TLDR: This research paper demonstrates how Symbolic Regression (SR) can significantly speed up the analysis of Beyond the Standard Model (BSM) physics, specifically the Constrained Minimal Supersymmetric Standard Model (CMSSM). SR generates accurate, interpretable analytical expressions for observables like Higgs mass and dark matter density, enabling much faster global fits and differentiable analyses compared to traditional computational methods or neural networks, which often require more focused data.
Exploring the mysteries of the universe beyond what our current understanding of particle physics, known as the Standard Model, can explain is a monumental task. Scientists are constantly searching for “Beyond the Standard Model” (BSM) theories to address phenomena like dark matter and the anomalous magnetic moment of the muon. However, testing these complex theories against experimental data is incredibly challenging due to the immense computational power required.
Traditionally, physicists use sophisticated computer programs to simulate BSM models. These programs take a set of input parameters and calculate various observable quantities, such as the mass of the Higgs boson, the density of dark matter in the universe, or the subtle magnetic properties of the muon. This process involves a long chain of intricate calculations, often including multi-loop corrections and threshold effects, making it very slow. For instance, determining the dark matter relic density in a model like the Constrained Minimal Supersymmetric Standard Model (CMSSM) involves solving multiple non-linear differential equations and analyzing complex particle interactions. This computational bottleneck makes it difficult to thoroughly explore the vast parameter spaces of BSM theories.
The ideal solution would be to have simple, analytical mathematical formulas that directly link the input parameters of a BSM model to its observable predictions. While such formulas could theoretically be derived, their complexity would render them impractical for most real-world scenarios. This is where a powerful technique called Symbolic Regression (SR) comes into play. Unlike traditional machine learning methods like neural networks, which often act as “black boxes,” SR aims to discover the underlying mathematical expressions that best describe the relationship between inputs and outputs.
Symbolic Regression: A New Approach to BSM Physics
A recent research paper, titled “Symbolic Regression and Differentiable Fits in Beyond the Standard Model Physics,” demonstrates the effectiveness of SR in probing BSM models. The authors, Shehu AbdusSalam, Steven Abel, Deaglan Bartlett, and Miguel Crispim Romão, focused their study on the CMSSM, a well-known BSM model with four key input parameters: m1/2, m0, A0, and tanβ. These parameters dictate the experimental signals and cosmological observables.
The study shows that SR can generate remarkably accurate analytical expressions for crucial observables like the Higgs mass, the supersymmetric contribution to the muon’s anomalous magnetic moment, and the dark matter relic density. These expressions, derived from data, can then be used to analyze the model’s phenomenology much faster than conventional methods. The researchers also developed a “classifier” function using SR, which can quickly determine if a given set of parameters is physically viable, rejecting unphysical points based on criteria such as the absence of charge and color breaking minima or the presence of a neutral dark matter particle.
Accelerating Global Fits and Differentiable Analysis
One of the major advantages highlighted in the paper is the ability of SR expressions to significantly accelerate “global fits.” Global fits are statistical analyses used to determine the most probable values of a model’s parameters by comparing its predictions with experimental data. Using SR-derived expressions, these fits can be performed orders of magnitude faster than when relying on the full, time-consuming physics calculations. The results obtained with SR expressions were found to be in good agreement with those performed using conventional, package-based methods.
Furthermore, the paper explores the benefit of using differentiable symbolic expressions. By restricting SR to use only differentiable mathematical operators or by replacing non-differentiable ones with smooth approximations (like a ‘softmax’ function for ‘max’), scientists can leverage gradient-based optimization techniques. This opens doors for more efficient parameter optimization, easier error propagation calculations, and a better understanding of how sensitive observables are to changes in input parameters. This sensitivity, often referred to as “fine-tuning,” is a critical indicator of a BSM model’s credibility.
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SR vs. Neural Networks
The researchers also compared SR with Neural Networks (NNs), another machine learning approach. While NNs can also provide differentiable models, the study found that SR produced more globally robust results. NNs often required a much larger and more focused dataset, “zoomed in” on promising regions of the parameter space, to achieve comparable accuracy. SR, on the other hand, was more sample-efficient and provided a better global picture across the entire range of observables without such stringent data focusing. This suggests that SR offers a unique balance of interpretability, speed, and global robustness for BSM physics research.
In conclusion, symbolic regression presents a powerful new tool for particle physicists. It offers a way to bypass the computationally intensive chain of physical calculations, providing analytical formulas that are both accurate and interpretable. This not only speeds up the analysis of BSM models but also enhances our ability to understand the underlying physics. The expressions generated by SR can be reused for future analyses, amortizing the initial computational cost of data generation and training. This work paves the way for more efficient and insightful exploration of theories beyond the Standard Model. You can find the full research paper here.
