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

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.

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