On the origin of the dressing phase in N=4 super-Yang-Mills

We derive the phase factor proposed by Beisert, Eden and Staudacher for the S-matrix of planar N=4 super-Yang-Mills, from the all-loop Bethe ansatz equations without the dressing factor. We identify a configuration of the Bethe roots, from which the closed integral formula of the phase factor is reproduced in the thermodynamic limit. This suggests that our configuration describes the 'physical vacuum' in the sense that the dressing phase is nothing but the effective phase for the scattering of fundamental excitations above this vacuum, providing an interesting clue to the physical origin of the dressing phase