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Common origin of θ 13 and ∆ m 2 12 in a model of neutrino mass with quaternion symmetry

The smallness of the 1 − 3 lepton mixing angle θ13 and of the neutrino mass-squared-difference ratio ∆m12/∆m 2 23 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 ∆m12 to the value of θ13. In such a model we predict 0.12 . sin θ13 . 0.2 if the ordering of the neutrino mass spectrum is normal, and sin θ13 . 0.12 if it is inverted.