Combinatorial Dyson-Schwinger equations in noncommutative field theory

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.

29 pages, 15 figures

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• Curing the UV/IR mixing for field theories with translation-invariant $\star$ products
• On the Origin of the Harmonic Term in Noncommutative Quantum Field Theory
• Renormalization and Hopf Algebraic Structure of the 5-Dimensional Quartic Tensor Field Theory
• Renormalization in combinatorially non-local field theories: the Hopf algebra of 2-graphs
• Translation-Invariant Noncommutative Renormalization
• Recipe theorem for the Tutte polynomial for matroids, renormalization group-like approach
• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
• A word Hopf algebra based on the selection/quotient principle
• Mulitgraded Dyson-Schwinger systems
• A selection-quotient process for packed word Hopf algebra
• star-Cohomology, Third Type Chern Character and Anomalies in General Translation-Invariant Noncommutative Yang-Mills
• Renormalization group-like proof of the universality of the Tutte polynomial for matroids
• A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory
• Noncommutative field theories with harmonic term