Mathematical aspects of scattering amplitudes

TL;DR

This paper explores the mathematical structures, especially multiple polylogarithms and their Hopf algebra properties, that underpin the computation of multi-loop Feynman integrals in quantum field theory.

Contribution

It introduces the Hopf algebra structure of multiple polylogarithms and demonstrates their utility in simplifying and deriving functional equations for loop integrals.

Findings

Hopf algebra structure aids in analytic computations

Functional equations among polylogarithms are derived

Mathematical structures improve understanding of multi-loop integrals

Abstract

In these lectures we discuss some of the mathematical structures that appear when computing multi-loop Feynman integrals. We focus on a specific class of special functions, the so-called multiple polylogarithms, and discuss introduce their Hopf algebra structure. We show how these mathematical concepts are useful in physics by illustrating on several examples how these algebraic structures are useful to perform analytic computations of loop integrals, in particular to derive functional equations among polylogarithms.