Massive hopfions

Abstract

The Skyrme-Faddeev model is a ($3+1$)-dimensional model which has knotted, stringlike, soliton solutions. In this paper we investigate a Skyrme-Faddeev model with an $SO(3)$ symmetry breaking potential. We then rescale this model and take the mass to infinity. This infinite mass model is found to have compact knotted solutions. In all of the investigated massive models we find similar charged solutions as in the usual $m=0$, model. We also find that their energies follow a similar $E\ensuremath{\sim}{Q}^{3/4}$ power growth as the $m=0$ model.