Ralf Rapp Cyclotron Institute + Physics Department

Texas A&M University College Station, USA

With: F. Riek (Texas A&M), H. van Hees (Giessen), V. Greco (Catania), M. Mannarelli (Barcelona)

Nonperturbative Heavy-Quark Interactions

in the QGP

1.) Introduction

• “Large” scale mQ >> QCD , T

• Low-energy/-momentum interactions:

- heavy-quark diffusion ↔ elastic scattering, Fokker-Planck

- quarkonia ↔ potential QCD

→ uniform framework

• s expansion inadequate

→ resummations, bound + scattering states

• theo. / pheno. constraints essential

(baselines prior to applications in heavy-ion collisions)

1.) Introduction

2.) T-Matrix Approach with Heavy Quarks

Potential Approach + Lippmann Schwinger Eq. Vacuum and pQCD Limits In-Medium Potentials Q-Q and Q-q Scattering in Medium

3.) Heavy-Quark Diffusion in QGP Fokker-Planck Equation Transport coefficients Electron Spectra at RHIC

4.) Conclusions

Outline

• 2-body potential VL in medium? Color-Magnetic Interaction?

Lippmann-Schwinger Equation

In-Medium Q-Q T-Matrix: -

2.) T-Matrix Approach with Heavy Quarks

)'q,k;E(T)k,E(G)k,q(Vdkk)'q,q(V)'q,q;E(T LQQLLL02

[Mannarelli+RR ’05, Cabrera+RR ‘06]

- Q-Q propagator: - bound + scattering states

• HQ potential concept well established in vacuum (EFT, lattice)

• 3-D reduction of Bethe-Salpeter Eq.

])(s/[)s(G QQkkQQ20 24 -

• Born approx. TQq = VQq recovers pQCD within ~20%

2.2 Color Magnetic Interaction and Constraints

Vacuum “Spectroscopy” Perturbative Q-q Scattering

• Color-Magnetic “Breit” Interaction VQ1Q2(r) → VQ1Q2(r) ( 1 – v1 · v2 )

[G.E. Brown ’52, Brown et al ‘04]

[van Hees et al ‘09][Riek et al ‘09]

• Q-Q and Q-q states ~ o.k.• spin-interactions O (1/mQ)

mc0 =1.4 GeV

- -

-

F1(r,T) = U1(r,T) – T S1(r,T)

• V1(r,T) ≡ X1(r,T) X1(r=∞,T) (X1

∞ / 2 : in-medium quark-mass?!)

(a) X1=F1 : relax << tint

(b) X1=U1 : relax >> tint

(c) Landau-Zener “mixing”

X1 = P U1 + (1-P) F1

P = exp[- 2 |H12|2 / vrel d/dr (F1-U1)]

|H12| ~ 1/relax

2.3 Lattice QCD Free Energy + In-Medium Potential

[Kaczmarek +Zantow ’05]

[Shuryak ‘08, Riek et al ‘09]

2.4 Charmonium T-Matrix in QGP

• ground state bound to ~ 2 Tc for V = U, VLZ

~ 1.2Tc for V = F

2.5 Heavy-Light Quark Scattering in QGP

• threshold S-wave resonances (meson+diquark) close to TC

QmDT

2

2

p

fD

p)pf(

tf

• Brownian

Motion:

thermalization rate diffusion coefficient

3.) Heavy-Quark Transport in the QGP

Fokker Planck Eq.[Svetitsky ’88,…]Q

k)p,k(wkdp Q3

23

21 k)p,k(wkdD Q

• Transition rate: wQ(p,k) ~ ∑ q,g ∫ fq,g(E;T) |TQq|2

• Heavy-quark selfenergy: Q

3.2 Charm-Quark Selfenergy + Drag

• charm quark widths c = -2 Imc ~ 250 MeV close to TC

• friction coefficients increase(!) with decreasing T→ TC!

Selfenergy Thermalization Rate

)kp(T)(fkd)p( a,LQqk

qa,LQ 3 k|)p,k(T|Fkdp 23

3.3 Comparison of Drag Coefficients

(Thermal Relaxation Rate)

• T-matrix rate ~ constant (melting resonances)• relax = 1/ ~ 7 fm/c

T [GeV]

[1

/fm

]

[Gubser ’06]

[Peshier ‘06; Gossiaux+Aichelin ’08]

[van Hees+RR ’04]

[van Hees,Mannarelli, Greco+RR ’07]

3.4 T-Matrix Approach vs. e± Spectra at RHIC

• max. interaction at ~Tc

↔ hadronic correlations ↔ quark coalescence

[van Hees,Mannarelli,Greco+RR ’07]

Spatial Diffusion Coeff.

4.) Summary and Conclusions

• In-Medium Q-q + Q-Q T-Matrix → heavy-quark diffusion and quarkonia in QGP on same footing

• Constraints essential: - lQCD based potential (F-U relaxation), Eucl. correlators - vacuum, pQCD

• “hadronic” correlations close to Tc ↔ quark coalescence

↔ max. coupling strength at ~Tc ↔ min. /s !?

• Radiative diffusion? Light-quark sector? Non-pert. gluons? …

• RHIC non-photonic e± Ds (2T) ≈ 5

- v2 - RAA correlation essential

- scrutinize medium evolution, Fokker-Planck, d-Au …

3.1 HQ Langevin Simulations: Hydro vs. Fireball Elastic pQCD + Hydrodynamics [Moore+Teaney ’05]

• b=6.5 fm Tc=165 MeV ≈ 9 fm/c

•Tc=180 MeV bulk-v2 ~5.5%QGP ≈ 5 fm/c

Resonance Model + Expanding Fireball

[van Hees,Greco +RR ’05]

Ds (2T) ≈ 6

v2max ~ 5-6%RAA~ 0.3

2.3 AdS/CFT-QCD Correspondence

[Gubser ‘07]

pdtdp 2

2 SYMc

CFT/ADS Tm

cCFT/ADS m

T)..(2

5012

• match energy density (d.o.f = 120 vs. ~40) and coupling constant (heavy-quark potential) to QCD

3-momentum independent

[Herzog et al, Gubser ‘06]

≈ (4-2 fm/c)-1 at T=180-250 MeV

Lat-QCD

TQCD ~ 250 MeV

3.) Phenomenology at RHIC• Medium evolution - hydrodynamics or parameterizations thereof

- realistic bulk-v2 (~5-6%)

- stop evolution after QGP; hadronic phase?

• Hadronization - fragmentation: c → D + X

- coalescence: c + q → D, adds momentum and v2

- chemistry (e.g. c enhancement)

• Semileptonic electron decays - approx. conserve v2 and RAA of parent meson

- charm/bottom composition in p-p

[Hirano et al ’06]

[Martinez et al, Sorensen et al ‘07]

[Greco et al, Dong et al ‘04]

3.3 Heavy-Quark Spectra at RHIC

• T-matrix approach ≈ effective resonance model • other mechanisms: radiative (2↔3), …

• relativistic Langevin simulation in thermal fireball background

pT [GeV]

Nuclear Modification Factor Elliptic Flow

pT [GeV]

[Wiedemann et al.’05,Wicks et al.’06, Vitev et al.’06, Ko et al.’06]

4.) Maximal “Interaction Strength” in the sQGP• potential-based description ↔ strongest interactions close to Tc

- consistent with minimum in /s at ~Tc

- strong hadronic correlations at Tc ↔ quark coalescence

• semi-quantitative estimate for diffusion constant:

[Lacey et al. ’06]

weak coupl. s ≈n <p> tr=1/5 T Ds

strong coupl.s≈ Ds= 1/2 T Ds

s≈ close toTc

3.2.2 The first 5 fm/c for Charm-Quark v2 + RAA Inclusive v2

• RAA built up earlier than v2

Time Evolution

2.2.2 “Lattice QCD-based” Potentials• accurate lattice “data” for free energy: F1(r,T) = U1(r,T) – T S1(r,T)• V1(r,T) ≡ U1(r,T) U1(r=∞,T)

[Cabrera+RR ’06; Petreczky+Petrov’04]

[Wong ’05; Kaczmarek et al ‘03]

• (much) smaller binding for V1=F1 , V1 = (1-U1 + F1

2.4 Single-e± at RHIC: Effect of Resonances• hadronize output from Langevin HQs (-fct. fragmentation, coalescence)• semileptonic decays: D, B → e++X

• large suppression from resonances, elliptic flow underpredicted (?)• bottom sets in at pT~2.5GeV

Fragmentation only

• less suppression and more v2 • anti-correlation RAA ↔ v2 from coalescence (both up) • radiative E-loss at high pT?!

2.4.2 Single-e± at RHIC: Resonances + Q-q Coalescence

frag2

2333

)p(f)p(f|)q(|qd)(

pdg

pd

dNE ccqqDD

D fq from , K

Nuclear Modification Factor Elliptic Flow

[Greco et al ’03]

2.1.3 Thermal Relaxation of Heavy Quarks in QGP

• factor ~3 faster with resonance interactions!

Charm: pQCD vs. Resonances

pQCD

“D”

• ctherm ≈ QGP ≈ 3-5 fm/c

• bottom does not thermalize

Charm vs. Bottom

3.2 Model Predictions vs. PHENIX Data

Single-e± Spectra [PHENIX ’06]

• coalescence increases both RAA and v2

• pQCD radiative E-loss with upscaled transport coeff.

• Langevin with elastic pQCD + resonances + coalescence

• Langevin with upscaled pQCD elastic