T-Matrix Approach to Heavy-Quark Diffusion and Quarkoniawxie/HF_workshop/slide/H_vanHees.pdfsQGP c q hard production ofHQs described by PDF's + pQCD (PYTHIA) Hadronization toD,B mesonsvia

T-Matrix Approach toHeavy-Quark Diffusion and Quarkonia

Hendrik van Hees

Justus-Liebig Universität Gießen

January 4, 2011

Institut für

Theoretische Physik

JUSTUS-LIEBIG-

UNIVERSITÄT

GIESSEN

Hendrik van Hees (JLU Gießen) Heavy-Quark Kinetics January 4, 2011 1 / 22

Outline

1 Heavy-quark interactions in the sQGPHeavy quarks in heavy-ion collisionsHeavy-quark diffusion: The Langevin EquationElastic pQCD heavy-quark scattering

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

3 Non-photonic electrons at RHIC

4 T-matrix approach to Quarkonium-Bound-State Problem

5 Summary and Outlook


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 )


Relativistic Langevin process

Langevin process: friction force + Gaussian random force

in the (local) rest frame of the heat bath

d~x =~p

Epdt,

d~p = −A~pdt+√

2dt[√B0P⊥ +

√B1P‖]~w

~w: normal-distributed random variable

A: friction (drag) coefficient

B0,1: diffusion coefficients

dependent on realization of stochastic process


Local-Equilibrium Limit

to guarantee correct (local) equilibrium limit:

use Hänggi-Klimontovich calculus, i.e., use B0/1(t, ~p+ d~p)Einstein dissipation-fluctuation relation B0 = B1 = EpTA.

to implement flow of the mediumuse Lorentz boost to change into local “heat-bath frame”use update rule in heat-bath frameboost back into “lab frame”

Realizes Milekhin-freeze-out distribution for t→∞dN

d3xd3p=

g

(2π)3p · u(x)E

exp

[−p · u(x)

T (x)

].

E = p0 =√m2 + ~p2

u(x): four-velocity-flow field of the mediumT (x): temperature field of the mediumthermal fireball adjusted such as to lead to correct light-meson andbaryon observablesusing the same coalescence/fragmentation model for hadronizationbulk v2 sensitive to freeze-out description (see next talk by P. Gossiaux)


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 January 4, 2011 6 / 22

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


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 January 4, 2011 8 / 22

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?


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 January 4, 2011 10 / 22

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


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±


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


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]


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)]


T-matrix approach for quarkonium-bound-state problemT-matrix Brückner approach for heavy quarkonia as for HQ diffusionconsistency between HQ diffusion and Q̄Q suppression!

T= + VT

c

V

= Σglu +Σ

Q̄

T

q k q′

4D Bethe-Salpeter equation → 3D Lippmann-Schwinger equationrelativistic interaction → static heavy-quark potential (lQCD)Tα(E; q

′, q) =Vα(q′, q) +

2

π

∫ ∞0

dk k2Vα(q′, k)GQQ̄(E; k)Tα(E; k, q)

× {1− nF [ω1(~k)]− nF [ω2(k)]}q, q′, k relative 3-momentum of initial, final, intermediate Q̄Q state[F. Riek, R. Rapp, PRC 82, 035201 (2010)]


The potential

Q

Q̄ Q̄

Q

non-perturbative static gluon propagator

D00(~k) = 1/(~k2 + µ2D) +m

2G/(

~k2 + m̃2D)2

finite-T HQ color-singlet-free energy from Polyakov loops

exp[−F1(r, T )/T ] =〈

Tr[Ω(x)Ω†(y)]/Nc

〉= exp

[g2

2NcT 2〈A0,α(x)A0,α(y)−A20,α(x)

〉]+O(g6)

identify 〈A0,α(x)A0,α(y)〉 = D00(x− y)color-singlet free energy

F1(r, T ) = −4

3αs

{exp(−mDr)

r+

m2G2m̃D

[exp(−m̃Dr)− 1] +mD}

in vacuo mD, m̃D → 0F1(r) = −

4

3

αsr

+ σr, σ =2αsm

2G

3[F. Riek, R. Rapp, arXiv:1005.0769 [hep-ph]]


Heavy quarkonia

fit parameters, αs(T ), mD(T ), m̃D(T ), m̃G(T ) to lQCDcalculate internal energy U(r, T ) = F (r, T )− T ∂∂T F (r, T )solve Lippmann-Schwinger equation ⇒ adjust mQ to get s-wavecharmonia/bottomonia masses in vacuumin the following

potential 1: Nf = 2 + 1 [O. Kaczmarek]potential 2: Nf = 3 [P. Petreczky]BbS: Blancenblecler-Sugar reduction schemeTh: Thompson reduction scheme

vacuum-mass splittingsuncertainty for charmonia 50-100 MeVuncertainty for bottomonia 30-70 MeVoverall uncertainty ' 10%

melting temperatures with U and F

s-wave (ηc, J/ψ): 2-2.5Tc, & 1.3Tc,Υ: > 2Tc, & 1.7Tc, 1Tc, & 2Tc, 1Tc, 1Tcp-wave (χc): & 1.2Tc, & 1Tc, χb: & 1.7Tc, 1.2Tc, all & 1Tc

[F. Riek, R. Rapp, arXiv:1005.0769 [hep-ph]]


Quarkonium-spectral functions in the vacuum


In-medium charmonium-spectral functions (s states)


In-medium charmonium-spectral functions (p states)


Summary and Outlook

SummaryHeavy 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!same model also used for quarkonia in medium

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?

Outlookuse more realistic bulk-medium description (→ following talks by P.Gossiaux and M. He)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 (→ talk by X. Zhao on Thursday)


Heavy-quark interactions in the sQGPHeavy quarks in heavy-ion collisionsHeavy-quark diffusion: The Langevin EquationElastic pQCD heavy-quark scattering

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

Non-photonic electrons at RHICT-matrix approach to Quarkonium-Bound-State ProblemSummary and Outlook

Documents

T-Matrix Approach to Heavy-Quark Diffusion and Quarkoniawxie/HF_workshop/slide/H_vanHees.pdfsQGP c q hard production ofHQs described by PDF's + pQCD (PYTHIA) Hadronization toD,B mesonsvia