Increasing Potentials in Non-Abelian and Abelian Gauge Theories

An exact solution for an SU(2) Yang-Mills field coupled to a scalar field is given. This solution has potentials with a linear and Coulomb part. This may have some physical importance since many phenomenological QCD studies assume a linear plus Coulomb potential. Usually the linear potential is motivated with lattice gauge theory arguments. Here the linear potential is an exact result of the field equations. We also show that in the Nielsen-Olesen Abelian model there is an exact solution in the BPS limit which has a Coulomb-like electromagnetic field and a logarithmically rising scalar field. Both of these solutions must be cut off from above to avoid infinite field energy.