Evolution equation for soft physics at high energy

TL;DR

This paper models the energy evolution and saturation effects in high-energy proton-proton and antiproton-proton elastic scattering using a non-linear logistic equation, predicting distinctive features for LHC energies.

Contribution

It introduces a semi-quantitative evolution equation based on a non-linear logistic model to describe high-energy elastic scattering and predicts novel features at LHC energies.

Findings

Geometrical scaling occurs at the black disk limit.

Scaling develops first for small values of the scaling variable.

Predicted differential cross-section at LHC shows two zeros and a large-|t| minimum.

Abstract

Based on the non-linear logistic equation we study, in a qualitative and semi-quantitative way, the evolution with energy and saturation of the elastic differential cross-section in $pp(\bar{p}p)$ collisions at high energy. Geometrical scaling occurs at the black disk limit, and scaling develops first for small values of the scaling variable $|t|\sigma_{tot.}$. Our prediction for $d \sigma / \ d t$ at LHC, with two zeros and a minimum at large $|t|$ differs, as far as we know, from all existing ones.