The diagrammatic coaction beyond one loop

TL;DR

This paper extends the diagrammatic coaction framework for Feynman graphs beyond one loop, providing a systematic approach to understand their discontinuities and differential equations, especially for two-loop topologies involving polylogarithms.

Contribution

It generalizes the diagrammatic coaction from one loop to higher loops, establishing properties and applying it to two-loop topologies with polylogarithmic integrals.

Findings

Established properties of the coaction at any loop order.

Determined master integrals and cuts for two-loop topologies.

Demonstrated the coaction encodes discontinuities and differential equations.

Abstract

The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the $\epsilon$ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.