$\mu \text{−}\tau$ symmetry breaking and $CP$ violation in the neutrino mass matrix

The $\mu \text{−}\tau$ exchange symmetry in the neutrino mass matrix and its breaking as a perturbation are discussed. The exact $\mu \text{−}\tau$ symmetry restricts the 2-3 and 1-3 neutrino mixing angles as ${\theta }_{23}=\pi /4$ and ${\theta }_{13}=0$ at a zeroth-order level. We claim that the $\mu \text{−}\tau$ symmetry breaking prefers a large $CP$ violation to realize the observed value of ${\theta }_{13}$ and to keep ${\theta }_{23}$ nearly maximal, though an artificial choice of the $\mu \text{−}\tau$ breaking can tune ${\theta }_{23}$, irrespective of the $CP$ phase. We exhibit several relations among the deviation of ${\theta }_{23}$ from $\pi /4$, ${\theta }_{13}$, and the Dirac $CP$ phase $\delta$, which are useful to test the $\mu \text{−}\tau$ breaking models in near-future experiments. We also propose a concrete model to break the $\mu \text{−}\tau$ exchange symmetry spontaneously, and its breaking is mediated by the gauge interactions radiatively in the framework of the extended gauge model with $B-L$ and $L_{\mu }-L_{\tau }$ symmetries. As a result of the gauge-mediated $\mu \text{−}\tau$ breaking in the neutrino mass matrix, the artificial choice is unlikely, and a large Dirac $CP$ phase is preferable.