Article,
Non-Abelian Discrete Symmetries in Particle Physics
(Apr 15, 2010)
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.
