Hadronic forward scattering: Predictions for the Large Hadron Collider and cosmic rays

The status of hadron-hadron interactions is reviewed, with emphasis on the forward and near-forward scattering regions. From analyticity, Finite Energy Sum Rules are introduced from which new analyticity constraints are derived that exploit the many very accurate low energy experimental cross sections, i.e., they constrain the values of the asymptotic cross sections and their derivatives at low energies just above the resonance regions, allowing us new insights into duality. A new robust fitting technique is introduced in order to `clean up' large data samples that are contaminated by outliers. Using our analyticity constraints, new methods of fitting high energy hadronic data are introduced which result in much more precise estimates of the fit parameters, allowing accurate extrapolations to much higher energies. It's shown that the $\gamma p$, $\pi^\pm p$ and nucleon-nucleon cross sections {\em all} go asymptotically as $\ln^2s$, saturating the Froissart bound, while conclusively ruling out $\ln s$ and $s^{\alpha}$ ($\alpha\sim 0.08$) behavior. Implications of this saturation for predictions of $\sigma_{pp}$ and $\rho_{pp}$ at the LHC and for cosmic rays p-air cross sections are given.