1/J^2 corrections to BMN energies from the quantum long range Landau-Lifshitz model

TL;DR

The paper analyzes quantum corrections to the BMN spectrum at order $1/J^{2}$ using both gauge- and string-theory long-range Landau-Lifshitz (LL) actions in the perturbative regime ${\tilde{\lambda}}=\lambda/J^{2}$. The gauge LL action is reconstructed to ${\tilde{\lambda}}^{3}$ by matching to circular-string Bethe ansatz results, and the subsequent $1/J^{2}$ corrections are computed by quantizing the LL Hamiltonian, yielding exact agreement with the gauge Bethe ansatz. Similarly, the string LL action, obtained as a fast-string limit, is confronted with the string Bethe ansatz (AFS) and yields $1/J^{2}$ corrections agreeing up to ${\tilde{\lambda}}^{3}$. The results reinforce integrability, clarify the gauge/string mismatch at 3 loops, and illuminate non-analytic ${\tilde{\lambda}}$ contributions that require careful regularization and may involve additional string modes beyond the LL subsector.

Abstract

In a previous paper (hep-th/0509071), it was shown that quantum 1/J corrections to the BMN spectrum in an effective Landau-Lifshitz (LL) model match with the results from the one-loop gauge theory, provided one chooses an appropriate regularization. In this paper we continue this study for the conjectured Bethe ansatz for the long range spin chain representing perturbative planar N=4 Super Yang-Mills in the SU(2) sector, and the ``quantum string" Bethe ansatz for its string dual. The comparison is carried out for corrections to BMN energies up to 3rd order in the effective expansion parameter $\tl=λ/J^2$. After determining the ``gauge-theory'' LL action to order $\tl^3$, which is accomplished indirectly by fixing the coefficients in the LL action so that the energies of circular strings match with the energies found using the Bethe ansatz, we find perfect agreement. We interpret this as further support for an underlying integrability of the system. We then consider the ``string-theory'' LL action which is a limit of the classical string action representing fast string motion on an S^3 subspace of S^5 and compare the resulting $\tl^3/J^2$ corrections to the prediction of the ``string'' Bethe ansatz. As in the gauge case, we find precise matching. This indicates that the LL Hamiltonian supplemented with a normal ordering prescription and zeta-function regularization reproduces the full superstring result for the $1/J^2$ corrections, and also signifies that the string Bethe ansatz does describe the quantum BMN string spectrum to order $1/J^2$. We also comment on using the quantum LL approach to determine the non-analytic contributions in $λ$ that are behind the strong to weak coupling interpolation between the string and gauge results.