Energy dependence of slope parameter in elastic nucleon-nucleon scattering

The study of slope parameter is presented for elastic proton-proton and antiproton-proton scattering with taking into account the resent experimental data at high energies. The expanded logarithmic approximations allow the description of the experimental slopes in all available energy range reasonably. Accounting for the LHC results leads to the dramatic change of behavior of the quadratic in logarithm approximation at high energies and to the closer trends for all fitting functions under study in comparison with the analysis at collision energies up to the 200 GeV. The estimations of the asymptotic shrinkage parameter $\alpha'_{\cal{P}}$ are discussed. Predictions for diffraction slope parameter are obtained for some proton-proton and antiproton-proton facilities.


Introduction
The one of the important quantity for nucleon elastic scattering is the slope parameter B which is defined in accordance with the following equation: where ∂ t ≡ ∂/∂t. The B is determined experimentally by fitting the differential cross section dσ/dt at some collision energy √ s. The study of B parameter is important, in particular, for reconstruction procedure of full set of helicity amplitudes for elastic nucleon-nucleon scattering [1]. In the last 20-30 years, high-energypp colliders have extended the maximumpp collision energy from √ s ∼ 20 GeV to √ s ∼ 2 TeV, the LHC facility allows one to obtain pp data up to √ s = 8 TeV so far. As consequence, the available collection of pp andpp slope data from literature has extended. The B(s) for elastic pp andpp reactions is under consideration below. Within the classical Regge model the Pomeron trajectory, α P (t), is linear function of momentum transfer, i.e. α P (t) = α P (0) + α ′ P t. Therefore using the definition (1) the following relation can be obtained for B(s) at some fixed t: where ε ≡ s/s 0 , s 0 = 1 GeV 2 . In general case for Pomeroninspired models the asymptotic shrinkage parameter α ′ P can be written as follows: 2α ′ P (s) = ∂ ln ε B(s, t).
where free parameters B 0 , a i , i = 1−3 depend on range of |t| which is used for approximation. There are the relations α ′ P = a 1 and α ′ P = a 1 + a 2 ln ε for parameterizations (2a) and (2d) inspired by the Pomeron exchange models. Experimental values of slope parameter collected at initial energies √ s ≤ 1.8 TeV are from [4]. Additional experimental results from Tevatron and the LHC are from [5] and [6,7,8,9], respectively. The full data sample consists of 490 experimental points. The number of experimental points is equal 145/138 (137/70) for pp/pp scattering at low (intermediate) |t| respectively. It should be noted that for intermediate |t| range the experimental results for B(s) obtained with help of linear parametrization for logarithm of differential cross-section, ln (dσ/dt) ∝ (−B|t|), are discussed below because the new experimental data at intermediate |t| [5,6] with respect to the [4] are obtained for such parametrization of ln (dσ/dt) namely. From the exponential parametrization with index quadratic in t for differential cross-section, ln (dσ/dt) ∝ −B|t| ± Ct 2 , one may calculate the local slope as following b (s, t)| |t|=0.2GeV 2 = B ± C ln |t|, B, C > 0.
In accordance with [4] the mean value of |t| (|t|) is calculated with taking into account the approximation of experimental dσ/dt distribution; errors of experimental points include available clear indicated systematic errors added in quadrature to statistical ones. The fitting algorithm is described in detail in [4]. As previously the points with n χ ≥ 2 are excluded from fit in our algorithm, where [4] Here B i m is the measured value of nuclear slope at s i with experimental error σ i , B (s i ; a) is the expected value from the fitting function with best χ 2 /n.d.f. among (2a) -(2d) for approximation of all range of available energies, the parameters α j are from N -dimensional vector α = {α 1 , ..., α N }. We consider the estimates of fit parameters as the final results if there are no excluded points for present data sample. The fraction of excluded points is about 2% for pp as well as forpp elastic scattering for low |t| domain. The maximum relative amount of rejected points is about 3%/12% for linear ln dσ/dt parametrization at intermediate |t| values for pp/pp scattering respectively.

Low |t| domain
The experimental dependence B(s) and corresponding fits by (2a) -(2d) are shown in Fig.1 for elastic pp scattering, Table 1 contains values of fit parameters. As seen the fitting functions (2a), (2d) describe the pp experimental data statistically acceptable only for √ s ≥ 5 GeV. The RHIC point at √ s = 200 GeV does not contradict the common trend within a large error bars and can't discriminate the approximations under study. In general the LHC results [7,8,9] added in fitting sample result in better agreement of fits (2a) -(2d) in comparison with previous study [4]. All fits are very close to each other in energy domain 10 GeV √ s 1 TeV, the quadratic in logarithm function (2d) shows some faster increasing of slope and noticeable difference from another fit functions in multi-TeV region only. It seems the ultra-high energy domain is suitable for separation of various parameterizations. Fitting functions (2b), (2c) allow us to describe experimental data at all energies with reasonable fit quality for pp ( Table 1). The functions (2a) -(2c) agrees very well for √ s ≥ 5 GeV, furthermore there is no visible difference between modifications (2b) and (2c) in all experimentally available energy domain. The function (2c) demonstrate some better quality for fit of full data sample (Table 1) than function (2b) in contrast with previous analysis [4]. The mean value of (4) for excluded points (n χ ) is equal 4.7 for pp data sample for parametrization (2c). The accounting for LHC data leads to some decreasing of values of B 0 and increasing of a 1 parameters for all fitting functions (2a) -(2d) under study in comparison with values of corresponding parameters in previous investigation [4]. This behavior of a 1 with collision energy agrees well with predicted growth of α ′ P with increasing of √ s [10,11]. As seen from Table 1 the third term in both the (2b) and the (2c) gives the main contribution at √ s < 5 GeV in the case of elastic protonproton scattering, i.e. describe the sharp changing of slope in the low energy domain. Therefore B pp ∝ ln ε at high √ s in accordance with (2b) and (2c). Such asymptotic behavior is in qualitative agreement with energy dependence of the slope parameter forpp collisions [4]. The increasing of a 1 exhibits that B pp growth some faster in multi-TeV region than one can expect from the trend based on data sample at √ s ≤ 200 GeV. This suggestion is confirmed by improvement of fit quality for one fitting function (2d) for present data sample in comparison with fit qualities for experimental points at √ s ≤ 200 GeV [4]. Values of a 2 obtained in present study and for fit at √ s ≤ 200 GeV are close within errors for functions (2b) and (2d) but accounting for the LHC data results in some increasing of the absolute value of a 3 in fit by (2b). The absolute values of a 2 and a 3 are the same within error bars for function (2c) for Table 1 and for fit in energy domain √ s ≤ 200 GeV [4]. The dot curve is the fit of experimental slope by the function (2a), thick solid -by the (2b), dashed -by the (2c), thin solid -by the (2d). The shaded band corresponds to the spread of fitting functions for previous analysis [4].
The comparison with earlier studies [12,13] confirms the conclusion above that accounting for high energy data points leads to increasing of value of a pp 1 parameter and growth of slope parameter seems to be faster in TeV-region than that at lower energies. Accordingly the a pp 1 value for function (2d) is closer significantly to the α ′ P ≈ 0.25 GeV −2 than that for previous analysis in energy range √ s ≤ 200 GeV [4]. On the other hand the values of the a pp 1 parameter obtained for fitting function (2a) is some larger than the prediction for α ′ P from Pomeron inspired model for TeV-energy domain [11]. Furthermore the proton-proton results from fit by function (2d) allow the estimation 2α ′ P (s)| √ s=8TeV = 0.86 ± 0.08 which is almost twice larger than the corresponding prediction from [11].
Predictions for B are obtained for some facilities based on the fit results shown above for pp and on the results from [4] forpp. Estimations at low |t| for different intermediate energies of the projects FAIR and NICA are shown in the Table 2 and for high energy domain are presented in the Table 3. As expected the functions (2b) and (2c) predicts for FAIR the B values coincide with each other within errors. The approximation functions (2a) and (2d) can predict for √ s ≥ 5 GeV only. The Pomeron inspired function (2a) predicts slope parameter values smaller significantly than that for modified fitting functions (2b) and (2c) at FAIR energies. Forpp the predictions with help of quadratic in ln ε function (2d) are equal with estimations based on any another fitting function under study within large error bars at √ s ≤ 14.7 GeV. In the case of elastic pp scattering the functions (2a) -(2c) predict just the equal values of B within error bars at any collision energy √ s discussed here. The difference between estimations from (2a) -(2c) and from (2d) onsets at the final energy of the LHC project √ s = 14 TeV only. Our prediction with (2d) function for RHIC energy is equal with early prediction for close energy based only on slope data in the region 5 < √ s < 62 GeV [15] within errors. But (2d) underestimates the B values in ultra-high energy domain √ s > 40 TeV in comparison with results based on the approach without odderons [15]. It should be emphasized that in contrast with previous analysis [4] the present fits by functions (2a) -(2d) of data sample included of the LHC results predict the similar increasing of B with energy as most of phenomenological models [16]. The B value predicted for the LHC at √ s = 14 TeV by (2a) only is close with errors to the predictions from [17,18], 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 [18] at √ s = 14 TeV coincides with estimation of B within error bars from fit function (2d) with fastest growth of B with √ s in multi-TeV region. But most of estimations of B at √ s = 14 TeV from Table 3 agree well within errors with model prediction from [19]. However the model estimates at √ s = 14 TeV described above were obtained for B (t = 0) and the t-dependence of slope shows the slight decreasing of B for the model with threepomeron exchange [18] and faster decreasing of B for the model from [19] at growth of momentum transfer up to |t| ≈ 0.1 GeV 2 . Therefore one can expect that the model with hadronic amplitude corresponding to the exchange of three pomerons [18] will be in the better agreement with values of B from Table 3 [21] which is larger noticeably than the predictions from Table 3 at the same √ s. Therefore the saturation regime will not be reached, at least, at the LHC energy √ s = 14 TeV as suggested, for example, in the model from [22] or simple saturation can not be enough in order to describe the LHC data at quantitative level.

Intermediate |t| domain
As indicated in [4] the situation is more complicated for intermediate |t| domain. Figure 2 shows the experimental data and corresponding fits for energy dependence of slope parameter at intermediate |t| for exponential approximation with index linear in |t| of differential cross-sections in elastic pp andpp scattering. The parameter values for fitting are indicated in Table 4 for various interaction types.
Usually the fit qualities are poorer for intermediate |t| values than that for low |t| range in pp elastic collisions for linear parametrization of ln(dσ/dt). The fitting functions (2a) and (2d) agree with experimental points qualitatively for √ s ≥ 5 GeV only. Furthermore in the first case there is significant discrepancy between experimental point and fit curve at the LHC energy also (Fig.2a). The "expanded" functions (2b), (2c) approximate experimental data at all energies reasonably with close fit qualities (Table 4), but these functions show a slow growth of slope parameter with energy increasing at √ s ≥ 100 GeV (Fig.2a). It should be stressed that the experimental point at the LHC energy leads to the dramatic change of behavior of the fitting function (2d) in comparison with previous analysis [4]. At present the fitting function (2d) 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 (2d) of experimental data sample at √ s ≤ 200 GeV [4]. As shown in previous analysis [4] the thepp experimental data admit the approximation by (2a), (2d) for all energy range but not only for √ s ≥ 5 GeV (Fig.2b). Thus the parameter values are shown in Table 4 for approximation by (2a), (2d) of all available experimental data. The fit curves show (very) close behaviors for both the present and previous analyses in the case of the Fig.2b. Thepp data disagreement with Pomeron inspired fitting function (2a) very significantly (Fig.2b). Functions (2b) and (2c) show a close behavior at all energies forpp data from linear parametrization of ln dσ/dt. These fitting functions have a better fit quality than (2d) but fits by functions (2b), (2c) are still statistically unacceptable. The n χ = 2.9 for excluded pp data with (2c) function and n χ = 18.3 for points excluded frompp fitted data sample for (2b) fitting function. As seen from Fig.2b behavior of the energy dependence of slope parameter for elastic pp scattering is close to the quadratic in logarithm function Bp p ∝ ln 2 ε at high √ s as well as for elastic pp collisions. But in the last case it is difficultly to make the unambiguous conclusion because there is only the LHC point in the asymptotic region. The estimations of asymptotic shrinkage parameter are following for fitting functions (2a) Table 5 for different energies of FAIR, NICA, and in the     12 ± 5 9 ± 9 8 ± 10 7 ± 11 6 ± 12 4 ± 13 2 ± 14 * The ultimate energy upgrade of LHC project [14].
value predicted for LHC at √ s = 14 TeV by (2d) is most close to the some model expectations [18,19]. Taking into account predictions in Table 2 based on the fitting functions (2a) -(2d) for low |t| one can suggest that the model with hadronic amplitude corresponding to the exchange of three pomerons [16,18]

∆B and N N data analysis
In accordance with rules from [4] the difference of slopes (∆B) for antiproton-proton and proton-proton elastic scattering is calculated for each function (2a) -(2d) under study with parameters from correspondedpp and pp fits: The energy dependence of ∆B is shown at Fig.3a and Fig.3b for low and intermediate |t| respectively. One can see that the difference of slopes decreasing with increasing of energy for low |t| domain (Fig.3a) as well as in the previous analysis [4]. The fitting functions (2b), (2c) demonstrate much faster decreasing of ∆B with increasing of √ s than that the functions (2a) and (2d). At present the proton-proton experimental data at highest available energy 8 TeV don't contradict with fast (square of logarithm of energy) increasing of slope at high energies in general case. Such behavior could be agreed with the asymptotic growth of total cross section. Furthermore in contrast with the previous analysis [4], here the quadratic in ln ε function (2d) leads to much smaller difference ∆B forpp and pp scattering in high energy domain for both low (Fig.3a) and intermediate (Fig.3b) values of |t|. The Pomeron inspired function (2a) only predicts the decreasing of ∆B with energy growth at intermediate |t| (Fig.3b) for any values of √ s. The parameterizations (2b) -(2d) predict the decreasing of difference of slopes at low and intermediate energies and fast increasing of ∆B at high energies for intermediate |t| domain (Fig.3b). As expected the most slow changing of ∆B is predicted by Pomeron inspired function (2a) at asymptotic energies. All fitting functions with experimentally inspired parameters don't predict the constant zero values of ∆B at high energies. But it should be emphasized that only separate fits were made for experimental data for pp andpp elastic reactions above. These results indicate on the importance of investigations at ultra-high energies both pp andpp elastic scattering for study of many fundamental questions and predictions related to the general asymptotic properties of hadronic physics.
Also we have analyzed general data samples for pp andpp elastic scattering. Slope parameters (B and b) shows a different energy dependence at √ s < 5 GeV in proton-proton and antiproton-proton elastic reactions in any |t| domains under study. Thus slopes for nucleonnucleon data are investigated only for √ s ≥ 5 GeV below. We have included in fitted samples only pp andpp points which have been included in corresponding final data samples at separate study pp andpp elastic reactions above. We did not exclude any points from N N data sample, we change only the low energy boundary for fitted domain. Experimental data for slope in nucleon-nucleon elastic scattering against collision energy are shown in Fig.4a at low |t| and in Fig.4b for intermediate |t| together with fits by functions (2a) -(2d). We have fitted the general nucleon-nucleon data sample at range of low energy   boundary s min = 25, 100, 225 and 400 GeV 2 . The fit parameter values are indicated in Table 7 on the first line for low boundary of the fitted energy domain s min = 25 GeV 2 and on the second line -for s min = 400 GeV 2 . The fit quality improves for most parameterizations under consideration at increasing of s min , thus fitting functions (2a) -(2d) are shown at Fig.4 for s min = 400 GeV 2 . As seen from Fig.4a there is no experimental data forpp between √ s = 5 GeV and √ s = 10 GeV at low |t|. This energy domain will available for further FAIR facility. One need to emphasize the fit quality is some poorer χ 2 /n.d.f. ≃ 2.3 − 2.9 at √ s ≥ 10 GeV than that for √ s ≥ 5 GeV for functions (2a) and (2c). For all cases of s min indicated above the value of the a 1 parameter obtained with the function (2a) agrees qualitatively with the prediction within the framework of Pomeron model, but the value of the asymptotic shrinkage parameter (2α ′ P = 0.662 ± 0.010) obtained at s min = 400 GeV 2 is some larger than the prediction for α ′ P from Pomeron inspired model for TeV-energy domain [11]. Also results from fit by function (2d) with acceptable quality at s min = 400 GeV 2 allow the estimation 2α ′ P (s)| √ s=8TeV = 0.74 ± 0.12 which is some larger than the corresponding prediction from [11]. Furthermore the estimations for 2α ′ P (s)| √ s=8TeV do not depend on s min within error bars. At low |t| all functions (2a) -(2d) are close to each other at energies up to √ s ∼ 1 TeV at least and shows quasi-linear behavior for parameter values obtained by fits with s min = 25 GeV 2 and s min = 400 GeV 2 . This observation confirms the suggestion that B N N ∝ ln ε at high √ s at low |t| values. 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 (2d) in comparison with previous analysis [4] due to experimental points at the LHC energies. At present the fitting function (2d) 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 (2d) of experimental data sample at √ s ≤ 1.8 TeV [4]. We have analyzed the nucleon-nucleon data for slope parameter B at intermediate |t| values for exponential parametrization with index linear in |t| of dσ/dt (Fig.4b). As seen experimental pp andpp data for B differ significantly up to √ s ≃ 10 GeV at least that results in unacceptable fit qualities for all functions under study (χ 2 /n.d.f. ≃ 28.9 for best fit by quadratic in logarithm function). The Pomeron inspired function (2a) contradicts with experimental data at any s min . We would like to emphasize that the best fit quality for (2a) is obtained at s min = 100 GeV 2 χ 2 /n.d.f. ≃ 9.45 but it is statistically unacceptable too. Functions (2b) -(2d) agree with experimental dependence B(s) reasonably and have a close fit qualities. Furthermore the functions (2b) and (2c) demonstrate very close behaviors in all range of √ s under consideration. Best fit is "expanded" function (2c) in difference with previous analysis [4]. As seen from Fig.4b Fig.4b shows that the experimental point at the LHC energy leads to faster increasing of most of fitting functions in multi-TeV region in comparison with previous analysis [4]. Comparison between Fig.4a and Fig.4b  One can conclude the slope parameters for pp andpp elastic scattering show universal behavior at √ s ≥ 20 GeV and "expanded" functions represent the energy dependence for both the low and the intermediate |t| ranges for this energy domain. Thus quantitative analysis of slopes at different |t| allows us to get the following estimation of low energy boundary: √ s ≃ 20 GeV for universality of elastic nucleon-nucleon scattering. This estimation agrees qualitatively with both the results for differential cross-sections of pp andpp elastic reactions based on the crossing-symmetry and derivative relations [1] and the results for global scattering parameters [22].

Conclusions
The present status of diffraction slope parameter for elastic pp andpp scattering is analyzed over the full energy domain as well as predictions for some facilities. The "expanded" parameterizations allow us to describe experimental B at all available energies in low |t| domain for pp as well as for intermediate |t| values for pp andpp quite reasonably. The similar situation is observed for fits of data samples joined for elastic N N scattering. Therefore "expanded" functions can be used as a reliable fits for wide range of momentum transfer at all energies. The new LHC data lead to dramatic change of behavior of quadratic in logarithm function and usually to better agreement between various fitting function in comparison with the analysis of pp data at √ s ≤ 200 GeV. At low values of |t| the "standard" approximation differs from expanded ones and experimental data mostly in the low energy region; at intermediate |t| this function is unacceptable for fitting of experimental N N data at all. The intermediate |t| range is preferable with respect to the low |t| values for discrimination of various phenomenological parameterizations for B(s) dependence. Based on the nuclear slope the low energy boundary for universality of elastic nucleon-nucleon scattering is estimated as √ s ≃ 20 GeV for both the low and the intermediate |t| values that is in agreement with the hypothesis of a universal shrinkage of the hadronic diffraction cone at high energies. The difference of slopes for antiproton-proton and proton-proton elastic scattering (∆B) shows the opposite behaviors at high energies for low and intermediate |t| domains (decreasing / increasing, respectively) for all fitting functions with the exception of Pomeron inspired one. All underlying fitting functions with experimentally inspired values of parameters don't predict the zero value for ∆B at both the low and the intermediate |t| ranges at high energies. Slop analysis of joined N N data samples allows us to estimate the asymptotic shrinkage parameter parameter for various domains of |t|. The estimation of the α ′ P obtained with quadratic in logarithm function for N N data at √ s = 8 TeV for low |t| is noticeably larger than the expectation from the Pomeron theory. But the growth of α ′ P with increasing of √ s is observed from the comparison of the fit results from present study and our earlier analysis for √ s ≤ 1.8 TeV. The such behavior of α ′ P agrees with Pomeron inspired model. The growth of α ′ P is observed with increasing of momentum transfer. Based on the fit results the predictions for slope parameters B and b are obtained for elastic pp andpp scattering in energy domains of some facilities. It seems the phenomenological model with hadronic amplitude corresponding to the exchange of three pomerons describes the B some closer to the experimental fit inspired values at the LHC energy both at low and intermediate |t| than other models.