Diffeomorphism Invariance Demands Conformal Anomalies

TL;DR

This paper explores how conformal anomalies are essential for maintaining diffeomorphism invariance in quantum field theories on curved spacetime, with detailed calculations in QED, QCD, and conformal gravity.

Contribution

It demonstrates that conformal anomalies generate necessary nonlocal actions ensuring diffeomorphism invariance at higher loops in various quantum theories.

Findings

Effective actions depend on physical momentum Q^2 = q^2/e^{2 extphi}

Confirmed at 3-loop level in QED using dimensional regularization

Effective QCD action relates to the reciprocal of the running coupling squared

Abstract

We study a series of the Wess-Zumino actions obtained by repeatedly integrating conformal anomalies with respect to the conformal-factor field that appear at higher loops. We show that they arise as physical quantities required to make nonlocal loop correction terms diffeomorphism invariant. Specifically, in a conformally flat spacetime $ds^2=e^{2\phi}(-d\eta^2 + d{\bf x}^2)$, we find that effective actions are described in terms of momentum squared expressed as a physical $Q^2 = q^2/e^{2\phi}$ for $q^2$ measured by the flat metric, which recalls the relationship between physical momentum and comoving momentum in cosmology. It is confirmed by calculating the effective action of QED in such a curved spacetime at the 3-loop level using dimensional regularization. The same applies to the case of QCD, in which we show that the effective action can be summarized in the form of the reciprocal of a running coupling constant squared described by the physical momentum. We also see that the same holds for renormalizable quantum conformal gravity and that conformal anomalies are indispensable for formulating the theory.