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We study a three-body system, formed by two identical heavy bosons and alight particle, in the Born-Oppenheimer approximation for an arbitrarydimension $D$. We restrict $D$ to the interval $2\,<\,D\,<\,4$, and derive theheavy-heavy $D$-dimensional effective potential proportional to $1/R^2$ ($R$ isthe relative distance between the heavy particles), which is responsible forthe Efimov effect. We found that the Efimov states disappear once the criticalstrength 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 correlationbetween the energies of two consecutive Efimov states as a function of $D$ andthe heavy-light binding energy $E^{D}_2$. In addition, we found that the energyof the $(N+1)^{\rm th}$ excited state reaches the two-body continuumindependently 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.

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