J/ψ production in an equilibrating partonic system
Xiao-Ming Xu
Institute for Nuclear Theory, University of Washington, Box 351550,
Seattle, WA 98195
Nuclear Physics Division, Shanghai Institute of Nuclear Research
Chinese Academy of Sciences, P.O.Box 800204, Shanghai 201800, China
Abstract
Any color singlet or octet cc̄ pair is created at short distances and then expands
to a full size of J/ψ. Such a dynamical evolution process is included here in cal-
culations for the J/ψ number distribution as a function of transverse momentum
and rapidity in central Au-Au collisions at both RHIC and LHC energies. The cc̄
pairs are produced in the initial collision and in the partonic system during the
prethermal and thermal stages through the partonic channels ab→ cc̄[2S+1LJ ] and
ab→ cc̄[2S+1LJ ]x, and then they dissociate in the latter two stages. Dissociation of
cc̄ in the medium occurs via two reactions: (a) color singlet cc̄ plus a gluon turns to
color octet cc̄ , (b) color octet cc̄ plus a gluon persists as color octet. There are mod-
est yields of cc̄ in the prethermal stage at RHIC energy and through the reactions
ab → cc̄[2S+1LJ ] at LHC energy for partons with large average momentum in the
prethermal stage at both collider energies and in the thermal stage at LHC energy.
Production from the partonic system competes with the suppression of the initial
yield in the deconfined medium. Consequently, a bulge within −1.5 < y < 1.5 has
been found for the J/ψ number distribution and the ratio of J/ψ number distribu-
tions for Au-Au collisions to nucleon-nucleon collisions. This bulge is caused by the
partonic system and is thus an indicator of a deconfined partonic medium. Based
on this result we suggest the rapidity region worth measuring in future experiments
at RHIC and LHC to be −3 < y < 3.
PACS codes: 24.85.+p, 12.38.Mh, 25.75.Dw, 25.75.Gz
Keywords: Ultrarelativistic nucleus-nucleus collisions, Equilibrating partonic
system, J/ψ number distribution, Survival probability
http://arxiv.org/abs/0704.0668v1
1. Introduction
A hot deconfined medium favors the dissociation of J/ψ since enough hard gluons
can overcome the large energy gap between the J/ψ and a continuum state of cc̄ [1].
Models based on perturbative QCD have shown that a dense partonic system can be pro-
duced in central Au-Au collisions at RHIC and LHC energies [2-6] and then evolve toward
thermal equilibrium and likely chemical equilibrium [7-11]. Such parton plasmas will be
searched for soon in experiments at Brookhaven National Laboratory Relativistic Heavy
Ion Collider (RHIC). The J/ψ suppression has been taken as a thermometer to identify
the evolution history of a parton plasma by showing transverse momentum dependence
of the survival probability in the central rapidity region [12].
Charmonium melting inside a hot medium, which leads to J/ψ suppression, was
proposed by Matsui and Satz to probe the existence of the quark-gluon plasma [13].
Before the complete formation of charmonium is achieved, a pre-resonant cc̄ is expanding
from a collision point. Dominance of the color octet plus a collinear gluon configuration
in the pre-resonance state [14] may account for the same suppression of ψ′ and J/ψ
production in proton-nucleus collisions [15]. The growth of the color octet configuration
and its interaction with nucleons along its trajectory in a nucleus are essential ingredients
in explaining measured J/ψ production cross sections. In addition, the importance of color
octet configurations has been verified in pp̄ collisions at center-of-mass energy
s = 1.8
TeV with the CDF detector at Fermilab [16]. Theoretically, the color-octet production
at short distances and its evolution into physical resonances has been well formulated in
nonrelativistic QCD [17]. At the collision energies of RHIC and LHC we can reasonably
expect considerable contributions from the color octet mechanism.
The evolution of ultrarelativistic nucleus-nucleus collisions, e.g. central Au-Au colli-
sions at both RHIC and LHC energies, has been divided into three stages in Refs. [9,18,19]:
(a) an initial collision where a parton gas is produced; (b) a prethermal stage where elas-
tic scatterings among partons lead to local momentum isotropy [20]; (c) a thermal stage
where parton numbers increase until freeze-out. The term ’partonic system’ refers to
the assembly of partons in the prethermal and thermal stages. The parton plasma only
denotes the assembly of partons in the thermal stage. The cc̄ pairs are produced in the
initial collision, prethermal and thermal stages but disintegrate in the latter two stages. In
order to understand and make predictions for J/ψ yields of RHIC and LHC experiments,
the following physical processes are taken into account. (a) In the initial collision, cc̄ pairs
are produced in hard and semihard scatterings between partons from incoming nuclei by
2 → 2 processes which start at order α3s through the partonic channels ab→ cc̄[2S+1LJ ]x.
In the prethermal and thermal stages, cc̄ pairs can also be produced in 2 → 1 collisions
which start at order α2s via the partonic channels ab → cc̄[2S+1LJ ] since partons in the
deconfined medium have large transverse momenta. (b) The cc̄ produced at short distance
is in a color singlet (cc̄)1 or color octet (cc̄)8 configuration which has a certain probability
to evolve nonperturbatively into a color singlet state. This J/ψ production process is
formulated in nonrelativistic QCD. (c) Since the color octet to singlet transition of (cc̄)8
takes time, gluons in the partonic medium couple to the color octet state and destroy this
transition process. Normally, dissociation cross sections for g+ cc̄→ (cc̄)8 depend on the
pair size. Expansion of the cc̄ from a collision point to a full J/ψ size has to be taken into
account.
A physical resonance formed by a cc̄ pair may be one of J/ψ, χcJ , ψ
′ and others.
Since the radiative transition from a higher charmonium state to the J/ψ takes a much
longer time, the transition of such a state with nonzero pT takes place outside the partonic
system. Since the Fermilab Tevatron experiments have been able to separate direct J/ψ’s
from those produced in radiative χcJ decays [16], in this work we assume that the direct
J/ψ production can also be extracted in heavy ion measurements. If χcJ and ψ
′ are
considered, suppression factors for χcJ and ψ
′ in a deconfined medium are included in
prompt J/ψ production. The identification of J/ψ suppression in the medium becomes
impossible for any prompt J/ψ production data. Therefore, no contributions from higher
charmonium states are taken into account in this work.
The purpose of this work is to study the dependence of the J/ψ survival probability
and number distributions produced in central Au-Au collisions at RHIC and LHC energies
on the transverse momentum and also rapidity which will be measured in RHIC experi-
ments [21]. The J/ψ number distributions corresponding to production of cc̄ in the initial
collision are given in Section 2. Since nuclear shadowing has been shown to influence J/ψ
production in proton-nucleus collisions [22], the nuclear modification of parton distribu-
tions is considered. The J/ψ number distributions due to cc̄ production in the prethermal
and thermal stages are given in Sections 3 and 4. Section 5 contains dissociation cross
sections for gluon-(cc̄)1 and gluon-(cc̄)8. Numerical results for nucleon-cc̄ cross sections,
J/ψ number distributions and four ratios including survival probability are presented in
Section 6. Conclusions are summarized in the final section.
2. Initial production of cc̄
Intrinsic transverse momenta of partons inside a nucleon result in the production of
J/ψ with typical momenta comparable to the QCD scale via 2 → 1 partonic scattering
processes [23]. Since we want to study J/ψ productions with pT > 2 GeV, contributions
from 2 → 1 partonic reactions are not considered in the initial nucleon-nucleon colli-
sion. The effect of intrinsic transverse momentum smearing is rather modest for large
transverse momentum J/ψ data from the Tevatron [24]. Upon omission of the intrinsic
transverse momentum, differential cross section for J/ψ production in nucleon-nucleon
collision resulting only from 2 → 2 partonic processes is given as
dydyxdp⊥
= 2p⊥
xaxbfa/N (xa)fb/N(xb)
(ab→ cc̄[2S+1L(1)J ]x → J/ψ)
(ab → cc̄[2S+1L(8)J ]x → J/ψ)] (1)
where the summation
abx is over partons labeled by a, b, x,
(1) for all possible color-
singlet states and
(8) for all possible color-octet states. Here
denotes the partonic
differential cross section for producing a cc̄[2S+1LJ ] and evolving to a J/ψ with spectro-
scopic notation for quantum numbers and superscripts for singlet and octet [23, 25], and
fa/N is the parton distribution function of the species a in a free nucleon. The longitudinal
momentum fractions carried by initial partons, xa and xb, are related to rapidities of cc̄
and x, y and yx, by
y + p⊥e
yx), xb =
−y + p⊥e
where
s, p⊥ and m⊥ are the center-of-mass energy of nucleon-nucleon collision, trans-
verse momentum and transverse mass of the J/ψ. The conditions xa < 1 and xb < 1
restrict yx to a region of
s−m⊥e−y
< yx < ln
s−m⊥ey
These 2 → 2 processes at order α3s , gg → cc̄[2S+1LJ ]g, qq̄ → cc̄[2S+1LJ ]g, gq → cc̄[2S+1LJ ]q
and gq̄ → cc̄[2S+1LJ ]q̄, start in initial nucleus-nucleus collisions and proceed with the
expansion of the heavy pair. While a cc̄ propagates inside a prethermal or thermal partonic
system, gluons hit and excite it to continuum states. Let σ
gcc̄[13S
be the cross section for
g+(cc̄)[13S
1 ] → (cc̄)8, σgcc̄[S(8)] for g+(cc̄)[S(8)] → (cc̄)8 and σgcc̄[P (8)] for g+(cc̄)[P (8)] →
(cc̄)8 respectively. The cross sections are calculated in Section 5. The probability for
dissociation of a small-size cc̄ into a free state relies on the relative velocity between the
gluon and cc̄, vrel, and gluon number densities in the prethermal and thermal stages, ng(x)
and ng(τ), respectively. Here the variables x and τ are individually space-time coordinates
and proper time. In the prethermal stage, parton distributions depend on the correlation
between momentum and space-time coordinates [18, 19]. The dependence of the gluon
number density ng(x) on x characterizes the partonic system in nonequilibrium. In the
thermal stage, thermal parton distributions can be approximated by Jüttner distributions
where the temperature and parton fugacities depend only on the proper time [9, 18, 20]. As
a consequence, the gluon number density is only a function of τ . Including cc̄ suppression
in the partonic system, the finally-formed number distribution of J/ψ resulting from cc̄
pairs produced in the initial central A+B collision is given by
dN2→2ini
dyd2p⊥
s−m⊥e
s−m⊥e
xafa/A(xa, m
⊥, ~r)xbfb/B(xb, m
⊥,−~r)
(ab→ cc̄[3S(1)1 ]x → J/ψ)
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[13S(1)
(k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[13S(1)1 ]
(k · u) >the θ(d− VT∆t)]
(ab→ cc̄[3S(8)1 ]x → J/ψ)
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[S(8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
(ab→ cc̄[1S(8)0 ]x → J/ψ)
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[S(8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
(ab→ cc̄[3P (8)J ]x → J/ψ)
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[P (8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[P (8)](k · u) >the θ(d− VT∆t)]}
where fa/A is the parton distribution function of a nucleus,
fa/A(x,Q
2, ~r) = TA(~r)Sa/A(x,~r)fa/N (x,Q
2) (3)
with the thickness function TA and nuclear parton shadowing factor Sa/A. Here, RA
is the nuclear radius. The symbols < · · · >pre and < · · · >the denote averages over
gluon distributions in the prethermal and thermal stages, respectively. Along the track of
nucleus-nucleus collisions, a deconfined partonic gas is produced from scatterings among
primary partons at τ0, then reaches thermalization at τiso and finally freezes out at τf .
Here, d is the shortest distance which a cc̄ travels from a production point ~r to the surface
of the partonic medium with transverse velocity VT [12]. Suppose a cc̄ is produced at a
proper time τ and a spatial rapidity η. The time ∆t for the partonic system to evolve to
another proper time τ ′ is
(V‖ sinh η − cosh η)τ +
(sinh η − V‖ cosh η)2τ 2 + (1− V 2‖ )τ ′2
1− V 2‖
where V‖ is the longitudinal component of the cc̄ velocity. The disappearance of medium
interactions on the cc̄ is ensured by the step function θ while this pair escapes from the
partonic medium.
3. Production of cc̄ in the prethermal stage
To order α2s, a cc̄ in a color singlet state is produced only through gluon fusion
gg → cc̄[2S+1L(1)J ]. For the cc̄[3S
1 ], this fusion does not occur. In contrast, color octet
states result from both channels gg → cc̄[2S+1L(8)J ] and qq̄ → cc̄[2S+1L
J ]. Nevertheless,
the number densities of quarks and antiquarks are so small that they are neglected in es-
timating the production of cc̄ in the prethermal stage where gluons dominate the partonic
system. Four momenta of the two initial partons and final cc̄ are denoted by k1 = (ω1, ~k1),
k2 = (ω2, ~k2) and p = (E, ~p) = (m⊥ cosh y, ~p⊥, m⊥ sinh y). The differential production rate
for gg → cc̄[2S+1L(8)J ] → J/ψ in the prethermal stage is
d3A2→1pre
8(2π)5
δ(4)(k1 + k2 − p)
g2Gfg(k1, x)fg(k2, x)
| M(gg → cc̄[2S+1L(8)J ] → J/ψ) |2
where gG is the degeneracy factor for gluons and the fg(k, x) is the correlated phase-space
distribution function given in Ref. [18]. The squared amplitudes | M |2 for cc̄ in color
singlet and color octet are calculated individually in Refs. [23, 25]. To order α2s, the
allowed color octet states are 1S
0 and
0,2 through the gluon fusion channel. Taking
into account the suppression of cc̄ in the prethermal and thermal stages, the finally-formed
number distribution of J/ψ resulting from cc̄ pairs produced through 2 → 1 processes in
the prethermal stage is given by
dN2→1pre
dyd2p⊥
16(2π)5
∫ τiso
τdτdηdφk1dyk1
g2Gfg(k1, x)fg(k2, x)
{| M(gg → cc̄[1S(8)0 ] → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[S(8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
+ | M(gg → cc̄[3P (8)J ] → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[P (8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[P (8)](k · u) >the θ(d− VT∆t)]}
where φki is the angle between
~k⊥i and ~p⊥ for i = 1, 2 and mc is the charm quark mass.
The kinematic variables k⊥1, k⊥2, φk2 and yk2 are expressed in terms of
k⊥1 =
m⊥ cosh(y − yk1)− p⊥ cosφk1
k⊥2 =
p2⊥ + k
⊥1 − 2p⊥k⊥1 cosφk1
sin φk2 = −
sinφk1
sinh yk2 =
(m⊥ sinh y − k⊥1 sinh yk1)
To order α3s, the differential production rate gets contributions from the processes
gg → cc̄[2S+1L(1,8)J ]g in the prethermal stage,
d3A2→2pre
16(2π)8
δ(4)(k1 + k2 − p− px)
g2Gfg(k1, x)fg(k2, x){
| M(gg → cc̄[2S+1L(1)J ]x → J/ψ) |2
| M(gg → cc̄[2S+1L(8)J ]x → J/ψ) |2} (7)
where px = (Ex, ~px) = (p⊥x cosh yx, p⊥x cosφx, p⊥x sin φx, p⊥x sinh yx) is the four momen-
tum of the massless parton x. Taking into account the suppression of cc̄ in the prethermal
and thermal stages, the finally-formed number distribution of J/ψ resulting from cc̄ pairs
produced through 2 → 2 processes in the prethermal stage is given by
dN2→2pre
dyd2p⊥
16(2π)8
∫ τiso
τdτdηp⊥xdp⊥xdφxdyxdφk1dyk1
2k2⊥1
g2Gfg(k1, x)fg(k2, x)
{| M(gg → cc̄[3S(1)1 ]x → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[13S(1)1 ]
(k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[13S(1)1 ]
(k · u) >the θ(d− VT∆t)]
+ | M(gg → cc̄[3S(8)1 ]x → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[S(8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
+ | M(gg → cc̄[1S(8)0 ]x → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[S(8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
+ | M(gg → cc̄[3P (8)J ]x → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[P (8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[P (8)](k · u) >the θ(d− VT∆t)]}
where ŝ = (k1 + k2)
2 and some kinematic variables are given by
k⊥1 = {4m2c + 2m⊥p⊥x cosh(y − yx)− 2p⊥p⊥x cosφx}/
{2[m⊥ cosh(y − yk1) + p⊥x cosh(yx − yk1)− p⊥ cosφk1 − p⊥x cos(φx − φk1)]}
k2⊥2 = m
⊥ + p
⊥x + 2m⊥p⊥x cosh(y − yx) + k2⊥1
−2k⊥1[m⊥ cosh(y − yk1) + p⊥x cosh(yx − yk1)]
sinh yk2 =
[m⊥ sinh y + p⊥x sinh yx − k⊥1 sinh yk1]
The J/ψ number distribution resulting from cc̄ pairs produced in the prethermal
stage becomes
dNpre
dyd2p⊥
dN2→1pre
dyd2p⊥
dN2→2pre
dyd2p⊥
4. Production of cc̄ in the thermal stage
In the thermal stage, parton distributions are approximated by thermal phase-space
distributions fi(k;T, λi) in which the temperature T and nonequilibrium fugacities λi are
functions of the proper time τ [9, 18]. While the partonic system evolves, quark and
antiquark number densities increase. To order α2s, both gg → cc̄[2S+1L
J ] → J/ψ and
qq̄ → cc̄[2S+1L(8)J ] → J/ψ contribute to the J/ψ number distribution in the thermal stage
dN2→1the
dyd2p⊥
16(2π)5
τdτdηdφk1dyk1
g2Gfg(k1;T, λg)fg(k2;T, λg) | M(gg → cc̄[1S
0 ] → J/ψ) |2
exp[−
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
g2Gfg(k1;T, λg)fg(k2;T, λg) | M(gg → cc̄[3P
J ] → J/ψ) |2
exp[−
dτ ′ng(τ
′) < vrelσgcc̄[P (8)](k · u) >the θ(d− VT∆t)]
+gqgq̄fq(k1;T, λq)fq̄(k2;T, λq̄) | M(qq̄ → cc̄[3S(8)1 ] → J/ψ) |2
exp[−
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]}
where gq and gq̄ are the degeneracy factors for quarks and antiquarks, respectively. In the
channel of quark-antiquark annihilation, only the squared amplitude for 3S
1 does not
vanish.
All lowest-order 2 → 2 reactions gg → cc̄[2S+1LJ ]g, qq̄ → cc̄[2S+1LJ ]g, gq →
cc̄[2S+1LJ ]q and gq̄ → cc̄[2S+1LJ ]q̄ contribute to the J/ψ number distribution in the ther-
mal stage
dN2→2the
dyd2p⊥
16(2π)8
τdτdηp⊥xdp⊥xdφxdyxdφk1dyk1
2k2⊥1
fa(k1;T, λa)fb(k2;T, λb)gagb
| M(ab→ cc̄[2S+1L(1)J ]x → J/ψ) |2
exp[−
dτ ′ng(τ
′) < vrelσgcc̄[12S+1L(1)
(k · u) >the θ(d− VT∆t)]
| M(ab→ cc̄[2S+1L(8)J ]x → J/ψ) |2
exp[−
dτ ′ng(τ
′) < vrelσgcc̄[L(8)](k · u) >the θ(d− VT∆t)]}
The J/ψ number distribution resulting from cc̄ pairs produced in the thermal stage
becomes
dNthe
dyd2p⊥
dN2→1the
dyd2p⊥
dN2→2the
dyd2p⊥
5. Gluon-cc̄ dissociation cross sections
A dissociation cross section of a full-size J/ψ induced by a gluon is given in Refs. [1,
26]. Since an initially-created cc̄ has a radius of about r0 =
and proceeds by expanding
to a full-size object, the dissociation cross section of cc̄ by a gluon has a size dependence.
By this we mean the dissociation of cc̄ into free states via this process g + cc̄ → (cc̄)8.
Cross sections are calculated with chromoelectric dipole coupling between gluon and cc̄ in
the procedure for gluon-J/ψ dissociation in Ref. [26]. The wave function of an expanding
cc̄ is needed for this purpose, but it has not been investigated in the partonic medium even
though some attempts have been made in studies of the color transparency phenomenon
[27]. We proceed with the construction of wave functions in a simple one-gluon-exchange
potential model.
In a parton plasma, the internal motion of J/ψ is obtained [12] from the attractive
Coulomb potential, V1 = −gs2/3πr. The quantum-mechanical interpretation of the cc̄
radius is
< r2 >, the square root of the radius-square expectation value of the relative-
motion wave function. For the 1S color singlet, its wave function in momentum space
normalized to the radius of cc̄[13S
1 ] is
[~rψ1s](~k) = 32
πa2.50
(1 + (ka0)2)3
where the variable a0 =
< r2 > /3 is the Bohr radius for a full-size J/ψ. The velocity-
square expectation value of the J/ψ wave function is < v2 >= 0.428. Then the radius
of cc̄ is assumed to expand according to
< r2 > =
< v2 >t+ r0. The gluon-cc̄[1
dissociation cross section is
gcc̄[13S
128gs
2m2.5c a
0 − ǫ0)1.5Q0
9[mca
0 − ǫ0) + 1]6
where Q0 is the gluon energy, ǫ0 the binding energy of J/ψ and gs the strong coupling
constant.
While the cc̄ is in a color octet state, it is not a bound state but rather a scattering
state. Its relative-motion wave function is determined by the repulsive potential V8 =
g2s/24πr. The radial part of the S wave function is
SR(r) + iSI(r) = e
iqrF (1 + iη, 2,−2iqr) (15)
and the radial part of the P wave function is
PR(r) + iPI(r) = qre
iqrF (2 + iη, 4,−2iqr) (16)
with q = mc
< v2 > and η = g2s/24π
< v2 >. The function F is the confluent hyper-
geometric function. Wave functions in momentum space are obtained by performing a
Fourier transform of the wave functions in space coordinates. Normalization constants of
the momentum-space wave functions, CS and CP , are determined by fitting the cc̄ radius.
Dissociation cross sections of the S-wave and P -wave color-octet states by a gluon are
σgcc̄[S(8)] =
g2s (mcQ
0)1.5
dr1dr2r
mcQ0r1)j1(
mcQ0r2)[SR(r1)SR(r2) + SI(r1)SI(r2)]
σgcc̄[P (8)] =
g2s (mcQ
0)1.5
dr1dr2
mcQ0r1)j0(
mcQ0r2) + 2j2(
mcQ0r1)j2(
mcQ0r2)]
[PR(r1)PR(r2) + PI(r1)PI(r2)]
where the j0, j1 and j2 are spherical Bessel functions. The b is determined so that the
square root of the r2 expectation value of the relative wave function in Eq. (15) or (16) is
the color-octet radius. Relations b = 1.435
< r2 > for S-wave and b = 1.3
< r2 > for
P -wave approximately hold for color-octet size less than normal hadron size.
6. Numerical results and discussions
Results for five aspects are presented in the following subsections. The first aspect
is the nucleon-cc̄ dissociation cross sections shown in the next subsection. The second one
in Subsection 6.2 is J/ψ number distributions versus transverse momentum at y = 0 and
rapidity at pT = 4 GeV with nuclear effect on parton distributions and cc̄ dissociation
in the partonic system. The third one in Subsection 6.3 is to define and calculate four
ratios including survival probability with y = 0 or pT = 4 GeV at both RHIC and LHC
energies. The fourth one is given in Subsection 6.4 to show J/ψ number distributions
without nuclear effect on parton distributions and cc̄ dissociation in the partonic system.
The fifth one concerns some uncertainties on the above results.
6.1. Nucleon-cc̄ dissociation cross sections
In the parton model of the nucleon, the gluon is a dominant ingredient. Whereas
the cross section for cc̄ dissociated directly by a real gluon is of order αs, the cross section
for the quark-cc̄ dissociation through a virtual gluon is of order α2s. With the gluon-(cc̄)1
cross section given in the last section, the nucleon-cc̄[13S
1 ] cross section driven mainly
by the gluon ingredient becomes
Ncc̄[13S
dxfg/N(x,Q
gcc̄[13S
where x
min =
with pN being the proton momentum in the rest frame of the J/ψ. The
gluon distribution function fg/N is that Glück-Reya-Vogt (GRV) result at leading order
in Ref. [28]. The cross section is drawn in Fig. 1 to show the energy and renormalization-
scale dependence while the cc̄ radius is the J/ψ radius in the attractive Coulomb poten-
tial, rJ/ψ = 0.348 fm. In Fig. 1, gluon field operators are renormalized at three scales
2ǫ0, Q
0, respectively. The coupling constant has the value αs =
correspond-
ing to the scale ǫ0 while it varies for the other two scales. Values of the cross section
s = 10 GeV are a little lower than the nucleon-J/ψ dissociation cross section ob-
tained by the subtraction of quasi-elastic cross section in Ref. [29] from the total cross
section given in Ref. [30]. In high-temperature hadronic matter or J/ψ photoproduction
reaction, a typical value of the center-of-mass energy for nucleon-J/ψ (or preresonance)
dissociation is around
s = 6 GeV [1]. At this energy, Fig. 2 is drawn to show the size
dependence of σ
Ncc̄[13S
, with fg/N (x,Q
2) at Q2 = ǫ20.
For the S-wave color octet the nucleon-cc̄[S(8)] cross section is
σNcc̄[S(8)] =
dxfg/N(x, (q +Q
0)2)σgcc̄[S(8)] (20)
where x
min =
. For the P -wave color octet the nucleon-cc̄[P (8)] cross section is
σNcc̄[P (8)] =
dxfg/N(x, (q +Q
0)2)σgcc̄[P (8)] (21)
Since the gluon momentum in a confining medium is bigger than the QCD scale [14], the
lowest value of x is set by ΛQCD = 0.2322 GeV used in the leading order GRV parton
distribution functions. Dependences of σNcc̄[S(8)] and σNcc̄[P (8)] on the center-of-mass energy√
s are depicted in Fig. 3 while the size of (cc̄)8 is the full size of J/ψ. The dot-dashed
line is obtained with the nucleon-J/ψ cross section given by Eq. (24) in Ref. [1] where
another gluon distribution function evaluated at Q2 = ǫ20 is used. While the (cc̄)8 has
small momentum in a nucleus, the cross section for nucleon-(cc̄)8 production is lower than
the absorption cross section determined by Gerschel and Hüfner [31] or the two-gluon
exchange result [32]. It was proposed by Kharzeev and Satz that the color octet plus a
gluon configuration is a dominant component produced in the proton-nucleus collisions
[14]. In fact, the present cross section for a nucleon and a bare (cc̄)8 is one part of the
nucleon-g(cc̄)8 cross section.
A (cc̄)8 pair produced at a collision point expands before becoming color singlet to
a size which may be larger or smaller than the full size of J/ψ. We then show in Fig. 4
the dependence of the nucleon-(cc̄)8 cross section on the color-octet pair radius.
We do not want to address proton-nucleus collisions in terms of nucleon-cc̄ cross
sections [33, 34] since only the gluon-cc̄ cross sections are needed to study cc̄ suppression
in the prethermal and thermal stages. In proton-nucleus collisions, once a color-octet
(cc̄)8 pair is produced, it picks up a collinear gluon to form a colorless configuration [14].
However, in central Au-Au collisions at RHIC and LHC energies, the accompanying gluon
scatters with other hard gluons in the dense partonic system and is driven away. Therefore
the bare cc̄ is the object that we want to study in the partonic system.
6.2. J/ψ number distributions with suppression
J/ψ number distributions versus transverse momentum at y = 0 and rapidity at
pT = 4 GeV for central Au-Au collisions at RHIC energy
s = 200AGeV are calculated
with respect to the initial collision, prethermal and thermal stages. Initial productions
of cc̄ are calculated with GRV parton distribution functions at renormalization scale µ =
p2⊥ + 4m
c . Evolution of color-octet states
1S0 and
3PJ toward the J/ψ is specified
by nonperturbative matrix elements < OJ/ψ8 (3S1) >, < O
1S0) > and < OJ/ψ8 (3P0) >
in nonrelativistic QCD [17]. In the nonperturbative evolution, a gluon from the partonic
system hits and prevents the color octet from color neutralizing via g + (cc̄)8 → (cc̄)8.
This medium effect has been expressed by exponentials in Eqs. (2), (6), (8), (10) and
(11). Therefore, the nonperturbative matrix elements are assumed to be invariant while
the medium effect is factorized into exponential forms. Values of these matrix elements
are well determined by fitting the CDF measurements for pp̄ collisions at
s = 1.8 TeV
in Ref. [35],
< OJ/ψ8 (3S1) >= (1.12± 0.14)× 10−2GeV3 (22)
< OJ/ψ8 (1S0) > +
< OJ/ψ8 (3P0) >= (3.90± 1.14)× 10−2GeV3 (23)
In pp̄ collisions, differential cross sections of direct J/ψ production depend on the com-
bination of < OJ/ψ8 (1S0) > and < O
3P0) >. However, since the dissociation cross
section for the S-wave color-octet state is different from that for the P -wave color-octet
state, such a dependence on the combination is destroyed. In calculations, values are
taken as follows,
< OJ/ψ8 (1S0) >= 4× 10−2GeV3, < O
3P0) >= −
× 10−2GeV3 (24)
The value of < OJ/ψ8 (3P0) > is positive at tree level and negative after renormalization
[36]. Eq. (23) is still satisfied by the values in Eq. (24). These values of nonperturbative
matrix elements are supposed to be universal for any center-of-mass energy
Various contributions to the J/ψ number distributions including initial collisions,
prethermal and thermal stages, 2 → 1 and 2 → 2 collisions, are drawn separately in Figs.
5 and 6. The dashed curve resulting from cc̄ production in the initial collision is obtained
by calculating Eq. (2) where the nuclear parton shadowing factor is given in Ref. [4]
throughout this subsection. The upper and lower dot-dashed curves resulting from cc̄
production in the prethermal stage are obtained by individually calculating Eq. (6) for
2 → 1 collisions and Eq. (8) for 2 → 2 collisions. The upper and lower dotted curves
resulting from cc̄ production in the thermal stage are obtained by calculating Eq. (10)
for 2 → 1 collisions and Eq. (11) for 2 → 2 collisions, respectively. To exclude the effect
of intrinsic transverse momentum smearing, only the region pT > 2 GeV is considered.
Consequently, no 2 → 1 collisions contribute in the initial collision. The J/ψ number
distribution resulting from the initial collision shown by the dashed line has a plateau
similar to that in proton-proton collision [37]. Both Figs. 5 and 6 show that cc̄ pairs
produced from the thermal stage can be neglected compared to the initial production, but
the contributions from the prethermal stage are important in the transverse momentum
region 2GeV < pT < 8GeV and rapidity region 0 < y < 1.2. Productions of cc̄ in the
prethermal and thermal stages bulge up the J/ψ number distribution shown by the solid
line in this rapidity region. Nevertheless, the dot-dashed and dotted lines fall rapidly as
the rapidity gets large. This bulging characterizes the formation of a deconfined medium
because the medium has average momentum limited but big enough to produce extra cc̄
and thus J/ψ. The 2 → 1 collisions in the partonic system have bigger contributions than
the 2 → 2 collisions.
Each of Figs. 7 and 8 contains two sets of lines to show contributions from the
color-singlet and color-octet pairs produced at short distance. Any set has a dashed line
obtained from Eq. (2) for the initial collision, a dot-dashed line from Eq. (9) for the
prethermal stage and a dotted line from Eq. (12) for the thermal stage, respectively. A
line in the upper (lower) set for the color-octet (color-singlet) contributions stems from
the terms for (cc̄)8 ((cc̄)1) states. The color octet states dominate productions of J/ψ at
RHIC energy. However, the ratio of color-octet to color-singlet contributions shown by the
two solid lines at pT = 6 GeV is reduced from about 70 at CDF collider energy
s = 1.8
TeV to about 40 at RHIC energy. Both contributions of color-singlet and color-octet
states have similar dependence on transverse momentum and rapidity.
Figs. 9 and 10 show transverse momentum and rapidity dependence of J/ψ number
distributions for central Au-Au collisions at LHC energy
s = 5.5ATeV. A prominent
feature is that the J/ψ number produced from the thermal stage is comparable to that
from the prethermal stage. Compared to the initial production, cc̄ and J/ψ produced
through 2 → 2 reactions may be neglected. A bulge is observed on the plateau in the
rapidity region 0 < y < 1.5. Such a bulge can be taken as a signature for the existence
of a parton plasma at the LHC energy. Figs. 11 and 12 depict contributions from the
color singlet and color octet at LHC energy. The ratio of color-octet to color-singlet
contributions shown by the two solid lines at pT = 6GeV reaches about 150. This indicates
the color octet states become more important with the increase of
6.3. Ratios including J/ψ survival probability
Nuclear shadowing results in a modification of gluon distribution functions inside
a nucleus [38] and such a nuclear effect is represented by the shadowing factor Sa/A in
Eq. (3). If the Sa/A = 1 for no shadowing, the dN
ini /dyd
2p⊥ is proportional to the
product of atomic masses of the two colliding nuclei. If Sa/A 6= 1 and depends on the
longitudinal momentum fraction x, the production of cc̄ is reduced in the shadowing region
and enhanced for the anti-shadowing region. Irrespective of interactions of J/ψ with the
partonic system, J/ψ number distributions produced in the initial central A+B collision
is obtained by putting all exponentials equal to 1 in Eq. (2),
dN2→20
dyd2p⊥
(Sa/A) = 2
s−m⊥e
s−m⊥e
xafa/A(xa, m
⊥, ~r)xbfb/B(xb, m
⊥,−~r)
(ab→ cc̄[3S(1)1 ]x → J/ψ)
(ab→ cc̄[3S(8)1 ]x → J/ψ)
(ab→ cc̄[1S(8)0 ]x → J/ψ)
(ab→ cc̄[3P (8)J ]x → J/ψ)}
To characterize the influence of nuclear parton shadowing on the J/ψ production from
the initial collision, a ratio is defined as
Rini =
dN2→20
dyd2p⊥
(Sa/A 6= 1)/
dN2→20
dyd2p⊥
(Sa/A = 1) (26)
Here the Sa/A 6= 1 from Ref. [4] applies throughout this subsection.
The initially produced J/ψ originates from the cc̄ pairs produced in the initial colli-
sion. Its dependence on the transverse momentum and rapidity is obtained by calculating
Eq. (25). Some cc̄ pairs produced in the initial collision may dissociate by gluons from the
partonic system. As a consequence, the J/ψ number is reduced. The survival probability
for the cc̄ transiting into a J/ψ is defined as the ratio
Splasma =
dN2→2ini
dyd2p⊥
(Sa/A 6= 1)/
dN2→20
dyd2p⊥
(Sa/A 6= 1) (27)
We have calculated the J/ψ number distributions produced in the prethermal and
thermal stages in Subsection 6.2. The J/ψ yield may be bigger than the reduced amount
of initially produced J/ψ due to the cc̄ dissociation by gluons in the partonic system. The
partonic system has two roles. One is to produce cc̄ pairs and another is to dissociate cc̄
pairs. To see the roles, a ratio is defined by
Rplasma = (
dN2→2ini
dyd2p⊥
(Sa/A 6= 1) +
dNpre
dyd2p⊥
dNthe
dyd2p⊥
dN2→20
dyd2p⊥
(Sa/A 6= 1) (28)
To understand the nuclear effect on parton distributions and the roles of the partonic
system, we need to compare J/ψ production in the central A+B collision with that in the
nucleon-nucleon collision. To this end, a ratio is defined as
R = (
dN2→2ini
dyd2p⊥
(Sa/A 6= 1) +
dNpre
dyd2p⊥
dNthe
dyd2p⊥
dN2→20
dyd2p⊥
(Sa/A = 1) (29)
which is also written as
R = RiniRplasma (30)
The ratios Rini, Splasma, Rplasma and R versus transverse momentum and rapidity
are depicted as dashed, dotted, dot-dashed and solid lines, respectively, in Figs. 13 and 14
for the RHIC energy and Figs. 15 and 16 for the LHC energy. In contrast to Rini < 1, the
value of Rplasma is larger than 1 for all transverse momenta in Fig. 13 and 0 < y < 1.5 in
Fig. 14 and 0.5 < y < 1.5 in Fig. 16. This results in prominent bulges on the solid lines
of R in Figs. 14 and 16. In contrast, the survival probability Splasma shown by the dotted
lines has no such bulge. Therefore, the bulges are present in Figs. 6 and 10 when the J/ψ
yield resulting from cc̄ pairs produced in the partonic system overwhelms the reduced
amount of initially produced J/ψ. We conclude that in the rapidity region 0 < y < 1.5
a bulge observed in the ratio R is an indicator for the existence of the partonic system.
For Rini < 1 and Splasma < 1, J/ψ suppression arises from the nuclear parton shadowing
found in HIJING and cc̄ dissociation in the partonic system.
6.4. J/ψ number distributions with no suppression
In Subsection 6.2, J/ψ number distributions have been presented while the cc̄ re-
duction due to the nuclear parton shadowing in the initial collision and cc̄ dissociation in
the partonic system are taken into account. In this subsection, the suppression including
both the reduction and dissociation is omitted in calculations of J/ψ number distributions
by setting to 1 all exponentials in Eqs. (2), (6), (8), (10) and (11). Figs. 17-20 depict
these distributions versus transverse momentum at y = 0 and rapidity at pT = 4 GeV at
both RHIC and LHC energies. We are now ready to explain the dip within y < 1 in Fig.
10. This dip disappears in Fig. 20 where suppression is not considered. Since Rini shown
by the dashed line in Fig. 16 is flat with respect to the rapidity y < 2 and Splasma shown
by the dotted line has a steep rise in 0.5 < y < 1, the dip phenomenon is solely due to the
cc̄ dissociation in the partonic system. Such a dip phenomenon is not obvious but still
can be observed in the prethermal and thermal stages when the J/ψ number distributions
with suppression are compared to those without suppression. The comparison is indicated
in Fig. 21 for the prethermal stage and Fig. 22 for the thermal stage. The solid and
dot-dashed lines for no suppression begin to fall from y = 0.5 to y = 1, but change to
rising as shown by the dashed and dotted lines when the cc̄ dissociation is switched on.
This change occurs because of the steep rise of Splasma. A relatively weak dependence of
Splasma on pT is shown by the dotted line in Fig. 15. The dip phenomenon thus cannot
be observed in the pT dependence of J/ψ number distributions. Referring back to Eqs.
(2), (6), (8), (10) and (11), exponentials there have sensitive dependence on the rapidity
in 0 < y < 1.5. The dip is more obvious in the color singlet channel as shown by the
lower solid, dashed, dot-dashed and dotted lines in Fig. 12. This is so because the cross
section for the 13S
1 -state dissociation has a narrower peak with respect to the incident
gluon energy [12] than the color octet states.
6.5. Uncertainties
Since gluon shadowing in nuclei has not been studied experimentally, theoretical
estimates of the nuclear gluon shadowing factor involve uncertainties. The nuclear parton
shadowing factor found in HIJING [4] is a result of the assumption that there is no Q-
dependence on the shadowing factor and the shadowing effect for gluons and quarks is the
same. Nevertheless, the shadowing factor has been shown by Eskola et al. to evolve with
momentum Q [39]. The difference between the latter and the former indicates uncertainty.
The ratio Rini defined in Eq. (26) is calculated with Eskola et al.’s parametrization [39]
and results are depicted in Fig. 23 showing momentum dependence at y = 0 and Fig.
24 showing rapidity dependence at pT = 4 GeV. Compared to the dashed lines in Figs.
13-16, the change of Rini at RHIC energy greater than 1 is prominent. This implies that
the anti-shadowing effect of Eskola et al.’s parametrization is quite important at RHIC
energy. Measurements on Rini in RHIC experiments are needed to confirm this nuclear
enhancement [21].
The ratio Rini is always flat within the rapidity region -1.5< y < 1.5 for parametriza-
tions given in HIJING and by Eskola et al., and the flatness seems to be independent of
parametrizations. If the partonic system does not come into being, the ratio R is flat, too,
since R = Rini. If the partonic system dissociates cc̄ pairs, the solid curve of R undergoes
bulging, dipping and then bulging from y = −1.5 to y = 1.5. Any such twist of R in
−1.5 < y < 1.5 observed in experiments is nontrivial, because only a deconfined medium
generates it.
Upon inclusion of uncertainties on the formation and evolution of parton plasma
arising from other factors, for instance, the dependence on the coupling constant αs
[40] and transverse flow [41], the J/ψ number distribution and the four ratios including
survival probability will change. In the partonic system considered here gluons dominate
the evolution and gluon-cc̄ interactions break the pairs. In a system where quarks and
antiquarks are abundant, interactions between quarks (antiquarks) and cc̄ may account
for a suppression of J/ψ [42]. Additional suppression caused by energy loss of the initial
state has not been considered since there is a controversy on the influence of the energy
loss [43, 44]. Some uncertainties are expected to be fixed by upcoming experiments at
RHIC.
7. Conclusions
We have studied J/ψ production through both color-singlet and color-octet cc̄ chan-
nels with various stages of central Au-Au collisions at both RHIC and LHC energies. In
addition to the scattering processes ab → cc̄[2S+1LJ ]x, contributions of the reactions
ab→ cc̄[2S+1LJ ] are also calculated in the prethermal stage and thermal stage. The effect
of the medium on an expanding cc̄ involves a gluon interacting with the cc̄ to prevent it
from a transition into a color singlet. Cross sections for g + cc̄ → (cc̄)8 are calculated
with internal wave functions of (cc̄)1 in an attractive potential and (cc̄)8 in a repulsive
potential. Furthermore, nucleon-cc̄ cross sections for color singlet, S- and P - wave color
octets as a function of
s or cc̄ radius are evaluated by assuming that the nucleon domi-
nantly contains gluons. Momentum and rapidity dependence of J/ψ number distribution
with various contributions are calculated for central Au-Au collisions at both RHIC and
LHC energies. Color octet contributions are one order of magnitude larger than the color
singlet contributions. Yields of cc̄ are large in the prethermal stage at RHIC energy and
through the 2 → 1 collisions ab → cc̄[2S+1LJ ] at LHC energy. Since the partonic system
offers fairly large amounts of cc̄, a bulge in 0 < y < 1.5 at RHIC energy and 0.5 < y < 1.5
at LHC energy can be observed in the rapidity dependence of the J/ψ number distribu-
tion and the ratio R of J/ψ number distributions for Au-Au collisions to nucleon-nucleon
collisions. Such a bulge is a signature for the existence of a deconfined partonic medium.
We suggest that RHIC and LHC experiments measure J/ψ number distributions and
the ratio R in the rapidity region 0 < y < 3 to observe a bulge. While the yield of cc̄
from the medium is larger than the reduced amount of initial production in the medium,
the ratio Rplasma is larger than 1. The competition between production and suppression
determines the values of Rplasma, which relies on the evolution of parton number density
and temperature of the partonic system [12]. A dip in the rapidity dependence of the J/ψ
number distributions at LHC energy may exist and this amounts to a suppression effect
of cc̄ in the partonic system. So far, we have obtained results and conclusions for positive
rapidity. It is stressed that the same contents for negative rapidity can be obtained from
the positive region by symmetry.
Acknowledgements
I thank the [Department of Energy’s] Institute for Nuclear Theory at the University
of Washington for its hospitality and the Department of Energy for partial support during
the completion of this work. I thank the Nuclear Theory Group at LBNL Berkeley for
their hospitality during my visit. I also thank X.-N. Wang, C.-Y. Wong and M. Asakawa
for discussions, K. J. Eskola for offering Fortran codes of nuclear parton shadowing factors,
H. J. Weber for careful reading through the manuscript. This work was also supported in
part by the project KJ951-A1-410 of the Chinese Academy of Sciences and the Education
Bureau of Chinese Academy of Sciences.
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http://arxiv.org/abs/hep-ph/9807297
http://arxiv.org/abs/hep-ph/9802350
Figure 1: Solid, dashed and dot-dashed lines are nucleon-cc̄[13S
1 ] cross sections for
fg/N (x,Q
2) evaluated at Q2 = ǫ20, 2ǫ
0, (Q
0)2, respectively. The (cc̄)1 has the same size as
Figure 2: Cross section for nucleon-cc̄[13S
1 ] at
s = 6 GeV as a function of the cc̄ radius
is calculated with fg/N (x,Q
2) evaluated at Q2 = ǫ20.
Figure 3: The solid and dashed lines are cross sections for nucleon-cc̄[S(8)] and nucleon-
cc̄[P (8)] collisions as a function of
s, respectively. The dot-dashed line is the nucleon-J/ψ
cross section calculated with Eq. (24) in Ref. [1]. The corresponding (cc̄)8 and J/ψ have
the same radius.
Figure 4: The solid and dashed lines show radius dependence of cross sections for nucleon-
(cc̄)8[S
(8)] and nucleon-(cc̄)8[P
(8)] collisions, respectively.
Figure 5: J/ψ number distributions versus transverse momentum at y = 0 and RHIC
energy with suppression. The dashed curve corresponds to cc̄ production in the initial
collision. The upper and lower dot-dashed (dotted) curves correspond to cc̄ produced
through 2 → 1 and 2 → 2 reactions in the prethermal (thermal) stage, respectively. The
solid curve is the sum of all contributions.
Figure 6: The same as Fig. 5, except for rapidity distribution at pT = 4 GeV.
Figure 7: J/ψ number distributions versus transverse momentum at y = 0 and RHIC
energy with suppression. The upper and lower dashed (dot-dashed, dotted and solid)
lines correspond to cc̄ in color octet and color singlet, respectively, produced in the initial
collision(prethermal stage, thermal stage and the all three stages).
Figure 8: The same as Fig. 7, except for rapidity distribution at pT = 4 GeV.
Figure 9: J/ψ number distributions versus transverse momentum at y = 0 and LHC
energy with suppression. The dashed curve corresponds to cc̄ productions in the initial
collision. The upper and lower dot-dashed (dotted) curves correspond to cc̄ produced
through 2 → 1 and 2 → 2 reactions in the prethermal (thermal) stage, respectively. The
solid curve is the sum of all contributions.
Figure 10: The same as Fig. 9, except for rapidity distribution at pT = 4 GeV.
Figure 11: J/ψ number distributions versus transverse momentum at y = 0 and LHC
energy with suppression. The upper and lower dashed (dot-dashed, dotted and solid)
lines correspond to cc̄ in color octet and color singlet, respectively, produced in the initial
collision (prethermal stage, thermal stage and all three stages).
Figure 12: The same as Fig. 11, except for rapidity distribution at pT = 4 GeV.
Figure 13: Ratios versus transverse momentum at y = 0 and RHIC energy. The solid,
dashed, dot-dashed and dotted lines are R, Rini, Rplasma and Splasma, respectively.
Figure 14: The same as Fig. 13, except for rapidity distribution at pT = 4 GeV
Figure 15: Ratios versus transverse momentum at y = 0 and LHC energy. The solid,
dashed, dot-dashed and dotted lines are R, Rini, Rplasma and Splasma, respectively.
Figure 16: The same as Fig. 15, except for rapidity distribution at pT = 4 GeV.
Figure 17: J/ψ number distributions versus transverse momentum at y = 0 and RHIC
energy without suppression. The dashed curve corresponds to cc̄ productions in the initial
collision. The upper and lower dot-dashed (dotted) curves correspond to cc̄ produced
through 2 → 1 and 2 → 2 reactions in the prethermal (thermal) stage, respectively. The
solid curve is the sum of all contributions.
Figure 18: The same as Fig. 17, except for rapidity distribution at pT = 4 GeV.
Figure 19: J/ψ number distributions versus transverse momentum at y = 0 and LHC
energy without suppression. The dashed curve corresponds to cc̄ productions in the initial
collision. The upper and lower dot-dashed (dotted) curves correspond to cc̄ produced
through 2 → 1 and 2 → 2 reactions in the prethermal (thermal) stage, respectively. The
solid curve is the sum of all contributions.
Figure 20: The same as Fig. 19, except for rapidity distribution at pT = 4 GeV.
Figure 21: J/ψ number distributions versus rapidity at pT = 4 GeV in the prethermal
stage of LHC energy. The dashed and solid lines individually correspond to cc̄ produced
through 2 → 1 reactions with and without suppression. The dotted and dot-dashed lines
through 2 → 2 reactions with and without suppression, respectively.
Figure 22: The same as Fig. 21, except for the thermal stage.
Figure 23: Ratio Rini versus transverse momentum at y = 0 is calculated with Eskola’s
parametrization.
Figure 24: The same as Fig. 23, except for rapidity distribution at pT = 4 GeV.

  Any color singlet or octet ccbar pair is created at short distances and then
expands to a full size of J/psi. Such a dynamical evolution process is included
here in calculations for the J/psi number distribution as a function of
transverse momentum and rapidity in central Au-Au collisions at both RHIC and
LHC energies. The ccbar pairs are produced in the initial collision and in the
partonic system during the prethermal and thermal stages through the partonic
channels ab to ccbar [{2S+1}L_J] and ab to ccbar [{2S+1}L_J]x, and then they
dissociate in the latter two stages. Dissociation of ccbar in the medium occurs
via two reactions: (a) color singlet ccbar plus a gluon turns to color octet
ccbar, (b) color octet ccbar plus a gluon persists as color octet. There are
modest yields of ccbar in the prethermal stage at RHIC energy and through the
reactions ab to ccbar [{2S+1}L_J] at LHC energy for partons with large average
momentum in the prethermal stage at both collider energies and in the thermal
stage at LHC energy. Production from the partonic system competes with the
suppression of the initial yield in the deconfined medium. Consequently, a
bulge within -1.5<y<1.5 has been found for the J/psi number distribution and
the ratio of J/psi number distributions for Au-Au collisions to nucleon-nucleon
collisions. This bulge is caused by the partonic system and is thus an
indicator of a deconfined partonic medium. Based on this result we suggest the
rapidity region worth measuring in future experiments at RHIC and LHC to be
-3<y<3.

Introduction
A hot deconfined medium favors the dissociation of J/ψ since enough hard gluons
can overcome the large energy gap between the J/ψ and a continuum state of cc̄ [1].
Models based on perturbative QCD have shown that a dense partonic system can be pro-
duced in central Au-Au collisions at RHIC and LHC energies [2-6] and then evolve toward
thermal equilibrium and likely chemical equilibrium [7-11]. Such parton plasmas will be
searched for soon in experiments at Brookhaven National Laboratory Relativistic Heavy
Ion Collider (RHIC). The J/ψ suppression has been taken as a thermometer to identify
the evolution history of a parton plasma by showing transverse momentum dependence
of the survival probability in the central rapidity region [12].
Charmonium melting inside a hot medium, which leads to J/ψ suppression, was
proposed by Matsui and Satz to probe the existence of the quark-gluon plasma [13].
Before the complete formation of charmonium is achieved, a pre-resonant cc̄ is expanding
from a collision point. Dominance of the color octet plus a collinear gluon configuration
in the pre-resonance state [14] may account for the same suppression of ψ′ and J/ψ
production in proton-nucleus collisions [15]. The growth of the color octet configuration
and its interaction with nucleons along its trajectory in a nucleus are essential ingredients
in explaining measured J/ψ production cross sections. In addition, the importance of color
octet configurations has been verified in pp̄ collisions at center-of-mass energy
s = 1.8
TeV with the CDF detector at Fermilab [16]. Theoretically, the color-octet production
at short distances and its evolution into physical resonances has been well formulated in
nonrelativistic QCD [17]. At the collision energies of RHIC and LHC we can reasonably
expect considerable contributions from the color octet mechanism.
The evolution of ultrarelativistic nucleus-nucleus collisions, e.g. central Au-Au colli-
sions at both RHIC and LHC energies, has been divided into three stages in Refs. [9,18,19]:
(a) an initial collision where a parton gas is produced; (b) a prethermal stage where elas-
tic scatterings among partons lead to local momentum isotropy [20]; (c) a thermal stage
where parton numbers increase until freeze-out. The term ’partonic system’ refers to
the assembly of partons in the prethermal and thermal stages. The parton plasma only
denotes the assembly of partons in the thermal stage. The cc̄ pairs are produced in the
initial collision, prethermal and thermal stages but disintegrate in the latter two stages. In
order to understand and make predictions for J/ψ yields of RHIC and LHC experiments,
the following physical processes are taken into account. (a) In the initial collision, cc̄ pairs
are produced in hard and semihard scatterings between partons from incoming nuclei by
2 → 2 processes which start at order α3s through the partonic channels ab→ cc̄[2S+1LJ ]x.
In the prethermal and thermal stages, cc̄ pairs can also be produced in 2 → 1 collisions
which start at order α2s via the partonic channels ab → cc̄[2S+1LJ ] since partons in the
deconfined medium have large transverse momenta. (b) The cc̄ produced at short distance
is in a color singlet (cc̄)1 or color octet (cc̄)8 configuration which has a certain probability
to evolve nonperturbatively into a color singlet state. This J/ψ production process is
formulated in nonrelativistic QCD. (c) Since the color octet to singlet transition of (cc̄)8
takes time, gluons in the partonic medium couple to the color octet state and destroy this
transition process. Normally, dissociation cross sections for g+ cc̄→ (cc̄)8 depend on the
pair size. Expansion of the cc̄ from a collision point to a full J/ψ size has to be taken into
account.
A physical resonance formed by a cc̄ pair may be one of J/ψ, χcJ , ψ
′ and others.
Since the radiative transition from a higher charmonium state to the J/ψ takes a much
longer time, the transition of such a state with nonzero pT takes place outside the partonic
system. Since the Fermilab Tevatron experiments have been able to separate direct J/ψ’s
from those produced in radiative χcJ decays [16], in this work we assume that the direct
J/ψ production can also be extracted in heavy ion measurements. If χcJ and ψ
′ are
considered, suppression factors for χcJ and ψ
′ in a deconfined medium are included in
prompt J/ψ production. The identification of J/ψ suppression in the medium becomes
impossible for any prompt J/ψ production data. Therefore, no contributions from higher
charmonium states are taken into account in this work.
The purpose of this work is to study the dependence of the J/ψ survival probability
and number distributions produced in central Au-Au collisions at RHIC and LHC energies
on the transverse momentum and also rapidity which will be measured in RHIC experi-
ments [21]. The J/ψ number distributions corresponding to production of cc̄ in the initial
collision are given in Section 2. Since nuclear shadowing has been shown to influence J/ψ
production in proton-nucleus collisions [22], the nuclear modification of parton distribu-
tions is considered. The J/ψ number distributions due to cc̄ production in the prethermal
and thermal stages are given in Sections 3 and 4. Section 5 contains dissociation cross
sections for gluon-(cc̄)1 and gluon-(cc̄)8. Numerical results for nucleon-cc̄ cross sections,
J/ψ number distributions and four ratios including survival probability are presented in
Section 6. Conclusions are summarized in the final section.
2. Initial production of cc̄
Intrinsic transverse momenta of partons inside a nucleon result in the production of
J/ψ with typical momenta comparable to the QCD scale via 2 → 1 partonic scattering
processes [23]. Since we want to study J/ψ productions with pT > 2 GeV, contributions
from 2 → 1 partonic reactions are not considered in the initial nucleon-nucleon colli-
sion. The effect of intrinsic transverse momentum smearing is rather modest for large
transverse momentum J/ψ data from the Tevatron [24]. Upon omission of the intrinsic
transverse momentum, differential cross section for J/ψ production in nucleon-nucleon
collision resulting only from 2 → 2 partonic processes is given as
dydyxdp⊥
= 2p⊥
xaxbfa/N (xa)fb/N(xb)
(ab→ cc̄[2S+1L(1)J ]x → J/ψ)
(ab → cc̄[2S+1L(8)J ]x → J/ψ)] (1)
where the summation
abx is over partons labeled by a, b, x,
(1) for all possible color-
singlet states and
(8) for all possible color-octet states. Here
denotes the partonic
differential cross section for producing a cc̄[2S+1LJ ] and evolving to a J/ψ with spectro-
scopic notation for quantum numbers and superscripts for singlet and octet [23, 25], and
fa/N is the parton distribution function of the species a in a free nucleon. The longitudinal
momentum fractions carried by initial partons, xa and xb, are related to rapidities of cc̄
and x, y and yx, by
y + p⊥e
yx), xb =
−y + p⊥e
where
s, p⊥ and m⊥ are the center-of-mass energy of nucleon-nucleon collision, trans-
verse momentum and transverse mass of the J/ψ. The conditions xa < 1 and xb < 1
restrict yx to a region of
s−m⊥e−y
< yx < ln
s−m⊥ey
These 2 → 2 processes at order α3s , gg → cc̄[2S+1LJ ]g, qq̄ → cc̄[2S+1LJ ]g, gq → cc̄[2S+1LJ ]q
and gq̄ → cc̄[2S+1LJ ]q̄, start in initial nucleus-nucleus collisions and proceed with the
expansion of the heavy pair. While a cc̄ propagates inside a prethermal or thermal partonic
system, gluons hit and excite it to continuum states. Let σ
gcc̄[13S
be the cross section for
g+(cc̄)[13S
1 ] → (cc̄)8, σgcc̄[S(8)] for g+(cc̄)[S(8)] → (cc̄)8 and σgcc̄[P (8)] for g+(cc̄)[P (8)] →
(cc̄)8 respectively. The cross sections are calculated in Section 5. The probability for
dissociation of a small-size cc̄ into a free state relies on the relative velocity between the
gluon and cc̄, vrel, and gluon number densities in the prethermal and thermal stages, ng(x)
and ng(τ), respectively. Here the variables x and τ are individually space-time coordinates
and proper time. In the prethermal stage, parton distributions depend on the correlation
between momentum and space-time coordinates [18, 19]. The dependence of the gluon
number density ng(x) on x characterizes the partonic system in nonequilibrium. In the
thermal stage, thermal parton distributions can be approximated by Jüttner distributions
where the temperature and parton fugacities depend only on the proper time [9, 18, 20]. As
a consequence, the gluon number density is only a function of τ . Including cc̄ suppression
in the partonic system, the finally-formed number distribution of J/ψ resulting from cc̄
pairs produced in the initial central A+B collision is given by
dN2→2ini
dyd2p⊥
s−m⊥e
s−m⊥e
xafa/A(xa, m
⊥, ~r)xbfb/B(xb, m
⊥,−~r)
(ab→ cc̄[3S(1)1 ]x → J/ψ)
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[13S(1)
(k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[13S(1)1 ]
(k · u) >the θ(d− VT∆t)]
(ab→ cc̄[3S(8)1 ]x → J/ψ)
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[S(8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
(ab→ cc̄[1S(8)0 ]x → J/ψ)
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[S(8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
(ab→ cc̄[3P (8)J ]x → J/ψ)
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[P (8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[P (8)](k · u) >the θ(d− VT∆t)]}
where fa/A is the parton distribution function of a nucleus,
fa/A(x,Q
2, ~r) = TA(~r)Sa/A(x,~r)fa/N (x,Q
2) (3)
with the thickness function TA and nuclear parton shadowing factor Sa/A. Here, RA
is the nuclear radius. The symbols < · · · >pre and < · · · >the denote averages over
gluon distributions in the prethermal and thermal stages, respectively. Along the track of
nucleus-nucleus collisions, a deconfined partonic gas is produced from scatterings among
primary partons at τ0, then reaches thermalization at τiso and finally freezes out at τf .
Here, d is the shortest distance which a cc̄ travels from a production point ~r to the surface
of the partonic medium with transverse velocity VT [12]. Suppose a cc̄ is produced at a
proper time τ and a spatial rapidity η. The time ∆t for the partonic system to evolve to
another proper time τ ′ is
(V‖ sinh η − cosh η)τ +
(sinh η − V‖ cosh η)2τ 2 + (1− V 2‖ )τ ′2
1− V 2‖
where V‖ is the longitudinal component of the cc̄ velocity. The disappearance of medium
interactions on the cc̄ is ensured by the step function θ while this pair escapes from the
partonic medium.
3. Production of cc̄ in the prethermal stage
To order α2s, a cc̄ in a color singlet state is produced only through gluon fusion
gg → cc̄[2S+1L(1)J ]. For the cc̄[3S
1 ], this fusion does not occur. In contrast, color octet
states result from both channels gg → cc̄[2S+1L(8)J ] and qq̄ → cc̄[2S+1L
J ]. Nevertheless,
the number densities of quarks and antiquarks are so small that they are neglected in es-
timating the production of cc̄ in the prethermal stage where gluons dominate the partonic
system. Four momenta of the two initial partons and final cc̄ are denoted by k1 = (ω1, ~k1),
k2 = (ω2, ~k2) and p = (E, ~p) = (m⊥ cosh y, ~p⊥, m⊥ sinh y). The differential production rate
for gg → cc̄[2S+1L(8)J ] → J/ψ in the prethermal stage is
d3A2→1pre
8(2π)5
δ(4)(k1 + k2 − p)
g2Gfg(k1, x)fg(k2, x)
| M(gg → cc̄[2S+1L(8)J ] → J/ψ) |2
where gG is the degeneracy factor for gluons and the fg(k, x) is the correlated phase-space
distribution function given in Ref. [18]. The squared amplitudes | M |2 for cc̄ in color
singlet and color octet are calculated individually in Refs. [23, 25]. To order α2s, the
allowed color octet states are 1S
0 and
0,2 through the gluon fusion channel. Taking
into account the suppression of cc̄ in the prethermal and thermal stages, the finally-formed
number distribution of J/ψ resulting from cc̄ pairs produced through 2 → 1 processes in
the prethermal stage is given by
dN2→1pre
dyd2p⊥
16(2π)5
∫ τiso
τdτdηdφk1dyk1
g2Gfg(k1, x)fg(k2, x)
{| M(gg → cc̄[1S(8)0 ] → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[S(8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
+ | M(gg → cc̄[3P (8)J ] → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[P (8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[P (8)](k · u) >the θ(d− VT∆t)]}
where φki is the angle between
~k⊥i and ~p⊥ for i = 1, 2 and mc is the charm quark mass.
The kinematic variables k⊥1, k⊥2, φk2 and yk2 are expressed in terms of
k⊥1 =
m⊥ cosh(y − yk1)− p⊥ cosφk1
k⊥2 =
p2⊥ + k
⊥1 − 2p⊥k⊥1 cosφk1
sin φk2 = −
sinφk1
sinh yk2 =
(m⊥ sinh y − k⊥1 sinh yk1)
To order α3s, the differential production rate gets contributions from the processes
gg → cc̄[2S+1L(1,8)J ]g in the prethermal stage,
d3A2→2pre
16(2π)8
δ(4)(k1 + k2 − p− px)
g2Gfg(k1, x)fg(k2, x){
| M(gg → cc̄[2S+1L(1)J ]x → J/ψ) |2
| M(gg → cc̄[2S+1L(8)J ]x → J/ψ) |2} (7)
where px = (Ex, ~px) = (p⊥x cosh yx, p⊥x cosφx, p⊥x sin φx, p⊥x sinh yx) is the four momen-
tum of the massless parton x. Taking into account the suppression of cc̄ in the prethermal
and thermal stages, the finally-formed number distribution of J/ψ resulting from cc̄ pairs
produced through 2 → 2 processes in the prethermal stage is given by
dN2→2pre
dyd2p⊥
16(2π)8
∫ τiso
τdτdηp⊥xdp⊥xdφxdyxdφk1dyk1
2k2⊥1
g2Gfg(k1, x)fg(k2, x)
{| M(gg → cc̄[3S(1)1 ]x → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[13S(1)1 ]
(k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[13S(1)1 ]
(k · u) >the θ(d− VT∆t)]
+ | M(gg → cc̄[3S(8)1 ]x → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[S(8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
+ | M(gg → cc̄[1S(8)0 ]x → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[S(8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
+ | M(gg → cc̄[3P (8)J ]x → J/ψ) |2
exp[−
∫ τiso
dτ ′ng(x
′) < vrelσgcc̄[P (8)](k · u) >pre θ(d− VT∆t)
dτ ′ng(τ
′) < vrelσgcc̄[P (8)](k · u) >the θ(d− VT∆t)]}
where ŝ = (k1 + k2)
2 and some kinematic variables are given by
k⊥1 = {4m2c + 2m⊥p⊥x cosh(y − yx)− 2p⊥p⊥x cosφx}/
{2[m⊥ cosh(y − yk1) + p⊥x cosh(yx − yk1)− p⊥ cosφk1 − p⊥x cos(φx − φk1)]}
k2⊥2 = m
⊥ + p
⊥x + 2m⊥p⊥x cosh(y − yx) + k2⊥1
−2k⊥1[m⊥ cosh(y − yk1) + p⊥x cosh(yx − yk1)]
sinh yk2 =
[m⊥ sinh y + p⊥x sinh yx − k⊥1 sinh yk1]
The J/ψ number distribution resulting from cc̄ pairs produced in the prethermal
stage becomes
dNpre
dyd2p⊥
dN2→1pre
dyd2p⊥
dN2→2pre
dyd2p⊥
4. Production of cc̄ in the thermal stage
In the thermal stage, parton distributions are approximated by thermal phase-space
distributions fi(k;T, λi) in which the temperature T and nonequilibrium fugacities λi are
functions of the proper time τ [9, 18]. While the partonic system evolves, quark and
antiquark number densities increase. To order α2s, both gg → cc̄[2S+1L
J ] → J/ψ and
qq̄ → cc̄[2S+1L(8)J ] → J/ψ contribute to the J/ψ number distribution in the thermal stage
dN2→1the
dyd2p⊥
16(2π)5
τdτdηdφk1dyk1
g2Gfg(k1;T, λg)fg(k2;T, λg) | M(gg → cc̄[1S
0 ] → J/ψ) |2
exp[−
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]
g2Gfg(k1;T, λg)fg(k2;T, λg) | M(gg → cc̄[3P
J ] → J/ψ) |2
exp[−
dτ ′ng(τ
′) < vrelσgcc̄[P (8)](k · u) >the θ(d− VT∆t)]
+gqgq̄fq(k1;T, λq)fq̄(k2;T, λq̄) | M(qq̄ → cc̄[3S(8)1 ] → J/ψ) |2
exp[−
dτ ′ng(τ
′) < vrelσgcc̄[S(8)](k · u) >the θ(d− VT∆t)]}
where gq and gq̄ are the degeneracy factors for quarks and antiquarks, respectively. In the
channel of quark-antiquark annihilation, only the squared amplitude for 3S
1 does not
vanish.
All lowest-order 2 → 2 reactions gg → cc̄[2S+1LJ ]g, qq̄ → cc̄[2S+1LJ ]g, gq →
cc̄[2S+1LJ ]q and gq̄ → cc̄[2S+1LJ ]q̄ contribute to the J/ψ number distribution in the ther-
mal stage
dN2→2the
dyd2p⊥
16(2π)8
τdτdηp⊥xdp⊥xdφxdyxdφk1dyk1
2k2⊥1
fa(k1;T, λa)fb(k2;T, λb)gagb
| M(ab→ cc̄[2S+1L(1)J ]x → J/ψ) |2
exp[−
dτ ′ng(τ
′) < vrelσgcc̄[12S+1L(1)
(k · u) >the θ(d− VT∆t)]
| M(ab→ cc̄[2S+1L(8)J ]x → J/ψ) |2
exp[−
dτ ′ng(τ
′) < vrelσgcc̄[L(8)](k · u) >the θ(d− VT∆t)]}
The J/ψ number distribution resulting from cc̄ pairs produced in the thermal stage
becomes
dNthe
dyd2p⊥
dN2→1the
dyd2p⊥
dN2→2the
dyd2p⊥
5. Gluon-cc̄ dissociation cross sections
A dissociation cross section of a full-size J/ψ induced by a gluon is given in Refs. [1,
26]. Since an initially-created cc̄ has a radius of about r0 =
and proceeds by expanding
to a full-size object, the dissociation cross section of cc̄ by a gluon has a size dependence.
By this we mean the dissociation of cc̄ into free states via this process g + cc̄ → (cc̄)8.
Cross sections are calculated with chromoelectric dipole coupling between gluon and cc̄ in
the procedure for gluon-J/ψ dissociation in Ref. [26]. The wave function of an expanding
cc̄ is needed for this purpose, but it has not been investigated in the partonic medium even
though some attempts have been made in studies of the color transparency phenomenon
[27]. We proceed with the construction of wave functions in a simple one-gluon-exchange
potential model.
In a parton plasma, the internal motion of J/ψ is obtained [12] from the attractive
Coulomb potential, V1 = −gs2/3πr. The quantum-mechanical interpretation of the cc̄
radius is
< r2 >, the square root of the radius-square expectation value of the relative-
motion wave function. For the 1S color singlet, its wave function in momentum space
normalized to the radius of cc̄[13S
1 ] is
[~rψ1s](~k) = 32
πa2.50
(1 + (ka0)2)3
where the variable a0 =
< r2 > /3 is the Bohr radius for a full-size J/ψ. The velocity-
square expectation value of the J/ψ wave function is < v2 >= 0.428. Then the radius
of cc̄ is assumed to expand according to
< r2 > =
< v2 >t+ r0. The gluon-cc̄[1
dissociation cross section is
gcc̄[13S
128gs
2m2.5c a
0 − ǫ0)1.5Q0
9[mca
0 − ǫ0) + 1]6
where Q0 is the gluon energy, ǫ0 the binding energy of J/ψ and gs the strong coupling
constant.
While the cc̄ is in a color octet state, it is not a bound state but rather a scattering
state. Its relative-motion wave function is determined by the repulsive potential V8 =
g2s/24πr. The radial part of the S wave function is
SR(r) + iSI(r) = e
iqrF (1 + iη, 2,−2iqr) (15)
and the radial part of the P wave function is
PR(r) + iPI(r) = qre
iqrF (2 + iη, 4,−2iqr) (16)
with q = mc
< v2 > and η = g2s/24π
< v2 >. The function F is the confluent hyper-
geometric function. Wave functions in momentum space are obtained by performing a
Fourier transform of the wave functions in space coordinates. Normalization constants of
the momentum-space wave functions, CS and CP , are determined by fitting the cc̄ radius.
Dissociation cross sections of the S-wave and P -wave color-octet states by a gluon are
σgcc̄[S(8)] =
g2s (mcQ
0)1.5
dr1dr2r
mcQ0r1)j1(
mcQ0r2)[SR(r1)SR(r2) + SI(r1)SI(r2)]
σgcc̄[P (8)] =
g2s (mcQ
0)1.5
dr1dr2
mcQ0r1)j0(
mcQ0r2) + 2j2(
mcQ0r1)j2(
mcQ0r2)]
[PR(r1)PR(r2) + PI(r1)PI(r2)]
where the j0, j1 and j2 are spherical Bessel functions. The b is determined so that the
square root of the r2 expectation value of the relative wave function in Eq. (15) or (16) is
the color-octet radius. Relations b = 1.435
< r2 > for S-wave and b = 1.3
< r2 > for
P -wave approximately hold for color-octet size less than normal hadron size.
6. Numerical results and discussions
Results for five aspects are presented in the following subsections. The first aspect
is the nucleon-cc̄ dissociation cross sections shown in the next subsection. The second one
in Subsection 6.2 is J/ψ number distributions versus transverse momentum at y = 0 and
rapidity at pT = 4 GeV with nuclear effect on parton distributions and cc̄ dissociation
in the partonic system. The third one in Subsection 6.3 is to define and calculate four
ratios including survival probability with y = 0 or pT = 4 GeV at both RHIC and LHC
energies. The fourth one is given in Subsection 6.4 to show J/ψ number distributions
without nuclear effect on parton distributions and cc̄ dissociation in the partonic system.
The fifth one concerns some uncertainties on the above results.
6.1. Nucleon-cc̄ dissociation cross sections
In the parton model of the nucleon, the gluon is a dominant ingredient. Whereas
the cross section for cc̄ dissociated directly by a real gluon is of order αs, the cross section
for the quark-cc̄ dissociation through a virtual gluon is of order α2s. With the gluon-(cc̄)1
cross section given in the last section, the nucleon-cc̄[13S
1 ] cross section driven mainly
by the gluon ingredient becomes
Ncc̄[13S
dxfg/N(x,Q
gcc̄[13S
where x
min =
with pN being the proton momentum in the rest frame of the J/ψ. The
gluon distribution function fg/N is that Glück-Reya-Vogt (GRV) result at leading order
in Ref. [28]. The cross section is drawn in Fig. 1 to show the energy and renormalization-
scale dependence while the cc̄ radius is the J/ψ radius in the attractive Coulomb poten-
tial, rJ/ψ = 0.348 fm. In Fig. 1, gluon field operators are renormalized at three scales
2ǫ0, Q
0, respectively. The coupling constant has the value αs =
correspond-
ing to the scale ǫ0 while it varies for the other two scales. Values of the cross section
s = 10 GeV are a little lower than the nucleon-J/ψ dissociation cross section ob-
tained by the subtraction of quasi-elastic cross section in Ref. [29] from the total cross
section given in Ref. [30]. In high-temperature hadronic matter or J/ψ photoproduction
reaction, a typical value of the center-of-mass energy for nucleon-J/ψ (or preresonance)
dissociation is around
s = 6 GeV [1]. At this energy, Fig. 2 is drawn to show the size
dependence of σ
Ncc̄[13S
, with fg/N (x,Q
2) at Q2 = ǫ20.
For the S-wave color octet the nucleon-cc̄[S(8)] cross section is
σNcc̄[S(8)] =
dxfg/N(x, (q +Q
0)2)σgcc̄[S(8)] (20)
where x
min =
. For the P -wave color octet the nucleon-cc̄[P (8)] cross section is
σNcc̄[P (8)] =
dxfg/N(x, (q +Q
0)2)σgcc̄[P (8)] (21)
Since the gluon momentum in a confining medium is bigger than the QCD scale [14], the
lowest value of x is set by ΛQCD = 0.2322 GeV used in the leading order GRV parton
distribution functions. Dependences of σNcc̄[S(8)] and σNcc̄[P (8)] on the center-of-mass energy√
s are depicted in Fig. 3 while the size of (cc̄)8 is the full size of J/ψ. The dot-dashed
line is obtained with the nucleon-J/ψ cross section given by Eq. (24) in Ref. [1] where
another gluon distribution function evaluated at Q2 = ǫ20 is used. While the (cc̄)8 has
small momentum in a nucleus, the cross section for nucleon-(cc̄)8 production is lower than
the absorption cross section determined by Gerschel and Hüfner [31] or the two-gluon
exchange result [32]. It was proposed by Kharzeev and Satz that the color octet plus a
gluon configuration is a dominant component produced in the proton-nucleus collisions
[14]. In fact, the present cross section for a nucleon and a bare (cc̄)8 is one part of the
nucleon-g(cc̄)8 cross section.
A (cc̄)8 pair produced at a collision point expands before becoming color singlet to
a size which may be larger or smaller than the full size of J/ψ. We then show in Fig. 4
the dependence of the nucleon-(cc̄)8 cross section on the color-octet pair radius.
We do not want to address proton-nucleus collisions in terms of nucleon-cc̄ cross
sections [33, 34] since only the gluon-cc̄ cross sections are needed to study cc̄ suppression
in the prethermal and thermal stages. In proton-nucleus collisions, once a color-octet
(cc̄)8 pair is produced, it picks up a collinear gluon to form a colorless configuration [14].
However, in central Au-Au collisions at RHIC and LHC energies, the accompanying gluon
scatters with other hard gluons in the dense partonic system and is driven away. Therefore
the bare cc̄ is the object that we want to study in the partonic system.
6.2. J/ψ number distributions with suppression
J/ψ number distributions versus transverse momentum at y = 0 and rapidity at
pT = 4 GeV for central Au-Au collisions at RHIC energy
s = 200AGeV are calculated
with respect to the initial collision, prethermal and thermal stages. Initial productions
of cc̄ are calculated with GRV parton distribution functions at renormalization scale µ =
p2⊥ + 4m
c . Evolution of color-octet states
1S0 and
3PJ toward the J/ψ is specified
by nonperturbative matrix elements < OJ/ψ8 (3S1) >, < O
1S0) > and < OJ/ψ8 (3P0) >
in nonrelativistic QCD [17]. In the nonperturbative evolution, a gluon from the partonic
system hits and prevents the color octet from color neutralizing via g + (cc̄)8 → (cc̄)8.
This medium effect has been expressed by exponentials in Eqs. (2), (6), (8), (10) and
(11). Therefore, the nonperturbative matrix elements are assumed to be invariant while
the medium effect is factorized into exponential forms. Values of these matrix elements
are well determined by fitting the CDF measurements for pp̄ collisions at
s = 1.8 TeV
in Ref. [35],
< OJ/ψ8 (3S1) >= (1.12± 0.14)× 10−2GeV3 (22)
< OJ/ψ8 (1S0) > +
< OJ/ψ8 (3P0) >= (3.90± 1.14)× 10−2GeV3 (23)
In pp̄ collisions, differential cross sections of direct J/ψ production depend on the com-
bination of < OJ/ψ8 (1S0) > and < O
3P0) >. However, since the dissociation cross
section for the S-wave color-octet state is different from that for the P -wave color-octet
state, such a dependence on the combination is destroyed. In calculations, values are
taken as follows,
< OJ/ψ8 (1S0) >= 4× 10−2GeV3, < O
3P0) >= −
× 10−2GeV3 (24)
The value of < OJ/ψ8 (3P0) > is positive at tree level and negative after renormalization
[36]. Eq. (23) is still satisfied by the values in Eq. (24). These values of nonperturbative
matrix elements are supposed to be universal for any center-of-mass energy
Various contributions to the J/ψ number distributions including initial collisions,
prethermal and thermal stages, 2 → 1 and 2 → 2 collisions, are drawn separately in Figs.
5 and 6. The dashed curve resulting from cc̄ production in the initial collision is obtained
by calculating Eq. (2) where the nuclear parton shadowing factor is given in Ref. [4]
throughout this subsection. The upper and lower dot-dashed curves resulting from cc̄
production in the prethermal stage are obtained by individually calculating Eq. (6) for
2 → 1 collisions and Eq. (8) for 2 → 2 collisions. The upper and lower dotted curves
resulting from cc̄ production in the thermal stage are obtained by calculating Eq. (10)
for 2 → 1 collisions and Eq. (11) for 2 → 2 collisions, respectively. To exclude the effect
of intrinsic transverse momentum smearing, only the region pT > 2 GeV is considered.
Consequently, no 2 → 1 collisions contribute in the initial collision. The J/ψ number
distribution resulting from the initial collision shown by the dashed line has a plateau
similar to that in proton-proton collision [37]. Both Figs. 5 and 6 show that cc̄ pairs
produced from the thermal stage can be neglected compared to the initial production, but
the contributions from the prethermal stage are important in the transverse momentum
region 2GeV < pT < 8GeV and rapidity region 0 < y < 1.2. Productions of cc̄ in the
prethermal and thermal stages bulge up the J/ψ number distribution shown by the solid
line in this rapidity region. Nevertheless, the dot-dashed and dotted lines fall rapidly as
the rapidity gets large. This bulging characterizes the formation of a deconfined medium
because the medium has average momentum limited but big enough to produce extra cc̄
and thus J/ψ. The 2 → 1 collisions in the partonic system have bigger contributions than
the 2 → 2 collisions.
Each of Figs. 7 and 8 contains two sets of lines to show contributions from the
color-singlet and color-octet pairs produced at short distance. Any set has a dashed line
obtained from Eq. (2) for the initial collision, a dot-dashed line from Eq. (9) for the
prethermal stage and a dotted line from Eq. (12) for the thermal stage, respectively. A
line in the upper (lower) set for the color-octet (color-singlet) contributions stems from
the terms for (cc̄)8 ((cc̄)1) states. The color octet states dominate productions of J/ψ at
RHIC energy. However, the ratio of color-octet to color-singlet contributions shown by the
two solid lines at pT = 6 GeV is reduced from about 70 at CDF collider energy
s = 1.8
TeV to about 40 at RHIC energy. Both contributions of color-singlet and color-octet
states have similar dependence on transverse momentum and rapidity.
Figs. 9 and 10 show transverse momentum and rapidity dependence of J/ψ number
distributions for central Au-Au collisions at LHC energy
s = 5.5ATeV. A prominent
feature is that the J/ψ number produced from the thermal stage is comparable to that
from the prethermal stage. Compared to the initial production, cc̄ and J/ψ produced
through 2 → 2 reactions may be neglected. A bulge is observed on the plateau in the
rapidity region 0 < y < 1.5. Such a bulge can be taken as a signature for the existence
of a parton plasma at the LHC energy. Figs. 11 and 12 depict contributions from the
color singlet and color octet at LHC energy. The ratio of color-octet to color-singlet
contributions shown by the two solid lines at pT = 6GeV reaches about 150. This indicates
the color octet states become more important with the increase of
6.3. Ratios including J/ψ survival probability
Nuclear shadowing results in a modification of gluon distribution functions inside
a nucleus [38] and such a nuclear effect is represented by the shadowing factor Sa/A in
Eq. (3). If the Sa/A = 1 for no shadowing, the dN
ini /dyd
2p⊥ is proportional to the
product of atomic masses of the two colliding nuclei. If Sa/A 6= 1 and depends on the
longitudinal momentum fraction x, the production of cc̄ is reduced in the shadowing region
and enhanced for the anti-shadowing region. Irrespective of interactions of J/ψ with the
partonic system, J/ψ number distributions produced in the initial central A+B collision
is obtained by putting all exponentials equal to 1 in Eq. (2),
dN2→20
dyd2p⊥
(Sa/A) = 2
s−m⊥e
s−m⊥e
xafa/A(xa, m
⊥, ~r)xbfb/B(xb, m
⊥,−~r)
(ab→ cc̄[3S(1)1 ]x → J/ψ)
(ab→ cc̄[3S(8)1 ]x → J/ψ)
(ab→ cc̄[1S(8)0 ]x → J/ψ)
(ab→ cc̄[3P (8)J ]x → J/ψ)}
To characterize the influence of nuclear parton shadowing on the J/ψ production from
the initial collision, a ratio is defined as
Rini =
dN2→20
dyd2p⊥
(Sa/A 6= 1)/
dN2→20
dyd2p⊥
(Sa/A = 1) (26)
Here the Sa/A 6= 1 from Ref. [4] applies throughout this subsection.
The initially produced J/ψ originates from the cc̄ pairs produced in the initial colli-
sion. Its dependence on the transverse momentum and rapidity is obtained by calculating
Eq. (25). Some cc̄ pairs produced in the initial collision may dissociate by gluons from the
partonic system. As a consequence, the J/ψ number is reduced. The survival probability
for the cc̄ transiting into a J/ψ is defined as the ratio
Splasma =
dN2→2ini
dyd2p⊥
(Sa/A 6= 1)/
dN2→20
dyd2p⊥
(Sa/A 6= 1) (27)
We have calculated the J/ψ number distributions produced in the prethermal and
thermal stages in Subsection 6.2. The J/ψ yield may be bigger than the reduced amount
of initially produced J/ψ due to the cc̄ dissociation by gluons in the partonic system. The
partonic system has two roles. One is to produce cc̄ pairs and another is to dissociate cc̄
pairs. To see the roles, a ratio is defined by
Rplasma = (
dN2→2ini
dyd2p⊥
(Sa/A 6= 1) +
dNpre
dyd2p⊥
dNthe
dyd2p⊥
dN2→20
dyd2p⊥
(Sa/A 6= 1) (28)
To understand the nuclear effect on parton distributions and the roles of the partonic
system, we need to compare J/ψ production in the central A+B collision with that in the
nucleon-nucleon collision. To this end, a ratio is defined as
R = (
dN2→2ini
dyd2p⊥
(Sa/A 6= 1) +
dNpre
dyd2p⊥
dNthe
dyd2p⊥
dN2→20
dyd2p⊥
(Sa/A = 1) (29)
which is also written as
R = RiniRplasma (30)
The ratios Rini, Splasma, Rplasma and R versus transverse momentum and rapidity
are depicted as dashed, dotted, dot-dashed and solid lines, respectively, in Figs. 13 and 14
for the RHIC energy and Figs. 15 and 16 for the LHC energy. In contrast to Rini < 1, the
value of Rplasma is larger than 1 for all transverse momenta in Fig. 13 and 0 < y < 1.5 in
Fig. 14 and 0.5 < y < 1.5 in Fig. 16. This results in prominent bulges on the solid lines
of R in Figs. 14 and 16. In contrast, the survival probability Splasma shown by the dotted
lines has no such bulge. Therefore, the bulges are present in Figs. 6 and 10 when the J/ψ
yield resulting from cc̄ pairs produced in the partonic system overwhelms the reduced
amount of initially produced J/ψ. We conclude that in the rapidity region 0 < y < 1.5
a bulge observed in the ratio R is an indicator for the existence of the partonic system.
For Rini < 1 and Splasma < 1, J/ψ suppression arises from the nuclear parton shadowing
found in HIJING and cc̄ dissociation in the partonic system.
6.4. J/ψ number distributions with no suppression
In Subsection 6.2, J/ψ number distributions have been presented while the cc̄ re-
duction due to the nuclear parton shadowing in the initial collision and cc̄ dissociation in
the partonic system are taken into account. In this subsection, the suppression including
both the reduction and dissociation is omitted in calculations of J/ψ number distributions
by setting to 1 all exponentials in Eqs. (2), (6), (8), (10) and (11). Figs. 17-20 depict
these distributions versus transverse momentum at y = 0 and rapidity at pT = 4 GeV at
both RHIC and LHC energies. We are now ready to explain the dip within y < 1 in Fig.
10. This dip disappears in Fig. 20 where suppression is not considered. Since Rini shown
by the dashed line in Fig. 16 is flat with respect to the rapidity y < 2 and Splasma shown
by the dotted line has a steep rise in 0.5 < y < 1, the dip phenomenon is solely due to the
cc̄ dissociation in the partonic system. Such a dip phenomenon is not obvious but still
can be observed in the prethermal and thermal stages when the J/ψ number distributions
with suppression are compared to those without suppression. The comparison is indicated
in Fig. 21 for the prethermal stage and Fig. 22 for the thermal stage. The solid and
dot-dashed lines for no suppression begin to fall from y = 0.5 to y = 1, but change to
rising as shown by the dashed and dotted lines when the cc̄ dissociation is switched on.
This change occurs because of the steep rise of Splasma. A relatively weak dependence of
Splasma on pT is shown by the dotted line in Fig. 15. The dip phenomenon thus cannot
be observed in the pT dependence of J/ψ number distributions. Referring back to Eqs.
(2), (6), (8), (10) and (11), exponentials there have sensitive dependence on the rapidity
in 0 < y < 1.5. The dip is more obvious in the color singlet channel as shown by the
lower solid, dashed, dot-dashed and dotted lines in Fig. 12. This is so because the cross
section for the 13S
1 -state dissociation has a narrower peak with respect to the incident
gluon energy [12] than the color octet states.
6.5. Uncertainties
Since gluon shadowing in nuclei has not been studied experimentally, theoretical
estimates of the nuclear gluon shadowing factor involve uncertainties. The nuclear parton
shadowing factor found in HIJING [4] is a result of the assumption that there is no Q-
dependence on the shadowing factor and the shadowing effect for gluons and quarks is the
same. Nevertheless, the shadowing factor has been shown by Eskola et al. to evolve with
momentum Q [39]. The difference between the latter and the former indicates uncertainty.
The ratio Rini defined in Eq. (26) is calculated with Eskola et al.’s parametrization [39]
and results are depicted in Fig. 23 showing momentum dependence at y = 0 and Fig.
24 showing rapidity dependence at pT = 4 GeV. Compared to the dashed lines in Figs.
13-16, the change of Rini at RHIC energy greater than 1 is prominent. This implies that
the anti-shadowing effect of Eskola et al.’s parametrization is quite important at RHIC
energy. Measurements on Rini in RHIC experiments are needed to confirm this nuclear
enhancement [21].
The ratio Rini is always flat within the rapidity region -1.5< y < 1.5 for parametriza-
tions given in HIJING and by Eskola et al., and the flatness seems to be independent of
parametrizations. If the partonic system does not come into being, the ratio R is flat, too,
since R = Rini. If the partonic system dissociates cc̄ pairs, the solid curve of R undergoes
bulging, dipping and then bulging from y = −1.5 to y = 1.5. Any such twist of R in
−1.5 < y < 1.5 observed in experiments is nontrivial, because only a deconfined medium
generates it.
Upon inclusion of uncertainties on the formation and evolution of parton plasma
arising from other factors, for instance, the dependence on the coupling constant αs
[40] and transverse flow [41], the J/ψ number distribution and the four ratios including
survival probability will change. In the partonic system considered here gluons dominate
the evolution and gluon-cc̄ interactions break the pairs. In a system where quarks and
antiquarks are abundant, interactions between quarks (antiquarks) and cc̄ may account
for a suppression of J/ψ [42]. Additional suppression caused by energy loss of the initial
state has not been considered since there is a controversy on the influence of the energy
loss [43, 44]. Some uncertainties are expected to be fixed by upcoming experiments at
RHIC.
7. Conclusions
We have studied J/ψ production through both color-singlet and color-octet cc̄ chan-
nels with various stages of central Au-Au collisions at both RHIC and LHC energies. In
addition to the scattering processes ab → cc̄[2S+1LJ ]x, contributions of the reactions
ab→ cc̄[2S+1LJ ] are also calculated in the prethermal stage and thermal stage. The effect
of the medium on an expanding cc̄ involves a gluon interacting with the cc̄ to prevent it
from a transition into a color singlet. Cross sections for g + cc̄ → (cc̄)8 are calculated
with internal wave functions of (cc̄)1 in an attractive potential and (cc̄)8 in a repulsive
potential. Furthermore, nucleon-cc̄ cross sections for color singlet, S- and P - wave color
octets as a function of
s or cc̄ radius are evaluated by assuming that the nucleon domi-
nantly contains gluons. Momentum and rapidity dependence of J/ψ number distribution
with various contributions are calculated for central Au-Au collisions at both RHIC and
LHC energies. Color octet contributions are one order of magnitude larger than the color
singlet contributions. Yields of cc̄ are large in the prethermal stage at RHIC energy and
through the 2 → 1 collisions ab → cc̄[2S+1LJ ] at LHC energy. Since the partonic system
offers fairly large amounts of cc̄, a bulge in 0 < y < 1.5 at RHIC energy and 0.5 < y < 1.5
at LHC energy can be observed in the rapidity dependence of the J/ψ number distribu-
tion and the ratio R of J/ψ number distributions for Au-Au collisions to nucleon-nucleon
collisions. Such a bulge is a signature for the existence of a deconfined partonic medium.
We suggest that RHIC and LHC experiments measure J/ψ number distributions and
the ratio R in the rapidity region 0 < y < 3 to observe a bulge. While the yield of cc̄
from the medium is larger than the reduced amount of initial production in the medium,
the ratio Rplasma is larger than 1. The competition between production and suppression
determines the values of Rplasma, which relies on the evolution of parton number density
and temperature of the partonic system [12]. A dip in the rapidity dependence of the J/ψ
number distributions at LHC energy may exist and this amounts to a suppression effect
of cc̄ in the partonic system. So far, we have obtained results and conclusions for positive
rapidity. It is stressed that the same contents for negative rapidity can be obtained from
the positive region by symmetry.
Acknowledgements
I thank the [Department of Energy’s] Institute for Nuclear Theory at the University
of Washington for its hospitality and the Department of Energy for partial support during
the completion of this work. I thank the Nuclear Theory Group at LBNL Berkeley for
their hospitality during my visit. I also thank X.-N. Wang, C.-Y. Wong and M. Asakawa
for discussions, K. J. Eskola for offering Fortran codes of nuclear parton shadowing factors,
H. J. Weber for careful reading through the manuscript. This work was also supported in
part by the project KJ951-A1-410 of the Chinese Academy of Sciences and the Education
Bureau of Chinese Academy of Sciences.
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Figure 1: Solid, dashed and dot-dashed lines are nucleon-cc̄[13S
1 ] cross sections for
fg/N (x,Q
2) evaluated at Q2 = ǫ20, 2ǫ
0, (Q
0)2, respectively. The (cc̄)1 has the same size as
Figure 2: Cross section for nucleon-cc̄[13S
1 ] at
s = 6 GeV as a function of the cc̄ radius
is calculated with fg/N (x,Q
2) evaluated at Q2 = ǫ20.
Figure 3: The solid and dashed lines are cross sections for nucleon-cc̄[S(8)] and nucleon-
cc̄[P (8)] collisions as a function of
s, respectively. The dot-dashed line is the nucleon-J/ψ
cross section calculated with Eq. (24) in Ref. [1]. The corresponding (cc̄)8 and J/ψ have
the same radius.
Figure 4: The solid and dashed lines show radius dependence of cross sections for nucleon-
(cc̄)8[S
(8)] and nucleon-(cc̄)8[P
(8)] collisions, respectively.
Figure 5: J/ψ number distributions versus transverse momentum at y = 0 and RHIC
energy with suppression. The dashed curve corresponds to cc̄ production in the initial
collision. The upper and lower dot-dashed (dotted) curves correspond to cc̄ produced
through 2 → 1 and 2 → 2 reactions in the prethermal (thermal) stage, respectively. The
solid curve is the sum of all contributions.
Figure 6: The same as Fig. 5, except for rapidity distribution at pT = 4 GeV.
Figure 7: J/ψ number distributions versus transverse momentum at y = 0 and RHIC
energy with suppression. The upper and lower dashed (dot-dashed, dotted and solid)
lines correspond to cc̄ in color octet and color singlet, respectively, produced in the initial
collision(prethermal stage, thermal stage and the all three stages).
Figure 8: The same as Fig. 7, except for rapidity distribution at pT = 4 GeV.
Figure 9: J/ψ number distributions versus transverse momentum at y = 0 and LHC
energy with suppression. The dashed curve corresponds to cc̄ productions in the initial
collision. The upper and lower dot-dashed (dotted) curves correspond to cc̄ produced
through 2 → 1 and 2 → 2 reactions in the prethermal (thermal) stage, respectively. The
solid curve is the sum of all contributions.
Figure 10: The same as Fig. 9, except for rapidity distribution at pT = 4 GeV.
Figure 11: J/ψ number distributions versus transverse momentum at y = 0 and LHC
energy with suppression. The upper and lower dashed (dot-dashed, dotted and solid)
lines correspond to cc̄ in color octet and color singlet, respectively, produced in the initial
collision (prethermal stage, thermal stage and all three stages).
Figure 12: The same as Fig. 11, except for rapidity distribution at pT = 4 GeV.
Figure 13: Ratios versus transverse momentum at y = 0 and RHIC energy. The solid,
dashed, dot-dashed and dotted lines are R, Rini, Rplasma and Splasma, respectively.
Figure 14: The same as Fig. 13, except for rapidity distribution at pT = 4 GeV
Figure 15: Ratios versus transverse momentum at y = 0 and LHC energy. The solid,
dashed, dot-dashed and dotted lines are R, Rini, Rplasma and Splasma, respectively.
Figure 16: The same as Fig. 15, except for rapidity distribution at pT = 4 GeV.
Figure 17: J/ψ number distributions versus transverse momentum at y = 0 and RHIC
energy without suppression. The dashed curve corresponds to cc̄ productions in the initial
collision. The upper and lower dot-dashed (dotted) curves correspond to cc̄ produced
through 2 → 1 and 2 → 2 reactions in the prethermal (thermal) stage, respectively. The
solid curve is the sum of all contributions.
Figure 18: The same as Fig. 17, except for rapidity distribution at pT = 4 GeV.
Figure 19: J/ψ number distributions versus transverse momentum at y = 0 and LHC
energy without suppression. The dashed curve corresponds to cc̄ productions in the initial
collision. The upper and lower dot-dashed (dotted) curves correspond to cc̄ produced
through 2 → 1 and 2 → 2 reactions in the prethermal (thermal) stage, respectively. The
solid curve is the sum of all contributions.
Figure 20: The same as Fig. 19, except for rapidity distribution at pT = 4 GeV.
Figure 21: J/ψ number distributions versus rapidity at pT = 4 GeV in the prethermal
stage of LHC energy. The dashed and solid lines individually correspond to cc̄ produced
through 2 → 1 reactions with and without suppression. The dotted and dot-dashed lines
through 2 → 2 reactions with and without suppression, respectively.
Figure 22: The same as Fig. 21, except for the thermal stage.
Figure 23: Ratio Rini versus transverse momentum at y = 0 is calculated with Eskola’s
parametrization.
Figure 24: The same as Fig. 23, except for rapidity distribution at pT = 4 GeV.
