  Moving from Beisert-Staudacher equations, the complete set of Asymptotic
Bethe Ansatz equations and $S$-matrix for the excitations over the GKP vacuum
is found. The resulting model on this new vacuum is an integrable spin chain of
length $R=2\ln s$ ($s=$ spin) with particle rapidities as inhomogeneities, two
(purely transmitting) defects and $SU(4)$ (residual R-)symmetry. The
non-trivial dynamics of ${\cal N}=4$ SYM appears in elaborated dressing factors
of the 2D two-particle scattering factors, all depending on the 'fundamental'
one between two scalar excitations. From scattering factors we determine bound
states. In particular, we study the strong coupling limit, in the
non-perturbative, perturbative and giant hole regimes. Eventually, from these
scattering data we construct the $4D$ pentagon transition amplitudes
(perturbative regime). In this manner, we detail the multi-particle
contributions (flux tube) to the MHV gluon scattering amplitudes/Wilson loops
(OPE or BSV series) and re-sum them to the Thermodynamic Bubble Ansatz.
