Closed vortices and the Hopf index in classical field theory

Closed-vortex configurations in classical field theory are investigated here. It is shown in detail that in the Abelian Higgs model such configurations are unstable by collapse. Closed-ring configurations having unit Hopf index and that explicitly exhibit the features of a twisted vortex are constructed for theories where a three-component scalar field is present. However, it is shown that in renormalizable theories the Hopf charge does not ensure the existence of stable solutions. It is proved that a nonlinear $\sigma$ model where an interaction which has fourth-power field derivatives is present has a twisted-ring solution. A lower bound for the mass and estimates for the radius and mass are given.