Unitary interaction geometries in few-body systems

We consider few-body systems in which only a certain subset of the particle-particle interactions is resonant. We characterize each subset by a {\it unitary graph} in which the vertices represent distinguishable particles and the edges resonant 2-body interactions. Few-body systems whose unitary graph is connected will collapse unless a repulsive 3-body interaction is included. We find two categories of graphs, distinguished by the kind of 3-body repulsion necessary to stabilize the associated system. Each category is characterized by whether the graph contains a loop or not: for tree-like graphs (graphs containing a loop) the 3-body force renormalizing them is the same as in the 3-body system with two (three) resonant interactions. We show numerically that this conjecture is correct for the 4-body case as well as for a few 5-body configurations. We explain this result in the 4-body sector qualitatively by imposing Bethe-Peierls boundary conditions on the pertinent Faddeev-Yakubovsky~decomposition of the wave function.