Approximate formula for total cross section for moderately small eikonal function

TL;DR

This paper derives an approximate formula for the total cross section in particle scattering using the eikonal approximation, valid for small eikonal functions, and demonstrates its accuracy through numerical tests.

Contribution

It introduces a new series-based approximation for the total cross section that avoids oscillatory integrals and relates eikonal behavior to the Born amplitude.

Findings

The formula accurately approximates the total cross section with very small error.

The series involves multiple integrals of the Born amplitude without oscillatory Bessel functions.

Numerical tests confirm the formula's effectiveness for specific Born amplitudes.

Abstract

The eikonal approximation for the total cross section for the scattering of two unpolarized particles is studied. The approximate formula in the case when the eikonal function chi(b) is moderately small, |chi(b)| < 0.1, is derived. It is shown that the total cross section is given by the series of multiple improper integrals of the Born amplitude A_B. Its advantage compared to standard eikonal formulas is that the integrals contain no rapidly oscillating Bessel functions. Two theorems which allow one to relate large-b behavior of chi(b) with analytical properties of the Born amplitude are proved. Several examples of these theorems are given. To check the efficiency of the main formula, it is applied for numerical calculations of the total cross section for a number of particular expressions of A_B. Only those Born amplitudes are chosen which result in moderately small eikonal functions and lead to the correct asymptotics of chi(b). The numerical calculations show that our formula approximates the total cross section with the relative error of O(10^(-5)), provided that the first three non-zero terms in it are taken into account.