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 $\sin(\delta+\rho)$. There is another redundant phase $\gamma$ to be removed by the rotation $\nu_\mu\to e^{i\gamma}\nu_\mu$ and $\nu_\tau\to e^{-i\gamma}\nu_\tau$. The perturbative result suggests how to calculate exact results of $\tan 2\theta_{12}e^{i\rho}$, and $\tan 2\theta_{13}e^{-i\delta}$. It is further shown that $\theta_{23}$ is completely determined to show that, for $\sin\theta_{13}\neq 0$, the maximal atmospheric neutrino mixing is necessarily linked to the maximal CP violation if ${\rm\bf M}_{\mu\mu}={\rm\bf M}_{\tau\tau}$