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The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals

Samuel Abreu, Ruth Britto, Claude Duhr

TL;DR

This SAGEX chapter surveys the deep mathematical structure underlying dimensionally regulated Feynman integrals, emphasizing how differential equations, parametric representations, and modern cohomological approaches illuminate multiloop analytic structure. It details IBP and dimension-shift relations that reduce complex integral families to finite master bases and explains how differential equations, particularly in canonical dlog form, enable systematic ε-expansions via iterated integrals. The text then introduces iterated integrals, their homotopy invariance, regularisation, and their connections to MPLs, eMPLs, and modular forms, highlighting practical pathways to analytic and numerical evaluation. Finally, intersection theory with twisted (co)homology provides a unifying framework for alternative reduction methods, differential equations, and coactions, illustrating how these abstract tools inform concrete computations of Feynman integrals and their discontinuities.

Abstract

Dimensionally-regulated Feynman integrals are a cornerstone of all perturbative computations in quantum field theory. They are known to exhibit a rich mathematical structure, which has led to the development of powerful new techniques for their computation. We review some of the most recent advances in our understanding of the analytic structure of multiloop Feynman integrals in dimensional regularisation. In particular, we give an overview of modern approaches to computing Feynman integrals using differential equations, and we discuss some of the properties of the functions that appear in the solutions. We then review how dimensional regularisation has a natural mathematical interpretation in terms of the theory of twisted cohomology groups, and how many of the well-known ideas about Feynman integrals arise naturally in this context. This is Chapter 3 of a series of review articles on scattering amplitudes, of which Chapter 0 [arXiv:2203.13011] presents an overview and Chapter 4 [arXiv:2203.13015] contains closely related topics.

The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals

TL;DR

This SAGEX chapter surveys the deep mathematical structure underlying dimensionally regulated Feynman integrals, emphasizing how differential equations, parametric representations, and modern cohomological approaches illuminate multiloop analytic structure. It details IBP and dimension-shift relations that reduce complex integral families to finite master bases and explains how differential equations, particularly in canonical dlog form, enable systematic ε-expansions via iterated integrals. The text then introduces iterated integrals, their homotopy invariance, regularisation, and their connections to MPLs, eMPLs, and modular forms, highlighting practical pathways to analytic and numerical evaluation. Finally, intersection theory with twisted (co)homology provides a unifying framework for alternative reduction methods, differential equations, and coactions, illustrating how these abstract tools inform concrete computations of Feynman integrals and their discontinuities.

Abstract

Dimensionally-regulated Feynman integrals are a cornerstone of all perturbative computations in quantum field theory. They are known to exhibit a rich mathematical structure, which has led to the development of powerful new techniques for their computation. We review some of the most recent advances in our understanding of the analytic structure of multiloop Feynman integrals in dimensional regularisation. In particular, we give an overview of modern approaches to computing Feynman integrals using differential equations, and we discuss some of the properties of the functions that appear in the solutions. We then review how dimensional regularisation has a natural mathematical interpretation in terms of the theory of twisted cohomology groups, and how many of the well-known ideas about Feynman integrals arise naturally in this context. This is Chapter 3 of a series of review articles on scattering amplitudes, of which Chapter 0 [arXiv:2203.13011] presents an overview and Chapter 4 [arXiv:2203.13015] contains closely related topics.
Paper Structure (46 sections, 13 theorems, 171 equations)

This paper contains 46 sections, 13 theorems, 171 equations.

Table of Contents

  1. Background and definitions
  2. Some properties of Feynman integrals
  3. Feynman graphs.
  4. Mass dimension of a Feynman integral.
  5. Symmetries.
  6. Scaleless integrals.
  7. Parametric representations
  8. Schwinger parametrisation.
  9. Feynman parametrisation.
  10. Cutkosky-Baikov parametrisation.
  11. Linear relations among Feynman integrals
  12. Total derivatives in dimensional regularisation
  13. Integration-by-parts relations
  14. Solving IBP relations
  15. Example: The one-loop bubble integral
  16. ...and 31 more sections

Key Result

Proposition 1

In dimensional regularisation, all scaleless integrals vanish.

Theorems & Definitions (52)

  • Remark 1
  • Proposition 1
  • proof
  • Corollary 1
  • proof
  • Remark 2
  • Proposition 2
  • Proposition 3
  • proof
  • Remark 3
  • ...and 42 more