We consider the nonrelativistic four-boson system with short-range forces and large scattering length in an effective quantum mechanics approach. We construct the effective interaction potential at leading order in the large scattering length and compute the four-body binding energies using the Yakubovsky equations. Cutoff independence of the four-body binding energies does not require the introduction of a four-body force. This suggests that two- and three-body interactions are sufficient to renormalize the four-body system. We apply the equations to $^{4}\mathrm{H}\mathrm{e}$ 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.We consider the nonrelativistic four-boson system with short-range forces and large scattering length in an effective quantum mechanics approach. We construct the effective interaction potential at leading order in the large scattering length and compute the four-body binding energies using the Yakubovsky equations. Cutoff independence of the four-body binding energies does not require the introduction of a four-body force. This suggests that two- and three-body interactions are sufficient to renormalize the four-body system. We apply the equations to $^{4}\mathrm{H}\mathrm{e}$ 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.

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

The three-body problem with short-range interactions

E Nielsen

Physics ReportsJun 1, 2001482 citations

Effective theories of scattering with an attractive inverse-square potential and the three-body problem

Thomas Barford, Michael C. Birse

A distorted-wave version of the renormalization group is applied to scattering by an inverse-square potential and to three-body systems. In attractive three-body systems, the short-distance wavefunction satisfies a Schrödinger equation with an attractive inverse-square potential, as shown by Efimov. The resulting oscillatory behaviour controls the renormalization of the three-body interactions, with the renormalization-group flow tending to a limit cycle as the cut-off is lowered. The approach used here leads to single-valued potentials with discontinuities as the bound states are cut off. The perturbations around the cycle start with a marginal term whose effect is simply to change the phase of the short-distance oscillations, or the self-adjoint extension of the singular Hamiltonian. The full power counting in terms of the energy and two-body scattering length is constructed for short-range three-body forces.A distorted-wave version of the renormalization group is applied to scattering by an inverse-square potential and to three-body systems. In attractive three-body systems, the short-distance wavefunction satisfies a Schrödinger equation with an attractive inverse-square potential, as shown by Efimov. The resulting oscillatory behaviour controls the renormalization of the three-body interactions, with the renormalization-group flow tending to a limit cycle as the cut-off is lowered. The approach used here leads to single-valued potentials with discontinuities as the bound states are cut off. The perturbations around the cycle start with a marginal term whose effect is simply to change the phase of the short-distance oscillations, or the self-adjoint extension of the singular Hamiltonian. The full power counting in terms of the energy and two-body scattering length is constructed for short-range three-body forces.

arXiv:nucl-th/0406008Journal of Physics A Mathematical and GeneralDec 24, 200448 citations

A cluster expansion applicable to the three-body problem

A. Osman

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoNov 1, 19850 citations

A model study of finite range three-body interactions

Rutgers University

Annals of PhysicsMay 1, 19900 citations

Effective Three-Boson Interactions using a Separable Potential

Corinne Beckers, Jacques Tempere, Jeff Maki +1 more

Effective field theories (EFTs) are widely used to study many-body systems by describing two-body interactions using zero-ranged contact potentials. However, when extended to three-body processes, these contact interactions lead to divergences due to the absence of an intrinsic length scale. In EFT, this is typically resolved by introducing a zero-ranged three-body interaction, which can be renormalized to make the low-energy physics independent of the short-distance physics. However, when the two-body potential has a finite range, such as in separable potentials, there is no need for such renormalization. In this work, we derive the integral equation for the three-body scattering amplitude for separable potentials, and solve it in the strongly-interacting regime. With our model, we retrieve the known analytic form of the scattering amplitude for inelastic scattering processes and formulate a new scaling law for elastic three-body scattering processes.Effective field theories (EFTs) are widely used to study many-body systems by describing two-body interactions using zero-ranged contact potentials. However, when extended to three-body processes, these contact interactions lead to divergences due to the absence of an intrinsic length scale. In EFT, this is typically resolved by introducing a zero-ranged three-body interaction, which can be renormalized to make the low-energy physics independent of the short-distance physics. However, when the two-body potential has a finite range, such as in separable potentials, there is no need for such renormalization. In this work, we derive the integral equation for the three-body scattering amplitude for separable potentials, and solve it in the strongly-interacting regime. With our model, we retrieve the known analytic form of the scattering amplitude for inelastic scattering processes and formulate a new scaling law for elastic three-body scattering processes.

arXiv:2602.24226Feb 27, 20260 citations

^{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.