Relativistic scalar Aharonov-Bohm scattering

Full text follows: We study the scattering of a charged spin zero relativistic particle from a fixed, thin, infinitely long solenoid in the framework of the first quantization comparing with the results obtained using the field theory approach. The scattering amplitude, within the viewpoint of the relativistic quantum mechanics, can be calculated exactly as a mimic of the nonrelativistic case, either in the original Aharonov-Bohm way or by using the Berry's magnetization scheme. To implement the perturbative analysis within the first-quantized treatment, we consider the Feshbach-Villars two-component formalism of the Klein-Gordon equation. It is shown that the first Born approximation gives an incomplete result while the second order is divergent as occurs in the nonrelativistic counterpart. We search for additional interactions which might provide the appropriated renormalization of the amplitude. It is shown that the addition of a phenomenological magnetic interaction (naively inspired in the spin half case) generates the correct first order term in the the nonrelativistic leading order but it does not give reasonable results in higher orders. We demonstrate that an external delta potential is necessary to reproduce the correct perturbation expansion as happens in the nonrelativistic situation. However, it does not produce the same effect of the quartic self-interaction in the second quantized treatment, which corresponds to the two body sector of a scalar Chern-Simons theory. In the later case, additional contributions coming from vacuum polarization and vertex corrections spoil the scale invariance characteristic of the nonrelativistic Aharonov-Bohm scattering. (author)