Learning the S-matrix from data: Rediscovering gravity from gauge theory via symbolic regression

1. Problem Statement

The paper addresses the challenge of autonomously reconstructing fundamental analytic structures in quantum field theory (QFT) scattering amplitudes directly from numerical data, without relying on prior knowledge of the underlying algebraic formulas.

• The Context: Scattering amplitudes (specifically in Yang-Mills theory and General Relativity) possess rich, hidden analytic structures, such as the Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations, and the Kawai-Lewellen-Tye (KLT) double-copy relations which express gravity amplitudes as products of gauge theory amplitudes.
• The Gap: While Deep Neural Networks (DNNs) can predict numerical outcomes, they lack interpretability and cannot "discover" the underlying physical laws or algebraic forms. Standard symbolic regression (SR) struggles with the combinatorial explosion of features in high-multiplicity scattering problems.
• The Goal: To demonstrate that modern machine learning, specifically Symbolic Regression (SR) combined with linear algebraic feature selection, can rediscover these flagship relations (KK, BCJ, KLT) and the Parke-Taylor formula using only numerical on-shell data and minimal theoretical priors.

2. Methodology

The authors propose a data-driven pipeline that integrates linear algebraic dimensionality reduction with symbolic regression. The workflow consists of four main stages:

A. Data Generation

• Kinematics: Random on-shell kinematics are generated in the center-of-mass frame for massless bosons in 4D.
• Signature: To ensure real-valued data (avoiding complex spinor products), the authors analytically continue to (2, 2) signature, where spinor products are real.
• Targets:
  • Inputs: Color-ordered Yang-Mills partial amplitudes ($A_n$) and Mandelstam invariants ($s_{ij}$).
  • Outputs: Graviton amplitudes ($M_n$) computed independently via Hodges' formula (to avoid biasing the search toward KLT forms).
• Preprocessing: Data is rescaled to be dimensionless and $O(1)$ to improve numerical stability.

B. Feature Selection via CPQR (Column-Pivoted QR)

Instead of using Principal Component Analysis (PCA), which produces non-interpretable linear combinations, the authors use Column-Pivoted QR (CPQR) factorization.

• Mechanism: CPQR selects a minimal subset of original columns (features) that spans the data space.
• Discovery of Redundancies:
  • When applied to the matrix of color-ordered amplitudes, CPQR identifies the rank as $(n-2)!$, automatically revealing the Kleiss-Kuijf (KK) relations as the linear dependencies removed.
  • When applied to composite features (products of Mandelstam invariants and amplitudes), CPQR identifies Bern-Carrasco-Johansson (BCJ) relations (degree-1 syzygies) as linear dependencies where coefficients are linear polynomials in $s_{ij}$.
• Benefit: This reduces the feature space from a combinatorial explosion to a minimal, independent basis without imposing group-theoretic constraints a priori.

C. Symbolic Regression (SR)

• Algorithm: The authors use PySR, searching for expressions in a space generated by operators $\{+, -, \times, /\}$.
• Feature Engineering: To constrain the search space, they incorporate minimal physical priors:
  • Mass Dimension: Ensuring the output matches the dimension of the target amplitude.
  • Little-Group Scaling: Ensuring the correct helicity weights.
  • Composite Features: Instead of raw inputs, they feed bilinears of amplitudes ($A \tilde{A}$) multiplied by Mandelstam polynomials, as dictated by the double-copy structure.
• Basis Selection: A decision-tree model is used to scan different orderings of gluon amplitudes to find the "basis" that minimizes the complexity of the resulting kernel function.

D. Benchmarking

The method is compared against a recent Transformer-based Neural Network approach (Cheung et al.) designed for symbolic simplification.

3. Key Contributions & Results

A. Rediscovery of Linear Relations (KK & BCJ)

• The CPQR step successfully identified the KK relations (reducing $(n-1)!$ to $(n-2)!$ amplitudes) and BCJ relations (reducing to $(n-3)!$) purely from numerical data.
• Significance: This proves that linear algebraic dependencies in scattering data correspond directly to known physical symmetries, recoverable without any group-theoretic input.

B. Rediscovery of the Parke-Taylor Formula

• The pipeline successfully rediscovered the closed-form Parke-Taylor formula for MHV gluon amplitudes.
• Performance: With physics-motivated priors (restricting to angle brackets $\langle ij \rangle$ and enforcing chirality), the SR found the correct expression in $O(10^3)$ seconds for $n=4,5,6$, whereas a raw search failed.

C. Rediscovery of KLT Relations

• 4-Point & 5-Point: The method successfully rediscovered the KLT relations (e.g., $M_4 = -s_{12} A_4 \tilde{A}_4$) with high numerical accuracy ($O(10^{-16})$).
• 6-Point Challenge: The method encountered a combinatorial explosion at 6 points. The search space for the KLT kernel (a degree-3 polynomial in Mandelstams) became too large for current SR algorithms to converge within reasonable compute time (8 hours), even with aggressive feature engineering.
• Insight: The difficulty arises because different bases of amplitudes can make the same physical object look like a simple polynomial or a complex rational function with spurious poles. The SR struggles to guess the "simple" basis without further guidance.

D. Comparison with Neural Networks

• Neural Networks (Transformers): Excel at symbolic-to-symbolic rewriting. They can take a complex 298-term expression and simplify it to a 2-term expression by learning algebraic identities (Schouten, etc.). However, they risk "hallucinating" incorrect expressions that must be validated.
• Symbolic Regression: Excels at numeric-to-symbolic discovery. It infers the function form directly from numerical evaluations. It is more robust (immediate verification on held-out data) but heavily dependent on the quality of the feature set.
• Conclusion: The approaches are complementary. Neural networks can act as a pre-processor to simplify inputs or suggest promising feature combinations, while SR performs the final compression and discovery of analytic forms.

4. Significance and Future Outlook

• Data-Driven Physics: The paper establishes a framework for "uncovering hidden relations in general theories" using only numerical data, suggesting a path to discovering new physics in regimes where analytic solutions are unknown.
• Interpretability: Unlike black-box DNNs, this method produces human-readable algebraic expressions, bridging the gap between machine learning and theoretical discovery.
• Limitations: The primary bottleneck is the combinatorial growth of the search space at higher multiplicities ($n \geq 6$).
• Future Directions:
  • Factorization Bootstrap: Learning amplitudes by probing near physical poles ($s_I \to 0$) to isolate residues, simplifying the learning task.
  • Hybrid Pipelines: Combining neural simplification (to reduce input complexity) with symbolic regression (to find the final analytic form).
  • Extensions: Applying the method to loop-level amplitudes (introducing transcendental functions) and string theory KLT relations.

In summary, this work demonstrates that Symbolic Regression, when guided by linear algebraic feature selection (CPQR) and minimal physical priors, is a powerful tool for reconstructing the analytic structure of the S-matrix, successfully rediscovering foundational relations in gauge and gravity theories from numerical data alone.