RAA of J/ψ near mid-rapidity
in RHIC at √sNN=200 GeV
and in LHC
Taesoo Song, Woosung Park,
and Su Houng Lee
(Yonsei Univ.)
QCD phase diagram
There are many probes to investigate the properties of hot nuclear matter.
One of them is J/ψ
Brief history of J/ψ
in ultrarelativistic heavy ion collision
• J/ψ suppression due to the Debye screening of color
charge between c and anti-c pair was first suggested as
a signature of quark-gluon plasma (QGP) formation in
relativistic heavy ion collision by T. Matsui, H. Satz in
1986.
T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986)
• Recently, lattice calculations
support that J/ψ survives
above Tc
M. Asakawa, T. Hatsuda, PRL
92, 012001 (2004), …
• However, J/ψ is still a good
probe to investigate the
property of hot nuclear matter
created from ultrarelativistic
heavy ion collision, because its
melting point is believed to
exist between Tc and the initial
temperature of hot nuclear
matter created through the
relativistic heavy ion collisions
in RHIC.
Phenomenological models
to describe J/ψ production
in relativistic heavy ion collision
1. Thermal model (P. Braun-Munzinger et al.)
Initially produced J/Ψ does not survive. All J/Ψs are
formed at hadronization stage by recombination of
charm and anti-charm.
2. Two component model (R. Rapp et al.)
Observed J/Ψ comes from at initial collisions or at
hadronization stage.
3. Simultaneous dissociation and recombination of J/Ψ
during the fireball expansion (P. Zhuang et al.)
Two-component model
Initially
produced
J/ψ through
binary N-N
collisions
Thermalizatioin
Hadronization
Chemical
(QGP formation)
T≈ 170 MeV
freeze-out
≈ 0.3~1 fm
T≈ 161 MeV
Nuclear
absorption
Thermal decay
in QGP
detector
Recombination
of J/ψ
Thermal decay
in hadronic matter
Glauber model
1. Woods - Saxon distributi on
 (r) 
0
1  e (r -r0 ) / C
2. Number of participan ts (wounded nucleons)

 
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
 

, where TA, B (b ) 


b-s
s
b



  A,B (b , x)dx ; thickness function.

The quantity of bulk matter is proportional to # of participants
3. Nuclear overlap function of two colliding nuclei

 

2
TAB (b )   d sdzd z   A ( s , z )  B (b  s , z )
As an example, the number of J/ produced in two nuclei collision

with impact parameter b is  J / TAB (b ).
The quantity of hard particles such as J/ψ is proportional to # of binary collisions
# of participants vs. # of binary collisions
with σin=42 mb
1200
3
1000
2.5
800
2
N_coll
600
ratio
3.5
number
1400
N_coll/N_part
1.5
N_part
400
1
200
0.5
0
0
0
5
10
15
20
25
impact parameter b (fm)
0
5
10
impact parameter b (fm)
15
What is RAA?
R
J /
AA
1
1
N AJ / A

 J /
N coll N n  n
0.9
0.8
If RAA is below 1, it is called that J/ is suppressed .
0.7
If RAA is above 1, it is called that J/ is enhenced.
0.6
0.5
From RHIC with
s  200 GeV at mid - rapidity,
RAA
0.4
0.3
0.2
0.1
0
0
200
No. of participants
400
1. Nuclear absorption
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 absorption
cross section is
obtained from pA
collision
σdiss=1.5mb
Comparison with experimental data of RHIC
(√s=200 GeV at midrapidity)
1
After nuclear absorption
0. 8
0. 6
R AA
0. 4
0. 2
0
0
100
200
300
No. of participants
400
2. Thermal decay in fireball
QGP phase
Mixed phase
(Assuming 1st order
phase transition)
HG phase
J/ψ
J/ψ
J/ψ
2.1. Thermal model
  Ej  j  
2
  1
nj 
dpp
exp 
2 2 
  Tcfo  
 j   B B j   I 3 I 3   s S j  C C j
1
dj
A.Andronic, P. Braun-Munzinger, K.
Redlich, J. Stachel
NPA 772, 167 (2006)
Thermal model successfully
describes particle ratios ni/nj at
chemical freeze-out stage
,where nj (μb , μI3 , μS , μC , Tcfo )
is particle number density in
grand canonical ensemble
  Ej  j  
dj
2
  1
nj 
dpp exp 
2 

2
  Tcfo  
 j   B B j   I 3 I 3   s S j  C C j
1
The latest result for RHIC data at mid-rapidity (√s=200 GeV) :
Tcfo=161 MeV, μb=22.4 MeV
A.Andronic, P. Braun-Munzinger, K. Redlich, J. Stachel
NPA 789, 334 (2007)
Z, N and other quantities μI3 , μS , μC, at chemical freeze-out are obtained from
below constraints.
Baryon number conservation V  n j B j  Z  N
j
Isospin conservation
V  n j I3 j 
j
Strangeness conservation
ZN
2
V  njS j  0
j
Charm conservation
V  n jC j  0
j
,where Z is the net number of wounded protons and N is that of wounded
neutrons in the fireball at mid-rapidity
Absolute hadron yields can also be
reproduced in thermal model
At midrapidity (|y|<1) with √s=200 GeV & Npart=350, Vcf≈2400 fm3
Here we parameteri ze total entropy as the combinatio n of N part and N coll
N part


S  21.51  x 
 xNcoll ,
2


where x  0.11 at s NN  200 GeV, and 0.09 at s NN  130 GeV
Local entropy density at initial stage is assumed to be
n part


S
s   21.5(1  x)
 xncoll ,
V
2


where n part  N part / V , n coll  N coll / V
Multiplicities of charged particles
vs. # of participants
4. 5
(dNch/dη)/(2Npart)
4
3. 5
3
2. 5
2
1. 5
1
0. 5
0
0
50
100
150
200
Npart
250
300
350
400
The latest result for RHIC data at mid-rapidity (√s=200 GeV) :
Tcfo=161 MeV, μb=22.4 MeV
A.Andronic, P. Braun-Munzinger, K. Redlich, J. Stachel
NPA 789, 334 (2007)
Z, N and other quantities μI3 , μS , μC, at chemical freeze-out are obtained from
below constraints.
Baryon number conservation V  n j B j  Z  N
set at about Npart/20 in |y|<0.35
j
Isospin conservation
V  n j I3 j 
j
Strangeness conservation
ZN
2
V  njS j  0
j
Charm conservation
V  n jC j  0
j
,where Z is the net number of wounded protons and N is that of wounded
neutrons in the fireball at mid-rapidity
Ratios of proton and anti-proton
vs. # of participants
Ratio of p/antip
1. 8
1. 6
1. 4
1. 2
1
0. 8
0. 6
0. 4
0. 2
0
0
100
200
Npart
300
400
Thermal mass of partons in QGP
Peter Levai & Ulrich Heinz PRC 57, 1879 (1998)
Strongly interacting massless partons
→ Noninteracting massive partons, reproducing well
thermal quantities obtained from LQCD
g 2 (T ) T 2  N c N f 

,
m 

2
3
6


g 2 (T ) T 2
m 
3
2
g
2
q
48 2
, where g (T ) 
(11N c  2 N f ) ln F 2 (T , Tc ,  )
2
F (T , Tc ,  ) 
18
18.4e 0.5(T / TC )
2
T Tc
 1 Tc 
N f  0, Tc  260 MeV , Tc /   1.03
N f  2, Tc  140 MeV , Tc /   1.03
N f  4, Tc  170 MeV , Tc /   1.05
Nf=2
Nf=4
gluon
quark
0.8
0.7
0.7
0.6
0.5
0.5
Mass (GeV)
Mass (GeV)
0.6
0.4
0.3
pure gloun
0.2
0.4
0.3
pure gloun
0.2
N_f=2
N_f=2
N_f=4
N_f=4
0.1
0.1
0
0
1
1.5
2
T/Tc
2.5
3
1
1.5
2
T/Tc
2.5
3
Entropy density in QGP & in HG
in HG,
1
1 

s
V T V ,  6 2

k 4 ( Ei  i ) e ( Ei  i ) / T
i  dk E T 2
2
( Ei   i ) / T
1
e
i
0
All mesons below 1.5 GeV, &
All baryons below 2.0 GeV
in QGP,
1
1 

s
V T V ,  6 2


 k 4 ( Ei  i ) e ( Ei  i ) / T
i  dk  E T 2
2
e ( Ei   i ) / T  1

i
0



 mi2   1
e ( Ei  i ) / T 
1
1


 

2 
2
3 ( Ei   i ) / T
T
/
)


E
(

i
i
 1 2 Ei T e
 1 
 T   2 Ei e


Thermal quark/anti-quark & gluon
Entropy density vs. temperature
Temperature (GeV)
0. 3
0. 25
0. 2
HG
QGP( Nf =2)
0. 15
QGP( Nf =4)
0. 1
0
10
20
30
40
50
60
Entropy density (fm-3)
70
80
Temperature distribution on
transverse plane at formation time
8
impact parameter b=0 (fm)
7
T=170 MeV
y (fm)
6
T=200 MeV
5
T=230 MeV
4
T=260 MeV
3
T=290 MeV
2
T=320 MeV
1
T=350 MeV
0
T=380 MeV
0
1
2
3
4
5
6
7
x (fm)
8
9
10
8
7
impact parameter b=4 (fm)
T=170 MeV
6
T=200 MeV
y (fm)
5
T=230 MeV
4
T=260 MeV
3
T=290 MeV
2
T=320 MeV
1
T=350 MeV
0
0
1
2
3
4
5
6
7
x (fm)
8
9
10
T=380 MeV
8
y (fm)
7
impact parameter b=8 (fm)
6
T=170 MeV
5
T=200 MeV
4
T=230 MeV
3
T=260 MeV
2
T=290 MeV
1
T=320 MeV
T=350 MeV
0
0
1
2
3
4
5
6
7
x (fm)
8
9
10
8
impact parameter b=12 (fm)
7
6
y (fm)
5
T=170 MeV
4
T=200 MeV
3
T=230 MeV
2
T=260 MeV
1
0
0
1
2
3
4
5
6
7
x (fm)
8
9
10
Temperature profiles with various impact
parameters
0.42
b=0 (fm)
Temperature (GeV)
b=1 (fm)
0.37
b=2 (fm)
0.32
b=4 (fm)
b=3 (fm)
b=5 (fm)
b=6 (fm)
0.27
b=7 (fm)
b=8 (fm)
0.22
b=9 (fm)
b=10 (fm)
0.17
b=11 (fm)
0
2
4
y (fm)
6
8
b=12 (fm)
Assuming isentropic expansion of fireball, with the below timedependent volume
2
1

2
VFB ( )  2  y  0      r0  a  ,
2


“we can deduce temperatures and chemical potentials at
midrapidity before reaching chemical freeze-out stage.”
From hydrodynamics simulation,
1.τ0 is the thermalization time of fireball ≈ 0.6 fm/c
2. a⊥ is transverse acceleration , which was set at 0.1 c2/fm
X. Zhao, R. Rapp, PLB664, 253 (2008)
terminal transverse velocity was set at 0.6 c
3. r0 is initial transverse radius of QGP
Fireball expansion (b=0 fm)
0.35
Temperature (GeV)
0.3
0.25
QGP
Mixed
0.2
HG
0.15
0.1
0.05
0
0
2
4
6
Time (fm/c)
8
10
Fireball expansion (b=0 fm)
Chemical potentials (GeV)
0.12
0.1
0.08
QGP
μb
μI3
μs
Mixed
0.06
0.04
0.02
HG
0
0
-0.02
2
4
6
Time (fm/c)
8
10
2.2. survival rate from the thermal decay
Considering feed-down from χc , Ψ’ to J/ψ,
survival rate of J/ψ from the thermal decay is
c
J /
'
SQGP HG  0.67 SQGP

0
.
25
S

0
.
08
S
 HG
QGP HG
QGP HG
,where
j
S QGP
 HG
  j

 exp    ( ')d '
 0

and Γj (τ) is thermal width of charmonia j
Γj (τ)= ΓQGPj (τ)
Γj (τ)= f ΓQGPj (τ) +(1-f) ΓHGj (τ)
Γj (τ)= ΓHGj (τ)
(T>Tc) in QGP phase
(T=Tc) in mixed phase
(T<Tc) in HG phase
Thermal decay widths in QGP & HG
Thermal width
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/
(in QGP, j=quark, antiquark, gluon
in hadronic matter, j=pion, kaon, …)
In order to calculate thermal width Γ, we must know
1) Thermodynamic quantities such as T, μ
2) Dissociation cross section σdiss
Dissociation cross section σdiss
Dissociation cross section σdiss is a crucial quantity to calculate
thermal width.
Most studies use two different models for QGP dissociation and
hadronic dissociation.
(As an example,
for the decay in QGP, quasi-free particle approximation,
and for that in HG, meson exchange model,…)
Here we use the same approach, pQCD, in QGP and in HG.
Bethe-Salpeter amplitude to describe the
bound state of heavy quarkonia
Definition ;
Solution is 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)
Wavefunctions of charmonia at finite T
Modified Cornell potential
J/ψ (1S)
F. Karsch, M.T. Mehr, H. Satz, Z phys. C. 37, 617 (1988)
V (r , T ) 


1  e   (T ) r 
 (T )
   (T ) r

r
Screening
mass
289 MeV
298 MeV
306 MeV
315 MeV
323 MeV
332 MeV
340 MeV
e
σ=0.192 GeV2 : string tension
α=0.471 : Coulomb-like potential
constant
μ(T) ~gT is the screening mass
In the limit μ(T)→0,
V (r , T )   r 

r
GeV
χc (1P)
Ψ’(2S)
Screening
mass
289 MeV
298 MeV
306 MeV
315 MeV
323 MeV
332 MeV
340 MeV
GeV
Screening
mass
289 MeV
298 MeV
306 MeV
315 MeV
323 MeV
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
In QGP
In hadronic matter
σdiss= σpQCD
σdiss(p)= ∫dx σpQCD (xp)D(x)
: factorization formula
1. partons with thermal mass ~gT,
2. temperature-dependent
wavefunction is used.
D(x) is parton distribution Ft. of
hadrons(pion, here) interacting with
charmonia
1. Massless partons
2. (mass factorization, loop diagrams,
and renormalization are required
to remove collinear divergence,
infrared divergence, and ultraviolet
divergence)
2. Coulomb wavefunction is used.
The role of coupling constant ‘g’
1. ‘g’ determines the thermal width of J/ψ
(in LO, Г∼g2, and in NLO, Г∼g4)
2. ‘g’ determines the screening mass, that is, the melting
temperature of charmonia (screening mass μ=gT)
TJ/ψ=377 MeV, Tχc =221 MeV, TΨ’=179 MeV
Comparison with experimental data of RHIC
(√s=200 GeV at midrapidity)
1
After nuclear absorption &
thermal decay g=1.85
0. 8
0. 6
R AA
0. 4
0. 2
0
0
100
200
300
No. of participants
400
3. Recombination of J/ψ at hadronization
If the number of cc pair is completely thermalized,
1

N cAB

n

n

c
open C
hidden C  V
2

However, the cross section for ‘cc pair → others’ or ‘others → cc pair’ is very small.
The life time of fireball is insufficient for the thermalization of the number of cc pair.
→ corrected with fugacity γ
N
AB
cc
1
  nopen CV   2 nhiddenCV
2
If produced cc pairs are very few, GCE must turn to CE
→ canonical ensemble suppression
N
AB
cc
I1 ( nopen CV )
1
  nopen CV
  2 nhiddenCV
2
I 0 ( nhiddenCV )
If the number of cc pair initially produced in AB collision is conserved, it scales with
the number of binary collision between nucleons in colliding nuclei.
N
AB
cc

 

NN
2
(b )   cc AB  d s  dz A  A (s , z A )  dz B  B (b  s , z B )
, where dσccNN/dy=63.7(μb) from pQCD.
fugacity
canonical suppression
45
1.2
40
1
35
30
0.8
25
0.6
20
15
0.4
10
0.2
5
0
0
0
100
200
300
No. of participant
400
0
100
200
300
No. of participant
400
Relaxation factor for kinematical
equilibrium
  H d 

R  1  exp   
   eq 
 0

, where a thermaliz ation time  eq  1 /( n )
n : the total density of quark/gluo n in the system
 : the elastic scattering cross section of charm/anti - charm
 H : the time at hadronizat ion
• Finally, the number of
recombined J/ψ is
VRγ2 {nJ/ψ+Br(χc)*nχc+ Br(ψ’) *nψ’}
< Reference for RAA>
• J/ψ production in pp
collisions at √s=200 GeV
PHENIX Collaboration, PRL 98,
232002 (2007)
The role of coupling constant ‘g’
3. ‘g’ determines relaxation factor of charm/anti-charm
quarks (relaxation time∼g2)
Comparison with experimental data of RHIC
(√s=200 GeV at midrapidity)
1
g=1.85
0. 8
sum
nuclear absorption &
thermal decay
0. 6
R AA
0. 4
recombination
0. 2
0
0
100
200
300
No. of participants
400
If there is no initial melting of J/ψ
1
0. 8
0. 6
R AA
0. 4
0. 2
0
0
100
200
300
No. of participants
400
Cu+Cu in RHIC at √sNN=200 GeV
1. 2
1
sum
0. 8
R AA
0. 6
0. 4
nuclear absorption &
thermal decay
recombination
0. 2
0
0
20
40
60
80
No. of participants
100
120
For LHC prediction
• By extrapolation,
Entropy S= 21.5{(1-x)Npart/2+xNcoll}
to 55.7{(1-x)Npart/2+xNcoll}, where x=0.11
J/ψ production cross section in p+p collision per rapidity
dσJ/ψpp/dy= 0.774 μb to 6.4 μb
• By pQCD,
cc production cross section in p+p collision per rapidity
dσccpp/dy= 63.7 μb to 639 μb
Pb+Pb in LHC at √sNN=5.5 TeV
1
sum
0. 8
0. 6
recombination
R AA 0. 4
nuclear absorption &
thermal decay
0. 2
0
0
100
200
No. of participants
300
400
Summary
•
•
1.
2.
3.
RAA of J/ψ near midrapidity in Au+Au collision at √sNN=200
GeV is well reproduced with almost no free parameter.
Something new different from other models are followings:
It is assumed that the sudden drop of RAA around Npart=190
is caused by that the maximum temperature of the fireball
begins to be over the melting temperature of J/ψ there.
Thermal masses of partons extracted from LQCD are used
to obtain thermal quantities of expanding fireball and to
calculate dissociation cross sections of charmonia
From this, g is determined, because the screening mass is
assumed to be gT.
• The same method was applied to Cu+Cu collision at the same
energy, and the result is not bad.
• With some modified parameters, RAA of J/ψ in LHC was
calculated. Different from RHIC, recombination effect is
dominant, because most charmonia produced at initial stage
are melt and much larger number of charm quark are
produced in LHC.
•
1.
2.
3.
4.
The future plan is to reproduce or predict
RAA at forward rapidity
The dependence of RAA on transverse momentum of J/ψ
RAA of heavier system such as Upsilon
…