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We discuss the structure of nonlocal effective action generating theconformal anomaly in classically Weyl invariant theories in curved spacetime.By the procedure of conformal gauge fixing, selecting the metric representativeon a conformal group orbit, we split the renormalized effective action intoanomalous and Weyl invariant parts. A wide family of thus obtained anomalousactions is shown to include two special cases of Riegert--Fradkin--Tseytlin andFradkin--Vilkovisky actions. Both actions are shown to be contained in thefirst three orders of the curvature expansion for a generic one-loop effectiveaction obtained by covariant perturbation theory. The complementary Weylinvariant part of the action is given by the ``conformization'' of the fulleffective action -- restricting its argument to the conformally invariantrepresentative of the orbit of the conformal group. This is likely to resolve along-standing debate between the proponents of the Riegert action and adherentsof the perturbation expansion for the effective action with typical nonlocallogarithmic form factors. We derive the relation between quantum stress tensorson conformally related metric backgrounds, which generalizes the knownBrown-Cassidy equation to the case of nonzero Weyl tensor, and discussapplications of this relation in the cosmological model driven by conformalfield theory. We also discuss the issue of renormalization group running forthe cosmological and gravitational coupling constants and show that it exhibitsa kind of a metamorphosis to the nonlocal form factors of the so-calledpartners of the cosmological and Einstein terms -- nonlocal curvature squaredterms of the effective action.

Notes on conformal anomaly, nonlocal effective action and the metamorphosis of the running scale