Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes

TL;DR

This paper demonstrates how the Hopf algebra structure of multiple polylogarithms can simplify multi-loop quantum field theory amplitudes, providing a more informative alternative to the symbol approach, exemplified by Higgs boson plus three gluons.

Contribution

It introduces a coproduct-based method leveraging Hopf algebra to simplify and analyze multi-loop amplitudes, capturing zeta value information often missed by symbols.

Findings

Simplified two-loop Higgs boson amplitudes using classical polylogarithms

Showed coproducts retain zeta value information unlike symbols

Provided a compact, more manageable amplitude expression

Abstract

We show how the Hopf algebra structure of multiple polylogarithms can be used to simplify complicated expressions for multi-loop amplitudes in perturbative quantum field theory and we argue that, unlike the recently popularized symbol-based approach, the coproduct incorporates information about the zeta values. We illustrate our approach by rewriting the two-loop helicity amplitudes for a Higgs boson plus three gluons in a simplified and compact form involving only classical polylogarithms.