Mass spectrum in a six-dimensional SU(n) gauge theory on a magnetized torus

We examine six-dimensional SU(n) gauge theories compactified on a two-dimensional torus with a constant magnetic flux background to obtain a comprehensive low-energy mass spectrum. We introduce general background configurations including the magnetic flux and continuous Wilson line phases, consistent with classical equations of motion. Under the standard gauge fixing procedure, the complete mass spectrum in low-energy effective theory for the SU(n) case is newly presented without imposing restrictions on the gauge fixing parameter. Our analysis confirms the inevitable existence of tachyonic modes, which neither depend on the background configurations of Wilson line phases nor are affected by the gauge fixing parameter. Masses for some low-energy modes exhibit dependence on the gauge fixing parameter, and these modes are identified as would-be Goldstone bosons that are absorbed by massive four-dimensional vector fields. We discuss the phenomenological implications associated with stabilization or condensation of the tachyonic states. Various mass spectra and symmetry-breaking patterns are expected with flux backgrounds in the SU(n) case. They are helpful for constructing phenomenologically viable models beyond the standard model, such as gauge-Higgs unification and grand unified theories.

Automated summary

We study six-dimensional SU(n) gauge theories compactified on a two-dimensional torus with a constant magnetic flux background and provide a comprehensive low-energy mass spectrum. We consider general background configurations including the magnetic flux and continuous Wilson line phases, and derive the complete mass spectrum without imposing restrictions on the gauge fixing parameter. Our analysis reveals the presence of tachyonic modes that are independent of the background configurations and gauge fixing parameter. We also discuss the implications of stabilizing or condensing these tachyonic states in terms of phenomenology and the potential for constructing beyond the standard model scenarios.