Lattice QCD

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Lattice QCD

• By Arjen van Vliet

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Outline

• Introduction QCD
• Lattice QCD basics
• Scalar field calculation
• Monte Carlo Method
• Wilson loops and Wilson action
• Quenched Approximation
• Results

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QCD

• Quantum field theory of quarks and gluons
• Based on symmetry group SU(3)
• Complex because of gluon-gluon interactions
• At high energies
• - small coupling constant
• - perturbation theory applies
• - very good quantitative predictions
• At low energies
• - large coupling constant
• - perturbation theory does not apply
• - no good quantitative predictions

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QCD Lagrangian

• with the gluon field strength tensor
• and the gauge covariant derivative
• where is the gluon field, g is the strong
  coupling constant and f denotes the quark flavor.
  Looks very similar to QED, except for the last
  term in the second equation.

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Perturbation Theory

• Calculate Feynman diagrams.
• Stop at certain order.
• Order corresponds to number of vertices.
• Proportional to coupling constant, only
  applicable for small coupling constant.

I. Allison, Matching the Bare and MS Charm Quark
Mass using Weak Coupling Simulations,
presentation at Lattice 2008
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Intrinsic QCD Scale

• Running coupling constant.
• Intrinsic QCD scale in the order of 1 GeV.
• Scale below which the coupling constant becomes
  so large that standard perturbation theory no
  longer applies.
• Many unresolved question about low-energy QCD.
• This is where Lattice QCD comes in!

R. Timmermans, D. Bettoni and K. Peters, Strong
interaction studies with antiprotons
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Lattice QCD

• Proposed by Wilson, 1974.
• Nonperturbative low-energy solution of QCD.
• E.o.m. discretized on 4d Euclidean space-time
  lattice.
• Quarks and gluons can only exist on lattice
  points and travel over connection lines.
• Solved by large scale numerical simulations on
  supercomputers.

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Set up LQCD action

• From continuum to discretized lattice
• n four-vector that labels the lattice site, a
  lattice constant
• Check, take an appropriate continuum limit (a?0)
  to get back the continuum theory.

http//globe-meta.ifh.de8080/lenya/hpc/live/APE/p
hysics/lattice.html
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Scalar field action

• Scalar field , action of continuum field
  theory in Euclidean space
• Discretize to a lattice
• Result

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Expectation value calculation

• Feynman path integral formalism
• Expectation value of an operator
• where
• Rescale fields
• Lattice action becomes

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Statistical Mechanics

• Rescaled expectation value
• Recognizable?
• Statistical mechanics partition function with
• Similar for fermion fields

R. Gupta, Introduction to Lattice QCD,
arXivhep-lat/9807028
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Monte Carlo Method

• Method from statistical mechanics to calculate
  expectation value numerically.
• Generate random distribution.
• Calculate expectation value for this
  distribution.
• Repeat this process very many times.
• Average over results.
• Results have statistical errors.
• A lot of computational power needed!

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Supercomputers

• Largest computing power in Japan, especially for
  LQCD
• Combination of Hitachi SR11000 model K1 (peak
  performance 2.15 TFlops) and IBM Blue Gene
  Solution (peak performance 57.3 TFlops)
• IBM-Blue Gene/L in Groningen peak performance
  27.5 TFlops

http//www.kek.jp/intra-e/press/2006/supercomputer
_e.html
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Wilson Loops

• Closed paths on the 4d Euclidean space-time
  lattice
• matrices defined on the links that connect
  the neighboring sites and
• Traces of product of such matrices along Wilson
  loops are gauge invariant
• Plaquette the elementary building block of the
  lattice, the 1 x 1 lattice square

R. Gupta, Introduction to Lattice QCD,
arXivhep-lat/9807028
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Wilson action

• Simplest discretized action of the Yang-Mills
  part of the QCD action
• Agrees with the QCD action to order O(a2).
• Proportional to the gauge-invariant trace of the
  sum over all plaquettes.

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From Wilson to Yang Mills

• Matrices U given by
• The simplest Wilson loop, the 1x1 plaquette given
  by
• Expanding about gives
• The Taylor series of the exponent gives
• From this we derive

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Method of operation

• Six unknown input parameters, coupling constant
  and the masses of the up, down, strange, charm
  and bottom quark.
• Top quark too short lived to form bound states at
  the energies we are looking at.
• Fix in terms of six precisely measured masses of
  hadrons.
• Masses and properties of all the other hadrons
  can be obtained this way.
• They should agree with experiment.

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Lattice constant

• Lattice constant a should be small to approach
  continuum limit, but not too small or the
  computation time becomes too long.
• Size nucleon in the order of 1 Fermi (1 Fermi
  1.0x1015 m).
• a between 0.05 and 0.2 Fermi
• Results also have systematic errors due to this
  lattice discretization.

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Quenched Approximation

• Quarks fully dynamical degrees of freedom that
  can be produced and annihilated.
• In the quenched approximation vacuum polarization
  effects of quark loops are turned off.
• Very popular approximation, reduces computation
  time by a factor of about 103-105.

R. Gupta, Introduction to Lattice QCD,
arXivhep-lat/9807028
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Consequence of QQCD

• Difference with QCD for large distances.
• In QCD separated quarks split up by forming a
  quark anti-quark pair.
• At smaller distances a reasonable but not great
  approximation, as can be seen from this picture.

R. Timmermans, D. Bettoni and K. Peters, Strong
interaction studies with antiprotons
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Goals of LQCD

• Test whether QCD is the correct theory of strong
  interactions in the nonperturbative regime.
• Improve understanding of low-energy aspects of
  QCD.
• Determine quark masses and the value of the
  strong coupling constant in this energy range.
• Determine hadron spectra and masses.

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Results glueball spectrum

• LQCD glueball spectrum.
• Glueball strongly interacting particle without
  any valence quarks.
• Entirely composed of gluons and quark-antiquark
  pairs.

R. Timmermans, D. Bettoni and K. Peters, Strong
interaction studies with antiprotons
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Multiple QQCD results

• QQCD predictions for the charmonium, the
  glueball, and the spin-exotic cc-glue hybrids
  spectrum.

R. Timmermans, D. Bettoni and K. Peters, Strong
interaction studies with antiprotons
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Results

• The lattice QCD prediction of the mass of the Bc
  meson.
• Approaching the precision of the value measured
  at Fermilab.

Fermilab Today, Precise Prediction of Particle
Mass, Confirmed by Experiment, May 11, 2005
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Thanks for listening!

• Any question left?