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We present a study of the correlation function of three stress-energy tensorsin $d$ dimensions using free field theory realizations, and compare them to theexact solutions of their conformal Ward identities (CWI's) obtained by ageneral approach in momentum space. The identification of the correspondingform factors is performed within a reconstruction method, based on theidentification of the transverse traceless components $(A_i)$ of the samecorrelator. The solutions of the primary CWI' s are found by exploiting theuniversality of the Fuchsian indices of the conformal operators and are-arrangement of the corresponding inhomogenous hypergeometric systems. Weconfirm the number of constants in the solution of the primary CWI's ofprevious analysis. In our comparison with perturbation theory, we discussscalar, fermion and spin 1 exchanges at 1-loop in dimensional regularization.Explicit checks in $d=3$ and $d=5$ prove the consistency of thiscorrespondence. By matching the 3 constants of the CFT solution with the 3 freefield theory sectors available in d=4, the general solutions of the conformalconstraints is expressed just in terms of ordinary scalar 2- and 3-pointfunctions $(B_0,C_0)$. We show how the renormalized $d=4$ TTT vertex separatesnaturally into the sum of a traceless and an anomaly part, the latterdetermined by the anomaly functional and generated by the renormalization ofthe correlator in dimensional regularization. The result confirms the emergenceof anomaly poles and effective massless exchanges as a specific signature ofconformal anomalies in momentum space, directly connected to therenormalization of the corresponding gravitational vertices, generalizing thebehaviour found for the $TJJ$ vertex in previous works.

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