Eikonal fit to $pp$ and $\bar{p}p$ scattering and the edge in the scattering amplitude

TL;DR

This paper performs a detailed eikonal fit to proton-proton and antiproton-proton scattering data across a wide energy range, revealing a fixed-shape impact-parameter edge in the scattering amplitude and providing analytic expressions for extrapolation.

Contribution

It introduces a comprehensive eikonal analysis of scattering data, characterizes the impact-parameter edge structure, and extends analytic models for high-energy extrapolations.

Findings

The edge region has a fixed shape with a peak near the black disk radius.

The edge width remains approximately 1 fm across energies.

Analytic expressions accurately describe cross sections and rho values, enabling reliable high-energy predictions.

Abstract

We make a detailed eikonal fit to current data on the total and elastic scattering cross sections, the ratios $\rho$ of the real to the imaginary parts of the forward elastic scattering amplitudes, and the logarithmic slopes $B$ of the differential cross sections $d\sigma/dt$ at $t=0$, for proton-proton and antiproton-proton scattering at center-of-mass energies $W$ from 5 GeV to 57 TeV. The fit allows us to investigate the structure of the eikonal amplitudes in detail, including the impact-parameter structure of the energy-independent edge in the scattering amplitude shown to exist by Block {\em et al.} \cite{edge}. We show that the edge region has an essentially fixed shape with a peak at approximately the "black disk" radius $R_{\rm tot}=\sqrt{\sigma_{\rm tot}/2\pi}$ of the scattering amplitude, a constant width $t_{\rm edge}\approx 1$ fm, and migrates to larger impact parameters with increasing energy proportionally to $R_{\rm tot}$. We comment on possible physical mechanisms which could lead to the edge. We show that the eikonal results for the cross sections and $\rho$ values are described to high accuracy by analytic expressions of the forms used in earlier analyses by Block and Halzen, and extend the result to the elastic-scattering slope parameter $B$. These expressions provide simple extrapolations of the results to much higher energies. Finally, we calculate the survival probabilities for large rapidity gaps in the scattering.