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Classical limit for the scattering of Dirac particles in a magnetic field

We present a relativistic quantum calculation at first order in perturbation theory of the differential cross section for a Dirac particle scattered by a solenoidal magnetic field. The resulting cross section is symmetric in the scattering angle as those obtained by Aharonov and Bohm (AB) in the string limit and by Landau and Lifshitz (LL) for the non relativistic case. We show that taking pr_0\|sin(θ/2)|/\hbar<<1 in our expression of the differential cross section it reduces to the one reported by AB, and if additionally we assume θ<< 1 our result becomes the one obtained by LL. However, these limits are explicitly singular in \hbar as opposed to our initial result. We analyze the singular behavior in \hbar and show that the perturbative Planck's limit (\hbar -> 0) is consistent, contrarily to those of the AB and LL expressions. We also discuss the scattering in a uniform and constant magnetic field, which resembles some features of QCD.