Geometric scaling in elastic $pp$ collisions

Geometric scaling was conjectured and observed at the ISR more than 50 years ago. We argue that it still holds at the LHC. We show that the dip-bump structures of the differential elastic cross-sections exhibit a remarkable regularity, namely that the ratio of the bump to dip positions is constant from 23 GeV to 13 TeV. Using crossing symmetry, analicity and the optical theorem, we identify the imaginary and real parts of the scattering amplitude. This allows us to compute the $\rho$ parameter and the ratio of the bump to dip values of the differential $pp$ cross-section. Finally, we discuss the energy dependence of the total elastic cross-section as compared to the total cross-section, and the violation of geometrical scaling outside the dip-bump region at the LHC.