  The Aharonov-Bohm elastic scattering with incident particles described by
plane waves is revisited by using the phase-shifts method. The formal
equivalence between the cylindrical Schr\"odinger equation and the
one-dimensional Calogero problem allows us to show that up to two scattering
phase-shifts modes in the cylindrical waves expansion must be renormalized. The
renormalization procedure introduces new length scales giving rise to
spontaneous breaking of the conformal symmetry. The new renormalized
cross-section has an amazing property of being non-vanishing even for a
quantized magnetic flux, coinciding with the case of Dirac delta function
potential. The knowledge of the exact beta function permits us to describe the
renormalization group flows within the two-parametric family of renormalized
Aharonov-Bohm scattering amplitudes. Our analysis demonstrates that for
quantized magnetic fluxes a BKT-like phase transition at the coupling space
occurs.
