论文标题
$ a_4 $夸克质量层次结构的模块化风味模型靠近固定点$τ=ω$
$A_4$ Modular Flavour Model of Quark Mass Hierarchies close to the Fixed Point $τ= ω$
论文作者
论文摘要
我们研究了描述夸克质量层次结构的可能性以及CKM夸克混合矩阵，而无需在模块化$ a_4 $对称的夸克风味模型中进行微调。夸克质量层次结构在固定点$τ=ω\ equiv \ exp（{i \，2π/3}）$（模块化群的基本域的左孔）的附近考虑，$τ$是模量的VEV。该模型涉及3级和权重6、4和2的模块化形式，并包含八个常数，其中只有两个（$ g_u $ and $ g_d $）可能是违反CP的来源，除了模量的VEV，$τ=ω+ε$，$（+ε）我们发现，如果是真实的（CP支持）$ G_U $和$ G_D $和$τ$（$ε$）在Quark和Up-Quark扇区中，则可以在不使用$ |ε|的情况下复制下型夸克质量层次结构\ cong 0.03 $，所有其他常数为$ {\ cal o}（1）$，大约对应于$ 1：|ε| ：|ε|^2 $。可以通过相同的$ |ε|可以实现上夸克质量层次结构。 \ cong 0.03 $，但允许$ g_u \ sim {\ cal o}（10）$，对应于$ 1：|ε|/| g_u | ：|ε|^2/| g_u |^2 $。在这种情况下，我们讨论了CKM夸克混合和CP违规。
We investigate the possibility to describe the quark mass hierarchies as well as the CKM quark mixing matrix without fine-tuning in a quark flavour model with modular $A_4$ symmetry. The quark mass hierarchies are considered in the vicinity of the fixed point $τ= ω\equiv \exp({i\,2π/3})$ (the left cusp of the fundamental domain of the modular group), $τ$ being the VEV of the modulus. The model involves modular forms of level 3 and weights 6, 4 and 2, and contains eight constants, only two of which, $g_u$ and $g_d$, can be a source of CP violation in addition to the VEV of the modulus, $τ= ω+ ε$, $(ε)^* \neq ε$, $|ε|\ll 1$. We find that in the case of real (CP-conserving) $g_u$ and $g_d$ and common $τ$ ($ε$) in the down-quark and up-quark sectors, the down-type quark mass hierarchies can be reproduced without fine tuning with $|ε| \cong 0.03$, all other constants being of ${\cal O}(1)$, and correspond approximately to $1 : |ε| : |ε|^2$. The up-type quark mass hierarchies can be achieved with the same $|ε| \cong 0.03$ but allowing $g_u\sim {\cal O}(10)$ and correspond to $1 : |ε|/|g_u| : |ε|^2/|g_u|^2$. In this setting, we discuss the CKM quark mixing and CP violation.
