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

Samuel Abreu, Ruth Britto+1 more|arXiv (Cornell University)|Mar 24, 2022

Black Holes and Theoretical Physics4 citations

TL;DR

This paper reviews advanced mathematical structures underlying multiloop Feynman integrals in dimensional regularization, focusing on differential equations, iterated integrals, and twisted cohomology. It establishes a rigorous framework using intersection theory and canonical forms, enabling systematic computation of scattering amplitudes via coaction principles and motivic structures in special functions like MPLs and hypergeometric functions.

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.

Motivation & Objective

• To systematize the mathematical foundations of multiloop Feynman integrals in dimensional regularization.
• To clarify how differential equations and canonical forms enable analytic computation of scattering amplitudes.
• To establish a connection between Feynman integrals and twisted cohomology, providing a geometric interpretation of integration cycles and forms.
• To explore the role of intersection theory in reducing integrals and constructing coaction maps on period matrices.
• To extend coaction structures to new classes of functions, including elliptic and hypergeometric functions arising in the ε-expansion.

Proposed method

• Utilizes dimensional regularization with D = D₀ − 2ε to define Feynman integrals as functions of external invariants and masses.
• Applies integration-by-parts (IBP) relations and total derivative identities to derive linear dependencies among integrals.
• Transforms Feynman integrals into systems of differential equations using a change of basis to achieve canonical form.
• Employs iterated integrals and homotopy invariance to represent solutions in terms of multiple polylogarithms (MPLs).
• Uses twisted cohomology and intersection theory to define pairing structures between integration cycles and differential forms.
• Constructs coaction maps on period matrices via intersection numbers, enabling diagrammatic and motivic interpretations of results.

Experimental results

Research questions

• RQ1 How can Feynman integrals be systematically reduced and solved using differential equations in dimensional regularization?
• RQ2 What is the role of canonical forms in simplifying the solution of differential equations for multiloop amplitudes?
• RQ3 How does twisted cohomology provide a geometric and algebraic foundation for Feynman integral computations?
• RQ4 In what way do intersection numbers and coaction structures unify the analytic and motivic properties of Feynman integrals?
• RQ5 Can coaction maps be generalized beyond MPLs to include elliptic and hypergeometric functions in the ε-expansion?

Key findings

• Canonical differential equations allow for the direct integration of Feynman integrals in terms of iterated integrals, with solutions expressible as multiple polylogarithms.
• The coaction on period matrices is derived from intersection numbers, and takes a simple form when canonical bases are chosen such that the intersection matrix is the identity.
• For the hypergeometric function ₂F₁, the coaction structure is explicitly computed and shown to decompose into products of solutions of Euler’s differential equation.
• The method of intersection theory enables a systematic reduction of Feynman integrals and provides a geometric origin for integration-by-parts identities.
• The framework naturally incorporates the ε-expansion and allows for a motivic treatment of special functions, including Lauricella and elliptic MPLs.
• The coaction on the ₂F₁ function includes both the function itself and a second solution, reflecting the monodromy structure of the hypergeometric equation.