Models of maximal atmospheric neutrino mixing and leptogenesis

We discuss two extensions of the Standard Model based on the seesaw mechanism and on non-abelian family symmetry groups O(2) and $D_4$, respectively. Both models have a twofold-degenerate neutrino Dirac mass matrix $M_D$, a Majorana mass matrix invariant under a $\mu$--$\tau$ interchange symmetry and the predictions of maximal atmospheric neutrino mixing and vanishing mixing angle $\theta_{13}$. Leptogenesis can naturally be incorporated if $10^{-3} \lesssim m_1 \lesssim 10^{-2}$ eV where $m_1$ the mass of the lightest neutrino and if the relevant heavy neutrinos are in the range $10^{11}$ to $10^{12}$ GeV. The $D_4$ model is more constrained and leptogenesis requires $m_1$ to be in the vicinity of $4 \times 10^{-3}$ eV.