Magnetic monopoles in Yang-Mills-Higgs theory with impurities

TL;DR

The paper extends Yang-Mills-Higgs monopole theory by coupling localized impurities that transform in the adjoint or as scalar sources, preserving half the BPS sector and yielding a Bogomol'nyi bound $E\ge 4\pi|N|$ with deformed BPS equations $B_k+D_k\phi=-S_k$. It develops a spherical (hedgehog) reduction, derives an explicit radial system for $h(r),k(r)$ with impurity profiles, and constructs a perturbative scheme around the Prasad–Sommerfield monopole to approximate solutions. Through several explicit impurity models, the authors demonstrate impurity-induced internal monopole structure, modified cores, and a range of asymptotic behaviors including long-range and super long-range tails. The work also discusses abelianization to a U(1) sector and remarks on dyons, offering a versatile framework for impurity effects in monopole dynamics with potential applications to defect scattering and extended topological solitons.

Abstract

In this work, BPS models built from the coupling of Yang-Mills-Higgs Lagrangian to impurities are investigated. We first consider scalar impurities, which in the BPS limit generate monopoles similar to those obtained in a previously considered class of $\mathrm{SU(2)}\times\mathrm{Z}_2$ or $\mathrm{SU(2)}\times\mathrm{SU(2)}$ models. We then focus on coupling with nonabelian impurities, defined as fixed backgrounds produced from fields transforming under the adjoint representation of SU(2), with a coupling chosen to preserve half of the BPS sectors. The nature of this coupling, the ensuing Bogomol'nyi bound and BPS equations, as well as the effect of these impurities in the abelianization that leads to the emergence of a U(1) gauge group are investigated. We study in greater detail impurities with spherical symmetry, and examine the manner in which impurity coupling changes the asymptotic behavior and range of monopole interactions. Moreover, we introduce a method that can be used to approximate solutions with the use of small perturbations around the Prasad-Sommerfield monopole, and discuss the possibility of extending the aforementioned results to dyons. In order to exemplify the most important properties of the theory, several specific impurity models are presented, with the respective monopole solutions are found numerically. These solutions present novel internal structure and multiple features that would not be possible in the original theory.

Figures (6)

• Figure 1: Function $\lambda(r)$, given by \ref{['lambdafunc']}.
• Figure 2: Solution $h(r)$ (red, solid line), $k(r)$ (blue, solid line) of equations \ref{['FO']} with $\alpha(r)$ and $\beta(r)$ given by \ref{['exponentialImp']}, with $A=2$ (left), $3$ (middle) and $4$ (right). Dashed lines of these same colors represent the Prasad-Sommerfield solution.
• Figure 3: Solution $h(r)$ (red, solid line), $k(r)$ (blue, solid line) of equations \ref{['FO']} with $\alpha(r)$ and $\beta(r)$ given by \ref{['Imp1']}, with $\epsilon=0.1$ (left), $0.5$ (middle) and $0.8$ (right). Dashed lines of these same colors represent the Prasad-Sommerfield solution, while the black dashed lines correspond to the approximate solution \ref{['approx']}.
• Figure 4: Solution $h(r)$ (red, solid line), $k(r)$ (blue, solid line) of equations \ref{['FO']} with $\alpha=0$ and $\beta=\left(1-e^{-r^n}\right)/2r^n$. Line thickness increases with $n$, which takes the values $2$, $4$ and $8$, and dashed lines correspond to the Prasad-Sommerfield solution.
• Figure 5: Solution $h(r)$ (red, solid line), $k(r)$ (blue, solid line) of equations \ref{['FO']} with $\alpha(r)$ and $\beta(r)$ given by \ref{['longrangeimp']}, with $A=1$ (left) $A=2$ (right) and $n=2$. Dashed lines represent the Prasad-Sommerfield solution.