Parameterization of Pontecorvo-Maki-Nakagawa-Sakata mixing matrix based on CP-violating bipair neutrino mixing

Abstract

CP violation in neutrino interactions is described by three phases contained in Pontecorvo-Maki-Nakagawa-Sakata mixing matrix ($U_{PMNS}$). We argue that the phenomenologocally consistent result of the Dirac CP violation can be obtained if $U_{PMNS}$ is constructed along bipair neutrino mixing scheme, namely, requiring that $ | U_{12} | = | U_{32} | {\rm and} | U_{22} | = | U_{23} | (\rm case 1)$ and $ | U_{12} | = | U_{22} | {\rm and} | U_{32} | = | U_{33}| (\rm case 2)$, where $U_{ij}$ stands for the $i$-$j$ matrix element of $U_{PMNS}$. As a results, the solar, atmospheric and reactor neutrino mixing angles $\theta_{12}$, $\theta_{23}$ and $\theta_{13}$, respectively, are correlated to satisfy $\cos 2{\theta_{12}} = \sin^2\theta_{23} - \tan^2\theta_{13}$ (case 1) or $\cos 2{\theta_{12}} = \cos^2\theta_{23} - \tan^2\theta_{13}$ (case 2). Furthermore, if Dirac CP violation is observed to be maximal, $\theta_{23}$ is determined by $\theta_{13}$ to be: $\sin^2\theta_{23} \approx ({\sqrt 2 - 1})({\cos^2\theta_{13} + \sqrt 2 \sin^2\theta_{13}})$ (case 1) or $\cos^2\theta_{23} \approx ({\sqrt 2 - 1})({\cos^2\theta_{13} + \sqrt 2 \sin^2\theta_{13}})$ (case 2). For the case of non-maximal Dirac CP violation, we perform numerical computation to show relations between the CP-violating Dirac phase and the mixing angles.