The unsuppressed CP violation in QCD is a problem in the standard model. If we have some mechanism to guarantee real determinants of the quark mass matrices, the vanishing physical vacuum angle θ indicates the CP invariance at the fundamental level. Thus, the small θ is technically natural, since we have an enhanced CP symmetry in the limit of the vanishing θ=0. In fact, it was proved that the vacuum angle is never renormalized up to the four-loop level once it is fixed at 0 value at some high energy scale. The purpose of this paper is to construct a model which guarantees the real determinants of the quark mass matrices assuming a non-invertible symmetry.
Kobayashi et al. (Thu,) studied this question.