Quantum chromodynamics on the lattice: an introductory presentation
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2010
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| Series: | Lecture notes in physics
788 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XV, 343 S. graph. Darst. |
| ISBN: | 9783642018497 |
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| 100 | 1 | |a Gattringer, Christof |d 1966- |e Verfasser |0 (DE-588)1032555025 |4 aut | |
| 245 | 1 | 0 | |a Quantum chromodynamics on the lattice |b an introductory presentation |c C. Gattringer ; C. B. Lang |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2010 | |
| 300 | |a XV, 343 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Lecture notes in physics |v 788 | |
| 650 | 4 | |a Quantenchromodynamik - Gittereichtheorie | |
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| 689 | 0 | 1 | |a Gittereichtheorie |0 (DE-588)4157377-8 |D s |
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Contents
The path integral on the lattice
. 1
1.1
Hubert space and propagation in Euclidean time
. 2
1.1.1
Hubert spaces
. 2
1.1.2
Remarks on Hubert 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
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
X
Contents
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
Contents
XI
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
.
Ill
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
75-riermiticity
.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
XII Contents
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
Contents XIII
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 chiral
fermions
.252
XIV Contents
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
(Γ, 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
Contents
XV
A
Appendix.327
A.I The Lie groups
SU(iV)
.327
A.I.I Basic properties
.327
A.I.
2
Lie algebra
.327
A.1.3 Generators for SU(2) and SV(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 |
| any_adam_object | 1 |
| author | Gattringer, Christof 1966- Lang, Christian B. 1948- |
| author_GND | (DE-588)1032555025 (DE-588)120209411 |
| author_facet | Gattringer, Christof 1966- Lang, Christian B. 1948- |
| author_role | aut aut |
| author_sort | Gattringer, Christof 1966- |
| author_variant | c g cg c b l cb cbl |
| building | Verbundindex |
| bvnumber | BV035815592 |
| classification_rvk | UD 8220 |
| classification_tum | PHY 413f |
| ctrlnum | (OCoLC)467889570 (DE-599)DNB995195447 |
| dewey-full | 539.7548 530.14 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 539 - Modern physics 530 - Physics |
| dewey-raw | 539.7548 530.14 |
| dewey-search | 539.7548 530.14 |
| dewey-sort | 3539.7548 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| format | Book |
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| id | DE-604.BV035815592 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T10:23:07Z |
| institution | BVB |
| isbn | 9783642018497 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018674437 |
| oclc_num | 467889570 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-703 DE-634 DE-384 DE-11 DE-91G DE-BY-TUM |
| owner_facet | DE-355 DE-BY-UBR DE-703 DE-634 DE-384 DE-11 DE-91G DE-BY-TUM |
| physical | XV, 343 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series | Lecture notes in physics |
| series2 | Lecture notes in physics |
| spelling | Gattringer, Christof 1966- Verfasser (DE-588)1032555025 aut Quantum chromodynamics on the lattice an introductory presentation C. Gattringer ; C. B. Lang Berlin [u.a.] Springer 2010 XV, 343 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Lecture notes in physics 788 Quantenchromodynamik - Gittereichtheorie Gittereichtheorie (DE-588)4157377-8 gnd rswk-swf Quantenchromodynamik (DE-588)4128082-9 gnd rswk-swf Quantenchromodynamik (DE-588)4128082-9 s Gittereichtheorie (DE-588)4157377-8 s DE-604 Lang, Christian B. 1948- Verfasser (DE-588)120209411 aut Erscheint auch als Online-Ausgabe 978-3-642-01850-3 Lecture notes in physics 788 (DE-604)BV000003166 788 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3330318&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018674437&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gattringer, Christof 1966- Lang, Christian B. 1948- Quantum chromodynamics on the lattice an introductory presentation Lecture notes in physics Quantenchromodynamik - Gittereichtheorie Gittereichtheorie (DE-588)4157377-8 gnd Quantenchromodynamik (DE-588)4128082-9 gnd |
| subject_GND | (DE-588)4157377-8 (DE-588)4128082-9 |
| title | Quantum chromodynamics on the lattice an introductory presentation |
| title_auth | Quantum chromodynamics on the lattice an introductory presentation |
| title_exact_search | Quantum chromodynamics on the lattice an introductory presentation |
| title_full | Quantum chromodynamics on the lattice an introductory presentation C. Gattringer ; C. B. Lang |
| title_fullStr | Quantum chromodynamics on the lattice an introductory presentation C. Gattringer ; C. B. Lang |
| title_full_unstemmed | Quantum chromodynamics on the lattice an introductory presentation C. Gattringer ; C. B. Lang |
| title_short | Quantum chromodynamics on the lattice |
| title_sort | quantum chromodynamics on the lattice an introductory presentation |
| title_sub | an introductory presentation |
| topic | Quantenchromodynamik - Gittereichtheorie Gittereichtheorie (DE-588)4157377-8 gnd Quantenchromodynamik (DE-588)4128082-9 gnd |
| topic_facet | Quantenchromodynamik - Gittereichtheorie Gittereichtheorie Quantenchromodynamik |
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| volume_link | (DE-604)BV000003166 |
| work_keys_str_mv | AT gattringerchristof quantumchromodynamicsonthelatticeanintroductorypresentation AT langchristianb quantumchromodynamicsonthelatticeanintroductorypresentation |