Correlation between Leptonic CP Violation and mu-tau Symmetry Breaking

Considering the $\mu$-$\tau$ symmetry, we discuss a direct linkage between phases of flavor neutrino masses and leptonic CP violation by determining three eigenvectors associated with ${\rm\bf M}=M^\dagger_\nu M_\nu$ for a complex flavor neutrino mass matrix $M_\nu$ in the flavor basis. Since the Dirac CP violation is absent in the $\mu$-$\tau$ symmetric limit, leptonic CP violation is sensitive to the $\mu$-$\tau$ symmetry breaking, whose effect can be evaluated by perturbation. It is found that the Dirac phase ($\delta$) arises from the $\mu$-$\tau$ symmetry breaking part of ${\rm\bf M}_{e\mu,e\tau}$ and an additional phase ($\rho$) is associated with the $\mu$-$\tau$ symmetric part of ${\rm\bf M}_{e\mu,e\tau}$, where ${\rm\bf M}_{ij}$ stands for an $ij$ matrix element ($i,j$=$e,\mu,\tau$). The phase $\rho$ is redundant and can be removed but leaves its effect in the Dirac CP violation characterized by $\sin (\delta + \rho)$. The perturbative results suggest the exact formula of mixing parameters including that of $\delta$ and $\rho$, 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 $\sin\theta_{13}=0$ or $\cos(\delta+\rho)=0$, the latter of which indicates maximal CP violation, where $\theta_{13}$ is the $\nu_e$-$\nu_\tau$ mixing angle.