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

TL;DR

This work develops an explicit rephasing transformation to the Kobayashi–Maskawa parametrization, providing a complete mapping of CP phases to matrix-element arguments. By applying the transformation to fermion diagonalization matrices, it expresses observable CP phases as fermion-specific KM phases and their relative phases, clarifying the origin of CP violation. In the KM basis, Majorana phases become simpler and, under a simplifying limit where certain elements vanish, the KM phase δ_KM depends on only two relative phases, yielding a compact two-term expression. The results offer a systematic, rephasing-invariant framework for CP-violation analyses across quark and lepton sectors, with potential impact on neutrino physics, neutrinoless double beta decay, grand unification, and leptogenesis.

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 $δ^{ν, e}_{\rm KM}$ and their relative phases. In particular, by neglecting the 3-1 elements $U_{31}^{ν,e}$ of the diagonalization matrices for the two fermions, the KM phase $δ_{\rm KM}$ is concisely expressed by fermion-specific rephasing invariants involving two relative phases $δ_{\rm KM} = \arg \left [1 + ({U^{e * }_{21} U^ν_{21} / U^{e * }_{11} U^ν_{11} }) \right ] + \arg \left [ - { U_{32}^{e *} U_{32}^ν / U^{e * }_{22} U^ν_{22} } \right] $.