Asymptotic Bethe Ansatz S-matrix and Landau-Lifshitz type effective 2-d actions

TL;DR

This paper establishes a connection between the asymptotic Bethe ansatz S-matrix in AdS/CFT and an effective Landau-Lifshitz type field theory, deriving exact interaction terms that match known spin chain S-matrices and exploring their string theory implications.

Contribution

It derives the exact form of the Landau-Lifshitz type action corresponding to the low-energy limit of the Bethe ansatz S-matrix in AdS/CFT, generalizing previous results to all orders in the 't Hooft coupling.

Findings

Exact quartic interaction terms in the generalized LL action match the BDS S-matrix.

Extension of the LL model to include string phase modifications and AFS phase.

Confirmation of universality of the dressing phase across sectors.

Abstract

Motivated by the desire to relate Bethe ansatz equations for anomalous dimensions found on the gauge theory side of the AdS/CFT correspondence to superstring theory on AdS_5 x S5 we explore a connection between the asymptotic S-matrix that enters the Bethe ansatz and an effective two-dimensional quantum field theory. The latter generalizes the standard ``non-relativistic'' Landau-Lifshitz (LL) model describing low-energy modes of ferromagnetic Heisenberg spin chain and should be related to a limit of superstring effective action. We find the exact form of the quartic interaction terms in the generalized LL type action whose quantum S-matrix matches the low-energy limit of the asymptotic S-matrix of the spin chain of Beisert, Dippel and Staudacher (BDS). This generalises to all orders in the `t Hooft coupling an earlier computation of Klose and Zarembo of the S-matrix of the standard LL model. We also consider a generalization to the case when the spin chain S-matrix contains an extra ``string'' phase and determine the exact form of the LL 4-vertex corresponding to the low-energy limit of the ansatz of Arutyunov, Frolov and Staudacher (AFS). We explain the relation between the resulting ``non-relativistic'' non-local action and the second-derivative string sigma model. We comment on modifications introduced by strong-coupling corrections to the AFS phase. We mostly discuss the SU(2) sector but also present generalizations to the SL(2) and SU(1|1) sectors, confirming universality of the dressing phase contribution by matching the low-energy limit of the AFS-type spin chain S-matrix with tree-level string-theory S-matrix.