Correlation between leptonic $CP$ violation and $\mu \mathrm{\text{−}}\tau$ symmetry breaking

Considering the $\mu \mathrm{\text{−}}\tau$ symmetry, we discuss a direct linkage between phases of flavor neutrino masses and leptonic $CP$ violation by determining three eigenvectors associated with $\mathbf{M}=M_{\nu }^{†}{M}_{\nu }$ for a complex flavor neutrino mass matrix $M_{\nu }$ in the flavor basis. Since the Dirac $CP$ violation is absent in the $\mu \mathrm{\text{−}}\tau$ symmetric limit, leptonic $CP$ violation is sensitive to the $\mu \mathrm{\text{−}}\tau$ symmetry breaking, whose effect can be evaluated by perturbation. It is found that the Dirac phase ($\delta$) arises from the $\mu \mathrm{\text{−}}\tau$ symmetry breaking part of ${\mathbf{M}}_{e\mu ,e\tau }$ and an additional phase ($\rho$) is associated with the $\mu \mathrm{\text{−}}\tau$ symmetric part of ${\mathbf{M}}_{e\mu ,e\tau }$, where ${\mathbf{M}}_{ij}$ 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 ${\nu }_e\mathrm{\text{−}}{\nu }_{\tau }$ mixing angle.