The diagrammatic coaction and the algebraic structure of cut Feynman integrals

TL;DR

This paper introduces a new diagrammatic coaction formula for cut Feynman integrals, revealing their algebraic structure and simplifying the extraction of properties like discontinuities and differential equations.

Contribution

It provides a novel diagrammatic representation of the coaction for one-loop cut Feynman integrals, enabling easier analysis of their algebraic and differential properties.

Findings

New formula for the coaction of integrals

Diagrammatic representation using graph pinches and cuts

Simplifies derivation of differential equations for Feynman integrals

Abstract

We present a new formula for the coaction of a large class of integrals. When applied to one-loop (cut) Feynman integrals, it can be given a diagrammatic representation purely in terms of pinches and cuts of the edges of the graph. The coaction encodes the algebraic structure of these integrals, and offers ways to extract important properties of complicated integrals from simpler functions. In particular, it gives direct access to discontinuities of Feynman integrals and facilitates a straightforward derivation of the differential equations they satisfy, which we illustrate in the case of the pentagon.