In the framework of the representation theory of finite groups, it was recently shown that a fully constrained complex-symmetric mass matrix can be conveniently mapped into a sextet of $Σ(72\times3)$. In this paper, we introduce an additional flavor group $X_{24}$ in the model so that the vacuum alignment of the $Σ(72\times3)$ sextet is determined not only by the symmetries of $Σ(72\times3)$ but also by that of $X_{24}$. We define several flavons which transform as multiplets under $Σ(72\times3)$ as well as $X_{24}$. The vacuum alignment of each of these flavons is obtained as a simultaneous invariant eigenstate of specific elements of the groups $Σ(72\times3)$ and $X_{24}$; i.e.,~the vacuum alignment is fully determined by its residual symmetries. These flavons couple together uniquely resulting in the fully constrained sextet of $Σ(72\times3)$. Through this work we propose a general formalism in which the flavor symmetry group ($G_f$) is obtained as the direct product, $G_f=G_r \times G_x$. Fermions transform nontrivially only under $G_r$ while they remain invariant under $G_x$. Flavons, on the other hand, transform nontrivially under both $G_r$ and $G_x$. The vacuum alignment of each flavon multiplet transforming irreducibly under $G_r \times G_x$ is uniquely identified by its corresponding residual symmetry (a subgroup of $G_r \times G_x$). Several such flavons couple together to form an effective multiple of $G_r$ which remains invariant under $G_x$. This effective multiplet couples to the fermions.
Lepton Mixing and Cancellation of the Dirac Mass Hierarchy in SO(10) GUTs with Flavor Symmetries T7 and Sigma(81)
在SO（10）大统一理论中，中微子的狄拉克质量项中的层次一般与上型夸克质量项中的层次一样强。我们提出了一种机制，部分或完全取消这种层次结构的轻中微子质量矩阵中的跷跷板上下文中。该抵消机制的两个主要成分是三个费米子规范单态的存在和在比SO（10）更高的尺度上破缺的离散味对称G_f。提出了取消机制的两种实现方式。基于Frobenius群T7 = Z7 x Z3的实现导致了层次的部分取消，并将最大2-3轻子混合与上夸克质量的几何层次联系起来。在实现与组西格玛（81）的抵消是完全的，三双极大轻子混合再现在最低阶。在这两种情况下，为了完全适应轻子数据，我们考虑到额外的影响，如高维运营商涉及一个以上的flavon的影响。考虑了重中性费米子的质量谱。对于这两种实现，我们分析的flavon潜在的可重正化的水平，以及如何产生的Cabibbo角。
Residual Symmetries and Scalar Multiplet Vacuum Alignment in Non-Abelian Flavour Models
Instituto Superior Técnico, Universidade de Lisboa（里斯本大学高等理工学院）；Homerton College, University of Cambridge（剑桥大学霍顿学院）；Department of Physics, SUNY at Buffalo（纽约州立大学布法罗分校物理系）；Department of Physics, University of Cincinnati（辛辛那提大学物理系）；Institute for Defense Analyses（国防分析研究所）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
$U(1)\otimes BRST$ symmetry, of on-shell T-matrix elements and (1-$ϕ$-I) Green's functions, determines the vacuum state of the Abelian Higgs Model from symmetry alone: minimization of the scalar-sector effective potential is unnecessary
