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Predicting CP violation in Deviation from Tri-bimaximal mixing of Neutrinos

We study the CP violation in the deviation from the tri-bimaximal mixing (TBM) of neutrinos. We examine non-trivial relations among the mixing angles and the CP violating Dirac phase in the typical four cases of the deviation from the TBM. The first two cases are derived by the additional rotation of the 2-3 or 1-3 generations of neutrinos in the TBM basis. The other two cases are given by the additional rotation of the 1-3 or 1-2 generations of charged leptons with the TBM neutrinos. These four cases predict different relations among three mixing angles and the CP violating Dirac phase. The rotation of the 2-3 generations of neutrinos in the TBM basis predicts $\sin ^2θ_{12}<1/3$, and the CP violating Dirac phase to be $\pm (0.09π\sim 0.76π)$ for NH ($\pm (0.15π\sim 0.73π) \ \text{for IH}$) depending on $\sin ^2θ_{23}$. The rotation of the 1-3 generations of neutrinos in the TBM basis gives $\sin ^2θ_{12}>1/3$. The CP violating Dirac phase is not constrained by the input of the present experimental data. For the case of the 1-3 and 1-2 rotations of charged leptons in the TBM basis, the CP violating Dirac phase is predicted in $\pm(0.35π\sim 0.60π)$ depending on $\sin ^2θ_{12}$ for both NH and IH cases. We also discuss the specific case that $θ_{13}$ is related with the Cabibbo angle $λ$ such as $\sinθ_{13}=λ/\sqrt{2}$, in which the maximal CP violation is preferred. The CP violating Dirac phase can distinguish the lepton flavor mixing patterns at T2K and NO$ν$A experiments in the future.