The twisted story of world-sheet scattering in $η$-deformed AdS$_\mathsf{5} \, \times \,{}$S$^\mathsf{5}$

We study the world-sheet scattering theory of the $\eta$ deformation of the AdS$_\mathsf{5} \, \times \,{}$S$^\mathsf{5}$ superstring corresponding to the purely fermionic Dynkin diagram. This theory is a Weyl-invariant integrable deformation of the AdS$_\mathsf{5} \, \times \,{}$S$^\mathsf{5}$ superstring, with trigonometric quantum deformed symmetry. We compute the two-body world-sheet S matrix of this string in the light-cone gauge at tree level to quadratic order in fermions. The result factorizes into two elementary blocks, and solves the classical Yang-Baxter equation. We also determine the corresponding exact factorized S matrix, and show that its perturbative expansion matches our tree level results, once we correctly identify the deformed light-cone symmetry algebra of the string. Finally, we briefly revisit the computation of the corresponding S matrix for the $\eta$ deformation based on the distinguished Dynkin diagram, finding a tree level S matrix that factorizes and solves the classical Yang-Baxter equation, in contrast to previous results.