论文标题
夸克和Lepton风味，带有$ a_4 $模块化对称性的通用模量$τ$
Quark and lepton flavors with common modulus $τ$ in $A_4$ modular symmetry
论文作者
论文摘要
我们在$ a_4 $模块化对称性中使用公共模量$τ$研究夸克和Lepton Mass矩阵。可行的夸克质量矩阵由模块化形式的权重$ 2 $，$ 4 $和$ 6 $组成。据指出，模量$τ$接近$ i $，这是SL $（2，Z）$的基本区域中的一个固定点，并且不违反CP对称性。实际上，观察到的CP违规行为以$τ$从$τ= i $有点偏离。还使用模块化的权重$ 2 $，$ 4 $和6 $的模块化形式，在其中检查了五个案例。中微子质量矩阵是根据Weinberg操作员的重量$ 4 $形式生成的。 Lepton Mass矩阵也与观测到的混合角度在$τ$接近$ i $的NH中性中微子群众一致。允许的夸克$τ$的区域在所有带电的Lepton Mass Matrix的情况下相互重叠。但是，中微子质量的总和对于测试夸克和叶子的共同$τ$至关重要。中微子质量的最小总和$ \ sum m_i $是$ 140 $ MEV，$ 140 $ MEV。 中微子群众的倒立层次结构在我们的框架中是不利的。人们强调的是，我们的结果表明残留的对称$ \ mathbb {z} _2^{s} = \ {i，s \} $在Quark和Lepton Mass Matrices中。
We study quark and lepton mass matrices with the common modulus $τ$ in the $A_4$ modular symmetry. The viable quark mass matrices are composed of modular forms of weights $2$, $4$ and $6$. It is remarked that the modulus $τ$ is close to $i$, which is a fixed point in the fundamental region of SL$(2,Z)$, and the CP symmetry is not violated. Indeed, the observed CP violation is reproduced at $τ$ which is deviated a little bit from $τ=i$. The charged lepton mass matrix is also given by using modular forms of weights $2$, $4$ and $6$, where five cases have been examined. The neutrino mass matrix is generated in terms of the modular forms of weight $4$ through the Weinberg operator. Lepton mass matrices are also consistent with the observed mixing angles at $τ$ close to $i$ for NH of neutrino masses. Allowed regions of $τ$ of quarks and leptons overlap each other for all cases of the charged lepton mass matrix. However, the sum of neutrino masses is crucial to test the common $τ$ for quarks and leptons. The minimal sum of neutrino masses $\sum m_i$ is $140$meV at the common $τ$. The inverted hierarchy of neutrino masses is unfavorable in our framework. It is emphasized that our result suggests the residual symmetry $\mathbb{Z}_2^{S}=\{ I, S \}$ in the quark and lepton mass matrices.
