Comments on the Monopole-Antimonopole Pair Solutions

Recently, the monopole-antimonopole pair and monopole-antimonopole chain solutions are solved with internal space coordinate system of $θ$-winding number $m$ greater than one. However, we notice that it is also possible to solve these solutions numerically in terms of $θ$-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 $θ$-winding and $ϕ$-winding number one is parameterized by $s=0$ at small $r$ and $s=1$ at large $r$.

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