Energy correlations and Planckian collisions
Hao Chen Robin Karlsson Alexander Zhiboedov









^aZhejiang Institute of Modern Physics, School of Physics, Zhejiang University, Hangzhou, 310027, China
^bCERN, Theoretical Physics Department, CH-1211 Geneva 23, Switzerland

Energy correlations characterize the energy flux through detectors at infinity produced in a collision event. Remarkably, in holographic conformal field theories, they probe high-energy gravitational scattering in the dual anti-de Sitter geometry. We use known properties of high-energy gravitational scattering and its unitarization to explore the leading quantum-gravity correction to the energy-energy correlator at strong coupling. 
We find that it includes a part originating from large impact parameter scattering that is non-analytic in the angle between detectors and is log N_c enhanced compared to the standard 1/N_c perturbative expansion. It is sensitive to the full bulk geometry, including the internal manifold, providing a refined probe of the emergent holographic spacetime. Similarly, scattering at small impact parameters leads to contributions that are further enhanced by extra powers of the 't Hooft coupling assuming it is corrected by stringy effects. We conclude that energy correlations are sensitive to the UV properties of the dual gravitational theory and thus provide a promising target for the conformal bootstrap.


    


[c]40mmCERN-TH-2024-050



Energy correlations and Planckian collisions

    Alexander Zhiboedov^b
    Version of April 23, 2024
==============================================



 






§ INTRODUCTION

In a collider experiment, energy correlations are familiar observables that characterize energy flux at infinity <cit.>. Originally computed using scattering amplitudes, they were later expressed in terms of integrated correlation functions <cit.>. This latter definition in terms of correlation functions readily generalizes to theories where the amplitude approach is not available, such as in strongly coupled holographic CFTs <cit.>, which are also the subject of the present paper.

From a more formal point of view, energy correlations present an interesting example of the matrix elements of light-ray operators <cit.>, allowing exploration of their properties, such as positivity <cit.>, the operator product expansion <cit.>, renormalization and mixing <cit.>, connection to dispersion relations <cit.>. They also received much attention recently in QCD as they occupy a middle ground between scattering amplitudes and correlation functions exhibiting properties of both <cit.>. One particularly interesting property of the energy correlators is their universal scaling behavior in the collinear limit, which is the consequence of QCD factorization <cit.> and the light-ray OPE <cit.>. This theoretical understanding and further phenomenological studies <cit.> have recently led to an experimental measurement of energy correlations inside a jet at the LHC <cit.>.





In this paper, we explore a fascinating connection between energy correlations and high-energy gravitational scattering (as well as  its unitarization) <cit.> in the context of holography <cit.>. Throughout the paper, we focus on the simplest example of this observable, namely the two-point energy correlator, which involves two detectors at infinity, see Figure <ref>. Weakly coupled dynamics in this context typically produces jets <cit.>, or energy fluxes collimated in a small angular cone, while a hallmark of strong coupling dynamics is a homogeneous energy distribution  <cit.>.[One can get approximately homogeneous energy distribution at weak coupling as well by considering states with many particles <cit.>. It was also observed in QCD data at extremely small angles <cit.>.] There is another possibly less familiar feature of energy correlators at strong coupling, something we can call the enhancement property. It states that perturbative corrections to the leading homogeneous distribution of energy are larger than naively expected from the perturbative expansion of the correlator of local operators from which the energy correlators are computed.  Let us illustrate this with a simple example of the four-point function used to compute the two-point energy correlator (EEC) in N=4 SYM. At strong coupling, we schematically have for the connected part

⟨O^†T_μν T_ρσ O ⟩_c = Tree-level Sugra +  1 λ^3/2 Stringy + 1 c_T One-loop Sugra + ...  ,
where λ≫ 1 is the 't Hooft coupling, and c_T≫1 is the central charge.[We define c_T∼ 1/G_N, where G_N is Newton's constant of the dual gravitational theory in AdS, in Appendix <ref>. For SU(N_c) N=4 SYM it is c_T=40(N_c^2-1).]^,[The 1/c_T correction that we wrote is universal, but not entirely precise. In the explicit examples dual to string/M-theory UV completions of gravity, the leading 1/c_T correction comes from an AdS contact term, see e.g. <cit.>. We will focus in this paper on the universal corrections that capture scattering at non-zero impact parameters in AdS for which (<ref>)  applies.] 

Let us next introduce the detector operator which is given by the light-transform of the stress tensor at future null infinity

E(n) ≡1 4∫_-∞^∞ d u  lim_r →∞ r^d-2 T_μν(u,r n⃗) n̅^μn̅^ν,
where u = t-r is the retarded time at future null infinity, and we defined the null vectors n^μ = (1, n⃗), n̅^μ = (1, - n⃗), and the unit vector n⃗^2 = 1 to specify the position of the detector on the celestial sphere S^d-2, see Figure <ref>.

The main object studied in this paper is the energy-energy correlator in a unit-normalized momentum eigenstate (source) |p = 1/√( N_p)∫ d^dx e^ip · x O(x)| Ω⟩:

E(n_1) E(n_2)  ≡p|E(n_1)E(n_2)|p.
Compared to four-point function of local operators (<ref>), the leading correction to the energy-energy correlator, computed from (<ref>), takes the form <cit.>

E(n_1) E(n_2)  = (p^0 4 π )^2 [ 1 + 1 λ 4 π^2 (1-6 z + 6 z^2) + ... ],
where z = 1 - cosθ 2 encodes the angle between the energy calorimeters cosθ≡n⃗_1 ·n⃗_2 and we assumed that the source carries momentum p^μ = (p^0, 0⃗) for simplicity. We see that the leading stringy correction to the energy-energy correlator is enhanced by an extra factor of √(λ).[There are further nonperturbative stringy corrections to the energy-energy correlator of the type ∼ z^#λ^1/4 which are exponentially suppressed at strong coupling λ≫ 1.]

The enhancement of corrections to the energy correlations originates from the fact that they probe high-energy scattering in the dual AdS geometry. Sensitivity to the high-energy behavior of the scattering amplitude in the bulk makes the computation of the correction to the energy correlations at strong coupling an interesting and subtle problem. By naively using the first stringy correction to the correlator (<ref>) to compute the energy-energy correlator, one  gets infinity! To get a finite result, one must effectively perform the computation at finite λ first — it makes the high-energy behavior of the scattering amplitude in AdS softer — and only then perform the large λ expansion <cit.>. We can call it stringy unitarization, and it is related to scattering at small impact parameters. In contrast, this paper explores the enhancement phenomenon in the context of gravitational unitarization, i.e. unitarization of scattering at large impact parameters by resummation of gravitational loops in AdS.

At the level of the correlation function of local operators, gravitational loops are suppressed by the inverse number of colors in the gauge theory or, equivalently, the central charge 1/c_T <cit.>. Again, trying to use the leading 1/c_T correction to the correlator (<ref>) recently found in <cit.> to compute the leading 1/c_T correction to the energy-energy correlator leads to infinity due to the high-energy behavior of gravitational scattering in AdS. One can trace the divergence to the integral over energies of the squared tree-level phase shift δ^2_tree(S,L), i.e. the integral ∫^∞dS  S^3δ_tree^2(S,L), where S is the scattering energy and L is the impact parameter. Indeed, it is a characteristic feature of gravitational theories that δ_tree(S,L) ∼ G_N S which produces the logarithmic divergence in the integral over energies.[For general detectors of spin J_1 and J_2 the measure becomes ∫d S  S^J_1 + J_2 -1.] It is, however, a well-known fact that such a divergence is an artifact of perturbation theory, and in the simplest case of elastic scattering regime the amplitude `unitarizes' such that we instead get ∫^∞dS  S^3sin^2 ( δ_tree(S,L)  2). Performing the integral first and expanding the result as G_N → 0, we get a finite contribution G_N^2 log G_N. 

The main subtlety of the argument above comes from the fact that it is a small, but finite G_N computation. We present arguments based on unitarity and expected properties of a gravitational EFT that the computation is still under control. Another important point is that the computation of the energy-energy correlator involves integration over all impact parameters (including zero impact parameter which is sensitive to the details of the UV completion of a gravitational theory) for which the simple model of elastic scattering does not apply. We will see that in general the small impact parameter contribution leads to an additional enhancement of the correction to the energy-energy correlator, which we model using stringy effects. Scattering at zero impact parameter instead contributes a polynomial in z correction to the energy-energy correlator (similar to the stringy corrections in (<ref>)). In contrast,  large impact parameter scattering leads to non-analytic at small z contributions.

Performing the computation, we get for the universal quantum-gravity correction to the energy-energy correlator at small angles z≈θ^2/4: 

E(θ) E(0)  _QG =γ_d (p^0/Ω_d-1)^2
logc_T c_T log1 θ^2 ,      θ→0, 
where Ω_d=2π^d/2Γ(d/2) is the volume of a (d-1)-dimensional unit sphere and γ_d is given in (<ref>).  
Compared to the stringy correction (<ref>), it is notably non-analytic around θ=0. As we explain below, this is related to the fact that it comes from large impact parameter scattering in the bulk and the universality of the eikonal phase shift in gravity <cit.>. Notice also that the strong coupling expression above does not depend on the source, expressing the universal nature of gravity. 

Making further assumptions about the dual geometry, we can compute the leading non-planar correction to the energy-energy correlator exactly. For example, for pure Einstein gravity in 
AdS_4 dual to a putative CFT_3 we get

⟨E(n_1)E(n_2) ⟩_AdS_4=    - 96/π^2 (p^0 2 π )^2logc_T c_T[(5-8 z) log(4 z) + 8 (1-z)] .
In the same way, we get for pure Einstein gravity in AdS_5 dual to a a putative CFT_4

   ⟨E(n_1)E(n_2)⟩_AdS_5=45/64 ( p^0/4 π )^2 logc_T c_T 

   ×1 z^2(6 (-3+z-65 z^2+67 z^3)+(9-z-1475 z^2/3+1225
   z^3) logz .    . +(9-4z+54 z^2-900 z^3+1225 z^4)
   (Li_2(√(z)) -Li_2(-√(z))+1/2 log(1-√(z)/1+√(z)) log(z))/√(z) )  , where the √(z) dependence is spurious and the small z dependence is again of the type ∑_n z^n(a_n+b_n log z). Let us note that the underlying four-point correlation functions of local operators which could be used to compute (<ref>) and (<ref>) are not known explicitly, rather we obtained the results above by arguing that they come solely from the regime dual to high-energy scattering at fixed impact parameter in the bulk.

The result in general d for AdS_d+1/CFT_d can be found in Section <ref>, see (<ref>) and the discussion around it. An important novelty in d>4 is that in addition to the high-energy enhancement, there is an extra enhancement that comes from physics at small impact parameters. In this paper, we model it using stringy effects which, similar to (<ref>) and (<ref>), lead to contributions that are enhanced by the string scale Δ_gap∼R_AdSℓ_s. For D-dimensional bulk geometries of the type AdS_d+1× X, the same statement about the additional enhancement coming from unitarization of small impact parameter scattering holds as long as D>5. In particular, it holds for string/M-theory UV completions of gravity for which we have D=10 and D=11 correspondingly.  

For the case of N=4 SYM, for which the dual geometry is AdS_5 × S^5 we instead get

E(n_1) E(n_2)  _AdS_5 ×S^5    = -384 (p^0 4 π )^2 logc_T c_T logz ( 1-36 z+ 216 z^2 - 400 z^3 + 225 z^4 ) + ⋯ ,  
which up to the precise coefficient was anticipated in <cit.>. In addition to the `non-local' ∼log z contribution (<ref>), there are also polynomial corrections denoted by ⋯, enhanced by powers of √(λ), similar to the stringy corrections (<ref>), which we omit here and discuss in the main text. The leading log z term of (<ref>) as z→0 agrees with the one of (<ref>) and (<ref>) due to the universality of the graviton exchange in the bulk at large impact parameters. The terms z^nlog z are, on the other hand, different due to the fact that in the case of N=4 there is an additional contribution from the S^5 KK modes. Therefore, by measuring the coefficient of z^n log z log c_T  c_T term in the energy correlator we can probe the size of the internal manifold.

To make the connection to the problem of high-energy scattering in the bulk more explicit, we notice that there is an exact symmetry of the correlator that relates the energy correlation computation to the bulk scattering computation

E(n_1) E(n_2) ≡O^†E(n_1) E(n_2) O _fold O^†O _fold  = O^†E(n_1) E(n_2) O _unfold O^†O _unfold  ,
as depicted in fig:EEC. Here by `fold' we mean that the correlator is evaluated on the Schwinger-Keldysh contour, or a time-fold, see fig:image13. The `unfold' stands for the normal time-ordered correlator, see fig:image3. Unfolding, where an operator acting on the vacuum is moved a Poincaré patch forward leaves the conformal cross-ratios invariant. The conformal tensor structures that multiply the non-trivial functions of conformal cross-ratios in CFT correlation functions <cit.>, on the other hand, acquire a nontrivial overall phase which however cancels between the numerator and the denominator in (<ref>) leaving the ratio invariant, see <cit.> and Appendix <ref>.[More precisely, (<ref>) holds in even dimensions and for conformal parity-even structures in odd dimensions. For parity-odd conformal structures in odd dimensions, there could be an extra minus sign under the unfold transformation, see Appendix <ref>. In this paper, we will only consider parity-even structures.] We then notice that there is a region of the retarded time integration in (<ref>) that is sensitive to high-energy bulk scattering. It corresponds to the limit when the detector integration times u_i →±∞ with opposite signs, see fig:image3 and Figure 6 in <cit.>. For weakly coupled theories, the contribution from the high-energy scattering region cannot be easily isolated from the rest of the energy correlator. However, as we show in this paper, in holographic theories, high-energy bulk scattering produces a distinctive contribution to the energy-energy correlator which is enhanced compared to the naive perturbative expansion of the correlator.

Conceptually, the difference between our work and the computation of the leading stringy correction is that  (<ref>) requires considering the high-energy behavior of string tree-level amplitudes, whereas the leading 1  c_T correction requires considering high-energy gravitational scattering at finite G_N. A convenient way to characterize this kinematical regime is via the bulk phase shift δ(S,L) <cit.>, where S is the scattering energy and L is the impact parameter. In terms of the phase shift, the high-energy scattering contribution to the energy-energy correlator takes the following form[We use mostly-plus signature.]

E(n_1)E(n_2)= 8ρ_T c_TΓ(d+1)^2 (-p^2)^d/(-p·n_1)^d-1(-p·n_2)^d-1 ∫_0^1 d u K_d (u,z) Re ⟨1 - e^i δ⟩,
where u=tanh L and we introduced the notation 

⟨X ⟩= ∫_S_0^∞d S S^3X(S,L) |_tanhL = u  .
Here K_d (u,z) is a kinematical kernel that is given in Appendix <ref> and ρ_T is given in (<ref>).[Curiously, 𝒦_d (u,z) turns out to be closely related to the Mandelstam kernel that appears in the study of double discontinuity in gapped theories <cit.>.] Our results are independent of the arbitrary fixed lower integration limit S_0, which leads to the sub-leading O(1/c_T) contribution to the energy-energy correlator. The correction (<ref>) automatically satisfies the  momentum conservation Ward identities <cit.>, see Appendix <ref>, thanks to the following properties of the kernel K_d (u,z)

∫_0^1 d z (z (1-z))^d-4 2 K_d (u,z) = ∫_0^1 d z (z (1-z))^d-4 2  z  K_d (u,z) = 0  . 
The structure of the kernel K_d (u,z) is such that the expansion of the phase shift at large impact parameter L →∞ is mapped to terms which are non-analytic when z → 0. This is in contrast to the terms that are encoding point-like scattering at zero impact parameter, such as the stringy correction in (<ref>), which produce terms that are polynomials in z. 

At high energies and large impact parameters, the phase shift takes the universal and simple form

δ(S,L) = 4 πG_N S  Π(L) ,
where Π(L) depends on the dual geometry and details of the scattered states. 
Inserting this universal behavior into the formula (<ref>) leads to the log c_T/c_T correction to the energy-energy correlator reported above. 

Let us also briefly discuss the regime of validity of the results above upon inclusion of the stringy corrections. We can characterize them by the parameter Δ_gap, which is defined as the scaling dimension of the lightest spin-four stringy state <cit.>. In this case, the leading high-energy behavior of the phase shift is given by the contribution of the leading Regge pole, see e.g. <cit.>. We then find that the relevant parameter for our discussion is a characteristic impact parameter L_*^2 ≡log c_T Δ_gap^2. Our results then apply in the regime
L_* ≪1      (gravitational unitarization) .
In contrast, when L_* ≫ 1 the leading correction to the energy-energy correlator is mostly unitarized by stringy effects
L_* ≫1      (stringy unitarization) .
In this case, the bulk scattering is effectively unitarized at the string scale, and the leading enhanced quantum-gravity correction to the energy-energy correlator is ∼Δ_gap^2  c_T instead of log c_T  c_T. In this paper, we focus on the more universal case of gravitational unitarization. Throughout, we also assumed that the source is a light field, namely that its dimension does not scale with c_T ≫ 1.

The plan of the paper is the following. In Section <ref>, we derive an expression for the contribution of the high-energy bulk scattering to the energy-energy correlators in holographic CFTs in terms of the bulk phase shift. In Section <ref>, we use this representation to obtain the leading quantum-gravity correction to the EEC in pure Einstein gravity on AdS_d+1. In Section <ref>, we consider N=4 SYM dual to AdS_5× S^5 and derive the relation between the phase shift and the EEC which is in perfect agreement with the general representation in Section <ref>. In Section <ref>, we present the relation between the representation in terms of the phase shift and the light-ray OPE. In Section <ref>, we discuss in more detail various corrections to our results and their regime of validity.  We conclude in Section <ref> with a discussion. Appendix <ref> contains our conventions for the stress tensor two-point function, Appendix <ref> reviews properties of correlators on the Lorentzian cylinder, Appendix <ref> explores the kernel of the representation of the EEC in terms of the phase shift, Appendix <ref> relates properties of our representation of the EEC to the light-ray OPE, in Appendix <ref> we comment on the tensor structures that are relevant in the case of pure Einstein gravity and in Appendix <ref> we discuss the stringy phase shift in d=4 in detail.



§ ENERGY CORRELATORS AND HIGH-ENERGY SCATTERING IN ADS




In this section, we derive a representation of the energy-energy correlator in terms of the bulk phase shift δ(S,L) that characterizes high-energy scattering in AdS, see fig:image33. Here (S,L) is the scattering energy and impact parameter in the bulk respectively, see fig:image33. In holographic CFTs with classical gravity duals, this representation allows us to compute the leading quantum-gravity correction to the energy-energy correlator using known properties of high-energy scattering in gravity. In the following sections, Section <ref> and <ref>, we will use this representation to derive the EEC in pure Einstein gravity in AdS_d+1 and N=4 SYM dual to AdS_5× S^5. We discuss further corrections and the regime of validity of our formulas in Section <ref>.



 §.§ Two-point energy correlator and bulk scattering


We want to study the conformal collider setup of <cit.> where we probe a localized perturbation using energy detectors ℰ(n⃗) sitting far away and measuring the energy flux in the direction n⃗∈ S^d-2 on the celestial sphere, see fig:EEC. The energy detectors can be defined in terms of the stress tensor operator integrated over retarded time, see 
(<ref>). Equivalently, it can be defined as the light transform of the stress tensor E(n)=2𝐋[T](x_∞,n) at spatial infinity x_∞ defined in <cit.>. It is clear from fig:image3 that spatial infinity x_∞ of the Poincaré patch P_1 coincides with the origin of the Poincaré patch P_2. We can then write the following equivalent expression for the energy detector

E(n) = 2∫_-∞^∞dα/(-α)^d+2T(-n/α,n) ,
where we have introduced index-free notation T(-n/α,n)= T_μν(-n/α)n^μ n^ν, and T_μν(0) on the right-hand side corresponds to the origin of the Poincaré patch P_2.

We now consider the energy-energy correlator in a momentum eigenstate |p of unit norm produced by a primary operator acting on the vacuum
|p   =1 √(N_p) ∫d^dx e^ip ·xO(x)| Ω⟩,  N_p = 2π^1+d/2/Γ(Δ_O)Γ(Δ_O-d-2/2)(-p^2/4)^Δ_O-d/2,
with p in the future light cone. In the formula above, we used a scalar primary operator with scaling dimension Δ_ O to create the state and we comment on the generalization to the spinning case at the end of the section.

The EEC, defined in (<ref>), can be equivalently written as
 
E(n_1)E(n_2)=∫d^dx e^ip ·x ⟨Ω| O^†(0) E(n_1) E(n_2) O(x)| Ω⟩∫d^dx e^ip ·x ⟨Ω| O^†(0) O(x)| Ω⟩ ,
where operators are ordered as written.[Note that in this section, ℰ is represented in the Poincaré patch 𝒫_2 (see (<ref>)), while the coordinates of source operators 𝒪^†(0), 𝒪(x) is in the first Poincaré patch 𝒫_1.] Given that the detector operators are inserted at future null infinity, this correlator is effectively evaluated on a time-fold, see Figure <ref>.

By Lorentz invariance and homogeneity with respect to rescaling n_i^μ→λ_i n_i^μ, which can be easily derived for example from (<ref>), we have 
E(n_1)E(n_2)= (-p^2)^d/(-p·n_1)^d-1(-p·n_2)^d-1F(z),
with the cross-ratio 0 ≤ z ≤ 1 given by

 z = p^2 (n_1·n_2)/2(p·n_1)(p·n_2).
In the rest frame of the source z is given by z=1/2(1-n⃗_1·n⃗_2)=sin^2(θ/2) where θ is the angle between the detectors. 

Given the light transform (<ref>) and (<ref>), the EEC can be written as follows 
⟨E(n_1)E(n_2)⟩= 1 N_p ∫_-∞^∞4dα_1dα_2/(-α_1)^d+2(-α_2)^d+2⟨O^†(0) T(-n_1/α_1,n_1)T(-n_2/α_2,n_2) O (p)⟩.
To proceed it is convenient to introduce the following notation
G(α_1,α_2) ≡1 N_p ⟨Ω| O^†(0) T(n_1/α_1,n_1)T(n_2/α_2,n_2) O (p) | Ω⟩,
and we split the integrals into four regions depending on the signs of α_i

       ⟨E(n_1)E(n_2)⟩= ∫_0^∞4dα_1dα_2/(α_1)^d+2(α_2)^d+2    ( G_33(-α_1,-α_2)+ G_22(α_1,α_2) +  G_23(α_1,-α_2)+ G_32(-α_1,α_2) ) ,
where the index represents the Poincaré patch where the stress tensor is inserted, see Figure <ref>.  To obtain this, we have rewritten the light-transform as ∫_-∞^∞dα_1/(-α_1)^d+2 T(-n_1/α_1,n_1) =∫_0^∞dα_1/α_1^d+2( T(0_2+n_1/α_1,n_1) + T(0_3-n_1/α_1,n_1)) with 0_i denoting the origin of the Poincaré patch P_i.

We next introduce the following approximation
G_jk(α_1,α_2) ≃∫d^d q (2 
π)^d e^i q ·(n_1 α_1 - n_2 α_2) G̃_jk(p,q,n_1,n_2) + ⋯,
which should correctly capture the large q-limit of the correlator. 

Let us, for example, consider G_22(α_1,α_2). At large q the leading contribution to it is given by the unit operator so that
G̃_22(p,q,n_1,n_2) ≃ρ_T c_T θ(q^0) θ(-q^2) G_0(q,n_1,n_2)
with the vacuum Wightman two-point function of the stress tensors given by  

    G_0(q,n_1,n_2)    = (q·n_1)^2(q·n_2)^2(-q^2)^d-4/2(4χ^2-4χ+d-2/d-1)

    χ   = q^2 (n_1·n_2)/2(q·n_1)(q·n_2)

    ρ_T    = π^d/2+1/2^d-1(d+1)Γ(d-1)Γ(d+2/2)Ω_d^2.The overall normalization of ρ_T c_T follows from the normalization of the stress tensor two-point function in the vacuum TT∼ c_T in momentum space, see Appendix <ref>. 

Consider next G_33(- α_1, - α_2 ). This correlator is closely related to G_22(α_1, α_2). To see it, we can translate the in- and the out- source one Poincaré patch forward. Following <cit.>, we can call this operation T and we notice the following transformation property of the correlator
⟨Ω| O^†(T^a x T^-a) …O (T^b y T^-b) | Ω⟩= e^i πΔ(a-b) ⟨Ω| O^†(x) …O (y) | Ω⟩,
that we review in Appendix <ref>.

Using this property, we see that in our approximation

G_33(- α_1, - α_2 ) = G_22^*(α_1, α_2) ,
where the complex conjugation on the right comes from flipping the phase sign in the Fourier transform and the fact that G_0(q,n_1,n_2) is real.

Next we consider the G_23 and G_32 terms. By moving the out-source one Poincaré patch forward we see that these terms are related to bulk scattering <cit.>. We therefore modify the formula above by introducing the bulk phase shifts
G̃_23(p,q,n_1,n_2)    ≃(-1)^d ρ_T c_T θ(q^0) θ(-q^2) G_0(q,n_1,n_2) e^i δ_23(S,L) , 

  G̃_32(p,q,n_1,n_2)    ≃(-1)^d ρ_T c_T θ(q^0) θ(-q^2) G_0(q,n_1,n_2) e^i δ_32(S,L) ,
where the factor (-1)^d = e^± i π d comes from the trivial evolution of G_0 (recall that for the stress tensor Δ = d), see Appendix <ref>, and we introduced

S    =√(-q^2)√(-p^2), 

coshL    = -p ·q √(-q^2)√(-p^2),
which are identified with the scattering energy and the impact parameter in the bulk. Physically, we expect the phase shift formulas above to apply at high energies S L^2 ≫ 1, such that the Compton wavelength of the scattered particles is much less than the impact parameter.

By exchanging the ingoing and outgoing stress-energy tensors, we get the following crossing relation[To see it, notice that 2 ↔ 3 exchange here corresponds to two ways of approaching the Regge limit z, z̅→ 1 of the four-point function. In the space of cross-ratios it corresponds to continuation of z around the origin: clockwise z → z e^2 π i and counterclockwise z → z e^-2 π i. This operation introduces opposite phases in the source×detector OPE discussed earlier, which then becomes (<ref>). ]

e^i δ_32(S,L)  = ( e^i δ_23 (S,L) )^*. 
Using this relationship, we have 
G_23(α_1, - α_2 ) = G_32^*(-α_1, α_2) ,
and we can thus rewrite the energy-energy correlator above as follows

       ⟨E(n_1)E(n_2)⟩= ∫_0^∞4dα_1dα_2/(α_1)^d+2(α_2)^d+2 2 Re( G_22(α_1,α_2)+ G_23(α_1,-α_2)) .


We next plug the Fourier-transformed expressions above into this formula and perform the α-integrals to finally get the following representation for the contribution of the high-energy scattering to the energy-energy correlator

F(z) = 
8ρ_T c_TΓ(d+1)^2∫_0^1 d u K_d (u,z) Re ⟨1 - e^i δ⟩,
where we dropped the index on the phase shift δ_23(S,L) and introduced the notation

⟨e^i δ ⟩= ∫_S_0^∞d S S^3 e^iδ(S,L) |_tanhL = u  ,
where S_0 is an arbitrary fixed energy scale.
The kinematical kernel 𝒦_d (u,z) is defined as follows
(-p^2)^d 𝒦_d(u,z) (p·n_1)^d-1 (p·n_2)^d-1  ≡∫_M^+ d^dq (2π)^d G_0(q,n_1,n_2) δ(1-√(-q^2)√(-p^2)) (q·n_1)^d+1(q·n_2)^d+1  δ(u - √(1-p^2 q^2 (p·q)^2))
where M^+ is the future Milne wedge defined by q^2<0 and q^0>0. The explicit expression for the kernel 𝒦_d (u,z) is given in (<ref>).

Let us notice that in agreement with the expected momentum conservation Ward identities satisfied by the energy-energy correlator, the kernel has the following properties

∫_0^1 dz (z(1-z))^d-4 2 K_d (u,z) = ∫_0^1 d z (z(1-z))^d-4 2  z K_d (u,z) = 0 . 


In this paper we will use (<ref>) to explore the quantum gravity or 1/c_T corrections to the energy-energy correlator. However, before proceeding with this, we need to discuss the regime of validity of this formula. Physically, we only expect the formula (<ref>) to correctly capture the high-energy large enough impact parameter contribution to the energy-energy correlator. As we explain below, this translates to a potential polynomial in z ambiguity in some of our computations related to the small impact parameter physics. More details can be found in Section <ref>.

A complete treatment of the bulk scattering for the four-point function ⟨ O^† T_μν T_ρσ O⟩ can be found in <cit.>. It involves two additional tensor structures compared to the ones considered above. These additional structures however are suppressed in theories where the stringy modes are heavy, or, equivalently, which have a large gap in the spectrum of single trace higher spin operators Δ_gap≫ 1, see    <cit.>. In particular, in this case the three-point function of gravitons is the one of general relativity. For the same reason, we expect that our formula applies to scattering of spinning states as well because in general relativity the phase shift is diagonal in polarization space. It would be interesting to generalize our formulas to include the `non-minimal' tensor structures.



 §.§ High-energy limit and the enhancement property


We would like next to understand the properties of the integral over S above which leads to the enhancement property discussed in the introduction. A characteristic feature of the gravitational scattering is that the phase shift exhibits a linear growth with energy

δ(S,L) = 1 c_T S Π(L) ,
where we have substituted G_N in terms of the central charge of the CFT dual. 

We therefore need to understand the following integral

∫_S_0^∞d S S^3 e^iδ(S,L)   .
Now comes the key point of our analysis. Looking at the finite integral (<ref>) we see that the integral over energies and the expansion at large c_T do not commute. Indeed, expanding under the integral we get the UV divergence at order 1  c_T^2 (which corresponds to the one-loop supergravity computation)
∫_S_0^∞d S S^3 (1 + i S Π(L) c_T - 1 2 (S Π(L) c_T)^2 + ... ) ,
whereas if we do the integral over S first we get

∫_S_0^∞d S S^3 e^i 1 c_T S Π(L) = c_0 + c_1 c_T -1 2 Π^2(L) logc_T S_0 Π(L) c_T^2 + … , 
where log c_T  c_T^2 is enhanced compared to the naive expansion of the correlator. This is true as long as S_0 does not scale with c_T. Physically, this condition translates to the statement that the source operator is light, namely its scaling dimension does not scale with c_T. Moreover, we will focus only on the term enhanced in c_T which is correctly captured by the high-energy regime (<ref>). This is substantiated by bounding the contribution of other regimes in Section <ref>. 

Focusing on the term enhanced in c_T we can perform the integral over S in (<ref>) with the result

F(z) = 
 4 ρ_T logc_Tc_TΓ(d+1)^2∫_0^1 d u K_d (u,z) Π^2(u),
where we denoted by Π(u) ≡Π(L(u)) and we hope that this will not cause confusion. Let us also comment that in the presence of extra dimensions the graviton mixes with KK modes at high energies. In this case the eikonal phase operator should be diagonalized before exponentiation and instead of e^i δ we will get ∑_k |⟨in| k ⟩|^2 e^i δ_k and ∑_k |⟨in| k ⟩|^2 Π_k^2 correspondingly. We will see how it works very explicitly in the example of N=4 SYM.
    
We have concluded that the enhanced term of the EEC is determined by the bulk phase shift according to (<ref>) and all that is left to be done is the integral over impact parameters. 



 §.§ Integral over impact parameters


To better understand the consequences of the representation (<ref>), it is useful to consider the following integral

R_Δ(z) ≡∫_0^1 du K_d (u,z) H_Δ- 1(L(u)),
where H_Δ-1(L) is the hyperbolic space propagator in (d-1)-dimensions

    H_Δ-1(L)=π^1-d/2Γ(Δ-1)/2Γ(Δ-d-2/2)e^-(Δ-1)L_2F_1(d/2-1,Δ-1,Δ-d/2+1;e^-2L).
We show in Appendix <ref> that R_Δ(z) is given by

R_Δ(z)   =((d-Δ)^2-1) Γ(Δ-1/2) Γ(Δ+3/2)
   Γ(d-Δ/2-1/2) Γ(1/2 (d+Δ-1))/2^d+2 π^d Γ(d+1)^2 Γ(-d/2+Δ+1) f_Δ(z)      +(d-2) (2 π)^-d/(d-1) (-2 d+Δ+1) (d+Δ-1)g_Δ(z),
where f_Δ(z) is a celestial conformal block that appears in the light-ray OPE <cit.>

f_Δ(z) = z^Δ-(2d-1) 2  _2 F_1(Δ-1 2, Δ-1 2, Δ- d 2+1, z) 
and g_Δ(z) is analytic at z=0 and is given explicitly in (<ref>).[The connection between the phase shift formulas and the light-ray OPE will be discussed further in Section <ref>.] 

In our computations we will encounter Δ=(2d-1)+2n with n∈ℤ coming from the graviton exchange in the bulk. In this case the coefficient of f_Δ(z) in (<ref>) develops a pole which cancels with a similar pole coming from g_Δ(z) producing a log z:
 
   R_(2d-1)+2n(z)=r_d,nf_(2d-1)+2n,d(z)logz+…,   r_d,n=(-1)^n+1 ((d+2 n-1)^2-1) Γ(d+n-1) Γ(d+n+1) Γ(3 d/2+n-1)/2^d+2 π^d n! Γ(d+1)^2 Γ(3
   d/2+2 n),
and the ellipses denote analytic terms. The universal (n=0) leading small-angle contribution to the energy-energy correlator takes the form
E(n_1) E(n_2) _graviton = - logc_T/c_T2^3 d-5 (d-2) d  Γ(d-1/2)^4 Γ(d+3/2) Γ(3 d/2-1)/π^d+1/2Γ(d/2)^2 Γ(2 d-2)logz,
where the non-analyticity in z arises from the large impact parameter regime.

We thus see that the large impact parameter physics produces small angle non-analyticities at z=0. This is in contrast to the contribution coming from finite impact parameters: if we cut off the impact parameter integral in (<ref>) so that we have ∫_0^u_0 du, the result is analytic at small z. The conclusion of this discussion is that large impact parameter expansion of the phase shift controls non-analytic in z contributions to the energy-energy correlator. 



 §.§ The OPE perspective


We can also understand the discussion above starting from the source×detector OPE that was worked out in <cit.>. By inserting a complete set of states spanned by primary operators we get the following representation of the energy-energy correlator in the scalar state

F(z)   =∑_Δ,J|λ_Δ,J|^2 2 sin^2(πΔ- J - d - Δ_O 2) P_Δ,J(z) , 
where P_Δ,J(z) is a polynomial of degree J in z. In writing the formula above we used the fact that ⟨ O T_μν O_Δ, J⟩ admits a single conformal invariant structure. We can rewrite the formula above as follows

F(z)    = Re ∑_Δ, J |λ_Δ,J|^2 |λ_Δ,J^GFF|^2 (1 - e^- i πγ_Δ,J ) ( |λ_Δ,J^GFF|^2 P_Δ,J(z) ),
where we have defined γ_Δ,J≡Δ - J - d - Δ_ O. 

We can now try to connect this expression to the phase shift discussion. The finite J contributions correspond to zero impact parameter scattering in the language of the previous section and they produce polynomial contributions in z. To get the non-analytic contributions, we take the limit Δ, J →∞ with Δ J kept fixed, see e.g. <cit.>. 

Exchanging the sum for the integral, and identifying δ(S,L) = -πγ_Δ,J, see e.g. <cit.>, we should recover the previous formula, where the kinematical kernel arises from the large Δ,J limit of |λ_Δ,J^GFF|^2  P_Δ,J(z). It would be interesting to check this statement explicitly. In the case of 𝒩=4 SYM we do it in Section <ref>, where we see in detail how the phase shift formula emerges from the source×detector OPE. Notice also that the leading contribution to the phase shift (<ref>) arises from the contribution of the double trace operators [ OT_μν]_n,J <cit.> with n∼ J ≫1. 

Finally, let us comment on inelasticity. In the standard eikonal approach to the gravitational scattering inelasticity is encoded in the imaginary part of the phase shift Imδ(S,L). From the OPE point of view this arises from averaging over effectively degenerate operators e^i δ(S,L) =1  k∑_j=1^k e^- i πγ_j.



 §.§ Relation to the quadruple discontinuity


In the discussion above we saw a finite number of operators in the  source×detector OPE produces a polynomial contribution to the energy-energy correlator. Equivalently, any non-analyticity in z can only emerge from the sum over infinitely many operators.

The non-analytic terms in z can be conveniently captured by taking the discontinuity around 0

Disc_z F(z) ≡F(z + i 0) - F(z - i 0) 2 i .
The fact that the s-channel blocks in (<ref>) are polynomials in z can be restated by saying that they are annihilated by taking the discontinuity around z=0. 

A related and more familiar fact is that taking the t-channel double discontinuity annihilates the s-channel blocks <cit.>. In the formula above, the s-channel double discontinuity has already been taken, which can be seen due to the presence of the sin^2 factor in the OPE. It originates from the fact that the energy calorimeters annihilate the vacuum and thus we can equally compute the energy-energy correlator starting from ⟨ [ O, T_μν] [T_ρσ,  O] ⟩.



We conclude that Disc_z  F(z) (the same statement applies to the discontinuity around z=1) is captured by the quadruple discontinuity of the correlator. This statement is in agreement with the earlier discussion in this section, which related non-analyticity in z to the large impact parameter physics in the bulk. The same bulk diagrams that produce non-trivial large impact parameter behavior of the phase shift, see e.g. Figure <ref>, are those that have nontrivial double discontinuity in AdS that corresponds to the quadruple discontinuity of the correlator in CFT. 

In the explicit perturbative computations of the energy-energy correlators, see e.g. <cit.>, one can, in principle, write a dispersive representation of the energy-energy correlator by considering complex angles z. We see that such a representation is closely related to the Mandelstam representation of the AdS correlators. It would be interesting to explore this connection further.  




§ EINSTEIN GRAVITY

In the previous section we have developed the general expression for the enhanced 
log c_T/c_T correction  to the EEC in holographic CFTs. In particular, it comes from high-energy scattering in the bulk, dual to the Regge limit from the CFT point of view. This region is captured by the bulk phase shift <cit.> whose characteristic linear growth in energy, intrinsic to gravitational scattering, leads to the logarithmic enhancement in the central charge c_T≫1 that we explained. In this section, we derive the precise prediction for the non-analytic in c_T term in theories dual to pure AdS_d+1 Einstein gravity.  


 §.§ Bulk phase shift in pure Einstein gravity

The holographic theories that we consider are dual to d+1 pure Einstein gravity

S_grav    =1/16πG_N∫d^d+1x√(-g)(R-Λ),
where Λ=-d(d-1)/R_AdS^2 is the cosmological constant and R_AdS the AdS scale which we set to 1. Here G_N is Newton's constant in D=d+1 dimensions and it is related to the central charge of the CFT by (see e.g. <cit.>[Their definition of c_T differs by a factor of two with ours.])
c_T =d+1/d-1π^d/2Γ(d+1)/Γ(d/2)^3R_AdS^d-1/2πG_N,
and see Appendix <ref> for further conventions. We then want to measure the EEC in a momentum eigenstate |p⟩ created by a local operator O such as O∼ϵ^μϵ^ν T_μν or another external probe field added to the gravity action (<ref>).

The phase shift in theories dual to pure AdS gravity was first studied in <cit.>, see also <cit.> for related work on the Regge limit in holographic CFTs. It is determined by the graviton exchange in terms of the scattering energy S and impact parameter L defined by (<ref>) and it is given by

    δ^(GR)(S,L)=4πG_NS H_d-1(L),
where H_d-1 is defined in (<ref>). In particular, the large impact parameter limit in any holographic CFT is determined by that of the graviton exchange in (<ref>) and will lead to a universal prediction for the small-angle behaviour of the energy-energy correlator. From the CFT point of view, this is due to the universality of the stress tensor exchange and using the relation between the phase shift and the anomalous dimensions of double-trace operators <cit.> this can also be found using bootstrap methods, see e.g. <cit.>. 

To align with our previous representation it is further convenient to define Π^(GR)(L) by 
δ^(GR)(S,L) = 1 c_T S Π^(GR)(L)
which explicitly is given by 

    Π^(GR)(L) = 2π^d/2Γ(d+2)/(d-1)Γ(d/2)^3H_d-1(L).
The enhanced correction to the energy-energy correlator is given by (<ref>) and for pure Einstein gravity in AdS_d+1 we get 

⟨E(n_1)E(n_2) ⟩_AdS_d+1   = 
logc_T/c_T4ρ_TΓ(d+1)^2 (-p^2)^d (-p ·n_1)^d-1 (-p ·n_2)^d-1  ∫_0^1 d u K_d (u,z)Π^(GR)(u)^2 .
Let us note that in the small impact parameter limit u→ 0, the kernel is given by 
K_d(u,z)∝u^d-2(d-2/d-1-4z+4z^2).
The pure Einstein gravity phase shift on the other hand behaves in AdS_d+1 as (Π^(GR)(u))^2∼ u^2(3-d) (log^2 u in d=3). In d≥ 5 we therefore see explicitly a small impact parameter divergence which as we discuss in Section <ref> is regulated by stringy corrections to the phase shift. 



 §.§ AdS_4/CFT_3

Consider the phase shift in d=3 where from (<ref>) Π^(GR)(L) is given by 
 
    Π^(GR)(L)= 48 ( e^-L (e^2 L+1) log(e^L+1/e^L-1)-2)/π.
Besides the phase shift, we need the kernel which can be obtained from Appendix <ref>. We present it explicitly here for d=3:

K_3(u,z)=   u (1-u^2)/16 π^2 (u^2 (z-1)+1)^7[u^8 (z-1)^3 (8 z^2+24 z+3)   -u^6 (z-1)^2 (16 z^3+136 z^2+30 z-7)+3 u^4 (48 z^4-24 z^3-62 z^2+39 z-1)   -u^2 (144 z^3-184 z^2+42
   z+3)+2 (8 z^2-8 z+1)].


We digress slightly to point out explicitly some salient features of the kernel (<ref>), already alluded to and which generalizes straightforwardly to general d. First, expanding the kernel at small angles z→ 0 the behaviour of the kernel at u→ 1 gets worse order-by-order in z
K_3(u,z)=u (2+3 u^2)/16 π^2 (1-u^2)^3-u (4+22
   u^2+9 u^4) z/4 π^2 (1-u^2)^4+O(z^2).
On the other hand the behaviour of the (Π^(GR)(L))^2∼ (1-u)^2 as u→1. This leads to the non-analytic log z term in the EEC as was also seen in (<ref>). A more detailed asymptotic expansion when z→0 and u→ 1 keeping 1-u/z–fixed is given in Appendix <ref> in general dimension. 
 
After having showcased some general properties of the kernel, it is clear that the integral in d=3 is well-behaved. Given the kernel (<ref>) and the phase shift (<ref>) we obtain the enhanced term in c_T in theories dual to pure Einstein gravity in AdS_4 by performing the integral in (<ref>):

⟨E(n_1)E(n_2) ⟩_AdS_4=    - 480/π^2 (p^0 2 π )^2logc_T c_T[(1-8/5 z) log(4 z) + 8/5 (1-z)].
As expected from the general properties of the kernel the result (<ref>) satisfy the momentum conservation Ward identities 
   ∫_0^1 dz/√(z(1-z))⟨E(n_1)E(n_2) ⟩_AdS_4=0,   ∫_0^1 zdz/√(z(1-z))⟨E(n_1)E(n_2) ⟩_AdS_4=0.




 §.§ AdS_5/CFT_4

Let us now consider instead d=4, still in pure Einstein gravity. As for the three-dimensional case, much of the work is laid out for us. To be explicit, the phase shift is determined by Π^(GR)(L) 
Π_d=4^(GR)(L)   =40 πe^-3 L/1-e^-2 L= 20 π(1-u)^3/2/u √(1+u),
and we have written it both in terms of L and u=tanh(L). The kernel on the other hand takes a form similar to the d=3 case (<ref>) but we refrain from writing it down in detail. It can easily be constructed as explained in Appendix <ref>. 

Explicitly performing the integral in (<ref>), we find the following result for the term non-analytic in the central charge c_T 

   ⟨E(n_1)E(n_2)⟩_AdS_5=45/64 ( p^0/4 π )^2 logc_T c_T 

   ×1 z^2(6 (-3+z-65 z^2+67 z^3)+(9-z-1475 z^2/3+1225
   z^3) logz .    . +(9-4z+54 z^2-900 z^3+1225 z^4)
   (Li_2(√(z)) -Li_2(-√(z))+1/2 log(1-√(z)/1+√(z)) log(z))/√(z) )  , where the √(z) cancels in the small z expansion of the energy-energy correlator and for simplicity we have set p^μ=(p^0,0⃗).
The result (<ref>) is displayed in fig:EEC4d and can be verified explicitly to satisfy the momentum conservation Ward identities (<ref>) and (<ref>), again as expected by the properties of the kernel (<ref>). 

For later convenience let us isolate the part that is non-analytic in z in (<ref>). Its small angle expansion is

⟨E(n_1)E(n_2) ⟩_AdS_5 =- 384 (p^0/4 π )^2 logc_T/c_T(1-27/7z+12/7z^2+…)logz+….
In particular, the leading term in (<ref>) as z→ 0 is universal for theories dual to AdS_5× X for any internal manifold X. This will be verified explicitly in the case of AdS_5× S^5 in Section <ref>. In the latter case, we will see how subleading terms are altered compared to AdS_5 in (<ref>) due to the presence of KK-modes.




 §.§ AdS_d+1/CFT_d

As we have argued, non-analyticities in z arise from the large impact parameter regime and a convenient method to extract these is to study the discontinuity of the kernel K_d(u,z). A useful way to do this is by utilizing the connection to the Mandelstam kernel <cit.> and we refer to Appendix <ref> for further details. In particular, the term non-analytic in the central charge is given by 
(<ref>)

Disc_z ⟨E(n_1)E(n_2) ⟩|_logc_T    = θ(-z)
4ρ_Tlogc_T/c_TΓ(d+1)^2 (p^0)^2 ∫_1/√(1-z)^1 d uDisc_z  K_d (u,z)Π^(GR)(u)^2,
where for simplicity we have set p^μ=(p^0,0⃗). Moreover, the discontinuity of the kernel is simpler than the kernel itself and vanishes for u<1/√(1-z). See Appendix <ref> for more details on the discontinuity of the kernel. Taking the discontinuity of the EEC therefore effectively removes ambiguities from small impact parameters. Given Π^(GR)(u) in (<ref>) and (<ref>), we need to perform the integral over the impact parameter. Performing this integral we find the leading quantum-gravity correction to the EEC in theories dual to pure Einstein gravity in AdS_d+1: 

   ⟨E(n_1)E(n_2) ⟩=-γ_dlogc_Tc_T(p^0/Ω_d-1)^2[P_d(z)logz+R_d(z)],   P_d(z)= _3F_2(1-d/2,d-1,d-1 ;  d-1/2,1 ;z),   γ_d =2^3 d-3 d(d-2)Γ(d-1/2)^2 Γ(d+3/2) Γ(3 d/2-1)/π^3/2 Γ(d/2)^2 Γ(2 d-2),
where R_d(z) denotes analytic terms in z and will be computed below. It is easy to verify that this reproduces the explicit results in d=3,4 obtained above. In particular, for odd d, P_d(z) truncates and the non-analytic terms are given by log z times a polynomial of degree (d-1)/2:

   ⟨E(n_1)E(n_2)⟩_AdS_4 = -480/π^2logc_T c_T(p^0/2π)^2(1-8z/5)logz+…

   ⟨E(n_1)E(n_2)⟩_AdS_6 = -21120/π^2logc_Tc_T(p^0/2π^2)^2(1-64 z/9+800 z^2/99)logz+….


More generally, from (<ref>), we see that the discontinuity of the EEC grows at large |z| as |z|^D-2/2 where D is the number of large bulk dimensions. Here we suggestively write it in terms of the bulk dimension D instead since we will see in Section <ref> that this is consistent with D=10 in the case of AdS_5× S^5.

Having obtained the general d-result (<ref>) for the non-analytic term, we can reconstruct the full result by a dispersive integral with subtractions; the result for R(z) in (<ref>) is  

   R_d(z) = c_d^(0) + c_d^(1) z

   - . ∑_k=0^∞z^k Γ(d-1/2)  (-H_-d/2+k-1/2-2
   H_d+k-2+H_d+k-3/2+2 H_k) Γ(-d/2+k+1/2) Γ(d+k-1)^2/Γ(1/2-d/2) Γ(d-1)^2 Γ(k+1)^2 Γ(d+k-1/2)where the analytic part refers to the terms that are polynomial at small z and H_k are the harmonic numbers. The coefficients c_d^(0) and c_d^(1) are determined by imposing the momentum conservation Ward identities (<ref>) and (<ref>). The analytic term in (<ref>) further has simple poles when d is odd, these should be removed and the finite piece will be a polynomial of degree (d-1)/2. 

For d>4 there is a polynomial ambiguity in the result above related to the small impact parameter physics coming from the fact that
K_d(u,z)(Π^(GR)(u))^2∼u^d-2 ( p_2(z) + u^2 p_3(z)+...)1/u^2(d-3)∼( p_2(z) + u^2 p_3(z)+...)/u^d-4,
for small u where p_k(z) stands for a degree k polynomial. In the computation above we regulated this divergence by analytically continuing in d. In the case of pure Einstein gravity, the ambiguity is a polynomial of degree ⌊d-1/2⌋ for d≥ 5. More generally, for a D-dimensional bulk we will have u^d-2( p_2(z) + u^2 p_3(z)+...)  u^2(D-4) in the formula above. As we discuss in Section <ref>, the divergence at small impact parameters can be regulated by the stringy effects, which leads to further stringy enhanced terms polynomial in z.



§ ADS_5 × S^5 SUPERGRAVITY

In the section above, we analyzed a universal non-analytic contribution to the energy-energy correlation in a theory with the Einstein gravity dual in AdS_d+1. In this section, we consider instead the canonical example of N=4 SYM, where the dual geometry is AdS_5× S^5, and the phase shift interpolates between the universal form used in the previous section at large impact parameters, and the 10d flat space one at small impact parameters. It is instructive to repeat the analysis of the previous section in this case. 

We proceed in the following steps: a) we use SUSY Ward identities to relate the energy-energy correlator to the scalar four-point function as in <cit.>, see also <cit.>; b) we write down the OPE in the source×detector channel; c) we use the results for the diagonalization of the dilatation matrix of the double trace operators obtained in <cit.> to effectively `eikonalize' them. The result of these steps is an OPE, which we can use to compute the universal log c_T  c_T contribution to the energy-energy correlator. Moreover, we will see explicitly how the representation of the EEC in terms of the bulk phase shift arises from this point of view. 

Let us recall that in the case of N=4 SYM and with our conventions reviewed in Appendix <ref>

c_T = 40 (N^2 - 1),
where N is the rank of the gauge group which we take to be SU(N). It will be convenient for us, following <cit.>, to introduce instead
a = 1 N^2 - 1 ,
and switch to c_T at the very end.

We consider the four-point function of 20' half-BPS operators and we choose the R-symmetry polarizations as described in <cit.>. We choose for the source operators y_14^2=1 so that
O^†(x_1) O(x_4) = 1 x_14^4  . 
For detector operators, we choose y_23^2=0. We also choose y_12^2 = y_34^2 = y_13^2 = y_24^2 = ( 1  8 (2 π)^4 a)^1/4 which simplifies the relationship to the energy-energy correlator.

With this choice, we have the following nonperturbative relationship due to superconformal symmetry

E(n_2) E(n_3) = 4 p^4 (n_2 ·n_3)^2 Õ(n_2) Õ(n_3) , 
which is strictly speaking valid only for n_2^μ≠ n_3^μ, see also <cit.> for the complete relationship. At strong coupling this subtlety is not relevant because the energy-energy correlator does not contain any contact terms at n_2^μ = n_3^μ.



 §.§ The source×detector OPE


We are interested in writing the OPE expansion of this correlator in the source×detector channel. Focusing on the relevant contribution of long (non-protected) superconformal multiplets, we get

O^†(x_1) Õ(x_2) Õ(x_3) O(x_4) =1 8 (2 π)^4 a 1 (x_12^2 x_34^2)^2 ∑_t, ℓ λ^2_t, ℓ u^2 v^2 F_t, ℓ(z, z̅)  ,
where as usual u = z z̅ and v = (1-z)(1-z̅). To get the formula above, we noticed that the projections described above correspond to y=y̅=1 for the R-symmetry cross-ratios in <cit.>. Here t =Δ - ℓ 2 is half the twist of exchanged superconformal primaries. The superconformal blocks F_t, ℓ(z, z̅) that appear in the formula above are simply related to the usual conformal blocks
u^2 v^2 F_t, ℓ(z, z̅) = v^2 u^2 G_Δ+4, ℓ(z, z̅)  . 


In <cit.> the matrix of double trace operators in the OPE expansion above was diagonalized using one-loop supergravity. We will use it in the computation of the energy-energy correlator. Hence, using the diagonalization result of <cit.>, we can rewrite the formula above as follows
O^†(x_1) Õ(x_2) Õ(x_3) O(x_4) =1 8 (2 π)^4 a 1 (x_14^2 x_23^2)^2 ∑_t=2^∞∑_ℓ= 0^∞∑_i=1^t-1 λ^2_t, ℓ, i v^4 u^4 G_Δ_t, ℓ, i+4, ℓ(z, z̅)  ,
where Δ_t, ℓ, i = 2 t + ℓ + a γ_t, ℓ, i and i correspond to a sum over families of operators that are degenerate to leading order in a.

The contribution of interest comes from high energy scattering in the bulk closely related to the Regge limit. To approach the Regge limit we need to continue z → e^- 2 π i z around the origin and then take z, z̅→ 1 with (1-z)/(1-z̅) kept fixed. Upon doing this continuation, the s-channel conformal blocks acquire a simple but important phase
   O^†(x_1) Õ(x_2) Õ(x_3) O(x_4) _z →e^- 2 πi z    = 1 8 (2 π)^4 a 1 (x_14^2 x_23^2)^2 ∑_t=2^∞∑_ℓ= 0^∞∑_i=1^t-1 λ^2_t, ℓ, i v^4 u^4 e^- i πa γ_t, ℓ, i G_Δ_t, ℓ, i+4, ℓ(z, z̅)  .
This phase is directly related to the phase shift in the previous section. 

As discussed earlier, the combination relevant for the computation of the energy-energy correlation is 2  Re(e^- i π a γ_t, ℓ, i  - 1 ). To proceed, it is convenient to first take the detector limit x_23^2 → -2 r_2 r_3 (n_2 · n_3), see <cit.> for details. In this way we get
   O^†(x_1) Õ(u_2, n_2) Õ(u_3, n_3) O(x_4) _Regge    =1 16 (2 π)^4 a 1 x_14^4 1 (n_2 ·n_3)^2 ∑_t=2^∞∑_ℓ= 0^∞∑_i=1^t-1 λ^2_t, ℓ, i v^4 u^4 Re(e^- i πa γ_t, ℓ, i  - 1 ) G_Δ_t, ℓ, i+4, ℓ(z, z̅),
and the cross-ratios become
u    = ((-x_1 ·n_2)-u_2)(u_3 - (-x_4 ·n_3)) ((-x_1 ·n_3)-u_3)(u_2 - (-x_4 ·n_2)) ,    v = x_14^2 (n_2 ·n_3) 2 ((-x_1 ·n_3)-u_3)(u_2 - (-x_4 ·n_2))  . 
The Regge limit then corresponds to u_2, u_3 →∞ with u_2 u_3 <0. The same limit with u_2 u_3 > 0 corresponds to the Euclidean OPE limit and is the source of -1 in the formula above.

To leading order in the Regge limit we have
u    = 1 - (-x_14 ·n_2) u_2 +  (-x_14 ·n_3) u_3 + …,     v =x_14^2 (n_2 ·n_3) (-2 u_2 u_3) + …. 
Consider for example u_3<0 and u_2>0. We then introduce u_2 = 1 α_2 and u_3 = - 1 α_3. In this way we get to the light-ray transform ∫_0^∞d α_2 d α_3 α_2^2 α_3^2, where
u    = 1 -(-x_14 ·[α_2 n_2 + α_3 n_3]) ,    v =x_14^2 (α_2 n_2 ·α_3 n_3) 2  . 
Our next computation step is to derive the `eikonal' representation for the conformal blocks. The Regge limit now is α_i → 0 with α_2/α_3 held fixed. The integration measure becomes

∫_0^∞d α_2 d α_3 α_2^2 α_3^2 1 x_14^4 1 (n_2 ·n_3)^2 v^4 u^4 =x_14^4 (n_2 ·n_3)^2 16 ∫_0^∞d α_2 d α_3  α_2^2 α_3^2 . 


The key aspect of the computation of interest is noticing that the quantum numbers that dominate as α_i → 0 are the following i, t, ℓ∼1 √(α). Let us introduce the following variables
h+ h̅    = Δ,    h - h̅ = ℓ . 
We then have the following representation for the blocks, see e.g. <cit.>,
lim_z,z̅ →1λ^2_GFF(h,h̅) G_h, h̅(z, z̅)    =(2 π^3)^2 ∫_M^+ dp (2 π)^4 d p̅ (2 π)^4 e^i p ·x- i p̅ ·x̅ δ(p^2 p̅^2 16 - h^2 h̅^2) δ(p ·p̅ 2 + h^2 + h̅^2) 
 
   ×4 h h̅ (h^2 - h̅^2),
where effectively we kept (1-z)h^2 and (1-z)h̅^2 fixed.

In the formula above, we chose the three-point function in the generalized free field theory of non-identical operators of dimension 2 (which means that the sum goes effectively over both odd and even spins).

Because in our case (<ref>) the sum goes over even spins only we need to consider the ratio
λ(h,h̅, i) = 1 2λ^2_t, ℓ, i λ_GFF(t+ℓ+2, t+2) =(16)^2 6 π (h+i)^3/2(h̅ - i)^3/2 i^3/2(h-h̅+i)^3/2 (h - h̅ + 2 i) h^4 h̅^4 ,
where λ_GFF(t, ℓ) is the square of the three-point function in the generalized free field theory of nonidentical operators <cit.>. The shift t+2 in the denominator is due to the abovementioned shift Δ+4 in the conformal block expansion.

In the language of the bulk phase shift analysis in <cit.>, the formulas above arise if we identify
x^μ   = x_14^μ,     x̅^μ= 1 2 ( α_2 n_2^μ+ α_3 n_3^μ)  .
The utility of the eikonal representation is that the dependence on x and x̅ is completely factorized. 

As an immediate result, we can trivially perform the Fourier transform with respect to x_14^μ. Similarly, the integral over α_i trivializes and we get
   Õ(n_2) Õ(n_3) = -1 8 πa (n_2 ·n_3)^2 ∫_0^∞d h ∫_0^h d h̅ ∫_0^h̅ d i λ(h,h̅, i) 

   ∫_M^+ d p̅ (2 π)^4  Re(e^- i πa γ_t, ℓ, i  - 1 )  (p̅ ·n_2)^3 (p̅ ·n_3)^3  4 h h̅ (h^2 - h̅^2) ∂_p^4 [ δ(p^2 p̅^2 16 - h^2 h̅^2) δ(p ·p̅ 2 + h^2 + h̅^2) ] , where ∂_p^4 comes from x_14^4 in the measure (<ref>) that we conveniently re-expressed via the derivatives over p^μ and integrated by parts.

Next by noticing that δ-functions only depend on p^2 p̅^2 and p ·p̅, we can exchange ∂_p^4 = p̅^4  p^4∂_p̅^4 and again integrate by parts to get
   Õ(n_2) Õ(n_3) =- 1 8 πa (n_2 ·n_3)^2 p^4 ∫_0^∞d h ∫_0^h d h̅ ∫_0^h̅ d i λ(h,h̅, i) 

   ∫_M^+ d p̅ (2 π)^d ∂_p̅^4 (  p̅^4  (p̅ ·n_2)^3 (p̅ ·n_3)^3  ) Re(e^- i πa γ_t, ℓ, i  - 1 )  4 h h̅ (h^2 - h̅^2) δ(p^2 p̅^2 16 - h^2 h̅^2) δ(p ·p̅ 2 + h^2 + h̅^2) . We can take the derivative, and we find that
∂_p̅^4(  p̅^4  (p̅ ·n_2)^3 (p̅ ·n_3)^3  ) = 576 (p̅ ·n_2)^3 (p̅ ·n_3)^3 (2 3 - 4 χ+ 4 χ^2 ),
where χ=p̅^2 (n_2 · n_3)  2 (p̅· n_2) (p̅· n_3). 

We next consider the following change of variables
h    = 1 2 √(S) e^L/2,    
h̅ = 1 2 √(S) e^-L/2,    
i = 1 2 √(S) e^-L/2 α  ,
so that S=√(p^2 p̅^2), cosh L = -p ·p̅√(p^2 p̅^2), and we then get for the anomalous dimension
γ(S,L,α) = -  e^3 L S (e^L-1+2 α)^6.  


After performing straightforward but slightly tedious algebra and using the identity (<ref>), we finally get the following result
E(n_2) E(n_3) =- 36 c_T 5 π p^4 (p ·n_2)^3 (p ·n_3)^3 ∫_0^1 d u K_4(u,z) Re⟨e^i δ - 1 ⟩ , 
where we have defined
⟨e^i δ ⟩   = 64 3 π ∫_0^1 d αe^-5L sinhL (e^2 L-1) ((1-α) α(α+e^L-1) (α+e^L))^3/2
   (2 α+e^L-1) 

      ×∫_S_0^∞d S S^3 e^i δ(S,L)|_tanhL = u  .
The phase shift in the formula above is taken to be
δ(S,L,α) = - 40 πc_T γ(S,L,α) , 
where we used (<ref>). Let us notice the presence of the extra α integral in the formula above compared to a more naive representation (<ref>) used in the previous sections. Its origin is in the fact that we have five extra dimensions due to the fact that geometry is AdS_5 × S^5. This leads to the fact that we have to consider the eigenstates of the eikonal operator which are different from a simple graviton state. The sum over these eigenstates becomes the α integral in the formula above.

We can now perform the integral over energies to get 
∫_S_0^∞d S S^3  Re(e^i δ(S,L)-1) = - 1 2 logc_T c_T^2 (40 π)^2 [e^3 L (e^L-1+2 α)^6 ]^2 + ...  . 
Notice that for α=0 the phase shift behaves as 1/L^6 at small impact parameters. This is behavior is expected because in this limit we reproduce ten-dimensional flat space geometry. 

Next, we compute the α integral to get

Re⟨e^i δ - 1 ⟩= - 1 2 logc_T c_T^2 (40 π)^2 
(1-u^2)^3 64 u^7  . 
We finally perform the u-integral to get 
E(n_2) E(n_3) =-24 π^2 logc_T logz c_T p^4 (p ·n_2)^3 (p ·n_3)^3 (1-36z+216 z^2-400 z^3 + 225 z^4) + … . 
When performing the u-integral we encounter divergence at small u which as we discuss in the next section is regularized by the stringy effects and leads to stringy enhanced polynomial contributions to the expression above. 

As expected to leading order at small z our N=4 SYM result agrees with the computation in the previous section because it comes from the universal graviton exchange, however, the subleading corrections probe the geometry of the S^5. One diagnostic of large extra dimensions in the bulk in terms of CFT data was studied in <cit.> and we here see that the enhanced contribution to the EEC provides another sharp sign of internal manifolds in the bulk dual.  



 §.§ The light-ray OPE

In the previous section, we used the OPE in the source×detector channel to obtain the EEC in N=4 SYM at strong coupling. In this section, we study the same EEC instead using the OPE in the detector×detector channel following <cit.> and utilizing the known expression for the OPE coefficients analytically continued in spin C^+(Δ,J) obtained <cit.>. The key point is that the E (n_1)× E (n_2) OPE is well defined when the Regge intercept J_0<3 which holds non-perturbatively in c_T^-1 but is violated perturbatively at one-loop. This is equivalent to the divergence at large energies that one gets from expanding the eikonal result ∼ e^i SG_NΠ(L). From the point of view of E (n_1)× E (n_2) OPE this is reflected by a divergence of the one-loop OPE data at J=3.[In N=4 SYM we use supersymmetry as before to relate the EEC to SSC of the scalar 20' operators which effectively shifts J=3 to J=-1.] The aim of this section is to explore this divergence and by regulating it reproduce the result obtained above.

The EEC can be written as <cit.>

   E (n_2) E(n_3) =(-p^2)^4/(- p·n_2)^3(-p·n_3)^3 ∫_2-i∞^2+i∞dΔ/2πi C^+(Δ,-1)π^2Γ(Δ-2)/2 Γ(Δ-1/2)^3Γ(3-Δ/2)f_Δ(z) .
Here C^+(Δ,-1) are spin J=-1 conformal partial waves analytically continued from even spins (that is the + signature), and the celestial blocks are given by (<ref>). We close the contour to the right picking up the poles of C^+(Δ,-1) which are given by 
C^+(Δ,J) ∼-a_i/Δ-Δ_i
where a_i and Δ_i can be expanded in 1/c_T[Here we use convention where a_i^(0)∼ c_T which is natural when relating the scalar correlator to the stress tensor correlator.]
a_i    = a_i^(0)(1+c_T^-1a_i^(1)+c_T^-2a_i^(2)+…),

Δ_i    = Δ_i^(0)+c_T^-1γ_i^(1)+c_T^-2γ_i^(2)+….
Here Δ_i^(0)=4+2i+J correspond to the double-trace operators  O_p□^n∂_μ_1…∂_μ_J O_p where again mixing will be important. 

Let us focus on the part that is non-analytic in z. At strong coupling, it first arises from the term ⟨ a^(0) (γ^(1))^2⟩ which contributes ∂_Δ f_Δ(z) |_Δ = 4+2i-1∼log z to the energy-energy correlator, where the averaging goes over the degenerate operators. This quantity has been computed in supergravity in <cit.> with the following result close to the relevant spin J=-1

⟨a^(0) (γ^(1))^2⟩_n, J  ∼1 c_T (n+1)_4^3 (J+1) + ... ,
where (a)_n is the Pochammer symbol. Compared to the tree-level supergravity computation, the new feature of the result above is a pole at J=-1, which effectively makes the formula (<ref>) inapplicable. The origin of this pole is the same enhanced perturbative Regge behavior that we discussed before. This pole is, therefore, an artifact of the large c_T perturbative expansion.

Compared to the phase shift analysis, we do not have a first principle way to see how this pole gets regularized at finite c_T. We proceed by noting the following simple fact. Let us assume that the pole is regularized as 1  J+1→ c_0 log c_T at finite c_T. We then get the following prediction
F(z)    = c_0 logc_T logz 8 π^2 c_T (-1 6 )∑_n=0^∞(-1)^n (n+1)_4^3 Γ(n+3)^2 Γ(5+2n) f_7+2n(z)    =- 768 c_0 logc_T logz (4 π)^2  c_T ( 1-36 z +216 z^2 -400 z^3 + 225 z^4 )  ,
where recall that E (n_2)  E(n_3) =(-p^2)^4/(- p· n_2)^3(-p·  n_3)^3 F(z) and (a)_m is the standard Pochhammer symbol. Our result agrees with the analysis in the dual channel. By matching the overall coefficient we can fix c_0 = 1/2.

The complete structure of the OPE data in the 1/c_T perturbation theory involves many more finite spin singularities beyond the simple pole (and its higher order in 1/c_T analogs) discussed in this section, see e.g. <cit.> for explicit results. It would be very interesting to understand in detail how all these singularities get resolved at finite coupling. Among other possible effects, it would require understanding the unitarization of the zero impact parameter scattering.



 §.§ The ω-deformation

In this section we review yet another way to regulate the divergence at large energies, or equivalently large detector times, in the c_T^-1 expansion of the EEC, again considering the specific example of N=4 SYM. Instead of considering energy correlators which measure the total energy flux deposited on the celestial sphere in <cit.> a generalized energy detector was introduced
E(ω̂,n) =1 4∫_-∞^∞ d u e^-iω̂u lim_r →∞ r^d-2 T_μν(u,r n⃗) n̅^μn̅^ν
See also <cit.> where similar operators are studied. The generalized energy detectors have an effective time resolution τ∼1/ω̂ and when ω̂→ 0 the measurement over all times as in the original energy detectors are recovered. 

In <cit.> the one-loop generalized energy-energy correlation in N=4 SYM at strong coupling was calculated at one-loop[We took the formula (7.6) in <cit.> and noticed that c_T^there = c_T  160.]

E(ω_1,n_1)E(0,n_2)= 25/c_T (p^0)^2 π^2[1-6z+6 z^2/6ω_1^2+3/22-15z+24z^2-10z^3/ω_1+12/25(1-36z+216z^2-400z^3+225z^4)log|ω_1|log(z^2|ω_1|)],
where the dimensionless frequency ω_1=2ω̂_1(p· n_1)/p^2 was introduced. This diverges when taking the energy detector limit ω→ 0. Cutting of the detector time at times τ∼ c_T∼1/ω one obtains in particular the non-analytic in z term 
E(ω_1,n_1)E(0,n_2)= -24 (p^0)^2π^2 logc_T logz c_T (1-36z+216z^2-400z^3+225z^4),
in agreement with the previous results. Other divergences are related to the small impact parameter physics and we expect these to get resolved by the stringy effects as we discuss in the next section.



§ BULK PHASE SHIFT AND THE LIGHT-RAY OPE


In the context of the energy-energy correlator, its small angle expansion and the discontinuity (<ref>) are conveniently captured by the light-ray OPE <cit.>, which we can schematically write as

F(z)  = ∑_i  λ_i f_Δ_i(3)(z),
where the sum goes over signature-plus operators of spin J=3 (for spinning targets, higher spin operators contribute as well <cit.>), λ_i are related to the corresponding three-point functions, and we recall that the celestial block f_Δ(z) was given in (<ref>). Notice that the sum in the formula above goes both over the single- and double-trace operators of spin three. It is interesting to ask how this connects to the regime of the EEC studied in this paper.



 §.§ Planar correlator

Let us first consider the case of the planar correlator, the context in which the formula (<ref>) has been applied and analyzed in detail. Notice that our analysis based on the phase shift and approximations we made when deriving the formula (<ref>) are centered around the high-energy enhanced contributions. In the planar theory, the contribution of the high-energy/Regge limit is not enhanced compared to the rest of the integral over the retarded time. Therefore we do not expect our phase shift formulas to capture the full result. It is still interesting to proceed and see what we get.


In the planar theory, the phase shift formula for the energy-energy correlator takes the following form

F(z)   = 8ρ_T c_TΓ(d+1)^2  ∫_0^1 d u K_d (u,z) ∫_S_0^∞d S S^3  Im δ(S,L(u) )+…,
where the ellipses refer to the contribution not captured by the phase shift formula.

Next, we consider the contribution of a single-trace Regge pole to the phase shift δ(S,L). To this extent, we introduce the single Regge pole model (see <cit.>):

δ_stringy(S,L)    = 1 Niπ^d/2/4∫_-∞^∞dνe^-iπj(ν)/2/sinπj(ν)/2r(Δ(j(ν)),j(ν))j'(ν)S^j(ν)-1H_iν+d/2-1(L), 

r(Δ,J)   = λ_TTO_Δ,Jλ_O_sO_sO_Δ,JK_Δ,J,
where, as throughout this paper, we only consider the simplest diagonal in the polarization phase shift. Notice that at finite λ other tensor structures are expected to appear, see e.g. <cit.>, and it would be interesting to analyze the phase shift including them. This however goes beyond the scope of the discussion in the present section, which we expect to be qualitatively the same in the presence of these extra tensor structures. For strongly coupled theories the expected leading correction to the Regge intercept takes the form

j(ν) ≃2-2ν^2+(d/2)^2/Δ_gap^2  ,
In this formula Δ_gap encodes the mass of the lightest higher-spin operator so that j(i Δ_gap) ≃ 4. In (<ref>) λ's are the OPE coefficients and K_Δ,J is a ratio of gamma functions and can be found e.g. in Eq. (41) in <cit.>. Moreover we have used the symmetry ν→-ν to write it in terms of the propagator H_iν+d/2-1(L).

Performing the integral over S in (<ref>) produces a factor -S_0^j(ν)-3 j(ν)-3. Let us now close the ν-contour. We are interested in the contributions from the pole j(ν)=3. Computing the residue, we see that the dependence on S_0 drops out and we get
∫_- ∞^∞d ν( -S_0^j(ν)-3 j(ν)-3 ) j'(ν) = 2 πi  |_j(ν_*)=3  . 
In other words, the spin-3 operator on the single-trace Regge trajectory contributes. Extra poles coming from e^-iπ j(ν)/2/sinπj(ν)/2 do not contribute to the imaginary part of the phase shift and therefore we can neglect them. 

Let us now discuss the u integral. On the pole above we are left with an integral of the following form

R_Δ(z) ≡∫_0^1 du K_d (u,z) H_Δ- 1(L(u)) .
This integral is computed in (<ref>) and the result is that up to analytic in z terms it reproduces precisely the celestial block that appeared in the light-ray OPE <cit.>
R_Δ(z) ∼f_Δ(z) + (analytic in  z),
where the second term describes the contribution which is analytic around z=0 and can be thought of as the contribution of the double-trace operators to the light-ray OPE. Moreover, R_Δ(z) is regular around z=1 as opposed to f_Δ(z). Another interesting difference is the behavior when Δ = 2 d - 1 + 2 n in which case f_Δ(z) becomes analytic in z, whereas R_Δ(z) ∼log z.

On the energy-energy correlator side there is no reason to expect that the result is analytic around z=1. The opposite is true, and there is a finite coupling formula for the expected behavior of the energy-energy correlator close to z=1 in the planar N=4 SYM <cit.>. In the detector×source OPE channel these arise from twist-two operators as opposed to the double twist operators studied in the present paper. In this regime, the correlator is controlled by the light-like Wilson loop and the detectors are localized close to the light cone of the source far from the endpoints for which we expect the phase shift formula to work.

To summarize, we get that the Regge pole expansion of the phase shift produces the following contribution to the energy-energy correlator

F(z)  = ∑_i, single-trace  λ̃_i R_Δ_i(3)(z) + ...  .
Comparing this formula to the exact result (<ref>), we see that the phase shift formula correctly captures terms which are non-analytic around z=0, but misses some analytic terms. In particular, it does not capture the z=1 non-analyticity as discussed above.  



 §.§ Non-planar correlator

Let us consider next non-planar correlators for which we expect the high-energy enhanced terms to be correctly captured by our phase shift formula. A good toy model for the unitarization discussed in the present paper is the following integral
I(J) =∫_1^∞dS S^J sin^2 S Πc_T.  
This integral is analytic for Re  J > 1. We can rewrite it as follows

I(J) =-12sinπJ 2 Γ(1-J) (2 Πc_T)^J-1 -∫_0^1 dS S^J sin^2 S Πc_T .
The second term generates the familiar 1/c_T^n perturbative expansion with poles at J=3,5,7,…. The first term is a `background' integral, which cancels these poles. 

It is also instructive to write the inverse of the integral above, namely 
θ(S>1) sin^2 S Πc_T = ∫_1- i ∞^1+ i ∞ d J_L 2 πi S^J_L - 1 I(J_L) ,
where the contour traverses the singularity at J=1 from the right.
For S<1 we can close the contour to the right and we get zero. For S>1, we close the contour to the left in the second term of (<ref>) and to the right in the first term, reproducing the LHS.

We can now model Re(e^i δ-1) by the formula above. Doing the integral over S we get the following representation for the energy-energy correlator

F(z)   = 8ρ_T c_TΓ(d+1)^2  ∫_0^1 d u K_d (u,z) 

   ∫_1- i ∞^1+ i ∞ d J_L 2 πi 12sinπJ_L 2 Γ(1-J_L) (2 Π(L(u)) c_T)^J_L-1 1 J_L-3  .
To perform the u integral we can use the following formula
( Π(L) )^J_L - 1 = ∑_n=0^∞c_n(J_L) H_[(J_L-1)(d-1)+2n+1]-1(L) ,
where the c_n's can be iteratively found by expanding both sides at large L.

We can now perform the u integral using (<ref>) to finally get

F(z)= 4ρ_T c_TΓ(d+1)^2 ∑_n=0^∞   ∫_1- i ∞^1+ i ∞ d J_L 2 πi sinπJ_L 2 Γ(1-J_L) (2 c_T)^J_L-1 

   ×c_n(J_L) R_[(J_L-1)(d-1)+2n+1]-1(z) J_L-3  .
As expected we get a finite result at finite c_T. If we now however try to perform the large c_T expansion and close the contour to the right, we see that we get a double pole at J_L=3 whose residue contains the term

F(z)= 8ρ_T Γ(d+1)^2 logc_T c_T∑_n=0^∞c_n(3) R_[2(d-1)+2n+1]-1(z)  ,
which is the focus of the present paper. 

Let us consider (<ref>) explicitly in d=4. Decomposing the square of the phase shift in terms of H_(7+2n)-1(L) we find

(Π_d=4^(GR)(L))^2 = 3200π^2∑_n=0^∞H_(7+2n)-1(L).
Combining the decomposition (<ref>) together with (<ref>) we reproduce the expected result. While analytic terms in z get contribution from all n in (<ref>), the non-analytic terms z^klog z are only sensitive to terms in (<ref>) with n≤ k.  

In a related context, the representation of the correlator in the Regge limit as an integral over complex spins is very natural when discussing the unitarization of gravitational scattering. It was discussed in the context of the conformal partial wave expansion in <cit.>. 



§ CORRECTIONS

In this section we discuss various corrections to the results obtained in the previous sections. Let us recapitulate its basic logic. We consider the energy-energy correlator at strong coupling. 
It is convenient to write it using the source×detector OPE channel. The leading non-planar correction, as captured by the one-loop computation in gravity, is encoded in the contribution of the double-trace operators to the OPE. They, however, produce a divergent answer at order 1/c_T. This divergence is an artifact of the perturbative expansion and can be regularized by `unitarizing' (or eikonalizing <cit.>) the anomalous dimensions of the double trace operators, which effectively changes γ_tree^2 →sin^2 πγ_tree 2. This produces a finite result computed in the paper (up to additional subtleties related to scattering at small impact parameters).

In this section, we discuss various corrections to this simple universal picture. From the point of view of the OPE, it is related to the presence of infinitely many other operators in the OPE, both single- and multi-traces. Our goal here is to provide some rationale and physical intuition for why we believe that the computation done in the previous sections does capture the leading quantum-gravity correction to the energy-energy correlator.

Let us start by recalling the general OPE formula

F(z)    = Re ∑_Δ, J |λ_Δ,J|^2 |λ_Δ,J^GFF|^2 (1 - e^- i πγ_Δ,J ) ( |λ_Δ,J^GFF|^2 P_Δ,J(z) ).
This paper concerns with the high-energy or large Δ part of the sum — the only part that could produce log c_T enhancement in the energy-energy correlator. The reason is that it is that part that corresponds to high-energy scattering in the bulk and it is that part that produces the contribution which gets enhanced. The behavior of the CFT three-point functions at large Δ is universal on average in any CFT <cit.>, as it is responsible to correctly reproducing the unit operator in the dual channel. In other words, we have |λ_Δ,J|^2  |λ_Δ,J^GFF|^2=1, where the averaging can be made very precise. The behavior of the phase, on the other hand, is non-universal and depends on the nature of the Regge limit of the four-point function, see e.g. <cit.> and the discussion of opaque versus transparent theories there. The phase shift description should capture the large (Δ,J) behavior of the sum on average: ( |λ_Δ,J^GFF|^2  P_Δ,J(z) ) becomes the kinematical kernel against which the phase shift is integrated; |λ_Δ,J|^2  |λ_Δ,J^GFF|^2 is related to mixing between the particle dual to the source and other degrees of freedom; e^- i πγ_Δ,J becomes the phase shift itself. We then use the phase shift description to characterize the large Δ behavior of the OPE sum.

Our basic conclusion is that extra contributions produce subleading terms compared to the log c_T  c_T term computed in the previous sections, up to polynomial in z ambiguities due to zero impact parameter scattering. 
Recall the relationship between S and u=tanh L and the quantum numbers of operators that appear in the source×detector OPE channel

S    =Δ^2 - J^2  ,    coshL = Δ^2+J^2 Δ^2 - J^2  .
In the following sections, we discuss the contribution from various regions of (Δ,J), or using (<ref>), (S,L).





 §.§ Scattering at small impact parameters: fixed spin operators


From the formulas above, we immediately see that the contribution of fixed J_0 and arbitrary Δ operators to the OPE corresponds to the zero impact parameter scattering in the limit S →∞. A slightly more refined statement using the formulas above is that S L^2 ≃ 4 J_0^2 as we take S ≫ 1. The phase shift discussion in terms of the impact parameters only makes sense when the Compton wavelength is much smaller than the impact parameter λ_C ∼1 √(S)≪ L which translates to J_0 ≫ 1.

The contribution of finite J_0 operators to the energy-energy correlator is given by a degree J_0 polynomial in z. Without having an explicit UV completion of a given gravitational theory, we do not have any control over such contributions. Therefore, the S L^2 ≲ 1 small impact parameter region of the (S,L) plane is UV sensitive, leading to polynomial in z ambiguity of our results, which we cannot control. At leading order in 1/c_T this ambiguity is related to the one-loop renormalization of the four-point correlator ambiguity which is fixed in the complete theory. We mark this region by I (brown) in fig:regions.

In holographic theories, such terms arise from polynomial contributions to the Mellin amplitude underlying the four-point function. They have been explicitly computed using localization in both string/M-theory UV completions of gravity, see e.g.  <cit.>. 


At high enough energies we expect black hole formation to control the small impact parameter scattering <cit.>. We can model this process by a simple black disc model such that Re (e^i δ(S,L)-1) = - 1 for L<L_Sch(S) ∼log√(S) c_T, where L_Sch(S) is the Schwarzschild radius in AdS, see e.g. <cit.>. Plugging this into (<ref>), we get the following formula for the contribution of black hole production at order 1  c_T^3 

F_BH(z) = 
8ρ_T c_TΓ(d+1)^2∫_0^1 d u K_d (u,z) ∫_c_T^2 e^2 (d-1) L^∞dS S^3 (-1) ∼z^d-1 logz c_T^3 ,
where we only kept the leading non-analyticity at small z. It would be interesting to understand if this contribution can be unambiguously isolated from other effects. 



 §.§ Scattering at low energies: fixed twist operators


Next, we can consider scattering of fixed twist and possibly large spin operators. In terms of (S,L) we have
τ≡Δ- J = √(S) e^-L/2 . 
The contribution of such operators can be computed by simply expanding the OPE data in 1  c_T and in agreement with the usual perturbation theory of local operators, we will get only integer powers of 1  c_T^n. 
To complete the argument we need to make sure that summing over spins these perturbatively expanded results does not produce a divergence. To leading order in 1/c_T relevant for this paper, the operators contributing are the double-twist operators for which one can use light-cone bootstrap techniques <cit.> to analyze the large spin tails explicitly. In this case, the sum over spins corresponds to scattering at very large impact parameters, and while it can produce enhanced terms in the energy-energy correlator at small z after integration against our kernel, its c_T dependence is not affected. We expect that the same statement applies to the multi-twist operators at higher orders in 1/c_T^n>1, and it would be interesting to check it explicitly using the higher-point version of the light-cone bootstrap <cit.>, see also <cit.>. Therefore, we conclude that operators with a bounded twist cannot produce an enhanced log c_T/c_T contribution and instead produce a subleading contribution, which is O(1/c_T). We mark this region by II (yellow) in fig:regions.



 §.§ Non-phase shift contributions: outlier Regge trajectories


In the picture above we assumed that the large (Δ,J) limit of the OPE sum that controls the correlator is smooth and exists, an assumption that we certainly cannot rigorously justify. If we imagine an isolated family of operators, which is not part of this smooth limit our formulas will certainly miss it. We can call such contributions non-phase shift contributions. To be concrete, we can imagine the contribution of the twist-two operators to the sum above which are responsible for non-analytic behavior of the energy-energy correlator around z=1.[In <cit.>, it has been shown that the discontinuity of energy-energy correlator in the z → 1 limit corresponds to the double lightcone limit of the underlying four-point function of local operators, which is controlled by the large spin tail of the twist-2 operators in the OPE.
] 
As we discussed in the previous section, we do not expect it to be captured by the approximation used to derive the phase shift formula. We however do not expect such contributions to be enhanced, and instead follow the usual 1/c_T perturbative pattern. For that reason we expect them to be sub-leading compared to the enhanced contribution analyzed in the present paper.



 §.§ Planckian energies


We would like next to understand the contribution from Planckian energies S ≤ S_eik(ϵ, L), where we define S_eik(ϵ, L) through the following relation
δ_tree( S_eik,L) = ϵ,
where ϵ is a fixed number which we can also take to be small;  δ_tree(S,L) is the tree-level phase shift ∼1  c_T. We mark this region by III (blue) in fig:regions.

Here comes our main assumption: in the regime where S L^2 ≫ 1 and energies such that δ_tree(S,L) ≤ϵ≪ 1 the full phase shift can be bounded as

| δ(S,L) | ≤c_0 δ_tree(S,L) ,
where c_0 is an O(1) numerical coefficient that does not depend on S and L. Physically, we think of the condition (<ref>) as the statement that as long as the tree-level gravitational interaction is very weak δ_tree(S,L) ≤ϵ≪ 1 quantum-gravity corrections stay small as well. 

Let us first discuss this statement in D-dimensional asymptotically flat spacetime.[It was used in <cit.> to derive an analogue of the Froissart-Martin bound on the growth of the amplitude with energy in gravitational theories.] In this case we can rewrite the tree-level phase shift as follows δ_tree(S,L) ∼ (√(S)L) (L_Sch(S)  L)^D-3, where we used that in flat space the Schwarzschild radius L_Sch(S)^D-3∼ G_N √(S). The theory is expected to become strongly coupled for the collisions with L ∼ L_Sch(S) when a black hole is expected to form <cit.>. Given that S L^2 ≫ 1 and that we keep δ_tree(S,L) ≤ϵ≪ 1, we see that the scattering takes place at L ≫ L_Sch(S), therefore we expect it to be weakly coupled and well-controlled by the tree-level result.

Consider next the situation in AdS_d+1. The condition for the black hole formation above could have been obtained by imposing that the transferred momentum as measured by ∂_L δ_tree(S,L) becomes ∼√(S). Applying the same condition for L ≫ 1 we get that in AdS we expect that the black hole formation corresponds to e^(d-1)L_Sch(S)∼ G_N √(S). Numerical analysis of the formation of a trapped surface in the shock wave collisions in AdS was performed in <cit.>, see also <cit.> for analytic results, and it supports this conclusion, see formula (3.19) in <cit.>. We can therefore rewrite the tree-level phase shift as δ_tree(S,L) ∼ (√(S)L) e^(d-1)(L_Sch(S) - L) and thus our regime of interest is again L ≫ L_Sch(S).

Using the inequality (<ref>), we can estimate

   | ∫_S_0^S_eik(ϵ, L) d S S^3 Re ( 1 + i δ- δ^2 2 - e^i δ) | ≤∑_k=3^∞c_0^k k! ∫_S_0^S_eik(ϵ, L) d S S^3 δ_tree^k (S,L) 

   ≤g(c_0,ϵ) Π_d-1^2(L) c_T^2 ,
where g(c_0,ϵ) can be easily computed explicitly but does not play any role, and we also used the Einstein gravity result for δ_tree(S,L). As a result, the higher-order terms produce non-enhanced 1/c_T contributions to the EEC.

We are thus left with the following terms
∫_S_0^S_eik(ϵ, L) d S S^3 ( Im δ+ 1 2 ( Re δ^2 - Im δ^2   ) ) . 


To discuss the enhanced terms in the integral above we can consider the perturbative expansion of the phase shift
Re δ(S,L) = δ_tree(S,L) + ∑_i=2^∞∑_k = 0^∞(S c_T )^i (logS)^k c_i,k(L) ,
where we only kept the corrections that could produce the enhanced terms, namely ∼ c_i,k(L) log^k c_T  c_T^2. Notice, however, that the terms with k>0 necessarily violate our basic assumption (<ref>). 

Similarly, we get for the imaginary part that the only term that is consistent with (<ref>) that can produce the enhanced contribution is 
Im δ(S,L) = (S c_T )^2 c̃(L) .
From the bulk diagrammatics, we only expect the imaginary part of the phase shift to start at two loops in the bulk or, equivalently, starting from 1  c_T^3, see fig:hDiagram for an example of the H-diagram that produces the leading gravitational waves production effect. Therefore c̃(L)=0. We do not expect the stringy corrections to change this basic picture, see also the discussion below.

Let us comment also on the expected form of the higher-order corrections that do appear in the expansion of the phase shift. The simplest one is related to the emission of gravity waves, see fig:hDiagram. This effect has been studied in flat space and we expect that the G_N and S dependence in AdS are the same and are given by Imδ(S,L) ∼ c_T^-3 S^2 log S, see also formula (7.5) in <cit.>. A similar diagram has been discussed in <cit.>.
This contribution to the phase shift would produce terms ∼log^2 c_T c_T^2 in the expansion of the energy-energy correlator, which is subleading compared to the ones studied in the present paper. It would be very interesting to understand them in greater detail.



We therefore conclude that only δ_tree^2(S,L) used in the bulk of the paper produces the enhanced log c_T  c_T contribution at intermediate energies.

A careful reader might have noticed the following interesting subtlety of the argument above which relied on setting δ_tree(S,L)=δ_GR(S,L). If we plug the estimate (<ref>) into the expression for the energy-energy correlator we will get a divergence coming from small impact parameters for d>4. In the present paper, we address this issue by considering δ_tree(S,L)=δ_stringy(S,L), and leave the exploration of more general scenarios to the future. As we will see explicitly below, the stringy phase shift stays finite at small impact parameters and therefore this problem does not arise.



 §.§ Trans-Planckian energies


Finally, we need to address the case of high-energy scattering at trans-Planckian energies S > S_eik(ϵ, L). We mark this region by IV (red) in fig:regions. To estimate this contribution we will use AdS scattering unitarity <cit.>, which states that Imδ(S,L) ≥ 0, and therefore  | 1 - e^i δ | ≤ 2.

We then get

| ∫_S_eik(ϵ, L)^∞d S S^3 Re ( 1 - e^i δ) | ≤2 ∫_S_eik(ϵ, L)^∞d S S^3 = Π_d-1^2(L) ϵ^2 c_T^2,
where we used δ_tree(S,L)=δ_GR(S,L). When combined with the pre-factor it produces a 1  c_T contribution to the energy-energy correlator. Notice that in performing the exercise above we assumed that there is no mixing between the graviton and other possible eigenstates of the eikonal operator. Including mixing, however, does not affect the estimate above. 

Another subtlety that was already discussed in the previous section is that the integral over impact parameters of the estimate above diverge in d>4. As before, in this paper we assume that the small impact parameter physics in this case can be regularized using the stringy effects, or by taking δ_tree(S,L)=δ_stringy(S,L), which we describe next.



 §.§ Stringy corrections


In this section, we consider the corrections due to stringy effects. It also helps to understand the role of strong coupling in our discussion. 

At large impact parameters the phase shift (<ref>) matches with the tree-level gravity phase shift used in the previous sections

δ(S,L) = δ^(GR)(S,L) + ... ,      L ≫1 . 
When L ∼log S Δ_gap^2, the integral develops a saddle point at ν_* = - i L  4 log S Δ_gap^2. The saddle point is a good approximation when log S Δ_gap^2≫ 1. To reproduce the behavior (<ref>) at large L we deform the ν contour into the lower half-plane and pick the stress tensor pole j(ν)=2. This is always the correct asymptotic at large L (independently of the validity of the saddle point approximation and the value of log S Δ_gap^2).

The stringy phase shift, however, has a very different small impact parameter asymptotic. To see this, we can expand

H_iν+d/2-1(L(u))    = (2^2-d π^1-d/2 u^3-d Γ(d-3)/Γ(d/2-1)+…)   +2^1-d π^1-d/2 Γ(3/2-d/2) Γ(d/2+i ν-1)/Γ(-d/2+i ν+2)+ O(u^2), 
where the first line contains all the singular terms and they are even in ν→-ν. These integrate to zero because the rest of the integrand is odd. The regularity of the phase shift (<ref>) can equivalently be seen by writing it in terms of Ω_iν(L)=iν/2π( H_iν+d/2-1(L)- H_-iν+d/2-1(L)) which is regular as L→ 0:

Ω_iν(L)=   Γ(d-2/2+iν)Γ(d-2/2-iν)ν/2^d-1 π^d+1/2 Γ(d-1/2)sinh(πν)[1-L^2ν^2+(d-2)^2/4/2(d-1)+…],
at small L. Below we further need the large ν limit of (<ref>) which is given by

Γ(d-2/2+iν)Γ(d-2/2-iν)νsinh(πν)∼ν^d-2.
Inserting the small L-behaviour (<ref>) in (<ref>), together with (<ref>), we get 

δ(S,L(u))   ∼r(d,2)S^1-d^2/2Δ^2_gap∫_0^∞ dνe^-logS/Δ_gap^2ν^2ν^d-4(1+# ν^2u^2+…)   ∼r(d,2)S^1-2(d 2)^2 Δ_gap^2 ( Δ_gap^2 logS )^d-3 2 (1+ #' Δ_gap^2logSu^2 + ... )
where we used Δ_gap^2≫ν^2≫ 1 and r(Δ(j(ν)),j(ν))≈ r(d,2). The latter condition was discussed in <cit.> and follows from assuming that r(ν) has a well-defined limit as Δ_gap^2→∞ keeping ν^2/Δ_gap^2 fixed such that 
r(ν) = r(0)+∑_n=1^∞ f_n(ν^2)/Δ_gap^2n,
where r(ν)≡ r(Δ(j(ν)),j(ν)) and f_n(ν^2) is a polynomial with a maximum degree of 2n and f_n(0)=0. To leading order in Δ_gap^-2 this is determined by the stress tensor r(0)≈ r(d,2). 

We now want to compute the leading enhanced correction to the energy-energy correlator using the phase shift (<ref>).
The basic effect is that after doing the integral over S, the phase shift is evaluated at energies S ∼ c_T. By comparing the small L expansion (<ref>) to the large L expansion (<ref>), we find that the interpolation between two different regimes takes place at a scale L_* defined as follows
L_*^2 ≡logc_T Δ_gap^2 . 
When L_* ≪ 1, stringy corrections become important only for u ≲ u_* ≪ 1. On the other hand, if L_* ≫ 1, stringy corrections dominate for u≲ 1-e^-16L_*^2 in d=4. The transition between the stringy phase shift and the pure gravity phase shift as a function of L_* can be seen explicitly in d=4 by evaluating the integral over ν; this is done in Appendix <ref>.

We consider now L_*≪ 1. In this case we see that the naive IR divergence is substituted by the following integral when we include the string corrections
   logc_T c_T L_*^2(3-d)∫_0^L_* d u K_d (u,z) ( c_0 + c_1 1 L_* u^2 + ... )  

   =(L_*^5-d Poly_2(z) + L_*^7-dPoly_3(z) + ... ),
where c_i ∼ O(1) are coefficients and Poly_n(z) are polynomials in z of degree n. 

If we approach the stringy regime from the large L direction we get the same estimate for the contribution to the energy-energy correlator

∫_L_*^1 d u K_d (u,z) ⟨Π^2 (L(u)) ⟩= (L_*^5-d Poly_2(z) + L_*^7-dPoly_3(z) + ... ) ,
where here in d=5 we get log L_*.

Applying this prescription to the N=4 SYM computation, using (<ref>) for ⟨Π^2 (L(u)) ⟩, we get the following enhanced terms

   logc_T c_T ( L_*^-4 (1-6z + 6 z^2) + L_*^-2 Poly_3(z)+Poly_4(z)  . 

   . + logL_* (1-36 z +216 z^2 -400 z^3 + 225 z^4) )    ,     L_* ≪1 .
This is precisely the structure that we observed in the ω-regularization of the energy-energy correlator (<ref>). It would be interesting to compute these enhanced polynomial corrections exactly. Comparing the enhancement (<ref>) to that in (<ref>) the presence of the large S^5 is reflected in the leading power of L_* which coincides with d=9 in (<ref>).

The conclusion is that the small impact parameter divergence of the phase shift in d>4 observed in the previous sections is regulated by stringy corrections which leads to enhanced polynomial contributions in the regime L_* ≪ 1. 

Finally, let us comment on the expected contribution from the zero impact parameter physics. The complete form of the one-loop bulk correlator, including polynomial terms in the Mellin amplitude, in N=4 SYM was computed using localization in <cit.>. It includes a constant term √(λ) c_T which assuming stringy unitarization leads to an additional term λ c_T(1-6 z+6 z^2) in the energy-energy correlator. In the language of the formulas above it corresponds to the contribution Δ_gap^4  c_T∼(log c_T)^2  c_T L_*^-4. We notice that at this order, the zero impact parameter contribution in string theory is more enhanced compared to the non-zero impact parameter contribution computed above.



 §.§ Stringy unitarization


Consider next the case L_* ≫ 1 and fixed. In this regime, the tree-level gravitational computation used in the previous sections is still valid at large impact parameters L ≳ L_*^2, which means that the non-analytic part of the answer ∼log z is not affected. However, the part which is analytic in z changes. 

To see what happens more explicitly, we plug the stringy phase shift into the integral ∫d S  S^3δ^2(S,L) to get

∫_S_0^S_eik(ϵ,L) d S S^3 S^j(ν) + j(ν')-2 = (S_eik(ϵ,L) )^j(ν) + j(ν')-4 - S_0^j(ν) + j(ν')-4 j(ν) + j(ν')-4 ,
where recall that S_eik(ϵ,L) ∼ c_T.

One can easily check that the computations of the previous sections arise from considering (<ref>) and evaluating the ν and ν' integral by taking the residue at the graviton j(ν)=j(ν')=2 pole. The factor 1  j(ν) + j(ν')-4 which is singular when j(ν) = j(ν') = 2 is responsible for an extra factor of log c_T.

To understand the analytic in z part for L_* ≫ 1, we can simply take L_* →∞. In this case, the ν and ν' integrals in the first term are effectively localized close to the origin. As a result, we get an extra suppression factor S_eik^j(0)∼ e^- 1  2d^2 L_*^2→ 0 for the first term in (<ref>). The main contribution therefore comes from the term - S_0^j(ν) + j(ν')-4 which is again dominated by the graviton pole and produces  Δ_gap^2  c_T contributions to the energy-energy correlator. Effectively the enhancement by the Δ_gap^2 factor happens because in this case the unitarization happens at the stringy energy scale log S_0∼Δ_gap^2.[This is the energy at fixed L at which the saddle point ν_* approach the graviton pole.] 





§ DISCUSSION


Unitarization of high-energy gravitational scattering requires all-loop re-summation. In flat space, unitarity of the four-point amplitude implies that at finite G_N, all partial waves must satisfy |S_J| ≤ 1: the scattering probability cannot be larger than one. In a unitary CFT, a somewhat analogous statement is that the nonperturbative Regge intercept of the four-point correlator is J_np≤ 1 <cit.>. This statement contrasts with perturbative gravity computations in AdS, which exhibit a Regge intercept that grows with the number of gravitational loops: J_tree=2, J_1-loop=3,… .[Here we only talk about the part which is non-local in the impact parameter space. Contact terms exhibit an even larger Regge intercept.] The same phenomenon also leads to a rich structure of poles at integer J>1 in the conformal partial waves and the OPE data, see e.g. <cit.>. Understanding in detail how these singularities disappear at finite G_N and the nonperturbative Regge bound is satisfied is an interesting open problem. 

In this paper, we explored imprints of unitarization of high-energy gravitational scattering in the energy-energy correlator at strong coupling. For the leading quantum-gravity correction, this in particular amounts to understanding the finite-c_T resolution of the universal 1  c_T# J-3 pole in the one-loop gravity OPE data of double trace operators <cit.>. We restated this problem in the language of high-energy fixed impact parameter scattering in AdS <cit.>. We then argued that the eikonalization of the tree-level gravitational scattering removes the pole and leads to an enhanced contribution to the energy-energy correlator ∼log c_T  c_T. We also showed that for D-dimensional bulk theories with D>5, there is an additional enhancement coming from small impact parameter scattering which we modeled using stringy effects. We computed the log c_T  c_T correction to the energy-energy correlator explicitly in various examples. 

The result for the log c_T  c_T correction to the energy-energy correlator encodes the bulk geometry via the dependence of the phase shift δ(S,L) on the impact parameter L, see (<ref>) and  (<ref>). The leading small angle contribution log c_T log z/c_T is universal (<ref>), whereas higher-order terms encode the full bulk geometry, including the internal dimensions. We have explicitly computed the log c_T/c_T correction to the energy-energy correlator in momentum eigenstates of pure Einstein gravity in AdS_d+1, see (<ref>), and N=4 SYM dual to AdS_5× S^5 (<ref>). In the latter case, the graviton mixes with KK modes at high energies, and we used the results of <cit.> to identify the eigenstates of the eikonal operator. More generally, we have observed that non-local scattering in AdS (scattering at large impact parameter) maps to non-analyticity in the angle variable z on the energy correlation side.

An important part of our argument is the analysis of various corrections in Section <ref> based on the source× detector OPE channel. While eikonalization (or exponentiation of anomalous dimensions) is completely automatic in this language, we used gravity-based arguments to estimate possible corrections to our result. In particular, we assumed that when the tree-level interaction is weak and the impact parameter is much larger than the Schwarzschild radius, loop corrections to the tree-level phase shift stay small, see (<ref>) for the precise statement. It would be, of course, much better to have a rigorous CFT proof of this fact.  Our analysis is blind to UV-sensitive zero impact parameter terms in the bulk scattering amplitude. These cannot be computed on general grounds and lead to an extra polynomial contribution to our results.[For example, in N=4 SYM it leads to an extra correction ∼λ c_T(1-6z+6z^2) to the energy-energy correlator.] While most of the paper focused on computations in pure gravity, in Section <ref> we also discussed the effect of stringy corrections on our discussion. We modeled the phase shift using the full leading Regge trajectory while not changing the simple tensor structure expected in pure general relativity. We have found that the presence of stringy corrections does not modify our discussion when Δ_gap^2/log c_T ≫ 1, where Δ_gap∼ m_HS R_AdS is related to the mass of the lightest spin-four stringy excitation. In addition to that, when the number of bulk dimensions D is larger than 5, we get enhanced by Δ_gap^2/log c_T polynomial corrections in z coming from small impact parameter scattering in the bulk. It would further be interesting to incorporate the additional tensor structures.

In this paper, we focused on `light' states, namely we assumed that their scaling dimension does not scale with c_T. It would be interesting to understand energy-energy correlators in `heavy' states, e.g., dual to black holes, for which Δ∼ c_T. In this case, we can use the Eigenstate Thermalization Hypothesis (ETH) <cit.> in the source× detector channel. The diagonal term is then responsible for the homogeneous distribution of energy, whereas the connected thermal two-point function related to the off-diagonal terms produces corrections on top of it. At high energies of the probe particle, the gravitational phase shift, in this case, encodes the propagation of a particle on top of a classical black hole background <cit.>. It would be interesting to explore this connection in more detail and compute the leading correction explicitly, see <cit.> for a related discussion.

We have also focused on the energy-energy correlators but similar effects should be possible to observe in other event shapes. In general we expect the detectors of spins J_1 and J_2 to probe the ∫^∞d S  S^J_1 + J_2 -1 moment of the phase shift e^i δ-1. In this way different event shapes measure different high-energy moments of the gravitational scattering in AdS. Similarly, the connection between high-energy gravitational scattering and energy correlators should generalize beyond two detectors. In this case, we expect that the multi-point Regge theory recently developed in the CFT context in <cit.> could be useful. The three- and higher-point energy correlators have been recently explored in <cit.>. It would be also interesting to generalize our discussion to the case of holographic gapped theories <cit.>.

Finally, it would be very interesting to explore the connection between energy correlators and high-energy AdS scattering using non-perturbative bootstrap methods <cit.>. For example, if we schematically write
E(n_1) E(n_2)  = (p^0/Ω_d-1 )^2 [ 1 + enhanced(z) + ... ],
one can wonder if the average null energy condition E(n_1)  E(n_2)  ≥ 0 imposes interesting bounds on the enhanced term and the underlying theory.[See <cit.> for related work.] Indeed, since the enhanced correction to 1 satisfies the momentum conservation Ward identities (which state that it integrates to zero with a pair of non-negative kernels, see (<ref>)), it has to become negative somewhere, and therefore by making the enhancement big enough we risk violating the ANEC condition. For example, in the context of the present paper, this argument suggests the existence of a bound Δ_gap^# c_T≲ 1.[See also a discussion in <cit.>.] To the best of our knowledge the condition E(n_1)  E(n_2)  ≥ 0 has not yet been explored in the bootstrap studies of the CFT correlators of stress tensors, see e.g. <cit.>. We leave it here to the reader's imagination and future work.





§ ACKNOWLEDGMENTS

We thank Alex Belin, James Drummond, Gregory Korchemsky, Petr Kravchuk, Vasco Goncalves, Andrei Parnachev, Hynek Paul, Joao Penedones, Emery Sokatchev, Gabriele Veneziano and Zahra Zahraee for helpful discussions. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement number 949077). 



§ CONVENTIONS

The vacuum stress-tensor two-point function in position space is given by 
⟨T_μν(x)T_ρσ(0)⟩_vac    =c_T/Ω_d^21/ x^2d(1/2!(I_μρ(x)I_νσ(x)+I_μσ(x)I_νρ(x))-1/dη_μνη_ρσ)
 
I_μν(x)   =η_μν-2x_μx_ν/x^2,
where Ω_d=2π^d/2/Γ(d/2) is the volume of a (d-1)-dimensional unit sphere. Performing the Fourier transform of the expression above, we get for the Wightman function
⟨T_μν(q)T_ρσ(-q)⟩= ρ_Tc_Tθ(q^0)θ(-q^2)(-q^2)^d/2Π_μν,ρσ(q),
where Π_μν,ρσ(q) is given in terms of π_μν(q)=η_μν-q_μ q_ν/q^2 as follows
Π_μν,ρσ(q) = 1/2(π_μρπ_νσ+π_μσπ_νρ)-1/d-1π_μνπ_ρσ
and 
ρ_T = π^d/2+1/2^d-1(d+1)Γ(d-1)Γ(d+2/2)Ω_d^2.


Defining χ=q^2 n_1· n_2/2 (q· n_1)(q· n_2) we have n_1^μπ_μν n^ν_2=n_1· n_2/2χ(2χ-1) and n^μ_iπ_μνn^ν_i=-(q· n_i)^2/q^2 and therefore

⟨T(q;n_1)T(-q;n_2)⟩=   ρ_Tc_Tθ(q^0)θ(-q^2)(q·n_1)^2 (q·n_2)^2 (-q^2)^d-4/2

   ×[4χ^2-4χ+d-2/d-1].




§ POINCARÉ SHIFTS AND CORRELATION FUNCTIONS



Here we review some standard material regarding CFT correlation functions in the Lorentzian signature. Lorentzian correlation functions in conformal field theory are naturally defined on the Lorentzian cylinder ℝ× S^d-1. To get to it, one can start from the usual Euclidean correlators on ℝ^d and map them to the Euclidean cylinder x^μ = e^τ𝐞^μ, with 𝐞·𝐞=1, such that
d s^2 = δ_μνdx^μd x^ν= e^2 τ ( d τ^2 + d𝐞^2) = e^2 τ d s_cyl^2 . 
Under this transformation, we have
⟨O_1(x_1) ... O_n(x_n) ⟩_ℝ^d = ∏_i=1^n e^-Δ_i τ ⟨O_1(τ_1, 𝐞_1) ... O_n(τ_n, 𝐞_n) ⟩_ℝ ×S^d-1 .
We can then perform the Wick rotation τ→ - i t and arrive at the Lorentzian cylinder. 

The Lorentzian cylinder can be thought of as a universal cover of the null cone in ℝ^2,d modded by rescaling ℝ_+
X^2 = -(X^-1)^2 - (X^0)^2 + ∑_i=1^d (X^i)^2 = 0  . 
Indeed, introducing the coordinates
X^-1 = cost,     X^0 = sint,     X^i = 𝐞^i ,     i = 1, .. , d ,
we see that the equation above is satisfied. 

With each point on the Lorentzian cylinder we can associate a Poincaré patch. To achieve it, we split ℝ^2,d = ℝ^1,1×ℝ^1,d-1, where ℝ^1,1 is spanned by a pair of orthonormal null vectors K^A and K̅^A that satisfy K^2=
K̅^2 =0 and -2 K ·K̅ = 1. We then write
X^A = K̅^A + K^A y^2 + y^A ,
where y^A are the usual coordinates in Minkowski space ℝ^1,d-1. Indeed, with this choice X^2 = 0.
The above choice corresponds to the gauge condition -2 X · K = 1 for rescaling of X^A.

For example, consider the gauge condition X^-1+X^d = 1 for which the Minkowski space coordinates are
y^μ= X^μX^-1+X^d ,     μ= 0, ..., d-1.
In terms of the global coordinates this becomes
y^0    = sint cost + 𝐞^d ,     y^i = 𝐞^i cost + 𝐞^d  . 


We can then write the following relationship between the correlation functions in the Minkowski space Poincaré patch and the Lorentzian cylinder
⟨O_1(y_1) ... O_n(y_n) ⟩_ℝ^1,d-1 = ∏_i=1^n (cost_i + 𝐞_i^d)^Δ_i ⟨O_1(t_1, 𝐞_1) ... O_n(t_n, 𝐞_n) ⟩_ℝ ×S^d-1 ,
such that, for example, the two-point function of scalar primary operators takes the form ⟨ O(t_1, 𝐞_1)  O(t_2, 𝐞_2) ⟩ = 2^-Δ(cos(t_1-t_2 - i ϵ ) - 𝐞_1 ·𝐞_2 )^-Δ. 

We next introduce the operation T of translating a point one Poincaré patch forward, such that T(t,𝐞) T^-1 = (t+π, -𝐞). In the embedding space coordinates, it is simply X^A → - X^A. Applying this to the two-point function, we immediately get
e^i πΔ⟨O(t_1+π, -𝐞_1) O(t_2, 𝐞_2) ⟩=e^-i πΔ ⟨O(t_1, 𝐞_1) O(t_2+π, -𝐞_2) ⟩=⟨O(t_1, 𝐞_1) O(t_2, 𝐞_2) ⟩.
In other words, translating a point one Poincaré patch forward produces a trivial overall phase. 

Consider next four points on the Lorentzian cylinder, out of which we can build a cross-ratio d_i j^2 d_kl^2 
 d_i k^2 d_jl^2. If we now translate a given point one Poincaré patch forward, for example, i, it sends d_ij^2 → - d_ij^2, d_ik^2 → - d_ik^2 leaving the cross-ratio invariant. This, however, does not yet mean that the monodromy of this transformation is trivial. It could be that the cross-ratio has been continued around the origin in a nontrivial way. To avoid this let us consider an operator that acts on the in- or out-vacuum inside the Wightman function. In this case, all the distances have the same i 
ϵ prescription, e.g. t_i - i ϵ if we have ⟨Ω|  O(t_i,𝐞_i) .... Therefore, the phase acquired by the numerator and the denominator of the cross-ratio precisely cancel so that the monodromy of the cross-ratio is trivial. 

We, therefore, conclude that Poincaré translation applied to operators acting on the vacuum acts trivially on cross-ratios. To see its overall effect, we need only to check its effect on the pre-factor and possibly tensor structures. We already have seen above that it produces a nontrivial phase when acting on the two-point function. 

Let us next consider the four-point function of symmetric tensor operators relevant for this paper. We want to act with the Poincaré translation on a spin J operator. In the embedding space, it acts on the polarization tensor as Z^A → - Z^A. The general form of the correlator was considered in <cit.> and it involves the basic tensor structures H_ij and V_i,jk. These structures are invariant under (X_i,Z_i) → -(X_i, Z_i). Taking into account the pre-factor, we therefore conclude that parity-even tensor conformal structures transform as

⟨Ω| O_J(-X,-Z) ...    = e^- i π(Δ+J) ⟨Ω| O_J(X,Z) ... , 

... O_J(-X,-Z) | Ω⟩   =e^i π(Δ+J) O_J(X,Z) | Ω⟩ . 


Including the parity odd structures containing the ϵ-tensor slightly changes this conclusion. Consider, for example, the following parity-odd tensor structure in d=3 ϵ(Z_1, Z_2, X_1, X_2, X_3).[For a detailed discussion of the embedding formalism in this context, see <cit.>.] This structure is odd under the T-transformation Z_i → - X_i, X_i → - X_i. More generally, it is clear that in an odd number of dimensions, parity-odd conformal structures that involve the ϵ-tensor acquire an additional minus sign under this transformation. In an even number of dimensions (<ref>) holds.



§ DETAILS ON THE KERNEL K_D(U,Z)


In this Appendix, we provide details and useful expressions for the kernel K_d(u,z). The definition of the kernel 𝒦_d(u,z) is
𝒦_d(u,z) ≡∫_M^+ d^dq (2π)^d G_0(q;n_1,n_2)(q·n_1)^d+1(q·n_2)^d+1 δ(1-√(-q^2)√(-p^2)) δ(u - √(1-p^2 q^2 (p·q)^2)) , 
where G_0(q;n_1,n_2) is the vacuum stress-tensor two-point function

		G_0(q;n_1,n_2)	   =(q ·n_1)^2 (q ·n_2)^2 (-q^2)^d-4 2 ( 4χ^2-4χ+d-2/d-1 ) .
To eliminate the delta functions in (<ref>), we use the Baikov representation <cit.> to rewrite the integral measure as
∫_M^+ d^dq  =Ω_d-3 2  ∫_R d(q·n_1) d(q·n_2) d(q·p) d(q^2) [-G(q,n_1,n_2,p)]^d-5 4 [-G(n_1,n_2,p)]^d-4 2
where G(k_1,⋯, k_n) is the Gram determinant (k_i· k_j) and the integral region ℛ is defined by the algebraic constraint G(q,n_1,n_2,p)<0.
As a consequence, the kernel 𝒦_d(u,z) can be decomposed into
𝒦_d(u,z)=1 (- p^2)^d-2 2Ω_d-3 (2π)^d   u (1-u^2)^-3/2 [-G(n_1,n_2,p)]^d-4 2[(n_1·n_2)^2 (-p^2)^2 I^d-52_d+1,d+1

   +2(n_1·n_2) -p^2 I^d-52_d,d+ d-2 d-1 I^d-52_d-1,d-1],
by introducing the integral family
I^γ_α_1, α_2 = ∫_R d(q·n_1) d(q·n_2) [-G(q,n_1,n_2,p)]^γ(-q·n_1)^α_1 (-q·n_2)^α_2.


The Gram determinants are

    G(n_1,n_2, p)    =    -4z(1-z) (p · n_1)^2 (p · n_2)^2  -p^2 ,
    
    G(q,n_1,n_2,p)    =   (p · n_1)^2 (p · n_2)^2  p^4      (y_1^2+y_2^2-2(1-2z) y_1 y_2 + 4 z √(1-u^2)(y_1+y_2)+4z(1 + u^2 z  1-u^2) )  ,

where y_i=p^2  (p · n_i) q· n_i. For convenience, we can choose a special configuration for scalar products
p^2 = p·n_1 = p·n_2=-1 ,    n_1·n_2 = -(1-cosθ)=-2 z .
The full dependence is then easy to restore using the dimensionality and the transformation properties of the object. 

We map the integration region to a unit disk using the change of variables:
y_1 = √(u^2 1-u^2) x_1 - 1 √(1-u^2) ,  y_2 = √(u^2 1-u^2) (cosθx_1 + sinθx_2) - 1 √(1-u^2) ,
followed by the polar coordinate parametrization x_1+i x_2 = r e^iϕ. The integral I^γ_α_1,α_2 then becomes
I^γ_α_1, α_2 =    [4z(1-z)]^γ+1/2 (u^2 1-u^2)^γ+1-α_1+α_2 2

       ×∫_0^1 r dr ∫_0^2πdϕ(1-r^2)^γ(1u-rcosϕ)^α_1 (1u- rcos(ϕ-θ))^α_2 ,
which belongs to the same category as the Mandelstam kernel in the massive 2→ 2 scattering process <cit.>

Ĩ_γ(η_1,η_2) = ∫_0^1 r dr ∫_0^2πdϕ(1-r^2)^γ(η_1-rcosϕ) (η_2- rcos(ϕ-θ)).
For α_i taking positive integer values, Ĩ_γ can serve as a generating function through taking derivatives with respect to η_1 and η_2:
I^γ_α_1, α_2 = [4z(1-z)]^γ+1/2 (u^2 1-u^2)^γ+1-α_1+α_2 2 (-1)^α_1+α_2 (α_1-1)!(α_2-1)! Ĩ_γ^(α_1-1, α_2-1)(u^-1, u^-1) .


We can now restore the dependence on p^2 and (p · n_i), which produce the expected factor (-p^2)^d  (-p · n_1)^d-1 (-p · n_2)^d-1. In this way, we get the following expression for the kernel

𝒦_d(u,z)    =Ω_d-3 (2π)^d u (1-u^2)^-3/2  
 
   ×[ 4 z^2  Ĩ^d-52_d+1,d+1(u,z) - 4 z Ĩ^d-52_d,d(u,z) + d-2 d-1 Ĩ^d-52_d-1,d-1(u,z) ] ,
where we defined the integral Ĩ^γ_α_1, α_2 as follows
Ĩ^γ_α_1, α_2(u,z)    = (u^2 1-u^2)^γ+1-α_1+α_2 2∫_0^1 r dr ∫_0^2πdϕ(1-r^2)^γ(1u-rcosϕ)^α_1 (1u- rcos(ϕ-θ))^α_2 ,

   = (u^2 1-u^2)^γ+1-α_1+α_2 2 (-1)^α_1+α_2 (α_1-1)!(α_2-1)! Ĩ_γ^(α_1-1, α_2-1)(u^-1, u^-1) .


The differential representation is useful to calculate the full kernel 𝒦_d(u,z) at small integer-valued d, while the integral representation is easier to study in generic d. One example is the small u expansion, where the expansion and integration commute.
Therefore, at small u the kernel can be expanded as follows

   𝒦_d(u,z) =2 πu^d-2 (d (1-2 z)^2-4 z^2+4 z-2)/(d-3) (d-1) 

   -πu^d (d^2 (4 z-3) (1-2 z)^2+d (32 z^3-36
   z^2+6)+8 z (2 z^2-4 z+1))/(d-3) (d-1) + O(u^d+2)  . 




 §.§ Kernel in four dimensions


Here we provide some explicit detail on the kernel in d=4. The basic ingredient that we need to compute in d=4 is [For the generic case, Ĩ_d-5 2 can be expressed in terms of AppellF1 hypergeometric functions.]
Ĩ_-1/2(η_1, η_2, z) = πlog(-i η_1 η_2+√(-(η_1-η_2)^2-4 z^2+(4-4 η_1 η_2) z)+2 i
   z-i/-i η_1 η_2+√(-(η_1-η_2)^2-4 z^2+(4-4 η_1 η_2) z)-2 i
   z+i)/√((η_1-η_2)^2+4 z^2+4 (η_1 η_2-1) z) . 
After this the kernel can be obtained using (<ref>). 

In this way we get the 4d expression for the kernel
𝒦_4(u,z)    =(-p^2)^4 (-p ·n_1)^3 (-p ·n_2)^31 8 π^4 K̃(u,z)  , 
where
K̃(u,z) = 4/3 πu^2 (6 z^2-6 z+1)-8/3 πu^4 (50 z^3-78 z^2+33 z-3) +O(u^6) . 
This is consistent with the small u expansion (<ref>) from the integral representation.



 §.§ Ward identities


Based on the definition (<ref>), we can show the kernel 𝒦_d(u,z) satisfies the Ward identities
0   =∫_0^1 dz  (z(1-z))^d-4 2 𝒦_d(u,z) , 

0   =∫_0^1 dz  z(z(1-z))^d-4 2 𝒦_d(u,z) . 
These Ward identities are satisfied by perturbative corrections to the EEC as we explain below, which is the consequence of energy-momentum conservation. 

Let us first explain the reason for the EEC case. To measure the total energy (as the eigenvalue of the Hamiltonian Ĥ) at future null infinity ℐ_+, we need to integrate the energy detector operator ℰ(n⃗) over the whole S^d-2 sphere
Ĥ = ∫_S^d-2 d^d-2n⃗ ℰ(n⃗) .
As a result, the EEC satisfies the following Ward identity

1(p^0)^2∫_S^d-2d^d-2n⃗_1 d^d-2 n⃗_2 ⟨ℰ(n⃗_1) ℰ(n⃗_2) ⟩_p = 1 ,
where p^μ denotes the momentum of the scalar source 𝒪. For convenience, we choose the center of mass frame p^μ = (Q,0⃗) so that ⟨ℰ(n⃗_1) ℰ(n⃗_2) ⟩_p=Q^2 f(-n_1· n_2  2). The Ward identity (<ref>) reduces to a constraint on f(z)

1=∫_0^1 dz ∫_S^d-2d^d-2n⃗_1 d^d-2 n⃗_2 δ(z+n_1·n_2/2)f(z) =2^d-3Ω_d-1Ω_d-2 ∫_0^1 dz  (z(1-z))^d-4 2 f(z) .
This result holds non-perturbatively and is independent of any coupling constant. Therefore, if we consider a perturbative expansion f(z)=∑_n=0^∞ g^n f^(n)(z), all higher order corrections must obey the identity

∫_0^1 dz  (z(1-z))^d-4 2 f^(n≥1)(z) =0 .
To derive the second Ward identity recall that the spatial momentum generator takes the following form
P̂^i = ∫_S^d-2 d^d-2n⃗ n^i ℰ(n⃗)  .
Let us consider next the source with zero momentum p⃗=0. We can then write 

1(p^0)^2∫_S^d-2d^d-2n⃗_1 d^d-2 n⃗_2 (n⃗_1 ·n⃗_2) ⟨ℰ(n⃗_1) ℰ(n⃗_2) ⟩_p^0, p⃗=0 = 0  .
Repeating the same manipulations as above and remembering that in the center-of-mass frame n⃗_1 ·n⃗_2 = 1 - 2 z we get
∫_0^1 dz  (z(1-z))^d-4 2 (1-2z)f(z) = 0 .
When applied to the perturbative expansion together with (<ref>) we get

∫_0^1 dz  (z(1-z))^d-4 2 z f^(n≥1)(z) =0 .


Now, we turn back to the discussion of 𝒦_d(u,z) and mathematically derive the corresponding Ward identities (<ref>) and (<ref>). Instead of directly integrating over variable z, it is more convenient to consider the integration of n⃗_1 and n⃗_2 on the celestial sphere S^d-2. Again, we emphasize that we have followed the previous convention to choose p^μ =(Q, 0⃗). From the definition (<ref>), we notice that all the n⃗_1 and n⃗_2 dependence is contained in ((q· n_1)(q· n_2))^-(d-1)(4χ^2 - 4χ + d-2  d-1). Our aim is to prove the following identities
0   =∫_0^1 dχ  (4χ^2 - 4χ+ d-2 d-1)
∫_S^d-2d^d-2n⃗_1 d^d-2 n⃗_2 δ(χ- (n_1·n_2) q^2 2(q·n_1)(q·n_2)) ((q·n_1)(q·n_2))^d-1  ,

0   =∫_0^1 dχ  (4χ^2 - 4χ+ d-2 d-1)
∫_S^d-2d^d-2n⃗_1 d^d-2 n⃗_2 
δ(χ- (n_1·n_2) q^2 2(q·n_1)(q·n_2)) ((q·n_1)(q·n_2))^d-1 (-n_1·n_2) .
If q^μ has vanishing spatial components, these will reduce to the previous EEC discussion in the center-of-mass frame because polynomial 4χ^2 - 4χ + d-2  d-1 satisfies the Ward identities. However, q is an integration variable in 𝒦_d(u,z) and hence these identities are not evident to be true. We can verify these identities by directly performing the spherical integration and we find the following results
   ∫_S^d-2dn⃗_1 d n⃗_2 δ(χ- (n_1·n_2) q^2 2(q·n_1)(q·n_2)) ((q·n_1)(q·n_2))^d-1
=2^d-3Ω_d-1Ω_d-2 (-q^2)^d((q^0)^2 + 1-2χ/d-1|q⃗|^2)(χ(1-χ))^d-4 2 , 

   ∫_S^d-2dn⃗_1 d n⃗_2 
δ(χ- (n_1·n_2) q^2 2(q·n_1)(q·n_2)) ((q·n_1)(q·n_2))^d-1 (-n_1·n_2)
=2^d-2Ω_d-1Ω_d-2 χ(χ(1-χ))^d-4 2 (-q^2)^d-1 .
They are linear combination of (χ (1-χ))^d-4  2 and χ (χ (1-χ))^d-4  2 and therefore annihilate the χ integral.

We can also obtain (<ref>) and (<ref>) by promoting these integrals in a Lorentzian invariant way
   W_1(χ, (p·q)^2 p^2 q^2) = (-q^2)^d-1/-p^2∫D^d-2n_1 D^d-2n_2 (p·n_1)(p·n_2) ((q·n_1)(q·n_2))^d-1 δ(χ- (n_1·n_2) q^2 2(q·n_1)(q·n_2)) ,

   W_2(χ) = (-q^2)^d-1∫D^d-2n_1 D^d-2n_2 -n_1·n_2 ((q·n_1)(q·n_2))^d-1 δ(χ- (n_1·n_2) q^2 2(q·n_1)(q·n_2)) ,
where the measure D^d-2n_i ≡d^d n_i θ(n_i^0) δ(n_i^2)  2vol ℝ_+ (see e.g. Eq. (2.27) in <cit.> for a detailed discussion, but with a different normalization). For W_2(χ), we can choose the center of mass frame q⃗=0 so that it is similar to (<ref>)
W_2(χ) = 2χ∫_S^d-2 d^d-2n⃗_1 d^d-2n⃗_2 δ(χ- 1-n⃗_1·n⃗_2/2)=2^d-2Ω_d-1Ω_d-2 χ(χ(1-χ))^d-4 2 .


W_1 is more complicated due to the presence of an extra variable y=(p· q)^2  p^2 q^2. By acting with the differential operator (p^μ∂_p^ν - p^ν∂_p^μ)^2 on W_1, we can construct a differential equation
(2(1-y)y ∂_y^2 + (1-d y)∂_y + d)W_1(χ,y) =  W_2(y) .
A general solution is 
W_1(χ, y) = W_2(χ)/d + c_1(χ) y d-1 2d - 1 + c_2(χ) (y-1)^3-d 2∑_k=0^∞#_k (y-1)^k .
The simplest boundary condition is that W_1(χ,y) should be regular at y=1 (or equivalently, q=p)
W_1(χ,1)= ∫_S^d-2 d^d-2n⃗_1 d^d-2n⃗_2 δ(χ- 1-n⃗_1·n⃗_2/2)=2^d-3Ω_d-1Ω_d-2 (χ(1-χ))^d-4 2  ,
which constrains the coefficient functions
c_2(χ)=0 ,    c_1(χ) = (2d-1)(d-2χ) 2d(d-1)χ W_2(χ) .
The final result for W_1(χ, y) is
W_1(χ,y) = 2^d-3Ω_d-1Ω_d-2 (χ(1-χ))^d-4 2 
y d -1 -2(y-1)χd-1 ,
which is the same as (<ref>).




 §.§ Asymptotic expansion


The large impact parameter physics in AdS high-energy scattering is imprinted in the non-analytic terms at small z in the EEC. The kernel 𝒦_d(u,z) plays an important role of converting the u→ 1 limit (or large impact parameter limit L→∞) to non-analyticity at small z. In terms of a region expansion, the non-analytic terms are generated from the integration region 1-u ∼ z in (<ref>). Therefore, in this subsection, we consider the leading asymptotic expansion of 𝒦_d(u,z) in the limit z,1-u→ 0 while keeping the ratio ξ = 1-u z fixed. This result can be used to determine the leading coefficient of the non-analytic part in (<ref>). Though using the discontinuity of the kernel is a more convenient method to extract the subleading coefficients, considering asymptotic expansion is intuitive and provides an independent check for the overall normalization.

Let us consider the integral u(1-u^2)^-3/2Ĩ^d-5  2_α,α in (<ref>). The dominant contribution in the expansion comes from the region 1-r^2∼ z and ϕ∼√(z). To perform the asymptotic expansion, we change variables from (r,ϕ) to 
ρ= 1-r^2z ,  Φ= ϕ/√(z) ,
such that the leading term of the integral simplifies
    u(1-u^2)^-3/2Ĩ^d-5 2_α,α 

=    z^-1-α[
2^3α-d/2-1ξ^α-d/2∫_0^∞dρ∫_-∞^∞dΦρ^d-52(Φ^2+ρ+2ξ)^α ((Φ-2)^2+ρ+2ξ)^α
+𝒪(z)] .
Using the Feynman parametrization 
   (Φ^2+ρ+2ξ)^-α ((Φ-2)^2+ρ+2ξ)^-α

=   Γ(2α)Γ(α)^2∫_0^1 dx (x(1-x))^α-1((Φ-2(1-x))^2+ρ+2ξ+ 4x(1-x))^2α  ,
the integration becomes straightforward and we obtain
    u(1-u^2)^-3/2Ĩ^d-5 2_α,α 

=   1z^α+1[
2^α-2√(π)Γ(d-32)Γ(2α+1-d2)
ξ^α+1 Γ(2α)
 _2F_1(α,2α+1-d2,α+1/2;-12ξ)
+𝒪(z)] .
Therefore, the asymptotic expansion of the d-dimensional kernel in this limit is 
    𝒦_d(u=1-zξ,z) (-p^2)^d (-p ·n_1)^d-1 (-p ·n_2)^d-1  = Ω_d-3(2π)^d2^d√(π)Γ(d-32) z^d ξ^d  [
(d-2) Γ(3 d/2-1)   _2F_1(d-1,3 d/2-1;d-1/2;-1/2 ξ)/4 Γ(2 d-1)    . 

    . +2 Γ(3 d/2+3)   _2F_1(d+1,3 d/2+3;d+3/2;-1/2 ξ)/ξ^2 Γ(2 d+2) -Γ(3 d/2+1)   _2F_1(d,3 d/2+1;d+1/2;-1/2 ξ)/ξΓ(2 d)   ] 





 §.§ Discontinuity of the kernel


The integral over ϕ in the master integral Ĩ_γ(η_1,η_2) in (<ref>) can be done explicitly and it is given by
Ĩ_γ(η_1,η_2) = 2π∫_0^1rdr(1-r^2)^d-5/2[η_1/√(η_1^2-r^2)+η_2/√(η_2^2-r^2)]/η_1η_2+√((η_1^2-r^2)(η_2-r^2))-(1-2z)r^2.
and changing variables to η=r^-2(η_1η_2+√((η_1^2-r^2)(η_2^2-r^2))) one obtains the representation <cit.>

Ĩ_γ(η_1,η_2)   =2π∫_η_+^∞dη/η-(1-2z)(η^2-1)^4-d/2/(η-η_+)^5-d/2(η-η_-)^5-d/2,η_±   = η_1η_2±√(η_1^2-1)√(η_2^2-1).
We now rescale further η→η_+ s and inserting (<ref>) into the representation of the kernel (<ref>) we obtain
𝒦_d(u,z)=∫_1^∞ds K̂_d(u,z,s),
with a computable function K̂_d(u,z,s) given the equations above. 

The singularity is then obtained by taking the residue at s=u^2 (1-2 z)/2-u^2 and we can write it as follows
Disc_z 𝒦_d(u,z) =- πθ(u-1/√(1-z)) Res_s=u^2 (1-2 z)/2-u^2K̂_d(u,z,s) .
We did not include the contribution of higher order poles because they produce terms localized at u=1 √(1-z). Their effect can be correctly included by applying dimensional regularization to ∫_1/√(1-z)^1 du  Disc_z 𝒦_d(u,z) which is formally divergent at u=1/√(1-z). Equivalently, the discontinuity of the kernel is given by ∮_1/√(1-z)^1 du, where the integral is taken along the `keyhole' contour, see e.g. <cit.>, which makes it manifestly regular.



§ RELATING THE REGGE EXPANSION AND THE LIGHT-RAY OPE


In this Appendix, we give a derivation of the structure

R_Δ(z)≡∫_0^1du 𝒦_d(u,z) ℋ_Δ-1(L(u))∼f_Δ(z)+(analytic in z) ,
where f_Δ(z) is the celestial block for the EEC <cit.> and it is an eigenfunction of the celestial quadratic Casimir C_2=-1/2 M_μν M^μν, where M_μν are Lorentz generators.

Let us briefly review the concept of celestial blocks. In conformal theories, based on dimensionality and homogeneity, the EEC has the following functional form
⟨ℰ(n_1) ℰ(n_2) ⟩= (-p^2)^d (-p·n_1)^d-1 (-p·n_2)^d-1 f(z) .
Now let us define the differential operator
ℒ_x^μν=x^μ∂_x^ν- x^ν∂_x^μ ,
and apply the celestial Casimir differential operator
𝒞_2=-1/2(ℒ_n_1^μν+ℒ_n_2^μν)^2 ,
to the energy correlator ⟨ℰ(n_1) ℰ(n_2) ⟩. The result is
𝒞_2 ⟨ℰ(n_1) ℰ(n_2) ⟩=    (-p^2)^d (-p·n_1)^d-1 (-p·n_2)^d-1[
-4 (z-1) z^2 f”(z) 
 
    + 2 z (d (3-4 z)+2 z) f'(z)-2 (d-1) (2 (d-1) z-d) f(z)]
A light-ray operator 𝕆_Δ,J(n), corresponding to a local (fictitious) operator with dimension Δ and spin J, is an eigenfunction to the celestial Casimir operator
[𝒞_2, 𝕆_Δ,J(n)] = (Δ-1)(Δ-d+1) 𝕆_Δ,J(n) .
Therefore, the block structure associated with 𝕆_Δ,J(n) in the light-ray OPE of ℰ(n_1)ℰ(n_2) should satisfy the celestial Casimir equation
-4 (z-1) z^2 f_Δ”(z) + 2 z (d (3-4 z)+2 z) f_Δ'(z)-2 (d-1) (2 (d-1) z-d) f_Δ(z) 

=(Δ-1)(Δ-d+1)f_Δ(z) ,
whose solution is
f_Δ(z) = z^Δ-(2d-1) 2  _2 F_1(Δ-1 2, Δ-1 2, Δ- d 2+1, z)  .


To show (<ref>), we follow the same trick and apply the celestial Casimir differential operator 𝒞_2 to R_Δ(z)
R_Δ(z)=∫_M^+ d^d q(2π)^d G_0(q;n_1,n_2) (q·n_1)^d+1 (q·n_2)^d+1 δ(1-√(-q^2)√(-p^2))ℋ_Δ-1(arccosh(-p·q √(p^2 q^2))) .
Notice that all the n_1,n_2 dependence is contained in the Lorentz-invariant integrand G_0(q;n_1,n_2)  (q· n_1)^d+1 (q· n_2)^d+1. So the action of 𝒞_2 is equivalent to
𝒞_2 (G_0(q;n_1,n_2) (q·n_1)^d+1 (q·n_2)^d+1) = -1/2 (ℒ_q^μν)^2(G_0(q;n_1,n_2) (q·n_1)^d+1 (q·n_2)^d+1) .
Given that the integration region is also Lorentz-invariant, we can use integration by parts and move the action of -1/2 (ℒ_q^μν)^2 onto the function ℋ_Δ-1:
𝒞_2 R_Δ(z) 
= -1/2∫_M^+ d^d q(2π)^d G_0(q;n_1,n_2) (q·n_1)^d+1 (q·n_2)^d+1 δ(1-√(p^2 q^2))  (ℒ_q^μν)^2ℋ_Δ-1(arccosh(-p·q √(p^2 q^2))) .
On the other hand, the propagator ℋ_Δ-1 satisfies
(-1/2(ℒ_q^μν)^2 -(Δ-1)(Δ-d+1))
ℋ_Δ-1 = -δ(p-q) ,
which leads to the differential equation
[𝒞_2 - (Δ-1)(Δ-d+1)]R_Δ(z) = - 1(2π)^d G_0(p;n_1,n_2) (p·n_1)^d+1 (p·n_2)^d+1 .
Again, we set p^2=p· n_i=-1 for  convenience and then the differential equation becomes

𝒟_2 R_Δ(z)= -1(2π)^d (4z^2 - 4z +d-2 d-1) ,
where 𝒟_2 is given by
D_2=   4(1-z) z^2∂^2_z-2z (2 (2 d-1) z-3 d)∂_z   + ((-d-Δ+1) (-2 d+Δ+1)-4 (d-1)^2 z) .


Neglecting the inhomogeneous term on the r.h.s,  (<ref>) coincides with the celestial Casimir equation. Therefore the solution for R_Δ(z) is

R_Δ(z) =    ((d-Δ)^2-1) Γ(Δ-1/2) Γ(Δ+3/2)
   Γ(d-Δ/2-1/2) Γ(1/2 (d+Δ-1))/2^d+2 π^d Γ(d+1)^2 Γ(-d/2+Δ+1) f_Δ(z)      +(d-2) (2 π)^-d/(d-1) (-2 d+Δ+1) (d+Δ-1)g_Δ(z)  .
Here the coefficient of the celestial block f_Δ(z) cannot be determined by (<ref>). However, this term fully captures non-analytic terms as z → 0, so the coefficient can be calculated from the asymptotic expansion using (<ref>). The function g_Δ(z) is analytic at z=0 and the explicit result is 

g_Δ(z)    = 1-z 4 (d-1) (d^2+d Δ-Δ^2-1)/(d-2) (2 d-Δ+1) (d+Δ+1) 

   + z^2 4 (d-1) (Δ^2-1)( (d-Δ)^2-1)   _3F_2(1,d+1,d+1;d-Δ/2+5/2,d/2+Δ/2+5/2;z)/(d-2) (2 d-Δ+1) (2 d-Δ+3) (d+Δ+1) (d+Δ+3)  . 


Note that the coefficient of f_Δ(z) in (<ref>) has simple poles at Δ=2d-1+2n for n=0,1,… which cancel with the second term involving g_Δ(z) and ends up producing a log z term 

∫_0^1    du K_d(z) H_(2d-1+2n)-1(L(u))|_logz 
 
   = (-1)^n+1 ((d+2 n-1)^2-1) Γ(d+n-1) Γ(d+n+1) Γ(3 d/2+n-1)/2^d+2 π^d n! Γ(d+1)^2 Γ(3
   d/2+2 n)f_2d-1+2n,d(z).
In particular these are the relevant dimensions from decomposing (Π^(GR)(L))^2 in terms of blocks H_(2d-1+2n)-1(L). 



§ TENSOR STRUCTURES OF THE FOUR-POINT FUNCTION OT_ΜΝT_ΡΣ O


In this appendix, we spell out the assumption about the possible tensor structures of the correlator OT_μνT_ρσ O made in the paper. 

We consider the T × T OPE channel. Given that we consider as a source a scalar primary operator O, only symmetric traceless operators contribute to the OPE. The relevant three-point function takes the form <cit.>
⟨T(P_1,Z_1)T(P_2,Z_2)O_Δ,J(P_3,Z_3)⟩= ∑_i=1^10λ_i Q_i/(P_12)^d+2-Δ+J/2(P_23)^Δ+J/2(P_13)^Δ+J/2,
where P_ij=-2P_i· P_j and Q_i are defined in terms of the building blocks 
H_ij    = -2((Z_i·Z_j)(P_i·P_j)-(Z_i·P_j)(Z_j·P_i)),V_i;jk    = (Z_i·P_j)(P_i·P_k)-(Z_i·P_k)(P_i·P_j)/P_j·P_k,
and (V_1≡ V_1;23,V_2≡ V_2;31,V_3≡ V_3;12)
Q_1   =V_1^2V_2^2V_3^JQ_2   =(H_23V_1^2V_2+H_13V_1V_2^2)V_3^J-1Q_3   =H_12V_1V_2 V_3^JQ_4   = H_12(H_13V_2+H_23V_1)V_3^J-1Q_5   = H_13H_23V_1V_2V_3^J-2Q_6   = H_12^2 V_3^JQ_7   =(H_13^2V_2^2+H_23^2V_1^2)V_3^J-2Q_8   =H_12H_13H_23V_3^J-2Q_9   =(H_13^2H_23V_2+H_13H_23^2V_1)V_3^J-3Q_10   =H_13^2H_23^2V_3^J-4.
Imposing the conservation condition reduces the number of structures to be at most three, see Table 1 in <cit.>.[The counting of possible structures is different in d=3. In this case one can use the ϵ-tensor when constructing conformal three-point functions <cit.>.] Three structures are possible for J ≥ 4 or for conserved spin-two current. Notice that J is necessarily even in our case.

When translated to the phase shift, these structures lead to three possible polarization dependence as explained in detail in <cit.>. In this paper, we have effectively considered a simplified case when the phase shift representation of the correlator is diagonal in the polarization space. In other words, contracting the stress tensor with ϵ^μϵ^νT_μν(p) and setting ϵ· p = 0, we assumed that the correlator is diagonal in the polarization space ∼ϵ^μνϵ^*_μν.

In the language of <cit.>, we set β_2 = β_3 = 0. This picks a unique `holographic' three-point function, or, equivalently, the one that produces a diagonal phase shift, the same as the one in semi-classical Einstein gravity. In N=4 SYM, this choice is dictated by supersymmetry. When O_Δ,J = T, the holographic three-point function corresponds to zero value of the Hofman-Maldacena parameters t_2 = t_4 = 0.



§ STRINGY PHASE SHIFT

Let us consider d=4 and compute the stringy phase shift explicitly. To leading order at high energies, we get the following integral
δ(S,L) ∼i S^1-8 Δ_gap^2 e^-L∫_- ∞^∞ d νS^-2 ν^2 Δ_gap^2νe^-i νLν^2 + 4 (1 + L) 
Performing the integral we get

δ(S,L) = δ^(GR)(S,L)/2(1+erf(L-8 L_*^2/2 √(2) L_*)-e^4 L erfc(L+8 L_*^2/2 √(2)
  L_*)),
where erf(x)=2/√(π)∫_0^x e^-t^2dt and erfc(x)=1-erf(x). For L≳ 8L^2_*≫ 1 this approaches the pure gravity phase shift while for smaller impact parameters the stringy corrections are important. On the other hand, for L_*≪ 1 we see that for L ≳  L_* the pure gravity phase shift is a good approximation but for smaller L, stringy corrections are important as shown in fig:ExplStringyPhase. 



JHEP

  
