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

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.