Explicit rephasing transformation to PDG parameterization and simplified expression of Dirac CP phase by fermion-specific invariants

TL;DR

This paper introduces an explicit rephasing transformation to convert any mixing matrix into the PDG standard form, simplifying the expression of CP phases using fermion-specific invariants, which enhances understanding of CP violation in fermion mixing.

Contribution

It provides a systematic explicit rephasing transformation to express mixing matrix phases in terms of matrix element arguments and extends this to fermion diagonalization matrices, clarifying CP phase relations.

Findings

Explicit rephasing transformation to PDG parameterization.

Simplified expression of Dirac CP phase using invariants.

Framework applicable to different fermion sectors.

Abstract

In this letter, we present an explicit rephasing transformation that maps an arbitrary mixing matrix $U$ to the PDG standard parameterization $ U^{0} = {\rm diag} ( e^{ - i \arg U_{e1} } \, , \, e^{ i \arg [ {U_{e2} U_{\tau 3} \over \det U } ]} \, , \, e^{ i \arg [ { U_{e2} U_{\mu 3} \over \det U } ] } )$ $U \, {\rm diag} ( 1 \, , \, e^{ - i \arg [{U_{e2} \over U_{e1} } ] } \, , \, e^{ - i \arg [ { U_{e2} U_{\mu 3} U_{\tau 3} \over \det U } ] })$. By this procedure, which has remained largely conceptual for more than four decades, six independent phases of the mixing matrix are expressed systematically in terms of arguments of matrix elements and the determinant. We further extend this framework to the diagonalization matrices of fermions, allowing the observable CP phases to be expressed by fermion-specific rephasing invariants between different fermion sectors.