Monopole-antimonopole chains and vortex rings

We consider static axially symmetric solutions of SU(2) Yang-Mills-Higgs theory. The simplest such solutions represent monopoles, multimonopoles and monopole-antimonopole pairs. In general such solutions are characterized by two integers, the winding number $m$ of their polar angle, and the winding number $n$ of their azimuthal angle. For solutions with $n=1$ and $n=2$, the Higgs field vanishes at $m$ isolated points along the symmetry axis, which are associated with the locations of $m$ monopoles and antimonopoles of charge $n$. These solutions represent chains of $m$ monopoles and antimonopoles in static equilibrium. For larger values of $n$, totally different configurations arise, where the Higgs field vanishes on one or more rings, centered around the symmetry axis. We discuss the properties of such monopole-antimonopole chains and vortex rings, in particular, their energies and magnetic dipole moments, and we study the influence of a finite Higgs self-coupling constant on these solutions.

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