CP violation in bipair neutrino mixing

There are experimentally determined two best-fit points for the atmospheric neutrino mixing angle $\theta_{23}$: $\sin^2 \theta_{23} = 0.413$ (case A) and $\sin^2 \theta_{23} = 0.594$ (case B). In the bipair neutrino mixing scheme, we predict $\sin^2\theta_{23} = \sqrt{2}-1$ (case 1) to be consistent with the case A and $\sin^2\theta_{23} = 2-\sqrt{2}$ (case 2) to be consistent with the case B. If the case B is realized in nature, the bipair neutrino mixing provides a unique neutrino model consistent with the observation $\sin^2 \theta_{23} = 0.594$. However, the reactor neutrino mixing angle $\theta_{13}$ is predicted to be $\sin^2\theta_{13} = 0$, which is inconsistent with the observation. We propose a new modification scheme to yield $\sin^2\theta_{13} \neq 0$ utilizing the charged lepton contribution and study its effect on both of CP-violating Dirac and Majorana phases, which is numerically estimated. It is found that there appear striking differences between the case 1 and the case 2 in their phase structure.