Table Of ContentTexts and Monographs in Physics
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http:// www.springer.de/phys/
Vincenzo Barone Enrico Predazzi
High-Energy
Particle Diffraction
With 188 Figures
~
Springer
Dr. Vincenzo Barone
Dipartimento di Seienze
Universita del Piemonte Orientale
and INFN, Gruppo di Alessandria
Corso Borsalino 54
15100 Alessandria
Ita!y
Professor Dr. Enrico Predazzi
Dipartimento di Fisiea Teoriea
Universita di Torino
and INFN, Sezione di Torino
Via P. Giuria 1
10125 Torino
Italy
Library of Congress Cataloging-in-Publication Data
Barone, Vincenzo.
High-energy particle diffraction / Vincenzo Barone, Enrico Predazzi.
p. em. -- (Texts and monographs in physics, ISSN 1864-5879)
Includes bibliographical references and index.
ISBN 978-3-642-07567-4 ISBN 978-3-662-04724-8 (eBook)
DOI 10.1007/978-3-662-04724-8
1. Deep inelastic collisions. 2. Hadron interactions. 3. Particles (Nuclear
physics)--Diffraction. 1. Predazzi, Enrico. II. Title. III. Series.
QC794.6.C6 B37 2002
539.7'57--dc21
2001049856
ISSN 1864-5879
ISBN 978-3-642-07567-4
This work is subject to copyright. AII rights are reserved. whether the whole or par! of the material is
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OriginalIy published by Springer-Verlag Berlin Heidelberg New York in 2002
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To our families
Preface
High-energy diffraction has become a hot and fashionable subject in recent
years due to the great interest triggered by the HERA and Tevatron data.
These data have helped to show the field from a different perspective paving
the road to a hopefully more complete understanding than hitherto achieved.
The forthcoming data in the next few years from even higher energies (LHC)
promise to sustain this interest for a long time.
We believe that it is therefore necessary to summarize the main devel
opments which have marked the growth of high-energy diffractive physics in
recent decades, and to assess the present state of the art. This is the purpose
of the present book, which is especially aimed at the young researchers who
are entering the field and want to get acquainted with the relevant results
and the main theoretical techniques. The "new" diffraction has started to
bridge the gap between the hard and soft regimes of strong interactions. A
modern account of the subject, in our opinion, should reflect this situation,
covering both the traditional approaches to soft processes which are still alive
and useful, and the modern treatment of hard dynamics in the framework of
perturbative QCD.
The book is divided into three parts. The first part (Chaps. 1-3) contains
some introductory material: the systematics of diffractive processes, some
historical remarks, the optical analogy, the eikonal approximation of quantum
mechanics, and high-energy kinematics.
In the second part (Chaps. 4-7), the "old" diffraction is reviewed, with
a focus on the theoretical and experimental results which are still of some
relevance nowadays. Topics covered in this part include: S-matrix properties,
Regge theory, s-channel models, and the phenomenology of soft diffractive
reactions.
Finally, the third part (Chaps. 8-11) is devoted to the "new" diffraction.
Our present understanding of the pomeron in perturbative QCD (the cele
brated BFKL theory) is presented in some detail. Deep inelastic scattering
at low x, diffractive deep inelastic scattering and diffractive hadron-hadron
processes are analysed phenomenologically, and their QCD description is out
lined. One of the main theoretical approaches, the color dipole picture, 1s
discussed at various stages.
VIII Preface
This book grew out of lectures that the authors delivered at the Centro
Brasileiro de Pesquisas Fisicas (Rio de Janeiro, Brazil) in February 1998
(V.B.) and at the Florianopolis Workshop on Hadron Physics (Florianopolis,
Santa Catarina, Brazil) in March 1998 (E.P.). We thank our Brazilian friends,
Ignacio Bediaga, Francisco Caruso, Alberto Santoro, Erasmo Ferreira, for
their hospitality and for giving us the opportunity to talk to an interested
and stimulating audience.
We are deeply indebted to the countless number of colleagues and friends
who have been discussing with us various aspects of DIS and diffraction in
the past decade, and from whom we learned much of what we know on these
subjects. Apologizing for any possible omission, we mention, in alphabeti
cal order: Halina Abramowicz, Michele Arneodo, Bj. Bjorken, John Dainton,
Umberto D'Alesio, Vittorio DelDuca, Victor Fadin, Roberto Fiore, Stefano
Forte, Marco Genovese, Maurice Giffon, Dino Goulianos, Laszlo Jenkovszky,
Boris Kopeliovich, Eduard Kuraev, Elliot Leader, Genya Levin, Aharon Levy,
Lev Lipatov, Uri Maor, Genya Martynov, Giorgio Matthiae, Kolya Nikolaev,
Francesco Paccanoni, Alessandro Papa, Cristiana Peroni, Irina Potashnikova,
Bogdan Povh, Misha Ryskin, Alberto Santoro, Ada Solano, Amedeo Staiano,
Slava Zakharov and Fabian Zomer. It is a pleasure to acknowledge a lon
glasting and fruitful collaboration with Boris Kopeliovich, Marco Genovese,
Kolya Nikolaev and Slava Zakharov.
We are grateful to Umberto D'Alesio, Alexei Prokudin and Ada Solano
for a careful reading of some chapters of the book, to Vittorio Del Duca, Dino
Goulianos, Elliot Leader and Kolya Nikolaev for useful suggestions and com
ments, to Giorgio Matthiae and Erasmo Ferreira for providing some figures,
and to Igor Pesando for his computer expertise which proved very helpful in
an emergency situation.
We dedicate this book to our families.
Torino, Vincenzo Barone,
December 2001 Enrico Predazzi
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Diffractive Phenomena: a Definition . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Brief Historical Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Kirchhoff Theory.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Fraunhofer Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.3 Cross-sections.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.5 Scattering of Light by a Sphere . . . . . . . . . . . . . . . . . . 19
2.2 Potential Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 The Schrodinger Equation Approach.............. 21
2.2.2 Partial Wave Expansion . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 The S-Matrix Approach. . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2.4 Born Approximation............................ 25
2.2.5 Central Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 The Eikonal Approximation............................. 29
3. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Scattering Processes: Generalities. . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Two-Body Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.1 Mandelstam Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.2 The Center-of-Mass System. . . . . . . . . . . . . . . . . . . . . . 38
3.2.3 The Laboratory System . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.4 Physical Domains of the s, u and t Channels....... 43
3.3 Single-Inclusive Processes.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3.1 Feynman's XF Variable . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3.2 Rapidity and Rapidity Gaps . . . . . . . . . . . . . . . . . . . . . 47
3.3.3 Diffractive Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . 49
4. The Relativistic S-Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 General Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Scattering Amplitudes and Cross-sections. . . . . . . . . . . . . . . . . 52
X Contents
4.2.1 Two-Body Exclusive Reactions................... 54
4.2.2 Single-Particle Inclusive Reactions. . . . . . . . . . . . . . . . 56
4.3 Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 The Optical Theorem........................... 60
4.3.2 Other Consequences of Unitarity . . . . . . . . . . . . . . . . . 60
4.3.3 Elastic Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4 Analyticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Crossing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4. 7 The Froissart-Gribov Representation. . . . . . . . . . . . . . . . . . . . . 72
4.8 Mueller's Generalized Optical Theorem................... 74
4.9 Rigorous Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.9.1 The Froissart-Martin Bound . . . . . . . . . . . . . . . . . . . . . 75
4.9.2 The Pomeranchuk Theorems..................... 78
5. Regge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.1 The Regge Pole Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2 Meson Exchange vs. Reggeon Exchange . . . . . . . . . . . . . . . . . . 85
5.3 Convergence of the Partial-Wave Expansion. . . . . . . . . . . . . . . 86
5.4 Complex Angular Momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5 Regge Poles in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 93
5.6 Regge Poles in Relativistic Scattering . . . . . . . . . . . . . . . . . . . . 94
5. 7 Regge Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.8 Regge Phenomenology ................................. 101
5.8.1 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.8.2 Total and Elastic Cross-sections .................. 103
5.8.3 The Pomeron .................................. 104
5.8.4 The Odderon .................................. 106
5.8.5 Regge Trajectories and Hadron Scattering ......... 106
5.8.6 Diffractive Dissociation ......................... 107
5.9 Regge Poles in Field Theory ............................ 111
5.10 Regge Cuts ........................................... 115
6. Geometrical and s-Channel Models . . . . . . . . . . . . . . . . . . . . . . . 123
6.1 The Eikonal Picture ................................... 124
6.2 A Model for the Eikonal Amplitude ...................... 126
6.3 The Structure of Hadrons in the Impact-Parameter Space .. 128
6.4 s-Channel Models ..................................... 131
6.4.1 The Geometrical Model ......................... 131
6.4.2 The Impact Picture ............................. 132
6.4.3 Geometrical Scaling ............................ 133
6.4.4 Other Models .................................. 135
6.5 Duality ............................................... 135
6.6 The Veneziano Model .................................. 137
Contents XI
7. Soft Diffraction: a Phenomenological Survey .............. 139
7.1 Total Cross-sections .................................... 139
7.2 The Real Part of the Forward Elastic Amplitude .......... 144
7.3 Elastic Cross-sections .................................. 147
7.3.1 The Forward Peak .............................. 149
7.4 The Dip-Shoulder Region ............................... 151
7.4.1 Odderon Phenomenology ........................ 152
7.4.2 The Large-It! Region ............................ 154
7.5 Diffractive Dissociation ................................. 155
7.6 Soft Diffraction at RHIC and LHC ....................... 160
8. The Pomeron in QCD .................................... 163
8.1 Early Approaches ...................................... 163
8.2 The Perturbative QCD Pomeron ........................ 165
8.3 Quark-Quark Scattering in Leading ln s Approximation .... 166
8.3.1 One-Gluon Exchange ........................... 167
8.3.2 Digression: Gluon-Gluon Scattering .............. 169
8.3.3 Two-Gluon Exchange ........................... 171
8.3.4 The Simplest Gluon Ladder ..................... 177
8.3.5 Higher Orders: the BFKL Ladder ................ 187
8.4 The BFKL Equation ................................... 194
8.5 Color-Octet Exchange .................................. 195
8.6 Color-Singlet Exchange ................................. 198
8. 7 Solution of the BFKL Equation for t = 0 ................. 200
8.8 Parton-Parton Scattering: Total Cross-sections ............ 205
8.9 Diffusion ............................................. 206
8.10 Running Coupling ..................................... 208
8.11 Soft vs. Hard Pomeron ................................. 208
8.12 Non-perturbative Effects ............................... 210
8.13 The Perturbative QCD Odderon ......................... 210
8.14 Solution of the BFKL Equation fort =f. 0 ................. 211
8.15 Parton-Parton Elastic Scattering ........................ 214
8.16 Hadron-Hadron Scattering .............................. 217
8.17 The BFKL Pomeron at Next-to-Leading Order ............ 218
9. Deep Inelastic Scattering ................................. 221
9.1 Kinematics ........................................... 221
9.2 Parton Model ......................................... 225
9.3 Structure Functions in QCD ............................ 231
9.4 Phenomenology of DIS ................................. 238
9.5 DIS at Low-x ......................................... 241
9.5.1 Regge Theory Predictions ....................... 244
9.5.2 Resummations in QCD ......................... 246
9.5.3 DLLA and Double Scaling ....................... 248
9.5.4 kJ. -Factorization ............................... 253
