LATTICE QCD SIMULATIONS, SOME RECENT RESULTS (END OF 2006) ITEP 7 February 2007

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

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

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

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


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


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


qmfm 22

Example of extrapolationExample of extrapolation


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

03

02

0 )/()()( raDrmCrmBAM N

SU(2) glue;SU(3) glue;2qQCD;(2+1)QCDSU(2) glue;SU(3) glue;2qQCD;(2+1)QCD

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

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)

SU(2) glueSU(2) glue SU(3) glueSU(3) glue 2qQCD (2+1)QCD 2qQCD (2+1)QCD

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!


Usually the teams are rather big, 5 - 10 -15 people


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


The Nuclear Force from Lattice QCDN. Ishii, S. Aoki and T. Hatsuda; nucl-th/0611096 hep-lat/0610002

Latticecalculations

with m/m=0.595


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


Potential between two B-mesonsJ. Savage et al. hep-lat/0611038

2qQCD (2+1)QCD2qQCD (2+1)QCD

u,d,s virtual quarks

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

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

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

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

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

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

SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDG. Schierholz (2006) (Trento)

SU(2) glue SU(3) glueSU(2) glue SU(3) glue 2qQCD 2qQCD (2+1)QCD (2+1)QCDG. Schierholz (Trento 2006)


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

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)

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

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

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

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

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)


DIK RESULTS


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 ;);(

;

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)

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.

(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

(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)

(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

(2+1)QCD(2+1)QCD JLQCD, CP-PACS JLQCD, CP-PACS

RESULTS

(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)

Instead of ConclusionsI did not discuss a number of important topics

Formfactors

Heavy-Light mesonsHeavy – Heavy mesonsand many others

LATTICE QCD SIMULATIONS, SOME RECENT RESULTS (END OF 2006) ITEP 7 February 2007

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