Feynman diagrams and polylogarithms: shuffles and pentagons

TL;DR

This paper explores the algebraic structures underlying Feynman diagrams, focusing on Hopf algebras, shuffles, and pentagon relations, to deepen understanding of their mathematical properties in quantum field theory.

Contribution

It reveals new algebraic structures in Feynman graphs, highlighting the role of Hopf algebras, shuffles, and pentagon relations in their mathematical framework.

Findings

Identification of Hopf algebra structures in Feynman diagrams

Connection between shuffles and algebraic relations in graphs

Insight into pentagon relations in the algebraic context

Abstract

We summarize the Hopf algebra structure on Feynman diagrams and emphasize the interest in further algebraic structures hidden in Feynman graphs.