Quantum Landau-Lifshitz model at four loops: 1/J and 1/J^2 corrections to BMN energies

TL;DR

This paper extends the effective Landau-Lifshitz model to four loops, computing 1/J and 1/J^2 corrections to BMN energies, and finds perfect agreement between gauge theory, string theory, and Bethe ansatz results up to this order.

Contribution

It provides the four-loop effective Landau-Lifshitz Lagrangian and confirms the matching of corrections from gauge and string theory with Bethe ansatz predictions.

Findings

Perfect matching of corrections up to four loops between gauge theory and Bethe ansatz.

Perfect matching of corrections up to four loops between string theory and Bethe ansatz.

Extension of the Landau-Lifshitz model to four loops in the SU(2) sector.

Abstract

In a previous paper (hep-th/0510080) the effective Landau-Lifshitz (LL) Lagrangians in the SU(2) sector coming from string theory and gauge theory have been found to three loops in the effective expansion parameter $\tilde\lambda=\lambda/J^2$. In this paper we continue this study and find the effective Landau-Lifshitz Lagrangians to four loops. We extend to four-loops $\tilde\lambda^4$ the computations of 1/J and $1/J^2$ corrections to BMN energies done in hep-th/0510080 to three-loops. We compare these corrections obtained from quantum ``gauge-theory'' LL action with the corrections obtained from the conjectured Bethe ansatz for the long range spin chain representing perturbative {large $N$} $\NN=4$ Super Yang-Mills in the SU(2) sector and find perfect matching to four loops $\tilde\lambda^4$. We compare also the 1/J and $1/J^2$ corrections obtained from quantum ``string-theory'' LL action with those obtained from the "quantum string" Bethe ansatz and again find perfect matching to four-loops.