Euler Lagrange Equation

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Lattice Quantum Chromodynamics
By Leila Joulaeizadeh 19 Oct. 2005
1- Literature Lattice QCD , C. Davis
Hep-ph/0205181 2- Burcham and Jobes
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• Outline
• - Introduction
• Hamilton principle
• Local gauge invariance and QED
• Local gauge invariance and QCD
• Lattice QCD calculations
• Some results
• Conclusion

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What is Quantum Chromodynamics and why LQCD?

• Strong interaction between coloured quarks by
  exchange of coloured gluon
• Gluons carry colour so they have self
  interaction
• Self interaction of gluons , nonabelian group
  SU(3)
• - QCD is a nonlinear theory so there is no
  analytical solution and we should use numerical
  methods

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Euler Lagrange Equation
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For motion of a point like particle with mass m
in a central potential
Hamilton Principle

• Physical systems will evolve in such a way to
  minimize the action

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In Quantum Field Theory
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Examples

• Scalar field (spin 0 particle)

Spinor field(spin 1/2 particle)
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Local Gauge Invariance and QED
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Example
Massless vector field(spin 1)
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Local Gauge Invariance and QCD
Non-Abelian nature of SU(3)
Gluon self interaction term
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Diagrams representing propagation of free quark
and gluon and their interaction
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O operator whose expectation value we want to
calculate
Lattice QCD
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Lattice gauge theory for gluons
x
x
X1
X1
x
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Lattice gauge theory for gluons
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Fermion doubling problem of quarks on the lattice
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Solutions of Fermion doubling problem
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Action with quarks
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Relating lattice results to physics
Make the correlators of quarks by using ? matrices
r
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1- choose the lattice spacing - close to the
continuum - computation costs2- Choose a
quark formulation and number of quark flavors3-
generating an ensemble of gluon configurations
- Try to go near small masses -
computation costs4- calculation of quark
propagators on each gluon configuration5-
combination of quark propagators to form hadron
correlators6- Determination of lattice spacing
in Gev(lattice calibration) 7- extrapolation of
hadron masses as a function of bare quark
masses8- repeat the calculation using several
lattice spacing to compare with physical results
at the limit of a 09- compare with
experiment or give a prediction for experiment
Steps of typical lattice calculation
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Some results of lattice QCD calculations
The spectrum of light mesons and baryons in the
quenched approximation
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The ratio of inverse lattice spacing
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(No Transcript)
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?c ? ?
JPC
Charmonium spectrum in quenched approximation
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• Summary
• Photons dont carry any colour charge, so QED is
  analytically solvable.
• Gluons do carry colour charge,so to solve the
  QCD theory, approximations are proposed
• (e.g. Lattice calculation method ).
• - There is a fermion doubling problem in lattice
  which can be solved by various methods.
• In order to obtain light quark properties, we
  need bigger computers and the
• calculation costs will be increased.
• Quenched approximation is reasonable in order to
  decrease the computation costs.