Neutrino mixing in SO(10) GUTs with non-abelian flavor symmetry in the hidden sector

The relation between the mixing matrices of leptons and quarks: $U_{\rm{PMNS}} \approx V_{\rm{CKM}}^\dagger U_0$, where $U_0$ is a matrix of special forms (e.g. BM, TBM), can be a clue for understanding the lepton mixing and neutrino masses. It may imply the Grand unification and existence of a hidden sector with certain symmetry which generates $U_0$ and leads to the smallness of neutrino masses. We apply the residual symmetry approach to obtain $U_0$. The residual symmetries of both the visible and hidden sectors are $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$. Their embedding in a unified flavor group is considered. We find that there are only several possible structures of $U_0$, including the BM mixing and matrices with elements determined by the golden ratio. Realization of the BM scenario based on the $SO(10)$ GUT with the $S_4$ flavor group is presented. Generic features of this scenario are discussed, in particular, the prediction of CP phase $144^{\circ}\lesssim\delta_{\rm CP}\lesssim 210^{\circ}$ in the minimal version.