The complex sinh-Gordon model: form factors of descendant operators and current-current perturbations

We study quasilocal operators in the quantum complex sinh-Gordon theory in the form factor approach. The free field procedure for descendant operators is developed by introducing the algebra of screening currents and related algebraic objects. We work out null vector equations in the space of operators. Further we apply the proposed algebraic structures to constructing form factors of the conserved currents $T_k$ and $\Theta_k$. We propose also form factors of current-current operators of the form $T_kT_{-l}$. Explicit computations of the four-particle form factors allow us to verify the recent conjecture of Smirnov and Zamolodchikov about the structure of the exact scattering matrix of an integrable theory perturbed by a combination of irrelevant operators. Our calculations confirm that such perturbations of the complex sinh-Gordon model and of the $\mathbb Z_N$ symmetric Ising models result in extra CDD factors in the $S$ matrix.