格点量子色动力学导论=QUANTUM CHROMODYNAMICS ON THE LATTICE：AN INTRODUCTORY PRESENTATION 影印版 英文版【2025|PDF下载-Epub版本|mobi电子书|kindle百度云盘下载】

格点量子色动力学导论=QUANTUM CHROMODYNAMICS ON THE LATTICE：AN INTRODUCTORY PRESENTATION 影印版 英文版PDF格式电子书版下载

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1 The path integral on the lattice1

1.1 Hilbert space and propagation in Euclidean time2

1.1.1 Hilbert spaces2

1.1.2 Remarks on Hilbert spaces in particle physics3

1.1.3 Euclidean correlators4

1.2 The path integral for a quantum mechanical system7

1.3 The path integral for a scalar field theory10

1.3.1 The Klein-Gordon field10

1.3.2 Lattice regularization of the Klein-Gordon Hamiltonian11

1.3.3 The Euclidean time transporter for the free case14

1.3.4 Treating the interaction term with the Trotter formula15

1.3.5 Path integral representation for the partition function16

1.3.6 Including operators in the path integral17

1.4 Quantization with the path integral19

1.4.1 Different discretizations of the Euclidean action19

1.4.2 The path integral as a quantization prescription20

1.4.3 The relation to statistical mechanics22

References23

2 QCD on the lattice-a first look25

2.1 The QCD action in the continuum25

2.1.1 Quark and gluon fields26

2.1.2 The fermionic part of the QCD action26

2.1.3 Gauge invariance of the fermion action28

2.1.4 The gluon action29

2.1.5 Color components of the gauge field30

2.2 Naive discretization of fermions32

2.2.1 Discretization of free fermions32

2.2.2 Introduction of the gauge fields as link variables33

2.2.3 Relating the link variables to the continuum gauge fields34

2.3 The Wilson gauge action36

2.3.1 Gauge-invariant objects built with link variables36

2.3.2 The gauge action37

2.4 Formal expression for the QCD lattice path integral39

2.4.1 The QCD lattice path integral39

References41

3 Pure gauge theory on the lattice43

3.1 Haar measure44

3.1.1 Gauge field measure and gauge invariance44

3.1.2 Group integration measure45

3.1.3 A few integrals for SU(3)46

3.2 Gauge invariance and gauge fixing49

3.2.1 Maximal trees49

3.2.2 Other gauges51

3.2.3 Gauge invariance of observables53

3.3 Wilson and Polyakov loops54

3.3.1 Definition of the Wilson loop54

3.3.2 Temporal gauge55

3.3.3 Physical interpretation of the Wilson loop55

3.3.4 Wilson line and the quark-antiquark pair57

3.3.5 Polyakov loop57

3.4 The static quark potential58

3.4.1 Strong coupling expansion of the Wilson loop59

3.4.2 The Coulomb part of the static quark potential62

3.4.3 Physical implications of the static QCD potential63

3.5 Setting the scale with the static potential63

3.5.1 Discussion of numerical data for the static potential64

3.5.2 The Sommer parameter and the lattice spacing65

3.5.3 Renormalization group and the running coupling67

3.5.4 The true continuum limit69

3.6 Lattice gauge theory with other gauge groups69

References70

4 Numerical simulation of pure gauge theory73

4.1 The Monte Carlo method74

4.1.1 Simple sampling and importance sampling74

4.1.2 Markov chains75

4.1.3 Metropolis algorithm-general idea78

4.1.4 Metropolis algorithm for Wilson's gauge action79

4.2 Implementation of Monte Carlo algorithms for SU(3)80

4.2.1 Representation of the link variables81

4.2.2 Boundary conditions82

4.2.3 Generating a candidate link for the Metropolis update83

4.2.4 A few remarks on random numbers84

4.3 More Monte Carlo algorithms84

4.3.1 The heat bath algorithm85

4.3.2 Overrelaxation88

4.4 Running the simulation89

4.4.1 Initialization91

4.4.2 Equilibration updates91

4.4.3 Evaluation of the observables92

4.5 Analyzing the data93

4.5.1 Statistical analysis for uncorrelated data93

4.5.2 Autocorrelation94

4.5.3 Techniques for smaller data sets96

4.5.4 Some numerical exercises98

References100

5 Fermions on the lattice103

5.1 Fermi statistics and Grassmann numbers103

5.1.1 Some new notation103

5.1.2 Fermi statistics104

5.1.3 Grassmann numbers and derivatives105

5.1.4 Integrals over Grassmann numbers106

5.1.5 Gaussian integrals with Grassmann numbers108

5.1.6 Wick's theorem109

5.2 Fermion doubling and Wilson's fermion action110

5.2.1 The Dirac operator on the lattice110

5.2.2 The doubling problem111

5.2.3 Wilson fermions112

5.3 Fermion lines and hopping expansion114

5.3.1 Hopping expansion of the quark propagator114

5.3.2 Hopping expansion for the fermion determinant117

5.4 Discrete symmetries of the Wilson action117

5.4.1 Charge conjugation117

5.4.2 Parity and Euclidean reflections119

5.4.3 γ5-hermiticity121

References121

6 Hadron spectroscopy123

6.1 Hadron interpolators and correlators123

6.1.1 Meson interpolators124

6.1.2 Meson correlators127

6.1.3 Interpolators and correlators for baryons129

6.1.4 Momentum projection131

6.1.5 Final formula for hadron correlators132

6.1.6 The quenched approximation133

6.2 Strategy of the calculation135

6.2.1 The need for quark sources135

6.2.2 Point source or extended source?136

6.2.3 Extended sources137

6.2.4 Calculation of the quark propagator138

6.2.5 Exceptional configurations141

6.2.6 Smoothing of gauge configurations142

6.3 Extracting hadron masses143

6.3.1 Effective mass curves144

6.3.2 Fitting the correlators146

6.3.3 The calculation of excited states147

6.4 Finalizing the results for the hadron masses150

6.4.1 Discussion of some raw data150

6.4.2 Setting the scale and the quark mass parameters151

6.4.3 Various extrapolations152

6.4.4 Some quenched results154

References155

7 Chiral symmetry on the lattice157

7.1 Chiral symmetry in continuum QCD157

7.1.1 Chiral symmetry for a single flavor157

7.1.2 Several flavors159

7.1.3 Spontaneous breaking of chiral symmetry160

7.2 Chiral symmetry and the lattice162

7.2.1 Wilson fermions and the Nielsen-Ninomiya theorem162

7.2.2 The Ginsparg-Wilson equation163

7.2.3 Chiral symmetry on the lattice164

7.3 Consequences of the Ginsparg-Wilson equation166

7.3.1 Spectrum of the Dirac operator166

7.3.2 Index theorem168

7.3.3 The axial anomaly170

7.3.4 The chiral condensate172

7.3.5 The Banks-Casher relation175

7.4 The overlap operator177

7.4.1 Definition of the overlap operator177

7.4.2 Locality properties of chiral Dirac operators178

7.4.3 Numerical evaluation of the overlap operator179

References183

8 Dynamical fermions185

8.1 The many faces of the fermion determinant185

8.1.1 The fermion determinant as observable186

8.1.2 The fermion determinant as a weight factor186

8.1.3 Pseudofermions187

8.1.4 Effective fermion action188

8.1.5 First steps toward updating with fermions189

8.2 Hybrid Monte Carlo190

8.2.1 Molecular dynamics leapfrog evolution191

8.2.2 Completing with an accept-reject step194

8.2.3 Implementing HMC for gauge fields and fermions195

8.3 Other algorithmic ideas199

8.3.1 The R-algorithm199

8.3.2 Partial updates200

8.3.3 Polynomial and rational HMC200

8.3.4 Multi-pseudofermions and UV-filtering201

8.3.5 Further developments202

8.4 Other techniques using pseudofermions203

8.5 The coupling-mass phase diagram205

8.5.1 Continuum limit and phase transitions205

8.5.2 The phase diagram for Wilson fermions206

8.5.3 Ginsparg-Wilson fermions208

8.6 Full QCD calculations209

References210

9 Symanzik improvement and RG actions213

9.1 The Symanzik improvement program214

9.1.1 A toy example214

9.1.2 The framework for improving lattice QCD215

9.1.3 Improvement of interpolators218

9.1.4 Determination of improvement coefficients219

9.2 Lattice actions for free fermions from RG transformations221

9.2.1 Integrating out the fields over hypercubes222

9.2.2 The blocked lattice Dirac operator223

9.2.3 Properties of the blocked action226

9.3 Real space renormalization group for QCD227

9.3.1 Blocking full QCD228

9.3.2 The RG flow of the couplings231

9.3.3 Saddle point analysis of the RG equation232

9.3.4 Solving the RG equations233

9.4 Mapping continuum symmetries onto the lattice236

9.4.1 The generating functional and its symmetries236

9.4.2 Identification of the corresponding lattice symmetries238

References241

10 More about lattice fermions243

10.1 Staggered fermions243

10.1.1 The staggered transformation243

10.1.2 Tastes of staggered fermions245

10.1.3 Developments and open questions248

10.2 Domain wall fermions249

10.2.1 Formulation of lattice QCD with domain wall fermions250

10.2.2 The 5D theory and its equivalence to 4D chiralfermions252

10.3 Twisted mass fermions253

10.3.1 The basic formulation of twisted mass QCD254

10.3.2 The relation between twisted and conventional QCD256

10.3.3 O(a)improvement at maximal twist258

10.4 Effective theories for heavy quarks260

10.4.1 The need for an effective theory260

10.4.2 Lattice action for heavy quarks261

10.4.3 General framework and expansion coefficients263

References264

11 Hadron structure267

11.1 Low-energy parameters267

11.1.1 Operator definitions268

11.1.2 Ward identities270

11.1.3 Naive currents and conserved currents on the lattice274

11.1.4 Low-energy parameters from correlation functions278

11.2 Renormalization279

11.2.1 Why do we need renormalization?279

11.2.2 Renormalization with the Rome-Southampton method281

11.3 Hadronic decays and scattering284

11.3.1 Threshold region284

11.3.2 Beyond the threshold region287

11.4 Matrix elements289

11.4.1 Pion form factor290

11.4.2 Weak matrix elements294

11.4.3 OPE expansion and effective weak Hamiltonian295

References297

12 Temperature and chemical potential301

12.1 Introduction of temperature301

12.1.1 Analysis of pure gauge theory303

12.1.2 Switching on dynamical fermions307

12.1.3 Properties of QCD in the deconfinement phase310

12.2 Introduction of the chemical potential312

12.2.1 The chemical potential on the lattice312

12.2.2 The QCD phase diagram in the(T,μ)space317

12.3 Chemical potential:Monte Carlo techniques318

12.3.1 Reweighting319

12.3.2 Series expansion321

12.3.3 Imaginary μ321

12.3.4 Canonical partition functions322

References323

A Appendix327

A.1 The Lie groups SU(N)327

A.1.1 Basic properties327

A.1.2 Lie algebra327

A.1.3 Generators for SU(2)and SU(3)329

A.1.4 Derivatives of group elements329

A.2 Gamma matrices330

A.3 Fourier transformation on the lattice332

A.4 Wilson's formulation of lattice QCD333

A.5 A few formulas for matrix algebra334

References336

Index337