A Method for BPS Equations of Vortices

TL;DR

This paper introduces a new method to derive BPS equations for static vortices using an energy function approach, simplifying the process and unifying the derivation of known vortex BPS equations.

Contribution

The paper presents a novel energy function-based method for obtaining BPS equations of vortices, streamlining and unifying the derivation process.

Findings

Successfully derives known BPS vortex equations using the new method.

Provides a simple and systematic procedure for obtaining BPS equations.

In some cases, also derives constraint equations alongside BPS equations.

Abstract

We develop a new method for obtaining the BPS equations of static vortices motivated by the results of the \textit{On-Shell} method on the standard Maxwell-Higgs model and its Born-Infeld-Higgs model~\cite{Atmaja:2014fha}. Our method relies on the existence of what we shall call an energy function, $Q$, which is a mere function of the (effective) fields. The total energy of BPS vortices, $E_{BPS}$, are simply given by a difference between the boundaries value of $Q$ at $r\to\infty$ and at $r=0$, $E_{BPS}=Q(r\to\infty)-Q(r=0)$. Imposing a condition that these (effective) fields are independent, we may define a BPS Lagrangian, $\mathcal{L}_{BPS}$, derived by taking integral of differential $Q$, $\mathcal{L}_{BPS}=-\int dQ$. Matching the Lagrangian $\mathcal{L}_{BPS}$ with the corresponding effective Lagrangian, we can extract several equations. Solving these equations yields the desired BPS equations and, in some cases, also constraint equations. With our method, the various known BPS equations of vortices are derived in a relatively simple procedure.