Predicting neutrino parameters from SO(3) family symmetry and quark-lepton unification

We show how the neutrino mixing angles and oscillation phase can be predicted from tri-bimaximal neutrino mixing, corrected by charged lepton mixing angles which are related to quark mixing angles via quark-lepton unification. The tri-bimaximal neutrino mixing can naturally originate from the see-saw mechanism via constrained sequential dominance (CSD), where CSD can result from the vacuum alignment of a non-Abelian family symmetry such as SO(3). We construct a realistic model of quark and lepton masses and mixings based on SO(3) family symmetry with quark-lepton unification based on the Pati-Salam gauge group. The atmospheric angle is predicted to be approximately maximal $\theta_{23}= 45^\circ$, corrected by the quark mixing angle $\theta_{23}^{\mathrm{CKM}}\approx 2.4^\circ$, with the correction controlled by an undetermined phase in the quark sector. The solar angle is predicted by the tri-bimaximal complementarity relation: $\theta_{12}+ \frac{1}{\sqrt{2}}\frac{\theta_{\mathrm{C}}}{3} \cos (\delta - \pi) \approx 35.26^\circ$, where $\theta_{\mathrm{C}}$ is the Cabibbo angle and $\delta$ is the neutrino oscillation phase. The reactor angle is predicted to be $\theta_{13} \approx \frac{1}{\sqrt{2}}\frac{\theta_{\mathrm{C}}}{3}\approx 3.06^\circ$. The MNS neutrino oscillation phase $\delta$ is predicted in terms of the solar angle to be $\cos (\delta - \pi) \approx (35.26^\circ - \theta_{12}^\circ)/3.06^\circ$. These predictions can all be tested by future high precision neutrino oscillation experiments, thereby probing the nature of high energy quark-lepton unification.