On resonances and bound states of the 't Hooft-Polyakov monopole

TL;DR

This paper develops a quaternionic approach to analyze the spectral properties of linearized Yang-Mills-Higgs equations around a 't Hooft-Polyakov monopole, revealing bound states and resonances in various coupling regimes.

Contribution

It introduces a quaternionic formulation for the spectral analysis of monopole linearized equations, extending previous studies to include non-zero Higgs self-coupling.

Findings

Identification of zero-energy bound states and Coulomb bound states.

Numerical analysis of Feshbach resonances in different coupling regimes.

Confirmation of resonances with non-vanishing Higgs self-coupling.

Abstract

We present a systematic approach to the linearised Yang-Mills-Higgs equations in the background of a 't Hooft-Polyakov monopole and use it to unify and extend previous studies of their spectral properties. We show that a quaternionic formulation allows for a compact and efficient treatment of the linearised equations in the BPS limit of vanishing Higgs self-coupling, and use it to study both scattering and bound states. We focus on the sector of vanishing generalised angular momentum and analyse it numerically, putting zero-energy bound states, Coulomb bound states and infinitely many Feshbach resonances into a coherent picture. We also consider the linearised Yang-Mills-Higgs equations with non-vanishing Higgs self-coupling and confirm the occurrence of Feshbach resonances in this situation.