Lattice QCDLattice QCDBy Arjen van VlietBy Arjen van Vliet

OutlineOutline Introduction QCDIntroduction QCD Lattice QCD basicsLattice QCD basics Scalar field calculationScalar field calculation Monte Carlo MethodMonte Carlo Method Wilson loops and Wilson actionWilson loops and Wilson action Quenched ApproximationQuenched Approximation ResultsResults

QCDQCD Quantum field theory of quarks and gluonsQuantum field theory of quarks and gluons Based on symmetry group SU(3)Based on symmetry group SU(3) Complex because of gluon-gluon interactionsComplex because of gluon-gluon interactions At high energies: At high energies:

- small coupling constant- small coupling constant- perturbation theory applies- perturbation theory applies- very good quantitative predictions- very good quantitative predictions

At low energies: At low energies: - large coupling constant- large coupling constant- perturbation theory does not apply- perturbation theory does not apply- no good quantitative predictions- no good quantitative predictions

QCD LagrangianQCD Lagrangian

with the gluon field strength tensorwith the gluon field strength tensor

and the gauge covariant derivativeand the gauge covariant derivative

where is the gluon field, where is the gluon field, g g is the strong coupling is the strong coupling constant and constant and ff denotes the quark flavor. Looks very similar to denotes the quark flavor. Looks very similar to QED, except for the last term in the second equation.QED, except for the last term in the second equation.

f

fffa

a qmDiqGG ][41

LQCD

AAfgAAG cbbcaaaa

aaA

giD2

Aa

Perturbation TheoryPerturbation Theory Calculate Feynman Calculate Feynman

diagrams.diagrams. Stop at certain order.Stop at certain order. Order corresponds to Order corresponds to

number of vertices.number of vertices. Proportional to Proportional to

coupling constant, coupling constant, only applicable for only applicable for small coupling small coupling constant.constant.

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

Intrinsic QCD ScaleIntrinsic QCD Scale Running coupling Running coupling

constant.constant. Intrinsic QCD scale Intrinsic QCD scale

in the order of 1 GeV.in the order of 1 GeV. Scale below which the Scale below which the

coupling constant coupling constant becomes so large that becomes so large that standard perturbation standard perturbation theory no longer applies.theory no longer applies.

Many unresolved Many unresolved question about low-question about low-energy QCD.energy QCD.

This is where Lattice This is where Lattice QCD comes in!QCD comes in!

QCD 4

)(2

)( sgs

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

Lattice QCDLattice QCD Proposed by Wilson, 1974.Proposed by Wilson, 1974. Nonperturbative low-energy solution of Nonperturbative low-energy solution of

QCD.QCD. E.o.m. discretized on 4d Euclidean space-E.o.m. discretized on 4d Euclidean space-

time lattice.time lattice. Quarks and gluons can only exist on lattice Quarks and gluons can only exist on lattice

points and travel over connection lines.points and travel over connection lines. Solved by large scale numerical Solved by large scale numerical

simulations on supercomputers.simulations on supercomputers.

Set up LQCD actionSet up LQCD action

From continuum to discretized lattice:From continuum to discretized lattice:

n n four-vector that labels the lattice site, four-vector that labels the lattice site, a a lattice lattice constantconstant

Check, take an appropriate continuum limit Check, take an appropriate continuum limit ((aa→0) to get back the continuum theory.→0) to get back the continuum theory.

n

axd 44

http://globe-meta.ifh.de:8080/lenya/hpc/live/APE/physics/lattice.html

Scalar field actionScalar field action Scalar field , action of continuum Scalar field , action of continuum

field theory in Euclidean space:field theory in Euclidean space:

Discretize to a lattice:Discretize to a lattice:

Result:Result:

)(x

])([)( 44

22212

214

mxdS

n

axd 44

nx )()()( ˆ

1nnax

4

1

44

22

42ˆ2 }{ )()()( 22

nn

mnn

n

a aS

Expectation value Expectation value calculationcalculation

Feynman path integral formalismFeynman path integral formalism Expectation value of an operatorExpectation value of an operator

wherewhere

Rescale fields:Rescale fields: Lattice action becomes: Lattice action becomes:

n

SnnnnZnnn eOdO

ll

)(1 ),...,,(][0|),...,,(|02121

n

Sn edZ )(][

nn '

)()( ''1 SS

4

1

44

22

42ˆ2 }{ )()()( 22

nn

mnn

n

a aS

Statistical MechanicsStatistical Mechanics Rescaled expectation valueRescaled expectation value

Recognizable?Recognizable? Statistical mechanics partition function Statistical mechanics partition function

withwith

Similar for fermion fieldsSimilar for fermion fields

nn

nZ

SdZ

SOdOlnnnnlnnn

)}(exp{][

)}(exp{)...(][0|)...(|0

''1''

''1''''1'''

21'

21

kT11

R. Gupta, “Introduction to Lattice QCD”, arXiv:hep-lat/9807028

Monte Carlo MethodMonte Carlo Method Method from statistical mechanics to Method from statistical mechanics to

calculate expectation value numerically.calculate expectation value numerically. Generate random distribution.Generate random distribution. Calculate expectation value for this Calculate expectation value for this

distribution.distribution. Repeat this process very many times.Repeat this process very many times. Average over results.Average over results. Results have statistical errors. Results have statistical errors. A lot of computational power needed!A lot of computational power needed!

SupercomputersSupercomputers Largest computing power Largest computing power

in Japan, especially for in Japan, especially for LQCDLQCD

Combination of Hitachi Combination of Hitachi SR11000 model K1 (peak SR11000 model K1 (peak performance 2.15 TFlops) performance 2.15 TFlops) and IBM Blue Gene and IBM Blue Gene Solution (peak Solution (peak performance 57.3 TFlops)performance 57.3 TFlops)

IBM-Blue Gene/L in IBM-Blue Gene/L in Groningen: peak Groningen: peak performance 27.5 TFlopsperformance 27.5 TFlops

http://www.kek.jp/intra-e/press/2006/supercomputer_e.html

Wilson LoopsWilson Loops Closed paths on the 4d Euclidean space-Closed paths on the 4d Euclidean space-

time latticetime lattice matrices defined on the links that matrices defined on the links that

connect the neighboring sitesconnect the neighboring sites and and Traces of product of such matrices along Traces of product of such matrices along

Wilson loops are gauge invariantWilson loops are gauge invariant Plaquette: the elementary building block Plaquette: the elementary building block

of the lattice, the 1 x 1 lattice squareof the lattice, the 1 x 1 lattice square

,xUx ̂ax

R. Gupta, “Introduction to Lattice QCD”, arXiv:hep-lat/9807028

Wilson actionWilson action

Simplest discretized action of the Yang-Simplest discretized action of the Yang-Mills part of the QCD actionMills part of the QCD action

Agrees with the QCD action to order Agrees with the QCD action to order O(O(aa22).).

Proportional to the gauge-invariant trace Proportional to the gauge-invariant trace of the sum over all plaquettes.of the sum over all plaquettes.

,

††,ˆ,ˆ,2

11 )1(TrRe2

xx,νaxaxxgW UUUUS

)()(441 xGxGxdS a

aYM

From Wilson to Yang From Wilson to Yang MillsMills

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

))(exp()( 2̂

xiagAxU

)()ˆ()ˆ()( ††11 xUxUxUxUW x

)])()ˆ()ˆ()([exp( 2ˆ

2ˆ

2ˆ

2ˆ

xAxAxAxAiag

2ˆˆ x

...])()(exp[ 3312

2 4

AAAAgia gia

...)(1 62

2 24

aOFFgFia ga

...)1(TrRe 211 24

FFW gax

Method of operationMethod of operation Six unknown input parameters, coupling Six unknown input parameters, coupling

constant and the masses of the up, down, constant and the masses of the up, down, strange, charm and bottom quark.strange, charm and bottom quark.

Top quark too short lived to form bound Top quark too short lived to form bound states at the energies we are looking at.states at the energies we are looking at.

Fix in terms of six precisely measured Fix in terms of six precisely measured masses of hadrons.masses of hadrons.

Masses and properties of all the other Masses and properties of all the other hadrons can be obtained this way.hadrons can be obtained this way.

They should agree with experiment. They should agree with experiment.

Lattice constantLattice constant Lattice constant Lattice constant a a should be small to should be small to

approach continuum limit, but not too approach continuum limit, but not too small or the computation time becomes small or the computation time becomes too long.too long.

Size nucleon in the order of 1 Fermi (1 Size nucleon in the order of 1 Fermi (1 Fermi = 1.0x10Fermi = 1.0x10–15 –15 m).m).

aa between 0.05 and 0.2 Fermi between 0.05 and 0.2 Fermi Results also have systematic errors Results also have systematic errors

due to this lattice discretization.due to this lattice discretization.

Quenched ApproximationQuenched Approximation Quarks fully dynamical Quarks fully dynamical

degrees of freedom that degrees of freedom that can be produced and can be produced and annihilated.annihilated.

In the quenched In the quenched approximation vacuum approximation vacuum polarization effects of polarization effects of quark loops are turned quark loops are turned off.off.

Very popular Very popular approximation, reduces approximation, reduces computation time by a computation time by a factor of about 10factor of about 1033-10-1055..

R. Gupta, “Introduction to Lattice QCD”, arXiv:hep-lat/9807028

Consequence of QQCDConsequence of QQCD Difference with QCD Difference with QCD

for large distances.for large distances. In QCD separated In QCD separated

quarks split up by quarks split up by forming a quark anti-forming a quark anti-quark pair.quark pair.

At smaller distances a At smaller distances a reasonable but not reasonable but not great approximation, great approximation, as can be seen from as can be seen from this picture.this picture.

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

Goals of LQCDGoals of LQCD Test whether QCD is the correct theory Test whether QCD is the correct theory

of strong interactions in the of strong interactions in the nonperturbative regime.nonperturbative regime.

Improve understanding of low-energy Improve understanding of low-energy aspects of QCD.aspects of QCD.

Determine quark masses and the value Determine quark masses and the value of the strong coupling constant in this of the strong coupling constant in this energy range.energy range.

Determine hadron spectra and masses.Determine hadron spectra and masses.

Results glueball Results glueball spectrumspectrum

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”

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”

ResultsResults The lattice QCD The lattice QCD

prediction of the prediction of the mass of the Bmass of the Bcc meson.meson.

Approaching the Approaching the precision of the precision of the value measured at value measured at Fermilab. Fermilab.

Fermilab Today, “Precise Prediction of Particle Mass, Confirmed by Experiment”, May 11, 2005

Thanks for Thanks for listening!listening!Any question left?Any question left?