Models for differential cross section in neutron-proton scattering and their implications

TL;DR

The study develops five purely phenomenological models to describe the neutron-proton elastic differential cross section $\frac{d\sigma}{dt}$ over $\sqrt{s}=3.36$–$26.02$ GeV. Each model embeds a Regge-inspired forward diffractive core with energy-dependent slopes and localized $t$-dependent corrections, complemented by additive C-odd and isospin terms to reproduce the dip–bump structures. Fitting to NP data via $\chi^2$ minimization, the models accurately describe the forward peak, dip evolution, and large-$|t|$ behavior, while predicting total, elastic, and inelastic cross sections and the slope parameter $B(s)$ in agreement with Regge expectations. The results yield analytically well-behaved parameterizations that capture diffractive shrinkage and can be extended to other NN channels and higher energies, contributing a compact phenomenological tool for elastic scattering analyses.

Abstract

A few analytic exponential models of elastic differential cross section, constructed as purely phenomenological models, are proposed and tested. The models incorporate energy-dependent exponential slopes, power-law prefactors, and localized Gaussian modifications which are built to reproduce the observed dip region, supplemented in some cases by logarithmic $t$-dependent slopes. Simple additive sub-leading exponential contributions that represent charge conjugation and isospin roles are introduced in the models to increase applicability and quality of fit across elastic differential cross section data of $np$, $n\bar{p}$, $pp$, and $p\bar{p}$ elastic scattering. The models reproduce the characteristic features of the elastic scattering data such as the dip-bump structure, shrinkage of the forward peak, and controlled curvature that is localized around the dip. Parameters of the models are found by fitting the experimental data of elastic $np$ differential cross section in an energy range of $\sqrt{s}$ = 3.36 GeV to 26.02 GeV, across a momentum range of $0.065 \leq \mid t\mid \leq 5.341 \textrm{GeV}^{2}$. The parameter values and their ranges, obtained by $χ^{2}$ minimization are found within their assumed expected bounds with the $np$ data fitting. The total cross section, the slope parameter, the interaction radius, the total elastic cross section, the inelastic cross section, the ratios $σ_{el}/σ_{tot}$, and $σ_{inel}/σ_{tot}$ are predicted by the models for the $np$ scattering at all the energies, which show accurate quantitative agreement with their reference values. The results show that the proposed models not only provide accurate quantitative description of $np$ elastic differential cross section but also yield estimates of the observables that are consistent with theoretical expectations from Regge phenomenology and high-energy scattering constraints.

Figures (9)

• Figure 1: (a) Model 1 fits on the $np$ elastic differential cross section data of Dataset 1 ($3.363 \leq \sqrt{s} \leq 4.935 \textrm{GeV}$) of Table \ref{['tab:expdata']}. Filled squares, filled rectangles, filled up triangles, filled circles, filled right triangles, filled down triangles, filled diamonds, and filled six-sided stars represent the experimental data of Dataset of \ref{['tab:expdata']} at 4.935, 4.741, 4.54, 4.329, 4.109, 3.876 , 3.628, and 3.363 GeV respectively. The data and model values are multiplied by $10^{-3(n-1)}$, where $n$ is the number of curve and corresponding data set starting from the top. The solid line represents the fit of our model to the data. (b) Model 1 fits on the $np$ elastic differential cross section data of Dataset 2 ($4.74 \leq \sqrt{s} \leq 6.704 \textrm{GeV}$) of Table \ref{['tab:expdata']}. Filled squares, filled rectangles, filled up triangles, filled circles, filled right triangles, filled down triangles, and filled diamonds represent the experimental data of Dataset of Table \ref{['tab:expdata']} at 6.704, 6.418, 6.119, 5.805, 5.473, 5.12, and 4.74 GeV, respectively. The data and model values are multiplied by $10^{-3(n-1)}$, where $n$ is the number of curve and corresponding data set starting from the top. The solid line represents the fit of our model to the data. (c) Model 1 fits on the $np$ elastic differential cross section data of Dataset 3 ($3.466-4.409 \leq \sqrt{s} \leq 7.024-7.538 \textrm{GeV}$) of Table \ref{['tab:expdata']}. Filled squares, filled rectangles, filled up triangles, filled circles, filled right triangles, and filled down triangles, represent the experimental data of Dataset of Table \ref{['tab:expdata']} at 7.024-7.538, 6.47-7.024, 5.863-6.47, 5.187-5.863, 4.409-5.187, and 3.466-4.409 GeV, respectively. Data and model values are multiplied by $10^{-3(n-1)}$, where $n$ is the number of curve and corresponding data set starting from the top. The solid line represents the fit of our model to the data. (d) Model 1 fits on the $np$ elastic differential cross section data of Dataset 4 ($13.748 \leq \sqrt{s} \leq 26.019 \textrm{GeV}$) of Table \ref{['tab:expdata']}. Filled squares, filled rectangles, filled up triangles, filled circles, filled right triangles, filled down triangles, and filled diamonds represent the experimental data of Dataset of Table \ref{['tab:expdata']} at 26.019, 24.536, 22.956, 21.471, 19.416, 16.823, and 13.748 GeV, respectively. Data and model values are multiplied by $10^{-3(n-1)}$, where $n$ is the number of curve and corresponding data set starting from the top. The solid line represents the fit of our model to the data.
• Figure 2: Model 2 fits on the $np$ elastic differential cross section data of Dataset 1 of Table \ref{['tab:expdata']}. Same legend for parts (a), (b), (c), and (d) as in the Figure 1.
• Figure 3: Model 3 fits on the $np$ elastic differential cross section data of Dataset 1 of Table \ref{['tab:expdata']}. Same legend for parts (a), (b), (c), and (d) as in the Figure 1
• Figure 4: Model 4 fits on the $np$ elastic differential cross section data of Dataset 1 of Table \ref{['tab:expdata']}. Same legend for parts (a), (b), (c), and (d) as in the Figure 1.
• Figure 5: Model 5 fits on the $np$ elastic differential cross section data of Dataset 1 of Table \ref{['tab:expdata']}. Same legend for the parts (a), (b), (c), and (d) as in the Figure 1.