Lepton mixing angle ${\theta }_{13}=0$ with a horizontal symmetry $D_4$

Abstract

We discuss a model for the lepton sector based on the seesaw mechanism and on a $D_4$ family symmetry. The model predicts the mixing angle ${\theta }_{13}$ to vanish. The solar mixing angle ${\theta }_{12}$ is free--it will in general be large if one does not invoke finetuning. The model has an enlarged scalar sector with three Higgs doublets, together with two real scalar gauge singlets ${\chi }_i$ ($i=1,2$) which have vacuum expectation values < χ_i >_0$attheseesawscale.Theatmosphericmixingangle$θ_{23}$isgivenby$\tan θ_{23} = <χ_2>_0$/<{\chi }_1{>}_0$, and it is maximal if the Lagrangian is $D_4$-invariant; but $D_4$ may be broken softly, by a term of dimension two in the scalar potential, and then < χ_2_0$ becomes different from < χ_1_0$. Thus, the strength of the soft $D_4$ breaking controls the deviation of ${\theta }_{23}$ from $\pi /4$. The model predicts a normal neutrino mass spectrum ($m_3>m_2>m_1$) and allows successful leptogenesis if $m_1\sim 4\times 10^{-3}\mathrm{eV}$; these properties of the model are independent of the presence and strength of the soft $D_4$ breaking.

13 pages, one figure