Quantum Chromodynamics on the Lattice : An Introduction for Beginners

by Gattringer, Christof; Lang, Christian B.

ISBN13: 9783642018497

ISBN10: 3642018491

Format: Hardcover

Pub. Date: 2009-12-04

Publisher(s): Springer Verlag

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Summary

The lattice formulation is at present the most successful approach to Quantum Chromodynamics - the theory of quarks and gluons. This book is intended for newcomers to the field and presents a clear and easy-to-follow path from the first principles all the way to actual calculations. It focusses on QCD and discusses mainly SU(3) lattice gauge theory, both with and without fermions. Numerical calculations in lattice field theory have now become the most effective approach for obtaining quantitative results, and thus three chapters include sections describing numerical techniques, as used in pure gauge theory, in quenched spectroscopy and in treating dynamical fermions.

The path integral on the lattice	p. 1
Hilbert space and propagation in Euclidean time	p. 2
Hilbert spaces	p. 2
Remarks on Hilbert spaces in particle physics	p. 3
Euclidean correlators	p. 4
The path integral for a quantum mechanical system	p. 7
The path integral for a scalar field theory	p. 10
The Klein-Gordon field	p. 10
Lattice regularization of the Klein-Gordon Hamiltonian	p. 11
The Euclidean time transporter for the free case	p. 14
Treating the interaction term with the Trotter formula	p. 15
Path integral representation for the partition function	p. 16
Including operators in the path integral	p. 17
Quantization with the path integral	p. 19
Different discretizations of the Euclidean action	p. 19
The path integral as a quantization prescription	p. 20
The relation to statistical mechanics	p. 22
References	p. 23
QCD on the lattice - a first look	p. 25
The QCD action in the continuum	p. 25
Quark and gluon fields	p. 26
The fermionic part of the QCD action	p. 26
Gauge invariance of the fermion action	p. 28
The gluon action	p. 29
Color components of the gauge field	p. 30
Naive discretization of fermions	p. 32
Discretization of free fermions	p. 32
Introduction of the gauge fields as link variables	p. 33
Relating the link variables to the continuum gauge fields	p. 34
The Wilson gauge action	p. 36
Gauge-invariant objects built with link variables	p. 36
The gauge action	p. 37
Formal expression for the QCD lattice path integral	p. 39
The QCD lattice path integral	p. 39
References	p. 41
Pure gauge theory on the lattice	p. 43
Haar measure	p. 44
Gauge field measure and gauge invariance	p. 44
Group integration measure	p. 45
A few integrals for SU(3)	p. 46
Gauge invariance and gauge fixing	p. 49
Maximal trees	p. 49
Other gauges	p. 51
Gauge invariance of observables	p. 53
Wilson and Polyakov loops	p. 54
Definition of the Wilson loop	p. 54
Temporal gauge	p. 55
Physical interpretation of the Wilson loop	p. 55
Wilson line and the quark-antiquark pair	p. 57
Polyakov loop	p. 57
The static quark potential	p. 58
Strong coupling expansion of the Wilson loop	p. 59
The Coulomb part of the static quark potential	p. 62
Physical implications of the static QCD potential	p. 63
Setting the scale with the static potential	p. 63
Discussion of numerical data for the static potential	p. 64
The Sommer parameter and the lattice spacing	p. 65
Renormalization group and the running coupling	p. 67
The true continuum limit	p. 69
Lattice gauge theory with other gauge groups	p. 69
References	p. 70
Numerical simulation of pure gauge theory	p. 73
The Monte Carlo method	p. 74
Simple sampling and importance sampling	p. 74
Markov chains	p. 75
Metropolis algorithm - general idea	p. 78
Metropolis algorithm for Wilson's gauge action	p. 79
Implementation of Monte Carlo algorithms for SU(3)	p. 80
Representation of the link variables	p. 81
Boundary conditions	p. 82
Generating a candidate link for the Metropolis update	p. 83
A few remarks on random numbers	p. 84
More Monte Carlo algorithms	p. 84
The heat bath algorithm	p. 85
Overrelaxation	p. 88
Running the simulation	p. 89
Initialization	p. 91
Equilibration updates	p. 91
Evaluation of the observables	p. 92
Analyzing the data	p. 93
Statistical analysis for uncorrelated data	p. 93
Autocorrelation	p. 94
Techniques for smaller data sets	p. 96
Some numerical exercises	p. 98
References	p. 100
Fermions on the lattice	p. 103
Fermi statistics and Grassmann numbers	p. 103
Some new notation	p. 103
Fermi statistics	p. 104
Grassmann numbers and derivatives	p. 105
Integrals over Grassmann numbers	p. 106
Gaussian integrals with Grassmann numbers	p. 108
Wick's theorem	p. 109
Fermion doubling and Wilson's fermion action	p. 110
The Dirac operator on the lattice	p. 110
The doubling problem	p. 111
Wilson fermions	p. 112
Fermion lines and hopping expansion	p. 114
Hopping expansion of the quark propagator	p. 114
Hopping expansion for the fermion determinant	p. 117
Discrete symmetries of the Wilson action	p. 117
Charge conjugation	p. 117
Parity and Euclidean reflections	p. 119
¿₅-hermiticity	p. 121
References	p. 121
Hadron spectroscopy	p. 123
Hadron interpolators and correlators	p. 123
Meson interpolators	p. 124
Meson correlators	p. 127
Interpolators and correlators for baryons	p. 129
Momentum projection	p. 131
Final formula for hadron correlators	p. 132
The quenched approximation	p. 133
Strategy of the calculation	p. 135
The need for quark sources	p. 135
Point source or extended source?	p. 136
Extended sources	p. 137
Calculation of the quark propagator	p. 138
Exceptional configurations	p. 141
Smoothing of gauge configurations	p. 142
Extracting hadron masses	p. 143
Effective mass curves	p. 144
Fitting the correlators	p. 146
The calculation of excited states	p. 147
Finalizing the results for the hadron masses	p. 150
Discussion of some raw data	p. 150
Setting the scale and the quark mass parameters	p. 151
Various extrapolations	p. 152
Some quenched results	p. 154
References	p. 155
Chiral symmetry on the lattice	p. 157
Chiral symmetry in continuum QCD	p. 157
Chiral symmetry for a single flavor	p. 157
Several flavors	p. 159
Spontaneous breaking of chiral symmetry	p. 160
Chiral symmetry and the lattice	p. 162
Wilson fermions and the Nielsen-Ninomiya theorem	p. 162
The Ginsparg-Wilson equation	p. 163
Chiral symmetry on the lattice	p. 164
Consequences of the Ginsparg-Wilson equation	p. 166
Spectrum of the Dirac operator	p. 166
Index theorem	p. 168
The axial anomaly	p. 170
The chiral condensate	p. 172
The Banks-Casher relation	p. 175
The overlap operator	p. 177
Definition of the overlap operator	p. 177
Locality properties of chiral Dirac operators	p. 178
Numerical evaluation of the overlap operator	p. 179
References	p. 183
Dynamical fermions	p. 185
The many faces of the fermion determinant	p. 185
The fermion determinant as observable	p. 186
The fermion determinant as a weight factor	p. 186
Pseudofermions	p. 187
Effective fermion action	p. 188
First steps toward updating with fermions	p. 189
Hybrid Monte Carlo	p. 190
Molecular dynamics leapfrog evolution	p. 191
Completing with an accept-reject step	p. 194
Implementing HMC for gauge fields and fermions	p. 195
Other algorithmic ideas	p. 199
The R-algorithm	p. 199
Partial updates	p. 200
Polynomial and rational HMC	p. 200
Multi-pseudofermions and UV-filtering	p. 201
Further developments	p. 202
Other techniques using pseudofermions	p. 203
The coupling-mass phase diagram	p. 205
Continuum limit and phase transitions	p. 205
The phase diagram for Wilson fermions	p. 206
Ginsparg-Wilson fermions	p. 208
Full QCD calculations	p. 209
References	p. 210
Symanzik improvement and RG actions	p. 213
The Symanzik improvement program	p. 214
A toy example	p. 214
The framework for improving lattice QCD	p. 215
Improvement of interpolators	p. 218
Determination of improvement coefficients	p. 219
Lattice actions for free fermions from RG transformations	p. 221
Integrating out the fields over hypercubes	p. 222
The blocked lattice Dirac operator	p. 223
Properties of the blocked action	p. 226
Real space renormalization group for QCD	p. 227
Blocking full QCD	p. 228
The RG flow of the couplings	p. 231
Saddle point analysis of the RG equation	p. 232
Solving the RG equations	p. 233
Mapping continuum symmetries onto the lattice	p. 236
The generating functional and its symmetries	p. 236
Identification of the corresponding lattice symmetries	p. 238
References	p. 241
More about lattice fermions	p. 243
Staggered fermions	p. 243
The staggered transformation	p. 243
Tastes of staggered fermions	p. 245
Developments and open questions	p. 248
Domain wall fermions	p. 249
Formulation of lattice QCD with domain wall fermions	p. 250
The 5D theory and its equivalence to 4D chiral fermions	p. 252
Twisted mass fermions	p. 253
The basic formulation of twisted mass QCD	p. 254
The relation between twisted and conventional QCD	p. 256
O(a) improvement at maximal twist	p. 258
Effective theories for heavy quarks	p. 260
The need for an effective theory	p. 260
Lattice action for heavy quarks	p. 261
General framework and expansion coefficients	p. 263
References	p. 264
Hadron structure	p. 267
Low-energy parameters	p. 267
Operator definitions	p. 268
Ward identities	p. 270
Naive currents and conserved currents on the lattice	p. 274
Low-energy parameters from correlation functions	p. 278
Renormalization	p. 279
Why do we need renormalization?	p. 279
Renormalization with the Rome-Southampton method	p. 281
Hadronic decays and scattering	p. 284
Threshold region	p. 284
Beyond the threshold region	p. 287
Matrix elements	p. 289
Pion form factor	p. 290
Weak matrix elements	p. 294
OPE expansion and effective weak Hamiltonian	p. 295
References	p. 297
Temperature and chemical potential	p. 301
Introduction of temperature	p. 301
Analysis of pure gauge theory	p. 303
Switching on dynamical fermions	p. 307
Properties of QCD in the deconfinement phase	p. 310
Introduction of the chemical potential	p. 312
The chemical potential on the lattice	p. 312
The QCD phase diagram in the (T, ¿) space	p. 317
Chemical potential: Monte Carlo techniques	p. 318
Reweighting	p. 319
Series expansion	p. 321
Imaginary ¿	p. 321
Canonical partition functions	p. 322
References	p. 323
Appendix	p. 327
The Lie groups SU(N)	p. 327
Basic properties	p. 327
Lie algebra	p. 327
Generators for SU(2) and SU(3)	p. 329
Derivatives of group elements	p. 329
Gamma matrices	p. 330
Fourier transformation on the lattice	p. 332
Wilson's formulation of lattice QCD	p. 333
A few formulas for matrix algebra	p. 334
References	p. 336
Index	p. 337
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Quantum Chromodynamics on the Lattice : An Introduction for Beginners

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