from the string and gauge theory sides – see the review articles [@Arutyunov:2009ga; @Beisert:2010jr] and references therein. However, the perturbative study of scattering amplitudes computed from the path integral defined by the classical action is still of interest. This is true not only because such calculations serve as a test of the proposed exact quantum S-matrix, but also because they provide insights into the structure of the amplitudes and the manifestation of symmetries, and confirm the integrable setup [@Klose:2006dd; @Roiban:2006yc; @Klose:2006zd; @Klose:2007wq; @Klose:2007rz]. The same is true for other integrable theories, including certain 2-d sigma-models [@Zamolodchikov:1978xm; @Ogievetsky:1987vv] and similar massive theories (see e.g. [@Dorey:1996gd]), for which exact quantum S-matrices are known. It is important to note that in the example of the light-cone gauge-fixed sigma-model for the AdS$_5\times S^5$ superstring standard perturbation theory has thus far not been a viable way – due to regularization issues [^1] – to evaluate the S-matrix beyond the leading order.

The aim of this work is to initiate the use of unitarity-based methods in the perturbative study of the S-matrix for *massive two-dimensional* field theories [^2]. Confining ourselves to the use of *standard* unitarity (where only two internal lines are placed on shell, subdividing 