Bulk phase shift, CFT Regge limit and Einstein gravity

Journal of High Energy Physics, Jun 2018

Abstract The bulk phase shift, related to a CFT four-point function, describes two-to-two scattering at fixed impact parameter in the dual AdS spacetime. We describe its properties for a generic CFT and then focus on large N CFTs with classical bulk duals. We compute the bulk phase shift for vector operators using Regge theory. We use causality and unitarity to put bounds on the bulk phase shift. The resulting constraints bound three-point functions of two vector operators and the stress tensor in terms of the gap o the theory. Similar bounds should hold for any spinning operator in a CFT. Holographically this implies that in a classical gravitational theory any non-minimal coupling to the graviton, as well as any other particle with spin greater than or equal to two, is suppressed by the mass of higher spin particles.

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Bulk phase shift, CFT Regge limit and Einstein gravity

JHE Bulk phase shift, CFT Regge limit and Einstein gravity Manuela Kulaxizi 0 2 Andrei Parnachev 0 2 Alexander Zhiboedov 0 1 Dublin 0 Ireland 0 0 Cambridge , MA 02138 , U.S.A 1 Department of Physics, Harvard University 2 School of Mathematics, Trinity College Dublin The bulk phase shift, related to a CFT four-point function, describes twoto-two scattering at fixed impact parameter in the dual AdS spacetime. We describe its properties for a generic CFT and then focus on large N CFTs with classical bulk duals. We compute the bulk phase shift for vector operators using Regge theory. We use causality and unitarity to put bounds on the bulk phase shift. The resulting constraints bound three-point functions of two vector operators and the stress tensor in terms of the gap of the theory. Similar bounds should hold for any spinning operator in a CFT. Holographically this implies that in a classical gravitational theory any non-minimal coupling to the graviton, as well as any other particle with spin greater than or equal to two, is suppressed by the mass of higher spin particles. AdS-CFT Correspondence; Conformal Field Theory - HJEP06(218) 1 Introduction 2 Bulk phase shift 2.1 Phase shift for spinning operators 3.1 3.2 3.3 4.1 4.2 4.3 5.1 5.2 5.3 5.4 5.5 5.6 5.7 3 Conformal blocks and particle exchanges in AdS Basic kinematics Limits of the t-channel conformal block Exchange Witten diagram 3.4 Impact parameter space 3.5 Double trace operators and the bulk phase shift 4 Conformal Regge theory in the impact parameter space Conformal Regge theory in the t-channel: scalars Conformal Regge theory in the t-channel: vectors Contribution of the stress tensor pole 5 Constraints on the bulk phase shift and leading Regge trajectory Finite subtractions and particles that do not lie on Regge trajectories Constraints from Rindler positivity Causality in the impact parameter space Scattering bound from unitarity Bound on the three-point coupling Bound on the three-point coupling from inelastic effects Other Regge trajectories 6 Conclusions A Harmonic functions B Choice of coordinates D Vectors from scalars C Generalized free fields in d = 2 in the Regge limit E The Regge limit of the correlation function hJμψψJν i F The Regge limit of hJμψψJν i in the impact parameter space G Polynomial terms in the Mellin amplitude H Diagonalization of the bulk phase shift – i – Introduction A useful way to chart distinct physical regimes in a gravitational theory is to study scattering as a function of energy S and impact parameter L [1–3], see also figure 1 in [ 4 ]. In flat space the relevant quantity, namely the phase shift δ(S, L), is related to the Fourier transform of the four-point scattering amplitude A(s, t = −~q2) with respect to the transferred momentum ~q, see also [5, 6]. In AdS the phase shift is given by the convolution of the four-point function of local operators with proper external wave functions [7–11]. In this paper we adopt the approach of [11] where the phase shift δ(S, L) was shown to be given by a certain double Fourier transform of the Lorentzian four-point function to be discussed in detail below. One motivation to study the phase shift comes from its relation to causality. Indeed, in AdS/CFT boundary observers can exchange signals either through the bulk or along the boundary. Microscopic CFT causality guarantees that the signals sent along the boundary are the fastest. In Einstein theory an analogous statement in the bulk is guaranteed due to the theorem of Gao and Wald [12]. More generally, signals in gravitational theories are delayed with respect to the asymptotic causal structure [13]. This fact could be used to constrain effective theories of gravity [14]. Another motivation is to have a more direct connection between the physics of bulk scattering and CFT correlators. This connection could be potentially useful in both directions. One can hope to use bulk intuition to get some insight into the CFT dynamics and conversely use non-perturbative CFT methods to say something nontrivial about physics in the bulk. In this paper we will be interested in regimes where the interaction is weak δ(S, L) ≪ 1. The behavior of δ(S, L) at large energies S ≫ 1 describes propagation of energetic particles through AdS and as such should respect the asymptotic causal structure. For interacting CFTs in d > 2 at large distances the interaction is generically controlled by the graviton exchange [15, 16].1 One can show that this implies the following behavior for the phase shift: δ(S, L) = cSe−(d−1)L + . . . , L ≫ 1 (1.1) which is simply the large impact parameter L asymptotic of the graviton propagator in the impact parameter space Hd−1.2 The coefficient c is fixed in terms of the three-point coupling between the propagating probe and the graviton. Asymptotic causality implies that c ≥ 0, namely that gravity is attractive. When applied to scalar particles this is automatically satisfied due to unitarity. When applied to particles with spin this condition implies the positivity of the energy flux [20]. Recently this statement was proved using very different-looking methods [21–26]. 1In free theories the dependence on impact parameter is trivial, namely at high energies it is just a function of S. This is an extreme example where locality is absent even on super-AdS scales. 2For CFTs with scalar operators of dimensions d−2 < Δ < d − 2 this might not be true. To avoid this we 2 can imagine considering a target that does not couple to such light scalars. Also note that in the language of double twist operators O(∂2)n∂μ1 . . . ∂μs O (1.1) corresponds to s ≫ n ≫ 1 [17–19]. – 1 – For special theories the simple behavior (1.1) of the phase shift persists for much smaller impact parameters L > L∗. These are theories with a large central charge hTµν Tρσi ∼ cT ≫ 1 and a large gap Δgap ≫ 1 in the spectrum of higher spin operators [27]. We are interested precisely in this regime, namely large S and L such that the graviton exchange dominates δ(S, L) = Sδ(L) + . . . , L ≥ L∗ . (1.2) This regime was considered in [14] where it was shown that positivity of δ(L) ≥ 0 for L ≥ L∗ implies stronger bounds on the coupling of matter with spin to the graviton. In the limit L∗ → 0 the only allowed coupling is the one of Einstein gravity. Moreover, [14] argued that classically only higher spin particles could fix the problem. The same problem was addressed using CFT methods in [28] under the assumption that double trace operators could be neglected. Here we find that they are non-negligible in the Regge limit but drop out when we consider scattering in the impact parameter space. There are two alternative ways in which the phase shift δ(S, L) could be computed in a CFT with a holographic dual: either using the shock wave geometry, or using the four-point correlation function. In the AdS analysis of [14] only the shock wave method was used; this was a computation done entirely on the gravity side. The purpose of the present paper is to provide a CFT computation of the phase shift, which is similar to the scattering amplitude computation in the case of flat space. A purely CFT description of the phase shift will facilitate computing corrections and will advance our understanding of the implications of [14] for generic CFTs. In particular, using CFT methods we can compute the correction to the phase shift due to the tower of higher spin particles. This is achieved using conformal Regge theory [29] and does not have a simple bulk counterpart. The rest of this paper is organised as follows. In section 2 we review the construction of [11] for the phase shift δ(S, L). To define it, it is convenient to describe the wave function of each operator in its own Poincare patch. The corresponding conformal transformation is standard in the context of Regge theory and appeared in several works, for example [11, 20, 30]. In section 3 we consider a simple example of the Witten exchange diagram and discuss its different limits. In particular we describe in detail the structure of the correlator before and after the Fourier transform and the role played by the double trace operators. The result for the phase shift in this case is what one gets from the shock wave computation in the bulk [14]. In section 4 we use Conformal Regge Theory to compute the high energy limit of the phase shift δ(S, L), which is related to the Regge limit of the four point function in CFT [29]. We consider a four-point function which involves spin one operators Jµ . We give an expression for the phase shift, assuming that the Regge limit is dominated by a Regge pole.3 Next, we compute the contribution of the stress tensor to the phase shift; the result is effectively the same as the spinning Shapiro time delay of [14] that was previously computed using shock waves in AdS. This time, however, we can compute the correction to it due to the tower of higher spin particles and explicitly see how the gravity computation breaks down. 3This is believed to be the case in large N CFTs. – 2 – of [14]. Similarly to previous work, we derive a parametric bound on hJµ TρσJν i when the gap is finite. This is related to the fact that the minimal impact parameter for which one can run the argument to constrain the three-point coupling depends on the energy S∗ for which our approximation is reliable, i.e., Δ2gap ∼ Δ2gap 1 order Δgap requires S ≫ Δ2gap. In this way we get4 L2∗ log S∗ . Considering impact parameters of where β4 is the combination of the coefficients in the three-point function hJµ TρσJν i which corresponds to the non-minimal coupling to the graviton. We also obtain the corresponding expressions for non-conserved, spin one operators. The bound for any spinning operator should be similar. 2 Bulk phase shift In this section we introduce the bulk phase shift. It describes 2 → 2 scattering at fixed impact parameter in AdS. In Minkowski space the phase shift δ(s,~b) is given by the Fourier transform of the scattering amplitude 1s A(s, t = −~q2) with respect to the transferred momentum, ~q. Here both the transfer momentum, ~q, and the impact parameter ~b, belong to the transverse space Rd−2. We describe the analogous AdS kinematics below. Let us first review AdSd+1 space in the embedding coordinates: The conformal group generators are simply the SO(2, d) rotations Mˆ AB = −i(ZA∂ZB − ZB∂ZA ), while, δZA = iMˆ ABZB. The embedding coordinates of (2.1) are related to the usual, AdS, global coordinates by Z0 + iZ−1 = eiτ coshρ, Zi = nisinhρ, ~n2 = 1, i = 1, . . . , d. (2.2) To approach the boundary we take ρ → ∞. Then (τ, ~n) label points in the boundary CFT. In the embedding space points on the boundary correspond to Z2 = 0 together with the identification Z ∼ λZ. We imagine a situation where two excitations start at τ = − π2 on the antipodal points on the sphere, collide at τ = 0,5 and reach the boundary at τ = π2 at the antipodal points on the sphere see figure 1. the CFT. 4If we were to use the eikonal methods we would get log S∗ ∼ log cT , where cT is the central charge of 5Notice that this statement does not assume bulk locality. The fact that τ = 0 is the first instance when excitations can interact follows from boundary causality [31]. – 3 – operator is inserted in the vicinity of the corresponding origin with the wavefunction which is a plane wave. The scattering occurrs in Hd−1 which is marked by a blue line. It stands for the intersection of two Poincare horizons which are depicted in shaded gray. Each Poincare horizon is generated by future or past null geodesics emanated from points xi. More precisely, the four points on the boundary are Point 1 : Point 2 : Point 3 : Point 4 : Z−1 = −1, Z1 = −1, Z−1 = −1, Z1 = 1, Z−1 = 1, Z−1 = 1, Z1 = −1, Z1 = 1, where we only wrote the non-zero ZA components. For the points above, Z2 = 0, since all of them belong to the AdS boundary. It is useful to introduce different Poincare coordinates around the boundary points above. It helps to understand symmetries of the problem. Choosing a Poincare patch for each point, corresponds to splitting R2,d into M 2 × M d [11, 17], where M 2 is specified by a pair of orthonormal null vectors KA and K¯ A, such that K2 = K¯ 2 = 0 and −2K.K¯ = 1, then we have ZA = K¯ A + y2KA + y A where yA are the usual coordinates in Minkowski space. Again for (2.4) we have Z2 = 0 as it should be. By contracting both sides with KA we get the gauge condition If we are to introduce the AdS Poincare coordinates, then Z.K = 0 corresponds to the Poincare horizon. − 2Z.K = 1. – 4 – (2.3) (2.4) (2.5) The Poincare horizon corresponds to K.Z = 0 where the generator simplifies and the momentum eigenstate has the wave function where Z~ refers to Zi with i = 2, . . . , d. The eigenvalue is Pµ = 2i (K.Z∂Zμ − Zµ K.∂Z ) . φ(Z) ∼ e−iP K¯ .Z δ(d−1)(Z~ − W~ ) Pµ φ(Z) = P Wµ φ(Z). (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) Here the indices µ, ν span M d. We are now ready to describe the Poincare patches associated to points (2.3). It is convenient to describe them using (2.5) which take the form and P2 intersect along Z−1 = Z1 = 0. This is Hd−1 Past Poincare horizons of the patches P3 and P4 and future Poincare horizons of P1 d i=2 − (Z0)2 + X(Zi)2 = −1 and describes the transverse impact parameter space in which we separate the colliding particles. We are interested in the wave functions for the boundary operators such that the state in the bulk is localized in Hd−1. As explained in [20] these are just plane waves in the corresponding Poincare patches. Indeed, let us consider for example the wave function of an eigenstate of the bulk isometry associated to the four-momentum Pµ Symmetry transformations in a given Poincare patch are related to the transformations in global AdS as follows [17, 32] Mµν = Mˆ µν , Pµ = −2KAMˆ Aµ , D = 2KAK¯ BMˆ AB . Kν = −2K¯ AMˆ Aν , four-momentum. We will consider time-like momenta p2 < 0. In terms of boundary CFT this corresponds to the four-momentum pµ = P W µ , where p 2 = −P 2 and W µ is the unit vector (2.8) W 2 = −1 that specifies the direction of the Notice that the isometry that we call Pµ depends on the choice of KA and is different for Poincare patches 1, 3 and 2, 4. For example, translations in patches 1, 3, that are identified as Pµ, 1 = −Pµ, 3, become special conformal transformations in patches 2, 4 and vice versa. On the other hand, Lorentz transformations x → Λx are the same in all four Poincare patches. – 5 – For two particles localized at different points on Hd−1 the geodesic distance between them is equal to introduction. p1.p2 −p12p −p22 coshL = W1.W2 = − p , 0 ≤ L < ∞ . (2.12) We will refer to L as the impact parameter and it is the same L that appeared in the Similarly, we can introduce energy of the collision S = P1P2 = We now imagine elastic scattering of two particles at fixed impact parameter space in AdS, see [11]. In light of kinematics described above a natural object to consider is the following. We start with the four-point function and operators are ordered as written. u = where the subscript stands for the operator insertion yi with i = 1, . . . , 4. This coordinate transformation is common in considerations of the Regge limit. After doing the coordinate transform we set x1,4 = ± x2 , x2,3 = ± 2 . Notice that in this way the correlator (2.14) is x¯ time-ordered. For reader’s convenience let us present the explicit transformation formulas y y 1+ = −y4+ = − x+ , 1− = −y4− = − 2x+ , 2 x 2 ~y1 = ~y4 = ~x x+ , y y 2− = −y3− = − x¯+ , 2+ = −y3+ = − 2x¯+ 2 x¯ 2 ~y2 = ~y3 = ~ x¯ x¯+ . In these formulas we take xµ and x¯µ to be future-directed time-like vectors. This implies that x+, x¯+ > 0 and x 2 < 0 and x¯ 2 < 0. Moreover, the spacelike yi2j in the original we will consider more general x and x¯ which could be both spacelike and timelike. x¯ where in the l.h.s. each operator is inserted in its own Poincare patch as described above. The cross ratios in terms of new variables take the form Next, as reviewed above, we do the transformation to the impact parameter space B(p, p¯) = ddxddx¯eip.xeip¯.x¯A(x, x¯). similarly for x¯2. In doing the Fourier transform we have to specify the iǫ prescription. The prescription is dictated by the ordering of operators and in this case takes the form x 2 → x2 − iǫx0 and Let us understand the symmetries of B(p, p¯). The transform is manifestly invariant under Lorentz transformations, thus, it can only depend on p2, p¯2 and p.p¯. Moreover, we have the following transformation of A(x, x¯) hφ(y4)ψ(y3)ψ(y2)φ(y1)i = A(u, v) (−x2)Δφ (−x¯2)Δψ , u = v = , Z 1 λ 1 λ C(Δ) = 2d+1−2Δπ1+ d2 Γ(Δ)Γ Δ − 2 which translates using (2.20) into This simmetry together with Lorentz invariance implies that A λx, x¯ = λ2(Δψ−Δφ)A(x, x¯) B λp, p¯ = λ−2(Δψ−Δφ)B(p, p¯). B(p, p¯) = B0(p, p¯) (1 + iδ(S, L)) where S and L were defined in (2.13) and (2.12). B0(p, p¯) is the Fourier transform of the disconnected part hφφihψψi Z B0(p, p¯) = ddxddx¯eip.xeip¯.x¯ 1 (−x2 + iǫx0)Δφ (−x¯2 + iǫx¯0)Δψ , = θ(p0)θ(−p2)θ(p¯0)θ(−p¯2)e−iπ(Δφ+Δψ)C(Δφ)C(Δψ)(−p2)Δφ− d2 (−p¯2)Δψ− d2 , The unity in (2.23), therefore, describes a free propagation in AdS. The phase shift δ(S, L), on the other hand, describes interaction. We will be interested in the scattering of energetic probes, namely S ≫ 1. The physical picture in this case depends on the impact parameter L. In this paper we focus on impact parameters for which the dominant contribution to the phase shift is due to the graviton exchange δ(S, L) = cSΠd−1(L) + . . . (2.25) where Πd−1(L) is the propagator in Hd−1 and . . . stands for subleading corrections. For a generic CFT we expect L∗ > RAdS, but for theories with large gap or local bulk dual we HJEP06(218) could have L∗ ≪ RAdS. In the present paper we analyze (2.25) using CFT methods. The high energy limit S ≫ 1 of the AdS phase shift δ(S, L) is related to the Regge limit and we assume that the relevant singularity in the J plane is a pole. In this way the corrections to the simple gravity formula (2.25) are dictated by the properties of the leading Regge trajectory. 2.1 Phase shift for spinning operators Below we will be interested in correlation functions of operators with spin as well. For simplicity we focus on the case of two spin one operators and two scalars. A natural object to consider in this case is the following Amn(x, x¯) = hJ m x ψ x¯ ψ x¯ 2 J n x 2 i , where the polarization tensors are defined in the corresponding Poincare patch. To relate to the original coordinates we get the following expression for the polarization tensors ǫµ (y1) = ∂xm ∂y1µ , ǫν (y4) = ∂xn ∂y4µ . So that in the original coordinates we have Amn(x, x¯) = 2 As before it will be convenient to separate the contribution of the disconnected piece B0mn. We set B0mn(p, p¯) = ηmn − g(Δ) B0(p, p¯) , pmpn p2 where g(Δ) = 2 ΔΔJJ−−d1/2 and study the bulk phase shift matrix δmn(p, p¯) defined by Bmn(p, p¯) − B0mn(p, p¯) B0(p, p¯) = iδmn(p, p¯) , where ηmn denotes the Minkowski metric. ǫµ (y1)ǫν (y4)hJ µ (y4)φ(y3)φ(y2)J ν (y1)i. 2 Z – 8 – Conformal blocks and particle exchanges in AdS In this section we consider a simple example of a particle exchange in AdS that allows us to illustrate some ideas which we develop further in the paper in more generality. We review notions of the Regge and the bulk point limits. We compare the behavior of the corresponding Witten diagram to the behavior of a single conformal block in these limits.6 The block and the AdS diagram differ by the contribution of an infinite series of double trace operators, see e.g. [33], which is non-negligible in the regime of our interest. We then transform the result to the impact parameter space and reproduce the high energy limit of the phase shift that is obtained from the bulk considerations [14]. The result is that both the block and the Witten diagram develop the same singularity in the Regge limit. However, the bulk point limit behavior is very different. The conformal block is singular in the bulk point limit, whereas the Witten diagram is regular. The difference is important to us because as we review below the result for the Regge limit of the correlator that is produced by a Regge pole is similar to the exchange diagram and is regular in the bulk point limit. This fact will be important for the discussion of bounds on the Regge limit coming from causality and unitarity. Next, we consider the transform to the impact parameter space in AdS. This was done in [11] and we review it below. The Regge limit is related to the high energy behavior of the phase shift. The crucial observation is that upon the transformation to the impact parameter space the contribution of double trace operators effectively cancels and the phase shift is controlled solely by the quantum numbers of the particle that is being exchanged in AdS. In this way the simple result of the bulk computation is reproduced. 3.1 Basic kinematics Consider the four point function of scalar operators It is convenient to discuss the kinematics in terms of the partial amplitude A(z, z¯) or rather G(z, z¯) ≡ hφ(∞)ψ(1)ψ(z, z¯)φ(0)i = ([1 − z][1 − z¯]))−Δψ A(z, z¯) , (3.1) (3.2) where ψ(∞) = limy→∞ y2Δψ ψ(y). coupling in AdS [29]. 6The result for the Witten diagram in the Regge limit does not depend on the details of the three-point – 9 – HJEP06(218) The correlator G(z, z¯) admits an expansion in terms of conformal blocks in three different channels [21, 23] s − channel : G(z, z¯) = (zz¯)− 21 (Δφ+Δψ) X λφψOΔ,J λψφOΔ,J gΔ,J Δφψ,−Δφψ (z, z¯) , Δφψ = Δφ − Δψ, |z| < 1. t − channel : G(z, z¯) = ([1 − z][1 − z¯])−Δψ X λφφOΔ,J λψψOΔ,J gΔ0,0,J (1 − z, 1 − z¯), u − channel : G(z, z¯) = (zz¯) 12 (Δφ+Δψ) X λφψOΔ,J λψφOΔ,J gΔ,J Δφψ,−Δφψ where the sum runs over an infinite set of primary operators that appear in the corresponding OPE channel. The expressions for conformal blocks can be found for example in [34]. There are several limits of this correlator that are usually discussed. These include light-cone, Regge and bulk point limits. We first review the behavior of a single conformal block in each of these limits. Let us describe the limits of interest in detail. The simplest limit to consider is the light-cone limit, Lightcone : z → 1, z¯ − fixed. In the t-channel OPE this limit is controlled by the lowest twist operators that contribute τ as (1 − z) 2 , where τ = Δ − J . To approach the Regge limit we continue z around 0 and then send both z, z¯ → 1 Regge : ze−2πi, z¯ → 1, z z¯ − fixed , where ze−2πi indicates that we have to analytically continue G(z, z¯) before taking the limit. To analyze the Regge limit, it is convenient to introduce the following coordinates following [29] 1 − z = σeρ, 1 − z¯ = σe−ρ. The Regge limit corresponds to σ → 0 limit with ρ held fixed. Notice as well that our choice of points corresponds to σ < 0. In this limit both the single trace and double trace operators are important. By the bulk point limit we loosely mean Bulk point : z → z¯, or ρ → 0 in the notations of (3.8). When talking about (3.9) we have to specify on which sheet and along which path we approach z = z¯ point. We will be interested in (3.9) after analytic continuation (3.7). The limit considered in [31] corresponds to the same contrinuation for z as above in the consideration of Regge limit supplemented by the continuation ρ → ρ ± iπ which makes z and z¯ negative see. OΔ,J (3.3) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) Imagine that in (3.2) there is an operator with dimension Δ and spin J being exchanged in the t-channel φφ ∼ OΔ,J ∼ ψψ. We would like to evaluate the contribution of one t-channel block in the Regge limit. Notice that the t-channel OPE does not converge in the Regge limit and at the end we will have to use Regge theory to analyze the t-channel physics. Let us first consider this exercise in d = 2, 4. The conformal blocks in the t-channel expansion (3.2) are given by7 kβ(z) = 2F1 , , β, z , gΔd=,J2(z, z¯) = z 2 z¯ 2 Δ−J gΔd=,J4(z, z¯) = Δ−J Δ−J z 2 z¯ 2 z − z¯ β β Notice the behavior of the blocks before analytic continuation. In the light-cone limit of the operator. In the limit z, z¯ → 0 the block behaves as gΔ,J (z, z¯) ∼ z Δ+2J z¯ 2 z → 0, z¯ fixed, the block behaves as gΔ,J (z, z¯) ∼ z 2 and is controlled by the twist τ = Δ−J Δ−J + c.c and is controlled by the dimension of the operator Δ. In the bulk point limit z → z¯ on the principal sheet the blocks are regular. To approach the Regge limit it is useful to recall the analytic continuation of the hypergeometric function8 kβ(1 − ze−2πi) = kβ(1 − z) + 2πi Γ(β) Γ( β2 )2 2F1 β β Using this identity we can find the leading order behavior of the conformal block in d = 2 and d = 4. The result is gΔd=,J2 = i gΔd=,J4 = i where we suppressed the constant of proportionality which is irrelevant for us. Let us pause for a moment and contemplate the result (3.12). First, these examples show the generic feature of the behavior of the conformal block in the Regge limit Regge : gΔ,J (σ, ρ) = i f Δ,J (ρ) σJ−1 . This is what makes it hard to approach the Regge limit using the standard OPE technique. The contribution of a given primary is controlled by its spin J and operators with higher spin dominate. 8See (4.27) in [21]. 7For convenience we use zhere = 1 − z in (3.2). We hope it will not cause any confusion. (3.10) HJEP06(218) (3.11) (3.12) (3.13) Second, the behavior in the bulk point limit ρ → 0 depends on the number of dimensions d. With some hindsight the behavior of the block is where in d = 3 the singularity becomes logarithmic. The expression in general d could be found using the Casimir equation [34].9 The result is that f Δ,J (ρ) = cΔ,J ΠΔ−1(ρ), where ΠΔ−1(ρ) is the Euclidean propagator in hyperbolic space Hd−1,10 ΠΔ−1(ρ) = cΔ,J = 2Γ(Δ − We see that each conformal block is singular at ρ = 0. This singularity is purely kinematical. It could be thought as coming from an infinite volume of Hd−1 in AdS which describes the center of the collision and not from the small impact parameter. The full correlator is regular in this limit [31]. The Regge limit of the stress tensor conformal block with external spinning operators was analyzed in detail in [28]. This conformal block alone, however, does not control the Regge limit of the correlator as we will see shortly. 3.3 Exchange Witten diagram We would like to contrast the behavior we found above for one conformal block with the behavior of the corresponding Witten diagram figure 2. We imagine an exchange of a particle with quantum numbers the same as above, namely dimension Δ and even spin J . As discussed in detail in [33] an exchange in the bulk when decomposed in terms of conformal blocks (3.2) contains two types of contributions: single-trace operator conformal block gΔ,J ; corrections to the anomalous dimensions of double trace operators that, schematically, could be written as φ∂2n∂µ 1 . . . ∂µ s φ and ψ∂2n∂µ 1 . . . ∂µ s ψ with s ≤ J . The precise result for the Witten diagram depends on the details of the three-point coupling in the bulk. However, in the Regge limit the answer is universal (see for example, 9This has two solutions which correspond to the usual conformal block and its shadow. The two can be distinguished by the transformation properties under ρ → ρ + 2πi at large ρ [35]. The usual conformal block transforms with the e−2πiΔ phase, whereas the shadow block picks e2πi(Δ−d). This allows one to pick the correct solution. 10The propagator satisfies the following equations (3.14) (3.16) HJEP06(218) √1g ∂i√ggij∂jΠΔ−1(ρ) − (Δ − 1)(Δ + 1 − d)ΠΔ−1(ρ) = δHd−1 (ρ) √1g ∂i√ggij∂jΠΔ−1(ρ) = Π¨Δ−1(ρ) + (d − 2) coth ρΠ˙ Δ−1(ρ) , ds2Hd−1 = dρ2 + sinh2ρ dΩ2Sd−2 , (3.15) where dots stand for the ∂ρ derivatives. Regge : A AdS(σ, ρ) = i f AdS(ρ) σJ−1 the appendix A.3 of [29]). The relevant expressions in [29] are given in terms of the Mellin amplitude associated to a given Witten diagram, but they can easily be rewritten as follows (3.17) (3.18) (3.20) (3.21) where the harmonic function Ωiν (ρ) in the hyperbolic space Hd−1 is defined in appendix A and KΔ,J could be found, for example, in eq. (41) of [29]. It is important to note that Ωiν (ρ) is regular at ρ = 0. To see this one can simply expand Ωiν (ρ) at small ρ Ωiν (ρ) ∼ Γ d − 2 2 + iν Γ d − 2 2 − iν ν sinh[πν] 1 − ρ 2 ν2 + (d−2)2 4 2(d − 1) ! 11For the odd spins we would also have the factor e−iπJ depending on the ordering of the external where A the reduced amplitude A(σ, ρ) and11 AdS(σ, ρ) denotes the contribution of the Witten diagram depicted in figure 2 to f AdS(ρ) = c0 γ(ν) = Γ Z ∞ −∞ dν γ(ν)γ(−ν) ν2 + (Δ − d2 )2 Ωiν (ρ) , , c0 = λψψOΔ,J λφφOΔ,J 41+J π d2 KΔ,J , γ(ν)γ(−ν) ∼ e−π|ν|, Γ d − 2 2 + iν Γ d − 2 2 − iν ν sinh[πν] ∼ νd−2. large ν and the result is an expansion of the type P∞ Due to the exponential suppression produced by the γ(ν)γ(−ν) the integral converges at n=0 cnρ2n. In contrast to (3.14) we where further corrections are again polynomial in ν2 and ρ2. We can now estimate the behavior of the integrand (3.18) at large ν, we have have in this case operators. To connect with the previous section notice that One can, in principle, substitute this into (3.18) and compute the integral. The contribution due to the pole at ν 2 = (Δ − 2 d )2 in (3.18) gives precisely the conformal block of the exchanged operator (3.12). Note that it is singular in the ρ → 0 limit. At the same time, contributions due to the double trace poles contribute corresponding conformal blocks as well (with some non-vanishing coefficients). The total result is regular in the ρ → 0 limit as shown above. Hence, the double trace operators precisely cancel the singular ρ → 0 behavior of the stress-tensor conformal block. More generally this cancelation follows from the s-channel OPE, see section 6.3 in [31]. In contrast to a single conformal block the exchange Witten diagram generates a consistent solution to the large N crossing [27]. We can think of the effective Lagrangian which contains two scalar fields dual to φ and ψ and the cubic coupling of this field to the dual of the exchanged operator OΔ,J . It also serves as a correct model for the Regge pole in a CFT as we will review in the next section. Another little comment is to notice that if we consider the bulk point singularity region by analytically continue (3.17) ρ → ρ ± iπ it does become singular for small ρ in accord with general arguments presented in [31]. To see it explicitly let us consider d = 4. In this case Ωiν (ρ) ∼ νssininhνρρ . This develops a singularity at ρ = 0 upon continuation ρ → ρ ± iπ. 3.4 Impact parameter space As explained in the previous section, to make a more direct connection with the bulk it is useful to transform the correlator to the impact parameter space [11]. The starting point is the correlation function (2.18) obtained from (2.14) via (2.17). To determine the reduced amplitude holographically, we simply need to add the contribution of the disconnected piece to the Witten diagram result (3.17) of the previous section. Before performing the Fourier transform we must relate (σ, ρ) or (z, z¯) with (x, x¯) in the Regge limit which corresponds to small (x, x¯). With the help of (2.19) one finds that (1 − z)(1 − z¯) = σ2 = x2x¯2 + . . . , (1 − z) + (1 − z¯) ≃ 2σcoshρ = 2x.x¯ + . . . , (3.23) where (x, x¯) label points in Minkowski space and . . . in the formula above stands for terms that are higher order in (x, x¯).12 As a result, the reduced amplitude in the Regge limit, including the contribution of the disconnected piece, is A(x, x¯) = 1 + ic0 coshρ = − √x2x¯2 . x.x¯ −∞ dν γ(ν)γ(−ν) ν2 + (Δ − d2 )2 (− 1 Ωiν (ρ) √x2x¯2)J−1 (3.24) In the formula above both x and x¯ are future-oriented time-like vectors. Next we perform the Fourier transform (2.20) with A(x, x¯) obtained from (2.18) after substitutution 12We use mostly plus convention. HJEP06(218) of the partial amplitude A(x, x¯), from the formula above. To perform the integral, it is useful to recall the following identity, for the derivation see appendix A, γ(ν)γ(−ν)Ωiν(e.e¯) where a = 2Δφ + J − 1, b = 2Δψ + J − 1 and M + stands for the future Milne wedge, defined as follows M + = pµ : p0 > 0, p2 < 0 . Using the representation (3.25) it is trivial to do the Fourier transform. Notice that HJEP06(218) ¯ h ∼ h ∼ √σ . (3.25) (3.26) (3.27) (3.29) (3.30) the factors (−1)J−1 cancel out. We get δ(S, L) = c0πd+242+d−J−Δφ−Δψ SJ−1 Z ∞ dν C(Δφ)C(Δψ) −∞ 1 ν2 + (Δ − 2 ) d 2 Ωiν(L) . Neatly all the double trace operator poles coming from the γ(ν)γ(−ν) are gone. As we explained above without the γ(ν)γ(−ν) pre-factor we cannot expand the integrand for small L. Indeed, in this limit the integral diverges at large ν. This signals that the phase shift is singular for small L. Similarly, we cannot close the contour at infinity because Ωiν(L) grows exponentially both in the upper and lower half-plane. A better way to do the integral is to use (3.22). The advantage of this representation is that Πiν+ d2 −1(ρ) decays in the lower half-plane. We can use the symmetry of the integrand ν → −ν to see that both terms contribute equally and then we can close the contour in the complex plane by 1 picking the contribution from the pole in the ν2+(Δ− d2 )2 . The result is δ(S, L) ∝ λφφOΔ,J λψψOΔ,J SJ−1ΠΔ−1(L), (3.28) which is exactly what one gets in the bulk [14]. 3.5 Double trace operators and the bulk phase shift It is natural to pose a question how the asymptotic of the correlator from the previous section is reproduced using the OPE in the φψ s-channel. This problem was solved in [8]. Let us quickly review their construction. On general grounds we know that the result should be reproduced from an infinite sum over double trace operators that we can schematically denote as φ∂2n∂µ 1 . . . ∂µ sψ which have spin s and dimension Δn,s = Δφ +Δψ +2n+s+γ(n, s). Let us introduce new variables for the quantum numbers of double trace operators h = The result of [8] is that the operators that reproduce the result of the previous section have the following property where Ih,h¯(z, z¯) is the expression for the conformal block to together with the tree-level three-point couplings in the limit (3.29) see appendix A. One can show that the correct identification that relates this expression to the previous section is 16h2h¯2 = p2p¯2, h h ¯ + ¯ h h 2p.p¯ = − |p||p¯| In these variables the s-channel conformal block weighted by the three-point function of generalized free field theory takes the following form, see appendix C, Ih,h¯(z, z¯) = (−x2)Δφψ C(Δψ)C(Δφ) Z dp dp¯ M+ (2π)d (2π)d (−p2)Δψ− d2 (−p¯2)Δφ− d2 ep.xep¯.x¯ In other words the Regge limit is reproduced by the large spin and large twist operators with their ratio related to the impact parameter in the bulk. Conformal blocks simplify in this limit and the leading contribution to the correlation function takes the form13 In this form it is trivial to take integrals over h and h¯ Z ∞ 0 dh Z h 0 dh¯ 4hh¯(h2 − h¯2)δ As as result we get the following expression for the correlator (3.31) (3.32) (3.33) (3.34) (3.35) (3.36) A(x, x¯) = C(Δφ)C(Δψ) Z dp dp¯ M+ (2π)d (2π)d (−p2)Δφ− d2 (−p¯2)Δψ− d2 ep.xep¯.x¯ (1 − iπγ(S, L)) where S and L are related to the original quantum numbers through (3.32). To relate to the phase shift we can rotate p → −ip and p¯ → −ip¯, so that S → e−iπS. In this way we get γ(S, L) = δ(eiπS, L) π For the simple case of Witten diagram (3.36) simply becomes γ(S, L) = − microscopic analysis of AdS exchanges can be found in a recent paper [36]. This example gives a nice interpretation for the phase shift in case of a simple exchange in the bulk in terms of anomalous dimensions of double trace operators. Notice that in this particular example the phase shift is purely real which corresponds to the fact that the scattering is elastic. In this case in the s-channel it is reproduced via double trace operators only. More generally, the phase shift will be complex and in the s-channel new single trace operators will be important as well [37]. It corresponds to inelastic effects in the bulk. It would be very interesting to understand the mapping between the phase shift and the s-channel data beyond the simple elastic regime considered in this section. 13The corrections to the three-point functions are subleading in the Regge limit. See for example the discussion in section 6.1 in [31]. δ(Sπ,L) . A detailed Conformal Regge theory in the impact parameter space In this section we start by briefly reviewing conformal Regge theory. We take the usual steps reviewed in detail in [29, 38, 39]. We consider a large N CFT and focus on a planar correlator. We assume that the leading singularity in the J plane is a pole. The outcome of this analysis is an expression for the contribution of the Regge pole to the correlation function. We then consider spin one operators and generalize the usual conformal Regge analysis to this case. For a previous discussion see [11]. Conformal Regge theory in the t-channel: scalars On the principal sheet the t-channel blocks have the usual asymptotic, but the continuation 1 z → e−2πiz has a dramatic effect on the t-channel blocks which develop (1−z)J singularity in the z → 1 limit. The t-channel OPE becomes divergent and is not reliable away from the light-cone limit z → 1 with z¯ held fixed. The way to deal with this is the subject of Regge theory and we briefly review it here, using our notations. Our starting point is the scalar correlator where again As explained above, it will be useful to use different Poincare patches (2.16) for each operator insertion. The conformal mapping (2.16) yields h φ x − 2 ψ x¯ − 2 ψ φ x¯ 2 x 2 i = and the cross ratios take the form u = (1 − σe−ρ)(1 − σeρ) = v = σ2 = x2x¯2 (1 + x126x¯2 + x2.x¯ )2 . , It will be useful to keep in mind that with the specific kinematics, when x2 and x¯2 are future timelike, σ < 0. The leading term in this equation gives (3.23), but we will need subleading terms as well which could be easily computed using (4.4). We now quickly review the standard argument of [29, 38, 39]. A convenient starting point is the partial wave expansion of the correlation function A(u, v)= X∞ Z J=0 dνbJ (ν2)Fν,J (u, v) (4.1) (4.2) (4.3) (4.4) (4.5) where the partial waves are given by a sum of the corresponding conformal block and its shadow with a particular relative coefficient: κν,J = (0,0) iν 2πK d2 +iν,J . Fν,J (u, υ) = κν,J g d2 +iν,J (1 − z, 1 − z¯) + κ−ν,J g d2 −iν,J (1 − z, 1 − z¯), (0,0) Here KΔ,J is a known function and its explicit form could be found, for example, in eq. (41) of [29]. In the Regge limit the partial waves take the following form (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) Fν,J (σ, ρ) ≃ i41+J π d2 σ1−J γ(ν, J )γ(−ν, J )Ωiν (ρ) + . . . , γ(ν, J ) = Γ where we wrote only the leading contribution in the small σ limit. Notice that (4.5) together with (4.7) are identical to the Regge limit of the Witten diagram (3.18) if we set 1 bJ (ν2) ∼ ν2+(Δ− d2 )2 . into (4.5) we get At the same time the partial wave expansion is closely related to the OPE and integral over ν is what substitutes the usual sum over dimensions. Indeed, substituting (4.6) A(u, v) = 2 X Z J (0,0) dνbJ (ν2)κν,J g d2 +iν,J (1 − z, 1 − z¯), where we used the symmetry of the integrand under ν → −ν which explains the origin of the factor of 2 in (4.8). Next we notice that g(d20+,0i)ν,J (1 − z, 1 − z¯) decays exponentially in the lower ν-plane, which means that we can close there the contour of integration. We recover the usual OPE, provided that partial amplitudes, bJ (ν2), have the following singularities with residues given by: bJ (ν2) ≈ ν2 + (Δ − d2 )2 , r[Δ, J ] r[Δ, J ] ≡ λ12OΔ,J λ34OΔ,J KΔ,J . There are also poles in κν,J at the following positions14 ν2 + 2Δψ + 2n − 2 d 2 + J = 0, ν2 + 2Δφ + 2n − 2 d 2 + J = 0 , where n ≥ 0 is an integer. In finite N CFTs these should be canceled by zeros in bJ (ν2). On the other hand, in large N CFTs poles (4.11) correspond to the contribution of double trace operators and are physical. In this way, for planar correlators in the large N limit, 14Our discussion of the possible singularities here is far from exhaustive. For more details, see for example [29, 39]. in the J plane is a pole, the phase shift is given by (5.45), where βk(ν) are related in a known way to the three-point couplings of the leading Regge trajectory operators to the external operators. Importantly, the phase shift δ(S, L) does not acquire contributions from the double trace operators which are important in the Regge limit. The leading Regge trajectory re-sums the higher spin states that were discussed in [14]. In particular, the phase shift δ(S, L) develops an imaginary part which corresponds to the production of stringy states which are dual to heavy single trace operators in a CFT. We then analyzed constraints on the phase shift due to causality and unitarity. First, we focussed on the regime where the phase shift takes the simple form (5.15). Eq. (5.15) is expected to be true in a generic CFT for large enough impact parameters L and is dual to a single graviton exchange in the bulk. Assuming that we can exponentiate the phase shift, we derived the bound (5.19). This bound is identical to the one discussed in [14]. Next, we considered the bound on the phase shift due to unitarity. The result is (5.29) obtained for real S. This bound should be valid in any CFTd>2 and for arbitrary impact parameters. This agrees with the semi-classical unitarity analysis of [14]. In [14] it was, moreover, concluded that (5.29) should hold in the upper S half-plane. It would be very interesting to see if the argument that led us to (5.29) could be extended in this way. We used these bounds on the phase shift to constrain three-point functions of spinning operators which led to the parametric bound (5.40). The conclusion is that any nonminimal coupling to the graviton, as well as any other particle of spin two or higher, should be suppressed by the gap in the spectrum of higher spin particles.30 Having a non-perturbative, purely CFT definition of the bulk phase shift may lead to new insights into the gravitational physics at energies S and impact parameters L that are not easily accessible using bulk effective theory. Conversely, having a simple bulk intuition for δ(S, L) may shed new light into microscopic properties of CFTs. For example, one would like to study the bulk phase shift at smaller impact parameters (see e.g. the discussion in [3, 11]). At smaller impact parameters inelastic effects become important. The imaginary part of the phase shift that we found above corresponds to tidal excitations of the string. At even smaller impact parameters, gravitational or N1 corrections become important and a black hole is potentially produced in the bulk [50].31 It would be interesting to see if bootstrap methods, see e.g. [52, 53] and references therein, could be used to chart various physical regimes of the bulk phase shift δ(S, L) in an abstract CFT. Acknowledgments We are grateful to S. Caron-Huot, M. Costa, T. Dumitrescu, T. Hartman, Z. Komargodski, J. Penedones, M. Strassler, and Xi Yin for useful discussions. M.K. is supported in part by a Marie-Curie fellowship with project no 203972 within the European Research and Innovation Programme EU H2020/2014-2020. The work of AZ is supported by a Simons Investigator Award from the Simons Foundation. We would like to thank GGI for hospi30These could be either classical stable resonances as in tree-level string theory, or two-particle states that appear in the loops as in QED. 31In theories with j(0) < 32 this regime is believed to be absent [49, 51]. tality during the workshop “Conformal Field Theories and Renormalization Group Flows in Dimensions d > 2.” MK and AP are also grateful to the Simons Center for Geometry and Physics and NORDITA for hospitality. A Harmonic functions Here we collect some useful formulas for harmonic function on Hd−1 [17]. The harmonic function Ωiν (ρ) in the hyperbolic space Hd−1 can be written as follows Γ d d 2 − 1 + iν Γ d 2 − 1 − iν 2F1 d and is regular for ρ = 0. Consider the product of two time-like unit vectors which define the scalar product in Hd−1 coshr = −e.e¯. It is useful to consider the following decomposition 1 (−e.w)a = Z ∞ 21−a π d −22 Γ a− d−22 +iν 2 Γ a− d−22 −iν 2 Γ(a) Ωiν (e.w). (A.3) Using this formula and some formulas from harmonic analysis [17] one can check the following identity γ(ν)γ(−ν)Ωiν (e.e¯) = πd−2 22−a−b Z 1 Γ(a)Γ(b) Hd−1 dwdw¯ (−w.e)a(−w¯.e¯)b Ωiν (w.w¯), a = 2ΔO + J − 1, b = 2Δψ + J − 1. We can introduce an additional transformation Z M+ dpdp¯ ep.xep¯.x¯ (−p2) d−2 a (−p¯2) d−2b = Z Hd−1 dwdw¯ Γ(a)Γ(b) (−w.x)a(−w¯.x¯)b , where we substituted p = tw used that w0 > 0 (and similarly for x¯) and integrated R0∞ dt td−1. In this way we get γ(ν)γ(−ν)Ωiν (e.e¯) (−p2) d−2 a (−p¯2) d−2b Ωiν (pˆ.p¯ˆ) which we derived for x2 future-directed and timelike. We can rescale and rotate momenta in the r.h.s. p → −ip, p¯ → −ip¯ to get γ(ν)γ(−ν)Ωiν (e.e¯) (−x2)ΔO+ J −21 (−x¯2)Δφ+ J −21 = (−p2) d−2 a (−p¯2) d−2b Ωiν (L) . (A.4) (A.5) (A.6) (A.7) Let us briefly comment in the iǫ prescription in the formula above. We derived the formula above first for timelike x2, x¯2 when analytically continuing to spacelike distances we have to specify the prescription how to go around the branch points. As reviewed in the bulk of the paper we shift x0 → x0 + iǫ, x¯0 → x¯0 + iǫ. In the r.h.s. the effect of this shift is e−ǫp0 (and similarly for p¯0) which makes the integral convergent when we integrate over p’s in the future Milne wedge M + for any x and x¯. B Choice of coordinates It is convenient to think of a CFT as defined on a universal cover of the projective space HJEP06(218) Let us describe different coordinate systems. The global coordinate system correin R2,d sponds to d i=1 − (Z−1)2 − (Z0)2 + X(Zi)2 = 0, Z ∼ λZ. Z−1 + iZ0 = eiτ , Zi = ni, (Z0)2 + (Z1)2 = 1, where ~n is the unit vector ~n2 = 1 on the Sd. The original y coordinates can be described as follows Let us review different coordinate systems that we used in the bulk of the paper [11]. yµ = Zµ y± = y0 ± y1 = ZZ−01+±ZZd1 . This choice corresponds to In other words, we can think of this patch as gauge fixing Z0 + Zd = 1. Different “gauges” describe CFTs on spaces that differ by Weyl rescaling. We also introduce The four Poincare patches discussed in the main text correspond to the following coordinates. For the transverse coordinates we get Whereas the light-cone coordinates take the following form K¯yA = δ0A + δdA, KyA = δ 0 − δdA . A (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) Let us understand how symmetries act. Clearly Lorentz transformations x → Λx are the same in all four Poincare patches. They are Mµˆνˆ where µ,ˆ νˆ = 0, d, 2, . . . , d − dilatation operator in y-coordinates. Similarly, boost M+− in the y-coordinates acts as a dilatation operator in each patch. Translations in patches 1, 3 become special conformal transformations in patches 2, 4 and conversely. Let us also recall how the AdS coordinate is introduced in each patch. In the embedding coordinates the AdS is given by In the original Poincare patch the radial coordinate is given by z = 1 and in each Poincare patch we have 1 1 z1 = − Z−1 + Z1 = − y+ z2 = − Z−1 − Z1 = − y− z z , , z3 = z4 = 1 1 Z−1 − Z1 = Z−1 + Z1 = z , . C Generalized free fields in d = 2 in the Regge limit Here we derive necessary formulas to understand the Regge limit for a small correction to generalized free fields. We have for double trace operators of spin s Δn,s = Δφ + Δψ + 2n + s, h = We would like to understand the limit of the s-channel blocks when h, h¯ → z, z¯ → 1 with (1 − z)h2 and (1 − z¯)h¯2 held fixed. A useful identity in this context is the ∞ and following lim z→1,(1−z)h2−fixed 1 2F1 (h+x, h+y, 2h, z) = √π 22h√h(1−z)− x+2y Kx+y 2h√1−z +O Let us now consider conformal blocks in two dimensions. These are explicitly known and could be found for example in [33]. We take Δ and J of the exchanged operator to be (C.1). In the limit ze−2πi, z¯ → 1 such that (1 − z)h2, hh¯ and 11−−zz¯ are fixed conformal blocks take the form ze−2πi,z¯l→im1,h,h¯→∞ gΔΔ=φψΔ,−n,Δs,φJψ=s(z, z¯) = e−2πih¯ 22hπ+2h¯ √hph¯(1−z) 2 (1−z¯) 2 Δφψ Δφψ KΔφψ 2h√1−z KΔφψ 2h¯√1−z¯ +KΔφψ 2h√1−z¯ KΔφψ 2h¯√1−z +. . . . (C.3) (B.7) (B.8) (B.9) (C.1) 1 h (C.2) HJEP06(218) We next apply it to the limit of the generalized free fields. In this case e−2πih¯ = e−iπ(ΔO+Δφ)(−1)s and the three-point coupling coefficients take the form, see formula (43) The s-channel OPE takes the form Ih,h¯ +. . . , X h¯<h,h Ih,h¯ = 4 Γ(Δφ)Γ(Δψ) 2 hΔφ+Δψ−1h¯Δφ+Δψ−1(1−z) 2 (1−z¯) 2 Δφψ Δφψ KΔφψ 2h√1−z KΔφψ 2h¯√1−z¯ +KΔφψ 2h√1−z¯ KΔφψ 2h¯√1−z . We can now go from the sum to the integral to get Z ∞ 0 Z h 0 dh¯ Ih,h¯(z, z¯) = 1 Z ∞ Z ∞ 0 dh¯ Ih,h¯(z, z¯) = 1 We also got that the contribution from the δ(h − h¯) which is the spin zero operators is subleading in the z, z¯ → 1 limit as expected. It is instructive to rewrite the two-dimensional result (C.5) as an integral over the future Milne wedge M +. It is easy to check that Ih,h¯(z, z¯) = (−x2)Δφψ C(Δφ)C(Δψ) Z dp dp¯ M+ (2π)2 (2π)2 (−p2)Δφ−1 (−p¯2)Δψ−1ep.xep¯.x¯ hh¯(h2 − h¯2)δ 16 − h2h¯2 . Indeed, let us write RM+ dp = R0∞ dp+dp− and −p2 = p+p−, we have then (C.4) (C.5) (C.6) (C.7) (C.8) . (C.9) HJEP06(218) δ = p.p¯ 2 16 1 −p¯2 h2 − h¯2 Ih,h¯(z, z¯) = (−x2)Δφψ C(Δψ)C(Δφ) where the integral is now over the d-dimensional future Milne wedge. δ p+ − p¯− 4h2 δ p− − p¯+ 4h¯2 + δ p+ − p¯− 4h¯2 δ p− − p¯+ 4h2 A beautiful result of [8] is that the formula (C.7) could be trivially generalized to d Z dp dp¯ M+ (2π)d (2π)d (−p2)Δφ− d2 (−p¯2)Δψ− d2 ep.xep¯.x¯ Indeed, noting that Z ∞ 0 dh Vectors from scalars We start by considering the following correlator of vector operator J µ of dimension ΔJ and a scalar operator of dimension Δψ. HJEP06(218) hJ (ǫ1, y1)J (ǫ2, y2)ψ(y3)ψ(y4)i = Aµν (y1, y2, y3, y4)ǫ1µ ǫ2ν . This correllator can be expressed as a sum over conformal blocks in the s-channel as follows: hJ (ǫ1, y1)J (ǫ2, y2)ψ(y3)ψ(y4)i = WO(y1, y2, y3, y4, ǫ1, ǫ2) where the sum runs over operators O of specific conformal dimension and spin, and WO denotes the partial wave of the exchanged operator, including all the prefactors and the conformal blocks, normalized according to (D.1). The partial wave32 is generally given by a sum of five terms33 WO(y1, y2, y3, y4, ǫ1, ǫ2) = 1 with the Dij as defined in [40, 41], and D1 ≡ D11D22, Dˆ2 = D12D21, Dˆ3 = −H12, Dˆ4 = D21D11, Dˆ5 = D12D22 (D.4) G1,O = G2,O = G3,O ≡ gO(u, v), G4,O = O +2,0(u, v), G5,O = y24 2 y124 g−2,0 O with gΔ12,Δ34 the corresponding conformal block for scalar operators and Δij ≡ Δi − Δj . O Here (u, v) denote the conformal cross-ratios defined by: X O Z h 0 dh¯ Ih,h¯(z, z¯) = (−x2)Δφψ C(Δφ)C(Δψ) × Z dp¯ (−x2)Δφ (−x¯2)Δψ = 1 (x2x¯2)Δψ = 1 (C.11) u = y122y324 , y123y242 u = y124y2223 , yij = yi − yj . Every term in the sum of (D.3) can be expanded as follows 32Here we only consider the contribution of symmetric and traceless operators in the OPE. 33Recall that Dij (y122)ΔJ +1(y324)Δψ = 0 for any i, j = 1, 2. 1 Dˆ kGk,O = X QiDˆ kigOa,b , 5 i=1 (D.1) (D.2) (D.3) (D.5) (D.6) (D.7) V1 = V1,23, V1′ = V1,24, V2 = V2,31, V2′ = V2,14 . (D.10) (D.8) (D.9) (D.11) (D.12) (D.13) where the Qi denote the conformally covariant tensor structures defined as follows: with Qi = {H12, V1V2, V1′V2, V1V2′, V1′V2′ } , H12 = −2ǫj · y12 ǫi · yij + yi2jǫi · ǫj Vi,jk = yi2jǫi · yik − yi2kǫi · yij 2 yjk Generally, only four out of five coefficients λJJO,k are independent. In particular, λJJO,4 = λJJO,5 as a result of considering identical vector operators. To simplify the notation in what follows we will denote by λk the OPE coefficients λJJO,k. We hope that this will not create any confusion. When the operator O in the OPE is conserved, where by unitarity its conformal dimension and spin are related: Δ − J = d − 2, then λ4 = λ5 = 0 and we are left with only three independent parameters. On the other hand when the external vector operators are conserved, ΔJ = d − 1 and only two independent coefficients (Δ − j − d)(Δ + j − 2)λ4 − 2(d − 2)λ2 − 4λ3 Δ(Δ − d) + j(j + d − 2) λ4 = λ5 = − Δ(Δ − d) + j(j + d − 2) λ2 . (Δ + j)(Δ − j − d + 2) The parameters (λ1, λ2, λ3, λ4) introduced here are linearly related to the (α, β, γ, η) which define the three-point function hJ J Oi in the basis of the conformally covariant tensor structures defined in (D.9): hJ J OΔ,ji = V3j−2 αV1V2V32 + βV3 (V1H23 + V2H13) + γH12V32 + ηH13H23 . Δ+j Δ+j (y122)ξ+1− Δ2+j (y223) 2 (y123) 2 The precise mapping between the parameters is given below: 1 1 1 1 1 1 1 1 1 α = j − 4 (j + Δ)2 λ1 − 4 (j − Δ)2λ2 + 2 (j − Δ)(j + Δ − 2)λ4 1 β = 4 j(Δ + j − 2)λ1 + 4 j (j − Δ)λ2 − 2 j(j − 1)λ4 γ = 4 (j − Δ)λ1 + 4 (j − Δ)λ2 − λ3 − 2 (j + Δ − 2)λ4 η = − 4 j(j − 1)(λ1 + λ2 − 2λ4) . In particular, for the exchange of an operator O of conformal dimension Δ and spin j ≥ 2, the conformal partial wave takes the following form WO = 1 f1H12 + f2V1V2 + f3V1′V2 + f4V1V2′ + f5V1′V2′ . (D.14) with coefficient functions fi f1 λψψO = 2 (−2λ3 +(λ1 +λ2)u∂u) gO+ = (λ1 +λ2)v∂v +(λ1 +λ2)v2∂v2 +(λ1 +λ2)vu∂u∂v gO+ +λ5v −2v∂v −v2∂v2 −2u∂u −2uv∂v∂u −u2∂u2 gO,(−2) −λ4 v∂v2gO,(+2) = (vλ1 +λ2)(−∂v −v∂v2)−λ1(u∂u +2uv∂u∂v +u2∂u2) gO+ +λ5 2v∂v +v2∂v2 +u∂u +uv∂u∂v gO,(−2) +λ4 v∂v2 +u∂u∂v gO,(+2) = v(λ1 +λ2v)∂v +v2(λ1 +vλ2)∂v2 +λ2(uv∂u +2uv2∂u∂v +u2v∂u2 gO+ +λ5v −2v∂v −v2∂v2 −u∂u −uv∂u∂v gO,(−2) +λ4v −v∂v2 −u∂u∂v gO,(+2) = − (λ1 +λ2)v∂v +(λ1 +λ2)v2∂v2 +(λ1 +λ2)vu∂u∂v gO+ +λ5v 2∂v +v∂v2 gO,(−2) +λ4 v2∂v2 +2uv∂u∂v +u2∂u2 gO,(+2) , (D.15) and λ4 = λ5. For the specific case of interest, where the exchanged operator is the stresstensor operator, (D.14) can be also written as: f1H12 + f2V1V2 + f3V1′V2 + f4V1V2′ + f5V1′V2′ , (D.16) WO = 1 (y122)ΔJ +1(y324)Δψ with coefficient functions fi 1 4 f2 = − (d − 1)(d − 2) f3 = 2 ns f4 = − (d − 1)(d − 2) 1 2 ns f5 = −f2 . f1 = − (d − 1) (d − 1)(λ3 + 4nf ) + ns − d − 2 ns u∂u gO 2ns v∂v + v2∂v2 + uv∂u∂v gO ∂v + v∂v2 gO + d − 1 4 nf (1 − v)(∂v + v∂v2) − u∂u − 2uv∂u∂v − u2∂u2 gO v v∂v + v2∂v2 + u∂u + 2uv∂u∂v + u2∂u2 gO+ In (D.16) we found it convenient to use as independent parameters, the constants ns and nf which determine the three point function hJ J Oi in the “free field” basis [55]. λ3 is an additional parameter, related via the Ward Identity of the stress-tensor in hJ J T i to the other two parameters ns, nf as follows [24]: n3 = ΔJ − d + 1 (4(d − 2)nf + ns) (D.17) (D.18) which clearly vanishes when the vector operators are conserved. To be specific, (ns, nf ) here are defined as in appendix B.1 of [22], where the partial wave for the stress-tensor operator in d = 4 was also quoted. The OPE coefficients λi are linearly related to the λ2 = − 2(ns + (2d − 4)nf ) , λ3 = (d − 1)n3 + 4(d − 1)nf + ns (d − 1) (D.19) Note that f3 and f4 in (D.17) differ from those quoted in (C.3) of [22], which contain a The Regge limit of the correlation function hJμψψJν i In this appendix we consider the correlation function hJµ ψψJν i expressed as a sum over conformal partial waves and follow the steps outlined in section 4.1 to arrive at eq. (4.23) of section 4.2. Our starting point will be the correlation function (D.1). To avoid confursion due to the different ordering, we will restate here all the relevant transformations and notations. The final result will of course be the same. Starting from (D.1) we introduce the Poincare patches (E.1) (E.2) (E.3) J µ (y1)ψ(y3)ψ(y4)J ν (y2)i = where the subscript stands for the operator insertion yi with i = 1, . . . , 4. After doing the coordinate transform we set x1,2 = ∓ x2 , x3,4 = ∓ 2 . Note that this is a time-ordered x¯ correlator. The explicit transformation formulas are xi = (xi+, xi−, ~xi) = − + (1, yi2, ~yi), xj = (xj+, xj−, ~xj ) = − − (1, yj2, ~yj ), y 1 i = 1, 2 , j = 3, 4 , y y 1+ = −y2+ = 1− = −y2− = ~y1 = ~y2 = 2 , , y y 3− = −y4− = 3+ = −y4+ = ~y3 = ~y4 = , 2 2x¯+ , ~ x¯ We take xµ and x¯µ to be future-directed time-like vectors, i.e., x+, x¯+ > 0 and x2 < 0 and x¯2 < 0. The conformal mapping leads to ǫ1mǫ2nAmn(x, x¯) = hJ m = = 2 ǫ∗mǫnA mn(u, v) (−x2)ΔJ +1(−x¯2)Δψ . x − 2 ψ x¯ − 2 x¯ 2 J n x 2 i = x¯+ −2Δψ ∂xm ∂xn 2 ψ ∂y1µ ∂y2ν h u = v = 1 + x126x¯2 − 2 1 + x126x¯2 + x2.x¯ !2 1 + x126x¯2 − 2 2 x.x¯ 2 = σ , = 1 − 2σcoshρ + σ2 . Notice that here σ > 0 as opposed to the main text. Using (D.2) we can express (E.3) as: (E.4) (E.5) (E.6) ǫ1mǫ2nAmn(x, x¯) = ǫ1mǫ2n X Δ,J P5 k=1 Q˜(k)mn f Oi(u, v) , where the f Oi are defined in (D.15) and (u, v) are the conformal cross-ratios expressed in terms of (x, x¯) in eq. (E.4). Here Q˜(k)mn = Q(k)µν ∂xm ∂xn ∂y1µ ∂y2ν = {H˜ 1µν2 , V˜1µ V˜2ν , (V˜1′)µ V˜2ν , V˜1µ (V˜2′)ν , (V˜1′)µ (V˜2′)ν } . To proceed we should rewrite the sum over the conformal dimensions of the exchanged operators as an integral over ν. To this end, one needs to define the corresponding partial waves Fν,J for vector operators. This is a straightfoward exercise; one simply replaces the scalar conformal blocks in (D.15) with the respective partial waves Fν,J (u, v). The corresponding partial amplitudes bj (ν2) have the same exact form as in (4.9). For notational simplicity, we will henceforth denote by f˜Oi the f Oi resulting from replacing the conformal blocks by rk[Δ, J ]σ1−J Ωiν (ρ). Repeating the procedure outlined in section 4.1 we arrive at: Amn(x, x¯) ≃ A0mn + 2πi X Z dν α(ν) eiπj(ν) Q˜(k)mnf˜Oi(σ, ρ) where α(ν) is defined in (4.16) and the index Regge denotes the Regge limit of the corresponding expression. We also seperated the contribution of the disconnected part which cannot be studied using Regge theory. Note the overall factor of e−iπ(1−j(ν)) compared to the main text, which can be traced to the opposite sign of σ. The factor can be completely fixed from the iǫ-prescription in σ and the identity (4.13) evaluated for the analytically continued blocks. To obtain the explicit form of Q˜(k)mnf˜Oi(σ, ρ) in the Regge limit, we first replace the scalar partial waves with their leading Regge behavior, eq. (4.6), obtained after the appropriate analytic continuation.34 The correct analytic continuation here involves transporting z around 1 counter-clockwise. The behavior of the scalar blocks and the partial waves will therefore be opposite to eq. (3.11). This is another way to understand the plus sign in front 34Note that the leading behavior of the scalar conformal blocks and their shadows is independent of the external operators’ dimensions. In other words, the Regge behavior of gOa,b is the same for any a, b. H˜12 = −4 x2ηmn − 2xmxn V˜1m = V˜2′m = −2xm 1 − σcoshρ + σ2 3 + x¯mx2 1 − 2 1 σcoshρ + σ2 V˜′m = −V˜2m = −2 xm 1 + σcoshρ − σ 1 3 16 − 4 1 1 8 − 2 cosh2ρ cosh2ρ + . . . 2 5 3 8 − 2 cosh2ρ 16 − 4 + x¯mx2 σcoshρ + σ2 cosh2ρ We must keep subleading terms in Q˜(k)mn up to O(σ2) because generally, f˜Oi ∼ σ−1−j in the Regge limit. A straightforward computation then yields Amn(x, x¯) ≃ A0mn + 2πi Z dν X αˆk(ν) eiπj(ν) σ1−j(ν) Lk mnΩiν(ρ) (−x2)ΔJ (−x¯2)Δψ of the second term of (E.7). Next we evaluate Q˜(k)mn in the Regge limit, or equivalently (E.8) (E.9) (E.10) (E.11) (E.12) αˆk(ν) = rˆk[ν, j(ν)]α(ν) . The differential operators Lkmn are defined as: x L1 2 mn = x2ηmn 2 mn = xmxn x L3 x L4 2 mn = x2(xm∂n + xn∂m) 2 mn = (x2)2∂m∂n , and the rˆk[ν, j(ν)] denote the analytic continuation of the residues rˆk[Δ, J ] which are explicitly given by: −1 − 2 1 4 k=1 Z Ωiν(ρ) 4 k=1 rˆ1[Δ, J ] = ((J − 1)(λ1 + λ2) − 4λ3 + 2(J − 1)λ4) λψψO KΔ,J rˆ2[Δ, J ] = − (J 2 − 1)(λ1 + λ2) − 8λ3 + 2(J − 1)(J + 3)λ4 λψψO KΔ,J rˆ3[Δ, J ] = (−(J + 1)λ1 + (J − 1)λ2 + 2λ4) λψψO KΔ,J rˆ4[Δ, J ] = (−λ1 − λ2 + 2λ4)λψψO KΔ,J . Observe that the differential operators only depend on x since x2 , − x2 denote the positions of the vector operators. Moving to the impact parameter representation, requires taking the Fourier transform of (E.9). This is most conveniently done after a change of basis such that: Amn(x, x¯) ≃ A0mn(x, x¯) − 2πi dν X αk(ν) eiπ j(ν) x Dk 2 mn × (−x2)ΔJ +(j(ν)+1)/2(−x¯2)Δψ+(j(ν)−1)/2 αk(ν) = rk[ν, j(ν)] α(ν) 1 where the new differential operators x2Dkmn commute with x2 and are equal to: x D1 mn = (x2)2∂m∂n − x2 xq(xm∂n + xn∂m)∂q− 1 d − 1 (x2ηmn + (d − 2)xmxn)xqxs∂q∂s , and the rk[ν, j(ν)] are the analytic continuations of the rk[Δ, J ] given by: HJEP06(218) − 4λ3 − 4λ4 + J (λ1 + λ2 + 2λ4) λψψO KΔ,J r2[Δ, J ] = [J (1 − J )(λ1 + λ2) − 2J (1 + J )λ4 + 4(λ3 + λ4))] λψψO KΔ,J (E.14) r3[Δ, J ] = J (λ2 − λ1)λψψO KΔ,J r4[Δ, J ] = (−λ1 − λ2 + 2λ4)λψψO KΔ,J . Technically, the above change of basis is possible due to the following identities: xm∂mΩiν(ρ) = 0 xqxs∂q∂sΩiν = 0 x2∂2Ωiν(ρ) = −Ω¨ iν − (d − 2) coth ρ Ω˙iν(ρ) x2 xq(xm∂n + xn∂m)∂qΩiν = −x2(xm∂n + xn∂m)Ωiν and the differential equation Ωiν satisfies (see also appendix A): Ω¨ iν(ρ) + (d − 2) coth ρ Ω˙ iν(ρ) + ν2 + Ωiν = 0 (E.16) where the dots denote differentiation with respect to ρ. The equations above allow us to find a simple relation between the differential operators Lmn and D mn: D1 mn = L1 mn + L2 mn D2 mn = L2 mn D3mnΩiν(ρ) = L3mnΩiν(ρ) D4mnΩiν(ρ) = L4 mn + D3 − d − 1 mn 1 ν2 + d − 2 2# 2 D1 mn Ωiν(ρ) . We are now ready to discuss the Fourier transform to the impact parameter space. Before doing so, it would be useful to revert to notations of the main text. The correlator in the main text is time-ordered as here, the only difference being the definition of σ. To obtain the correct expression for the position space correlator in the Regge limit, we thus need to replace σhere by eiπ(1−j(ν))σthere. For the correlator in the main text we thus have: Amn(x, x¯) ≃ A0mn(x, x¯) + 2πi αk(ν) = rk[ν, j(ν)] α(ν) . Z 4 dν X k=1 αk(ν) x Dk 2 mn (−√x2x¯2)1−j(ν) Ωiν(ρ) (E.13) (E.15) (E.17) and then integrates by parts to effectively replace all the derivatives in x with x µ → −i ∂pµ , ∂ ∂ ∂xm → −ipm . Afterwards, one can use the impact parameter representation of Ωiν (ρ) eq. (A.7) with The Regge limit of hJμψψJν i in the impact parameter space Here we discuss how to obtain the Fourier transform of the correlator Amn(x, x¯) in the impact parameter space: Z ddx ddx¯eip·x eip¯·x¯ Amn(x, x¯) . Our starting point is eq. (4.23) from section 4. Substituting (4.23) into (F.1), one first (F.1) (F.2) (F.3) (F.4) (F.5) (F.6) replaces all the powers of x with derivatives in p a → a + 2 to write: δmn(p, p¯) ≃ (−p2)d/2−ΔJ (−p¯2)d/2−Δψ × π d2 e− iπj2(ν) 2 sin πj(ν) rk[ν, j(ν)]β(ν) where the differential operators Lˆk are given by: −L1 ˆmn = ηmn∂2 − ∂m∂n ˆmn = ∂4pmpn − ∂2∂s(∂mpn + ∂npm)ps− 1 1 d − 1 ∂s∂t ηmn∂2 + (d − 2)∂m∂n pspt . Note that here the derivatives ∂m denote differentiation with respect to pm. To proceed, it is convenient to rewrite the differential operators above in the following form: −p2Lˆ3mn = −(pm∂n +pn∂m)(p2∂2)−2 ηmn −2 p2∂2 +2p2∂m∂n (p·∂ +d−2) pmpn p2 −p2Lˆ1mn = ηmn(p2∂2)−p2∂m∂n −p2Lˆ2mn = p2∂m∂n pmpn p2 1 1 − d−1 −(pm∂n +pn∂m)p2∂2 p·∂ +p2∂2 (p2∂2)− d2 −5d+10 2(d−1) d+2 2(d−1) · p ∂ + p2∂m∂n 6(d−2)+p2∂2 +2(d+1)p·∂ +2(p·∂)2 − ηmnp2∂2 6+(5−d)p·∂ +p2∂2 −(p·∂)2 , Ωiν (L) The impact parameter representation then takes the form p · ∂ → 2ΔJ + j(ν) − (d − 1) p ∂ 2 2 → (2ΔJ + j(ν) − (d − 1)) (2ΔJ + j(ν) − 1) + ν2 + . δmn(p, p¯) ≃ −4c˜0 sj(ν)−1 X βk(ν) DˆkmnΩiν (L) , Z π d2 e− iπj2(ν) sin πj(ν) 2 s = pp2p¯2, 4 k=1 coshL = − pp2p¯2 . so that the commutting operators p · ∂ and p2∂2 can be replaced by: The differential operators Dˆkmn are equal to D1 ˆmn = ηmn D2 ˆmn = pmpn p2 D3 ˆmn = pm∂n + pn∂m pmpn p2 and the βk(ν) denote the linear combinations D4 ˆmn = p2∂m∂n + (pm∂n + pn∂m) − d − 1 1 ηmn pmpn p2 p2∂2 , βk(ν) = −β(ν) X bik ri[ν, j(ν)] with with: b11 = b21 = b41 = − 3d2 − 4 b12 = b32 = b43 = − 1 2 b14 = −1, b44 = (d−2)2 +4ν2 16(d−1)2 4(d−1) × d(−3d+4j +8ΔJ +4)−4 j −ν2 +2ΔJ , d3 +d2(−4j −8ΔJ +2)+4d (j +2ΔJ )2 +ν2 −1 +4 j −2ν2 +2ΔJ −4(j +2ΔJ ) 2 4(d−1) b31 = − (d−2)2 +4ν2 (d−j −2ΔJ ) 2(d−1) × d3 −2d2(2j +4ΔJ +5)+4d j2 +j(4ΔJ +6)+ν2 +4(ΔJ +1)(ΔJ +2) − −8 4j +ν2 +8ΔJ −16(j +2ΔJ ) 2 +d(j +2ΔJ +1)−j +ν2 −2ΔJ , b22 = ((d−j −2ΔJ −1)(d−j −2ΔJ )) (F.7) (F.8) (F.9) (F.10) (F.11) (F.12) (F.13) + (F.14) (F.15) b42 = −(d−1) b14 b13 = (d−j −2ΔJ ) , b23 = −b3, b33 = 1 4 −3d(j +2ΔJ +2)+(j +2ΔJ )2 +6j −ν2 +12ΔJ " d3(j +2ΔJ −4)−4d2 (j +2ΔJ )2 −7 +4d j3 +j2(6ΔJ +4)+j ν2 +4(ΔJ (3ΔJ +4)−1) 4d 2ν2(ΔJ −2)+8(ΔJ −1)(ΔJ +1)(ΔJ +2) +16 4j +3ν2 +8ΔJ −16(j +2ΔJ )3 # b24 = 1, b34 = −2(d−3−j −2ΔJ ) d2 −4d(3j +6ΔJ −2)+4 3j2 +6j(2ΔJ +1)+ν2 +12Δ2J +12ΔJ −8 7d2 4(d−1) 8(d−1) 8(d−1) Polynomial terms in the Mellin amplitude Here we consider how do polynomial corrections to the Mellin amplitude affect the bulk phase shift. Due to the chaos bound (or Rindler positivity), we will restrict ourselves to considering terms of the form Mpol(s, t) = Mmnsmtn with m, n ≤ 2 , where M0 is some arbitrary coefficient. We will see that polynomial terms in the Mellin amplitude yield δ-function-terms in the phase shift as expected. For simplicity, we will focus on the scalar correlator hφφψψi and closely follow the appendix C of [29]. The case of spinning operators is a straightforward generalization which will not alter the main result. Starting from the expression for the reduced amplitude A(u, v) in the Mellin (G.1) (G.2) (G.3) (G.4) (G.5) (G.6) (G.7) (G.8) dt ds (4πi)2 M (s, t)u 2t v− (s+t) 2 Γ s − 2 Γ 2 t + s 2 1 (2coshρ)z = g(ν, z) = π d −22 Z ∞ Γ 2 −∞ z− d −22 +iν 2 dν g(ν, z)Ωiν (ρ) Γ v − (s+2t) → v − (s+2t) e−iπ(s+t) , Γ(a + i x )Γ(b − i x2 ) ≃ 2πei π2 (a−b) x a+b−1 e− π2 x , 2 2 −i∞ 4i Z i∞ dt t −t u 2 v 2 Γ dx (ix)m tn x t−2 v− 2i x . 2 Further approximating v ≃ 1 − 2σcoshρ and recalling that u = σ2, we proceed to evaluate 2 Γ e−iπ 2t +im π2 2m tnMm,n Z ∞ 0 dx 2 x m+t−2 eixσcoshρ , −i∞ 4i Z i∞ dt σ1−mMm,nΓ e−i π2 (−1)m 2m+1 tn Γ(m + t − 1) (2coshρ)t+m−1 . To simplify the presentation, we henceforth set n = 0. We will discuss the n 6= 0 case separately at the end. We are primarily interested in the impact parameter representation of these terms, we will thus procceed by using the identity: An=0 ≃ (−1)m+12m−2π d −22 (2πi)Mm,0 σ1−m Z dν Z i∞ dt −i∞ 2πi Γ t+m−1− d −22 −iν ! t+m−1− d −22 +iν ! Γ 2 2 Γ Let us recall an integral representation of the hypergeometric function, i.e., Γ(a)Γ(b)Γ(c−a)Γ(c−b) Z i∞ dt −i∞ 2πi Γ(t)Γ(c−a−b+t)Γ(a−t)Γ(b−t) (1−z)−t . Changing variables in (G.9) according to t → 12 (2Δφ − t) allows us to make use of the identity above by setting: a = − d2 + 2Δφ + m − iν , b = − d2 + 2Δφ + m − iν , The result is 2 c = Δφ + Δψ + m − 2 d . (G.11) An=0 = (−1)m+12m−2π d −22 (2πi)Mm,0 Γ(2Δφ + m − 1)Γ(2Δψ + m − 1) Γ(Δφ + Δψ + m − 2) × dν g(2Δφ + m − 1, ν)g(2Δψ + m − 1, ν) Ωiν (ρ) 2 × σ1−m Z (G.9) (G.10) (G.12) (G.13) (G.14) (G.15) where we used the fact that 2F1[a, b, c, 0] = 1. To obtain the impact parameter representation of (G.12) we use the following identity g(2Δφ +m−1, ν)g(2Δψ +m−1, ν)Ωiν (e·e¯) = (−x2)Δφ+ m2−1 (−x2)Δψ+ m2−1 M which leads to: e2x·pe2x¯·p¯Ωiν (−p2) d−2Δ1−m+1 2 (−x2)Δφ+Amn2−=10((−x,xx¯2))Δψ+ m2−1 = (−1)m+12mπ −d2+2 (2πi)Mm,0 Γ(Δφ +Δψ +m−2) × Z M dpdp¯ (−p2) d−22Δψ e2x·pe2x¯·p¯ dν Ωiν p p−p2 p−p¯2 p¯ From eq. (G.12) we can read off the impact parameter representation of the reduced correlation function: δn=0(S, L) ∼ Mm,0Sm−1δHd−1 (L) . We see that the addition of an sm polynomial term in the Mellin amplitude, produces a phase shift which is proportional to a δ-function as expected on general grounds [27]. Using the identification between anomalous dimensions of double trace operators in the s-channel and the phase shift we see that the δ-function in the impact parameter space corresponds to h = h¯, or equivalently to the s-channel exchange of spin zero double trace operators. Let us now move onto the case n 6= 0. Starting from (G.7) for n = 0 and using the identity allows us to simply express the result for n = 1 as: Γ Γ t 2Δφ − 2 − t 2 An=1 Δφ,Δψ = 2Δφ An=0 Δφ,Δψ − 2An=0 Δφ+1,Δψ . To obtain the impact parameter representation of An=1, we use the Γ function identities (−x2)Δφ+Amn2−=11((−x,xx¯2))Δψ+ m2−1 = (−1)m+12mπ −d2+2 (2πi)Mm,1 2Γ(Δφ +Δψ +m−1) × Z M (−p2) d−22Δφ e2x·pe2x¯·p¯ dν (aˆ−ν2)Ωiν (w, w¯) , with Eq. (G.18) leads to the following impact parameter representation aˆ = 4Δφ(Δφ + Δψ + m − 2) − 2Δφ + m − 1 − δn=1(S, L) ∝ Mm,1 Sm−1 aˆ + 2 + ∇H δH (L) where to arrive at (G.20) we used the differential equation Ωiν satisfies. Here ∇2H represent the Laplacian on the Hyperbolic space Hd−1. One can easily work out higher powers of n the same way. They all lead to δ-function-like terms, as expected. H Diagonalization of the bulk phase shift In this appendix we will address the implications of imposing positivity on the phase shift δmn(S, L). Our focus will be on a large-N theory with an infinite gap in the spectrum. As discussed in section 3.4, to perform the impact parameter integral (5.45) one must express it in terms of the propagator in hyperbolic space, Πiν+d/2−1, defined in (A.1), and close the contour below the real axis. In the infinite gap limit, the integral picks up a pole from d the contribution of the stress-tensor operator for ν = −i 2 . The result for the phase shift is then proportional to the matrix and positivity requires that its eigenvalues are positive definite. 4 k=1 X βk(ν)DˆkmnΠd−1(L) , (G.16) (G.17) (G.18) (G.19) (G.20) (H.1) To perform the diagonalization, one needs explicit expressions for (H.1), which can be obtained by differentiation. In particular, we have that: Dˆ3mnΠd−1 = − Π˙d−1cschL pmp¯n + p¯mpn ! − coth L Π˙d−1 2pmpn p2∂m∂nΠd−1 = −csch2L Π˙d−1 coth L − Π¨d−1 + cschL coth LΠ¨ d−1 − csch2L Π˙iν + coth L Π¨ iν coth L + Π˙ iν(2 − csch2L) p¯mp¯n p¯2 − ηmn Π˙ iν coth L pmp¯n + p¯mpn ! pmpn p2 . 4 k=1 with: With the help of (H.2) the phase shift can be expressed as follows: X βk(ν)DˆkmnΠd−1(L) = ηmnA1 + pmpn p2 A2 + A3 pmp¯n + p¯mpn ! pp2p¯2 + A4 p¯2 p¯mp¯n q 2 2 a1 − 4(a2 − a3) , λi = A1(L), i = 2, · · · , d a1 = −A4 − A2cosh2L + 2A3coshL, a2 = (A1 − A2sinh2L)(−A1 − A2cosh2L + 2A3coshL − A4), a3 = A3 − 2 A2 coshL sinhL . X βk(ν)DˆkmnΠd−1(L) and its eigenvalues are: λ0,1 = 0 · · · 0 a1 ± A1 = β1Πd−1 −β4 coth L Π˙d−1 −Πd−1 A2 = −β1Πd−1 +β2Πd−1 −2β3 coth L Π˙d−1 +β4 coth2 L Π¨d−1 −coth L csch2LΠ˙ d−1 −Πd−1 A3 = −β3 cschL Π˙d−1 +β4 cschL coth LΠ¨ d−1 −coth2 L Π˙d−1 A4 = β4 csch2L Π¨ d−1 −coth L Π˙d−1 . p·p¯ Without loss of generality, we set p¯0 = 1, p~¯= 0 and use coshL = − √p2p¯2 to express (p0, p~) as: p0 = p−p2 coshL, p1 = p −p2 sinhL, p2 = p3 = · · · = pd−1 = 0 . The phase shift is completely determined in terms of the hyperbolic space coordinate L, −A1 − A2 cosh2L + 2 A3 coshL − A4 −A2sinhL coshL + A3 sinhL0 · · · 0 −A2sinhL coshL + A3 sinhL A1 − A2 sinh2L · · · 0 0 · · · 0  A100  (H.4) (H.6) (H.7) HJEP06(218) We proceed by analyzing the bahaviour of the eigenvalues for small and large impact parameters. To do that, it is useful to recall that Πd−1 behaves as: Πd−1(L) =  Γ(d/2 + 1) dΓ( d2 ) u π1−d/2 Γ(d − 1) 1 + π1−d/2 (d − 1)Γ(d) 4Γ(d/2 + 2) u = e−L When the impact parameter is small L ≪ 1, then the leading behaviour of the eigenvalues turns out to be: λ1,2 ∝ −c0 β4(d − 2) Ld−1 , λi ∝ c0 Ld−1 β4 where c0 = 2−dπ1−d/2 Γd(Γd(/d2−+11)) is equal to the proportionality coefficient of the leading term in the small L expansion and is positive definite. Causality then requires that β4 = 0 for both conserved and non-conserved vector operators J . Setting β4 = 0 is enough to guarantee positivity of the phase shift at small impact parameters for unitary theories satisfying ΔJ ≥ d − 1, when ns ≥ 0. It is interesting to check whether the positivity of the phase shift reduces to the Hofman-Maldacena bounds for large values of the impact parameter. The leading behaviour of the eigenvalues (H.7) for large impact parameters L, or equivalently, 0 < u ≪ 1 is: λ1,2 = c1ud−3 (β1 − β2 − d(d − 2)β4) + O(u2) , λi = c2ud−5(β1 + dβ4) + · · · , (H.10) where (c1, c2) are two positive constants and we have used the fact that β3 = 0 when d evaluated at the stress-tensor Regge pole ν = −i 2 . Positivity of the phase shift leads to a single condition, namely that, L ≪ 1 L ≫ 1 . (H.8) (H.9) HJEP06(218) β1 − β2 − d(d − 2)β4 ≥ 0 . (H.11) For conserved currents, (H.11) simply reduces to nf ≥ 0 for ns ∈ R. For non-conserved vector operators, the result is the inequality obtained in deep inelastic scattering (eq. (4.29) of [24]) but for any real ns.35 In search of the energy-flux constraints we repeat the analysis above, this time fixing pm = (1, ~0) and solving for p¯m through coshL = − √p2p¯2 p·p¯ . The diagonalization problem is exactly the same as before, with the replacement: A2 → A4 and A4 → A2. 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Manuela Kulaxizi, Andrei Parnachev, Alexander Zhiboedov. Bulk phase shift, CFT Regge limit and Einstein gravity, Journal of High Energy Physics, 2018, 121, DOI: 10.1007/JHEP06(2018)121