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Physical Review C

Efimov effect in the four-body case

H. Kröger, R. Perne

Published Jul 1, 1980• 17 citations
Publisher

Abstract

Introducing a suitable cutoff parameter ${r}_{0}$ for the effective potential in the three-body subsystem, we show that the number of four-body bound states is roughly $k\mathrm{ln}[\ensuremath{\epsilon}a({r}_{0})]$, where $\ensuremath{\epsilon}$ is a momentum cutoff and $\frac{2}{3\ensuremath{\pi}}\ensuremath{\le}k\ensuremath{\le}\frac{1}{3}$, if the scattering length $a({r}_{0})$ for a two-body bound state and a third particle tends to infinity. In a numerical calculation we found that the value ${r}_{0}\ensuremath{\approx}1.82$ fm, which provides a sufficiently resonant effective interaction in the three-body subsystem, is independent of the applied two-body potentials within the numerical accuracy.NUCLEAR REACTIONS Efimov effect.

DOI
10.1103/physrevc.22.21

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Introducing a suitable cutoff parameter r0{r}_{0}r0​ for the effective potential in the three-body subsystem, we show that the number of four-body bound states is roughly kln[\ensuremathϵa(r0)]k\mathrm{ln}[\ensuremath{\epsilon}a({r}_{0})]kln[\ensuremathϵa(r0​)], where \ensuremathϵ\ensuremath{\epsilon}\ensuremathϵ is a momentum cutoff and 23\ensuremathπ\ensuremath≤k\ensuremath≤13\frac{2}{3\ensuremath{\pi}}\ensuremath{\le}k\ensuremath{\le}\frac{1}{3}3\ensuremathπ2​\ensuremath≤k\ensuremath≤31​, if the scattering length a(r0)a({r}_{0})a(r0​) for a two-body bound state and a third particle tends to infinity. In a numerical calculation we found that the value r0\ensuremath≈1.82{r}_{0}\ensuremath{\approx}1.82r0​\ensuremath≈1.82 fm, which provides a sufficiently resonant effective interaction in the three-body subsystem, is independent of the applied two-body potentials within the numerical accuracy.NUCLEAR REACTIONS Efimov effect.
By studying the eigenvalue spectrum of the Faddeev kernel in a certain singular limit, we give an independent proof of an effect recently deduced by Efimov: When three identical particles interact via short-range pairwise potentials, the number of three-body bound states grows without limit when the pairwise scattering length aaa becomes large. [The number of bound states is then roughly $(\frac{1}{\ensuremath{\pi}})\mathrm{ln}(\ensuremath{\Lambda}|a|)$, where $\ensuremath{\Lambda}$ is a momentum cutoff]. We extend our proof to the case where only two particles are identical and show that Efimov's effect persists in the special limiting cases with two heavy and one light particle, and with two light and one heavy particle.
The possibility of a zero-energy Efimov effect is investigated in a model consisting of three identical heavy particles and a lighter one, when the light-heavy interaction leads to a zero-energy bound state for the two-heavy---one-light subsystem. The model is solved in the Born-Oppenheimer approximation with the light-heavy interaction taken to be a separable sss-wave potential of Yamaguchi form. The heavy-heavy interaction is short range and, if attractive, is to be taken weak enough to support no two- or three-heavy-particle bound states. The relevant parameter is the potential strength $\ensuremath{\lambda}$ of the light-heavy interaction which for $\ensuremath{\lambda}={\ensuremath{\lambda}}_{c}$ supports a single zero-energy light-heavy bound state. If the first zero-energy bound state of the two-heavy---one-light subsystem occurs for $\ensuremath{\lambda}={\ensuremath{\lambda}}^{\ensuremath{'}}$ such that $|{\ensuremath{\lambda}}^{\ensuremath{'}}|<|{\ensuremath{\lambda}}_{c}|$, there are no Efimov four-body bound states. On the contrary, if the chosen heavy-heavy potential is repulsive enough to prevent the existence of three-body bound states for $|\ensuremath{\lambda}|<|{\ensuremath{\lambda}}_{c}|$, then as $\ensuremath{\lambda}\ensuremath{\rightarrow}{\ensuremath{\lambda}}_{c}$ a few four-body Efimov states may emerge but their number remains finite. These states disappear for $|\ensuremath{\lambda}|\ensuremath{\gtrsim}|{\ensuremath{\lambda}}_{c}|$ as the three-body cut overrides them.
Efimov physics relates to 3-body systems with large 2-body scattering lengths a and small effective ranges r. For many systems in nature the assumption of a small effective range is not valid. The present report shows binding energies E of three identical bosons calculated with 2-body potentials that are fitted to scattering data and momentum cut-offs (L) by inverse scattering. Results agree with previous works in the case of r<0 when L=~10/r. With 1/a=0 the converged energies are given by E(n) =C(n)/r*r with n labeling the energy-branch and calculated values C(0)=0.77, C(1)=.0028. This gives a ratio ~278 thus differing from the value ~515 in the Efimov case. Efimov's angular dependent function is calculated. Good agreement with previous works is obtained when r<< a. With the increased values of effective range the shallow states still appear Efimov-like. For deeper states the angular dependence differs but is independent of the effective range.
Armed with a new solution to the four-body problem, we reanalyze the recent Innsbruck observation of an Efimov resonance in three-body collisions of cesium atoms, and show that it provides evidence of new, universal four-body physics as well. Our theoretical treatment of four bosonic atoms in the limit of strong two-body attraction suggests a ubiquitous result: namely, that resonantly-bound states of all four atoms should occur whenever the two-body scattering length is a specific fraction (0.43) of the value where a three-body Efimov resonance occurs. A hitherto unanalyzed feature in the 2006 Innsbruck experiment supports this universal prediction, and it provides the first known evidence to date that a direct four-body recombination process plays a role. While we confirm the conclusion reached previously by some studies that no true Efimov effect exists for four particles, we demonstrate the existence of a universal class of four-body states that are intimately connected with three-body Efimov states.