Non-Abelian Discrete Symmetries in Particle Physics

TL;DR

This paper provides a comprehensive review of non-Abelian discrete groups, their mathematical properties, and their applications in particle physics model building, including symmetry breaking and anomaly considerations.

Contribution

It offers detailed group-theoretical methods and explicit examples for various non-Abelian discrete groups used in particle physics models, including flavor symmetries.

Findings

Explicit derivation of group properties like conjugacy classes and representations.

Application of groups such as $A_4$, $S_4$, and $ riangle(54)$ in flavor models.

Discussion of symmetry breaking patterns and anomaly considerations.

Abstract

We review pedagogically non-Abelian discrete groups, which play an important role in the particle physics. We show group-theoretical aspects for many concrete groups, such as representations, their tensor products. We explain how to derive, conjugacy classes, characters, representations, and tensor products for these groups (with a finite number). We discussed them explicitly for $S_N$, $A_N$, $T'$, $D_N$, $Q_N$, $\Sigma(2N^2)$, $\Delta(3N^2)$, $T_7$, $\Sigma(3N^3)$ and $\Delta(6N^2)$, which have been applied for model building in the particle physics. We also present typical flavor models by using $A_4$, $S_4$, and $\Delta (54)$ groups. Breaking patterns of discrete groups and decompositions of multiplets are important for applications of the non-Abelian discrete symmetry. We discuss these breaking patterns of the non-Abelian discrete group, which are a powerful tool for model buildings. We also review briefly about anomalies of non-Abelian discrete symmetries by using the path integral approach.