论文标题
$ \ rm a_4 $的模块化不变风味模型和附近固定点的分层结构
Modular invariant flavor model of $\rm A_4$ and hierarchical structures at nearby fixed points
论文作者
论文摘要
在$ \ rm a_4 $的模块化不变风味模型中，我们在附近的$τ= i $和$τ=ω$的附近固定点上研究了模量的层次结构，该点在$ \ rm psl的基本领域（2，\ rm psl（\ rm psl）（\ rm psl（\ rathbbbb））$。这些固定点对应于残差对称$ \ mathbb {z} _2^{s} = \ {i，s \} $和$ \ Mathbb {z} _3^{s} = \ {s} = \ {i，st，st，st，st，（st），（st）无限$τ= i \ infty $还保留了子组$ \ mathbb {z}^t_3 = \ {i，t，t，t，t^2 \} $的残留对称性。我们研究了典型的两种质量矩阵，该典型的重量为2 $，$ 4 $和6 $的模块化形式，而中微子质量矩阵则具有Weinberg Operator的中微子质量矩阵。线性模块形式大约是通过在固定点围绕模块化形式的泰勒扩展而获得的。通过使用它们，在附近的固定点检查了Lepton和Quark质量矩阵的风味结构。这些质量矩阵的层次结构清楚地显示在$ s $，$ t $和$ st $的对角线基础中。在某些质量矩阵的情况下，可以在附近的固定点复制观察到的PMN和CKM混合矩阵。通过在附近的固定点上进行数值扫描模型参数，我们的讨论均已确认正常层次结构和倒置的中微子质量。对中微子质量的总和和CP违反了附近每个固定点的lapt子的总和。
In the modular invariant flavor model of $\rm A_4$, we study the hierarchical structure of lepton/quark flavors at nearby fixed points of $τ=i$ and $τ=ω$ of the modulus, which are in the fundamental domain of $\rm PSL(2,\mathbb{Z})$. These fixed points correspond to the residual symmetries $\mathbb{Z}_2^{S}=\{I, S \}$ and $\mathbb{Z}_3^{S}=\{ I, ST, (ST)^2 \}$ of $\rm A_4$, where $S$ and $T$ are generators of the $A_4$ group. The infinite $τ= i \infty$ also preserves the residual symmetry of the subgroup $\mathbb{Z}^T_3=\{ I,T,T^2 \}$ of $\rm A_4$. We study typical two-type mass matrices for charged leptons and quarks in terms of modular forms of weights $2$, $4$ and $6$ while the neutrino mass matrix with the modular forms of weight $4$ through the Weinberg operator. Linear modular forms are obtained approximately by performing Taylor expansion of modular forms around fixed points. By using them, the flavor structure of the lepton and quark mass matrices are examined at nearby fixed points. The hierarchical structure of these mass matrices is clearly shown in the diagonal base of $S$, $T$ and $ST$. The observed PMNS and CKM mixing matrices can be reproduced at nearby fixed points in some cases of mass matrices. By scanning model parameters numerically at nearby fixed points, our discussion are confirmed for both the normal hierarchy and inverted one of neutrino masses. Predictions are given for the sum of neutrino masses and the CP violating Dirac phase of leptons at each nearby fixed point.
