Is our vacuum global in a 331 model with three triplets?

This paper systematically analyzes the vacuum stability of a specific 331 model with three triplets and a softly broken $\mathbb{Z}_2$ symmetry by identifying all potential minima, deriving conditions for a global electroweak vacuum, and calculating metastability bounds to map the viable parameter space in terms of physical quantities.

The Gist

Imagine the universe as a giant, complex building. For decades, physicists have been trying to understand the blueprint of this building, specifically the "Standard Model," which describes the fundamental particles and forces. But there's a missing piece in the blueprint: Why are there exactly three families of particles? (Think of them as three generations of a family: parents, children, and grandchildren, all looking similar but with different weights).

This paper explores a new, more elaborate blueprint called the 331 Model. It suggests that the building's structure is actually based on a more complex symmetry (a 3D shape instead of a 2D one), which naturally explains why there are three families.

However, building a new blueprint is risky. If the design is flawed, the building might collapse, or worse, it might be stable for a while but then suddenly crumble into a different, lower state. The authors of this paper are the structural engineers checking if this new 331 blueprint is safe to live in.

Here is a breakdown of their work using simple analogies:

1. The Three "Tripods" (The Triplets)

In this model, the stability of the universe relies on three specific pillars, called scalar triplets. Imagine these as three tripods holding up a tent.

• The Problem: The tent (our universe) has a specific shape (the "Electroweak Vacuum") that allows us to exist. But if you push the tripods too hard or arrange them wrong, the tent could snap into a completely different shape where physics as we know it doesn't work.
• The Goal: The authors wanted to map out every possible way these tripods could be arranged to see if our current "tent shape" is the strongest one, or if there's a deeper, more stable shape hiding somewhere else.

2. The "Orbit Space" (The Map of Possibilities)

Usually, checking every possible arrangement of these tripods is like trying to find a needle in a haystack the size of a galaxy. There are too many variables.

• The Trick: The authors used a mathematical tool called "Orbit Space." Imagine taking a 3D sculpture and squashing it down into a 2D map that shows all the unique shapes it can make, ignoring the rotations and spins.
• The Result: They discovered that this "map" isn't a simple circle or square. It's a weird, curved, 3D shape (like a rounded pyramid with a curved face). By studying this shape, they could easily see all the possible "valleys" (stable states) and "peaks" (unstable states) without getting lost in the math.

3. The "Global Minimum" vs. The "Local Minimum"

Think of a ball rolling on a hilly landscape.

• Local Minimum: The ball rolls into a small dip and stops. It looks stable, but if you push it hard enough, it might roll over a small hill into a much deeper valley.
• Global Minimum: The deepest valley in the entire landscape. This is the most stable state possible.
• The Danger: Our universe is currently sitting in a "Local Minimum" (the Electroweak Vacuum). The authors asked: "Is there a deeper valley nearby?"
  • If Yes: Our universe is Metastable. It's like a ball balanced on a small hill. It might stay there for billions of years, but eventually, it could tunnel through the hill and fall into the deep valley. If that happens, the laws of physics would change instantly, and everything would be destroyed.
  • If No: We are in the Global Minimum. We are safe forever.

4. The Findings: "It's Complicated"

The authors ran massive computer simulations (like testing the building with thousands of different wind speeds and earthquake scenarios) to see where the "deeper valleys" might be.

• The Good News: In many parts of the blueprint, our current vacuum is indeed the Global Minimum. The building is solid.
• The Bad News: In other parts of the blueprint, there are deeper valleys. In these scenarios, our universe is Metastable.
  • The Twist: Even if we are in a metastable state, it doesn't mean we are doomed today. The "tunnel" to the deep valley might be so high and thick that it would take longer than the age of the universe for the ball to roll over. So, we are safe for now, but the design is technically "risky."

5. The "Mixing" Factor

The paper found a specific condition that determines safety: How much do the tripods mix?

• If the tripods (the particles) interact with each other in a specific, messy way (non-zero mixing), the risk of falling into a deeper valley increases.
• If they stay mostly separate (negligible mixing), the model is perfectly safe, and our vacuum is the global champion.

The Bottom Line

The authors have created a safety manual for the 331 Model. They have:

1. Mapped the terrain: They found all the possible shapes the universe could take in this model.
2. Checked the stability: They calculated exactly which combinations of particle masses and forces keep our universe safe.
3. Set the limits: They told future physicists, "If you build a 331 model with these specific numbers, you are safe. If you use these other numbers, you might be living on borrowed time."

In short, they didn't just say "this model might work." They said, "Here is exactly how to make it work without the universe collapsing." It's a crucial step in deciding if this exotic theory is a viable candidate for the true laws of nature.

Technical Summary

Here is a detailed technical summary of the paper "Is our vacuum global in a 331 model with three triplets?" by Kristjan Kannike, Niko Koivunen, and Aleksei Kubarski.

1. Problem Statement

The Standard Model (SM) leaves fundamental questions unanswered, such as the origin of the three fermion generations. 331 models, based on the gauge group $SU(3)_C \times SU(3)_L \times U(1)_X$, offer a solution by linking the number of generations to anomaly cancellation. However, these models possess complex scalar sectors.

Specifically, the paper addresses the vacuum stability of a 331 model with $\beta = -1/\sqrt{3}$ and three scalar triplets ($\eta, \rho, \chi$). While previous studies have examined boundedness-from-below (BFB) conditions or specific limits, a systematic determination of all potential minima and the conditions under which the electroweak (EW) vacuum is the global minimum has been lacking. If the EW vacuum is not the global minimum, the universe could be in a metastable state, potentially decaying into a deeper, charge-breaking, or symmetry-breaking vacuum. The authors aim to map the full vacuum structure and determine the parameter space where the EW vacuum is stable or metastable.

2. Methodology

The authors employ a rigorous geometric approach to minimize the scalar potential, reducing the complexity of the field space.

• Model Setup: The study focuses on a potential with three $SU(3)_L$ triplets and a softly broken $Z_2$ symmetry ($\chi \to -\chi$). The potential includes quartic couplings, a trilinear term (softly breaking $Z_2$ to avoid axions), and specific cross-terms allowed by the symmetry.
• Orbit Space Formalism: Instead of minimizing the potential with respect to the many real field components directly, the authors utilize orbit space methods and P-matrix formalism.
  • They define a basis of gauge invariants ($p_i$) constructed from the triplets.
  • They separate the potential into radial field norms and dimensionless orbit variables ($\vartheta_i$), which represent the relative angles and volumes between the triplet vectors.
  • This reduces the problem from a high-dimensional field space to a low-dimensional geometric space (the orbit space).
• Geometric Analysis:
  • The shape of the orbit space is determined using the P-matrix (the Jacobian of the invariants). The authors prove the orbit space is a convex set, specifically bounded by an elliptope (a specific semialgebraic set).
  • Because the potential is linear in the orbit variables (specifically $\vartheta_i^2$ and $\vartheta_4$), the global minima must lie on the boundary (the convex hull) of the orbit space.
• Extrema Classification: The authors systematically identify 20 possible extrema (vacua) by analyzing the vertices, edges, and faces of the orbit space boundary. These include the desired EW vacuum, charge-breaking vacua, and vacua with different symmetry breaking patterns.
• Constraints & Stability:
  • Perturbative Unitarity: Constraints are derived from $2 \to 2$ scalar scattering amplitudes.
  • Boundedness from Below (BFB): Necessary and sufficient conditions are derived using copositivity criteria on the quartic coupling matrix, evaluated at the origin, vertices, and the elliptope surface of the orbit space.
  • Metastability: For regions where the EW vacuum is not the global minimum, the authors use the FindBounce code to calculate the Euclidean action ($S_E$) and the tunneling rate to ensure the vacuum lifetime exceeds the age of the universe.

3. Key Contributions

1. Complete Vacuum Structure: This is the first study to systematically determine all 20 extrema of the scalar potential for the 331 model with three triplets and a soft $Z_2$ breaking term.
2. Orbit Space Geometry: The authors explicitly determine the shape of the orbit space for this specific model, proving its convexity and identifying the boundary as an elliptope. This provides a geometric criterion for identifying the global minimum without solving complex coupled equations for every field configuration.
3. Analytical Conditions for Globality: They derive analytical approximations and conditions under which the EW vacuum ($V_{EW}^+$) is deeper than competing vacua (specifically $V_\eta, V_\rho, V_{\eta\rho}$).
4. Parameter Space Mapping: They provide a comprehensive parametrisation of the scalar couplings in terms of physical observables (VEVs, masses, mixing angles) and map the allowed parameter space considering unitarity, BFB, and vacuum stability.

4. Key Results

• Vacuum Classification: The analysis reveals that only two extrema are phenomenologically viable (containing three non-zero neutral VEVs to generate tree-level fermion masses): the standard EW vacuum ($V_{EW}^+$) and a specific edge vacuum ($V_{edge}^1$). All other extrema break electromagnetic $U(1)_{EM}$ or are otherwise unphysical.
• Global Stability Conditions:
  • The EW vacuum is not always global. It can be surpassed by vacua where only one or two triplets acquire VEVs ($V_\eta, V_\rho$) or mixed configurations ($V_{\eta\rho}$).
  • The stability is highly sensitive to the mixing between the triplets. Specifically, if the mixing of the $\eta$ and $\rho$ triplets with the $\chi$ triplet is non-zero, the EW vacuum may become a local minimum rather than the global one.
  • Analytical conditions (Eq. 6.4, 6.7) show that large values of couplings like $\lambda_{\rho\chi}$ and $\lambda_{\eta\chi}$ relative to the trilinear parameter $f$ can destabilize the EW vacuum.
• Metastability: Even when the EW vacuum is not global, a significant portion of the parameter space remains metastable (tunneling time $> 10^{10}$ years). The "hatched yellow" regions in their figures represent these safe metastable zones.
• Parameter Space Constraints:
  • Unitarity and BFB constraints are generally satisfied in the high-mass decoupling limit ($v_\chi \approx f \gg v_{\eta,\rho}$).
  • The globality constraint is the most restrictive. In regions where the mixing between triplets is negligible, the model is unitary, bounded, and the EW vacuum is global.
  • Figure 3 and 4 illustrate that for specific mass scales ($m_{331} \sim 10$ TeV) and mixing angles, large swathes of the parameter space are excluded if absolute stability is required, though they may be allowed under metastability.

5. Significance

This work provides a critical theoretical foundation for phenomenological studies of 331 models.

• Phenomenological Safety: Many previous studies assumed the EW vacuum was stable without rigorous proof. This paper demonstrates that assuming stability can lead to incorrect conclusions about the allowed parameter space.
• Methodological Advancement: The successful application of orbit space and P-matrix methods to a 331 model with three triplets demonstrates a powerful technique for handling complex scalar sectors in BSM physics, which can be extended to other models with multiple representations.
• Experimental Implications: The results constrain the mass spectrum and mixing angles of the new scalars. If future collider experiments (like the LHC or FCC) discover 331-like signatures, the vacuum stability analysis presented here will be essential to determine if the observed parameters are consistent with a stable universe or require a metastable interpretation.

In summary, the paper establishes that while the 331 model with three triplets is a viable extension of the SM, the global stability of the electroweak vacuum is not guaranteed and imposes strict constraints on the scalar sector's mixing and couplings, necessitating careful analysis of the parameter space for any phenomenological application.