On matrix differential equations in the Hopf algebra of renormalization

TL;DR

This paper formulates matrix differential equations for counter terms and renormalized characters within the Hopf algebra framework of renormalization, generalizing Sakakibara's equations to include Feynman rules.

Contribution

It extends Sakakibara's differential equations to a matrix setting in the context of Hopf algebras, encompassing Feynman rules in perturbative renormalization.

Findings

Established matrix differential equations for counter terms and renormalized characters.

Unified Sakakibara's equations with Hopf algebra structures in renormalization.

Applicable to Feynman rules in perturbative quantum field theory.

Abstract

We establish Sakakibara's differential equations in a matrix setting for the counter term (respectively renormalized character) in Connes-Kreimer's Birkhoff decomposition in any connected graded Hopf algebra, thus including Feynman rules in perturbative renormalization as a key example.