Common origin of ${\theta }_{13}$ and $\Delta m_{12}^2$ in a model of neutrino mass with quaternion symmetry

The smallness of the 1–3 lepton mixing angle ${\theta }_{13}$ and of the neutrino mass-squared-difference ratio $\Delta m_{12}^2/\Delta m_{23}^2$ 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 $\Delta m_{12}^2$ to the value of ${\theta }_{13}$. In such a model we predict $0.12\lesssim \sin {\theta }_{13}\lesssim 0.2$ if the ordering of the neutrino mass spectrum is normal, and $\sin {\theta }_{13}\lesssim 0.12$ if it is inverted.

Figure 1

The 1–3 mixing as a function of $\Delta m_{12}^2/\Delta m_{23}^2$ for the matrix $m_{\nu }$ in Eq. (9), assuming no $CP$ violation. The left (right) panel corresponds to the case of normal (inverted) ordering of the mass spectrum. We imposed the constraints $0.24<{sin}^2{\theta }_{12}<0.40$ ($3\sigma$ allowed interval) as well as $\sum _i{m}_i<1\mathrm{eV}$. The dashed rectangle indicates the $3\sigma$ allowed region defined by $0.022<\Delta m_{12}^2/\Delta m_{23}^2<0.045$ and $\sin {\theta }_{13}<0.2$. The gray points are those where $d$ is significantly smaller (less than one third) than $a$, $b$, and $|c|$.

Figure 2

The 1–3 mixing as a function of the absolute mass scale for the matrix $m_{\nu }$ in Eq. (9), assuming no $CP$ violation. The left (right) panel shows the allowed region in the $m_1-\sin {\theta }_{13}$ ($m_3-\sin {\theta }_{13}$) plane for the case of normal (inverted) ordering of the mass spectrum. We imposed the constraints $0.022<\Delta m_{12}^2/\Delta m_{23}^2<0.045$ and $0.24<{sin}^2{\theta }_{12}<0.40$ ($3\sigma$ allowed intervals). The dashed line indicates the $3\sigma$ upper bound $\sin {\theta }_{13}<0.2$. The gray points are those where $d$ is significantly smaller (less than one third) than $a$, $b$, and $|c|$.

Figure 3

The allowed region in the plane ${sin}^2{\theta }_{12}-\sin {\theta }_{13}$ for the matrix $m_{\nu }$ in Eq. (9) with the most general $CP$ violating phases. The left (right) panel corresponds to the case of normal (inverted) ordering of the mass spectrum. We imposed the constraints $0.022<\Delta m_{12}^2/\Delta m_{23}^2<0.045$ ($3\sigma$ allowed interval) as well as $\sum _i{m}_i<1\mathrm{eV}$. The dashed rectangle indicates the $3\sigma$ allowed region defined by $0.24<{sin}^2{\theta }_{12}<0.4$ and $\sin {\theta }_{13}<0.2$. The gray points are those where $|d|$ is significantly smaller (less than one third) than $|a|$, $|b|$, and $|c|$.