In Greiner's Classical Mechanics, vol. 2 (Springer 2010, p. 59), he mentions that differential cross-section in Rutherford scattering diverges as scattering angle goes to 0, i.e. when impact parameter goes to zero. Then he writes: "This is due to long-range nature of the Coulomb force. If one uses potentials which decrease faster than 1/r, this singularity disappears." I figure this singularity is the reason why the total cross-section in Rutherford scattering in Coulomb field is infinite. So, if, according to this text, the singularity is weakened by other type of potential (e.g. Yukawa), then the total cross-section might become finite?
I do not understand, however, how it can be finite. As long as the potential only goes asymptotically to zero (and never becomes zero), the total cross-section should be infinite (since the particle will feel the effect of the target at any distance and will be deflected), even if the potential decays very fast with r.
