J/ψ production and elliptic flow in
relativistic heavy-ion collisions
Taesoo Song
(Texas A&M Univ., USA)
Reference : T. Song, C. M. Ko, S. H. Lee and J. Xu,
arXiv:1008.2730
Contents
1.
2.
3.
4.
5.
6.
Introduction
Schematic model for fireball expansion
Thermal properties of charmonia
Charmonia in heavy-ion collisions
Results
Summary
1. introduction
QCD phase diagram
J/ψ suppression
• Long time ago, J/ψ suppression was suggested
by Matsui and Satz as a signature of QGP
formation in heavy-ion collisions. (due to color
screening between c and anti-c)
• The suppression was observed at SPS & RHIC.
• LQCD suggests the dissociation temperature of
J/ψ higher than Tc.
• J/ψ is still one of the promising diagnostic
probes for hot nuclear matter created by
heavy-ion collisions.
Phenomenological models
1. Statistical model (P. Braun-Munzinger)
Low dissociation temperature of J/ψ
Most J/ψ in heavy-on collisions are regenerated
ones.
2. Two-component model (R. Rapp)
High dissociation temperature of J/ψ
Some of J/ψ come from regeneration, some of
them come from initial production.
NJ/ψ vs. Npart
statistical model
two-component model
NJ/ψ vs. Pt
statistical model
two-component model
Questions
• How can both models successfully
describe experimental data?
• How can both models be discriminated?
2. Schematic model for
expanding fireball
• Initial condition
• Equation of state (EoS)
• modeling
2. 1. Glauber model
1. Number of participan ts

 
B

N part (b )  A TA ( s ) 1  1  TB (b  s ) in d 2 s
 

A
 B  TB (b  s ) 1   1  TA ( s ) in  d 2 s
 



b-s
s
2. Number of binary collisions

 

2
N coll (b )   in  d sdzdz   A ( s , z )  B (b  s , z )
where  (r) 
0
; Woods - Saxon distributi on
1 e



TA, B (b )    A, B (b , x)dx ; thickness function.

(r - r0 ) / C
b
2. 2. Initial condition
N part

dN chA B dN chp  p 

 xNcoll 
(1  x)
d
d 
2

x  0.09 at s NN  130 GeV
x  0.11 at s NN  200 GeV
Local entropy density in initial stage
is assumed to be
n part


s
S

 30.3(1  x)
 xncoll ,
 V
2


Charged particle multiplicities
where n part  dN part /  0 dA,
PRC65, 061901 (2002)
n coll  dN coll /  0 dA
EoS of QGP
• Quasiparticle picture
Noninteracting massive
partons to reproduce
thermal quantities
extracted from LQCD
Strongly interacting
massless partons
g 2 (T ) T 2  N c N f 
g 2 (T ) T 2
2

, mq 
m 

,
2
6 
3
 3
48 2
2
where g (T ) 
,
2
(11N c  2 N f ) ln F (T , Tc ,  )
2
g
F (T , Tc ,  ) 
18
T Tc
,
 1 Tc 
18.4e 0.5(T / TC )
N f  3, Tc  170 MeV , Tc /   1.05
2
P. Levai & U. Heinz
PRC 57, 1879 (1998)
EoS of HG
• Resonance gas model
1. all mesons of masses lighter than 1.5
GeV & all baryons of masses lighter
than 2.0 GeV are considered in HG
phase.
2. They are assumed to have constant
masses and to be noninteracting.
Energy density and pressure
16
d3p
 (T )  
2 
4
e/T , p/T
4
12
m2  p 2
e
 m 2  p 2    / T


 B (T )
1
4
e/T
4
p/T
8
1 d3p
p2
p (T )  
3 2  m 2  p 2 
e
 B(T )
4
0
0.15
0.20
0.25
T (GeV)
0.30
0.35
1
m 2  p 2    / T

1
Isothermal lines on transverse plane
at τ0=0.6 fm/c
y (fm)
8
7
6
5
4
3
2
1
0
impact parameter b=0 (fm)
T=170
MeV
T=240
MeV
0 1 2 3 4 5 6 7 8
x (fm)
Temperature (GeV)
Temperature profiles
at various impact parameters
0.42
b=0 (fm)
0.37
0.32
b=4 (fm)
0.27
b=8 (fm)
0.22
0.17
0
2
4
y (fm)
6
8
2. 3. fireball expansion
• Radial acceleration in central collision
ar 
( p  pf )A
M
p f : pressure at freeze - out
A : cylindrica l area ~ R 2
.24
M : inertia mass  0.36 N 1part
Parameterized to fit
experimental data <pt> of π,
K, p at freeze-out
Assuming isentropic expansion, s(τ)=s0*v0/V(τ)
vT (c)
0.8
0.6
0.4
0.2
T (GeV)
0.4
0.3
0.2
0.1
0
5
10
t (fm/c)
15
20
• Radial acceleration in non-central collision
Parameter to fit
a x  ar 1  z 
a y  ar 1  z ,

R y  Rx
R y  Rx
experimental data v2 of
π, K, p at freeze-out
, z  2.2
Fireball expanding in elliptic shape,
 cos  sin  
R ( )   2 
2 
R y 
 Rx
2
2
1 / 2
,
 v x cos 2  v y sin 2  
v( )  R ( ) 


3
3
R y 
 Rx
T (GeV)
Mixed
0.2
HG
QGP
vx, vy (c)
0.1
vx
0.6
0.4
vy
0.2
Rx, Ry (fm)
b=9 fm,
0.3
10
8
Rx
6
Ry
4
0
2
4
6
 (fm/c)
8
10
12
Blast wave model
by using Cooper - Frye formula,
dN
 mT
 pT sinh    mT cosh  

rdrd

I
K1 
0
2 
2


dydpT 2 
T
T

 
where   tanh 1 vT .
dN
dydpT2

dN
2
dp
 T dydpT2
2
dp
T
 pT
pT
v2  cos 2 p 
dN
dyd 2 pT
dN
2
d
p
 T dyd 2 pT
2
d
 pT cos 2 p
30
20
10
<pT> (MeV/c)
v2 (%)
1000
800
600
0
30
20
400
pions
kaons
protons
200
pions
kaons
protons
10
0
0
0
100
200
Npart
300
400
0
1
2
pT (GeV/c)
3
4
(bottom) after multiplyin g
exp  cpT / n, c  1.14GeV 1
n  2 for meson, 3 for baryon
3. Thermal properties of charmonia
• Dissociation temperatures
• Dissociation cross section in QGP and in
HG
3. 1. wavefunctions & binding energies
& radii of charmonia at finite T
Modified Cornell potential
F. Karsch, M.T. Mehr, H. Satz, Z phys. C. 37, 617 (1988)
V (r , T ) 



1  e   (T ) r   e   (T ) r
 (T )
r
σ=0.192 GeV2 : string tension
α=0.471 : Coulomb-like potential constant
μ(T) =√(Nc/3+Nf/6) gT : screening mass in pQCD
In the limit μ(T)→0,
V (r , T )   r 

r
J/ψ (1S)
χc (1P)
Ψ’(2S)
Screening
mass
289 MeV
298 MeV
306 MeV
315 MeV
323 MeV
332 MeV
340 MeV
GeV
GeV
GeV
Binding energies & radii of charmonia
0.25
9
J/psi
0.2
chi_c
7
J/psi
0.15
chi_c
Psi'(2S)
0.1
Radius (fm)
Binding energy (GeV)
8
Psi'(2S)
6
5
4
3
2
0.05
1
0
0
300
500
700
Screening mass (MeV)
300
500
Screening mass (MeV)
700
3. 2. dissociation cross section
• Bethe-Salpeter amplitude
Definition ;
Solution in NR limit ;
Leading Order (LO)
quark-induced
Next to Leading Order (qNLO)
gluon-induced
Next to Leading Order (gNLO)
Leading Order (LO)
quark-induced
Next to Leading Order (qNLO)
gluon-induced
Next to Leading Order (gNLO)
In QGP
In hadronic matter
σdiss= ∑j σ jpQCD
Factorization formula:
σdiss(p)= ∑ j ∫dx σ ipQCD (xp)Dj i(x)
Dj i(x) is PDF of parton i in hadron j
interacting with charmonia
1. partons with thermal mass
2. temperature-dependent
wavefunctions from
modified Cornell potential
are used.
1. Massless partons
mass factorization, loop diagrams
and renormalization remove
collinear, infrared and UV
divergence respectively
2. Coulomb wavefunctions are used.
4. Charmonia in heavy-ion collisions
•
•
•
•
Cronin effect
Nuclear absorption (nuclear destruction)
Thermal decay and leakage effect
Regeneration
Two-component model
Thermalization
Hadronization Kinetic
Before
freeze-out
cc production (QGP formation) T≈ 170 MeV
T≈ 120 MeV
Cronin effect ≈ 0.6 fm/c
Nuclear
Initial
absorption
Thermal decay
production
in QGP
of J/ψ
Thermal decay
through
in hadronic matter
binary N-N
detector
collisions
Regenerated
J/ψ
Thermal decay
in hadronic matter
4. 1. Cronin effect
1.
2.
3.
4.
Charmonia are produced mainly through g+g fusion
Different from in p+p collision, gluon in A+B collision
can get additional Pt through g+N collision
It broadens Pt distribution of gluons
Subsequently, it broadens Pt distribution of J/ ψ in A+B
collision, compared with in p+p collision
Pt
2 AB
J /
 Pt
2 pp
J /

Pt
2 gN
gN
LAB
g
Primordial J/ψ is
produced
Nucleus B
Nucleus A
4. 2. Nuclear destruction
Primordial J/ψ is
produced
Nucleus A
Nucleus B
 

1
2
S nuc (b,  nuc ) 
d sdzdz '  A ( s , z )  B (b  s , z ' )

TAB (b)




 exp  ( A  1)  dz A  A ( s , z A ) nuc 
z



 


 exp  ( B  1)  dz B  B (b  s , z B ) nuc 
z'


Nuclear destruction
cross section is
obtained from pA
collision
σdiss=1.5mb
4. 3. Thermal decay
QGP phase
Mixed phase
(Assuming 1st order
phase transition)
HG phase
J/ψ
J/ψ
J/ψ
Thermal decay widths in QGP & HG
d 3k
  gj
n j (T ,  )vrel (T ) diss (T )
3
(2 )
j
n j : density of parton or hardon dissociati ng J/
vrel : relative velocity between j and J/
 diss : dissociati on cross section of J/
( )  QGP ( )
in QGP phase
( )  f * QGP ( )  (1  f ) * HG ( ) in mixed phase
( )  HG ( )
in HG phase
J/ψ (1S)
Ψ’(2S)
χc (1P)
The leakage effect
Thermal decay width =0
Thermal decay width ≠0
Thermal decay width : Γ→Γ*θ[R(τ)-r(τ)]
Survival probability from thermal decay
SQGP HG
 

 exp    ( ')d '
 0

Considering feed-down from χc , Ψ’ to J/ψ,
c
J /
'
SQGP HG  0.67 SQGP

0
.
25
S

0
.
08
S
 HG
QGP HG
QGP HG
4. 4. Regeneration
• From Glauber model (dσccNN/dy=63.7(μb) from pQCD),

 

AB
NN
2
N cc (b )   cc AB  d s  dz A  A (s , z A )  dz B  B (b  s , z B )
• From Statistical model,
1

N   nopen C  nhiddenC  V
2

• Discrepancy between them is corrected with
fugacity
1
AB
N cc   nopen CV   2 nhiddenCV
2
AB
cc
• GCE is converted to CE because of small # of pairs
N
AB
cc
I1 ( nopen CV )
1
  nopen CV
  2 nhiddenCV
2
I 0 ( nopen CV )
Canonical suppression
canonical suppression
25
1
0.9
20
0.8
0.7
fugacity
15
0.6
0.5
10
0.4
0.3
5
0.2
0.1
0
0
100
200
Npart
300
400
0
0
100
200
300
Npart
400
Relaxation factor for kinetic equilibrium
  H d 

R  1  exp   
   relax. 
 0

, where relaxation time  relax  1 /(  ni i vrel )
i
ni : number density of parton i
 i : elastic scattering cross section of charm/anti - charm
by parton i
 H : the time at hadronizat ion
the number of regenerated J/ψ
NJ/ψrec=
VRγ2 {nJ/ψSJ/ψHG +Br(χc)*nχc *SχcHG + Br(ψ’) *nψ’* Sψ’HG }
• nJ/ψ, nχc , nψ’ : number densities of charmonia
• SJ/ψHG, SχcHG , Sψ’HG : survival rate of charmonia in HG
• Br(χc), Br(ψ’) : branching ratios of χc, ψ’ to J/ψ
• R : relaxation factor
• γ : fugacity
5. Results
•
•
•
•
•
RAA vs. Npart
RAA vs. pT
<pT>
V2
Higher-order corrections in pQCD
5. 1. RAA of J/ψ
Nuclear modificati on factor
J /
RAA
1
N AJ / A

 J /
N coll N n  n
RAA  1, J/ is suppressed .
RAA  1, J/ is enhenced.
From RHIC near midrapidty
at √sNN=200 GeV
RAA of J/ψ as a function of Npart
(near midrapidity in Au+Au collision at √s=200 GeV)
Regeneration
The role of coupling constant g in our model
1. ‘g’ determines dissociation temperatures of
charmonia
(screening mass μ=√(Nc/3+Nf/6) gT)
TJ/ψ=386 MeV, Tχc =199 MeV, TΨ’=185 MeV with
g=1.5
2. ‘g’ determines the thermal widths of charmonia
(Г∼g2 in LO, and Г∼g4 in NLO)
3. ‘g’ determines the relaxation factor of charm quarks
W/O initial dissociation of J/ψ
1
0.9
0.8
0.7
0.6
RAA
without
0.5
0.4
0.3
with
0.2
0.1
0
0
100
200
Npart
300
400
RAA of J/ψ as a Function of pt
(For J/ψ, Tf=160 MeV)
<Pt2> of J/ψ
v2 of J/ψ (b=9 fm)
<Assumption>
1. Elastic cross section of
J/ψ(color singlet) in
QGP is much smaller
than that of charm
quark.
2. For J/ψ, inelastic
collision is more
effective than elastic
collision in QGP
because of its small
binding energy and
large radius at high T.
RAA of J/ψ as a function of Npart
(near midrapidity in Cu+Cu collision at √s=200 GeV)
Regeneration
Applying to Pb+Pb collision at √sNN=5.5
TeV (LHC) with the modified parameters
• by extrapolation,
Entropy dS/dη= 30.3{(1-x)Npart/2+xNcoll}
to 78.5{(1-x)Npart/2+xNcoll}, where x=0.11
J/ψ production cross section per rapidity in p+p collision
dσJ/ψpp/dy= 0.774 μb to 6.4 μb
7.36 μb at 7 TeV (Nov.
2010)
• from pQCD,
cc production cross section per rapidity in p+p collision
dσccpp/dy= 63.7 μb to 639 μb
Ref. is NPA 789, 334 (2007)
RAA of J/ψ as a function of Npart
(near midrapidity in Pb+Pb collision at √s=5.5 TeV)
1
Total
0.8
0.6
RAA0.4
Recombination
Regeneration
Nuclear absorption &
Thermal decay in QGP & HG
0.2
0
0
100
200
Npart
300
400
5. 2. Higher-order corrections
• Dissociation cross section of charmonia
σ [J/ψ+q(g)→c+c+q(g)] *A
; enhances decay of charmonia
• Elastic cross section of charm quarks
σ [c+q(g)→c+q(g)] *B
; enhances regeneration of charmonia
Fractions of regenerated J/ψ
=(A,B)
RAA of J/ψ as a function of Npart
(near midrapidity in Au+Au collision at √s=200 GeV)
RAA of J/ψ as a Function of pt
<Pt2> of J/ψ
v2 of J/ψ (b=9 fm)
5. Summary
Summary of nuclear modification of
charmonia in heavy-ion collision
• Before production
; Cronin effect (pt↑)
• After production
; nuclear destruction (NJ/ψ↓)
; initial dissociation (NJ/ψ↓)
• After thermalization
; thermal decay (NJ/ψ↓)
; leakage effect (NJ/ψ↑, pt↑)
; regeneration (NJ/ψ↑)
; flow effect (pt↑)
Summary of results
• We reproduced successfully RAA of J/ψ in Au+Au and Cu+Cu
collisions at RHIC and estimated RAA in Pb+Pb collision at LHC by
using 2-component model.
• There seems to be a kink in RAA vs. Npart curve in Au+Au collision.
→ initial temperature begins to be over TJ/ψ?
• 2-component model vs. statistical model
The number of J/ψ : the excessive number of J/ψ in 2-component
model is reduced by multiplying relaxation factor to regenerated
J/ψ.
pt of J/ψ : In 2-component model, Cronin effect mainly enhances
pt while in the statistical model, flow effect mainly enhances.
→ both models successfully describe RAA and pt of J/ψ in RHIC.
• Only v2 of J/ψ seems to be able to discriminate two models.
→ Precise measurement of v2 of J/ψ will reveal the fraction of
regenerated J/ψ