Lattice Methods for Quantum Chromodynamics

The following sections are included:

• Preface

• Contents

Strongly interacting elementary particles such as the proton and neutron are not fundamental particles, but are bound states of spin −1/2 fermions, the quarks, and spin-1 gauge bosons, the gluons. The strong interactions are described by a quantum field theory called Quantum Chromodynamics (QCD). It is believed that quarks and gluons are permanently confined, i.e., it is not possible to separate an isolated quark a macroscopic distance from an antiquark or another quark pair. At short distances the effective coupling constant of QCD becomes small, and it is possible to compare results from experiments with theoretical predictions based on perturbation theory. However, at long distances the strong interactions are truly strong, and perturbation theory breaks down. Some nonperturbative technique is necessary to perform calculations of quantities that are sensitive to the long distance behavior of QCD. One of those techniques is the subject of this book: it is called “lattice QCD.”…

The following sections are included:

• The Lagrangian and QCD at short distance

• The nonrelativistic quark model
  • Explanations and puzzles
  • Quark model interpolating operators
  • Glueballs, hybrids, and exotics
  • Flavor and glueball mixing example: the anomaly
  • Quark model hadron masses

• Heavy quark systems
  • Quarkonium
  • Heavy quark symmetries

• Chiral symmetry and chiral symmetry breaking
  • Chiral symmetry
  • Linear sigma model
  • Nonlinear sigma model
  • Nonlinear effective chiral Lagrangian for QCD

• A technical aside: Ward identities

• The axial anomaly and instantons
  • Nonconservation of the flavor-singlet axial current
  • Witten-Veneziano formula
  • Suppression of the topological susceptibility
  • QCD vacuum
  • The large N_c limit

The following sections are included:

• Lattice Schwinger model
  • Free fermions in one dimension
  • Species doubling

• Hamiltonian with gauge fields

• Feynman path integral
  • Pure gauge theory
  • Generalization to SU(3) pure gauge theory
  • Static quark potential
    • Polyakov loops and potential energy of point charges
    • Wilson loops and point charge potential

• Free fermions
  • Grassmann calculus
  • Grassmann path integral for a two-level system
  • Fermion propagator
  • Path integral with particles and antiparticles
  • Generalization to higher dimensions

• The interacting theory
  • Functional integral representation for the lattice Schwinger model

The following sections are included:

• Blocking transformations

• Renormalization group equations
  • Renormalization in scalar field theory

• Renormalization group equations for the scalar field

• Effective field theories

The following sections are included:

• Gauge invariance on the lattice

• Yang-Mills actions

• Gauge fixing
  • Maximal tree gauge
  • Landau and Coulomb gauge

• Strong coupling
  • Wilson loop and confinement
  • Glueball mass
  • Polyakov loop

The following sections are included:

• Naive fermions

• Wilson-type fermions
  • Twisted-mass fermions

• Staggered fermions

• Lattice fermions with exact chiral symmetry

• Exact chiral symmetry from five dimensions
  • Five dimensions in the continuum
  • Five dimensions on the lattice

• Heavy quarks
  • Heavy quark effective theory
  • Nonrelativistic QCD
  • Heavy, relativistic quarks

The following sections are included:

• Importance sampling
  • Monte Carlo methods
    • Metropolis et al. method
    • Heat bath method
  • Molecular dynamics method
  • Refreshed molecular dynamics
  • Hybrid Monte Carlo
  • Leapfrog algorithm and improvements

• Special methods for the Yang-Mills action
  • Heat bath
  • Overrelaxed updates
  • Molecular dynamics

The following sections are included:

• Taming the fermion determinant: the Φ algorithm

• Taming the fermion determinant: the R algorithm

• The fourth root approximation

• An exact algorithm for the fourth root: rational hybrid Monte Carlo

• Refinements
  • Sexton-Weingarten scheme
  • Hasenbusch method
  • Schwarz alternating method

• Special considerations for overlap fermions

• Monte Carlo methods for fermions
  • Multiboson method
  • Ratio of determinants

• Conjugate gradient and its relatives
  • Even-odd preconditioning
  • The conjugate gradient algorithm
  • Biconjugate gradient
  • Stabilized biconjugate gradient
  • Shifted solvers
  • Computing fermion eigenmodes

The following sections are included:

• Correlations in simulation time

• Correlations among observables
  • Correlated least chi square
  • Truncating the correlation matrix
  • Jackknife and bootstrap methods

• Fitting strategies
  • Fitting range
  • Signal to noise ratio
  • Interpolating operator
  • Asymmetric lattice
  • Bayesian methods

The following sections are included:

• Motivation

• Symanzik improvement
  • Gauge action improvement
  • Fermion action improvement
  • Nonperturbative improvement

• Tadpole improvement

• Renormalization-group inspired improvement

• “Fat link” actions

The following sections are included:

• Computing propagators and correlation functions

• Sewing propagators together
  • Mesonic correlators – Wilson fermion formalism
  • Baryonic correlators – Wilson fermion formalism
  • Mesonic correlators – staggered fermion formalism
    • Flavor nonsinglet mesons
    • Flavor singlet mesons
  • Baryonic correlators – staggered fermion formalism

• Glueballs

• The string tension

The following sections are included:

• Motivation

• Technology
  • Free fermion and gluon propagators
  • Quark-gluon vertices
  • Fat links
  • Quark self energy and tadpole
  • Vertex graph
  • Ultraviolet divergences
  • Automation

• The scale of the coupling constant

The following sections are included:

• Perturbative techniques for operator matching

• Nonperturbative techniques for operator matching
  • Methods for approximations to conserved currents
  • Regularization-independent scheme
  • Schrödinger functional methods

The following sections are included:

• Minimal introduction to chiral perturbation theory

• Quenching, partial quenching, and unquenching
  • The eta prime correlator
  • Chiral Lagrangian with partial or complete quenching

• Chiral perturbation theory for staggered fermions
  • Staggered chiral perturbation theory
  • Rooted staggered chiral perturbation theory

• Computing topological charge

The following sections are included:

• Finite volume effects in chiral perturbation theory

• The ɛ-regime
  • Banks-Casher formula
  • Chiral Lagrangian in the epsilon regime
  • Random matrix theory
  • Further applications

• Finite volume, more generally
  • Single particle states
  • Two particle states
    • Scattering lengths from the two-particle ground state
    • Decay widths from level repulsion

• Miscellaneous comments

The following sections are included:

• Overview

• Strong renormalization of weak operators
  • Effective Hamiltonian
  • An example:
  • Lattice vs continuum renormalization of the effective Hamiltonian
  • Mixing with operators of higher and lower dimension

• Lattice discrete symmetries

• Some simple examples
  • Leptonic decay constants of mesons
  • Leptonic decay constants in the heavy quark limit.
  • Electromagnetic widths of vector mesons
  • Form factors
  • Meson B parameters
  • Other purely hadronic weak interactions

• Evading a no-go theorem

The following sections are included:

• Simulating high temperature

• Introducing a chemical potential

• High quark mass limit and chiral limit

• Locating and characterizing the phase transition

• Simulating in a nearby ensemble

• Dimensional reduction and nonperturbative behavior

• Miscellaneous observables
  • Quark number susceptibilities
  • Quarkonium potential
  • Equation of state

• Nonzero density

• Spectral functions and maximum entropy

The following sections are included:

• Bibliography

• Index