  The standard unitarity-cut method is applied to several massive
two-dimensional models, including the world-sheet AdS$_5\times S^5$
superstring, to compute $2\to 2$ scattering S-matrices at one loop from tree
level amplitudes. Evidence is found for the cut-constructibility of
supersymmetric integrable models, while for models without supersymmetry (but
integrable) the missing rational terms can be interpreted as a shift in the
coupling.
