Explicit rephasing transformation of four-generation mixing matrix and formulae for CP phases

TL;DR

This paper introduces an explicit rephasing transformation for four-generation mixing matrices, providing formulas for all CP phases and unphysical phases, which simplifies to known three-generation results when the fourth generation is decoupled.

Contribution

It presents a systematic method to derive all CP and unphysical phases in four-generation models using rephasing-covariant minors and determinants.

Findings

Derived explicit formulas for all CP phases in four-generation mixing matrices.

Provided a unified framework that reduces to three-generation results in the decoupling limit.

Enhanced understanding of phase structure in extended flavor mixing models.

Abstract

In this letter, we present an explicit rephasing transformation that maps a general $4\times4$ flavor mixing matrix to the standard parametrization of four-generation models. By combining rephasing-covariant minors and the determinant systematically, we derive expressions for all three Dirac-type CP phases, three Majorana phases, and four unphysical phases by arguments of matrix elements. The resulting formulae smoothly reduce to the three-generation results in the limit where the fourth generation decouples.