Symbolic Classification-Enabled LHC Limits Online BSM Global Fits

This paper demonstrates that Large Hadron Collider limits can be efficiently incorporated into online global fits of the phenomenological Minimal Supersymmetric Standard Model by using symbolic regression to derive fast, accurate mathematical approximations of ATLAS exclusion constraints.

The Gist

Imagine you are a detective trying to solve a massive mystery: Is there a hidden world of particles beyond what we currently know?

In the world of physics, scientists have a "rulebook" called the Standard Model. But many suspect there are more characters in the story, like "Supersymmetry", an example BSM physics. Here, BSM stands for Beyond the Standard Model. To find them, they use a giant particle collider called the LHC (Large Hadron Collider), which smashes particles together to see if anything new pops out.

The Problem: The "Slow-Motion" Detective

The paper describes a major headache scientists face. They have a massive list of possible "suspects" (theoretical models with millions of different settings). To check if a specific suspect is guilty, they have to run a simulation:

1. Smash the particles.
2. Watch how they break apart.
3. Simulate the detector seeing them.
4. Compare the result to real data from the LHC.

The problem is that this simulation takes hours for just one suspect. Since there are billions of suspects to check, doing this one by one is impossible. It's like trying to find a needle in a haystack by building a new, full-scale factory to test every single piece of straw.

So, usually, scientists do a "quick scan" to find the most likely suspects, and then they run the slow, expensive factory simulation on just the winners. This is called "post-processing." It's slow and inefficient because they might waste time on suspects that would have been ruled out immediately.

The Solution: The "Magic Cheat Sheet"

This paper introduces a clever shortcut using a technique called Symbolic Regression. Think of this as teaching a computer to write a simple math formula that acts like a cheat sheet.

Instead of running the full, slow factory simulation for every suspect, the researchers:

1. Took a huge dataset of past LHC results (specifically from the ATLAS experiment).
2. Fed this data into a computer program (using an Artificial Intelligence tool called Feyn) that looked for patterns.
3. The computer discovered a single, compact mathematical equation that could predict, with 97% accuracy, whether a specific set of settings would be "allowed" or "excluded" by the LHC.

It's like having a magic spell that instantly tells you, "No, that suspect is innocent," without needing to build the factory.

The "Online" Upgrade

The biggest breakthrough is how they use this cheat sheet.

• Old Way (Post-Processing): "Let's guess a million suspects, pick the best ones, and then check if the LHC rules them out."
• New Way (Online): "Let's check the LHC rules while we are guessing."

By plugging that simple math formula directly into the search process, the computer can instantly reject bad suspects the moment they are generated. It's like a bouncer at a club who checks your ID before you even get in line, rather than letting you in and kicking you out later.

What They Found

The researchers tested this on a set of theoretical particles called "electroweakinos." They ran two searches:

1. One without the LHC rules (the old way).
2. One with the LHC rules applied instantly (the new way).

The Result:
When they applied the "instant check," the list of possible suspects shrank dramatically. The "allowed" area of the mystery became much smaller and tighter.

• They found that the LHC limits (the rules) and a concept called "naturalness" (a rule about how the universe should look) are working together to squeeze the possibilities into very specific corners.
• Essentially, the LHC is getting very good at ruling out these theories, even if it hasn't found the particles yet.

The Bottom Line

This paper doesn't claim to have found new particles. Instead, it claims to have found a faster, smarter way to look for them. By turning complex, slow computer simulations into a simple math formula, they can now check the LHC's rules in real-time. This makes the search for new physics much more efficient and helps scientists focus their energy on the most promising areas of the universe.

Technical Summary

Technical Summary: Symbolic Classification-Enabled LHC Limits Online BSM Global Fits

Problem Statement
Global fits of Beyond the Standard Model (BSM) physics, such as the phenomenological Minimal Supersymmetric Standard Model (pMSSM), require a rigorous interplay between theoretical parameter scans and experimental constraints. A significant bottleneck in this process is the incorporation of Large Hadron Collider (LHC) exclusion limits. Traditional methods rely on "a posteriori" post-processing, where collider constraints are applied only after global fit sampling is complete. This is necessary because directly embedding LHC limits "online" (during the parameter scan) is computationally prohibitive. Standard reinterpretation tools require full simulation pipelines—including Monte Carlo event generation, parton showering, hadronization, detector simulation, and event reconstruction—for every parameter point, a process too slow for high-dimensional Bayesian inference. While machine learning (ML) approaches like SUSY-AI have attempted to approximate these limits, they often suffer from integration difficulties, reliance on specific simplified models, or restriction to older datasets (e.g., LHC Run-1).

Methodology
The authors propose a framework to incorporate LHC limits directly into the global fit sampling process using symbolic regression. The methodology proceeds in two main stages:

1. Symbolic Classification of ATLAS Limits:

   • Dataset: The study utilizes a dataset of 12,280 pMSSM electroweakino models generated for ATLAS Run-2 analyses. These models cover eight specific search channels targeting chargino-neutralino pair production ($\tilde{\chi}^\pm_1 \tilde{\chi}^0_2$ and $\tilde{\chi}^+_1 \tilde{\chi}^-_1$) with various decay topologies (leptonic, hadronic, Higgs, and disappearing tracks).
   • Labeling: Each model point is assigned a binary label based on the combined ATLAS likelihood ($CL_s$): "excluded" ($CL_s < 0.05$) or "allowed" ($CL_s > 0.05$).
   • Regression Technique: The authors employ the Feyn symbolic regression package. Unlike black-box ML classifiers (e.g., Random Forests), symbolic regression searches the space of mathematical formulae to find an analytic expression $y(\mathbf{x})$ that maps pMSSM parameters $\mathbf{x}$ to the exclusion label.
   • Optimization: The process uses a multi-stage "warm-start" strategy to refine solutions, optimizing for a binary cross-entropy loss function. The resulting expression is a compact mathematical formula capable of classifying parameter space points as allowed or excluded.

2. Online Global Fit Implementation:

   • Framework: A global fit of the pMSSM is performed using the MultiNest nested sampling algorithm.
   • Likelihood Construction: The likelihood function $\mathcal{L}(\theta)$ combines three standard observables (muon anomalous magnetic moment $\delta(g-2)_\mu$, neutralino relic density $\Omega_{CDM}h^2$, and Higgs mass $m_H$) with the LHC constraints.
   • Integration: The symbolic expression derived in the first stage replaces the computationally expensive simulation pipeline. For any sampled parameter point $\theta$, the symbolic formula $y(\theta)$ outputs a score. If the score indicates the point is "allowed," the LHC likelihood term is set to 1; if "excluded," it is set to 0. This allows the LHC constraint to be evaluated instantaneously during the sampling process.

Key Contributions

• First Online Integration: To the authors' knowledge, this is the first demonstration of incorporating LHC direct search limits for new physics directly "online" within a pMSSM global fit, rather than as a post-processing step.
• Symbolic Approximation: The paper successfully derives a compact, analytic mathematical expression from complex ATLAS exclusion limits that achieves a high degree of accuracy (Area Under the Curve $\approx 0.97$) in classifying the pMSSM parameter space.
• Computational Efficiency: By replacing full simulation chains with a fast analytic function, the framework makes the inclusion of collider constraints computationally feasible within the global fit loop, bypassing the need for repeated, expensive event-level recasting.

Results
The authors compare two global fits: one including the ATLAS limits via the symbolic classifier and one excluding them.

• Parameter Space Contraction: The inclusion of ATLAS limits significantly tightens the posterior distributions for electroweakino-related parameters ($M_1, M_2, \mu, A_t$), pushing allowed regions into specific corners of the parameter space.
• Correlation Analysis: The study reveals that ATLAS constraints sharpen the correlation between the muon anomalous magnetic moment contribution $\delta(g-2)_\mu$ and the parameter $\tan\beta$. Without the limits, the theoretical proportionality $\delta(g-2)_\mu \propto \tan\beta$ breaks down in moderate-to-high $\tan\beta$ regions; with the limits, this relationship is preserved.
• Naturalness Interplay: The results highlight a tension between the "naturalness line" (defined by $m_A \sim \frac{1}{\sqrt{2}} m_Z \tan\beta$) and the LHC limits. While naturalness suggests an upper bound on $m_A$ for a given range of $\tan\beta$ values, the ATLAS limits push the allowed lower bound of $m_A$ higher towards the naturalness upper bound. The combined effect squeezes the viable parameter space, potentially excluding large regions of the MSSM.

Significance and Claims
The paper positions this work as a proof of principle. The primary significance lies in demonstrating that symbolic regression can effectively translate discrete, complex experimental exclusion boundaries into explicit mathematical formulas. This approach offers a principled, interpretable, and computationally efficient pathway to integrate collider information into global fits.

The authors modestly claim that this framework is general and can, in principle, be extended to other BSM models and LHC analyses. They emphasize that this method clarifies which regions of BSM parameter space are supported or disfavoured by combining complementary experimental data, thereby guiding future theoretical and experimental efforts. The study does not claim to definitively rule out the MSSM but illustrates how the interplay of specific constraints (LHC limits vs. naturalness) can effectively reduce the viable parameter space.