Planar N=4 gauge theory and the Inozemtsev long range spin chain

TL;DR

This paper explores the potential of the Inozemtsev long-range spin chain to model the planar N=4 gauge theory dilatation operator, finding good agreement at two loops but discrepancies at three and four loops, indicating limits of integrability.

Contribution

It demonstrates the applicability and limitations of the Inozemtsev spin chain in describing N=4 gauge theory beyond two loops, highlighting where perturbative and non-perturbative approaches diverge.

Findings

Excellent match with string theory at two loops.

Breakdown of agreement at three loops.

Failure of BMN scaling at four loops.

Abstract

We investigate whether the (planar, two complex scalar) dilatation operator of N=4 gauge theory can be, perturbatively and, perhaps, non-perturbatively, described by an integrable long range spin chain with elliptic exchange interaction. Such a chain was introduced some time ago by Inozemtsev. In the limit of sufficiently ``long'' operators a Bethe ansatz exists, which we apply at the perturbative two- and three-loop level. Spectacular agreement is found with spinning string predictions of Frolov and Tseytlin for the two-loop energies of certain large charge operators. However, we then go on to show that the agreement between perturbative gauge theory and semi-classical string theory begins to break down, in a subtle fashion, at the three-loop level. This corroborates a recently found disagreement between three-loop gauge theory and near plane-wave string theory results, and quantitatively explains a previously obtained puzzling deviation between the string proposal and a numerical extrapolation of finite size three-loop anomalous dimensions. At four loops and beyond, we find that the Inozemtsev chain exhibits a generic breakdown of perturbative BMN scaling. However, our proposal is not necessarily limited to perturbation theory, and one would hope that the string theory results can be recovered from the Inozemtsev chain at strong 't Hooft coupling.