Advances in High Energy Physics
Volume 2015 (2015), Article ID 914170, 14 pages
http://dx.doi.org/10.1155/2015/914170

Research Article

Energy Dependence of Slope Parameter in Elastic Nucleon-Nucleon Scattering

V. A. Okorokov

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 SCOAP³.

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 quantity

in 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.

Table 1: Values of parameters for fitting of slope energy dependence in low domain.

Function
Parameter

, GeV
, GeV
, GeV

/n.d.f.

Proton-proton scattering

(4a)

—
—

(4b)

(4c)

(4d)

—

Proton-antiproton scattering

(4a)

—
—

(4b)

(4c)

(4d)

—

Figure 1: Energy dependence of the elastic slope parameter for proton-proton scattering for low domain. Experimental points from fitted samples are indicated as •; unfitted points are indicated as . The dot curve is the fit of experimental slope by the function (4a), thick solid—by (4b), dashed—by (4c), thin solid—by (4d). The shaded band corresponds to the spread of fitting functions for previous analysis [9].

Figure 2: Energy dependence of the elastic slope parameter for antiproton-proton scattering for low domain. Experimental points from fitted samples are indicated as ■, unfitted points are indicated as . The correspondence of curves to the fit functions is the same as well as in Figure 1. The inner picture shows the experimental data and fits for the same scale of slope as well as in Figure 1.

The RHIC point for collisions does not contradict the common trend within large error bars and cannot discriminate the approximations under study. In general the LHC results [12–14] added in fitting sample result in better agreement of fits (4a)–(4d) in comparison with previous study [9]. All fits are very close to each other in energy domain ; the quadratic function (4d) shows somewhat faster increasing of slope and noticeable difference from other fit functions in multi-TeV region only. It seems the ultra-high energy domain is suitable for separation of various parameterizations. Fitting functions (4b) and (4c) allow us to describe experimental data at all energies with reasonable fit quality for (Table 1). The functions (4a)–(4c) agree very well for  GeV; furthermore there is no visible difference between modifications (4b) and (4c) in all experimentally available energy domains. The function (4c) demonstrates some better quality for fit of full data sample (Table 1) than function (4b) in contrast with previous analysis [9]. The accounting for LHC data leads to some decreasing of values of and increasing of parameters for all fitting functions (4a)–(4d) under study in comparison with values of corresponding parameters in previous investigation [9]. This behavior of with collision energy agrees well with predicted growth of with increasing of [17, 18]. As seen from Table 1 the third term in both (4b) and (4c) gives the main contribution at  GeV in the case of elastic proton-proton scattering, that is, describing the sharp changing of slope in the low energy domain. Therefore at high in accordance with (4b) and (4c). The increasing of exhibits that grows somewhat faster in multi-TeV region than one can expect from the trend based on data sample at  GeV. This suggestion is confirmed by improvement of fit quality for one fitting function (4d) for present data sample in comparison with fit qualities for experimental points at  GeV [9]. Values of obtained in the present study and for fit at  GeV are close within errors for functions (4b) and (4d) but accounting for the LHC data results in some increasing of the absolute value of in fit by (4b). The absolute values of and are the same within error bars for function (4c) for Table 1 and for fit in energy domain  GeV [9].

As seen from Table 1 values of parameter obtained for slope in elastic scattering are significantly larger than thaose for elastic reactions for any approximations (4a)–(4d) under study. The parameter for Regge-inspired function (4a) is close to estimation for Pomeron parameter for . The values from Table 1 coincided within error bars for fitting function (4a)–(4c). These values are larger than earlier experimental estimations of “true” Pomeron shrinkage parameter [19]. Also the value of parameter for fitting function (4c) exceeds significantly the value of corresponding parameter for such fit of slope momentum dependence but in much narrower energy range GeV [8]. Thus the comparison with earlier estimations confirms the conclusion made above that the including of high energy data points into fitted sample leads to the increasing of value of the parameter and growth of slope parameter seems to be faster in TeV-region than that at lower energies. Accordingly the value for function (4d) is significantly closer to the than that for previous analysis in energy range  GeV [9]. On the other hand the values of the parameter obtained for fitting function (4a) are somewhat larger than the prediction for from Pomeron-inspired model for TeV-energy domain [18]. Furthermore the proton-proton results from fit by function (4d) allow the estimation which is almost twice larger than the corresponding prediction from [18].

The qualities of (4b) and (4c) approximation functions for elastic scattering data are much poorer because of very sharp behavior of experimental data near the low energy boundary. But one can see that the functions (4b) and (4c) agree with experimental points at qualitative level and are (very) close to each other for all energy range. In contrast with elastic scattering the fit quality is somewhat better for function (4b) than for parametrization (4c) for data. Additional study of antiproton-proton data shows that the increasing of low boundary of range of fitted energies leads to the better fit quality for functions (4b) and (4c) but at the same time, obviously, to the loss of some low-energy data. The fit quality changes dramatically at small increasing of from low boundary value to 3.72 . It was obtained for function (4b)/(4c), respectively, for fit range . On the other hand the corresponding data sample is about from the maximum one in this case. Thus it seems that the is one of the optimum values from point of view of both the fit quality and the closing to the threshold . The average value of (6) for excluded points is equal to 4.7 for and 65.8 for data sample for parametrization (4c).

Predictions for nuclear slope parameter values for some facilities have been obtained based on the fit results shown above. The estimations at low for different intermediate energies of the projects FAIR and NICA are shown in Table 2 and those for high energy domain are presented in Table 3. As expected the functions (4b) and (4c) predict for FAIR that the values coincide with each other within errors. The approximation functions (4a) and (4d) can be predicted for  GeV only. The Regge-inspired function (4a) predicts slope parameter values significantly smaller than those for modified fitting functions (4b) and (4c) at FAIR energies. Predictions with help of quadratic function (4d) of are equal with estimations based on any other fitting function under study within large error bars at  GeV. In the case of elastic scattering the functions (4a)–(4c) predict just the equal values of within error bars at any collision energy discussed here. The difference between estimations from (4a)–(4c) and from (4d) onsets at the final energy of the LHC project  TeV only. Our prediction with (4d) function for RHIC energy is equal to early prediction for close energy based only on slope data in the region  GeV [7] within errors. But (4d) underestimates the values in ultra-high energy domain  TeV in comparison with results based on the approach without Odderons [7]. It should be emphasized that in contrast with previous analysis [9] the present fits by functions (4a)–(4d) of data sample included results from the LHC predict similar increasing of with energy to most of phenomenological models [20]. The value predicted for the LHC at  TeV by (4a) only is close to errors to the predictions from [21, 22], in particular, with estimation from model with hadronic amplitude corresponding to the exchange of two Pomerons. Prediction of phenomenological model with hadronic amplitude corresponding to the exchange of three Pomerons [22] at  TeV coincides with estimation of within error bars from fit function (4d) with the fastest growth of with in multi-TeV region. But most of estimations of at  TeV from Table 3 agree well within errors with model prediction from [23]. However the model estimates at  TeV described above were obtained for and the -dependence of slope shows the slight decreasing of for the model with three-Pomeron exchange [22] and faster decreasing of for the model from [23] at growth of momentum transfer up to . Therefore one can expect that the model with hadronic amplitude corresponding to the exchange of three Pomerons [22] will be in better agreement with values of from Table 3 predicted for finite (nonzero) low values. But the model from [24] overestimates the value at  TeV in comparison with the corresponding predictions from fitting functions (4a)–(4d) despite the sharp decreasing of at growth of momentum transfer up to in this model. As suggested sometimes the saturation regime, Black Disk Limit (BDL), may be reached at the LHC. One of the models in which such effects appear, namely, Dubna Dynamical Model (DDM), predicts the slope at  TeV [25] which is noticeably larger than the predictions from Table 3 at the same . Therefore the saturation regime will not be reached, at least, at the LHC energy  TeV as suggested, for example, in the model from [26], or simple saturation can not be enough in order to describe the LHC data at quantitative level.

Table 2: Predictions for slope in nucleon-nucleon elastic scattering at intermediate energies for low domain.

Fitting function
Facility energies , GeV

FAIR
NICA

3
5
6.5
14.7
20
25

(4a)
—

(4b)

(4c)

(4d)
—

Table 3: Predictions for slope in elastic scattering at high energies for low domain.

Fitting function
Facility energies , TeV

RHIC
LHC
FCC/VLHC

0.5
14
28
42
100
200
500

(4a)

(4b)

(4c)

(4d)

The ultimate energy upgrade of LHC project [4].

2.2. Intermediate Domain

As indicated above the situation is more complicated for intermediate domain. Differential cross-section is approximated by linear, , and/or quadratic, , function in various experiments; ranges used for approximations differ significantly, and so forth. For quadratic exponential parametrization the and parameters are highly correlated by fits.

Figure 3 shows the experimental data and corresponding fits for energy dependence of slope parameter at intermediate for and elastic scattering. Figures 3(a) and 3(c) correspond to the linear exponential approximation of differential cross-section for and , respectively. Experimental data obtained at quadratic exponential fit of and fitting functions (4a)–(4d) are presented in Figure 3(b) for and in Figure 3(d) for collisions. The fitting parameter values are indicated in Table 4 for various interaction types and for different parameterizations. Usually the fit qualities are poorer for intermediate values than those for low range in elastic collisions for linear exponential parametrization of . The fitting functions (4a) and (4d) agree with experimental points qualitatively both for linear (Figure 3(a)) and quadratic (Figure 3(b)) exponential parameterizations of for  GeV only. The “expanded” functions (4b) and (4c) approximate experimental data at all energies reasonably with close fit qualities, especially for quadratic exponential parametrization of (Table 4). But (4b) function shows a very slow growth of slope parameter with energy increasing at  GeV (Figure 3(a)). It should be stressed that the experimental point at the LHC energy leads to the dramatic change of behavior of the fitting function (4d) in comparison with previous analysis [9]. At present the fitting function (4d) predicts increasing of the nuclear slope in high energy domain as well as all other fitting functions under study. Such behavior is opposite to the result of fit by function (4d) of experimental data sample at  GeV [9]. In the case of linear exponential approximation of mean value of characteristic (6) is about 2.9 for excluded data points with (4c) function and for points excluded from fitted data sample for (4b) fitting function.

Table 4: Values of parameters for fitting of energy dependence of slope at intermediate .

Function
Parameter

, GeV
, GeV
, GeV

/n.d.f.

Proton-proton scattering, experimental data for

(4a)

—
—

(4b)

(4c)

(4d)

—

Proton-proton scattering, experimental data for

(4a)

—
—

(4b)

(4c)

(4d)

—

Proton-antiproton scattering, experimental data for

(4a)

—
—

(4b)

(4c)

(4d)

—

Proton-antiproton scattering, experimental data for

(4a)

—
—

(4b)

(4c)

(4d)

—

(a)

(b)

(c)

(d)

Figure 3: Energy dependence of in proton-proton (a, b) and proton-antiproton (c, d) elastic scattering for linear (a, c) and quadratic (b, d) exponential parametrization of differential cross-section. Experimental points from fitted samples are indicated as close/open circles (squares) for    for (a, c)/(b, d); unfitted points are indicated as . The dot curve is the fit of experimental slope by the function (4a), thick solid—by (4b), dashed—by (4c), and thin solid—by (4d). The shaded band (a) corresponds to the spread of fitting functions for previous analysis [9].

One can see that the experimental data admit the approximation by (4d) for all energy range but not only for  GeV. Indeed the fit quality for the first case is much better than that for second one. Additional analysis demonstrated just the same behavior of fit quality for function (4a) too. Thus experimental points from linear exponential parametrization of differential cross-section are fitted by (4a) and (4d) at all energies not only for  GeV. The parameter values are shown in Table 4 for approximation by (4a) and (4d) of all available experimental data. The fit curves show (very) close behaviors for both the present and previous analyses in the case of Figure 3(c). The data disagree with Regge-inspired fitting function very significantly (Figure 3(c)). Functions (4b) and (4c) show a very close behavior at all energies for data from linear parametrization of . These fitting functions have a better fit quality than (4d) but fits by functions (4b) and (4c) are still statistically unacceptable. As previously experimental data at Figure 3(d) allow the approximation by (4a) and (4d) for all energy range but not only for  GeV. The fit qualities are better in the first case of energy range and fitting parameters are indicated in Table 4, namely, for this energy range. In contrast with the case of linear parametrization of for here the functions (4c) and (4d) show a close fit quality which is somewhat better than this parameter for (4b) fitting function. One can see the fit qualities for (4a)–(4d) are better significantly for data from quadratic exponential parametrization of differential cross-sections than for data from linear exponential approximation of . Thus fitting functions (4c) and (4d) agree with data points at quantitative level for quadratic parametrization of proton-antiproton (Figure 3(d)) and these fits are statistically acceptable. Excluded points are characterized by for data with (4b) fitting function and by for data sample at (4d) function.

From the quadratic exponential parametrization of differential cross-section one may calculate the local slope at a certain -value via the following relation:

This characteristic can be useful for elastic scattering for study of in wider range. It is suggested according to definition (7). The -parameter is named slope too; it is evaluated at usually. One of the advantages of this characteristic is the expectation of more smooth energy (momentum) dependence than that for -parameter discussed above. Indeed we have included the 100% of available experimental points in fitted sample for elastic scattering. But the number of points is somewhat smaller than that for -parameter because of absent -parameter values for some low energy measurements from [15]. We excluded one point at  TeV [27] from fitted sample for elastic reaction because there are unacceptably large errors (relative error is ) for this point.

Experimental values of depend on collision energy and corresponding fits are shown in Figure 4 for elastic scattering and in Figure 5 for collisions. In the last case fit qualities for (4a) and (4d) functions are better for fitting at  GeV only than for fitting of all available energy domain. The fit parameter values are shown in Table 5. Fit qualities are significantly better than those for corresponding fits of -parameter with the exception of (4a) for data. Functions (4a) and (4d) approximate for data statistically acceptable for  GeV only. Functions (4b) and (4c) show acceptable close fit qualities and difference at high energies only. The shrinkage parameter for approximation function (4c) with best fit quality is in a good agreement with the early results [8]. Function (4b) shows a best fit quality for data. Thus the “expanded” parameterizations (4b) and (4c) suppose statistically acceptable representation of all available experimental data for -parameter both in and elastic reactions.

Table 5: Values of parameters for fitting of the energy dependence.

Function
Parameter

, GeV
, GeV
, GeV

/n.d.f.

Proton-proton scattering

(4a)

—
—

(4b)

(4c)

(4d)

—

Proton-antiproton scattering

(4a)

—
—

(4b)

(4c)

(4d)

—

Figure 4: Energy dependence of the slope parameter at for proton-proton scattering. The correspondence of curves to the fit functions is the same as that in Figure 1.

Figure 5: Energy dependence of the slope parameter at for antiproton-proton scattering. The correspondence of curves to the fit functions is the same as that in Figure 1.

We have obtained predictions for nuclear slope parameters and for some facilities and intermediate based on the fit results shown above. The predicted values at intermediate are calculated on the base of fitting parameters obtained for linear exponential parametrization of . Slope values are shown in Table 6 for different energies of FAIR and NICA and in Table 7 for RHIC, LHC, and FCC/VLHC. According to the fit range function (4a) can predict the value for scattering at all energies under study not in  GeV domain only. As expected the functions (4b)–(4d) predicted very close values for slope parameter for FAIR. These fitting functions, especially (4b) and (4c), predict the close values for nuclear slope in NICA energy domain too. Functions (4a)–(4c) predict smaller values for in high energy collisions than (4d) approximation especially for FCC/VLHC energy domain. Perhaps, the future more precise RHIC results will be useful for discrimination of fitting functions under study for intermediate values. In contrast with previous analysis [9], here the function (4d) with obtained parameters predicts fast growth of values at energies of future experiments. This behavior of estimations calculated for functions (4a)–(4d) contradicts with earlier predictions from some phenomenological models. It should be emphasized that various phenomenological models predict a very sharp decreasing of nuclear slope in the range at LHC energy  TeV [20]. Just the positive value predicted for LHC at  TeV by (4d) is most close to the some model expectations [22, 23]. Taking into account predictions in Table 2 based on the fitting functions (4a)–(4d) for low one can suggest that the model with hadronic amplitude corresponding to the exchange of three Pomerons [20, 22] describes the nuclear slope somewhat closer to the experimentally inspired values at LHC energy both at low and intermediate than other models. The situation with predictions for at intermediate energies is similar to that for : functions (4b)–(4d) predict close values for within large errors for both the FAIR and the NICA. Furthermore all fitting functions predict the close values of at  GeV. The value of -parameter obtained from function (4a) differs significantly from estimations with other fitting functions for elastic scattering at low energies. Therefore the -parameter seems more perspective for distinguishing of Regge-inspired function (4a) from “expanded” parameterizations (4b)-(4c) at  GeV than . One can see the functions (4b) and (4c) predict very close values of in high energy domain. Function (4d) shows a decreasing of at high energies in the contrast with -parameter. In general estimations for -parameter in Table 7 agree within errors for all fitting functions under study up to the  TeV. But the large errors for function (4d) do not allow the unambiguous physics conclusion especially at the LHC energies and above.

Table 6: Predictions for slopes in nucleon-nucleon elastic scattering at intermediate energies for intermediate domain.

Fitting function
Facility energies , GeV

FAIR
NICA

3
5
6.5
14.7
20
25

-parameter

(4a)

(4b)

(4c)

(4d)

-parameter

(4a)

(4b)

(4c)

(4d)

Table 7: Predictions for slopes in elastic scattering at high energies for intermediate domain.

Fitting function
Facility energies , TeV

RHIC
LHC
FCC/VLHC

0.5
14
28
42
100
200
500

-parameter

(4a)

(4b)

(4c)

(4d)

-parameter

(4a)

(4b)

(4c)

(4d)

The ultimate energy upgrade of LHC project [4].

2.3. and Data Analysis

Phenomenological models predict the zero difference of slopes for proton-antiproton and proton-proton elastic scattering at asymptotic energies. Here the difference is calculated for all functions (4a)–(4d) under study with parameters corresponding to and fits: , = (4a)–(4d). (Obviously, one can suggest various combinations of fitting functions for calculations, for example, the difference between fitting functions with best fit qualities, etc.) It should be stressed that the equal energy domain is used in and fits for calculations; that is, the parameter values obtained by (4d) fitting function for data from linear exponential fit of for  GeV are used for corresponding definition. The energy dependence of is shown in Figures 6(a) and 6(b) for low and intermediate , respectively. One can see that the difference of slopes decreases with increasing of energy for low domain (Figure 6(a)) as well as in the previous analysis [9]. The fitting functions (4b) and (4c) demonstrate much faster decreasing of with increasing of than that of the functions (4a) and (4d). At present the proton-proton experimental data at highest available energy 8 TeV do not contradict with fast (square of logarithm of energy) increasing of slope at high energies in the general case. Such behavior could agree with the asymptotic growth of total cross-section. Furthermore, in contrast with the previous analysis [9], here the quadratic logarithmic function (4d) leads to much smaller difference for and scattering in high energy domain for both low (Figure 6(a)) and intermediate (Figure 6(b)) values of . The only Regge-inspired function (4a) predicts the decreasing of with energy growth at intermediate (Figure 6(b)) for any values of . The parameterizations (4b)–(4d) predict the decreasing of difference of slopes at low and intermediate energies and fast increasing of at high energies for intermediate domain (Figure 6(b)). As expected the slowest changing of is predicted by Regge-inspired function (4a) at asymptotic energies. All fitting functions with experimentally inspired parameters do not predict the constant zero values of at high energies. But it should be emphasized that only separate fits were made for experimental data for and elastic reactions above. These results indicate on the importance of investigations at ultra-high energies both and elastic scattering for study of many fundamental questions and predictions related to the general asymptotic properties of hadronic physics.

(a)

(b)

Figure 6: The energy dependence of the difference of elastic slopes for proton-antiproton and proton-proton scattering in low domain (a) and in intermediate range for linear exponential fit of differential cross-section (b). The correspondence of curves to the fit functions is the same as that in Figure 1.

Also we have analyzed general data samples for and elastic scattering. Slope parameters ( and ) show a different energy dependence at  GeV in proton-proton and antiproton-proton elastic reactions in any domains under study. Thus slopes for nucleon-nucleon data are investigated only for  GeV below. We have included in fitted samples only and points which were included in corresponding final data samples at separate study and elastic reactions above. We did not exclude any points from data sample; we change only the low energy boundary for fitted domain. Figure 7 shows the experimental data for slope in nucleon-nucleon elastic scattering against collision energy at low . As seen from Figure 7 there is no experimental data for between  GeV and  GeV. This energy domain will be available for further FAIR facility. We have fitted the general nucleon-nucleon data sample at the range of low energy boundary and . The fitting parameter values are indicated in Table 8 on the first line for low boundary of the fitted energy domain and on the second line for . The fit quality improves for most parameterizations under consideration at increasing of . One need to emphasize the fit quality is some poorer at  GeV than that for  GeV for functions (4a) and (4c). For these cases of the fitted energy domain the value of the parameter obtained with the function (4a) agrees qualitatively with the Regge model prediction, but the value of the obtained at is somewhat larger than the prediction for from Pomeron-inspired model for TeV-energy domain [18]. Also results from fit by function (4d) with acceptable quality at allow the estimation which is some larger than the corresponding prediction from [18]. Furthermore the estimations for do not depend on within error bars. Fitting functions (4a)–(4d) are shown at Figure 7 for . All functions (4a)–(4d) are close to each other at energies up to  TeV at least and show quasi-linear behavior for parameter values obtained by fits with and . As seen from comparison between the present fits and the spread of previous fit functions (shaded band) there is a dramatic change of behavior of the fitting function (4d) in comparison with previous analysis [9] due to experimental points at the LHC energies. At present the fitting function (4d) predicts increasing of the nuclear slope in high energy domain as well as all other fitting functions under study. Such behavior is opposite to the result of fit by function (4d) of experimental data sample at  TeV [9]. We have analyzed the nucleon-nucleon data for slope parameters and at intermediate values for linear and quadratic exponential parametrization of , respectively. Experimental and data for differ significantly up to  GeV; at least that results in unacceptable fit qualities for all functions under study ( for best fit by quadratic logarithmic function). Additional analysis demonstrates the improving of fit quality for (4b)–(4d) with increasing of low energy boundary from up to . The values of fit parameters for the last case are shown in Table 9. The Regge-inspired function (4a) contradicts with experimental data. We would like to emphasize that the best fit quality for (4a) is obtained at but it is statistically unacceptable too. Functions (4b)–(4d) agree with experimental dependence reasonably and have very close fit qualities. One can see from Table 9 that the statistically acceptable fits have been obtained for -parameter at only. Experimental data and fit functions are presented at Figure 8. Functions (4b)–(4d) show close fit qualities. Best fit is (4d) but “expanded” parameterizations agree with data too. One needs to emphasize that the significant errors and absence of experimental points at  TeV do not allow one to get the more clear conclusion. The RHIC as well as LHC data for nucleon-nucleon differential cross-section at intermediate will be helpful for distinguishing of various fit functions.

Table 8: Fitting parameters for slope energy dependence in low domain for elastic scattering.

Function
Parameter

, GeV
, GeV
, GeV

/n.d.f.

(4a)

—
—

—
—

(4b)

(4c)

(4d)

—

—

Table 9: Fitting parameters for energy dependence of slope parameters at intermediate for elastic scattering.

Function
Parameter

, GeV
, GeV
, GeV

/n.d.f.

-parameter

(4a)
5.17 ± 0.11
0.369 ± 0.006
—
—
353/36

(4b)
10.7 ± 0.2
−0.019 ± 0.011
(1.04 ± 0.17) × 10⁻⁴
4.08 ± 0.08
107/34

(4c)
10.5 ± 0.2
−0.165 ± 0.019
0.66 ± 0.06
0.187 ± 0.004
106/34

(4d)
14.9 ± 0.6
−0.61 ± 0.06
0.089 ± 0.006
—
118/35

-parameter

(4a)

—
—

—
—

(4b)

(4c)

(4d)

—

—

Figure 7: Energy dependence of the elastic slope parameter for nucleon-nucleon scattering for low domain. Experimental points from fitted samples are indicated as • for and as ■ for . Fits are shown for  GeV. The dot curve is the fit of experimental slope by the function (4a), thick solid—by (4b), dashed—by (4c), and thin solid—by (4d). The shaded band corresponds to the spread of fitting functions for previous analysis [9].

Figure 8: Energy dependence of the elastic slope parameter at for nucleon-nucleon scattering for intermediate domain. Experimental fitted points are indicated as ▲ for and as ▼ for . Fits are shown for  GeV. The correspondence of curves to the fit functions is the same as that in Figure 1.

One can conclude that the slope parameters for and elastic scattering show universal behavior at  GeV and “expanded” functions represent the energy dependencies for both low and intermediate ranges for this energy domain. Thus quantitative analysis of slopes at different allows us to get the following estimation of low energy boundary:  GeV for universality of elastic nucleon-nucleon scattering. This estimation agrees with results for differential cross-sections of and elastic reactions based on the crossing-symmetry and derivative relations [2, 3].

3. Conclusions

The main results of this paper are the following. Energy dependence for several slope parameters is analyzed quantitatively for elastic nucleon-nucleon scattering in various domains. Most of all available experimental data samples for slope parameters in elastic nucleon collisions are approximated by different analytic functions.

The suggested new parameterizations allow us to describe experimental nuclear slope at all available energies in low domain for quite reasonably. The new approximations agree with experimental data at qualitative level but these fits are still statistically unacceptable because of the very sharp behavior of near the low energy limit. The best fit qualities are obtained for suggested “expanded” functions both for and data. The obtained values of asymptotic shrinkage parameter for elastic scattering are larger than values for elastic reactions for the Pomeron-inspired fitting function. Various approximations differ from each other both in the low energy and very high energy domains. Predictions for slope parameter are obtained for elastic proton-proton and proton-antiproton scattering in energy domains of some facilities at low momentum transfer. Our predictions based on the all available experimental data do not contradict the phenomenological model estimations qualitatively. The situation is more unclear at intermediate values than that at low domain. Only the qualitative agreement is observed between approximations and experimental points both for and collisions for linear exponential parametrization of because of poorer quality of data. The “expanded” functions describe data for -parameter for any differential cross-section parametrization reasonably. Best fit quality is obtained for quadratic function of logarithm for the data from any exponential parametrization of . One needs to emphasize that this fit function allows us to describe data at all available energies and shows a statistically acceptable fit quality for data sample obtained from quadratic exponential parametrization of . Slope parameter calculated at shows more smooth energy dependence. We have obtained acceptable fit qualities for “expanded” functions both for and data at all initial energies. The obtained values of shrinkage parameter for elastic scattering are close to the early results for intermediate domain. As well as for low domain predictions for slope parameters and are obtained for elastic proton-proton and proton-antiproton scattering in energy domains of some facilities. It seems the phenomenological model with hadronic amplitude corresponding to the exchange of three Pomerons describes the nuclear slope somewhat closer to the experimental fit inspired values at the LHC energy both at low and intermediate than other models.

The energy dependence of difference of slopes for proton-antiproton and proton-proton elastic scattering was obtained for fitting functions under study. The parameter shows the opposite behaviors at high energies for low and intermediate domains (decreasing/increasing, resp.) for all fitting functions with the exception of Regge-inspired one. The last function predicts the slow decreasing of with energy growth. It should be emphasized that all underlying empirical fitting functions with experimentally inspired parameter values do not predict the zero difference of slopes for proton-antiproton and proton-proton elastic scattering both at low and intermediate for high energy domain. We have analyzed general nucleon-nucleon data samples for slopes at  GeV. The “expanded” functions show the statistically acceptable fit qualities at  GeV for low domain. Slop analysis allows us to find the following value for parameter. The estimation of asymptotic shrinkage parameter obtained with quadratic function of logarithm for data at  TeV is noticeably larger than the expectation from the Pomeron theory. But the growth of with increasing of is observed from the comparison of the fit results from present analysis and our earlier analysis for  TeV. Such behavior of agrees with Pomeron-inspired model. Functions (4c) and (4d) represents experimental data for and slope parameters, respectively, at intermediate with best quality. But the function (4b) shows close qualities and agrees with data reasonably. Therefore the suggested “expanded” functions can be used as reliable fits for wide range of momentum transfer at all energies. The universal behavior was found for available experimental and slopes at  GeV both for low and intermediate that is in agreement with the hypothesis of a universal shrinkage of the hadronic diffraction cone at high energies.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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