Renormalisation of \phi^4-theory on noncommutative R^4 in the matrix base

TL;DR

This paper proves the all-order renormalisability of the four-dimensional noncommutative -model, highlighting the necessity of an harmonic oscillator term and analyzing the flow equations influenced by graph topology and propagator behavior.

Contribution

It demonstrates the renormalisability of the noncommutative --model and introduces a modified action with an harmonic oscillator term, using flow equations in the matrix base.

Findings

Renormalisability proven to all orders in perturbation theory.

Additional harmonic oscillator term is necessary in the action.

Flow equations depend on graph topology and propagator asymptotics.

Abstract

We prove that the real four-dimensional Euclidean noncommutative \phi^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.