Common origin ofθ13andΔm122in a model of neutrino mass with quaternion symmetry

Abstract

The smallness of the 1--3 lepton mixing angle ${\ensuremath{\theta}}_{13}$ and of the neutrino mass-squared-difference ratio $\ensuremath{\Delta}{m}_{12}^{2}/\ensuremath{\Delta}{m}_{23}^{2}$ can be understood as the departure from a common limit where they both vanish. We discuss in general the conditions for realizing the mass degeneracy of a pair of neutrinos and show that the vanishing of a $CP$ violating phase is needed. We find that the discrete quaternion group $Q$ of eight elements is the simplest family symmetry which correlates the smallness of $\ensuremath{\Delta}{m}_{12}^{2}$ to the value of ${\ensuremath{\theta}}_{13}$. In such a model we predict $0.12\ensuremath{\lesssim}\mathrm{sin}{\ensuremath{\theta}}_{13}\ensuremath{\lesssim}0.2$ if the ordering of the neutrino mass spectrum is normal, and $\mathrm{sin}{\ensuremath{\theta}}_{13}\ensuremath{\lesssim}0.12$ if it is inverted.