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

Abstract

We show that the CP phases $\phi_{1,2}$ appearing in the TM$_{1,2}$ mixing are directly identified with rephasing invariants $\phi _{1} = - \arg \left[ { U_{e2} U_{e3} U_{\mu 1} U_{\tau 1} / U_{e 1} \det U } \right]$, $\phi_{2} = - \arg \left[ { U_{e1} U_{e3} U_{\mu 2} U_{\tau 2} / U_{e 2} \det U} \right]$. Furthermore, we demonstrate identities $\phi_{i} = \delta - \arg [ U_{\mu i}^{0} U_{\tau i}^{0} ]$ among $\phi_{1,2}$, the Dirac CP phase $\delta$ and matrix elements in the PDG parametrization $U^{0}_{\alpha i}$. These relations of CP phases are interpreted as specific elements of general sum rules represented by phase matrices.