23}$$ which is used to express $\theta_{23}$ in terms of $M_{ij}$. The maximal CP violation can be induced if $$M_{\tau \tau } - M_{\mu \mu } = {\rm imaginary},
\label{Eq:Maximal-CP1}$$ as well as $$M_{\mu \tau } = {\rm imaginary},
\quad
M_{e\mu}\cos\theta_{23} - M_{e\tau}\sin\theta_{23}= {\rm real},
\label{Eq:Maximal-CP2}$$ for $\cos 2\theta_{23}\neq 0$, or $$M_{e\mu}-\sigma M_{e\tau}= {\rm real},
\label{Eq:Maximal-CP2 and ATM}$$ for $\cos 2\theta_{23}= 0$ indicating the maximal atmospheric neutrino mixing, where $\sigma=\pm 1$ takes care of the sign of $\sin\theta_{23}$.

From the recent result of the Planck [@Planck2015arXiv], the upper limit of the neutrino masses is given by $\sum m_\nu \le 0.17$ eV. On the other hand, the neutrino oscillation experiments measure $\Delta m_{31}^2 = m_3^2-m_1^2$ and $\Delta m_{32}^2 = m_3^2-m_2^2$. Choosing $\Delta m_{31}^2= 2.46\times 10^{-3}$ eV$^2 (\sim m_3^2)$ for NH with $m^2_3\gg m^2_2\gg m^2_1$ and $\Delta m_{32}^2 = -2.45\times 10^{-3}$ eV$^2 (\sim -m_2^2)$ for IH with $m^2_2> m^2_1 \gg m^2_3$ [@NuData], we obtain that the heaviest neutrino mass, either $m_3$ or $m_2$, is approximately estimated to be $0.05$ eV. Why neutrinos are so light is a puzzling question to be solved. One of the promising theoretical ideas is the one based on the seesaw mechanism [@Seesaw], which utilizes right-handed neutrinos. The 