An algebraic Birkhoff decomposition for the continuous renormalization group

TL;DR

This paper develops an algebraic framework for the continuous renormalization group using Hopf algebras, connecting perturbative renormalization with Birkhoff decomposition of rooted trees.

Contribution

It introduces an algebraic formulation of the Exact Renormalization Group Equation within the Hopf algebra setting, linking renormalization to Birkhoff decomposition.

Findings

Establishes a connection between renormalization and algebraic Birkhoff decomposition.

Provides algebraic preliminaries for perturbative renormalization in differential equations.

Lays groundwork for a Hopf algebra approach to the continuous renormalization group.

Abstract

This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited.