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电子书下载

1 The path integral on the lattice 1

1.1 Hilbert space and propagation in Euclidean time 2

1.1.1 Hilbert spaces 2

1.1.2 Remarks on Hilbert spaces in particle physics 3

1.1.3 Euclidean correlators 4

1.2 The path integral for a quantum mechanical system 7

1.3 The path integral for a scalar field theory 10

1.3.1 The Klein-Gordon field 10

1.3.2 Lattice regularization of the Klein-Gordon Hamiltonian 11

1.3.3 The Euclidean time transporter for the free case 14

1.3.4 Treating the interaction term with the Trotter formula 15

1.3.5 Path integral representation for the partition function 16

1.3.6 Including operators in the path integral 17

1.4 Quantization with the path integral 19

1.4.1 Different discretizations of the Euclidean action 19

1.4.2 The path integral as a quantization prescription 20

1.4.3 The relation to statistical mechanics 22

References 23

2 QCD on the lattice-a first look 25

2.1 The QCD action in the continuum 25

2.1.1 Quark and gluon fields 26

2.1.2 The fermionic part of the QCD action 26

2.1.3 Gauge invariance of the fermion action 28

2.1.4 The gluon action 29

2.1.5 Color components of the gauge field 30

2.2 Naive discretization of fermions 32

2.2.1 Discretization of free fermions 32

2.2.2 Introduction of the gauge fields as link variables 33

2.2.3 Relating the link variables to the continuum gauge fields 34

2.3 The Wilson gauge action 36

2.3.1 Gauge-invariant objects built with link variables 36

2.3.2 The gauge action 37

2.4 Formal expression for the QCD lattice path integral 39

2.4.1 The QCD lattice path integral 39

References 41

3 Pure gauge theory on the lattice 43

3.1 Haar measure 44

3.1.1 Gauge field measure and gauge invariance 44

3.1.2 Group integration measure 45

3.1.3 A few integrals for SU(3) 46

3.2 Gauge invariance and gauge fixing 49

3.2.1 Maximal trees 49

3.2.2 Other gauges 51

3.2.3 Gauge invariance of observables 53

3.3 Wilson and Polyakov loops 54

3.3.1 Definition of the Wilson loop 54

3.3.2 Temporal gauge 55

3.3.3 Physical interpretation of the Wilson loop 55

3.3.4 Wilson line and the quark-antiquark pair 57

3.3.5 Polyakov loop 57

3.4 The static quark potential 58

3.4.1 Strong coupling expansion of the Wilson loop 59

3.4.2 The Coulomb part of the static quark potential 62

3.4.3 Physical implications of the static QCD potential 63

3.5 Setting the scale with the static potential 63

3.5.1 Discussion of numerical data for the static potential 64

3.5.2 The Sommer parameter and the lattice spacing 65

3.5.3 Renormalization group and the running coupling 67

3.5.4 The true continuum limit 69

3.6 Lattice gauge theory with other gauge groups 69

References 70

4 Numerical simulation of pure gauge theory 73

4.1 The Monte Carlo method 74

4.1.1 Simple sampling and importance sampling 74

4.1.2 Markov chains 75

4.1.3 Metropolis algorithm-general idea 78

4.1.4 Metropolis algorithm for Wilson's gauge action 79

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

4.2.1 Representation of the link variables 81

4.2.2 Boundary conditions 82

4.2.3 Generating a candidate link for the Metropolis update 83

4.2.4 A few remarks on random numbers 84

4.3 More Monte Carlo algorithms 84

4.3.1 The heat bath algorithm 85

4.3.2 Overrelaxation 88

4.4 Running the simulation 89

4.4.1 Initialization 91

4.4.2 Equilibration updates 91

4.4.3 Evaluation of the observables 92

4.5 Analyzing the data 93

4.5.1 Statistical analysis for uncorrelated data 93

4.5.2 Autocorrelation 94

4.5.3 Techniques for smaller data sets 96

4.5.4 Some numerical exercises 98

References 100

5 Fermions on the lattice 103

5.1 Fermi statistics and Grassmann numbers 103

5.1.1 Some new notation 103

5.1.2 Fermi statistics 104

5.1.3 Grassmann numbers and derivatives 105

5.1.4 Integrals over Grassmann numbers 106

5.1.5 Gaussian integrals with Grassmann numbers 108

5.1.6 Wick's theorem 109

5.2 Fermion doubling and Wilson's fermion action 110

5.2.1 The Dirac operator on the lattice 110

5.2.2 The doubling problem 111

5.2.3 Wilson fermions 112

5.3 Fermion lines and hopping expansion 114

5.3.1 Hopping expansion of the quark propagator 114

5.3.2 Hopping expansion for the fermion determinant 117

5.4 Discrete symmetries of the Wilson action 117

5.4.1 Charge conjugation 117

5.4.2 Parity and Euclidean reflections 119

5.4.3 γ5-hermiticity 121

References 121

6 Hadron spectroscopy 123

6.1 Hadron interpolators and correlators 123

6.1.1 Meson interpolators 124

6.1.2 Meson correlators 127

6.1.3 Interpolators and correlators for baryons 129

6.1.4 Momentum projection 131

6.1.5 Final formula for hadron correlators 132

6.1.6 The quenched approximation 133

6.2 Strategy of the calculation 135

6.2.1 The need for quark sources 135

6.2.2 Point source or extended source? 136

6.2.3 Extended sources 137

6.2.4 Calculation of the quark propagator 138

6.2.5 Exceptional configurations 141

6.2.6 Smoothing of gauge configurations 142

6.3 Extracting hadron masses 143

6.3.1 Effective mass curves 144

6.3.2 Fitting the correlators 146

6.3.3 The calculation of excited states 147

6.4 Finalizing the results for the hadron masses 150

6.4.1 Discussion of some raw data 150

6.4.2 Setting the scale and the quark mass parameters 151

6.4.3 Various extrapolations 152

6.4.4 Some quenched results 154

References 155

7 Chiral symmetry on the lattice 157

7.1 Chiral symmetry in continuum QCD 157

7.1.1 Chiral symmetry for a single flavor 157

7.1.2 Several flavors 159

7.1.3 Spontaneous breaking of chiral symmetry 160

7.2 Chiral symmetry and the lattice 162

7.2.1 Wilson fermions and the Nielsen-Ninomiya theorem 162

7.2.2 The Ginsparg-Wilson equation 163

7.2.3 Chiral symmetry on the lattice 164

7.3 Consequences of the Ginsparg-Wilson equation 166

7.3.1 Spectrum of the Dirac operator 166

7.3.2 Index theorem 168

7.3.3 The axial anomaly 170

7.3.4 The chiral condensate 172

7.3.5 The Banks-Casher relation 175

7.4 The overlap operator 177

7.4.1 Definition of the overlap operator 177

7.4.2 Locality properties of chiral Dirac operators 178

7.4.3 Numerical evaluation of the overlap operator 179

References 183

8 Dynamical fermions 185

8.1 The many faces of the fermion determinant 185

8.1.1 The fermion determinant as observable 186

8.1.2 The fermion determinant as a weight factor 186

8.1.3 Pseudofermions 187

8.1.4 Effective fermion action 188

8.1.5 First steps toward updating with fermions 189

8.2 Hybrid Monte Carlo 190

8.2.1 Molecular dynamics leapfrog evolution 191

8.2.2 Completing with an accept-reject step 194

8.2.3 Implementing HMC for gauge fields and fermions 195

8.3 Other algorithmic ideas 199

8.3.1 The R-algorithm 199

8.3.2 Partial updates 200

8.3.3 Polynomial and rational HMC 200

8.3.4 Multi-pseudofermions and UV-filtering 201

8.3.5 Further developments 202

8.4 Other techniques using pseudofermions 203

8.5 The coupling-mass phase diagram 205

8.5.1 Continuum limit and phase transitions 205

8.5.2 The phase diagram for Wilson fermions 206

8.5.3 Ginsparg-Wilson fermions 208

8.6 Full QCD calculations 209

References 210

9 Symanzik improvement and RG actions 213

9.1 The Symanzik improvement program 214

9.1.1 A toy example 214

9.1.2 The framework for improving lattice QCD 215

9.1.3 Improvement of interpolators 218

9.1.4 Determination of improvement coefficients 219

9.2 Lattice actions for free fermions from RG transformations 221

9.2.1 Integrating out the fields over hypercubes 222

9.2.2 The blocked lattice Dirac operator 223

9.2.3 Properties of the blocked action 226

9.3 Real space renormalization group for QCD 227

9.3.1 Blocking full QCD 228

9.3.2 The RG flow of the couplings 231

9.3.3 Saddle point analysis of the RG equation 232

9.3.4 Solving the RG equations 233

9.4 Mapping continuum symmetries onto the lattice 236

9.4.1 The generating functional and its symmetries 236

9.4.2 Identification of the corresponding lattice symmetries 238

References 241

10 More about lattice fermions 243

10.1 Staggered fermions 243

10.1.1 The staggered transformation 243

10.1.2 Tastes of staggered fermions 245

10.1.3 Developments and open questions 248

10.2 Domain wall fermions 249

10.2.1 Formulation of lattice QCD with domain wall fermions 250

10.2.2 The 5D theory and its equivalence to 4D chiralfermions 252

10.3 Twisted mass fermions 253

10.3.1 The basic formulation of twisted mass QCD 254

10.3.2 The relation between twisted and conventional QCD 256

10.3.3 O(a)improvement at maximal twist 258

10.4 Effective theories for heavy quarks 260

10.4.1 The need for an effective theory 260

10.4.2 Lattice action for heavy quarks 261

10.4.3 General framework and expansion coefficients 263

References 264

11 Hadron structure 267

11.1 Low-energy parameters 267

11.1.1 Operator definitions 268

11.1.2 Ward identities 270

11.1.3 Naive currents and conserved currents on the lattice 274

11.1.4 Low-energy parameters from correlation functions 278

11.2 Renormalization 279

11.2.1 Why do we need renormalization? 279

11.2.2 Renormalization with the Rome-Southampton method 281

11.3 Hadronic decays and scattering 284

11.3.1 Threshold region 284

11.3.2 Beyond the threshold region 287

11.4 Matrix elements 289

11.4.1 Pion form factor 290

11.4.2 Weak matrix elements 294

11.4.3 OPE expansion and effective weak Hamiltonian 295

References 297

12 Temperature and chemical potential 301

12.1 Introduction of temperature 301

12.1.1 Analysis of pure gauge theory 303

12.1.2 Switching on dynamical fermions 307

12.1.3 Properties of QCD in the deconfinement phase 310

12.2 Introduction of the chemical potential 312

12.2.1 The chemical potential on the lattice 312

12.2.2 The QCD phase diagram in the(T,μ)space 317

12.3 Chemical potential:Monte Carlo techniques 318

12.3.1 Reweighting 319

12.3.2 Series expansion 321

12.3.3 Imaginary μ 321

12.3.4 Canonical partition functions 322

References 323

A Appendix 327

A.1 The Lie groups SU(N) 327

A.1.1 Basic properties 327

A.1.2 Lie algebra 327

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

A.1.4 Derivatives of group elements 329

A.2 Gamma matrices 330

A.3 Fourier transformation on the lattice 332

A.4 Wilson's formulation of lattice QCD 333

A.5 A few formulas for matrix algebra 334

References 336

Index 337