Where is the black disc limit?

By using realistic models for elastic hadron scattering, we demonstrate that at present accelerator energies the s-channel unitarity bound is safe, not to be reached before \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$10^5$\end{document} GeV, while the black disc limit is saturated around 6 TeV. By increasing the energy, a larger transparency of the scattered particles near the center, surrounded by a black ring, is expected. Unitarity effects are incorporated and their effects are studied both in a phenomenological and formal, analytical way.