Bumblebee field in a Topological Framework

TL;DR

This paper investigates a topological framework for the Bumblebee model, a vector field theory with spontaneous Lorentz violation, revealing vortex solutions, stability conditions, and the influence of topological terms on the spectrum and vacuum structure.

Contribution

It introduces a topological approach to the Bumblebee model in (1+2)D, deriving vortex solutions and analyzing the effects of Lorentz violation and Chern-Simons terms on stability and spectrum.

Findings

Found vortex solutions in the Bumblebee model with Lorentz violation.

Demonstrated stability of magnetic vortex configurations.

Showed topological mass influences the dynamical mass poles.

Abstract

A vector field coming from spontaneous Lorentz violation mechanism, namely Bumblebee model is analysed in a topological framework in a $(1+2)D$ Minkowski space-time. Taking a $(1+2)D$ nonlinear Bumblebee vector matter field dynamics where we include topological like Chern-Simons type terms, a vector version of a soliton state, or vortex was found. The Nielsen-Olesen procedure was used in order to derive a Lorentz-violation vector parameter which characterizes, via Spontaneous Symmetry Breaking mechanism, the non-trivial vacuum. We verify the stability of the model as much as the magnetic vortex, and noticed that the soliton modes with polarized direction generated can be associated with local anisotropy of vacuum energy. The vortex equations of motion and the asymptotic behaviour is presented. We have obtained that the effect of the Lorentz symmetry violation expressed by the a time-like Bumblebee vector field vacuum could be shown as kind of pulse at a fixed point $r_0$ in a limitless universe, or as a barrier at $r_0$ which can represent a boundary in the universe, if the Bumblebee vector field vacuum has space-like characteristic. We also analyse the spectrum via propagators where we note that the topological mass contributes as well to the dynamical mass poles. We obtain that the Chern-Simons type terms, in fact, indicates the "speed" of the field to saturate the asymptotic limit and that the vortex core can not be dimension zero.