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We present a general study of 3-point functions of conformal field theory inmomentum space, following a reconstruction method for tensor correlators, basedon the solution of the conformal Ward identities (CWI' s), introduced in recentworks by Bzowski, McFadden and Skenderis (BMS). We investigate and detail thestructure of the CWI's, their non-perturbative solutions and the transition tomomentum space, comparing them to perturbation theory by taking QED as anexample. We then proceed with an analysis of the $TJJ$ correlator, presentingindependent and detailed re-derivations of the conformal equations in thereconstruction method of BMS, originally formulated using a minimal tensorbasis in the transverse traceless sector. A careful comparison with a secondbasis introduced in previous studies shows that this correlator is affected byone anomaly pole in the graviton (T) line, induced by renormalization. Theresult shows that the origin of the anomaly, in this correlator, should benecessarily attributed to the exchange of a massless effective degree offreedom. Our results are then exemplified in massless QED at one-loop in$d$-dimensions, expressed in terms of perturbative master integrals. Anindependent analysis of the Fuchsian character of the solutions, which bypassesthe 3K integrals, is also presented. We show that the combination of fieldtheories at one-loop - with a specific field content of degenerate masslessscalar and fermions - is sufficient to generate the complete non-perturbativesolution, in agreement with a previous study in coordinate space. The resultshows that free conformal field theories, in specific dimensions, arrested atone-loop, reproduce the general result for the $TJJ$. Analytical checks of thiscorrespondence are presented in $d=3,4$ and $5$ spacetime dimensions[..].

Exact Correlators from Conformal Ward Identities in Momentum Space and the Perturbative $TJJ$ Vertex