Effective action for the Yukawa model in curved spacetime

We consider the one-loop renormalization of a real scalar field interacting with a Dirac spinor field in curved spacetime. A general Yukawa interaction is considered which includes both a scalar and a pseudoscalar coupling. The scalar field is assumed to be non-minimally coupled to the gravitational field and to have a general quartic self-interaction potential. All of the one-loop renormalization group functions are evaluated and in the special case where there is no mass scale present in the classical theory (apart from the fields) we evaluate the one-loop effective action up to and including order $R^2$ in the curvature. In the case where the fermion is massive we include a chiral term in $\gamma_5$ and we show that although the $\gamma_5$ term can be removed by a redefinition of the spinor field an anomaly in the effective action arises that is related to the familiar axial current anomaly.