1 The path integral on the lattice

1.1 Hilbert space and propagation in Euclidean time

1.1.1 Hilbert spaces

1.1.2 Remarks on Hilbert spaces in particle physics

1.1.3 Euclidean correlators

1.2 The path integral for a quantum mechanical system

1.3 The path integral for a scalar field theory

1.3.1 The Klein-Gordon field

1.3.2 Lattice regularization of the Klein-Gordon Hamiltonian

1.3.3 The Euclidean time transporter for the free case

1.3.4 Treating the interaction term with the Trotter formula

1.3.5 Path integral representation for the partition function

1.3.6 Including operators in the path integral

1.4 Quantization with the path integral

1.4.1 Different discretizations of the Euclidean action

1.4.2 The path integral as a quantization prescription

1.4.3 The relation to statistical mechanics

References

2 QCD on the lattice-a first look

2.1 The QCD action in the continuum

2.1.1 Quark and gluon fields

2.1.2 The fermionic part of the QCD action

2.1.3 Gauge invariance of the fermion action

2.1.4 The gluon action

2.1.5 Color components of the gauge field

2.2 Naive discretization of fermions

2.2.1 Discretization of free fermions

2.2.2 Introduction of the gauge fields as link variables

2.2.3 Relating the link variables to the continuum gauge fields

2.3 The Wilson gauge action

2.3.1 Gauge-invariant objects built with link variables

2.3.2 The gauge action

2.4 Formal expression for the QCD lattice path integral

2.4.1 The QCD lattice path integral

References

3 Pure gauge theory on the lattice

3.1 Haar measure

3.1.1 Gauge field measure and gauge invariance

3.1.2 Group integration measure

3.1.3 A few integrals for SU(3)

3.2 Gauge invariance and gauge fixing

3.2.1 Maximal trees

3.2.2 Other gauges

3.2.3 Gauge invariance of observables

3.3 Wilson and Polyakov loops

3.3.1 Definition of the Wilson loop

3.3.2 Temporal gauge

3.3.3 Physical interpretation of the Wilson loop

3.3.4 Wilson line and the quark-antiquark pair

3.3.5 Polyakov loop

3.4 The static quark potential

3.4.1 Strong coupling expansion of the Wilson loop

3.4.2 The Coulomb part of the static quark potential

3.4.3 Physical implications of the static QCD potential

3.5 Setting the scale with the static potential

3.5.1 Discussion of numerical data for the static potential

3.5.2 The Sommer parameter and the lattice spacing

3.5.3 Renormalization group and the running coupling

3.5.4 The true continuum limit

3.6 Lattice gauge theory with other gauge groups

References

4 Numerical simulation of pure gauge theory

4.1 The Monte Carlo method

4.1.1 Simple sampling and importance sampling

4.1.2 Markov chains

4.1.3 Metropolis algorithm-general idea

4.1.4 Metropolis algorithm for Wilson's gauge action

4.2 Implementation of Monte Carlo algorithms for SU(3)

4.2.1 Representation of the link variables

4.2.2 Boundary conditions

4.2.3 Generating a candidate link for the Metropolis update

4.2.4 A few remarks on random numbers

4.3 More Monte Carlo algorithms

4.3.1 The heat bath algorithm

4.3.2 Overrelaxation

4.4 Running the simulation

4.4.1 Initialization

4.4.2 Equilibration updates

4.4.3 Evaluation of the observables

4.5 Analyzing the data

4.5.1 Statistical analysis for uncorrelated data

4.5.2 Autocorrelation

4.5.3 Techniques for smaller data sets

4.5.4 Some numerical exercises

References

5 Fermions on the lattice

5.1 Fermi statistics and Grassmann numbers

5.1.1 Some new notation

5.1.2 Fermi statistics

5.1.3 Grassmann numbers and derivatives

5.1.4 Integrals over Grassmann numbers

5.1.5 Gaussian integrals with Grassmann numbers

5.1.6 Wick's theorem

5.2 Fermion doubling and Wilson's fermion action

5.2.1 The Dirac operator on the lattice

5.2.2 The doubling problem

5.2.3 Wilson fermions

5.3 Fermion lines and hopping expansion

5.3.1 Hopping expansion of the quark propagator

5.3.2 Hopping expansion for the fermion determinant

5.4 Discrete symmetries of the Wilson action

5.4.1 Charge conjugation

5.4.2 Parity and Euclidean reflections

5.4.3 γ5-hermiticity

References

6 Hadron spectroscopy

6.1 Hadron interpolators and correlators

6.1.1 Meson interpolators

6.1.2 Meson correlators

6.1.3 Interpolators and correlators for baryons

6.1.4 Momentum projection

6.1.5 Final formula for hadron correlators

6.1.6 The quenched approximation

6.2 Strategy of the calculation

6.2.1 The need for quark sources

6.2.2 Point source or extended source?

6.2.3 Extended sources

6.2.4 Calculation of the quark propagator

6.2.5 Exceptional configurations

6.2.6 Smoothing of gauge configurations

6.3 Extracting hadron masses

6.3.1 Effective mass curves

6.3.2 Fitting the correlators

6.3.3 The calculation of excited states

6.4 Finalizing the results for the hadron masses

6.4.1 Discussion of some raw data

6.4.2 Setting the scale and the quark mass parameters

6.4.3 Various extrapolations

6.4.4 Some quenched results

References

7 Chiral symmetry on the lattice

7.1 Chiral symmetry in continuum QCD

7.1.1 Chiral symmetry for a single flavor

7.1.2 Several flavors

7.1.3 Spontaneous breaking of chiral symmetry

7.2 Chiral symmetry and the lattice

7.2.1 Wilson fermions and the Nielsen-Ninomiya theorem

7.2.2 The Ginsparg-Wilson equation

7.2.3 Chiral symmetry on the lattice

7.3 Consequences of the Ginsparg-Wilson equation

7.3.1 Spectrum of the Dirac operator

7.3.2 Index theorem

7.3.3 The axial anomaly

7.3.4 The chiral condensate

7.3.5 The Banks-Casher relation

7.4 The overlap operator

7.4.1 Definition of the overlap operator

7.4.2 Locality properties of chiral Dirac operators

7.4.3 Numerical evaluation of the overlap operator

References

8 Dynamical fermions

8.1 The many faces of the fermion determinant

8.1.1 The fermion determinant as observable

8.1.2 The fermion determinant as a weight factor

8.1.3 Pseudofermions

8.1.4 Effective fermion action

8.1.5 First steps toward updating with fermions

8.2 Hybrid Monte Carlo

8.2.1 Molecular dynamics leapfrog evolution

8.2.2 Completing with an accept-reject step

8.2.3 Implementing HMC for gauge fields and fermions

8.3 Other algorithmic ideas

8.3.1 The R-algorithm

8.3.2 Partial updates

8.3.3 Polynomial and rational HMC

8.3.4 Multi-pseudofermions and UV-filtering

8.3.5 Further developments

8.4 Other techniques using pseudofermions

8.5 The coupling-mass phase diagram

8.5.1 Continuum limit and phase transitions

8.5.2 The phase diagram for Wilson fermions

8.5.3 Ginsparg-Wilson fermions

8.6 Full QCD calculations

References

9 Symanzik improvement and RG actions

9.1 The Symanzik improvement program

9.1.1 A toy example

9.1.2 The framework for improving lattice QCD

9.1.3 Improvement of interpolators

9.1.4 Determination of improvement coefficients

9.2 Lattice actions for free fermions from RG transformations

9.2.1 Integrating out the fields over hypercubes

9.2.2 The blocked lattice Dirac operator

9.2.3 Properties of the blocked action

9.3 Real space renormalization group for QCD

9.3.1 Blocking full QCD

9.3.2 The RG flow of the couplings

9.3.3 Saddle point analysis of the RG equation

9.3.4 Solving the RG equations

9.4 Mapping continuum symmetries onto the lattice

9.4.1 The generating functional and its symmetries

9.4.2 Identification of the corresponding lattice symmetries

References

10 More about lattice fermions

10.1 Staggered fermions

10.1.1 The staggered transformation

10.1.2 Tastes of staggered fermions

10.1.3 Developments and open questions

10.2 Domain wall fermions

10.2.1 Formulation of lattice QCD with domain wall fermions

10.2.2 The 5D theory and its equivalence to 4D chiralfermions

10.3 Twisted mass fermions

10.3.1 The basic formulation of twisted mass QCD

10.3.2 The relation between twisted and conventional QCD

10.3.3 O(a)improvement at maximal twist

10.4 Effective theories for heavy quarks

10.4.1 The need for an effective theory

10.4.2 Lattice action for heavy quarks

10.4.3 General framework and expansion coefficients

References

11 Hadron structure

11.1 Low-energy parameters

11.1.1 Operator definitions

11.1.2 Ward identities

11.1.3 Naive currents and conserved currents on the lattice

11.1.4 Low-energy parameters from correlation functions

11.2 Renormalization

11.2.1 Why do we need renormalization?

11.2.2 Renormalization with the Rome-Southampton method

11.3 Hadronic decays and scattering

11.3.1 Threshold region

11.3.2 Beyond the threshold region

11.4 Matrix elements

11.4.1 Pion form factor

11.4.2 Weak matrix elements

11.4.3 OPE expansion and effective weak Hamiltonian

References

12 Temperature and chemical potential

12.1 Introduction of temperature

12.1.1 Analysis of pure gauge theory

12.1.2 Switching on dynamical fermions

12.1.3 Properties of QCD in the deconfinement phase

12.2 Introduction of the chemical potential

12.2.1 The chemical potential on the lattice

12.2.2 The QCD phase diagram in the(T,μ)space

12.3 Chemical potential:Monte Carlo techniques

12.3.1 Reweighting

12.3.2 Series expansion

12.3.3 Imaginary μ

12.3.4 Canonical partition functions

References

A Appendix

A.1 The Lie groups SU(N)

A.1.1 Basic properties

A.1.2 Lie algebra

A.1.3 Generators for SU(2)and SU(3)

A.1.4 Derivatives of group elements

A.2 Gamma matrices

A.3 Fourier transformation on the lattice

A.4 Wilson's formulation of lattice QCD

A.5 A few formulas for matrix algebra

References

Index