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Description
This work presents an analysis of the scaling properties of total and elastic cross sections in proton–proton and neutron–proton collisions from ISR to LHC and cosmic-ray energies. Using the gray disk model, we introduce two energy-dependent functions, R(s) and f(s), that describe the evolution of the effective interaction radius and the gluon saturation density, respectively. The parametrization is consistent with the Froissart bound and provides an excellent description of the experimental data across several decades in energy, including results from Bevatron, CERN, Fermilab, and Pierre Auger Observatory. The comparison between pp and np systems reveals a universal asymptotic behavior, suggesting that both approach the black disk limit at ultra-high energies. These findings contribute to a unified understanding of hadronic scattering in the pre-saturation regime.
