Spontaneous $CP$-Violating Electroweak Baryogenesis and Dark Matter from a Complex Singlet Scalar

CP non-invariance is strongly limited by present experiments, while extra sources of CP-violation are needed for a successful baryogenesis. Motivated by those observations we consider a model which predicts spontaneous violation of CP at high temperature and restoration of CP at present temperature of the Universe. In addition we propose a dark matter (DM) candidate that meets all known properties of DM. Looking for a minimal model that satisfies the above conditions leads us to extending the Standard Model (SM) of fundamental interactions by adding a complex singlet scalar $S$. We impose the $CP$ and $Z_2$ symmetries on the scalar potential. With the complex vacuum expectation value of $S$ at the temperature higher than the EW phase transition, the $CP$ symmetry is spontaneously broken and a strong first-order electro-weak phase transition is easily realized. Introducing a dimension-6 effective operator that gives new complex contributions to the top quark mass, we show that it is easy to yield the observed baryon asymmetry in our Universe. On the other hand, the $CP$ and $Z_2$ symmetries are recovered after the EW phase transition so that the present strong constraints on $CP$ violation can be satisfied and the lighter of $\Re S$ or $\Im S$ can be the dark matter candidate. By scanning the parameter space, we find regions where the model can explain the dark matter relic abundance and the baryon asymmetry simultaneously while satisfying all other experimental constraints. Finally, we discuss the explicit $CP$ symmetry breaking in the scalar potential that can help dynamically eliminate the domains producing the negative baryon asymmetry. It is found that this can be achieved by a tiny explicit $CP$-violating phase of ${\cal O}(10^{-15})$.