Examination of pairs in neutrino mixing matrix

We exam the pairs of neutrino mixing matrix and suggest pairs that can be used in the construction of new mixing patterns, with “pair” denoting the equality of the modulus of a pair of matrix elements. The results show that the tri-maximal mixing in ${\nu }_2$ and the $\mu$-$\tau$ interchange symmetry are good choices under current experimental results. The two cases of bi-pair mixing pattern depend on the mass hierarchy of neutrinos. We also derive constraints on the CP-phase by the pairs. The results are compatible with the maximal CP-violation in most cases that are both self-consistent and consistent with experimental results.