Abstract
Extra-dimensional components of gauge fields in higher-dimensional gauge theories will play a role of the Higgs field and become tachyonic after Kaluza-Klein compactifications on internal spaces with (topologically nontrivial) gauge field backgrounds. Its condensation is then expected to break gauge symmetries spontaneously. But, contrary to the expectation, some models exhibit restoration of gauge symmetries. In this paper, by considering all the massive Kaluza-Klein excitations of gauge fields, we explicitly show that some of them indeed become massless at the minimum of the Higgs potential and restore (a part of) the gauge symmetries which are broken by gauge field backgrounds. We particularly consider compactifications on ${S}^{2}$ with monopolelike fluxes and also on ${\mathbb{CP}}^{2}$ with instanton and monopolelike fluxes. In some cases, the gauge symmetry is fully restored, as argued in previous literatures. In other cases, there is a stable vacuum with a partial restoration of the gauge symmetry after Higgs condensation. Topological structure of the gauge field configurations prevent the gauge symmetries from being restored.
Highlights
The dynamics of gauge symmetry breaking is yet to be investigated, especially when it is caused by the elementary Higgs scalar field with a nontrivial potential
The background gauge fluxes explicitly break some of the original gauge symmetries, and often provide tachyonic scalar fields whose vacuum expectation value further breaks the remaining gauge symmetries in the four-dimensional effective field theory
We revisit such higher-dimensional models including all the massive Kaluza-Klein modes to investigate their roles in the gauge symmetry breaking
Summary
The dynamics of gauge symmetry breaking is yet to be investigated, especially when it is caused by the elementary Higgs scalar field with a nontrivial potential. The original gauge symmetry in higher dimensions is explicitly broken by the background gauge fluxes in the compact spaces, and the Higgs vacuum expectation value is expected to further break some part of the remaining gauge symmetries spontaneously in four-dimensional effective theory. The Higgs vacuum expectation value itself breaks a part of the remaining gauge symmetries and the corresponding gauge bosons become massive, some of the massive Kaluza-Klein modes will become massless and gauge symmetries are recovered in the four-dimensional effective theory. Such a possibility was pointed out in [36,37]. In Appendix H, we prove that the symmetric Higgs field satisfies the condition of the symmetric field on G=H
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