Impact of a complex scalar spectator field on baryon asymmetry within spontaneous baryogenesis

We extend the framework of spontaneous baryogenesis by investigating the generation of baryon asymmetry when the inflaton, $\theta$, is minimally coupled with a complex spectator scalar field $\phi$, as $\theta^2|\phi|^2$. To do so, we also consider $\phi$ non-minimally coupled with the Ricci scalar curvature $R$ through a Yukawa-like interaction. We do not consider further interactions of the spectator field with the fermions of the Standard Model, considering it \emph{de facto} as a dark scalar field. In evaluating the violation of the baryon-number conservation during the reheating epoch, in a perfectly homogeneous and isotropic universe, we follow a semiclassical approach, where $\theta$, $\phi$ and gravity are considered as classical fields, whereas the fermions are quantized. We solve the equations of motion for the inflaton and spectator fields, respectively at first and zero-order in perturbation theory, neglecting at first stage the expansion of the universe. Afterwards, we quantify how the spectator field modifies the inflationary dynamics and thus find the baryon asymmetry produced via the inflaton decays into fermion-antifermion pairs by computing the corresponding decay amplitudes. We therefore obtain small first order correction to standard spontaneous baryogenesis and finally discuss the mass-mixing between fermions. Accordingly, the effects of considering the universe expansion are accounted, showing when the coupling between $\phi$ and $R$ becomes noticeable in altering the overall baryon asymmetry.