Integrable Renormalization II: the general case

TL;DR

This paper generalizes the Hopf algebra framework for renormalization, extending previous results to decorated rooted trees and connecting with Rota-Baxter algebra, providing a broader mathematical foundation for quantum field theory renormalization.

Contribution

It introduces a comprehensive Hopf algebra of decorated rooted trees and derives the Birkhoff decomposition for characters with Rota-Baxter target space, extending previous work.

Findings

Generalization to decorated rooted trees

Derivation of Birkhoff decomposition using Rota-Baxter double construction

Outline of extension to Feynman graph Hopf algebra

Abstract

We extend the results we obtained in an earlier work. The cocommutative case of rooted ladder trees is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the Rota-Baxter double construction, respectively Atkinson's theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees.