  Continuing the program initiated in arXiv:1304.1798 we investigate unitarity
methods applied to two-dimensional integrable field theories. The one-loop
computation is generalized to encompass theories with different masses in the
asymptotic spectrum and external leg corrections. Additionally, the
prescription for working with potentially singular cuts is modified to cope
with an ambiguity that was not encountered before. The resulting methods are
then applied to three light-cone gauge string theories; i) $AdS_3 \times S^3
\times T^4$ supported by RR flux, ii) $AdS_3 \times S^3 \times S^3 \times S^1$
supported by RR flux and iii) $AdS_3 \times S^3 \times T^4$ supported by a mix
of RR and NSNS fluxes. In the first case we find agreement with the exact
result following from symmetry considerations and in the second case with
one-loop semiclassical computations. This agreement crucially includes the
rational terms and hence supports the conjecture that S-matrices of integrable
field theories are cut-constructible, up to a possible shift in the coupling.
In the final case, under the assumption that our methods continue to give all
rational terms, we give a conjecture for the one-loop phases.
