Combinatorial Dyson-Schwinger equations in noncommutative field theory

TL;DR

This paper develops a Hopf algebra framework for noncommutative renormalization in quantum field theory on Moyal space, introducing combinatorial Dyson-Schwinger equations and illustrating them with explicit loop calculations.

Contribution

It introduces a novel Hopf algebra structure and Hochschild cocycles to formulate Dyson-Schwinger equations in noncommutative quantum field theory.

Findings

Explicit one- and two-loop examples demonstrated

New combinatorial Dyson-Schwinger equations derived

Framework applicable to noncommutative renormalization

Abstract

We give here the Hopf algebra structure describing the noncommutative renormalization of a recently introduced translation-invariant model on Moyal space. We define Hochschild one-cocyles $B_+^\gamma$ which allows us to write down the combinatorial Dyson-Schwinger equations for noncommutative quantum field theory. One- and two-loops examples are explicitly worked out.