Four-Point Functions in Momentum Space: Conformal Ward Identities in the Scalar/Tensor case

TL;DR

This paper derives and analyzes conformal Ward identities for four-point tensor correlators involving stress-energy tensors and scalars in momentum space, extending reconstruction methods and exploring their solutions in various limits.

Contribution

It extends the reconstruction method for tensor correlators to four-point functions and analyzes the conformal Ward identities in momentum space for specific tensor-scalar correlators.

Findings

Derived the structure of conformal Ward identities for TOOO correlators.

Expressed solutions in terms of Lauricella functions and Bessel integrals.

Compared systems for TOOO and OOOO correlators in different conformal cases.

Abstract

We derive and analyze the conformal Ward identities (CWI's) of a tensor 4-point function of a generic CFT in momentum space. The correlator involves the stress-energy tensor $T$ and three scalar operators $O$ ($TOOO$). We extend the reconstruction method for tensor correlators from 3- to 4-point functions, starting from the transverse traceless sector of the $TOOO$. We derive the structure of the corresponding CWI's in two different sets of variables, relevant for the analysis of the 1-to-3 (1 graviton $\to$ 3 scalars) and 2-to-2 (graviton + scalar $\to$ two scalars) scattering processes. The equations are all expressed in terms of a single form factor. In both cases, we discuss the structure of the equations and their possible behaviors in various asymptotic limits of the external invariants. A comparative analysis of the systems of equations for the $TOOO$ and those for the $OOOO$, both in the general (conformal) and dual-conformal/conformal (dcc) cases, is presented. We show that in all the cases the Lauricella functions are homogenous solutions of such systems of equations, also described as parametric 4K integrals of modified Bessel functions.