Broken scale invariance and the regularization of a conformal sector in gravity with Wess-Zumino actions

We elaborate on anomaly induced actions of the Wess-Zumino (WZ) form and their relation to renormalized effective action, which is defined by an ordinary path integral over a conformal sector, in external gravitational background. In - actions, issue scale breaking usually not addressed, since these are obtained only solving trace constraint determined invariant functionals. investigate changes structure such once identified dimensional renormalization (DR) when ϵ=d−4→0 limit accompanied reduction (DRed) field dependencies. show that operatorial nonlocal modifications (∼□ϵ) counterterms unnecessary justify anomaly. this case, finite subtractions play critical role determination breaking. This illustrated for WZ action its consistency condition, as seen from procedure. Logarithmic corrections also pure d=4 (cutoff) scheme. The interplay between two schemes, one based regularization second cutoff d=4, illustrates ambiguities DR handling quantum curved Therefore, using background, anomalies can both be Weyl at d=4.