⚛️ phenomenology

System of excited monopole-antimonopole pair in the Weinberg-Salam model

This paper presents a numerical study of 300 axially symmetric excited monopole-antimonopole pair solutions in the Weinberg-Salam model, confirming their energy range of 13.1690–21.0221 TeV, their magnetic charges of $\pm\frac{4\pi}{e}\sin^2\theta_W$ , and an oscillatory pattern revealed through symmetry-based data analysis.

Original authors: Dan Zhu, Khai-Ming Wong, Guo-Quan Wong

Published 2026-10-21

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CC BY 4.0

Original authors: Dan Zhu, Khai-Ming Wong, Guo-Quan Wong

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, invisible architecture of the universe, there are particles that carry electric charge, like the electrons that power our lights. But for decades, physicists have wondered if there are also particles that carry a magnetic charge, known as magnetic monopoles. While magnets we use every day always come in pairs with a north and a south pole, a monopole would be a particle that is purely north or purely south. The idea was first proposed nearly a century ago, but finding one has remained one of the greatest challenges in physics. These hypothetical particles are not just curiosities; if they exist, they would help explain why the universe has the structure it does and could even serve as a final test for our most fundamental theories of how matter and forces interact.

A team of researchers has now taken a significant step toward understanding what these particles might look like if they were created in the high-energy environment of the early universe. Working within the standard model of particle physics, which describes how the basic forces of nature operate, the scientists used powerful computer simulations to construct a specific type of magnetic system. Instead of looking for a single isolated particle, they modeled a pair consisting of a magnetic monopole and its opposite, an antimonopole, bound together. In this theoretical setup, the two particles are held in place not by a physical string, but by a concentrated line of energy field that connects them, much like a tether made of pure force.

The researchers focused on a scenario that matches the physical constants of our actual universe, using the most recent data available for the masses of the particles involved. They generated three hundred distinct numerical solutions to describe how this pair could behave. By analyzing these simulations, they determined that the energy required to create such a system would be immense, falling between 13.1690 and 21.0221 tera-electronvolts. This places the potential mass of these objects squarely in the range that modern particle accelerators, like the Large Hadron Collider, are designed to probe, making the search for them a realistic goal for experimental physicists.

One of the most striking discoveries in this work is the behavior of the system itself. Although the computer model used to create these solutions was static, meaning it did not include the variable of time, the resulting data revealed a pattern that strongly suggests movement. When the researchers arranged their results based on the symmetry of the system, a clear, rhythmic pattern emerged. The energy within the system appeared to shift back and forth between the two particles. At one moment, the energy would be concentrated near the top particle, and in the next, it would migrate to the bottom one. This suggests that the pair is not a frozen object but is instead in a state of constant, internal oscillation, with energy flowing between the two ends of their magnetic tether.

The study also confirmed the magnetic strength of these theoretical particles. The researchers calculated the magnetic charge for each solution and found that the values matched a specific prediction made decades ago. The charge was found to be exactly what theory suggests it should be, confirming that the mathematical structure of the model holds up under scrutiny. Furthermore, they observed that the electric currents swirling around the pair moved in sync with the shifting energy, creating a complex, dynamic dance of fields even within a static calculation.

Perhaps the most surprising aspect of the findings is the nature of the movement itself. In a typical magnetic pair, one might expect the two poles to move toward and away from each other, like a spring compressing and expanding. However, in these simulations, both the monopole and the antimonopole appeared to shift in the same direction along their axis of connection. This unusual behavior, where the entire system seems to slide back and forth together rather than stretching apart, hints at a deeper, more complex dynamic that the current static model cannot fully capture. The researchers emphasize that to truly understand this motion, one would need to solve equations that include time as a variable, a task they plan to tackle in future work.

This research does not claim to have found a monopole in a laboratory, but it has provided a detailed map of what such a system would look like if it were to exist. By confirming the energy range and the magnetic properties, the study narrows the search for experimentalists. It also introduces a new way of looking at these systems, suggesting that they are far more active and fluid than previously thought. The discovery of these oscillating patterns in a static model serves as a compelling clue that the universe may be hiding dynamic secrets within its most fundamental building blocks, waiting for the right tools to reveal them.

Technical Summary: System of Excited Monopole-Antimonopole Pair in the Weinberg-Salam Model

Problem Statement
While magnetic monopoles have been extensively studied since Dirac's introduction of the concept, and finite-energy solutions like the 't Hooft-Polyakov monopole exist in SU(2) Yang-Mills-Higgs (YMH) theory, the realistic physical context lies within the SU(2)×U(1) Weinberg-Salam model. Previous numerical studies of axially symmetric monopole-antimonopole pairs (MAP) in this model, such as those in Ref. [21], utilized unphysical parameters (specifically $\theta_W = \pi/4$ ) and lacked dimensional analysis, rendering their results physically insignificant. Furthermore, the conventional numerical approach of using a solution at one parameter value as an initial guess for the next fails for the 1-MAP configuration in the Weinberg-Salam model due to the system's sensitivity. This paper addresses the need for a rigorous investigation of the 1-MAP configuration using physical Higgs self-coupling ( $\beta = 0.7782$ ) and the physical Weinberg angle ( $\tan \theta_W = 0.5358$ ), derived from the latest Particle Data Group values.

Methodology
The authors employ a novel data sampling approach to overcome the limitations of standard numerical continuation methods.

Model and Ansatz: The study utilizes the standard Weinberg-Salam Lagrangian with an axially symmetric magnetic ansatz (Ref. [21]) characterized by winding numbers $n=1$ and $p=2$ .
Dimensionless Transformation: The equations of motion are reduced to seven coupled second-order partial differential equations. A dimensionless transformation is applied using $x = m_W r$ , with physical parameters fixed to $\beta = 0.7782$ and $\tan \theta_W = 0.5358$ .
Novel Data Sampling: Instead of the standard "solution-as-initial-guess" method, the authors construct 300 initial guesses by taking a known 1-MAP solution from the SU(2) YMH theory and introducing minute variations (entries differing by $< 10^{-11}$ ) to the profile functions. This technique exploits the fundamental nature of the 1-MAP in the Weinberg-Salam model, where such minute changes in the initial guess yield a wide spectrum of distinct numerical solutions.
Symmetry Quantification: Since the flux string binding the monopole and antimonopole prevents accurate measurement of pole separation ( $d_z$ ), a new algorithm is devised to quantify the degree of symmetry ( $D$ ) of the Higgs modulus $|\Phi|$ between the upper and lower hemispheres.
Physical Quantities: The total energy is calculated by integrating the energy density, and magnetic charges are computed using a definition that accounts for the mixing of U(1) and SU(2) fields, specifically utilizing the 't Hooft electromagnetic tensor.

Key Results

Energy Range: The study obtained 300 distinct numerical solutions. The total energy of these 1-MAP configurations resides in the range of 13.1690 TeV to 21.0221 TeV.
Energy-Symmetry Correlation: When data are arranged by the symmetry parameter $D$ , a clear pattern emerges: solutions with higher asymmetry (higher $D$ ) correspond to higher total energy. The minimum peak separation in the SU(2) energy density ( $d_{SU(2)}^z$ ) is found to be 0.9038.
Magnetic Charge: The numerical calculation confirms that the magnetic charge carried by the monopole and antimonopole is indeed $\pm \frac{4\pi}{e \sin^2 \theta_W}$ , validating Y. Nambu's prediction [15]. The U(1) contribution to the magnetic charge is found to be negligible (close to zero), indicating the charge is solely contributed by the SU(2) gauge field.
Oscillating Pattern: A unique, non-trivial pattern is observed across the 300 solutions. As the symmetry parameter $D$ $D$ varies, the system exhibits behavior resembling oscillation:
- The energy density "bump" (weighted energy density) shifts along the z-axis.
- The magnetic charge distribution shifts in the same direction for both the monopole and antimonopole, contrasting with traditional dipole oscillations where charges move in opposite directions.
- The electric current in the U(1) gauge field shifts in accordance with the magnetic charge distribution.
- The oscillations of the Higgs field energy density ( $\epsilon_H$ ) appear to be out of phase with the gauge field energy densities ( $\epsilon_{U(1)}$ and $\epsilon_{SU(2)}$ ).

Significance and Claims
The paper claims to provide the first physically significant numerical analysis of the 1-MAP configuration in the Weinberg-Salam model using correct physical parameters. The primary significance lies in the discovery of a "unique pattern" within a static model that suggests time-dependent, oscillating features. The authors emphasize that while the solutions are derived from a static model, the ubiquity of this pattern across all profile functions and physical quantities warrants further investigation. They explicitly state that a definitive confirmation of this oscillatory behavior requires solving the full time-dependent equations (functions of $t, r, \theta$ ), which is beyond the scope of the current static study.

The work also confirms the theoretical prediction regarding the magnetic charge magnitude and establishes the energy scale of these configurations (13–21 TeV), which is relevant for experimental searches for electroweak monopoles. The authors note that the 1-MAP configuration is an unstable saddle point solution in the Weinberg-Salam model under the given physical conditions. Finally, the paper suggests that future work could extend this model to include electric charges (dyons) to explore interactions between the dyon's charge and the reported U(1) electric current.

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