On axially symmetrical solitons in Abelian–Higgs models

On vortices and rings in extended Abelian models

“…That model, is a continuation of previous work [8][9][10]. In the second part of this paper, we examine a modified version of the model in [18]. We add higher derivative terms which might help in stabilizing the torus-shaped soliton.…”

“…It could also be relatively helpful numerically, as we would have four fields for minimization in the energy functional instead of five we had in [18]. This is not as easy as it might seem, since there are two instability modes.…”

“…The vortex itself is not necessarily stable while forming a torus, and the latter has the tendency to shrink due to its tension. The first instability can be avoided in a U (1) × U (1) model as the one presented in [18], where a numerical search for bosonic superconducting static vortex rings in a U (1) A × U (1) W model is done. There, the existence of straight strings is ensured for topological reasons.…”

“…We work in the A 0 = 0 = W 0 gauge. We follow the ansatz of [18]. A more general choice for φ would be φ(ρ, ϕ, z) = F(ρ, z)e iMΘ(ρ,z)+iχ(ρ,z)…”

“…This can happen by taking a piece of such straight string and periodically connect its ends together. These string loops are examined in [18] but in the framework of a U (1) × U (1) model, where the existence of the defect is ensured for topological reasons [19,23]. That model, is a continuation of previous work [8][9][10].…”