Rotated $\mu$\,--\,$\tau$ Symmetry for One Generic Neutrino Mixing Angle: An analytical Study

We find a realization of the $Z_2$-symmetry in the neutrino mass matrix which expresses a rotation of the $\mu-\tau$ symmetry and is able to impose a generic smallest mixing angle, in contrast to a zero-value predicted by the usual non-rotated form of the $\mu-\tau$ symmetry. We extend this symmetry for the lepton sector within type-I seesaw scenario, and show it can accommodate the mixing angles, the mass hierarchies and the lepton asymmetry in the universe. We then study the effects of perturbing the specific form of the neutrino mass matrix imposed by the symmetry and compute the resulting mixing and mass spectrum. We trace back this "low-scale" perturbation to a "high-scale" perturbation, and find realizations of this latter one arising from exact symmetries with an enriched matter content.