格子 QCD シミュレーションによる QGP 媒質中のクォーク間ポテンシャルの研究

格子 QCD シミュレーションによるQGP 媒質中のクォーク間ポテンシャルの研究

格子 QCD シミュレーションによるQGP 媒質中のクォーク間ポテンシャルの研究

前沢祐前沢祐 (( 理研理研 ))

in collaboration within collaboration with

青木慎也青木慎也 , , 金谷和至金谷和至 , , 大野浩史大野浩史 (( 筑波大物理筑波大物理 ))浮田尚哉浮田尚哉 (( 筑波大計算セ筑波大計算セ ))

初田哲男初田哲男 ,, 石井理修石井理修 (( 東京大学東京大学 ))江尻信司江尻信司 (BNL)(BNL)

梅田貴士梅田貴士 (( 広島大学広島大学 ))

WHOT-QCD CollaborationWHOT-QCD Collaboration

原子核・ハドロン物理 @ KEK, 2009 年 8 月 11—13 日

WHOT-QCD Coll. arXive:0907.4203WHOT-QCD Coll. PoS LAT2007 (2007) 207

WHOT-QCD Coll. Phys. Rev. D75 (2007) 074501

2

IntroductionIntroductionStudy of Quark-Gluon Plasma (QGP)

• Early universe after Big Bang• Relativistic heavy-ion collision

Theoretical study based on first principle (QCD) Lattice QCD simulation at finite (T, q)

Bulk properties of QGP (p, , Tc,…) Internal properties of QGP

are well investigated. are still uncertain.

Big Bang

RHICT

q

QGP

nucleusCSC

s-QGP

Big Bang

RHICT

q

QGP

nucleusCSC

s-QGP

/T 4

T / Tpc

CP-PACS 2001 (Nf = 2, Tc ~ 170 MeV)

/T 4

T / Tpc

CP-PACS 2001 (Nf = 2, Tc ~ 170 MeV)

qq qqqq

Properties of quarks and gluons in QGP Heavy-quark potential: free energy btw. static charged quarks in QGP

1. Relation of inter-quark interaction between zero and finite T2. Temperature dependence of various color-channels3. Heavy-quark potential at finite density (q) in Taylor expansion method

33

Heavy-quark potential at finite THeavy-quark potential at finite THeavy-quark potential interaction btw. static quark (Q) and antiquark (Q) in QCD matter

T = 0,

T > 0, string tension () decreases,

string breaking at rc

At T > Tc, screening effect in QGP

rr

rV )(

rTmDer

TTrV )(eff )(

),(

CTT CTT

CTT CTT

Lattice QCD simulations

Properties of heavy-quark potential in quark-gluon plasma

• Heavy-quark bound state (J/, Υ) in QGP

• Screening effect in QGP

• Inter-quark interaction btw. QQ and QQ

44

)y()x(Trln),(1 †TTrF

Heavy-quark potential Brown and Weisberger (1979)

Wilson-loop operator

Heavy-quark free energyHeavy-quark free energy

r

r

WrV

);y,x(lnlim)( 1

Heavy-quark free energy Nadkarni (1986)x

r

);y,x( W

Correlation b/w Polyakov-lines projected to a color singlet channel in the Coulomb gauge

Heavy-quark free energy may behave in QGP as:

Polyakov-line : static quark at x

tN

U1

4 ),x()x(

T = 0

T > 0

To characterize an interaction b/w static quarks,

r0

)x(† )y(T/1

x

Operator similar to the Wilson-loop

at finite T

short r: F 1(r,T ) ~ V(r) (no effect from thermal medium)

mid r : screened by plasma long r : two single-quark free energies w/o interactions

CTT CTT

CTT CTT

5

Fixed Nt approach

Fixed scale approach WHOT-QCD, PRD 79 (2009) 051501.

Approaches of lattice simulations at finite TApproaches of lattice simulations at finite TTo vary temperature on the lattice,

• Changing a (controlled by the gauge coupling ( = 6/g2)), fixing Nt

• Changing Nt , fixing a

Advantages: T can be varied arbitrarily, because is an input parameter.

Advantages: investigation of T dependence w/o changing spatial volume possible.

renormalization factor dose not change when T changes.

tNaT 1 a: lattice spacing

Nt: lattice size in E-time direction

tN

ax

T/1

T

T/1

Tx

66

Results of numerical simulations

1. Heavy-quark potential at T = 0 and T > 0

2. Heavy-quark potential for various color-channels

3. Heavy-quark potential at finite density (q)

77

Simulation details

Nf = 2+1 full QCD simulations

Lattice size: Ns3 x Nt = 323 x 12, 10, 8, 6, 4

Temperature: T ~ 200-700 MeV (5 points)

Lattice spacing: a = 0.07 fm

Light quark mass*: m/m = 0.6337(38)

Strange quark mass*: m/m = 0.7377(28)

Scale setting: Sommer scale, r0 = 0.5 fm

*CP-PACS & JLQCD Coll., PRD78 (2008) 011502.

88

Heavy-quark potential at T = 0

00)( Vr

rrV

Data calculated by

CP-PACS & JLQCD Coll.on a 283 x 58 lattice

Phenomenological potential

)(rV

[fm] r

GeV .

GeV .

.

232

4340

4410

0

0

V

Our fit results

= Coulomb term at short r + Linear term at long r + const. term

9

),(1 TrF )(rV

[fm] r

Heavy-quark free energy at T > 0

At short distance

F 1(r,T ) at any T

converges to

V(r) = F 1(r,T=0 ).

Short distance physics is

insensitive to T.

Note:

In the fixed Nt approach, this property is used to adjust the constant term of F 1(r,T ).

In our fixed scale approach, because the renormalization is common to all T. no further adjustment of the constant term is necessary.

We can confirm the expected insensitivity!

10

),(1 TrF )(rV

[fm] r


At short distance

F 1(r,T ) at any T

converges to

V(r) = F 1(r,T=0 ).

Short distance physics is

insensitive to T.

T ~ 200 MeV: F 1 ~ V up to 0.3--0.4 fm

T ~ 700 MeV: F 1 = V even at 0.1 fm

Range of thermal effect is T-dependent

Debye screening effect

WHOT-QCD Coll. arXive:0907.4203WHOT-QCD Coll. Phys. Rev. D75 (2007) 074501

11

),(1 TrF )(rV

[fm] r


Long distanceF

1(r,T ) becomes flat and has no increase linealy.

Confinement is destroyed due to a thermal medium effect.

Qr FTTTrF 2Trln)y()x(Trln),(

21 †

At large r, correlation b/w Polyakov-lines will disappear,

F 1 converges to 2x(single-quark free energy) at long distance

2FQ calculated from

Polyakov-line operator

1212




3. Heavy-quark potential at finite density q

Heavy-quark potential for various color-channels

qq qqqqSeparation to each color channel

Projection of Polyakov-line correlators Nadkarni (1986)

Normalized free energy

Interaction for various color-channels are induced in QGP

CCC*C 8133

C*CCC 6333

• QQ correlator:

• QQ correlator:

0),(2),(),( r

MQ

MM TrVFTrFTrV

6 ,3 ,8 ,1 C*CCCM

Heavy-quark potential for various color-channels

C1

C8

*C3

C6

Temperature increases becomes weak

1c, 3*c: attractive force, 8c, 6c: repulsive force

Single gluon exchange ansatz

Consistent with Casimir scaling e.g.)

),( TrV M

)x(† )y(

4A 4A

effg effg

at at

)()( yxtr †

mD

)x(† )y()x(† )y(

4A 4A

effg effg

at at

)()( yxtr †

mDrTmM

Der

TCTrV )(eff )(

),(

3

1,

3

2,

6

1,

3

46*381 CCCC

M

a

a

aM ttC 2

8

1 1

Casimir factor

*CC 31 2~ VV

QQ potential QQ potential

1515





16

Heavy-quark potential at finite qBig Bang

RHICT

q

QGP

nucleusCSC

s-QGP

Big Bang

RHICT

q

QGP

nucleusCSC

s-QGP

Motivation

Taylor expansion method in terms of q /T

Properties of QGP at 0 < q /T << 1

Expectation values at finite q

At q /T << 1, Taylor expansion of quark determinant detD(q)

where

• Early universe after Big Bang• Relativistic heavy-ion collisions

Low density

e.g. q /Tc ~ 0.1 at RHIC

c.f.) sign problem

detD becomes complex when q≠0

aq Re

17

Heavy-quark potential at finite q

Heavy-quark potential up to 2nd order

with expansion coefficients:

QQ potential (1c, 8c) QQ potential (3*c, 6c)

Odd term of QQ potential vanishes because of symmetry under q→ -q

18

QQ potentialMeV 186~,80.0/,416,2 pc

3 TmmN f

1c channel: attractive force

8c channel: repulsive forcebecomes weak

at 0 < q /T << 1

QQ potential

3*c channel: attractive force

6c channel: repulsive force

becomes strong

at 0 < q /T << 1

20

Heavy-quark potential at finite q

QQ potential (1c, 8c)

QQ potential (3*c, 6c)

Attractive channel in heavy-quark potential

Heavy-meson correlation (1c) weak

at 0 < q /T << 1

Diquark correlation (3*c) strong

21




SummaryProperties of quark-gluon plasma

Heavy-quark potential (F 1(r,T )) defined

by free energy btw. static charged quarks

At short r: F 1 at any T converges to heavy-quark potential at T = 0

Short distance physics is insensitive to T. At mid r : screening effect appears At long r: F

1 becomes flat and has no linear behavior.

Correlation b/w Polyakov-lines disappears and F 1 converges to 2FQ

),(1 TrF )(rV

[fm] r

rTmM

Der

TCTrV )(eff )(

),(

3

1,

3

2,

6

1,

3

46*381 CCCC

1c, 3*c: attractive force, 8c, 6c: repulsive force

Consistent with Casimir scaling based on single gluon exchange ansatz

Taylor expansion in terms of q /T

Heavy-meson correlation (1c) weak Diquark correlation (3*c) strong

at 0 < q /T << 1

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格子 QCD シミュレーションによる QGP 媒質中のクォーク間ポテンシャルの研究