The General 3-Graviton Vertex ($TTT$) of Conformal Field Theories in Momentum Space in $d=4$

TL;DR

This paper analyzes the three-stress-energy tensor correlator in conformal field theories in four dimensions, deriving explicit solutions, confirming their consistency with perturbation theory, and elucidating the role of anomalies and massless exchanges in momentum space.

Contribution

It provides the first comprehensive momentum-space solution for the general 3-graviton vertex in conformal field theories, connecting conformal constraints with free field theory sectors and anomaly structures.

Findings

Explicit form factors for the 3-graviton vertex in d=4.

Confirmation of anomaly poles and massless exchanges.

Matching CFT solutions with free field theory sectors.

Abstract

We present a study of the correlation function of three stress-energy tensors in $d$ dimensions using free field theory realizations, and compare them to the exact solutions of their conformal Ward identities (CWI's) obtained by a general approach in momentum space. The identification of the corresponding form factors is performed within a reconstruction method, based on the identification of the transverse traceless components $(A_i)$ of the same correlator. The solutions of the primary CWI' s are found by exploiting the universality of the Fuchsian indices of the conformal operators and a re-arrangement of the corresponding inhomogenous hypergeometric systems. We confirm the number of constants in the solution of the primary CWI's of previous analysis. In our comparison with perturbation theory, we discuss scalar, fermion and spin 1 exchanges at 1-loop in dimensional regularization. Explicit checks in $d=3$ and $d=5$ prove the consistency of this correspondence. By matching the 3 constants of the CFT solution with the 3 free field theory sectors available in d=4, the general solutions of the conformal constraints is expressed just in terms of ordinary scalar 2- and 3-point functions $(B_0,C_0)$. We show how the renormalized $d=4$ TTT vertex separates naturally into the sum of a traceless and an anomaly part, the latter determined by the anomaly functional and generated by the renormalization of the correlator in dimensional regularization. The result confirms the emergence of anomaly poles and effective massless exchanges as a specific signature of conformal anomalies in momentum space, directly connected to the renormalization of the corresponding gravitational vertices, generalizing the behaviour found for the $TJJ$ vertex in previous works.