Proton-proton, pion-proton and pion-pion diffractive collisions at   ultra-high energies
  The LHC energies are those at which the asymptotic regime in hadron-hadron diffractive collisions ($pp,\pi p,\pi\pi$) might be switched on. Based on results of the Dakhno-Nikonov eikonal model which is a generalization of the Good-Walker eikonal approach for a continuous set of channels, we present a picture for transformation of the constituent quark mode to the black disk one. In the black disk mode ($\sqrt s \geq 10$ TeV) we have a growth of the logarithm squared type for total and elastic cross sections, $\sigma_{tot}\sim\ln^2s$ and $\sigma_{el}\sim\ln^2s$, and $(\tau={\bf q}_\perp^2\sigma_{tot})$-scaling for diffractive scattering and diffractive dissociation of hadrons. The diffractive dissociation cross section grows as $\sigma_{D}\sim\ln{s}$, $\sigma_{DD}\sim\ln{s}$, and their relative contribution tends to zero: $\sigma_{D}/\sigma_{tot}\to 0$, $\sigma_{DD}/\sigma_{tot}\to 0$. Asymptotic characteristics of diffractive and total cross sections are universal, and this results in the asymptotical equality of cross sections for all types of hadrons (the Gribov's universality). The energy scale for switching on the asymptotic mode is estimated for different processes. 
V.V. Anisovich, K.V. Nikonov, V.A. Nikonov, J. Nyiri
hep-ph