The asymptotic Hopf Algebra of Feynman Integrals

TL;DR

This paper introduces the asymptotic Hopf algebra framework for analyzing Feynman integrals' asymptotic expansions, connecting combinatorial graph operations with algebraic structures to better understand divergence subtraction.

Contribution

It formulates the graph combinatorial operations in Feynman integral expansions as an asymptotic Hopf algebra, extending the motic Hopf algebra and relating it to the R* operation.

Findings

Established the connection between the asymptotic Hopf algebra and the motic Hopf algebra.

Linked Bogoliubov's R operation with the Hopf algebraic structure of series remainders.

Introduced a Hopf monoid formulation for higher power expansions.

Abstract

The method of regions is an approach for developing asymptotic expansions of Feynman Integrals. We focus on expansions in Euclidean signature, where the method of regions can also be formulated as an expansion by subgraph. We show that for such expansions valid around small/large masses and momenta the graph combinatorial operations can be formulated in terms of what we call the asymptotic Hopf algebra. This Hopf algebra is closely related to the motic Hopf algebra underlying the $R^*$ operation, an extension of Bogoliubov's $R$ operation, to subtract both IR and UV divergences of Feynman integrals in the Euclidean. We focus mostly on the leading power, for which the Hopf algebra formulation is simpler. We uncover a close connection between Bogoliubov's $R$ operation in the Connes-Kreimer formulation and the remainder $\mathcal{R}$ of the series expansion, whose Hopf algebraic structure is identically formalised in the corresponding group of characters. While in the Connes-Kreimer formulation the UV counterterm is formalised in terms of a twisted antipode, we show that in the expansion by subgraph a similar role is played by the integrand Taylor operator. To discuss the structure of higher power expansions we introduce a novel Hopf monoid formulation.