Table of Contents
1 Introduction 1
1.1 Main Properties of QCD 1
1.2 Principal Tools to Work with QCD in the High-Energy Regime 2
2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory 5
2.1 Quantum Mechanics 5
2.1.1 Time Evolution of Classical Systems 5
2.1.2 Hilbert Space and Operators 9
2.1.3 From Classical to Quantum Mechanics: The Heisenberg Picture 10
2.1.4 The Schrödinger Picture 11
2.1.5 Time Evolution in the Dirac Picture 13
2.1.6 Scattering Matrix in the Dirac Picture, Time and Path Ordering 14
2.1.7 Path Ordering 17
2.1.8 Connection Between the Heisenberg and Dirac Pictures 19
2.2 Correlation Functions in Quantum Field Theory 20
2.2.1 Correlation Functions in the Heisenberg Picture 20
2.2.2 Correlation Functions in the Dirac Pictures 23
2.2.3 Positive and Negative Frequency Decomposition 24
2.2.4 Quantum Harmonic Oscillator: Particle Number Representation 27
2.2.5 Creation and Annihilation Operators and Normal Ordering 29
2.2.6 Wick's Theorems: Normal and Time Ordering 30
2.3 Summary 32
3 Geometry of Quantum Field Theories 33
3.1 Parallel Transport and Wilson Lines 33
3.1.1 The Parallel Transporter 33
3.1.2 Non-Abelian Paths 36
3.1.3 The Covariant Derivative 40
3.2 The Gauge Field Tensor and Wilson Loops 41
3.3 Summary 47
4 Basics of Wilson Lines in QCD 49
4.1 A Wilson Line Along a Path 49
4.1.1 Properties of Wilson Lines 50
4.1.2 Path Ordering 50
4.2 Piecewise Wilson Lines 53
4.3 Wilson Lines on a Linear Path 58
4.3.1 Bounded from Below 58
4.3.2 Bounded from Above 60
4.3.3 Path Reversal 61
4.3.4 Finite Wilson Line 65
4.3.5 Infinite Wilson Line 69
4.3.6 External Momenta 73
4.4 Relating Different Path Topologies 74
4.5 Piecewise Linear Wilson Lines 77
4.5.1 Path Functions 81
4.5.2 Diagrams with Final-State Cuts 83
4.6 Eikonal Approximation 87
5 Gauge-Invariant Parton Densities 91
5.1 Revision of Deep Inelastic Scattering 91
5.1.1 Kinematics 91
5.1.2 Invitation: The Free Parton Model 93
5.1.3 The Parton Model 95
5.1.4 Parton Distribution Functions 100
5.1.5 Operator Definition for PDFs 104
5.1.6 Gauge-Invariant Operator Definition 107
5.2 Semi-inclusive Deep Inelastic Scattering 111
5.2.1 Conventions and Kinematics 111
5.2.2 Structure Functions 113
5.2.3 Transverse Momentum-Dependent PDFs 115
5.2.4 Gauge-Invariant Definition for TMDs 118
5.3 Evolution of TMDs 123
5.3.1 About the Rapidity Cut-offs 126
6 Simplifying Wilson Line Calculations 129
6.1 Advanced Colour Algebra 129
6.1.1 Calculating Products of Fundamental Generators 130
6.1.2 Calculating Traces in the Adjoint Representation 135
6.2 Self-Interaction Blobs 138
6.2.1 2-Gluon Blob 138
6.2.2 3-Gluon Blob 141
6.3 Wick Rotations 144
6.3.1 Regular Wick Rotation 144
6.3.2 Wick Rotation with Wilson Lines 147
6.3.3 Light-Cone Coordinates: Double Wick Rotation 150
6.4 Wilson Integrals 151
6.4.1 2-Gluon Blob Connecting Two Adjoining Segments 154
A Brief Literature Guide 163
B Conventions and Reference Formulae 164
B.1 Notational Conventions 164
B.3 Vectors and Tensors 165
B.3 Spinors and Gamma Matrices 166
B.4 Light-Cone Coordinates 169
B.5 Fourier Transforms and Distributions 171
B.6 Lie Algebra 174
B.6.1 Representations 174
B.6.2 Properties 175
B.6.3 Useful Formulae 178
B.7 Summary of the Noether Theorems 180
B.8 Feynman Rules for QCD 181
C Integrations 184
C.1 Reference Integrals 184
C.1.1 Algebraic Integrals 184
C.1.2 Logarithmic Integrals 185
C.1.3 Cyclometric Integrals 186
C.1.4 Gaussian Integrals 186
C.1.5 Discrete Integrals 187
C.2 Special Functions and Integral Transforms 188
C.2.1 Gamma Function 188
C.2.2 Beta Function 190
C.2.3 Polylogarithms 190
C.2.4 Elliptic K-Function 192
C.2.5 Integral Transforms 192
C.3 Dimensional Regularization 195
C.3.1 Euclidian Integrals 195
C.3.2 Wick Rotation and Minkowskian Integrals 197
C.4 Path Integrals 198
C.4.1 Properties 198
Bibliography 201
Index 211
