Symbolic Classification-Enabled LHC Limits Online BSM Global Fits

Abstract: Global fits of Beyond the Standard Model (BSM) physics often involve a two-way interplay between theory and experiment. Theoretical models provide guidance for experimental searches, while experimental results, in turn, constrain theoretical frameworks. A crucial aspect of this feedback loop is the direct inclusion of measurements and exclusion limits online'' global fits, i.e. during the parameter scans aspects of the global fits. However, incorporating the Large Hadron Collider (LHC) limits into such analyses has been computationally prohibitive, often due to time taken per parameter point exceeding the scales acceptable for global fit frameworks. In this study, we show that LHC limits can be incorporatedonline'' global fits by leveraging approximations derived from symbolic regression techniques. We utilize a dataset of ATLAS constraints from searches for electroweakino productions to derive a mathematical expression capable of classifying the phenomenological Minimal Supersymmetric Standard Model (pMSSM) parameter space as allowed or excluded. This is subsequently incorporated for making a global fit of the pMSSM to data, including the LHC Run-2 limits.

• The paper develops a symbolic regression model to encode LHC exclusion boundaries as analytic functions within pMSSM global fits.
• It demonstrates computational efficiency by achieving an AUC of 0.97 and enabling online incorporation of collider constraints during Bayesian sampling.
• The approach significantly contracts the viable parameter space, enhancing the interplay between experimental limits and theoretical models in BSM inference.

Symbolic Regression for Online LHC Constraints in pMSSM Global Fits

Introduction

The integration of Large Hadron Collider (LHC) constraints into global fits of Beyond Standard Model (BSM) scenarios, particularly within the phenomenological Minimal Supersymmetric Standard Model (pMSSM), has long been stifled by computational limitations. Traditional approaches typically rely on either computationally expensive simulations for each parameter point or treat LHC bounds as an afterthought, applying them a posteriori to samples generated without these constraints. The work "Symbolic Classification-Enabled LHC Limits Online BSM Global Fits" (2605.22330) addresses this bottleneck by developing symbolic regression models that encode the LHC exclusion boundaries as analytic functions of the theory parameters. This innovation enables the imposition of LHC constraints "online," that is, dynamically during sampling, thus fundamentally altering the computational landscape for BSM global fits.

Motivation and Background

Incorporating collider constraints within global fits is essential for robust BSM inference, as these limits frequently provide the most stringent exclusions on viable parameter space. Physical pipelines for LHC recasts entail event generation, detector simulation, and likelihood calculations, making per-point evaluation prohibitively slow for high-dimensional scans. While approaches based on simplified models or ML classifiers (e.g., SUSY-AI) offer partial mitigation, the former lack generality for realistic BSM spectra and the latter are often restricted to outdated datasets and suffer from integration friction within fit frameworks. Recent advances in symbolic regression—a machine learning technique targeting explicit, interpretable mathematical relationships—provide a promising direction: analytic surrogates that deliver fast classification with controlled fidelity.

Symbolic Regression Methodology

The study employs the Feyn symbolic regression package to translate ATLAS Run-2 exclusion results for electroweakinos into compact analytic classifiers over the pMSSM parameter space. The procedure is as follows:

• Dataset Construction: 12,280 pMSSM points, compatible with LHC preselection, are labeled as allowed or excluded based on combined ATLAS analyses.
• Model Training: Symbolic regression is performed using a balanced set of features and a cross-entropy loss. The model search is regularized via the Akaike Information Criterion to avoid undue complexity.
• Optimization Strategy: Iterative, multi-stage warm starts are adopted, refining expressions by seeding new generations with the best candidates from previous runs.
• Computational Scale: Approximately $10^7$ candidate expressions were explored, consuming roughly 840 CPU core-hours.
• Evaluation: The resulting classifier achieves an area under the ROC curve (AUC) of 0.97 on the held-out test set. The Youden threshold ($t \approx 0.506$) is used to map pseudo-probabilities to binary allowed/excluded labels.

This demonstrates that the complex, simulation-based separation boundary can be efficiently encapsulated in analytic form, enabling its use as a computationally trivial surrogate in subsequent statistical fits.

Integration of LHC Limits in Global Fits

The derived symbolic classifier is incorporated into a Bayesian global-fit pipeline for the pMSSM. The fit includes both standard observables (anomalous magnetic moment of the muon $\delta a_\mu$, neutralino relic density $\Omega_{\mathrm{CDM}} h^2$, and Higgs mass $m_H$) and LHC exclusion limits encapsulated through the symbolic expression. Parameter sampling is performed using MultiNest with flat priors over physically motivated intervals, and unphysical points are culled via spectrum-generator consistency checks.

The likelihood function for each parameter point is a product of Gaussian terms for the standard observables and a binary classifier for the LHC constraint, ensuring that points violating the LHC-derived analytic exclusion boundary are assigned zero likelihood.

A direct comparison is made between fits performed with and without the "online" LHC constraint to evaluate its quantitative effect on the inferred posterior distributions.

Results

• Parameter Space Contraction: Incorporation of the LHC symbolic constraint systematically restricts the allowed parameter regions, pushing high-probability posteriors towards smaller volumes and excluding substantial portions of parameter space accessible in unconstrained fits.
• Correlations With Observables: The impact of the LHC cut is most pronounced in its interplay with the anomalous magnetic moment of the muon and naturalness considerations. Posterior correlations between $\tan\beta$ and $\delta a_\mu$ become sharper, and parameter combinations violating the "naturalness line" $m_A / \tan\beta \sim 1$ are efficiently removed when both constraints are imposed.
• Viable Space Under Competing Constraints: The results reveal an antagonistic squeezing of parameter space by the LHC constraints (setting lower bounds, e.g., $m_A > 500$ GeV) and theoretical conditions (naturalness, setting upper bounds), suggesting that the parameter volume for viable supersymmetry is being sharply reduced. The analysis underscores the possibility that, under continued null results, the LHC can "rule out the MSSM" in large swaths of the parameter space.

Implications and Future Directions

The methodology establishes that high-fidelity analytic surrogates for collider constraints can be extracted and integrated directly into probabilistic inference workflows, allowing true dynamical incorporation of LHC information during parameter sampling. This obviates the need for post-processing filters and removes a key computational bottleneck limiting the scale and precision of BSM global fits.

The approach is generalizable. Although this study focuses on ATLAS electroweakino searches in the pMSSM, the technique is readily extendable to other collider analyses and BSM scenarios, including those with higher-dimensional or more intricate exclusion boundaries. The transparency of symbolic classifiers may additionally enhance interpretability and facilitate rapid error propagation or robustness tests.

As the LHC and future colliders accumulate more data and explore new signatures, the need for scalable, modular inference tools will continue to intensify. Symbolic regression, combined with expanding datasets and improved algorithmic sophistication, is poised to become an essential component of the global-fit paradigm in particle phenomenology. Further developments may target multi-class or probabilistic constraint surrogates, joint handling of multiple experimental analyses, and integration with simulation-based inference and active-learning strategies.

Conclusion

This study demonstrates the first integration of LHC direct search limits into a pMSSM global fit using analytic symbolic classifiers. Leveraging symbolic regression, experimental exclusion contours are rapidly and accurately encoded as mathematical functions, enabling "online" enforcement of collider constraints during global fits. This framework delivers substantial computational gains, maintains high fidelity (AUC ≈ 0.97), and leads to more stringent and physically consistent inference of BSM parameter spaces. The approach is flexible and extensible, offering a practical path toward comprehensive, collider-informed BSM statistics as the LHC data set grows.