Correlation between Leptonic CP Violation and mu-tau Symmetry Breaking

Abstract

Considering the $μ$-$τ$ symmetry, we discuss a direct linkage between phases of flavor neutrino masses and leptonic CP violation by determining three eigenvectors associated with ${\rm\bf M}=M^\dagger_νM_ν$ for a complex flavor neutrino mass matrix $M_ν$ in the flavor basis. Since the Dirac CP violation is absent in the $μ$-$τ$ symmetric limit, leptonic CP violation is sensitive to the $μ$-$τ$ symmetry breaking, whose effect can be evaluated by perturbation. It is found that the Dirac phase ($δ$) arises from the $μ$-$τ$ symmetry breaking part of ${\rm\bf M}_{eμ,eτ}$ and an additional phase ($ρ$) is associated with the $μ$-$τ$ symmetric part of ${\rm\bf M}_{eμ,eτ}$, where ${\rm\bf M}_{ij}$ stands for an $ij$ matrix element ($i,j$=$e,μ,τ$). The phase $ρ$ is redundant and can be removed but leaves its effect in the Dirac CP violation characterized by $\sin (δ+ ρ)$. The perturbative results suggest the exact formula of mixing parameters including that of $δ$ and $ρ$, which turns out to be free from the effects of the redundant phases. As a result, it is generally shown that the maximal atmospheric neutrino mixing necessarily accompanies either $\sinθ_{13}=0$ or $\cos(δ+ρ)=0$, the latter of which indicates maximal CP violation, where $θ_{13}$ is the $ν_e$-$ν_τ$ mixing angle.