Motivated by the recent construction of a translation-invariant renormalizable non-commutative model for a scalar field [1], we introduce models for non-commutative U (1) gauge fields along the same lines. More precisely, we include some extra terms into the action with the aim of getting rid of the UV/IR mixing.

When considering quantum field theories on non-commutative spaces one inevitably encounters the infamous UV/IR mixing problem. So far, only very few renormalizable models exist and all of them describe non-commutative scalar field theories on fourdimensional Euclidean Groenewold-Moyal deformed space, also known as 'θ-deformed space' R 4 θ . In this work we discuss some major obstacles of constructing a renormalizable non-commutative gauge field model and sketch some possible ways out.

In this paper we elaborate on the translation-invariant renormalizable φ 4 theory in 4-dimensional non-commutative space which was recently introduced by the Orsay group. By explicitly performing Feynman graph calculations at one loop and higher orders we illustrate the mechanism which overcomes the UV/IR mixing problem and ultimately leads to a renormalizable model. The obtained results show that the IR divergences are also suppressed in the massless case, which is of importance for the gauge field theoretic generalization of the scalar field model.

Abstract. The aim of this review is to present an overview over available models and approaches to non-commutative gauge theory. Our main focus thereby is on gauge models formulated on flat Groenewold-Moyal spaces and renormalizability, but we will also review other deformations and try to point out common features. This review will by no means be complete and cover all approaches, it rather reflects a highly biased selection.

Motivated by the success of the non-commutative scalar Grosse-Wulkenhaar model, a non-commutative U ⋆ (1) gauge field theory including an oscillator-like term in the action has been put forward in [1]. The aim of the current work is to analyze whether that action can lead to a fully renormalizable gauge model on non-commutative Euclidean space. In a first step, explicit one-loop graph computations are hence presented, and their results as well as necessary modifications of the action are successively discussed.