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Theoretical and Mathematical Physics

, Volume 201, Issue 1, pp 1484–1502 | Cite as

Approximate Formula for the Total Cross Section for a Moderately Small Eikonal Function

  • A. V. KisselevEmail author
Article
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Abstract

We study the eikonal approximation of the total cross section for the scattering of two unpolarized particles and obtain an approximate formula in the case where the eikonal function χ(b) is moderately small, |χ(b)| ≲ 0.1. We show that the total cross section is given by a series of improper integrals of the Born amplitude AB. The advantage of this representation compared with standard eikonal formulas is that these integrals contain no rapidly oscillating Bessel functions. We prove two theorems that allow relating the large-b asymptotic behavior of χ(b) to analytic properties of the Born amplitude and give several examples of applying these theorems. To check the effectiveness of the main formula, we use it to calculate the total cross section numerically for a selection of specific expressions for AB, choosing only Born amplitudes that result in moderately small eikonal functions and lead to the correct asymptotic behavior of χ(b). The numerical calculations show that if only the first three nonzero terms in it are taken into account, this formula approximates the total cross section with a relative error of O(10−5).

Keywords

eikonal approximation total cross section Bessel function Hankel transform 

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Notes

Conflicts of interest. The author declares no conflicts of interest.

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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Logunov Institute for High Energy PhysicsNational Research Center “Kurchatov Institute,”Protvino, Moscow OblastRussia

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