论文标题
部分可观测时空混沌系统的无模型预测
Conditions of general $Z_{2}$ symmetry and TM$_{1,2}$ mixing for the minimal type-I seesaw mechanism in an arbitrary basis
论文作者
论文摘要
在本文中，使用$ ldl^{t} $（或广义cholesky）分解的最小型I SeeSAW机制，将一般$ Z_ {2} $的条件 - 中性质量矩阵$ M $不变性。发现条件为$（m_ {22} a_ {i}^{+} - m_ {12} b_ {i}^{+}） \，b_ {j}^{ - } $用于$ z_ {2} $ - 对称和-Antisymmetric部分的Yukawa Matrix $ y_ {ij}^{\ pm}^{\ pm} \ equiv（y \ pm t y） b_ {j}^{\ pm}）$和右手中微子质量矩阵$ m_ {ij} $。换句话说，$ b_ {i} $的对称和反对称部分必须与数量$ \ tilde a_ {i} \ equiv a_ {i} {i} - {m_ {12}} \ yom_ {22}} b_ {i} $成比例。它们等同于以下条件：$ m $是由发电机$ t $的特征向量对角线对角线的。 这些结果应用于三个$ z_ {2} $对称，$μ-τ$对称性，tm $ _ {1} $混合以及预测TM $ _ {2} $混合的魔术对称性。对于tm $ _ {1,2} $的情况，对称条件变为$ m_ {22}^{2} \，\ tilde {a} _ {1}^{\ rm tbm}^{\ rm tbm} \ tilde a_ _ _ _ {2}}} tbm} b_ {2}^{\ rm tbm} $和$ m_ {22}^{2} {2} \，\ tilde {a} _ {1,2}^{\ rm tbm} b_ {1,2}^{\ rm tbm} b_ {3}^{\ rm tbm} $带有组件$ \ tilde a_ {i}^{\ rm tbm} $和$ b_ {\ rm tbm} $和$ b_ {i} $ \ mathbf {v} _ {1,2,3} $。特别是，对于tm $ _ {2} $混合，魔术（抗）对称的Yukawa矩阵，带有$ s_ {2} y = \ pm y $在现象学上被排除在现象学上，因为它可以预测$ M_ {2} = 0 $或$ M_ {1}，m_ {1}，m_ {1}，m_ {3} = 0 $。如果Yukawa不是（反）对称的情况，则显示质量奇异值没有根标志。
In this paper, using a formula for the minimal type-I seesaw mechanism by $LDL^{T}$ (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 $Y_{ij}^{\pm} \equiv (Y \pm T Y )_{ij} /2 \equiv (a_{j}^{\pm}, b_{j}^{\pm})$ 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 $\tilde a_{i} \equiv a_{i} - {M_{12} \over M_{22}} b_{i}$. 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 $μ-τ$ 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 $\tilde a_{i}^{\rm TBM}$ and $b_{i}^{\rm TBM}$ 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.
