Lattice methods for quantum chromodynamics

Thomas Degrand, Carleton DeTar9812567275, 9789812567277, 9789812773982

Numerical simulation of lattice-regulated QCD has become an important source of information about strong interactions. In the last few years there has been an explosion of techniques for performing ever more accurate studies on the properties of strongly interacting particles. Lattice predictions directly impact many areas of particle and nuclear physics theory and phenomenology. This book provides a thorough introduction to the specialized techniques needed to carry out numerical simulations of QCD: a description of lattice discretizations of fermions and gauge fields, methods for actually doing a simulation, descriptions of common strategies to connect simulation results to predictions of physical quantities, and a discussion of uncertainties in lattice simulations. More importantly, while lattice QCD is a well-defined field in its own right, it has many connections to continuum field theory and elementary particle physics phenomenology, which are carefully elucidated in this book.

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Table of contents :
Contents……Page 10
Preface……Page 8
1. Introduction……Page 18
2. Continuum QCD and its phenomenology……Page 22
2.1 The Lagrangian and QCD at short distance……Page 23
2.2 The nonrelativistic quark model……Page 24
2.3 Heavy quark systems……Page 30
2.4 Chiral symmetry and chiral symmetry breaking……Page 32
2.5 A technical aside: Ward identities……Page 38
2.6 The axial anomaly and instantons……Page 41
2.7 The large Nc limit……Page 44
3.1 Lattice Schwinger model……Page 50
3.2 Hamiltonian with gauge fields……Page 53
3.3 Feynman path integral……Page 55
3.4 Free fermions……Page 65
3.5 The interacting theory……Page 74
4.1 Blocking transformations……Page 78
4.2 Renormalization group equations……Page 88
4.3 Renormalization group equations for the scalar field……Page 94
4.4 Effective field theories……Page 96
5.1 Gauge invariance on the lattice……Page 104
5.2 Yang-Mills actions……Page 106
5.3 Gauge fixing……Page 107
5.4 Strong coupling……Page 109
6.1 Naive fermions……Page 118
6.2 Wilson-type fermions……Page 123
6.3 Staggered fermions……Page 132
6.4 Lattice fermions with exact chiral symmetry……Page 139
6.5 Exact chiral symmetry from five dimensions……Page 144
6.6 Heavy quarks……Page 151
7.1 Importance sampling……Page 160
7.2 Special methods for the Yang-Mills action……Page 167
8.1 Taming the fermion determinant: the $ algorithm……Page 172
8.2 Taming the fermion determinant: the R algorithm……Page 176
8.3 The fourth root approximation……Page 177
8.4 An exact algorithm for the fourth root: rational hybrid Monte Carlo……Page 179
8.5 Refinements……Page 180
8.6 Special considerations for overlap fermions……Page 182
8.7 Monte Carlo methods for fermions……Page 187
8.8 Conjugate gradient and its relatives……Page 190
9. Data analysis for lattice simulations……Page 200
9.1 Correlations in simulation time……Page 201
9.2 Correlations among observables……Page 203
9.3 Fitting strategies……Page 208
10.1 Motivation……Page 214
10.2 Symanzik improvement……Page 216
10.3 Tadpole improvement……Page 221
10.4 Renormalization-group inspired improvement……Page 226
10.5 “Fat link” actions……Page 228
11.1 Computing propagators and correlation functions……Page 232
11.2 Sewing propagators together……Page 236
11.3 Glueballs……Page 249
11.4 The string tension……Page 250
12.2 Technology……Page 252
12.3 The scale of the coupling constant……Page 261
13. Operators with anomalous dimension……Page 266
13.1 Perturbative techniques for operator matching……Page 267
13.2 Nonperturbative techniques for operator matching……Page 270
14.1 Minimal introduction to chiral perturbation theory……Page 278
14.2 Quenching partial quenching and unquenching……Page 282
14.3 Chiral perturbation theory for staggered fermions……Page 287
14.4 Computing topological charge……Page 292
15.1 Finite volume effects in chiral perturbation theory……Page 296
15.2 The e-regime……Page 299
15.3 Finite volume more generally……Page 303
15.4 Miscellaneous comments……Page 310
16.2 Strong renormalization of weak operators……Page 312
16.3 Lattice discrete symmetries……Page 320
16.4 Some simple examples……Page 323
16.5 Evading a no-go theorem……Page 331
17.1 Simulating high temperature……Page 334
17.3 High quark mass limit and chiral limit……Page 335
17.5 Simulating in a nearby ensemble……Page 337
17.6 Dimensional reduction and nonperturbative behavior……Page 339
17.7 Miscellaneous observables……Page 340
17.8 Nonzero density……Page 342
17.9 Spectral functions and maximum entropy……Page 343
Bibliography……Page 346
Index……Page 358