Three-parameter deformation of $\mathbb{R}\times S^3$ in the Landau-Lifshitz limit

In this article we construct the effective field theory associated to the $\mathbb{R}\times S^3$ sector of the three-parameter deformation of $AdS_3 \times S^3 \times T^4$ in the Landau-Lifshitz approximation. We use this action to compute the dispersion relation of excitations around the BMN vacuum and the perturbative $S$-matrix associated to them. We are able to compute and sum all the different loop contributions to the $S$-matrix in this limit.