# Combinatorial Hopf algebras from renormalization

## TL;DR

This paper explores the structure of three non-commutative Hopf algebras relevant to quantum field theory renormalization, providing new methods to define their associative products.

## Contribution

It introduces the right-sided combinatorial Hopf structures of three specific non-commutative Hopf algebras in renormalization, with two novel approaches to defining their associative products.

## Key findings

- Describes the non-commutative Faà di Bruno Hopf algebra
- Details the non-commutative charge renormalization Hopf algebra on planar binary trees
- Introduces recursive and grafting/shuffling methods for associative product definition

## Abstract

In this paper we describe the right-sided combinatorial Hopf structure of three Hopf algebras appearing in the context of renormalization in quantum field theory: the non-commutative version of the Fa\`a di Bruno Hopf algebra, the non-commutative version of the charge renormalization Hopf algebra on planar binary trees for quantum electrodynamics, and the non-commutative version of the Pinter renormalization Hopf algebra on any bosonic field. We also describe two general ways to define the associative product in such Hopf algebras, the first one by recursion, and the second one by grafting and shuffling some decorated rooted trees.