Renormalization of noncommutative phi 4-theory by multi-scale analysis

TL;DR

This paper presents a more efficient proof of the renormalizability of the four-dimensional noncommutative phi 4-theory on the Moyal plane, using multi-scale analysis and bounds on the propagator to improve previous methods.

Contribution

It introduces rigorous bounds on the propagator that enable a powerful multi-scale analysis, simplifying the renormalization proof for noncommutative phi 4-theory.

Findings

Proved renormalizability to all orders for the model.

Developed bounds that facilitate multi-scale analysis of ribbon graphs.

Reduced power-counting to a balance of propagators and inner vertices.

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 r\^ole 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.