极小I型跷跷板机构的$LDL^{T}$分解公式及任意基上$CP$对称性条件
A formula by $LDL^{T}$ decomposition for the minimal type-I seesaw mechanism and conditions of $CP$ symmetry in an arbitrary basis
摘要：
本文通过对最小I型跷跷板机制的$LDL^{T}$分解，定义了一个公式，得到了中微子质量矩阵$m$在任意基上具有$CP$对称性的条件.对于Yukawa矩阵$Y_{ij} = (a_{j}, b_{j})$和右手中微子质量矩阵$M_{ij}$，条件为${\rm Re\,} (M_{22} a_{i} - M_{12} b_{i}) \, {\rm Im \,} ( M_{22} a_{j} - M_{12} b_{j}) = - \det M \, {\rm Re\,} b_{i} \, {\rm Im \,} b_{j}$或$ = - \det M \, {\rm Im \,} b_{i} \, {\rm Re\,} b_{j}$.换句话说，$b_{i}$的实部或虚部必须与量$(M_{22} a_{i} - M_{12} b_{i})$的实部或虚部成比例。
英文摘要：
In this paper, defining a formula by $LDL^{T}$ decomposition for the minimal type-I seesaw mechanism, we obtain conditions of $CP$ symmetry for the neutrino mass matrix $m$ in an arbitrary basis. The conditions are found to be ${\rm Re\,} (M_{22} a_{i} - M_{12} b_{i}) \, {\rm Im \,} ( M_{22} a_{j} - M_{12} b_{j}) = - \det M \, {\rm Re\,} b_{i} \, {\rm Im \,} b_{j}$ or $ = - \det M \, {\rm Im \,} b_{i} \, {\rm Re\,} b_{j}$ for the Yukawa matrix $Y_{ij} = (a_{j}, b_{j})$ and the right-handed neutrino mass matrix $M_{ij}$. In other words, the real or imaginary part of $b_{i}$ must be proportional to the real or imaginary part of the quantity $(M_{22} a_{i} - M_{12} b_{i})$.
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