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

TL;DR

This work analyzes the general $TTT$ vertex of four-dimensional conformal field theories in momentum space, solving conformal Ward identities and reconstructing the full vertex from a transverse-traceless basis of form factors $A_i$. By matching perturbative free-field sectors (scalars, fermions, gauge) to the CFT solution, the authors show that in $d=3,5$ the solution is saturated by free-field content, while in $d=4$ renormalization separates the vertex into a traceless part and an anomaly part, with anomaly poles arising from the renormalization of local counterterms. The reconstruction framework leverages Appell functions $F_4$ and reduces the problem to scalar two- and three-point functions $B_0$ and $C_0$, clarifying the role of the anomaly action and the emergence of massless exchanges. The results unify cft-based constraints with perturbative renormalization, providing a clear pathway to higher-point correlators and to the conformal anomaly action in momentum space.

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.

Figures (5)

• Figure 1: Vertices used in the Lagrangian realization of the $TTT$ correlator.
• Figure 2: One-loop scalar diagrams for the three-graviton vertex.
• Figure 3: One-loop fermion diagrams for the three-graviton vertex.
• Figure 4: One-loop gauge diagrams for the three-graviton vertex.
• Figure 5: Anomaly interactions mediated by the exchange of one, two or three poles. The poles are generated by the renormalization of the longitudinal sector of the $TTT$.