It has been shown that, in the infinite length limit, the magnons of the
gauge theory spin chain can form bound states carrying one finite and one
strictly infinite R-charge. These bound states have been argued to be
associated to simple poles of the multi-particle scattering matrix and to world
sheet solitons carrying the same charges. Classically, they can be mapped to
the solitons of the complex sine-Gordon theory.
Under relatively general assumptions we derive the condition that simple
poles of the two-particle scattering matrix correspond to physical bound states
and construct higher bound states ``one magnon at a time''. We construct the
scattering matrix of the bound states of the BDS and the AFS S-matrices. The
bound state S-matrix exhibits simple and double poles and thus its analytic
structure is much richer than that of the elementary magnon S-matrix. We also
discuss the bound states appearing in larger sectors and their S-matrices. The
large 't Hooft coupling limit of the scattering phase of the bound states in
the SU(2) sector is found to agree with the semiclassical scattering of world
sheet solitons. Intriguingly, the contribution of the dressing phase has an
independent world sheet interpretation as the soliton-antisoliton scattering
phase shift. The small momentum limit provides independent tests of these
identifications.Comment: 25 pages, Latex V2: clarifying comments added to footnote 1 and
footnote 10; references added V3: typos correcte