Spontaneous breaking of the rotational symmetry induced by monopoles in extra dimensions

TL;DR

This paper presents a theoretical model where extra-dimensional monopoles induce spontaneous breaking of rotational symmetry, leading to vortex formations that reduce symmetry groups, with stability confirmed against higher-order corrections.

Contribution

It introduces a new field theoretical model with extra dimensions and monopoles that demonstrates spontaneous symmetry breaking and vortex formation, extending understanding of symmetry dynamics in higher-dimensional theories.

Findings

Vortices form when the sphere's radius exceeds a critical value.

Symmetry is broken from U(1)×SU(2) to smaller groups depending on monopole charge.

Vortex configurations remain stable under higher-order perturbations.

Abstract

We propose a field theoretical model that exhibits spontaneous breaking of the rotational symmetry. The model has a two-dimensional sphere as extra dimensions of the space-time and consists of a complex scalar field and a background gauge field. The Dirac monopole, which is invariant under the rotations of the sphere, is taken as the background field. We show that when the radius of the sphere is larger than a certain critical radius, the vacuum expectation value of the scalar field develops vortices, which pin down the rotational symmetry to lower symmetries. We evaluate the critical radius and calculate configurations of the vortices by the lowest approximation. The original model has a $U(1) \times SU(2)$ symmetry and it is broken to U(1), U(1), D_3 for each case of the monopole number q = 1/2, 1, 3/2, respectively, where D_3 is the symmetry group of a regular triangle. Moreover, we show that the vortex configurations are stable against higher corrections of the perturbative approximation.