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The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals

Johannes Blümlein, Carsten Schneider

TL;DR

The SAGEX Chapter 4 review surveys a comprehensive toolkit for analytic and symbolic treatment of multi-loop Feynman integrals, emphasizing ε-expansions and representations in families of special functions. It covers guessing methods, solving linear recurrences and differential equations, transforming integrals and sums, symbolic summation and integration, the large-moment method, and the role of advanced function spaces (e.g., harmonic sums, polylogarithms, and elliptic structures) in zero-, single-, and double-scale QFT calculations. The work highlights how coupled systems arising from master integrals are tackled via uncoupling or direct solvers, and how modern difference-ring and holonomic approaches enable closed forms or structured representations for challenging problems, including massive OMEs and two-scale setups. The discussion underscores a synergistic, non-straightforward toolbox, where method interplay is essential to achieve precision predictions for current and future collider physics, while also pointing to emerging mathematical structures (Calabi–Yau motives, elliptic and beyond) and numerical strategies as ongoing frontiers.

Abstract

The analytic integration and simplification of multi-loop Feynman integrals to special functions and constants plays an important role to perform higher order perturbative calculations in the Standard Model of elementary particles. In this survey article the most recent and relevant computer algebra and special function algorithms are presented that are currently used or that may play an important role to perform such challenging precision calculations in the future. They are discussed in the context of analytic zero, single and double scale calculations in the Quantum Field Theories of the Standard Model and effective field theories, also with classical applications. These calculations play a central role in the analysis of precision measurements at present and future colliders to obtain ultimate information for fundamental physics.

The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals

TL;DR

The SAGEX Chapter 4 review surveys a comprehensive toolkit for analytic and symbolic treatment of multi-loop Feynman integrals, emphasizing ε-expansions and representations in families of special functions. It covers guessing methods, solving linear recurrences and differential equations, transforming integrals and sums, symbolic summation and integration, the large-moment method, and the role of advanced function spaces (e.g., harmonic sums, polylogarithms, and elliptic structures) in zero-, single-, and double-scale QFT calculations. The work highlights how coupled systems arising from master integrals are tackled via uncoupling or direct solvers, and how modern difference-ring and holonomic approaches enable closed forms or structured representations for challenging problems, including massive OMEs and two-scale setups. The discussion underscores a synergistic, non-straightforward toolbox, where method interplay is essential to achieve precision predictions for current and future collider physics, while also pointing to emerging mathematical structures (Calabi–Yau motives, elliptic and beyond) and numerical strategies as ongoing frontiers.

Abstract

The analytic integration and simplification of multi-loop Feynman integrals to special functions and constants plays an important role to perform higher order perturbative calculations in the Standard Model of elementary particles. In this survey article the most recent and relevant computer algebra and special function algorithms are presented that are currently used or that may play an important role to perform such challenging precision calculations in the future. They are discussed in the context of analytic zero, single and double scale calculations in the Quantum Field Theories of the Standard Model and effective field theories, also with classical applications. These calculations play a central role in the analysis of precision measurements at present and future colliders to obtain ultimate information for fundamental physics.
Paper Structure (33 sections, 54 equations, 6 figures, 1 table)

This paper contains 33 sections, 54 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Guessing methods
  3. Guessing integer relations
  4. Guessing recurrences and differential equations
  5. Solving linear recurrences and differential equations
  6. Ordinary linear equations
  7. Ordinary linear difference equations
  8. Ordinary linear differential equations
  9. Partial linear equations
  10. Solving coupled systems of linear differential equations
  11. Uncoupling algorithm and scalar solvers
  12. Direct solver
  13. Transformation to special integral and sum representations
  14. Symbolic summation
  15. Simplification of indefinite nested sums
  16. ...and 18 more sections

Figures (6)

  • Figure 1: Solving systems.
  • Figure 2: Sigma's summation spiral; see Schneider:04c.
  • Figure 3: 2-mass 3-loop diagram.
  • Figure 4: A 3-loop ladder diagram with a central triangle Ablinger:2015tua.
  • Figure 5: The large moment engine.
  • ...and 1 more figures