Renormalisation of Noncommutative ϕ 4-Theory by Multi-Scale Analysis

Abstract

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.

Non-commutative Renormalization

“…We have also extended the Grosse-Wulkenhaar results to theφ φ φ φ LSZ model [43]. Our proof relies on a multiscale analysis analogous to [42] but in direct space. It allows a more transparent understanding of the Moyality of the counterterms for planar subgraphs at higher scales when seen through external propagators at lower scales.…”

“…This is a real tour de force! The matrix based Mehler kernel is quasi-diagonal, and they were able to use their computation to prove perturbative renormalizability of the theory, up to some estimates which were finally proven in [42].…”

“…By replacing these cutoffs by smoother cutoffs which cut directly the Mehler parameter into slices, we could derive rigorously the estimates that were only numerically checked in [40] hence close the last gaps in the BPHZ theorem for vulcanized non commutative φ 4 4 [42]. We could also replace the somewhat cumbersome recursive use of the Polchinski equation [46] by more direct and explicit bounds in a multiscale analysis.…”

“…In [42], a multi-scale analysis was introduced to complete the rigorous study of the power counting of the non-commutative Φ 4 theory.…”

Constructive matrix theory

“…To fully construct φ 4 4 we have also to generalize the single N analysis of this paper to a multiscale analysis such as the one of [25]. This requires to optimize as usually the tree expansion over all the scales so that connected functions of higher scales are always correctly connected through the tree.…”