Scattering in integrable pp-wave backgrounds: S-matrix and absence of particle production

Particle production in integrable field theories may exist depending on the vacuum around which excitations are defined. To tackle this and analogous issues with conventional field theoretical tools, we consider the integrable $\lambda$-deformed model for $SU(2)$ together with a timelike coordinate. We construct the corresponding four-dimensional plane wave background keeping also post-plane wave corrections, as well as all the non-trivial $\lambda$-dependence. After imposing the light-cone gauge and the Virasoro constraints, we obtain an interacting field theory for the transverse physical modes which are massive. We explicitly demonstrate the absence of particle production to leading order in the large $k$-expansion. This is based crucially on the form of the interaction vertices and their dependence on the $\lambda$-deformation parameter. In addition, we compute the $S$-matrix for the two-particle elastic scattering exactly in $\lambda$ and to leading order in the large $k$-expansion. Our method can be applied to any integrable theory with at least one isometry.