Vanishing of Beta Function of Non Commutative $Φ^4_4$ Theory to all orders

TL;DR

The paper proves that the beta function of the non-commutative $\phi^4_4$ theory on Moyal space vanishes to all orders at the Langmann–Szabo dual point $\Omega=1$ by deriving Ward identities and analyzing the Dyson equation for the planar, single-border sector. The central result is the all-orders relation $\Gamma^{4}(0,0,0,0)=\lambda (1-\partial_{L}\Sigma(0,0))^2$, established for both bare and renormalized theories, with the mass-renormalized framework used to control divergences. This demonstrates a bounded ultraviolet RG flow and excludes a Landau pole in this NC setting, marking a key step toward a non-perturbative construction and offering insights for extending these techniques to gauge theories and quantum Hall contexts.

Abstract

The simplest non commutative renormalizable field theory, the $φ_4$ model on four dimensional Moyal space with harmonic potential is asymptotically safe up to three loops, as shown by H. Grosse and R. Wulkenhaar, M. Disertori and V. Rivasseau. We extend this result to all orders.

Figures (3)

• Figure 1: The Ward identity
• Figure 2: The Dyson equation
• Figure 3: Two point insertion and opening of the loop with index $p$