Conformal anomalies for interacting scalar fields in curved spacetime

The occurrence of anomalies in the trace of the energy-momentum tensor for scalar field theories in curved space-time is discussed. For the special case of spherical space-time, an O(n+1)-covariant formalism is used to rederive the standard free-field anomaly in four dimensions, and to calculate the anomaly in six dimensions. It is then shown that for an interacting scalar field theory there is a further contribution to the trace anomaly proportional to the renormalization group β function. This assertion is then checked by explicit calculations in φ⁴ theory in four dimensions and φ³ theory in six dimensions and values for the anomaly found to fourth order in the renormalized coupling constants λ and g. Finally, these results are generalized to the case of an arbitrary background space-time, where it is shown that the introduction of a position-dependent coupling constant λ(x) enables the relation between the trace anomaly and the β function to be expressed in the form ${T}_{I{μ}}^{{μ}}={-}{{{β}({λ}){δ}{W}I}{{δ}{λ}(x)}|}_{{λ}(x)={λ}}$. where ${W}I$ is the sum over vacuum bubble diagrams with interactions.