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Physical Review C

Effective interaction of three resonantly interacting particles and the force range

V.N. Efimov

Published May 1, 1993• 58 citations
Publisher

Abstract

The effect of the force range ${\mathit{r}}_{0}$ on an effective long-range interaction in a system of three resonantly interacting particles is considered. By employing a boundary-condition approach, it is explicitly shown that the account of the force range gives rise to a correction ${\mathit{r}}_{0}$/${\mathit{R}}^{3}$ in the effective long-range interaction. The correction is universal (i.e., independent of detail structure of the force), and is attractive for symmetric ${0}^{+}$ three-particle states. The effect of the correction on the well-known correlation between the three-particle binding energy and the particle-pair scattering length is calculated.

DOI
10.1103/physrevc.47.1876

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The effect of the force range r0{\mathit{r}}_{0}r0​ on an effective long-range interaction in a system of three resonantly interacting particles is considered. By employing a boundary-condition approach, it is explicitly shown that the account of the force range gives rise to a correction r0{\mathit{r}}_{0}r0​/R3{\mathit{R}}^{3}R3 in the effective long-range interaction. The correction is universal (i.e., independent of detail structure of the force), and is attractive for symmetric 0+{0}^{+}0+ three-particle states. The effect of the correction on the well-known correlation between the three-particle binding energy and the particle-pair scattering length is calculated.
The scattering and bound states of a system of three resonantly interacting particles are analyzed. The characteristics of the scattering and the bound states are shown to be independent of the particular nature of the two-particle forces; instead, these characteristics are governed simply by the fact that resonances occur. The resonances lead to an effective long-range interaction among the three particles. The physical characteristics of the system obey a scaling law. Explicit equations are derived for the particle pair scattering length, for the particle pair scattering cross section at and above the threshold for disintegration of the three-particle system, and for the ternary-collision amplitude at zero energy. The properties of the system are compared with those of a system of two resonantly interacting particles.
A nonrelativistic system of three identical particles will display a rich set of universal features known as Efimov physics if the scattering length aaa is much larger than the range lll of the underlying two-body interaction. An appropriate effective theory facilitates the derivation of both results in the $\ensuremath{\mid}a\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ limit and finite-l∕al∕al∕a corrections to observables of interest. Here we use such an effective-theory treatment to consider the impact of corrections linear in the two-body effective range, rs{r}_{s}rs​, on the three-boson bound-state spectrum and recombination rate for $\ensuremath{\mid}a\ensuremath{\mid}⪢\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$. We do this by first deriving results appropriate to the strict limit $\ensuremath{\mid}a\ensuremath{\mid}\ensuremath{\rightarrow}\ensuremath{\infty}$ in coordinate space. We then extend these results to finite aaa using once-subtracted momentum-space integral equations. We take the cutoff on these equations to be large compared to $1∕\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$, and find that the first-order effects of $\ensuremath{\mid}{r}_{s}\ensuremath{\mid}$ are independent of the cutoff in this regime. We also discuss the implications of our results for experiments that probe three-body recombination in Bose-Einstein condensates near a Feshbach resonance.
We consider a system of three identical bosons near a Feshbach resonance in the universal regime with large scattering length usually described by model-independent zero-range potentials. We employ the adiabatic hyperspherical approximation and derive the rigorous large-distance equation for the adiabatic potential for finite-range interactions. The effective range correction to the zero-range approximation must be supplemented by a new term of the same order. The nonadiabatic term can be decisive. Efimov physics is always confined to the range between effective range and scattering length. The analytical results agree with numerical calculations for realistic potentials.
We study charged particles in three dimensions interacting via a short-range potential in addition to the Coulomb potential. When the Bohr radius and the scattering length are much larger than the potential range, low-energy physics of the system becomes independent from details of the short-range potential. We develop the zero-range theory to describe such universal physics in terms of the Bohr radius and the scattering length by generalizing the Bethe-Peierls boundary condition, which is then applied to two charged particles to reveal their bound states and resonances. Infinite resonances are found for a repulsive Coulomb potential, one of which turns into a bound state with increasing inverse scattering length, whereas infinite bound states exist for an attractive Coulomb potential with no resonances at any scattering length. The zero-range theory is also applied to three equally charged particles at infinite scattering length under the variational Born-Oppenheimer approximation. We find that the effective potential between two heavy particles induced by a light particle is an inverse-square attraction at distances shorter than the Bohr radius, leading to infinite deep bound states, whereas shallow ones successively turn into resonances with increasing Coulomb repulsion.
We discuss the impact of a finite effective range on three-body systems interacting through a large two-body scattering length. By employing a perturbative analysis in an effective field theory well suited to this scale hierarchy we find that an additional three-body parameter is required for consistent renormalization once range corrections are considered. This allows us to extend previously discussed universal relations between different observables in the recombination of cold atoms to account for the presence of a finite effective range. We show that such range corrections allow us to simultaneously describe the positive and negative scattering-length loss features observed in recombination with 7 Li atoms by the Bar-Ilan group. They do not, however, significantly reduce the disagreement between the universal relations and the data of the Rice group on 7 Li recombination at positive and negative scattering lengths.