Rephasing invariant CP phases and sum rules in TM$_{1,2}$ mixing

Abstract

We show that the CP phases $ϕ_{1,2}$ appearing in the TM$_{1,2}$ mixing are directly identified with rephasing invariants $ϕ_{1} = - \arg \left[ { U_{e2} U_{e3} U_{μ1} U_{τ1} / U_{e 1} \det U } \right]$, $ϕ_{2} = - \arg \left[ { U_{e1} U_{e3} U_{μ2} U_{τ2} / U_{e 2} \det U} \right]$. Furthermore, we demonstrate relations $ϕ_{i} = δ- \arg [ U_{μi}^{0} U_{τi}^{0} ]$ among $ϕ_{1,2}$, the Dirac CP phase $δ$ and matrix elements in the PDG parametrization $U^{0}$. These relations of CP phases are interpreted as specific elements of general sum rules among physical quantities.