The Weyl anomaly in interacting quantum field theory on curved spacetimes

TL;DR

The paper develops a rigorous locally covariant framework to define Weyl (scale) anomalies for interacting quantum field theories on curved spacetimes and clarifies their relation to the trace anomaly. It introduces Weyl anomalies $A_f$ and analyzes their locality, cohomology, and behavior under renormalization group flow, culminating in a quantum notion of Weyl transformations and a criterion for quantum conformality. The $\phi^4$ theory is used as a testbed, with the Weyl anomaly computed up to second order in the coupling; conformal-violating terms like $\Box\phi^2$ and $\Box R$ can be removed by finite renormalizations, yielding an explicit on-shell trace anomaly in terms of ${\mathcal{E}}_4$, $C^2$, and $\phi^4$ contributions, subject to scheme dependence in $C^2$. While in agreement with prior results in key aspects, the work also highlights perturbative removability of certain terms at higher orders and points toward extensions to gauge theories via BRST/BV formalisms and more general field content.

Abstract

We define the notion of Weyl anomalies, measuring the violation of local scale invariance, in interacting quantum field theory on curved spacetimes in the framework of locally covariant field theory. We discuss some general properties of Weyl anomalies, such as their relation to the trace anomaly. We give a criterion for a theory to be conformal at the quantum level, and show that even for a conformal theory Weyl transformations in general obtain quantum corrections. We study the trace anomaly in detail for the $φ^4$ theory, in particular determining it up to second order in the interaction. We also show that at third order in the interaction a potential $\Box φ^2$ term can be removed by finite renormalization.