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a). Introductiona). Introductionb). Quenched calculationsb). Quenched calculationsc). Calculations with 2 light dynamical quarksc). Calculations with 2 light dynamical quarksd). (2+1) QCDd). (2+1) QCD
LATTICE QCD SIMULATIONS, SOME RECENT RESULTS
(END OF 2006)
ITEP 7 February 2007
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INTRODUCTIONINTRODUCTION
ff
f mDFg
L )(Tr 1 _
22
Main Problems: starting from Lagrangianstarting from Lagrangian
(1)(1) obtain hadron spectrum, obtain hadron spectrum, (2)(2) calculate various matrix elements,calculate various matrix elements,(3) describe phase transitions, and (3) describe phase transitions, and
phase diagramphase diagram(4) explain confinement of color(4) explain confinement of color
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INTRODUCTIONINTRODUCTIONThe main difficulty is the absence of analytical methods, the interactions are strong and only computer simulations give results starting from the first principles.
The force The force between between quark and quark and antiquark is antiquark is 12 tones12 tones
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INTRODUCTIONINTRODUCTIONMethodsMethods
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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INTRODUCTIONINTRODUCTION
The statistical theory in four dimensions can be simulated The statistical theory in four dimensions can be simulated by Monte-Carlo methodsby Monte-Carlo methods
The typical multiplicities of integrals are The typical multiplicities of integrals are 101066-10-101010
We have to invert matrices 10We have to invert matrices 108 8 x x 101088
The cost of simulation of one configuration The cost of simulation of one configuration is:is:
yearTeraflops
GevafmLm
m
75
6
6 ][][104
Improved Wilson fermions
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INTRODUCTIONINTRODUCTION
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 valuesTypical valuesExtrapolationExtrapolation
++
Chiral perturbation Chiral perturbation theorytheory
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INTRODUCTIONINTRODUCTION
qmfm 22
Example of extrapolationExample of extrapolation
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INTRODUCTIONINTRODUCTION
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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SU(2) glue;SU(3) glue;2qQCD;(2+1)QCDSU(2) glue;SU(3) glue;2qQCD;(2+1)QCD
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SU(2) glueSU(2) glue SU(3) glue 2qQCD (2+1)QCD SU(3) glue 2qQCD (2+1)QCD
Theory of color confinementTheory of chiral symmetry breakingMonopolesVorticesInstantons and caloronsLocalization of Dirac eigenmodes
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SU(2) glueSU(2) glue SU(3) glue 2qQCD (2+1)QCD SU(3) glue 2qQCD (2+1)QCD
Theory of color confinementTheory of chiral symmetry breakingMonopolesVorticesInstantons and caloronsLocalization of Dirac eigenmodes
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SU(2) glueSU(2) glue SU(3) glue 2qQCD (2+1)QCD SU(3) glue 2qQCD (2+1)QCD
Theory of color confinementTheory of chiral symmetry breakingMonopolesVorticesInstantons and caloronsLocalization of Dirac eigenmodes
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SU(2) glueSU(2) glue SU(3) glue 2qQCD (2+1)QCD SU(3) glue 2qQCD (2+1)QCDTheory of color confinementTheory of chiral symmetry breakingMonopolesVorticesInstantons and caloronsLocalization of Dirac eigenmodes (Anderson localistion)
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SU(2) glueSU(2) glue SU(3) glueSU(3) glue 2qQCD (2+1)QCD 2qQCD (2+1)QCD
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SU(2) glueSU(2) glue SU(3) glue SU(3) glue 2qQCD (2+1)QCD2qQCD (2+1)QCD
Study of the complicated systems:
a)Structure of gluon fields inside hadronb)Nucleon-Nucleon potential Three body forces!
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SU(2) glueSU(2) glue SU(3) glue SU(3) glue 2qQCD (2+1)QCD2qQCD (2+1)QCD
Usually the teams are rather big, 5 - 10 -15 people
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SU(2) glueSU(2) glue SU(3) glue SU(3) glue 2qQCD (2+1)QCD2qQCD (2+1)QCD
The Nuclear Force from Lattice QCDN. Ishii, S. Aoki and T. Hatsuda; nucl-th/0611096; hep-lat/0610002
)(
)(21
)(r
rmErV
From lattice calculations(six quark matrix element)
Phenomenological potential
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SU(2) glueSU(2) glue SU(3) glue SU(3) glue 2qQCD (2+1)QCD2qQCD (2+1)QCD
The Nuclear Force from Lattice QCDN. Ishii, S. Aoki and T. Hatsuda; nucl-th/0611096 hep-lat/0610002
Latticecalculations
with m/m=0.595
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SU(2) glueSU(2) glue SU(3) glue SU(3) glue 2qQCD (2+1)QCD2qQCD (2+1)QCD
Viscosity of quark gluon plasma
A. Nakamura, S. Sakai, hep-lat/0510039
RHIC result at 1.4<T/Tc<1.8: quark-gluon plasma is not a gas but rather a kind of liquid with low viscosity
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SU(2) glueSU(2) glue SU(3) glue SU(3) glue 2qQCD (2+1)QCD2qQCD (2+1)QCD
Potential between two B-mesonsJ. Savage et al. hep-lat/0611038
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2qQCD (2+1)QCD2qQCD (2+1)QCD
u,d,s virtual quarks
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDNew effect: String Breaking New effect: String Breaking
2qQCD
QQQQ
QqQq QqQq
Glue
Dynamical quarks
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDString Breaking String Breaking (DIK collaboration)(DIK collaboration)
MESONMESON
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDString Breaking String Breaking (DIK collaboration)(DIK collaboration)
BARYONBARYON
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCD
Partition function of QCD with one Partition function of QCD with one flavor at temperature T is:flavor at temperature T is:
T
qmAgiFxddtAS
ASDDDAZ
/1
0
23 })ˆˆ(){(],,[
]},,[exp{
MMdd det}exp{ In computerIn computer
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDTypes 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
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD(2+1)QCD
G. Schierholz (Trento 2006)
2006
yearTeraflops
GevafmLm
m
75
6
6 ][][104
OLD2001
NEW2006
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDG. Schierholz (2006) (Trento)
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDG. Schierholz (2006) (Trento)
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDG. Schierholz (Trento 2006)
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCD
Phase diagram(F.Karsch)
Four plenary talks at Lattice 2006!Color superconductivity in ultra-dense quark matter. Mark G. Alford; hep-lat/0610046
Lattice QCD at finite density. C. Schmidt; hep-lat/0610116
Recent progress in finite temperature lattice QCD. Urs M. Heller; hep-lat/0610114
QCD phase diagram: an overview. M.A. Stephanov; hep-lat/0701002
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDPhase diagram THEORY
3
2-nd or 1-st order for =0?Di Giacomo –first order (2006)
First order (Pisarski, Wilczek)
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDPhase diagram: numerical calculations are very difficult, since we have a complex Monte-Carlo weight
}
})ˆˆ(){(],,[
]},,[exp{
03
/1
0
/1
0
23
xddt
mAgiFxddtAS
ASDDDAZ
T
T
q
MMdd det}exp{
)}ˆˆdet(ln][exp{ 0 qmAgiASDAZ
COMPLEX
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD(2+1)QCD
)}ˆˆdet(ln][exp{ qmAgiASDAZ
Phase diagram: numerical calculations are very difficult, since we have a complex Monte-Carlo weight
Various numerical tricks: analytical continuations, ->i
QCD critical point in T- plane
RED – RHIC experimentBLACK – phenomenological modelsGREEN – Lattice calculations
M.A. Stephanov;hep-lat/0701002
)}ˆˆdet(ln][exp{ 0 qmAgiASDAZ
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDPure 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(
Critical temperature, =0
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDTc by DIK (DESY-ITEP-Kanazawa) collaborationV.G. Bornyakov, M.N. Chernodub, Y. Mori , S.M. Morozov, Y. Nakamura,M.I. Polikarpov, G. Schierholz, A.A. Slavnov, H. Stüben, T. Suzuki (2006)
Russian (JSCC)Russian (JSCC)supercomputer supercomputer
M15000M15000
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDTc by DIK (DESY-ITEP-Kanazawa) collaborationV.G. Bornyakov, M.N. Chernodub, Y. Mori , S.M. Morozov, Y. Nakamura,M.I. Polikarpov, G. Schierholz, A.A. Slavnov, H. Stüben, T. Suzuki (2006)
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCD
DIK RESULTS
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCD
Plasma thermodynamicsPlasma thermodynamics
Free energy densityFree energy density
energy, entropy, velocity of sound, energy, entropy, velocity of sound, . pressure . pressure
),(ln VTZVT
f
s scp
ddp
cT
pTs
TP
dTd
TT
p
fp
s
24344 ;);(
;
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDPlasma thermodynamics, example: pressure
F. Karsch (2001-2005)F. Karsch (2001-2005)
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SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDQuark condensateQuark condensate
F.Karsch et al.F.Karsch et al.
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(2+1)QCD(2+1)QCD JLQCD, CP-PACS JLQCD, CP-PACSThe description of the meson mass spectrum is good, but not
excellent for lattice QCD with two dynamical quarks
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(2+1)QCD(2+1)QCD JLQCD, CP-PACS JLQCD, CP-PACSThe description of the meson mass spectrum is good, but not
excellent for lattice QCD with two dynamical quarks
meson mass vs lattice spacing (the mass of the s-quark is fitted from the mass of the K meson)
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(2+1)QCD(2+1)QCD JLQCD, CP-PACS JLQCD, CP-PACSAlmost three years of gauge field trajectories generation at Earth Simulator; Lattice spacial volume is (2 fm)^3, a=0.07, 0.1, 0.12 fm
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(2+1)QCD(2+1)QCD JLQCD, CP-PACS JLQCD, CP-PACS
RESULTS
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(2+1)QCD(2+1)QCD JLQCD, CP-PACS JLQCD, CP-PACS
RESULTS
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(2+1)QCD(2+1)QCD MILC configurations, MILC configurations,staggered dynamical fermions, staggered dynamical fermions, NPLQCD CollaborationNPLQCD Collaboration
Hyperon-NucleonHyperon-Nucleon phase shifts (hep-lat/0612026) phase shifts (hep-lat/0612026)
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Instead of ConclusionsI did not discuss a number of important topics
Formfactors
Heavy-Light mesonsHeavy – Heavy mesonsand many others