Investigation of Nuclear Compression in the Ampt Model

Home , Nuclear density

INVESTIGATION OF NUCLEAR COMPRESSION IN THE AMPT MODEL

OF NUCLEUS-NUCLEUS COLLISIONS

Athesissubmittedto Kent State University in partial Fulﬁllment of the requirements for the Degree of Master of Science

Huda Alalawi

December, 2018

Except for previously published materials Thesis written by

Huda Alalawi

B.S., Umm Al-Qura University, 2011

M.S., Kent State University, 2018

Approved by

, Advisor Dr. Declan Keane

, Advisor Dr. Spyridon Margetis

,Chair,DepartmentofPhysics Dr. James T. Gleeson

, Dean, College of Arts and Sciences Dr. James L. Blank Table of Contents

Table of Contents ...... iii

List of Figures ...... v

List of Tables ...... vii

Acknowledgments ...... viii

1 Introduction ...... 1

1.1 Standard Model (SM) ...... 1

1.2 Quantum Chromodynamics QCD ...... 4

1.3 The QCD Phase Diagram ...... 5

1.3.1 Phase transition ...... 6

1.4 Bag Model of a Hadron ...... 6

1.5 Quark-Gluon Plasma-Hadron Phase Transition in the Bag

Model ...... 8

1.5.1 Equation of state ...... 8

1.6 High-Energy Heavy -Ion Collisions ...... 10

2 AMPT Model ...... 13

2.1 Components of The AMPT Model ...... 14

2.2 Modiﬁed AMPT Code ...... 17

3 Findings Using the Modiﬁed AMPT Code and Discussion ..... 18

iii 4 Summary and Suggestions for Future Research ...... 27

References ...... 29

iv List of Figures

1.1 The Standard Model of elementary particles with the three generations

of matter, gauge bosons in the fourth column, and the Higgs boson in

the ﬁfth...... 3

1.2 A sketch of a possible phase diagram of QCD matter...... 5

1.3 MIT bag model[14] ...... 7

1.4 Collisions of two heavy nuclei in relativistic heavy ion collisions. The

collisions are not always head-on, thus sometimes some of the nucleons

become spectators while the rest become participants...... 10

1.5 The space-time evolution of heavy ion collision...... 11

1.6 The space-time evolution of heavy ion collision...... 12

2.1 Illustration of the structure of the default AMPT model...... 15

2.2 Illustration of the structure of the AMPT model with string melting. 16

3.1 A 238U+238Usemi-centralcollision,showingthechangeofbaryonden-

sity versus time...... 19

3.2 Snapshots of a Au + Au collision at 11.6 A GeV from AMPT/ART,

showing (a) combined local baryon density for target and projectile,

and (b) the density for the projectile alone [34]. Each panel plots

3 the baryon density (fm )onagrayscalewithinay z spatial grid over a projected area of 20 fm 20 fm, where the horizontal (z)axis ⇥ corresponds to the beam direction. Comparison of (a) and (b) suggests

thatcompressionisobserver–systemdependent...... 20

v 3.3 Time evolution of central baryon density for Au + Au in the CMS sys-

tem (blue circles) and the target (lab) system (red circles) at psNN =

8 GeV in the AMPT model. The vertical scale gives relative density in

units of ⇢ .Thehorizontalscaleistimesteps 10 fm/c...... 21 0 ⇥

3.4 Central baryon density in multiples of ⇢0 as a function of time for 1000

Au + Au central collisions at psNN = 8 GeV. At each time-step, each

magenta dot represents the density for one event. The overlaid black

circles are the average densities for each time-step...... 23

3.5 Average central baryon density as a function of time for di↵erent colli-

sion energies. The highest value is achieved for psNN =13GeV(black

circles) ...... 24

3.6 Maximum average central baryon density, in multiples of ground-state

nuclear density, as a function of beam energy for central Au + Au

collisions...... 26

4.1 Average central baryon density as a function of time as predicted by

ﬁve di↵erent transport model calculations for central Au+Au collision

at 5 A GeV (left panel) and at 10 A GeV (right panel)...... 28

vi List of Tables

3.1 Maximum average central baryon density as a function of CM energy

forcentralAu+Aucollisions...... 25

vii Acknowledgments

First of all, I express our deep gratitude to the Almighty Allah for giving me an opportunity to do this project. This thesis could not have been possible without precious support and guidance of many people. I would like to express my special thanks of gratitude to Prof. Declan Keane. He opened the door to this study by being my research adviser. His sincere guidance and valuable advice enabled me to achieve this project. Besides my advisor, I would like to extend my deepest appreciation to Dr. Spyridon Margetis who helped me with the AMPT model and ROOT. His instruction, patience, and advice have been the cornerstone to this work. Also, I would like to thank prof. Mina Katramatou for being a member of my thesis committee.

Thanks also go to my mom and my dad. Their constant prays, and encouragement reminded me I was not alone. I would like to thank my sisters and my brothers whose their support has followed me throughout my education. Finally, I can never thank enough my husband Naif. His warmth, understanding, and unwavering faith in me sustained me more than I can ever say.

viii Chapter 1

Introduction

All particles in the Universe are governed by four fundamental forces. In the early twentieth century, physicists believed that protons, neutrons, and electrons are the fundamental particles that constituted all matter and make up all atoms. By the mid-1960’s, physicists were beginning to realize that their prior understanding of the fundamental particles was not adequate to explain many new discovries, such as point-like particles (protons) inside the nucleon[1]. In 1964, Murray Gell-Mann and

George Zweig proposed that these new pointlike objects are the subatomic particles known as quarks which as far as we know have no substructure. In the early 1970s,

Murray Gell-Mann and George Zweig proposed a theory that is now an important part of the Standard Model (SM) of particle physics[2].

1.1 Standard Model (SM)

The Standard Model (SM) contains the handful of elementary particles shown in

Figure 1.1. It describes a theory of fundamental particles and how they interact. Ac- cording to the Standard Model, the elementary particles exist in two primary groups.

1. Fermions: Fermions that have half-integer spin and obey the Fermi–Dirac

statistics, i.e., fermions follow Pauli’s exclusion principle[3]. The fermions can

be divided into three sets.

Quarks : • 1 Quarks have fractional color charge and come in six ﬂavors which refer to set of quantum numbers. These ﬂavors with their charge are u (up), c (charm), and t

2 (top) have a charge of + 3 .Inaddition,d (down), s (strange), and b (bottom) have a charge of 1 .Theybindtogethertomakeahadronandcannotexist 3 freely. They are a↵ected by the strong, weak, electromagnetic, and gravitational force.

Leptons : •

Leptons (light particles) carry integer charge. They do not have color charge and are a↵ected by all forces except the strong force[4]. The leptons consist of e the electron, µ muon, ⌧ tau, ⌫e the electron neutrino, ⌫µ muon neutrino and

⌫⌧ tau neutrino.

Baryons : •

Baryons (heavy particles) are not fundamental particles because they are composed of three quarks that can be any combination of the six quarks as long as they combine to have a baryon number of 1. Examples of baryons are protons

(uud), neutrons (udd),++(uuu)etc.

Quarks and leptons are 12 particles in total, and these particles are further classiﬁed into three generations. Two quarks and two leptons are paired into one generation. The ﬁrst generation has the lightest and most stable particles, i.e. the quarks in this generation are u (up) and d (down) and leptons are e

(electron) ⌫e (electron neutrino). The particles in the ﬁrst generation make up

2 all stable matter in the universe[5]. In contrast, the second and third gener-

ations have more massive and unstable particles. The second generation has

the two quarks c (charm) and s (strange) and the two leptons µ (muon) and

⌫µ(muon neutrino) whereas the third generation has the t (top) and b (bottom)

quarks and ⌧ (tau) and ⌫⌧ ( tau neutrino ) leptons. All fermions have antimatter

particles with equal mass but opposite sign of electric charge.

Figure 1.1: The Standard Model of elementary particles with the three generations of matter, gauge bosons in the fourth column, and the Higgs boson in the ﬁfth.

2. Boson:

Bosons have integer spin and obey the Bose-Einstein statistics, i.e. they do not follow

Pauli’s exclusion principle. They are responsible for mediating the particle interactions, they are the force carriers. We, so far, veriﬁed the existence of the following bosons:

3 1- Photons do not carry a charge and are the mediators of the electromagnetic interactions.

+ 0 2- W ,W and Z bosons are the mediators of the weak interactions.

3- The gluons are the mediators of the strong (nuclear) interactions.

4- The Higgs Boson as shown in Figure 1.1 was discovered at CERN in 2012[6]. It has charge and spin zero[7] and it is responsible for the mass of the quarks.

5- Mesons are not fundamental particles as they are made up of a quark–antiquark pair. Examples are the ⇡ and K-meson.

The Standard Model classiﬁes all four fundamental forces that govern our universe.

In the Standard Model, a force is described as an exchange of bosons. These fundamental forces are the strong force mediated by gluons, the electromagnetic force mediated by photons, the weak force mediated by the W and Z,andthegravitational force mediated by gravitons.

1.2 Quantum Chromodynamics QCD

Quantum Chromodynamics (QCD) is a quantum ﬁeld theory of strong interactions between quarks and gluons. The only fundamental particles in this theory are quarks and gluons which carry the color charge of strong interactions[8]. Moreover, it is an essential part of the Standard Model of particle physics. QCD exhibits two main properties:

-Conﬁnement sometimes also called “color conﬁnement” which is the phenomenon that the quarks do not exist individually. In the ordinary matter, the quarks clump together to form hadrons only as baryons or mesons.

-Asymptotic freedom generally is a feature that causes the interaction between particles to become asymptotically weaker at high energy, or as the distance between

4 them gets shorter. At high energies quarks interact weakly, but their interaction becomes stronger at low energies and, in this latter case, conﬁnement will happen and the quarks and gluons will form hadrons[9].

1.3 The QCD Phase Diagram

Figure 1.2 shows a schematic plot of the QCD phase diagram that has along the x-axis the net baryon density while along y-axis temperature T is plotted[11]. The early universe started at zero net baryon density and high temperature T > Tc,where

Tc is the critical temperature at which the quark matter becomes deconﬁned. As it expanded and cooled which is T < Tc,ahadronphaseappearedthatisconsideredthe normal state where quarks are observed only within hadrons (baryons, antibaryons, and mesons)[10]. Furthermore, at high temperature T > Tc or/and high net baryon

Figure 1.2: A sketch of a possible phase diagram of QCD matter. density, a new phase transition possibly occurs from hadrons to matter in the form

5 of deconﬁned quarks and gluons which is called the Quark-Gluon Plasma (QGP). At small temperature and high net baryon density, there is another interesting region corresponding to neutron stars.

1.3.1 Phase transition

AsystemsuchaswaterisdescribedbythevariablespressureP and temperature

T .ForagivenP and T ,theequilibriumstateofthewaterdoesnothavetobe uniform. Solid, liquid, and gas are all possible states for water. These states are called phases while a phase transition is a change from one phase to another. The discontinuity at the transition determines the order of phase transition. That means if the ﬁrst derivative of the energy density with respect to some thermodynamic variable is discontinuous, the phase transition is called a ﬁrst-order phase transition.

An example of a ﬁrst order phase transition is ice to liquid water[12]. As shown in

Fig 1.2, theorists predict that a ﬁrst -order phase transition for QCD matter, and they also predict a critical point. In the next section, I will introduce the Bag Model of a hadron to ﬁnd an expression for the pressure of QCD matter since it is a variable that is relevant in this thesis.

1.4 Bag Model of a Hadron

To understand why new phases of quark matter are expected or to understand the conditions of how quarks can become deconﬁned in a new phase of quark matter, there is a useful phenomenological description which is called the bag model. This model has many di↵erent versions, but I will present the MIT bag model which is the version of the model that contains the essential characteristics of the phenomenology of quark conﬁnement[13].

6 Figure 1.3: MIT bag model[14]

In the MIT bag model, hadron is considered as a bag which has a finite dimension, usually spherical shape and is filled with quarks. Inside the bag, the quarks are treated as massless particles, and their pressure is known as the bag pressure B or quark pressure which is directed inward. Also, the kind of vacuum within the bag is perturbative QCD-vacuum. Outside the bag, the quarks are infinitely massive.

Here, the kinetic energy of the quarks causes a vacuum pressure directed outward.

Moreover, this side has a normal QCD-vacuum (non-perturbative QCD-Vacuum). If the bag pressure B balances the pressure from outside, the result is a conﬁned state.

Confinement prevents isolated quarks from being produced, and only color-neutral hadrons are observed[15]. If the quarks are confined in the bag, the gluons are also confined in the bag. In contrast, when the outside pressure is greater than the inward bag pressure, the bag pressure cannot balance the outward quark matter pressure.

7 Then the bag cannot conﬁne the quark matter contained inside. As a result of this, a new phase of matter containing the quarks and the gluons in an unconﬁned state is then possible. We conclude that the main condition for a new phase of quark matter is an imbalance between the pressure inside and outside the bag, and this imbalance happens when the temperature of the matter is high, and/or when the baryon density is large. After that, new phases of quark matter are then expected.

1.5 Quark-Gluon Plasma-Hadron Phase Transition in the Bag Model

1.5.1 Equation of state

From theoretical considerations, QGP is deﬁned as an extended deconﬁned medium of quarks and gluons. Indeed, QGP is considered as a phase of QCD matter where

QGP is a transient state, and it usually occurs at a high temperature or high-density phase. For theoretical treatment, because of the extended medium, the QGP is considered as a macroscopic system which is dealing with a few macroscopic or thermodynamic variables. From the relation between these kinds of variables, the equation of state for the system must be considered[16].

QGP at High Temperatures

To get the equation of state, we consider a gas made up of the quark-gluon system in thermal equilibrium at high temperature, T, within a volume where quarks and gluons are non-interacting (the potential energy is zero) and massless and the number of quarks equal to the number of antiquarks (net baryon number=0 )[17].

By calculating the degeneracies of quarks, gluons, we obtain :

For boson (gluon): g =numberofcolors number of spins = 8 2=16 g ⇥ ⇥ For fermion (quark and antiquark): g =numberofcolors number of spins q ⇥ ⇥

8 number of ﬂavors = 3 2 2 2=24 ⇥ ⇥ ⇥ The energy density of this system is :

⇡2 " =37 T 4 QGP 30

Because the pressure is equal to "/3, the total pressure of this system at temperature

T may be expressed as: ⇡2 P =37 T 4 QGP 90

QGP with a High Baryon Density

The energy density of quark gas in a volume V for the same system except at

T =0andhighbaryondensityis:

g " = µ4 8⇡2 q where µq is the baryonic chemical potential. From the relation between the pressure and the energy density, we get :

" g P = = µ4 3 24⇡2 q

By combining of these two equations of pressure at high temperature and high baryon density, we can find the pressure of the QCD matter. If the first derivative of the energy density with respect to temperature is discontinuous, the phase trans- formation is called a first-order transition. A good example of a first-order phase transition occurs when ice changes to liquid water, or when liquid water changes to vapor. However, an example of a second-order phase transition occurs when a fer- romagnet is heated above the Curie temperature where its ferromagnetic properties disappear[18].

9 Figure 1.4: Collisions of two heavy nuclei in relativistic heavy ion collisions. The collisions are not always head-on, thus sometimes some of the nucleons become spectators while the rest become participants.

1.6 High-Energy Heavy -Ion Collisions

Arelativisticheavyioncollisionisthoughttoproceedasfollows:

1- Before the collision, the two incoming nuclei are accelerated to move at near the speed of light. At this time, in the center of mass frame, the stationary observer sees the target and projectile nuclei look disk-shaped due to Lorentz contraction as shown in Figure 1.4[19]. Lorentz contraction or length contraction is the phenomenon that amovingobject’slengthismeasuredtobeshorterthanitsproperlength,whichis the length as measured in the object’s rest frame.

2-Attime⌧ =0andz = 0, the two accelerated nuclei hit each other, and the target and the projectile nucleons lose a signiﬁcant fraction of incoming kinetic energy which is deposited in the central region and then the interactions start developing. These nuclei do not always collide head-on. The collision is called non-central if it is not head-on.

The impact parameter b is deﬁned as the distance between the centers of the nuclei

10 when the colliding nuclei are passing. Therefore, small impact parameter collisions are called central collisions while large impact parameter collisions b R + R are ⇠ P T called peripheral collisions where RP and RT are radii of the projectile and target nuclei respectively. Also, the region of the interacted nuclei is called the overlap region while the reaction plane is the plane that contains the impact parameter and the beam axis[20]. The spectators refer to the part of the nucleons that are outside the overlapping region, and they do not interact and continue moving essentially una↵ected. The nucleons at the overlapping region are called participants.

3 - In the ﬁrst moments of the reaction, the hard quark and gluon scattering

Figure 1.5: The space-time evolution of heavy ion collision. process dominates and can be modeled by perturbative QCD (pQCD). Afterward, pre-equilibrium stage occurs when the gluons and quarks become deconﬁned because the temperature and energy density of the participants are increasing[21].

4 - Just after being liberated, the super dense and hot state of quark-gluon matter is formed. Then the temperature of the phase exceeds the critical temperature, and the outcome is the Quark-Gluon Plasma (QGP).

5-DuringtheexpansionofQGP(about10-20fm/c),thetemperaturedecreases

11 Figure 1.6: The space-time evolution of heavy ion collision. and the ﬁreball gets into a mixed phase. This phase has a gas of quarks, gluons, and hadrons[22].

6 -The temperature of the hadron gas continues to fall due to expansion. As a result, we get the freeze-out stage when the inelastic (ﬁrst) and elastic (later) interactions among the hadrons stop.

12 Chapter 2

AMPT Model

Quarks and gluons are tiny particles, and they are the building blocks of all matter that feels the strong nuclear force. Ordinary nuclear matter is composed of protons and neutrons, which in turn are composed of u and d quarks that are bound together by gluons. When nuclear matter is subject to extreme conditions of temperature and density [23], the protons and neutrons “melt”. Then the quarks and gluons become deconﬁned, and are able to move freely inside the region of high temperature and density. In this case, the excited matter is the QGP. Scientists working in this ﬁeld believe that QGP existed in the early Universe, a few microseconds after its creation in the Big Bang. To create QGP in the laboratory, such as at the Relativistic Heavy

Ion Collider (RHIC) at Brookhaven National Laboratory, and at the Large Hadron

Collider (LHC) at the CERN laboratory in Switzerland [24], two beams of heavy nuclei are accelerated in opposite directions to almost the speed of light. After they collide, a hot and dense QGP region is created, but it lasts only for a very short time

23 (a few times 10 seconds). To study the properties and formation of this matter, large detectors detect the particles that are emitted after the QGP expands, cools and returns to ordinary matter in the form of baryons like protons and neutrons, and mesons like pions and kaons.

To simulate what happens experimentally and to study the hot and dense matter produced in relativistic nuclear collisions, a multi-phase transport model (AMPT) was developed. The AMPT (A Multi-Phase Transport) model is a Monte Carlo transport

13 model for heavy ion collisions at relativistic energies [25]. This model is designed to describe nuclear collisions ranging from hadron-hadron, hadron-nucleus, and nucleus- nucleus (p+p, p+A, and A+A)systemsoverawiderangeofcenter-of-massenergies from psNN of a few GeV up to 5500 GeV, which is the maximum energy of the Large

Hadron Collider. The AMPT model includes both initial interactions between quarks and gluons in the QGP state of matter, as well as later interactions between baryons and mesons, after the QGP has decayed. As a result, the AMPT model has many components, and a speciﬁc code models each component in AMPT. These di↵erent components work together to give a cohesive description of relativistic heavy ion collisions. In this chapter, I give a brief description of the components of the AMPT model that are used in this thesis project .

2.1 Components of The AMPT Model

Initial conditions:

In the default AMPT model, when two nucleons collide, they produce new particles if the energy is high enough. These particles are described in terms of a hard and soft component. The hard component takes into account perturbative processes with momentum transfer above a cuto↵momentum p0 .Thesehardprocessesproduce energetic minijet partons [26]. On the other hand, the soft component involves processes in which the momentum transfer is less than a cuto↵momentum p0.These soft processes lead to the formation of “strings”. According to the Lund JETSET fragmentation model[27], the excited strings decay independently. The AMPT model has the option of string melting for the e↵ect of excited strings in the region of high energy density, when all excited strings convert to partons. Therefore, to generate the initial conditions, the AMPT model uses hard minijet partons and soft strings

14 Figure 2.1: Illustration of the structure of the default AMPT model. from the HIJING model[28].

Parton cascade:

In the default AMPT model, only minijets are included in the partonic stage. How- ever, the AMPT model with string melting has both excited strings and minijets in the partonic stage. To describe scatterings among partons, AMPT uses the ZPC

Parton Cascade[29].

Hadronization:

This stage describes the passage from partonic to hadronic matter. Two di↵erent hadronization mechanisms correspond to the two types of initial conditions. In the default AMPT model, after partonic interactions, new excited strings are formed from

15 Figure 2.2: Illustration of the structure of the AMPT model with string melting. minijets and the remaining part of their parent nucleons [30]. The Lund string model describes the hadronization of these strings. On the other hand, with string melting,

AMPT uses the quark coalescence model to describe the hadronization of soft partons which are produced from these strings.

Hadron cascade:

AMPT uses the ART (A Relativistic Transport) hadronic transport model to describe scatterings among the ﬁnal hadrons. It was originally developed for describing heavy ion collisions at beam energy/A 15 GeV. This model has elastic and in-  elastic scatterings for baryon-baryon, baryon-meson, and meson-meson interactions.

Also, it has mean-ﬁeld potentials which are suitable for studying the e↵ect due to the

16 hadronic equation of state. Resonances such as can be produced from pion-nucleon scattering. Possible changes to the masses and widths of the medium are neglected

[31].

2.2 Modiﬁed AMPT Code

We changed the input–parameter ﬁle and also the ART1.0 codes in this model as follow:

Modiﬁed in input ﬁle:

We chose to run the code in the center of mass frame (CMS) because of its nice symmetry and applicability to colliders. The range for the impact parameter b was set from bmin =0tobmax =0.1 to select very central (head–on) collisions. Also, we simulated one thousand events at each energy, and the number of timesteps was set to 150 in 0.2 fm/c each, i.e. the total time we recorded was 30 fm/c. Because we concentrate the project in range of relatively low energies, we chose Default AMPT model that means no string melting.

Modiﬁed in ART1.0 code:

ART stands for A Relativistic Transport, which is a hadronic transport model. We modiﬁed, wrote and debugged the code to calculate the baryon density in a 1 fm3 cell centered at (x, y, z)=(0, 0, 0) coordinates (where the maximum achieved baryon density is expected).Then we built some redundancy to check our coding and its performance .

The analysis and histogramming was done using the ROOT analysis framework developed at CERN.

17 Chapter 3

Findings Using the Modiﬁed AMPT Code and Discussion

This chapter summarizes the results of this project. We carried out simulations based on a widely-used model of nucleus-nucleus collisions, the AMPT model, which is described in Chapter 2. The speciﬁc aim of this thesis work was to map-out the change in baryon compression as the beam energy is varied over a range where other models indicate that high compressions are reached. It is of particular interest to determine

AMPT’s prediction about the range of energies where the baryon compression reaches its maximum.

The nuclear collisions studied in this project are with Gold (197Au) beams. This stable isotope has 197 nucleons; 79 protons and 118 neutrons. 197Au was chosen because it is one of the two most commonly-studied heavy nuclei (197Au at RHIC and

208Pb (lead) at CERN). For most aspects of low-energy nuclear collisions, changing the number of nucleons up or down by a couple of percent can change the nuclear structure and its internal properties. The situation in the study of high-energy nuclear collisions is the complete opposite. Adding or removing a single nucleon from any nucleus makes almost no di↵erence in the properties of bulk nuclear matter, e.g. achieved compression or energy density.

In this study, the maximum impact parameter was set to b =0.1fmatallenergies, since we are interested in investigating the maximum compression in head-on collisions. The center-of-mass energies per nucleon pair (psNN) studied are 2.6, 2.8, 3.5,

5, 6.5, 8, 10, 12, 13, 14, 15, 20 and 30 GeV. The energies were chosen after an initial

18 scan that allowed us to roughly determine the energy of maximum baryon density.

We have modiﬁed the standard AMPT code, version v1.26t7/v2.26t7 as downloaded from the OSCAR website in 2016 [32].

Figure 3.1 shows an example of how the baryon density changes over the course of a

Figure 3.1: A 238U+238Usemi-centralcollision,showingthechangeofbaryondensity versus time. collision [33]. Here two 238U+238U nuclei collide at the low ﬁxed-target beam energy of 1.0A GeV and in a semi-central collision. As the initial inter-penetration of the two nuclei progresses, they heat and compress each other. The density reaches a maxi-

3 3 mum value which in this case is ⇢ 0.45 fm ,orabout2.8⇢ where ⇢ =0.16 fm ⇠ 0 0 is the value of ground state nuclear density used by AMPT and us here. As the

19 “ﬁreball” expands and cools, the temperature and baryon density decrease rapidly.

Figure 3.2: Snapshots of a Au + Au collision at 11.6 A GeV from AMPT/ART, showing (a) combined local baryon density for target and projectile, and (b) the 3 density for the projectile alone [34]. Each panel plots the baryon density (fm )ona gray scale within a y z spatial grid over a projected area of 20 fm 20 fm, where the horizontal (z)axiscorrespondstothebeamdirection.Comparisonof(a)and(b) ⇥ suggests that compression is observer–system dependent.

Figure 3.2 shows local baryon densities in a Au + Au 11.6 A GeV collision in the center of mass frame (CM) frame (a) and the beam (projectile) frame (b). Twelve time slices, from t = 3fm/ctot =8fm/careselectedfromeachframecollision. Time t =0fm/cisthemomentwhentwonucleicompletelyoverlapwitheachother.

The gray scales indicate the local baryon density achieved at each step. Let us examine the Au + Au collision in the CM frame (left panel). We see that the density distribution remains always symmetric with respect to the origin. This is not the

20 case in the projectile frame (right panel). Also the areas where the highest densities achieved are di↵erent in the two frames. This ﬁgure demonstrates in a visual way the fact that the values of achieved densities are observer-system dependent. This is an unphysical artifact that is present in many nuclear collision model implementations, and is normally minimized by running the code in the CM frame.

Figure 3.3: Time evolution of central baryon density for Au + Au in the CMS system (blue circles) and the target (lab) system (red circles) at psNN = 8 GeV in the AMPT model. The vertical scale gives relative density in units of ⇢0.Thehorizontalscaleis timesteps 10 fm/c. ⇥

We discuss this in more detail below, but in order to check it with our AMPT model, we performed the following test. We ran the same collision energy but in two modes, CM and LAB frames. In both runs, we estimated the achieved densities in a cell centered around coordinates (x, y, z)=(0, 0, 0). In the CM-run (collider mode)

21 this system is at rest relative to the observer/lab frame. In the LAB-run the CM of the collision (and the point (0,0,0)) is moving relative to the ﬁxed target (lab) frame.

Figure 3.3 shows the resulting densities as a function of time (fm/c 10). We also ⇥ see that the shape of the distribution is stretched in the CMS-run most likely due to the time-dilation.

At t = 2 fm/c we see in ﬁg. 3.2 that each Au nucleus is a Lorentz-contracted disc. For large nuclei such as Pb or Au, we calculate the radius using the expression R =

1 r0A 3 ,wherer0 =1.12 fm. So the diameter of the disc is about 14 fm and its thickness is about 14/ fm where the relativistic factor = Eparticle/m is approximately 100 and 2500 at the highest beam energies achievable at RHIC and LHC respectively

[35]. The local baryon density can be evaluated by ﬁnding the baryon density at each location in the “selected frame” (CM in our case). The local baryon density in the rest frame of each Au nuclei before the collision is the normal (ground state) nuclear matter density ⇢0. But, viewed from the CM rest frame each nucleus has an apparent density ⇢ = ⇢0.Attimet = 0 fm/c, when the two Au nuclei completely overlap, the baryon density in the overlapping region, even without any compression, is going to be times higher the actual baryon density. So the increased baryon density at this time is not due to compression but simply due to the relativistic factor. In lab-frame collisions the situation is even more complex, therefore we chose to do our studies in the CM frame.

We have considered central collisions (impact parameters b in the range of 0

0.1fm) of Au nuclei at energies psNN =2.6, 2.8, 3.5, 5, 6.5, 8, 10, 12, 13, 14, 15,

20 and 30GeV. For each energy we ran the model and sampled the baryon density at di↵erent time steps from 0 to 30 fm/c. Our time-step size is 0.2fm/c.Weran

22 1000 events at each energy. We have considered a volume cell of 1 1 1fm3 centered ⇥ ⇥ at the origin of the coordinate system (0,0,0). The central baryon density in each time-step is obtained by counting the baryons (initial and produced) in the cell. We then divide this apparent density by the cell’s factor to obtain the proper density,

0 ⇢ = ⇢ /cell where the cell Lorentz factor is calculated as

particles Ecell i Ei cell = = m particles particles cell ( PE2) ( p2) i i i i q where i runs over all particles in the cell.P P

Figure 3.4: Central baryon density in multiples of ⇢0 as a function of time for 1000 Au + Au central collisions at psNN = 8GeV. At each time-step, each magenta dot represents the density for one event. The overlaid black circles are the average densities for each time-step.

Figure 3.4 shows the central baryon density versus time-step in head-on Au + Au

23 central collisions at psNN = 8GeV. One thousand collisions were simulated. We see that the achieved values in a given time ﬂuctuate a lot. Therefore for our analysis, we used the average value for each time step. This is shown as black ﬁlled circle in the

ﬁgure. The net baryon density starts from a small value, reaches a maximum when

Figure 3.5: Average central baryon density as a function of time for di↵erent collision energies. The highest value is achieved for psNN =13GeV(blackcircles) the two nuclei pass through each other and then drops again as the system expands.

Figure 3.5 shows the average central baryon density as a function of time for selected collision energies. We observe that as the energy increases the maximum is reached at slightly earlier times and also the width of the resulting shape becomes narrower. The maximum value reached is about 8 times the value of ground state nuclear matter (⇢0)attheenergyofpsNN =13GeV(blackcirclesinﬁgure).

For each energy we can ‘read’ from the resulting distribution the maximum value

24 psNN GeV Max. Baryon Density/⇢0 2.6 2.37 2.8 2.56 3.5 3.26 5 4.62 6.5 5.96 8 7.02 10 7.3 12 7.54 13 7.8 14 7.64 15 7.57 20 7.0 30 5.88

Table 3.1: Maximum average central baryon density as a function of CM energy for central Au+Au collisions. of achieved average central baryon density. Table 3.1 and ﬁgure 3.6 summarize the results. We observe that the peak of baryon density, about 8 times the ground state nuclear matter density, is achieved for energies in the 10–15GeV range (13GeV here).

25 Figure 3.6: Maximum average central baryon density, in multiples of ground-state nuclear density, as a function of beam energy for central Au + Au collisions.

26 Chapter 4

Summary and Suggestions for Future Research

In summary, this thesis has used AMPT/ART, a relativistic transport model, to predict properties of collisions at the Relativistic Heavy Ion Collider. The model runs over all stages of the collision process (both initial partonic and ﬁnal hadronic interactions and the transition between these two phases of matter). Results presented in chapter 3 o↵er estimates of the peak baryon density in central collisions (b =0.1 fm) in the center of mass frame for 197Au +197Au collisions. Test volumes 1 fm3 have been used for calculating densities over a range of beam energies. The maximum central baryon density, which is about 8 times the ground-state matter density, has been found at a beam energy ofpsNN = 13 GeV. In addition, I was found that the peak baryon density decreases with increasing collision energy beyond 13 GeV. Last but not least, at later times than the maximum as the beam energy increases, the shape of the time evolution of the central baryon density is modiﬁed, with the tail becoming much more stretched-out.

Finally, we compare our results from AMPT model with some recent calculations of heavy-ion collisions more relevant for the FAIR future project at GSI facility in

Germany. Figure 4.1 presents the predicted achieved central baryon density in the center of the ﬁreball in central Au+Au collisions (b =0)at5(left)and10(right)A

GeV (LAB energy) from several models[38]. We observe that the model predictions are varying by about a factor of two in the predicted values. We also note that at 5

3 (10) A GeV energy the average maximum baryon density achieved is about ⇢ =1fm

27 Figure 4.1: Average central baryon density as a function of time as predicted by ﬁve di↵erent transport model calculations for central Au+Au collision at 5 A GeV (left panel) and at 10 A GeV (right panel).

3 (⇢ =1.4fm ), or about 6 (8) times the ground state nuclear density, a result similar to our ﬁndings with the AMPT model.

28 References

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