利用$e^+e^-$对撞机可见截面数据获取玻恩截面的新方法
A new method for obtaining a Born cross section using visible cross section data from $e^+e^-$ colliders
摘要：
在这篇文章中，我们提出了一种新的方法来获得玻恩截面使用可见截面数据。假设在可见光截面中考虑了初态辐射，而在玻恩截面中忽略了初态辐射的影响。由于连接玻恩截面和可见截面的方程是第一类积分方程，因此求其数值解的问题是不适定的。各种基于正则化的方法通常用于解决不适定问题，因为直接方法通常不会导致可接受的结果。然而，在这篇文章中，它表明，一个直接的方法可以成功地用于数值求解所考虑的方程的条件下，一个小的光束能量分散和不确定性。这种朴素的方法是基于通过将积分方程简化为线性方程组来找到积分方程的数值解。朴素方法工作得很好，因为积分算子的核是变量$x$的快速递减函数。核的这一性质导致了线性方程组矩阵的条件数是单位阶的事实，这使得当满足上述条件时可以忽略问题的不适定性。朴素方法的优点是它的模型独立性和以简单的方式获得玻恩截面的协方差矩阵的可能性。
英文摘要：
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.
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