Thermal Kinetic Equation Approach to Charmonium Production in Heavy-Ion Collision

Xingbo Zhaowith Ralf Rapp

Department of Physics and Astronomy

Iowa State University Ames, USA

Brookhaven National Lab, Upton, NY, Jun. 14th 2011

2

Outline

Thermal rate-equation approach• Dissociation rate in quasi-free approximation• Regeneration rate from detailed balance• Connection with lattice QCD

Numerical results compared to exp. data• Collision energy dependence (SPS->RHIC->LHC)• Transverse momentum dependence (RHIC)• Rapidity dependence (RHIC)

3

Motivation: Probe for Deconfinement• Charmonium (Ψ): a probe for deconfinement– Color-Debye screening reduces binding energy -> Ψ dissolve

• Reduced yield expected in AA collisions relative to superposition of individual NN collisions

• Other factors may also suppress Ψ yield in AA collision- Quantitative calculation is needed

[Matsui and Satz. ‘86]

4

Motivation: Eq. Properties Heavy-Ion Coll.

• Equilibrium properties obtained from lattice QCD– free energy between two static quarks ( heavy quark

potential)– Ψ current-current correlator ( spectral function)

• Kinetic approach needed to translate static Ψ eq. properties into production in the dynamically evolving hot and dense medium

?

?

5

Picture of Ψ production in Heavy-Ion Coll.

• 3 stages: 1->2->31. Initial production in hard collisions2. Pre-equilibrium stage (CNM effects)3. Thermalized medium

• 2 processes in thermal medium:1. Dissociation by screening & collision 2. Regeneration from coalescence

• Fireball life is too short for equilibration - Kinetic approach needed for off-equilibrium system

J/ψ D

D-

J/ψc-c

6

Thermal Rate-Equation• Thermal rate-equation is employed to describe

production in thermal medium (stage 3)

– Loss term for dissociation Gain term for regeneration– Γ: dissociation rate Nψ

eq: eq. limit of Ψ– Detailed balance is satisfied by sharing common Γ in the

loss and gain term– Main microscopic inputs: Γ and Nψ

eq

7

Kinetic equations

lQCD potential

diss. & reg. rate: Γ

Initial conditions Experimental observables

lQCD correlator

Link between Lattice QCD and Exp. Data

Ψ eq. limit: NΨeq

εBΨ mΨ, mc

8

Kinetic equations

lQCD potential

diss. & reg. rate: Γ

Initial conditions Experimental observables

lQCD correlator

Link between Lattice QCD and Exp. Data

Ψ eq. limit: NΨeq

εBΨ mΨ, mc

9

In-medium Dissociation Mechanisms

[Bhanot and Peskin ‘79][Grandchamp and Rapp ‘01]

• Gluo-dissociation is not applicable for reduced εBΨ<T

quasifree diss. becomes dominant suppression mechanism

- strong coupling αs~ 0.3 is a parameter of the approach

• Dissociation cross section σiΨ

- gluo-dissociation: quasifree dissociation:

g+Ψ→c+ g(q)+Ψ→c+ +g(q)

VS.

• Dissociation rate:

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Kinetic equations

lQCD potential

diss. & reg. rate: Γ

Initial conditions Experimental observables

lQCD correlator

Link between Lattice QCD and Exp. Data

Ψ eq. limit: NΨeq

εBΨ mΨ, mc

11

Charmonium In-Medium Binding•

• Potential model employed to evaluate

• V(r)=U(r) vs. F(r)? (F=U-TS)

• 2 “extreme” cases:

• V=U: strong binding

• V=F: weak binding

[Cabrera et al. ’07, Riek et al. ‘10]

[Riek et al. ‘10]

[Petreczky et al ‘10]

12

T and p Dependence of Quasifree Rate

• Gluo-dissociation is inefficient even in the strong binding scenario (V=U)• Quasifree rate increases with both temperature and ψ momentum• Dependence on both is more pronounced in the strong binding scenario

13

Kinetic equations

lQCD potential

diss. & reg. rate: Γ

Initial conditions Experimental observables

lQCD correlator

Link between Lattice QCD and Exp. Data

Ψ eq. limit: NΨeq

εBΨ mΨ, mc

14

Kinetic equations

lQCD potential

diss. & reg. rate: Γ

Initial conditions Experimental observables

lQCD correlator

Link between Lattice QCD and Exp. Data

Ψ eq. limit: NΨeq

εBΨ mΨ, mc

15

Model Spectral Functions• Model spectral function = resonance + continuum

• At finite temperature:

• Z(T) reflects medium induced change of resonance strength

Tdiss=2.0Tc V=U

Tdiss=1.25Tc V=FZ(Tdiss)=0

• In vacuum:

• Z(T) is constrained from matching lQCD correlator ratio

width ΓΨ

threshold 2mc*

pole mass mΨ

• Regeneration is possible only if T<Tdiss

quasifree diss. rate

TdissTdiss

16

Correlators and Spectral Functions

• Peak structure in spectral function dissolves at Tdiss • Model correlator ratios are compatible with lQCD results

weak binding strong binding

[Petreczky et al. ‘07]

17

Kinetic equations

lQCD potential

diss. & reg. rate: Γ

Initial conditions Experimental observables

lQCD correlator

Link between Lattice QCD and Exp. Data

Ψ eq. limit: NΨeq

εBΨ mΨ, mc

18

Regeneration: Inverse Dissociation

• For thermal c spectra, NΨeq follows from statistical model

- charm quarks distributed over open charm and Ψ states according to their mass and degeneracy

- masses for open charm and Ψ are from potential model

• Realistic off-kinetic-eq. c spectra lead to weaker regeneration:

[Braun-Munzinger et al. ’00, Gorenstein et al. ‘01]

• Gain term dictated by detailed balance:

• Charm relaxation time τceq is our second parameter: τc

eq~3/6fm/c

19

Kinetic equations

lQCD potential

diss. & reg. rate: Γ

Initial conditions Experimental observables

lQCD correlator

Link between Lattice QCD and Exp. Data

Ψ eq. limit: NΨeq

εBΨ mΨ, mc

1. shadowing2. nuclear

absorption3. Cronin

20

Kinetic equations

lQCD potential

diss. & reg. rate: Γ

Initial conditions Experimental observables

lQCD correlator

Link between Lattice QCD and Exp. Data

Ψ eq. limit: NΨeq

εBΨ mΨ, mc

1. Coll. energy dep.2. Pt dep.3. Rapidity dep.

1. shadowing2. nuclear

absorption3. Cronin

21

Compare to data from SPS NA50 weak binding (V=F) strong binding (V=U)

incl

. J/p

si y

ield

• Different composition for different scenarios

• Primordial production dominates in strong binding scenario

• Significant regeneration in weak binding scenario

• Large uncertainty on σcc

22

J/Ψ yield at RHIC weak binding (V=F) strong binding (V=U)

• Larger primordial (regeneration) component in V=U (V=F)

• Compared to SPS regeneration takes larger fraction in both scenarios

• Formation time effect and B meson feeddown are included

incl

. J/p

si y

ield

See also [Thews ‘05],[Yan et al. ‘06],[Andronic et al. ‘07]

23

J/Ψ yield at LHC (w/o Shadowing) weak binding (V=F) strong binding (V=U)

• Parameter free prediction – both αs and τceq fixed at SPS and RHIC

• Regeneration component dominates except for peripheral collisions

• RAA<1 for central collisions (with , )

• Comparable total yield for V=F and V=U

24

With Shadowing Included

• Shadowing suppresses both primordial production and regeneration• Regeneration dominant in central collisions even with shadowing• Nearly flat centrality dep. due to interplay between prim. and reg.

25

Compare to Statistical Model weak binding (V=F) strong binding (V=U)

• Regeneration is lower than statistical limit:- statistical limit in QGP phase is more relevant for ψ regeneration

- statistical limit in QGP is smaller than in hadronic phase

- charm quark kinetic off-eq. reduces ψ regeneration

- J/ψ is chemically off-equilibrium with cc (small reaction rate)

26

High pt Ψ at LHC

• Negligible regeneration for pt > 6.5 GeV• Strong suppression for prompt J/Ψ• Significant yield from B feeddown• Similar yields and composition between V=U and V=F

27

Pt Dependence at RHIC Mid-Rapidity

see also [Y.Liu et al. ‘09]

V=UV=U

• Primordial production dominant at pt>5GeV• Regeneration concentrated at low pt due to c quark thermalization• Formation time effect and B feeddown increase high pt production [Gavin and Vogt ‘90, Blaizot and Ollitrault ‘88, Karsch and Petronzio ‘88]

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RAA(pT) at RHIC Mid-RapidityV=FV=F

• At low pt regeneration component is larger than V=U

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J/ψ v2(pT) at RHIC

• Small v2(pT) for entire pT range

- At low pt v2 from thermal coalescence is small

- At high pt regeneration component is gone

• Even smaller v2 even in V=F

- Small v2 does not exclude coalescence component

strong binding (V=U) weak binding (V=F)

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J/Ψ Yield at RHIC Forward Rapidity weak binding (V=F) strong binding (V=U)

• Hot medium induced suppression and reg. comparable to mid-y

• Stronger CNM induced suppression leads to smaller RAA than mid-y

• Larger uncertainty on CNM effects at forward-y

incl

. J/p

si y

ield

See also [Thews ‘05],[Yan et al. ‘06],[Andronic et al. ‘07]

31

RAA(pT) at RHIC Forward RapidityV=UV=U

• Shadowing pronounced at low pt & fade away at high pt

• Large uncertainty on CNM effects

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RAA(pT) at RHIC Forward RapidityV=FV=F

• At low pt reg. component is larger than V=U (similar to mid-y)

3333

Summary and Outlook• A thermal rate-equation approach is employed to describe

charmonium production in heavy-ion collisions• Dissociation and regeneration rates are compatible with lattice QCD

results • J/ψ inclusive yield consistent with experimental data from collision

energy over more than two orders of magnitude• Primordial production (regenration) dominant at SPS (LHC)• RAA<1 at LHC (despite dominance of regeneration) due to incomplete

thermalization (unless the charm cross section is really large)

• Calculate Ψ regeneration from realistic time-dependent charm phase space distribution from e.g., Langevin simulations

34

Thank you!

based on X. Zhao and R. Rapp Phys. Lett. B 664, 253 (2008)

X. Zhao and R. Rapp Phys. Rev. C 82, 064905 (2010)

X. Zhao and R. Rapp Nucl. Phys. A 859, 114 (2011)

35

Compare to data from SPS NA50 weak binding (V=F) strong binding (V=U)

incl

. J/p

si y

ield

tran

s. m

omen

tum

• primordial production dominates in strong binding scenario

36

J/ψ v2(pT) at RHIC

• Small v2(pT) for entire pT range

strong binding (V=U)

37

Explicit Calculation of Regeneration Rate

• in previous treatment, regeneration rate was evaluated using detailed balance

• was evaluated using statistical model assuming thermal charm quark distribution

• thermal charm quark distribution is not realistic even at RHIC ( )

• need to calculate regeneration rate explicitly from non-thermal charm distribution

[van Hees et al. ’08, Riek et al. ‘10]

38

3-to-2 to 2-to-2 Reduction

• reduction of transition matrix according to detailed balance

2 2

gcc g gc gcM M ( )2c

pp

dissociation: regeneration:

• g(q)+Ψ c+c+g(q)diss.

reg.

39

Thermal vs. pQCD Charm Spectra

• regeneration from two types of charm spectra are evaluated:

1) thermal spectra: 2 2( ) exp /c cf p m p T

2) pQCD spectra:

22

( )1 /

c

p Af p

p B

[van Hees ‘05]

40

Reg. Rates from Different c Spectra

• thermal : pQCD : pQCD+thermal = 1 : 0.28 : 0.47

• introducing c and angular correlation decrease reg. for high pt Ψ

• strongest reg. from thermal spectra (larger phase space overlap)

See also, [Greco et al. ’03, Yan et al ‘06]

41

Ψ Regeneration from Different c Spectra

• strongest regeneration from thermal charm spectra

• c angular correlation lead to small reg. and low <pt2>

• pQCD spectra lead to larger <pt2> of regenerated Ψ

• blastwave overestimates <pt2> from thermal charm spectra

4242

V=F V=U

• larger fraction for reg.Ψ in weak binding scenario• strong binding tends to reproduce <pt

2> data

J/Ψ yield and <pt2> at RHIC forward y

incl

. J/p

si y

ield

tran

s. m

omen

tum

4343

J/Ψ suppression at forward vs mid-y

• comparable hot medium effects• stronger suppression at forward rapidity due to CNM effects

44

RAA(pT) at RHIC

• Primordial component dominates at high pt (>5GeV)

• Significant regeneration component at low pt

• Formation time effect and B-feeddown enhance high pt J/Ψ

• See also [Y.Liu et al. ‘09]

V=F V=U

[Gavin and Vogt ‘90, Blaizot and Ollitrault ‘88, Karsch and Petronzio ‘88]

4545

J/Ψ Abundance vs. Time at RHIC V=F V=U

• Dissoc. and Reg. mostly occur at QGP and mix phase

• “Dip” structure for the weak binding scenario

4646

J/Ψ Abundance vs. Time at LHC V=F V=U

• regeneration is below statistical equilibrium limit

47

Ψ Reg. in Canonical Ensemble

• Integer charm pair produced in each event

• c and anti-c simultaneously produced in each event,c c c cf f f f

• c and anti-c correlation volume effect further increases local c (anti-c) density

48

Ψ Reg. in Canonical Ensemble

• Larger regeneration in canonical ensemble

• Canonical ensemble effect is more pronounced for non-central collisions

• Correlation volume effect further increases Ψ regeneration

4949

Fireball Evolution• , {vz,at,az} “consistent” with: - final light-hadron flow - hydro-dynamical evolution

• isentropical expansion with constant Stot (matched to Nch) and

s/nB (inferred from hadro-chemistry)• EoS: ideal massive parton gas in QGP, resonance gas in HG

2 2 20 0

1 1( ) ( ) ( )

2 2FB z zV z v a r a

[X.Zhao+R.Rapp ‘08]

( )( )tot

FB

Ss T

V

50

Primordial and Regeneration Components • Linearity of Boltzmann Eq. allows for decomposition of primordial and

regeneration components

;tot prim regf f f

/ ;prim prim primf t v f f

/ ;reg reg regf t v f f

00regf

0 0

prim totf f

• For primordial component we directly solve homogeneous Boltzmann Eq.

• For regeneration component we solve a Rate Eq. for inclusive yield and estimate its pt spectra using a locally thermal distribution boosted by medium flow.

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Rate-Equation for Reg. Component

/eqN G

/reg reg regf v f f •

3 3,p G d pd x

/reg regdN d N G

/reg reg eqdN d N N

• For thermal c spectra, Neq follows from charm conservation: 21 1

=2 2

tot eqcc oc c oc FB c FBN N +N n V n V

• Non-thermal c spectra lead to less regeneration:

[1 exp( / )]eq eq eq eqcN R N N

(Integrate over Ψ phase space)

typical 3 10 fm/eqc c

[van Hees et al. ’08, Riek et al. ‘10]

[Braun-Munzinger et al. ’00, Gorenstein et al. ‘01]

[Grandchamp, Rapp ‘04]

[Greco et al. ’03]

52

• follows from Ψ spectra in pp collisions with Cronin effect applied

Initial Condition and RAA

• is obtained from Ψ primordial production0( , , )f x p t

0 0 0( , , ) ( , ) ( , )f x p t f x t f p t

• follows from Glauber model with shadowing and nuclear absorption parameterized with an effective σabs

0( , )f x t

assuming

0( , )f p t

• nuclear modification factor:AAΨ

AA ppcoll Ψ

NR

N N

Ncoll: Number of binary nucleon-nucleon collisions in AA collisions

RAA=1, if without either cold nuclear matter (shadowing, nuclear absorption, Cronin) or hot medium effects

53

Correlators and Spectral Functions

†( , ) ( , ) (0,0) ,G r j r j

pole mass mΨ(T), width Ψ(T)

threshold 2mc*(T),

• two-point charmonium current correlation function:

• charmonium spectral function: 0

cosh[ ( 1/ 2 )]( , ) ( , )

sinh[ / 2 ]

TG T d T

T

• lattice QCD suggests correlator ratio ~1 up to 2-3 Tc:

( , )

( , )Grec

G TR

G T

[Aarts et al. ’07, Datta te al ’04, Jakovac et al ‘07]

5, 1, , ...j q q

54

Initial Conditions• cold nuclear matter effects included in initial conditions• nuclear shadowing: • nuclear absorption:• Cronin effect:

• implementation for cold nuclear matter effects:• nuclear shadowing• nuclear absorption• Cronin effect Gaussian smearing with smearing width

guided by p(d)-A data

Glauber model with σabs from p(d)-A data

55

Kinetic equations

lQCD potential

diss. & reg. rates

Initial conditions

Experimental observables

lQCD correlator

(Binding energy)

Link between Lattice QCD and Exp. Data