Advances in High Energy Physics

Volume 2015, Article ID 914170, 14 pages

http://dx.doi.org/10.1155/2015/914170

## Energy Dependence of Slope Parameter in Elastic Nucleon-Nucleon Scattering

National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe Shosse 31, Moscow 115409, Russia

Received 25 June 2014; Revised 28 October 2014; Accepted 3 November 2014

Academic Editor: Bhartendu K. Singh

Copyright © 2015 V. A. Okorokov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The diffraction slope parameter is investigated for elastic proton-proton and proton-antiproton scattering based on all the available experimental data at low and intermediate momentum transfer values. Energy dependence of the elastic diffraction slope is approximated by various analytic functions. The expanded “standard” logarithmic approximations with minimum number of free parameters allow description of the experimental slopes in all the available energy range reasonably. The estimations of asymptotic shrinkage parameter are obtained for various domains based on all the available experimental data. Various approximations differ from each other both in the low energy and very high energy domains. Predictions for diffraction slope parameter are obtained for elastic proton-proton scattering from NICA up to future collider (FCC/VLHC) energies, for proton-antiproton elastic reaction in FAIR energy domain for various approximation functions.

#### 1. Introduction

Elastic hadron-hadron scattering, the simplest type of hadronic collision process, remains one of the topical theoretical problems in the physics of fundamental interactions at present. Forward elastic scattering process is an excellent test for some fundamental principles (unitarity, analyticity, and asymptotic theorems) of modern approaches. In the case of and elastic scattering, although many experiments have been made over an extended range of initial energies and momentum transfer, these reactions are still not well understood. One can suggests that, by the time the accelerator complexes like RHIC, LHC, and so forth are operating, the interest in the soft physics increases significantly. In the absence of a pure QCD description of the elastic and these large-distance scattering states (soft diffraction), an empirical analysis based on model-independent fits to the physical quantities involved plays a crucial role [1]. Therefore, empirical fits of energy dependencies of global scattering parameters have been used as an important source of the model-independent information. This approach for and was used in [2, 3]. The third important quantity for nucleon elastic scattering is the slope parameter. The nuclear slope for elastic scattering is of interest in its own right. This quantity defined according to the following equation, ,is determined experimentally by fitting the differential cross-section at some collision energy . On the other hand the study of -parameter is important, in particular, for reconstruction procedure of full set of helicity amplitudes for elastic nucleon scattering [2, 3]. In the last 20–30 years, high energy colliders have extended the maximum collision energy from GeV to TeV, the LHC facility allows one to obtain data up to TeV so far. As consequence, the available collection of and slope data from literature has extended. The present status of slope for elastic and scattering is discussed over the full energy domain. Predictions for further facilities are obtained based on the available experimental data.

#### 2. Experimental Slope Energy Dependence

We have attempted to describe the energy behavior of the elastic nuclear slopes for and reactions. The classical Pomeron theory gives in the first approximation the following expression for the differential cross-section of elastic scattering at asymptotically high energies: where is a Pomeron trajectory. If is linear function of momentum transfer, that is, , then for the slope parameter at some using the definition (1) one can obtain where , . In general case for Pomeron-inspired models the asymptotic shrinkage parameter can be written as follows: . Indeed the ensemble of experimental data for slope for elastic nucleon collisions can be approximated reasonably by many phenomenological approaches, at least for GeV. But models contradict with experimental data at lower energies and/or phenomenological approaches have a significant number of free parameters as usual. On the other hand it is apparent from previous investigations that the experimental data for slope parameter do not follow the straight lines at any initial energies when plotted as function of . The new “expanded” logarithmic parameterizations with small number of free parameters have been suggested in [2, 3, 5, 6] for description of the elastic slope at all available energies. Thus taking into account standard Regge parametrization and quadratic function of logarithm from [7] the following analytic equations are under study here:In general parameters , , depend on range of which is used for approximation. There are the relations and for parameterizations (4a) and (4d) inspired by the Pomeron exchange models. As seen the function (4d) is the special case of (4b) at fixed value . Additional terms in (4b)–(4d) take into account the nonlogarithmic part of the energy dependence of the elastic nuclear slopes. Approximation function (4c) is analogy of parametrization of momentum slope dependence from [8]. One can see that the parametrization (4c) is compatible to first order with the Regge pole model where the additional term represents the interference between the Pomeron and secondary trajectories [8].

Most of experimental investigations as well as theoretical models are focused on the diffraction region . In this paper we study all available experimental data for nuclear slope parameter up to . Experimental values of slope parameter collected at initial energies TeV are from [9]. Additional experimental results from Tevatron and the LHC are from [10] and [11–14], respectively. The full data sample consists of 490 experimental points. The number of experimental points equals for scattering at low , respectively. In the intermediate domain experimental data set is for reaction, respectively. Thus the experimental sample is significantly larger than that in some early investigations [2, 3, 5–8, 15, 16]. The careful analysis of data sample allows us to suggest that the influence of double counting in the experimental data is negligible. It should be emphasized that the experimental data for intermediate range are separated on two samples which correspond to the various parametrization types for differential cross-section, namely, linear, , and quadratic, , function. Here are suggested. As known the measurements of nuclear slope, especially at intermediate , do not form a smooth set in energy, in contrast with the situation for global scattering parameters and , where there is a good agreement between various group data [7]. Detailed comparisons of slope data from different experiments are difficult because the various experiments cover different ranges, use various fitting procedures, and treat systematic errors in different ways, and, moreover, some experimental details are lost, especially, for very early data. We have tried to use as much as possible data for fit from available samples. But some of the values were not further used, either due to internal inconsistencies in the fitting procedure or as redundant in view of a better determination at a nearby initial energy. Thus the data samples for approximations are somewhat smaller because of exclusion of points which, in particular, differ significantly from the other experimental points at close energies. The choice of the range over which the fit of is performed is important significantly for a consistent determination of slope parameters. It seems that both the mean value of and -boundaries of corresponding measurements are important for separation of experimental results by different domains. Here the is calculated taking into account the approximation of experimental distribution instead of identifying of with mean point of -range as previously [5, 6]. Errors of experimental points include available clear indicated systematic errors added in quadrature to statistical ones. One needs to emphasize that the systematic errors caused by the uncertainties of normalization (total or/and differential cross-sections) are not taken into account if these uncertainties are not included in the systematic errors in the original papers.

Let us describe the fitting algorithm in more detail. We use the fitting procedure with standard likelihood function for this investigation of nuclear slope parameter. In accordance with [16] let us define the quantity where is the measured value of nuclear slope at , is the expected value from the one of the fitting functions under study, and is the experimental error of the th measurement. The parameters are given by the -dimensional vector . Our fitting algorithm is somewhat similar to the “sieve” algorithm from [16] with the following modification. We reject the points which* a priori* differ significantly from other experimental data at close energies. The step allows us to get a first estimation of . with minimum number of rejected points. The fit quality is improved at the next steps consequently. As indicated above smoothness of experimental slope energy dependence differs significantly for data samples in various -domains and for various parameterizations of (see below). The absolute value of the can be large for one data sample but it can be acceptable for another sample at the same time. Therefore we suggest using the relative quantityin order to reject the outliers (points far off from the fit curve for the certain data sample) instead of constant cut value from the “sieve” algorithm [16]. One needs to emphasize that the fit function with best . for description of all range of available energies among (4a)–(4d) is used for calculation of the expected value in (5) at each algorithm step and then for estimation of (6) quantity. The points with are excluded from future study in our algorithm, where the is some empirical cut value. The conventional fit is made to the new “sifted” data sample. We consider the estimates of fit parameters as the final results if there are no excluded points for present data sample. We use the one value for all data samples considered in this paper below. The fraction of excluded points is about for as well as for elastic scattering for low domain. The maximum relative amount of rejected points is about for linear parametrization and for quadratic one at intermediate values for scattering, respectively.

##### 2.1. Low Domain

The energy dependence for experimental slopes and corresponding fits by (4a)–(4d) is shown at Figures 1 and 2 for and , respectively. The values of fit parameters are shown in Table 1. One can see that the fitting functions (4a) and (4d) describe the (Figure 1) and (Figure 2) experimental data statistically acceptable only for GeV. Additional study demonstrates that the extension of approximation domain down to the lower energies for parameterizations (4a) and (4d) results in significant increasing of . for both and data samples. Thus these fit functions allow one to get reasonable fit qualities only at GeV for scattering as well as for elastic reaction.