  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.
