A partial $\mu$-$\tau$ symmetry and its prediction for leptonic CP violation

We find that the lepton flavor mixing matrix $U$ should possess a partial $\mu$-$\tau$ permutation symmetry $|U^{}_{\mu 1}| = |U^{}_{\tau 1}|$, and the latter predicts a novel correlation between the Dirac CP-violating phase $\delta$ and three flavor mixing angles $\theta^{}_{12}$, $\theta^{}_{13}$ and $\theta^{}_{23}$ in the standard parametrization. Inputting the best-fit values of these angles reported by Capozzi {\it et al}, we obtain the prediction $\delta \simeq 255^\circ$ in the normal neutrino mass ordering, which is in good agreement with the best-fit result $\delta \simeq 250^\circ$. In this connection the inverted neutrino mass ordering is slightly disfavored. If this partial $\mu$-$\tau$ symmetry is specified to be $|U^{}_{\mu 1}| = |U^{}_{\tau 1}| =1/\sqrt{6}~$, one can reproduce the phenomenologically-favored relation $\sin^2\theta^{}_{12} = \left(1 - 2\tan^2\theta^{}_{13}\right)/3$ and a viable two-parameter description of $U$ which were first uncovered in 2006. Moreover, we point out that the octant of $\theta^{}_{23}$ and the quadrant of $\delta$ can be resolved thanks to the slight violation of $|U^{}_{\mu 2}| = |U^{}_{\tau 2}|$ and $|U^{}_{\mu 3}| = |U^{}_{\tau 3}|$ either at the tree level or from radiative corrections.