Broken $\mu \mathrm{\text{−}}\tau$ symmetry and leptonic $CP$ violation

We propose that leptonic $CP$ violation arises from a breaking of the $\mu \mathrm{\text{−}}\tau$ exchange symmetry in the neutrino mass matrix which in turn is indicated by the near maximal atmospheric neutrino mixing and the near zero ${\theta }_{13}$. We find that for the case of a normal hierarchy, present data already restrict the way $CP$ violation may appear in the neutrino mass matrix and there is an interesting correlation between the mixing angle ${\theta }_{13}$, the solar mixing angle ${\theta }_{12}$ and the Dirac $CP$ phase. In the inverted hierarchy only ${\theta }_{13}$ and the Dirac phase are linked. We also discuss the impact of this kind of $CP$ violation on the deviation of the atmospheric mixing from its maximal value. Moreover, if corrections to $\mu \mathrm{\text{−}}\tau$ symmetry arise from the charged lepton sector, where $\mu \mathrm{\text{−}}\tau$ symmetry is known to be broken anyway, we find interesting connections between the $CP$-even and $CP$-odd terms as well. Our predictions are testable in the proposed long baseline neutrino experiments.

Figure 1

Behavior of $|U_{e3}|$, $\sin \delta$ and ${sin}^2{\theta }_{12}$ as a function of the $\mu \mathrm{\text{−}}\tau$ symmetry breaking phase $\alpha$, as it follows from Eq. (4). For this particular example we chose $a=1.4$ and $ϵ=0.16$. We also indicated the current best-fit value and $3\sigma$ range of ${sin}^2{\theta }_{12}$ from 19.

Figure 2

Behavior of ${sin}^2{\theta }_{12}$ and $J_{CP}$ as a function of the phase $\varphi$, as it follows from Eq. (21). In this example $|U_{e3}|\simeq 0.16$ and $\psi ={60}^0$, $B=0.8$ and $\omega ={90}^0$.