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

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 $(M_{22} a_{i}^{+} - M_{12} b_{i}^{+}) , ( M_{22} a_{j}^{-} - M_{12} b_{j}^{-}) = - \det M , b_{i}^{+} , b_{j}^{-}$forthe$Z_{2}$-symmetricand-antisymmetricpartofaYukawamatrix$Y_{ij}^{\pm} \equiv (Y \pm T Y ){ij} /2 \equiv (a{j}^{\pm}, b_{j}^{\pm})$andtheright-handedneutrinomassmatrix$M_{ij}$. 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}^{\rm TBM} \tilde a{2}^{\rm TBM} = - \det M , b_{1}^{\rm TBM} b_{2}^{\rm TBM}$and$ M_{22}^{2} , \tilde {a}{1,2}^{\rm TBM} \tilde a{3}^{\rm TBM} = - \det M , b_{1,2}^{\rm TBM} b_{3}^{\rm TBM}$ with components ${\tilde{a}}_i^{\mathrm{T}\mathrm{B}\mathrm{M}}$ and $b_i^{\mathrm{T}\mathrm{B}\mathrm{M}}$ 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.