Comments on the Monopole-Antimonopole Pair Solutions

Abstract: Recently, the monopole-antimonopole pair and monopole-antimonopole chain solutions are solved with internal space coordinate system of $\theta$-winding number $m$ greater than one. However, we notice that it is also possible to solve these solutions numerically in terms of $\theta$-winding number $m=1$ instead. When $m=1$, the exact asymptotic solutions at small and large distances are parameterized by a single integer parameter $s$. Here we once again study the monopole-antimonopole pair solution of the SU(2) Yang-Mills-Higgs theory which belongs to the topological trivial sector numerically in its new form. This solution with $\theta$-winding and $\phi$-winding number one is parameterized by $s=0$ at small $r$ and $s=1$ at large $r$.