Dynkin operators and renormalization group actions in pQFT

TL;DR

This paper explores the role of Dynkin operators within the Hopf algebra framework in perturbative quantum field theory, highlighting their significance in renormalization and renormalization group equations through a simplified toy model.

Contribution

It extends Dynkin operators to the Hopf algebra setting and demonstrates their natural appearance in renormalization group analysis, introducing a toy model that captures key features of RG in pQFT.

Findings

Dynkin operators are crucial in the Hopf algebra approach to renormalization.

The toy model illustrates the natural emergence of these operators in RG equations.

A universal Galois group for quantum field theories is constructed.

Abstract

Renormalization techniques in perturbative quantum field theory were known, from their inception, to have a strong combinatorial content emphasized, among others, by Zimmermann's celebrated forest formula. The present article reports on recent advances on the subject, featuring the role played by the Dynkin operators (actually their extension to the Hopf algebraic setting) at two crucial levels of renormalization, namely the Bogolioubov recursion and the renormalization group (RG) equations. For that purpose, an iterated integrals toy model is introduced to emphasize how the operators appear naturally in the setting of renormalization group analysis. The toy model, in spite of its simplicity, captures many key features of recent approaches to RG equations in pQFT, including the construction of a universal Galois group for quantum field theories.