Parton Densities in Quantum Chromodynamics: Gauge Invariance, Path-dependence and Wilson Lines (De Gruyter Studies in Mathematical Physics) (de Gruyter Studies in Mathematical Physics, 37)

The purpose of this book is to give a systematic pedagogical exposition of the quantitative analysis of Wilson lines and gauge-invariant correlation functions in quantum chromodynamics. Using techniques from the previous volume (Wilson Lines in Quantum Field Theory, 2014), an ab initio methodology is developed and practical tools for its implementation are presented. Emphasis is put on the implications of gauge invariance and path-dependence properties of transverse-momentum dependent parton density functions. The latter are associated with the QCD factorization approach to semi-inclusive hadronic processes, studied at currently operating and planned experimental facilities. Contents: Introduction Particle Number Operators in Quantum Mechanics and in Quantum Field Theory Geometry of Quantum Field Theories Basics of Wilson Lines in QCD Gauge-Invariant Parton Densities Simplifying Wilson Line Calculations Brief Literature Guide Conventions and Reference Formulae Integrations Bibliography Index Preface Contents 1 Introduction 1.1 Main Properties of QCD 1.2 Principal Tools to Work with QCD in the High-Energy Regime 2 Particle Number Operators in Quantum Mechanics and in Quantum Field Theory 2.1 Quantum Mechanics 2.1.1 Time Evolution of Classical Systems 2.1.2 Hilbert Space and Operators 2.1.3 From Classical to Quantum Mechanics: The Heisenberg Picture 2.1.4 The Schrödinger Picture 2.1.5 Time Evolution in the Dirac Picture 2.1.6 Scattering Matrix in the Dirac Picture, Time and Path Ordering 2.1.7 Path Ordering 2.1.8 Connection Between the Heisenberg and Dirac Pictures 2.2 Correlation Functions in Quantum Field Theory 2.2.1 Correlation Functions in the Heisenberg Picture 2.2.2 Correlation Functions in the Dirac Pictures 2.2.3 Positive and Negative Frequency Decomposition 2.2.4 Quantum Harmonic Oscillator: Particle Number Representation 2.2.5 Creation and Annihilation Operators and Normal Ordering 2.2.6 Wick’s Theorems: Normal and Time Ordering 2.3 Summary 3 Geometry of Quantum Field Theories 3.1 Parallel Transport and Wilson Lines 3.1.1 The Parallel Transporter 3.1.2 Non-Abelian Paths 3.1.3 The Covariant Derivative 3.2 The Gauge Field Tensor and Wilson Loops 3.3 Summary 4 Basics of Wilson Lines in QCD 4.1 A Wilson Line Along a Path 4.1.1 Properties of Wilson Lines 4.1.2 Path Ordering 4.2 Piecewise Wilson Lines 4.3 Wilson Lines on a Linear Path 4.3.1 Bounded from Below 4.3.2 Bounded from Above 4.3.3 Path Reversal 4.3.4 Finite Wilson Line 4.3.5 Infinite Wilson Line 4.3.6 External Momenta 4.4 Relating Different Path Topologies 4.5 Piecewise Linear Wilson Lines 4.5.1 Path Functions 4.5.2 Diagrams with Final-State Cuts 4.6 Eikonal Approximation 5 Gauge-Invariant Parton Densities 5.1 Revision of Deep Inelastic Scattering 5.1.1 Kinematics 5.1.2 Invitation: The Free Parton Model 5.1.3 The Parton Model 5.1.4 Parton Distribution Functions 5.1.5 Operator Definition for PDFs 5.1.6 Gauge-Invariant Operator Definition 5.2 Semi-inclusive Deep Inelastic Scattering 5.2.1 Conventions and Kinematics 5.2.2 Structure Functions 5.2.3 Transverse Momentum-Dependent PDFs 5.2.4 Gauge-Invariant Definition for TMDs 5.3 Evolution of TMDs 5.3.1 About the Rapidity Cut-offs 6 Simplifying Wilson Line Calculations 6.1 Advanced Colour Algebra 6.1.1 Calculating Products of Fundamental Generators 6.1.2 Calculating Traces in the Adjoint Representation 6.2 Self-Interaction Blobs 6.2.1 2-Gluon Blob 6.2.2 3-Gluon Blob 6.3 Wick Rotations 6.3.1 Regular Wick Rotation 6.3.2 Wick Rotation with Wilson Lines 6.3.3 Light-Cone Coordinates: Double Wick Rotation 6.4 Wilson Integrals 6.4.1 2-Gluon Blob Connecting Two Adjoining Segments A. Brief Literature Guide B Conventions and Reference Formulae B.1 Notational Conventions B.2 Vectors and Tensors B.3 Spinors and Gamma Matrices B.4 Light-Cone Coordinates B.5 Fourier Transforms and Distributions B.6 Lie Algebra B.6.1 Representations B.6.2 Properties B.6.3 Useful Formulae B.7 Summary of the Noether Theorems B.8 Feynman Rules for QCD C Integrations C.1 Reference Integrals C.1.1 Algebraic Integrals C.1.2 Logarithmic Integrals C.1.3 Cyclometric Integrals C.1.4 Gaussian Integrals C.1.5 Discrete Integrals C.2 Special Functions and Integral Transforms C.2.1 Gamma Function C.2.2 Beta Function C.2.3 Polylogarithms C.2.4 Elliptic K-Function C.2.5 Integral Transforms C.3 Dimensional Regularization C.3.1 Euclidian Integrals C.3.2 Wick Rotation and Minkowskian Integrals C.4 Path Integrals C.4.1 Properties Bibliography Index "The book consists of six chapters. In Introduction we briefly discuss the conceptual issues inherent to the theory of strong interaction in the high energy regime and present the motivation to the use of parton density functions within the QCD factorization framework. In Chapter 2 we introduce particle number functions in general form. We show how the normal-, time-, and path-ordering operations make the multiple products of local quantum operators well-defined, and how Wick's theorems enable practical calculations of quantum correlation functions. Chapter 3 outlines geometrical approach to the construction of a general nonabelian gauge field theory. The idea of local gauge invariance is used to derive a Wilson line as a parallel transporter associated to field operators defined in different space-time points. In Chapter 4 we give a detailed description of the properties of piecewise linear finite and infinite Wilson lines and present the basis for a new methodology of perturbative calculations with such Wilson lines. Chapter 5 is devoted to the principles of gauge-invariant path-dependent parton density functions (PDFs), which are intrinsic to QCD factorization approaches to the analysis of fully inclusive and semi-inclusive hadronic processes at high energy. We discuss geometrical structure of the Wilson lines which make the PDFs gauge-invariant, paying special attention to the operator definition of transverse momentum dependent PDFs and their evolution"-- Provided by publisher "The book consists of six chapters. In Introduction we briefly discuss the conceptual issues inherent to the theory of strong interaction in the high energy regime and present the motivation to the use of parton density functions within the QCD factorization framework. In Chapter 2 we introduce particle number functions in general form. We show how the normal-, time-, and path-ordering operations make the multiple products of local quantum operators well-defined, and how Wick's theorems enable practical calculations of quantum correlation functions. Chapter 3 outlines geometrical approach to the construction of a general nonabelian gauge field theory. The idea of local gauge invariance is used to derive a Wilson line as a parallel transporter associated to field operators defined in different space-time points. In Chapter 4 we give a detailed description of the properties of piecewise linear finite and infinite Wilson lines and present the basis for a new methodology of perturbative calculations with such Wilson lines. Chapter 5 is devoted to the principles of gauge-invariant path-dependent parton density functions (PDFs), which are intrinsic to QCD factorization approaches to the analysis of fully inclusive and semi-inclusive hadronic processes at high energy. We discuss geometrical structure of the Wilson lines which make the PDFs gauge-invariant, paying special attention to the operator definition of transverse momentum dependent PDFs and their evolution"-- Résumé de l'éditeur The purpose of this book is to give a systematic pedagogical exposition of the quantitative analysis of Wilson lines and loops in quantum chromodynamics. Using techniques from the previous volume (Wilson Lines in Quantum Field Theory, 2014), ab initio techniques are developed and practical tools for their implementation presented. An emphasis is put on their renormalization and on implications on processes observable at experimental facilities