LATTICE QCD is FUN 　

Tetsuo Hatsuda, Univ. Tokyo

Second Berkeley School on Collective Dynamics May 21-25, 2007

[1] Lattice QCD basics　　 [2] Nuclear force on the lattice ( dense QCD)

[3] In-medium hadrons on the lattice ( hot QCD)

[4] Summary

In-medium Hadrons

SB CSC

QGP

　

SB

CSC

QGP

B

● 　 Asakawa & Yazaki, Nuc. Phys A504 (‘89) 668● 　 Yamamoto, Tachibana, Baym & T.H., 　 Phys. Rev. Lett. 97 (2006)122001

1/T

a

L

Lattice setup at finite TLattice setup at finite T

1/T = Nt a L = Ns a

1/T fixed, Nt/Ns small, Nt large “a” small

continuum limit

Bulk Thermodynamics in full QCDBulk Thermodynamics in full QCD

Critical temperature

Tc ： 160 – 190 MeV ～ 1012 [K]

Critical energy density

εc ： ～　 2 GeV/fm3

～ 10 εnm

Order of the transition

　 2nd order 　 (u,d; m=0)　 1st order (u,d,s; m=0) crossover (real world)

MILC Coll., hep-lat/061001

(2+1)-flavor, O(a2) improved action, Ns/Nt=2

Karsch, hep-lat/0608003 Wuppertal-Budapest Coll., hep-lat/0510084 stout, Ccond/Cnt correction by hand Ns/Nt=3

What is Phase Transition ? What is Phase Transition ?

Susceptibilities

n-th order transition: non-analiticity starts from

e.g. 1st order: P smooth, dP/dT=s discontinuous 2nd order: P smooth, dP/dT=s smooth, (d/dT)2P=ds/dT=cV/T divergent

crossover: P(K) is everywhere analytic

Order of the transition in full QCD (Nf=2+1) Order of the transition in full QCD (Nf=2+1)

Fluctuation: chiral susceptibility

Wuppertal-Budapest Coll., Nature 443 (2006)

1/T

m/T

2

m/T

2

　　　　　　　　　　　　　　 2nd order transition

•Relation between and , e.g. (3-dimension)

x

xh

Z

Vht )0()(

ln1)0,(

2

2

r

er

r

~)0()(

2000

00

2

)0()()0,(

rrr

r

edrere

drrexdrrht

Wuppertal-Budapest Coll., Nature 443 (2006)

Pseudo critical temperature Tpc Pseudo critical temperature Tpc

n-th order transition: non-analiticity starts from

e.g. 1st order: P smooth, dP/dT=s discontinuous 2nd order: P smooth, dP/dT=s smooth, (d/dT)2P=ds/dT=cV/T divergent

crossover: P(K) is everywhere analytic

Intrinsic ambiguity to define Tpc

m/T

2

MILC Coll., hep-lat/0405029169(12)(4)(5) MeV

Asqtad, Nt=4,6,8, Ns/Nt=2, r_1=0.317(7) fm

RBC-Bielefeld Coll., hep-lat/0608013192(7)(4) MeV

P4fat3, Nt=4,6 Ns/Nt=2-4, r_0=0.469(7) fm

Wuppertal-Budapest Coll., hep-lat/0609068 151(3)(3) MeV + 9 MeV

stout, Nt=6,8,10, Ns/Nt=4, F_K scale

WHOT-QCD Coll., preliminary 175(4)(2) MeV (Nf=2, Nt=6, Polyakov-loop sus.)

clover, Nt=4, 6, Ns/Nt=3-4, m_V scale

Wilson fermion

Staggered fermion

Tpc (a 0) in full QCD (Nf=2+1) from m/T2 Tpc (a 0) in full QCD (Nf=2+1) from m/T2

[MeV]

[MeV]

Tpc on the lattice from chain rule Tpc on the lattice from chain rule

r0=0.469 (7) fm, HPQCD-UKQCD Coll. hep-lat/0507013

from bottomonium mass splitting (Nf=2+1, staggered)

r0=0.516 (21) fm, CP-PACS-JLQCD Coll., hep-lat/0610050

from ρ-meson mass (Nf=2+1, Wilson)

Sommer scales

Critical point Critical point

Cf. Asakawa & Yazaki, NPA504 (1989) 668 Klimt, Lutz & Weise, PLB249 (’90) 386

de Forcrand and Phillipsen, hep-lat/0607017 Nf=2+1, Nt=4, standard staggered

SB CSC

QGP

Spectral Properties of Hot QCD

What are the elementary excitations in the plasma?

T ΛQCD

pz

px

py

DeTar’s conjecturePhys.Rev.D32 (1985) 276

T/Tc

/s pQCD

AdS/CFT

Shear viscosity in quenched QCD

Quenched Lattce QCD: 24x24x24x8 Nakamura & Sakai, Phys.Rev.Lett.94:072305,2005 & hep-lat/0510100

Heavy probes Heavy probes

Matsui & Satz, PLB178 (’86) Miyamura et al., PRL57 (’86)

Dynamic probeStatic probe

Gluon matter (quenched QCD)Quark-gluon matter (full QCD)

Singlet free energy in full QCD (Nf=2+1)Singlet free energy in full QCD (Nf=2+1)

RBC-Bielefeld Coll., hep-lat/0610041

163x4, p4fat3 action, mud/ms=0.1

g,u,d,s

r

Charmonium “wave function” （ quenched QCD)

Charmonium “wave function” （ quenched QCD)

QCD-TARO Coll., Phys. Rev. D63 (’01)

Matsui & Satz, PLB178 (’86) Miyamura et al., PRL57 (’86)

gr r

(G

eV-1)

2

3

5

4

free

T/Tc=1.53

T/Tc=0.93

t (GeV-1)

0.5fm

),(ˆIm1

),( R pDpA

~

Imaginary-time (Matsubara) correlation

xdeJxJpD xpi 3)0,0(),(T),0(

dpAe

eT

),(10

/

dpAK ),(),(

0

Dynamic correlation & The spectral function (SPF) Dynamic correlation & The spectral function (SPF)

)0,0(),(R),(R JxtJixtD

Real-time (Retarded) correlation

dpAK

xdeJxJpD xpi

),(),(

)0,0(),(T),( 3

Maximum Entropy Method

LatticeQCD data

“Laplace” kernel

Maximum Entropy Method (MEM)Maximum Entropy Method (MEM)

P[A|D] ~ P[D|A] P[A]

T. Bayes C.E. Shannon (1702-1761) (1916-2001)

dmAAmAS

S

))/ln((

)exp(P[A]

Review + proofs : Asakawa, Nakahara & T.H., Prog. Part. Nucl. Phys. 46 (’01) 459

First application of MEM to LQCD: Asakawa, Nakahara & T.H, Phys. Rev. D60 (’99) 091503

Review + proofs : Asakawa, Nakahara & T.H., Prog. Part. Nucl. Phys. 46 (’01) 459 　

1. No parameterization necessary for A

2. Unique solution D A

3. Error estimate for A possible

Why MEM is so powerful ?Why MEM is so powerful ?

P[A|D] ~ P[D|A] P[A]

D = K×A

D A D A

Image reconstruction by MEM

MEM: mock dataMEM: mock data

Asakawa, Nakahara & T.H.,

PRD60 (‘99) 091503

’

’

Wilsondoubler

Wilsondoubler

MEM for mesons at T=0MEM for mesons at T=0

MEM

MEM

JP=1/2+

JP=1/2-

N N’

N* N*’

WD1 WD2

WD1 WD2

Sasaki, Sasaki and T. H., Phys. Lett. B623 (’05) 208MEM for baryons at T=0MEM for baryons at T=0

Sp

ectr

al f

un

ctio

n ρ

(ω)

J/ψ(3.1GeV)

1. J/ψ survives even up to 1.6 Tc

2. J/ψ disappears in 1.6 Tc < T < 1.7 Tc

see also,• Umeda et al, hep-lat/0401010 • Datta et al., PRD 69 (’04) 094507• Jakovac et al., hep-lat/0611017

Asakawa & T.H., PRL 92 (’04) 012001

MEM: charmonium above Tc (quenched) MEM: charmonium above Tc (quenched)

Sp

ectr

al f

un

ctio

n ρ

(ω)

J/ψ(3.1 GeV)

ηc(3.0GeV)

J/ψ and ηc above Tc (quenched)J/ψ and ηc above Tc (quenched)

Asakawa, Nakahara & Hatsuda, [hep-lat/0208059]

mud << ms~Tc << mc < mbA

(ω）

/ω2

mφ(T=0)=1.03 GeV at T/Tc= 1.4ss-channel

Light meson spectra in quenched QCDLight meson spectra in quenched QCD

1. Strong correlations

in JP=0+ (σ) and JP=0- (π) channels above Tc ? Kunihiro and T.H., Phys. Rev. Lett. 55 (’85) 88

2. Dynamical confinement in all color singlet channels above Tc ? DeTar, Phys. Rev. D32 (’85) 276

Possible mechanisms of

supporting “hadrons” above Tc

Possible mechanisms of

supporting “hadrons” above Tc

3. Strong Coulomb interaction in color singlet and non-singlet channels above Tc ?

Shuryak and Zahed, Phys. Rev. D70 (2004) 054507 Brown, Lee, Rho and Shuryak, Nucl. Phys.A740 (’04) 171

anisotropic lattice, 323 x (96-32)=4.0, at=0.01 fm, (Ls=1.25fm)Asakawa & Hatsuda, hep-lat/0308034

g

J/ c

Charmonium spectra in quenched QCDCharmonium spectra in quenched QCD

anisotropic lattice, 243 x (160-34)=4.0, at=0.056 fm, (Ls=1.34 fm)Jakovac, Petreczky, Petrov & Velytsky hep-lat/0611017

c

isotropic lattice, 483 x(24-12), a=0.04 fm (Ls=1.9 fm)

J/ c

Datta, Karsch, Petreczky & Wetzorke, hep-lat/0312034

g

Hatsuda, hep-ph/0509306 Net dissociation rate may even be smaller in full QCD

g,u,d

Charmonium spectra in full QCD (Nf=2)Charmonium spectra in full QCD (Nf=2)

Hamber-Wu, stout, ξ=6, at=0.025fm, 83 x (16,24,32), m/m=0.5Aarts et al., hep-lat/0610065, 0705.2198 [hep-lat]

Jc

Datta, Karsch, Wissel, Petreczky & Wetzorke, [hep-lat/0409147]

Aarts, Allton, Foley, Hands & Kim, [hep-lat/0610061]

g

J/ moving in the plasma in quenched QCD J/ moving in the plasma in quenched QCD

g

quenched, a = 0.02 fmDatta, Jakovac, Karsch & Petreczky, [hep-lat/0603002]

Bottomonium spectra in quenched QCDBottomonium spectra in quenched QCD

anisotropic lattice, 243 x (160-34)=4.0, at=0.056 fm, (Ls=1.34 fm)Jakovac, Petreczky, Petrov & Velytsky hep-lat/0611017

Hot QCD -- a “paradigm” -- Hot QCD -- a “paradigm” -- viscou

s fluid

p

erfect fluid

visco

us flu

id Pion gas

Resonance gas

q + g +”extra”plasma ?

weakly int.　 q + g plasma

q + g plasma

0

f

Tc

2Tc

10 Tc3

T

T*~T ΛQCD

pz

px

py

Chen, Stajic, Tan & Levin,

Phys. Rep. (’05)

High Tc superconductor

SummarySummary

2. Progress in bulk thermodynamics Equation of state, Pseudo-critical temperature, Susceptibilities precision science

3. Progress in spectral analysis elementary excitations in QGP still exploratory

1. Progress in lattice QCD Improved action, Faster algorithm, Faster computer simulations of the REAL world

RHIC LATTICE

AdS/CFT HTS/BEC

4. Progress in finite density no conclusion yet

Back up slides

Inter-particledistance

Electric screening

Magneticscreening

1/T

1/gT

1/g2T

QGP for g << 1 ( T >> 100 GeV )QGP for g << 1 ( T >> 100 GeV )

Relativistic plasma :

“Coulomb” coupling parameter :

Debye number :

S. Ichimaru, Rev. Mod. Phys. 54 (’82) 1071

QCD is non-perturbative even at T = ∞

Non-Abelian magnetic problem Non-Abelian magnetic problem

　 μ 　 ν

EOS : A. Linde, Phys. Lett. B96 (’80) 289

“Debye” screening :

Kraemmer & Rebhan, Rept.Prog.Phys.67 (’04)351

magnetic screening :

(m~ g2T)

soft magnetic gluons are always non-perturbative even if g 0 (T ∞)

pertubation theory from O(g6)