# The algebraic structure of cut Feynman integrals and the diagrammatic coaction

## TL;DR

This paper introduces a diagrammatic coaction operation for Feynman integrals that generalizes known mathematical structures, enabling direct access to discontinuities and simplifying the derivation of differential equations for one-loop integrals.

## Contribution

It proposes a new algebraic and diagrammatic coaction for Feynman integrals that applies broadly, including to hypergeometric functions and generic one-loop integrals, enhancing analytic understanding.

## Key findings

- The coaction reduces to known structures on multiple polylogarithms.
- It provides a diagrammatic representation using graph operations.
- It simplifies the derivation of differential equations for one-loop integrals.

## Abstract

We study the algebraic and analytic structure of Feynman integrals by proposing an operation that maps an integral into pairs of integrals obtained from a master integrand and a corresponding master contour. This operation is a coaction. It reduces to the known coaction on multiple polylogarithms, but applies more generally, e.g. to hypergeometric functions. The coaction also applies to generic one-loop Feynman integrals with any configuration of internal and external masses, and in dimensional regularization. In this case, we demonstrate that it can be given a diagrammatic representation purely in terms of operations on graphs, namely contractions and cuts of edges. The coaction gives direct access to (iterated) discontinuities of Feynman integrals and facilitates a straightforward derivation of the differential equations they admit. In particular, the differential equations for any one-loop integral are determined by the diagrammatic coaction using limited information about their maximal, next-to-maximal, and next-to-next-to-maximal cuts.