We prove that the real four-dimensional Euclidean noncommutative φ 4 -model is renormalisable to all orders in perturbation theory. Compared with the commutative case, the bare action of relevant and marginal couplings contains necessarily an additional term: an harmonic oscillator potential for the free scalar field action. This entails a modified dispersion relation for the free theory, which becomes important at large distances (UV/IR-entanglement). The renormalisation proof relies on flow equations for the expansion coefficients of the effective action with respect to scalar fields written in the matrix base of the noncommutative R 4 . The renormalisation flow depends on the topology of ribbon graphs and on the asymptotic and local behaviour of the propagator governed by orthogonal Meixner polynomials.

Solving the exact renormalisation group equationà la Wilson-Polchinski perturbatively, we derive a power-counting theorem for general matrix models with arbitrarily non-local propagators. The power-counting degree is determined by two scaling dimensions of the cut-off propagator and various topological data of ribbon graphs. As a necessary condition for the renormalisability of a model, the two scaling dimensions have to be large enough relative to the dimension of the underlying space. In order to have a renormalisable model one needs additional locality properties-typically arising from orthogonal polynomials-which relate the relevant and marginal interaction coefficients to a finite number of base couplings. The main application of our power-counting theorem is the renormalisation of field theories on noncommutative R D in matrix formulation.According to Polchinski's derivation of the exact renormalisation group equation we now consider a (at first sight) different problem than (2.6). Via a cut-off function K[m, Λ], which is smooth in Λ and satisfies K[m, ∞] = 1, we modify the weight of a matrix index m as a function of a certain scale Λ:7) S[φ, J, Λ] = V D m,n,k,l 1 2 φ mn G K mn;kl (Λ) φ kl + L[φ, Λ] + m,n,k,l φ mn F mn;kl [Λ]J kl + m,n,k,l 1 2 J mn E mn;kl [Λ]J kl + C[Λ] , (2.8) G K mn;kl (Λ) := i∈m,n,k,l K[i, Λ] −1 G mn;kl , (2.9)

We consider an external gauge potential minimally coupled to a renormalisable scalar theory on 4-dimensional Moyal space and compute in position space the oneloop Yang-Mills-type effective theory generated from the integration over the scalar field. We find that the gauge invariant effective action involves, beyond the expected noncommutative version of the pure Yang-Mills action, additional terms that may be interpreted as the gauge theory counterpart of the harmonic oscillator term, which for the noncommutative ϕ 4 -theory on Moyal space ensures renormalisability. The expression of a possible candidate for a renormalisable action for a gauge theory defined on Moyal space is conjectured and discussed.

We show that the photon self-energy in quantum electrodynamics on noncommutative R 4 is renormalizable to all orders (both in θ andh) when using the Seiberg-Witten map. This is due to the enormous freedom in the Seiberg-Witten map which represents field redefinitions and generates all those gauge invariant terms in the θ-deformed classical action which are necessary to compensate the divergences coming from loop integrations.

In this paper we give a much more efficient proof that the real Euclidean φ 4 -model on the four-dimensional Moyal plane is renormalizable to all orders. We prove rigorous bounds on the propagator which complete the previous renormalization proof based on renormalization group equations for non-local matrix models. On the other hand, our bounds permit a powerful multi-scale analysis of the resulting ribbon graphs. Here, the dual graphs play a particular rôle because the angular momentum conservation is conveniently represented in the dual picture. Choosing a spanning tree in the dual graph according to the scale attribution, we prove that the summation over the loop angular momenta can be performed at no cost so that the power-counting is reduced to the balance of the number of propagators versus the number of completely inner vertices in subgraphs of the dual graph.