lattice qcd results on heavy quark potentials bound …€¦ · 0.2 0.4 0.6 0.8 1 1.2 ... -...

Lattice QCD results on heavy quark potentialsand

bound states in the quark gluon plasma

Olaf Kaczmarek Felix Zantow

Universität Bielefeld

April 5, 2006

Lattice QCD results on heavy quark potentials and bound states in the quark gluon plasma Exploration of Hadron Structureand Spectroscopy, Seattle, April, 2006 – p.1/27

”Machines” for QCD - Experiments

RHIC@BNL LHC@CERN

STARdetector

=⇒

only indirect signalsfrom collision region

⇓theoretical input needed


”Machines” for QCD - lattice theory

QCDOC apeNEXT

Installation in Bielefeld

1999/2001 144 Gflops APEmille

2005/2006 5 Tflops apeNEXT


Charmonium suppression

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 2 4 6 8 10L (fm)

Mea

sure

d / E

xpec

ted

J/ψ / DY4.2-7.0Pb-Pb

ψ , / DY4.2-7.0Pb-Pb

S-U

S-U

p-A

p-A

observed charmonium yields at SPS

nuclear absorption length L

using Bjorken formula (τ = 1 f m) + lattice EOS:

ε =dET/dη

τ AT

L [ f m] ε [GeV/ f m3] T/Tc

4 ∼ 1 ∼ 1.0

8 ∼ 3 ∼ 1.16

10 ∼ 4 ∼ 1.25

suppression patterns in a narrow temperature regime

detailed (lattice) equation of state needed

detailed knowledge on heavy quark bound states needed

What are the processes of charmonium formation/suppression?


Charmonium suppression

0.25

0.50

0.75

1.00

1 2 3 4 5

In−In, SPS

Au−Au, RHIC, |y| < 0.35

Pb−Pb, SPS

Au−Au, RHIC, |y|=[1.2,2.2]

ψS(J/ )

ε (GeV/fm )3

[F. Karsch,D. Kharzeev,H. Satz,hep-ph/0512239]

observed charmonium yields at SPS

nuclear absorption length L

using Bjorken formula (τ = 1 f m) + lattice EOS:

ε =dET/dη

τ AT

L [ f m] ε [GeV/ f m3] T/Tc

4 ∼ 1 ∼ 1.0

8 ∼ 3 ∼ 1.16

10 ∼ 4 ∼ 1.25

suppression patterns in a narrow temperature regime

detailed (lattice) equation of state needed

detailed knowledge on heavy quark bound states needed

What are the processes of charmonium formation/suppression?


Motivation - Heavy quark bound states above deconfinement

Quarkonium suppression as a probe for thermal properties of hot and dense matter [Matsui and Satz]

- heavy quark potential gets screened

- screening radius related to parton density

rD ∼ 1

g√

n/T

- at high T screening radius smaller than size of a quarkonium state

Typical length scales of heavy quark bound states: 1/ΛQCD ∼ 1 fm

- screening has to be strong enough to modify short distance behaviour

- detailed analysis of ”heavy quark potentials”

- temperature and r dependence

- screening properties above deconfinement

- What is the correct effective potential at finite temperature ?

F1(r,T) = U1(r,T)−T S1(r,T)


Heavy quark bound states above deconfinement

Strong interactions in the deconfined phase T>∼Tc

Possibility of heavy quark bound states?

Charmonium (χc,J/ψ) as thermometer above Tc

Suppression patterns of charmonium/bottomonium

=⇒ Potential models

→ heavy quark potential (T=0)

V1(r) = −43

α(r)r

+σ r

→ heavy quark free energies (T > Tc)

F1(r,T) ≃−43

α(r,T)

re−m(T)r

→ heavy quark internal energies (T 6=0)

F1(r,T) = U1(r,T)−T S1(r,T)

=⇒ Charmonium correlation functions/spectral functions


The lattice set-up

Polyakov loop correlation function and free energy:L. McLerran, B. Svetitsky (1981)

ZQQ̄

Z (T)≃ 1Z (T)

Z

D A... L(x) L†(y) exp

(

−Z 1/T

0dt

Z

d3xL [A, ...]

)

−FQQ̄(r ,T)

TQQ̄ = 1, 8, av

Lattice data used in our analysis:

Nf = 0:323×4,8,16-lattices( Symanzik )

O. Kaczmarek,F. Karsch,P. Petreczky,F.Z. (2002, 2004)

Nf = 2:163×4-lattices( Symanzik, p4-stagg. )

mπ/mρ ≃ 0.7 (m/T = 0.4)

O. Kaczmarek, F.Z. (2005),O. Kaczmarek et al. (2003)

Nf = 3:163×4-lattices

mπ/mρ ≃ 0.4

P. Petreczky,K. Petrov (2004)


The lattice set-up

Polyakov loop correlation function and free energy:L. McLerran, B. Svetitsky (1981)

ZQQ̄

Z (T)≃ 1Z (T)

Z

D A... L(x) L†(y) exp

(

−Z 1/T

0dt

Z

d3xL [A, ...]

)

−FQQ̄(r ,T)

TQQ̄ = 1, 8, av

O. Philipsen (2002)O. Jahn, O. Philipsen (2004)

− ln(

〈T̃rL(x)T̃rL†(y)〉)

=Fq̄q(r,T)

T

− ln(

〈T̃rL(x)L†(y)〉)

∣

∣

∣

∣

∣

GF

=F1(r,T)

T

− ln

(

98〈T̃rL(x)T̃rL†(y)〉− 1

8〈T̃rL(x)L†(y)〉

∣

∣

∣

∣

∣

GF

)

=F8(r,T)

T

1/T =Nτ

1/3σV =N

a

a

a


Heavy quark free energy

-500

0

500

1000

0 0.5 1 1.5 2 2.5 3

r [fm]

F1 [MeV]

0.76Tc0.81Tc0.90Tc0.96Tc1.00Tc1.02Tc1.07Tc1.23Tc1.50Tc1.98Tc4.01Tc

Renormalization of F(r,T)

bymatching with T=0 potential

e−F1(r,T)/T =(

Zr (g2))2Nτ 〈Tr (LxL

†y)〉



-500

0

500

1000

0 0.5 1 1.5 2 2.5 3

r [fm]

F1 [MeV]


-1

-0.5

0

0.5

1

1.5

0.2 0.4 0.6 0.8 1 1.2 1.4

r σ1/2

F(r,T)/σ1/2

T/Tc0.961.001.071.161.231.501.98

T-independentr ≪ 1/

√σ

F(r,T) ∼ g2(r)/r

Renormalization of F(r,T)

bymatching with T=0 potential

e−F1(r,T)/T =(

Zr (g2))2Nτ 〈Tr (LxL

†y)〉



-500

0

500

1000

0 0.5 1 1.5 2 2.5 3

r [fm]

F1 [MeV]


0

500

1000

1500

0 1 2 3 4T/Tc

F∞ [MeV] Nf=0Nf=2Nf=3


√σ

F(r,T) ∼ g2(r)/r

String breakingT < Tc

F(r√

σ ≫ 1,T) < ∞



-500

0

500

1000

0 0.5 1 1.5 2 2.5 3

r [fm]

F1 [MeV]


0

500

1000

1500

0 1 2 3 4T/Tc



√σ

F(r,T) ∼ g2(r)/r


F(r√

σ ≫ 1,T) < ∞

high-T physics

rT ≫ 1 ; screeningµ(T) ∼ g(T)T

F(∞,T) ∼−T



-500

0

500

1000

0 0.5 1 1.5 2 2.5 3

r [fm]

F1 [MeV]



√σ

F(r,T) ∼ g2(r)/r


F(r√

σ ≫ 1,T) < ∞

high-T physics

rT ≫ 1 ; screeningµ(T) ∼ g(T)T

F(∞,T) ∼−T

Complex r and T dependence

Small vs. large distance behaviorRunning coupling vs. screeningWhere do temperature effects set in?


Temperature depending running coupling

0

0.5

1

1.5

2

2.5

0.1 0.5 1

αqq(r,T)

r [fm]

T/Tc0.810.900.961.021.071.231.501.984.01

Free energy in perturbation theory:

F1(r,T) ≡V(r) ≃−43

α(r)r

for rΛQCD ≪ 1

F1(r,T) ≃−43

α(T)

re−mD(T)r for rT ≫ 1

QCD running coupling in the qq-scheme

αqq(r,T) =34

r2 dF1(r,T)

dr



0

0.5

1

1.5

2

2.5

0.1 0.5 1

αqq(r,T)

r [fm]

T/Tc0.810.900.961.021.071.231.501.984.01

non-perturbative confining part for r>∼0.4 fm

αqq(r) ≃ 3/4r2σ

present below and just above Tc

temperature effects set in at smaller r with increasing T

maximum due to screening


F1(r,T) ≡V(r) ≃−43

α(r)r

for rΛQCD ≪ 1

F1(r,T) ≃−43

α(T)



αqq(r,T) =34

r2 dF1(r,T)

dr



0

0.5

1

1.5

2

2.5

0.1 0.5 1

αqq(r,T)

r [fm]

T/Tc0.810.900.961.021.071.231.501.984.01

non-perturbative confining part for r>∼0.4 fm

αqq(r) ≃ 3/4r2σ

present below and just above Tc

temperature effects set in at smaller r with increasing T

maximum due to screening

=⇒ At which distance do T-effects set in ?

=⇒ definition of the screening radius/mass

=⇒ definition of the T-dependent coupling


F1(r,T) ≡V(r) ≃−43

α(r)r

for rΛQCD ≪ 1

F1(r,T) ≃−43

α(T)



αqq(r,T) =34

r2 dF1(r,T)

dr


Heavy quark bound states aboveTc?

0

0.5

1

1.5

0 1 2 3 4T/Tc

r [fm]

J/ψ

ψ’χc

rmedrD

bound states above deconfinement?

first estimate:

mean charge radii of charmonium statescompared to screening radius: V(rmed) ≡ F∞(T)

thermal modifications on ψ′ and χc already at Tc

J/ψ may survive above deconfinement

(a) (c)(b)

cloud2r r

〈rn〉 ≡ 1N

Z

d3r r n F1(r ,T)

High T: Using a screened Coulomb form〈r〉 ≃ 2/mD

T = 0: Using parameterization of V(r) at T = 0: 〈r〉T=0 ≃ 0.74 − 0.85 fm

0.01

0.1

1

0 5 10 15 20 25

rD [fm]

T/Tc

<r>rmed

rentropy2/mD


Heavy quark bound states aboveTc?

0

0.5

1

1.5

0 1 2 3 4T/Tc

r [fm]

J/ψ

ψ’χc

rmedrD

bound states above deconfinement?

first estimate:

mean charge radii of charmonium statescompared to screening radius: V(rmed) ≡ F∞(T)

thermal modifications on ψ′ and χc already at Tc

J/ψ may survive above deconfinement

Better estimates:

effective potentials in Schrödinger Equation

direct calculation using correlation functions

Better estimates:

Potential models, effective potential Ve f f(r,T)

Maximum entropy method → spectral function

But: Free energies vs. internal energies F(r,T) = U(r,T)−TS(r,T)


Free energy vs. Entropy at large separations

0

500

1000

1500

0 1 2 3 4T/Tc


Free energies not only determinedby potential energy

F∞ = U∞ −TS∞

Entropy contributions play a role at finite T

S∞ = − ∂F∞∂T


Free energy vs. Entropy at large separations

0

1000

2000

3000

4000

0 1 2 3 4

U∞ [MeV]

T/Tc

S∞ = −∂F∞

∂T

U∞ = −T2 ∂F∞/T∂T

0

1000

2000

3000

4000

0 1 2 3

TS∞ [MeV]

T/Tc

Nf=0Nf=2Nf=3

High temperatures:

F∞(T) ≃ −43

mD(T)α(T) ≃ −O (g3T)

TS∞(T) ≃ −43

mD(T)α(T)

U∞(T) ≃ −4mD(T)α(T)β(g)

g

≃ −O (g5T)

(a) (c)(b)

cloud2r r

The large distance behavior of the fi-nite temperature energies is related toscreeningrather than to the temperaturedependence of masses of correspondingheavy-light mesons!


r-dependence of internal energies

0

500

1000

0 0.5 1 1.5

TS1 [MeV]

r [fm]

(b)

(a) (c)(b)

cloud2r r

The large distance behavior of the fi-nite temperature energies is related toscreeningrather than to the temperaturedependence of masses of correspondingheavy-light mesons!

F1(r,T) = U1(r,T)−TS1(r,T)

S1(r,T) =∂F1(r,T)

∂T

U1(r,T) = −T2 ∂F1(r,T)/T∂T

Entropy contributions vanish in the limit r → 0

F1(r ≪ 1,T) = U1(r ≪ 1,T) ≡V1(r)

important at intermediate/large distances


r-dependence of internal energies

0

1000

2000

3000

4000

0 0.5 1 1.5 2 2.5 3

U1 [MeV]

r [fm]

0.88Tc0.93Tc0.98Tc

F1(r,T) = U1(r,T)−TS1(r,T)

S1(r,T) =∂F1(r,T)

∂T

U1(r,T) = −T2 ∂F1(r,T)/T∂T

Entropy contributions vanish in the limit r → 0

F1(r ≪ 1,T) = U1(r ≪ 1,T) ≡V1(r)

important at intermediate/large distances

0

500

1000

1500

0 0.5 1 1.5 2

U1 [MeV]

r [fm]

1.09Tc1.13Tc1.19Tc1.29Tc1.43Tc1.57Tc1.89Tc

=⇒ Implications on heavy quark bound states?

=⇒ What is the correct Ve f f(r,T)?


Heavy quark bound states

0

500

1000

0 0.5 1 1.5

Veff(r,T) [MeV]

r [fm]

V(∞)

∆EJ/ψ(T=0)

steeper slope of Ve f f(r,T) = U1(r,T)

=⇒ J/ψ stronger bound using Ve f f = U1(r,T)

=⇒ dissociation at higher temperatures compared to Ve f f(r,T) = F1(r,T)


Estimates on bound states from Schrödinger equation

Schrödinger equation for heavy quarks:

[

2mf +1

mf∆2 +Ve f f(r,T)

]

Φ fi = E f

i (T)Φ fi , f = charm,bottom

T-dependent color singlet heavy quark potential mimics in-medium modifications of qq̄ interaction

reduction to 2-particle interaction clearly too simple, in particular close to Tc

recent analysis:

using Ve f f = F1: S.Digal, P.Petreczky, H.Satz, Phys. Lett. B514 (2001)57using Ve f f = V1: C.-Y. Wong, hep-ph/0408020using Ve f f = V1: W.M. Alberico, A. Beraudo, A. De Pace, A. Molinari, hep-ph/0507084

state J/ψ χc ψ′ ϒ χb ϒ′ χ′b ϒ′′

Eis[GeV] 0.64 0.20 0.005 1.10 0.67 0.54 0.31 0.20

r0[ f m] 0.50 0.72 0.90 0.28 0.44 0.56 0.68 0.78

Td/Tc 1.1 0.74 0.1-0.2 2.31 1.13 1.1 0.83 0.74

Td/Tc ∼ 2.1 ∼ 1.2 ∼ 1.2 ∼ 5.0 ∼ 1.95 ∼ 1.65 - -

Td/Tc 1.75-1.95 1.13-1.15 1.10-1.11 4.4-4.7 1.5-1.6 1.4-1.5 ∼ 1.2 ∼ 1.2


Correlation functions of hadronic states

1

10

100

1000

0 0.2 0.4 0.6 0.8 1

τ T

G(τ,T)/T3

0.6 Tc

pseudo-scalarscalar

1

10

100

1000

0 0.2 0.4 0.6 0.8 1

τ T

G(τ,T)/T3

1.5 Tc

pseudo-scalarscalar

Direct lattice calculations on bound states

Thermal correlation functions:

GH(τ, r,T) = 〈JH(τ, r)J†H(0,0)〉 with JH = q̄(τ, r)ΓHq(τ, r)

Example using light quarks, scalar (δ) and pseudo-scalar (π):

splitting below Tc due to chiral and axial U(1) symmetry breaking

symmetry restauration at high temperatures


Spectral functions of charmonium states

1

2

3

4

0.1 0.2 0.3 0.4 0.5

G(τ)/Grecon(τ)

τ[fm]

0.75Tc

AX+1.0 SC

1.12Tc1.5Tc

0.8

1

1.2

1.4

0.1 0.2 0.3 0.4

G(τ)/Grecon(τ)

τ[fm]

0.9Tc

PS+0.2 VC

1.5Tc2.25Tc

3Tc

Where do thermal effects on charmonium states set in?

χc,1

χc,0

ηc

J/ψ

strong effects already near Tc no effect in ηc up to 1.5Tc

modifications on J/ψ only of order 10% at 1.5Tc

bound states above Tc possible?

=⇒ reconstruction of spectral functions



spectral representation of correlators =⇒ in-medium properties of hadrons

spectral representation of Euclidean correlation functions:

GβH(τ,~r) =

Z ∞

0dω

Z

d3~p(2π)3 σH(ω,~p,T) ei~p~r cosh(ω(τ−1/2T))

sinh(ω/2T)

calculate the most probablespectral function

Bayesian data analysis, e.g. Maximum Entropy Method (MEM)



0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

ω[GeV]

σ(ω)/ω2

0.9Tc1.5Tc

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

ω[GeV]

σ(ω)/ω2

2.25Tc3Tc

ω [GeV]

0 5 10 15 20 25 30

T = 0.78Tc

T = 1.38Tc

T = 1.62Tc

0

0.5

1.5

2

2.5

0

0.5

1

1.5

2

1

T = 1.87Tc

T = 2.33Tc

J/ψ spectral function

M. Asakawa, T. Hatsuda,Phys.Rev.Lett.92 (2004) 012001

S. Datta et al.,Phys.Rev.D69 (2004) 094507

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

ω[GeV]

σ(ω)/ω2

0.9Tc1.5Tc

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

ω[GeV]

σ(ω)/ω2

2.25Tc3Tc

ω [GeV]

0 5 10 15 20 25 30

T = 0.78Tc

T = 1.38Tc

T = 1.62Tc

0

0.5

1.5

2

2.5

0

0.5

1

1.5

2

1

T = 1.87Tc

T = 2.33Tc



0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

ω[GeV]

σ(ω)/ω2

0.9Tc1.5Tc

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

ω[GeV]

σ(ω)/ω2

2.25Tc3Tc

ω [GeV]

0 5 10 15 20 25 30

T = 0.78Tc

T = 1.38Tc

T = 1.62Tc

0

0.5

1.5

2

2.5

0

0.5

1

1.5

2

1

T = 1.87Tc

T = 2.33Tc

J/ψ spectral function

M. Asakawa, T. Hatsuda,Phys.Rev.Lett.92 (2004) 012001

S. Datta et al.,Phys.Rev.D69 (2004) 094507

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20 25 30

ω[GeV]

σ(ω)/ω2

0.9Tc1.5Tc

0

0.2

0.4

0.6

0.8

0 5 10 15 20 25 30

ω[GeV]

σ(ω)/ω2

2.25Tc3Tc

ω [GeV]

0 5 10 15 20 25 30

T = 0.78Tc

T = 1.38Tc

T = 1.62Tc

0

0.5

1.5

2

2.5

0

0.5

1

1.5

2

1

T = 1.87Tc

T = 2.33Tc

J/ψ dissociates for 1.6Tc<∼T<∼1.9Tc

rather abrupt disappearance of J/ψJ/ψ gradually disappears for T>∼1.5Tc

strength reduced by 25% at T = 2.25Tc


Heavy quark bound states above deconfinement - Diquarks [OK,K. Hübner, O. Vogt]

Coloured bound states speculated just above Tc (SQGP)

- Is the interaction strong enough to support diquarks?

- Potential models for coloured bound states

- Diquark free and internal energies

Colour antitriplet (anti-symmetric) or color sextet (symmetric) state:

3⊗3 = 3̄⊕6.

C3qq(R,T) =

32〈Tr L(0)Tr L(R)〉− 1

2〈Tr L(0)L(R)〉

C6qq(R,T) =

34〈Tr L(0)Tr L(R)〉+ 1

4〈Tr L(0)L(R)〉

F3,6qq (R,T) = −T lnC3,6

qq (R,T)

[E.V. Shuryak,I. Zahed (2004/05)]


Z(3) symmetry and volume scaling

0.010

0.100

0.80 1.00 1.20 1.40 1.60 1.80

<|Tr L|>

T/Tc

Nτ=4

Nσ16243248

- Polyakov loop and diquark correlation functions not Z(3) symmetric

- In the confined phase all non Z(3) symmetric observables vanish

- Z(3) rotation such that ReL > 0

- At β = 0 the Polyakov loop scales like

〈|L|〉 = 〈| 1V ∑

~x

Tr L~x|〉 ∼1√V

.

- We assume this scaling behaviour for L and C3,6qq (r,T) for T < Tc

C3qq(r,T) ∝

(

1Nσ

)3ν(r,T)

= V−ν(r,T).







〈|L|〉 = 〈| 1V ∑

~x

Tr L~x|〉 ∼1√V

.


C3qq(r,T) ∝

(

1Nσ

)3ν(r,T)

= V−ν(r,T).

3.0

4.0

5.0

6.0

7.0

8.0

9.0

8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0

-ln C3qq

-ln(1/N3σ)

RT0.360.490.620.901.031.30

-ln <|TrL|> 0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.90 0.92 0.94 0.96 0.98 1.00 1.02

ν

T/Tc

RT0.360.490.620.901.03

<|TrL|>







〈|L|〉 = 〈| 1V ∑

~x

Tr L~x|〉 ∼1√V

.


C3qq(r,T) ∝

(

1Nσ

)3ν(r,T)

= V−ν(r,T).

0.055

0.060

0.065

0.070

0.075

0.080

0.085

0.090

0.095

0.100

0.0e+00 5.0e-05 1.0e-04 1.5e-04 2.0e-04 2.5e-04 3.0e-04 3.5e-04

C3qq

1/V

RT0.360.490.620.901.031.302.00

c<|TrL|>

- Below Tc the diquark free energies diverge

- Above Tc the Z(3) symmetry is spontanously broken

- In full QCD the Z(3) symmetry is explicitly broken at all T

- All observables are finite in the thermodynamic limit


Diquark free energies -qqvs.qq̄ in the deconfined phase

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5 1.0 1.5 2.0 2.5 3.0

∆F1q—q(R,T)/T

RT

2∆F3qq(R,T)/T T/Tc

1.0131.0311.1491.6822.9955.000

normalized free energy

∆Fi(r,T) := Fi(r,T)−Fi(r → ∞,T)

is a renormalization invariant quantity.

perturbative relation between diquark and quark-antiquark free energies

∆F3qq(r,T) ≃ ∆

12

F1qq̄(r,T)

good approximation above Tc.


Diquark free energies -qqvs.qq̄ in the deconfined phase

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5 1.0 1.5 2.0 2.5 3.0

∆F1q—q(R,T)/T

RT

2∆F3qq(R,T)/T T/Tc

1.0131.0311.1491.6822.9955.000

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0.5 1 1.5 2 2.5

F1q—q(R,T)/σ1/2

Rσ1/2

F3qq(R,T)/σ1/2

T/Tc1.0051.0311.1491.6822.9955.000

normalized free energy

∆Fi(r,T) := Fi(r,T)−Fi(r → ∞,T)

is a renormalization invariant quantity.

perturbative relation between diquark and quark-antiquark free energies

∆F3qq(r,T) ≃ ∆

12

F1qq̄(r,T)

good approximation above Tc.

temperature dependence for all separations

=⇒ entropy contributions play a role for all r.

same asymptotic value for qq and qq̄

=⇒ quarks in both systems are screened

independently by the medium


3-quark free energies [ K. Hubner, OK, O. Vogt., hep-lat/0408031 ]

-8

-6

-4

-2

0

2

0 0.1 0.2 0.3 0.4 0.5 0.6

Rσ(1/2)

F(R,T)/σ(1/2)

6Tc

Tqqq=0average

singletoctet

decuplet

Different color channels: 3⊗3⊗3 = 10⊕8′⊕8⊕1

exp(

−F1qqq/T

)

=16

⟨

27Tr L1Tr L2Tr L3−9Tr L1Tr (L2L3)

−9Tr L2Tr (L1L3)−9Tr L3Tr (L1L2)

+3Tr (L1L2L3)+3Tr (L1L3L2)⟩

exp(

−F8qqq/T

)

=124

⟨

27Tr L1Tr L2Tr L3 +9Tr L1Tr (L2L3)

−9Tr L3Tr (L1L2)−3Tr (L1L3L2)⟩

exp(

−F8′qqq/T

)

=124

⟨


−9Tr L1Tr (L2L3)−3Tr (L1L2L3)⟩

exp(

−F10qqq/T

)

=160

⟨


+9Tr L2Tr (L1L3)+9Tr L3Tr (L1L2)

+3Tr (L1L2L3)+3Tr (L1L3L2)⟩



-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

F1qqq(R,T)/σ1/2

Rσ1/2

3(F3—

qq(R,T)-Fq(T))/σ1/2

T/Tc1.0051.0311.1491.6842.8586.000

-8

-6

-4

-2

0

2

0 0.1 0.2 0.3 0.4 0.5 0.6

Rσ(1/2)

F(R,T)/σ(1/2)

6Tc

Tqqq=0average

singletoctet

decuplet

- Renormalization of 3-quark Polyakov loop correlation functions

Fqqq(r,T) =(

ZR(g2))3Nτ 〈 f (Lx,Ly,Lz)〉

- Renormalization of n-point functions with the same ZR(g2)

- Comparison with qq free energies

Fqqq(r) ≃ ∑〈qq〉

Fqq(r,T)−3Fq(T) ≃ 32

Fqq̄(r,T) with Fq(T) = 1/2Fqq̄(r → ∞)

- Baryonic interaction dominated by two-particle interactions



-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

F1qqq(R,T)/σ1/2

Rσ1/2

3(F3—


T/Tc1.0051.0311.1491.6842.8586.000

-3

-2

-1

0

1

2

3

0.5 1 1.5 2 2.5 3 3.5 4 4.5

F1qqq(R,T)/σ1/2

Rσ1/2

3(F3—


T/Tc0.880.910.971.011.031.071.161.371.66

QuenchedFull QCD

- Renormalization of 3-quark Polyakov loop correlation functions

Fqqq(r,T) =(

ZR(g2))3Nτ 〈 f (Lx,Ly,Lz)〉

- Renormalization of n-point functions with the same ZR(g2)

- Comparison with qq free energies

Fqqq(r) ≃ ∑〈qq〉

Fqq(r,T)−3Fq(T) ≃ 32

Fqq̄(r,T) with Fq(T) = 1/2Fqq̄(r → ∞)

- Baryonic interaction dominated by two-particle interactions


qq and qqq entropy and internal energies

The r-dependence of static q̄q, qq and qqq interactions are related by Casimir factors

The relations for entropy and internal energy follow directly (Fq(T) = 1/2Fqq̄(r → ∞)):

F3qq(r,T) ≃ 1

2F1

q̄q(r,T)+Fq(T)

S3qq(r,T) ≃ 1

2S1

q̄q(r,T)+Sq(T)

U3qq(r,T) ≃ 1

2U1

q̄q(r,T)+Uq(T)

and for the baryonic system:

F1qqq(r,T) ≃ 3

2F1

q̄q(r,T)

S1qqq(r,T) ≃ 3

2S1

q̄q(r,T)

U1qqq(r,T) ≃ 3

2U1

q̄q(r,T)

Potential models based on this relations possible


SQGP - Strongly Coupled QGP ??? J. Liao, E.V. Shuryak, hep-ph/0508035

1 1.25 1.5 1.75 2 2.25 2.5T

�

T c

0�2�4�6Eb

�T c

� � � � � � � � � � � ��

� � � � � � �� meson bound states� diquark bound states baryon bound states 3 �gluon bound states

channel rep. charge factor no. of states

gg 1 9/4 9s

gg 8 9/8 9s∗16

qg+ q̄g 3 9/8 3c ∗6s∗2∗Nf

qg+ q̄g 6 3/8 6c ∗6s∗2∗Nf

q̄q 1 1 8s∗N2f

qq+ q̄q̄ 3 1/2 4s∗3c ∗2∗N2f

Bound states of quasi-particles above Tc?(here m∗ = 800 MeV)?

Is the interaction in the different channels strongenough to support bound states above Tc?


SQGP - Strongly Coupled QGP ??? J. Liao, E.V. Shuryak, hep-ph/0508035

1 1.25 1.5 1.75 2 2.25 2.5T

�

T c

0 2 4 6

Eb

�T c

� � � � � � � � � � � ��

� � � � � � �� meson bound states� diquark bound states� baryon bound states� 3 �gluon bound states

(a)(b)

(c)

(d)

T<Tc

T=Tc

Tc<T<1.5Tc

T > 3Tc

Polymer Chains are speculative

channel rep. charge factor no. of states

gg 1 9/4 9s

gg 8 9/8 9s∗16

qg+ q̄g 3 9/8 3c ∗6s∗2∗Nf

qg+ q̄g 6 3/8 6c ∗6s∗2∗Nf

q̄q 1 1 8s∗N2f

qq+ q̄q̄ 3 1/2 4s∗3c ∗2∗N2f

Bound states of quasi-particles above Tc?(here m∗ = 800 MeV)?

Is the interaction in the different channels strongenough to support bound states above Tc?

Binding energy Eb of the order of T already at Tc!

Coloured bound states and


What are the constituents of matter at high temperature? [S.Ejiri,F.Karsch,K.Redlich]

0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

R4,2

T/T0

SB

q

HRG

qq

0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

R4,2

T/T0

SB

Q

HRG

qq

Recent discussion of quark number and charge fluctuations inthe QGP

“Specific ratios of cummulants of (net) quark number and charge can characterize the

particular degrees of freedom which are relevant in the low and high temperature phase”

dx2 ≡

1VT3

∂2 lnZ∂(µx/T)2

∣

∣

∣

∣

µu=µd=0

dx4 ≡

1VT3

∂4 lnZ∂(µx/T)4

∣

∣

∣

∣

µu=µd=0

quark number

Rq4,2 =

dq4

dq2

charge

RQ4,2 =

dQ4

dQ2


What are the constituents of matter at high temperature? [S.Ejiri,F.Karsch,K.Redlich]

0.6 0.8 1 1.2 1.4 1.6 1.8 20

5

10

15

R4,2

T/T0

SB

q

HRG

qq

0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.5

1

1.5

2

R4,2

T/T0

SB

Q

HRG

qq

Recent discussion of quark number and charge fluctuations inthe QGP

“Specific ratios of cummulants of (net) quark number and charge can characterize the

particular degrees of freedom which are relevant in the low and high temperature phase”

dx2 ≡

1VT3

∂2 lnZ∂(µx/T)2

∣

∣

∣

∣

µu=µd=0

dx4 ≡

1VT3

∂4 lnZ∂(µx/T)4

∣

∣

∣

∣

µu=µd=0

quark number

Rq4,2 =

dq4

dq2

charge

RQ4,2 =

dQ4

dQ2

Already for T > 1.5Tc quark number and charge are predominantly carried bystates with quantum numbers of quarks

See also V. Koch, A. Majumder, J. Randrup (2005)Lattice QCD results on heavy quark potentials and bound states in the quark gluon plasma Exploration of Hadron Structureand Spectroscopy, Seattle, April, 2006 – p.23/27

Renormalized Polyakov loop

0

500

1000

1500

0 1 2 3 4T/Tc


-500

0

500

1000

0 0.5 1 1.5 2 2.5 3

r [fm]

F1 [MeV]


Lren = exp

(

−F(r = ∞,T)

2T

)

Defined by long distance behaviour of F(r,T).

Renormalized by renormalization of F(r,T)

at small distances.

Lren =(

ZR(g2))Nt Llattice



0

0.5

1

0.5 1 1.5 2 2.5 3 3.5 4

Lren

T/Tc

2F-QCD: Nτ=4SU(3): Nτ=4

Nτ=8

0

500

1000

1500

0 1 2 3 4T/Tc


-500

0

500

1000

0 0.5 1 1.5 2 2.5 3

r [fm]

F1 [MeV]


Lren = exp

(

−F(r = ∞,T)

2T

)



0

0.5

1

0.5 1 1.5 2 2.5 3 3.5 4

Lren

T/Tc


Nτ=8

Lren = exp

(

−F(r = ∞,T)

2T

)

Quenched QCD:

Lren = 0 for T < Tc.

Finite gap at Tc

Full QCD:

Lren finite for all T.

Strong increase near Tc.



0

0.5

1

0.5 1 1.5 2 2.5 3 3.5 4

Lren

T/Tc


Nτ=8

Lren = exp

(

−F(r = ∞,T)

2T

)

Quenched QCD:

Lren = 0 for T < Tc.

Finite gap at Tc

Full QCD:

Lren finite for all T.

Strong increase near Tc.

Renormalized Polyakov loop- well defined in quenched and full QCD- non-zero for finite quark mass- strong increase near Tc



0

0.5

1

1.5

0 5 10 15 20 25

Lren

T/Tc

Nτ=4Nτ=8

Lren = exp

(

−F(r = ∞,T)

2T

)

perturbation theory:Lren(T) approaches

high temperature limitfrom above

Lren(T) ≃ 1+23

mD(T)

Tα(T)


Polyakov loops in higher representations [S. Gupta,K. Hübner,OK]

use character property of direct product rep: χp⊗q(g) = χp(g)χq(g)

Z(3) symmetry: LD → eitφLD

triality (Z(3) charge): t ≡ p−q mod 3

adjoint link variable[

UD=8]

i j := 12Tr

[

UD=3λi(UD=3)†λ j]

D (p,q) t C2(r) dD = CD/CF LD(x)

3 (1,0) 1 4/3 1 L3

6 (2,0) 2 10/3 5/2 L23−L∗

3

8 (1,1) 0 3 9/4 |L3|2−1

10 (3,0) 0 6 9/2 L3L6−L8

15 (2,1) 1 16/3 4 L∗3L6−L3

15′ (4,0) 1 28/3 7 L3L10−L15

24 (3,1) 2 25/3 25/4 L∗3L10−L6

27 (2,2) 0 8 6 |L6|2−L8−1



perturbation theory:

FD(r,T) = −CDαs(r)

rfor rΛQCD ≪ 1

renormalization of the Polykov loop:

〈LrenD 〉 =

(

ZD(g2))NτdD 〈Lbare

D 〉,

i.e. the renormalization constants are related by Casimir [G.Bali, Phys.Rev.D62 (2000) 114503]

-5

-4

-3

-2

-1

0

1

2

3

0.5 1 1.5 2 2.5

Fsing8 (r,T)/σ1/2

rσ1/2

T/Tcd8 V3(T=0)

1.0131.0311.1491.6823.000



perturbation theory:

FD(r,T) = −CDαs(r)

rfor rΛQCD ≪ 1

renormalization of the Polykov loop:

〈LrenD 〉 =

(

ZD(g2))NτdD 〈Lbare

D 〉,

i.e. the renormalization constants are related by Casimir [G.Bali, Phys.Rev.D62 (2000) 114503]

1.1

1.2

1.3

1.4

1.5

1.6

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5

ZD(g2)

g2

r3 r6 r8



Does Casimir scaling hold beyond leading order?

Singlet free energies of static quarks in representation D=3,8:

FsingD (r,T)

T= − ln

(

〈T̃rLD(x)L†D(y)〉

)

∣

∣

∣

∣

∣

GF

with LD made up of UD=3 and[

UD=8]

i j := 12Tr

[

UD=3λi(UD=3)†λ j]

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

∆FsingD (r,T)/C2(D)/T

D=8 D=3

rTc

T/Tc1.0311.1491.6823.0005.000



Does Casimir scaling hold beyond leading order? Yes !

Even the Polyakov loops are related by Casimir

〈Ld〉 ≃ 〈Lr 〉dD ,

with dD = C2(D)/C2(3).

This relation holds for bare and renormalized Polykov loops:

0

0.1

0.2

0.3

0.4

0.5

0.6

5 10 15 20 25

LDbare

T/Tc

D=3 D=6 D=8 D=10D=15



Does Casimir scaling hold beyond leading order? Yes !

Even the Polyakov loops are related by Casimir

〈Ld〉 ≃ 〈Lr 〉dD ,

with dD = C2(D)/C2(3).

This relation holds for bare and renormalized Polykov loops:

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 2 3 4 5 6 7 8 9 10

Lren

T/Tc

L3 L6 L8 L10L15


Conclusions

- Heavy quark free and internal energies

Complex r and T dependence

Running coupling shows remnants of confinenement above Tc

Entropy contributions play a role at finite T

Separation of entropy and internal energy contributions possible

Potential energy larger than free energy

- Charmonium in the quark gluon plasma

First estimates from potential models

Higher dissociation temperature using V1

(directly produced) J/ψ exist well above Tc


Outlook

- Heavy quark free and internal energies

Smaller quark masses

Smaller lattice spacing to analyze small r dependence

Better understanding of the different contributions

- Charmonium in the quark gluon plasma

What is the correct Ve f f in potential models

Full QCD calculations of correlation/spectral functions

Relevant processes for charmonium production/dissociation ?


lattice qcd results on heavy quark potentials bound …€¦ · 0.2 0.4 0.6 0.8 1 1.2 ... -...

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