A Novel Long Range Spin Chain and PlanarN=4 Super Yang-Mills

Abstract

We probe the long-range spin chain approach to planar N = 4 gauge theory at high loop order. A recently employed hyperbolic spin chain invented by Inozemtsev is suitable for the su(2) subsector of the state space up to three loops, but ceases to exhibit the conjectured thermodynamic scaling properties at higher orders. We indicate how this may be bypassed while nevertheless preserving integrability, and suggest the corresponding all-loop asymptotic Bethe ansatz. We also propose the local part of the all-loop gauge transfer matrix, leading to conjectures for the asymptotically exact formulae for all local commuting charges. The ansatz is finally shown to be related to a standard inhomogeneous spin chain. A comparison of our ansatz to semi-classical string theory uncovers a detailed, non-perturbative agreement between the corresponding expressions for the infinite tower of local charge densities. However, the respective Bethe equations differ slightly, and we end by refining and elaborating a previously proposed possible explanation for this disagreement.

Planar plane-wave matrix theory at the four loop order: Integrability without BMN scaling

“…The remaining freedom has been fixed in order to match the eigenvalue formulas (D.2). These formulas are in accordance with the general all-loop expressions for q i given in [8,14,25].…”

“…Curiously our five spin interaction terms do agree with the proposed four-loop dilation operator of N = 4 super Yang-Mills theory [8]. However, they differ in the two, three and four spin interaction terms.…”

“…D. Let us now compare this form to the other known perturbative long-range Bethe ansätze in the literature. We note that the dispersion relation (4.11) and the phase function (4.13) are exactly the same as for the Beisert-Dippel-Staudacher spin-chain [8]. The difference lies in the presence of the exponential factor in the Bethe ansatz for the PWMT.…”

“…A similar breakdown is by now firmly established in the "near" plane-wave superstring [11] and in the matching of energies of rotating solitonic AdS 5 × S 5 string solutions [12] at three-loop order in perturbative gauge theory. This discrepancy has been attributed to an order of limits problem, stating that a perturbative agreement should not be expected [8]. Hence, based on these results there presently is no reason to expect perturbative BMN scaling in N = 4 super Yang-Mills!…”

“…"Singular" means that the Bethe equations become singular for the lowest order solution of the Bethe momenta p i . In these cases one should use the inhomogeneous Bethe ansatz as in [8].…”