A new method for obtaining a Born cross section using visible cross section data from $e^+e^-$ colliders

TL;DR

This paper introduces a straightforward, model-independent method to extract the Born cross section from visible cross section data in $e^+e^-$ colliders, leveraging the properties of the integral kernel under specific conditions.

Contribution

It demonstrates that a direct numerical solution to the integral equation is feasible and effective when the beam energy spread and uncertainties are small, simplifying the problem without regularization.

Findings

The naive method yields accurate results under small energy spread conditions.

The integral kernel's properties ensure a well-conditioned linear system.

The approach allows easy computation of the covariance matrix.

Abstract

In this article, we propose a new method for obtaining a Born cross section using visible cross section data. It is assumed that the initial state radiation is taken into account in a visible cross section, while in a Born cross section this effect is ommited. Since the equation that connects Born and visible cross sections is an integral equation of the first kind, the problem of finding its numerical solution is ill-posed. Various regularization-based approaches are often used to solve ill-posed problems, since direct methods usually do not lead to an acceptable result. However, in this article it is shown that a direct method can be successfully used to numerically solve the considered equation under the condition of a small beam energy spread and uncertainty. This naive method is based on finding a numerical solution to the integral equation by reducing it to a system of linear equations. The naive method works well because the kernel of the integral operator is a rapidly decreasing function of the variable $x$. This property of the kernel leads to the fact that the condition number of the matrix of the system of linear equations is of the order of unity, which makes it possible to neglect the ill-posedness of the problem when the above condition is satisfied. The advantages of the naive method are its model independence and the possibility of obtaining the covariance matrix of a Born cross section in a simple way.