Conformal `anomalies' and renormalized stress-tensor correlators for non-conformal theories

TL;DR

This paper investigates the definition of Weyl anomalies in non-conformal theories, specifically a scalar field with quartic interaction, proposing a renormalized approach that addresses ambiguities and extends to interacting cases.

Contribution

It introduces a renormalized prescription for Weyl anomalies in non-conformal theories, overcoming limitations of previous regularization methods in interacting quantum field theories.

Findings

The original regularized prescription cannot be extended to interacting theories.

The proposed renormalized approach reproduces known results at one-loop.

Extension to interactions includes non-local contributions.

Abstract

We analyse the proposal of defining the Weyl anomaly for classically non-conformal theories as $g^{mn} \langle T_{mn}\rangle - \langle g^{mn} T_{mn} \rangle$, originally put forward by M. Duff, in the case of a scalar field with quartic self-interaction in 4d. We work in the context of dimensional regularization in curved background to two-loops (first order in the coupling). We review the original regularized but not renormalized prescription and its ambiguities; we argue that it cannot be extended to the interacting theory as it fails to provide a finite result. We then propose an alternative prescription via renormalized expectation values. At one-loop our candidate reproduces the local heat kernel result, while its extension to interacting theories contains non-local contributions.