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  • Laboratori Nazionali di Frascati

    ISTITUTO NAZIONALE DI FISICA NUCLEARE

    INFN-16-13/LNF 3rd November 2016

    Introduction to the physics of the total cross-section at LHC A Review of Data and Models

    Giulia Pancheri1 and Yogendra N. Srivastava2

    1)INFN-Laboratori Nazionali di Frascati Via E. Fermi 40, Frascati, Italy

    and Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MASS USA 2) Physics Department, U. of Perugia, Via A. Pascoli 6, Perugia 06123, Italy

    and Physics Department, Northeastern University, Boston, MASS 02115, USA

    Abstract

    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 sixty 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 difiractive models. As the c.m. energy increases, the total cross-section also probes the transition into hard scattering describable with perturba-tive QCD, the so-called mini-jet region. Further tests are provided by cross-section measurements of γp, γ*p and γ*γ* for models based on vector meson dominance, scaling limits of virtual photons at high Q2 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 SUNYand Stony Brook.

    Pubblicato da SIDS–Pubblicazioni Laboratori Nazionali di Frascati

  • Introduction to the physics of the total cross-section at LHC

    A Review of Data and Models

    Giulia Pancheri1 and Yogendra N. Srivastava2

    1 INFN Frascati National Laboratory, Via E. Fermi 40, I00044 Frascati, Italy and Center for Theoretical Physics, Massachusetts Institute of Technology, Cambridge, MASS USA

    2 Physics Department, U. of Perugia, Via A. Pascoli 6, Perugia 06123, Italy and Physics Department, Northeastern University, Boston, MASS 02115, USA

    Abstract. 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 sixty 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 perturba- tive QCD, the so-called mini-jet region. Further tests are provided by cross-section measurements of γp, γ∗p and γ∗γ∗ for models based on vector meson dominance, scaling limits of virtual photons at high Q2

    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 [1].

    PACS. 13.85.Lg Total cross sections – 13.85.Dz Elastic scattering

    Contents

    1 The theoretical framework from unitarity and analyt- icity . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1 General principles . . . . . . . . . . . . . . . . 5 1.2 Kinematics of elastic scattering . . . . . . . . . 6 1.3 Unitarity and the scattering amplitude . . . . . 6 1.4 The optical theorem and the total cross-section 7 1.5 The elastic scattering amplitude and its partial

    wave expansion . . . . . . . . . . . . . . . . . . 7 1.6 Asymptotic behaviour and Regge theory . . . . 8 1.7 Constraints from FESR and Duality for the to-

    tal cross-sections . . . . . . . . . . . . . . . . . 10 1.8 The Froissart-Martin bound . . . . . . . . . . . 12

    1.8.1 Froissart’s derivation of the asymptotic behaviour of the scattering amplitude . 12

    1.8.2 André Martin’s derivation . . . . . . . 12 1.8.3 Eikonal Picture derivation . . . . . . . . 14 1.8.4 Gribov’s derivation . . . . . . . . . . . . 14

    1.9 The Pomeranchuk theorem . . . . . . . . . . . 15

    1.10 Determination of the ρ parameter through Coulomb Interference and soft radiation . . . . . . . . . . 16

    1.10.1 Coulomb interference . . . . . . . . . . . 16

    1.10.2 Soft photon radiation as a possible tool for measurements of the total cross-section 18

    2 Non-accelerator experiments . . . . . . . . . . . . . 18

    2.1 Heisenberg and cosmic radiation . . . . . . . . 19

    2.2 The Glauber model for high energy collisions . 20

    2.2.1 Scattering with bound particles . . . . . 21

    2.2.2 The Glauber model for high energy scat- tering of protons by nuclei . . . . . . . . 22

    2.3 Cosmic rays: measurements and extraction of pp data . . . . . . . . . . . . . . . . . . . . . . . . 22

    2.3.1 Cosmic ray experiments and the extrac- tion of energy dependence of σpptotal up to 10 TeV after the ISR data . . . . . . . 23

  • 2 Giulia Pancheri, Yogendra N. Srivastava: Introduction to the physics of the total cross-section at LHC

    2.3.2 Prescriptions for more precise extraction of σpptot after the advent of the CERN Spp̄S data . . . . . . . . . . . . . . . . . 25

    2.3.3 The Durand and Pi mini-jet model for p− air interactions . . . . . . . . . . . 26

    2.3.4 More about uncertainties in extracting σpptot from cosmic ray data, after the Teva- tron . . . . . . . . . . . . . . . . . . . . 28

    2.3.5 Extracting information from cosmic ray showers . . . . . . . . . . . . . . . . . . 29

    2.3.6 Air shower modeling . . . . . . . . . . . 29 2.3.7 Block, Halzen and Stanev: models vs. mea-

    sured attenuation length . . . . . . . . 30 2.4 The extraction of p−air cross-section from cos-

    mic rays . . . . . . . . . . . . . . . . . . . . . 31 2.4.1 Extraction of σpptot in Block and Halzen

    model . . . . . . . . . . . . . . . . . . . 32 2.4.2 The inelastic cross-section and model un-

    certainties, including diffraction . . . . . 33 2.5 Modeling the cosmic ray flux and energy distri-

    bution of particles . . . . . . . . . . . . . . . . 33 2.5.1 Power law flux and critical indices of cos-

    mic radiation . . . . . . . . . . . . . . . 33 2.5.2 Evaporation of fluid particles . . . . . . 33 2.5.3 Cosmic ray particle production . . . . . 34 2.5.4 The critical exponent for classical and

    quantum particles . . . . . . . . . . . . 34 2.6 Cosmic ray results after start of the LHC . . . 34

    2.6.1 A recent analysis of Glauber theory with inelastic scattering . . . . . . . . . . . . 35

    2.6.2 The Telescope-Array measurement at 95 TeV c.m. energy . . . . . . . . . . . . . 36

    2.7 Eikonal models for inelastic p− air scattering. 37 2.7.1 A multichannel model inclusive of diffrac-

    tion and triple Pomeron coupling . . . . 37 2.7.2 A single channel model with QCD mini-

    jets . . . . . . . . . . . . . . . . . . . . . 38 2.8 Conclusions . . . . . . . . . . . . . . . . . . . . 38

    3 The measurement of σtotal before the LHC: descrip- tion of experiments and their results . . . . . . . . . 39 3.1 Fixed target experiments . . . . . . . . . . . . 40 3.2 The ISR measurement and the rise of the total

    cross-section . . . . . . . . . . . . . . . . . . . . 40 3.2.1 ISR measurements for the total cross-

    section and the elastic scattering ampli- tude . . . . . . . . . . . . . . . . . . . . 41

    3.2.2 Radiative corrections to the determina- tion of the ρ parameter . . . . . . . . . 42

    3.2.3 The four methods used at ISR . . . . . 43 3.2.4 A final analysis of ISR results . . . . . . 44 3.2.5 Measurements of ρ and the slope param-

    eter . . . . . . . . . . . . . . . . . . . . 44 3.3 Measurements at the Spp̄S . . . . . . . . . . . 45