Approximately $\mu$-$\tau$ Symmetric Minimal Seesaw Mechanism and Leptonic CP Violation

Abstract

We estimate the effect of leptonic CP violating phases in the minimal seesaw model with two heavy neutrinos $N$ by imposing the approximately $\mu-\tau$ symmetry on the model. For $N$ subject to the $\mu-\tau$ symmetry, we find that neutrinos show the normal mass hierarchy and obtain the general phase structure that contains only two phases which arise from $\mu-\tau$ symmetry breaking terms. To perform similar consideration for $N$ blind to the $\mu-\tau$ symmetry necessarily giving the inverted mass hierarchy, we assume the same phase structure. Our phases consist of six phases that need to take care of generic phases of flavor neutrino masses, from which one can obtain one Dirac phase and one Majorana phase as observable phases (because one neutrino is massless). The Dirac CP violating phase $\delta_{CP}(=\delta+\rho)$ depends on the $\nu_e-\nu_{\tau}$ ($\nu_e-\nu_{\mu}$) mixing phase $\delta$ ($\rho$). The Majorana CP violating phase is suppressed because it depend on $\rho$, which is found to be associated with almost real quantity.