A Simple Model for Describing the Minimum Differential Cross-Section of Elastic Proton Scattering on Protons at High Energies

Abstract

Relevance. The most modern and widely applied phenomenological theory, which well describes an entire range of physical characteristics of such processes as elastic and inelastic proton-proton and antiproton-proton scattering at high energies, is the Regge poles method in relativistic theory. Based on a simple amplitude, such as the dipole pomeron, where the pomeron is the Regge pole, the researchers add different terms to it. Using such more complex amplitudes, it is possible to explain well or satisfactorily together not only the experimental data for these reactions obtained at the end of the last century, but also the latest ones obtained at the Large Hadron Collider. Purpose. The purpose of this study is to find numerical values of the amplitude parameters at which the circumferences of the differential cross-sections of elastic proton scattering on protons at high energies are satisfactorily described, and to answer whether the parameters satisfy the obtained constraints. Methods. To find the amplitude parameters, the least squares method is used and minima equations are obtained for the squares of differences between the experimental and theoretical values of these cross-sections. These equations are transcendental, and therefore, they are solved by approximation. Within the framework of successive approximations, a method is selected that ensures fast convergence of the process, namely: the faster descent method or the gradient method. Parameter errors are calculated using a covariance matrix. The statistical acceptability of the model is determined according to the Fischer criterion. Results. Numerical calculations from experimental data of differential cross-sections of elastic pp-scattering were used to find the values of the parameters and the scale multiplier of the amplitude. Differential cross-sections are calculated from the amplitude near the minima. The corresponding graphs of these cross-sections are presented. The coincidence with the experiment is satisfactory in most cases, and in some cases, it is of high quality. A covariance matrix is obtained, from which the errors of the model parameters are found. Under general conditions, restrictions on the found approximate values of parameters are derived. It is shown that they satisfy the constraint. According to the Fischer criterion, the statistical acceptability of the model was verified, which turned out to be positive. Conclusions. The proposed simple amplitude ensures that the Froissard constraint and unitarity are met. It will serve as a seed for constructing more complex amplitudes that will describe a wide range of experiments on proton-proton and antiproton-proton scattering at high energies.