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Uncovering Singularities in Feynman Integrals via Machine Learning

Yuanche Liu, Yingxuan Xu, Yang Zhang

TL;DR

This work addresses extracting the symbol alphabet of multi-loop Feynman integrals by leveraging symbolic regression to uncover the analytic structure encoded in canonical differential equations. By formulating the problem as a mixed discrete–continuous optimization and using PySR to identify analytic expressions for CDE matrix elements from numerical data, the authors obtain the complete symbol letters, including square-root structures, without relying on explicit integral reductions. Demonstrations on planar three-loop four-point one-mass and non-planar two-loop three-point examples validate the method’s ability to reconstruct alphabets across diverse integral families. The approach offers a systematic, interpretable route to analytic structure discovery with potential impact on automating symbol extraction, bootstrap approaches, and higher-loop amplitude analyses in quantum field theory.

Abstract

We introduce a machine-learning framework based on symbolic regression to extract the full symbol alphabet of multi-loop Feynman integrals. By targeting the analytic structure rather than reduction, the method is broadly applicable and interpretable across different families of integrals. It successfully reconstructs complete symbol alphabets in nontrivial examples, demonstrating both robustness and generality. Beyond accelerating computations case by case, it uncovers the analytic structure universally. This framework opens new avenues for multi-loop amplitude analysis and provides a versatile tool for exploring scattering amplitudes.

Uncovering Singularities in Feynman Integrals via Machine Learning

TL;DR

This work addresses extracting the symbol alphabet of multi-loop Feynman integrals by leveraging symbolic regression to uncover the analytic structure encoded in canonical differential equations. By formulating the problem as a mixed discrete–continuous optimization and using PySR to identify analytic expressions for CDE matrix elements from numerical data, the authors obtain the complete symbol letters, including square-root structures, without relying on explicit integral reductions. Demonstrations on planar three-loop four-point one-mass and non-planar two-loop three-point examples validate the method’s ability to reconstruct alphabets across diverse integral families. The approach offers a systematic, interpretable route to analytic structure discovery with potential impact on automating symbol extraction, bootstrap approaches, and higher-loop amplitude analyses in quantum field theory.

Abstract

We introduce a machine-learning framework based on symbolic regression to extract the full symbol alphabet of multi-loop Feynman integrals. By targeting the analytic structure rather than reduction, the method is broadly applicable and interpretable across different families of integrals. It successfully reconstructs complete symbol alphabets in nontrivial examples, demonstrating both robustness and generality. Beyond accelerating computations case by case, it uncovers the analytic structure universally. This framework opens new avenues for multi-loop amplitude analysis and provides a versatile tool for exploring scattering amplitudes.

Paper Structure

This paper contains 14 sections, 14 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Canonical Differential Equations and Symbol Alphabets
  3. From Feynman Integrals to Canonical Basis
  4. Symbol Alphabet from Canonical Differential Equations
  5. Symbolic Regression Frameworks and PySR
  6. PySR: A High-Performance Symbolic Regression Engine
  7. Build Symbol Letters From Numerical Evaluations
  8. Examples
  9. Planar three-loop four-point one-mass integrals
  10. Non-planar two-loop three-point Feynman integrals
  11. Conclusion
  12. Acknowledgment
  13. Supplemental Material
  14. Options and Tuning of PySR Regressor

Figures (4)

  • Figure 1: PySR Workflow. Dataset characters will be learned by evolution of searching tree, which combines operators and operands to a test expression. After considering the Pareto Frontiers of expressions in respect to their complexity, size, and accuracy, PySR will return several best results to its user.
  • Figure 2: The workflow consists of three layers. In the Pre-processing Layer, Feynman integrals of a given family are analyzed and IBP reductions are performed at multiple numerical points. These results serve as input for the Regression Layer, where symbolic regression constructs the CDE matrix. In the Post-processing Layer, the resulting symbolic expressions are exponentiated and factorized, and all candidate symbol letters are collected to assemble the complete symbol alphabet.
  • Figure 3: Tripple box Feynman diagrams.
  • Figure 4: Non-planar two-loop three-point Feynman diagram.