Efimov effect in a $D$-dimensional Born-Oppenheimer approach

We study a three-body system, formed by two identical heavy bosons and a light particle, in the Born-Oppenheimer approximation for an arbitrary dimension $D$. We restrict $D$ to the interval $2\,<\,D\,<\,4$, and derive the heavy-heavy $D$-dimensional effective potential proportional to $1/R^2$ ($R$ is the relative distance between the heavy particles), which is responsible for the Efimov effect. We found that the Efimov states disappear once the critical strength of the heavy-heavy effective potential $1/R^2$ approaches the limit $-(D-2)^2/4$. We obtained the scaling function for the $^{133}$Cs-$^{133}$Cs-$^6$Li system as the limit cycle of the correlation between the energies of two consecutive Efimov states as a function of $D$ and the heavy-light binding energy $E^{D}_2$. In addition, we found that the energy of the $(N+1)^{\rm th}$ excited state reaches the two-body continuum independently of the dimension $D$ when $\sqrt{E^{D}_2/E_3^{(N)}}=0.89$, where $E_3^{(N)}$ is the $N^{\rm th}$ excited three-body binding energy.