Renormalization of noncommutative phi 4-theory by multi-scale analysis
V. Rivasseau, F. Vignes-Tourneret, R. Wulkenhaar
TL;DR
This work provides a rigorous, efficient renormalization proof for the four-dimensional noncommutative $\phi^4$ theory on the Moyal plane by establishing robust bounds on the propagator and applying a multi-scale analysis to ribbon graphs. By formulating the theory in the matrix base and using dual graphs to exploit angular-momentum conservation, the authors implement a scale decomposition and tree-based momentum routing that reduces power-counting to propagator decay balanced against inner-vertex contributions. A discrete Taylor subtraction around vanishing external indices yields four base functions for the marginal/relevant divergences, fixed by normalizations of the coupling, mass, field, and harmonic oscillator frequency; the one-loop beta-function analysis further shows a bounded bare coupling via the constant ratio $\lambda/\Omega^2$. The approach not only proves renormalizability to all orders but also advances toward a constructive program, providing analytical propagator bounds and a more efficient framework than prior RG-based proofs, especially for $\Omega$ away from the critical value. Although the Omega=1 case requires separate treatment, the methodology lays the groundwork for a constructive version of the Grosse–Wulkenhaar model and strengthens the understanding of UV/IR behavior in noncommutative QFTs.
Abstract
In this paper we give a much more efficient proof that the real Euclidean phi 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.