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

Abstract

We show that the CP phases ${\varphi }_{1,2}$ appearing in the TM${}_{1,2}$ mixing are directly identified with rephasing invariants ${\varphi }_1=-\arg \left[U_{e2}{U}_{e3}{U}_{\mu 1}{U}_{\tau 1}/U_{e1}\det U\right]$, ${\varphi }_2=-\arg \left[U_{e1}{U}_{e3}{U}_{\mu 2}{U}_{\tau 2}/U_{e2}\det U\right]$. Furthermore, we demonstrate identities ${\varphi }_i=\delta -\arg [U_{\mu i}^0{U}_{\tau i}^0]$ among ${\varphi }_{1,2}$, the Dirac CP phase $\delta$ and matrix elements in the PDG parametrization $U_{\alpha i}^0$. These relations of CP phases are interpreted as specific elements of general sum rules represented by phase matrices.

Comments: 7 pages