Heavy quark free energies and screening in lattice QCD · Heavy quark free energies and screening in lattice QCD Hard Probes 2008, Illa da Toxa, June 10, 2008 – p.1/15 Heavy quark

Heavy quark free energies and screening in latticeQCD

Olaf Kaczmarek

Universität Bielefeld

June 10, 2008

RBC-Bielefeld collaboration

O. Kaczmarek, PoS CPOD07 (2007) 043

RBC-Bielefeld, Phys.Rev.D77 (2008) 014511

O. Kaczmarek, F. Zantow, Phys.Rev.D71 (2005) 114510

O. Kaczmarek, F. Zantow, hep-lat/0506019

Heavy quark free energies and screening in lattice QCD Hard Probes 2008, Illa da Toxa, June 10, 2008 – p.1/15

Heavy quark bound states above deconfinement

Strong interactions in the deconfined phase T>∼Tc

Possibility of heavy quark bound states?

Suppression patterns of charmonium/bottomonium

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

=⇒ 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


Zero temperature potential -nf =2+1

-2

0

2

4

6

8

0 0.5 1 1.5 2 2.5 3

r/r0

r0 Vqq(r)-

3.3823.43 3.51 3.57 3.63 3.69 3.76 3.82 3.92

Vstring

1.0

1.1

1.2

1.3

1.4

1.5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0 0.05 0.1 0.15 0.2 0.25 0.3

a/r0

a [fm]

r0/r1r0σ1/2

1.30

1.35

1.40

1.45

1.50

1.55

1.60

3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1

β

Zren(β)

renormalization: VT=0(r) = − log(

(Zren(β))2 W(r,τ)W(r,τ+1)

)

cut-off effects are small

r2 dVq̄q(r)

dr

∣

∣

∣

∣

r=r0

= 1.65

r2 dVq̄q(r)

dr

∣

∣

∣

∣

r=r1

= 1.0

(r0 = 0.469(7) fm)

2+1 flavor QCD

highly improved p4-staggered

almost realistic quark masses

mπ ≃ 220 MeV, physical ms

Large distance behaviourconsistent with string model prediction:

V(r) = − π12r +σ r , for large r

−→ used for renormalization


Zero temperature potential -nf =2+1

-2

0

2

4

6

8

0 0.5 1 1.5 2 2.5 3

r/r0

r0 Vqq(r)-

3.3823.43 3.51 3.57 3.63 3.69 3.76 3.82 3.92

Vstring

2+1 flavor QCD

highly improved p4-staggered

almost realistic quark masses

mπ ≃ 220 MeV, physical ms

Large distance behaviourconsistent with string model prediction:

V(r) = − π12r +σ r , for large r

−→ used for renormalization

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

r0V(r)

r/r0

VstringCornell

Cornell+1/r2

Short distance behaviourdeviations at small r

enhancement of the running coupling

fit: V(r) = −0.392(6)/r +σ r

r-dependent running coupling α(r)


The lattice set-up

log() =⇒ −FQQ̄(r ,T)

TQQ̄ = 1, 8, av

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, ...]

)

Lattice data used in our analysis:

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

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

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

mπ/mρ ≃ 0.7 (m/T = 0.4)

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

Nf = 3:163×4-lattices( stagg., Asqtad )hybrid-R

mπ/mρ ≃ 0.4

P. Petreczky,K. Petrov (2004)

Nf = 2+1:244×6-lattices( Symanzik, p4fat3 )RHMC

mπ ≃ 220MeV, phys.ms

O. Kaczmarek (2007),RBC-Bielefeld (2008)


The lattice set-up

log() =⇒ −FQQ̄(r ,T)

TQQ̄ = 1, 8, av

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, ...]

)

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 - 2+1-flavors

-1000

-500

0

500

1000

0 0.5 1 1.5 2 2.5

F(r,T) [MeV]

r [fm]

T[MeV] 175186197200203215226240259281326365424459548

Renormalization of F(r,T)

using Zren(g2) obtained at T=0

e−F1(r,T)/T =(

Zr (g2))2Nτ 〈Tr (LxL

†y)〉

alternative renormalization proceduresall equivalent!(O. Kaczmarek et al.,PLB543(2002)41,

S. Gupta et al., Phys.Rev.D77 (2008) 034503)



-1000

-500

0

500

1000

0 0.5 1 1.5 2 2.5

F(r,T) [MeV]

r [fm]

T[MeV] 175186197200203215226240259281326365424459548

xxxxxxxxx

-1200

-1000

-800

-600

-400

-200

0

200

400

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

F(r,T) [MeV]

r [fm]

T-independentr ≪ 1/

√σ

F(r,T) ∼ g2(r)/r

Renormalization of F(r,T)

using Zren(g2) obtained at T=0

e−F1(r,T)/T =(


†y)〉

alternative renormalization proceduresall equivalent!(O. Kaczmarek et al.,PLB543(2002)41,

S. Gupta et al., Phys.Rev.D77 (2008) 034503)



-1000

-500

0

500

1000

0 0.5 1 1.5 2 2.5

F(r,T) [MeV]

r [fm]

T[MeV] 175186197200203215226240259281326365424459548

xxxxxxxxx

-400

-200

0

200

400

600

800

1000

1200

0 1 2 3 4 5

F∞ [MeV]

T/Tc

SU(3): Nτ=4SU(3): Nτ=8

2F-QCD: Nτ=42+1F-QCD: Nτ=42+1F-QCD: Nτ=6

T-independentr ≪ 1/

√σ

F(r,T) ∼ g2(r)/r

String breakingT < Tc

F(r√

σ ≫ 1,T) < ∞

high-T physics

rT ≫ 1 ; screeningµ(T) ∼ g(T)T

F(∞,T) ∼−T


Renormalized Polyakov loop

-400

-200

0

200

400

600

800

1000

1200

0 1 2 3 4 5

F∞ [MeV]

T/Tc

SU(3): Nτ=4SU(3): Nτ=8


-1000

-500

0

500

1000

0 0.5 1 1.5 2 2.5

F(r,T) [MeV]

r [fm]

T[MeV] 175186197200203215226240259281326365424459548

Lren = exp

(

−F(r = ∞,T)

2T

)

Using short distance behaviour of free energies

Renormalization of F(r,T) at short distances

e−F1(r,T)/T =(


†y)〉

Renormalization of the Polyakov loop

Lren =(

ZR(g2))Nt Llattice

Lren defined by long distance behaviour of F(r,T)



0.0

0.2

0.4

0.6

0.8

1.0

100 150 200 250 300 350 400 450

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

T [MeV]

Tr0 Lren

Nτ=468

-400

-200

0

200

400

600

800

1000

1200

0 1 2 3 4 5

F∞ [MeV]

T/Tc

SU(3): Nτ=4SU(3): Nτ=8


0

0.5

1

0.5 1 1.5 2 2.5 3 3.5 4

Lren

T/Tc

SU(3): Nτ=4SU(3): Nτ=8


Lren = exp

(

−F(r = ∞,T)

2T

)



0

0.5

1

1.5

0 5 10 15 20 25

Lren

T/Tc

Nτ=4Nτ=8

SU(3) pure gauge results

Lren = exp

(

−F(r = ∞,T)

2T

)

High temperature limit, Lren = 1,reached from above as expected from PT

Clearly non-perturbative effects below 5Tc


Temperature depending running coupling

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.70.60.50.40.30.20.1

αeff(r,T)

r[fm]

T[MeV] 156186197200203215226240259281326365424459548

non-perturbative confining part for r>∼0.4 fm

αqq(r) ≃ 3/4r2σ

present below and just above Tc

remnants of confinement at T>∼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

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


Temperature depending running coupling

0

0.2

0.4

0.6

0.8

1

1.2

1 1.5 2 2.5 3

αmax(T)

T/Tc

nf=0nf=2

nf=2+1 Nt=6nf=2+1 Nt=4

define α̃qq(T) by maximum of αqq(r,T):

α̃qq(T) ≡ αqq(rmax,T)

perturbative behaviour at high T:

g−22-loop(T) = 2β0 ln

(

µTΛMS

)

+β1

β0ln

(

2ln

(

µTΛMS

))

,

non-perturbative large values near Tc

not a large Coulombic coupling

remnants of confinement at T>∼Tc

string breaking and screening difficult to separate

slope at high T well described by perturbation theory

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

=⇒ calculation of the screening mass/radius


Screening mass - perturbative vs. non-perturbative effects

0

1

2

3

4

5

1 1.5 2 2.5 3 3.5 4

T/Tc

mD/T Nf=2Nf=0

Screening masses obtained from fits to:

F1(r,T)−F1(r = ∞,T) = −4α(T)

3re−mD(T)r

at large distances rT >∼ 1

leading order perturbation theory:

mD(T)

T= A

(

1+Nf

6

)1/2

g(T)



0

1

2

3

4

1 2 3 4T/Tc

Nf=0 Nf=2 Nf=2+1

ααmax

0

1

2

3

4

5

1 1.5 2 2.5 3 3.5 4

T/Tc

mD/T Nf=2+1

Nf=2Nf=0

ANf =2+1 = 1.66(2)

ANf =2 = 1.42(2)

ANf =0 = 1.52(2)


F1(r,T)−F1(r = ∞,T) = −4α(T)

3re−mD(T)r



mD(T)

T= A

(

1+Nf

6

)1/2

g(T)

perturbative limit reached very slowly



0

1

2

3

4

5

1 1.5 2 2.5 3 3.5 4

T/Tc

mD/T Nf=2+1

Nf=2Nf=0

0

1

2

3

0 5 10 15 20 25

T/Tc

mD/T

ANf =2+1 = 1.66(2)

ANf =2 = 1.42(2)

ANf =0 = 1.52(2)

ANf =0 = 1.39(2)


F1(r,T)−F1(r = ∞,T) = −4α(T)

3re−mD(T)r



mD(T)

T= A

(

1+Nf

6

)1/2

g(T)

perturbative limit reached very slowly

T depencence qualitativelydescribed by perturbation theory

But A≈ 1.4 - 1.5 =⇒ non-perturbative effects

A→ 1 in the (very) high temperature limit


Screening mass - density dependence (Nf = 2) [M.Doring et al., Eur.Phys.J. C46 (2006) 179]

0

0.5

1

1.5

2

2.5

3

3.5

4

1 1.5 2 2.5 3 3.5 4

m10/T

T/Tc 0

0.5

1

1.5

2

1 1.5 2 2.5 3 3.5 4

T/Tc

m12/T


mD(T,µq)

T= g(T)

√

1+Nf

6+

Nf

2π2

(µq

T

)2

Taylor expansion:

mD(T) = m0(T)+m2(T)(µq

T

)2+O (µ4

q)

m2(T) agrees with perturbation theory for T>∼1.5Tc

non-perturbative effects dominated by gluonic sector


Diquark free energies - Screening and string breaking[M.Doring et al., Phys.Rev.D75 (2007) 054504]

-1

-0.5

0

0.5

0 0.5 1 1.5 2

rT

NQQ T/Tc = 0.87 T/Tc = 0.90 T/Tc = 0.96

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 0.5 1 1.5 2

rT

NQQ T/Tc = 1.002T/Tc = 1.02 T/Tc = 1.07 T/Tc = 1.16 T/Tc = 1.50

Net quark number induced by a qq-pair:

N(c)QQ(r,T) =

⟨

Nq⟩

QQ =

⟨

NqL(c)QQ(r,T)

⟩

⟨

L(c)QQ(r,T)

⟩ ,

where Nq is the quark number operator,

Nq =12

Tr

[

D−1(m̂,0)

(

∂D(m̂,µ)

∂µ

)

µ=0

]

.

Net quark number induced by a single staticquark source,

NQ(T) =⟨

Nq⟩

Q =

⟨

NqTr P(~0)⟩

⟨

Tr P(~0)⟩ .



-1

-0.5

0

0.5

0 0.5 1 1.5 2

rT

NQQ T/Tc = 0.87 T/Tc = 0.90 T/Tc = 0.96

-2

-1.5

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2

T/Tc

NQQ(r0)

NQQ(∞)


N(c)QQ(r,T) =

⟨

Nq⟩

QQ =

⟨

NqL(c)QQ(r,T)

⟩

⟨

L(c)QQ(r,T)

⟩ ,


Nq =12

Tr

[

D−1(m̂,0)

(

∂D(m̂,µ)

∂µ

)

µ=0

]

.


NQ(T) =⟨

Nq⟩

Q =

⟨

NqTr P(~0)⟩

⟨

Tr P(~0)⟩ .

Diquark is neutralized by quarks or antiquarks

from the vacuum to be color neutral overall

limT→0

NQQ(r,T) =

1 , r < rc

−2 , r > rc

,



-1

-0.5

0

0.5

0 0.5 1 1.5 2

rT

NQQ T/Tc = 0.87 T/Tc = 0.90 T/Tc = 0.96

-2

-1.5

-1

-0.5

0

0.5

1

0 0.5 1 1.5 2

T/Tc

NQQ(r0)

NQQ(∞)


N(c)QQ(r,T) =

⟨

Nq⟩

QQ =

⟨

NqL(c)QQ(r,T)

⟩

⟨

L(c)QQ(r,T)

⟩ ,


Nq =12

Tr

[

D−1(m̂,0)

(

∂D(m̂,µ)

∂µ

)

µ=0

]

.


NQ(T) =⟨

Nq⟩

Q =

⟨

NqTr P(~0)⟩

⟨

Tr P(~0)⟩ .

Diquark is neutralized by quarks or antiquarks

from the vacuum to be color neutral overall

limT→0

NQQ(r,T) =

1 , r < rc

−2 , r > rc

,


Free energy vs. Entropy at large separations

-400

-200

0

200

400

600

800

1000

1200

0 1 2 3 4 5

F∞ [MeV]

T/Tc

SU(3): Nτ=4SU(3): Nτ=8


Free energies not only determinedby potential energy

F∞ = U∞ −TS∞

Entropy contributions play a role at finite T

S∞ = − ∂F∞∂T



-500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 0.5 1 1.5 2 2.5 3

T/Tc

U∞[MeV] F∞[MeV]

-500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 0.5 1 1.5 2 2.5 3

T/Tc

U∞[MeV] F∞[MeV]

S∞ = −∂F∞

∂T

U∞ = −T2 ∂F∞/T∂T

-500

0

500

1000

1500

2000

2500

3000

3500

4000

0 0.5 1 1.5 2 2.5 3

T/Tc

TS∞[MeV] F∞[MeV]

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 finitetemperature energiesis rather related toscreeningthan to t he temperature depen-dence of masses of corresponding heavy-light mesons!



-500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 0.5 1 1.5 2 2.5 3

T/Tc

U∞[MeV] F∞[MeV]

-500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0 0.5 1 1.5 2 2.5 3

T/Tc

U∞[MeV] F∞[MeV]

S∞ = −∂F∞

∂T

U∞ = −T2 ∂F∞/T∂T

0

1000

2000

3000

4000

0 1 2 3

TS∞ [MeV]

T/Tc

Nf=0 Nf=2 Nf=3 Nf=2+1

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 finitetemperature energiesis rather related toscreeningthan to t he temperature depen-dence of masses of corresponding heavy-light mesons!


r-dependence of internal energies (Nf = 2+1)

-1000

-500

0

500

1000

1500

2000

0 0.2 0.4 0.6 0.8 1 1.2

TS(r,T) [MeV]

r[fm]

T[MeV] 0

191

-100

0

100

200

300

400

500

600

700

800

900

0 0.2 0.4 0.6 0.8 1

TS(r,T) [MeV]

r[fm]

T[MeV] 0

233249345394

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 (Nf = 2+1)

-2000

-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

0 0.2 0.4 0.6 0.8 1 1.2

U(r,T) [MeV]

r[fm]

T[MeV] 0

166191199

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

-2000

-1500

-1000

-500

0

500

1000

1500

0 0.2 0.4 0.6 0.8 1

U(r,T) [MeV]

r[fm]

T[MeV] 0

233249345394

=⇒ Implications on heavy quark bound states?

=⇒ What is the correct Ve f f(r,T)?


Heavy quark bound states from potential models

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)


Conclusions

- Heavy quark free energies, internal energies and entropy

Results for almost physical quark masses, nf = 2+1

Complex r and T dependence

Running coupling shows remnants of confinenement above Tc

Entropy contributions play a role at finite T

Non-perturbative effects in mD up to high T

Non-perturbative effects dominated by gluonic sector

- Bound states in the quark gluon plasma

Estimates from potential models?

What is the correct potential for such models?

Higher dissociation temperature using V1

(directly produced) J/ψ may exist well above Tc

Full QCD calculations of correlation/spectral functions needed

What are relevant processes for charmonium?


Documents

Heavy quark free energies and screening in lattice QCD · Heavy quark free energies and screening in lattice QCD Hard Probes 2008, Illa da Toxa, June 10, 2008 – p.1/15 Heavy quark