arXiv:2606.00854

Three- and four-boson systems expanded around the unitarity limit: Application to $^4$He molecules

Feng Wu, Xincheng Lin, U. van Kolck, S. König

Published May 30, 2026• 0 citations

Abstract

The three- and four-boson systems with a large scattering length and a short effective range in the two-body sector are studied in the framework of Short-Range Effective Field Theory (SREFT). The starting point (leading order) of the EFT is taken to be the universal unitarity limit, where the two-body sector is parameter-free and only one three-body parameter enters. In this limit, physical systems manifest discrete scale invariance. Deviations from universality arising from finite scattering-length and effective-range corrections, as well as a four-body force required by renormalization, are included perturbatively at next-to-leading order. The three-body ground state and its associated four-body ground and first-excited states are studied using the Faddeev-Yakubovsky (FY) formalism and a complementary diagrammatic approach. By employing techniques to remove contributions from deep trimers in tetramer calculations, we extend our analysis to larger cutoffs than previously accessible within the FY approach to SREFT. Our results for binding energies and radii of $^4$He three- and four-atom systems converge well to results obtained with sophisticated phenomenological potentials. These successes suggest that the physics of $^4$He atomic clusters is governed by only small deviations from discrete scale invariance.

DOI: 10.48550/arxiv.2606.00854

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arXiv:2606.00854

Three- and four-boson systems expanded around the unitarity limit: Application to $^4$ He moleculesThree- and four-boson systems expanded around the unitarity limit: Application to $^4$He molecules

Feng Wu, Xincheng Lin, U. van Kolck, S. König

Published May 30, 2026• 0 citations

arXiv HTML PDF

Abstract

^4

He three- and four-atom systems converge well to results obtained with sophisticated phenomenological potentials. These successes suggest that the physics of

^4

He atomic clusters is governed by only small deviations from discrete scale invariance.

DOI: 10.48550/arxiv.2606.00854

Related papers

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arXiv:cond-mat/0404313Physical Review ANov 5, 2004201 citations

Effective field theory for few-boson systems

Betzalel Bazak, Moti Eliyahu, U. van Kolck

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 $^{4}\mathrm{He}$ 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.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 $^{4}\mathrm{He}$ 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.

arXiv:1607.01509Physical Review ANov 1, 201647 citations

Three- and four-boson systems expanded around the unitarity limit: Application to

^4

He molecules

^4

He three- and four-atom systems converge well to results obtained with sophisticated phenomenological potentials. These successes suggest that the physics of

^4

He atomic clusters is governed by only small deviations from discrete scale invariance.

We develop an effective field theory (EFT) for strongly interacting bosonic clusters, using

^4

He as a paradigmatic example of universality in systems with large scattering length. At leading order (LO), two- and three-body zero-range interactions are entirely determined by the dimer and trimer ground-state energies. We show that ground-state energies for up to

N=15

particles converge to cutoff-independent limits with extrapolation coefficients of natural size. At next-to-leading order (NLO), corrections stemming from the two-body interaction range and a four-body force, calibrated to the tetramer ground-state energy, reduce cutoff sensitivity. Close agreement with results from a realistic potential is found at LO and improved at NLO, demonstrating systematic convergence with few parameters at each order. The resulting EFT is directly applicable to larger clusters and bulk helium.

Small Clusters of

^4

He Atoms in Finite-Cutoff Effective Field Theory

Small clusters of

^4

He atoms are benchmark systems for universal few-body physics near the unitary limit. We study these systems using a finite-cutoff effective field theory calibrated to low-energy observables from the realistic LM2M2 potential. The chosen two-body cutoff reproduces both the atom--atom scattering length and effective range, while a regulated three-body interaction is adjusted to the trimer and tetramer ground-state energies. We distinguish total binding energies from the one-atom separation energies

S_A^*=B_A^*-B_{A-1}

of the shallow excited states. Ground-state energies through

A=8

are reproduced at the few-percent level, but threshold-sensitive observables are less accurate. In particular, the calculated tetramer separation energy is

S_4^*=2.85

~mK, compared with

0.92

0.96

~mK in converged LM2M2 calculations, and the atom--trimer scattering length differs substantially from modern LM2M2 benchmarks. For

A\ge5

, the available excited-state benchmarks are sparse and strongly method dependent. Comparison with another soft-core interaction suggests that the discrepancies primarily reflect short-distance physics and higher-order operators omitted from the finite-cutoff Hamiltonian rather than numerical convergence of the stochastic variational calculation. Finite-cutoff EFT therefore provides an efficient description of ground-state systematics, whereas shallow excited states and atom--cluster scattering require a controlled higher-order and cutoff-dependence analysis.

^{4}\mathrm{He}

atoms, binding energies of 526.1(5) and 128.517(1) mK are obtained at leading order for the tetramer ground and excited states, respectively, where errors come from the truncation of three-body partial waves. Tetramer binding energies and decay widths are also computed approaching the unitary limit. In the unitary limit, there are two tetramers associated with each trimer of binding energy

{B}_{3}^{(0)}

. The binding energy and decay width for the associated tetramer ground state are

{E}_{4}^{(0)}=4.60(1){B}_{3}^{(0)}

and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(0)}/2=0.0160(1){B}_{3}^{(0)}$, respectively, and for the associated tetramer excited state,

{E}_{4}^{(1)}=1.0022(3){B}_{3}^{(0)}

and ${\mathrm{\ensuremath{\Gamma}}}_{4}^{(1)}/2=2.57(2)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}{B}_{3}^{(0)}$, respectively. This calculation is a gateway to higher-order and/or more-body calculations in nuclear and atomic systems.

^{4}\mathrm{H}\mathrm{e}

atoms and calculate the binding energy of the

^{4}\mathrm{H}\mathrm{e}

tetramer. We observe a correlation between the trimer and tetramer binding energies similar to the Tjon line in nuclear physics. Over the range of binding energies relevant to

^{4}\mathrm{H}\mathrm{e}

atoms, the correlation is approximately linear.

^{4}\mathrm{He}

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.