analogous to the cut-constructibility of one-loop amplitudes in massless, supersymmetric theories in four dimensions [@Bern:1994zx; @Bern:1994cg]. We also apply our procedure to the AdS$_5 \times S^5$ light-cone gauge-fixed world-sheet superstring finding agreement, at one-loop in the near-BMN expansion, with the S-matrix known from integrability techniques.

The paper proceeds as follows. In Section \[sec:general\] we set out the general formalism, and obtain the expressions for the cut-constructible part of the one-loop $2\to 2$ S-matrix. We then apply this result to various relativistic integrable models in Section \[sec:relativistic\] and to string theory on AdS$_5\times S^5$ in Section \[sec:strings\]. Concluding remarks are given in Section \[sec:conclusions\].

Two-particle one-loop S-matrix from unitarity cuts {#sec:general}
==================================================

In this section we derive a candidate expression for the one-loop two-particle S-matrix in terms of the tree-level one, as follows from the standard application of cutting techniques (see for example  [@Dixon:1996wi]) to the two-dimensional case. While due to the expected absence of rational terms this formula can never be completely general, we investigate its validity in later sections on various examples.\
The two-body scattering process of a field theory invariant under space and time translations \[eqn:4ptampl\] \^[P]{}(p\_3)\^[Q]{}(p\_4)||\_[M]{}(p\_1)\_[N]{}(p\_2)=\_[[M]{}[N]{}]{}\^[[P]{}[Q]{}]{}(p\_1,p\_2,p\_3,p\_4) is described via the four-point amplitude \[eqn:ampcons\] \_[[M]{}[N]{}]{}\^[[P]{}[Q]{}]{}(p\_1,p\_2,p\_3,p\_4)=(2)\^2 \^[(d)]{}(p\_1+p\_2-p\_3-p\_4) \_[[M]{}[N]{}]{}\^[[P]{}[Q]{}]{}(p\_1,p\_2,p\_3,p\_4) . 