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

TL;DR

This work derives and analyzes conformal Ward identities for the TOOO four-point function in momentum space, constructing the full correlator from its transverse-traceless sector via a reconstruction approach and expressing all equations in terms of a single form factor. It develops primary and secondary CWIs in two formulations (1→3 and 2→2), reveals permutation symmetry structures, and connects the solutions to hypergeometric families, notably Lauricella functions and 3K integrals, including dual-conformal/conformal (dcc) cases. The paper compares tensor and scalar (OOOO) cases, explores asymptotic limits (large s,t, IR, equal-mass), and demonstrates that Lauricella-type homogeneous solutions underlie the asymptotics, providing a bridge between CFT constraints and perturbative master integrals. These results lay groundwork for a momentum-space bootstrap of higher-point correlators and have potential applications to cosmology, condensed matter, and holography, where anomalies and conformal symmetry play central roles.

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.

Figures (5)

• Figure 1: Orbits of the primary CWI's of the $TOOO$ under $P_{23}$ and $P_{24}$
• Figure 2: Equivalent pictorial representation of the orbits as in Fig. \ref{['tikzz']}.
• Figure 3: Orbits of the CWI's for the $TOOO$ under $P_{34}$.
• Figure 4: Orbits of the secondary CWI's under permutations.
• Figure 5: The $TOOO$ in a kinematical region in which can be described as a $1\to 3$ process.