the implementation of unitary cuts in fixed dimension, are not necessarily expected to give the full answer, possibly missing rational terms (terms with only a rational dependence on momentum invariants) that can arise from the $\epsilon$-expansion in $d=2-2\epsilon$ dimensions [@vanNeerven:1985xr]. A thorough discussion of these issues is contained in Section \[sec:general\].

The main part of this paper focusses on testing the validity of our procedure for several models of interest, and therefore exploring the *cut-constructibility* (via standard unitarity methods) of S-matrices in two dimensions. For bosonic theories with *integrability*, the result obtained is rather intriguing – we find close agreement with perturbation theory, the only price to pay being a finite shift in the coupling. Incidentally, a hint of a connection between unitarity and integrability can be seen, for example, in the sine-Gordon model – the only non-vanishing contribution to the connected part of the $3 \to 3$ tree-level amplitude is obtained by setting the intermediate particles on-shell [@Dorey:1996gd]. These results seem to suggest a relationship between quantization preserving integrability and unitarity techniques which would be interesting to investigate further.

Possibly more expected is that one-loop amplitudes in *integrable, supersymmetric* theories appear to be cut-constructible via standard unitarity. This is 