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

Abstract In this paper, 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...

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Autores principales:	S. S. Gribanov, A. S. Popov
Formato:	article
Lenguaje:	EN
Publicado:	SpringerOpen 2021
Materias:	e+-e- Experiments Nuclear and particle physics. Atomic energy. Radioactivity QC770-798
Acceso en línea:	https://doaj.org/article/2747bf363c9848609b0362d9e747173d
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spelling	oai:doaj.org-article:2747bf363c9848609b0362d9e747173d2021-12-05T12:25:09ZA new method for obtaining a Born cross section using visible cross section data from e + e − colliders10.1007/JHEP11(2021)2031029-8479https://doaj.org/article/2747bf363c9848609b0362d9e747173d2021-11-01T00:00:00Zhttps://doi.org/10.1007/JHEP11(2021)203https://doaj.org/toc/1029-8479Abstract In this paper, 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 paper 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. It should be noted that there are already a number of methods for obtaining a Born cross section using visible cross section data, which are commonly used in e + e − experiments. However, at least some of these methods have various disadvantages, such as model dependence and relative complexity of obtaining a Born cross section covariance matrix. It should be noted that this paper focuses on the naive method, while conventional methods are hardly covered. The paper also discusses solving the problem using the Tikhonov regularization, so that the reader can better understand the difference between regularized and non-regularized solutions. However, it should be noted that, in contrast to the naive method, regularization methods can hardly be used for precise obtaining of a Born cross section. The reason is that the regularized solution is biased and the covariance matrix of this solution do not represent the correct covariance matrix of a Born cross section.S. S. GribanovA. S. PopovSpringerOpenarticlee+-e- ExperimentsNuclear and particle physics. Atomic energy. RadioactivityQC770-798ENJournal of High Energy Physics, Vol 2021, Iss 11, Pp 1-36 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	e+-e- Experiments Nuclear and particle physics. Atomic energy. Radioactivity QC770-798
spellingShingle	e+-e- Experiments Nuclear and particle physics. Atomic energy. Radioactivity QC770-798 S. S. Gribanov A. S. Popov A new method for obtaining a Born cross section using visible cross section data from e + e − colliders
description	Abstract In this paper, 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 paper 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. It should be noted that there are already a number of methods for obtaining a Born cross section using visible cross section data, which are commonly used in e + e − experiments. However, at least some of these methods have various disadvantages, such as model dependence and relative complexity of obtaining a Born cross section covariance matrix. It should be noted that this paper focuses on the naive method, while conventional methods are hardly covered. The paper also discusses solving the problem using the Tikhonov regularization, so that the reader can better understand the difference between regularized and non-regularized solutions. However, it should be noted that, in contrast to the naive method, regularization methods can hardly be used for precise obtaining of a Born cross section. The reason is that the regularized solution is biased and the covariance matrix of this solution do not represent the correct covariance matrix of a Born cross section.
format	article
author	S. S. Gribanov A. S. Popov
author_facet	S. S. Gribanov A. S. Popov
author_sort	S. S. Gribanov
title	A new method for obtaining a Born cross section using visible cross section data from e + e − colliders
title_short	A new method for obtaining a Born cross section using visible cross section data from e + e − colliders
title_full	A new method for obtaining a Born cross section using visible cross section data from e + e − colliders
title_fullStr	A new method for obtaining a Born cross section using visible cross section data from e + e − colliders
title_full_unstemmed	A new method for obtaining a Born cross section using visible cross section data from e + e − colliders
title_sort	new method for obtaining a born cross section using visible cross section data from e + e − colliders
publisher	SpringerOpen
publishDate	2021
url	https://doaj.org/article/2747bf363c9848609b0362d9e747173d
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A new method for obtaining a Born cross section using visible cross section data from e + e − colliders

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