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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

There Is No Efimov Effect for Four or More Particles

Ralph D. Amado, F. C. Greenwood

Published Apr 15, 1973• 62 citations
Publisher

Abstract

We show that the existence of a zero-energy ($n\ensuremath{-}1$)-body bound state will not produce an infinite number of $n$-body bound states for $n\ensuremath{\ge}4$. We show this in particular for $n=4$, and sketch the proof for $n>4$.

DOI
10.1103/physrevd.7.2517

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Efimov effect in the four-body case

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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.

Physical Review C•Jul 1, 1980•17 citations
We show that the existence of a zero-energy (n\ensuremath−1n\ensuremath{-}1n\ensuremath−1)-body bound state will not produce an infinite number of nnn-body bound states for n\ensuremath≥4n\ensuremath{\ge}4n\ensuremath≥4. We show this in particular for n=4n=4n=4, and sketch the proof for n>4n>4n>4.
We consider a system of N pairwise interacting particles described by the Hamiltonian H, where σess(H) = [0, ∞) and none of the particle pairs has a zero energy resonance. The pair potentials are allowed to take both signs and obey certain restrictions regarding the fall off. It is proved that if N ⩾ 4 and none of the Hamiltonians corresponding to the subsystems containing N − 2 or less particles has an eigenvalue equal to zero then H has a finite number of negative energy bound states. This result provides a positive proof to a long-standing conjecture of Amado and Greenwood stating that four bosons with an empty negative continuous spectrum have at most a finite number of negative energy bound states. Additionally, we give a short proof to the theorem of Vugal'ter and Zhislin on the finiteness of the discrete spectrum and pose a conjecture regarding the existence of the “true” four-body Efimov effect.
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.
Amado and Greenwood have shown that there is no Efimov effect for four or more particles. The present authors use an (n-1)-fold successive separable approximation (nssa) to the general n-body operator identities. They find that for the conditions considered the necessary condition for the existence of an infinite number of n-body bound levels (n > or = 4) is not violated. The nssa reduces the n-body to an effective two-body phase space. It is concluded that the appearance of the Efimov effect is not restricted to three-body systems. (RWR)
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 $k\mathrm{ln}[\ensuremath{\epsilon}a({r}_{0})]$, 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(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 ${r}_{0}\ensuremath{\approx}1.82$ 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.