$μ$-$τ$ symmetry breaking and CP violation in the neutrino mass matrix

The $μ$-$τ$ exchange symmetry in the neutrino mass matrix and its breaking as a perturbation are discussed. The exact $μ$-$τ$ symmetry restricts the 2-3 and 1-3 neutrino mixing angles as $θ_{23} = π/4$ and $θ_{13} = 0$ at a zeroth order level. We claim that the $μ$-$τ$ symmetry breaking prefers a large CP violation to realize the observed value of $θ_{13}$ and to keep $θ_{23}$ nearly maximal, though an artificial choice of the $μ$-$τ$ breaking can tune $θ_{23}$, irrespective of the CP phase. We exhibit several relations among the deviation of $θ_{23}$ from $π/4$, $θ_{13}$ and Dirac CP phase $δ$, which are useful to test the $μ$-$τ$ breaking models in the near future experiments. We also propose a concrete model to break the $μ$-$τ$ 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 $L_μ- L_τ$ symmetries. As a result of the gauge mediated $μ$-$τ$ breaking in the neutrino mass matrix, the artificial choice is unlikely, and a large Dirac CP phase is preferable.

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