Four-body Efimov effect in a Born-Oppenheimer model

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 $s$-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 $\lambda$ of the light-heavy interaction which for $\lambda ={\lambda }_c$ 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 $\lambda ={\lambda }^{\prime }$ such that $\left|{\lambda }^{\prime }\mathrm{}\right|<\mathrm{}\left|\mathrm{}{\lambda }_c\right|$, 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 $\left|\lambda \mathrm{}\right|<\mathrm{}\left|\mathrm{}{\lambda }_c\right|$, then as $\lambda \to {\lambda }_c$ a few four-body Efimov states may emerge but their number remains finite. These states disappear for $\left|\lambda \right|\gtrsim \left|{\lambda }_c\right|$ as the three-body cut overrides them.