Vanishing beta function for Grosse-Wulkenhaar model in a magnetic field

TL;DR

This work proves that the $eta$-function of the complex Grosse-Wulkenhaar model in a magnetic field vanishes at all orders of perturbation, by combining $(q,p)$-Ward identities with Dyson equations to express the amputated four-point function in terms of self-energy derivatives. It introduces and analyzes the $(q,p)$-deformation with $q=1+B$, $p=1-B$ under the constraint $|B|<1$, and computes the one-loop renormalization effects leading to a constrained RG flow for the left and right wave-function renormalizations, yielding a constant product $qp=K$ and an IR non-Gaussian fixed point on the line $p=q$. The results demonstrate asymptotic safety for this NC model with magnetic field and hint at connections to NC quantum Hall physics, where IR fixed points may capture plateau-like behavior. The approach leverages Ward identities, Dyson equations, and careful treatment of left/right insertions to establish the all-orders vanishing of the $eta$-function and to map the one-loop flow of the deformation parameters, offering a blueprint for studying similar noncommutative renormalization phenomena.

Abstract

We prove that the beta function of the Grosse-Wulkenhaar model including a magnetic field vanishes at all order of perturbations. We compute the renormalization group flow of the relevant dynamic parameters and find a non-Gaussian infrared fixed point. Some consequences of these results are discussed.

Figures (5)

• Figure 1: $q$-Ward identity for a $2\ell$ point function with insertion on the left face.
• Figure 2: The left Dyson equation.
• Figure 3: Two point left insertion and opening of the loop with index $k$.
• Figure 4: The self energy.
• Figure 5: RG flow of $q(i)$ and $p(i)$ versus $i$ with cutoff $\Lambda=100$ and $p_{uv}=10^{-6}$.

Theorems & Definitions (3)

• Theorem 3.1
• Lemma 3.1
• Lemma 3.2