Explicit Rephasing to PDG Standard Form and Origin of CP-Violating Phases

Abstract: In this letter, we present a fully explicit rephasing transformation that maps an arbitrary mixing matrix to the PDG standard parameterization, a procedure that had remained conceptual for more than four decades. As a result, all six phases of the mixing matrix, including the Dirac and Majorana phases are expressed systematically using only the 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 Dirac-like phases, Majorana-like phases, and relative phases between different fermion sectors. This formulation provides a transparent and universal description of CP violation, with direct applications to both low-energy experiments and high-energy theories of flavor.