The Phillips - Barger model for the elastic cross section and the Odderon.

Inspired by the recent TOTEM data for the elastic proton -- proton ($pp$) scattering at $\sqrt{s} =$ 8 and 13 TeV, we update previous studies of the differential cross sections using the Phillips -- Barger (PB) model, which parametrizes the amplitude in terms of a small number of free parameters. We demonstrate that this model is able to describe the recent $pp$ data on a statistically acceptable way. Additionally, we perform separate fits of the $pp$ data for each center - of - mass energy and propose a parametrization for the energy dependence of the parameters present in the PB model. As a consequence, we are able to present the PB predictions for the elastic proton - proton cross section at $\sqrt{s} = 546$ GeV and $1.8$ TeV, which are compared with the existing antiproton -- proton ($\bar{p}p$) data. We show that the PB predictions, constrained by the $pp$ data, are not able to describe the $\bar{p}p$ data. In particular, the PB model predicts a dip in the differential cross section that is not present in the $\bar{p}p$ data. Such result suggests the contribution of the Odderon exchange at high energies.