Lattice QCD (INTRODUCTION)
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Lattice QCDLattice QCD(INTRODUCTION)(INTRODUCTION)
DUBNA WINTER SCHOOL 1-2 FEBRUARY DUBNA WINTER SCHOOL 1-2 FEBRUARY 20052005
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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/
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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
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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
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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
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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
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Chiral limitChiral limit
Quark masses Pion massQuark masses Pion mass
qmfm 22
ExpExp
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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
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SpectrumSpectrum
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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
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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
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Types of FermionsTypes of Fermions
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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{
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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(?).
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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)
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Earth simulatorEarth simulator
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2020
Earth simulatorEarth simulator
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Earth simulatorEarth simulator
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Earth simulatorEarth simulator
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Earth simulatorEarth simulator
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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
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ExampleExample
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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
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Quark condensateQuark condensate
Fermion condensate vs. T, F.Karsch et al.Fermion condensate vs. T, F.Karsch et al.
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Phase diagram mPhase diagram mqq-T, one flavor-T, one flavor
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Quark mass dependence of TcQuark mass dependence of Tc
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Three quarksThree quarks
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Critical temperature for Critical temperature for pure glue and for various pure glue and for various
dynamical quarksdynamical quarks
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PurePure
SU(3)SU(3)
glueglue
(simplest(simplest
case)case)
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Temperature of the phase transitionTemperature of the phase transitionPure glue SU(3)Pure glue SU(3) F. KarschF. Karsch
MevTc )2271(
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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(
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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(
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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(
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Example of extrapolation (DIK 2005)Example of extrapolation (DIK 2005)
)()(),( 2
01 amCra
CTamT cc
Russian (JSCC)Russian (JSCC)
supercomputer M1000supercomputer M1000
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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 ;);(
;
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Example: pressureExample: pressure
F. Karsch (2001-2005)F. Karsch (2001-2005)
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FULL PROBLEMFULL PROBLEM
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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{
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АнекдотАнекдот
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Mu-T diagramme, example of calculation
(weighted expantion on mu)
C.R. Aallton et al.C.R. Aallton et al.
(2005)(2005)
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Full diagram (theory)Full diagram (theory) C.R. Aallton et al. (2005)C.R. Aallton et al. (2005)
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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
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Special thanks toSpecial thanks toVALERY SCHEKOLDINVALERY SCHEKOLDIN
(photo )(photo )