Thermodynamics and heavy-quark free energy

at finite temperature in full-QCD lattice simulations with Wilson quarks

Thermodynamics and heavy-quark free energy

at finite temperature in full-QCD lattice simulations with Wilson quarks

Yu Maezawa (RIKEN)

in collaboration with

S. Aoki, K. Kanaya, H. Ohno, H. Saito (Univ. of Tsukuba)

S. Ejiri (Niigata univ.)

T. Hatsuda (Univ. of Tokyo)

T. Umeda (Hiroshima univ.)

WHOT-QCD Collaboration

NFQCD @ YITP, Mar. 16, 2010

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

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

2

ContentsContents 1. Introduction

i. Heavy-quark free energy and potential on lattice

ii. Fixed scale approach

2. Results of lattice QCD simulations

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

ii. Color-channel dependence and Debye screening effect in QGP

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

3. Summary

3

IntroductionStudy 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)

/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

r

44

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 )(

),(

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

T >Tc

T <Tc

55

)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

6

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

Kanaya san’s talk, yesterday

77

Results of lattice QCD simulations

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

2. Color-channel dependence and Debye screening effect in QGP

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

88

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.

99

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

10

),(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!

11

),(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.

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

discussion, later

12

),(1 TrF )(rV

[fm] r

Heavy-quark free energy at T > 0

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

1313

Results of lattice QCD simulations

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

2. Color-channel dependence and Debye screening effect in QGP

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

Heavy-quark potential for various color-channels

qq qqqqDecomposition of color channel

Projection of Polyakov-line correlators Nadkarni (1986)

Normalized free energy (heavy-quark potential)

Nf = 2 lattice QCD simulations (fixed Nt approach)

Various color-channels of interaction 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

MeV 186~,80.0,65.0/,416 pc33 TmmNN ts

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

extract eff(T) and mD(T) by fitting V(r,T)

),( TrV M

)x(† )y(

4A 4A

effg effg

at at

)()( yxtr †

mD

)x(† )y()x(† )y(

4A 4A

effg effg

at at

)()( yxtr †

mD

)x(† )y(

4A 4A

effg effg

at at

)()( yxtr †

mD

)x(† )y()x(† )y(

4A 4A

effg effg

at at

)()( yxtr †

mD

rTmM

Der

TCTrV )(eff )(

),(

3

1,

3

2,

6

1,

3

46*381 CCCC

M

a

a

aM ttC 2

8

1 1

Casimir factor

QQ potential QQ potential80.0/ mm

r [fm] r [fm]

disappears at T > 2Tpc

becomes large at T ~Tpc

~

Results of eff(T) and mD(T) for each color channel

Channel dependence of eff and mD

Universal descriptions byrTm

MDe

r

TCTrV )()(

),( eff

Single-gluon exchange ansatz dose not work well.

Debye screening effect in QGP65.0/ mm

Comparison with perturbative screening mass

Perturbative screening mass

)(ln)(),;(

)(SM2loop

2

6

6 2

12

1

1

2

311 go

m

mTgTg

T

Tm

mag

DN

NNLOD

f

f

LO NLO Rebhan, PRD 48 (1993) 48

Lattice screening mass is not reproduced by LO-type screening mass.

Contribution of NLO-type corrects the LO-type screening mass.

MeV 2613 fN

SMTT 3~

TTgmmag )(48.0~ 2

NLO

LO

pcTT

T

TmD )(

Results of mD(T) for color singlet channel

1818

Results of lattice QCD simulations

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

2. Color-channel dependence and Debye screening effect in QGP

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

19

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

20

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

21

QQ potential80.0/ mm

1c channel: attractive force

8c channel: repulsive force

r [fm]

becomes weak

at 0 < q /T << 1

r [fm]

QQ potential

3*c channel: attractive force

6c channel: repulsive force

80.0/ mm

r [fm]

1

1

r [fm]

becomes strong

at 0 < q /T << 1

2

2

r [fm]

23

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

• Channel dependence disappear

at T > 2Tpc

• mD is not reproduced by

LO thermal perturbation

24

Screened Coulomb form:

with the Debye mass,

rTmM

Der

TCTrV ),(),(

),,( qqeffq

)()()0,(),( 42

2 qqq

OTcTmTm TDD

650./ VPS mm

)(2 Tc Similar properties at q = 0

Debye screening mass at finite q

~

25

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

2. Color channel dependence and Debye screening effect in QGP

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

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 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

Taylor expansion in terms of q /T

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

at 0 < q /T << 1

rTmM

Der

TCTrV )(eff )(

),(

single-gluon exchange ansatz works well at T > 2Tpc :

mD is not reproduced by LO thermal perturbation

mD has higher order or non-perturbative contribution.

~