Connection between nonlocal one-body and local three-body correlations of the Lieb-Liniger model

We derive a connection between the fourth coefficient of the short-distance Taylor expansion of the one-body correlation function, and the local three-body correlation function of the Lieb-Liniger model of δ-interacting spinless bosons in one dimension. This connection, valid at arbitrary interaction strength, involves the fourth moment of the density of quasimomenta. Generalizing recent conjectures, we propose approximate analytical expressions for the fourth coefficient covering the whole range of repulsive interactions, validated by comparison with accurate numerics. In particular, we find that the fourth coefficient changes sign at interaction strength γc≃3.816, while the first three coefficients of the Taylor expansion of the one-body correlation function retain the same sign throughout the whole range of interaction strengths.