Neutrino mass, mixing and discrete symmetries

Abstract

Status of the discrete symmetry approach to explanation of the lepton masses and mixing is summarized in view of recent experimental results, in particular, establishing relatively large 1-3 mixing. The lepton mixing can originate from breaking of discrete flavor symmetry $G_f$ to different residual symmetries $G_{\ell}$ and $G_\nu$ in the charged lepton and neutrino sectors. In this framework the {\it symmetry group condition} has been derived which allows to get relations between the lepton mixing elements immediately without explicit model building. The condition has been applied to different residual neutrino symmetries $G_\nu$. For generic (mass independent) $G_\nu = {\bf Z}_2$ the condition leads to two relations between the mixing parameters and fixes one column of the mixing matrix. In the case of $G_\nu = {\bf Z}_2 \times {\bf Z}_2$ the condition fixes the mixing matrix completely. The non-generic (mass spectrum dependent) $G_\nu$ lead to relations which include mixing angles, neutrino masses and Majorana phases. The symmetries $G_{\ell}$, $G_\nu$, $G_f$ are identified which lead to the experimentally observed values of the mixing angles and allow to predict the CP phase.