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

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 $\mathcal{O}(10^{-5})$.