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TMF, 2019, Volume 201, Number 1, Pages 84–104 (Mi tmf9721)  

Approximate formula for the total cross section for a moderately small eikonal function

A. V. Kisselev

Logunov Institute for High Energy Physics, National Research Center "Kurchatov Institute", Protvino, Moscow Oblast, Russia

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 $\chi(b)$ is moderately small, $|\chi(b)|\lesssim0.1$. We show that the total cross section is given by a series of improper integrals of the Born amplitude $A_{\mathrm{B}}$. 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 $\chi(b)$ 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 $A_{\mathrm{B}}$, choosing only Born amplitudes that result in moderately small eikonal functions and lead to the correct asymptotic behavior of $\chi(b)$. 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 $\mathcal{O}(10^{-5})$.

Keywords: eikonal approximation, total cross section, Bessel function, Hankel transform.

DOI: https://doi.org/10.4213/tmf9721

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English version:
Theoretical and Mathematical Physics, 2019, 201:1, 1484–1502

Bibliographic databases:

Received: 14.03.2019
Revised: 12.04.2019

Citation: A. V. Kisselev, “Approximate formula for the total cross section for a moderately small eikonal function”, TMF, 201:1 (2019), 84–104; Theoret. and Math. Phys., 201:1 (2019), 1484–1502

Citation in format AMSBIB
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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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