Analytic structure of the MNS matrix with diagonal reflection symmetries

In this paper, we survey the analytic structure of the MNS matrix with diagonal reflection symmetries. If the mass matrix of charged leptons $m_{e}$ is hierarchical (i.e., $|(m_{e})_{33}| \simeq m_{\tau} \gg |(m_{e})_{1i , j1}|$), by neglecting the 1-3 mixing of $m_{e}$, the MNS matrix is represented by four parameters and several sign degrees of freedom. By substituting the three observed mixing angles $\theta_{ij}$ as input parameters, the Dirac phase $\delta$ and the Majorana phases $\alpha_{2,3}$ are represented by functions of the 1-2 mixing of charged leptons $s_{e}$. The effective mass of double beta decay $m_{ee}$ is also displayed as a function of $s_{e}$ and the lightest neutrino mass $m_{1 \, \rm or \, 3}$. Because the generalized CP symmetries restrict the effective mass to near the maximum or minimum value in the whole parameter region, several scenarios are suggested to be excluded by the latest limit of the KAMLAND-Zen collaboration.