EMMI Workshop: Mining for gluon saturation at the LHC and the future EIC

Scaling amplitudes describing pp elastic scattering differential cross-sections in the dip-bump region at the LHC have been recently derived. We check that this scaling is also verified by the $p\bar p$ cross-section at the highest energy of the Tevatron. Using the energy-phase relations for given plus or minus signature coming from the Regge formalism and deriving the scaling amplitude with only positive signature (i.e. the Pomeron), lead phase predictions which are in agreement with the $p\bar p $ data. An evidence for a negative signature component (i.e. the Odderon) of the scaling amplitudes is thus found. Its amplitude and contribution to the cross-sections are extracted and discussed.