  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.
