Local correlations in the 1D Bose gas from a scaling limit of the XXZ chain

We consider the K -body local correlations in the (repulsive) 1D Bose gas for general K , both at finite size and in the thermodynamic limit. Concerning the latter we develop a multiple integral formula which applies for arbitrary states of the system with a smooth distribution of Bethe roots, including the ground state and finite-temperature Gibbs states. In the cases K ≤ 4 we perform the explicit factorization of the multiple integral. In the case of K = 3 we obtain the recent result of Kormos et al , whereas our formula for K = 4 is new. Numerical results are presented as well.