The Diagrammatic Coaction

TL;DR

This paper explores the algebraic structure called the diagrammatic coaction, which relates Feynman integrals, their cuts, and differential equations, revealing dualities and structures useful for understanding polylogarithmic integrals.

Contribution

It reviews the current understanding of the diagrammatic coaction and its manifestation in dimensionally-regularized Feynman integrals, highlighting dualities between forms and contours.

Findings

Demonstrates duality between forms and contours in Feynman integrals

Shows correspondence between local and global coactions in hypergeometric functions

Provides examples with one- and two-loop integrals

Abstract

The diagrammatic coaction underpins the analytic structure of Feynman integrals, their cuts and the differential equations they admit. The coaction maps any diagram into a tensor product of its pinches and cuts. These correspond respectively to differential forms defining master integrals, and integration contours which place a subset of the propagators on shell. In a canonical basis these forms and contours are dual to each other. In this talk I review our present understanding of this algebraic structure and its manifestation for dimensionally-regularized Feynman integrals that are expandable to polylogarithms around integer dimensions. Using one- and two-loop integral examples, I will explain the duality between forms and contours, and the correspondence between the local coaction acting on the Laurent coefficients in the dimensional regulator and the global coaction acting on generalised hypergeometric functions.