Natural neutrino mass matrix

The naturalness of the neutrino mass hierarchy and mixing is studied. First we select a few patterns from among 12 neutrino mixing patterns, which could form the natural neutrino mass matrix. Further we show that if the Dirac neutrino mass matrix is taken as the natural one in the quark sector, then only two mixing patterns without large mixing lead to the natural right-handed Majorana mass matrix. The rest of the chosen patterns with three degenerate mass solutions lead to the unnatural right-handed Majorana mass matrix in the seesaw mechanism. Notice, however, that for the chosen two natural patterns there could be a huge mass hierarchy such as $\mathcal{O}{(10\mathrm{}}^4–{10}^6)$ in order to reproduce the inverse mass hierarchy of the light neutrinos.