Universal Hopf Algebra of Renormalization and Hopf Algebras of Rooted Trees

TL;DR

This paper explores the universal Hopf algebra of renormalization, representing it through rooted trees, and uncovers new relations with other Hopf algebras, providing insights into quantum field theory renormalization.

Contribution

It introduces a rooted tree representation of the universal Hopf algebra of renormalization and establishes new connections with Hopf algebras of rooted trees and symmetric functions.

Findings

New rooted tree representation of the universal Hopf algebra.

Relations between this Hopf algebra and other Hopf algebras of rooted trees.

A novel interpretation of the universal singular frame related to physical counterterms.

Abstract

In this paper we are going to find a rooted tree representation from universal Hopf algebra of renormalization (in Connes-Marcolli's approach in the study of renormalizable Quantum Field Theories under the scheme minimal subtraction in dimensional regularization). With attention to this new picture, interesting relations between this specific Hopf algebra and some important Hopf algebras of rooted trees and also Hopf algebra of (quasi-) symmetric functions are obtained. And moreover a new interpretation from universal singular frame, based on Hall rooted trees, is deduced such that it can be applied to the physical information of a renormalizable theory such as counterterms.