Explicit rephasing transformation to PDG parameterization and simplified expression of Dirac CP phase by fermion-specific invariants

In this letter, we present an explicit rephasing transformation that maps an arbitrary mixing matrix $U$ to the PDG standard parameterization $U^0=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(e^{-i\arg U_{e1}},e^{i\arg [\frac{U_{e2}{U}_{\tau 3}}{\det U}]},e^{i\arg [\frac{U_{e2}{U}_{\mu 3}}{\det U}]})$ $U\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(1,e^{-i\arg [\frac{U_{e2}}{U_{e1}}]},e^{-i\arg [\frac{U_{e2}{U}_{\mu 3}{U}_{\tau 3}}{\det U}]})$. By this procedure, which has remained largely conceptual for more than four decades, six independent phases of the mixing matrix are expressed systematically in terms of arguments of matrix elements and the determinant. We further extend this framework to the diagonalization matrices of fermions, allowing the observable CP phases to be expressed by fermion-specific rephasing invariants between different fermion sectors.