PHYSICAL REVIEW C 72, 064901 (2005)
Multiphase transport model for relativistic heavy ion collisions
Zi-Wei Lin
∗
Physics Department, The Ohio State University, Columbus, Ohio 43210, USA
Che Ming Ko
Cyclotron Institute and Physics Department, Texas A&M University, College Station, Texas 77843, USA
Bao-An Li and Bin Zhang
Department of Chemistry and Physics, Arkansas State University, State University, Arkansas 72467, USA
Subrata Pal
Department of Physics, Michigan State University, East Lansing, Michigan 48824, USA
(Received 30 November 2004; published 8 December 2005)
We describe in detail how the different components of a multiphase transport (
AMPT) model that uses the
heavy ion jet interaction generator (
HIJING) for generating the initial conditions, Zhang’s parton cascade (ZPC)
for modeling partonic scatterings, the Lund string fragmentation model or a quark coalescence model for
hadronization, and a relativistic transport (
ART) model for treating hadronic scatterings are improved and combined
to give a coherent description of the dynamics of relativistic heavy ion collisions. We also explain the way
parameters in the model are determined and discuss the sensitivity of predicted results to physical input in
the model. Comparisons of these results to experimental data, mainly from heavy ion collisions at the BNL
Relativistic Heavy Ion Collider, are then made in order to extract information on the properties of the hot dense
matter formed in these collisions.
DOI: 10.1103/PhysRevC.72.064901 PACS number(s): 25.75.−q, 12.38.Mh, 24.10.Lx
I. INTRODUCTION
Colliding heavy ions at relativistic energies makes it
possible to subject nuclear matter to the extreme condition
of large compression, leading to energy densities that can
exceed that for producing a plasma of deconfined quarks and
gluons, which is believed to have existed during the first
microsecond after the Big Bang. Experiments at the BNL
Relativistic Heavy Ion Collider (RHIC) with center-of-mass
energy up to
√
s
NN
= 200 GeV in Au+Au collisions thus
provide the opportunity to study the properties of this so-called
quark-gluon plasma (QGP). At the future Large Hadron
Collider (LHC) at CERN, which will allow Pb+Pb collisions
at
√
s
NN
= 5.5 TeV, the produced quark-gluon plasma will
have an even higher temperature and a nearly vanishing net
baryon chemical potential.
Many observables have been measured at RHIC, such as the
rapidity distributions of various particles and their transverse
momentum spectra up to very high transverse momentum, the
centrality dependence of these observables, and the elliptic
flows of various particles, as well as both identical and
nonidentical two-particle correlations. To understand these
extensive experimental results, many theoretical models have
been introduced. They range from thermal models [1–4] based
on the assumption of global thermal and chemical equilibrium,
to hydrodynamic models [5–11] based only on the assumption
of local thermal equilibrium, to transport models [12–26]
that treat nonequilibrium dynamics explicitly. The thermal
∗
Present address: 301 Sparkman Drive, VBRH E-39, University of
Alabama in Huntsville, Huntsville, AL 35899.
models have been very successful in accounting for the yield
of various particles and their ratios, while the hydrodynamic
models are particularly useful for understanding the collective
behavior of low transverse momentum particles such as the
elliptic flow [8–11]. Since transport models treat chemical
and thermal freeze-out dynamically, they are also natural and
powerful tools for studying the Hanbury-Brown-Twiss inter-
ferometry of hadrons. For hard processes that involve large
momentum transfer, approaches based on the perturbative
quantum chromodynamics (pQCD) using parton distribution
functions in the colliding nuclei have been used [27,28].
Also, the classical Yang-Mills theory has been developed
to address the evolution of parton distribution functions in
nuclei at ultrarelativistic energies [29–31] and used to study
the hadron rapidity distribution and its centrality dependence
at RHIC [32–34]. These problems have also been studied
in the pQCD-based final-state saturation model [35–37].
Although studies based on the pQCD [38] have shown that
thermalization could be achieved in collisions of very large
nuclei and/or at extremely high energy, even though the strong
coupling constant at the saturation scale is asymptotically
small, the dense matter created in heavy ion collisions at
RHIC may, however, not achieve full thermal or chemical
equilibrium as a result of its finite volume and energy. To
address such nonequilibrium many-body dynamics, we have
developed a multiphase transport (
AMPT) model that includes
both initial partonic and final hadronic interactions and the
transition between these two phases of matter [39–50]. The
AMPT model is constructed to describe nuclear collisions
ranging from p + A to A + A systems at center-of-mass
energies from about
√
s
NN
= 5 to 5500 GeV at LHC, where
0556-2813/2005/72(6)/064901(29)/$23.00 064901-1 ©2005 The American Physical Society
LIN, KO, LI, ZHANG, AND PAL PHYSICAL REVIEW C 72, 064901 (2005)
strings and minijets dominate the initial energy production
and effects from final-state interactions are important. For
the initial conditions, the
AMPT model uses the hard minijet
partons and soft strings from the heavy ion jet interaction
generator (
HIJING) model. Zhang’s parton cascade (ZPC)is
then used to describe scatterings among partons, which is
followed by a hadronization process based on the Lund string
fragmentation model or by a quark coalescence model. The
latter is introduced for an extended
AMPT model with string
melting in which hadrons, which would have been produced
from string fragmentation, are converted instead to their
valence quarks and antiquarks. Scatterings among the resulting
hadrons are described by a relativistic transport (
ART) model.
With parameters, such as those in the string fragmentation,
fixed by the experimental data from heavy ion collisions at
the CERN super proton synchrotron (SPS), the
AMPT model
has been able to reasonably describe many of the experimental
observations at RHIC.
In this paper, we give a detailed description of the different
components of the
AMPT model, discuss the parameters in the
model, show the sensitivity of its results to the input to the
model, and compare its predictions with experimental data.
The paper is organized as follows. In Sec. II, we describe
the different components of the
AMPT model: The HIJING
model and string melting, the ZPC model, the Lund string
fragmentation model, the quark coalescence model used for
the scenario of string melting, and the extended
ART model.
Tests of the
AMPT model against data from pp and p
¯
p reactions
are given in Sec. III. Results from the
AMPT model for
heavy ion collisions at SPS energies are discussed in Sec. IV
for hadron rapidity distributions and transverse momentum
spectra, baryon stopping, and antiproton production. In Sec. V,
we show results at RHIC for hadron rapidity distributions
and transverse momentum spectra, particle ratios, baryon and
antibaryon production, and the production of multistrange
baryons as well as J/ψ. We further show results from the
AMPT model with string melting on hadron elliptic flows and
two-pion interferometry at RHIC. In Sec. VI, we present
the
AMPT predictions for hadron rapidity and transverse
momentum distributions in Pb+Pb collisions at the LHC
energy. Discussions on possible future improvements of the
AMPT model are presented in Sec. VII, and a summary is
finally given in Sec. VIII.
II. THE AMPT MODEL
The AMPT model consists of four main components: the
initial conditions, partonic interactions, conversion from
the partonic to the hadronic matter, and hadronic interactions.
The initial conditions, which include the spatial and momen-
tum distributions of minijet partons and soft string excitations,
are obtained from the
HIJING model [51–54]. Currently, the
AMPT model uses the HIJING model version 1.383 [55], which
does not include baryon junctions [56]. Scatterings among
partons are modeled by
ZPC [18], which at present includes
only two-body scatterings with cross sections obtained from
the pQCD with screening masses. In the default
AMPT model
[39–44,46,47,49], partons are recombined with their parent
strings when they stop interacting, and the resulting strings
FIG. 1. (Color online) Structure of the default AMPT model.
are converted to hadrons using the Lund string fragmentation
model [57–59]. In the
AMPT model with string melting
[45,48,50], a quark coalescence model is used instead to
combine partons into hadrons. The dynamics of the subsequent
hadronic matter is described by a hadronic cascade, which
is based on the
ART model [14,25] and extended to include
additional reaction channels that are important at high energies.
These channels include the formation and decay of K
∗
resonance and antibaryon resonances and baryon-antibaryon
production from mesons and their inverse reactions of annihi-
lation. Final results from the
AMPT model are obtained after
hadronic interactions are terminated at a cutoff time t
cut
when
observables under study are considered to be stable, i.e., when
further hadronic interactions after t
cut
will not significantly
affect these observables. We note that two-body partonic
scatterings at all possible times have been included because
the algorithm of
ZPC, which propagates partons directly to the
time when the next collision occurs, is fundamentally different
from the fixed time step method used in the
ART model.
In Figs. 1 and 2, we show, respectively, the schematic
structures of the default
AMPT model [39–44,46,47] and the
AMPT model with string melting [45,48,50] described above.
The full source code of the
AMPT model in the FORTRAN
77 language and instructions for users are available online
at the OSCAR [60] and EPAPS [61] websites. The default
AMPT model is named version 1.x, and the AMPT model
with string melting is named version 2.y;thevalueof
the integer extension x or y increases whenever the source
code is modified. Current versions of the
AMPT models are
1.11 for the default model and 2.11 for the string melting
model. In the following, we explain in detail each of the
four components of the
AMPT model and the way they
are combined to describe relativistic heavy ion collisions.
A. Initial conditions
1. The default
ampt
model
In the default AMPT model, initial conditions for heavy
ion collisions at RHIC are obtained from the
HIJING model
[51–54]. In this model, the radial density profiles of the two
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MULTIPHASE TRANSPORT MODEL FOR RELATIVISTIC . . . PHYSICAL REVIEW C 72, 064901 (2005)
FIG. 2. (Color online) Structure of the AMPT model with string
melting.
colliding nuclei are taken to have Woods-Saxon shapes, and
multiple scatterings among incoming nucleons are treated in
the eikonal formalism. Particle production from two colliding
nucleons is described in terms of a hard and a soft compo-
nent. The hard component involves processes in which the
momentum transfer is larger than a cutoff momentum p
0
and
is evaluated by the pQCD using the parton distribution function
in a nucleus. These hard processes lead to the production
of energetic minijet partons and are treated via the
PYTHIA
program. The soft component, on the other hand, takes into
account non-perturbative processes with momentum transfer
below p
0
and is modeled by the formation of strings. The
excited strings are assumed to decay independently according
to the Lund
JETSET fragmentation model.
From the pp and p
¯
p total cross sections and the ratio of
σ
el
/σ
tot
in the energy range 20 <
√
s<1800 GeV, it has
been found that the experimental data can be fitted with a
nucleon-nucleon soft cross section σ
s
(s) = 57 mb at high
energies and p
0
= 2GeV/c [51]. The independence of these
two parameters on the colliding energy is due to the use of the
Duke-Owens set 1 for the parton distribution function [62] in
the nucleon. With different parton distribution functions, an
energy-dependent p
0
may be needed to fit the same pp and p
¯
p
data [63,64]. We note that since the number of hard collisions
in an A + A collision roughly scales as A
4/3
and grows fast
with colliding energy while the number of strings roughly
scales as A, minijet production becomes more important as the
energy of heavy ion collisions increases [51,65].
Because of nuclear shadowing, both quark [66] and gluon
[67] distribution functions in nuclei are different from the
simple superposition of their distributions in a nucleon. This
effect has been included in the
HIJING model via the following
impact-parameter-dependent but Q
2
(and flavor)-independent
parametrization [52]:
R
A
(x,r) ≡
f
A
a
(x,Q
2
,r)
Af
N
a
(x,Q
2
)
= 1 + 1.19 ln
1/6
A(x
3
− 1.2x
2
+ 0.21x)
−
α
A
(r)−
1.08(A
1/3
− 1)
√
x
ln(A + 1)
e
−x
2
/0.01
, (1)
where x is the light-cone momentum fraction of parton a, and
f
a
is the parton distribution function. The impact-parameter
dependence of the nuclear shadowing effect is controlled by
α
A
(r) = 0.133(A
1/3
− 1)
1 − r
2
/R
2
A
, (2)
with r denoting the transverse distance of an interacting
nucleon from the center of the nucleus with radius R
A
=
1.2A
1/3
. Note that there is a modified HIJING model which
uses a different parametrization for the nuclear shadowing that
is also flavor dependent [63].
To take into account the Lorentz boost effect, we have
introduced a formation time for minijet partons that depends
on their four momenta [68]. Specifically, the formation time
for each parton in the default
AMPT model is taken to have a
Lorentzian distribution with a half width t
f
= E/m
2
T
, where E
and m
T
are the parton energy and transverse mass, respectively.
Initial positions of formed minijet partons are calculated from
those of their parent nucleons using straight-line trajectories.
2. The
ampt
model with string melting
Although the partonic part in the default AMPT model
includes only minijets from the
HIJING model, its energy
density can be very high in heavy ion collisions at RHIC.
As shown in Fig. 3 for the time evolutions of the energy
and number densities of partons and hadrons in the central
cell of central (b = 0 fm) Au+Au collisions at
√
s
NN
=
200 GeV in the center-of-mass frame, the partonic energy
density during the first few fm/c’s of the collision is more
than an order of magnitude higher than the critical energy
density (∼1GeV/fm
3
) for the QCD phase transition, similar
to that predicted by the high-density QCD approach [69]. The
sharp increase in energy and number densities at about 3 fm/c
is due to the exclusion of energies that are associated with
the excited strings in the partonic stage. Keeping strings in
the high-energy-density region [70] thus underestimates the
10
–1
10
0
10
1
10
2
t (fm/c)
10
–2
10
–1
10
0
10
1
10
2
n (fm
–3
), ε (GeV/fm
3
)
n
ε
parton
stage
Au+Au (s
1/2
=200A GeV,b=0)
hadron stage
FIG. 3. Energy and number densities of minijet partons and
formed hadrons in the central cell as functions of time for central
(b = 0fm)Au+Au collisions at
√
s
NN
= 200 GeV from the default
AMPT model, where the energy stored in the excited strings is absent
in the parton stage and is released only when hadrons are formed.
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LIN, KO, LI, ZHANG, AND PAL PHYSICAL REVIEW C 72, 064901 (2005)
partonic effect in these collisions. We note that the central cell
in the above calculation is chosen to have a transverse radius
of 1 fm and a longitudinal dimension between −0.5t and 0.5t,
where time t starts when the two nuclei are fully overlapped
in the longitudinal direction.
To model the above effect in high-energy-density regions,
we extend the
AMPT model to include the string melting
mechanism [45,48,50], i.e, all excited strings that are not
projectile and target nucleons without any interactions are
converted to partons according to the flavor and spin structures
of their valence quarks. In particular, a meson is converted to
a quark and an antiquark, while a baryon is first converted
to a quark and a diquark with weights according to relations
from the SU(6) quark model [71], and the diquark is then
decomposed into two quarks. The quark and diquark masses
are taken to be the same as in the
PYTHIA program [59],
e.g., m
u
= 5.6,m
d
= 9.9, and m
s
= 199 MeV/c
2
. We further
assume that the above two-body decomposition is isotropic in
the rest frame of the parent hadron or diquark, and the resulting
partons do not undergo scatterings until after a formation time
given by t
f
= E
H
/m
2
T,H
, with E
H
and m
T,H
denoting the
energy and transverse mass of the parent hadron. Similar
to the case of minijet partons in the default
AMPT model,
initial positions of the partons from melted strings are cal-
culated from those of their parent hadrons using straight-line
trajectories.
The above formation time for partons is introduced to
represent the time needed for their production from strong
color fields. Although we consider hadrons before string
melting as a convenient step in modeling the string melting
process, choosing a formation time that depends on the
momentum of the parent hadron ensures that partons from the
melting of the same hadron would have the same formation
time. The advantage of this choice is that the
AMPT model
with string melting reduces to
HIJING results in the absence
of partonic and hadronic interactions as these partons would
then find each other as closest partners at the same freeze-out
time and thus coalesce back to the original hadron. We
note that the typical string fragmentation time of about 1
fm/c is not applied to the melting of strings because the
fragmentation process involved here is considered as just an
intermediate step in modeling parton production from the
energy field of the strings in an environment of high energy
density.
B. Parton cascade
In the transport approach, interactions among partons are
described by equations of motion for their Wigner distribution
functions, which describe semiclassically their density distri-
butions in phase space. These equations can be approximately
written as the following Boltzmann equations:
p
µ
∂
µ
f
a
(x, p,t) =
m
b
1
,b
2
,···,b
m
m
i=1
d
3
p
b
i
(2π)
3
2E
b
i
f
b
i
x, p
b
i
,t
×
n
c
1
,c
2
,···,c
n
n
j=1
d
3
p
c
j
(2π)
3
2E
c
j
|M
m→n
|
2
×(2π)
4
δ
4
m
k=1
p
b
k
−
n
l=1
p
c
l
×
−
m
q=1
δ
ab
q
δ
3
p − p
b
q
+
n
r=1
δ
ac
r
δ
3
p − p
c
r
. (3)
In the above, f
a
(x, p,t) is the distribution function of parton
type a at time t in the phase space, and M
m→n
denotes the
matrix element of the multiparton interaction m → n. If one
considers only two-body interactions, these equations reduce
to
p
µ
∂
µ
f (x, p,t) ∝
σf (x
1
, p
1
,t)f (x
2
, p
2
,t), (4)
where σ is the cross section for partonic two-body scattering,
and the integral is evaluated over the momenta of the other
three partons with the integrand containing factors such as a
δ function for momentum conservation.
The Boltzmann equations are solved using
ZPC [18], in
which two partons undergo scattering whenever they approach
each other with a closest distance smaller than
√
σ/π.At
present,
ZPC includes only parton two-body scattering such as
gg → gg with cross sections calculated from the pQCD. For
gluon elastic scattering, the leading-order QCD gives
dσ
gg
dt
=
9πα
2
s
2s
2
3 −
ut
s
2
−
us
t
2
−
st
u
2
9πα
2
s
2
1
t
2
+
1
u
2
, (5)
where α
s
is the strong coupling constant, and s, t and u are
standard Mandelstam variables for elastic scattering of two
partons. The second line in the above equation is obtained by
keeping only the leading divergent terms. Since the scattering
angle ranges from 0 to π/2 for identical particles, one then
has [18]
dσ
gg
dt
9πα
2
s
2t
2
, (6)
if the scattering angle is between 0 and π.
The singularity in the total cross section can be regulated
by a Debye screening mass µ, leading to
dσ
gg
dt
9πα
2
s
2(t −µ
2
)
2
,
(7)
σ
gg
=
9πα
2
s
2µ
2
1
1 + µ
2
/s
.
The screening mass µ is generated by medium effects and is
thus related to the parton phase-space density. For the partonic
system expected to be formed in Au+Au collisions at RHIC,
the value of µ is on the order of one inverse fermi [18].
For massless partons in a plasma at temperature T, their
average colliding energy is
√
s ∼
√
18T , thus µ<
√
s for µ =
3fm
−1
leads to the requirement T>141 MeV. Since s>µ
2
generally holds in hot QGP, the following simplified relation
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MULTIPHASE TRANSPORT MODEL FOR RELATIVISTIC . . . PHYSICAL REVIEW C 72, 064901 (2005)
between the total parton elastic scattering cross section and
the screening mass is used in the
ZPC [72]
σ
gg
≈
9πα
2
s
2µ
2
. (8)
A value of 3 fm
−1
for the screening mass µ thus leads to a total
cross section of about 3 mb for the elastic scattering between
two gluons. By changing the value of the screening mass µ,
different cross sections can be obtained, and this will be used
in studying the effect of parton cross sections in heavy ion
collisions at RHIC. This cross section is used in
AMPT not
only in the default model, which includes only scatterings of
minijet gluons, but also in the string melting model, which
only includes scatterings of quarks/antiquarks of all flavors.
We have therefore neglected in the latter case the difference
between the Casimir factors for quarks and gluons.
We note that minijet partons produced from hard scatterings
in the
HIJING model can lose energy by gluon splitting and
transfer their energies to nearby soft strings. In the
AMPT
model, this so-called jet quenching in the HIJING model is
replaced by parton scatterings in
ZPC. Since only two-body
scatterings are included in
ZPC, higher-order contributions to
the jet energy loss are still missing in the
AMPT model.
C. Hadronization
Two different hadronization mechanisms are used in the
AMPT model for the two different initial conditions introduced
in Sec. II A. In the default
AMPT model, minijets coexist with
the remaining part of their parent nucleons, and together
they form new excited strings after partonic interactions.
Hadronization of these strings are described by the Lund string
model. In the
AMPT model with string melting, these strings
are converted to soft partons, and their hadronization is based
on a simple quark coalescence model, similar to that in the
ALCOR model [73].
1. Lund string fragmentation for the default
ampt
model
Hadron production from the minijet partons and soft strings
in the default
AMPT model is modeled as follows. After minijet
partons stop interacting, i.e., after they no longer scatter with
other partons, they are combined with their parent strings to
form excited strings, which are then converted to hadrons
according to the Lund string fragmentation model [57,58]. In
the Lund model as implemented in the
JETSET/PYTHIA routine
[59], one assumes that a string fragments into quark-antiquark
pairs with a Gaussian distribution in transverse momentum.
A suppression factor of 0.30 is further introduced for the
production of strange quark-antiquark pairs relative to that
of light quark-antiquark pairs. Hadrons are formed from these
quarks and antiquarks by using a symmetric fragmentation
function [57,58]. Specifically, the transverse momentum of a
hadron is given by those of its constituent quarks, while its
longitudinal momentum is determined by the Lund symmetric
fragmentation function [74]
f (z) ∝ z
−1
(1 − z)
a
exp(−bm
2
⊥
/z), (9)
with z denoting the light-cone momentum fraction of the
produced hadron with respect to that of the fragmenting string.
The average squared transverse momentum is then given by
p
2
⊥
=
p
2
⊥
f (z)d
2
p
⊥
dz
f (z)d
2
p
⊥
dz
=
z
max
0
z(1 −z)
a
exp(−bm
2
/z)dz
b
z
max
0
(1 − z)
a
exp(−bm
2
/z)dz
. (10)
For massless particles, it reduces to
p
2
⊥
=
1
b
1
0
z(1 −z)
a
dz
1
0
(1 − z)
a
dz
=
1
b(2 + a)
. (11)
Since quark-antiquark pair production from string fragmen-
tation in the Lund model is based on the Schwinger mechanism
[75] for particle production in strong field, its production
probability is proportional to exp(−πm
2
⊥
/κ), where κ is the
string tension, i.e., the energy in a unit length of string. Due
to its large mass, strange quark production is suppressed by
the factor exp[−π(m
2
s
− m
2
u
)/κ], compared to that of light
quarks. Also, the average squared transverse momentum of
produced particles is proportional to the string tension, i.e.
p
2
⊥
∝κ. Comparing this with Eq. (11), one finds that the
two parameters a and b in the Lund fragmentation function are
approximately related to the string tension by
κ ∝
1
b(2 + a)
. (12)
After production from string fragmentation, hadrons are
given an additional proper formation time of 0.7 fm/c [76].
Positions of formed hadrons are then calculated from those of
their parent strings by following straight-line trajectories.
2. Quark coalescence for the
ampt
model with string melting
After partons in the string melting scenario stop interacting,
we model their hadronization via a simple quark coalescence
model by combining the two nearest partons into a meson
and the three nearest quarks (antiquarks) into a baryon
(antibaryon). Since the invariant mass of combined partons
forms a continuous spectrum instead of a discrete one, it
is generally impossible to conserve four-momentum when
partons are coalesced into a hadron. At present, we choose
to conserve the three-momentum during coalescence and
determine the hadron species according to the flavor and
invariant mass of coalescing partons [77]. For pseudoscalar
and vector mesons with same flavor composition, the meson
with mass closer to the invariant mass of coalescing quark
and antiquark pair is formed. For example, whether a π
−
or
a ρ
−
is formed from the coalescence of a pair of
¯
u and d
quarks depends on whether the invariance mass of the quarks
is closer to the π
−
mass or the centroid of ρ mass. The same
criterion applies to the formation of octet and decuplet baryons
that have same flavor composition. It is more complicated to
treat the formation probabilities of flavor-diagonal mesons
such as π
0
and η in the pseudoscalar meson octet, ρ
0
and ω in the vector meson octet. Neglecting the mixing of
η meson with the s
¯
s state, we take the following approach for
these flavor-diagonal mesons within the SU(2) flavor space.
064901-5
