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Lattice QCD

Lattice QCD. By Arjen van Vliet. Outline. Introduction QCD Lattice QCD basics Scalar field calculation Monte Carlo Method Wilson loops and Wilson action Quenched Approximation Results. QCD. Quantum field theory of quarks and gluons Based on symmetry group SU(3)

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Lattice QCD

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  1. Lattice QCD By Arjen van Vliet

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

  3. 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

  4. 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.

  5. 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

  6. 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”

  7. 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.

  8. 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.de:8080/lenya/hpc/live/APE/physics/lattice.html

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

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

  11. Statistical Mechanics • Rescaled expectation value • Recognizable? • Statistical mechanics partition function with • Similar for fermion fields R. Gupta, “Introduction to Lattice QCD”, arXiv:hep-lat/9807028

  12. 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!

  13. 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

  14. 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”, arXiv:hep-lat/9807028

  15. 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.

  16. 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 From Wilson to Yang Mills

  17. 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.

  18. 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 = 1.0x10–15 m). • a between 0.05 and 0.2 Fermi • Results also have systematic errors due to this lattice discretization.

  19. 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”, arXiv:hep-lat/9807028

  20. 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”

  21. 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.

  22. 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”

  23. 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”

  24. 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

  25. Thanks for listening! Any question left?

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