The β-function in duality-covariant noncommutative φ^4-theory

TL;DR

The paper investigates the renormalisation group flow of a real, four-dimensional, duality-covariant noncommutative $\phi^4$ theory by computing the one-loop $\beta$-functions for the coupling $\lambda$ and oscillator frequency $\Omega$ using a matrix-base formulation. It shows that the one-loop $\beta$-functions are independent of the noncommutativity scale $\theta$ and that $\beta_\lambda \ge 0$ with $\beta_\lambda=0$ at $\Omega=1$, while $\beta_\Omega$ also vanishes at $\Omega=1$ and in the limit $\Omega\to 0$. The results indicate a potential all-order vanishing of $\beta_\lambda$ at the duality-invariant point and a controlled definition of the standard noncommutative $\phi^4$ theory as a limit of the duality-covariant family. The work relies on a renormalisation proof for all-orders renormalisability and uses planar-graph dominance to derive explicit one-loop expressions, connecting the model to exactly solvable structures.

Abstract

We compute the one-loop β-functions describing the renormalisation of the coupling constant λand the frequency parameter Ωfor the real four-dimensional duality-covariant noncommutative φ^4-model, which is renormalisable to all orders. The contribution from the one-loop four-point function is reduced by the one-loop wavefunction renormalisation, but the β_λ-function remains non-negative. Both β_λand β_Ωvanish at the one-loop level for the duality-invariant model characterised by Ω=1. Moreover, β_Ωalso vanishes in the limit Ω\to 0, which defines the standard noncommutative φ^4-quantum field theory. Thus, the limit Ω\to 0 exists at least at the one-loop level.