Small $\theta_{13}$ and solar neutrino oscillation parameters from $\mu-\tau$ symmetry

Exchange $\mu-\tau$ symmetry in the effective Majorana neutrino mass matrix does predict a maximal mixing for atmospheric neutrino oscillations asides to a null mixing that cannot be straightforwardly identified with reactor neutrino oscillation mixing, $\theta_{13}$, unless a specific ordering is assumed for the mass eigenstates. Otherwise, a non zero value for $\theta_{13}$ is predicted already at the level of an exact symmetry. In this case, solar neutrino mixing and scale, as well as the correct atmospheric mixing, arise from the breaking of the symmetry. I present a mass matrix proposal for normal hierarchy that realizes this scenario, where the smallness of $\tan\theta_{13}$ is naturally given by the parameter $\epsilon\sim\sqrt{\Delta m^2_{sol}/\Delta m^2_{ATM}}$ and the solar mixing is linked to the smallness of $\Delta m^2_{sol}$. The proposed matrix remains stable under renormalization effects and it also allows to account for CP violation within the expected region without further constrains.