Gauge symmetry breaking with $S^2$ extra dimensions

TL;DR

This paper investigates gauge symmetry breaking in a six-dimensional model with a two-sphere extra dimension, deriving the mass spectrum of Kaluza-Klein modes and identifying mechanisms for symmetry breaking unique to $S^2$.

Contribution

It introduces a novel method for gauge symmetry breaking using non-trivial gauge field backgrounds on $S^2$, analyzing the resulting KK spectrum and scalar modes.

Findings

Gauge fields commuting with the background remain symmetry operators.

Non-commuting gauge fields do not preserve symmetry.

Identified physical scalar and Nambu-Goldstone modes.

Abstract

We consider symmetry breaking of arbitrary gauge groups on a six-dimensional space-time which consists of a four-dimensional Minkowski space-time $M^4$ and a two-dimensional sphere $S^2$. We expand the gauge fields in the presence of a non-trivial background unique to $S^2$. We analyze Kaluza-Klein(KK) modes of the gauge fields and derive the mass spectrum of the KK modes. We found that the gauge fields (not) commuting with the background fields (do not) remain symmetry operators in four dimensions. We also discuss the mass spectrum of the extra-dimensional components of the gauge fields and identify a physical scalar $\phi$ and a Nambu-Goldstone mode $\chi$. As a result, we obtain a method to break gauge symmetry due to the nontrivial solution for gauge fields which is a unique feature of $S^2$.