Non-Abelian Discrete Groups from the Breaking of Continuous Flavor Symmetries

TL;DR

This paper explores how non-abelian discrete flavor symmetries can emerge from the spontaneous breaking of continuous SU(2) or SU(3) symmetries, specifically resulting in the quaternion group D_2' under certain conditions.

Contribution

It systematically analyzes all cases where continuous flavor symmetries are broken by small representations, identifying the quaternion group D_2' as the unique residual non-abelian discrete symmetry.

Findings

Only the quaternion group D_2' can arise as a residual symmetry.

Small representations lead to limited possible discrete groups.

The analysis constrains flavor symmetry breaking scenarios.

Abstract

We discuss the possibility of obtaining a non-abelian discrete flavor symmetry from an underlying continuous, possibly gauged, flavor symmetry SU(2) or SU(3) through spontaneous symmetry breaking. We consider all possible cases, where the continuous symmetry is broken by small representations. "Small" representations are these which couple at leading order to the Standard Model fermions transforming as two- or three-dimensional representations of the flavor group. We find that, given this limited representation content, the only non-abelian discrete group which can arise as a residual symmetry is the quaternion group D_2'.