W. Greiner· S. Schramm· E. Stein QUANTUM CHROMODYNAMICS

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Walter Greiner· Stefan Schramm Eckart Stein

QUANTUM CHROMODYNAMICS

With a Foreword by D. A. Bromley

Second Revised and Enlarged Edition With 161 Figures,

and 66 Worked Examples and Exercises

Springer

Professor Dr. Walter Greiner Institut fiir Theoretische Physik der Johann Wolfgang Goethe-Universitlit Frankfurt Postfach 111932 60054 Frankfurt am Main Germany

Street address:

Robert-Mayer-Strasse 8-10 60325 Frankfurt am Main Germany

email: greiner@th.physik.uni-frankfurt.de

Dr. Stefan Schramm Argonne National Laboratory 9700 S. Cass Avenue Argonne, IL 60439 USA

email: schramm@theory.phy.anl.gov

Dr. Eckart Stein Physics Department Maharishi University of Management 6063 NP Vlodrop The Netherlands

email: EStein@maharishi.net

Title of the original German edition: Theoretische Physik, Ein Lehr- und Obungsbuch, Band 10: Quantenchromodynamik © Verlag Harri Deutsch, Thun, 1984, 1989

Library of Congress Cataloging·in-Publication Data applied for.

Die Deutsche Bibliothek - CIP-Einheitsaufnahme

Greiner, Walter: Quantum chromodynamicslWalter Greiner; Stefan Schramm; Eckart Stein. ISBN 978-3-540-66610-3 ISBN 978-3-662-04707-1 (eBook) DOI 10.1007/978-3-662-04707-1

ISBN 978-3-540-66610-3

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Foreword to Earlier Series Editions

More than a generation of German-speaking students around the world have worked their way to an understanding and appreciation of the power and beauty of modem theoretical physics - with mathematics, the most fundamental of sciences - using Walter Greiner's textbooks as their guide.

The idea of developing a coherent, complete presentation of an entire field of science in a series of closely related textbooks is not a new one. Many older physicists remember with real pleasure their sense of adventure and discovery as they worked their ways through the classic series by Sommerfeld, by Planck and by Landau and Lifshitz. From the students' viewpoint, there are a great many obvious advantages to be gained through use of consistent notation, logi­cal ordering of topics and coherence of presentation; beyond this, the complete coverage of the science provides a unique opportunity for the author to convey his personal enthusiasm and love for his subject.

The present five volume set, Theoretical Physics, is in fact only that part of the complete set of textbooks developed by Greiner and his students that presents the quantum theory. I have long urged him to make the remaining volumes on classical mechanics and dynamics, on electromagnetism, on nuclear and particle physics, and on special topics available to an English-speaking audience as well, and we can hope for these companion volumes covering all of theoretical physics some time in the future.

What makes Greiner's volumes of particular value to the student and profes­sor alike is their completeness. Greiner avoids the all too common "it follows that ... "which conceals several pages of mathematical manipulation and con­founds the student. He does not hesitate to include experimental data to illumi­nate or illustrate a theoretical point and these data, like the theoretical content, have been kept up to date and topical through frequent revision and expansion of the lecture notes upon which these volumes are based.

Moreover, Greiner greatly increases the value of his presentation by includ­ing something like one hundred completely worked examples in each volume. Nothing is of greater importance to the student than seeing, in detail, how the theoretical concepts and tools under study are applied to actual problems of inter­est to a working physicist. And, finally, Greiner adds brief biographical sketches to each chapter covering the people responsible for the development of the the­oretical ideas and/or the experimental data presented. It was Auguste Comte (1798-1857) in his Positive Philosophy who noted, "To understand a science it is necessary to know its history". This is all too often forgotten in modem

VI Foreword to Earlier Series Editions

physics teaching and the bridges that Greiner builds to the pioneering figures of our science upon whose work we build are welcome ones.

Greiner's lectures, which underlie these volumes, are internationally noted for their clarity, their completeness and for the effort that he has devoted to mak­ing physics an integral whole; his enthusiasm for his science is contagious and shines through almost every page.

These volumes represent only a part of a unique and Herculean effort to make all of theoretical physics accessible to the interested student. Beyond that, they are of enormous value to the professional physicist and to all others working with quantum phenomena. Again and again the reader will find that, after dipping into a particular volume to review a specific topic, he will end up browsing, caught up by often fascinating new insights and developments with which he had not previously been familiar.

Having used a number of Greiner's volumes in their original German in my teaching and research at Yale, I welcome these new and revised English trans­lations and would recommend them enthusiastically to anyone searching for a coherent overview of physics.

Yale University New Haven, CT, USA 1989

D. Allan Bromley Henry Ford II Professor of Physics

Preface to the Second Edition

The theory of strong interactions, quantum chromodynamics (QCD), was for­mulated 30 years ago and has since been a very active field of research. The underlying equations of motion for the gauge degrees of freedom are nonlin­ear and minimally coupled to fermions with global and local SU(3) charges. This leads to spectacular problems compared with those of QED since the gauge bosons themselves interact with each other. On the other hand, it is exactly the self-interaction of the gluons which leads to asymptotic freedom and the pos­sibility to calcuate quark-gluon interaction at small distances in the framework of perturbation theory. We discover one of the most complicated but most beau­tiful gauge theories which poses extremely challenging problems on modem theoretical and experimental physics today.

Quantum chromodynamics is the quantum field theory that allows us to cal­culate the propagation and interaction of colored quarks and gluons at small distances. Today's experiments do not allow these colored objects to be detected directly; instead one deals with colorless hadrons: mesons and baryons seen far away from the actual interaction point. The hadronization itself is a complicated process and not yet understood from first principles. Therefore one may won­der how the signature of quark and gluon interactions can be traced through the process of hadronization.

Very advanced analytical and numerical techniques have been developed in order to analyze the world of hadrons on the grounds of fundamental QCD. To­gether with a much improved experimental situation we seem to be ready to answer the question wether QCD is the correct theory of strong interactions at all scales or just an effective high-energy line of a yet undiscovered theory.

In this book, we try to give a self-consistent treatment of QCD, stressing the practitioners point of view. For pedagogical reasons we review quantum elec­trodynamics (QED) (Chap. 2) after an elementary introduction. In Chap. 3 we study scattering reactions with emphasis on lepton-nucleon scattering and intro­duce the language for describing the internal structure of hadrons. Also the MIT bag model is introduced, which serves as an illustrative and successful example for QCD-inspired models.

In Chap. 4 the general framework of gauge theories is described on the basis of the famous Standard Model of particle physics. We then concentrate on the gauge theory of quark-gluon interaction and derive the Feynman rules of QCD, which are very useful for pertubative calculations. In particular, we show explic­itily how QCD is renormalized and how the often-quoted running coupling is obtained.

VIII Preface to the Second Edition

Chapter 5 is devoted to the application of QCD to lepton-hadron scatter­ing and therefore to the state-of-the-art description of the internal structure of hadrons. We start by presenting two derivations of the Dokshitzer-Gribov­Lipatov-Altarelli-Parisi equations. The main focus of this chapter is on the indispensable operator product expansion and its application to deep inelastic lepton-hadron scattering. We show in great detail how to perform this expan­sion and calculate the Wilson coefficients. Furthermore, we discuss perturbative corrections to structure functions and perturbation theory at large orders, i.e. renormalons.

After analyzing lepton-hadron scattering we switch in Chap. 6 to the case of hadron-hadron scattering as described by the Drell-Yan processes. We then tum to the kinematical sector where the so-called leading-log approximation is no longer sufficient. The physics on these scales is called small-x physics.

Chapter 7 is devoted to two promising nonperturbative approaches, namely QCD on the lattice and the very powerful analytical tool called the QCD sum rule technique. We show explicitely how to formulate QCD on a lattice and discuss the relevant algorithms needed for practical numerical calculations, including the lattice at finite temperature. This is very important for the physics of hot and dense elementary matter as it appears, for example, in high-energy heavy ion physics. The QCD sum rule technique is explained and applied to the calculation of hadron masses.

Our presentation ends with some remarks on the nontrivial QCD vacuum and its modification at high temperature and/or baryon density, including a sketch of current developments concerning the so-called quark-gluon plasma in Chap. 8. Modem high-energy heavy ion physics is concerned with these issues.

We have tried to give a pedagogical introduction to the concepts and tech­niques of QCD. In particular, we have supplied over 70 examples and exercises worked out in great detail. Working through these may help the practitioner in perfoming complicated calculations in this challenging field of theoretical physics.

In writing this book we profited substantially from a number of exist­ing textbooks, most notably J.J.R. Aitchison and A.J.G. Hey: 'Gauge Theo­ries in Particle Physics', O. Nachtmann: 'Elementarteilchenphysik', B. Milller: 'The Physics of the Quark-GIuon Plasma', P. Becher, M. Boehm and H. Joos: 'Eichtheorien', J. Collins: 'Renormalization', R.D. Field: 'Application of Per­turbative QCD', and M. Creutz: 'Quarks, GIuons and Lattices', and several re­view articles, especially: L.V. Gribov, E.M. Levin and M.G. Ryskin: 'Semihard processes in QCD', Phys. Rep. 100 (1983) 1, Badelek, Charchula, Krawczyk, Kwiecinski, ' Small x physics in deep inelastic lepton hadron scattering', Rev. Mod. Phys. 927 (1992), L.S. Reinders, H. Rubinstein, S. Yazaki: 'Hadron properties from QCD sumrules', Phys. Rep. 127 (1985) 1.

We thank many members of the Institute of Theoretical Physics in Frank­furt who added in one way or another to this book, namely Dr. J. Augustin, M. Bender, Ch. Best, S. Bernard, A. Bischoff, M. Bleicher, A. Diener, A. Dumi­tru, B. Ehrnsperger, U. Eichmann S. Graf, N. Hammon C. Hofmann, A. Jahns, J. Klenner, O. Martin, M. Massoth, G. Plunien, D. Rischke, and A. Scherdin.

Preface to the Second Edition

Special thanks go to Mrs. A. Steidel who drew the figures, Dr. Henning We­ber for supervising and solving numerous technical problems and, in particularly to Dr. Stefan Hofmann who supervised the editing process and gave hints for many improvements.

We hope that our readers will enjoy learning about quantum chromodynam-ics.

Frankfurt am Main, May 2002

Walter Greiner Stefan Schramm

Eckart Stein

IX

Contents

1. The Introduction of Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The Hadron Spectrum ............................. 1

2. Review of Relativistic Field Theory . . . . . . . . . . . . . . . . . . . . . . .. 17 2.1 Spinor Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . .. 17

2.1.1 The Free Dirac Equation and Its Solution ......... 17 2.1.2 Density and Current Density .................. 20 2.1.3 Covariant Notation ......................... 21 2.104 Normalization of Dirac Spinors ................ 21 2.1.5 Interaction with a Four-Potential All. . . . . . . . . . . .. 23 2.1.6 Transition Amplitudes ....................... 24 2.1.7 Discrete Symmetries ........................ 24

2.2 Scalar Quantum Electrodynamics ..................... 25 2.2.1 The Free Klein-Gordon Equation and its Solutions .. 25 2.2.2 Interaction of a rr+ with a Potential All .......... 28 2.2.3 rr+K+ Scattering ........................... 31 2.204 The Cross Section .......................... 34 2.2.5 Spin-l Particles and Their Polarization ........... 45 2.2.6 The Propagator for Virtual Photons .............. 51

2.3 Fermion-Boson and Fermion-Fermion Scattering . . . . . . . .. 56 2.3.1 Traces and Spin Summations .................. 58 2.3.2 The Structure of the Form Factors

from Invariance Considerations ................ 68

3. Scattering Reactions and the Internal Structure of Baryons ..... 75 3.1 Simple Quark Models Compared ..................... 75 3.2 The Description of Scattering Reactions ................ 78 3.3 The MIT Bag Model .............................. 123

4. Gauge Theories and Quantum-Chromodynamics ............. 153 4.1 The Standard Model: A Typical Gauge Theory ........... 153 4.2 The Gauge Theory of Quark-Quark Interactions .......... 163 4.3 Dimensional Regularization ......................... 182 404 The Renormalized Coupling Constant of QCD ........... 200 4.5 Extended Example: Anomalies in Gauge Theories ......... 217

4.5.1 The Schwinger Model on the Circle ............. 217 4.5.2 DiracSea ................................ 219

XII Contents

4.5.3 Ultraviolet Regularization .................... 221 4.5.4 Standard Derivation ......................... 225 4.5.5 Anomalies in QCD ......................... 227 4.5.6 The Axial and Scale Anomalies ................ 228 4.5.7 Multiloop Corrections ....................... 234

5. Perturbative QeD I: Deep Inelastic Scattering ............... 237 5.1 The Gribov-Lipatov-Altarelli-Parisi Equations ........... 237 5.2 An Alternative Approach to the GLAP Equations ......... 263 5.3 Common Parametrizations of the Distribution Functions

and Anomalous Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 279 5.4 Renormalization and the Expansion Into Local Operators .... 290 5.5 Calculation of the Wilson Coefficients ................. 326 5.6 The Spin-Dependent Structure Functions ............... 354

6. Perturbative QeD II: The Drell-Van Process and Small-x Physics 385 6.1 The Drell-Van Process ............................. 385 6.2 Small-x Physics .................................. 428

7. Nonperturbative QeD . ................................. 439 7.1 Lattice QCD Calculations ........................... 440

7.1.1 The Path Integral Method ..................... 440 7.1.2 Expectation Values ......................... 445 7.1.3 QCD on the Lattice ......................... 447 7.1.4 GIuons on the Lattice ........................ 449 7.1.5 Integration in SU(2) ......................... 452 7.1.6 Discretization: Scalar and Fermionic Fields ....... 454 7.1.7 Fermionic Path Integral ...................... 459 7.1.8 Monte Carlo Methods ....................... 461 7.1.9 Metropolis Algorithm ....................... 463 7.1.10 Langevin Algorithms ........................ 464 7.1.11 The Microcanonical Algorithm ................ 466 7.1.12 Strong and Weak Coupling Expansions ........... 472 7.1.13 Weak-Coupling Approximation ................ 474 7.1.14 Larger Loops in the Limit

of Weak and Strong Coupling ................. 475 7.1.15 The String Tension ......................... 477 7.1.16 The Lattice at Finite Temperature ............... 480 7.1.17 The Quark Condensate ....................... 483 7.1.18 The Polyakov Loop ......................... 485 7.1.19 The Center Symmetry ....................... 486

7.2 QCD Sum Rules ................................. 488

8. Phenomenological Models for Nonperturbative QeD Problems .. 513 8.1 The Ground State of QCD .......................... 513 8.2 The Quark-GIuon Plasma .......................... 528

Contents XIII

Appendix ............................................... 539 A The Group SU(N) ................................ 539 B Dirac Algebra in Dimension d ....................... 541 C Some Useful Integrals ............................. 544

Subject Index . ........................................... 547

Contents of Examples and Exercises

1.1 The Fundamental Representation of a Lie Algebra . . . . . . . . . . .. 10 1.2 Casimir Operators of SU(3) ............................ 15

2.1 The Matrix Element for a Pion Scattered by a Potential ........ 30 2.2 The Flux Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 36 2.3 The Mandelstam Variable s ............................ 37 2.4 The Lorentz-Invariant Phase-Space Factor. . . . . . . . . . . . . . . . .. 38 2.5 Jt+Jt+ and Jt+Jt- Scattering. . . . . . . . . . . . . . . . . . . . . . . . . . .. 39 2.6 The Cross Section for Pion-Kaon Scattering . . . . . . . . . . . . . . .. 41 2.7 Polarization States of a Massive Spin-l Particle . . . . . . . . . . . . .. 47 2.8 Compton Scattering by Pions ........................... 52 2.9 Elastic e-Jt+ Scattering (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56 2.10 Features of Dirac Matrices ............................. 61 2.11 Electron-Pion Scattering (II) ........................... 64 2.12 Positron-Pion Scattering .............................. 67 2.13 Electron-Muon Scattering ............................. 71

3.1 Normalization and Phase Space Factors ................... 85 3.2 Representation of WJLV by Electromagnetic Current Operators ... 86 3.3 The Nucleonic Scattering Tensor with Weak Interaction. . . . . . .. 88 3.4 The Inclusive Weak Lepton-Nucleon Scattering ............. 92 3.5 The Cross Section as a Function of x and y . . . . . . . . . . . . . . . .. 97 3.6 The Breit System .................................... 110 3.7 The Scattering Tensor for Scalar Particles .................. 111 3.8 Photon-Nucleon Scattering Cross Sections

for Scalar and Transverse Photon Polarization . . . . . . . . . . . . . . . 113 3.9 A Simple Model Calculation

for the Structure Functions of Electron-Nucleon Scattering ... . . 116 3.10 Antiquark Solutions in a Bag ........................... 125 3.11 The Bag Wave Function for Massive Quarks ................ 133 3.12 Gluonic Corrections to the MIT Bag Model ................. 141 3.13 The Mean Charge Radius of the Proton .................... 144 3.14 The Magnetic Moment of the Proton ...................... 147

4.1 The Geometric Formulation of Gauge Symmetries. . . . . . . . . . . . 156 4.2 The Feynman Rules for QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 4.3 Fadeev-Popov Ghost Fields ............................ 176 4.4 The Running Coupling Constant . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.5 The d-Dimensional Gaussian Integral ..................... 186

XVI Contents of Examples and Exercises

4.6 The d-Dimensional Fourier Transform .................... 195 4.7 Feynman Parametrization .............................. 198

5.1 Photon and GIuon Polarization Vectors .................... 242 5.2 More about the Derivation of QCD Corrections

to Electron-Nucleon Scattering ......................... 243 5.3 The Bremsstrahlung Part of the GLAP Equation ............. 258 5.4 The Maximum Transverse Momentum .................... 265 5.5 Derivation of the Splitting Function PGq ................... 267 5.6 Derivation of the Splitting Function Pqq ................... 274 5.7 Derivation of the Splitting Function PqG ................... 275 5.8 Calculation of Moments of Splitting Functions

(Anomalous Dimensions) .............................. 286 5.9 Decomposition Into Vector and Axial Vector Couplings ........ 294 5.10 The Proof of (5.148) ................................. 298 5.11 The Lowest-Order Terms of the f3 Function . . . . . . . . . . . . . . . . . 306 5.12 The Moments of the Structure Functions ................... 313 5.13 Calculation of the Gluonic Contribution to Fdx, Q2) ......... 336 5.14 Calculation of Perturbative Corrections to Structure Functions

with the Cross-Section Method . . . . . . . . . . . . . . . . . . . . . . . . . . 344 5.15 Calculation of the Gluonic Contribution to FL

with the Cross-Section Method .......................... 352 5.16 Higher Twist in Deep Inelastic Scattering .................. 364 5.17 Perturbation Theory in Higher Orders and Renormalons ....... 368

6.1 The Drell-Yan Cross Section ........................... 407 6.2 The One-GIuon Contribution to the Drell-Yan Cross Section .... 408 6.3 The Drell-Yan Process as Decay of a Heavy Photon .......... 409 6.4 Heavy Photon Decay Into Quark, Antiquark, and GIuon ........ 411 6.5 Factorization in Drell-Yan ............................. 417 6.6 Collinear Expansion and Structure Functions

in Deep Inelastic Lepton-Nucleon Scattering ............... 425

7.1 Derivation of the Transition Amplitude (7.5) ................ 443 7.2 The Average Link Value ............................... 468 7.3 PCAC and the Quark Condensate ........................ 496 7.4 Calculation of QCD Sum-Rule Graphs with

Dimensional Regularization ............................ 502

8.1 The QCD Vacuum Energy Density ....................... 519 8.2 The QCD Ground State and the Renormalization Group ....... 525 8.3 The QGP as a Free Gas ............................... 527