Adiabatic regularization with a Yukawa interaction

We extend the adiabatic regularization method for an expanding universe to include the Yukawa interaction between quantized Dirac fermions and a homogeneous background scalar field. We give explicit expressions for the renormalized expectation values of the stress-energy tensor $\langle T_{\mu\nu} \rangle$ and the bilinear $\langle \bar\psi\psi\rangle$ in a spatially flat FLRW spacetime. These are basic ingredients in the semiclassical field equations of fermionic matter in curved spacetime interacting with a background scalar field. The ultraviolet subtracting terms of the adiabatic regularization can be naturally interpreted as coming from appropriate counterterms of the background fields. We fix the required covariant counterterms. To test our approach we determine the contribution of the Yukawa interaction to the conformal anomaly in the massless limit and show its consistency with the heat kernel method using the effective action.