Diagrammatic coaction of two-loop Feynman integrals

Abstract

When evaluating Feynman integrals as Laurent series in the dimensional regulator epsilon one encounters families of iterated integrals, the simplest of which are the multiple polylogarithms. These functions are known to possess a structure called the coaction which captures their analytic properties and the set of functional relations they obey. It has been found that this coaction, when applied to a one-loop Feynman integral, may be expressed using integrals corresponding to subgraphs, as well as cut integrals. In the present work we will explore how this diagrammatic coaction generalises to two-loop Feynman integrals and related questions. Expressing Feynman integrals using generalised hypergeometric functions is a useful alternative to considering them in Laurent series form. The properties of these functions have been well studied and can be invoked in the study of Feynman integrals. Importantly, we will see that hypergeometric functions also possess a coaction which may be used in computing coactions of Feynman integrals. We will compute the coactions for a range of two-loop graphs and establish how they differ from one-loop cases. Specifically, the correspondence between subgraphs and cuts observed at one loop will be preserved while multiple master integrals for a given graph can appear at two loops, as can multiple cuts associated with a particular subset of propagators. The appropriate generalisation of deformation terms in the diagrammatic coaction will also be considered. Given the important role cut integrals play in this picture, we will also examine their calculation. There are also many subtle features involved in specifying how these cuts are defined, and in creating elegant dual bases of master integrals and cuts, which will be explored.