论文标题
模块化$ S_4 $的双重封面用于风味型号
Double Cover of Modular $S_4$ for Flavour Model Building
论文作者
论文摘要
我们开发了有限模块化组$γ'_4\ equiv s'_4 $的形式主义，这是模块化排列组$γ_4\ simeq S_4 $的双重封面，用于风味的理论。 4级模块化形式的整数$ k> 0 $对于形式主义必不可少的形式可能是偶数或奇怪的。我们明确构建了两个Jacobi Theta常数的最低重量（$ k = 1 $）模块化表格，称为$ \ varepsilon（τ）$和$θ（τ）$，$τ$是模量。我们表明，这些表格提供了$ S_4 $不存在的3D表示形式。在得出$ S'_4 $乘法规则和Clebsch-Gordan系数之后，我们构建了最高$ k = 10 $的模块化形式的多形。这些以$ \ varepsilon $和$θ$表示为多项式，绕过了搜索非线性约束的需求。我们进一步表明，在$ S'_4 $中，有两个选择可以定义（广义）CP变换，我们讨论了基于模块化和CP不变性理论中可能的残差对称性。最后，我们提供了两个应用结果的示例，构建了现象学上可行的Lepton风味模型。
We develop the formalism of the finite modular group $Γ'_4 \equiv S'_4$, a double cover of the modular permutation group $Γ_4 \simeq S_4$, for theories of flavour. The integer weight $k>0$ of the level 4 modular forms indispensable for the formalism can be even or odd. We explicitly construct the lowest-weight ($k=1$) modular forms in terms of two Jacobi theta constants, denoted as $\varepsilon(τ)$ and $θ(τ)$, $τ$ being the modulus. We show that these forms furnish a 3D representation of $S'_4$ not present for $S_4$. Having derived the $S'_4$ multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to $k=10$. These are expressed as polynomials in $\varepsilon$ and $θ$, bypassing the need to search for non-linear constraints. We further show that within $S'_4$ there are two options to define the (generalised) CP transformation and we discuss the possible residual symmetries in theories based on modular and CP invariance. Finally, we provide two examples of application of our results, constructing phenomenologically viable lepton flavour models.
