Binding and structure of tetramers in the scaling limit

TL;DR

This paper investigates the universal properties and structure of weakly-bound tetramers at the unitary limit, revealing a scaling function relating successive tetramer energies and the influence of short-range four-body scales.

Contribution

It demonstrates a universal scaling relation for tetramer energies and analyzes the momentum-space structure of FY components at the unitary limit using a renormalized zero-range interaction.

Findings

Tetramer energies follow a universal ratio of approximately 4.6 relative to the trimer energy.

Both FY channels exhibit high momentum tails indicating short-range four-body effects.

The H-channel dominates at low momentum when the four-body scale exceeds the three-body scale.

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 $B_4^{(N)}/B_3 = B_4^ {(N)}/B_4^{(N+1)} \simeq 4.6$, 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, which reflect the short-range four-body scale. We also found that the $H-$channel is favored over $K-$channel at low momentum when the four-body momentum scale largely overcomes the three-body one.