J/
c
suppression in an equilibrating parton plasma
Xiao-Ming Xu,
1,2
D. Kharzeev,
3,4
H. Satz,
3,4
and Xin-Nian Wang
1
1
Nuclear Science Division, Mailstop 70A-3307, Lawrence Berkeley National Laboratory, Berkeley, California 94720
2
Theory Division, Shanghai Institute of Nuclear Research, Chinese Academy of Sciences, P.O. Box 800204, Shanghai 201800, China
3
Theory Division, CERN, CH-1211 Geneva, Switzerland
4
Fakulta
¨
tfu
¨
r Physik, Universita
¨
t Bielefeld, D-33501 Bielefeld, Germany
~Received 27 November 1995!
Short-distance QCD is employed to calculate the J/
c
survival probability in an equilibrating parton gas
whose evolution is governed by a set of master rate equations. Partons in the early stage of high-energy nuclear
collisions may initially not be in equilibrium, but their average transverse momentum is sufficiently high to
break up a QQ
¯
bound state. Such a breakup during the evolution of the parton gas is shown to cause a
substantial J/
c
suppression at both RHIC and LHC energies, using realistic estimates of the initial parton
densities. The transverse momentum dependence of the suppression is also shown to be sensitive to the initial
conditions and the evolution history of the parton plasma. @S0556-2813~96!06205-X#
PACS number~s!: 24.85.1p, 12.38.Mh, 25.75.Dw, 25.75.Gz
I. INTRODUCTION
It is generally believed, and confirmed by lattice QCD
calculations @1#, that hadronic matter under extreme condi-
tions will form a plasma in which quarks and gluons are no
longer confined to individual hadrons and are in both thermal
and chemical equilibrium. To search for such a quark-gluon
plasma, it is proposed to study collisions of heavy nuclei at
extremely high energies. Recent work using models based on
perturbative QCD indeed shows that at high energy a dense
partonic system can be produced @2–6#. Though it is not all
clear if such partonic systems will reach thermal and chemi-
cal equilibrium before hadronization @7–11#, the partons,
mainly gluons, are certainly in a deconfined state at such
high densities @12#.
Many signals could arise from this deconfined state, such
as charm quark enhancement @13–15# or enhanced photon
and dilepton production @16,17#. Charm quarks, for example,
cannot be easily produced during the mixed and hadronic
phases of dense matter, due to their large masses compared
to the temperature. They can be readily produced only during
the early stage of the evolution, when partonic degrees of
freedom are relevant. In this paper, we will discuss how pre-
equilibrium J/
c
suppression can be used to probe the early
deconfined state of the partonic system and the dynamics
governing its evolution toward equilibrium.
J/
c
suppression due to color screening has been proposed
to probe deconfinement @18#. This requires that the interac-
tions of J/
c
with hadrons and deconfined partons be differ-
ent @19#. Because of its small size, a heavy quarkonium can
probe the short-distance properties of light hadrons. It is thus
possible to make a parton-based calculation of the
J/
c
-hadron cross section via an operator product expansion
method similar to that used in deeply inelastic lepton-hadron
scatterings @19–21#. The resulting J/
c
-hadron cross section
can be related to the distribution function of gluons inside a
hadron. The energy dependence of the cross section near the
threshold of the breakup of a J/
c
is determined by the large-
x behavior of the gluon distribution function, giving rise to a
very small breakup cross section at low energies. Only at
very high energies will this cross section reach its asymptotic
value of a few mb. In other words, dissociation can only
occur if the gluon from the light hadron wave function is
hard in the J/
c
rest frame, i.e., its energy is high enough to
overcome the binding energy threshold. A hadron gas with
temperature below 0.5 GeV certainly cannot provide such
energetic gluons to break up the J/
c
. Therefore, a slow
J/
c
is very unlikely to be absorbed inside a hadron gas of
reasonable temperature @19#. This conclusion does not seem
to be affected substantially by nonperturbative effects, ana-
lyzed in Ref. @22#.
On the other hand, a deconfined partonic system contains
much harder gluons which can easily break up a J/
c
@19,23#.
A study of the energy dependence of the gluon-J/
c
inelastic
cross section @23# shows a strong peak just above the
breakup threshold of the gluon energy,
e
0
5 2M
D
2 M
J/
c
,
where M
J/
c
and M
D
are the J/
c
and D meson masses, re-
spectively. In the preequilibrium stage, i.e., before the par-
tons have reached equilibrium, the average parton transverse
momentum is sufficiently large @24# to break up a J/
c
, pro-
vided the partons are deconfined. The dissociation of J/
c
will continue during the whole equilibration process until the
effective temperature drops below a certain value or the be-
ginning of hadronization, whichever takes place first. There-
fore measurements of J/
c
suppression can probe the decon-
finement of the early partonic system and shed light on the
subsequent equilibration process, provided that possible
nuclear effects on the production of QQ
¯
pairs and on pre-
resonance charmonium states are understood and taken into
account.
In the following we will first calculate the thermal gluon-
J/
c
dissociation cross section at different temperatures and
for different J/
c
transverse momenta. We then follow the
evolution of an initially produced parton gas toward equilib-
rium and calculate the resulting total survival probability of a
J/
c
and its P
T
dependence.
II. J/
c
DISSOCIATION BY GLUONS
The operator product expansion allows one to express the
hadron-J/
c
inelastic cross section in terms of the convolu-
PHYSICAL REVIEW C JUNE 1996VOLUME 53, NUMBER 6
53
0556-2813/96/53~6!/3051~6!/$10.00 3051 © 1996 The American Physical Society
tion of the gluon-J/
c
dissociation cross section with the
gluon distribution inside the hadron @19#. The gluon-J/
c
dis-
sociation cross section is given by @23#
s
~
q
0
!
5
2
p
3
S
32
3
D
2
S
16
p
3g
s
2
D
1
m
Q
2
~
q
0
/
e
0
2 1
!
3/2
~
q
0
/
e
0
!
5
, ~1!
where g
s
is the coupling constant of a gluon and c quark,
m
Q
the c quark mass, and q
0
the gluon energy in the J/
c
rest
frame; its value must be larger than the J/
c
binding energy
e
0
. Since for the tightly bound ground state of quarkonium
the binding force between the heavy quark and antiquark is
well approximated by the one-gluon-exchange Coulomb po-
tential, the QQ
¯
bound state is hydrogenlike and the Coulomb
relation holds,
e
0
5
S
3g
s
2
16
p
D
2
m
Q
. ~2!
The cross section thus can be rewritten as
s
~
q
0
!
5
2
p
3
S
32
3
D
2
1
m
Q
~
e
0
m
Q
!
1/2
~
q
0
/
e
0
2 1
!
3/2
~
q
0
/
e
0
!
5
. ~3!
As shown in Monte Carlo simulations @24#, the parton
density in the early stage of high-energy heavy-ion collisions
has an approximate Bjorken-type @25# scaling behavior. We
will only consider J/
c
suppression in the central rapidity
region (y
J/
c
.0). In this case, the J/
c
will move in the
transverse direction with a four-velocity
u5
~
M
T
,P
W
T
,0
!
/M
J/
c
, ~4!
where M
T
5
A
P
T
2
1 M
J/
c
2
is defined as the J/
c
transverse
mass. A gluon with a four-momentum k5 (k
0
,k
W
) in the rest
frame of the parton gas has an energy q
0
5 k• u in the rest
frame of the J/
c
. The thermal gluon-J/
c
dissociation cross
section is then defined as
^
v
rel
s
~
k• u
!
&
k
5
*
d
3
k
v
rel
s
~
k• u
!
f
~
k
0
;T
!
*
d
3
kf
~
k
0
;T
!
, ~5!
where the gluon distribution in the rest frame of the parton
gas is defined as
f
~
k
0
;T
!
5
l
g
e
k
0
/T
2 l
g
, ~6!
with l
g
<1 specifying the deviation of the system from
chemical equilibrium. For large momentum gluons which are
responsible for the J/
c
dissociation, we can approximate the
above distribution by a factorized Bose-Einstein distribution
f
~
k
0
;T
!
'
l
g
e
k
0
/T
2 1
. ~7!
The relative velocity
v
rel
between the J/
c
and a gluon is
v
rel
5
P
J/
c
• k
k
0
M
T
5 12
k
W
• P
W
T
k
0
M
T
. ~8!
Changing the variable to the gluon momentum, q5(q
0
,q
W
),
in the rest frame of the J/
c
, the integral in the numerator of
Eq. ~5! can be rewritten as
E
d
3
q
M
J/
c
M
T
s
~
q
0
!
f
~
k
0
;T
!
, ~9!
where
k
0
5
~
q
0
M
T
1 q
W
• P
W
T
!
/M
J/
c
. ~10!
One can carry out the integral in the denominator,
*
d
3
kf(k
0
;T)58
pz
(3)l
g
T
3
, and the angular part in the nu-
merator, to get
^
v
rel
s
~
k• u
!
&
k
5
S
8
3
D
3
p
z
~
3
!
M
J/
c
2
P
T
M
T
T
3
S
e
0
m
Q
D
3/2
(
n5 1
`
T
n
3
E
1
`
dx
~
x21
!
3/2
x
4
~
e
2 a
n
2
x
2 e
2 a
n
1
x
!
,
~11!
with T
n
5 T/n and
a
n
6
5
e
0
T
n
M
T
6 P
T
M
J/
c
. ~12!
In order to understand the temperature and P
T
depen-
dence of the thermal gluon-J/
c
dissociation cross section,
we first plot in Fig. 1 the cross section
s
(q
0
) of Eq. ~3! as a
function of the gluon energy in the J/
c
rest frame. It de-
creases strongly toward the threshold and is broadly peaked
around q
0
5 10
e
0
/75 0.92 GeV, with a maximum value of
about 3 mb. Low-momentum gluons do not have the resolu-
tion to distinguish the heavy constituent quarks or the energy
to excite them to the continuum. On the other end, high-
momentum gluons also have small cross section with a J/
c
since they cannot see the large size.
We can also express the cross section as a function of the
center-of-mass energy of gluons and the J/
c
,
s
(q
0
)5
s
(s/2M
J/
c
2 M
J/
c
/2), where s5 (k1 P
J/
c
)
2
. One
FIG. 1. Gluon-J/
c
dissociation cross section as a function of the
gluon energy q
0
in the rest frame of the J/
c
.
3052 53
XU, KHARZEEV, SATZ, AND WANG
can thus translate the energy dependence in Fig. 1 into tem-
perature and P
T
dependences after thermal average, since the
thermally averaged
^
s
&
is proportional to both P
T
and tem-
perature T. In Fig. 2 we plot the thermally averaged gluon-
J/
c
dissociation cross section as a function of temperature
for different values of the J/
c
transverse momentum P
T
.
We observe the same kind of peak structure, with a de-
creased maximum value due to the thermal average. The
position of the peak also shifts to smaller values of T when
P
T
is increased, corresponding to a fixed value of the aver-
aged center-of-mass energy
^
s
&
. A similar behavior is ex-
pected if one plots the thermal cross section as a function of
P
T
at different temperatures, as done in Fig. 3. However, in
this case, the peak simply disappears at high enough tem-
peratures, because the averaged
^
s
&
will be above the thresh-
old value even for P
T
5 0. These features will have consid-
erable consequences for the survival probability of a J/
c
in
an equilibrating parton gas, especially the P
T
dependence.
We should also mention that the use of the factorized ap-
proximation of the Bose-Einstein distribution function in Eq.
~7! has an effect of about 20% on the thermal cross section,
compared to that obtained with a Boltzmann distribution
@23#.
III. J/
c
SUPPRESSION IN AN EQUILIBRATING
PARTON GAS
Using the thermal cross section just obtained, we can now
calculate the survival probability of J/
c
in an equilibrating
parton plasma. In this paper, we will neglect the transverse
expansion and consider only longitudinal expansion. We will
also only consider J/
c
suppression in the central rapidity
region. A J/
c
produced at point r
W
with velocity
v
W
in the
transverse direction will travel a distance
d52rcos
f
1
A
R
A
2
2 r
2
~
12 cos
2
f
!
~13!
in the time interval t
c
5 M
T
d/P
T
before it escapes from a
gluon gas of transverse extension R
A
; here, cos
f
5
v
W
ˆ
•r
W
ˆ
. Sup-
pose the system evolves in a deconfined state until the tem-
perature drops below a certain value, which we assume to be
200 MeV. The total amount of time the J/
c
remains inside a
deconfined parton gas is the smaller one of the two times
t
c
and t
f
, the lifetime of the parton gas. Assume that the
initial production rate of the J/
c
is proportional to the num-
ber of binary nucleon-nucleon interactions at impact param-
eter r, N
A
(r)5A
2
(12r
2
/R
A
2
)/2
p
R
A
2
. The survival probabil-
ity of the J/
c
averaged over its initial position and direction
in an equilibrating parton gas is
S
~
P
T
!
5
*
d
2
r
~
R
A
2
2 r
2
!
exp
@
2
*
0
t
min
d
t
n
g
~
t
!
^
v
rel
s
~
k• u
!
&
k
#
*
d
2
r
~
R
A
2
2 r
2
!
,
~14!
where
t
min
5 min
~
t
c
,t
f
!
, ~15!
and n
g
(
t
) is the gluon number density at a given time
t
.
In Eq. ~14!, both the gluon number density n
g
(
t
) and the
thermal cross section
^
v
rel
s
(k• u)
&
k
depend on the tempera-
ture, which in turn is a function of time. In addition, n
g
is
proportional to gluon fugacity which also evolves with time.
To evaluate the survival probability we need to know the
entire evolution history of the parton system. Roughly speak-
ing, one can divide this history into two stages: ~1! First
there is kinetic thermalization, mainly through elastic scat-
terings and expansion. The kinematic separation of free-
streaming partons gives us
t
0
;0.5–0.7 fm/c as an estimate
of the time when local isotropy in momentum distribution is
reached @9,24#. ~2! The parton gas now further evolves to-
ward chemical equilibrium through parton proliferation and
gluon fusion; it does so until hadronization or freeze-out,
whichever happens first. This evolution can be determined
by a set of master rate equations which give us the time
dependence of the temperature and fugacities.
In this paper we will not address the question of prereso-
nance J/
c
suppression, which has been discussed and shown
to be responsible for the J/
c
suppression observed in p-A
and S-U collisions @26–29#. We will here consider the sup-
FIG. 2. The thermal-averaged gluon-J/
c
dissociation cross sec-
tion
^
v
rel
s
&
as a function of the temperature at different transverse
momenta P
T
.
FIG. 3. The thermal-averaged gluon-J/
c
dissociation cross sec-
tion
^
v
rel
s
&
as a function of the transverse momentum P
T
at differ-
ent temperatures.
53
3053
J/
c
SUPPRESSION IN AN EQUILIBRATING PARTON PLASMA
pression of fully formed physical J/
c
states as it should take
place if nuclear collision produces dense partonic system.
Following Ref. @9#, we characterize the nonequilibrium of
the system by gluon and quark fugacities which are less than
unity. The dominant reactions leading to chemical equilib-
rium are assumed to be the following two processes:
gg↔ggg, gg↔qq
¯
. ~16!
Assuming that elastic parton scatterings are sufficiently rapid
to maintain local thermal equilibrium, the evolution of the
parton densities can be given by the master rate equations.
Combining these master equations together with one-
dimensional hydrodynamic equation, one can get the follow-
ing set of equations @9#:
l
˙
g
l
g
1 3
T
˙
T
1
1
t
5 R
3
~
12 l
g
!
2 2R
2
S
12
l
q
2
l
g
2
D
, ~17!
l
˙
q
l
q
1 3
T
˙
T
1
1
t
5 R
2
a
1
b
1
S
l
g
l
q
2
l
q
l
g
D
, ~18!
S
l
g
1
b
2
a
2
l
q
D
3/4
T
3
t
5 const, ~19!
where a
1
5 16
z
(3)/
p
2
'1.95, a
2
5 8
p
2
/15'5.26, b
1
5 9
z
(3)N
f
/
p
2
'2.20, and b
2
5 7
p
2
N
f
/20'6.9. The density
and velocity weighted reaction rates
R
3
5
1
2
^
s
gg→ggg
v
&
n
g
, R
2
5
1
2
^
s
gg→qq
¯
v
&
n
g
~20!
can be found in Refs. @9,13#. After taking into account parton
screening @12# and the Landau-Pomeranchuck-Migdal effect
in induced gluon radiation @30,31#, R
3
/T and R
2
/T are
found to be functions of only l
g
. Solving the above rate
equations as shown in Ref. @9#, one finds that the parton gas
cools considerably faster than predicted by Bjorken’s scaling
solution (T
3
t
5 const!, because the production of additional
partons approaching the chemical equilibrium state con-
sumes an appreciable amount of energy. The accelerated
cooling, in turn, slows down the chemical equilibration pro-
cess. For the initial conditions given by
HIJING Monte Carlo
simulations @4#, at RHIC energy the parton system can
hardly reach its equilibrium state, since the effective tem-
perature here drops below T
c
'200 MeV in a time of some
1–2 fm/c. At LHC energy, however, the parton gas comes
very close to equilibrium, since the system may exist in a
deconfined state for as long as 4–5 fm/c. In the following we
will use the numerical results for the time evolution of the
temperature and fugacities obtained from the above master
equations to calculate the survival probability of a J/
c
in
such an equilibrating parton system.
Shown in Fig. 4~a! are the J/
c
survival probabilities in
the deconfined and equilibrating parton plasma at RHIC and
LHC energies with initial conditions given by
HIJING Monte
Carlo simulations @4#, denoted as set 1 in Table I. We find
that there is stronger J/
c
suppression at LHC than at RHIC
energy, due both to the higher initial parton densities and
longer lifetime of the parton plasma. The increase of the
survival probabilities with J/
c
transverse momentum is a
consequence of the decrease of the thermal cross section
with increasing P
T
at high temperatures, as shown in Fig. 3,
and the shorter time spent by a higher-P
T
J/
c
inside the
parton plasma, an effect first considered in Ref. @32#. J/
c
dissociation during the late stage of the evolution should
have a peak and increase a little with P
T
when the tempera-
ture drops below 0.3 GeV, as illustrated in Fig. 3. This be-
havior flattens the total survival probability at small values of
P
T
when we integrate over the entire history of the evolution
from high initial temperatures. For a parton system with a
low initial temperature ~below 300 MeV!, the P
T
depen-
dence of the survival probability should be even flatter. One
can therefore use the P
T
dependence to shed light on the
initial temperature and the evolution history of the system.
To demonstrate the effects of the chemical nonequilib-
rium in the initial system, we also show in Fig. 4~b! the
survival probabilities in an ideal parton gas with the same
initial temperatures as before, but with full chemical equilib-
rium ~unit initial fugacities!. In this case, the temperature
simply decreases like T(
t
)5 T(
t
0
)(
t
0
/
t
)
1/3
. We see that the
J/
c
is now much more suppressed than in the case of an
equilibrating parton plasma, because of both the higher par-
ton density and the longer lifetime of the system.
FIG. 4. ~a! The survival probability of J/
c
in an equilibrating
parton plasma at RHIC and LHC energies with initial conditions
given as set 1 in Table I and ~b! for an initially equilibrated plasma
at the same temperatures.
TABLE I. Different sets of initial conditions of the temperature,
fugacities, and parton number densities at
t
0
5 0.7 fm/c for RHIC
and
t
0
5 0.5 fm/c for LHC.
RHIC~1! LHC~1! RHIC~2! LHC~2! RHIC~3! LHC~3!
T ~GeV! 0.55 0.82 0.55 0.82 0.4 0.72
l
g
0.05 0.124 0.2 0.496 0.53 0.761
l
q
0.008 0.02 0.032 0.08 0.083 0.118
n
g
~fm
2 3
) 2.15 18 8.6 72 8.6 72
n
q
~fm
2 3
) 0.19 1.573 0.76 6.29 0.76 6.29
3054 53
XU, KHARZEEV, SATZ, AND WANG
Since there is considerable uncertainty in the estimate of
the initial parton production by
HIJING Monte Carlo simula-
tions, as discussed in Ref. @9#, we would like to test the
sensitivity of J/
c
suppression to this. We therefore multiply
initial parton number densities by a factor of 4, thus increas-
ing initial parton fugacities. We denote such initial condi-
tions as set 2 in Table I. If the uncertainties in initial condi-
tions are caused by soft parton production from the color
mean fields, the initial effective temperature will decrease.
Therefore, we can alternatively increase the initial parton
density by a factor of 4 and at the same time decrease T
0
to
0.4 and 0.72 GeV at RHIC and LHC energies, respectively.
This leads to higher initial fugacities, listed as set 3 in Table
I. The corresponding survival probabilities calculated with
these two sets of initial conditions are shown in Fig. 5. We
can see that the J/
c
suppression is much stronger if the
initial parton densities are higher. Comparing the solid and
dashed lines, however, shows that the J/
c
suppression is less
sensitive to the variation of the initial temperature and
fugacities as far as the parton densities are fixed.
IV. CONCLUSIONS
To summarize, we have used the cross section of J/
c
dissociation by gluons to calculate the J/
c
suppression in an
equilibrating parton gas produced in high-energy nuclear col-
lisions. The large average momentum in the hot gluon gas
enables gluons to break up the J/
c
, while hadron matter at
reasonable temperature does not provide sufficiently hard
gluons. We find a substantial J/
c
suppression in such a non-
equilibrium partonic medium; however, it is smaller than that
in a fully equilibrated parton plasma. In particular, in an
equilibrating plasma the behavior of the J/
c
-gluon cross sec-
tion at high gluon momenta reduces the J/
c
suppression at
large P
T
.
In addition to the J/
c
dissociation during the equilibra-
tion of the parton plasma, there are other possible sources of
suppression for the actually observed J/
c
’s. As already
noted, nuclear modifications of the QQ
¯
production process,
e.g., through modified gluon distributions in a nucleus
@33,34#, multiple scattering accompanied by energy loss
@35#, or a suppression of the nascent J/
c
before it forms an
actual physical resonance @29# must be taken into account.
Such effects would cause J/
c
suppression in addition to
what we have obtained from the equilibrating parton plasma
and modify the transverse momentum dependence of J/
c
suppression @36#. Moreover, interactions between gluons and
cc
¯
bound states before the kinetic thermalization of the par-
tons could also lead to a substantial J/
c
suppression; this
would depend on the time needed to achieve local momen-
tum isotropy. On the other hand, gluon fusion could also
result in J/
c
production during the evolution of the parton
system, similar to the enhancement of open charm. Although
studies of preequilibrium open charm production indicate
@13,15# that J/
c
production during the parton evolution is
not significant compared to primary production, a consistent
study of J/
c
suppression should include the preequilibrium
production in a form of a master rate equation.
ACKNOWLEDGMENTS
X.X. thanks the Nuclear Theory Group at LBL Berkeley
for their hospitality during his visit, when this work was
carried out. D.K., H.S., and X.-N.W. thank K. J. Eskola for
stimulating discussions. This work was supported by the
U.S. Department of Energy under Contract No. DE-AC03-
76SF00098 and by the German Research Ministry ~BMBW!
under Contract No. 06 BI 721. X.-N.W. was also supported
by the U.S.-Hungary Science and Technology Joint Fund
J.F. No. 378.
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FIG. 5. The survival probability of J/
c
in an equilibrating par-
ton plasma with initial conditions given as set 2 ~solid! and set 3
~dashed! in Table I.
53
3055
J/
c
SUPPRESSION IN AN EQUILIBRATING PARTON PLASMA
