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

TL;DR

The paper develops an explicit rephasing transformation for a general $U$ in four-generation models to reach a standard form $U_{std}$, enabling a transparent decomposition of CP violation. By combining rephasing-invariant minors and the determinant $det(U)$, it derives closed-form expressions for all three Dirac-type CP phases, three Majorana phases, and four unphysical phases, reducing correctly to the three-generation results when the fourth generation decouples. The authors provide a complete mapping from an arbitrary basis to the standard parametrization and express all ten phases as functions of matrix-element arguments and minors, including consistency sum rules. This approach yields a practical tool for analyzing CP violation in four-generation and sterile-neutrino scenarios and extends directly to fermion mass diagonalization matrices $U_f$.

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.