SEARCH FOR MAXIMUM NUCLEAR COMPRESSION IN A MODEL OF
NUCLEUS-NUCLEUS COLLISIONS
A thesis submitted to
Kent State University in partial
fulfillment of the requirements for the
degree of Master of Science
by
Sultan Alhomaidhi
December, 2015
c Copyright
All rights reserved
Except for previously published materials
i
Thesis written by
Sultan Alhomaidhi
B.S., King Saud University, 2009
M.S., Kent State University, 2015
Approved by
Declan Keane
James T. Gleeson
James L. Blank
, Advisor
, Chair, Department of Physics
, Dean, College of Arts and Sciences
ii
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2 UrQMD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
3 Findings Using the Modified UrQMD Code . . . . . . . . . . . . . .
12
4 Summary and Suggestions for Future Research . . . . . . . . . . . .
43
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
iii
LIST OF FIGURES
1.1
The elementary particles and forces of the Standard Model . . . . . .
2
1.2
The phase diagram of the nuclear matter . . . . . . . . . . . . . . . .
4
2.1
Two heavy nuclei colliding at impact parameter b . . . . . . . . . . .
√
Two nuclei at a beam energy of sN N = 20 GeV before collision with
11
b=0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.1
3.2
Example 1 of a single 200 Hg + 200 Hg collision event, with baryon information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Example 1 of a single
200
Hg +
200
Hg collision event, with antibaryon
information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
15
16
Example 2 of a single 200 Hg + 200 Hg collision event, with baryon information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
200
Hg +
200
17
3.5
Example 2 of a single
Hg collision event, with antibaryon
18
3.6
information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
Average number of baryons inside a sphere of radius R0 at sN N = 2
20
3.7
GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
Average number of baryons inside a sphere of radius R0 at sN N = 3
21
3.8
GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
Average number of baryons inside a sphere of radius R0 at sN N = 5
22
3.9
GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
Average number of antibaryons inside a sphere of radius R0 at sN N
= 5 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
iv
3.10 Average number of baryons inside a sphere of radius R0 at
√
sN N = 10
GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.11 Average number of antibaryons inside a sphere of radius R0 at sN N
24
= 10 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.12 Average number of baryons inside a sphere of radius R0 at sN N = 20
25
GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.13 Average number of antibaryons inside a sphere of radius R0 at sN N
26
= 20 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.14 Average number of baryons inside a sphere of radius R0 at sN N = 40
27
GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.15 Average number of antibaryons inside a sphere of radius R0 at sN N
28
= 40 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.16 Average number of baryons inside a sphere of radius R0 at sN N = 80
29
GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.17 Average number of antibaryons inside a sphere of radius R0 at sN N
30
= 80 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.18 Average number of baryons inside a sphere of radius R0 at sN N =
31
160 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.19 Average number of antibaryons inside a sphere of radius R0 at sN N
32
= 160 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.20 Average number of baryons inside a sphere of radius R0 at sN N =
33
320 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
√
3.21 Average number of antibaryons inside a sphere of radius R0 at sN N
34
= 320 GeV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
v
3.22 Maximum number of baryons inside a sphere of radius R0 . . . . . . .
36
3.23 Maximum number of net baryons inside a sphere of radius R0 . . . .
38
3.24 The maximum net baryon density . . . . . . . . . . . . . . . . . . . .
39
3.25 The maximum net-baryon density with a 2nd-order polynomial fit . .
40
vi
ACKNOWLEDGMENTS
First of all, I would like to take this opportunity to express my sincere gratitude
to my advisor Prof. Declan Keane for the continuous support of my MS study. His
guidance helped me in all the time of research and writing of this thesis. Besides my
advisor, I would like to thank Prof. Spyridon Margetis and Prof. Veronica Dexheimer,
for being members of my thesis committee. My thanks also goes to Dr. Wei-Ming
Zhang, who gave me access to compute resources and helped me with the UrQMD
model. Without their precious support, this thesis would not have been possible.
Also, I would like to thanks all my friends who helped me get on the road to coding
and LATEX .
Last but not the least, I would like to thank my family: my parents, my brothers
and sisters for the unceasing encouragement, support and attention.
vii
CHAPTER 1
Introduction
Fundamental discoveries since quarks were proposed in 1963 by physicists Murray
Gell-Mann and George Zweig prove that quarks are the most elementary constituents
of the nucleus, since there is strong evidence that quarks themselves have no substructure (they are not made of other particles) [1, 2]. The Standard Model is a unified
description of matter in terms of quarks, particles in the electron family (leptons), and
the fields mediating their interactions. According to the Standard Model, there are
three generations of elementary constituents among the quarks and among the leptons, with two quarks and two leptons in each generation. All of the above have spin
1
,
2
and are called fermions. Interactions among fermions are mediated by emission
and absorption of force field quanta, which have integer spin (bosons). The forces are
comprised of the strong nuclear force mediated by gluons, the electromagnetic force
mediated by photons, the weak nuclear force mediated by the W and Z bosons, and
the gravitational force mediated by gravitons.
1
Fig. 1.1: The elementary particles and forces of the Standard Model[3].
The final elementary particle in the Standard Model is the Higgs boson. In 2012,
it was discovered at the LHC (Large Hadron Collider) [4, 5]. The Higgs has zero
spin, and no electric charge or color charge — see Fig. 1.1. Flavor refers to set a of
quantum numbers that are carried by quarks. There are three quarks with an electric
charge of − 31 units, and these three have flavors d, s and b, while the remaining three
quark types carry an electric charge of + 32 units, and these three have flavors u, c
and t. Leptons have either electric charge -1, namely e− (electron), µ− (muon) or τ −
(tau), or zero charge in the case of νe (electron neutrino), νµ (muon neutrino) or ντ
(tau neutrino). The baryon number B is + 31 for every quark, and is zero for leptons.
The lepton number L is +1 for leptons and zero for quarks. All of the above fermions
have antimatter partner particles with equal mass and opposite sign of quantum
2
numbers, including electric charge. Quarks are never detected individually; in normal
matter, we observe only baryons (three quarks bound together), or antibaryons (three
antiquarks) or mesons (quark-antiquark pair). The part of the Standard Model that
deals with quarks and their interactions is called Quantum Chromodynamics (QCD).
Experiments at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Lab, on Long Island, New York, study the particles emerging from very energetic collisions between heavy nuclei like gold (each made up of 197 protons and
neutrons), and offer unique means for investigating the properties of quarks and the
particles associated with the forces between quarks. The most high-profile scientific
breakthrough so far at the RHIC accelerator was the discovery of quark-gluon plasma
(QGP) [6, 7, 8, 9, 10, 11]. Fig 1.2 shows the locations of the QGP phase in the phase
diagram of nuclear matter.
3
Fig. 1.2: A schematic sketch of the phase diagram of QCD matter [12]. The QGP phase is
indicated by the brown region, while the white region represents the normal state where quarks are
observed only within hadrons (baryons, antibaryons and mesons). If this sketch based on theoretical
hypotheses is correct, the phase transitions observed so far at RHIC took place on the far left side of
this diagram and were continuous transitions. However, a transition crossing the yellow band would
be a first-order phase transition.
The existence of this new phase of matter had been predicted on theoretical
grounds, and it is believed to have existed fleetingly during the first microsecond
after the Big Bang. Its discovery was further confirmation of QCD and the Standard
Model.
Until recently, the available evidence suggested that the transition between an
ordinary matter phase and QGP phase is a continuous transition[13, 14, 15]. To
explain phase transitions [16, 17], I will introduce the pressure of QCD matter, since
it is a variable that is relevant in this thesis. The pressure P due to QCD matter at
4
a high temperature T with zero net baryon density is
P =
E
π2
= 37 T 4 ,
3V
90
(1.1)
E
where V is the energy density. The pressure due to a high baryon density with zero
temperature is
P =
E
gq 4
=
µ ,
3V
24π 2 q
(1.2)
where gq is the degeneracy number of quarks and µq is the quark Fermi momentum.
The pressure of the QCD matter can be found as combination of these two equations at
high temperature and high baryon density. If the first derivative of the energy density
with respect to temperature is discontinuous, the phase transformation is called a
first-order transition. A familiar example of a discontinuous phase transition occurs
when ice changes to liquid water, or when liquid water changes to vapor. However,
if the first derivative mentioned above is continuous but the second derivative is not,
the phase transition is dubbed a second-order transition or a continuous transition.
An example of a continuous transition occurs when a ferromagnet is heated above
the Curie temperature where its ferromagnetic properties disappear.
Theorists had anticipated that the normal maximum energy of the RHIC accelerator is too high to observe a first-order phase transition, and they predicted that
it might be seen at a lower energy[18, 19, 20, 21]. Without any doubt, we know
that the peak net-baryon density increases in nuclear collisions as the beam energy is
lowered below the maximum energy of the RHIC accelerator at Brookhaven. When
we refer to peak net-baryon density, we recognize that net-baryon density rises and
falls during the early stages of a collision, and the peak refers to the maximum value
reached during the time evolution of the collision. The above-mentioned beam-energy
5
dependence of peak net-baryon density happens because most baryons emitted from
collisions at high energies come from baryon-antibaryon pairs excited from the vacuum, and thus those baryons contribute zero net-baryon density. However, at lower
energies, there are more baryons from the initial colliding nuclei that are slowed-down
or “stopped” by the collision process. On the other hand, as the beam energy drops,
there might be insufficient energy to reach or cross the yellow transition region depicted in Fig. 1.2. These insights and questions motived scientists at RHIC to conduct
a series of experiments known as the Beam Energy Scan (BES)[22, 23, 24]. These
√
BES experiments investigated a very wide range of beam energies: sN N values are
200 GeV, 130 GeV, 62.4 GeV, 39 GeV, 27 GeV, 19.6 GeV, 14.5 GeV, 11.5 GeV and
7.7 GeV.
The Beam Energy Scan is a major project that requires a lot of resources: a large
number of scientists need to put several years of effort into it, and the total cost
to operate the RHIC accelerator is over $100 million per year. Detailed theoretical
models are essential for interpreting the measurements from the Beam Energy Scan.
Nuclear transport models are an important category, and they are implemented as
event-generating computer codes using Monte Carlo techniques which simulate the
random processes in a relativistic collision between two nuclei. An example of such a
model is described in chapter 2. The codes need to be run many thousands of times
in order to build up enough model statistics. With a sufficient number of models
events, the theory can be tested in comparisons with experimental results.
The transport models that are available at the present time do not include the
physics of a phase transition to QGP, but assume that the excited matter of the
colliding nuclei stays in the ordinary phase of hadronic matter throughout the collision
6
process. Although we believe that a phase transition to QGP does indeed happen
during the early stage of these collisions, it is still essential to study models without a
phase transition. This allows us to understand the “background” hadronic processes,
which can be quite complicated, and therefore helps us to identify what features of the
experimental measurements cannot be explained unless a phase transition to QGP
happens.
One important question that models need to answer relates to stopping in relativistic nuclear collisions. The relevance of nuclear stopping was outlined earlier.
In high-energy nucleus-nucleus collisions in a head-on configuration (zero impact parameter), a low amount of nuclear stopping would be consistent with a large fraction
of the nucleons from the initial nuclei retaining their original momentum, or close
to it, after the collision is finished. Besides revealing information about the reaction
mechanism, the level of nuclear stopping offer information about whether there is
sufficient excitation of the QCD matter to form quark-gluon plasma.
We know that stopping decreases at the beam energy of relativistic nuclear collisions increases, and decreased stopping in turn leads to less nuclear compression. On
the other hand, it is obvious that more energy is available for exciting the colliding
nuclear matter as the beam energy increases, and some of this excitation must be in
the form of compressional potential energy. Therefore, it has been speculated [25, 22]
that there is a particular beam energy, or range of beam energies, where the competition between these two opposing factors might cause a maximum in the net-baryon
compression. If so, it is important to estimate the beam energy region of this maximum net-baryon compression. Such an energy region, if it exists, would be an ideal
location for the study of phase transition effects. Furthermore, if we know the beam
7
energy region where the peak net-baryon density is the highest, then in this region,
we stay as far as possible to the right in Fig. 1.2 while still exciting the QCD matter
sufficiently to reach the quark-gluon plasma phase. In that scenario, we would have
the best chance of exploring the region of a first-order phase transition QGP.
8
CHAPTER 2
UrQMD Model
The UrQMD (Ultra-relativistic Quantum Molecular Dynamics) model [26] is a
microscopic transport theory based on phase space description of the reaction [27, 28,
29]. It is suitable for simulating heavy ion collisions over a wide range of energies,
from a few GeV to hundreds of GeV. It is intended to describe many aspects of the
collision process like rates and spectra of all produced particles, correlations, etc.
One of the strengths of UrQMD is that a variety of options and parameters can be
changed. In this chapter, I summarize how the UrQMD model initializes the two
colliding nuclei, and introduce the assumptions behind the reactions and the collision
terms.
First of all, the two incident nuclei are constituted based on the Fermi-gas approach [27]. The wave functions of the nucleons are represented by Gaussian-shaped
density distributions. There are many restrictions and constraints that every initialization of a nucleus has to fulfill. The nucleus has to be at rest and centered around
zero. That means the summation of the nucleon velocity must be zero and the summation of the distances between nucleons must also be zero. Also, the nucleus has to
be in the ground state. The radius of the nucleus is given by
1
R(A) = r0 ( [A + (A1/3 − 1)3 ])1/3 .
2
(2.1)
A further constraint is that the binding energy of the two nuclei should agree with
the binding energy calculated by the Bethe-Weizsäcker formula. The initial momenta
9
of the nucleons are randomly chosen between zero and the local Thomas-Fermi momentum. The code uses the system clock for each event to generate a random seed
for all probabilistic calculations. This ensures that the output result of each collision
of two nuclei is different.
In the UrQMD model, the collision between nucleons is based on a non-relativistic
density-dependent Skyrme potential plus Yukawa potential and Coulomb potential.
The Pauli potential can be optionally added to the formula where a momentumdependent potential is not included. The model has 55 different baryon species and
32 different meson species, not counting antiparticles. For a particle-particle collision
to happen, the distance between the two particles dtrans must be within
dtrans 6 d0 =
√
σtot = σ( s, type) ,
p
σtot /π,
(2.2)
where σtot is the total cross section for that binary particle interaction. The total
cross sections depend on the particle species in the initial and final state of the
binary interaction, on the center-of-mass energy, and on isospins. The trajectory and
evolution of each individual particle is followed, and some aspects of the collision
environment are recalculated every time step, where the interval between time steps
is a parameter chosen by the user of the code. A typical interval between time steps
is 0.1 fm/c, which corresponds to 3.33 × 10−25 seconds.
For the model to be executed properly, a set of parameters and options has to
be defined according to the study one wants to conduct. The code will read these
parameters and options first, and then starts simulating the collisions. One such
example is impact parameter b. The impact parameter is defined as the minimum
distance between the centers of the two nuclei if they were to pass each other without
any interaction (see Fig. 2.1). For exactly head-on collisions, the impact parameter
10
is b = 0.
Fig. 2.1: Left: two heavy nuclei at impact parameter b before they collide. Right: spectators and
participants after the two nuclei collide [30]. Note that the left side of this diagram depicts Lorentzcontracted nuclei, whereas on the right, in order to represent the participant zone with better clarity,
no Lorentz contraction is applied.
The left side of this figure gives a cartoon-like illustration of two heavy nuclei
approaching each other from opposite directions, as in the laboratory frame at a
collider. Each nucleus is assumed to have a speed v that is a significant fraction of
the speed of light (c), and therefore there is a Lorentz contraction effect along the
p
direction of motion by a factor 1/γ = 1 − v 2 /c2 (also shown in Fig. 3.1). On the
right side of Fig. 2.1, the Lorentz contraction is not applied, since this allows the
details of the excited participant zone to be shown more clearly.
11
CHAPTER 3
Findings Using the Modified UrQMD Code
In this chapter, I show the result of my work. I carried out a simulation based on
a widely-used model of nucleus-nucleus collisions: UrQMD. As explained in Chapter
1, the main goal of this thesis is to map-out the change in net-baryon compression as
the beam energy is varied over the range available to current experiments at RHIC.
It is of particular interest to determine if UrQMD predicts that there is a definite
beam energy region where the net-baryon compression is a maximum.
The initial nuclei studied in this work are Mercury on Mercury (200 Hg +
This stable isotope has 200 nucleons: 80 protons and 120 neutrons.
200
200
Hg and
Hg).
200
Hg
are the two most abundant isotopes present in terrestrial ores. The exact number
of proton or neutron is not important in the present work. Protons and neutrons
can be considered to be the same particle state (the nucleon) as far as the strong
interaction is concerned, and they differ only slightly in that their electromagnetic
interactions are different. Mercury-200 was chosen because it is close enough to the
two most commonly-studied heavy nuclei at accelerators —
US facilities and
208
197
Au (Gold) at most
Pb (Lead) at CERN. For most aspects of high-energy nuclear
collisions, changing the number of nucleons up or down by a couple of percent makes
almost no difference. The situation in the study of low-energy nuclear structure is the
complete opposite, because in such cases, adding or removing a single nucleon from
any nucleus can dramatically change the nuclear structure and its internal properties.
In this study, the impact parameter was set to zero (b = 0) at all energies, since
12
I am just interested in investigating the maximum compression. My studied centerof-mass energies per nucleon pair are 2, 3, 5, 10, 20, 40, 80, 160 and 320 GeV, which
covers a range that is a little broader than what is available using the RHIC facility at
Brookhaven. I have modified the standard UrQMD code, version 3.3 as downloaded
from the UrQMD website [26] in 2014, to allow the number of baryons and antibaryons
inside three volumes to be counted and recorded as a function of the time step in the
UrQMD collision evolution. The baryon and antibaryon numbers are counted inside
three spheres, with radii R0 = 1.2, 1.5 and 2.0 fm. The center of each sphere is always
at the mid-point of the line joining the centers of the two colliding nuclei.
13
8
7
Z
6
X
5
4
3
2
1
0
!8
!7
!6
!5
!4
!3
!2
!1
0
1
2
3
4
5
6
7
8
!1
!2
radius 2.02fm
!3
radius 1.52fm
!4
radius 1.22fm
!5
!6
!7
!8
Fig. 3.1: A representation of two
200
Hg nuclei at a beam energy of
√
sN N = 20 GeV, before a
collision at b = 0. The semi-random distribution of nucleons within the nuclear volume can be seen.
The plot also indicates the three spheres within which baryons and antibaryons are counted.
At the beginning of the collision, when the nuclei first touch, there are no antibaryons yet, and the baryons counted are nucleons from the relevant volumes of the
initial-state nuclei. After a few time steps, the baryon counts increase because the
two nuclei are interacting.
14
60
Sample 1
Radius 1.2 fm
Radius 1.5 fm
Radius 2.0 fm
50
# of Baryon
40
30
20
10
2.925
2.825
2.725
2.625
2.525
2.425
2.325
2.225
2.125
2.025
1.925
1.825
1.725
1.625
1.525
1.425
1.325
1.225
1.125
1.025
0.925
0.825
0.725
0.625
0.525
0.425
0.325
0.225
0.125
0.025
0
Time (fm/c)
Fig. 3.2: Example 1 of a single 200 Hg + 200 Hg collision event. This shows the number of baryons
inside a sphere of radius R0 as a function of the time step (in units of fm/c) during the collision
process at a center-of-mass beam energy of 20 GeV per nucleon pair in the UrQMD model, at zero
impact parameter. Three values of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot describes
a single nucleus-nucleus collision (event), and therefore statistical fluctuations can be seen.
15
7
sample 1 of Anti-Baryon
Radius 1.2 fm
6
Radius 1.5 fm
Radius 2.0 fm
# of Anti-Baryon
5
4
3
2
1
2.925
2.825
2.725
2.625
2.525
2.425
2.325
2.225
2.125
2.025
1.925
1.825
1.725
1.625
1.525
1.425
1.325
1.225
1.125
1.025
0.925
0.825
0.725
0.625
0.525
0.425
0.325
0.225
0.125
0.025
0
Time (fm/c)
Fig. 3.3: Example 1 of a single 200 Hg + 200 Hg collision event. This shows the number of antibaryons
inside a sphere of radius R0 as a function of the time step (in units of fm/c) during the collision
process at a center-of-mass beam energy of 20 GeV per nucleon pair in the UrQMD model, at zero
impact parameter. Three values of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot describes
a single nucleus-nucleus collision (event), and therefore statistical fluctuations can be seen.
16
60
Sample 2
50
Radius 1.2 fm
Radius 1.5 fm
Radius 2.0 fm
# of Baryon
40
30
20
10
2.925
2.825
2.725
2.625
2.525
2.425
2.325
2.225
2.125
2.025
1.925
1.825
1.725
1.625
1.525
1.425
1.325
1.225
1.125
1.025
0.925
0.825
0.725
0.625
0.525
0.425
0.325
0.225
0.125
0.025
0
Time (fm/c)
Fig. 3.4: Example 2 of a single 200 Hg + 200 Hg collision event. This shows the number of baryons
inside a sphere of radius R0 as a function of the time step (in units of fm/c) during the collision
process at a center-of-mass beam energy of 20 GeV per nucleon pair in the UrQMD model, at zero
impact parameter. Three values of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the
same as shown in Fig. 3.2 except that a different event, with different random fluctuations, is shown.
17
9
sample 2 of Anti-Baryon
8
Radius 1.2 fm
Radius 1.5 fm
7
Radius 2.0 fm
# of Anti-Baryon
6
5
4
3
2
1
2.925
2.825
2.725
2.625
2.525
2.425
2.325
2.225
2.125
2.025
1.925
1.825
1.725
1.625
1.525
1.425
1.325
1.225
1.125
1.025
0.925
0.825
0.725
0.625
0.525
0.425
0.325
0.225
0.125
0.025
0
Time (fm/c)
Fig. 3.5: Example 2 of a single 200 Hg + 200 Hg collision event. This shows the number of antibaryons
inside a sphere of radius R0 as a function of the time step (in units of fm/c) during the collision
process at a center-of-mass beam energy of 20 GeV per nucleon pair in the UrQMD model, at zero
impact parameter. Three values of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the
same as shown in Fig. 3.3 except that a different event, with different random fluctuations, is shown.
Figs. 3.6 to 3.21 show similar plots as Figs. 3.2 and 3.3, except now the baryon
and antibaryon fluctuations are suppressed by taking an average over 100 independent
events for each of the nine studied beam energies. There are several conclusions that
can be reached on the basis of these plots:
First, the time of the peak in baryon and antibaryon density is almost independent
of the radius R0 of the test spheres.
18
Second, at the time steps of interest, the number of baryons and antibaryons inside
each of the three spheres is not proportional to the sphere’s volume, probably because
the two larger spheres include some regions of empty space (an example of this can
be seen from Fig. 3.1).
Third, as expected, the time of the peak in baryon and antibaryon density shifts
earlier as the beam energy increases and the time needed to reach the fully overlapped
configuration of the two nuclei becomes shorter.
Fourth, also as expected, the height of the peak in baryon and antibaryon density
increases as the beam energy increases. This is mostly caused by the increasing
production of baryon-antibaryon pairs at higher beam energies.
Fifth, the shape of the time evolution of the baryon and antibaryon density changes
significantly with beam energy, and at later times than the peak, the tail becomes
much more stretched-out as the beam energy increases. This is also to be expected,
because baryons from baryon-antibaryon pairs produced nearly at rest in the centerof-mass frame become increasingly dominant at the higher beam energies.
19
25
Energy2GeV
#ofBaryon
20
15
Radius1.2fm
Radius1.5fm
10
Radius2.0fm
5
39.5
38.0
36.5
35.0
33.5
32.0
30.5
29.0
27.5
26.0
24.5
23.0
21.5
20.0
18.5
17.0
15.5
14.0
12.5
9.5
11.0
8.0
6.5
5.0
3.5
2.0
0.5
0
Time(fm/c)
Fig. 3.6: Average number of baryons inside a sphere of radius R0 as a function of the time step (in
units of fm/c) during the collision process for
200
Hg +
200
Hg in the UrQMD model. Three values
of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. To suppress statistical fluctuations, the number of
√
baryons has been averaged over 100 events. The center-of-mass energy sN N for this collision is 2
GeV per nucleon pair, and the impact parameter is zero (central collisions).
20
40
Energy3GeV
35
30
#ofBaryon
25
20
Radius1.2fm
Radius1.5fm
Radius2.0fm
15
10
5
19.5
18.5
17.5
16.5
15.5
14.5
13.5
12.5
11.5
10.5
9.5
8.5
7.5
6.5
5.5
4.5
3.5
2.5
1.5
0.5
0
Time(fm/c)
Fig. 3.7: Average number of baryons inside a sphere of radius R0 as a function of the time step (in
units of fm/c) during the collision process for
200
Hg +
200
Hg in the UrQMD model. Three values
of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.6 except
√
that the center-of-mass energy sN N for this collision is 3 GeV per nucleon pair.
21
50
Energy 5 GeV
45
40
# of Baryon
35
30
25
Radius 1.2 fm
Radius 1.5 fm
20
Radius 2.0 fm
15
10
5
7.00
6.70
6.40
6.10
5.80
5.50
5.20
4.90
4.60
4.30
4.00
3.70
3.40
3.10
2.80
2.50
2.20
1.90
1.60
1.30
1.00
0.70
0.40
0.10
0
Time (fm/c)
Fig. 3.8: Average number of baryons inside a sphere of radius R0 as a function of the time step (in
units of fm/c) during the collision process for
200
Hg +
200
Hg in the UrQMD model. Three values
of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.6 except
√
that the center-of-mass energy sN N for this collision is 5 GeV per nucleon pair.
22
0.045
Energy"5"GeV
0.04
0.035
#"of"Anti)Baryon
0.03
0.025
Radius21.22fm
0.02
Radius21.52fm
Radius22.02fm
0.015
0.01
0.005
7.00
6.70
6.40
6.10
5.80
5.50
5.20
4.90
4.60
4.30
4.00
3.70
3.40
3.10
2.80
2.50
2.20
1.90
1.60
1.30
1.00
0.70
0.40
0.10
0
Time"(fm/c)
Fig. 3.9: Average number of antibaryons inside a sphere of radius R0 as a function of the time step
(in units of fm/c) during the collision process for 200 Hg + 200 Hg in the UrQMD model. Three values
of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. To suppress statistical fluctuations, the number of
antibaryons has been averaged over 100 events. However, quantization and statistical fluctuations
still can be seen because of the low probability for production of antibaryons at this center-of-mass
√
energy. The center-of-mass energy sN N for this collision is 5 GeV per nucleon pair, and the impact
parameter is zero (central collisions).
23
60
Energy 10 GeV
50
# of Baryon
40
30
Radius 1.2 fm
Radius 1.5 fm
Radius 2.0 fm
20
10
3.95
3.80
3.65
3.50
3.35
3.20
3.05
2.90
2.75
2.60
2.45
2.30
2.15
2.00
1.85
1.70
1.55
1.40
1.25
1.10
0.95
0.80
0.65
0.50
0.35
0.20
0.05
0
Time (fm/c)
Fig. 3.10: Average number of baryons inside a sphere of radius R0 as a function of the time step
(in units of fm/c) during the collision process for 200 Hg + 200 Hg in the UrQMD model. Three values
of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.6 except
√
that the center-of-mass energy sN N for this collision is 10 GeV per nucleon pair.
24
1.4
Energy"10"GeV
1.2
#"of"Anti)Baryon
1
0.8
Radius21.22fm
Radius21.52fm
0.6
Radius22.02fm
0.4
0.2
3.95
3.80
3.65
3.50
3.35
3.20
3.05
2.90
2.75
2.60
2.45
2.30
2.15
2.00
1.85
1.70
1.55
1.40
1.25
1.10
0.95
0.80
0.65
0.50
0.35
0.20
0.05
0
Time"(fm/c)
Fig. 3.11: Average number of antibaryons inside a sphere of radius R0 as a function of the time
step (in units of fm/c) during the collision process for
200
Hg +
200
Hg in the UrQMD model. Three
values of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.9
√
except that the center-of-mass energy sN N for this collision is 10 GeV per nucleon pair.
25
60
Energy 20 GeV
50
Radius 1.2 fm
Radius 1.5 fm
Radius 2.0 fm
# of Baryon
40
30
20
10
0.025
0.100
0.175
0.250
0.325
0.400
0.475
0.550
0.625
0.700
0.775
0.850
0.925
1.000
1.075
1.150
1.225
1.300
1.375
1.450
1.525
1.600
1.675
1.750
1.825
1.900
1.975
2.050
2.125
2.200
2.275
2.350
2.425
2.500
2.575
2.650
2.725
2.800
2.875
2.950
0
Time (fm/c)
Fig. 3.12: Average number of baryons inside a sphere of radius R0 as a function of the time step
(in units of fm/c) during the collision process for 200 Hg + 200 Hg in the UrQMD model. Three values
of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.6 except
√
that the center-of-mass energy sN N for this collision is 20 GeV per nucleon pair.
26
6
Energy"20"GeV
5
Radius21.22fm
Radius21.52fm
Radius22.02fm
#"of"Anti)Baryon
4
3
2
1
0.025
0.100
0.175
0.250
0.325
0.400
0.475
0.550
0.625
0.700
0.775
0.850
0.925
1.000
1.075
1.150
1.225
1.300
1.375
1.450
1.525
1.600
1.675
1.750
1.825
1.900
1.975
2.050
2.125
2.200
2.275
2.350
2.425
2.500
2.575
2.650
2.725
2.800
2.875
2.950
0
Time"(fm/c)
Fig. 3.13: Average number of antibaryons inside a sphere of radius R0 as a function of the time
step (in units of fm/c) during the collision process for
200
Hg +
200
Hg in the UrQMD model. Three
values of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.9
√
except that the center-of-mass energy sN N for this collision is 20 GeV per nucleon pair.
27
70
Energy 40 GeV
Radius 1.2 fm
60
Radius 1.5 fm
Radius 2.0 fm
# of Baryon
50
40
30
20
10
2.00
1.94
1.88
1.82
1.76
1.70
1.64
1.58
1.52
1.46
1.40
1.34
1.28
1.22
1.16
1.10
1.04
0.98
0.92
0.86
0.80
0.74
0.68
0.62
0.56
0.50
0.44
0.38
0.32
0.26
0.20
0.14
0.08
0.02
0
Time (fm/c)
Fig. 3.14: Average number of baryons inside a sphere of radius R0 as a function of the time step
(in units of fm/c) during the collision process for 200 Hg + 200 Hg in the UrQMD model. Three values
of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.6 except
√
that the center-of-mass energy sN N for this collision is 40 GeV per nucleon pair.
28
14
Energy"40"GeV
Radius21.22fm
12
Radius21.52fm
Radius22.02fm
#"of"Anti)Baryon
10
8
6
4
2
2.00
1.94
1.88
1.82
1.76
1.70
1.64
1.58
1.52
1.46
1.40
1.34
1.28
1.22
1.16
1.10
1.04
0.98
0.92
0.86
0.80
0.74
0.68
0.62
0.56
0.50
0.44
0.38
0.32
0.26
0.20
0.14
0.08
0.02
0
Time"(fm/c)
Fig. 3.15: Average number of antibaryons inside a sphere of radius R0 as a function of the time
step (in units of fm/c) during the collision process for
200
Hg +
200
Hg in the UrQMD model. Three
values of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.9
√
except that the center-of-mass energy sN N for this collision is 40 GeV per nucleon pair.
29
90
Energy 80 GeV
80
70
Radius 1.2 fm
# of Baryon
60
Radius 1.5 fm
Radius 2.0 fm
50
40
30
20
10
1.00
0.97
0.94
0.91
0.88
0.85
0.82
0.79
0.76
0.73
0.70
0.67
0.64
0.61
0.58
0.55
0.52
0.49
0.46
0.43
0.40
0.37
0.34
0.31
0.28
0.25
0.22
0.19
0.16
0.13
0.10
0.07
0.04
0.01
0
Time (fm/c)
Fig. 3.16: Average number of baryons inside a sphere of radius R0 as a function of the time step
(in units of fm/c) during the collision process for 200 Hg + 200 Hg in the UrQMD model. Three values
of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.6 except
√
that the center-of-mass energy sN N for this collision is 80 GeV per nucleon pair.
30
25
Energy"80"GeV
#"of"Anti)Baryon
20
Radius21.22fm
15
Radius21.52fm
Radius22.02fm
10
5
1.00
0.97
0.94
0.91
0.88
0.85
0.82
0.79
0.76
0.73
0.70
0.67
0.64
0.61
0.58
0.55
0.52
0.49
0.46
0.43
0.40
0.37
0.34
0.31
0.28
0.25
0.22
0.19
0.16
0.13
0.10
0.07
0.04
0.01
0
Time"(fm/c)
Fig. 3.17: Average number of antibaryons inside a sphere of radius R0 as a function of the time
step (in units of fm/c) during the collision process for
200
Hg +
200
Hg in the UrQMD model. Three
values of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.9
√
except that the center-of-mass energy sN N for this collision is 80 GeV per nucleon pair.
31
90
Energy 160 GeV
80
70
Radius 1.2 fm
# of Baryon
60
Radius 1.5 fm
Radius 2.0 fm
50
40
30
20
10
1.00
0.97
0.94
0.91
0.88
0.85
0.82
0.79
0.76
0.73
0.70
0.67
0.64
0.61
0.58
0.55
0.52
0.49
0.46
0.43
0.40
0.37
0.34
0.31
0.28
0.25
0.22
0.19
0.16
0.13
0.10
0.07
0.04
0.01
0
Time (fm/c)
Fig. 3.18: Average number of baryons inside a sphere of radius R0 as a function of the time step
(in units of fm/c) during the collision process for 200 Hg + 200 Hg in the UrQMD model. Three values
of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.6 except
√
that the center-of-mass energy sN N for this collision is 160 GeV per nucleon pair.
32
40
Energy"160"GeV
35
#"of"Anti)Baryon
30
Radius21.22fm
25
Radius21.52fm
Radius22.02fm
20
15
10
5
1.00
0.97
0.94
0.91
0.88
0.85
0.82
0.79
0.76
0.73
0.70
0.67
0.64
0.61
0.58
0.55
0.52
0.49
0.46
0.43
0.40
0.37
0.34
0.31
0.28
0.25
0.22
0.19
0.16
0.13
0.10
0.07
0.04
0.01
0
Time"(fm/c)
Fig. 3.19: Average number of antibaryons inside a sphere of radius R0 as a function of the time
step (in units of fm/c) during the collision process for
200
Hg +
200
Hg in the UrQMD model. Three
values of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.9
√
except that the center-of-mass energy sN N for this collision is 160 GeV per nucleon pair.
33
120
Energy 320 GeV
100
Radius 1.2 fm
# of Baryon
80
Radius 1.5 fm
Radius 2.0 fm
60
40
20
0.49
0.47
0.45
0.43
0.41
0.39
0.37
0.35
0.33
0.31
0.29
0.27
0.25
0.23
0.21
0.19
0.17
0.15
0.13
0.11
0.09
0.07
0.05
0.03
0.01
0
Time (fm/c)
Fig. 3.20: Average number of baryons inside a sphere of radius R0 as a function of the time step
(in units of fm/c) during the collision process for 200 Hg + 200 Hg in the UrQMD model. Three values
of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.6 except
√
that the center-of-mass energy sN N for this collision is 320 GeV per nucleon pair.
34
60
Energy"320"GeV
50
Radius11.21fm
#"of"Anti)Baryon
40
Radius11.51fm
Radius12.01fm
30
20
10
0.49
0.47
0.45
0.43
0.41
0.39
0.37
0.35
0.33
0.31
0.29
0.27
0.25
0.23
0.21
0.19
0.17
0.15
0.13
0.11
0.09
0.07
0.05
0.03
0.01
0
Time"(fm/c)
Fig. 3.21: Average number of antibaryons inside a sphere of radius R0 as a function of the time
step (in units of fm/c) during the collision process for
200
Hg +
200
Hg in the UrQMD model. Three
values of the radius R0 are shown: 1.2, 1.5 and 2.0 fm. This plot is the same as shown in Fig. 3.9
√
except that the center-of-mass energy sN N for this collision is 320 GeV per nucleon pair.
35
TheMax#ofBaryon
radius1.2fm
radius1.5fm
radius2.0fm
120
100
#ofBaryon
80
60
40
20
0
2
3
5
10
20
40
80
160
320
BeamEnergy(GeV)
Fig. 3.22: The maximum number of baryons inside a sphere of radius R0 as a function of beam
energy. This maximum refers to the peak vertical axis values (average number of baryons) in Figs.
3.6 through 3.20.
At
√
sN N = 5 GeV, the available energy is well above the threshold for antiproton
production. However, 5 GeV is still too low to produce antibaryons except with a
low probability, and the initial-state baryons dominate. Antibaryons are created with
steeply increasing probability as the beam energy is increased above 5 GeV. Baryon
number is always conserved, so the baryons produced from the vacuum are always
exactly balanced by equal numbers of antibaryons produced from the vacuum. The
only way to build up a high density of net baryons is to slow down the incoming
baryons from the initial Hg nuclei and compress them. This is the compression effect
36
that needs to be exploited in order to move to the right in the QCD phase diagram,
as explained in Chapter 1. In particular, we want to find out what is the best beam
energy for maximizing the peak net-baryon density, according to the UrQMD model.
Two different methods for calculating the peak net-baryon density have been
tested:
In method #1, the number of net baryons is calculated for each time step. Then, the
average number of net-baryons is computed based on the sample of 100 events for
each beam energy. Finally, the maximum number of net baryons is determined from
the result of the prior step.
In method #2, the number of baryons and antibaryons is separately counted for
each time step. Then, the average number of baryons and the average number of
antibaryons are computed based on the sample of 100 events for each beam energy.
Finally, the maximum number of net baryons is determined from the result of the
prior step.
37
TheMax#ofNetBaryon
radius1.2fm
radius1.5fm
radius2.0fm
60
#ofNetBaryon
50
40
30
20
10
0
2
3
5
10
20
40
80
160
320
BeamEnergy(GeV)
Fig. 3.23: The maximum number of net baryons inside a sphere of radius R0 as a function of beam
energy. This plot is similar to that shown in Fig. 3.22 except that net baryons, defined as baryons
minus antibaryons, are shown here. A large net baryon density combined with sufficient thermal
excitation of the QCD medium is needed to explore an important region of the QCD phase diagram
(see Chapter 1).
These two methods yield similar results, and lead to net-baryon densities that
agree within a few percent. This difference is so small that it can be neglected. The
first method has been used in this study.
38
TheNetBaryonDensity
Radius1.2fm
Radius1.5fm
Radius2.0fm
0.9
0.8
NetBaryonDensity(fm-3)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
3
5
10
20
40
80
160
320
BeamEnergy(GeV)
Fig. 3.24: The maximum net baryon density, in units of fm−3 , computed for a sphere of radius R0 ,
as a function of beam energy. All densities are divided by γ =
√
sN N /2m, where m is the nucleon
mass. This factor 1/γ corrects for the apparent compression caused by Lorentz contraction. The
error bars plotted here are statistical. There are systematic uncertainties that probably are larger
for the larger radii, arising from integrating over a relatively large volume within which the true
density may fluctuate as low as zero (see text). This result is based on
200
Hg +
200
Hg collisions in
the UrQMD model, at zero impact parameter.
It is possible to carry out a polynomial fit to the data presented in Fig. 3.24 and
in this way infer a beam energy associated with the maximum at each of the three R0
values. This has been explored by assigning an integer x = 1, 2, 3, 4, 5, 6, 7, 8 and 9
to the nine beam energies, and the data for R0 = 1.2, 1.5 and 2.0 fm are denoted by
(y1 , x1 ), (y2 , x2 ) and (y3 , x3 ), respectively. Then a 2nd-order polynomial fit gives
39
y1 = 0.0003x51 − 0.0097x41 + 0.1162x31 − 0.6365x21 + 1.3957x1 − 0.2645
(3.1)
y2 = 0.0002x52 − 0.0076x42 + 0.0944x32 − 0.5279x22 + 1.1538x2 − 0.1467
(3.2)
y3 = 0.0003x53 − 0.0094x43 + 0.1086x33 − 0.5684x23 + 1.1695x3 − 0.1608
(3.3)
TheNetBaryonDensity
Radius1.2fm
Radius1.5fm
Radius2.0fm
0.9
0.8
y1 =0.0003x1 5 - 0.0097x1 4 +0.1162x1 3 - 0.6365x1 2 +1.3957x1 - 0.2645
NetBaryonDensity(fm-3)
0.7
y2 =0.0002x2 5 - 0.0076x2 4 +0.0944x2 3 - 0.5279x2 2 +1.1538x2 - 0.1467
0.6
y3 =0.0003x3 5 - 0.0094x3 4 +0.1086x3 3 - 0.5684x3 2 +1.1695x3 - 0.1608
0.5
0.4
0.3
0.2
0.1
0
2
3
5
10
20
40
80
160
320
BeamEnergy(GeV)
Fig. 3.25: The maximum net-baryon density with a 5th-order polynomial fit.
The derivatives of Eqs. (3.1), (3.2) and (3.3) are
dy1
= 0.0015x41 − 0.0388x31 + 0.3486x21 − 1.273x1 + 1.3957
dx1
(3.4)
dy2
= 0.001x42 − 0.0304x32 + 0.2832x22 − 1.0558x2 + 1.1538
dx2
(3.5)
40
dy3
= 0.0015x43 − 0.0376x33 + 0.3258x23 − 1.1368x3 + 1.1695
dx3
(3.6)
From Eqs. (3.4), (3.5) and (3.6), the values of xi where the slopes go to zero are
x1 = 1.866
x2 = 1.813
x3 = 1.719
There is a relationship between beam energy and the integer x as defined above; it
can be described by E = 0.025x5i − 0.375x4i + 2.4583x3i − 7.125x2i + 10.017xi − 3,
and therefore the three inferred beam energies E peak associated with the maximum
net-baryon density are
ERpeak
= 2.87 GeV
0 =1.2fm
ERpeak
= 2.83 GeV
0 =1.5fm
ERpeak
= 2.75 GeV
0 =2.0fm
In Fig. 3.24, the net-baryon density in units of fm−3 as function of the beam energy
shows the peak net-baryon density within each of the three spheres. The uncertainties given by the plotted error bars are from Poisson statistics only. The difference
in density for the three different radii are mostly artifacts of averaging over quite
large volumes in the case of the two bigger radii, and so the results associated with
two larger R0 values ought to be considered to have relatively large systematic uncertainties. With the larger radii, the true density at different positions within the
test sphere may fluctuate as low as zero, which biases the mean density towards lower
values. However, the systematic uncertainties when comparing different beam energies at the same R0 are likely much smaller than the systematic uncertainties when
comparing densities at different R0 values at fixed beam energy. Taking UrQMD at
face value, Figs. 3.24 and 3.25 tell us that the maximum net-baryon density occurs
just slightly less than 3 GeV, and still remains quite close to the maximum up to a
few GeV higher than that energy. In other words, the maximum net-baryon density
41
lies below the lower end of the Beam Energy Scan (BES) region at RHIC, but still
we can infer that the lower end of the BES region is not much below the point of
maximum net-baryon density.
42
CHAPTER 4
Summary and Suggestions for Future Research
This thesis has used the UrQMD nuclear transport model to probe certain questions with a connection to future experiments at the Relativistic Heavy Ion Collider.
The UrQMD model has a long track record of at least reasonable qualitative agreement with a wide range of experimental measurements over the range of beam energies under consideration. However, an important caveat is that this model assumes
a hadronic phase of matter at all times during the collision, and therefore possible
signatures of a phase transition will never be present. The quantities of particular
interest here are baryonic compression during the early stages of the collision process.
To be even more specific, net-baryon density during the early times of a collision is
especially of interest because it is a crucial variable (of course it is a purely theoretical
quantity, not accessible via experiment) in describing the phase diagram of Quantum
Chromodynamics (QCD).
Results presented in Chapter 3 offer estimates of baryon and antibaryon density
at the center of the collision zone in the center of mass frame, for exactly head-on
(zero impact parameter) 200 Hg + 200 Hg collisions. A spherical test volume, with three
different radii, has been used for calculating densities. Due to Lorentz contraction,
which varies widely over the beam energies of interest, a spherical volume might
not be ideal, but it is the natural and obvious first step in an investigation such as
this one. Future investigations might benefit from using a volume shaped like an
oblate ellipsoid with its shortest axis along the beam direction. Much higher model
43
statistics might also be of benefit, and would allow even smaller test volumes for
density calculations.
Generally, most of the overall trends observed in the output from UrQMD conform
to normal expectations. The main result of interest is the peak net-baryon density,
which happens anywhere from about 0.08 to 28 fm/c after the nuclei begin to touch,
depending on the beam energy. Peak baryon densities rise steadily with beam energy.
Peak net-baryon densities rise with beam energy up to about 3 GeV per nucleon pair,
then drop significantly. If the results presented in Chapter 3 are taken at face value, it
√
suggests that a broad range of beam energies centered near sN N ∼ 2.82 GeV is best
for maximizing net-baryon density. This conclusion is consistent with our knowledge
that stopping decreases significantly, both in experiment [6, 7, 8, 9, 10, 22] and in
UrQMD and similar models[18, 21, 27, 28, 29], over the studied beam energy range.
44
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