  We investigate the emergence of a conformal anomaly pole in conformal field
theories in the case of the $TJJ$ correlator. We show how it comes to be
generated in dimensional renormalization, using a basis of 13 form factors (the
$F$-basis), where only one of them requires renormalization $(F_{13})$,
extending previous studies. We then combine recent results on the structure of
the non-perturbative solutions of the conformal Ward identities (CWI's) for the
$TJJ$ in momentum space, expressed in terms of a minimal set of 4 form factors
($A-$ basis), with the properties of the $F$-basis, and show how the singular
behaviour of the corresponding form factors in both basis can be related. The
result proves the centrality of such massless effective interactions induced by
the anomaly, which have recently found realization in solid state, in the
theory of topological insulators and of Weyl semimetals. This pattern is
confirmed in massless abelian and nonabelian theories (QED and QCD)
investigated at one-loop.
