Determination of the forward slope in $p~p$ and $bar p~p$ elastic scattering up to LHC energy

In the analysis of experimental data on $p p$ (or $bar p p$) elastic differential cross section it is customary to define an average forward slope $b$ in the form $exp{(-b|t|)}$, where $t$ is the momentum transfer. Taking as working example the results of experiments at Tevatron and SPS, we will show with the help of the impact picture approach, that this simplifying assumption hides interesting information on the complex non-flip scattering amplitude, and that the slope $b$ is not a constant. We investigate the variation of this slope parameter, including a model-independent way to extract this information from an accurate measurement of the elastic differential cross section. An extension of our results to the LHC energy domain is presented in view of future experiments.