⚛️ high-energy theory

Electroweak Monopole-antimonopole Pair in the Standard Model

This paper presents the first numerical solution for a finite-separation Cho-Maison monopole-antimonopole pair in the Standard Model's Weinberg-Salam theory, analyzing their magnetic properties and separation across various parameters while noting that the system's total energy remains infinite due to point singularities at the monopole locations.

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

For decades, physicists have chased a ghost that might actually be real: the magnetic monopole. In our everyday world, magnets always come in pairs, with a north pole and a south pole stuck together. If you break a magnet in half, you do not get a lone north or south pole; you simply get two smaller magnets, each with both poles. This rule has held up in every experiment ever performed. However, the equations that describe the fundamental forces of nature, specifically the theory that unifies electricity and magnetism with the weak nuclear force, suggest that a single, isolated magnetic pole should be able to exist. If such a particle were found, it would be a monumental discovery, the first elementary particle to carry a magnetic charge, and it could help explain how the universe formed its large-scale structures. The challenge is that these theoretical particles are predicted to be incredibly heavy and difficult to create, and until now, no one had successfully modeled a pair of them sitting close together in the specific framework of the Standard Model of particle physics.

A team of researchers from the School of Physics at Universiti Sains Malaysia has now taken a significant step toward understanding how these particles might behave. They have produced the first numerical solution that describes a pair of these theoretical magnetic poles, one positive and one negative, existing as a distinct unit within the standard laws of physics. In their work, they constructed a model where a magnetic monopole and its opposite, an antimonopole, are separated by a finite distance but are not connected by a string of energy, a configuration that had previously been elusive in this specific theory. The researchers found that these two poles, carrying opposite magnetic charges, can sit apart from each other in a configuration where the positive pole resides in the upper half of their mathematical space and the negative pole in the lower half, though whether this arrangement represents a stable bound state remains to be determined.

The scientists focused on a specific type of monopole known as the Cho-Maison monopole, which is a hybrid version of the theoretical particles predicted by the standard model. Unlike other theoretical monopoles that are connected by a flux tube, or a string of energy, the pair they modeled exists as two separate entities. To find this solution, the team used powerful computers to solve a complex set of equations that describe how the fields of force and the Higgs field interact. They did not just look for any solution; they specifically searched for a configuration where the two poles are separated by a measurable distance. They tested their model across a wide range of conditions, varying the strength of the interaction between the particles and the angle that determines how the electromagnetic and weak forces mix.

One of the most striking findings was how the distance between the two poles changes depending on the properties of the universe they are modeled in. When the researchers adjusted the strength of the Higgs field's self-interaction, they observed that the distance between the poles would shrink and then expand again. They found that there is a specific limit to how strong this interaction can be before the solution simply disappears, a feature that distinguishes these particles from other similar theoretical objects. Similarly, when they varied the mixing angle of the forces, they discovered a lower limit below which the pair could not exist. In their most realistic simulation, using the known physical values for the masses of the particles involved, they calculated that the magnetic dipole moment of this pair is approximately 8.55 in their specific units, and the distance separating the two poles is about 8.20 units.

Despite these successes, the researchers were careful to note a significant limitation in their model. Because the mathematical description of these monopoles includes a point of infinite density at the very center of each pole, the total energy of the system comes out as infinite. This mathematical singularity means that, in this specific simulation, it is not yet possible to determine if the pair constitutes a bound state, and their total mass cannot be calculated directly. However, the researchers point out that this does not necessarily mean the particles do not exist; it simply means the current mathematical tools need refinement, perhaps by introducing a smoothing function to handle the infinite point. Their work confirms that a configuration of two separated Cho-Maison monopoles is mathematically possible within the standard model, providing a concrete target for future experiments at particle colliders like the Large Hadron Collider, where scientists are actively searching for evidence of these elusive magnetic charges.

Technical Summary: Electroweak Monopole-Antimonopole Pair in the Standard Model

Problem Statement
While magnetic monopoles have been a subject of theoretical inquiry since Dirac, and finite-energy solutions like the 't Hooft-Polyakov monopole exist in SU(2) Yang-Mills-Higgs (YMH) theory, the existence of monopoles within the Standard Model's SU(2)×U(1) Weinberg-Salam (WS) theory remains a critical open question. The Cho-Maison monopole, an electroweak generalization of the Dirac monopole, is theoretically required if the Standard Model is correct. However, previous investigations have primarily focused on single monopoles or Nambu monopole-antimonopole pairs (MAPs) connected by a $Z^0$ flux string. A significant gap exists in the numerical understanding of a finitely separated Cho-Maison MAP, which lacks a connecting flux string and carries the specific magnetic charge characteristic of the Cho-Maison solution ( $\pm 4\pi/e$ ). Furthermore, the infinite energy of these configurations due to point singularities has historically complicated the analysis of their physical properties.

Methodology
The authors present the first numerical solution for a Cho-Maison MAP in the SU(2)×U(1) WS theory. The study utilizes an axially symmetric magnetic ansatz within the bosonic sector of the WS Lagrangian.

Governing Equations: The system is reduced from the full field equations to seven coupled second-order partial differential equations by substituting the specific ansatz for the gauge fields ( $A^a_\mu, B_\mu$ ) and the Higgs doublet ( $\phi$ ).
Dimensionless Formulation: The equations are rescaled using the W-boson mass ( $m_W$ ) and the Higgs vacuum expectation value ( $H_0$ ), introducing dimensionless parameters: the Weinberg angle ( $\tan \theta_W = g'/g$ ) and the Higgs self-coupling ( $\beta = \lambda/g^2$ ).
Numerical Implementation: The equations are solved using a finite difference method (central difference approximation) on a non-equidistant grid ( $M=70, N=60$ ) covering the entire space via a compactified radial coordinate.
Boundary Conditions: Solutions are constrained by specific boundary conditions at the origin, along the z-axis ( $\theta=0, \pi$ ), and at spatial infinity to ensure the correct topological structure and asymptotic behavior.
Parameter Space: The investigation covers a range of physical parameters: $\tan \theta_W \in [0.4675, 10]$ and $\beta \in [0, 1.7704]$ , with specific focus on physical values ( $\tan \theta_W \approx 0.5356$ , $\beta \approx 0.7782$ ).

Key Contributions

First Numerical Solution: The paper provides the first numerical construction of a finitely separated Cho-Maison monopole-antimonopole pair. Unlike the Nambu MAP, this configuration does not possess a $Z^0$ flux string connecting the poles.
Charge Verification: The authors rigorously calculate the magnetic charge enclosed within Gaussian surfaces surrounding the poles. The results confirm that the upper pole carries a charge of $+4\pi/e$ and the lower pole $-4\pi/e$ , verifying their identity as Cho-Maison monopoles rather than Nambu monopoles (which carry charge $\pm 4\pi \sin^2 \theta_W / e$ ).
Topological Distinction: The study distinguishes the Cho-Maison MAP from SU(2) YMH MAPs and Nambu MAPs (electroweak sphalerons) through 3D Higgs modulus plots, highlighting the unique separation and lack of flux string in the Cho-Maison configuration.

Results

Energy and Singularities: The total energy of the system is found to be infinite due to point singularities located at the monopoles. This is evidenced by the divergence of the weighted energy density ( $\epsilon_W$ ) at the pole locations.
Pole Separation ( $d_z$ ): The separation between the monopole and antimonopole was measured as a function of $\beta$ $β$ (at fixed physical $\tan \theta_W$ $tan θ_{W}$ ) and $\tan \theta_W$ $tan θ_{W}$ (at fixed physical $\beta$ $β$ ).
- At physical $\beta = 0.7782$ , $d_z \approx 8.2012$ (in units of $1/m_W$ ).
- As $\beta$ increases from 0, $d_z$ decreases to a local minimum of $8.0920$ at $\beta \approx 1.25$ , then increases until a critical upper bound $\beta_c = 1.7704$ is reached, beyond which no solutions exist. This upper bound is a feature not observed in SU(2) YMH MAPs.
Magnetic Dipole Moment ( $\mu_m$ ): The magnetic dipole moment was calculated for the same parameter ranges.
- At physical parameters, $\mu_m \approx 8.5514$ (in units of $1/(e \cdot m_W)$ ).
- A lower bound for $\tan \theta_W$ was identified at $0.4675$, below which no solutions exist.
Physical Values: The paper reports precise numerical values for the physical configuration: $d_z = 8.2012$ and $\mu_m = 8.5514$ .

Significance and Claims
The authors assert that this work demonstrates the feasibility of constructing a finitely separated Cho-Maison MAP within the Standard Model. The existence of such a solution in the topologically trivial sector suggests that Cho-Maison monopoles could exist in nature, despite the triviality of the second homotopy group of the vacuum manifold in the standard formulation. The paper emphasizes that while the infinite energy prevents a definitive conclusion on whether these pairs are bound states, the solutions are mathematically consistent within the classical field theory. The authors note that the solutions can be regularized (rendering the energy finite) by introducing a non-trivial U(1) hypercharge permeability, a topic reserved for future work. The study establishes a baseline for the magnetic dipole moment and pole separation of these objects, providing necessary data for potential future experimental searches at facilities like the LHC (MoEDAL and ATLAS), though the paper itself does not propose new experimental protocols.

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