A method for BPS equations of vortices

We develop a new method for obtaining the BPS equations of static vortices motivated by the results of the On-Shell method on the standard Maxwell-Higgs model and its Born-Infeld-Higgs model [1]. Our method relies on the existence of what we shall call a BPS energy function Q as such the total energy of BPS vortices EBPS are simply given by an integral of total differential of the BPS energy function, EBPS=∫dQ. Imposing a condition that the effective fields are independent of each other, we may define a BPS Lagrangian LBPS by EBPS≡−∫d2xLBPS. Equating this BPS Lagrangian with the corresponding effective Lagrangian, the equation is expected to be a sum of positive-semidefinite functions Leff−LBPS=∑iNAi2=0, where N is the number of effective fields. Solving this equation by parts would yields the desired BPS equations. With our method, the various known BPS equations of vortices are derived in a relatively simple procedure. We show that in all models considered here, the BPS energy function is given by a general formula Q=2πaF(f), where a and f are the effective fields for the gauge field and scalar field, and F′(f)=±2fw(f), with w is an overall coupling of the scalar field's kinetic term.