The black disk and the dip in the differential elastic cross section at asymptotic energy

We test the validity of the black disk limit in elastic scattering by studying the evolution of the dip in the scaling variable $\tau =-t_D{\sigma }^{tot}$, where $t_D$ is the transverse momentum squared at the dip and ${\sigma }_{tot}$ the total cross section. As $s\to \mathrm{\infty }$ and $-t_D\to 0$, $\tau$ may consistently be approaching the black disc value, $\tau \underset{\sqrt{s}\to \mathrm{\infty }}{\overset{}{\to }}{\tau }_{BD}=35.92$ GeV${}^2$ mb.