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 function. This assertion is then checked by explicit calculations in ^4 theory in four dimensions and ^3 theory in six dimensions and values for the anomaly found to fourth order in the renormalized coupling constants 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 (x) enables the relation between the trace anomaly and the function to be expressed in the form T₈^=-{ () {W₈} (x) |} (ₗ) =. where W₈ is the sum over vacuum bubble diagrams with interactions.