Heavy-Quark Kineticsin the Quark-Gluon Plasma

Hendrik van Hees

Justus-Liebig Universität Gießen

October 28, 2010

Institut für

Theoretische Physik

JUSTUS-LIEBIG-

UNIVERSITÄT

GIESSEN

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 1 / 18

Outline

1 Heavy-quark interactions in the sQGPHeavy quarks in heavy-ion collisionsHeavy-quark diffusion: The Langevin EquationElastic pQCD heavy-quark scatteringNon-perturbative interactions: Resonance Scattering

2 Non-photonic electrons at RHIC

3 Microscopic model for non-perturbative HQ interactionsStatic heavy-quark potentials from lattice QCDT-matrix approach

4 Summary and Outlook

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 2 / 18

Heavy Quarks in Heavy-Ion collisions

q

K

e±

νe

c,b quark

cg

sQGP

c

q̄

hard production of HQsdescribed by PDF’s + pQCD (PYTHIA)

Hadronization to D,B mesons viaquark coalescence + fragmentationV. Greco, C. M. Ko, R. Rapp, PLB 595, 202 (2004)

HQ rescattering in QGP: Langevin simulationdrag and diffusion coefficients frommicroscopic model for HQ interactions in the sQGP

semileptonic decay ⇒“non-photonic” electron observablesRe

+e−AA (pT ), v

e+e−2 (pT )

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 3 / 18

Relativistic Langevin process

Langevin process: friction force + Gaussian random forcein the (local) rest frame of the heat bath

d~x =~p

Epdt,

d~p = −A~pdt+√

2dt[√B0P⊥ +

√B1P‖]~w

~w: normal-distributed random variableA: friction (drag) coefficientB0,1: diffusion coefficientsdependent on realization of stochastic processto guarantee correct equilibrium limit: Use Hänggi-Klimontovichcalculus, i.e., use B0/1(t, ~p+ d~p)Einstein dissipation-fluctuation relation B0 = B1 = EpTA.to implement flow of the medium

use Lorentz boost to change into local “heat-bath frame”use update rule in heat-bath frameboost back into “lab frame”

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 4 / 18

Elastic pQCD processes

Lowest-order matrix elements [Combridge 79]

Debye-screening mass for t-channel gluon exch. µg = gT , αs = 0.4not sufficient to understand RHIC data on “non-photonic” electrons[Moore, Teaney 2005]Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 5 / 18

Non-perturbative interactions: Resonance Scattering

General idea: Survival of D- and B-meson like resonances above Tcelastic heavy-light-(anti-)quark scattering

q̄

c q̄

c

D,D′, Ds

s

cq

qc

u D,D′, Ds

D- and B-meson like resonances in sQGP

c

q

D,D′, Ds D,D′, Ds

k k

parametersmD = 2 GeV, ΓD = 0.4 . . . 0.75 GeVmB = 5 GeV, ΓB = 0.4 . . . 0.75 GeV

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 6 / 18

Cross sections

1.5 2 2.5 3 3.5 4√s [GeV]

0

2

4

6

8

10

12

σ [

mb

]

Res. s-channelRes. u-channelpQCD qc scatt.

pQCD gc scatt.

total pQCD and resonance cross sections: comparable in size

BUT pQCD forward peaked ↔ resonance isotropicresonance scattering more effective for friction and diffusion

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 7 / 18

Time evolution of the fire ball

Elliptic fire-ball parameterizationfitted to hydrodynamical flow pattern [Kolb ’00]

V (t) = π(z0 + vzt)a(t)b(t), a, b: semi-axes of ellipse,

va,b = v∞[1− exp(−αt)]∓∆v[1− exp(−βt)]

Isentropic expansion: S = const (fixed from Nch)

QGP Equation of state:

s =S

V (t)=

4π2

90T 3(16 + 10.5n∗f ), n

∗f = 2.5

obtain T (t) ⇒ A(t, p), B0(t, p) and B1 = TEAfor semicentral collisions (b = 7 fm): T0 = 340 MeV,QGP lifetime ' 5 fm/c.simulate FP equation as relativistic Langevin process

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 8 / 18

Initial conditions

need initial pT -spectra of charm and bottom quarks

(modified) PYTHIA to describe exp. D meson spectra, assumingδ-function fragmentationexp. non-photonic single-e± spectra: Fix bottom/charm ratio

0 1 2 3 4 5 6 7 8p

T[GeV]

10-7

10-6

10-5

10-4

10-3

10-2

1/(

2 π

pT)

d2N

/dp

T d

y [a

.u.]

STAR D0

STAR prelim. D* (×2.5)

c-quark (mod. PYTHIA)

c-quark (CompHEP)

d+Au √sNN

=200 GeV

0 1 2 3 4 5 6 7 8p

T [GeV]

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

1/(

2π

pT)

dN

/dp

T [

a.u.]

STAR (prel, pp)

STAR (prel, d+Au/7.5)

STAR (pp)

D → e

B → esum

σbb

/σcc

= 4.9 x10-3

e±

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 9 / 18

Spectra and elliptic flow for heavy quarks

0 1 2 3 4 5p

T [GeV]

0

0.5

1

1.5

RA

A

c, reso (Γ=0.4-0.75 GeV) c, pQCD, α

s=0.4

b, reso (Γ=0.4-0.75 GeV)

Au-Au √s=200 GeV (b=7 fm)

0 1 2 3 4 5p

T [GeV]

0

5

10

15

20

v2 [

%]

c, reso (Γ=0.4-0.75 GeV) c, pQCD, α

s=0.4

b, reso (Γ=0.4-0.75 GeV)

Au-Au √s=200 GeV (b=7 fm)

µD = gT , αs = g2/(4π) = 0.4

resonances ⇒ c-quark thermalizationwithout upscaling of cross sections

Fireball parametrization consistent with hydro

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 10 / 18

Comparison to single-electron spectra @ RHIC

AA

R

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

= 200 GeVNNsAu+Au @

0−10% central(a)

Moore &

Teaney (III)T)π3/(2

T)π12/(2

van Hees et al. (II)

Armesto et al. (I)

[GeV/c]T

p0 1 2 3 4 5 6 7 8 9

HF

2v

0

0.05

0.1

0.15

0.2

PRLPHENIX Collaboration

98 172301 (2007)

(b)

minimum bias

> 4 GeV/cT

, pAA R0π

T, p2 v

0π

HF2

v±, eAA

R±e

> 2 GeV/c

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 11 / 18

Microscopic model: Static potentials from lattice QCD

V

Kaczmarek et al

lQCD

color-singlet free energy from latticeuse internal energy

U1(r, T ) = F1(r, T )− T∂F1(r, T )

∂T,

V1(r, T ) = U1(r, T )− U1(r →∞, T )Casimir scaling for other color channels [Nakamura et al 05; Döring et al 07]

V3̄ =1

2V1, V6 = −

1

4V1, V8 = −

1

8V1

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 12 / 18

T-matrix

Brueckner many-body approach for elastic Qq, Qq̄ scattering

T= + VTc

q, q̄V

= Σglu + TΣ

reduction scheme: 4D Bethe-Salpeter → 3D Lipmann-SchwingerS- and P waves

same scheme for light quarks (self consistent!)

Relation to invariant matrix elements∑|M(s)|2 ∝

∑q

da(|Ta,l=0(s)|2 + 3|Ta,l=1(s)|2 cos θcm

)Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 13 / 18

Microscopic justification for resonances: T-matrixcalculation

0

2

4

6

8

10

12

0 1 2 3 4

-Im

T (

GeV

-2)

Ecm (GeV)

s wave

color singlet

T=1.1 TcT=1.2 TcT=1.3 TcT=1.4 TcT=1.5 TcT=1.6 TcT=1.7 TcT=1.8 Tc

0

1

2

3

0 1 2 3 4

-Im

T (

GeV

-2)

Ecm (GeV)

s wave

color triplet

T=1.1 TcT=1.2 TcT=1.3 TcT=1.4 TcT=1.5 TcT=1.6 TcT=1.7 TcT=1.8 Tc

use static heavy-quark potentials from lQCD

resonance formation at lower temperatures T ' Tcmelting of resonances at higher T ! ⇒ sQGPmodel-independent assessment of elastic Qq, Qq̄ scattering

problems: uncertainties in extracting potential from lQCDin-medium potential V vs. F?

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 14 / 18

Transport coefficients

0

0.05

0.1

0.15

0 1 2 3 4 5

Α (

1/f

m)

p (GeV)

T-matrix: 1.1 TcT-matrix: 1.4 TcT-matrix: 1.8 Tc

pQCD: 1.1 TcpQCD: 1.4 TcpQCD: 1.8 Tc

0

10

20

30

40

0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34

2π

T D

s

T (GeV)

pQCD+T-matpQCD

from non-pert. interactions reach Anon−pert ' 1/(7 fm/c) ' 4ApQCDA decreases with higher temperature

higher density (over)compensated by melting of resonances!

spatial diffusion coefficient

Ds =T

mA

increases with temperatureHendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 15 / 18

Non-photonic electrons at RHICsame model for bottomquark coalescence+fragmentation → D/B → e+X

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1 2 3 4 5

RA

A

pT (GeV)

Au-Au √s=200 GeV (central)

pQCD, αs=0.4T-matrix

0

5

10

15

0 1 2 3 4 5

v2 (

%)

pT (GeV)

Au-Au √s=200 GeV

b=7 fm

pQCD, αs=0.4T-matrix 0

0.5

1

1.5

RA

A

theory [Wo]

frag. only [Wo]

theory [SZ]

PHENIX

STAR

0 1 2 3 4 5p

T [GeV]

0

0.05

0.1

0.15

v2

PHENIXPHENIX QM08

0-10% central

minimum bias

Au+Au √s=200 AGeV

coalescence crucial for description of dataincreases both, RAA and v2 ⇔ “momentum kick” from light quarks!“resonance formation” towards Tc ⇒ coalescence natural [Ravagli, Rapp 07]

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 16 / 18

Transport properties of the sQGP

spatial diffusion coefficient: Fokker-Planck ⇒ Ds = TmA = T2

Dmeasure for coupling strength in plasma: η/s

η

s' 1

2TDs (AdS/CFT),

η

s' 1

5TDs (wQGP)

-0.5

0

0.5

1

1.5

2

0.2 0.25 0.3 0.35 0.4

η/s

T (GeV)

charm quarks

T-mat + pQCDreso + pQCDpQCDpQCD run. αsKSS bound

[Lacey, Taranenko (2006)]

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 17 / 18

Summary and Outlook

Summary

Heavy quarks in the sQGPnon-perturbative interactions

mechanism for strong coupling: resonance formation at T & TclQCD potentials parameter freeres. melt at higher temperatures ⇔ consistency betw. RAA and v2!

also provides “natural” mechanism for quark coalescenceresonance-recombination model [L. Ravagli, HvH, R. Rapp, Phys. Rev. C 79, 064902 (2009)]problems

potential approach at finite T : F , V or combination?

Outlook

use more realistic bulk-medium description (real hydro)include inelastic heavy-quark processes (gluo-radiative processes)take into account D/B-meson rescattering in the hadronic phaseother heavy-quark observables like charmoniumsuppression/regeneration

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics October 28, 2010 18 / 18

Heavy-quark interactions in the sQGPHeavy quarks in heavy-ion collisionsHeavy-quark diffusion: The Langevin EquationElastic pQCD heavy-quark scatteringNon-perturbative interactions: Resonance Scattering

Non-photonic electrons at RHICMicroscopic model for non-perturbative HQ interactionsStatic heavy-quark potentials from lattice QCDT-matrix approach

Summary and Outlook