Common origin of θ₁₃ and Δm₁₂² 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 Δm₁₂²/Δm₂₃² 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 Δm₁₂² 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.