GUT-motivated non-invertible symmetry as a solution to the strong CP problem and the neutrino CP-violating phase

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 $\bar{\theta }$ indicates the CP invariance at the fundamental level. Thus, the small $\bar{\theta }$ is technically natural, since we have an enhanced CP symmetry in the limit of the vanishing $\bar{\theta }=0$. 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.