Optical Theorem, Crossing Property and Derivative Dispersion Relations: Implications on the Asymptotic Behavior of $\sigma_{tot}(s)$ and $\rho(s)$

TL;DR

This paper analyzes the asymptotic behavior of total cross sections and the rho parameter in high-energy proton-proton and antiproton-proton collisions using fundamental theoretical principles, providing parameterizations that fit experimental data and predict the onset of asymptotic regimes.

Contribution

It introduces new parameterizations based on crossing, dispersion relations, and the optical theorem that effectively describe scattering parameters at multi-TeV energies and suggest the asymptotic regime begins around 25.5-130 TeV.

Findings

Asymptotic behavior likely begins between 25.5 and 130 TeV.

The Pomeranchuk theorem's generalized form is more consistent with data.

The energy dependence of scattering parameters is well approximated in the multi-TeV region.

Abstract

In this paper, one presents some results concerning the behavior of the total cross section and $\rho$-parameter at asymptotic energies in proton-proton ($pp$) and antiproton-proton ($\bar{p}p$) collisions. For this intent, we consider three of the main theoretical results in high energy physics: the crossing property, the derivative dispersion relation, and the optical theorem. The use of such machinery allows the analytic formulas for wide set of the measured global scattering parameters and some important relations between them. The suggested parameterizations approximate simultaneously the energy dependence for total cross section and $\rho$-parameter for $pp$ and $\bar{p}p$ with statistically acceptable quality in multi-TeV region. Also the qualitative description is obtained for important interrelations, namely difference, sum and ratio of the antiparticle-particle and particle-particle total cross sections. Despite the reduced number of experimental data for the total cross section and $\rho$-parameter in TeV-scale, which turns any prediction for the beginning of the asymptotic domain a hard task, the fitting procedures indicates that asymptotia lies in the energy range 25.5-130 TeV. Moreover, in the asymptotic regime, one obtains $\alpha_{\mathbb{P}}=1$. Detailed quantitative study of energy behavior of measured scattering parameters and their combinations in ultra-high energy domain indicates that the scenario with the generalized formulation of the Pomeranchuk theorem is more favorable with respect to the original formulation of this theorem.