Trace anomalies and λφ4 theory in curved space

It is shown for a conformally invariant λφ⁴ theory in a weakly curved background, how to extend previous results to obtain full information about the trace anomaly in perturbation theory, including the “topological” term in the gravitational part of the anomaly. There is a strong connection among renormalisability of the curved space theory, finiteness of the energy-momentum tensor, and the role of normal products. Combined with a renormalisation-group analysis this provides an efficient means of calculating some terms in the anomaly to high orders of perturbation theory. In particular, the first λ-dependent coefficient of the topological part of the anomaly appears at O(λ⁴) and can be deduced from simple flat-space results without the calculation of any further Feynman diagrams. Some techniques based on an absorptive-part argument are developed in order to compute other anomalous coefficients, and a direct 5-loop calculation confirms the indirect renormalisation-group derivation of a non-vanishing R² anomaly at O(λ⁵). All the essential information can be obtained from the massless theory. The underlying ideas are applicable to other theories, and similar results for massless QED are obtained in a subsequent paper.