$CP$ issues in the SM from a view point of spontaneous $CP$ violation

The standard model (SM) has several issues related to the $CP$ violation which could give clues to search physics beyond the SM. They are a $CP$ phase in the CKM matrix, the strong $CP$ problem, $CP$ phases in the PMNS matrix and $CP$ asymmetry in lepton number violating processes related to baryon number asymmetry. We consider a model which could give a unified explanation for them in a framework of spontaneous $CP$ violation. It is an extension of the SM with vector-like fermions and singlet scalars. In this model, they are explained by a common complex phase caused in the spontaneous $CP$ violation. We present concrete examples for them and also discuss some relevant phenomenology.