Geometric scaling of elastic $pp$ cross section at the LHC

Abstract: We show that geometric scaling conjectured and observed at the ISR more than 50 years ago, still holds at the LHC. We discuss regularities of the dip-bump structures of the differential elastic cross sections, emphasizing the fact that the ratio of bump to dip positions is constant from the ISR to the LHC. Applying crossing and analyticity we identify imaginary and real parts of the scattering amplitude and compute the $\rho$ parameter and the ratio of bump to dip values of the differential $pp$ cross sections. We also discuss the energy dependence of the total elastic cross section and the violation of geometrical scaling outside the dip-bump region at the LHC.