Explicit rephasing to Kobayashi-Maskawa representation and fundamental phase structure of CP violation

Abstract

In this letter, we construct an explicit rephasing transformation that converts an arbitrary unitary matrix into the Kobayashi--Maskawa (KM) parameterization and identify all independent CP phases in the mixing matrix as the arguments of its matrix elements. Furthermore, by applying this rephasing transformation to the fermion diagonalization matrices $U^{f}$, we show that the Majorana phases are represented by fermion-specific phases $\delta^{\nu, e}_{\rm KM}$ and their relative phases. In particular, by neglecting the 3-1 elements $U_{31}^{\nu ,e}$ of the diagonalization matrices for the two fermions, the KM phase $\delta_{\rm KM}$ is concisely expressed by fermion-specific rephasing invariants involving two relative phases $\delta_{\rm KM} = \arg \left [1 + ({U^{e * }_{21} U^{\nu}_{21} / U^{e * }_{11} U^{\nu}_{11} }) \right ] + \arg \left [ - { U_{32}^{e *} U_{32}^{\nu} / U^{e * }_{22} U^{\nu}_{22} } \right] $.