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

In this paper, one presents some results concerning the behavior of the total cross section and $ρ$-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 $ρ$-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 $ρ$-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 $α_{\mathbb{P}}=1$. 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.