Monopole-Antimonopole and Vortex Rings

The SU(2) Yang-Mills-Higgs theory supports the existence of monopoles, antimonopoles, and vortex rings. In this paper, we would like to present new exact static antimonopole-monopole-antimonopole (A-M-A) configurations. The net magnetic charge of these configurations is always negative one, whilst the net magnetic charge at the origin is always positive one for all positive integer values of the solution's parameter $m$. However, when $m$ increases beyond one, vortex rings appear coexisting with these A-M-A configurations. The number of vortex rings increases proportionally with the value of $m$. They are located in space where the Higgs field vanishes along rings. We also show that a single point singularity in the Higgs field does not necessarily corresponds to a structureless 1-monopole at the origin but to a zero size monopole-antimonopole-monopole (\textcolor{blue}{MAM}) structure when the solution's parameter $m$ is odd. This monopole is the Wu-Yang type monopole and it possesses the Dirac string potential in the Abelian gauge. These exact solutions are a different kind of BPS solutions as they satisfy the first order Bogomol'nyi equation but possess infinite energy due to a point singularity at the origin of the coordinate axes. They are all axially symmetrical about the z-axis.