Yang-Mills Yukawa model in curved spacetime

The one-loop effective action for the scalar field part of a non-Abelian gauge theory based on a general gauge group of the form $G\times U(1)$, where the gauge group $G$ is arbitrary, is calculated. A complex scalar field, both Abelian and non-Abelian gauge fields and Dirac fermions coupled to gauge and scalar fields are included. A general mass term for the Dirac fields that includes a pseudoscalar term as well as both scalar and pseudoscalar Yukawa couplings is considered. The background field method is used in its manifestly gauge condition independent and gauge invariant form to isolate the divergent part of the one-loop effective action and to calculate the associated renormalisation group functions. Terms in the renormalised effective action up to and including those quadratic in the curvature are calculated using renormalisation group methods. The background scalar field is not assumed to be constant, so the second order derivative terms in the effective action can be calculated, and the gravitational background is kept arbitrary. The difference between the gauge condition independent approach and the more standard approach where gauge condition dependent results are found is demonstrated by explicit calculations. The anomaly that can arise if the pseudoscalar mass term for the fermions is transformed away from the classical theory is noted.