  Conformal back-reaction generates cosmological models where the trace anomaly
reflects the breaking of Weyl invariance. Analyzing these actions yields a
dynamic approach to dark energy through anomaly-induced actions (AIAs), that
are variational solutions of the trace anomaly functional constraint. Expanded
around Minkowski space, they produce semiclassical correlators subject to
hierarchical anomalous Ward identities, tied to conformal symmetry and
diffeomorphism invariance. We focus on comparing the hierarchy of a specific
4-point function, particularly the 2-gravitons-2-photons correlator $(TTJJ)$,
generated by AIAs, to free field theory realizations of the same correlator. We
observe that the free field theory original hierarchy splits into one ordinary
and one anomalous hierarchy, both satisfying the conservation Ward identities
from diffeomorphism invariance. However, we find that the anomalous hierarchy
derived from ordinary AIAs in both the Riegert or Fradkin-Vilkovisky gauges,
are either affected by double poles or violate the hierarchy of the trace Ward
identity, respectively. We show that correct forms of the anomalous hierarchies
of 4-point functions (for the $TTTT$ and $TTJJ$), identified in a perturbative
free field theory expansion around flat space, are characterised by anomaly
poles, corresponding to a curvature expansion in $R\Box^{-1}$, together with
Weyl invariant terms. We derive the effective action that generates the correct
form of the hierarchy for the $TTJJ$.
