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Quantum chromodynamics on the lattice: an introductory presentation

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Bibliographic Details
Main Authors:	Gattringer, Christof 1966- (Author), Lang, Christian B. 1948- (Author)
Format:	Book
Language:	English
Published:	Berlin [u.a.] Springer 2010
Series:	Lecture notes in physics 788
Subjects:	Quantenchromodynamik - Gittereichtheorie Gittereichtheorie Quantenchromodynamik
Links:	http://deposit.dnb.de/cgi-bin/dokserv?id=3330318&prov=M&dok_var=1&dok_ext=htm 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
Physical Description:	XV, 343 S. graph. Darst.
ISBN:	9783642018497

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adam_text	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
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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

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