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

TL;DR

This paper develops an explicit rephasing method to transform any unitary matrix into the Kobayashi-Maskawa form, clarifying the fundamental phase structure of CP violation and expressing the KM phase through fermion-specific rephasing invariants.

Contribution

It introduces a new explicit rephasing transformation to connect arbitrary unitary matrices with the KM parameterization and identifies the fundamental CP phases as matrix element arguments.

Findings

Explicit rephasing transformation to KM form.

Representation of Majorana phases by fermion-specific phases.

Concise expression of KM phase via rephasing invariants.

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] $.