Introduction to the physics of the total cross section at LHC

This review describes the development of the physics of hadronic cross sections up to recent LHC results and cosmic ray experiments. We present here a comprehensive review – written with a historical perspective – about total cross sections from medium to the highest energies explored experimentally and studied through a variety of methods and theoretical models for over 60 years. We begin by recalling the analytic properties of the elastic amplitude and the theorems about the asymptotic behavior of the total cross section. A discussion of how proton–proton cross sections are extracted from cosmic rays at higher than accelerator energies and help the study of these asymptotic limits, is presented. This is followed by a description of the advent of particle colliders, through which high energies and unmatched experimental precisions have been attained. Thus the measured hadronic elastic and total cross sections have become crucial instruments to probe the so called soft part of QCD physics, where quarks and gluons are confined, and have led to test and refine Regge behavior and a number of diffractive models. As the c.m. energy increases, the total cross section also probes the transition into hard scattering describable with perturbative QCD, the so-called mini-jet region. Further tests are provided by cross section measurements of $$\gamma p$$, $$\gamma ^* p$$ and $$\gamma ^* \gamma ^*$$ for models based on vector meson dominance, scaling limits of virtual photons at high $$Q^2$$ and the BFKL formalism. Models interpolating from virtual to real photons are also tested. It seems to us to be a necessary task to explore bit-by-bit the rigorous consequences of analyticity, unitarity and crossing. Who knows if someday one will not be able to reassemble the pieces of the puzzle. – A. Martin and F. Cheung, based on 1967 A.M. Lectures at Brandeis Summer School and Lectures at SUNY and Stony Brook (Martin and Cheung in Analyticity properties and bounds of the scattering amplitudes. Gordon and Breach Science, New York, 1970)