Vortices in the extended Skyrme-Faddeev model

TL;DR

This paper develops analytical and numerical vortex solutions in an extended Skyrme-Faddeev model, incorporating a quartic kinetic term and symmetry-breaking potential, revealing finite energy vortices with light-speed propagating waves.

Contribution

It introduces a new extended Skyrme-Faddeev model with added quartic and potential terms, and constructs vortex solutions using an ansatz reducing to an ODE, including analytical and numerical results.

Findings

Vortices have finite energy per unit length.

Vortices support waves propagating at the speed of light.

Solution spectrum depends on coupling constants.

Abstract

We construct analytical and numerical vortex solutions for an extended Skyrme-Faddeev model in a $(3+1)$ dimensional Minkowski space-time. The extension is obtained by adding to the Lagrangian a quartic term, which is the square of the kinetic term, and a potential which breaks the SO(3) symmetry down to SO(2). The construction makes use of an ansatz, invariant under the joint action of the internal SO(2) and three commuting U(1) subgroups of the Poincar\'e group, and which reduces the equations of motion to an ODE for a profile function depending on the distance to the $x^3$-axis. The vortices have finite energy per unit length, and have waves propagating along them with the speed of light. The analytical vortices are obtained for special choice of potentials, and the numerical ones are constructed using the Successive Over Relaxation method for more general potentials. The spectrum of solutions is analyzed in detail, specially its dependence upon special combinations of coupling constants.