Scaling properties of elastic proton-proton scattering at LHC energies

The TOTEM Collaboration has measured the differential cross section of elastic proton-proton scattering $\mathrm{d}\sigma/\mathrm{d}|t|$ at $\sqrt{s} = 2.76$, 7, 8, and 13 TeV. It is observed that all $\mathrm{d}\sigma/\mathrm{d}|t|$ data points vary in the same way as a function of the center-of-mass energy in the so-called "dip" and "bump" regions. These features hint at possible universal properties of elastic scattering. Based on these empirical observations, and taking inspiration from saturation models, we propose a simple scaling law for proton-proton elastic scattering at LHC energies. We find that the $\mathrm{d}\sigma/\mathrm{d}|t|$ at LHC energies fall onto a universal curve when they are mapped to the scaling variables $ \mathrm{d}\sigma/\mathrm{d}|t| \times (s/\text{TeV}^2)^{-0.305} $ versus $(s/\text{TeV}^2)^{0.065} (|t|/\text{GeV}^2)^{0.72}$. Some implications of this scaling law in the impact parameter picture of the scattering amplitudes are explored.