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Gauge symmetry breaking with S2 extra dimensions

Kento Asai1,2,*, Yuki Honda3,†, Hiroki Ishikawa4,‡, Joe Sato3,§, and Yasutaka Takanishi4,∥

  • *Contact author: kento.asai@yukawa.kyoto-u.ac.jp
  • †Contact author: honda-yuki-wy@ynu.jp
  • ‡Contact author: h.ishikawa.404@ms.saitama-u.ac.jp
  • §Contact author: sato-joe-mc@ynu.ac.jp
  • ∥Contact author: sci56439@mail.saitama-u.ac.jp
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Phys. Rev. D 112, 075015 – Published 6 October, 2025

DOI: https://doi.org/10.1103/8khc-q9ly

Abstract

We consider symmetry breaking of arbitrary gauge groups on a six-dimensional space-time which consists of a four-dimensional Minkowski space-time M4 and a two-dimensional sphere S2. We expand the gauge fields in the presence of a nontrivial background unique to S2. 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 ϕ and a Nambu-Goldstone mode χ. 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 S2.

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Phys. Rev. D 112, 075015– Published 6 October, 2025
  • Received 8 July 2025
  • Accepted 12 September 2025
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