Determining the Proximity of \Gamma^{\ast} N Scattering to the Black Body Limit Using DIS and J/ \psi Production

Abstract

We use information about DIS and $J/ \psi$ production on hydrogen to model the $t$-dependence of the $\gamma^{\ast} N$ scattering amplitude. We investigate the profile function for elastic scattering of hadronic components of the virtual photon off both a nucleon and heavy nuclear target, and we estimate the value of the impact parameter where the black body limit is reached. We also estimate the fraction of the cross section that is due to hadronic configurations in the virtual photon wave function that approach the unitarity limit. We extract, from these considerations, approximate lower limits on the values of $x$ where the leading twist approximation in DIS is violated. We observe that the black body limit may be approached within HERA kinematics with $Q^{2}$ equal to a few GeV$^2$ and $x \sim 10^{-4}$. Comparisons are made with earlier predictions by Munier {\it et al.}, and the longitudinal structure function is compared with preliminary HERA data. The principle advantage of our method is that we do not rely solely on the $t$-dependence of $\rho$-meson production data. This allows us to extend our analysis down to very small impact parameters and dipole sizes. Finally, we perform a similar calculation with a $^{208}$Pb target, and we demonstrate that the black body limit is already approached at $Q^{2} \sim 20$ GeV$^{2}$ and $x \sim 10^{-4}$.