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HU-EP-13/17\

Lorenzo Bianchi, Valentina Forini, Ben Hoare\

**Abstract**

Using unitarity methods, we compute, for several massive two-dimensional models, the cut-constructible part of the one-loop $2\to 2$ scattering S-matrices from the tree-level amplitudes. We apply our method to various integrable theories, finding evidence that for supersymmetric models the one-loop S-matrix is cut-constructible, while for models without supersymmetry (but with integrability) the missing rational terms are proportional to the tree-level S-matrix and therefore can be interpreted as a shift in the coupling. Finally, applying our procedure to the world-sheet theory for the light-cone gauge-fixed AdS$_5\times S^5$ superstring we reproduce, at one-loop in the near-BMN expansion, the S-matrix known from integrability techniques.

Introduction {#sec:intro}
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The remarkable efficiency of unitarity-based methods for the calculation of space-time scattering amplitudes in non-abelian gauge theories (see e.g. [@Roib_Review]) motivates the application of similar techniques to perturbative regimes of other interesting models. This is certainly the case for the AdS$_5\times S^5$ superstring world-sheet theory. The S-matrix for the scattering of its excitations is known exactly, fixed by symmetry and integrability up to a phase [@Beisert:2005tm], with the latter determined from a non-relativistic generalization of crossing symmetry as well as perturbative data both 