  A thorough analysis is presented of the class of central fields of force that
exhibit: (i) dimensional transmutation and (ii) rotational invariance. Using
dimensional regularization, the two-dimensional delta-function potential and
the $D$-dimensional inverse square potential are studied. In particular, the
following features are analyzed: the existence of a critical coupling, the
boundary condition at the origin, the relationship between the bound-state and
scattering sectors, and the similarities displayed by both potentials. It is
found that, for rotationally symmetric scale-invariant potentials, there is a
strong-coupling regime, for which quantum-mechanical breaking of symmetry takes
place, with the appearance of a unique bound state as well as of a logarithmic
energy dependence of the scattering with respect to the energy.
