[Paper Review] Diagrammatic Coaction of Two-Loop Feynman Integrals

Samuel Abreu, Ruth Britto+3 more|arXiv (Cornell University)|Dec 13, 2019

Advanced Topics in Algebra7 references4 citations

TL;DR

This paper extends the diagrammatic coaction formalism—previously known for one-loop Feynman integrals and hypergeometric functions—to two-loop Feynman integrals. It introduces a coaction structure that preserves the pairing between contracted and cut graphs, while introducing new features such as sums over multiple master integrals and deformation terms. The key contribution is a systematic framework for computing two-loop coactions using differential forms and integration contours, generalizing the one-loop diagrammatic coaction to higher loops.

ABSTRACT

It is known that one-loop Feynman integrals possess an algebraic structure encoding some of their analytic properties called the coaction, which can be written in terms of Feynman integrals and their cuts. This diagrammatic coaction, and the coaction on other classes of integrals such as hypergeometric functions, may be expressed using suitable bases of differential forms and integration contours. This provides a useful framework for computing coactions of Feynman integrals expressed using the hypergeometric functions. We will discuss examples where this technique has been used in the calculation of two-loop diagrammatic coactions.

Motivation & Objective

• To generalize the diagrammatic coaction formalism, known for one-loop integrals and hypergeometric functions, to two-loop Feynman integrals.
• To investigate whether the coaction structure—featuring cuts and deformation terms—persists beyond one loop.
• To explore the role of multiple master integrals and their corresponding cuts in the two-loop coaction.
• To determine the conditions under which deformation terms appear in two-loop coactions.
• To assess the viability of a diagrammatic coaction for integrals beyond multiple polylogarithms.

Proposed method

• The paper uses a basis of differential forms and integration contours to express the coaction in the form Δ∫γω = ∑i,j cij ∫γ ωi ⊗ ∫γj ω.
• It applies this formalism to two-loop integrals by identifying appropriate bases of forms and contours that encode the analytic structure of the integrals.
• The coaction is computed via residue operations on integration contours, generalizing the one-loop case where cuts are defined by encircling poles.
• The method leverages intersection theory to compute the coefficients cij in the coaction expansion.
• It draws analogies with hypergeometric functions, where similar coaction structures have been successfully implemented.
• The framework is applied to explicit two-loop examples, revealing new features such as multiple master integrals for the same graph.

Experimental results

Research questions

• RQ1 Can the diagrammatic coaction formalism, valid at one loop, be extended to two-loop Feynman integrals?
• RQ2 What new structural features emerge in the two-loop coaction compared to the one-loop case?
• RQ3 How do deformation terms arise in the two-loop coaction, and what determines their coefficients?
• RQ4 Do homology relations between cut contours—responsible for deformation terms at one loop—generalize to two loops?
• RQ5 Is a diagrammatic coaction possible for two-loop integrals not expressible in terms of multiple polylogarithms?

Key findings

• The two-loop coaction preserves the diagrammatic structure of one-loop coactions, with pairings between contracted and cut graphs.
• New features appear, such as sums over multiple master integrals corresponding to the same graph and its cuts.
• Deformation terms are present in the two-loop coaction, similar to the one-loop case, but their origin and structure remain to be fully understood.
• The coaction coefficients are determined by intersection theory applied to differential forms and contours.
• The framework successfully computes coactions for explicit two-loop examples, demonstrating consistency with known structures.
• The paper leaves open the question of whether the coaction formalism generalizes to non-MPL integrals, such as those involving elliptic functions.