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We elaborate on the structure of the conformal anomaly effective action up to4-th order, in an expansion in the gravitational fluctuations $(h)$ of thebackground metric, in the flat spacetime limit. For this purpose we discuss therenormalization of 4-point functions containing insertions of stress-energytensors (4T), in conformal field theories in four spacetime dimensions with thegoal of identifying the structure of the anomaly action. We focus on aseparation of the correlator into its transverse/traceless and longitudinalcomponents, applied to the trace and conservation Ward identities (WI) inmomentum space. These are sufficient to identify, from their hierarchicalstructure, the anomaly contribution, without the need to proceed with acomplete determination of all of its independent form factors. Renormalizationinduces sequential bilinear graviton-scalar mixings on single, double andmultiple trace terms, corresponding to $R\square^{-1}$ interactions of thescalar curvature, with intermediate virtual massless exchanges. Thesedilaton-like terms couple to the conformal anomaly, as for the chiral anomalousWIs. We show that at 4T level a new traceless component appears afterrenormalization. We comment on future extensions of this result to more generalbackgrounds, with possible applications to non local cosmologies.

The Conformal Anomaly Action to Fourth Order (4T) in $d=4$ in Momentum Space