The non-topological $Z^\prime$ string in the 331 model and its classical stability

This paper investigates the classical stability of a non-topological $Z^\prime$ string in the minimal 331 model derived from an $SU(6)$ toy model and concludes that the string is only stable near the semilocal limit, suggesting such defects are unlikely to exist in unified theories based on $SU(N>5)$ Lie algebras.

The Gist

Imagine the universe as a giant, complex machine made of invisible threads and gears. Physicists call these threads "fields" and the gears "particles." Sometimes, when this machine is built or when it changes its state (like water freezing into ice), it can get stuck in a weird shape. These stuck shapes are called defects.

One specific type of defect is a cosmic string. Think of it like a tiny, infinitely long, super-tight rubber band stretched across the universe. It's not made of rubber, but of energy and magnetic fields.

This paper is about investigating a very specific, hypothetical rubber band called the $Z'$ string within a theoretical model called the 331 model. Here is the breakdown of what the authors did, using simple analogies:

1. The Setup: Building a New Machine

The Standard Model (our current best description of the universe) is like a well-tested car engine. But physicists suspect there might be a bigger, more complex engine underneath it, like a Grand Unified Theory (GUT).

The authors started with a "toy model" based on a mathematical structure called $SU(6)$. Think of this as a blueprint for a massive, 6-cylinder engine. They wanted to see what happens when this big engine breaks down into smaller, familiar parts (like our current 3-cylinder engine).

• The Result: When they broke the big engine down, a new type of force carrier appeared, called the $Z'$ boson.
• The Goal: They wanted to see if this new force could create a stable cosmic string (a rubber band) that could survive in the universe.

2. The Problem: The Rubber Band is Wobbly

In physics, some rubber bands are naturally stable because of their shape (like a knot you can't untie). These are called topological strings.
However, the $Z'$ string they found is non-topological. This is like a rubber band that isn't tied in a knot; it's just held together by tension. If you wiggle it too much, it will snap or unravel.

The authors asked: "Can this wobbly rubber band stay intact, or will it collapse?"

3. The Experiment: Shaking the String

To test stability, the authors acted like engineers shaking a prototype.

• They built a mathematical simulation of the string.
• They introduced tiny "perturbations" (little shakes or wiggles) to the string's energy and shape.
• They solved complex equations (like the "Helmholtz equations") to see if these wiggles would die out (stable) or grow into a giant crash (unstable).

4. The Findings: It Only Works in a "Sweet Spot"

The results were surprising and restrictive. They found that the string is unstable in almost every scenario, unless the universe is tuned to a very specific, extreme setting.

• The Analogy: Imagine trying to balance a pencil on its tip. It's almost impossible. But if you have a very specific, weird wind blowing from a specific direction, maybe, just maybe, it stays up.
• The "Sweet Spot": The string only stays stable if the mixing angle (a parameter that determines how different forces mix, like mixing paint colors) is pushed to the extreme limit of $\pi/2$ (90 degrees).
• The Catch: In this extreme limit, the physics changes so much that the "local" rules of the universe effectively become "global" rules. It's a very unnatural state for our universe to be in.

5. The Conclusion: Don't Hold Your Breath

The authors concluded that because the universe we observe doesn't seem to be in this extreme "sweet spot" (the mixing angles don't match the required values), these specific $Z'$ strings probably don't exist in nature.

Furthermore, they suggest that if you try to build even bigger, more complex theories (using $SU(N)$ algebras where $N > 5$), these non-topological strings will likely be even less stable.

Summary in a Nutshell

The authors built a theoretical model to see if a new kind of cosmic "energy rubber band" could exist. They shook it, tested it, and found that it's too wobbly to survive unless the universe is tuned to a very weird, extreme setting that doesn't match reality. Therefore, these specific cosmic strings are likely just mathematical curiosities, not real things floating in space.

Technical Summary

1. Problem Statement

The paper investigates the classical stability of non-topological Z′ strings within the context of the minimal 331 model.

• Context: The 331 model is an extension of the Standard Model (SM) where the gauge group $SU(3)_c \times SU(3)_W \times U(1)_X$ breaks down to the SM group $SU(3)_c \times SU(2)_W \times U(1)_Y$. This model often arises as a subalgebra of Grand Unified Theories (GUTs) based on $SU(N)$ Lie algebras with $N > 5$ (specifically $SU(6)$ in this work).
• The Issue: While topological defects (like ANO strings) are protected by homotopy groups, non-topological strings (like the electroweak Z-string) rely on specific parameter ranges for stability. In the SM, stable Z-strings only exist in the "semilocal limit" (Weinberg angle $\theta_W \to \pi/2$) with a light Higgs boson, a condition ruled out by experimental data.
• Goal: The authors aim to determine if the Z′ string in the 331 model can be classically stable under realistic parameters, or if it requires an unphysical semilocal limit, thereby constraining the viability of such strings in unified theories based on $SU(N > 5)$.

2. Methodology

The authors employ a systematic field-theoretic approach to analyze the stability of the string solution:

• Model Construction:

  • They derive the minimal 331 model from a one-generation $SU(6)$ toy model.
  • Fermion Sector: Utilizes the maximal symmetry breaking pattern $SU(6) \to SU(3)_c \times SU(3)_W \times U(1)_X \to SU(3)_c \times SU(2)_W \times U(1)_Y$.
  • Higgs Sector: Introduces two Higgs triplet fields ($\Phi_{3,1}, \Phi_{3,2}$) arising from the $SU(6)$ adjoint and fundamental representations to facilitate sequential symmetry breaking.
  • Gauge Sector: Defines the mixing between the $SU(3)_W$ generator $W^8$ and the $U(1)_X$ boson $X$ via a mixing angle $\vartheta_S$, defining the massive $Z'$ boson.

• String Solution:

  • They construct the classical Z′ string solution with unit winding number in cylindrical coordinates $(r, \phi)$.
  • The solution involves profile functions for the gauge field ($\bar{\zeta}(r)$) and the two Higgs triplets ($\bar{f}_1(r), \bar{f}_2(r)$).
  • These profiles are obtained by solving coupled non-linear Euler-Lagrange differential equations numerically.

• Stability Analysis:

  • Perturbation: Small perturbations are applied to both the gauge fields ($\delta W'$) and scalar fields ($\delta \pi$) around the string background.
  • Gauge Fixing: A specific gauge-fixing term is introduced to eliminate unphysical degrees of freedom and linear derivative terms.
  • Stability Matrix: The perturbed string tension is expanded to quadratic order. The problem is reduced to an eigenvalue equation $\hat{O}\delta\Theta = \omega^2 \delta\Theta$, where $\hat{O}$ is a stability matrix acting on the perturbation vector $\delta\Theta$.
  • Numerical Solution: The authors solve for the eigenvalues $\omega^2$ of this matrix. A negative eigenvalue ($\omega^2 < 0$) indicates an instability (a mode that grows exponentially).

3. Key Contributions

• Derivation of the 331 Z′ String: Explicitly constructed the string profile functions for the minimal 331 model derived from an $SU(6)$ GUT, including the effects of two Higgs triplets.
• Generalized Stability Matrix: Derived a comprehensive stability matrix that accounts for:
  • Couplings between two distinct Higgs triplets (off-diagonal terms $D_{12}$).
  • Interactions between off-diagonal gauge bosons ($W'^\pm, N, \bar{N}$) and the string background.
  • Spin-magnetic coupling terms arising from the $Z'$ magnetic field.
• Identification of Instability Sources: Isolated two primary sources of instability analogous to the SM Z-string but modified for the 331 context:
  1. Off-diagonal Gauge Boson Condensate: Instability driven by the magnetic field strength coupling to the spin of off-diagonal gauge bosons ($W'^\pm$).
  2. Higgs Potential Instability: Instability driven by the self-couplings of the Higgs fields ($\beta_1, \beta_2$) and the mixing parameter $\beta_3$.

4. Results

The numerical analysis yields the following critical findings:

• Semilocal Limit Requirement: The Z′ string is unstable for generic parameter choices. It becomes stable only when the 331 mixing angle $\vartheta_S$ approaches the semilocal limit $\vartheta_S \to \pi/2$ (where $c_{\vartheta_S} \to 0$).
• Critical Coupling Ratios:
  • In the limit where the Higgs self-coupling $\lambda_3$ is tuned to minimize instability, stability requires $s^2_{\vartheta_S} \gtrsim 0.9$ to $0.95$.
  • Translating this to gauge couplings, stability requires the ratio $g_X / g_{3W} \gtrsim 5.20$ to $7.55$.
• Incompatibility with Unification:
  • Conventional $SU(6)$ unification requires $g_{3W}(v_U) = g_1(v_U)$ (where $g_1$ is related to $g_X$).
  • Affine $SU(6)$ unification requires $2g_{3W} = g_1$.
  • The stability requirement ($g_X \gg g_{3W}$) is incompatible with these standard unification relations.
• Role of Higgs Parameters: Even with tuning the Higgs self-coupling $\lambda_3$ to be negative (to counteract the destabilizing effect of $\lambda_1, \lambda_2$), the string remains unstable unless the mixing angle is in the semilocal limit.

5. Significance

• Constraint on Unified Theories: The results suggest that non-topological Z′ strings are unlikely to exist as stable classical solutions in unified theories based on $SU(N > 5)$ Lie algebras. The parameter space required for stability contradicts the gauge coupling unification conditions typically expected in such models.
• Extension of SM Z-String Physics: The paper confirms that the mechanism of instability (dependence on the mixing angle and Higgs mass/couplings) observed in the SM Z-string persists and is generalized in the 331 model, despite the presence of additional Higgs fields and gauge bosons.
• Future Directions: The authors suggest that while non-topological strings are unstable in this framework, future research should focus on topological strings arising from the spontaneous breaking of global $U(1)$ symmetries in extended Higgs sectors, which may remain stable.

In summary, the paper provides a rigorous numerical proof that the non-topological Z′ string in the minimal 331 model is classically unstable under realistic unification conditions, effectively ruling out their existence in standard $SU(N > 5)$ GUT scenarios.