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.
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References
E.A. Kuraev and V.S. Fadin, On Radiative Corrections to e+e− Single Photon Annihilation at High-Energy, Sov. J. Nucl. Phys. 41 (1985) 466 [INSPIRE].
V.V. Danilov, P.M. Ivanov, I.A. Koop, I.N. Nesterenko, E.A. Perevedentsev, D.N. Shatilov et al., The concept of round colliding beams in 5th European Particle Accelerator Conference, Sitges, Barcelona, Spain, 10–14 June 1996, pp. 1149 [https://cds.cern.ch/record/860802].
D. Berkaev et al., VEPP-2000 operation with round beams in the energy range from 1-GeV to 2-GeV, Nucl. Phys. B Proc. Suppl. 225-227 (2012) 303 [INSPIRE].
A.N. Tikhonov and V.Y. Arsenin, Solutions of ill-posed problems, V. H. Winston & Sons, Scripta Series in Mathematics, Washington D.C. U.S.A. (1977).
J. Hadamard, Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Franklin Classics Trade Press (2018) [ISBN: 9780344862403].
V. Serov, Fourier series, fourier transform and their applications to mathematical physics 197 Applied Math. Sci. (2017) 33.
P. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM (2010) [DOI].
A. Neumaier, Introduction to numerical analysis, Cambridge University Press (2001) p. 99 [DOI].
R. Angst, The condition of a system of linear equations: Alternative derivation, https://www2.math.ethz.ch/education/bachelor/lectures/hs2014/other/linalg_INFK/matrix-condition-number.pdf.
A.N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Sov. Math. Dokl. 4 (1963) 1035.
P.C. Hansen and D.P. O’Leary, The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems, SIAM J. Sci. Comput. 14 (1993) 1487.
A.N. Tikhonov, A. Goncharsky, V.V. Stepanov and A.G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Math. Appl., Springer Dordrecht (1995) [DOI].
S.I. Kabanikhin, Inverse and Ill-posed Problems Theory and Applications 55 of Inverse and Ill-Posed Problems Series, De Gruyter (2011) [DOI].
J. Dahl, P. Hansen, S. Jensen and T. Jensen, Algorithms and software for total variation image reconstruction via first-order methods, Num. Algorithms 53 (2010) 67.
R.A. Sadek, Svd based image processing applications: State of the art, contributions and research challenges, IJACSA 3 (2012) [arXiv:1211.7102].
T.A. Hearn and L. Reichel, Application of denoising methods to regularizationof ill-posed problems, Num. Algorithms 66 (2014) 761.
T. Tirer and R. Giryes, Image restoration by iterative denoising and backward projections, IEEE Trans. Image Processing 28 (2019) 1220 [arXiv:1710.06647].
L. Ballani, H. Greiner-Mai and D. Stromeyer, Determining the magnetic field in the core-mantle boundary zone by non-harmonic downward continuation, Geophys. J. Int. 149 (2002) 374.
G.P. Deidda, E. Bonomi and C. Manzi, Inversion of electrical conductivity data with tikhonov regularization approach: some considerations, Ann. Geophys. Italy 46 (2003) .
M. Abdelazeem Mohamed, Solving ill-posed magnetic inverse problem using a parameterized trust-region sub-problem, Contrib. Geophys Geodes 43 (2013) 99.
T. Brufati, S. Oliveira and A. Bassrei, Conjugate gradient method for the solution of inverse problems: Application in linear seismic tomography, Trends in Computational and Applied Mathematics 16 (2016) 185.
A. Hocker and V. Kartvelishvili, SVD approach to data unfolding, Nucl. Instrum. Meth. A 372 (1996) 469 [hep-ph/9509307] [INSPIRE].
M. Kuusela, Statistical issues in unfolding methods for high energy physics, MSc Thesis, Aalto University, Espoo Finland (2012) [http://urn.fi/URN:NBN:fi:aalto-201210043224].
F. Spanò, Unfolding in particle physics: a window on solving inverse problems, EPJ Web Conf. 55 (2013) 03002.
M. Kuusela and V. Panaretos, Statistical unfolding of elementary particle spectra: Empirical bayes estimation and bias-corrected uncertainty quantification, Ann. Appl. Stat. 9 (2015) 1671.
M. Kuusela and P.B. Stark, Shape-constrained uncertainty quantification in unfolding steeply falling elementary particle spectra, Ann. Appl. Stat. 11 (2017) 1671.
G. Guennebaud, B. Jacob et al., Eigen v3 (2010) [http://eigen.tuxfamily.org].
S.S. Gribanov et al., Measurement of the e+e− → ηπ+π− cross section with the CMD-3 detector at the VEPP-2000 collider, JHEP 01 (2020) 112 [arXiv:1907.08002] [INSPIRE].
R.R. Akhmetshin et al., Study of ϕ → π+π−π0 with CMD-2 detector, Phys. Lett. B 642 (2006) 203 [INSPIRE].
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Gribanov, S.S., Popov, A.S. A new method for obtaining a Born cross section using visible cross section data from e+e− colliders. J. High Energ. Phys. 2021, 203 (2021). https://doi.org/10.1007/JHEP11(2021)203
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DOI: https://doi.org/10.1007/JHEP11(2021)203