(I2k+3(2µ)+ I2k+2(2µ)) ∞ 4µ2 2µ = + = k 0 2k 1 . (5.38) − 2k+1 − ∞ I2k+1(2µ)(Ik;1(2µ) 2µ Ik;0(2µ)) I2k+2(2µ)Ik;0(2µ) 4µ k=0 2k+1 A numerical evaluation of (5.38) in the limit µ → 0 establishes precise agreement with (2.14) in this case.

5.2.3. Nf = 4 + 2 Let us now consider the case Nf = 2n with generic odd ﬂavour rank n = 2 + 1, 1. The calculation proceeds analogously to that of the even rank case, except that now one needs to keep careful track of the single unpaired integration. Starting from (5.21) we derive the determinant formula

(4+2) Z4 (µ,...,µ) θ2a 2µ cos θ2a 2µ cos θ2a−1 = dθ2a e dθ2a−1 e a=1 Dθ θ2a−2 2+1 π j−1 2µ cos θ + × + 2 1 det cos θi i,j=1,...,2+1 dθ2 1 e θ  2  cosj−1 θ 2b  θ −  = dθ e2µ cos θ2a det  2b dθ e2µ cos θ cosj 1 θ  . (5.39) 2a θ2b−2 = π − Dθ a 1 dφ e2µ cos φ cosj 1 φ b=1,..., 2+1 θ2 j=1,...,2+1 Using symmetry of the integrand in (5.39) to extend the integration ranges as before, expansion of the determinant into a sum over permutations yields

(4+2) Z4 (µ,...,µ) π 2µ cos θ2a w(ˆ 2a)−1 = sgn(w)ˆ dθ2a e cos θ2a ˆ ∈ = w S2+1 a 1 0

θ2a π ˆ − − ˆ + − × dθ e2µ cos θ cosw(2a 1) 1 θ dφ e2µ cos φ cosw(2 1) 1 φ 0 0 194 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

π θ2a 2µ cos θ2a 2µ cos θ = sgn(w)ˆ dθ2a e dθ e ˆ ∈ > a= w S2+1 1 0 0 w(ˆ 2a)−1 w(ˆ 2a−1)−1 w(ˆ 2a−1)−1 w(ˆ 2a)−1 × cos θ2a cos θ − cos θ2a cos θ π ˆ + − × dφ e2µ cos φ cosw(2 1) 1 φ (5.40) 0 which again has the standard form of a Pfaffian of dimension (2 + 2) × (2 + 2).Thelast integral in (5.40) can be expressed in terms of derivatives of the modified Bessel function I0(2µ). With the same functions (5.36), we thereby find the equal mass finite volume partition function in the explicit form

(4+2) Z4 (µ,...,µ) (i−1) (j−1) ∞ I (2µ)I ; − (2µ)−I (2µ)I ; − (2µ) − 2k+1 k j 1 2k+1 k i 1 I (i 1)(2µ) = Pfaff k=0 2k+1 0 . − (j−1) I0 (2µ) 0 i,j=1,...,2+1 (5.41) The (rather cumbersome) sum rules may now be extracted from (5.41) exactly as described above for the even rank case.

5.2.4. Nf = 4 + 1 For odd numbers of ﬂavours Nf = 2n + 1 the equal mass limit of the ﬁnite volume partition function (5.19) is given by

(2n+1) Z4 (µ,...,µ) n − − 2 = (2n 1)µ 4µκi 2 2 + − ND/ { →− } e dκi e ∆ κ1 ,...,κn ( 1) µ µ . (5.42) Dκ i=1 n The calculation of the integrals in (5.42) is identical to that of (5.21), with the obvious → 2 modiﬁcation of the integration domain and the replacements cos θi 2κi everywhere in the previous formulas. For n = 2, one may in this way bring (5.42) into the Pfafﬁan form

(4+1) Z4 (µ,...,µ)

−(4−1)µ ND/ (4−1)µ = e Pfaff E2(4µ) + (−1) e Pfaff E2(−4µ), (5.43) where E2(x) is the 2 × 2 antisymmetric matrix deﬁned by the elements

1 1 xκ2 2 i−1 xκ 2 2 j−1 E2(x)ij = dκ e κ dκ sgn(κ − κ )e κ . (5.44) 0 0 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 195

In this case it is more convenient to use the Fourier integral representation of the sign function given by ∞ 1 sin qy sgn(y) = dq , (5.45) π q + iε −∞ for y ∈ R, where the inﬁnitesimal parameter ε → 0+ regulates the ambiguity in sgn(y) at y = 0. Then the matrix elements (5.44) can be written in terms of the functions

1 xκ2+iqκ Ξq (x) = dκ e (5.46) 0 which may be integrated explicitly in terms of error functions. In this we may write the finite volume partition function (5.43) explicitly as (4+1) Z4 (µ,...,µ) ∞ (i−1) (j−1) ∗ (j−1) (i−1) ∗ − − i Ξq (4µ)Ξq (4µ) − Ξq (4µ)Ξq (4µ) = e (4 1)µ Pfaff dq 4π q + iε −∞ i,j=1,...,2 + (−1)ND/ {µ →−µ}, (5.47) where 1 π q2 q q − 2ix Ξq (x) = e 2x erf √ − erf √ . (5.48) 2i x 2 x 2 x The matrix elements in (5.47) can be made somewhat more explicit by using the derivative identity k d erf(x) k−1 2 −x2 = (−1) √ e Hk− (x), (5.49) dxk π 1 where Hm(x) are the Hermite polynomials. As before, this expression includes the one- flavour case Nf = 1( = 0), which is obtained from (5.47) by omitting the Pfaffians. These relations can also be used to now extract the massive spectral sum rules in these instances.

5.2.5. Nf = 4 + 3 The partition function (5.42) for n = 2 + 1 is evaluated identically as in (5.40) with → 2 the obvious change of integration domain and the substitutions cos θi 2κi as above. The single unpaired integration in this case can be expressed in terms of derivatives of the function 1 π √ Ξ (x) = erf(i x) (5.50) 0 2i x and we can thereby write the equal mass ﬁnite volume partition function as the (2 + 2) × (2 + 2) Pfafﬁan formula 196 R.J. Szabo / Nuclear Physics B 723 (2005) 163–197

+ Z(4 3)(µ,...,µ) 4   (i−1) (j−1) ∗ (j−1) (i−1) ∗ i ∞ Ξq (4µ)Ξq (4µ) −Ξq (4µ)Ξq (4µ) (i−1) − + −∞ dq Ξ (4µ) = e (4 1)µ Pfaff 4π q+iε 0  − (j−1) Ξ0 (4µ) 0 i,j=1,...,2+1 + (−1)ND/ {µ →−µ}. (5.51) This expression encompasses the case of three flavours ( = 0) by omitting the (2 + 1) × (2 + 1) blocks in (5.51). The Goldstone manifold (5.1) in this case is diffeomorphic to ∼ ∼ 2 ∼ 2 adisk,G4(3) = SO(3)/SO(2) × Z2 = S /Z2 = D , and the partition function is equal to the function (5.50) evaluated at x = 4µ. This simple evaluation follows of course from the explicit forms of the angular integrals (5.16) in this case. The corresponding massive spectral sum rule (2.13) is given by √ √ √ 1 4 µe4µ − π erf(2i µ) = √ √ . (5.52) 2 2 2 ζ + µ = 12 πµ erf(2i µ) ζ>0 4 Nf 3 In addition to the sum rules, the expressions (2.8), (5.7), (5.35), (5.41), (5.47) and (5.51) suffice to determine the spectral k-point correlation functions in closed form for an arbitrary number of flavours Nf in the limit where the collection of equal quark mass and (imaginary) Dirac operator eigenvalues is completely degenerate. In particular, they may be used to determine the correlators at the spectral origin for massless quarks, as well as the smallest eigenvalue distribution functions (2.10), (2.11).

Acknowledgements

The author is grateful to G. Akemann, P. Damgaard, D. Johnston and G. Vernizzi for helpful discussions and correspondence. This work was supported in part by a PPARC Ad- vanced Fellowship, by PPARC Grant PPA/G/S/2002/00478, and by the EU-RTN Network Grant MRTN-CT-2004-005104.

References

[1] J. Gasser, H. Leutwyler, Phys. Lett. B 188 (1984) 477; P.H. Damgaard, Nucl. Phys. B (Proc. Suppl.) 128 (2004) 47, hep-lat/0310037. [2] H. Leutwyler, A. Smilga, Phys. Rev. D 46 (1992) 5607. [3] C. Bernard, M.F.L. Golterman, Phys. Rev. D 46 (1992) 853, hep-lat/9204007; C. Bernard, M.F.L. Golterman, Phys. Rev. D 49 (1994) 486, hep-lat/9306005; M.F.L. Golterman, Acta Phys. Pol. B 25 (1994) 1731, hep-lat/9411005. [4] L. Giusti, C. Hoelbing, M. Lüscher, H. Wittig, Comput. Phys. Commun. 153 (2003) 31, hep-lat/0212012; L. Giusti, M. Lüscher, P. Weisz, H. Wittig, JHEP 0311 (2003) 023, hep-lat/0309189. [5] E.V. Shuryak, J.J.M. Verbaarschot, Nucl. Phys. A 560 (1993) 306, hep-th/9212088; J.J.M. Verbaarschot, I. Zahed, Phys. Rev. Lett. 70 (1993) 3852, hep-th/9303012. [6] G. Akemann, P.H. Damgaard, U. Magnea, S.M. Nishigaki, Nucl. Phys. B 487 (1997) 721, hep-th/9609174. [7] R.D. Pisarski, Phys. Rev. D 29 (1984) 2423. [8] J.J.M. Verbaarschot, I. Zahed, Phys. Rev. Lett. 73 (1994) 2288, hep-th/9405005. [9] M.A. Halasz, J.J.M. Verbaarschot, Phys. Rev. D 52 (1995) 2563, hep-th/9502096. R.J. Szabo / Nuclear Physics B 723 (2005) 163–197 197

[10] U. Magnea, Phys. Rev. D 61 (2000) 056005, hep-th/9907096. [11] U. Magnea, Phys. Rev. D 62 (2000) 016005, hep-th/9912207. [12] T. Nagao, K. Slevin, J. Math. Phys. 34 (1993) 2075. [13] P.H. Damgaard, U.M. Heller, A. Krasnitz, T. Madsen, Phys. Lett. B 440 (1998) 129, hep-th/9803012. [14] P.H. Damgaard, S.M. Nishigaki, Phys. Rev. D 57 (1998) 5299, hep-th/9711096. [15] G. Akemann, P.H. Damgaard, Nucl. Phys. B 598 (1998) 411, hep-th/9801133; G. Akemann, P.H. Damgaard, Phys. Lett. B 432 (1998) 390, hep-th/9802174. [16] J. Christiansen, Nucl. Phys. B 547 (1999) 329, hep-th/9809194. [17] C. Hilmoine, R. Niclasen, Phys. Rev. D 62 (2000) 096013, hep-th/0004081. [18] T. Nagao, S.M. Nishigaki, Phys. Rev. D 63 (2001) 045011, hep-th/0005077. [19] R.J. Szabo, Nucl. Phys. B 598 (2001) 309, hep-th/0009237. [20] G. Akemann, D. Dalmazi, P.H. Damgaard, J.J.M. Verbaarschot, Nucl. Phys. B 601 (2001) 77, hep- th/0011072. [21] T. Andersson, P.H. Damgaard, K. Splittorff, Nucl. Phys. B 707 (2005) 509, hep-th/0410163. [22] G. Akemann, P.H. Damgaard, Nucl. Phys. B 576 (2000) 597, hep-th/9910190. [23] H.W. Braden, A. Mironov, A. Morozov, Phys. Lett. B 514 (2001) 293, hep-th/0105169. [24] M.R. Zirnbauer, J. Math. Phys. 37 (1996) 4986, math-ph/9808012. [25] S. Helgason, Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York, 1978. [26] S. Helgason, Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators and Spherical Functions, Academic Press, New York, 1984. [27] M. Caselle, U. Magnea, Phys. Rep. 394 (2004) 41, cond-mat/0304363. [28] P.H. Damgaard, Phys. Lett. B 425 (1998) 151, hep-th/9711047. [29] S.M. Nishigaki, P.H. Damgaard, T. Wettig, Phys. Rev. D 58 (1998) 087704, hep-th/9803007; P.H. Damgaard, S.M. Nishigaki, Phys. Rev. D 63 (2001) 045012, hep-th/0006111. [30] A. Smilga, J.J.M. Verbaarschot, Phys. Rev. D 51 (1995) 829, hep-th/9404031; A.D. Jackson, M.K.Sener, ¸ J.J.M. Verbaarschot, Phys. Lett. B 387 (1996) 355, hep-th/9605183; T. Guhr, T. Wettig, Nucl. Phys. B 506 (1997) 589, hep-th/9704055; T. Nagao, S.M. Nishigaki, Phys. Rev. D 62 (2000) 065006, hep-th/0001137. [31] A.N. Redlich, Phys. Rev. Lett. 52 (1984) 18. [32] M.L. Mehta, Random Matrices, Academic Press, San Diego, 1991. [33] U. Magnea, Phys. Rev. D 64 (2001) 018902, hep-th/0009208. [34] M.R. Zirnbauer, F.D.M. Haldane, Phys. Rev. B 52 (1995) 8729, cond-mat/9504108. [35] Harish-Chandra, Amer. J. Math. 79 (1957) 87. [36] R.J. Szabo, Equivariant Cohomology and Localization of Path Integrals, Springer-Verlag, Berlin, 2000. [37] C. Itzykson, J.-B. Zuber, J. Math. Phys. 21 (1980) 411. [38] A.B. Balentekin, P. Cassak, J. Math. Phys. 43 (2002) 604, hep-th/0108130. [39] B. Schlittgen, T. Wettig, J. Phys. A 36 (2003) 3195, math-ph/0209030. [40] T. Nagao, P.J. Forrester, Nucl. Phys. B 509 (1998) 561. Nuclear Physics B 723 (2005) 201–202

Erratum Erratum to: “Factorization and the soft overlap contribution to heavy-to-light form factors” [Nucl. Phys. B 690 (2004) 249]

Björn O. Lange a, Matthias Neubert b

a Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA b Institute for High-Energy Phenomenology, Newman Laboratory for Elementary-Particle Physics, Cornell University, Ithaca, NY 14853, USA Received 10 June 2005 Available online 1 July 2005

Abstract

In Section 6 entitled “Operator mixing and Sudakov logarithms” of our work [1],we (M) (M) stated incorrectly that only linear combinations of the operators Ok that include O5 can contribute to the B → M form factors. The argument that led to this conclusion, (M) presented after Eq. (47), is erroneous. Linear combinations of the operators Ok with k = 1,...,4 may also contribute, provided that the corresponding Wilson coefﬁcients are free of endpoint singularities. As a result, Eq. (48) in [1] is incorrect, and Eq. (50) must be

DOI of original article: 10.1016/j.nuclphysb.2004.04.025. E-mail address: [email protected] (M. Neubert).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.06.019 202 B.O. Lange, M. Neubert / Nuclear Physics B 723 (2005) 201–202 generalized to read Ci(E, µI)ζM (µI,E) =Ci(E, µ)ζM (µ, E) SCETI SCETII 4 + (M) Ci(E, µhc)Dk (E, µhc,µ) = k 1 − (M) ⊗ (M) Ci(E, µ) dk (E, µ) ξk (µ), (50) (M) → (M) where ξk (µ) denote the factorizable B M matrix elements of the operators Ok in Eqs. (21) and (22), which can be expressed in terms of twist-2 and twist-3, two- and three-particle light-cone distribution amplitudes of the B meson and the light meson M. The main conclusion of our paper, that there exists an unsuppressed “soft overlap” contribution to the form factors (the first term√ on the right-hand side in the above equation) for which the hard-collinear scale µhc ∼ EΛ is irrelevant, remains unchanged. The coefficient functions in the second and third lines of Eq. (50), on the other hand, in general do contain logarithms of the hard-collinear scale. An explicit calculation of these coefficients could help to shed light on the question about the numerical size of the soft overlap piece.

Acknowledgements

We are grateful to M. Beneke and I. Stewart for useful discussions.

References

[1] B.O. Lange, M. Neubert, Nucl. Phys. B 690 (2004) 249, hep-ph/0311345. Nuclear Physics B 723 [FS] (2005) 205–233

Dynamical symmetries of semi-linear Schrödinger and diffusion equations

Stoimen Stoimenov a,b, Malte Henkel a

a Laboratoire de Physique des Matériaux 1, Université Henri Poincaré Nancy I, B.P. 239, F-54506 Vandœuvre lès Nancy Cedex, France b Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Soﬁa, Bulgaria Received 11 April 2005; received in revised form 18 May 2005; accepted 10 June 2005 Available online 1 July 2005

Abstract Conditional and Lie symmetries of semi-linear 1D Schrödinger and diffusion equations are studied if the mass (or the diffusion constant) is considered as an additional variable. In this way, dynamical symmetries of semi-linear Schrödinger equations become related to the parabolic and almost-parabolic subalgebras of a three-dimensional conformal Lie algebra (conf3)C. We consider non-hermitian representations and also include a dimensionful coupling constant of the non-linearity. The corresponding representations of the parabolic and almost-parabolic subalgebras of (conf3)C are classiﬁed and the complete list of conditionally invariant semi-linear Schrödinger equations is obtained. Possible applications to the dynamical scaling behaviour of phase-ordering kinetics are discussed.  2005 Elsevier B.V. All rights reserved.

PACS: 05.10.Gg; 02.20.Sv; 02.30.Jr; 11.25.Hf

Keywords: Non-linear Schrödinger equation; Non-linear diffusion equation; Schrödinger group; Conformal group; Conditional symmetry; Phase-ordering kinetics

E-mail address: [email protected] (M. Henkel). 1 Laboratoire associé au CNRS UMR 7556.

1. Introduction

Symmetries have since a long time played a major rôle in physics. For example, starting with Lie in 1881, the maximal kinematic invariance group of the free diffusion equation, where M is a constant M − 2 = 2 ∂t Φ ∂r Φ 0 (1.1) has been studied, which is the so-called Schrödinger group Sch(d) [61] which also arises a dynamical symmetry of the non-relativistic free particle [54,41,4]. We shall denote its Lie = algebra by schd .Ford 1 space dimensions, sch1 is spanned by the generators

X−1 =−∂t ,Y− 1 =−∂r , 2 Y 1 =−t∂r − Mr, 2 1 x X =−t∂ − r∂ − , 0 t 2 r 2 M X =−t2∂ − tr∂ − r2 − xt, 1 t r 2 M0 =−M, (1.2) which allows to write the non-vanishing commutators compactly as [Xn,Xn ]=(n − n )Xn+n , [Xn,Ym]=(n/2 − m)Yn+m, [Y 1 ,Y− 1 ]=M0 where n, n ∈{±1, 0} and m ∈ 2 2 {± 1 } 2 (see [45] for generalizations to d>1). The free Schrödinger equation is recovered from the analytical continuation M = im. It is a well-known fact that the same group also acts as kinematic invariance group of certain non-linear Schrödinger equations of the form − 2 = ∗ 2mi∂t Φ ∂r Φ F t,r,Φ,Φ . (1.3) If the potential is chosen in the form F = c(ΦΦ∗)2/d Φ, where c is a constant, then Eq. (1.3) is invariant under schd provided the scaling dimension of Φ is taken to be x = d/2, see e.g. [11,34,67]. Non-linear Schrödinger equations arise in many physical applications, for recent reviews see [3,70]. The Schrödinger equation shares with the Dirac equation the feature that its solutions have a probability interpretation as a wave function which can be understood from a subtle connection between the conformal and the Schrödinger groups [5]. We also mention the recently established Schrödinger-invariance of non-relativistic fluid dynamics [42,62]. Mathematically, there has been a lot of effort to establish the existence of solutions with certain regularity properties, see e.g. [10], and on the other hand the group classification of non-linear Schrödinger equations has been inten- sively studied, see [11,67,34,35,40,16,17,66,18] and references therein. Spectral properties are studied in [7]. A great deal is known about the representations of schd [63,60,27,25,31] and this can be applied to find symbolic solutions of non-linear Schrödinger equations.2

2 Related questions include the dynamical symmetries in de Sitter space [24,33], q-deformations [32],Chern– Simons theory [30,23], difference equations [26,57] or even the integrable quantum non-linear Schrödinger equation, see e.g. [15]. S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 207

Very similar equations have been studied in attempts to understand the coarsening process which systems undergo after having been quenched from a disordered initial state to below their critical point, see e.g. [12,9,13,22,38,20,49,51] for reviews. In its most simple setting, one considers a ferromagnetic system (e.g. described by an Ising model) which from some initial high-temperature state is rapidly quenched into its ordered phase below its critical temperature Tc > 0 where there are at least two competing equilibrium states. Although the system relaxes locally towards one of the equilibrium states, a global relax- ation is not possible and this leads to a very slow evolution of the macroscopic observables. From a microscopic point of view, the system’s evolution is characterized by the formation of correlated domains of time-dependent linear size L(t) and one typically finds a power-law behaviour L(t) ∼ t1/z where z is called the dynamical exponent. This in turn signals a dynamical scale-invariance in the system’s evolution. A coarse-grained description is usually given in terms of the order parameter Φ = Φ(t,r), which in the absence of any macroscopic conservation law is assumed to satisfy [52,12] ∂Φ dV(Φ) = Γ ∇2Φ − , (1.4) ∂t dΦ where Γ is a kinetic coefficient and V(Φ)is the potential which enters into the Ginzburg– Landau free-energy functional and which is assumed to have a double-well structure, i.e. V(Φ)= (1 − Φ2)2 [12,13]. For simplicity, we dropped here the thermal noise which is known [12] to be irrelevant for the long-time behaviour we are interested in. The disordered initial state enters through ‘white-noise’ initial conditions and one looks for the long-time behaviour of the solutions of (1.4). It has turned out to be convenient to characterize this evolution in terms of the two-time autocorrelation C(t,s) and the associated two-time autoresponse R(t,s) defined as δΦ(t) C(t,s) := Φ(t)Φ(s) ,R(t,s):= , (1.5) δh(s) h=0 where h(s) is the time-dependent magnetic field conjugate to Φ and · denotes the average over the fluctuations in the initial state (and thermal histories). Here, t is referred to as observation time and s as waiting time. By definition, the system undergoes ageing if C or R depend on both t and s and not merely on the difference τ = t − s. It is well- accepted (although still unproven) that in the ageing regime, that is for times t,s tmicro and t − s tmicro, where tmicro is some microscopic time scale, dynamical scaling holds true such that −b −1−a C(t,s) = s fC(t/s), R(t, s) = s fR(t/s) (1.6) and one would like to be able to compute the scaling functions fC,R(y). The free diffusion equation is the simplest example of a system undergoing ageing [21]. Motivated by a formal analogy with the well-known conformal invariance in equilibrium critical phenomena, it has been suggested that the Schrödinger group might play a similar rôle in certain ageing systems [43,45].3 If this working hypothesis is made, explicit forms for the scal-

3 Here we only consider the phase-ordering kinetics of ferromagnetic systems without any macroscopic conservation laws and quenched to below their critical point. Then z = 2 has been derived [68] and a necessary condition for the applicability of Schrödinger-invariance is satisﬁed. 208 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 ing function fR(y) [44,45] and more recently also for fC(y) [65,50] can been derived. These agree with the exact results in several soluble models, such as the spherical model [37,71,19,64], the 1D Glauber–Ising model [36,58,48], the critical voter model [28,29], the bosonic versions of the contact and the pair-contact processes [6] and the free random walk [21]. Furthermore, these forms also agree with the results of numerical simulations in the 2D and 3D kinetic Ising [47] and XY models [1,2] and with recent results in the 2D three-states Potts model [59]. In order to understand whether these observations may be viewed as manifestations of a dynamical symmetry we remark the following:

1. In ageing phenomena, time-translation invariance is broken. Therefore, one should a priori not expect full Schrödinger-invariance but at best invariance under some subalgebra of schd which does not contain time-translations. For a classification of the Lie subalgebras of sch1 see [11]. 2. If one considers equations as (1.4) as the classical equations of motion of a field-theory, sufficient conditions for the validity of Schrödinger-invariance can be formulated. If that field-theory is local and taking the analogous case of conformal invariance as a guide, one may show from a consideration of the energy–momentum tensor that the special Schrödinger-invariance (generated by X1) follows provided that dynamical scaling and in addition Galilei-invariance are satisfied [46]. 3. If the deterministic part of the field-theory is Galilei-invariant, then both C(t,s) and R(t,s) in the presence of noise can be expressed in terms of quantities calculable from the noiseless part [65]. Hence it is enough to study the symmetries of the deterministic part, which reduces the problem to understanding the symmetries, especially the Galilei-invariance, of a non-linear partial differential equation, such as (1.4).

At first sight, the problem of finding the Galilei-invariant semi-linear Schrödinger equations (1.3) seems to have been answered long ago and only allows the specific potential F quoted above [34]. Furthermore, complex-valued solutions φ of Eq. (1.3) are required, which apparently excludes applications to kinetic equations such as (1.4) with real-valued solutions. In this paper, we shall propose a way around these difficulties. We begin by recalling that the Schrödinger algebra schd in d spatial dimensions is a + subalgebra of the complexified conformal algebra (confd+2)C in d 2 dimensions [14, 46]. A simple way of seeing this is to treat the ‘mass’ M = im as a further dynamical variable and to introduce a new wave function [39,46] 1 −imζ Φ = Φm(t, r) = √ dζe Ψ(ζ,t,r). (1.7) 2π R For notational simplicity, we restrict from now on to the case d = 1. Then the generators (1.2) become

X−1 =−∂t ,Y− 1 =−∂r , 2 Y 1 =−t∂r − r∂ζ , 2 1 x X =−t∂ − r∂ − , 0 t 2 r 2 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 209

1 X =−t2∂ − tr∂ − r2∂ − xt, 1 t r 2 ζ M0 =−∂ζ . (1.8) − 2 = The free Schrödinger equation (1.1) then becomes (2∂ζ ∂t ∂r )Ψ 0 which through a further change of variables can be brought into a massless Klein–Gordon/Laplace equa- ∼ C ∼ tion in three dimensions which has the simple Lie algebra (conf3)C = so(5, ) = B2 as dynamical symmetry. It is useful to illustrate this in terms of a root diagram, see Fig. 1(a), from which the correspondence between the roots and the generators of sch1 can be read off. Four additional generators should be added in order to get the full conformal algebra (conf3)C whichwetakeintheform[46]

N =−t∂t + ζ∂ζ ,

V− =−ζ∂r − r∂t , 2 1 2 W =−ζ ∂ζ − ζr∂r − r ∂t − xζ, 2 2 V+ =−2tr∂t − 2ζr∂ζ − r + 2ζt ∂r − 2xr. (1.9)

In applications to ageing phenomena, it turned out that the parabolic subalgebras of conf3 play an important rôle.4 The complete list of non-isomorphic parabolic subalgebras of conf3 is as follows [46]

{ } 1. sch1, spanned by the set X−1,0,1,Y− 1 , 1 ,M0,N . { 2 2 } 2. age1, spanned by the set X0,1,Y− 1 , 1 ,M0,N . 2 2 3. alt1, spanned by the set {D,X1,Y− 1 1 ,M0,N,V+}. 2 , 2

Here we used the generator D of the full dilatations

D := 2X0 − N =−t∂t − r∂r − ζ∂ζ − x. (1.10) In Fig. 1(b)–(d), we illustrate the deﬁnition of these three parabolic subalgebras through their root diagrams. One can also consider the corresponding subalgebras without the generator N, namely

{ } 1. sch1, spanned by the set X−1,0,1,Y− 1 , 1 ,M0 . { 2 2 } 2. age1, spanned by the set X0,1,Y− 1 1 ,M0 . 2 , 2 3. alt1, spanned by the set {D,X1,Y− 1 1 ,M0,V+}. 2 , 2

We call almost-parabolic subalgebras those subalgebras of a parabolic subalgebra s which are obtained by leaving out one of the generators of the Cartan subalgebra h of s. Therefore, sch1, age1 and alt1 are almost parabolic, because we merely have to add the generator

4 The minimal standard parabolic subalgebra s0 of a simple complex Lie algebra g is spanned by the Cartan subalgebra h of g and the set of positive roots. A standard parabolic subalgebra s ⊂ g is a subalgebra of g which contains s0 [55]. 210 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233

Fig. 1. (a) Root diagram of the complex Lie algebra B2 and the identiﬁcation of the generators (1.8), (1.9) of the complexiﬁed conformal Lie algebra (conf3)C ⊃ (sch1)C. The double circle in the center denotes the Cartan subalgebra. The generators belonging to the three non-isomorphic parabolic subalgebras [46] are indicated by the full points, namely (b) sch1,(c)age1 and (d) alt1.

N ∈ h in order to make them parabolic subalgebras. In view of possible applications to 5 ageing phenomena, we shall be mainly interested in the algebras age1 and alt1,aswellas age1 and alt1 because they do not contain the time-translation generator X−1. After these preparations, we can now formulate the questions studied in this paper. For notational simplicity, we restrict ourselves to d = 1 space dimension and look for a semilinear and non-derivative extension of the free “Schrödinger equation” of the form Sˆ := − 2 = ∗ Ψ 2∂ζ ∂t ∂r Ψ F ζ,t,r,Ψ,Ψ , (1.11) which is invariant under one of the subalgebras of (conf3)C as sketched in Fig. 1(b)– (d). We shall initially take Ψ to be a complex-valued function as is appropriate for the physical context of Eq. (1.3) and shall return later to the question whether non-trivial symmetries involving only real-valued functions Ψ are possible, as would appear more natural for Eq. (1.4). Our main tool will be the construction of new differential-operator representations of the algebras defined above. In Section 2, we shall relax the condition of having skew-hermitian representations of the Schrödinger group and of the other algebras of Fig. 1. More general representations will be constructed and we shall then find the most general semi-linear equation of the form (1.11) invariant under these. On the other hand, non-linear equations of the forms (1.3), (1.4), (1.11) imply the existence of a coupling constant with the non-linearities, which in existing studies is tacitly admitted to be dimensionless. This hidden assumption leads to rather restricted forms of admissible potentials. Hence for a given form of the potential, Schrödinger-invariance should only hold for one special value of the spatial dimensionality d, which is in disagreement with the existing numerical and analytical results on ageing quoted above. We shall inquire into the consequences of treating dimensionful coupling constants, which consequently will transform under scaling and special transformations. In Section 3, we construct the representations with a dimensionful coupling constant and in Section 4 we find the non-linear Schrödinger equations invariant under these new representations. Section 5 presents our conclusions.

5 The name of age1 comes of course from ageing, while alt1 is inspired by its German equivalent, altern. S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 211

2. Non skew-hermitian representations

2.1. General remarks

We recall first the basic method used in finding non-linear equations with a given symmetry group, as outlined e.g. in [11]. Consider first the linear equation SΦ(t,r)ˆ = 0, (2.1) and let G be an one-dimensional Lie group which acts on Eq. (2.1) such that solutions are transformed into solutions. In this paper we are going to consider a projective action of the elements g ∈ G on the solutions Φ(t,r) such that T(g)Φ (t, r) = µg(t, r)Φ t ,r , (2.2) where (t,r) = g(t,r).ForG to be symmetry group one requires Sˆ T(g)Φ (r, t) = 0 (2.3) for all Φ satisfying (2.1). It is convenient to consider instead of G its Lie algebra g and to expand T(g)= 1 + εX +···, XΦ = a(t,r)∂t + b(t,r)∂r + c(t,r) Φ. (2.4) Now, Eq. (2.3) implies the operator equation [S,Xˆ ]=λ(t, r)S.ˆ (2.5) In the same way, for the non-linear equation ∗ SΦ(t,r)ˆ = F t,r,Φ,Φ (2.6) one arrives at the following conditions ∗ Sˆ T(g)Φ (t, r) = F t,r, T(g)Φ (t, r), T(g)Φ (t, r) , ˆ ∗ S(XΦ) = (XΦ)∂Φ F + (XΦ) ∂Φ∗ F. (2.7) Combining (2.5), (2.7) with the explicit form (2.4) of X gives [11] ∗ ∗ a(t,r)∂ + b(t,r)∂ − c(t,r)Φ∂ − c (t, r)Φ ∂ ∗ t r Φ Φ ∗ + c(t,r) + λ(t, r) F t,r,Φ,Φ = 0. (2.8) This linear equation can be solved with standard techniques. For Lie algebras with dim g > 1, one has one such equation for each independent generator. In the same way, further independent variables can be included straightforwardly. Evidently, the symmetry algebra of the quasi-linear equations considered here must be a subalgebra of the symmetry algebra of the linear part. Throughout, we shall work with the linear Schrödinger operator ˆ := − 2 S 2M0X−1 Y− 1 , (2.9) 2 212 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 which satisfies ˆ ˆ ˆ ˆ ˆ [S,M0]=[S,Y− 1 ]=[S,Y1 ]=[S,X−1]=[S,N]=0, 2 2 ˆ ˆ ˆ ˆ 1 [S,X0]=−S, [S,X1]=−2tS − 2 x − M0, 2 ˆ ˆ 1 [S,V+]=−4rS − 4 x − Y− 1 , (2.10) 2 2 from which we can read off the eigenvalues λ(t, r). We also see that invariance under the algebras sch1, age1 or alt1 (or their parabolic extensions) holds on the space of solutions of SΦˆ = 0, provided x = 1/2.

2.2. Representation with a generalized scaling behaviour

ˆ = For the free Schrödinger equation SΦ 0, invariance under sch1 implies that the scal- = 1 ing dimension of the solution Φ is x 2 . In order to relax this constraint while conserving the symmetry, we follow [65] and modify the generator X1 in (1.8) by adding a term −kt, where k is some constant. Then we should further add a further generator Z := −kZ0, where Z0 is central. From now on, we shall work with a variable ‘mass’ and go over to the conjugate variable ζ . We require that the commutators which are not in the centre are unchanged and ﬁnd that with respect to Eqs. (1.8), (1.9) only the following generators are modiﬁed

2 1 2 X1 =−t ∂t − tr∂r − r ∂ζ − (x + k)t, 2 2 V+ =−2tr∂t − 2ζr∂ζ − r + 2ζt ∂r − 2(x + k)r,

Z =−kZ0. (2.11) The only commutators which are modiﬁed are the central ones

[X1,X−1]=2X0 + kZ0,

[V+,Y− 1 ]=4X0 − 2N + 2kZ0, (2.12) 2 and we see that Z0 can be absorbed into a redeﬁnition of the generators X0 or D,as expected on general grounds [53]. However, the presence of the parameter k is important for the determination of invariant non-linear equations, as we shall see now. Consider the non-linear Schrödinger equation (1.11) − 2 = ˜ ∗ 2∂ζ ∂t ∂r Ψ(ζ,t,r) F ζ,t,r,Ψ,Ψ . (2.13) Eq. (2.8) then gives the following conditions on F˜ , for each of the generators: ˜ Y− 1 : ∂r F = 0, (2.14) 2 ˜ X−1: ∂t F = 0, (2.15) ˜ M0: ∂ζ F = 0, (2.16) ˜ Y 1 : (t∂r + r∂ζ )F = 0, (2.17) 2 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 213

1 x ∗ x X : t∂ + r∂ − Ψ∂ + Ψ ∂ ∗ + + 1 F˜ = 0, (2.18) 0 t 2 r 2 Ψ Ψ 2 ∗ ˜ D: t∂t + r∂r + ζ∂ζ − x Ψ∂Ψ + Ψ ∂Ψ ∗ + (x + 2) F = 0, (2.19) ˜ N: (t∂t − ζ∂ζ )F = 0, (2.20)

2 r ∗ X : t2∂ + tr∂ + ∂ − (x + k)t(Ψ ∂ + Ψ ∂ ∗ ) 1 t r 2 ζ Ψ Ψ

+ (x + k + 2)t F˜ = 0, (2.21) 2 ∗ V+: 2tr∂t + 2tζ + r ∂r + 2rζ∂ζ − 2r(x + k) Ψ∂Ψ + Ψ ∂Ψ ∗

+ 2r(x + k + 2) F˜ = 0. (2.22)

We point out that already the invariance of the linear Schrödinger equation under X1 + = 1 requires x k 2 . Consequently, the scaling dimension of Ψ which has to be ﬁxed for an invariance is given by x + k and not by x alone. This is a necessary condition for the invariance of the non-linear equation. Our results are as follows.

1. The algebra sch1.WehavetosolvethesystemofEqs.(2.14)–(2.18), (2.21).Asolu- = = 1 tion only exists in the well-known case k 0, hence x 2 . This in indeed quite analogous to Eq. (1.3) with a ﬁxed mass where the same power-law for the potential was found, see e.g. [11,4,34,67]. One has

∗ Ψ F˜ = F˜ Ψ,Ψ = Ψ 5f , (2.23) sch1 sch1 Ψ ∗ where f denotes an arbitrary differentiable function. Furthermore, adding the generator N by taking Eq. (2.20) into account does not give anything new. Hence the result for the ˜ ˜ parabolic subalgebra sch is the same as before, viz. F = F .6 1 sch1 sch1 In consequence, no new form of an invariant non-linear equation is found. 2. The algebra age1. We now have to solve the system of Eqs. (2.14), (2.16)–(2.18), (2.21). There is no further constraint on k and the solution is, using again x + k = 1/2

∗ − + + + Ψ F˜ = F˜ Ψ,Ψ ; t = t 4k/(2k 1)Ψ (2k 5)/(2k 1)f , (2.24) age1 age1 Ψ ∗ and we see that there also appears an explicit dependence on time (only the special case = k 0 was considered in [11]). On the other hand, if we go to the parabolic subalgebra age1 = ˜ = ˜ by adding the generator N we must have k 0 and we ﬁnd Fage1 Fsch1 , that is the same result as in the Schrödinger case.

6 In view of the embedding schd ⊂ (confd+2)C one might also consider the semi-linear conformally invariant + − Klein–Gordon equation φ = F(φ)= φ(n 2)/(n 2) in n = d + 2 dimensions, e.g. [34,69], which reproduces the form F ∼ φ5 for the case d = 1. 214 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233

3. The algebra alt1. Here we must solve Eqs. (2.14), (2.16), (2.17), (2.19), (2.21), (2.22). There are two distinct solutions. The ﬁrst one exists for all values of k and reads

∗ − Ψ F˜ (1) = F˜ (1) Ψ,Ψ ; t = t 2Ψf . (2.25) alt1 alt1 Ψ ∗ However, for k = 0 there is a second solution ∗ ∗ F˜ (2) = F˜ (2) Ψ,Ψ ; t = F˜ Ψ,Ψ . (2.26) alt1 alt1 sch1 Finally, if we go over to alt1 by adding the generator N, a solution exists only for k = 0, ˜ ˜ hence F = F . alt1 sch1

A few observations are in order. First, we see that we can trade in some cases the dependence of the potential F˜ on Ψ in favour of an explicit dependence on t such that the total dimension of F˜ is unchanged. Second, we find that this freedom does not exist for the parabolic subalgebras but only for the almost-parabolic subalgebras age1 and alt1 which do not contain the time-translations. Third, and in contrast to the well-known results where the ‘masses’ m (or M) are kept fixed [11,34,67], the dependence on the phase Ψ/Ψ ∗ is not determined in these representations. In particular, if we take f = cste., we even have the possibility to consider a Schrödinger equation with a real solution Ψ and a non-trivial symmetry, which is impossible if the masses are kept fixed. While we are not aware of immediate physical applications of the equations found here, these examples suggest that it may be useful to consider the presence of other dimensionful quantities, such as coupling constants, in the non-linear equations. We shall take this up in the next section.

3. Representations with an additional dimensionful coupling

In our search for more general quasi-linear Schrödinger equations with a non-trivial symmetry we now consider explicitly a dimensionful coupling constant g in the non- linear term. In what follows, we do consider that g also changes under the local scale- transformations. Hence the dilatation generator is taken to be of the form 1 x X =−t∂ − r∂ − yg∂ − , (3.1) 0 t 2 r g 2 where y is the scaling dimension of the coupling g. In addition, we also assume throughout that the generator of space translations Y− 1 =−∂r is unchanged. 2 In this section we construct the remaining generators. Their explicit form will be used in Section 4 to ﬁnd the invariant semilinear equations.

3.1. Subalgebras age1 and age1

Starting form Eqs. (1.8), (1.9), we look for extensions of the usual generators and write

M0 =−∂ζ − L(t, r, ζ, g)∂g, S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 215

Y 1 =−t∂r − r∂ζ − Q(t, r, ζ, g)∂g, 2 1 X =−t2∂ − tr∂ − r2∂ − P(t,r,ζ,g)∂ − xt, 1 t r 2 ζ g N =−t∂t + ζ∂ζ − K(t,r,ζ,g)∂g. (3.2) The unknown functions L, Q, P , K are determined from the following two requirements:

1. The commutators of the standard realizations of age1 and age1 are assumed to remain valid. 2. Invariance of the linear Schrödinger equation under the new representations.

First, we consider the almost-parabolic subalgebra age1. Imposing the commutator relations, our results are as follows. The commutators 1 [X0,X1]=−X1, [X0,Y1 ]=− Y 1 , [X0,M0]=0 (3.3) 2 2 2 give

y+1 y y−1 P = p0(ζ )t p(u,v), Q = q0(ζ )t q(u,v), L = l0(ζ )t l(u,v), (3.4) where u = r2/t and v = ty/g.Nextweuse

[X1,Y− 1 ]=Y 1 , [Y 1 ,Y− 1 ]=M0, 2 2 2 2 [Y− 1 ,M0]=0, [Y 1 ,M0]=0 (3.5) 2 2 and obtain

1 1 q(u,v) = 2u 2 ∂up(u,v), l(u,v) = 2u 2 ∂uq(u,v),

∂ul(u,v) = 0,p0(ζ ) = q0(ζ ) = l0(ζ ). (3.6) Hence l = l(v) and u q(u,v) = u1/2l(v) + n(v), p(u, v) = l(v) + u1/2n(v) + m(v), (3.7) 2 where m(v), n(v) are two functions of v to be determined. The last remaining commutators

[X1,Y1 ]=0, [X1,M0]=0 (3.8) 2 lead to the following system for the yet unknown functions l(v), n(v), m(v), p0(ζ ), where the prime denotes the derivative with respect to the argument p (ζ )y l(v) + vl (v) + p2(ζ )v2 l(v)m (v) − m(v)l (v) − p (ζ )m(v) = 0, (3.9) 0 0 0 (2y − 1)n(v) + 2yvn (v) + 2p (ζ )v2 n(v)m (v) − m(v)n (v) = 0, (3.10) 0 − 2 2 − = p0(ζ )n(v) p0(ζ )v l(v)n (v) n(v)l (v) 0. (3.11) Performing a separation of the variables v and ζ in Eq. (3.11), we see that two cases must be distinguished, depending on whether p0(ζ ) vanishes or not. 216 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233

Case (a) The NMG-representation with a non-modiﬁed mass generator M0. In this case, one has p0(ζ ) = p01 = cste. We have n(v) = l(v) = 0, m(v) = 0 and consequently: y+1 L = 0,Q= 0,P= p01t m(v) (3.12) and the function m(v) remains arbitrary.

Case (b) The MMG-representation with a modiﬁed mass generator M0. In this case, we have p (ζ ) =−2y where l is a constant. We ﬁnd the solutions l(v) = l v−1, n(v) = 0 l0ζ 0 0 (1−2y)/(2y) n0v and m(v) = cn(v) and set h0 := n0/l0. Consequently 2y 2y − L =− g, Q =− rg + h g(2y 1)/(2y) , ζ ζ 0

2 2y r g − − P =− + h trg(2y 1)/(2y) + ch t3/2g(2y 1)/(2y) . (3.13) ζ 2 0 0

We have hence established the existence of two distinct representations of the algebra =− age1. Explicitly, we always have Y−1/2 ∂r and further in the case (a) of the NMG- representation

M0 =−∂ζ ,

Y 1 =−t∂r − r∂ζ , 2 1 x X =−t∂ − r∂ − yg∂ − , 0 t 2 r g 2 1 + X =−t2∂ − tr∂ − r2∂ − p ty 1m ty/g ∂ − xt. (3.14) 1 t r 2 ζ 01 g

The NMG-representation of age1 is parametrized by the three constants x, y, p01 and an arbitrary function m(v). In the case (b) of the MMG-representation we have 2y M =−∂ + g∂ , 0 ζ ζ g 2y (2y−1)/(2y) Y 1 =−t∂r − r∂ζ + rg + h0g ∂g, 2 ζ 1 x X =−t∂ − r∂ − yg∂ − , 0 t 2 r g 2 2 1 2 X1 =−t ∂t − tr∂r − r ∂ζ 2 2 2y r g − − + + h trg(2y 1)/(2y) + ch t3/2g(2y 1)/(2y) ∂ − xt. (3.15) ζ 2 0 0 g

The MMG-representation of age1 is parametrized by the four constants x, y, h0, c. We now look for the most general representation of the parabolic subalgebra age1.From the commutators

[X0,N]=0, [Y− 1 ,N]=0 (3.16) 2 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 217 we obtain y K = k0(ζ )t k(u,v), ∂uk(u,v) = 0, (3.17) hence k(u,v) = k(v). Next, from the commutators

[X1,N]=X1, [Y 1 ,N]=Y 1 , [M0,N]=M0 (3.18) 2 2 we obtain that the functions l(v), n(v), m(v), k(v), p0(ζ ) and k0(ζ ) must satisfy the system: yk (ζ ) k(v) + vk (v) − yp (ζ ) m(v) + vm (v) + ζp (ζ )m(v) 0 0 0 − k (ζ )p (ζ )v2 m(v)k (v) − m (v)k(v) = 0, (3.19) 0 0 2yp (ζ ) n(v) + vn (v) − p (ζ )n(v) − 2ζp (ζ )n(v) 0 0 0 + 2k (ζ )p (ζ )v2 n(v)k (v) − n (v)k(v) = 0, (3.20) 0 0 yp (ζ ) l(v) + vl (v) − p (ζ ) + ζp (ζ ) l(v) 0 0 0 + 2 − − = k0(ζ )p0(ζ )v l(v)k (v) l (v)k(v) k0(ζ )k(v) 0. (3.21)

Our results from above for the functions l(v), n(v), m(v) found for the algebra age1 lead to: Case (a) NMG-representation = The non-trivial equations are (3.19) and (3.21). The last leads to k0(ζ )k(v) 0 which is −1 satisﬁed if k0(ζ ) = k0 = cste. (The other case k(v) = 0 leads to K = 0, m(v) = m0v , m0 = cste., which one can equivalently obtain by putting k0 = 0in(3.19).) =− =− We ﬁnd, besides Y−1/2 ∂r and M0 ∂ζ for the NMG-representation of age1

Y1/2 =−t∂r − r∂ζ , 1 x X0 =−t∂t − r∂r − yg∂g − , 2 2 y 2 1 2 y+1 t X1 =−t ∂t − tr∂r − r ∂ζ − p01t m ∂g − xt, 2 g ty N =−t∂ + ζ∂ − k tyk ∂ . (3.22) t ζ 0 g g

This representation is characterized by the constants x, y, p01, k0 and the functions m(v), k(v) which must satisfy the condition

d d d m(v) yk vk(v) − yp vm(v) − k p v2k(v)2 = 0. (3.23) 0 dv 01 dv 0 01 dv k(v)

Case (b) MMG-representation In this case Eqs. (3.19) and (3.20) lead to yk0(ζ ) k(v) + vk (v) + p0(ζ )m(v) = 0. (3.24)

Two cases must be considered: 218 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233

−1 1. k0(ζ ) = k0 = cste. This leads to k(v) = v , n(v) = m(v) = 0 and the ﬁrst MMG- realization of the algebra age1 is 2y M =−∂ + g∂ , 0 ζ ζ g 2y Y 1 =−t∂r − r∂ζ + (rg) ∂g, 2 ζ 1 x X0 =−t∂t − r∂r − yg∂g − , 2 2 1 2y r2g X =−t2∂ − tr∂ − r2∂ + ∂ − xt, 1 t r 2 ζ ζ 2 g N =−t∂t + ζ∂ζ − k0g∂g. (3.25) = = = 2. k0(ζ ) 0. We divide Eq. (3.24) by (k0(ζ )p0(ζ )) (since k0(ζ ) 0, p0(ζ ) 0), denote K = 1/k0(ζ ) and take the derivative with respect to ζ for obtaining (k(v) + vk (v)) K (ζ ) = l =: C = cste. (3.26) 0 m(v) 0 with solutions

1 C0 (1−2y)/(2y) k0(ζ ) = ,k(v)= 2y m(v) = v , = 2yC0ch0. (3.27) C0ζ l0 The second MMG-realization of the algebra age1 is 2y M =−∂ + g∂ , 0 ζ ζ g 2y (2y−1)/(2y) Y 1 =−t∂r − r∂ζ + rg + h0g ∂g, 2 ζ 1 x X =−t∂ − r∂ − yg∂ − , 0 t 2 r g 2 2 1 2 X1 =−t ∂t − tr∂r − r ∂ζ 2 2 2y r g − − + + h trg(2y 1)/(2y) + ch t3/2g(2y 1)/(2y) ∂ − xt, ζ 2 0 0 g

ch0 1 (2y−1)/(2y) N =−t∂t + ζ∂ − t 2 g ∂ . (3.28) ζ ζ g

Having exhausted the constraints from the commutation relation, we now require that the solution space of the linear Schrödinger equation is invariant under the action of these representations. This means ˆ ˆ [S,Xi]=λiS, (3.29) for each of the generators, where Sˆ is in the representation-independent form (2.9).We must distinguish the two cases S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 219

1. A ﬁxed scaling dimension x = 1/2 for the wave function Ψ . 2. Wave functions with arbitrary scaling dimensions.

First, for the NMG-realization (3.14) of the algebra age1 we have ˆ ˆ [S,X0]=−S, ˆ ˆ ˆ [S,M0]=[S,Y− 1 ]=[S,Y1 ]=0, 2 2 ˆ ˆ ˆ [S,X1]=−2tS − (1 − 2x)M0 + M0Q, (3.30) where ˆ y Q := 2p01t (y + 1)m(v) + yvm (v) ∂g. (3.31) In order to satisfy the condition (3.29) one must have ˆ (1 − 2x)M0 − M0Q Ψ = 0. (3.32) In general, we want to satisfy the (3.32) on the operator level and shall relax this to a condition on the functions in the representation space only if a non-trivial solution cannot be found otherwise. 1 ˆ y We begin with the case x = where we must have QΨ = (2p01t ((y + 1)m(v) + 2 yvm (v)))∂gΨ = 0. This leads to −(y+1)/y m(v) = m0v . (3.33)

The ﬁnal NMG-realization of age1 is in the case at hand

Y− 1 =−∂r , 2 M0 =−∂ζ ,

Y 1 =−t∂r − r∂ζ , 2 1 1 X =−t∂ − r∂ − yg∂ − , 0 t 2 r g 4 1 + 1 X =−t2∂ − tr∂ − r2∂ − p m g(y 1)/y∂ − t, (3.34) 1 t r 2 ζ 01 0 g 2 − 2 = and the invariant linear Schrödinger equation is simply (2∂ζ ∂t ∂r )Ψ 0. In the other case of arbitrary scaling dimension x = 1/2wehave QΨˆ = (1 − 2x)Ψ (3.35) to be satisﬁed. Here, because of the arbitrary value of m(v) the condition (3.35) is written in integral form (using Ψ(t,r,ζ,g)→ Ψ(t,r,ζ,v)where v = ty/g): 1 − 2x dv Ψ(t,r,ζ,v)= Ψ (t,r,ζ) 0 2 (3.36) 2p01 v ((y + 1)m(v) + yvm (v)) and the NMG-realisation of the algebra age1 is given by Eq. (3.14). We remember now, that “almost-parabolic subalgebras” were deﬁned through parabolic ones and we anticipate a result of the parabolic subalgebra age1. In the NMG- representation, we shall show below that invariance of the linear Schrödinger equation 220 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 leads to k(v) = v−1. Hence, because of (3.23) we have m(v) = v−1 and ˆ Q = 2p01g∂g (3.37) = 1 = = = 1 for x 2 and Q 0 (p01 0) in the case of canonical scaling dimension x 2 . Invariance of the linear Schrödinger equation implies the following condition on the wave function ˆ QΨ(t,r,ζ,g)= 2p01g∂gΨ(t,r,ζ,g)= (1 − 2x)Ψ(t,r,ζ,g), (3.38) − which means Ψ(t,r,ζ,g)= g(1 2x)/(2p01)ψ(t,r,ζ) and the dependence of Ψ on g is determined. There is another possibility, namely M0Ψ = 0. In this case the wave functions do not depend on ζ . So we have seen that instead of a simple Lie symmetry, we rather have a so-called conditional symmetry as introduced in [34,56]. Finally, the NMG-representation of the algebra age1 for arbitrary scaling dimensions, subject to the auxiliary condition (3.38) is

Y− 1 =−∂r , 2 M0 =−∂ζ ,

Y 1 =−t∂r − r∂ζ , 2 1 x X =−t∂ − r∂ − yg∂ − , 0 t 2 r g 2 1 X =−t2∂ − tr∂ − r2∂ − p tg∂ − xt. (3.39) 1 t r 2 ζ 01 g

Second, we consider the MMG-realization of (3.15) of the algebra age1. From the conditions (3.29) we have h0 = 0 and hence ˆ ˆ [S,X0]=−S, ˆ ˆ ˆ [S,M0]=[S,Y− 1 ]=[S,Y1 ]=0, 2 2 ˆ ˆ [S,X1]=−2tS − (1 − 2x)M0, (3.40) = 1 which are satisﬁed automatically in the case x 2 .

Remark 1. The subalgebra of age1 obtained when leaving out the generator X1 leaves the linear Schrödinger equation invariant for arbitrary h0.

In the case of an arbitrary scaling dimension we must have

2y M Ψ(t,r,ζ,g)= ∂ − g∂ Ψ(t,r,ζ,g)= 0, (3.41) 0 ζ ζ g which implies Ψ(t,r,ζ,g)= Ψ(t,r,ζg1/2y ) such that the dependence on ζ and g merely enters through the scaling variable u := g1/2yζ . Again, we merely ﬁnd a conditional symmetry. S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 221

Our ﬁnal result for the MMG-representation of age1 reads, for any value of x

Y− 1 =−∂r , 2 2y M =−∂ + g∂ , 0 ζ ζ g 2y Y 1 =−t∂r − r∂ζ + rg∂g, 2 ζ 1 x X =−t∂ − r∂ − yg∂ − , 0 t 2 r g 2 1 y X =−t2∂ − tr∂ − r2∂ + r2g∂ − xt. (3.42) 1 t r 2 ζ ζ g We ﬁnish by extending our results to the parabolic subalgebra age1.FortheNMG- representation we must also satisfy the condition ˆ y−1 [S,N]=yt k0 k(v) + vk (v) = 0, (3.43) which gives k(v) = v−1 and hence, from (3.23) we ﬁnd m(v) = v−1 for the general case = = 1 and m(v) 0 in the special case x 2 . The generators of the NMG-representation of age1 read in the general case

Y− 1 =−∂r , 2 M0 =−∂ζ ,

Y 1 =−t∂r − r∂ζ , 2 1 x X =−t∂ − r∂ − yg∂ − , 0 t 2 r g 2 1 X =−t2∂ − tr∂ − r2∂ − p tg∂ − xt, 1 t r 2 ζ 01 g N =−t∂t + ζ∂ζ − k0g∂g. (3.44)

= 1 = Remark 2. The case x 2 is obtained from the above realization by setting p01 0. The MMG-representations of age1 which leave invariant the linear Schrödinger equation coïncide with Eq. (3.25).Forx = 1/2 there is no restriction on Ψ while the condition (3.41) must be satisﬁed if x = 1/2. We shall need later that for the NMG-representation the linear Schrödinger equation reads simply − 2 = 2∂ζ ∂t ∂r Ψ(t,r,ζ,g) 0, (3.45) while for the MMG-representation there arises an additional term

4y 2∂ ∂ − g∂ ∂ − ∂2 Ψ(t,r,ζ,g)= 0. (3.46) ζ t ζ g t r This follows from the explicit representations and the form (2.9) of the Schrödinger operator Sˆ. 222 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233

Table 1 Representations of the almost-parabolic and parabolic subalgebras g ⊂ (conf3)C and their (conditionally) invariant linear Schrödinger equation SΨˆ = 0. The form of Sˆ and the auxiliary conditions only depend on the value of x but are independent of whether a parabolic or an almost-parabolic subalgebra is considered Case g Representation x Auxiliary conditions Sˆ = − 2 0 age1 NMG 1/22∂ζ ∂t ∂r L = 0, Q = 0, (y+1)/y P = p01m0g = − 2 NMG 1/2seeEq.(3.36) 2∂ζ ∂t ∂r L = 0, Q = 0, = y+1 y = 2 P p01t m(t /g) ∂ζ Ψ 0 ∂r = − 2 1 age1 NMG 1/22∂ζ ∂t ∂r = = = + − = − 2 L 0, Q 0, 1/2 (2p01g∂g (2x 1))Ψ 02∂ζ ∂t ∂r = = 2 P p01tg ∂ζ Ψ 0 ∂r 2 age1 K = k0g = − −1 − 2 3 age1 MMG 1/22∂ζ ∂t 4ygζ ∂g∂t ∂r L =−2yg/ζ =− = − = − 2 = Q 2y gr/ζ 1/2 (∂ζ 2y(g/ζ)∂g)Ψ 0 (2∂ζ ∂t ∂r )Ψ 0 P =−ygr2/ζ 4 age1 K = k0g = = = + −1 − 2 5 alt1 L sg/ζ, Q srg/ζ 1/22∂ζ ∂t 2sgζ ∂g∂t ∂r P = sr2g/2ζ −1 F = 2srg = 1/2 (∂ζ + sgζ ∂g)Ψ = ∂r Ψ = 0 ∂ζ ∂t = 6 alt1 K k0g = = = = = − 2 7 sch1 L Q P 0 1/2 ∂gΨ 02∂ζ ∂t ∂r = 2 ∂ζ Ψ 0 ∂r = = = = = 2 L Q 0, P 2ytg 1/2 ∂ζ Ψ 0 ∂r + − = − 2 (4yg∂g (2x 1))Ψ 02∂ζ ∂t ∂r 8 sch1 K = k0g

We summarize the results of the classification of this and the following subsections in Table 1.Forage1 and age1 we distinguish five cases. Case 1 is obtained from case 0 = −1 by setting m(v) v and only then there is an extension from age1 to age1 which is case 2. The cases 0, 1 and 3 refer to age1 and distinguish between the NMG and MMG representations as characterized by the explicit functions L, Q, P . For each of them, we give for both x = 1/2 and x = 1/2 the explicit form of the linear Schrödinger operator Sˆ.Forx = 1/2 there may be one or several auxiliary conditions which have to be met as well and in each case will lead to a modified form of the linear Schrödinger operator Sˆ. In these cases, we have hence not found a Lie symmetry but rather a non-classical one, usually referred to as Q-conditional symmetry [8,34]. Finally, the cases 2 and 4 apply to ˆ age1. Here the only new information is the explicit form of K whereas the form of S and the auxiliary conditions remain unchanged. As an example, consider case 2. It refers to the NMG-representation of age1 and the four functions P , Q, L, K can be read off. Furthermore, one sees that for x = 1/2, there S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 223

ˆ = − 2 is no further condition on Ψ and the linear Schrödinger operator is S 2∂ζ ∂t ∂r .Onthe other hand, for x = 1/2, there are two distinct auxiliary conditions. For each of them, one reads off the corresponding invariant linear Schrödinger operator Sˆ. The entry for case 1 can be read in the same way but the function K is of course not speciﬁed. The other cases are understood similarly.

3.2. Subalgebras alt1 and alt1

For these algebras we ﬁx the generators of dilatations D and space translations Y−1/2:

Y−1/2 =−∂r ,D=−t∂t − r∂r − ζ∂ζ − sg∂g − x, (3.47) where the exponent s describes the scaling behaviour of the coupling g. The remaining generators are taken in the following general form

M0 =−∂ζ − L(t, r, ζ, g)∂g,

Y1/2 =−t∂r − r∂ζ − Q(t, r, ζ, g)∂g, 1 X =−t2∂ − tr∂ − r2∂ − P(t,r,ζ,g)∂ − xt, 1 t r 2 ζ g N =−t∂ + ζ∂ − K(t,r,ζ,g)∂ , t ζ g 2 V+ =−2tr∂t − 2ζr∂ζ − r + 2ζt ∂r − F(t,r,ζ,g)∂g − 2xr. (3.48)

Since this is a subalgebra of (conf3)C, we obtain the following results for alt1. The commutators

[D,X1]=−X1, [D,Y1/2]=0,

[D,M0]=M0, [D,V+]=−V+ (3.49) give + P = ts 1p(u,v,w), Q = tsq(u,v,w), − + L = ts 1l(u,v,w), F = ts 1f(u,v,w) (3.50) where

r − ζ u = ,v= t sg, w = . (3.51) t t Next we use

[X1,Y− 1 ]=Y 1 , [Y 1 ,Y− 1 ]=M0, [Y− 1 ,M0]=0, 2 2 2 2 2 [Y 1 ,M0]=0, [V+,Y− 1 ]=2D, (3.52) 2 2 and ﬁnd

q(u,v,w) = u∂up(u,v,w), l(u,v,w)= u∂uq(u,v,w),

∂ul(u,v,w) = 0, q(u,v,w)= ul(v, w) + n(v, w), 224 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233

u2 p(u,v,w) = l(v,w) + un(v, w) + m(v, w), 2 f(u,v,w)= 4suv + z(v, w), (3.53) hence l = l(v,w). The remaining commutators

[X1,Y1 ]=0, [V+,M0]=2Y 1 , 2 2 [V+,Y1 ]=2X1, [V+,X1]=0 (3.54) 2 give the following system for the unknown functions l(v,w), n(v, w), m(v, w) and z(v, w)

∂wn + l∂vn − n∂vl = 0,

(s − 1)l − sv∂vl − w∂wl − ∂wm + m∂vl − l∂vm = 0,

(s − 1)n − sv∂vn − w∂wn + m∂vn − n∂vm = 0,

2n + z∂vl − l∂vz = 0,

2n + z∂vl − l∂vz − ∂wz = 0,

2m + 2wl − 2sv + z∂vn − n∂vz = 0,

2wn − (s + 1)z + sv∂vz + w∂wz + z∂vn − n∂vz = 0. (3.55) The solution is readily found

z = 0,n= 0,l(v,w)= vl1(w), m(v, w) = sv − wvl1(w), (3.56) and we arrive at the most general representation of the algebra alt1

D =−t∂t − r∂r − ζ∂ζ − sg∂g − x,

Y− 1 =−∂r , 2 −1 M0 =−∂ζ − t l1(w)g∂g, −1 Y 1 =−t∂r − r∂ζ − t rl1(w)g∂g, 2 1 t−1r2 X =−t2∂ − tr∂ − r2∂ − l (w) + t(s − wl (w)) g∂ − xt, 1 t r 2 ζ 2 1 1 g 2 V+ =−2tr∂t − 2ζr∂ζ − r + 2ζt ∂r − 2srg∂g − 2xr, (3.57) which depends on the parameters x,s and arbitrary function l1(w). For the algebra alt1 one must satisfy also the commutators

[D,N]=0, [Y− 1 ,N]=0, [M0,N]=M0, 2 [Y 1 ,N]=Y 1 , [X1,N]=X1, [V+,N]=0, (3.58) 2 2 s which gives K = t k(u,v,w) and ∂uk(u,v,w) = 0 which means that k(u,v,w) = k(v,w) and furthermore we have the set of equations

(2 − s)l + sv∂vl + 2w∂wl + ∂wk + l∂vk − k∂vl = 0,

(1 − s)n + sv∂vn + 2w∂wn + n∂vk − k∂vn = 0, S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 225

sk − sv∂vk − w∂wk − sm + sv∂vm + 2w∂wm + m∂vk − k∂vm = 0,

(s + 1)z − sv∂vz − 2w∂wz + z∂vk − k∂vz = 0. (3.59) Using Eq. (3.56) the system (3.59) reduces to the single equation

2vl1 + 2vw∂wl1 + ∂wk − kl1 + vl1∂vk = 0, (3.60) and if we recall that D = 2X0 − N we obtain the ﬁnal result = + = = −1 = k(v,w) k0v, k0 s 2y, l1(w) l0w ,Kk0g, (3.61) where k0 and l0 are constants. We therefore have the following realization of alt1

D =−t∂t − r∂r − ζ∂ζ − sg∂g − x,

Y− 1 =−∂r , 2 l M =−∂ − 0 g∂ , 0 ζ ζ g l0 Y 1 =−t∂r − r∂ζ − rg∂g, 2 ζ 1 l X =−t2∂ − tr∂ − r2∂ − 0 r2 + (s − l )t g∂ − xt, 1 t r 2 ζ 2ζ 0 g 2 V+ =−2tr∂t − 2ζr∂ζ − r + 2ζt ∂r − 2srg∂g − 2xr, =− + − N t∂t ζ∂ζ k0g∂q . (3.62) Next we impose the invariance of the linear Schrödinger equation. We have ˆ ˆ [S,X0]=−S, ˆ ˆ ˆ [S,M0]=[S,Y− 1 ]=[S,Y1 ]=0, 2 2 ˆ ˆ [S,X1]=−2tS − (1 − 2x)M0, ˆ ˆ [S,V+]=−4rS + 2(1 − 2x)Y− 1 . (3.63) 2 The conditions (3.29) are satisﬁed if l0 = s. ˆ Remark 3. If the generator X1 is left out from alt1 and alt1, the equation SΨ = 0is invariant even for arbitrary l0.

= 1 In the case with ﬁxed scaling dimension x 2 the following realization of algebra alt1 is a dynamical symmetry of the linear Schrödinger equation 1 D =−t∂ − r∂ − ζ∂ − sg∂ − , t r ζ g 2 Y− 1 =−∂r , 2 s M =−∂ − g∂ , 0 ζ ζ g s Y 1 =−t∂r − r∂ζ − rg∂g, 2 ζ 226 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233

1 s 1 X =−t2∂ − tr∂ − r2∂ − r2g∂ − t, 1 t r 2 ζ 2ζ g 2 2 V+ =−2tr∂t − 2ζr∂ζ − r + 2ζt ∂r − 2syrg∂g − 2xr (3.64) without any restrictions on the wave function. For arbitrary x the conditions (3.63) lead to very strong restrictions on the wave function

s M Ψ(ζ,t,r,g)= ∂ + g∂ Ψ(ζ,t,r,g)= 0, 0 ζ ζ g Y− 1 Ψ(ζ,t,r,g)=−∂r Ψ(ζ,t,r,g)= 0, (3.65) 2 which means that Ψ(ζ,t,r,g)= ψ(t,g1/sζ). Finally, we add the generator N in order to obtain the representation of alt1.Itisof =− + − [ ˆ ]= the form N t∂t ζ∂ζ k0g∂g and the condition S,N 0 is satisﬁed automatically. Note that for this algebra the linear Schrödinger equation has the form:

2s 2∂ ∂ + g∂ ∂ − ∂2 Ψ(ζ,t,r,g)= 0. (3.66) ζ t ζ g t r These results are also included in Table 1. 3.3. Subalgebras sch1 and sch1

In these algebras we must add to age1 and age1 the generator of time translation X−1 =−∂t and satisfy the commutator [X1,X−1]=2X0. Inserting this condition into −1 −y the generators Eq. (3.39) we ﬁnd p01 = 2y and m(v) = v = gt . The representation of sch1 is

X−1 =−∂t ,Y− 1 =−∂r , 2 M0 =−∂ζ ,

Y 1 =−t∂r − r∂ζ , 2 1 x X =−t∂ − r∂ − yg∂ − , 0 t 2 r g 2 1 X =−t2∂ − tr∂ − r2∂ − 2ytg∂ − xt, (3.67) 1 t r 2 ζ g = and for sch1 we have K k0g, hence (3.67) holds true, together with

N =−t∂t + ζ∂ζ − k0g∂g. (3.68) For the invariance of the linear Schrödinger equation we merely have to consider a single commutator ˆ ˆ ˆ ˆ [S,X1]=−2tS − (1 − 2x)M0 + M0Qsch, Qsch = 4yg∂g. (3.69) The linear Schrödinger equation is invariant if either

M0Ψ = 0, Ψ(ζ,t,r,g)= ψ(t,r,g), (3.70) S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 227 which means that either the wave function does not depend on ζ or else ˆ (1−2x)/4y QschΨ = (1 − 2x)Ψ, Ψ(ζ,t,r,g) = g ψ(ζ,t,r). (3.71) = 1 Even in the case x 2 we must impose an auxiliary condition on the wave functions. Again, we include our results in Table 1.

4. Invariant non-linear equations

Having classiﬁed in the previous section the representations of the parabolic and almost-parabolic subalgebras of (conf3)C which leave the linear Schrödinger equation SΨˆ = 0 invariant, we now construct systematically all semilinear invariant equations SΨˆ = F(ζ,t,r,g; Ψ,Ψ ∗), along the lines recalled in Section 2.1.

4.1. Subalgebras age1 and age1

First we take the almost-parabolic subalgebra (3.14). The potential F must satisfy the system, see Eq. (2.8) ∂ F = 0,∂F = 0, r ζ ∗ 2t∂ + 2yg∂ − x Ψ∂ + Ψ ∂ ∗ + (x + 2) F = 0, t g Ψ Ψ y ∗ t∂t + p01t m(v)∂g − x Ψ∂Ψ + Ψ ∂Ψ ∗ + (x + 2) F = 0. (4.1) In the general case the solution is: x+2 dv p01vm(v) − 2y Ψ 0 = x + Fage Ψ f0 ln Ψ , ∗ . (4.2) 1 v p01vm(v) − y Ψ −k In the special case m(v) = v this leads to the following form of f0

− − − x Ψ f = f ln tx/2g x/(2y) p − yt(k 1)yg1 k 2y(k−1) Ψ , . (4.3) 0 0 01 Ψ ∗ Note that the case x = 1/2 can be obtained by putting k = (y + 1)/y. When the operator Qˆ is taken in form (3.37) the solution in the NMG-representation is

− x+2 x 1 p −2y − 1 p −2y − Ψ F = t 01 g 2(p01 y) f t 01 g 2(p01 y) Ψ, . (4.4) age1 age1 Ψ ∗ So, the non-linear Schrödinger equation is 2∂ ∂ − ∂2 Ψ(ζ,t,r,g)= F 1 . (4.5) ζ t r age1 For the extension to the parabolic algebra age1, we must add the equation

(t∂t + k0g∂g)F = 0, (4.6) 1 which leads to the following equation for fage 1 − + p01 2y k0 + − ∂ 1 = x 2 xu fage (u, v) 0, (4.7) 2(p01 − y) ∂u 1 228 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233

x − − ∗ where u = (tp01 yg) 2(p01 y) Ψ and v = Ψ/Ψ . There are two cases:

1. The generic solution p01 = 2y − k0

+ 1 x 2 Ψ F = Ψ x fsch , (4.8) age1 1 Ψ ∗ which is the same as for the non-modiﬁed representation of the Schrödinger algebra. 2. A non-generic solution p01 = 2y − k0. Here the form of the potential is the same as in (4.4) but one has an additional constraint on the parameters of the algebra.

= 1 = Note that in the canonical case x 2 one must also put p01 0 in the above results. We consider now the MMG-realisation (3.42) of the algebra age1. The potential is found from

g ∂ F = 0, ∂ − 2y ∂ F = 0, r ζ ζ g ∗ 2t∂ + 2yg∂ − x Ψ∂ + Ψ ∂ ∗ + (x + 2) F = 0, t g Ψ Ψ ∗ t∂t − x Ψ∂Ψ + Ψ ∂Ψ ∗ + (x + 2) F = 0, (4.9) which has the solution

+ − Ψ − F 2 = bx 2f 2 b xΨ, ,b= t 1ζg1/(2y). (4.10) age1 age1 Ψ ∗ The non-linear equation invariant under the same algebra is

4y 2 2 2∂ζ ∂t − g∂g − ∂ Ψ(ζ,t,r,g)= F . (4.11) ζ r age1 When we extend this to the parabolic subalgebra age1 by including the generator N (see (3.25)), we have

k ∂ 0 − 2 x + 2 − xu f 2 (u, v) = 0, (4.12) 2y ∂u age1 where u = bxΨ and v = Ψ/Ψ ∗. Our results give

1. The generic solution k0 = 4y. We recover the same result as for non-modiﬁed Schrödinger algebra (4.8). 2. A non-generic solution k0 = 4y. The result is the same as in (4.10), but the central generator N =−t∂t + ζ∂ζ − 4yg∂g is speciﬁed.

= 1 These results are valid for arbitrary scaling dimension, but for the case x 2 there is no restriction on the wave functions. The results for the non-linear potential F are collected in Table 2. We give the generic solutions for the cases as deﬁned in Table 1. For each of the cases, we had the linear equation SΨˆ = 0 together with eventual auxiliary condition(s) for a given value of x.Now S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 229

Table 2 Solutions for the invariant semilinear potential of the equation SΨˆ = F . The cases are the ones from Table 1 and f is a generic symbol for an arbitrary function Case Subalgebra Potential F Condition x+2 p vm(v)−2y ∗ 0 age Ψ x f ln Ψ + dv 01 ,Ψ/Ψ m(v) arbitrary v = ty /g 1 v p01vm(v)−y x+2 x ∗ p −2y 1/(2(p −y)) 1 age1 a f(a Ψ,Ψ/Ψ )a=[t 01 g] 01 (x+2)/x ∗ 2 age1 Ψ f(Ψ/Ψ )p01 = 2y − k0 x+2 x ∗ −k 1/2(y−k ) age1 a f(a Ψ,Ψ/Ψ )p01 = 2y − k0 a =[gt 0 ] 0 (x+2) −x ∗ −1 1/2y 3 age1 b f(b Ψ,Ψ/Ψ )b= t ζg (x+2)/x ∗ 4 age1 Ψ f(Ψ/Ψ )k0 = 4y (x+2) −x ∗ −1 1/2y age1 b f(b Ψ,Ψ/Ψ )k0 = 4yb= t ζg −x−2 −s x ∗ 5 alt1 t f(ζ g,t Ψ,Ψ/Ψ ) −x−2 x ∗ s −1 1/(s+k ) 6 alt1 c f(c Ψ,Ψ/Ψ )c= (ζ g ) 0 t −(x+2)/2y x/2y ∗ 7 sch1 g f(g Ψ,Ψ/Ψ ) (x+2)/x ∗ 8 sch1 Ψ f(Ψ/Ψ )k0 = 0 −(x+2)/2y x/2y ∗ sch1 g f(g Ψ,Ψ/Ψ )k0 = 0 the associated non-linear equation is simply SΨˆ = F , where F is read from the table. In Table 2 we also list the non-generic solutions which are distinguished by the conditions mentioned and for some of which there are modiﬁed scaling variables to be used. 4.2. Subalgebras alt1 and alt1

For the realization (3.64) of alt1 (note that for this realization of the algebra one has + = k0 s 2y by construction), the potential is found from ∂ F = 0,(ζ∂+ sg∂ )F = 0, r ζ g ∗ t∂ + ζ∂ + sg∂ − x Ψ∂ + Ψ ∂ ∗ + (x + 2) F = 0, t ζ g Ψ Ψ ∗ t∂t − x Ψ∂Ψ + Ψ ∂Ψ ∗ + (x + 2) F = 0, (4.13) to have the form

− − − Ψ F = t x 2f txΨ,ζ sg, . (4.14) alt1 alt1 Ψ ∗ When we want to have also invariance under alt1 the N operator gives (t∂t − ζ∂ζ + k g∂g)Falt = 0. So in this case we have 0 1 −x−2 x Ψ s/(s+k ) −1/(s+k ) F = a f a Ψ, ,a= tζ 0 g 0 . (4.15) alt1 alt1 Ψ ∗ The non-linear equation invariant under the same algebra is

g 2 2∂ζ ∂t + 4y ∂g∂t − ∂ Ψ(ζ,t,r,g)= Falt (F ), (4.16) ζ r 1 alt1 and the results are again included in Table 2. We point out that the form of F agrees with the special solution obtained before for age1-MMG representation. 230 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233

4.3. Subalgebras sch1 and sch1

Now the system is ∂ F = 0,∂F = 0,∂F = 0, r ζ t ∗ 2yg∂g − x Ψ∂Ψ + Ψ ∂Ψ ∗ + (x + 2) F = 0, (4.17) which has the solution

− + Ψ F = g (x 2)/(2y)f gx/(2y)Ψ, . (4.18) sch1 sch1 Ψ ∗ The result for the algebra sch1 we obtain by adding the invariance under generator N, which leads to the following equation

k ∂ ∗ 0 x + 2 − xu f (u, v) = 0,u= gx/(2y)Ψ, v = Ψ/Ψ , (4.19) 2y ∂u sch1 with solutions:

(i) In the case k0 = 0 like (4.8). (ii) In the case k0 = 0 like (4.18), but with a non-modiﬁed central generator N.

The non-linear equation has the generic form − 2 = 2∂ζ ∂t ∂r Ψ(t,r,ζ,g) Fsch1 . (4.20)

5. Consequences and conclusion

Motivated by the problem to understand the form of the scaling functions of the two- time observables in phase-ordering kinetics, we have been led to reconsider the question of finding semilinear Schrödinger equations which are invariant under a conveniently chosen representation of the Schrödinger group. The main difficulty is that if one considers the ‘mass’ as a fixed constant, Galilei- together with spatial-translation invariance implies that such an equation should be invariant under simple phase shifts which severely restricts the possible form of a non-linear potential and in general is only possible for complex wave functions Φ. As we have seen, a possible way out of this difficulty is to consider the ‘mass’ as an additional dynamical variable and then to go over to a ‘dual’ formulation with a wave = function Ψ Ψ(ζ,t,r),seeEq.(1.7). In this way the Schrödinger algebra schd is actually embedded into a conformal algebra (confd+2)C which naturally leads to the question of finding all semilinear Schrödinger equations SΨˆ = F(Ψ,Ψ∗) conditionally invariant under some parabolic or almost-parabolic subalgebra of (confd+2)C,seeFig. 1(b)–(d). We then considered two extensions by further giving up some other property which is habitually admitted:

1. Non-hermitian representations were constructed in Section 2 and the corresponding non-linear equations are found, see Eqs. (2.23)–(2.26). S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 231

2. A dimensionful coupling g was explicitly introduced into the potential. The classiﬁ- cation of the differential operator representations which also leave the linear equation SΨˆ = 0 invariant is given in Table 1 and the corresponding semi-linear Schrödinger equations SΨˆ = F are listed in Table 2.

Besides this classiﬁcation, we think the most remarkable result is that quite generally, real- valued solutions of these invariant equations are obtained. To illustrate the possible impact on the understanding of phase-ordering kinetics, we reconsider Eq. (4.4) with p01 = 2y or else Eq. (4.18) together with their auxiliary condition. We write the wave function as Ψ(ζ,t,r,g)= g(1−2x)/(4y)ψ(ζ,t,r) where ψ is a real-valued function which satisﬁes the equation − 2 = −5/(4y) 1/(4y) = 5 ¯ 4y 2∂ζ ∂t ∂r ψ g f g ψ ψ f gψ , (5.1) where f is an arbitrary function and f(x) = x5f(¯ x). Up to the Fourier/Laplace transform with respect to ζ , this is of the same form as the coarse-grained kinetic equation (1.4) habitually used to describe coarsening. Since quite different functions f can be described in terms of the same symmetry, this might provide an explanation for the well-established fact [12] that the long-time behaviour of correlators and response functions in phase-ordering kinetics is quite independent of the precise form of the potential V(Φ). We shall elaborate on this elsewhere. On the other hand, one may use these symmetries to reduce the non-linear Schrödinger equation to a linear equation and hence obtain new explicit solutions, along the lines of [31]. We hope to return to this elsewhere.

Acknowledgements

We thank R. Cherniha and R. Schott for useful conversations. S.S. was supported by the EU Research Training Network HPRN-CT-2002-00279.

References

[1] S. Abriet, D. Karevski, Eur. Phys. J. B 37 (2004) 43. [2] S. Abriet, D. Karevski, Eur. Phys. J. B 41 (2004) 79. [3] I.S. Aranson, L. Kramer, Rev. Mod. Phys. 74 (2002) 100. [4] A.O. Barut, R. Raczka, Theory of Group Representations and Applications, Polish Science Publications, Warsaw, 1980. [5] A.O. Barut, B.W. Xu, Phys. Lett. A 82 (1981) 218. [6] F. Baumann, M. Henkel, M. Pleimling, J. Richert, cond-mat/0504243, J. Phys. A 38 (2005), in press. [7] V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov, Adv. Theor. Math. Phys. 7 (2004) 711. [8] G.W. Bluman, J.D. Cole, J. Math. Mech. 18 (1969) 1025. [9] J.P. Bouchaud, in: M.E. Cates, M.R. Evans (Eds.), Soft and Fragile Matter, IOP Press, Bristol, 2000. [10] J. Bourgain, Global Solutions of Non-Linear Schrödinger Equations, Amer. Math. Soc., Providence, 1999. [11] C.D. Boyer, R.T. Sharp, P. Winternitz, J. Math. Phys. 17 (1976) 1439. [12] A.J. Bray, Adv. Phys. 43 (1994) 357. 232 S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233

[13] A.J. Bray, in: M.E. Cates, M.R. Evans (Eds.), Soft and Fragile Matter, IOP Press, Bristol, 2000. [14] G. Burdet, M. Perrin, P. Sorba, Commun. Math. Phys. 34 (1973) 85; G. Burdet, J. Patera, M. Perrin, P. Winternitz, Lie subalgebras and Schrödinger algebra, preprint CRM-689 (fév 1997) (in French). [15] V. Caudrelier, M. Minchev, E. Ragoucy, J. Phys. A 37 (2004) L367. [16] R. Cherniha, J.R. King, J. Phys. A 33 (2000) 267; R. Cherniha, J.R. King, J. Phys. A 33 (2000) 7839. [17] R. Cherniha, J.R. King, J. Phys. A 36 (2003) 405. [18] R. Cherniha, M. Henkel, J. Math. Anal. Appl. 298 (2004) 487. [19] F. Corberi, E. Lippiello, M. Zannetti, Phys. Rev. E 65 (2002) 046136. [20] A. Crisanti, F. Ritort, J. Phys. A 36 (2003) R181. [21] L.F. Cugliandolo, J. Kurchan, G. Parisi, J. Physique I 4 (1994) 1641. [22] L.F. Cugliandolo, Dynamics of glassy systems, cond-mat/0210312. [23] M. de Montigny, F.C. Khanna, A.E. Santana, E.S. Santos, J.D.M. Vianna, Ann. Phys. 277 (1999) 144. [24] P.A.M. Dirac, J. Math. Phys. 4 (1963) 901. [25] V.K. Dobrev, H.D. Doebner, C. Mrugalla, Rep. Math. Phys. 39 (1997) 201. [26] V.K. Dobrev, H.D. Doebner, C. Mrugalla, Mod. Phys. Lett. A 14 (1999) 1113. [27] H.-D. Doebner, H.-J. Mann, J. Math. Phys. 36 (1995) 3210. [28] I. Dornic, H. Chaté, J. Chave, H. Hinrichsen, Phys. Rev. Lett. 87 (2001) 045701. [29] I. Dornic, Thèse de doctorat, Nice et Saclay, 2002. [30] G.V. Dunne, R. Jackiw, C.A. Trugenberger, Ann. Phys. 194 (1989) 197. [31] P. Feinsilver, Y. Kocik, R. Schott, Fortschr. Phys. 52 (2004) 343. [32] R. Floreanini, L. Vinet, Lett. Math. Phys. 32 (1994) 37. [33] C. Fronsdal, Rev. Mod. Phys. 37 (1965) 221; C. Fronsdal, Phys. Rev. D 10 (1974) 589; C. Fronsdal, Phys. Rev. D 12 (1975) 3819; C. Fronsdal, Phys. Rev. D 26 (1982) 1988; M. Flato, C. Fronsdal, Phys. Lett. B 97 (1980) 236. [34] W.I. Fushchich, W.M. Shtelen, N.I. Serov, Symmetry Analysis and Exact Solutions of Equations of Nonlin- ear Mathematical Physics, Kluwer, Dordrecht, 1993. [35] W.I. Fushchich, R.M. Cherniha, J. Phys. A 28 (1995) 5569. [36] C. Godrèche, J.M. Luck, J. Phys. A 33 (2000) 1151. [37] C. Godrèche, J.M. Luck, J. Phys. A 33 (2000) 9141. [38] C. Godrèche, J.M. Luck, J. Phys.: Condens. Matter 14 (2002) 1589. [39] D. Giulini, Ann. Phys. 249 (1996) 222. [40] F. Güngör, J. Phys. A 32 (1999) 977. [41] C.R. Hagen, Phys. Rev. D 5 (1972) 377. [42] M. Hassaïne, P.A. Horváthy, Ann. Phys. 282 (2000) 218; M. Hassaïne, P.A. Horváthy, Phys. Lett. A 279 (2001) 215. [43] M. Henkel, J. Stat. Phys. 75 (1994) 1023. [44] M. Henkel, M. Pleimling, C. Godrèche, J.-M. Luck, Phys. Rev. Lett. 87 (2001) 265701. [45] M. Henkel, Nucl. Phys. B 641 (2002) 405. [46] M. Henkel, J. Unterberger, Nucl. Phys. B 660 (2003) 407. [47] M. Henkel, M. Pleimling, Phys. Rev. E 68 (2003) 065101(R). [48] M. Henkel, G.M. Schütz, J. Phys. A 37 (2004) 591. [49] M. Henkel, Adv. Solid State Phys. 44 (2004) 389. [50] M. Henkel, A. Picone, M. Pleimling, Europhys. Lett. 68 (2004) 191. [51] M. Henkel, cond-mat/0503739. [52] P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49 (1977) 435. [53] V.G. Kac, A.K. Raina, Bombay Lectures on Heighest-Weight Representations of Inﬁnite-Dimensional Lie Algebras, World Scientiﬁc, Singapore, 1987. [54] H.A. Kastrup, Nucl. Phys. B 7 (1968) 545. [55] A.W. Knapp, Representation Theory of Semisimple Groups: an Overview Based on Examples, Princeton Univ. Press, Princeton, 1986. S. Stoimenov, M. Henkel / Nuclear Physics B 723 [FS] (2005) 205–233 233

[56] D. Levi, P. Winternitz, J. Phys. A 22 (1989) 2915. [57] D. Levi, P. Winternitz, nlin.SI/0502004. [58] E. Lippiello, M. Zannetti, Phys. Rev. E 61 (2000) 3369. [59] E. Lorenz, W. Janke, 2005, in preparation. [60] A. Medina, P. Revoy, Ann. Sci. École Norm. Sup. (4) 18 (1985) 533. [61] U. Niederer, Helv. Phys. Acta 45 (1972) 802. [62] L. O’Raifeartaigh, V.V. Sreedhar, Ann. Phys. 293 (2001) 215. [63] M. Perroud, Helv. Phys. Acta 50 (1977) 233. [64] A. Picone, M. Henkel, J. Phys. A 35 (2002) 5575. [65] A. Picone, M. Henkel, Nucl. Phys. B 688 (2004) 217. [66] R.O. Popovich, N.M. Ivanova, H. Eshraghi, J. Math. Phys. 45 (2004) 3049. [67] G. Rideau, P. Winternitz, J. Math. Phys. 34 (1993) 558. [68] A.D. Rutenberg, A.J. Bray, Phys. Rev. E 51 (1995) 5499. [69] J. Shatah, M. Struwe, Geometric Wave Equations, Courant Lecture Notes in Mathematics, vol. 2, Amer. Math. Soc., New York, 2000. [70] C. Sulem, P.-L. Sulem, The Non-Linear Schrödinger Equation, Applied Mathematical Sciences, vol. 139, Springer, New York, 1999. [71] W. Zippold, R. Kühn, H. Horner, Eur. Phys. J. B 13 (2000) 531. Nuclear Physics B 723 [FS] (2005) 234–254

Perturbation theory for O(3) topological charge correlators

Miguel Aguado, Erhard Seiler

Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, D-80805 Munich, Germany Received 11 May 2005; received in revised form 14 June 2005; accepted 20 June 2005 Available online 11 July 2005

Abstract To check the consistency of positivity requirements for the two-point correlation function of the topological charge density, which were identified in a previous paper, we are computing perturbatively this two-point correlation function in the two-dimensional O(3) model. We find that at the one-loop level these requirements are fulfilled.  2005 Elsevier B.V. All rights reserved.

1. Introduction

In a preceding paper [1] we analyzed the implications of physical positivity (‘unitarity’) and positivity of the topological susceptibility for the two-point correlation function of the topological density. In particular we found that the short distance singularity has to be softer than tree level perturbation theory would predict. In this article we verify for the two-dimensional O(3) nonlinear sigma model that one-loop perturbation theory is indeed in agreement with this requirement. Such a softening of the short distance singularity is already predicted by including the running of the coupling constant in tree level perturbation theory [2,3]. But since we are dealing here with a composite operator, this check of internal consistency of perturbation theory is quite nontrivial.

E-mail address: [email protected] (M. Aguado).

2. Topological charge density correlator

The action of the (ferromagnetic) O(3) spin model on a two-dimensional square lattice Λ ⊆ Z2 is S =−β SX · SY , (1) X,Y where the sum is over nearest neighbor pairs XY of lattice sites. The partition function reads −S Z = [DS]e , (2) with measure [ ]= 3 2 − DS d SX δ SX 1 , (3) x imposing the unit norm constraint for all spins, i.e. S ∈ S2 ⊂ R3 ∀X ∈ Λ. We will consider the thermodynamic limit as the L →∞limit of the model on an L×L square lattice with periodic boundary conditions, so that translation invariance holds. The origin 0 is then located at the center of Λ. Unit lattice vectors are called 1 and 2.

2.1. Deﬁnitions of the topological charge

We want to study the two-point function of the topological charge density qX,tobe −→ denoted by Gx (with x = 0X), 1 −S G = G =q q = [DS]q q e . (4) x 0,X X 0 Z X 0 The topological susceptibility is deﬁned by χt = G0,X. (5) X

In this paper we will consider two lattice definitions of qX: a ‘field theoretical’ one and the geometric one due to Berg and Lüscher. Both make use of a triangulation of the square lattice Z2 as in Fig. 1. In both cases, the topological charge density is written as = a + b qX qX qX, (6) where a = + + b = + + + + qX f(X,X 1,X 2), qX f(X 1 2,X 2,X 1), (7) with f : Λ3 → R a certain function. In [1] we used a symmetrization of this definition by introducing in addition a ‘mirror’ triangulation c = + + + d = + + + qX f(X,X 1,X 1 2), qX f(X,X 1 2,X 2) (8) 236 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254

Fig. 1. Triangulation of the lattice plane.

= 1 a + b + c + d and deﬁning qX 2 (qX qX qX qX); this was necessary there in order to maintain reﬂection symmetry. Here, however, this symmetrization is unnecessary, because it has no effect in perturbation theory. The 2-point function Gx = G0,X thus has the form G = = a + b a + b = Gij x qXq0 qX qX q0 q0 x (9) i,j=a,b with Gij = j i = x qXq0 ,i,j a,b. (10) To make the notation more transparent, call

(Z1,Z2,Z3) = (0, 0 + 1, 0 + 2),

(W1,W2,W3) = (X, X + 1,X+ 2), = + + + + (Z1,Z2,Z3) (0 1 2, 0 2, 0 1), = + + + + (W1,W2,W3) (X 1 2,X 2,X 1) (11) = and denote Zi SZi and analogously for W , Z , W (see Fig. 1). The indices just introduced are denoted by i, j, k,etc. a Gaa In the following sections, only formulae for qX and x will be presented. The full topological charge density can be recovered by means of (6) and (7), and the full two-point function by adding the other three contributions in (9), obtained by substituting Z W , ZW and Z W for ZW, respectively. Moreover, a numerical factor will be introduced in the deﬁnition of the correlator so as to work with simpler expressions:

˜ 2 Gx = (8π) Gx. (12) M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 237

2.1.1. Field theoretical definition We first consider a symmetrized version of the field theoretical (FT) definition [4] of the topological charge density

FT,a 1 q = S · (S + ∧ S + ), (13) X 8π X X 1 X 2 which gives rise to the two-point function G˜FT,aa = · ∧ · ∧ = · x Z1 (Z2 Z3) W1 (W2 W3) det(Zi Wj ) . (14) i,j

The perturbative treatment of the problem begins with the O(3) → O(2)z decomposition of the spins, 2 w 2 zi zi wj j Zi = √ , + 1 − , Wj = √ , + 1 − . (15) β β β β Substituting in the determinant, we get G˜FT,aa = 1 · x det(zi wj ) β3 i,j 3 2 w 2 1 i+j zi j + (−1) 1 − 1 − det (zk ·w) β2 β β k, =i, j i,j=1 1 = εij k εmnzi ·wzj ·wmzk ·wn 6β3 ij kmn 2 2 1 zi w + εij k εmn 1 − 1 − zj ·wmzk ·wn . (16) 2β2 β β ij kmn The perturbative expansion of (16) consists of a Taylor expansion in β−1/2. Only integral powers of β−1 contribute. For our purposes, it will be enough to keep terms up to and including order β−3. Then ˜FT,aa;pert. 1 (0) G = εij k εmnzj ·wmzk ·wn x 2β2 ij kmn 1 (1) + εij k εmn zj ·wmzk ·wn 2β3 ij kmn 1 − − 3z2 − 2z ·w + 3w 2 z ·w z ·w (0) + O β 4 , (17) 6 i i j m k n where, in general, perturbative contributions to magnitude M are denoted by 1 Mpert. ∼ M(n). (18) βn n 238 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254

2.1.2. Berg and Lüscher’s definition Berg and Lüscher’s (BL) definition of the topological charge density was introduced in [5]. With each elementary triangle (X,Y,Z) of the chosen triangulation one associates a multiple of the signed area of the minimal spherical triangle determined by (SX, SY , SZ) on the unit sphere. Explicitly, · ∧ BL,a = 1 −1 SX (SX+1 SX+2) qX tan . (19) 2π 1 + SX · SX+1 + SX+1 · SX+2 + SX+2 · SX The corresponding perturbative two-point function is, up to and including order β−3, G˜BL,aa;pert. X deti,j (Zi · Wj ) = 16 + · + · + · + · + · + · (1 Z1 Z2 Z2 Z3 Z3 Z1)(1 W1 W2 W2 W3 W3 W1) − + O β 4 . (20) Substituting (15) in (20) and Taylor expanding the result, we arrive at G˜BL,aa;pert. = G˜FT,aa;pert. + Gãa x x x , (21) where 3 (0) ãa 1 2 2 G = εij k εmn z +w − (zu ·zv +wu ·wv) x 8β3 u u ij kmn u=1 u

In this section we use the BL topological charge density (19) and will comment on the difference with the FT deﬁnition when appropriate. Therefore we drop the BL labels and write 1 1 − G˜pert. = G˜T + G˜L + O β 4 . (23) x β2 x β3 x For bookkeeping purposes we do the perturbative computation of the density correlator out keeping factors N − 1 explicit, as for an O(N) problem, although the topological density has a meaning only for N = 3. The only vertices relevant for Feynman diagrams up to and including order β−3 are order β−1:

• Two-π vertex: 1 N − 1 V k = 1 − δkδ . (24) ZW β V ZW M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 239

• Four-π vertex: 1 V kmn = −n δ δkδmn + δkmδn + δknδm ZWT U β Z ZWT U + δn.n.δ δ δkδmn + δn.n. δ δ δkmδn ZT ZW TU ZW ZT WU + n.n. kn m δZW δZU δWT δ δ , (25) where subindices are lattice points, superindices are O(N) indices, nZ = 4 is the number of nearest neighbors of point Z, two-index deltas are Kronecker, 1, if Z = W = T = U, δ = (26) ZWT U 0, otherwise, and 1, if Z,W nearest neighbors, δn.n. = (27) ZW 0, otherwise. The explicitly volume-dependent factor in (24) comes from ﬁxing one spin using the Faddeev–Popov method as in [6] to eliminate the divergent zero mode that would otherwise appear in the Gaussian measure deﬁning tree level perturbation theory with periodic boundary conditions.

2.2.1. Green’s functions In order to compute the Green’s functions relevant to the correlators being studied, we −→ start from the corresponding quantity on the L × L square lattice and put x = 0X = (x, y), −L/2 x,y < L/2. a2 cos(ap · x) G ,X = , (28) 0 2L2 2 − cos(ap ) − cos(ap ) p x y where the momenta 2π p = (p ,p ) = (n ,n ), n ,n ∈ Z (29) x y L x y x y are summed over the ﬁrst Brillouin zone − L/2 nx,ny

+π +π 1 cos(q · x) − 1 G˜ ≡ G − G = q q 0,X 0,X 0 2 d x d y (30) 2(2π) 2 − cos qx − cos qy −π −π 240 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254

(another possible source of infrared divergence is the sum over intermediate lattice sites, ˜ ˜ ˜ which will be addressed in Section 2.4). Obviously G0 = 0, G(x,y) = G(y,x). This subtracted propagator was studied by other methods in [8], which we will use as a check for our results. Due to parity, the numerator in the integrand of (30) can be replaced by cos(xqx) × cos(yqy)−1. Integration in qx can be performed by the calculus of residues, with the result +1 √ − + − + 2 −x − ˜ 1 dt (2 t 3 4t t ) Ty(t) 1 G0,X = √ √ . (31) 2π 1 − t2 3 − 4t + t2 −1

Here Ty(t) is Chebyshev’s polynomial of the yth kind. It is straightforward to check that (31) behaves asymptotically as 1 d − G˜ =− ln |x|−c − + O |x| 2 (32) 0,X 2π |x| with c, d constants. According to [8], 2γ + 3ln2 c = E ,d= 0, (33) 4π where γE is Euler’s constant. Similarly it can be seen that 2 · 2 ˜ ˜ ˜ 1 v (v x) −3 G ,X+ + G ,X− − 2G ,X = − + O |x| (34) 0 v 0 v 0 π 2 x2 |x|4 and ˜ ˜ ˜ ˜ G0,X+v − G0,X+w + G0,X−v − G0,X+w 2 2 2 2 1 v − w − 2(v · x) + 2(w · x) − = + O |x| 3 (35) 2π |x|4 for large |x| and |v|, |w|=O(1). To see this, one subtracts from the Fourier representation of the lhs of (34) and (35) the corresponding continuum expression with a suitable smooth high momentum cutoff; this difference, being the Fourier transform of a smooth (C∞) function, will decay faster than any power of |x|. The cutoff on the continuum expression, on the other hand, does not affect the leading long distance behavior, which is obtained straightforwardly. Expressions (34) and (35), in which the constants c and d in (32) do not appear, will be useful in the computation of the charge correlator. In the following, we will denote Green’s functions by G0,X even when subtracted.

2.2.2. Tree level G˜11 The tree level contribution to x is given by 1 G˜T,aa = ε ε z ·w z ·w (0), (36) x 2 ij k mn j m k n ij kmn the correlator in the rhs being evaluated in the Gaussian measure with one spin ﬁxed. The corresponding diagrams have the structure of Fig. 2. M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 241

Fig. 2. Tree level diagram structure.

Contracting spin indices, 1 G˜T,aa = (N − 1)2 ε ε G G x 2 ij k mn Zj Wm ZkWn ij kmn 1 + (N − 1) ε ε G G 2 ij k mn Zj Zk WmWn ij kmn 1 + (N − 1) ε ε G G . (37) 2 ij k mn Zj Wn ZkWm ij kmn

The 3 × 3 matrices GZ·Z· and GW·W· are symmetric, and therefore the second term in the rhs vanishes. Playing with indices, we can recast equation (37) in the form 1 G˜T,aa = (N − 1)2 − (N − 1) ε ε G G . (38) x 2 ij k mn Zj Wm ZkWn ij kmn √ This can be written in terms of differences Gv − Gw with |v − w| 2(whichre- stricts IR divergences to summation effects, the IR divergences of Green’s functions being cancelled in the differences): G˜T,aa =−1 − 2 − − x (N 1) (N 1) 2 × − − εij k εmn(GZj Wm GZkWm )(GZkWm GZkWn ). (39) ij kmn

We can now perform the sums over all indices: in general, for a 3 × 3matrixaij , = 2 − 2 + 2 − εij k εmnajmakn (tr a) tr a δi 2 a i 2(tr a)ai ij kmn i = 3 (tr a)2 − tr a2 + 2s a2 − 2(tr a)s(a), (40) where s(a) denotes the sum of all entries in a. This identity can be obtained as follows: one starts from the well-known identity ln det(1 + zA) = tr ln(1 + zA) (41) and expands it to order z3 to obtain 1 1 1 tr A ∧ A ∧ A = tr A3 − tr A tr A2 + (tr A)3. (42) 3 2 6 By ‘polarization’, i.e. the replacement of A by xA+ yB + zC on both sides and comparing the coefﬁcient of xyz Eq. (40) follows if we put B = C = a and take for A the matrix with all entries equal to 1. 242 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254

Fig. 3. One-loop diagram structures.

In our case, G˜T,aa =−1 − 2 − − x (N 1) (N 1) 2 × 2 − 2 + 2 − 3 (tr GZ·W· ) tr GZ·W· 2s GZ·W· 2(tr GZ·W· )s(GZ·W· ) . (43)

2.2.3. One loop The one-loop (order β−3) correction to the correlator can be decomposed as G˜L,aa = aa + aa + aa + aa + aa x (I) (II) (III) (IV) (V) , (44) according to the structures depicted in Fig. 3.

• (I) contains the contributions of the 4-legged, disconnected graphs with one 2-vertex: N − 1 (I)aa = 1 − (N − 1)2 − (N − 1) V × εij k εmnGZj Wm GZkP GWnP . (45) ij kmn P • (II) contains the X-shaped contributions: 1 (II)aa =− (N − 1)2 − (N − 1) ε ε 2 ij k mn ij kmn × − − (GZj P GWmP GZj QGWmQ)(GZkP GWnP GZkQGWnQ). P,Q (46) • (III) gathers the contributions of 4-legged graphs with tadpoles: N − 1 (III)aa =− (N − 1)2 − (N − 1) ε ε G 2 ij k mn Zj Wm ij kmn × − − (GPP GQQ)(GZkP GWnP GZkQGWnQ) P,Q − − 2 − − (N 1) (N 1) εij k εmnGZj Wm ij kmn × + GPPGZkP GWnP GQQGZkQGWnQ P,Q − + GPQ(GZkP GWnQ GZkQGWnP ) . (47) M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 243

The ﬁrst term of (III) vanishes by translation invariance. • (IV) consists of the six-legged graphs contributing to the correlator according to the FT deﬁnition of the charge density: N − 3 (IV)aa = (N − 1)2 − (N − 1) det G 6 Z·W· N − 1 − (N − 1)2 − (N − 1) ε ε G G G 2 ij k mn 00 Zj Wm ZkWn ij kmn − (N − 1)2 − (N − 1) × + εij k εmnGZj Wm (GZi Zk GZi Wn GZkW GWWn ). (48) ij kmn The determinant term in (IV)aa has a vanishing prefactor. • (V) is the BL difference, also consisting of six-legged graphs: N − 1 (V)aa =− (N − 1)2 − (N − 1) (G − G ) 8 ZuZv 00 u,v × εij k εmnGZj Wm GZkWn ij kmn 1 − (N − 1)2 − (N − 1) ε ε G 4 ij k mn Zj Wm ij kmn × − + − GZuZk (GZvWn GZuWn ) GZkWu (GWnWv GWnWu ) . (49) u,v

As happened with the tree level expression, the√ one-loop correction can be rewritten in terms of differences Gv − Gw with |v − w| 2. This works independently for (I), (II) and (V): 1 N − 1 (I )aa = 1 − (N − 1)2 − (N − 1) ε ε d 3 V ij k mn ij kmn × − − − (GZj Wm GZkWm ) (GZkP GZi P )(GWnP GWmP ), (50) P

1 (II )aa = (N − 1)2 − (N − 1) ε ε d 16 ij k mn ij kmn × − − − (GZj P GZj Q) (GWmP GWmQ) P,Q × (G − G ) − (G − G ) ZkP ZkQ WnP WnQ × (GZ P − GZ P ) + (GZ Q − GZ Q) − (GW P − GW P ) j k j k m n − − 2 (GWmQ GWnQ) , (51) 244 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 N − 1 (V )aa = (N − 1)2 − (N − 1) (G − G ) d 8 ZuZv 00 u,v × − − εij k εmn(GZj Wm GZkWm )(GZkWm GZkWn ) ij kmn 1 − (N − 1)2 − (N − 1) ε ε (G − G ) 8 ij k mn Zj Wm ZkWm ij kmn × − − (GZuZk GZvZk )(GZvWn GZuWn ) u,v + − − (GZkWu GZkWv )(GWnWv GWnWu ) , (52) while (III)aa and (IV)aa cannot be brought to that form. However, the sum (III)aa +(IV)aa can, because, using translation invariance and the fact that N = 3, aa = − − (IIId ) 2G00 εij k εmn(GZj Wm GZkWm )(GZkWm GZj Wn ) ij kmn − − − 2(G01 G00) εij k εmn(GZj Wm GZkWm ) ij kmn × − − (GZkP GZkQ)(GWnP GWnQ) P,Q 8 + (G − G ) ε ε (G − G ) 3 01 00 ij k mn Zj Wm ZkWm ij kmn × − − (GZkP GZi P )(GWnP GWmP ) (53) P,Q and aa =− − − (IVd ) 2G00 εij k εmn(GZj Wm GZkWm )(GZkWm GZj Wn ) ij kmn − − − − εij k εmn(GZi Zk G00)(GZj Wm GZj Wn )(GZi Wn GZkWn ) ij kmn − − − − εij k εmn(GWWn G00)(GZj Wm GZkWm )(GZj W GZkWn ). ij kmn (54) The ﬁrst terms in the rhs of (53) and (54) cancel.

2.3. ζab notation ˜ Green’s functions appearing (up to 1-loop order) in the perturbative expansion of Gx are of the form Gv+w, where

−→ −→ −→ • v is one of ζ ≡ 0X, ξ ≡ P 0 (where P is summed over all lattice points), η ≡ PX and −→ κ ≡ 0 ≡ 00 , M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 245

• w is a linear combination n11 + n22 of unit lattice vectors with ni integers (|ni| 2 up to 1-loop order).

The following notation proves useful when applying computer techniques to this perturbative problem:

Gv+w → vword. (55) Here v is one of the letters ζ , ξ, η or κ according to the vector v. The subscript is the n n n n minimal word of the form a a c c b b d d , ni 0, matching vector w under the rule

na nc nb nd a c b d → (na − nc)1 + (nb − nd )2 (56) (obviously, only words of the forms ana bnb , ana dnd , cnc bnb , cnc dnd can be minimal). For instance,

G0,X+1+2 → ζab,GP,0+1−2−2 → ξadd,G0,0−1 → κc. (57) This notation is way more readable than the standard notation, and allows us to cope with the full form of the density correlator, including all four terms in (9). A C program was used to generate the full analytic forms in ζab notation for each contribution (tree level and one-loop contributions I through V), its output being simpliﬁed by means of a computer algebra program. The simpliﬁcation thus achieved is dramatic, witness the results in next section.

2.3.1. The correlator in ζab notation The full tree level 2-point topological charge correlator between 0 and X, both in standard and ζab notation, reads: ˜T G = 2 (G + + + G − − − 2G )(G + − + G − + − 2G ) x 0,X 1 2 0,X 1 2 0,X 0,X 1 2 0,X 1 2 0,X 2 − (G + − G + + G − − G − ) 0,X 1 0,X 2 0,X 1 0,X 2 2 = 2 (ζab + ζcd − 2ζ )(ζad + ζcb − 2ζ)− (ζa − ζb + ζc − ζd ) . (58) From this expression and the asymptotics (34) and (35) of the differences of Green’s G˜T functions, the leading term of the asymptotic behavior for x can be easily computed: 2 1 − G˜T =− + O |x| 5 . (59) x π 2 |x|4

The full 1-loop contributions will only be presented in ζab notation (cancelling terms in (III) and (IV) are omitted):

• the graphs with 2-vertices: 2 (I ) = 2 1 − ζ V × (ζa − ζb + ζc − ζd ) (ξa − ξb)(ηab − η) + (ξab − ξ)(ηa − ηb) P 246 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 − (ζab + ζcd − 2ζ) (ξa − ξb)(ηa − ηb) P − (ζad + ζcb − 2ζ) (ξab − ξ)(ηab − η) , (60) P • the X-shaped graphs: (IIζ ) = 2 (ξab − ξ)(ξab − ξaa) − (ξa − ξb)(ξa − ξaab) P × (ηab − η)(ηab − ηaa) − (ηa − ηb)(ηa − ηaab) + 2 (ξab − ξ)(ξab − ξbb) + (ξa − ξb)(ξb − ξabb) P × (ηab − η)(ηab − ηbb) + (ηa − ηb)(ηb − ηabb) , (61) • the tadpole graphs:

(III ) = (κ − κ) ζ a × (ζab + ζcd − 2ζ) −(ξa − ξb)(ηaa + ηad − ηbb − ηcb) P − (ξaa + ξad − ξbb − ξcb)(ηa − ηb) + (ζad + ζcb − 2ζ) −(ξab − ξ)(ηaab + ηabb − ηc − ηd ) P − (ξaab + ξabb − ξc − ξd )(ηab − η) + (ζa − ζb + ζc − ζd ) (ξab − ξ)(ηaa + ηad − ηbb − ηcb) P + (ξaa + ξad − ξbb − ξcb)(ηab − η) + − + − − (ξa ξb)(ηaab ηabb ηc ηd ) + (ξaab + ξabb − ξc − ξd )(ηa − ηb) , (62)

• graphs with six external legs in the FT deﬁnition: (IVζ ) = 4(κa + κab − 2κ) (ζab + ζcd − 2ζ )(ζad + ζcb − 2ζ) 2 − (ζa − ζb + ζc − ζd ) , (63) • the BL difference: (Vζ ) =−4(2κa + κab − 3κ) (ζab + ζcd − 2ζ )(ζad + ζcb − 2ζ) 2 − (ζa − ζb + ζc − ζd ) . (64)

Note that (IVζ ) and (Vζ ) do not involve sums over lattice points, and that they are proportional to the tree level expression (58). Their sum reads M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 247 (IV ) + (V ) =−4(κ − κ) (ζ + ζ − 2ζ )(ζ + ζ − 2ζ) ζ ζ a ab cd ad cb 2 − (ζa − ζb + ζc − ζd ) . (65) The simplicity of these expressions might be a hint of the existence of a more direct derivation.

2.4. Numerical evaluation

The procedure we employ to evaluate the correlator numerically follows the conven- tional philosophy adopted in formal perturbation theory: the contributions to each order are first evaluated in a finite volume and then the thermodynamic limit is taken order by order. It is hoped (though far from proven) that by this procedure one obtains an asymptotic expansion of the infinite volume correlator. For a more detailed discussion of the difficult issues involved see for instance [9]. ˜ The various perturbative contributions to the charge density correlator G0,X were computed, by means of C programs, for X in a finite 400 × 400 sublattice of the whole R2. Lattice symmetry allows to reconstruct all correlators in the 2L × 2L square lattice (in our case, L = 200) from those in the shaded region. As explained in the next sections, the computation of Gx involves sums over Green’s functions Gx with |x| arbitrarily large, and a cutoff LO on the summation range is imposed. Also, since we have an asymptotic expression for Gx, we only need to compute Green’s functions exactly for |x| in a smaller region determined by another cutoff LI; for larger |x| the asymptotic expression is used. Fig. 4 shows the ranges L, LI, LO involved in the computation.

Fig. 4. Ranges for the computation: L determines the range of |x| for which topological charge correlators Gx are computed. LO is a cutoff on the sum over Green’s functions Gx. LI determines the range for which exact values of Gx, as opposed to asymptotic approximations, are used. 248 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254

2.4.1. Computation of Green’s functions It is impractical to compute and store an array of all possible values of (subtracted) Green’s functions Gx according to the exact expression (31). It is far more convenient to store only such values as differ noticeably from the approximate expression (32).We chose to compute exact values for Gx with |x| less than a certain value LI (see Fig. 4), and approximate values for the remaining ones as they were needed. The constant appearing in (32) must be computed for this procedure to be useful. Working with LI = 1000, exact Green’s functions G(x,y) were computed for the trian- gular region 0 x LI,0 y x. All those in the region |x| LI can be obtained from them by symmetry. Exact values of LI at the edge x = LI of the triangle were used to ﬁt the Green’s function space dependence to (32), since this edge is the furthest removed from the centre and therefore provides the best agreement with the asymptotic form. A value c = 0.257343 is obtained for the constant (see Fig. 5), which agrees with the analytical expression (33) taken from [8]. For LI = 1000, the numerical results for the correlators are stable with respect to 10% changes in LI.

2.4.2. Computation of correlators The tree level correlator (58), as well as one-loop contributions (63) and (64), do not involve sums over lattice points, and are readily computed for each X.

− 1 | | = = Fig. 5. Fit of Green’s functions to const 2π ln x for x (Li ,y),0 y LI 1000. The dashed line corresponding to the ﬁtted dependence is indistinguishable from the computed points in this range. M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 249

One-loop contributions (60), (61) and (62) involve sums over points arbitrarily far away from this sublattice, and a cut-off must be introduced. We choose to restrict the sum to points within the circle of radius LO about the origin. At LO = 2000, results are stable with respect to 10% changes in LO or in the shape of the summation domain (circle vs. square). This indicates the absence (or cancellation) of any infrared divergences connected with the summation over intermediate lattice points. Results are presented for the computations outlined in the previous sections. Range parameters are L = 200, LI = 1000, LO = 2000. Fig. 6 shows the (nonuniversal) near-origin tree level and one loop contributions to the ˜ topological charge density correlator G0,(x,0) as a function of x. Note that both contributions are positive at the origin, negative everywhere else, and are suppressed for large x. The peak at the origin results, in the continuum limit, in the singular contact term discussed e.g. in [10]. The values of the peak at tree level and one loop level are G˜T ≈ G˜L ≈ 0 0.810569, 0 0.899007. (66) G˜T = The tree level peak can also be calculated analytically from (31) and (58), yielding 0 8/π 2 ≈ 0.810570. Since this is a perturbative analysis, the topological susceptibility (5) should vanish order by order in β−1. Summing the computed correlators over the whole 2L × 2L region, the following results are obtained: G˜T ≈ × −5 G˜L ≈ × −5 x 1.3 10 , x 3.7 10 , (67) x x which are compatible with zero and serve as a check of our computations. Note, in particular, that the problems deemed to arise with χT in the continuum limit of the O(3) model do not show up in this perturbative treatment. Fig. 7 shows the tree level contribution multiplied by x4, as a function of x. This product approaches for large x the constant −0.2026, compatible with (59), so the tree level result (including all numerical factors) is 1 GT ∼− (68) 0,(x,0) 32π 4x4 for x 1. The one-loop contributions (60) through (64) to the correlator were computed and their behavior for large x analyzed:  a  (Iζ ) ∼− ,  x2   b ln x  (II ) ∼− ,  ζ x4 x 1, (69) a  (IIIζ ) ∼+ ,  x2   d  (IV + V ) ∼− ,  ζ ζ x4 250 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254

˜ Fig. 6. Near-origin behavior of correlator G0,(x,0), at tree level and one loop, as a function of x. with a = 0.101, b = 0.0653, d = 0.101. These Ansätze were obtained by inspection of the plots and trial and error, except for the last one, which stems directly from the proportion- ality of (65) to (58) and the fact that κ = 0, κa =−1/4, which results in the asymptotics in (69) with d = 1/π 2. M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 251

˜ 4 Fig. 7. Tree level G0,(x,0) multiplied by x as a function of x.

−4 The leading terms in (Iζ ) and (IIIζ ) cancel, but subleading terms O(x ) and O(x−4 ln x) remain. The sum of all one-loop contributions (see Fig. 8) behaves as c + c ln x G˜L ∼− 0 1 ,x 1, (70) 0,(x,0) x4 with constants

c0 = 2.206,c1 = 0.0645. (71) Note that these are not identical to b and d in (69), because the subleading terms in the asymptotic behavior of (Iζ ) and (IIIζ ) contribute. The same analysis was performed along the diagonal of points (x, x) with identical results for the asymptotic forms of the correlator: full rotational invariance is restored in this regime.

2.4.3. Renormalization and continuum limit We now discuss the continuum limit, again in the spirit of formal perturbation theory, i.e. termwise in the perturbative expansion; no claim can be made that the resulting expansion is asymptotic to a nonperturbatively deﬁned continuum limit. Up to one loop we have found for the correlator on the unit lattice and large lattice distances |x|1 1 1 1 1 1 G ∼− − c + c ln |x| + O . (72) x β2 32π 4|x|4 β3 64π 2|x|4 0 1 β4 252 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254

˜ 4 Fig. 8. One-loop level G0,(x,0) multiplied by x as a function of ln x, together with a linear ﬁt for large x.

To obtain a continuum limit we have to introduce the lattice spacing a, rescale x = y/a and make β dependent on the cutoff a. We also have to rescale the correlator G by a 4 Ga factor a according to its engineering dimension, to obtain the correlator y in continuum normalization

a 4 1 1 G = a G a ∼− y x β(a)2 32π 4|y|4 1 1 |y| 1 − c + c ln + O , (73) β(a)3 64π 2|y|4 0 1 a β(a)4 valid asymptotically for |y|a. A continuum limit should exist if we let β depend logarithmically on a according to the one-loop Callan–Symanzik β function for the O(3) model [11]: 1 1 − 1 1 − = + O β 3 = 1 + ln(µa) + O β 2 . (74) β(a) − 1 0 β 2πβ 0 β0 2π ln(µa) 0 0 −3 Inserting this in (73) and re-expanding to order β0 we see that the terms proportional 3 to ln a cancel if c1 = 2/π , which is consistent with the value 0.0645 produced by our numerical computation; the continuum limit of the correlator is then 3 1 1 1 π c0 − Ga ≡ G =− + | | 2 + 4 lim y y 1 ln y µe O β . (75) a→0 2 4| |4 0 β0 32π y πβ0 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254 253

3 − π c0 By choosing a renormalization scale µ = µ0e 2 we obtain Ga ≡ G =− 1 1 + 1 | | + −4 lim y y 1 ln µ0 y O β . (76) a→0 2 4| |4 0 β0 32π y πβ0 If we had chosen the FT instead of the BL definition, the only difference would have been a change in c0, corresponding to different renormalization scale, so to the order considered the two definitions are related by a finite renormalization. Eq. (76) is now valid for all y, since in the limit a → 0 only the asymptotic behavior of Gx survives. We can compare this result with the renormalization group improved tree level result [2,3] 1 1 G =− , (77) y 32π 4β(y)2 |y|4 −3 if we re-expand (77) to order β0 , using the one-loop RG flow 1 |y| β(y) = β(y0) − ln . (78) 2π |y0| −3 = −3 The expressions (76) and (77) agree to order β0 β(y0) if the renormalization scales are chosen appropriately. So we have established that our one-loop calculation supports the softening of the short distance behavior predicted by the RG improved tree level result.

3. Conclusions

We computed the two-point functions of the topological charge density perturbatively to one loop and found consistency with the RG improved tree level perturbative result, indicating a softening of the short distance singularity compared to the naive tree level result. This means that at the level of formal one-loop perturbation theory the requirements of the two positivities analyzed in [1] are indeed satisfied. But we should point out again that a mathematical justification of the procedures of formal perturbation theory (interchange of the weak coupling limit with the thermodynamic and continuum limits) does not exist, therefore it would be very interesting to see if this softening is really present at the nonperturbative level. Numerical checks using Monte Carlo simulations are presumably very difficult, however, since the check requires identifying logarithmic corrections to a rather large power.

Acknowledgements

We are grateful to Peter Weisz for a critical reading of the manuscript and for drawing our attention to reference [8]. 254 M. Aguado, E. Seiler / Nuclear Physics B 723 [FS] (2005) 234–254

References

[1] M. Aguado, E. Seiler, The clash of positivities in topological density correlators, MPP-2005-18, hep-lat/ 0503015. [2] J. Balog, M. Niedermaier, Nucl. Phys. B 500 (1997) 421. [3] E. Vicari, Nucl. Phys. B 554 (1999) 301. [4] A. Di Giacomo, F. Farchioni, A. Papa, E. Vicari, Phys. Lett. B 276 (1992) 148; A. Di Giacomo, F. Farchioni, A. Papa, E. Vicari, Phys. Rev. D 46 (1992) 4630. [5] B. Berg, M. Lüscher, Nucl. Phys. B 190 (1981) 412. [6] P. Hasenfratz, Phys. Lett. B 141 (1984) 385. [7] F. David, Phys. Lett. B 96 (1980) 371. [8] D.-S. Shin, Nucl. Phys. B 525 (1998) 457. [9] A. Patrascioiu, E. Seiler, Phys. Rev. D 57 (1998) 1394. [10] E. Seiler, I.O. Stamatescu, Some remarks on the Witten–Veneziano formula for the eta-prime mass, MPI- PAE/PTh 10/87. [11] E. Brézin, J. Zinn-Justin, Phys. Rev. B 14 (1976) 3110. Nuclear Physics B 723 [FS] (2005) 255–280

The Meissner effect in a two ﬂavor LOFF color superconductor

Ioannis Giannakis a, Hai-Cang Ren a,b

a Physics Department, Rockefeller University, New York, NY 10021, USA b Institute of Particle Physics, Central China Normal University, Wuhan 430079, China Received 10 May 2005; accepted 6 June 2005 Available online 28 June 2005

Abstract We calculate the magnetic polarization tensor of the photon and of the gluons in a two ﬂavor color superconductor with a LOFF pairing that consists of a single plane wave. We show that at zero temperature and within the range of the values of the Fermi sea displacement that favors the LOFF state, all the eigenvalues of the magnetic polarization tensor are non-negative. Therefore the chromomagnetic instabilities pertaining to a gapless color superconductor disappear.  2005 Elsevier B.V. All rights reserved.

1. Introduction

The LOFF (Larkin–Ovchinnikov–Fudde–Ferrell) state [1,2] was introduced as a candidate superconducting state of an alloy containing paramagnetic impurities with a ferromagnetic spin alignment. The presence of a nonzero expectation value for the spins of the impurities and its corresponding coupling to the electrons gives rise to a displacement of the Fermi surfaces of the pairing electrons since they have opposite spins. The characteristic feature of this state is that the corresponding order parameter is not constant but depends on the coordinates. Though the experimental evidence for the LOFF state is still inconclu-

E-mail addresses: [email protected] (I. Giannakis), [email protected] (H.-C. Ren).

0550-3213/$ – see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2005.06.008 256 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 sive, the subject has experienced a revived interest due to its relevance to the physics of color superconductors and compact stars [3]. The interaction between two quarks that renders quark matter at ultra high baryon density unstable to the formation of Cooper pairs and leads to the emergence of a color superconducting state, results from the perturbative one-gluon exchange in the color antisymmetric diquark channel. All quark flavors, u, d and s can be treated as massless since the Fermi energies of the quarks are much higher than ΛQCD. In contrast at moderately high baryon densities, say the density inside a compact star, where the Fermi energies of the quarks are only few hundred MeV, instanton effects are expected to dominate the interaction responsible for the pairing of quarks [4], again in the color antisymmetric channel. Since the main pairing channel is between quarks of different flavors, factors such as the largemassofthes quark and the different magnitudes of electric charges of the u and d quarks together with the color/electric neutrality condition induce a considerable displacement between the Fermi momenta of the pairing quarks. This makes the search for a realistic color superconductivity phase challenging. The LOFF state is one of several states that were proposed as a possible ground state for quark matter at moderately high baryon density [5,6]. Others include the homogeneous gapless states (g2SC or gCFL) [7–9] and the heterogeneous mixture of BCS states without Fermi sea displacement and normal phases [12]. The gapless state is the analog of the Sarma state [10] for quark matter and a potential ground state since the Sarma instability can be removed by imposing the charge neutrality constraint [7,11]. But a subsequent calculation revealed that the squares of the Meissner masses of the gluons were all negative signaling a chromomagnetic instability in the presence of the gauge field [13–16].This instability represents a major problem that needs to be resolved before we would be able to identify the ground state of the quark matter. The mixed state on the other hand is free from chromomagnetic instabilities but it has the drawback that it is difficult to make a quantitative comparison of its free energy with one of the homogeneous CSC state. In a previous paper [15], we explored the relationship between the chromomagnetic instability and the LOFF state. For a g2SC, we showed explicitly that the chromomagnetic instability associated with the 8th gluon implies the existence of a LOFF state with a lower free energy and a positive Meissner mass square for the 8th gluon excitation. But the possibility remains that the chromomagnetic instability will emerge when we calculate the Meissner masses of the gluons that are associated with the 4–7th generators of the color group. In this paper we shall continue this project by calculating the Meissner masses of all the gluons of a LOFF state. Our result shows that the chromomagnetic instability is completely removed at zero temperature and in the LOFF window where the LOFF state is energetically favored for a given Fermi momentum displacement. This paper is organized as follows: the free energy function of a two flavor LOFF CSC will be derived in Section 2. A discussion of its properties will also be included. In Sec- tion 3, the Meissner masses of all the gluonic excitations will be calculated. The analytic proof of the absence of the chromomagnetic instabilities for a small gap parameter ∆,at T = 0 will be presented in Section 4 and the numerical results for an arbitrary ∆ will be discussed in Section 5, where we extend the chromomagnetic instability free region to the entire LOFF window at T = 0. Finally, Section 6 will summarize our results. Some tech- I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 257 nical details will be presented in Appendices A–C. Unless explicitly stated, we shall work exclusively with the single plane wave ansatz for which analytical investigation can be carried out quite far. We shall also designate the symbol 2SC-LOFF for a two flavor LOFF color superconducting state.

2. The free energy function of a 2SC-LOFF

We consider the pairing of two quark flavors, u and d. The NJL effective Lagrangian in the massless limit is given by [17] ∂ψ L =−ψγ¯ + ψγ¯ µψ + G (ψψ)¯ 2 + (ψ¯ τψ) 2 µ ∂x 4 S µ ¯ c ¯ c + GD ψCγ5 τ2ψ ψγ5 τ2ψC , (1) ˜¯ where ψ represents the quark fields and ψC = Cψ is its charge conjugate with C = iγ2γ4. All gamma matrices are hermitian, (c)mn = cmn is a 3 × 3 matrix acting on the red, green and blue color indices and the Pauli matrices τ act on the u and d flavor (isospin) indices. The chemical potential is also a matrix in color-flavor space, i.e.,

µ =¯µ − δτ3 + δ λ8 (2) with λl being the lth Gell-Mann matrix. The most general expression of the order parameter of 2SC-LOFF is a superposition of plane waves of diquark condensate, i.e., ¯ 2iq·r ψ(r)γ 5λ2τ2ψC(r) = Φqe . (3) q In this article, we shall be mainly working with the type of the LOFF state that contains only one plane wave, more speciﬁcally the form of the order parameter will be given by ¯ 2iq·r ψ(r)γ 5λ2τ2ψC(r) = Φe . (4) The condensate Eq. (4) is invariant under a translation accompanied by a phase shift. Since the free energy depends on the magnitude of the condensate only, it is translationally invariant. But this is not clearly the case with the general form of the 2SC-LOFF condensate Eq. (3). Let us introduce a transformed quark ﬁeld

− · χ(r) = e iq r ψ(r). (5) The NJL Lagrangian in terms of the new quark ﬁeld reads ∂χ L =−¯χγ − iχ¯ γ ·qχ +¯χγ µχ + G (χχ)¯ 2 + (χ¯ τχ) 2 µ ∂x 4 S µ c c + GD χ¯Cγ5 τ2χ χγ¯ 5 τ2χC , (6) 258 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 while the condensate (4) takes the form χ(¯ r)γ 5λ2τ2χC(r) = Φ. (7) By ignoring the chiral condensate and expanding the NJL Lagrangian to linear order in the ﬂuctuation

χγ¯ 5λ2τ2χC − Φ, (8) we derive the following mean ﬁeld expression for the NJL Lagrangian ∆2 ∂χ LMF =− −¯χγµ − iχ¯ γ ·qχ +¯χγ4µχ 4GD ∂xµ 1 + ∆(−¯χ γ λ τ χ +¯χγ λ τ χ ), (9) 2 C 5 2 2 5 2 2 C where the gap parameter is given by ∆ = 2GDΦ>0. The free energy F is given by the Euclidean path integral ¯ 4 F =−kB T ln [dψ dψ] exp d x LMF (10)

−1 with 0 0, (∂∆ ) ∂∆ ∂q > 0. (13) (∂∆2)2 ∂2Γ ∂2Γ ∂∆2∂q ∂q2 It follows then that ∂2Γ > 0 (14) ∂q2 at the minimum. In terms of the Nambu–Gorkov representation of the quark ﬁeld

χ ¯ Ψ = , Ψ = (χ,¯ χ¯Cτ2) (15) τ2χC and its Fourier transform

d3p Ψ(r) = k T eiPxΨ (16) B ( π)3 P ν 2 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 259 we ﬁnd that 2 3 ∆ 1 d p − d4x L = Ω − + Ψ¯ S 1(P )Ψ , (17) MF 4G 2 ( π)3 P P D ν 2 where S−1(P ) is the momentum representation of the inverse quark propagator

−1 P/ − iγ ·q +¯µγ4 − δγ4τ3 + δ γ4λ8 ∆γ5λ2 S (P ) = −∆γ5λ2 P/ + iγ ·q −¯µγ4 − δγ4τ3 − δ γ4λ8 = P/ − δγ4τ3 + (−iγ ·q +¯µγ4 + δ γ4λ8)ρ3 + i∆γ5λ2ρ2. (18)

P/ =−iγµPµ, P = (p, −ν) with ν = (2n + 1)πkB T being the Matsubara frequency, and we have introduced the Pauli matrices with respect to NG blocks, i.e.,

0 I 0 −iI I 0 ρ = ,ρ= ,ρ= . (19) 1 I 0 2 iI 0 3 0 −I The product of Dirac, color, flavor and NG matrices is a direct product and the total dimensionality of S−1(P ) is 4(Dirac) × 3(color) × 2(flavor) × 2(NG) = 48. Carrying out the path integration in Eq. (10), we find

∆2 k T d3p det S−1(P ) Γ = − B ln , (20) 4G 2 ( π)3 S−1(P ) D ν 2 det where S(P) ≡ S(P )|∆=0 is the free quark propagator. The typical energy scale where the pairing occurs is characterized by the zero temperature gap parameter ∆0 in the absence of a displacement, i.e., δ = δ = 0. At weak coupling, ∆0 ¯µ and the pairing interaction extends over a width of ω0 on both sides of the Fermi level with ∆0 ω0 ¯µ. Furthermore the displacement δ that is induced by the condensate, is of higher order and can be neglected. As a result we have the only displacement parameter δ ∼ ∆0. Under this approximation, the quark propagator takes the approximate form (−iγ ·ˆp + ρ γ ) [|p − ρ q|−µ ¯ + ρ (iν − δτ ) + ∆λ γ γ ] S(P ) − 3 4 3 3 3 2 5 4 2 2 2 (21) 2 (p −¯µ) − (iν − δτ3 −ˆp ·q) + ∆ while the determinant in Eq. (20) is given by − det S 1(P ) = (2µ)¯ 24 (p −¯µ)2 − (iν + sδ −ˆp ·q)2 + (λ∆)2 2, (22) s,λ where s =±1 and λ =±1, 0 refer to the eigenvalues of τ3 and λ2 respectively. A more detailed derivation of Eqs. (21) and (22) can be found in Appendix A. It follows from the analytic continuation of (21) to the real energy, iν → p0, that the low-lying excitation spectrum reads ± = −¯ 2 + 2 ± −ˆ· Ep (p µ) ∆ δ p q (23) for red and green colors and Ep =|p −¯µ|±δ −ˆp ·q for blue color. The excitations (23) + − | − | + are fully gapped if ∆>δ q. Ep becomes gapless when δ q <∆<δ q or q<δ, − ± − + ∆<δ q and both Ep become gapless when q>δand q δ<∆

Substituting (22) into (20) we obtain that

1 ∞ ∆2 µ¯ 2k T ξ 2 − z2(ν,θ,∆) Γ = − B d cos θ dξ ln (24) 2 2 − 2 4GD π | | ξ z (ν, θ, 0) ν <ω0−1 −∞ with ξ = p −¯µ and z(ν, θ, ∆) = (iν + δ − q cos θ)2 − ∆2. The contributions from the eigenvalues of τ3 with opposite sign are equal if we reverse the signs of the Matsubara frequency and simultaneously perform the transformation θ → π − θ. This will also be the case with the calculation of the Meissner masses in Section 3. We shall employ the phase conventions that are determined by

Im z(ν, θ, ∆) > 0forν>0, (25) with ζ(−ν,θ,∆) = ζ (ν,θ,∆). Let us now proceed and perform the ξ integral by closing the contour around one of the branch cuts of the logarithm. As a result we obtain the expression

1 ∆2 4µ¯ 2k T Γ = − B Im d cos θ z(ν, θ, ∆) − z(ν, θ, 0) . (26) 4GD π −1 0<ν<ω0 At zero temperature, T = 0, we have

ω0 dν k T (···) = (···) (27) B 2π 0<ν<ω0 0+ and

∆2 2µ¯ 2 2ω 1 Γ = − ∆2 ln 0 + 4G π 2 ∆ 2 D 1 2 | |+ 2 − 2 µ¯ δθ δθ ∆ + d cos θ θ |δ |−∆ ∆2 ln π 2 θ ∆ −1 −| | 2 − 2 + 2 δθ δθ ∆ δθ , (28) where δθ = (1 − ρ cos θ)δ with ρ = q/δ, being the normalized diquark momentum. The limit of the angular integral depends on the gapless region of the excitation spectrum (23) and will be discussed in detail in Appendix B. In terms of the gap parameter ∆0 at δ = 0, given by the standard BCS gap equation

1 2µ¯ 2 2ω = 0 , 2 ln (29) 4GD π ∆0 we have I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 261 2µ¯ 2 ∆ 1 Γ = ∆2 ln − π 2 ∆ 2 0 (ρ + 1)3 1 + x 2 + δ2 1 − x2 ln 1 + 2x3 − 3x + 1 4ρ 1 1 − x 3 1 1 1 − 3 + + (ρ 1) 2 − 2 1 x2 + 2 3 − + δ 1 x2 ln 2x2 3x2 1 , (30) 4ρ 1 − x2 3 where x1 and x2 are the same dimensionless parameters introduced in Ref. [2], i.e., ∆ ∆2 x = θ 1 − 1 − (31) 1 (ρ + 1)δ (ρ + 1)2δ2 and ∆ ∆2 x = θ 1 − 1 − . (32) 2 |ρ − 1|δ (ρ − 1)2δ2

Differentiating Eq. (30) with respect to ∆2 and ρ, we derive the equilibrium equations ∂Γ 2µ¯ 2 ∆ ρ + 1 1 − x ρ − 1 1 − x = − 1 + x − 2 + x = 2 2 ln ln 2 1 ln 2 2 0 ∂∆ π ∆0 4ρ 1 + x1 4ρ 1 + x2 (33) and ∂Γ 4µ¯ 2δ2 (ρ + 1)2 1 + x = ρ 1 − 3 1 − x2 ln 1 + 4(ρ + 1)x3 − 6x ∂ρ 3π 2 8ρ3 1 1 − x 1 1 1 (ρ − 1)2 1 + x − − − x2 2 + (ρ − )x3 + x = . 3 3 1 2 ln 4 1 2 6 2 0 (34) 8ρ 1 − x2 If we substitute into (30) the solution to the gap equation (33), we ﬁnd 2µ¯ 2 1 1 1 Γ = δ2 + ρ2δ2 − ∆2 − (ρ + 1)3x3 + (ρ − 1)3x3 δ2 . (35) π 2 3 2 6ρ 1 2 Furthermore, the second order derivatives provide us with the expressions ∂2Γ µ¯ 2 1 1 = + , 2 2 2 2 (36) (∂∆ ) 2π ρδ (ρ + 1)(1 + x1) (ρ − 1)(1 + x2) ∂2Γ 4µ¯ 2δ2 3(ρ + 1)2 1 + x 3(ρ − 1)2 1 + x = + − x2 1 − − x2 2 2 2 1 3 1 1 ln 3 1 2 ln ∂ρ 3π 4ρ 1 − x1 4ρ 1 − x2 (ρ + 1)3 3(ρ + 1)(ρ2 + ρ + 1) (ρ − 1)3 + x3 − x + x3 ρ3 1 2ρ3 1 ρ3 2

3(ρ − 1)(ρ2 − ρ + 1) − x (37) 2ρ3 2 262 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 and ∂2Γ µ¯ 2 1 − x 1 − x = 1 − 2 + (ρ + )x + (ρ − )x . 2 2 2 ln ln 2 1 1 2 1 2 (38) ∂∆ ∂ρ 2π ρ 1 + x1 1 + x2 Our expression of the free energy function Eq. (35) coincides with the expression derived in [2] if we perform the following modifications: replace the Fermi velocity in their formula with one and multiply it by a factor of four. This factor accounts for the four pairing configurations, (r, uR) − (g, dR), (r, uL) − (g, dL), (r, dR) − (g, uR) and (r, dL) − (g, uL), per momentum for quarks versus one pairing configuration, spin up-spin down, per momentum for electrons, where the subscripts R and L refer to the helicities of the pairing quarks.

3. The Meissner masses

In this section we shall consider the coupling of the quark fields to external gauge fields. The mean field NJL Lagrangian is modified

2 L =− ∆ −¯ ∂ − l − −¯ +¯ χγµ igAµTl ieQAµ χ χγ qχ χγ4µχ 4GD ∂xµ + ∆(−¯χCγ5λ2τ2χ +¯χγ5λ2τ2χC). (39) l Aµ and Aµ represent the gluon and photon gauge potentials, respectively. The Lagrangian is invariant under SU(3)c × U(1)em transformations. The SU(3)c generators and the U(1)em charge operator can be written as 1 1 1 T = λ ,Q= + τ . (40) l 2 l 6 2 3 The squared Meissner masses are the eigenvalues of the Meissner tensors, deﬁned by 2 l l = ll M lim Πij (K), (41) ij K→0 ll where Πij (K) is the gluon/photon magnetic polarization tensor given by

−1 (kB T) 1 − · Π l l(K) = dx d3re iK x J l (r,τ)J l (0, 0) ij 2 4 i j C 0 k T d3p =− B Tr Γ l S(P )Γ lS(P + K), (42) 2 ( π)3 i j ν 2 where ··· C stands for the connected Green’s functions, i.e., AB C = AB − A B and K = (kz,ˆ −ω) is the Euclidean four momentum either of the gluon or of the photon. The l = 1 ¯ l current operator reads J 2 Ψ Γ Ψ , where the quark–gluon/photon vertex with respect to the NG basis (15) takes the form

T l = γµ l 0 Γµ ˜ , (43) 0 −γµτ2Tlτ2 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 263 where l = 1, 2,...,9 and the tilde denotes transpose. Here we have introduced a new set of generators, Tl, that mixes the gluons with photon, in accordance with the residual symmetry in the presence of the condensate (4) [18], i.e.,

Tl = Tl (44) for l = 1,...,7, e T = T cos θ + Q sin θ (45) 8 8 g and g T =− T sin θ + Q cos θ (46) 9 e 8 √ with cos θ = √ 3g and sin θ = √ e . The condensate is invariant under the residual 3g2+e2 3g2+e2 gauge transformations SU(2) × U(1), where SU(2) is generated by T1, T2, T3 and U(1) by T9. We shall refer to the photon–gluon mixture associated to T8 as the 8th gluon. Each component of the Meissner tensor is a matrix with respect to the generators of the gauge group, Eqs. (44)–(46). A simpliﬁcation follows from the observation that the isospin part of the quark–photon vertex is the unit matrix with respect to the NG blocks (−τ2τ˜3τ2 = τ3) and consequently it contributes to the integrand of the Meissner tensor a total derivative, i.e.,

d3p d3p ∂ γ τ S(P )MS(P ) = i τ MS(P ), 3 Tr j 3 3 Tr 3 (47) (2π) (2π) ∂pj where M is an arbitrary matrix. Therefore the isospin term of the charge operator (40) can = 1 be omitted and we have effectively Q 6 [15,16]. Next we shall prove that the Meissner tensor is diagonal with respect to the indices l and l, more speciﬁcally that it has the following form 2 11 = 2 22 = 2 33 = 2 99 = M ij M ij M ij M ij 0, (48) 2 44 = 2 55 = 2 66 = 2 77 ≡ 2 M ij M ij M ij M ij m ij , (49) and 2 88 ≡ 2 M ij m ij . (50) The relations above follow in a straightforward manner from symmetry arguments, in particular the residual SU(2) symmetry. We proceed to divide the set of the nine generators into three subsets. Subset I includes the unbroken generators T1, T2 and T3, subset II includes the broken generators T4, T5, T6 and T7 while subset III includes the broken T8 and † the unbroken T9 generators. Under an SU(2) transformation, Tl → uTlu , the generators in the subset I transform as the adjoint representation, the generators in the subset II as the fundamental one while the generators in the subset III remain invariant. Therefore the indices l and l which correspond to generators from different subsets do not mix in the 264 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280

2 ll expression for (M )ij . The mass matrix vanishes within the subset I because of the unbroken gauge symmetry. As for generators of the subset II, we proceed to relabel its members J = J = J = J = according to 1 T4, 1¯ T5, 2 T6 and 2¯ T7. Introducing

1 0 φ = ,φ= , (51) 1 0 2 1 we have

0 φ J = 1 α α † , (52) 2 φα 0 and similar expressions for Jα¯ with φα¯ =−iφα, where the block decomposition is with respect to color indices. Since Jα is symmetric and Jα¯ is antisymmetric, there is no mix- J J = ing between α and β¯, following from the properties of the matrix C iγ2γ4, i.e., −1 Cγ˜µC =−γµ and

˜ −1 ρ3CS(P )(ρ3C) =−S(P ), (53) a consequence of the symmetry under a combined transformation of a space inversion, a time reversal operation and a γ5 multiplication (see Appendix C for details). The identity ¯ of the matrix elements (M2)αβ and (M2)α¯ β follows from the equivalence of the doublet representation supported by Tα and the antidoublet one by Tα¯ . Finally in the subset III, the color matrices never mix the red and green components with the blue ones. The latter do not carry a condensate. It follows then that we need only to work within the red–green sub- T T 1 2 + 2 space, in which 9 does not contribute and 8 may be replaced by a factor 6g 3g e . The above argument on the group theoretic structure of the Meissner tensor leads also to its relation with the momentum susceptibility introduced in Ref. [15],

2 2 2 = 1 2 + e ∂ Γ (m )ij g . (54) 12 3 ∂qi∂qj 2 Because of the single plane wave ansatz, Eq. (4), each of the nontrivial tensors (m )ij 2 and (m )ij can further be decomposed into its transverse and longitudinal components, i.e., 2 = −ˆ ˆ + ˆ ˆ m ij A(δij qiqj ) Bqiqj , (55) 2 = −ˆ ˆ + ˆ ˆ m ij C(δij qiqj ) Dqiqj . (56) In particular, Eq. (54) implies that

g2 e2 ∂Γ g2 e2 ∂Γ C = 1 + = 1 + (57) 12ρδ2 3g2 ∂ρ 6δ2 3g2 ∂ρ2 and

g2 e2 ∂2Γ D = 1 + . (58) 12δ2 3g2 ∂ρ2 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 265

It follows from Eqs. (57) and (34) at ρ = 0 that the Meissner tensor of the 8th gluon is longitudinal at the LOFF equilibrium. So is, the Meissner tensor of an electronic LOFF superconductor. This conclusion of ours disagrees with that of [2], where a nonzero transverse Meissner mass was reported. The above property does not contradict with the δij structure of the Meissner tensor of a homogeneous superconductor since it corresponds to the trivial solution, ρ = 0ofEq.(34), where ∂Γ = 0. ∂ρ2 In order to calculate the quantities A, B, C and D we follow the same procedure that we used to calculate the free energy in the previous section. In order to improve the UV convergence, we employ the technique of [19] and subtract from the polarization tensor of the superconducting phase, (42), the corresponding expression of the normal phase with the same µ¯ and δ, which is zero at K = 0, and write k T d3p Π l l(0) =− B Tr Γ l S(P )Γ lS(P ) − Γ l S(P)Γ lS(P) , (59) ij 2 ( π)3 i j i j ν 2 where S(P) = S(P )|∆=0. Then the weak coupling approximation of the propagator, Eq. (21), is inserted. We take the z-axis to be parallel to q and carry out the integration over the azimuthal angle and the magnitude of p. We are then left with the integration over the meridian angle and the summation over the Matsubara energies. Finally, we project out the transverse and longitudinal components and obtain the expressions 1 3 A = d cos θ sin2 θF (θ, ∆), (60) 4 −1

1 3 B = d cos θ cos2 θF (θ, ∆), (61) 2 −1

1 3 C = d cos θ sin2 θG(θ, ∆), (62) 4 −1 and 1 3 D = d cos θ cos2 θG(θ, ∆), (63) 2 −1 where 2g2µ¯ 2k T 1 iν + δ − z(ν, θ, ∆) F(θ,∆)= B Im θ (64) 3π z(ν, θ, ∆) iν + δθ + z(ν, θ, ∆) ν>0 and

2g2µ¯ 2∆2k T e2 1 G(θ, ∆) = B 1 + Im . (65) 9π 3g2 z3(ν,θ,∆) ν>0 266 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280

At T = 0, the summation over ν becomes an integral and we ﬁnd g2µ¯ 2 1 δ2 δ2 ∆2 F(θ,∆)= − θ + θ |δ |−∆ θ − 2 2 θ 2 1 2 (66) 3π 2 ∆ ∆ δθ and g2µ¯ 2 e2 θ(|δ |−∆) G(θ, ∆) = + − θ . 2 1 2 1 (67) 9π 3g − 2 2 1 ∆ /δθ We recognize that Eqs. (66) and (67) are precisely the expressions that Huang and Shovkovy obtained for the Meissner masses [13] when their parameter δµ is replaced by an angle-dependent one, |δθ |. By carrying out the angular integration as it is described in detail in Appendix B, we ﬁnd g2µ¯ 2 2δ2 1 3∆2 1 + x 1 + x A = − + ρ2 − 1 − 2 2 1 2 1 3 2 ln ln 6π ∆ 5 16ρ δ 1 − x1 1 − x2 δ2 δ2 + (ρ + 1)5x5 − (ρ + 1)4x 5x2 − 3 (68) 5ρ3∆2 1 8ρ3∆2 1 1

δ2 δ2 + (ρ − 1)5x5 + (ρ − 1)4x 5x2 − 3 , 5ρ3∆2 2 8ρ3∆2 2 2 g2µ¯ 2 2δ2 3 3∆2 1 + x 1 + x B = − + ρ2 + 1 − 2 2 1 2 1 3 2 ln ln 6π ∆ 5 8ρ δ 1 − x1 1 − x2 δ2 2δ2 δ2 + (ρ + 1)3x3 − (ρ + 1)5x5 + (ρ + 1)4x 5x2 − 3 ρ∆2 1 5ρ3∆2 1 4ρ3∆2 1 1

δ2 2δ2 δ2 + (ρ − 1)3x3 − (ρ − 1)5x5 − (ρ − 1)4x 5x2 − 3 , ρ∆2 2 5ρ3∆2 2 4ρ3∆2 2 2 (69) g2µ¯ 2 e2 (ρ + 1)2 1 + x C = 1 + 1 − 3 1 − x2 ln 1 + 4(ρ + 1)x3 − 6x 9π 2 3g2 8ρ3 1 1 − x 1 1 1 (ρ − 1)2 1 + x − − − x2 2 + (ρ − )x3 + x 3 3 1 2 ln 4 1 2 6 2 (70) 8ρ 1 − x2 and g2µ¯ 2 e2 ρ + 1 1 + x D = 1 + 1 + 3(ρ + 1) 1 − x2 ln 1 2 2 3 1 − 9π 3g 4ρ 1 x1 ρ − 1 1 + x + 4(ρ + 1)2x3 − 6 ρ2 + ρ + 1 x + −3(ρ − 1) 1 − x2 ln 2 1 1 3 2 − 4ρ 1 x2 + − 2 3 − 2 − + 4(ρ 1) x2 6 ρ ρ 1 x2 . (71) Comparing (70) and (71) with (34) and (37), we conﬁrm explicitly the relation between the Meissner tensor of the 8th gluon and the momentum susceptibility, Eq. (54). I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 267

4. The Meissner effect for a small gap parameter

In the last two sections we derived the expressions for the free energy and the Meissner masses of the gluons for a 2SC-LOFF of arbitrary ∆, δ and ρ. In this section we shall proceed to address the issue of the chromomagnetic instability at equilibrium. Initially we shall consider the case ∆ δ,—the gap parameter much smaller than the displacement. This is the situation near a second order phase transition to the normal phase. For a LOFF state consisting of a single plane wave at T = 0 this occurs at [2]

δ = δc 0.754∆0, (72) which is also the maximum displacement for the LOFF condensate, Eq. (4). Although it has been argued that the single plane wave ansatz is not energetically favorable compared with the multi plane wave ansatz in this neighborhood and that the transition to the normal phase may be a ﬁrst order one [21], we shall still pursue the small gap expansion since it provides a full analytic treatment and consequently can be very instructive. We shall focus on the LOFF state at T = 0. Let us expand the free energy, Eq. (30) to quartic power of the small parameter, ∆ˆ ≡ ∆/δ, 2µ¯ 2δ2 1 Γ = α∆ˆ2 + β∆ˆ4 (73) π 2 2 with 2δ ρ + 1 ρ − 1 α = ln − 1 + ln(ρ + 1) + ln |ρ − 1| (74) ∆0 2ρ 2ρ and 1 β = . (75) 4(ρ2 − 1) The corresponding expansions of the quantities A and B read g2µ¯ 2 A = a ∆ˆ2 + a ∆ˆ4 +··· , (76) 6π 2 0 1

g2µ¯ B = b ∆ˆ2 + b ∆ˆ4 +··· , (77) 6π 2 0 1 where

3 ρ + 1 a = 2ρ − ln , (78) 0 8ρ3 ρ − 1 1 1 1 1 1 1 a =− + − − , (79) 1 16ρ3 ρ − 1 ρ + 1 32ρ3 (ρ − 1)2 (ρ + 1)2

3 2 − ρ2 1 ρ + 1 b = + ln (80) 0 4ρ2 ρ2 − 1 ρ ρ − 1 268 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 and 1 1 1 1 1 1 b =− + − + 1 3 ρ − ρ + 3 − 2 + 2 8ρ 1 1 16 ρ (ρ 1) (ρ 1) 1 1 1 + + . (81) 16ρ (ρ + 1)3 (ρ − 1)3 Similarly, we ﬁnd

g2µ¯ 2 e2 C = 1 + c ∆ˆ2 + c ∆ˆ4 +··· , (82) 9π 2 3g2 0 1

g2µ¯ 2 e2 D = 1 + d ∆ˆ2 + d ∆ˆ4 +··· , (83) 9π 2 3g2 0 1 where

3 ρ + 1 c = 2ρ − ln = 2a , (84) 0 4ρ3 ρ − 1 0 3 2 2 1 1 c = − − + − , (85) 1 32ρ3 ρ + 1 ρ − 1 (ρ + 1)2 (ρ − 1)2

3 2 − ρ2 1 ρ + 1 d = + ln = 2b (86) 0 2ρ2 ρ2 − 1 ρ ρ − 1 0 and 3 3 3 3 3 1 1 d = + − + + + . (87) 1 16ρ3 ρ + 1 ρ − 1 (ρ + 1)2 (ρ − 1)2 (ρ + 1)3 (ρ − 1)3

The validity of the relations c0 = 2a0 and d0 = 2b0 is not restricted to zero temperature and the weak coupling approximation employed here. If we expand the integrand of the ˆ2 Meissner mass tensor (59) to the order ∆ , we ﬁnd that the loop integrals for a0 and b0 are related to those for c0 and d0 through integration by parts [15]. The same argument also leads to identical criteria for the chromomagnetic instability of the 4–7th gluons and the 8th gluon near Tc for g2SC. Minimization of the free energy with respect to ∆ and ρ yields

ˆ2 = 2 − − δ ∆ 4 ρc 1 1 , (88) δc and 1 ρ ρ = ρ + c ∆ˆ2, (89) c 2 − 4 ρc 1 where ρc 1.20 is the solution of the equation [2] 1 ρ + 1 ln c = 1 (90) 2ρc ρc − 1 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 269 and δ 1 c = 0.754. (91) 2 − ∆0 2 ρc 1 The free energy at the minimum reads

4µ¯ 2 Γ =− ρ2 − 1 (δ − δ )2 (92) min π 2 c c and both branches of Eq. (23) are gapless here. The solution satisﬁes the stability condition (13). Substituting it into Eqs. (76), (77), (82) and (83), we obtain

g2µ¯ 2 ∆ˆ4 g2µ¯ 2 ∆ˆ2 A = 0,B= 0 (93) 2 2 − 2 2 2 − 96π (ρc 1) 8π ρc 1 and

g2µ¯ 2 e2 ∆ˆ2 C = 0,D= 1 + 0. (94) 2 2 2 − 6π 3g ρc 1 Therefore the chromomagnetic instabilities are removed completely for small gap parameters. The situation for a multi-plane wave ansatz for the LOFF state will be better if the small gap expansion remains numerically approximate. As we have seen for the single plane wave ansatz, the Meissner tensors are purely longitudinal to ∆2 and the longitudinal components are positive at the equilibrium. It follows that if the LOFF state (3) contains three linearly independent momenta, the Meissner tensor will be positive deﬁnite, since it is additive with respect to different terms of Eq. (3) to order ∆2.

5. Numerical results

In the last section, we have shown analytically that the chromomagnetic instability is absent in the region near the outer edge of the LOFF window at zero temperature. In this section, we shall explore the interior region in order to determine if the absence of the chromomagnetic instability can be extended to the entire LOFF region at T = 0. Before presenting our numerical results, we shall emphasize some general features of the solutions of Eqs. (33) and (34). A trivial solution to Eq. (34), ρ = 0, corresponds to a homogeneous 2SC and there is always a solution to Eq. (33) with ∆ = ∆0 for 0 <δ<∆0. The expression for the free energy then becomes

2µ¯ 2 1 Γ = δ2 − ∆2 . (95) π 2 2 0 Since the excitation spectrum, Eq. (23), is fully gapped in this case, we shall refer to the corresponding state as the BCS state even in the presence of a displacement, following the convention of Ref. [2]. We observe that the BCS state becomes meta-stable relative to the 270 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 normal phase for 1 δ>√ ∆0. (96) 2 ∆0 = For the displacement within the interval 2 <δ<∆0, there is another solution at ρ 0 with ∆ = ∆0(2δ − ∆0) (97) which leads to the following expression for the free energy µ¯ 2 Γ = (2δ − ∆ )2 > 0. (98) π 2 0 This solution corresponds to the Sarma state that is unstable in the grand canonical ensemble for a given δ. In a canonical ensemble, it may be the only charge neutral state for certain pairing strength [7] but suffers from chromomagnetic instabilities [13]. In what follows we shall restrict the terminology ‘LOFF state’ to the nontrivial solution to Eq. (34) that satisﬁes (ρ + 1)2 1 + x 1 − 3 1 − x2 ln 1 + 4(ρ + 1)x3 − 6x 8ρ3 1 1 − x 1 1 1 (ρ − 1)2 1 + x − − − x2 2 + (ρ − )x3 + x = 3 3 1 2 ln 4 1 2 6 2 0 (99) 8ρ 1 − x2 together with Eq. (33). It follows from Eq. (33) that

∆<∆0 (100) at the solution and Eq. (99) implies that ∆ < 1 + ρ. (101) δ Otherwise, x1 = x2 = 0 and Eq. (99) cannot hold. Physically, Eq. (100) is the consequence of the reduced phase space available for pairing and Eq. (101) is the condition for the existence of gapless excitations, which is necessary to make the net baryon number current vanish. Combining Eqs. (100) and (101), we find δ<∆0 for the LOFF state. For a small gap parameter, an analytical solution was obtained in the last section, which corresponds to a stable state with lower energy compared with the BCS and the normal states and is free of chromomagnetic instabilities. For a general gap parameter, however, Eqs. (33) and (99) can only be solved numerically. Our strategy is to find solutions of (99) for ∆/δ for a given ρ and subsequently to calculate the corresponding δ/∆0 in terms of (33). Concurrently, we examine the local stability according to Eq. (13) and the competi- tion between the LOFF state we find and the BCS state of the same δ/∆0. The Meissner tensors are calculated as well. We observe that the quantities ρ, δ/∆0, the free energy difference, the Meissner masses and the stability determinant of Eq. (13), at the LOFF solution are single-valued functions of the normalized gap parameter ∆/∆0. These functions are plotted in Figs. 1–7. I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 271

Fig. 1. The dimensionless diquark momentum ρ of the LOFF state as a function of the normalized gap parameter ∆/∆0.

The diquark momentum of a LOFF state, normalized by the displacement parameter versus the normalized gap parameter is shown in Fig. 1 and the corresponding ratio of the displacement parameter to the BCS gap is shown in Fig. 2.InFig. 3, we display the free energy difference between a LOFF state and a BCS state at the same δ, normalized by the magnitude of the BCS energy (95) at δ = 0. Combining the three ﬁgures, we ﬁnd the LOFF window, in which the LOFF state is of the lowest energy,

δmin <δ<δmax, (102) where δmax = δc 0.754∆0 and δmin 0.706∆0, slightly below the threshold (96).The corresponding momentum of the diquark increases monotonically from ρ 1.20 at δmax to ρ 1.28 at δmin. Our results conﬁrm those of Ref. [2]. In addition, we found that the normalized gap parameter, ∆/∆0 varies monotonically from 0 at δmax to 0.242 at δmin. So the LOFF window on all ﬁgures corresponds to the domain 0 <∆/∆0 < 0.242 of the abscissa. The numerical results of the Meissner tensors, Figs. 4–6 are most interesting. The numbers shown are normalized by the Meissner mass square of the BCS state at δ = 0, i.e., 2 = m ij A0δij (103) and 2 = m ij D0δij (104) 272 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280

Fig. 2. The normalized displacement parameter, δ/∆0, of a LOFF state as a function of the normalized gap parameter ∆/∆0.

Fig. 3. The free energy difference between a LOFF state and a BCS state as a function of the normalized gap parameter ∆/∆0. The coordinate is normalized to the magnitude of the free energy of the BCS state, Eq. (94),at δ = 0 and the LOFF state is favored when the value is negative. I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 273

Fig. 4. The normalized transverse Meissner mass square of the 4–7th gluons, A/A0, as a function of the normalized gap parameter ∆/∆0 of a LOFF state.

Fig. 5. The normalized longitudinal Meissner mass square of the 4–7th gluons, B/A0, as a function of the normalized gap parameter ∆/∆0 of a LOFF state. 274 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280

Fig. 6. The normalized longitudinal Meissner mass square of the 8th gluon, D/D0, as a function of the normalized gap parameter ∆/∆0 of a LOFF state. with

g2µ¯ 2 g2µ¯ 2 e2 A = ,D= 1 + . (105) 0 6π 2 0 9π 2 3g2 All squared Meissner tensors are nonnegative within the LOFF window and the region free from the chromomagnetic instabilities extends beyond it. While the Meissner tensor associated to the eighth gluon is aways longitudinal, Fig. 6, the transverse component of the Meissner tensor associated to the 4–7th gluons, Fig. 4 is aways much smaller than its longitudinal component, Fig. 5. The stability of the solution against virtual displacements of ∆ and ρ within the LOFF window is guaranteed by the positivity of D, plotted in Fig. 6 and the positivity of the determinant in Eq. (13), plotted in Fig. 7. The non-smooth behavior of the function D and the determinant of Figs. 6 and 7 correspond to the transition of the LOFF state from the region where both branches of (23) are gapless, x2 > 0, to the − = region where only the Ep branch is gapless, x2 0. The derivative of x2 with respect to both ∆ and ρ become singular there. When the gap parameter is expressed in terms of ρ and x2 > 0 according to Eq. (32) and the function D and the determinant are expanded according to the powers of x2, the linear term is sensitive to the singularity and gives rise to the nonsmooth behavior. On the other hand, the same type of expansion of the functions A and B does not contain a linear term in x2 and thus the transition behavior of A and B is much more smooth. As the LOFF momentum goes to zero, ρ → 0, we have ∆ → ∆0 and δ → ∆0,aresult can be established analytically. Therefore the LOFF state, the BCS state and the Sarma state joins at the point (∆0, ∆0, 0) in the three-dimensional parameter space of ∆, δ and ρ.We I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 275

Fig. 7. The stability determinant in Eq. (13) (arbitrary unit) of a LOFF state as a function of the normalized gap parameter ∆/∆0. also observe from Figs. 4–6 that the chromomagnetic instability and the instability against a small variation of parameters show up for sufﬁciently low values of ρ. Therefore the instability free region of a LOFF state is disconnected from the g2SC state in the parameter space.

6. Concluding remarks

In this paper, we have systematically examined the Meissner effect in a two-flavor color superconductor with a LOFF pairing. We found that within the LOFF window at T = 0, that is within the range of the displacement parameter where the LOFF state is energetically favored, all the squares of the Meissner masses are nonnegative. Although the positivity of the longitudinal component of the Meissner mass tensor associated with the eighth gluon follows from the stability of the LOFF state against a variation of the LOFF momentum, ρ, the positivity of the Meissner tensor for the 4–7th gluons appears accidental regarding the pairing ansatz (4). We have also found that the Meissner tensor of the 8th gluon is always longitudinal. This is also the case with the Meissner effect in an electronic LOFF superconductor. The charge neutrality condition has to be implemented before we can claim that the LOFF state is the ground state of quark matter at moderately high baryon density. In the previous letter [15], we showed that the chromomagnetic instability of the 8th gluon pertaining to g2SC implies a state of lower free energy with a nonzero diquark momentum and without offsetting the charge neutrality. This is not sufficient since the LOFF window, 276 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 with the diquark momentum comparable to the Fermi momentum displacement, is quite far away from the g2SC and is very narrow in the three-dimensional parameter space. It remains to be seen if quark matter within the LOFF window can be made neutral. In a hypothetical world where the sum of the electric charges of the u and d quarks is a small number, we can find within the framework of the weak coupling approximation an analytic expression for an electrically neutral LOFF phase that is located near the upper edge of the LOFF window and is energetically favored over both the neutral normal phase and the neutral BCS phase. Also it is free of chromomagnetic instabilities. The details will be presented in a forthcoming publication [20]. Notice that the neutral phases being compared do not have the same displacement parameter δ since the charge neutrality constraint adjusts δ differently for different phases. In the real world, the sum of the charges of the u and d quarks is 1/3. The weak coupling approximation may be marginal and one may have to seek numerical solutions of Eqs. (33), (99) and the charge neutrality constraint simultaneously. We also notice that, if the pairing force is mediated by the long range one-gluon exchange of QCD instead of the point coupling of NJL model, the LOFF window will be at least five times wider under the same ∆0 (δmax 0.968∆0 and δmin 0.706∆0) [6], making the implementation of the charge neutrality more likely. In this paper we have considered only the simplest LOFF ansatz that consists of one plane wave and pairs u and d quarks. It was suggested that multi plane wave condensates are favored more near the upper edge of the LOFF window [21]. Numerical analysis also suggests that the s quark cannot be ignored in pairing at T = 0. A more complicated structure of the condensate is required for a realistic LOFF pairing in quark matter and may not be accessible via analytical means. In any case, the absence of chromomagnetic instabilities for the simple 2SC-LOFF is encouraging and promotes the LOFF state as a potential candidate of the ground state of quark matter at moderately baryon densities.

Acknowledgements

Hai-cang Ren thanks DeFu Hou for valuable discussions. We are indebted to I. Shovkovy for an interesting communication. This work is supported in part by the US Department of Energy under grants DE-FG02-91ER40651-TASKB.

Appendix A

In this appendix, we shall ﬁrst derive the exact expression of the quark propagator in the superconducting phase for δ = 0. Subsequently we shall perform the weak coupling expansion. Finally, we shall approximate the determinant of the quark propagator which enters the free energy function (20) in a similar manner. As τ3 and λ2 are the only isospin and color matrices appearing in the inverse quark propagator (18) at δ = 0, they can be treated as a c-numbers. Taking the square of the inverse propagator, we have I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 277 − S 1(P ) 2 = (iν − δτ )2 +¯µ2 − p2 − q2 − ∆2λ2 3 2 + 2ρ3 µ(iν¯ − δτ3) −p ·q + 2∆ρ1γ5(−iγ ·q +¯µγ4)λ2. (A.1) In terms of the matrix N(P)= (iν − δτ )2 +¯µ2 − p2 − q2 − ∆2λ2 3 2 − 2ρ3 µ(iν¯ − δτ3) −p ·q − 2∆ρ1γ5(−iγ ·q +¯µγ4)λ2, (A.2) we obtain the exact expression of the quark propagator S−1(P )N(P ) S(P ) = , (A.3) D(P) where D(P) = (iν − δτ )2 +¯µ2 − p2 − q2 − ∆2λ2 2 − 4 µ(iν¯ − δτ ) −p ·q 2 3 2 3 + 2 ¯ 2 − 2 2 4∆ µ q λ2, (A.4) 2 and λ2 projects into the subspace of the red and green colors. In order to perform the weak coupling approximation, we regard p −¯µ, ν, δ and q as small quantities of comparable magnitudes and maintain only the leading order terms in the numerator and the denominator of (A.3). We ﬁnd that ¯ 2 − − −ˆ· 2 + −¯ 2 + 2 2 D(P) 4µ (iν δτ3 p q) (p µ) ∆ λ2 (A.5) and − S 1(P )N(P ) −2µ¯ 2(−iγ ·ˆp + ρ γ ) 3 4 × |p − ρ3q|−µ ¯ + ρ3(iν − δτ3) + ∆λ2γ5γ4 . (A.6) Upon substitution of (A.5) and (A.6) into (A.3), we derive the approximate quark propagator expression (21) that appeared in Section 2. The determinant of S−1(P ) may be calculated similarly. Let us write − − det S 1(P ) = det1/2 S 1(P ) 2 (2µ)¯ 24det1/2d(P), (A.7) where

d(P) =¯µ − p + ρ3(iν − δτ3 −ˆp ·q)+ ∆λ2ρ1γ5γ4. (A.8)

Now by decomposing the 48×48 matrix d(P), according to the eigenvalues of τ3 and λ2, i.e.,

det d(P) = det d(P) (A.9) τ3=±1,λ2=±1,0 we have | = −¯ 2 + 2 − ∓ −ˆ· 2 4 det d(P) τ3=±1,λ2=1 (p µ) ∆ (iν δ p q) , (A.10) | = −¯ 2 + 2 − ∓ −ˆ· 2 4 det d(P) τ3=±1,λ2=−1 (p µ) ∆ (iν δ p q) (A.11) 278 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 and | = −¯ 2 − ∓ −ˆ· 2 4 det d(P) τ3=±1,λ2=0 (p µ) (iν δ p q) . (A.12) Eq. (22) is then established.

Appendix B

The integrand of the angular integrals for the free energy function (28) and that for the Meissner tensor, Eqs. (66) and (67), contain terms with θ(|δθ |−∆). It conﬁnes the integral within the region in the phase space where either or both branches of the excitation ± + spectrum, Ep of (23), become gapless. This occurs when ∆<(1 ρ)δ. Otherwise the integration domain speciﬁed by θ(|δθ |−∆) is null. There are three different cases to be considered:

| − | + − (1) 1 ρ δ<∆<(1 ρ)δ. Only the Ep branch may be gapless and we have

δ−∆ 1 ρδ d cos θθ |δθ |−∆ (···) = d cos θ(···). (B.1) −1 −1 − − (2) ρ<1 and ∆<(1 ρ)δ. Again we may have only one gapless branch, Ep ,but instead 1 1 d cos θθ |δθ |−∆ (···) = d cos θ(···) (B.2) −1 −1 δ−∆ because ρδ > 1. − + ± (3) ρ>1 and (ρ 1)δ < ∆ < (ρ 1)δ. Both branches, Ep may be gapless and we have

δ−∆ 1 ρδ 1 d cos θθ |δθ |−∆ (···) = d cos θ(···) + d cos θ(···). (B.3) −1 −1 δ+∆ ρδ

The indeﬁnite integrals of the integrands of the free energy function and the Meissner tensors are all elementary functions.

Appendix C

In this appendix, we shall supply the details supporting the statement in Section 3 that the identity (53) for the quark propagator follows from the invariance of the 2SC-LOFF I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280 279 under a combined transformation of a space inversion, a time reversal and a γ5 multiplication. We shall work with the canonical formulation at T = 0 though the generalization to T = 0 is straightforward. The Hamiltonian operator corresponding to the mean ﬁeld NJL action (9) reads

∆2 H = Ω + d3r ψ¯ γ · ∇ ψ − ψγ¯ µψ MF 4G 4 D ¯ −2iqr ¯ 2iqr − ∆ −ψCγ5λ2τ2ψe + ψγ5λ2τ2ψCe . (C.1) Introduce an antiunitary operator U which implements a space inversion, a time reversal and a γ5 multiplication. The transformation laws of the ﬁeld operators in Heisenberg representation are given by

−1 Uψ(r,t) U = iγ2ψ(−r,−t), † −1 † Uψ (r,t) U =−iψ (−r,−t)γ2. (C.2) The corresponding transformations of the NG spinors are

−1 UΨ(r,t) U = iρ3γ2Ψ(−r,−t), † −1 † UΨ (r,t) U =−iΨ (−r,−t)γ2ρ3, (C.3) following (5) and (15). The phase factor associated to the time reversal has been ﬁxed in accordance with phase convention of the condensate such that

−1 UHMFU = HMF. (C.4) For a state | that is invariant under U, say the ground state and an arbitrary operator O,we have [22]

− |O†| = |UOU 1| . (C.5)

The Hamiltonian (C.1) is also invariant under a unitary transformation, U, that generates a translation and a rephasing according to

− · Uψ(r,t)U † = e iq aψ(r +a,t), · Uψ†(r,t)U † = eiq aψ†(r +a,t). (C.6)

Consider the quark propagator in coordinate space, ¯ ¯ Sαβ (r,t) = |Ψα(r,t) Ψβ (0, 0)| θ(t)− |Ψβ (0, 0)Ψα(r,t) | θ(−t). (C.7) It follows from (C.3), (C.5), (C.6) and time translation invariance that

Sαβ (r,t) =−(ρ3C)αγ Sσγ(r, t)(Cρ3)βσ, (C.8) which, upon a Fourier transformation, gives rise to the identity (53). 280 I. Giannakis, H.-C. Ren / Nuclear Physics B 723 [FS] (2005) 255–280

References

[1] A. Larkin, Yu.v Ovchinnikov, Zh. Eksp. Teor. Fiz. 47 (1964) 1136, Sov. Phys. JETP 20 (1965) 762; P. Fulde, R.A. Ferrell, Phys. Rev. A 135 (1964) 550. [2] S. Takada, T. Izuyama, Prog. Theor. Phys. 41 (1969) 635. [3] K. Rajagopal, F. Wilczek, in: M. Shifman (Ed.), B.L. Ioffe Festschrift, At the Frontier of Particle Physics/Handbook of QCD, World Scientific, Singapore, 2001, p. 2061; M. Alford, Annu. Rev. Nucl. Part. Sci. 51 (2001) 131; T. Schäfer, hep-ph/0304281; D.H. Rischke, Prog. Part. Nucl. Phys. 52 (2004) 197; H.-C. Ren, hep-ph/0404074; M. Huang, hep-ph/0409167; I. Shovkovy, nucl-th/0410091, and references therein. [4] R. Rapp, T. Schaefer, E.V. Shuryak, M. Velkovsky, Phys. Rev. Lett. 81 (1998) 53. [5] M.G. Alford, J.A. Bowers, K. Rajagopal, Phys. Rev. D 63 (2001) 074016; A.K. Leibovich, K. Rajagopal, E. Shuster, Phys. Rev. D 64 (2001) 094005; R. Casalbuoni, G. Nardulli, Rev. Mod. Phys. 76 (2004) 263. [6] I. Giannakis, J.T. Liu, H.C. Ren, Phys. Rev. D 66 (2002) 031501. [7] I. Shovkovy, M. Huang, Phys. Lett. B 564 (2003) 205; M. Huang, I. Shovkovy, Nucl. Phys. A 729 (2003) 835; M. Huang, P.F. Zhuang, W.Q. Chao, Phys. Rev. D 67 (2003) 065015; J. Liao, P. Zhuang, Phys. Rev. D 68 (2003) 114016; A. Mishra, H. Mishra, Phys. Rev. D 69 (2004) 014014. [8] M. Alford, C. Kouvaris, K. Rajagopal, Phys. Rev. Lett. 92 (2004) 222001. [9] K. Iida, T. Matsuura, M. Tachibana, T. Hatsuda, Phys. Rev. Lett. 93 (2004) 132001; S. Ruester, I. Shovkovy, D. Rischke, Nucl. Phys. A 743 (2003) 127; S. Ruester, V. Werth, M. Buballa, I. Shovkovy, D. Rischke, The phase diagram of neutral quark matter: Self-consistent treatment of quark masses, hep-ph/0503184; D. Blaschke, S. Freriksson, H. Grigorian, A.M. Oztas, F. Sandin, The phase diagram of three-flavor quark matter under compact star constraint, hep-ph/0503194; P. Zhuang, Phase structure of color superconductivity, hep-ph/0503250; H. Abuki, M. Kitazawa, T. Kunihiro, How does the dynamical chiral condensation affect the three flavor neutral quark matter?, hep-ph/0412382. [10] G. Sarma, Phys. Chem. Solids 24 (1963) 1029. [11] W.V. Liu, F. Wilczek, Phys. Rev. Lett. 90 (2003) 047002; E. Gubankova, W.V. Liu, F. Wilczek, Phys. Rev. Lett. 91 (2003) 032001. [12] P.F. Bedaque, H. Caldas, G. Rupak, Phys. Rev. Lett. 91 (2003) 247002; H. Caldas, Phys. Rev. A 69 (2004) 063602; S. Reddy, G. Rupak, Phys. Rev. C 71 (2005) 025201. [13] M. Huang, I. Shovkovy, Phys. Rev. D 70 (2004) 094030; M. Huang, I. Shovkovy, Phys. Rev. D 70 (2004) 051501(R); M. Huang, Spontaneous current generation in the g2SC phase, hep-ph/0504235. [14] R. Casalbuoni, R. Gatto, M. Mannarelli, G. Nardulli, M. Raggieri, Phys. Lett. B 605 (2005) 362. [15] I. Giannakis, H.C. Ren, Phys. Lett. B 611 (2005) 137. [16] M. Alford, Q.H. Wang, Photons in gapless color-flavor locked quark matter, hep-ph/0501078. [17] M. Huang, P.F. Zhuang, W.Q. Chao, Phys. Rev. D 65 (2002) 076012. [18] M. Alford, J. Berges, K. Rajagopal, Nucl. Phys. B 571 (2000); E.V. Gorbar, Phys. Rev. D 62 (2000) 014007. [19] E.M. Lifshitz, L.P. Pitaevskii, Statistical Physics, Pergamon, Elmsford, NY, 1980, Chapter V. [20] I. Giannakis, H.-C. Ren, in preparation. [21] J.A. Bowers, K. Rajagopal, Phys. Rev. D 66 (2002) 065002. [22] T.D. Lee, Particle Physics and Introduction to Field Theory, Harwood Academic, Reading, UK, 1981, Chap- ter 13. Nuclear Physics B 723 (2005) 281–282

CUMULATIVE AUTHOR INDEX B721–B23

Aguado, M. B723 (2005) 234 Galante, A. B723 (2005) 77 Ananth, S. B722 (2005) 166 Gao, P. B721 (2005) 287 Antonov, E.N. B721 (2005) 111 Gehrmann, T. B723 (2005) 91 Azcoiti, V. B723 (2005) 77 Genovese, L. B721 (2005) 212 Giannakis, I. B723 (2005) 255 Bakulev, A.P. B721 (2005) 50 Giménez, V. B721 (2005) 175 Bedford, J. B721 (2005) 98 Giovannangeli, P. B721 (2005) 1 Behrndt, K. B721 (2005) 287 Giovannangeli, P. B721 (2005) 25 Bekaert, X. B722 (2005) 225 Gutperle, M. B722 (2005) 81 Belitsky, A.V. B722 (2005) 191 Bellucci, S. B722 (2005) 297 Berman, D.S. B723 (2005) 117 Haefeli, C. B721 (2005) 136 Bernreuther, W. B723 (2005) 91 Heinesch, R. B723 (2005) 91 Bonciani, R. B723 (2005) 91 Henkel, M. B723 (2005) 205 Boucaud, P. B721 (2005) 175 Boulanger, N. B722 (2005) 225 Israël, D. B722 (2005) 3 Bourrely, C. B722 (2005) 149 Itoyama, H. B723 (2005) 33 Brandhuber, A. B721 (2005) 98 Ivanov, E. B722 (2005) 297 Brink, L. B722 (2005) 166 Jarczak, C. B722 (2005) 119 Caprini, I. B722 (2005) 149 Cherednikov, I.O. B721 (2005) 111 Colangelo, G. B721 (2005) 136 Kato, J. B721 (2005) 229 Copland, N.B. B723 (2005) 117 Kawamoto, N. B721 (2005) 229 Cvetic,ˇ M. B721 (2005) 287 Kim, S.-S. B722 (2005) 166 Korchemsky, G.P. B722 (2005) 191 Da Rold, L. B721 (2005) 79 Korthals Altes, C.P. B721 (2005) 1 Derkachov, S.É. B722 (2005) 191 Korthals Altes, C.P. B721 (2005) 25 D’Hoker, E. B722 (2005) 81 Kuraev, E.A. B721 (2005) 111 Di Carlo, G. B723 (2005) 77 Dorsner, I. B723 (2005) 53 Laliena, V. B723 (2005) 77 Dudas, E. B721 (2005) 309 Lange, B.O. B723 (2005) 201 Dürr, S. B721 (2005) 136 Lee, K.S.M. B721 (2005) 325 Eden, B. B722 (2005) 119 Leineweber, T. B723 (2005) 91 Lin, C.-J.D. B721 (2005) 175 Fileviez Pérez, P. B723 (2005) 53 Lipatov, L.N. B721 (2005) 111 Fujiwara, K. B723 (2005) 33 Lubicz, V. B721 (2005) 175

0550-3213/2005 Published by Elsevier B.V. doi:10.1016/S0550-3213(05)00642-5 282 Nuclear Physics B 723 (2005) 281–282

Manashov, A.N. B722 (2005) 191 Sabio Vera, A. B722 (2005) 65 Martinelli, G. B721 (2005) 175 Sachrajda, C.T. B721 (2005) 175 McLoughlin, T. B723 (2005) 132 Sakaguchi, M. B723 (2005) 33 Miyake, A. B721 (2005) 229 Seiler, E. B723 (2005) 234 Sieg, C. B723 (2005) 3 Nagi, J. B722 (2005) 249 Sokatchev, E. B722 (2005) 119 Neubert, M. B723 (2005) 201 Spence, B. B721 (2005) 98 Nikolov, N.M. B722 (2005) 266 Stanev, Ya.S. B721 (2005) 212 Stanev, Ya.S. B722 (2005) 119 Pakman, A. B722 (2005) 3 Stefanis, N.G. B721 (2005) 50 Papinutto, M. B721 (2005) 175 Stewart, I.W. B721 (2005) 325 Phong, D.H. B722 (2005) 81 Stoimenov, S. B723 (2005) 205 Pomarol, A. B721 (2005) 79 Sutulin, A. B722 (2005) 297 Swanson, I. B723 (2005) 132 Quiros, M. B721 (2005) 309 Szabo, R.J. B723 (2005) 163

Ramond, P. B722 (2005) 166 Todorov, I.T. B722 (2005) 266 Rehren, K.-H. B722 (2005) 266 Torrielli, A. B723 (2005) 3 Remiddi, E. B723 (2005) 91 Travaglini, G. B721 (2005) 98 Ren, H.-C. B723 (2005) 255 Troost, J. B722 (2005) 3