  We present a general study of 3-point functions of conformal field theory in
momentum space, following a reconstruction method for tensor correlators, based
on the solution of the conformal Ward identities (CWI' s), introduced in recent
works by Bzowski, McFadden and Skenderis (BMS). We investigate and detail the
structure of the CWI's, their non-perturbative solutions and the transition to
momentum space, comparing them to perturbation theory by taking QED as an
example. We then proceed with an analysis of the $TJJ$ correlator, presenting
independent and detailed re-derivations of the conformal equations in the
reconstruction method of BMS, originally formulated using a minimal tensor
basis in the transverse traceless sector. A careful comparison with a second
basis introduced in previous studies shows that this correlator is affected by
one anomaly pole in the graviton (T) line, induced by renormalization. The
result shows that the origin of the anomaly, in this correlator, should be
necessarily attributed to the exchange of a massless effective degree of
freedom. Our results are then exemplified in massless QED at one-loop in
$d$-dimensions, expressed in terms of perturbative master integrals. An
independent analysis of the Fuchsian character of the solutions, which bypasses
the 3K integrals, is also presented. We show that the combination of field
theories at one-loop - with a specific field content of degenerate massless
scalar and fermions - is sufficient to generate the complete non-perturbative
solution, in agreement with a previous study in coordinate space. The result
shows that free conformal field theories, in specific dimensions, arrested at
one-loop, reproduce the general result for the $TJJ$. Analytical checks of this
correspondence are presented in $d=3,4$ and $5$ spacetime dimensions[..].
