Radially separated monopole solutions in non-Abelian gauge models

Abstract

We consider a Yang-Mills-Higgs Lagrangian invariant under local gauge transformations belonging to an arbitrary compact group $G$. The Higgs fields are assumed to belong to a real representation of $G$. We analyze in detail the conditions imposed on the fields due to the requirement that the static Hamiltonian or the total energy of the system be finite. We then seek static finite-energy solutions for which the radial dependence of the fields is factorized. We show that for the coupled system of nonlinear equations emerging from the Lagrangian the equations for angular functions which carry the internal-symmetry labels and the equations for the radial functions separate into two separate systems of coupled nonlinear equations. We solve the equations for angular functions completely and show that the gauge fields vanish outside a fixed SO(3) subgroup of $G$ and that inside the SO(3) group they reduce to the 't Hooft-Polyakov solution with unit magnetic charge in appropriate units. The Higgs fields may belong to any integer representation of this SO(3) group. The static Hamiltonian and consequently the total energy or mass of the monopole depend on the representation of the Higgs field. Thus we obtain in principle a mass formula for the monopoles, the one with the lowest mass corresponding to the 't Hooft-Polyakov case.