Non-commutative Gross-Neveu model at large N

TL;DR

This paper investigates the large N limit of non-commutative Gross-Neveu models in two and three dimensions, revealing non-renormalizability in the large N expansion due to UV/IR mixing, despite renormalizable coupling expansions.

Contribution

It extends the analysis of non-commutative Gross-Neveu models to three dimensions and clarifies the limitations of large N renormalization due to UV/IR mixing effects.

Findings

Non-commutative 2D Gross-Neveu model is renormalizable in coupling expansion.

Large N expansion of the non-commutative theory is not renormalizable.

Large N limit in 3D is trivial when the cutoff is removed.

Abstract

The non-commutative O(N) Gross-Neveu model is solved in the large N limit in two and three space-time dimensions. The commutative version of the two dimensional model is a renormalizable quantum field theory, both in a coupling constant expansion and an expansion in 1/N. The non-commutative version has a renormalizable coupling constant expansion where ultraviolet divergences can be removed by adjusting counterterms to each order. On the other hand, in a previous work, we showed that the non-commutative theory is not renormalizable in the large N expansion. This is argued to be due to a combined effect of asymptotic freedom and the ultraviolet/infrared mixing that occurs in a non-commutative field theory. In the present paper we will elaborate on this result and extend it to study the large N limit of the three dimensional Gross-Neveu model. We shall see that the large N limit of the three dimensional theory is also trivial when the ultraviolet cutoff is removed.