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 d

DeGrand, Thomas

DeTar, Carleton

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HETHODS FORQUANTUM CHROMODYIVAMICSThomas DeGrandUniversity of Colorado, USACarleton DeTarUniversity of Utah, USA1> World ScientificNEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAIContentsPreface	 vii1. Introduction	 12. Continuum QCD and its phenomenology 	 52.1 The Lagrangian and QCD at short distance 	 	 62.2 The nonrelativistic quark model 	 	 72.2.1 Explanations and puzzles 	 	 72.2.2 Quark model interpolating operators 	 	 92.2.3 Glueballs, hybrids, and exotics 	  102.2.4 Flavor and glueball mixing example: the anomaly . 	 112.2.5 Quark model hadron masses 	  112.3 Heavy quark systems 	  132.3.1 Quarkonium 	  132.3.2 Heavy quark symmetries 	  142.4 Chiral symmetry and chiral symmetry breaking 	  152.4.1 Chiral symmetry 	  152.4.2 Linear sigma model 	  162.4.3 Nonlinear sigma model 	  182.4.4 Nonlinear effective chiral Lagrangian for QCD . 	 202.5 A technical aside: Ward identities 	  212.6 The axial anomaly and instantons 	  242.6.1 Nonconservation of the flavor-singlet axial current . 	 242.6.2 Witten-Veneziano formula 	  252.6.3 Suppression of the topological susceptibility 	  262.6.4 QCD vacuum 	  27ixx	 Lattice Methods for Quantum Chromodynamics2.7 The large Nc limit 	  273. Path integration	 333.1 Lattice Schwinger model 	  333.1.1 Free fermions in one dimension 	  333.1.2 Species doubling 	  353.2 Hamiltonian with gauge fields 	  363.3 Feynman path integral 	  383.3.1 Pure gauge theory 	  393.3.2 Generalization to SU(3) pure gauge theory 	  433.3.3 Static quark potential 	  453.3.3.1 Polyakov loops and potential energy of pointcharges 	  453.3.3.2 Wilson loops and point charge potential 	 463.4 Free fermions 	  483.4.1 Grassmann calculus 	  483.4.2 Grassmann path integral for a two-level system	 503.4.3 Fermion propagator 	  523.4.4 Path integral with particles and antiparticles	 543.4.5 Generalization to higher dimensions 	  563.5 The interacting theory 	 573.5.1 Functional integral representation for the latticeSchwinger model 	  574. Renormalization and the renormalization group	614.1 Blocking transformations 	 614.2 Renormalization group equations 	 714.2.1 Renormalization in scalar field theory	 724.3 Renormalization group equations for the scalar field	774.4 Effective field theories 	  795. Yang-Mills theory on the lattice 	 875.1 Gauge invariance on the lattice 	 875.2 Yang-Mills actions 	 895.3 Gauge fixing 	 905.3.1 Maximal tree gauge 	  905.3.2 Landau and Coulomb gauge 	 915.4 Strong coupling 	  92Contents	 xi5.4.1 Wilson loop and confinement 	 935.4.2 Glueball mass 	 965.4.3 Polyakov loop 	 986. Fermions on the lattice	 1016.1 Naive fermions 	 1016.2 Wilson-type fermions 	 1066.2.1 Twisted-mass fermions 	 1136.3 Staggered fermions 	 1156.4 Lattice fermions with exact chiral symmetry 	 1226.5 Exact chiral symmetry from five dimensions 	 1276.5.1 Five dimensions in the continuum 	 1276.5.2 Five dimensions on the lattice 	 1306.6 Heavy quarks 	 1346.6.1 Heavy quark effective theory 	 1356.6.2 Nonrelativistic QCD 	 1376.6.3 Heavy, relativistic quarks 	  1417. Numerical methods for bosons 	 1437.1 Importance sampling 	  1437.1.1 Monte Carlo methods 	  1447.1.1.1 Metropolis et al. method 	  1457.1.1.2 Heat bath method	 1457.1.2 Molecular dynamics method 	  1467.1.3 Refreshed molecular dynamics 	  1477.1.4 Hybrid Monte Carlo 	  1487.1.5 Leapfrog algorithm and improvements 	  1487.2 Special methods for the Yang-Mills action 	  1507.2.1 Heat bath 	  1507.2.2 Overrelaxed updates 	  1517.2.3 Molecular dynamics 	  1528. Numerical methods for fermions 	 1558.1 Taming the fermion determinant: the e algorithm 	 1558.2 Taming the fermion determinant: the R algorithm 	 1598.3 The fourth root approximation 	 1608.4 An exact algorithm for the fourth root: rational hybridMonte Carlo 	  162xii	 Lattice Methods for Quantum Chromodynamics8.5 Refinements 	  1638.5.1 Sexton-Weingarten scheme 	  1638.5.2 Hasenbusch method 	  1638.5.3 Schwarz alternating method 	  1648.6 Special considerations for overlap fermions 	  1658.7 Monte Carlo methods for fermions 	  1708.7.1 Multiboson method 	  1718.7.2 Ratio of determinants 	  1718.8 Conjugate gradient and its relatives 	  1738.8.1 Even-odd preconditioning 	  1748.8.2 The conjugate gradient algorithm 	  1758.8.3 Biconjugate gradient 	  1788.8.4 Stabilized biconjugate gradient 	  1798.8.5 Shifted solvers 	  1808.8.6 Computing fermion eigenmodes 	 1819. Data analysis for lattice simulations 	 1839.1 Correlations in simulation time 	  1849.2 Correlations among observables 	 1869.2.1 Correlated least chi square 	  1869.2.2 Truncating the correlation matrix 	  1879.2.3 Jackknife and bootstrap methods 	 1899.3 Fitting strategies 	  1919.3.1 Fitting range 	  1919.3.2 Signal to noise ratio 	  1929.3.3 Interpolating operator 	  1929.3.4 Asymmetrie lattice 	  1939.3.5 Bayesian methods 	  19410. Designing lattice actions	 19710.1 Motivation 	 19710.2 Symanzik improvement 	 19910.2.1 Gauge action improvement	 20010.2.2 Fermion action improvement	 20110.2.3 Nonperturbative improvement 	 20210.3 Tadpole improvement 	 20410.4 Renormalization-group inspired improvement 	 20910.5 "Fat link" actions 	 211Contents	 xiii11. Spectroscopy	 21511.1 Computing propagators and correlation functions 	 21511.2 Sewing propagators together	 21911.2.1 Mesonic correlators – Wilson fermion formalism . .	 21911.2.2 Baryonic correlators – Wilson fermion formalism .	 22211.2.3 Mesonic correlators – staggered fermion formalism	 22411.2.3.1 Flavor nonsinglet mesons 	 22511.2.3.2 Flavor singlet mesons 	 22911.2.4 Baryonic correlators – staggered fermion formalism	 22911.3 Glueballs 	 23211.4 The string tension 	 23312. Lattice perturbation theory	 23512.1 Motivation 	 23512.2 Technology 	 23512.2.1 Free fermion and gluon propagators 	 23612.2.2 Quark-gluon vertices 	 23812.2.3 Fat links 	 23912.2.4 Quark seif energy and tadpole 	  24112.2.5 Vertex graph 	 24212.2.6 Ultraviolet divergences 	  24312.2.7 Automation 	  24312.3 The scale of the coupling constant 	  24413. Operators with anomalous dimension	 24913.1 Perturbative techniques for operator matching 	  25013.2 Nonperturbative techniques for operator matching 	  25313.2.1 Methods for approximations to conserved currents	 25313.2.2 Regularization-independent scheme 	  25413.2.3 Schrödinger functional methods 	  25814. Chiral symmetry and lattice simulations	 26114.1 Minimal introduction to chiral perturbation theory 	  26114.2 Quenching, partial quenching, and unquenching 	 26514.2.1 The eta prime correlator 	  26614.2.2 Chiral Lagrangian with partial or complete quenching 26814.3 Chiral perturbation theory for staggered fermions 	 27014.3.1 Staggered chiral perturbation theory 	 270xiv	 Lattice Methode for Quantum Chromodynamics14.3.2 Rooted staggered chiral perturbation theory 	  27314.4 Computing topological charge 	  27515. Finite volume effects 	 27915.1 Finite volume effects in chiral perturbation theory 	  27915.2 The e-regime 	  28215.2.1 Banks-Casher formula 	  28215.2.2 Chiral Lagrangian in the epsilon regime 	  28315.2.3 Random matrix theory 	  28415.2.4 Further applications 	  28615.3 Finite volume, more generally 	  28615.3.1 Single particle states 	 28615.3.2 Two particle states 	  28815.3.2.1 Scattering lengths from the two-particleground state 	 28915.3.2.2 Decay widths from level repulsion 	  29015.4 Miscellaneous comments 	 29316. Testing the standard model with lattice calculations	 29516.1 Overview 	  29516.2 Strong renormalization of weak operators 	  29516.2.1 Effective Hamiltonian 	  29516.2.2 An example: c --> sdu	 29716.2.3 Lattice vs continuum renormalization of the effectiveHamiltonian 	  30016.2.4 Mixing with operators of higher and lower dimension 30116.3 Lattice discrete symmetries 	 30316.4 Some simple examples 	 30616.4.1 Leptonic decay constants of mesons 	 30616.4.2 Leptonic decay constants in the heavy quark limit .	 30716.4.3 Electromagnetic widths of vector mesons 	 30816.4.4 Form factors 	 30816.4.5 Meson B parameters 	  31016.4.6 Other purely hadronic weak interactions 	 31416.5 Evading a no-go theorem 	 31417. QCD at high temperature and density	 31717.1 Simulating high temperature 	 317Contents	 xv17.2 Introducing a chemical potential 	 31817.3 High quark mass limit and chiral limit 	  31817.4 Locating and characterizing the phase transition 	 32017.5 Simulating in a nearby ensemble 	  	  32017.6 Dimensional reduction and nonperturbative behavior	 32217.7 Miscellaneous observables 	 32317.7.1 Quark number susceptibilities 	 32317.7.2 Quarkonium potential 	 32317.7.3 Equation of state 	 32317.8 Nonzero density 	 32517.9 Spectral functions and maximum entropy 	 326Bibliography	 329Index	 341

Lattice Methods for Quantum Chromodynamics

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