The Noncommutative Supersymmetric Nonlinear Sigma Model

TL;DR

This paper investigates the effects of noncommutativity on supersymmetric nonlinear sigma models, showing that while noncommutativity destroys renormalizability in some models, it can preserve it in the supersymmetric O(N) model in three dimensions, restoring dynamical mass generation and avoiding UV/IR mixing.

Contribution

It demonstrates that noncommutative supersymmetric O(N) nonlinear sigma models are renormalizable in D=3 up to subleading order, unlike their non-supersymmetric counterparts.

Findings

Noncommutativity destroys renormalizability of certain models.

Supersymmetry restores renormalizability in D=3 for the O(N) sigma model.

Divergences in four-point functions in D=2 cannot be renormalized with Moyal product counterterms.

Abstract

We show that the noncommutativity of space-time destroys the renormalizability of the 1/N expansion of the O(N) Gross-Neveu model. A similar statement holds for the noncommutative nonlinear sigma model. However, we show that, up to the subleading order in 1/N expansion, the noncommutative supersymmetric O(N) nonlinear sigma model becomes renormalizable in D=3. We also show that dynamical mass generation is restored and there is no catastrophic UV/IR mixing. Unlike the commutative case, we find that the Lagrange multiplier fields, which enforce the supersymmetric constraints, are also renormalized. For D=2 the divergence of the four point function of the basic scalar field, which in D=3 is absent, cannot be eliminated by means of a counterterm having the structure of a Moyal product.