Simulations of Magnetic Monopole Collisions

TL;DR

This paper investigates the collisions of BPS magnetic monopoles using numerical simulations to test and extend the moduli space approximation. By constructing approximate analytic multi-monopole configurations and performing planar and non-planar scattering simulations, it verifies characteristic scattering angles (90°, 60°, 45°) and reveals toroidal, tetrahedral, and cubic intermediate states. The study also probes relativistic and non-BPS regimes, showing that the moduli-space description remains robust while exposing deviations due to radiation and inner-structure effects. Overall, the work demonstrates the feasibility of detailed field-theoretic investigations of multi-monopole dynamics and outlines future directions for analytic initial data and non-planar dynamics, with implications for broader topological soliton systems.

Abstract

In this paper, we investigate the scattering of BPS magnetic monopoles through numerical simulations. We present an ansatz for various multi-monopole configurations suitable for analyzing monopole scattering processes. Our study includes planar scattering scenarios involving two, three, and four monopoles, as well as non-planar processes where three and four monopoles form intermediate tetrahedral and cubic states, respectively. Our observations align with the theoretical predictions of the moduli space approximation. Furthermore, we extend our analysis to relativistic velocities and explore parameters beyond the BPS limit.

Figures (9)

• Figure 1: These two figures sketch the moduli space structure of two magnetic monopoles. The cone with the smoothed-out tip describes the motion of the monopoles in the $x$-$y$-plane. The trumped describes the motion in the other two planes.
• Figure 2: This contour plot illustrates the energy density within the interval $[0.04\, m_v^4/g^2, 0.065\, m_v^4/g^2]$ for a charge-$2$ magnetic monopole in the non-BPS case with $m_h/m_v=0.1$. The high-energy contours (yellow) form a toroidal shape. However, a low energy contribution (red) appears in the center of the torus, which we didn't observe in the BPS case, shown in Figure \ref{['fig:N-monopole-solution']}. The coordinates are given in units of $m_v^{-1}$.
• Figure 3: These contour plots illustrate the toroidal shape of the energy density for charge -$2$, -$3$, and -$4$ monopoles in the BPS limit $m_h \rightarrow 0$. We can observe that the radius of the torus increases for higher charges. The contours illustrate energy densities within the intervals $[0.045,0.07]$, $[0.032,0.053]$, and $[0.03,0.045]$ (in units of $m_v^4/g^2$), respectively. The length values are given in units of $m_v^{-1}$.
• Figure 4: The scalar field direction $(\hat{\phi}^1, \hat{\phi}^2)^T$ in the $x$-$y$-plane (top) and $(\hat{\phi}^1, \hat{\phi}^3)^T$ in the $x$-$z$-plane (bottom) is illustrated in a vector plot.
• Figure 5: These contour plots show three frames for the energy densities bigger than $0.06 m_v^4/g^2$ for the two monopoles right-angle scattering. As we can see, the energy density forms a toroidal structure during the scattering process. The length and time values are given in units of $m_v^{-1}$.