Empirical parametrization of the nucleon-nucleon elastic scattering amplitude at high beam momenta for Glauber calculations and Monte Carlo simulations

A parametrization of the nucleon-nucleon elastic scattering amplitude is needed for future experiments with nucleon and nuclear beams in the beam momentum range of 2–50 $\mathrm{GeV}/c/\text{nucleon}$. There are many parametrizations of the amplitude at $P_{\mathrm{lab}}>$ 25–50 $\mathrm{GeV}/c$, and at $P_{\mathrm{lab}}\le 5$ $\mathrm{GeV}/c$. Our paper is aimed at covering the range between 5–50 $\mathrm{GeV}/c$. The amplitude is used in Glauber calculations of various cross sections and Monte Carlo simulations of nucleon-nucleon scatterings. Usually, the differential nucleon-nucleon elastic scattering cross sections are described by an exponential expression. Corresponding experimental data on $pp$ interactions at $\left|t\right|>$ 0.005 $(\mathrm{GeV}/c){}^2$ and $\left|t\right|\le 0.125$ $(\mathrm{GeV}/c){}^2$ have been fit. We propose formulas to approximate the beam momentum dependence of these parameters in the momentum range considered. The same was done for $np$ interactions at $\left|t\right|\le 0.5$ ($\mathrm{GeV}/c$)${}^2$. Expressions for the momentum dependence of the total and elastic cross sections, and the ratio of real to imaginary parts of the amplitude at zero momentum transfer are also given for $pp$ and $np$ collisions. These results are sufficient for a first approximation of the Glauber calculations. For more exact calculations we fit the data at $\left|t\right|>0.005$ $(\mathrm{GeV}/c){}^2$ without restrictions on the maximum value of $\left|t\right|$ using an expression based on two coherent exponentials. The parameters of the fits are found for the beam momentum range 2–50 $\mathrm{GeV}/c$.

Figure 1

(a), (b) Fit parameters $A$ and $B$ as functions of the projectile momentum, $P_{\mathrm{lab}}$. Open circles are results of the fit to the data [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60]. Filled circles are the results for the latest COSY data [37]. Stars present results of the fit to the EDDA data [48]. Diamonds are data from Ref. [49] [see Fig. 1]. Dashed (magenta) curves are approximations of the dependencies. Descriptions of the solid curves are given in the text. (c) Elastic cross section as a function of $P_{\mathrm{lab}}$. The data points are from the PDG database [34].

Figure 2

${\sigma }_{pp}^{\mathrm{tot}},{\sigma }_{pp}^{el}$, and ${\rho }_{pp}$ as functions of $P_{\mathrm{lab}}$. Points are data from the PDG database [34]. Dashed lines are extrapolations of the PDG parametrizations. The solid lines are our approximations.

Figure 3

Fitted parameters $A_1,A_2,B_1$, and $B_2$ as functions of projectile momentum. Solid points (blue) are results of the fitting of the data [50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63] with constant $\varphi$. Open points (red) are results for the EDDA data [48]. Arrows mark the results for $A_1$ and $B_1$ at 5.5 and 10 $\mathrm{GeV}/c$. Solid lines are approximations (see below).

Figure 4

Differential cross sections at various momentum transfers. The solid points are reference data [52, 56], and the open points have been fit [52, 53, 58]. The blue solid lines are the results of the fit.

Figure 5

${\sigma }^{\mathrm{tot}},{\sigma }^{el}$, and $\rho$ of $pn$ and $np$ interactions as functions of $P_{\mathrm{lab}}$. The data points are from [34]. Solid (black) lines are approximations for $pn$ and $np$ interactions. Dotted (magenta) lines are our approximations for $pp$ interactions. Dashed (blue) lines are calculations (see the text).

Figure 6

Fit results of Eq. (1) to $np$ differential elastic scattering cross sections [64, 65, 66, 67, 68, 69, 70, 71] at $\left|t\right|<0.5$ ($\mathrm{GeV}/c$)${}^2$. Points are the fit results. Solid lines are approximations [Eqs. (25) and (26)]. For a description of the dashed line see the text.

Figure 7

Data points are from Refs. [64, 67, 69] for (a), (b), and (c), respectively. Lines are fit results described in the text.

Figure 8

(a) Differential cross section distributions of $np$ elastic scattering at $P_{\mathrm{lab}}=5.1$, 6.12, and 6.77 $\mathrm{GeV}/c$ [66] (open points) and at $P_{\mathrm{lab}}=5$, 6, and 7 $\mathrm{GeV}/c$ [67] (closed points). Dashed and solid lines are fit results by Eq. (27) with $A_1=A_2=0$ to the data (Perl 1970: [66]) and (Stone 1977: [67]), correspondingly. (b) $A_3$ values from the fits to the data. The solid line is Eq. (28).

Figure 9

Fit results of Eq. (27) to the data [66, 67, 68, 69, 70]. Points are the fit results. Stars (green) are the results of fitting of Eq. (27) with $A_2=0$ to the data [64]. Solid lines are approximations (see below). Dashed lines are approximations of the $A$ and $B$ parameters of the one exponential expression.

Figure 10

Description of $np$ elastic differential cross sections by Eq. (27) with approximations (29, 30, 31, 32). Points are experimental data [66, 67, 69, 70]. The experimental errors displayed are statistical only. Lines are calculation results.