Topological defect with nonzero Hopf invariant in Yang–Mills–Higgs model

We propose a topological defect or instanton solution with nonzero Hopf invariant to the 3+1D non-Abelian gauge theory coupled with scalar fields. This solution, which we call Hopf defect, represents a spacetime event that makes a 2π   rotation of vacuum manifold of the monopole. Although the action of this Hopf defect is logarithmically divergent, it may still give relevant contributions in a finite-sized system. Since the Chern–Simons term for the unbroken U(1)U(1) gauge field may appear in the low energy effective theory, the Hopf defect may possibly generate a phase factor change for the monopoles.