A Detailed Study of Bogomol'nyi Equations in Two-Dimensional Generalized Maxwell-Higgs Model Using \textit{On-Shell} Method

TL;DR

This paper applies a recent on-shell method to derive and classify Bogomol'nyi equations in a two-dimensional generalized Maxwell-Higgs model, revealing new equations and conditions for their existence beyond the BPS limit.

Contribution

It introduces a systematic on-shell approach to generate and classify Bogomol'nyi equations in the generalized Maxwell-Higgs model, including new equations for non-zero constant parameters.

Findings

Classified Bogomol'nyi equations into two types based on a constant C_0.

Identified that known equations correspond to C_0=0.

Discovered new Bogomol'nyi equations for C_0≠0 related to energy density differences.

Abstract

We use a recent {\it on-shell} Bogomol'nyi method, developed in~\cite{Atmaja:2014fha}, to construct Bogomol'nyi equations of the two-dimensional generalized Maxwell-Higgs model~\cite{Bazeia:2012uc}. The formalism can generate a large class of Bogomol'nyi equations parametrized by a constant $C_0$. The resulting equations are classified into two types, determined by $C_0=0$ and $C_0\neq0$. We identify that the ones obtained by Bazeia {\it et al}~\cite{Bazeia:2012uc} are of the type $C_0=0$. We also reveal, as in the case of ordinary vortex, that this theory does not admit Bogomol'nyi equations in the Bogomol'nyi-Prasad-Sommerfield limit in its spectrum. However, when the vacuum energy is lifted up by adding some constant to the energy density then the existence of such equation is possible. Another possibility whose energy is equal to the vacuum is also discussed in brief. As a future of the \textit{on-shell} method, we find another new Bogomol'nyi equations, for $C_0\neq0$, which are related to a non-trivial function defined as a difference between energy density of potential term of the scalar field and kinetic term of the gauge field.