Lattice QCDLattice QCD(INTRODUCTION)(INTRODUCTION)

DUBNA WINTER SCHOOL 1-2 FEBRUARY DUBNA WINTER SCHOOL 1-2 FEBRUARY 20052005

22

Main ProblemsMain Problems

ff

f mDFg

L )(Tr 1 _

22

Starting from Lagrangian Starting from Lagrangian

(1) obtain hadron spectrum, (1) obtain hadron spectrum,

(2) describe phase transitions,(2) describe phase transitions,

(3) explain confinement of color(3) explain confinement of color

http://http://www.claymath.org/Millennium_Prize_Problems/www.claymath.org/Millennium_Prize_Problems/

33

The main difficulty is the absence of The main difficulty is the absence of analytical methods, the interactions analytical methods, the interactions are strong and only computer are strong and only computer simulations give results starting simulations give results starting from the first principles.from the first principles.

The force between The force between quark and antiquark quark and antiquark

is 12 tonesis 12 tones

44

MethodsMethods

Imaginary time Imaginary time tt→→itit

Space-time discretizationSpace-time discretization

Thus we get from functional integral Thus we get from functional integral the statistical theory in four dimensionsthe statistical theory in four dimensions

]}[exp{ SiDZ ]}[exp{ SDZ

x

xdxD )( ]}[exp{ SdZ xx

55

The statistical theory in four The statistical theory in four dimensions can be simulated by dimensions can be simulated by

Monte-Carlo methodsMonte-Carlo methods

The typical multiplicities of integrals are The typical multiplicities of integrals are 101066-10-1088

We have to invert matrices 10We have to invert matrices 106 6 x x 101066

The cost of simulation of one configuration The cost of simulation of one configuration is:is:

yearTeraflops

GevafmLm

m

75

6

6 ][][104

66

Three limitsThree limits

0

0

qm

L

a

Mevm

fmL

fma

q 100

42

1.0

Lattice spacingLattice spacing

Lattice sizeLattice size

Quark massQuark mass

Typical values nowTypical values now

ExtrapolationExtrapolation

++

Chiral perturbation theoryChiral perturbation theory

77

Chiral limitChiral limit

Quark masses Pion massQuark masses Pion mass

qmfm 22

ExpExp

88

Nucleon mass extrapolationNucleon mass extrapolation

Fit on the base of the chiral perturbation theoryFit on the base of the chiral perturbation theory 2

03

02

0 )/()()( raDrmCrmBAM N

99

SpectrumSpectrum

1010

Earth SimulatorEarth Simulator Based on the NEC SX architecture, 640 nodes, each node with 8 Based on the NEC SX architecture, 640 nodes, each node with 8 vector processors (8 Gflop/s peak per processor), 2 ns cycle time, vector processors (8 Gflop/s peak per processor), 2 ns cycle time, 16GB shared memory. – Total of 5104 total processors, 40 TFlop/s 16GB shared memory. – Total of 5104 total processors, 40 TFlop/s peak, and 10TB memory.peak, and 10TB memory. It has a single stage crossbar (1800 miles of cable) 83,000 copper It has a single stage crossbar (1800 miles of cable) 83,000 copper cables, 16 GB/s cross section bandwidth.cables, 16 GB/s cross section bandwidth.700 TB disk space, 1.6 PB mass store700 TB disk space, 1.6 PB mass storeArea of computer = 4 tennis courts, 3 floorsArea of computer = 4 tennis courts, 3 floors

Lattice QCD at finite Lattice QCD at finite temperaturetemperatureand density and density

(INTRODUCTION)(INTRODUCTION)

DUBNA WINTER SCHOOL 2-3 FEBRUARY DUBNA WINTER SCHOOL 2-3 FEBRUARY 20062006

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Finite Temperature in Field TheoryFinite Temperature in Field Theory

In imaginary time the partition function In imaginary time the partition function which defines the field theory is:which defines the field theory is:

The action is:The action is:

]}[exp{ SdZ xx

),(][/1

0

LdxdydzdtST

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QCD at Finite TemperatureQCD at Finite Temperature

Partition function of QCD with one Partition function of QCD with one flavor at temperature T is:flavor at temperature T is:

The action is:The action is:

T

qmAgiFxddtAS

ASDDDAZ

/1

0

23 })ˆˆ(){(],,[

]},,[exp{

MMdd det}exp{ In computerIn computer

1414

Types of FermionsTypes of Fermions

1515

Types of FermionsTypes of Fermions

WilsonWilson

Kogut-SuskindKogut-Suskind

Wilson improvedWilson improved

Wilson nonperturbatevely improvedWilson nonperturbatevely improved

Domain wallDomain wall

StaggeredStaggered

OverlapOverlap

1. Quark mass ->0 2. Fast algorithms1. Quark mass ->0 2. Fast algorithms

MMdd det}exp{

1616

Approximations to real QCDApproximations to real QCD

Quenched approximation (no fermion Quenched approximation (no fermion loops), gauge group SU(2), SU(3)loops), gauge group SU(2), SU(3)

Dynamical fermions, the realistic situation, Dynamical fermions, the realistic situation, heavy heavy ss quark and quark and 5Mev5Mev uu and and d d quarks quarks will be available on computers in 2015(?).will be available on computers in 2015(?).

1717

Types of AlgorithmsTypes of Algorithms

1. Hybrid Monte Carlo + Molecular dynamics, 1. Hybrid Monte Carlo + Molecular dynamics, leap-frogleap-frog

2. Local Boson Algorithm2. Local Boson Algorithm

3. Pseudofermionic Hybrid Monte Carlo3. Pseudofermionic Hybrid Monte Carlo

3. Two step multyboson3. Two step multyboson

4. Polynomial Hybrid Monte Carlo 4. Polynomial Hybrid Monte Carlo

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1. Earth (solid state) 2. Water (liquid) 3. Air (gas) 4. Fire (plasma) 5.….. (quark-gluon plasma)

1919

Earth simulatorEarth simulator

2020

Earth simulatorEarth simulator

2121

Earth simulatorEarth simulator

2222

Earth simulatorEarth simulator

2323

Earth simulatorEarth simulator

2424

How to find quark gluon plasma?How to find quark gluon plasma?

ORDER PARAMETERSORDER PARAMETERS

}Aexp{i line Polyakov1/T

0

00dxP

mq

condensateQuark

0qm

2525

ExampleExample

2626

Lattice calculation by DIK groupLattice calculation by DIK group

Polyakov loop susceptibility Polyakov loop susceptibility clower improved Wilson fermions clower improved Wilson fermions

16**3*8 lattice16**3*8 lattice

2727

Quark condensateQuark condensate

Fermion condensate vs. T, F.Karsch et al.Fermion condensate vs. T, F.Karsch et al.

2828

Phase diagram mPhase diagram mqq-T, one flavor-T, one flavor

2929

Quark mass dependence of TcQuark mass dependence of Tc

3030

Three quarksThree quarks

3131

Critical temperature for Critical temperature for pure glue and for various pure glue and for various

dynamical quarksdynamical quarks

3232

PurePure

SU(3)SU(3)

glueglue

(simplest(simplest

case)case)

3333

Temperature of the phase transitionTemperature of the phase transitionPure glue SU(3)Pure glue SU(3) F. KarschF. Karsch

MevTc )2271(

3434

3535

Temperature of the phase transitionTemperature of the phase transitionPure glue SU(3) Pure glue SU(3) F. KarschF. Karsch

Two flavor QCD, clover improved Wilson fermionsTwo flavor QCD, clover improved Wilson fermions C.Bernard (2005)C.Bernard (2005)

DIK collaboration (2005) DIK collaboration (2005)

MevTc )2271(

MevTc )4171(

MevTc )3166();3173(

3636

3737

Temperature of the phase transitionTemperature of the phase transitionPure glue SU(3) Pure glue SU(3) F. KarschF. Karsch

Two flavor QCD, clover improved Wilson fermionsTwo flavor QCD, clover improved Wilson fermions C.Bernard (2005)C.Bernard (2005)

DIK collaboration (2005) DIK collaboration (2005)

Two flavor QCD, improved staggered fermions Two flavor QCD, improved staggered fermions

F.Karsch (2000)F.Karsch (2000)

MevTc )2271(

MevTc )4171(

MevTc )8173(

MevTc )3166();3173(

3838

3939

Temperature of the phase transitionTemperature of the phase transitionPure glue SU(3) Pure glue SU(3) F. KarschF. Karsch

Two flavor QCD, clover improved Wilson fermionsTwo flavor QCD, clover improved Wilson fermions C.Bernard (2005)C.Bernard (2005)

DIK collaboration (2005) DIK collaboration (2005)

Two flavor QCD, improved staggered fermions Two flavor QCD, improved staggered fermions

F.Karsch (2000)F.Karsch (2000)

ThreeThree flavor QCD, improved staggered fermions! flavor QCD, improved staggered fermions! F.Karsch (2000)F.Karsch (2000)

MevTc )2271(

MevTc )4171(

MevTc )8173(

MevTc )3166();3173(

MevTc )8154(

4040

Example of extrapolation (DIK 2005)Example of extrapolation (DIK 2005)

)()(),( 2

01 amCra

CTamT cc

Russian (JSCC)Russian (JSCC)

supercomputer M1000supercomputer M1000

4141

Plasma thermodynamicsPlasma thermodynamics

Free energy densityFree energy density

energy, entropy, velosity of sound, energy, entropy, velosity of sound, . pressure . pressure

),( VTZVT

f

s scp

ddp

cT

pTs

TP

dTd

TT

p

fp

s

24344 ;);(

;

4242

Example: pressureExample: pressure

F. Karsch (2001-2005)F. Karsch (2001-2005)

4343

FULL PROBLEMFULL PROBLEM

4444

Non zero chemical potentialNon zero chemical potential

QCD partition function at finite temperature QCD partition function at finite temperature and chemical potentialand chemical potential

At finite chemical potential the fermionic At finite chemical potential the fermionic determinant is not positively defined! Thus determinant is not positively defined! Thus we have no interpretation of the partition we have no interpretation of the partition function as probability wait (big difficulties function as probability wait (big difficulties with MC).with MC).

}],,[exp{ 03

/1

0

xddtASDDDAZT

MMdd det}exp{

4545

АнекдотАнекдот

4646

Mu-T diagramme, example of calculation

(weighted expantion on mu)

C.R. Aallton et al.C.R. Aallton et al.

(2005)(2005)

4747

4848

Full diagram (theory)Full diagram (theory) C.R. Aallton et al. (2005)C.R. Aallton et al. (2005)

4949

5050

LiteratureLiterature

H. Satz hep-ph/0007209H. Satz hep-ph/0007209

F. Karsch hep-lat/0106019F. Karsch hep-lat/0106019

C.R. Allton et al. hep-lat/0504011C.R. Allton et al. hep-lat/0504011

F. Karsch hep-lat/0601013F. Karsch hep-lat/0601013

DIK (DESY-ITEP-Kanazawa) collaboration DIK (DESY-ITEP-Kanazawa) collaboration hep-lat/0509122hep-lat/0509122, , hep-lat/0401014 hep-lat/0401014

5151

5252

Special thanks toSpecial thanks toVALERY SCHEKOLDINVALERY SCHEKOLDIN

(photo )(photo )