Generalizations and UV completions of Cho-Maison monopole

This paper demonstrates that Cho-Maison-like monopole configurations can be constructed in a broad class of gauge theories with electroweak-type symmetry breaking and establishes that the Cho-Maison monopole serves as the low-energy effective description of an 't Hooft-Polyakov monopole, specifically emerging from the Pati-Salam model after integrating out heavy degrees of freedom.

The Gist

Imagine the universe is built on invisible rules called "symmetries." Sometimes, these rules break, much like a perfectly round snowflake melting into a puddle. When this happens, strange things can appear. One of these things is a magnetic monopole—a particle that acts like a magnet with only a North pole and no South pole.

For decades, physicists have known about two main types of these magnetic particles:

1. The "Perfect" Monopole: Discovered by 't Hooft and Polyakov, this is a smooth, stable, finite-energy ball of energy. It's like a perfectly formed marble.
2. The "Cho-Maison" Monopole: Discovered by Cho and Maison in the 1990s, this is a weird, jagged version that appears in our Standard Model of physics (the theory describing electricity and magnetism). It's like a marble that has a sharp, infinite spike right in the center.

This paper, written by Fukutaro Miya and Ryosuke Sato, tackles two big questions about the jagged Cho-Maison monopole: Where else can we find them? and Can we fix their sharp spike?

Here is the breakdown of their findings using simple analogies.

1. The "Sharp Spike" Problem

In the original Cho-Maison monopole, the energy at the very center (the origin) goes to infinity. Imagine trying to build a tower of blocks, but the very first block at the bottom is infinitely heavy. The whole structure becomes unstable and breaks the laws of physics.

The authors explain that this happens because the "spike" is essentially a leftover from a simpler, older theory (like the Dirac monopole) that wasn't fully smoothed out.

2. Finding More "Jagged" Monopoles

First, the authors asked: Is this jagged monopole unique to our specific universe, or can we find it elsewhere?

They built a "toy model" (a simplified theoretical playground) with a different set of symmetry rules: SU(3) × SO(3). Think of this as building a new type of Lego set with different colored bricks. They showed that even in this new, more complex set, you can still build a Cho-Maison-style monopole.

The Takeaway: The Cho-Maison monopole isn't a one-off accident. It's a general feature that appears whenever you have a specific type of symmetry breaking (where a big group breaks down into a smaller, diagonal group). It's like finding that a specific type of knot can be tied not just with red string, but with blue, green, or any color string, as long as you tie it the right way.

3. The "UV Completion": Smoothing the Spike

The second, and more exciting, part of the paper answers: How do we fix the infinite spike?

The authors propose that the jagged Cho-Maison monopole is actually just a low-resolution view of a smooth 't Hooft–Polyakov monopole.

The Analogy:
Imagine looking at a high-definition photo of a smooth, round apple on your phone screen.

• The 't Hooft–Polyakov Monopole is the real, high-definition apple. It is perfectly smooth everywhere, even under a microscope.
• The Cho-Maison Monopole is what you see if you zoom in too far on a low-resolution screen. The pixels get so big that the smooth curve of the apple looks like a jagged, blocky spike.

The paper shows that if you look at the Cho-Maison monopole through a "high-resolution lens" (a more fundamental theory called a Grand Unified Theory, or GUT), the spike disappears. It turns out the "spike" was just an illusion caused by ignoring the heavy, hidden particles that exist at very high energies.

4. The Pati–Salam Model: A Real-World Candidate

To prove this isn't just a toy trick, the authors applied this idea to a real, famous theory called the Pati–Salam model. This is a Grand Unified Theory that tries to combine the forces of nature.

They demonstrated that in the Pati–Salam model:

1. At very high energies (the "UV" or high-resolution view), there is a smooth, perfect 't Hooft–Polyakov monopole.
2. As you zoom out to the lower energies of our current universe (the "IR" or low-resolution view), the heavy particles disappear, and the smooth monopole looks exactly like the jagged Cho-Maison monopole.

The Result: The jagged, infinite-energy problem of the Cho-Maison monopole is solved because, in the full theory, the monopole is actually smooth and finite. The "spike" is just a shadow cast by the heavy particles we can't see at low energies.

Summary

• Generalization: The Cho-Maison monopole isn't unique to our current physics; it can appear in many different theoretical universes with similar symmetry-breaking patterns.
• The Fix: The "infinite energy" problem is solved by realizing the Cho-Maison monopole is just a low-energy shadow of a smooth, perfect 't Hooft–Polyakov monopole.
• Stability: Because the underlying "parent" monopole is stable, the Cho-Maison version inherits that stability, making it a physically viable object in these theories.

In short, the paper takes a weird, broken-looking particle and shows us that it's actually just a smooth, perfect particle seen through a blurry lens.

Technical Summary

Technical Summary: Generalizations and UV Completions of Cho–Maison Monopole

Problem Statement
The Cho–Maison monopole is a specific monopole configuration within the electroweak theory ($SU(2)_L \times U(1)_Y \to U(1)_{EM}$) constructed by Cho and Maison [1]. Unlike the regular, finite-energy 't Hooft–Polyakov monopole found in theories with non-trivial second homotopy groups ($\pi_2(G/H) \neq 0$), the Cho–Maison monopole arises in the Standard Model where $\pi_2((SU(2)_L \times U(1)_Y)/U(1)_{EM}) = 0$. Consequently, the Cho–Maison solution exhibits a singularity at the origin, leading to a divergent energy density due to the $U(1)_Y$ gauge field. Furthermore, the conditions under which such monopole configurations can arise in theories beyond the electroweak sector have not been systematically understood, unlike the well-established criteria for 't Hooft–Polyakov monopoles.

Methodology
The authors employ a combination of analytical construction and numerical simulation to address two primary objectives:

1. Generalization: The authors construct a generalized Cho–Maison monopole in a toy model with gauge symmetry $SU(3)_A \times SO(3)_B$ breaking diagonally to $SO(3)_{\text{diag}}$. They utilize a specific ansatz for the scalar and gauge fields, employing two gauge patches (analogous to the Wu–Yang construction) to handle the singularities at the poles of the sphere. They derive the equations of motion (EOM) for the profile functions and solve them numerically to verify the existence and behavior of the solution.
2. UV Completion: The authors propose that the singular Cho–Maison monopole is the low-energy effective description of a regular 't Hooft–Polyakov monopole in an ultraviolet (UV) theory. They demonstrate this via two-step symmetry breaking scenarios:
   • A toy model with $SU(2)_A \times SU(2)_B$ symmetry breaking first to $SU(2)_A \times U(1)_B$ and then to $U(1)_{\text{diag}}$.
   • The Pati–Salam model ($SU(4)_C \times SU(2)_L \times SU(2)_R$) breaking first to the Standard Model gauge group and then to $SU(3)_C \times U(1)_{EM}$.
     In these models, heavy degrees of freedom are integrated out to derive the effective low-energy Lagrangian, which is shown to match the electroweak Cho–Maison setup. The authors construct the full 't Hooft–Polyakov ansatz for the UV theory and numerically solve the coupled EOMs to observe the transition between the UV regular core and the IR Cho–Maison-like tail.

Key Contributions and Results

• Existence of Generalized Monopoles: The paper explicitly demonstrates that Cho–Maison-like configurations are not unique to the electroweak sector. In the $SU(3)_A \times SO(3)_B \to SO(3)_{\text{diag}}$ model, the authors construct a monopole solution where the scalar field $\rho(r)$ behaves as $r^{(\sqrt{3}-1)/2}$ near the origin, rather than the linear $r$ behavior of the 't Hooft–Polyakov monopole. This confirms that such configurations arise in theories with non-Abelian diagonal symmetry breaking patterns (e.g., $SU(N) \times SO(N) \to SO(N)_{\text{diag}}$). Like the original Cho–Maison monopole, these generalized solutions require two gauge patches for a consistent global definition and possess a divergent energy density at the origin in the effective theory.

• UV Completion Mechanism: The authors show that the energy divergence and singularity of the Cho–Maison monopole are artifacts of the low-energy effective description. By embedding the electroweak sector into a UV theory with a two-step symmetry breaking pattern (e.g., $SU(2)_A \times SU(2)_B \to SU(2)_A \times U(1)_B \to U(1)_{\text{diag}}$ or the Pati–Salam model), the monopole is realized as a regular 't Hooft–Polyakov monopole.

  • Profile Analysis: Numerical results (Figures 3 and 4) reveal a smooth transition in the scalar profile functions. In the deep UV region ($r \lesssim r_1$, where $r_1$ is the inverse mass of the heavy Higgs), the scalar fields behave linearly ($\propto r$), characteristic of a regular 't Hooft–Polyakov monopole. In the intermediate region ($r_1 \lesssim r \lesssim r_2$), where the heavy fields are integrated out, the profile matches the Cho–Maison solution.
  • Stability: The topological stability of the 't Hooft–Polyakov monopole in the UV theory (guaranteed by $\pi_2(G/H) \neq 0$) is inherited by the Cho–Maison configuration in the low-energy effective theory, providing a mathematical guarantee of stability that is absent in the pure electroweak description.

• Specific Application to Pati–Salam Model: The paper explicitly constructs the monopole solution in the minimal Pati–Salam model. It demonstrates that a 't Hooft–Polyakov monopole formed at the GUT scale ($v_\Phi$) reduces to an electroweak Cho–Maison monopole at distances larger than the GUT scale but smaller than the electroweak scale. This provides a concrete, realistic candidate for the UV completion of the electroweak monopole.

Significance and Claims
The paper claims to provide a systematic understanding of the conditions under which Cho–Maison monopoles arise, moving beyond the specific case of the Standard Model to a broad class of gauge theories with diagonal symmetry breaking. Furthermore, it resolves the issue of infinite energy and singularity by identifying the Cho–Maison monopole as a low-energy effective limit of a regular 't Hooft–Polyakov monopole. The authors emphasize that while the Cho–Maison monopole is singular in the effective theory, its embedding in a UV-complete theory renders it a physically well-motivated, finite-energy, and topologically stable object. The work suggests that the realization of Cho–Maison monopoles requires specific conditions on the gauge group and symmetry breaking pattern, distinct from those in standard GUTs like minimal $SU(5)$ where the Higgs doublet acquires a VEV at the origin, preventing the Cho–Maison behavior.