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In this paper, one presents some results concerning the behavior of the totalcross section and $\rho$-parameter at asymptotic energies in proton-proton($pp$) and antiproton-proton ($\bar{p}p$) collisions. For this intent, weconsider three of the main theoretical results in high energy physics: thecrossing property, the derivative dispersion relation, and the optical theorem.The use of such machinery allows the analytic formulas for wide set of themeasured global scattering parameters and some important relations betweenthem. The suggested parameterizations approximate simultaneously the energydependence for total cross section and $\rho$-parameter for $pp$ and $\bar{p}p$with statistically acceptable quality in multi-TeV region. Also the qualitativedescription is obtained for important interrelations, namely difference, sumand ratio of the antiparticle-particle and particle-particle total crosssections. Despite the reduced number of experimental data for the total crosssection and $\rho$-parameter in TeV-scale, which turns any prediction for thebeginning of the asymptotic domain a hard task, the fitting proceduresindicates that asymptotia lies in the energy range 25.5-130 TeV. Moreover, inthe asymptotic regime, one obtains $\alpha_{\mathbb{P}}=1$. Detailedquantitative study of energy behavior of measured scattering parameters andtheir combinations in ultra-high energy domain indicates that the scenario withthe generalized formulation of the Pomeranchuk theorem is more favorable withrespect to the original formulation of this theorem.

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