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E-raamat: Lattice Quantum Chromodynamics: Practical Essentials

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This book provides an overview of the techniques central to lattice quantum chromodynamics, including modern developments. 
The book has four chapters. The first chapter explains the formulation of quarks and gluons on a Euclidean lattice. The second chapter introduces Monte Carlo methods and details the numerical algorithms to simulate lattice gauge fields. Chapter three explains the mathematical and numerical techniques needed to study quark fields and the computation of quark propagators. The fourth chapter is devoted to the physical observables constructed from lattice fields and explains how to measure them in simulations. The book is aimed at enabling graduate students who are new to the field to carry out explicitly the first steps and prepare them for research in lattice QCD.
1 Quantum Field Theory (QFT) on the Lattice
1(34)
1.1 A Brief History of Quarks and Gluons
1(1)
1.2 Classical Fields and Gauge Invariance
2(1)
1.3 Hamiltonian and Path Integral Formulations of Quantum Mechanics
3(4)
1.3.1 One Bosonic Oscillator
4(2)
1.3.2 One Fermionic Oscillator
6(1)
1.4 Quantum Fields on a Lattice
7(16)
1.4.1 From One Oscillator to a Field
7(3)
1.4.2 Correlation Functions
10(1)
1.4.3 Analytic Continuation to Minkowski Space
11(2)
1.4.4 Gauge Invariance on the Lattice
13(1)
1.4.5 The Wilson Gauge Action
14(1)
1.4.6 Strong Coupling Expansions
15(2)
1.4.7 The Wilson Fermion Action
17(1)
1.4.8 Chiral Symmetry and the Nielsen-Ninomiya No-Go Theorem
18(3)
1.4.9 Other Fermion Formulations
21(2)
1.5 Recovering Continuum QCD
23(9)
1.5.1 Classical Continuum Limit
24(1)
1.5.2 Renormalisation Group Equation
25(2)
1.5.3 Isospin Symmetry and Ward Identities
27(1)
1.5.4 Improvement of the Continuum Limit
28(2)
1.5.5 Improvement of Wilson Fermions
30(1)
1.5.6 Twisted Mass Fermions
31(1)
1.6 Further Reading
32(3)
References
33(2)
2 Monte Carlo Methods
35(20)
2.1 Markov Chain Monte Carlo
35(2)
2.2 Sampling Yang-Mills Gauge Fields
37(6)
2.2.1 Random Numbers and the Rejection Method
38(1)
2.2.2 Heatbath Algorithms
39(3)
2.2.3 Overrelaxation Algorithms
42(1)
2.3 Hybrid Monte Carlo
43(5)
2.3.1 Detailed Balance Condition
44(2)
2.3.2 Hamilton's Equation of Motion
46(2)
2.4 Symplectic Integration Schemes
48(4)
2.5 Summary and Further Reading
52(3)
References
53(2)
3 Handling Fermions on the Lattice
55(42)
3.1 Wick Contractions
55(3)
3.2 Sparse Linear Algebra
58(15)
3.2.1 Krylov Subspace Techniques
59(2)
3.2.2 Iterative Solvers for Linear Systems
61(6)
3.2.3 Algebraic Multigrid Methods
67(1)
3.2.4 Other Uses of Krylov Subspaces
68(5)
3.3 Fermion Determinant
73(4)
3.3.1 Pseudofermions
73(2)
3.3.2 Factorisations
75(2)
3.4 HMC with Fermions Revisited
77(7)
3.4.1 Equations of Motion
77(3)
3.4.2 Multi-rate Integration Schemes
80(1)
3.4.3 A Single Dynamical Quark Flavour: The RHMC Algorithm
81(3)
3.5 The Quark Propagator from a Point Source
84(3)
3.5.1 Quark Observables from a Single Point Source
84(2)
3.5.2 Reducing Variance in Point Propagator Calculations
86(1)
3.6 All-to-All Quark Propagators
87(6)
3.6.1 Stochastic Estimators
87(5)
3.6.2 Exploiting Low Eigenmodes of the Dirac Operator
92(1)
3.7 Summary and Further Reading
93(4)
References
94(3)
4 Calculating Observables of Quantum Fields
97(38)
4.1 Symmetry Properties of Creation and Annihilation Operators
97(6)
4.1.1 Gauge-Invariant Observables
98(1)
4.1.2 Charge Conjugation, Isopin and Flavour Symmetry
98(2)
4.1.3 Spin and Parity
100(2)
4.1.4 Translation Invariance and Momentum
102(1)
4.1.5 Reducing Representations of Symmetries
102(1)
4.2 Techniques for Hadron Spectroscopy
103(8)
4.2.1 Variational Methods
104(1)
4.2.2 Scale Setting
105(1)
4.2.3 The Wilson Flow
106(2)
4.2.4 Scattering and the Luscher Method
108(3)
4.3 Gluons, Wilson Loops and Glueballs
111(7)
4.3.1 Gauge Smearing
111(2)
4.3.2 Glueballs
113(1)
4.3.3 The Static Potential and Strong Coupling Constant
114(3)
4.3.4 Topological Charge
117(1)
4.4 Quarks and Hadron Physics
118(8)
4.4.1 Quark Smearing
119(1)
4.4.2 Hadron Physics with Quarks
120(3)
4.4.3 Hadron Scattering in Lattice Monte Carlo Calculations
123(2)
4.4.4 The Static Potential with Light Quarks: String Breaking
125(1)
4.5 Statistical Data Analysis
126(5)
4.5.1 Controlling Bias, Covariance and Autocorrelations in Data from a Markov Chain
127(3)
4.5.2 Comparing Monte Carlo Data to a Model
130(1)
4.6 Summary
131(4)
References
131(4)
Appendix A Notational Conventions 135
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