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 m Q >>  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 V L in medium? Color-Magnetic Interaction? Lippmann-Schwinger Equation In-Medium Q-Q T-Matrix: - 2.) T-Matrix Approach with Heavy Quarks [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. -

Born approx. T Qq = V Qq recovers pQCD within ~20% 2.2 Color Magnetic Interaction and Constraints Vacuum “Spectroscopy” Perturbative Q-q Scattering Color-Magnetic “Breit” Interaction V Q1Q2 (r) → V Q1Q2 (r) ( 1 – v 1 · v 2 ) [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/m Q ) m c 0 =1.4 GeV - - -

F 1 (r,T) = U 1 (r,T) – T S 1 (r,T) V 1 (r,T) ≡ X 1 (r,T)  X 1 (r=∞,T) (X 1 ∞ / 2 : in-medium quark-mass?!) (a) X 1 =F 1 :  relax << t int (b) X 1 =U 1 :  relax >> t int (c) Landau-Zener “mixing” X 1 = P U 1 + (1-P) F 1 P = exp[- 2  |H 12 | 2 / v rel d/dr (F 1 -U 1 )] |H 12 | ~ 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 T c for V = U, V LZ ~ 1.2T c for V = F

2.5 Heavy-Light Quark Scattering in QGP threshold S-wave resonances (meson+diquark) close to T C

Brownian Motion: thermalization rate diffusion coefficient 3.) Heavy-Quark Transport in the QGP Fokker Planck Eq. [Svetitsky ’88,…] Q Transition rate: w Q (p,k) ~ ∑ q,g ∫ f q,g (E;T) |T Qq | 2 Heavy-quark selfenergy: Q

3.2 Charm-Quark Selfenergy + Drag charm quark widths  c = -2 Im  c ~ 250 MeV close to T C friction coefficients increase(!) with decreasing T→ T C ! Selfenergy Thermalization Rate

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 ~T c ↔ 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 T c ↔ quark coalescence ↔ max. coupling strength at ~T c ↔ min.  /s !? Radiative diffusion? Light-quark sector? Non-pert. gluons? … RHIC non-photonic e ±  D s (2  T) ≈ 5 - v 2 - R AA 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 T c =165 MeV  ≈ 9 fm/c T c =180 MeV bulk-v 2 ~5.5%  QGP ≈ 5 fm/c Resonance Model + Expanding Fireball [van Hees,Greco +RR ’05] D s (2  T) ≈ 6  v 2 max ~ 5-6% R AA ~ 0.3

2.3 AdS/CFT-QCD Correspondence [Gubser ‘07] 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= MeV Lat-QCD T QCD ~ 250 MeV

3.) Phenomenology at RHIC Medium evolution - hydrodynamics or parameterizations thereof - realistic bulk-v 2 (~5-6%) - stop evolution after QGP; hadronic phase? Hadronization - fragmentation: c → D + X - coalescence: c + q → D, adds momentum and v 2 - chemistry (e.g.  c enhancement) Semileptonic electron decays - approx. conserve v 2 and R AA 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 p T [GeV] Nuclear Modification Factor Elliptic Flow p T [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 T c - consistent with minimum in  /s at ~T c - strong hadronic correlations at T c ↔ quark coalescence semi-quantitative estimate for diffusion constant: [Lacey et al. ’06] weak coupl.  s ≈  n tr =1/5 T D s strong coupl.  s  ≈  D s  = 1/2 T D s   s  ≈  close to  T c

3.2.2 The first 5 fm/c for Charm-Quark v 2 + R AA Inclusive v 2 R AA built up earlier than v 2 Time Evolution

2.2.2 “Lattice QCD-based” Potentials accurate lattice “data” for free energy: F 1 (r,T) = U 1 (r,T) – T S 1 (r,T) V 1 (r,T) ≡ U 1 (r,T)  U 1 (r=∞,T) [Cabrera+RR ’06; Petreczky+Petrov’04] [Wong ’05; Kaczmarek et al ‘03] (much) smaller binding for V 1 =F 1, V 1 = (1-  U 1 +  F 1

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 p T ~2.5GeV Fragmentation only

less suppression and more v 2 anti-correlation R AA ↔ v 2 from coalescence (both up) radiative E-loss at high p T ?! Single-e ± at RHIC: Resonances + Q-q Coalescence f q 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”  c therm ≈  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 R AA and v 2 pQCD radiative E-loss with upscaled transport coeff. Langevin with elastic pQCD + resonances + coalescence Langevin with upscaled pQCD elastic