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 $⟨T_{\mu \nu }⟩$ and the bilinear $⟨\overline{\psi }\psi ⟩$ in a spatially flat Friedmann-Lemaitre-Robertson-Walker (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.

Figure 1

The red line shows $\frac{1}{m^3}|⟨\overline{\psi }\psi {⟩}_{\mathrm{ren}}|$ as a function of time for $\mu =1$ given by Eq. (b7). For $mt\lesssim -0.4$ we have $⟨\overline{\psi }\psi {⟩}_{\mathrm{ren}}<0$ (red continuous line), while for $mt\gtrsim -0.4$ we have $⟨\overline{\psi }\psi {⟩}_{\mathrm{ren}}>0$ (red dotted line). The purple dashed line shows the corresponding approximation at fourth adiabatic order given in Eqs. (b10).