Describing the Dynamics of the Quark-Gluon Plasma Using Relativistic Viscous Hydrodynamics

DESCRIBING THE DYNAMICS OF THE QUARK-GLUON PLASMA USING RELATIVISTIC VISCOUS HYDRODYNAMICS

A Thesis Submitted to the Kent State University, College of Arts and Sciences in partial fulfillment of the requirements for the degree of Master of Science.

Mohammad N. Yaseen

August 2016

A thesis written by

Mohammad N. Yaseen

B.S., University of Anbar, 2008

M.S., Kent State University, 2016

Approved by

______, Director, Master’s Thesis Committee Michael Strickland

______, Member, Master’s Thesis Committee Bjorn Lussem

______, Member, Master’s Thesis Committee Spyridon Margetis

Accepted by

______, Chair, Department of Physics James T. Gleeson

______, Dean, College of Arts and Sciences James L. Blank

iii

YASEEN, MOHAMMAD N., M.S., August 2016 QGP Produced in Ultra Relativistic Heavy Ion Collisions

DESCRIBING THE DYNAMICS OF THE QUARK-GLUON PLASMA USING RELATIVISTIC VISCOUS HYDRODYNAMICS (83 pp.)

Director of Thesis: Michael Strickland, Ph.D.

When heavy nuclei collide at ultra-relativistic energies, their nuclear matter will melt producing what is known as the Quark-Gluon Plasma (QGP); a new state of matter that has been produced at the Relativistic Heavy Ion Collider

(RHIC) at Brookhaven National Laboratory (BNL) and Super Proton Synchrotron

(LHC) in European Organization for Nuclear Research (CERN). Scientists now think that this matter filled the entire universe during the first micro second after the Big Bang. According to the experimental data, this matter acts like a nearly perfect liquid. This study requires a quantitatively precise theoretical framework to describe the dynamical evolution of the fireball produced by the collision. The equations that control the fireball expansion cannot be solved analytically. As a result, scientists must solve these equations numerically. The main goal of this thesis is to find precise numerical solutions for these equations. This is complicated by the fact that when using fluctuating initial conditions, discontinuities may be present which cause problems for standard centered- differences schemes. To fix this problem, we will use the following two numerical methods: LAX and weighted LAX.

ACKNOWLEDGEMENTS

Here, I mainly want to thank my advisor Dr. Michael Strickland for his continuous support and patience. He showed great knowledge and interest in theoretical physics and, he is really a great educator who can introduce deep theoretical concepts in simplified way. I also want to thank my friend Ammar Kirmani who helped me to understand the principles of programming. I also want to thank my thesis defense committee members Spyridon Margetis and Bjorn Lussem for the time they granted to me.

Preface

When heavy nuclei collide at ultra-relativistic energies, their nuclear matter will melt producing what is known as the Quark-Gluon Plasma (QGP). Since QGP behaves like a nearly perfect relativistic liquid, one can use viscous hydrodynamics equations to simulate the QGP hot bulk expansion. The viscous hydrodynamics equations cannot be solved analytically. Therefore, I use the numerical method to solve them and simulate the

QGP expansion and this is the main goal of this thesis.

In “chapter I’’, I introduce the reader to the basic important concepts that are necessary to understand what the QGP is and how it behaves. I introduce basic concepts of QCD after taking a historical glance. After that, I introduce the QGP concept and try to simplify the main concepts related to GGP as much as possible.

In “chapter II”, I derive the main equations needed in the numerical calculations.

I take the zeroth, first and second moment of the Boltzmann equation to derive the necessary number of independent equations needed to evaluate the necessary variables.

In “chapter III”, I introduce the numerical calculation results. To perform this simulation, I use the following two numerical methods: LAX and weighted LAX.

In “Chapter IV” I introduce the link between the astronomy studies and QGP and how QGP studies with gravitational waves can help us to understand the compact star inner structure. Finally, I discuss the results of the whole experimental and theoretical work done to simulate the QGP behavior and try to take a look at the future.

TABLE OF CONTENTS

Page ACKNOWLEDGMENTS ……………………………………………………………… iv

PREFACE ………………………………………………………………………………. v

LIST OF FIGURES ………………………………………………………………....… vii

CHAPTER I. INTRODUCTION ……………………………………………………………….… 1 Historical Glance …...………………………………………………………………. 1 Quarks, Gluons and Chromodynamics ……………………………………………... 3 Quark Model …………………………………………………………………...….3 The Experimental Evidence for Quarks ……………………………………………7 Quantum Chromodynamics (QCD), Color Charge and Strong Interaction ……… 10 Color Confinement ……………………………………………………………….. 15 Jets ……………………………………………………………………………….. 15 Jet Quenching ……………………………………………………………………. 18 Quark Gluon Plasma (QGP) ………………………………………………………... 19 What is QGP? ………………………………………………………………….…. 19 A Glance on Experimental Detection Techniques and Calculations ………….….. 19

II. ANISOTROPIC RELATIVISTIC HYDRODYNAMICS ………………………… 22 Elliptic Flow ……………………………………………..………………………… 22 Derivation of Ideal Hydrodynamics Equations from Kinetic Theory ……………... 28 Bjorken Hydrodynamics …………………………………………………………… 32 General 3+1d Anisotropic Hydrodynamics Equations for a Massless Hydrodynamics System …………………………………………………………… 34 Convention and Notation ……………………………………………………….. 35 Basic Vectors ……………………………………………………………………. 35 Distribution function …………………………………………………………….. 36 Dynamical Variables …………………………………………………………….. 37 Bulk Variables ………………………………………………………………….... 38 Dynamical Equations …………………………………………………………….. 40

III. NUMERICAL CALCULATIONS ……………………………………….………. 44 Weighted LAX ………………………………………………………….……….. 44 What is Weighted LAX? …………………………………………….……….. 44 Mathematical Formula ……………………………………………….……….. 44 The Best Value of λ ………………………………………………….………… 45 Numerical Results……………………………………………………….………... 46 Code constant Parameters and Initial Conditions …………………….……….. 46 Longitudinal Anisotropic Momentum Parameter Graph …………….………... 47 Effective Temperature Graphs ……………………………………….………... 47 Transverse Anisotropic Momentum Parameter Graph ……………….………... 52 The Graph of the Ratio of the Transverse Parameter to the Longitudinal Parameter ………………………………………….……… 54 Pion Differential Spectra Graph ………………………………………….……. 57 Freeze-out Hypersurface Graph ………………………………………….…….. 60

IV. APPLICATIONS, DISCUSIONS AND LOOK AT THE FUTURE ……………. 65 Theoretical Discussions and Conclusions ………………………………………... 65 Why Viscous Hydrodynamics ……………………………………………………. 67 The Experimental Results of Viscous Hydrodynamics ……………...…………… 68 QGP Existence in the Compact Stars and the Big Bang …………….………….…73

APPENDIX ……………………………………………………………………………77 REFERENCES ……………………………………………………………………….. 81

LIST OF FIGURES

Figure Page

1. The basic quark model scheme ……………………………………………………….. 4

2. The internal structure of proton and neutron in the original quark model ……………. 5

3. The interaction between incident electron and proton’s components ………………… 9

4. The three basic colors combined with each other …………………………………… 14

5. Detecting jets in laboratory ………………………………………………………...... 16

6. The basic jets and color confinement phenomenon illustration …………………...… 17

7. QGP evaluation stages ….……………………………………….………………….…21

8. Collision cross-section ………………………………………………………………. 23

9. The approximate QGP bulk shape and dimensions …………………………….…… 26

10. The asymmetry of the emerging quarks …………………………………………… 27

11. The longitudinal anisotropy parameter (훂z) as a function of position

at ( = 1.25 fm/c) ……………………………………………………………………. 48

12. The longitudinal anisotropy parameter (훂z) as a function of position at

( = 10.25 fm/c) …………………………………………………………………….. 49

13. The effective temperature as a function of position at ( = 1.25 fm/c) ….…………. 50

14. The effective temperature as a function of position at ( = 10.25 fm/c) …………… 51

15. The transverse anisotropy parameter (훂x) as a function of position

at ( = 0.65 fm/c) …………………………………………………………………… 53

16. The transverse anisotropy parameter (훂x) and the longitudinal anisotropy

parameter (훂z) as a function of position at ( = 1.25 fm/c) ……………………...… 55

vii

17. The transverse anisotropy parameter (훂x) and the longitudinal anisotropy

parameter (훂z) as a function of position at ( = 10.25 fm/c) ……………………… 56

18. The pion differential spectra graph for large number of events

against to the transvers angular momentum ……………………………………….. 58

19. The pion differential spectra graph for one event against to the

transvers angular momentum ………………………………………………………. 59

20. The temperature spatial invariant inside QGP bulk ………………………………... 62

21. The temperature invariant with time for every point inside the QGP bulk ………… 63

22. The freeze-out hypersurface graph …………………………………………………. 64

23. Multiplicity versus centrality (%) …………………………………………………... 69

24. Average transverse momentum versus centrality (%) ….……………………..…… 70

25. Flow hadronic coefficients versus centrality (%) ………….……………………….. 71

26. Transverse momentum spectra graph (experimental and simulation data) …………. 72

27. The nowadays estimated universe history …………………………………………... 76

viii

CHAPTER I INTRODUCTION Historical Glance Like many other discoveries in physics, the Quark Gluon Plasma (QGP) was theoretically predicted many decades before it was experimentally realized. There was a problem that if Quantum Chromodynamics (QCD) were like Quantum Electrodynamics

(QED), the interaction force would become stronger as the distance between two particles becomes smaller. With this contradiction, not only would QGP formation be impossible, but even QCD was doubtable. When this contradiction was solved when asymptotic freedom was discovered, it took theorists less than a year to predict the existence of the

QGP and describe some of its features and formation conditions.

Attempting to produce a QGP in the laboratory, scientists have conducted many experiments since the mid-1980s. Progress was achieved due to high beam energies and advanced detection techniques that have been used in the modern era. Regarding the thermalization and flow hydrodynamics, these experiments showed good agreement between the QGP theory predictions and the data collected. But as the detection techniques were still not efficient enough to give firm results, there were still doubts and the results were still controversial.

A second set of experiments was conducted in the era 1995 – 2003 by both the

European Organization for Nuclear Research (CERN) and Brookhaven National

Laboratory (BNL). At this time, the CERN beam energy reached up to 11 GeV (per nucleon). These collisions generated thousands of particles per event with energy densities exceeding 1 GeV/fm3, emphasizing that the community had produced small

samples of a new form of “condensed matter’’. Contrary to what QGP theorists expected before the experiment, the properties of this new matter implied that the QGP was strongly rather than weakly coupled as was expected. The data could be successfully interpreted according to hypothesis of QGP formation as a nearly perfect liquid, not a weakly-interacting gas. However, there was still a “skeptical camp’’. It seemed that the experiments needed to be performed with even higher beam energies and more advanced detection techniques.

Experiments at the Relativistic Heavy Ion Collider (RHIC) in BNL began taking data in the summer of 2000, with Au + Au collisions at energies approximately 20 times higher than achievable Super Proton Synchrotron (LHC) in CERN. Now, with more advanced detection techniques and higher beam energies, there was a clear evidence that the BNL team succeeded to create matter in nearly local thermal equilibrium that expands collectively according to the laws of relativistic fluid dynamics, not a gaseous plasma.

The higher beam energies generated matter that had a temperature exceeding the critical temperature (Tc) required to create QGP (Tc ~ 155 MeV). Temperatures of this magnitude and the evidence for a large degree of thermalization already at less than about

1 fm/c after nuclear impact, made it unavoidable that QGP had been created in these collisions (1).

Another experiment, with even higher collision energy (√푠푁푁 = 2.76 TeV), was conducted at CERN using Large Hadron Collider (LHC) starting in 2010. The goal of this still ongoing experiment is to precisely determine the quantitative thermodynamic and transport properties of the QGP. The higher the collision energy, the more particles

will be produced in an event and the less viscous the QGP flow will be. This will make it easier to study the high temperature QGP thermal and dynamical properties. This study also requires a quantitatively precise theoretical framework to describe the dynamical evolution of the fireball produced by the collision. The equations that control the fireball expansion cannot be solved analytically. As a result, scientists must solve these equations numerically. The main goal of this thesis is to find precise numerical solutions for those equations.

Quarks, Gluons and Chromodynamics

Quark Model

In 1964, both Murray Gell-Mann and George Zweig independently introduced the

Quark Model. It was a quite successful theoretical work as it successfully predicted other hadrons that were not yet observed. Later, these predicted hadrons were observed experimentally. The Quark Model achievement reminds us of the Mendeleev Periodic

Table and how it was able to predict unknown elements before they were experimentally realized. Gell-Mann made a baryonic scheme consisting of three dimensions. On every dimension axis, he chose four integer numbers. Every integer number on this dimension axis would represent the number of the identical quarks in the baryon (quarks from the same type). The values began with zero and ended with three as Gell-Mann proposed that there were just three types of quarks: strange, up and down, and there were three quarks in every baryon. In this scheme, every crossing point represents a specific baryon,

See Figure 1. This scheme later has expanded to include all of the six known quarks and later the quark model was absorbed into the standard model (2).

Figure 1. The basic quark model scheme: it has been built on the fact that there are just three quarks, up, down and strange (4).

According to the standard model, the heavier quarks tend to decay to produce the two lighter quarks, up and down, which are more stable. That explains why the up and down quarks are the most common in nature, and hence, why the neutrons and protons are the most common hadrons. This scheme was enough to explain the structure and properties for the most important two hadrons in nature, protons and neutrons.

According to this model, the proton consists of two up quarks and one down quark. The additive quantum numbers of the constituent quarks of proton give it its observed properties. From Table 1 you can easily conclude the following:

Proton charge = + 2/3 + 2/3 -1/3 = +1

Neutron charge = + 2/3 - 1/3 - 1/3 = 0

See Figure 2 for a picture of the internal structure of proton and neutron in the original quark model (3).

Table 1

The three quarks and their quantum numbers according to the basic Quark Model that was constructed by Gell-Mann (3).

Quark Name Spin Baryon Number Lepton Number Charge

Up ½ 1/3 0 2/3

Down ½ 1/3 0 - 1/3

Strange ½ 1/3 0 - 1/3

Figure 2. The internal structure of proton and neutron in the original quark model (5).

The Experimental Evidence for Quarks

The first real evidence came from analyzing the data that was collected by

Stanford Linear Accelerator Center (SLAC) in 1960s. It was an enhanced version of the

Rutherford scattering experiment that was performed by Giger and Marsden in 1909.

Trying to study the positive charge distribution inside the atom, Marsden bombarded gold foils with α particles emitted from natural radioactive isotope. In the Marsden experiment, the collision between α particles and the gold nuclei was elastic because the

α particle energy was low.

At SLAC, scientists used the electron as a probe because the electron does not experience strong interaction. The electron also has a lighter mass and that makes it easier to accelerate; and it also can be produced in large amount comparatively easily.

The experiment goal was to study the charge distribution of the proton. The results of the experiment were as follows: when the electron energy was low, the electrons deflected slightly and the protons recoiled. When the electron energy exceeded specific threshold, the electron suddenly began to deflect at large angles and proton started to split into a shower of particles instead recoiling by itself.

There are many ways to explain these results. Feynman has explained this according to QED as follows: When a low speed electron passes through a proton, it will take comparatively more time to pass. Hence, while the electron passing near the proton, the quarks will move covering more distance. Therefore, the electron will see a perturbing electromagnetic field over the whole proton volume; and as a result, this electromagnetic field will be of low intensity. Since the photon emission is dependent on

its subsequent absorption (according to QED), the electromagnetic field will trigger the electron to emit a photon with low intensity. Then the photon will be absorbed by the whole proton and that will cause the proton to deflect from its path.

When a high-speed electron passes through the proton, it will take comparatively less time to pass. Hence, while the electron passes near the proton, the quarks will cover less distance. Therefore, the electron will see a perturbing electromagnetic field over the quark volume; and as a result, this electromagnetic field will be of high intensity. Since the photon emission is dependent on its subsequent absorption (according to QED), the electromagnetic field will trigger the electron to emit a photon with high intensity. Then the photon will be absorbed by individual quark and that will cause the quark to deflect from its path but the quark will not be able to propagate for long distances. As the quark tries to go farther from the other two quarks, according to QCD, it will form other mesons

(see “color confinement’’). Therefore; a shower of particles will appear in the hadron detectors and the number of these hadrons will depend on the speed of the probe electron.

Figure 3 shows a carton picture of electron-proton scattering at different energies.

For their work, the leaders of SLAC project, Friedman, Taylor and Kendal were awarded the 1990 Nobel Prize in Physics. Other experiments were performed similar to the SLAC experiment but with even higher collision energies. With these high-energy experiments, it was easy to collect the evidence for the existence of gluons and sea quarks. Therefore, more and more evidence of the existence of quarks and gluons and

QCD theory was confirmed.

Figure 3. The interaction between incident electron and proton’s components (a) The upper part of the figure shows interaction between the low-energy electron and the whole proton. (b) The intermediate part of the figure shows interaction between a high-energy electron and the quark. (c) The down part of the figure showing interaction between a higher-energy electron and the gluon and “sea quarks” (6).

Quantum Chromodynamics (QCD), Color Charge and Strong Interaction

Why color charge? The original quark model discussed above faced two theoretical problems:

1. According to Pauli Exclusion Principle, no two fermionic particles or more can

share the same quantum state in some system. Hence, as there were two quarks or

more of the same type in the same ground quantum state in some baryon, it had

become a theoretical contradiction. For example, ∆++ consists of three up quarks

(uuu) which exist in the same ground quantum state. Therefore, there are at least

two quarks in the same quantum ground state with the same spin direction.

2. The only experimentally observed hadrons were 푞 푞 푞 (Baryons) 푞̅ 푞̅ 푞̅

(Antibaryons), and 푞̅푞 (mesons). Nothing outside these three groups had been

detected. For example, why a hadron with a charge +4/3 (uu) had never been

detected (푞푞) (3)? Or, in general, why (qq) states are never experimentally

observed.

In 1964, shortly after Gell-Mann introduced the Quark Model, Oscar W. Greenberg introduced the concept of color charge to solve these two puzzles. Since then, chromodynamics began to develop gradually. Until now, there is some computational and conceptual puzzles facing scientists when dealing with QCD.

According to Greenberg’s proposal, there are three types of color charge. The quarks interact with each other through color exchange. The color carrier is the gluon.

The role of the gluon in QCD is analogous to the role of photon in QED but the case in

QCD is more complicated due to the color charges. Let me explain how this color

exchange between quarks happens through the gluons. Every quark should carry one of the three fundamental colors; red, green and blue (These are not real colors. They are named in this way because the algebra of the three color charges and the three anti-color charges can be successfully explained according to color theory roles) (7).

First notice that; although the color charge of quark or gluon changes frequently, it is still conserved quantity, as the whole net color charge for the hadron will stay fixed.

According to QCD, the Heisenberg uncertainty relation allows the gluons to be created but for tiny period according to the following equation:

ℎ ∆퐸 ∆푡 ≤ . 2휋

See Figure 4 so that you can easily understand the following:

1. By basic colors, I mean the following colors: blue, green and red.

2. When the three basic colors are combined with each other (or two color, anti-

color quarks) the hadron will be colorless (or white). Quarks or anti-quarks do

not have to be colorless. The whole observed hadron is colorless and it remains

colorless regardless of the color exchange happening inside it between the quarks,

as this exchange obeys the conservation laws.

3. The following symbols refer to the following colors:

Red: 푅

Blue: 퐵

Green: 퐺

Anti-Red (Cyan): 푅̅

Anti-Blue (Yellow): 퐵̅

Anti-Green (Magenta): 퐺̅

4. Gluons can be created with the following color possibilities:

푅푅̅ 퐵푅̅ 퐺푅̅

푅퐵̅ 퐵퐵̅ 퐺퐵̅

푅퐺̅ 퐵퐺̅ 퐺퐺̅

Notice that the analogy that we introduced between color quantum number and

color concept is not perfect. According to QCD mathematical predictions, the

gluons 푅푅̅, 퐵퐵̅, 퐺퐺̅ forms the following three states: 푅푅̅ − 퐵퐵̅ , 푅푅̅ + 퐵퐵̅ −

2퐺퐺̅ and 푅푅̅ + 퐵퐵̅ + 퐺퐺̅. The only state that is not affected by 푅, 퐵, 퐺 color

space rotation is 푅푅̅ + 퐵퐵̅ + 퐺퐺̅ and hence, it forms an observed meson (휋

meson). Therefore, we are left with only the following 8 gluons:

푅푅̅ − 퐵퐵̅ 퐵푅̅ 퐺푅̅

푅퐵̅ 푅푅̅ + 퐵퐵̅ − 2퐺퐺̅ 퐺퐵̅

푅퐺̅ 퐵퐺̅

5. According to color charge conservation law, when one of the eight color/anti-

color combinations are created, the basic color created will be the original basic

color of the quark before the gluon created, and the anti-color created would be

the anti-color of the new basic color of the quark (after the gluon is created). For

example, when a 퐵퐺̅ gluon leave the blue quark, the quark will turn from blue to

green; thus the net creation is; green + anti-green. Therefore, we can see that the

three

basic colors should be created with each other equally; as the positive and negative electric charge should be created equally. And accordingly, when one of the following gluons; 푅푅̅, 퐵퐵̅ and 퐺퐺̅, leaves the quark, the quark color will not change.

Figure 4. The three basic colors combined with each other (8).

Color Confinement

The strong force behaves like a simple ideal spring force. When pulling a spring beyond the elastic limit, it will break into two separate springs. In similar way, when pulling two quarks away from each other, they will produce another pair of quark- antiquark. The energy of the new pair creation will come from the potential energy of the original pair. This is because the force between two quarks increases with distance as some exerted force tries to separate them from each other. The net color produced will be white because color is conserved quantity. Therefore, quarks are never seen separated.

This phenomenon is called as “Color Confinement”. This is why the QGP produces hadrons and baryons when it is freezes out. In QGP, color deconfinement happens just within the QGP bulk itself and still no quark can escape alone from the QGP bulk (2)(7).

See Figure 6.

Jets

When two quarks moving with high energy collide with each other, they will bounce and move away from each other. During their motion out of each other, they will produce a strong flux tube as they both have color charge. As they move away, they will lose kinetic energy and the tube will gain potential energy. Eventually the tube will break up into a shower of mesons. This shower of mesons will not move exactly in the same line, there is some divergence, but the quarks will retain much of the initial quarks motion. These showers can be recorded by appropriate equipment producing quite striking computer displays as shown in Figure 5. The mathematical calculations that related to jets have shown very good agreement with experiment and it was considered

Figure 5. Detecting jets in laboratory (CERN) (9).

Figure 6. The basic jets and color confinement phenomenon illustration (10).

further evidence for the existence of quarks (2)(7). Figure 6 shows how the flux tube breaks up.

Jet Quenching

As we have seen above, when two quarks collide with each other at ultra-high energy, they will produce a jet. When two heavy nuclei collide with each other at ultra- high energy, some of these nuclei’s quarks will collide at ultra-high energy. These colliding pairs of quarks (or gluons) may form jets. When two nuclei collide at ultra-high energy, the QGP will form after about 0.3 fm/c. During that time, the jet particles will not travel more than about 0.3 fm. The QGP bulk dimensions are mostly larger than this distance in about order of 1. That means the jets quarks will spend much time moving inside the QGP. Since they are moving inside the QGP, they will interact with the QGP quarks and, hence, will lose some energy. This loss of energy is called “Jet Quenching”.

Jet Quenching in the QGP is analogous to X-ray imaging in medicine. Physicists are working hard at CERN and BNL to study jet quenching. They hope that jet quenching can tell us much about what happens inside the QGP.

Quark Gluon Plasma (QGP)

What is QGP?

QGP is an ensemble of many deconfined colored quarks with gluons acting as a force mediator. The interacting force between the quarks (and on-shell gluons) is strong enough to keep the quarks in a nearly perfect liquid phase.

As we mentioned above, the goal of the most recent experiments was to describe the quantitative thermal and dynamic properties of the QGP fireball expansion as accurately as possible. As physicists are not yet able to determine non-equilibrium QGP properties using lattice QCD simulations, they use analytical QCD to describe its properties assuming that the force between the particles can be treated perturbatively.

According to this simulation, The QGP should expands like a weakly-interacting gas.

Contrary to what had been expected, the experiments have shown that the QGP expanded like a nearly perfect liquid, not a gas as theorists expected (1). Reacting to these experimental results, theorists around the world worked hard to find simulations and computational models by assuming that the QGP expands according to the viscous relativistic fluid model, and this is the main goal of this thesis.

A Glance on Experimental Detection Techniques and Calculations

As we mentioned above, we do not detect the quark and gluon soup directly due to color confinement. What we detect are the hadrons, photons, and other kinds of particles that form during the expansion of the fireball, which reduces the temperature, and energy density of the QGP. The temperature will reduce due to the pressure resulting from the force between the particles, and the energy density will decrease as the

temperature decreases and particle density decreases due to the expansion. This will allow the hadrons to form. Every particle will form in the stage where the circumstances

(Temperature and energy density) will be appropriate, see Figure 7. Viscous hydrodynamics predicts the angular distribution and final kinetic energy for every type of particle formed. Since the actual data for the angular distribution for the resulting particles agrees with theoretical predictions for a nearly perfect liquid QGP at both

CERN and BNL, physicists now think that QGP exists and behaves as a nearly perfect liquid. That is how we know about the QGP and most of the important microscale physical facts.

One of the challenges facing QGP researchers is that the partial differential equations, that govern the viscus relativistic fluid dynamics, cannot be solved analytically. In practice, theorists must solve these equations numerically. The agreement between the experimental data and theory was very good but particle physics theorists still work hard to find solutions that are more accurate and efficient. We should take into account that the numerical solution is not the only source of inaccuracy. Any experiment can deliver a limited degree of accuracy. Thus, what we need to do is to find a numerical solution that has more accuracy than the experiment itself, after that; adding more accuracy to the numerical solution will be not necessary, as its effect will be negligible comparing to the experimental error. The main goal of this thesis is to shed light on this point and to try to find accurate numerical solutions for the relativistic fluid equations and make accurate thermal and dynamical flow estimation of the fireball expansion.

Figure 7. QGP evaluation stages: this scheme has been build according to Cartesian Milne coordinates. It represents a timed evaluation of the QGP expansion from the beginning of the collision until freeze out (11).

CHAPTER II

ANISOTROPIC RELATIVISTIC HYDRODYNAMICS

In this chapter, we will study the relativistic hydrodynamics needed in QGP theorization. We will begin with the derivation of the simple case of the ideal hydrodynamic equations that are derived from kinetic theory. After that we will derive the dynamical equations of massless relativistic hydrodynamics system based on taking the zeroth, first and second moment of Boltzmann equations. Then, we will use these dynamical equations in our final numerical calculations.

Elliptic Flow?

For an off-center collision, the angular hadronic distribution of detected hadrons was not as expected for quarks that behave like a gas. When the nuclei move in v ~ c, they will contract in length along their direction of motion according to relativity.

Therefore, at high energies, an observer in the lab frame will see the nuclei as they were approximately disk-shaped. When these two disk-shaped nuclei collide at high energy, most of their quarks will not collide. They will resolve each other and there is less probability that they will collide. The two nuclei will pass each other leaving approximately the cross section of two overlapping cylinders composed of gluonic matter bulk behind them (maybe with some high energy quark jets), see Figure 8. This gluonic matter will not take more than 1/3 fm/c to form non-equilibrium QGP. Let us begin a rough evaluation of the QGP bulk dimensions before this QGP bulk begins to expand significantly. The area of this overlapping cross-section depends on the

Figure 8. Collision cross-section: (a) The two disk-shaped nuclei colliding. (b) The QGP bulk is formed within about 0.3 fm/c after collision (12).

overlapping area; this is mostly comparable to the colliding nuclei surface area (for example, A = π (7.3)2 fm2 for a gold nucleus). Regarding the thickness of this QGP bulk, we should first estimate the approximate disk-shaped nucleus thickness. According to relativity, we can calculate the contraction in length as following:

Lₒ 1 푚 ɤ ≡ = = L 푣2 푚 √1−( ) 0 푐2

Lₒ: rest length. mₒ: rest mass.

L: relativistic length. m: relativistic mass.

In recent, still-ongoing, experiments at BNL the collision energy has reached 100

GeV/nucleon for each beam. The rest mass of the proton is about 0.94 GeV so:

100GeV ɤ = ≈ 100 0.94GeV and at LHC in CERN the energy has reached 2.76/2 TeV for each beam so:

2760GeV ɤ = ≈ 1500. 2∗0.94GeV

Therefore, we can calculate L as following:

7푓푚 퐿 ~ ~ 0.07 푓푚 (at RHIC) 100

7푓푚 or: 퐿 ~ ~ 0.005 푓푚 (at LHC). 1500

Therefore, we can easily see that the contraction in length is enough to neglect L and consider the nuclei as a thin-disk-shaped. During the QGP formation time t ~ 0 - 0.3 fm/c, the nuclei will travel a distance d that can be calculated as following:

h ~ 2*c* 0.3 (fm/c) ~ 0.6 fm where we will consider, as an approximation, that the QGP will begin to expand significantly after 0.3 fm/c (in fact it will begin to expand before this time). The bulk will be, approximately elliptic, see Figure 9. In Figure 9, d, c and h are the estimated

QGP bulk dimensions. The elliptic area depends on the overlapping area from the two colliding nucleus. In an off-center collision, we easily notice that d ˃ c, therefore, the flow for a viscous fluid will be anisotropic with more flow in the x direction than in y direction, see Figure 10.

Regarding the flow in the z direction, it depends on the overlapping area. When the overlapping area is small enough so that h will be significantly bigger than c and d, then:

The flowx ˃ flowy ˃ flowz (the least probable case) when the overlapping area is small enough so that: h ≈ c, then:

flowz ≈ flowx ˃ flowy. when the overlapping area is big enough so that h will be significantly smaller than c and d, then:

flowz ˃ flowx ˃ flowy (the most probable case as h ≈ 0.6 fm as we calculated

above and r ≈ 7.5 fm for gold nucleus)

This is known as elliptic flow. Scientists have observed this dramatic asymmetry in the expansion of the particles. There were more particles emerging along the reaction plane than perpendicular to it (see Figure 10.), as expected for a nearly perfect liquid, not as supposed to an ideal gas.

Figure 9. The approximate QGP bulk shape and dimensions.

Figure 10. The asymmetry of the emerging quarks (10).

Derivation of Ideal Hydrodynamics Equations from Kinetic Theory (11)

In this section, I will introduce the derivation of the equations that govern relativistic ideal hydrodynamics. Here we will consider that all chemical potentials are zero to simplify the calculations. The starting point for the derivation is the Boltzmann equation:

µ 푝 휕µ 푓(푥, 푝) = −퐶 [푓(푥, 푝)] (1)

µ µ where: x = (t, x), p = (Ep, p), 휕µ = (휕푡, − ∇), C is the collisional kernel which includes particle scattering effects. We can obtain the bulk equations of motion by taking the moments of the Boltzmann equation and multiplying the left and right sides of the equation by an integral operator:

̂푣1 푣2……푣푛 푛 푣푖 퐼 ( . ) = ∫ 푑푃 ∏𝑖=1 푝 ( . ) (2)

푑3푝 where: 푑푃 = is the Lorentz-invariant phase space measure. To obtain the zeroth 퐸 (2휋)3 moment of Boltzmann equation, we will apply this operator at zeroth order:

µ ∫ 푑푃 푝 휕µ 푓 = − ∫ 푑푃 퐶[푓]

µ µ Or: 휕µ[ ∫ 푑푃 푝 푓 ] = 푗 = − ∫ 푑푃 퐶[푓] (3) where: [ ∫ 푑푃 푝µ 푓 ] is the particle four-current 푗µ = (푛, 풋), ∫ 푑푃 퐶[푓] is the zeroth moment of the collision kernel.

For simplicity, we will introduce a notation for the nth-moment of the collisional

푛 µ푖 kernel, 퐶𝑖 ≡ ∫ 푑푃 ∏𝑖=1 푝 퐶[푓] , hence we can write the zeroth moment equation as follow:

µ 휕µ 푗 = −퐶0 (4)

For number-conserving theories, C0 is zero. Hence, we can write equation (4) as follows:

µ 휕µ 푗 = 0 (5)

Equation (5) is simply the relativistic continuity equation. Now, we will introduce a tensor basis for the particle current. The two four-vectors available to us are:

uµ: which is the four-velocity of the local rest frame (fluid four velocity).

Vµ: which is transverse particle current.

µ Here: u Vµ = 0 (by definition)

µ µ Note that u is normalized such that u uµ =1. From this we find that the four velocity has only three independent components. In general, we can write the current in the following form:

jµ = nuµ + Vµ (6)

Here:

n: is the net charge density.

µ µ v V = ∆ vj : is the diffusion current.

∆µv = gµv + uµuv: is the transfer projector which projects out the component of a

u µ v four-vector that are orthogonal to u and obeys ∆ vu = 0.

For an ideal fluid, we can assume that particle flow and energy flow are the same.

Therefore, we can take Vµ → 0 and we can write Eq. (6) as:

jµ = nuµ (7)

Substituting equation (7) in equation (5) we obtain:

µ 휕µ (n푢 ) = 0 (8) and we can write:

µ µ 푢 휕µ푛 + 푛휕µ푢 = 0. (9)

I will introduce the following quantities:

µ 퐷 ≡ 푢 휕µ (10)

µ 휃 ≡ 휕µ푢 (11) where D is the comoving derivative and 휃 is the expansion scalar. Inserting (10) and (11) in (9), we can write Eq. (9) as follow:

D푛 + 푛휃 = 0 (12)

Next, let us consider the first moment of Boltzmann Equation:

푣 µ ∫ 푑푃 푝 푝 휕µ 푓 = − 퐶1

µ 푣 휕µ[∫ 푑푃 푝 푝 푓] = − 퐶1

µ푣 휕µ푇 = − 퐶1 (13)

where 푇µ푣 = [∫ 푑푃 푝µ푝푣 푓] is the energy momentum tensor. In quantum field theory one has an energy conserving-collisional kernel which implies that C1 = 0.

Therefore, Eq. (13) can be written as:

µ푣 휕µ푇 = 0. (14)

This equation is a statement of energy-momentum conservation. Now, we will establish a basis for 푇µ푣. For Ideal hydrodynamics we will assume that the system is locally isotropic. Therefore, there are only two structures that represent a rank-two tensor. These two structures are 푔µ푣 and 푢µ푢푣. Hence, we can write:

푇µ푣 = 퐴 푢µ푢푣 + 퐵 푔µ푣 . (15)

Where A and B are Lorentz-invariant coefficients. Let us now try to find A and B. We

µ푣 휇 can achieve this by evaluating 푇 in the local rest frame where one has 푢퐿푅퐹 = (1,0,0,0)

휇푣 휇푣 and 푇퐿푅퐹 = 푑푖푎푔(휀, 푃, 푃, 푃). The 00-component evaluation of 푇퐿푅퐹 give us: 휀 = 퐴 + 퐵.

휇푣 The three space-like ii-components of 푇퐿푅퐹 give us 푃 = −퐵. As a result, we can write A and B as follows: 퐴 = 휀 + 푃, 퐵 = −푃. Substituting this into Equation (15), we obtain:

푇µ푣 = (휀 + 푃)푢µ푢푣 − 푃 푔µ푣 = 휀푢µ푢푣 − 푃 ∆µ푣 (16)

By substituting Equation (16) in Equation (14) we get:

µ 푣 µ푣 휕µ[(휀 + 푃)푢 푢 − 푃 푔 ] = 0

푣 µ 푣 µ µ 푣 푣 푢 푢 휕µ(휀 + 푃) + (휀 + 푃)(푢 휕µ푢 + 푢 휕µ푢 ) − 휕 푃 = 0

푢푣퐷(휀 + 푃) + (휀 + 푃)(푢푣휃 + 퐷푢푣) − 휕푣푃 = 0 (17)

푣 푣 Now, let us project these equations by 푢푣 , taking into account that 푢 푢푣 = 1, 푢푣퐷푢 =

푣 푣 퐷(푢 푢푣) = 0 and 퐷 ≡ 푢푣휕 :

푣 푣 푣 푣 푢푣푢 퐷(휀 + 푃) + (휀 + 푃)(푢푣푢 휃 + 푢푣퐷푢 ) − 푢푣휕 푃 = 0

퐷(휀 + 푃) + (휀 + 푃)휃 − 퐷푃 = 0 (18)

By simplifying we obtain:

퐷휀 + (휀 + 푃)휃 = 0 (19)

훼 By projecting equation (17) with the transverse projector; ∆푣 , we obtain:

훼 푣 훼 푣 (휀 + 푃) ∆푣 퐷푢 − ∆푣 휕 푃 = 0 (20)

훼 훼 푣 Let us now introduce the spatial gradient: ∆ ≡ ∆푣 휕 . We also can use

훼 푣 훼 푣 푣 훼 푣 훼 훼 퐷(∆푣 푢 ) = 0 to obtain ∆푣 퐷푢 = − 푢 퐷∆푣 = 푢 퐷(푢 푢푣) = 퐷푢 and replace the spacelike index 훼 with 푖. Then, we can write Eq. (20) as:

(휀 + 푃) 퐷푢𝑖 − ∆𝑖푃 = 0 (21)

One can write a relation between 휀 and 푃 to reduce the number of unknowns.

µ This relation is called the equation of state. This can be achieved as follows: 푇µ = 휀 −

3푃 ≡ 퐼, 퐼 is the trace-anomaly and its value is zero for an ideal gas. Therefore, for an ideal fluid one has 휀 = 3푃. As a result, for three dimensions and with Eq. (21) and Eq.

(19), we will end up with four equations and four unknowns; 휀 and 푢𝑖 with 푖 ∈ {1,2,3}.

Bjorken Hydrodynamics (11)

Let us consider a simple case of hydrodynamics where the system is boost- invariant and transversally homogeneous (no transverse dynamics). This special case was originally presented by Bjorken. The main goal of studying this case is to get a feeling for how the temperature evolves after a heavy ion collision. In this case, it is convenient to use comoving ‘’Milne’’ coordinates

푡 = 𝜏 cosh 𝜍

푧 = 𝜏 sinh 𝜍

(22)

For a boost-invariant system, the four-velocity in Minkowski Space is:

푡 푧 푢µ = ( , 0, 0, ) = (cosh𝜍, 0,0, sinh𝜍) (23) 𝜏 𝜏

Where here µ 휖 {푡, 푥, 푦, 푧}. Transforming this to Milne coordinate one finds

푢̃µ = (1,0,0,0),

(24)

Now, with µ 휖 {𝜏, 푥, 푦, 𝜍} we have:

퐷 = 휕𝜏,

1 휃 = (25) 𝜏

Applying Eq. (25) to the zeroth moment of Boltzmann given in equation (12), we obtain:

푛 휕 푛 = − (26) 𝜏 𝜏 which has a solution of the form:

𝜏 푛(𝜏) = 푛 0 (27) 0 𝜏

If we now apply Eq. (25) to the first moment of Boltzmann equation given in Eq.

(20) and Eq. (21) one finds:

휀 +푃 휕 휀 + = 0 (28) 𝜏 𝜏

As we mentioned before, the equation of state for the ideal fluid is 휀 = 3푃, and by applying it in Eq. (25) we obtain:

4 휀 휕 휀 = − (29) 𝜏 3 𝜏

By solving Eq. (29) we get:

4 𝜏 ⁄3 휀 = 휀 ( 0) (30) 𝑖푑푒푎푙 푓푙푢𝑖푑 0 𝜏

If the system has an equation of state that corresponds to a constant speed of

2 2 sound, i.e. 푑푃⁄푑휀 = 푐푠 or one can write 푃 = 푐푠 휀. Therefore, Eq. (30) can be written as:

2 𝜏 1+푐푠 휀 = 휀 ( 0) (31) 𝑖푑푒푎푙 푓푙푢𝑖푑 0 𝜏

2 notice that Eq. (31) reduce to ideal case when 푐푠 = 1/3.

General 3+1d Anisotropic Hydrodynamics Equations for A Massless

Hydrodynamics System

Now, we will derive the general 3+1d relativistic hydrodynamics equations for a massless system based on taking moments of Boltzmann equation. We will continue take moments until second moment as this enough to find the number of equations that is equal to the number of variables needed to be found. To accomplish this, we use the general basis vectors to contract the equations obtained from Boltzmann equation and also expand the bulk variables of the system over. The general basis vectors are obtained by performing a set of Lorentz transformation from local rest frame (LRF) to lab frame

(LF), releasing any assumptions related to the symmetry of the system. Bulk variables

are obtained by taking moments of anisotropic distribution function defined in the following section.

Convention and Notation

In this thesis, the metric is taken to be “mostly minus” such that in Minkowski space with 푥휇 = (푡, 푥, 푦, 푧), the line element is:

2 휇 푣 2 2 2 2 푑푠 = 푔휇푣푑푥 푑푥 = 푑푡 − 푑푥 − 푑푦 − 푑푧 . (32)

Cartesian Milne coordinates are defined by 푥휇 = (𝜏, 푥, 푦, 𝜍), where 𝜏 = √푡2 − 푧2 is the longitudinal proper time, 𝜍 = 푡푎푛ℎ−1(푧/푡) is the longitudinal space time rapidity.

Please see Figure 7 to take a picture about Cartesian Milne coordinates. In some place

휇 we donate the scalar product between two four-vectors with a dot, i.e. 푎휇푏 ≡ 푎. 푏. In all

휇 휇 cases, the flow velocity 푢 is normalized as: 푢휇푢 = 1.

Basis Vectors

The general basis vectors in LRF can be parametrized as following:

휇 푢퐿푅퐹 ≡ (1, 0, 0, 0),

휇 푋퐿푅퐹 ≡ (0, 1, 0, 0),

휇 푌퐿푅퐹 ≡ (0, 0, 1, 0),

휇 푍퐿푅퐹 ≡ (0, 0, 0, 1). (33)

One can define the general basis vectors in LF by performing a parametrization comprising a set of Lorentz transformations on LRF basis vectors. The transformation required can be constructed using a longitudinal boost 휗 along the beam axis, followed

by a rotation 휑 around the beam axis. And, finally, a transverse boost by 휃⊥ along the x- axis (14) (15). This parametrization gives:

휇 푢 ≡ (cosh 휃⊥ cosh 휗 , sinh 휃⊥ cos 휑, sinh 휃⊥ sin 휑 , cosh 휃⊥ sinh 휗 ),

휇 푋 ≡ (sinh 휃⊥ cosh 휗 , cosh 휃⊥ cos 휑 , cosh 휃⊥ sin 휑 , sinh 휃⊥ sinh 휗 ),

휇 푌 ≡ (0, − sin 휑 , cos 휑 , 0),

휇 푍 ≡ (sinh 휗 , 0, 0, cosh 휗). (34)

The three fields 휑, 휗 and 휃⊥ are functions of Cartesian Milne coordinates

(𝜏, 푥, 푦, 𝜍). Introducing a new set of variables based on temporal and transverse of flow velocity:

푢0 = cosh 휃⊥,

푢푥 = 푢⊥cos 휑 ,

푢푦 = 푢⊥sin 휑 . (35)

2 2 2 Where 푢⊥ ≡ √푢푥 + 푢푦 = √푢0 − 1 = sinh 휃⊥ , one obtains basis vectors as a function of three independent parameters 푢푥, 푢푦 and 휗:

휇 푢 ≡ (푢0 cosh 휗 , 푢푥, 푢푦, 푢0sinh 휗),

휇 푢0푢푥 푢0푢푦 푋 ≡ (푢⊥cosh 휗 , , , 푢⊥ sinh 휗 ), 푢⊥ 푢⊥

휇 푢푦 푢푥 푌 ≡ (0, − , , 0), 푢⊥ 푢⊥

휇 푍 ≡ (sinh 휗 , 0, 0, cosh 휗). (36)

Distribution Function

Generally, anisotropic hydrodynamics is defined through the introduction of an anisotropy tensor of the form (15) (16):

Ξµ푣 = uµu푣 + ξµ푣 − ∆µ푣Φ (37) where:

푢µ: is the four velocity.

휉µ푣: is the symmetric and traceless part.

훷: is associated with bulk viscous degree of freedom.

µ µ푣 µ The quantities 푢 , 휉 and 훷 are functions of spacetime and obey 푢 푢µ = 1,

휇 휇 µ푣 휇 휉 휇 = 0, ∆ 휇 = 3, and 푢휇ξ = 0; therefore one has ξ 휇 = 1 − 3훷. At leading order in the anisotropic hydrodynamics expansion one takes the one-particle distribution function of the form:

1 푓(푥, 푝) = 푓 ( 푝 Ξµ푣푝 ) (38) 𝑖푠표 휆 √ 휇 푣

Where λ has dimensions of energy and can be identified with temperature only in the

µ푣 (14) isotropic equilibrium limit (ξ = 0 and Φ = 0) . It is appropriate to take 푓𝑖푠표 to be a

푥 −1 thermal equilibrium distribution function of the form 푓𝑖푠표(푥) = 푓푒푞(푥) = (푒 + 푎) , where 푎 = ± 1 gives Fermi-Dirac or Bose-Einstein statistics, respectively, and a = 0 gives Boltzmann statistics. From here on, we assume that the distribution is of

Boltzmann form, i.e. a = 0.

Dynamical Variables

Since the most important viscous corrections are the diagonal components of the energy-momentum tensor, to good approximations one can assume that ξµ푣 =

𝑖 푑푖푎푔(0, 흃) with 흃 ≡ (휉풙, 휉풚, 휉풛) and 휉𝑖 = 0. In this case, expanding the argument of the square root appearing on the right-hand side of Eq. (38) in the LRF gives:

2 1 푝푖 2 푓(푥, 푝) = 푓𝑖푠표 ( √∑𝑖 2 + 푚 ) 휆 훼푖

For high energies, we can neglect the rest mass and write;

2 1 푝푖 푓(푥, 푝) = 푓𝑖푠표 ( √∑𝑖 2 ) (39) 휆 훼푖 where 푖 ∈ {푥, 푦, 푧} and the anisotropy parameters 훼𝑖 are:

−0.5 훼𝑖 ≡ (1 + 휉𝑖 + Φ) . (40)

Note that, for brevity, one can collect the three anisotropy parameters into vector

훼 ≡ (훼푥, 훼푦, 훼푧) . In the isotropic equilibrium limit, where 휉𝑖 = Φ = 0 and 훼𝑖 = 1, one

µ푣 2 2 has 푝휇Ξ 푝푣 = (푝. 푢) = 퐸 and λ → T and, Therefore:

퐸 푓(푥, 푝) = 푓 ( ) . (41) 푒푞 푇(푥)

Using Eq. (40) and the tracelessness of ξµ푣, one can write Φ in terms of

1 anisotropy parameters, Φ = ∑ 훼−2 − 1. This indicates that out of the four anisotropy 3 𝑖 𝑖 and bulk parameters there are only three independent ones. In practice, we use three

variables 훼𝑖 as the dynamical anisotropy parameters.

Bulk Variables

In this section, bulk variables, i.e. number density, energy density, and the pressure components are calculated by taking the projections of particle four-current and energy- momentum tensor. From here on, by taking the 푛푡ℎ-moment of the quantity Ƒ we mean calculating:

∫ 푑푃 푝휇1 … . . 푝휇푛 Ƒ (42) where dP = d3p / (2π)3/ E.

Particle current four-vector. The particle current four-vector 퐽휇 ≡ (푛, 퐉) is defined is first moment of distribution function:

퐽휇 ≡ ∫ 푑푃 푝휇푓(푥, 푝). (43)

One can expand 퐽휇 using the basis vectors as

휇 휇 휇 휇 휇 퐽 = 푛푢 + 퐽푥푋 + 퐽푦푌 + 퐽푧푍 . (44)

Using Eq. (39) and Eq. (43) one finds that the only surviving components of 퐽휇 are:

퐽휇 = (푛, 0) = 푛 푢휇 (45) where 푛 = 훼푛푒푞(휆) and 훼 ≡ 훼푥훼푦훼푧.

Energy-momentum tensor. The energy-momentum tensor 푇휇푣 is defined as the second moment of the distribution function

푇휇푣 = ∫ 푑푃 푝휇 푝푣 푓(푥, 푝). (46)

Expanding it over the basis vector one obtains:

휇푣 휇 푣 휇 푣 휇 푣 휇 푣 푇 = 휀 푢 푢 + 푃푥 푋 푋 + 푃푦 푌 푌 + 푃푧 푍 푍 (47)

Using Eq. (39), Eq. (46) and Eq. (47) and taking projections of 푇휇푣 one can obtain the energy density and the components of pressure:

휀 = 휀푒푞(휆) 퐻3(훂),

푃푥 = 푃푒푞(휆) 퐻3푥(훂),

푃푦 = 푃푒푞(휆) 퐻3푦(훂),

푃푧 = 푃푒푞(휆) 퐻3퐿(훂). (48)

where 휀푒푞 and 푃푒푞 are the isotropic energy density and pressure and H-functions take into account the deviation of anisotropic hydrodynamics bulk variables from isotropy.

The various 퐻-functions appearing above are defined in Appendix.

Equilibrium bulk variables. As mentioned before, our system follows the

Boltzmann statistics. The bulk variables for a Boltzmann gas in the equilibrium limit can be obtained by taking the limit 훼𝑖 → 1 and 휆 → 푇, as follows:

3 푛푒푞(푇) = 8휋푁̃ 푇 , (49)

3 푆푒푞(푇) = 32휋푁̃ 푇 , (50)

4 휀푒푞(푇) = 24휋푁̃ 푇 , (51)

4 푃푒푞(푇) = 8휋푁̃ 푇 . (52)

Equations of state. In order to propose a consistent way to implement equation of state in anisotropic hydrodynamics, we extract the isotropic part of energy density and pressures and relate them through EoS. The a realistic EoS used here is obtained from the KRAKOW parametrization of lattice QCD calculations (17).

Dynamical Equations

In this thesis, the necessary hydrodynamical equations are derived by taking the moments of Boltzmann equations. In what follows, we assume relaxation-time approximation (RTA) for collisional kernel, however, the general methods presented here can applied to any collisional kernel. In principle, RTA is known to be relevant for systems which evolve close to equilibrium. However, in practice, it works fairly will in far-from-equilibrium cases, too. The Boltzmann equation in flat space time is:

휇 푝 휕휇푓 = −퐶[푓] (53) with collisional kernel C[ f ] which in RTA is given by

휇 푝 푢휇 퐶[푓] = (푓 − 푓푒푞) (54) 𝜏푒푞 where 푓푒푞 denotes the equilibrium limit of one-particle distribution function Eq. (41) and

𝜏푒푞 is the relaxation time which depends on spacetime but is taken to be momentum- independent. To obtain a realistic model for 𝜏푒푞, one can assume that the shear viscosity

휂 to anisotropy density, ≡ 휂̅, is held fixed during the evaluation and then using the 푆푒푞 thermodynamic relation; 휀푒푞 + 푃푒푞 = 푇 푆푒푞 one has

5휂̅ 휀푒푞(푇) 𝜏푒푞(푇) = ( 1 + ) (55) 4푇 푃푒푞(푇)

Zeroth moment. Taking the zeroth moment of the Boltzmann equation and using Eq. (45) one finds an equation for the particle four-current in the RTA as:

1 퐷푢푛 + 푛휃푢 = (푛푒푞 − 푛). (56) 𝜏푒푞

First moment. The conservation of energy and momentum is enforced by

푢푣 휕휇 푇 = 0, which requires that the first moment of collisional kernel, appearing on the right-hand side of Boltzmann equation to vanish. This implies the Landau-matching condition as

휇푣 휇푣 푢휇푇 = 푢휇푇푒푞 (57) which using Eq. (47) simplifies to 휀 = 휀푒푞. Finally, using the factorization relevant for imposing realistic EoS one has:

퐻3(훼) 휀𝑖푠표(휆) = 휀𝑖푠표(푇) (58)

turning to the left hand side of Boltzmann equation, using Eq. (47) and taking U-, X-, Y-, and Z-projections, one obtains four independent equations:

휇 휇 휇 퐷푢휀 + 휀 휃푢 + 푃푥푢휇퐷푥 푋 + 푃푦푢휇퐷푦 푌 + 푃푧푢휇퐷푧 푍 = 0 ,

휇 휇 휇 퐷푥푃푥 + 푃푥휃푥 − 휀푋휇퐷푢푢 − 푃푦푋휇퐷푦 푌 + 푃푧푋휇퐷푧 푍 = 0,

휇 휇 휇 퐷푦푃푦 + 푃푦휃푦 − 휀푌휇퐷푢푢 − 푃푥푌휇퐷푥 푋 + 푃푧푌휇퐷푧 푍 = 0,

휇 휇 휇 퐷푧푃푧 + 푃푧휃푧 − 휀푍휇퐷푢푢 − 푃푥푍휇퐷푥 푋 + 푃푦푍휇퐷푦 푌 = 0 (59) where the necessary derivatives are defined in App. A.

Second moment. The second moment of Boltzmann equation in the RTA is:

휇푣휆 1 휇푣휆 휇푣휆 휕휇 퐼 = (푢휇 퐼푒푞 − 푢휇 퐼 ) (60) 𝜏푒푞 where 퐼휇푣휆 can be obtained from third moment of distribution function as:

퐼휇푣휆 ≡ ∫ 푑푃 푝휇 푝푣 푝휆 푓(푥, 푝). (61)

For a distribution function of the form specified in Eq. (39), surviving terms in

퐼휇푣휆 are:

퐼 = 퐼푢 [푢 ⊗ 푢 ⊗ 푢] + 퐼푥[ 푢 ⊗ 푋 ⊗ 푋 + 푋 ⊗ 푢 ⊗ 푋 + 푋 ⊗ 푋 ⊗ 푢]

+ 퐼푦[ 푢 ⊗ 푌 ⊗ 푌 + 푌 ⊗ 푢 ⊗ 푌 + 푌 ⊗ 푌 ⊗ 푢]

+ 퐼푧[ 푢 ⊗ 푍 ⊗ 푍 + 푍 ⊗ 푢 ⊗ 푍 + 푍 ⊗ 푍 ⊗ 푢]. (62)

Evaluating the necessary integrals using the distribution function Eq. (39), one finds

2 퐼푢 = (∑𝑖 훼𝑖 )훼 퐼푒푞(휆) (63)

2 퐼𝑖 = 훼훼𝑖 퐼푒푞(휆) (64) where:

5 퐼푒푞(휆) = 32휋푁̃휆 . (65)

Note that 퐼푢 = ∑𝑖 퐼𝑖. Expanding Eq. (60) and taking its uu-, XX-, YY- and ZZ- projections gives:

휇 1 퐷푢퐼푥 + 퐼푥 (휃푢 + 2푢휇퐷푥푋 ) = (퐼푒푞 − 퐼푥) 𝜏푒푞

휇 1 퐷푢퐼푦 + 퐼푦 (휃푢 + 2푢휇퐷푦푌 ) = (퐼푒푞 − 퐼푦) 𝜏푒푞

휇 1 퐷푢퐼푧 + 퐼푧 (휃푢 + 2푢휇퐷푧푍 ) = (퐼푒푞 − 퐼푧). (66) 𝜏푒푞

By now, we have had eight independent equations for eight variables. The variables are: 훼푥, 훼푦, 훼푧, 푢푥, 푢푦, 휗 (푤푖푙푙 푎푝푝푒푎푟 푖푛 퐴푝푝. 퐴), 휆 푎푛푑 푇. The equations are:

(58) (one equation) (59) (four Equations) (66) (three equations). These are called the dynamical equations. These equations govern the QGP expansion. They are the basis equations which we will use in our numerical calculations.

CHAPTER III

NUMERICAL CALCULATIONS

Weighted LAX

What is Weighted LAX?

In this chapter, we will show and discuss the results of numerical calculations.

We will introduce the Weighted LAX method and show the benefit of using this technique and this is the main goal of this chapter. This technique was first presented by

Michael Strickland (18). As you will see, this technique can help us get rid of the unwanted numerical fluctuations when using fluctuating event-by-event initial conditions. To demonstrate this method, we will use an anisotropic hydrodynamics code that was written using a standard centered difference algorithm supplemented by the weighted LAX smoothing technique (18). We will use a (1+1) d code (19) which models central collisions of heavy nuclei.

Mathematical Formula

In practice, the weighted LAX technique is performed by combining the current value of a given dynamical variable with a local spatial average over neighboring sites and using this as a stand in for the current value of the variable, this can be represented mathematically in following steps:

̃ 1. 푓 (푥, 푦, 푧, 푡표) → 푓 (푥, 푦, 푧, 푡1) [centered difference temporal update]

̃ ̃ ̃ 2. 푓퐿퐴푋 (푥, 푦, 푧, 푡1) = [푓 (푥 + ∆푥, 푦, 푧, 푡1) + 푓 (푥 − ∆푥, 푦, 푧, 푡1) + 푓 (푥, 푦 +

̃ ̃ ̃ ∆푦, 푧, 푡1) + 푓 (푥, 푦 − ∆푦, 푧, 푡1) + 푓 (푥, 푦, 푧 + ∆푧, 푡1) + 푓 (푥, 푦, 푧 − ∆푧, 푡1)]/6

̃ 3. 푓 (푥, 푦, 푧, 푡1) = 푓퐿퐴푋(푥, 푦, 푧, 푡1) 휆 + (1 − 휆)푓 (푥, 푦, 푧, 푡1) ,

where 푡1 = 푡0 + ∆푡. In step 1, we perform the usual centered-differences update appropriate for the dynamical partial differential equations obtained from anisotropic hydrodynamics. In step 2, we use the naively updated value 푓̃ to compute a local average over neighboring sites (LAX average). In step 3, we compute a weighted linear combination of the LAX-averaged value and the naively updated value 푓̃ to obtain the updated value of the variable. This procedure if then repeated in order to evolve the functions (in our case, set of functions) forward in time. In the update rules specified above, λ is a weight factor by which we can control the amount of smoothing that is required to get rid of the spurious numerical oscillations. These oscillations result from the weak coupling between the even and odd sites in the lattice inherent in centered- difference schemes. Note that in the code and figures we refer to λ as LAXFRAC.

The Best Value of λ

The main function of λ is to smear out the numerical fluctuations which can result if there are large gradients in the functions being evolved. This smoothing is particularly important if one uses fluctuating initial conditions that can have very large gradients.

The amount of smoothing depends on the value of λ which ranges between 0-1. The value of zero represents no smoothing or, in another words, turning off the LAX weighting. The value of one represents the total smoothing and it corresponds the original LAX technique (21). Now, you may wonder if this smoothing may affect the physical results or not. In fact, it can affect the physical results and the magnitude of the effect depends on the value of λ. If λ = 0 there are, by definition, no physical effects from the smoothing and, if λ = 1 one will find that there are large physical effects. If the

value is smaller than it should be, the smoothing will not be enough and the application of weighted LAX will not give its maximum benefit. If the value is larger than it should be, the smoothing may be enough to change the physical situation in significant magnitude and this is surely unwanted. Therefore, we did need to find as accurately as possible the optimal value of λ. By best value, we mean the value that shows the best smoothing with the lowest permitted change in the actual physical results. The application of code to case of fluctuating initial conditions has shown that the best value of λ is = 0.02. Thus, we have run the code with the following three values, 0 (no smoothing), 0.02 (the best value of weighted LAX) and 1 (standard LAX). This will enable us to compare between LAX, weighted LAX and running the code without either of these two techniques

Numerical Results

Code constant Parameters and Initial Conditions.

Here are statements about all relevant parameters used in the simulations. We consider Pb-Pb at 2.76 TeV/nucleon, 4 η /s = 3 and T0 = 0.6 GeV. In all plots shown, the evaluation begins at  = 0.25 fm/c after the nuclear impact. The first set of figures uses smooth Glauber initial conditions with the energy density set by the wounded nucleon density. The second set of figures uses Monte Carlo Glauber initial conditions which take into account event-by-event fluctuations which occur in individual collisions. In all cases, we considered central collisions and used a lattice with NUM=200 points in the radial direction. The lattice spacing was taken to 0.15 fm and the temporal step size was taken to be 0.01 fm/c. For temporal updates we used fourth-order Runge-Kutta updates.

Longitudinal Anisotropic Momentum Parameter Graph

Now let us consider the numerical implications of Figures 11 and 12. As you can see in the graphs, there is not a significant difference between the curves corresponding to λ (LAXFRAC) values of 0 and 0.02. This is reasonable because the graphs were generated using smooth initial conditions for which weighted LAX smoothing is not strictly necessary. These results demonstrate that the weighted LAX technique does not change the actual physical results in significant way when it used with the λ value of

0.02. On the other hand, when we use the value of 1 (total smoothing corresponding to the naïve LAX method) you can see how this value will significantly affect the actual physical results.

Effective Temperature Graphs

Let us now discuss the effective temperature graphs that are shown in Figures 13 and 14. These two graphs belong to the same collision but at two different times. As you can notice, there is not much difference in results between the curves that were generated with a λ value of 0.02 and 0. But there is also significant difference in the actual physical results for a λ value of 1. Notice that, for λ value of 1, there is more significant difference in Figure 13 (t = 1.25 fm/c) than in Figure 14 (t = 10.25 fm/c). This implies that the actual physical effect of λ value depends on the measurements range. The wider the measurements range, the more significant physical difference. This should be taken

Figure 11. The longitudinal anisotropy parameter (훂z) as a function of position at  = 1.25 fm/c. The result was generated with smooth Glauber initial

conditions.

Figure 12. The longitudinal anisotropy parameter (훂z) as a function of position at  = 10.25 fm/c. The result was generated with smooth Glauber initial conditions.

Figure 13. The effective temperature as a function of position at  = 1.25 fm/c. The result was generated with smooth Glauber initial conditions.

Figure 14. The effective temperature as a function of position at  = 10.25 fm/c. The result was generated with smooth Glauber initial conditions.

into account when performing such numerical calculations using weighted LAX technique.

Transverse Anisotropic Momentum Parameter Graph

Let us now introduce the transverse anisotropic momentum graph, see Figure 15.

This graph simulates one event using Monte Carlo Glauber fluctuating initial conditions.

The black curve is for λ = 0. In this curve, you can see the sharp numerical fluctuations explicitly at |x| ~ 10.5 fm. These fluctuations do have a physical origin but it is not completely physical because numerical instabilities amplify the actual physical oscillations causing a ``run away’’ and eventually the code will crash. For example, the λ

= 0 does not cause such fluctuations in the previous four graphs (Figures 10-14) because the initial conditions for these graphs were smooth. Since our goal is to make our simulations as close to reality as possible, we do not want to totally get rid of the fluctuations. Instead, we want to restore the initial actual physical oscillations. As you can see from Figure 15, λ = 1 causes total smoothing and this really changes the actual physical state. As you see above in the previous four graphs in this chapter, λ = 0.02 is an appropriate value in the sense that it does not significantly change the actual physical state. Hence, it is the best value to represent the actual physical state. This can be proven by performing the same plot for the curve of λ = 1, but taking ∆x much smaller than what we have taken to perform this graph. However, taking the spatial step size too small makes the algorithm inefficient and slow to execute. The benefit of weighted LAX compared to the standard LAX method is that one approaches the continuum limit (∆x →

0) faster and, as a result, one can obtain more accurate results on a coarser lattice with

Figure 15. The transverse anisotropy parameter (훂x) as a function of position at  = 0.65 fm/c. The result was generated using fluctuating Monte-Carlo Glauber initial

conditions.

less computational cost. Michael Strickland has proven this mathematically when he originally presented the weighted LAX technique.

The Graph of the Ratio of the Transverse Parameter to the Longitudinal Parameter

Now let us introduce and discuss the graph that shows a comparison between the transverse parameter and the longitudinal parameter (훂x /훂z). Figures 16 and 17 show these graphs. Let us discuss the physical implication of Figure 16. In these figures, x is the distance from the center of the collision (for a central collision). If (훂x /훂z) is less than one, then the momentum flow through x or y axis will be less than the flow in the z direction in the local rest frame of the matter (momentum anisotropy). If (훂x /훂z) is one, then the system is will be isotropic in momentum space. If (훂x /훂z) is more than one, then the momentum flow through x or y axis will be more than through z direction

(anisotropic flow). From Figures 16 & 17, one can easily see that this ratio’s value is more than one. Physically, this implies that the QGP flow is momentum-space anisotropic as a nearly perfect liquid, not a gas. The deviation from isotropy represents non-equilibrium dynamics of the QGP. If the system were described by ideal hydrodynamics one would find that (훂x /훂z) would be equal to unity at all times. As the magnitude of the shear viscosity to entropy density ratio increases (η / S), one finds that the degree of momentum-space anisotropy increases and, in the non-interacting limit, the system would be highly momentum-space anisotropic.

Figure 16. The transverse anisotropy parameter (훂z) and the longitudinal anisotropy parameter (훂x) as a function of position at  = 1.25 fm/c. The result was generated with smooth Glauber initial conditions.

Figure 17. The transverse anisotropy parameter (훂z) and the longitudinal anisotropy parameter (훂x) as a function of position at  = 10.25 fm/c. The result was generated with smooth Glauber initial conditions.

regions of the QGP, however, at late times and in the center of the QGP one sees that the system approaches isotropy in the local rest frame.

Pion Differential Spectra Graph

Figures 18 and 19 show two pion differential spectra graphs, one for smooth

Glauber initial conditions which represent an average over many events, the other represents one sampled event (collision). The pion differential spectra graph shows the pion transverse momentum distribution. It represents the number of pions produced as a function of transverse momentum. As you can see in Figure 18, there is not a significant difference between the results obtained with λ = 0 and λ = 0.02. Again, however, you can see a significant difference associated with the λ value of 1.

One can notice that there is not a curve generated according to the λ value of 0 in

Figure 19 (single event). When we choose the value of zero for the single event graph, the code crashes shortly after t=0.65 fm/c because of the numerical instability. As result, it is not even possible to compute the pion spectrum in this case. This represents an example where the weighted LAX technique is absolutely necessary. We are confident, however, based on the smooth (averaged) initial conditions, that for λ = 0.02 the physical effects are minimal.

Figure 18. The pion differential spectra versus transverse momentum. The result was generated with smooth Glauber initial conditions.

Figure 19. The pion differential spectra versus transverse momentum. The result was generated using fluctuating Monte-Carlo Glauber initial conditions.

Freeze-out Hypersurface Graph

Let us now introduce the freeze-out hypersurface graph that shown in Figure 22.

For this graph to be easily understood, there are some concepts that should be introduced first. The QGP temperature depends on the distance from the center of the QGP bulk.

The temperature increases as one goes closer to the center until it reaches about 600 MeV at the center, and decreases as one goes closer to the surface until it reaches about 150

MeV (freeze-out temperature) at the surface, see Figure 20. Therefore, the closer to the surface the QGP particles are, the faster they cool and freeze-out. Figure 21 shows, for every point inside the QGP bulk, how temperature changes with time until it reaches the freeze-out value (about 150 MeV). It shows the time that every point inside the QGP

bulk needs to reach freeze-out. This time is called the freeze-out time (FO). One can easily conclude from the Figures 20 & 21 that for smooth initial conditions the freeze-out time at a particular point depends on the distance of this point from the center of the QGP bulk.

Let us now discuss the freeze-out hypersurface graph shown in Figure 22. The area inside the curves represents the QGP phase. The outside area is the hadronic gas phase. From this graph, you can notice that the QGP bulk contracts a bit in the beginning of the collision because of cooling due to longitudinal expansion. Although the QGP phase is much denser than the nuclear phase, the QGP bulk does not contract more than the amount shown in the figure because the high temperature and pressure pushes it out quickly. Therefore, it expands in the beginning of the collision and this graph shows us this amount of expansion. As it expands it begins to freeze-out. The closer the QGP

particles are to the surface, the faster they freeze-out (as I have explained above when I discussed the Figures 20 and 21). The freeze-out means that the QGP quarks and gluons begin to form hadrons because no colored particle can leave the QGP bulk as an independent entity (according to the color confinement phenomenon). The more particles leave the QGP bulk, the less its volume becomes. As a result, a bit after the expansion, the QGP bulk begins to contract in volume and becomes smaller and smaller until it vanishes. If one looks at the graph, one may notice that the QGP bulk contracts faster and faster with time. This is intuitive because the ratio of the surface area to the volume of a sphere goes higher as the volume of this sphere goes smaller. That means the rate of the relative contraction for a small volume bulk is higher than such a rate of a bigger bulk.

Notice also that the value of λ = 1 has significantly changed the actual physical state of the system (e.g. QGP size and temporal duration). The weighted LAX method did not significantly change the actual physical system state.

Figure 20. The effective temperature profile inside QGP bulk: x = 0 represents the center of the QGP bulk. The result was generated with smooth Glauber initial conditions.

600 T0 600 MeV 500 T0 500 MeV T0 400 MeV

400 T0 300 MeV

TFO 150 MeV

MeV 300 T 200

100

0 5 10 15 20

fm c

Figure 21. The temperature variation with time for a sample of four points inside the QGP bulk: each vertical line determines the freeze-out time for the corresponding point inside the QGP which depends on its initial temperature. The result was generated with smooth Glauber initial conditions.

Figure 22. The freeze-out hypersurface graph: the freeze-out time versus the radial position inside the QGP bulk, or the freeze-out radius (rFO). The result was generated with smooth Glauber initial conditions.

CHAPTER IV

APPLICATIONS, DISCUSSIONS AND LOOK AT THE FUTURE

Theoretical Discussions and Conclusions

Let us now discuss what can the experimental results of QGP experiments tell us about QCD. Before that, I think it is necessary to first introduce some essential concepts related to the formulation of physical theories. In general, there are two roads for a theorist to get a theory that describes some specific phenomenon. The first road and the stronger one is to put a theory before looking at any experimental results. In this way, the only thing that the theorist takes from the practical world is the hypothesis of the theory.

Then, with free brain and full confidence, he goes through the right logic and math until he gets the final results and conclusions and make his full expectations accordingly.

After that, if the experimentalists test the theory, and if they found that all its expectations were right; that means this theory is very close to reality (and may represent reality). The wider measurements scope and the more phenomenon this theory will successfully explain, the more probability it will represent reality (or the closer to reality). An example for such a road is how Einstein obtained relativity theory. This is why it has been a strong theory in a sense that it is still working fine until these days. The last prediction of this theory was the gravitational waves that has recently been emphasized experimentally by LIGO. With the success of LIGO, many relativity predictions are now verified. Such theories always have long age and work successfully in a wide range of measurements. In fact, there is never correct or wrong theory in our modern understanding. Instead, there is a theory close to reality and there is another closer. No

theory can be guaranteed 100% to be right. There are only two ways to make sure that some theory represents the exact reality. The first one is to make the experimental measurements with 0% error and find that the theory was success. The other way is to test the theory and find that it can explain an infinite number of phenomena. Both of these two options are impossible. This mean that the more number of phenomena the theory can explain and the less experimental error the experiments would be performed within, the closer to reality the theory would be (or there is more probability it will represent reality).

The second road is to follow the experiment and try to find some explanation that can explain the phenomenon. This is the weaker way because, mostly, there are more than one theory that can explain some limited phenomenon in some limited measurements ranges. Maybe, the most important example of that is what Newton did to get his mechanical theory. Such theories, always do not successfully explain what happens exactly in the microscopic world. However, they are very useful to explain the macroscopic world phenomenon and they are very useful to perform applications. An example of this kind of theory is Hooke’s law. This law was built according to the experimental data that Hooke collected. According to this law, the tension of spring is linearly proportional to the change happens in its length, that is; F=K∆L. This law works correctly when ∆L is small enough for some specific experimental error. For this law to be true, the less experimental error there is, the less magnitude ∆L should be.

Why Viscous Hydrodynamics

To explain the experimental results of heavy ion collisions experiments, unfortunately, the second road was taken. That was because in the beginning of this field of study, some physicists took the first road assuming that the QGP behaves as weakly- interacting gas, but this road has failed. The reason that this road has been considered as failed is that its prediction did not agree with the experimental results. As we have seen in Chapter III, the QGP flow is anisotropic, like a nearly perfect liquid, not highly anisotropic as supposed for a weakly-interacting gas, as these theorists expected. It seems as we still do not fully understand the nature of QCD interactions in the right way.

Regarding this theoretical failure, contemporary physicists are divided into two teams.

The first team think that the QCD is not a right theory and it should be edited or replaced.

They think that the QGP expansion, with its terminus variable’s measurements, is a phenomenon that outside the QCD limits. They see the experimental success of the QCD to explain many of the nuclear phenomenon as the success of Hooke’s law to explain the spring behavior in specified experimental error and maximum value of ∆L. The other team think that there is no evidence that the QCD has failed and it may still be true. They think that the source of failure may be the computational difficulties, not necessarily the

QCD theory. However contemporary particle physicists are working hard these days trying to figure out the reason that the QCD simulation predictions did not agree with experimental results.

To deal with the case, and as physicists do always in such cases, they tried to look at the situation from macroscopic scale depending on the experimental data gained from

LHC and RHIC. From the data of the anisotropic flow of QGP, they found that these data can successfully explained by treating QGP as a viscous hydrodynamic fluid with η /

S ≈ 0.1, regardless what the nature of interactions between QGP particles is.

Experimentally, this theory is very successful.

But to understand why taking η / S ≈ 0.1 was successful it is still the mission of theorists. The value that was recently predicted by leading-logarithm perturbative QCD simulations is η / S = 0.5-0.8. As you see, this is still a failure. Some theorists are working to develop the computational techniques using QCD to perform reliable simulations. Others try to find if there is problem with QCD itself. QCD interactions are very complicated and what exactly happens inside QGP bulk is still thoughts, rather than facts. I estimate it will take theorists no less than a decade to solve this problem.

The Experimental Results of Viscous Hydrodynamics

There is very good agreement between the predictions of numerical simulations that use viscous hydrodynamics and the experimental results gained from experiments in

ALICE, CERN, CMS and BNL. That is shown in the Figures 23-26. These graphs show comparisons between the experimental results and the simulated curves.

Experimentalists are working to perform the experiments with as wide range of measurements as possible and trying also to lessen the experimental error as much as possible. This may be gained by increasing the collision energy and enhancing the detection techniques as much as possible. They also try to enhance the computational operations by using more efficient computers and enhancing the computational algorithms (i.e. weighted LAX development).

Figure 23. Multiplicity versus centrality (%) (20).

Figure 24. Average transverse momentum versus centrality (%) (20).

Figure 25. Flow hadronic coefficients versus centrality (%) (20).

Figure 26. Transverse momentum spectra graph (experimental and simulation data): transverse momentum spectra (upper panels) of pions, kaons, and protons and harmonic flow coefficients (lower panels) as a function of the transverse momentum. Two centrality classes are considered: 0 – 5 % (left panels) and 30- 40% (right panels). The bands denote the statistical uncertainty of the calculation. The full and open symbols correspond to measurements by the ALICE and CMS collaboration respectively, with bars denoting the experimental uncertainty (20).

QGP Existence in the Compact Stars and the Big Bang

I want to discuss the link between the QGP and some kinds of compact stars.

Some scientists think that, if the mass of compact star is bigger than the maximum mass of a neutron star, the star will suffer the quark matter degeneration. Until now, theorists do not have much information about quark stars, as they should not emit many particles

(according to dominating degeneration theories). However, with the last great news that came from LIGO, now we have the ability to collect gravitational waves data to enhance our understanding of compact star structure. Gravitational waves can provide us information about the equation of state that governs the compact star and that will help us know what the best theory that describes the compact star is. There are also great benefits that may be gained from using gravitational waves. It may inspire us to unify the four nature forces, as we will be forced to describe a system where the four forces work effectively, with similar magnitude. We cannot produce this system in the laboratory.

All that we able to produce is a spot of QGP matter where the mass is too small to take into account the gravitational effect. Compact star formation is a ‘’free’’ experiment done in nature and all we need to do is to collect the data and reap the benefit of this free experiment.

The laboratory QGP production does not fully simulate the Big Bang for the following two reasons:

1. All that we are able to produce is a spot of QGP where the mass is very small to

take into account its gravitational effect.

2. The pre-quark epochs and the post-hadronic epochs will not be simulated. We

will begin in the quark epoch and end up with the hadrons and other kinds of

particles passing through the particle detectors.

However, a successful QGP model will help us find the best model for the QGP star. It will help us test models as a special case where the gravitational force is negligible. Regarding early universe simulation, QGP production has two aspects that do not exist in compact star formation:

1. The QGP production will help us understand the system’s transition from the

quark epoch to the hadronic epoch. We cannot understand that by collecting data

from the compact stars where there are not many particles emitted.

2. The QGP expands like the early universe, does not collapse.

3. The QGP has tremendous temperature and energy density, far more than the

quark matter in compact stars do.

Therefore, by combining the data from the both gravitational waves and the data of QGP production at BNL and CERN, we may obtain a much better understanding of compact stars and the Big Bang.

Testing the theories, that suppose the quark matter existence in the compact stars, may be achieved by the following three steps:

1. Finding a theoretical evaluation for equation of state for the compact stars under

study. This evaluation will be achieved according to the theory that we want to

test. We will estimate the mass distribution of compact stars under study

according to theory testified.

2. Finding evaluation of the gravitational waves propagation for this compact stars

according to its structure (mass distribution) that was predicted in step -1-. We can

evaluate the waves’ properties using general relativity with programming

simulation help if needed. In other words, we will predict the wave properties that

should reach the detector from some stars according to the theory under test.

3. Comparing the theoretically predicted data (Step-2- ) with data collected by LIGO

and other gravitational detectors.

Researchers may extend the study trying to correct the theories that have predicted data near that collected by LIGO. We may benefit from the QGP model to determine this correction. We may also extend the study by trying to simulate predictions of the data that should have come from the early universe. This will help us test the predictions coming from the QGP formation in the Lab.

Scientists hope that understanding the QGP thermalization and dynamical flow will help us build the best universe age stages scheme. That’s because they think that the early universe in some epoch was just a QGP hot bulk that expanded and froze out to form this universe. Figure 27 represents a modern universe history scheme as estimated by theoretical physicists and astronomers.

Figure 27. The nowadays estimated universe history (17).

APPENDIX

DERIVATIVES

Appendix

Derivatives Cross Terms

In this appendix, first we introduce the notations used in derivation of the general moment-based hydrodynamics equations and then, by taking the appropriate limits, we simplify them for the transversally-homogeneous 0+1d case. Using the definitions:

1 퐷 ≡ cosh(휗 − 𝜍) 휕 + sish(휗 − 𝜍) 휕 𝜏 𝜏 𝜍

∇⊥. u⊥ = 휕푥u푥 + 휕푦u푦 ,

u⊥ . ∇⊥= u푥 휕푥 +u푦 휕푦,

u⊥ × ∇⊥= u푥 휕푦 − u푦 휕푥.

And from four-vectors defined in Eq. (36) one has:

휇 퐷푢 ≡ 푢 휕휇 = 푢0퐷 + u⊥ . ∇⊥ ,

휇 푢0 퐷푥 ≡ 푋 휕휇 = u⊥퐷 + (u⊥ . ∇⊥), u⊥

휇 1 퐷푦 ≡ 푌 휕휇 = (u⊥ × ∇⊥), u⊥

휇 퐷푧 ≡ 푍 휕휇 = 퐷̃.

The divergences are defined as:

휇 휃푢 ≡ 휕휇푢 = 퐷푢0 + 푢0퐷̃휗 + ∇⊥. u⊥,

휇 ̃ 푢0 1 휃푥 ≡ 휕휇푋 = 퐷u⊥ + u⊥퐷휗 + (∇⊥. u⊥) − 2 (u⊥. ∇⊥)u⊥, u⊥ u⊥푢⊥ 78

휇 1 휃푦 ≡ 휕휇푌 = − (u⊥. ∇⊥)휑, u⊥

휇 휃푧 ≡ 휕휇푍 = 퐷휗.

푢 Where 휑 = 푡푎푛−1 푦 . Some terms of the following form appear in the second moment 푢푥 equations:

휇 1 u휇퐷훼푋 = 퐷훼 u⊥, u0

휇 u휇퐷훼푌 = u⊥퐷훼휑 ,

휇 u휇퐷훼푍 = u0퐷훼휗 ,

휇 X휇퐷훼푌 = u0퐷훼휑 ,

휇 X휇퐷훼푍 = u⊥퐷훼휗 ,

휇 Y휇퐷훼푍 = 0.

Where 훼 ∈ {푢, 푥, 푦, 푧}.

Now, we will provide definitions of the special functions appearing in the body of the text. We are starting by introducing:

2 푦 −1 푦 −1 2 퐻2(퐲) ≡ [푡푎푛ℎ √ + 푦 √푦 − 1 ], √푦2−1 푦2

푦 푦2−1 퐻 (퐲) ≡ [(2푦2 − 1)푡푎푛ℎ−1√ − 푦 √푦2 − 1 ], 2푇 (푦2−1)3/2 푦2

푦3 푦2−1 퐻 (퐲) ≡ [푦√푦2 − 1 − 푡푎푛ℎ−1√ ]. 2퐿 3 푦2 (푦2−1)2

In terms of this special functions, the 퐻3 functions used in the definition of bulk variables are:

훼푥훼푦 2휋 2 훼푧 퐻3(훂) = ∫ 푑∅ 훼⊥ 퐻2 ( ), 4휋 0 훼⊥

3 3훼푥훼푦 2휋 2 훼푧 퐻3푥(훂) = ∫ 푑∅ 푐표푠 ∅ 퐻2푇 ( ), 4휋 0 훼⊥

3 3훼푥 훼푦 2휋 2 훼푧 퐻3푦(훂) = ∫ 푑∅ 푠푖푛 ∅ 퐻2푇 ( ), 4휋 0 훼⊥

3훼푥훼푦 2휋 2 훼푧 퐻3퐿(훂) = ∫ 푑∅ 훼⊥ 퐻2퐿 ( ). 4휋 0 훼⊥

REFERENCES

REFERENCES 1- Heinz, Ulrich, “Quark gluon soup the perfectly liquid phase of QCD’’,

International Journal of Modern Physics A, 30, 1-6 (2015).

2- Allday, Jonathan, “Quarks, Leptons and Big Bang’’, Institute of Physics

Publishing, UK, (2002).

3- Francis, Halzen & Martin, Alan, “Quarks and Leptons, an introductory course in

modern particle physics’’, John Wiley & Sons, New York, (1984)

4- https://en.wikipedia.org/wiki/Baryon.

5- http://nebulosacabulosa.blogspot.com/2015/10/o-mundo-da-fisica-quantica- parte-1.html

6- https://inspirehep.net/record/1128300/plots?ln=en.

7- Griffiths, David, ‘’Introduction to Elementary Particles’’, Wiley-VCH Verlag

GmbH & Co. KGaA, Germany, (2008).

8- http://delenalexy.blogspot.com/2015/05/primary-colors-of-light-subject-of.html.

9- http://cds.cern.ch/record/39449.

10- http://web.science.uu.nl/itf/Teaching/2005/Corstanje.pdf.

11- Strickland, Michael, “Anisotropic Hydrodynamics: Three lectures”, arXiv:

1410.5786v2 [nucl-th] (2014).

12- http://nextbigfuture.com/2009/10/attoseconds-zeptoseconds-and.html.

13- http://www.nap.edu/read/13438/chapter/4#85.

14- W. Florkowski and R. Ryblewski, Phys.Rev. C85, 044902 (2012), arXiv:1111.5997

[nucl-th].

15- M. Martinez, R. Ryblewski, and M. Strickland, Phys.Rev. C85, 064913 (2012),

arXiv:1204.1473 [nucl-th].

16- M. Nopoush, R. Ryblewski and M. Strickland, Phys.Rev. C90, 014908 (2014),

arXiv:1405.1355 [hep-ph].

17- M. Chojnacki and W. Florkowski, Acta Phys. Polon. B38, 3249 (2007), arXiv: nucl-

th/0702030 [nucl-th].

18- M. Martinez, R. Ryblewski, and M. Strickland, Boost-Invariant (2+1)-

dimensional Anisotropic Hydrodynamics, Phys. Rev. C 85, 064913, PDF,

Abstract, (2012).

19- http://personal.kent.edu/~mstrick6/code/.

20- C. Gale, S. Jeon, and B. Schenke, M. Gyulassy and L. McLerran, “The

importance of the bulk viscosity of QCD in ultra relativistic heavy-ion collisions”,

arXiv:1502.01675v2 [nucl-th] 12 Sep 2015.

21- P. Lax & K. Friedrichs, “Communications on Pure and Applied

Mathematics 7” (1954).