  We develop the derivation we proposed in hep-th/0703177 of the dressing phase
of the S-matrix in the AdS/CFT correspondence in the framework of the
underlying bare integrable model. We elaborate the configuration of the Bethe
roots describing the physical vacuum, which consists of a long Bethe string
stretched along the imaginary axis and stacks distributed along the real axis.
We determine the distribution of all Bethe roots in the thermodynamic limit. We
then directly compute the scattering phase of the fundamental excitations over
the physical vacuum and reproduce the BHL/BES dressing phase.
