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

TL;DR

The paper tests whether planar N=4 SYM's dilatation operator can be captured by the Inozemtsev long-range spin chain, using an asymptotic Bethe ansatz to reproduce known two- and three-loop results for BMN-type operators. It finds striking agreement with spinning-string predictions at two loops and a subtle but definite breakdown at three loops, consistent with independent discrepancies between three-loop gauge theory and near plane-wave string results. Inozemtsev's chain reproduces perturbative BMN scaling up to three loops, but exhibits a generic breakdown at four loops, challenging the view of a simple all-order integrable description. In the thermodynamic limit for long operators, the formalism yields integral equations for rapidity densities that reproduce two-loop gauge energies but again show three-loop divergences from FT-string predictions, though non-perturbative connections at strong coupling remain plausible.

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.