We propose and explore the Regge limit for correlation functions of five local primary operators in conformal field theories. After reviewing some features of Regge theory for flat-space scattering amplitudes, we analyze the analytic structure of conformal blocks both in position and Mellin space in the Regge limit and propose an extension of conformal Regge theory for five-point functions. As a byproduct of our analysis we also introduce a new basis of three-point correlation functions for operators with spin and the associated Euclidean conformal blocks.

Portugal

Miguel Costa

Vasco Gonçalves

Aaditya Salgarkar

Switzerland

João Boas

Conformal multi-Regge theory

Higher-point functions of scalar operators are a rich observable in CFTs, as they contain OPE data involving multiple spinning operators. We derive the lightcone blocks for five- and six-point functions in the snowflake channel and use them to bootstrap these correlators in the lightcone limit. As a result we determine the large spin expansion of OPE coefficients involving two or three spinning operators. We verify our results by comparing to the block decomposition of higher-point functions in generalized free theory and in theories with a cubic coupling.

António Antunes

Lightcone bootstrap at higher points

The boundary correlation functions for a Quantum Field Theory (QFT) in an Anti-de Sitter (AdS) background can stay conformally covariant even if the bulk theory undergoes a renormalization group (RG) flow. Studying such correlation functions with the numerical conformal bootstrap leads to non-perturbative constraints that must hold along the entire flow. In this paper we carry out this analysis for the sine-Gordon RG flows in AdS2, which start with a free (compact) scalar in the UV and end with well-known massive integrable theories that saturate many S-matrix bootstrap bounds. We numerically analyze the correlation functions of both breathers and kinks and provide a detailed comparison with perturbation theory near the UV fixed point. Our bounds are often saturated to one or two orders in perturbation theory, as well as in the flat-space limit, but not necessarily in between.

João Penedones

France

Balt Rees

Towards bootstrapping RG flows: sine-Gordon in AdS

We initiate the study of Regge theory in a bottom-up holographic model for QCD in the Veneziano limit, where the backreaction of the quarks to the gluon dynamics is included. We determine the parameters of the model by carrying out a precise fit to the meson spectrum in QCD. The spectrum for spin-one and pseudoscalar mesons is well reproduced. We then generalise the model to incluce higher spin fields in the bulk trajectories dual to the Pomeron and meson Regge trajectories at the boundary. With this setting, we fit the masses of the mesons with spins J = 2, 3, and 4, as well as the experimental data of the total cross-sections σ(γγ → X), σ(γp → X) and σ(pp → X). For the cross sections we obtain a   <math> <msubsup> <mi>χ</mi> <mrow> <mi>d</mi> <mo>.</mo> <mi>o</mi> <mo>.</mo> <mi>f</mi> </mrow> <mn>2</mn> </msubsup> </math>  $$ {\chi}_{\mathrm{d}.\mathrm{o}.\mathrm{f}}^2 $$  of 0.74 for a total of 199 experimental points.

Artur Amorim

South Korea

Matti Järvinen

Regge theory in a holographic dual of QCD in the Veneziano limit

We derive an optical theorem for perturbative CFTs which computes the double discontinuity of conformal correlators from the single discontinuities of lower order correlators, in analogy with the optical theorem for flat space scattering amplitudes. The theorem takes a purely multiplicative form in the CFT impact parameter representation used to describe high-energy scattering in the dual AdS theory. We use this result to study four-point correlation functions that are dominated in the Regge limit by the exchange of the graviton Regge trajectory (Pomeron) in the dual theory. At one-loop the scattering is dominated by double Pomeron exchange and receives contributions from tidal excitations of the scattering states which are efficiently described by an AdS vertex function, in close analogy with the known Regge limit result for one-loop string scattering in flat space at finite string tension. We compare the flat space limit of the conformal correlator to the flat space results and thus derive constraints on the one-loop vertex function for type IIB strings in AdS and also on general spinning tree level type IIB amplitudes in AdS.

Sweden

United Kingdom

Tobias Hansen

Sourav Sarkar

The perturbative CFT optical theorem and high-energy string scattering in AdS at one loop

We construct a new class of differential operators that naturally act on AdS harmonic functions. These are weight shifting operators that change the spin and dimension of AdS representations. Together with CFT weight shifting operators, the new operators obey crossing equations that relate distinct representations of the conformal group. We apply our findings to the computation of Witten diagrams, focusing on the particular case of cubic interactions and on massive, symmetric and traceless fields. In particular we show that tree level 4-point Witten diagrams with arbitrary spins, both in the external fields and in the exchanged field, can be reduced to the action of weight shifting operators on similar 4-point Witten diagrams where all fields are scalars. We also show how to obtain the conformal partial wave expansion of these diagrams using the new set of operators. In the case of 1-loop diagrams with cubic couplings we show how to reduce them to similar 1-loop diagrams with scalar fields except for a single external spinning field (which must be a scalar in the case of a two-point diagram). As a bonus, we provide new CFT and AdS weight shifting operators for mixed-symmetry tensors.

AdS weight shifting operators

We consider the Regge limit of the CFT correlation functions JJOO $$ \left\langle \mathcal{JJOO}\right\rangle $$ and T T O O $$ \left\langle TT\mathcal{O}\mathcal{O}\right\rangle $$ , where J $$ \mathcal{J} $$ is a vector current, T is the stress tensor and O $$ \mathcal{O} $$ is some scalar operator. These correlation functions are related by a type of Fourier transform to the AdS phase shift of the dual 2-to-2 scattering process. AdS unitarity was conjectured some time ago to be positivity of the imaginary part of this bulk phase shift. This condition was recently proved using purely CFT arguments. For large N CFTs we further expand on these ideas, by considering the phase shift in the Regge limit, which is dominated by the leading Regge pole with spin j ( ν ), where ν is a spectral parameter. We compute the phase shift as a function of the bulk impact parameter, and then use AdS unitarity to impose bounds on the analytically continued OPE coefficients C J J j ν $$ {C}_{\mathcal{JJ}j\left(\nu \right)} $$ and C TTj(ν ) that describe the coupling to the leading Regge trajectory of the current J $$ \mathcal{J} $$ and stress tensor T . AdS unitarity implies that the OPE coefficients associated to non-minimal couplings of the bulk theory vanish at the intercept value ν = 0, for any CFT. Focusing on the case of large gap theories, this result can be used to show that the physical OPE coefficients C J J T $$ {C}_{\mathcal{JJT}} $$ and C TTT , associated to non-minimal bulk couplings, scale with the gap Δ g as Δ g − 2 or Δ g − 4 . Also, looking directly at the unitarity condition imposed at the OPE coefficients C J J T $$ {C}_{\mathcal{JJT}} $$ and C TTT results precisely in the known conformal collider bounds, giving a new CFT derivation of these bounds. We finish with remarks on finite N theories and show directly in the CFT that the spin function j ( ν ) is convex, extending this property to the continuation to complex spin.

Bounds for OPE coefficients on the Regge trajectory

We develop a formalism where the hard and soft pomeron contributions to high energy scattering arise as leading Regge poles of a single kernel in holographic QCD. The kernel is obtained using effective field theory inspired by Regge theory of a 5-d string theory. It describes the exchange of higher spin fields in the graviton Regge trajectory that are dual to glueball states of twist two. For a specific holographic QCD model we describe Deep Inelastic Scattering in the Regge limit of low Bjorken x , finding good agreement with experimental data from HERA. The observed rise of the effective pomeron intercept, as the size of the probe decreases, is reproduced by considering the first four pomeron trajectories. In the case of soft probes, relevant to total cross sections, the leading hard pomeron trajectory is suppressed, such that in this kinematical region we reproduce an intercept of 1.09 compatible with the QCD soft pomeron data. In the spectral region of positive Maldelstam variable t the first two pomeron trajectories are consistent with current expectations for the glueball spectrum from lattice simulations.

Brazil

Alfonso Ballon-Bayona

Robert Quevedo

Unity of pomerons from gauge/string duality

We numerically construct asymptotically AdS 4 solutions to Einstein-Maxwell-dilaton theory. These have a dipolar electrostatic potential turned on at the conformal boundary S 2 × ℝ t $$ {S}^2\times {\mathrm{\mathbb{R}}}_t $$ . We find two classes of geometries: AdS soliton solutions that encode the full backreaction of the electric field on the AdS geometry without a horizon, and neutral black holes that are “polarised” by the dipolar potential. For a certain range of the electric field ℰ $$ \mathrm{\mathcal{E}} $$ , we find two distinct branches of the AdS soliton that exist for the same value of ℰ $$ \mathrm{\mathcal{E}} $$ . For the black hole, we find either two or four branches depending on the value of the electric field and horizon temperature. These branches meet at critical values of the electric field and impose a maximum value of ℰ $$ \mathrm{\mathcal{E}} $$ that should be reflected in the dual field theory. For both the soliton and black hole geometries, we study boundary data such as the stress tensor. For the black hole, we also consider horizon observables such as the entropy. At finite temperature, we consider the Gibbs free energy for both phases and determine the phase transition between them. We find that the AdS soliton dominates at low temperature for an electric field up to the maximum value. Using the gauge/gravity duality, we propose that these solutions are dual to deformed ABJM theory and compute the corresponding weak coupling phase diagram.

Lauren Greenspan

Jorge Santos

Polarised black holes in ABJM

This paper develops a method to compute any bosonic conformal block as a series expansion in the optimal radial coordinate introduced by Hogervorst and Rychkov. The method reduces to the known result when the external operators are all the same scalar operator, but it allows to compute conformal blocks for external operators with spin. Moreover, we explain how to write closed form recursion relations for the coefficients of the expansions. We study three examples of four point functions in detail: one vector and three scalars; two vectors and two scalars; two spin 2 tensors and two scalars. Finally, for the case of two external vectors, we also provide a more efficient way to generate the series expansion using the analytic structure of the blocks as a function of the scaling dimension of the exchanged operator.