CP violating phase sum rule $δ^q_{\rm KM}+δ^l_{\rm KM}=0$ for CKM and PMNS matrices

Abstract: The non-zero Dirac phases $\delta<sup>q$ and $\delta<sup>l$ in the CKM and PMNS mixing matrices signify CP violation. In general they are independent. Experimental data including recent T2K results show, however, that in the original KM parameterization for the mixing matrix, the sum $$\delta^q_{\rm KM} + \delta^l_{\rm KM}$$ is close to zero with $\delta^q_{\rm KM}$ to be approximately $\pi/2$. The KM parameterization may have provided some hints that these phases are actually related and CP is maximally violated. We show that this sum rule can be accommodated in models with spontaneous CP violation where both phases originate from a non-trivial common spontaneous CP violating maximized phase in the Higgs potential. We find some interesting phenomenological consequences for flavor changing neutral current and CP violation for such a model. In particular, data from $B_s - \bar B_s$ mixing provide very strong constraints on the mass scale for the new neutral scalars in the model, yet the model still allows the electric dipole moments of electron and neutron to reach to their current upper bounds. The model can be tested by near future experiments.