Efimov pathology in effective two-body system

Heinrich Kroeger, R. Perne

Published Jan 1, 1980• 0 citations

Abstract

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)

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Efimov pathology in effective two-body system

Heinrich Kroeger, R. Perne

Published Jan 1, 1980• 0 citations

OpenAlex

Abstract

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a

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.

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

s

-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.