The electroweak magnetic monopole in the presence of KSVZ axion

This paper investigates the implications of the KSVZ axion on the electroweak Cho-Maison monopole by deriving the equations of motion and numerically analyzing how the axion-photon coupling alters the monopole's mass, electromagnetic charges, and potential energy.

The Gist

Imagine the universe as a giant, complex machine. For a long time, physicists have been trying to understand two very strange, hypothetical parts of this machine: Axions and Magnetic Monopoles.

• Axions are like invisible, ghostly particles proposed to solve a specific puzzle about why the universe behaves the way it does regarding time and symmetry. They are also a top candidate for "Dark Matter," the invisible stuff holding galaxies together.
• Magnetic Monopoles are particles that act like a magnet with only a North pole (no South pole). While we usually break a magnet in half and get two smaller magnets (North and South), a monopole would be a single, isolated pole.

This paper asks a simple but profound question: What happens if these two ghostly particles meet? specifically, how does an Axion change the behavior of a "Cho-Maison Monopole," which is a theoretical version of a magnetic monopole that fits inside our current understanding of particle physics (the Standard Model).

Here is the breakdown of their findings using everyday analogies:

1. The Setup: A Heavy Magnet and a Ghostly Wind

Think of the Cho-Maison Monopole as a very heavy, dense, spherical magnet sitting in space. It's so heavy it weighs about as much as 11,000 proton-masses (a "TeV scale" mass), which is why scientists hope to find it in giant particle colliders like the LHC.

Now, imagine a KSVZ Axion (a specific type of axion) as a "ghostly wind" blowing through the universe. This wind doesn't just blow past the magnet; it interacts with the magnet's magnetic field.

2. The Interaction: The "Witten Effect"

The paper relies on a concept called the Witten Effect. You can think of this like a magical rule: If a magnetic monopole sits in a field of axions, the monopole suddenly gains an electric charge.

Normally, a magnetic monopole is just magnetic. But because of the axion "wind," the monopole starts to act like it has an electric charge too. It becomes a "dyon" (a particle with both magnetic and electric properties).

3. The Experiment: Simulating the Collision

The authors didn't smash particles in a lab; they built a detailed mathematical simulation. They created a model where:

• The Monopole is the core structure.
• The Axion is a field wrapping around it.
• They used a "spherical" shape for everything to keep the math manageable (like assuming a planet is a perfect sphere).

They solved complex equations to see how the axion field behaves when it is trapped inside the magnetic field of the monopole.

4. The Results: What Changed?

When they turned on the axion in their simulation, three main things happened to the monopole:

• It got slightly heavier: Just like a backpack gets heavier when you add a book, the monopole's mass increased slightly (by about 0.2% to 6%) because of the energy added by the axion interaction.
• Its electric charge changed: This is the big one. The axion wind altered the amount of electric charge the monopole carries. Depending on the specific settings of their model, the charge changed by up to 30%.
  • Analogy: Imagine a magnet that usually has a specific "static cling" (electric charge). The axion wind changes how sticky it is, making it either stickier or less sticky than before.
• The Axion itself got pushed away: The monopole's magnetic field is so intense that it actually repels the axion field near the very center. The axion field stays "suppressed" (flat) near the core and only starts to rise further out. It's like a strong wind pushing a light feather away from a spinning fan; the feather can't get close to the center.

5. Why Does This Matter?

The paper concludes that if we ever find a magnetic monopole in a particle collider, we shouldn't just look for a magnet. We need to look for a magnet with a specific, slightly altered electric charge.

• If we find a monopole with a charge that doesn't match our old predictions, it might be proof that axions exist.
• Conversely, if we know axions exist, it changes how we calculate the mass and charge of monopoles, which helps us know exactly what to look for in experiments.

Summary

In simple terms, the paper says: "If you put a magnetic monopole in a sea of axions, the axions will nudge the monopole, making it slightly heavier and changing its electric charge. If we find these monopoles in the future, these tiny changes could be the smoking gun that proves axions are real."

Technical Summary

Technical Summary: The Electroweak Magnetic Monopole in the Presence of KSVZ Axion

Problem Statement
The paper addresses the theoretical interplay between two candidates for physics beyond the Standard Model (SM): the axion and the magnetic monopole. Specifically, it investigates the implications of the Kim-Shifman-Vainshtein-Zakharov (KSVZ) axion model on the properties of the Cho-Maison (CM) electroweak monopole. The CM monopole is a topological solution within the Weinberg-Salam theory with a mass scale of approximately 11 TeV, making it a target for collider searches (e.g., ATLAS, MoEDAL). While the Witten effect establishes a relationship between axions and magnetic monopoles by inducing electric charges, this work seeks to determine how the axion-photon coupling modifies the specific topological solutions, mass, and electromagnetic charges of the electroweak monopole.

Methodology
The authors construct a spherically symmetric ansatz for the electroweak dyon within the KSVZ framework. The effective Lagrangian is decomposed into three components: the Cho-Maison monopole sector, the axion kinetic energy, and the axion-photon interaction term ($\frac{1}{4} g_{a\gamma\gamma} a F_{\mu\nu} \tilde{F}^{\mu\nu}$).

1. Field Parameterization: The Higgs doublet, gauge fields ($W, B$), and the complex scalar singlet (containing the axion $a$) are parameterized using radial functions $\rho(r)$, $f(r)$, $A(r)$, $B(r)$, and $a(r)$.
2. Equations of Motion (EoM): The authors derive a coupled system of five non-linear differential equations for these radial functions. The system includes modifications due to the axion-photon coupling $g_{a\gamma\gamma}$.
3. UV Regularization: Recognizing that the classical energy of the CM monopole is infinite due to the $U(1)_Y$ singularity, the authors employ a UV regularization scheme. This involves introducing a hypercharge permittivity function $\epsilon(\rho) = (\rho/\rho_0)^n$ to render the monopole mass finite.
4. Numerical Solution: The coupled differential equations are solved numerically using the relaxation method for the region $x_r = M_W r > x_{min}$, with analytical approximations used near the core ($x < x_{min}$). Boundary conditions are set for the Higgs, gauge, and axion fields, with the asymptotic axion value $a(\infty)$ treated as a parameter characterizing the background.

Key Contributions and Results

• Topological Solutions: The paper presents the first numerical solutions for the Cho-Maison electroweak monopole in the presence of a KSVZ axion field. The results show that for small axion-photon couplings, the monopole profile remains compatible with the pure CM solution, but significant deviations occur for the axion field profile and the gauge fields when the regularization parameter $n$ is large.
• Monopole Mass: The study calculates the total energy (rest mass) of the dyon. The presence of the KSVZ axion induces a slight increase in the monopole mass. Depending on the regularization parameter $n$ and the axion coupling, the mass increases by approximately 0.2% to 6% compared to the axion-free case. For instance, at $n=36$ (constrained by LHC Higgs data), the mass increases by roughly 2.64%.
• Electromagnetic Charges: The axion-photon coupling modifies the electric charge of the monopole via the Witten effect. The magnetic charge $q_m$ remains unchanged ($-4\pi/e$), but the electric charge $q_e$ is altered. The presence of the KSVZ axion changes the magnitude of the electric charge by up to ~31% for specific values of $n$ (e.g., $n=60$). The shift is non-monotonic with respect to $n$.
• Axion Potential Energy: The authors analyze the energy cost associated with the axion field in the monopole background. They compare the numerical solution of the axion profile against the classical analytical approximation ($a(r) \sim e^{-r_0/r}$). The results indicate that the monopole background suppresses the axion field near the core, delaying its growth to larger radii. Consequently, the integrated axion potential energy derived from the numerical solution is significantly larger (by orders of magnitude) than the minimum energy calculated from the classical analytical formula.

Significance and Claims
The paper claims that the interaction between the axion and the electroweak monopole leads to observable modifications in the monopole's physical characteristics, specifically its mass and electric charge. These modifications are directly linked to the axion-photon coupling strength and the axion background.

The authors assert that these findings have dual implications:

1. For Axion Physics: The properties of the monopole could serve as a probe to confirm the nature of the axion if the monopole is observed.
2. For Monopole Searches: The presence of axions alters the expected mass and charge of TeV-scale electroweak monopoles, which must be accounted for in dedicated searches at colliders like the LHC.

The work emphasizes that the regularization parameter $n$ plays a critical role in the stability of the solutions and the magnitude of these effects, noting that as $n \to \infty$, the axion and $Z$-boson fields are suppressed to zero, reverting the system to the standard SM dyon solution. The paper concludes that the axion background effectively confines the axion potential energy and modifies the electroweak monopole's topological characteristics, providing a new theoretical framework for interpreting potential future discoveries of electroweak monopoles.