Combinatorial Hopf algebraic description of the multiscale renormalization in quantum field theory

TL;DR

This paper introduces several Hopf algebras to describe the combinatorics of multi-scale renormalization in quantum field theory, providing a new algebraic framework for analyzing scale-dependent couplings.

Contribution

It defines new Hopf algebras on assigned graphs and Gallavotti-Nicolò trees, linking them with the Connes-Kreimer algebra for multi-scale renormalization.

Findings

Hopf algebras on assigned graphs and trees are constructed.

Morphisms between these algebras and the Connes-Kreimer algebra are established.

Scale-dependent couplings are analyzed within this algebraic framework.

Abstract

We define in this paper several Hopf algebras describing the combinatorics of the so-called multi-scale renormalization in quantum field theory. After a brief recall of the main mathematical features of multi-scale renormalization, we define assigned graphs, that are graphs with appropriate decorations for the multi-scale framework. We then define Hopf algebras on these assigned graphs and on the Gallavotti-Nicol\`o trees, particular class of trees encoding the supplementary informations of the assigned graphs. Several morphisms between these combinatorial Hopf algebras and the Connes-Kreimer algebra are given. Finally, scale dependent couplings are analyzed via this combinatorial algebraic setting.