Integrable Renormalization I: the Ladder Case

TL;DR

This paper explores the algebraic structures behind renormalization in quantum field theory, specifically using Hopf algebras and classical r-matrices to understand Birkhoff factorizations in the ladder case.

Contribution

It demonstrates how Birkhoff factorization in renormalization can be formulated via classical r-matrices derived from Rota-Baxter structures, focusing on the ladder trees case.

Findings

Birkhoff factorization relates to classical r-matrices.

Rota-Baxter structures underlie the algebraic formulation.

Detailed analysis of the ladder trees Hopf algebra.

Abstract

In recent years a Hopf algebraic structure underlying the process of renormalization in quantum field theory was found. It led to a Birkhoff factorization for (regularized) Hopf algebra characters, i.e. for Feynman rules. In this work we would like to show that this Birkhoff factorization finds its natural formulation in terms of a classical r-matrix, coming from a Rota-Baxter structure underlying the target space of the regularized Hopf algebra characters. Working in the rooted tree Hopf algebra, the simple case of the Hopf subalgebra of ladder trees is treated in detail. The extension to the general case, i.e. the full Hopf algebra of rooted trees or Feynman graphs is briefly outlined.