Multiple bifurcations and transitions for electrically charged monopole-antimonopole chain and vortex-ring solutions

TL;DR

This paper explores the complex bifurcation and transition phenomena in electrically charged monopole-antimonopole chains with varying winding numbers, revealing intricate solution structures and transitions as the Higgs self-coupling constant changes.

Contribution

It provides a detailed analysis of bifurcations and transitions in monopole-antimonopole chain solutions for multiple winding numbers, extending previous studies from two-pole to three-pole configurations.

Findings

No bifurcation for n=2 within the studied interval.

Multiple bifurcations and transitions observed for n=3, 4, 5.

Complex solution patterns with multiple branches and energy levels for higher n.

Abstract

The dependence of physical properties of the electrically charged two-poles monopole-antimonopole pair (MAP) solutions in the Higgs self-coupling constant is previously investigated. In this paper we study the three-poles monopole-antimonopole chain (MAC) solutions. The study includes $\phi$-winding number $n=2,3,4$, and $5$. For the case of $n=2$, there is no bifurcating branch along with the fundamental solution. Also no transition happens for this solution for the Higgs self-coupling interval of $0\leq \lambda \leq 144$. For the case of $n=3$, two transitions happen along the fundamental solution. Also at a higher energy, there are two bifurcating branches. The lower energy branch of these bifurcating branches, merges with the fundamental solution and both terminate at the convergence point and do not survive for larger values of $\lambda$. For $n=4$, a bifurcation is observed at higher energy in comparison with the fundamental solution. Here there are three transitions. One is observed along the fundamental solution and the others happen along the higher energy bifurcating branch. For the case of $n=5$, the pattern is more complex. A bifurcation in $\lambda = \lambda_{b1}$ happens with a higher energy than the fundamental solution. A second bifurcation is observed at $\lambda = \lambda_{b2}$. The two branches of the second bifurcation are both very close in energy to the lower energy branch of the first bifurcation, but they have different electrical and geometrical properties. Therefore, for the case of $n=5$, we have three distinct solution for the interval of $\lambda_{b1}\leq \lambda \leq \lambda_{b2}$ and five distinct solutions for $\lambda_{b2}\leq \lambda \leq 300$. Also two transitions are observed in the higher energy branch of the first bifurcation.