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Introduction to the Physics of the Total Cross Section at LHC

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Introduction to the Physics of the Total Cross Section at LHC Eur. Phys. J. C (2017) 77:150 DOI 10.1140/epjc/s10052-016-4585-8 Review Introduction to the physics of the total cross section at LHC A review of data and models Giulia Pancheri1,2,a, Yogendra N. Srivastava3,4 1 INFN Frascati National Laboratory, Via E. Fermi 40, 00044 Frascati, Italy 2 Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MA, USA 3 Physics Department, University of Perugia, Via A. Pascoli 6, 06123 Perugia, Italy 4 Physics Department, Northeastern University, Boston, MA 02115, USA Received: 15 August 2016 / Accepted: 23 August 2016 / Published online: 9 March 2017 © The Author(s) 2017. This article is published with open access at Springerlink.com Abstract This review describes the development of the lyticity properties and bounds of the scattering amplitudes. physics of hadronic cross sections up to recent LHC results Gordon and Breach Science, New York, 1970). and cosmic ray experiments. We present here a comprehen- sive review – written with a historical perspective – about total cross sections from medium to the highest energies Contents explored experimentally and studied through a variety of methods and theoretical models for over 60 years. We begin 1 Introduction ..................... 4 by recalling the analytic properties of the elastic amplitude 2 The theoretical framework from unitarity and and the theorems about the asymptotic behavior of the total analyticity ...................... 5 cross section. A discussion of how proton–proton cross sec- 2.1 General principles ............... 6 tions are extracted from cosmic rays at higher than accel- 2.2 Kinematics of elastic scattering ........ 6 erator energies and help the study of these asymptotic lim- 2.3 Unitarity and the scattering amplitude .... 7 its, is presented. This is followed by a description of the 2.4 The optical theorem and the total cross section 8 advent of particle colliders, through which high energies and 2.5 The elastic scattering amplitude and its partial unmatched experimental precisions have been attained. Thus wave expansion ................ 8 the measured hadronic elastic and total cross sections have 2.6 Asymptotic behaviour and Regge theory ... 9 become crucial instruments to probe the so called soft part 2.7 Constraints from FESR and duality for the of QCD physics, where quarks and gluons are confined, and total cross sections .............. 11 have led to test and refine Regge behavior and a number of 2.8 The Froissart–Martin bound ......... 13 diffractive models. As the c.m. energy increases, the total 2.8.1 Froissart’s derivation of the asymptotic cross section also probes the transition into hard scattering behaviour of the scattering amplitude .13 describable with perturbative QCD, the so-called mini-jet 2.8.2 André Martin’s derivation ....... 13 region. Further tests are provided by cross section measure- 2.8.3 Eikonal picture derivation ....... 15 γ γ ∗ γ ∗γ ∗ ments of p, p and for models based on vector 2.8.4 Gribov’s derivation .......... 15 meson dominance, scaling limits of virtual photons at high 2.9 The Pomeranchuk theorem .......... 16 2 Q and the BFKL formalism. Models interpolating from vir- 2.10 Determination of the ρ parameter through tual to real photons are also tested. Coulomb interference and soft radiation ... 18 It seems to us to be a necessary task to explore bit-by-bit 2.10.1 Coulomb interference ......... 18 the rigorous consequences of analyticity, unitarity and cross- 2.10.2 Soft photon radiation as a possible tool ing. Who knows if someday one will not be able to reassemble for measurements of the total cross the pieces of the puzzle. – A. Martin and F. Cheung, based on section ................. 19 1967 A.M. Lectures at Brandeis Summer School and Lec- 3 Non-accelerator experiments ............ 20 tures at SUNY and Stony Brook (Martin and Cheung in Ana- 3.1 Heisenberg and cosmic radiation ....... 21 3.2 The Glauber model for high energy collisions 21 a e-mail: [email protected] 3.2.1 Scattering with bound particles .... 23 123 150 Page 2 of 178 Eur. Phys. J. C (2017) 77 :150 3.2.2 The Glauber model for high energy 4.2.2 The four methods used at ISR ..... 45 scattering of protons by nuclei .... 23 4.2.3 A final analysis of ISR results ..... 46 3.3 Cosmic rays: measurements and extraction of 4.2.4 Measurements of ρ and the slope pp data ..................... 24 parameter ............... 47 3.3.1 Cosmic ray experiments and the extrac- 4.3 Measurements at the SppS¯ .......... 48 σ pp tion of energy dependence of total up 4.3.1 Early total cross section measurements: to 10 TeV after the ISR data ...... 25 UA1 and UA4 ............. 48 3.3.2 Prescriptions for more precise extrac- 4.3.2 UA4 and UA2 ............. 49 σ pp tion of tot after the advent of the 4.3.3 The ramping run and UA5 measurement 49 CERN SppS¯ data ........... 27 4.4 Reaching the TeV region ........... 49 3.3.3 The Durand and Pi mini-jet model for 4.4.1 Measurements at the TeVatron .... 50 p-air interactions ........... 29 4.4.2 A comment on the black disk model .51 3.3.4 More about uncertainties in extracting 4.4.3 The ρ parameter at the Tevatron .... 52 σ pp tot from cosmic ray data, after the 4.5 Conclusions .................. 52 Tevatron ................ 31 5 Theoretical scenarios and phenomenological 3.3.5 Extracting information from cosmic applications ..................... 52 ray showers .............. 32 5.1 Molière theory of multiple scattering ..... 55 3.3.6 Air shower modelling ......... 32 5.2 The Heisenberg model ............ 55 3.3.7 Block, Halzen and Stanev: models vs. 5.3 A general observation about the various ways measured attenuation length ...... 32 to obtain the Froissart bound ......... 57 3.4 The extraction of p-air cross section from cos- 5.4 The impact picture .............. 58 mic rays .................... 33 5.4.1 Cheng and Wu description of high σ pp 3.4.1 Extraction of tot in Block and Halzen energy scattering, including work with model ................. 35 Walker ................. 58 3.4.2 The inelastic cross section and model 5.4.2 The impact-parameter description by uncertainties, including diffraction .. 35 Soffer, Bourrely and Wu ........ 59 3.5 Modelling the cosmic ray flux and energy dis- 5.5 The universal Regge and Pomeron pole descrip- tribution of particles .............. 36 tion by Donnachie and Landshoff ....... 60 3.5.1 Power-law flux and critical indices of 5.6 Hadronic matter distribution ......... 61 cosmic radiation ............ 36 5.7 Role of resummation in QED ......... 62 3.5.2 Evaporation of fluid particles ..... 36 5.7.1 The Rutherford singularity ...... 62 3.5.3 Cosmic ray particle production .... 36 5.7.2 Infra-red catastrophe and the Bloch– 3.5.4 The critical exponent for classical and Nordsieck cure ............. 63 quantum particles ........... 37 5.7.3 Covariant formalism by Touschek and 3.6 Cosmic ray results after start of the LHC ... 37 Thirring ................ 64 3.6.1 A recent analysis of Glauber theory 5.7.4 Schwinger’s ansatz on the exponen- with inelastic scattering ........ 38 tiation of the infrared factor and the 3.6.2 The Telescope-Array measurement at appearance of double logarithms ... 64 95 TeV c.m. energy .......... 39 5.7.5 The Sudakov form factor ....... 65 3.7 Eikonal models for inelastic p-air scattering .40 5.7.6 Status of the field in the early 1960s .65 3.7.1 A multichannel model inclusive of 5.7.7 A semi-classical approach to radiative diffraction and triple Pomeron coupling 40 corrections ............... 66 3.7.2 A single-channel model with QCD 5.7.8 Reggeisation of the photon ...... 67 mini-jets ................ 40 5.7.9 Comments on the reggeisation of the 3.8 Conclusions .................. 42 photon ................. 69 4 The measurement of σtotal before the LHC: descrip- 5.8 High energy behaviour of QCD scattering tion of experiments and their results ........ 42 amplitudes in the Regge limit ......... 69 4.1 Fixed target experiments ........... 43 5.8.1 Non abelian gauge theory with Higgs 4.2 The ISR measurement and the rise of the total symmetry breakdown and the BFKL cross section .................. 43 integral equation ............ 70 4.2.1 ISR measurements for the total cross 5.8.2 The Odderon .............. 72 section 5.8.3 Odderons in QCD ........... 73 and the elastic scattering amplitude .. 44 5.8.4 Gribov–Levin–Ryskin (GLR) model .73 123 Eur. Phys. J. C (2017) 77 :150 Page 3 of 178 150 5.8.5 KMR model with BFKL Pomeron .. 75 6.3.2 Soft and hard Pomeron exchanges in 5.9 Mini-jet models ................ 77 Donnachie and Landshoff model ...109 5.9.1 Non-unitary mini-jet model by Gaisser 6.3.3 The model by Schegelsky and Ryskin . 111 and Halzen ............... 77 6.4 Analyses with Pomeron, Odderon and Regge 5.9.2 Eikonalisation of mini-jet models ... 78 exchanges ...................112 5.9.3 QCD-inspired models, Aspen model .79 6.4.1 Phenomenological analyses with and 5.9.4 Resummation and mini-jets ...... 80 without the Odderon contribution ...112 5.9.5 Hadronic matter distribution and QCD 6.4.2 Jenkovszky’s Pomeron/Odderon dipole soft kt distribution ........... 81 model .................115 5.9.6 Bloch and Nordsieck inspired model 6.5 Eikonal models driven by Pomeron exchanges, for the total cross section ....... 83 parton dynamics and QCD-inspired inputs . 116 5.9.7 Soft gluon kt -resummation and the 6.5.1 Quarks and gluons in the Islam Froissart bound ............ 85 model .................117 5.10 AdS/CFT correspondence and the total cross 6.5.2 The eikonal model by Bourrely, Soffer section ..................... 88 and Wu .................118 5.11 Phenomenological fits to the total cross 6.5.3 Many Pomeron structures in eikonal section ....................
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