Binding and structure of tetramers in the scaling limit

Abstract

The momentum-space structure of the Faddeev-Yakubovsky (FY) components of weakly bound tetramers is investigated at the unitary limit using a renormalized zero-range two-body interaction. The results, obtained by considering a given trimer level with binding energy ${B}_{3}$, provide further support to a universal scaling function relating the binding energies of two successive tetramer states. The correlated scaling between the tetramer energies comes from the sensitivity of the four-boson system to a short-range four-body scale. Each excited $N$th tetramer energy ${B}_{4}^{(N)}$ moves as the short-range four-body scale changes, while the trimer properties are kept fixed, with the next excited tetramer ${B}_{4}^{(N+1)}$ emerging from the atom-trimer threshold for a universal ratio ${B}_{4}^{(N)}/{B}_{3}={B}_{4}^{(N)}/{B}_{4}^{(N+1)}\ensuremath{\simeq}4.6$, which does not depend on $N$. We show that both channels of the FY decomposition [atom-trimer ($K$ type) and dimer-dimer ($H$ type)] present high-momentum tails that reflect the short-range four-body scale. We also found that the $H$ channel is favored over the $K$ channel at low momentum, when the four-body momentum scale largely overcomes the three-body scale.