not be the case for the original bubble integrals before cutting – due to factors of loop-momentum in the numerators. These divergences, along with those coming from tadpole graphs, which we did not consider, should be taken into account for the renormalization of the theory. In this paper we do not investigate this issue, however all the theories we consider in Section \[sec:relativistic\] and \[sec:strings\] are either UV-finite or renormalizable. Furthermore, whether the cut-constructible contribution / to the S-matrix gives the full result is a priori unclear as rational terms following from non-trivial cancellations in the regularization procedure may be missing.

In the following sections we will make use of (\[eqn:final\]) and to construct the one-loop S-matrix from its tree-level form for some relativistic models (Section \[sec:relativistic\]) and for the non-relativistic string world-sheet field theory on AdS$_5\times S^5$ (Section \[sec:strings\]). For all these theories we will compare with known results (either from perturbation theory or integrability techniques) and analyze the effectiveness of the method of unitarity cuts in two dimensions.

Relativistic models {#sec:relativistic}
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To explore the validity of the formula we will initially focus on some relativistic models. In relativistic theories it is natural to write the on-shell momenta 