$AdS_3 \times S^3 \times M^4$ string S-matrices from unitarity cuts

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.