# Renormalization: a quasi-shuffle approach

## TL;DR

This paper introduces a universal semi-group framework based on quasi-shuffle algebras to interpret renormalization in quantum field theory, providing a new algebraic perspective on the process.

## Contribution

It constructs a universal semi-group associated with Rota-Baxter algebras that captures the renormalization operations via quasi-shuffle bialgebras, offering a novel algebraic approach.

## Key findings

- Defines a universal semi-group acting on Feynman graph amplitudes.
- Shows the semi-group reproduces known renormalization operations.
- Highlights the role of quasi-shuffle bialgebras in encoding renormalization.

## Abstract

In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate in this paper to any such algebra a universal semi-group (different in nature from the Connes-Marcolli "cosmical Galois group"). Its action on the physical amplitudes associated to Feynman graphs produces the expected operations: Bogoliubov's preparation map, extraction of divergences, renormalization. In this process a key role is played by commutative and noncommutative quasi-shuffle bialgebras whose universal properties are instrumental in encoding the renormalization process.