The book is a self-contained introduction to perturbative and nonperturbative Quantum chromodynamics (QCD) with worked-out exercises for students of theoretical physics. It will be useful as a reference for research scientists as well. Starting with the hadron spectrum, the reader becomes familiar with the representations of SU(N). Relativistic quantum field theory is recapitulated, and scattering theory is discussed in the framework of scalar quantum electrodynamics. Then the gauge theory of quarks and gluons is introduced. In the more advanced chapters, perturbative and nonperturbative techniques in state-of-the-art QCD are discussed in great detail.

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

      2.1.4 Normalization of Dirac Spinors 22

      2.1.5 Interaction with a Four-Potential Aµ 24

      2.1.6 Transition Amplitudes 25

      2.1.7 Discrete Symmetries 25

2.2 Scalar Quantum Electrodynamics 26

      2.2.1 The Free Klein–Gordon Equation and its Solutions 26

      2.2.2 Interaction of a π+ with a Potential Aµ 29

      2.2.3 π+K+ Scattering 32

      2.2.4 The Cross Section 35

      2.2.5 Spin-1 Particles and Their Polarization 47

      2.2.6 The Propagator for Virtual Photons 53

2.3 Fermion–Boson and Fermion–Fermion Scattering 58

      2.3.1 Traces and Spin Summations 60

      2.3.2 The Structure of the Form Factors from Invariance Considerations 70

3. Scattering Reactions and the Internal Structure of Baryons 77

3.1 Simple Quark Models Compared 77

3.2 The Description of Scattering Reactions 80

3.3 The MIT Bag Model 125

4. Gauge Theories and Quantum-Chromodynamics 155

4.1 The Standard Model: A Typical Gauge Theory 155

4.2 The Gauge Theory of Quark–Quark Interactions 165

4.3 Dimensional Regularization 184

4.4 The Renormalized Coupling Constant of QCD 203

4.5 Extended Example: Anomalies in Gauge Theories 220

      4.5.1 The Schwinger Model on the Circle 220

      4.5.2 Dirac Sea 222

      4.5.3 Ultraviolet Regularization 225

      4.5.4 Standard Derivation 228

      4.5.5 Anomalies in QCD 231

      4.5.6 The Axial and Scale Anomalies 232

      4.5.7 Multiloop Corrections 238

5. Perturbative QCD I: Deep Inelastic Scattering 239

5.1 The Gribov–Lipatov–Altarelli–Parisi Equations 239

5.2 An Alternative Approach to the GLAP Equations 265

5.3 Common Parametrizations of the Distribution Functions and Anomalous Dimensions 281

5.4 Renormalization and the Expansion Into Local Operators 292

5.5 Calculation of the Wilson Coefficients 328

5.6 The Spin-Dependent Structure Functions 356

6. Perturbative QCD II: The Drell–Yan Process and Small-x Physics 387

6.1 The Drell–Yan Process 387

6.2 Small-x Physics 430

7. Nonperturbative QCD 441

7.1 Lattice QCD Calculations 442

      7.1.1 The Path Integral Method 442

      7.1.2 Expectation Values 447

      7.1.3 QCD on the Lattice 449

      7.1.4 Gluons on the Lattice 451

      7.1.5 Integration in SU(2) 454

      7.1.6 Discretization: Scalar and Fermionic Fields 456

      7.1.7 Fermionic Path Integral 461

      7.1.8 Monte Carlo Methods 463

      7.1.9 Metropolis Algorithm 465

      7.1.10 Langevin Algorithms 466

      7.1.11 The Microcanonical Algorithm 468

      7.1.12 Strong and Weak Coupling Expansions 474

      7.1.13 Weak-Coupling Approximation 476

      7.1.14 Larger Loops in the Limit of Weak and Strong Coupling 477

      7.1.15 The String Tension 479

      7.1.16 The Lattice at Finite Temperature 482

      7.1.17 The Quark Condensate 485

      7.1.18 The Polyakov Loop 487

      7.1.19 The Center Symmetry 488

7.2 QCD Sum Rules 490

8. Phenomenological Models for Nonperturbative QCD Problems 515

8.1 The Ground State of QCD 515

8.2 The Quark–Gluon Plasma 530

Appendix 541

A The Group SU(N) 541

B Dirac Algebra in Dimension d 543

C Some Useful Integrals 546

Subject Index 549

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