Higher-order effective interactions for bosons near a two-body zero crossing

We develop the perturbation theory for bosons interacting via a two-body potential $V$ of vanishing mean value. We find that the leading nonpairwise contribution to the energy emerges in the third order in $V$ and represents an effective three-body interaction, the sign of which in most cases (although not in general) is anticorrelated with the sign of the long-range tail of $V$. Explicit results are obtained for a few particular two-body interaction potentials and we perform a detailed perturbative analysis of tilted dipoles in quasi-low-dimensional geometries.