Conditions of general $Z_2$ symmetry and TM${}_{1,2}$ mixing for the minimal type-I seesaw mechanism in an arbitrary basis

Abstract

In this paper, using a formula for the minimal type-I seesaw mechanism by $LD{L}^T$ (or generalized Cholesky) decomposition, conditions of general $Z_2$-invariance for the neutrino mass matrix $m$ is obtained in an arbitrary basis. The conditions are found to be $(M_{22}{a}_i^{+}-M_{12}{b}_i^{+})(M_{22}{a}_j^{-}-M_{12}{b}_j^{-})=-\det M{b}_i^{+}{b}_j^{-}$ for the $Z_2$-symmetric and -antisymmetric part of a Yukawa matrix $Y_{ij}^{\pm }\equiv (Y\pm TY{)}_{ij}/2\equiv (a_j^{\pm },b_j^{\pm })$ and the right-handed neutrino mass matrix $M_{ij}$. In other words, the symmetric and antisymmetric part of $b_i$ must be proportional to those of the quantity ${\tilde{a}}_i\equiv a_i-\frac{M_{12}}{M_{22}}{b}_i$. They are equivalent to the condition that $m$ is block diagonalized by eigenvectors of the generator $T$. These results are applied to three $Z_2$ symmetries, the $\mu -\tau$ symmetry, the TM${}_1$ mixing, and the magic symmetry which predicts the TM${}_2$ mixing. For the case of TM${}_{1,2}$, the symmetry conditions become $M_{22}^2{\tilde{a}}_1^{\mathrm{TBM}}{\tilde{a}}_2^{\mathrm{TBM}}=-\det M{b}_1^{\mathrm{TBM}}{b}_2^{\mathrm{TBM}}$ and $M_{22}^2{\tilde{a}}_{1,2}^{\mathrm{TBM}}{\tilde{a}}_3^{\mathrm{TBM}}=-\det M{b}_{1,2}^{\mathrm{TBM}}{b}_3^{\mathrm{TBM}}$ with components ${\tilde{a}}_i^{\mathrm{TBM}}$ and $b_i^{\mathrm{TBM}}$ in the TBM basis ${\mathbf{v}}_{1,2,3}$. In particular, for the TM${}_2$ mixing, the magic (anti-)symmetric Yukawa matrix with $S_{2}Y=\pm Y$ is phenomenologically excluded because it predicts $m_2=0$ or $m_1,m_3=0$. In the case where Yukawa is not (anti-)symmetric, the mass singular values are displayed without a root sign.

Comments: 22 pages, published in Nuclear Physics B