Rephasing invariant structure of CP phase for simplified mixing matrices in Fritzsch--Xing parametrization

TL;DR

The paper addresses how Dirac CP phases depend on the chosen mixing-parametrization by constructing an explicit rephasing transformation that maps any unitary mixing matrix to the Fritzsch--Xing (FX) form. It analyzes the rephasing-invariant structure of the FX phase $\delta_{\rm FX}$ under simplifying assumptions, deriving a compact expression in the limit $U_{13}^{e}=0$ and $U_{13}^{\nu}=0$, and extends to the general case with a perturbative treatment for small $|U^{e}_{23}|$. The results show that $\delta_{\rm FX}$ can be decomposed into two independent relative phases between $U^{e}$ and $U^{\nu}$, with a clean generalization to finite $U_{13}^{\nu}$, and establish explicit links between FX and PDG CP phases via determinant-based invariants. This framework offers a transparent interpretation of CP violation across parametrizations and provides a tool for flavor-model-building that leverages rephasing invariants to constrain CP-violating phases.

Abstract

In this paper, we construct an explicit rephasing transformation that converts an arbitrary unitary mixing matrix into the Fritzsch--Xing (FX) parametrization, which is obtained by trivializing arguments of the matrix elements in the third row and third column. We further analyzed the rephasing invariant structure of the FX phase $δ_{\rm FX}$ under the approximation where the 1-3 element of the diagonalization matrix of charged leptons $U^{e}$ is neglected, $U_{13}^{e} = 0$. As a result, in the limit where the 1-3 element of diagonalization of neutrinos $U^ν$ is also neglected, we obtained a compact expression for the FX phase $δ_{\rm FX} = \arg [ { U^{e}_{11} U^{e}_{12} U^{e}_{33} } { U^{ν*}_{11} U^{ν*}_{12} U^{ν*}_{33} / \det U^{e} \det U^{ν*} } ] - \arg [ 1 + ({U^{e *}_{23} U^ν_{23} / U^{e *}_{33} U^ν_{33}} ) ] $. In this limit, the FX phase is controlled by two independent relative phases between $U^{ν, e}$, and the FX phase for finite $U_{13}^ν$ is understood as a generalization to the compact expression.