Searching for BPS vortices with nonzero stress tensor in the generalized Born–Infeld–Higgs model

Abstract

In this article, we show how the new Bogomolny’s equations for BPS vortices, with nonzero stress tensor, in the three-dimensional generalized Maxwell–Higgs model can be derived rigorously using the BPS Lagrangian method. Particularly, we add into the original BPS Lagrangian, which is a total derivative term, two additional terms that are proportional to square of the first derivative of scalar effective field and to a function that depends only on the scalar effective field. These additional terms imply additional constraint equations which are the Euler–Lagrange equations of the BPS Lagrangian. We employ this procedure to the generalized Born–Infeld–Higgs model and obtain new Bogomolny’s equations for BPS vortices. We show their static energy could be finite if the scalar potential $$0<V< 2b^2$$ and $$V\ne b^2$$ , with b being the Born–Infeld parameter. We also show their stress tensor is nonzero and parameterized by a constant $$C_0$$ . It turns out the total static energy is proportional to the topological charge N and also the constant $$C_0$$ . We do the numerical analysis and find that solutions of the scalar and gauge effective fields behave nicely near the origin, but unfortunately they diverge near the boundary at the order of $$\mathcal{O}(r^4)$$ . We suggest incorporating gravity or considering BPS vortex in higher-dimensional models might resolve this problem. Although there are no regular solutions, the BPS Lagrangian method could be used to find Bogomolny’s equations, with nonzero stress tensor, for other BPS solitons whose existence might play a role in the Kibble–Zurek mechanism.