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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 actionto compute the dispersion relation of excitations around the BMN vacuum and theperturbative $S$-matrix associated to them. We are able to compute and sum allthe different loop contributions to the $S$-matrix in this limit.

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