Rephasing invariant structure of CP phase for simplified mixing matrices in Fritzsch--Xing parametrization

In this paper, we construct an explicit rephasing transformation that converts an arbitrary unitary mixing matrix into the Fritzsch–Xing (FX) parametrization, which is obtained by trivializing arguments of the matrix elements in the third row and third column. We further analyze rephasing invariant structure of the FX phase ${\delta }_{\mathrm{F}\mathrm{X}}$ under an approximation $U_{13}^e=0$, where the 1-3 element of the diagonalization matrix of charged leptons $U^e$ is neglected. With an additional approximation $U_{23}^e=0$, the FX phase becomes highly simplified, reducing to a sum of the neutrino-intrinsic FX phase ${\delta }_{\mathrm{F}\mathrm{X}}^{\nu }$ and the contribution from the relative phase ${\rho }_1^{\mathrm{\prime }}-{\rho }_2^{\mathrm{\prime }}$ between the lighter 1-2 generations. The phase ${\delta }_{\mathrm{F}\mathrm{X}}$ for finite $U_{23}^e$ is understood as a generalization of the compact expression. This result covers almost all perturbative calculations of CP phases for the CKM and MNS matrices with hierarchical charged fermions.