Indirect evidence of the Odderon from the LHC data on elastic proton-proton scattering

A simple multipole Pomeron and Odderon model for elastic hadron scattering, reproducing the structure of the first and second diffraction cones is used to analyze $pp$ and $\bar{p}p$ scattering. The main emphasis is on the delicate and non-trivial dynamics in the dip-bump region, at $|t| \approx 1$ GeV$^2$ and at the second cone. The simplicity of the model and the expected smallness of the absorption corrections enables one the control of various contributions to the scattering amplitude, in particular the interplay between the C-even and C-odd components of the amplitude, as well as their relative contribution, changing with $s$ and $t$. The role of the non-linearity of the Regge trajectories is verified. A detailed analysis of the LHC energy region, where most of the exiting models may be either confirmed or ruled out, is presented.