Symmetry and Entropy of Black Hole Horizons

Nuclear Physics B 744 (2006) 1–13

Symmetry and entropy of black hole horizons

Olaf Dreyer, Fotini Markopoulou, Lee Smolin ∗

Perimeter Institute for Theoretical Physics, 35 King Street North, Waterloo, ON N2J 2W9, Canada Department of Physics, University of Waterloo, Waterloo, ON N2L 3G1, Canada Received 31 August 2005; accepted 3 February 2006 Available online 31 March 2006

Abstract We argue, using methods taken from the theory of noiseless subsystems in quantum information theory, that the quantum states associated with a Schwarzschild black hole live in the restricted subspace of the Hilbert space of horizon boundary states in which all punctures are equal. Consequently, one value of the Immirzi parameter matches both the Hawking value for the entropy and the quasi normal mode spectrum of the Schwarzschild black hole. The method of noiseless subsystems allows us to understand, in this example and more generally, how symmetries, which take physical states to physical states, can emerge from a diffeomorphism invariant formulation of quantum gravity. © 2006 Published by Elsevier B.V.

1. Introduction

This paper is concerned with two distinct problems in background independent formulations of quantum gravity. First, how one can recover the semiclassical Bekenstein–Hawking results [1,2] in the context of results about the entropy of horizons and surfaces in loop quantum gravity [3–6]. Second, how the description of spacetime in terms of classical general relativity emerges from the quantum theory. Up till now, these problems have been treated in isolation. Here we propose a perspective that relates them, and also addresses a technical issue concerning the black hole entropy. This technical issue concerns the value of a certain free parameter, γ , in the theory called the Immirzi parameter [7]. This parameter, not present in the classical theory, arises from an

* Corresponding author. E-mail addresses: [email protected] (O. Dreyer), [email protected] (F. Markopoulou), [email protected] (L. Smolin).

0550-3213/$ – see front matter © 2006 Published by Elsevier B.V. doi:10.1016/j.nuclphysb.2006.02.045 2 O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13 ambiguity in the quantization procedure because there is a choice in the connection variable that is used in the construction of the quantum observable algebra. As it does not appear in the classical theory, its value should be fixed by some physical requirement. In previous work [3–5], it was shown that γ is fixed by the requirement that the quantum gravity computation reproduce the Hawking value of the entropy. Since the story is very simple and physical, we review it for non-experts. The Immirzi parameter comes into the formula for the area of a surface. When a surface is punctured by a set of spin network edges with spin labels {jα}, then the area of the surface is given by A {jα} = 8πγ jα(jα + 1), (1) α 2 which we take here to be measured in units of the Planck area lPl. In loop quantum gravity, there is an exact description of the quantum geometry of black hole horizons [5], as well as a more general class of boundaries [3]. The entropy of the horizon is defined in terms of the Hilbert space of the boundary theory. For reasons that we spell out shortly, it is taken to be the logarithm of the dimension of the Hilbert space of boundary states. The Immirzi parameter does not come into the entropy. Hence it appears in the ratio of the entropy S[A] of a black hole to its area: S[A] c R = = , (2) A[{jα}] γ where c is a constant to be determined by calculating the entropy. At the same time, we believe from Hawking’s semiclassical calculation of black hole entropy that [2] 1 R = . (3) 4 For the two equations to agree, γ = 4c. (4) Recently, one of us, following a clue uncovered by Hod [8], described a semiclassical argument which fixes γ by an argument that appears to be independent of considerations of entropy or radiation but rather makes use of the quasinormal mode spectrum [6]. As it is relevant, we repeat here the basic argument. The quasinormal mode spectrum of the Schwarzschild black hole turns out to have an asymptotic form ln(3) i 1 Mω = + n + , (5) 8π 4 2 for arbitrarily large integer n.Asn goes to infinity, the decay times of the modes goes to zero in units of the light crossing time of the horizon. This means that the excitations involve more and more local regions of the horizon. At the same time, the real part of the frequency goes to the asymptotic value [9] ln(3) ω = . (6) qnm 8πM This frequency is then associated with an arbitrarily short lived, and hence local, excitation of the horizon. By the correspondence principle, this must correspond to an energy ln(3)h¯ M = hω¯ = . (7) qnm qnm 8πM O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13 3

In the classical theory, the region of the horizon excited may be arbitrarily small, but in the quantum theory there is a smallest region that can be excited, which is a minimal puncture with minimal area A = 8πγ jmin(jmin + 1). (8) However, for a Schwarzschild black hole,

A = 16πM2, (9) from which it follows that, for a quasinormal mode, = = 2 Aqnm 32πMM 4ln(3)lPl. (10) The result is a prediction for the Immirzi parameter, which is ln(3) γ qnm = √ . (11) 2π jmin(jmin + 1) Does this match the value needed to get R = 1/4? It turns out to depend on what we take for the Hilbert space of the black hole horizon. By arguments given in [5], we know that the horizon Hilbert space, for fixed area A in Planck units, is related to the Hilbert space of the 2 { } V 1 invariant states of U(1) Chern–Simons theory on an S with punctures jα , denoted by {jα}. The level k is related to the mass of the black hole, and is assumed here to be large. Given a set of punctures {j}, V = H {jα} Inv jα (12) α where Hj is the (2j + 1)-dimensional spin space for spin j and Inv means the Hilbert states contains only states with total spin zero. We know that the Schwarzschild black hole has definite area A[{j}] and definite energy:

A({j}) M {j} = . (13) 16π As the area is quantized, we see that the energy is quantized as well. V We then expect that at least some of the Hilbert spaces {jα} contain states which correspond to Schwarzschild black holes. The state is expected to be thermal, because it is entangled with the states of radiation that has left the black hole. We are, however, ignorant of the exact state so, instead, the black hole entropy is usually taken to be Schwarzscild = HSchwarzschild S ln dim A , (14) HSchwarzschild Schwarzscild where A is a Hilbert space on which the density matrix ρA describing the ensemble of Schwarzschild black holes may be expected to be non-degenerate. One proposal is to take HSchwarzschild = HAll = V A A {j}, (15) A({j})=A

1 There is an equivalent description in terms of SU(2) Chern–Simons theory [3]. 4 O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13 where we use the superscript All to remind us that in this case we sum over all sets of punctures that give an area between A and A + A, for some small A. However, this proposal can be criticized on the basis that the only information about the Schwarzschild black hole that is used is its area. We may expect that the actual state of the Schwarzschild black hole is determined by additional physical considerations and hence is non- degenerate only in a subspace HSchwarzschild HAll A A (16) that is picked out by additional physical input. In this article, we will argue that the physical Hilbert space associated to the horizon of a Schwarzschild black hole is dominated by the V{j} for which all punctures have equal minimal spin jmin. The difference is important, because in the ﬁrst case we have ln dim HAll RAll = A , (17) A({j}) from which recent calculations [10] have deduced the value2

γ All = 0.23753295796592 ... . (18) On the other hand, in the second case

ln dim V{j ,...,j } Rmin = min min . (19) A({j}) This is easy to compute [3,6] and leads to a value ln(3) γ min = √ . (20) 2π jmin(jmin + 1) We note that

γ qnm = γ min = γ All. (21) Hence, we have the following situation. If indeed the right choice for the Hilbert space of a Schwarzschild black hole consists only of states with equal and minimal punctures, then there is a remarkable agreement between the two computations. This supports the case that there is something physically correct about the description of black holes in loop quantum gravity. HAll On the other hand, if A is the right Hilbert space for the horizon of a Schwarzschild black hole, loop quantum gravity is in deep trouble, because there is no choice of γ that will agree with both the Hawking entropy and the quasi-normal mode spectrum. It is clear that to resolve this problem we need additional physical input. The only property of the Schwarzschild black hole used in previous work to constrain the corresponding quantum state is its area. But a more complete treatment should take into account other characteristics of the Schwarzschild black hole such as its symmetry and stability. To do so, we need to understand how those classical properties can emerge from a quantum state, described so far in the background independent language of loop quantum gravity.

2 But see [17] for a criticism of that calculation. O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13 5

Recently, one of us proposed a new perspective on the general problem of the emergence of particles and other semiclassical states from background independent approaches to quantum gravity [13]. This makes use of the concept of noiseless subsystems, developed in the context of quantum information theory to describe how particle-like states may emerge there [12]. We shall see in the following that the black hole problem provides a nice example of the general strategy proposed in [13], while it resolves the present problem.3

2. Symmetry and noiseless subsystems

We begin by asking two questions:

HSchwarzschild (1) How do we ﬁnd the subspace A corresponding to a Schwarzschild black hole, as opposed to a general surface of a given area that satisﬁes the appropriate boundary conditions? (2) How do we recognize in that subspace the excitations that, in the classical limit, correspond to the quasinormal modes?

We should here make an important comment, which clariﬁes the sense in which these questions are asked. We note that the boundary conditions we require for our analysis are more general and apply to all horizons, not just Schwarzschild black holes. Thus, our framework differs from the isolated horizon framework [5] in one crucial aspect. We posit that the states of all quantum black holes live in a single Hilbert space, so that angular momentum and other multiple moments are to be measured by quantum operators. In the isolated horizon picture the angular momentum and multiple moments are treated classically, and the Hilbert space is deﬁned for each value of them. Only in the our framework does it make sense to ask how the states corresponding to a non-rotating black hole emerge from a more general Hilbert space of horizon states. Our two questions are then analogous to problems concerned with the emergence and stability of persistent quantum states in condensed matter physics and quantum information theory. For the present purposes, we will use the following idea from the theory of noiseless subsystems in quantum information theory. We have a complete quantum experiment, which we want to divide into the system S and environment E. We want to understand what quantum properties of the system may survive stably in spite of continual and uncontrollable interactions with the environment. The joint Hilbert space decomposes into the product of system and environment, S E Htotal = H ⊗ H , (22) while the Hamiltonian decomposes into the sum S E H = H + H + H int (23) where H S acts only on the system, H E acts only on the environment and all the interactions int between them are contained in H . The reduced dynamics of S is given by a completely positive operator φ : HS → HS , S = S = S ⊗ E † ρf φ ρi trE U ρi ρi U , (24)

3 While this paper was in draft, a paper was posted that reaches a similar conclusion by a different argument [11].More recently, additional arguments for it have been given by [18]. 6 O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13 where the joint state of S and E evolves unitarily and then the environment is traced out. While a generic “noise” φ will affect the entire state space HS , there may be a noiseless subsystem of S, and possibly even a subspace of HS that is left unchanged by φ, i.e., evolves unitarily. What follows is a necessary and sufﬁcient condition for the existence of a noiseless subsystem. Eq. (24) can be rewritten as S = S † φ ρ Akρ Ak, (25) k where

ψi|Ak|ψf =ψi|⊗ek|U|ψf ⊗|ek , (26) | | ∈HS {| } HE for any ψi , ψf and ek an orthonormal basis on . One can check that † = k AkAk 1. If B(HS ) is the algebra of all operators acting on HS ,theinteraction algebra Aint ⊆ B(HS ) int is the subalgebra generated by the Ak (assuming that A is closed under † and φ is unital). int Up to a unitary transformation, A can be written as a direct sum of dj × dj complex matrix algebras, each of which appear with multiplicity μj :

Aint ⊗ B dj 1μj C , (27) j

μj where 1μj is the identity operator on C . The commutant, Aint of Aint is the set of all operators in B(HS ) that commute with every element of Aint, i.e., Aint B μj ⊗ C 1dj . (28) j This decomposition induces a natural decomposition of HS :

S H = Cμj ⊗ Cdj . (29) j

int Note now that any state ρ in A is a fixed point of φ since it commutes with all the Ak: S = S † = † S = S φ ρ Akρ Ak AkAkρ ρ . (30) k k It can be shown [12] that the reverse also holds, i.e., S φ ρ = ρ ⇔ ρ ∈ Aint . (31) Hence, the noiseless subsystem can be identified with the Cμj in Eq. (29). These are relevant for the physical description of the system, because the interactions with the environment will not disturb them and hence are useful for describing the long term behavior of the system, because they are conserved. The relevance of this argument for quantum gravity was proposed in [13]. If we divide the quantum state of the gravitational field arbitrarily into subsystems, those properties which are conserved under interactions between the subsystems are going to characterize the low energy limit of the spacetime geometry. If classical spacetime physics emerges in the low energy limit, in which these regions are arbitrarily large on Planck O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13 7 scale, the commutant of the interaction algebra should include the symmetries that characterize classical spacetime in the ground state (presumably the Poincaré or the de Sitter group, or some deformation of them). In the case of vanishing cosmological constant, we then expect that the low energy limit is characterized by representations of the Poincaré group. But these, of course, are the elementary particles, as described in quantum field theory on Minkowski spacetime. The noiseless subsystem idea is then a way to understand how particle states, and the low energy vacuum could emerge from a background independent quantum theory of gravity. In the following section, the system state space is that of the states of loop quantum gravity that correspond to a black hole of a given area. This is coupled to an environment of bulk spin network states and, while we do not know the quantum dynamics, we do know that a Schwarzschild black hole has a classical SO(3) symmetry. In the next section, we make the reasonable assumption that there is a microscopic analogue of SO(3) that acts on the system, a symmetry that should be contained in the commutant of the interaction algebra for black holes.

3. Application to black holes

HSchwarzschild In this section we shall look for the subspace A of states that correspond to a HSchwarzschild classical Schwarzschild black hole. We want to show that A is a proper subspace of HAll ± the space A of states of horizons with area A A, i.e., HSchwarzschild HAll A A . (32) The geometry of a horizon is only partially fixed by its area A. A Schwarzschild horizon is a very special example of such a geometry that is distinguished by its SO(3) symmetry. Since all the HAll other horizon geometries have quantum counterparts in A as well, we expect that the Hilbert HSchwarzschild HAll space A corresponding to a Schwarzschild black hole is a proper subspace of A . In contrast to the smooth horizon geometry of a classical Schwarzschild black hole, the geometry of the corresponding quantum horizon is concentrated at the punctures and is thus discrete. HSchwarzschild Consequently, we cannot expect an action of the smooth Lie group SO(3) on A . What we can expect is an action of some discrete symmetry group Gq that only on the classical level coincides with the action of SO(3) (see Fig. 1). What can reasonably be assumed about the group Gq ? Since Gq is a group of symmetries we assume it commutes with the action of the Hamiltonian H int on the horizon Hilbert space HS , i.e., int Gq ⊂ A . (33) We do not know the precise action of the Hamiltonian H int but it is possible to restrict it sufficiently to compute the commutant Gq . We first review the basics of the description of black hole horizons in loop quantum gravity [5]. The Hilbert space is at the kinematical level of the form (22), where the environmental Hilbert space has a basis consisting of all spin networks that end on the horizon, in any number of punctures. Given that the energy of a black hole is a function of its area, it makes sense to decompose the Hilbert space in eigenspaces of the area operator. The eigenvalues are given by sets of spins at the punctures {ji}, where i labels the punctures. Gauss’s law acting at the boundary enforces that the labels on the punctures on the horizon match the spins of the edges of spin network in the bulk, incident on the horizon. We then have S E Htotal = H ⊗ H . (34) {ji } {ji } {ji } 8 O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13

Fig. 1. The geometry of a quantum horizon is concentrated at a discrete set of punctures. The classical symmetry group SO(3) will thus not act directly on the quantum states representing a Schwarzschild black hole. A discrete group Gq ,on the other hand, is expected to act on these states. In the classical limit, the action of this discrete group coincides with that of SO(3). For a large number of punctures, the discrete group action shown above would approximate a rotation of the horizon.

The environment (sometimes called “bulk”) Hilbert spaces HE have bases consisting of dif- {ji } feomorphism classes of spin networks in the bulk, with edges ending on the boundary at the punctures with the speciﬁed spin labels. We note that the diffeomorphisms on the surface are ﬁxed. (There may also be an exterior boundary at which the diffeomorphisms are also restricted.) The system, or surface, Hilbert spaces HS are direct sums of the one-dimensional Hilbert {ji } spaces of U(1) Chern–Simons theory on the sphere, with punctures labeled by charges mi at 2 points σi ∈ S , subject to the conditions | | mi ji, (35) mi = 0. (36) i

The level k of each Chern–Simons theory depends on the ji ’s and is given by [5] a [{j }] k = 0 i (37) 4πγ where a0[{ji}] is the nearest number to A[{ji}] such that the level k is an integer. We note that the level k is very large for black holes large in Planck units, so that for some estimates the limit k →∞can be taken. This is normally done in the computation of the entropy, for which the difference between the classical and quantum dimensions may be neglected. The U(1) connection on the boundary satisﬁes 4π F (σ ) = m δ2(σ, σ ). (38) ij k i i i We then can use a basis of the boundary theory, which is labeled (j1,m1), (j2,m2), . . . , (jN ,mN ) . (39) We can now turn to the speciﬁcation of the interaction algebra. We begin by specifying a general interaction algebra, which consists of all possible interactions between an environment and a quantum horizon. Then we will specialize to a subalgebra which excludes the case of an O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13 9 asymmetric environment or external geometry. The general interaction algebra, Agint,mustbe generated by operators that

(1) act simultaneously on the environment and system Hilbert space; (2) act locally.

A sufﬁcient set of generators for Agint consists of:

• addition or removal of a puncture; • braidings of the punctures.

The Gauss’ law constraint that ties the labels on punctures to the labels on edges of spin networks that meet them implies that each of these involve changes to both the surface and environment state. We note that the braidings generate a group, called the braid group and together with adding and removing punctures, these generate the tangle algebra discussed by Baez in [15]. Let us consider, for example, a generator in Agint that corresponds to braiding two punctures, which we will label i = 1, 2. It has the effect of rotating, with a positive orientation, the two punctures around each other, returning them to their original positions. This braids the two edges of the bulk spin network, which changes the diffeomorphism class of the bulk state. The action of H int on HS due to a braiding of the punctures can be shown to modify each basis state (39) by a phase. As the punctures are unchanged, it is represented by an operator Bˆ in HS . A simple computation in the quantum Chern–Simons theory [14] shows that this is 12 {ji } realized by ˆ B12 ◦ (j1,m1), (j2,m2),...,(jN ,mN ) + 2πı(m1 m2 = e k (j1,m1), (j2,m2),...,(jN ,mN ) . (40) The general interaction algebra does not take into account any symmetries, hence it describes the general case, in which the black hole horizon may have non-vanishing multiple moments.4 If we want to specialize to the case of a black hole with symmetries, we must impose conditions on the interaction algebra, for a symmetric subsystem cannot exist stably in an asymmetric environment. In principle we could code various kinds of symmetries in the choice of interaction algebra. But it sufﬁces to consider the simplest choice, which is the subalgebra Aint ⊂ Agint which consists of adding and removing punctures plus a symmetric sum of braiding operations, ˆT ˆ B = Bij , (41) i

P1. The elements of Gq do not change the area of the horizon.

4 In the isolated horizon approach [5] angular momentum and multiple moments are coded as classical parameters. Here we seek instead to deﬁne different subspaces of the horizon Hilbert space associated with black holes with different macroscopic properties. 10 O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13

5 We take this to mean that the elements of Gq commute with the area operator. This is trivially true for the action of SO(3) on the classical spacetime and we assume that it is also true for the action of G . Because of the property P1, the elements of G will map the spaces HS into q q {ji } themselves. A necessary condition for Eq. (33) to hold, i.e., for Gq to commute with the action of H int, is that G commutes with the action of H int on each of the spaces HS . q {ji } On each HS the braiding generators in H int act as shown in Eq. (40). The commutant of {ji } int A is then easy to ﬁnd. It is generated by all those operators that just permute the mi values for given spins ji : ...,(j,m1), . . . (j, mK ), . . . −→ ...,(j,mπ(1)),...(j,mπ(K)),..., (42) for some π ∈ PK , the permutation group of K objects. Assuming Gq to be unitary we then have ⊂ Gq PKj , (43) j where the product is over all the different j’s in {ji}. We now assume that Gq also shares the following property with SO(3):

P2. Gq acts transitively on the punctures, i.e., for every two punctures there is an element in Gq that connects the two.

Because of Eq. (43), we know that Gq is a subgroup of = P PKj . (44) j

For Gq to act transitively, it is necessary that the bigger group P acts transitively on the punctures. Since P is the product of permutation groups, this is only possible if P coincides with just one permutation group. It follows that there is one j such that

Kj = N. (45) HSchwarzschild V All the punctures have the same spin j and the Hilbert space A consists of those {j} for which all the j’s coincide. It is then straightforward to show that the dimension of the V{j} varies sharply with the spin j and is dominated by the lowest spin jmin.

4. Conclusions

In this paper, we used a microscopic analogue of the classical SO(3) symmetry of a Schwarz- schild black hole to restrict the loop quantum gravity state space HA of black holes of area A to a smaller subspace, left invariant by the symmetry. The method we used is the noiseless subsystems of quantum information theory, in which the symmetry of the dynamics implies a non-trivial commutant of the interaction algebra of the system. The construction of the state space HA in loop quantum gravity is a hybrid construction that requires the imposition of black hole horizon conditions at the classical level. One eventually

5 In principle we could consider a weaker requirement which is that the elements of Gq do not change the expectation value of the area. This would be an interesting extension of the usual problem to study. O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13 11 wants to be able to identify black hole states directly in the quantum theory and derive the classical geometry in the appropriate classical limit. While the present work has the same hybrid character (we start from HA), it indicates that properties of the quantum states can be inferred from the dynamics algebraically, without need for a classical geometry. This approach also clarifies the important but subtle question of how symmetries can arise from a diffeomorphism invariant state. All the states discussed here are invariant under spatial diffeomorphisms. The point is that the commutant, Gq takes physical, diffeomorphism invariant states, to other distinct physical, diffeomorphism invariant, states. The transitive permutations that we discuss translate the black hole horizon with respect to the spin networks that define the external geometry. This is physically meaningful because these transformations are not subgroups of the diffeomorphisms, instead they act on the space of diffeomorphism invariant states. Thus, we have a realization of the proposal in [13] that in a diffeomorphism invariant theory a symmetry can only arise as a motion of one subsystem with respect to another, and its generators must live in the commutant of the interaction algebra, defined by the splitting of the universe into subsystems. This proposal has ramifications that might lead to a better understanding of how the classical properties of black holes arise from the quantum geometry of the horizon. We also see in a simple example, how properties of classical spacetime geometries can emerge from exact quantum geometries, by making use of the insights gained from the study of similar questions in quantum information theory. We close by mentioning several queries that may be made regarding the construction used here.

• What about the connection to quasi-normal modes? Can we understand it as other than a coincidence? It may be possible to understand how the quasi-normal modes arise from the states which transform non-trivially under those generators of the commutant that become rotations in the A →∞limit. • What about rotating black holes? It is of interest to use these methods to characterize the states of a rotating black hole, or a horizon with multiple moments, in terms of the same language. We expect that this will involve other choices of the interaction algebra than that made here. We may note that attempts to extend the connection between quasi-normal modes and entropy to rotating black holes have not succeeded; this is something that needs explanation. Note also that while we employ here the same boundary conditions as the isolated horizon formalism, the intension is different. In particular, in the isolated horizon picture, rotation and other multiple moments are ﬁxed in the classical description. The isolated horizon boundary conditions do not change, but the functions on the phase space that correspond to different observables will depend on the multiple moments. In our picture, we seek to recover the generators of rotation, and hence the conjugate conserved quantities, as operators in a single Hilbert space. Hence here, unlike the pure isolated horizon case, we expect that a single Hilbert space contains the states corresponding to black holes with all values of angular momentum. • HAll HSchwarzchild What about the states in A that are not in A ? These are states whose pictures give the right area, but are not in the noiseless subsystem we identify. But they arose from a quantization of the isolated horizon boundary conditions. We cannot give a deﬁnitive answer without more dynamical input, particularly a Hamiltonian on the horizon states. But we can argue that they must correspond to either non-static, classical metrics or they correspond to no classical geometry. 12 O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13

Given that the boundary conditions do not distinguish rotating from non-rotating black holes, some of theses states must be associated with rotating black holes. Apart from these, it must be that most of these states are associated with semiclassical states of non-stationary horizons, corresponding to distorted black holes. These will be excited, transient states of the black hole, which classically decay to stationary states, emitting gravitational radiation. How do we know HAll HAll that such states are included in A ? All that distinguishes states in A is that there is a horizon of a particular area. This will include all such configurations, whether belonging to stationary states, semiclassical states corresponding to non-stationary horizons as well as quantum states that have no semiclassical approximation. In particular, these must include states corresponding to quasi normal modes. Beyond those categories, our argument implies that the remaining states couple strongly to the random noise coming from the environment of the black hole. They are then states that cannot play a role in the semiclassical limit. The only definitive way to distinguish the different kinds of states is dynamically. If we had an effective Hamiltonian for the horizon and near horizon geometry, then the non-stationary states would have to have higher energy than the stationary states, corresponding to their potential for radiating energy to infinity in gravitational radiation. Hence, we can deduce from the fact that in the classical theory non-stationary states radiate, that those states would be suppressed in a HAll Boltzman weighted, equilibrium ensemble. In the ensemble A they appear unsuppressed, but that is only because no dynamics are invoked, so states of different energies are counted as having the same weight. Given that there are many more non-stationary classical configurations than stationary ones, it is reasonable that, for a given horizon area, those states that correspond to static black holes must HAll HSchwarzchild appear to be a minority, as is the case if we compare A with its subspace A .But, HAll HSchwarzchild it is incorrect to argue just on the basis of the fact that A is larger than A , that a typical state corresponding to an equilibrium, static black hole will be in the latter rather than the former, because dynamics has yet to be invoked. The point of the construction of this paper is then that, even in the absence of an explicit Hamil- tonian, we can use symmetry properties of the Hamiltonian to apply an argument from quantum information to deduce which subspace of states will emerge in the low energy limit as corresponding to stationary configurations. The fact that, in the absence of dynamics, these are a minority of the ensemble of states with a given area must be expected and is not an objection against the construction presented here. • Finally, we note that the same method could be applied within quantum cosmology, to study how states that correspond to symmetric universes emerge out of a more general Hilbert space, corresponding to the full theory [16].

Acknowledgements

We are very grateful to Paolo Zanardi for pointing out an oversight in an earlier version of this paper. We also thank Abhay Ashtekar and Jerzy Lewandowski for very useful discussions clarifying the treatment of symmetries in the isolated horizon formalism.

References

[1] J.D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333. [2] S.W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 (1975) 199. O. Dreyer et al. / Nuclear Physics B 744 (2006) 1–13 13

[3] L. Smolin, Linking topological quantum ﬁeld theory and nonperturbative quantum gravity, J. Math. Phys. 36 (1995) 6417, gr-qc/9505028. [4] K. Krasnov, On quantum statistical mechanics of a Schwarzschild black hole, Gen. Relativ. Gravit. 30 (1998) 53–68, gr-qc/9605047; C. Rovelli, Black hole entropy from loop quantum gravity, gr-qc/9603063. [5] A. Ashtekar, J. Baez, K. Krasnov, Quantum geometry of isolated horizons and black hole entropy, gr-qc/0005126; A. Ashtekar, J. Baez, A. Corichi, K. Krasnov, Quantum geometry and black hole entropy, Phys. Rev. Lett. 80 (1998) 904–907, gr-qc/9710007. [6] O. Dreyer, Quasinormal modes, the area spectrum, and black hole entropy, Phys. Rev. Lett. 90 (2003) 081301, gr-qc/0211076. [7] G. Immirzi, Nucl. Phys. B (Proc. Suppl.) 57 (1997) 65, gr-qc/9701052. [8] S. Hod, Phys. Rev. Lett. 81 (1998) 4293, gr-qc/9812002. [9] L. Motl, An analytical computation of asymptotic Schwarzschild quasinormal frequencies, gr-qc/0212096; L. Motl, A. Neitzke, Asymptotic black hole quasinormal frequencies, hep-th/0301173. [10] K.A. Meissner, Black hole entropy in loop quantum gravity, gr-qc/0407052; M. Domagala, J. Lewandowski, Black hole entropy from quantum geometry, gr-qc/0407051. [11] S. Alexandrov, On the counting of black hole states in loop quantum gravity, gr-qc/0408033. [12] P. Zanardi, M. Rasetti, Phys. Rev. Lett. 79 (1997) 3306; E. Knill, R. Laﬂamme, L. Viola, Phys. Rev. Lett. 84 (2000) 2525; J. Kempe, D. Bacon, D.A. Lidar, K.B. Whaley, Phys. Rev. A 63 (2001) 042307. [13] F. Markopoulou, D. Poulin, in preparation. [14] D.P. Arovas, R. Schrieffer, F. Wilczek, A. Zee, Statistical mechanics of anyons, Nucl. Phys. B 251 (1985) 117; S. Rao, Anyon primer, hep-th/9209066. [15] J. Baez, Quantum gravity and the algebra of tangles, Class. Quantum Grav. 10 (1993) 673–694, hep-th/9205007. [16] E. Poisson, personal communication. [17] T. Tamaki, H. Nomura, Ambiguity of black hole entropy in loop quantum gravity, hep-th/0508142. [18] E.R. Livine, D.R. Terno, Quantum black holes: Entropy and entanglement on the horizon, gr-qc/0508085. Nuclear Physics B 744 (2006) 14–33

Finite volume effects for the pion mass at two loops

Gilberto Colangelo ∗, Christoph Haefeli

Institut für Theoretische Physik, Universität Bern, Sidlerstr. 5, 3012 Bern, Switzerland Received 16 February 2006; accepted 9 March 2006 Available online 30 March 2006

Abstract We evaluate the pion mass in finite volume to two loops within chiral perturbation theory. The results are compared with a recently proposed extension of the asymptotic formula of Lüscher. We find that contributions, which were neglected in the latter, are numerically very small at the two-loop level and conclude that for Mπ L 2, L 2 fm the finite volume effects in the meson sector are analytically well under control. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

Numerical simulations in lattice QCD are bound to rather small volumes. When one determines the hadron spectrum and other low energy observables one has to understand and properly account for the volume dependence in order to correctly interpret the numerical data. Analytical methods which allow one to predict the size of the finite volume effects are particularly useful in this respect. In the case of hadron masses, there are two different methods to analytically evaluate the finite volume effects: the asymptotic formula derived by Lüscher [1], and chiral perturbation theory (ChPT) in finite volume [2–4]. Lüscher’s formula relates the volume dependence of the mass of a hadron to an integral over the π-hadron forward scattering amplitude. Knowledge of the latter scattering amplitude immediately translates into an estimate of finite volume effects for the hadron mass. Since the integral is dominated by the low-energy region, one can rely on the chiral representation of the scattering amplitude in order to numerically evaluate the integral. In this manner one makes use of ChPT only in infinite volume and obtains an estimate only of the leading exponential term in the finite volume dependence, the term of the order exp(−Mπ L), where L is the box size. Alternatively one can perform the calculation of the hadron mass in

* Corresponding author. E-mail address: [email protected] (G. Colangelo).

ChPT in finite volume and obtains an estimate also of the terms which are exponentially subleading. A detailed study and comparison of the two approaches has been performed in [5,6] for the case of the pion mass. This case is particularly interesting both because it is a simple hadron to be studied on the lattice and because the ππ scattering amplitude, which is needed in the Lüscher formula, is now known to next-to-next-to-leading order [7–10] in the chiral expansion. It was therefore possible to study numerically how well the leading exponential term dominates the series, and how fast the chiral expansion (of the leading exponential term) converges. The somewhat surprising result was that the leading term in both series receives large corrections both from subleading exponentials as well as from the next-to-leading order in the chiral expansion. The next-to-next-to-leading order chiral correction, on the other hand, was found to be rather small, which indicates that—at this level—the chiral series has started converging well. The fact that the Lüscher formula appeared to give the numerically dominating term only for volumes so large that the finite volume correction itself has become negligible appeared to be more worrisome. At first sight this might have lead one to conclude that in cases of practical interest one could not rely on this extremely convenient formula in order to evaluate reliably the finite volume effects. We believe, however, that such a negative conclusion is unjustified and have proposed a resummation of the Lüscher formula [11,12] which retains all the convenience of the original one but does not suffer from the same large corrections. This resummation can be understood in simple terms: Lüscher has shown that the leading exponential correction comes from a radiative correction in which the emitted virtual light particle (the pion, in QCD) goes around the world once (thanks to the periodic boundary conditions) before being reabsorbed. The resummed formula takes into account all other possible ways which the pion has to go around the world (go along the diagonal, or go around more than once, etc.). As was shown in [12] this resummed formula exactly reproduces the one-loop ChPT calculation of the pion mass in finite volume if one inserts in the integral the leading chiral representation of the ππ scattering amplitude. Inserting the next-to-leading chiral representation of the scattering amplitude one reproduces the full two-loop calculation√ of the pion√ mass in ChPT in finite ¯ ¯ volume up to corrections of order exp(−ML) with M ( 3 + 1)/ 2Mπ . Despite this improvement of the algebraic accuracy of the resummed formula with respect to the Lüscher’s one, it is essential to show that the resummation is numerically effective and that the corrections to it are small. In order to make this check we have now performed a full two loop calculation of the pion mass in finite volume in ChPT. As we will demonstrate in this article, the corrections which are not captured by the resummed asymptotic formula are negligibly small for Mπ L 2. The results of the present two-loop calculation were anticipated in [13,14]. Here, we give further details about the calculation and the results. One of the reasons for performing this further investigation of finite volume effects for hadron masses is that one may view Lüscher’s formula (or its resummed version) as a way to determine on the lattice a scattering amplitude, cf. [12]. While it is true that the scattering amplitude is seen here in an exponentially suppressed effect, it is also true that the direct calculation of a scattering amplitude on the lattice is much more difficult than that of a mass. This indirect method to extract a scattering amplitude may in some cases turn out to be more practical, and we find it worthwhile to discuss its theoretical feasibility. Indeed if the part of the finite volume corrections which is related to the scattering amplitude is not strongly dominating with respect to the rest, this is not a viable method to determine the scattering amplitude. The conclusion reached in this paper is therefore encouraging in this respect and can be used to estimate in which region of the (Mπ ,L) 16 G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 plane the finite volume correction to a hadron mass is given to a good approximation by the integral containing the scattering amplitude. The two-loop calculation is interesting in its own right, also from a technical viewpoint. To date, a number of finite volume calculations have been performed at one-loop order [15–21],but as far as we know the only finite volume two-loop calculation which has been performed until now is for the quark condensate [22], which is a much simpler calculation. In the related context of finite temperature field theory there are a few examples of calculations beyond one loop [23, 24]. We wish to briefly mention related work. Most notably, the asymptotic formula may also be applied to the nucleon mass [25], see Koma and Koma [26] as well as [14] for recent work. Braun, Pirner and Klein have evaluated the volume dependence of the pion mass based on a quark–meson model [27]. In the framework of a lattice regularized ChPT finite volume effects have been addressed by Borasoy et al. in Ref. [28]. The outline of the article is as follows. In Section 2 we set the notation and remind of the basic assumptions for an application of ChPT in finite volume. Section 3 is devoted to outline the calculation and state the main results. In Section 4 we show the explicit expressions which have been used for the numerical analysis in Section 5. We conclude with a summary.

2. Preliminaries

In this section we shall set the notation and the basic deﬁnitions for the two-loop calculation.

2.1. ChPT in ﬁnite and in inﬁnite volume

Chiral perturbation theory (ChPT) is the effective theory for QCD at low energies. If we first restrict ourselves to the infinite volume case, the effective Lagrangian of QCD for two light flavours at low energies consists of an infinite number of terms [7],

Leff = L2 + L4 + L6 +···. (1)

As we wish to calculate the pion mass, an on-shell quantity, external ﬁelds can be dropped in Leff. We work in the isospin symmetry limit mu = md in Euclidean space–time, and for the choice of the pion ﬁelds we use the nonlinear sigma model parameterization. We have

F 2 L = u u − χ+, (2) 2 4 μ μ with

φ φ2 U = σ + i ,σ2 + = , 2 1 F √ F π 0 2π + φ = √ = φiτ i, 2π − −π 0 = † † =− † = † = † † + † uμ iu ∂μUu iu∂μU u uμ,χ+ u χu uχ u, 1 χ = 2Bmˆ 1, mˆ = (m + m ), (3) 2 u d G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 17

2 with u = U. The symbol A denotes the trace of the two-by-two matrix A. The Lagrangian L4 can be written as [7] 4 L4 = iPi +···, (4) i=1 where 1 1 P =− u u 2,P=− u u 2, 1 4 μ μ 2 4 μ ν 1 2 i P =− χ+ ,P= u χ− , (5) 3 16 4 4 μ μ with † † † χ−μ = u ∂μχu − u∂μχ u. (6) The ellipsis in Eq. (4) denotes terms that do not contribute to the pion mass. The low energy constants i are divergent and remove the ultraviolet divergences generated by one-loop graphs from L2. The complete effective Lagrangian L6 with its divergence structure at d = 4 has been constructed in [29,30]. As will be discussed in Section 3, terms from L6 merely renormalize the pion mass in infinite volume and do not contribute to finite volume corrections which we are interested in. Thus, we refrain from showing it here. Given the effective Lagrangian and the parameterization for the pion fields, it is straightforward to calculate the pion mass to two-loops. We refer to [9], where one also finds a detailed discussion of the renormalization procedure (two-loop diagrams in ChPT are discussed in [31]). The effective framework is still appropriate, when the system is enclosed by a large box of size V = L3. We refer to the literature for the foundations [2–4] and a recent review [11]. Here, we only recall a few fundamental results which guided the present calculation: the volume has to be large enough, such that ChPT can give reliable results, 2Fπ L 1. The value of the parameter Mπ L determines the power counting for the perturbative calculation: if Mπ L 1 one is in the “p-regime” in which 1/L counts as a small quantity of order Mπ .IfMπ L 1 one is in the “ - 2 regime” and 1/L is a quantity of order Mπ . In both cases the effective Lagrangian is the same as in the infinite volume. In this article we only consider the “p-regime”, where the system is distorted mildly and the only change brought about by the finite volume is a modification of the pion propagator due to the periodic boundary conditions of the pion fields1 0 0 G x , x = G0 x , x + nL , (7) n∈Z3 with G0(x) the propagator in infinite volume.

2.2. Basic deﬁnitions

In Euclidean space–time the propagator is deﬁned through the connected correlation function ab = a b G(x)δ φ (x)φ (0) L, (8)

1 Throughout we denote by the volume the three-dimensional volume V = L3, whereas the time direction is not compactified. 18 G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 where a, b stand for isospin indices of the pion fields and the subscript L in the correlation function denotes that it is evaluated in finite volume. We also have

0 − dp φ1(x)φ1(0) = L 3 eipx G p0, p , L 2π p 0 −1 2 2 0 G p , p = M + p − ΣL p , p , (9) where the momenta p can only have discrete values 2π p = n, n ∈ Z3, (10) L and M2 = 2Bmˆ is the tree-level pion mass in infinite volume. The pion mass in finite volume MπL is now defined by the pole equation −1 G(pˆL) = 0, for pˆL = (iMπL, 0). (11)

3. Outline of calculation and statement of results

For a large volume, the ﬁnite size effects are expected to be small, such that the pole equation can be solved perturbatively. We outline the calculation and proceed with the main results. More details about the calculation are relegated to later sections or Appendices A and B.

3.1. One-loop result

Since the effective Lagrangian remains unchanged when going to the ﬁnite volume, we can immediately write down the Feynman diagrams which contribute to the self-energy at two-loops, see Fig. 1, the only difference with respect to an inﬁnite volume calculation is that the propagators

Fig. 1. Self-energy graphs to two-loops in ChPT. A spline corresponds to a periodified propagator, whereas those without correspond to an infinite volume propagator, cf. Eq. (7). Normal vertices come from L2, squared vertices from L4 and the circle-crossed from L6. G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 19 need to be periodified, cf. Eq. (7). At leading order, the graphs 1(a) and 1(e) need to be evaluated and the self-energy admits the form 1 3 M4 1 Σ p0, = G( ) − M2 − p2 − π + O , L p 2 0 π 2 3 2 4 (12) Fπ 2 Fπ Fπ with G(0) the value of the finite volume propagator at the origin. It contains a logarithmic divergence due to the contribution from the term n = 0 in Eq. (7). In dimensional regularization

1 1 (1 − d/2) − G (0) = dd p = Md 2. (13) 0 (2π)d p2 + M2 (4π)d/2 The remaining terms with n = 0 are finite and may be expressed in terms of a kinematical func- 2 2 tion g1(M , 0,L), 2 G(0) = G0(0) + g1 M , 0,L , ∞ 2 2 dτ − 2 n L g M2, ,L = e τM − . 1 0 d/2 exp (14) (4πτ) = 4τ 0 n 0 For a derivation of Eq. (14), we refer to [32].InAppendix A we provide a different derivation, based on a contour integration analysis. The pole of the gamma function in Eq. (13) in four dimensions is absorbed in a renormalization of the low-energy constant 3. One readily verifies that inserting Eq. (14) into Eq. (12) yields for the leading finite volume shift [2] 1 1 M = M + g M2 , ,L + O . πL π 1 2 1 π 0 4 (15) 4Fπ Fπ The separation of the cut-off and the volume-dependence is as expected: finite volume corrections do not generate new UV-divergences. At leading order the finite volume corrections could be isolated immediately. This will not be the case at the two-loop level. The graph (d) in Fig. 1 does not factorize in pure one-loop integrals and further steps need to be performed. We refer to Appendix A for further details.

3.2. Minimal set of periodiﬁed propagators

In the evaluation of the Feynman diagrams, it is mandatory to make use of the following result. In order to evaluate a graph consisting of L loops, one needs to consider only a certain set of L propagators as the periodified, finite-volume ones—the others can be taken as infinite-volume propagators. In the following we briefly discuss this statement (see also p. 18ff in Ref. [1]). Consider an arbitrary self-energy graph with L loops, I internal lines and V vertices. Since the number of loops is the number of independent integrations over momenta, we have

L = I − V + 1. (16) For every line the propagator is an infinite sum of terms characterized by an integer vector n(). The loop graph itself is an infinite sum of terms which can be identified by a set of integer vectors {n()} assigned to all the internal lines. As remarked by Lüscher this structure can be

2 The second argument of g1 is the temperature, which we keep zero. Notation as in Ref. [2]. 20 G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 viewed as a gauge ﬁeld on the self-energy graph. Gauge transformations are deﬁned as3

n () = n() + f() − i() , (17) where (v) is some field of integer vectors and i() (f ()) is the initial (final) vertex of the line . One verifies immediately that two contributions of a Feynman graph which differ only by a gauge transformation yield the same mathematical expression. In the calculation of the loop graph what matters is the sum over representatives of gauge equivalence classes. A convenient representative can be found, e.g., by adjusting (i()) and (f ()) iteratively, such that n() = 0 for as many internal lines as possible. This can be achieved for V − 1 lines. Therefore, there remain I − V + 1 internal lines where the periodified propagator has to be inserted—which coincides with the number of loops of the graph. This shall be our minimal set of periodified propagators. In Fig. 1, we have attached a spline to a periodified propagator, whereas lines without a spline correspond to an infinite volume propagator.

3.3. Two-loop result

It is convenient to split the sum over the equivalence classes into three parts4:

Σ(0): n() = 0 ∀ (pure gauge), Σ(1): n() = 0 ∀ except for one line ¯ (simple gauge), (2) ¯ ¯ Σ : n() = 0 ∀ except for two lines 1, 2. (18) The pion mass in ﬁnite volume to two-loops then admits the form 2 = 2 − (1) − (2) MπL Mπ Σ Σ , 2 = 2 − (0) Mπ M Σ , (19) where, using λπ = Mπ L, we get (1) 3 Σ = Ip + Ic + O ξ , (20) ∞ 2 ∞ √ M m(n) − + 2 I = π √ dy F (iy)e n(1 y )λπ , p 2 π (21) 16π λπ n n=1 −∞ ∞ ∞ √ ∞ − ˜+ 2 iM2 m(n) e n(s y )λπ I =− π √ dy ds˜ disc F (s,˜ 1 + iy) , (22) c 3 ˜ + π 32π λπ = n s 2iy n 1 −∞ 4 2 with m(n) ≡ number of integer vectors z with z = n. Ip denotes the pole and Ic the cut contribution, whose meaning and precise deﬁnitions will be explained below. The expression for Σ(2) is more cumbersome:

(2) = 2 2 9 ˜ 2 − 1 ˜ ∂ ˜ + + O 3 Σ Mπ ξ g1(λπ ) λπ g1(λπ ) g1(λπ ) Δ ξ , (23) 8 8 ∂λπ

3 We borrow the notation from Ref. [1]. 4 Note the slight difference in the deﬁnition of simple ﬁelds with respect to [1]. We do not require |n()|=1. G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 21

˜2 and we merely note that it can be split into products of one-loop contributions terms with g1 and a pure two-loop part, denoted by Δ. We have introduced the abbreviation

M2 ξ = π . 2 2 (24) 16π Fπ

A detailed derivation of these results will be given in the subsequent sections and we confine ourselves to a few comments at this stage: In Ip one recovers the asymptotic formula of Lüscher, if one restricts the sum to the first addendum. Its extension to the present form has already been suggested in [11] and was applied in [12]. The function Fπ (ν) denotes the isospin zero ππ scattering amplitude in the forward kinematics. It contains cuts due to the two-pion intermediate state. In the derivation of the asymptotic formula, Lüscher consistently dropped contributions arising from the cuts, since in a large volume expansion they are beyond the order of accuracy he aimed at. Here, we take them into account up to two-loops, relying on the chiral representation of ππ scattering amplitude. The outcome of the analysis is summarized in Ic in which the same symbol Fπ , this time with two arguments appears: with the latter we mean the scattering amplitude of two on-shell pions into two off-shell pions in the forward kinematics configuration. The contributions from the cuts can still be written as integrals over the ππ scattering amplitude with an exponential weight, as in the asymptotic formula. Notice however, that this formula explicitly relies on the chiral representation of the scattering amplitude and is only valid up to two-loops, whereas the Lüscher formula holds to all orders. Contributions from two pion propagators in finite volume are ultimately captured in Σ(2), being expressed in terms of a dimensionless function g˜1(λπ ) and a numerically small correction Δ arising from graph 1(d). Both are explicitly given in Appendix A and Eq. (44), respectively. The pure gauge contributions are not volume dependent and merely renormalize the pion mass, cf. Eq. (19). A detailed discussion of this calculation can be found in [33,9], with which we agree—this was a useful, nontrivial check for our calculation.

Fig. 2. Skeleton diagrams representing (a) Eq. (25) and (b) Eq. (B.3). The blob in (a) stands for the 4-point function of ππ scattering in infinite volume and in (b) for a subtracted 6-point function of πππ scattering in infinite volume. The 2 double-line with the spline is a finite volume propagator with the physical pion mass Mπ . 22 G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33

3.4. Self-energy to ﬁrst order: Σ(1)

The simple ﬁelds can be summed up in closed form and may be represented by a skeleton diagram, see Fig. 2(a) ∞ 1 d4q √ (1) iq1 nL 2 Σ = m(n)e G q Γππ(p,q,ˆ −ˆp,−q), (25) 2 (2π)4 0 n=1 with pˆ = (iMπ , 0) and Γππ(p,q,ˆ −ˆp,−q) the 4-point function of ππ scattering in the forward scattering kinematics. Note that the 4-point function in Eq. (25) is an off–shell amplitude and that (2) we evaluate it at pˆ and not at pˆL. The difference is taken into account in Σ . The representation of Σ(1) in Eq. (25) has already been used by Lüscher and eventually led him to the asymptotic 2 formula [1]. He then discussed the contribution of the pole of the propagator G0(q ) that one meets at = 2 + 2 + 2 = q1 i Mπ q⊥ q0 ,q⊥ (q2,q3). (26) Above this pole the singularities come from the cuts of the propagator and the 4-point function Γππ(p,q,ˆ −ˆp,−q). These start from =− ˆ + 2 2 =− ˆ − 2 2 − 2 2 s (p q) 4Mπ ,u (p q) 4Mπ , q 9Mπ . (27) Lüscher showed that the pole contribution is dominating with respect to those coming from the cuts and neglected the latter. In fact, his discussion involved no further assumptions about the 4-point function and remains true at every order of the perturbative expansion. Since our goal is to test the asymptotic formula beyond the leading exponentials, we wish to work out the impact of the contributions that were dropped by Lüscher. Doing this at the two-loop level is rather straightforward, as we will now show. Up to O(p4), the four-point function can be decomposed into a combination of functions which have either a singularity in s or in u. Since Σ(1) is symmetric in s and u, we may write ¯ Γππ(p,q,ˆ −ˆp,−q) in terms of a function Γππ(s, pq)ˆ , which depends only on the variables s ˆ 2 and pq, and which has cuts at s 4Mπ , 1 Γ (p,q,ˆ −ˆp,−q) = Γ¯ (s, pq)ˆ + O . ππ ππ 6 (28) Fπ

Fig. 3. Integration contour in the complex q1 plane with the pole from the pion propagator and the branch cut from the ππ scattering amplitude. G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 23

= + ∈ R 2 We write q1 x iy and for the domain s , s 4Mπ we have 2 − 2 − 2 − 2 2 =− y x q0 q⊥ 3Mπ ,xyq0Mπ . (29)

In particular we observe that in the complex q1 plane the cut starts above the pole of Eq. (26), as illustrated in Fig. 3. Performing now the contour integration in the upper half plane we obtain two terms: the ﬁrst one, Ip, comes from the pole and is simply its residuum. The other comes from the integral along the new integration path. The latter contribution vanishes as we push the integration lines to inﬁnity, except for the path which goes around the cut, to be denoted by Ic in the following,

(1) Σ = Ip + Ic. As was shown by Lüscher [1], the former can be simplified considerably and can be brought in the usual form of the asymptotic formula ∞ 2 ∞ √ M m(n) − + 2 I = π √ dy F (iy)e n(1 y )λπ , p 2 π 16π λπ n n=1 −∞ with Fπ (iy) the ππ forward scattering amplitude. Restricting the sum to the first term we recover Lüscher’s formula [1].ForIc we find ∞ √ 2 iq1 nL 1 dq d q⊥ e − I = m(n) 0 dq disc Γ¯ (s, pq)ˆ + O F 6 , (30) c 4 1 2 + 2 ππ π 2 = (2π) Mπ q n 1 [ 2 ] s 4Mπ ¯ ¯ where disc[Γππ(s, pq)ˆ ] denotes the discontinuity of Γππ along the cut. It is now convenient to shift the integration path in q0 from Im(q0) = 0toIm(q0) =−iMπ . Along this path we have =¯ − =−¯2 − 2 − 2 ¯ ∈ R q0 q0 iMπ ,sq0 q1 q⊥, q0 , (31) and the integration over q1 = x + iy falls onto the imaginary axis, = = 2 +¯2 + 2 x 0,yy0 4Mπ q0 q⊥, ∞ ∞ ¯ 2 √ [ ¯ ˆ ] i dq0 d q⊥ −y nL disc Γππ(s, pq) Ic = m(n) dy e . (32) 2 (2π)4 M2 + q2 n=1 π y0

Next, we change the integration variable from q1 to s, ∞ ∞ − +¯2+ 2 n(s q q⊥)L ¯ i dq¯ d2q⊥ e 0 disc[Γ (s, pq)ˆ ] I =− m(n) 0 ds ππ , (33) c 4 +¯2 + 2 1/2 + ¯ 2 = (2π) 2(s q q⊥) s 2iMπ q0 n 1 2 0 4Mπ and make use of 2 √ d q⊥ 1 − 2+ 2 1 − e n(μ q⊥)L = √ e μ nL, (34) 2 2 2 1/2 (2π) 2(μ + q⊥) 4π nL to carry through the integration over q⊥ and to end up with Eq. (22). 24 G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33

3.5. Self-energy to second order: Σ(2)

The term Σ(2) is more complicated to manipulate, although it is suggestive to think that it may be related to an integral over the 3π → 3π amplitude. A similar question has been asked in the framework of ﬁnite temperature QCD [23]. We discuss the situation in ﬁnite volume in Appendix B. Since these considerations have not lead us to a nice and compact representation for Σ(2) we have written it as in (23) where we only have split the factorizable two-loop contributions from the rest. The latter, which we have denoted by Δ we have evaluated numerically. We conclude this section with a remark concerning the large L behaviour of the two-loop results. In the large volume limit, the contributions from Eq. (21) behave according to 1 −λπ lim Ip ∼ e . (35) L→∞ 3/2 λπ Similarly, we may evaluate the large volume behaviour of the terms occurring in Eqs. (23), (22).It turns out that√ these, besides√ being exponentially suppressed and behaving at least as exp(−αλπ ), with α = ( 3 + 1)/ 2, are also suppressed√ by a power of λπ . Unfortunately this suppression is not particularly strong, being of 1/ λπ with respect to Eq. (35). While we can conclude that√ the resummed Lüscher formula is dominating in comparison to other two-loop diagrams in a 1/ λπ expansion, i.e., ¯ −αλπ MπL = MπL 1 + O e / λπ , ¯ 1 MπL = Mπ − Ip, (36) 2Mπ we are convinced that the most important test of the resummed formula is the numerical one, which we will discuss in the following.

4. Summary of analytical results

In this section we shall give the analytical results in explicit form, i.e., insert the chiral representation of the amplitudes which appear in the formulae given in the previous section and express our results in terms of a few basic integrals. In Eqs. (19)–(24) we have split the finite size (1) (2) effects of the pion mass into Σ = Ip + Ic and Σ . For the former two we find ∞ = 2 m(n)√ 1 (2) + (4) Ip Mπ ξ IM ξIM , n λπ π π n=1 ∞ = 2 m(n)√ 1 2 ˜ Ic Mπ ξ Ic, (37) n λπ n=1 where ξ is the chiral expansion parameter defined in Eq. (24) and the expressions I (2) , I (4) have Mπ Mπ already been given in Refs. [6,12], and we reproduce them here for completeness:

I (2) =−B0, Mπ (4) 0 55 ¯ 8 ¯ 5 ¯ ¯ 2 112 8 ¯ 32 ¯ I = B − + 41 + 2 − 3 − 24 + B − 1 − 2 Mπ 18 3 2 9 3 3 13 16 40 + R0 − R1 − R2, (38) 3 0 3 0 3 0 G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 25 where √ 0,2 0,2 0 2 B ≡ B ( nλπ ), B (x) = 2K1(x), B (x) = 2K2(x)/x, (39)

k and the integrals R0 are deﬁned as

∞ √ Re − +˜2 k even, Rk = dy˜ y˜k e n(1 y )λπ g(2 + 2iy)˜ for (40) 0 Im k odd, −∞

¯ 2 where g(x) is related to the standard one-loop function J(xMπ ) through = 2 ¯ 2 g(x) 16π J xMπ , (41)

¯ 2 and J(xMπ ) given in Eq. (A.10). For instance for x<0, we have

σ − 1 g(x) = σ log + 2, with σ = 1 − 4/x, (42) σ + 1 and elsewhere deﬁned through analytic continuation. In Ref. [12] also the next-to-next-to-leading order term in I , namely ξ 2I (6) has been given and we will use it in our numerical analysis. The p Mπ ˜ coefﬁcient Ic can be expressed as a combination of basic integrals

1 I˜ = 112C0,2 + 37C1,0 − 40C1,2 − 4C2,0 , c 3 ∞ ∞ √ − ˜+ 2 e n(s y )λπ 4 1/2 Cj,k = dy ds˜ 1 − s˜j yk. (43) s˜2 + 4y2 s˜ −∞ 4 Notice that the latter expression is obtained from the chiral representation of the off-shell ππ scattering amplitude for forward kinematics. This is parameterization dependent, and the result given here is obtained for the parameterization discussed in Section 2.1. The dependence on the parameterization must cancel in the full result. The self-energy to second order has already been introduced in Eq. (23) and reproduced here for convenience

(2) = 2 2 9 ˜ 2 − 1 ˜ ∂ ˜ + + O 3 Σ Mπ ξ g1(λπ ) λπ g1(λπ ) g1(λπ ) Δ ξ , 8 8 ∂λπ with

ˆ 2 2 7 p Δ = 16π 4p˜μp˜νHμν + 4p˜μHμ + H , p˜ = , (44) 6 Mπ where H , Hμ and Hμν are related to the sunset-type integrals of Fig. 1(d). We have not been able to ﬁnd a compact representation for these integrals and only elaborate on their numerical analysis in Appendix A in some detail. 26 G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33

5. Numerics

5.1. Setup

The numerical analysis is performed in line with the setup of Ref. [12]. The quantity of interest is − ≡ MπL Mπ RMπ , (45) Mπ whose quark mass dependence shall be evaluated numerically for different sizes L. The parameters of RMπ are(seeEqs.(19)–(24) and Eqs. (37)–(44)) the pion mass Mπ and the pion decay constant F in inﬁnite volume as well as (implicitly in I (4/6))theSU(2) low energy constants. π Mπ The quark mass dependence of the pion decay constant may be taken into account by expressing Fπ as a function of the pion mass Mπ (see, e.g., [6]). Regarding the low energy constants, we use the ones determined in [9,10] which are the same as in our previous ﬁnite size studies [6,12].

5.2. Results

= We plot our results for RMπ in Figs. 4 and 5 both for L 2, 3, 4 fm as a function of Mπ and for Mπ = 100, 300, 500 MeV as a function of L. We show the one-loop result (LO) as well as the two-loop result (NLO). These shall be compared with the resummed asymptotic formula with LO/NLO/NNLO input for the ππ scattering amplitude. Note that the one-loop result and the resummed asymptotic formula to LO coincide. The best estimate for RMπ is ﬁnally obtained by adding to the asymptotic pure three-loop contribution the two-loop result (NNLO asympt. + full NLO). At NLO, the ﬁnite size effects contain low energy constants, see, e.g., diagrams (f) and (g) in Fig. 1, leading to a nonnegligible error band. We take up a point which was already observed in [5], namely the large contributions when going from LO to NLO in the asymptotic formula (dotted to thin-dash-dotted). Compared to this gap, the additional contributions from the full two-loop result (thick-dash-dotted) are very small. The two-loop and the NLO result from the asymptotic formula only drift away, when we go beyond the region where the p-regime can be safely applied. In Table 1 we wish to underline these statements with a numerical example: we show the relative numerical impact of the LO, the pure

NLO and the pure NNLO contribution for RMπ . In the second column we give the source of the effect. The fifth line, e.g., contains only the contribution of the dispersive terms Ic to RMπ , once the kinematical prefactors are properly accounted for. We observe that Ic is strongly suppressed, irrespective of the value Mπ L. Consider the column with Mπ L 1.4. Although the bulk of the subleading effects is still due to the asymptotic contributions, the terms of Σ(2) play a significant role and cannot be neglected. This behaviour was expected, since with Mπ L 1.4 we might have already crossed the border of the p-regime. As we increase this parameter to Mπ L 2, the asymptotic regime begins to set in. The contributions from the resummed asymptotic formula (2) are now dominating with respect to those from Σ . The numerical results for Mπ L 2.5 confirm this trend. The fact that even the additional NNLO asymptotic terms (a partial three-loop result) are larger than the NLO nonasymptotic contributions is in nice agreement with the analytical expectation found in Eq. (36). However, we find that the suppression seen in the numbers is stronger than what one could have expected on the basis of the latter analytical argument. Finally, the discussion of the subleading effects allows us to give a reliable estimate for the lower bound G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 27

= − = Fig. 4. RMπ MπL/Mπ 1vs.Mπ for L 2, 3, 4 fm. The result of the resummed asymptotic Lüscher formula (21) with LO/NLO/NNLO chiral input (with attribute “asympt.” for NLO and NNLO in legenda) is compared to the one-loop

(LO) and two-loop (NLO) result. The best estimate for RMπ is obtained by adding the pure three loop contribution from the asymptotic formula to the two-loop result (NNLO asympt. + full NLO). The error band comes from the uncertainties in the low energy constants and is only shown for the best estimate. In the region above the Mπ L = 2 line, one is not safely in the p-regime and our results should not be trusted. of Mπ L for the p-regime,

Mπ L 2: lower bound for p-regime. (46)

6. Summary

(i) We have evaluated the finite volume corrections for the pion mass to two-loops within the framework of chiral perturbation theory (ChPT) in the p-regime (Mπ L 1, L>2fm, Mπ < 500 MeV). (ii) We have compared the two-loop result with the resummed version of the asymptotic formula of Lüscher. We have found that whenever the effects are calculated for masses and volumes such that Mπ L 1, such that one is safely within the p-regime of ChPT, the contributions which are not included in the resummed asymptotic formula are very small. The result gives us confidence in the claim that the resummed asymptotic formula is a convenient and efficient way to reliably calculate finite volume effects for hadronic masses [12]. (iii) The derivation of the asymptotic formula for decay constants [34] follows closely the original one for the masses [1]—the only new, subtle point, concerns the amplitude which appears in the integrand of the asymptotic formula, which has a pole in the integration region which first needs to be subtracted (see [34] for details). This is, however, a technical point—the physics 28 G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33

= − = Fig. 5. RMπ MπL/Mπ 1vs.L for Mπ 100, 300, 500 MeV. The rest of the legenda is as in Fig. 4.

Table 1 = − = RMπ MπL/Mπ 1 for a 2 fm volume and pion masses Mπ 140, 200, 250 MeV. We show the relative numerical impact of the LO, the pure NLO and the pure NNLO contribution for RMπ . In the second column we give the source of the effect. The ﬁfth line, e.g., contains the contribution to RMπ of the dispersive terms Ic alone, once kinematical prefactors are added accordingly. The numerical results for the Ip contributions, to LO, NLO and NNLO have already been given in [6,12] = = = RMπ Mπ 140 MeV Mπ 200 MeV Mπ 250 MeV L = 2fm Mπ L 1.4 Mπ L 2.0 Mπ L 2.5 Ip LO 0.0463 0.0199 0.0105 Ip ΔNLO 0.0291 0.0149 0.0088 Ic ΔNLO 0.0005 0.0001 0.0000 Σ(2) ΔNLO 0.0206 0.0038 0.0011 Ip ΔNNLO 0.0076 0.0053 0.0035 Total 0.1041(55) 0.0440(50) 0.0239(42)

of the ﬁnite volume corrections for masses and decay constants appears to be rather similar. In view of this we believe that the present results speak also in favour of the resummed asymptotic formula for decay constants [12]. A check of this claim could be obtained by evaluating the pion decay constant to two loops. (iv) The two-loop calculation performed here allows us to better estimate the region of validity of ChPT in the p-regime. Our conclusion is that in the case of the pion mass it is necessary to have Mπ L 2. Again, this serves as a guideline also for decay constants and masses of other hadrons. G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 29

Acknowledgements

We thank Stephan Dürr for useful discussions and a careful reading of the manuscript. This work has been supported by the Schweizerischer Nationalfonds and partly by the EU “Euridice” program under code HPRN-CT2002-00311.

Appendix A. Finite volume integrals

In this appendix we give further details on the finite volume integrals which occurred in the two-loop calculation. Throughout we applied dimensional regularization, as is common in ChPT. Since the finite box breaks Lorentz invariance, tensor simplifications have to be performed with care. Consider, e.g., the integral

dd q eiqnL pˆ pˆ A = (pq)ˆ 2, (A.1) μ ν μν (2π)d M2 + q2 n∈Z3 π which arises when evaluating diagram (f) in Fig. 1. Remember pˆ = (iMπ , 0). The prime in the sum denotes that the term with n = 0 is excluded. The ansatz

Aμν = δμνA (A.2) with A a scalar integral which is valid in inﬁnite volume, leads to an incorrect result in ﬁnite volume. A direct calculation of the integral with the help of Eq. (A.15) yields ∞ (M2 )1+d/2 m (n) √ pˆ pˆ A =− π d K λ n . μ ν μν d/2 d/4 d/2 π (A.3) (2πλπ ) n n=1 In four dimensions the result agrees with the derivation outlined in the next section, where one proceeds along the same lines as in Section 3.4.

A.1. Tadpole

In Fig. 1, the ﬁnite volume corrections of the diagrams (a)–(c), (f) and (g) factorize into one- loop integrals which are of the generic form

4 iqnL d q e P pqˆ ; q2 , (A.4) (2π)4 (M2 + q2)k n∈Z3 π with k an integer positive number and P(pq,qˆ 2) a polynomial of pqˆ and q2. It sufﬁces to discuss the case for k = 1, since k = 2, 3,... are obtained through appropriate derivatives with respect 2 to Mπ . The only singularity of the integrand is the pole of the propagator. Therefore, the contour integration analysis applied in Section 3.4 yields the result ∞ ∞ 2 √ m(n) M − + 2 √ π dy e n(1 y )λπ P iM2 y;−M2 . 2 π π (A.5) n 8π λπ n=1 −∞

In particular, in the text we have used the dimensionless function g˜1(λπ ), 16π 2 g˜ (λ ) = g M2 , ,L , 1 π 2 1 π 0 Mπ 30 G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33

dd q eiqnL g M2 , 0,L = , (A.6) k π (2π)d (M2 + q2)k n∈Z3 π = 2 evaluated at d 4, and where g1(Mπ , 0,L) was introduced a long time ago by Gasser and Leutwyler [2,3], see also Eq. (14).

A.2. Sunset

The only real two-loop diagram is the sunset graph Fig. 1(d), and we will comment on it in some detail. It is convenient to split the finite volume integrals as in Eq. (18) only after the tensor simplifications. As alluded in the beginning of the appendix, tensor simplifications in finite volume have to be performed with care. Even though one cannot rely on Lorentz invariance, the sunset tensor integrals may still be reduced to the structures

d ipx 2 {H, Hμ, Hμν}= d xe G(x) {1,i∂μ, −∂μ∂ν}G(x) , (A.7) with G(x) from Eq. (7). A rather direct way to perform these steps is to work in coordinate space and to make use of partial integrations, i.e.,

1 dd xeipx ∂ G(x)∂ G(x)G(x) =− dd xeipx G(x)2 p i∂ G(x) + ∂ ∂ G(x) . μ ν 2 μ ν μ ν (A.8) Notice that the same identities in momentum space are derived with the help of translational invariance which is still respected in ﬁnite volume due to the periodic boundary conditions. In the following, we will only elaborate on the scalar integral H in more detail. We expand the integral in terms of number of ﬁnite volume propagators as motivated in Section 3.3,

H = H(0) + 3H(1) + H(2), (A.9) where the ﬁrst (second) addend corresponds to the pure (simple) gauge ﬁelds contribution. For H(0) we refer to [31]. Further, √ ∞ d4q eiq1 nL H(1) = m(n) J¯ (pˆ − q)2 + g M2 , 0,L J(0), (A.10) (2π)4 M2 + q2 1 π n=1 π with

dd 1 1 J k2 = = J¯ k2 + J(0). (A.11) d [ 2 + 2] [ 2 + − 2] (2π) Mπ Mπ (k ) The first term of Eq. (A.10) is finite, the second carries an UV-divergence which is absorbed by a counterterm. This shows that although finite volume effects do not generate new UV-divergences, they still appear at intermediate steps of the calculation. It is a thorough check on our calculation that these nonanalytic divergences cancel. Finally,

d4q d4k eiqnL eikrL 1 H(2) = . (A.12) (2π)4 (2π)4 [M2 + q2] [M2 + k2] [M2 + (pˆ − q − k)2] n,r∈Z3\0 π π π n=r G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 31

In the text, we used its dimensionless version

4 4 ˜ iqn˜ λ ikr˜ λ ˜ ˜ ˜ d q˜ d k e π e π {1; kμ; kμkν} {H ; Hμ; Hμν}= , (A.13) (2π)4 (2π)4 [1 +˜q2] [1 + k˜2] [1 + (p˜ −˜q − k)˜ 2] n,r∈Z3\0 n=r which are UV-finite and do not need to be renormalized. We only have to find a convenient representation for the numerical analysis. We shall restrict ourselves to the scalar integral in the following. Introducing a Feynman parameter by combining the second and third denominator, we find

1 ˜ − + ˜ d4q˜ d4k˜ 1 eiq(r nx)λπ inkλπ H = dx . (A.14) (2π)4 (2π)4 [1 +˜q2] [1 + (p˜ −˜q)2x(1 − x) + k˜2]2 0 n,r∈Z3\0 n=r We use Schwinger’s trick for both remaining denominators ∞ 1 − + 2 = dαe α(1 x ). (A.15) 1 + x2 0

The integrals over k˜ and q˜ are then of the Gaussian type and can be performed analytically. We are then left with three integrations over a rather lengthy expression which shall not be written down here. Despite their unhandy form, the integrations may still safely be performed numerically. The accuracy of the determination of Eq. (A.14) is not limited by the integration routine, but by the rather slow convergence of the sum in n and r for moderate λπ . Consequently, the evaluation of the sunset integrals going into Δ is restricted to three signiﬁcant digits. (The last digit given for Σ(2) in Table 1 is not signiﬁcant.) Note that the uncertainty of the H -type integrals is not a serious matter. Firstly, it could be lowered by brute force and secondly it only plays a (minor) role, in case when the p-regime cannot be safely applied anymore.

Appendix B. Self-energy to second order: Σ(2)

In this section we ask ourselves whether the self-energy to second order can be represented in a similar compact form as in the case of the self-energy to first order in Eq. (25). As will be discussed, it is indeed possible to relate the self-energy to second order to a 3π scattering amplitude in the forward scattering kinematics, as illustrated in Fig. 2(b). This has already been observed by Schenk in a related context [23]. He investigated the dynamics of pions in a cold heat bath of temperature T and examined the effective mass of the pion Mπ (T ) within the framework of ChPT. The close relation between finite temperature field theory and finite volume effects becomes apparent in the imaginary time formalism, where one treats the inverse temperature as a finite extension in the imaginary time direction. The expansion in terms of the number of finite volume propagators is then in one-to-one correspondence with the expansion in terms of the number of finite temperature propagators. In the following, Schenk’s approach shall be adapted to the finite volume scenario. We first establish the relation between Σ(2) and the three-to-three particle scattering amplitude and proceed with various remarks. 32 G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33

Consider the 6-point function in d dimensions in the nonlinear sigma model parameterization Eq. (3) in the forward kinematics 3 − − − − − − dx ···dx e ip(x1 x4) ik(x2 x5) iq(x3 x6)0|Tϕ1 ϕa ϕb ϕ1 ϕa ϕb |0 1 5 x1 x2 x3 x4 x5 x6 a,b=1 Z3 = T (p,k,q), (B.1) 2 + 2 2 2 + 2 2 2 + 2 2 πππ (Mπ p ) (Mπ k ) (Mπ q ) with Z the wave function renormalization constant and ϕx ≡ ϕ(x). The amplitude Tπππ(p,k,q) ˆ2 =− 2 contains a pole at p Mπ which needs to be subtracted R(p,k,q) T (p,k,q)= Tˆ (p,k,q)+ . (B.2) πππ πππ 2 + 2 Mπ p The self-energy to second order can then be written in terms of the subtracted amplitude ˆ Tπππ(p,k,q)ˆ 4 4 ˆ 1 d q d k + T (p,k,q)ˆ 1 Σ(2) = eiqnL ikrL πππ + O , (B.3) 8 (2π)4 (2π)4 (M2 + q2)(M2 + k2) F 6 n,r π π π where the prime restricts the sum to integer vectors n and r obeying n = 0 = r, and for diagram (c) in Fig. 6 in addition n = r. The latter restriction avoids double counting as this term is already accounted for in a simple gauge ﬁeld of Σ(1).

(i) The reason for the subtraction is easily accounted for. In Fig. 6 we decompose the 3π–3π scattering amplitude in terms of 1-particle irreducible parts. The subtraction removes the contribution from the diagram in Fig. 6(e) which does not correspond to a 1-particle irreducible self-energy diagram, once the external legs are appropriately closed. A thorough discussion on the physical interpretation of the pole term can be found in [23]. (ii) Notice that Eq. (B.2) defines a subtracted off-shell amplitude which depends on the regularization scheme as well as on the parameterization of the pion fields. At the order we are work- ˆ ing, the regularization dependence is not an issue, since the subtracted amplitude Tπππ(p,k,q)ˆ is only needed at tree level. However, the dependence on the parameterization of the pion fields is of concern. While the momentum integrations in Eq. (B.3) for the diagrams Fig. 6(b) and (d) put the momenta k and q on-shell and the ambiguity due to the parameterization therefore drops out, this does not happen in the case of diagram 6(c). Note that the same parameterization ambiguity already occurred in Ic in the dispersive analysis (1) 2 =− 2 of Σ (cf. Eq. (31) where q Mπ ). In order to understand the close relation between these

Fig. 6. Decomposition of the 3π–3π amplitude in terms of 1 particle irreducible parts. The characters 1, a and b on the external legs denote isospin indices. G. Colangelo, C. Haefeli / Nuclear Physics B 744 (2006) 14–33 33 two terms, we first note that only diagram Fig. 1(d) contributed to Ic. Further, the simple gauge field of Fig. 1(d) with n = 0 for one propagator can immediately be written as a contribution with two finite volume propagators: one periodifies a second propagator, however with the same n = 0 as already for the first one. Since MπL does not depend on the parameterization of the pion fields, the ambiguities of the two terms have to cancel each other. To explicitly show this is however nontrivial. (2) In summary, while our representations for Ic in Eq. (22), respectively Eq. (37) and for Σ in Eq. (B.3), respectively Eq. (23) do depend on the off-shell dependence of the scattering amplitudes, the sum in Eq. (19) does not. (iii) Even though the self-energy to second order can be expressed in a compact form, further simplifications (similar to those performed in Section 3.4) seem not to be straightforward. Instead, for the (numerical) integrations we had to discuss the various terms contributing to Σ(2) one by one. A general discussion of finite volume integrals has been provided in Appendix A. After treating the finite volume integrals, we end up with the basic functions given in Eq. (23).

References

[1] M. Luscher, Commun. Math. Phys. 104 (1986) 177. [2] J. Gasser, H. Leutwyler, Phys. Lett. B 184 (1987) 83. [3] J. Gasser, H. Leutwyler, Phys. Lett. B 188 (1987) 477. [4] J. Gasser, H. Leutwyler, Nucl. Phys. B 307 (1988) 763. [5] G. Colangelo, S. Durr, R. Sommer, Nucl. Phys. B (Proc. Suppl.) 119 (2003) 254, hep-lat/0209110. [6] G. Colangelo, S. Durr, Eur. Phys. J. C 33 (2004) 543, hep-lat/0311023. [7] J. Gasser, H. Leutwyler, Ann. Phys. 158 (1984) 142. [8] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M.E. Sainio, Phys. Lett. B 374 (1996) 210, hep-ph/9511397. [9] J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M.E. Sainio, Nucl. Phys. B 508 (1997) 263, hep-ph/9707291; J. Bijnens, G. Colangelo, G. Ecker, J. Gasser, M.E. Sainio, Nucl. Phys. B 517 (1998) 639, Erratum. [10] G. Colangelo, J. Gasser, H. Leutwyler, Nucl. Phys. B 603 (2001) 125, hep-ph/0103088. [11] G. Colangelo, Nucl. Phys. B (Proc. Suppl.) 140 (2005) 120, hep-lat/0409111. [12] G. Colangelo, S. Durr, C. Haefeli, Nucl. Phys. B 721 (2005) 136, hep-lat/0503014. [13] C. Haefeli, hep-lat/0509078. [14] G. Colangelo, A. Fuhrer, C. Haefeli, hep-lat/0512002. [15] S.R. Sharpe, Phys. Rev. D 46 (1992) 3146, hep-lat/9205020. [16] D. Becirevic, G. Villadoro, Phys. Rev. D 69 (2004) 054010, hep-lat/0311028. [17] QCDSF-UKQCD Collaboration, A. Ali Khan, et al., Nucl. Phys. B 689 (2004) 175, hep-lat/0312030. [18] D. Arndt, C.J.D. Lin, Phys. Rev. D 70 (2004) 014503, hep-lat/0403012. [19] S.R. Beane, Phys. Rev. D 70 (2004) 034507, hep-lat/0403015. [20] S.R. Beane, M.J. Savage, Phys. Rev. D 70 (2004) 074029, hep-ph/0404131. [21] P.F. Bedaque, I. Sato, A. Walker-Loud, hep-lat/0601033. [22] J. Bijnens, N. Danielsson, K. Ghorbani, T. Lahde, hep-lat/0509042. [23] A. Schenk, Phys. Rev. D 47 (1993) 5138. [24] D. Toublan, Phys. Rev. D 56 (1997) 5629, hep-ph/9706273. [25] M. Luscher, DESY 83/116, Lecture given at Cargese Summer Institute, Cargese, France, 1–15 September 1983. [26] Y. Koma, M. Koma, Nucl. Phys. B 713 (2005) 575, hep-lat/0406034; Y. Koma, M. Koma, hep-lat/0504009. [27] J. Braun, B. Klein, H.J. Pirner, Phys. Rev. D 71 (2005) 014032, hep-ph/0408116. [28] B. Borasoy, G.M. von Hippel, H. Krebs, R. Lewis, hep-lat/0509007. [29] J. Bijnens, G. Colangelo, G. Ecker, JHEP 9902 (1999) 020, hep-ph/9902437. [30] J. Bijnens, G. Colangelo, G. Ecker, Ann. Phys. 280 (2000) 100, hep-ph/9907333. [31] J. Gasser, M.E. Sainio, Eur. Phys. J. C 6 (1999) 297, hep-ph/9803251. [32] P. Hasenfratz, H. Leutwyler, Nucl. Phys. B 343 (1990) 241. [33] U. Burgi, Nucl. Phys. B 479 (1996) 392, hep-ph/9602429. [34] G. Colangelo, C. Haefeli, Phys. Lett. B 590 (2004) 258, hep-lat/0403025. Nuclear Physics B 744 (2006) 34–58

Pseudoscalar meson decay constants and couplings, the Witten–Veneziano formula beyond large Nc, and the topological susceptibility ✩

G.M. Shore

Department of Physics, University of Wales, Swansea, Swansea SA2 8PP, UK Received 13 January 2006; received in revised form 9 March 2006; accepted 10 March 2006 Available online 31 March 2006

Abstract The QCD formulae for the radiative decays η,η → γγ, and the corresponding Dashen–Gell-Mann– Oakes–Renner relations, differ from conventional PCAC results due to the gluonic U(1)A axial anomaly. This introduces a critical dependence on the gluon topological susceptibility. In this paper, we revisit our earlier theoretical analysis of radiative pseudoscalar decays and the DGMOR relations and extract explicit experimental values for the decay constants. This is our main result. The flavour singlet DGMOR relation is the generalisation of the Witten–Veneziano formula beyond large Nc, so we are able to give a quantitative assessment of the realisation of the 1/Nc expansion in the U(1)A sector of QCD. Applications to other aspects of η physics, including the relation with the first moment sum rule for the polarised photon structure γ function g , are highlighted. The U(1)A Goldberger–Treiman relation is extended to accommodate SU(3) 1 flavour breaking and the implications of a more precise measurement of the η and η -nucleon couplings are discussed. A comparison with the existing literature on pseudoscalar meson decay constants using large-Nc chiral Lagrangians is also made. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

The phenomenology of the pseudoscalar mesons opens a window on many interesting aspects of the non-perturbative dynamics of QCD, including spontaneous chiral symmetry breaking, the electromagnetic and gluonic axial anomalies, the OZI rule, the gluon topological susceptibility,

✩ This research is supported in part by PPARC grants PP/G/O/2002/00470 and PP/D507407/1. E-mail address: [email protected] (G.M. Shore).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.03.011 G.M. Shore / Nuclear Physics B 744 (2006) 34–58 35 and so on. Nevertheless, until comparatively recently, phenomenological analyses did not take fully into account the role of the gluonic U(1)A anomaly in this sector, with the result that most of the existing determinations of the pseudoscalar meson decay constants and couplings are based on over-simplified theoretical formulae which miss the most interesting anomaly-sensitive physics. In a previous paper [1] (see also [2]) we have analysed in detail the radiative decays of the neutral pseudoscalar mesons, π 0,η,η → γγ, and provided a set of formulae describing these processes together with modified Dashen–Gell-Mann–Oakes–Renner (DGMOR) [3,4] relations which fully include the effect of the anomaly and the gluon topological susceptibility in the flavour singlet sector as well as explicit flavour SU(3) breaking. In this paper, we confront these formulae with experimental data to extract a set of results for the four pseudoscalar meson decay constants, f 0η ,f0η,f8η and f 8η. Our approach, which is based on a straightforward generalisation of conventional PCAC to include the anomaly in a renormalisation group consistent way, is unique in that it is theoretically consistent yet does rely on using the 1/Nc expansion. The decay formulae and DGMOR relations we derive are valid for all Nc. At some point, however, the fact that the η is not a Nambu– Goldstone (NG) boson means that to make predictions from our formulae we need to augment the standard dynamical approximations of PCAC or chiral perturbation theory with a further dynamical input. Here, we make a judicious use of 1/Nc ideas to justify the use of the lattice calculation of the topological susceptibility in pure Yang–Mills theory as an input into our flavour singlet DGMOR relation. The self-consistency of this approach allows us to test the validity of 1/Nc methods in the U(1)A sector against real experimental data. In particular, the flavour singlet DGMOR relation is the finite-Nc generalisation of the well-known Witten–Veneziano formula [5,6], and we can use our results to study how well the large-Nc limit is realised in real QCD. This is especially important in view of the extensive use of large Nc in modern duality-based approaches to non-perturbative gauge theories such as the AdS–CFT correspondence. The relation of our analysis of the low-energy decays η(η) → γγ to high-energy physics in γ the form of the first moment sum rule for the polarised photon structure function g1 [7,8] is also discussed. The dependence of this sum rule on the photon momentum encodes many aspects of non-perturbative QCD physics and its full measurement is now within the reach of planned high-luminosity e+e− colliders [9]. Our methods can also be readily extended to a variety of other reactions involving the pseudoscalar mesons, including η(η) → Vγ, where V is one of the flavour singlet vector mesons ρ,ω,φ, η(η) → π +π −γ , a variety of electro and photoproduction reactions such as γp → pη(η) and γp → pφ, and η and η production in pp collisions, pp → ppη(η). The latter are relevant for determining the couplings gηNN and gηNN which occur in gen- eralisations of the Goldberger–Treiman relation [10]. In this paper, we present a new version of the U(1)A Goldberger–Treiman relation [11–13] incorporating flavour octet–singlet mixing in the η–η sector. We are then able to use our results for the decay constants to show how the physical interpretation of this relation, which is closely related to the ‘proton spin’ problem in p high-energy QCD (i.e., the first moment sum rule for the polarised proton structure function g1 ), depends critically on the experimental determination of gηNN. Finally, in Appendix A, we give a brief comparison of the similarities and differences between our approach and an alternative theoretically consistent framework for describing η and η physics, viz. the large-Nc chiral Lagrangian formalism of Refs. [14–16] (see also [17–21]). A comprehensive review of the phenomenology of the pseudoscalar mesons, incorporating the chiral Lagrangian results, is given in Ref. [22] 36 G.M. Shore / Nuclear Physics B 744 (2006) 34–58

2. Radiative decay and DGMOR formulae

In this section, we present the radiative decay and DGMOR formulae and discuss how to extract phenomenological quantities from them, emphasising their 1/Nc dependence. A brief review of their derivation in the language of ‘U(1)A PCAC’ is given in Section 3, but we refer to the original papers [1,2,23] for full details, including a more precise formulation using functional methods. Throughout, we consider the case of three light flavours, nf = 3. The assumptions made in deriving these formulae are the standard ones of PCAC. The formulae are based on the zero-momentum chiral Ward identities. It is then assumed that the decay aα 2 2 ‘constants’ (f (k ) in the notation below) and couplings (gηαγγ(k )) are approximately constant functions of momentum in the range from zero, where the Ward identities are applied, to the appropriate physical particle mass. It is important that this approximation is applied only to pole- free quantities which have only an implicit dependence on the quark masses. This is equivalent to assuming pole dominance of the propagators for the relevant operators by the (pseudo-)NG bosons, and naturally becomes exact in the chiral limit. Notice that these dynamical approximations are equally necessary in the chiral Lagrangian approach, where they are built in to the basic structure of the model—the effective fields are chosen to be in one-to-one correspondence with the NG bosons and pole dominance is implemented through the low-momentum expansion with momentum-independent parameters corresponding to the decay constants. The relations for the π0 decouple from those for the η and η in the realistic approximation of SU(2) flavour invariance of the quark condensates, i.e., ¯uu=dd¯ . The decay formula for π0 → γγ is the standard one, α f g = a3 em (2.1) π πγγ em π 3 = 1 with aem 3 Nc, while the DGMOR formula is simply 2 2 =− ¯ + ¯ fπ mπ mu uu md dd . (2.2)

The coupling gπγγ is defined as usual from the decay amplitude: 0 =− α β λ ρ γγ π igπγγ λραβ p1 p2 (p1) (p2) (2.3) in obvious notation [1]. The r.h.s. of Eq. (2.1) arises from the electromagnetic U(1)A anomaly and is of course sensitive to the number of colours, Nc. The decay constant has a simple theoretical interpretation as the coupling of the pion to the flavour triplet axial current, | 3 | = 0 Jμ5 π ikμfπ , which is conserved in the chiral limit. In the η,η sector, however, explicit SU(3) flavour breaking means there is mixing. The decay constants therefore form a 2 × 2 matrix: 0η 0η aα f f f = , (2.4) f 8η f 8η where the index a is an SU(3) × U(1) flavour index (including the singlet a = 0) and α denotes the physical particle states π 0,η,η. These four decay constants are independent [14,15,24,25], in contradiction to the original phenomenological parametrisations which erroneously expressed f aα as a diagonal decay constant matrix times an orthogonal η–η mixing matrix, giving a total of only three independent parameters. Sometimes the four decay constants are expressed in terms of two constants and two mixing angles but, while we also express our results in this form in Section 4, there is no particular reason to parametrise in this way. G.M. Shore / Nuclear Physics B 744 (2006) 34–58 37

The other new feature in the η,η sector is the presence of the gluonic U(1)A anomaly in the flavour singlet current. This means that even in the chiral limit the η is not a true NG boson and therefore the direct analogue of the PCAC formula (2.1) does not hold. Nonetheless, we can still write an analogous formula but with an extra term related to the gluon topological susceptibility. The decay formulae are1: √ αem f 0η g + f 0ηg + 6Ag = a0 , (2.5) η γγ ηγ γ Gγ γ em π αem f 8η g + f 8ηg = a8 , (2.6) η γγ ηγ γ em π √ 0 2√2 8 √1 where a = Nc and a = Nc, and the corresponding DGMOR relations are: em 3 3 em 3 3 0η 2 2 + 0η 2 2 =−2 ¯ + ¯ + ¯ + f mη f mη mu uu md dd ms ss 6A, (2.7) 3 √ 2 f 0η f 8η m 2 + f 0ηf 8ηm2 =− m ¯uu+m dd¯ −2m ¯ss , (2.8) η η 3 u d s 1 f 8η 2m 2 + f 8η 2m2 =− m ¯uu+m dd¯ +4m ¯ss . (2.9) η η 3 u d s There are several features of these equations which need to be explained. First, because of the anomaly, the decay constants in the flavour singlet sector cannot be identified as the couplings of 0 | 0 | = the η,η to the singlet axial-vector current. In particular, the object Fη defined as 0 Jμ5 η 0 ikμFη is not a renormalisation group invariant and so is not to be identified as a physical decay constant. The derivation of the radiative decay and DGMOR relations in the next section makes 0 0η it clear that Fη plays no role in these formulae—certainly it is not f . Instead, our decay constants are defined in terms of the couplings of the π,η,η to the pseudoscalar currents through aα | b| α= c the relations f 0 φ5 η dabc φ (for notation, see Section 3). This coincides with the usual definition except in the flavour singlet case. The coefficient A appearing in the flavour singlet equations is the non-perturbative constant that determines the topological susceptibility in QCD. Recall that the topological susceptibility χ(0) is defined as χ(0) = d4xi0|T Q(x)Q(0)|0, (2.10)

= αs μν ˜ where Q 8π tr G Gμν is the gluon topological charge. The anomalous chiral Ward identities determine the dependence of χ(0) on the quark masses and condensates up to a single non- perturbative parameter [1,26],viz. − 1 1 χ(0) =−A 1 − A (2.11) m ¯qq q=u,d,s q

1 Notice that compared to Refs. [1,2,9,23] we have changed the normalisation of the flavour singlets by a factor √ = a b = 1 a = × 1/ 2nf 1/ 6. The normalisations here are tr T T 2 δab for generators T (a 0, 1,...,8) of SU(3) U(1), i.e., T i = 1 λi (i = 1,...,8) where λi are the Gell-Mann matrices, and T 0 = √1 1 for the flavour singlet. The d-symbols 2 6 { a b}= c = = = = = 2 = =− = 1 are defined by T ,T dabcT and include d000 d033 d088 d330 d880 3 , d338 d383 d888 3 . 38 G.M. Shore / Nuclear Physics B 744 (2006) 34–58 or alternatively, − ¯ ¯ ¯ = Amumd ms uu dd ss χ(0) ¯ ¯ ¯ . mumd ms¯uudd¯ss−A(mumd ¯uudd+mums¯uu¯ss+md msdd¯ss) (2.12) Notice how the well-known result that χ(0) = 0 if any of the quark masses is zero is realised in this expression. The final new element in the flavour singlet decay formula is the coupling parameter gGγ γ . This is unique to our approach. It takes account of the fact that, because of mixing with the pseudoscalar gluon operator Q due to the anomaly, the physical η is not a NG boson. gGγ γ is not a physical coupling, although it may reasonably be thought of as the coupling of the photons to the gluonic component of the η. (A justification for this picture is given later.) Its precise field-theoretic definition is given in Section 3 and Refs. [1,2,23]. We should emphasise, however, that an analogous parameter must appear in any correct current algebra analysis involving the η, such as the 1/Nc chiral Lagrangian approach of Refs. [14–16]. Phenomenologically, it may be found to be relatively small (we test this numerically in Section 4), but any formulae which leave out such a term completely are inevitably theoretically inconsistent. Of course, in addition to these formulae, we also have the DGMOR relations for the remaining pseudoscalars. For example, for the K+, 2 2 =− ¯ + ¯ fK mK mu uu ms ss . (2.13) If we assume exact SU(2) flavour symmetry, we can then use Eqs. (2.2) and (2.13) to eliminate 2 2 the quark masses and condensates in favour of fπ ,fK ,mπ and mK in the DGMOR formulae for the η and η. We find2: 0η 2 2 + 0η 2 2 = 1 2 2 + 2 2 + f mη f mη fπ mπ 2fK mK 6A, (2.14) 3 √ 2 2 f 0η f 8η m 2 + f 0ηf 8ηm2 = f 2m2 − f 2 m2 , (2.15) η η 3 π π K K 1 f 8η 2m 2 + f 8η 2m2 =− f 2m2 − 4f 2 m2 . (2.16) η η 3 π π K K A quick look at the full set of decay and DGMOR formulae (2.5), (2.6) and (2.14)–(2.16) now 0η 0η 8η 8η shows that we have five equations for six parameters, viz. f ,f ,f ,f ,A and gGγ γ , assuming that fπ , fK and the physical masses are known along with the experimental values for the couplings gηγγ and gηγ γ . It is not surprising that this set is under-determined. In particular, the necessary presence of the unphysical coupling gGγ γ in the flavour singlet decay equation essentially removes its predictivity. At this stage, the best we can do is to evaluate the singlet decay constants and the gluonic coupling gGγ γ as functions of the topological susceptibility parameter A. This is done in Section 4. In order to make more progress, we need a further, dynamical input.

2 Notice that since our definitions of the flavour non-singlet decay constants coincide with the standard ones, Eq. (2.16) coincides with the usual formula at leading order in the quark mass expansion, O(m). In the chiral Lagrangian approach, the decay constants are usually fitted to the data using the equivalent of Eq. (2.16) valid to O(m2), in which the meson masses do not appear and chiral logarithms are included. (See Appendix A,Eq.(A.15).) In our phenomenological analysis in Section 4, we have chosen to fit the full set of DGMOR formulae self-consistently to O(m). This explains why our determinations of f 8η and f 8η do not exactly coincide with the usual chiral Lagrangian results. G.M. Shore / Nuclear Physics B 744 (2006) 34–58 39

As yet, everything we have done has been entirely independent of the 1/Nc expansion. The 1/Nc expansion (or the OZI limit—see Ref. [27] for a careful discussion of the differences) is known to give a good approximation to many aspects of the dynamics of QCD and could provide the required extra input, but its application to the U(1)A sector needs to be handled with great care. For example, in the chiral limit, the mass of the η, which arises due to the anomaly, is 2 = formally mη O(1/Nc); however, numerically (allowing for the quark masses going to zero) this is of the same order of magnitude as a typical meson such as the ρ and is certainly not small. Generally, it is not clear that quantities which are formally suppressed in 1/Nc are in fact numerically suppressed in real QCD, so we must be extremely careful in applying 1/Nc methods here. Despite these caveats, we will see that 1/Nc can play a useful role in analysing the decay formulae and DGMOR relations. Conventional√ large Nc counting gives the following orders aα = for the various√ quantities: f O( Nc) for all the decay constants; the η and η couplings = = 2 2 2 = gηαγγ O( Nc) but gGγ γ O(1); mη and mη are both O(1), but note that mη O(1/Nc) in the chiral limit—the numerically dominant contribution to its mass from the anomaly is formally 1/Nc suppressed relative to the O(1) contribution from the explicit chiral symmetry breaking ¯ = a = quark masses; the condensates qq O(Nc); the anomaly coefficients aem O(Nc); while finally the coefficient in the topological susceptibility is A = O(1). It follows that at large Nc, χ(0) −A = O(1). Moreover, it is clear from looking at planar diagrams that at leading order in 1/Nc, χ(0) in QCD coincides with the topological susceptibility in pure Yang–Mills theory, χ(0)|YM. We therefore have

A = χ(0)|YM + O(1/Nc) (2.17) a result that plays an important role in the Witten–Veneziano formula (see below). Now consider the flavour singlet DGMOR relation (2.7) or (2.14). Each term is O(Nc) apart from A, which is O(1). Naively, we might think that this sub-leading term would be small so could be neglected. However, we know that it contributes at the same order as the sub-leading 0η 2 2 2 term in (f ) mη given by the large anomaly-induced O(1/Nc) contribution to mη .Sothe topological susceptibility contribution to this relation is crucial, even though it is formally suppressed in 1/Nc. However, we may reasonably expect that the further O(1/Nc) corrections to A are genuinely small and that a good numerical approximation to Eqs. (2.7) or (2.14) is obtained by keeping only the terms up to O(1). This means that we may sensibly approximate the parameter A by χ(0)|YM in the DGMOR relation. This is the crucial simplification. A reliable estimate of χ(0)|YM at O(1) is available from lattice calculations in pure Yang–Mills theory [28]. This extra dynamical input allows us to determine the four decay constants from Eqs. (2.6) and (2.14)–(2.16). We can then analyse the flavour singlet decay formula (2.5) to determine gGγ γ and see how important this new term actually is numerically. We already know that the AgGγ γ contribution to Eq. (2.12) is suppressed by one power of 1/Nc compared to the other terms in the formula. Moreover, gGγ γ is renormalisation group invariant (see Refs. [2,23]). Now, in previous work we have developed an intuition as to when it is likely to be reliable to assume that 1/Nc suppressed terms are actually numerically small. The argument is based on the idea that violations of the OZI rule are associated with the U(1)A anomaly, so that we can identify OZI-violating quantities by their dependence on the anomalous 0 dimension associated with the non-trivial renormalisation of Jμ5 due to the anomaly. RG non- 40 G.M. Shore / Nuclear Physics B 744 (2006) 34–58 invariance can therefore be used as an indicator of which quantities we expect to show large OZI violations. In this case, gGγ γ is RG invariant, so we would expect the OZI rule to be good. This means that since it is OZI suppressed (essentially, higher order in 1/Nc) relative to the η decay 3 coupling gηαγγ, it should be numerically smaller as well. We will test this conjecture with the experimental data in Section 4. The issue is an important one. If gGγ γ can be neglected, then the naive current algebra formulae will turn out phenomenologically to be a good approximation to data, even though they are theoretically inconsistent. This would apply not just to the radiative pseudoscalar decays, but to a whole range of current algebra processes involving the η (see, for example, Refs. [22,23]). We have also used this conjecture in our analysis of the closely related ‘proton spin’ problem in polarised deep inelastic scattering where it plays an important role in our prediction of the first moment of the polarised p proton structure function g1 [29]. Finally, to close this section, we show in detail how these DGMOR relations are related to the well-known Witten–Veneziano formula for the mass of the η , which is derived in the large-Nc limit of QCD. In fact, this is simply the Nc →∞limit of the flavour singlet DGMOR formula, which we emphasise is valid for all Nc. To see this in detail, recall that the Witten–Veneziano formula for non-vanishing quark masses is [6]

2 2 2 6 m + m − m =− χ( )| . η η 2 K 2 0 YM (2.18) fπ

2 Of course, only the mη term on the l.h.s. is present in the chiral limit. Now add the DGMOR formulae (2.14) and (2.16). We ﬁnd 0η 2 2 + 0η 2 2 + 8η 2 2 + 8η 2 2 − 2 2 = f mη f mη f mη f mη 2fK mK 6A. (2.19)

To reduce this to its Witten–Veneziano approximation, we impose the large-Nc limit to identify the full QCD topological charge parameter A with −χ(0)| according to Eq. (2.17). We then YM set the ‘mixed’ decay constants f 0η and f 8η to zero and all the remaining decay constants 0η 8η f ,f and fK equal to fπ . With these approximations, we recover Eq. (2.18). In Section 4, when we ﬁnd the explicit experimental values for all these quantities in real QCD, we will be able to judge how good an approximation the large-Nc Witten–Veneziano formula is to our general DGMOR relation.

3. Theory

We now sketch the ‘U(1)A PCAC’ derivation of the decay formulae and DGMOR relations. For a more precise treatment in terms of functional chiral Ward identities and a complete renormalisation group analysis, as well as an effective Lagrangian formulation, we refer to our original papers [1,2,23].

√ 3 To be precise, the conjecture is that the contribution 6AgGγ γ in the decay formula will be small compared to 0η the dominant term f gηγγ. Note that as deﬁned the dimensions of gGγ γ and gηγγ are different, so they cannot be directly compared. G.M. Shore / Nuclear Physics B 744 (2006) 34–58 41

The starting point is the U(1)A chiral anomaly equation in pure QCD with nf flavours of massive quarks, viz.4 μ a = b + ∂ Jμ5 Mabφ5 2nf Qδa0, (3.1) a =¯ a a =¯ a where the axial vector current is Jμ5 qγμγ5T q and the pseudoscalar is φ5 qγ5T q.The corresponding chiral Ward identities for the two-point Green functions of interest are therefore (in momentum space): μ a c ik J Q − 2nf δa0QQ−Mac φ Q = 0, (3.2) μ5 5 μ a b − b − c b = ik Jμ5φ5 2nf δa0 Qφ5 Mac φ5φ5 Φab. (3.3) Since there are no massless pseudoscalar mesons in the theory, the terms in these identities involving the axial current vanish at zero momentum because of the explicit factors of kμ.The zero-momentum chiral Ward identities simply comprise the remaining terms. In particular, we have b d =− + MacMbd φ5 φ5 (MΦ)ab (2nf ) QQ δa0δb0. (3.4) We can also derive the result quoted in Eq. (2.11) for the topological susceptibility. In this notation, −A χ(0) ≡QQ= . (3.5) − −1 1 (2nf )A(MΦ)00 α = 0 0 a The physical mesons η (π ,η ,η ) couple to the pseudoscalar operators φ5 and Q,so their properties may be deduced from the two-point Green functions above. In the PCAC approximation, as explained in Section 2, the zero-momentum Ward identities are sufficient. In order to make the correspondence between the QCD operators and the physical mesons as close as possible, it is convenient to redefine linear combinations such that the propagator (two-point Green function) matrix is diagonal and properly normalised. We therefore define operators ηa and G such that b QQQφ GG 0 5 → , (3.6) a a b 0 ηαηβ φ5 Q φ5 φ5 This is achieved by taking = − a −1 b G Q Qφ5 φ5φ5 ab φ5 (3.7)

4 We use the following SU(3) notation for the quark masses and condensates: ⎛ ⎞ mu 00 ⎝ ⎠ a a 0 md 0 = m T 00ms a=0,3,8 and ⎛ ⎞ ¯uu 00 ⎝ 0 dd¯ 0 ⎠ = 2 φa T a, 00¯ss a=0,3,8 where φa is the VEV of φa =¯qT aq. It is also very convenient to introduce the compressed notation c c Mab = dacbm ,Φab = dabc φ . 42 G.M. Shore / Nuclear Physics B 744 (2006) 34–58 which reduces at zero momentum to = + −1 b G Q 2nf AΦ0b φ5 (3.8) and deﬁning α = Tαa −1 b η f Φab φ5 . (3.9) With this choice, the GG propagator at zero momentum is simply

GG=−A (3.10) and we demand that the ηαηβ propagator has the canonical normalisation 1 ηαηβ = δαβ . (3.11) 2 − 2 k mηα This normalisation fixes the decay constants introduced in Eq. (3.9). The DGMOR relations then follow immediately from the zero-momentum chiral Ward identity (3.4) and the expression (3.5) for the topological susceptibility. We find aα 2 Tβb = −1 =− + f mαβ f Φac φ5φ5 cd Φdb (MΦ)ab (2nf )Aδa0δb0. (3.12) Unwrapping the condensed SU(3) notation then shows that this matrix equation is simply the set of DGMOR relations (2.7)–(2.9). The next step is the PCAC calculation of ηα → γγ. We implement PCAC by the identification μ a → aα 2 β + ∂ Jμ5 f mαβ η 2nf Gδa0 (3.13) which follows from the anomaly Eq. (3.1) and the definitions of the fields G and ηα.Tobe precise, the → notation in Eq. (3.13) means that the identification can be made for insertions of the operators into zero-momentum Green functions and matrix elements only. Notice that it is not valid at non-zero momentum, in particular for on-shell matrix elements. It is the natural μ 3 → 2 generalisation of the familiar PCAC relation ∂ Jμ5 fπ mπ π, defining the phenomenological pion field π. The other input is the full axial anomaly for QCD coupled to electromagnetism, viz. α ∂μJ a = M φb + 2n Qδ + aa F μνF˜ , (3.14) μ5 ab 5 f a0 em 8π μν a where aem are the anomaly coefficients given in Section 2. Implementing the PCAC relation (3.13) together with the full anomaly equation, we therefore find μ | a | ik γγ Jμ5 0 α = f aαm2 γγ|ηβ |0+ 2n γγ|G|0δ + aa γγ|F μνF˜ |0 αβ f a0 em 8π μν α = f aαm2 ηβ ηγ γγηγ + 2n GGγγ|Gδ + aa γγ|F μνF˜ |0 (3.15) αβ f a0 em 8π μν at zero momentum, where we have used the fact that the propagators are diagonal in the ηα, G basis. The l.h.s. vanishes at zero momentum as there is no massless particle coupling to the axial current. Then, using the explicit expressions (3.10) and (3.11) for the G and ηα propagators, G.M. Shore / Nuclear Physics B 744 (2006) 34–58 43 we find the decay constant formulae: aα a α f g α + 2n Ag δ = a . (3.16) η γγ f Gγ γ a0 em π

The novel coupling gGγ γ is precisely defined from the matrix element γγ|G in analogy with Eq. (2.3) for the conventional couplings. Finally, notice that the mixing of states is conjugate to the mixing of fields. In particular, the mixing for the states corresponding to Eqs. (3.8) and (3.9) for the fields G and ηα is

|G=|Q (3.17) and α = −1 αa b − | η f Φab φ5 2nf Aδa0 Q . (3.18) In this sense, we see that we can regard the physical η (and, with SU(3) breaking, the η)asan admixture of quark and gluon components, while the unphysical state |G is purely gluonic. This is why we can usefully picture the unphysical coupling gGγ γ as the coupling of the photons to the anomaly-induced gluonic component of the η, as already mentioned in Section 2.

4. Phenomenology

In this section, we use the experimental data on the radiative decays η,η → γγ to deduce values for the pseudoscalar meson decay constants f 0η , f 0η, f 8η and f 8η from the set of decay formulae (2.5), (2.6) and DGMOR relations (2.14)–(2.16). We will also ﬁnd the value of the unphysical coupling parameter gGγ γ and test the realisation of the 1/Nc expansion in real QCD. The two-photon decay widths are given by 3 mη(η) Γ η (η) → γγ = |g |2. (4.1) 64π η (η)γ γ The current experimental data, quoted in the Particle Data Group tables [30],are Γ(η → γγ)= 4.28 ± 0.19 keV (4.2) which is dominated by the 1998 L3 data [31] on the two-photon formation of the η in e+e− → e+e−π +π −γ , and

Γ(η→ γγ)= 0.510 ± 0.026 keV (4.3) which arises principally from the 1988 Crystal Ball [32] and 1990 ASP [33] results on e+e− → e+e−η. (Notice that we follow the note in the 1994 PDG compilation [34] and use only the two-photon η production data.) From this data, we deduce the following results for the couplings gηγγ and gηγ γ :

−1 gηγγ = 0.031 ± 0.001 GeV (4.4) and

−1 gηγ γ = 0.025 ± 0.001 GeV (4.5) which may be compared with gπγγ = 0.024 ± 0.001 GeV. 44 G.M. Shore / Nuclear Physics B 744 (2006) 34–58

We also require the pseudoscalar meson masses:

mη = 957.78 ± 0.14 MeV,mη = 547.30 ± 0.12 MeV,

mK = 493.68 ± 0.02 MeV,mπ = 139.57 ± 0.00 MeV (4.6) and the decay constants fπ and fK . These are deﬁned in the standard way, so we take the following values (in our normalisations) from the PDG [30]:

fK = 113.00 ± 1.03 MeV,fπ = 92.42 ± 0.26 MeV (4.7) giving fK /f π = 1.223 ± 0.012. The ﬁnal input, as explained in Section 2, is the lattice calculation of the topological susceptibility in pure Yang–Mills theory. The most recent value, obtained in Ref. [28],is

4 −3 4 χ(0)|YM =−(191 ± 5MeV) =−(1.33 ± 0.14) × 10 GeV . (4.8) 4 This supersedes the original value χ(0)|YM −(180 MeV) obtained some time ago [35].Sim- ilar estimates are also obtained using QCD spectral sum rule methods [36]. Using the argument explained in Section 2, we therefore adopt the value − A = (1.33 ± 0.14) × 10 3 GeV4 (4.9) for the non-perturbative parameter determining the topological susceptibility in full QCD. The strategy is now to solve the set of ﬁve simultaneous equations (2.5), (2.6) and (2.14), 0η 0η 8η η 5 (2.15), (2.16) for the ﬁve remaining unknowns f , f , f , f and gGγ γ . The results are : f 0η = 104.2 ± 4.0MeV,f0η = 22.8 ± 5.7MeV, f 8η =−36.1 ± 1.2MeV,f8η = 98.4 ± 1.4MeV (4.10) that is f 0η f 0η = 1.13 ± 0.04, = 0.25 ± 0.06, fπ fπ f 8η f 8η =−0.39 ± 0.01, = 1.07 ± 0.02 (4.11) fπ fπ and

−4 gGγ γ =−0.001 ± 0.072 GeV . (4.12) These are the main results of this paper. Before going on to consider their signiﬁcance, we can re-express the results for the decay constants in terms of one of the two-angle parametrisations used in the literature. We reiterate that in our view there is no particular virtue in parametrising in this way, but in order to help

5 Note that this analysis includes the errors from the experimental inputs and the lattice evaluation of χ(0)|YM,butnot the systematic effect of the approximation A =−χ(0)|YM. The errors on the singlet decay constants are dominated by the error on A. Isolating this, we have

f 0η = 104.2 ± 0.3 ± 4.0 MeV,f0η = 22.8 ± 3.5 ± 4.5 MeV.

The octet decay constants are of course unaffected by the value of A. G.M. Shore / Nuclear Physics B 744 (2006) 34–58 45 comparison with other analyses it may be helpful to present our results in this form as well. Refs. [14,15] deﬁne f 0η f 0η f cos θ −f sin θ = 0 0 0 0 8η 8η . (4.13) f f f8 sin θ8 f8 cos θ8 Translating from Eq. (4.11), we ﬁnd

f0 = 106.6 ± 4.2MeV,f8 = 104.8 ± 1.3MeV,

θ0 =−12.3 ± 3.0deg,θ8 =−20.1 ± 0.7deg (4.14) that is f f 0 = 1.15 ± 0.05, 8 = 1.13 ± 0.02. (4.15) fπ fπ We emphasise, however, that since our definitions of the decay constants differ from those in Refs. [14,15], any comparison of the numbers above should be made with care. The novel feature of this analysis is the use of the flavour singlet DGMOR relation. We emphasise again that this is, within the framework of PCAC or chiral Lagrangians, an exact result in the sense of being entirely independent of the 1/Nc expansion. Its large-Nc limit is the well- established Witten–Veneziano formula. This determination of the decay constants is therefore on firm theoretical ground and provides a sound basis for phenomenology.6 In fact, the only way in which the 1/Nc approximation is involved here is in justifying the use of the Yang–Mills topological susceptibility to provide the input value for the non-perturbative parameter A. In time, lattice calculations should be able to determine this parameter accurately from simulations in full QCD with massive, dynamical quarks, making our analysis entirely independent of 1/N . In the meantime, to illustrate the dependence of the decay constants on the c topological susceptibility, we have plotted the singlet decay constants f 0η and f 0η against A in Fig. 1. It is clear from the DGMOR and radiative decay relations that the octet decay constants f 8η and f 8η are themselves independent of A. The most striking numerical feature of the results (4.10) for the decay constants is how close 0η 8η the diagonal ones, f and f ,aretofπ , even the singlet. Predictably, the off-diagonal ones are strongly suppressed, especially f 0η. This supports the approximations used in Section 2 in deriving the large Nc Witten–Veneziano limit of the DGMOR relations. To see this in more detail, it is interesting to compare the magnitudes of the various terms appearing in the DGMOR relations, together with their formal orders in 1/Nc. We find (all terms in units of 10−3 GeV4): 0η 2 2 [ ; ]+ 0η 2 2[ ; ] f mη Nc 9.96 f mη Nc 0.15 1 2 = f 2m2 [N ; 0.06]+ f 2 m2 [N ; 2.07]+6A[1; 7.98], (4.16) 3 π π c 3 K K c f 0η f 8η m 2[N ;−3.45]+f 0ηf 8ηm2[N ; 0.67] √ η c √ η c 2 2 2 2 = f 2m2 [N ; 0.16]− f 2 m2 [N ;−2.94], (4.17) 3 π π c 3 K K c

6 Note that the PDG tables currently do not quote values for the flavour singlet decay constants because of the subtleties in their definition. A good case can therefore be made for adopting the definitions and experimental numbers presented here as a simple and theoretically well-motivated parametrisation of the data. 46 G.M. Shore / Nuclear Physics B 744 (2006) 34–58

Fig. 1. The decay constants f 0η and f 0η as functions of the non-perturbative parameter A = (x MeV)4 which determines the topological susceptibility in QCD.

8η 2 2[ ; ]+ 8η 2 2[ ; ] f mη Nc 1.19 f mη Nc 2.90 1 4 =− f 2m2 [N ;−0.06]+ f 2 m2 [N ; 4.15]. (4.18) 3 π π c 3 K K c The interesting feature here is the explicit demonstration that the dominant term in the ﬂavour singlet DGMOR (Witten–Veneziano) relation is the topological susceptibility factor 6A,even though it is formally suppressed by a power of 1/Nc relative to the others. It is matched by the 0η 2 2 subdominant (O(1)) contribution to (f ) mη , which arises because of the numerically large 2 but O(1/Nc) anomaly-induced part of the mη which survives in the chiral limit. The numerical results therefore conﬁrm the theoretical intuition expressed in Section 2. To emphasise this point further, we can summarise the numerical magnitudes in the combined singlet–octet relation (2.19), which reduces to the full Witten–Veneziano formula (2.18): 0η 2 2 [ ; ]+ 0η 2 2[ ; ]+ 8η 2 2[ ; ] f mη Nc 9.96 f mη Nc 0.15 f mη Nc 1.19 + 8η 2 2[ ; ]− 2 2 [ ;− ]= [ ; ] f mη Nc 2.90 2fK mK Nc 6.22 6A 1 7.98 . (4.19) G.M. Shore / Nuclear Physics B 744 (2006) 34–58 47

Fig. 2. This shows the relative sizes of the contributions to the ﬂavour singlet radiative decay formula (2.5)√ expressed as functions of the topological susceptibility parameter A = (x MeV)4. The dotted (black) line denotes 2√ 2 αem .The 3 π 0η dominant contribution comes from the term f gη γγ, denoted by the long-dashed√ (green) line, while the short-dashed 0η (blue) line denotes f gηγ γ . The contribution from the gluonic coupling, 6AgGγ γ , is shown by the solid (red) line.

The validity of the large Nc limit leading to Eq. (2.18) is particularly transparent in this form. Numerically, the surprising accuracy of the approximate formula (2.18) is seen to be in part due 8η to a cancellation between the underestimates of f (taken to be 0) and fK (set equal to fπ ). Now consider the radiative decay formulae themselves. Here, because the η is not a NG boson even in the chiral limit, the flavour singlet PCAC formula (2.5) acquires an extra term. In our formulation, this is the parameter gGγ γ which we have argued may reasonably, though certainly non-rigorously, be interpreted as the coupling of the photons to the anomaly-induced gluonic component of the η, i.e., the component which removes its NG boson status. To illustrate the relative magnitudes of the various terms contributing to Eq. (2.5),wehave first plotted them in Fig. 2 as functions of the topological susceptibility parameter A. Inserting the input value (4.9) for A, the numerical magnitudes and 1/Nc orders of the various contributions to the decay formulae are (in units of 10−3): √ 0η 0η f gηγγ[Nc; 3.23]+f gηγ γ [Nc; 0.57]+ 6AgGγ γ [1;−0.005 ± 0.23] α = a0 em [N ; 3.79] (4.20) em π c and αem f 8η g [N ;−1.12]+f 8ηg [N ; 2.46]=a8 [N ; 1.34]. (4.21) η γγ c ηγ γ c em π c The interest here is in the realisation of the 1/Nc approximation in the flavour singlet decay 7 formula. As explained in Section 2, the coupling gGγ γ is renormalisation group invariant and O(1/Nc) suppressed and our conjecture is that such terms would indeed be relatively small. Remarkably, evaluated at the central value of the topological susceptibility found in Ref. [28], the coupling gGγ γ is essentially zero. This is probably a numerical coincidence, since we cannot think of a dynamical reason why this coupling should vanish identically. What is more reasonable

7 The topological susceptibility parameter A is also renormalisation group invariant. 48 G.M. Shore / Nuclear Physics B 744 (2006) 34–58 is to consider its value across the range of error of the topological susceptibility. In this case, we see from Fig. 2 that the suppression is numerically still under 10%, which is closer to that expected for a typical OZI correction although still remarkably small. This is a very encouraging result. First, it increases our confidence that we are able to identify quantities where the OZI, or leading 1/Nc, approximation is likely to be numerically good. It also shows that gGγ γ gives a contribution to the decay formula which is entirely consistent with its picturesque interpretation as the coupling of the photons to the anomaly-induced gluonic component of the η. A posteriori, the fact that its contribution is at most 10% explains the general success of previous theoretically inconsistent phenomenological parametrisations of η decays in which the naive current algebra formulae omitting the gluonic term are used. Nevertheless, while the flavour singlet decay formula is well defined and theoretically consistent, it is necessarily non-predictive. To be genuinely useful, we would need to find another process in which the same coupling enters. The problem here is that, unlike the decay constants which are universal, the coupling gGγ γ is process-specific just like gηγγ or gηγ γ . There are of course many other processes to which our methods may be applied such as η(η) → Vγ, where V is a flavour singlet vector meson ρ,ω,φ,orη(η) → π+π −γ . The required flavour singlet formulae may readily be written down, generalising the naive PCAC formulae. However, each will introduce its own gluonic coupling, such as gGV γ . Although strict predictivity is lost, our experience with the two-photon decays suggests that these extra couplings will give relatively small, at most O(10–20%), contributions if like gGγ γ they can be identified as RG invariant and 8 1/Nc suppressed. This observation restores at least a reasonable degree of predictivity to the use of PCAC methods in the U(1)A sector. Interestingly, these novel gluonic couplings may also arise in high-energy processes. For example, the standard analysis of the two-photon deep-inelastic scattering process e+e− → e+e−X γ reduces the problem of finding the first moment of the polarised photon structure function g1 to | 0 | a non-perturbative evaluation of the off-shell matrix element γ Jμ5 γ . The difference with η(η) → γγ is that in the DIS scenario, the photons are off-shell and the interest in the first γ moment sum rule for g1 is precisely how it depends on the target photon momentum [7–9].Nev- 2 2 ertheless, the problem may be formulated in terms of form factors gη(η)γ γ (k ) and gGγ γ (k ) of which the couplings discussed here are simply the k2 = 0 limit [9]. It has also been suggested that the gluonic couplings gGφγ and gGNN could play a dominant role in the photoproduction process γN → φN [37]. This interpretation is less clear, but it is an interesting subject for future work to look at a variety of electro or photoproduction experiments in the light of the PCAC methods developed here.

5. Pseudoscalar meson couplings of the nucleon and the U(1)A Goldberger–Treiman relation

A further particularly interesting application of these ideas is to the pseudoscalar couplings of the nucleon. For the pion, the relation between the axial-vector form factor of the nucleon and the pion–nucleon coupling gπNN is the well-known Goldberger–Treiman relation. Here, we are concerned with its generalisation to the ﬂavour-singlet sector, which involves the anomaly and

8 Notice though that this must be used carefully. For example, the coupling gGNN which enters the U(1)A Goldberger– Treiman relation alongside the η -nucleon coupling gηNN is 1/Nc suppressed but not RG invariant and is expected to display large deviations from its large Nc or OZI limit. See Section 5. G.M. Shore / Nuclear Physics B 744 (2006) 34–58 49 gluon topology. This U(1)A Goldberger–Treiman relation was first developed in Refs. [11–13]. In this case, the corresponding high-energy process involves the measurement of the first moment N of the polarised structure function of the nucleon g1 in deep-inelastic scattering. In the flavour- singlet sector, this is the so-called ‘proton spin’ problem. (For reviews, see, e.g., Refs. [38,39].) The axial-vector form factors are defined from | a | = a 2 + a 2 N Jμ5 N 2mN GA k sμ GP k k.skμ , (5.1) where sμ =¯uγμγ5u/2mN is the covariant spin vector. In the absence of a massless pseudoscalar, only the form factors GA(0) contribute at zero momentum. Using the ‘U(1)A PCAC’ substitution μ a (3.13) for ∂ Jμ5 and repeating the steps explained in Section 3 (particularly at Eq. (3.15)), we straightforwardly find the following generalisation of the Goldberger–Treiman relation: a = aα + 2mN GA(0) f gηαNN 2nf AgGNN δa0 (5.2) with the obvious definition of the gluonic coupling gGNN in analogy to gηαNN. a = a For the individual flavour components, this reads (abbreviating GA(0) GA): 3 = 2mN GA fπ gπNN, (5.3) 8 = 8η + 8η 2mN GA f gη NN f gηNN , (5.4) √ 0 = 0η + 0η + 2mN GA f gη NN f gηNN 6AgGNN . (5.5) 9 Eq. (5.5) is the U(1)A Goldberger–Treiman relation. Notice that the flavour-singlet coupling 0 GA is not renormalisation group invariant and so depends on the RG scale. This is reflected in the RG non-invariance of the gluonic coupling gGNN [13]. In the notation that has become standard in the DIS literature, the axial couplings are 1 1 1 G3 = a3,G8 = √ a8,G0 = √ a0 (5.6) A 2 A 2 3 A 6 and have the following interpretation in terms of parton distribution functions:

a3 = u − d, a8 = u + d − 2s, 3α a0 = u + d + s − g. (5.7) 2π Experimentally,

a3 = 1.267 ± 0.004,a8 = 0.585 ± 0.025 (5.8)

9 The original form as quoted in Ref. [13] applies to the chiral limit and reads, in the notation of [13] but allowing for our different normalisation of the singlet,

1 0 = + 2 2 2mN GA Fgη NN F mη gGNN , 2nf where F is a RG invariant decay constant defined from the two-point Green function of the pseudoscalar field φ .Inthe 5 chiral limit, where there is no SU(3) mixing, this is reproduced by the definition (3.12) of f 0η . The off-diagonal decay constant f 0η vanishes. The final term is reproduced by Eq. (5.5) by virtue of the flavour-singlet DGMOR relation (2.7) in the chiral limit. 50 G.M. Shore / Nuclear Physics B 744 (2006) 34–58 from low-energy data, while the latest result for a0 quoted by the COMPASS Collaboration [40, 41] is 0 = +0.024 a Q2=4 GeV2 0.237−0.029. (5.9) It is the fact that a0 is much less than a8, as would be predicted on the basis of the simple quark model (the Ellis–Jaffe sum rule [42]), that is known as the ‘proton spin’ problem. For a careful analysis of the distinction between the angular momentum (spin) of the proton and the axial coupling a0, see however Refs. [43,44]. From the standard Goldberger–Treiman relation (5.3), we immediately find the following result for the (dimensionless) pion–nucleon coupling,

gπNN = 12.86 ± 0.06 (5.10) consistent to within ∼ 5% with the experimental value 13.65(13.80) ± 0.12 (depending on the dataset used) [45]. In an ideal world where gηNN and gηNN were both known, we would now verify the octet formula (5.4) then determine the gluonic coupling gGNN from the singlet Goldberger–Treiman relation (5.5). However, the experimental situation with the η- and η-nucleon couplings is far less clear. (See Refs. [46,47] for reviews of the relevant experimental literature and recent results.) One would hope to determine these couplings from the near threshold production of the η and η in nucleon–nucleon collisions, i.e., pp → ppη and pp → ppη, measured for example at COSY- ∗ II [48]. However, the η production is dominated by the S11 nucleon resonance N (1535) which decays to Nη, and as a result very little is known about gηNN itself. The detailed production mechanism of the η is not well understood. However, since there is no known baryonic resonance decaying into Nη, we may simply assume that the reaction pp → ppη is driven by the direct coupling supplemented by heavy-meson exchange. This allows an upper bound to be placed on gηNN and on this basis Ref. [49] quotes gηNN < 2.5. This is supported by an analysis [50] of very recent data from CLAS [51] on the photoproduction reaction γp → pη. Describing the cross-section data with a model comprising the direct coupling together with t-channel meson exchange and s- and u-channel resonances, it is found that equally good ﬁts can be obtained for several values of gηNN covering the whole region 0

10 More precisely, we expect the OZI approximation to be unreliable for quantities which have a different RG behaviour in QCD itself and in the OZI limit. The complicated RG non-invariance of gGNN in QCD is induced by the axial G.M. Shore / Nuclear Physics B 744 (2006) 34–58 51

Fig. 3. These figures show the dimensionless η-nucleon coupling gηNN and the gluonic coupling gGNN in units of −3 GeV expressed as functions of the experimentally uncertain η -nucleon coupling gηNN, as determined from the flavour octet and singlet Goldberger–Treiman relations (5.4) and (5.5). radiative decay formula, we would not be surprised if gGNN makes a sizeable numerical contri- 11 bution to the U(1)A GT relation. This is quantified in Fig. 4, where we have plotted the contribution of each term in the U(1)A 0η GT relation as a function of gηNN. We see immediately that the contribution from f gηNN is 0 ∼ relatively constant around 0.08, compared with 2mN GA 0.18. This means that the variation 0η of √f gηNN over the experimentally allowed range is compensated entirely by the variation of 6AgGNN . For generic values of gηNN, there is no sign of a significant suppression of the contribution of the gluonic coupling gGNN relative to the others. This should be contrasted with the corresponding plot for gGγ γ in the radiative decay formula (Fig. 2). To see this in more detail, consider a representative value, gηNN = 2.0, which would correspond to the direct coupling contributing substantially to the cross sections for pp → ppη and −3 γp → pη . In this case, gηNN = 3.96±0.16 and gGNN =−14.6±4.3GeV . The contributions

0 0 anomaly, since GA itself is required to scale with the anomalous dimension γ of the ﬂavour-singlet current Jμ5 and all the other terms in the U(1)A GT relation are RG invariant. The anomaly, and thus the anomalous dimension γ ,vanishes in the large-Nc limit leaving gGNN RG invariant as Nc →∞. 11 −3 Of course, since gGNN is deﬁned with dimension GeV whereas gηNN and gηNN are dimensionless, we cannot make a direct comparison of the couplings themselves. 52 G.M. Shore / Nuclear Physics B 744 (2006) 34–58

Fig. 4. This shows the relative sizes of the contributions to the U(1)A Goldberger–Treiman relation from the individ- 0 ual terms in Eq. (5.5), expressed as functions of the coupling gη NN. The dotted (black) line denotes 2mN GA.The 0η 0η long-dashed (green) line is f gη NN and√ the short-dashed (blue) line is f gηNN . The solid (red) line shows the contribution of the novel gluonic coupling, 6AgGNN ,whereA determines the QCD topological susceptibility. to the U(1)A GT relation are then (in GeV): 2m G0 [N ; 0.18] N A c √ 0η 0η = f gηNN[Nc; 0.21]+f gηNN [Nc; 0.09]+ 6AgGNN O(1);−0.12 . (5.11) 0 8 The anomalously small value of GA compared to GA is due to the partial cancellation of the sum of the two meson-coupling terms, which together contribute close to the expected OZI 8 = value (2mN GA 0.32), by the gluonic coupling gGNN . Although this is formally O(1/Nc) sup- 0 pressed, numerically it gives the dominant contribution to the large OZI violation in GA.Thisis in line with our expectations and would provide further evidence that the insights developed in our body of work on both low-energy η and η physics and related high-energy phenomena such as the ‘proton spin’ problem are on the right track. However, it may still be that gηNN is significantly smaller, implying a relatively small contribution to the production reactions pp → ppη and γp → pη from the direct coupling. In particular, there is a range of gηNN around 0.7–1.3 where the gluonic coupling gGNN only contributes to the U(1)A GT relation at the level expected of a typical OZI-suppressed quantity, despite its RG non-invariance. In the extreme case where gηNN = 1.0, we have gηNN = 3.59 ± 0.15, −3 gGNN = 0.5 ± 3.8GeV , and the contributions to the U(1)A GT formula become (in GeV): 2m G0 [N ; 0.18] N A c √ 0η 0η = f gηNN[Nc; 0.10]+f gηNN [Nc; 0.08]+ 6AgGNN O(1);−0.004 . (5.12) This scenario would be similar to the radiative decays, where we found that the corresponding coupling gGγ γ 0 using the central value of the lattice determination of the topological susceptibility and contributes only at O(10%) within the error bounds on A. This would then suggest that RG non-invariance is not critical after all and the O(1/Nc) suppressed gluonic couplings are indeed numerically small. It would also leave open the possibility that all couplings of type gGXX are close to zero, which in the picturesque interpretation discussed earlier would imply 0 that the gluonic component of the η wave function is small. The suppression of GA relative to 8 GA would then not be due to gluonic, anomaly-induced OZI violations but rather to the particular G.M. Shore / Nuclear Physics B 744 (2006) 34–58 53 nature of the flavour octet–singlet mixing in the η–η sector. In the parton picture of the ‘proton spin’ problem, this would be a hint that the suppression in a0 is not primarily due to the polarised gluon distribution g but also involves a strong contribution from the polarised strange quark distribution s. Although we would consider this alternative scenario rather surprising and prefer the more theoretically motivated interpretation of the flavour singlet sector in which gηNN 2 and gluon topology plays an important role, ultimately the decision rests with experiment. Clearly, a reliable determination of gηNN, or equivalently gηNN , would shed considerable light on the U(1)A dynamics of QCD.

Acknowledgements

I would like to thank G. Veneziano for interesting comments and collaboration on the original investigations of U(1)A physics on which this paper is based. This work is supported in part by PPARC grants PPA/G/O/2002/00470 and PP/D507407/1.

Appendix A. Comparison with chiral Lagrangians

It is useful to compare the results presented in this paper with those arising in extensions of the chiral Lagrangian formalism to include the η and low-energy flavour singlet physics. The large-Nc expansion plays a crucial role in this approach, since it is only in the limit Nc →∞that the anomaly disappears, the chiral symmetry is enlarged to U(3)L × U(3)R and the η appears as a light NG degree of freedom to be included in the fundamental fields ϕa (a = 0, 1,...,8) of the chiral Lagrangian. Large-Nc chiral Lagrangians have been developed by a number of authors, notably Ref. [21], though here we shall focus on the results obtained by Kaiser and Leutwyler [14–16]. The elegance of chiral Lagrangians should not obscure the fact that all the dynamical approximations we have made in deriving our results, such as pole dominance by NG bosons, weak momentum dependence of pole-free quantities like decay constants and couplings and a judicious use of the 1/Nc expansion, are necessarily also made in the chiral Lagrangian formalism, where they are built in to the structure of the initial Lagrangian. Indeed, our results can also be systematised into an effective Lagrangian (see Ref. [1] for details). The principal merit of the chiral/effective Lagrangian approach in general is in providing a systematic way of calculating higher-order corrections. The physical results obtained using the two methods should therefore be equivalent. However, a number of our definitions, notably of the flavour singlet decay constants, differ from those made by Kaiser and Leutwyler so the comparison is not straightforward. However, we would argue that in many ways the formalism developed in this paper provides a better starting point for the description of U(1)A phenomenology, especially in the use of RG invariant decay constants and our natural generalisation of the Witten–Veneziano formula as the flavour singlet DGMOR relation. The fundamental fields in the KL chiral Lagrangian are assembled into matrices U = exp[iϕaT a] so that, up to mixing, the fields are in one-to-one correspondence with the NG 54 G.M. Shore / Nuclear Physics B 744 (2006) 34–58

12 bosons in the chiral and large-Nc limits. The Lagrangian is a simultaneous expansion in three parameters—momentum (p), quark mass (m) and 1/Nc. For bookkeeping purposes, KL consider 2 these to be related as follows: p = O(δ), m = O(δ),1/Nc = O(δ), and expand consistently in the small parameter δ. It will be clear, however, that this is mere bookkeeping and should be treated with considerable caution. As we have seen, the realisation of the 1/Nc expansion in the singlet sector is extremely delicate and it cannot simply be assumed that quantities, especially RG non-invariant ones, that are 1/Nc suppressed are necessarily numerically small or indeed of O(p2,m). Nevertheless, arranging the allowed terms in the chiral Lagrangian according to their order in δ, KL ﬁnd

L = L0 + L1 + L2 +···, (A.1) where 1 1 3 L = F 2 tr ∂ U †∂μU + F 2B tr M†U + U †M − τ ϕ0 2 (A.2) 0 4 μ 2 4 and L = 2BL tr ∂ U †∂μU M†U + U †M + 4B2L tr M†U 2 + U †M 2 1 5 μ 8 1 i 3 + F 2Λ ∂μϕ0∂ ϕ0 + F 2BΛ ϕ0 tr M†U − U †M . (A.3) 8 1 μ 6 2 2 Most of the physics we are interested in can be derived from L0 + L1. However, we also en- counter some of the terms in the next order, † μ † † L2 = 2BL4 tr ∂μU ∂ U tr M U + U M + 4B2L tr M†U + U †M 2 + 4B2L tr M†U − U †M 2 6 7 3 0 † μ μ † − 2i BL18∂μϕ tr M ∂ U − ∂ U M 2 3 − 4i B2L ϕ0 tr M†U 2 + U †M 2 +···. (A.4) 2 25 Here, M is the quark mass matrix, B sets the scale of the quark condensate, F is the leading-order decay constant before SU(3) breaking, and τ is a parameter which is identified at leading order in 2 1/Nc with the topological susceptibility of pure Yang–Mills. Their 1/Nc orders are: F = O(Nc) and B,τ = O(1). The coefficients entering at higher order have the following dependence [16, 20]: Λ1,Λ2 = O(1/Nc), L5,L8 = O(Nc) and L4,L6,L7,L18,L25 = O(1). a KL define their decay constants FP (P = π, K, η, η ) in the conventional way as the couplings to the axial-vector currents: | a | = a 0 Jμ5 P ikμFP . (A.5) 0 0 As a consequence, the singlet decay constants Fη and Fη are not RG invariant but scale with the usual anomalous dimension γ corresponding to the multiplicative renormalisation of the singlet

12 We use the same normalisation√ for the generators as in the rest of this paper, so our singlet ﬁeld differs from the ψ of Refs. [14–16] by ϕ0 = 2/3ψ. The normalisation of the singlet currents and decay constants is, however, the same. We assume isospin symmetry and represent the quark mass matrix by M = diag(mu,md ,ms ) with mu = md = m. G.M. Shore / Nuclear Physics B 744 (2006) 34–58 55 current. It follows that the parameters (1 + Λ1) and τ are also not RG invariant, scaling with anomalous dimension 2γ .13 This means that while τ coincides with A (the non-perturbative parameter determining the QCD topological susceptibility) at O(1), it differs beyond leading order in 1/Nc. The SU(3) breaking which distinguishes the decay constants arises ﬁrst from the terms in L1. Beyond this order, in addition to the direct contributions from the new couplings in the chiral Lagrangian, there are also contributions from loop diagrams calculated using L0. These give rise m2 m2 to the ‘chiral logarithms’ μ = p ln P , which KL have calculated explicitly. To present P 32π2F 2 μ2 their results, we again use the two-angle parametrisation (cf. Eq. (4.13)) in the octet–singlet sector: 0 0 Fη Fη F cos ϑ −F sin ϑ = 0 0 0 0 . (A.6) 8 8 F8 sin ϑ8 F8 cos ϑ8 Fη Fη To this order, the KL decay constants are then [15]: 4B F = F 1 + 2 m L + 2mL + O(μ ) , (A.7) π F 2 q 4 5 P 4B F = F 1 + 2 m L + (m + m )L + O(μ ) , (A.8) K F 2 q 4 s 5 P 4B 2 F = F 1 + 2 m L + (m + 2m )L + O(μ ) . (A.9) 8 F 2 q 4 3 s 5 P For the singlet, ¯ F0 = 1 + Λ1F0, (A.10) where the scale-invariant part is 4B 2 F¯ = F 1 + 2 m L + (2m + m )(−L + L ) , (A.11) 0 F 2 q 4 3 s 5 A √ where LA = (2L5 + 3L18)/ 1 + Λ1 = 2L5 + O(1). Notice there are no loop corrections to F0. Finally, the difference in the angles ϑ0 and ϑ8 is determined from √ 0 8 0 8 8 2 F F + F F =−F F sin(ϑ − ϑ ) = B(m − m )(2L + 3L ) (A.12) η η η η 0 8 0 8 3 s 5 18 and is proportional to the SU(3) breaking m, ms mass difference. Also recall [18] that at lowest 2 = 2 = + 2 = 2 + order, the pseudoscalar masses are mπ 2mB, mK (m ms)B, mη 3 (m 2ms)B and 2 = 2 + mη 3 (2m ms)B.

13 Explicitly, the parameters renormalise according to [16]

2 2 τR = Z τB , 1 + Λ1R = Z (1 + Λ1B ), 0 = 0 where Z is the usual multiplicative renormalisation factor for the axial current, Jμ5R ZJμ5B . In the KL chiral Lagrangian formalism, the singlet ﬁeld ϕ0 is itself renormalised. Allowing for a non-zero vacuum angle θ,thisﬁeld renormalisation is − 2 − ϕ0 = Z 1ϕ0 + Z 1 − 1 θ. R 3 56 G.M. Shore / Nuclear Physics B 744 (2006) 34–58

These decay constants satisfy a set of relations which are closely analogous, but not identical, to the DGMOR relations (2.14), (2.15) and (2.16). In fact, up to terms involving chiral logarithms, the decay constants shown above satisfy (see also Ref. [22]) ¯ 0 ¯ 0 + ¯ 0 ¯ 0 = ¯ 0 2 = 1 2 + 2 Fη Fη Fη Fη F Fπ 2FK , (A.13) 3 √ ¯ 0 8 ¯ 0 8 ¯ 0 8 2 2 2 2 F F + F F =−F F sin(ϑ − ϑ ) = F − F , (A.14) η η η η 0 8 3 π K 8 8 8 8 8 2 1 2 2 F F + F F = F =− F − 4F . (A.15) η η η η 3 π K The corresponding DGMOR relations, i.e., including the appropriate pseudoscalar mass terms, are broken also by the terms proportional to L7 and L8 (the L6 contributions cancel). These are just the expected O(m2) corrections to the leading-order PCAC relations. (See also the footnote following Eq. (2.16).) Our main interest, as always, is in the ﬂavour singlet sector. Notice that the relation (A.13) is written for the scale-invariant part of the singlet decay constants only, omitting the factor involving the OZI-violating coupling Λ1. This does not involve the topological susceptibility in any way and is not related to the Witten–Veneziano formula. This enters the formalism as follows. In the chiral limit and working to O(δ), the relevant part of the chiral Lagrangian is just 1 3 L ∼ F 2(1 + Λ )∂ ϕ0∂μϕ0 − τ ϕ0 2 (A.16) 8 1 μ 4 from which it follows immediately that 0 2 2 = 0 = + Fη mη 6τ, Fη 1 Λ1F. (A.17)

At leading order in 1/Nc, this is the original Witten–Veneziano formula, since Λ1 = O(1/Nc) and we may interpret τ =−χ(0)|YM +O(1/Nc). However, beyond leading order and away from the chiral limit, it does not have a straightforward generalisation in terms of the QCD topological susceptibility, since τ is not identical to A. Fundamentally, this originates from the use of an RG non-invariant field ϕ0 in the formulation of the chiral Lagrangian. In contrast, the corresponding formula (2.14), i.e., the flavour singlet DGMOR relation written with our RG invariant definition of f 0η and involving the parameter A determining the topological susceptibility in QCD itself, is the appropriate generalisation of the Witten–Veneziano relation and provides a more suitable basis for describing U(1)A phenomenology. These formulae allow a numerical determination of the decay constants and mixing angles in the octet–singlet sector. The results quoted in Ref. [25] are

F0 = 1.25fπ ,F8 = 1.28fπ ,

θ0 =−4deg,θ8 =−20.5deg. (A.18) These should be compared with our results, Eq. (4.14). Since the definitions of the octet decay constants are the same, we expect F8 f8 and ϑ8 θ8. Indeed, the angles agree while F8 is a little higher than f8, by around 10%. This is explicable by the fact that the KL fit incorporates the next-to-leading order corrections in the chiral expansion but not the radiative decays, whereas we have used the leading-order DGMOR relation together with the octet radiative decay formula. The definitions of the flavour singlet decay constant and mixing angle are different in the two approaches, so cannot be directly compared. G.M. Shore / Nuclear Physics B 744 (2006) 34–58 57

Radiative decays are described in the chiral Lagrangian approach by the Wess–Zumino– Witten term, which encodes the anomalous Ward identities. The relevant part, constructed by Kaiser and Leutwyler [16],is14 αN 1 L =− c tr eˆ2ϕ + Λ tr eˆ2 tr ϕ + 2BΛ tr eˆ2Mϕ F F˜ μν, (A.19) WZW 4π 3 3 4 μν = a a ˆ = 2 − 1 − 1 where ϕ ϕ T and e diag( 3 , 3 , 3 ) is the quark charge matrix. Λ3 is O(1/Nc) and scales in the same way as Λ1, while Λ4 is O(1) and RG invariant. It is then straightforward to derive the following formulae (omitting the O(m) corrections arising from the Λ4 term for simplicity)

0 0 0 α F g + F g = (1 + Λ )a , (A.20) η η γγ η ηγ γ 3 em π 8 8 8 α F g + F g = a . (A.21) η η γγ η ηγ γ em π These are the analogues of our Eqs. (2.5) and (2.6). The key point is that in each case, the flavour singlet formula has to incorporate OZI breaking by the inclusion of a new, process- specific parameter, Λ3 in the KL formalism and gGγ γ in our approach, which must be determined from the experimental data. In each case, this removes the predictivity of the formula unless, as we have discussed, we can bring theoretical arguments to bear to argue that the OZI- violating terms are small. In the KL case, this would presumably require the RG-invariant ratio (1 + Λ3)/(1 + Λ1) to be close to 1. However, once again, we consider that the flavour singlet decay formula presented here has the added virtue of explicitly showing the link with the topological susceptibility and giving a physical interpretation of the new OZI-violating parameter in terms of the gluonic component of the physical η meson.

References

[1] G.M. Shore, Nucl. Phys. B 569 (2000) 107, hep-ph/9908217. [2] G.M. Shore, G. Veneziano, Nucl. Phys. B 381 (1992) 3. [3] M. Gell-Mann, R.J. Oakes, B. Renner, Phys. Rev. 175 (1968) 2195. [4] R.F. Dashen, Phys. Rev. 183 (1969) 1245. [5] E. Witten, Nucl. Phys. B 156 (1979) 269. [6] G. Veneziano, Nucl. Phys. B 159 (1979) 213. [7] S. Narison, G.M. Shore, G. Veneziano, Nucl. Phys. B 391 (1993) 69. [8] G.M. Shore, G. Veneziano, Mod. Phys. Lett. A 8 (1993) 373. [9] G.M. Shore, Nucl. Phys. B 712 (2005) 411, hep-ph/0412192. [10] M.L. Goldberger, S.B. Treiman, Phys. Rev. 111 (1958) 354. [11] G. Veneziano, Mod. Phys. Lett. A 4 (1989) 1605. [12] G.M. Shore, G. Veneziano, Phys. Lett. B 244 (1990) 75. [13] G.M. Shore, G. Veneziano, Nucl. Phys. B 381 (1992) 23. [14] H. Leutwyler, Nucl. Phys. B (Proc. Suppl.) 64 (1998) 223, hep-ph/9709408. [15] R. Kaiser, H. Leutwyler, hep-ph/9806336. [16] R. Kaiser, H. Leutwyler, Eur. Phys. J. C 17 (2000) 623, hep-ph/0007101. [17] K. Kawarabayashi, N. Ohta, Nucl. Phys. B 175 (1980) 477; K. Kawarabayashi, N. Ohta, Prog. Theor. Phys. 66 (1981) 1789. [18] J. Gasser, H. Leutwyler, Ann. Phys. 158 (1984) 142; J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 465.

14 In Ref. [16], the constants Λ3 and Λ4 are called K1 and K2, respectively. The Λ3 notation is used in Refs. [14] and [22]. The radiative decay formulae (A.20), (A.21) are identical to those quoted in Ref. [22], Eqs. (39) rewritten in our notation. 58 G.M. Shore / Nuclear Physics B 744 (2006) 34–58

[19] J. Schechter, A. Subbaraman, H. Weigel, Phys. Rev. D 48 (1993) 339, hep-ph/9211239. [20] S. Peris, E. de Rafael, Phys. Lett. B 348 (1995) 539, hep-ph/9412343. [21] P. Herrera-Siklody, J.I. Latorre, P. Pascual, J. Taron, Nucl. Phys. B 497 (1997) 345, hep-ph/9610549; P. Herrera-Siklody, J.I. Latorre, P. Pascual, J. Taron, Phys. Lett. B 419 (1998) 326, hep-ph/9710268. [22] T. Feldmann, Int. J. Mod. Phys. A 15 (2000) 159, hep-ph/9907491. [23] G.M. Shore, Phys. Scr. T 99 (2002) 84, hep-ph/0111165. [24] T. Feldmann, P. Kroll, Eur. Phys. J. C 5 (1998) 327, hep-ph/9711231. [25] T. Feldmann, P. Kroll, B. Stech, Phys. Rev. D 58 (1998) 114006, hep-ph/9802409. [26] P. Di Vecchia, G. Veneziano, Nucl. Phys. B 171 (1980) 253. [27] G. Veneziano, Invited Talk Given at Okubofest: From Symmetries to Strings: Forty Years of Rochester Conferences, Rochester, NY, 1990, CERN-TH-5840-90. [28] L. Del Debbio, L. Giusti, C. Pica, Phys. Rev. Lett. 94 (2005) 032003, hep-th/0407052. [29] S. Narison, G.M. Shore, G. Veneziano, Nucl. Phys. B 546 (1999) 235, hep-ph/9812333. [30] Particle Data Group, Review of Particle Properties, Phys. Lett. B 592 (2004) 1. [31] L3 Collaboration, M. Acciarri, et al., Phys. Lett. B 418 (1998) 399. [32] Crystal Ball Collaboration, D.A. Williams, et al., Phys. Rev. D 38 (1988) 1365. [33] ASP Collaboration, N.A. Roe, et al., Phys. Rev. D 41 (1990) 17. [34] Particle Data Group, Review of Particle Properties, Phys. Rev. D 50 (1994) 1451. [35] A. Di Giacomo, Nucl. Phys. B (Proc. Suppl.) 23 (1991) 191. [36] S. Narison, Phys. Lett. B 255 (1991) 101; S. Narison, Z. Phys. C 26 (1984) 209. [37] N.I. Kochelev, V. Vento, Phys. Lett. B 515 (2001) 375; N.I. Kochelev, V. Vento, Phys. Lett. B 541 (2002) 281. [38] G.M. Shore, in: From the Planck Length to the Hubble Radius, Erice, 1998, pp. 79–105, hep-ph/9812355. [39] S.D. Bass, The spin structure of the proton, hep-ph/0411005. [40] G. Mallot, S. Platchkov, A. Magnon, CERN-SPSC-2005-017, SPSC-M-733. [41] COMPASS Collaboration, E.S. Ageev, et al., Phys. Lett. B 612 (2005) 154, hep-ex/0501073. [42] J.R. Ellis, R.L. Jaffe, Phys. Rev. D 9 (1974) 1444; J.R. Ellis, R.L. Jaffe, Phys. Rev. D 10 (1974) 1669, Erratum. [43] G.M. Shore, B.E. White, Nucl. Phys. B 581 (2000) 409, hep-ph/9912341. [44] G.M. Shore, Nucl. Phys. B (Proc. Suppl.) 96 (2001) 171, hep-ph/0007239. [45] D.V. Bugg, Eur. Phys. J. C 33 (2004) 505. [46] P. Moskal, Hadronic interaction of η and η mesons with protons, hep-ph/0408162. [47] S.D. Bass, Phys. Scr. T 99 (2002) 96, hep-ph/0111180. [48] P. Moskal, et al., Int. J. Mod. Phys. A 20 (2005) 1880, hep-ex/0411052. [49] P. Moskal, et al., Phys. Rev. Lett. 80 (1998) 3202, nucl-ex/9803002. [50] K. Nakayama, H. Haberzettl, Analyzing η photoproduction data on the proton at energies of 1.5–2.3 GeV, nucl-th/ 0507044. [51] CLAS Collaboration, M. Dugger, S = 0 pseudoscalar meson photoproduction from the proton, nucl-ex/0512005. Nuclear Physics B 744 (2006) 59–79

Resummation for polarized semi-inclusive deep-inelastic scattering at small transverse momentum

Yuji Koike a,∗, Junji Nagashima a, Werner Vogelsang b

a Department of Physics, Niigata University, Ikarashi, Niigata 950-2181, Japan b Physics Department and RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA Received 22 February 2006; accepted 13 March 2006 Available online 3 April 2006

Abstract We study the transverse-momentum distribution of hadrons produced in semi-inclusive deep-inelastic scattering (SIDIS). We consider cross sections for various combinations of polarizations of the initial lepton and nucleon or the produced hadron, for which we perform the resummation of large double-logarithmic perturbative corrections arising at small transverse momentum. We present phenomenological results for the processes lp → lπX with longitudinally polarized leptons and protons. We discuss the impact of the perturbative resummation and of estimated non-perturbative contributions on the corresponding cross sections and their spin asymmetry. Our results should be relevant for ongoing studies in the COMPASS experiment at CERN, and for future experiments at the proposed eRHIC collider at BNL. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

Our knowledge about the structure of hadrons has been vastly improved by experiments with polarized high-energy lepton beams scattering off polarized nucleon targets. Spin observables in deeply-inelastic lepton–nucleon collisions allow us to extract the spin-dependent parton distributions of the nucleon. At the same time, they challenge our understanding of the reaction mechanism within QCD, and our ability to perform reliable theoretical calculations of the relevant cross sections for each process and kinematic region of interest. Of particular interest in this respect are observables in semi-inclusive deeply-inelastic scattering (SIDIS), lp → lhX, for which a hadron h is detected in the ﬁnal state. Depending on the type

* Corresponding author. E-mail address: [email protected] (Y. Koike).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.03.009 60 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 of hadron considered, various different aspects of nucleon structure may be probed. Two current lepton scattering experiments, HERMES at DESY and COMPASS at CERN, employ this method extensively. Measurements of spin asymmetries for longitudinally polarized beam and target, integrated over all transverse momenta of the produced hadron, have served to allow conclusions about the helicity-dependent up, down, and strange quark and anti-quark distributions in the nucleon [1]. It has also been recognized that distributions in the hadron’s transverse momentum can be of great interest, in particular when the nucleon target is transversely polarized [2]. The associated experimental investigations by HERMES [3],COMPASS[4], the SMC [5], and CLAS [6] have been remarkably productive and have opened windows on novel QCD phenomena such as the Sivers [7] and Collins [8] effects. It is hoped that experiments at a possibly forthcoming polarized electron–proton collider, eRHIC, would carry on and extend these studies [9]. Theoretically, the most interesting kinematic regime is characterized by large virtuality Q2 of the photon exchanged in the DIS process, and relatively small transverse momentum, qT Q. This regime also provides for the bulk of the events in experiment. It is precisely here, for example, that effects related to intrinsic transverse momenta of partons in the nucleon may become visible, potentially offering new insights into nucleon structure. At the same time, the theoretical analysis of hard-scattering in this regime is fairly involved, but well-understood. In particular, the emission of gluons from the DIS Born process γ ∗q → q also leads to non-vanishing transverse- momentum of the ﬁnal-state hadron and needs to be taken into account appropriately. It is the goal of this paper to present state-of-the-art calculations for the transverse-momentum dependence of some SIDIS observables. For this study, we will focus entirely on the set of leading-twist double- spin reactions,

(i) e + p → e + π + X, (ii) e +p → e + Λ + X, ↑ ↑ (iii) e + p → e + Λ + X, (iv) e +p → e + π + X, (v) e + p → e + Λ + X. (1) Here arrows to the right (upward arrows) denote longitudinal (transverse) polarization. Needless to say that the ﬁnal-state pion could be replaced by any hadron. The same is true for the Λ, as long as the observed hadron is spin-1/2 and its polarization can be detected experimentally. In our study, we will make use of several ingredients available in the literature. In an earlier publication [10], two of us presented results for the 2 → 3 partonic reactions lq → lqg and lg → lqq¯, for all polarizations of interest. In the perturbative expansion and using collinear factorization, these processes are the ﬁrst to yield a non-vanishing transverse momentum of the produced hadron. In terms of the strong coupling αs they are of order O(αs). We therefore refer to them as “leading order (LO)” processes for the hadron transverse-momentum distribution. They are expected to be adequate (at least qualitatively) for achieving a good theoretical description at large transverse momentum, qT ∼ Q. In the unpolarized case, the complete next-to-leading O 2 (NLO) ( (αs )) corrections to the qT -distribution have been calculated [11–13] which will lead 1 to an improvement of the theoretical calculation at large qT .

1 The study [10] may be viewed as an extension of previous work on the qT -integrated polarized SIDIS cross section, ∗ for which the LO process is γ q → q. LO calculations were performed in [14] and NLO ones in [15,16] (for initial work on the NNLO corrections to the unpolarized qT -integrated SIDIS cross section, see [17]). Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 61

At low transverse momentum, qT Q, ﬁxed-order calculations are bound to fail. The reason for this is well understood: when qT → 0, gluon emission is inhibited, and the cancellation of infra-red singularities between real and virtual diagrams in the perturbative series leaves behind logarithmic remainders of the form

lnm(Q2/q2 ) αk T S 2 (2) qT 2 = − in the cross section dσ/dqT at the kth order of perturbation theory, where m 1,...,2k 1. Ultimately, when qT Q, αs will not be useful anymore as the expansion parameter in the perturbative series since the logarithms will compensate for the smallness of αs . Accordingly, in order to obtain a reliable estimate for the cross section, one has to sum up (“resum”) the large logarithmic contributions to all orders in αs . Techniques for this resummation are well established, starting with pioneering work mostly on the Drell–Yan process in the late 1970’s to mid 1980’s [18–22]. The “Collins–Soper–Sterman” (CSS) formalism [22] has become the standard method for qT resummation. It is formulated in impact-parameter (b) space, which guarantees conservation of the soft-gluon transverse momenta. The formalism has also been applied to the unpolarized SIDIS cross section [23,24], and it was found [24] that data for qT distributions from the HERA ep collider [25,26] are satisfactorily described. Resummations at small transverse momentum qT have also been developed for spin observables. In Refs. [27,28] the CSS formalism was applied to longitudinal and transverse double-spin asymmetries in the Drell–Yan process. Early resummation studies on qT -distributions in jet production in polarized DIS were performed in [29].InRefs.[30,31] leading-logarithmic (LL) resummation effects were investigated for spin asymmetries that involve transverse-momentum dependent distributions, in particular the Collins functions mentioned above. In Ref. [32],resummation formulas for polarized SIDIS were derived, based on a factorization theorem at low transverse momentum [33]. A main result of [32] is that the CSS evolution equation is the same for the spin-dependent cases as in the unpolarized one. Knowing the expression for the unpolarized resummed cross section, and the complete polarized LO cross sections [10],itis then relatively straightforward to determine the resummed expressions for the polarized case. These will be provided in explicit form in this paper, for the cases listed in (1). We shall present results corresponding to resummation to next-to-leading logarithmic (NLL) accuracy, which corresponds to resummation of the towers with 2k − 1, 2k − 2, 2k − 3in(2). We shall also present numerical estimates for the reaction ep → eπX at COMPASS and eRHIC, in order to study the general features of the resummation and its impact on the cross sections and the spin asymmetry. We note that in the numerical evaluation one needs to specify a recipe for treating the integration over the impact parameter at very large b ∼ 1/ΛQCD, in order to avoid the Landau pole present in the resummed expression. This is closely related to non-perturbative effects generated by resummation [22,34–39]. We will use two different methods for treating the large b-region. We stress that we will not address single-transverse spin phenomena in this paper like those related to the Sivers and Collins effects mentioned above. For these, the analysis of QCD radiative effects is rather more involved [31] than for the double-spin case we consider, in particular regarding the connection of the behavior at small and large transverse momentum qT [40].Also, as we shall see below, the qT -differential cross section in general contains terms depending on the angle φ between the hadron and lepton planes. We shall only consider the resummation of the φ-independent pieces in the cross section, which dominate at small qT . It is an interesting topic by itself to study the resummation of the terms that depend on φ [41]. 62 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79

The remainder of this paper is organized as follows: Section 2 provides all formulas needed for the qT -differential SIDIS cross section, at LO and for the NLL resummed case. In Section 3,we present numerical estimates for ep → eπX at COMPASS and eRHIC, based on the resummation formulas. We discuss in particular the effect of the resummation on the spin asymmetry. We also study the impact of the treatment of the large-b region and of non-perturbative corrections on the resummed cross sections and the spin asymmetry. We conclude in Section 4.InAppendix A,we list the complete O(αs) cross sections for the processes in (1), correcting some typos in [10].

2. The qT -differential SIDIS cross section

2.1. Kinematics

We are interested in the cross section for the process l(k) + A(pA,SA) → l(k ) + B(pB ,SB ) + X, (3) where SA, SB are the spin vectors for the initial nucleon and the produced final-state hadron, which can be longitudinal or transverse, as indicated in (1). The lepton can be unpolarized or longitudinally polarized; transverse-spin effects for the lepton are suppressed by ml/Q because of chirality conservation at the lepton–photon vertex and hence are negligible. From now on, we will for definiteness take hadron A to be a proton and the lepton l to be an electron. We define 2 2 five Lorentz invariants, denoted Sep ,xbj ,Q ,zf , and qT , to describe the process. The center of mass energy squared, Sep , for the initial electron and the proton is

2 Sep = (pA + k) 2pA · k, (4) ignoring masses. The conventional DIS variables are deﬁned in terms of the virtual photon momentum q = k − k as

2 Q 2 2 2 xbj = ,Q=−q =−(k − k ) , (5) 2pA · q and they may be determined experimentally by observing the scattered electron. For the ﬁnal- state hadron B, we introduce the scaling variable

pA · pB zf = . (6) pA · q

Finally, we deﬁne the “transverse” component of q, which is orthogonal to both pA and pB : · · μ = μ − pB q μ − pA q μ qt q pA pB . (7) pA · pB pA · pB qt is a space-like vector, and we denote its magnitude by = − 2 qT qt . (8) To completely specify the kinematics, we need to choose a reference frame. We shall work in the so-called hadron frame [42,43], which is the Breit frame of the virtual photon and the initial proton:

qμ = (0, 0, 0, −Q), (9) Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 63 μ = Q Q pA , 0, 0, . (10) 2xbj 2xbj Further, in this frame the outgoing hadron B istakentobeinthexz plane: 2 2 zf Q q 2q q pμ = 1 + T , T , 0, T − 1 . (11) B 2 Q2 Q Q2

As one can see, the transverse momentum of hadron B is in this frame given by zf qT .Thisistrue for any frame in which the 3-momenta of the virtual photon and the initial proton are collinear. By introducing the angle φ between the hadron plane and the lepton plane, the lepton momentum can be parameterized as Q kμ = (cosh ψ,sinh ψ cos φ,sinh ψ sin φ,−1), (12) 2 and one finds 2xbj SeA cosh ψ = − 1. (13) Q2 For the case (iii) in (1) [ep ↑ → eΛ↑X] when the initial proton is transversely polarized and the transverse polarization of an outgoing spin-1/2 hadron is observed, we need to parameterize the spin vectors SA⊥ and SB⊥ to lie in the planes orthogonal to pA and pB , respectively. In the hadron frame, they can be written as μ = SA⊥ (0, cos ΦA, sin ΦA, 0), μ = − SB⊥ (0, cos ΘB cos ΦB , sin ΦB , sin ΘB cos ΦB ), (14) so that ΦA,B are the azimuthal angles of SA,B⊥ around pA,B as measured from the hadron plane and ΘB is the polar angle of pB measured with respect to pA. One finds: q2 − Q2 2q Q cos Θ = T , sin Θ = T . (15) B 2 + 2 B 2 + 2 qT Q qT Q We note that the polarization state depends on the frame we choose. For example, a state that is transversely polarized in the hadron frame becomes a mixture of longitudinally polarized and transversely polarized states in the laboratory frame where the initial electron and proton are collinear. With the above definitions, the cross section for (3) can be expressed in terms of Sep , 2 2 xbj , Q , zf , qT and φ in the hadron frame. Note that φ is invariant under boosts in the q-direction, so that is the same in the hadron frame and, for example, in the photon–proton center-of-mass frame.

2.2. Structure of the lowest-order cross section

The complete set of the O(αs) spin-dependent partonic cross sections for large-qT hadron production in SIDIS have been derived in Ref. [10]. They are obtained from the Feynman diagrams for the reactions lq → lqg and lg → lqq¯. For completeness, and for the reader’s convenience, we list them in Appendix A. Here we summarize the main characteristics of the hadronic cross section. The cross section can be decomposed into several pieces with different φ-dependences: d5σ = σ + cos(φ)σ + cos(2φ)σ , (16) 2 2 0 1 2 dQ dxbj dzf dqT dφ 64 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 for processes (i) and (ii) in (1), d5σ = σ + cos(φ)σ , (17) 2 2 0 1 dQ dxbj dzf dqT dφ for (iv) and (v), and 5 d σT 2 2 dQ dxbj dzf dqT dφ = − − T + − − T + − T cos(ΦA ΦB 2φ)σ0 cos(ΦA ΦB φ)σ1 cos(ΦA ΦB )σ2 , (18) T 2 2 2 for (iii). In the hadron frame, σ0 and σ0 in these decompositions behave as αs ln(Q /qT )/qT at small qT . This is the manifestation of the large logarithmic corrections described in the introduc- k 2k 2 2 2 tion at this order. At yet higher orders, corrections as large as αS ln (Q /qT )/qT in the cross section arise. In the following, we will study the NLL resummation of these large logarithmic corrections within the CSS formalism. We note that the other components of the cross sections → T 2 2 in Eqs. (16)–(18) are less singular as qT 0: σ1 and σ1 behave as αs ln(Q /qT )/qT , and σ2 T 2 2 and σ2 as αs ln(Q /qT ). These pieces too, however, receive large higher-order corrections at small qT [23] and would need to be resummed [41]. In this paper we will only focus on the T resummation for σ0 and σ0 .

2.3. Asymptotic part of the lowest-order cross section

We will now discuss the behavior of the cross sections at small qT in more detail. This is a first step in arriving at the resummed cross section. Following [23,24], we write the cross section at O(αs) as d5σ LO d5σ asymp = + Y(S ,Q,q ,x ,z ), (19) 2 2 2 2 ep T bj f dQ dxbj dzf dqT dφ dQ dxbj dzf dqT dφ T where the first term on the right-hand side is the “asymptotic” part in σ0 and σ0 containing the 2 2 2 2 pieces that are singular as ln(Q /qT )/qT or 1/qT at small qT , while Y collects the remainder which is at most logarithmic as qT → 0. It is straightforward to derive the asymptotic part of the LO unpolarized SIDIS cross section from the expressions in Appendix A. One finds: d5σ asymp dQ2 dx dz dq2 dφ bj f T 2 2 2 αemαs 2Q 2 Q 3 = A1 e 2fq (xbj ,μ)Dq (zf ,μ) CF ln − CF 8πx2 S2 Q2 q2 q q2 2 bj ep T q,q¯ T + f (x ,μ)⊗ P in,(0) + f (x ,μ)⊗ P in,(0) D (z ,μ) q bj qq g bj qg q f + out,(0) ⊗ + out,(0) ⊗ fq (xbj ,μ) Pqq Dq (zf ,μ) Pgq Dg(zf ,μ) , (20) where αem is the electromagnetic coupling constant, eq the fractional quark charge, and CF = 4/3. A1 is defined in (A.2).InEq.(20), fq (x, μ) and fg(x, μ) denote, respectively, the unpolarized quark and gluon distribution functions for the proton at scale μ, and Dq (z, μ) and Dg(z, μ) are quark and gluon fragmentation functions for the produced hadron. Furthermore, for the un- in,(0) out,(0) polarized cross section, the Pij and Pij are identical, and they are equal to the customary Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 65

in,(0) = out,(0) ≡ (0) LO unpolarized splitting functions: Pij Pij Pij , where + 2 (0) = 1 x + 3 − Pqq (x) CF δ(1 x) , (1 − x)+ 2 1 + (1 − x)2 P (0)(x) = C ,P(0)(x) = T x2 + (1 − x)2 . (21) gq F x qg R Here, TR = 1/2, and the “+”-prescription in the ﬁrst line acts in an integral from x to1as 1 1 f(y) f(y)− f(1) dy = dy + f(1) ln(1 − x), (22) (1 − y)+ 1 − y x x for any suitably regular function f . Finally, the symbol ⊗ denotes convolutions of the forms 1 dx x f (x ,μ)⊗ P in,(0) ≡ f (x, μ)P in,(0) bj , q bj qq x q qq x xbj 1 dz zf P out,(0) ⊗ D (z ,μ)≡ P out,(0)(z)D ,μ . (23) qq q f z qq q z zf For the other processes listed in (1), the asymptotic parts are obtained in the same manner from the cross sections given in Appendix A. They can be cast in the form (20) with the following replacements:

(ii) ep → eΛX :

fq → Δf q ,fg → Δf g,Dq → ΔDq ,Dg → ΔDg, (24)

where the Δf q , Δf g and ΔDq , ΔDg denote the helicity-dependent quark and gluon parton densities and fragmentation functions, respectively (for deﬁnitions, see [2]). Furthermore, in,(0) → (0) out,(0) → (0) Pij ΔPij ,Pij ΔPij , (25) with (0) = (0) (0) = [ − ] ΔPqq (x) Pqq (x), ΔPgq (x) CF 2 x , (0) = [ − ] ΔPqg (x) TR 2x 1 . (26) (iii) ep ↑ → eΛ↑X: 2 A1 → sinh ψ cos(ΦA − ΦB − 2φ),

fq → δfq ,fg → 0,Dq → δDq ,Dg → 0, in,(0) → (0) out,(0) → (0) Pqq δPqq ,Pqq δPqq . (27)

Here, the δfq and δDq denote the transversity distribution and fragmentation functions, respectively. As is well known, gluons do not contribute to transversity at leading twist [2]. The LO transversity splitting function is given by (0) = 2x + 3 − δPqq (x) CF δ(1 x) . (28) (1 − x)+ 2 66 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79

(iv) ep → eΛX:

A1 →−A6,fq → Δf q ,fg → Δf g, in,(0) → (0) out,(0) → (0) Pij ΔPij ,Pij Pij . (29) (v) ep → eΛX :

A1 →−A6,Dq → ΔDq ,Dg → ΔDg, in,(0) → (0) out,(0) → (0) Pij Pij ,Pij ΔPij . (30)

2.4. NLL resummed cross section

Resummation of the small-qT logarithms is achieved in impact-parameter (b) space, where k 2k−1 2 2 2 the large corrections exponentiate. In b-space, the leading logarithms αS ln (Q /qT )/qT turn − k 2k = γE into αs ln (bQ/b0), where b0 2e with γE being the Euler constant. Subleading logarithms are down by one or more powers of ln(bQ/b0). In the following, we will present the resummation formulas for the processes (i)–(v) in (1) we are interested in, based on the CSS formalism [22] (see also [23,24,32]). We will make some standard choices [22] for the parameters and scales in that formalism, which render the resulting expressions as simple and transparent as possible. For the unpolarized resummed cross one has

res 2 2 dσ πα d b · = em A eib qT W(b,Q,x ,z ), (31) 2 2 2 2 1 2 bj f dxbj dzf dQ dqT dφ 2xbj Sep (2π) where the b-space expression for the resummed cross section W is given by

W(b,Q,xbj ,zf ) = 2 ⊗ in × S(b,Q) × out ⊗ ej fi(xbj ,b0/b) Cji e Ckj Dk(zf ,b0/b) , (32) j=q,q¯ i,k=q,q,g¯ the convolutions being deﬁned as in Eqs. (23). As indicated, the scale in the parton distributions is given by μ = b0/b. We will return to this in the next subsection. The Sudakov form factor S(b,Q) in Eq. (32) reads

Q2 dk2 Q2 S(b,Q) =− T A α (k ) + B α (k ) , 2 s T ln 2 s T (33) kT kT 2 2 b0/b where the functions A and B have perturbative expansions of the form ∞ ∞ α k α k A(α ) = A s ,B(α) = B s . (34) s k π s k π k=1 k=1

For the resummation at NLL, one needs the coefﬁcients A1,2 and B1, which read [34,44]: 1 67 π 2 5 3 A = C ,A= C C − − N ,B=− C . (35) 1 F 2 2 F A 18 6 9 f 1 2 F Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 67

in out The functions Cij and Cij in Eq. (32) are also perturbative. Their expansions read ∞ α (μ) k Cin/out x,α (μ) = Cin/out,(k)(x) s . (36) ij s ij π k=0 Here μ is a renormalization scale of order Q. The LO coefﬁcients are given by

in,(0) out,(0) = − = − Cqq (x) δqq δ(1 x), Cqq (z) δqq δ(1 z), in,(0) = out,(0) = Cqg Cgq 0. (37)

in/out,(1) The first-order coefficients Cij could be obtained by expanding Eq. (32) to O(αs), performing an inverse Fourier transform, and comparing to the fixed-order (O(αs)) qT distribution. The asymptotic part we have given in Eq. (20) alone is not sufficient for this because it does 2 not contain the contributions ∝ δ (qT ) at O(αs). Fortunately, the coefficients are known in the literature, and have a very transparent structure in the MS scheme [24,43] (see also work on the coefficients in the related Drell–Yan case in [21,34,45]). One has: 1 Cin,(1)(x) =− P in, (x) + δ C δ(1 − x), (38) ij 2 ij ij δ 1 Cout,(1)(z) =− P out, (z) + ln(z)P out,(0)(z) + δ C δ(1 − z). (39) ij 2 ij ij ij δ in, out, ∝ = − Here the Pij (x), Pij (x) are the terms in the LO splitting functions in d 4 2 dimen- in/out,(0) sions, Pij (x, ), and are defined through

in/out,(0) = in/out,(0) + in/out, Pij (x, ) Pij (x) Pij (x). (40) The usual splitting functions in d = 4 dimensions have been given above in Eqs. (21).One has [46] 1 P (x) =−C 1 − x − δ(1 − x) , qq F 2 =− − =− Pqg(x) 2TRx(1 x), Pgq(x) CF x. (41)

Furthermore, the coefficient Cδ in Eqs. (38), (39) arises from hard virtual corrections. It is therefore only present for the case ij = qq, where Cδ =−7CF /4. Finally, for the “final-state” out,(1) ∝ coefficient Cij there is an extra contribution ln z [24,43] times the LO splitting function. It is of kinematic origin and contributed by the phase space for collinear gluon radiation off the quark leg in the final state [49]. With this, the formulas for the resummation of the unpolarized SIDIS cross section in b-space are complete. It is now straightforward to extend them to the various spin-dependent cross sections in (1). The Sudakov form factors are entirely related to soft-gluon emission and hence are spin-independent and the same in each case [32]. For each process, we first need to make the appropriate replacements as described above in Eqs. (24)–(30). In addition, we obtain the respective in/out,(1) coefficients Cij from Eqs. (38), (39), substituting appropriately all splitting functions, and in/out, ∝ = − also the Pij by the terms in the d 4 2 -dimensional versions of the LO splitting functions that (24)–(30) direct us to use. For this we need the ∝ terms in the spin-dependent 68 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 splitting functions. For longitudinal polarization we have [47] 1 ΔP (x) =−C 1 − x − δ(1 − x) , qq F 2 =− − = − ΔPqg(x) 2TR(1 x), ΔPgq(x) 2CF (1 x). (42) In case of transversity, the splitting function in 4 − 2 dimensions is for x<1 identical to that in four dimensions [48], so that C δP (x) = F δ(1 − x). (43) qq 2 The coefficient Cδ for the case ij = qq in Eqs. (38), (39) is the same for all polarized cases, since it comes from virtual diagrams. (0),(1) Knowledge of the coefficients A1,2, B1 and Cij in Eqs. (31), (32) is sufficient for the resummation to NLL accuracy. For consistency, we also need to use the two-loop expression for the strong running coupling, and the NLO evolution for the parton densities and fragmentation functions. Let us finally give an explicit formula for the NLL expansion of the Sudakov form factor. Defining = + 2 2 2 L ln 1 Q b /b0 , (44) and choosing the renormalization scale μ,wehave[50]: 1 S(Q,b) = f0 αs(μ)L + f1 αs(μ)L , (45) αs(μ) where

A1 f0(y) = β0y + ln(1 − β0y) , (46) πβ2 0 A β 1 β y ln(1 − β y) B f (y) = 1 1 2( − β y) + 0 + 0 + 1 ( − β y) 1 3 ln 1 0 ln 1 0 πβ 2 1 − β0y 1 − β0y πβ0 0 1 Q2 A β y + A − 2 ( − β y) + 0 , 1 ln 2 ln 1 0 (47) πβ0 μ πβ0 1 − β0y with the coefﬁcients of the QCD β-function β0 = (33 − 2Nf )/(12π) and β1 = (153 − 2 2 19NF )/(24π ). We observe that as y → 0, f0(y) = O(y ) and f1(y) = O(y). Note that in + 2 2 2 2 2 2 Eq. (44) we have followed Ref. [51] to choose ln(1 Q b /b0) rather than ln(Q b /b0) as the large logarithm in b-space. As far as the large-b behavior and NLL resummation are concerned, the two are equivalent, of course. With the choice in (44), however, the Sudakov exponent vanishes at b = 0, as it should [19,21,22], whereas for the other choice it gives large logarithms not only at large b (small qT ) but also artiﬁcially at small b (large qT ). For further discussion, see [51].

2.5. Evolution of parton distributions and fragmentation functions

As we have seen in Eq. (32), in the CSS formalism the parton distributions and fragmentation functions are evaluated at the factorization scale μ = b0/b. As discussed in [37], it then becomes a great convenience to treat them in Mellin-moment space, because this enables one to explicitly express their evolution between a large scale ∼ Q and b0/b. In this way, one avoids the problem Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 69 normally faced in qT resummation that one needs to call the parton densities or fragmentation functions at scales far below their range of validity (see below), so that some sort of “freezing” (or related prescription) for handling them is required. We also anticipate that below we will choose the impact parameter b to be complex-valued, so that it becomes desirable to separate the complex scale b0/b from the scale at which the parton densities are explicitly evaluated. To be more speciﬁc, we write the resummed cross section as W(b,Q,xbj ,zf ) in Eq. (32) as an inverse Mellin transform of moments in xbj and zf : 2 1 − − W(b,Q,x ,z ) = e2 dN dM z M x N bj f j 2πi f bj j=q,q¯ C C i,k=q,q,g¯ N M × N in,N S(b,Q) out,M M fi (b0/b)Cji e Ckj Dk (b0/b), (48) where the moment of each function is deﬁned as 1 1 N ≡ N−1 in,N ≡ N−1 in fi (μ) x fi(x, μ), Cji αs(μ) x Cji x,αs(μ) , (49) 0 0 and so forth. The integrations over N and M in Eq. (48) are over contours in the complex plane, to be chosen in such a way that they lie to the right of the rightmost singularities of the parton distributions and the fragmentation functions. In moment space, it is possible to express the N M ∼ moments fi (b0/b) and Dk (b0/b) by the corresponding moments at scale Q. The evolution “factor” between the scales b0/b and Q (which in general is a matrix because of singlet mixing) can be written in closed analytical form and can be expanded to NLL accuracy in L.One then only needs the parton densities and fragmentation functions at scale Q. For further details, see [37].

2.6. Inverse Fourier transform and non-perturbative corrections

As shown in Eq. (31), the resummed cross section in qT -space is obtained by an inverse Fourier transform of W(b,Q,xbj ,zf ). This involves an integration over b =|b| from 0 to ∞. Because of the Landau pole of the perturbative strong coupling in the Sudakov exponent (33) at kT = ΛQCD, this integration is ill-defined. In the expansion (45) for the Sudakov form factor, the singularity occurs at β0αs(μ)L = 1. In other words, the b-integration extends over both perturbative (b 1/ΛQCD) and non-perturbative (b 1/ΛQCD or even larger) regions. In order to define a perturbative resummed cross section, a prescription for the b integration is required that avoids the Landau pole. The ambiguity in the perturbative series implied by the presence of the Lan- dau pole corresponds to a non-perturbative correction. One may therefore use resummation to examine the structure of non-perturbative corrections in hadronic cross sections [52]. Typically, for the qT -differential Drell–Yan or SIDIS cross sections, one expects Gaussian non-perturbative effects in b-space [22,39]: − 2 eS(b,Q) → eS(b,Q) gb , (50) where g is a coefficient that may be determined by comparison to data. It will depend in general on Q and on xbj and zf . This non-perturbative Gaussian form factor may be thought of to partly incorporate (or, at least, mimic) the effect of smearing due to the partons’ intrinsic transverse momenta. 70 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79

Several prescriptions for treating the inverse Fourier transform have been proposed in the ∗ literature. The earliest one, known as “b -prescription” [22,34], is to prevent b from becoming → ∗ = + 2 2 larger than a certain bmax in the Sudakov form factor by replacing b b b/ 1 b /bmax. This evidently introduces a new parameter, bmax. The prescription was refined recently in [36]. Extensive phenomenological studies using this prescription and a non-perturbative term as in (50) have been carried out in particular for the qT -distribution in the Drell–Yan process [34–36]. The method has also been used in studies of the qT -distribution in unpolarized SIDIS [24,43,53]. Another method, proposed in [54] and applied to the Drell–Yan cross section in [37],isto deform the b integration in Eq. (31) into a contour in the complex b-plane. In this way, the Landau pole is avoided since it lies far out on the real-b axis. This procedure does not introduce any new parameter and is identical to the original b-integration in (31) for any finite-order expansion of the Sudakov exponent. For all details of the complex-b prescription, see [37]. This method was also recently used in a study of the transverse-momentum distribution of Higgs bosons at the LHC [51,55]. In the present paper we will use both the b∗ and the complex-b methods and compare the results and their dependence on the non-perturbative parameters chosen. We note that for the complex-b method the b-integral converges even for g = 0inEq.(50), while for the b∗ prescription a (non-perturbative) suppression of the integrand is required [37]. We finally mention that other methods for dealing with the behavior of the resummed exponent at large b have been discussed and used [38]. Also resummations directly in qT -space have been studied [56].

2.7. Matching to ﬁnite order

In order to obtain an adequate theoretical description also at large qT ∼ Q, we “match” the resummed cross section to the ﬁxed-order (LO, O(αs)) one. This is achieved by subtracting from the resummed expression in Eq. (31) its expansion to O(αs),

πα2 d2b em ib·qT A1 e W(b,Q,xbj ,zf ) − W(b,Q,xbj ,zf ) , (51) 2 2 2 O(αs ) 2xbj Sep (2π) and then adding the full O(αs) cross section, given by Eq. (19). We use this matching procedure, which avoids any double-counting of higher perturbative orders, in an identical way for the various polarized cross sections.

3. Numerical analysis for ep → eπX

In order to study the effect of resummation on the polarized SIDIS cross section and spin asymmetry, we will perform a numerical calculation for one case considered in (1), lp → lπX. This calculation is relevant for the ongoing COMPASS experiment and for experiments at the proposed polarized ep collider eRHIC. As typical values of the kinematical parameters, 4 2 2 2 2 we choose Sep = 10 GeV , Q = 100 GeV , xbj = 0.012 for eRHIC and Sep = 300 GeV , 2 2 Q = 10 GeV , xbj = 0.04 for COMPASS. Clearly, these choices are only meant to represent the typical kinematics; future detailed comparisons to experimental data will likely require to study a much broader range of values. We integrate over zf > 0.2. Our two sets of the parameters give an identical cosh ψ in Eqs. (13) and (A.2). For the NLO parton densities and the fragmentation functions, we use the GRV unpolarized distributions [57], the GRSV “standard” polarized distributions [58], and the pion fragmentation functions by Kretzer [59]. We have each of these available in Mellin-moment space, which is very helpful in the light of the discussion in Section 2.5. Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 71

In addition, we will use a simple Gaussian non-perturbative factor as in Eq. (50).Weem- phasize again the more illustrative character of our study and will therefore choose only some representative values for g, in order to investigate the sensitivity of the results to g.Forthe complex-b prescription, we will use g = 0.6, 0.8 GeV2 for both the eRHIC and the COMPASS cases. The value g = 0.8GeV2 was found in [37] in the context of the “joint” resummation for Z production at the tevatron. This value therefore applies to the scale MZ. g is expected to have logarithmic dependence on Q [35] and should be smaller at lower energy. This√ motivates our 2 ∗ lower choice of g = 0.6GeV .Fortheb method, we will choose bmax = 1/( 2GeV) and the values g = 0.8, 1.3GeV2 for the eRHIC case and g = 0.4, 0.8GeV2 for COMPASS. We note that in the analysis [24,43] of the HERA data [25,26] for SIDIS observables, a detailed phenomenological study of non-perturbative effects was performed, using the b∗ prescription. From the energy ﬂow observable in SIDIS it was found that the non-perturbative effects appear to become larger at small xbj . If we boldly assumed that the non-perturbative form of [24,43] also applied for the kinematics and the observable we are considering here, we would ﬁnd g ≈ 0.4GeV2 for the COMPASS situation, and g ≈ 1.3GeV2 at eRHIC. We are also interested in the comparison of the two methods with the same non-perturbative Gaussian, hence the additional choice of g = 0.8GeV2. Finally, we shall always use the same values of g in the unpolarized and the polarized cases. This assumption is motivated by the fact that the perturbative Sudakov form factors are independent of polarization. Figs. 1(a) and (b), respectively, show the unpolarized and polarized cross sections

max zf 1 d(Δ)σ dz dφ f 2 (52) 2π dxbj dzf dQ dqT dφ 0.2 max for eRHIC kinematics, where zf is given in Eq. (A.6). Note that we have multiplied by a factor qT , as compared to the cross section we considered in (19). The LO cross sections rapidly diverge as qT → 0. For the matched cross section using the complex-b method with g = 0, one obtains an enhancement at intermediate qT and the expected reduction at small qT . The inclusion of the non-perturbative Gaussian form factor makes this tendency stronger. However, the results for different choices of the Gaussian, g = 0.6 and 0.8 GeV2, are not very different and just have a slightly smaller normalization and are shifted to√ the right. Also shown in these figures are the ∗ 2 curves for the b prescription with bmax = 1/( 2GeV) and g = 0.8 and 1.3 GeV .Forthe eRHIC case, the choice of g = 0.8 gives a result relatively close to the one with g = 0inthe complex-b method, while g = 1.3 gives a result very close to those with g = 0.6 and 0.8 in the complex-b method. One should note that these resummed curves all give a very similar qT - integrated cross section, close to the full NLO one, due to our choice of L in Eq. (44) and our matching procedure described in the previous subsection [51]. Fig. 1(c) shows the corresponding spin asymmetries, defined by the ratios of the polarized and unpolarized cross sections for all the curves shown in Figs. 1(a) and (b). Although the effects of resummation and the non-perturbative Gaussians are significant in each cross section, they cancel to a large degree in the spin asymmetry. If one looks in more detail, resummation somewhat enhances the asymmetry compared to LO in the range of small to intermediate qT , where also the cross sections shown in Figs. 1(a) and (b) receive enhancements due to the resummation. Figs. 2(a)–(c) show the same quantities as Figs. 1(a)–(c), now for the COMPASS kinematics described above. In this case, the complex-b method without Gaussian smearing (g = 0) turned out to be difficult to control numerically at small qT , and we do not show the result for it. We 72 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79

Fig. 1. (a) Unpolarized SIDIS cross section for eRHIC kinematics. We show the ﬁxed-order (LO) result, and resummed 2 ∗ results for the complex-√ b method with non-perturbative parameters g = 0andg = 0.6, 0.8GeV , and for the b method 2 with bmax = 1/( 2GeV) and g = 0.8, 1.3GeV . (b) Same for the longitudinally polarized case. (c) Spin asymmetries corresponding to the various cross sections shown in (a) and (b).

find that resummation leads to a significant enhancement of the cross section at qT 1GeV.In both unpolarized and polarized cross sections, the resummed results we show, for the√ complex-b 2 ∗ method with g = 0.6 and 0.8 GeV , and for the b prescription with bmax = 1/( 2GeV) and g = 0.8GeV2, turn out to be very similar, while the b∗ prescription with g = 0.4GeV2 gives a higher peak that is shifted to the left compared to the other three resummed results. All these resummed results give a very similar spin asymmetry for the process lp → lπX for COMPASS kinematics; we find that resummation just leads to a moderate decrease of the asymmetry.

4. Summary and conclusions

We have carried out a study of the soft-gluon resummation for the transverse-momentum (qT ) distribution in semi-inclusive deeply-inelastic scattering. Resummation is crucial at small transverse momenta, qT Q, where it takes into account large double-logarithmic corrections to all orders in the strong coupling constant. We have considered all relevant leading-twist double- spin cross sections, focusing on the terms that are independent of the angle between the lepton and the hadron planes, and have presented the resummation formulas for each. Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 73

∗ Fig. 2. Same as in Figs. 1(a)–(c), but for COMPASS kinematics. For the b prescription, we have chosen here the non-perturbative parameters g = 0.4, 0.8GeV2.

We have performed phenomenological studies for the process lp → lπX at COMPASS and at a possible future polarized ep collider, eRHIC. Here we have chosen two different prescriptions for treating the region of very large impact parameters in the Sudakov form factor, which is related to the onset of non-perturbative phenomena. We have used simple estimates for the non- perturbative term suggested by the resummed formula. Our results indicate that resummation effects as well as non-perturbative effects cancel to a large extent in the spin asymmetry.

Acknowledgements

We are grateful to D. Boer, D. de Florian and F. Yuan for very useful discussions and comments. W.V. thanks RIKEN, Brookhaven National Laboratory (BNL) and the US Department of Energy (contract number DE-AC02-98CH10886) for providing the facilities essential for the completion of his work. The major part of this work was done while Y.K. stayed at BNL during 2003–2004 by the support from Monbu-Kagaku-sho. Y.K. is grateful to Monbu-Kagaku-sho for the ﬁnancial support and to the BNL Physics Department, in particular to Larry McLerran, for the warm hospitality during his stay. 74 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79

Appendix A. Analytic formulas for the LO cross sections

Here we summarize the LO cross section formulas for the processes in (1), which were derived in [10]. The differential cross sections are given by 1 1 d5σ α2 α dx dz = em s A [f ◦ D ◦ˆσ ] 2 2 2 2 2 k k dxbj dQ dzf dq dφ 8πx Sep Q x z T bj k x z min min q2 1 1 × δ T − − 1 − 1 , (A.1) Q2 xˆ zˆ where the Ak are deﬁned as [10,42] 2 A1 = 1 + cosh ψ,

A2 =−2,

A3 =−cos φ sinh 2ψ, 2 A4 = cos 2φ sinh ψ,

A6 =−2 cosh ψ,

A7 = 2 cos φ sinh ψ,

A8 =−sin φ sinh 2ψ, 2 A9 = sin 2φ sinh ψ, (A.2) and the summation over k is k = 1,...,4 for processes (i) and (ii) in (1), k = 1,...,4, 8, 9 2 for process (iii), and k = 6, 7 for processes (iv) and (v) (see below). αem = e /4π is the QED coupling constant, and we have introduced the variables xbj zf xˆ = , zˆ = , (A.3) x z and z q2 x q2 x = x + f T ,z= z + bj T . min bj 1 2 min f 1 2 (A.4) 1 − zf Q 1 − xbj Q 2 For a given Sep , Q and qT , the kinematic constraints for xbj and zf are Q2

The term [f ◦ D ◦ˆσk] in Eq. (A.1) takes the following form for the processes in (1):

(i) e + p → e + π + X [60]:

[ ◦ ◦ˆ ]= 2 ˆ k + 2 ˆ k f D σk eq fq (x)Dq (z)σqq eq fg(x)Dq (z)σqg q,q¯ q,q¯ + 2 ˆ k eq fq (x)Dg(z)σgq, (A.9) q,q¯

where (CF = 4/3) 1 Q4 σˆ 1 = 2C xˆzˆ + Q2 − q2 2 + 6 , qq F 2 2 ˆ2 ˆ2 T Q qT x z ˆ 2 = ˆ qq = ˆ ˆ σqq 2σ4 8CF xz, ˆ 3 = ˆ ˆ 1 2 + 2 σqq 4CF xz Q qT , (A.10) QqT Q2 1 2 2 2 σˆ 1 =ˆx(1 −ˆx) − + 2 + 10 − − , qg 2 ˆ2 ˆ2 ˆ ˆ ˆ ˆ qT x z xz x z σˆ 2 = 2σˆ 4 = 8x(ˆ 1 −ˆx), qg qg 2 ˆ 3 =ˆ −ˆ 2 2 + 2 − Q σqg x(1 x) 2 Q qT , (A.11) QqT xˆzˆ 1 Q4 (1 −ˆz)2 zˆ2q2 2 σˆ 1 = 2C x(ˆ 1 −ˆz) + Q2 − T + 6 , gq F 2 2 ˆ2 ˆ2 ˆ2 −ˆ 2 Q qT x z z (1 z) ˆ 2 = ˆ 4 = ˆ −ˆ σgq 2σgq 8CF x(1 z), 1 zˆ2q2 σˆ 3 =− C x(ˆ −ˆz)2 Q2 + T . gq 4 F 1 2 (A.12) zQqˆ T (1 −ˆz) (ii) e +p → e + Λ + X: [ ◦ ◦ˆ ]= 2 ˆ k + 2 ˆ k f D σk eq Δf q (x)ΔDq (z)ΔLσqq eq Δf g(x)ΔDq (z)ΔLσqg q,q¯ q,q¯ + 2 ˆ k eq Δf q (x)ΔDg(z)ΔLσgq, (A.13) q,q¯ where ˆ k =ˆk = ΔLσqq σqq (k 1, 2, 3, 4), (A.14)

(2xˆ − 1){Q4(xˆ − 1)2 − q4 xˆ2} Δ σˆ 1 =− T , L qg 2 2 ˆ ˆ − Q qT x(x 1) ˆ 2 = ˆ gq = ΔLσqg ΔLσ4 0, 76 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79

2{Q2(xˆ − 1) − q2 xˆ} ˆ 3 =− T ΔLσqg , (A.15) QqT ˆ − ˆ ˆ + 4 ˆ2 − ˆ + 2 1 x 2 x(x 1) qT 2(2x 2x 1) qT ΔLσˆ = 2CF xˆzˆ + + , gq xˆ − 1 (xˆ − 1)2 Q4 (xˆ − 1)2 Q2 xˆ2zˆ q2 Δ σˆ 2 = 2Δ σˆ qg = 8C T , L gq L 4 F xˆ − 1 Q2 ˆ ˆ ˆ2 2 3 4CF xz 2 x qT qT ΔLσˆ = (xˆ − 1) + . (A.16) gq (xˆ − 1)2 Q2 Q (iii) e + p↑ → e + Λ↑ + X: For this process, it is more transparent to write the cross section by including the factors Ak in (A.1): A [ ◦ ◦ˆ ]≡ 2 ˆ qq k f D σk eq δfq (x)δDq (z)δσ , (A.17) k=1,...,4,8,9 q,q¯ where Q δσˆ qq = 4C 1 + cosh2 ψ cos(Φ − Φ ) − sinh 2ψ cos(Φ − Φ − φ) F A B q A B T Q2 + 2 ψ (Φ − Φ − φ) . 2 sinh cos A B 2 (A.18) qT

Here the terms with A3 and A8 in Eq. (A.1) have been combined to give the cos(ΦA − ΦB − φ) term in (A.18). Likewise, those with A4,9 give the term with cos(ΦA − ΦB − 2φ). (iv) e +p → e + π + X: [ ◦ ◦ˆ ]= 2 ˆ k + 2 ˆ k f D σk eq Δf q (x)Dq (z)ΔLO σqq eq Δf g(x)Dq (z)ΔLO σqg q,q¯ q,q¯ + 2 ˆ k eq Δf q (x)Dg(z)ΔLO σgq, (A.19) q,q¯ where 1 Q2 xˆzqˆ 2 Δ σˆ 6 =−2C +ˆxzˆ − T , LO qq F ˆ ˆ 2 2 xz qT Q Q2 − q2 ˆ 7 =− ˆ ˆ T ΔLO σqq 4CF xz , (A.20) QqT 2xˆ − 1 xˆ − 1 Q2 Δ σˆ 6 = 2xˆ + , LO qg ˆ ˆ2 2 x z qT ˆ − ˆ − ˆ 7 = 2Q (x 1)(2z 1) ΔLO σqg , (A.21) qT zˆ ˆ ˆ4 4 6 2CF z 1 2 x qT ΔLO σˆ = − (xˆ − 1) + , gq xˆ − 1 zˆ2 (xˆ − 1)2 Q4 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 77 4C xˆzˆ xˆ q Δ σˆ 7 = F 1 − T . (A.22) LO gq xˆ − 1 zˆ Q (v) e + p → e + Λ + X: [ ◦ ◦ˆ ]= 2 ˆ k + 2 ˆ k f D σk eq fq (x)ΔDq (z)ΔOLσqq eq fg(x)ΔDq (z)ΔOLσqg ¯ ¯ q,q q,q + 2 ˆ k eq fq (x)ΔDg(z)ΔOLσgq, (A.23) q,q¯ where ˆ 6,7 = ˆ 6,7 ΔOLσqq ΔLO σqq , (A.24) 2xˆ2 − 2xˆ + 1 Q2 Δ σˆ 6 = xˆ + (xˆ − 1) , OL qg ˆ ˆ 2 xz qT ˆ − ˆ − ˆ 7 = 2Q (x 1)(2x 1) ΔOLσqg , (A.25) qT zˆ 2C zˆ 1 xˆ4 q4 Δ σˆ 6 = F + (xˆ − 1)2 − T , OL gq ˆ − ˆ2 ˆ − 2 4 x 1 z (x 1) Q 4C xˆzˆ xˆ q Δ σˆ 7 =− F 1 − T . (A.26) OL gq xˆ − 1 zˆ Q

References

[1] A. Airapetian, et al., HERMES Collaboration, Phys. Rev. Lett. 92 (2004) 012005, hep-ex/0307064; A. Airapetian, et al., HERMES Collaboration, Phys. Rev. D 71 (2005) 012003, hep-ex/0407032; S. Platchkov, COMPASS Collaboration, Talk given at the Particles and Nuclei International Conference (PANIC05), Santa Fe, NM, October 2005; For earlier data, see: B. Adeva, et al., Spin Muon Collaboration, Phys. Lett. B 420 (1998) 180, hep-ex/9711008. [2] For reviews, see: M. Anselmino, A. Efremov, E. Leader, Phys. Rep. 261 (1995) 1; M. Anselmino, A. Efremov, E. Leader, Phys. Rep. 281 (1997) 399, Erratum; V. Barone, A. Drago, P.G. Ratcliffe, Phys. Rep. 359 (2002) 1; S.D. Bass, Rev. Mod. Phys. 77 (2005) 1257, hep-ph/0411005. [3] A. Airapetian, et al., HERMES Collaboration, Phys. Rev. Lett. 94 (2005) 012002, hep-ex/0408013; M. Diefenthaler, HERMES Collaboration, AIP Conf. Proc. 792 (2005) 933, hep-ex/0507013. [4] V.Y. Alexakhin, et al., COMPASS Collaboration, Phys. Rev. Lett. 94 (2005) 202002, hep-ex/0503002. [5] D. Adams, et al., Spin Muon Collaboration, Phys. Lett. B 336 (1994) 125; A. Bravar, Spin Muon Collaboration, Nucl. Phys. A 666 (2000) 314. [6] H. Avakian, CLAS Collaboration, Talk presented at the RBRC workshop Single-Spin Asymmetries, Brookhaven National Laboratory, Upton, NY, 1–3 June 2005. [7] D.W. Sivers, Phys. Rev. D 41 (1990) 83; D.W. Sivers, Phys. Rev. D 43 (1991) 261. [8] J.C. Collins, Nucl. Phys. B 396 (1993) 161, hep-ph/9208213. [9] A. Deshpande, R. Milner, R. Venugopalan, W. Vogelsang, Annu. Rev. Nucl. Part. Sci. 55 (2005) 165, hep-ph/ 0506148. [10] Y. Koike, J. Nagashima, Nucl. Phys. B 660 (2003) 269, hep-ph/0302061. [11] P. Aurenche, R. Basu, M. Fontannaz, R.M. Godbole, Eur. Phys. J. C 34 (2004) 277, hep-ph/0312359. [12] A. Daleo, D. de Florian, R. Sassot, Phys. Rev. D 71 (2005) 034013, hep-ph/0411212. [13] B.A. Kniehl, G. Kramer, M. Maniatis, Nucl. Phys. B 711 (2005) 345, hep-ph/0411300; B.A. Kniehl, G. Kramer, M. Maniatis, Nucl. Phys. B 720 (2005) 231, Erratum. 78 Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79

[14] F.E. Close, R.G. Milner, Phys. Rev. D 44 (1991) 3691; L.L. Frankfurt, et al., Phys. Lett. B 230 (1989) 141; X.D. Ji, Phys. Rev. D 49 (1994) 114, hep-ph/9307235. [15] D. de Florian, C.A. Garcia Canal, R. Sassot, Nucl. Phys. B 470 (1996) 195, hep-ph/9510262; D. de Florian, R. Sassot, Nucl. Phys. B 488 (1997) 367, hep-ph/9610362. [16] D. de Florian, M. Stratmann, W. Vogelsang, Phys. Rev. D 57 (1998) 5811, hep-ph/9711387. [17] A. Daleo, C.A. Garcia Canal, R. Sassot, Nucl. Phys. B 662 (2003) 334, hep-ph/0303199; A. Daleo, R. Sassot, Nucl. Phys. B 673 (2003) 357, hep-ph/0309073. [18] Y.L. Dokshitzer, D. Diakonov, S.I. Troian, Phys. Lett. B 79 (1978) 269. [19] G. Parisi, R. Petronzio, Nucl. Phys. B 154 (1979) 427. [20] J.C. Collins, D.E. Soper, Nucl. Phys. B 193 (1981) 381; J.C. Collins, D.E. Soper, Nucl. Phys. B 213 (1983) 545, Erratum; J.C. Collins, D.E. Soper, Nucl. Phys. B 197 (1982) 446. [21] G. Altarelli, R.K. Ellis, M. Greco, G. Martinelli, Nucl. Phys. B 246 (1984) 12. [22] J.C. Collins, D.E. Soper, G. Sterman, Nucl. Phys. B 250 (1985) 199. [23] R. Meng, F.I. Olness, D.E. Soper, Phys. Rev. D 54 (1996) 1919, hep-ph/9511311. [24] P. Nadolsky, D.R. Stump, C.P. Yuan, Phys. Rev. D 61 (2000) 014003, hep-ph/9906280; P. Nadolsky, D.R. Stump, C.P. Yuan, Phys. Rev. D 64 (2001) 059903, Erratum; P. Nadolsky, D.R. Stump, C.P. Yuan, Phys. Rev. D 64 (2001) 114011, hep-ph/0012261; P. Nadolsky, D.R. Stump, C.P. Yuan, Phys. Lett. B 515 (2001) 175, hep-ph/0012262. [25] I. Abt, et al., H1 Collaboration, Z. Phys. C 63 (1994) 377; S. Aid, et al., H1 Collaboration, Z. Phys. C 70 (1996) 609, hep-ex/9602001; S. Aid, et al., H1 Collaboration, Z. Phys. C 72 (1996) 573, hep-ex/9608011; I. Abt, et al., H1 Collaboration, Phys. Lett. B 328 (1994) 176; C. Adloff, et al., H1 Collaboration, Nucl. Phys. B 485 (1997) 3, hep-ex/9610006; S. Aid, et al., H1 Collaboration, Phys. Lett. B 356 (1995) 118, hep-ex/9506012; C. Adloff, et al., H1 Collaboration, Eur. Phys. J. C 12 (2000) 595, hep-ex/9907027. [26] M. Derrick, et al., ZEUS Collaboration, Z. Phys. C 70 (1996) 1, hep-ex/9511010; J. Breitweg, et al., ZEUS Collaboration, Phys. Lett. B 481 (2000) 199, hep-ex/0003017. [27] A. Weber, Nucl. Phys. B 382 (1992) 63; A. Weber, Nucl. Phys. B 403 (1993) 545. [28] H. Kawamura, J. Kodaira, H. Shimizu, K. Tanaka, hep-ph/0512137. [29] A. Weber, Thesis, Dortmund University, DO-TH-92-28, unpublished. [30] D. Boer, Phys. Rev. D 62 (2000) 094029, hep-ph/0004217. [31] D. Boer, Nucl. Phys. B 603 (2001) 195, hep-ph/0102071. [32] A. Idilbi, X.D. Ji, J.P. Ma, F. Yuan, Phys. Rev. D 70 (2004) 074021, hep-ph/0406302. [33] X.D. Ji, J.P. Ma, F. Yuan, Phys. Lett. B 597 (2004) 299, hep-ph/0405085. [34] C.T.H. Davies, W.J. Stirling, Nucl. Phys. B 244 (1984) 337. [35] C.T.H. Davies, B.R. Webber, W.J. Stirling, Nucl. Phys. B 256 (1985) 413; P.B. Arnold, R.P. Kauffman, Nucl. Phys. B 349 (1991) 381; G.A. Ladinsky, C.P. Yuan, Phys. Rev. D 50 (1994) 4239, hep-ph/9311341; F. Landry, R. Brock, G. Ladinsky, C.P. Yuan, Phys. Rev. D 63 (2001) 013004, hep-ph/9905391; F. Landry, R. Brock, P.M. Nadolsky, C.P. Yuan, Phys. Rev. D 67 (2003) 073016, hep-ph/0212159; C. Balazs, C.P. Yuan, Phys. Rev. D 56 (1997) 5558, hep-ph/9704258. [36] A.V. Konychev, P.M. Nadolsky, hep-ph/0506225. [37] A. Kulesza, G. Sterman, W. Vogelsang, Phys. Rev. D 66 (2002) 014011, hep-ph/0202251. [38] J.W. Qiu, X.F. Zhang, Phys. Rev. D 63 (2001) 114011, hep-ph/0012348. [39] For further discussion, see also: S. Tafat, JHEP 0105 (2001) 004, hep-ph/0102237. [40] X.D. Ji, J.W. Qiu, W. Vogelsang, F. Yuan, in preparation. [41] D. Boer, W. Vogelsang, in preparation. [42] R. Meng, F. Olness, D. Soper, Nucl. Phys. B 371 (1992) 79. [43] For a clear discussion, see also: P.M. Nadolsky, hep-ph/0108099. [44] J. Kodaira, L. Trentadue, Phys. Lett. B 112 (1982) 66; J. Kodaira, L. Trentadue, Phys. Lett. B 123 (1983) 335. [45] D. de Florian, M. Grazzini, Phys. Rev. Lett. 85 (2000) 4678, hep-ph/0008152; D. de Florian, M. Grazzini, Nucl. Phys. B 616 (2001) 247, hep-ph/0108273. Y. Koike et al. / Nuclear Physics B 744 (2006) 59–79 79

[46] R.K. Ellis, D.A. Ross, A.E. Terrano, Nucl. Phys. B 178 (1981) 421. [47] W. Vogelsang, Nucl. Phys. B 475 (1996) 47, hep-ph/9603366. [48] W. Vogelsang, Phys. Rev. D 57 (1998) 1886, hep-ph/9706511. [49] M. Stratmann, W. Vogelsang, Nucl. Phys. B 496 (1997) 41, hep-ph/9612250. [50] S. Frixione, P. Nason, G. Ridolﬁ, Nucl. Phys. B 542 (1999) 311, hep-ph/9809367. [51] G. Bozzi, S. Catani, D. de Florian, M. Grazzini, Nucl. Phys. B 737 (2006) 73, hep-ph/0508068. [52] See: G. Sterman, W. Vogelsang, Phys. Rev. D 71 (2005) 014013, hep-ph/0409234, and references therein. [53] S. Berge, P. Nadolsky, F. Olness, C.P. Yuan, Phys. Rev. D 72 (2005) 033015, hep-ph/0410375. [54] E. Laenen, G. Sterman, W. Vogelsang, Phys. Rev. Lett. 84 (2000) 4296, hep-ph/0002078; E. Laenen, G. Sterman, W. Vogelsang, Phys. Rev. D 63 (2001) 114018, hep-ph/0010080. [55] A. Kulesza, G. Sterman, W. Vogelsang, Phys. Rev. D 69 (2004) 014012, hep-ph/0309264. [56] R.K. Ellis, S. Veseli, Nucl. Phys. B 511 (1998) 649, hep-ph/9706526; A. Kulesza, W.J. Stirling, Nucl. Phys. B 555 (1999) 279, hep-ph/9902234; A. Kulesza, W.J. Stirling, JHEP 0001 (2000) 016, hep-ph/9909271; A. Kulesza, W.J. Stirling, Eur. Phys. J. C 20 (2001) 349, hep-ph/0103089. [57] M. Glück, E. Reya, A. Vogt, Eur. Phys. J. C 5 (1998) 461, hep-ph/9806404. [58] M. Glück, E. Reya, M. Stratmann, W. Vogelsang, Phys. Rev. D 63 (2001) 094005, hep-ph/0011215. [59] S. Kretzer, Phys. Rev. D 62 (2000) 054001, hep-ph/0003177. [60] A. Mendez, Nucl. Phys. B 145 (1978) 199. Nuclear Physics B 744 (2006) 80–95

Spin-glass model of QCD near saturation

Robi Peschanski

Service de Physique Théorique, CEA/Saclay, 91191 Gif-sur-Yvette Cedex, France 1 Received 16 March 2006; accepted 22 March 2006 Available online 17 April 2006

Abstract We establish a connection between the cascading of gluon momenta modeled within the diffusive approximation of the Balitsky–Fadin–Kuraev–Lipatov kernel and the thermodynamics of directed polymers on a tree with disorder. Using known results on the low-temperature spin-glass phase of this statistical mechanic problem we describe the dynamical phase space of gluon transverse momenta near saturation including its ﬂuctuation pattern. It exhibits a nontrivial clustering structure, analogous to “hot spots”, whose distributions are derived and possess universal features in common with other spin-glass systems. © 2006 Elsevier B.V. All rights reserved.

1. Saturation in QCD is expected to occur when parton densities inside an hadronic target are so high that their wave-functions overlap. This is expected from the high-energy evolution of deep-inelastic scattering governed by the Balitsky–Fadin–Kuraev–Lipatov (BFKL) kernel [1]. The BFKL evolution equation is such that the number of gluons of fixed size increases exponentially and would lead, if not modified, to an average violation of unitarity. Saturation [2,3] is characterized by a typical transverse momentum scale Qs(Y ), depending on the overall rapidity of the reaction, at which the unitarity bound is reached by the BFKL evolution of the amplitude. The problem we want to address here is the characterization of the gluon-momentum distribution near saturation. This study may help analyzing the color glass condensate (CGC) which has been proposed [4] as a description of the QCD phase at full saturation. As we shall see, our aim in the present paper is to fully understand the spectrum of transverse momentum fluctuations among the gluons which are generated by the BFKL evolution in rapidity, near saturation. Indeed, one of the major recent challenges in QCD saturation is the problem of taking into account the rôle of fluctuations which are present in the BFKL evolution, i.e. in the dilute regime.

E-mail address: [email protected] (R. Peschanski). 1 URA 2306, unité de recherche associée au CNRS.

Quite surprisingly, the fluctuations in this region modify the overall approach to saturation by a mechanism which has been understood as the formation of traveling-wave solutions of nonlinear differential equations. If at first one neglects the fluctuations (in the mean-field approximation), the effect of saturation on a dipole-target amplitude is described by the nonlinear Balitsky– Kovchegov [5] (BK) equation, where a nonlinear damping term adds to the BFKL equation. As shown in [6], this equation falls into the universality class of the Fisher and Kolmogorov– Petrovsky–Piscounov (F-KPP) nonlinear equation [7] which admits asymptotic traveling-wave solutions of “pulled-front” type [8]. In the QCD case, this means that they are driven by the gluons of higher momentum. Hence the overall behavior of the solution is driven by the high- momentum tail, which is precisely in the “dilute” domain of the BFKL regime, far from the range which is dominated by saturation. The exponential behavior of the BFKL evolution quickly enhances the effects of the tail towards a region where finally the nonlinear damping regulates both the traveling-wave propagation and structure. In these conditions it was realized, for “pulled-front” traveling waves [9] and in QCD [10], that the fluctuations inherent of the dilute regime may have a surprisingly large effect on the overall solution of the nonlinear equations of saturation. Indeed, a fluctuation in the dilute regime may grow exponentially and thus modify its contribution to the overall amplitude. Hence, in order to enlarge our understanding of the QCD evolution with rapidity, it seems important to give a quantitative description of the fluctuation pattern generated by the BFKL evolution equations for the set of cascading dipoles (and thus gluons) near saturation, which is the goal of our paper. We shall work in the leading order in 1/Nc, where the QCD dipole framework is valid [11,12]. The equations for the generating functionale of moments of the QCD dipole distributions have been derived already some time ago [11] and they have been used in instructive numerical simulations of onium–onium scattering [13] and also applied recently in diffractive and non- diffractive deep-inelastic scattering [14]. However, to our knowledge, a thorough investigation of their solutions for the pattern of dipoles or equivalently for the event-by-event properties of gluon transverse momenta has not yet been considered. We will provide an answer to this question in the QCD dipole approximation (and moreover in the diffusive approximation of the 1-dimensional BFKL kernel, see further on), and the results appear quite rich already within this approximation scheme. Many aspects we find show “universality” features and thus are expected to be valid also in the dipole framework beyond the approximations (going beyond the dipole approximation is still a conceptual challenge [15]). The strategy of the present paper is to determine the structure of the event-by-event gluon transverse-momentum distributions. In the present paper, and as a first step towards a more complete solution, we will simplify the problem, focusing on modeling the QCD gluon cascading using the diffusive approximation of the BFKL kernel and aiming at a quantitative understanding of the transition-to-saturation phase within this scheme. We show that one can build a mapping between the energy evolution of the set of gluon momenta (in the 1-dimensional approximation) and the time evolution of directed polymers on a tree with disorder. This is obtained through the matching of the partition function of the polymer problem with a related event-by-event physical quantity defined from the cascading gluon process. Then we use the relation between the cascading evolution around the unitarity limit of the amplitude and the saturation mechanism to find that a spin glass structure of the gluon phase space exists at saturation, in direct analogy with the low-temperature phase of the directed polymers. We then derive the physical consequences of the spin-glass phase for the distribution of gluon transverse momenta at saturation in our model of cascading. 82 R. Peschanski / Nuclear Physics B 744 (2006) 80–95

The plan of our study is the following. In Section 2, we shall recall the main ingredients of the rapidity evolution of the cascading dipoles and gluon-momentum distributions as driven by the BFKL evolution in the 1-dimensional and diffusive approximations. In Section 3,we introduce the mapping between the cascading of gluons and the propagation of directed polymers on random trees. In Section 4 we recall the nontrivial properties of the directed polymer problem, characterized by a spin-glass transition at low temperature. In Section 5, we provide a check of the consistency of the gluon/polymer mapping with the saturation mechanism in the mean ﬁeld approximation. In Section 6, we derive the gluon phase space properties at large rapidity from the spin-glass features. The ﬁnal Section 7 is devoted to conclusion and outlook.

2. In this section, we present some known results on the equations governing the dipole distributions in the BFKL framework together with those concerning the saturation amplitude in the mean-ﬁeld approximation and relevant for our study.

QCD rapidity evolution

Let us start by briefly describing the QCD evolution of the dipole distributions in the Balitsky– Fadin–Kuraev–Lipatov BFKL framework [11,13,14]. Formally, let us denote Zvw[Y,U] the generating functional of moments for the N-uple dipole-probability distributions after a rapidity evolution range Y . The dipole probabilities can be deduced from Zvw[Y,U] by functionally differentiating with respect to the source fields U and then letting U → 0. The evolution equation satisfied by Zvw[Y,U] reads ∂Z vw ≡ d2z K(v, w; z){Z Z − Z }, (1) ∂Y vz zw vw z where

α N (v − w)2 K(v, w; z) = s c (2) π (v − z)2(z − w)2 is the BFKL kernel [1] describing the dissociation vertex of one dipole (v, w) into two dipoles at (v, z) and (z, w), where v, w, z are arbitrary 2-dimensional transverse space coordinates. From Eq. (1), we will mainly retain its physical meaning. The functionale structure of Eq. (1) describes a 2-dimensional tree structure of dipoles in transverse position space evolving with rapidity. Let us, for instance, focus on the rapidity evolution starting from one massive qq¯ pair or onium [11],seeFig. 1. At each branching vertex, the wave function of the onium-projectile is described by a collection of color dipoles. The dipoles split with a probability per unit of rapidity defined by the BFKL kernel (2). The BFKL rapidity evolution generates a cascade of dipoles. Each vertex of the cascade is governed in coordinate space by the 2-dimensional BFKL kernel (2), determining both the energy and transverse position space evolution of the cascade, as sketched in Fig. 1. As we shall see later on, this cascading property can be converted in a similar cascade of gluons in momentum space. At this stage, it is interesting to note [14] the mathematical similarity of (1) with the BK equation [5], as verified by the S-matrix element of the dipole-target amplitude S(v, w,Y)in the mean-field approximation. The equation is indeed formally the same if one replaces term by term the functional Zvw[Y,U] by the function S(v, w,Y)representing the S-matrix element. R. Peschanski / Nuclear Physics B 744 (2006) 80–95 83

Fig. 1. BFKL cascading and saturation. The QCD branching process in the BFKL regime and beyond is represented along the rapidity axis (upper part). Its 2-dimensional counterpart in transverse position space is displayed at two different rapidities (lower part). The interaction region is represented by a shaded disk of size 1/Q. At rapidity Y1,the interaction still probes individual dipoles (or gluons), which corresponds to the exponential BFKL regime. There exists a smooth transition to a regime where the interaction only probes groups of dipoles or gluons, e.g. at rapidity Y2.This gives a description of the near-saturation region corresponding to a mean-ﬁeld approximation, where correlations can be neglected. Further in rapidity, Y>Y2, other dynamical effects, such that merging and correlations appear, leading eventually to the CGC.

Gluons in the 1-dimensional and diffusive approximation

As an approximation of the 2-dimensional formulation of the BFKL kernel (2), obtained when one neglects the impact-parameter dependence, we shall restrict our analysis in the present pa- 2 per to the 1-dimensional reduction of the problem to the transverse-momenta squared ki of the cascading gluons. After Fourier transforming to transverse-momentum space, the leading-order BFKL kernel [1] deﬁning the rapidity evolution in the 1-dimensional approximation is known [5] to act in transverse momentum space as a differential operator of inﬁnite order

χ(−∂l) ≡ 2ψ(1) − ψ(−∂l) − ψ(1 + ∂l), (3) = 2 αs Nc where l log k and Y is the rapidity in units of the ﬁxed coupling constant π . In the sequel, we shall restrict further our analysis to the diffusive approximation of the BFKL kernel. We thus expand the BFKL kernel to second order around some value γc

1 χ(γ)∼ χ + χ (γ − γ ) + χ (γ − γ )2 = A − A γ + A γ 2, (4) c c c 2 c c 0 1 2 84 R. Peschanski / Nuclear Physics B 744 (2006) 80–95 where γc will be defined in such a way to be relevant for the near-saturation region of the BFKL regime. Within this diffusive approximation, it is easy to realize that the first term (A0) is responsible for the exponential increase of the BFKL regime while the third term (A2) is a typical diffusion term. The second term (A1) is a “shift” term since it amounts to a rapidity-dependent redefinition of the kinematic variables, as we shall see. In Eq. (4), γc is chosen in order to ensure the validity of the kernel (4) in the transition region from the BFKL regime towards saturation. Indeed, the derivation of asymptotic solutions of the BK equation [6] leads to consider the condition χ(γc) = γcχ (γc) (5) whose solution determines γc. √ This condition applied to the kernel formula (3) gives γc = A0/A2 ≈ 0.6275 ... and {A0,A1,A2}≈{9.55, 25.56, 24.26}. It is worth noting that a more general choice of the Ai ’s can be considered for the forthcoming discussion, leaving the possibility of considering also “effective” kernels [16], eventually taking into account next leading corrections and thus other values of γc.

Saturation in the mean-ﬁeld approximation

As already noticed [14], the 2-dimensional BK equation describing saturation in the mean- ﬁeld approximation is formally similar to (1) by substituting Z by S, the S-matrix element of the scattering amplitude. In the 1-dimensional approximation which we will use, the BK equation reads

2 ∂Y T(l,Y)= χ(−∂l)T (l, Y ) − T (l, Y ), (6) with the same notations as in (3). In the 1-dimensional approximation, the Fourier transform of the dipole-target scattering amplitude T(l,Y) is related to the unintegrated gluon distribution in the target (by a simple convolution). Applying only the linear operator (3) to T(l,Y)leads to an exponential increase of the gluon density with rapidity characteristic of the BFKL regime. The damping term in Eq. (6) is responsible for the saturation mechanism in the mean-field approximation. There exists an already long literature on the properties of the BK equation (see, for instance, [3,5,6]). For further purpose, we remark that many results which can be obtained from the solutions of the BK equation can already be derived from the BFKL evolution when only the linear operator in Eq. (6) is retained. The idea [17,18] is to consider the extension of the BFKL evolution in the rapidity region when the unitarity limit is reached and assume a smooth transition to saturation. This amounts to study the BFKL regime alongside the saturation line defined by the 2 saturation scale Qs(Y ) in the 2-dimensional k , Y plane. Indeed, one obtains a significant part [17,18] of the results obtained [6] by a direct derivation of the asymptotic solutions of the BK equation. Let us list some of the most relevant results of this approach based on the unitarity bound: The gluon momentum distribution can be derived at high rapidity and large enough k. One finds √ − k2 k2 A0/A2 1 k2 T(l,Y)∝ − 2 log 2 2 exp log 2 . (7) Qs (Y ) Qs (Y ) A2Y Qs (Y ) R. Peschanski / Nuclear Physics B 744 (2006) 80–95 85

2 The saturation scale Qs (Y ) is given by

2 ∝ − − √ 3 + O log Qs (Y ) 2 A0A2 A1 Y log Y (1). (8) 2 A0/A2 Furthermore, at large rapidity, the third term of (7) and the subasymptotic correction in (8) become negligible, leaving a gluon momentum distribution exhibiting geometric scaling [19] 2 √ k − T(l,Y)→ T τ = ∝ τ A0/A2 τ; Q2(Y ) ∝ A A − A Y. 2 log log s 2 0 2 1 Qs (Y ) (9) The asymptotic results (9) can be directly obtained from the unitarity bound applied to the solution of BFKL equation [17], while the subasymptotic results can also be obtained from the BFKL equation, but with more care taken about the boundary conditions [18].

3. Now we introduce the mapping from the BFKL regime of QCD to the thermodynamics of directed polymers. As we have mentioned earlier, the BFKL rapidity evolution generates a cascade of gluons [11]. Each vertex of the cascade is governed in coordinate space by the 2-dimensional BFKL kernel, determining together the energy and the transverse space evolution of the cascade, see Fig. 1. In transverse-momentum space and within the 1-dimensional diffusive approximation (4), we already noticed that the BFKL kernel models boils down to a branching, shift and diffusion operator acting in the gluon transverse-momentum-squared space. Hence the cascade of gluons can be put in correspondence with a continuous branching, velocity-shift and diffusion probabilistic process whose probability by unit of rapidity is defined by the coefficients Ai of (4). Let us first introduce the notion of gluon-momenta “histories” ki(y). They register the evolution of the gluon-momenta starting from the unique initial gluon momentum k(0) and terminating with the specific ith momentum ki , after successive branchings. They define a random function of the running rapidity y, with 0 y Y , the final rapidity range when the evolution ends up (say, for a given total energy). It is obvious that two different histories ki(y) and kj (y) are equal before the rapidity when they branch away from their common ancestor, see Fig. 1. We then introduce random paths xi(t) using formal space and time coordinates x,t,0 t t which we relate to gluon-momenta “histories” as follows:

= t ; 2 ≡− − + − y log ki (y) β xi(t) x(0) (A0 A1)y, (10) A0 where (A0 − A1)y is a conveniently chosen and deterministic “drift term”, x(0) is an arbitrarily ﬁxed origin of an unique initial gluon and thus the same for all subsequent random paths. The random paths xi(t) are generated by a continuous branching and Brownian diffusion process in space–time (cf. Fig. 2). As we shall determine later on, the important parameter β, which plays the rôle of an inverse temperature for the Brownian process, will be ﬁxed to 2A β = 2 . (11) A0 In fact, the relation (11) will be required by the condition that the stochastic process of random paths describes the BFKL regime of QCD near saturation. Another choice of β would eventually 86 R. Peschanski / Nuclear Physics B 744 (2006) 80–95

Fig. 2. Branching diffusion model for polymers. The coordinates x1(t) ···x7(t) correspond to the random paths along the tree in the x, t phase-space. The oblique axis is for L = βx + A1/A0t, which takes into account the “time-drift” in the mapping to the QCD problem. describe the same branching process but in other conditions. Hence the condition (11) will be crucial to determine the QCD phase at saturation (within the diffusive approximation). Let now introduce the tree-by-tree random function deﬁned as the partition function of the random paths system n n 2 − − + = k (y) Z(t) ≡ e βxi (t) = e βx0 A1y × i 1 i . (12) eA0y i=1 Using the fact that the total number of momenta in one event at rapidity y is n ≈ n =eA0y at large y, (12) takes the form

Z = λeA1y × k¯2(y), (13) 1 n 2 ≡ ¯2 where n i=1 ki (y) k (y) is the event-by-event average over gluon momenta at rapidity y − and λ = e βx0 is an arbitrary constant. Note that one has to distinguish ··· ,i.e.thesample-to- sample average, from ···which denotes the average made over only one event. Z(t) is an event-by-event random function. The physical properties are obtained by averaging various observables over the events. Note that the distinction between averaging over one event and the sample-to-sample averaging appears naturally in the statistical physics problem in terms of “quenched” disorder: the time scale associated with the averaging over one random tree structure is different (much shorter) than the one corresponding to the averaging over random trees. In parallel with the statistical physics problem [20], it is convenient to introduce the generating functional G(t, x) of its event-averaged moments Zp , namely 1 − G(t, x) ≡ −eβxZ(t) p = exp −e βxZ(t) = exp −λk¯2(y)/κ2 , (14) p! p where t A y = , log κ2 = βx + 1 t. (15) A0 A0 The moments Zp(t) are easily recovered by differentiation over x or log κ2 at ﬁxed t. R. Peschanski / Nuclear Physics B 744 (2006) 80–95 87

It is interesting to note the ﬁnal form of (14) showing a similarity [14] with a unitary S-matrix amplitude if κ is interpreted as the scale of a transverse-momentum probe, such as depicted in Fig. 1. This similarity will be useful for the derivation of saturation properties from the BFKL regime in the unitarity limit.

4. Now that we have established the connection (10) between the model of gluon-momenta submitted to branching, shift and diffusion, and the random paths defined by xi(t), let us rederive the properties of QCD gluon cascading near saturation from the point-of-view of the statistical random-path system. For this sake, we recall the corresponding model for directed polymers on a tree, as defined in Ref. [20]. Directed polymers on a random tree are modeled, see Fig. 2, by the time evolution of random paths in 1-dimensional space with continuous diffusion and branching (with probability per unit 1 of time fixed respectively to 2 and 1, by convention). Indeed, contemplating the definitions (12) and (14) it is straightforward to identify Z as the sample-to-sample partition function and G(t, x) as the generating functional of its sample- averaged moments for the model [20] of directed polymers on a random tree. Let us list some relevant properties of the polymer problem [20].

• u(t, x) ≡ 1 − G(t, x) veriﬁes the Fisher and Kolmogorov–Petrovsky–Piscounov (F-KPP) equation [7]

∂ 1 ∂2u(t, x) u(t, x) = + u(t, x) − u2(t, x). (16) ∂t 2 ∂x2 u(t, x) admits traveling-wave solutions at large time t [8]

u(t, x) ∼ u x − mβ (t) ≈ u x − c(β)t , (17)

−βx where the wave speed c(β) is governed by the initial condition u(t0,x)∼ e √at large x. The range of values of c(β) = 1/β + β/2 is bounded√ from below by c(βc) ≡ c = 2, where the wave velocity remains ﬁxed whenever β βc = 2. c and βc are usually called, respectively, the critical velocity and the critical slope of the waves. In the “supercritical” regime β βc, the wave front and its speed remain “frozen” and 3 m (t) = c t − log t + O(1). (18) β 4 • The free energy of the polymer system at equilibrium, deﬁned by 1 F =− log Z(t) ≡−m (t), (19) β β √ has a “frozen” density F/t →−c for β βc = 2 indicating the existence of a phase transition at βc. This is related to the dynamical transition c(β) → c in the traveling-wave properties [8]. • Introducing the tree-by-tree free energy spectrum 1 1 f(t)=− log Z(t) + log Z(t) , (20) β β 88 R. Peschanski / Nuclear Physics B 744 (2006) 80–95

its probability distribution Pβ (f ) around the average free energy F is found by deriving asymptotic expressions for the moments [20] to satisfy √ 2f Pβ (f ) ∼−fe for f →−∞, √ −[(2− 2)f ] Pβ (f ) ∼ e for f →∞, (21)

where we quoted only the distribution obtained for the supercritical regime β βc.This means that the fluctuations of the free energy around its event average remain finite even at asymptotic time, i.e. there is no thermodynamical self-averaging, demonstrating that the phase transition is not similar to an ordinary liquid-gas phase transition. • The corresponding phase transition is towards a spin-glass phase characterized by a non- trivial structure in “valleys” or ground state minima, which can be explored quite in detail. We shall describe this structure in the next section, since it will give, through matching with the gluon cascading problem, a description of the gluon-momentum phase at saturation within the diffusive mean-field scheme.

5. Let us make now the connection of the directed-polymer properties with the description of the gluon-momentum phase near saturation. Our aim is to study the structure of the gluon-momentum spectrum when the system reaches saturation. We thus start with a simple initial state (e.g. an onium or one gluon) and, the rapidity increasing, the state develops into a cascading structure of gluons governed by the model that we assume. It consists in random tree structures generated by branching, shifting and diffusing, as previously described. As we have discussed in introduction, the rapid exponential increase of the number of gluons generates reinteractions which damp the exponential increase and cause the saturation phenomenon [3]. In the simple mean-field approximation which neglects the dynamical correlations [5],the saturation mechanism is known to be studied from the BFKL properties in the following approximation [17]. When the exponential increase of the gluon density reaches the unitarity limit of the amplitude, the saturation effects set in. This defines a kinematical region in the k, Y plane around the saturation scale Qs(Y ) where the BFKL regime is expected to smoothly connect the saturation region. Indeed, this expectation has been verified for the amplitude, where many aspects of the saturation features rigorously found [6] for the solutions of the mean field BK saturation equation have been already derived from the BFKL regime considered near the unitarity limit [17]. We shall now try and extend this approach of saturation, which provided already valuable results on averaged observables, to the event-by-event spectrum of gluon transverse momenta. 2 = O 2 In other words, we study momenta of the order of the saturation scale, i.e. ki (Qs ).This is nothing else than the neighborhood of the kinematic region where “universal” traveling-wave asymptotic solutions of the BK equation exist [6]. We thus expect from the studies of Ref. [6] that the properties we shall derive will be independent of initial conditions (provided they are leading to a “supercritical” regime), of the precise form of the kernel (beyond the diffusive approximation) and even of the precise form of the nonlinearities [8]. A comment of caution is nevertheless in order here, concerning the domain of validity of our approach. The properties in the region of smaller momenta ki/Qs 1 will probably depend more on the specific structure of the BFKL kernel beyond the diffusive approximation while the region with ki/Qs 1 will keep the memory of the initial conditions. They should require a different treatment, which is not included in our present work and deserves later study. R. Peschanski / Nuclear Physics B 744 (2006) 80–95 89

Inserting the QCD definitions (13) in (19) for observables after a final rapidity evolution- range Y (or t in time) gives ¯2 log k ≡ log Z −log λ − A1Y = βmβ ( t) − A1Y − log λ = A2 − 3 − + O 2 2 A0Y log Y A1Y (1) log Qs . (22) A0 4 The second line of Eq. (22) is obtained by identifying the inverse temperature β with its value ¯2 ≈ 2 defined by the relation (11). This is the condition to verify k Qs , the identification coming from formula (8). This condition is crucial to check the main saturation property that the sample averaging gluon momentum stays at the value of the saturation scale. Hence, the relations (22), important for the QCD saturation problem, determines the equivalent temperature of the polymer√ system. Starting from (4) and (11), the inverse temperature obtains β = 1/μ ≈ 2.25 >βc = 2, hence larger than the critical value. As a consequence, we are led to consider and study the low- temperature phase of the equivalent random-path system in order to get information on the QCD phase of gluons near saturation. As a further check of the validity of the random cascading model in the “unitarity limit” region, where the average momentum is consistently found to be determined by the saturation scale, let us discuss the distribution around the average. The event-by-event mean momentum k¯2 at rapidity Y can be read off from Eq. (20) as being k¯2 2A ≡−βf ( T ) =− 2 f(A Y) log 2 0 (23) Qs(Y ) A0 −βf ∼ ¯2 2 and, from the first line of (21), when e k /Qs (Y ) is large, its probability distribution is given by k¯2 k¯2 A k¯2 P ∝ − 0 . log 2 log 2 exp log 2 (24) Qs Qs A2 Qs By comparison with formula (9), the interpretation of Eq. (24) is clear. The probability law (24) for the event-by-event partition function coincides with the gluon distribution in the saturation region and follows geometric scaling. This gives the confirmation, through the properties of the directed polymers, that the gluon-cascading model near the unitarity limit gives the correct saturation scale and gluon distribution. In mathematical terms, this comes from the fact that the equation for the generating function of moments for the model (16) is in the same (F-KPP) universality class than the BK equation [6]. Both properties (22), (24) prove the consistency of the model with the properties expected from saturation. We shall then look for other properties of the cascading gluon model. It is important to realize that this consistency fails for any other choice of the parameter β different from (11). This justifies a-posteriori the identification of the equivalent temperature of the system in the gluon/polymer mapping framework.

6. Let us now come to the main topics concerning the determination of structure of the gluon-momentum phase at saturation in the diffusive model. The striking property of the directed polymer problem on a random tree is the spin-glass structure of the low temperature phase. As we shall see this will translate directly into a speciﬁc 90 R. Peschanski / Nuclear Physics B 744 (2006) 80–95

··· ··· Fig. 3. The clustering structure. The drawing represents the s1 s4 clusters near momenta ks1 ks4 . They branch either near t 1or( t − t) 1, where t is the total amount of time evolution. clustering structure of gluon transverse momenta in their phase near the “unitarity limit”. Let us divide the discussion in a few subsections.

(i) The equivalent temperature. Following the relation (11), one is led to consider the polymer system at temperature T with

Tc − T βc ≡ 1 − = 1 − γc, (25) Tc β where γc is the critical exponent. We are thus naturally led to consider the√ low-temperature phase (T

Translating these results in terms of gluon momenta (in modulus), the phase space landscape consists in event-by-event distribution of clusters of momenta around some values k2 ≡ si 1/(n ) k2, where n is the cluster multiplicity. The probability weights to ﬁnd a cluster s i i∈si i i i after the whole evolution range Y is deﬁned by 2 ∈ k W = i si i , si 2 i ki where the summation in the numerator is over the momenta of gluons within the si th cluster (see

Fig. 3). The normalized distribution of weights Wsi thus allows one to study the probability distribution of clusters. The clustering tree structure (called “ultrametric” in statistical mechanics) is the most prominent feature of spin-glass systems [23]. R. Peschanski / Nuclear Physics B 744 (2006) 80–95 91

Note again that, for the QCD problem, this property is proved for momenta in the region of the “unitarity limit”, or more concretely in the momentum region around the saturation scale. This means that the cluster average-momentum is also such that k2 = O(Q2). Hence the clustering si s structure is expected to appear in the range which belongs to the traveling-wave front [6] or, equivalently, of clustering with ﬁnite ﬂuctuations around the saturation scale.

(iii) The “overlap” function. There exists in statistical physics of spin-glasses a well-known function which reveals the space–time structure of a spin-glass state in a quantitative way: the “overlap” function [23]. Translating the definitions (cf. [20]) in terms of the QCD problem, one introduces the overlap between two final gluon momenta ki and kj , using their “histories” ki(y) and kj (y). We recall that ki(y) and kj (y) defined as function of the running rapidity y correspond to the two (not always disjoint) paths histories which finally lead to the momenta ki and kj observed at final (and large) rapidity Y .

The overlap is defined by the quantity Qij , the fraction of the total rapidity Y for which ki(y) = kj (y),0 Qij 1. It is a measure of the rapidity interval (“time”) during which they belong to the same branch before splitting. For a given event, the probability Y˜(q) of finding an overlap q can be defined by n ˜ 1 2 2 Y(q) dq = k k Θ q

− − − − 2 g(ρ) = dvv T/Tc 1 1 − e v ρv . (29) 0 92 R. Peschanski / Nuclear Physics B 744 (2006) 80–95

Fig. 4. The probability distribution of overlaps Π(Y). The ﬁgure (the simulation is by courtesy from [28],usingthe method of Ref. [25]) is drawn both for the theoretical QCD value γc = 0.7, and for γc = 0.3 for comparison. The statistics 8 3 6 2 used for the simulation is 10 events in 10 bins for γc = 0.3and3.25 × 10 events in 10 bins for γc = 0.7.

The overlap distribution depends only on T/Tc, which confirms its high degree of universality. One finds, in particular, Y =1 − T/Tc, which is the average probability of having overlap 1, while T/Tc is the probability of overlap 0. The rather involved probability distribution Π(Y) is quite intriguing, as obtained by inversion of moments Yν [24,25]. It possesses a priori an infinite number of singularities at Y = 1/n,n integer. It can be seen when the temperature is significantly lower from the critical value, see for instance the curve for 1 − T/Tc = 0.7inFig. 4. However, the predicted curve for the QCD value 1 − T/Tc = 0.3 is smoother and shows only a final cusp at Ws = 1 within the considered statistics. It thus seems that configurations with only one cluster can be more prominent than the otherwise smooth generic landscape. However, it is also a “fuzzy” landscape since many clusters of various sizes seem to coexist in general.

(iv) The cluster distribution. The sample-to-sample distribution of “valleys” can be explic- W , which itly computed by a generalization of the overlap function [24,25] to any power , si translates into a well-deﬁned distribution of cluster-weight probability. For instance one determines:

− − − W γc 1(1 − W)γc 1 Φ(W) = , (γc)(1 − γc) − − − (W W ) γc 1(1 − W − W )2γc 1 Φ(W ,W ) = γ 1 2 1 2 × θ( − W − W ), 1 2 c 2 1 1 2 (2γc) (1 − γc)

Φ(W1,W2,...,Wp) − − p − p (W W ···W ) γc 1(1 − W )pγc 1 = p 1 2 p 1 i × − γc (p) p θ 1 Wi , (30) (pγc) (1 − γc) 1 R. Peschanski / Nuclear Physics B 744 (2006) 80–95 93

≡ for, respectively, the average number Φ(W) of clusters of weight Wsi W , the average number ≡ of pairs Φ(W1,W2) of clusters in the same event with weights Wsi W1,W2, and more generally ≡ Φ(W1,W2,...,Wp) the average number of p-uples of clusters with weights Wsi W2,...,Wp.

It is interesting to note that smaller is γc = T/Tc, smaller is the probability to ﬁnd Ws ∼ 0, i.e. no clustering, and higher is the probability for having one big cluster Ws ∼ 1. As an illustration, the phenomenological value γc ≈ 0.3 leads to a more pronounced clustering structure than the BFKL value γc ≈ 0.6. The sharp structuration in clusters expected for small γc can already be seen in the overlap distribution of Fig. 4.

7. Let us summarize the main points of our approach. We investigated the landscape of transverse momenta in gluon cascading around the saturation scale at asymptotic rapidity. Limiting our study to a diffusive 1-dimensional modelization of the BFKL regime of gluon cascading, we make use of a mapping on a statistical physics model for directed polymers propagating along random tree structures at fixed temperature. We then focus our study on the region near the unitarity limit where information can be obtained on saturation, at least in the mean-field approximation. Our main result is to find a low-temperature spin-glass structure of phase space, characterized by event-by-event clustering of gluon transverse momenta (in modulus) in the vicinity of the rapidity-dependent saturation scale. The weight distribution of clusters and the probability of momenta overlap during the rapidity evolution are derived. Interestingly enough the clusters at asymptotic rapidity are branching either near the beginning (“overlap 0” or y/Y 1) or near the end (“overlap 1” or 1 − y/Y 1) of the cascading event. The probability distribution of overlaps is derived and shows a rich singularity structure. Phenomenological and theoretical questions naturally emerge from this study. On a phenomenological ground, it is remarkable that saturation density effects are not equally spread out on the event-by-event set of gluons; our study suggests that there exists random spots of higher density whose distribution may possess some universality properties. In fact it is natural to expect this clustering property to be present not only in momentum modulus (as we could demonstrate) but also in momentum azimuth-angle. This is reminiscent of the “hot spots” which were some time ago [26] advocated from the production of forward jets in deep-inelastic scattering at high energy (small Bjorken variable). The observability of the cluster distribution through the properties of “hot spots” is an interesting possibility. On the theoretical ground, the questions arise how the pattern of fluctuations we found influ- ence the onset of saturation and its properties, given the fact that fluctuations are of primordial importance in the development of the whole QCD process, as we have stressed in the introduction. First, the question arises to extend the study to the full BFKL kernel (3), and in two dimensions, as given initially by Eq. (1). Universality properties of the solutions of the BK equations in the traveling-wave formalism [6] and in the 2-dimensional case [27] would encourage such a generalization. However, some care must be taken of the potential problems which can arise from an event-by-event description of the cascading regime. The suggestion is [28] to formulate the problem directly in the full 2-dimensional position space, for which the gluon cascading and the unitarity constraints are well defined. Some preliminary numerical results are encouraging. Second, it would be instructive to incorporate the pattern of fluctuations we found in the calculation of the contribution of Pomeron loops to the scattering amplitude. The Pomeron loops are found to be important to consider and are the subject of many studies (see, e.g. [10]). The Pomeron loops are obtained by merging the splitting of cascading gluons of the projectile with 94 R. Peschanski / Nuclear Physics B 744 (2006) 80–95 the merging in the target. We think that the spin-glass structure of the cascading gluons could lead to useful hints for dealing with the Pomeron loops. In conclusion, we think that the already rich structure of the phase space landscape of gluon clusters found with the diffusive model of gluon cascading opens an interesting way towards a deeper understanding of the QCD phase in the saturation regime and could lead to a deeper description the color glass condensate.

Acknowledgements

I would like to thank Edmond Iancu for inspiring and stimulating remarks and Cyrille Mar- quet and Gregory Soyez for many fruitful suggestions and a careful reading of the manuscript. Special thanks are due to Gregory Soyez for providing the simulation of Fig. 4. I am indebted to B. Derrida for his seminal work on traveling waves, directed polymers and spin glasses.

References

[1] L.N. Lipatov, Sov. J. Nucl. Phys. 23 (1976) 338; E.A. Kuraev, L.N. Lipatov, V.S. Fadin, Sov. Phys. JETP 45 (1977) 199; I.I. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978) 822. [2] L.V. Gribov, E.M. Levin, M.G. Ryskin, Phys. Rep. 100 (1983) 1. [3] L. McLerran, R. Venugopalan, Phys. Rev. D 49 (1994) 2233; L. McLerran, R. Venugopalan, Phys. Rev. D 49 (1994) 3352; L. McLerran, R. Venugopalan, Phys. Rev. D 50 (1994) 2225; A. Kovner, L. McLerran, H. Weigert, Phys. Rev. D 52 (1995) 6231; A. Kovner, L. McLerran, H. Weigert, Phys. Rev. D 52 (1995) 3809; R. Venugopalan, Acta Phys. Pol. B 30 (1999) 3731; E. Iancu, A. Leonidov, L. McLerran, Nucl. Phys. A 692 (2001) 583; E. Iancu, A. Leonidov, L. McLerran, Phys. Lett. B 510 (2001) 133; E. Iancu, L. McLerran, Phys. Lett. B 510 (2001) 145; E. Ferreiro, E. Iancu, A. Leonidov, L. McLerran, Nucl. Phys. A 703 (2002) 489; H. Weigert, Nucl. Phys. A 703 (2002) 823; For comprehensive reviews on saturation and other references, see e.g. A.H. Mueller, Parton saturation: An overview, hep-ph/0111244; E. Iancu, R. Venugopalan, The color glass condensate and high energy scattering in QCD, hep-ph/0303204. [4] L. McLerran, What is the evidence for the color glass condensate?, hep-ph/0402137. [5] I.I. Balitsky, Nucl. Phys. B 463 (1996) 99, hep-ph/9509348; Y.V. Kovchegov, Phys. Rev. D 60 (1999) 034008, hep-ph/9901281. [6] S. Munier, R. Peschanski, Phys. Rev. Lett. 91 (2003) 232001, hep-ph/0309177; S. Munier, R. Peschanski, Phys. Rev. D 69 (2004) 034008, hep-ph/0310357; S. Munier, R. Peschanski, Phys. Rev. D 70 (2004) 077503, hep-ph/0310357. [7] R.A. Fisher, Ann. Eugenics 7 (1937) 355; A. Kolmogorov, I. Petrovsky, N. Piscounov, Moscow Univ. Math. Bull. A 1 (1937) 1. [8] M. Bramson, Mem. Amer. Math. Soc. 44 (1983) 285; E. Brunet, B. Derrida, Phys. Rev. E 56 (1997) 2597; U. Ebert, W. van Saarloos, Physica D 146 (2000) 1. [9] E. Brunet, B. Derrida, Phys. Rev. E 57 (1997) 2597, cond-mat/0005362; E. Brunet, B. Derrida, Comput. Phys. Commun. 121 (1999) 376, cond-mat/0005364. [10] A.H. Mueller, A.I. Shoshi, Nucl. Phys. B 692 (2004) 175, hep-ph/0402193; E. Iancu, A.H. Mueller, S. Munier, Phys. Lett. B 606 (2005) 342, hep-ph/0410018; S. Munier, Nucl. Phys. A 755 (2005) 622, hep-ph/0501149; E. Iancu, D.N. Triantafyllopoulos, Nucl. Phys. A 756 (2005) 419, hep-ph/0411405; E. Iancu, D.N. Triantafyllopoulos, Phys. Lett. B 610 (2005) 253, hep-ph/0501193; C. Marquet, R. Peschanski, G. Soyez, Consequences of strong ﬂuctuations on high-energy QCD evolution, hep- ph/0512186. R. Peschanski / Nuclear Physics B 744 (2006) 80–95 95

[11] A.H. Mueller, Nucl. Phys. B 415 (1994) 373; A.H. Mueller, B. Patel, Nucl. Phys. B 425 (1994) 471, hep-ph/9403256; A.H. Mueller, Nucl. Phys. B 437 (1995) 107, hep-ph/9408245. [12] N.N. Nikolaev, B.G. Zakharov, Z. Phys. C 49 (1991) 607. [13] G.P. Salam, Nucl. Phys. B 449 (1995) 589, hep-ph/9504284; G.P. Salam, Nucl. Phys. B 461 (1996) 512, hep-ph/9509353; G.P. Salam, Comput. Phys. Commun. 105 (1997) 62, hep-ph/9601220; A.H. Mueller, G.P. Salam, Nucl. Phys. B 475 (1996) 293, hep-ph/9605302. [14] For recent discussion and applications of the dipole generating functionale, see Y. Hatta, E. Iancu, C. Marquet, G. Soyez, D.N. Triantafyllopoulos, Diffusive scaling and the high-energy limit of deep inelastic scattering in QCD at large N(c), hep-ph/0601150. [15] A. Kovner, M. Lublinsky, Nucl. Phys. A 767 (2006) 171, hep-ph/0510047; E. Iancu, G. Soyez, D.N. Triantafyllopoulos, On the probabilistic interpretation of the evolution equations with Pomeron loops in QCD, hep-ph/0510094. [16] R. Peschanski, Phys. Lett. B 622 (2005) 178, hep-ph/0505237; C. Marquet, R. Peschanski, G. Soyez, Phys. Lett. B 628 (2005) 239, hep-ph/0509074. [17] E. Iancu, K. Itakura, L. McLerran, Nucl. Phys. A 708 (2002) 327, hep-ph/0203137. [18] A.H. Mueller, D.N. Triantafyllopoulos, Nucl. Phys. B 640 (2002) 331, hep-ph/0205167. [19] A.M. Stasto,´ K. Golec-Biernat, J. Kwiecinski, Phys. Rev. Lett. 86 (2001) 596, hep-ph/0007192. [20] B. Derrida, H. Spohn, J. Stat. Phys. 51 (1988) 817. [21] B. Derrida, Phys. Rev. Lett. 45 (1980) 79; B. Derrida, Phys. Rev. B 24 (1980) 2613. [22] D. Sherrington, S. Kirkpatrick, Phys. Rev. Lett. 35 (1975) 1972; D. Sherrington, S. Kirkpatrick, Phys. Rev. B 17 (1978) 4384. [23] The classical book on spin-glass theory and the overlap function is: M. Mézard, G. Parisi, M.A. Virasoro, Spin Glass Theory and Beyond, World Scientiﬁc Lecture Notes in Physics, vol. 9, World Scientiﬁc, Singapore, 1987. [24] B. Derrida, G. Toulouse, J. Physique Lett. 46 (1985) L223. [25] B. Derrida, H. Flyvbjerg, J. Phys. A: Math. Gen. 20 (1987) 5273. [26] A.H. Mueller, Nucl. Phys. B (Proc. Suppl.) 18 (3) (1990) 125; A.H. Mueller, J. Phys. G 17 (1991) 1443; J. Bartels, M. Besancon, A. De Roeck, J. Kurzhofer, Measurement of hot spots at HERA, in: Hamburg 1991, Proceedings, Physics at HERA, vol. 1, 1991, pp. 203–213 (see High energy physics index 30 (1992) No. 12988); J. Bartels, Measurement of hot spots in the proton at HERA, Prepared for International Workshop on Quark Cluster Dynamics, Bad Honnef, Germany, 29 June–1 July 1992; J. Bartels, H. Lotter, Phys. Lett. B 309 (1993) 400. [27] C. Marquet, R. Peschanski, G. Soyez, Nucl. Phys. A 756 (2005) 399, hep-ph/0502020; C. Marquet, G. Soyez, Nucl. Phys. A 760 (2005) 208, hep-ph/0504080. [28] G. Soyez, private communication. Nuclear Physics B 744 (2006) 96–120

Large N orbifold ﬁeld theories on the twisted torus

C. Hoyos

Instituto de Física Teórica UAM/CSIC, Facultad de Ciencias, Cantoblanco, 28049 Madrid, Spain Received 16 February 2006; accepted 23 March 2006 Available online 17 April 2006

Abstract We study the planar equivalence of orbifold ﬁeld theories on a small three-torus with twisted boundary conditions, generalizing the analysis of [J.L.F. Barbón, C. Hayos, JHEP 0601 (2006) 114, hep-th/0507267]. The non-supersymmetric orbifold models exhibit different large N dynamics from their supersymmetric “parent” counterparts. In particular, a moduli space of Abelian zero modes is lifted by an O(N2) potential in the “daughter” theories. We also ﬁnd disagreement between the number of discrete vacua of both theories, due to fermionic zero modes in the parent theory, as well as the values of semiclassical tunneling contributions to fermionic correlation functions, induced by fractional instantons. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

Although there have been considerable advances in the strong-coupling description of gauge theories since their formulation, this remains an unresolved matter. A line of attack to this problem has been to analyze related systems with simpler dynamics. Specially successful has been the study of supersymmetric theories, that in some cases can be solved exactly. However, the applicability of the methods employed usually relay on supersymmetry and cannot be used safely for non-supersymmetric theories. A different and older approach is ’t Hooft’s large N expansion [1], where each term in the expansion is non-perturbative in the coupling constant. Although it gives a good qualitative description of many features of strongly coupled gauge theories, extracting quantitative results for realistic theories has been demonstrated to be a difﬁcult task. Recently, a new combination of both lines has been proposed under the name of planar equivalence [2]. Starting from a supersymmetric theory, called ‘parent’, it is projected to a non-

E-mail address: [email protected] (C. Hoyos).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.03.027 C. Hoyos / Nuclear Physics B 744 (2006) 96–120 97 supersymmetric theory, called ‘daughter’, so that they are equivalent in the large N limit. The projections are inspired in string constructions and they are named after orbifolds [3] and orientifolds [4]. Orbifold projection relates a N = 1 SU(kN) supersymmetric gauge theory with a (SU(N))k gauge theory with fermions in bifundamental representations under two adjacent gauge group factors. Orientifold projection relates a N = 1 SU(N) supersymmetric gauge theory with a SU(N) gauge theory with fermions in two-index symmetric or antisymmetric representations. For these theories, planar equivalence has been proved at the perturbative level, and there are formal proofs of exact equivalence [5,6]. A remarkable example of this program is the computation of the quark condensate in terms of N = 1 gaugino condensate using orientifold theories [7]. Further works on finite N computations in orientifold theories using string duals can be found in [8]. Theories with other possible representations of fermions and their planar properties are studied in [9]. Although the results are encouraging, the exact equivalence is not completely established, specially for orbifold theories, that have been the subject of much discussion [10–14], although there are also some concerns about orientifold theories [14]. We want to clarify this point by studying these theories in a dynamically controlled regime, where we can analyze the behavior of both parent and daughter theory and compare them. In order to do that, we will introduce them in a small spatial torus R×T3, where the theory is in a weak coupling regime, so perturbative and semiclassical methods can be used. The first analysis of planar equivalence using finite volume is [13], where a single spatial direction was compactified in a circle. It was shown that the dynamics of the orbifold daughter breaks down planar equivalence for small enough radius, because the sector where it would hold becomes tachyonic. However, it is not clear that this will remain true when the circle is decompactified. The extension to a total compactification of space was made in [14]. In this case non-trivial flat connections in the torus are zero modes that generate a moduli space. In the small volume limit, zero modes control the dynamics of the theory, so we can concentrate on the study of the moduli space. In principle, quantum corrections can generate a potential of order O(N2) over the moduli space, but supersymmetry guarantees that a potential is not generated. Planar equivalence boils down to the vanishing of the daughter effective potential to O(N2), but it is shown that this is violated locally at some points of the moduli space due to non-linear effects, although it is satisfied on the average. Also, the O(N) potential makes the physics of parent and daughter moduli space very different. It seems that the signals of a possible failure of planar equivalence are a consequence of having scalar degrees of freedom that arise when we compactify the theory, and that their effects may disappear when we go to the infinite volume limit. Then, it is interesting to get rid of the moduli space in order to avoid this problem. This can be done using twisted boundary conditions, that will constitute the frame of this paper. Orbifold theories are invariant under a global ZN group of symmetry, the center, that can be used to introduce twisted boundary conditions and lift most or all of the non-Abelian moduli space. Orientifold theories are invariant only under a Z2 center group, that is not enough to significantly change the moduli space. For this reason, this paper will concentrate on orbifold theories. Twisted boundary conditions will allow the construction of a precise mapping of topological sectors between parent and daughter, so we can compare the vacuum sectors. Moreover, we will be able to compare fermionic correlation functions induced by tunneling between vacua in each theory. Ordinary instantons can contribute to such quantities, but they are not useful to answer questions about planar equivalence because their semiclassical contributions vanish ex- 2 ponentially in the N →∞limit. The reason is that their action scales as Sinst = 8π N/gt , where = 2 gt gYMN is the ’t Hooft coupling. However, when twisted boundary conditions are introduced, new vacua and configurations associated to tunneling appear. This new class of tunneling con- 98 C. Hoyos / Nuclear Physics B 744 (2006) 96–120

ﬁgurations can give a non-vanishing contribution to fermionic correlation functions, since their 2 action scales as a fractional instanton Stun = 8π /gt . The outline of the paper is as follows. In Section 2 we introduce twisted boundary conditions and show how they can be used to make a complete map between different gauge bundles of parent and daughter theory. In Section 3 we use this map to describe and compare vacuum states of both theories. In Section 4 we extend the analysis to include effects induced by tunneling. Finally, we give a geometrical interpretation in terms of bound states of D-branes and conclude.

2. Embedding of the orbifold theory

We are interested in finding the map between physical configurations of parent and daughter theories relevant for planar equivalence. According to [5], we should compare the vacuum sectors of both theories. However, this is a difficult task, specially for the daughter theory, because we cannot use perturbation theory and we do not have non-perturbative effects under control in the IR limit. In order to avoid these problems, we introduce parent and daughter theories in a small finite volume. This has been an extensively used tool to study asymptotically free gauge theories, since they are in a weak coupling regime where perturbative and semi-classical methods are applicable. The orbifold theories that we are considering have a (U(N))k gauge group and Weyl fermi- ¯ ons that transform as the bifundamental representation (Ni, Ni+1) of two adjacent gauge groups. These theories can be obtained from a N = 1 U(kN) supersymmetric gauge theory after performing an “orbifold projection”. ⎛ ⎞ 1 ⎛ ⎞ Aμ 0 λ1,2 ⎜ 2 ⎟ ⎜ ⎟ ⎜ Aμ ⎟ .. .. → ⎜ ⎟ → ⎜ . . ⎟ Aμ ⎝ . ⎠ ,λ⎝ ⎠ (2.1) .. 0 λk−1,k k λ 0 Aμ k,1 i in general, when summing over different orbifold components Aμ, λi,i+1, we will always assume that indices are defined modulo k. For this kind of theories, it has been proved that parent and daughter planar diagrams give the same result, if the coupling constants of both theories are 2 = 2 related by kgp gd . We should point that the gauge group of the daughter theory is a subgroup of the gauge group of the parent theory of the same rank. This is quite useful if we try to establish a map between both theories. For instance, every representation of the parent theory has a unique decomposition in representations of the daughter theory. Notice that if we ignore Abelian groups in the daughter theory, the rank would be different. We will see that Abelian groups play a relevant role in the mapping. We will ignore only the diagonal U(1) group of the daughter theory, that maps trivially to the Abelian part of the parent group. All fields are uncharged under these groups, so they decouple trivially.

2.1. Twisted boundary conditions in the torus

Twisted boundary conditions were introduced by ’t Hooft [15] and successfully applied by Witten to study supersymmetric theories [16,17]. The topics we review brieﬂy here can be found more thoroughly studied in these references. An extensive classical analysis of twisted boundary conditions in T4 can also be found in [18]. C. Hoyos / Nuclear Physics B 744 (2006) 96–120 99

When we compactify a gauge theory in a three-torus R×T3, we have some freedom to specify the boundary conditions of the fields on the sides of the torus. For a SU(M) gauge group, the fields must be periodic up to gauge transformations. In order to be in a configuration with zero field strength, the gauge transformations on each side must commute. We can always change the boundary conditions by making gauge transformations and redefining our field, so all these configurations are equivalent. In particular, all can be reduced to the case with trivial periodic boundary conditions, where there are zero modes for the gauge fields given by constant Abelian connections. For the theories we are interested, the fields are invariant under a discrete subgroup of the gauge group, that we call the invariant center. For instance, for a (supersymmetric) SU(M) gauge theory, the center is ZM . As a consequence, we can consider our theory as a SU(M)/ZM theory and impose non-trivial boundary conditions on the fields. The fields are periodic up to a gauge transformation, but now the transformations on each side need to commute only up to an element of the center. Therefore, for each side we can give an element of the center that defines non-equivalent boundary conditions. The information is encoded in the so-called magnetic flux m , which is a three-vector of integers defined modulo M. Formally, we are constructing gauge bundles of different topology, using the mapping between non-trivial one-cycles of the group SU(M)/ZM to the torus. Since in general we cannot reduce these configurations to the trivial case, there are no zero modes for gauge fields. Imagine that we are in a sector of definite magnetic flux, and we choose to work in the A0 = 0 gauge. This gauge only fixes time-dependent transformations. When we make a time- independent gauge transformation, it has to satisfy the same boundary conditions as the fields, up to an element of the center. Then, we can introduce an element of the center for each direction that characterizes the twisted gauge transformation. We can group them in a single three-vector k of integers. Twisted gauge transformations are not continuously connected to the identity, so physical states do not need to be invariant under them. As a matter of fact, each time we make a twisted gauge transformation we can reach a different sector of the Hilbert space. Using Fourier transformation, we can build states invariant under the action of a twisted gauge transformation, labelled by the electric flux e, a three-vector of integers defined modulo M. The information carried by electric and magnetic flux can be encoded in a single quantity nμν , the twist tensor, that emerges when we study gauge bundles in an Euclidean T4.AsinT3,we use gauge transformations to glue the sides of the torus. The gauge transformations on the sides μ and ν must commute up to an element of the center given by exp(2πinμν/M). 2πi Ω(μ,x + l )Ω(ν, x) = exp n Ω(ν,x + l )Ω(μ, x). (2.2) ν M μν μ The twist tensor is an antisymmetric tensor of integers defined modulo M. It is related to electric and magnetic flux in a similar way as electric and magnetic components are related to field strength

nij = ij k mk,n0i = ki. (2.3) In general, the Euclidean gauge bundles will have a curvature. In this case, the action does not vanish, but it has a minimum value given by the Pontryagin number or topological charge. Usu- ally, contributions from ordinary instantons are considered. They come from bundles associated to the wrapping of the gauge group over a S3. Then, this contribution is an integer known as the winding number. For gauge bundles associated to the torus, the contribution to topological charge can be fractional, because we are using a representation that is not faithful for SU(M)/ZM .In 100 C. Hoyos / Nuclear Physics B 744 (2006) 96–120 the sector of zero winding number, the topological charge is given by 1 ˜ 4 1 1 Q = tr Fμν(x)Fμν(x) d x =− nμνn˜μν =− k ·m (2.4) 16π 2 4M M T4 ˜ where Aμν = (1/2)μνρσ Aρσ .

2.2. Twisted boundary conditions and orbifold theories

The parent theory is a SU(M) supersymmetric gauge theory with M = kN. All the previous discussion can be applied directly to this case. In order to study the relation between parent and daughter theories in this setup, we follow the construction that ’t Hooft used to find self-dual solutions of fractional topological charge [19], called torons, and generalize it to our case. The analysis is made in the Euclidean T4. We split rows and columns in k diagonal boxes of N × N size. Then, we work in the subgroup (SU(N))k × U(1)k−1 ⊂ SU(kN), which is the orbifold projection on the gauge sector. If we had considered only a (SU(N))k theory, without the Abelian part, then the possible bundles in the k daughter theory would have been characterized by a ZN diagonal subgroup of (ZN ) .Thisisso because of the fermions in the bifundamental representations. If twist tensors of different SU(N) groups were different, boundary conditions for fermions will not be consistent, according to (2.2). However, Abelian fields can be used to absorb extra phases and enlarge the number of possible bundles. This will be evident in our construction. Let ωi , i = 1,...,k be the U(1) generators ωi = 2π diag −N1N ,...,N(k− 1)1N ,...,−N1N , (2.5) where N(k− 1)1N is located at the ith position and1N denotes the N × N identity matrix. Note − k = that only k 1 generators are independent, since i=1 ωi 0. Define the following twist matrices 2πi iω V W = W V exp + i . (2.6) i i i i kN kN2

All other pairs commute. Notice that although Vi and Wi are matrices deﬁned in principle to make a SU(kN) twist, in fact the terms in the exponent are such that the non-trivial twist is made only over the ith N × N box. A possible representation is (i) Vi = diag 1N ,...,P ,...,1N , N = (i) Wi diag 1N ,...,QN ,...,1N , (2.7) (i) (i) = 2πi/N where QN and PN are the SU(N) matrices of maximal twist PN QN QN PN e associated to the ith gauge group. Let us try the ansatz k i i k = aμ bμ i Ω(μ,x) Vi Wi exp i ωiαμνxν/lν , (2.8) i=1 i=1 i i i where summation over ν is understood. The numbers aμ, bμ are arbitrary integers. αμν is an antisymmetric real tensor which will be associated to the Abelian ﬂuxes that are necessary to cancel out phases in the case of non-diagonal twist. C. Hoyos / Nuclear Physics B 744 (2006) 96–120 101

Inserting (2.8) in the consistency conditions (2.2) we find that k = i i = i i − i i nμν nμν,nμν aμbν aνbμ (2.9) i=1 and the k conditions k k k j − i = 2 i − j i = nμν knμν 2kN kαμν αμν , αμν 0 (2.10) j=1 j=1 i=1 i i which can be used to obtain the αμν tensors from the nμν tensors. The second equation in (2.10) comes from the fact that there are only k −1 independent U(1) groups. Any other linear condition will be valid. This establishes a map between Euclidean bundles in parent and daughter theories, identifying i nμν as the twist tensors of the daughter theory. This implies that many inequivalent twists in the daughter theory map to the same twist in the parent. A second consequence relies on the fact that i αμν are constant Abelian field strengths. This implies that configurations like ’t Hooft’s torons in the parent theory can be mapped to Abelian electric and magnetic fluxes in the daughter, showing that Abelian groups play a relevant role in the mapping. Generically, physically inequivalent configurations in the daughter theory, like Abelian fluxes and torons, map to the same configuration (up to gauge transformations) in the parent theory. The map can be one-to-many, and it is given by the possible decompositions of the parent twist tensor nμν (defined mod kN)ink daughter’s i twist tensors nμν (each defined mod N). A remark is in order. Although in principle we can turn on arbitrary fluxes of Abelian fields in the daughter theory, we must take into account the presence of charged fermions under these. Since bifundamentals correspond to off-diagonal boxes in a SU(kN) matrix, the value of Abelian fluxes is determined modulo N by consistency conditions of the twist. This implies that fractional contributions of Abelian fluxes to the topological charge are determined completely by (2.10). On the other hand, integer contributions to the topological charge in the parent theory can map to non-Abelian instantonic configurations, Abelian fluxes of order N or a mixture of both. Since the energy of this kind of configurations is a factor N larger than fractional contributions, they are irrelevant in the large N limit and we will not worry about them anymore. We can be more precise with the map of Abelian configurations. The U(1) groups under consideration follow from the decomposition of SU(kN) gauge connections in boxes Aμ = 1 N diag(Aμ,...,Aμ ) so that U(1) connections are given by i = i − i+1 Bμ tr Aμ tr Aμ , (2.11) where i = 1,...,k − 1. Then, following [20], we can relate topological invariants of Abelian i groups of the daughter theory to the twist tensor of the parent theory. Let Fμν be the field strength of the ith Abelian group. We have the following topological invariants 1 ci = dx ∧ dx tr F i , (2.12) μν 2π μ ν μν given by the integral of the first Chern class over non-contractible surfaces of the torus and 1 Qi = tr F i F˜ i d4x, (2.13) 16π 2 μν μν T4 102 C. Hoyos / Nuclear Physics B 744 (2006) 96–120 which is the Pontryagin number and it is related to first and second Chern classes. We can relate Qi i to the topological charge of the parent theory, while cμν (modN) will be a twist tensor nμν ⊂ i − i+1 − i for the SU(N) SU(kN) subgroup of gauge connection Aμ Aμ Bμ. It contributes to a fractional topological charge through (2.4).

3. Vacuum structure

We have been able to construct a map between configurations of both theories, showing that they are physically inequivalent, although this is not a surprise. As a matter of fact, we are interested only in large N equivalence. According to [5], planar equivalence will hold non- perturbatively in the sector of unbroken Zk orbifold symmetry. The necessary and sufficient condition is that operators related by orbifold projection, and invariant under orbifold transformations, have the same vacuum expectation values. Therefore, it is enough to check the vacuum sector. In infinite volume it is argued that large N equivalence does not hold for k>2 because of spontaneous breaking of orbifold symmetry by the formation of a gaugino condensate in the parent theory [11].Fork = 2 the condensate does not break orbifold symmetry and the question remains open. In finite volume in principle there is no spontaneous breaking, so any failure of large N orbifold equivalence must show itself in a different way. In the following analysis we will restrict to the sector of vanishing vacuum angles for simplicity.

3.1. Ground states in the daughter theory

We will start by comparing the bosonic sector of ground states of parent and daughter theories in a three-torus. The first question we can ask ourselves is what are the relevant configurations we must consider. We assume that the orbifold projection has been made at a scale of energy much larger than the typical scale of the torus μ 1/l. At this point, all the coupling constants of the orbifold theory are equal. The running of coupling constants of non-Abelian groups is given at leading order by the renormalization group of the parent theory, by perturbative planar equivalence. However, the running for Abelian groups is different since they are not asymptotically free. As a matter of fact, the coupling constants of Abelian groups will be smaller than the coupling constants of non-Abelian groups in our torus, and the difference will increase with the volume. We can now consider what are the contributions to energy of Abelian configurations, that depend on electric (Ei = α0i ) and magnetic (Bi = ij k αjk) fluxes as e2(l) 1 H = E2 + B2 . (3.1) 2 2e2(l) 3 Tl In the quantized theory B2 acts as a potential, while E2 plays the role of kinetic energy. We see that the magnetic contribution is proportional to the inverse of the coupling, so the ground states of the theory will be in the sectors of zero Abelian magnetic flux. From (2.9),this forces us to introduce equal magnetic flux on all non-Abelian gauge factors,

m p = km d , (3.2) where m p(d) is the magnetic ﬂux in the parent (daughter) theory. This kind of boundary conditions preserve the orbifold symmetry of interchanging of SU(N) groups. C. Hoyos / Nuclear Physics B 744 (2006) 96–120 103

The twist of the daughter theory is determined by a ZN diagonal subgroup. The elements of this group are actually equal to the elements of a ZN subgroup of the center of the parent theory 2πil/N 2πil(N−1)/N Zl = diag 1,e ,...,e . (3.3) k We are now interested in imposing twisted boundary conditions in the daughter theory such that the moduli space of non-Abelian gauge groups is maximally lifted. We can introduce a unit of magnetic ﬂux in the same direction for each of the gauge groups, so we are performing a maximal twist of the theory and thus lifting all the non-Abelian moduli space. However, the kind of boundary conditions we are imposing does not prevent the existence of constant connections for Abelian groups. Bifundamental fermions are charged under these U(1) groups, so it is not possible to gauge away constant connections. Moreover, they become periodic variables. Therefore, the Abelian moduli space of the daughter theory is a (k − 1)-torus (for each spatial direction). In the A0 = 0 gauge we have freedom to make gauge transformations depending on spatial coordinates. Abelian gauge transformations must be such that they compensate the phases that can appear on fermionic ﬁelds if we make twisted gauge transformations that are not equal for all non-Abelian gauge groups. Only transformations with k ·m = 0modN lead to a different vacuum, so there are N k possible non-Abelian vacua. The total moduli space consists on N k disconnected T3(k−1) tori. The Abelian part of the vacuum is a wavefunction over the torus. In our setup, orbifold symmetry is preserved in the vacua that are reached by twisted gauge transformations of the same topological class for all non-Abelian gauge groups. For these, Abelian gauge transformations are trivial. In Section 3.5 we will see that these vacua are dynamically selected by the physics on Abelian moduli space.

3.2. The mapping of vacua

We would like to study planar equivalence in a physical setup that is equivalent for the parent theory. From (3.2) this corresponds to the introduction of k units of magnetic flux in the same direction as in the daughter theory. In this case we can reach N disconnected sectors making twisted gauge transformations. We would like to see how these vacua can be mapped to the orbifold-preserving vacua of the daughter theory. In order to do that, we should be able to construct the operators associated to twisted gauge transformations in the daughter theory from the corresponding ones in the parent theory. A second condition is that we should be able to identify the moduli spaces of both theories. Let us examine the first condition. Suppose that we introduce k units of magnetic flux in the x3 direction in the parent theory: + = ˜ ˜ −1 Aμ(x1 L,x2,x3) VkAμ(x1,x2,x3)Vk , + = ˜ ˜ −1 Aμ(x1,x2 L,x3) WkAμ(x1,x2,x3)Wk , Aμ(x1,x2,x3 + L) = Aμ(x1,x2,x3), (3.4)

˜ ˜ ˜ ˜ 2πik/kN where twist matrices must satisfy VkWk = WkVke . We can go from one vacuum state to other by making a twisted gauge transformation U, that satisﬁes the above conditions (3.4), except in the x3 direction, where

2πik/kN U(x1,x2,x3 + L) = e U(x1,x2,x3). (3.5) 104 C. Hoyos / Nuclear Physics B 744 (2006) 96–120

˜ ˜ A good election for the twist matrices is Vk = 1k ⊗ PN , Wk = 1k ⊗ QN . Moreover, we can use the SU(k) group that is not broken by the boundary conditions to rewrite U as a box-diagonal matrix, that can be mapped to the twisted gauge transformations of the daughter theory, if we arrange them in a single matrix. In general, the transformations in the daughter theory will include Abelian phases, so the map of vacua will be from one in the parent to many in the daughter. We now turn to the second condition. The twisting is not enough to lift completely the moduli space of the parent theory, but a torus T3(k−1) remains. As a matter of fact, it is an orbifold, since we should mod out by the Weyl group of SU(k). In the daughter theory, the Abelian moduli space is the same, except that there are no Weyl symmetries acting on this space. However, in the daughter theory there are discrete global symmetries associated to the permutations of factor groups that can be mapped to Weyl symmetries of the parent theory. They do not modify the moduli space, but we can make a projection to the invariant sector of the Hilbert space to study planar equivalence. There will be a wavefunction over the moduli space associated to the vacuum. In the parent theory, supersymmetry implies that the wavefunction is constant. In the daughter theory there is no supersymmetry, so a potential can be generated that will concentrate the wavefunction at the minima. Another consequence of supersymmetry is that there are fermionic zero modes over the moduli space, which give rise to more zero-energy states. We will investigate both questions in the next sections.

3.3. Electric ﬂuxes and vacuum angles

The vacuum states we have studied in the parent theory are generated from the trivial vacuum |0p , associated to the configuration of magnetic flux m = (0, 0,k) and classical gauge connection Aμ = 0, by operators that implement the minimal twisted gauge transformations on the ˆ parent Hilbert space Up, characterized by k = (0, 0, −1) as the electric component of the twist tensor (2.3). | = ˆ l | l Up 0p . (3.6) We can build Fourier transformed states of (3.6) that only pick up a phase when we make a twisted gauge transformation. We must take into account that the operators associated to twisted transformations depend on the vacuum angle θp, so we restrict to a sector of the Hilbert space with definite vacuum angle. Then,

ˆ N = iθp ˆ Up e 1. (3.7) The states we are considering are labelled by the electric flux e = (0, 0,e) N−1 1 − 2πil + θp | =√ kN (e 2π k) ˆ l | e e Up 0p (3.8) N l=0 and transform as θ ˆ 2πi (e+ p k) Up|e =e kN 2π |e . (3.9) Notice that e must be a multiple of k, in order to be consistent with (3.7). The theory must be invariant under 2π rotations of the vacuum angle θp → θp + 2π.The states (3.8) are not invariant under this transformation, but it generates a spectral flow that moves a state to another one e → e + k. In our particular case we can define a e = kep electric flux such C. Hoyos / Nuclear Physics B 744 (2006) 96–120 105 that the spectral flow acts as ep → ep + 1 N−1 1 − 2πil + θp | =√ N (ep 2π ) ˆ l | ep e Up 0p . (3.10) N l=0

In the daughter theory, the vacua are obtained from the trivial vacuum |0d , associated to i = i = magnetic ﬂuxes m (0, 0, 1) and zero classical gauge connections Aμ 0, by the set of oper- ˆ ators that implement minimal twisted gauge transformations on the daughter Hilbert space Ui , characterized by ki = (0, 0, −1) k | = ˆ li | l1,...,lk Ui 0d . (3.11) i=1 We will concentrate on states that are reached by diagonal twisted transformations, so that they are characterized by a single integer li = l, i = 1,...,k. Again, we can build the Fourier transformed states, taking into account that we are in a sector of the Hilbert space with deﬁnite vacuum angles θi ,so

ˆ N = iθi ˆ Ui e 1. (3.12)

The states are labelled by electric ﬂuxes ei

N−1 k 1 − 2πil + i θi | = √ N ( i ei 2π ) ˆ l| e1,...,ek D e Ui 0d . (3.13) N l=0 i=1 If we had considered transformations other than diagonal, this would be shown in the exponent, where different combinations of electric ﬂuxes and vacuum angles will appear. If we make a diagonal twisted transformation, these states change as

k 2πi i θi ˆ ( ei + ) Ui|e1,...,ek D = e N i 2π |e1,...,ek D. (3.14) i=1 If the twisted transformation is non-diagonal, there are extra phases appearing on the boundary conditions of the ﬁelds that must be compensated by Abelian transformations, so the state will change to a different vacuum in the Abelian sector. We can deﬁne subsectors of the Hilbert space that are invariant under diagonal transformations and that are shifted to other sectors by non- diagonal transformations. The labels for such sectors will be given by the Abelian part, so there will be N k−1 of them.

3.4. Fermionic zero modes

Fermionic zero modes are spatially constant modes that satisfy the Dirac equation with zero eigenvalues. In the parent theory the gaugino has k − 1 zero modes a = a ⊗ λ0 λk×k 1N , (3.15) where a = 1,...,k− 1 runs over the Cartan subalgebra of the unlifted SU(k) subgroup. If we go to a Hamiltonian formulation, as in [16], they do not contribute to the energy. We can deﬁne ∗α α = creation and annihilation operators for these modes: aa and aa where α 1, 2 refers to spin 106 C. Hoyos / Nuclear Physics B 744 (2006) 96–120 and a = 1,...,k− 1 is the gauge index. Physical states must be gauge invariant. For zero modes gauge invariance appears in the form of Weyl invariance, a discrete symmetry that stands after ﬁxing completely the gauge. The set of Weyl-invariant operators that we can construct with creation operators is limited to contract the gauge indices with δab. Thus, the only allowed operators that create fermions over the gauge vacuum are powers of

U = ∗α ∗β αβ aa aa . (3.16) Fermi exclusion principle limits the number of times we can apply U over the vacuum, so the set of possible vacua is

k−1 |0B , U|0B ,...,U |0B . (3.17) Some remarks are in order. First, the total number of vacua in the parent theory is kN, in agreement with the Witten index. Second, U acquires a phase under general chiral transformations, so the chiral symmetry breaking vacua of infinite volume might be constructed from appropriate linear combinations of the states (3.17). What is the situation in the daughter theory? The only possible candidates to give fermionic zero modes in spite of twisted boundary conditions are the trace part of bifundamental fields. However, those fields are charged under Abelian groups, so it is not possible to find zero modes over the whole moduli space. We are then confronted to the same kind of concern as in [11], the number of vacuum states is different in each theory, even in the sector of vacua that preserve orbifold symmetry. A necessary condition for planar equivalence to hold is that the vacuum expectation values of orbifold invariant quantities are the same up to factors of k given by the orbifold mapping [5]. In infinite volume, this could happen only for k = 2. Otherwise, the vacua of the parent theory spontaneously break orbifold symmetry [11]. However, in the Z2 orbifold the number of parent vacua is twice the number of daughter vacua so planar equivalence would be true only if there is a two-to-one mapping between vacua of parent and daughter theories. Although we cannot establish whether such a mapping exists or not, a similar mapping for the vacua we have found in the finite volume analysis will not work, because expectation values of operators involving fermionic fields take a different value depending on the number of fermionic zero modes present. Notice that this result is the same for all values of k, the orbifold correspondence does not work even though orbifold symmetry is not broken. For instance, if we consider the operator ∞ ∞ ∗ ∗ −2πinx 2πinx O = †α α ∼ α + α L α + α L p tr λ λ aa aa (n)e aa aa (n)e (3.18) n=1 n=1 we are counting the number of fermions in the state, so there is a diagonal non-vanishing contribution

l l 0B |U OpU |0 ∼2l δ δ . (3.19) B l,l 0B ,0B However, the orbifold projection of this operator O = † α α +···+ †α α d tr λ12 λ12 λk1 λk1 (3.20) will have a vanishing diagonal contribution, since there are no fermionic zero modes over the vacuum state. C. Hoyos / Nuclear Physics B 744 (2006) 96–120 107

3.5. Potential over moduli space

We now turn to the question of whether a potential appears over the Abelian moduli space of the daughter√ theory due to quantum effects. For simplicity, we will work with U(1) fields Aμ = aμ 1/N1N . Gauge fields are uncharged, so they cannot give rise to a potential. Fermionic fields do couple through k N2 L = ¯ a + i i − i+1 i+1 a λa λi,i+1 /∂ e a/ e a/ λi,i+1, (3.21) i=1 a=1 where ei is the Abelian coupling. In principle ei = ei+1 = e, since all Abelian gauge groups enter symmetrically and we have applied the renormalization group from a point where all couplings 2 = 2 = 2 were equal. At that point e gt /N , where gt gYMN is the non-Abelian ’t Hooft coupling. Let us comment several aspects of the Abelian moduli space. We can scale the fields so that the coupling constants come in front of the action. Then, when we make an Abelian gauge transformation that shifts the gauge field by a constant, the fermionic field charged under that Abelian group picks up a phase depending on the position. For instance, if + + i − i 1 → i − i 1 + ⇒ → ic3x3 a3 a3 a3 a3 c3 λi,i+1 e λi,i+1, (3.22) ic l then, when we move a period along the x3 direction, the phase of the field changes by e 3 3 .We 1 conclude that the periods of the S components of the Abelian moduli space are 2π/l3. Would 1 be twisted gauge transformations correspond to 2π/Nl3 translations along the S , although fixed boundary conditions for charged fields imply that they are no longer a symmetry of the theory. Since there are no fermionic zero modes with support over the whole moduli space, we can integrate out the fermionic fields in the path integral. The regions of zero measure where fermionic zero modes have support will be localized at conical singularities of the effective potential obtained this way. We proceed now to give the fermionic potential. At the one-loop level, it comes from the determinant that results in the path integral when we integrate out fermionic fields. k i i i+1 N2 Veff a =−log det /∂ + a/ − a/ . (3.23) i=1 This potential can be computed using the methods of [21]. The result, after rescaling by the length of the spatial torus ai → ai/l,is k i 2 i i+1 Veff a =−N V a − a , (3.24) i=1 where the shape of the potential is given by i − i+1 + 1 cos(nμ(a a )) V ai − ai 1 =− μ μ . (3.25) π 2l (n2)2 n∈Z4 Notice that the potential depends only on the differences ai − ai+1, so no potential is generated for the diagonal U(1), only for the U(1)k−1 groups coming from the non-Abelian part of the parent group after the orbifold projection. This potential has its minima at non-zero values of the differences, in the twisted sector. Since it is of order N 2, it cannot be ignored in the large N limit. 108 C. Hoyos / Nuclear Physics B 744 (2006) 96–120

Fig. 1. The wavefunction of the ground state is localized at the minimum of the potential in the Abelian moduli space. A translation produced by a twisted transformation will lift the state to a conﬁguration with more energy.

i − Examining more closely the potential, we can see that there is only one minimum at aμ i+1 = aμ π. Usually, wavefunctions over the moduli space are characterized by the electric flux, that we can interpret as momentum along the moduli space. However, kinetic energy on the moduli space is proportional to e2 (3.1), that is very small and becomes smaller as we increase the volume. On the other hand, the effective potential is very large and does not depend on the coupling. Therefore, the wavefunction is very localized, even for translations given by twisted transformations, and it is more convenient to use a position representation over the moduli space. The first consequence of all this is that if we make an Abelian twisted transformation, we are moving the system to a configuration with more energy, see Fig. 1. Therefore, most of the vacua of the theory are lifted. Only when we make diagonal non-Abelian twisted gauge transformations, the system remains in a ground state. This leaves N vacua. On the other hand, the wavefunctions over the moduli space of parent and daughter theories are very different. The first one can be seen as a zero-momentum state, while the last is more a position state. This suggests that in the infinite volume limit the daughter theory will fall into an Abelian twisted configuration, breaking orbifold symmetry in that sector. Another difference is that the ground state has a negative energy of order O(N2). These results are of the same kind as the ones found in [13,14], although in the other cases the moduli space was non-Abelian for the daughter theory. We have found many differences between the vacua of parent and daughter theories in finite volume with maximal twist. We were looking to lift the non-Abelian moduli space that in previous works pointed towards non-perturbative failure of planar equivalence. We have seen that properly taking into account the Abelian groups in the daughter theory, the situation does not really improve. Thanks to Abelian groups we are able to map the moduli space of parent and daughter theories, but the generation of a potential by fermionic fields in the daughter theory makes the physical behavior of both theories quite different, although it lifts many of the unex- pected non-Abelian vacua. The moduli space of the parent theory is also a problem, because it implies that there are fermionic zero modes that generate vacua that cannot be mapped to the daughter theory. C. Hoyos / Nuclear Physics B 744 (2006) 96–120 109

4. Tunneling effects

We have shown that parent and daughter theories have physically inequivalent vacua in this context due to the potential generated over the moduli space in the daughter theory and to the mismatch of vacua. However, some quantities do not depend on the moduli space, so the wavefunction will just give a normalization factor that can be chosen to be the same. The orbifold conjecture may still be useful to make some computations in the common vacua, up to kinematical factors. One of the main applications of twisted boundary conditions is the computation of fermion condensates [22] generated by tunneling between different vacua. These fermion condensates do not depend on the moduli space and tunneling can be studied using self-dual solutions of Euclid- ean equations of motion. In the case where the tunneling is between vacua related by a twisted gauge transformation, the relevant configurations are of fractional topological charge, which we have associated to the twist tensor (2.4). Notice that to map parent and daughter theories we are using (2.9) and (2.10). If we want to make the mapping between purely non-Abelian configurations, we should take αμν = 0, otherwise we will be introducing self-dual Abelian fluxes in the daughter theory. In this case, all the twist tensors in the daughter theory must be equal because of fermions in the bifundamental representations. This implies that parent and daughter twist tensors are proportional p = d nμν knμν. (4.1) Under these conditions, the fractional contribution to topological charge (2.4) by tunneling is the same in both theories 1 1 Q =− np n˜p = k − nd n˜d . (4.2) kN μν μν N μν μν

4.1. Tunneling in parent theory

p In the SU(kN) parent theory with nμν = 0modk twist, the solution of minimal charge has Q = 1/N, k times the minimal possible topological charge of the theory. It contributes to matrix elements of the form 0|OUˆ |0 for some operator O. The operator Uˆ acts over the trivial gauge vacuum |0 by making a twisted gauge transformation (3.5). We can compute this quantity using a Euclidean path integral 1 ˆ −HT ˆ ¯ −SE 0|OU|0 =− lim 0|Oe U|0 → DA Dλ Dλ OEe . (4.3) T →∞ T Q=1/N Although we do not know the explicit Euclidean solution that contributes in the saddle point approximation, we know what should be the vacuum expectation value of some operators. Be- cause of supersymmetry, only zero modes contribute. There are 4k real zero modes of both bosonic and fermionic ﬁelds. The existence of fermionic zero modes implies that the tunneling contributes only to operators involving a product of 2k fermionic ﬁelds. In particular, there is no contribution to the vacuum energy. Therefore,

1 − E ≡− lim ln0|e HTUˆ |0 =0 (4.4) T →∞ T 110 C. Hoyos / Nuclear Physics B 744 (2006) 96–120 and k 1 k −HT ˆ 1 k trkN(λλ) ≡−lim 0| trkN(λλ) e U|0 = trkN λλ (4.5) T →∞ T k! which is a tunneling contribution to a 2k-fermionic correlation function. By trkN λλ we refer to the value of the gaugino condensate in the parent theory. This does not mean that a gaugino condensate is generated by these configurations, we are just referring to the numerical value of (4.5). This is based on the use of torons to estimate the gaugino condensate [22] and on the coincidence of the instanton measure with the measure of a superposition of torons [23].The factorial factor comes from considering the configuration of charge Q = 1/N as a superposition of k equal configurations of charge Q = 1/kN. Another peculiar fact about this configuration is that it is transformed into itself under a Nahm transformation [24]. A Nahm transformation identifies moduli spaces of different self- dual configurations of different gauge theories in different spaces. From a stringy perspective it is a remnant of T-duality [25]. We will use techniques developed in [26,27] for Nahm transformations with twisted boundary conditions. Given the rank r, the topological charge Q, and the twist 0 Ξ n = , (4.6) −Ξ 0 where Ξ = diag(q1,q2), the Nahm transformed quantities are given by

p1 = N/gcd(q1,N), p2 = N/gcd(q2,N),

s1q1 =−gcd(q1,N) mod N, s2q2 =−gcd(q2,N) mod N, Q = r/p1p2,r= Qp1p2, (4.7) and the twist Ξ = diag (p1 − s1)p2Q,(p2 − s2)p1Q . (4.8) Our tunneling conﬁguration can be determined by m = (0, −k,0), k = (0, 1, 0) for instance. Then, we can write the twist tensor as Ξ = diag(1,k).Using(4.7) and (4.8), it is straightforward to see that r = r = kN, Q = Q = 1/N and Ξ = Ξ = diag(1,k).

4.2. Tunneling in daughter theory

Self-dual solutions contributing to tunneling in the daughter theory do not coincide with the projection from the parent theory, as given by (4.1). Tunneling between two adjacent vacua is given by a configuration where each gauge factor contributes by Qd = 1/N to the topological charge. Then, the total topological charge is Q = k/N, while the topological charge in the parent theory is Qp = 1/N, in disagreement with (4.2). The configurations we are using for tunneling are not related by the map we have proposed, but notice that we are comparing semiclassical contributions of the same order, since the classical p = 2 2 d = 2 2 Euclidean action in parent Scl 8π /gpN and daughter Scl 8π k/gd N theories have the same 2 = 2 value according to the orbifold projection kgp gd . When we examine the moduli space of parent and daughter tunneling configurations, we find further evidence that we are comparing the correct quantities. The bosonic moduli space is 4k-dimensional for both theories, and the moduli space of the daughter theory also transforms into itself under a Nahm transformation. C. Hoyos / Nuclear Physics B 744 (2006) 96–120 111

Q = Q = = = This can be seen using (4.7) for each of the gauge factors: d d 1/N, rd rd N, = = Ξd Ξd diag(1, 1). Even using this ‘improved’ mapping, disagreement emerges from fermionic fields. In the daughter theory there is no supersymmetry in general, because fermions are in a representation different to bosons. However, bifundamental representations of the daughter theory can be embedded in the adjoint representation of the parent theory, and this will be useful. In order to make an explicit computation, we need a formula for the self-dual gauge configuration of fractional topological charge. We do not have an analytic expression in general, but ’t Hooft found a set of solutions [19], called torons, that are self-dual when the sizes of the torus lμ satisfy some relations. We will assume that we are in the good case and that, although in other cases the relevant configurations will be different, the physics will be the same. Torons are the solutions of minimal topological charge in a situation with maximal twist. For a SU(N) gauge theory, this means that Qtoron = 1/N. Therefore, they can be used in the daughter theory, where we will have a toron for each of the non-Abelian groups. We can compare torons to the most known self-dual solutions, instantons. Instanton solutions can be characterized by their size, orientation in the algebra and center position. The size of torons is fixed by the size of the spatial torus, and they have a fixed orientation along a U(1) subgroup. However, they still have a center that can be put at any point of the torus. Therefore, the moduli space of torons is a four-dimensional torus. An explicit expression for torons in the daughter theory in the presence of magnetic flux in the x3 direction is i = iπ 3 − i = Aμ ημν x z νω, i 1,...,k, (4.9) 2glμlν a = ⊂ where ημν , a 1, 2, 3 are ’t Hooft’s self-dual eta symbols [28], ω is a generator of a U(1) SU(N) and zi are the center positions. Notice that we have chosen to work with anti-hermitian and canonically normalized gauge fields. Now that we are armed with an analytic expression, we can calculate what happens with fermions in the background of k torons. The Euclidean Dirac operator is 0 −iσμDμ γμDμ = , (4.10) iσ¯μDμ 0 = ¯ = † + where σμ (12,iσ), σμ σμ and σi are the Pauli matrices. In a self-dual background Fμν ,the square of the Dirac operator is positive definite for negative chirality fields

2 (iσ¯μDμ)(iσνDν) =−D 12, (4.11) while for positive chirality ﬁelds, it has a potentially negative contribution

1 + (iσ D )(iσ¯ D ) =−D21 − gσ F , (4.12) μ μ ν ν 2 2 μν μν where σμν = σ[μσ¯ν]. This means that there are no fermionic zero modes of negative chirality, although there can be zero modes of positive chirality. In the supersymmetric case, where fermions are in the adjoint representation, there are two zero modes in the background of a single toron. The covariant derivative acting over bifundamental ﬁelds is

= + i − i+1 Dμλi,i+1 ∂μλi,i+1 gAμλi,i+1 gλi,i+1Aμ . (4.13) 112 C. Hoyos / Nuclear Physics B 744 (2006) 96–120

Then, introducing (4.9) and (4.13) in (4.12), we ﬁnd the operator − 2 − 1 + − i + 2 2 ⊗ D 12 gσμνFμν 2gAμ(Dμ)adj12 M (z)ω 12, (4.14) 2 adj where “adj” denotes that the operator is equal to the case where it acts over the adjoint representation of SU(N). Extra contributions are given by the separation between torons of different groups

i = i+1 − i i = iπ 3 i zμ zμ zμ,Aμ ημνzνω. (4.15) 2glμlν The coefﬁcient of the positive ‘mass’ contribution is π 2 (zi )2 + (zi )2 (zi )2 + (zi )2 M2(z) = 0 3 + 1 2 . 2 2 2 2 (4.16) 4 l0 l3 l1 l2 The physics of fermions is clear now. When two torons of adjacent groups coincide at the same space–time point, there are zero modes for bifundamental fermions transforming under these groups. Zero modes are identical to the case of adjoint representation (supersymmetric case). When we separate the torons, the zero modes acquire a mass proportional to the distance of separation. Therefore, we can split the moduli space of torons Mk in sectors where a different number of torons are coincident. We can use quiver diagrams to represent different sectors. Each node is a toron position. Links joining different nodes represent massive fermions, while links coming back to the same node represent fermions effectively in the adjoint representation, so they give rise to supersymmetric contributions. The expansion will look like

Mk = + + +···. (4.17)

A sector with n adjoint links can contribute only to vacuum expectation values of quantities involving at least n factors of the form λi,i+1λi,i+1. The contribution is determined by the expectation value of the gaugino condensate of a N = 1 SU(N) supersymmetric gauge theory, λλ susy. As an example, consider the pure supersymmetric contribution k = λλ susy (4.18) k i=1 trN λi,i+1λi,i+1 that receives non-supersymmetric corrections from other sectors of the moduli space, for instance

k i=1 trN λi,i+1λi,i+1 − 1 1 k 2 1 2 ˜ 4 1 2 −2Scl = k λλ susy dA1 z dA2 z M z − z e tr tr (4.19) D/ 12 D/ 21 i ˜ 2 we denote by Ai the bosonic zero modes, depending on the toron position z . The factors M come from fermionic would-be zero modes, and make the contribution to vanish in the regions of C. Hoyos / Nuclear Physics B 744 (2006) 96–120 113 moduli space where the separated toron coincide with the others. We have taken care of bosonic and fermionic (would be) zero modes, so they have been subtracted from the computation of the −1 −1 correlation function involving the fermion propagators trD/ 12 trD/ 21 . The classical action is the action of the toron solution 8π 2 S = . (4.20) cl g2N Coming back to (4.4), we see that there is a non-zero contribution of tunneling to energy. When two torons are coincident, there are fermionic zero modes, so contributions to E come from the sector of the moduli space of torons where all are separated. This contribution is completely non-supersymmetric k k det (D/ + (A ,A + )) i ˜ 2 i i+1 −kScl i=1 i,i 1 i i 1 E1-loop = dAi z M z − z e . (4.21) k − 2 1/2 i=1 i=1(det ( Di (Ai))) We have taken care of bosonic and fermionic (would be) zero modes, so they have been subtracted from the determinants det. With the notation we are employing, the contribution to E will come from a “ring” diagram Fig. 2. When we make the orbifold projection of (4.5), we find k → ··· 2 = k + k−2 (1) +··· trkN(λλ) trN (λ1 λk) λλ susy k λλ susy FNS . (4.22) (1) Where the first non-supersymmetric factor FNS is the same as in (4.19), and the dots indicate that there are more contributions from other sectors of moduli space, all involving different 2 = non-supersymmetric factors. Notice that the orbifold mapping kgp gd , implies that the renormalization invariant scales of parent and daughter theories are the same at the planar level. −8π 2 −8π 2 λλ ∼Λ3 = μ3 = μ3 = Λ3 ∼λλ . trkN p exp 2 exp 2 d susy (4.23) gpkN gd N As a consequence, the first term in (4.22) coincides with (4.5), up to the factorial factor, that in the daughter theory does not appear because torons belonging to different groups are distinguishable. So we must conclude that there are no simple relations between (4.4) and (4.21) or between (4.5) and (4.22) although both sets of quantities are invariant under orbifold symmetry.

Fig. 2. The ring sector produces no supersymmetric factors. 114 C. Hoyos / Nuclear Physics B 744 (2006) 96–120

4.3. Semiclassical dependence on the vacuum angles

The results of the previous sections can be used to analyze the dependence on the vacuum angles when we deform the theories adding a mass m for the fermions. This is necessary in order to have such a dependence, otherwise massless fermions will erase it through the chiral anomaly. Notice that we can make the fermions of the daughter theory massive only for the Z2 orbifold, so we will restrict to this case. In the parent theory, when we introduce a mass for the fermions, the states previously associated to fermionic zero modes (3.17) are lifted by an energy that is roughly the mass times the number of fermionic modes. The energy density is obtained dividing by the volume of the box V = l3.Sowehavek = 2 levels, each with N states labelled by the electric flux. As we will see in a moment, tunneling between states of the same level produces a lifting in the energy density of order m4. If we are in the small volume (small mass) limit ml 1, then the lifting produced by fermionic modes is larger than tunneling contributions, so the states in the highest level decouple. However, as we increase the volume there will be level-crossings between states of different levels. In the daughter theory, a mass for the fermions modifies the effective potential over the Abelian moduli space (3.25).Ifml 1, then only high frequency contributions are suppressed, but increasing the volume will exponentially suppress the whole potential, so non-diagonal vacua could become relevant. We are interested in estimating the dependence of the vacuum energy on the vacuum angles and electric fluxes

1 − E(e, θ) = lim − lne,θ|e HT|e,θ . (4.24) T →∞ T In a semiclassical expansion, the relevant contributions come from fractional instantons that encode tunneling processes between different vacua. We will ﬁrst study the leading term given by a single (anti)toron of minimal topological charge, and afterwards we will estimate the contribution of inﬁnite many (anti)torons using a dilute gas approximation [29]. In the parent theory, the lowest order contribution is 2 − 8π det (iD/ + m) 0 |e HTUˆ |0 =T dA z1,z2 m4 exp − F p p p 2 − 2 1/2 gpN detB ( D )

4 −2Scl = Tm Kp(m)e , (4.25) where A(z1,z2) are the zero modes of the SU(2N) gauge field, depending on eight parameters of the Q = 1/N toron configuration. The fermionic mass is m, so the factor m4 comes from the would be fermionic zero modes that appear when we turn m to zero. Therefore, they have been subtracted from the determinants detF . The bosonic determinant detB includes only physical polarizations of non-zero modes of vector bosons, so any ghost contribution has been taken into 2 = 2 account in it. In the last equality, we have used the orbifold map gp gd /2. Kp(m) is a m- dependent constant that will be obtained after making the integral over the moduli space of the (anti)toron. For a general transition amplitude between different twisted vacua, there can be contributions of an arbitrary number of torons and anti-torons, as long as the total topological charge has the C. Hoyos / Nuclear Physics B 744 (2006) 96–120 115 correct value. Since we are in a dilute gas approximation, we neglect possible interactions, ∞ ∞ +¯ ˆ + l+ −HT ˆ l− 1 4 −2S n n 0 |(U ) e (U ) |0 = Tm K (m)e cl δ −¯− + . (4.26) p p p p n!¯n! p n n l+ l− n=0 n¯=0 We are ready now to compute the vacuum energy in the state (3.10), it is straightforward to see that − 2π θp E (e ,θ ) =−2m4K (m)e 2Scl cos e + . (4.27) p p p p N p 2π A similar analysis can be done for the daughter theory. The lowest order contribution is, in the diagonal case, − 0 |e HTUˆ Uˆ |0 d 1 2 d 2 2 4 − det (−D + m ) = T dA z1 dA z2 m + M(z)˜ e 2Scl F 12 1 2 − 2 1/2 − 2 1/2 detB ( D1) detB ( D2) 4 −2Scl = Tm Kd (m)e . (4.28) We have argued above that it is plausible that the bosonic measures of parent dA(z1,z2) and 1 2 4 daughter dA1(z )dA2(z ) are the same. However, there are extra contributions to the m factor, givenbythemassM(z) (4.16) that bifundamental modes acquire when the two torons are separated. We can also see that the one-loop determinants in (4.25) and (4.28) are different. All this imply that Kp(m) and Kd (m) are not simply related, as we would have expected from the orbifold projection. Then, a general transition between diagonal vacua is given by | ˆ + ˆ + l+ −HT ˆ ˆ l− | 0d (U2 U1 ) e (U1U2) 0d ∞ ∞ 1 4 −2S n+¯n = Tm K (m)e cl δ −¯− + . (4.29) n!¯n! d n n l+ l− n=0 n¯=0 So the vacuum energy of the state (3.13) will be − 2π θ1 + θ2 ED(e ,e ,θ ,θ ) =−2m4K (m)e 2Scl cos e + e + . (4.30) d 1 2 1 2 d N 1 2 2π We can study the spectral flows induced by a change in the vacuum angles [30] in both theories and compare the results. In Fig. 3, we illustrate the discussion with a simple example. In the parent theory, the energy of a state of definite electric flux is invariant only under shifts θp → θp + 2πN. However, the whole spectrum is invariant under a 2π rotation of θp, so there is a non-trivial spectral flow that may become non-analytic and produce oblique confinement in the infinite volume limit. Notice also that there are two well-separated levels in the small volume (ml 1) limit, where the lifted states have fermionic constant Abelian modes. In the opposite limit, states of the same electric flux number become nearly degenerate. This could be a hint for a two-to- one mapping between parent and daughter states of equivalent electric flux, in a finite volume version of the statement made in [11]. Notice that the parent and daughter “vacuum angles” should be mapped as 2θd = θp at the orbifold point θ1 = θ2 = θd , as the orbifold correspondence dictates. Indeed, we find that the spectrum of the daughter theory is invariant under θd → θd + π. However, this does not look as the right transformation for a vacuum angle and, in view of (4.30), the daughter vacuum angle seems rather to be θ1 + θ2, that has the value 2θd at the orbifold point. 116 C. Hoyos / Nuclear Physics B 744 (2006) 96–120

Fig. 3. We show a picture of the spectral flow of Ep/Kp (dark lines) and Ed /Kd (light lines) for different states of electric fluxinthecaseN = 3. The horizontal axis corresponds to θp for the parent theory and to θd for the daughter theory. We assume ml 1, so the parent theory presents two separated levels. Notice that the periodicity of the spectral flow of the daughter is half the periodicity of the parent.

On the other hand, the lifted vacua of the parent theory are unsuitable for a correspondence with daughter states because some matrix elements involving fermions will be different, as we have argued before. Regarding the remaining vacua, there is a quantitative disagreement between parent (4.27) and daughter energies (4.30), given by the difference in the values of the tunneling coefﬁcients Kp(m) and Kd (m), so it seems that planar equivalence does not work at this level.

5. D-brane interpretation

We can give a geometrical interpretation of our results based on a construction with D-branes in the torus, as in [31]. In the parent theory we will have a set of kN D4-branes on T4, the 01234 directions including time. The low energy theory is a U(kN) supersymmetric gauge theory. We are interested in studying possible vacuum configurations, that are characterized by different bundles of D-branes over the torus. We will ignore the scalars associated to the transverse dimensions to the torus, anyway, they will be spectators in the analysis below. The twist of the configuration associated to tunneling in the parent theory, Section 4.1,is realized diluting D2-brane charge in the worldvolume of D4s. The reason is the coupling of D2 charge with U(1) first Chern class of D4 gauge group

C3 ∧ tr F (5.1) D4 the twist induced in the U(1) must be compensated by the non-Abelian part, in order to have a well-defined U(kN) bundle. So we can introduce k units of magnetic flux in the 3 direction and C. Hoyos / Nuclear Physics B 744 (2006) 96–120 117 one unit of electric flux in the same direction by introducing k D2 branes in 34 direction and a D2 brane in 12 directions. Since the intersection number is non-zero we can interpret also this configuration as having D2 at some angle inside the torus.

0 1 2 3 4 kN D4 × × × × × k D2 × • • × × 1D2 × × × • •

After making a T-duality along 12 directions

0 1 2 3 4 kN D2 × • • × × k D4 × × × × × 1D0 × • • • • this conﬁguration is equivalent to an instanton for a U(k) theory with kN units of magnetic ﬂux. However, if the T-duality transformation is made along 34 directions

0 1 2 3 4 kN D2 × × × • • k D0 × • • • • 1D4 × × × × × we will have k ‘instantons’ in a U(1) theory with kN units of electric flux. In the daughter theory, we have a set of N D4 branes wrapped around a T4 but sitting at an orbifold point in transverse space. The twist associated to the tunneling configuration in the daughter theory, Section 4.2, is realized diluting a D2-brane both in 34 and 12 directions. The orbifold action is responsible of the multiples copies of the gauge group. So physically we are introducing what in an unorbifolded theory will be N D4 branes and not kN branes, although the field content is constructed projecting the last.

0 1 2 3 4 (k)N D4 × × × × × (k)1D2 × • • × × (k)1D2 × × × • •

We can make a T-duality transformation along 12 directions 118 C. Hoyos / Nuclear Physics B 744 (2006) 96–120

0 1 2 3 4 (k)N D2 × • • × × (k)1D4 × × × × × (k)1D0 × • • • • so we have an ‘instanton’ in the presence of (k)N units of magnetic ﬂux in a U(1)k theory. If the T-duality transformation is made along 34 directions

0 1 2 3 4 (k)N D2 × × × • • (k)1D0 × • • • • (k)1D4 × × × × × now the flux is electric instead of magnetic. In this two cases, we can interpret the ‘untwisted’ fractional instanton as a D0 brane in the presence of a D2 brane. This is the sector where all torons are at the same point. The sectors where torons are at different points correspond to the splitting of the D0 in fractional D0s that can move independently along the torus directions. The gauge group is the same for k fractional D0s as for a regular D0, however the strings joining different fractional D0s acquire a mass proportional to the separation. This is reflected in the mass we have computed for fermionic modes (4.16). So in fact, the quiver diagrams we have used to label different sectors of the moduli space correspond to the gauge theory living in the D0s at different points of the moduli space. From this point of view we also have a geometric picture of why the configuration of the daughter theory should be mapped to a configuration of the parent theory with k units of both electric and magnetic flux. The matching is between the number of fractional branes in the orbifolded theory and the number of branes of the same dimensions in the ‘parent’. If we make another T-duality transformation along 1234 directions,

0 1 2 3 4 (k)N D0 × • • • • (k)1D2 × × × • • (k)1D2 × • • × × the D4s become D0s that can move along the torus. The separation in fractional D0s is encoded in the Abelian part of the daughter theory. The diagonal U(1) correspond to the center of mass position, while the U(1)k−1 group that form the Abelian moduli space are the relative separations of D0s in this T-dual interpretation. From (3.24) we know that D0s tend to separate, a signal of the orbifold tachyon.

6. Summary

We extend the study of planar equivalence in a small volume [14] introducing twisted boundary conditions for orbifold field theories. We have found several sources of disagreement with orbifold large N equivalence. First of all, a ‘kinematical’ map seems not to work properly. When C. Hoyos / Nuclear Physics B 744 (2006) 96–120 119 we try to embed the vacuum of the daughter theory in the vacuum of the parent theory, we find difficulties because the number of vacua does not match due to the presence (absence) of fermionic zero modes in the parent (daughter) theory. We also find difficulties in identifying the relevant configurations for tunneling using only algebraic arguments. An important ‘dynamical’ indication of disagreement with planar equivalence is the formation of a potential over the Abelian moduli space of the daughter theory, which makes the ground wavefunction of both theories very different and shifts the vacuum energy at leading order. These results are analogous to the behavior found for the non-Abelian moduli space in the case of periodic boundary conditions [14]. When we try to compute contributions produced by tunneling effects in the daughter theory, we find a remarkable relation with a supersymmetric theory. We can split the contributions in sectors where some factors are identical to quantities of a supersymmetric theory (to be precise, to the gaugino condensate). However, the relation is with a N = 1 SU(N) gauge theory (or k copies of it) and not with the parent theory, that has group SU(kN). We also compute the semiclassical dependence of the energy on the electric flux for the case k = 2, but we fail in establishing a quantitative mapping between states of parent and daughter theories. In view of these results, it seems that the ‘supersymmetric’ properties of the orbifold daughter stem from the fact that a N = 1 SU(N) supersymmetric sector is ‘wrapping’ it, the sector of diagonal configurations. This is in agreement with previous results [14] and suggests that planar equivalence of orbifold theories could be traced to the fact that the parent theory and N = 1 SU(N) gauge theory are also planar equivalent [12]. From the point of view of finite volume physics, the fact that the orbifold theory can be embedded in a supersymmetric theory, the parent theory, apparently has no more meaning except that it is probably a necessary condition for having a supersymmetric sector. It would be interesting to check whether the properties of the orientifold theory relies on having a SO(N) supersymmetric sector or work differently to the orbifold case.

Acknowledgements

I would like to thank J.L.F. Barbón for reading the manuscript and making many comments and suggestions and for many useful discussions. This work was supported by Spanish MEC FPU grant AP2002-0433.

References

[1] G. ’t Hooft, Nucl. Phys. B 72 (1974) 461. [2] M.J. Strassler, hep-th/0104032. [3] S. Kachru, E. Silverstein, Phys. Rev. Lett. 80 (1998) 4855, hep-th/9802183; A.E. Lawrence, N. Nekrasov, C. Vafa, Nucl. Phys. B 533 (1998) 199, hep-th/9803015; M. Bershadsky, Z. Kakushadze, C. Vafa, Nucl. Phys. B 523 (1998) 59, hep-th/9803076; M. Bershadsky, A. Johansen, Nucl. Phys. B 536 (1998) 141, hep-th/9803249; M. Schmaltz, Phys. Rev. D 59 (1999) 105018, hep-th/9805218. [4] Z. Kakushadze, Nucl. Phys. B 529 (1998) 157, hep-th/9803214; Z. Kakushadze, Phys. Rev. D 58 (1998) 106003, hep-th/9804184; F. Sannino, M. Shifman, Phys. Rev. D 69 (2004) 125004, hep-th/0309252; A. Armoni, M. Shifman, G. Veneziano, hep-th/0403071; A. Armoni, A. Gorsky, M. Shifman, Nucl. Phys. B 702 (2004) 37, hep-th/0404247; P. Di Vecchia, A. Liccardo, R. Marotta, F. Pezzella, Fortschr. Phys. 53 (2005) 450, hep-th/0412234; F. Sannino, hep-th/0507251. 120 C. Hoyos / Nuclear Physics B 744 (2006) 96–120

[5] P. Kovtun, M. Unsal, L.G. Yaffe, JHEP 0507 (2005) 008, hep-th/0411177. [6] A. Armoni, M. Shifman, G. Veneziano, Nucl. Phys. B 667 (2003) 170, hep-th/0302163; A. Armoni, M. Shifman, G. Veneziano, Phys. Rev. D 71 (2005) 045015, hep-th/0412203. [7] A. Armoni, M. Shifman, G. Veneziano, Phys. Rev. Lett. 91 (2003) 191601, hep-th/0307097; A. Armoni, M. Shifman, G. Veneziano, Phys. Lett. B 579 (2004) 384, hep-th/0309013. [8] A. Armoni, E. Imeroni, hep-th/0508107. [9] T.A. Ryttov, F. Sannino, hep-th/0509130. [10] A. Gorsky, M. Shifman, Phys. Rev. D 67 (2003) 022003, hep-th/0208073; R. Dijkgraaf, A. Neitzke, C. Vafa, hep-th/0211194; J. Erlich, A. Naqvi, JHEP 0212 (2002) 047, hep-th/9808026; P. Kovtun, M. Unsal, L.G. Yaffe, JHEP 0312 (2003) 034, hep-th/0311098; A. Armoni, A. Gorsky, M. Shifman, Phys. Rev. D 72 (2005) 105001, hep-th/0505022. [11] P. Kovtun, M. Unsal, L.G. Yaffe, hep-th/0505075. [12] M. Shifman, hep-th/0510169. [13] D. Tong, JHEP 0303 (2003) 022, hep-th/0212235. [14] J.L.F. Barbón, C. Hoyos, JHEP 0601 (2006) 114, hep-th/0507267. [15] G. ’t Hooft, Nucl. Phys. B 153 (1979) 141. [16] E. Witten, Nucl. Phys. B 202 (1982) 253. [17] E. Witten, Adv. Theor. Math. Phys. 5 (2002) 841, hep-th/0006010. [18] A. Gonzalez-Arroyo, hep-th/9807108. [19] G. ’t Hooft, Commun. Math. Phys. 81 (1981) 267. [20] D.R. Lebedev, M.I. Polikarpov, A.A. Rosly, Sov. J. Nucl. Phys. 49 (1989) 1113, Yad. Fiz. 49 (1989) 1799. [21] M. Luscher, Nucl. Phys. B 219 (1983) 233. [22] E. Cohen, C. Gomez, Phys. Rev. Lett. 52 (1984) 237; E. Cohen, C. Gomez, Nucl. Phys. B 223 (1983) 183. [23] A.R. Zhitnitsky, Nucl. Phys. B 340 (1990) 56. [24] M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld, Y.I. Manin, Phys. Lett. A 65 (1978) 185; W. Nahm, Phys. Lett. B 90 (1980) 413; P.J. Braam, P. van Baal, Commun. Math. Phys. 122 (1989) 267. [25] E. Witten, J. Geom. Phys. 15 (1995) 215, hep-th/9410052; E. Witten, Nucl. Phys. B 460 (1996) 541, hep-th/9511030; D.E. Diaconescu, Nucl. Phys. B 503 (1997) 220, hep-th/9608163; A. Astashkevich, N. Nekrasov, A. Schwarz, Commun. Math. Phys. 211 (2000) 167, hep-th/9810147; M. Hamanaka, H. Kajiura, Phys. Lett. B 551 (2003) 360, hep-th/0208059. [26] H. Schenk, Commun. Math. Phys. 116 (1988) 177; A. Gonzalez-Arroyo, C. Pena, JHEP 9809 (1998) 013, hep-th/9807172; P. van Baal, Phys. Lett. B 448 (1999) 26, hep-th/9811112; M. Garcia Perez, A. Gonzalez-Arroyo, C. Pena, P. van Baal, Nucl. Phys. B 564 (2000) 159, hep-th/9905138. [27] A. Gonzalez-Arroyo, Nucl. Phys. B 548 (1999) 626, hep-th/9811041. [28] G. ’t Hooft, Phys. Rev. D 14 (1976) 3432; G. ’t Hooft, Phys. Rev. D 18 (1978) 2199, Erratum. [29] S. Coleman, Aspects of Symmetry: Selected Erice Lectures, Cambridge Univ. Press, 1985. [30] P. van Baal, Twisted boundary conditions: a non-perturbative probe for pure non-Abelian gauge theories, PhD Thesis, Utrecht, 1984; P. van Baal, hep-ph/0008206. [31] Z. Guralnik, S. Ramgoolam, Nucl. Phys. B 499 (1997) 241, hep-th/9702099; Z. Guralnik, S. Ramgoolam, Nucl. Phys. B 521 (1998) 129, hep-th/9708089. Nuclear Physics B 744 (2006) 121–135

QCD decoupling at four loops

K.G. Chetyrkin a,∗,1,J.H.Kühna,C.Sturmb

a Institut für Theoretische Teilchenphysik, Universität Karlsruhe, D-76128 Karlsruhe, Germany b Dipartimento di Fisica Teorica, Università di Torino and INFN, Sezione di Torino, Italy Received 19 January 2006; accepted 27 March 2006 Available online 6 April 2006

Abstract

(nf ) We present the matching condition for the strong coupling constant αs at a heavy quark threshold to four loops in the modiﬁed minimal subtraction scheme. Our results lead to further decrease of the theoretical uncertainty of the evolution of the strong coupling constant through heavy quark thresholds. Using a low energy theorem we furthermore derive the effective coupling of the Higgs boson to gluons (induced by a virtual heavy quark) in four- and (partially) through ﬁve-loop approximation. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

The masses of the known quark species differ vastly in their magnitude. As a result of this, in many QCD applications√ the mass of a heavy quark h is much larger than the characteristic momentum scale s which is intrinsic to the physical process. In such a situation there appear two interrelated problems when using an MS-like renormalization scheme.√ First, given the two large but in general quite different mass scales, s and m, two different types of potentially dangerously large logarithms may arise. The standard trick of a proper choice of the renormalization scale μ is no longer effective; one cannot set one parameter μ equal to two very different mass scales simultaneously.

* Corresponding author. E-mail address: [email protected] (K.G. Chetyrkin). 1 On leave from Institute for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia.

Second, according to the Appelquist–Carazzone theorem [1] the effects due to heavy particles should eventually “decouple” from the low-energy physics.2 However, a peculiarity of mass- independent renormalization schemes is that the decoupling theorem does not hold in its naive form for theories renormalized in this framework. The effective QCD action to appear will not be canonically normalized. Even worse, potentially large mass logarithms in general appear, when one calculates a physical observable. The well-known way to bring the large mass logarithms under control is to construct an effective field theory by “integrating out” the heavy quark field h [2–6]. By construction, the resulting effective QCD Lagrangian will not include the heavy quark field. In order to be specific, let us consider QCD with nl = nf − 1 light quarks and one heavy quark h with mass m. The quark– gluon coupling constants in both theories, the full nf -flavor QCD and the effective nl-flavor (nf ) (nl ) one, αs and αs are related by the so-called matching condition of the form [2,3] (n ) (n ) (nl ) = 2 f f αs (μ) ζg μ,αs (μ), m αs (μ). (1)

Here m = m(μ) is the (running) mass of the heavy quark; the decoupling function ζg is equal to one at the leading (tree) level but receives nontrivial corrections in higher orders. The matching point μ in Eq. (1) should be chosen in such a way to minimize the effects of logarithms of the heavy quark mass, e.g., μ = O(m). An important phenomenological application of Eq. (1) appears in the determination of αs(MZ) at the Z-boson scale through evolution with the renormalization group equation, starting from the measured value αs(mτ ) at the τ -lepton scale. A careful analysis of the effects of four-loop run- (5) (3) ning and three-loop matching in extracting αs (MZ) with 5 active quark flavors from αs (mτ ) with 3 active quark flavors has been recently performed in Ref. [7]. We will demonstrate that the inclusion of the newly computed four-loop matching condition leads to further reduction of the theoretical error from the evolution. The second aspect, mentioned above, is of importance for Higgs-boson production in hadronic collisions. The dominant subprocess for the production of the Standard Model (SM) Higgs boson at the CERN Large Hadron Collider (LHC) will be the one via gluon fusion. Therefore, an important ingredient for the Higgs-boson search will be the effective coupling of the Higgs boson to gluons, usually called C1.InRef.[8] a low-energy theorem, valid to all orders, was established, which relates the effective Higgs-boson–gluon coupling, induced by the virtual presence of a heavy quark, to the logarithmic derivative of ζg with respect to the heavy quark mass. In that paper the method was used to squeeze from the three-loop decoupling function ζg the analytical result for C1 in three- and even in four-loop approximation (the latter in a indirect way through a sophisticated use of the RG evolution equations and the four-loop QCD β-function). With our new full four-loop result for ζg we are able to confirm the result of Ref. [8] in a completely independent way, without using the four-loop contribution to the QCD β-function. In addition a five-loop prediction for C1 can be obtained (modulo yet unknown contribution from the five- loop nf -dependent term in the β-function). The outline of the paper is as follows. In Section 2 we recall the main formulae from [8] which reduce the evaluation of the decoupling function ζg to the calculation of vacuum integrals. Then we discuss shortly the technique used to compute these integrals. Section 3 describes our four- loop results for the decoupling function. In Section 4 we make use of the low energy theorem to

2 It should be stressed that the statement is literally valid only if power-suppressed corrections of order (s/m2)n with n>0 are neglected. K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135 123 derive the effective coupling of the Higgs boson to gluons (due to the virtual presence of a heavy quark) through four and (partially) through ﬁve loops. Section 5 deals with phenomenological applications of our result which lead to a reduction of the theoretical error due to evolution of the coupling constant αs through heavy quark thresholds. Summary and conclusions are presented in the ﬁnal Section 6.

2. Formalism

2.1. Decoupling for the gauge coupling constant

Let us consider QCD with nl = nf − 1 massless quarks ψ ={ψl | l = 1,...,nl} and one heavy quark h with mass m. The corresponding bare QCD Lagrangian reads L ; 0,a 0,a g0,m0,ξ0 ψ0,Gμ ,c ,h0 1 =− G0,a 2 + iψ¯ Dψ/ + h¯ (iD/ − m )h 4 μν 0 0 0 0 0 0 + terms with ghost fields and the gauge-fixing term, (2) 0,a 0,a where the index ‘0’ marks bare quantities, Gμ and c are the gluon and ghost field, respectively. The relations between the bare and renormalized quantities read 0 = ε 0 = 0 − = − gs μ Zggs,mZmm, ξ 1 Z3(ξ 1), 0 = 0,a = a 0,a = ˜ a ψq Z2ψq ,Gμ Z3Gμ,c Z3c , (3) √ where gs = 4παs is the (renormalized) QCD gauge coupling, μ is the renormalization scale, d = 4 − 2ε is the dimensionality of space–time. The gauge parameter ξ is defined through the gluon propagator in lowest order, i qμqν −gμν + ξ . (4) q2 + i q2 O 4 All renormalization constants appearing in (3) are known by now through order (αs ) from [9–13]. Integrating out the heavy quark transforms the full QCD Lagrangian (2) into the one corre- = − = sponding to the effective massless QCD with nf nf 1 nl quark flavors (plus additional higher dimension interaction terms suppressed by powers of the heavy mass and neglected in what follows). Denoting the effective fields and parameters by an extra prime, we write the effective Lagrangian as follows: L = L 0 0; 0 0,a 0,a gs ,ξ ψq ,Gμ ,c . (5) Here the primed quantities are related to nonprimed ones through g0 = ζ 0g0,ξ0 − 1 = ζ 0 ξ 0 − 1 ,ψ0 = ζ 0ψ0, s gs 3 q 2 q 0,a = 0 0,a 0,a = ˜ 0 0,a Gμ ζ3 Gμ ,c ζ3 c , (6) where the primes mark the quantities of the effective nl-flavor theory. As was first demonstrated in [8] the bare decoupling constants can all be expressed in terms of massive Feynman integrals without any external momenta (so-called massive tadpoles). The 124 K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135

0 corresponding relation for ζg reads:

ζ˜ 0 0 = 1 ζg , (7) ˜ 0 0 ζ3 ζ3 where ˜ 0 = + 0h ζ1 1 ΓGcc¯ (0, 0), (8) 0 = + 0h ζ3 1 ΠG (0), (9) ˜ 0 = + 0h ζ3 1 Πc (0). (10) 2 2 Here ΠG(p ) and Πc(p ) are the gluon and ghost vacuum polarization functions, respectively. 2 2 ΠG(p ) and Πc(p ) are related to the gluon and ghost propagators through gμν δab + terms proportional to pμpν p2[1 + Π 0 (p2)] G

· = i dxeip x TG0,aμ(x)G0,bν (0) , (11)

ab δ · − = i dxeip x Tc0,a(x)c¯0,b(0) , (12) 2[ + 0 2 ] p 1 Πc (p ) 0 respectively. The vertex function ΓGcc¯ (p, k) is deﬁned through the one-particle-irreducible (1PI) part of the amputated Gcc¯ Green function as pμg0 −if abc 1 + Γ 0 (p, k) + other color structures s Gcc¯

· + · = i2 dxdyei(p x k y) Tc0,a(x)c¯0,b(0)G0,cμ(y) 1PI, (13) where p and k are the outgoing four-momenta of c and G, respectively, and f abc are the structure constants of the QCD gauge group. 0 Finally, in order to renormalize the bare decoupling constant ζg we combine Eq. (7) with = 2 = 2 Eqs. (3) and arrive at (αs gs /(4π),αs (gs) /(4π)) Z 2 = g 0 = 2 αs(μ) ζg αs(μ) ζg αs(μ). (14) Zg Direct application of this equation is rather clumsy as the constant Zg on its right side depends on the renormalized effective coupling constant which we are looking for. Of course, within perturbation theory, one can always solve Eq. (14) by iteration. A simpler way, which we have used, reduces to inverting ﬁrst the series 0 = = 2 = + i αs Zα(αs, )αs,Zα(αs, ) Zg 1 Zα,i( )αs (15) i1 to express the renormalized coupling constant αs in terms of the bare one: = 0 0 0 0 = + 0 0 i αs Zα αs , αs ,Zα(αs, ) 1 Zα,i( ) αs , (16) i1 K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135 125 where, up to four loops, 0 =− 0 =− + 2 Zα,1 Zα,1,Zα,2 Zα,2 2 Zα,1, 0 =− + − 3 Zα,3 Zα,3 5Zα,1Zα,2 5Zα,1, 0 =− + + 2 − 2 + 4 Zα,4 Zα,4 6Zα,1Zα,3 3Zα,2 21Zα,1Zα,2 14Zα,1. (17) With the use of Eq. (17) one now could conveniently transform all the primed (that is effective quantities) in Eq. (14) into the nonprimed ones. After this the renormalization can be done directly.

2.2. Vacuum integrals

At the end of the day we have to evaluate a host of four-loop massive tadpoles entering the definitions of the bare decoupling constants. The evaluation of these massive tadpoles in three- loop approximation has been pioneered in Ref. [14] and automated in Ref. [15]. Similar to the three-loop case, the analytical evaluation of four-loop tadpole integrals is based on the traditional integration-by-parts (IBP) method. In contrast to the three-loop case the man- ual construction of algorithms to reduce arbitrary diagrams to a small set of master integrals is replaced by Laporta’s algorithm [16,17]. In this context the IBP identities are generated with numerical values for the powers of the propagators and the irreducible scalar products. In the next step, the resulting system of linear equations is solved by expressing systematically complicated integrals in terms of simpler ones. The resulting solutions are then substituted into all the other equations. This reduction has been implemented in an automated FORM3 [18,19] based program in which partially ideas described in Refs. [17,20,21] have been implemented. The rational functions in the space–time dimension d, which arise in this procedure, are simplified with the program FERMAT [22]. The automated exploitation of all symmetries of the diagrams by reshuffling the powers of the propagators of a given topology in a unique way strongly reduces the number of equations which need to be solved. In general, the tadpole diagrams contributing to the decoupling constants contain both massive and massless lines. In contrast, the computation of the four-loop β-function can be reduced to the evaluation of four-loop tadpoles composed of completely massive propagators. These special cases have been considered in [9,13,21]. All four-loop tadpole diagrams encountered during our calculations were expressed through the set of 13 master integrals shown in Figs. 1 and 2. The first four integrals of Fig. 1 have analytical solution in terms of gamma functions for generic values of . The first nine terms of the -expansion of the fifth integral (T52) can be obtained from results of Refs. [14,23] (for more details see Ref. [28]). As for the less simple master integrals pictured in Fig. 2 one finds that for the case of ζg all necessary ingredients are known analytically [24–29] except for T54,3, T62,2

Fig. 1. Analytically known master integrals. 126 K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135

Fig. 2. Master integrals where only a few terms of their -expansion are known analytically. The solid (dashed) lines denote massive (massless) propagators. The three numbers in brackets (n1,n2,n3) are decoded as follows: n1 is the maximal power of the spurious pole in which might appear in front of the integral, n2 is the maximal power of the 0 spurious pole in which happens to enter into the decomposition of the bare decoupling constant ζg in terms of the master integrals, n3 is the maximal analytically known power of the -expansion of the same integral as determined in [28] and conﬁrmed in [29]. and T91,0, where we have denoted 4−2 (−4) d p 1 = i Tn · · → Tn,i. (18) π 2− (p2 + 1)2 0 i−4 These three integrals are known numerically from [28,29] and read

T54,3 =−8445.8046390310298, (19)

T62,2 =−4553.4004372195263, (20)

T91,0 = 1.808879546208. (21)

In fact, in Refs. [29,30] an analytical result for T91,0 (in terms of T62,2 and T54,3) has been also derived. Thus, the results of the next section will contain only two numerical constants, viz. T62,2 and T54,3.

3. Decoupling at four loops: results

We have generated the relevant diagrams with the help of the program QGRAF [31]. The total 0 ˜ 0 0 number of diagrams at four loops amount to 6070, 765 and 9907 for ζ3 , ζ3 and ζ1 , respectively. All calculations were performed in a general covariant gauge with the gluon propagator as given in Eq. (4). However, since extra squared propagators lead to signiﬁcant calculational complications, only linear terms in ξ were kept. The bare results are rather lengthy and will be made available (in computer-readable form) in http://www-ttp.physik.uni-karlsruhe.de/Progdata/ttp05/ ttp05-27. After constructing the bare decoupling constant from Eq. (7) followed by the renormalization as described in Section 2 we arrive at the following gauge independent3 result for the decoupling MS function ζg : MS 2 = + i ζg 1 as(μ)dMS,i, (22) i1

3 In fact, the gauge dependence on ξ disappears already for the bare decoupling constant as it should be. K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135 127 where we use the notation (n ) α f (μ) a (μ) = s s π and the coefﬁcients dMS,i read 1 d =− , (23) MS,1 6 μm 11 11 1 d = − + 2 , (24) MS,2 72 24 μm 36 μm = 564731 − 82043 − 955 + 53 2 − 1 3 dMS,3 ζ3 μm μm μm 124416 27648 576 576 216 2633 67 1 + n − + − 2 , (25) l 31104 576 μm 36 μm = + 7391699 − 2529743 + 2177 2 − 1883 3 + 1 4 dMS,4 ΔMS,4 μm ζ3μm μm μm μm 746496 165888 3456 10368 1296 + −110341 + 110779 − 1483 2 − 127 3 nl μm ζ3μm μm μm 373248 82944 10368 5184 + 2 6865 − 77 2 + 1 3 nl μm μm μm , (26) 186624 20736 324 134805853579559 254709337 151369 18233772727 Δ = − π 4 − π 6 − ζ MS,4 43342154956800 783820800 30481920 783820800 3 151369 4330717 9869857 121 2057 + ζ 2 + ζ + a − a − π 4 ln 2 544320 3 207360 5 272160 4 36 5 51840 9869857 121 9869857 121 − π 2 ln2 2 − π 2 ln3 2 + ln4 2 + ln5 2 6531840 2592 6531840 4320 82037 151369 + T54,3 − T62,2 30965760 11612160 4770941 541549 4 3645913 115 685 + nl − − π + ζ3 + ζ5 + a4 2239488 14929920 995328 576 5184 685 685 271883 167 − π 2 ln2 2 + ln4 2 + n2 − + ζ . (27) 124416 124416 l 4478976 5184 3

2 In Eqs. (23)–(27) = ln μ , m(μ) is the (running) heavy quark mass in the MS-scheme μm m2(μ) and μ represents the renormalization scale. Furthermore, ζn = ζ(n) is Riemann’s zeta function = = ∞ i n and an Lin(1/2) i=1 1/(2 i ). For another convenient deﬁnition of the quark mass—the so-called scale-invariant mass, de- = = μ2 ﬁned through the relation μh m(μh)—Eqs. (22)–(27) are transformed to (μh ln 2 ): μh SI 2 = + i ζg 1 as(μ)dSI,i, (28) i1 where 1 d =− , (29) SI,1 6 μh 128 K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135

11 19 1 d = − + 2 , (30) SI,2 72 24 μh 36 μh = 564731 − 82043 − 6793 − 131 2 − 1 3 dSI,3 ζ3 μh μh μh 124416 27648 1728 576 216 2633 281 + n − + , (31) l 31104 1728 μh 2398621 2483663 14023 d = Δ − − ζ − 2 SI,4 MS,4 746496 μh 165888 3 μh 3456 μh − 8371 3 + 1 4 μh μh 10368 1296 + 190283 + 133819 + 983 2 + 107 3 nl μh ζ3μh μh μh 373248 82944 3456 1728 8545 79 + n2 − 2 . (32) l 186624 μh 6912 μh For practical applications also the inverted formulae are needed: 1 = 1 + a (μ) id , (33) (ζ MS)2 s MS,i g i1 and 1 = 1 + a (μ) id , (34) (ζ SI)2 s SI,i g i1 where we use the notation

(nl ) α (μ) a (μ) = s s π and the coefﬁcients dMS,i, dSI,i read

1 d = , (35) MS,1 6 μm 11 11 1 d =− + + 2 , (36) MS,2 72 24 μm 36 μm =−564731 + 82043 + 2645 + 167 2 + 1 3 dMS,3 ζ3 μm μm μm 124416 27648 1728 576 216 2633 67 1 + n − + 2 , (37) l 31104 576 μm 36 μm 121 11093717 3022001 1837 d = − Δ − + ζ + 2 MS,4 1728 MS,4 746496 μm 165888 3 μm 1152 μm + 2909 3 + 1 4 μm μm 10368 1296 + 141937 − 110779 + 277 2 + 271 3 nl μm ζ3μm μm μm 373248 82944 10368 5184 6865 77 1 + n2 − + 2 − 3 , (38) l 186624 μm 20736 μm 324 μm K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135 129

1 d = , (39) SI,1 6 μh 11 19 1 d =− + + 2 , (40) SI,2 72 24 μh 36 μh =−564731 + 82043 + 2191 + 511 2 + 1 3 dSI,3 ζ3 μh μh μh 124416 27648 576 576 216 2633 281 + n − , (41) l 31104 1728 μh 121 1531493 2975921 33887 d = − Δ − + ζ + 2 SI,4 1728 MS,4 746496 μh 165888 3 μh 3456 μh + 14149 3 + 1 4 μh μh 10368 1296 158687 133819 515 107 + n − − ζ − 2 − 3 l 373248 μh 82944 3 μh 1152 μh 1728 μh 8545 79 + n2 − + 2 . (42) l 186624 μh 6912 μh Numerically, Eqs. (22) and (28) read MS 2 = 2 3 ζ = 1 + 0.152778a (μh) + a (μh)(0.972057 − 0.0846515nl) g μ μh s s + 4 − − 2 as (μh) 5.17035 1.00993nl 0.0219784nl , (43) 1 = 2 3 = 1 − 0.152778a (μ ) + a (μ )(−0.972057 + 0.0846515n ) (ζ MS)2 μ μh s h s h l g + 4 − + + 2 as (μh) 5.10032 1.00993nl 0.0219784nl . (44) Using the three-loop relation between the pole and MS masses [32–34] one could easily express the decoupling relations in terms of the pole mass of the heavy quark. As the resulting expressions are rather lengthy they are relegated to Appendix A.

4. Coupling of the Higgs boson to gluons

Within the Standard Model (SM) the scalar Higgs boson is responsible for the mechanism of the mass generation. Its future (non)discovery will be of primary importance for all the particle physics. The SM Higgs boson mass is constrained from below, Mh > 114 GeV, by experiments at LEP and SLC [35,36]. Indirect constraints coming from precision electroweak measurements [37] lead to an upper limit of about 200 GeV. With the SM Higgs boson mass within this range, its coupling to a pair of gluons is mediated by virtual quarks [38] and plays a crucial rôle in Higgs phenomenology. Indeed, with Yukawa couplings of the Higgs boson to quarks being proportional to the respective quark masses, the ggH coupling of the SM is essentially generated by the top quark only. The ggH coupling strength becomes independent of the top quark mass Mt in the limit MH 2Mt . In general, the theoretical description of such interactions is very complicated because there are two different mass scales involved, MH and Mt . However, in the limit MH 2Mt ,the situation may be greatly simpliﬁed by integrating out the top quark, i.e., by constructing a heavy- top-quark effective Lagrangian [39,40]. 130 K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135

This Lagrangian is a linear combination of certain dimension-four operators acting in QCD with five quark flavors, while all Mt dependence is contained in the coefficient functions. The final renormalized version of Leff is (GF is Fermi’s constant) L =− 1/4 1/2 [ ] eff 2 GF HC1 O1 . (45) [ ] = 0 0μν Here, O1 is the renormalized counterpart of the bare operator O1 GaμνGa , where Gaμν is the color field strength, the superscript 0 denotes bare fields, and primed objects refer to the five- flavor effective theory. C1 is the corresponding renormalized coefficient function, which carries all Mt dependence. In Ref. [8] a low-energy theorem was established, valid to all orders, which relates the effective coupling of the Higgs boson to gluons, induced by a presence of heavy quark h,tothe logarithmic derivative of ζg w.r.t. m. The theorem states that 1 ∂ C =− m2 ln ζ 2. (46) 1 2 ∂m2 g Another, equivalent, but more convenient form of (46) was also derived in [8] by exploiting evolution equations in full and effective theories. It reads = π ˆ − ˆ − ˆ ∂ 2 C1 β (as) β(as) β(as)as ln ζg , (47) 2[1 − 2γm(as)] ∂as 2 = 2 where γm is the quark mass anomalous dimension, ζg ζg (μ, as,m) and the β-function is defined as follows: ∞ d (n ) (n ) (n ) + μ2 a f = β(nf ) a f =− β f aN 1, dμ2 s s N−1 s N=1 (n ) = f = (nl ) ˆ = (nf ) ˆ = (nl ) as as ,as as ,asβ(as) β (as), asβ (as) β (as). (48)

An important feature of the low energy theorem (47) is the fact that the decoupling function ζg appears there multiplied by at least one power of as . It means that in order to compute C1,say,at four loops one should know the QCD β-function to four loops (only nf dependent part) but the quark anomalous dimension γm and the decoupling function ζg only to three loops. It is this fact which allowed to find C1 at four loops in [8] long before the four-loop result for the decoupling function would be available. Now, armed with the newly computed four-loop term in ζg, we could easily check that the old result for C1 of [8] by a direct use of Eq. (46). We have done this simple exercise and found full agreement. In fact, one could even use Eq. (47) in order to construct C1 at five loops in terms of the known four-loop QCD decoupling function, the quark anomalous dimension γm and the β-function and the only yet unknown parameter: the nf -dependent piece of the five-loop contribution to the β-function. The result reads 1 11 2821 67 C =− a (μ ) 1 + a (μ ) + a2(μ ) + n − 1 12 s h 4 s h s h 288 l 96 4004351 1305893 115607 110779 + a3(μ ) − + ζ(3) + n − ζ(3) s h 62208 13824 l 62208 13824 6865 + n2 − l 31104 K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135 131 + 4 −13546105 − 91 − 31 4 − 853091 + 475 as (μh) ΔMS,4 π ζ3 ζ5 41472 2 432 6912 9 28598581 29 4 3843215 575 + nl + 3ΔMS,4 − π + ζ3 − ζ5 497664 432 110592 36 3503 1 3 83 1 + n2 − + π 4 − ζ + n3 − ζ + 6Δβ l 62208 216 4 3 l 7776 54 3 4 ≈− 1 + + − 2 as(μh) 1 2.7500as(μh) (9.7951 0.6979nl)as (μh) 12 + 49.1827 − 7.7743n − 0.2207n2 a3(μ ) l l s h + − + − 2 − 3 + 4 662.507 137.601nl 2.53666nl 0.077522nl 6Δβ4 as (μh) , (49) where − = (nf 1) − (nf ) Δβ4 β4 β4 .

To save space we have written Eq. (49) for the value of μ = μh. Unfortunately, simple es- timations show that the β-function dependent contribution to Eq. (49) could be numerically important.

5. Application: αs(MZ) from αs(Mτ )

A central feature of QCD is the possibility to describe measurements performed at very different energy scales with the help of only one coupling constant. The customary choice is the (MS) coupling constant αs(MZ) which can be measured precisely in Z-boson decays. (For a review see, e.g., [41].) On the other hand the dependence of the τ -decay rate on the strong coupling αs has been used for a determination of αs at lower energies, with the most recent results of 0.340 ± 0.005exp ± 0.014th and 0.348 ± 0.009exp ± 0.019th by the ALEPH [42] and OPAL [43] Collaborations. After evolution up to higher energies (taking into account the threshold effects due to the c- and b-quarks) these results agree remarkably well with determinations based on the hadronic Z decay rate, which provides a genuine test of asymptotic freedom of QCD. Very recently an analysis of the evolution has been performed in [7]. They start from an updated determination of αs(mτ )

αs(mτ ) = 0.345 ± 0.004exp ± 0.008th (50) based on most recent experimental results of the ALEPH Collaboration [42]. Their result for the evolution of αs(mτ ) giveninEq.(50), based on the use of the four-loop running and three-loop quark-ﬂavor matching reads

as(MZ) = 0.1215 ± 0.0004exp ± 0.0010th ± 0.0005evol, = 0.1215 ± 0.0012, (51) where the ﬁrst two errors originate from the αs(mτ ) determination given in Eq. (50), and the last error stands for the ambiguities in the evolution due to uncertainties in the matching scales of the quark thresholds. That evolution error received contributions from the uncertainties in the c-quark mass (0.00020, μc = 1.31 ± 0.1 GeV) and the b-quark mass (0.00005, μb = 4.13 ± 0.1 GeV), the matching scale (0.00023, μ varied between 0.7μq and 3.0μq ), the three-loop 132 K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135 truncation in the matching expansion (0.00026) and the four-loop truncation in the RGE equation (0.00031). (For the last two errors the size of the shift due the highest known perturbative term was treated as systematic uncertainty.) The errors had been added in quadrature. We have repeated this analysis including the newly computed four-loop approximation for the 4 matching. As a result the value of as(MZ) has been marginally increased (by 0.0001) and both errors from the choice of the matching scale and from the four-loop truncation in the matching equation have been halved. The updated version of (51) now reads

as(MZ) = 0.1216 ± 0.0004exp ± 0.0010th ± 0.0004evol, = 0.1216 ± 0.0012. (52)

6. Summary and conclusions

We have computed the decoupling relation for the QCD quark–gluon coupling constant in four-loop approximation. The new contribution leads to a decrease of the matching related uncertainties in the process of the evolution of the αs(mτ ) to αs(MZ) by a factor of two. As a by-product we have directly confirmed the long available result for C1, the effective coupling of the Higgs boson to gluons at four loops, and (partially) extended it to five loops. It remains only to find the QCD β-function at five loops to get the full result for C1.Inthe light of recent significant progress in the multiloop technology [45,46] the completely analytical evaluation of the former seems to be possible in not too far distant future. We would like to mention that the result for the decoupling function at four loops as well as for the Higgs boson to gluons effective coupling at five loops have been obtained by a completely independent calculation in [30]. The results are in full mutual agreement.

Acknowledgements

The authors are grateful to York Schröder and Matthias Steinhauser for useful discussions and information about [30]. K.Ch. thanks Michail Kalmykov for some useful advice about integral T52. The work was supported by the Deutsche Forschungsgemeinschaft in the Sonderforschungs- bereich/Transregio SFB/TR-9 “Computational Particle Physics”. The work of C.S. was also partially supported by MIUR under contract 2001023713_006.

Appendix A. ζ g for the heavy quark mass in the on-shell scheme

The relevant formulas for the case of the heavy quark mass renormalized in the on-shell scheme (denoted as M)are: OS 2 = + i ζg 1 as(μ)dOS,i (A.1) i1 and 1 = 1 + a i(μ)d , (A.2) (ζ OS)2 s OS,i g i1

4 We have used the package RunDec [44]. K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135 133

2 where = ln μ and μM M2 1 d =− , (A.3) OS,1 6 μM 7 19 1 d =− − + 2 , (A.4) OS,2 24 24 μM 36 μM 58933 1 2 80507 1 2 8521 dOS,3 =− − π − ζ3 − π ln 2 − μM 124416 9 27648 27 1728 131 1 2479 1 409 − 2 − 3 + n + π 2 + , (A.5) 576 μM 216 μM l 31104 54 1728 μM 19696909 29 2439119 29 d = Δ − − π 2 − ζ − π 2 ln 2 OS,4 OS,4 746496 μM 54 μM 165888 3 μM 162 μM − 7693 2 − 8371 3 + 1 4 μM μM μM 1152 10368 1296 1110443 41 2 132283 1 2 + nl μM + π μM + ζ3μM + π ln 2μM 373248 324 82944 81 + 6661 2 + 107 3 μM μM 10368 1728 1679 1 493 + n2 − − π 2 − 2 , (A.6) l 186624 μM 162 μM 20736 μM 2180918653146841 697121 231357337 Δ =− − π 2 − π 4 OS,4 43342154956800 116640 783820800 151369 18646246327 1439 151369 − π 6 − ζ + π 2ζ + ζ 2 30481920 783820800 3 1296 3 544320 3 3698717 10609057 121 1027 + ζ + a − a + π 2 ln 2 207360 5 272160 4 36 5 972 2057 9278497 121 10609057 − π 4 ln 2 − π 2 ln2 2 − π 2 ln3 2 + ln4 2 51840 6531840 2592 6531840 121 5 82037 151369 + ln 2 + T54,3 − T62,2 4320 30965760 11612160 1773073 557 2 697709 4 4756441 115 + nl + π − π + ζ3 + ζ5 746496 972 14929920 995328 576 173 11 2 1709 2 2 173 4 + a4 + π ln 2 − π ln 2 + ln 2 5184 243 124416 124416 140825 13 19 + n2 − − π 2 − ζ , (A.7) l 1492992 972 1728 3 1 d = , (A.8) OS,1 6 μM 7 19 1 d = + + 2 , (A.9) OS,2 24 24 μM 36 μM = 58933 + 1 2 + 80507 + 1 2 + 8941 dOS,3 π ζ3 π ln 2 μM 124416 9 27648 27 1728 511 1 2479 1 409 + 2 + 3 + n − − π 2 − , (A.10) 576 μM 216 μM l 31104 54 1728 μM 134 K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135

49 21084715 35 2922161 d = − Δ + + π 2 + ζ OS,4 192 OS,4 746496 μM 54 μM 165888 3 μM + 35 2 + 47039 2 + 14149 3 + 1 4 π ln 2μM μM μM μM 162 3456 10368 1296 1140191 47 2 132283 1 2 + nl − μM − π μM − ζ3μM − π ln 2μM 373248 324 82944 81 − 9115 2 − 107 3 μM μM 10368 1728 1679 1 493 + n2 + π 2 + 2 , (A.11) l 186624 μM 162 μM 20736 μM OS 2 = 2 3 ζ = 1 − 0.291667a (M) + a (M)(−5.32389 + 0.262471nl) g μ M s s + 4 − + − 2 as (M) 85.875 9.69229nl 0.239542nl , (A.12) 1 = 2 3 = 1 + 0.291667a (M) + a (M)(5.32389 − 0.262471n ) (ζ OS)2 μ M s s l g + 4 − + 2 as (M) 86.1302 9.69229nl 0.239542nl . (A.13)

References

[1] T. Appelquist, J. Carazzone, Phys. Rev. D 11 (1975) 2856. [2] S. Weinberg, Phys. Lett. B 91 (1980) 51. [3] B.A. Ovrut, H.J. Schnitzer, Phys. Lett. B 100 (1981) 403. [4] W. Wetzel, Nucl. Phys. B 196 (1982) 259. [5] W. Bernreuther, W. Wetzel, Nucl. Phys. B 197 (1982) 228. [6] W. Bernreuther, Ann. Phys. 151 (1983) 127. [7] M. Davier, A. Hocker, Z. Zhang, hep-ph/0507078. [8] K.G. Chetyrkin, B.A. Kniehl, M. Steinhauser, Nucl. Phys. B 510 (1998) 61, hep-ph/9708255. [9] T. van Ritbergen, J.A.M. Vermaseren, S.A. Larin, Phys. Lett. B 400 (1997) 379, hep-ph/9701390. [10] J.A.M. Vermaseren, S.A. Larin, T. van Ritbergen, Phys. Lett. B 405 (1997) 327, hep-ph/9703284. [11] K.G. Chetyrkin, Phys. Lett. B 404 (1997) 161, hep-ph/9703278. [12] K.G. Chetyrkin, Nucl. Phys. B 710 (2005) 499, hep-ph/0405193. [13] M. Czakon, Nucl. Phys. B 710 (2005) 485, hep-ph/0411261. [14] D.J. Broadhurst, Z. Phys. C 54 (1992) 599. [15] M. Steinhauser, Comput. Phys. Commun. 134 (2001) 335, hep-ph/0009029. [16] S. Laporta, E. Remiddi, Phys. Lett. B 379 (1996) 283, hep-ph/9602417. [17] S. Laporta, Int. J. Mod. Phys. A 15 (2000) 5087, hep-ph/0102033. [18] J.A.M. Vermaseren, math-ph/0010025. [19] J.A.M. Vermaseren, Nucl. Phys. B (Proc. Suppl.) 116 (2003) 343, hep-ph/0211297. [20] P. Mastrolia, E. Remiddi, Nucl. Phys. B (Proc. Suppl.) 89 (2000) 76. [21] Y. Schröder, Nucl. Phys. B (Proc. Suppl.) 116 (2003) 402, hep-ph/0211288. [22] R.H. Lewis, Fermat’s User Guide, http://www.bway.net/~lewis/. [23] D.J. Broadhurst, hep-th/9604128. [24] S. Laporta, Phys. Lett. B 549 (2002) 115, hep-ph/0210336. [25] K.G. Chetyrkin, J.H. Kuhn, P. Mastrolia, C. Sturm, Eur. Phys. J. C 40 (2005) 361, hep-ph/0412055. [26] Y. Schröder, M. Steinhauser, Phys. Lett. B 622 (2005) 124, hep-ph/0504055. [27] B.A. Kniehl, A.V. Kotikov, hep-ph/0508238. [28] Y. Schröder, A. Vuorinen, JHEP 0506 (2005) 051, hep-ph/0503209. [29] K. Chetyrkin, et al., -ﬁnite basis of master integrals for the integration-by-parts method, in preparation. [30] Y. Schröder, M. Steinhauser, preprint BI-TP 2005/44, SFB/CCP-05-70, TTP05-21, hep-ph/0512058. [31] P. Nogueira, J. Comput. Phys. 105 (1993) 279. [32] K.G. Chetyrkin, M. Steinhauser, Phys. Rev. Lett. 83 (1999) 4001, hep-ph/9907509. K.G. Chetyrkin et al. / Nuclear Physics B 744 (2006) 121–135 135

[33] K.G. Chetyrkin, M. Steinhauser, Nucl. Phys. B 573 (2000) 617, hep-ph/9911434. [34] K. Melnikov, T. van Ritbergen, Phys. Lett. B 482 (2000) 99, hep-ph/9912391. [35] ALEPH Collaboration, R. Barate, et al., Phys. Lett. B 565 (2003) 61, hep-ex/0306033. [36] LEP Collaborations, hep-ex/0412015. [37] M. Carena, H.E. Haber, Prog. Part. Nucl. Phys. 50 (2003) 63, hep-ph/0208209. [38] F. Wilczek, Phys. Rev. Lett. 39 (1977) 1304. [39] T. Inami, T. Kubota, Y. Okada, Z. Phys. C 18 (1983) 69. [40] K.G. Chetyrkin, B.A. Kniehl, M. Steinhauser, Phys. Rev. Lett. 79 (1997) 353, hep-ph/9705240. [41] K.G. Chetyrkin, J.H. Kühn, A. Kwiatkowski, Phys. Rep. 277 (1996) 189. [42] ALEPH Collaboration, S. Schael, et al., Phys. Rep. 421 (2005) 191, hep-ex/0506072. [43] OPAL Collaboration, K. Ackerstaff, et al., Eur. Phys. J. C 7 (1999) 571, hep-ex/9808019. [44] K.G. Chetyrkin, J.H. Kühn, M. Steinhauser, Comput. Phys. Commun. 133 (2000) 43, hep-ph/0004189. [45] P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, Phys. Rev. Lett. 95 (2005) 012003, hep-ph/0412350. [46] P.A. Baikov, K.G. Chetyrkin, J.H. Kühn, hep-ph/0511063. Nuclear Physics B 744 (2006) 136–155

Semilocalized instability of the Kaluza–Klein linear dilaton vacuum

Oren Bergman ∗, Shinji Hirano 1

Department of Physics, Technion, Israel Institute of Technology, Haifa 32000, Israel Received 26 December 2005; accepted 7 March 2006 Available online 7 April 2006

Abstract The Kaluza–Klein linear dilaton background of the bosonic string and the Scherk–Schwarz linear dilaton background of the superstring are shown to be unstable to the decay of half of spacetime. The decay proceeds via a condensation of a semilocalized tachyon when the circle is smaller than a critical size, and via a semiclassical instanton process when the circle is larger than the critical size. At criticality the two pictures are related by a duality of the corresponding two-dimensional conformal ﬁeld theories. This provides a concrete realization of the connection between tachyonic and semiclassical instabilities in closed string theory, and lends strong support to the idea that nonlocalized closed string tachyon condensation leads to the annihilation of spacetime. © 2006 Elsevier B.V. All rights reserved.

Keywords: Tachyon condensation; Sine-Liouville model; Cigar

1. Introduction

In a now classic paper, Witten demonstrated that the Kaluza–Klein (KK) vacuum of five- dimensional gravity is unstable to decay semiclassically via a “bubble of nothing” [1]. The bubble is nucleated by an instanton process described by the five-dimensional Euclidean Schwarzschild black hole, and then expands at a speed approaching the speed of light. Witten’s result has been generalized to other cases, such as the KK magnetic field background [2], supergravity fluxbranes

* Corresponding author. E-mail addresses: [email protected] (O. Bergman), [email protected] (S. Hirano). 1 Address starting 10 October 2006: Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen, Denmark.

[3], twisted circle compactifications in supergravity [4], and in particular Scherk–Schwarz (SS) compactification. The semiclassical description of the decay in terms of a gravitational instanton is valid as long as R lP . As the radius of the circle decreases the (super)gravity description eventually breaks down, and must be replaced with string theory. However one does not in general know the exact string backgrounds corresponding to these instantons. String theory in nonsupersymmetric backgrounds often exhibits another type of instability as well, namely one associated with tachyonic modes. The stable background in these cases is given by a tachyon condensate. Since this corresponds to an off-shell state in the original background, we need to use off-shell techniques like string field theory and worldsheet renormalization group analysis to study the stable background. This program has been most successful for open string tachyons living on unstable D-branes, where both open string field theory and boundary RG analysis support the intuitive conjecture that the true vacuum corresponds to the closed string vacuum without the branes [5]. Closed string tachyons have been more challenging. On the one hand closed string field theory is much more complicated than open string field theory due to its nonpolynomiality,2 and on the other hand the worldsheet RG approach is generally hampered by Zamolodchikov’s c-theorem. Conceptually, the question of closed string tachyons is expected to be harder since closed string theories are theories of gravity, and the dynamics of the tachyon is strongly correlated with the structure of spacetime. A simple analogy with open string tachyons suggests that the condensation of closed string tachyons should lead to the decay of spacetime itself. However we do not know what this means in the context of the original string theory. The above discussion raises two questions. The first is whether we can get a better handle on the fate of backgrounds with closed string tachyons, and the second is whether there is a connection between tachyonic and semiclassical instabilities in closed string theory. Recently some progress has been made on the first question in the context of closed string tachyons which are localized on lower-dimensional hypersurfaces in space. There are a few examples of backgrounds with localized closed string tachyons for which we have strong evidence regarding the nature of the tachyon condensate [7,8]. One example is the cone orbifold C/ZN [9]. Tachyons in the twisted sector of the orbifold are localized at the tip of the cone, and it has been argued, based on a D-brane probe analysis, that the condensation of these tachyons leads to a reduction in the rank N, and thereby to a smoothing out of the singularity. Further support for this picture comes from worldsheet RG analysis [9–11], and closed string field theory [12,13].An- other example consists of a SS circle which shrinks locally in the transverse directions, leading to a localized winding tachyon [14]. RG analysis suggests that the condensation of this tachyon leads to a topology change in space, whereby the circle pinches and the transverse space splits at that point. Localized winding tachyons may also play a role in resolving cosmological singularities [15,16], black hole physics [17–19], and chronology protection [20]. In both examples the effect of the tachyon is felt only locally, and the bulk of spacetime remains intact. It would be nice to have an example with a nonlocalized tachyon.3 The second question is interesting since, if true, it would demystify somewhat the role of closed string tachyons. Semiclassical instabilities provide a clear geometrical picture for the decay of an unstable background, at least in the regime where the low-energy approximation is valid.4 De Alwis and Flournoy have conjectured that when continued past this regime, the

2 See [6] for recent progress however. 3 Attempts were made in [21,22] for the bulk rolling tachyon, [23] for the bulk bouncing tachyon. See [24] for more general observations. 4 This is also true in the open string case [25,26]. 138 O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155

Fig. 1. Tachyonic and semiclassical instabilities in λφ4 (λ<0) field theory. instability would eventually become tachyonic [4]. As evidence for this they list several examples of string backgrounds, such as the SS circle, which exhibit a semiclassical instability in one regime of the moduli space (R 1), and a tachyonic instability in another regime (R<1). In field theory we are used to such a transition. Consider for example λφ4 field theory with λ<0. 2 If m > 0√ the origin φ = 0 is a local minimum, which is unstable to decay by tunneling out to φ =±m/ −λ, and then rolling down. If m2 < 0 the origin is a local maximum, i.e., tachyonic, and the field decays by rolling (Fig. 1). In both cases the post-decay state is the same. In closed string theory the picture is less clear because on the one hand we do not have a good description of the tachyon condensate, and on the other hand the semiclassical regime is very far from the tachyonic regime. An approach which may shed some light on both questions is to focus on the point in moduli space where the tachyon becomes massless. At this point the tachyon condensate would correspond to an on-shell state (if the corresponding vertex operator is exactly marginal), and the true vacuum would be described by an exact two-dimensional CFT. In fact the first hard evidence for Sen’s conjecture about open string tachyons came from this direction. Sen considered an unstable Dp-brane wrapped on a circle of radius R, and a unit KK momentum “kink” mode of the open string tachyon [27]. This mode becomes massless when R = 1/2, and the corresponding operator is exactly marginal. The condensate of this mode is described by an exact CFT given by the boundary sine-Gordon model, whose solution interpolates between Neumann and Dirichlet boundary conditions on the circle [28,29], which in turn implies that the tachyon kink corresponds to a D(p − 1)-brane. A simple closed string example of this is the winding tachyon in the SS compactification of type II string theory at R = 1, or the bosonic string at R = 2. The world- sheet field theory of the tachyon condensate is given by the massless limit of the sine-Gordon model. However as far as we can tell, it is not known whether this theory is an exact CFT. There exists a related theory which is an exact CFT, given by the sine-Liouville model. Motivated by this, and by the conjectured duality of the sine-Liouville model and the cigar (SL(2, C)/SU(2))/U(1) CFT [30,31], we set out to study the instability of the linear dilaton background compactified on a circle (with SS boundary√ conditions in type II).5 The sine- Liouville model describes the massless limit, at R = k, where k is the level of SL(2) current algebra in the dual CFT, of a tachyon which condenses in an exponential profile, i.e., a semilocalized tachyon. The dual cigar background clearly shows that the condensate corresponds geometrically to a smooth truncation of the relevant two-dimensional part of the space from an infinite cylinder to a semiinfinite cigar (Fig. 2). A similar decay was argued to occur in finite tem-

5 A related topic was discussed in [32]. O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 139

Fig. 2. Semilocalized tachyon condensation turns the cylinder into a cigar. perature string theory on AdS below the Hagedorn temperature [33,34]. An open string version of this was recently discussed in [19]. Interestingly, this example also gives an explicit connection between a tachyonic instability and a semiclassical instability. We will show that the cigar is a type of bubble spacetime which results√ from a bounce, and that this can be generalized to larger radii (but not smaller). For R> k the KK (or SS) linear dilaton background√ is therefore unstable to decay semiclasically into a cigar-like spacetime, and for√R< k it is unstable due to a semilocalized tachyon. The two instabilities are equivalent at R = k. Furthermore, since k can be made large the supergravity approximation remains valid all the way down to the transition point. The rest of the paper is organized as follows. In Section 2 we review the relevant features of the cigar, the duality between the cigar and the sine-Liouville model, and a three-dimensional deformation of the cigar. In Section 3 we put the sine-Liouville model and the cigar in the context of the instability of the KK (or SS) linear dilaton background. We show that the cigar can be interpreted as a type of bubble of nothing resulting from a gravitational bounce, and generalize this to larger radii using a deformed cigar background. We then compute the classical decay rate due to bubble nucleation. In Section 4 we show, using a generalized cigar–sine-Liouville duality, that the semiclassical decay of the KK linear dilaton via the gravitational bounce can also be described as a ﬁeld-theoretic tunneling√ process of the massive scalar ﬁeld corresponding to the semilocalized tachyon for R> k. In Section 5 we address the question of the negative mode in the Euclidean backgrounds, which is relevant for the quantum-corrected decay rate. Section 6 contains our conclusions. We have also included an appendix containing a description of the spectrum in the post-decay background.

2. Review of some known results

We begin by reviewing some known results which we will use in the following sections. These include the two-dimensional cigar, the conjectured duality between the cigar and sine-Liouville theory, and a certain three-dimensional deformation of the cigar.

2.1. The cigar

The two-dimensional cigar is described by the metric and dilaton [35–37] ds2 = k dr2 + tanh2 rdφ2 ,

Φ − Φ0 =−log cosh r, (2.1) where φ is a periodic coordinate with φ ∼ φ + 2π. In string theory the cigar corresponds to an exact conformal ﬁeld theory given by the coset (SL(2, C)/SU(2))/U(1), where the parameter k in the metric corresponds to the level of the SL(2) current algebra. The central charge in the bosonic string case is given by 3k c = − 1. (2.2) k − 2 140 O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155

There exists also a supersymmetric version of the cigar CFT [38] with a central charge c = 3(k + 2)/k, and an odd spin structure at r →∞. The basic set of local operators (ignoring the oscillators) is labeled by three quantum numbers {j,m,m¯ }, where j is associated with the dependence on r, and m and m¯ are related to the momentum and winding around the cigar as 1 1 m = (n + kw ), m¯ =− (n − kw ). (2.3) 2 φ φ 2 φ φ The conformal dimensions of these operators are given by + 2 + ¯ 2 j(j 1) m ¯ j(j 1) m Δ ¯ =− + , Δ ¯ =− + . (2.4) j,m,m k − 2 k j,m,m k − 2 k In the supersymmetric case k − 2 is replaced by k in the j dependent part. There are two kinds of states in this background. First, there are bulk states which can propagate along the cigar. These are ordinary normalizable states which are the same as the states in the asymptotic linear dilaton background. In particular, they have a continuous spectrum with j =−1/2 + is, where s ∈ R. Second, there are states which are localized at the tip of the cigar with j ∈ R [39]. Unitarity places a bound on j given by j<(k− 2)/2(j−1/2. These states are nonnormalizable in general, however as argued in [38,40,41] a subset of them given by 1 k − 3 −

2.2. FZZ duality

It has been conjectured that the cigar CFT is equivalent to the sine-Liouville model [30,31] (see also [43]) given by 1 L = (∂x )2 + (∂x )2 + QRxˆ + λebx1 cos Rx , (2.6) 4π 1 2 1 2 ≡ − where x2 x2L x2R. This is a linear dilaton CFT with

Φ − Φ0 =−Qx1, (2.7) and a sine-Liouville interaction. The central charge of this theory is c = 2 + 6Q2. Equating this with the central charge of the cigar (2.2) gives Q2 = 1/(k − 2). An analogous conjecture relates N = 2 Liouville theory [44] to the supersymmetric version of the cigar coset CFT [45–48],in 2 which case Q = 1/k. In the sine-Liouville√ model x2 is a periodic coordinate with period 2πR. The radius is taken to be R = k, in agreement with the asymptotic radius of the cigar (2.1). Requiring the sine-Liouville potential to be marginal then ﬁxes b =−1/Q. In the asymptotic weakly-coupled region both the cigar and sine-Liouville model look like a cylinder√ with a linear dilaton, and the coordinates are identiﬁed (for large k)asr ∼ Qx1 and φ ∼ x2/ k. O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 141

The local operators discussed above correspond in the sine-Liouville model to + + O ∼ ax1 ip2Lx2L ip2Rx2R a,p2 e (2.8) with scaling dimensions 1 1 1 1 Δ =− a(a + 2Q) + p2 , Δ¯ =− a(a + 2Q) + p2 , (2.9) a,p2 4 4 2L a,p2 4 4 2R where6

nφ + kwφ nφ − kwφ a =−2Q(j + 1), p2L = √ ,p2R = √ . (2.10) k k

Bulk states therefore correspond to operators with a =−Q + ip1, exactly as in the free linear- dilaton theory, and the normalizable localized states (localized near the Liouville wall) are given by a −(Q + 1/Q) < a < −Q, n + kw =± − + l − 1 ,l∈ Z+, (2.11) φ φ Q or the same with wφ →−wφ. In particular the sine-Liouville potential itself corresponds to the localized state with a =−1/Q (or equivalently j = (k − 4)/2 in the bosonic string, and j = (k − 2)/2 in the superstring), nφ = 0 and wφ = 1.

2.3. Deformed cigar

A background which generalizes the cigar can be obtained by the J 3J¯3 deformation of SL(2, R) deﬁned by the coset [49,50]7 SL(2, R) × U(1) . (2.12) U(1) The corresponding metric, dilaton and B ﬁeld are given (in the string frame) by α tanh2 r dt2 ds2 = k dr2 + dφ2 − , α − tanh2 r α − tanh2 r − α − 1 e2(Φ Φ0) = , cosh2 r(α − tanh2 r) tanh2 r B = k dφ ∧ dt, (2.13) α − tanh2 r where α is the deformation parameter. The original SL(2, R) corresponds to α = 1. In the limit α →∞the background reduces to cigar × R, and for α → 0 it becomes trumpet × R.Forα<0 the geometry is Euclidean; it can be continued to a Lorentzian spacetime by replacing t → it, but then B becomes imaginary, so the background is not physical. We will be interested in the geometries with α>1, which can be regarded as deformations 2Φ of cigar × R. The classical supergravity approximation is√ valid for k 1√ and e 0 1. The asymptotic geometry is that of a cylinder with radius R = kα/(α − 1)> k, a linear dilaton

6 This differs from [31] by the replacement j →−j − 1. 7 The related SU(2) case was discussed in [51–53]. 142 O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155

∼− ∼ k ∧ 8 Φ(r) r and B α−1 dφ dt. Note that the nontrivial asymptotic B field will affect the spectrum of winding states. In our application we will consider a closely related background with a B field whose (φ, t) component vanishes asymptotically, so that the asymptotic background is the ordinary KK (or SS) linear dilaton. The background (2.13) actually corresponds to an infinite cover of the coset (2.12).Forthe single cover t is periodic with t ∼ t + 2π. We can define the Nth cover by replacing t → Nt, and the infinite cover corresponds to N →∞with noncompact time [54]. The spectrum of this background is most easily obtained by T-dualizing along t [54]. Since T-duality of a timelike coordinate is problematic we will define it by T-dualizing the Euclidean geometry with α<0, and analytically continuing to α>1.9 The T-dual background is given by10 2 2 dt dt ds 2 = k dr2 + tanh2 r dφ − − α , kN kN − =− Φ Φ0 log cosh r, (2.14) and a trivial B field. Redefining φ˜ = φ − t /(kN) and t˜ = t /(kN), we see that this is basically a cigar × S1, but with the identifications 2πn 2πn (φ,˜ t˜ ) ∼ φ˜ + 2πn− , t˜ + ,n,n∈ Z. (2.15) kN kN

This is the orbifold (cigar ×U(1)−αk)/ZkN . The spectrum is obtained by tensoring the cigar part of the spectrum {j,n˜φ, w˜ φ} with the U(1)−αk part, labeled by momentum and winding numbers {˜ ˜ } ∈ Z Z nt , wt , including twisted sectors labeled by γ kN , and projecting onto the kN invariant sector. The invariant states must satisfy ˜ =˜ + ∈ Z nt nφ kNp, p . (2.16) The twist γ corresponds to a fractional winding in φ˜, as can be seen from the relations between the winding numbers in the φ˜, t˜ and φ, t coordinates

1 1 w˜ = w − w , w˜ = w , (2.17) φ φ kN t t kN t = and from γ wt mod kN.

3. The decay of the KK linear dilaton

3.1. The background

1,1 1 Our initial background is R × S × MD−3 with a linear dilaton 2 = 2 + 2 − 2 + 2 ds dx1 dx2 dt dsD−3, Φ − Φ0 =−Qx1, (3.1)

8 The coefficient for the dilaton in (2.13) was chosen so that the dilaton is independent of α in the limit r →∞. 9 Alternatively, we can perform T-duality with respect to the coordinate it. 10 This differs by a sign from [54]. O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 143 where x2 ∼ x2 + 2πR. We will consider both the bosonic string and type II strings. For type II strings we impose antiperiodic boundary conditions on the fermions, i.e., SS compactification, so that spacetime supersymmetry is broken. The three-dimensional part of the background given by (x1,x2,t)corresponds to a free linear-dilaton CFT 1 L = (∂x )2 + (∂x )2 − (∂t)2 + QRxˆ , (3.2) 4π 1 2 1 with a central charge c = 3 + 6Q2 for the bosonic string, and c = 9/2 + 6Q2 for type II strings. 2 The (D − 3)-dimensional space MD−3 corresponds to a CFT with c = 23 − 6Q for the bosonic 2 string, and to a SCFT with c = 21/2 − 6Q for type II strings.√ For example this can be a sub- D−3 critical bosonic string theory with MD−3 = R and√Q = (26 − D)/√6, or critical bosonic or D−6 3 type II strings with MD−3 = R × S and Q = 1/ N + 2(Q = 1/ N in type II), where N is the level of the SU(2) current algebra which defines the S3. In the type II case this is the CHS model [55], which corresponds to the near-horizon geometry of N extremal NS5-branes [56]. The spectrum of local operators in the oscillator ground state is given by − O ∼ ip0t O O e a,p2 D−3, (3.3) O O M where a,p2 is the same as (2.8), and D−3 is a ground state local operator in D−3.Bulk states, as before, correspond to operators with a =−Q + ip1. Operators with a ∈ R correspond to localized (or more properly semilocalized) states, which in the absence of a potential are always nonnormalizable. The background described by (3.1) is not perturbatively well defined since the string coupling gs = exp(Φ) grows exponentially, and there is a strong coupling singularity (in the Einstein frame metric) at x1 →−∞. The usual way in which this problem is evaded is by turning on a potential, corresponding to a nonnormalizable state, that prevents strings from propagating into the strong coupling region.

3.2. A semilocalized tachyon

Consider the localized operator with the sine-Liouville potential at an arbitrary radius R (and b =−1/Q), − O ∼ x1/Q a,p2 e cos Rx2. (3.4) This can be thought of as a particular mode of the winding number one ground state of the string. In type II string theory the SS periodicity conditions imply that the ground state is projected out at even winding number, but kept at odd winding number. We can compute the mass of this state by requiring that the total scaling dimension be 1 in the bosonic case and 1/2 in the type II case. Using Q2 = 1/(k − 2) for the bosonic string and Q2 = 1/k for the type II string (since we will want to relate this to the (deformed) cigar) we ﬁnd M2 = R2 − k. (3.5) √ The state associated with the sine-Liouville potential is therefore tachyonic for R< k, indicating that the KK (or√ SS in the type II case) linear dilaton background becomes perturbatively unstable below R = k.11

11 Of course the bosonic string is perturbatively unstable for all R due to the winding number 0 tachyon. We are ignoring this mode here. 144 O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155

The semilocalized tachyon (3.4) describes a particular non-uniform mode of the closed string tachyon, in√ which the tachyon condenses in the strong coupling regime x1 < 0. At the critical radius R = k this mode is exactly marginal, and the deformed CFT which describes the corresponding condensate is given by the sine-Liouville model (2.6) tensored with R × MD−3. The semilocalized tachyon background provides the potential which prevents strings from propagating into the strong coupling region. In this sense it effectively eliminates that part of the spacetime in which it condenses, which includes the singularity at x1 →−∞. A more precise spacetime picture will emerge from the dual cigar CFT in the next section. √ 3.3. Semiclassical instability at R = k √ FZZ duality implies that the semilocalized tachyon condensate at the critical radius R = k in the KK linear dilaton background is equivalent to the background given by cigar × R × MD−3: 2 = 2 + 2 2 − 2 + 2 ds k dr tanh rdφ dt dsD−3, Φ − Φ0 =−log cosh r. (3.6) This provides a clear spacetime interpretation of the semilocalized tachyon condensate, whereby the half of spacetime containing the strong coupling region is excised. But what is even more intriguing is that in the dual picture the background is actually a type of “bubble of nothing”. To see this we introduce a new radial coordinate − − − 2 ρ ≡ e 2(Φ Φ0)/(D 2) = (cosh r)D−2 (3.7) and change to the Einstein frame − 2 2 D 2 k 2 2 2−D 2 2 2 2 2 ds = dρ + kρ 1 − ρ dφ − ρ dt + ρ ds − . (3.8) E 2 1 − ρ2−D D 3 The size of the φ-circle shrinks to zero as ρ → 1, so the region 0 ρ<1 is excised from the space, forming a static “bubble of nothing”. The usual expanding bubble picture is obtained by changing to the coordinates √ (D − 2) k 2t u = ρ cosh √ , 2 (D − 2) k √ (D − 2) k 2t v = ρ sinh √ , (3.9) 2 (D − 2) k in terms of which the ρ,t part of the metric is asymptotically flat. In these coordinates the surface of the bubble is at u2 − v2 = (D − 2)2k/4. The bubble appears at time v = 0, and expands with time v at a speed approaching the speed of light (Fig. 3). This strongly suggests that the spacetime described by (3.6) can be identified as√ the post-decay spacetime resulting from a bounce in the KK linear dilaton background at R = k. The bounce is described by the Euclidean geometry 2 = 2 + 2 2 + 2 + 2 ds k dr tanh rdφ dτ dsD−3, (3.10) where τ = it. Single-valuedness in the (u, v) plane requires a definite periodicity for τ , √ τ ∼ τ + π(D − 2) k. (3.11) O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 145

Fig. 3. Bubble nucleation and expansion.

Fig. 4. Two copies of the cylinder decay into two cigars.

The bubble corresponds to the Lorentzian√ continuation of the Euclidean geometry at the turning points τ1 = 0 and τ2 = π(D − 2) k/2(Fig. 3). Note that unlike the ordinary KK bubble [1] the Lorentzian geometry here actually consists of two identical disconnected components. We should therefore think of the original background as two copies of the KK linear dilaton, each of which decays into one of these components (Fig. 4). √ 3.4. Generalization to R> k √ For R> k the sine-Liouville operator (3.4) is irrelevant, corresponding to a massive state. In this case the KK linear dilaton background is perturbatively stable. On the other hand we can deform√ the three-dimensional cigar × R part of the spacetime (3.6) to a spacetime (2.13) with R> k, which suggests a generalization of the cigar bounce instability. We are actually 146 O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 interested in a slight variation of the deformed background given by12 2 2 2 2 α tanh r 2 N 2 2 ds = k dr + dφ − dt + ds − , α − tanh2 r α − tanh2 r D 3 − α − 1 e2(Φ Φ0) = , cosh2 r(α − tanh2 r) tanh2 r 1 B = Nk − dφ ∧ dt, (3.12) α − tanh2 r α − 1 where φ ∼ φ + 2π and t ∼ t + 2π. As we explained in the previous section, this corresponds to the N-fold cover of the coset (SL(2, R) × U(1))/U(1). We will eventually take N →∞to get noncompact time. We have also added a constant term to the (φ, t) component of the B field so →∞ R × 1 × 1 × M that it vanishes at r . This background asymptotes to S St D−3 with a linear 2 = − 2 = 2 − dilaton, R kα/(α 1) and Rt kN /(α 1). The Einstein metric is given by D − 2 2 (α − 1)2k ds2 = dρ2 E 2 (α − 1 − αρ 2−D)(α − 1 − ρ2−D) αk − + ρ2 α − 1 − αρ 2 D dφ2 (α − 1)2 2 kN 2 2−D 2 2 2 − ρ α − 1 − ρ dt + ρ ds − , (3.13) (α − 1)2 D 3 where −2/(D−2) − − − (α − 1) ρ ≡ e 2(Φ Φ0)/(D 2) = , (3.14) cosh2 r(α − tanh2 r) so the surface of the bubble is now at ρ =[α/(α − 1)]1/(D−2). We can again define 2d asymptotically flat coordinates √ (D − 2) k 2Nt u = ρ cosh √ , 2 (D − 2) α − 1 √ (D − 2) k 2Nt v = ρ sinh √ , (3.15) 2 (D − 2) α − 1 such that the surface of the bubble expands in time v. The bounce is given by the Euclidean continuation of (3.12) with τ = iNt, 2 2 2 α tanh r 2 1 2 2 ds = k dr + dφ + dτ + ds − , α − tanh2 r α − tanh2 r D 3 tanh2 r 1 B =−ik − dφ ∧ dτ, (3.16) α − tanh2 r α − 1 and the same dilaton. Single-valuedness in the (u, v) coordinates now requires a periodicity √ (D − 2) α − 1 τ ∼ τ + 2πβ, β ≡ . (3.17) 2

12 We are suppressing the components of the B field on the S3. O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 147 √ Note that the asymptotic radius in the τ direction is given by Rτ = (D − 2) k/2, independent of α. Here the continuation makes B complex (the S3 part is real, and the φ,t part is imaginary), but this is acceptable in a Euclidean configuration [57–59]. We therefore√ claim that (3.12) corresponds to the post-decay background of the KK linear dilaton for R> k. Note that the Euclidean√ geometry (3.12) with α<0 is also√ asymptotic to a cylinder, this time with radius R = k|α|/(|α|+1), which is smaller than k. Upon analytically continuing to Lorentzian√ signature by replacing t → it, this seems to give a deformation of the cigar to R< k, and therefore a bounce instability in the regime where the semilocalized state (3.4) is tachyonic. Now, however, the Lorentzian background has a complex B field,√ so this is not a physical background. This is consistent with our expectation that for R< k there should not be a bounce instability.

3.5. Classical decay rate

The classical rate of decay is estimated as exp(−I), where I is the Euclidean action of the bounce [60]. To be specific we will consider the cases of the bosonic and type II strings on R1,1 × S1 × S3 × T D−6, where the (D − 6)-dimensional part is compactified on a torus with 3 volume VD−6 in order to have a finite action. The radius of the S isgivenbythelevelN of the SU(2), which for large k is N ∼ k. The Euclidean metric which defines the bounce is given in this case by

− α tanh2 r 1 D 1 ds2 = k dr2 + dφ2 + dτ2 + dΩ2 + dy2, (3.18) α − tanh2 r α − tanh2 r 3 i i=D−6

3 where φ ∼ φ + 2π, τ ∼ τ + 2πβ, and dΩ3 is the volume form of S . The total Euclidean three form ﬁeld strength is given by

2kα tanh r sech2 r H3 = 2kdΩ3 + i dr ∧ dφ ∧ dτ. (3.19) (α − tanh2 r)2 Since there are no background RR ﬁelds in the type II case the bulk part of the action is given by √ 1 − 1 I =− dDx Ge 2Φ R + 4|∂Φ|2 − |H |2 (3.20) bulk 2κ2 2 · 3! 3 for both the bosonic and type II strings. In evaluating the Euclidean action for the bounce two kinds of boundary terms need to be included. The ﬁrst is the Gibbons–Hawking term [61,62], which cancels the second derivative terms in the Einstein action,

1 − I =− dD 1x h K − K(0) . (3.21) GH κ2 E E E

The subscript E denotes that these quantities are deﬁned with respect to the Einstein frame. hE is the induced metric on the boundary at r →∞, KE is the extrinsic curvature of the boundary √ ∂ h = rr √r E KE GE , (3.22) hE 148 O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155

(0) and KE is the extrinsic curvature of the boundary in the background metric (the metric for the asymptotic geometry). The second boundary term comes from the Euclidean action of the back- 13 ground metric Ibg, which serves as a regularization term (see for example [63]). The regularized Euclidean action is thus

I = Ibulk − Ibg + IGH. (3.23) The computation is simplified somewhat using the equations of motion, which give 5 R + 4|∂Φ|2 = |H |2. (3.24) 12 3 The different terms are given by 4 − 2 ∞ 8π (D 2)VD−6k α 2 α Ibulk =− cosh r − , (3.25) κ2 α − 1 α − tanh2 r 0 2π4(D − 2)V − k2 α ∞ I =− D 6 e2r , (3.26) bg κ2 α − 1 −∞ 4π 4(2D − 3)V − k2 α α + 1 I = D 6 . (3.27) GH (D − 2)κ2 α − 1 α − 1 Putting it all together we find that the rate of decay is given by 4π4(D − 1)2V − k2 α α + 1 Γ ∼ exp − D 6 ∼ exp −cR 2R2 − k . (3.28) (D − 2)κ2 α − 1 α − 1 We would like to make two comments about this result. First, the decay rate vanishes in the limit α → 1, which corresponds to an infinite radius. This is consistent with the fact that the background is stable in this limit; for type II strings the background becomes supersymmetric, and therefore absolutely stable, and for the bosonic string it is stable up to the usual winding number 0 tachyon.14 √ Second, in the limit α →∞, corresponding to the critical radius R = k, the decay rate is still exponentially small. This is somewhat surprising since this is the point where the semilocalized tachyon appears, and we would expect an O(1) decay rate. The same thing happens in the decay of the ordinary KK vacuum [1], namely the decay rate remains exponentially small at the critical radius. But there the semiclassical description breaks down at the critical radius, which is R ∼ O(1), whereas in our case the semiclassical picture continues to hold as long as k 1. We will come back to this point in the next section.

4. Tachyon bounce

The picture that emerges is that the√ KK linear dilaton background is unstable to the decay of half of spacetime for√ all R.ForR< k the instability is described by a semilocalized tachyonic mode, and for R> k it corresponds to a gravitational bounce. The two descriptions meet at

13 That this is a boundary term follows from the equations of motion. For the same reason Ibulk also gets a contribution only from the boundary. 14 Note that the two limits r →∞and α → 1 do not commute. The limit α → 1of(3.12) with r finite is AdS3 ×MD−3. Here we first take r →∞and then α → 1, which corresponds to our interpretation of the α → 1 limit as the post-decay 1,1 1 geometry for R × S × MD−3 with R →∞. O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 149 √ R = k, where they are related by FZZ duality. The situation is analogous to the λφ4 field theory model with λ<0. The φ = 0 state is unstable to decay via rolling for m2 < 0, and via tunneling for m2 > 0. In the related setting of finite temperature string theory, Barbon and Rabinovici have argued that at temperatures below the Hagedorn temperature, the background would decay into a bubble of nothing by a tunneling process of the massive thermal tachyon [33]. This can be put on firm footing in the context of the KK linear dilaton.√ Consider first the critical radius R = k. In this case the dual descriptions of the post-decay background are the cigar-bubble (3.6) on one side, and the static tachyon configuration √ − ∼ x1/Q T e cos( kx2) (4.1) on the other. For large x1 we can express this configuration in terms of the asymptotically flat coordinates (u, v) as

−(D−2) √

∼ 2 − 2 2Q2 T u v cos( kx2), (4.2) which is a time-dependent conﬁguration. The bounce corresponds to v → ivE

−(D−2) √

∼ 2 + 2 2Q2 TE u vE cos( kx2). (4.3)

This is a tunneling process. At vE →−∞the field TE vanishes, and at the turning point vE = 0it takes its maximal value. This is where the Lorentzian evolution (4.2) takes over, and the field T proceeds to increase with time v. In this case the field is actually massless, so apparently this describes some sort of barrier-less tunneling (see [64] for field theory examples of barrier-less tunneling). This is also consistent with the fact that the classical rate of decay (3.28) was exponentially suppressed√ even at the critical radius. For R> k the post-decay background is given by (3.12). Since in the T-dual picture this can again be described as a cigar, modulo a certain identification (see Appendix A), we can use FZZ duality again to get a dual tachyon configuration √ − ∼ x1/Q ˜ T e cos( k x2), (4.4) √ ˜ ˜ where the coordinate x˜2 is related asymptotically to the cigar coordinate φ as x˜2 ∼ k φ.Us- ing (A.2) we can express this in terms of the asymptotic flat KK linear dilaton (and time) coordinates as − αk k T ∼ e x1/Q cos x − x , (4.5) α − 1 2 α − 1 0 √ 2 where x2 ∼ Rφ = kα/(α − 1)φ, x0 ∼ Rt t = kN /(α − 1)t. In terms of the (u, v) coordinates this becomes −(D−2) αk D − 2 k − v T ∼ u2 − v2 2Q2 cos x − √ tanh 1 . (4.6) α − 1 2 2 α − 1 u The Euclidean continuation is complex −(D−2) − 2 2 2 D 2 k −1 vE TE ∼ u + v 2Q cos Rx − i √ tan . (4.7) E 2 2 α − 1 u 150 O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155

This can also be seen in the original deformed background (3.12), where the B ﬁeld becomes complex after Wick-rotation, or in the T-dual background (A.1), where the metric becomes complex. The bounce in this case is not time-reversal symmetric, since the imaginary part of (4.7) is odd under time reversal. This is similar to a system with friction. The friction term in the action breaks time-reversal symmetry, and in the Euclidean theory it becomes imaginary, rendering the solution complex. The post-tunneling evolution is given by analytically continuing the Euclidean solution at vE = 0 to the Lorentzian solution (4.6) at v = 0. An unusual property in this case is that the evolution begins with a nonzero velocity, which is associated with the imaginary part of the Euclidean solution.15

5. Where is the negative mode? √ We have seen that the background (3.12) is asymptotic to the KK linear dilaton with R k, and we have computed the action of its Euclidean continuation (3.16). However a crucial requirement for the Euclidean background to correspond to a bounce is the existence of a negative mode in its spectrum. This is because the rate of decay including the one-loop contribution is [65] 1/2 det Δ − Γ ∼ bounce e I , Im 1/2 (5.1) det Δ0 where Δbounce and Δ0 are the quadratic forms of the ﬂuctuations in the bounce and in the Euclidean continuation of the pre-decay background, respectively. So we need to determine if the Euclidean background (3.16) possesses a negative mode, which the Euclidean continuation of the KK linear dilaton background does not. A negative mode corresponds to an operator with (holomorphic) scaling dimension less than 1 in the bosonic case, and less than 1/2 in the type II case. The spectrum of the Euclidean background can be determined formally as in the Lorentzian case (see Appendix A), by relating it via T-duality along τ to a cigar. The resulting scaling dimensions can be read off from (A.7) by replacing →− → nt iβnτ /N, wt iwτ N/β, (5.2) which gives 2 2j(j + 1) α − 1 αk k(D − 2) 2 Δ + Δ¯ =− + n2 + w2 + n 2 + w 2. k − 2 2αk φ 2(α − 1) φ 8 τ k(D − 2)2 τ (5.3) For the bulk states, i.e. the states with j =−1/2 + is, this is the correct thing to do. In particular this gives normalizable states with real scaling dimensions. This is exactly the same spectrum of bulk states as in the Euclidean continuation of the pre-decay KK linear dilaton background, and so will not contribute in (5.1). The required negative mode must therefore come from a localized state. The problem is that the replacements (5.2) generally give complex scaling dimensions for the localized states, as can be inferred from the relations (A.9), (A.5) and (A.6).16

15 A similar thing happens in the instanton process representing the Schwinger pair production in the presence of constant electric and magnetic ﬁelds. The instanton solution in that case is also complex, and the electron–positron pair are created with a nonzero velocity due to the magnetic ﬁeld. 16 A similar thing happens to the discrete series in the Euclidean continuation of AdS3 [66].TheSL(2, R) quantum numbers are related as m =±(j + l) and m¯ =±(j + l)¯ ,wherel,l¯ ∈ Z+. On the other hand m and m¯ are related to the O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 151

We have not resolved this problem, and it is not completely clear to us whether and how the negative mode appears in the spectrum of localized states. However let us outline a number of proposals which may lead to a resolution.

= = (1) States with trivial momentum and winding in τ (nτ wτ 0) have real scaling dimensions. For α 1, and j taking its maximal allowed value17 − k 4 + k for the bosonic string, j ∼ 2 2α (5.4) k−2 + k 2 2α for the type II string, the ground state with nφ = 0 and wφ = 1 has the scaling dimension − 1 − k 2 for the bosonic string, Δ ∼ 4α (5.5) 1 − k 2 4α for the type II string. This corresponds to a negative mode. In the cigar limit α →∞it becomes a zero mode, so the one-loop decay rate vanishes. We expect that higher loop corrections render the decay rate finite in this limit. (2) For α ∼ O(1) the above mode is positive, since the winding contribution to the scaling dimension grows indefinitely as α → 1, whereas the j dependent part is O(k). There can however be negative modes with wφ = 0 at the first excited level (in the bosonic string there are additional negative modes at the ground state). For example the state with nφ = k and j = (k − 2)/2inthe type II case has a scaling dimension k Δ = 1 − , (5.6) 4α which corresponds to a negative mode if α

6. Conclusions

The Kaluza–Klein compactiﬁcation of the bosonic string, and similarly the Scherk–Schwarz compactiﬁcation of type II strings, in a linear dilaton background, exhibits√ a semilocalized instability, described√ by a semilocalized√ winding tachyon when R k, and by a gravitational bounce when R k.AtR = k the tachyon condensate is described by the sine-Liouville model, and the bounce corresponds to the cigar CFT. The conjectured equivalence of the two

energy and angular momentum in AdS3 as m = (E + Pθ )/2andm¯ = (E − Pθ )/2. The replacement E → iE therefore makes j, and with it the scaling dimension, complex for these states. 17 More precisely, one needs α>k 1 for the value of j not to exceed the upper bound. For smaller α, one must choose larger l in (A.9).Foragivenl, α has to be greater than k/(2l − 1). 152 O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 models provides the first explicit connection between a tachyonic instability and a semiclassical instability, and implies that the tachyon condensate (at the critical radius) corresponds√ to a smooth semiinfinite truncation of space. The geometrical picture extends to R> k, where the bounce corresponds to a three-dimensional deformation of the√ cigar. In this paper we have not discussed much the regime R< k, where the decay proceeds via a condensation of the semilocalized tachyon. The dynamical evolution of the condensation√ will be described by a rolling tachyon. The tachyon profile can√ be guessed from the√ R> k case (4.5). − ∼ x1/Q − − − + We propose that it takes the form T λe cos( αk/(α 1)x2 k/(α 1)x0) c.c. with α<0. However, this represents an s-brane-like√[67] process rather than a rolling process. To obtain a rolling process, one shifts x0 → x0 +log λ/ k/(|α|+1) and then takes the limit λ → 0. We believe that T is not only marginal but an exactly marginal operator, and the corresponding CFT is tractable by properly extending the results of [68]. It is not clear if FZZ duality extends to this case. However, it is tempting to speculate that the rolling tachyon does have a spacetime dual description where the spacetime is time dependent and consists of two copies of a cigar whose tips recede as time goes by. The entire spacetime eventually disappears in the infinite future, which is where the tachyon condensate becomes infinite.

Acknowledgements

We would like to thank Amit Giveon, Carlos Herdeiro, Dan Israel and Harald Nieder for useful discussions. This work is supported in part by the Israel Science Foundation under grant No. 101/01-1.

Appendix A. The post-decay spectrum

The spectrum of our deformed cigar background (3.12) can be determined along the lines of [54], by T-dualizing along t. The T-dual background is given by 2 2 2 2 2 αdφ dt dt dφ 2 ds = k dr + tanh r − − α − + ds − , α − 1 kN kN α − 1 D 3 − 2(Φ −Φ ) 1 α e 0 = , (A.1) kN2 cosh2 r and a trivial B field. This differs a bit from the T-dual background in Section 2.3 (2.14) due to the difference in the original B field, but it can again be related to a cigar × U(1) CFT by defining

αφ t t φ φ˜ ≡ − , t˜ ≡ − , (A.2) α − 1 kN kN α − 1 which now have the identiﬁcations 2παn 2πn 2πn 2πn (φ,˜ t˜ ) ∼ φ˜ + − , t˜ + − ,n,n∈ Z. (A.3) α − 1 kN kN α − 1 With these identiﬁcations the background does not correspond to a simple orbifold of the cigar × U(1) CFT, but we can still determine the spectrum by imposing invariance of the cigar × U(1) spectrum. By FZZ duality we know that the vertex operators are asymptotically given by

−2(j+1)r i(n˜ +kw˜ )φ˜ +i(n˜ −kw˜ )φ˜ −i(n˜ +kαw˜ )t˜ −i(n˜ −kαw˜ )t˜ e e φ φ L φ φ R e t t L t t R , (A.4) O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 153

˜ ˜ ˜ ˜ ˜ where nφ and wφ are the KK momentum and winding number, respectively, in φ, and nt and wt are the KK momentum and winding number in t˜ . Invariance under the identiﬁcations in (A.3) requires

kNn αkNn n˜ = n − t , n˜ =−n + t , (A.5) φ φ α − 1 t φ α − 1 ∈ Z where nφ,nt . As we shall soon see nφ and nt will be identiﬁed as the KK momenta in φ and t , respectively. From (A.2) we see that the winding numbers are related as

α 1 1 1 w˜ = w − w , w˜ = w − w . (A.6) φ α − 1 φ kN t t kN t α − 1 φ

Note in particular that in the limit α →∞this reduces to the ZkN orbifold, where the twist is ∈ Z identiﬁed with wt mod kN (2.17).Soforwt ,wφ we believe this describes the complete spectrum of the theory for any α>1 (including the twisted sectors). The total scaling dimension in the cigar × U(1) is given by (in the bosonic string) 2 2j(j + 1) α − 1 αk kN α − 1 Δ + Δ¯ =− + n2 + w2 − n 2 − w 2 (A.7) k − 2 2αk φ 2(α − 1) φ 2(α − 1) t 2kN2 t and − ¯ = − Δ Δ nφwφ nt wt . (A.8)

We can therefore identify nφ as the KK momentum in φ, and nt as the KK momentum in t , or equivalently as the winding number in the original t. So the compact part of√ the spectrum is 1 × 1 = − just what one would expect for the asymptotic S St geometry with R kα/(α 1) and 2 Rt = kN /(α − 1). The spectrum includes bulk and localized states. In terms of the cigar quantum numbers the bulk states correspond to j =−1/2 + is, and arbitrary n˜φ, w˜ φ, and the localized states correspond to real j in the range −1/2

n˜φ + kw˜ φ =±2(j + l) or n˜φ − kw˜ φ =±2(j + l), l ∈ Z+. (A.9)

References

[1] E. Witten, Instability of the Kaluza–Klein vacuum, Nucl. Phys. B 195 (1982) 481. [2] F. Dowker, J.P. Gauntlett, G.W. Gibbons, G.T. Horowitz, The decay of magnetic fields in Kaluza–Klein theory, Phys. Rev. D 52 (1995) 6929, hep-th/9507143. [3] M.S. Costa, M. Gutperle, The Kaluza–Klein Melvin solution in M-theory, JHEP 0103 (2001) 027, hep-th/0012072. [4] S.P. De Alwis, A.T. Flournoy, Closed string tachyons and semiclassical instabilities, Phys. Rev. D 66 (2002) 026005, hep-th/0201185. [5] A. Sen, Tachyon dynamics in open string theory, hep-th/0410103. [6] H. Yang, B. Zwiebach, A closed string tachyon vacuum?, hep-th/0506077; H. Yang, B. Zwiebach, Dilaton deformations in closed string field theory, JHEP 0505 (2005) 032, hep-th/0502161; H. Yang, B. Zwiebach, Testing closed string field theory with marginal fields, JHEP 0506 (2005) 038, hep- th/0501142. [7] E.J. Martinec, Defects, decay, and dissipated states, hep-th/0210231. [8] M. Headrick, S. Minwalla, T. Takayanagi, Closed string tachyon condensation: An overview, Class. Quantum Grav. 21 (2004) S1539, hep-th/0405064. [9] A. Adams, J. Polchinski, E. Silverstein, Don’t panic! Closed string tachyons in ALE space–times, JHEP 0110 (2001) 029, hep-th/0108075. 154 O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155

[10] C. Vafa, Mirror symmetry and closed string tachyon condensation, hep-th/0111051. [11] J.A. Harvey, D. Kutasov, E.J. Martinec, G.W. Moore, Localized tachyons and RG flows, hep-th/0111154. [12] Y. Okawa, B. Zwiebach, Twisted tachyon condensation in closed string field theory, JHEP 0403 (2004) 056, hep- th/0403051. [13] O. Bergman, S.S. Razamat, On the CSFT approach to localized closed string tachyons, JHEP 0501 (2005) 014, hep-th/0410046. [14] A. Adams, X. Liu, J. McGreevy, A. Saltman, E. Silverstein, Things fall apart: Topology change from winding tachyons, hep-th/0502021. [15] J. McGreevy, E. Silverstein, The tachyon at the end of the universe, JHEP 0508 (2005) 090, hep-th/0506130. [16] M. Berkooz, Z. Komargodski, D. Reichmann, V. Shpitalnik, Flow of geometries and instantons on the null orbifold, hep-th/0507067. [17] G.T. Horowitz, Tachyon condensation and black strings, JHEP 0508 (2005) 091, hep-th/0506166. [18] S.F. Ross, Winding tachyons in asymptotically supersymmetric black strings, hep-th/0509066. [19] D. Kutasov, Accelerating branes and the string/black hole transition, hep-th/0509170. [20] M.S. Costa, C.A.R. Herdeiro, J. Penedones, N. Sousa, Hagedorn transition and chronology protection in string theory, hep-th/0504102. [21] A. Strominger, T. Takayanagi, Correlators in timelike bulk Liouville theory, Adv. Theor. Math. Phys. 7 (2003) 369, hep-th/0303221. [22] V. Schomerus, Rolling tachyons from Liouville theory, JHEP 0311 (2003) 043, hep-th/0306026. [23] S. Hirano, Energy quantisation in bulk bouncing tachyon, JHEP 0507 (2005) 017, hep-th/0502199. [24] T. Suyama, On decay of bulk tachyons, hep-th/0308030; T. Suyama, Closed string tachyons and RG flows, JHEP 0210 (2002) 051, hep-th/0210054; T. Suyama, Target space approach to closed string tachyons, hep-th/0302010. [25] C.G. Callan, J.M. Maldacena, Brane dynamics from the Born–Infeld action, Nucl. Phys. B 513 (1998) 198, hep- th/9708147. [26] K. Hashimoto, Dynamical decay of brane–antibrane and dielectric brane, JHEP 0207 (2002) 035, hep-th/0204203. [27] A. Sen, Descent relations among bosonic D-branes, Int. J. Mod. Phys. A 14 (1999) 4061, hep-th/9902105. [28] J. Polchinski, L. Thorlacius, Free fermion representation of a boundary conformal field theory, Phys. Rev. D 50 (1994) 622, hep-th/9404008. [29] C.G. Callan, I.R. Klebanov, A.W.W. Ludwig, J.M. Maldacena, Exact solution of a boundary conformal field theory, Nucl. Phys. B 422 (1994) 417, hep-th/9402113. [30] V. Fateev, A. Zamolodchikov, Al. Zamolodchikov, unpublished. [31] V. Kazakov, I.K. Kostov, D. Kutasov, A matrix model for the two-dimensional black hole, Nucl. Phys. B 622 (2002) 141, hep-th/0101011. [32] T. Suyama, Deformation of CHS model, Nucl. Phys. B 641 (2002) 341, hep-th/0206171. [33] J.L.F. Barbon, E. Rabinovici, Touring the Hagedorn ridge, hep-th/0407236. [34] M. Kruczenski, A. Lawrence, Random walks and the Hagedorn transition, hep-th/0508148. [35] E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314. [36] G. Mandal, A.M. Sengupta, S.R. Wadia, Classical solutions of two-dimensional string theory, Mod. Phys. Lett. A 6 (1991) 1685. [37] S. Elitzur, A. Forge, E. Rabinovici, Some global aspects of string compactifications, Nucl. Phys. B 359 (1991) 581. [38] A. Giveon, D. Kutasov, Little string theory in a double scaling limit, JHEP 9910 (1999) 034, hep-th/9909110. [39] R. Dijkgraaf, H.L. Verlinde, E.P. Verlinde, String propagation in a black hole geometry, Nucl. Phys. B 371 (1992) 269. [40] O. Aharony, A. Giveon, D. Kutasov, LSZ in LST, Nucl. Phys. B 691 (2004) 3, hep-th/0404016. [41] D. Kutasov, D.A. Sahakyan, Comments on the thermodynamics of little string theory, JHEP 0102 (2001) 021, hep-th/0012258. [42] J.M. Maldacena, H. Ooguri, Strings in AdS3 and the SL(2,R) WZW model, I, J. Math. Phys. 42 (2001) 2929, hep-th/0001053; J.M. Maldacena, H. Ooguri, J. Son, Strings in AdS3 and the SL(2,R)WZW model, II: Euclidean black hole, J. Math. Phys. 42 (2001) 2961, hep-th/0005183. [43] T. Fukuda, K. Hosomichi, Three-point functions in sine-Liouville theory, JHEP 0109 (2001) 003, hep-th/0105217. [44] D. Kutasov, N. Seiberg, Noncritical superstrings, Phys. Lett. B 251 (1990) 67. [45] K. Hori, A. Kapustin, Duality of the fermionic 2d black hole and N = 2 Liouville theory as mirror symmetry, JHEP 0108 (2001) 045, hep-th/0104202. [46] D. Tong, Mirror on the wall: On two-dimensional black holes and Liouville theory, JHEP 0304 (2003) 031, hep- th/0303151. O. Bergman, S. Hirano / Nuclear Physics B 744 (2006) 136–155 155

[47] T. Eguchi, Y. Sugawara, Conifold type singularities, N = 2 Liouville and SL(2,R)/U(1) theories, JHEP 0501 (2005) 027, hep-th/0411041; T. Eguchi, Y. Sugawara, SL(2,R)/U(1) supercoset and elliptic genera of non-compact Calabi–Yau manifolds, JHEP 0405 (2004) 014, hep-th/0403193; T. Eguchi, Y. Sugawara, Modular bootstrap for boundary N = 2 Liouville theory, JHEP 0401 (2004) 025, hep- th/0311141. [48] C. Ahn, C. Kim, C. Rim, M. Stanishkov, Duality in N = 2 super-Liouville theory, Phys. Rev. D 69 (2004) 106011, hep-th/0210208. [49] J.H. Horne, G.T. Horowitz, Exact black string solutions in three dimensions, Nucl. Phys. B 368 (1992) 444, hep- th/9108001; J.H. Horne, G.T. Horowitz, A.R. Steif, An equivalence between momentum and charge in string theory, Phys. Rev. Lett. 68 (1992) 568, hep-th/9110065. [50] D. Israel, C. Kounnas, M.P. Petropoulos, Superstrings on NS5 backgrounds, deformed AdS3 and holography, JHEP 0310 (2003) 028, hep-th/0306053. [51] S.F. Hassan, A. Sen, Marginal deformations of WZNW and coset models from O(d,d) transformation, Nucl. Phys. B 405 (1993) 143, hep-th/9210121. [52] A. Giveon, E. Kiritsis, Axial vector duality as a gauge symmetry and topology change in string theory, Nucl. Phys. B 411 (1994) 487, hep-th/9303016. [53] S.K. Yang, Marginal deformation of minimal N = 2 superconformal field theories and the Witten index, Phys. Lett. B 209 (1988) 242. [54] D. Israel, D-branes in Lorentzian AdS3, hep-th/0502159. [55] C.G. Callan, J.A. Harvey, A. Strominger, Supersymmetric string solitons, hep-th/9112030. [56] O. Aharony, M. Berkooz, D. Kutasov, N. Seiberg, Linear dilatons, NS5-branes and holography, JHEP 9810 (1998) 004, hep-th/9808149. [57] J.D. Brown, E.A. Martinez, J.W. York, Complex Kerr–Newman geometry and black hole thermodynamics, Phys. Rev. Lett. 66 (1991) 2281; J.D. Brown, J.W. York, The path integral formulation of gravitational thermodynamics, gr-qc/9405024. [58] S.W. Hawking, S.F. Ross, Duality between electric and magnetic black holes, Phys. Rev. D 52 (1995) 5865, hep- th/9504019. [59] I.S. Booth, R.B. Mann, Complex instantons and charged rotating black hole pair creation, Phys. Rev. Lett. 81 (1998) 5052, gr-qc/9806015. [60] S.R. Coleman, The fate of the false vacuum, 1: Semiclassical theory, Phys. Rev. D 15 (1977) 2929; S.R. Coleman, Phys. Rev. D 16 (1977) 1248, Erratum. [61] G.W. Gibbons, S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752. [62] J.W. York, Role of conformal three geometry in the dynamics of gravitation, Phys. Rev. Lett. 28 (1972) 1082. [63] S. Ng, M. Perry, Brane splitting via quantum tunneling, Nucl. Phys. B 634 (2002) 209, hep-th/0106008. [64] K.M. Lee, E.J. Weinberg, Tunneling without barriers, Nucl. Phys. B 267 (1986) 181. [65] C.G. Callan, S.R. Coleman, The fate of the false vacuum, 2: First quantum corrections, Phys. Rev. D 16 (1977) 1762. [66] J.M. Maldacena, H. Ooguri, Strings in AdS3 and the SL(2,R) WZW model, III: Correlation functions, Phys. Rev. D 65 (2002) 106006, hep-th/0111180. [67] M. Gutperle, A. Strominger, JHEP 0204 (2002) 018, hep-th/0202210. [68] P. Baseilhac, V.A. Fateev, Expectation values of local fields for a two-parameter family of integrable models and related perturbed conformal field theories, Nucl. Phys. B 532 (1998) 567, hep-th/9906010. Nuclear Physics B 744 (2006) 156–179

Split extended supersymmetry from intersecting branes

Ignatios Antoniadis a,1, Karim Benakli b,∗, Antonio Delgado a, Mariano Quirós c, Marc Tuckmantel a,d

a TH-Division, CERN, CH-1211 Geneva, Switzerland b Laboratoire de Physique Théorique et Hautes Energies, Universités de Paris VI et VII, France c Institució Catalana de Recerca i Estudis Avançats (ICREA), Theoretical Physics Group, IFAE/UAB, E-08193 Bellaterra, Barcelona, Spain d Institut für Theoretische Physik, ETH Höngerberg, CH-8093 Zürich, Switzerland Received 5 January 2006; accepted 9 March 2006 Available online 3 April 2006

Abstract We study string realizations of split extended supersymmetry, recently proposed in [I. Antoniadis, et al. hep-ph/0507192]. Supersymmetry is broken by small () deformations of intersection angles of D-branes 2 ∼ 2 giving tree-level masses of order m0 Ms ,whereMs is the string scale, to localized scalars. We show through an explicit one-loop string amplitude computation that gauginos acquire hierarchically smaller D ∼ 2 Dirac masses m1/2 m0/Ms. We also evaluate the one-loop higgsino mass, μ, and show that, in the ∼ 4 3 absence of tree-level contributions, it behaves as μ m0/Ms . Finally we discuss an alternative suppression of scales using large extra dimensions. The latter is illustrated, for the case where the gauge bosons appear in N = 4 representations, by an explicit string model with Standard Model gauge group, three generations of quarks and leptons and gauge coupling uniﬁcation. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

Implementing the idea of split supersymmetry [1] in string theory is straightforward [2].The appropriate framework is type I theory [3] compactiﬁed in four dimensions in the presence of constant internal magnetic ﬁelds [4,5], or equivalently D-branes intersecting at angles [6,7] in the

* Corresponding author. E-mail address: [email protected] (K. Benakli). 1 On leave of absence from CPHT, Ecole Polytechnique, UMR du CNRS 7644.

T -dual picture. However, in simple brane constructions the gauge group sector comes in multiplets of extended supersymmetry, while matter states are in N = 1 representations. In Ref. [8] we showed that these economical string-inspired brane constructions reconcile with unification of gauge couplings at scales safe from proton decay problems, and provide a natural dark matter candidate. Indeed a simple way to break supersymmetry in the above context is by deforming the intersection angles of the Standard Model branes from their special values corresponding to a supersymmetric configuration. A small deformation of these angles by breaks supersymmetry via a D-term vacuum expectation value (VEV), associated to a magnetized Abelian gauge = 2 group factor in the T -dual picture, D Ms with Ms the string scale [4,9]. This leads to 2 ∼ 2 mass shifts of order m0 Ms for all charged scalar fields localized at the intersections, such as squarks and sleptons, while gauginos (and higgsinos) remain massless. Alternatively supersymmetry breaking can be communicated to the scalar observable sector by radiative corrections from a supersymmetric messenger sector, with D-breaking triggered by a magnetized Abelian subgroup, or a non-supersymmetric sector with large extra dimensions. In all cases all previously massless scalars in the observable sector are expected to acquire large masses by radiative corrections and a fine-tuning is needed in the Higgs sector in order to keep the hierarchy between the electroweak scale and m0, as required in split supersymmetry. On the other hand fermion (gaugino and higgsino) masses are protected by a chiral R- symmetry and the magnitude of radiative corrections depends on the mechanism of its breaking. In fact R-symmetry is in general broken in the gravitational sector by the gravitino mass but its value, as well as the mediation of the breaking to the brane (Standard Model) sector, is model dependent and brings further uncertainties. Here we will restrict ourselves to possible sources of fermion mass generation due to brane effects described by open string propagation within only global supersymmetry, assuming that gravitational (closed strings) corrections are negligible. Indeed, R-symmetry is in general broken by α-string corrections and gaugino Majorana masses can be induced by a dimension-seven effective operator which is the chiral F -term [10]: d2θ W2 Tr W 2, where W and W are the magnetic U(1) and non-Abelian gauge superfields, respectively. Its moduli dependent coefficient is given by the topological partition function F(0,3) on a world-sheet with no handles and three boundaries.2 From the effective field theory point of view, it corresponds to a two-loop correction involving massive open string states. Upon a VEV W=θD, the above F -term generates Majorana gaugino masses that are hierarchically M ∼ 4 3 smaller than the scalar masses and behave as m1/2 m0/Ms . In models where the gauge bosons come in multiplets of extended supersymmetry, there exists the possibility of generating Dirac gaugino masses that do not require the breaking of R-symmetry. Such a mass can be induced at one-loop via the effective chiral dimension-five operator [8,11]: d2θ W Tr(WA), where A denotes the N = 1 chiral superfield(s) containing the additional gaugino(s). Upon the D-auxiliary VEV W this term generates Dirac gaugino D ∼ 2 M masses that scale as m1/2 m0/Ms , and are thus much higher than the Majorana masses m1/2. In Ref. [8], we studied the renormalization group evolution and showed that this scenario is compatible with one-loop gauge coupling unification at high scale for both cases where the gauge sector is N = 2 and N = 4 supersymmetric. The low energy sector of these models contains, besides the Standard Model fields, just some fermion doublets (higgsinos) and eventually two singlets, the binos, if the corresponding cor-

2 Two of them correspond to W and W gauge groups while the third one can be an orientifold. 158 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 rections to their Dirac mass vanish. In fact, the higgsinos must acquire a mass, μ, of order the electroweak scale in order to provide a dark matter candidate. This can be induced by the fol- 2 2 ¯ 2 ¯ ¯ lowing dimension-seven operator, generated at one-loop level [8,10]: d θ W D H1H2, where H1,2 are the two N = 1 Higgs supermultiplets. It follows that the induced higgsino mass is of the ∼ M ∼ 4 3 same order as the gaugino Majorana masses, μ m1/2 m0/Ms . The appearance of gauginos in multiplets of extended supersymmetry is common in previous attempts to embed the Standard Model in intersecting brane constructions [12]. There are several examples in the literature of such models with gauge sectors forming multiplets of either N = 4 [13] or N = 2 extended supersymmetry [14]. In this work we describe the general string framework realizing the above scenario of split supersymmetry with extended supersymmetric gauge sector and perform an explicit one-loop computation of the dimension-five and seven effective operators needed to produce Dirac gaugino and higgsino masses, respectively. The relevant world-sheet diagram is the annulus involving two D-brane stacks on its boundaries (or three in the case of higgsinos). We find that both Dirac gaugino and higgsino masses are in general non-vanishing when the two brane-stacks are parallel in one of the three internal compactification planes. The leading behavior in the supersymmetric limit, m0/Ms → 0, gives the coefficient of the corresponding effective operator, which thus receives non-trivial contributions only from N = 2 supersymmetric sectors. Moreover, we find that in this limit the result simplifies and becomes topological; the non-zero mode determinants cancel and the effective couplings depend only on the momentum lattice of the plane where the two brane-stacks are parallel. In the higgsino case the two fermions should come from an N = 2 supersymmetry preserving intersection, localized in the remaining two internal planes. Finally, for concreteness, we present an explicit string construction with the Standard Model gauge group, precisely three generations of quarks and leptons, and sharing the desired features described above. Moreover, it emerges from an SU(5) grand unified group and thus satisfies gauge coupling unification, realizing a particular D-brane configuration proposed in Ref. [2]. Standard Model particles live in the intersection of supersymmetric branes while there is a non- supersymmetric brane such that particles living in its intersection with the observable branes are non-chiral and act as messengers of gauge mediated supersymmetry breaking. In this case the hierarchy between the string scale and the masses of supersymmetric partners can be triggered by extra dimensions hierarchically larger than the string length. Our paper is organized as follows. In Section 2, we describe the general framework of supersymmetry breaking in D-brane models intersecting at angles. In Section 3, we perform the one-loop computation of the induced Dirac gaugino masses and extract the coefficient of the relevant dimension-five effective operator by evaluating the behavior in the supersymmetric limit. In Section 4, we perform a similar computation for the higgsino masses and the corresponding dimension-seven effective operator. In Section 5, we present an explicit construction of the Stan- dard Model spectrum in this framework with the desired features. Our conclusions are drawn in Section 6 and finally some relevant formulae are presented in Appendix A and some technical computational details about the bosonic correlation function of higgsinos in Appendix B.

2. Supersymmetry breaking from intersecting branes

We will consider a set of intersecting stacks of branes that can be divided into two subsets: the ﬁrst one, denoted as O (for observable), gives rise in its light spectrum to the observable sector, i.e. a supersymmetric version of the Standard Model with the gauge sector in N = 2or I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 159

Fig. 1. Rectangular (left panel) and tilted (right panel) tori.

N = 4 representations. The second subset, which we denote as M (for messenger), provides the supersymmetry breaking messengers through its intersections with the branes in O. In order to perform explicit computations, we consider a compactiﬁcation on a six torus fac- 2 ⊗ 2 ⊗ 2 torizable as T1 T2 T3 with appropriate projections and orientifold planes to provide the desired supersymmetric framework. A basis cycles [a(i)] and [b(i)] of the corresponding homol- 2 ogy classes is deﬁned for every torus Ti . Every stack a of D6-branes in type IIA wraps a 3-cycle [Πa] factorizable into the product of 1-cycles: [ ]= (i) = (i) (i) + (i) (i) Πa Πa na a ma b (2.1) i i and forms angles3 with the cycles [a(i)] given by:

(i) (i) (i) (i) (i) ma R /R + na cot ϕ θ (i) = 2 1 tan a (i) (2.2) na

(i) (i) (i) (i) (i) (i) where R1 and R2 are the radii along the horizontal X (a -cycles) and vertical Y (b - (i) 2 cycles) axes, respectively, and ϕ is the angle of the tilted torus Ti . In the presence of an orientifold plane along the X(i)-axis, the angle ϕ(i) is ﬁxed to either (i) = (i) = (i) (i) cot ϕ 0, which corresponds to rectangular tori, or to cot ϕ R2 /2R1 which corresponds (i) to tilted tori (see Fig. 1). In these cases, we can write the angles θa as

m˜ (i) R(i) θ (i) = a 2 , tan a (i) (i) (2.3) na R1 (i) (i) (i) where we deﬁne m˜ a = ma + bina and bi = 0(bi = 1/2) for rectangular (tilted) tori. From (i) here on and for simplicity we will remove the tilde from m˜ a . Given a generic couple (a, b) of (i) stacks of branes they intersect with angles πθab in the ith torus:

(i) (i) θa − θ θ (i) = b . (2.4) ab π

3 (i) Here we choose −π/2 θa π/2. 160 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179

Of special importance is the number of such intersections as it measures the number of chiral fermions. It is given by =[ ]·[ ]= (i) (i) − (i) (i) Iab Πa Πb na mb ma nb (2.5) i and it corresponds in the T -dual picture to the index theorem for compactification with internal magnetic fields. In the special case where the brane stack b is the image of a under the orientifold action (b = a∗), the chiral states given by this formula fall in two categories: they transform either in the antisymmetric (A)orinthesymmetric(S) representation of the gauge group, due to the orientifold projection. Their respective multiplicities are given by: A,S (i) 1 (j) I ∗ = 2m n ∓ 1 . (2.6) aa a 2 a i j In simple toroidal compactifications, states arising from open strings with both ends on the same stack of branes form representations of N = 4 supersymmetry. For more complicated compactifications, part of these states can be projected out and one remains with representations of lower supersymmetry. In this work, we will focus on the cases where these states are in N = 4 or N = 2 multiplets as already stated. Open strings stretching between two non-parallel stacks of Na and Nb branes give rise to states ¯ in the bifundamental representation (Na, Nb) of U(Na) ⊗ U(Nb). The lightest modes contain, in addition to massless fermions, scalars with masses:

−|θ (1)|+|θ (2)|+|θ (3)| |θ (1)|−|θ (2)|+|θ (3)| m2 = ab ab ab M2,m2 = ab ab ab M2, ab,1 2 s ab,2 2 s

|θ (1)|+|θ (2)|−|θ (3)| |θ (1)|+|θ (2)|+|θ (3)| m2 = ab ab ab M2,m2 = 1 − ab ab ab M2. (2.7) ab,3 2 s ab,4 2 s For parallel branes the ﬁrst three states become the N = 4 scalar partners of the gauge vector bosons. On the other hand, if some of them are massless there is a boson–fermion degeneracy (i) that indicates that part of the original supersymmetry is preserved. Moreover, if any of the θab angles vanishes, a supersymmetric mass iMs can be generated through separation of the branes −1 by a distance iMs in the corresponding torus. Finally, the last mass in (2.7) contains the contribution of a massive string oscillator that could become the lightest state for some values of the angles. There are three cases corresponding to three different kinds of intersections that must be discussed:

• The case where branes a and b are in the observable set O with intersections giving rise to chiral N = 1 multiplets. The lightest states φab are identiﬁed with quarks, leptons and their supersymmetric partners. This can be achieved with the choice

2 2 2 2 m =|γ2|M ,m=|γ1|M , ab,1 s ab, 2 s 2 = 2 = −| | 2 mab,3 0,mab,4 1 γ3 Ms (2.8) which is such that for generic values of γ1 and γ2 the system only preserves one supersymmetry. (i) = | |=| |+| | For simplicity, we are using here the notation θab γi and choosing γ3 γ1 γ2 . I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 161

• The case where branes a and b are in the observable set O, and states in their intersection give rise to N = 2 supermultiplets which are identiﬁed with supersymmetric pairs of Higgs doublets. This corresponds to Eq. (2.8) with γ1 = 0, i.e. 2 2 2 m =|γ2|M ,m= 0, ab,1 s ab, 2 2 = 2 = −| | 2 mab,3 0,mab,4 1 γ2 Ms . (2.9)

We assume that nH Higgs chiral multiplets remain light while the other N = 2 multiplets get large (∼ Ms ) supersymmetric masses. Actually, this is not the only way to obtain Higgs multiplets. One of the two doublets may emerge from a chiral intersection together with the leptons, say between a U(2) and a U(1) brane, while the other one could arise as a chiral “antidoublet” from the intersection of U(2) with the mirror of U(1) brane.

• The case where brane a is in the observable set O and brane b in sector M. The states φab living at such intersections are assumed to be non-chiral and we will call them “messengers” for reasons that will be apparent below. They are non-supersymmetric because of mass splitting between fermionic and bosonic modes. Through loop effects, they will induce supersymmetry (i) = breaking to the observable sector. We will use here the notation θab αi and choose again |α3|=|α1|+|α2|. Two different kinds of models can arise at this level. They correspond to the following two possibilities: (i) The ﬁrst possibility is that the states in the intersection φab originate as a perturbation around an N = 2 supersymmetric solution. In fact, in order to have a scale of supersymmetry breaking hierarchically small compared to the string scale we perform a tiny deformation of the intersection angles (preserving N = 2 supersymmetry):

|αi|→|αi|+i. (2.10) In one of the tori, chosen to be the ﬁrst one, the branes should remain parallel and separated4 by −1 a distance 1Ms ,i.e.

α1 = 1 = 0,α2 + α3 = = 0 (2.11) leading to the localized scalar masses: 2 2 2 2 2 2 m |α2|+ M ,m − + M , ab,1 1 s ab,2 1 s 2 + 2 2 2 −| |+ 2 2 mab,3 1 Ms ,mab,4 1 α2 1 Ms . (2.12) These states will induce at one-loop Dirac masses for gauginos and higgsinos, as we will show in next sections. For instance, the Dirac gaugino masses turn out to be α mD M , (2.13) 1/2 4π s where α is the gauge coupling. This provides a stringy realization of gauge mediated supersymmetry breaking in the absence of R-symmetry breaking. Scalars localized at supersymmetric intersections, as those discussed in (2.8), can be given a small supersymmetry breaking through another tiny deformation of the corresponding angles

4 Note that this supersymmetric mass will prevent the appearance of tachyons for scalars in (2.12) and will make the conﬁguration stable. 162 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 by . This can be described at the effective ﬁeld theory level as a Fayet–Iliopoulos D-term break- ∼ 2 ing corresponding to the presence of an anomalous U(1) factor D Ms [4,9]. As a result, scalar masses in the observable sector can acquire different masses depending on whether matter is charged under this U(1) or not. In particular:

• If the observable sector is charged, bosons of the matter multiplets acquire tree-level masses, 2 ∼ 2 m0 Ms . • Otherwise the bosons of the observable sector receive masses at the two-loop level [15]

α 2 m2 M . (2.14) 0 4π s

In both cases the transmission of supersymmetry breaking to the gaugino and higgsino sector is mediated through the “messengers” since the quark and lepton sectors are chiral (and do not contribute at this order) while the Higgs sector (with masses tuned to remain in the TeV range) will contribute negligibly. Note that the brane b does not have to be necessarily in sector M.It could also be part of the “observable” sector. Consider, for instance, the example where chiral quark doublets come from the intersection of a U(3) and a U(2) brane stack. Non-chiral antiquark doublets, coming from the intersection of U(3) with the orientifold image of U(2) can play the role of messengers generating Dirac gaugino masses for both U(3) and U(2) gauge groups. Moreover, the image of U(2) may have non-trivial intersection with another U(1) stack producing leptons, and thus it is part of sector O. (ii) The second possibility corresponds to the case where a supersymmetry is preserved by the subset of branes in O for some choice of the compactification moduli, but will never be conserved by the whole set of branes O ⊕M. This is, for instance, the case for the toroidal compactification discussed in Section 5 below. Let us denote by ci =±1 the relevant coefficients that define in (2.7) the supersymmetry preserved by the observable sector. The intersection between the O and M branes leads to states φab with a supersymmetry breaking mass (i) (i) (i) (i) 1 ma R m R m2 c arctan 2 − arctan b 2 M2 ≡ M2. (2.15) ab 2π i (i) (i) (i) (i) s s i na R1 nb R1

Keeping the branes (a, b) parallel in one of the tori allows to use again the states φab at their intersection as messengers to produce gaugino and higgsino Dirac masses, as in Sections 3 and 4. However now the desired suppression of will be generated by the presence of a large hierarchy between the size of the compact dimensions. More precisely for

R(i) 2 (i) 1 (2.16) R1 one can obtain an parameter hierarchically smaller than one

(i) (i) (i) 1 ma mb R2 ci − (2.17) 2π (i) (i) (i) i na nb R1 where the sum obviously goes over the tori where branes intersect. In the absence of tree level supersymmetry breaking, superpartners in the observable sector will acquire two-loop mass splitting given by (2.14). A toy model based on these ideas will be presented in Section 5. I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 163

3. String computation of the Dirac mass

As the gauginos lie in representations of extended supersymmetry (N 2), they come in copies that can pair up to receive a Dirac mass at one-loop order in the string genus expansion. The corresponding world-sheet has the topology of a cylinder stretching between two stacks of (1) (2) Na and Nb branes, respectively (see Fig. 2). The vertex operators V and V associated with the gauginos are inserted on one boundary, for example, that corresponding to the Na branes. The mass is given by the integrated two-point correlation function of the relevant vertex operators: A(1, 2) = dz dw V (1)(z)V (2)(w) , (3.1) where the integrals are along the boundary of the annulus. The N = 2 supersymmetry which relates the gauge vector and gauginos is chosen to be associated with the supercharges that preserve the (undeformed) N = 2 preserving brane intersection (see Eq. (2.12) with = 0). These supercharges are given by α˙ α˙ Qα,−++,Q,+−− , Qα,−−−,Q,+++ , (3.2) where ± denotes helicities in the internal directions and α, α˙ are Weyl spinor indices in four dimensions. The supercharges have been grouped into Majorana spinors in four dimensions. As the gauginos of N = 2 supersymmetry have the same internal helicities as the supercharges (3.2), − 1 the vertex operators in the 2 -ghost picture are given by: (1) − φ i · i + + · = 1 2 2 S H 2 (H1 H2 H3) ik1 X V− 1 (z) g0λ e e e e (z), 2 (2) − φ − i · − i − + + · = 2 2 2 S H 2 ( H1 H2 H3) ik2 X V− 1 (z) g0λ e e e e (z), (3.3) 2 together with their CPT conjugates which are obtained by simply reversing helicities in the internal directions and flipping the space–time chirality. Here φ is the bosonized two-dimensional μ reparametrization ghost field, X are the target space coordinates, k1 = k2 = 0 are the space– time momenta, Hi arise from bosonization of the spin fields and S is the helicity vector in four

Fig. 2. Non-supersymmetric states stretching between the two-branes induce at one-loop masses for the gauginos on each brane. 164 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 dimensions which is constrained by the GSO projection to be either

S = (+, +) or S = (−, −). (3.4) i Also g0 is some normalization constant and λ are Chan–Paton factors of the associated gauge group factor. On the cylinder, the total φ-charge must vanish. Therefore, we must use picture-changing to + 1 transform one of the vertex operators to the 2 picture. This is done by operating with the BRST charge:

V (2)(w) = lim eφT (z)V (2)(w), (3.5) 1 → F − 1 2 z w 2 where TF (z) is the world-sheet supercurrent: 2 1 T (z) = i ∂Xμψ + √ ∂ZiΨ¯ i + ∂Z¯ iΨ i . (3.6) F α μ i=1,2,3 2 The ﬁelds ψμ are real world-sheet fermions while in the second and third terms of (3.6) we have used complex coordinates for the internal directions:

Zi = X(i) + iY(i), Z¯ i = X(i) − iY(i),i= 1, 2, 3 (3.7) i ¯ i i ¯ i and analogously for Ψ and Ψ . Note that Ψ Ψ = i∂Hi . (2) After picture-changing, the vertex operator V− 1 contains several terms, but only one of them 2 contributes to the two-point function V (1)(z)V (2)(w). Indeed, the internal helicities must cancel between the two vertex operators to yield a non-vanishing correlator. For the choice made in (3.3), TF must induce a ﬂip of the sign of H1. We can therefore set

(2) 1 φ − i · − i + + → √ 2 2 2 S H 2 (H1 H2 H3) ik2X 1 V 1 (w) i g0λ e e e e ∂Z . (3.8) 2 α

Inserting these operators into a path integral on the cylinder, with k1 = k2 = 0, leads to (1) (2) 1 2 1 2 V (z)V (w) = i √ g tr λ λ Nb[BC]×[FC], (3.9) α 0 where the factor Nb comes from the trace over the Chan–Paton degrees of freedom of the second boundary of the cylinder. The factors [BC] and [FC] denote, respectively, correlation functions of world-sheet bosons and world-sheet fermions together with the associated ghosts of opposite statistics. They are given by 1 [ ]=I I μ I I BC bc X ∂Z Z1 Z2 Z3 (3.10) and a − φ φ i S·H − i S·H i H − i H [FC]= C e 2 e 2 e 2 e 2 e 2 i e 2 i (3.11) b a,b i=1,2,3 I ··· where is the identity operator. In the bosonic correlator, Zi denote bosonic path-integrals i over the complex coordinates Z while ···bc is a path integral over the parametrization bc- ghosts. In the fermionic correlation function the sum is over the spin structures for which we use the same notation a, b as for brane stacks but the difference is obvious for the reader. I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 165

3.1. The bosonic correlator

The one-point functions of identity operators are given by the appropriate piece of the partition function on the cylinder for the two stacks of intersecting D6-branes. The bc-ghosts cancel exactly the path-integral over two of the four space–time coordinates Xμ, and we are left with the simple expression5:

1 η(it) η(it) [BC]= ∂Z1(w) , (3.12) 2 Z1 1 + 1 + η(it) 2 α2 2 α3 ϑ 1 ϑ 1 2 2 where α2, α3 are the non-vanishing intersection angles of the D6-branes in the Z2 and Z3 planes. Here, t is the open string proper-time parametrizing the world-sheet annulus. To compute the correlation function of ∂Z1(w), consider the coordinate domain of the annulus to be a square parametrized by two coordinates (τ, σ ) with ranges 0 σ π,0 τ 2πt and the identiﬁcation τ ∼ τ + 2πt. The operator Z1 is located on the boundary, thus − − − ∂Z1(w) = Tr e 2π(t τ)H∂Z1(τ, 0)e 2πτH , (3.13) where H is the Hamiltonian for the Z1 coordinate ﬁeld. Only the zero-modes of the mode expansion of Z1 contribute to the trace in (3.13) and, therefore, if the branes would intersect with a non-trivial angle in the Z1 plane, the correlation function would vanish. We therefore need to impose

α1 = 0 (3.14) as anticipated in (2.11). This is also a necessary condition for the D-brane intersection to preserve the supercharges (3.2). As the two stacks are parallel in the ﬁrst torus, they can be separated at a distance 2π.For simplicity, this separation is taken along the X(1) direction. The mode expansions read:

√ (1) α X(1) = x(1) + 2α p(1)τ − i 2α 1,n einτ cos nσ, 0 n n =0 √ α(1) Y (1) = y(1) + 2nR(1) + 2 σ + 2α 2,n einτ sin nσ, (3.15) 0 2 n n =0

(1) (1) where n is the winding number. The momentum p is quantized in units of 1/R1 .Usingthe expression for the Hamiltonian, 1 2 H = α p(1) 2 + 2nR(1) + 2 2 + α(1) α(1) − (3.16) 4α 2 i,−n i,n 24 n>0,i=1,2 shows, as stated above, that the oscillators in (3.15) do not contribute to the trace:

5 The deﬁnitions of the various modular theta functions appearing in this section are given in Appendix A. 166 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179

(1) m nR ∂Z1(τ, 0) = 2iα + + 2 (1) α α m,n R1 2 (1) 2 m nR 1 × exp −2πα t + 2 + . (3.17) (1)2 α α η(it)2 R1 The sum over the quantized momenta does not contribute either, since the different terms cancel pairwise. However, the sum over windings is non-vanishing provided that the two stacks are non-coincident, i.e. = 0. Inserting (3.17) into (3.12) yields the complete bosonic correlator: 2iα [BC]= 1 + 1 + 2 2 α2 2 α3 η ϑ 1 (0)ϑ 1 (0) 2 2

(1) (1) 2 nR nR × 2 + exp −2πα t 2 + . (3.18) α α α α n

3.2. The fermionic correlator

To compute the fermionic correlator, we will make use of the elementary two-points functions:

1 φ − φ ϑ1(z − w) 4 η 2 2 = e e a z−w , (3.19) ϑ1(0) ϑ b ( 2 ) 1 4 − i H − i H ϑ1(0) a z w 1 e 2 J e 2 J = ϑ ,J= ξ,1, (3.20) ϑ1(z − w) b 2 η 1 4 − i H − i H ϑ1(0) a + αi z w 1 e 2 i e 2 i = ϑ ,i= 2, 3, (3.21) ϑ1(z − w) b 2 η ∈{ 1 } where ξ labels the two non-compact complexified space–time dimensions and a, b 0, 2 .The difference between (3.20) and (3.21) comes from the fact that in the Z2 and Z3 planes, the brane intersection angles α2 and α3 are non-vanishing. The z − w dependence was determined, for instance, in Ref. [16] and the correct normalization is inferred by matching the short distance limit z → w on both sides. For example, from the operator product expansion (OPE) of bosonized fermions we have a i − i 1 1 ϑ (0) 2 Hξ 2 Hξ → I= b e e 1 1 (3.22) (z − w) 4 (z − w) 4 η as the correlator of the identity operator is the spin-structure dependent partition function of a complex fermion. Taking the same limit on the right-hand side of (3.20) and comparing with (3.22) allows to check that the dependence on the cylinder modulus is correctly reproduced. Inserting (3.19)–(3.21) into (3.11), we obtain: 1 ϑ (0) [FC]= 1 Σ, 4 (3.23) η(it) ϑ1(z − w) where 2 a a z − w a + α z − w a + α z − w Σ = C ϑ ϑ 2 ϑ 3 . (3.24) b b 2 b 2 b 2 a,b I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 167 a The coefficients C b can be obtained by the same method as the one used above to deduce the normalization of the elementary correlators. From the OPE we deduce that the fermionic contribution [FC] has a pole in z − w whose residue, being the correlator of the identity, should be the full fermionic partition function on the annulus: 2 a a+α2 a+α3 1 1 − + ϑ (0) ϑ (0) ϑ (0) [FC]→ Z = η e 2πib(α2 α3) b b b (z − w) F z − w a,b η2 η η a,b (3.25) where as usual η0,0 = η 1 1 =−η 1 =−η 1 = 1. Comparing with the limit z → w of the right- 2 , 2 0, 2 2 ,0 hand side of (3.23), we obtain a − + C = η e 2πib(α2 α3). (3.26) b a,b The spin-structure sum in Σ can now be performed by using the Riemann theta-identity (A.10). To apply this identity we must first use the rearrangement 2 a + α 2πi(b+z)α α a ϑ (z) = e q 2 ϑ (z + αit) (3.27) b b a to put the theta functions in the appropriate form. The αi -dependent phase factor in C b then cancels out. After applying the Riemann identity, we can use (3.27) again to put α2 and α3 back into the argument of the theta function. The result is: 1 (1 + α + α ) 1 (1 − α + α ) = 2 2 3 − 2 2 3 Σ 2ϑ 1 (z w)ϑ 1 (0) 2 2 1 (1 + α − α ) 1 (1 − α − α ) × 2 2 3 2 2 3 ϑ 1 (0)ϑ 1 (0). (3.28) 2 2 We can now expand this result around the N = 2 supersymmetric configuration of D-branes using = α2 + α3. Expanding (3.28) to lowest order in gives 1 + α 1 + α [ ]=− 2 2 2 2 2 3 FC 8π η ϑ 1 (0)ϑ 1 (0), (3.29) 2 2 2 = 2 6 where we made use of the identity ϑ1(0) 4π η .

3.3. The two-gaugino amplitude

Inserting (3.18) and (3.29) into (3.9) we see that, to lowest order in , all contributions from the string oscillators cancel and only the classical part of the correlation function remains. This piece is independent of z − w and the integrals over the locations of the vertex operators in (3.1) only contribute a factor (2πt)2. The result is then simply

√ (1) (1) 2 nR nR A(1, 2) = 64π 4t2 αg2 tr λ1λ2 N 2 + exp −2πtα 2 + . 0 b α α α α n (3.30) From this we can obtain the two-gaugino amplitude as ∞ 1 V dt A(1, 2) = 4 A(1, 2), (3.31) 4 (8π 2α)2 t3 0 168 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 where the integration measure has been obtained by comparing with the partition function for intersecting D6-branes. The infinite volume factor V4 arises from momentum conservation. Had 4 (4) we chosen k1, k2 = 0, it would be replaced by a δ-function, V4 → (2π) δ (k1 + k2).The normalization g0 can in principle be determined. However for our purposes we only need to 2 ∼ = ϕ know that g0 is proportional to the open string coupling and therefore g0 gs , where gs e is the closed string coupling, determined by the VEV of the string dilaton ϕ. For the Dirac mass of the gauginos we then obtain the final result: ∞ (1) (1) 2 dt nR nR mD ∼ g N 2 + exp −2πtα 2 + . (3.32) 1/2 s 2 t α α α α n 0 By a Poisson resummation, it can easily be checked that the integral is finite. Note also that in the limit where → 0, the mass vanishes. To interpret the meaning of this result, we consider the mass spectrum of open strings which stretch between the two stacks of D-branes. In the limit where → 0, these string modes form N = 2 hypermultiplets. The mass of the lightest multiplet is given by αm2 = 2/α. When is non-vanishing, supersymmetry is completely broken and there is a mass splitting between fermi- 2 = 2 ons and scalars. The lightest fermions remain at α mF /α , while their scalar superpartners 2 = 2 + 1 have their√ masses lifted to α mS α mF 2 .Theresult(3.32) then implies the relation, in the limit α:

m2 − m2 mD ∼ g ∼ g S F m , 1/2 s s 2 F (3.33) α Ms 2 = ∼ where Ms 1/α is the string scale. For string size brane separation, such that Ms 1, we recover the anticipated behavior (2.13).

4. Computation of the higgsino mass

(i) Consider two stacks of D6-branes: the first stack of Na branes aligned with the horizontal X 6 axes and the second stack of Nc branes intersecting the horizontal axes in the three tori at angles (i) θc = βi , i = 1, 2, 3. For β1 = 0 and β2 = β3, this configuration preserves N = 2 supersymmetry and the lightest string modes on the intersection describe hypermultiplets. They are made of two N = 1 chiral multiplets, identified as two Higgs bosons and their N = 1 superpartners, the higgsinos. A simple way to generate a mass term for these higgsinos is to separate the branes in the first torus where they are parallel. This corresponds to switching on a vacuum expectation value for a particular scalar that parametrizes the location of the brane in the direction X(1). The mass is then given by the tension times the minimal length of the string, which is just the brane separation. This mass generation is not always possible, as for specific constructions such as orbifold models, the corresponding scalar is projected out of the light spectrum. In this section we will investigate the possibility that the higgsino mass is generated at one- loop in a way similar to the Dirac gaugino mass. The appropriate string amplitude corresponds

6 This can be achieved by a rotation of the subgroup U(1)3 ⊂ SO(6). I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 169

Fig. 3. Higgsinos are localized at the intersection of branes a and c. Non-supersymmetric states stretching between the branes (a, b) and (c, b) induce at one-loop masses for the higgsinos. to a world-sheet with the topology of an annulus. On the ﬁrst boundary two vertex operators that create the higgsinos are inserted. The higgsinos arise from open strings with the two endpoints located on different stacks of D-branes. This means that part of the ﬁrst boundary of the annulus lies on the stack of Na branes and the other part lies on the other stack of Nc branes (see Fig. 3).

This change of boundary conditions is implemented by the insertion of twist ﬁelds σβi and σ1−βi . The other boundary of the annulus must be located on a third stack of Nb branes. The two stacks Nb and Na intersect at angles αi . As this additional stack must preserve the same supersymmetries as the other stacks, we will start from the simplest possibility αi = 0 and make a small deformation of the angles to break supersymmetry, as described in Section 2. The generalization to αi = 0 is straightforward. − 1 The vertex operators of the higgsinos in the 2 -ghost picture are:

(1) 1 − φ i S·H i H −i(β − 1 )H −i(β − 1 )H ik ·X = 2 2 2 1 2 2 2 3 2 3 − − 1 V− 1 (z) g0λ e e e e e σ1 β2 σ1 β3 e (z), 2 (2) − φ − i · i − 1 − 1 · = 2 2 2 S H 2 H1 i(β2 2 )H2 i(β3 2 )H3 ik2 X V− 1 (z) g0λ e e e e e σβ2 σβ3 e (z). 2 Note the similarities with the vertex operators of the gauginos (3.3). The helicities have been shifted by the intersection angles. In addition we have inserted the bosonic twist operators as ±i(β − 1 )H world-sheet superpartners of the operators e i 2 i . In the limit where βi → 0, we recover precisely the gaugino vertices since in this limit the bosonic twists become the identity operators. Since the annulus has vanishing ghost-charge we need again to transform one of the operators to 1 the 2 -picture. This is done precisely in the same way as in Section 3. To obtain a non-vanishing correlator, the world-sheet supercurrent TF (z) must be used to ﬂip the sign of H1 in one of the exponents. The result is

(2) φ − i · − i − 1 − 1 → 2 2 2 S H 2 H1 i(β2 2 )H2 i(β3 2 )H3 ik2X 1 V 1 (z) g0λ e e e e e σβ2 σβ3 e ∂Z . (4.1) 2 Inserting the vertex operators at zero momentum in the correlator leads to an expression of the same form as (3.9), (1) (2) 1 2 1 2 V (z)V (w) = i √ g tr λ λ Nb[BC]×[FC], (4.2) α 0 where now 170 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 1 [ ]=I I μ − − BC bc X ∂Z Z1 σ1 β2 σβ2 Z2 σ1 β3 σβ3 Z3 , (4.3) a − φ φ i S·H − i S·H i H − i H [FC]= C e 2 e 2 e 2 e 2 e 2 1 e 2 1 b a,b −i(β − 1 )H i(β − 1 )H × e i 2 i e i 2 i . (4.4) i=2,3 1 [ ] Because of the factor ∂Z Z1 in BC (which was computed in Section 3), we must impose that (1) α1 = 0 and that the branes are separated by a distance = 0intheX direction in order to obtain a non-vanishing contribution.

4.1. The fermion correlator

The computation of the fermion correlator is similar to the one in Section 3.2. The correlator is computed for general α2, α3 then expanded around the (N = 4) supersymmetric conﬁguration αi = 0. A slight generalization of (3.21) is given by: a2 − ϑ (0) a + α 1 eiaHk e iaHk = 1 ϑ k a(z − w) ,k= 2, 3 (4.5) ϑ1(z − w) b η ∈{ 1 } with a, b 0, 2 . The remaining elementary correlators are unchanged and are given in (3.19) and (3.20). Inserting these in (4.4) we obtain

− − − − 1 ϑ (0) 1 β2(1 β2) β3(1 β3) [FC]= 1 × Σ, 4 (4.6) η ϑ1(z − w) where a a z − w a + α 1 Σ = C ϑ2 ϑ 2 β − (z − w) b b 2 b 2 2 a,b a + α 1 × ϑ 3 β − (z − w) . (4.7) b 3 2 a The coefficients C b are given as before by (3.26). Using the Riemann identity (A.10) allows to perform the spin-structure sum and put Σ into a simpler form. After imposing the supersymmetry condition β2 = β3 ≡ β, one finds: 1 (1 + α + α ) 1 (1 + α − α ) = 2 2 3 − × 2 2 3 Σ 2ϑ 1 β(z w) ϑ 1 (0) 2 2 1 (1 − α + α ) 1 (1 − α − α ) × 2 2 3 × 2 2 3 − − ϑ 1 (0) ϑ 1 (1 β)(z w) . (4.8) 2 2 This can now be expanded around the supersymmetric configuration αi = 0. For simplicity, we take α2 = 0 and α3 = , and expand in powers of : 1 1 =− 2 − 2 − − 2 2 Σ 2ϑ 1 β(z w) ϑ 1 (1 β)(z w) ϑ1(0) . (4.9) 2 2 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 171

4.2. The bosonic correlator

The correlator of world-sheet fermions, Eq. (4.9), is already of order 2. Since we are interested only in the leading behavior, for the purpose of computing the bosonic correlation function [BC], we can keep the leading order corresponding to αi = 0. Most pieces of [BC] have already been computed in Section 3. The additional complication is the appearance of correlators of bosonic twist ﬁelds. These have been studied, for instance, for arbitrary genus and any number of twist ﬁelds in Ref. [17]. The complete calculation is presented in Appendix B where only the relevant results will be quoted. In fact using Eq. (B.23) in (4.3), leads to

(2) (3) − −2β(1−β) R1 R1 ϑ1(z w) 1 1 [BC]=i (2iα ) α ϑ (0) det W(β)η8 1 (1) (1) nR 2 nR2 −2πtα 2 + × + e α α α α n

−S (v(2),v(2)) −S (v(3),v(3)) × e cl 1 2 × e cl 1 2 , (4.10) (2) (2) (3) (3) v1 ,v2 v1 ,v2 (i) where W(β) is deﬁned as in (B.5) for the twist β and v1,2 are deﬁned in Eq. (B.16).

4.3. The higgsino two-point correlation function

Finally inserting (4.6), (4.9) and (4.10) into (4.2) gives for the two-point function: √ R(2)R(3) V (1)(z)V (2)(w) = 32iπ3 α2g2 tr λ1λ2 N 1 2 0 b α

(1) (1) nR 2 nR2 −2πtα 2 + × + e α α f(z− w), (4.11) α α n where ϑ (β(z − w))ϑ ((1 − β)(z − w)) 1 f(z− w) = 1 1 ϑ (z − w)η3 det W(β) 1 −S (v(2),v(2)) −S (v(3),v(3)) × e cl 1 2 × e cl 1 2 . (4.12) (2) (2) (3) (3) v1 ,v2 v1 ,v2 The higgsino mass μ is proportional to the two-higgsino amplitude obtained upon integration of the correlation function (4.11) over the position of the vertex operators and inserting the result into (3.31). Since the correlator depends only on z − w, one of the integrals is trivial and contributes simply a factor 2πt. The ﬁnal result reads ∞ (1) (1) nR 2 2 dt nR2 −2πtα 2 + μ ∼ g N + e α α I, (4.13) s 1 t α α n 0 where 2πt 1 I = dxf(ix) (4.14) 2πt 0 172 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 and it is of order 2 as expected.

5. A toy model

In this section we will present a simple model as a framework to implement the realization of the extended split supersymmetry scenario described above with gauge bosons in representations of N = 4 supersymmetry. It consists on a toroidal orientifold based on the factorized 2 (i) six-dimensional torus i Ti with an orientifold plane along the X axes. This implies for each D6a-brane with wrapping numbers (na,ma) the presence of the image brane D6a∗ with wrapping numbers (na, −ma). The minimal setup is given by three intersecting stacks of D6-branes giving rise to the gauge group

⊗ ⊗ ≡ ⊗ ⊗ U(Na1 ) U(Na2 ) U(Nb) U(5) U(1) U(Nb). (5.1)

The branes D6a1 ,D6a2 and their images, with supersymmetric intersections, provide the observable set O while the brane D6b and its image stand for the (messenger) sector M. The Standard Model gauge group is embedded in U(5) in the following way:

• Open strings localized at the intersection of D6 and D6 ∗ describe three generations trans- a1 a1 forming as 10 of SU(5). Their massless modes transform according to the antisymmetric representation and come in three generations, i.e. (i) n = 1,I∗ = 3. (5.2) a1 a1a1 i It is easy to see that an odd number of generations requires the use of tilted tori as those described in Fig. 1. • Open strings localized at the intersection of D6 and D6 ∗ provide three generations of 5¯, a1 a2 i.e.

∗ =− Ia1a2 3. (5.3) • The D6a2 brane being parallel to the D6a1 in one torus, the massless modes at their intersections will be identiﬁed with N = 2 hypermultiplets that contain the Supersymmetric Standard Model Higgs doublets.

The supersymmetry breaking messengers arise from strings stretching between brane D6a1 and ∗ D6b on one side and between D6a2 and D6b on the other side. These branes are parallel (and separated) along the second and ﬁrst torus, respectively, and their intersections contain non-chiral matter with non-supersymmetric masses. We will discuss here a simple example given in Table 1 where the wrapping numbers of different D6I -branes (I = a1,a2,b) in the three tori are listed. It is easy to see that the conditions for cancellation of RR-charge tadpole are satisﬁed. The association of any stack of branes D6I (I = a1,a2,b) with its orientifold image D6I ∗ preserves one of the four supersymmetries, that we label as Sα with α = 1,...,4, if the angles (i) θI satisfy one of the corresponding relations: I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 173

Table 1 Wrapping numbers for the three stacks in the model (1) (1) (2) (2) (3) (3) NI nI ,mI nI ,mI nI ,mI = − − Na1 5 (1, 1/2)(1, 1/2)(1, 3/2) = − Na2 1 (1, 1/2)(1, 5/2)(1, 5/2) (1) (2) (3) = (1) (2) (3) Nbnb nb nb 10 nb (1, 1/2)nb (1, 1/2)nb (1, 1/2)

: (1) + (2) + (3) = :−(1) + (2) + (3) = S1 θI θI θI 0,S2 θI θI θI 0, : (1) − (2) + (3) = : (1) + (2) − (3) = S3 θI θI θI 0,S4 θI θI θI 0. (5.4) It is easy to check that none of these equations can be simultaneously satisfied by the three set of branes. Instead, we will look for ratios of radii R(i) A = 2 (i = , , ) i (i) 1 2 3 (5.5) R1 that allow one of the relations, chosen to be S1, to be satisfied by the stacks a1, a2 and their images. This fixes two of the ratios as functions of the third one. For instance, in the limit of small ratios Ai 1, the supersymmetric conditions read as 5 A A , (5.6) 1 2 2 1 A A . (5.7) 3 2 2 ∩ ∗ ∩ ∗ ∩ ∗ ∩ As a result, the intersections a1 a1 , a2 a2 and a1 a2 preserve S1, the intersections a1 a2 ∗ ∩ ∗ ⊕ ∩ ⊕ and a1 a2 preserve S1 S2, the intersections a1 b preserve S3 S4, and finally the other intersections involving b and/or b∗ do not preserve any supersymmetry. At these intersections the lightest states have supersymmetry breaking masses of order: 2 ∼ 2 Ms A2Ms . (5.8) ∗ ¯ In particular, the states at the intersection a1 ∩ b could act as messengers (a number of 5 + 5 of SU(5)) to generate at one-loop level Dirac masses for fermions, gauginos and higgsinos, and at two-loop supersymmetry breaking masses for bosons of the observable sector. Note that the gauge symmetry U(5) can be broken to the Standard Model one by a discrete or continuous Wilson line without introducing extra fields. Unification of gauge couplings is thus guaranteed in such particular constructions. Actually, this model realizes the particular D-brane configuration of Ref. [2], in which the U(5) stack is replaced by two separate brane collections, U(3) and U(2), describing strong and weak interactions. Moreover, the hypercharge is the linear combination Y =−Q3/3 + Q2/2 of the two corresponding U(1) factors Q3 and Q2.The antisymmetric representation of SU(5) is then decomposed in terms of the quark doublets, the up antiquarks and the right-handed lepton, the latter arising as antisymmetric representations of = A = A = U(3) and U(2), respectively. Imposing now I32 I33∗ I22∗ 3, the absence of symmetric rep- (i) = ∗ = resentations i n 1 for the two groups, as well as the absence of antiquark doublets I32 0, one finds that the strong and weak branes are parallel in all three planes and thus correspond to a Wilson line breaking of the GUT group U(5) studied above.7

7 Other constructions of a similar model with or without extended supersymmetric gauge sector are given in Ref. [18]. 174 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179

6. Conclusions

Small deformations of the brane intersection angles provide a simple mechanism to break supersymmetry, which in the T -dual picture corresponds to the introduction of appropriate combinations of magnetic fluxes. At the effective field theory level, this can be described as Fayet–Iliopoulos D-term breaking corresponding to the presence of anomalous U(1) factor(s). 2 ∼ 2 Charged scalars localized at their intersections acquire tree-level masses of order m0 Ms . Scalars in non-chiral sectors can transmit supersymmetry breaking to the observable sector through gauge mediated loop corrections. However, since the D-term does not break R-symmetry no gaugino Majorana masses are expected (at least to the lowest order). Phenom- enologically interesting models can appear when the gauge sector has an extended supersymmetry in which case gauginos can get Dirac masses. A phenomenological application of this scenario is given in Ref. [8] where a minimal model based on extended split supersymmetry was shown to be consistent with unification and the presence of dark matter. Dirac masses for gauginos and higgsinos where obtained from the D-breaking assuming the presence of some effective higher-dimensional operators. Even if such operators are absent at tree-level, the present work shows how they arise at one-loop. The precise string calculation points out important features of the messenger sector in the simplest realization. In particular, the fact that the messengers arise from strings stretching between branes that are separated in one direction. They correspond to deformations of N = 2 sectors. A simple toroidal compactification, with an orientifold along one direction, does not allow non-trivial supersymmetric intersecting branes. A vacuum to be considered as starting point for the above mentioned deformation is then missing, so we instead propose a different scenario. We identify a subset of supersymmetric intersecting branes with the observable sector. The other branes lead to some non-supersymmetric intersections. The matter living at such non-chiral intersections will, as in the previous scenario, play the role of the supersymmetry breaking messenger sector. In order to obtain masses hierarchically smaller than the string scale, small ratios of radii are required and they make all intersection angles small. We have presented a GUT model to illustrate this possibility. Finally note that for both scenarios there are two possibilities from the phenomenological point of view: (i) To introduce an deformation breaking supersymmetry in the observable sector at tree-level. (ii) To keep the observable sector supersymmetric at tree-level. Non-supersymmetric branes such that the matter living on their intersection with the observable branes are non-chiral will transmit supersymmetry breaking by gauge interactions. In case (ii) both gauginos, squarks, sleptons and higgses have masses of the same order of magnitude (as in the usual gauge mediated models) opening up the possibility of having a supersymmetric spectrum in the TeV range. This will give rise to a rich phenomenology accessible at LHC energies.

Acknowledgements

Work supported in part by the European Commission under the RTN contract MRTN-CT- 2004-503369, in part by CICYT, Spain, under contracts FPA2005-02211 and FPA2002-00748, in part by the INTAS contract 03-51-6346 and in part by IN2P3-CICYT under contract Pth 03-1. We wish to thank A. Uranga for useful discussions. K.B. wishes to thank the CERN Theory Unit for hospitality. A.D. and M.Q. wish to thank the LPTHE of “Université de Paris VI et VII” for hospitality. I.A. and K.B. acknowledge the hospitality of the “Universidad Autonoma de Barcelona” and IFAE. I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 175

Appendix A. Theta functions

In this short appendix, we establish our conventions for the modular theta functions and list a few useful properties. The theta functions are defined by: ∞ a + 2 + + + ϑ (z, τ) = eπi(n a) τ 2πi(n a)(z b), (A.1) b n=−∞ where τ is the (complex) modular parameter of the torus, not to be confused with the world-sheet coordinate used in the text. On the cylinder, this parameter is purely imaginary and in the main text we use the definition τ = it. Alternatively, the theta functions can be defined as an infinite product: a ϑ (z, τ) b i2πa(b+z) 1 a2 n+a− 1 2πi(b+z) n−a− 1 −2πi(b+z) n = e q 2 1 + q 2 e 1 + q 2 e 1 − q , (A.2) n1 where q = e2πiτ. They satisfy the following periodicity conditions: a 2πi(a− 1 ) a ϑ (z + 1,τ)=−e 2 ϑ (z, τ), (A.3) b b a −2πi(b− 1 ) −πiτ−2πiz a ϑ (z + τ,τ) =−e 2 e ϑ (z, τ), (A.4) b b which is the reason why they are well suited to describe correlators on the torus. Defining 1 ≡ 2 ϑ1(z) ϑ 1 (z, τ) (A.5) 2 we can see from the product representation that ∞ i π τ m m −1 m ϑ1(z) =−2e 4 sin(πz) 1 − q 1 − zq 1 − z q . (A.6) m=1 In particular = +··· ϑ1(z) zϑ1(0) , (A.7) a fact that was used repeatedly in the main text. We also need the Dedekind eta function, which is defined as 1 n η(τ) = q 24 1 − q . (A.8) n1

It is related to the function θ1(z) by the simple identity =− 3 θ1(0) 2πη(τ) . (A.9) Finally, the theta functions satisfy the following Riemann identity: 176 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 a a a a η ϑ (z )ϑ (z )ϑ (z )ϑ (z ) a,b b 1 b 2 b 3 b 4 a,b 1 1 1 1 = 2 2 2 2 2ϑ 1 (z1)ϑ 1 (z2)ϑ 1 (z3)ϑ 1 (z4) (A.10) 2 2 2 2 with

1 1 z = (z + z + z + z ), z = (z − z + z − z ), 1 2 1 2 3 4 2 2 1 2 3 4 1 1 z = (z − z − z + z ), z = (z + z − z − z ). 3 2 1 2 3 4 4 2 1 2 3 4 Appendix B. Bosonic correlator for higgsino mass

Let us consider the correlator on the torus with twists of an arbitrary angle θ. This correlator has a quantum piece Zqu and a classical piece which takes into account the contributions from world-sheet instantons: −Scl(i) σ1−θ (z1)σθ (z2) = Zqu e , (B.1) i where the sum is over classical solutions of the equations of motion. Both pieces are expressed in terms of the so-called cut-differentials, which are holomorphic one-forms on a branched covering of the punctured torus. When expressed as functions of the torus coordinate z, they become multi-valued with branch-cut singularities at the location of the punctures. In the present context, where the torus has only two punctures corresponding to the insertions of the two twist ﬁelds, these cut-differentials are simply given by: − − − ω(z) = ϑ (z − z ) θ ϑ (z − z ) (1 θ)ϑ z − z − θ(z − z ) , (B.2) 1 1 1 2 1 2 1 2 −(1−θ) −θ ω (z) = ϑ1(z − z1) ϑ1(z − z2) ϑ1 z − z1 + θ(z1 − z2) (B.3) and satisfy ω(z+ 1) = ω(z+ τ)= ω(z), ω (z + 1) = ω (z + τ)= ω (z), (B.4) where τ is the complex modulus of the torus. The singularities at the insertions zi depend on the twist-angle θ. i From these cut-differentials, we construct a “period matrix” Wa deﬁned as 1 = 2 = ¯ ¯ ¯ Wa dzω(z), Wa dz ω (z), (B.5)

γa γa where the paths γa, a = 1, 2, denote the canonical homology basis of the torus. From [17] we obtain the quantum part of the correlation function:

1 − − Z = ϑ (z − z ) 2θ(1 θ). (B.6) qu det W 1 1 2 To determine the classical part of the correlator, we need to evaluate the action for all possible classical solutions of the equations of motion. Each contribution will have the form 1 S = d2z(∂Z ∂¯Z¯ + ∂Z¯ ∂Z¯ ), (B.7) cl 4πα cl cl cl cl I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 177

¯ ¯ where Zcl and Zcl are classical solutions, i.e. Zcl solves ∂∂Zcl = 0 on the punctured torus with appropriate boundary conditions (i.e. branch cut singularities) at the location of the punctures. In particular, under parallel transport along the canonical homology basis of the torus, Zcl is expected to shift as ¯ ΔaZcl = dz∂Zcl + dz¯ ∂Zcl = va, (B.8)

γa γa 1 2 where va are given complex numbers. For instance, for a toroidal compactiﬁcation Z = X +iX with

1 1 2 2 X ∼ X + 2πR1,X∼ X + 2πR2 (B.9) when Zcl is transported along a closed loop on the world-sheet torus, it must return to its original value modulo a lattice vector of the space–time torus. This implies

v1 = 2π(m1R1 + in1R2), v2 = 2π(m2R1 + in2R2), (B.10) where the integers mi and ni correspond to the winding numbers of the closed string propagating in the loop. The classical part of the correlator is written as

−Scl(v1,v2) Zcl = e (B.11) v1,v2 where Scl(v1,v2) is now expressed in terms of the period matrix (B.5): i − − S = v v¯ W¯ 1 bW¯ 1Mad + W 1 aW 2Mbc . (B.12) cl 4πα a b 1 d 2 c The two-point correlation function of twist ﬁelds on the torus is then given by 1 −2θ(1−θ) −S (v ,v ) σ − (z )σ (z ) = ϑ (z − z ) e cl 1 2 , (B.13) 1 θ 1 θ 2 T det W 1 1 2 v1,v2 where the sum over the vi corresponds to a sum over all possible winding modes. The case we are interested here, the annulus two-point correlation function, is obtained from (B.13) by taking the “square root” and replacing the complex modulus τ by τ → it. As a result, the twist correlator on the annulus takes the form

− −θ(1−θ) 1 ϑ1(z1 z2) −S (v ,v ) − = cl 1 2 σ1 θ (z1)σθ (z2) A K(τ) 1 e , (B.14) 2 ϑ (0) det W 1 v1,v2 where now i − − S = v v¯ W¯ 1 bW¯ 1Mad + W 1 aW 2Mbc (B.15) cl 8πα a b 1 d 2 c and K(τ) is a normalization factor which can depend on the modulus. It is now understood that the twist ﬁelds are located on one of the boundaries of the cylinder, zj = iyj with yj real. The values of va must also be appropriately modiﬁed. Instead of (B.10),wenowhave

v1 = 2πR1m, v2 = i4π(R2n + ), (B.16) 178 I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 where R1, R2 are the compactiﬁcation radii of the space–time torus and is the separation between the D-branes along the X2 direction. Taking the limit y1 → y2, allows to check that this expression is correct and to determine the normalization constant K(τ). In this limit, the operator product expansion of the twist ﬁelds behaves as: 1 σ −θ (z )σθ (z ) → I 1 1 2 (z − z )θ(1−θ) 1 2 2 2 1 1 m nR = exp −2πα t + 2 + , − θ(1−θ) η2 2 α α (z1 z2) m,n R1 (B.17) wherewehaveusedthatI is the partition function on the annulus of the complex boson Z. The factor 1/η2 is the contribution of the oscillators while the sum is over the momentum and winding modes of the string. The same result should be obtained by taking the limit z1 → z2 on the right-hand side of (B.14). The expression (B.3) for the cut-differential implies that ω(z) → 1,ω(z) → 1. (B.18) In this limit the components of the period matrix and its inverse can be explicitly evaluated:

1 1 W W it 1 − 1 1 −1 W = 1 2 = ,W1 = . (B.19) 2 2 −it 1 it it W1 W2 2it Inserting this in the action (B.15) leads to π 2πt S =− R2m2 − (R n + )2 (B.20) cl 2αt 1 α 2 so that the classical piece of the correlator becomes

2 2 − πR R2 e Scl = exp − 1 m2 exp −2πtα + n 2αt α α v1,v2 m n 1 2 2 2α t 2 m nR = exp −2πα t + 2 + , (B.21) 2 2 α α R1 m,n R1 where the second equality is obtained by a Poisson resummation. Using (A.7) and (B.19), the limit z1 → z2 of the quantum part is

K(τ) 1 1/2 Z = . qu θ(1−θ) (B.22) |z1 − z2| 2it Inserting (B.21) and (B.22) into (B.1) and comparing with (B.17) shows that both expressions = 2 1/2 2 agree to each other provided that K(τ) (iR1/α ) 1/η . The ﬁnal result for the correlation function of twist–antitwist on the annulus is 1 − − 2 2 − θ(1 θ) iR1 1 1 ϑ1(z w) −S (v ,v ) σ − (z)σ (w) = e cl 1 2 , (B.23) 1 θ θ 2 1 α η(it) 2 ϑ (0) det W 1 v1,v2 where v1 and v2 are given in Eq. (B.16). I. Antoniadis et al. / Nuclear Physics B 744 (2006) 156–179 179

References

[1] N. Arkani-Hamed, S. Dimopoulos, JHEP 0506 (2005) 073, hep-th/0405159; G.F. Giudice, A. Romanino, Nucl. Phys. B 699 (2004) 65, hep-ph/0406088; G.F. Giudice, A. Romanino, Nucl. Phys. B 706 (2005) 65, Erratum; N. Arkani-Hamed, S. Dimopoulos, G.F. Giudice, A. Romanino, Nucl. Phys. B 709 (2005) 3, hep-ph/0409232. [2] I. Antoniadis, S. Dimopoulos, Nucl. Phys. B 715 (2005) 120, hep-th/0411032. [3] C. Angelantonj, A. Sagnotti, Phys. Rep. 371 (2002) 1, hep-th/0204089; C. Angelantonj, A. Sagnotti, Phys. Rep. 376 (2003) 339, Erratum. [4] C. Bachas, hep-th/9503030. [5] C. Angelantonj, I. Antoniadis, E. Dudas, A. Sagnotti, Phys. Lett. B 489 (2000) 223, hep-th/0007090. [6] M. Berkooz, M.R. Douglas, R.G. Leigh, Nucl. Phys. B 480 (1996) 265, hep-th/9606139. [7] R. Blumenhagen, L. Goerlich, B. Kors, D. Lust, JHEP 0010 (2000) 006, hep-th/0007024. [8] I. Antoniadis, A. Delgado, K. Benakli, M. Quiros, M. Tuckmantel, hep-ph/0507192. [9] I. Antoniadis, E. Gava, K.S. Narain, T.R. Taylor, Nucl. Phys. B 511 (1998) 611, hep-th/9708075; S. Kachru, J. McGreevy, Phys. Rev. D 61 (2000) 026001, hep-th/9908135; M. Mihailescu, I.Y. Park, T.A. Tran, Phys. Rev. D 64 (2001) 046006, hep-th/0011079; E. Witten, JHEP 0204 (2002) 012, hep-th/0012054; D. Cremades, L.E. Ibanez, F. Marchesano, JHEP 0207 (2002) 009, hep-th/0201205. [10] I. Antoniadis, K.S. Narain, T.R. Taylor, Nucl. Phys. B 729 (2005) 235, hep-th/0507244. [11] P.J. Fox, A.E. Nelson, N. Weiner, JHEP 0208 (2002) 035, hep-ph/0206096. [12] See, for example, A.M. Uranga, Class. Quantum Grav. 22 (2005) S41, and references therein. [13] L.E. Ibanez, F. Marchesano, R. Rabadan, JHEP 0111 (2001) 002, hep-th/0105155; M. Cvetic, G. Shiu, A.M. Uranga, Nucl. Phys. B 615 (2001) 3, hep-th/0107166. [14] R. Blumenhagen, L. Goerlich, B. Kors, JHEP 0001 (2000) 040, hep-th/9912204; G. Aldazabal, S. Franco, L.E. Ibanez, R. Rabadan, A.M. Uranga, J. Math. Phys. 42 (2001) 3103, hep-th/0011073. [15] G.F. Giudice, R. Rattazzi, Phys. Rep. 322 (1999) 419, hep-ph/9801271. [16] J.J. Atick, A. Sen, Nucl. Phys. B 286 (1987) 189. [17] J.J. Atick, L.J. Dixon, P.A. Grifﬁn, D. Nemeschansky, Nucl. Phys. B 298 (1988) 1. [18] R. Blumenhagen, D. Lust, S. Stieberger, JHEP 0307 (2003) 036, hep-th/0305146; T.P.T. Dijkstra, L.R. Huiszoon, A.N. Schellekens, Phys. Lett. B 609 (2005) 408, hep-th/0403196; T.P.T. Dijkstra, L.R. Huiszoon, A.N. Schellekens, Nucl. Phys. B 710 (2005) 3, hep-th/0411129; A.N. Schellekens, private communication. Nuclear Physics B 744 (2006) 180–206

Intrinsic CPT violation and decoherence for entangled neutral mesons

J. Bernabéu a, N.E. Mavromatos b, J. Papavassiliou a,∗, A. Waldron-Lauda b

a Departamento de Física Teórica and IFIC, Centro Mixto, Universidad de Valencia-CSIC, E-46100 Burjassot, Valencia, Spain b King’s College London, University of London, Department of Physics, Strand WC2R 2LS, London, UK Received 22 November 2005; received in revised form 9 March 2006; accepted 31 March 2006 Available online 18 April 2006

Abstract We present a combined treatment of quantum-gravity-induced decoherence and intrinsic CPT violation in entangled neutral-kaon states. Our analysis takes into consideration two types of effects: first, those associated with the loss of particle–antiparticle identity, as a result of the ill-defined nature of the CPT operator, and second, effects due to the nonunitary evolution of the kaons in the space–time foam. By studying a variety of φ-factory observables, involving identical as well as general final states, we derive analytical expressions, to leading order in the associated CPT violating parameters, for double-decay rates and their time-integrated counterparts. Our analysis shows that the various types of the aforementioned effects may be disentangled through judicious combinations of appropriate observables in a φ factory. © 2006 Elsevier B.V. All rights reserved.

PACS: 11.30.Er; 13.25.Es; 03.65.Ud

1. Introduction

To date, the special theory of relativity, the established theory of ﬂat space–time physics based on Lorentz symmetry, is very well tested. In fact, during the current year, it completes a century of enormous success, having passed very stringent and diverse experimental tests. On the other hand, a quantum theory of gravity, that is, a consistent quantized version of Einstein’s general

* Corresponding author. E-mail address: [email protected] (J. Papavassiliou).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.03.028 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 181 relativity, still eludes us. This may be partially attributed to the lack of any concrete observational evidence on the structure of space–time at the characteristic scale of quantum gravity (QG), the 19 Planck mass-scale MP ∼ 10 GeV. One would hope that, like any other successful physical theory, a physically relevant quantum theory of gravity should lead to experimental predictions testable in the foreseeable future. In the QG case, however, such predictions may be both numerically suppressed and experimentally difficult to isolate. This is mainly due to the extremely weak nature of the gravitational interaction as compared with the rest of the known forces in nature. = 2 Specifically, the dimension-full coupling constant of gravity, the Newton constant GN 1/MP , appears as a very strong suppression factor of any physical observable that could be associated with predictions of quantum gravity. If the feebleness of the gravitational interaction were to be combined with the exact conservation of all symmetries and properties in particle physics, one would arrive at the inescapable conclusion that a true “phenomenology of QG” should be considered wishful thinking. In recent years, however, physicists have ventured into the idea that laws, such as Lorentz invariance and unitarity, which are characteristic of a flat space–time quantum field theory, may not be valid in a full quantum theory of curved space–time. For instance, the invariance of the laws of Nature under the CPT symmetry [1] (i.e., the combined action of charge conjugation (C), parity- reflection (P), and time-reversal (T)) may not be exact in the presence of QG effects. Indeed, the possibility of a violation of CPT invariance (CPTV) by QG has been raised in a number of theoretical models, that go beyond conventional local quantum field theoretic treatments of gravity [2–8]. CPT violating scenarios may be classified into two major categories, according to the specific way the symmetry is violated. The first category contains models in which CPTV is associated with (spontaneous) breaking of (linear) Lorentz symmetry [6] and/or locality of the interactions [7]. In these cases the generator of the CPT symmetry is a well-defined quantum mechanical operator, which however does not commute with the (effective) Hamiltonian of the system. In the second major category, the CPT operator, due to a variety of reasons, is ill-defined. The most no- table example of such a type of CPTV is due to the entanglement of the matter theory with an “environment” of quantum gravitational degrees of freedom, leading to decoherence of matter [2,4,5,8]. This decoherence, in turn, implies, by means of a powerful mathematical theorem due to Wald [9], that the quantum mechanical operator generating CPT transformations cannot be consistently defined. Wald refers to this phenomenon as “microscopic time irreversibility”, which captures the essence of the effect. We shall refer to such a failure of CPT through decoherence as “intrinsic”, in contradistinction to “extrinsic” violations, related to the noncommutativity of the matter Hamiltonian with a well-defined CPT generator [6,7]. As emphasized recently in [8], this intrinsic CPTV leads to modifications to the concept of an “antiparticle”. The resulting loss of particle–antiparticle identity in the neutral-meson system induces a breaking of the Einstein–Podolsky–Rosen (EPR) correlation imposed by Bose statistics. Specifically, in the CPT invariant case, the antiparticle of a given particle is defined as the state of equal mass (and life-time) and opposite values for all charges. If the CPT operator is well defined, such a state is obtained by the action of this operator on the corresponding particle state. If, however, the operator is ill-defined, the particle and antiparticle spaces should be thought of as independent subspaces of matter states. In such a case, the usual operating assumption that the electrically neutral meson states (kaons or B mesons) are indistinguishable from (“identical” to) their respective antiparticles (K¯ 0 or B¯ 0) is relaxed. This, in turn, modifies the symmetry properties in the description of (neutral) meson entangled states, and may bring about deviations to their EPR correlations [8,10]. We emphasize, though, that the associated intrinsic CPTV effects 182 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 are treated as small perturbations, being attributed to quantum-gravitational physics. From now on, we will refer to this effect as the “ω effect”, due to the complex parameter ω used in its parametrization. It should be stressed at this point that the ω effect is distinct, both in its origin as well as the physical consequences, from analogous effects generated by the evolution of the system in the decoherence-inducing medium of QG, discussed in [5]. As already mentioned in [8], and will be discussed in detail in the next section, the ω effect appears immediately after the φ-decay (at time t = 0) due to the violation of the symmetry properties of the entangled kaon state wave-function under permutations, assuming conservation of angular momentum. In other words, the ω effect provides an answer to the question: “What is the (symmetry of the) initial state of the K0K¯ 0 system in the presence of a QG-decoherence inducing background?” This is to be contrasted to the effects of [5], stemming from the decoherent evolution of the system in the QG medium, which have been interpreted as a violation of angular momentum conservation. Our working hypothesis is that the QG medium conserves angular momentum, at least as far as the ω effect is concerned, extrapolating from no-hair theorems of classical macroscopic black holes. In [8] it was assumed for simplicity that the intrinsic CPTV manifested itself only at the level of the wave-function describing the entangled two-meson state immediately after the decay of the initial resonance, whereas the subsequent time evolution of the system was taken to be due to an effective Hamiltonian. However, given that the origin of this intrinsic CPTV is usually attributed to the decoherent nature of QG, in the spirit discussed originally in [9], a complete description would necessitate the introduction of such “medium” effects in the time evolution as well. To accomplish such a description, at least for the purpose of providing a phenomenological framework for decoherent QG, no detailed knowledge of the underlying microscopic quantum gravity theory is necessary. This can be achieved following the so-called Lindblad or mathematical semi-groups approach to decoherence [11], which is a very efficient way of studying open systems in quantum mechanics.1 The time irreversibility in the evolution of such semigroups, which is linked to decoherence, is inherent in the mathematical property of the lack of an inverse in the semigroup. This approach has been followed for the study of quantum-gravity decoherence in the case of neutral kaons [2,4], as well as other probes, such as neutrinos [12].2 The Lindblad approach to decoherence does not require any detailed knowledge of the environment, apart from energy conservation, entropy increase and complete positivity of the (reduced) density matrix ρ(t) of the subsystem under consideration. The basic evolution equation for the (reduced) density matrix of the subsystem in the Lindblad approach is linear in ρ(t). We notice that the Lindblad part, expressing interaction with the “environment” cannot be written as a commutator

1 The authors are well aware of the limitations of a linear master equation, which may not represent the full quantum- gravity situation. The possibility that quantum gravity may be nonlinear has already been contemplated in earlier papers (see, for instance, [19]), but any non-linearity may not be visible in a situation like neutral kaons, where a mean-field result, parametrized by a (linear) Lindblad equation, should suffice. 2 We would like to stress at this point that such parametrizations in general may also refer to conventional decoherence effects, as a result, for instance, of the passage of the probe through ordinary matter. One therefore should be extremely careful when interpreting such effects, should the latter arise in an experimental situation. For instance, as argued in [13], an uncertainty in the energy of the neutrino beam suffices to induce damping modifications in the respective oscillation formulae, which resemble those of decoherence. In such a case, the disentanglement of such conventional effects may come, if not from their size, which is comparable to that of QG-induced decoherence in optimistic scenaria, definitely from the details of the energy and oscillation-length dependence of the damping decoherence factors, see [14].For instance, ordinary matter effects mimicking CPTV decrease with the energy of the probe, in contrast to genuine QG effects. J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 183

(of a Hamiltonian function) with ρ. Environmental contributions that can be cast in Hamiltonian evolution (commutator form) are absorbed in an “effective” Hamiltonian. The purpose of this article is to extend the analysis of [8], where the intrinsic CPTV manifested itself only in the initial entangled meson state, to include QG environmental entanglement, by resorting to the above-mentioned Lindblad evolution formalism. Speciﬁcally, we shall compute various observables relevant for precision decoherence tests in a φ-factory, placing emphasis on the possibility of disentangling the decoherent evolution parameters α, β, and γ of [2] from the ω parameter of [8], expressing the loss of particle–antiparticle identity mentioned above. The structure of the article is as follows: in Section 2 we present our formalism, including notations and conventions, which agree with those used in [5]. In Section 3 we describe the various observables of a φ factory that constitute sensitive probes of our ω and decoherence α, β, γ effects, and explain how the various effects (or bounds thereof) can be disentangled experimentally. In this section we restrict ourselves only to leading order corrections in the CPTV and decoherence effects, and ignore corrections. Such corrections do not affect the functional form of the CPTV and decoherent evolution terms of the various observables, and their inclusion is discussed in Appendix A. Conclusions and discussion are presented in Section 4. Particular emphasis in the discussion is given in the contamination of the observable giving traditionally (i.e., ignoring decoherent CPTV evolution) the effects, by terms depending on the decoherence coefﬁcients. Finally, as already mentioned, in Appendix A we give the complete formulae for the observables of Section 3, including corrections.

2. Formalism

In this section we shall follow the notation and conventions of [5,15]. The CP violating parameters are deﬁned as S = M + , L = M − , according to which the physical mass- eigenstates are expressed as NS ¯ |KS=√ (1 + S)|K0+(1 − S)|K0 , 2 NL ¯ |KL=√ (1 + L)|K0−(1 − L)|K0 , (2.1) 2 where NS,L are positive normalization constants, M is odd under CP, but even under CPT, and is odd under both CP and CPT. Furthermore,

± β = ± , i i d ±∗ ∗ β = ± ,i= L,S, (2.2) i i d∗ with

i − d = m + Γ =|d|ei(π/2 φSW) 2 − = (3.483 ± 0.006) + i(3.668 ± 0.003) × 10 15 GeV (2.3) ◦ and the super-weak angle [16] φSW = (43.5 ± 0.7) . To ﬁnd the correct expressions for the observables in the density matrix formalism of [2,4,5], matching the standard phenomenology in the CPT-conserving quantum mechanical case, we use 184 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206

+ − the following parametrization for the amplitudes of the decay of |K1, |K2→|π π [5]: √ + − M K → π π = 2A eiδ0 + Re A eiδ2 , 1 0 2 + − iδ2 M K2 → π π = i Im A2 ∗ e , (2.4) M → + − = √i Im A2 iδ ≡ − | |2 ≡| (KL π π ) |2 =| + where A0 is real, and e , δ δ2 δ0, and η+− M → + − L 2 A0 √ (KS π π ) M → 0 0 √1 i Im A2 iδ|2 | |2 ≡| (KL π π ) |2 =| − i Im A2 iδ|2 e , η00 0 0 L 2 e . We also introduce the standard 2 A0 M(KS →π π ) A0 + − + − iφ+− parametrization η+− = π π |KL/ π π |KS=|η+−|e , where [16] |η+−|=(2.288 ± − ◦ 0.014) × 10 3, φ+− = (43.4 ± 0.7) . Similar parametrizations can be used for the corresponding rations representing kaon decays to two neutral as well as three pions. The corresponding phases are denoted φ00 and φ3π . We remind the reader that within quantum mechanics η+− = L + , η00 = L − 2 . All the above numbers are extracted without assuming CPT invariance. In the presence of CPTV decoherence parameters (β) this relation is modified, given that now L,S are replaced ± − ¯ by L,S (2.2). In that case [5], L is related to η+−, which replaces the quantum-mechanical |¯ | iφ+− = − + = η+− defined above, through η+− e L Y+−, where Y+− includes the effects, Y+− + − + − √1 0 ¯ 0 √1 0 ¯ 0 π π |K2/ π π |K1, with K1 = (|K +|K ) and K2 = (|K −|K ) the standard 2 2 CP eigenstates. We will also use the abbreviations SL = S − L and SL = S + L, and ¯ = 1 + Γ 2 (ΓL ΓS), as well as the definition [5] − − γ β d R =| |2 + + 4 Im L , (2.5) L L Γ Γ d∗ √which we shall make use of in the present article. We also note the current experimental values = ± × −3 + = ± × −3 RL (2.30 0.035) 10 ,2ReL (3.27 0.12) 10 . Finally, the relevant density =| | =| | =| | =| | matrices read ρL KL KL , ρS KS KS , ρI KS KL , ρI¯ KL KS . In the present article we shall use values for the decoherence parameters α, β, and γ , which saturate the experimental bounds obtained by the CPLEAR experiment [17]: − − − α<4.0 × 10 17 GeV, |β| < 2.3 × 10 19 GeV,γ<3.7 × 10 21 GeV. (2.6) We note at this stage that in certain models [18], where complete positivity of the density matrix of the entangled kaon state has been invoked, there is only one nonvanishing decoherence parameter (α = γ ) in the formalism of [2], which we adopt here. However, as the analysis of [18] has demonstrated, in such completely positive entangled models there are other parametrizations, with more decoherence parameters. Our intrinsic CPT-violation ω-effects [8] can be easily incorporated in those formalisms, which however we shall not follow here. For our purposes, complete positivity is achieved as the special case of only one decoherence parameter (γ = α) in the formulae that follow. We emphasize, however, that the issue of complete positivity is not entirely clear in a quantum-gravity context, where the relevant master equation may even be nonlinear [19]. In conventional formulations of entangled meson states [20–22] one imposes the requirement of Bose statistics for the state K0K¯ 0 (or B0B¯ 0), which implies that the physical neutral meson–antimeson state must be symmetric under the combined operation CP, with C the charge conjugation and P the operator that permutes the spatial coordinates. Specifically, assuming conservation of angular momentum, and a proper existence of the antiparticle state (denoted by a bar), one observes that, for K0K¯ 0 states which are C-conjugates with C = (−1) (with the J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 185 angular momentum quantum number), the system has to be an eigenstate of P with eigenvalue (−1). Hence, for = 1, we have that C =−, implying P =−. As a consequence of Bose statistics this ensures that for = 1 the state of two identical bosons is forbidden [20]. However, these assumptions may not be valid if CPT symmetry is intrinsically violated, in the sense of the loss of particle–antiparticle identity. In such a case K¯ 0 cannot be considered as identical to K0, and thus the requirement of CP =+, imposed by Bose-statistics, is relaxed. Therefore, the initial state after the φ decay, when expressed in terms of mass-eigenstates contains, in addition to the standard KLKS terms, otherwise forbidden terms of the type KLKL, KSKS : |i=C K (k),K (−k) − K (k),K (−k) S L L S + ω KS(k),KS(−k) − KL(k),KL(−k) , (2.7) where C is to be computed, and ω is a complex parameter. We hasten to emphasize at this stage that the genuine, quantum gravity induced ω-effect, due to the loss of particle–antiparticle identity, should not be confused with the ordinary C-even-background effects that have been studied in [18]. In fact the ω-effect can be easily disentangled from background effects, as has been discussed in [8]. Under the nonunitary decoherent Lindblad evolution [11], appropriately tailor to the present problem [4,5] † ∂t ρ(t)= iρ(t)H − iH ρ(t)+ δHρ(t), (2.8) where H denotes the (non-Hermitian) Hamiltonian of the system (taking decay into account) and δH contains the decoherent effects, the initial state evolves to a mixed state, which assumes the general structure [5]

ρ = Aij ρi ⊗ ρj ,i,j= S,L, (2.9) i,j where the constants Aij depend on the decoherence parameters. The relevant observables are computed by means of double-decay rates with the following generic structure

− − P ; = [ O ] [ O ] λi τ1 λj τ2 (f1,τ1 f2,τ2) Aij tr ρi f1 tr ρj f2 e , (2.10) ij where the constants λi , λj also depend on the decoherence parameters. The double decay rate P(f1,τ; f2,τ) interpolated at equal times τ1 = τ2 = τ is a quantity that pronounces the unusual time dependences of the decoherent evolution, which will also be calculated in this work. Finally, of particular experimental interest are also integrated distributions at ﬁxed time inter- vals τ = τ2 − τ1 > 0 (which we can assume for our purposes here): ∞ 1 P¯ (f ; f ; τ > 0) = d(τ + τ ) P(f ,τ ; f ,τ ), (2.11) 1 2 2 1 2 1 1 2 2 τ 1 → + where the factor 2 originates from the Jacobian when changing variables (τ1,τ2) (τ1 τ2, τ). In the particular case of a φ factory that we analyze here, the pertinent density matrix describing the decay of one kaon of momentum k at time τ1 to a ﬁnal state f1 and the other of 186 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206

momentum −k at time τ2 to a final state f2 is given by ∗ ∗ − − ρ = ρ ⊗ ρ 1 + ω + ω e ΓLτ2 ΓS τ1 S L SL SL ∗ ∗ −ΓLτ1−ΓS τ2 + ρL ⊗ ρS 1 − ω SL − ω e SL ∗ ∗ − − − ⊗ + − i m(τ1 τ2) ρI ρI¯ 1 ω SL ω SL e ∗ ∗ + − − ¯ + − + + ⊗ − + i m(τ1 τ2) (Γ α γ )(τ1 τ2) ρI¯ ρI 1 ω SL ω SL e e iα 2 −i m(τ1+τ2) − |ω| − ρI ⊗ ρI e m iα + + − ¯ + − + + | |2 + ⊗ i m(τ1 τ2) (Γ α γ )(τ1 τ2) ω ρI¯ ρI¯e e m 2γ 2γ 2 −ΓS (τ1+τ2) 2 −ΓL(τ1+τ2) + |ω| − ρS ⊗ ρSe + |ω| + ρL ⊗ ρLe Γ Γ ¯ 2β −i mτ1−ΓS τ2−(Γ +α−γ)τ1 + ρI ⊗ ρSe ω − d ¯ 2β −i mτ2−ΓS τ1−(Γ +α−γ)τ2 − ρS ⊗ ρI e ω + d + − − ¯ + − ∗ 2β + ⊗ i mτ1 ΓS τ2 (Γ α γ)τ1 − ρI¯ ρSe ω ∗ d + − − ¯ + − ∗ 2β − ⊗ i mτ2 ΓS τ1 (Γ α γ)τ2 + ρS ρI¯e ω ∗ d − − − ¯ + − ∗ 2β + ⊗ i mτ1 ΓLτ2 (Γ α γ)τ1 + ρI ρLe ω ∗ d − − − ¯ + − ∗ 2β − ⊗ i mτ2 ΓLτ1 (Γ α γ)τ2 − ρL ρI e ω ∗ d + − − ¯ + − 2β + ⊗ i mτ1 ΓLτ2 (Γ α γ)τ1 + ρI¯ ρLe ω d +i mτ −Γ τ −(Γ¯ +α−γ)τ 2β − ρ ⊗ ρ ¯e 2 L 1 2 ω − (2.12) L I d from the which the expression for the various constants in Eq. (2.10) may be gleaned. An important remark is now in order. Notice that above we have treated |ω|2 effects as being of comparable order to decoherence α, γ , and have kept only terms at most linear in the decoherence parameters. This is a very simplifying assumption, which, due to a lack of a microscopic theory of QG, cannot be made rigorous. However, as we shall discuss below, a reasonable a posteriori justification of this approximation may come from the fact that such terms appear on an equal footing as medium-generated corrections to certain physically important terms of (2.12) to be analyzed below. In Eq. (2.12) we notice that, starting from the third line, terms of entirely novel type make their appearance. Such terms are due to both the intrinsic CPTV loss of particle–antiparticle identity (ω-related) and decoherent evolution (α, β, γ -related). Such terms persist in the limit τ1 = τ2 = 0, due to a specific choice of boundary conditions in the solution of the appropriate density-matrix evolution equation (2.8), expressing the omnipresence of the QG space–time J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 187 foam effects. In particular, terms linear in ω are always accompanied by the parameter β, whereas terms quadratic in ω always combine with the α or γ parameters. Of particular physical importance are the otherwise forbidden terms ρL ⊗ ρL and ρS ⊗ ρS .Atfirstsight,itistemptingto interpret [5] such terms as signaling a subtle breakdown of angular momentum conservation, in addition to the CPTV introduced by the “medium”. That seems to be an inescapable conclusion, if one were to begin the time evolution from a purely antisymmetric wave function (i.e., with ω = 0in(2.7)), as assumed in [5], which is an eigenstate of the orbital angular momentum L with eigenvalue = 1. Since, then, the medium generates in the final state forbidden terms violating this last property, one interprets this as nonconservation of L. However, if one accepts [8] that the intrinsic CPTV affects not only the evolution but also the concept of the antiparticle, such terms are present already in the initial state. In such a case, there is no issue of nonconservation of angular momentum; the only effect of the time evolution is to simply modify the relative weights between ρL ⊗ ρL and ρS ⊗ ρS terms. One can therefore say that QG may behave as a CPTV medium differentiating between particles and antiparticles. From a physical point of view one may even draw a vague analogy between QG media and “regenerators”, in the following sense: exactly as the regenerator differ- entiates between K0 and K¯ 0, by means of different relative interactions, in a similar way QG acts differently on particles and antiparticles; however, unlike the regenerator case, where the experimentalist can control its position, due to their universal nature, QG effects are present even in the initial state after the φ decay.

3. Observables at φ factories

In this section we present a detailed study of a variety of observables measurable at φ factories, and determine their dependence on both intrinsic CPTV (ω) and decoherence (α, β, γ ) parameters, in an attempt to disentangle and separately constrain the two possible effects. Specifically, we will derive expressions for the double-decay rates of Eq. (2.10) and their integrated counterparts of Eq. (2.11), for the cases of identical as well as general (nonidentical) final states. We will restrict ourselves to linear effects in α, β, γ , while keeping terms linear and quadratic in ω,for the reasons explained in the previous section. For our purposes here we shall be interested in the + − 0 decays of KL,S to final states consisting of two pions, π π or π , three pions, as well as semileptonic decays to +π −ν or −π +ν¯. In this section we shall also ignore explicit effects. Such effects appear in the branching ratios η of various pion channels, and their inclusion affects the form of the associated observables. We present complete formulae for the relevant double-decay time distributions, including such effects, in Appendix A. As we note there, the inclusion of such effects does not affect significantly the functional form of the decoherent/CPTV evolution. In the density-matrix formalism of [2,5] the relevant observables are given by X+− Y+− O+− = , Y∗ | Im A eiδ|2 +− 2 X Y O = 00 00 , 00 Y∗ |−2i Im A eiδ|2 00 2 |a|2 11 O + = , 2 11 |a|2 1 −1 O − = , 2 −11 188 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 00 O =|X |2 , (3.1) 3π 3π 01 √ √ √ iδ 2 −iδ iδ where X+− =| 2A0 + Re√ A2e | , Y+− = ( 2A0 + Re A2e )(i Im A2e ), X00 =| 2A0 − iδ 2 −iδ iδ 2ReA2e | and Y00 = ( 2A0 − 2ReA2e )(−2i Im A2e ), and X3π = 3π|K2,inanobvi- ous shorthand notation Of , where f denotes the observable associated with a kaon decay leading to f final state, for instance f =+−denotes decay to π+π −, f = 00 denotes decay to two π 0, f = + denotes semileptonic decay +π −ν etc. Notice that if effects are ignored, which we shall do here for brevity, then we do not distinguish between the observables for the two cases of the two-pion final states. Note that above we ignored CPT violating effects in the decay amplitudes to the final states. If such effects are included then the observables acquire the form [5]: 2 1 Y+− O+− =|X+−| , ∗ | |2 Y+− Y+− 1 Y O =|X |2 00 , 00 00 Y ∗ |Y |2 00 00 |a + b|2 11 O + = , 2 11 |a − b|2 1 −1 O − = , −11 2 | |2 ∗ 2 Y3π Y3π O3π =|X3π | , (3.2) Y3π 1 ππ|K− Re(B ) Re B where X = ππ|K+, Y = , and more specifically Y+− = ( 0 ) + , Y+− = ( 0 ) ππ|K+ A0 A0 3π|K1 − 2 , Y π = , and Re(a/b) and Re(B )/A parametrize CPT violation in the decay am- 3 3π|K2 0 0 plitudes to the appropriate final states.

3.1. Identical ﬁnal states

Consider ﬁrst the case where the kaons decay to π+π −. Ignoring CPTV in the decays and effects, the relevant observable, O+−, is obtained from the ﬁrst of Eq. (3.1) by setting A2 = 0, namely 2A2 0 O+− = 0 . 00 The corresponding double-decay rate is then + − + − P π π ,τ ; π π ,τ 1 2 − − − − = 4 ΓLτ2 ΓS τ1 + ΓLτ1 ΓS τ2 2A0 RL e e 2 −(Γ¯ +α−γ )(τ +τ ) − 2|¯η+−| cos m(τ − τ ) e 1 2 1 2 2 2γ 4β sin φ+− −Γ (τ +τ ) + |ω| − − |¯η+−| e S 1 2 Γ |d| cos φSW |¯ | 4β η+− −(Γ¯ +α−γ)τ −Γ τ + sin( mτ + φ+− − φ )e 1 S 2 |d| 1 SW J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 189

|¯ | 4β η+− −(Γ¯ +α−γ)τ −Γ τ + sin( mτ + φ+− − φ )e 2 S 1 |d| 2 SW −Γ (τ )−(Γ¯ +α−γ)τ + 2|ω||η ¯+−| cos(τ + φ+− − 2Ω)e S 2 1 1 ¯ −ΓS τ1−(Γ +α−γ)τ2 − cos(τ2 + φ+− − Ω)e . (3.3) ¯ The integrated observable P(f1,f2, τ)is obtained by integrating over (τ1 +τ2) keeping τ fixed. The result is + − + − P¯ π π ; π π ; τ − − − ¯ + − τΓL + τΓS (Γ α γ) τ 4 e e 2 e = A RL −|¯η+−| cos( m τ) 0 Γ + Γ (Γ¯ + α − γ) L S −ΓS τ 2 2γ 4β sin φ+− e + |ω| − − |¯η+−| Γ |d| cos φSW 2ΓS + 1 m2 + (Γ¯ + α − γ)2 + 2(Γ¯ + α − γ)Γ + Γ 2 S S |¯ | 4β η+− ¯ − τΓ × mcos(φ+− − φ ) + (Γ + α − γ + Γ ) sin(φ+− − φ ) e S |d| SW S SW − τ(Γ¯ +α−γ) + e mcos( m τ + φ+− − φ ) SW ¯ + (Γ + α − γ + Γ ) sin( m τ + φ+− − φ ) S SW − τΓS ¯ + 2|ω||η ¯+−|e (Γ + α − γ + ΓS) cos(φ+− − Ω)− msin(φ+− − Ω) (Γ¯ +α−γ) τ ¯ − 2|ω||η ¯+−|e (Γ + α − γ + Γ ) cos( m τ + φ+− − Ω) S − msin( m τ + φ+− − Ω) (3.4) whichweplotinFigs. 1 and 2. In the figures we assume rather large vales of |ω|, of order |¯η+−|. As evidenced by looking at the curves for relatively large values of τ , this assumption, together with the assumed values of the decoherence parameters (2.6), allows for a rather clear disentanglement of the decoherence plus ω situation from that with only ω effects present (i.e., unitary evolution). Indeed, in the latter case, the result for the pertinent time-integrated double-decay rate quickly converges to the quantum-mechanical situation, in contrast to the decoherent case. Moreover, the nonzero value of the time-integrated asymmetries at τ = 0, is another clear deviation from the quantum mechanical case, which in the (unitary evolution) case with only ω = 0 is pronounced, due to the presence of |¯η+−| factors [8]. We notice at this stage, though, that this may not be the case in an actual quantum gravity foam situation. For instance, it has been argued [18] that if complete positivity is imposed for the density matrix of entangled states, then the only nontrivial decoherence parameter would be α = γ>0, whilst β = 0. It is not inconceivable that |ω|2 are of the same order as 2γ/ Γ, in which case, for all practical purposes the integrated curve for the two-pion double decay rate (3.4) would pass through zero for τ = 0. In such a situation, one should also look at the linear in ω interference terms for τ = 0, in order to disentangle the ω from γ effects. Lacking a fundamental microscopic theory underlying these phenomenological considerations, however, which would make definite predictions on the relative order of the various CPTV effects, we are unable to make more definite statements at present. 190 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206

± ± Fig. 1. Plots of P¯ (π ,π , τ) vs. τ extended to a large range of τ . The dashed curves are with α, β, γ, ω all non-zero, the long-dashed curve has only ω = 0, whilst the solid curve represents α, β, γ, ω = 0. Here and in all subsequent ﬁgures the values for α, β,andγ saturate the CPLEAR bound of Eq. (2.6).Thevaluesforω and Ω are (from left to right, and top to bottom) (i) |ω|=|¯η+−|, Ω = φ+− − 0.16π, (ii) |ω|=|¯η+−|, Ω = φ+− + 0.95π, (iii) |ω|=0.5|¯η+−|, = +− + | |= |¯+−| = +− P¯ |¯+−|2 4 Ω φ 0.16π,(iv) ω 1.5 η , Ω φ . is in units of η τs 4A0 and τ is in units of τS .

Fig. 2. As in Fig. 1, but shown in detail for small values of τ near the origin, to illustrate the difference between the various cases. Again, the dashed curves are with α, β, γ, ω all non-zero, the long-dashed curve has only ω = 0, whilst the solid curve represents α, β, γ, ω = 0.

We next look at the semileptonic decays of kaons. First we consider the case where both kaons + decay to . As discussed in the beginning of the section, (3.1) the relevant observable is Ol+, |a|2 11 O + = . l 2 11 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 191 = − = + + ∗ = Recalling that L M and S M so (SL SL) 4ReM we have for the double-decay rate: ± ± P l ,τ1; l ,τ2 4 a − − − − = 1 ± 4Re( ) e ΓLτ2 ΓS τ1 + e ΓLτ1 ΓS τ2 8 M ¯ −(Γ +α−γ )(τ1+τ2) − 2 1 ± 4Re(M ) cos m(τ1 − τ2) e ¯ 2 −(Γ +α−γ )(τ1+τ2) 2 −ΓS (τ1+τ2) −ΓL(τ1+τ2) − 2|ω| cos m(τ1 + τ2) e +|ω| e + e 2γ 2α ¯ −ΓL(τ1+τ2) −ΓS (τ1+τ2) −(Γ +α−γ )(τ1+τ2) + e − e + sin m(τ1 + τ2) e Γ m 4β − ¯ + − − ± sin( mτ − φ ) + 2|ω| cos( mτ − Ω) e (Γ α γ)τ1 e ΓS τ2 |d| 1 SW 1 4β − ¯ + − − ± sin( mτ − φ ) − 2|ω| cos( mτ − Ω) e (Γ α γ)τ2 e ΓS τ1 |d| 2 SW 2 4β − ¯ + − − ± sin( mτ + φ ) + 2|ω| cos( mτ + Ω) e (Γ α γ)τ1 e ΓLτ2 |d| 1 SW 1 4β − ¯ + − − ± sin( mτ + φ ) − 2|ω| cos( mτ + Ω) e (Γ α γ)τ2 e ΓLτ1 (3.5) |d| 2 SW 2 and the integrated quantity reads ± ± P¯ l ; l , τ − − a4 e ΓL τ + e ΓS τ = 2(1 ± 4ReM ) 16 ΓS + ΓL ¯ e−(Γ +α−γ) τ − (1 ± 4ReM )2 cos( m τ) (Γ¯ + α − γ) ¯ − ¯ + − (Γ + α − γ)cos( m τ) − msin( m τ) − 2|ω|2e τ(Γ α γ) (Γ¯ + α − γ)2 + m2 − − − − e ΓS τ e ΓL τ 2γ e ΓL τ e ΓS τ +|ω|2 + + − ΓS ΓL Γ ΓL ΓS ¯ 2α − ¯ + − ((Γ + α − γ)sin( m τ) + mcos( m τ)) + e (Γ α γ) τ m (Γ¯ + α − γ)2 + m2 ± 4 m2 + (Γ¯ + α − γ)2 + 2(Γ¯ + α − γ)Γ + Γ 2 S S 2β − × e τΓS mcos(φ ) − (Γ¯ + α − γ + Γ ) sin(φ ) |d| SW S SW − τΓS ¯ +|ω|e msin(Ω) + (Γ + α − γ + ΓS) cos(Ω) 2β − ¯ + − + e (Γ α γ) τ |d| ¯ × mcos( m τ − φSW) + (Γ + α − γ + ΓS) sin( m τ − φSW) 192 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206

− ¯ + − +|ω|e (Γ α γ) τ ¯ × − msin( m τ − Ω)+ (Γ + α − γ + ΓS) cos( m τ − Ω)

± 4 2 ¯ 2 ¯ 2 m + (Γ + α − γ) + 2(Γ + α − γ)ΓL + Γ L 2β − × e τΓL mcos(φ ) + (Γ¯ + α − γ + Γ ) sin(φ ) |d| SW L SW − τΓL ¯ +|ω|e − msin(Ω) + (Γ + α − γ + ΓL) cos(Ω)

2β − ¯ + − + e (Γ α γ) τ |d| ¯ × mcos( m τ − φSW) + (Γ + α − γ + ΓL) sin( m τ − φSW) − ¯ + − +|ω|e (Γ α γ) τ ¯ × − msin( m τ + Ω)+ (Γ + α − γ + ΓL) cos( m τ + Ω) . (3.6)

We plot the corresponding integrated double-decay rate for ++ in Fig. 3. Next we consider the case in which one of the kaons decays to + and the other to −. On not- − ing that the relevant observable corresponding to the case is associated with the operator Ol− of (3.1) |a|2 1 −1 O − = l 2 −11

+ + Fig. 3. Plots of P¯ (l ,l , τ), in units of τa4/4vs. τ with α, β, γ, ω nonzero as dashed curve and α, β, γ, ω = 0is the solid curve, and the long-dashed curve denotes the case where only ω = 0. J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 193

the relevant double-decay rate reads: + − P l ,τ ; l ,τ 1 2 4 a − − − − = 1 + 4Re( − β/d) e ΓLτ2 ΓS τ1 + 1 + 4Re(β/d − ) e ΓLτ1 ΓS τ2 8 − ¯ + − + + 2 cos( m τ) + 4Im( + β/d)sin( m τ) e (Γ α γ )(τ1 τ2) 2α ¯ 2 −(Γ +α−γ )(τ1+τ2) + 2|ω| cos m(τ1 + τ2) − sin m(τ1 + τ2) e m 2γ − + 2γ − + + |ω|2 − e ΓS (τ1 τ2) + |ω|2 + e ΓL(τ1 τ2) Γ Γ 4β − ¯ + − − + ( mτ − φ ) + |ω| ( mτ − Ω) e (Γ α γ)τ1 e ΓS τ2 | | sin 1 SW 2 cos 1 d 4β − ¯ + − − − ( mτ − φ ) − |ω| ( mτ − Ω) e (Γ α γ)τ2 e ΓS τ1 | | sin 2 SW 2 cos 2 d 4β − ¯ + − − + ( mτ + φ ) + |ω| ( mτ + Ω) e (Γ α γ)τ1 e ΓLτ2 | | sin 1 SW 2 cos 1 d 4β − ¯ + − − − sin( mτ + φ ) − 2|ω| cos( mτ + Ω) e (Γ α γ)τ2 e ΓLτ1 (3.7) |d| 2 SW 2 and the integrated over time distribution is + − P¯ l ; l , τ − − a4 2e ΓL τ 2e ΓS τ = 1 + 4Re( − β/d) + 1 + 4Re(β/d − ) 16 ΓS + ΓL ΓS + ΓL ¯ e−(Γ +α−γ) τ + 2 cos( m τ) + 4Im( + β/d)sin( m τ) (Γ¯ + α − γ) ¯ e−(Γ +α−γ) τ + 2|ω|2 (Γ¯ + α − γ)cos( m τ) − msin( m τ) (Γ¯ + α − γ)+ m2 2α − mcos( m τ) + (Γ¯ + α − γ)sin( m τ) m − − 2γ e ΓS τ 2γ e ΓL τ + |ω|2 − + |ω|2 + Γ ΓS Γ ΓL + 4 m2 + (Γ¯ + α − γ)2 + 2(Γ¯ + α − γ)Γ + Γ 2 S S 2β − × e τΓS mcos(φ ) − (Γ¯ + α − γ + Γ ) sin(φ ) |d| SW S SW − τΓS ¯ +|ω|e msin(Ω) + (Γ + α − γ + ΓS) cos(Ω) 2β − ¯ + − − e (Γ α γ) τ m ( m τ − φ ) | | cos SW d ¯ + (Γ + α − γ + ΓS) sin( m τ − φSW) − ¯ + − −|ω|e (Γ α γ) τ − msin( m τ − Ω) 194 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 ¯ + (Γ + α − γ + ΓS) cos( m τ − Ω)

+ 4 m2 + (Γ¯ + α − γ)2 + 2(Γ¯ + α − γ)Γ + Γ 2 L L 2β − × e τΓL mcos(φ ) + (Γ¯ + α − γ + Γ ) sin(φ ) |d| SW L SW − τΓL ¯ +|ω|e − msin(Ω) + (Γ + α − γ + ΓL) cos(Ω) 2β − ¯ + − − e (Γ α γ) τ mcos( m τ − φ ) |d| SW ¯ + (Γ + α − γ + ΓL) sin( m τ − φSW) − ¯ + − +|ω|e (Γ α γ) τ − msin( m τ + Ω) ¯ + (Γ + α − γ + ΓL) cos( m τ + Ω) . (3.8)

We next give the three pion decay channels using the relevant observable (3.1), | |2 ∗ 2 Y3π Y3π O3π =|X3π | . (3.9) Y3π 1

Here we take |Y3π |=0, since we ignore the associated CP and CPT violation in the decay and obtain 00 O =|X |2 . (3.10) 3π 3π 01

4 P¯ +; − |a| = Fig. 4. Plots of (l l , τ), in units of τS 4 ,vs. τ with α, β, γ, ω nonzero as dashed curve and α, β, γ, ω 0is the solid curve, and the long-dashed curve denotes the case where only ω = 0. J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 195

Fig. 5. As in Fig. 4, but shown in detail to better illustrate the difference between the various cases. Again, the dashed curves are with α, β, γ, ω all nonzero, the long-dashed curve has only ω = 0, whilst the solid curve represents α, β, γ, ω = 0.

|X |4 P¯ ; |¯+−|2 3π Fig. 6. Plots of (3π 3π, τ), in units of τL η 2 (for convenience), vs. τ (also in units of τL), with α, β, γ, ω non zero as dashed curve and α, β, γ, ω = 0 is the solid curve, and the long-dashed curve denotes the case where only |ω|=|¯η+−| = 0 (this value is chosen for concreteness). We see that the CPTV effects are pronounced near the origin and are easily disentangled from the quantum mechanical case (which coincides with the horizontal axis).

This leads to the following expression for the associated double-time distribution

P(3π,τ ; 3π,τ ) 1 2 4 |X3π | − − − − = R e ΓLτ2 ΓS τ1 + R e ΓLτ1 ΓS τ2 2 S S − ¯ + − + − 2|¯η |2 cos m(τ − τ ) e (Γ α γ )(τ1 τ2) 3π 1 2 2γ 4β sin φ 2 3π −ΓL(τ1+τ2) + |ω| + + |¯η3π | e Γ |d| cos φSW ¯ −ΓLτ1−(Γ +α−γ)τ2 +|¯η3π |e 196 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 4β × −2|ω| cos( mτ − φ + Ω)+ sin( mτ − φ + φ ) 2 3π |d| 2 3π SW − − ¯ + − +|¯ | ΓLτ2 (Γ α γ)τ1 η3π e 4β × 2|ω| cos( mτ − φ + Ω)+ sin( mτ − φ + φ ) , (3.11) 1 3π |d| 1 3π SW which integrated over time gives ¯ P(3π,τ1; 3π,τ2) − − − ¯ + − |X |4 2R (e ΓL τ + e ΓS τ ) e (Γ α γ) τ = 3π S − 2|¯η |2 cos( m τ) 4 Γ + Γ 3π ¯ + − S L Γ α γ −ΓL τ 2 2γ 4β sin φ3π e + |ω| + + |¯η3π | Γ |d| cos φSW ΓL |¯ | −(Γ¯ +α−γ) τ + 4 η3π e 2 + ¯ + − 2 + ¯ + − + 2 m (Γ α γ) 2(Γ α γ)ΓL ΓL −(Γ¯ +α−γ) τ ¯ × |ω|e −(Γ + α − γ + ΓL) cos( m τ − φ3π + Ω) + msin( m τ − φ3π + Ω) 2β − ¯ + − + e (Γ α γ) τ (Γ¯ + α − γ + Γ ) ( m τ − φ + φ ) | | L sin 3π SW d + mcos( m τ − φ + φ ) 3π SW − +|ω|e ΓL τ (Γ¯ + α − γ + Γ ) (Ω − φ ) − m (Ω − φ ) L cos 3π sin 3π 2β − + e ΓL τ (Γ¯ + α − γ + Γ ) sin(φ − φ ) + mcos(φ − φ ) . |d| L SW 3π SW 3π (3.12) We plot this function vs. τ in Fig. 6. We use units of τL for convenience. The quantum- mechanical case with ω = α = β = γ = 0 coincides with the horizontal axis. As we observe from the graph, the CPTV and decoherence effects are pronounced near τ = 0. Thus, in principle this is the cleanest way of bounding such effects, but unfortunately in practice this is a very difﬁcult channel to measure experimentally. This completes the analysis of identical-ﬁnal-states observables.

3.2. General ﬁnal states

We consider in this subsection observables for the case where the final states are different. We first consider the case in which one kaon decays to π+π − and the other to π 0π 0.Forthe double-decay rate we find + − P π π ,τ ; π 0π 0,τ 1 2 − − +− − − = 4 00 ΓLτ2 ΓS τ1 + ΓLτ1 ΓS τ2 2A0 RL e RL e −(Γ¯ +α−γ )(τ +τ ) − |¯η+−||η ¯ | m(τ − τ ) + φ+− − φ e 1 2 2 00 cos 1 2 00 2γ 2β (|¯η+−| sin φ+− +|¯η00| sin φ00) − + + |ω|2 − − e ΓS (τ1 τ2) Γ |d| cos φSW J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 197

− − ¯ + − +|¯ | ΓS τ2 (Γ α γ)τ1 η+− e 4β × 2|ω| cos( mτ + φ+− − Ω)+ sin( mτ + φ+− − φ ) 1 |d| 1 SW − − ¯ + − +|¯ | ΓS τ1 (Γ α γ)τ2 η00 e 4β × −2|ω| cos( mτ + φ − Ω)+ sin( mτ + φ − φ ) . (3.13) 2 00 |d| 2 00 SW The integrated-over-time distribution is given by + − P¯ π π ; π 0π 0; τ −ΓL τ −ΓS τ = 4 00 e + +− e 2A0 RL RL ΓL + ΓS ΓL + ΓS ¯ e−(Γ +α−γ) τ −|¯η+−||η ¯00| cos( m τ + φ+− − φ00) Γ¯ + α − γ − 2γ 2β (|¯η+−| sin φ+− +|¯η | sin φ ) e ΓS τ + |ω|2 − − 00 00 Γ |d| cos φSW 2ΓS + 2 m2 + (Γ¯ + α − γ)2 + 2(Γ¯ + α − γ)Γ + Γ 2 S S −ΓS τ ¯ × |¯η+−|e |ω| (Γ + α − γ + ΓS) cos(φ+− − Ω)− msin(φ+− − Ω) 2β ¯ + (Γ + α − γ + Γ ) (φ+− − φ ) + m (φ+− − φ ) | | S sin SW cos SW d −(Γ¯ +α−γ) τ ¯ +|¯η00|e |ω| −(Γ + α − γ + ΓS) cos( m τ + φ00 − Ω) + msin( m τ + φ00 − Ω) 2β + (Γ¯ + α − γ + Γ ) ( m τ + φ − φ ) | | S sin 00 SW d + mcos( m τ + φ00 − φSW) . (3.14)

Finally we consider the case in which one of the ﬁnal states consists of two pions, and the other is the result of a kaon semileptonic decay. The relevant double-decay rate is given by ± ± P π ,τ1; l ,τ2 2 2 A a γ − − − − = 0 1 ± δ + e ΓLτ2 ΓS τ1 + R e ΓLτ1 ΓS τ2 4 L Γ L −(Γ¯ +α−γ )(τ +τ ) ∓ 2|¯η+−| cos m(τ − τ ) + φ+− e 1 2 1 2 2γ 4β|¯η+−| sin φ+− − + + |ω|2 − − e ΓS (τ1 τ2) | | Γ d cos φSW 4β − − ¯ + − ± sin( mτ − φ ) − 2|ω| cos( mτ − Ω) e ΓS τ1 (Γ α γ)τ2 , (3.15) |d| 2 SW 2 = + where we used the deﬁnition δL 2ReL . 198 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206

+ − |X+−|2|X |2 P¯ ; 0 0; |¯+−|2 00 Fig. 7. Plots of (π π π π τ), in units of τS η 2 ,vs. τ with α, β, γ, ω non zero as dashed curve and α, β, γ, ω = 0 is the solid curve, and the long-dashed curve denotes the case where only ω = 0.

The integrated double-decay rate reads ± ± P¯ π ; l , τ 2 2 −ΓL τ −ΓS τ A0a γ e e = 1 ± δL + + RL 4 Γ ΓL + ΓS ΓL + ΓS ¯ e−(Γ +α−γ) τ ∓|¯η+−| cos( m τ + φ+−) (Γ¯ + α − γ) − 2γ 4β|¯η+−| sin φ+− e ΓS τ + |ω|2 − − Γ |d| cos φSW 2ΓS ± 2 2 ¯ 2 ¯ 2 m + (Γ + α − γ) + 2(Γ + α − γ)ΓS + Γ S 2β − ¯ + − + e (Γ α γ) τ mcos( m τ − φ ) |d| SW ¯ − (Γ + α − γ + ΓS) sin( m τ − φSW) − ¯ + − −|ω|e (Γ α γ) τ − msin( m τ − Ω) ¯ + (Γ + α − γ + ΓS) cos( m τ − Ω) . (3.16)

This completes our analysis of observables in a φ factory that can be used as sensitive probes for tests of possible CPTV and quantum decoherence, to leading order in the small parameters parametrizing the effects. J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 199

2 2 P¯ ±; ± |X+−| |a| Fig. 8. Plots of (π l , τ), in units of τS 8 ,vs. τ with α, β, γ, ω non zero as dashed curve and α, β, γ, ω = 0 is the solid curve, and the long-dashed curve denotes the case where only ω = 0. Notice that in the scale chosen, the dashed and long-dashed curves overlap completely.

4. Discussion and conclusions

In this work, we have embarked into a combined treatment of decoherent and intrinsic CPTV effects in a φ factory. By studying a variety of observables, involving identical as well as general final states, we have derived analytical expressions for double-decay rates and their integrated counterparts (over time τ1 + τ2, keeping τ = τ1 − τ2 fixed), which we plotted as functions of τ . Our analysis included ω, as well as decoherence α, β, γ effects. We presented the results to leading order in these small parameters. Although the pertinent formulae are algebraically rather complex, nevertheless from our general study it became evident that one may disentangle the ω from the decoherent evolution effects by looking simultaneously at different regimes of τ , namely, (i) τ = 0, (ii) intermediate values of τ (as compared with τS , which sets a convenient characteristic time scale in the problem), with emphasis on interference terms with sinusoidal time dependence, and (iii) rather large values of τ , with emphasis on the exponential damping of the relevant quantities, which is affected (slowed down) by the presence of decoherence terms α − γ ,inanω-independent way. In addition to these results in a φ factory, one can obtain valuable information on the decoherence parameters α, β, and γ by looking at experiments involving single neutral kaon beams, such as CPLEAR [17], which is achieved through flavour tagging of the neutral kaon by means of the electric charge of the pion on the complementary side. As discussed in [4], the decoherence parameters α, β, γ can be separately disentangled by combined studies of several kaon asymmetries, which are again time-dependent functions. Notice though that, if complete positivity of entangled state density matrices is imposed [18], then only one decoherence parameter γ>0 survives in the model of [2], which facilitates the situation enormously. We would like at this point to make a clarification concerning the important differences between proposed experiments here, using entangled kaon states, and some existing proposals, 200 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 notably by the CPLEAR Collaboration [23], to measure EPR correlations. The reader should notice that the accuracy for measuring the appropriate observables in [23] appears to be of order 10−1; however, this accuracy pertains only to an asymmetry built from pp¯ → K0K¯ 0 in CPLEAR, looking for the dilepton decay channel, between “like” (K0K0 or K¯ 0K¯ 0) and “unlike” (K0K¯ 0 or K¯ 0K0) decays after time evolution. As shown in our paper, the most interesting observable for the ω-effect comes from identical decay channels, i.e., in this case the “intra” asymmetry between K0K0 and K¯ 0K¯ 0, which was not discussed in Ref. [23]. Furthermore, our observation [8] that the channel (π+π −)(π +π −) is automatically enhanced by three orders of magnitude, due to the relative (ω/η+−) amplitude between the “wrong” and the “right” symmetry states suggests that even a 10−1 experimental effect would represent a 10−4 effect in ω.In fact, the expectation for the channel e+e− → K0K¯ 0 in an upgraded DANE is definitely much better, anticipating an experimental effect of the order of 10−4–10−5. In our analysis we saturated the bounds for the decoherence parameters α, β, γ obtained from CPLEAR [17], to disentangle the ω effect by looking at decoherent evolution of observables in a φ factory. As mentioned above, the most sensitive probe appears at first sight to be the two- charged-pion channels, as a result of enhancement factors of η¯+−. In fact the clearest test of deviation from quantum mechanics is to look at the behaviour of the pertinent time-integrated decay rate (3.4) near the τ = 0 regime, where the novel CPTV effects would yield a nonzero result for the integrated asymmetry. However, in the presence of decoherence there is a reduction of the value of the asymmetry as compared to the pure ω = 0 unitary case of [8], by shifting the value of the asymmetry at τ = 0 2 2 from |ω | to |ω |−2γ/ Γ (in appropriate√ units). This reduction may be significant in the case where the |ω| effects are of order γ/ Γ. Of course, for a reliable order-of-magnitude estimate of the ω-parameter, one needs to resort to detailed microscopic models of QG space–time foam, a task we hope to undertake in the near future. The fact that the interference terms for τ = 0of the asymmetry (3.4), which depend sinusoidally on τ , are proportional to |ω|, and they do not depend (apart from damping factors) on decoherence parameters, allows in principle for a disentanglement of ω from decoherence (γ,...) effects. Notice also that another clear (in principle) probe of such effects would be the time-integrated observable (3.12), associated with three-pion decay channels, near τ = 0. Unfortunately, however, the current experimental sensitivity for this observable is low. Before closing we would like to make a few comments on the contamination of the actual observable for the measurement of by decoherence [5] and intrinsic CPTV effects. The traditional observable for the measurement of is the asymmetry based on charged-pion/neutral-pion (3.14)-type observables:

¯ + − 0 0 ¯ 0 0 + − + − P(π π ; π π ; τ) − P(π π ; π π ; τ) A π ,π ; π 0,π0; δτ = P¯ (π +π −; π 0π 0; τ) + P¯ (π 0π 0; π +π −; τ) = 3Re I − 3Im I . (4.1) 1 2 In particular we ﬁnd 1 β −ΓL τ I1 = e 1 + 2 sin(φSW − φ+−) D |d||η ¯+−| β −ΓS τ − e 1 + 2 sin(φSW − φ+−) −|z| sin(φSW − φ+− + φz) |d||η ¯+−| J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 201 |ω| + |z| cos(Ω − φ+− + φz) |¯η+−| −(Γ¯ +α−γ) τ β + e 2 |z| sin( m τ + φ+− − φSW − φz) |d||η ¯+−| |ω| − |z| cos( m τ − Ω + φ+− − φz) , (4.2) |¯η+−| 1 β −ΓL τ I2 = e 2 cos(φSW − φ+−) D |d||η ¯+−| β −ΓS τ − e 2 cos(φSW − φ+−) −|z| cos(φSW − φ+− + φz) |d||η ¯+−| |ω| + |z| sin(Ω − φ+− + φz) |¯η+−| −(Γ¯ +α−γ) τ β + e 2sin( m τ) − 2 |z| cos( m τ + φ+− − φSW − φz) |d||η ¯+−| |ω| − |z| sin( m τ + Ω − φSW − φz) , (4.3) |¯η+−| and − γ β sin(2φSW − φ+−) D = e ΓL τ 1 + + 2 Γ |¯η+−|2 |d||η ¯+−| cos(φ ) SW − γ ΓL + e ΓS τ 1 + Γ |¯η+−|2 Γ S β sin(2φSW − φ+−) + 2 − 2|z| sin(φSW + φz − φ+−) |d||η ¯+−| cos(φ ) SW |ω|2Γ¯ |ω| + − 2 |z| cos(Ω + φz − φ+−) |¯η+−|2Γ |¯η+−| S −(Γ¯ +α−γ) β − e 2 cos( m τ) − 4 |z| sin( m τ + φ+− − φSW − φz) |d||η ¯+−| 2|ω| + |z| cos( m τ + φ+− − Ω − φz) , (4.4) |¯η+−|

iφ 2Γ¯ where we have defined [5]: |z|e z = ¯ . ΓS +Γ +i m It therefore becomes clear that, although in conventional situations, where the foam and intrinsic CPTV ω-effects are absent, the quantities Re(/) and Im(/) can be extracted from a measurement of A(π +,π−; π 0,π0; τ) by a simple two-parameter fit, in the presence of the quantum-gravity effects this is no longer true. The coefficients Ii,i = 1, 2 are modified in such a case by decoherence [5] and ω-dependent terms. It is worthy of pointing out at this stage that the ω-dependent terms in the expression for the −Γ τ above asymmetry are always accompanied by factors e S . Therefore, in the limit ΓS τ 1 such terms are suppressed. We remind the reader that, in this limit, in conventional situations, A + −; 0 0; → one has simply that (π ,π π ,π τ) 3Re . In contrast, in the CPTV foamy situation the same limit contains, to leading order, terms proportional to both real and imaginary parts 202 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 of /, with coefficients dependent on the decoherence parameters, β,γ, but independent of ω. It is also worthy of noting the form of I2 in this limit: I2 = O(β) to leading order in the small parameters. In complete positivity models [18], therefore, which require β = 0, such I2 terms are absent from the right-hand-side of the observable (4.1), and the (limiting) result for the −|O |¯ |2 | asymmetry involves, in such a case, only the factor 3 Re (1 (γ / Γ η+− ) ) to leading order. The above issues are important to bear in mind in experimental derivations of the parameter, which in the general case require disentanglement of the (possible) decoherence and ω effects by means of a combined study of the φ-factory observables described in this article and in [5].

Acknowledgements

This work has been partially supported by the Grant CICYT FPA2002-00612. N.E.M. wishes to thank the University of Valencia, Department of Theoretical Physics and IFIC for the hospitality during the ﬁnal stages of the collaboration. A.W.-L. also thanks the University of Valencia, Department of Theoretical Physics for the hospitality and support during the early stages of this work.

Appendix A. Complete formulae with corrections

In this appendix we give the corrections to the pertinent double-decay rates. The inclusion of such effects is accounted for by the Y+− parts of the observables for the kaon decays to two and three pions in (3.1). For the two-charged-pion decay we have O =| |2 1 Y+− +− X+− ∗ 2 Y+− |Y+−| | | = ππ K1 with X+− ππ K2 Yππ ππ|K and we remind the reader that the quantum mechanical am- 2 plitudes η+− = + , η00 = − 2 , should be replaced now by their barred counterparts η¯ that include decoherent contributions as well. In particular, above we have used the relations [5] = γ +|¯ |2 + β [¯ ∗ − ] |¯ | iφ+− = − + +− RL Γ η+− 4 Γ Im η+−d/d Y+− and η+− e L Y (the reader should − = − β recall that L L d ). The relevant double-decay time distribution reads then + − + − P π π ,τ1; π π ,τ2 4 |X+−| − − − − = R e ΓLτ2 ΓS τ1 + e ΓLτ1 ΓS τ2 2 L ¯ 2 −(Γ +α−γ )(τ1+τ2) − 2|¯η+−| cos m(τ1 − τ2) e |¯ | 4β η+− −(Γ¯ +α−γ)τ −Γ τ + sin( mτ + φ+− − φ )e 1 S 2 |d| 1 SW |¯ | 4β η+− −(Γ¯ +α−γ)τ −Γ τ + sin( mτ + φ+− − φ )e 2 S 1 |d| 2 SW ¯ −ΓS (τ2)−(Γ +α−γ)τ1 + 2|ω||η ¯+−| cos( mτ1 + φ+− − Ω)e J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 203 −Γ τ −(Γ¯ +α−γ)τ − cos( mτ + φ+− − Ω)e S 1 2 2 2 2γ β −Γ (τ +τ ) + |ω| − − 8 Im[¯η+− − Y+−] e S 1 2 . (A.1) Γ Γ The time integrated distribution reads + − + − P¯ π π ; π π ; τ − − − ¯ + − | |4 τΓL + τΓS (Γ α γ) τ = X+− e e −|¯ |2 e RL η+− cos( m τ) ¯ 2 ΓL + ΓS (Γ + α − γ) + 1 m2 + (Γ¯ + α − γ)2 + 2(Γ¯ + α − γ)Γ + Γ 2 S S |¯ | 4β η+− ¯ − τΓ × mcos(φ+− − φ ) + (Γ + α − γ + Γ ) sin(φ+− − φ ) e S |d| SW S SW − τ(Γ¯ +α−γ) + e mcos( m τ + φ+− − φ ) SW ¯ + (Γ + α − γ + Γ ) sin( m τ + φ+− − φ ) S SW − τΓS ¯ + 2|ω||η ¯+−|e (Γ + α − γ + ΓS) cos(φ+− − Ω)− msin(φ+− − Ω) (Γ¯ +α−γ) τ ¯ − 2|ω||η ¯+−|e (Γ + α − γ + Γ ) cos( m τ + φ+− − Ω) S − msin( m τ + φ+− − Ω) −ΓS τ 2 2γ β e + |ω| − − 8 Im[¯η+− − Y+−] . (A.2) Γ Γ 2ΓS For the neutral-pion decay channel we use the observable: 1 Y O =|X |2 00 00 00 ∗ | |2 Y00 Y00 and the associated double-decay rate involving charged- and neutral-pion decay channels has the form + − 0 0 P π π ,τ1; π π ,τ2 2 2 |X+−| |X00| − − +− − − = R00e ΓLτ2 ΓS τ1 + R e ΓLτ1 ΓS τ2 2 L L ¯ −(Γ +α−γ )(τ1+τ2) − 2|¯η+−||η ¯00| cos m(τ1 − τ2) + φ+− − φ00 e −Γ τ −(Γ¯ +α−γ)τ +|¯η+−|e S 2 1 4β × 2|ω| cos( mτ + φ+− − Ω)+ sin( mτ + φ+− − φ ) 1 |d| 1 SW − − ¯ + − +|¯ | ΓS τ1 (Γ α γ)τ2 η00 e 4β × −2|ω| cos( mτ + φ − Ω)+ sin( mτ + φ − φ ) 2 00 |d| 2 00 SW 2 2γ β −Γ (τ +τ ) + |ω| − − 4 Im[¯η+− − Y+− +¯η − Y ] e S 1 2 . (A.3) Γ Γ 00 00 204 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206

± ± Fig. 9. Plots of P¯ (π ,π , τ) vs. τ , including corrections. The long-dashed curve has only ω = 0, whilst the solid curve represents α, β, γ, ω = 0. The values for ω and Ω are (from left to right, and top to bottom) (i) |ω|=|¯η+−|, Ω = φ+− − 0.16π, (ii) |ω|=|¯η+−|, Ω = φ+− + 0.95π, (iii) |ω|=0.5|¯η+−|, Ω = φ+− + 0.16π,(iv)|ω|=1.5|¯η+−|, = +− P¯ |¯+−|2 4 Ω φ . is in units of η τs 4A0 and τ is in units of τS .

The double-decay time distribution involving charged-pion and dilepton channels is given by ± ± P π ,τ ; l ,τ 1 2 2 2 |X+−| |a| γ − − − − = 1 ± δ + e ΓLτ2 ΓS τ1 + R e ΓLτ1 ΓS τ2 4 L Γ L −(Γ¯ +α−γ )(τ +τ ) ∓ 2|¯η+−| cos m(τ − τ ) + φ+− e 1 2 1 2 4β − − ¯ + − ± ( mτ − φ ) − |ω| ( mτ − Ω) e ΓS τ1 (Γ α γ)τ2 | | sin 2 SW 2 cos 2 d 2 2γ 8β −Γ (τ +τ ) + |ω| − − Im[¯η+−] e S 1 2 . (A.4) Γ Γ Finally for the three-pion decay channel the relevant observable is | |2 ∗ 2 Y3π Y3π O3π =|X3π | Y3π 1

3π|K1 with 3π|K Y π = . 2 3 3π|K2 ∗ =− γ +|¯ |2 − β [¯ − ] |¯ | iφ3π = In the following we use RS Γ η3π 4 Γ Im η3π d/d Y3π where η3π e + + S Y3π . The pertinent three-pion double-decay time distribution is P(3π,τ ; 3π,τ ) 1 2 4 |X3π | − − − − = R e ΓLτ2 ΓS τ1 + R e ΓLτ1 ΓS τ2 2 S S ¯ 2 −(Γ +α−γ )(τ1+τ2) − 2|¯η3π | cos m(τ1 − τ2) e J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206 205

¯ −ΓLτ1−(Γ +α−γ)τ2 +|¯η3π |e 4β × −2|ω| cos( mτ − φ + Ω)+ sin( mτ − φ + φ ) 2 3π |d| 2 3π SW ¯ −ΓLτ2−(Γ +α−γ)τ1 +|¯η3π |e 4β × 2|ω| cos( mτ − φ + Ω)+ sin( mτ − φ + φ ) 1 3π |d| 1 3π SW 2γ 8β − + + |ω|2 + + Im[¯η − Y ] e ΓL(τ1 τ2). (A.5) Γ Γ 3π 3π

This completes our analysis. We observe that the inclusion of the corrections does not affect the functional form of the decoherence and CPTV effects.

References

[1] R.F. Streater, A.S. Wightman, PCT, Spin and Statistics, and All That, Addison–Wesley, Redwood City, 1989, p. 207. [2] J. Ellis, J. Hagelin, D.V. Nanopoulos, M. Srednicki, Nucl. Phys. B 241 (1984) 381. [3] S. Hawking, Commun. Math. Phys. 43 (1975) 199; S. Hawking, Commun. Math. Phys. 87 (1982) 395; J. Bekenstein, Phys. Rev. D 12 (1975) 3077. [4] J.R. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Phys. Lett. B 293 (1992) 142, hep-ph/9207268; J.R. Ellis, J.L. Lopez, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. D 53 (1996) 3846, hep-ph/9505340. [5] P. Huet, M.E. Peskin, Nucl. Phys. B 434 (1995) 3, hep-ph/9403257. [6] A. Kostelecky, hep-ph/0403088, and references therein. [7] G. Barenboim, J. Lykken, Phys. Lett. B 554 (2003) 73, hep-ph/0210411. [8] J. Bernabeu, N.E. Mavromatos, J. Papavassiliou, Phys. Rev. Lett. 92 (2004) 131601, hep-ph/0310180. [9] R. Wald, Phys. Rev. D 21 (1980) 2742. [10] E. Alvarez, J. Bernabeu, N.E. Mavromatos, M. Nebot, J. Papavassiliou, Phys. Lett. B 607 (2005) 197, hep- ph/0410409. [11] G. Lindblad, Commun. Math. Phys. 48 (1976) 119; R. Alicki, K. Lendi, in: Lecture Notes in Physics, vol. 286, Springer-Verlag, Berlin, 1987. [12] E. Lisi, A. Marrone, D. Montanino, Phys. Rev. Lett. 85 (2000) 1166, hep-ph/0002053; F. Benatti, R. Floreanini, JHEP 0002 (2000) 032, hep-ph/0002221. [13] T. Ohlsson, Phys. Lett. B 502 (2001) 159. [14] G. Barenboim, N.E. Mavromatos, Phys. Rev. D 70 (2004) 093015, hep-ph/0406035; N.E. Mavromatos, gr-qc/0411067. [15] L. Maiani, G. Pancheri, N. Paver (Eds.), The Second DANE Physics Handbook, 1995. [16] Particle Data Group, S. Eidelman, et al., Phys. Lett. B 592 (2004) 1. [17] CPLEAR Collaboration, R. Adler, et al., Phys. Lett. B 364 (1995) 239, hep-ex/9511001. [18] F. Benatti, R. Floreanini, Phys. Lett. B 389 (1996) 100, hep-th/9607059; F. Benatti, R. Floreanini, Nucl. Phys. B 511 (1998) 550, hep-ph/9711240; F. Benatti, R. Floreanini, Phys. Lett. B 468 (1999) 287, hep-ph/9910508. [19] J.R. Ellis, N.E. Mavromatos, D.V. Nanopoulos, Phys. Rev. D 63 (2001) 024024, gr-qc/0007044. [20] H.J. Lipkin, Phys. Rev. 176 (1968) 1715; H.J. Lipkin, Phys. Lett. B 219 (1989) 474; I. Dunietz, J. Hauser, J.L. Rosner, Phys. Rev. D 35 (1987) 2166. 206 J. Bernabéu et al. / Nuclear Physics B 744 (2006) 180–206

[21] J. Bernabeu, F.J. Botella, J. Roldan, Phys. Lett. B 211 (1988) 226; J. Bernabeu, F.J. Botella, J. Roldan, in: Proc. 25th Anniversary of the Discovery of CP Violation, Blois, France, 22–26 May 1989, Blois CP Violations, 1989, pp. 0389–0400, QCD 161 (1989) I44. [22] L. Wolfenstein, Nucl. Phys. B 246 (1984) 45; M.C. Banuls, J. Bernabeu, Phys. Lett. B 464 (1999) 117; M.C. Banuls, J. Bernabeu, Nucl. Phys. B 590 (2000) 19. [23] CPLEAR Collaboration, A. Apostolakis, et al., Phys. Lett. B 422 (1998) 339. Nuclear Physics B 744 (2006) 207–220

Fractional branes and the gravity dual of partial supersymmetry breaking

Paolo Merlatti

II. Institut für Theoretische Physik Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany Received 8 February 2006; accepted 30 March 2006 Available online 18 April 2006

Abstract In type IIB string theory, we consider fractional D3-branes in the orbifold background dual to four- dimensional N = 2 supersymmetric Yang–Mills theory. We find the gravitational dual description of the generation of a non-trivial field theory potential on the Coulomb branch. When the orbifold singularity is softened to the smooth Eguchi–Hanson space, the resulting potential induces spontaneous partial supersymmetry breaking. We study the N = 1 theory arising in those vacua and we see how the resolved conifold geometry emerges in this way. It will be natural to identify the size of the blown-up two-cycle of that geometry with the inverse mass of the adjoint chiral scalar. We finally discuss the issue of geometric transition in this context. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

In recent years supersymmetric Yang–Mills theories have been much investigated and a number of non-trivial results about their non-perturbative dynamics have been reached. The motivation of this intensive investigation is string theory. Indeed it has become clear that the embedding of gauge theories in string theory (via D-branes) is a powerful tool to explore various interesting features of their dynamics. Within this framework, several approaches have been developed, all going under the rather generic name of gauge/string correspondence. The most celebrated example is the N = 4 AdS–CFT duality [1] (see [2] for an excellent review). Many interesting results have been also found for less (super)-symmetric theories (see [3] for some extensive reviews). In this paper we make contact with a nice and fruitful way of looking at this duality.

E-mail address: [email protected] (P. Merlatti).

Such perspective has been introduced in [4] for the topological string. There, it has been shown that in topological string theory the deformed conifold background, with N 3-brane wrapped on S3 (equivalent to the S3 Chern–Simons theory [5]), is dual to the same topological string but without string modes and on a different background. The new background is the resolved conifold, whose parameters depend now on N. Such gauge/geometry correspondence has been successfully embedded in superstring theory in [6]. The central idea is the concept of geometric transition. The field theory has to be engineered via D-branes wrapped over certain cycles of a non-trivial Calabi–Yau geometry. The low energy dual arises from a geometric transition of the Calabi–Yau itself, where the branes have disappeared and have been replaced by fluxes. The simplest example [6,7] is pure N = 1 super-Yang–Mills theory. Before the transition the field theory is engineered on N D5 branes wrapped on the blown-up S2 of a resolved conifold. After the transition we are instead left with N units of R–R flux through the S3 of a deformed conifold [8]. In this paper we discuss the gauge/string correspondence starting from N = 2 supersymmetric theories and we find how partial supersymmetry breaking can be described on the supergravity side. Within this purely ten dimensional superstring context, we see then how the resolved conifold background emerges naturally as the proper one to engineer N = 1 super-Yang–Mills theory. In a first part we discuss the string dual of N = 2 supersymmetric SU(N) Yang–Mills theories with non-trivial potential for the adjoint scalar. We move from the general observation that the same supergravity solution can be given in terms of forms of different degree, that are related to each other by electric–magnetic duality. If we consider for example type II supergravity, the same Dp-brane solution can be written with an electric ansatz (the relevant R–R field in the solution is the potential Cp+1) or with a magnetic one (the relevant R–R potential is C7−p). The two solutions should be equivalent. Indeed they are, from a bulk perspective. It is perhaps surprising that, as we show in this paper for the specific N = 2 case, they correspond to different gauge dynamics of the dual Yang–Mills theory. To see this, we add probe-branes in the background geometry, and the boundary action describing the Yang–Mills theory living on the world-volume of such branes will give different results according to which kind of ansatz (electric or magnetic) one makes for the bulk solution. To be more specific, we consider the orbifold background C2/Z2 with just one kind of fractional branes and no bulk ones. The complete explicit bulk solution for this N = 2 supersymmetric background has been found in [9]. Such solution can be described equivalently in terms of the R–R twisted four-form potential A4 or of its six-dimensional magnetic dual, the twisted scalar c. In this paper we study the dual Yang–Mills theory and we find that, according to which bulk field is used to describe the solution, its low energy properties are different. Mak- ing a probe analysis, we find that if the solution is given in terms of A4 (electric ansatz), there is no potential for the U(1) gauge theory living on the world-volume of the probe. Contrary, if the solution is given in terms of c (magnetic ansatz), a non-trivial potential is generated. Thus, the magnetic ansatz corresponds to switching on an internal flux that generates a gauge theory potential. We will see such potential corresponds to the generation of electric-magnetic Fayet– Iliopoulos terms of the kind discussed in [10]. Even if both the results are consistent with N = 2 quantum field theory, we find it surprising that the Yang–Mills dual physics depends crucially on the kind of ansatz one makes to describe the bulk geometry. In this paper this observation is just empirical, but this phenomenon could be more general and it deserves further investigation. We are indeed currently studying the origin of such discrepancy via a detailed stringy analysis. In the second, more speculative, part of the paper we see directly in type II superstring theory how the resolved conifold geometry emerges naturally as the proper geometry to engineer N = 1 P. Merlatti / Nuclear Physics B 744 (2006) 207–220 209 supersymmetric Yang–Mills theories. This part is a development of the first one. Starting in fact from the orbifold background, particularly from the analysis we make for the magnetic case, we consider its generalization to the case where the singular orbifold geometry is softened to the smooth Eguchi–Hanson space. We find vacua where partial supersymmetry breaking occurs and we study the description of N = 1 super-Yang–Mills theory in such vacua. This study leads us to argue that the natural background where N = 1 Yang–Mills theories should be engineered is the resolved conifold. Besides we gain information on the way such engineering works. In particular it turns out to be quite natural to identify the mass of the chiral adjoint superfield with the inverse of the size of the blown-up two-cycle. Shrinking this two-cycle to zero size corresponds thus to decouple the adjoint scalar and to get pure N = 1 super-Yang–Mills in the singular conifold background. We know eventually the rest of the story: to cure the singularity of this geometry, the S3 has to be blown up [6,8]. This corresponds to generate the infrared dynamically generated scale of the dual gauge theory (in this case coming from gaugino condensation). The picture we get in this way has an immediate translation in the type IIA T-dual scenario of rotated NS-branes [11–13]. We will comment on this relation in Section 4. Another way to make contact with the N = 1 conifold theory starting from a more supersymmetric theory in the ultraviolet is to start from a quiver N = 2 supersymmetric field 5 theory [14]. In that case one starts from the orbifold background S /Z2, that is dual to the N = 2 super-conformal Yang–Mills U(N) × U(N) theory with hypermultiplets transforming in (N, N)¯ ⊕ (N,N)¯ .FromanN = 1 perspective, the hypermultiplets correspond to chiral multiplets and other chiral multiplets in the adjoint representations of the two U(N)’s are present. There is also a non-trivial superpotential for the multiplets in the bi-fundamental representations of the gauge group. One can add now a relevant term to this superpotential. To integrate out the adjoint multiplets in presence of this relevant perturbation corresponds to blow up the orbifold 5 singularity of S /Z2 [14]. Remarkably, in [14] it has also been shown that such blown-up space is topologically equivalent to the coset space T 1,1, the base of the conifold. This way of flowing to the conifold has some points in common with our construction. The main difference is that we start from a simpler gauge theory (we have no matter and just one gauge group from the very beginning) and we follow all the steps in terms of explicit supergravity solutions.

2. Brief review of N = 2 supersymmetric Lagrangians

In this section we briefly review the field theory analysis of N = 2 supersymmetric gauge theories (with gauge group U(1)), emphasizing the role of electric and magnetic Fayet–Iliopoulos (FI) terms [10]. The general form of a N = 2 Lagrangian is determined by the analytic prepo- tential F(A) (A being the N = 2 vector superfield): i L = d2θ d2θ F(A) + c.c. (1) 0 4 1 2 In N = 1 language, the Abelian vector multiplet A contains the N = 1 gauge multiplet A and a neutral chiral superfield Φ, whose complex scalar component we denote by a.InN = 1 super- space the Lagrangian (1) reads: 1 L = d2θf(Φ)W2 + c.c. + d2θd2θK(Φ,¯ Φ),¯ (2) 0 4 where W is the standard gauge field strength superfield and i ¯ K(a,a)¯ = (aF¯ −¯aF ), f (a) =−iF , (3) 2 a a aa 210 P. Merlatti / Nuclear Physics B 744 (2006) 207–220 where the a (a¯) sub-script denotes derivative with respect to a (a¯). It is well known that this Lagrangian can be supplemented by a FI term that preserves N = 2 supersymmetry [15]: √ LD = 2ξD, (4) with ξ a real constant. More generally, it has been shown in [10] that the following superpotential

W = ea + mFa (5) is also compatible with N = 2 supersymmetry. The resulting scalar potential is:

|e + mτ|2 + ξ 2 VN =1 = , (6) τ2 where τ ≡ τ1 + iτ2 ≡ Faa(a). For m = 0 the superpotential (5) is equivalent to a FI term (4) with ξ = e [15]. It is now quite natural to interpret the term mFa in (5) as a magnetic FI term (it is related to the electric one by S duality). In [10] it has been shown how such interpretation becomes even more clear if a manifestly N = 2 supersymmetric formalism is used. In that case the ﬁeld theory analysis shows also that a constant shift to the potential (6) is still compatible with N = 2 supersymmetry. The resulting potential is:

|e + mτ|2 + ξ 2 V = VN =1 + 2mp = + 2mp, (7) τ2 where p is an arbitrary real constant.

2.1. Stable vacua

Again following [10], we now turn to the minimization of the scalar potential (7). We ﬁnd a vacuum at e ξ τ1=− , τ2= . (8) m m From this relations a vacuum expectation value for the scalar ﬁeld (a) can be found. Now there are different cases to consider. Two of them are specially interesting for us:

(1) ξm = 0: in this case at the point (8) we end up with an N = 1 vacuum. Therefore partial supersymmetry breaking does occur and we get here a massless U(1) vector multiplet plus m2 a massive chiral multiplet (with mass 2ξ τa ). (2) m = 0, ξ = 0: in this case the metric Kaa¯ ≡τ2 is singular. The singularity is due to the appearance of a new massless dyonic state with quantum numbers (m0,e0) at a. It can further be shown that

(m, e) = c(m0,e0), (9) for some constant c. This dyon does condense, breaking the U(1) gauge symmetry while N = 2 supersymmetry is unbroken. P. Merlatti / Nuclear Physics B 744 (2006) 207–220 211

3. Orbifold geometry

In the framework of the gauge-string correspondence it is possible to give dual supergravity descriptions of N = 2 supersymmetric Yang–Mills theories [9,16–18]. The approach we follow 1,5 here is to study a stack of N fractional D3-branes in the IIB orbifold background (R × C2/Z2) [9,17]. On their world volume a N = 2 supersymmetric SU(N) Yang–Mills theory lives. Non- trivial information on such theory can be extracted from a low energy (i.e. supergravity) analysis of this background [9,17]. We start reviewing the solution found in [9]1 (with slightly different boundary conditions) and then move to its ﬁeld theoretic interpretation. We will ﬁnd some results that have been overlooked in the past and we will show how they are compatible with the generation of an N = 2 potential of the kind we discussed in the previous section. The solution we are discussing can be written in terms of the metric, a R–R four form potential ˜ (C4), a NS–NS twisted scalar (b) and a R–R twisted one (c): − ds2 = H(r,ρ) 1/2η dxα dxβ + H(r,ρ)1/2δ dxi dxj , αβ ij −1 0 3 C4 = H(r,ρ) − 1 dx ∧···∧dx , ρ c = NKθ, b˜ = NKlog , (10)

i j where α, β = 0,...,3; i, j = 4,...,9; r = δij x x ; K = 4πgsα ; is a regulator; ρ and θ are polar coordinates in the plane x4, x5. The self-duality constraint on the R–R five-form field strength has to be imposed by hand. The precise functional form of the function H(r,ρ) has also been determined [9] but it is not relevant for our purposes. It is enough for us to note that the spacetime (10) has a naked singularity. This is a quite general feature of those gravitational backgrounds that are dual to non-conformal gauge theories. Luckily, in many cases the singularities are not actually present but they are removed by various mechanisms. In theories N = 2 supersymmetric the relevant mechanism is the enhançon [19]. It removes a family of time-like singularities by forming a shell of (massless) branes on which the exterior geometry terminates. As a result, the interior singularities are “excised” and the spacetime becomes acceptable, even if the geometry inside the shell has still to be determined. For the case at hand, the appearance of such shell, its field-theoretic interpretation and its consistency at the supergravity level have been discussed in [9,17,20].It turns out that the enhançon shell is simply a ring in the (ρ, θ)-plane. Its radius is

−π/(2Ngs ) ρe = e . (11)

The geometry (10) is thus valid outside the enhançon shell, i.e. for ρ ρe. On the gauge theory side the enhançon locus corresponds to the locus where the Yang–Mills coupling constant diverges [9,17].

3.1. Electric case

The solution (10) can also be written following an electric ansatz, i.e. in terms of the R–R twisted ﬁeld A4, the ﬁeld to which a fractional D3-brane couples electrically. A4 replaces its

1 We refer to that paper also for the notation. 212 P. Merlatti / Nuclear Physics B 744 (2006) 207–220 six-dimensional magnetic dual c. Following [9], it is easy to determine it:

ρ 0 3 2 ρ 0 3 A4 = NKlog dx ∧···∧dx + 2π α = NKlog dx ∧···∧dx , (12) ρe where, with respect to [9], we have changed the boundary conditions in such a way that A4 is positive for every value of ρ>ρe. The world-volume action of a fractional D3-brane in terms of such field reads: T3 4 1 ˜ S =−√ d x − det Gαβ 1 + b bdy 2π 2α 2korb T 1 T 1 + √ 3 C + b˜ + √ 3 A , 4 1 2 2 4 (13) 2korb 2π α 2korb 2π α √ 7/2 2 where T3 = π and korb = (2π) gsα . Substituting the classical solution in the boundary action we can make the so-called probe computation (for a review of this method see, for example, [21]) and get the static result T V − KN ρ S =−√ 3 4 H 1 1 + log probe 2π 2α 2korb − KN ρ 1 ρ − H 1 − 1 1 + log − KN log + 2π 2α = 0, (14) 2π 2α 2π 2α where V4 is the four-dimensional volume filled by the brane. The fact that all position dependent terms exactly cancel indicates that there is no potential felt by the probe-brane. The no-force condition is thus respected to all orders. This was expected because of the BPS properties of the system. When we identify the coordinates xa = 2πα Φa with the chiral scalar field of the Yang– Mills theory living on the world-volume of the brane, the vanishing result of (14) corresponds to have no potential in the gauge model. This is consistent with the field theory analysis of the previous section, being V = 0 a special case of the most general N = 2 potential in (7) (i.e. with e = m = ξ = 0). A constant (different from zero) result, as it has been obtained in [9], would be an equally good check of the no-force condition, but according to (7) it would not be consistent with N = 2 supersymmetry. The fact we get precisely zero is thus a check of the boundary condition chosen in (12). Following [9], we can also include the world-volume fields fluctuations and study the motion of the probe-brane in the geometry (10). This is done by inserting the classical solution in the world-volume action (13) and expanding it in powers of the velocity of the probe D3-brane in the two transverse directions x4 and x5. We find that the terms quadratic in the velocity of the probe survive and allow to define a non-trivial two-dimensional metric on the moduli space. It is easy to see [9] that this non-trivial metric corresponds holographically to the one-loop running of the Yang–Mills coupling constant, that in terms of the gravitational degrees of freedom reads: 1 T (2πα )2 1 = √ 3 + b˜ . 2 1 2 (15) gY 2korb 2 2π α

3.2. Magnetic case

We now repeat the same probe analysis directly in terms of the fields appearing in (10),i.e. without dualizing the R–R twisted scalar c, the field to which a fractional D3-brane couples P. Merlatti / Nuclear Physics B 744 (2006) 207–220 213 magnetically. We should thus consider (instead of (13)) the following boundary action T3 4 1 ˜ S =−√ d x − det Gαβ + 2πα Fαβ 1 + b bdy 2π 2α 2korb T 1 T α + √ 3 C + b˜ + √ 3 cF ∧ F, 4 1 2 (16) 2korb 2π α 2korb 2 where we have also included the world-volume vector fields fluctuations. We make now the probe computation with such boundary action and get: 2 T3 4 (2πα ) 1 α a b 1 αβ a NK ρ S =−√ d x δab∂ Φ ∂αΦ + F F 1 + log probe 2 2 4 a αβ 2π 2α 2korb NK ρ α + V 1 + log − KNθ F ∧ F . (17) 4 2π 2α 2 One immediate consequence of doing the probe computation in this way is that we respect holo- morphicity getting also the expected term reproducing the one-loop chiral anomaly [22]. Besides we see also that a non-trivial potential is generated, namely: T NK ρ 1 V(Φ)= √ 3 + = τ , 1 2 log 3 2 Y 2 (18) 2korb 2π α (2π) α ≡ + ≡ 4πi + θY where, consistently with the electric result (15), we have defined τY τY 1 iτY 2 2 2π as gY the holographic complexified Yang–Mills coupling [22]: 1 2 ˜ τY = √ c + i 2π α + b . (19) 2 (2π α ) gs This computation shows that, when the solution is written in term of a magnetic ansatz, the no-force property of the BPS system is lost. This could seem quite surprising as the configuration is still N = 2 supersymmetric and usually the no-force condition is respected with such amount of supersymmetry (on the field theory side such condition reflects the flatness of the Coulomb branch). Nevertheless, at least from a purely field theoretical point of view, we have seen in the previous section that such flatness can be violated by electric–magnetic Fayet–Iliopoulos terms and as a result the potential (6) is still compatible with N = 2 supersymmetry. Luckily, it turns out the potential (18) is just a specific case of (6), namely with mθ 1 e =− Y ,ξ= 0 and m2 = . (20) 2π (2π)3α 2 We see in this way that the violation of the no-force condition does not contradict supersymmetry, but it corresponds to the generation of a purely magnetic FI term. We call it purely magnetic be- + mθ cause a dyon with quantum numbers (e, m) has an effective electric charge equal to e 2π [23] and with the values in (20) it vanishes identically. Such values ensure also that the probe-brane is in a proper vacuum in the θˆ direction, but for generic values of ρ it is not and it moves toward ρ = ρe. This situation holographically corresponds to the second type of vacua discussed in Sec- tion 2.1, where we have learned it preserves all the supersymmetries but it signals a singularity. This continues to be in perfect agreement with the gravity picture we just got: the probe-brane is pushed to the enhançon and there stays. From the gauge theory it has been moreover possible to argue at that point of the moduli space extra massless states appear (i.e. monopoles) and to give a physical interpretation of the singularity. On the gravity side it corresponds to the well-known 214 P. Merlatti / Nuclear Physics B 744 (2006) 207–220 fact that at the enhançon extra massless objects appear [19] and it enforces the interpretation of the enhançon locus as the field theoretic curve of marginal stability (we mean the curve in moduli space where extra massless objects, typically dyons, appear [25]).

4. Away from the orbifold limit

In this section we soften the orbifold singularity and consider the resulting (smooth) Eguchi– Hanson space [26]. We generalize the probe analysis that we have done in the orbifold for the magnetic case. We find partial supersymmetry breaking. We are then led to consider N = 1 supersymmetric Yang–Mills theory and the resolved conifold geometry. This could look at first sight surprising. It is indeed well known that, in presence of fractional branes, it is not possible to soften the orbifold geometry to a smooth ALE (asymptotically locally Euclidean) space in a supersymmetric way. A nice way to understand this statement is to go to the T-dual type IIA set-up [13], where the configuration corresponding to fractional branes on the orbifold is that of two parallel NS5-branes with a stack of D4-branes stretched between them. The blowing up of the orbifold corresponds to moving the NS5-branes in the directions transverse to the D4-branes and this cannot be done without some cost in energy. In our case there is a caveat to this no-go theorem. In the magnetic case we have discussed in Section 3.2, the fractional brane is not at the orbifold point (background BNS field flux through the cycle equal to 1/2 in string units), but it sits at the singular point (with zero flux) where the no-go theorem [13] does not apply.

4.1. Eguchi–Hanson space

The dual field theory analysis of the previous section has been entirely performed in an orbifold background and we have found, in the magnetic case, purely magnetic FI terms. It is well known [24] that a way to introduce standard (i.e. electric) FI terms is via the blow up of the orbifold singularity. In such case the resulting space is no longer an orbifold but it is a (smooth) Eguchi–Hanson (EH) space [26]. This is an ALE space. It is characterized by three moduli that correspond to three different ways of blowing-up the relevant two cycle when going away from the orbifold limit (via the triplet of massless NS–NS twisted scalars). Analogously to the field theory description of N = 2 FI terms, it is possible to use the global SU(2)R symmetry and to reduce to the case of just one modulus. The one-parameter family of metrics we get in this way can be written as: − a 4 1 a 4 ds2 = 1 − dr2 + r2 1 − σ 2 + r2 σ 2 + σ 2 , (21) EH r r 3 1 2 2 where a r ∞, a is proportional to the size of the blown-up cycle, the σi are defined in terms of the S3 Euler angles (θ,φ,β):

2σ1 =−sin βdθ+ cos β sin θdφ, (22)

2σ2 = cos βdθ+ sin β sin θdφ, (23)

2σ3 = dβ + cos θdφ (24) = ∧ 2 + 2 2 and they satisfy dσa εabcσb σc. Note also that 4(σ1 σ2 ) is the standard unit radius S 2 + 2 + 2 3 metric and σ1 σ2 σ3 is the S one. As the angular variables have periodicity 0 <θ π, 3 0 <φ, β 2π, this space at inﬁnity looks like S /Z2. P. Merlatti / Nuclear Physics B 744 (2006) 207–220 215

The solution corresponding to fractional branes in the Eguchi–Hanson background has been found in [27] and it is very similar to the orbifold one. It can be written in terms of a metric 2 = −1/2 α β + 1/2 2 + 2 2 + 1/2 2 ds H(r,ρ) ηαβ dx dx H(r,ρ) dρ ρ dθ H(r,ρ) dsEH, (25) a self dual R–R five-form −1 0 3 −1 0 3 F5 = d H(r,ρ) dx ∧···∧dx + d H(r,ρ) dx ∧···∧dx , (26) and a complex three form (a linear combination of the NS–NS one and the R–R one) valued only in the transverse space: H ≡ NS + R = ∧ F3 iF3 dγ(ρ,θ) ω, (27) where γ(ρ,θ)is an analytic function on the transverse R2, ω is the harmonic anti-self-dual form on the Eguchi–Hanson space and r is now the radial Eguchi–Hanson coordinate. Even if the microscopic interpretation of this solution is quite obscure [27], we assume that it corresponds to the standard geometrical interpretation of fractional D3-branes as D5-branes wrapped on the non-trivial cycle that shrinks to zero size in the orbifold limit [28]. The form of the solution is indeed consistent with the interpretation of twisted fields as fields coming from wrapping of higher degree forms [28], as it was already confirmed in [29] by explicit computations of string scattering amplitudes. The main difference with respect to the orbifold solution is the explicit functional form of the warp factor H(r,ρ), whose physical implication is that there is still a naked singularity screened by an enhançon mechanism, but contrary to the orbifold case, in the internal directions (those along the ALE space) the singularity is cured by the blowing-up of the cycle and there is no enhançon there [27]. Unluckily, the Eguchi–Hanson background lacks of a conformal field theory description. This make it difficult to determine the correct form of the boundary actions one would need to get gauge theory information. It looks nevertheless quite reasonable to assume the proper boundary action is the same as the orbifold one with the inclusion of the new FI term corresponding to having blown up the cycle. This can be motivated by the knowledge of the explicit solution in the Eguchi–Hanson case, that makes it clear the close similarity to its orbifold limit. Besides, this is very analogous in spirit to the successful assumption made in [30]. The potential the brane fills will be thus the same we computed in the orbifold (we refer to the magnetic case (18))but now with an extra term corresponding to have ξ = 0 and proportional to the size of the blown-up cycle (a2): 2 2 ξ V(Φ)= m τ2 + . (28) τ2 We see again that the no-force condition is violated but now there is an interesting minimum where the probe-brane can sit, i.e. at

π ξ ρ∗ = ρee N m , (29) where the enhançon radius ρe has been defined in (11).Asρ∗ >ρe, ρ = ρ∗ defines a regular spacetime locus. From the field theory analysis we know this is a vacuum of the first type we discussed in Section 2.1 and as such it preserves just half of the supersymmetry. This means that the gauge theory living on the probe is N = 1 supersymmetric with a massless vector multiplet and a massive chiral one, with mass M given by: 2 m M = τ . (30) 2ξ 216 P. Merlatti / Nuclear Physics B 744 (2006) 207–220

From the supergravity point of view we have again a solution with bulk N = 2 supersymmetry (as shown in [27]) but the embedding of the brane probe seems to break half of it. It would be interesting to see this purely in a stringy context by making a proper K-supersymmetry study of this geometry. However, our ﬁeld theory analysis already shows that the probe-brane sees a geometry that is locally N = 1 supersymmetric. In the T-dual IIA set-up the chiral adjoint scalar gets a mass if the two NS5-branes are relatively rotated [11]. As this type IIA geometry is eventually dual to the conifold one [12],we expect the conifold geometry to appear now.

4.2. Resolved conifold

We want now to analyze better the geometry seen by the probe brane. We start noting that, according to the interpretation of fractional D3-branes as D5-branes wrapped on the non-trivial two-cycle of the Eguchi–Hanson space, the probe-brane naturally tends to go toward the point of the Eguchi–Hanson space where the cycle has minimal area, i.e. towards r → a in (21).Evenif the point r = a looks a singularity in the metric (21), it is well known that the limit r → a + for small is well deﬁned and it gives:

1 a ds2 ∼ d 2 + 4 2(dβ + cos θdφ)2 + a2 dθ2 + sin2 θdφ2 , (31) 4 which is topologically looking locally as R2 × S2 (globally there is a further fibered structure due to the dβ dφ cross term). To see this more clearly, we make the further change of coordinate = v2/a and the metric becomes: a2 ds2 = dv2 + v2(dβ + cos θdφ)2 + dθ2 + sin2 θdφ2 . (32) 4 This makes it clear as, having β periodicity 2π, the putative singularity at r = a is simply an artifact of the coordinates. In the R2 directions (parameterized by ρ and θ in (25)) the probe-brane sits instead at the circle defined by ρ = ρ∗ and the value of ρ∗ (given in (29)) varies with the parameters of the solution. Finally, the geometry seen by the probe-brane is determined by the R2 × EH metric, in the r → a limit. This space looks locally as R4 × S2: a2 ds2 = dx2 + dy2 + dv2 + v2(dβ + cos θdφ)2 + dθ2 + sin2 θdφ2 , (33) 4 where now we have chosen to parameterize R2 with Cartesian coordinate x and y. Making now the coordinate redefinition: θ˜ α + φ˜ x = u cos cos , (34) 2 2 θ˜ α + φ˜ y = u cos sin , (35) 2 2 θ˜ v = u sin , (36) 2 α − φ˜ β = , (37) 2 P. Merlatti / Nuclear Physics B 744 (2006) 207–220 217 where 0 <α 4π,0< φ˜ 2π,0< θ˜ π, we get u2 ds2 = du2 + (dα + cos θd˜ φ˜ + cos θdφ)2 + dθ˜2 + sin2 θd˜ φ˜2 4 a2 − 2 cos θdφ(dφ˜ + cos θdα)˜ + dθ2 + sin2 θdφ2 , (38) 4 where terms involving higher power of u with two differentials in the θ,φ directions have been neglected. As shown in Appendix A, this metric is locally the same as the “near horizon” of the resolved conifold one (see Eq. (A.3) in Appendix A). We note that globally there is a small difference between the two, being the fibration of the blown-up S2 not identical (due to the cross term cos θdφ(dφ˜ + cos θdα)˜ in (38)). It is perhaps not surprising we find the resolved conifold geometry appearing here: it is indeed known the resolved conifold geometry is the proper one to engineer N = 1 pure supersymmetric Yang–Mills theory in the type IIB set-up [6]. In this section we have just followed this engineering entirely in terms of supergravity solutions and we clearly see its relation with the parent N = 2 orbifold theory. A non-trivial information we immediately have (with the help of the field theory analysis and particularly thanks to the formula (30)) is that the size of the blown-up two-cycle is directly related to the inverse mass of the chiral multiplet. The assumption about the microscopic interpretation of the probe as wrapped D5-branes cannot be proven here, but, as we will see, it makes all the picture consistent. Under this assumption, we have learned that, if a D5-brane is wrapped on the non-trivial two-cycle of a resolved conifold, close to the apex of such cone the gauge theory living on its world-volume is N = 1 supersymmetric. The embedding of N branes in such background will then give rise to N = 1su- persymmetric U(N) Yang–Mills theory. The fact we reach such conclusion via a probe analysis means that we are studying open strings (and then branes) in the given background. This matches perfectly with the picture of geometric transition we discussed in the introduction: being talking of open strings and branes, we are before the transition and accordingly the background is the resolved conifold! From [6,8], we know how the story goes on, but we briefly review it here following the perspective we have just developed. To get pure N = 1 super-Yang–Mills, we take the zero size limit of the blown-up cycle. In this way the chiral adjoint field acquires infinite mass (see Eq. (30)) and it is decoupled. The relevant geometry is now the conifold one. This geometry is still singular [31] and according to Eq. (29) the brane will tend to go to the enhançon locus. Consistently, it has been shown that at least from a supergravity point of view the enhançon is a consistent mechanism also in the conifold case [20]. In principle there are two ways now to de-singularize the geometry (or we could also say to determine the geometry inside the enhançon shell): the S2 or the S3 can be blown-up. As we do not want to go back to the original case, the only remaining possibility is to look for a consistent deformed conifold solution. From [8] we know this is indeed the right thing to do: the deformed conifold is a non-singular solution and its geometry is the proper one to describe the infrared N = 1 Yang–Mills dynamics.

5. Conclusions

In this paper we have described how partial supersymmetry breaking can be obtained in the context of the gauge/string correspondence. This has been reached studying the low energy dynamics of open strings in given backgrounds. The tool we used to perform such study is the probe-analysis. We have seen in this way that a potential, due to the internal flux, is generated 218 P. Merlatti / Nuclear Physics B 744 (2006) 207–220 in the N = 2 orbifold. This potential drives the probe-brane to the enhançon locus. Instead, when the orbifold singularity is softened to the smooth Eguchi–Hanson space, the resulting potential drives the probe-brane to regular space–time points (i.e. outside the enhançon shell). We have found that at those points (corresponding to minima of the potential) partial supersymmetry breaking does occur. We have further analyzed the geometry seen by the probe-brane close to those minima and we have found it is the resolved conifold one, in the vicinity of the blown-up two-cycle. We have finally showed that shrinking the two-cycle to zero size corresponds, on the field theory side, to decouple the adjoint chiral scalar and thus to get pure N = 1 super-Yang– Mills. An issue deserving more investigation is that our findings seem to suggest that the T-dual type IIA configuration of rotated NS5-branes corresponds, for generic angle of rotation, to the resolved conifold geometry. The singular conifold would be the relevant one just for the case of perpendicular NS-branes.

Acknowledgements

I thank M. Billò, P. Di Vecchia, M. Frau, G. Vallone and especially A. Lerda for many useful discussions and exchange of ideas. This work is partly supported by the EU contracts MRTN- CT-2004-503369 and MRTN-CT-2004-512194, DAAD, and the DFG grant SA 1336/1-1.

Appendix A. Resolved conifold metric

We review here brieﬂy the resolved conifold case [32].2 Its metric is:

− 1 ds2 = K 1(u) du2 + K(u)u2(dψ + cos θ dφ + cos θ dφ )2 9 1 1 2 2 1 1 + u2 dθ2 + sin2 θ dφ2 + u2 + b2 dθ2 + sin2 θ dφ2 , (A.1) 6 1 1 1 6 2 2 2 where

u2 + 9b2 K(u) = . (A.2) u2 + 6b2 For small u we get now: 2 1 ds2 ∼ du2 + u2(dψ + cos θ dφ + cos θ dφ )2 3 4 1 1 2 2 1 b2 + u2 dθ2 + sin2 θ dφ2 + dθ2 + sin2 θ dφ2 . (A.3) 4 1 1 1 4 2 2 2

We note ﬁnally that this is locally the same metric as R2 × EH near the two cycle (38). Globally there is a small difference between the two, as the ﬁbration of the two S2 are not identical (see the last “cross” terms in (38)).

2 See also [33] for a way to embed the non-supersymmetric solution of [32] in a supersymmetric set-up. P. Merlatti / Nuclear Physics B 744 (2006) 207–220 219

References

[1] J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231, Int. J. Theor. Phys. 38 (1999) 1113, hep-th/9711200. [2] O. Aharony, S.S. Gubser, J.M. Maldacena, H. Ooguri, Y. Oz, Large N field theories, string theory and gravity, Phys. Rep. 323 (2000) 183, hep-th/9905111. [3] O. Aharony, The non-AdS/non-CFT correspondence, or three different paths to QCD, hep-th/0212193; M. Bertolini, Four lectures on the gauge-gravity correspondence, Int. J. Mod. Phys. A 18 (2003) 5647, hep- th/0303160; F. Bigazzi, A.L. Cotrone, M. Petrini, A. Zaffaroni, Supergravity duals of supersymmetric four-dimensional gauge theories, Riv. Nuovo Cimento 25 (12) (2002) 1, hep-th/0303191; E. Imeroni, The gauge/string correspondence towards realistic gauge theories, hep-th/0312070; P. Di Vecchia, A. Liccardo, R. Marotta, F. Pezzella, On the gauge/gravity correspondence and the open/closed string duality, Int. J. Mod. Phys. A 20 (2005) 4699, hep-th/0503156. [4] R. Gopakumar, C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415, hep-th/9811131. [5] E. Witten, Chern–Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637, hep-th/9207094. [6] C. Vafa, Superstrings and topological strings at large N, J. Math. Phys. 42 (2001) 2798, hep-th/0008142. [7] F. Cachazo, K.A. Intriligator, C. Vafa, A large N duality via a geometric transition, Nucl. Phys. B 603 (2001) 3, hep-th/0103067. [8] I.R. Klebanov, M.J. Strassler, Supergravity and a confining gauge theory: Duality cascades and χSB-resolution of naked singularities, JHEP 0008 (2000) 052, hep-th/0007191. [9] M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta, I. Pesando, Fractional D-branes and their gauge duals, JHEP 0102 (2001) 014, hep-th/0011077. [10] I. Antoniadis, H. Partouche, T.R. Taylor, Spontaneous breaking of N = 2 global supersymmetry, Phys. Lett. B 372 (1996) 83, hep-th/9512006; H. Partouche, B. Pioline, Partial spontaneous breaking of global supersymmetry, Nucl. Phys. B (Proc. Suppl.) 56 (1997) 322, hep-th/9702115. [11] J.L.F. Barbon, Rotated branes and N = 1 duality, Phys. Lett. B 402 (1997) 59, hep-th/9703051; K. Hori, H. Ooguri, Y. Oz, Strong coupling dynamics of four-dimensional N = 1 gauge theories from M theory fivebrane, Adv. Theor. Math. Phys. 1 (1998) 1, hep-th/9706082. [12] A.M. Uranga, Brane configurations for branes at conifolds, JHEP 9901 (1999) 022, hep-th/9811004; K. Dasgupta, S. Mukhi, Brane constructions, conifolds and M-theory, Nucl. Phys. B 551 (1999) 204, hep- th/9811139. [13] K. Dasgupta, S. Mukhi, Brane constructions, fractional branes and anti-de Sitter domain walls, JHEP 9907 (1999) 008, hep-th/9904131. [14] I.R. Klebanov, E. Witten, Superconformal field theory on threebranes at a Calabi–Yau singularity, Nucl. Phys. B 536 (1998) 199, hep-th/9807080. [15] P. Fayet, Fermi–Bose hypersymmetry, Nucl. Phys. B 113 (1976) 135. [16] S. Kachru, E. Silverstein, 4d conformal theories and strings on orbifolds, Phys. Rev. Lett. 80 (1998) 4855, hep- th/9802183. [17] J. Polchinski, N = 2 gauge-gravity duals, Int. J. Mod. Phys. A 16 (2001) 707, hep-th/0011193. [18] M. Billo, L. Gallot, A. Liccardo, Classical geometry and gauge duals for fractional branes on ALE orbifolds, Nucl. Phys. B 614 (2001) 254, hep-th/0105258; J.P. Gauntlett, N. Kim, D. Martelli, D. Waldram, Wrapped fivebranes and N = 2 super-Yang–Mills theory, Phys. Rev. D 64 (2001) 106008, hep-th/0106117; F. Bigazzi, A.L. Cotrone, A. Zaffaroni, N = 2 gauge theories from wrapped five-branes, Phys. Lett. B 519 (2001) 269, hep-th/0106160; P. Di Vecchia, A. Lerda, P. Merlatti, N = 1andN = 2 super-Yang–Mills theories from wrapped branes, Nucl. Phys. B 646 (2002) 43, hep-th/0205204. [19] C.V. Johnson, A.W. Peet, J. Polchinski, Gauge theory and the excision of repulson singularities, Phys. Rev. D 61 (2000) 086001, hep-th/9911161; C.V. Johnson, A.W. Peet, J. Polchinski, Gauge theory and the excision of repulson singularities, Phys. Rev. D 61 (2000) 086001, hep-th/9911161. [20] P. Merlatti, The enhancon mechanism for fractional branes, Nucl. Phys. B 624 (2002) 200, hep-th/0108016. [21] C.V. Johnson, D-brane primer, hep-th/0007170. 220 P. Merlatti / Nuclear Physics B 744 (2006) 207–220

[22] M. Bertolini, P. Di Vecchia, M. Frau, A. Lerda, R. Marotta, More anomalies from fractional branes, Phys. Lett. B 540 (2002) 104, hep-th/0202195. [23] E. Witten, Dyons of charge eθ/2π, Phys. Lett. B 86 (1979) 283. [24] M.R. Douglas, G.W. Moore, D-branes, quivers, and ALE instantons, hep-th/9603167. [25] A. Fayyazuddin, Some comments on N = 2 supersymmetric Yang–Mills, Mod. Phys. Lett. A 10 (1995) 2703, hep-th/9504120; F. Ferrari, A. Bilal, The strong-coupling spectrum of the Seiberg–Witten theory, Nucl. Phys. B 469 (1996) 387, hep-th/9602082. [26] T. Eguchi, A.J. Hanson, Asymptotically flat selfdual solutions to Euclidean gravity, Phys. Lett. B 74 (1978) 249. [27] M. Bertolini, V.L. Campos, G. Ferretti, P. Fré, P. Salomonson, M. Trigiante, Supersymmetric 3-branes on smooth ALE manifolds with flux, Nucl. Phys. B 617 (2001) 3, hep-th/0106186. [28] M.R. Douglas, Enhanced gauge symmetry in M(atrix) theory, JHEP 9707 (1997) 004, hep-th/9612126; D.E. Diaconescu, M.R. Douglas, J. Gomis, Fractional branes and wrapped branes, JHEP 9802 (1998) 013, hep- th/9712230. [29] P. Merlatti, G. Sabella, World volume action for fractional branes, Nucl. Phys. B 602 (2001) 453, hep-th/0012193. [30] J. Polchinski, Tensors from K3 orientifolds, Phys. Rev. D 55 (1997) 6423, hep-th/9606165. [31] I.R. Klebanov, A.A. Tseytlin, Gravity duals of supersymmetric SU(N) × SU(N + M) gauge theories, Nucl. Phys. B 578 (2000) 123, hep-th/0002159. [32] L.A. Pando Zayas, A.A. Tseytlin, 3-branes on resolved conifold, JHEP 0011 (2000) 028, hep-th/0010088. [33] M. Becker, K. Dasgupta, A. Knauf, R. Tatar, Geometric transitions, flops and non-Kaehler manifolds. I, Nucl. Phys. B 702 (2004) 207, hep-th/0403288; S. Alexander, K. Becker, M. Becker, K. Dasgupta, A. Knauf, R. Tatar, In the realm of the geometric transitions, Nucl. Phys. B 704 (2005) 231, hep-th/0408192; M. Becker, K. Dasgupta, S. Katz, A. Knauf, R. Tatar, Geometric transitions, flops and non-Kaehler manifolds. II, hep-th/0511099. Nuclear Physics B 744 (2006) 221–237

Twist ﬁeld as three string interaction vertex in light-cone string ﬁeld theory

Isao Kishimoto a, Sanefumi Moriyama b,∗, Shunsuke Teraguchi c

a High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801, Japan b Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan c National Center for Theoretical Sciences, Taiwan Received 22 March 2006; accepted 31 March 2006 Available online 18 April 2006

Abstract It has been suggested that matrix string theory and light-cone string field theory are closely related. In this paper, we investigate the relation between the twist field, which represents string interactions in matrix string theory, and the three-string interaction vertex in light-cone string field theory carefully. We find that the three-string interaction vertex can reproduce some of the most important OPEs satisfied by the twist field. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

Retrospecting recent progress in understanding various interesting effects in string theory, we are led to the desire of constructing a complete off-shell formulation of string theory. As will be explained below, at present there are two formulations for light-cone quantization of type IIB closed superstring theory. However, neither of them is considered to be complete. One of the formulations is the light-cone superstring ﬁeld theory (LCSFT) [1,2] constructed from supersymmetry algebra. The starting point is the Green–Schwarz action for free strings. We can construct the Hamiltonian and the supercharges satisfying the supersymmetry algebra out of it. The interaction terms are added to these charges by requiring that the total charges satisfy the

* Corresponding author. E-mail addresses: [email protected] (I. Kishimoto), [email protected] (S. Moriyama), [email protected] (S. Teraguchi).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.03.035 222 I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237 supersymmetry algebra perturbatively. The ﬁrst order interaction term is given as

|H = ZiZ¯ j vij (Λ)|V , (1) 1 123 123 α˙ ¯ i iα˙ Q = Z s (Λ)|V 123, (2) 1 123 ˜ α˙ = i ˜iα˙ | Q1 123 Z s (Λ) V 123. (3)

Here |V 123 is the three-string interaction vertex constructed by the overlapping condition and Zi (Z¯ i) is the holomorphic (anti-holomorphic) part of the bosonic momentum at the interaction point, whose divergence is regularized as follows: i 1 i 1 i P + X (σ )|V 123 ∼ √ Z |V 123, (4) 2πα σ − σI + with α = p and σI being the interaction point. Λ is the regularization of the fermionic momentum at the interaction point and vij (Λ), siα˙ (Λ) and s˜iα˙ (Λ) are known but intricate functions of Λ. The program of constructing the interaction terms is successful at the first order, though it is too complicated to proceed to higher orders. The other formulation is matrix string theory (MST) [3,4], which stems from the matrix formulation of light-cone quantization of M-theory [5] and takes the form of (1 + 1)-dimensional super-Yang–Mills theory. To relate MST to the perturbative string, we first note√ that the Yang– −1 = Mills√ coupling gYM is related to the string coupling gs and the string length α by gYM gs α . Hence, the free string limit corresponds to the IR limit and the first order interaction term to the least irrelevant operator. From the requirement of the dimension counting and the locality of the interaction, we expect that the first order interaction term is written as dimension-three operator constructed essentially out of the twist field. The interaction term of MST is proposed to be [4] = i i ¯j ¯ j H1 dσ τ Σ τ Σ m,n, (5) m,n where τ i is the excited twist field defined as 1 ∂Xi(z) · σ(0) ∼ √ τ i(0), (6) z i with σ(z,z)¯ being the Z2 twist field and Σ (z) being the spin field for the Green–Schwarz fermions. The indices m and n of the twist fields denote the string bits where the “exchange” interaction takes place. These indices have to be summed over in calculating the string amplitude. The expression (5) in MST seems somewhat formal compared with that in LCSFT (1), though it is more promising to go beyond the first order in MST than in LCSFT [6]. Hopefully we can obtain some information in LCSFT from MST. For this purpose, we would like to relate LCSFT to MST carefully. We can easily find a close analogy between (1) and (5) and between i i (4) and (6),ifweregardσ(z,z)¯ as |V 123 and τ (z, z)¯ as Z |V 123. Following this analogy between LCSFT and MST, two supercharges of MST were written down explicitly in [7]. These arguments of supercharges are consistent with the pioneering but primitive argument in [4] and with the relation between LCSFT and MST proposed in [6]. In this paper, we would like to proceed further to investigate the relation between the twist field σ(z,z)¯ and the three-string interaction vertex |V 123 scrupulously. In particular, in addition to the defining OPE of the excited twist field (6), we would like to realize the OPE of two twist I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237 223

fields [8,9] (for each dimension) 1 σ(z,z)¯ · σ(0) ∼ , (7) |z|1/4(ln |z|)1/2 in terms of the three-string interaction vertex |V 123. To realize the OPE (7), we identify the interaction point σI of |V 123 with the insertion point z of the twist field σ(z,z)¯ . We multiply two string interaction vertices with a short intermediate time T to see whether the effective interaction vertex reproduces the reflector (which corresponds to the identity operator in CFT) with the suitable singularity. A natural question arises here. In LCSFT the three-string interaction vertex |V 123 is always accompanied by the level-matching projection

dθ iθ(L(r)−L¯ (r)) P = e 0 0 , (8) r 2π on each string r = 1, 2, 3. We would like to see which one corresponds to the twist field; the interaction vertex with projections, P1P2P3|V 123, or the vertex without them, |V 123. Our answer to this question is as follows. To calculate the amplitude in LCSFT, we need to integrate over the intermediate string length (α = p+) and perform the level-matching projection at each string. The level-matching projection is equivalent to integrating over the diagrams by shifting the interaction point by an angle. These two integrations are combined into a simple summation over string bits m and n in (5). Since the twist field by itself is the expression before the summations, the corresponding interaction vertex in LCSFT should not contain any summations. Hence, the interaction vertex corresponding to the twist field is the one |V 123 without the level-matching projection and the intermediate string length integration. There are two ways to realize (7) because the interaction vertices can be connected in two different ways. One of them is the four-point tree diagram connecting the long string of two interaction vertices (Fig. 1) and the other is the two-point 1-loop diagram connecting two short strings (Fig. 2). We shall evaluate these two diagrams in the next section to see that both of the results are proportional to the reflector with the same singularity as that in (7). Note that it is desirable to perform all the computations of the above diagrams in the superstring theory, if we want to relate LCSFT to MST. However, here we shall utilize the bosonic string theory for simplicity. It should not be too difficult to generalize our computation to the supersymmetric case. Also note that the separation of the two interaction points in the above diagrams is in the worldsheet time direction, while we separate two insertion points in the space direction in (4). Since OPE does not depend on in which direction one operator approaches the other, we assume that the results do not depend on the directions. The content of this paper is as follows. In the next section, we shall first recapitulate some necessary ingredients of LCSFT. The first subsection is devoted to the computation of the tree diagram and in the second and third subsections we compute the 1-loop diagram. Finally, we conclude with some further directions. A short review of Neumann coefficients is given in Ap- pendix A. A somewhat related result about free field realization of boundary changing operators is given in Appendix B.

2. LCSFT computation

In this section, we would like to evaluate the two diagrams mentioned in the introduction to see the correspondence between the twist field σ(z,z)¯ and the interaction vertex |V 123. For this 224 I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237 purpose, let us briefly review the closed LCSFT here. Three-string interaction vertex for LCSFT with α = p+ fixed is given as = 2 E(1,2,3)+E(¯ 1,2,3)| | | V(1α1 , 2α2 , 3α3 ) μ(α1,α2,α3) δ(1, 2, 3)e p1 1 p2 2 p3 3, (9) where α1 + α2 + α3 = 0, and 3 1 3 μ(α1,α2,α3) = exp −τ0 ,τ0 = αr log |αr |, (10) = αr = r 1 r 1 d−2 d−2 d−2 d p d p d p − − δ(1, 2, 3) = 1 2 3 (2π)d 2δd 2(p + p + p ), (11) (2π)d−2 (2π)d−2 (2π)d−2 1 2 3 3 3 = 1 r,s i(r)† i(s)† + r i(r)†Pi − τ0 P2 E(1, 2, 3) Nmnam an Nnan 123 123, (12) 2 2α1α2α3 r,s=1 m,n1 r=1 n1 with Pi = α pi − α pi (i = 1,...,d − 2). We define the normalized left-moving oscillators 123√ 1 2 2 1√ i = i i† = i [ i j†]= i,j i | = ¯i an αn/ n, an α−n/ n for n 1 satisfying an,am δn,mδ and an p 0. an is the ¯ i ¯i right-moving cousin and E(1, 2, 3) is defined by replacing an by an in E(1, 2, 3). For later convenience, let us note that E(1, 2, 3) can be recast into the following form: 1 1 E(1, 2, 3) = a(3)†TN 3,3a(3)† + a(3)†TN 3,12a(12)† + a(12)†TN 12,12a(12)† 2 2 + (3)†T 3 + (12)†T 12 P − τ0 P2 a N a N 123 123, (13) 2α1α2α3 if we adopt the matrix notation for the indices of infinite oscillation modes (r) = (r) (r)† = (r)† r = r r,s = r,s a m am , a m am , N m Nm, N mn Nmn, (14) and define a(1)† N 1 a(12)† = , N 12 = , a(2)† N 2 N 1,3 N 1,1 N 1,2 N 3,12 = N 3,1 N 3,2 ,N12,3 = ,N12,12 = . (15) N 2,3 N 2,1 N 2,2 In terms of the matrix notation, the Neumann coefficients satisfy among others1 N 3,12N 12,3 + N 3,3N 3,3 = 1,N3,12N 12,12 + N 3,3N 3,12 = 0, N 12,12N 12,12 + N 12,3N 3,12 = 1, N 3,12N 12 + N 3,3N 3 =−N 3,N12,12N 12 + N 12,3N 3 =−N 12, − τ N 3T N 12,3 1N 12 =− 0 , (16) α1α2α3

1 One may wonder why N12,3 is invertible because it does not look like a square matrix. To clarify this point, let us regularize the size of the infinitely-dimensional Neumann matrices by truncating it at a finite level. Due to the expression of the mode expansion (A.2), the UV cutoff of the worldsheet σr of string r is related to the truncation level Lr by Lr σr /|αr |∼1. If we fix the UV cutoff of the worldsheet to a constant and choose string 3 to be the long string, 12,3 |α3|=|α1|+|α2|, the truncation levels of three strings have to be related by L3 = L1 + L2.InthissenseN is a square matrix. I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237 225 which play important roles in our later computation. More details about the Neumann coefficients can be found in Appendix A. The reflector for closed LCSFT is given as −(a(1)Ta(2)+a¯ (1)Ta¯ (2)) R(1, 2) = δ(1, 2)1p1|2p2|e , (17) with d−2 d−2 d p d p − − δ(1, 2) = 1 2 (2π)d 2δd 2(p + p ), (18) (2π)d−2 (2π)d−2 1 2

2.1. Tree diagram

We now turn to the evaluation of the four-point tree diagram with two incoming strings 1 (with length α1 (>0)) and 2 (with length α2 (>0)), joining and splitting again into two outgoing strings 4 and 5 of the same length as 1 and 2, respectively. (See Fig. 1.) Note that since only the two twist ﬁelds which exchange the same string bits enjoy the OPE (7), we have to choose α4 =−α1 and α5 =−α2 so that the exchange interactions take place at the same string coordinate. The effective four-string interaction vertex of this diagram is given as (α = 2)

T (3) ¯ (3) − | | (L +L ) = α3 0 0 A(1, 2, 4, 5) R(3, 6) e V(1α1 , 2α2 , 3α3 ) V(4−α1 , 5−α2 , 6−α3 ) , (20) = i i + i† i − ¯ = i i + ¯i† ¯i − with L0 p p /2 n1 nan an 1 and L0 p p /2 n1 nan an 1. Note that under r,s r the simultaneous change of signs α1 →−α1, α2 →−α2, α3 →−α3, N is even while N | is odd. Hence the Neumann coefficients in V(4−α1 , 5−α2 , 6−α3 ) can be written in terms of | | those of V(1α1 , 2α2 , 3α3 ) .FromtheOPE(7) we expect that A(1, 2, 4, 5) is proportional to the reflector |R(1, 4)|R(2, 5) with the coefficient divergent as (T −1/4(ln T)−1/2)24 = (T (ln T)2)−6 if we take the limit T →+0. Here we identify the coordinate |z| of the twist field σ(z,z)¯ with the intermediate propagation time T up to a numerical factor when we are interested only in the short

Fig. 1. Four-string tree diagram in the ρ-plane. 226 I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237 distance behavior of two operators. Though generally z may be related to T in a complicated way, the relation is approximately linear at a short distance. In order to perform our calculation simply, let us ﬁrst rewrite the proper time expression of the T (3) ¯ (3) T (3) ¯ (3) (6) ¯ (6) − | | (L +L ) − | | (L +L +L +L ) propagator e α3 0 0 into e 2 α3 0 0 0 0 . By applying the formula (19) with 3,3 0 −1 N 0 a = a(36),M= ,N= T/2 , −10 3,3 0 NT/2 3,12 3 N 0 a(12)† N P123 k = 0, l = T/2 + T/2 , (21) 3,12 a(45)† − 3 P 0 NT/2 N T/2 456 T T T T 3,3 − | | C − | | C 3,12 − | | C − | | C = α3 3,3 α3 = α3 3,12 3 = α3 3 with NT e N e , NT e N and N T e N , we can compute |A(1, 2, 4, 5) without difﬁculty: FT (1,2,4,5) A(1, 2, 4, 5) = AT δ(1, 2, 4, 5)e |p11|p22|p44|p55, (22) with 2 = 2 −(d−2)/2 − 3,3 3,3 AT μ(α1,α2,α3) det 1 NT/2NT/2 , (23) dd−2p dd−2p dd−2p dd−2p δ(1, 2, 4, 5) = 1 2 4 5 (2π)d−2 (2π)d−2 (2π)d−2 (2π)d−2 d−2 d−2 × (2π) δ (p1 + p2 + p4 + p5). (24)

The exponent FT (1, 2, 4, 5) takes a complicated expression. However, in the limit T →+0we can evaluate it formally with the use of (16) and P123 − P456 = α3(p1 + p4) which holds because d−2 d−2 of the momentum conservation (2π) δ (p3 + p6). The formal result of it is as follows: (12)†T (45)† (12)†T (45)† lim FT (1, 2, 4, 5) =− a a + a¯ a¯ T →+0 3T 12,3 −1 (12)† (45)† (12)† (45)† − (p1 + p4)α3N N a + a + a¯ + a¯ − + 2 2 3T − 3,3 2 −1 3 (p1 p4) α3N 1 N N . (25) Note that the first term gives the nonzero modes of the reflector. This is the first sign that our d−2 expectation works. Also the last term will give the zero mode part of the reflector δ (p1 + p4) = 2 3T − 3,3 2 −1 3 if the quantity b0 α3N (1 (N ) ) N is divergent. Actually, this seems to be true from numerical analysis. We can fit as b0 (1–3) log L + (constant), where L is the size of Neumann matrices which we used in our numerical computation. To regularize it properly, let us retrieve the intermediate time T in our calculation. We find = 2 3T − 3,3 2 −1 3 that, instead of the divergent quantity b0,wehavebT α3N T/2(1 (NT/2) ) N T/2 which, according to (C.18), (C.20) and (C.21) in [10], is identified to be = 2 3T − 3,3 3,3 −1 3 =− − bT α3N T 1 N NT N log(1 Z5). (26) Here we have mapped the worldsheet in the “light-cone” type ρ-plane into the whole complex z-plane by the Mandelstam map ρ(z)= α1 log(z − Z1) − log(z − Z4) + α2 log(z − Z2) − log(z − Z5) , (27) and fixed the gauge by choosing Z1 =∞,Z2 = 1,Z4 = 0 and 0

Since the limit T →+0 corresponds to the t-channel limit, Z5 should approach Z2 = 1inthis limit. After an explicit calculation, we ﬁnd T α α ∼ 4 1 2 1 − Z , (29) | | 2 5 α3 α3 and hence bT ∼ 2log(|α3|/T). This is consistent with the numerical result if we identify the regularization parameter by |α3|/T ∼ L. Therefore, the contribution of the exponential of the last term in (25) is d−2 2 π 2 −bT (p1+p4) d−2 e ∼ δ (p1 + p4). (30) 2log(|α3|/T) This implies that

2 d−2 d−2 −bT (p1+p4) (2π) δ (p1 + p2 + p4 + p5)e d−2 d−2 d−2 d−2 ∝ (2π) δ (p1 + p4)(2π) δ (p2 + p5), (31) which gives the zero mode part of the reﬂector. The determinant factor AT in (22) is already evaluated as (d = 26) − α α 2 T 6 A ∼ 210 1 2 , (32) T 2 | | α3 α3 in [10,12]. (See also [13, Appendix B].) Combining all the contributions, we ﬁnd that in the limit T →+0, 2 −6 − − T T A(1, 2, 4, 5) ∼ 2 26π 12 log R(1, 4) R(2, 5) , (33) |α123| |α123| 1/3 with α123 = (α1α2α3) . This is consistent with our expectation.

2.2. 1-loop diagram

In the previous subsection, we have computed one realization of the OPE (7). Here we would like to proceed to the other realization via the 1-loop diagram: the incoming string 6 splits into two short strings and join again into the outgoing string 3. (See Fig. 2.) For this purpose, let us calculate (α1,α2 > 0) (1) (1) (2) (2) − T (L +L¯ )− T (L +L¯ ) B(3, 6) = R(2, 5) R(1, 4) e α1 0 0 α2 0 0 × V(1α1 , 2α2 , 3α3 ) V(4−α1 , 5−α2 , 6−α3 ) . (34) The calculation is parallel to the previous case of the tree diagram. Using (19) we obtain dd−2p 1 FT (3,6,p1) B(3, 6) = BT δ(3, 6)e |p |p , (35) (2π)d−2 3 3 6 6 228 I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237

Fig. 2. Two-string 1-loop diagram in the ρ-plane. with 2 = 2 −(d−2)/2 − 12,12 12,12 BT μ(α1,α2,α3) det 1 N NT , (36)

12,12 − T C − T C − T C − T C = α1 α2 12,12 α1 α2 and NT diag(e ,e )N diag(e ,e ). Again, the exponent FT (3, 6,p1) can be evaluated in the limit T →+0 using various formulas of Neumann coefﬁcients (16) as (3)† (6)† (3)† (6)† lim FT (3, 6,p1) =− a a + a¯ a¯ . (37) T →+0 Namely, (35) is proportional to |R(3, 6) including the zero mode sector. However, the integration d−2 of the loop momentum p1 gives a divergent constant δ (0) for T = 0. Therefore, we need to regularize |B(3, 6) by the intermediate time T again: 2 = + − α1 FT (3, 6,p1) FT (3, 6,p1) osc cT p1 p3 α3 + T †(3) − †(6) + ¯ †(3) − ¯ †(6) − α1 CT a a a a p1 p3 , (38) α3 with FT (3, 6,p1)|osc being the oscillator bilinear part of FT (3, 6,p1) and cT and CT being τ − c = 2α2 − 0 + N 12T 1 − N 12,12N 12,12 1 N 12 + N 12,12N 12 , (39) T 3 α α α T T T 1 2 3 = 3 + 3,12 − 12,12 12,12 −1 12 + 12,12 12 CT α3 N N 1 NT N N T NT N , (40) − T C − T C P = P = − 12 = α1 α2 12 wherewehaveused 123 456 α3p1 α1p3 and deﬁned N T diag(e ,e )N . Note that our previous result (37) is equivalent to the following statement: (3)† (6)† (3)† (6)† lim FT (3, 6,p1) =− a a + a¯ a¯ , T →+0 osc lim cT = 0, lim CT = 0. (41) T →+0 T →+0

After we perform the loop momentum p1 integration, the result |B(3, 6) for T = 0 becomes − d−2 B(3, 6) = B (4πc ) 2 T T 1 T †(3)− †(6)+¯ †(3)−¯ †(6) 2 4c (CT (a a a a )) FT (3,6,p1)|osc × δ(3, 6)e T e |p33|p66. (42)

If we can further prove that (n, m 1) (C ) (C ) lim T m T n = 0, (43) T →+0 cT I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237 229 we have B(3, 6) ∼ KT R(3, 6) , (44) −(d−2)/2 with KT = BT (4πcT ) for T →+0. This assumption (43) seems to be true from our numerical analysis, though it is still desirable to prove it algebraically. The numerical analysis strongly suggests that our result is proportional to the reﬂector |R(3, 6). In the next subsection, we would like to turn to the evaluation of the leading order of KT for d = 26, to see the singular behavior of the OPE (7).

2.3. Evaluation of KT

It is difficult to calculate KT in (44) directly using the Neumann coefficients. For the evaluation of KT let us contract |B(3, 6) with two tachyon states. Since the full propagator including the light-cone directions is given as ∞ 1 dTr − Tr (−2p+p−+L(r)+L¯ (r)) Δ = = e αr r r 0 0 , (45) r + − (r) ¯ (r) −2pr pr + L + L αr 0 0 0 the total amplitude (without applying the level-matching projections P1P2 on string 1 and 2) is given as S = −k | −k | RLC(2, 5) RLC(1, 4) 36 3 3 6 6 × LC LC Δ1Δ2 V (1α1 , 2α2 , 3α3 ) V (4−α1 , 5−α2 , 6−α3 ) ∞ ∞ − 2T1k d d e 3 = (2π) δ (k3 + k6) dα1 dT1 dT2 δ(T2 − T1) KT , (46) 4πα1α2 0 0 where T = T1 = T2 and the light-cone directions are included in the tachyon state k|, the reflector RLC| and the interaction vertex |V LC. According to [14] this light-cone expression can be calculated in the α = p+ HIKKO string field theory: (1) ¯(1) (2) ¯(2) α=p+ α=p+ b0 b0 b0 b0 S36 = 3−k3| 6−k6| R (2, 5) R (1, 4) Ltot(1) + L¯ tot(1) Ltot(2) + L¯ tot(2) 0 0 0 0 × α=p+ α=p+ V (1α1 , 2α2 , 3α3 ) V (4−α1 , 5−α2 , 6−α3 ) . (47) + + Here the reflector Rα=p | and the interaction vertex |V α=p are those of the α = p+ HIKKO tot ¯ tot string field theory and L0 and L0 are the total Virasoro operators including the light-cone directions of the matter part and the ghost part. We can calculate it with CFT by mapping the light-cone ρ-plane into the torus u-plane, where the incoming string 6 and the outgoing string 3 in the ρ-plane are mapped to U6 and U3 in the u-plane, respectively. (See Fig. 3.) The Mandelstam map ρ(u) [15], which corresponds to Fig. 3, is given as

ϑ1(u − U6|τ) ρ(u)=|α3| log − 2πiα1u. (48) ϑ1(u − U3|τ) Here τ and U6 − U3 are pure imaginary because we do not insert the level-matching projections P1P2 in (47). The moduli parameter τ of the torus is related to T in Fig. 2 by ﬁnding the 230 I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237

Fig. 3. The Mandelstam map from the ρ-plane to the u-plane. The left ﬁgure is the light-cone ρ-plane corresponding to Fig. 2 while the right one represents the u-plane of the torus with periods 1 and τ . They are related by ρ(U6) =−∞, ρ(U3) =∞, ρ(u+) = ρ+ and ρ(u−) = ρ−. stationary points of the Mandelstam map as in (28), dρ T = ρ(u−) − ρ(u+), = 0. (49) du u=u± In the degenerating limit T →+0, we have [16]:

− iπ T e τ ∼ . (50) 8|α3| sin(πα1/|α3|) tot(r) + ¯ tot(r) Using the Mandelstam map (48) and the proper time representation of 1/(L0 L0 ), (47) can be put into the CFT expression on the torus u-plane: ∞ ∞ = −1 2 ¯ ¯ ; ¯ ; ¯ C S36 dT1 dT2 (α1α2) (α1α2) bT1 bT1 bT2 bT2 V(k3 U3, U3)V (k6 U6, U6) τ . (51) 0 0 −1 Note that we have to put the measure (α1α2) inside the CFT correlator because the Mandelstam map (48) depends on αr .HereT1 and T2 are the worldsheet proper time of the propagating strings ¯ 1and2whilebT , bT and V(k; u, u)¯ are defined as i i du du du¯ du¯ b = b(u), b¯ = b(¯ u),¯ (52) Ti 2πi dρ Ti 2πi dρ¯ ¯ Ci Ci μ ¯ V(k; u, u)¯ = c(u)c(¯ u)¯ :eikμX (u,u):, (53) ¯ + with the integration contours Ci and Ci shown in Fig. 3. Note that in the α = p HIKKO string ¯ tot + ¯ tot field theory, the propagator b0b0/(L0 L0 ) is originally defined on the light-cone ρ-plane, but the vertex operator V(k; u, u)¯ comes from that constructed on the unit disk. Therefore, we have 2 to take the conformal factor into account. The factor (α1α2) is from mapping four of b0 in the ρ-plane of each string into the total ρ-plane, while the factor C is the conformal factor for the tachyon vertices mapped from the local unit disk wr of each string into the torus u-plane: k2+k2−4 3 6 du du C = , (54) dw dw 3 u=U3 6 u=U6 where 2 du du ϑ (U − U |τ) α1 − − T 1 6 3 2πi |α | (U6 U3) |α | =− e 3 3 , (55) dw dw ϑ (0|τ) 3 u=U3 6 u=U6 1 I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237 231

| = | with ϑ1(ν τ) ∂νϑ1(ν τ) and α3 log w3 + T/2, Re ρ>T/2, ρ = (56) −α3 log w6 − T/2, Re ρ<−T/2. Now all we have to do is to evaluate each sector of (51). This was done explicitly in [17] for the open string case. The ghost sector and the nonzero mode contribution of the matter sector are exactly the square of the open string case. The only difference comes from the zero mode contribution of the matter sector and still it can be evaluated similarly to the open string case. The contribution from the ghost part is. (See [17, Eq. (6.5)].) 2 ¯ ¯ ¯ ¯ R bT bT bT bT c(U3)c(¯ U3)c(U6)c(¯ U6) = G , (57) 1 1 2 2 τ 2π where R and G are deﬁned as −1 =| | − | − − | G = 2πi 2 R α3 g1(u+ U6 τ) g1(u+ U3 τ) , η(τ) , (58) |α3| πiτ ∞ | = 2[ | ] = 12 − 2πinτ with g1(ν τ) ∂ν log ϑ1(ν τ) and η(τ) e n=1(1 e ). The contribution from the matter part is

¯ ¯ ik3X(U3,U3) ik6X(U6,U6) :e ::e :α1α2C τ πk2 d 3 ¯ ¯ 2 d − − ((U3−U6)−(U3−U6)) = (2π Im τ) dα1 δ(T2 − T1)δ (k3 + k6)(Im τ) 2 e Im τ −2k2 − ϑ (U − U |τ) 3 × η(τ) 2d 1 3 6 α α C. | 1 2 (59) ϑ1(0 τ) The derivation of the ﬁrst line from the zero mode sector of the matter part is technical. See [17] for more details. Note that δ(T2 − T1) = δ(ρ(u + τ)− ρ(u)) implies that the correlator gives a nonzero result only when

α1 U6 − U3 = τ, (60) |α3| is satisfied. Combining the ghost, matter contribution and the conformal factor, we find (d = 26) ¯ ¯ ; ¯ ; ¯ C α1α2bT1 bT1 bT2 bT2 V(k3 U3, U3)V (k6 U6, U6) τ 2 −6 − − α α T T ∼ (2π)d δd (k + k ) dα δ(T − T )2 15π 13 1 2 log , (61) 3 6 1 2 1 4 | | | | α3 α3 α3 for T = T1 = T2 →+0. Therefore, we find |B(3, 6) is proportional to the reflector with the singular coefficient as expected from (7): 2 −6 − − T T B(3, 6) ∼ 2 13π 12 log R(3, 6) , (62) |α123| |α123| if we compare the result of LCSFT (46) and that of the α = p+ HIKKO string field theory (61). 232 I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237

3. Discussion

In this paper, we investigated the correspondence between the twist ﬁeld σ(z,z)¯ and the three- string interaction vertex |V 123 in LCSFT. We evaluated two diagrams corresponding to the OPE (7) and found that both of them, (33) and (62), showed the same behavior including the log factor as expected from the calculation of the twist ﬁeld. We would like to list several further directions.

• Due to some technical difficulties, our computation of the Neumann matrices is not completely satisfactory. First of all, we only perform the numerical analysis for (43) instead of proving it algebraically. Secondly, to evaluate KT we have to detour to the CFT techniques and the α = p+ HIKKO string field theory. We hope we will have more direct computation tools in the future. • One of our original motivations comes from construction of LCSFT. After relating the twist field σ(z,z)¯ with the three-string vertex |V 123 carefully in this paper, we would like to see how matrix string theory can help in the construction of LCSFT. As explained in the introduction, it is difficult to proceed to construction of higher-order contact terms in LCSFT. We would like to see whether we can construct higher order terms explicitly with the help of MST. Our realization of the twist field via the three-string interaction vertex has been considered in the bosonic string theory in this paper. The first step should be to generalize our computation to the superstring case. • In the fermionic sector, it is a popular fact that the spin fields can be realized by the fundamental free bosons. It is interesting to see whether the free field realization has anything to do with our realization via the three-string interaction vertex in LCSFT. Though there is no simple free field realization for the twist field, we can construct one for its open string cousin, the boundary changing operator. Since the boundary changing operator changes the boundary conditions between the Neumann type and the Dirichlet type, or in other words, changes the signs of the anti-holomorphic part, it can be regarded as the twist field in the open string sector. We present the free field realization of the boundary changing operator in Appendix B by applying a result of [20]. Hopefully, we can relate the free field realization with our current realization via the three-string vertex in the future.

Acknowledgements

We would like to thank A. Hashimoto, P. Ho, Y. Kazama, E. Watanabe and especially H. Hata for valuable discussions and comments. The work of S.T. was supported by the National Center for Theoretical Sciences (North), (NSC 94-2119-M-002-001). S.M. would like to thank Univer- sity of Tokyo (Komaba) and KEK for hospitality where part of this work was done. Part of this work was done in 2005 Taipei Summer Institute on Strings, Particles and Fields, Summer Insti- tute String Theory 2005 at Sapporo and YITP workshop (YITP-W-05-21). We are grateful to the organizers of these workshops.

Appendix A. Neumann coefﬁcients

In this appendix, we would like to briefly review the Neumann coefficients. The Neumann coefficient matrix are constructed from the overlapping condition I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237 233 (1) (2) P (σ )Θ(−πα1 <σ <πα1) + P (σ − πα1)Θ(πα1 <σ)

(2) (3) + P (σ + πα1)Θ(σ < −πα1) + P (−σ) |V 123 = 0, (A.1) of the momentum function of each string ∞ √ (r) = 1 (r) + (r)c nσ + (r)s nσ P (σ ) p 2n pn cos pn sin , (A.2) 2π|αr | |αr | |αr | n=1 which states that in the string interaction process the momentum is conserved along the string worldsheet. Θ(inequality) denotes the step function, which takes value 1 if the inequality holds and otherwise 0. Our next task is to rewrite the overlapping condition in terms of the mode expansion. If we normalize the momentum π so that the trigonometric functions have the unit norm, (i) √ (i)c = 1 √ p (i)s = 1 (i)s π (i)c , π Cp , (A.3) 2π|αi| Cp 2π|αi| the overlapping condition can be recast into the following form: − α1 π (1)c − α1 π (1)s (3)c α3 (3)s α3 π = (U1 U2 ) , π = (V1 V2 ) , (A.4) − α2 π (2)c − α2 π (2)s α3 α3 where U1, U2, V1, V2 are deﬁned as √ − − T α1/α3 0 U = √ , 1 − α1 − C C − − α1 − C (1) √1 α α2( 1) 2 B α ( 1) CA 3 √ 3 C − − T α2/α3 0 U = √ , 2 − − α2 − C C − − α2 − C (2) √1 α α1( 1) 2 B α ( 1) CA 3 3 C √ √ α3 C 1 (1) α3 C 1 (2) V1 = − (−1) √ A C, V2 = − (−1) √ A C, (A.5) α1 C α2 C with A(1), A(2), B and C being

πα − m 1 (1) = n ( 1) nσ mσ A mn 2 cos cos dσ m πα1 α1 α3 0 πα1 m (−1)m nσ mσ = 2sin sin dσ, (A.6) n πα3 α1 α3 0 π(α +α ) − m 1 2 − (2) = n ( 1) n(σ πα1) mσ A mn 2 cos cos dσ m πα2 α2 α3 πα1

π(α1+α2) m (−1)m n(σ − πα ) mσ = 2sin 1 sin dσ, (A.7) n πα3 α2 α3 πα1 234 I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237

πα1 π(α1+α2) 2(−1)m+1 mσ 2(−1)m mσ (B)m = √ cos dσ = √ cos dσ, (A.8) mπα1α2 α3 mπα1α2 α3 0 πα1 (C)mn = mδmn. (A.9) Since the overlapping condition relates the incoming string momentum with the outgoing string momentum, it does not drop any information. Therefore, the transformation matrices (U1 U2 ) and (V1 V2 ) should be unitary [18]. By requiring the unitarity of these matrices, we have (r, s = 1, 2)

αr (r)T (s) (r)T − A CA = δrsC, A CB = 0, α3 1 T α3 (r)T 1 (s) 1 α1α2B CB = 1, − A A = δrs , 2 αr C C 3 3 (t) 1 (t)T 1 T 1 (r) (r)T αt A A = α1α2α3BB , A CA = 0, (A.10) C 2 αr t=1 r=1 (3) if we define (A )mn = δmn in addition. In fact, these identities are proved in [1]. As in [1], the three-string interaction vertex can be constructed by matching the momentum eigenstates. After the Gaussian integration, we find the result is given by (9) with the Neumann coefficient matrices given as − − N rs = δrs − 2A(r)TΓ 1A(s), N r =−A(r)TΓ 1B, Γ = 1 + A(1)A(1)T + A(2)A(2)T. (A.11) With this formal expression of the Neumann coefficients, we can prove all the formulas we need in this paper (16),aswellas[19]

3 3 3 r,t t,s r,t t r tT t 2τ0 N N = δr,s , N N =−N , N N = . (A.12) α1α2α3 t=1 t=1 t=1

Appendix B. Free ﬁeld realization of boundary changing operators

In this appendix we would like to present a free field realization of the boundary changing operators. In [21] a certain class of boundary deformations 1 iX(θ) −iX(θ) Sint =− dθ g exp √ +¯g exp √ , (B.1) 2 2 2 was solved exactly by the boundary state |= | − + +¯ − B N exp iπ(gr J0 gr J0 ) , (B.2) when the target space is compactified at the self-dual radius. Here gr is the renormalized coupling constant, which equals to 0 when the boundary interaction satisfies the Neumann boundary condition, g = 0, and to 1/2 when it satisfies the Dirichlet boundary condition, g =∞.In[20] the relation between the bare coupling constant g and the renormalized one gr was worked out explicitly and the coupling constant g is further generalized into an external source g(θ) depending I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237 235 on the boundary coordinate θ. The result is 1 √ 1 √ i 2XL(θ) −i 2XL(θ) B|=N| exp dθ gr g(θ),g(θ)¯ e + g¯r g(θ),g(θ)¯ e , (B.3) 2 2 2 g(θ) π g g(θ),g(θ)¯ = arctan tanh g(θ) . (B.4) r π |g(θ)| 2 If we choose g(θ) to be

g(θ) = gΘ(φ1 <θ<φ2), (B.5) and take the limit g →∞finally, we can realize a situation with one part of the boundary satisfying the Neumann boundary condition while the other satisfying the Dirichlet boundary condition. This boundary is the same as that with two boundary changing operators inserted. Hence we can consider the realization of the boundary changing operators using the result of [20]. With this choice of g(θ) we have 1 g g(θ),g(θ)¯ = Θ(φ <θ<φ ). (B.6) r 2 1 2 a = ∞ a m+1 In terms of the mode expansion J (z) m=−∞ Jm/z , our result is expressed by ∞ − − (w n − w n) B|=N| exp 1 2 J 1 , (B.7) 2n n n=−∞ −iφ with wi = e i . Here we have used the Fourier expansion of the step function, ∞ − − 2 ieinθ(e inφ2 − e inφ1 ) Θ(φ <θ<φ ) = . (B.8) 1 2 π 4n n=−∞ Note that we have used the bookkeeping notation for the zero mode n = 0. More correctly, the zero mode should be spelled out as (φ2 − φ1)/4. Therefore, a naive candidate for the boundary changing operators is −m w 1 log w 1 σ±(w) exp ± J − J . (B.9) 2m m 2 0 m =0 The sign ± is chosen depending whether the boundary operator changes the Neumann type into the Dirichlet type or vice versa. In the expression of (B.9) we are not careful about the zero mode and the normal ordering. This ambiguity can be fixed by requiring that the boundary changing operators are the primary fields with dimension (1/16, 0). Since we have w−m log w i iX (w) J 3 − J 3 − √ x =− L√ , (B.10) 2m m 2 0 L m =0 2 2 2 2 if we replace the direction 1 by the direction 3 in the exponent of (B.9), our boundary changing operators should be written as −iπJ2/2 iXL(w) iπJ2/2 σ±(w) = e 0 :exp ± − √ :e 0 , (B.11) 2 2 which are (1/16, 0) primary fields. 236 I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237

It is difﬁcult to proceed to simplify the expression (B.11) of the boundary changing operators but we can check the OPEs 1 1 σ+(z) · σ−(0) ∼ ,∂X(z) · σ±(0) ∼ √ τ±(0). (B.12) z1/8 L z

The latter one is shown by transforming the operator ∂XL(z) by the SU(2) rotation, instead√ of 2 3 = rotating the boundary changing√ operators. Since under the J0 rotation J (z) i∂XL(z)/ 2is 1 transformed into J (z) =:cos 2XL(z):, the latter one of (B.12) is easily found by noting √ √i 3√i − XL(0) 1 XL(0) :ei 2XL(z):·:e 2 2 :∼√ :e 2 2 :. (B.13) z

Here we have used XL(z) · XL(0) ∼−ln z.

References

[1] M.B. Green, J.H. Schwarz, Superstring interactions, Nucl. Phys. B 218 (1983) 43. [2] M.B. Green, J.H. Schwarz, L. Brink, Superfield theory of type II superstrings, Nucl. Phys. B 219 (1983) 437. [3] L. Motl, Proposals on nonperturbative superstring interactions, hep-th/9701025; S. Sethi, L. Susskind, Rotational invariance in the M(atrix) formulation of type IIB theory, Phys. Lett. B 400 (1997) 265, hep-th/9702101; T. Banks, N. Seiberg, Strings from matrices, Nucl. Phys. B 497 (1997) 41, hep-th/9702187. [4] R. Dijkgraaf, E.P. Verlinde, H.L. Verlinde, Matrix string theory, Nucl. Phys. B 500 (1997) 43, hep-th/9703030. [5] T. Banks, W. Fischler, S.H. Shenker, L. Susskind, M theory as a matrix model: A conjecture, Phys. Rev. D 55 (1997) 5112, hep-th/9610043. [6] R. Dijkgraaf, L. Motl, Matrix string theory, contact terms, and superstring field theory, hep-th/0309238. [7] S. Moriyama, Comments on supersymmetry algebra and contact term in matrix string theory, JHEP 0409 (2004) 013, hep-th/0405091. [8] L.J. Dixon, D. Friedan, E.J. Martinec, S.H. Shenker, The conformal field theory of orbifolds, Nucl. Phys. B 282 (1987) 13; M. Bershadsky, A. Radul, Conformal field theories with additional Z(N) symmetry, Int. J. Mod. Phys. A 2 (1987) 165. [9] Y. Okawa, B. Zwiebach, Twisted tachyon condensation in closed string field theory, JHEP 0403 (2004) 056, hep- th/0403051. [10] H. Hata, K. Itoh, T. Kugo, H. Kunitomo, K. Ogawa, Covariant string field theory 2, Phys. Rev. D 35 (1987) 1318. [11] M.B. Green, J.H. Schwarz, E. Witten, Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenol- ogy, Cambridge Univ. Press, Cambridge, 1987. [12] E. Cremmer, J.L. Gervais, Infinite component field theory of interacting relativistic strings and dual theory, Nucl. Phys. B 90 (1975) 410. [13] I. Kishimoto, Y. Matsuo, E. Watanabe, A universal nonlinear relation among boundary states in closed string field theory, Prog. Theor. Phys. 111 (2004) 433, hep-th/0312122. [14] T. Kugo, B. Zwiebach, Target space duality as a symmetry of string field theory, Prog. Theor. Phys. 87 (1992) 801, hep-th/9201040. [15] S. Mandelstam, The interacting string picture and functional integration, in: M.B. Green, D.J. Gross (Eds.), Unified String Theories, World Scientific, Singapore, 1986, p. 46. [16] I. Kishimoto, Y. Matsuo, Cardy states, factorization and idempotency in closed string field theory, Nucl. Phys. B 707 (2005) 3, hep-th/0409069. [17] T. Asakawa, T. Kugo, T. Takahashi, One-loop tachyon amplitude in unoriented open–closed string field theory, Prog. Theor. Phys. 102 (1999) 427, hep-th/9905043. [18] J. Gomis, S. Moriyama, J.w. Park, Open + closed string field theory from gauge fields, Nucl. Phys. B 678 (2004) 101, hep-th/0305264. [19] T. Yoneya, String coupling constant and dilaton vacuum expectation value in string field theory, Phys. Lett. B 197 (1987) 76. I. Kishimoto et al. / Nuclear Physics B 744 (2006) 221–237 237

[20] H. Kogetsu, S. Teraguchi, Massless boundary sine-Gordon model coupled to external fields, JHEP 0501 (2005) 048, hep-th/0410197. [21] C.G. Callan, I.R. Klebanov, A.W.W. Ludwig, J.M. Maldacena, Exact solution of a boundary conformal field theory, Nucl. Phys. B 422 (1994) 417, hep-th/9402113; J. Polchinski, L. Thorlacius, Free fermion representation of a boundary conformal field theory, Phys. Rev. D 50 (1994) 622, hep-th/9404008. Nuclear Physics B 744 [FS] (2006) 239–276

Reﬂection factors and exact g-functions for purely elastic scattering theories

Patrick Dorey a, Anna Lishman a, Chaiho Rim b, Roberto Tateo c,∗

a Department of Mathematical Sciences, University of Durham, Durham DH1 3LE, United Kingdom b Department of Physics and Research Institute of Physics and Chemistry, Chonbuk National University, Chonju 561-756, South Korea c Dipartimento di Fisica Teorica and INFN, Università di Torino, Via P. Giuria 1, 10125 Torino, Italy Received 2 February 2006; accepted 23 February 2006 Available online 24 March 2006

Abstract We discuss reflection factors for purely elastic scattering theories and relate them to perturbations of specific conformal boundary conditions, using recent results on exact off-critical g-functions. For the nonunitary cases, we support our conjectures using a relationship with quantum group reductions of the sine- Gordon model. Our results imply the existence of a variety of new flows between conformal boundary conditions, some of them driven by boundary-changing operators. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

There are two general approaches to the study of integrable boundary quantum ﬁeld theories: infrared and ultraviolet. From the infrared point of view, solving the boundary Yang–Baxter and bootstrap constraints allows sets of reﬂection factors to be associated with various bulk scattering theories. However, there are many ambiguities left by these constraints, which cannot be lifted without the introduction of more selective criteria. This is to be expected, since a given bulk theory may admit many different integrable boundary conditions.

* Corresponding author. E-mail addresses: [email protected] (P. Dorey), [email protected] (A. Lishman), [email protected] (C. Rim), [email protected] (R. Tateo).

Alternatively, a theory might be defined via a classical Lagrangian, or as a perturbation of a boundary conformal field theory. In either case, it is natural to ask for a ‘UV/IR dictionary’, giving the relationship between the infrared and ultraviolet specifications of the model. In particular, this would reveal which among the infinite families of solutions to the boundary Yang–Baxter and bootstrap equations are physically realised, and to which boundary conditions they correspond. Progress on this issue includes the relationship between the parameters in the boundary reflection factors and the couplings in the boundary sine-Gordon Lagrangian derived by Aliosha Zamolodchikov [1] (checked using the boundary TCSA [2] in [3]), and various perturbative results for cases where the theory admits a Lagrangian definition (see for example [4,5]). However, these methods cannot be readily adapted to the study of general perturbed boundary conformal field theories. In particular, a boundary analogue of the thermodynamic Bethe ansatz (TBA) c- function, which allows bulk S-matrices to be identified with specific perturbed conformal field theories [6,7], has been lacking. An obvious candidate is the g-function defined by Affleck and Ludwig in [8], but the initial proposal for a TBA-like equation for a fully off-critical version of g, taking as input just the bulk S-matrix and the boundary reflection factors [9], turned out to be incorrect [10]. Recently it has been shown that the situation can be remedied [11]. The detailed work in [11] concentrated on the boundary scaling Lee–Yang model, for which the relationship between microscopic boundary conditions and reflection factors had already been established [2].Inthis paper we extend the investigations of [11] to a collection of theories for which boundary UV/IR relations have yet to be found, namely the minimal purely elastic scattering theories associated with the ADET series of diagrams [7,12–17]. The bulk S-matrices of these models have long been known, but less progress has been made in associating solutions of the boundary bootstrap equations with specific perturbed boundary conditions. We present a collection of minimal reflection factors for the ADET theories, and test them by checking the g-function flows that they imply. We also show how these reflection factors can be modified to incorporate a free parameter, generalising a structure previously observed in the Lee–Yang model [2]. This enables us to predict a number of new flows between conformal boundary conditions. In more detail, the rest of the paper is organised as follows. Bulk and boundary ADET scattering theories are discussed in Sections 2 and 3, with Section 3 containing a full set of reflection factors fulfilling certain minimality criteria. These solutions have no free parameters, but in Sec- tion 4 the minimality requirement is relaxed and nonminimal one-parameter families of solutions are proposed to describe simultaneous bulk and boundary perturbations of boundary conformal field theories by relevant operators. In Section 5 the one-parameter families of reflection amplitudes for the Tr models are alternatively deduced by considering special reductions of the boundary sine-Gordon model. After a discussion of the exact off-critical g-function in Section 6, the physical consistency of the minimal solutions is checked in Section 7, by comparing the predictions for the ultraviolet values of the g-function obtained from the reflection factors with conformal field theory results. An additional check in first order perturbation theory is performed for the 3-state Potts model in Section 8. Section 9 summarises the g-function predictions for the one-parameter families of reflection factors and discusses their UV/IR relations. Our conclusions are given in Section 10, and Appendix A gathers together various results concerning the fields and g-function values in the diagonal gˆ1 × gˆ1/gˆ2 coset models which are needed in the main text. P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 241

2. The ADET family of purely elastic scattering theories

Purely elastic scattering theories are characterised by the property that their S-matrices are diagonal. The scattering of particles a and b with relative rapidity θ is then described by a single function Sab(θ), which is a pure phase for physical rapidities. The unitarity and crossing symmetry conditions simplify to

Sab(θ)Sab(−θ)= 1, (2.1) = − Sab(θ) Sab¯(iπ θ), (2.2) respectively, where b¯ is the antiparticle of b. The Yang–Baxter equation is trivially satisfied, but, as stressed by Zamolodchikov [18,19], the bootstrap still provides a useful constraint. This states that whenever an on-shell three-point coupling Cabc is nonzero, the S-matrix elements of particles a, b and c¯ with any other particle d satisfy = − ¯ b + ¯ a Sdc¯(θ) Sda θ iUac Sdb θ iUbc , (2.3) c where the ‘fusing angles’ Uab are related to the masses of particles a, b and c by 2 = 2 + 2 + c Mc Ma Mb 2MaMb cos Uab (2.4) and U¯ = π − U. These angles also appear as the locations of certain odd-order poles in the S- matrix elements. More details of the workings of the bootstrap can be found in [18,19] and, for example, [20]. The purely elastic scattering theories that we shall treat in this paper fall into two classes. The first class associates an S-matrix to each simply-laced Lie algebra g, of type A, D or E [7,12–14]. These S-matrices are minimal, in that they have no zeros on the physical strip 0 m θ π, and one-particle unitary, in that all on-shell three-point couplings, as inferred from the residues of forward-channel S-matrix poles, are real. They describe particle scattering in the perturbations of the coset conformal field theories gˆ1 × gˆ1/gˆ2 by their (1, 1, ad) operators, where gˆ is the affine algebra associated with g. The unperturbed theories have central charge 2r c = g , (2.5) (h + 2) where rg is the rank of g, and h is its Coxeter number. These UV central charges can be recovered directly from the S-matrices, using the thermodynamic Bethe ansatz [6,7]. The ADE S-matrices describe the diagonal scattering of rg particle types, whose masses together form the Perron–Frobenius eigenvector of the Cartan matrix of g. This allows the particles to be attached to the nodes of the Dynkin diagram of g. Each S-matrix element can be conveniently written as a product of elementary blocks [12] sinh( θ + iπx ) {x}=(x − 1)(x + 1), (x)(θ) = 2 2h (2.6) θ − iπx sinh( 2 2h ) as Sab = {x}, (2.7) x∈Aab for some index set Aab. Note that − (0) = 1,(h)=−1,(−x) = (x) 1,(x± 2h) = (x). (2.8) 242 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

Fig. 1.

The notation (2.6) has been so arranged that the numbers x are all integers. The sets Aab are tabulated in [12]; a universal formula expressing them in geometrical terms was found in [14], and is further discussed in [15,16]. The S-matrices of the second class [21–23] are labelled by extending the set of ADE Dynkin diagrams to include the ‘tadpole’ Tr . They encode the diagonal scattering of r particle types, and can be written in terms of the blocks (2.6) with h = 2r + 1 [24]:

a+b−1 Sab = {l}{h − l}. (2.9) l=|a−b|+1 step 2 These S-matrix elements are again minimal, but they are not one-particle unitary, reflecting the fact that they describe perturbations of the nonunitary minimal models M2,2r+3, with central charge c =−2r(6r + 5)/(2r + 3). The perturbing operator this time is φ13.TheTr S-matrices are quantum group reductions of the sine-Gordon model at coupling β2 = 16π/(2r + 3) [21], a fact that will be relevant later. A self-contained classification of minimal purely elastic S-matrices is still lacking, but the results of [17] single out the ADET theories as the only examples having TBA systems for which all pseudoenergies remain finite in the ultraviolet limit. The ADET diagrams are shown in Fig. 1, with nodes giving our conventions for labelling the particles in each theory. For the Dr theories, particles r − 1 and r are sometimes labelled s and s,ors and s¯,forr even or odd respectively.

3. Minimal reﬂection factors for purely elastic scattering theories

The scattering of particles by a boundary in an integrable quantum field theory is described by a set of reflection factors. In this paper we shall restrict ourselves to situations where not only the bulk, but also the boundary scattering amplitudes are purely elastic. The reflection factors are then rapidity-dependent functions Ra(θ), one for each particle type a in the theory. Unitarity [25,26] and crossing-unitarity [26] constrain these functions as

Ra(θ)Ra(−θ)= 1, (3.1)

Ra(θ)Ra¯ (θ − iπ) = Saa(2θ). (3.2) P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 243

In addition, whenever a bulk three-point coupling Cabc is nonzero, a boundary bootstrap equation holds [25,26]: = + ¯ b − ¯ a + ¯ b − ¯ a Rc¯(θ) Ra θ iUac Rb θ iUbc Sab 2θ iUac iUbc . (3.3) For a given bulk S-matrix there are infinitely-many distinct sets of reflection factors consistent with these constraints, because any solution can be multiplied by a solution of the bulk bootstrap, unitarity and crossing equations to yield another solution [27]. To identify a set of reflection factors as physically relevant some more information is needed. A common basis for the con- jecturing of bulk scattering amplitudes is a ‘minimality hypothesis’, that in the absence of other requirements one should look for solutions of the constraints with the smallest possible number of poles and zeros. In this section we show how the same principle can be used to find sets of boundary amplitudes, one for each purely elastic S-matrix of type A, D or E. The amplitudes given below were at first conjectured [28] as a natural generalisation of the minimal versions of the Ar affine Toda field theory amplitudes found in [29,30]. The reasoning behind these conjectures will be explained shortly. More recently, Fateev [31] proposed a set of reflection factors for the affine Toda field theories. These were in integral form, and not all matrix elements were given. However, modulo some overall signs and small typos, the coupling-independent parts of Fateev’s conjectures match ours. (This has also been found by Zambon [32], who obtained equivalent formulae to those recorded below taking Fateev’s integral formulae as a starting point.) Unitarity and crossing-unitarity together imply that the reflection factors must be 2πi periodic; unitarity then requires that they be products of the blocks (x) introduced in the last section. The crossing-unitarity constraint is then key for the analysis of minimality. Each pole or zero of Saa(2θ) on the right-hand side of (3.2) must be present in one or other of the factors on the left-hand side of that equation. This sets a lower bound on the number of poles and zeros for the reflection factor Ra(θ). To exploit this observation, it is convenient to work at the level of the larger blocks {x},the basic units for the bulk bootstrap equations [12,14]. We define two complementary ‘square roots’ of these blocks as − − x − 1 2h − x − 1 1 x + 1 2h − x + 1 1 x= , ˜x= (3.4) 2 2 2 2 and record their basic properties x(θ) =˜x(θ + iπ) (3.5) and x(θ)˜x(θ) ={x}(2θ). (3.6) The crossing-unitarity equation can then be solved, in a minimal fashion, by any product Ra = fx, (3.7) x∈Aaa where each factor fx can be freely chosen to be x or ˜x, modulo one subtlety: since (0) = 1, ˜ minimality requires that any factor f1 be taken to be 1 rather than 1. In fact there is always exactly one such factor for the diagonal S-matrix elements relevant here [15].1

1 = x−1 x+1 ˜= 2h−x−1 −1 2h−x+1 −1 Wecouldalsoset x ( 2 )( 2 ), x ( 2 ) ( 2 ) throughout, but this would break the pattern previously seen for the Ar theories, and turns out not to ﬁt with the g-function calculations later. 244 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

There remain the bootstrap equations (3.3). To treat these we need to know how the blocks x(θ) and ˜x(θ) transform under general shifts in θ.Asin[14,15], these shifts are best discussed by deﬁning θ iπx (x)+(θ) = sinh + (3.8) 2 2h and then − − x − 1 2h − x − 1 1 x + 1 2h − x + 1 1 x+ = , ˜x+ = (3.9) 2 + 2 + 2 + 2 + so that (x) = (x)+/(−x)+, x=x+/−x+ and ˜x=˜x+/−x+. Once a putative set of reﬂection factors (3.7) has been decomposed into these blocks, it is straightforward to implement the boundary bootstrap equations (3.3) using the properties

x+(θ + iπy/h) =x + y+(θ), ˜x+(θ + iπy/h) =x+ y+(θ). (3.10) For the ADE theories it turns out that the boundary bootstrap equations can all be satisfied by minimal conjectures of the form (3.7), and that the choice of blocks is then fixed uniquely. (From the point of view of the bootstrap equations alone, an overall swap between x and ˜x is possible, but the requirement that f1 =1 fixes even this ambiguity.) The final answers are recorded below.

The reﬂection factors for this case were given in [29]. In the current notation they are 2a−1 Ra = x. (3.11) x=1 x odd

The reﬂection factors, Ra,fora = 1,...,r − 2are 2a−1 2a−1 2a−5 Ra = x h − x h − x − 2, for a odd, x=1 x=1 x=1 x odd step 4 step 4 2a−1 2a−3 Ra = x h − xh − x − 2, for a even (3.12) x=1 x=1 xodd step 4 while for a = r − 1 and r we have 2r−5 Rr−1 = Rr = x, for r odd, x=1 step 4 2r−3 Rr−1 = Rr = x, for r even. (3.13) x=1 step 4 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 245

= = R1 R1¯ 1 7 , R2 =15711, = = ˜ R3 R3¯ 1 3 5 7 7 9 , 2 2 ˜ 2 ˜ ˜ R4 =13 5 57 79911. (3.14)

R1 =1917, R2 =171117, R3 =1579111317, ˜ R4 =13799111517, ˜ 2 R5 =135779 1111131517, 2 ˜ 2 ˜ 2 R6 =135 779 9111113 1517, 2 2 ˜ 3 ˜ 2 ˜ 2 3 2 2 R7 =13 5 57 79 9 11 111313 15 17. (3.15)

R1 =1111929, R2 =17111317192329, R3 =139111113171919212729, R4 =1579111113151517191921232529, ˜ 2 2 2 R5 =13579911 11131315 171719 192121 2325 2729, ˜ 2 2 2 2 2 R6 =135779 11 1113 13151517 171919 21223232527 29, 2 2 ˜ 2 ˜ 2 2 3 2 2 3 2 R7 =135 7 79 911 11 13 1315 15 17 1719 19 2212212323 22522729, 2 2 ˜ 3 ˜ 3 ˜ 2 4 2 3 3 4 2 3 3 R8 =13 5 57 79 9 11 11 13 13 15 15 17 17 19419 221221323323 2525227229 . (3.16) For the T series the story is different: it is not possible to satisfy the boundary bootstrap equations with a conjecture of the form (3.7). This means that the minimal reﬂection factors for these models are forced by the bootstrap to have extra poles and zeros beyond those required by crossing-unitarity alone. Our general proposal will be given in Eq. (5.4), but the situation can be understood using the boundary T1, or Lee–Yang, model: the minimal reﬂection factor found in 246 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

[2] for the single particle in this model bouncing off the |1 boundary is −1 | 1 3 4 R 1 = . (3.17) 2 2 2

1 4 −1 The simpler function ( 2 )( 2 ) would have been enough to satisfy crossing-unitarity, but then the boundary bootstrap would not have held, and so (3.17) really is a minimal solution. This observation fits nicely with the g-function calculations to be reported later: had the minimal reflection factors for the Tr theories fallen into the pattern seen for other models, there would have been a mismatch between the predicted UV values of the g-functions and the known values from conformal field theory.

4. One-parameter families of reﬂection factors

The minimal reflection factors introduced in the last section have no free parameters. However, combined perturbations of a boundary conformal field theory by relevant bulk and boundary operators involve a dimensionless quantity—the ratio of the induced bulk and boundary scales— on which the reflection factors would be expected to depend. To describe such situations, we must drop the minimality hypothesis, and extend our conjectures. A first observation, rephrasing that of [27], is that given any two solutions Ra(θ) and Ra(θ) of the reflection unitarity, crossing-unitarity and bootstrap relations (3.1), (3.2) and (3.3), their ratios ≡ Za(θ) Ra(θ)/Ra(θ) automatically solve one-index versions of the bulk unitarity, crossing and bootstrap equations (2.1), (2.2) and (2.3):

Z (θ)Z (−θ)= 1,Z(θ) = Z¯ (iπ − θ), (4.1) a a a a = − ¯ b + ¯ a Zc¯(θ) Za θ iUac Zb θ iUbc . (4.2)

The minimal reﬂection factors Ra(θ) can therefore be used as multiplicative ‘seeds’ for more = −1 general conjectures Ra(θ) Ra(θ)/Za(θ), with the parameter-dependent parts Za(θ) con- |d strained via (4.1) and (4.2). An immediate solution is Za (θ) = Sda(θ) for any (ﬁxed) particle type d in the theory, where we have used the ket symbol to indicate that the label |d might ultimately refer to one of the possible boundary states of the theory. However, this does not yet introduce a parameter. Noting that a symmetrical shift in θ preserves all the relevant equations, one possibility is to take

|d,C = + − Za (θ) Sda(θ C)Sda(θ C), (4.3) with C at this stage arbitrary. This is indeed the solution adopted by the boundary scaling Lee–Yang example studied in [2]. This model is the r = 1 member of the Tr series described earlier, and corresponds to the perturbation of the nonunitary minimal model M25 by its only = ¯ =−1 relevant bulk operator, ϕ, of conformal dimensions Δϕ Δϕ 5 . The minimal model has two conformally-invariant boundary conditions which were labelled |1 and |Φ in [2].The|1 boundary has no relevant boundary fields, and has a minimal reflection factor, given by (3.17). On the other hand, the |Φ boundary has one relevant boundary field, denoted by φ, and gives rise to a one-parameter family of reflection factors

| | | R b (θ) = R 1 (θ)/Z b (θ), (4.4) P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 247 where the factor Z|b(θ) has exactly the form mentioned above: | π π Z b (θ) = S θ + i (b + 3) S θ − i (b + 3) , (4.5) 6 6 where S(θ) is the bulk S-matrix, and the parameter b can be related to the dimensionless ratio μ2/λ of the bulk and boundary couplings λ and μ [2,10]. (Since there is only one particle type in the Lee–Yang model, the indices a, d and so on are omitted. We also changed the notation slightly from that of [2] to avoid confusing the parameter b with a particle label.)2 As an initial attempt to extend (4.4) to the remaining ADET theories one could therefore try |d,C = |d,C Ra (θ) Ra(θ)/Za (θ), (4.6) |d,C with Za (θ) as in (4.3). This maneuver certainly generates mathematically-consistent sets of reflection amplitudes, but in the more general cases it is not the most economical choice. Consider instead the functions obtained by replacing the blocks {x} in (2.7) by the simpler blocks (x) [33]: F = Sab (x). (4.7) x∈Aab For the Lee–Yang model, SF (θ) coincides with S(θ), but for other theories it has fewer poles and zeros. The bootstrap constraints are only satisfied up to signs, but if SF is used to define a |d,C function Za as | π π Z d,C (θ) = SF θ + i C SF θ − i C (4.8) a da h da h then these signs cancel, and thus (4.8) provides a more ‘minimal’ generalization of the family of Lee–Yang reflection factors which nevertheless preserves all of its desirable properties. Setting d equal to the lightest particle in the theory generally gives the family with the smallest number of additional poles and zeros, but we shall see evidence later that all cases have a role to play. The normalisation of the shift was changed in passing from (4.3) to (4.8); this is convenient because, as a consequence of the property π π (x − C)(θ) × (x + C)(θ) = (x) θ + i C × (x) θ − i C , (4.9) h h we have |d,C = − + Za (x C)(x C). (4.10) x∈Aad

|d,C −1 The factors (Za ) therefore coincide with the coupling-constant dependent parts of the afﬁne Toda ﬁeld theory S-matrices of [7,12], with C related to the parameter B of [12] by C = 1 − B.

5. Boundary Tr theories as reductions of boundary sine-Gordon

The reﬂection factors presented so far are only conjectures, and no evidence has been given linking them to any physically-realised boundary conditions. The best signal in this respect will

2 There are also boundary-changing operators, but these were not considered in [2,10]. 248 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 come from the exact g-function calculations to be reported in later sections. However, for the Tr theories, alternative support for our general scheme comes from an interesting relation with the reflection factors of the sine-Gordon model. In the bulk, the Tr theories can be found as particularly-simple quantum group reductions of the sine-Gordon model at certain values of the coupling, in which the soliton and antisoliton states are deleted leaving only the breathers [21]. Quantum group reduction in the presence of boundaries has yet to be fully understood, but the simplifications of the Tr cases allow extra progress to be made. This generalises the analysis of [2] for T1, the Lee–Yang model, but has some new features. The boundary S-matrix for the sine-Gordon solitons was found by Ghoshal and Zamolod- chikov in [26], and extended to the breathers by Ghoshal in [34]. To match the notation used above for the ADE theories, we trade the sine-Gordon bulk coupling constant β for 16π h = − 2. (5.1) β2 Ghoshal and Zamolodchikov expressed their matrix solution to the boundary Yang–Baxter equation for the sine-Gordon model3 in terms of two parameters ξ and k. However for the scalar part (which is the whole reflection factor for the breathers) they found it more convenient to use η and ϑ, related to ξ and k by 1 1 cos η cosh ϑ =− cos ξ, cos2 η + cosh2 ϑ = 1 + . (5.2) k k2 Ghoshal–Zamolodchikov’s reflection factor for the ath breather on the |η,ϑ boundary can then be written as |η,ϑ = (a) (a) Ra (θ) R0 (θ)R1 (θ), (5.3) where − ( h )(a + h) a1 (l)(l + h) R(a) = 2 (5.4) 0 + 3h + 3h 2 (a 2 ) l=1 (l 2 ) is the boundary-parameter-independent part and (a) = (a) (a) R1 (θ) S (η, θ)S (iϑ, θ) (5.5) with − a1 ( 2ν − h + l) S(a)(ν, θ) = π 2 (5.6) 2ν + h + l=1−a ( π 2 l) step 2 contains the dependence on η and ϑ. In these formulae we used the same blocks (x) as defined in (2.6) for the minimal ADE theories, with h now given by (5.1). Now for the quantum group reduction. In the bulk, suppose that β2 is such that

h = 2r + 1,r∈ N. (5.7)

3 Also found by de Vega and Gonzalez Ruiz [35]. P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 249

At these values of the coupling, Smirnov has shown [21] that a consistent scattering theory can be = sin(πa/h) obtained by removing all the solitonic states, leaving r breathers with masses Ma sin(π/h) M1. The scattering of these breathers is then given by the Tr S-matrix (2.9). In order to explain all poles without recourse to the solitons, extra three-point couplings must be introduced, some of which are necessarily imaginary, consistent with the Tr models being perturbations of nonunitary conformal field theories. In the presence of a boundary, these extra couplings give rise to extra boundary bootstrap equations, which further constrain the boundary reflection factors and impose a relation between the parameters η and ϑ. This can be seen by a direct analysis of the boundary bootstrap equations but it is more interesting to take another route, as follows. We first recall the observation of [2], that the reflection factors for the boundary Lee–Yang model match solutions of boundary Yang–Baxter equation for the sine-Gordon model at ξ → i∞. (5.8) The Lee–Yang case corresponds to r = 1, h = 3, but we shall suppose that the same constraint should hold more generally (see also [9]). The condition must be translated into the (η, ϑ) parametrisation. One degree of freedom can be retained by allowing k to tend to infinity as the limit (5.8) is taken. The relations (5.2) then become cos η cosh ϑ = A, cos2 η + cosh2 ϑ = 1. (5.9) The constant A can be tuned to any value by taking ξ and k to infinity suitably, and so the first equation in (5.9) is not a constraint; solving the second for iϑ, π iϑ = η + + (r − d)π, d ∈ Z. (5.10) 2 (The shift by r is included for later convenience.) This appears to give a countable infinity of one-parameter families of reflection factors, but this is not so: the blocks (x) in (5.5) and (5.6) depend on the boundary parameters only through the combination 2iϑ 2η = + 2(r − d)+ 1. (5.11) π π Since (x + 2h) = (x), the reflection factors for d are therefore the same as those for d + h.In addition, the freedom to redefine the parameter η gives an extra invariance of the one-parameter families under d → 2r − d. Thus the full set of options is realised by d = 0, 1,...,r. (5.12) (a) = Note also that R1 (θ), the coupling-dependent part of the reflection factor, is trivial if d 0. Thus the limit (5.8) corresponds to r one-parameter families of breather reflection factors, and one ‘isolated’ case. This matches the counting of conformal boundary conditions (the set of bulk Virasoro primary fields [36])fortheM2,2r+3 minimal models, and the fact that of these boundary conditions, all but one have the relevant φ13 boundary operator in their spectra. To see how this fits in with our earlier ideas, we swap η for C, defined by 2η h = d − + C. (5.13) π 2 Then, starting from (5.5) and relabelling, − a1 (−h + d + l + C) (−d + l + C) R(a) = 1 (d + l + C) (h − d + l + C) l=1−a step 2 250 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

a−1 = (−d − l − C)(−d − l + C)(−h + d + l + C)(−h + d + l − C) l=1−a step 2 a+d−1 = (−l − C)(−l + C)(−h + l + C)(−h + l − C). (5.14) l=1−a+d step 2 − (a) Since the ﬁrst a d terms in the last product cancel if a>d, it is easily seen that R1 coincides |d,C −1 with (Za ) , as expressed by (4.10),fortheTr S-matrix (2.9).

6. Some aspects of the off-critical g-functions

Our main tool in linking reflection factors to specific boundary conditions will be the formula for an exact off-critical g-function introduced in [11]. In this section we begin by reviewing the main formula, and then discuss some further properties and modifications which will be relevant later.

6.1. The exact g-function for diagonal scattering theories

The ground-state degeneracy, or g-function, was introduced by Affleck and Ludwig as a useful characterisation of the boundary conditions in critical quantum field theories [8]. In a massive model, the definition can proceed along similar lines [2,9,11]. One starts with the partition function Zα|α[L,R] for the theory on a finite cylinder of circumference L, length R and boundary conditions of type α at both ends. In the R-channel description, time runs along the length of the cylinder, and the partition function is represented as a sum over an eigenbasis {|ψk} of H circ(M, L), the Hamiltonian which propagates states along the cylinder: ∞ − circ (k) 2 − circ [ ]= | RH (M,L)| = G REk (M,L) Zα|α L,R α e α |α (l) e . (6.1) k=0 Here l = ML, M is the mass of the lightest particle in the theory, and α|ψ G(k)(l) = k , |α 1/2 (6.2) ψk|ψk where |α is the (massive) boundary state [26] for the α boundary condition. At finite values of l any possible infinite-volume vacuum degeneracy is lifted by tunneling effects, making the ground state |ψ0 unique. This gives Zα|α the following leading and next-to-leading behaviour in the large-R limit:

[ ]∼− circ + G(0) ln Zα|α L,R RE0 (M, L) 2ln |α(l). (6.3) It is the second, sub-leading term which characterises the massive g-function. Subtracting a linearly-growing piece −f|αL,theg-function for the boundary condition α at system size l is deﬁned to be = G(0) + ln g|α(l) ln |α(l) f|αL. (6.4) P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 251

The constant f|α is equal to the constant (boundary) contribution to the ground-state energy strip strip ≡ strip E0 (R) of the L-channel Hamiltonian H (R) Hα|α(R), which propagates states living on a segment of length R and boundary conditions α at both ends. Although this definition is in principle unambiguous, the difficulty of characterising the finite- size states |α and |ψ0 limits its utility in practice. An alternative expression for g can be { strip } obtained by comparing (6.3) with the L-channel representation, a sum over the full set Ek (R) of eigenvalues of H strip(R): ∞ −LEstrip(M,R) Zα|α[L,R]= e k . (6.5) k=0 As R is sent to infinity, we have ∞ 1 − strip = LEk (M,R) + + circ ln g|α(l) lim ln e 2f|αL RE0 (M, L) . (6.6) 2 R→∞ k=0 In theories with only massive excitations in the bulk and no infinite-volume vacuum degeneracy, ln g|α(l) tends exponentially to zero at large l, while in the UV limit l → 0 it reproduces the value of the g-function in the unperturbed boundary conformal field theory. The subleading nature of the term being extracted makes it hard to evaluate (6.6) directly. In [11], considerations of the infrared (large-l) asymptotics and a conjectured resummation led to the proposal of a general formula for the exact g-function of a purely elastic scattering theory. However in Section 6.2 we shall argue that the result of [11] needs to be modified by the introduction of a simple symmetry factor whenever there is coexistence of vacua at infinite volume. This happens in the low-temperature phases of the E6, E7, A and D models. The more-general result is rg 1 |α − (θ) ln g| (l) = ln C| + dθ φ (θ) − δ(θ) − 2φ (2θ) ln 1 + e a + Σ(l), α α 4 a aa a=1 R (6.7) |α where C|α is the symmetry factor to be discussed shortly, and the functions φa (θ) and φab(θ) |α are related to the bulk and boundary scattering amplitudes Sab(θ) and Ra (θ) by

| i d | φ α (θ) =− ln R α (θ), (6.8) a π dθ a i d φ (θ) =− ln S (θ). (6.9) ab 2π dθ ab The boundary-condition-independent piece Σ(l) can be expressed in terms of solutions to the bulk thermodynamic Bethe ansatz (TBA) equations as

∞ r 1 g 1 dθ dθ Σ(l)= 1 ··· n ε (θ ) ε (θ ) 2 n 1 + e a1 1 1 + e an n n=1 a1...an=1 Rn × + − ··· − φa1a2 (θ1 θ2)φa2a3 (θ2 θ3) φanan+1 (θn θn+1) (6.10) 252 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 with θn+1 = θ1, an+1 = a1. The functions εa(θ), called pseudoenergies, solve the bulk TBA equations rg εa(θ) = MaL cosh θ − dθ φab(θ − θ )Lb(θ ), a = 1,...,rg, (6.11) b=1 R −ε (θ) where La(θ) = ln(1 + e a ). We also recall that the ground-state energy of the theory on a circle of circumference L is given in terms of these pseudoenergies by r g dθ Ecirc(M, L) =− M cosh θL (θ) + ELM2, (6.12) 0 2π a a a=1 R where ELM2 is the bulk contribution to the energy. In [11] the formula (6.7) was checked in detail, but only for the rg = 1, C|1 = 1 case corresponding to the Lee–Yang model. Our results below will conﬁrm that it holds in more general cases, provided C|α is appropriately chosen.

6.2. Models with internal symmetries

The scaling Lee–Yang model, on which the analysis of [11] was mostly concentrated, is a massive integrable quantum field theory with fully diagonal scattering and a single vacuum. However many models lack one or both these properties. In this section we shall extend our analysis to models possessing, in infinite volume, N equivalent vacua but still described by purely elastic scattering theories.4 For the ADE-related theories the N equivalent vacua are related by a global symmetry Z. We shall see that when these systems are in their low-temperature phases, the formula for g(l) as originally proposed in [11] should be slightly corrected, by including a ‘symmetry factor’ C|α. Low-temperature phases are common features of two-dimensional magnetic spin systems below their critical temperatures. A discrete symmetry Z of the Hamiltonian H is spontaneously broken, and a unique ground state is singled out from a multiplet of equivalent degenerate vacua. This contrasts with the high-temperature phase, where the ground state is Z-invariant. The prototype of two-dimensional spin systems is the Ising model, and we shall treat this case first. The Ising Hamiltonian for zero external magnetic field is invariant under a global spin reversal transformation (Z = Z2), and at low temperatures T

The Z2 symmetry of H is preserved under renormalisation and it also characterizes the continuum field theory version of the model: in the low-temperature phase the bulk field theory has two degenerate vacua {|+, |−} with excitations, the massive kinks K[+−](θ), K[−+](θ), corresponding to field configurations interpolating between these vacua. In infinite volume the ground state is either |+ or |−, and the transition from |+ to |− (or vice versa) can only happen in an infinite interval of time. On the other hand in a finite volume V ,

4 Integrable models with non-equivalent vacua are typically associated to non-diagonal scattering like, for example, the φ13 perturbations of the minimal Mp.q models [41]. P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 253 tunneling is allowed: a kink with finite speed can span the whole volume segment in a finite time interval. We are interested in the scaling limit of the Ising model on a finite cylinder and we interpret the open-segment direction of length R as the space coordinate. Time is then periodic with period L. If the boundary conditions are taken to be fixed, of type +, at both ends of the segment, then the only multi-particle states which can propagate have an even number of kinks, of the form

K[+−](θ1)K[−+](θ2) ···K[−+](θn)(neven). (6.14)

For R large the rapidities {θj } are quantised according to the Bethe ansatz equations |+ r sinh θj − i ln R (θj ) = πj (r = MR, j = 1, 2,...), (6.15) where R|+(θ) is the amplitude describing the scattering of particles off a wall with ﬁxed boundary conditions of type +. At low temperatures Z+|+[L,R] therefore receives contributions only from states with an even number of particles. It is conveniently written in the form (see for example [42])

1 (0) 1 (1) Z+|+[L,R]= Z [L,R]+ Z [L,R], (6.16) 2 +|+ 2 +|+ where

strip (b) −LE (M,R) b −l cosh θj Z+|+[L,R]=e 0 1 + (−1) e (l = ML), (6.17) j>0 strip { } E0 (M, R) is the ground-state energy, and the set θj is quantised by (6.15). In order to extract the subleading contributions to Z+|+,asin[11] we ﬁrst set

1 ln Z(0) [L,R] 1 ln Z(1) [L,R] Z+|+[L,R]= e +|+ + e +|+ , (6.18) 2 2 and in the limit R →∞use Newton’s approximation to transform sums into integrals. The result is 1 r |+ − ln Z(b) [L,R]∼ dθ cosh(θ) + φ (θ) − δ(θ) ln 1 + (−1)be l cosh θ +|+ 2 π R = (b) − circle,b = 2lng|+(l) RE0 (M, L) (b 0, 1). (6.19) From (6.19) we see that at ﬁnite values of l as R →∞

(0) (1) ln Z+|+[L,R]−ln Z+|+[L,R]→+∞ (6.20) and therefore 2 −REcircle(M,L) 1 (0) Z+|+[L,R]∼ g|+(l) e 0 ∼ Z [L,R] 2 +|+ 1 (0) 2 −REcircle(M,L) = g (l) e 0 , (6.21) 2 |+ circle ≡ circle,0 where E0 (M, L) E0 (M, L) is the ground state energy for the system with periodic boundary conditions. 254 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

(0) Notice that Z+|+[L,R] takes contributions from states with both even and odd number of (0) = Ref. [11] particles. In fact, following [11],wehaveg|+ gﬁxed and

1 Ref. [11] g|+(l) = g|−(l) = √ g . (6.22) 2 fixed 1 In conclusion, the appearance of the extra symmetry factor C|+ = √ is related to the kink 2 selection rule restricting the possible multi-particle states. In the Ising model this was seen in the exact formula for the low-temperature partition function Z+|+[L,R], by its being written as an (0) (1) averaged sum of Z+|+[L,R] and Z+|+[L,R]. In more general theories with discrete symmetries, similar considerations apply. Consider a set of i = 1, 2,...,N ‘fixed’ boundary conditions, matching the i = 1, 2,...,N vacua. A derivation of Zi|i as in [11], including all multiparticle states satisfying the Bethe momentum quantisation conditions, will give an incorrect answer, since some states will be forbidden by the kink structure. Instead, let us construct a new boundary state N |U= |j, (6.23) j=1 and consider the partition function ZU|i. Since boundary scattering does not mix vacua it is clear that the only effect of replacing |i by |U is to eliminate the kink condition on allowed multiparticle states. Thus the counting of states on an interval with U at one end and i at the other is exactly the same as it would be in a high-temperature phase with the reflection factor at both ends being R|i(θ), and the derivation in [11] goes through to find at large R [ ]∼− circ + G(0) ln ZU|i L,R RE0 (M, L) 2ln (l), (6.24) (0) |α |i where G is—up to a linear term—given by (6.7) with φa (θ) = φa (θ) and C|α = 1. On the other hand, since all vacua are related by the discrete symmetry Z and the finite-volume vacuum state |ψ0 must be symmetrical under this symmetry, ψ0|i=ψ0|j∀i, j, and so | | | 2 G(0) 2 = U ψ0 ψ0 i = ψ0 i = G(0) 2 (l) N N |i (l) . (6.25) ψ0|ψ0 ψ0|ψ0 Hence 1 g|i(l) = √ g(l) (6.26) N 1 and g| (l) is given by the formula (6.7) with C| = √ . i i N An immediate consequence is that in the infrared, g|i(l) does not tend to one. Instead, 1 lim g|i(l) = C|i = √ . (6.27) l→∞ N This perhaps-surprising claim can be justified independently. In infinite volume, boundary states can be described using a basis of scattering states [26]: ∞ 1 |B= 1 + Kab(θ)A (−θ)A (θ) +··· |0, (6.28) 2 a b −∞ P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 255

Table 1 Data√ for the ADET purely elastic scattering theories. The symmetry factor C|i for ﬁxed-type boundary conditions is 1/ N in each case, where N is the number of degenerate vacua in the low-temperature phase

Model ceff(0) Coxeter number h Global symmetry Z N 2r + Z + Ar r+3 r 1 r+1 r 1 Dr (r even) 1 2r − 2 Z2 × Z2 4 Dr (r odd) 1 2r − 2 Z4 4 6 Z E6 7 12 3 3 7 Z E7 10 18 2 2 1 Z E8 2 30 1 1 2r + Z Tr 2r+3 2r 1 1 1 where |0 is the bulk vacuum for an infinite line. In finite but large volumes, the same expression provides a good approximation to the boundary state, the only modification normally being that the momenta of the multiparticle states must be quantised by the relevant Bethe ansatz equations. However a subtlety arises when there is a degeneracy among the bulk vacua, in situations where the boundary condition distinguishes between these vacua. (Similar issues are encountered in comparing exact bulk VEVs of fields with those obtained from finite-volume approximations [43,44].) Take the case of a fixed boundary condition which picks out one of the degenerate vacua, say i: then the state |0 on the RHS of (6.28) is |i. However when calculating the finite- volume g-function, we must consider ψ0|B, where ψ0| is the finite-volume vacuum. (Note that the bulk energy is normalised to zero when considering (6.28), so this inner product will give us g directly, rather than G.) In large but finite volumes, the tunneling amplitude between the infinite-volume bulk vacua is non-zero and so the appropriately-normalised finite-volume ground state is the symmetric combination

1 N ψ0|=√ j|. (6.29) N j=1 √ The limiting value of g in the infrared is therefore not 1, but 1/ N, as found in the exact calculation earlier. Table 1 lists the central charge, the Coxeter number, the symmetry group Z, and the number of degenerate vacua for the ADET models. For the Ar , Dr , E6 and E7 theories at low temperatures there is a coexistence of N vacuum states |i, i = 1, 2,...,N, with N = r + 1, 4, 3 and 2, respectively. The corresponding symmetry factor√ C|i, for ﬁxed-type boundary conditions which single out a single bulk vacuum, is always 1/ N. (For more general boundary conditions one can anticipate other symmetry factors, but we shall leave a detailed discussion of this point for another time.)

6.3. Further properties of the exact g-function

An important property of the sets of TBA equations that we are considering concerns the so-called Y -functions [37],

εa(θ) Ya(θ) = e . (6.30) 256 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

These satisfy a set of functional relations called a Y -system:

rg [G] iπ iπ A Y θ + Y θ − = 1 + Y (θ) ba , (6.31) a h a h b b=1 [G] where Aba is the incidence matrix of the Dynkin diagram G labelling the TBA system. Deﬁning an associated set of T -functions through the relations rg [G] π π Y (θ) = T (θ) Aca , 1 + Y (θ) = T θ + i T θ − i (6.32) a c a a h a h c=1 we have

rg [G] π π − − T θ + i T θ − i T (θ) Aca = 1 + Y 1(θ). (6.33) a h a h c a c=1 In addition the T -functions satisfy

rg [G] iπ iπ A T θ + T θ − = 1 + T (θ) ba . (6.34) a h a h b b=1 Fourier transforming the logarithm of Eq. (6.33), solving taking the large θ asymptotic into account, and transforming back recovers the (standard) formula rg LM ln T (θ) = a cosh θ − dθ χ (θ − θ )L (θ ), a = 1,...,r , (6.35) a 2 cos π ab b g h b=1 R where [17,37] dk [ ] − i d χ (θ) =− eikθ 2 cosh(kπ/h)1 − A G 1 =− SF (θ). (6.36) ab 2π ab 2π dθ ab R The result (6.35) allows us to make a connection between the exact g-functions for our parameter- dependent reflection factors and the T -functions. Taking the reflection factors defined by (4.8) and (4.6) | π π R b,C = R(a)(θ) SF θ − i C SF θ + i C (6.37) a ab h ab h and using (6.7), we find rg 1 π ln g| (l) = ln g(l) − dθ χ θ − i C L (θ ) b,C 2 ab h b a=1 R rg 1 π − dθ χ θ + i C L (θ ). (6.38) 2 ab h b a=1 R

Comparing this result with (6.35) and using the property Tb(θ) = Tb(−θ), = + π − LMb π ln g|b,C(l) ln g(l) ln Tb i C π cos C (6.39) h 2 cos h h P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 257 or, subtracting the linear contribution π G(0) (l) = G(0)(l)T i C . (6.40) |b,C b h Exact relations between g- and T -functions in various situations where the bulk remains critical were observed in [38–40]. These were extended off-criticality to a relation between the G- and T -function of the Lee–Yang model in [10]. The generalisation of [10] proposed here relies on the specific forms of our one-parameter families of reflection factors, and thus provides some further motivation for their introduction. It is sometimes helpful to have an alternative representation for the infinite sum Σ(l) in (6.7), the piece of the g-function which does not depend on the specific boundary condition. It is readily checked that, for values of l such that the sum converges, ∞ ∞ 1 1 Σ(l)= Tr Kn − Tr H n, (6.41) n 2n n=1 n=1 where = 1 √ 1 + + − 1 Kab(θ, θ ) φab(θ θ ) φab(θ θ ) (6.42) 2 1 + eεa(θ) 1 + eεb(θ ) and = √ 1 − 1 Hab(θ, θ ) φab(θ θ ) . (6.43) 1 + eεa(θ) 1 + eεb(θ ) The identity ∞ 1 ln Det(I − M)= Tr ln(I − M)=− Tr Mn (6.44) n n=1 then allows Σ(l) to be rewritten as 1 I − H Σ(l)= ln Det . (6.45) 2 (I − K)2 This formula makes sense even when the original sum diverges, a fact that will be used in the next section.

7. UV values of the g-function

Taking into account the effect of vacuum degeneracies described above, a first test of the reflection factors found earlier is to calculate the l → 0 limit of (6.7).Thevalueofg|α(0) should match the value of a conformal field theory g-function, either of a Cardy state or of a superposition of such states. As l → 0, the pseudoenergies εa(θ) tend to constants, which we shall denote as a. Their values were tabulated by Klassen and Melzer in [7], who also observed an elegant formula for the integrals of the logarithmic derivatives of the bulk S-matrix elements, later proved in [15]:

dθ φab(θ) =−Nab, (7.1) R 258 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

−1 5 where Nab is related to the Cartan matrix C of the associated Lie algebra by N = 2C − 1. We shall also need the integrals of the logarithmic derivatives of the reflection factors. Writing Ra = (x) (7.2) x∈A for some set A, the required integrals are | x dθ φ α (θ) =−2 1 − sign[x/h] (7.3) a h R x∈A with sign[0]=0. For the minimal ADE reflection factors found in Section 3 these evaluate to |α = − N dθ φa (θ) 1 aa, (7.4) R while for the Tr reflection factors (5.4), |α =− N dθ φa (θ) 2 aa. (7.5) R To calculate the UV limit of Σ(l), we define a matrix M whose elements are

Nab Mab =− , (7.6) 1 + ea in terms of which ∞ 1 1 1 Σ(0) = en +···+en =− ln (1 − e ) ···(1 − e ) , (7.7) 4 n 1 r 4 1 r n=1 where e1,...,er are the eigenvalues of M. When one of these eigenvalues is larger than 1, the inﬁnite sum does not converge (this is the case for Ar , r 5, Dr , r 4 and Er )buttheRHS of (7.7) gives the correct analytic continuation, since it follows from (6.45).

7.1. Tr

We start with the Tr theories, whose UV limits are the non-unitary minimal models M2,2r+3. There are r + 1 ‘pure’ conformal boundary conditions, and the corresponding values of the conformal g-functions can be found using the formula

S(1,1+r);(1,1+d) g + = ,d= ,...,r, (1,1 d) 1/2 0 (7.8) |S(1,1+r);(1,1)| where (1, 1 + r) is the ground state of the bulk theory, with lowest conformal weight, (1, 1) is the ‘conformal vacuum’—with conformal weight 0—and S is the modular S-matrix (see for

⎛ ⎞ 2 −1 ⎜ ⎟ ⎜ −12−1 ⎟ ⎜ ⎟ 5 ⎜ .. ⎟ The Cartan matrix for Tr is taken to be ⎜ . ⎟. ⎝ ⎠ 2 −1 −11 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 259 example [45]). The components of S for a general minimal model Mpp are 2 1+mρ+nσ p p S ; = 2 (−1) sin π nρ sin π mσ , (7.9) (n,m) (ρ,σ ) pp p p − − p p where 1 n, ρ p 1, 1 m, σ p 1 and, to avoid double-counting, m< p n and ρ

7.2. Ar , Dr and Er

The ADE cases exhibit an interestingly uniform structure: substituting (7.1) and (7.4) into (6.7) shows that the UV limit of the second, reflection-factor-dependent, term of (6.7) is always zero when the minimal reflection factors from Section 3 are used. This result can be confirmed by examining the contributions of the blocks (x) of S(2θ), and x (or ˜x)ofR|α(θ),as follows. Since the bulk S-matrix can be written as Sab(θ) = {x}(θ), (7.13) x∈Aab 260 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 where {x}=(x + 1)(x − 1), Sab(2θ) can be written similarly: Sab(2θ)= [x](θ), (7.14) x∈Aab where − − x + 1 x − 1 2h − x + 1 1 2h − x − 1 1 [x]= . (7.15) 2 2 2 2 Summing the contributions of each block [x], it is easily seen that 2x 2 dθ φ (2θ)= + δ − 2 . (7.16) aa h x,1 R x∈Aaa The minimal reflection factors discussed in Section 3 can be written in a similar way: Ra = fx, (7.17) x∈Aaa |α where each fx is either x or ˜x . The contribution to dθ φa (θ) from each x is −2+2x/h+ 2δx,1 and from each ˜x is −2 + 2x/h. Noting that every minimal reflection factor contains the block 1 exactly once, we find | 2x dθ φ α (θ) = + 2δ − 2 , (7.18) a h x,1 R x∈Aaa and so |α − − = dθ φa (θ) δ(θ) 2φaa(2θ) 0 (7.19) R for every A, D and E theory, as claimed. Working backwards, this gives a general proof of the formula (7.4),given(7.1). With the reflection-factor-dependent term giving zero, the sum of Σ(0) and the symmetry factor should correspond to a conformal g-function value. To find these values in a uniform way, we shall use the diagonal coset description gˆ1 × gˆ1/gˆ2 where gˆl is the affine Lie algebra at level l associated to one of the A, D or E Lie algebras [46]. Coset fields are specified by triples {μ, ν; ρ} of gˆ weights at levels 1, 1 and 2, and the g-function for the corresponding conformal boundary condition can be written in terms of the modular S-matrix of the coset model S{0,0;0}{μ,ν;ρ} as

S{0,0;0}{μ,ν;ρ} g{μ,ν;ρ} = . (7.20) S{0,0;0}{0,0;0}

The integrable highest weights μ of gˆ can be expressed in terms of the fundamental weights i of gˆ as rg μ = nii, (7.21) i=0 where the nonnegative integers ni , i = 0,...,rg, are the Dynkin labels. The fundamental weights can be written as

0 = (0, 1, 0), P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 261

i = (λi, aˇi, 0), (7.22) where λi are the fundamental weights of g and the colabels, aˇi , are given for each case below:

Ar :ˇai = 1, for all i,

Dr :ˇa1 =ˇar−1 =ˇar = 1, all others are 2,

E6:ˇai = (1, 1, 2, 2, 2, 3),

E7:ˇai = (1, 2, 2, 2, 3, 3, 4),

E8:ˇai = (2, 2, 3, 3, 4, 4, 5, 6). (7.23)

The highest weights μ0 of the corresponding ﬁnite-dimensional Lie algebra, g, can be written in a similar way, with Dynkin labels, n1,...,nrg , taken from the same set:

rg = μ0 niλi. (7.24) i=1 Once the level l of gˆ has been speciﬁed, the Dynkin labels must satisfy

rg l = n0 + niaˇi (7.25) i=1 so the representation of gˆl is completely determined by the Dynkin labels of the corresponding representation of g. For example, the label 0 is given to the vacuum representation, where ni = 0, i = 1,...,rg, for both gˆ1 and gˆ2. In Appendix A we show among other things that the coset conformal g-function (7.20) depends only on the level 2 weight, via the level 2 modular S-matrix6: (2) S0ρ g{μ,ν;ρ} = gρ = . (7.26) (2) S00 Clearly, this will not fix the coset field in general. For example, the number of boundary conditions with g-function equal to g0 is r + 1, 4, 3, 2, 2 for the Ar , Dr , E6, E7 and E8 models respectively. More details on this point are given in Appendix A. Algorithms for computing the modular S-matrices are given by Gannon in [47]; Schellekens has also produced a useful program for their calculation [48]. The level 2 modular S-matrix elements for A, D and E theories needed to calculate the UV values of the g-functions using Eq. (7.26) are given in Table 2. The representations are labelled by the Dynkin labels, ni , i = 0,...,r. The number of coset fields corresponding to each label is equal to the order of the orbit of that level 2 weight, under the outer automorphism group O(gˆ). Note for Dr , r even, the weights ρr−1 and ρr are in different orbits, each with order 2, whereas for r odd they are in the same orbit with order 4, so in both cases there are 4 coset fields with the same g-function value. For each coset, the possible CFT values of the g-function can now be calculated using Eq. (7.26) and the modular S-matrix elements given in Table 2. Working case-by-case, we checked these numbers against the sums ln C|α + Σ(0), the UV limits of the off-critical g- functions g(l) for the minimal reflection factors described in Section 3. In every case, we found

6 We’d like to thank Daniel Roggenkamp for his help in explaining this observation. 262 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

Table 2 Level 2 modular S-matrix elements for A, D and E models S-matrix element Labels h+3 [ h+1 ] (2) 2 2 √ π 2 kπ Ar S = sin sin 0: n = 2, n = 0, for i = 1,...,r 00 (h+2) h h+2 k=1 h+2 0 i 0,i= j, ρ : n = 1,n = j 0 i 1,i= j, h+3 [ h+1 ] (2) 2 2 √ (j+1)π 2 kπ S = sin + = sin + for j = 1,...,[h/2], 0ρj (h+2) h h 2 k 1 h 2 where [x] is the integer part of x and h is the Coxeter number. (2) = 1 2 : = = = Dr S00 4 r 0 n0 2,ni 0, for i 1,...,r : = = = ρ1 n0 n1 1, all other ni 0 (2) (2) (2) 0,i= j, S = S = 2S ρj : n0 = 0,ni = 0ρ1 0ρj 00 1,i= j, for j = 2,...,r/2 (2) (2) √1 S = S = ρ − : n0 = nr−1 = 1, all other ni = 0 0ρr−1 0ρr 2 2 r 1 : = = = ρr n0 nr 1, all other ni 0 (2) √2 2π ¯ ¯ E6 S = sin + 0: n0 = 2,ni = 0fori = 1, 1, 2, 3, 3, 4 00 21 h 2 (2) √2 2(4−j)π S = sin + ,forj = 1, 2 ρ : n0 = 1,n1 = 1, all other ni = 0 0ρj 21 h 2 1 : = = = ρ2 n0 0,n2 1, all other ni 0 E S(2) = √ 2 sin 4π 0: n = 2,n = 0, for i = 1,...,7 7 00 h+2 h+2 0 i (2) = 2 8π : = = = S0ρ 2 h+2 sin h+2 ρ1 n0 1,n1 1, all other ni 0 1 (2) √ 2 8π S = sin + ρ : n0 = 0,n2 = 1, all other ni = 0 0ρ2 h+2 h 2 2 (2) 2 4π S = 2 + sin + ρ : n0 = 0,n3 = 1, all other ni = 0 0ρ3 h 2 h 2 3 (2) (2) 1 E8 S = S = 0: n0 = 2,ni = 0, for all i = 1,...,8 00 0ρ2 2 : = = = ρ2 n0 0,n2 1, all other ni 0 (2) √1 S = ρ : n0 = 0,n1 = 1, all other ni = 0 0ρ1 2 1 that

ln g(l)|l=0 = ln C|α + Σ(0) = ln g0 (7.27) provided that the symmetry factors C|α were assigned as in Table 1. The explicit g-function values are given in Table 3, and discussed further in Appendix A. For some minimal models, we can compare g0 to the values of the g-function corresponding to known cases, thereby matching our reflection factors to physical boundary conditions. For the three-state Potts model (A2), the tricritical Ising model (E7) and the Ising model (E8) the corresponding boundary condition is the ‘fixed’ condition in each case [8,49,50]. Note the number of coset fields with g-function equal to g0 corresponds to the number of degenerate vacua, and hence to the number of possible ‘fixed’ boundary conditions for all A, D and E models.

8. Checks in conformal perturbation theory

The off-critical g-functions only match boundary conformal ﬁeld theory values in the far ultraviolet. Moving away from this point one expects a variety of corrections, some of which P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 263

Table 3 UV g-function values calculated from the minimal reﬂection factors for the A, D and E models

C|α g0 Number of coset fields h+3 [ h+1 ] √ 1 2 2 √ π 2 kπ 1/2 Ar sin sin r + 1 r+1 (h+2) h h+2 k=1 h+2 1 1 2 1/4 Dr 4 2 2 r E √1 √2 sin 2π 1/2 3 6 3 21 h+2 √1 √ 2 4π 1/2 E7 sin 2 2 h+2 h+2 E 1 √1 2 8 2 were analysed using conformal perturbation theory in [10]. The expansion provided by our exact g-function result is instead about the infrared, but convergence is sufficiently fast that the first few terms of the UV expansion can be extracted numerically, allowing a comparison with conformal perturbation theory to be made. In [11] this was done for the boundary Lee–Yang model; in this section we test our more general proposals for the case of the three-state Potts model. Since the treatment of conformal perturbation theory in [10] concentrated on the nonunitary Lee–Yang case, we begin with a general discussion of the leading bulk-induced correction to the g-function in the (simpler to treat) unitary cases. Consider a unitary conformal field theory on a circle of circumference L, perturbed by a bulk ¯ spinless primary field ϕ with scaling dimension xϕ = Δϕ + Δϕ. The perturbed Hamiltonian is then ˆ ˆ ˆ H = H0 + λH1, (8.1) where 2π c Hˆ = L + L¯ − (8.2) 0 L 0 0 12 and − L 1 xϕ Hˆ = ϕ eiθ dθ. (8.3) 1 2π For λ real in the ADE models we also expect a λ–M relation of the form 2−xϕ − M M(λ)= κ|λ|1/(2 xϕ ), λ(M) = (8.4) κ with κ a model-dependent constant. Leaving the boundary unperturbed, we have a conformal boundary condition α, with boundary | ≡ 0 = | ϕ = | | | state α . Set g|α g|α α 0 and g|α α ϕ , where 0 and ϕ are the states corresponding to the fields 1 and ϕ. Since the theory is unitary, |0 is also the unperturbed ground state; and since ϕ is primary, we have 0|ϕ|0=0. The aim is to calculate ln G|α(λ, L) = lnα|Ω where |α is the unperturbed CFT boundary state, and |Ω is the PCFT vacuum. This will be a power series in the dimensionless quantity | 2−xϕ α λL ; here, we shall obtain the coefficient of the linear term, d1 . The calculation follows [10], but is a little different (and in fact simpler) because the theory is unitary, so that the ground state is the conformal vacuum |0. 264 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

First-order perturbation theory implies |Ω=|0+λ Ωa|ψa+···, (8.5) a where the sum is over all states excluding |0, ψa|0=0 and | ˆ | = ψa H1 0 Ωa ˆ ˆ . (8.6) 0|H0|0−ψa|H0|ψa | ˆ | − | ˆ | =− | 2π + ¯ | Since the theory is unitary, 0 H0 0 ψa H0 ψa ψa L (L0 L0) ψa . Using rotational invariance as well, − 2 xϕ | | | | =− λL ψa ψa ϕ(1) 0 λ Ωa ψa − . (8.7) 1 xϕ | + ¯ | a (2π) a ψa L0 L0 ψa Hence − λL2 xϕ 1 |Ω=| − ( − P) ( − P)ϕ( )| +···, 0 1−x 1 ¯ 1 1 0 (8.8) (2π) ϕ L0 + L0 1 where P =|00| is the projector onto the ground state. Using the formula ¯ = L0+L0 ¯ 1 L0+L0−1 0 q dq, 1 2−x λL ϕ dq + ¯ | = | − | − L0 L0 − | +··· α Ω α 0 − α (1 P) q (1 P)ϕ(1) 0 . (8.9) (2π)1 xϕ q 0 + ¯ + ¯ − − ¯ Since 0|ϕ|0=0 and qL0 L0 ϕ(1)|0=qL0 L0 ϕ(1)q L0 L0 |0=qxϕ |0 this last expression simpliﬁes to

− 1 λL2 xϕ | = | − xϕ−1 | | +··· α Ω α 0 − dq q α ϕ(q) 0 . (8.10) (2π)1 xϕ 0 Now α|ϕ(q)|0 is a disc amplitude, and by Möbius invariance it is given by − | | = ϕ − 2 xϕ α ϕ(q) 0 g|α 1 q . (8.11) (This is a signiﬁcant simpliﬁcation over the nonunitary case discussed in [10], where the corresponding amplitude had to be expressed in terms of hypergeometric functions.) Taking logarithms

This is a general result. As a nontrivial check we specialise to the 3-state Potts model, described by the A2 scattering theory. There are three possible values A, B and C of the microscopic spin variable, related by an S3 symmetry. At criticality the model corresponds to a c = 4/5 conformal field theory. The primary fields are the identity I , a doublet of fields {ψ,ψ†} ¯ ¯ of dimensions Δψ = Δψ = 2/3, the energy operator ε of dimensions Δε = Δε = 2/5 and a sec- † ¯ ond doublet of fields {σ,σ } with dimensions Δσ = Δσ = 1/15. The bulk perturbing operator ϕ ¯ which leads to the A2 scattering theory is ε, and so xϕ = Δε + Δε = 4/5. Boundary conditions and states for the unperturbed model are discussed in [49,55]. One of the three ‘fixed’ boundary states, say |A, can be written in terms of W3-Ishibashi states as [55] |A=|I˜≡K |I + X|ε + | ψ + ψ† + X|σ + Xσ † , (8.14) √ √ where K4 = (5 − 5)/30 and X2 = (1 + 5)/2. Hence = =− ϕ = = ln g|A ln K 0.5961357674 ..., g|A/g|A X 1.2720196495 .... (8.15) Putting everything into (8.13), the CPT prediction for the coefficient of the first perturbative correction to the g-function for fixed boundary conditions in the three-state Potts model is |A =− d1 3.011357884 .... (8.16) The 3-state Potts model also admits ‘mixed’ boundary conditions AB, BC and CA [56].For later use we recall from [55] that the corresponding boundary state is − − − |AB=K X2|I − X 1|ε + X2|ψ + X2ψ† − X 1|σ − X 1σ † (8.17) so that, for the AB boundary, we find instead

| 1 | d AB =− d A . (8.18) 1 X4 1 The results (8.15) and (8.16) can be compared to the numerical evaluation of our exact g- function result, written in terms of λL6/5 using Fateev’s formula [51] 3(4/3) κ = (2π)5/6 γ(2/5)γ (4/5) 5/12 = 4.504307863 ..., (8.19) 2(2/3) 6/5 6/5 where γ(x)= (x)/(1 − x). Setting x = λL =−(l/κ) (remembering that (T − Tc) ∝ λ<0) a ﬁt to our numerical data yielded ln g(l) =−0.596135768 − 0.49999991l − 3.0113570x − 0.909937x2 + 0.3982x3 + 1.0x4 +··· (8.20) which agrees well with (8.15) and (8.16). The match of the constant term in (8.20) with ln g|A from (8.15) is guaranteed by our exact formula, and so serves as a check on the accuracy of our numerics. A calculation of the coefﬁcient of the irregular (in x) term, proportional to l,asin[10], predicts the value 0.5, again in good agreement with our numerical results.

9. One-parameter families and RG ﬂows

Just as the minimal reflection factors were tested in Section 7, we do the same now with |d,C the one-parameter families of reflection factors. These reflection factors, Ra , depend on the parameter C and are given in (4.6) and (4.10). 266 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

9.1. The ultraviolet limit

If we use the reﬂection factors as input to calculate the g-function, Eq. (6.7), and take the limit l → 0 then

|d g ≡ g|d,C(l)|l=0 = g0Td , (9.1) where Td = Td (θ)|l=0 is a θ-independent constant. From (6.32) and (6.30) we have √ Td = 1 + e d , (9.2) where the d values can be found in [7]. For every ADET theory, Eq. (9.1) leads to a possible CFT g-function value. For M>0 there will be many massive bulk flows as the parameter C is varied. However, it should be possible to tune C, as the limit l → 0 is taken, so as to give a massless boundary flow between the conformal g-function g|d, corresponding to the UV limit of the reflection factor |d,C Ra , and g, corresponding to the UV limit of the minimal reflection factor. These flows are depicted in Fig. 2. Note that the UV g-function corresponding to the minimal reflection factors (g0 for the ADE cases and g(1,1) for Tr ) is the smallest conformal value in all cases so, by Affleck and Ludwig’s g-theorem [8], this is a stable fixed point of the boundary RG flow. For the Tr case we find |d g = g(1,d+1) for d = 1,...,r. (9.3) This is consistent with a simple pattern of flows

(1,d + 1) → (1, 1) for d = 1,...,r. (9.4) The conformal values corresponding to the UV limits of g|d for the A, D and E cases are given in Tables 4 and 5. Again we expect there to be boundary ﬂows

|d g → g0 (9.5) in each case. Notice that in many cases the UV g-function values are sums of ‘simple’ CFT values, corresponding to ﬂows from superpositions of Cardy boundary conditions, driven by boundary-changing operators.

Fig. 2. The expected RG ﬂow pattern. P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 267

Table 4 UV g-function values for A and D models

Ar Dr |1 = |r = |i = + = − g g gρ1 g (i 1)g0,fori 1,...,r 2 |2 = |r−1 = |r−1 = |r = = g g gρ2 g g gρr−1 gρr . . |r/2 = |r/2+1 = g g gρr/2 for r even |h/2 = g gρh/2 for r odd

Table 5 UV g-function values for E6, E7 and E8 models

→ The conjectured flow for the three-state Potts model (A2), gρ1 g0, corresponds to the ‘mixed-to-fixed’ flow (AB → A) found by Affleck, Oshikawa and Saleur [49], and by Freden- hagen [52]. → Similarly, the flow gρ1 g0 in the tricritical Ising model (E7) corresponds to the ‘degenerate- → − → + + → to-fixed’ ((d) ( ) or (d) ( ))flowsof[52,53]. Notice that g0 gρ2 g0 in E7 matches the CFT g values for the flow (−) ⊕ (0+) → (−) conjectured in [52]. However this latter flow is driven by the boundary field with scaling dimension 3/5, which is inappropriate for the E7 coset description. A more careful analysis shows that our flow, which is driven by a boundary field with scaling dimension 1/10, must start from either (+) ⊕ (0+) or (−) ⊕ (−0), and flow to (+) or (−). This serves as a useful reminder that the g-function values alone do not pin down a boundary condition, and that this ambiguity can be physically significant in situations involving superpositions of boundaries.

9.2. On the relationship between the UV and IR parameters

The one-parameter families of boundary scattering theories introduced above should describe simultaneous perturbations of boundary conformal ﬁeld theories by relevant bulk and boundary operators. The action is

A = ABCFT + ABULK + ABND |α |α |α , (9.6) ABCFT where |α is the unperturbed boundary CFT action. We suppose that the boundary condition is imposed at x = 0; in general it might correspond to a superposition of n|α Cardy states. 268 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

Denoting these by |c, c = 1, 2,...,n|α, the boundary is in the state

n|α |α= n|c|c, (9.7) c=1 with n|c ∈ N. The bulk perturbing term is

0 ∞ ABULK = λ dx dy ϕ(x,y) (9.8) −∞ −∞ while the boundary perturbing part is ∞ n|α ABND = |α μc|d dy φc|d(y). (9.9) c,d=1 −∞

The operators φc|c live on a single Cardy boundary, while the φc|d with c = d are boundary = † changing operators. Since φc|d φd|c, in a unitary theory we also expect (see for example [54]) = ∗ μc|d μd|c. (9.10) Using periodicity arguments to analyse the behaviour of the ground-state energy on a strip with perturbed boundaries as in [37] and [2] (see also (9.15) and (9.16)), one can argue that the scaling dimensions of the fields φc|d must, in these integrable cases, be half that of the bulk perturbing operator ϕ: x 2 x = ϕ = (9.11) φ 2 h + 2 for the ADE systems and x 2 − h x = ϕ = (9.12) φ 2 h + 2 for the Tr models. The results recorded in Eq. (9.3) and Tables 4 and 5 provide information about the conformal boundary conditions associated with the one-parameter families of reflection factors. We see that for Tr and Ar the boundary is always in a pure Cardy state, while for Dr , E6 and E7 this is true only in one case per model, and in the E8-related theories the UV boundary always corresponds to a nontrivial superposition of the states |± and |free. This observation fits nicely with the conformal field theory results for the Ising model [55]: from (9.11) we see that for the E8 coset description the integrable boundary perturbation must have dimension xφ = 1/16, and indeed the only boundary operators with this dimension in the Ising model are φ±|free and φfree|±. For simplicity, we shall only discuss the cases involving a single Cardy boundary, where ∞ ABND = |α μ dy φ(y). (9.13) −∞ As in the Ising and Lee–Yang examples of [2,26], a simple formula is expected to link the couplings λ and μ of bulk and boundary fields to the parameter C in the reflection factors. However, P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 269 without a precise identification of the operator φ it is hard to see how such relation can be determined. Even so, a general argument combined with a numerically-supported conjecture allows the relation formula to be fixed up to a single overall dimensionless constant. This goes as follows. From Section 6.3 we know that in all cases π G(0) (l) = G(0)(l)T i C , (9.14) |d,C d h (0) where Td is the TBA-related T -function, and G (l) is the CPT G-function corresponding to the minimal reflection factor, for which there is no boundary perturbation. In addition, Td (θ) ≡ = − + h+2 = Td (θ, l) is even in θ, Td (θ) Td ( θ), and periodic, Td (θ iπ h ) Td (θ), and so it can be Fourier expanded as ∞ π 2πk T i C = c (l) + c (l) cos C . (9.15) d h 0 k h + 2 k=1

We can now use the observation of [24] that Ta(θ, l) admits an expansion with ﬁnite domain of ± 2h convergence in the pair of variables a± = (le θ ) h+2 to see that

4h 2h 4h + + + c0(l) = c0 + O l h 2 ,c1(l) = c1l h 2 + O l h 2 . (9.16) We also know that the minimal G-function has an expansion ∞ (0) (0) k(2−2xφ ) ln G (l) = ln G + gkl (9.17) k=1 G(0) while the conformal perturbation theory expansion of |d,C has the form (see [10]) ∞ − m − n G(0) = 1 xφ 2 2xφ ln |d,C(λ,μ,L) cmn μL λL . (9.18) m,n=1

Comparing (9.14)–(9.17) with (9.18) we conclude that, so long as c10 = 0, the relationship between C and μ must have the form 2π 2h 2π −x μ =ˆμ cos C M h+2 =ˆμ cos C M1 φ , (9.19) 0 h + 2 0 h + 2 where μˆ 0 is an unknown dimensionless constant. However the result (9.19) can only be correct if 1 − xφ = 2h/(h + 2). This is true only in the nonunitary Tr models, and indeed it reproduces the Lee–Yang result of [11] when specialised to T1.FortheADE theories, 1 − xφ = h/(h + 2) and we conclude that c10, the ﬁrst μ-dependent correction to G, must be zero. (This is not surprising disk = since in a unitary CFT this correction is proportional to φ |α 0.) The ﬁrst contribution is − then at order O(μ2) = O(M2 2xφ ), and at this order there is an overlap between the expansions of T (θ, l) and G(0)(l). This leads to the less-restricted result d 2 ˆ 2π 2h ˆ 2π 2−2x μ = k zˆ + cos C M h+2 = k zˆ − cos C M φ , (9.20) 0 h + 2 0 h + 2 ˆ where now both k0 and zˆ are unknown constants. If we now consider the Ising model, then the μ − C formula is known [26]. Written in terms of C it becomes π √ π hRef. [26] 2 = μ2 = 2M 1 + cos C ⇒ μ = 2 M cos C . (9.21) 2 4 270 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

Thus the boundary magnetic field is an even function of C. It is then tempting to conjecture that zˆ = 1 for all the g|1,C cases in the Ar models, and, to preserve the perfect square property, that zˆ is either 1 or −1 in all other ADE single boundary condition situations: πC − μ − πC zˆ = : μ =ˆμ M1 xφ ↔ √ =ˆμ κ1 xφ , 1 0 cos + 0 cos + (9.22) h 2 λ h 2 πC μ πC 1−xφ 1−xφ zˆ =−1: μ =ˆμ0 sin M ↔ √ =ˆμ0κ sin . (9.23) h + 2 λ h + 2 We now return to the physical picture of flows parametrised by C depicted in Fig. 2.Atλ = 0the bulk mass is zero, and the only scale in the problem is that induced by the boundary coupling μ. The massless boundary flow down the left-hand edge of the diagram therefore corresponds to varying |μ| from 0 to ∞.Ifλ is instead kept finite and nonzero while |μ| is sent to infinity, the flow will collapse onto the lower edge of the diagram, flowing from g0 in the UV. For the g-function calculations to reproduce this behaviour, the reflection factor should therefore reduce to its minimal version as |μ|→∞.For(9.22), |μ|→∞corresponds to C → i∞, which does indeed reduce the reflection factor as required. On the other hand, taking |μ|→∞in (9.23), requires C → (h + 2)/2 + i∞. While the reduction is again achieved in the limit, real analyticity of the reflection factors is lost at intermediate values of μ. For this reason option (9.22) might be favoured, but more detailed work will be needed to make this a definitive conclusion. In fact, the proposal (9.22) can be checked at μ = 0 in the 3-state Potts model (h = 3), as follows. Consider the results (8.16) and (8.18) and set − | | − 1 | δ =−κ 6/5 d AB − d A = κ 6/5 1 + d A =−0.683763720 .... (9.24) 1 1 1 X4 1 According to the conclusions of Section 9.1 and Eq. (9.22), G(0) = G(0) π |AB(l) |A(l)T1 C , (9.25) 3 C=5/2 6/5 and δ1 should match the coefficients t1 of l in the expansion of the function T1(θ, l)|θ=0 about l = 0. Noticing that T1(θ, l) = T2(θ, l) = TLY (θ, l) we can use Table 6 of [10]:lnTLY (iπ(b + −6/5 3)/6) = ε(iπ(b + 3)/6), C = 5/2 corresponds to b = 2, hM = hc =−0.6852899839, and we find

t1 ∼ 0.9977728224hc =−0.683763721 ... (9.26) which within numerical accuracy is equal to δ1. The study of the cases where the theory is perturbed by boundary changing operators is more complicated. Our results are very preliminary, and we have decided to postpone the discussion to another occasion.

10. Conclusions

Since the initial studies of integrable boundary scattering theories, surprisingly little progress has been made on the identification of the many known boundary reflection factors with particular perturbations of conformal field theories. One aim of this paper has been to use the recently- discovered exact expression for the ground-state degeneracy g to begin to fill this gap. Working within the framework of the minimal ADET models we were able to identify a special class P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 271 of reflection factors possessing a very simple conformal field theory interpretation: in the ultraviolet limit they describe the reflection of massless particles against a wall with fixed boundary conditions |i matching the low-temperature vacua. A collection of one-parameter families of amplitudes was also proposed, and partially justified by a suggested quantum group reduction of the boundary sine-Gordon model at β2 = 16π/(2r + 3). The consistency of these novel families of boundary scattering theories was supported by our g-function calculations: they appear to correspond to perturbations of Cardy boundary states, or of superpositions of such states, by relevant boundary operators, with each boundary flow ending at a fixed boundary condition |i. While our results indicate that these models all have interesting interpretations as perturbed boundary conformal field theories, the checks performed so far should be considered as preliminary, and more work will be needed to put these results on a firmer footing. Two immediate open problems are fixing completely the boundary UV/IR relations, and determining the boundary bound-state content of each ADET model. The effort involved should be rewarded both by the well-established role in condensed matter physics taken by some of these theories, and by the mathematical insights into general integrable boundary theories their further study should bring.

Acknowledgements

We would like to thank Daniel Roggenkamp for his helpful explanations of coset conformal ﬁeld theories, and Zoltan Bajnok, Davide Fioravanti, Gerard Watts, Cristina Zambon and Aliosha Zamolodchikov for useful discussions. P.E.D. and A.R.L. thank APCTP, Pohang, for hospitality during the Focus Program ‘Finite-Size Technology in Low-Dimensional Quantum Field Theory’, where much of this material was presented, and INFN, Torino University for hospitality in the ﬁnal stage of the project. The work was partly supported by the EC network “EUCLID”, contract number HPRN-CT-2002-00325, and partly by a NATO grant PST.CLG.980424. P.E.D. was also supported in part by JSPS/Royal Society grant and by the Leverhulme Trust, and the work by C.R. was partly supported by the Korea Research Foundation 2002-070-C00025, and by the Science Research Center Program of the Korea Science and Engineering Foundation through the Center for Quantum Spacetime (CQUeST) of Sogang University with grant number R11-2005- 021.

Appendix A. Diagonal gˆ 1 × gˆ 1/gˆ 2 coset data

In general, to extract the gˆ/hˆ coset conformal theory from the gˆ WZW model, we need to decompose the representations, μ,ofgˆ into a direct sum of representations, ν,ofhˆ: μ → bμνν. (A.1) ν This decomposition corresponds to the character identity

χμ(τ) =˜χν(τ)bμν(τ), (A.2) ˆ where χμ and χ˜ν are the characters of the gˆ and h representations μ and ν respectively and the branching function, bμν(τ), is the character of the coset theory. Let Π denote the projection matrix giving the explicit projection of every weight of g onto a weight of h. Clearly for a coset 272 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 character to be nonzero we must have − ∈ Πμ0 ν0 ΠQ, (A.3) where Q is the root lattice of g. For our diagonal cosets, since Π(Q⊕ Q) = Q this selection rule is particularly simple: + − ∈ μ0 ν0 ρ0 Q, (A.4) where μ0, ν0 and ρ0 are weights of g. The group of outer automorphisms of gˆ, O(gˆ), permutes the fundamental weights in such a way as to leave the extended Dynkin diagram invariant. The action of an element, A ∈ O(gˆ),on a modular S-matrix element of gˆ at some level l is

2πi(A0,μ ) S(Aμ)μ = Sμμ e 0 . (A.5) The modular S-matrix for a diagonal coset theory has the form

(1) (1) (2)∗ = S{μ,ν;ρ}{μ ,ν ;ρ } Sμμ Sνν Sρρ . (A.6) Under the O(gˆ) action it transforms as

2πi(A0,μ +ν −ρ ) S{Aμ,Aν;Aρ}{μ,ν;ρ} = e 0 0 0 S{μ,ν;ρ}{μ,ν;ρ} = S{μ,ν;ρ}{μ,ν;ρ} (A.7) + − ∈ since μ0 ν0 ρ0 Q [57]. This suggests that S is a degenerate matrix, which cannot be the case since it represents a modular transformation. We must therefore remove this degeneracy by identifying the ﬁelds

{Aμ,Aν; Aρ}≡{μ, ν; ρ}∀A ∈ O(gˆ). (A.8) For these diagonal cosets, the orbit of every element A of O(gˆ) has the same order, N, which is simply the order of the global symmetry group of the model. With this field identification, we find that every field in the theory appears with multiplicity N. To remedy this, the partition function must be divided by this multiplicity, which has the effect of introducing N into the coset modular S-matrix [58]:

(1) (1) (2) = S{μ,ν;ρ}{μ ,ν ;ρ } NSμμ Sνν Sρρ . (A.9) More information on field identifications can be found, for example, in [59]. For the level 1 representations of simply laced affine Lie algebras, the characters have a particularly simple form [60,61]: 1 χμ(q) = θμ(q), (A.10) ηrg (q) where q = exp(2πiτ) and the generalised θ and Dedekind η functions are 1 (μ ,μ ) θμ(q) = q 2 0 0 , (A.11) ∈ μ0 Q ∞ 1 n η(q) = q 24 1 − q . (A.12) n=1 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 273

→−1 Under the modular transformation τ τ the theta function transforms as 1 rg/2 −2πi(ν0,μ ) θμ → (−iτ) √ e 0 θν, (A.13) |P/Q| ν0∈P/Q where P and Q are the weight lattice and root lattice respectively [61,62], and the η function becomes 1 η − = (−iτ)1/2η(τ). (A.14) τ The modular S-matrix for level 1 is therefore

1 − S(1) = √ e 2πi(ν0,μ0). (A.15) μν |P/Q| (1) For the coset description of the g-function, Eq. (7.20), we need to compute S0μ which is simply 1 S(1) = √ . (A.16) 0μ |P/Q| It is important to note that P/Q is isomorphic to the centre of the group under consideration [59,63]. The group of ﬁeld identiﬁcations of the coset is also isomorphic to the centre of this group so consequently Eq. (A.16) becomes 1 S(1) = √ , for all μ. (A.17) 0μ N The g-function can now be written in terms of the level 2 modular S matrices only:

(2) S0ρ g{μ,ν;ρ} = . (A.18) (2) S00 As we can see from (A.18), identifying the conformal g-function value will only pin down the level 2 representation. For E8, since there is only one possible level 1 representation (with Dynkin labels ni = 0, i = 1,...,8), by specifying the level 2 representation we fix the coset field. However, for other cases, although the coset selection and identification rules (A.4), (A.8) do constrain the possible coset representations we are still left with some ambiguity in general. For example, for the Ar cases the selection and identification rules are quite simple and the possible cosets, given a fixed level 2 weight, are shown in Table 6. The notation is as follows: = = μj , νj are level 1 weights and ρj is the level 2 weight with Dynkin labels ni 1fori j and = = ni 0, i 1,...,r otherwise. The labels μ0, ν0 and ρ0 now represent 0 with Dynkin labels ni = 0, i = 1,...,r. The coset fields for the A2 (three-state Potts model) and E7 (tricritical Ising model) cases, along with the corresponding boundary condition labels from [49] and [50] respectively, are given in Tables 7 and 8 as concrete examples. It is useful to note that for the Ar , Dr , E6 and E7 models, S00 = S0ρ only when ρ = A0 for some A ∈ O(gˆ) [64] and for each such ρ there is a unique coset field, so the number of fields with g-function equal to g{μ,ν,0} is equal to the size of the orbit of 0. On the other hand, E8 has = no diagram symmetry, but it is also exceptional in that S00 S0ρ2 where ρ2 has Dynkin labels n2 = 1, all other ni = 0. Physically, this is to be expected as the two fields correspond to the two 274 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

Table 6 Ar coset fields, indicating the number of distinct fields for each level 2 weight; in the first column, [x] denotes the integer part of x Fixedlevel2weight Level 1 weight Level 1 weight Number of coset ρ μ ν fields {μ, ν; ρ} 000 μi νr+1−i , i = 1,...,r r + 1 ρj μi νj−i , i = 0,...,j j = 1,...,[(r + 1)/2] μj+k νr+1−k, k = 1,...,r − jr+ 1

Table 7 A2 coset ﬁelds with the corresponding boundary labels and level 2 weight labels from Table 2. The weights are given in terms of Dynkin labels [n0,n1,n2,...]

A2 Coset ﬁeld Boundary label Level 2 weight label from [49] from Table 2 {[1, 0, 0], [1, 0, 0]; [2, 0, 0]} A {[0, 0, 1], [0, 1, 0]; [2, 0, 0]} B 0 {[0, 1, 0], [0, 0, 1]; [2, 0, 0]} C {[1, 0, 0], [0, 1, 0]; [1, 1, 0]} AB {[ ] [ ]; [ ]} 0, 0, 1 , 0, 0, 1 1, 1, 0 BC ρ1 {[0, 1, 0], [1, 0, 0]; [1, 1, 0]} AC

Table 8 E7 coset ﬁelds with the corresponding boundary labels and level 2 weight labels from Table 2. The weights are given in terms of Dynkin labels [n0,n1,n2,...]

E7 Coset ﬁeld Boundary label Level 2 weight label from [50] from Table 2 {[1, 0,...,0], [1, 0,...,0], [2, 0,...,0]} (−) 0 {[0, 1, 0,...], [0, 1, 0,...]; [2, 0,...,0]} (+) {[ ] [ ]; [ ]} 1, 0,...,0 , 0, 1, 0,... 1, 1, 0,...,0 (d) ρ1 {[ ] [ ]; [ ]} − 1, 0,...,0 , 1, 0,...,0 0, 0, 1, 0,... ( 0) ρ2 {[0, 1, 0,...], [0, 1, 0,...]; [0, 0, 1, 0,...]} (0+) {[ ] [ ]; [ ]} 1, 0,...,0 , 0, 1, 0,... 0, 0, 0, 1, 0,... (0) ρ3

ﬁxed boundary conditions (−) and (+), which clearly must have equal g-function values. (This exceptional equality is discussed from a more mathematical perspective in, for example, [64].) From (A.5) it is clear that S0(Aρ) = S0ρ is also true for ρ = 0.FortheA and E models at = level 2, we ﬁnd that these are the only cases where S0ρi S0ρj ;fortheDr models there is more degeneracy.

References

[1] Al.B. Zamolodchikov, Talk at the 4th Bologna workshop on conformal and integrable models, 3 July 1999, unpublished. [2] P. Dorey, A. Pocklington, R. Tateo, G.M.T. Watts, TBA and TCSA with boundaries and excited states, Nucl. Phys. B 525 (1998) 641, hep-th/9712197. P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276 275

[3] Z. Bajnok, L. Palla, G. Takacs, Spectrum and boundary energy in boundary sine-Gordon theory, Nucl. Phys. B 622 (2002) 565, hep-th/0108157. [4] E. Corrigan, A. Taormina, Reflection factors and a two-parameter family of boundary bound states in the sinh- Gordon model, J. Phys. A 33 (2000) 8739, hep-th/0008237. [5] M. Kormos, L. Palla, Some semi-classical issues in boundary sine-Gordon model, J. Phys. A 35 (2002) 5471, hep- th/0201230. [6] Al.B. Zamolodchikov, Thermodynamic Bethe ansatz in relativistic models. Scaling three state Potts and Lee–Yang models, Nucl. Phys. B 342 (1990) 695. [7] T.R. Klassen, E. Melzer, Purely elastic scattering theories and their ultraviolet limits, Nucl. Phys. B 338 (1990) 485. [8] I. Affleck, A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161. [9] A. LeClair, G. Mussardo, H. Saleur, S. Skorik, Boundary energy and boundary states in integrable quantum field theories, Nucl. Phys. B 453 (1995) 581, hep-th/9503227. [10] P. Dorey, I. Runkel, R. Tateo, G.M.T. Watts, g-function flow in perturbed boundary conformal field theories, Nucl. Phys. B 578 (2000) 85, hep-th/9909216. [11] P. Dorey, D. Fioravanti, C. Rim, R. Tateo, Integrable quantum field theory with boundaries: The exact g-function, Nucl. Phys. B 696 (2004) 445, hep-th/0404014. [12] H.W. Braden, E. Corrigan, P. Dorey, R. Sasaki, Affine Toda field theory and exact S-matrices, Nucl. Phys. B 338 (1990) 689. [13] V.A. Fateev, A.B. Zamolodchikov, Conformal field theory and purely elastic S-matrices, Int. J. Mod. Phys. A 5 (1990) 1025. [14] P. Dorey, Root systems and purely elastic S-matrices, Nucl. Phys. B 358 (1991) 654. [15] P. Dorey, Root systems and purely elastic S-matrices, 2, Nucl. Phys. B 374 (1992) 741, hep-th/9110058. [16] A. Fring, D.I. Olive, The fusing rule and the scattering matrix of affine Toda theory, Nucl. Phys. B 379 (1992) 429. [17] F. Ravanini, R. Tateo, A. Valleriani, Dynkin TBAs, Int. J. Mod. Phys. A 8 (1993) 1707, hep-th/9207040. [18] A.B. Zamolodchikov, Integrals of motion and S-matrix of the (scaled) T = T(C)Ising model with magnetic field, Int. J. Mod. Phys. A 4 (1989) 4235. [19] A.B. Zamolodchikov, Integrable field theory from conformal field theory, in: Proceedings of Taniguchi Symposium, Kyoto, 1988. [20] P. Dorey, Exact S-matrices, in: Proceedings of the 1996 Eötvös Graduate School, hep-th/9810026. [21] F.A. Smirnov, The perturbated c<1 conformal field theories as reductions of sine-Gordon model, Int. J. Mod. Phys. A 4 (1989) 4213. [22] J.L. Cardy, G. Mussardo, S-matrix of the Yang–Lee edge singularity in two dimensions, Phys. Lett. B 225 (1989) 275. [23] P.G.O. Freund, T.R. Klassen, E. Melzer, S-matrices for perturbations of certain conformal field theories, Phys. Lett. B 229 (1989) 243. [24] P. Dorey, R. Tateo, Excited states in some simple perturbed conformal field theories, Nucl. Phys. B 515 (1998) 575, hep-th/9706140. [25] A. Fring, R. Koberle, Factorized scattering in the presence of reflecting boundaries, Nucl. Phys. B 421 (1994) 159, hep-th/9304141. [26] S. Ghoshal, A.B. Zamolodchikov, Boundary S-matrix and boundary state in two-dimensional integrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841, hep-th/9306002; S. Ghoshal, A.B. Zamolodchikov, Int. J. Mod. Phys. A 9 (1994) 4353, Erratum. [27] R. Sasaki, Reflection bootstrap equations for Toda field theory, in: W. Nahm, J.-M. Shen (Eds.), Proceedings of the Conference, Interface between Physics and Mathematics, hep-th/9311027. [28] P. Dorey, unpublished. [29] E. Corrigan, P. Dorey, R.H. Rietdijk, R. Sasaki, Affine Toda field theory on a half line, Phys. Lett. B 333 (1994) 83, hep-th/9404108. [30] E. Corrigan, P. Dorey, R.H. Rietdijk, Aspects of affine Toda field theory on a half line, Prog. Theor. Phys. Suppl. 118 (1995) 143, hep-th/9407148. [31] V.A. Fateev, Normalization factors, reflection amplitudes and integrable systems, hep-th/0103014. [32] C. Zambon, PhD thesis, York, 2004, unpublished. [33] P. Dorey, F. Ravanini, Staircase models from affine Toda field theory, Int. J. Mod. Phys. A 8 (1993) 873, hep- th/9206052. [34] S. Ghoshal, Bound state boundary S-matrix of the sine-Gordon model, Int. J. Mod. Phys. A 9 (1994) 4801, hep- th/9310188. 276 P. Dorey et al. / Nuclear Physics B 744 [FS] (2006) 239–276

[35] H.J. de Vega, A. Gonzalez Ruiz, Boundary K-matrices for the six vertex and the n(2n − 1)A(n− 1) vertex models, J. Phys. A 26 (1993) L519, hep-th/9211114. [36] J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (1984) 514. [37] Al.B. Zamolodchikov, On the thermodynamic Bethe ansatz equations for reflectionless ADE scattering theories, Phys. Lett. B 253 (1991) 391. [38] P. Fendley, H. Saleur, N.P. Warner, Exact solution of a massless scalar field with a relevant boundary interaction, Nucl. Phys. B 430 (1994) 577, hep-th/9406125. [39] V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov, Integrable structure of conformal field theory, quantum KdV theory and thermodynamic Bethe ansatz, Commun. Math. Phys. 177 (1996) 381, hep-th/9412229. [40] V.V. Bazhanov, A.N. Hibberd, S.M. Khoroshkin, Integrable structure of W(3) conformal field theory, quantum Boussinesq theory and boundary affine Toda theory, Nucl. Phys. B 622 (2002) 475, hep-th/0105177. [41] F.A. Smirnov, Reductions of the sine-Gordon model as a perturbation of minimal models of conformal field theory, Nucl. Phys. B 337 (1990) 156. [42] R. Chatterjee, Exact partition function and boundary state of 2D massive Ising field theory with boundary magnetic field, Nucl. Phys. B 468 (1996) 439, hep-th/9509071. [43] V. Fateev, S.L. Lukyanov, A.B. Zamolodchikov, Al.B. Zamolodchikov, Expectation values of local fields in Bullough–Dodd model and integrable perturbed conformal field theories, Nucl. Phys. B 516 (1998) 652, hep- th/9709034. [44] R. Guida, N. Magnoli, Vacuum expectation values from a variational approach, Phys. Lett. B 411 (1997) 127, hep-th/9706017. [45] P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory, Springer, Berlin, 1997. [46] P. Goddard, A. Kent, D.I. Olive, Virasoro algebras and coset space models, Phys. Lett. B 152 (1985) 88. [47] T. Gannon, Algorithms for affine Kac–Moody algebras, hep-th/0106123. [48] B. Schellekens, http://www.nikhef.nl/~t58/kac.html. [49] I. Affleck, M. Oshikawa, H. Saleur, Boundary critical phenomena in the three-state Potts model, cond-mat/9804117. [50] L. Chim, Boundary S-matrix for the tricritical Ising model, Int. J. Mod. Phys. A 11 (1996) 4491, hep-th/9510008. [51] V.A. Fateev, The exact relations between the coupling constants and the masses of particles for the integrable perturbed conformal field theories, Phys. Lett. B 324 (1994) 45. [52] S. Fredenhagen, Organizing boundary RG flows, Nucl. Phys. B 660 (2003) 436, hep-th/0301229. [53] I. Affleck, Edge critical behaviour of the 2-dimensional tri-critical Ising model, J. Phys. A 33 (2000) 6473, cond- mat/0005286. [54] K. Graham, On perturbations of unitary minimal models by boundary condition changing operators, JHEP 0203 (2002) 028, hep-th/0111205. [55] J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581. [56] H. Saleur, M. Bauer, On some relations between local height probabilities and conformal invariance, Nucl. Phys. B 320 (1989) 591. [57] D. Gepner, Field identification in coset conformal field theories, Phys. Lett. B 222 (1989) 207. [58] A.N. Schellekens, S. Yankielowicz, Field identification fixed points in the coset construction, Nucl. Phys. B 334 (1990) 67. [59] C.R. Ahn, M.A. Walton, Field identifications in coset conformal theories from projection matrices, Phys. Rev. D 41 (1990) 2558. [60] I.B. Frenkel, V.G. Kac, Basic representations of affine Lie algebras and dual resonance models, Invent. Math. 62 (1980) 23. [61] V.G. Kac, D.H. Peterson, Infinite-dimensional Lie algebras, theta functions and modular forms, Adv. Math. 53 (1984) 125. [62] V.G. Kac, Infinite-Dimensional Lie Algebras, Cambridge Univ. Press, Cambridge, 1990. [63] D. Bernard, String characters from Kac–Moody automorphisms, Nucl. Phys. B 288 (1987) 628. [64] J. Fuchs, Simple WZW currents, Commun. Math. Phys. 136 (1991) 345. Nuclear Physics B 744 [FS] (2006) 277–294

Effective electron–electron and electron–phonon interactions in the Hubbard–Holstein model

G. Aprea, C. Di Castro, M. Grilli ∗, J. Lorenzana

INFM-CNR SMC Center, and Dipartimento di Fisica, Università di Roma “La Sapienza”, piazzale Aldo Moro 5, I-00185 Roma, Italy Received 17 January 2006; accepted 15 March 2006 Available online 6 April 2006

Abstract We investigate the interplay between the electron–electron and the electron–phonon interaction in the Hubbard–Holstein model. We implement the ﬂow-equation method to investigate within this model the effect of correlation on the electron–phonon effective coupling and, conversely, the effect of phonons in the effective electron–electron interaction. Using this technique we obtain analytical momentum-dependent expressions for the effective couplings and we study their behavior for different physical regimes. In agreement with other works on this subject, we ﬁnd that the electron–electron attraction mediated by phonons in the presence of Hubbard repulsion is peaked at low transferred momenta. The role of the characteristic energies involved is also analyzed. © 2006 Elsevier B.V. All rights reserved.

PACS: 71.10.Fd; 63.20.Kr; 71.10.-w

Keywords: Electron–phonon interaction; Electron–electron interaction; Flow equations

1. Introduction

The electron–phonon (e–ph) interaction can play an important role in many physical systems, where it can be responsible for instabilities of the metallic state leading, e.g., to charge density waves and superconductivity. When the e–ph is particularly strong polaronic effects can also be responsible for a huge increase in the effective electronic masses. On the other hand

* Corresponding author. E-mail address: [email protected] (M. Grilli).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.03.021 278 G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 the issue of strong electron–electron (e–e) correlations is of relevance for many solid state systems like, e.g., (doped) Mott insulators, heavy fermions, and high-temperature superconductors. Therefore the interplay between the e–e correlation and the e–ph coupling is far from being academic and its understanding is of obvious pertinence in several physically interesting cases. In particular, in the superconducting cuprates the e–e correlations are strong and are custom- arily related to the marked anomalies of the metallic state, including the linear temperature dependence of the resistivity [1,2], where phonons are not apparent. This led to believe that the e–ph interaction plays a secondary role in these materials, and that the main physics is ruled by electron correlation. However, the presence of polaronic spectroscopic features [3] as well as various isotopic effects [4] in underdoped superconducting cuprates shows that the e–ph coupling is sizable in these systems. Recent results obtained with angle resolved photo-emission spectroscopy (ARPES) [5] have also been interpreted in terms of electrons substantially coupled to phononic modes. These contradictory evidences naturally raise the issue of why correlations can reduce the e–ph coupling in some cases and not in other. In this regard it has been often advocated [7–9,11] that strong electron correlation suppresses the e–ph large angles scattering, which mainly enters in transport properties, much more than the forward scattering, which is instead relevant in determining charge instabilities [8]. Therefore the momentum dependence of the e–ph interaction can give rise to important effects and select relevant screening processes due to correlations. In this paper we study the effect of (a weak) electronic correlation on a two-dimensional Holstein e–ph system using the flow-equation technique developed by Wegner [12,13]. The flow- equations describe the evolution of the parameters of the Hamiltonian under an infinite series of infinitesimal canonical transformations labeled by a parameter l that rules the flow. The generator of the canonical transformation is chosen in such a way to eliminate some interactions in favor of other interactions. Shall one solve the flow equations exactly the initial Hamiltonian at l = 0 and the final Hamiltonian at l =∞are guaranteed to have the same eigenvalues. Of course in general the flow-equations are not solvable, but suitable approximations can be introduced to obtain qualitative trends. Under certain approximations one can map the interacting problem into a noninteracting one. In this case the method can be roughly seen as a way to derive a Fermi liquid theory where bare fermions evolve into quasiparticles under the flow. Since one can choose what parts of the Hamiltonian remain, the method can also be exploited to get effective interactions. Of course, in getting a fully diagonal Hamiltonian one is naturally led to discover the “low-energy” phsysics of the problem. This is the approach (and the spirit of it) previously adopted by Wegner and coworkers to analyze the instabilities of various forms of the Hubbard model [14–16]. Another application of the method is the derivation of effective interactions. For example this scheme was used to obtain an effective e–e interaction by eliminating phononic degrees of freedom in the absence of the initial e–e interaction [17]. Rather than obtaining the low-energy physics one obtains an interacting Hamiltonian which still has to be solved. Ana- lyzing the structure of the resulting effective interaction we will discuss the physics of the e–ph interaction in the presence of the e–e interaction and vice versa. We will see that this procedure provides the correct energy scales for the phonons dressing the effective e–e interaction and, conversely, of the e–e interaction dressing the e–ph interaction. The paper is organized as follow: first, in Section 2, we introduce the Hubbard–Holstein Hamiltonian and, in Section 3, we present the formalism of the flow equations technique. Then in Section 4.1 we validate the method by applying it to a two atom system. In Section 4.2 we compute the effective e–ph interaction in a two-dimensional system and in Section 5 we eliminate the e–ph interaction to get an effective e–e coupling, thus providing a model Hamiltonian suitable G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 279 to discuss different physical regimes. Finally the summary and the conclusions are provided in Section 6.

2. Model

We consider a system of interacting electrons and phonons described by the Hubbard–Holstein model, where the electron density is locally coupled with a dispersionless phonon. For the sake of simplicity and since this case is relevant in physical systems of broad interest like the cuprates and the manganites, we consider the model in a two-dimensional lattice = † − + + + † + † H ti,j ci,σ cj,σ μ niσ U ni↑ni↓ g0 ni ai ai ω0 ai ai, (1) i,j,σ iσ i i i ≡ + ≡ † † † where ni ni↑ ni↓, with niσ ci,σ ci,σ . a(c)and a (c ) are the phonon (electron) annihilation and creation operators. For simplicity we consider only the nearest-neighbor hopping ti,j =−t for i and j nearest-neighbor lattice sites. The above Hamiltonian defines the starting point of our flow. Since under the flow new interactions will be generated, in order to describe the evolution of the Hubbard–Holstein parameters, we have to write the Hamiltonian in a more general form = + + H T V G, (2) = : † :+ − : †τ τ : T ωq aq aq (k μ) ck ck , (3) q k,τ 1 †τ †−τ −τ τ V = V :c c c c :, (4) 2N k,k ,q k+q k −q k k k,k,q,τ = √1 † + ∗ : †τ τ : G gk,qa−q gk+q,−q aq ck+q ck . (5) N k,q,τ Fig. 1 shows the diagrammatic representation of the e–ph interaction. In the following we assume the coupling gk,q to be real, while ωq and k represent the phonon and electron dispersions respectively. Vk,k,q is the e–e interaction. N is the number of lattice sites. Colons indicate normal ordering of the operators with respect to the Fermi sea. The expression (2) represents the Hamiltonian at a generic value of the flow parameter l whereas Eq. (1) provides the initial values of the parameters at l = 0, i.e. gk,q(l = 0) = g0, Vk,k,q (l = 0) = U, ωq (l = 0) = ω0, and k =−2t(cos(kx) + cos(ky)) (here we use a unit lattice spacing).

Fig. 1. Bare e–ph vertex. Straight lines represent electron propagators while wavy lines are for phonons. With our notation, in the vertex with an outgoing phonon the ﬁrst index of g represents the incoming electron momentum, while in the vertex with incoming phonon it represents the outgoing electron momentum; in both cases the second index of g represents the opposite of the phonon momentum. 280 G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294

3. Method

The flow-equation method [12,13] is based on an algebraic technique, the Jacobi method, to (block) diagonalize Hamiltonians. A series of infinitesimal rotations is performed such that at each step the non-diagonal elements get smaller while the others asymptotically approach a finite limit. If l is the parameter ruling the flow, this technique performs a unitary transformation U(l) such that H(l)= U(l)H(l = 0)U †(l). One can absorb all the evolution in the parameters of the Hamiltonian generating a new H(l =∞). This transformed Hamiltonian has the property that, when expressed in terms of the original operators, it has the desired diagonal (or block-diagonal) form with renormalized couplings. The spectrum naturally stays unchanged. The evolution of the Hamiltonian is controlled by dH(l) = η(l),H(l) , (6) dl = dU(l) † =− † where the anti-Hermitian generator η(l) dl U (l) η (l). Proper identification of the numerical coefficients in this operator equation yields the flow equations. In a finite Hilbert space, one can use a matrix representation of the H and η operators bringing Eq. (6) toaclosedsys- tem of equations. On the other hand, working in second quantization, in general the commutator in Eq. (6) will generate N-body operators, which are not necessarily included in the starting Hamiltonian. In this case a truncation is needed at some N in order to obtain a closed form. For operators of conserved particles, this truncation is naturally justified in the low-density limit. On the other hand, for phonons, this truncation is only allowed within a weak-coupling approximation. The first step is to find a generator that produces the convergence of the flow to the desired form of the Hamiltonian. We split the Hamiltonian in H0 plus a part that has to be eliminated V . Wegner has shown that the following choice produces the desired flow [12,13] η(l) = H0(l), V (l) = H0(l), H (l) = H(l),V(l) . (7) This method has been used to eliminate one kind of interaction (electron–phonon) in favor of an other effective (electron–electron) interaction [17]. The method was also adopted to select a specific channel in the electron–electron Hubbard interaction to see how this is dressed by all other channels in order to study the different possible instabilities of the model [14–16]. Here, as a first step starting from the model Hamiltonian (2), in the weak correlation limit, we use this technique to eliminate the e–e interaction and study how the resulting effective e–ph interaction behaves. Since this use is somehow unconventional we will first test if the method is suitable to this purpose in a toy model with two sites and a truncated phonon Fock space and then proceed to the general case. Subsequently we will consider the (more standard [17]) opposite procedure to eliminate the e–ph interaction in favor of an effective e–e interaction.

4. Effective electron–phonon interaction

4.1. Two-site lattice

As mentioned above, our ﬁrst aim is to eliminate the e–e interaction term dressing the e–ph one. In order to achieve this result we use the generator η(l) =[T + G, V ]. (8) G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 281

In this case we are not properly trying to diagonalize the Hamiltonian but we want to eliminate some off-diagonal elements corresponding to the e–e interaction to see how other off-diagonal elements, the ones corresponding to the e–ph interaction, renormalize. Thus it is necessary to check if the generator given above yields a solution with vanishing Vk,k,q and finite effective coupling gk,q. An analytical proof is beyond the purposes of this paper and therefore we will proceed constructively by analyzing a two-site Hubbard–Holstein model as a first non-trivial playground to apply this technique. One first remark is in order: The in-phase ionic displacement corresponds to a rigid shift of the two-site “lattice” and decouples from the electronic degrees of freedom. Therefore in momentum space the phonons at zero momentum can be disregarded and we only will consider phonons at momentum π. Accordingly only the e–ph couplings gk,q with q = π total transferred momentum will be considered. Specifically we consider a toy model consisting in a two-site lattice with two electrons which can occupy a truncated Hilbert space made by the following Sz = 0 states: ↑ ↓ ↑ ↓ |1 =c† c† 0 , |2 =c† c† 0 , π π 0 0 | = † †↑ †↓ | = † †↓ †↑ 3 aπ c0 cπ 0 , 4 aπ c0 cπ 0 . The crucial simplification of this model is that the phonon Hilbert space is truncated to the subspace with only zero or one phonon. Of course this is a severe limitation if one aims to apply the model to the case of strong e–ph couplings, but is acceptable, instead, for the general analysis of the method and for the investigation of the weak e–ph coupling regime. We use the following single-electron dispersion

q =−2t cos(q). (9) In matricial representation (2) takes the form: ⎛ ⎞ ˜π (l) V0,0,π (l) gπ,π (l) −gπ,π (l) ⎜ V (l) ˜ (l) g (l) −g (l) ⎟ H(l)= ⎝ 0,0,π 0 0,π 0,π ⎠ , (10) gπ,π (l) g0,π (l) λ(l) −V0,π,π (l) −gπ,π (l) −g0,π (l) −V0,π,π (l) λ(l) where

˜π (l = 0) = 2π + U, (11)

˜0(l = 0) = 20 + U, (12)

λ(l = 0) = π + 0 + U + ω0. (13) Since we aim to eliminate the e–e interaction and see how the e–ph interaction changes, we must implement a transformation to eliminate the matrix elements (1, 2), (2, 1), (3, 4) and (4, 3) of H . With the generator η(l) given by Eq. (8) the ﬂow equations read ˜ dπ (l) 2 =−4g0,π (l)gπ,π (l)V0,0,π (l) + 4gπ,π (l) V0,π,π (l) dl 2 + 2V0,0,π (l) ˜π (l) −˜0(l) , (14) ˜ d0(l) 2 =−4g0,π (l)gπ,π (l)V0,0,π (l) + 4g0,π (l) V0,π,π (l) dl 2 + 2V0,0,π (l) ˜0(l) −˜π (l) , (15) dλ(l) = 4g (l)g (l)V (l) − 2g (l)2V (l) − 2V (l)g (l)2, (16) dl 0,π π,π 0,0,π 0,π 0,π,π 0,π,π π,π 282 G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294

dV (l) 0,0,π =− ˜ (l) −˜ (l) 2V (l) − 2g (l)2V (l) dl 0 π 0,0,π 0,π 0,0,π 2 + 4g0,π (l)gπ,π (l)V0,π,π (l) − 2gπ,π (l) V0,0,π (l), (17) dV (l) 0,π,π =−2g (l)2V (l) + 4g (l)g (l)V (l) − 2g (l)2V (l), (18) dl 0,π 0,π,π 0,π π,π 0,0,π π,π 0,π,π dg (l) π,π = g (l)V (l) 2˜ (l) −˜ (l) − λ(l) + g (l)V (l) −˜ (l) + λ(l) dl 0,π 0,0,π π 0 π,π 0,π,π π 2 2 − 2g0,π (l)V0,0,π (l)V0,π,π (l) + gπ,π (l)V0,0,π (l) + gπ,π (l)V0,π,π (l) , (19) dg (l) 0,π = g (l)V (l) −˜ (l) + λ(l) + g (l)V (l) −˜ (l) + 2˜ (l) − λ(l) dl 0,π 0,π,π 0 π,π 0,0,π π 0 2 2 − 2gπ,π (l)V0,0,π (l)V0,π,π (l) + g0,π (l)V0,0,π (l) + g0,π (l)V0,π,π (l) . (20) These equations can be solved numerically. As a consistency check we have compared the eigenvalues of both H(l= 0) and H(l =∞) for different values of the starting couplings and obtained an exact match within our numerical precision. In the many-site case we cannot deal with the full form of the flow equations. Therefore it is useful to test which approximation can be done without generating neither too big errors in the asymptotic values of the effective e–ph coupling nor a divergence in the case of degenerate levels. The simplest choice is to make a weak-U approximation stopping at first order in U. Aiming to the effective e–ph interaction, we can neglect the flow equations for the energies because they would give higher order corrections when introduced in the flow-equations for the g’s: d˜ (l) d˜ (l) dλ(l) π = 0 = = 0, (21) dl dl dl dV (l) 0,0,π =− ˜ (l) −˜ (l) 2V (l) − 2g (l)2V (l) dl 0 π 0,0,π 0,π 0,0,π 2 + 4g0,π (l)gπ,π (l)V0,π,π (l) − 2gπ,π (l) V0,0,π (l), (22) dV (l) 0,π,π =−2g (l)2V (l) + 4g (l)g (l)V (l) dl 0,π 0,π,π 0,π π,π 0,0,π 2 − 2gπ,π (l) V0,π,π (l), (23) dgπ,π (l) = g0,π (l)V0,0,π (l) 2˜π (l) −˜0(l) − λ(l) dl + gπ,π (l)V0,π,π (l) −˜π (l) + λ(l) , (24) dg0,π (l) = g0,π (l)V0,π,π (l) −˜0(l) + λ(l) dl + gπ,π (l)V0,0,π (l) −˜π (l) + 2˜0(l) − λ(l) . (25) We analyzed the behavior of the approximated effective e–ph couplings comparing them to the exact case for different values of g0 and U ∈[0, 3], ω0 = 0.02. For the sake of convenience, we have used energy units such that t = 1/2. As far as ω0 is concerned, we consider the most frequent adiabatic regime with a typical value corresponding to a hundredth of the maximum electron bandwidth. In Fig. 2 we show a comparison between approximated and exact asymptotic values of gk,q for various values of U. In this example strong renormalizations of gk,q at various U are present, both in the exact and in the approximated expressions. The approximated solutions are in good agreement with the exact ones only for values of electronic repulsion U smaller than t.Both G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 283

Fig. 2. Exact (thick) and approximate (thin) numerical results for g0,π (green) and gπ,π (red) vs U.Hereweassume gk,q(l = 0) = 0.3, ω0 = 0.02. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) these phenomena, the strong renormalizations of g and the sizable differences between the exact and the approximate results, are due to the big energy differences between the states with and without phonons, basically controlled by the energy-level difference of order t and to the small value of g0 we have been using. As far as the second point is concerned, the approximated and 2 the exact values are different because the vanishing of V0,0,π is controlled by t, whereas g0 controls the vanishing of V0,π,π , which for small values of g0 goes to zero slowly in the flow 2 equations. The presence of g0 is necessary in the equation for V0,π,π to get an asymptotically vanishing solution. Otherwise, due to the degeneracy of the levels λ, V0,π,π would not flow at all. The quadratic terms in U, which are missing in our approximated expressions, modify strongly in this discrete case the asymptotic values of g when U is not very small. To further clarify the physical interpretation of the various terms appearing in the exact asymptotic form of the Hamiltonian, we rewrite in real space the e–ph interaction. By disregarding the zero-momentum phonons, the e–ph coupling for a fixed value of the spin takes the form

(g A† +¯g A)δ + (g¯ A − g A†)iJ δ δ J J , (26) 2 = † − † = † − † = − where δ c1c1 c2c2,J i(c1c2 c2c1), A a1 a2 and g + g g − g g = 0,π π,π ,g= 0,π π,π . (27) δ 2 J 2

We explicitly checked that at very large U, gJ grows monotonically with U while gδ decreases vanishing asymptotically in U. This result naturally arises from the correlation-induced suppression of charge fluctuations. Intuitively, as U increases the charge fluctuations become stiffer and 284 G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 this reflects in a decrease of the e–ph coupling that couples with the charge inbalance. Previ- ous analyses already found that at small transferred momenta, the vertex corrections to g arising + s s from the e–e interactions suppress the e–ph coupling by a factor (1 F0 ), where F0 is the standard Landau parameter entering the compressibility and is an increasing function of U [8,9]. Indeed the out-of-phase phonon A couples with the charge unbalance between the two sites. Since charge-density fluctuations are hindered by the local repulsion U, this is mirrored at large U in the suppression of the related coupling gδ. On the other hand, the flow equations implement a transformation, which obviously keeps the spectrum unchanged. Since the energy scale U is present in the starting Hamiltonian, the same scale must appear in the transformed Hamiltonian as well. gδ vanishes for large U s, and the interacting system keeps the scale U via the dressed e–ph interaction gJ . In particular, in the large-U limit (which can be dealth with exactly in the t = 0 limit) one can show that the local repulsion√ U is replaced by a nearest-neighbor attraction ∼ 2 ∼ V gJ /ω0. This leads to the relation gJ ω0U, which we numerically tested in the large-U limit. The appearance of a non-local e–ph coupling is also obtained by a standard Lang–Firsov transformation g exp S = exp − − n a† − a LF ω iσ i i 0 iσ which is usually implemented to eliminate the linear local e–ph coupling in favor of an effective 2 local e–e attraction −2g /ω0 and a coupling between the phonons and the hopping term [18]. By expanding to first order in the phonon fields this last term one can easily check that a linear non-local e–ph coupling 2 te−α α g =− c† c − c† c a† − a† − (a − a ) J 2 iσ jσ jσ iσ i j i j i,j,σ √ is obtained. Then the specific value g = Uω0/2 also leads to the elimination of the e–e repulsion in the transformed Hamiltonian. Therefore for this very specific choice, the effect of the Lang–Firsov transformation is similar to the one generically obtained via our flow equations at large values√ of U, where the local e–ph coupling gδ vanishes together with V , while gJ (l =∞) 1 tends to Uω0/2. Before going on analyzing the 2D continuum case, we reassert the two relevant indications of this section: The first is that it is indeed possible to eliminate the e–e interaction matrix elements obtaining a convergent solution for the effective e–ph couplings. The second indication is that the presence of an effective phonon-mediated interaction introduces the quadratic terms in e– ph coupling in the flow equations of the e–e coupling, allowing to get a convergent asymptotic solution with a vanishing effective V already at first order in U.

4.2. Effective e–ph interaction

So far we discussed the generalities of the method and showed the possibility to have a convergent solution while eliminating some matrix elements in favor of others, using the generator

1 2 We also ﬁnd in the Lang–Firsov-generated e–ph coupling a proportionality to the exponential factor exp[−(g/ω0) ]. This factor is somewhat misleading since it should disappear in the effective four-fermion interaction as a result of the summation over intermediate multi-phononic states (see, e.g., the effective superexchange coupling in the Hubbard– Holstein model of Ref. [19] or the effective Kondo coupling in the Anderson–Holstein model of Ref. [20]). Here the limitation of 0 or 1 phonon states prevents this cancellation to occur. G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 285

(8) within our toy model. Here we proceed to the calculation of the effective e–ph coupling in the more general case of a two-dimensional continuum. In this work, for simplicity we limit our analysis to the first order in the bare e–e coupling so that the flow equations are obtained from the following commutators dg (l) k,q → [G, V ],T + [T,V],G + o U 2 , dl dV (l) k,k ,q → [T,V],T + [G, V ],G + o U 2 , dl d (l) k → [G, V ],G + o U 2 , dl dω (l) q → [G, V ],G + o U 2 . dl Since our aim is the determination of an effective e–ph coupling while eliminating U at first order, we do not consider the flow equations for the energies because they would add higher order terms in U when inserted in the equations for the g’s. Having neglected the U 2 terms, 2 corresponds to the t U and g /t U conditions. Thus the flow equations for gk,q(l) and Vk,k,q (l) read

dVk,k ,q (l) 2 =− V (l) + “ [G, V ],G ”, (28) dl k,k ,q k,k ,q dgk,q(l) 1 = g − (l)V (l)(n − − n )[2 + α ], (29) dl N p q,q k,p,q p q p k,k ,q k,q p where

k,k,q ≡ k+q + k−q − k − k,αk,q ≡ k+q − k + ω0 and “[[G, V ],G]” stands for “part of flow equations coming from the operatorial term [[G, V ],G]” and makes it very difficult to get an analytical solution. Nevertheless, as in the toy = ¯ = ¯ =¯ = model, this term is essential for those values of k k, k k and q q such that k¯,k¯q¯ 0: For these momenta, without this term, Vk,¯ k¯,q¯ (l) would not flow to zero and Eq. (29) would be non-convergent. The term [[G, V ],G] is cumbersome, but for our purposes it will be enough to keep the simplest diagonal part and neglect additional non-diagonal ones with one of the three momentum indices of V under summation [ ] − 2 “ G, V ,G ” 8g0Vk,k ,q (l). (30)

Working at ﬁrst order in U, we have also replaced g(l) by its bare value g0. The solution for Vk,k,q (l) is then

−[ 2 + 2] 8g0 l Vk,k,q (l) = Ue k,k ,q . (31)

Again we can substitute gp−q,q in the sum in Eq. (29) with its initial value g0 and solve it −[ 2 +8g2]l U e k,p,q 0 − 1 g (l) = g 1 − (n − − n )[2 + α − ] (32) k,q 0 N [ 2 + 2] p q p k,p,q p q,q p k,p,q 8g0 286 G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 so that gk,q(∞) = g0 1 + Uχ(k,q,ω˜ 0) , (33) where 1 (np−q − np)[2 k,p,q + αp−q,q] χ(k,q,ω˜ ) ≡ . (34) 0 N [ 2 + 2] p k,p,q 8g0

To start with we assume k+q − k + ω0 = 0 (elastic scattering) so that the k dependence in the asymptotic solution disappears and χ(k,q,ω˜ 0) becomes χ(q,ωˆ 0) 1 (np−q − np)(p−q − p − ω ) χ(q,ωˆ ) ≡ 0 . (35) 0 N − − 2 + 2 p (p−q p ω0) 8g0

2 If one further neglects the g0 in the denominator the asymptotic solution becomes U np−q − np gq (∞) = g0 1 + Uχ(q,ω0) = g0 1 + , (36) N − − − ω p p q p 0 where χ(q,ω0) is the usual Lindhardt polarization bubble. It is worth noting that in the limit − of small ω , since np−q np < 0, we have g (∞)

gk,q(∞)−g0 Fig. 3. The relative variation of the e–ph coupling. Here E = + − q = 0, ω = 0.02. We recall that g0×U k,q k q 0 with our units, N = 1/2. units, χ =−N =−1/2) and then vanishes. As a result g reaches a plateau and finally recovers its bare value at large q’s. The momentum dependence given in Fig. 3 shows the important feature of the e–ph interaction in the presence of correlation of being peaked at small momenta. The reduction of g at large transferred momenta has been already found in previous treatments of the Hubbard–Holstein model with slave bosons [7,8,10], Hubbard operators [9] and quantum Monte Carlo [11], where the momentum scale at which g decreases is set by kF . Within a two-dimensional Monte Carlo approach [11], it was found that the decrease of g is largest at intermediate U’s, and becomes smaller at large U. Of course this non-monotonic result is not found with Eq. (33) here because of the weak coupling approximation. Having obtained an effective e–ph interaction, which is reduced by the Hubbard repulsion U and acquires a momentum dependence, it is of interest to study its effects on the electronic dispersion. One should, however, notice that the present first-order approximation limitates our possibility of obtaining substantial quantitative changes in the electronic properties with respect to the standard Engelsberg–Schrieffer results in Ref. [23]. We calculated the electron spectral density A(k, ω) =−(1/π) Im G(k, ω), where the electron Green’s function contains the self- energy corrections at second order in g. Fig. 4 displays the dispersions of the spectral-function peaks for various values of U at ω0 = 0.02 and g0 = 0.3. The overall behavior is obviously similar to that found in [23]. Specifically the e–ph coupling mixes the electron with the phonon and bends the low-energy part of the dispersion on the phonon dispersion at E ∼ ω0.Onthe other hand at very large energy the electron recovers its bare dispersion. It is clearly apparent that the e–ph interaction produces a break in the electron dispersion, which is naturally more pronounced for larger e–ph coupling (i.e. smaller U). The presence of weak interaction U introduces an additional interesting feature: Upon increasing U, the quantity 2 λ ∝ g /(ω0t) is suppressed thereby reducing the electronic mass m ≈ m0(1 + λ) near the Fermi level. This is made visible in Fig. 4 by the increased velocity of the electrons at low energy. This decrease of the mass upon increasing the interaction U is a generic feature, which was 288 G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294

Fig. 4. Dispersion of the peaks in the electronic spectral function for different values of U,andg0 = 0.3, ω0 = 0.02. We report ωˆ vs k where ωˆ is the solution of ωˆ − k = Re[Σ(k,ω)ˆ ] (ﬁlled symbols). We also report the position of the maximum of A(k, ω) for ω>ω0 (unﬁlled symbols).

Fig. 5. e–e vertex from renormalized e–ph interaction. also found within a DMFT treatment of the Hubbard–Holstein model [21]. We also notice that a difference in the low-energy and high-energy dispersions as well as the s-shaped dispersion of Fig. 5 (which displaces upon tuning the e–e interaction), are reminiscent of the kinks [5,24] and of the breaks [25] in the electronic dispersion of the cuprates experimentally observed along the (1, ±1) directions and around the (±1, 0), (0, ±1) regions of the Brillouin zone respectively. These breaks are a rather generic feature of electrons coupled to collective modes [26–29].

5. Effective electron–electron interaction

An effective e–e coupling in the static limit can be obtained from the effective e–ph coupling of Eq. (33) considering the following two contributions diagrammatically depicted in Fig. 5

g (l =∞)g − (l =∞) g + − (l =∞)g − (l =∞) eff =− k,q k q,q − k q, q k , q Vk,k,q . (38) ω0 ω0 G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 289

If one restricts the external momenta k, k to the domain I such that the condition of energy − = − eff conservation k k −q k+q k is satisﬁed, one obtains a simple expression for Vk,k,q 2 2 − [ + − − + − ] eff 2g0 4g0U (np−q np) 2(k+q p−q p k) (p p−q ) V =− − , k,k ,q ω ω N + − − 2 + 2 0 0 p (k+q p−q p k) 8g0 k,k ∈ I. (39)

In this way we obtain the first correction of order U/t to the purely phononic limit Ueff ≈ − 2 2 − = − 2g0/ω0. We also notice that, imposing the elastic-scattering condition k −q k k+q = eff =− 2 + ˆ + ˆ k 0 one retrieves the perturbative result Vq-elastic (2g0/ω0)(1 Uχ(q,ω0) Uχ(q, − =− 2 + ˆ ˆ ω0)) (2g0/ω0)(1 2Uχ(q,0)) where χ is the modified “polarization” bubble of Eq. (35). We have obtained a reduced attractive interaction modified according to the incipient effect of correlation on the e–ph interaction. In order to get further insight on the interplay between electronic correlations and the e–ph interaction we turn to the study of the effective e–e coupling by reversing the previous procedure: We use the flow equation technique to eliminate the e–ph interaction in favor of the effective V(k,k,q). The generator of the unitary transformation is

η(l) =[:T :+:V :, :G:] (40) leading to the ﬂow equations dG → [:T :, :G:], :T : + [:T :, :G:], :V : + [:V :, :G:], :T : + [:V :, :G:], :V : , (41) dl dV → [:T :, :G:], :G: + [:V :, :G:], :G: . (42) dl

Here we do not consider the ﬂow equations for the energies under the condition that (t, ω0) g0. Moreover, when inserted in the equations for the effective Vk,k,q , they would add higher-order 2 corrections in g0. In the equation for G we neglect terms of order U g0 (the last commutator 2 in the r.h.s.). In the equation for V we disregard terms of order Ug0 (the last commutator in the r.h.s.). Both these approximations are valid under the condition t U. When rewritten in terms of couplings and with the above approximations, the ﬂow equations read dgk,q(l) 2 1 =−α g (l) − g − (l)V (l)(2α − + )(n − − n ), dl k,q k,q N p q,q k,p,q p q,q k,p,q p q p p (43) dVk,k,q (l) =−(α − + α )g − (l)g (l) dl k q,q k,q k q,q k,q − (αk+q,−q + αk,−q )gk+q,−q (l)gk,−q (l), (44) where, again, αk,q = k+q −k +ω0, k,k,q = k+q +k−q −k −k, and εk ≡−2t[cos(kx)+ cos(ky)].TheU = 0 case was previously considered in Ref. [17]. In this limit, Eqs. (43) and (44)

2 The use of the perturbative diagrammatic approach in Fig. 5 implies the smallness of the e–ph coupling g0 t, since otherwise higher order phononic corrections (self-energy and vertex) should be considered. It is worth noticing that this limitation is instead not required in Section 4.2, where the elimination of the coupling V with the neglect of some commutators imposes conditions on the smallness of V coupling only. 290 G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 can be solved without any further approximation yielding = − 2 gk,q(l) g0 Exp αk,ql (45) for the e–ph coupling and

−αk−q,q − αk,q −αk+q,−q − αk,−q V (l =∞) = g2 + g2 (46) k,k ,q 0 2 + 2 0 2 + 2 αk−q,q αk,q αk+q,−q αk,−q as the new effective e–e coupling. As in Ref. [17] Eq. (46) in the BCS channel (k =−k)gives

2 2ω0 2 2ω0 V − (l =∞) =−g − g k, k,q 0 − 2 + 2 0 − 2 + 2 2(k k+q ) 2ω0 2(k k+q ) 2ω0 2 2g ω0 =− 0 (47) − 2 + 2 (k k+q ) ω0 while in the case of scattering with energy conservation (k+q − k =−k−q + k )itgives g2 g2 =∞ =− 0 − 0 = 2 2ω0 Velastic(l ) g0 . (48) + − − 2 − 2 αk,q αk q, q (k+q k) ω0 As discussed in Ref. [17], the effective interaction in the BCS channel is always attractive, with a noticeable difference with respect to the result of Frölich [6]. In the generic U>0 case Eqs. (43) and (44) are not analytically tractable and one should treat them numerically. Here instead we aim to get an insight on the energy scales involved in the phonons dressing the effective e–e interaction and therefore we consider an approximate treatment. We expect that the dependence of the effective couplings on the momenta of the fermionic legs is less important than the dependence on the transferred momentum. This is particularly reasonable if one is mostly interested in the physically relevant region of scattering processes around the Fermi surface. According to this expectation, we assume here that gk,q mostly depends on the transferred momenta q and we neglect its k dependence. Under this assumption the electronic parts of the α’s in Eq. (44) cancel leaving ω0 as the only scale and the k and k dependence of V drops. With the further assumption that in Eq. (43) V can be replaced by its bare value U,Eqs.(43) and (44) become dgq (l) 2 Ugq (l) =−α g (l) − ( − − )(n − − n ), (49) dl k,q q N p p q p q p p

dVq (l) =−4ω g2(l). (50) dl 0 q + − − In deriving the equation for g we exploited the obvious property p(2ω0 (k k−q ))(np−q np) = 0. Moreover to eliminate the k dependence from the r.h.s., we consider the average quan- 2 ≈ 2 tity αk,q αk,q over the momenta k around the Fermi surface. The resulting expression for α2 can be cast in the form 2 = 2 + + 2 αk,q t βq ω0tγq ω0, (51) where the functions βq and γq depend on the speciﬁc average procedure (e.g., from the width of the shell around the Fermi surface over which the average is carried out). In particular, the average could be taken strictly on the Fermi surface, thereby getting explicit expressions. In general one G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 291

→ 2 can notice that both βq and γq vanish for q 0 and that αk,q is obviously a non-negative quantity. 2 2 We have not reported here the cumbersome expressions of the O(U g0) and O(g0U) terms, but we explicitly checked that under the conditions Vk,p,q U, adopted in the evolution for g and gk,q(l) ∼ gq (l), these additional terms vanish and no further corrections should be considered in the ﬂow equations. Therefore within this approximation, the limitation of small U can be somewhat relaxed. The solution of Eq. (49) is 2 2 2tU g (l) = g Exp − t β + tω γ + ω − (¯ −¯ − )(n − − n ) l . (52) q 0 q 0 q 0 N p p q p q p p

Where we use the notation p = 2t¯p. Here we note that the exponential argument is manifestly negative and it implies the vanishing of gq (l) for l →∞. We can now calculate the effective electron–electron coupling 2 2g (ω0) V (l =∞) = U − 0 . (53) k,k ,q 2 + + 2 + 2tU ¯ −¯ − t βq tω0γq ω0 N p(p p−q )(np−q np)

The appearance of ω0 only in the numerator is a consequence of the above-mentioned cancellation of the electronic scale in the α’s strictly valid under the assumption that gq does not depend on k. The effective coupling has four non-negative contributions in the denom- 2 2 inator, with purely phononic, purely electronic and mixed character: ω0, t βq , tω0γq and ≡ 2Ut ¯ −¯ − = 2Utηq N p(p p−q )(np−q np). The last three expressions vanish at q (0, 0) and − 2 one finds in this limit 2g0/ω0 as a correction to U. The presence of correlation generically reduces this attractive contribution to Vk,k,q when q is finite. Depending on the relative importance of the various terms appearing in the denominator of Eq. (53) as determined by the parameters t, U and ω0, the effective interaction acquires different limiting forms. First of all we notice that the correct anti-adiabatic limit (ω0/t), (ω0/U) →∞is recovered 2 2g0 Vk,k,q (l =∞) = U − ω0 which was also obtained in the limit q → 0. When ω0 t, a further momentum dependent correction γq t/ω0 is obtained. → = 2 In the opposite limit, (ω0 0, but with λ 2g0/(ω0 t) finite) we find that the whole contribution of the e–ph attraction vanishes for ω0 → 0. This is reasonable, since an instan- taneous effective interaction cannot be affected by a static lattice deformation. We notice that in Ref. [21], where the effective U was also calculated within the DMFT approach, a phonon correction linearly vanishing with ω0 was also found. Numerically the correction involved the scales U ∼ t and ω0. An effective U was also obtained by means of a slave-boson-Lang–Firsov treatment of the Hubbard–Holstein model [22], where in the adiabatic limit t ω0, it was found 2 Ueff ≈ U − g /4t for a square lattice in two dimensions. In our case, in the same adiabatic limit, our Eq. (53) results in 2 2g ω0 Ueff ≈ U − . (54) tβq + 2Uηq t Our present crude approximate treatment therefore reproduces the results of the DMFT and of the Lang–Firsov-slave-boson approaches with an additional ω0/t factor. This factor in this limit 292 G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 provides a substantial reduction of the phononic corrections at finite q’s, while this reduction is less and less severe at small transferred momenta, owing to the vanishing of βq and ηq in this limit. On the other hand, going back to Eq. (50), one should remember that relaxing the assumption of k independence of g, an additional electronic scale coming from the αk,q would appear together with ω0. As long as ω0 is larger than t, these corrections are unessential and the present approximation works well in the anti-adiabatic limit. On the contrary, in the adiabatic limit and finite q these electronic corrections should appear in the numerator of Eq. (53).Asa 2 consequence U would be corrected by a term of order g /t without any additional ω0/t factor. This analysis therefore suggests that the flow equations correctly capture the proper energy scales and provide non-trivial momentum dependencies.

6. Conclusions

This paper considered two main issues related to the interplay between e–e correlations and e–ph coupling. We first investigated the effect of the e–e Hubbard repulsion on the Holstein e– ph coupling, showing that this latter is generically suppressed. This finding is consistent with previous results based on general Fermi-liquid arguments [8], on large-N expansions based on the slave-boson [7,8] or Hubbard-operator [9] techniques and on numerical quantum Monte Carlo approaches [11]. Remarkably, although we limited our discussion to the weak-correlation limit, we also find an agreement as far as the momentum dependencies are concerned. Specifically we find that the Hubbard repulsion more severely suppresses the e–ph coupling when large momenta are transferred. As already mentioned in the introduction this finding could explain the lack of phonon features in transport properties (which mostly involve large transferred momenta) while charge instabilities and d-wave superconductivity can arise from (mostly) forward scattering and still be induced by a less suppressed e–ph coupling. The suppression of the e–ph coupling by e–e correlations is also found within the dynamical mean-field theory technique, which provides an exact treatment of the dynamics. However, although this technique can be applied for any strength of the e–e and e–ph couplings, it neglects the space correlations and provides poor informations on the momentum structure of the renormalized couplings. In this sense our work based on the flow- equation technique explores the correlation-induced renormalization of the e–ph coupling from a complementary point of view. A second outcome of our research is provided by the elimination of the e–ph coupling to obtain an effective e–e interaction. Remarkably our procedure, even though it has been discussed within crude approximations, produces a reliable effective model for correlated electron systems in the presence of the e–ph interaction. All the energy scales, namely ω0,g0,t and U are explicitly included and provide proper momentum dependence corrections in the different physical regimes in agreement with previous analyses carried out with the DMFT and with the Lang– Firsov-slave-boson approaches. The phonons naturally introduce an effective attractive interaction, which reduces the repulsive U. Again we find that the phonon-mediated interaction is more sizable at small transferred momenta. This allows to infer some preliminary conclusions on the effects of phonons on the instabilities of the Hubbard model. This issue had previously been addressed within the flow- equation technique [14,16], where a phase diagram was proposed for the various instability lines as a function of U and doping x. Here we qualitatively remark that the phonons induce an additional attraction, such that the effective repulsion U(q) is smaller at smaller momenta. Therefore, the instabilities occurring at large momenta (like anti-ferromagnetism or flux-phase) G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294 293 are only weakly affected by the phonon attractive part, while the instabilities occurring at small momenta (phase separation, Pomeranchuk, superconductivity) will occur at larger values of the bare repulsion parameter U.

Acknowledgements

One of us (C.D.C.) gratefully acknowledges stimulating discussions with F. Wegner. We all acknowledge interesting discussions with M. Capone, C. Castellani, and R. Raimondi. This work was ﬁnancially supported by the Ministero Italiano dell’Università e della Ricerca with the Projects COFIN 2003 (prot. 2003020230006) and COFIN 2005 (prot. 205022492001). One of us (C.D.C.) also thanks the von Humboldt Fundation.

References

[1] See, e.g., B. Batlogg, et al., Physica C 235–240 (1994) 130. [2] M. Calandra, O. Gunnarsson, Phys. Rev. B 66 (2002) 205105, and references therein. [3] See, e.g., in: Y. Bar-Yam, T. Egami, J. Mustre de Leon, A.R. Bishop (Eds.), Lattice Effects in Cuprates, World Scientiﬁc, Singapore, 1992; J.P. Falck, et al., Phys. Rev. Lett. 69 (1992) 1109; P. Calvani, M. Capizzi, S. Lupi, G. Belestrino, Europhys. Lett. 31 (1995) 473, and references therein. [4] S. Andergassen, et al., Phys. Rev. Lett. 87 (2001) 056401, and references therein; H. Keller, Structure and Bonding, Springer-Verlag, Berlin, 2004, and references therein. [5] A. Lanzara, et al., Nature (London) 412 (2001) 510. [6] H. Frölich, Proc. R. Soc. London A 215 (1952) 291. [7] J.H. Kim, Z. Tesanovic, Phys. Rev. Lett. 71 (1993) 4218. [8] M. Grilli, C. Castellani, Phys. Rev. B 50 (1994) 16880. [9] R. Zeyher, M.L. Kulic, Phys. Rev. B 53 (1996) 2850; For a recent review see also M.L. Kulic, cond-mat/0404287. [10] E. Cappelluti, et al., Phys. Rev. B 69 (2004) 161101. [11] Z.B. Huang, W. Hanke, E. Arrigoni, D.J. Scalapino, Phys. Rev. B 68 (2003) 220507. [12] F. Wegner, Ann. Pys. (Leipzig) 3 (1994) 77. [13] F.J. Wegner, Phys. Rep. 348 (2001) 77. [14] I. Grote, E. Körding, F. Wegner, J. Low Temp. Phys. 126 (2002) 1385, cond-mat/0106604. [15] V. Hankevych, I. Grote, F. Wegner, Phys. Rev. B 66 (2002) 094516, cond-mat/0205213. [16] V. Hankevych, F. Wegner, Eur. Phys. J. B 31 (2003) 333. [17] P. Lenz, F. Wegner, Nucl. Phys. B 482 (1996) 693. [18] See, e.g., G. Mahan, Many-Particle Physicas, Plenum, New York, 1990. [19] W. Stephan, M. Capone, M. Grilli, C. Castellani, Phys. Lett. A 227 (1997) 120. [20] P.S. Cornaglia, H. Ness, D.R. Grempel, Phys. Rev. Lett. 93 (2004) 147201; P.S. Cornaglia, D.R. Grempel, H. Ness, Phys. Rev. B 71 (2005) 075320. [21] G. Sangiovanni, M. Capone, C. Castellani, M. Grilli, Phys. Rev. Lett. 94 (2005) 026401. [22] P. Barone, R. Raimondi, M. Capone, C. Castellani, cond-mat/0509090. [23] S. Engelsberg, J.R. Schrieffer, Phys. Rev. 131 (1963) 993. [24] T. Valla, et al., Science 285 (1999) 2110; P.V. Bogdanov, et al., Phys. Rev. Lett. 85 (2000) 2581; A. Kaminski, et al., Phys. Rev. Lett. 86 (2001) 1070; P.D. Johnson, et al., Phys. Rev. Lett. 87 (2001) 177007; F. Ronning, et al., Phys. Rev. B 67 (2003) 165101; X.J. Zhou, et al., Nature (London) 423 (2003) 398. 294 G. Aprea et al. / Nuclear Physics B 744 [FS] (2006) 277–294

[25] T. Sato, et al., Phys. Rev. Lett. 91 (2003) 157003; M.R. Norman, et al., Phys. Rev. B 64 (2001) 184508. [26] M. Eschrig, M.R. Norman, Phys. Rev. Lett. 85 (2000) 3261; A.V. Chubukov, M.R. Norman, Phys. Rev. B 70 (2004) 174505, and references therein. [27] G. Seibold, M. Grilli, Phys. Rev. B 63 (2001) 224505. [28] Z.-X. Shen, A. Lanzara, S. Ishihara, N. Nagaosa, cond-mat/0108381. [29] M.L. Kulic, cond-mat/0404287, and references therein. Nuclear Physics B 744 [FS] (2006) 295–311

Dual form of the paperclip model

Sergei L. Lukyanov a,b,∗, Alexander B. Zamolodchikov a,b,c

a NHETC, Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855-0849, USA b L.D. Landau Institute for Theoretical Physics, Chernogolovka 142432, Russia c Chaire Internationale de Recherche Blaise Pascal, Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, Paris cedex 05, France Received 13 February 2006; accepted 15 March 2006 Available online 6 April 2006

Abstract The “paperclip model” is 2D model of quantum field theory with boundary interaction defined through a special constraint imposed on the boundary values of massless bosonic fields [S.L. Lukyanov, et al., hep-th/0312168]. Here we argue that this model admits equivalent “dual” description, where the boundary constraint is replaced by special interaction of the boundary values of the bosonic fields with an additional boundary degree of freedom. The dual form involves the topological θ-angle in explicit way. © 2006 Published by Elsevier B.V.

1. Introduction

In this work we describe the dual form of the “paperclip model” of boundary interaction in 2D quantum field theory. The paperclip model was introduced in Ref. [1], where its basic properties are discussed. The model involves two-component Bose field X(σ, τ) = (X(σ, τ), Y (σ, τ)) which lives on a semi-infinite cylinder with Cartesian coordinates (σ, τ), σ 0, τ ≡ τ + 1/T.1 In the bulk, i.e. at σ>0, the dynamics is described by the free-field action: 1/T ∞ 1 A [X, Y ]= dτ dσ (∂ X)2 + (∂ Y)2 . (1) bulk 4π ν ν 0 0

* Corresponding author. E-mail address: [email protected] (S.L. Lukyanov). 1 Euclidean formulation of the theory is implied. Due to the compactiﬁcation τ ≡ τ + 1/T it is equivalent to the Matsubara representation of the (1 + 1)-dimensional theory at thermodynamic equilibrium at the temperature T .

The interaction takes place at the boundary at σ = 0, due to a non-linear boundary constraint: the boundary values XB = (XB ,YB ) ≡ X|σ =0 of the field X are restricted to the “paperclip curve” X Y r cosh B − cos B = 0, |Y | πa. (2) 2b 2a B Here a, b, and r are real positive parameters, the first two being related as follows,2 1 a2 − b2 = . (3) 2 As usual, the non-linear constraint requires renormalization, but one can check (up to two loops) that the RG transformation affects the curve (2) only through renormalization of the parameter r, which “flows” according to the equation

E∗ 2 = 4b2 1 − r2 r4b , (4) E where E is the RG energy, and E∗ is the integration constant of the RG equation, which sets up the “physical scale” in the model. As in [1], in what follows we always take the scale E proportional to the temperature T , namely

E = 2πT. (5)

It is useful to introduce also the external ﬁeld h = (hx,hy) which couples to the boundary values XB , i.e. to add the boundary term

1/T

Ah[XB ,YB ]= dτ(hxXB + hyYB ) (6) 0 to the action (1). Then, the ﬁrst object of interest is the partition function,

−Abulk[X,Y ]−Ah[XB ,YB ] Z0 = DX DY e , (7) where the functional integration is over all fields X(σ, τ) obeying the boundary constraint (2). General definition of the model involves additional parameter, the topological angle θ. Topo- logically, the paperclip curve (2) is a circle, hence the configuration space for the field X(σ, τ) consists of sectors characterized by an integer w, the number of times the boundary value XB winds around the paperclip curve when one goes around the boundary at σ = 0. The contributions from the topological sectors can be weighted with the factors eiwθ . Thus, in general ∞ iwθ (w) Zθ = e Z , (8) w=−∞

2 The parameter n used in [1] is related to a and b as √ √ n + 2 n a = ,b= . 2 2 S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 297

Fig. 1. The paperclip formed by junction of two hairpins. where Z(w) is the functional integral (7) taken over the fields from the sector w only. Physics of the model, in particular its infrared (i.e. low temperature) behavior, depends on θ in a significant way (see [1,2] for details). The ultraviolet (high-T ) limit of the paperclip model is understood in terms of the conformally invariant “hairpin model” of boundary interaction [1]. In this limit the parameter r tends to zero, and the paperclip curve becomes a composition of two “hairpins”, as shown in Fig. 1. The hairpin model is defined by replacing the paperclip constraint (2) by the non-compact r ± XB = YB “hairpin” curve 2 exp( 2b ) cos( 2a ). We refer to this model as the “left” or the “right” hairpin, depending on the sign in the exponential; the two models are related by simple field transformation (X, Y ) → (−X, Y ). Note that the left hairpin corresponds to the right part in Fig. 2, and vice versa (just like the way a human brain is wired to the rest of the body). The left = X =−X (right) hairpin model is conformally invariant with the linear dilaton D(X) 2b (D(X) 2b ). More details on the hairpin model can be found in [1] and in Section 2 below. The paperclip model has many features in common with the so-called “sausage” sigma model studied in Ref. [3]. The UV splitting of the paperclip into the hairpins is analogous to the UV splitting of the sausage into two semi-infinite “cigars” (see [3]). Like the sausage sigma model, the paperclip model seems to be integrable at two values of the topological angle, θ = 0 and θ = π. The sausage model is known to admit a dual description, where the non-trivial metric is replaced by certain potential (“tachion”) term in the action [4]. This analogy is one of the reasons to expect that a similar dual representation exists for the paperclip model. The aim of this work is to introduce the dual representations of both the hairpin model and the paperclip model. In this paper we argue that the paperclip model is equivalent (or “dual”) to another model of boundary interaction. The dual model also involves a two-component Bose field (X(σ, τ), Y(σ,τ))˜ (where Y˜ is interpreted as the T-dual3 of Y ) on the semi-infinite cylinder, which has free-field dynamics in the bulk, and obeys no constraint at the boundary σ = 0; instead it interacts with an additional boundary degree of freedom. It is best to discuss the dual model in terms of its Hamiltonian representation, with the cyclic coordinate τ ≡ τ + 1/T taken as the Euclidean (or Matsubara) time. In this picture the partition function (8) admits the dual representation as the trace

ˆ − H Zθ = TrH˜ e T , (9) H˜ = H ⊗ C2 H taken over the space X,Y˜ , where X,Y˜ is the space of states of the two-component boson (X(σ ), Y(σ))˜ on the half-line σ 0 (with no constraint at σ = 0) and C2 is the two- dimensional space representing the new boundary degree of freedom. The dual Hamiltonian in ˆ ˆ ˆ (9) consists of the bulk and the boundary parts, H = Hbulk + Hboundary. The bulk part is just the

3 ˜ ˜ The T-dual of the free massless field is defined as usual, through the relations: ∂τ Y = i∂σ Y and ∂σ Y =−i∂τ Y . 298 S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 free-field Hamiltonian ∞ 1 Hˆ = dσ Π 2 + Π 2 + (∂ X)2 + (∂ Y)˜ 2 , (10) bulk 4π X Y˜ σ σ 0 ˜ 4 H ˆ where (ΠX,ΠY˜ ) are momenta conjugated to the field operators (X, Y) acting in X,Y˜ (Hbulk acts as identity in the C2 component of H˜ ). The boundary term describes coupling of the bound- ˜ ˜ ary values (XB , YB ) ≡ (X, Y)|σ =0 of the fields to the additional boundary degree of freedom 2 2 represented by C (σ± and σ3 are the Pauli matrices acting in C ), ˆ ˆ Hboundary = hxXB + πahyσ3 + V, (11) where5: ˆ θ iaY˜ θ −iaY˜ V = μ σ+ cosh bX − i e B + σ− cosh bX + i e B , (12) B B 2 B 2 with μB related to the scale E∗ in (4) as

2E∗ μ = . (13) B π In Eqs. (11), (12), a,b and (hx,hy) are the same as in (2), (6).

2. Dual form of the hairpin

In this section we describe the dual form of the conformal hairpin model. To be deﬁnite, throughout this section we concentrate attention on the left hairpin model; the right hairpin is obtained by reﬂection X →−X. The left hairpin boundary constraint has the form r∗ X Y exp B = cos B , |Y | πa. (14) 2 2b 2a B Here r∗ relates to the energy scale (5) in a simple way

E∗ 2 4b2 = 4b (r∗) . (15) 2πT The boundary state of the (left) hairpin model was described in [1], 2 B⊃|= d P B⊃(P)IP|, (16) P where IP| are the Ishibashi states associated with the W -algebra of the hairpin model (see [1] and Section 2.1 below), P is the zero-mode momentum of the free boson X, and the amplitude B⊃(P) has the following explicit form in terms of the components (P, Q) of the vector P + P 2 −2ibP 2b (2ibP) (1 i 2b ) B⊃(P, Q) = g r , (17) D ∗ 1 − + 1 + + (2 aQ ibP) (2 aQ ibP)

4 Normalization is such that, for instance, [X(σ),ΠX(σ )]=2πiδ(σ − σ ). 5 We assume canonical conformal normalization of the boundary vertex operators (see e.g. Eq. (48)) involved in this interaction term. S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 299

−1/4 where gD = 2 . The amplitude B⊃(P, Q) coincides with the partition function of the hairpin model on the semi-inﬁnite cylinder of circumference 1/T :

Z⊃(hx,hy) = B⊃(P, Q), (18) where the dependence of Z⊃ on the parameters hx , hy is brought about through the linear boundary term (6), and P , Q in the right-hand side are related to the these parameters as follows

h hy P = i x ,Q= i . (19) T T 2.1. W -algebra and dual potential

As was explained in [1], the hairpin boundary condition is conformally invariant (with linear dilaton X/b), and moreover it has extended conformal symmetry with respect to certain W - algebra. Here we use the notation W⊃ for the W -algebra of the left hairpin model. It is generated by a set of local currents {Ws(z), s = 2, 3, 4,...} (built from derivatives of the free ﬁelds X, Y ) which commute with two “screening operators”,6

bX±iaY˜ dwWs(z) e (w, w)¯ = 0. (20) z Then, it is almost trivial observation that this W -symmetry is present in any model of boundary interaction which has no boundary constraint, but instead whose action has an additional boundary potential term 1/T ˜ ˜ bXB +iaYB bXB −iaYB ABP = dτ S+ e + S− e . (21) 0 Here S± may be either c-numbers, or more generally any non-trivial boundary degrees of freedom whose own dynamics is “topologically invariant”, i.e. invariant with respect to any diffeomorphism τ → f(τ)of the boundary. Since such boundary interaction has the hairpin W -algebra as its symmetry, it seems natural to expect that under appropriate choice of the boundary degree of freedom S± the model with the boundary potential (21) is equivalent (or, in modern speak, “dual”) to the left hairpin model. Because of the essentially quantum nature of the boundary degree of freedom S± (see below), it will be convenient to discuss in terms of the Hamiltonian representation of the model, as was mentioned in Introduction. The partition function of the left hairpin model is written as the − Hˆ ˆ ˆ ˆ trace Tr[e T ], where H is the sum Hbulk + V⊃ of the bulk free-ﬁeld Hamiltonian (10), and the boundary term

ˆ bX +iaY˜ bX −iaY˜ V⊃ = S+ e B B + S− e B B , (22) corresponding to the term (21) in the action. Here S± are the operators associated with the boundary degree of freedom S±(τ) in (21). The latter operators must commute with the ﬁeld operators,

6 Note that we write the screening operators in terms of the T-dual field Y˜ . This is done in preparation to the discussion of the dual hairpin below. At this point the distinction makes no difference in the definition of the W⊃-algebra, since Eq. (20) is sensitive to the holomorphic parts of the fields X and Y only. 300 S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 i.e. [S±, X(σ )]=[S±, Π(σ )]=0atanyσ , lest the W -symmetry of the theory be violated, but the commutation relations among the boundary observables S± themselves are not fixed a priori. Our goal here is to identify the algebra of these operators in the dual hairpin model, as well as its representation ρ. Some relations can be inferred from the following simple argument. The boundary potential term (21) vanishes in the limit X →−∞. In the absence of this term both fields X and Y˜ would obey the von Neumann (free) boundary conditions. Equivalently, in the absence of the potential term the field Y would obey the Dirichlet (fixed) boundary condition, i.e. the boundary values (XB ,YB ) would lay on a straight brane parallel to the X-axis. Note that in the same limit X → −∞ the right hairpin curve (14) becomes a composition of two parallel branes,

YB →±πa as XB →−∞, (23) separated by the distance 2πa. In the full hairpin curve, these two “legs” are bridged at the right, allowing for a passage from the upper leg to the lower leg and vice versa. In the dual representation of the hairpin model such passages should be attributed to two terms in the boundary ± ˜ potential (21). The boundary vertex operators e iaYB (τ) create jumps in the boundary value of Y exactly of the desired magnitude, YB (τ + 0) − YB (τ − 0) =±2πa. The fact that the hairpin has only two “legs” (23) clearly suggests that the operators S± in (22) must obey the “fermionic” relations7

2 2 S+ = S− = 0. (24) So far in the discussion of the dual hairpin we have ignored the coupling (6) to the external field (hx,hy). Adding the term hxXB to the dual hairpin Hamiltonian is straightforward, but ˜ in the description in terms of the T-dual field Y the term hyYB would be non-local. One can circumvent this difficulty by introducing the “fermion number” operator N, associated with the boundary degrees of freedom S±, which satisfies with them the following commutation relations:

[N, S+]=2S+, [N, S−]=−2S−. (25)

Then one can check that the sum YB − πaN commutes with the Hamiltonian (22). Therefore, in the presence of the external ﬁeld h, expected form of the full Hamiltonian of the dual hairpin model is ˆ ˆ ˆ H = H0 + V⊃, (26) ˆ where V⊃ is the boundary potential operator (22), and ˆ ˆ H0 = Hbulk + hxXB + πahyN, (27) with the operators N and S± in (22) satisfying the relations (24) and (25).

7 It is important at this point that our definition of the hairpin (and of the paperclip) model involves uncompactified field Y . Also, the bound |YB | <πa in (14) (and in (2)) is essential. Without the bound, Eq. (14) (as well as Eq. (2)) would define a series of disconnected curves, Y → Y + 4πaZ copies of the original hairpin (or paperclip). Although the models of boundary interaction which involve more then one copy also deserve attention, they are different from the hairpin (paperclip) model as defined in [1], and we do not address them here. S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 301

2.2. Singularities of the hairpin boundary amplitude

Looking at the right hairpin amplitude (17) as a function of complex variables P and Q, one observes two sets of singularities. First one is a sequence of poles at 2ibP= 0, −1, −2,...due to the ﬁrst gamma-factor in the numerator in (17). These poles admit straightforward interpretation in terms of potential divergences of the functional integral (7), with the non-compact boundary constraint (14), at the inﬁnite end of the hairpin X →−∞(see Ref. [1]). The second set is a string of poles at

Pk = 2ibk, k = 1, 2, 3,... (28) due to the second gamma-factor in (17). Note that in the weak-coupling limit of the hairpin model, i.e. at b →∞, these poles depart to infinity. Clearly, in terms of the hairpin functional integral these singularities represent non-perturbative effects. Instead, the poles (28) become most visible at small b, which is the weak coupling domain in the dual representation of the hairpin model, and indeed they admit simple interpretation in terms of the dual hairpin model. Since the boundary potential (21) vanishes in the limit X →−∞, there is a potential divergence of the functional integral associated with the dual hairpin.8 Following [6], one can first integrate out the constant mode of the field X. This integration produces poles in P exactly at the points (28) whose residues Rk = iResP =2ibk Z⊃(−iPT,−iQT ) (k = 0, 1, 2,...) (29) are expressed through the integrals of the 2k-point correlation functions nk2 = T ··· ˆ⊃ ··· ˆ⊃ Rk (2πT) dτ2k dτ1 V (τ2k) V (τ1) 0, (30) where the τ -ordering symbol T signifies that the integration is performed over the domain ˆ τHˆ ˆ −τHˆ 1/T τ2k ··· τ1 0. In (30) V⊃(τ) = e 0 V⊃e 0 is the unperturbed Matsubara operator ˆ − H0 associated with the boundary potential (22), and · · · 0 ≡ Tr[···e T ]. Since the unperturbed ˆ ˜ Hamiltonian H0 involves no interaction between the boundary variables S± and the fields X, Y , the expectation value in (30) factorizes in terms of these two parts of the system. In view of (22), it can be written in the form = S S ···S iπaQN T ··· V 2k ···V 1 Rk Trρ 2k 2k−1 1 e dτ2k dτ1 B (τ2k) B (τ1) 0, (31) i =± which involves the traces over the space of states ρ of the boundary degrees of freedom S±, ··· V± as well as the free-field expectation values 0 of the boundary values B (τ) of the vertex operators (20). Thanks to the relations (24), there are only two non-vanishing contributions to the sum in (31), with ( 1 ··· 2k) = (+−+−···+−) and ( 1 ··· 2k) = (−+−+···−+). With this observation, and using explicit form of the free-field correlators in (31), this expression

8 It is not difficult to write down the full functional integral for the dual hairpin model, which involves integration ˜ over the fields X and Y , as well as the integral over the boundary spin S = (S+,S−,S3) with the Wess–Zumino term (see, e.g., Ref. [5]). Such expression is not very useful for our analysis, except for the observation that the only terms ˜ = in the full action which involve the constant mode of X are the boundary potential term (21),andthetermAh 1/T [ + ] 0 dτ hx XB (τ) πahy S3(τ) responsible for the coupling to the external field (hx ,hy ). 302 S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 can be brought to the form: 2 g N R = D Tr eiπaQ , 0 2π ρ 2 g − R = D (2πT) kF G (k = 1, 2,...), (32) k 2π k k where iπaQ iπaQN k −iπaQ iπaQN k Fk = e Trρ e (S−S+) = e Trρ e (S+S−) , (33) and Gk are given by the 2k-fold integrals

2π uk vk v2 u1

Gk = duk dvk duk−1 ··· du1 dv1 0 0 0 0 0 k k −4b2−1 uj − ui vj − vi uj − vi × 4sin sin 2sin 2 2 2 j>i ji k −4b2−1 k vj − ui × 2sin 2 cos aQ π + (v − u ) . (34) 2 i i j>i i=1 2 gD The overall factor 2π in (32) appears because (31) involves unnormalized free-ﬁeld correlation −1/4 functions; here gD = 2 is well-known “boundary entropy” factor [7] associated with the Dirichlet and von Neumann boundary conditions for a free boson.9 Obviously, the way they are written above, the integrals (34) diverge for all positive b2.Asis common in conformal perturbation theory, we assume here a version of “analytic regularization”, where the expressions are understood as analytic continuations of these integrals from the domain e b2 < 0.10 This procedure is performed in Appendix A. Remarkably, the integrals (34) are evaluated in a closed form (the calculations are presented in Appendix A), (2π)k+1(−4b2)−k (1 − 4b2k) Gk = (k 1). (35) ! 1 − 2 − 1 − 2 + k (2 2b k aQ) (2 2b k aQ) Note that the Q-dependence of (35) is exactly as expected from (17), and moreover (32) coincide with the residues of (18), (17) at the points (28) provided iπaQN Trρ e = 2 cos(πaQ), (36) and k k k E∗ Tr (S−S+) = Tr (S+S−) = (k = 1, 2,...). (37) ρ ρ 2π

9 The g-factor of uncompactified boson X with the von Neumann boundary condition diverges. Formally, it involves gD the factor 2π dX0,whereX0 is the zero mode of X [8].In(30) the integration over X0 is already performed—this ˜ integration was the origin of the poles (28). Additional factor gD comes from the unperturbed partition function of Y . ˜ ˜ Since it is the T-duality transform of Y , its partition function equals to dQ BN |Q =gD,whereBN | is the boundary state associated with the von Neumann boundary condition for the field Y˜ . 10 More generally, these divergences have to be canceled by adding a counterterm Me2bXB (with the cutoff-dependent coefficient M) to the Hamiltonian (27). S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 303

Eqs. (36) and (37) are sufﬁcient to identify the representation ρ of the algebra (24), (25) of the boundary degrees of freedom. From (36) one ﬁnds

dim(ρ) = 2, (38) and

2 N = I, Trρ[N]=0. (39) Next, the two-dimensional representation of the algebra must satisfy additional relation

E∗ {S+, S−}= × I. (40) 2π Indeed, ρ is necessarily an irreducible representation of the algebra. As follows from Eqs. (24), (25), the anticommutator {S+, S−} commutes with S± and N, therefore it should be a constant in ρ. With Eq. (37) this implies the condition (40). There is a unique (up to equivalence) two-dimensional representation of (24), (25), (40) which satisﬁes (39):

E∗ E∗ ρ: S+ = σ+, S− = σ−, N = σ , (41) 2π 2π 3 where σa are conventional Pauli matrices. Thus we identify the boundary degree of freedom of the dual hairpin model with the spin s = 1/2. Note that two eigenvalues of σ3 are associated with two legs of the hairpin. Note also that according to (41) the operators S± have dimension [energy]1/2, as required by the balance of dimensions in Eq. (21) (in view of (3) the vertex ± ˜ operators ebXB iaYB have dimensions [energy]1/2).

3. Dual to the paperclip model

The dual representation for the paperclip model can be identiﬁed using similar line of arguments. The idea of the paperclip (2) being the composition of the left and right hairpins makes it natural to look for the Hamiltonian of the dual paperclip model in the form ˆ ˆ ˆ H = Hbulk + hxXB + πahyN + V, (42) ˆ where Hbulk is the same free-ﬁeld bulk Hamiltonian (10), and

ˆ E∗ bX +iaY˜ bX −iaY˜ −bX +iaY˜ −bX −iaY˜ V = A+ e B B + A− e B B B+ e B B + B− e B B . (43) 2π Here A±, B± and N are operators representing boundary degrees of freedom, which√ commute with the ﬁeld operators X(σ ), Π(σ ). Note that we have explicitly put the factor ∝ E∗ in (43),so that the operators A±, B± are dimensionless. Obviously, the ﬁrst two terms in (43) are associated with the left hairpin component of the paperclip, i.e. the operators A± play the same role as S E∗ ± 2π in (22). The last two terms in (43) are associated in a similar way with the dual form of the right hairpin. This correspondence suggests that the operators B±,aswellasA±, satisfy the fermionic relations analogous to (24)

2 2 A± = 0, B± = 0. (44) 304 S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311

In addition, they have to satisfy the commutation relations with N

[N, A±]=±2A±, [N, B±]=±2B±, (45) and the anticommutation relations

{A+, A−}={B+, B−}=I. (46) Intuitively, these relations can be advocated by the same arguments that were considered in the previous section. The paperclip curve in Fig. 1 can be regarded as the combination of two nearly straight parallel D-branes connected to each other by the left and the right hairpin curves. Then, ± ˜ the presence of the vertex operators ebXB iaYB in (43) is the way how the dual representation reflects the possibility of passages from one straight brane to another via the connection at the −bX ±iaY˜ right, i.e. at sufficiently large positive XB . Likewise, the operators e B B , which become significant at large negative XB , describe the transitions between the nearly parallel branes via the connection at the left. This picture suggests that the space of states associated with the boundary degrees of freedom in (42), i.e. the supporting space of the representation ρ of the above algebra, is two-dimensional, with two basic vectors (the eigenvectors of N) corresponding to the two constituent straight branes. Then, Eq. (44) simply express the statement that there is s single copy of the paperclip curve (as is specified by the YB bound in Eq. (2)). Also, since ρ is irreducible, and since in the limit E∗ → 0wehavetorecover(40), the relations (46) follow. It is easy to check that any two-dimensional representation ρ of the algebra (44), (45) and (46) is equivalent to the following one:

±i θ ∓i θ ρ: A± = e 2 σ±, B± = e 2 σ±, (47) where θ is an arbitrary complex parameter. It is possible to show that this parameter must be real and, moreover, it coincides with the θ-angle of the paperclip model. The easiest way to verify this identiﬁcation of θ is to analyze speciﬁc logarithmic divergences generated by the interaction (43). The divergences appear due to the singular term in the operator product expansions

± ˜ − ∓ ˜ 1 ebXB iaYB (τ) e bXB iaYB (τ ) = + regular terms, (48) |τ − τ | and it is easy to check that in view of (47) they can be absorbed by local boundary counterterm dτ −2E∗ cos(θ) log(Λ/E∗) , (49) 2π where Λ is the UV cut-off. Exactly the same counterterm, with θ being the topological angle, is required in the original formulation of the paperclip model, where its role is to compensate for “small instanton” divergences (see [1] for details). The above observation suggests that the instanton contributions of the original paperclip model are reproduced by certain terms of conformal perturbation theory of the dual model (42), (43). To be sure, the conformal perturbation theory, understood as a regular expansion in powers of E∗, cannot be literally valid for this model. The model has many properties in common with the boundary sinh-Gordon model, and in view of the analysis in [9], one expects rather complicated structure of the UV expansion in (42). For example, in the case hx = 0 and at | | T E∗ the partition function Z(hx,hy T)hx =0 of the model (42) is expected to develop a large S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 305 logarithmic term, 1 Z(0,h |T)= G(Q|κ)log + F(Q|κ), (50) y κ ∼ −1 1 which derives from large ( b log( κ )) fluctuations of the zero mode of the field X. The coef- = hy ficient G depends on Q i T and E∗ κ = . (51) 2πT Up to an overall factor κ2κ cos(θ) (whose origin could be traced down to the singular term in 1 Eq. (48)), it admits a small-κ expansion in double series in powers of κ and κ 2b2 (the term F has a similar expansion). The term ∼ κ of this expansion is given by the integral 2 gD iπaQN ˆ G(Q|κ) = Trρ e 1 + T dτ2 dτ1 V⊃(τ2)V⊂(τ1) 2πb + ˆ⊂ ˆ⊃ +··· V (τ2)V (τ1) 0 , (52) ˆ ˆ where V⊃ and V⊂ correspond to the first two terms and the second two terms on Eq. (43), respectively. Putting in explicit correlation functions of the exponential fields in (43) and evaluating the trace, one can bring this expression to a more explicit form g2 G(Q|κ) = D 2 cos(πaQ) 1 + κD(Q)cos(θ) +··· , (53) 2πb where the factor cos(θ) in the second term appears as the result of evaluation of the trace using Eqs. (47), and D(Q) is the integral

2π u2 1 cos(aQ(π − u2 + u1)) D(Q) = du2 du1 − + counterterm, (54) 2π u2 u1 cos(πaQ) sin( 2 ) 0 0 where u1 and u2 differ from τ1 and τ2 in (52) by a factor of 2πT. The integral logarithmically diverges when u1 → u2, but its divergent part cancels with the counterterm (49), and the Q- = d dependent ﬁnite part is evaluated explicitly, in terms of Euler’s function ψ(x) dx log (x).As the result, the expansion of G(Q|κ) has the form (53) with 1 1 D(Q) = 2log(κ) − ψ − aQ − ψ + aQ . (55) 2 2 With this, the coefﬁcient in front of cos(θ) in (53) exactly matches one-instanton contribution to the paperclip partition function (see Eq. (110) of [1]).

Acknowledgements

The authors are grateful to Vladimir V. Bazhanov, Vladimir A. Fateev and Alexei B. Zamolod- chikov for discussions and interest to this work. The research is supported in part by DOE grant #DE-FG02-96 ER 40959. A.B.Z. gratefully acknowledges kind hospitality of Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, and generous support from Foundation de l’Ecole Normale Superieure,´ via chaire Internationale de Recherche Blaise Pascal. 306 S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311

Appendix A. Calculation of the integral Gk (34)

A.1. Contour integral representation of Gk

Up to overall factor, the integrand in (34) coincides with the free-field expectation value of product of 2k chiral vertex operators 2bX ±2iaY V± = e R R , (A.1) where XR = (XR,YR) in the exponent stands for the holomorphic part of the Bose field X(σ, τ) = XR(τ + iσ)+ XL(τ − iσ). As usual, the free-field expectation values are fully determined by the two-point functions, 1 X (τ)X (τ ) = Y (τ)Y (τ ) =− log sin πT(τ − τ ) . (A.2) R R 0 R R 0 2 It is useful to introduce the following set of integrated products

τ0+1/T τp τ2 ··· = ··· ··· Jp( 1 p) dτp dτp−1 dτ1 V p (τp) V 1 (τ1), (A.3)

τ0 τ0 τ0 where ( 1 ··· p) is a set of signs, τ0 is ﬁxed real parameter, and integration is along the real axis. Clearly, the integral (34) is certain linear combinations of expectation values of J2k( 1 ··· 2k) with ( 1 ··· 2k) being one of two alternating sign sequences (+−+···+−) and (−+−···−+). As the ﬁrst step in calculation we transform (A.3) into contour integrals. To do this we introduce the “screening operators”, the integrals of the chiral vertex operators:

τ0+1/T 1 2bX ±2iaY x± = dτ e R R , (A.4) q − q−1 τ0 where11 q = eiπn. (A.5) Here and bellow in this appendix we use the notation from Ref. [1]: n ≡ 4b2. (A.6) The following relations, −1 k q − q x+J − (−···−) = J (+···−) − q J (−···+), 2k 1 2k 2k −1 q − q x−J − (−···−) = 0, 2k 1 −1 q − q x+J (−···+) = J + (+···+), 2k 2k 1 −1 k q − q x−J2k(−···+) = q J2k+1(−···−), (A.7) can be easily established by rearranging the integration domains. Additional set of relations is obtained from these by replacing x± → x∓ and simultaneously changing all signs, Jp( 1 ··· p) → Jp(− 1 ···− p). Recursively applying (A.7), one can prove that

11 τ +1/T As it is known, the screening operators x±, together with the two zero mode operators, 0 dτ(a∂τ X ± τ0 R ib∂τ YR), form the Chevelley basis of the Borel subalgebra of quantum superalgebra Uq,q(sl(2|1)) [10]. S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 307

Fig. 2. The integration contour in Eq. (A.11).

+ − k − k(k 1) −1 k ( 1) q 2 k k k J2k(+···−) = q − q q (x−x+) + (x+x−) , [k]q ! + − k − k(k 1) −1 k ( 1) q 2 k k k J2k(−···+) = q − q q (x+x−) + (x−x+) , (A.8) [k]q ! where the standard notations, qk − q−k [k] !=[1] [2] ···[k] , [k] = , (A.9) q q q q q q − q−1 are used. To make the next step more transparent we change to the new coordinate z = e2πiTτ. (A.10) The screening charges (A.4) become contour integrals in the variable z. More precisely, the action of the operators x± on any state created by a set of local insertions in the z-plane is written as the integral 1 x±(···) = dzV±(z)(···), (A.11) q − q−1 C 2πiTτ where the contour C starts from the point ζ0 = e 0 , goes around all the insertions in the counterclockwise direction, and then returns back to ζ0,asisshowninFig. 2. The integrand in (A.11) should be understood in terms of free-ﬁeld operator product expansions. In general, the contour is not closed since operator product in the integrand is multivalued function of z.Using (A.8) the integral Gk can be rewritten in the form: − 2 − 1 k k k (q q ) nk k G = 2(−1) q 2 cos π aQ + (x+x−) V (0) k [k] ! 2 λ q nk k + cos π aQ − (x−x+) V (0) , (A.12) 2 λ or more explicitly −k k − iπ nk2 21 i e 2 = ··· Gk k dζk dzk dζ1 dz1 j=1 sin(πnj) Ck Sk C1 S1 nk × cos π aQ + V−(ζk)V+(zk) ···V−(ζ1)V+(z1)Vλ(0) 2 nk + cos π aQ − V+(ζ )V−(z ) ···V+(ζ )V−(z )V (0) , (A.13) 2 k k 1 1 λ 308 S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311

Fig. 3. The integration contours in Eq. (A.13).

where the integration is over a set of contours Cj , Sj (j = 1, 2,...,k) each starting at ζ0 and returning to the same point after going around the point 0, and arranged so that Ck lays entirely (except for the point ζ0 itself) inside Sk, Sk lays inside Ck−1, Ck−1 lays inside Sk−1, etc., as is depicted in Fig. 3. The expectation values in the integrand in (A.13) involves a certain vertex operator, n(λ+ + λ−) n(λ− − λ+) V = exp − X + i Y , (A.14) λ 2b R 2a R with λ± to be speciﬁed below, inserted at the origin z = 0, so that V−(ζk)V+(zk) ···V+(z1)Vλ(0) k − − iπ 2 nλ+ nλ− = k 2 nk − − i e zj ζj (zj zi)(ζi ζj ) j=1 j>i k −n−1 −n−1 × (ζj − zi)(zj − ζi) (ζj − zj ) . (A.15) j>i j=1

We assume that the brunches of the power functions in (A.15) are chosen to give real positive values at

In Eqs. (A.14), (A.15) the following notations are used

kn − 1 kn − 1 nλ+ = + aQ + ε, nλ− = − aQ, (A.17) 2 2 where ε is a complex parameter, which is assumed to be small, and eventually will be sent to zero. S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 309

Fig. 4. The contour of analytic continuation in Eq. (A.19).

A.2. Combinatorics of the contour integrals

Now let us introduce another set of integrated products of vertex operators,

0 zp z2 ··· = ··· ··· Ip( 1 p) dzp dzp−1 dz1V 1 (z1) V p (zp)Vλ(0). (A.18)

ζ0 ζ0 ζ0

Here Vλ is the vertex operator (A.14). Although the ﬁnal result of the calculations below does not depend on a choice of the point ζ0, for convenience we choose it to be real and negative, and we assume that all the integrations in (A.18) are along the real axis. The operator products in (A.18) are multivalued functions of the integration variables, and we assume the same choice of the branch as in (A.15). Note that the integrals (A.18) are similar but different from (A.3),the main difference being in the form of integration contours. The monodromies of the operator products in (A.18) are determined by symbolic relations 2λ± AC V±(z)Vλ(ζ ) = q V±(z)Vλ(ζ ), (A.19) and, in particular, −2 AC V+(z)V−(ζ ) = q V+(z)V−(ζ ), (A.20) where the symbol AC[···] denotes analytic continuation in the variable z along the contour C shown on Fig. 4. Using these relations one can derive the following identities:

λ+−k x+I2k(+···−) =−q [λ+]q I2k+1(+···+), λ+−k x+I2k(−···+) =−q [λ+ − k]q I2k+1(+···+),

x+I2k−1(+···+) = 0, λ+−k λ+−k x+I2k−1(−···−) =−q [λ+ − k]q I2k(+···−) − q [λ+]q I2k(−···+), (A.21) as well as similar identities obtained from these by simultaneous change of the signs: {x± → x∓,λ± → λ∓,Ip( 1 ··· p) → Ip(− 1 ···− p)}. Here the action of the screening charges x± is deﬁned by Eq. (A.11). The recursion (A.21) allows one to express the ordered integrals (A.18) in terms of the screening charges. In particular, one ﬁnds 1 k k I2k(−···+) = [λ+ − k]q (x−x+) +[λ+]q (x+x−) Vλ(ζ ), (A.22) ck[λ+]q where [ ] ![ + − ] ! k k(λ++λ−−k) k q λ+ λ− 1 q ck = (−1) q . (A.23) [λ+ + λ− − k − 1]q ! 310 S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311

Now assume that λ± are given by (A.17), and consider the limit ε → 0. It is easy to check that at ε = 0 the expression (A.18) becomes invariant with respect to simultaneous rescaling of all the integration variables, hence the integral develops logarithmic divergence at zi → 0. Thus, as the function of ε, I2k(−+···+) is expected to have a simple pole at ε = 0. The appearance of the pole is very explicit in (A.22), since the coefﬁcient ck vanishes at ε = 0,

2πi c → ε (−1)k[k] ![k − 1] !. (A.24) k q − q−1 q q

The representation (A.22) also makes it easy to isolate the residue at this pole,

lim εI2k(−···+) ε→0 − k − −1 = ( 1) (q q ) [ ] ![ − ] ! + kn 2πi k q k 1 q cos(π(aQ 2 )) kn k kn k × cos π aQ − (x−x+) + cos π aQ + (x+x−) V (ζ ). (A.25) 2 2 λ

Comparing (A.25) with (A.12) we observe that the desired integrals Gk can be expressed through such residues,

k−1 + iπ 2 nk = k 1 k 2 nk + −···+ Gk 2 πi e sin(πnj) cos π aQ lim ε I2k( ) = . (A.26) 2 ε→0 ζ 0 j=1

Furthermore, it is easy to see that −···+ lim ε I2k( ) = ε→0 ζ 0

1 z3 z2 1 = dz − ··· dz dz V−(1)V+(z − ) ···V+(z )V (0) , (A.27) k 2k 1 2 1 2k 1 1 λ 0 0 0 and hence

− 2k+1π k1 nk G = sin(πnj) cos π aQ + k k 2 j=1

1 zk ζ1 − k1 − kn 1 −aQ × ··· 2 − − dzk dζk−1 dz1 ζj (1 ζj ) (ζj ζi) = 0 0 0 j 1 j>i

k − kn 1 +aQ × 2 − −n−1 − | − |−n−1 zj (1 zj ) (zj zi) zi ζj . (A.28) j=1 j>i j,i S.L. Lukyanov, A.B. Zamolodchikov / Nuclear Physics B 744 [FS] (2006) 295–311 311

A.3. Final step of the calculation

Integrations over the variables z1,...,zk in (A.28) can be eliminated by using the following identity12:

ζ − ζk k 2 ζ1 k k αj dzk dzk−1 ··· dz1 (zj − zi) |ζj − zi| = = ζ − ζ − ζ j>i j 0 i 1 k 1 k 1 0 k + j=0 (1 αj ) + + = (ζ − ζ )αi αj 1, (A.29) + + k j i (k 1 j=0 αj ) j>i where ζk >ζk−1 > ···>ζ0 is an ordered set of real numbers. In the case under consideration ζ0 = 0,ζk = 1 and kn − 1 α = + aQ, α =−n − 1 (j = 1,...,k). (A.30) 0 2 j Thus + k−1 2k 1π 2D − G = k 1 k(−n) sin(πnj), (A.31) k 1−kn − 1−kn + k ( 2 aQ) ( 2 aQ) j=1 where Dk−1 is the Selberg integral [12,13]:

ζ − 1 k 1 ζ2 k−1 = ··· (k−1)n−1 − −2n − −2n Dk−1 dζk−1 dζk−2 dζ1 ζj (1 ζj ) (ζj ζi) = 0 0 0 j 1 j>i π k−1(−n)−k (1 − kn) 1 = . (A.32) (k − 1)! k(−n) k−1 j=1 sin(πnj) Combining Eqs. (A.31) and (A.32) one arrives to (35).

References

[1] S.L. Lukyanov, E.S. Vitchev, A.B. Zamolodchikov, Nucl. Phys. B 683 (2004) 423. [2] S.L. Lukyanov, A.M. Tsvelik, A.B. Zamolodchikov, Nucl. Phys. B 719 (2005) 103. [3] V.A. Fateev, E. Onofri, Al.B. Zamolodchikov, Nucl. Phys. B 406 (1993) 521. [4] V.A. Fateev, Phys. Lett. B 357 (1995) 397. [5] A.M. Tsvelik, Quantum Field Theory in Condensed Matter Physics, Cambridge Univ. Press, Cambridge, 1995. [6] M. Goulian, M. Li, Phys. Rev. Lett. 66 (1991) 2051. [7] I. Afﬂeck, A. Ludwig, Phys. Rev. Lett. 67 (1991) 161. [8] P. Fendley, H. Saleur, N.P. Warner, Nucl. Phys. B 430 (1994) 577. [9] Al.B. Zamolodchikov, unpublished notes, 2001. [10] R.B. Zhang, J. Phys. A: Math. Gen. 27 (1994) 817. [11] P. Baseilhac, V.A. Fateev, Nucl. Phys. B 532 (1998) 567. [12] A. Selberg, Norsk. Mat. Tidsskr. 26 (1944) 71. [13] V. Dotsenko, V. Fateev, Nucl. Phys. B 251 (1985) 691.

12 This identity can be interpreted as the Dotsenko–Fateev representation [13] of the simple conformal block of Virasoro algebra with the central charge c =−2. A similar identity was used in Ref. [11]. Nuclear Physics B 744 [FS] (2006) 312–329

T –Q relation and exact solution for the XYZ chain with general non-diagonal boundary terms

Wen-Li Yang a,b,∗, Yao-Zhong Zhang b

a Institute of Modern Physics, Northwest University, Xian 710069, PR China b Department of Mathematics, University of Queensland, Brisbane, QLD 4072, Australia Received 22 February 2006; received in revised form 11 March 2006; accepted 16 March 2006 Available online 6 April 2006

Abstract We propose that the Baxter’s Q-operator for the quantum XYZ spin chain with open boundary conditions is given by the j →∞limit of the corresponding transfer matrix with spin-j (i.e., (2j + 1)-dimensional) auxiliary space. The associated T –Q relation is derived from the fusion hierarchy of the model. We use this relation to determine the Bethe Ansatz solution of the eigenvalues of the fundamental transfer matrix. The solution yields the complete spectrum of the Hamiltonian. © 2006 Elsevier B.V. All rights reserved.

PACS: 75.10.Pq; 04.20.Jb; 05.50.+q

Keywords: Spin chain; Reﬂection equation; Bethe Ansatz; Q-operator; Fusion hierarchy

1. Introduction

In this paper we are interested in the exact solution of the quantum XYZ quantum spin chain with most general non-diagonal boundary terms, deﬁned by the Hamiltonian

N−1 = x x + y y + z z + (−) x + (−) y + (−) z H Jxσj σj+1 Jyσj σj+1 Jzσj σj+1 hx σ1 hy σ1 hz σ1 j=1 + (+) x + (+) y + (+) z hx σN hy σN hz σN , (1.1)

* Corresponding author. E-mail address: [email protected] (W.-L. Yang).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.03.025 W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 313 where σ x , σ y , σ z are the usual Pauli matrices, N is the number of spins, the bulk coupling constants Jx , Jy , Jz are related to the crossing parameter η and modulus parameter τ by the following relations, eiπησ(η+ τ ) eiπησ(η+ 1 + τ ) σ(η+ 1 ) J = 2 ,J= 2 2 ,J= 2 , x σ(τ ) y 1 + τ z 1 2 σ(2 2 ) σ(2 ) and the components of boundary magnetic ﬁelds associated with the left and right boundaries are given by

3 (∓) 3 (∓) ∓ − τ − 1 (∓) −iπ( 3 α( )− τ ) σ(η) σ(αl 2 ) (∓) σ(η) σ(αl 2 ) h =±e l=1 l 2 ,h=± , x σ(τ ) (∓) z 1 (∓) 2 l=1 σ(αl ) σ(2 ) l=1 σ(αl ) 3 (∓) ∓ − 1 − τ (∓) −iπ( 3 α( )− 1 − τ ) σ(η) σ(αl 2 2 ) h =±e l=1 l 2 2 . (1.2) y 1 + τ (∓) σ(2 2 ) l=1 σ(αl ) { (∓)} Here σ(u) is the σ -function defined in the next section and αl are free boundary parameters which specify the boundary coupling (equivalently, the boundary magnetic fields). In solving the closed XYZ chain (with periodic boundary condition), Baxter constructed a Q-operator [1], which has now been proved to be a fundamental object in the theory of exactly solvable models [2]. However, Baxter’s original construction of the Q-operator was ad hoc and its connection with the quantum inverse scattering method was not clear. It was later argued [3] that the Q-operator for the XXZ spin chain with periodic boundary condition may be constructed from the Uq (sl2) universal L-operator whose associated auxiliary space is taken as an infinite- dimensional q-oscillator representation of Uq (sl2). (See also [4] for recent discussions on the direct construction of the Q-operator of the closed chain.) In [5], a natural set-up to define the Baxter’s Q-operator within the quantum inverse scattering method was proposed for the open XXZ chain. However, the generalization of the construction to off-critical elliptic solvable models [6] (including the XYZ chain) is still lacking even for the closed chains. In this paper, we generalize the construction in [5] to Baxter’s eight-vertex model and propose that the Q-operator Q(u)¯ for the XYZ quantum spin chain (closed or open) is given by the j →∞limit of the corresponding transfer matrix t(j)(u) with spin-j (i.e., (2j +1)-dimensional) auxiliary space, Q(u)¯ = lim t(j)(u − 2jη). (1.3) j→∞ This relation together with the fusion hierarchy of the transfer matrix leads to the T –Q relation. We then use the T –Q relation, together with some additional properties of the transfer matrix, to determine the complete set of the eigenvalues and Bethe Ansatz equations of the transfer matrix associated with the Hamiltonian (1.1) under certain constraint of the boundary parameters (see (5.16) below). The paper is organized as follows. In Section 2, we introduce our notation and some basic ingredients. In Section 3, we derive the T –Q relation from (1.3) and the fusion hierarchy of the open XXZ chain. In Section 4, some properties of the fundamental transfer matrix are obtained. By means of these properties and the T –Q relation, we in Section 5 determine the eigenvalues of the transfer matrix and the associated Bethe Ansatz equations, thus giving the complete spectrum of the Hamiltonian (1.1) under the constraint (5.16) of the boundary parameters. We summarize our conclusions in Section 6. Some detailed technical calculations are given in Appendices A–C. 314 W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329

2. Fundamental transfer matrix

Let us ﬁx τ such that Im(τ) > 0 and a generic complex number η. Introduce the following elliptic functions ∞ a θ (u, τ) = exp iπ (m + a)2τ + 2(m + a)(u + b) , (2.1) b =−∞ m 1 ∂ σ(u)= θ 2 (u, τ), ζ(u) = ln σ(u) . (2.2) 1 ∂u 2 Among them the σ -function1 satisﬁes the following identity:

σ(u+ x)σ(u − x)σ(v + y)σ(v − y) − σ(u+ y)σ(u − y)σ(v + x)σ(v − x) = σ(u+ v)σ(u − v)σ(x + y)σ(x − y), (2.3) which will be useful in deriving equations in the following. The well-known eight-vertex model R-matrix R(u) ∈ End(C2 ⊗ C2) is given by ⎛ ⎞ a(u) d(u) ⎜ b(u) c(u) ⎟ R(u) = ⎝ ⎠ . (2.4) c(u) b(u) d(u) a(u) The non-vanishing matrix elements are [2] 1 1 0 2 + 2 0 + θ 1 (u, 2τ)θ 1 (u η,2τ) θ 1 (u, 2τ)θ 1 (u η,2τ) a(u) = 2 2 ,b(u)= 2 2 , 1 1 0 2 0 2 θ 1 (0, 2τ)θ 1 (η, 2τ) θ 1 (0, 2τ)θ 1 (η, 2τ) 2 2 2 2 1 1 0 0 2 2 + θ 1 (u, 2τ)θ 1 (u + η,2τ) θ 1 (u, 2τ)θ 1 (u η,2τ) c(u) = 2 2 ,d(u)= 2 2 . (2.5) 0 0 0 0 θ 1 (0, 2τ)θ 1 (η, 2τ) θ 1 (0, 2τ)θ 1 (η, 2τ) 2 2 2 2 Here u is the spectral parameter and η is the so-called crossing parameter. The R-matrix satisﬁes the quantum Yang–Baxter equation

R1,2(u1 − u2)R1,3(u1 − u3)R2,3(u2 − u3) = R2,3(u2 − u3)R1,3(u1 − u3)R1,2(u1 − u2), (2.6) and the properties,

= = t1t2 PT-symmetry: R1,2(u) R2,1(u) R1,2 (u), (2.7) i i = i i = Z2-symmetry: σ1σ2R1,2(u) R1,2(u)σ1σ2, for i x,y,z, (2.8)

1 Our σ -function is the ϑ-function ϑ (u) [7]. It has the following relation with the Weierstrassian σ -function if denoted 1 η u2 2 1 ∞ nq2n iτ it by σw(u): σw(u) ∝ e 1 σ(u), η = π ( − 4 ) and q = e . 1 6 n=1 1−q2n W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 315

σ(u+ η)σ(u − η) Unitarity relation: R (u)R (−u) =−ξ(u)id,ξ(u)= , (2.9) 1,2 2,1 σ(η)σ(η) = t2 − − =− y Crossing relation: R1,2(u) V1R1,2( u η)V1,V iσ . (2.10)

Here R2,1(u) = P12R1,2(u)P12 with P12 being the usual permutation operator and ti denotes transposition in the ith space. Throughout this paper we adopt the standard notations: for any 2 2 2 matrix A ∈ End(C ), Aj is an embedding operator in the tensor space C ⊗ C ⊗···, which acts as A on the jth space and as identity on the other factor spaces; Ri,j (u) is an embedding operator of R-matrix in the tensor space, which acts as identity on the factor spaces except for the ith and jth ones. Integrable open spin chains can be constructed as follows [8]. Let us introduce a pair of K-matrices K−(u) and K+(u). The former satisﬁes the reﬂection equation (RE)

− − + − R1,2(u1 u2)K1 (u1)R2,1(u1 u2)K2 (u2) = − + − − K2 (u2)R1,2(u1 u2)K1 (u1)R2,1(u1 u2), (2.11) and the latter satisﬁes the dual RE [8,9]

− + − − − + R1,2(u2 u1)K1 (u1)R2,1( u1 u2 2η)K2 (u2) = + − − − + − K2 (u2)R1,2( u1 u2 2η)K1 (u1)R2,1(u2 u1). (2.12) Then the transfer matrix t(u) of the open XYZ chain with general integrable boundary terms is given by [8] = + − ˆ t(u) tr0 K0 (u)T0(u)K0 (u)T0(u) , (2.13) ˆ where T0(u) and T0(u) are the monodromy matrices ˆ T0(u) = R0,N (u) ···R0,1(u), T0(u) = R1,0(u) ···RN,0(u), (2.14) and tr0 denotes trace over the “auxiliary space” 0. In this paper, we consider the most general solutions K∓(u) [10] to the associated reﬂection equation and its dual, (−) −iπu (−) −iπu (−) − σ(2u) c σ(u)e cy σ(u)e c σ(u) K (u) = I + x σ x + σ y + z σ z , (2.15) 2σ(u) σ(u+ τ ) σ(u+ 1+τ ) σ(u+ 1 ) 2 2 2 + = − − − K (u) K ( u η) { (−)}→{ (+)}, (2.16) cl cl × { (∓)} where I is the 2 2 identity matrix and the constants cl are expressed in terms of boundary { (∓)} parameters αl as follows:

3 (∓) 3 (∓) ∓ − τ − 1 (∓) −iπ( α( )− τ ) σ(αl 2 ) (∓) σ(αl 2 ) c = e l l 2 ,c= , x (∓) z (∓) l=1 σ(αl ) l=1 σ(αl ) 3 (∓) ∓ − 1 − τ (∓) −iπ( α( )− 1 − τ ) σ(αl 2 2 ) c = e l l 2 2 . y (∓) (2.17) l=1 σ(αl ) 316 W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329

Sklyanin has shown that the transfer matrices with different spectral parameters commute with each other: [t(u),t(v)]=0. This ensures the integrability of the open XYZ chain. The Hamil- tonian (1.1) can be expressed in terms of the transfer matrix = σ(η) ∂ − − + H ln t(u) (N 1)ζ(η) 2ζ(2η) , (2.18) σ (0) ∂u u=0 = ∂ | where σ (0) ∂uσ(u) u=0.

3. Fusion hierarchy and T –Q relation

We shall use the fusion procedure, which was ﬁrst developed for R-matrices [11] and then later generalized for K-matrices [12,13], to obtain the Baxter T –Q relation. The fused spin- 1 = 1 3 (j, 2 ) R-matrix (j 2 , 1, 2 ,...)is given by [11] = (+) + ··· + − (+) R 1...2j ,2j+1(u) P1...2j R1,2j+1(u)R2,2j+1(u η) R2j,2j+1 u (2j 1)η P1...2j , (3.1) (+) where P1...2j is the completely symmetric projector. Following [12,13], the fused spin-j − K-matrix K 1...2j (u) is given by − = (+) − + − + K 1...2j (u) P1...2j K2j (u)R2j,2j−1(2u η)K2j−1(u η) × + + − + ··· R2j,2j−2(2u 2η)R2j−1,2j−2(2u 3η)K2j−2(u 2η) × + − − + − (+) R2,1 2u (4j 3)η K1 u (2j 1)η P1...2j . (3.2) + The fused spin-j K-matrix K 1...2j (u) is given by + = | − − − K 1...2j (u) F(u 2j)K 1...2j ( u 2jη) { (−)}→{ (+)}, (3.3) cl cl | | = q−1 l + + where the scalar functions F(u q) are given by F(u q) 1/( l=1 k=1 ξ(2u lη kη)),for q = 1, 2,.... The fused transfer matrix t(j)(u) constructed with a spin-j auxiliary space is given by (j) = + − ˆ + − t (u) tr1...2j K 1...2j (u)T 1...2j (u)K 1...2j (u)T 1...2j u (2j 1)η , 1 j = , 1,..., (3.4) 2 where

T 1...2j (u) = R 1...2j ,N (u) ···R 1...2j ,1(u), ˆ T 1...2j u + (2j − 1)η = R 1...2j ,1(u) ···R 1...2j ,N (u). (3.5)

1 = 1 = ( 2 ) The transfer matrix (2.13) corresponds to the fundamental case j 2 , i.e., t(u) t (u).The fused transfer matrices constitute commutative families, namely, t(j)(u), t(k)(v) = 0. (3.6) W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 317

They satisfy a so-called fusion hierarchy relation [12,13] (j) (j− 1 ) (j−1) t u − (2j − 1)η = t 2 u − (2j − 1)η t(u)− δ(u)t u − (2j − 1)η , 3 j = 1, ,.... (3.7) 2 In the above hierarchy, we have used the convention t(0)(u) = id. The coefﬁcient function δ(u) can be expressed in terms of the quantum determinants of the monodromy matrices T(u), T(u)ˆ and K-matrices [12,13],

Det{T(u)} Det{T(u)ˆ } Det{K−(u)} Det{K+(u)} δ(u) = . (3.8) ρ˜1,1(2u − η) ˜ = σ(−u)σ (u+2η) Here ρ1,1(u) σ(η)σ(η) and the corresponding determinants are given by [13] N − − σ(u+ η)σ(u − η) Det T(u) id = tr P ( )T (u − η)T (u)P ( ) = id, (3.9) 12 12 1 2 12 σ(η)σ(η) N − − σ(u+ η)σ(u − η) Det T(u)ˆ id = tr P ( )T (u − η)T (u)P ( ) = id, (3.10) 12 12 1 2 12 σ(η)σ(η) − (−) − − Det K (u) = tr12 P K (u − η)R12(2u − η)K (u) , (3.11) 12 1 2 + = (−) + − − + − Det K (u) tr12 P12 K2 (u)R12( 2u η)K1 (u η) , (3.12) (−) (−) = 1 − where P12 is the completely antisymmetric project: P12 2 (id P12). Using the crossing relations of the K-matrices (see (4.9)–(4.12) below) and after a tedious calculation (for details see Appendix A), we ﬁnd that the quantum determinants of the most general K-matrices K±(u) given by (2.15) and (2.16) respectively are

3 (−) (−) − σ(2u − 2η) σ(α + u)σ (α − u) Det K (u) = l l , (3.13) σ(η) (−) (−) l=1 σ(αl )σ (αl ) 3 (+) (+) + σ(2u + 2η) σ(u+ α )σ (u − α ) Det K (u) = l l . (3.14) σ(η) (+) (+) l=1 σ(αl )σ (αl ) Substituting (3.9)–(3.14) into (3.8), one has that σ(u+ η)σ(u − η) 2N σ(2u − 2η)σ(2u + 2η) δ(u) = σ(η)σ(η) σ(2u − η)σ(2u + η) 3 σ(u+ α(γ ))σ (u − α(γ )) × l l . (γ ) (γ ) (3.15) γ =± l=1 σ(αl )σ (αl ) For generic η, the fusion hierarchy does not truncate (cf. the roots of unity case in the trigonometric limit [14,15]). Hence {t(j)(u)} constitute an infinite hierarchy, namely, j taking values 1 3 { (j) } 2 , 1, 2 ,.... The commutativity (3.6) of the fused transfer matrices t (u) and the fusion relation (3.7) imply that the corresponding eigenvalue of the transfer matrix t(j)(u), denoted by Λ(j)(u), satisfies the following hierarchy relation 318 W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 (j) (j− 1 ) 1 (j−1) Λ (u + η − 2jη)= Λ 2 u − 2 j − η Λ(u) − δ(u)Λ u − η − 2(j − 1)η , 2 3 j = 1, ,.... (3.16) 2 ( 1 ) Here we have used the convention Λ(u) = Λ 2 (u) and Λ(0)(u) = 1. Dividing both sides − 1 (j 2 ) − − 1 of (3.16) by Λ (u 2(j 2 )η),wehave Λ(j)(u + η − 2jη) Λ(j−1)(u − η − 2(j − 1)η) Λ(u) = + δ(u) . (3.17) − 1 − 1 (j 2 ) − − 1 (j 2 ) − − 1 Λ (u 2(j 2 )η) Λ (u 2(j 2 )η) We now consider the limit j →∞. We make the fundamental assumption (1.3) (in particular, that the limit exists), which implies for the corresponding eigenvalues Q(u)¯ = lim Λ(j)(u − 2jη), (3.18) j→+∞ where we have used the same notation for the operator Q¯ and its eigenvalues (cf. Eq. (1.3)). It follows from (3.17) that Q(u¯ + η) Q(u¯ − η) Λ(u) = + δ(u) . (3.19) Q(u)¯ Q(u)¯ Assuming the function Q(u)¯ has the decomposition Q(u)¯ = f (u)Q(u) with M Q(u) = σ(u− uj )σ (u + uj + η), (3.20) j=1 where M is certain non-negative integer and {uj } are some parameters which will be specified later (see (5.17) and (5.19) below), then Eq. (3.19) becomes Q(u + η) Q(u − η) Λ(u) = H (u) + H (u) . (3.21) 1 Q(u) 2 Q(u) = f(u+η) = f(u−η) { | = Here H1(u) f(u) and H2(u) δ(u) f(u) . It is easy to see that the functions Hi(u) i 1, 2} satisfy the relation

H1(u − η)H2(u) = δ(u) (3.22) with δ(u) given by (3.15). In summary, the eigenvalue Λ(u) of the fundamental transfer matrix t(u) (2.13) has the decomposition form (3.21), where the coefﬁcient functions {Hi(u)} satisfy the constraint (3.22).In the following, we shall use certain properties of the eigenvalue Λ(u) derived from the transfer matrix to determine the functions {Hi(u)} and therefore the eigenvalue Λ(u).

4. Properties of the fundamental transfer matrix

Here we will derive ﬁve properties of the fundamental transfer matrix t(u), which together with the T –Q relation (3.21) enable us to determine the functions {Hi(u)}. In addition to the Riemann identity (2.3),theσ -function enjoys the following identity: 2σ(u)σ(u+ 1 )σ (u + τ )σ (u − 1 − τ ) σ(2u) = 2 2 2 2 , (4.1) 1 τ − 1 − τ σ(2 )σ ( 2 )σ ( 2 2 ) W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 319

−2iπ(u+ τ ) σ(u+ 1) =−σ(u), σ(u+ τ)=−e 2 σ(u). (4.2) Moreover, from the deﬁnition of the elliptic function (2.1), one may show that 1 0 − + 1 + τ + = iπ(u 2 2 ) 2 θ 1 (u τ,2τ) e θ 1 (u, 2τ), (4.3) 2 2 1 − + 1 + τ 0 2 + = iπ(u 2 2 ) θ 1 (u τ,2τ) e θ 1 (u, 2τ). (4.4) 2 2 The property (4.1) of the σ -function implies that the matrix elements of the K-matrices K∓(u) given by (2.15)–(2.16) are analytic functions of u. Thus the commutativity of the transfer matrix t(u) and the analyticity of the R-matrix and K-matrices lead to the following analytic property:

The eigenvalue of t(u) is an analytic function of u at ﬁnite u. (4.5) The quasi-periodicity of the elliptic functions, (4.2)–(4.4), allows one to derive the following properties of the R-matrix and K-matrices: + =− z z =− z z ∓ + =− z ∓ z R1,2(u 1) σ1 R1,2(u)σ1 σ2 R1,2(u)σ2 ,K(u 1) σ K (u)σ , − + η + τ − + η + τ + =− 2iπ(u 2 2 ) x x =− 2iπ(u 2 2 ) x x R1,2(u τ) e σ1 R1,2(u)σ1 e σ2 R1,2(u)σ2 , − −2iπ(3u+ 3 τ) x − x K (u + τ)=−e 2 σ K (u)σ , + −2iπ(3u+3η+ 3 τ) x + x K (u + τ)=−e 2 σ K (u)σ . From these relations one obtains the quasi-periodic properties of t(u), − + + + t(u+ 1) = t(u), t(u+ τ)= e 2iπ(N 3)(2u η τ)t(u). (4.6) − + The initial conditions of the R-matrix R12(0) = P12 and K-matrices: K (0) = id, tr K (0) = σ(2η)/σ(η), imply that the initial condition of the fundamental transfer matrix t(u) is given by σ(2η) t(u) = × id . (4.7) u=0 σ(η) The identity 1 0 2 θ 1 (u, 2τ)θ (u, 2τ) σ(u) 1 = 2 2 , σ(τ ) 1 2 0 τ 2 τ θ 1 ( 2 , 2τ)θ 1 ( 2 , 2τ) 2 2 suggests the semi-classical property of the R-matrix limη→0{σ(η)R12(u)}=σ(u)id. This enables us to derive the semi-classical property of t(u), + − lim σ 2N (η)t(u) = σ 2N (u) lim tr K (u)K (u) × id . (4.8) η→0 η→0 Now, we derive the crossing relation of t(u). For this purpose, we introduce − − K¯ (u) = tr P R (−2u − 2η) K (u) t2 , (4.9) 1 2 12 12 2 ¯ + = + t2 K1 (u) tr2 P12R12(2u) K2 (u) . (4.10) 320 W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329

Through a straightforward calculation (see Appendix A for details), we ﬁnd

¯ − σ(2u) y − y − K (u) =− σ K (−u − η)σ ≡ f−(u)V K (−u − η)V, (4.11) σ(η) + ¯ + σ(2u 2η) y + y + K (u) = σ K (−u − η)σ ≡ f+(u)V K (−u − η)V, (4.12) σ(η) × = σ(2u) =−σ(2u+2η) where the 2 2matrixV is given in (2.10), f−(u) σ(η) and f+(u) σ(η) . The PT- symmetry (2.7) and crossing relation (2.10) of the R-matrix implies t0 − − = − − ··· − − t0 T0 ( u η) R0,N ( u η) R0,1( u η) (2.10) N−1 tN t1 t0 = (−1) V0R (u) ···R (u)V0 0,N 0,1 N−1 t1...tN t0 = (−1) V0 R0,N (u) ···R0,1(u) V0 N−1 t0 t1...tN = (−1) V0 R0,N (u) ···R0,1(u) V0 (2.7) N−1 (2.7) N−1 ˆ = (−1) V0R0,1(u) ···R0,N (u) V0 = (−1) V0T0(u)V0. (4.13) Similarly, we have another “duality relations” between the monodromy matrices T(u)and T(u)ˆ ,

ˆ t0 − − = − N−1 T0 ( u η) ( 1) V0T0(u)V0. (4.14) The crossing relations of the K-matrices (4.11) and (4.12), and the “duality relations” (4.13) and (4.14), imply that + − ˆ t(−u − η) = tr0 K (−u − η)T0(−u − η)K (−u − η)T0(−u − η) 0 0 + t0 − ˆ t0 = tr0 K (−u − η)T0(−u − η) K (−u − η)T0(−u − η) 0 0 = t0 − − + − − t0 ˆ t0 − − − − − t0 tr0 T0 ( u η) K0 ( u η) T0 ( u η) K0 ( u η) (4.13),(=4.14) ˆ + − − t0 − − − t0 tr0 T0(u) V0 K0 ( u η) V0 T0(u) V0 K0 ( u η) V0 (4.11),(=4.12) ˆ ¯ + t0 ¯ − t0 tr0 T0(u) K0 (u) T0(u) K0 (u) /f +(u)f−(u) + − tr tr tr (Tˆ (u)P R (2u)K (u)T (u)P R (−2u − 2η)K (u)) (4.9),(=4.10) 0 1 2 0 01 0,1 1 0 02 0,2 2 f+(u)f−(u) − + tr tr tr (P Tˆ (u)R (2u)T (u)P R (−2u − 2η)K (u)K (u)) = 0 1 2 01 1 0,1 0 02 0,2 2 1 f+(u)f−(u) + − ˆ tr0 tr1 tr2(K (u)T1(u)P01R0,1(2u)P02R0,2(−2u − 2η)K (u)T1(u)) = 1 2 . f+(u)f−(u) (4.15) In deriving the second last equality of the above equation, we have used the relation, ˆ ˆ T1(u)R0,1(2u)T0(u) = T0(u)R0,1(2u)T1(u), which is a simple consequence of the quantum Yang–Baxter equation (2.6). Then, let us consider − tr tr P R (2u)P R (−2u − 2η)K (u) /f +(u)f−(u) 0 2 01 0,1 02 0,2 2 = − − − tr0 P01R0,1(2u) tr2 P02R0,2( 2u 2η)K2 (u) /f +(u)f−(u) (4=.9) ¯ − t0 tr0 P01R0,1(2u) K0 (u) /f +(u)f−(u) W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 321 (4=.11) − − − t0 tr0 V0P01R0,1(2u)V0 K0 ( u η) /f +(u) (2=.8) − − − t0 tr0 V1P01R0,1(2u)V1 K0 ( u η) /f +(u) − − (4.9) ¯ − (4.11) f−( u η) − − = V K (−u − η)V /f +(u) = K (u) = K (u). 1 1 1 f + (u) 1 1 Substituting the above equation into (4.15), we have the following crossing relation of t(u): − − = + − ˆ = t( u η) tr1 K1 (u)T1(u)K1 (u)T1(u) t(u). (4.16) We remark that our above proof of the crossing relation (4.16) generalizes the previous proofs given in [16].

5. Eigenvalues and Bethe Ansatz equations

It follows from (4.5)–(4.8) and (4.16) that the eigenvalue Λ(u), as a function of u, has the following properties, Analyticity: Λ(u) is an analytic function of u at finite u, (5.1) Quasi-periodicity: Λ(u + 1) = Λ(u), − + + + Λ(u + τ)= e 2iπ(N 3)(2u η τ)Λ(u), (5.2) Crossing symmetry: Λ(−u − η) = Λ(u), (5.3) σ(2η) Initial condition: Λ(0) = , (5.4) σ(η) + − Semi-classic property: lim σ 2N (η)Λ(u) = σ 2N (u) lim tr K (u)K (u) . (5.5) η→0 η→0 The T –Q relations (3.21) and (3.22), together with the above properties (5.1)–(5.5), can be used to determine {Hi(u)} and the eigenvalues of the fundamental transfer matrix. { (γ ) =± | =± = } For this purpose, we define 6 discrete parameters l 1 γ ,l 1, 2, 3 and fix (−) =+ { (γ )} 1 1 in the following. Associated with l , we introduce

2N 3 (γ ) (γ ) ± σ (u)σ (2u) σ(u+ η ± α ) H ( ) u (γ ) = l l , (5.6) 1 l σ 2N (η)σ (2u + η) (γ ) (γ ) γ =± l=1 σ(l αl ) 2N 3 (γ ) (γ ) ± σ (u + η)σ(2u + 2η) σ(u∓ α ) H ( ) u (γ ) = l l . (5.7) 2 l σ 2N (η)σ (2u + η) (γ ) (γ ) γ =± l=1 σ(l αl ) (±) |{ (γ )} (±) |{ (γ )} One may readily check that the functions H1 (u l ) and H2 (u l ) indeed satisfy (3.22), namely, (±) − (γ ) (±) (γ ) = H1 u η l H2 u l δ(u). (5.8) The general solution to (3.22) can be written as follows: = (±) (γ ) = (±) (γ ) H1(u) H1 u l g1(u), H2(u) H2 u l g2(u), (5.9) where {gi(u)} satisfy the following relations,

g1(u − η)g2(u) = 1,g1(u + 1) = g1(u), g2(u + 1) = g2(u). (5.10) 322 W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329

The solutions to (5.10) have the following form, N2 − + N1 − − − j=1 σ(u uj ) 1 j=1 σ(u uj η) g1(u) = λ ,g2(u) = , (5.11) N1 − − λ N2 − + − j=1 σ(u uj ) j=1 σ(u uj η) where N1 and N2 are integers such that N1,N2 0, and λ is an non-zero constant. In the

− =∓ (γ ) (γ ) − + =∓ (γ ) (γ ) − above equation, we assume that uj l αl η and uj l αl η, otherwise the { (±) |{ (γ )} } { (±) |{ (γ )} } corresponding factors in gi(u) make transitions among H1 (u l ) ( H2 (u l ) ,respectively). Then the analyticity of Λ(u) (5.1) requires that g1(u) and g2(u) have common poles; = − = + + i.e., N1 N2, and uj uj η. This means − − N1 − + N1 − − σ(u uj η) 1 σ(u uj η) g1(u) = λ − ,g2(u) = − . (5.12) σ(u− u ) λ σ(u− u ) j=1 j j=1 j (±) |{ (γ )} = (±) − − |{ (γ )} Since H2 (u l ) H1 ( u η l ), the crossing symmetry of Λ(u) (5.3) implies that H2(u) = H1(−u − η). Hence, N1 is even and

N1 − − 2 − + + + σ(u uj η)σ(u uj 2η) g1(u) = λ − − , (5.13) σ(u− u )σ (u + u + η) j=1 j j

N1 − − 2 − − + σ(u uj η)σ(u uj ) g2(u) = λ − − ,λ=±1. (5.14) σ(u− u )σ (u + u + η) j=1 j j This is equivalent to having additional Bethe roots; and the corresponding factors, except λ, can be absorbed into those of Q(u) (3.20). Moreover the initial condition (5.4) implies λ =+1. Therefore the eigenvalues of the transfer matrix take the following forms:

± Q(u + η) ± Q(u − η) H ( ) u (γ ) + H ( ) u (γ ) . (5.15) 1 l Q(u) 2 l Q(u) In the above expression, we still have an unknown non-negative integer M defined in (3.20) to be fixed. Moreover, Λ(u) must fulfill the properties (5.2) and (5.5). This provides the following { (γ )} constraint to the boundary parameters αl : 3 3 (γ ) (γ ) = (γ ) =− l αl kη mod(1), l 1, (5.16) γ =± l=1 γ =± l=1 where k is an integer such that |k| N − 1 and N − 1 + k being even, and yields two different (±) (±) |{ (γ )} values M for M in (3.20), corresponding respectively to Hi (u l ),

± 1 M( ) = (N − 1 ∓ k). (5.17) 2 The proof of the above constraint is relegated to Appendix B. { (γ )} Finally, we have that if the boundary parameters αl satisfy any of the constraints (5.16), the eigenvalues of the fundamental transfer matrix t(u) are given by (±) (±) ± ± Q (u + η) ± Q (u − η) Λ( )(u) = H ( ) u (γ ) + H ( ) u (γ ) , (5.18) 1 l Q(±)(u) 2 l Q(±)(u) W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 323

± (±) = M( ) − (±) + (±) + { (±)} where Q (u) j=1 σ(u uj )σ (u uj η) and the parameters uj respectively satisfy the Bethe Ansatz equations, (±) (±)|{ (γ )} (±) (±) + H (uj l ) Q (uj η) ± 2 =− ,j= 1,...,M( ). (5.19) (±) (±)|{ (γ )} (±) (±) − H1 (uj l ) Q (uj η) Indeed one can verify that both Λ(±)(u) given by (5.18) have the desirable properties (5.1)– (5.5) provided that the constraint (5.16) and the Bethe Ansatz equations (5.19) are satisﬁed. In the trigonometric limit τ →+i∞, we recover the results in [5] (for details see Appendix C). The completeness [5,17] of the eigenvalues {Λ(±)(u)} in the trigonometric case suggests that for a given set of bulk and boundary parameters satisfying the constraint (5.16), the eigenvalues Λ(−)(u) and Λ(+)(u) together constitute the complete set of eigenvalues of the transfer matrix t(u) of the open XYZ spin chain. Therefore the complete set of the energy eigenvalues E(±) of the Hamiltonian (1.1), when the boundary parameters satisfy the constraint (5.16), are respectively given by

± M( ) 3 ± σ(η) ± ± E( ) = 2 ζ u( ) − ζ u( ) + η + (N − 1)ζ(η) + ζ ∓(γ )α(γ ) , σ (0) j j l l j=1 γ =± l=1 (5.20) { (±)} where the parameters uj respectively satisfy the Bethe Ansatz equations (5.19). Some remarks are in order. Firstly, in [18] a generalized algebraic Bethe Ansatz [19] was used to diagonalize the open XYZ spin chain when boundary parameters satisfying a constraint. However, only partial eigenvalues of transfer matrix corresponding to our Λ(+)(u) were obtained there and it is not clear whether the approach in [18] can give all the eigenvalues of the transfer matrix. (See also [20], where partial eigenvalues of the transfer matrix of the open XXZ spin chain were obtained.) Indeed, it is found in this paper that both Λ(±)(u) are needed to constitute a complete set of eigenvalues. This implies that two sets of Bethe Ansatz equation (5.19) and therefore presumably two pseudo-vacua are required. Unfortunately, it is not yet clear how to construct the second pseudo-vacuum in the generalized algebraic Bethe Ansatz approach. Sec- 1 ondly, when the crossing parameter η is equal to p+1 with p being a non-negative integer (i.e. roots of unity case), the corresponding fusion hierarchy truncates as in the trigonometric case [14], and one may generalize the method developed in [15] to obtain the corresponding functional relations obeyed by the transfer matrices and therefore the eigenvalues of the fundamental transfer matrix t(u). If the boundary parameters satisfy the constraint (5.16), the complete set of eigenvalues and the associated Bethe Ansatz equations, in the roots of unity case, are expected to be still given by (5.18) and (5.19), respectively. Thirdly, it would be interesting to determine the conditions for which the limit (1.3) exists. We have seen that (5.18)–(5.19) solve the open XYZ chain with generic values of η if the constraint (5.16) is satisﬁed. This suggests that, for generic values of η, the constraint (5.16) may be a necessary condition for the existence of the limit (1.3). It would be interesting to explicitly evaluate the Q-operator directly from Eq. (1.3).

6. Conclusions

We have argued that the Baxter’s Q-operator for the open XYZ chain is given by the j →∞ limit of the transfer matrix with spin-j auxiliary space. This together with the fusion hierarchy leads to the Baxter’s T –Q relations (3.21) and (3.22). These T –Q relations, together with the 324 W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 additional properties (5.1)–(5.5), enable us to successfully determine the eigenvalues (5.18) of the fundamental transfer matrix and the associated Bethe Ansatz equations (5.19). For a given set of bulk and boundary parameters satisfying the constraint (5.16), the eigenvalues Λ(−)(u) and Λ(+)(u) together constitute the complete set of eigenvalues of the fundamental transfer matrix t(u), which leads to the complete spectrum (5.20) of the Hamiltonian (1.1). Our derivation for the Q-operator and associated T –Q relation can be applied to other models which share sl2- like fusion rule. Moreover, it would be interesting to generalize our method to integrable models associated with higher rank algebras [21–23].

Acknowledgements

We thank Ryu Sasaki for critical comments and useful discussions. Financial support from the Australian Research Council is gratefully acknowledged.

Appendix A. Determinants and crossing relations

In this appendix, we compute the determinants (3.11) and (3.12) of the K-matrices. In so doing, we will also prove the crossing relations (4.11) and (4.12). One rewrites K−(u) in (2.15) as 3 − α 0 1 x 2 y 3 z K (u) = fα(u)σ ,σ= I, σ = σ ,σ = σ ,σ = σ . α=0

It is easy checked that the coefﬁcient function {fα(u)} satisfy

f0(−u) = f0(u), fi(−u) =−fi(u), for i = 1, 2, 3. The above parity relations lead to the following unitarity relation of K−(u) 3 − − K (u)K (−u) = Δ2(u) × id,Δ2(u) = fα(u)fα(−u). (A.1) α=0 From the expression (2.15) of K−(u) and after direct calculation, we ﬁnd that − − 3 σ(α( ) + u)σ (α( ) − u) Δ (u) = l l . 2 (−) (−) (A.2) l=1 σ(αl )σ (αl ) − =− (−) We now follow a method similar to that developed in [24]. Noting R1,2( η) 2P12 and PT-symmetry (2.7) of the R-matrix, the reﬂection equation (2.11), when u1 = u, u2 = u + η, reads (−) − + − + = − + + − (−) P12 K1 (u)R1,2(2u η)K2 (u η) K2 (u η)R1,2(2u η)K1 (u)P12 . (−) Since the dimension of the image of P12 is equal to one, the above equation becomes − − (−) (−) − − (−) K (u + η)R1,2(2u + η)K (u)P = P K (u)R1,2(2u + η)K (u + η)P 2 1 12 12 1 2 12 = (−) − + − + (−) tr12 P12 K1 (u)R1,2(2u η)K2 (u η) P12 (3=.11) − + (−) Det K (u η) P12 . (A.3) W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 325

It is well known that the completely antisymmetric vector with unit normal in C2 ⊗ C2 is 1 1 2 √ |1 ⊗|2 −|2 ⊗|1 = (id ⊗V) √ |i ⊗|i , (A.4) 2 2 i=1 where {|1 , |2 } are the orthonormal basis of C2. Acting both sides of (A.3) on the vector (A.4) and after straightforward calculating, we ﬁnd that (A.3) is indeed equivalent to − − − − K¯ (u) = Det K (u + η) VK (u + η)V 1. Using (A.1),wehave − − Det{K (u + η)} − K¯ (u) = VK (−u − η)V. (A.5) Δ2(u + η) Taking trace of both sides of (A.5), we have, for the L.H.S. of the resulting relation, − L.H.S. = tr K¯ (u) (4=.9) − − − t2 tr12 P12R1,2( 2u 2η) K2 (u) − = tr tr P R (−2u − 2η) K (u) t2 2 1 12 1,2 2 = − − + − − − a( 2u 2η) c( 2u 2η) tr2 K2 (u) σ(2u) = a(−2u − 2η) + c(−2u − 2η) σ(u) σ(2u)σ (2u + 2η) =− . (A.6) σ(η)σ(u+ η) In the above deriving, we have used the following relations: tr1 P12R1,2(u) = a(u) + c(u) × id, σ(2u + 2η)σ(u) a(−2u − 2η) + c(−2u − 2η) =− , σ(η)σ(u+ η) which can be obtained from (2.3)–(2.5), and the following identities of the elliptic functions: 1 1 σ(u)σ(u+ 1 ) θ 2 (2u, 2τ)= θ 2 (τ, 2τ)× 2 , 1 1 τ 1 + τ 2 2 σ(2 )σ ( 2 2 ) τ 1 τ 0 0 σ(u− )σ (u + + ) θ (2u, 2τ)= θ (0, 2τ)× 2 2 2 . 1 1 − τ 1 + τ 2 2 σ( 2 )σ ( 2 2 ) On the other hand, for the R.H.S. of the resulting relation, we obtain − Det{K (u + η)} − R.H.S. = tr VK (−u − η)V Δ2(u + η) − Det{K (u + η)} − = tr V 2K (−u − η) Δ2(u + η) − Det{K (u + η)} − =− tr K (−u − η) Δ2(u + η) 326 W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329

Det{K−(u + η)} σ(2u + 2η) =− . (A.7) Δ2(u + η) σ(u+ η) Comparing with (A.6) and (A.7),wehave

Det{K−(u + η)} σ(2u) = . (A.8) Δ2(u + η) σ(η) Now (3.13) follows from (A.2) and (A.8). Similarly one can prove (3.14) using (2.16). Finally, (A.5) and (A.8) give rise to (4.11), and (4.12) can be similarly be proven from (2.16).

Appendix B. Constraint condition (5.16)

One can easily check that both forms (5.15) of eigenvalues of t(u) satisfy the ﬁrst quasi- periodicity of (5.2) as required. Now we investigate the second quasi-periodic property. The { (±) |{ (γ )} } deﬁnitions (5.6) and (5.7) of Hi (u l ) imply

(γ ) (γ ) (±) (γ ) − iπ((N+ )( u+τ)+ η) −2iπ(± 3 α ) (±) (γ ) H u + τ = e 2 3 2 4 e γ =± l=1 l l H u , 1 l 1 l (±) + (γ ) H2 u τ l − + + + + − ∓ 3 (γ ) (γ ) (±) (γ ) = 2iπ((N 3)(2u τ) (2N 2)η) 2iπ( γ =± l=1 l αl ) e e H2 u l . Keeping the above quasi-periodic properties and the deﬁnition (3.20) of Q(u) in mind, in order { (γ )} that (5.15) have the second property of (5.2) one needs that the boundary parameters αl satisfy the following constraint

3 (γ ) (γ ) = l αl kη mod(1), (B.1) γ =± l=1 where k is an integer such that |k| N − 1 and N − 1 + k being even. For such integer k, (±) (±) = 1 − ∓ let us introduce two non-negative integers M : M 2 (N 1 k). Moreover, associated (±) |{ (γ )} (±) respectively with Hi (u l ), one introduces Q (u),

± M( ) (±) = − (±) + (±) + Q (u) σ u uj σ u uj η . j=1 Hence the eigenvalues of the transfer matrix take the following forms:

(±) (±) ± Q (u + η) ± Q (u − η) H ( ) u (γ ) + H ( ) u (γ ) . (B.2) 1 l Q(±)(u) 2 l Q(±)(u) { (γ )} At this stage, we still have freedom to chose the discrete parameters l . Now we apply the semi-classical property (5.5) of Λ(u) to obtain a restriction to this freedom. For this purpose, let us introduce Λ(±)(u), corresponding to (B.2),

(±) (±) ± ± Q (u + η) ± Q (u − η) Λ( )(u) = H ( ) u (γ ) + H ( ) u (γ ) , (B.3) 1 l Q(±)(u) 2 l Q(±)(u) W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 327

¯ (γ ) = (γ ) { (γ )} and deﬁne αl limη→0 αl . Since the value of the discrete parameters l does not change when taking the limit of η → 0, the constraint (B.1) in this limit becomes

3 (γ ) ¯ (γ ) = l αl 0mod(1). (B.4) γ =± l=1 Substituting (B.3) into the semi-classical property (5.5) of the eigenvalues of the fundamental transfer matrix t(u), we have for the L.H.S. of the resulting relation, ± L.H.S. = lim σ 2N (η)Λ( )(u) η→0 3 σ(u± (γ )α¯ (γ )) 3 σ(u∓ (γ )α¯ (γ )) = σ 2N (u) l l + l l . (γ ) ¯ (γ ) (γ ) ¯ (γ ) γ =± l=1 σ(l αl ) γ =± l=1 σ(l αl ) Comparing both side of the resulting relation, (5.5) is equivalent to the following relation 3 (γ ) (γ ) 3 (γ ) (γ ) σ(u± α¯ ) σ(u∓ α¯ ) + − l l + l l = lim tr K (u)K (u) . (B.5) (γ ) ¯ (γ ) (γ ) ¯ (γ ) η→0 γ =± l=1 σ(l αl ) γ =± l=1 σ(l αl ) {¯(γ )} Keeping the constraint (B.4) among αl , after straightforward calculation, we find that (B.5) { (γ )} is satisfied if and only if the following constraint among the discrete parameters l were fulfilled 3 (γ ) =− l 1. (B.6) γ =± l=1 (B.1) and (B.6) are nothing but the constraints listed in (5.16).

Appendix C. Trigonometric limit

In this appendix, we consider the trigonometric limit τ →+i∞ of our result. The deﬁnition of the elliptic functions (2.1)–(2.2) imply τ −iπ(u+ 1 + τ ) 0 σ u + = e 2 4 θ 1 (u, τ), (C.1) 2 2 and the following asymptotic behaviors

iπτ lim σ(u)=−2e 4 sin πu+···, (C.2) →+ ∞ τ i 0 lim θ (u, τ) = 1 +···. (C.3) →+ ∞ 1 τ i 2 The above asymptotic behaviors lead to the well-known result: ⎛ ⎞ sin π(u+ η) 1 ⎜ sin πu sin πη ⎟ lim R(u) = ⎝ ⎠ . (C.4) τ→+i∞ sin πη sin πη sin πu sin π(u+ η) 328 W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329

After reparameterizing the boundary parameters

(∓) (∓) 1 (∓) 1 τ α =±α∓,α=∓β∓ + ,α=−θ∓ + + , (C.5) 1 2 2 3 2 2 and using (C.1)–(C.3), one has − − lim K (u) ≡ K (u) τ→+i∞ (t) 1 = 2sinπα− cos πβ− cos πuI 2sinπα− cos πβ− z x y + 2i cos πα− sin πβ− sin πuσ + sin 2πu cos πθ−σ − sin πθ−σ , (C.6) + ≡ + = − − − lim K (u) K (u) K ( u η) → − − . (C.7) τ→+i∞ (t) (t) (α−,β−,θ−) ( α+, β+,θ+) Hence our R-matrix and K-matrices, when taking the trigonometric limit, reduce to the corresponding trigonometric ones used in [5] after a rescaling of the corresponding parameters. In order for the constraint (5.16) to be still satisﬁed after the limit, one needs further to require that (−) =− (+) { (±) |{ (γ )} } 3 3 . Then, the resulting constraint becomes that in [5]. Moreover, our Hi (u l ) in the limit give rise to the corresponding trigonometric ones in [5] up to some constants (due to the constant factors appearing in (C.4), (C.6)–(C.7)). Therefore, our result recovers that of [5] in the trigonometric limit.

References

[1] R.J. Baxter, Phys. Rev. Lett. 26 (1971) 832. [2] R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, 1982. [3] V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov, Commun. Math. Phys. 190 (1997) 247; A. Antonov, B. Feigin, Phys. Lett. B 392 (1997) 115; G.P. Pronko, Commun. Math. Phys. 212 (2000) 687; M. Rossi, R. Weston, J. Phys. A 35 (2002) 10015. [4] D.E. Derkachov, G.P. Korchemsky, A.N. Manashov, JHEP 0310 (2003) 053; R.J. Baxter, J. Stat. Phys. 114 (2004) 43; C. Korff, J. Phys. A 38 (2005) 6641. [5] W.-L. Yang, R.I. Nepomechie, Y.-Z. Zhang, Phys. Lett. B 633 (2006) 664, hep-th/0511134. [6] P.A. Pearce, Phys. Rev. Lett. 58 (1987) 1502; P.A. Pearce, K.A. Seaton, Phys. Rev. Lett. 60 (1988) 1347; G.E. Andrews, R.J. Baxter, P.J. Forrester, J. Stat. Phys. 28 (1982) 1; M. Kashiwara, T. Miwa, Nucl. Phys. B 275 (1986) 121. [7] E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, fourth ed., Cambridge Univ. Press, Cambridge, 2002. [8] E.K. Sklyanin, J. Phys. A 21 (1988) 2375. [9] I.V. Cherednik, Theor. Math. Phys. 61 (1984) 35. [10] T. Inami, H. Konno, J. Phys. A 27 (1994) L913; B.Y. Hou, K.J. Shi, H. Fan, Z.-X. Yang, Commun. Theor. Phys. 23 (1995) 163. [11] M. Karowski, Nucl. Phys. B 153 (1979) 244; P.P. Kulish, N.Yu. Reshetikhin, E.K. Sklyanin, Lett. Math. Phys. 5 (1981) 393; P.P. Kulish, E.K. Sklyanin, Lecture Notes in Physics, vol. 151, Springer, 1982, p. 61; A.N. Kirillov, N.Yu. Reshetikhin, J. Sov. Math. 35 (1986) 2627; A.N. Kirillov, N.Yu. Reshetikhin, J. Phys. A 20 (1987) 1565. [12] L. Mezincescu, R.I. Nepomechie, J. Phys. A 25 (1992) 2533. [13] Y.K. Zhou, Nucl. Phys. B 458 (1996) 504, hep-th/9510095. [14] V.V. Bazhanov, S.L. Lukyanov, A.B. Zamolodchikov, Commun. Math. Phys. 177 (1996) 381, hep-th/9412229; A. Kuniba, K. Sakai, J. Suzuki, Nucl. Phys. B 525 (1998) 597. W.-L. Yang, Y.-Z. Zhang / Nuclear Physics B 744 [FS] (2006) 312–329 329

[15] R.I. Nepomechie, J. Phys. A 34 (2001) 9993, hep-th/0110081; R.I. Nepomechie, Nucl. Phys. B 622 (2002) 615, hep-th/0110116; R.I. Nepomechie, J. Stat. Phys. 111 (2003) 1363, hep-th/0211001; R.I. Nepomechie, J. Phys. A 37 (2004) 433, hep-th/0304092. [16] L. Mezincescu, R.I. Nepomechie, Nucl. Phys. B 372 (1992) 597; R.E. Behreud, P. Pearce, D.L. O’Brien, J. Stat. Phys. 84 (1996) 1. [17] R.I. Nepomechie, F. Ravanini, J. Phys. A 36 (2003) 11391, hep-th/0307095; R.I. Nepomechie, F. Ravanini, J. Phys. A 37 (2004) 1945, Addendum. [18] H. Fan, B.Y. Hou, K.J. Shi, Z.-X. Yang, Nucl. Phys. B 478 (1996) 723. [19] R.J. Baxter, Ann. Phys. (N.Y.) 76 (1973) 1; L.D. Faddeev, L.A. Takhtajan, Russian Math. Surveys 34 (1979) 11. [20] J. Cao, H.-Q. Lin, K.-J. Shi, Y. Wang, Nucl. Phys. B 663 (2003) 487. [21] W.-L. Yang, R. Sasaki, Nucl. Phys. B 679 (2004) 495, hep-th/0308127; W.-L. Yang, R. Sasaki, Y.-Z. Zhang, JHEP 0409 (2004) 046, hep-th/0409002. [22] W.-L. Yang, Y.-Z. Zhang, JHEP 0412 (2004) 019, hep-th/0411160; W.-L. Yang, Y.-Z. Zhang, JHEP 0501 (2005) 021, hep-th/0411190; W.-L. Yang, Y.-Z. Zhang, R. Sasaki, Nucl. Phys. B 729 (2005) 594, hep-th/0507148. [23] A. Doikou, Nucl. Phys. B 668 (2003) 447; A. Doikou, Nucl. Phys. B 725 (2005) 493; W. Galleas, M.J. Martins, Phys. Lett. A 335 (2005) 167; A. Nichols, J. Stat. Mech. 0509 (2005) P009; D. Arnaudon, N. Crampe, A. Doikou, L. Frappat, E. Ragoucy, math-ph/0512037. [24] B.Y. Hou, K.-J. Shi, W.-L. Yang, Commun. Theor. Phys. 30 (1998) 415, hep-th/9905202. Nuclear Physics B 744 [FS] (2006) 330–339

High temperature limit of the order parameter correlation functions in the quantum Ising model

S.A. Reyes, A.M. Tsvelik ∗

Department of Physics and Astronomy, SUNY at Stony Brook, Stony Brook, NY 11794-3840, USA Department of Condensed Matter Physics and Materials Science, Brookhaven National Laboratory, Upton, NY 11973-5000, USA Received 6 February 2006; received in revised form 21 March 2006; accepted 22 March 2006 Available online 5 April 2006

Abstract In this paper we use the exact results for the anisotropic two-dimensional Ising model obtained by Bugrii and Lisovyy [A.I. Bugrii, O.O. Lisovyy, Theor. Math. Phys. 140 (2004) 987] to derive the expressions for dynamical correlation functions for the quantum Ising model in one dimension at high temperatures. © 2006 Elsevier B.V. All rights reserved.

PACS: 71.10.Pm; 72.80.Sk

1. Introduction

The previous attempts to calculate correlation functions of integrable models has mostly concentrated on the low energy (temperature) limit where ﬁeld theory methods can be applied. In that case calculations are simpliﬁed due the emerging Lorentz symmetry and conformal symmetry (at criticality). Since such symmetries are absent for lattice theories at large temperatures, much less is known about the high temperature dynamics. In this paper we discuss the high temperature limit of the two-point correlation functions in the one-dimensional quantum Ising (QI) model. Though this model has been extensively studied [1], the high temperature asymptotics are known only for the one site correlation function [2].We extract these asymptotics from the general expressions for the correlation functions derived for the two-dimensional anisotropic classical Ising model (IM) in [4]. To translate them into quantum

* Corresponding author. E-mail address: [email protected] (A.M. Tsvelik).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.03.015 S.A. Reyes, A.M. Tsvelik / Nuclear Physics B 744 [FS] (2006) 330–339 331 language, we will exploit the well-known correspondence between a certain scaling limit of the two-dimensional classical IM and the one-dimensional QI model [3]. The energy of the 2D classical IM is, E =− Kxσ(rx,ry)σ (rx + 1,ry) + Kyσ(rx,ry)σ (rx,ry + 1) . (1) rx ,ry The critical line in the anisotropic 2D Ising model is determined by the equation

sinh 2Kx sinh 2Ky = 1. (2) Let us choose the y direction as the (imaginary) time direction. Then one-dimensional QI model arises when the following continuum limit in the y direction (Ky →∞) is taken:

−2Ky τ = ryτ0,ry →∞,τ0 = e ,

sinh 2Kx sinh 2Ky = λ. (3) In this limit the classical model gives rise to the following quantum Hamiltonian: 1 H =−J σ z(r )σ z(r + 1) + σ x(r ) . (4) x x λ x rx More speciﬁcally, the correlators of the order parameter ﬁeld in the classical model correspond z to correlators of the quantum operator σ (rx,τ).

2. Quantum limit of the correlation functions

It was shown in [4] that the 2D anisotropic Ising correlator in the ferromagnetic phase (λ>1) for the model of inﬁnite size in the x direction and length N in the y-direction, can be written in the following form: N/2 −1 −|rx |Λ σ(0, 0)σ (rx,τ) = ξξT e g2l(rx,τ) (5) l=0 where, − −1 2 n −| | ¯ + ¯ − ¯ e nΛ t (1 − t2) n /2 e rx γ(qj ) iτqj η(qj ) g = y x F 2[¯q], n n 2 n (6) n!N tx(1 − t ) sinh γ(q¯j ) y q¯ j=1 n 2 n2−n n 2 sin ((q¯iτ −¯qj τ )/2) τ (q¯i −¯qj ) F 2[¯q]= 0 0 = 0 (7) n 2 + 2 2 + i

−2 1/2 −1 ξ = 1 − λ , qτ¯ 0 = q, q¯ = 2πTm T = β = Nτ0 . (8) It is convenient to take the continuum limit as early as possible in the calculation, so we write the following, 2 ty(1 − t ) sinh 2Ky 1 x = → , (9) − 2 2 tx(1 ty ) sinh 2Kx 4λτ0 332 S.A. Reyes, A.M. Tsvelik / Nuclear Physics B 744 [FS] (2006) 330–339

cosh 2Ky cosh 2Kx − sinh 2Ky sinh 2Ky cosh γ(q) = + 2sin2(q/2) sinh 2Kx sinh 2Kx 2 q¯ 1 − → + λ 1 + λ , (10) 8λ 2

cosh 2Kx cosh 2Ky − sinh 2Kx sinh 2K cosh γ(q)¯ = + 2sin2(q/2) x sinh 2K sinh 2K y y 2λ γ(q)¯ 2 → 1 + 2 1 + λ2 1 − cos(q) τ 2 +···=1 + +··· (11) 1 + λ2 0 2 then, √ Nγ(q)¯ 1 + λ2 2λ 1/2 → 1 − cos(q) , (12) 2 T 1 + λ2 where the arrows indicate that the continuum limit has been taken. In other notations, q¯2 q¯2 sinh(γ /2) = M2 + , cosh(γ /2) = 1 + M2 + , 16λ 16λ 1 √ √ M2 = λ − 1/ λ 2. (13) 4 In the continuum (that is quantum) limit (6) becomes 2 n −| | ¯ + ¯ − ¯ n T n 1 n /2 e rx γ(qj ) iτqj α(qj ) (q¯ −¯q )2 g = i j , n 2 (14) n! 16λ sinh(γj /2) cosh(γj /2) ((γ + γ )/ ) q¯ j=1 i 1, (22) c ± cos x c2 − 1 0 π log(1 − 2a cos x) − 2 1+a2 = 2bπ (1 ab) 2 2 dx + 2 log ,a 1,b < 1 (23) ( 1 b − cos x) 1 − b2 1 + a2 0 2b we obtain

T − − ¯ α(q)¯ ≈ log − log 1 − λ 1e γ(q) . (24) λ Substituting these results into (14) we obtain n2/2 n −| | + ¯ λn 1 e rx γj iτqj −1 −γj gn ≈ 1 − λ e n! 16λ sinh(γj /2) cosh(γj /2) q¯ j=1 n 2 (q¯i −¯qj ) × . (25) 2 + i

3. Real time asymptotics

To study the real time properties of the correlation function we will need to transform this sum into a n-variable integral in the complex plane, as it was done in [6]. The details of this calculation are given in Appendix A. The ﬁnal result is |rx | λ − | | σ z(r ,t)σz(0, 0) ∼ θ(t) e R(t, rx ,λ), (26) x T where R isgivenbyEq.(A.18). It can also be expressed in terms of the dispersion of fermions (solitons) in the 1D quantum Ising model (4) 2J (k,λ) = 1 + λ2 − 2λ cos ka 1/2, (27) λ 334 S.A. Reyes, A.M. Tsvelik / Nuclear Physics B 744 [FS] (2006) 330–339 where a is the lattice spacing. Considering this and restoring dimensions according to (19),the result for R(t,|rx|,λ)can then be written as π/a 2 R(t,|r |,λ)= dk v(k,λ)t −|r |a θ v(k,λ)t −|r |a x π x x −π/a π/a 1 = dk x − v(k,λ)t −|x| , (28) π −π/a where x =|rx|a, and, d (k,λ) Jasin ka v(k,λ) = = √ . (29) dk λ(M2 + sin2(ka/2))1/2 It can be easily shown that, 2Ja max v(k,λ) = (30) λ for any value of λ. This means that, R(t,|rx|,λ) vanishes at t<λ|rx|/2J and has the following form: t 2Jt R t,|rx|,λ = tf ,λ θ −|rx| . (31) rx λ Hence, there is no time dependence in the long distance correlator at t<λ|rx|/2J . An analytic expression can be obtained for R, 2t + − |r |a + − 2Jt R t,|r |,λ = (k ,λ)− (k ,λ)− x (k − k ) θ −|r | , (32) x π 0 0 t 0 0 λ x where, 2 4 2 ± |r | |r | |r | cos k a = λ x ∓ λ2 x − 1 + λ2 x + 1, 0

× θ 2Jt −|rx| . (34)

Notice that the only difference in the time dependence in both regimes is that of a rescaling of time (t → t/λ). Remember now that all the results were obtained for the ferromagnetic phase (λ>1). For the paramagnetic phase, Eq. (5) becomes a sum over g2l+1. It turns out that this change has a little effect over the high temperature correlator. In fact, the form of R does not change and one can easily derive the following result for λ<1, |rx | J − | | J σ z(r ,t)σz(0, 0) ∼ θ(t) e R(t, rx ,λ),T . (36) x λT λ The “high temperature regime” has to be redeﬁned because in the paramagnetic phase the transverse ﬁeld becomes larger than the spin–spin coupling.

4. Conclusions

Let us now discuss qualitative features of Eq. (26). As expected, at zero time we get the standard result for the one-dimensional classical Ising model: z z |rx | σ (rx, 0)σ (0, 0) ∼ (J/T ) . (37)

At least in the framework of the adopted approximation |rx|1, this result persists at nonzero x time at times smaller than λ|rx|/2J . Recall that J/λ is the coefﬁcient in front of σ in the Hamiltonian. Without this term there is no dynamics. Hence the obtained time dependence is a quantum effect, as one expects. If 2tJ λ|rx| one gets the following approximation: T 8Jt λ|r |2 σ z(r ,t)σz(0, 0) ∼ θ(t)exp −|r | log − 2 − − x . (38) x x J πλ πJt

This is different from the Gaussian in t behavior which holds for T =∞for |rx|=0 [2].

Acknowledgements

This research was supported by US DOE under contract numbers DE-AC02-98 CH 10886. We are grateful to B. McCoy, R. Schrock and V.A. Fateev for valuable discussions. S.A.R. is grateful to R.M. Konik for help.

Appendix A. Transformation of sum (25) into a contour integral

In this appendix we demonstrate how to derive Eq. (26). We do it in a manner similar to the one described in [6], namely by transforming the sum (25) into a contour integral. In what follows we will take advantage of the following identities: 2 ¯2 ¯2 2 γi + γj q¯ qj qj q¯ sinh = M2 + i 1 + M2 + + M2 + 1 + M2 + i , (A.1) 2 16λ 16λ 16λ 16λ ¯2 ¯2 qj qj sinh γ = 2 M2 + 1 + M2 + , (A.2) j 16λ 16λ 336 S.A. Reyes, A.M. Tsvelik / Nuclear Physics B 744 [FS] (2006) 330–339 γ q¯2 j 2 j sinh = M + =−isgn( e αj ) cosh αj , (A.3) 2 16λ ¯2 γj qj cosh = 1 + M2 + =−isinhα , (A.4) 2 16λ j 2 2 q¯j =−sgn( e αj )4i λ cosh αj + M (A.5) with, ¯2 π − qj α = i ± sinh 1 M2 + . (A.6) j 2 16λ In order to be able to simplify notation we deﬁne the following, | | − [sinh α − sgn( e α)cosh α]2 rx {1 + λ 1[sinh α − sgn( e α)cosh α]2} F(α)≡−i √ , (A.7) esgn( e α)4β λ(cosh2 α+M2) − 1 cosh2 α + M2

2 2 2 2 2 sgn( e αi)i cosh αi + M − sgn( e αj )i cosh αj + M 2 TAN (αi,αj ) ≡ . sgn( e αi) cosh αi sinh αj + sgn( e αj ) cosh αj sinh αi (A.8) The discontinuity generated by sgn( e α) arises because we are choosing different values of the square root at each side of the imaginary axis. The points (A.6) in the plane, are the poles of (A.7).Theresult(25) can now be written as a multivariable contour integral: 2l √ λ 1 2 + 2 g ≈ dα F(α )e4τ sgn( e α1) λ(cosh α1 M ) ··· 2l 2πT (2l)! 1 1 C √ 2 2 4τ sgn( e α2l ) λ(cosh α2l +M ) 2 × dα2l F(α2l)e TANH (αi,αj ). (A.9) C i

We are interested in the situation when |rx|1, so the exponential in F makes the integrand decrease rapidly with |α|. This means that terms in g2l that include integrals along these contours will be at least of order O(1/|rx|) (smaller) when compared with terms that do not include them. S.A. Reyes, A.M. Tsvelik / Nuclear Physics B 744 [FS] (2006) 330–339 337

Fig. 1. The ﬁgure shows the integration contours in the α plane. The circles are the poles of F at αj given by (A.6).

Then, to calculate the leading term it is safe to neglect the part of the integral contained in these contours and keep only the contours C2 and C5. To evaluate the integrals along these contours we deﬁne:

∓i2|r |α −1 ∓i2α ± −ie x (1 + λ e ) F (α) ≡ √ √ , (A.12) e±4β λ(cos2 α+M2) − 1 cos2 α + M2 2 2 2 2 2 cos αi + M − cos αj + M 2 TAN±(αi,αj ) ≡ , (A.13) sin(αi + αj ∓ i2δ) cos2 α + M2 + cos2 α + M2 2 i j 2(α ,α ) ≡ . COTAN i j − − (A.14) sin(αi αj i2δ)

+ − In Eq. (A.12) F (F ) gives the value of F in (A.7) along C5 (C2). We can replace now τ for it to get the following approximation,

π 2l 2l p √ λ 1 + 2 + 2 g ≈ dα F (α )eit4 λ(cos αi M ) 2l ! − ! i i 2πT = p (2l p) = p 0 0 i 1 2 l−p −it4 λ(cos2 α +M2) × − j 2 dαj F (αj )e TAN+(αi,αk) j=1 i>k

× 2 2 TAN−(αj ,αm) COTAN (αi,αj ). (A.15) j>m i,j

The denominator of F ± can be easily simpliﬁed if we use the obvious approximation for β 1/λ. Let us then write down the result explicitly: 338 S.A. Reyes, A.M. Tsvelik / Nuclear Physics B 744 [FS] (2006) 330–339 √ √ π 2l 2l p − | | + 2 + 2 − − λ 1 e i2 rx αi it4 λ(cos αi M )(1 + λ 1e i2αi ) g ≈ dα 2l ! − ! i 2 + 2 8π = p (2l p) = cos αi M p 0 0 i 1 − | | − 2 + 2 2 l p i2 rx αj it4 λ(cos αj M ) −1 i2α e (1 + λ e j ) × dα j cos2 α + M2 j=1 j 2 + 2 − 2 + 2 2 × cos αi M cos αk M sin(α + α − i2δ) i>k i k 2 + 2 − 2 + 2 2 cos α M cos αm M × j sin(α + α + i2δ) j>m j m 2 + 2 + 2 + 2 2 cos αi M cos αj M × . (A.16) sin(α − α − i2δ) i,j i j

We are interested in |rx|1 and in this case the main contribution to the integral comes from − = ± regions around the divergencies. There are divergencies that originate from αi αj 0, π. Let us take into account only the terms in g2l which are maximally singular. This maximal singularity occurs when p = l and in regions where every α pairs with an α in order to approach a singularity. There are 2ll! of this regions. Finally we get the following approximation, √ √ π ∞ √ − | | + 2 + 2− 2 − + 2 1 λ e i4 rx 4it λ( cos α M cos (α 2 ) M ) g l ≈ dα d 2 l! 32π 2 ( − i0)2 0 −∞ √ l + −1 −i2α + −1 i2(α−2 ) 2 + 2 + 2 − + 2 2 × (1 λ e )(1 λ e )( cos α M cos (α 2 ) M ) (cos2 α + M2)(cos2(α − 2 ) + M2)

π √ 1 λ (1 + λ−1e−i2α)(1 + λ−1ei2α) 2t λ sin α cos α = − dα √ −|rx| l! π cos2 α + M2 cos2 α + M2 0 √ l 2t λ sin α cos α × θ √ −|rx| cos2 α + M2 1 ≡ −R t,|r |,λ l, (A.17) l! x where

π √ √ 4 2t λ sin α cos α 2t λ sin α cos α R ≡ dα √ −|rx| θ √ −|rx| . (A.18) π cos2 α + M2 cos2 α + M2 0 And considering that, S.A. Reyes, A.M. Tsvelik / Nuclear Physics B 744 [FS] (2006) 330–339 339

− T Λ 1 ≈ log (A.19) λ we get the ﬁnal result for the dynamical (real time) correlation function in the form of (26).

References

[1] One can ﬁnd a summary of the early results on the Ising model in: J.H.H. Perk, H.W. Capel, G.R.W. Quispel, F.W. Nijhoff, Physica A 123 (1984) 1. [2] M. Mohan, Phys. Rev. B 21 (1980) 1264; M. Mohan, Phys. Rev. B 23 (1981) 433. [3] E. Fradkin, L. Susskind, Phys. Rev. D 17 (1978) 2637. [4] A.I. Bugrii, O.O. Lisovyy, Theor. Math. Phys. 140 (1) (2004) 987. [5] I.S. Gradshteyn, I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, London, 1979. [6] B.L. Altshuler, R.M. Konik, A.M. Tsvelik, cond-mat/0508618. Nuclear Physics B 744 [FS] (2006) 340–357

AnewlargeN phase transition in YM2

A. Apolloni, S. Arianos, A. D’Adda ∗

Dipartimento di Fisica Teorica dell’Università di Torino and Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via P. Giuria 1, I-10125 Torino, Italy Received 31 August 2005; received in revised form 22 February 2006; accepted 22 March 2006 Available online 6 April 2006

Abstract Inspired by the interpretation of two-dimensional Yang–Mills theory on a cylinder as a random walk on the gauge group, we point out the existence of a large N transition which is the gauge theory analogue of the cutoff transition in random walks. The transition occurs in the strong coupling region, with the ’t Hooft coupling scaling as α log N, at a critical value of α (α = 4 on the sphere). The two phases below and above the transition are studied in detail. The effective number of degrees of freedom and the free energy are found 2− α to be proportional to N 2 below the transition and to vanish altogether above it. The expectation value of a Wilson loop is calculated to the leading order and found to coincide in both phases with the strong coupling value. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

Yang–Mills theory in two dimensions is at the intersection of many different ﬁelds of theoretical physics. It is one example of non-trivial completely solvable gauge theory [1,2], in which both perturbative and non-perturbative effects can be studied. The interpretation of YM2 in terms of string theories, namely a theory of branched coverings on a two-dimensional Riemann surface, was discovered for large N in [3], and for ﬁnite N in [4], for the intersecting Wilson loop on the plane. Its large N expansion has then been proved to describe a two-dimensional string theory on generic two-dimensional manifolds [5,6]. Non-trivial topological sectors in the unitary gauge also seem to be related to matrix string states [7]. Its partition function on a torus can be de-

* Corresponding author. E-mail addresses: [email protected] (A. Apolloni), [email protected] (S. Arianos), [email protected] (A. D’Adda).

0550-3213/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2006.03.017 A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 341 scribed in terms of a gas of free fermions [13–16], and the kernel on a cylinder by the evolution of a system of N free fermions on a circle, namely by the Sutherland model. Some new connections between two-dimensional gauge theories and statistical mechanical systems were pointed out in [8] where two-dimensional gauge theories of the symmetric group Sn in the large n limit were investigated. Gauge theories of Sn also describe n-coverings of a Riemann surface and hence they are closely related to two dimensional Yang–Mills theories; the relation being essentially provided by Frobenius formula that relates U(N) characters (in a representation with n boxes in the Young diagram) to the corresponding characters of the Sn group. It was shown in [8] that the partition function of a gauge theory on a disc or a cylinder can be interpreted in terms of random walks on the gauge group, whose initial and final positions are the holonomies at the ends of the cylinder1 and the number of steps is the area of the surface. Although the focus in [8] was on the discrete group Sn the argument can be trivially extended to continuous Lie groups. Similar results were independently obtained in [19–21]. It is well known in random walks theory that after a certain number of steps the end point of the walk becomes independent of the starting point: the walker has lost any memory of the point he started from. The critical number of steps after which that happens can be exactly calculated in a number of situations, and the corresponding transition is known as cutoff transition. Given the correspondence between random walks on the group and gauge theory on a cylinder, one expects to find the cutoff transition also in gauge theories. Indeed it was found in [8] that for an Sn gauge theory where the holonomy on each elementary plaquette is given by a single transposition,2 a cutoff transi- 1 tion occurs in the large n limit when the number of plaquettes (and hence the area) is 2 n log n, in agreement with previous results in random walks [17]. Models with more general Boltzmann weight for the plaquettes have a richer structure of phase diagrams [8]. The stringy interpretation of the cutoff transition in the Sn gauge theory is the following: beyond the transition the string world sheet is completely connected in the large n limit, while before the transition the world sheet consists of a large connected part and of a small fraction (in fact vanishing in the large n limit) of disconnected parts. Another well known correspondence relates random walks with random graphs [18], that is graphs obtained by randomly connecting n points with p links. These can be put in correspondence with random walks on Sn made of p steps, each step consisting of a simple transposition. Two types of transitions are known in the large n large p double scaling limit in random graphs: a percolation transition at a critical value β = βc when p = βn and the cutoff transition at α = 1/2 when p = αnlog n. Beyond the transition, namely for p>1/2n log n all the n points are connected whereas before the transition a vanishing fraction of disconnected points survive. It is rather natural at this point to look for a similar transition in the large N limit of U(N) gauge theories. A large N phase transition on a sphere and on a cylinder in two-dimensional Yang–Mills theories—the Douglas–Kazakov phase transition—has been known for quite some time. However this is not a cutoff transition. In a cutoff transition the partition function on a disc for instance becomes independent on the holonomy on the border of the disc, and this is not the case in the Douglas–Kazakov transition. Besides the Douglas–Kazakov transition occurs at a finite value of the ’t Hooft coupling whereas from the previous examples it appears that the cutoff transition occurs when the area scales as log N at large N. From this point of view the Douglas–

1 In a disc the starting point is the identity of the group, and on a sphere both starting and ending points are the identity. 2 In terms of the string interpretation this means that in each plaquette there is a single quadratic branch point connecting two of the n sheets of the world sheet. 342 A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357

Kazakov transition appears more similar to the percolation transition in random graphs, although a precise correspondence is still to be found. The existence of the cutoff transition in the large N limit of 2D Yang–Mills on the sphere when the ’t Hooft coupling scales as log N was proved by a simple argument in [8]. The present paper is devoted to study such transition further on both the sphere and the cylinder, in order to characterize its phases and give some physical interpretation. The paper is organized as follows: in Section 2 we review the large N Douglas–Kazakov transition and its physical interpretation. In Section 3 we introduce the cutoff transition on the sphere and study the phases above and below the transition. Section 4 is devoted to the transition on the disc and on the cylinder and Section 5 to the calculation of the expectation value of Wilson loops.

2. Large N transition

The partition function of a pure gauge theory on an arbitrary orientable two-dimensional manifold M of genus G, p boundaries and area A˜ has been known for many years [1,2]: √ 1 2 μν μ − M d x g Tr F Fμν ZM = DA e 4λ2 − − A 2 2G p − C2(r) = χr (g1) ···χr (gp)dr e 2N . (1) r The sum runs over all irreducible representations of the gauge group, λ is the gauge coupling, χr (gi) is the character of the holonomy gi in the representation r and C2(r) is the quadratic Casimir operator in the representation r. A is related to the actual area of M through A = λ2NA˜. We consider G = 0 manifolds with at most 2 boundaries; i.e. spheres, discs and cylinders. Moreover, we will conﬁne our analysis to the unitary groups U(N) and SU(N). A third order phase transition in the large N limit was discovered in the case of a sphere by Douglas and Kaza- kov [10] at a critical value A = π 2 of the rescaled area A. This transition appears to separate a weak coupling (A<π2) from a strong coupling (A>π2) regime. These results were generalized to the case of a cylinder in [11,12] where the phase transition was also interpreted as a result of instanton condensation. The partition function on the sphere can be written as a sum over the set of integers n1 >n2 > 3 ···>nN that label the irreducible representations of SU(N) and U(N) : A 2 A N 2 − (N −1) 2 − = (ni ) Z = e 24 (ni − nj ) e 2N i 1 . (2) n1>n2>···>nN i

A 4 ρ(λ) = − λ2, (3) 2π A

3 − N−1 In the case of U(N) the extra condition nN 2 must be imposed. 4 This approach for the study of YM2 was ﬁrst used by Rusakov [9]. A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 343

Fig. 1. Eigenvalue distribution for A π2.

Fig. 2. Eigenvalue distribution for A>π2 (i.e. DK transition). where the continuum variables i n x = ,λ(x)= i , N N and the corresponding density of eigenvalues ∂x ρ(λ)= (4) ∂λ have been introduced. In (2) however the would-be eigenvalues ni are distinct integers and as a consequence the corresponding density ρ(λ) in the large N limit is constrained by:

ρ(λ) 1 ∀λ. (5) Hence the Wigner semicircle solution is acceptable only in the weak coupling phase, namely for A π 2 where the condition (5) is fulfilled. In fact the maximum of |ρ(λ)| occurs at λ = 0, it increases with the area A and becomes equal to 1 at A = π 2, as easily seen from (3). The solution in the strong coupling phase A>π2 was found in [10] and is expressed in terms of elliptic integrals. In this phase a finite fraction of the eigenvalues condenses, namely the distribution ρ(λ) is flat and equal to one in a symmetric interval around λ = 0asshowninFigs. 1 and 2.

2.1. Conﬁguration space

It is well known [13–16] that the partition function of two-dimensional Yang–Mills theories with gauge group U(N) can be interpreted in terms of a gas of N free fermions on a circle described by a Sutherland–Calogero model. In particular, if we denote by K2(θ, φ; A) the kernel on a cylinder of scaled area A and with the U(N) holonomies at the two ends given by the 344 A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 invariant angles θi and φi , it was shown that K2(θ, φ; A) can be interpreted as the propagator in a time A from an initial configuration where the positions of the N fermions on the circle are given by θi and a final configuration with positions labeled by φi . The partition function on the sphere of area A is a particular case where the initial and final configurations are just θi = φi = 0for all i’s, namely the amplitude for a process where all fermions start at the origin and come back to the origin after a time A. By a modular transformation on the kernel of the cylinder one finds that the integers ni labeling the irreducible representations of U(N) are just the discrete momenta of the fermions on the circle. While in the momentum representation the Douglas–Kazakov phase transition can be interpreted as fermion condensation, in the configuration representation it can be seen in terms of instantons condensation [11,12]. In fact, while going from the initial θi = 0 configuration to the final φi = 0 configuration, a fermion can in principle wind an arbitrary number of times around the circle. These winding (instantons) configurations do not contribute in the large N limit to the weak coupling phase, as shown by the following simple argument. Consider the Wigner distribution (3) of momenta in the weak coupling phase. The maximum allowed momentum is n = √2 , hence the maximum shift in position for a single fermion in max A the time A is given by 2 √ θmax = A√ = 2 A. (6) A The existence of winding trajectories requires this shift in position to be at least 2π, namely A to be greater of π 2. Hence the critical value of A, where the Douglas–Kazakov phase transition occurs, marks the point where instantons condense, and contribute to the functional integral in the large N limit. A more detailed understanding of the Douglas–Kazakov phase transition can be achieved by introducing in the large N limit the density ρ(θ,t) of fermions in the position θ at a given time (= area) t. Due to the compact nature of the configuration space ρ(θ,t) is defined in the interval −π θ π with ρ(−π,t) = ρ(π,t). Matytsin proved [22] that if the evolution equation of the fermions is given by the Calogero–Sutherland model then the density ρ(θ,t) is governed by the Das–Jevicki equation [23] which admits Wigner semicircular distribution of radius r(t) as a solution: 2 ρ(θ,t)= r(t)2 − θ 2, |θ| r(t) (7) πr(t)2 2 with r(t) satisfying the differential equation d r(t) + 4 = 0. On a sphere of area A the bound- dt2 r(t)3 ary conditions are r(0) = r(A) = 0 and the differential equation has the solution:

t(A− t) r(t) = 2 . (8) A The solution given by (7) and (8) is valid provided the support of the density function ρ(θ,t) is in the interval [−π,π] at all t, namely√ provided r(t) π. The maximum value for the radius r(t) occurs for t = A/2 and is rmax = A. Hence the condition for the validity of the Wigner semicircular solution is A π2. Beyond the critical value A = π 2 the fermions “realize” that the space they live in is compact, instantons effects become important and (7) is not an acceptable solution any longer. On the cylinder a similar phase transition occurs in general, at a critical value of the area that depends on the holonomies at the boundaries.5 If the distribution of the

5 However with particular conditions at the boundary the phase transition may also be absent, see for instance [26]. A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 345

Fig. 3. Eigenvalues distribution for 0 <α<4. invariant angles of the holonomies at the boundaries is the Wigner semicircular distribution, then the critical area can be calculated exactly in the same way as for the sphere [11]. For a more general discussion see [12].

3. Large A(N) phase transition

It is apparent from the discussion in the previous section and from the explicit form (2) of the partition function that as the area of the sphere increases the distribution of the “momenta” ni (i.e. of the integers that label the U(N) representations) becomes more and more similar to a double step function, like the one drawn with dashed lines in Fig. 3. In fact for very large areas the attractive quadratic potential tends to dominate over the repulsive force produced by the Vandermonde determinant. The double step distribution corresponds to the trivial representation of U(N) in which all characters are identical irrespective of their argument. If this distribution dominates the functional integral then the kernel on a disc or on a cylinder becomes independent from the holonomies at the boundaries. As already mentioned in the introduction, the partition functions on the disc and on the cylinder may be interpreted in terms of random walks on the group manifold. From this point of view the very large area phase, where the sum over the irreducible representations is dominated by the trivial representation corresponds to a walk which is so long that the walker has lost any notion of the starting point. The transition where this situation sets in is known in random walk theory as “cutoff transition”, and the same term will be used here. The cutoff regime occurs for areas larger than a critical N dependent value Ac(N) which was found in [8]. The argument is very simple: consider the trivial double step representation R0 that 2 minimizes the Casimir term i ni N − 1 N − 1 R : {n ,...,n }= ,...,− , (9) 0 1 N 2 2 and compare its contribution to the partition function (2) with the one coming from a representa- = N−1 + tion R1 in which n1 has been increased by 1, namely in which n1 2 1. In other words we look for the value of A at which R0 ceases to be dominant. A simple calculation shows that the 346 A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 ratio between the two contributions is

A Z (A,N,R ) e 2 0 0 = . 2 (10) Z0(A,N,R1) N

This ratio is larger than 1 (hence R0 dominates and we are in the cutoff phase) if A>4logN. In order to study this phase transition in more detail, it is convenient to parametrize the area A by rescaling it with log N:

A = α log N + β. (11)

From the previous argument we expect the cutoff transition to occur at the critical value αc = 4, separating two distinct phases. So Yang–Mills theory on a sphere seems to have four phases altogether: the first two at α = 0 separated by the Douglas–Kazakov phase transition at β = π 2, the other two when the area is logarithmically rescaled with N that are separated by the cutoff transition. The phase before the cutoff is qualitatively different from the strong coupling phase à la Douglas–Kazakov, as we will show in the following. Nonetheless, since the rescaling of the area with log N is essentially introduced by hand, it is hard to say whether there is some higher-order phase transition between them or just a crossover. The aim of this section is to find the saddle point configuration and the free energy in the two phases above and below the cutoff point αc = 4: the cutoff phase α>αc has been discussed above and it is rather trivial, but the phase below αc appears as an interesting intermediate phase between the strong coupling phase in the Douglas–Kazakov transition and the cutoff phase. Hence we shall concentrate on this in the rest of the section. Let us consider again the partition function (2) and the corresponding action

N A N S = 2 log |n − n |− n2. (12) i j 2N i i>j=1 i=1 We want to ﬁnd the extremum of this action in the large N limit, when A is parametrized as in Eq. (11). Since we expect the saddle point distribution of the “momenta” ni to be symmetric with respect to the origin (ni →−ni ) we shall perform the variation only with respect to symmetric conﬁgurations, that is we set N − 1 −M i M, with M = and n− =−n . (13) 2 i i Using this symmetry one can restrict the sums to non-negative values of i (i = 1,...,M) and write the action as:

M M A M S = 2 log(n − n )2 + log(n + n )2 − n2. (14) i j i j 2M i i>j1 i>j1 i=1

While in the cutoff phase ni = i for i = 1,...,M below the cutoff transition we expect a conﬁguration of the type described in Fig. 3, namely:

ni = i, i = 1,...,M − l,

nM−l+α = M − l + α + rα,α= 1,...,l, (15) A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 347 where the value of l and the spectrum of the integers rα are to be determined. With these notations the action can be written as:

− l Ml r r S − S = 4 log 1 + α 1 + α 0 α + j M − l + α + j α=1 j=1 rα + rβ rα − rβ + 2 log 1 + + log 1 + α + β + 2(M − l) α − β α=β

A l − 2(M − l + α)r + r2 , (16) 2M α α α=1 where S0 represents the value of the action in the trivial representation R0. We shall assume that as M →∞also l →∞but at a slower rate than M, namely l/M → 0. We shall also assume that l and rα will be of the same order in the large M limit. These assumptions will be justiﬁed a posteriori, in the sense that they will provide a stable saddle point in the large M limit when the area is scaled like log M. They are also very reasonable assumptions: l is of order N in the strong coupling phase following the Douglas–Kazakov transition and one expects that with the logarithmic rescaling of the area it will shrink further by some power of N. The ﬁrst step in dealing with (16) is to make the dependence from M explicit. By using the identity

− M l r r (2M − 2l + α + r )! α! 1 + α 1 + α = α (17) j + α M − l + j + α (2M − 2l + α)! (α + rα)! j=1 we can rewrite the action as

l (2M − 2l + α + rα)! α! S − S0 = 4 log (2M − 2l + α)! (α + rα)! α=1 rα + rβ rα − rβ + 2 log 1 + + log 1 + α + β + 2(M − l) α − β α=β

A l − 2(M − l + α)r + r2 . (18) 2M α α α=1 (N+C)! →∞ →∞ All ratios of factorials in (18) can be reduced to the form log N! with N , C C → C → and N 0. By repeated use of Stirling formula one ﬁnds, up to terms that vanish as N 0: (N + C)! C log ∼ C log N + f (19) N! N where C M−l k C C log(1 + ) − z f = log 1 + + N − 1 = (−1)k 1 . (20) N N C k(k + 1) N k=1 348 A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357

By using this asymptotic behaviour, and introducing continuum variables in the large N limit, namely α r x = r(x) = α = l dx. (21) l l α The action ﬁnally takes the form

1 M S − S = 4l2 dx r(x) log − log x + r(x) + 1 + log 2 0 l 0 1 1 r(x)− r(y) + x log x − log x + r(x) + 2l2 dx dy log 1 + x − y 0 0 1 − Al2 dxr(x), (22) 0 l where subleading terms (by powers of M ) have been neglected. It is apparent from (22) that in the new regime M l 1 the quadratic term coming from the Casimir operator has been linearized. Thus our problem reduces to a particular case of a class of models characterized by a repulsive Vandermonde-like term and an attractive linear term, which have been widely studied in the literature starting from [24] (see also [25] and references therein). By the very same procedure showed at the beginning of the present section, Eqs. (9), (10), one can prove that a cutoff transition exists also for general linear models. Let us now parametrize the area A according to Eq. (11) and write the action as:

2 S − S0 = 4l [F0 log M − F1 log l + F2], (23) where F0, F1 and F2 are of order 1 in the large M and l limit and are given by:

1 α F = 1 − dxr(x), 0 4 0 1

F1 = dxr(x), 0 1 β F = dx r(x) − log x + r(x) + 1 − log 2 − + x log x − log x + r(x) 2 4 0 1 1 1 r(x)− r(y) + dx dy log 1 + . 2 x − y 0 0 In order to ﬁnd the conﬁguration that maximizes the functional integral in the large M limit we take the variation of (23) with respect to both l and r(x). The variation with respect to l gives the A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 349 equation: α F 1 1 − log M + 2 − = log l (24) 4 F1 2 1− α which shows that l grows like M 4 . This is consistent with what we expected: for α → 0it gives l ∼ M as in the strong phase beyond the Douglas–Kazakov transition, and at the cutoff point α = 4 the power vanishes as expected. The variation with respect to r(x) gives on the other hand

1 1 dy − log x + r(x) = C (25) x + r(x)− y − r(y) 0 with α β F 1 β C = log l − 1 − log M + + log 2 = 2 − + + log 2. (26) 4 4 F1 2 4 = + = dx If one introduces the new variable ξ x r(x) and the density function ρ(ξ) dξ with support in the interval [0,a],Eq.(25) becomes6: a ρ(η) dη − log ξ = C. (27) ξ − η 0 This is a standard type of equation for the density of eigenvalues in the large N limit of matrix models and can be solved by standard analytic methods (for a detailed discussion of this equation see for instance [27]). This same equation was obtained in [25], Eq. (66), for the large-N limit of models with a linear potential. The only seeming difference is the presence, in our case, of a logarithmic non-constant term; this is because we have explicitly performed the integral in the region where ρ = 1. The resolvent function, whose discontinuity across the cut gives the density ρ(ξ), is given by: √ √ ξ − a + ξ H(ξ)= log ξ + C − 2log √ (28) a with the additional condition that for large ξ 1 1 H(ξ)= + O . (29) ξ ξ 2 The corresponding density is given by 2 ξ ρ(ξ)= arccos . (30) π a This solution obviously describes, through the symmetry (13), both the positive and the negative region of ni .

6 The lowest extreme of the interval is r(0) which is zero by construction, the upper end a is, according with the deﬁnition of ξ, a = 1 + r(1). 350 A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357

The asymptotic condition (29) gives the two extra equations a = 2,C= log 2. (31) Eq. (30), together with the first of (31) define r(x) completely, although in an implicit way. Hence all integrals involved in the definition of F1 and F2 can be calculated. The calculation can actually be done analytically and gives:

1 F2 1 β F1 = , = − . (32) 4 F1 2 4 The second equation could have been derived independently from (26) and (31), so it constitutes a non trivial consistency check. We can now write explicitly l and the free energy F in terms of the area A(M):

− β 1− α − A(M) l = e 4 M 4 = Me 4 , 1 2 1 − β 2− α 1 2 − A(M) F = S − S = l = e 2 M 2 = M e 2 . (33) 0 2 2 2 Beyond the cutoff transition we have instead F = l = 0, as the dominant eigenvalue distribution is given by the double step function sketched in Fig. 3. The interpretation of these results from the point of view of the free fermion description is very clear: beyond the cutoff (α>4) we are effectively in a zero temperature situation where all fermions fill the Fermi sea with no holes. Below the cutoff instead (α<4) some excited fermions and the corresponding holes are present in proximity of the surface of the Fermi sea both on the positive and negative momentum side. The number of fermions above the sea level is given by l − α l 4 in (33). The ratio M vanishes like M in the large M limit. This distinguishes this phase from the strong coupling phase of the Douglas–Kazakov transition, where such ratio remains finite, namely the number of fermions above the Fermi sea level is of order M. In spite of being described in terms of N free fermions, the free energy is proportional (with the standard ’t Hooft scaling) to N 2, which reflects the original number of degrees of freedom in a unitary N × N matrix model.7 So it is not surprising that the free energy becomes proportional to l2 in presence of l effective fermionic degrees of freedom when the area is rescaled by a log M factor. It is as if the effective size of the original matrix had shrunk to l × l. A full understanding of the reduction of number of degrees of freedom from the point of view of the original gauge degrees of freedom is still wanted, although some light on it might be thrown by the study of the kernels on the disc and the cylinder in the following sections. 2− α It is apparent from (33) that the number of effective degrees of freedom M 2 is the actual order parameter for the cutoff transition. Incidentally, this unusual dependence of the number of degrees of freedom upon α, together with the choice of a large M limit, makes it rather delicate to classify such a transition according to standard terminology. Let us finally consider the representations of U(N) and/or SU(N) that correspond to the “momentum” distribution pictured in Fig. 3 and given in (30). With the group SU(N)8 this corresponds to a composite representation in the sense of Ref. [6], whose Young diagram is shown

7 The description in the terms of N fermions follows the integration over the angular variables that reduces the matrix model to an integral over the eigenvalues. The ensuing Vandermonde determinant makes the wave function describing the eigenvalues antisymmetric. As a consequence the total momentum of M left moving fermions is of order M2 rather than M, that is of the same order as the original number of bosonic degrees of freedom. 8 2 2 − 1 2 With SU(N) the term i ni in the action should be replaced by i ni N ( i ni ) which is invariant under ni → ni + a. However with suitable choice of a the extra term can be set equal to zero. A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 351

Fig. 4. Young tableau for a composite representation. in Fig. 4 where the constituent representations are denoted by R and S. The rows in the Young diagram of R and S (that coincide in this case) have lengths rα and their total number of boxes |r| is given in the large N limit by 1 |r|= r ∼ l2 dxr(x) = l2. (34) α 4 α If the group is U(N) an arbitrary number of columns of length N can be added or subtracted to the Young diagram of Fig. 4, and the composite representation can be seen as the direct product of the two constituent representations R and S with opposite U(1) charges. In fact the integers labeling the representation S are in this case negative, which corresponds to changing signs of † the invariant angles θi , namely to changing U into U . In the strong coupling regime of the Douglas–Kazakov transition (A>π2) the partition function is dominated in the large N limit by a composite representation of the same type but with the Young diagram of R and S made of rows and columns of lengths of order N, rather than l, and a total number of boxes of order N2 rather than l2.

4. Cutoff transition on the disc and on the cylinder

In this section we are going to consider the partition function on a disc and on a cylinder, with ﬁxed holonomies at the boundaries, in the regime where the area A is scaled as in (11). We shall show that the cutoff transition occurs also in this case at the same critical value of α, except for some particular holonomies at the boundaries. This is in analogy to what happens in the case of the Douglas–Kazakov transition which was proved to occur on a cylinder in [11,12], except possibly for some special conﬁgurations (see [26]). 352 A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357

We shall use the standard expression for the partition function on a cylinder with holonomies U and V at the boundaries, which is a particular case of (1): − A (N2−1) − A n2 Zcyl(U, V ) = e 24 χr (U)χr (V )e 2N i i . (35) r The partition function on a disc is obtained from (35) by taking for instance U → 1 and the partition function on the sphere is recovered by taking the double limit U → 1 and V → 1. The area A in (35) scales as in (11). The sum over the representations r is replaced in the large N limit by the saddle point, namely by a representation whose “momentum” distribution is of the type described in Fig. 3. This corresponds to a composite representation, whose constituent representations R and S have rows and columns of order l with l/N → 0inthelargeN limit. The ﬁrst step in evaluating (35) is then to calculate the characters in such limit.

4.1. Characters in the large N,largel limit

For simplicity, let us consider ﬁrst the case where only one constituent representation is present, which would be the case if only right (or left) moving fermions were present above the Fermi sea. This is given by:

ni = i for i = 1,...,N − l,

ni = N − l + α + rα for i = N − l + α and α = 1,...,l. (36)

The dimension Δr of this representation can be written in the form |(ni − nj )| α! rα − rβ (N − l + α + r )! Δ = = 1 + α r (i − j) (α + r )! α − β (N − l + α)! α α α i>j α>β d (N − l + α + r )! d l = r α = r N rα 1 + O , (37) |r|! (N − l + α)! |r|! N α where dr is the dimension of the representation of the symmetric group S|r| associated to the Young diagram r of rows rα. The order of the symmetric group is the total number of boxes in | |= the Young diagram: r α rα. The dependence on N in (37) is explicit: the ratio of factorials |r|−k in (37) is a polynomial in N of degree |r|.Ifl, α and rα are all of order l the coefficient of N in this polynomial is of order lk in the large N, large l limit. Hence we can write |r| k d | | l Δ = r N r 1 + c , (38) r |r|! k N k where the coefficients ck are smooth in the large l, large N limit. If the limit is taken keeping l/N finite, as in the strong coupling phase of the Douglas–Kazakov transition, all terms at the r.h.s. of (38) are of the same order and cannot be neglected. On the other hand if the double scaling limit is taken with l/N → 0, as in the previous section, then all terms after the 1 are subleading and can be neglected. The same argument holds if instead of the dimension of the representation we consider a character of U(N). In fact the celebrated Frobenius formula gives: d χˆ (σ ) Tr U sj r r kσ χr (U) = N , (39) |r|! dr N σ ∈S|r| j A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 353 where χˆr (σ ) is a character of the symmetric group S|r| in the representation labeled by the same Young diagram as χr (U), sj are the lengths of the cycles in the cycle decomposition of σ =| | → ( j sj r ) and kσ is the number of cycles in σ . By taking U 1 and comparing (38) and (39) ˆ | |− one finds that the contribution of χr (σ ) when σ consists of k cycles is ∼ l r kσ in the large l dr σ limit. So if we take the double scaling limit where both l and N go to infinity and the ratio l/N goes to zero, all terms in the sum over σ in (39) are subleading with respect to the one where σ is the identity. We obtain: |r| |r| d Tr U | | Tr U χ (U) = r N r 1 + O(l/N) = Δ 1 + O(l/N) . (40) r |r|! N r N Notice again that in the strong coupling phase of the Douglas–Kazakov transition, where in the large N limit l/N is kept constant, all terms in (39) coming from different permutations σ are of the same order and cannot be neglected.

4.2. Partition function on the cylinder and special boundary conditions

As a result of previous analysis we find that in the composite representations that are dominant in the region 0 <α<4 the characters χr (U) depend only from Tr U. Hence the partition function on the cylinder will depend only on the trace of the holonomies on the boundaries. In fact it is apparent from (40) that going from the sphere (U = 1) to the cylinder just amounts Tr U |r| to a multiplicative factor ( N ) . In the case of interest however the composite representation contains both chiral and anti-chiral component representations (that is R and S of Fig. 4) and not just one as in the simplified example discussed above. However in the large N limit the two component representations are decoupled [6] and the character of the composite representation becomes just the product of the characters of the component representations: 2|r| † 2 Tr U χcomposite(U) = χr (U)χr U = Δ . (41) r N By replacing (41) into (35) we find 2|r| Tr U Tr V Zcyl(U, V ) = Zsphere , (42) N N where of course the value of |r| is the one determined by the saddle point equations and the equality holds, in the large N limit, only in the regime where the area A is scaled as in (11) with = =|Tr U | =|Tr V | α 0. If we set u N and v N it is almost immediate to see that the multiplicative factor at the r.h.s. of (42) is equivalent to replace in the action (23) the constant term β in the area A with a β(u,v)ˆ given by:

β → β(u,v)ˆ = β − log u2 − log v2. (43) The partition function on the cylinder is then the same as the partition function on a sphere whose area is obtained from the area of the cylinder by adding the two terms log u2 and log v2. The latter can be interpreted as the areas of the two discs necessary to go, in the given momentum = = | Tr U |= conﬁguration, from U 1 (respectively V 1) to the holonomy at the boundary with N u | Tr V |= (respectively N v). The areas of the two discs are of order 1, so this correction does not affect the position of the cutoff transition that remains on the cylinder at α = 4. 354 A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357

The discussion above relies on the fact that the leading term in Frobenius formula (39) comes from the identical permutation. However this is not always true: if we take the large N and l limit keeping Tr U = 09 then all terms in (39) with σ containing cycles of length 1 would vanish. Assuming that Tr U 2 = 0, the term in Frobenius formula (39) with the highest power on N would |r| then come from permutations σ all made out of cycles of length 2 and would be of order N 2 instead of N |r|. Supposing that both the trace of U and of V vanish the coefﬁcient of log M in the ﬁrst term of (22) would be halved. Correspondingly the critical value of α at which the cutoff transition occurs would also be halved and become αc = 2. Let us make this argument general and more quantitative. Let us assume that Tr U k1 = 0 with j Tr U = 0forj

9 This can happen for instance if the holonomy at a boundary has a symmetry of some sort, for instance of the type θ → θ + π, that is preserved through the limiting process. 10 We assume here for simplicity that we take l →∞keeping the total number of boxes in the Young diagram multiple of k1 and k2. 11 A lot is known on the characters of permutations with cycles all of the same length, so an explicit expression for F2 is probably obtainable. A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 355 which generalizes the result on the sphere.

5. Wilson loops

In this section we calculate the expectation value of a Wilson loop in a Yang–Mills theory with gauge group U(N) in the large N limit. The space–time manifold has the topology of a two-dimensional sphere whose area scales like logN as in Eq. (11). We shall follow th approach of Daul and Kazakov [29] and Boulatov [30] who did the calculation for constant areas. The general solution for arbitrary self-intersecting Wilson loops was obtained in [28]. We may think of the sphere of area A as two discs of areas A1 and A2 (A1 + A2 = A)sewed along their common boundary and with holonomy on the boundary respectively U and U †.The Wilson loop is then given by 1 W(A1,A2) = Tr U N A A 1 1 † − 1 C − 2 C = d d dU Tr Uχ (U)χ U e 2N 1 2N 2 , (49) Z 1 2 N 1 2 R1,R2 where d1 and C1 are the dimension and the Casimir operator in the representation R1, referred † to the disc of area A1; likewise for d2 and C2. The quantity dU Tr Uχ1(U)χ2(U ) may be either 0 or 1, namely it is 1 when the Young diagram of R2 is obtained by adding one box to the diagram of R1 and 0 otherwise. That is, if R1 is labeled by the integers n1 >n2 > ···>nN , R2 is labeled by a set on integers where one of the ni is increased by one. Daul and Kazakov used this property to get rid of one summation and obtained + 1 1 1 − A1 A2 A2 = 2 + 2N C1 N ni W(A1,A2) d1 1 e e , (50) Z N nj − ni R1 i j,j=i where the sum over i corresponds to all the possible ways of adding one box to the diagram. This is not however the whole result, in fact the original expression is symmetric under exchange of 12 A1 and A2, so a term with A1 and A2 exchanged must be added to (50) and gives: + 1 1 1 − A1 A2 A1 A2 = 2 + 2N C1 N ni + N ni W(A1,A2) d1 1 e e e . (51) Z N nj − ni R1 i j,j=i

Moreover, Eq. (50),aswellas(51), is clearly not symmetric under ni ↔−ni , so we cannot restrict our considerations to ni > 0 any longer; instead we have to consider the whole interval −∞

12 † This term originates from the fact that dU Tr Uχ1(U)χ2(U ) is different from zero also if the representation conjugate to R1 is obtained from the representation conjugate to R2 by adding a box in the Young diagram. 356 A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357

Since A = A1 + A2 = α log N + β, this situation can occur in two ways:

− β2 (1) A2 = β2, A1 = α log N + β1: then W ∼ e 2 , − α2 (2) A1 = α1 log N + β1, A2 = α2 log N + β2 with α1 >α2: then W ∼ N 2 .

We proceed now to evaluate W(A1,A2) in the phase before the cutoff, namely for α<4. The sum over representations in (51) can be replaced by the contribution of the dominant representation in the large N limit, calculated on the sphere in Section 3. The saddle point is unaffected by the presence of the extra term A A 1 1 n 2 n 1 + e N i + e N i n − n i j,j=i j i which is subleading with respect to the action. After replacing in (51) the sum with the saddle point contribution, some simpliﬁcations occur and we get A A 1 1 1 n 2 n W(A ,A ) = 1 + e N i + e N i . (54) 1 2 N n − n i j,j=i j i The representation in (54) is of the type given in (15), and the sum over i describes all possible ways of adding a box to the Young diagram. However the replacement ni → ni + 1 is impossible in the region −l i l as the resulting sequence of integers would not be monotonic increasing. So the sum over i in (54) can be replaced by a sum over α with 1 α l. As a matter of fact we must consider only positive α’s, as adding a box to the adjoint representation amounts to symmetrize with respect to A1 and A2, and it has been taken already into account. Hence in (54) we must replace the index i with α, ni with M − l + α + rα while the index j goes from −M to M, namely it goes over both the condensed and the non-condensed regions. With these substitutions the expression for the Wilson loop becomes: l 1 l ξα W(A1,A2) = 1 − 1 − = ξα N l α 1 l 1 A2 l − ξα × exp log 1 + − 1 − 2 +{A2 → A1}, (55) ξα − ξβ 2 N β=1 where

ξα = α + rα. (56) = α = It is convenient as usual to use in the large N limit the continuum variables x l , ξ(x) ξα and the density function ρ(ξ) = dx . The crucial part of the calculation is the evaluation of l dξ l 1 = log(1 + ) which can be done following Ref. [29]. By expanding the logarithm one β 1 ξα−xiβ finds: l + 1 = 1 − 1 1 k log 1 dηρ(η) k ρ(ξ) . (57) ξα − ξβ ξ − η k (β − α) β=1 k=2 β The first term at the r.h.s. comes from the k = 1 term of the log expansion and can be evaluated sin πρ using Eq. (27), the other terms can be calculated as in [29] and give log πρ . By inserting these A. Apolloni et al. / Nuclear Physics B 744 [FS] (2006) 340–357 357 results into (55) and using the explicit form of the solution (30) one finally obtains (neglecting O( l ) terms): N sin πρ − A2 − A1 − A2 − A1 W(A ,A ) = 2 dξ e 2 + e 2 = e 2 + e 2 (58) 1 2 π which is exactly the same result as in the frozen phase. The result is not trivial, but it was somehow to be expected. Both phases, before and after the transition, are strong coupling phases and the expectation value of the Wilson loop should be in both of them obtained, to the leading order, by filling the loop with elementary plaquettes in the fundamental representation. The effects of ∼ l the transition are expected to appear only at the next-to-leading order ( N ) which is sensitive l to the O(N ) degrees of freedom which are not frozen below the cutoff.

Acknowledgements

A.A. wants to thank the “Service de Physique Theorique (SPhT)” at Saclay for the kind hospitality of these last months. He also would like to thank I. Kostov and G. Vernizzi for many useful discussions during this period. S.A. would like to thank the Niels Bohr Institute for warm hospitality during the early stage of this work.

References

[1] B.E. Rusakov, Mod. Phys. Lett. A 5 (1990) 693. [2] A.A. Migdal, Sov. Phys. JETP 42 (1975) 413, Zh. Eksp. Teor. Fiz. 69 (1975) 810. [3] V.A. Kazakov, I.K. Kostov, Nucl. Phys. B 176 (1980) 199. [4] V.A. Kazakov, Nucl. Phys. B 179 (1981) 283. [5] D.J. Gross, W.I. Taylor, Nucl. Phys. B 400 (1993) 181, hep-th/9301068. [6] D.J. Gross, W.I. Taylor, Nucl. Phys. B 403 (1993) 395, hep-th/9303046. [7] M. Billo, M. Caselle, A. D’Adda, P. Provero, Nucl. Phys. B 543 (1999) 141, hep-th/9809095. [8] A. D’Adda, P. Provero, Commun. Math. Phys. 245 (2004) 1, hep-th/0110243. [9] B. Rusakov, Phys. Lett. B 303 (1993) 95, hep-th/9212090. [10] M.R. Douglas, V.A. Kazakov, Phys. Lett. B 319 (1993) 219, hep-th/9305047. [11] M. Caselle, A. D’Adda, L. Magnea, S. Panzeri, hep-th/9309107. [12] D.J. Gross, A. Matytsin, Nucl. Phys. B 429 (1994) 50, hep-th/9404004. [13] M.R. Douglas, hep-th/9303159. [14] J.A. Minahan, A.P. Polychronakos, Phys. Lett. B 312 (1993) 155, hep-th/9303153. [15] J.A. Minahan, A.P. Polychronakos, Phys. Lett. B 326 (1994) 288, hep-th/9309044. [16] M. Caselle, A. D’Adda, L. Magnea, S. Panzeri, Nucl. Phys. B 416 (1994) 751, hep-th/9304015. [17] P. Diaconis, M. Shahshahani, Z. Wahrsch. Verw. Gebiete 57 (1981) 159. [18] I. Pak, V.H. Vu, Discrete Appl. Math. 110 (2001) 251. [19] S. de Haro, M. Tierz, Phys. Lett. B 601 (2004) 201, hep-th/0406093. [20] S. de Haro, JHEP 0408 (2004) 041, hep-th/0407139. [21] S. de Haro, hep-th/0412110. [22] A. Matytsin, Nucl. Phys. B 411 (1994) 805, hep-th/9306077. [23] S.R. Das, A. Jevicki, Mod. Phys. Lett. A 5 (1990) 1639. [24] A.M. Vershik, S.V. Kerov, Soviet. Math. Dokl. 18 (1977) 527. [25] E. Brézin, V. Kazakov, math-ph/9909009. [26] S. Zelditch, Commun. Math. Phys. 245 (2004) 611, hep-th/0305218. [27] K. Kostov, M. Staudacher, T. Wynter, Commun. Math. Phys. 191 (1998) 283, hep-th/9703189. [28] B. Rusakov, Phys. Lett. B 329 (1994) 338, hep-th/9310134. [29] J.M. Daul, V.A. Kazakov, Phys. Lett. B 335 (1994) 371, hep-th/9310165. [30] D.V. Boulatov, Mod. Phys. Lett. A 9 (1994) 365, hep-th/9310041. Nuclear Physics B 744 [FS] (2006) 358–379

Boundary renormalisation group ﬂows of unitary superconformal minimal models

Márton Kormos

Institute for Theoretical Physics, Eötvös University, 1117 Budapest, Pázmány Péter sétány 1/A, Hungary Received 21 December 2005; received in revised form 2 March 2006; accepted 22 March 2006 Available online 6 April 2006

Abstract In this paper we investigate renormalisation group flows of supersymmetric minimal models generated by ˆ the boundary perturbing field G−1/2φ1,3. Performing the Truncated Conformal Space Approach analysis the emerging pattern of the flow structure is consistent with the theoretical expectations. According to the results, this pattern can be naturally extended to those cases for which the existing predictions are uncertain. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

Conformal field theories with boundary attracted much interest recently, due to their relevance in condensed matter physics, e.g., in the Kondo problem [1] and their applications in describing D-branes in string theory [2,3]. In terms of string theory the renormalisation group flow generated by a boundary perturbing field corresponds to tachyon condensation and exploring these flows can help in understanding the decay of D-branes. Many papers appeared in the literature about the boundary perturbations and the corresponding renormalisation flows of unitary minimal models [4–8]. Up to now, a systematic charting of the boundary flows of the unitary superconformal minimal models has been missing. Although there may be flows within the perturbative domain, for a general study a nonperturbative tool is necessary. We choose the Truncated Conformal Space Approach (TCSA), originally proposed in the paper [9] and applied to boundary problems in [10] and [7]. The essence of the TCSA is to diagonalise the Hamiltonian of the system on a subspace of the infinite-dimensional Hilbert space.

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

The paper is organised as follows. In Section 2 we recall the basic facts that we need about superconformal minimal models, including the commutation rules of various fields. After briefly summarizing in Section 3 some main properties of these models in the presence of boundaries, we turn to determining the Hamiltonian of the system and defining the boundary flows in Sec- tion 4. We also motivate our choice for the perturbing operator in this section. Section 5 can be regarded as the main part of the paper: we introduce the TCSA method and give the details of the calculation of matrix elements. The theoretical expectations are briefly summarized in Section 6, while our results are listed and compared to these expectations in Section 7. Finally, in Section 8 we summarize the conclusions of the paper. Some numerical data in tables and figures can be found in Appendix A.

2. Superconformal minimal models

2.1. The N = 1 superconformal algebra

The algebra of superconformal transformations in the plane are generated by two fields, T(z) and G(z) whose mode expansions read −n−2 T(z)= Lnz , (2.1a) n −r−3/2 G(z) = Gr z . (2.1b) r T(z)is a spin 2 field, while G(z) has spin 3/2. Because of its fermionic character two kinds of boundary conditions are possible for G(z): G e2πiz = G(z) → Neveu–Schwarz sector, (2.2) G e2πiz =−G(z) → Ramond sector. (2.3) The N = 1 supersymmetric extension of the Virasoro algebra is defined by the following (anti)commutation relations: c 2 [Ln,Lm]=(n − m)Ln+m + n n − 1 δn,−m, (2.4a) 12 c 2 1 {G ,G }=2L + + r − δ − , (2.4b) r s r s 3 4 r, s n [L ,G ]= − r G + . (2.4c) n r 2 n r Here n, m denote integers, while according to the boundary conditions for G(z) the indices r, s can take half-integer (Neveu–Schwarz sector) or integer (Ramond sector) values. The highest weight representations of the algebra are defined by highest weight states |h satisfying

Ln|h=0,n>0, (2.5a)

L0|h=h, (2.5b)

Gr |h=0,r>0. (2.5c) 360 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379

The Verma module is generated by operators having a nonpositive index. The vectors ··· ··· | ··· ··· Ln1 Lnk Gr1 Grl h ,n1 nk < 0,r1 <

2.2. The minimal series

The irreducible highest weight representations are the quotients of the Verma modules by their maximal invariant submodules. In this paper we deal with the superconformal minimal models which contain only a ﬁnite number of Verma modules. In these models the Verma modules turn out to be highly reducible. The superconformal minimal models are indexed by two positive integers, p and q (we choose the convention p

2.3. Field representations

Since we will be interested in correlation functions, we have to investigate how fields transform. Our discussion follows Ref. [11]. We denote the field corresponding to the state |h by φh(z). By definition, φh(z)|0=exp(zL−1)|h, where |0 is the superconform-invariant vacuum state. M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 361

2.3.1. Virasoro properties The mode expansion (2.1a) and the deﬁning relations (2.5a), (2.5b) lead to the following operator product expansion: hφ(w) ∂φ(w) T(z)φ(w)= + +···. (2.9) (z − w)2 z − w Using standard contour integration technique one obtains the commutation relation n Lnφ(z)= z (n + 1)h + z∂ φ(z)+ φ(z)Ln. (2.10) Combining the relations for n = m and n = 0 one arrives at m m Lm − z L0,φ(z) = hmz φ(z). (2.11)

2.3.2. Neveu–Schwarz case From Eqs. (2.1b), (2.5c) and (2.4b) we conclude ˆ (G− / φ)(w) G(z)φ(w) = 1 2 +···, (2.12a) z − w ˆ 2hφ(w) ∂φ(w) G(z)(G− φ)(w) = + +···, (2.12b) 1/2 (z − w)2 z − w ˆ where limz→0(G−1/2φ)(z) = G−1/2|h. The commutation relations in this case turn out to be

r+1/2 ˆ G φ(z)= z (G− φ)(z) + η φ(z)G , (2.13a) r 1/2 φ r ˆ r−1/2 ˆ Gr (G−1/2φ)(z) = z (2r + 1)h + z∂ φ(z)− ηφ(G−1/2φ)(z)Gr (2.13b) with ηφ =±1 for ﬁelds of even and odd fermion number, respectively.

2.3.3. Ramond case There is a cut in the operator product of G(z) and a Ramond field and thus a Ramond field changes the moding of G(z) from integral to half-integral and vice versa. So it is impossible to obtain a commutation relation of Gr with a Ramond field for fixed r. Fortunately, we can avoid using any (anti)commutation relation for Ramond fields in our calculations. All we need is formally generalizing relations (2.13) for integral r. Then, just like in the Virasoro case in Section 2.3.1, combining the (2.13b) equations for r = m and r = 0 yields

ˆ m ˆ m−1/2 G (G− φ)(z) = z G (G− φ)(z) + 2mhz φ(z) m 1/2 0 1/2 ˆ m + ηφ(G−1/2φ)(z) z G0 − Gm . (2.14) We will come back to these issues later in Section 5.2.1.

3. Superconformal ﬁeld theory on the upper half plane

The upper half plane (UHP) is the prototype of geometries having nontrivial boundaries. The superconformal transformations then have two roles. Firstly, they connect the correlation functions on general geometries to those on the UHP. On the other hand, the transformations that leave the geometry invariant put constraints on the correlation functions on the UHP. 362 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379

In the following we give a lightning review of the superconformal boundary conditions and consistent boundary states based on paper [12]. The derivation of the Ward identity on the UHP leads to a new constraint on the operators of the chiral algebra. For the ordinary stress energy tensor one obtains Txy(x, 0) = 0, which means that there is no energy ﬂow across the boundary (the real axis). In complex coordinates this translates to the condition

T(z,z)¯ = T(¯ z,¯ z) for Re z = 0. (3.1) A strip of width L with boundary conditions α and β on the left and the right boundary, respectively, can be mapped to the UHP by the exponential mapping, under which the boundaries of the strip are mapped to the negative and positive part of the real axis. The boundary conditions on the strip become conformal boundary conditions on the plane. The discontinuity of the boundary condition along the real axis can be described by a boundary condition changing field ψab. These fields are in one to one correspondence with the operator content of the theory on the strip. Looking at the strip from the side (“closed channel”) and letting the time flow in the direction of L the boundary conditions turn into initial and final boundary states. Then the conformal invariance of the boundary conditions translate to the following condition on these boundary states: ¯ (Ln − L−n)|B=0 (B = α, β). (3.2) For a general chiral field W of spin s one finds s ¯ Wn − (−1) W−n |B=0, (3.3) which in our case means ¯ (Gr ± iG−r )|B=0. (3.4) This implies that the left-chiral and right-chiral part of the theory are linked together. Only one copy of the supersymmetric Virasoro algebra remains and all the calculations are chiral. It is also necessary to carry out the projection described in Section 2.1. The equations for the state |B are solved in terms of the so-called Ishibashi states [13], whose linear combinations give the physically consistent boundary states. The further constraints for these states, generalizations of the Cardy equations [14], come from the modular properties of the partition function. Now let us list the main results of this analysis. In the Ramond sector the consistent boundary states are in one-to-one correspondence with the irreducible highest weight representations of the algebra, so they can be labeled by the same index set: we will denote them by |kR. In the Neveu– Schwarz sector there are two physical boundary states for every irreducible representation, they are denoted by |kNS and |kNS. For the SM(p, p + 2) models (with p odd) the state content of the Hilbert space of the theory on the strip with boundary conditions α and β is given by − RHαβ = i NS trNS e nαβ χi (q), (3.5a) i∈Δ NS − RHαβ = ˜i NS trNS Γe nαβ χi (q), (3.5b) i∈ΔNS M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 363 − RHαβ = i R trR e mαβ χi (q), (3.5c) i∈ΔR −RHαβ trR Γe = 0, (3.5d) where q = e−πR/L and the characters are − NS = L0 c/24 ∈ χi (q) tri q ,iΔNS, (3.6a) − NS = L0 c/24 ∈ χi (q) tri Γq ,iΔNS, (3.6b) − R = L0 c/24 ∈ χi (q) tri q ,iΔR. (3.6c) i i Using a generalised Verlinde formula the coefficients nαβ and mαβ can be identified with the fusion rule coefficients in the fusion Φα × Φβ → Φi . Particularly, ni =ñi = δi ,mi = 0, (3.7a) 0NSkNS 0NSkNS k 0NSkNS ni =−ñi = δi ,mi = 0, (3.7b) 0NSkNS 0NSkNS k 0NSkNS ni =ñi = 0,mi = 2δi . (3.7c) 0NSkR 0NSkR 0NSkR k The field content of the (projected) theory is thus given by

1 − 1 − Z = tr (1 + Γ)e RHαβ + tr (1 + Γ)e RHαβ NS 2 R 2 1 1 = ni χNS(q) +˜ni χNS(q) + mi χR(q). (3.8) 2 αβ i αβ i 2 αβ i i∈ΔNS i∈ΔR

If we choose β corresponding to |0NS and α to |kNS (or |0NS and |kNS) then the right-hand side becomes 1 1 − Z = χNS(q) + χNS(q) = trNS (1 + Γ)qL0 c/24. (3.9) 2 k k k 2

In case we choose β corresponding to |0NS and α to |kNS (or |0NS and |kNS) we get 1 1 − Z = χNS(q) − χNS(q) = trNS (1 − Γ)qL0 c/24. (3.10) 2 k k k 2

Finally, if we choose β corresponding to |0NS (or |0NS) and α to |kR the result is = R Z χk (q). (3.11) The case of even p is more complicated due to the special supersymmetric representation p p+2 ( 2 , 2 ).

4. Boundary renormalisation group ﬂows

In this paper we examine RG flows of superconformal minimal models with boundary, generated by a boundary field. Since no bulk perturbing field is allowed and the boundary perturbation preserves supersymmetry, the RG flow takes place in the space of possible supersymmetric boundary conditions, in which the fixed points are the superconformal ones. To study flows starting from a Cardy boundary condition it is convenient to choose the boundary condition on one of the edges of the strip to correspond to the h = 0 primary field, that is 364 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379

β ∼|0NS or |0NS. Then (3.9) implies that if α and β are both NS or NS type boundary conditions, the Hilbert space contains only the bosonic half of a single Neveu–Schwarz module, the one corresponding to α. Similarly, according to (3.10),ifα is a NS type and β is a NS type boundary condition (or vice versa), the Hilbert space consists of the fermionic half of a single α module. Finally, as (3.11) shows, if α is a R type boundary condition, the Hilbert space contains a single Ramond module. At the endpoint of the flow the final boundary condition can be read off in a similar way simply from the spectrum of the Hamiltonian. Since the boundary condition β does not change along the flow (it is not perturbed), if we start with the bosonic half of a NS Verma module (NS–NS or NS– NS) and at the end of the flow we find bosonic part(s) of Verma module(s) then this is a NS → NS (or NS → NS) type flow. Similarly, if we identify fermionic part(s) then this indicates aNS→ NS (or NS → NS) type flow. We have to choose the perturbing field on the boundary α. We would like a perturbation that preserves supersymmetry off-critically. Ramond operators do not spoil supersymmetry but they are boundary condition changing operators, so they cannot live on a boundary. Neveu–Schwarz ˆ fields of the form (G−1/2φpert) also preserve supersymmetry and they can be located on the boundary, so we choose this kind of perturbation. The perturbing operator should be relevant, that is its scaling dimension must be less than one. Finally, since the Hamiltonian is bosonic, ˆ φpert must be fermionic. The choice φpert = φ1,3 satisfy all these conditions. (G−1/2φ1,3) has conformal weight p/q < 1 and φ1,3 is fermionic. The latter can be seen from the fact that in the free fermion limit p,q →∞(q − p = const) h1,3 → 1/2. Our choice also has the advantage that this operator can live on every boundary that has supersymmetric relevant perturbations. This can be seen from the following. An operator ψ living on the boundary of the strip is mapped under the exponential map to an operator living on the real axis. We want this operator to leave the boundary condition unchanged along the axis, which means that the coefficients in (3.5) for α = β should be nonzero. Since they can be related to the fusion coefficients this means that the representation hψ should appear in the fusion of φα 1 with itself. The field φ1,3 appears in the self fusion of all fields except for the ones in the first and the last column of the superconformal Kac table (φ(1,s), φ(p−1,s)), but the corresponding boundaries do not have any relevant supersymmetric perturbations at all. Also note that being a ˆ bosonic field, the fusion with (G−1/2φ1,3) maps the projected subspaces onto themselves. ˆ The perturbation (G−1/2φ1,3) is an integrable one [15], but we shall not use this property in our analysis. The Hamiltonian of the perturbed superconformal field theory takes the form

H = HCFT + Hpert, (4.1) where HCFT is the Hamiltonian of the unperturbed theory π c H = H = L − , (4.2) CFT α0 L 0 24 and Hpert comes from the perturbation on the strip:

strip = ˆ Hpert λ(G−1/2φ1,3)(0, 0). (4.3)

1 ˜i The coefﬁcients nαβ are not fusion coefﬁcients themselves, but this does not change the argument essentially. M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 365

The location of the left boundary is at x = 0 and we are free to choose t = 0 for calculating the spectrum. By the exponential map this on the z-plane becomes h +1/2 π 1,3 ˆ H = λ (G− φ )(1) . (4.4) pert L 1/2 1,3 z Thus the complete Hamiltonian on the plane can be written as 1/2−h π c L 1,3 ˆ H = L − + λ (G− φ )(1) . (4.5) L 0 24 π 1/2 1,3 Since during a numerical calculation one works with dimensionless quantities we have to measure the volume (L) and the energies in some typical mass or energy difference of the model, M. − In other words, we use the dimensionless quantities l = ML, ε = E/M, κ = λ/M1/2 h1,3 and h = H/M: 1/2−h π c l 1,3 ˆ h = L − + κ (G− φ )(1) . (4.6) l 0 24 π 1/2 1,3 The renormalisation group flow can be implemented by varying the volume l while keeping the coupling constant κ fixed. Equivalently—and we choose this way—one can keep l fixed and vary κ on some interval. Starting from κ = 0, which is the ultraviolet (UV) limit, the matrix h can be diagonalised at different positive and negative values of κ. In both directions the flow approaches a fixed point, that is a new supersymmetric conformal boundary condition. By (3.5) we should observe the eigenstates rearranging themselves into a direct sum of super-Verma modules (see the figures in Appendix A). Since by (3.7) the coefficients in this sum can only take values 0 and 1, we expect a simple sum of super-Verma modules at the end of the flows. From the degeneracy pattern of the eigenvalues we can identify the modules (the boundary conditions) using the characters and weight differences of the supersymmetric minimal model in question.

5. The truncated conformal space approach

The method we use for organising the boundary flows is the so-called truncated conformal space approach, or TCSA. In this approach the infinite-dimensional Hilbert space is truncated to a finite-dimensional vector space by using only those states whose energy is not greater than a threshold value, Ecut. This is equivalent to truncating the Hilbert space at a given level. The Hamiltonian is then diagonalised on this truncated space. One can think of this procedure as being equivalent to the variational method: the ansatzes for the energy eigenstates of the perturbed Hamiltonian are finite linear combinations of the eigenstates of the conformal Hamiltonian. It is important that the errors of the TCSA diagonalisation are not under control. For example, it may happen that before the flow reaches the scaling region the truncation errors start to dominate. If we use various cuts and find that the flow picture does not change drastically (only the precision of the result gets higher with higher cuts), then it means that the unpleasant case mentioned above does not happen. Of course, establishing the endpoint of the renormalisation group flow by TCSA cannot be regarded as a conclusive proof. What one can see is that the flow goes in the vicinity of some superconformal boundary condition. The exact infrared fixed point can never be reached by TCSA because of the truncation (this can be observed in Fig. 3(a)). We are looking for the range where the TCSA trajectory is closest to the fixed point. 366 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379

5.1. The matrix elements of the Hamiltonian

There are two ways for determining the eigenvalues of the total Hamiltonian. The ﬁrst is to use an orthonormal basis for calculating the matrix elements, which requires an orthogonalisation process. We can avoid this by choosing the second way, in which the basis is not orthogonal. But then the numerical matrix to be diagonalised is

hij = fi|h|ej , (5.1) where {fi} are the elements of the reciprocal basis, that is

fi|ej =δij . (5.2) This amounts to calculating the matrix element = −1 | | hij M ik ek h ej , (5.3) where M is the inner product matrix:

Mij = ei|ej . (5.4) Using the notation ˆ Bij = ei|(G−1/2φ1,3)(1)|ej (5.5) it can be expressed as − 1/2 h1,3 π c l − h = h − δ + κ M 1B . (5.6) ij l i 24 ij π ij

Here we made use of the fact that the basis elements {ei} are eigenvectors of H0.

5.2. Computing the matrix elements of the perturbing ﬁeld between descendant states

The task is now to calculate the matrices M and B and then to determine the eigenvalues of the resulting matrix h. The Hilbert space consists of the Verma module of φα (see Eq. (3.5)) and as mentioned above, we choose the following basis elements: ··· ··· | ··· ··· Ln1 Lnk Gr1 Grl h ,n1 nk < 0,r1 <

The calculation of the matrix B is more difﬁcult because it requires the usage of some commutation rules for reducing the matrix elements. However, these commutation rules are different for the Ramond and the Neveu–Schwarz case. Let us deal with the two cases in turn.

5.2.1. Ramond case In the Ramond case we can use the already mentioned commutational rules (2.11), (2.13a) and (2.14). We recall them because they lie at the heart of the algorithm. For the modes Ln we saw (cf. (2.11)) L φ(z)= znL φ(z)+ φ(z) nhzn − znL + L , (5.10a) n 0 0 n n n n φ(z)Ln = z φ(z)L0 + −nhz − z L0 + Ln φ(z). (5.10b) Similarly, ˆ n ˆ L (G− φ)(z) = z L (G− φ)(z) n 1/2 0 1/2 ˆ n n + (G−1/2φ)(z) n(h + 1/2)z − z L0 + Ln , (5.11a) ˆ n ˆ (G− φ)(z)L = z (G− φ)(z)L 1/2 n 1/2 0 n n ˆ + −n(h + 1/2)z − z L0 + Ln (G−1/2φ)(z). (5.11b) We make use of these formulae in reducing the three-point functions. Here is an example of the application of rule (5.10a):

L−nO1α|φ1,3(1)|O2α

= O1α|Lnφ1,3(1)|O2α

= (hα + nh1,3 − hα) O1α|φ1,3(1)|O2α+ O1α|φ1,3(1)|LnO2α, (5.12) where O1 and O2 are arbitrary strings of lowering operators. The counterpart of these relations for the modes Gr are (see (2.13a)) r+1/2 ˆ Gr φ(z)= z (G−1/2φ)(z) + ηφφ(z)Gr , (5.13a) r+1/2 ˆ φ(z)Gr =−ηφz (G−1/2φ)(z) + ηφGr φ(z), (5.13b) and (cf. (2.14)) ˆ m ˆ m−1/2 G (G− φ)(z) = z G (G− φ)(z) + 2mhz φ(z) m 1/2 0 1/2 ˆ m + ηφ(G−1/2φ)(z) z G0 − Gm , (5.14a) ˆ m ˆ m−1/2 (G− φ)(z)G = z (G− φ)(z)G + η 2mhz φ(z) 1/2 m 1/2 0 φ m ˆ + ηφ z G0 − Gm (G−1/2φ)(z). (5.14b)

These rules can be used in a similar way to (5.12), but only if φα is a Ramond field (G0φα is not defined for Neveu–Schwarz fields). Iteratively applying the rules (5.11) and (5.13), (5.14) to a matrix element leads to a linear combination of the following correlation functions: ˆ α|(G−1/2φ1,3)(1)|α, (5.15a)

α|φ1,3(1)|α, (5.15b)

G0α|φ1,3(1)|α, (5.15c)

α|φ1,3(1)|G0α. (5.15d) 368 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379

The second of these always equals to zero in our case, because φ1,3 is a fermion-like operator. =− This also means that ηφ1,3 1. Due to (5.13a) ˆ G0α|φ1,3(1)|α= α|(G−1/2φ1,3)(1)|α+ηφ α|φ1,3(1)|G0α. (5.16) † = Since G0 G0

G0α|φ1,3(1)|α= α|φ1,3(1)|G0α, (5.17) =− which implies for ηφ1,3 1

1 ˆ G α|φ (1)|α= α|(G− φ )(1)|α. (5.18) 0 1,3 2 1/2 1,3 We see that every matrix element can be traced back to the three-point function ˆ α|(G−1/2φ1,3)(1)|α, (5.19) which is related to a single structure constant. However, we do not need the numerical value of this constant, because it can be absorbed into the coupling constant κ. The physical scale is fixed by the value of κ where the crossover takes place. Since we are interested only in the asymptotic behaviour of the flows (the fixed points), we do not need to fix the real physical scale.

5.2.2. Neveu–Schwarz case In the Neveu–Schwarz case we can derive the rules for reducing the three-point functions using contour integration technique. From L A(∞) B(0) n

= A(∞) L− B(0) n dz − + dz − + = A(∞) z n 1T(z)B(0) = − z n 1T(z)A(∞) B(0) (5.20) 2πi 2πi 0 ∞ we obtain dz − + L Φ(∞) =− z n 1T(z)Φ(∞). (5.21) n 2πi ∞ Then for a three-point function:

L− A(∞) B(1) C(0) n dz + =− zn 1 T(z)A(∞) B(1) C(0) 2πi ∞ n dz n + 1 + = (z − 1)k 1 A(∞) T(z)B(1) C(0) 2πi k + 1 k=−1 1 dz + + zn 1 A(∞) B(1) T(z)C(0) 2πi 0 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 369

n n + 1 = A(∞) B(1) L C(0) + A(∞) L B(1) C(0) . (5.22) n k + 1 k k=−1

With similar manipulations one can obtain rules for the remaining two initial positions of L−n and their counterparts for the modes G−r . These formulae can be obtained from the algebraic commutation relations as well. Now they can be applied iteratively to the three-point function in the following manner. First we throw down all operators from the first field to the others: n n + 1 L− A|B(1)|C= A|B(1)|L C+ A|L B(1)|C, (5.23a) n n k + 1 k k=−1 n + 1 n 2 G−nA|B(1)|C=ηB A|B(1)|GnC+ A|GkB(1)|C. (5.23b) k + 1 k=−1/2 2 Then, exploiting that A is now primary we throw everything from the rightmost field to the middle one: ∞ −n + 1 A|B(1)|L− C=− A|L B(1)|C, (5.24a) n k + 1 k k=−1 ∞ − + 1 n 2 A|B(1)|G−nC=−ηB A|GkB(1)|C. (5.24b) k + 1 k=−1/2 2 Certainly, in practice the sums truncate at the level of B. Finally, we eliminate the operators from the field B (A and C are now primary): n n A|L−nB(1)|C=(−1) A|B(1)|L−1C+(−1) (n − 1) A|B(1)|L0C, (5.25a) n+1/2 A|G−nB(1)|C=ηB (−1) A|B(1)|G−1/2C. (5.25b)

Now we can throw this L−1 and G−1/2 back onto the middle field using (5.24) and repeat step (5.25). Eventually some L−1 and possibly one G−1/2 can accumulate on the middle field. The effect of operators L−1 is simply differentiating. Since conformal symmetry determines the functional form of the three-point function of quasi-primary fields we obtain: | m | = − − − − − ··· A L−1B(1) C (hA hB hC)(hA hB hC 1) × (hA − hB − hC − m + 1) A|B(1)|C. (5.26) The final result is—like in the Ramond case—a linear combination of matrix elements (5.15a), (5.15b) where the second one gives zero again. The reason for not using contour integral technique in the Ramond case is the presence of the cut in the operator product of G(z) with a Ramond field. It is interesting to note that albeit G0φα is not really well-defined for φα being a Neveu– Schwarz field, the Ramond method explained in Section 5.2.1 still works in the Neveu–Schwarz case and gives the same result as the contour integral method.

6. Previous results and expectations

Before turning to the result of the TCSA let us summarize the theoretical predictions for the boundary flows found in the literature. Our main resource is the paper of Fredenhagen [16], 370 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 where the author gives some predictions for boundary flows in general coset models. Here we only summarize the consequences for the special case of superconformal minimal models. The SM(p, p + 2) unitary superminimal models are realized by the su(2) ⊗ su(2) k 2 (6.1) su(2)k+2 coset models with p = k + 2. Let us label the sectors of su(2)k by l = 0, 1,...,k, the sectors of su(2)2 by m = 0, 1, 2, and the sectors of su(2)k+2 by l = 0, 1,...,k+2. The sectors of the direct product are thus labeled by the pairs (l, m). These sectors split up with respect to the su(2)k+2 subalgebra, Vl,m = Wl,m,l ⊗ Vl , (6.2) l where Wl,m,l are the sectors of the coset theory. Only those coset sectors are allowed for which l + m + l is even. (6.3) Furthermore, there are identifications between these admissible representations:

(l,m,l ) =∼ (k − l,2 − m, k + 2 − l ). (6.4) Using the p = k + 2 rule and the new variables (the Kac-indices)

r = l + 1,r= 1, 2,...,p− 1, s = l + 1,s= 1, 2,...,p+ 1, the precise relations with the superminimal ﬁelds are (see [17]):

(r, s)NS = (r − 1, 0,s− 1) ⊕ (r − 1, 2,s− 1), (6.6)

(r, s)R = (r − 1, 1,s− 1). (6.7) In the Neveu–Schwarz sector the direct sum in (6.6) corresponds to the sum of the Γ -projected subspaces. The identiﬁcation (6.4) translates to

(r, s) =∼ (p − r, p + 2 − s), (6.8) which is the well-known symmetry relation of the Kac table of the superminimal models. In the A-series of superminimal models the boundary conditions α are labeled by triplets (l,m,l) taking values in the same range as the sectors including selection and identification rules. The NS and NS boundary conditions correspond to the m = 0, 2 choices. The agreement between our results and the predictions for the flows justifies this correspondence. The boundary flows are predicted in the following way. First, we have to find a boundary condition α and a representation S so that + 0,S × α = (l,m,l )(l+ m + l is even). (6.9) su(2)k+2 Then a boundary flow is predicted between the following coset boundary configurations: + X := 0,S × α → (S, 0) × α =: Y. (6.10) su(2)k+2 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 371

If we want to study flows starting from the boundary condition (l,m,l) with 0 l k we can set α = (l, m, 0) and S = (l, 0). This choice corresponds to the flow → (k)J (l,m,l ) Nl,l (J, m, 0), (6.11) J (k)J where the Nl,l are the fusion coefficients in su(2)k. Similarly, in the case of 2 l k + 2 the choice α = (k − l,2 − m, 0) and S = (k + 2 − l, 0) leads to the flow → (k)J − (l,m,l ) Nl,l−2(J, 2 m, 0). (6.12) J The boundary conditions which are not covered by these rules are2 for l = k + 1, 1. These correspond to the (r, p) and (r, 2) states. (The boundary conditions (r, p + 1) and (r, 1) do not have relevant supersymmetric perturbations.) For l = k + 1 the suitable choice is S = (0, 1) and α = (l,m,k+ 2). Then (l, 0or2,k+ 1) → (l, 1,k+ 2) =∼ (k − l,1, 0), (6.13) (l, 1,k+ 1) → (k − l,0, 0) ⊕ (k − l,2, 0). (6.14) Finally, for l = 1 one should choose S = (0, 1) and α = (l, m, 0), which yields the flows (l, 0or2, 1) → (l, 1, 0), (6.15) (l, 1, 1) → (l, 0, 0) ⊕ (l, 2, 0). (6.16) Let us summarize the l = 0 case, when these results give (0,m,l ) → (l ,m,0), 0 l k, (6.17) (0, 0or2,k+ 1) → (k, 1, 0), l = k + 1, (6.18) (0, 1,k+ 1) → (k, 0, 0) ⊕ (k, 2, 0), (6.19)

(0,m,l ) → (l − 2, 2 − m, 0), 2 l k + 2, (6.20) (0, 1, 1) → (0, 0, 0) ⊕ (0, 2, 0). (6.21) Multiplying both sides of these rules by (l, 0, 0) we get back the rules for general l.IntheKac index language this is the “disorder line rule”, discussed in paper [18]: if there is a ﬂow between boundary conditions α and β, then there exists a ﬂow between α × γ and β × γ where × denotes the fusion product. For the boundary states of N = 1 superminimal models the fusion rule (2.8) should be used together with the rules3 NS × NS = NS × NS = NS, (6.22a) NS × NS = NS , (6.22b) R × NS = R × NS = R, (6.22c) R × R = NS ⊕ NS . (6.22d)

2 The author is thankful to S. Fredenhagen for the private discussion on these cases. 3 These are correct with the choice m = 0 ⇔ NS and m = 2 ⇔ NS. In case of the other choice a NS ↔ NS swap is needed. 372 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379

This means that we only need to determine the flows starting from states in the first row of the Kac table, all the other flows are given by the fusion rules of the operator algebra. The authors of paper [19] have also derived the flows for some Neveu–Schwarz states in the first row of the Kac table, but only for even p.

7. TCSA results

We were searching the answers for the following questions.

1. Does our TCSA analysis affirm or disprove the predictions of the Fredenhagen rules? 2. Do all the flows obey the disorder line rule? 3. What happens in the p even models, where both these rules and the classification of consistent boundary conditions are somewhat certain?

We have positive answers for the ﬁrst two questions. Namely, the following rules can be read out from the TCSA analysis of the boundary ﬂows of the unitary superconformal minimal model ˆ SM(p, p + 2) perturbed by the operator G−1/2φ1,3 (see also Appendix A): 1. For κ>0:

(a) For 2 s p − 1

(1,s)NS → (s, 1)NS, (7.1a) → (1,s)NS (s, 1)NS , (7.1b) (1,s)R → (s, 1)R. (7.1c) This agrees with the Fredenhagen result (6.17). According to the disorder line rule we found (cf. (6.11)): (r, s) → |s − r|+1, 1 ⊕ |s − r|+3, 1 ⊕··· ⊕ min s + r − 1, 2p − (s + r)− 1 , 1 . (7.2) These ﬂows do not change the type of the boundary condition so on the right-hand side all the boundary conditions are of the same type. Thus, if two of them have different fermion parity, then the half-integer levels of one should combine with the integer levels of the other. That is exactly what we observed. (b) For s = p

(1,p)NS → (p − 1, 1)R, (7.3a) → − (1,p)NS (p 1, 1)R, (7.3b) → − ⊕ − (1,p)R (p 1, 1)NS (p 1, 1)NS . (7.3c) Note that if the initial boundary condition is of Neveu–Schwarz type, then the result of the flow is a Ramond type boundary condition and vice versa. We also identified the flows related to the flow (7.3) by the disorder line rule:

(r, p)NS → (p − r, 1)R, (7.4a) → − (r, p)NS (p r, 1)R, (7.4b) → − ⊕ − (r, p)R (p r, 1)NS (p r, 1)NS . (7.4c) M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 373

2. For κ<0:

(a) For 3 s p → − (1,s)NS (s 2, 1)NS , (7.5a) → − (1,s)NS (s 2, 1)NS, (7.5b) (1,s)R → (s − 2, 1)R. (7.5c) This again gives back the Fredenhagen flow (6.20). We found that the disorder line rule applies: we identified the flows (r, s) → |r − s + 2|+1, 1 ⊕ |r − s + 2|+3, 1 ⊕··· ⊕ min s + r − 3, 2p − (s + r)+ 1 , 1 . (7.6) In the Neveu–Schwarz case these flows, just like their ancestors, change the type of the boundary conditions. The comment given in case 1(a) is true here as well. (b) For s = 2 → ⊕ (1, 2)R (1, 1)NS (1, 1)NS . (7.7) This is another example for a flow starting from a Ramond boundary condition running to a Neveu–Schwarz one. The disorder line rule is consistent with the flows

(r, 2)NS → (r, 1)R, (7.8a) → (r, 2)NS (r, 1)R, (7.8b) → ⊕ (r, 2)R (r, 1)NS (r, 1)NS . (7.8c)

It is interesting that despite the lack of a complete classiﬁcation of consistent boundary states for p even, our results indicate the same rules for these cases. The only exception is the bound- p p+2 ary condition corresponding to the supersymmetric ( 2 , 2 ) representation which could not be investigated directly by TCSA. It is also interesting to note that via the symmetry transformation of the Kac table (r → p − r, s → p + 2 − s)therules(7.2), (7.4) are mapped to the rules (7.6) and (7.8), respectively. This means that the two groups of rules (κ<0, κ>0) are not independent. We note that although one can only approach and never reach the ﬁxed point, the endpoints are (r, 1) type boundary conditions that have no relevant supersymmetric perturbations, so it is reasonable to take our observations to be strong indications for the real result.

8. Conclusions

In this work we investigated the renormalisation group flows of boundary conditions in N = 1 superconformal minimal models. Starting from a pure Cardy-type boundary condition, the ˆ (supersymmetry preserving) perturbating operator G−1/2Φ1,3 induces a renormalisation group flow whose infrared fixed points are again superconformal invariant boundary conditions. As we have shown, the Fredenhagen rules [16] for boundary flows in general coset theories predict the IR fixed points of our flows. The predicted flows obey a product rule called disorder line rule [18]. In this paper we used the TCSA method to check the Fredenhagen rules. We found that these rules held in every case that we investigated. It is interesting to note that there are flows which 374 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 go from a Neveu–Schwarz type boundary condition to a Ramond type, and vice versa. Finally, we showed that all the flows obey the disorder line rule. Although for p even both these predictions and the description of boundary states are flawed by the presence of the supersymmetric representation, our analysis implies that a natural extension of the rules are valid for these models. There are several possibilities for continuing this work. One of them is its extension to the study of the more complicated space of flows of non-Cardy type boundary conditions. Another possibility is to consider N = 2 models, which have direct relevance in string theory. However, although the TCSA analysis should be straightforward in principle, we expect too many states for reasonably high cuts, which can spoil the applicability of TCSA in practice. It would be interesting to include non-unitary models, too. The energies typically become complex in these models (like for ordinary minimal models, as reported in [10]), which makes the identification of the endpoint rather difficult.

Acknowledgements

I thank my supervisor Gábor Takács for his guidance, help and the interesting discussions. I am also grateful to Stefan Fredenhagen for his valuable remarks.

Appendix A

In the appendix we present the results of the TCSA for the first unitary models in detail. In the subsections we follow the classification of the flows given in Section 7. In the tables the “level” entry indicates the level up to which the agreement between the character and the calculated degeneracy pattern in the IR holds. In the fourth field of every row the dimension of the truncated Hilbert space is shown. The last field contains the number of the corresponding figure. In the figures collected at the end of the appendix we plotted the energy levels against the dimensionless coupling parameter κ. In these figures the starting boundary condition is unprojected for the Neveu–Schwarz ones (i.e., NS ⊕ NS), so these flows go to unprojected NS or R boundary conditions. In Fig. 1(a) the energy itself is plotted, while in all the other figures we plotted the normalised energy (ε − ε0)/(εi − ε0) for some appropriate i (usually 1).

A.1. First group of ﬂows: κ>0

Table 1 Flows starting from b.c. (r, s) when s p − 1 Model Flow Level dim Fig. SM(3, 5)(1, 2) → (2, 1) 11 302 1(c) SM(4, 6)(1, 3) → (3, 1) 16 454 2(a) (2, 2) → (1, 1) ⊕ (3, 1) 17, 15 536 2(b) (1, 2) → (2, 1) 11 414 SM(5, 7)(1, 3) → (3, 1) 13 388 (2, 2) → (1, 1) ⊕ (3, 1) 13, 11 311 (2, 4) → (3, 1) 12 601 3(b) (1, 2) → (2, 1) 11 341 (1, 4) → (4, 1) 7 357 3(c) (2, 3) → (2, 1) ⊕ (4, 1) 8, 6 303 3(d) M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 375

Table 1 (continued) Model Flow Level dim Fig. SM(6, 8)(1, 3) → (3, 1) 14 396 (1, 5) → (5, 1) 8 540 4(a) (2, 2) → (1, 1) ⊕ (3, 1) 13, 11 316 (2, 4) → (3, 1) ⊕ (5, 1) 12, 8 440 4(b) (1, 2) → (2, 1) 10 344 (1, 4) → (4, 1) 7 385 (2, 3) → (2, 1) ⊕ (4, 1) 8, 6 318 (2, 5) → (4, 1) 6 367 4(d) (3, 2) → (2, 1) ⊕ (4, 1) 9, 8 317 4(e)

Table 2 Flows starting from b.c. (r, s) when s = p Model Flow Level dim Fig. SM(3, 5)(1, 3) → (2, 1) 6 407 1(b) SM(4, 6)(1, 4) → (3, 1) 18 490 2(c) SM(5, 7)(1, 5) → (4, 1) 7 298 3(a) SM(6, 8)(1, 6) → (5, 1) 14 289 4(f) (2, 6) → (4, 1) 7 426 4(c)

A.2. Second group of ﬂows: κ<0

Table 3 Flows starting from b.c. (r, s) when s 3 Model Flow Level dim Fig. SM(3, 5)(1, 3) → (1, 1) 12 407 1(b) SM(4, 6)(1, 3) → (1, 1) 18 454 2(a) (1, 4) → (2, 1) 10 490 2(c) SM(5, 7)(1, 3) → (1, 1) 12 388 (1, 5) → (3, 1) 14 298 3(a) (2, 4) → (1, 1) ⊕ (3, 1) 15, 13 601 3(b) (1, 4) → (2, 1) 9 357 3(c) (2, 3) → (2, 1) 8 303 3(d) SM(6, 8)(1, 3) → (1, 1) 14 396 (1, 5) → (3, 1) 14 540 4(a) (2, 4) → (1, 1) ⊕ (3, 1) 13, 11 440 4(b) (2, 6) → (3, 1) ⊕ (5, 1) 17, 13 426 4(c) (1, 6) → (4, 1) 9 289 4(f) (1, 4) → (2, 1) 9 385 (2, 3) → (2, 1) 7 318 (2, 5) → (2, 1) ⊕ (4, 1) 7, 5 367 4(d) 376 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379

Table 4 Flows starting from b.c. (r, s) when s = 2 Model Flow Level dim Fig. SM(3, 5)(1, 2) → (1, 1) 9 302 1(c) SM(4, 6)(2, 2) → (2, 1) 9 536 2(b) (1, 2) → (1, 1) 22 414 SM(5, 7)(2, 2) → (2, 1) 7 311 (1, 2) → (1, 1) 20 341 SM(6, 8)(2, 2) → (2, 1) 7 316 (1, 2) → (1, 1) 20 344 (3, 2) → (3, 1) 16 317 4(e)

(a) (b)

(c)

Fig. 1. Flows in SM(3, 5). (a) Flows starting from b.c. (1, 3) in SM(3, 5). (b) Flows starting from b.c. (1, 3) in SM(3, 5). (c) Flows starting from b.c. (1, 2) in SM(3, 5). M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 377

(a) (b)

(c)

Fig. 2. Flows in SM(4, 6). (a) Flows starting from b.c. (1, 3) in SM(4, 6). (b) Flows starting from b.c. (2, 2) in SM(4, 6). (c) Flows starting from b.c. (1, 4) in SM(4, 6).

(a) (b)

Fig. 3. Flows in SM(5, 7). (a) Flows starting from b.c. (1, 5) in SM(5, 7). (b) Flows starting from b.c. (2, 4) in SM(5, 7). (c) Flows starting from b.c. (1, 4) in SM(5, 7). (d) Flows starting from b.c. (2, 3) in SM(5, 7). 378 M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379

(a) (b)

(e) (f)

Fig. 4. Flows in SM(6, 8). (a) Flows starting from b.c. (1, 5) in SM(6, 8). (b) Flows starting from b.c. (2, 4) in SM(6, 8). (c) Flows starting from b.c. (2, 6) in SM(6, 8). (d) Flows starting from b.c. (2, 5) in SM(6, 8). (e) Flows starting from b.c. (3, 2) in SM(6, 8). (f) Flows starting from b.c. (1, 6) in SM(6, 8).

References

[1] H. Saleur, Lectures on non-perturbative field theory and quantum impurity problems, cond-mat/9812110. [2] C. Angelantonj, A. Sagnotti, Open strings, Phys. Rep. 371 (2002) 1, hep-th/0204089. [3] J.A. Harvey, D. Kutasov, E.J. Martinec, On the relevance of tachyons, hep-th/0003101. [4] A. Recknagel, D. Roggenkamp, V. Schomerus, On relevant boundary perturbations of unitary minimal models, Nucl. Phys. B 558 (2000) 552, hep-th/0003110. [5]P.Dorey,I.Runkel,R.Tateo,G.Watts,g-function flow in perturbed boundary conformal field theories, Nucl. Phys. B 578 (2000) 85, hep-th/9909216. [6] F. Lesage, H. Saleur, P. Simonetti, Boundary flows in minimal models, Phys. Lett. B 427 (1998) 85, hep-th/9802061. [7] K. Graham, I. Runkel, G.M.T. Watts, Minimal model boundary flows and c = 1 CFT, Nucl. Phys. B 608 (2001) 527, hep-th/0101187. [8] K. Graham, I. Runkel, G.M.T. Watts, Renormalisation group flows of boundary theories, hep-th/0010082. M. Kormos / Nuclear Physics B 744 [FS] (2006) 358–379 379

[9] V.P. Yurov, Al.B. Zamolodchikov, Truncated conformal space approach to the scaling Lee–Yang model, Int. J. Mod. Phys. A 5 (1990) 3221. [10] P. Dorey, A. Pocklington, R. Tateo, G. Watts, TBA and TCSA with boundaries and excited states, Nucl. Phys. B 525 (1998) 641, hep-th/9712197. [11] G.M.T. Watts, Null vectors of the superconformal algebra: the Ramond sector, Nucl. Phys. B 407 (1993) 213, hep-th/9306034. [12] R.I. Nepomechie, Consistent superconformal boundary states, J. Phys. A 34 (2001) 6509, hep-th/0102010. [13] N. Ishibashi, The boundary and crosscap states in conformal field theory, Mod. Phys. Lett. A 4 (1989) 251. [14] J.L. Cardy, Boundary conditions, fusion rules and the Verlinde formula, Nucl. Phys. B 324 (1989) 581. [15] P. Mathieu, Integrability of perturbed superconformal minimal models, Nucl. Phys. B 336 (1990) 338. [16] S. Fredenhagen, Organizing boundary RG flows, Nucl. Phys. B 660 (2003) 436, hep-th/0301229. [17] P. Goddard, A. Kent, D.I. Olive, Unitary representations of the Virasoro and super-Virasoro algebras, Commun. Math. Phys. 103 (1986) 105. [18] K. Graham, G.M.T. Watts, Defect lines and boundary flows, JHEP 0404 (2004) 019, hep-th/0306167. [19] Ch. Ahn, Ch. Rim, Boundary flows in general coset theories, J. Phys. A 32 (1999) 2509, hep-th/9805101. Nuclear Physics B 744 [FS] (2006) 380–397

Transformations among large c conformal ﬁeld theories

Marcin Jankiewicz ∗, Thomas W. Kephart

Department of Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA Received 25 October 2005; received in revised form 15 March 2006; accepted 28 March 2006 Available online 18 April 2006

Abstract We show that there is a set of transformations that relates all of the 24 dimensional even self-dual (Niemeier) lattices, and also leads to non-lattice objects some of which can perhaps be interpreted as a basis for the construction of holomorphic conformal field theory. In the second part of this paper, we extend our observations to higher-dimensional conformal field theories build on extremal partition functions, where we generate c = 24k theories. We argue that there exists generalizations of the c = 24 models based on Niemeier lattices and of the non-Niemeier spin-1 theories. The extremal cases have spectra decompos- able into the irreducible representations of the Fischer–Griess Monster. This additional symmetry leads us to conjecture that these extremal theories, as well as the higher-dimensional analogs of the group lattice bases Niemeiers, will eventually yield to a full construction of their associated CFTs. We observe interesting periodicities in the coefficients of extremal partition functions and characters of the extremal vertex operator algebras. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

It was proven by Dixon et al. [1] that the partition function Z of an arbitrary c = 24 holomorphic conformal ﬁeld theory based on R24/Λ, where Λ could be any of the 24 even self-dual Niemeier lattices in 24 dimensions, can be written as follows

Z = J + 24(h + 1). (1)

* Corresponding author. E-mail addresses: [email protected] (M. Jankiewicz), [email protected] (T.W. Kephart).

Here J is the weight zero modular function (with constant term set equal to zero), and h is a Coxeter number for a lattice solution. We will show that any Niemeier lattice Λ1, represented in terms of the Θ-series related to the partition function (1), can be obtained from another Niemeier Λ2. We accomplish this using projections that rearrange the lattice points to form a new lattice. Only for the particular combination of the projection parameters corresponding to the Coxeter numbers of the Niemeier lattices, do we have a lattice as a solution. For other combinations non-lattice solutions are obtained. Since Θ-functions of the 24-dimensional lattices are modular forms of weight 12, and partition functions are modular invariant (weight zero), a natural question to ask is, which object is (more) physical? The answer to this question depends on the system being investigated. Any Niemeier lattice can be used as a starting point, i.e., any Θ-function corresponding to 0 a lattice can be used for the initial Θ-function Z0 . The role of the transformation parameters is simple, they rearrange vectors in a lattice, e.g., by rescaling. The number of transformation 0 parameters depends strictly on an initial choice of the Θ-function Z0 , and hence on the number of different conjugacy classes or the number of canonical sublattices in a lattice. For example, 3 for E8 , which is one of the Niemeier lattices, we have initially three parameters, one for each SO(16) spinor conjugacy class. In the case of D16 × E8 we will have 4 parameters, that build up the sublattice. However, in these examples, upon constructing the new Θ-series (or partition function), the initial number of parameters can be reduced to a single independent parameter, leading to the Θ-function related to (1) with h being represented by the last free transformation parameter. Parametrization of the twisted sector has been used to obtain new theories from 16 dimensional even self-dual lattices [2,3]. These include theories that were already known, like supersymmetric E8 × E8, non-supersymmetric but non-tachyonic SO(16) × SO(16), and non- supersymmetric and tachyonic E8 × SO(16), etc. We generalize the analysis of [2,3], to 24- dimensional lattices and also relax the constraints on the transformation parameters, i.e., we will no longer be working with just Z2 actions acting on the conjugacy classes, but rather with more complicated actions. The plan of the paper is as follows. In Section 2 we review modular transformations and 16D lattices. Section 3 introduces our transformation procedure for c = 24 theories. We rederive equation (1) and demonstrate some specific patterns of lattices that can be obtained by this method. Most of our new results are contained in Section 4 where we provide examples with c = 24k via a composition construction of k modular invariant c = 24 partition functions. The Leech lattice is an extremal case for k = 1 where the coefficient of q2 vanishes and where all other coefficients in the q-expansion decompose into irreducible representations of the Fischer–Griess Monster group. We show that when k>1, the coefficients of the analog extremal cases also decompose into Monster group representations. Furthermore, we demonstrate interesting periodicities in k. The symmetry due to this higher-dimensional Monster Moonshine leads us to conjecture that the extremal examples will correspond to lattices and corresponding CFTs. The final sections contains this conjecture and a discussion.

2. Modular transformations and a 16-dimensional example

We start with some basic definitions and a classic example already present in the literature [2,3]. A formal definition of a Θ-series of an even self-dual lattice Λ is m ZΛ = N(m)q , (2) x∈Λ 382 M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397 where N(m) is a number of vectors with length squared equal to an even number m.Froma mathematical point of view, ZΛ is a modular form of weight dim(Λ)/2. Let us recall the difference between ZΛ and a partition function Z. Namely, a partition function Z is a modular function (a modular form of weight zero). By looking at the modular properties of ZΛ we conclude that it dim(Λ) is related to Z by Z = ZΛ/η , where ∞ η(q) = q1/12 1 − q2m (3) m=1 is a modular form of weight 1/2 called the Dedekind η-function with q = eπiτ and τ is a modular parameter. Most of the time, we will focus on the lattices, hence we will work with Θ-functions. However we will make some remarks about partition functions as well. Let us investigate the relationship between SO(32) and E8 × E8 compactification lattices. The Θ-function of both can be expressed in terms of different conjugacy classes of the SO(16) × SO(16) lattice, which is a maximal common subgroup of both SO(32) and E8 × E8. SO(2N)groups have four conjugacy classes namely, the adjoint (IN ), the vector (VN ), the spinor (SN ), and the conjugate spinor (CN ). They can be expressed in terms of Jacobi-θ functions as follows [4]: 1 1 I ≡ θ N + θ N ,V≡ θ N − θ N , (4) N 2 3 4 N 2 3 4 1 1 S ≡ θ N ,C≡ θ N , (5) N 2 2 N 2 2 where N is the rank of SO(2N).Bothspinor and conjugate spinor have the same Θ-expansions. Before going further, let us recall some of the properties of the modular group Γ SL(2, Z). It is generated by 11 0 −1 T = and S = , (6) 01 10 n n − n and any element in this group can be written as T k ST k 1 S ···ST 1 where the ni are integers [5]. The transformation rules for Jacobi-θ functions are given in Appendix A by (A.2)–(A.7), and from them we obtain S-transformed conjugacy classes 1 1 S(I ) = (I + V + S + C ), S(V ) = (I + V − S − C ), (7) N 2 N N N N N 2 N N N N 1 1 S(S ) = (I − V ), S(C ) = (I − V ), (8) N 2 N N N 2 N N and T-transformations acting on S-transformed conjugacy classes of SO(2N) give 1 1 TS(I ) = (I − V + S + C ), TS(V ) = (I − V − S − C ), (9) N 2 N N N N N 2 N N N N 1 1 TS(S ) = (I + V ), TS(C ) = (I + V ). (10) N 2 N N N 2 N N We will use a TS transformation to restore the modular invariance of the new partition functions resulting from the construction presented below. Now consider the Θ-function of the SO(32) lattice, which is given in terms of SO(16) × SO(16) conjugacy classes by = 2 + 2 + 2 + 2 ZSO(32) I8 V8 S8 C8 , (11) M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397 383

Table 1 A Z2 transformation applied to ZSO(32)

I8 V8 S8 C8 First SO(16) ++ ++ Second SO(16) +− +−

where squares are just a short-hand notation for (I8,I8),etc.AZ2 action on the conjugacy classes means we have to multiply a given conjugacy class by ±1. If one chooses to act with transformations that ﬂip the sign of the vector and conjugate spinor representations of the second SO(16) [3],asshowninTable 1 one gets a new Θ-series = 2 − 2 + 2 − 2 Z1 I8 V8 S8 C8 . (12)

It is obvious that Z1 is not modular invariant as can be seen from its q-expansion. I.e., some of the coefﬁcients are negative. In order to restore wanted modular properties, one has to add the S and the TS transformed forms of Z1:

SZ1 = I8S8 + S8I8 + C8V8 + V8C8, (13)

TSZ1 = I8S8 + S8I8 − C8V8 − V8C8. (14) Adding partition functions (11)–(14), and also taking into account the overall normalization, one obtains the Θ-function of the dual E8 × E8 theory, namely

1 2 (Z + Z + SZ + TSZ ) = Z × = (I + S ) . (15) 2 SO(32) 1 1 1 E8 E8 8 8 3. Transformation model in 24 dimensions

Now we move on to 24-dimensional lattices. We begin by writing the D16 × E8 Niemeier 1 lattice represented in terms of D8 ×D8 ×D8 conjugacy classes. We follow the standard notation 0 1 0 = 1 (see, for example, [1]), where Z0 is the initial untwisted sector, Z0 is a projection, Z1 SZ0 and 1 = 0 Z1 TZ1 are modular transformed (twisted) sectors. Therefore 0 = 2 + 2 + 2 + 2 + Z0 I V S C (I S), (16) which, after evaluation of conjugacy classes in terms of Jacobi-θ functions, can be expanded into a q-series 0 = + 2 + 4 + O 6 Z0 1 720q 179280q q . (17) Now let us make a projection, which in the most general way, can be written as 1 = 2 + 2 + 2 + 2 + Z0 I aV bS cC (I eS), (18) where a, b, c and e are projection parameters, that change the signs and/or scale the conjugacy 1 classes. We also evaluate the q-expansion of Z0 to see the inﬂuence of the projection we have just made 1 = + + + 2 + O 4 Z0 1 16(21 16a 8e)q q . (19)

1 We omit subscript “8” in the notation. 384 M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397

This series is still even in powers of q, however self-duality may have been lost. Using transformation properties (7)–(8) we get twisted sectors 0 ≡ 1 = 1 + + + 2 + + − − 2 + − 2 + − 2 Z1 SZ0 (I V S C) a(I V S C) b(I V) c(I V) 8 × I + V + S + C + e(I − V) , (20) and 1 ≡ 1 = 1 − + + 2 + − − − 2 + + 2 + + 2 Z1 TSZ0 (I V S C) a(I V S C) b(I V) c(I V) 8 × I − V + S + C + e(I + V) . (21) New transformed (orbifolded under certain circumstances) theories can be represented in the following general form 1 Z = Z0 + Z1 + Z0 + Z1 . (22) new 2 0 0 1 1 We have chosen S and TS transformations to restore modular invariance of a Θ-function. Neither 0 1 0 Z1 nor Z1 is even. We can see this directly from the q-expansion of Z1 0 = 1 + + + + + − − + − − + + Z1 (1 a b c)(1 e) 2 2 b c e 3be 3ce a(3 e) q 8 + O q2 . (23)

0 However, when added to Z1,theTS transformed Θ-function eliminates terms with odd powers in q, i.e., its q-expansion is 1 = 1 + + + + − − − + − − + + Z1 (1 a b c)(1 e) 2 2 b c e 3be 3ce a(3 e) q 8 + O q2 . (24) An even self-dual lattice scaled by the appropriate power of Dedekind η-function is by definition a modular function, but modular invariance does not necessarily imply we have a lattice; therefore modular invariance is a more general concept.2 But as we see we are safe; the odd terms in the twisted sectors have cancelled. Note, Znew is modular invariant regardless of the values of the transformation parameters. However, a q-expansion of a theory transformed in a way described above has to be properly normalized. Normalization is not yet guaranteed, since the zeroth order term in the q-expansion is given by 1 1 + (1 + a + b + c)(1 + e). (25) 8 For e = 1, we are left with only one combination of the parameters a,b and c, such that normalization of the zeroth order term is fixed to 1, i.e., a + b + c =−1. As a result of this fixing, one is left with

3 Znew = (I + S) . (26)

2 This is important since only even and self-dual lattices can be used as a basis for a compactiﬁcation torus. M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397 385

3 The resulting lattice corresponds to the E8 × E8 × E8 isospectral partner of D16 × E8. But there is a more interesting case. For e =−1 the normalization condition is already ﬁxed, but this puts no constraints on the rest of the parameters. Moreover, the parameters (a,b,c) are found in a speciﬁc combination4 at every order in q2

x ≡ 3a − b − c. (27)

We show this by explicitly evaluating Znew

2 4 6 Znew = 1 + 48(13 + 2x)q − 144(−1261 + 16x)q + 4032(4199 + 6x)q +···, (28) which can be rewritten 2 4 6 2 4 6 Znew = 1 + 624q + 181584q + 16930368q +··· + 96x q − 24q + 252q +··· , (29) wherewehavedividedZnew into x-dependent and x-independent parts. The number of independent parameters (after one fixes e) is reduced to one. This parameter, x, can be related to the Coxeter number h of a given lattice. The relation is model dependent and depends on an initial 0 choice of Z0. For the case at hand h − 26 x = . (30) 4 We observe that the Θ-series in the first square bracket in (29) is an even self-dual (i.e. invariant under S and T) function. It does not correspond to a lattice solution. However it can be related to the J -invariant (see (A.8) in Appendix A). Terms in the second square bracket in (29) form a unique cusp form of weight 12 [5], which can be written as the 24th power of the Dedekind η-function. Using this knowledge we can write our solution in a more compact form 24 Znew = J + 24(h + 1) η , (31) where h is a positive integer and is equal to the Coxeter number of the 24 dimensional even self-dual lattice if Znew forms a Niemeier lattice. Only for specific values of x does one get a solution that corresponds to a lattice. However, for the majority of cases, representation in terms of group lattices is not possible. Physically this means that the primary fields of the CFT do not transform under any group (there are exceptions to be discussed below, see [22]), hence one is left with 24 · h singlets. We now classify all solutions that can be derived by this technique. Using (30) in (31) one finds the allowed values of x form a set of 8191 elements.5 This is true under two conditions, first we assume that all the coefficients in a q-expansion are positive integers [6,8]. This assumption is not only reasonable but also physical, since these coefficients give us the number of states at each string mass level from the partition function point of view, and from the lattice point of view they correspond to the number of sites in each layer. The second assumption is that the kissing number for both lattices and non-lattices is an integer number which can change by one.6

3 A pair of lattices is said to be isospectral if they have the same Θ-expansion. 4 For other values of e the analysis gets more complicated. 5 For what it is worth 8191 is a Marsenne prime. 6 One could assume that a kissing number for a non-lattice solution is still a multiple of 24, as with lattices. Then modular properties of the Θ-series are preserved, but the number of solutions is reduced to 342. 386 M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397

Table 2 Relatives of D16E8 lattice, and their parametrization Lattice hxabc

D24 46 5 1 −1 −1 3 E8 30 1 1 1 1 2 − − − − D12 22 1 1 1 1 2 − E7 D10 18 20 1 1 3 − − − D8 14 3 111

Table 3 Maximal subgroups of D16E8 lattice, and their parametrization Lattice hxabc 4 − − D6 10 4 110 6 − − D4 6 5 111 24 − − A1 2 6 211

Remembering that our starting point was the D16E8 lattice, one can immediately see that integer values of (a,b,c)parameters correspond to “relatives”7 of this lattice [6]. Table 2 shows 2 the transformation parameters for all five relatives of D16E8. All except E7 D10, can be gotten Z 3 = = = from the action of a 2. Since E8 and D16E8 are isospectral, we can set a b c 1in 4 6 that case. There are three other possible “integer combinations” for x, corresponding to D6 , D4 24 and A1 (Table 3), that are maximal subgroups of D16E8. Other integer values for x that reach Niemeier lattices cannot be expressed in terms of integer values of (a,b,c), which means that more complicated actions are needed. We list all the lattice solutions and the corresponding x parametrization in Table 4. Other values of (a,b,c) give the other Niemeier lattices and non- lattices solutions. 3 As a second example, we choose another Niemeier lattice E8 with a Θ-function 0 = + 3 Z0 (I S) , (32) where the projected sector is 1 = + + + Z0 (I aS)(I bS)(I cS). (33) 0 1 Again to restore modular invariance we introduce twisted sectors Z1 and Z1.Asaresultanew modular invariant Θ-function is obtained, and can be written in the form (31), except that the relation between Coxeter number and transformation parameters is different. For example, after fixing b = c =−1 we are left with one free parameter a = (h − 22)/8, which tells us that the space of the values of the a parameters reaches all 8191 solutions. If one restricts parameters (a,b,c)to ±1 one gets a family of simple Z2 actions transforming 3 the original lattice into the relatives D16E8 and D8D8D8 of E8 . Let us finish this section with the following simple observation. The Z2 orbifold actions are the actions which break/restore symmetry in a special way. If Λ1 and Λ2 have a common maximal subgroup then there is a Z2 action that transforms Λ1 into Λ2. This is not the case when

7 We call a pair of group lattices relatives if they share a common maximal subgroup. M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397 387

Table 4 All lattice solutions in 24 dimensions. The second column represents the Coxeter number h. The third column shows x 0 = parametrization when Z0 ZD16E8 . The glue code in the explicit form is given for most of the lattices and in a generator 8 12 24 + form for A3,A2 ,A1 as in [6]. Numbers in this column represent conjugacy classes, means combination of adjoint and spinor conjugacy classes, 0 stands for the adjoint, 1, 2, 3 are vector, spinor and conjugate spinor for Dn lattices (similarly for An with n − 1, and E6, E7 with two conjugacy classes). In the last three columns the ﬁrst three coefﬁcients in the Θ-series are listed with a2 being a kissing number for a given lattice (except for Leech)

Lattice Coxeter x Glue code a2 a4 a6

E8 30 + 240 2160 6720 2 ++ E8 30 480 61920 1050240 D16 30 + 480 61920 1050240

D24 46 5 + 1104 170064 17051328 D16E8 30 1 ++ 720 179280 16954560 3 +++ E8 30 1 720 179280 16954560 A24 25 −1/4 [0]+2([5]+[10]) 600 182160 16924320 2 − [ ]+ [ ]+[ ] D12 22 1 00 2 12 11 528 183888 16906176 A17E7 18 −2 [00]+2[31]+2[60]+[91] 432 186192 16881984 2 − [ ]+[ ]+[ ]+[ ] D10E7 18 2 000 110 310 211 432 186192 16881984 A15D9 16 −5/2 [00]+2([12]+[16]+[24]) +[08] 384 187344 16869888 3 − [ ]+ [ ]+[ ] +[ ] D8 14 3 000 3( 011 122 ) 111 336 188496 16857792 2 − [ ]+ [ ]+[ ]+[ ] A12 13 13/4 00 4( 15 23 46 ) 312 189072 16851744 A11D7E6 12 −7/2 [000]+2[111]+2[310]+2[401]+2[511]+[620] 288 189648 16845696 4 − [ ]+ [ ] E6 12 7/2 0000 8 1110 288 189648 16845696 2 − [ ]+ [ ]+[ ]+[ ]+[ ] + [ ]+[ ] A9D6 10 4 000 4( 121 132 240 341 ) 2 051 552 240 190800 16833600 4 − [ ]+ [ ]+[ ]+[ ]+[ ] D6 10 4 0000 12 0123 1111 2222 3333 240 190800 16833600 3 − [ ]+ [ ]+[ ]+[ ]+[ ] + [ ] A8 9 17/4 000 6( 114 330 122 244 ) 2 333 216 191376 16827552 2 2 − [ ]+ [ ]+[ ]+[ ]+[ ]+[ ] A7D5 8 9/2 0000 4( 0211 1112 2202 3312 2411 ) 192 191952 16821504 + 8[1301]+2[0422]+[4400] 4 − [ ]+ [ ]+ [ ]+[ ]+[ ] A6 7 19/4 0000 24 0123 8( 1112 1222 2223 ) 168 192528 16815456 4 − [ ]+ [ ]+ [ ]+[ ] A5D4 6 5 00000 6 00331 24( 00121 12231 ) 144 193104 16809408 + 8([02220]+[11130]) +[33330] 6 − [ ]+ [ ]+ [ ] D4 6 5 000000 45 000011 18 111111 144 193104 16809408 6 − [ ]+ [ ]+ [ ] A4 5 21/4 000000 60 001122 40 111222 120 193680 16803360 + 12[011111]+12[022222] 8 − [ ]+ A3 4 11/2 3(2001011) cyclic permutations of (2001011) 96 194256 16797312 12 − [ ]+ A2 3 23/4 2(11211122212) cyclic permutations of (11211122212) 72 194832 16791264 24 − [ ] A1 2 6 1(00000101001100110101111) 48 195408 16785216 + cyclic permutations of (00000101001100110101111) Leech 0 −13/2 None 0 196560 16773120

Λ1 and Λ2 do not have a common maximal subgroup. Therefore, this is possible for only a few pairs of Niemeier lattices (see Table 5). This result follows the lines of the procedure discussed by Dolan et al. in [7]. We want to emphasize that the main point of this section was to introduce transformations between extant lattices, and hence between known partition functions of their holomorphic 388 M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397

Table 5 Patterns of lattices obtained by Z2 actions → 3 E8D16 D8 ↓ 3 → 3 → 6 → 24 → E8 D8 D4 A1 Leech → 2 → 4 → 8 D24 D12 D6 A3 2 → 2 2 E7 D10 A7D5 conformal field theories. We do not claim that the other class of solutions we found (namely non-lattices) are new CFTs. They are however modular functions and can be generated from the choice of the projection parameter which corresponds to the analog of a Coxeter number of the non-existent Niemeier lattice, so they remain CFT candidates. We delay further discussion of these theories until we reach the discussion section. In the next section we discuss the generalization of these results to k>1. The examples in this section will serve as a basis for construction of potential higher-dimensional CFTs based on generalizations of Niemeier lattices, and in particular on generalized extremal Θ-functions or partition functions. The list of physical requirements which has to be imposed on a one-loop partition function of a conformal field theory was given, for example, in [8,9]. All of these constraints are satisfied by any ‘candidate CFT’ build on an even self-dual lattice presented in this section and known earlier in the literature [1]. Following this line of reasoning we proceed to focus on potential classes of CFTs build on extremal even self-dual lattices in dimensions 24k if and when they exist, or if not, on extremal even self-dual Θ-series with c = 24k. This avoids possible (or almost certain) complications of constructions based on non-lattice objects.

4. c = 24k extremal Θ series

In this section we generalize our procedure to higher dimensions. We concentrate on c = 24kΘseries which are in some cases 24k-dimensional lattices. Their Θ-functions can be ex- 8 pressed in terms of positive integer powers of Znew given in (31). For example, one can use them in the construction of the lattices with dense packing in 48 dimensions and the highest packing in 24. These lattices are build on the so-called extremal Θ-functions. For k = 1the kissing number (so the first non-zero coefficient in the q-expansion of the lattice) of the lattice with the highest packing is obtained as follows. The coefficients a2 and a4 in the q-expansion 2 4 1 + a2q + a4q +··· are constrained by the equation 24a2 + a4 = 196560 (see below). From this we see that the maximum packing corresponds to the choice a2 = 0 and a4 = 196560. In general, the equivalent of a 24 dimensional even self-dual lattice, i.e., modular invariant lattice, can be obtained from

24 2 4 6 η (J + 24 + a2) = 1 + a2q + (196560 − 24a2)q + 252(66560 + a2)q +···, (34) where a2 is a positive integer. The extremal a2 = 0 case is a Leech lattice. In order to preserve wanted properties we have to put constraints on values of the integer a2. It is easy to see that a2 ∈[0, 8190], generates q-expansion with positive entries. The same kind of constraint can be

8 However, one can generalize this procedure to other dimensions (8k) as well. M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397 389 imposed in 48 dimensions, where a modular invariant Θ-series is written in the general form η48(J + 24 + a)(J + 24 + b) = 1 + (a + b)q2 + 2 · 196560 − 24(a + b) + ab q4 + 12 2795520 + 16401(a + b) − 4ab q6 +···. (35) By rewriting the expression as

2 4 6 1 + a2q + a4q + (52416000 + 195660a2 − 48a4)q +··· (36) we see that a dense packing with kissing number 52 416 000 is obtained if one chooses a2 = a4 = 0. It is the P48 lattice [6]. In 72 dimensions we have 2 4 6 8 1 + a2q + a4q + a6q + (6218175600 + 57091612a2 + 195660a4 − 72a6)q +···. (37) In this case one would expect a lattice corresponding to the extremal Θ-series would be obtained by setting a2, a4 and a6 to zero so that the corresponding kissing number would be 6218175600 except for the fact that this Θ-series is not known to correspond to a lattice [10]. The extremal Θ-series in 24k dimensions obtained from this procedure is gotten by the requirement that all 9 of the coefficients a2,...,a2k vanish. We can find in principle the number of solutions with this parametrization, i.e., sensible Θ-functions in 24k dimensions. In 24 dimensions the values of a2 were constrained. In the rest of the cases, i.e., k>1, the number of independent parameters is k. However again the parameter space is finite. Using this information one can calculate the number of possible Θ-functions in any dimension. For example, in 48 dimensions we find 806022416786149 Θ-series. In this plethora of possibilities, we expect only some small fraction can be interpreted as lattices, which can be related to CFT.10 Finally the partition function for any 24k-dimensional theory contains a finite number of tachyons. For k = 1 there is a single tachyon with (m)2 =−1, for k>1wehavek tachyon levels in the spectrum. The most general formula of a partition function in 24k dimensions is

k ∞ 1 − J (x) ≡ (J + 24 + x ) = 1 + f (x ,...,x )q2m 2k , (38) k m q2k 2m 1 k m=1 m=(k−1) where x = (x1,...,xk) and f2m 0 are polynomials in the xi . The lowest (tachyonic) state with (m)2 =−k is always populated by a single tachyon, and higher states are functions of (x1,x2,...,xk).Thexi s can be chosen in such a way that all tachyon levels above the lowest level are absent, hence the next populated level would be occupied by massless states. This choice involves the elimination of k − 1 parameters. The ﬁnal series would then depend on a single parameter xk, more precisely the massless level is a polynomial in xk of the order k. The remainder of the spectrum does not depend on the choice of xk, in analogy with the 24-dimensional case. What is appealing in these models is that for k 1 we can have a single tachyonic state with arbitrarily large negative mass square that could potentially decouple from the spectrum leaving only states with (m2) 0, and the partition function of such a theory is still a well deﬁned modular function. This may be an alternative to tachyon condensation [12].

9 Alternatively, we can use the bound known in from the lattice theory [11], that the minimal norm of n-dimensional [ n ]+ unimodular lattice is μ 24 2. 10 For example, there are 242 even self-dual CFTs constructed out of 24-dimensional Niemeier lattices. This is a subset of the even self dual lattices in 48D. Even at 32D the number of even self dual lattices is known to be very large. 390 M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397

Let us evaluate a few examples with only a single tachyon coupled to the identity at q−2k for k = 1, 2, 3 and 4, which we define as Gk = Jk|extremal. These are: G (x ) = 1/q2 + (24 + x ) + 196884q2 +···, 1 1 1 G (x ) = 1/q4 + 393192 − 48x − x2 + 42987520q2 +···, 2 2 2 2 G (x ) = 1/q6 + 50319456 − 588924x + 72x2 + x3 + 2592899910q2 +···, 3 3 3 3 3 G = 8 + − − + 2 − 3 − 4 4(x4) 1/q 75679531032 48228608x4 784080x4 96x4 x4 + 80983425024q2 +···. (39) The allowed values of the polynomial coefficient of q0 are integers that run from zero to the value of the q2 coefficient. If one changes k, then the q2 coefficients are Fourier coefficients of the unique weight-2 normalized meromorphic modular form for SL(2, Z) with all poles at infinity [13]. The partition functions (39) are examples of the replication formulas [14]. There exists an interesting alternative set of possible CFTs with partition functions [15] that we will call Hk, where for different values of k we have 2 2 H1 = 1/q + 196884q +···, 4 2 H2 = 1/q + 1 + 42987520q +···, 6 2 2 H3 = 1/q + 1/q + 1 + 2593096794q +···, 8 4 2 2 H4 = 1/q + 1/q + 1/q + 1 + 81026609428q +···. (40) Again we fix the tachyon levels by appropriate choices of the xs. Note that the q2 coefficients for k = 1 and 2 coincide in (39) and (40) but not for larger k. These are characters of the extremal vertex operator algebra of rank 24k (if it exists) [15]. These characters were obtained by requiring + k = − − 24k m=1 xm 0 (so that the (k 1) state is empty), all other coefficients of tachyon levels up the to massless states are fixed to one. The extremal 24-dimensional case has been shown to be related to the Fischer–Griess Monster group. In fact G1(x1) is the modular function j when x1 =−24. j has the expansion j = 1/q2 + 196884q2 + 21493760q4 + 864299970q6 + 20245856256q8 +··· (41) and the coefficients of this expansion decompose into dimensions of the irreducible representations of the Monster (see Table 6), where we use the notation

2 2 4 j = 1/q + j2q + j4q +···. (42)

Table 6 Decomposition of the coefﬁcients of j into irreducible representations of the Monster group (for more see [16,17]) J -invariant Monster j2 196884 1 + 196883 j4 21493760 1 + 196883 + 21296876 j6 864299970 2 · 1 + 2 · 196883 + 21296876 + 842609326 j8 20245856256 3 · 1 + 3 · 196883 + 21296876 + 2 · 842609326 + 18538750076 j10 333202640600 4 · 1 + 5 · 196883 + 3 · 21296876 + 2 · 842609326 + 18538750076 + 19360062527 + 293553734298 j12 4252023300096 3 · 1 + 7 · 196883 + 6 · 21296876 + 2 · 842609326 + 4 · 19360062527 + 293553734298 + 3879214937598 M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397 391

Table 7 Coefﬁcients of 24k-dimensional extremal partition functions Gk in terms of coefﬁcients j2n of modular function j kg2 g4 g6 g8 g10 g12 22j4 2j8 + j2 2j12 2j16 + j4 2j20 2j24 + j6 33j6 3j12 3j18 + j2 3j24 3j30 3j36 + j4 44j8 4j16 + 2j4 4j24 4j32 + 2j8 + j2 4j40 4j48 + 2j12 55j10 5j20 5j30 5j40 5j50 + j2 5j60 66j12 6j24 + 3j6 6j36 + 2j4 6j48 + 3j12 6j60 6j72 + 3j18 + 2j8 + j2 ...... kg14 g16 g18 g20 g22 g24 22j28 2j32 + j8 2j36 2j40 + j10 2j44 2j48 + j12 33j42 3j48 3j54 + j6 3j60 3j66 3j72 + j8 44j56 4j64 + 2j16 + j4 4j72 4j80 + 2j20 4j88 4j96 + 2j24 + j6 55j70 5j80 5j90 5j100 + j4 5j110 5j120 66j84 6j96 + 3j24 6j108 + 2j12 6j120 + 3j30 6j132 6j144 + 3j36 + 2j16 + j4 ...... kg26 g28 g30 g32 g34 g36 22j52 2j56 + j14 2j60 2j64 + j16 2j68 2j72 + j18 33j78 3j84 3j90 + j10 3j96 3j102 3j108 + j12 44j104 4j112 + 2j28 4j120 4j128 + 2j32 + j8 4j136 4j144 + 2j36 55j130 5j140 5j150 + j6 5j160 5j170 5j180 66j156 6j168 + 3j42 6j180 + 2j20 6j192 + 3j48 6j204 6j216 + 3j54 + 2j24 + j6 ...... kg38 g40 g42 g44 g46 g48 22j76 2j80 + j20 2j84 2j88 + j22 2j92 2g96 + j24 33j114 3j120 3j126 + j14 3j132 3j138 3j144 + j16 44j152 4j160 + 2j40 + j10 4j168 4j176 + 2j44 4j184 4j192 + 2j48 + j12 55j190 5j200 + j8 5j210 5j220 5j230 5j240 66j228 6j240 + 3j60 6j252 + 2j28 6j264 + 3j66 6j276 6j288 + 3j72 + 2j32 + j8 ...... kg50 g52 g54 g56 g58 g60 22j100 2j104 + j26 2j108 2j112 + j28 2j116 2j120 + j30 33j150 3j156 3j162 + j18 3j168 3j174 3j180 + j20 44j200 4j208 + 2j52 4j216 4j224 + 2j56 + j14 4j232 4j240 + 2j60 55j250 + j10 5j260 5j270 5j280 5j290 5j300 + j12 66j300 6j312 + 3j78 6j324 + 2j36 6j336 + 3j84 6j348 6j360 + 3j90 + 2j40 + j10 ......

2n For 24k one can expand the q coefficients of Gk in terms of j coefficients which in turn can be expanded in terms of the dimensions of irreducible representations of the Monster. Table 7 demonstrates explicitly how the coefficients of the extremal 24k partition functions are decomposed into the coefficients of j. Observe that the pattern of the g2n coefficients in the kth row in Table 7 is periodic with period k. The first k rows of the table of g coefficients is overall k! periodic. As the periodicity holds for many coefficients, as shown in Tables 7 and 8, we conjecture it continues to hold for all k. The polynomial conditions to be satisfied to find the extremal partition functions for large k become increasingly more difficult to solve with increasing k,so 392 M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397

Table 8 Periodicity of the coefficients gn for c = 24k extremal partition functions Gk,andforhn coefficients of characters of the extremal vertex operator algebras Hk in terms of coefficients the j2n of the modular function j (k = 6 case for hn is not displayed since it is long but it has period 24) k = 2 k = 2 k = 2 k = 2 g4i+2 2j2(4i+2) h4i+2 2j2(4i+2) g4i+4 2j2(4i+4) + j2(2i+2) h4i+4 2j2(4i+4) + j2(2i+2) k = 3 k = 3 k = 3 k = 3 g6i+2 3j3(6i+2) h6i+2 3j3(6i+2) + j6i+2 g6i+4 3j3(6i+4) h6i+4 3j3(6i+4) + j6i+4 g6i+6 3j3(6i+6) + j2i+2 h6i+6 3j3(6i+6) + j2i+2 + j6i+6 k = 4 k = 4 k = 4 k = 4 g8i+2 4j4(8i+2) h8i+2 4j4(8i+2) + 2j2(8i+2) + j8i+2 g8i+4 4j4(8i+4) + 2j2(2i+4) h8i+4 4j4(8i+4) + 2j2(2i+4) + 2j2(8i+4) + j8i+4 + j4i+2 g8i+6 4j4(8i+6) h8i+6 4j4(8i+6) + 2j2(8i+6) + j8i+6 g8i+8 4j4(8i+8) + 2j(8i+8) + j2i+2 h8i+8 4j4(8i+8) + 2j(8i+8) + j2i+2 + 2j2(8i+8) + j8i+8 + j4i+4 k = 5 k = 5 k = 5 k = 5 g10i+2 5j5(10i+2) h12i+2 g12i+2 + 3j3(12i+2) + 2j2(12i+2) + j12i+2 g10i+4 5j5(10i+4) h12i+4 g12i+4 + 3j3(12i+4) + 2j2(12i+4) + j12i+4 + j6i+2 g10i+6 5j5(10i+6) h12i+6 g12i+6 + 3j3(12i+6) + 2j2(12i+6) + j12i+6 + j4i+2 g10i+8 5j5(10i+8) h12i+8 g12i+8 + 3j3(12i+8) + 2j2(12i+8) + j12i+8 + j6i+4 g10i+10 5j5(10i+10) + j2i+2 h12i+10 g12i+10 + 3j3(12i+10) + 2j2(12i+10) + j12i+10 h12i+12 g10i+12 + 3j3(12i+12) + 2j2(12i+12) + j12i+12 + j6i+6 + j4i+4 k = 6 k = 6 g12i+2 6j6(12i+2) g12i+4 6j6(12i+4) + 3j3(6i+2) g12i+6 6j6(12i+6) + 2j2(4i+2) g12i+8 6j6(12i+8) + 3j3(6i+4) g12i+10 6j6(12i+10) g12i+12 6j6(12i+12) + 3j3(6i+6) + 2j2(4i+4) + j2i+2 we do not have results for k>6. Table 8 give the general periodicity. These results are somewhat reminiscent of Bott periodicity for the stable homotopy of the classical groups. Here we are dealing with (the equivalent of) increasing level algebras. To summarize, when k = 1 it is known via standard Monster Moonshine that the coefficients of j decompose into Monster representations [14]. The related extremal lattices are Leech in 24 and P48 in 48 dimensions. The fact that all the higher k coefficients also decompose into Monster representations indicates that they have large symmetries containing the Monster and the fact that they have these symmetries may indicate that they are related to 24k-dimensional lattices, and this increases the probability that one can construct CFTs from these extremal partition functions.

5. Discussion and conclusions

In the discussion section of [1] it is argued that by using Z2 twists of the 23 Lie type Niemeier lattices one gets holomorphic conformal field theories that are not graded isomorphic to any of the untwisted theories based on Niemeier lattices. What we have shown is that there exists a family of transformations, not necessarily of the simple Z2 form, that connects the members M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397 393 of the class of holomorphic conformal field theories, i.e., the Niemeier lattices. Furthermore these transformations connect non-Niemeier modular invariant c = 24 Θ-functions. These results generalize to any c = 24k case (in particular, when the resulting parametrization corresponds to 24k-dimensional lattice). We also found interesting patterns of periodicity related to Monster Moonshine. It is not clear what some of the non-lattice structures are. For instance, if we transform a subcomponent of a lattice by multiplying by a constant C it changes the number of points in the layers of that subcomponent. If we require the number of points in a layer to be an integer (and the spacing between points in each layer to be uniform), then C must be a rational fraction. If we start with a lattice built from a set of Jacobi-θ functions, then we can change the number of points in sets of layers by multiplying the Jacobi-θ’s by Ci ’s, with the only requirement that the coefficients in a q-expansion of a lattice are positive integers. We can also change the number of points on a lattice by adding (discrete amounts of) curvature, positive (negative) curvature reduces (increases) the number of points. So the transformation of a subcomponent of a lattice by multiplying by a rational fractional is in some sense equivalent to adding curvature. This will in general not be compatible with maintaining the integrity of the lattice, although we could still generate a perfectly acceptable partition function. Indeed in 24 dimensions, if we start with a Niemeier lattice, we can make many fractional transformation that lead to an even self-dual partition function, but most of these are not lattices (since we know there are only 24 Niemeiers). We expect the same to be true if we start with a 24k-dimensional lattice and transform. There will be a large number of modular invariant partition functions generated, but only a limited number will correspond to even self-dual lattices. In 24 dimensions there is one extremal partition function, corresponding to a Leech lattice, whose q-expansion coefficients can be written in the form of a linear combinations of dimensions of the irreducible representations of the Monster group; this relationship is at the core of the Monster Moonshine. The decomposition is possible only in the extremal case, since in all other cases a non-zero constant term in the q-expansion would be present. The presence of this term in the expansion would imply the introduction of an enormous unnatural set of singlets in the decomposition. We believe a similar situation occures at 24k where we have extended this argument. Instead of a Leech lattice we have to deal with higher-dimensional extremal partition functions (but note that there is more than one type of extremal partition function in 24k dimensions). Existence of 24k extremal lattices for k>2 is however only a conjecture [6], but our results are consistant with and provides supporting evidence for this conjecture. One can use our generalized version of Monster Moonshine to postulate that we already have the q-expansion of higher-dimensional extremal lattices, and that the symmetry provided by the Monster decomposition can be used to learn more about these lattices. Some of the results obtained here (related to the transformations between c = 24 CFTs) are already present in the literature [7,21]. For example, the Z2 patterns (in Table 5) were explained in [21]. Also it was shown that from any even-self dual lattice it is always possible to construct one untwisted and one twisted conformal field theory. However in our case it is enough to start from just a single even self-dual lattice to obtain all other lattice solutions by a proper choice of projection parameters. Finally there are a number of interesting open questions. For instance, is there any relation between solutions found here (including non-lattice solutions) and other solutions corresponding to higher level Kac–Moody algebras classified in [22–24]. As for new CFTs, we suggest that the extremal G partition functions will generate new c = 24k CFTs. We know the first at k = 1 corresponds to the Leech lattice, the second k = 2 case also corresponds to the known lattice P48, and since the higher k cases all possess Monster symmetry 394 M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397

Table 9 Θ-functions (modular forms of weight 12) corresponding to 47 theories with a non-Abelian spin-1 algebra in Schellekens 0 = [22]. The second column represents our x parametrization of isospectral cases, when Z0 D16E8. Coefﬁcients ai are the ﬁrst three terms in the q-expansion of the corresponding Θ-function

Lattice xa2 a4 a6 E8,2B6,1 −11/4 360 187920 16863840 B12,2 −29/8 276 189936 16842672 C10,1B6,1 −15/4 264 190224 16839648 3 2 (A8,1) ,C8,1(F4,1) ,E7,2B5,1F4,1 −17/4 216 191376 16827552 D9,2A7,1 −9/2 192 191952 16821504 2 2 D8,2(B4,1) (C6,1) B4,1 −19/4 168 192528 16815456 E6,2C5,1A5,1,E7,3A5,1 −5 144 193104 16809408 2 (B6,2) −41/8 132 193392 16806384 6 4 2 (A4,1) ,(C4,1) ,D6,2C4,1(B3,1) ,A9,2A4,1B3,1 −21/4 120 193680 16803360 A8,2F4,2 −43/8 108 193968 16800336 8 2 2 3 2 − (A3,1) ,(D5,2) (A3,1) ,E6,3(G2,1) ,A7,2(C3,1) A3,1, 11/2 96 194256 16797312 D7,3A3,1G2,1,C7,2A3,1 3 (B4,2) −45/8 84 194544 16794288 12 2 4 2 2 2 − (A2,1) ,(D4,2) (C2,1) ,(A5,2) C2,1(A2,1) ,A8,3(A2,1) , 23/4 72 194832 16791264 E6,4C2,1A2,1 2 4 C4,2(A4,2) ,(B3,2) −47/8 60 195120 16788240 24 4 4 3 3 − (A1,1) ,(A3,2) (A1,1) ,A5,3D4,3(A1,1) ,A7,4(A1,1) , 6 48 195408 16785216 2 D5,4C3,2(A1,1) ,C5,3G2,2A1,1 6 4 (C2,2) ,D4,4(A2,2) ,F4,6A2,2 −49/8 36 195696 16782192 16 6 3 2 − (A1,2) (A2,3) ,(A3,4) A1,2,A5,6C2,3A1,2,(A4,5) , 25/4 24 195984 16779168 D5,8A1,2,A6,7 12 (A1,4) ,D4,12A6,C4,10 −51/8 12 196272 16776144 we conjecture that it is likely that they correspond to CFTs constructed on extremal lattices in 24k dimensions. To the best of our knowledge, Monster symmetry was not known to come into play except at k = 1. Finally, let us compare with [22] where all examples have c = 24 and N divisible by 12. The Niemeier cases obviously correspond to those in our work, so we only need to consider his 47 non-Niemeier cases. We make the tentative identiﬁcations shown in Table 9. To conclude, given these assignments, when we extend our analysis to larger k values we expect: (i) a set of even self dual group lattices in 24k dimensions which generalize the Niemeiers, (ii) an extremal 24k dimensions lattice that is a generalization of th Leech lattice, and (iii) a set of c = 24k CFTs that generalize the Schellekens spin-1 algebra cases. There exists a possible application of conformal ﬁeld theories (with high central charge) to →∞ 2 →− · 2 →−∞ cosmology, since for k ,(mtachyon) k MPlanck , which suggests this tachyon maybe the single tachyon of J (xk) and lead to some variant of a tachyon condensation [12].Also in the case of k →∞(hence divergent central charge [18,19]) we get, for example, a theory with a gauge lattice (G)k. Depending on the representation content of the gauge group one could potentially partially deconstruct (G)k [20] to go from 2D CFT to a 4D theory. The natural place for theories with c>24 is in condensed matter systems, since there are not enough ghosts to cancel a conformal anomaly. Nevertheless, we hope applications of the M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397 395 transformation techniques investigated here may lead to further/deeper understanding of dualities relating N = 2 heterotic string theories in 2D [25].

Acknowledgements

We beneﬁted greatly from several discussions with John Ratcliffe. We also thank Rich Hol- man, Burt Ovrut, Ashoke Sen and Hirosi Ooguri for discussions, Neil Sloane for an email correspondence, and Ralf Lehnert for careful reading of the manuscript. This work is supported in part by US DoE grant # DE-FG05-85ER40226.

Appendix A. Relations between modular functions and lattices

The ZΛ of any lattice Λ can be expressed in terms of the Jacobi θ-functions: ∞ 2miξ+πiτm2 θ3(ξ|τ)≡ e . m=−∞ For us it will be enough to work with the simpler theta functions, often called θ constants, deﬁned as: ≡ πiτ/4 πτ | ≡ | θ2(τ) e θ3 2 τ ,θ3(τ) θ3(0 τ), θ4(τ) ≡ θ3(τ + 1). (A.1) These functions have fantastic properties including a basically inﬁnite web of identities which will be used later on. Most important for us, is that they have very simple modular transformation properties. Here we show their behavior under the generators of the modular group, namely under S we have −1 τ 1/2 θ = θ (τ), (A.2) 2 τ i 4 −1 τ 1/2 θ = θ (τ), (A.3) 3 τ i 3 −1 τ 1/2 θ = θ (τ), (A.4) 4 τ i 2 and under T: √ θ2(τ + 1) = iθ2(τ), (A.5)

θ3(τ + 1) = θ4(τ), (A.6)

θ4(τ + 1) = θ3(τ). (A.7) These transformation rules are easily derived using the Poisson resummation formula [6].Fi- nally, we introduce the modular invariant function J , which plays a very important role in our considerations 1 J ≡ θ (τ)θ (τ) 12 + θ (τ)θ (τ) 12 − θ (τ)θ (τ) 12 + 24 η24 3 4 2 3 2 4 1 = + 196884q2 + 21493760q4 + O q6 +···. (A.8) q2 396 M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397

This function (sometimes called the J -invariant related to weight-zero modular function j by J = j − 744) is the modular form of weight zero, as can be easily proved using transformations (A.2)–(A.7). In the denominator of (A.8) we have the 24th power of Dedekind η-function

∞ 24 η24(τ) = q1/12 1 − q2m = q2 − 24q4 + 252q6 − 1472q8 +···, (A.9) m=1 which is the unique form of weight 12, with the following transformation rules under S and T 1 √ η24 − = −iτη(τ) 24,η24(τ + 1) = eiπ/12η(τ) 24. (A.10) τ

References

[1] L.J. Dixon, P.H. Ginsparg, J.A. Harvey, Beauty and the beast: Superconformal symmetry in a Monster module, Commun. Math. Phys. 119 (1988) 221. [2] W. Lerche, D. Lust, A.N. Schellekens, Ten-dimensional heterotic strings from Niemeier lattices, Phys. Lett. B 181 (1986) 71; A.N. Schellekens, Classification of ten-dimensional heterotic strings, Phys. Lett. B 277 (1992) 277, hep-th/9112006. [3] J.D. Blum, K.R. Dienes, Strong/weak coupling duality relations for non-supersymmetric string theories, Nucl. Phys. B 516 (1998) 83, hep-th/9707160; J.D. Blum, K.R. Dienes, Duality without supersymmetry: The case of the SO(16)×SO(16) string, Phys. Lett. B 414 (1997) 260, hep-th/9707148. [4] D. Lust, S. Theisen, Lectures on String Theory, Lecture Notes in Physics, vol. 346, 1989, p. 1; W. Lerche, A.N. Schellekens, N.P. Warner, Lattices and strings, Phys. Rep. 177 (1989) 1. [5] T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Graduate Texts in Mathematics, Springer- Verlag, 1990. [6] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, New York, 1993. [7] L. Dolan, P. Goddard, P. Montague, Conformal field theory, triality and the Monster group, Phys. Lett. B 236 (1990) 165. [8] J.A. Harvey, S.G. Naculich, Conformal field theory and genus zero function fields, Nucl. Phys. B 305 (1988) 417. [9] L. Dolan, P. Goddard, P. Montague, Conformal field theory of twisted vertex operators, Nucl. Phys. B 338 (1990) 529. [10] N.J.A. Sloane, My favorite integer sequences, http://www.research.att.com/~njas/doc/sg.pdf. [11] E.M. Rains, N.J.A. Sloane, The shadow theory of modular and unimodular lattices, J. Number Theor. 73 (1998) 359. [12] A.A. Gerasimov, S.L. Shatashvili, On exact tachyon potential in open string field theory, JHEP 0010 (2000) 034, hep-th/0009103. [13] C.L. Siegel, Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay, 1980, pp. 249– 268. [14] T. Gannon, Monstrous moonshine and the classification of CFT, math.QA/9906167. [15] G. Hoehn, Selbstduale Vertexoperatorsuperalgebren und das Babymonster, in: Bonner Mathematische Schriften, vol. 286, 1996, pp. 1–85, http://www.research.att.com/projects/OEIS?Anum=A028520. [16] R.E. Borcherds, A.J.E. Ryba, Duke Math. J. 83 (2) (1996) 435–459. [17] Y.H. He, V. Jejjala, Modular matrix models, hep-th/0307293. [18] A. Bilal, J.L. Gervais, Extended C =∞conformal systems from classical Toda field theories, Nucl. Phys. B 314 (1989) 646. [19] E. Verlinde, On the holographic principle in a radiation dominated universe, hep-th/0008140. [20] N. Arkani-Hamed, A.G. Cohen, H. Georgi, (De)constructing dimensions, Phys. Rev. Lett. 86 (2001) 4757, hep- th/0104005. [21] L. Dolan, P. Goddard, P. Montague, Conformal field theories, representations and lattice constructions, Commun. Math. Phys. 179 (1996) 61, hep-th/9410029. M. Jankiewicz, T.W. Kephart / Nuclear Physics B 744 [FS] (2006) 380–397 397

[22] A.N. Schellekens, Meromorphic C = 24 conformal field theories, Commun. Math. Phys. 153 (1993) 159, hep- th/9205072; A.N. Schellekens, S. Yankielowicz, Curiosities at C = 24, Phys. Lett. B 226 (1989) 285. [23] P.S. Montague, Orbifold constructions and the classification of selfdual c = 24 conformal field theories, Nucl. Phys. B 428 (1994) 233, hep-th/9403088. [24] P.S. Montague, Conjectured Z(2)-orbifold constructions of self-dual conformal field theories at central charge 24: The neighborhood graph, Lett. Math. Phys. 44 (1998) 105, hep-th/9709076. [25] H. Ooguri, C. Vafa, N = 2 heterotic strings, Nucl. Phys. B 367 (1991) 83. Nuclear Physics B 744 (2006) 398–399

CUMULATIVE AUTHOR INDEX B741–B744

Abe, H. B742 (2006) 187 Delgado, A. B744 (2006) 156 Agashe, K. B742 (2006) 59 Desrosiers, P. B743 (2006) 307 Ahl Laamara, R. B743 (2006) 333 Di Castro, C. B744 (2006) 277 Altarelli, G. B741 (2006) 215 Donini, A. B743 (2006) 41 Altarelli, G. B742 (2006) 1 Dorey, P. B744 (2006) 239 Antoniadis, I. B744 (2006) 156 Dreyer, O. B744 (2006) 1 Apolloni, A. B744 (2006) 340 Drissi, L.B. B743 (2006) 333 Aprea, G. B744 (2006) 277 Arianos, S. B744 (2006) 340 Eberle, H. B741 (2006) 441 Arkani-Hamed, N. B741 (2006) 108 Elias, V. B743 (2006) 104

Ball, R.D. B742 (2006) 1 Faisst, M. B742 (2006) 208 Ball, R.D. B742 (2006) 158 Fernández-Martínez, E. B743 (2006) 41 Balog, J. B741 (2006) 390 Feruglio, F. B741 (2006) 215 Barrau, A. B742 (2006) 253 Flohr, M. B741 (2006) 441 Becker, K. B741 (2006) 162 Flohr, M. B743 (2006) 276 Bellucci, S. B741 (2006) 297 Forrester, P.J. B743 (2006) 307 Benakli, K. B744 (2006) 156 Forte, S. B742 (2006) 1 Bergman, O. B744 (2006) 136 Forte, S. B742 (2006) 158 Bernabéu, J. B744 (2006) 180 Fré, P. B741 (2006) 42 Bertolini, M. B743 (2006) 1 Fré, P. B742 (2006) 86 Bi, X.-J. B741 (2006) 83 Froggatt, C. B743 (2006) 133 Billò, M. B743 (2006) 1 Bolognesi, S. B741 (2006) 1 Burrington, B.A. B742 (2006) 230 Gargiulo, F. B741 (2006) 42 Giudice, G.F. B741 (2006) 108 Caracciolo, S. B741 (2006) 421 Gomshi Nobary, M.A. B741 (2006) 34 Casteill, P.-Y. B741 (2006) 297 Grain, J. B742 (2006) 253 Chen, B. B741 (2006) 269 Grange, P. B741 (2006) 199 Chetyrkin, K.G. B742 (2006) 208 Grilli, M. B744 (2006) 277 Chetyrkin, K.G. B744 (2006) 121 Gudnason, S.B. B741 (2006) 1 Chishtie, F.A. B743 (2006) 104 Colangelo, G. B744 (2006) 14 Haefeli, C. B744 (2006) 14 Contino, R. B742 (2006) 59 Hamanaka, M. B741 (2006) 368 Cove, H.C.D. B741 (2006) 313 Harmark, T. B742 (2006) 41 He, Y.-L. B741 (2006) 269 D’Adda, A. B744 (2006) 340 Higaki, T. B742 (2006) 187 de Forcrand, Ph. B742 (2006) 124 Hirano, S. B744 (2006) 136 Delgado, A. B741 (2006) 108 Hoyos, C. B744 (2006) 96

0550-3213/2006 Published by Elsevier B.V. doi:10.1016/S0550-3213(06)00339-7 Nuclear Physics B 744 (2006) 398–399 399

Ilderton, A. B742 (2006) 176 Peschanski, R. B744 (2006) 80 Itou, E. B743 (2006) 74 Philipsen, O. B742 (2006) 124 Pope, C.N. B741 (2006) 17 Jacobsen, J.L. B743 (2006) 153 Jacobsen, J.L. B743 (2006) 207 Quirós, M. B744 (2006) 156 Jaffe, R.L. B743 (2006) 249 Jankiewicz, M. B744 (2006) 380 Randjbar-Daemi, S. B741 (2006) 236 Reyes, S.A. B744 (2006) 330 Kajiyama, Y. B743 (2006) 74 Richard, J.-F. B743 (2006) 153 Kephart, T.W. B744 (2006) 380 Rigolin, S. B743 (2006) 41 Kishimoto, I. B744 (2006) 221 Rim, C. B744 (2006) 239 Kobayashi, T. B742 (2006) 187 Rulik, K. B741 (2006) 42 Koike, Y. B742 (2006) 312 Russo, R. B743 (2006) 1 Koike, Y. B744 (2006) 59 Konishi, K. B741 (2006) 180 Saidi, E.H. B743 (2006) 333 Kormos, M. B744 (2006) 358 Salas, J. B743 (2006) 153 Korthals Altes, C.P. B742 (2006) 124 Saleur, H. B743 (2006) 207 Krohn, M. B743 (2006) 276 Salvio, A. B741 (2006) 236 Kubo, J. B743 (2006) 74 Scardicchio, A. B743 (2006) 249 Kühn, J.H. B744 (2006) 121 Schnitzer, H.J. B742 (2006) 295 Sepahvand, R. B741 (2006) 34 Lerda, A. B743 (2006) 1 Shaposhnikov, M. B741 (2006) 236 Lishman, A. B744 (2006) 239 Shore, G.M. B744 (2006) 34 Liu, J.T. B742 (2006) 230 Smolin, L. B742 (2006) 142 Livine, E.R. B741 (2006) 131 Smolin, L. B744 (2006) 1 Lorenzana, J. B744 (2006) 277 Steele, T.G. B743 (2006) 104 Lü, H. B741 (2006) 17 Stelle, K.S. B741 (2006) 17 Lukyanov, S.L. B744 (2006) 295 Sturm, C. B742 (2006) 208 Sturm, C. B744 (2006) 121 Mann, R.B. B743 (2006) 104 Szabo, R.J. B741 (2006) 313 Marino, E.C. B741 (2006) 404 Markopoulou, F. B744 (2006) 1 Takács, G. B741 (2006) 353 Marmorini, G. B741 (2006) 180 Tateo, R. B744 (2006) 239 Mavromatos, N.E. B744 (2006) 180 Tentyukov, M. B742 (2006) 208 McKeon, D.G.C. B743 (2006) 104 Meloni, D. B743 (2006) 41 Teraguchi, S. B744 (2006) 221 Merlatti, P. B744 (2006) 207 Terno, D.R. B741 (2006) 131 Minasian, R. B741 (2006) 199 Trigiante, M. B741 (2006) 42 Mognetti, B.M. B741 (2006) 421 Tseng, L.-S. B741 (2006) 162 Morales, J.F. B743 (2006) 1 Tsvelik, A.M. B744 (2006) 330 Moriyama, S. B744 (2006) 221 Tuckmantel, M. B744 (2006) 156

Naculich, S.G. B742 (2006) 295 Vogelsang, W. B744 (2006) 59 Nagashima, J. B742 (2006) 312 Nagashima, J. B744 (2006) 59 Wágner, F. B741 (2006) 353 Nevzorov, R. B743 (2006) 133 Waldron-Lauda, A. B744 (2006) 180 Niedermayer, F. B741 (2006) 390 Weisz, P. B741 (2006) 390 Nielsen, H.B. B743 (2006) 133 Nunes, L.H.C.M. B741 (2006) 404 Yang, W.-L. B744 (2006) 312 Yasui, Y. B742 (2006) 275 Obers, N.A. B742 (2006) 41 Yokoi, N. B741 (2006) 180 Oota, T. B742 (2006) 275 Zamolodchikov, A.B. B744 (2006) 295 Papavassiliou, J. B744 (2006) 180 Zhang, P. B741 (2006) 269 Pelissetto, A. B741 (2006) 421 Zhang, Y.-Z. B744 (2006) 312