Renormalization of Gauge theories and the Hopf Algebra of Diagrams

TL;DR

This paper extends the Hopf algebra framework for renormalization, originally applied to scalar field theories, to nonabelian gauge theories, providing a new algebraic approach to understanding gauge theory renormalization.

Contribution

It generalizes the Hopf algebra structure of Feynman diagrams to nonabelian gauge theories and demonstrates the gauge group action's compatibility with this algebraic structure.

Findings

Defined the gauge group action on the Hopf algebra

Proved the consistency of the gauge group action with the Hopf algebra

Outlined a new proof of the S-matrix using Hopf algebra

Abstract

In 1999 A. Connes and D. Kreimer have discovered a Hopf algebra structure on the Feynman graphs of scalar field theory. They have found that the renormalization can be interpreted as a solving of some Riemann - Hilbert problem. In this work the generalization of their scheme to the case of nonabelian gauge theories is proposed. The action of the gauge group on the Hopf algebra is defined and the proof that this action is in consistent with the Hopf algebra structure is given. The scetch of new proof of S-matrix, based on the Hopf algebra is given.