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Magnetic monopoles with an internal degree of freedom

Petr Beneš, Filip Blaschke

TL;DR

This work analyzes magnetic monopoles in spontaneously broken SU(2) gauge theories with adjoint scalars, focusing on non-canonical kinetic terms and the Bogomol'nyi–Prasad–Sommerfield (BPS) limit. It derives a general class of Lagrangians, establishes a BPS framework via a reduced set of form-functions, and obtains analytic monopole solutions under a hedgehog ansatz. A novel internal moduli parameter $\xi$ emerges in a particular non-invertible case, yielding a family of finite-energy monopoles with identical mass $M=\frac{4\pi v}{g}$ but distinct energy-density profiles, suggesting a physical zero mode on the monopole moduli space. The paper further illustrates explicit realizations via power-function Lagrangians (special and general) and discusses the regularity of gauge potentials, the role of redundancies in the Lagrangian description, and potential connections to a collapsed wormhole geometry, with directions for extending beyond the BPS limit.

Abstract

We consider a class of spontaneously broken $SU(2)$ gauge theories with adjoint scalar and look for exact magnetic monopole solutions in the Bogomol'nyi-Prasad-Sommerfield (BPS) limit. We find that some of the resulting solutions exhibit a new internal degree of freedom (a moduli space parameter) that controls the energy density profile of the monopole while keeping the total energy (mass) constant.

Magnetic monopoles with an internal degree of freedom

TL;DR

This work analyzes magnetic monopoles in spontaneously broken SU(2) gauge theories with adjoint scalars, focusing on non-canonical kinetic terms and the Bogomol'nyi–Prasad–Sommerfield (BPS) limit. It derives a general class of Lagrangians, establishes a BPS framework via a reduced set of form-functions, and obtains analytic monopole solutions under a hedgehog ansatz. A novel internal moduli parameter emerges in a particular non-invertible case, yielding a family of finite-energy monopoles with identical mass but distinct energy-density profiles, suggesting a physical zero mode on the monopole moduli space. The paper further illustrates explicit realizations via power-function Lagrangians (special and general) and discusses the regularity of gauge potentials, the role of redundancies in the Lagrangian description, and potential connections to a collapsed wormhole geometry, with directions for extending beyond the BPS limit.

Abstract

We consider a class of spontaneously broken gauge theories with adjoint scalar and look for exact magnetic monopole solutions in the Bogomol'nyi-Prasad-Sommerfield (BPS) limit. We find that some of the resulting solutions exhibit a new internal degree of freedom (a moduli space parameter) that controls the energy density profile of the monopole while keeping the total energy (mass) constant.

Paper Structure

This paper contains 15 sections, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The stage
  3. The general (non-BPS) Lagrangian
  4. Form-invariance and redundancy
  5. The BPS limit
  6. Spherical symmetry
  7. Solving the BPS equations
  8. The solution
  9. The parameter $\xi$
  10. Exploiting the redundancy
  11. The behavior of the gauge potential
  12. Examples
  13. Power-function Lagrangian -- Special case
  14. Power-function Lagrangian -- General case
  15. Summary and discussion

Figures (2)

  • Figure 1: The function $\lambda(\rho)$, Eq. \ref{['Ei_H0']}, for various values of the parameter $|\xi|$. Notice the different behavior for $|\xi| \leq 1$ and for $|\xi| > 1$.
  • Figure 2: Energy densities \ref{['edensitypownoninv']} for a single monopole solution of the power-function theory \ref{['LagPowSimp']} for $N = n/m = 0.1,\, 0.5,\, 1,\, 2$ and for various values of $|\xi| \leq 1$. In the case of $N < 1$ (top panels) the limit value $|\xi| = 1$ is included just to show that the energy density indeed diverges in this case. Notice that dependence of the energy density profile on $\xi$ is stronger for smaller $N$.