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arXiv:2206.06265 · Physical Review A

Subleading contributions to N -boson systems inside the universal window

Paolo Recchia, Alejandro Kievsky, Luca Girlanda, M. Gattobigio

Published Aug 23, 2022• 7 citations
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Abstract

We study bosonic systems in the regime in which the two-body system has a shallow bound state or, equivalently, a large value of the two-body scattering length. Using the effective field theory framework as a guidance, we construct a series of potential terms which have a decreasing importance in the description of the binding energy of the systems. The leading-order potential terms consist in a two-body term, usually attractive, plus a three-body term, usually repulsive; this last term is required to prevent the collapse of systems with more than two particles. At this order, the parametrization of the two-body potential is done to obtain a correct description of the scattering length, which governs the dynamics in this regime, whereas the three-body term fixes a three-body datum. We investigate the role of the cutoff in the leading-order description, and we extend the exploration beyond the leading order by including the next-to-leading-order terms in both the two- and three-body potentials. We use the requirement of the stability of the $N$-body system, whose energy is variationally estimated, to introduce the three-body forces. The potential parametrization, as a function of the cutoff, is fixed to describe the energy of $^{4}\mathrm{He}$ clusters up to seven particles within the expected accuracy. Finally, we also explore the possibility to describe at the same time the atom-dimer scattering length.

DOI
10.1103/physreva.106.022812

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In this work we investigate small clusters of bosons using the hyperspherical harmonic basis. We consider systems with A=2,3,4,5,6A=2,3,4,5,6A=2,3,4,5,6 particles interacting through a soft interparticle potential. In order to make contact with a real system, we use an attractive Gaussian potential that reproduces the values of the dimer binding energy and the atom-atom scattering length obtained with one of the most widely used 4{}^{4}4He-4{}^{4}4He interactions, the LM2M2 potential of Aziz and Slaman. The intensity of the potential is varied in order to explore the clusters' spectra in different regions with large positive and large negative values of the two-body scattering length. In addition, we include a repulsive three-body force to reproduce the trimer binding energy. With this model, consisting in the sum of a two- and three-body potential, we have calculated the spectrum of the four-, five-, and six-particle systems. In all the regions explored, we have found that these systems present two states, one deep and one shallow close to the $A\ensuremath{-}1$ threshold. Some universal relations between the energy levels are extracted; in particular, we have estimated the universal ratios between thresholds of the three-, four-, and five-particle continua using the two-body Gaussian potential. They agree with recent measurements and theoretical predictions.
We study universal bosonic few-body systems within the framework of effective field theory at leading order (LO). We calculate binding energies of systems of up to six particles and the atom-dimer scattering length. Convergence to the limit of zero-range two- and three-body interactions is shown, indicating that no additional few-body interactions need to be introduced at LO. Generalizations of the Tjon line are constructed, showing correlations between few-body binding energies and the binding energy of the trimer, for a given dimer energy. As a specific example, we implement our theory for 4He^{4}\mathrm{He}4He atomic systems and show that the results are in surprisingly good agreement with those of sophisticated $^{4}\mathrm{He}\text{\ensuremath{-}}^{4}\mathrm{He}$ potentials. Potential implications for the convergence of the EFT expansion are discussed.
The role of an intrinsic four-body scale in universal few-boson systems is the subject of active debate. We study these systems within the framework of effective field theory. For systems of up to six bosons we establish that no four-body scale appears at leading order (LO). However, we find that at next-to-leading order (NLO) a four-body force is needed to obtain renormalized results for binding energies. With the associated parameter fixed to the binding energy of the four-boson system, this force is shown to renormalize the five- and six-body systems as well. We present an original ansatz for the short-distance limit of the bosonic A-body wave function from which we conjecture that new A-body scales appear at N^{A-3} LO. As a specific example, calculations are presented for clusters of helium atoms. Our results apply more generally to other few-body systems governed by a large scattering length, such as light nuclei and halo states, the low-energy properties of which are independent of the detailed internal structure of the constituents.