Heavy Quark/onium

in Hot Nuclear Matter

Ralf Rapp Cyclotron Institute + Physics Department

Texas A&M University College Station, USA

INT Program (Week 7) on “Quantifying the Properties of Hot QCD Matter”

INT (Seattle), 06.-09.07.10

1.) Introduction: Virtues of Heavy Quarks (c,b)

• “Large” scale mQ >> QCD , T

- factorization in production; thermal medium: pth2 ~ 2mQ T >> T2

• Interactions spacelike (“low” pt):

- quarkonium: potential QCD - heavy-quark diffusion: Brownian motion

→ unified framework

• Beyond perturbation theory (s expansion)

→ resummations, bound + scattering states

• Constraints essential (latQCD, pQCD, vacuum spectrum,…)

• Heavy-ion collisions: - “initial-state” effects - medium effects: equilibrium properties, expansion collectivity

2220

2 kkkk

Q Q

1.2 Charm/onium Suppression at SPS + RHIC

• Same force operative for quarkonium (un)binding + heavy-quark transport?

Anomalous J/ Suppression Heavy-Quark Suppression+Flow

1.) Introduction

2.) T-Matrix for Heavy Quark/onium in QGP

Vacuum Spectroscopy, In-Medium Potentials Spectral + Correlation Functions

3.) Quarkonia in Heavy-Ion Collisions

Thermal Rate Equation Suppression vs. Regeneration

4.) Heavy-Quark Diffusion in QGP

Fokker-Planck + Thermalization Observables at RHIC

5.) Conclusions

Outline

• Lippmann-Schwinger equation

In-Medium Q-Q T-Matrix: -

2.) Heavy-Quark Potential + Thermal T-Matrix

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

[Mannarelli+RR ’05, Cabrera+RR ’06, Riek+RR ‘09]

- Q-Q propagator: - importance of threshold effects

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

• Quark-Gluon Plasma: bound+scattering states (quarkonia + HQ transport)

])(E/[)k,E(G QQkkQQ220 24 -

• 2-body potential VL at finite temperature?

2.2 Heavy-Quark Free Energy in Lattice QCD

F1(r,T) = U1(r,T) – T S1(r,T)

• Potential “Choices” : (a) Free energy F1 => weak potential, B(1.1Tc) ~ 50 MeV mQ(TF1(r=∞,T) small

(b) Internal Energy U1 ( U = ‹Hint› )

=> strong potential, B(1.1Tc) ~ 500 MeV mQ(T) ~ U1(r=∞,T) large

• approximate compensation in bound-state mass: E = 2mc

0 + 2mQ B

[Kaczmarek+Zantow ’05]• need improved ways to extract HQ potential

• Relativistic effects - kinematics - magnetic interaction → “Breit” correction: VQ1Q2(r) → VQ1Q2(r) ( 1 – v1 · v2 ) (↔ Poincaré-invariance, pQCD)

• Retardation effects - 4-D → 3-D reduction of Bethe-Salpeter equation - energy transfer fixed (q0=0), off-shell behavior ambiguous

• Gauge dependence of color-singlet free energy

• Field-theoretic ansatz: [Megias et al ‘07]

color-Coulomb: vector , string:

- fit color-average free energy to lat. QCD scalar

implement into “extended T-Matrix approach”

2.3 Corrections to Heavy-Quark Potential

[Riek+RR ‘10]

222

2

22001

)m~k(

m

mk)k(D

D

G

D

22

20 8

1G

c

NPa, m

g)N(

A

[Brown et al ‘52, ‘05]

[Philipsen ‘08]

In-Medium HQ Free Energies Model Parameters

2.3.2 Temperature Dependence of Fit Parameters

• s ~ 0.3

• screening of color-Coulomb + string term

• “Debye masses” ~ T

2.4.1 Constraints I: Vacuum Spectroscopy Quarkonia D-Mesons

• no hyperfine splitting• (bare) masses adjusted to ground state • ~ ±50 MeV accuracy

Born Approximation compared to Perturbative QCD

2.4.2 Constraints II: High-Energy Q-q Scattering

• Breit correction essential

2.5 Quarkonium Spectral Functions in Medium2.5.1 Lattice-QCD Correlators

]T/[)]T/([

)T,(d)T,(G2sinh

21cosh

0

• direct computation of Euclidean Correlation Fct.

spectral function

[Asakawa et al ’03, Iida et al ’06, Aarts et al ‘07, Jakovac et al ‘07]

[Datta et al ‘04]

• ~20% variation for S-wave charmonia ~ 0.9-3 Tc

• Bound states survive above Tc?!

c

J/

)T,(G

)T,(G)T,(RG

rec

2.5.2 T-Matrix Spectral Functions with Potential U

• S-wave ground state “melts” at Tdiss ≈ 2 Tc

• correlator ratios within 30% (3D reduction scheme)

S-Wave Spectral Function Euclidean Correlator Ratio(narrow-width

limit)

2.5.3 T-Matrix Spectral Functions with Potential F

• S-wave ground state “melts” at Tdiss ≈ 1.3 Tc

• reduced c-c threshold → low-energy strength

S-Wave Spectral Function Euclidean Correlator Ratio

[Cabrera+RR ’06, Riek+RR ‘09]

-

2.5.4 Importance of Confining Force

J/

Υ

2.6 Charmonium Widths in QGP

→ sensitive to binding energy (i.e., color screening)

• J/ lifetime ~ 1-4 fm/c

)s()T;(f)(

kd disspk

p

g,qp

3

3

2

s~0.25

q q

J/ Dissociation Rates

[Grandchamp+RR ’01]

J/

S-Wave Spectral Function

• accelerates “melting”: Tdiss ≈ 1.6 Tc

• correlator ratio temperature-stable

med=200MeV

• dashed lines: gluo-dissociation

• solid lines: quasifree dissociation

• similar to full NLO calculation

2.6.2 Momentum Dependence of Inelastic Width

_

[Zhao+RR ‘07][Park et al ‘07]

q q

2.6.3 Relation of Quarkonium Widths to EFT

•Landau damping

• Singlet-octet transition

q q

reaction rate equilibrium limit ( -width) )m,m,dp/dN( cTc

3.) Quarkonium Production in URHICs

J/ D

D-

J/c- c

)NN(d

dN eq

[PBM et al ’01, Gorenstein et al ’02,Thews et al ’01, Grandchamp+RR ’01, Ko et al ’02, Cassing et al ’03, Zhuang et al ’05, …]

J/ + g c + c + X←→ -• Regeneration in QGP + HG:

- detailed balance

mc*

B

• Input from Thermodynamic T-Matrix (weak/strong binding)

3.1 Inputs and Parameters• Input

- J/ (c, ’), c-c production cross sections [p-p data [PHENIX] ]

- “Cold Nuclear Matter”: shadowing, nuclear absorption, pt broadening [p-A data]

- Thermal fireball evolution: thermalization time (↔ initial T0),

expansion rate, lifetime, Tc , freezeout …

[hadron data, hydrodynamics]

• Parameters

- strong coupling s controls diss

- schematic relaxation for c-quark equilibration: N

eq ()~ Ntherm() · [1-exp(-/c

eq)] _

-

3.2 Centrality Dependence of J/ at SPS + RHIC

• regeneration controlled by c-quark relaxation time (ceq

= 6 vs. 3 fm/c)• similar total yield, but different composition

[Zhao+RR in prep]

Strong-Binding Scenario (U) Weak-Binding Scenario (F)

3.3 pT-Dependence of J/ at SPS + RHIC

• weak binding problematic with pt-dependence?!

Strong Binding (U) Weak Binding (F)

3.3.2 pT-Dependence II: Blast Wave at RHIC

• blast wave at ~Tc too soft?

• lever arm for direct prod. at high pT?

Regeneration only (Stat. Model) Rate-Equation (strong bind.)Au-Au

200AGeV

[Andronic et al. ‘07]

3.3.3 Charm-Quark pT-Spectra and Regeneration

• supports sensitivity to thermal relaxation time of c quarks

• microscopic calculation of gain term c + c + g → J/ + g-

QmDT

2

2

p

fD

p)pf(

tf

• Brownian

Motion:

thermalization rate diffusion coefficient

4.) Heavy-Quark Diffusion in the QGP

Fokker Planck Eq.[Svetitsky ’88,…]

Q

k)p,k(wkdp Q3

23

21 k)p,k(wkdD Q

• In-medium heavy-light T-matrix:

)E(T)E(GVdkkV)E(T LQqLLL 2

• pQCD elastic scattering:

1= therm ≥ 20 fm/c slow

q,g

c

2

2elast

D

scg ~

direct connection to quarkonia!

[Svetitsky ’88, Mustafa et al ’98, Molnar et al ’04, Zhang et al ’04, Hees+RR ’04, Teaney+Moore ’04, Peshier,Gossiaux+Aichelin ‘09]

[van Hees et al ’07, Riek+RR ‘10]

4.2 Charm-Quark T-Matrix + Thermalization

• meson/diquark resonances for T < 1.5 Tc

• factor 3-4 (~2) larger than pert. QCD for U (F) potential

Thermalization RateThermal Q-q T-Matrix

[Riek+RR ‘10]

T [GeV]

[1

/fm

]

4.3 e± Spectra at RHIC

• hadronic resonances at ~Tc ↔ quark coalescence

• connects 2 “pillars” of RHIC: strong coupl. + coalescence

[van Hees et al ‘07]

T-mat

T-mat

5.) Conclusions

• Thermodynamic T-matrix for heavy quarks + quarkonia - vacuum: spectroscopy + pQCD limit - in-medium potential from lattice QCD? U1 (Td

~2Tc) , F1 (Td~1.3Tc) , or else …

- confining force mandatory for realistic calculations

• Quarkonium phenomenology - “strong” vs. “weak” J/ binding (pt-data, lever arm, …)

- bottomonium suppression? (less regeneration …)

• Open heavy flavor - resonances close to Tc ? (strong coupling + coalescence …)

- RHIC non-photonic e±Ds (2T) ≈ 5 - scrutinize medium evolution, Fokker-Planck, d-Au …

3.2.3 Rapidity Dependence at RHIC

• regeneration yield sensitive to dNc/dy

• hot matter effects insufficient • additional shadowing at forward y (assuming constant abs)

Thermal Rate-Eq Approach

[Kharzeev et al. ‘07, Ferreiro et al. ‘08] [Zhao+RR in prep]

• regeneration at low pT

3.2.5 Momentum Spectra

• regeneration part → blast-wave at Tc

Au-Au200AGeV

• high pT: formation time ( ), bottom feeddown, …

[Karsch+Petronzio ’87, Blaizot+Ollitrault ‘87]

[Zhao+RR ’07, ‘08]

3.2.4 Momentum Spectra and Elliptic Flow

• regeneration at low pt → small v2

• direct component at high pt → small v2

[Zhao+RR ’08, Zhuang et al ‘06]

2.4.2 Example from “Extended T-Matrix Model”

c

• cc propagator with c= 100 MeV:

• S-wave “melting” Tdiss ≈ 1.5-2 Tc

• correlator ratio temperature-stable

- ])(s/[)s(G cckkcc20 24

S-Wave Spectral Function Euclidean Correlator Ratio

4.3 Thermalization Rate and Diffusion Coefficient

• Factor ~3-4 larger thermalization rates than in pert. QCD

•“different” approaches related, e.g. AdS/CFT ↔ Coulomb

T [GeV]

[1

/fm

]

T [GeV]

2.4 Mesonic Spectral Functions + Correlators

• Euclidean Correlation Function (precise lat-QCD data avail.!)

Correlator Ratio:

2.2.2 Potential Models in the QGP

U1 potential

• F1 low threshold (2mc~ 2.7GeV), ground state Tdiss ~ 1.2 Tc

• U1 decreasing threshold and B, Tdiss ~2.5Tc

both scenarios compatible with lat-QCD

~F1 potential [Cabrera +RR ‘06]

[Mocsy+ Petreczky ’05,‘08]

mc=1.7GeVmc=1.7GeVmc*

c

212

3 3

32

,N)T,m(f)(

qdVN cccFB

eq

3.1.3 Equilibrium Limit (Statistical Model)

eq

opc

opcD,copcFBcc N

)N(I)N(I

)T;m(nVN0

1

21• fixed c-c number:

• equilibrium number:

• (very) sensitive to open-charm spectrum

• thermal relaxation for c-quark spectra:

[Grandchamp et al ’03, Andronic et al ’07, …]

-

]/)T(N)(N eqc

eqeq exp[-

2.1.3 In-Medium Charm-Quark Mass in LQCD

• U: large variation close to Tc – mass interpretation?!

[Kaczmarek+Zantow ’05]

F

• fit quark-number fluctuations with zero-width quasiparticle model (T) ~ ∂2P / ∂2c

[Velytsky et al ’09]

3.3.4 Rapidity Dependence at RHIC

• reproduced in statistical hadronization model (GC ensemble) [Andronic et al. ’07]

• more problematic in dynamic approaches

• additional shadowing at forward y?

Statistical Model Thermal Rate-Eq Approach

[Capella et al. ’07, Zhao+RR ‘08]

[Kharzeev et al. ‘07, Ferreiro et al. ‘08]

3.4 Upsilon at RHIC

• (1S,2S) suppression unambiguous QGP signature ?!• NB: 50% feed-down on (1S)

No Color-Debye Screening With Color-Debye Screening

[Grandchamp et al. ’05]

3.3 Heavy-Quark Spectra at RHIC

• T-matrix approach ≈ effective resonance model • similar to “coll. dissoc.” [Adil+Vitev ’07]; radiative E-loss? (2↔3), …

• relativistic Langevin simulation in elliptic expanding fireball background

pT [GeV]

Nuclear Modification Factor Elliptic Flow

pT [GeV]

2.3.2 Bottomonium Reaction Rates in QGP

[Grandchamp et al. ’05]

• color-screening accelerates dissociation• significance at RHIC: Y ≈ 50 → 5 fm/c