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

TL;DR

This paper proposes a parameterization of the PMNS neutrino mixing matrix based on bipair neutrino mixing schemes, linking neutrino mixing angles and CP violation phases, and provides numerical relations for maximal and non-maximal CP violation cases.

Contribution

The study introduces a novel parameterization of the PMNS matrix using bipair mixing schemes, establishing specific correlations among mixing angles and CP phases, and offers numerical predictions for these parameters.

Findings

Correlation between mixing angles and CP phases derived

Relations for maximal CP violation case established

Numerical analysis of non-maximal CP violation conducted

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.