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 $\mathrm{\mid }{U}_{\mu 1}^{}\mathrm{\mid }=\mathrm{\mid }{U}_{\tau 1}^{}\mathrm{\mid }$, 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 $|U^{}{\mu 1}| = |U^{}{\tau 1}| =1/\sqrt{6}~$,onecanreproducethephenomenologically-favoredrelation$\sin^2\theta^{}{12} = \left(1 - 2\tan^2\theta^{}{13}\right)/3$ 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.