Large neutrino mixing and normal mass hierarchy: A discrete understanding

We discuss the possibility of flavor symmetries to explain the pattern of charged lepton and neutrino masses and mixing angles. We emphasize what are the obstacles for the generation of an almost maximal atmospheric mixing and what are the minimal ingredients to obtain it. A model based on the discrete symmetry $S_3$ is constructed, which leads to the dominant $\mu \tau$-block in the neutrino mass matrix, thus predicting normal hierarchy. This symmetry makes it possible to reproduce current data and predicts $0.01\lesssim {\theta }_{13}\lesssim 0.03$ and strongly suppressed neutrinoless $2\beta$-decay. Moreover, it implies a relation between lepton and quark mixing angles: ${\theta }_{23}^q\approx 2(\pi /4-{\theta }_{23})$. The Cabibbo mixing can also be reproduced and ${\theta }_{13}^q\sim {\theta }_{12}^q{\theta }_{23}^q$. $S_3$ is thus a candidate to describe all the basic features of standard model fermion masses and mixing.