Assessing boundedness from below in the $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetric three-Higgs-doublet model: algorithm and machine learning

This paper presents a Mathematica code named StableWein and a machine-learning tool to assess the boundedness from below of the $\mathbb{Z}_2 \times \mathbb{Z}_2$-symmetric three-Higgs-doublet model by iteratively applying necessary conditions and achieving over 99% classification accuracy.

The Gist

Imagine you are an architect designing a skyscraper. Before you pour a single drop of concrete, you need to make sure the building won't collapse into a bottomless pit. In the world of particle physics, the "skyscraper" is a model of the universe, and the "foundation" is something called the scalar potential.

If this foundation isn't stable, the universe's energy could drop forever into negative infinity, meaning the universe would have no stable state—it would just fall apart. Physicists call this requirement "Boundedness from Below" (BFB). It simply means: The energy floor must exist; it cannot go down forever.

This paper tackles a specific, complex type of building design called the Three-Higgs-Doublet Model (3HDM). Think of this model as a skyscraper with three distinct, interacting wings (the "doublets"). The authors are trying to figure out exactly which combinations of building materials (mathematical numbers called "couplings") will result in a stable structure.

Here is the breakdown of their work, explained through simple analogies:

1. The Problem: Too Many Variables

In the simplest version of the Standard Model (our current best theory of physics), checking if the building is stable is easy. It's like checking if a single number is positive.

But in this complex 3HDM model, there are 13 different numbers (couplings) that define how the three wings interact. It's like trying to balance a tower made of 13 different types of juggling balls. If you get the ratios wrong, the tower collapses.

For a long time, physicists only had "Sufficient Conditions."

• The Sufficient Condition Analogy: Imagine a safety inspector who says, "If your building uses only red bricks, it is definitely safe."
• The Problem: This is too strict! You might have a perfectly safe building made of blue bricks, but the inspector rejects it because it's not red. This means physicists were throwing away potentially valid and interesting models just because they didn't fit a simple, easy-to-check rule.

2. The Solution: "Necessary" Conditions (The Smart Filter)

Instead of looking for a perfect, easy rule that covers everything (which doesn't exist for this complex model), the authors decided to use "Necessary Conditions."

• The Necessary Condition Analogy: Imagine a series of increasingly strict security checkpoints.
  • Checkpoint 1: "Do you have a ticket?" (If no, you're out. If yes, you pass.)
  • Checkpoint 2: "Is your ticket valid?" (If no, you're out.)
  • Checkpoint 3: "Did you pass the metal detector?"
  • Checkpoint 4: "Did a human scan your entire body?"

The authors created a hierarchy of these checkpoints, which they call NCL1, NCL2, NCL3, and NCL4.

• NCL1 is a quick check. It catches the obvious disasters.
• NCL4 is a deep, slow, and thorough scan that catches almost everything.

The beauty of their approach is user choice. You can tell their software, "I want a quick answer" (run the fast, less accurate check) or "I want the absolute truth" (run the slow, super-accurate check).

3. The Tool: "StableWein"

The authors wrote a computer program (a Mathematica package) called StableWein.

• What it does: You feed it a set of numbers (your building materials).
• What it tells you: "Yes, this is likely stable," or "No, this will collapse."
• The Trade-off: If you want 99.9% certainty, it takes longer to compute. If you want 90% certainty, it's instant.

4. The Secret Weapon: Artificial Intelligence (Machine Learning)

Here is where it gets really cool. The authors realized that checking every single possibility with math is slow. So, they trained a Neural Network (a type of AI) to be a "super-inspector."

• How it works: They showed the AI millions of examples of "stable" and "unstable" buildings. The AI learned the patterns of stability without being explicitly told the math rules.
• The Result: The AI can look at a new set of numbers and guess if it's stable with 99.9% accuracy in a fraction of a second. It's like having a veteran architect who can glance at a blueprint and instantly know if it will stand, without doing the calculations.

5. The "Brute Force" Reality Check

To make sure their math and their AI were right, they also used a "Brute Force" method.

• The Analogy: Imagine trying to find the lowest point in a mountain range. The math rules are like a map. The AI is like a hiker with a good eye. The "Brute Force" method is like sending a drone to fly over every single inch of the mountain to find the absolute lowest point.
• The Finding: The drone (Brute Force) confirmed that the map (Math) and the hiker (AI) were almost perfectly accurate. The math rules they developed (NCL1–4) are so good that they catch almost every unstable model, and the AI is nearly as good as the drone but much faster.

Summary

This paper is a toolkit for physicists building complex models of the universe.

1. The Problem: Checking if complex models are stable is hard and usually requires throwing away good ideas because the rules are too strict.
2. The Fix: They created a flexible system of "checkpoints" (NCL1–4) that can be tuned for speed or accuracy.
3. The Innovation: They built a software package (StableWein) and trained an AI to do this checking instantly with near-perfect accuracy.

In short, they turned a slow, error-prone, and overly strict safety inspection into a fast, flexible, and highly accurate process, allowing physicists to explore a much wider range of universe-building possibilities without fear of the foundation collapsing.

Technical Summary

1. Problem Statement

In particle physics, the consistency of any model requires that its scalar potential is bounded from below (BFB). If the potential can tend to negative infinity, the theory lacks a stable vacuum state.

• Context: The paper focuses on the Three-Higgs-Doublet Model (3HDM), an extension of the Standard Model with three $SU(2)$ scalar doublets ($\Phi_1, \Phi_2, \Phi_3$).
• Specific Challenge: The general 3HDM quartic potential ($V_4$) contains 45 real couplings. Deriving necessary and sufficient mathematical conditions for BFB in terms of these couplings is analytically intractable.
• Target Model: The authors focus on the Weinberg model, a 3HDM with a specific internal symmetry group $Z^{(3)}_2 \times Z^{(2)}_2$. This symmetry reduces the number of independent real couplings in $V_4$ to 13.
• Limitation of Existing Methods: Previous work provided sufficient conditions for BFB, but these exclude large portions of the physically viable parameter space. There were no known necessary and sufficient conditions for this specific symmetry.

2. Methodology

The authors developed a hybrid approach combining rigorous mathematical derivation, numerical optimization, and machine learning.

A. Mathematical Derivation of Necessary Conditions

The authors reformulated the BFB problem as a copositivity condition on a real symmetric matrix $M$ derived from the quartic potential. They derived a hierarchy of increasingly stringent Necessary Conditions (NCL):

1. NCL1 (Level 1): Based on the copositivity of the diagonal elements and the 2-Higgs-Doublet Model (2HDM) sub-cases. These are standard conditions ($a_k \ge 0$, etc.).
2. NCL2 (Level 2): Enforces the copositivity inequality at specific "corner" points ($P_1, P_2, P_3$) of the orbit space.
3. NCL3 (Level 3): Evaluates the inequality at the center of the orbit space ($P_4$) and specific phase configurations, generating a set of 21 inequalities.
4. NCL4 (Level 4): Involves a numerical scan over the continuous orbit-space surface (defined by parameters $\varpi_k$ and $\vartheta_k$) to find the global minimum of the potential.

B. Numerical Minimization (Ground Truth)

To validate the accuracy of the NCLs and generate training data, the authors performed global minimization of $V_4$ using Mathematica's NMinimize.

• Strategies: Used multiple algorithms (Nelder-Mead, Differential Evolution), varied field ranges, and utilized multiple random seeds to avoid local minima.
• Role: This "brute-force" approach (Mode 4) serves as the ground truth to determine if a potential is truly BFB, against which the NCLs and Machine Learning models are benchmarked.

C. Machine Learning (Neural Networks)

The authors trained Deep Neural Networks (NNs) to classify potentials as BFB or non-BFB.

• Architecture: Fully connected feed-forward networks with 8 layers.
• Training Strategy: A "chain" of networks ($N_7 \to N_{10}$) was used, where smaller, faster networks filter data for larger, more accurate networks.
• Data: Trained on synthetic data generated by the NCLs and validated by the brute-force minimization.
• Goal: To create a classifier that achieves near-perfect accuracy without the computational cost of scanning or minimization.

3. Key Contributions: The StableWein Package

The primary output of this research is the StableWein Mathematica package, which implements the findings.

• Functionality:
  • Checks BFB for both the Weinberg model (general CP-violating) and the Branco model (CP-conserving).
  • Implements Unitarity (UNI) constraints (perturbative unitarity bounds).
  • Offers 4 Modes of BFB checking:
    • Mode 0: Sufficient conditions (fast, but incomplete).
    • Mode 1-3: Necessary conditions (NCL1–NCL4) with increasing accuracy.
    • Mode 4: Brute-force minimization (exact, but slow).
  • Neural Network Integration: Can generate or filter data using trained NNs for rapid classification.
• User Control: Users can trade computational time for precision by selecting the mode.

4. Results and Performance

The authors tested the package on datasets of $10^6$ quartic potentials using high-performance hardware (Intel Core i9 and Apple M4 Pro).

• Accuracy of Necessary Conditions:
  • NCL1 (Mode 1): Identifies ~75% of true BFB potentials.
  • NCL1–3 (Mode 2): Achieves ~99.9% accuracy.
  • NCL1–4 (Mode 3): Achieves ~99.999% accuracy.
  • Comparison: The sufficient conditions (Mode 0) reject nearly 20% of valid BFB points, whereas the NCL approach captures almost the entire valid region.
• Computational Efficiency:
  • Mode 3 (Scanning): ~100 seconds for $10^6$ points (on i9).
  • Mode 4 (Minimization): ~31,000 seconds for $10^6$ points.
  • Conclusion: Mode 3 is roughly $10^3$ times faster than the exact minimization while maintaining near-perfect accuracy.
• Machine Learning Performance:
  • The trained NNs achieve >99.9% accuracy in identifying BFB potentials.
  • Inference time is significantly faster than Mode 3 and orders of magnitude faster than Mode 4.
  • The NNs successfully reproduce the density distribution of the parameter space found by the rigorous methods.

5. Significance

• Solving an Open Problem: The paper provides a practical, high-precision solution to the BFB problem for the $Z_2 \times Z_2$-symmetric 3HDM, a model widely used in phenomenology but previously lacking precise stability criteria.
• Methodological Shift: Instead of seeking elusive sufficient conditions that miss valid physics, the authors demonstrate that a hierarchy of necessary conditions can approximate the true boundary with arbitrary precision.
• Tool Availability: The StableWein package democratizes this analysis, allowing physicists to easily test model stability without implementing complex minimization algorithms.
• Future Applicability: The success of the Machine Learning approach suggests that similar techniques can be applied to 3HDMs with no internal symmetries (where analytical conditions are even harder to derive), effectively bypassing the need for closed-form mathematical proofs.

In summary, this work bridges the gap between theoretical constraints and practical model building in Beyond Standard Model physics, offering a robust, open-source tool that combines analytical rigor with modern machine learning efficiency.