Almost general analysis of CKM and MNS matrices for hierarchical Yukawa structure and interpretation of Dirac CP phase probed by DUNE and T2HK

In this letter, we perform an almost general analysis of flavor-mixing matrices $V_{\mathrm{C}\mathrm{K}\mathrm{M}}$ and $U_{\mathrm{M}\mathrm{N}\mathrm{S}}$ to investigate the discriminative power of CP phases by next-generation neutrino oscillation experiments. As an approximation, we neglect the 1-3 mixing of diagonalization for more hierarchical fermions $u,e$. Thus there are two sources of CP violation in $V_{\mathrm{C}\mathrm{K}\mathrm{M}}$ and $U_{\mathrm{M}\mathrm{N}\mathrm{S}}$, the intrinsic CP phase ${\delta }_{d,\nu }$ in diagonalization of less hierarchical fermions $d,\nu$ and relative phases between two unitary matrices. By eliminating unphysical phases and imposing constraints of the three measured mixing angles, the flavor-mixing matrices are analytically displayed by two phases and the 1-2 mixing $s_{u,e}$ of more hierarchical fermions. For sufficiently small 1-2 mixing $s_e$ of charged leptons, the Dirac phase $\delta$ is mostly identical to the intrinsic phase of neutrinos ${\delta }_{\nu }$. Therefore, future detection of the Dirac phase indicates the observation of ${\delta }_{\nu }$. On the other hand, if such a CP violation is not observed, an upper limit is placed on a combination of ${\delta }_{\nu }$ and relative phases.