Wilsonian Renormalization of Noncommutative Scalar Field Theory

TL;DR

This paper develops a Wilsonian renormalization framework for noncommutative scalar field theory, introducing floating-points and normalized operators, and confirms the vanishing beta-function in the self-dual case through explicit calculations.

Contribution

It extends Wilsonian renormalization to noncommutative theories, introducing floating-points and a new scalar product for operator relevance, and reproduces known beta-function results.

Findings

Floating-points replace fixed-points in noncommutative theories.

The one-loop beta-function vanishes in the self-dual theory at large cutoff.

The framework clarifies the interpretation of vanishing beta-functions in a Wilsonian context.

Abstract

Drawing on analogies with the commutative case, the Wilsonian picture of renormalization is developed for noncommutative scalar field theory. The dimensionful noncommutativity parameter, theta, induces several new features. Fixed-points are replaced by `floating-points' (actions which are scale independent only up to appearances of theta written in cutoff units). Furthermore, it is found that one must use correctly normalized operators, with respect to a new scalar product, to define the right notion of relevance and irrelevance. In this framework it is straightforward and intuitive to reproduce the classification of operators found by Grosse & Wulkenhaar, around the Gaussian floating-point. The one-loop beta-function of their model is computed directly within the exact renormalization group, reproducing the previous result that it vanishes in the self-dual theory, in the limit of large cutoff. With the link between this methodology and earlier results made, it is discussed how the vanishing of the beta-function to all loops, as found by Disertori et al., should be interpreted in a Wilsonian framework.