The Higgs field as the Cheshire cat and his Yang-Mills "smiles"
  The well-known Bogomol'nyi-Prasad-Sommerfield (BPS) monopole is considered in the limit of the infinite mass of the Higgs field as a basis of the Yang-Mills field vacuum with the finite energy density. In this limit, the Higgs field disappears, but it leaves its traces as the BPS monopole transforms into the Wu-Yang monopole obtained in the pure Yang-Mills theory by a spontaneous scale symmetry breaking in the class of functions with the zero value of a topological charge. The topological degeneration of the BPS monopole manifests itself as the Gribov copies of the covariant Coulomb gauge in the form of the time integral of the Gauss constraint. We also show that in the theory there is a zero mode of the Gauss constraint leading to an electric monopole and an additional mass of eta_0-meson in QCD. The consequences of the monopole vacuum in the form of a rising potential and topological confinement are studied in the framework of the perturbation theory. An estimation of the vacuum expectation value of the square of the magnetic tension is given through the eta_0-meson mass, and arguments in favour of the stability of the monopole vacuum are considered. We also discuss why all these "smiles" of the Cheshire cat are kept by the Dirac fundamental quantization, but not the conventional Faddeev-Popov integral. 
L.D. Lantsman, V.N. Pervushin