Functional Renormalization of Noncommutative Scalar Field Theory

TL;DR

This paper applies the Functional Renormalization Group Equation to a non-commutative scalar field theory, deriving the flow equations and analyzing the theory's behavior, including asymptotic safety and beta-functions, without truncations.

Contribution

It introduces a non-truncated FRGE approach to non-commutative scalar field theory, demonstrating asymptotic safety and computing beta-functions in this context.

Findings

The theory is asymptotically safe for small  coupling.

The FRGE approach simplifies the calculation of one-loop beta-functions.

The analysis confirms previous results on the model's safety and renormalization properties.

Abstract

In this paper we apply the Functional Renormalization Group Equation (FRGE) to the non-commutative scalar field theory proposed by Grosse and Wulkenhaar. We derive the flow equation in the matrix representation and discuss the theory space for the self-dual model. The features introduced by the external dimensionful scale provided by the non-commutativity parameter, originally pointed out in \cite{Gurau:2009ni}, are discussed in the FRGE context. Using a technical assumption, but without resorting to any truncation, it is then shown that the theory is asymptotically safe for suitably small values of the $\phi^4$ coupling, recovering the result of \cite{disertori:2007}. Finally, we show how the FRGE can be easily used to compute the one loop beta-functions of the duality covariant model.