Explicit Rephasing Transformation to PDG Parameterization and Simplified Expression of the Dirac CP Phase by Fermion-Specific Invariants

Abstract

In this letter, we present an explicit rephasing transformation that maps an arbitrary mixing matrix $U$ to the PDG standard parameterization $U^{0} = {\rm diag} ( e^{ - i \arg U_{e1} } \, , \, e^{ i \arg [ {U_{e2} U_{τ3} \over \det U } ]} \, , \, e^{ i \arg [ { U_{e2} U_{μ3} \over \det U } ] } )$ $U \, {\rm diag} ( 1 \, , \, e^{ - i \arg [{U_{e2} \over U_{e1} } ] } \, , \, e^{ - i \arg [ { U_{e2} U_{μ3} U_{τ3} \over \det U } ] })$. 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 the arguments. We also apply this framework to the fermion diagonalization matrices $U^{ν,e}$ under the approximation $U^{ν,e}_{13}=0$. By partially employing the inversion formula, we factorize all CP phases through the explicit rephasing, thereby revealing the entire CP structure of the mixing matrix. The observable Dirac phase depends only on two relative phases between the fermion sectors and is determined by fermion-specific invariants.