Boundary correlators in WZW model on AdS$_2$

Boundary correlators of elementary fields in some 2d conformal field theories defined on AdS$_2$ have a particularly simple structure. For example, the correlators of the Liouville scalar happen to be the same as the correlators of the chiral component of the stress tensor on a plane restricted to the real line. Here we show that an analogous relation is true also in the WZW model: boundary correlators of the WZW scalars have the same structure as the correlators of chiral Ka\v{c}-Moody currents. This is checked at the level of the tree and one-loop Witten diagrams in AdS$_2$. We also compute some tree-level correlators in a generic $\sigma$-model defined on AdS$_2$ and show that they simplify only in the WZW case where an extra Ka\v{c}-Moody symmetry appears. In particular, the terms in 4-point correlators having logarithmic dependence on 1d cross-ratio cancel only at the WZW point. One motivation behind this work is to learn how to compute AdS$_2$ loop corrections in 2d models with derivative interactions related to the study of correlators of operators on Wilson loops in string theory in AdS.


Introduction
Study of σ-models in AdS 2 is of interest for several reasons (see, e.g., [1][2][3]). Here we will consider correlators of elementary σ-model fields in Euclidean AdS 2 with Poincare metric ds 2 = 1 z 2 (dt 2 + dz 2 ). While in flat space the scattering amplitudes of massless scalar fields in perturbative vacuum are ambiguous due to IR divergences (see, e.g., [4]) the coordinate-space boundary correlators in AdS 2 are well-defined and are constrained by 1d conformal invariance. One interesting question is how the structure of these correlators is further restricted by hidden symmetries of the σ-model and how to compute AdS 2 loop corrections in a way consistent with these underlying symmetries.
Since a classical σ-model in curved 2d space is Weyl-invariant (with the scalar field not transforming), defined on AdS 2 it is formally the same as on a half-plane ds 2 = dt 2 + dz 2 , z > 0. This is true also at the quantum level if the σ-model is UV finite: then dependence on the 1 z 2 conformal factor of the AdS 2 metric disappears. Compared to a generic boundary CFT set-up here we are interested in (i) the standard AdS 2 (or Dirichlet) boundary conditions ϕ(t, z) z→0 = z ∆ Φ(t) + · · · for an elementary field with mass m 2 = ∆(∆ − 1); (ii) correlators of elementary fields ϕ rather than composite operators with good 2d conformal transformation properties. The 1d boundary operators dual to the massless σ-model fields with Dirichlet b.c. will thus have ∆ = 1. In contrast to the Liouville theory case discussed in [5][6][7] it turns out that the classical 2d conformal invariance of the bulk σ-model theory does not sufficiently constrain the structure of the tree-level boundary correlators. For example, the tree-level boundary four-point functions are still non-trivial log functions of 1d cross-ratio. An important difference is that while the σ-model field is a scalar on which conformal symmetry acts trivially, the Liouville field transforms non-trivially under the conformal transformations. 1 Similar tree-level correlators (containing logs) were found also for the fields of the Nambu action in AdS 2 [1,2], but they appear already in the case of 2-derivative σ-model vertices.
To study the role of additional σ-model symmetries here we will consider the example of the WZW model [13,14] which has an infinite-dimensional Kac-Moody (KM) symmetry g = u(w) g v(w), w = t + iz. It appears for the special value of the ratio of the coefficients of the principal chiral model (PCM) and WZ terms in the action when the classical equations of motion admit a chiral decomposition (the resulting model is then conformal and KM invariant also at the quantum level). Like the Virasoro symmetry in the Liouville case here the KM symmetry will impose rigid constraints on the AdS 2 boundary correlators of the elementary fields ϕ a parametrizing g. In particular, the KM symmetry rules out the presence of log terms in the four-point correlators, both at the tree and the quantum level.
As we will argue below, the AdS 2 boundary correlators of the massless fields ϕ a defined in the standard way as Φ a1 (t 1 ) · · · Φ an (t n ) ≡ lim zi→0 n i=1 z −∆ i ϕ a1 (t 1 , z 1 ) · · · ϕ an (t n , z n ) AdS 2 , ∆ = 1 , (1.1) are constrained by the underlying KM symmetry so that they are equal, up to a universal prefactor, to the correlators of the chiral component of the WZW current J a (w) ∼ tr(t a ∂ w gg −1 ), w = t + iz, restricted to the boundary. This is formally equivalent to the "identification" of the boundary operator associated to ϕ a with the chiral component of the current J a (w → t) Here k is the WZW level. For comparison, in the Liouville theory case the role of the ∆ = 1 current J ≡ J w (the generator of KM symmetry) is played by the ∆ = 2 chiral stress tensor T ≡ T ww (the generator of the Virasoro symmetry) 2 and the proportionality coefficient was κ = −4 c−1 6c 2 where c = 1 + 6(b −1 + b) 2 is the Liouville central charge [7]. 3 In the WZW case the KM symmetry implies the Virasoro symmetry but is much stronger: as already mentioned above, the boundary correlators in conformal σ-models that do not have an extra KM symmetry have much more complicated structure. 4 Explicitly, the standard OPE relation for the chiral components of the KM current (see, e.g., [19]) determines all higher current correlators to be given by Below we will explicitly reproduce (1.4),(1.5),(1.6) with w i → t i as the expressions for the boundary correlators of the WZW fields (1.1) computed in the 1/k perturbation theory in AdS 2 with the identification (1.2). A semiclassical argument of why the boundary correlators of ϕ a are related to the restriction of the current correlators to the boundary of half-plane can be given as follows (for a similar though more involved argument in the Liouville theory case see [7]). Starting with the expression ) and using the boundary condition ϕ a (z, t) z→0 → z Φ a (t) + ... we find that (up to an overall normalization constant) J a z→0 → Φ a .
To demonstrate the correspondence (1.2) we shall start in section 2 with the example of the SL(2, R) WZW model on AdS 2 and compute boundary correlators of its fields in the leading treelevel approximation. We shall also consider the corresponding PCM q theory (i.e. the PCM with a WZ term with coefficient ∝ q), and show that the four-point correlators simplify (with logs of coordinates cancelling out) and thus can be matched with the correlators of the chiral currents only at the WZW point (q 2 = 1) when the model has an extra KM symmetry. In section 3 we shall repeat the computation of the tree boundary correlators for a generic σ-model including the case of PCM q for an arbitrary group G.
In section 4 we shall test the relation between the boundary correlators of the WZW fields and the chiral currents (1.2) beyond the classical (large k) limit by computing the one-loop corrections to the two-point and three-point boundary correlators. Like in similar computations in the Liouville and Toda theories in AdS 2 [7,11,12] this requires an explicit evaluation of loop integrals in AdS 2 which is subtle in the present case of the σ-model theory with two derivatives in the vertices. We shall argue 2 In the Liouville (or Toda) case the Virasoro symmetry becomes realized as a reparametrizations of the boundary and thus completely fixes the structure of the correlators modulo overall powers of the coordinate-independent factor κ. 3 This close analogy may not be accidental given that the Liouville theory may be obtained by a Hamiltonian reduction from the SL(2) WZW model [15,16]. 4 The key point is that the elementary σ-model field transforms non-trivially under the KM symmetry (like the Liouville field was transforming under the conformal symmetry). Note also that the simplification of the form of boundary correlators in the case of KM symmetry is analogous to what happens in the AdS/CFT examples when the bulk theory has higher symmetry thus constraining also the correlators of the dual boundary CFT. An example is provided by the vectorial AdS/CFT where the symmetry in question is a higher spin symmetry [17,18]. that there exists a particular computational scheme in which the one-loop terms in the WZW field boundary correlators vanish, implying that the proportionality coefficient κ in (1.2) does not receive 1/k correction and thus its expression in (1.2) is expected to be exact.
It is interesting to note that while in flat space the scattering amplitudes for the massless WZW fields vanish [20,4] their coordinate-space boundary correlators in AdS 2 are non-vanishing. Their structure, however, is simple being dictated by the KM symmetry. One may wonder if with some natural definition of the S-matrix in AdS they may actually correspond to trivial scattering in AdS 2 or on half-plane. We will address this question in section 5. There is a close analogy with what happens in the Liouville theory [5] where the full quantum S-matrix was argued to be trivial [21,22]. We shall discuss the idea of defining AdS 2 scattering amplitudes by Fourier transform of boundary correlators or using the prescription of [5] (cf. [23]) and argue that this leads to trivial three-point scattering amplitudes also in the present WZW case.
Some concluding remarks will be made in section 6. Appendix A will list our notation and conventions. In Appendix B we shall discuss a constraint imposed by global symmetry on boundary two-point functions in the SL(2, R) WZW model. In Appendix C we shall revisit the computation of the one-loop corrections to the two-point boundary correlators in SL(2, R) WZW model using an alternative form of the action and emphasizing some subtle scheme-dependence issues.
2 Boundary correlators in SL(2, R) WZW model on AdS 2 To demonstrate the correspondence (1.2) we shall first consider the example of the SL(2, R) WZW model and compute its boundary correlators on AdS 2 in the leading-order (tree) approximation. It is useful to view this WZW model as a special case of the PCM q , i.e. the principal chiral model with an additional WZ term. This allows one to investigate the consequences of the Kac-Moody symmetry appearing at the WZW point for the structure of the boundary correlators.

Action
The action for the PCM q may be written as where k is the coefficient of the WZ term, Σ is a Riemann surface and B 3 is the 3d extension of Σ such that ∂B 3 = Σ. When the action (2.1) reduces to the WZW model action.
Assuming k > 0, a generic SL(2, R) group element may be represented in the Gauss decomposition form (see, e.g., [24]) Then the action (2.1) (written on generic curved 2-space with metric g) becomes Here µν = ε µν √ g is the standard antisymmetric tensor. This action may be interpreted as that of a σ-model with AdS 3 target space and particular B-field coupling.
Specializing to the q = 1 WZW point and the Euclidean AdS 2 background (see Appendix A for our notation and conventions) we get the following expression for the corresponding SL(2, R) WZW action S =ˆd 2 w ∂φ∂φ+e bφ ∂ψ∂ψ =ˆd 2 w ∂φ∂φ+∂ψ∂ψ+b φ∂ψ∂ψ+ 1 As the conformal factor of the metric decouples, this is formally the same as the WZW action on a flat half-plane z > 0. However, it will be useful to phrase the computation of the boundary correlators in the AdS 2 language. We shall assume that the massless fields φ, ψ,ψ are subject to the standard (Dirichlet) boundary conditions and thus they should be dual to the boundary operators with dimension ∆ = 1, i.e. the asymptotic expansion of these fields near z = 0 is Our aim will be to compute the tree level boundary correlation functions (1.1) for the fields in (2.6) and then match them with the correlators of KM currents.

Propagators
The bulk-to-bulk propagator of a massless scalar in AdS 2 with a standard normalization 1 where the geodesic distance is defined as Hence, for the field φ in (2.6) we have (cf. (A.7)) 5 The bulk-to-bulk propagator of the pair of fields ψ,ψ is similarly Given the structure of the perturbative (small b or large k) expansion in (2.6), it is useful also to quote the propagators for the differentiated fields where we used the relations (A.6). The δ-function piece here will be important to account for below.
To compute the boundary correlators, we will also need the bulk-to-boundary propagators Considering the boundary-to-boundary case of the propagators (2.10), we get the following two-point functions (using the notation in (1.1)) (2.14) These have the same form as the boundary restriction of (1.4).
The only non-zero three-point function is ΦΨ Ψ , which, at the tree level (leading order in 1/k), is computed by the Witten diagram 6 .

(2.15)
We have Using the propagators in (2.13), we get (2.17) This integral can be done by first computing the residues in the t integration variable. Integrating then over z one finds This has again the same structure as the real-line limit of (1.5).

Four-point functions
We now turn to the four-point functions the computation of which is little more involved. The only non-vanishing cases are the correlators Ψ 2 Ψ 2 and Φ 2 Ψ Ψ .
At tree level this correlator is given by the following Witten diagrams (2.19) We can represent the result as To compute this integral we may first integrate by parts at the vertex w, where we ignored 2-derivative terms assuming Indeed, possible terms with δ (2) (t − w ) and its derivative may be neglected here as they localize the bulk point to the boundary, and hence give zero contributions after performing the bulk integral. The integral in (2.23) can be evaluated by applying the residue theorem Then from (2.20) we finally obtain Remarkably, all logarithmic (and sign function) terms present in (2.25) cancel out in the sum of the two exchange Witten diagrams. This cancellation is crucial in order to be able to match (2.26) with the correlators of the KM currents that are rational functions of the differences of points (cf. (1.6)).
Φ 2 Ψ Ψ : This correlator is given by the sum of the following three diagrams It can be written as where the explicit form of B 4 and C 4 is Here B reg 4 and C 4 are the contributions from the regular and singular parts of the internal propagators in (2.12), respectively (note that in (2.29) the singular δ-function part in the propagator (2.12) turns the exchange diagram into a contact diagram). Using the explicit form of the propagators we get Using the same method as for the previous four-point function, we obtain i log t 12 t 34 t 13 t 24 2 + π − sgn t 12 + sgn t 13 − sgn t 24 + sgn t 34 , (2.33) Inserting these results into (2.28) gives 7 As in the case of the correlator Ψ(t 1 ) Ψ(t 2 )Ψ(t 3 ) Ψ(t 4 ) in (2.26), both the logarithms and the sign functions again cancel out.

Matching AdS 2 boundary correlators with correlators of chiral currents
Let us now compare the above boundary correlators with the correlators of the chiral WZW currents on the plane restricted to the real line. The correlation functions of the currents are a direct consequence of the KM algebra (1.3). Adapted to the SL(2, R) case the OPEs of the three currents (H, J + , J − ) read (see, e.g., [24]) From (2.36) we conclude that: (i) the two-point functions are (ii) the only non-vanishing three-point function is and (iii) the non-trivial four-point functions are These four-point functions are non-trivial in the sense that in addition to the k 2 contribution they also have a term linear in k (cf. (1.6)). 8 Comparing to boundary correlators discussed above, the k 2 term is a counterpart with disconnected AdS 2 Witten diagram contribution, while the order k term corresponds to connected exchange and contact contributions due to non-trivial bulk interactions. In fact, it is possible to establish the precise matching between the tree-level 2-point and 3-point boundary correlators in (2.14),(2.18) and the current correlators (2.37),(2.38) restricted to the real line using the following identification (cf. (1.2)) A non-trivial consistency check is that the connected 4-point boundary correlators (2.26),(2.36) then also match with the non-trivial order k parts of the 4-current correlators in (2.39). The fact that there is just a single universal proportionality coefficient κ follows from the global group symmetry of the WZW model (this is true not only at the tree level but also to all orders in 1/k). 8 For instance, a four-point function which is non-vanishing but trivial in the above sense is

Boundary correlators in PCM q on AdS 2
Let us now go back to the SL(2, R) principal chiral model with a general coefficient q of the WZ term in (2.4) to emphasize that its boundary correlators have a complicated structure already at the tree level (containing, in particular, logarithmic terms found also in a similar σ-model context in [1,2]). In contrast to the Liouville and Toda theories discussed in [6,7,10] here the classical conformal symmetry of PCM q (2.4) is not enough to sufficiently constrain the boundary correlators. 9 The correlators simplify precisely at the WZW point q 2 = 1 and this may be attributed to the emerging KM symmetry (that implies chiral decomposition in flat space). The action (2.4) in AdS 2 expanded in powers of b reads (cf. (2.6)) 10 Repeating the calculation of the tree-level four-point Ψ 2 Ψ 2 correlator we find (cf. (2.20)) where A 4 is given by (2.21),(2.22), (2.25). Similarly, for the Φ 2 Ψ Ψ correlator we get where B 4 and C 4 are given by (2.29),(2.30),(2.33), (2.34). We conclude that these four-point functions contain logarithmic terms. These cancel only at the WZW point q 2 = 1 allowing one to relate these boundary correlators to the "connected" part of the correlators of the chiral WZW currents as explained above.

Action
Let us start with a general bosonic σ-model with coupling functions (G ab , B ab ) and expand it in normal coordinates near the origin using G ab (X) = δ ab − 1 . Then its Euclidean action may be written as 11 In what follows we will consider the leading terms in this action parametrized as where the constant real coupling functions P and Q are given by Thus Q is totally antisymmetric and P has algebraic symmetries of the curvature To account for the manifest symmetry of the 4-vertex in (3.2) in (a, c) it is useful to introduce also the corresponding symmetrization of P abcd Then specifying to the AdS 2 background the action (3.2) may be written as (cf. (2.6)) The action (3.2) represents as a particular case the expansion of the PCM q (2.1) for an arbitrary group G. Let us normalize the generators {t a } and the invariant bilinear form of the Lie algebra of G as 12 Then choosing the parametrization of the group field as 13 g = e −iλtaX a we find that in the PCM q case The WZW theory corresponds to the choice In what follows we shall assume that k > 0. 11 We ignore the overall coupling factor or 1 α that can be absorbed into a rescaling of X a and then appears in R and H. 12 We assume that ta are Hermitian and thus the structure constants f ab c are purely imaginary. The group indices are raised or lowered by δ ab , implying that f abc = f ab c is fully anti-symmetric. Repeated group indices are summed over, regardless of their positions. 13 Notice that this parametrization is different from (2.3) used in the SL(2, R) case.

Tree-level AdS 2 boundary correlation functions
The fields X a in (3.8) are massless and thus, assuming the Dirichlet boundary conditions, we have (cf. They should correspond to the boundary operators with dimension ∆ = 1. As in (2.10), their bulkto-bulk AdS 2 propagator is given by while the bulk-to-boundary propagator is (3.13) Then the boundary two-point function is (cf. (2.14)) . (3.14) Starting with (3.8) it is straightforward also to compute the three-point function The connected tree-level four-point function receives contributions from both exchange diagrams and contact diagrams Exchange diagrams. The exchange part contains contributions of the three different channels As these are related by permutations (crossing), we only need to compute one of them (3.18) Note that using integration by parts, one can always arrange so that the derivatives in the cubic vertex in (3.8) act only on the two external legs. 14 Then the 6 terms in the cubic vertex can be written as where the two terms arise from the two ways of acting by derivative on the external legs and the factor of 3 comes from rearranging other similar cubic terms. Using (3.19), we find for the exchange diagram HereH was defined in (2.22) and computed in (2.25).
Contact diagrams. Since the quartic vertex in (3.8) contains derivatives, the contact contribution may also be represented as a sum of the three contributions Here we have indicated explicitly the coupling tensors appearing from each diagram (the factor of 2 arises from two ways of contracting the two legs without derivative). Explicitly, we get where we used (3.7). As a result, where where in the r.h.s. we suppressed the indices (a, b, c, d) introducing the symbolic notation (s, t, u stand for different channels) 15 This implies that and also that 9 4 Q abe Q cde =P abcd −P abdc = P abcd + P cbad − P abdc − P dbac = 3P abcd , i.e. P abcd = 3 4 Q abe Q cde , (3.32) where we used (3.6) and symmetry properties of the curvature tensor in (3.4) and (3.5). Written in terms of R and H in (3.3) this reads Interestingly, the trace of this relation, i.e. R ac = 1 4 H abe H cbe , is the same as the vanishing of the one-loop beta-function [25] of the σ-model in (3.1).
In the group space case (3.10) the condition (3.31) is automatically satisfied due to the Jacobi identity for the structure constants. The condition (3.32) or (3.33) reduces to i.e. is valid only in the WZW model case (cf. (2.2),(3.10)).

WZW model case: matching with correlators of chiral currents
Thus the cancellation of the logarithmic terms in the four-point boundary correlators of a generic σ-model in AdS 2 happens only in the WZW model. This generalizes the observation made in section 2 in the SL(2, R) WZW case. Then the resulting expression for the connected four-point correlator (3.27) may be written as (using (3.10) with q = 1, i.e. λ 2 = 2 k ) As in the SL(2, R) case (cf. (2.26), (2.36)), we can now explicitly check the correspondence between AdS 2 boundary correlators and holomorphic correlation functions of Kac-Moody currents. 15 Note that the permutations of legs on the first and second lines of (3.27) are different.
The basic OPE relation for the WZW chiral currents on the plane (1.3) gives the two-point function (1.4). Higher point correlators can be obtained by repeatedly using the OPE (1.3). 16 In particular, one finds The "connected" part of (3.37) may be written as Here in the second line we wrote an equivalent expression (expressing crossing symmetry of the fourpoint function) that is a consequence of the Jacobi identity for the structure constants.
Restricting the points to the real line (w i → t i ) we can identify the two-point (1.4) and three-point (1.5) correlators of the currents with the corresponding boundary correlators in (3.14) and (3.15) up to an overall universal factor κ n where n = 2, 3, .... is the number of legs. Explicitly, this amounts to the formal identification (assuming k > 0) Indeed, the two-point functions match if κ 2 = 2 k , while the three-point functions match for Q abc in (3.15) related to f abc in (3.36) as in (3.10) and κ 3 = 2 k λ. Furthermore, the four-point correlator (3.35) is also in precise agreement with the boundary restriction of the connected part of the correlator of four currents in (3.38).
The mutual locality of the KM currents implies a trivial (meromorphic) singularity structure and the solution of the above relation is simply obtained by isolating poles as in (3.36)-(3.38).
currents (1.2) beyond the classical (large k) limit. This requires determining loop corrections to AdS 2 boundary correlators. Similar computations were done in the Liouville and Toda theories in [7,11,12] and it was found that the analogs of the coefficient κ in (1.2) that relate boundary correlators of elementary fields in AdS 2 to correlators of CFT currents (stress tensor and W-symmetry currents) restricted to real line receive quantum corrections.
In the present WZW model case, the simplicity of the semiclassical argument in (3.40) suggests instead that the relation (1.2) or (2.40),(3.39) may be exact. 17 To provide support to this conjecture below we shall consider the computation of one-loop corrections to the two-point and three-point boundary correlators (1.1) on the example of the SL(2, R) WZW model. A central issue will be the choice of a UV regularization and subtraction scheme consistent with underlying SL(2, R) symmetry of the model. It turns out to be possible to relate the scheme ambiguity to the definition of the propagator at the coinciding points, i.e. to the choice of the renormalized value of the self-contraction contributions.

One-loop corrections to the two-point correlators
Let us start with computing the one-loop corrections to the tree-level two-point functions (2.14) for the fields in the action (2.6), i.e. to the boundary correlators Ψ Ψ and ΦΦ .

Ψ Ψ
One-loop corrections to the ψ,ψ propagator in AdS 2 come from the following diagrams: Here we have separated the contributions of the regular and δ-function terms in (2.12) combining the latter with the self-contraction diagram corresponding to the vertex φ 2 ψψ in (2.6) as both are proportional to the free scalar propagator at the coinciding points, i.e. g(w, w) (cf. (2.10),(2.11)). The last tadpole diagram with a ψ loop is linearly divergent and may be removed by imposing the normalization condition φ = 0. The first contribution in (4.1) involving the regular part of the second derivative of the propagator in (2.12) (with legs taken to the boundary) is given by 17 One could wonder if the level k in (1.2) may get a familiar quantum shift by the dual Coxeter number of G (i.e. k → k + c G ) which is known to appear from the quantum jacobian transformation from the group fields g to currents and in the Sugawara construction of the stress tensor and related computation of the central charge. As we shall argue below, there exists a natural computation scheme in which this does not apparently happen in the present case of κ in (1.2).
By formal shifting and rescaling w, w one may try to argue that the integral J should be independent of t 1 , t 2 . However, it is IR divergent and thus requires a regularization. A regularization will then be expected to give J ∼ log(Λ −1 |t 1 − t 2 |) and thus a finite ∼ 1 t 2 12 contribution to E(t 12 ). Indeed, integrating by parts the formal expression in (4.4) we get (using (2.11)) 18 (4.5) This is divergent due to the contribution from the y → +∞ region where the integrand scales as ∼ 1/(x 2 y). A cutoff on the z, z integrals near zero in (4.4) translates into the modified integration range 0 < y < Λ t12 , Λ → ∞. Then we find for the regularized integral 19 and thus Including also the contribution of the square bracket terms in (4.1) which depend on regularized value of g(w, w) we finish with the following one-loop (i.e. order b 2 ∼ 1 k ) correction to the tree-level boundary correlator (2.14) Thus a particular scheme choice where g 0 = 1 would lead to the vanishing of the one-loop correction.
To put this in a more general context, while the WZW is UV finite in the sense that there is no coupling renormalization, there may still be a wave function renormalization (i.e. UV divergent Zfactor in the off-shell 2-point function). This should be accounted for in the definition of the S-matrix: the scattering amplitudes defined in terms of correlators with extra powers of Z will be automatically finite (see, e.g., a discussion in [27] and refs. there). Similar considerations should apply to the analog of S-matrix in AdS (see section 5) and thus to the boundary correlators. Here we will effectively by-pass this subtlety by simply assuming a particular subtraction under which the wave-function renormalization factor is trivial. 20

ΦΦ
The one-loop correction to the boundary two-point function Φ(t 1 )Φ(t 2 ) is given by the sum of two diagrams: a bubble and a self-contraction diagram.
Bubble. The bubble contribution is . (4.9) 18 The integral over z, z here may be split and turned into a double integral over z, Z with 0 < z < Z. Then setting Z = yt 12 , z = xyt 12 one is to integrate over 0 < x < 1 and 0 < y < ∞. 19 We first integrate over x and then add and subtract the leading term of the y → ∞ expansion. 20 For some recent discussions of one-loop self-energy corrections in AdS see [28][29][30].
Here we decomposed the derivatives of both propagators (2.12) in the loop into the regular and δfunction parts getting four terms: with no δ-function factors ( D), with one ( D ± ) and with two ( D cont ). Explicitly, (4.10) Integrating over t, t , z gives (4.11) Self-contraction. With the same decomposition of the two derivatives of the propagator in the loop (2.12) we get As a result, D cont here exactly cancels against the double δ-function part in the bubble diagram (4.9). The total expression for the one-loop correction is then where we used (4.11), i.e. D ± = − 1 π D. A more rigorous derivation of (4.13) requires introducing a regularization factor z ε in each (formally divergent) AdS integral. Then (4.14) Then expanding for small ε gives leading again to (4.13). Compared to (4.8) the vanishing result in (4.13) suggests that for consistency with global symmetry (see (B.6))) the value of g 0 = g(w, w) to be used in (4.8) should be indeed (4.16) Thus finally (using the notation g 0 in (4.8)) Self-energy corrections. The contribution of the corresponding diagrams (here gray circles stand for sums of relevant one-loop diagrams as in (4.1) and (4.9),(4.12)) may be represented as with the full 1-loop correction to three-point function thus given by In view of (4.22) to (4.8),(4.13) we conclude that for the special scheme choice (4.16) under which the two-point functions do not receive one-loop corrections the same is true also for the three-point function (4.25). Then comparing to the correlators of chiral currents in (2.37),(2.38) this suggests that the coefficient κ in (2.40) does not receive quantum corrections. 23 In Appendix C we will further elaborate on the issue of the scheme dependence of the one-loop corrections to the boundary correlators starting with a classically equivalent action in terms of redefined fields.

Boundary correlators and scattering amplitudes on AdS 2
While the scattering amplitudes for the massless WZW fields in flat space is known to vanish [20,4], we have seen that the coordinate-space boundary correlators for WZW fields AdS 2 are non zero. Their structure, however, is simple being dictated by the KM symmetry. One may wonder if with some natural definition of the AdS S-matrix they may actually correspond to trivial scattering in AdS 2 or on half-plane. Below we will attempt to clarify this issue.
It is useful first to recall what happened in the Liouville theory -how triviality of scattering in AdS 2 emerges in that case. The flat space scattering in this theory was argued to be trivial in [21], 23 We are assuming that the quantum theory is defined by the path integral with the WZW action (2.1),(2.3), (2.6) where the overall coefficient k or b in (2.6) has its classical value (an action with a shifted k would correspond to a different scheme choice). It is not clear if the quantum effective action [31,32] given by the WZW action with k → k + c G (that reproduces correlators of currents computed in perturbation theory on a plane) is a possible starting point in computing boundary correlators of elementary fields of the WZW theory in AdS 2 .
based on previous results about the energy-momentum eigenstates in finite volume [33][34][35][36]. 24 The scattering in a non-trivial Liouville vacuum or effectively in AdS 2 space was discussed in [5]. 25 Ref. [5] have shown that at the tree level there exists a perturbative expansion which is infrared safe and leads to trivial S-matrix. This conclusion was generalized and proved in more formal way in [22].
One may attempt to define S-matrix in AdS space by specifying suitable "in" and "out" states and computing amputated bulk correlators (as in flat space LSZ formula). In addition to the question of which asymptotic states to use (cf. [23]) a major technical problem is how to explicitly construct the Lorentzian AdS scattering amplitudes starting directly from the Euclidean coordinate-space boundary correlators.
Below we shall first outline the general relation between the AdS scattering amplitudes and the Lorentzian boundary correlators. Then we shall discuss the Euclidean → Lorentzian correlator reconstruction problem in the case of the Liouville theory relating it to the approach of [5]. Finally, we shall comment on the simplest scattering amplitude in the WZW theory in AdS 2 using an analogous method.

Massive scalar S-matrix on AdS 2
Let us start a scalar field theory in AdS 2 with mass parameter m 2 = ∆(∆ − 1). Let us consider a Witten diagram with one propagator connected to a bulk point (t, z) (here t is real Minkowski time, and z ≥ 0 is the radial AdS 2 Poincare coordinate). Ignoring dependence on other external points, it may be symbolically represented as 26 where D ≡ G ∆ is the Lorentzian massive scalar propagator with Dirichlet boundary conditions 27 and Γ stands for the rest of the diagram (i.e. with one line amputated). The propagator D may be written as where the functions {f ω (z)} ω>0 are eigenmodes of the kinetic operator for a scalar field in AdS 2 To avoid infrared problems, the theory may considered on a circle, where the Liouville field ϕ can be expressed in terms of a free field ϕ (0) by means of a quantum Bäcklund transformation. All energy-momentum eigenstates on the circle can be obtained by acting on the vacuum with the modes of the stress tensor T (0) mn of the Bäcklund field. In [21], it was argued that the dynamical properties of the infinite volume multi-particle states are equivalent to the large radius limit of the (free) T (0) mn eigenstates. This implies that the S-matrix is trivial. 25 As a normalizable translation-invariant ground state does not exist in Liouville theory in flat space, ref. [5] considered, following [8], the theory in a non-invariant domain-wall background that spontaneously breaks translation invariance and "semi-compactifies" space to a half-line. The resulting model can be identified with the Liouville theory in AdS 2 geometry. 26 Here t is Minkowski time related to Euclidean AdS 2 time t used above by t = it. 27 As in (2.8), this is for the standard normalization of the action, i.e. S = 1 They form a basis in z ∈ [0, ∞) with normalization f ω (z) can be identified with the wave function of the asymptotic state with energy ω created by the scalar field. Its explicit form for the Dirichlet boundary condition is where the normalization a(ω) is determined by (5.5). The corresponding scattering amplitude A(ω 1 , . . . , ω N ) may be formally defined as where in Γ we included the external leg labels and the subscript in f indicates the corresponding value of ∆ (in the case of multi-scalar scattering with different masses).

Comments on relation to boundary correlators
It is possible to formally "derive" a relation between (5.7) and a Fourier transform of the coordinatespace boundary correlators. Let us consider one leg in (5.1) taken to the boundary, i.e. define the boundary correlator where the circular line on the left denotes AdS 2 boundary. Substituting (5.3) into (5.1) and computing (5.8) using that f ω (ωz) ∼ z ∆ for z → 0, we find that the Fourier transform of A(t) is actually the same as the scattering amplitude in (5.7). Indeed, (here c ∆ is a coefficient dependent only on ∆) 28 Evaluating the integral over ω by picking (one-half of) the contribution from the pole at ω = ω, we obtain in agreement with (5.7). The same result is found by directly considering the boundary limit of (5.1). This amounts to replacing the bulk propagator D by the bulk-to-boundary expression 11) and taking the Fourier transform of (5.11) in Cauchy principal value sense (i.e. summing half of the two residues at t = t ± z ). 29 For ω > 0, it reads (5.12) and thus implies again (5.10). To summarize, we have shown that under a certain prescription, one can start with the N -leg boundary correlator for fields with dual conformal dimensions ∆ 1 , . . . , ∆ N take its Fourier transform in each leg and as result find an alternative representation for the scattering amplitude A(ω 1 , . . . , ω N ) in (5.7), i.e.
dt i e i ωi ti A(t 1 , . . . , t N ) . (5.14) Let us note that the amputated Green's function Γ in (5.7), as well as the boundary correlator in (5.14), are the Lorentzian ones. In general, the explicit analytical continuation of the boundary correlators from the Euclidean to the Lorentz signature should be done according to the general prescriptions based on reconstruction theorems [37] as discussed more recently in [38][39][40]. In particular, to compute the fully time-ordered Wightman function from the Euclidean correlators, one replaces t i → t i − iε i with ε i > ε j when t i > t j and then takes ε i → 0. 30 The Fourier transform of the resulting expression is expected to give the scattering amplitude and to match (5.7).
The on-shell condition is α 2 = ω 2 , namely α = ±ω. Besides, f −ω,ω (t, z) = f −ω,−ω (t, z), and we can simultaneously treat both signs of ω, i.e. "in" or "out" states. Up to irrelevant constants, the scattering amplitude for a 3-particle process may be written as wherē 29 At this stage this is just a formal prescription. More precisely, one should shift the integration contour by adding causal iε shifts, see below. 30 As first discussed in [41], the analytical continuation can be done at the level of Mellin amplitudes, see [42][43][44].
We may now use the known value of the following definite integral 31 is the area of a triangle with sides ω 1 , ω 2 , ω 3 (if ω 1 , ω 2 , ω 3 do not form a triangle, the integral is zero). From (5.18) ref. [5] found the following expression for (5.17) As the kinematically allowed 3-particle processes are associated with a degenerate triangle with vanishing area S = 0 one finds that A 3 = 0. This calculation has been extended in [5] to the 4-particle scattering processes that were also found to vanish. To try to recover (5.20) as a Fourier transform (5.14) of the boundary correlator we need first to analytically continue the Euclidean boundary 3-point function ∼ to the Lorenzian signature (to get the Lorentzian time-ordered 3-point function). Evaluating the associated Fourier transform seems far from trivial because the d 3 t integration region has to be split according to the time ordering and suitable ±iε shifts have to be introduced. 32 In principle, another approach is to look for an analytic continuation of the triple-K integral representation of the Euclidean 3-point function [47]. Such analytic continuations have been recently discussed in [48].
Let us note that continuation to Lorentzian signature and time-like momenta requires an analytic continuation of expressions involving the Appel function and this is known to be related to triple-J integrals for special arguments, see Eq. (7.1) of [49] and also [45,46]. This procedure is yet to be investigated in detail, but let us note that This relation shows that with a simple (although ad hoc) regularization of the triple-K integral, the triple-J integral (relevant for the scattering amplitude) shows up as the residue at the singular pole. The fact that leading singularities of divergent triple K integrals may contain physical objects has been discussed in the Euclidean context in [50]. It would be interesting to understand the relation between their analysis and relations like (5.21).

Massless scattering case
In view of the subtleties involved in extracting the scattering amplitudes from the Euclidean boundary correlators, here we shall consider massless scattering following the approach of [5] based on (5.7). Let us start with the simplest 1 → 2 process and emphasize the difference between models with derivativeindependent scalar φ 3 vertex and with φ(∂φ) 2 σ-model type (classically) conformally invariant vertices. For a massless scalar we have ∆ = 1 or ∆ − 1 2 = 1 2 in (5.6) and for a φ 3 interaction vertex the analog of the integral in (5.17) representing the tree level 1 → 2 particle production amplitude iŝ ∞ 0 dz √ z J 1/2 (ω 1 z)J 1/2 (ω 2 z)J 1/2 (ω 3 z) 31 Useful integrals involving three Bessel functions are discussed in [45,46]. 32 The Fourier representation of the Wightman Lorentzian 3-point function with fixed time ordering where we defined ω 12 = ω 1 + ω 2 − ω 3 , etc., and Ω = ω 1 + ω 2 + ω 3 . The integral (5.22) does not vanish on-shell. For instance, if ω 3 → ω 1 + ω 2 it has a finite non-zero limit. 33 In the σ-model case in flat space the 3-point amplitude vanishes due to on-shell kinematics. This is less automatic in the AdS 2 case. Let us consider the case of a general σ-model in the parametrization used in (3.2), (3.8) where the cubic vertex in the WZW case is ∼ f abc ∂X a ∂X b X c . Because of antisymmetry of f abc the vertex is effectively ∼ 1 2 (∂X a ∂X b − ∂X a ∂X b )X c . Let us first consider the contribution of the first term and then antisymmetrize in momenta. We will need the wave functions f ±ω,ω (t, z) = e ±iωt sin(zω) , Starting from (5.7), suppressing the group indices and defining α i = ±ω i we find (cf. (5.17),(5.22)) Here we used that´∞ 0 dz e iωz = πδ(ω) + iω −1 . We are still to antisymmetrize in ω 1 ↔ ω 2 , but since the expression in (5.24) is symmetric, the final result is thus zero. Thus the 3-point scattering amplitude vanishes also in AdS 2 . 34 As for the 4-particle scattering amplitude, in the Liouville theory in AdS 2 it was found to vanish in a non-trivial way, due to a cancellation of different contributions [5]. It would be interesting to see if it also vanishes in the WZW theory in AdS 2 . A possible reason of why this may happen is the absence of non-trivial structures in the corresponding Euclidean boundary correlators, i.e. the cancellation of logarithmic terms that happens in the Liouville theory [6,7] and that we also observed above for the WZW limit of a general σ-model. To establish this link it remains to derive the AdS 2 scattering amplitudes from Euclidean boundary correlators in a systematic way.

Concluding remarks
In this paper we considered boundary correlators of elementary fields of 2d σ-models in AdS 2 . Similar problem appears in the study of correlators of operators on a Wilson line in the strong-coupling description in terms of the AdS 5 × S 5 Nambu string action in the static gauge [1,2]. One motivation is to learn how to compute loop Witten diagrams in AdS 2 in models with derivative interactions. We have observed, in particular, that the structure of four-point correlators simplifies (with logarithmic 33 Notice that we can put an arbitrary scale µ in the logarithms in (5.22) since Ω − ω 12 − ω 13 − ω 23 = 0. 34 Let us note that dealing with massless 2d fields requires extra care. The wave function f in (5.23) is not vanishing for z → ∞. Thus, integration by parts is not a priori allowed in (5.24). That means that the starting form of the action may be important as the contribution of boundary terms (produced by integrations by parts) may be non-trivial.
terms of the 1d cross-ratio cancelling out) only in the WZW case when the σ-model has an extra KM symmetry. In that case the boundary correlation functions of the WZW fields are found to be the same as the correlators of the chiral WZW currents on the plane restricted to the real line.
Another possible motivation is related to the search for new integrable 2d σ-models using S-matrix based criteria as in the massive case. If one expands near a trivial σ-model vacuum in flat 2d space one gets massless scattering amplitudes which, in general, suffer from IR ambiguities [4,51]. If instead one considers the σ-model on AdS 2 then its coordinate-space boundary correlators are better defined and one may try to find the analogs of the standard integrability constraints (S-matrix factorization and no particle creation) directly in terms of them. As any 2d σ-model is classically Weyl invariant, the tree-level problem in AdS 2 is equivalent to the same problem on flat half-plane with particular (Dirichlet) boundary conditions. Hidden conserved charges that exist in a classically integrable σmodel on a plane should lead to constraints on the corresponding Euclidean boundary correlators and the associated S-matrix on half-plane. This should also extend to the quantum level if the σ-model is quantum scale invariant (like the WZW model).
where the geodesic distance η is defined in (2.9). The associated bulk-to-boundary propagator is g ∂ (t; w ) = lim The action (2.6) is readily checked to be invariant under (B.3) (using integration by parts). Using the boundary asymptotics (2.7) we get from the z → 0 limit of (B.3) the following transformation of the corresponding boundary fields Assuming the computational scheme preserves the global SL(2, R) symmetry, it then imposes constraints on the boundary correlators. In view of the symmetry rotating ψ intoψ one should have Φ(t 1 ) Ψ(t 2 ) = 0. Applying the variation (B.4) to this relation gives The SL(2, R) symmetry implies that the tadpole Φ should vanish (φ is shifted by the parameter θ 1 in (B.1)). 35 We thus find the following relation This relation is expected to hold at the quantum level assuming the above SL(2, R) symmetry is preserved by the computational scheme. This is a necessary condition for matching the correlation functions of chiral currents on which SL(2, R) acts linearly. 35 Note that the one-loop contribution to Φ given by the tadpole with (ψ,ψ) propagator computed with a cutoff z > ε is linearly divergent This divergence is to be subtracted in a SL(2, R) preserving scheme (see also discussion below (4.1)).
A more consistent approach should be to define the boundary correlator with the "wave-function" renormalization factors included and that should ensure the invariance of the result under field redefinitions. Then the expressions in this Appendix found starting with the redefined action (C.2) could be reconciled with the approach used in section 4. 39 This remains to be clarified further.