  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.
