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TURKISH JOURNAL OF PHYSICS

Consistency of the Born Approximation for Aharonov-Bohm Scattering with Massless spin-1/2 Particles

Müge Boz, Namik Kemal Pak

Published Oct 25, 2000• 0 citations
OpenAlex

Abstract

The second order contribution for the relativistic scattering of massless spin-\frac{1}{2} particles from an infinitely long solenoid, in the context of covariant perturbation theory, is calculated and shown to vanish. Thus, Born approximation is consistent for this case as well.

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We discuss the scattering of relativistic spin zero particles by an infinitely long and arbitrarily thin solenoid. The exact solution of the first-quantized problem can be obtained as a mimic of the nonrelativistic case, either in the original Aharonov-Bohm way or by using the Berry magnetization scheme. The perturbative treatment is developed in the Feshbach-Villars two-component formalism for the Klein-Gordon equation and it is shown that it also requires renormalization as in the Schrödinger counterpart. The results are compared with those of the field theoretical approach, which corresponds to the two-body sector of the scalar Chern-Simons theory.
The Born scattering amplitude of a non-relativistic spin one-half particle in an Aharonov-Bohm potential is calculated up to second order. It is demonstrated that perturbation theory works well for this model, in contrast with the case of scalar particles, thanks to the spin-magnetic moment interaction term. The first order amplitude is shown to coincide with the exact amplitude when expanded to the same order; the second order amplitude is finite and null. The polarized scattering cross section is found to be different from the unpolarized one only if the incident particle has a spin component perpendicular to the flux tube.
The exact solution to the problem of scattering of electrons by a solenoid of infinite length is studied in two limiting cases; firstly, when the radius r 0 to 0 with the enclosed flux phi held constant, and secondly, when phi to 0 with r 0 held constant. Previously reported discrepancies between the Born approximation and the Aharonov-Bohm solution of this problem are reconciled and conditions for the validity of each of these two approximations are established. The relevance of the results to a related scattering problem involving crystal dislocations is briefly discussed and in this case the Born approximation is shown to be valid over almost all the Fermi surface.
The Aharonov-Bohm effect is reconsidered as a scattering event of an electron by a magnetic field confined in an infinite solenoid of finite radius both in the situation where the solenoid is penetrable as well as impenetrable. We next discuss the validity of the Born approximation for the partial-wave scattering amplitudes and explain why for the cylindrically symmetric (m=0)(m=0)(m=0) partial wave the first Born approximation fails in the long-wavelength limit or as the radius of the solenoid shrinks to zero.