We use the functional renormalisation group to study the spectrum of three- and four-body states in bosonic systems around the unitary limit. Our effective action includes all energy-independent contact interactions in the four-atom sector and we introduce a running trimer field to eliminate couplings that involve the atom-atom-dimer channel. The results show qualitatively similar behaviour to those from exact approaches. The truncated action we use leads to overbinding of the two four-body states seen in those treatments. It also generates a third state, although only for a very narrow range of two-body scattering lengths.
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