Patryk

Julia Dorota

Asia Szporki

Izerki Karol

Keywords:

Beam Energy Scan dependance of elliptic and triangular flow of identified hadrons in the STAR experiment and the EPOS model

Studying the properties of the collective expansion of matter created in heavy-ion collisions provides a unique tool to understand better the nonperturbative aspect of QCD. Input from both the theoretical and the experimental side is needed. Hydrodynamic calculations predict anisotropies in particle production, as a consequence of asymmetries in the initial state of the system's evolution. Measurements of the systematics (energy-, system-dependence) of these anisotropies allow one not only to validate theoretical ideas, but also to determine the unknown elements, like plasma properties (EoS), themalization processes. The broadening of our knowledge in this topics is the main goal of the thesis. The experimental methods were used to provide the insight into the investigation of the anisotropies in expansion of the particles and antiparticles, while with the theoretical approach was used for the EoS studies.

Two-particle flow correlations are a powerful tool in studies of the expansion of the matter. They express the relation between the directions of the produced particles. The nonuniform distribution of final-state momenta is a consequence of the initial-stage anisotropy in the collision's geometry. It is transformed via pressure gradients into an asymmetric flow, providing finally a strong correlation of the direction of the particles' motion with the collision's participant plane. The shape of the expansion is described in terms of a Fourier decomposition. Each harmonic corresponds to the subsequent shape of expanding matter: second harmonic (v 2 ) is the elliptic flow, and third one (v 3 ) the triangular one. v 2 originates primarily from the eccentricity (ϵ 2 ) of the overlap collision. On the other hand, all odd coefficients are not sensitive to the ϵ, and the fluctuations and viscous effects dominate the measured signal. It is also less contaminated with the particle correlations not related with flow (non-flow). Possible, detailed investigation about all these ingredients makes the performed studies of both harmonics a comprehensive source of information about various aspects of the matter's initial-state properties.

Multiple interesting observations based on protons' and antiprotons' production are used as probes of studies of QCD phase diagram. One of the intriguing measurements showed the increasing elliptic flow difference between p and p with a decrease of the collision energy. There are several theoretical explanations of the observation. However, none of them has been proven. The STAR data were used to verify several scenarios and improve the experimental reference for theoretical studies. The similar trend of the differences between p and p is also observed for the triangular flow, which is mostly driven by fluctuations. Moreover, the n q (KE T ) scaling gave a piece of information about protons and antiprotons' possible different origins.

On the other hand, the created matter is expected to behave like a viscous fluid, and its expansion can be based on the hydrodynamical equations. Transitions between different states of matter and thermodynamical relations characterizing it are included in the Equation of State (EoS). It cannot be derived directly from first principles or measured by an experiment. Studies of the EoS are crucial in understanding the dynamics of the matter in different regions of the phase diagram of nuclear matter. It is not apparent that experimentally measurable observables could give a piece of information about EoS or its changes. Introducing the possibility of different EoS choices while developing the EPOS model gave a unique possibility for such studies. The family of EoS proposed by the BEST Collaboration was implemented. The Critical Point position and the strength of criticality variation were changed, and the effects on the final observables such as particle yield, transverse momentum spectra, flow, or moments of the net-proton distributions were studied.

The thesis consists of Chapter 1, where we present the concise review of the significant high energy physics aspects necessary for this thesis. The review of anisotropies in the expansion of the matter, flow harmonics and two particles cumulants is in Chapter 2. In Chapter 3, the STAR experiment at RHIC is described. The data selection and evaluation of systematic uncertainties are discussed in Chapter 4. The Chapter 5 is devoted to the experimental results. In subsequent Chapter 6, we included the EPOS event generator description and implemented developments (introduction of new EoS). The studies of the implemented EoS into the EPOS and comparison with the STAR results are presented in Chapter 7. Finally, we summarise the performed studies and provide the outlook for the impact on the future research in Chapter 7.2.5.

There are three Appendices consists of short summary of the my engagement into the iTPC software development Appendix 9, activities conducted in the frame of RIVET project Appendix 10, and list of useful thermodynamic equations in Appendix 8. This thesis includes the unique combination of both experimental and theoretical approaches to exploring the properties and dynamics of the matter.

STRESZCZENIE

Badania zależności od energii zderzenia przepływów eliptycznych oraz trójkątnych zidentyfikowanych hadronów w eksperymencie STAR oraz modelu EPOS Badanie właściwości kolektywnej ekspansji materii powstałej w zderzeniach ciężkich jonów jest unikalnym narzędziem wykorzystywanym do lepszego zrozumienia nieperturbacyjnego aspektu QCD. Potrzebny jest wkład zarówno ze strony teoretycznej, jak i eksperymentalnej. Obliczenia hydrodynamiczne przewidują anizotropie w produkcji cząstek, będącą konsekwencją asymetrii w stanie początkowym. Pomiary systematyki (zależności od energii, systemu) tych anizotropii pozwalają nie tylko na potwierdzenie modeli teoretycznych, ale także na określenie nieznanych elementów, takich jak właściwości plazmy (EoS) czy procesy termalizacji.

Dwucząstkowe korelacje przepływu są potężnym narzędziem w badaniach nad ekspansją materii. Wyrażają zależność między kierunkami produkowanych cząstek. Nierównomierny rozkład pędów stanu końcowego jest konsekwencją anizotropii etapu początkowego w geometrii zderzenia. Przekształcana jest poprzez gradienty ciśnienia w przepływ asymetryczny, zapewniając ostatecznie silną korelację kierunku ruchu cząstek z płaszczyzną uczestników zderzenia. Kształt ekspansji jest opisany za pomocą rozkładu Fouriera, gdzie v 2 odpowiada przepływowi eliptycznemu, a v 3 trójkątnemu. Druga harmoniczna pochodzi głównie z mimośrodu (ϵ 2 ) obszaru nakładania się jąder podczas kolizji. Z drugiej strony, wszystkie nieparzyste współczynniki nie są wrażliwe na zmiany ϵ, zaś ich sygnał jest zdominowany przez fluktuacje i efekty lepkości. Znieszktałcenie sygnału przez wkład tzw. "non-flow" jest mniejsze. Szczegółowe badanie wszystkich tych składników sprawia, że przeprowadzone studium obu harmonicznych jest fantastycznym źródłem informacji o różnych aspektach właściwości materii w stanie początkowym.

Wiele interesujących obserwacji opartych na produkcji protonów i antyprotonów jest wykorzystywanych w badaniach diagramu fazowego QCD. Jeden z intrygujących pomiarów pokazał rosnącą różnicę przepływu eliptycznego między p a p wraz ze spadkiem energii zderzenia. Istnieje kilka teoretycznych wyjaśnień tej obserwacji. Jednak żadne z nich nie zostało w pełni udowodnione. Dane eksperymentu STAR wykorzystałam do zw-eryfikowania wielu scenariuszy i udoskonalenia odniesienia doświadczalnego dla badań teoretycznych. Podobny trend różnic między p a p zaobserwowałam również dla przepływu trójkątnego, który jest głównie generowany w skutek fluktuacji. Co więcej, skalowanie n q (KE T ) dostarczyło informacji na temat możliwych różnych źródeł pochodzenia protonów i antyprotonów.

Z drugiej strony zakłada się, że utworzona materia będzie zachowywać się jak lepki płyn, a jej rozszerzanie można opisać równaniami hydrodynamicznymi. Przejścia między różnymi stanami skupienia materii i charakteryzujące je zależności termodynamiczne są zawarte w równaniu stanu (EoS). Nie można go wyprowadzić bezpośrednio z pierwszych zasad ani zmierzyć w eksperymencie. Badania EoS mają kluczowe znaczenie dla zrozumienia dynamiki materii w różnych obszarach diagramu fazowego materii jądrowej. Nie jest oczywiste, że obserwable mierzalne eksperymentalnie mogą dostarczyć informacji na temat EoS lub jego zmian. Wprowadzenie możliwości różnych wyborów EoS podczas tworzenia modelu EPOS dało wyjątkową możliwość takich badań. Rodzina EoS zaproponowana przez kolaborację BEST została zaimplementowana. Zmieniono położenie punktu krytycznego i poziom krytyczności, a także zbadano wpływ na końcowe obserwable, takie jak produkcje cząstek, rozkłady pędu poprzecznego, przepływy lub momenty rozkładów netto-protonów.

Praca składa się z Rozdziału 1, w którym przedstawiamy zwięzły przegląd istotnych dla tej pracy aspektów fizyki wysokich energii. Opis anizotropii w ekspansji materii, harmonicznych przepływu i kumulantów dwu-cząstkowych znajduje się w Rozdziale 2. W Rozdziale 3 opisano eksperyment STAR na akceleratorze RHIC. Wybór danych i ocena niepewności systematycznych omówiona została w Rozdziale 4. Rozdział 5 jest poświęcony wynikom eksperymentalnym. W kolejnym Rozdziale 6 zawarty został opis modelu EPOS i zaimplementowane w nim zmiany (wprowadzenie nowego EoS). Badania wprowadzonego EoS do modelu EPOS i porównanie z wynikami STAR przedstawiono w Rozdziale 7. Na koniec, w Rozdziale 7.2.5, podsumowano przeprowadzone badania i przedstawiono perspektywę wpływu na przyszłe badania.

Istnieją trzy załączniki zawierające krótkie podsumowanie mojego zaangażowania w: rozwój oprogramowania iTPC -Dodatek 9, działania przeprowadzone w ramach projektu Rivet -Dodatek 10 oraz lista użytecznych równań termodynamicznych w Dodatku 8.

Niniejsza rozprawa obejmuje unikalne połączenie podejść eksperymentalnych i teoretycznych do badania właściwości i dynamiki materii.

RÉSUMÉ

Étude de la dépendance en énergie des flux elliptique et triangulaire pour les particules identifiées dans le Beam Energy Scan avec l'expérience STAR et le générateur d'événements EPOS

L'étude des propriétés de l'expansion collective de la matière créée dans les collisions d'ions lourds est un outil unique, permettant de mieux comprendre les aspects non-perturbatifs de la QCD, et nécessitant un travail tant sur le plan théorique que sur le plan expérimental. D'une part, des calculs d'évolution hydrodynamique prédisent, en conséquence de l'asymétrie du système dans l'état initial, une anisotropie dans la production des particules finales. D'autre part, les mesures de cette anisotropie, en fonction du système et de l'énergie considérés, permettent non seulement de valider les prédictions théoriques, mais aussi de déterminer des éléments inconnus comme les propriétés du plasma créé lors de ces collisions (e.g. son équation d'état), ou bien les processus de thermalisation mis en jeu. Améliorer nos connaissances sur ce sujet est le principal objectif de cette thèse. Les méthodes expérimentales décrites ont pour objectif de donner un aperçu de la mesure de l'anisotropie de l'expansion des particules et anti-particules, tandis que les approches théoriques ont été utilisées pour étudier l'équation d'état de la matière.

Les corrélations du flux entre deux particules sont un outil très puissant dans l'étude de l'expansion de la matière: elles traduisent la relation entre les directions des particules produites. Une distribution non uniforme des impulsions de ces particules est la conséquence d'asymétrie dans la géométrie de l'état initial de la collision. Cette asymétrie est alors transformée, via des gradients de pression, en flux anisotrope, entraînant une forte corrélation entre les directions des particules finales par rapport au plan des participants. La forme de cette expansion peut être décrite par décomposition de Fourier, où chaque harmonique correspond à une forme particulière d'expansion de la matière : la seconde harmonique (v 2 ) correspond au flux elliptique, et la troisième (v 3 ) au flux triangulaire. Le v 2 est originellement créé par l'excentricité (ϵ 2 ) de la région de collision. D'autre part, les coefficients de Fourier impairs ne sont pas tous sensibles à l'excentricité, et les fluctuations ainsi que les effets de viscosité dominent les données mesurées. De plus, la mesure de ces coefficients impairs est aussi moins contaminée par les corrélations entre particules indépendantes du flux (effets non liés au flux). Ainsi, tous ces ingrédients font de l'étude détaillée de ces deux harmoniques une source d'information complète concernant de nombreux aspects des propriétés de l'état initial de la matière.

De nombreuses observations d'intérêt, basées sur la production de protons et d'antiprotons, sont utilisées comme sondes d'étude du diagramme de phase de la matière nucléaire. Une de ces mesures en particulier montre un résultat intrigant : la différence de flux elliptique entre p et p augmente lorsque l'énergie de la collision diminue. Nombreux sont les scénarios théoriques pour expliquer cette observation, cependant aucun d'entre eux n'a pu être prouvé jusque-là. Les données de l'expérience STAR ont donc été utilisées pour améliorer les références expérimentales pour les études théoriques, permettant ainsi de vérifier plusieurs de ces scénarios. Un comportement similaire a aussi pu être observé entre le flux triangulaire des p et p, qui est principalement causé par des fluctuations. Enfin, l'échelle de n q (KE T ) a permis de donner quelques informations sur les possibles origines de ces différences entre protons et antiprotons.

Par ailleurs, il est attendu que la matière créée lors de ces collisions se comporte comme un fluide visqueux, dont l'expansion est basée sur les équations de l'hydrodynamique relativiste. Les transitions entre les différents états de la matière ainsi que les relations thermodynamiques qui les caractérisent sont décrites par l'équation d'état, dont l'étude est donc cruciale pour comprendre la dynamique de la matière nucléaire dans différentes régions de son diagramme de phase. Elle ne peut cependant pas être dérivée directement de premiers principes physiques, et aucune observable mesurable expérimentalement ne semble pouvoir donner quelconque information à son sujet, ni sur ses propriétés.

Le fait d'implémenter, durant le développement du générateur d'événements EPOS, le choix de différentes équations d'état proposées par la collaboration BEST donna donc une possibilité unique de réaliser cette étude. La position du point critique, ainsi que l'intensité des variations de criticalité ont été modifiées, et les effets de ces changements sur certaines observables de l'état final comme le taux de production des particules, les spectres en impulsion transverse, les différents types de flux ou encore les moments des distributions en nombres nets de protons ont été étudiés.

Cette thèse est composée d'un Chapitre 1, où est présentée une revue synthétique des aspects de la physique à haute énergie nécessaires pour cette thèse. Les principes d'anisotropie dans l'expansion de la matière, les harmoniques de flux et les cumulants à deux particules sont exposées dans le Chapitre 2. Le Chapitre 3 contient la description de l'expérience STAR au RHIC. La sélection des données ainsi que l'évaluation des incertitudes systématiques sont discutées dans le Chapitre 4, et le Chapitre 5 est dédié aux résultats expérimentaux. Le Chapitre 6 suivant décrit quant à lui le générateur d'événements EPOS, et détaille les développements réalisés (inclusion d'une nouvelle équation d'état). L'étude de ces équations d'état implémentées dans EPOS et la comparaison avec les résultats STAR sont présentées dans le Chapitre 7. Enfin, les différentes études réalisées durant cette thèse sont résumées dans le Chapitre 7.2.5, où différentes perspectives sur l'impact de futures recherches sont données.

S'ajoutent à cela trois annexes comprenant un court résumé de mon implication dans le développement du logiciel iTPC (Annexe 9), les activités conduites dans le cadre du projet RIVET (Annexe 10), et une liste d'équations importantes de thermodynamique (Annexe 8).

Cette thèse inclus la combinaison unique d'approches à la fois théorique et expérimentale afin d'explorer les propriétés et la dynamique de la matière.

ACKNOWLEDGEMENT

The research presented in this thesis would never be completed without the help and support of numerous people. Here, I would like to take the opportunity to express my gratitude and acknowledge the support I have received.

First of all, I would like to thank my supervisors, prof. Hanna Zbroszczyk and prof. Klaus Werner. This journey was quite challenging, but you always supported me and helped with even tiny issues. I really appreciate the freedom you gave me, trusting that I will do my best without daily supervision. Although maybe sometimes I needed some "push", I had to become independent and responsible. Studying in the frame of BGF Cotutelle was an excellent adventure but also a tough challenge for my personal life. I always knew that I could count on both of you! The Covid-19 pandemic started the avalanche of complications, but you helped me even more than I could ever expect. Furthermore, both of you are excellent experts in the field. I have learned a lot, and I promise that I will never forget all the discussions and inspiring meetings! Prof. Hanna, you have taken care of me for already 8 years! You became my guide and role model as a woman and scientist. I highly admire all our informal discussions related to personal life, career, or just everyday situations. I know I can trust you and count on you; I can share my troubles with you as a friend.

Secondly, I would like to thank Niseem Magdy, the fantastic mentor and guide of my experimental research. You have spent hundreds of hours, just as a volunteer, to help me with my analysis. You really believed in me, and without you, there would be nothing except maybe a few pages of this thesis. You taught me how to stand for my rights and defend my opinions. The discussions with you and Prof. Roy Lacey are the most motivating ones I have ever had. The meetings with the group at Stony Brook University opened a new chapter of my science fascination. They gave me a new perspective on my work as a researcher. I would like to thank Doctor Zhangbu Xu, whose support allowed me to come to BNL and collaborate with Niseem, the SBU, and BNL groups.

I want to thank the Heavy-Ion Relativistic Group, Prof. Jan Pluta, Prof. Adam Kisiel, Prof. Kasia Grebieszkow, Dr Małgosia Janik, Dr Łukasz Graczykowski, Dr Georgy Kornakov, Paweł Szymański, Diana Pawłowska (our participation in lots of conferences were great adventures!). Our weekly meetings were very productive and educative! But mostly, I would like to thank Prof. Daniel Kikoła, who has never been my official supervisor. Still, he always finds time for discussion with me. I really appreciate all the feedback you gave me (also the critical one!), all the "short questions" I could've asked sitting and disturbing you in the same office.

I would like to express my gratitude to all the staff working at WUT's Faculty of Physics. You were very patient with all my delays and confusions I introduced.

I want to acknowledge the whole STAR Collaboration. Thanks to you, I had an opportunity to perform my studies on the data collected by the experiment, discuss my progress and get lots of feedback. I learned a lot by taking shifts or participating in the service work with the iTPC group.

During my studies, I have met many fantastic people who became my great friends.

Johannes, Mahbobeh, Andreas, Iuliia, Yurii, Gabriel, David, my stays in Nantes couldn't be better, and all of these thanks to you! Johannes, your contribution to my work with EPOS is substantial, and I highly appreciate it! Jurku i Ewo, zostaliście moimi francuskimi rodzicami. Nigdy nie poznałam ludzi tak ciepłych, zaangażowanych, dobrych i bezinteresownych. Dziękuję za to, że wyciągnęliście mnie z najróżniejszych tarapatów i zawsze mogłam po prostu z Wami być. Dzięki Wam, Nantes było moim "drugim domem". Daniel Kincses, thank you for all these jam sessions at Jazz Loft in Port Jefferson, crazy trips, and dozens of hours spent working on many scientific projects. But I am most grateful for your support 24h/7 with all my troubles. You are a fantastic friend! We are really great companions in this "scientific" journey, and I feel that this is just the beginning! Meg tudom csinálni! I want to thank my parents Dorota and Tomek. Od początku mnie wspieraliście w realizacji wszystkich moich szalonych pomysłów. Dzięki Wam mogłam wyjechać na studia do Warszawy, a później kontynuować karierę naukową. Mamo, pokazałaś mi, czym jest sztuka, muzyka, kino. Dzięki Tobie, nie stałam się robotem, a jestem osobą wrażliwą na piękno otaczającego mnie świata. Tato, dzięki Tobie wiem, że nigdy nie można odpuszczać. Stałam się osobą ambitną, wytrwałą w dążeniu do celu. Najbardziej mnie cieszy to, że pokazałeś mi czym jest przygoda i aktywność. Sport stał się ogromnie ważną częścią mojego życia, która kształtuje mój charatker i pozwala przeżyć niezapomniane chwile.

Siostro, Aniu, nigdy nie przestaniesz być najważniejszą osobą w moim życiu. Prawie trzydzieści lat się mną opiekujesz i razem przeżyłyśmy niesamowite historie. Było wiele

INTRODUCTION

Physics is the empirical answer, not leaving place for the "believes". The theory has to be proven by the experiment in the agreed range of uncertainty. However, some of the physics fields are focused on studies of such extremely small (∼ 1 fm) or located in the long-distance (many solar years form observer) objects that it becomes challenging to describe these phenomena both theoretically and experimentally.

One of the most intriguing puzzles is understanding of the beginning of the Universe. Scientists proposed as the answer the Big Bang model, which is nowadays broadly accepted. It explains the origin of the matter, but still, this description is not complete. We do not precisely understand the matter dynamics in the initial stages of evolution or what states and transitions matter goes through. This thesis broadens our knowledge in this topic and answers the questions: what are the properties of the expanding newly-created matter, how we can measure them and what we can improve in our theoretical description of processes occurring during the initial stages of the Universe existence. I am using both experimental and theoretical approach in the investigation of flowing nuclear matter.

Studying the properties of strongly interacting matter is one of the significant milestones for nuclear physics. Various experimental facilities have been designed to investigate the Quantum Chromo-Dynamical (QCD) phase diagram, such as the Relativistic Heavy Ion Collider (RHIC), the Super Proton Synchrotron (SPS), or the Large Ion Collider (LHC). There are also future Nuclotron-based Ion Collider fAcility (NICA) and Facility for Antiproton and Ion Research in Europe (FAIR) which scientific programs are dedicated to exploring high baryon density region. These programs are at the forefront of experimental efforts designed to map the thermodynamic and transport properties of strongly interacting QCD matter. Both theoretical and experimental approaches suggest a smooth cross-over phase transition from a hadronic state of matter to a Quark-Gluon Plasma (QGP) at low baryon chemical potential and high-temperature values, that is, in collisions at the LHC and top RHIC energies. At high baryon chemical potentials and low temperatures, a first-order phase transition is expected. Both transitions are foreseen to be connected by the Critical Point (CP), at which the phase transition is predicted to be second order. STAR experiment carried out an extensive research program to search for CP.

The experimental data suggest that QGP behaves almost like an ideal fluid. Due to the short lifetime of the plasma state, many of its properties can only be inferred indirectly by comparing the final state measurements with transport model calculations. Among the current phenomena of interest are the interdependences between two collective flow effects, elliptic v 2 and triangular flow v 3 . These are the observables characterising the shape and dynamics of the expanding matter. Both are very sensitive to the properties of the system at very early times of its evolution. The elliptic flow is mainly related to the initial geometry and collective expansion of the system, whereas triangular flow is sensitive to the initial state's fluctuations.

STAR is caring out an extensive research program to search for CP in the Beam Energy Scan program frame. The relevant part of the performed investigations concentrated on the measurements of v 1 , v 2 and fluctuations. So far results for protons and antiprotons are the most intriguing: observation of the non-monotonic net-proton fluctuations, change of sign for net-proton dv 1 /dy or large difference between v 2 of p and p. These observations were that significant and interesting for the physicist community that the Beam Energy Scan II is being carried out at RHIC, providing new data that are already available. Here, the possible sources of the studied phenomena are explored-the effect of the initial state fluctuations and how in-medium interactions (characterised by transport coefficients) affect the observables used as probes of the CP. Since the triangular flow is susceptible to these effects, the measurements of v 3 for particles and antiparticles (focusing on p and p) can provide an additional, vital experimental handle on initial-state fluctuations and shear viscosity.

The experimental research is complemented with developing the phenomenological model describing the matter's creation and its dynamics. The EPOS model is applicable to systems corresponding to small µ B . It needs to undergo significant adaptations to describe better physical phenomena present at high baryon densities that occur at relatively low collision energies at which the BES studies are carried. Different ingredients of the EPOS model: hydrodynamical evolution, Equation of State and processes of hadronisation, need to be revised and improved. Each of them can give essential information about the nature of the phase transition (i.e. cross-over vs first order), the location and proprieties of Critical Point. It will enable EPOS to become one of the basic theoretical models for the previously mentioned experimental programs.

The hydrodynamical evolution included in the model is based on the Equation of State (EoS). It characterises the relations between the thermodynamical quantities describing the evolving system. However, it cannot be studied directly in the experiment, and a phenomenological reference is needed. Although there are many models, it is challenging to acknowledge what experimental observables are sensitive to the changes in EoS and which as related to the other model ingredients. The possibility of changing only the EoS in the given model provides direct information about the searched effects.

In this thesis, a novel approach of heavy-ions studies is presented from both experimental and theoretical sides. The 2-particles cumulants of flow harmonics for identified hadrons have been studied at STAR for the first time as a function of transverse momentum and collision centrality. Moreover, a unique analysis of the impact on the several observables of the various EoS implemented in the EPOS model has been performed. Some of the studies performed as a part of this thesis has been already published in [START_REF] Stefaniak | [END_REF][2][3][4]. Additionally, the results related to the flow harmonics for Au+Au collisions at √ s N N = 200 GeV measured at STAR are already under the internal STAR committee review. Furthermore, the rest of the obtained results are now being prepared for the publishing process for two articles: one dedicated to the v n for experimental studies and the second related to the studies of EoS with EPOS model.

Chapter 1

HIGH-ENERGY NUCLEAR COLLISIONS

In this chapter, the theoretical background of high-energy physics is introduced. Firstly, the concept of the Standard Model of particle physics and quantum chromodynamics is highlighted.

Standard Model and Quantum Chromodynamics

Standard Model

From the 1930s onwards one received a comprehensive insight into the knowledge of the most elementary structure of matter and its interaction. The few fundamental particles grouped into blocks governed by four fundamental forces were found to be the only ingredients creating the Universe. The relations between the given groups of matter particles and three of the forces are encapsulated in the Standard Model of particle physics. This theory was developed in the early 1970s, successfully describing nearly all experimental results and predicting a huge range of phenomena where some of them are still tested. The all particles are divided into two groups: fermions (matter particles) and bosons (interaction carriers).

Matter particles

The matter which surrounds us consists of fundamental particles (no internal structure can be distinguished). They can be divided into two primary groups called quarks and leptons. There are six particle's species in each of them, and they are related to pairs called "generations". The first generation consists of the most stable and the lightest particles, while the second and the third particles are heavier and less stable. Consequently, the whole stable matter in the Universe is made from first-generation particles. The heavier ones nearly immediately decay to the more stable particles. The generations of the quarks are listed in the bound only in such ways as to create the colourless objects. This number was introduced to understand the possibility of obeying the statistical rule described by Enrico Fermi and Wolfgang Pauli called the exclusion principle. It states that no two fermions can occupy the same quantum state, be described by same quantum numbers [5]. The second group of fermions consists of six leptons, which are arranged into three generations. The tau, muon and electron have a sizable mass and an electric charge. The neutrinos are electrically neutral and were treated, for a very long time, as mass-less particles. Recently, it was discovered that they do have mass [START_REF] Nagataki | JPS Conf. Proc[END_REF].

Force carriers

In the Universe, the four fundamental forces can be distinguished:

-the electromagnetic force, -the strong force, -the weak force, -the gravitational force (not included in SM).

It is predicted each force is carried by gauge bosons. The integer spin characterises them, and they obey Bose-Einstein quantum statistics. Bosons transfer discreet energy amount between matter particles. The electromagnetic interactions are exchanged by photons (γ) between charged particles, and similarly to gravitational forces, their range is infinite. It is described by Quantum Electrodynamics (QED). Contrarily, weak and strong forces are effective at the subatomic level. Very massive particles carry the weak interactions: W + , W -, Z 0 . It is characterised by Quantum FlavourDynamics (QFD) and affects both quarks and leptons. Strong forces are mediated by eight gluons (g), and they are responsible for the colour exchange between quarks. The Quantum ChromoDynamics (QCD) is the theory describing this type of interactions (more in the next subsection). The gravitational force is carried by hypothetical "graviton", which existence is not yet proven.

Standard Model does include electromagnetic, strong and weak forces, all their carriers, and describes how they interact with particles of matter. However, gravity is not a part of SM. In the Figure 1.1 there are presented examples of SM interactions.

Both bosons W and Z have mass a hundred times bigger than protons. In 1964 theorists Peter Higgs, François Englert, and Robert Brout proposed the theory explaining this puzzle by the mechanism BEH (Brout-Englert-Higgs). The assumed the existence of an invisible field that reacts with bosons W and Z, giving them mass, while it does not interfere with photons, leaving them massless. Just after the Big Bang, the so-called "Higgs field" was equal to zero and all the particles were massless moving with the speed of light. After cooling down, the Universe reached the critical temperature, and the filed spontaneously increased and, via interactions, gave particles mass. The BEH mechanism implies that the value of the elementary particles' mass depends on the strength of their interaction with the Higgs field. The particle which mediates in this field is called the Higgs boson and is one of the Standard Model's ingredients.

Quantum Chromodynamics

Quantum Chromodynamics (QCD) is the theory describing the strong interactions between quarks. They are mediated by gluons, which are chargeless and massless. Similarly to photons, gluon is the vector particle, and its spin and parity are equal to J P = 1 -. In QCD, there are six types or the strong charges called colours -the extra internal degrees of freedom. Quark can carry only one out of three possible colours and antiquark one anticolour; they can exchange them via gluons. It is assumed that the colour symmetry is the strict symmetry and the strong forces are colour independent. The example of the interaction is presented in Fig. 1.2, where the red quark interacts with the blue one exchanging the red-blue gluon. The gluon has its own color charge (r b). As a consequence, the gluons can directly interact with each other.

The idea of introducing this extra quantum number was derived after observation of particles built up from three quarks of the same flavor: ∆ ++uuu, ∆ -ddd and Ω -sss.

To satisfy Pauli's exclusion principle, the new quantum number describing the systems had to be added. Greenberg introduced it in 1964 and by Han and Nambu in 1965 [10, 11]. Both baryons (qqq or q q q) or mesons (q q) have to be colour-singlets.

Potential of strong interactions

The potential between quarks in QCD can be expressed with the following equation:

V = - 4 3 α S r + kr (1.1)
where: α s is the strong coupling constant, r is the distance between quarks, and k is the parameter experimentally estimated to be equal 0.87 GeV/fm. The proposed potential correctly explains the energy levels of charmonium and bottomium. In Figure 1.3 the dependence of the QCD potential on the distance between quarks. For small r the first part of the V (eq. 1.1) dominates. For the bigger distance, the second part dominates, which depends linearly on radii. It is related to the confinement of quarks in hadrons.

The lines of the colour forces can visualise it pulled together by gluon-gluon interactions, creating a tube or a string. The stretching of the string causes the energy proportional to kr to reach the critical value where, for the energetic reason, it is more efficient to create the new pair of q q. As a consequence, instead of one long string, there are two shorter. This process is called the fragmentation of the strings.

The asymptotic freedom and quark confinement

In the QCD there is defined the coupling constant α s , which determines the strength of the force acting on quarks and gluons. In reality α s is not constant and scales with Q 2 and can be expressed by the following :

α s (Q 2 ) ∝ 1 β 0 ln(Q 2 /Λ 2 ) (1.2)
where β 0 is one-loop approximation for the function [13], Λ ≈ 127M eV -scale constant 1

In the Fig. 1.4 the α s dependance on the momentum exchange between particles is presented. There are two main scenarios possible for colored objects: confinement dominating in low energies [15][16][17] and asymptotic freedom at high energies [18][19][20].

Reactions with momentum exchange Q < 1 GeV are called the soft processes, which constitute the majority of the nuclear reactions. Thus the reactions with Q > 1GeV are hard processes, which are just a percent of the whole occurring reactions. It is worth mentioning that the strength of the interactions between quarks is inversely proportional to the momentum transfer: r ∝ 1 Q . As a consequence of the conservation of the coupling constant α s two effects can be deduced:

Asymptotic freedom: phenomena of dissipation of the force acting on quarks where the distance between them decrease. At the asymptotically high Q 2 (→ ∞) the coupling constant α s (Q 2 ) → 0, so at short distances (comparing to the radius of a nucleon), so the interactions between quarks and gluons are infinitely weak. Consequently, at high energies, quarks act like the "free" objects, not bounded. The predicted state where particles are in the asymptotic freedom and the system is in the thermal equilibrium is the strongly coupled Quark-Gluon Plasma (QGP). Confinement of quarks: a state where quarks and gluons are bounded into hadrons. They are not expected to exist as separate particles. The energy per length of string is constant, so the energy required for the separation of partons is infinite. A simple example is meson which consists of two: quark and antiquark. As a consequence, quarks and gluons are bounded into hadrons and cannot exist as independent objects. Several models are explaining the production of new particles; one of them is the String model [START_REF] Andersson | 7th European Symposium on Antiproton Interactions: From LEAR to the Collider and Beyond[END_REF]. The string is an object: quark-antiquark, or quark-diquark, which is filled with the colour field. The energy of the string is proportional to the distance of separation. The string formation is based on momentum or colour exchange (depends on the model). If there is enough energy accumulated and the quarks are pulled apart, the string fragments. The newly created quarks and antiquarks immediately bound into hadrons [START_REF] Grebieszkow | Lecture: Introduction to Heavy-Ion Physics[END_REF].

There are two, main approaches to solve the Quantum Chromodynamics equations:

perturbative QCD: used for r ∼ 1/Q and much lower the nucleon radius. The processes are hard, and it is possible to utilise the calculations similar for the QED. non-perturbative QCD: for large distances r and very small Q 2 (≤ 1GeV 2 ). It reports to the soft collisions. The calculations can be done based on the numeri-cal calculation techniques on the space-time matrices called lattice or using phenomenological models (described in Section 1.2.4)

Relativistic heavy-ion collisions

To reproduce the Big Bang in the laboratory, heavy-ions are accelerated close to the relativistic speeds. As a result of their collision, a created system is described by the very high energy density where the two nuclei's energy is transformed into a fluid, so mainly into kinetic energy.

There are two possible scenarios of the system's evolution showed in Figure 1.5: one considers the Quark-Gluon Plasma (QGP) phase (right side) while the other one does not (left side). It is related to the energy of the collision and the reactions happening just after it. However, for low energies the QGP phase is not expected, at most the unbound state of quarks and gluons.

For the higher collision energies, the initial phase of the collision system is in the thermodynamic non-equilibrium (pre-equilibrium), where the primary collisions occur, and the new matter is produced. Subsequently, the dynamics of partons (cascades) develop and the processes with low momentum transfer dominate. As a result of these interactions, after about 1 fm/c, the local equilibrium is reached. This state is called QGP. The system expands and cools. The so-called chemical freeze-out takes place where the inelastic interactions dissipate, and the chemical composition of the system is established -it reminds the hadronic gas. With the cooling, the particles stop to collide and exchange momentum -the kinetic freeze-out.

QCD Phase Diagram

As mentioned in Section 1.1.2, QCD is the theory of strong interactions. One of the crucial features of the theory is the self-coupling of gluons. It causes an increase of the coupling constant (α) with a decrease of momentum transfer, which gives rise to the phenomena of asymptotic freedom for high momentum exchange and confinement for low momentum exchange [24]. While confinement is responsible for all hadrons we observe in nature, the fundamental questions are (i) how a multi-parton system behaves at extremely high temperatures where the deconfinement regime is achieved and (ii) how and when the partons become bound into hadrons again. It is understood nowadays that Figure 1.5 -Space-time system evolution created in the collision of ions: a) with the QGP creation, b) without QGP phase [START_REF] Grebieszkow | Lecture: Introduction to Heavy-Ion Physics[END_REF] .

at the extremely high temperatures above 170 MeV the partons (quarks and gluons) are in a state of QGP.

The QCD phase diagram in Fig. 1.6 illustrates the phases of the strongly interacting matter at different temperatures T and baryochemical potentials µ B . The baryochemical potential is the partial derivative of the internal energy on a number of the particles and entropy of the system:

µ B = ∂U ∂N S,V =const (1.3)
This variable describes how internal energy will change if one baryon is added or subtracted. In the QCD Phase Diagram, µ B can be interpreted as a measure of the imbalance between baryons and antibaryons in the system, so µ B = 0 MeV corresponds to equity between baryons and antibaryons and µ B > 0 MeV the advantage of the matter.

The QCD phase diagram is still not well-understood. The possible structure of it is shown in Fig. 1.6. Except for the two states of matter described before -hadronic matter and Quark-Gluon Plasma, there is a phase of colour superconductors in the diagram. It is a hypothetical state where the quarks are bounded into Cooper pairs, and then they condensate and create the phase analogical to the electrical superconductors. At very high densities, the preferred phase is the colour-flavour-locked (CFL) with the equal number of Depending on the µ b between hadronic matter and QGP, there can occur various phase transitions:

-first-order phase transition: where the Gibbs Free Energy G is continuous while its derivatives do not; the entropy, volume, density, internal energy, enthalpy, baryonic and energy density change incrementally, -second-order phase transition: G is continuous, its derivatives as well, but its second derivatives do not, the heat capacity and thermal expansion factor changes rapidly, -"cross-over" transition: there are no discontinuities, but the parameters change drastically.

In the highest temperature and low µ B region, the phase of matter changes via a "crossover" transition. At the higher µ B , the first-order phase transition is assumed to take place. Between those two cases, there is a so-called Critical Point (CP) which correspond to the second-order phase transition [START_REF] Fodor | [END_REF]26]. Studies of the QCD phase diagram, the properties of strongly interacting matter, and the phase transitions' nature are the vital parts of the Beam Energy Scan (BES) program [27].

In heavy ion (Au+Au) collisions at BES energies, the created matter is characterised by higher baryon density and lower temperatures than those obtained at the LHC. The program covers a broad range of collision energies, vital for precision studies of the phase transition between QGP and hadronic matter in different areas of its phase diagram. Colliding gold nuclei with precisely calculated energies gives a possibility to "scan" parts of the phase diagram, study the properties of matter, and pinpoint the possible CP location. The increase of the collision energy moves the system's thermalisation to higher T and lower µ B . More horizontal shift on the phase diagram can be introduced colliding ions or nucleons with lower (more to the left side of diagram) or higher (oppositely) atomic mass.

Quark-Gluon Plasma

The QGP is the state of the matter characterised by thermal equilibrium, which exists at extremely high temperatures and densities, where the quarks and gluons are not confined. They can propagate in the whole volume occupied by the system. This state was predicted in the frame of QCD in the 70s. It is claimed that the QGP existed in the few milliseconds after the Big Bang. Nowadays, this state can be reproduced in high-energy experiments via collisions of ions.

Although in QGP, the quarks and gluons are not confined, the remnant interactions make it behave similarly to the dense liquid (plasma). It can be created via a) heating or b) squeezing the medium. Experimentally, squeezing is not possible, as well as heating with the external source. The increase of the temperature and density is gained via collisions of the atomic nucleus accelerated to the ultra-relativistic velocities.

The direct investigation of the QGP is impossible because it rapidly changes into the hadronic gas (after a few fm/s). Basing on the experimental measurements, one can study the signatures of the QGP, such as:

-direct photon production -prompt photons originating from the pQCD and thermal from QGP phase, interact weakly with the QGP, and they carry the information about the evolution of the system. The presence of QGP can be deduced in the investigation of the momentum spectra of direct photons. If there is a surplus of thermal photons over immediate ones, one can assume that the QGP phase took place. However, photons are produced rarely, and their identification is challenging, so this method is not highly effective. As the photons can also originate from the decays, their identification is very complicated. -dileptons production -similarly to photons, leptons interact with QGP weakly.

Studies of invariant mass spectra of various leptons created in the p-p or p-Pb collisions prove that they originate from decays of mesons (ρ, ϕ, ω), while in collisions of Pb-Pb the surplus of produced leptons is disproportional to the size of the collided system. It is explained by the meson characterisation (mass, ratios of decay branches) modification happening during the QGP phase, which finally causes the increase of decays to the dilepton's channels. The surplus of leptons is proportional to the duration of the QGP phase. -charm suppression -quarkonia (for example cc, b b) are produced at the beginning of the evolution of the system and originate from the lepton decay channels. In the presence of QGP, the colour charge of quarks is screened by other quarks and gluons (similarly to the Debye'a effect) so that the interactions between quarks cc are weaker, the dissociation of this pair takes place and the J/ψ(cc particle cannot be produced (in other words: the freely moving colour charges "hide" the coupling potential cc). The charm's suppression is studied in reference to p-p collisions or corresponding to the Drell-Yan effect. -strangeness production enhancement -strange quarks produced at the freeze-out surface are less suppressed than originating from string decays. A vast number of strange particles can be treated as a signature of the existence of the QGP phase. -elliptic flow -the flow of matter in the transverse direction in non-central collisions.

More information in Chapter 2, in Section 2.3. -jet suppression -jets strongly interact with QGP medium, and as a result, their energy relevantly decreases. The missing energy of the jet can state the existence of the QGP phase. -inelastic (dynamic) fluctuations -for example, the fluctuations in the kinematic distributions or global characteristics (like average transverse momentum or multiplicity of the collision) are studied using event-by-event methods. Such fluctuations can be treated as a signature of QGP. Moreover, the significant increase of the fluctuation can be related to the Critical Point.

Equation of State

Equation of State (EoS) is the formula describing the states of the matter and its transitions. It expresses the relations between various parameters such as pressure, tem-perature, energy density, speed of sound, and former. It is not trivial to characterise the whole phase diagram area. Consequently, such features of QCD as chiral symmetry breaking or confinement, are not fully understood.

The non-perturbative QCD is applied for the calculations of the dynamics of the matter at extreme densities and temperatures. For the description of the system where partons are expected to be in the deconfined state, the lattice calculations can be used. These are the numerical calculations performed on the discrete time-space grid [28]. The obtained EoS is derived from the first principles. It provides quantitative information on the deconfined state QGP and cross-over transition, the region of the QCD phase diagram characterised by the extremely high temperatures and low baryon chemical potential. In Figure 1.7 the relations between energy density/pressure and temperature are presented. The energy density rapidly changes at T ∼ 190 MeV, which is caused by the rapid increase of the effective degrees of freedom. The pressure does not change that rapidly because during the strong phase transition, the speed of sound, expressed with c S = ∂P/∂ϵ, is reduced. At the extremely high temperature, the ϵ nearly reaches the value corresponding to the ideal massless gas limitation.

Although the full potential of Lattice calculations for QCD is increasing by applying more advanced algorithms and hardware technology [30], the region of the diagram at non-zero µ B is still not completely understood. It is not possible to predict based on the first principles the existence and location of the CEP. In order to extend to the finite chemical potential, the Taylor series expansion in powers of µ B /T [28,31,32] is applied to the Lattice calculations. The following equation can express the expansion of the pressure: 1.4) the coefficients are:

P (T, µ B ) = T 4 Σ n c 2n (T )( µ B T ) 2n (
c n (T ) = 1 n! ∂ n P/T 4 ∂(µ B /T ) n µ B =0 = 1 n! χ n (T ) (1.5)
where χ n are the susceptibilities of the baryon number. However, only few of the expansion coefficients are known [33][34][35] and they do not allow to investigate the region of QCD phase diagram closer to CEP.

There are various attempts performed to generate the Equation of State, which will allow one to characterise the whole QCD phase diagram, starting from µ b = 0 finishing on cold neutron stars [START_REF]Neutron Stars[END_REF]. These EoS introduce the first-order phase transition for finite baryochemical potential and relatively lower temperatures -the region studied in the Figure 1.7 -Pressure 3P/T 4 (dashed curve) and energy density ϵ/T 4 (solid curve) in function of temperature T from lattice calculations [29].

BES program. Nevertheless, some of them provide information about the Critical Point location and properties of this transition [30,[START_REF] Parotto | [END_REF]. One such EoS is described in Section 6.6.1.

Transport coefficients

To describe the state and dynamical properties of the expanding matter, one should investigate the transport coefficients.

The QGP is treated as a nearly perfect fluid [38], and its motion is described with hydrodynamics. Several parameters called transport coefficients are investigated to characterise the expanding bulk. The speed of sound c S , which corresponds to the mechanical wave, is characteristic of the given state of matter. The shear viscosity η/s is expected generically to rise slowly above critical temperature (T c -where the CP is expected), drastically below and has a minimum at T c [39]. On the other hand, the bulk viscosity ζ/s peaks close to T c , as a consequence of long-range critical correlations breaking the conformal symmetry [40]. It decreases on both sides of T c , making the shear viscous effects dominate in given regions. Observation of the strong variations of bulk viscosity is expected close to CP. However, it is impossible to measure transport coefficients directly in the experiment. They can be extracted from the real data, comparing them with the viscous fluid dynamical, phenomenological simulations.

Monte Carlo Event Generators

The processes occurring just after the collisions are incredibly complex. Although the experimental potential is high, and the applied technologies are extraordinarily sophisticated and precise, they do not allow one to follow each step of the created system's evolution. It is still impossible to detect single partons in the deconfined state and measure their properties directly. The particles identified in the experiment already hadronised and cooled.

Multiple theories are describing the properties of strongly-interacting matter created in the collision. To reproduce such scenarios, various MC Event Generators are created [41,42]. These are the computer software that simulates the states and dynamics of the medium according to the given theory. The random number generators and MC methods are applied. For the soft interaction (low momentum transfers), the phenomenological models are incorporate, while in the case of hard processes, the perturbative solutions are utilised.

Heavy ion collision characterization -concepts and definitions

The most important definitions and concepts for these studies are outlined in this subsection: firstly, these corresponding to the collision characterisation, then related to the particle dynamics.

Collision Geometry

The accelerated to the relativistic speeds nuclei appear as flat discs (depending on the collision energy) due to the Lorentz contraction along the beam directions, referring to the centre of mass (CM) frame (left part of Fig. 1.8). If A is the atomic mass number of the nuclei, the transverse radius can be estimated as R = 1.2A 1/3 . During the collision, the incoming nuclei do not overlap completely. The quantity called centrality of the collision describes how big is this overlap region. The systems where the collisions are "headon", and these more peripheral are relevantly different (Fig. 1.9). The eccentricity (ϵ) is the parameter describing the distortion from the round overlap region of collided nuclei. Higher values of ϵ correspond to more peripheral collisions.

To characterise the newly-created matter, one needs to categorise the centrality of the collision. There are several ways to extract information about the overlap region's size and finally classify the given event. Centrality definition is crucial in the studies of the system's evolution as the dynamics of the matter depends on the number of participants or the size of anisotropy in the overlapping region.

There is an impact parameter b introduced, which is equal to a shorties distance between the centres of the collided nuclei, measured in the moment of the nucleus overlapping (left part of Fig. 1.8). The smaller b, the more central collision is. However, it is impossible to measure it directly in the experiment. On the right side of Fig. 1.8 the particles' classification is presented. The nucleons which take part in the interactions occurring during the collision are called participants, while these which continue unaffectedspectators.

Assuming that the multiplicity of particles is a monotonic function of b, in the experiment, the information about the centrality of the collision can be extracted, for example, As the shape of the overlap region is ellipsoidal, consequently, such anisotropy is transferred to the anisotropy in the emission of the created particles (Fig. 2.1).

Reaction and participant plane

In the studies of the anisotropies in the created matter dynamics, the coordinate system's choice is crucial. The same definition of the reference plane for the direction of the particle motion measurement has to be established to investigate all collisions. There are two possible choices: reaction and participant plane.

Reaction plane (Ψ RP ) -expanded between beam direction and impact parameter. As the impact parameter cannot be directly measured, the reaction plane angle cannot be derived straight from the experiment. It is estimated from the event-by-event azimuthal particle distribution.

Participant plane (Ψ P P ) -can differ from the reaction plane as the distribution of Figure 1.11 -The reaction and participant planes coordinate systems [38].

participants is not uniform in the overlap region.

Both systems are illustrated in Fig. 1.11. It is relevant in flow measurement to distinguish between these two coordinate systems.

Kinematics

To characterise the kinematics of the matter, the coordinate system needs to be established. At RHIC, the z-axis is parallel to the beam axis, x-and y-axes form the plane perpendicular to the z-axis, while y-axes points vertically up. Although the prime interaction point is located at (0, 0, 0) -coordinate, it does not necessarily correspond to the place of the collision. The primary vertex (position of the collision) has to determined.

Energy of the collision

In the CM frame the energy of the collision is calculated using the beams 4-vector

E(E, 0, 0, p z ): E CM = (2E) 2 = 2E (1.6)
The CM energy per nucleon pair is denoted as √ s N N .

Transverse momentum

A particle's measured momentum is decomposed to vectors corresponding to x, y, z direction. However, due to particles' relativistic motion, the Lorentz invariant variables are introduced, such as the transverse momentum. It can be expressed with following:

p T = p 2 x + p 2 y (1.7)

Rapidity

The momentum component parallel to the z-axis is rarely used, contrarily to the additively invariant under Lorentz transformations variable called rapidity. Rapidity is defined as:

y = ln[ 1 2 E + p z E -p z ] (1.8)

Pseudorapidity

To determine the rapidity of particle the particle mass has to be known. The variable which may be used in the charaterization of the direction of the particle's motion is pseudorapidity:

η = -ln[tan θ 2 ] (1.9)
where tan(θ) = √ x 2 + y 2 /z. For the determination of η only the (x, y, z) coordinates needed; consequently, the information about the particle's momentum is not required. For particles with mass relevantly smaller than momentum, the energy

E = √ m 2 + p 2 → p, therefore y → η.
Chapter 2

AZIMUTHAL ANISOTROPY IN HEAVY-ION

COLLISIONS

The heavy-ion collisions are invaluable for creating and investigating nuclear matter, dynamics, and phase transitions. Similarly to the early stages of the universe, the collision's newly-created, dense and extremely hot system will cool down and expand. The obtained medium goes through various phases changing its energy densities and temperatures, while quarks and gluons experience multiple interactions forming the QGP (if it is created). It expands collectively and finally becomes dilute enough to hadronise. Such collective expansion is named flow. Studies of flow have a relevant contribution to the experimental studies of EoS and transport properties of QGP. The particles are not produced uniformly in all directions. Such anisotropy is one of the most significant experimental signatures of the presence of flow in the system obtained in heavy-ion collision [38,[44][START_REF] Heinz | Relativistic Heavy Ion Physics[END_REF][START_REF] Huovinen | [END_REF]. Studies of this observation are crucial because anisotropies in flow are highly sensitive to the system properties at the first moments of its evolution -the spatial asymmetries immediately decrease. The flow can be built only during the first f m/c of the system's lifetime. In Fig. 2.1 the non-central collision is illustrated. Part of the ions passes by leaving the ellipsoidal overlap region. This anisotropy in the shape of the collided matter bulk results in the anisotropies in the final-state observables.

Source of collective flow

The azimuthal anisotropy characterises the expansion of highly-compressed matter created in non-central heavy-ion collisions. It is one of the most significant observables used in studies of hot and dense matter. The process of creation of collectivity can be described in several stages:

-During the non-central collisions, the shape of the overlapping nucleus in the beam axis direction resembles an almond or an ellipse. the maximum inside of it, showed in Fig. 2.2 (pressure is related to the energy density). The gradients are higher in perpendicular to the beam direction. -Consequently, particles in transverse direction have higher acceleration and, therefore, higher flow velocities in-plane. -The final result of the differences in values of gradients of pressure and scattering is the anisotropy in production and motion of the newly-created matter.

In Figure 2.2, the evolution of the initial transverse energy density profile is illustrated. The expansion of the almond-shaped system and buildup of the collective motion is governed by sound velocity (c s ) 1 . It is a significant feature because it varies between different EoS (see Section 1.2.3).

Once it was realised that the ideal hydro does not describe the system's evolution correctly, the corrections for the viscous effects were taken into account [39,[48][49][50]. The shear viscosity ( η s ) term was introduced. It defines how "imperfect" fluid is2 . Since then, it became an interesting probe in studies of the EoS (left panel of Fig. 2.3). [43,48] Consequently, the shape of the measured flow is dominated by two effects illustrated in Figure 2.4 (valid for n > 1):

-for high multiplicities it origins from eccentricity ϵ n -at more peripheral collisions the shear viscosity dominates, which is inversely proportional to the number of participants.

Non-flow contribution

The measured azimuthal correlations between particles also include contribution not solely originating from anisotropic flow, referred to as non-flow. These are correlations of particles not associated with the characterisation of the matter's dynamics in reference to the reaction plane. The most notable non-flow contributions correspond to such effects as jets, resonance decays, short-range correlations (Hanbury-Brown Twiss or momentum conservation) [51,52]. There are few methods proposed to suppress non-flow, although it is problematic to estimate its contribution quantitatively into performed measurements. For example, considering the rapidity gap between correlated particles or various charge combinations of them. In two-particle correlations studies non-flow can by quantified by

δ n parameters: ⟨cos[n(ϕ i -ϕ j )]⟩ = ⟨v 2 n ⟩ + δ n (2.1)
The non-flow contributions are the effects mainly concerning more than one particle, the δ n is nearly inversely proportional to the multiplicity.

To suppress the not negligible non-flow, one can use the methods based on the multiparticle correlations. Although they are almost insensitive to non-flow, they are biased by the interference of several flow harmonics [53]. The various methods of measurements of azimuthal anisotropy are affected differently by non-flow. Short-range correlations are characteristic for pairs of particles with similar η. In order to suppress this contribution of non-flow, the difference between pseudorapidity of particles is ensured by introduction of ∆η. In Figure 2.5 the ratio of measured v 2 {2} with ∆η 1 = 0.0 and 0.3 to v 2 {2} with ∆η 2 = 0.6 for Au+Au collisions at √ s N N = 200 GeV as a function of centrality is presented. The higher value of minimal difference in particles' η, the more non-flow contribution is eliminated.

Flow fluctuations and initial eccentricity

Even on the fix impact parameter one can observe the event-by-event fluctuations of flow in magnitude and direction. The flow fluctuations can be expressed by [38]:

σ 2 vn = ⟨v 2 n ⟩ -⟨v n ⟩ 2 (2.
2)

The geometry of the participant zone can variate in both the eccentricities ε n3 and the direction of the major axes (Fig. 1.11). The Figure 2.6 shows the various ε n with corresponding shapes.

Figure 2.6 -Eccentricities ε n with the corresponding participant zone shape [START_REF] Stefaniak | Beam-energy and collision-system size dependence of the anisotropic flow measurements from STAR[END_REF].

It is challenging to identify the fluctuations. However, higher-order cumulant measurements (v{n}, n ≥ 4) are not sensitive to both non-flow and fluctuations of eccentricity. They are precisely equal to v 2,RP -the flow projection on the reaction plane. Simultaneously, due to a lack of possibility of separating the non-flow contributions from flow fluctuations through correlation measurements, the participant plane cannot be measured [START_REF] Ollitrault | [END_REF].

Flow harmonics

The azimuthal distribution of particles is anisotropic in reference to the reaction plane. It can be expressed by the Fourier decomposition, where each of the coefficients reports to the shape of the matter flow: v 1 -directed flow, v 2 -elliptic flow (see Fig. 2.7), v 3 -triangular flow [43]:

E d 3 N d 3 p = 1 2π d 2 p T dp T dy (1 + Σ ∞ n=1 2v n cos(n(ϕ -Ψ RP ))) (2.3)
where: N -number of particles, n -denotes the harmonic ϕ -azimuthal angle, Ψ P R -reaction plane angle.The sinus terms are absent as a result of symmetry with respect to the Ψ P R .

The flow harmonic coefficients v n are influenced by eccentricities ε n , fluctuations, system size, speed of sound c s (µ B , T ) and transport coefficient η s (µ B , T ). Investigation of these parameters gives unique information about the states of the matter obtained during the evolution of the system (more in Section 1.2.3).

Directed flow

The hot and dense matter is created in the overlap region of the collision, and it deflects the remaining nuclear matter (see Fig. 2.8 and 2.9). The spectators in the positive [56] rapidity part are deflected to +x direction, so ⟨p x ⟩ > 0 and oppositely the remnants from negative rapidity are pushed to -x with ⟨p x ⟩ < 0. This process occurs during the whole time of the collision of heavy-ions. Consequently, the deflection is the excellent probe of the compressibility of the new matter. At low energies such as obtained at AGS, the directed flow is the leading mechanism and nearly linearly depends on the rapidity. On the other hand, at higher energies (top RHIC and LHC), the linearity of directed flow is predicted to break; the so-called "wiggle" can be created -the directed flow in the function of rapidity switches the sign three times (Fig 2.8). It can originate from various physical mechanisms, such as the QGP phase's presence in the created system's evolution. It can be treated as a signature of the QGP phase transition [38,57]. STAR conducts extensive studies in the broad range of the collision energies and systems of the directed flow in the function of rapidity [58,59]. The most intriguing results are observed for protons and antiprotons. The significant differences between v 1 of these particles are present for all studied energies for Au+Au collisions, while in the case of pions, this dissimilarity is negligible. The increasing difference oppositely to the collision energy can be explained by the interplay between the v 1 of p related to the baryon number originating from the initial beam rapidity and transported close to the mid-rapidity region, and protons' v 1 coming from the particle-antiparticle pairs created in the vicinity of the mid-rapidity.

Elliptic flow

The second harmonic of the Fourier decomposition corresponds to the expanding matter's elliptical shapev 2 . It depends mainly on the initial geometry, so the size of the overlap region of the collided nuclei. However, the tiny v 2 signal is also observed for central collisions, which is explained by the fluctuation in the initial state of the occurring processes. The strength of the elliptic flow signal gives information about the differences between particle yields in the direction "in-plane" and "out-of-plane".

Triangular flow

In simple models, all odd ε n coefficients vanish because of symmetry ((ϕ)&(-ϕ) or (ϕ)&(ϕ + π)) [60], whereas in a realistic event-by-event treatment, the fluctuations break this symmetry. They can originate inter alias from (Fig. 2.11):

-the shape of the incoming nuclei -the interaction mechanism -distribution of binary collisions and the degree of formalization [61].

The fluctuations strongly affect the observables of the QGP. Wounded nucleons (participants) are indicated as solid circles, while spectators are dotted circles [62]. For the better visualization with the yellow color the collided ions are selected, with blue the ellipsoidal area corresponding to ϵ 2 and with red triangularϵ 3 .

The third coefficient of the Fourier decomposition of the flow arises from event-byevent fluctuations (Section 2.1.2). It leads to a triangular anisotropy in azimuthal particle production through the collective expansion of the medium. It is also very sensitive to the viscosity of expanding medium [62]. What is more, it has been proven not to be dependent on the size of the system (Fig. 2.12) [63]. The triangular flow clarifies the fluctuations and viscous effects, while the measurements of v 2 are dominated by other components (already mentioned initial eccentricities).

The study of v 3 in the collision energy function provides information about the changes of transport coefficients in various points on the phase diagram. As a consequence, it becomes a unique tool in the investigation of matter transitions. The so-called mass ordering is visible up to p T = 2 GeV/c. It means that the lighter particles exhibit higher elliptic flow for the given p T range than heavier protons. For the intermediate transverse momentum (i.e. 2.0 < p T < 5.0GeV/c) particles are dividing into two groups: mesons (quark-antiquark) and baryons (3 (anti)quarks) (see Fig. 2.10). It is explained by the interplay between elliptic and radial flow, which strongly affects the particle spectra and by the shift of the heavier particles to higher p T . Consequently, the more extensive depletion of spectra is achieved for the directed along with the symmetry plane values of p T . As the elliptic flow expresses the relative difference between yields of particles in-plane to the out-of-plane, so described processes reduce that flow. Finally, the mixture of these components makes the v 2 higher with decreasing mass at a lower p T . Consequently, the studies of the flow in the function of p T and particle mass are excellent probes in investigating the relations in the dynamics creation of the matter and EoS.

Beam energy dependence

The anisotropic flow originates from the eccentricity-driven hydrodynamic evolution of the system. Consequently, measurements of this quantity are sensitive to the initial-state geometry and EoS or transport properties of the medium. Accordingly, investigation of anisotropic flow is at the forefront of ongoing studies of T and µ B dependence η/s, which is substantial transport coefficients in search of the matter transitions. The BES program gives the possibility to perform such studies as a function of beam energy providing the unique opportunity for understanding the relations between these quantities.

In [64] authors suggest that the related with T and µ B , η/s could have a minimum at the CP vicinity. In Fig. 2.13 the good agreement of theoretical calculations of azimuthal flow from [65] with the experimental data is shown. In these computations, the η/s increases with the lowering of the collision energy but is constant for the whole process of the fireball evolution at given √ s N N . Accordingly, the studies of v n as a function of beam energy give substantial constraints into the theoretical studies of relations η/s(T, µ B )

In higher energy collisions, the partonic phase is longer and consequently, the hadronic contribution to the elliptic flow is reduced. Figure 2.14 shows the elliptic flow's dependence on the collision energy. The apparent continuous increase from RHIC to LHC energies is visible. In the case of much cooler systems (lower collision energy), the matter is more trapped inside the created bulk and squeezed-in what results in negative elliptic flow -the push in out-of-plane direction is more substantial than in-plane one.

Collision Centrality

The flow measurements as a function of the centrality give direct information about the relation between eccentricity and the anisotropy in the matter expansion. The collided system's initial geometry is influenced by the eccentricity and distribution of nuclei in the overlap region. The flow originates from the anisotropies in initial geometry, so it should proportionally increase with the eccentricity. As it is directly related to the collision's centrality, the v n is expected to be proportional to the number of charged hadrons N ch . However, as mentioned before, the QGP is not a perfect fluid, but the viscous one. The effects of the viscosity dominate the low multiplicity events, which result in lowering the flow. While in the most central collisions, the created flow is mainly dominated by the eccentricity, so the N ch as illustrated in

Coalescence and NCQ-scaling

Particles can be created in the process of coalescence 4 of some primordial particles. Their distributions are expected to reflect the distributions of the primordial ones. If this process does not have relevant influence on the distributions of original particles, basing on the coalescence probability B A the following formula can be applied for the nucleon 4. Coalescence in high energy physics is the process where two or more particles merge during contact to create a single daughter particle [68]. [67] coalescence into light nuclei with A as an atomic number [38]:

E A d 3 n A dp A = B A E p d 3 n p d 3 p p A (2.4)
If one focuses on the momentum distribution and neglect the possibility of the anisotropy in the spatial distribution, they can conclude that the nuclei A distribution 2.4 gets less uniform. Consequently, there can be found, such flow scaling relation:

v n,A (p T,A ≈ Av n,p (p T,A /A) (2.5)
A few collaborations like E877 [69] (at AGS), or PHENIX [70] and STAR [71] (at RHIC) made an observation that deuterons' flow are indeed consistent with the coalescence model predictions. In [38] author states that the constituent quark coalescence is relevant in the process of production of particle at the intermediate p T .

Experimental studies were performed at RHIC [72,73] and were treated as proof of the QGP phase's presence. Nowadays, many physicists argue about that. STAR showed in [73] that at the intermediate p T (∼ 2 -4 GeV/c) the significant scaling with the number of constituent quarks (NCQ) is kept for both mesons and baryons. In the smaller scale by similarity to the light nucleons, it is assumed that if the hadrons are created via coalescence of constituent quarks, their particle yields would be proportional to the quark density to the power equivalent to the Number of Constituent Quarks (NCQ) forming the given hadron. As a result of the intensified production of particles in this intermediate p T region, one can conclude that the elliptic flow can also be scaled with NCQ: v 2 (p T ) ≈ nv 2 (p T /n) (where n correspond to ncq of baryon n = 3 or mesons n = 2) [74]. If the probability of the coalescence is small such scaling equations can be applied for the identified hadrons (first one for mesons, second for baryons):

d 3 n M d 3 p M ∝ [ d 3 n q d 3 p q (p q ≈ p M /2)] 2 and d 3 n B d 3 p B ∝ [ d 3 n q d 3 p q (p q ≈ p B /3)] 3 (2.6)
where: n q correspond to the given quark, and p q to its momenta. Indexes M -meson, and B -baryon. Although the violation of scaling can be present for lower p T , it does not guarantee that the hadrons are not formed by the process of coalescence in this region.

It can be caused by the break down of the formulas 2.6 describing the mechanism of coalescence. In 11.5, 14.5, 19.6, 27, and 39 GeV. This was complemented by earlier 62.4, 130 and200 GeV and later 54.4 GeV collected data sets [76]. There are four physical goals:

-search for the vanish of the signatures of the QGP -search for the possible first-order phase transition -search for the possible CP -investigate the transport properties of the strongly interacting matter as a function of T and µ B [77][78][79][80].

The important observations made in this program's frame are related to the anisotropies in flow of the matter. Several of them are described and discussed in this Section.

In Figure 2.16 the v 2 dependence on p T measured using four-particle cumulant method is shown for three centrality ranges and five collision energies (from √ s N N = 7.7 GeV -2.76 TeV). The differences in v 2 signal for the inclusive charged hadrons between all the examined collision energies are little. It is striking because the ranges of energies (changes by a factor of 400) and initial energy densities (factor ∼ 10) are wide.

Another observation done by STAR is illustrated in Fig. 2.17. It shows the ncq-scaling of v 2 vs. (m T -m 0 )/n q for identified hadrons in 0 -80% centrality range of Au+Au collisions at √ s N N = 7.7 -62.4 GeV [66]. The scaling holds within 10% -15% for most of the particles and antiparticles. Due to the lack of the required statistic, it cannot be explained why the ϕ meson does not follow the trend at lower examined energies. Using the hybrid model calculation, the weak dependence of v 2 (p T ) on the collision energy is understood as a consequence of the interplay of the hydrodynamic and hadronic transport phase [81].

As the v 3 is more sensitive to the viscous damping, it is treated as a crucial observable in probing the QGP and pressure gradients formation in the early phase of plasma. In Fig. 2.18 the variation of the v 2 3 2 for the charged hadrons in several centrality bins at various collision energies: from 7.7 GeV to 2.76 TeV. At energies below 14.5 GeV, v 2 3 2 is nearly consistent with zero at more peripheral collisions. This observation agrees with the assumption that in the low-energy, peripheral collisions, the low viscosity QGP phase is Figure 2.16 -The variation of v 2 (p T ) for charged hadrons produced at mid-rapidity from 7.7 GeV up to 2.76 TeV [66,77] Figure 2.17 -Quark-number (n q ) scaled elliptic flow, v 2 /n q as a function of (m Tm 0 )/n q , for identified particles produced in 0 -80% central Au+Au collisions at √ s N N = 7.7 -62.4 GeV, where n q is the constituent valence quark number of each hadron and (m T -m 0 ) is the transverse kinetic energy of the particle [73,77] Figure 2.18 -The variation of v 2 3 {2} for charged hadrons produced at mid-rapidity from 7.7 GeV up to 2.76 TeV [66] for different bins in collision centrality [77,82] absent [81]. In more central collisions, the v 2 3 2 is positive and does not change relevantly between collision energies < 20 GeV. For the higher energies, the measured triangular flow increases linearly with the log( √ s N N ) for the all examined centrality ranges [66].

The left panel of Figure 2.19 shows p T -integrated v n for 0 -40% central Au+Au collisions as a function of √ s N N [83]. The essentially monotonic with beam energy trend for v 2,3,4 was observed. It is expected since the temperature increase with the √ s N N .

However, the viscous coefficient ′′ [83] shows non-monotonic behaviour over the same beam energy range (right panel of Fig. 2.19). A similar non-monotonic trend is predicted by the hybrid viscous hydrodynamical calculations in [65].

The results of the BES I were so promising that the second phase of this program was planned (data taking in 2019-2020) [76]. The goals are similar but extended with the study of chiral symmetry restoration. BES II provides a substantial improvement in the data collection. The range of the gold ions beams energy is broadened down to √ s N N = 3.0 GeV by the successful implementation of the fixed-target (FXT) at the entrance of the STAR TPC. The collected data sets have one-order-of-magnitude more extensive statistics in collider mode and two-orders-of-magnitude higher statistics in FXT mode. BES II extends the STAR experiment's reach across a crucial energy regime of high baryon densities:

µ B ∼ 200 -720 MeV.
The more technical aspects, such as the introduction of the FXT [77,83] are described in Chapter 3.

Particle -antiparticle flow difference

As a part of the BES program, the different behaviour between particles and their anti-equivalents in elliptic flow analysis has been reported [59,84,85]. The substantial increase of v 2 ratio baryon -anti-baryon with the decrease of collision energy was observed. There are some indications that the growing contributions from hadronic interactions in the system evolution or the stopping power with decreasing energy of collision can be responsible for the differences between v 2 (X) and v 2 ( X) [86,87]. The discrepancies in the case of mesons (π and K) were suggested in [88] to be a consequence of the chiral magnetic effect induced by the strong magnetic field in non-central collisions.

Scenario 1: "Viscous Corrections"

In [89] authors propose that the new viscous corrections to v n at finite µ B obtained by solving the equations of viscous hydrodynamics coupled with conserved currents assuming conformal and boost-invariant symmetries. They are enhanced at higher baryon density and give the leading order contribution to the differences in v n between particles and

δv n ϵ n = K 256 Γ(3n) Γ(4n) 128 B 3 n Γ 2 n 2 n 3 (n -1) 3n -1 - 27 4 (3n 2 +3n+2)+9γ 3n 2 + 1 k - 3f ′ 4f
(2.7) where: K is the Knudsen number proportional to the η s , B is the parameter related to the (position-dependent) freeze-out time τ f , δf is the deviation from the equilibrium Boltzmann distribution. Concluding, the applied viscous corrections to the perfect fluid hydrodynamics explain the differences between v n of particles and antiparticles. Also, the correction are dependent on the µ B so consequently they explain the relation of differences with the energy of collision. Authors performed simulations using the corrections -the results are illustrated in Figure 2.21.

Scenario 2: "Mean field theory"

In [90], the researchers suggest that studied differences can be explained by the mean field theory assumptions. Attractive potential makes particles trapped in the system and Comparison of STAR published data [85] and simulations using the procedures proposed in [89]. pushed more in the direction perpendicular to the participant plane. On the other hand, the repulsive potential causes the motion out of the system in the direction along the plane. Protons (three quarks) are affected by the repulsive potential, while antiprotons (q q q) by attractive. Consequently, the first ones have a stronger push along the participant plane, making their elliptic flow relevantly higher than antiprotons trapped in the system. The described effects are more pronounced in lower energy collisions, which could explain the dependence of the observed differences in flow on the √ s N N .

Scenario 3: "Transported protons"

The another theoretical scenario [91,92] is based on the interplay between so-called transported and produced protons. Authors distinguish quarks that build the hadrons from collided ions -transported, and these created during the evolution of the system -produced. In the Figure 2.22 the different possible scenarios of the particle production using UrQMD [93,94] are presented. The produced protons p(0iq) (notation presented in Fig 2 .22)and antiprotons are expected to behave similarly; they are both created in the early stages of the system evolution characterised by the relatively high energy density in lower energy collisions. The transported protons p(3iq) are transferred from forward rapidity Figure 2.22 -Particle production in a string-excitation scheme in UrQMD. White circles, black circles and grey circles stand for initial quarks, produced quarks and produced antiquarks, respectively. In case [START_REF] Stefaniak | [END_REF], no quark pair is created. In case ( 2), one diquarkantidiquark pair and one quark-antiquark pair are spontaneously created in the colour flux-tube between the initial quark and the initial diquark. Similarly, in case ( 3), two quark-antiquark pairs and one diquark-antidiquark pair are created, and one meson is produced [91] to mid-rapidity. As a consequence of the effect of nuclear stopping, which is stronger for lower energies, their elliptic flow variate from v 2 of p(0iq) and p. Moreover, transported protons' dynamics is affected by the process of the system evolution -they are present when the anisotropy of the initial geometry of the overlap region is being transformed into the anisotropy in expanding the matter. In contrast, the produced protons can only experience a part of these processes. The elliptic flow of p(3iq) is expected to be higher than the flow of produced protons and anti-equivalents. The increasing power of nuclear stopping with the decrease of the collision energy explains the dependence of the observations on the √ s N N . Authors made simulations using UrQMD, and the results of the v 2 (p T ) are shown in Fig. 2.23, where transported protons flow is substantially higher than produced ones and the differences increase with the decrease of the collision energy.

As many proposed theoretical scenarios explain the observed differences in particles' elliptic flow and antiparticles, more experimental references are needed to verify and extend the theoretical suggestions.

Analysis method

In the flow analysis, the two-particle cumulants method is used. It is derived from the following calculations.

Direct cumulants

The flow measurements can be performed using equation:

v n = ⟨cos[n(ϕ -Ψ RP )]⟩, (2.8)
so the coefficients depend on the values of transverse momentum (p T ) and rapidity (y)differential flow. The integrated flow is determined by the weighted average (the weight is the invariant distribution, and the quantities with corresponding symbols are shown in Fig. 2.24) [53]:

v n ≡ ∞ 0 v n (p T ) dN dp T dpT ∞ 0 dN dp T dpT (2.9)
As mentioned in Section 1.3.2, the reaction plane Ψ R cannot be measured experi-mentally. However, it is possible to study flow using the azimuthal correlations between detected particles. The 2-particle cumulants (2PC) can be described by a Fourier decomposition 1 + 2v 2 n cos n∆ϕ. The v n corresponds to harmonics of the single-particle anisotropy. The measurements are strongly affected by the contribution of the so-called short-range non-flow correlations [95][96][97]. The multi-particle cumulants method cannot completely remove them, but it can significantly suppress them.

The Fourier expansion of the distribution of the probability P (ϕ) characterizes the particle production azimuthal not-uniformity:

P (ϕ) = 1 2π ∞ n=-∞
v n e inϕ , v n = v n e inΨ (2.10)with: v n corresponding to magnitude and Ψ to phase. In the heavy-ion collisions the event-by-event flow fluctuations are present and can be characterized by a probability distribution p(v n ), which shape can be similar to Gaussian or integrated out the angle ϕ Bessel-Gausssian v n distribution:

p( v n )= 1 2πδ 2 n e -|vn-v 0 n | 2 /(2δ 2 n ) ,p(vn)= vn δ 2 n e - (vn) 2 +(v 0 n ) 2 2δ 2 n I 0 v 0 n vn δ 2 n (2.11)
where I 0 is modifued Bessel function, andv 0 n is the driven by the average overlap region geometry component (for n = 2, v 0 n is sizable). As a consequence of the finite multiplicity M of the event, using the flow vector Q n and normalized by per-particle vector q n becomes a convenient method. They can be expressed by following: (2.12)where sum is over all particles in the event, and ϕ, Ψ n are illustrated in the Fig. 2.24. Vector's q n direction and magnitude are not same as the real flow, it consists of two additional ingredients:

Q n ≡ i e inϕ i = Q n e inΨn , q n =≡ i e inϕ i M = q n e inΨn
q n = v n + s stat n + s n (2.13)
where s stat n are fluctuations related to finite multiplicity of particles and s n is the non-flow contribution originating from short-range correlations. Both of them variate from event to event. The first one becomes negligible as a consequence of averaging over large number of Figure 2.24 -Schematic illustration of a non-central ion collision, where red circles correspond to newly-formed particle 1 and 2. This thesis events or because of introducing multi-particle correlation method of analysis. Cancelling of the second term is more challenging, there are numerous sources of the short-range correlations, such as: HBT5 , resonance decays or dijets.

The single-event average 2-particles azimuthal correlations are defined with given expression [53]:

⟨2⟩ = ⟨e in(ϕ 2i-1 -ϕ 2i ) ⟩ (2.14)
Following, using double brackets denoting the average over all selected events:

⟨⟨2⟩⟩ = ⟨⟨e in(ϕ 2j-1 -ϕ 2j ) ⟩⟩ ≡ events (W ⟨2⟩ ) i ⟨2⟩ i events (W ⟨2⟩ ) i (2.15)
To minimise the influence of the multiplicity variations, the weight of the given event W ⟨2⟩ are introduced. It is equal to the number of possible combinations of pairs of particles in the event of the multiplicity M :

W ⟨2⟩ = M (M -1) (2.16)
The cumulat c n {2} is equal to the averaged 2-particle azimuthal correlations (eq. 2.15). This leads to the following cumulant-based definition of harmonic flow v n :

v n {2} = c n {2}
(2.17)

Reference Flow

The calculation of the second order cumulant requires the separation of the diagonal and off-diagonal contributions of |Q n | 2 , what can be simply applied with [53]:

⟨2⟩ = |Q n | 2 -M M (M -1) (2.18)
Then the averaging over all events is performed using eq. 2.15, and the reference flow with eq. 2.17.

Differential flow

In calculations of differential flow the two labels describing particles are introduced [53]:

-Reference Flow Particle -RFP -Particle Of Interest -POI Firstly, the reference flow is calculated only for RFP. Secondly, the differential flow of POI is estimated, considering the RFP flow from the previous step as a reference.

Reduced 2-particle azimuthal correlations

The reduced 2-particle azimuthal correlations are described with following equations:

⟨2 ′ ⟩ ≡ ⟨e in(ψ 1 -ϕ 2 ) ⟩ ≡ 1 m p M -m q mp i=1 M j=1 ′ e in(ψ 1 -ϕ 2 )
(2. 19) where: m p corresponds to the total number of POI, m q to the number of both POI and RFP and M is the multiplicity of RFP. The ψ i and ϕ j are the azimuthal angles of i-th and j-th particles labeled accordingly PIO and RFP. The sum ′ is over all different indices. The reduced, averaged over all events 2PC are defined as:

⟨⟨2 ′ ⟩⟩ =≡ events (w ⟨2 ′ ⟩ ) i ⟨2 ′ ⟩ i events (w ⟨2 ′ ⟩ ) i (2.20)
where the weights of events are calculated as:

w ⟨2 ′ ⟩ = m p M -m q (2.21)

Differential cumulants

The two p-and q-vectors are introduced into the differential cumulats calculations (this q vector does not correspond to the same vector as in eq. 2.13). First one correspond to all m p POI. In order to reduce the impact of the autocorrelations, the second vector is formed and it corresponds only to the m q POI which are also labeled as RFP:

p n ≡ mp i=1 e inψi
(2.22)

q n ≡ mq i=1 e inψi (2.23)
Thus, the average reduced single and all-event 2PC can be expressed with:

⟨2 ′ ⟩ = p n Q * n -m q m p M -m q (2.24) ⟨⟨2 ′ ⟩⟩ = N i=1 (w ⟨2 ′ ⟩ ) i ⟨2 ′ ⟩ i N i=1 (w ⟨2 ′ ⟩ ) i (2.25)
If the detectors used in collecting the data have the uniform azimuthal acceptance, the differential second order cumulant is defined as:

d n {2} = ⟨⟨2 ′ ⟩⟩ (2.26)
Finally, the estimation of the differential flow v ′ n {2} is equivalent to:

v ′ n {2} = d n {2} c n {2}
(2.27)

Particle weights

As the detection of the dynamics of the particles is not ideal (more in Chapter 3), the calculations of 2PC using the real, experimental data have to be modified. The weighted Q-vector is introduced [53]:

Q n,k ≡ M i=1 w k i e inϕ i (2.28)
w i is the weight of the i-th particle and M is the multiplicity of RFP in the event. In case of two-particle azimuthal correlations the k = 2. Accordingly:

p n,k ≡ mp i=1 w k i e inψi (2.29)
In case of particles which are labeled only POI the w i = 1, while for the particles labeled as RFP have non-unit weight. In situation when the particle is labeled simulatanously POI and RFP (in total there are m q partciles), the q-vector is introduced:

q n,k ≡ mq i=1 w k i e inψi (2.30)
Then, for RFPs the next step of calculations is: .32) In case of particles labeled both POI and RFP:

S p,k ≡ M i=1 w k i p (2.31) M abcd... ≡ M i,j,k...=1 w a i w b j w c k ... ( 2
s p,k ≡ mq i=1 w k i p (2.33)
And anagously: .34) where the first sum is over all POIs and second over RFPs. The event weight is here:

M ′ abcd... ≡ mq i=1 ′M i,j,k...=1 w a i w b j w c k ... ( 2
W ⟨2⟩ ≡ M 11 (2.35)
so that the weighted single-event 2PC:

⟨2⟩ ≡ 1 M 11 ′M i,j=1
w i w j e in(ϕ i -ϕ j ) (2.36)

Continuing, the reduced single-event 2PC here are:

⟨2 ′ ⟩ ≡ 1 M ′ 01 mp i=1 ′M j=1 w j e in(ϕ i -ϕ j ) (2.37)
where the event weights are now:

w ⟨2 ′ ⟩ ≡ M ′ 01 (2.38)
The weighted average 2PC are defined as:

⟨2⟩ = |Q n,1 | 2 -S 1,2 S 2,1 -S 1,2 ⟨⟨2⟩⟩ = N i=1 (M 11 ) i ⟨2⟩ i N i=1 (M 11 ) i )M 11 ≡ i,j=1 ′ w i w j = S 2,1 -S 1,2 (2.39)
Finally, the 2-particle azimuthal correlations are expressed with the following equations:

⟨2 ′ ⟩ = p n,0 Q * n,1 -s 1,1 m p S 1,1 -s 1,1 ⟨⟨2 ′ ⟩⟩ = N i=1 (M ′ 01 ) i ⟨2 ′ ⟩ i N i=1 (M ′ 01 ) i M ′ 01 ≡ mp i=1 M j=1 ′ w j = m p S 1,1 -s 1,1 (2.40)

Subevent cumulants

The main goal of introduction of the Subevent cumulat method is the reduction of the non-flow contribution [53,59,99,100]. This method's general picture is illustrated in Fig. 2.25. The two vectors A and B corresponding to particles from a given pseudorapidity range ensure no self-correlations are taken into account. The average distance (in η) between correlated particles is higher than zero, which provides the reduction of the shortrange non-flow contribution originating from, for example, HBT (more in Sec. 2.1.1).

Each event is divided into subevent A and B, which consists of accordingly M a and M b particles. The two-particle correlator which considers event-by-event weights can be described using flow vectors:

⟨2⟩ a|b ≡ ⟨e in(ϕ a 1 -ϕ b 2 ) ⟩ = Q n,a Q * n,b M a M b (2.41)
where :

Q n,o ≡ Σ i e inϕ o i , o = a, b (2.42)
Particles in subevent A are correlated with particles from subevent B, ensuring no selfcorrelations in calculations.

The ∆η gap between subevents

As mentioned before and underlined in equation 2.13 the non-flow contribution is measured using described above methods. The more non-flow is suppressed, the more detailed studies of the anisotropies in particles' yields are. The short-range correlations are present for particles close in the pseudorapidity. To suppress their contribution in the flow measurements, the η gap between subevents is introduced. In this chapter the STAR experimental setup is described.

Relativistic Heavy-Ion Collider RHIC

In the 2000 year, RHIC, one of the biggest accelerators, was built in Brookhaven National Laboratory (BNL) in New York State [110][111][112]. The primary goal was to accelerate gold ions and collide them with the energy √ s N N = 200 GeV. Then the physics program called Beam Energy Scan (BES I) conducted at RHIC became more diverse, and new collision systems and energies were applied to explore a wider range of the QCD phase diagram. There were performed studies of collisions obtain at energies between √ s N N = 7.7 -200 GeV for systems including: Au (max 100 GeV/nucleon) , U, Ru, Al, d, BES II [76]. The statistics of performed collisions is increased and more collision energies (such as √ s N N = 54.4 GeV) are provided. However, there is also, an energy bottom limit of the accelerator performance in the collider system. To decrease the collision energy, the golden shield was located inside the STAR experiment replacing one of the beams.

The new Fix Target program allows studying even more diverse phase diagram regionscharacterized by lower temperature and higher baryochemical potential (µ B ≈ 720 MeV), where Au+Au collisions happen with √ s N N = 3.0 GeV.

RHIC consists of the two separate accumulation rings: "Blue" and "Yellow", each of length 3.8 km. There are 1740 superconducting magnets (dipoles, quadrupoles, sextupoles and corrector magnets) used in RHIC to provide high-precision collimation and maintenance of the beam trajectory. They are made of niobium and titanium alloy processing at 4.6 K, and in the arc, dipoles generating the 3.45 T magnetic field [114]. To stabilize the polarized-proton spin, the Siberian Snake devices are used [115].

In Figure 3.2 the whole structure of the acceleration complex is illustrated [116].

In order to prepare the proton beam, several stages can be listed: 1) Firstly, in the Optically Pumped Polarized Ion Source (OPPIS) [117] the polarized hydrogen ions are created, The preparation of heavy-ion beams can be described in such steps as: 1) It starts in Laser Ion Source (LION), where they are produced.

2) Next, they are transported to an Electron Beam Ion Source (EBIS) where after acceleration they achieve the energy of 17 keV/nucleon. 3) Their charge is also multiplied (+32 in case of Au) [118]. 4) The subsequent acceleration is performed in RFQ and Linac (up to 2 MeV/nucleon). 5) then in Booster (100 MeV/nucleon), where they are also stripped changing their charge (for Au +77). 6) The nominal charge is established in AGS (+79 for Au) where the ions continue being accelerated, reaching the energy of 8.87 GeV/nucleon. 7) In the end, the ion beams are injected into the RHIC, where they achieve the nominal energy.

Several experiments consisting of complexes of detectors are constructed at the RHIC. However, only one is currently operational -STAR (Solenoid Tracker at RHIC, see Subsection 3.1.1). PHOBOS [119]

STAR

Solenoid Tracker at RHIC was designed and constructed to study the properties of the QGP created in the heavy-ion collisions and investigate the data from the polarized proton collisions to explore cold-QCD. It is multi-purpose, mid-rapidity detector, which consists of over ten various subsystems used to investigate multiple observables, the structure is illustrated in Fig. 3 As the TPC, iTPC, TOF, and VPD detectors are the most vital for this analysis, they are described in details in the subsequent sections.

Time Projection Chamber

Time Projection Chamber (TPC) is one of the most relevant STAR detectors used in nearly all performed analysis. It is 4.2 m long, and its diameter is equal to 4 m. The Figure 3.5 shows the scheme of TPC [135]. The chamber is filled with the P10 mixture of gases: 90% argon Ar and 10% methane CH 4 . It is designed to register particles' tracks (normally it reconstructs the multiplicities equal more than 3000 tracks) and measure their momenta (in range of 100 MeV/c to 30 GeV/c). It covers the full azimuthal angle (0 < ϕ < 2π) and ±1.8 units of pseudo-rapidity (-1 < η < 1). The basic parameters of TPC are listed in Table 3.6.

The inner filed cage is located 0.5 m from the centre of the beam pipe, while the outer one in a distance of 2 m. The operating at a high voltage of 28 kV central membranes (CM) provides the uniform electric field of 135 V/cm. It is placed at the centre of TPC (z = 0). The solenoidal magnet gives the magnetic field of 0.5 T. Both fields are directed along the beam. The collision occurs close to the centre of the TPC. The primary charged where: 2πN a r 2 m e c 2 = 0.1535 MeV cm 2 /g and γ

= 1/ √ 1 -β 2 = 1/ 1 -(v/c) 2 .
N a : Avogadro's number r e : classical electron radius m e : electron mass ρ: density of absorbing material Z: atomic number of absorbing material A: atomic weight of absorbing material z: charge of incident particle [e] W max : maximum energy transfer in single collision

I: mean excitation potential

The calculations of the loss of the energy are performed in the function of the momentum transfer. Various species of particles having the same momentum, loose in the different way their ionization energy (dE/dx). In Figure 3.8 the example of dE/dx distribution in function of particles' momentum at √ s N N = 39 GeV with the fitted by the Bichsel function lines [138]. They correspond to given particle specie. It is visible that the identification with TPC of protons up to p < 1 GeV/c is clear. For the studies of the higher momentum range, the Time Of Flight detector is used.

Time of Flight

Time of Flight is the fast-timing detector, which main goal is to register the time of the motion of the particle. It is dedicated for the identification of the particles with momentum p ≥ 1 GeV/c. The TOF detector is placed outside of the TPC. The geometry of TOF's trays, modules and pads are illustrated in Figure 3.9. On each side of TPC (east and west), 60 trays cover -1 < η < 1 of pseudo-rapidity and full azimuthal angles. The tray is build of 32 Multi-gap Resistive Plate Chamber (MRPC) modules illustrated in Figure 3.10.

In the construction of MRPC the two electrodes with a voltage 7 kV and a stack of resistive glass plates are used. There are six uniform gas gaps between plates, which are filled with the uniform, high electric fields. The MRPC proceeds in the avalanche mode: when the particle transverse the module, they cause the simultaneous avalanches in the gas gaps. The created signal is the superposition of the avalanches. The distance (L) between TOF and the vertex of the collision is know, the speed (β) and mass (m) of particles can be calculated using following equations:

β = v c = L c∆t (3.2)

Vertex Position Detector

Vertex Position Detector (VPD) measures the position of the primary collision vertex and provides the "start-time" for the fast-timing detectors (TOF or MTD). VPD is the 2x19 channel detector, before the pseudo Vertex Position Detector (pVPD) consisted only of 2x3 channels. It is used as a minimum-bias trigger for Au+Au collisions.

VPD consists of the pair of detectors located symmetrically in the distance 5.7 m from the centre STAR, on the beam pipe on either side of the detectors' complex. The construction of VPD reminds the revolver -the 19 photo-multiplier tubes (PMTs) surround the beam-pipe, illustrated in Fig. 3.12. It registers the neutral remnants from the collided ions (spectators) and produced from π 0 photons. For the determination of the location of the collision vertex (Z vtx ) the measured times at east (T east ) and west (T west ) sides are used:

Z vtx = c(T east -T west )/2 (3.4)
Thus, the start-time for other detectors can be calculated with:

T start = (T east + T west )/2 -L/c (3.5)
where c is the speed of the light and L is the distance between the center of the STAR and assembly.

Chapter 4

ANALYSIS

The data's selections, events, tracks are described in this chapter, and the particles identification and applied corrections to the experimental analysis.

Data samples

The studies are performed on the data recorded by STAR experiment during Au+Au collisions at √ s N N = 27, 39, 54.4 and 200 GeV. The data sets are selected using the STAR software and subsequently stored in PicoDst root files [139]. PicoDst files contain only the most relevant to the analysis pieces of information. The main goals of usage of this type of files are:

-Have relevantly smaller size than previously used by STAR MuDst files -Contain most of the essential information or most of the physics analysis -Store data in plain ROOT -Allow working in STAR-independent environment -Compatible with ROOT5/6, Windows, Linux, macOS [140].

The events collected during the beam or detectors' calibration tests are rejected and when any miscalibration is observed. The example luminosity of the events registered in the run 17 by STAR for Au+Au collisions at √ s N N = 54.4 GeV are presented in Figure 4.1.

In the performed analysis, there were chosen three sets of produced data listed in Table 4.1. Several selection criteria were applied to remove events of low quality and/or suppress background.

Event selection

For the best quality event selection, the pileup (average number of collisions per bunch crossing) removal techniques were applied. The TPC detector's measured quantities had to be validated with corresponding measurements performed by fast-timing detectors such as TOF or VPD to avoid the pileup events. Firstly, the multiplicities of events detected by TPC (Refmult) are compared with the related distribution of these registered by TOF (TOFmatch). In Figure 4.2 the plots of the relation between TOFmatch and Refmult before and after cuts are presented. The green curve in the right-hand side plot corresponds to the difference between distributions of the Refmult before and after the corrections.

Secondly, the Vertex Z relation using data registered by TPC and VPD detectors is investigated (Fig. 4.3). The events selected in the final analysis have to be characterized by the exact value of V Z confirmed by both detectors.

Track selection

The most important selection criteria are listed in Table 4.3. The application of stricter limits can highly decrease the statistics so that there has to be a compromise between the quality of the selected data and the sufficient amount of data for the given analysis.

The Distance of Closest Approach (DCA) corresponds to the shortest distance between the particle's track's main vertex and the point where it was registered. If the value of DCA is close to 0 cm, the probability that the particle is primary does not originate from the resonance decay, is high. The bigger DCA, the further the primary vertex of the particle from the collision vertex. The condition of the minimum number of hits fitted with the one curve used to reconstruction the track of particle (nHitsFit) is applied to reject fake tracks.

The distributions of variables are illustrated in

Particle Identification

The identification of pions, kaons, and protons is accomplished by combining TPC and TOF detectors information. For the TPC data, the N σ method is applied, where σ is the standard deviation from the Bethe-Bloch parameterization of the energy loss signal dE/dx. Figure 4.6 shows the average dE/dx of measured charged particles by TPC plotted as a function of "rigidity" (i.e. momentum/charge) of the particles. The TOF identification is based on tracking inverse particle's velocity in a unit of the speed of light 1/β, which depends on the mass square of the given particle (Figure 4.7). Finally, the whole p T range is divided into bins, and the combination of these methods is applied for each bin accordingly. The selection of these criteria was based on the previous STAR

Efficiency and Acceptance corrections

The construction and calibration of the detectors are performed with high precision. However, there are present some imperfections in acceptance and reconstruction efficiency. Due to such inadequacies, some of the particles are lost in collecting the data.

Acceptance correction

The STAR detectors do not cover perfectly 100% of acceptance -particles are not reconstructed, for example, in the x -y plane along the gaps between the TPC sectors, or some of the particles are missing due to the possible failure in the reconstruction software. Several corrections are applied to the analysis of the already collected data to increase the investigated data quality.

In the proposed method, one has to create the histograms, including the particles' distributions in the η, ϕ plane. The histograms are divided into multiple groups depending on: -the Z vertex of the event -the centrality of the event Each cell refers to the particle's 1/weight value, which is applied in the final analysis. The Figure 4.8 illustrates the examples distributions of particles' η, ϕ, which are not uniform. Assigning the proper weight to the particle results in this distribution's flattening, which is treated as an acceptance correction.

The particles' ϕ distribution for the Au+Au collisions at √ s N N = 39 GeV for all centrality ranges before and after acceptance correction are shown in Figure 4.9. The application of particles' weights makes the distribution flat.

TPC Tracking Efficiency

Another issue is the imperfection of the tracking efficiency. For instance, particles with very low p T could not be taken into account by the reconstruction software. The method used to correct such effects is based on the embedding simulated events. They are propagated through the Starsim algorithms, which provides the same conditions as in the Figure 4.8 -The ϕ, η particle distribution for Au+Au collisions at √ s N N = 39 GeV. This thesis detector geometry and reconstruction procedure. The efficiency of the given detector (in this case, TPC) corresponds to the probability of detecting the given particle specie. It can be expressed as a ratio of "matched" tracks (a reconstructed track that matches one of the pre-reconstructed simulated tracks) to the numbers of embedded MC tracks.

The impact of the application of the TPC Tracking Efficiency as an additional to the Acceptance correction was investigated. In Figures 4.10 the v 2 {2} in function of p T (as the efficiency correction strongly depends on transverse momentum) for Au+Au collisions at √ s N N = 39 GeV is shown. In the bottom panels there are ratios between data corrected with efficiency+acceptance/acceptance. The parameters of the fitted function used for the correction can be found in [142].

The effect of the correction is small ∼ 1%. In conclusion, the TPC Tracking Efficiency correction has such a negligible effect that it is not applied in the analysis as an addition to the acceptance correction.

Systematic uncertainty analysis

As listed in previous sections, several data selection criteria are applied to prepare the data sets for analysis. The removal/suppression of undesired effects in the reconstruction of particles' characteristics strongly limits the statistics, and in various studies, the applied GeV. This thesis criteria differ. In order to consider all possible discrepancies between available, reasonable criteria application, systematic uncertainties are introduced.

In the performed analysis, there are four sources of systematic uncertainty studied: a) maximum value of DCA: The whole analysis was performed twice for: one default DCA event cut and the second value of default DCA±1 cm. b) events characterized only positive or both positive and negative V Z : The events chosen for the analysis had only V Z , or default their V Z was in range -V Z to +V Z . c) introduced ∆η:

The selection of the minimum difference between particles' η is not trivial. With that the part of non-flow contribution is compressed, but also it can significantly decrease the statistics. Sometimes, the selection of this criteria is treated as additional study. In case of this thesis it is considered as the systematic uncertainty. d) minimum number of hits in the reconstructed particle's track: Similarly, to the DCA selection criteria, in the systematic untertainty analysis, the two separate studies of flow was performed for default and slightly changed nHitsFit.

Other sources of systematic uncertainties were negligible (like originating from particle identification) in comparison to those listed above.

The example plots of v n {2}(p T ) for different data selection criteria are shown in Figures 4.11 and 4.12.

The final systematic uncertainty is calculated as weighted average of the deviation from the ratio equal unity:

σ sys = N i ratio i * 1/σ i,stat N i 1/σ i,stat (4.1)
Where: σ sys is the systematic error, N is the number of points in ratio, σ i,stat is the statistical uncertainty of given ratio point. The point's weight is inversely proportional to the σ i,stat of it. The Tables 4.4 and 4.5 consist of percentage contribution of each of source of the systematic uncertainties for v 2 and v 3 measurements.

The rounded sums of listed systematical uncertainty contributions are included in

BEAM ENERGY DEPENDENCE OF ELLIPTIC AND TRINGULAR FLOW

This is one of two most substantial chapters of this thesis. It consists the main results of the studies performed in frame of this PhD project. In this chapter, the studies of azimuthal anisotropies in newly-created matter's dynamics using experimental STAR data is described. Firstly, the comparisons with already published by STAR data are shown. Then, the flow studies for Au+Au at √ s N N = 200 GeV are reported. Subsequently, the measurements for low beam energies are shown and discussed in two options: integrated and p T -differential.

Comparison with published STAR results

The comparison with already published results for elliptic flow was performed. The Collaboration had already obtained results for v 2 of identified hadrons for various centralities [73]. As the purely technical aspects of different flow harmonics' calculations do not differ relevantly, such comparison could give information about possible mistakes in performed computations. In Figures 5.1,5.2,and 5.3

the comparisons between various centralities for Au+Au collisions at

√ s N N = 39 GeV are shown. In the bottom panels, there are ratios between data from this work and already published by STAR.

Both data sets are in a reasonable agreement. The methods used in both studies are different: in published data authors used the Event Plane method [73], while in this work the 2PC one. Another source of possible discrepancies is different binning of two data sets. Consequently, the ratio is between data obtained in this analysis and the fit to the published data. That is why the data sets are not expected to overlap, but the difference up to 10% is considered acceptable. [73] for Au+Au at √ s N N = 39 GeV at 0 -10% centrality. In the bottom panels there is a ratio between this work to fit to the STAR published data, where the dotted lines correspond to the 5% difference. This thesis 106 Plane method [73] for Au+Au at √ s N N = 39 GeV at 10% -40% centrality. In the bottom panels there is a ratio between this work to fit to the STAR published data, where the dotted lines correspond to the 5% difference. This thesis Plane method [73] for Au+Au at √ s N N = 39 GeV at 40% -80% centrality. In the bottom panels there is a ratio between this work to fit to the STAR published data, where the dotted lines correspond to the 5% difference. This thesis 108

Flow of identified hadrons for Au+Au collisions at

√ s N N = 200 GeV

Flow of identified hadrons for Au+Au collisions at √ s N N = 200 GeV

The top RHIC energy is the baseline for the beam energy studies of the flow harmonics at STAR. This section consists of the first results of the identified hadron v 3,4 measurements STAR, which are already under revision of the Collaboration and will be soon published. In Figure 5.4 the difference between particles and antiparticles flow are shown. As they are negligible, the rest of the analysis was performed for the hadrons of both signs.

Flow of identified hadrons for Au+Au collisions at BES energies

Since the increasing with lowering the collision energy difference between particles and antiparticles flow was observed, the studies of systems obtained at BES energies (below 200 GeV) requires the distinguishing particles' signs. The relations between positive and negative particles flow are described in this section in great details.

Integrated flow harmonics

Described in Section 2.2.5 integrated flow over transverse momenta provides information about the dependence of the flow on the multiplicity of the collision. The examined data (Fig. 5.7 and 5.8) show the clear particle species dependence of both v n for all the collision energies. As these are integrated by the p T measurements, such differences can be explained by looking into distributions of transverse momenta of given types of particles. Protons' spectra is dominated by the higher p T in comparison to remnant mesons [143], what results in the higher integrated v n . For the centralities up to ∼ 30% the v n {2} increase nearly linearly, from the mid-central collisions the curves flatten. Finally, in peripheral collisions, the v n {2} is dominated by the viscous effects, which causes a decrease of measured flow.

The triangular flow shows similar differences between v 3 {2} of various species. As it is a fluctuation-driven quantity, the non-zero signal corresponds to the possible overlap region's granularities' presence. However, the measured signal due to the high uncertainties does not provide the information about the dependence of v 3 {2} on the collision's centrality.

In Figures 5.9 and 5.10 the dependence of the v n {2} on the collision energy is pre- of centrality, all the differences originate from the transport proprieties of the matter.

The general trend shows that increase of both flow harmonics with the collision energy. However, the effect is not substantial.

p T -differential flow harmonics

The studies of p T -differential flow harmonics were performed for the 0 -60% centrality range. The triangular flow's signal is small, so the high statistics are needed for the reasonably performed investigation. Nonetheless, the measured flow in the peripheral collisions is dominated by the viscous effects (see the previous section), and in order to avoid such contribution, these centralities are eliminated. For all three energies √ s N N = 27, 39 and 54.4 GeV the mass ordering is visible. Up to p T ∼ 1.5 GeV/c, the elliptic flow curves are parallel and increasing inversely to the mass of particles. At higher transverse momenta there is a clear split between baryons (protons' flow is relevantly higher) and mesons (pions' and kaons' flow is similar, start to overlap).

v 2 {2} p T < 0.2, 4.0 > GeV /c Kaon- ∆η = 0.2 v 2 {2} √ s N N = 27GeV √ s N N = 39GeV √ s N N = 54.4GeV
v 2 {2} p T < 0.2, 4.0 > GeV /c Kaon+ ∆η = 0.2 v 2 {2} √ s N N = 27GeV √ s N N = 39GeV √ s N N = 54.4GeV
The triangular flow keeps similar trends as the elliptic one, although the uncertainties are high. The mass ordering for all the energies is visible for low p T , and since the last bins were merged, it is impossible to make a statement about the higher-p T region.

n q flow scaling
The number of constituent quarks scaling was performed on the data from all three energies for both elliptic and triangular flow. The negative particles do not break the 27, 39 and 54.4 GeV in centrality 0 -60%. The ∆η = 0.2. This thesis scaling, while there is a visible deviation in protons, especially in triangular flow.

0 1 2 3 p T [GeV/c] 0.0 0.1 0.2 0.3 v 2 {2} √ s N N = 27GeV centrality 0 -60% ∆η = 0.2 v 2 {2} π- K- p 0 1 2 3 p T [GeV/c] 0.0 0.1 0.2 0.3 v 2 {2} √ s N N = 27GeV centrality 0 -60% ∆η = 0.2 v 2 {2} π+ K+ p 0 1 2 3 p T [GeV/c] 0.0 0.1 0.2 0.3 v 2 {2} √ s N N = 39GeV centrality 0 -60% ∆η = 0.2 v 2 {2} π- K- p 0 1 2 3 p T [GeV/c] 0.0 0.1 0.2 0.3 v 2 {2} √ s N N = 39GeV centrality 0 -60% ∆η = 0.2 v 2 {2} π+ K+ p 0 1 2 3 p T [GeV/c] 0.0 0.1 0.2 0.3 v 2 {2} √ s N N = 54GeV centrality 0 -60% ∆η = 0.2 v 2 {2} π- K- p 0 1 2 3 p T [GeV/c] 0.0 0.1 0.2 0.3 v 2 {2} √ s N N = 54GeV centrality 0 -60% ∆η = 0.2 v 2 {2} π+ K+ p Figure 5.11 -2PC p T -differential v 2 {2} for Au+Au collisions at √ s N N =

Particle-antiparticle flow difference

Studies of differences in the flow between particles and antiparticles are still ongoing. The theoretical models are developing; however, authors need bigger constrain from the experimental side. In this project, several investigations are proposed, which could be used 27, 39 and 54.4 GeV in centrality 10% -60%. The ∆η = 0.2. This thesis to validate or reject some of the theoretical scenarios.

v 3 {2} √ s N N = 39GeV centrality 0 -60% ∆η = 0.2 v 3 {2} π- K- p 0 1 2 3 p T [GeV/c] 0.00 0.05 0.10 0.15 v 3 {2} √ s N N = 39GeV centrality 0 -60% ∆η = 0.2 v 3 {2} π+ K+ p 0 1 2 3 p T [GeV/c] 0.00 0.05 0.10 0.15 v 3 {2} √ s N N = 54GeV centrality 0 -60% ∆η = 0.2 v 3 {2} π- K- p 0 1 2 3 p T [GeV/c] 0.00 0.05 0.10 0.15 v 3 {2} √ s N N = 54GeV centrality 0 -60% ∆η = 0.2 v 3 {2} π+ K+ p Figure 5.12 -2PC p T -differential v 3 {2} for Au+Au collisions at √ s N N =

Triangular flow

Firstly, the 2-Particle Cumulants method was applied in the investigation of the differences in p T -differential elliptic flow of particles vs. antiparticles for Au+Au collisions at √ s N N = 27, 39 and 54.4 GeV and shown in Figure 5.15. As triangular flow is more sensitive to such effects than elliptic one, investigating the differences between particles and antiparticles v 3 {2} was performed. Fig 5.16, the differences for all energies between the flow of kaons positive and negative is close to zero. 27, 39 and 54.4 GeV in centrality 0 -60% scaled with number of constituent quarks (v n {2}/n n/2 q in function of transverse kinetic energy KE T . This thesis Pions' differences in v 3 {2} do not variate significantly (< 0.01). The only exception is √ s N N = 39 GeV, where π -v 3 is slightly higher than π + . In the case of protons' and antiprotons' triangular flow, the higher p T , the bigger difference observed. Figure 5.17 shows that the collision energy dependence of elliptic and triangular flow differences between particles and antiparticles. The same trends are visible: the growing difference between n q -scaling It is claimed that the n q scaling is the proof of the common origin of hadrons' flow [72,73]. It is built during the QGP phase where quarks are deconfined, and they are boosted as separate particles. Subsequently, they are bounded into hadrons, but their flow is already established. Due to such an approach, breaking the scaling means that the flow of given particle specie does not originate completely from the QGP phase.

v 3 {2}/n n/2 q √ s N N = 39GeV centrality 0 -60% , ∆η = 0.2 v 3 {2}/n n/2 q π- K- p 0.0 0.5 1.0 1.5 2.0 KE T /n q 0.00 0.01 0.02 0.03 0.04 v 3 {2}/n n/2 q √ s N N = 39GeV centrality 0 -60% , ∆η = 0.2 v 3 {2}/n n/2 q π+ K+ p 0.0 0.5 1.0 1.5 2.0 KE T /n q 0.00 0.01 0.02 0.03 0.04 v 3 {2}/n n/2 q √ s N N = 54GeV centrality 0 -60% , ∆η = 0.2 v 3 {2}/n n/2 q π- K- p 0.0 0.5 1.0 1.5 2.0 KE T /n q 0.00 0.01 0.02 0.03 0.04 v 3 {2}/n n/2 q √ s N N = 54GeV centrality 0 -60% , ∆η = 0.2 v 3 {2}/n n/2 q π+ K+ p Figure 5.14 -2PC p T -differential v 3 {2} for Au+Au collisions at √ s N N =
√ s N N = 27GeV , centrality 0% -60%, ∆η = 0.2 v 2 {2}(x) -v 2 {2}(x) π + -π- K + -K- p -p 0 1 2 3 p T (GeV/c) -0.02 0.00 0.02 0.04 √ s N N = 39GeV , centrality 0% -60%, ∆η = 0.2 v 2 {2}(x) -v 2 {2}(x) π + -π- K + -K- p -p 0 1 2 3 p T (GeV/c) -0.02 0.00 0.02 0.04 √ s N N = 54.4GeV , centrality 0% -60%, ∆η = 0.2 v 2 {2}(x) -v 2 {2}(x) π + -π- K + -K- p -p
In Figure 5.18 the range of breaking the scaling of protons and antiprotons is illustrated. In top row there is ratio of p and p v 2 {2}/n q to the fit of pions' v 2 {2}/n q and in bottom one to koans' v 2 {2}/n q . In both cases, protons break relevantly the scalingmore than 10% for KE T /n q < 0.6 GeV. While antiprotons violate scaling only up to 10% = 27, 39 and 54.4 GeV in centrality 0 -60%. This thesis for the Au+Au collisions at √ s N N = 54.4 GeV, and for lower energies it is up to 5%. The energy dependence is not clear, and could be caused by lack of consideration of the systematic uncertainties in the ratio computation.

√ s N N = 27GeV , centrality 0% -60%, ∆η = 0.2 v 3 {2}(x) -v 3 {2}(x) π + -π- K + -K- p -p 0
√ s N N = 39GeV , centrality 0% -60%, ∆η = 0.2 v 3 {2}(x) -v 3 {2}(x) π + -π- K + -K- p -p 0.0 0.5 1.0 1.5 2
√ s N N = 54.4GeV , centrality 0% -60%, ∆η = 0.2 v 3 {2}(x) -v 3 {2}(x) π + -π- K + -K- p -p
These studies provide the starting point for investigating protons flow's various origins, as suggested in Scenario 2 "Mean field theory".

Another theoretical approach explains the differences between particles' and antiparticles flow' using the Mean field theory. Kaon as s meson consists of quark and anti-quark, and is characterized by relatively high mass. According to the Scenario "Transported protons", the effect of the potential on kaons is negligible comparing to protons. In such case, if the mass factor is cancelled by scaling, the antiprotons behave oppositely to protons concerning the scaled flow of kaons. In Figure 5.19 the ratio of p and p v 2 {2}/n q to fitted kaons' v 2 {2}/n q is shown for Au+Au collisions at √ s N N = 39 GeV. The ratio for negative particles is equal to unity within the uncertainties, while for protons, it is more than 10% higher. If the ratios of p and p were the mirror reflection to K, it could be treated Figure 5.17 -The difference in Event Plane v 2 (data from [84]) and in 2PC v 3 {2} between particles (X) and the corresponding anti-particles ( X) as a function of √ s N N for 0 -80% central Au+Au collisions. This thesis as evidence favouring the explanation included in Scenario "Transported protons". However, this theoretical approach cannot be validated with performed studies.

0.0 0.2 0.4 0.6 0.8 1.0 KE T /n q 0.6 0.8 1.0 1.2 1.4 v2{2}/nq(p) f itv2{2}/nq(π+) centrality 0 -60% , ∆η = 0.2 Ratio to Pions fit: a + bx + cx 2 + dx 3 + ex 4 Protons √ s N N = 27GeV √ s N N = 39GeV √ s N N = 54.4GeV 0.0 0.2 0.4 0.6 0.8 1.0 KE T /n q 0.6 0.8 1.0 1.2 1.4 v2{2}/nq(p) f itv2{2}/nq(π-) centrality 0 -60% , ∆η = 0.2 Ratio to Pions fit: a + bx + cx 2 + dx 3 + ex 4 Antiprotons √ s N N = 27GeV √ s N N = 39GeV √ s N N = 54.4GeV 0.0 0.2 0.4 0.6 0.8 1.0 KE T /n q 0.6 0.8 1.0 1.2 1.4 v2{2}/nq(p) f itv2{2}/nq(π+) centrality 0 -60% , ∆η = 0.2 Ratio to Kaons fit: a + bx + cx 2 + dx 3 + ex 4 Protons √ s N N = 27GeV √ s N N = 39GeV √ s N N = 54.4GeV 0.0 0.2 0.4 0.6 0.8 1.0 KE T /n q 0.6 0.8 1.0 1.2 1.4 v2{2}/nq(p) f itv2{2}/nq(π-) centrality 0 -60% , ∆η = 0.2 Ratio to Kaons fit: a + bx + cx 2 + dx 3 + ex 4 Antiprotons √ s N N = 27GeV √ s N N = 39GeV √ s N N = 54.4GeV
Figure 5.18 -Collision energy dependence of ratios of protons' and antiprotons' v 2 {2}/n q to fit of (top row) fit to pions' v 2 {2}/n q and (bottom row) kaons' v 2 {2}/n q in function of transverse kinetic energy. This thesis

0.0 0.2 0.4 0.6 0.8 1.0 KE T /n q [GeV] 0.6 0.8 1.0 1.2 1.4 v 2 {2}/n q (p) f itv 2 {2}/n q (π-) √ s N N = 39GeV centrality 0 -60% , ∆η = 0.2 p p Figure 5
.19 -Collision energy dependence of ratios of protons' and antiprotons' v 2 {2}/n q to fit of kaons' v 2 {2}/n q in function of transverse kinetic energy. This thesis Chapter 6

EPOS MODEL

The second part of this thesis is focused on the development of the phenomenological MC model EPOS. The possibility of the choice of the family of the new EoS was introduced into the model. It allowed me to study the impact of changes of the EoS on the final experimental observables.

EPOS is an abbreviation of Energy conserving quantum mechanical multiple scattering approach, based on Partons (parton ladders), Off-shell remnants, and Saturation of parton ladders.

The EPOS is a phenomenological model based on the MC techniques. It gives a possibility to investigate various observables (like particle production, momenta distribution or flow correlations), providing a better understanding of the evolution of the system created in the elementary collisions such as proton-proton and during complex reactions with heavy-ions. The theoretical framework included in the model provides the coherent description of the space-time expansion of matter based on the precise spectrum of studies of both elementary processes such as electron-positron annihilation or lepton-nucleon scattering and more compound collisions of protons or nucleus'. This model was designed to describe processes present after collisions at µ B ≈ 0, at very high energies like top RHIC or obtained at LHC and for various systems, such as Au+Au, Pb+Pb or p+p.

The EPOS model consists of several phases of evolution:

-initial stage (based on the Parton Gribov-Regge theory), -core/corona division, -hydrodynamical evolution, -hadronization, -hadron rescattering, -resonance decays.

In this chapter, all of these stages are described as well as the new developments of the generator.

Initial Stage of the evolution

In the model's theoretical framework, the crucial element is the sophisticated treatment of the hadron-hadron scattering and the initial stage of the collisions at ultra-relativistic energies. It is highly relevant in the understanding of possible parton-hadron phase transition. In the EPOS model, the merged approach of Gribov-Regge Theory (GRT) and the eikonalised parton model is utilised to provide proper treatment of the first interactions happening just after the collision -satisfying conservation laws and equal treatment of subsequent Pomerons. This part of the thesis is based on the book [144].

Parton-Based Gribov-Regge Theory

Gribov-Regge Theory

The Gribov-Regge approach is an effective field theory focusing on the soft aspects of the particle interactions [145]. Numerous elementary interactions can happen simultaneously and they are represented by the complex, theoretical objects called "Pomerons". The single exchange of Pomeron is parametrized with the elastic amplitude T expressed by:

T (s, t) ∼ is α 0 +α ′ t (6.1)
where: s, t are the Mandelstam's variables1 , α Regge Pole [146]. Multiple Pomeron exchange occur in parallel (Fig. 6.1).

Elastic amplitude:

The n Pomeron exchange amplitude can be expressed with:

T n (s, t) ∝ d 2 b × e ikb - s × ω(s, b) n n! (6.2)
where b is the impact parameter, k is the transverse momentum of outgoing hadrons, while ω is:

ω(s, b) ∝ ye α 0 -1 R 2 + α ′ y × exp - 1 4 b 2 R 2 + α ′ y (6.3)
where y = ln s s 0 and R is a free parameter. The total elastic amplitude is:

T (s, t) = ∞ n=1 T n (s, t) ∝ d 2 b × e ikb × y(s, b) ⇒ y(s, b) = 1 -se ω(s,b) (6.4)
The multiple Pomeron exchanges respect the Froissard limit [147] and their total amplitude can be used in calculation of the total cross-section:

σ tot = 2ℜ[y(s, b)]d 2 b (6.5)

Inelastic amplitude:

The production of particles is governed by the field theory. Utilizing the cutting rules the inelastic cross section can be expressed by following:

σ h 1 h 2 inel = d 2 b{1 -exp(-G(s, b))} (6.6)
where G(s, b) is proportional to the Fourier transformation of T (s, t) and is an equivalent to the one elementary interaction (green line on the Fig. 6.1). Although the crosssections of all elementary interactions are respected identically, subsequent Pomerons are not treated equally in particle production. Moreover, energy conservation is not ensured.

Parton Model

Feynman proposed the Parton Model during his studies of proton-proton collisions [148]. It introduces the components of protons. Consequently, they stopped being treated as elementary particles.

The Parton Model considers quarks and gluons. The momentum distributions f h 1 and f h 2 represent the partons of projectile and target. It considers the hard processes and the inclusive cross-section of the parton jets' production with the squared transverse T bigger than some cutoff Q 2 0 can be expressed by following equation:

σ h 1 h 2 incl = Σ 2 ij T dx + dx -f i h 1 (x + , p 2 T )f i h 2 (x -, p 2 T ) dσ ij p 2 T (x + x -s)θ(p 2 T -Q 2 0 ) (6.7)
where i, j are the parton flavors and dσ ij p 2 T reports to the elementary parton-parton crosssection. However, similarly to the GRT, in this approach there is no clear solution for the way of energy sharing in the exclusive calculations -the generation of the event.

Parton-Based Gribov-Regge Theory

Combining these two approaches -Parton-based Gribov-Regge Theory (PBGRT)provides the solution for some of the imperfections from each of the model separately. It is a realistic treatment of both soft and hard processes occurring in the medium evolution.

Here, the rigorous approach to the multiple scattering aspect and particle production ensures the proper energy sharing and overall conservation.

In the Figure 6.2 the fundamental two-body elastic scattering with amplitude T is illustrated. Each of incoming particle has 4-momenta p and p ′ , and outgoing: p = p + q, p′ = p ′ + q respectively, where q correspond to transferred momentum. The total crosssection can be expressed as:

σ tot (s) = 1 2s 2ℑT (s, t = 0) (6.8)
The Fourier transform T of T is:

T (s, b) = 1 4π 2 d 2 q ⊥ e -i⃗ q ⊥ ⃗ b T (s, t) (6.9)
Figure 6.3 -Elementary interaction in PBGRT. This thesis

if t = -q 2 ⊥ , the profile function G(s, b) is: G(s, b) = 1 2s 2ℑ T (s, b) (6.10)
Consequently the total cross-section can be expressed as

σ tot (s) = d 2 bG(s, b) (6.11)
This profile function can be understood as a probability of interaction at the given impact parameter. It can be applied to partonic profile functions and hadronic or nuclear functions that can be derived from the partonic one.

In the Figure 6.3 the schematic elementary interaction ladder depicted, where blue lines corresponds to incoming and outgoing partons. The ladder consists of three contributions:

soft: non-perturbative QCD, the virtuality of partons is smaller than the saturation factor Q 2 s ; in the description and parametrization the phenomenological Regge theory is used, semi-hard: getting closer to the center of the ladder causes the increase of virtuality: Q 2 > Q 2 s ; consequently it becomes harder hard: perturbative QCD used in the discription, the DGLAP equations are used for the virtuality of partons [149]; in the center of the diagram there is a 2 → 2 interaction called Born proccess.

The contributions of the semi-hard combination of partons originate from the sea, or valance quark are considered as well.

Moreover, the equal treatment of subsequent pomerons in particle production is pro-Figure 6.4 -Proton-proton (left) and proton-nucleus (right) scattering. The total energy is conserved: from each incoming proton/nucleus, the energy is shared among several "constituents". Consequently, each Pomeron obtains only a part of the total energy. This thesis vided. In Figure 6.4, the proton-proton and proton-nucleus collisions are illustrated. The green lines correspond to the elementary interactions and are the sum of soft, semi-hard and hard contributions -the treatment is coherent.

Finally, in the EPOS model calculations of the cross-sections and particle production, the same formalism is used. It is based on the Feynman diagrams of the QCD-inspired effective field theory and provides energy conservation. The nucleons are divided into a certain number of "constituents" carrying the incident momentum fraction. The fractions sum to unity in order to ensure momentum conservation. For each nucleon, there can be (illustrated on the Fig. 6.5) :

-multiple or one participants, which take part in the elementary interactions (green lines) with constituents from the opposite side, -one spectator, or -one remnant, which does not interact (nucleon -participants)

The parton ladder is used for the representation of the evolution from the projectile to the target. There are two kinds of mention above ladders: open and closed. The first one corresponds to the interactions where the new particles are produced, while the closed one to elastic processes.

Applying the cutting rule techniques, the partial cross-section for exclusive event classes are obtained and simulated using the Markov chain techniques.

String Model and Flux Tube Approach

The particle production is based on the String Model approach [START_REF] Andersson | 7th European Symposium on Antiproton Interactions: From LEAR to the Collider and Beyond[END_REF]150], which describes the process of hadronisation. It uses strings as an analogy of the production of new particles. The QCD potential between quarks expressed with the eq. 1.1, is dominated at large distances by the linear part. It is used in the division (fragmentation) of the high-masses q q into multiple small-masses ones. The colour field stretched between the quarks is approximated with the one-dimensional string (Fig. 6.7). The potential energy increases with the distance of separation of the original quarks. When the invariant mass of the system is big enough, the production of the new pair of q q is more energetically favourable than a continuation of stretching the string. The new particles are created via so-called string fragmentation.

Flux Tubes

In EPOS model the parton ladders are recognized as a quasi-longitudinal color fieldelementary flux tubes and treated as classical strings [151]. The strings are divided into several segments [α k , α k+1 ] with a corresponding velocity:

{ V (α) ≡ ∂X ∂β (α, β = 0) = V k , in[α k , α k+1 ] (6.12)
where X(α, β) is the 2D surface in 3+1 dimensional space-time. The interval with a constant V k is called a kink. The length of the segment is given by the parton energy E k , the velocity is the parton velocity p µ k /E k . The intermediate gluons introduce the transverse motion into the kinky string string evolution. The schematic picture of the flux tube with the transverse kink is shown in the Fig. 6.8. The illustration is just two dimensional, the object is mainly longitudinal but there are pieces of the string moving transversely in the y-direction. The fragmentation of the flux tube occurs according to the string model via Figure 6.8 -The flux tube with the kink, moving transversely in R 3 . This thesis Figure 6.9 -Schematic illustration of the fragmentation of the string. The segments close to the kink will contribute to corona. This thesis q -q or qq -qq production (Fig. 6.10). This process happens with a given probability in the 3D+1 space-time, what is rigorously treated utilizing the area law hypothesis [152].

Core-Corona approach

If the density of the strings is very high, they cannot decay independently, and this situation is characteristic for the heavy-ions and the high-multiplicity p -p collisions. In EPOS the dynamical process of division of the strings segments into core and corona is introduced in order to deal with this issue [151,153,154].

The separation is based on the abilities of a given string segment to leave the "bulk matter". As the criteria for the decision, if it goes to a given group, the transverse momentum of the element and the local string density is taken into account. If the string segment belongs to the very dense area, it will not escape but will build the core, which will be driven in the next step by the hydrodynamical evolution. When the segment origi- nates from the part of the string close to the kink, is characterized by the high transverse momenta, it escapes the bulk matter and join the corona and consequently will show up as a hadron (jet-hadrons). There is also a possibility that the string segment is close to the surface of the dense part of the medium, and its momentum is high enough to leave it; it also becomes a corona particle. The following equation is used for the determination of core and corona: 6.13) where: is the trajectory of the segment, ρ density, f Eloss ̸ = 0 constant for p t < p t,1 ,

p new t = p t -f Eloss γ ρdL (
f Eloss = 0 for p t > p t,2 . If the p new t
is positive for a given segment, it escapes and becomes a corona particle; in the opposite case, it contributes to the core.

Viscous Hydrodynamical evolution

As it has been studied [39,40,43,155], the QGP does not expand like an ideal fluid, and the effect of the bulk viscosity has to be taken into account in the simulations. In EPOS, the 3D+1 hydrodynamics is applied, providing an excellent description of the collective expansion of the matter [151].

The thermalization occurs at the given proper time τ 0 and the core string segments provide the initial conditions. Their four-momenta is used in the computation of the flavour flow vector and energy-momentum tensor at given position x:

N µ q (x) = i δp µ i δp 0 i q i g(x -x i ), (6.14)
T µ,ν (x) = i δp µ i δp ν i δp 0 i g(x -x i ), (6.15)
where: g is a Gaussian smoothing kernel with a transverse width σ perp = 0.25fm, q ∈ u, d, s corresponds to the net flavor of string segment and its four-momenta is: (6.16) In order to convert the system into the coo-moving frame, the Lorentz transformation is used:

δp = ∂X(α, β) ∂β δα + ∂X(α, β) ∂α δβ
Λ α mu Λ ν T µν = T µν com (6.17)
The EoS is computed as: .18) where: p Q is the pressure of the ideal QGP (including bag pressure), and p H -of a resonance gas, and: .19) with: (6.20) where: δ = 0.24exp(-µ 2 b /0.4 2 ). The compatibility of equation of state with the lattice data [156] is ensured by λ.

p = p Q + λ(p H -p Q ) ( 6
λ = exp(-z -3z 2 )Θ(T -T C ) + Θ(T C -T ) ( 6
{ z = x/(1 + x/0.77), x = (T -T C )/δ
In EPOS there is applied the 3D+1 viscous hydrodynamics approach called vHLLE (viscous HLLE-based algorithm) [157]. The Minkowski coordinates {t, x, y, z} are used as

τ = √ t 2 -z 2 , η = 1 2 ln(t + z)/(t -z).
The hydrodynamic equations are:

∂ µν T µν = ∂ ν T µν + Γ µ νλ T νλ + Γ ν νλ T µλ = 0 (6.21)
where Γ is the Christoffel symbol. In order to avoid application of a higher order numer-ical time integration scheme, the T µν is introduced as: (6.22) so from eq. 6.3 τ T µν : (6.23) where:

{ T νµ = T µν , µ, ν ̸ = η, T νη = T µη /2, µ ̸ = η, T ηη = T ηη /τ 2 ,
{ ∂ ν (τ T τ ν ) + 1 τ (τ T ηη ) = 0, ∂ ν (xT τ ν ) = 0, ∂ ν (yT τ ν ) = 0, ∂ ν (ηT τ ν ) + 1 τ (τ T ητ ) = 0,
∂ ≡ (∂/∂τ, ∂/∂x, ∂ ∂y, (1/τ )∂/∂η) (6.24)
To solve the equations of the relativistic viscous hydrodynamics, the calculations are performed in the 159]. To satisfy the energy-momentum conservation in viscous case, the ideal-viscous splitting technique is used [160]. The Israel-Stewart equations are calculated simultaneously, and the correction of the momentum/energy expansion is performed according to the source terms in eq.6.3 and viscous fluxes.

Event-by-event treatment

In the simulations, the highly important is the definitive treatment of individual events -the generalization in considering smooth initial conditions for all events is not applied. The event-by-event (ebe) approach in hydrodynamical evolution is based on the random flux tube initial conditions [151]. It has a relevant impact on the final observables such as spectra or various harmonics of flow.

Equation of State

The viscous hydrodynamics uses Equation of State X3F ("cross-over" and "3 flavor conservation") which is compatible with Lattice QCD data (Fig. 6.11). It corresponds to µ B = 0 MeV so that it limits the possibility to describe the region of QCD phase diagram characterized by the finite baryon density [151]. To extend these limits, the new EoS has to be implemented.

Hadronization

The expanding medium in the processes of hydrodynamical evolution reaching the given freeze-out condition is transformed into the particle spectra. In EPOS 3, the criteria characterizing the hadronization hypersurface is the energy density equal 0.57Gev/f m 3 . The transformations are governed y the Cooper-Frye formula [161]:

E dN d 3 p = Σ dΣ µ p µ f (T, p µ u µ , π µν ) (6.25)
where f (T, p µ u µ , π µν ) corresponds to the particle distribution function, Σ is the hadronization hypersurface.

As a new development, the micro-canonical approach to the hadronization has been introduced.

Hadronic Cascades

The final part of the simulation uses so-called hadronic afterburner -UrQMD [162,163].

When the density is very high and the mean free paths of constituent particles are small about any macroscopic length scale, the hydrodynamic description can be used -in the initial phase of the QGP evolution. With the cooling of the system, the density and the mean free paths decrease; oppositely, the η/s increases. Finally, the differences in the mean free path of various particle species become relevant, and the collective description of the system becomes not adequate. When the density and the temperature are low enough, the kinetic theory is applied using the UrQMD code [162,163].

The particles can interact only when they leave the hyper-surface of the freeze-out. The 2 → n hadronic scattering is performed according to the measure reaction cross sections [164]. The 60 different baryonic species and their anti-particles, about 40 mesonic states are considered [162,163]. There are implemented such interactions between hadrons as [165]:

-elastic scattering, -string excitations, -resonance excitations, -strangeness exchange reactions

The hadronic scattering has a significant impact on the final observables [2].

New developments

One of the essential elements of this PhD project was to develop the code of the EPOS generator applying new Equations of State and make it possible to choose the one from the immense variety. The goal is to study the impact of the different EoS used in the evolution of the system on the final observables such as one-particle spectra or flow investigation. It is substantial to investigate which experimental measurements are sensitive to each implemented phase transition or strength of criticality of given EoS.

BEST EoS

Collaboration Beam Energy Scan Theory proposed the family of Equations of State describing same region of QCD phase diagram as studied in BES program [START_REF] Parotto | [END_REF]166]. It covers the region of chemical potential in range 0 ≤ µ B ≤ 450 MeV and T between 30 MeV and 800 MeV. The equations respect the lattice QCD results up to ′(µ 4 B ). They consider existence of cross-over transition, first order phase transition and give possibility to choose the location of Critical Point.

In the proposed EoS, the consistency with the derived from the first principle calculations is preserved. However, the problem with the limitation in the extraction of information about the medium characterized by the finite µ B (described in Section 1.2.3) is still present. Although the Taylor series used to expend the region studied with the QCD EoS never goes beyond the µ B higher than one at Critical Point, the universality class to which the CP belongs is known, and it is possible to study the behaviour of EoS in some limited neighbourhood of CP.

The calculations applied by BEST are based on such a strategy:

1. In the region close to CP, the suitable parametrization is used in the description of EoS universal scaling behaviour in the 3D Ising model;

2. The 3D Ising model phase diagram is mapped using a parametric change of variables (not universal) onto the QCD one;

3. Estimation of critical contribution to Lattice QCD expansion coefficients using the Ising model thermodynamics;

4. Reconstruction of the full pressure -compatible with Lattice QCD at µ B = 0 and consisting the proper critical behaviour.

Despite the possibility to locate the critical point on the thermodynamics and hydrodynamic simulations, the proposed approach depends on the choice of parameters in the mapping of Ising → QCD, which is not arbitrary. It is a consequence of a need to match QCD Lattice data which puts some limitations on the CP position.

Scaling and mapping 3D Ising model Equation of State to QCD

In these transformation close to the CP, firstly the Ising model EoS is parametrized and then mapped on the QCD diagram. The reduced temperature r = (T -T C )/T C , magnetic fields h, and magnetization M is introduced by following equations [167][168][169][170]:

M = M 0 RΘ (6.26) h = h 0 R σ h(Θ) (6.27) r = R(1 -Θ 2 ) (6.28)
where M 0 , h 0 report to normalization constants, α, β are the 3D Ising critical exponents [168], h(Θ) = Θ(1 + aΘ 2 + bΘ 4 ) (where a = -0.76201 and b = 0.00804). The Gibbs free energy density can be expressed as a function of (R, Θ) parameters [167,168]. Finally, considering the relation between Gibbs free energy density and pressure: G/V = -P , the scaling can be expressed with:

P Ising (R, Θ) = -h 0 M 0 R 2-α [g(Θ) -Θ h(Θ)] (6.29)
In the Fig. 6.12 the non-universal mapping procedure is illustrated. Using six parameters the critical thermodynamics is linearly transferred to QCD (Ising variables to QCD coordinates: (h, r) → (T, µ B )): .31) where: (T C , µ BC ) is the location of CP, α 1 and α 2 are the angles between r and h axes and T = const., w and rho reports to respectively global and relative rescaling of r and h. Such a description provides the possibility to locate the CP on the curve, set precise parameters characterizing the system, and satisfy the need for proper mapping from the Ising model to the QD phase diagram.

T -T C T C = w(rρ sin α 1 + h sin α 2 ) (6.30) µ B -µ BC T C = w(-rρ cos α 1 -h cos α 2 ) ( 6
With the globally invertible second step, the map is ready:

(R, Θ) → (h, r) ↔ (T, µ B ) (6.32) The QCD pressure's critical input is:

P QCD crit (T, µ B ) = f (T, µ B )P Ising (R(T, µ B ), Θ(T, µ B )) (6.33)
where f (T, µ B ) is a regular function with energy fourth dimension.

The coefficients derived from Lattice QCD are expressed by the ensured by matching with Lattice EoS at µ B = 0 "regular coefficients" (c reg (T )) and these including the contributions coming from Ising CP: (6.34) Finally, the full pressure can be calculated using the following equation:

c LAT n (T ) = c regn(T )+c crit n (T )
P (T, µ B ) = T 4 n=0 2 c reg 2n (T ) µ B T 2n + P QCD crit (T, µ B ) (6.35)
This equation ensures the matching with Lattice QCD at µ B = 0 and introduces the criticality in the correct universality class.

Microcanonical hadronization

The microcanonical hadronization is introduced as a consequence of difficulties in matching the dynamical properties of the bulk created in hydrodynamical evolution and the given hadronic system. Another reason is to ensure that the energy and flavor are conserved for small systems [171][172][173][174][175].

The hadronization hyper-surface x µ (τ, φ, η) is characterized by following equations: (6.36) respecting one of the chosen by the authors condition with given constant (FO):

{ x 0 = τ coshη, x 1 = rcosφ, x 2 = rsinφ, x 3 = rsinhη,
-temperature: T H = const.

-energy density: ϵ H = const. The element of the hyper-surface is: The flow of momentum vector dP µ (Fig. 6.13) and charges dQ A for each element are conserved: dP µ = T µν dΣ ν (6.42)

dΣ µ = ϵ µ,ν
dQ A = J ν A dΣ ν (6.43)
The effective mass is the sum of all the surface elements: M = surface area dM (6.44) where dM = dP µ dP µ . The volume and the four-velocity U µ (ONLY if the pressure is 0, it is equal to velocity of fluid u µ ) of each element are known: (6.45) dV = u µ dΣ µ (6.46) The constructed effective mass decays microcanonically: 6.47) where:

U µ = dP µ /dM
dP = C vol C deg C ident × δ(E -ΣE i )δ(Σ - → p i ) A δ Q A ,Σq Ai n i=1 d 3 p i (
C vol = V n 2π) 3n (volume), C deg = n i=1 g i (degeneracy), C ident = α∈S 1 n α !
(identical particles factor), n α correspond to the number of the species α of particles, and S reports to the set of the particle species. After the decay, the particles are boosted according to the velocities U µ .

Chapter 7

THE EQUATION OF STATE AT EPOS MODEL

This is second of the most substantial chapters including main results of author of this thesis.

In this chapter, the introduction of the new Equation of State into the EPOS model is described. The proposed EoS is characterized in details in 6.6.1. The main goal of the presented studies is to investigate the influence introduced changes onto the final observables.

Implementation of EoS into the code of generator

The EPOS model includes the hydrodynamical description of the part of the evolution of the simulated system. This part of calculations uses tables including the information about the dependence of the temperature and µ B on energy density (ϵ), speed of sound (c s ), pressure (P ) in a given range. Such tables are generated using the relations included in Equation of State. Concluding, the hydrodynamical evolution is based on the EoS.

The source code of the generator is developed using the Fortran and C++ programming languages. The main introduced modification is a simplification of the part of the code related to using the information in EoS tables. There is the possibility of choosing and quickly adding the EoS, which is used in the simulations. There were performed various successful tests in order to validate the introduced changes.

Selection of the EoS parameters

The primary studies possible due to the modifications and development of EPOS model allow one to study the impact on final observables of the changes between the variety of EoS. As mentioned in Section 6.6.1, the BEST EoS is, in reality, the "family" of EoS, the set of divers EoS tables. To obtain the EoS one needs to choose the set of parameters corresponding to mapping properties and locating the CP onto the QCD phase diagram.

The composition of the input parameters is crucial in setting the strength of the "criticality" of the transitions of the matter, moreover via changing the position of the CP one can expect the cross-over or the first-order transitions in the evolution of examined simulated systems. The structure of the parameter input file is:

MODE T 0 κ µ BC ∆ α 1 ,α 2 ω ρ
where:

-MODE -corresponds to the CP location and choice of following parameters. In this studies the CP lies on parabola parallel to the chiral transition line -what reports to MODE = PAR; -T 0 -the value of temperature at which the parabolic pseudo-critical line crosses the T axis; κ -the curvature of the transition line at the T axis; µ BC -the chemical potential at the critical point: since both T C and α 1 are determined by this choice, they do not need to be input; -∆ α 1 ,α 2 -the difference between two angles (see Fig. 6.12) ω -the global scaling parameter in the mapping (the higher ω the less criticality in transitions of matter); ρ -the relative scaling in the mapping [166];

Example: PAR 155 -0.0149 350 90 1 2

The critical temperature, α 1,2 can be easily calculated from the given parameters:

T C = T 0 + κ/T 0 µ 2 BC α 1 = 180/π arctan(- 2κ T 0 µ BC ) α 2 = α 1 + ∆α 1 , α 2 (7.1)

Critical Point location

In the performed studies various sets of parameters were chosen in order to investigate the wide range of properties of EoS. In order to study the critical behaviour the CP was placed on the T , µ B plane close to the average temperature of the system simulated by EPOS model. Due to the fact that these are approximate estimations the temperature was extracted from the positive pion's p T spectra distribution obtained for the simulated most central Au+Au collisions (0 -5%). The used modified Hagedorn function for the fit is described in [176,177], where A correspond to the so-called also inverse slope parameter, which can be treated as an approximate average temperature of the expanding system 7.2.

Comparison of various EoS at EPOS model

Thermodynamical relations

The only element of the whole simulation performed by EPOS model which changes is the Equation of State. Concluding, all the presented data sets can be used in the direct comparison of the proposed EoS. Firstly, all the basic relations between thermodynamical properties are illustrated in the Figures 7. 3, 7.4, 7.5, and 7.6. Also, the approximate CP's location is pointed on the plots for better visualisation. The "BEST 5" set of parameters was excluded due to the non-physical output. The lower the parameter ω, the more visible the critical behaviour -drastic change (pick or break-down) on the illustrated surfaces. The highest peack in the Fig. 7.6 for BEST 7 is present due to ω value 0.5, and they correspond to the critical behaviour of the system (rapid changes, fluctuations of thermodynamical properties of matter). Oppositely, in the BEST 4 √ s N N = 27 GeV, the approximate T C extracted from spectra fit is above the limit of the proposed BEST EoS. The simulated system is expected to go through, at least mostly, the cross-over transition.

Production of particles

The production mechanism highly depends on the potentials characterising the system. The higher the µ B , the more produced baryons than antibaryons. Similarly in case of µ q (charge) and µ s (strange). Such relations are illustrated in Figure 7.7, where the most central 0 -5% collisions of Au+Au simulated with EPOS 3.4. model are compared with STAR data published in [143]. Various EoS sets of parameters were used in performed simulations; the numbers of EPOS' data sets correspond to these listed in the Table 7.2. The relations between particles' and antiparticles' production is reflected using ratios in Figure 7.8. The criteria for selecting the theoretical data are the same as applied by the experimentalists: |y| < 0.1 and p T < 2.0 GeV/c. The higher number of produced baryons than antibaryons proves that in EPOS simulations, the impact of non-zero baryon potential is kept for all the proposed EoS. The This thesis model reflects the experimental data reasonably, except for pions. The number of these both negative and positive mesons is overestimated. Nonetheless, their ratio is kept.

The possible reason for such discrepancies could be the too wide rapidity distribution of simulated data. In the experimental analysis, the selection of particles characterised by the |y| < 0.1 is very narrow. In such a case, even a small deviation in the rapidity distribution strongly affects the performed investigation.

In both Fig. 7.7 and 7.8 no relevant differences between simulations obtained with various EoS are observed. Notwithstanding the X3F Equation of State corresponds to the cross-over transition, which is not expected to happen for cooling systems created in collisions of Au+Au at √ s N N = 7.7 GeV.

π + π - K + K - p p Λ Λ Ξ Ξ 10 -1

Dynamics of matter

In this subsection, the transverse momentum dependence of particle production is compared with experimental STAR data. Furthermore, simulations were performed for various EoS sets of parameters listed in Table 7.2. It gives a possibility to test the impact of different EoS introduced in the same generator framework on the final observables. This thesis

π - π + K - K + p p Λ Λ Ξ + Ξ K - π p π Λ π Ξ + π 0

Transverse momentum spectra

The fundamental observable studied in high-energy physics is the transverse momentum. One of the possible application is the calculation via proper fitting method the chemical freeze-out temperature of the examined system [143,176,177]. In these studies, there were two Au+Au collision energies considered:

√ s N N = 7.7 and 27 GeV. In the Figures 7. 9, 7.10,7.11, and 7.12 there are shown the particle spectra of identified hadrons from simulations with EPOS and comparisons with STAR data [143].

Au+Au at

√ s N N = 7.7 GeV The slopes of the p T spectra of the simulated data are similar to the experimental ones. It means that the relation between the amount of core and corona particles is correct. The more core particles, the p T distribution gets flatten. As studied in the previous subsection, too many mesons are generated in the events of 0 -5% centrality in rapidity range (|y| < 1.0). It is reflected in the "height" of the distributions of pions and kaons, while in case of protons the agreement between EPOS and STAR spectra is better. Furthermore, Figure 7.7 shows that positive kaons production is closer to the experimental one than the negative, what is also repeated in their p T spectra.

The data simulated using various EoS do not variate relevantly. The differences are present, what is visible in the bottom-ratio plots, but there is no clear trend showing the The plots of p T spectra for peripheral Au+Au collisions in Figure 7.10 show the much better agreement between simulated and experimental data sets. However, the statistical fluctuations prevent making a final statement about the differences between EoS -the In the bottom panels there is a ratio between the simulated and experimental data. This thesis collision energies (like in BES), such as the wrong magnitude of radial flow, resulting in a broader distribution of p T spectra. The plotted spectra are too flat for protons, which can be caused by the not adequate fraction of corona particles over core ones. On the other hand, the clear overproduction of particles in a given range of rapidity is visible in spectra for peripheral collisions.

For both centralities, the differences between data sets based on various EoS are similar to these at √ s N N = 7.7 GeV: no clear trend, the ratios fluctuate. Figure 7.12 -The p T spectra of the most peripheral 60% -80% Au+Au collisions at √ s N N = 27 GeV of Pion+, Pion-, Kaon+, Kaon-, Proton, and Antiproton, simulated with EPOS 3.4. using various EoS (Tab 7.2). Compared with STAR data [143]. In the bottom panels there is a ratio between the simulations based on BEST EoS and X3F. This thesis Figure 7.13 -Elliptic flow of pions, kaons and protons calculated using Event Plane method (described in [43]) for Au+Au collisions at √ s N N = 7.7 GeV, centrality 0 -80%.

Various data sets corresponds to simulated with EPOS 3.4. and experimental STAR data from [73]. This thesis -the mean (M ) -the standard deviation (σ) -the skewness (S), which corresponds to the asymmetry of the distribution -the kurtosis (κ), reflecting the degree to which the distribution is peaked relative to the normal distribution.

They are linked with the corresponding higher-order thermodynamic susceptibilities, and the system's correlation length [182,183], which are expected to deviate for large samples in equilibrium at the CP. As a result of slowing down in the vicinity of CP, in reality, the system is driven away from the thermodynamic equilibrium, and the maximum value of correlation length attains 1. Figure 7.19 shows the Sσ and κσ 2 integrated overall centralities as a function of the collision energy. The significant energy dependence is present for Sσ for all EoS. All the points at given energy are within the statistic errors, so no clear statement about the discrepancies between the EoS can be made. κσ 2 shows bigger variations between The shifts of the point on x-axis are applied for better visualisation. This thesis different EoS data sets. At √ s N N = 7.7 GeV the highest point corresponds to the EoS where the CP is located at high T and low µ B and the simulated system is expected to go through the first-order transition, while the negative value is related with BEST 4, where the criticality is the less pronounced. For data sets simulated at √ s N N = 27 GeV the differences are smaller; however, the BEST9 value is the highest. The energy dependence is not definite.

The measurements of the net-proton distributions' moments show the differences between data simulated using various Equations of State.

Two Particle Cumulants

This Section consists of comparison of the STAR data described in Section 4 (same data selection criteria) and simulated with EPOS 3.4 using EoS BEST1 (parameters: PAR 143 350 3 93 143 286). In Figures 7. 20 3.4 (line). This thesis The elliptic flow of pions and kaons, both signs, is well reproduced up to p T < 1 GeV/c. The trend of the mass splitting is kept in the model but for higher transverse momentum v 2 {2} is underestimated. The most relevant discrepancies are visible for protons, which significantly do not follow the experimental data in the whole range of p T . 3.4 (line). This thesis Triangular flow of both data sets are closer to the agreement than for the elliptic flow. The simulated with EPOS v 3 {2} of negative particles shows the trend of mass ordering, and reproduce the experimental data well. Triangular flow of π+ and K-is consistent with STAR, while, protons' one is not.

CONCLUSION

Summary and discussion of results

In this thesis, detailed studies of the dynamics of newly-created matter in the heavy-ion collisions were presented and discussed. Both an experimental and a theoretical approach were applied to broaden the knowledge in this subject.

The first part of the thesis consists of studies of v 2 {2} and v 3 {2} calculated basing on 2-Particle Cumulants for the real data collected by the STAR experiment. Both harmonics as a function of centrality show clear particle species dependence, where the protons flow is the highest. The v 2 {2} is susceptible to the eccentricity, which varies with the centrality of the collision. Oppositely, such a dependence is not observed for the v 3 {2}. The dominating viscous effects cause the elliptic flow attenuation at peripheral collisions (in EPOS approach the corona part dominates there and the jet contribution causes the vanishing of the flow). There is not a substantial enhancement of all particle species flow with increasing collision energy. The difference between the eccentricities in the given centrality bin at different collision energies in the studied range is negligible. Consequently, all the discrepancies between flow measured in collisions at various beam energies can be caused by matter reaction properties.

The p T -differential v n {2} for all energies show the mass ordering and split for the higher p T (≈ 2 GeV/c) to baryon and mesons. This observation is consistent with the studies at higher beam energies such as √ s N N = 200 GeV or LHC energies. This indicates a similar interplay between radial and higher-order flow harmonics. The scaling with the number of constituent quarks is kept for the Au+Au collisions at √ s N N = 200 GeV and for negative particles at lower examined energies. Although there is a small scale breaking for low KE T for positive particles at BES energies, all these observations imply that flow originates mostly from the partonic phase of the system evolution. However, hadrons are created at the freeze-out hypersurface from the flowing fluid, where the mass effect of not present yet. It originates from the following hadronic rescaterring, mainly at low p T .

Both elliptic and triangular flow do not differ between protons and antiprotons at √ s N N = 200 GeV. However, the relevant increase of such differences is present for lower collision energies. In this thesis, a detailed investigation of possible sources of these differences was performed. Firstly, no p T dependence of v 2 (p) -v 2 (p) relations is indicated in examined data. It implies the lack of the differences between flow harmonics interplay of particles and antiparticles. Subsequently, the possible relation of observed discrepancies with the initial geometry of the collision was examined.The triangular flow shows increasing differences between flow of protons and antiprotons with the lowering of √ s N N , similar as observed for elliptic flow. This observation is substantial for the theoretical studies from [89] of additional viscous corrections to the measured flow. The violation by protons and antiprotons n q -scaling of flow in the function of transverse kinetic energy was studied. Protons break the scaling relevantly, while antiprotons' do not. It can lead to the conclusion that these baryons' origin is not the same as it was suggested in [91,92]. However, there is no clear energy dependence of the size of protons' scaling violation. In the proposed theoretical scenario, the lower the collision energy, the more immense contribution of transported protons is into the measured flow. Finally, the kaons were used as a neutral reference for investigating the mean filed potentials' effect on the p and p flow [90]. The mass-independent scaled v 2 {2} of protons is expected to behave oppositely to antiprotons' concerning kaons. The performed measurements do not confirm this scenario.

In the second part of the thesis, studies of various Equations of State implemented in the EPOS model were described. The development of the generator's code by introducing new EoS gave a unique possibility to investigate the impact of EoS on the final observables. Apart of the EoS, the whole structure of the model remained unchanged. Studies of yields, transverse momentum spectra and elliptic flow did not show any substantial effect of switching the EoS. The moments of net protons distributions gave important information about the variations of EoS. The differences between data sets are visible for all studied moments: mean, standard deviation, skewness, and kurtosis. It makes the investigation of fluctuation of the particle distributions the most adequate observable for studies of the EoS.

Data simulated with EPOS 3.4 were compared in various ways with the STAR ones: particles' yields, p T -spectra, elliptic flow (Event Plane) and finally basing on 2-Particle Cumulants, v 2 {2} and v 3 {2}. The model has the biggest problem with a reproduction of higher-p T data and overestimates pions' production in a given range of rapidity. Implementation of the EoS considering the finite µ B made a a major step in the model's development. The agreement with experimental data was not expected because the model has not yet been adjusted to describe lower collision energies. At BES energies, the ac-celerated ions do not have such a flat shape as those at LHC. Consequently, the more spherical shape has to be considered in the description of the binary collisions. Furthermore, in the hydrodynamical evolution included EPOS model only the mean values are propagated. The extension of these calculation are needed to improve the reproduction of the large fluctuations in the CP vicinity.

In conclusion, the performed research significantly impact the studies of nuclear matter's dynamics and properties. Combining the experimental and theoretical investigation made it a unique work providing additional constraints in the model development, studies of thermodynamic relations in expanding matter, and factual information about the higher-order harmonics of flow.

Outlook

The research included in this thesis is a significant reference for both theoretical and experimental studies of the QCD phase diagram. The studies performed as a function of beam energy can be used to map the µ B , T space with the view to investigating properties of the matter and its transitions. The BES I and II are programs that introduced a unique possibility for studying a broad range of the QCD phase diagram. The experimental studies of flow harmonics are an essential reference for developing the hydrodynamical phases in the MC models. The triangular flow measurements provide unique information about the initial stage fluctuations and constrain the studies of viscous effects in the medium. The performed development of EPOS is one of the first steps in preparing the model describing the evolution of the various systems (N+N, A+A) obtained due to the collisions at a wide range of beam energies. Absence of the MC model applicable for the low energies ( √ s N N < 20 GeV) is one of the biggest problems for the characterization of the matter at finite baryonic densities or first-order phase transition.

On the other hand, investigation of the impact of the changes in the EoS withing a single model on the final observables is a milestone for the experimentalist looking for the signatures of phase transitions of the medium. I have tests the sensitivity to EoS of several experimental analysis types yet found only one susceptible to introduces variations in EoS. However, the hydrodynamical evolution included EPOS model only the mean values are propagated. The extension of these calculation are needed to improve the propagation of the large fluctuations in CP vicinity. More studies have to be performed to complete the research, such as studies of the space-time evolution using the HBT methods or extend studies of the particles distributions' moments. These results are crucial for planning the scientific programs for new, still-under-construction experimental facilities such as NICA or FAIR.

Summarizing, the performed research in the frame of this thesis, provide substantial constrain onto the development of our present knowledge and plans for the future studies of the properties of the strongly interacting matter and its transitions.

β = v c = L c∆t (8.6)
Mass square of particle:

m 2 = p 2 ( 1 β -1) (8.7)
where β = p/E and E = √ p 2 + M 2 , p is the measured by TPC momentum of the particle and ∆t is the time.

Figure 9.1 -The measured ADC in sector 24 (top) and 9 (bottom) in different pads. This thesis collider mode, are not expected to be present in the cosmic rays (negative q * p). The particles originating from cosmic rays traverse the detector from the "top" to the "bottom", not from the inner to outer direction as these coming from the ion collisions. The rotation caused by the magnetic field makes the reconstruction of part the tracks looking like they refer to the negatively charged particles. The In the top panels, the experimental data are illustrated with black markers, while two lines, red and blue, correspond to the event generator data, where two different methods of the centrality definitions were applied. One IMP is based on the impact parameter extracted from the simulated data, the GEN corresponds to the method using the multiplicity of the events. In the bottom panels, the ratios of MC data to experimental ones are shown. With the introduction of this analysis's code into the Rivet repository, the popularization of the Rivet in the heavy-ion STAR community was initiated.

Secondly, with my Master student Johannès JAHAN, we modified the EPOS code, adding the possibility of having the output file from simulations consistent with the Rivet software requirements.

This phase of the Rivet development was described and published in [START_REF] Stefaniak | [END_REF]. 11.5, 14.5, 19.6, 27, 39, 64.4 GeV Abstract: Studying the properties of the collective expansion of matter created in heavyion collisions provides a unique tool to understand better the nonperturbative aspect of QCD. Input from both the theoretical and the experimental side is needed. Hydrodynamic calculations predict anisotropies in particle production, as a consequence of asymmetries in the initial state of the system's evolution. Measurements of the systematics (energy-, system-dependence) of these anisotropies al-low one not only to validate theoretical ideas but also to determine the unknown elements, like plasma properties (EoS), thermalization processes. The broadening of our knowledge in these topics is the main goal of the thesis. The experimental methods were used to provide insight into the investigation of the anisotropies in the expansion of the particles and antiparticles, while the theoretical approach was used for the EoS studies.

Figure 1 . 1 -

 11 Figure 1.1 -Examples of interactions included in the Standard Model of particle physics. Names, what do they exchange and the carriers [8].

Figure 1 . 2 -

 12 Figure 1.2 -the QCD interaction between coloured quarks via a coloured gluon r b [9].

Figure 1 . 3 -

 13 Figure 1.3 -QCD potential. The dotted line shows the short distance potential; the solid line includes the long distance potential [12].

Figure 1 . 4 -

 14 Figure 1.4 -QCD Coupling constant α s dependence on the energy scale Q. The points refers to the pQCD calculations: NLO -next0to-leading order, NNLO -next-to-next-toleading order, N 3 NLO -next-to-NNLO [21].

Figure 1 . 6 -

 16 Figure 1.6 -Proposed structure of the QCD Phase Diagram of strongly interacting matter [23].

Figure 1 .

 1 Figure 1.8 -Left: Approaching ions just before the collision, impact parameter b, Right:Participants zone, where the new matter is created and unaffected spectators[43]

Figure 1 . 9 -Figure 1 . 10 -

 19110 Figure 1.9 -Three heavy-ions collisions: central, midcentral and peripheral (This thesis)

Figure 2 . 1 -

 21 Figure 2.1 -Heavy-ion collisin with depicted overlap region and reaction plane. Drawn by Thomas Ullrich

Figure 2 . 2 -

 22 Figure 2.2 -The evolution of the initial transverse energy density profile of the system created in non-central heavy-ion collision. The x-axis is defined by the parameter b and z-axis goes along the beam direction [47].

Figure 2 . 3 -

 23 Figure 2.3 -Elliptic flow in function of transverse momentum dependence on shear viscosity over entropy of charged hadrons compared with STAR experimental data[43,48]

Figure 2 . 4 -

 24 Figure 2.4 -The illustration of the relations between various effects dominating the creation of flow in function of centrality of the collision. (This thesis)

2

 2

Figure 2 . 5 -

 25 Figure 2.5 -Ratio of the centrality dependent v 2 {2} with ∆η 1 = 0.0 and 0.3 to v 2 {2} with ∆η 2 = 0.6 for Au+Au collisions at √ s N N = 200 GeV. The transverse momentum range: 0.2GeV/c < p T < 4.0GeV/c [4]

Figure 2 .

 2 Figure 2.7 -Illustrations of directed and elliptic flow [38].

Figure 2 .

 2 Figure 2.8 -The net-baryon density in the reaction plane (τ = 12f m/c), the velocity arrows at |y| < 0.5 corresponds to fluid elements of anti-flow -thin arrows and normal flow -bold arrows[56]

Figure 2 . 9 -

 29 Figure 2.9 -System dependence of directed flow v 1 (η) measured at STAR experiment for mid-central (30%60%) Au+Au and Cu+Cu at 200 GeV and 62.4 GeV [57].

Figure 2 .

 2 Figure 2.10v 2 {2} p T -differential of identified hadrons for Au+Au collisions at √ s N N = 200 GeV. Part of the article which is now under review in STAR Collaboration (as a Primary Author)

Figure 2 . 11 -

 211 Figure 2.11 -Distribution of nucleons on the transverse plane for a √ s N N = 200 GeV Au+Au collision event with ϵ 3 = 0.53 from Glauber Monte Carlo. The nucleons in the two nuclei are shown in gray and black.Wounded nucleons (participants) are indicated as solid circles, while spectators are dotted circles[62]. For the better visualization with the yellow color the collided ions are selected, with blue the ellipsoidal area corresponding to ϵ 2 and with red triangularϵ 3 .

Figure 2 .

 2 Figure2.12ϵ 2 and ϵ 3 dependence on the system size[63]

  Figure 2.4.

Figure 2 .

 2 Figure 2.13 -The left panel: p T integrated v 2 , 3 coefficients vs.√ s N N . The √ s N N dependent η/s. The experimental data from the STAR collaboration[66]. The right panel shows the effective values of η/s used to describe the experimental data at different collision energies. The green-band represents lines represents the results obtained using the estimated uncertainty[65]

Figure 2 .

 2 Figure 2.14 -Comparison of integrated elliptic flow in the centrality classes similar to 20 -30% in function of collision energy[67]

Figure 2 .

 2 15 there are results published by STAR collaboration showing the test of the NCQ-scaling of the v 2 . Anisotropic Flow Measurements at STAR in the frame of BES BES I is the first phase of a scientific program run at RHIC completed in 2011. It is based on Au+Au collision data collected in 2010-2014 at √ s N N = 7.7,

Figure 2 . 15 -

 215 Figure 2.15 -Test of the constituent quark number scaling of elliptic flow for minimum bias Au+Au collisions at √ s N N = 200 GeV. The dashed lines are polynomial fits [75]

Figure 2

 2 Figure 2.19 -√ s N N dependence of the p T -integrated v n (left panel) and the estimated viscous coefficient β ′′ ∝ ηs (right panel). Results are shown for 0 -40% central Au+Au collisions; the shaded lines are the systematic uncertainty[77,83]

Figure 2 .

 2 Figure 2.20 -Data published by STAR experiment in [84]. The difference in v 2 between particles (X) and the corresponding anti-particles ( X)) as a function of √ s N N for 0 -80% central Au+Au collisions. The dash lines in the plot are fits with a power-law functions

Figure 2 .

 2 Figure 2.21 -Particle's and antiparticle's elliptic flow difference as a function of √ s.Comparison of STAR published data[85] and simulations using the procedures proposed in[89].

Figure 2 .

 2 Figure 2.23 -Upper panel: The elliptic flow of p(3 iq), p(2 iq), p(1 iq), p(0 iq) and p as a function of the transverse momentum pT for 0 -80% central Au+Au collisions at √ s N N = 7.7, 11.5, 39200 GeV. The lower panels show the difference of v 2 (p T ) betweenp(3iq), p(2iq), p(1iq), p(0iq) and p, respectively[92]

Figure 2 . 25 -

 225 Figure 2.25 -An event divided into two subevents A and B with the η gap between them. This thesis

Figure 3 . 1 -

 31 Figure 3.1 -The QCD phase diagram with depicted regions studied in different experimental programs [109]

Figure 3 . 2 -

 32 Figure 3.2 -The RHIC accelerator complex [110].

  in 2005 and BRAHMS [120] in 2006 completed their scientific programs and operations. The last one, in 2015, PHENIX[121] was decommissioned for the development purpose, the sPHENIX is expected to start collecting data in 2023[122, 123].

. 3 .

 3 The STAR experiment consists of such detectors as:-Time Projection Chamber (TPC) -responsible for tracking[124], -Time of Flight (TOF) -used for particle identification[125, 126], -Barrel Electromagnetic Calorimeter (BEMC) -particle energy measurement [127], -Heavy-Flavor Tracker (HFT) -the short-lifetime decay vertices reconstruction [128], -Muon Telescope Detector (MTD) -identification of di-muon decays (from J/ψ and Υs [129]), -Vertex Position Detector (VPD) -position of the primary collision vertex, and start the time counting for the fast-timing detectors (TOF or MTD) [130]; In last two years the upgrade of the STAR experiment took place. The three new systems were installed (illustrated in the Figure 3.4 with the short description) [131]: -Event Plane Detector (EPD) [132] -endcap Time Of Flight (eTOF) [133] -Inner Time Projection Chamber (iTPC) [134] As part of this project, I was engaged in the iTPC software development, what is shortly described in Appendix 9.

Figure 3 . 3 -

 33 Figure 3.3 -STAR experiment schematic illustration with depicted detector subsystems. The TPC, TOF are the detectors of the special interest to this thesis. Figure courtesy of A. Schmah (STAR Collaboration).

Figure 3 . 4 -

 34 Figure 3.4 -STAR experiment schematic illustration with depicted detector subsystems. The TPC, TOF are the detectors of the special interest to this thesis. Figure courtesy of A. Schmah (STAR Collaboration).

Figure 3 . 5 -Figure 3 . 6 -

 3536 Figure 3.5 -The schematic profile of TPC detector at STAR [135]

Figure 3 .

 3 Figure 3.8 -The ionization energy loss distribution of charged particles at √ s N N = 39 GeV. The solid lines are from the Bichsel functions [138].

Figure 3 . 9 -Figure 3 . 10 -Figure 3 . 11 -

 39310311 Figure 3.9 -The details of TOF trays, modules and pads [126]

Figure 3 .

 3 11 shows the example distribution of mass square (Eq. 8.7) in function of ridigity at √ s N N = 39 GeV measured with Time of Flight detector.

Figure 3 .

 3 Figure 3.12 -The left panel is schematic drawing of one of the VPD assemblies, and the right panel is a photograph of both VPD assemblies [130]

Figure 4 . 1 -

 41 Figure 4.1 -The integrated luminosity in function of time of the events registered by STAR for Au+Au collisions at √ s N N = 54.4 GeV [141].

Figure 4 . 2 -

 42 Figure 4.2 -The Quick Analysis (QA) distributions of relations between multiplicity of events detected by TPC and TOF for Au+Au collisions at √ s N N = 54.4 GeV before and after the pileup removal technique application. On the right the distribution of Refmult before (red) and after (blue) applying cuts, and the difference (green). (The plot is normalized to unity) This thesis

  Figure 4.4 illustrates the particles' tracks, collision

Figure 4 . 3 -

 43 Figure 4.3 -The Quick Analysis (QA) TPC V Z distribution on the left, and on the right the relation between the Vertex Z from TPC and VPD. This thesis

  Figures 4.5.

Figure 4 . 4 -Figure 4 . 5 -

 4445 Figure 4.4 -Illustration of the collision vertex and reconstructed particle track with depicted Distance of Closest Approach. This thesis

Figure 4 . 6 -

 46 Figure 4.6 -The dE/dx of charged tracks at midrapidity (|y| < 1.0) plotted as function of rigidity (p/q) in Au+Au collisions at √ s N N = 39 GeV. This thesis

Figure 4 .

 4 Figure 4.7 -The combined information from TPC and TOF with depicted selected regions corresponding to identified hadrons species for collision Au+Au at √ s N N = 39 GeV. This thesis

Figure 4 . 9 -Figure 4 . 10 -

 49410 Figure 4.9 -The ϕ distributions for Au+Au collisions at √ s N N = 39 GeV at 9 centrality bins before and after application of acceptance correction. This thesis

Figure 4 . 11 -

 411 Figure 4.11 -First column consists of the v 2 {2}(p T ) (top) and v 3 {2}(p T ) (bottom) panels with different ∆η between sub-events, the right one correspond to two possible requirements considering DCA for Au+Au at √ s N N = 39 GeV. The bottom panels correspond to ratios of two data sets. This thesis

Figure 5 . 1 -

 51 Figure 5.1 -Comparison of 2PC flow and STAR published flow measurements using Event Plane method[73] for Au+Au at √ s N N = 39 GeV at 0 -10% centrality. In the bottom

Figure 5 . 2 -

 52 Figure 5.2 -Comparison of 2PC flow and STAR published flow measurements using EventPlane method[73] for Au+Au at √ s N N = 39 GeV at 10% -40% centrality. In the bottom panels there is a ratio between this work to fit to the STAR published data, where the dotted lines correspond to the 5% difference. This thesis 107

Figure 5 . 3 -

 53  flow and STAR published flow measurements using Event Plane method[73] for Au+Au at √ s N N = 39 GeV at 40% -80% centrality. In the bottom

Figure 5 . 4 -

 54 Figure 5.4 -The particles and anti-particles elliptic and triangular flow for 0 -80% central Au+Au collisions at √ s N N = 200 GeV, using the two-particle cumulant method.The lower panels (d)-(i) represent the v 2 and v 3 difference between particles and antiparticles. This thesis

Figure 5 . 5 -

 55 Figure 5.5 -The transverse momentum dependence identified particles v 2 , v 3 , and v 4 for 0-30% and 30˘80% central Au+Au collisions at √ s N N = 200 GeV. This thesis

Figure 5 . 6 -

 56 Figure 5.6 -The transverse mass dependence identified particles v 2 , v 3 , and v 4 for 0˘30% and 30˘80% central Au+Au collisions at √ s N N = 200 GeV. This thesis

Figure 5

 5 Figure 5.7 -2PC integrated over p T v 2 {2} for Au+Au collisions at √ s N N = 27, 39 and 54.4 GeV in function of centrality. The ∆η = 0.2 and the p T range is 0.2 to 4.0 GeV/c. This thesis

Figure 5

 5 Figure 5.8 -2PC integrated over p T v 3 {2} for Au+Au collisions at √ s N N = 27, 39 and 54.4 GeV in function of centrality. The ∆η = 0.2 and the p T range is 0.2 to 4.0 GeV/c. This thesis

  s N N = 27GeV √ s N N = 39GeV √ s N N = 54.4GeV

Figure 5 . 9 -

 59 Figure 5.9 -The energy dependence of 2PC integrated over p T v 2 {2} for Au+Au collisions at √ s N N = 27, 39 and 54.4 GeV in function of centrality. The ∆η = 0.2 and the p T range is 0.2 to 4.0 GeV/c. This thesis

Figure 5 . 10 -

 510 Figure 5.10 -The energy dependence of 2PC integrated over p T v 3 {2} for Au+Au collisions at √ s N N = 27, 39 and 54.4 GeV in function of centrality. The ∆η = 0.2 and the p T range is 0.2 to 4.0 GeV/c. This thesis

Figure 5 . 13 - 2 q

 5132 Figure 5.13 -2PC p T -differential v 2 {2} for Au+Au collisions at √ s N N = 27, 39 and 54.4 GeV in centrality 0 -60% scaled with number of constituent quarks (v n {2}/n n/2 q

Figure 5 . 15 -

 515 Figure 5.15 -Difference in 2PC p T -differential v 2 {2} of particles and anti-equivalents for Au+Au collisions at √ s N N = 27, 39 and 54.4 GeV in centrality 0 -60%. This thesis

Figure 5 . 16 -

 516 Figure 5.16 -Difference in 2PC p T -differential v 3 {2} of particles and anti-equivalents for Au+Au collisions at √ s N N = 27, 39 and 54.4 GeV in centrality 0 -60%. This thesis

Figure 6 . 1 -

 61 Figure 6.1 -Hadron-hadron scattering. Red incoming lines represents the hadrons and each green thick line Pomeron -the exchanges happen in parallel. This thesis

Figure 6 . 2 -

 62 Figure 6.2 -Elastic scattering amplitude T . This thesis

Figure 6 . 5 -

 65 Figure 6.5 -The illustration of the nucleon-nucleon rescattering. Two projectiles A and two targets B interaction. The split between participants and remnants shows the momentum sharing between constituents ensuring conservation of given valuable. This thesis

Figure 6 .

 6 Figure 6.10 -The Monte Carlo simulation: full circles correspond to string segments contributing to core and open to corona. The big circles are just for the eye-guidance showing the surface of collided nuclei [154]

Figure 6 .

 6 Figure 6.11 -The energy density (top line) and pressure (bottom line) relations with temperature for EPOS EoS [153] compared with Lattice data [156].

Figure 6 .

 6 Figure 6.12 -The linear transformation is used in mapping the 3D Ising model diagram on the QCD one [166].

Figure 6 . 13 -

 613 Figure 6.13 -Hydronization hyper-surface, with elements dΣ µ and flow momentum vector dP µ . This thesis

3 =

 3 153.4 ± 48.1 MeV ∼ 169 MeV 27 GeV 174.4 ± 80.7 MeV ∼ 184.3 MeV Table7.1 -Critical Temperature (T C ) and T 0 extracted from the fits from Fig.7A(e (ap T +bp 2 T ) + p T /p 0 ) -n (7.2)The spectra with the fits are shown in the Figures 7.1 and 7.2. The extracted T C and calculated with Eq. 7.1 T 0 are listed in the Table7.1.

Figure 7 . 1 -

 71 Figure 7.1 -The p T spectra of positive pions from the most central Au+Au collisions 0 -5% at √ s N N = 7.7 GeV simulated with EPOS 3.4 model with a modified Hagedron fit. This thesis

Figure 7 . 2 -

 72 Figure 7.2 -The p T spectra of positive pions from the most central Au+Au collisions 0 -5% at √ s N N = 27 GeV simulated with EPOS 3.4 model with a modified Hagedron fit. This thesis

Figure 7 . 3 -Figure 7 . 4 -Figure 7 . 5 -Figure 7 . 6 -

 73747576 Figure 7.3 -Energy density (ϵ/T 4 ) dependence on T and µ B . Some of the variables in sets of parameters are highlighted in order to underline the change which they introduce. This thesis

Figure 7

 7 Figure 7.8 -Particle ratios for Au+Au most central 0 -5% collisions at √ s N N = 7.7 GeV simulated with EPOS 3.4. model using various EoS and compared with STAR data [143]. This thesis

Figure 7 . 9 -

 79 Figure7.9 -The p T spectra of the most central 0 -5% Au+Au collisions at √ s N N = 7.7 GeV of Pion+, Pion-, Kaon+, Kaon-, Proton, and Antiproton, simulated with EPOS 3.4. using various EoS (Tab 7.2). Compared with STAR data[143]. In the bottom panels there is a ratio between the simulated and experimental data. This thesis

Figure 7 . 11 -

 711 Figure 7.11 -The p T spectra of the most central 0 -5% Au+Au collisions at √ s N N = 27 GeV of Pion+, Pion-, Kaon+, Kaon-, Proton, and Antiproton, simulated with EPOS 3.4. using various EoS (Tab 7.2). Compared with STAR data [143]. In the bottom panels there is a ratio between the simulated and experimental data. This thesis

Figure 7 . 16 -

 716 Figure 7.16 -Moments of particle distributions: mean, standard deviation, skewness and kurtosis for Au+Au collisions at √ s N N = 7.7 GeV as a function of number of participants.The shifts of the point on x-axis are applied for better visualisation. This thesis

Figure 7 . 17 -

 717 Figure 7.17 -Moments of particle distributions: mean, standard deviation, skewness and kurtosis for Au+Au collisions at √ s N N = 27 GeV as a function of number of participants.

  and 7.21 the elliptic and triangular flow for Au+Au

Figure 7 .Figure 7

 77 Figure 7.18 -The Sσ and κσ 2 for Au+Au collisions at √ s N N = 7.7 (full markers) and 27 GeV (empty markers) as a function of number of participants. The shifts of the point on x-axis are applied for better visualisation. This thesis

Figure 7

 7 Figure 7.21v 3 {2} for Au+Au collisions at √ s N N = 27 GeV for positive and negative particles. Comparison of experimental STAR results (points) and simulated with EPOS3.4 (line). This thesis

2 T

 2 P -2Sn B ∂ T ∂ µ B P + S 2 ∂ 2 µ B P (ϵ + P )(∂ 2 T P ∂ 2 mu B P -(∂ T ∂ µ B P )

Figure 9 .

 9 3 shows first example plots obtained in year 2019 during the 20 th and 25 th days of the data-taking.

Figure 9 . 2 -Figure 9 . 3 -

 9293 Figure 9.2 -The left panel: the measured ADC in sector 7 in different pads with the calculated for each row mean of values for normal and Landau distributions The right panel: the quality of the Landau fit to obtained distributions for each pad row. This thesis

Figure 10 . 1 -

 101 Figure 10.1 -Rivet comparison of v 3 {2} measured by STAR [82] and simulated with Pythia [186]. The most central collisions 0 -5% of Au+Au at √ s N N =11.5, 14.5, 19.6,27,39, 64.4 GeV

Titre:

  Étude de la dépendance en énergie des flux elliptique et triangulaire pour les particules identifiées dans le Beam Energy Scan avec l'expérience STAR et le générateur d'événements EPOS Mot clés : ions lourds, diagramme de phase, équation d'état, flux Résumé : L'étude des propriétés de l'expansion collective de la matière créée dans les collisions d'ions lourds est un outil unique, permettant de mieux comprendre les aspects non-perturbatifs de la QCD, et nécessitant un travail tant sur le plan théorique que sur le plan expérimental. D'une part, des calculs d'évolution hydrodynamique prédisent, en conséquence de l'asymétrie du système dans l'état initial, une anisotropie dans la production des particules finales. D'autre part, les mesures de cette anisotropie, en fonction du système et de l'énergie considérés, permettent non seule-ment de valider les prédictions théoriques, mais aussi de déterminer des éléments inconnus comme les propriétés du plasma créé lors de ces collisions (e.g. son équation d'état), ou bien les processus de thermalisation mis en jeu. Améliorer nos connaissances sur ce sujet est le principal objectif de cette thèse. Les méthodes expérimentales décrites ont pour objectif de donner un aperçu de la mesure de l'anisotropie de l'expansion des particules et anti-particules, tandis que les approches théoriques ont été utilisées pour étudier l'équation d'état de la matière. Title: Beam Energy Scan dependance of elliptic and triangular flow of identified hadrons in the STAR experiment and the EPOS model Keywords: heavy-ions, phase diagram, equation of state, flow

Rivet software development Bibliography

  5.3.3 Particle-antiparticle flow difference . . . . . . . . . . . . . . . . . . 116 5.3.4 Triangular flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 Initial Stage of the evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 126 6.1.1 Parton-Based Gribov-Regge Theory . . . . . . . . . . . . . . . . . . 126 6.1.2 String Model and Flux Tube Approach . . . . . . . . . . . . . . . . 132 6.2 Core-Corona approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.3 Viscous Hydrodynamical evolution . . . . . . . . . . . . . . . . . . . . . . 134 6.3.1 Event-by-event treatment . . . . . . . . . . . . . . . . . . . . . . . 136

	EPOS Model	125
	6.1

6.3.2 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 6.4 Hadronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.5 Hadronic Cascades . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 6.6 New developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.6.1 BEST EoS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.6.2 Microcanonical hadronization . . . . . . . . . . . . . . . . . . . . . 141 The Equation of State at EPOS model 145 7.1 Implementation of EoS into the code of generator . . . . . . . . . . . . . . 145 7.1.1 Selection of the EoS parameters . . . . . . . . . . . . . . . . . . . . 145 7.2 Comparison of various EoS at EPOS model . . . . . . . . . . . . . . . . . 148 7.2.1 Thermodynamical relations . . . . . . . . . . . . . . . . . . . . . . 148 7.2.2 Production of particles . . . . . . . . . . . . . . . . . . . . . . . . . 149 7.2.3 Dynamics of matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.2.4 Event-by-event Fluctuations . . . . . . . . . . . . . . . . . . . . . . 160 7.2.5 Two Particle Cumulants . . . . . . . . . . . . . . . . . . . . . . . . 165 Conclusion 169 7.3 Summary and discussion of results . . . . . . . . . . . . . . . . . . . . . . 169 7.4 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Appendix A: Useful equations 173 8.1 Thermodynamic quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Appendix B: iTPC software development 175 9.1 ADC tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 9.2 Data quality assurance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Appendix C:

table 1 .

 1 1. Nevertheless, each quark is characterised by a different quantum number called colour, which can come in the three various colours and

		Quarks			Leptons
	Generation Symbol Charge Mass	Symbol Charge Mass
	1	u	2/3e	1.7 -3.3M eV /c 2	e	-e	0.511M eV /c 2
		d	-1/3e 4.1 -5.8M eV /c 2	v e	0	2eV /c 2
	2	c	2/3e	1.18 -1.34GeV /c 2	µ -	-e	105M eV /c 2
		s	-1/3e 80 -130M eV /c 2	v mu	0	0.19M eV /c 2
	3	t	2/3e	169.8 -174.2GeV /c 2 τ -	-e	1.68GeV /c 2
		b	-1/3e 4.1 -4.4GeV /c 2	v τ	0	18.2M eV /c 2

Table 1 .

 1 1 -Elementary particles, their mass and charges as described in the Starndard Model of particles physics[7]

Table 4 .

 4

1 -The data sets used in analysis, and the basic selection criteria.

Table 4 .

 4 2 -Numbers of events after application of selection criteria.

	√ s N N	DCA	nHitsF it N hits/Hitmax p T [GeV /c] η
	27 GeV	< 3 cm > 20	> 0.52	⟨0.2, 4.0⟩	< 1
	39 GeV	< 3 cm > 20	> 0.52	⟨0.2, 4.0⟩	< 1
	54.4 GeV < 3 cm > 20	> 0.52	⟨0.2, 4.0⟩	< 1
	200 GeV < 3 cm > 15	> 0.52	⟨0.2, 4.0⟩	< 1
		Table 4.3 -Track selection criteria
	vertex with DCA depicted.

Table 4 .

 4

	6.

Table 4 .

 4 4 -Contributions to systematic uncertainties of v 2 of identified hadrons for Au+Au √ s N N = 39 GeV.

	particle DCA nHits V z	∆η
	pions	9.7% 2.7% 4.6%. 1.0%
	kaons	5.8% 5.0% 0.6%. 2.0%
	protons 9.5% 1.4% 8.7%. 0.7%

Table 4 .

 4 5 -Contributions to systematic uncertainties of v 3 of identified hadrons for Au+Au √ s N N = 39 GeV.

	v n pions kaons protons
	v 2 5.3% 4.4% 4.7%
	v 3 11.1% 7.9% 11.9%

Table 4 .

 4 6 -Systematic uncertainties of v n of identified hadrons for Au+Au √ s N N = 39 GeV.

Table 7 .

 7 the ω = 10, what suppresses the criticality and makes the illustrated 2 -Sets of input parameters for the construction of nine BEST EoS used in the EPOS simulations. surface in the Fig. 7.3 the most smooth one. The sets BEST 6 and 7 have CP placed in the approximate temperature of chemical freeze-out of system simulated by EPOS in Au+Au collision at √ s N N = 7.7 GeV. The BEST 8 is characterized by very high T C = 157 MeV and µ BC = 440 MeV (what is nearly the top available potential). On the other hand, the last, ninth set of parameters correspond to the highest studied T C = 170 MeV and lowest µ BC = 300 MeV, what places the system obtained at the examined Au+Au at √ s N N = 7.7 GeV approximately at the line of first-order phase transition. In case of simulations of Au+Au collisions at

	Number M ODE	T 0	κ	µ BC ∆ α1,2 ω	ρ T C µ BC α 1 α 2 ωT C	ωT C
	BEST 1:	PAR	155 -0.0149 350	90	1	2 143 350	3	93 143	286
	BEST 2:	PAR	155 -0.0149 350	90	4	1 143 350	3	93 572	572
	BEST 3:	PAR	155 -0.0149 420	90	0.75 2 138 420	4	94 103	207
	BEST 4:	PAR	155 -0.0149 350	90	10	1 143 350	3	93 1432 1432
	BEST 5:	PAR	169 -0.0149 420	90	0.25 2 165 3	86 179 23599 6749k
	BEST 6:	PAR	169 -0.0149 420	90	1	1 153 420	4	94 153	153
	BEST 7:	PAR	169 -0.0149 420	90	0.5 1 153 420	4	94 76	76
	BEST 8:	PAR	174 -0.0149 440	90	1	1 157 440	4	94 157	157
	BEST 9:	PAR	178 -0.0149 300	90	1	1 170 300	2	92 170	170

  = 27 GeV The mesons' spectra up to p T < 1 GeV/c are reasonably reproduced by the EPOS model. The model parameters are not yet adjusted to the lower

			10 3		10 3						10 3	10 3
						X3F	EoS: BEST 9			X3F	EoS: BEST 9
			10 2		10 2	X3F BEST par 1 BEST 1	STAR	BEST par 3 BEST par 4	10 2	10 2	X3F BEST par 1 BEST 1	STAR	BEST par 3 BEST par 4
	1 d 2 N	2π pT dydpT	10 -1 10 0 10 1	pT /dηdpT dN 2π	10 -1 10 0 10 1 AuAu@ centrality 0 -5% √ sNN = 27GeV pions+		√ sNN = 7.7GeV centrality 60 -80% pions+ BEST par 2 STAR d 2 N 1	pT dydpT 2π	10 -1 10 1 10 0	pT /dηdpT dN 2π	10 -1 10 0 10 1 AuAu@ centrality 0 -5% √ sNN = 27GeV pions-	√ sNN = 7.7GeV centrality 60 -80% pions-BEST par 2 STAR
	10 -2 0.00 -5 0 5 ST AR 10 -2 10 -1 10 0 10 1 10 2 10 3 pT /dηdpT dN 2π EoS 0.00 -5 0 5 EoS ST AR 10 -2 10 -1 10 0 10 1 10 2 10 3 pT /dηdpT dN 2π 0.00 -5 0 5 EoS ST AR Figure 7.10 -The p T spectra of the most peripheral 60% -80% Au+Au collisions at 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 pT (GeV/c) 10 -2 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 pT (GeV/c) -5 0 5 EoS ST AR √ sNN = 7.7GeV centrality 60 -80% kaons+ EoS: X3F BEST par 1 BEST par 2 BEST par 3 BEST par 4 STAR 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 pT 10 -2 10 -1 10 0 10 1 10 2 10 3 pT /dηdpT dN 2π √ sNN = 7.7GeV centrality 60 -80% Kaons-EoS: X3F BEST par 1 BEST par 2 BEST par 3 BEST par 4 STAR 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 pT (GeV/c) -5 0 5 EoS ST AR √ sNN = 7.7GeV centrality 60 -80% protons EoS: X3F BEST par 1 BEST par 2 BEST par 3 BEST par 4 STAR 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 pT (GeV/c) 10 -2 10 -1 10 0 10 1 10 2 pT /dηdpT dN 2π √ sNN = 7.7GeV centrality 60 -80% antiprotons EoS: X3F BEST par 1 BEST par 2 BEST par 3 BEST par 4 STAR 0.2 0.4 0.6 0.8 1.0 1.2 pT (GeV/c) -5 0 5 EoS ST AR √ s N N = 7.7 GeV of Pion+, Pion-, Kaon+, Kaon-, Proton, and Antiproton, simulated with EPOS 3.4. using various EoS (Tab 7.2). Compared with STAR data [143]. In the bottom panels there is a ratio between the simulations based on BEST EoS and X3F. This thesis points of ratio variate around unity, but they are marked with high uncertainties. Au+Au at √ 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 p T (GeV/c) 0.8 1.0 1.2 10 -2 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 p T (GeV/c) 0.8 1.0 1.2 EoS X3F 10 -2 X3F 10 -1 10 0 10 1 10 2 10 3 2π EoS 1 d 2 N pT dydpT AuAu@ √ sNN = 27GeV centrality 0 -5% kaons+ X3F BEST 1 BEST 9 STAR 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 p T (GeV/c) 0.8 1.0 1.2 EoS X3F 10 -2 10 -1 10 0 10 1 10 2 10 3 1 2π d 2 N pT dydpT AuAu@ √ sNN = 27GeV centrality 0 -5% kaons-X3F BEST 1 BEST 9 STAR 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 p T (GeV/c) 0.8 1.0 1.2 EoS X3F 10 -2 10 -1 10 0 10 1 10 2 10 3 1 2π d 2 N pT dydpT AuAu@ √ sNN = 27GeV centrality 0 -5% protons X3F BEST 1 BEST 9 STAR 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 p T (GeV/c) 0.8 1.2 10 -2 10 -1 10 0 10 1 10 2 1 2π d 2 N pT dydpT AuAu@ √ s N N = 27GeV centrality 0 -5% antiprotons X3F BEST 1 BEST 9 STAR 0.2 0.4 0.6 0.8 1.0 1.2 p T (GeV/c) 0.8 1.2 s N N 10 -2 1.0 EoS X3F 1.0 EoS X3F

  ). The differences are becoming more negligible with the increase in the number of participants. The low-multiplicity events consist mostly of the core particles, so going through the hydrodynamical evolution. The effect of changes of EoS, as it is expected, is mostly enhanced there. For collisions at √ s N N = 7.7 GeV the statistic error bars are relevantly bigger than in case of collisions at √ s N N = 27 GeV. The clear separation between the various data sets is visible for the net-

		22		0.035 0.040					BEST 9 BEST 6 6		0.035 0.040	Au+Au @	√ s N N = 7.7 GeV	BEST 9 BEST 6
		20		0.030	AuAu @	√ sNN = 7.7GeV			BEST 1 X3F		0.030	AuAu @	√ sNN = 7.7GeV	BEST 1 X3F
	Mean	18	dN/dvobs	0.015 0.020 0.025	π-centrality 0 -80% Au+Au @	√ s N N = 7.7 GeV	StDev	2 4	dN/dvobs	0.015 0.020 0.025	π+ centrality 0 -80%
		16		0.005 0.010		BEST 1 BEST 4 BEST 6	BEST 9 X3F		0		0.005 0.010	BEST 1 BEST 6 BEST 4	BEST 9 X3F
	Scewness	-4 2 4 0 -2	-1.00 -0.75 -0.50 -0.25 -1.00 -0.75 -0.50 -0.25 K-centrality 0 -80% AuAu @ √ sNN = 7.7GeV -1.00 -0.75 -0.50 -0.25 p centrality 0 -80% AuAu @ √ sNN = 7.7GeV Figure 7.15 -The event-by-event v obs 0.00 0.25 0.50 0.75 vobs 0.000 0.00 0.25 0.50 0.75 vobs 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 dN/dvobs 0.00 0.25 0.50 0.75 BEST 9 1.00 1.00 BEST 6 BEST 1 X3F 1.00 vobs 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 dN/dvobs BEST 9 BEST 6 BEST 1 X3F 2 distributions of pions, kaons, protons for Au + Au -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 vobs 0.000 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 vobs 0.0000 0.0025 0.0050 0.0075 0.0100 0.0125 0.0150 0.0175 0.0200 dN/dvobs K+ centrality 0 -80% AuAu @ √ sNN = 7.7GeV BEST 9 BEST 6 BEST 1 X3F -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 vobs 0.000 0.001 0.002 0.003 0.004 0.005 dN/dvobs p centrality 0 -80% AuAu @ √ sNN = 7.7GeV BEST 9 BEST 6 BEST 1 X3F collisions at √ s N N = 7.7. GeV for centrality 0 -80%. This thesis (corresponding to systems obtained at higher 100 200 300 400 Npart 0 100 200 300 400 Npart 0 100 200 300 400 Npart Au+Au @ √ s N N = 7.7 GeV 0 100 200 300 400 Npart 2 Au+Au @ √ s N N = 7.7 GeV √ s N N 0 BEST 1 BEST 9 0 Kurtosis -2 BEST 1 BEST 9 BEST 4 BEST 6 X3F BEST 6 -4 BEST 4 X3F

5-3 fm [182]

. During the fireball evolution after the hadronisation stage, the freeze-out signal information can dissipate

[184]

. However, if it survives,

In the case of a perfect fluid, the kinetic energy of the flow is not transformed into heat and has low viscosity

ε -parameter characterizing the shape of conic (ε = 0 -circle, ε > 0&ε < 1 ellipse, etc.)

He, Zr ions, and protons (max √ s N N = 510 GeV)[113]. Now, this program is extended -

improved versio of the Bethe-Bloch formula[14].

Sigitta of a track is the distance from the centre of the reconstructed track curvature to the centre of its base.

Mandelstam variables are Lorentz-invariant variables describing the kinematics of particle reactions, carring th einformation about the momenta of particles before and after collisions

Elliptic flow

The simulated flow for Au+Au collisions at √ s N N = 7.7 GeV (Fig. 7. 13) shows the good agreements with the STAR data. However, the high p T part is marked with big statistic uncertainties. In case of Au+Au collisions at √ s N N = 27 GeV (Fig. 7.14) the differences between simulated and experimental data are more vital. The biggest discrepancies are visible for protons.

The anisotropies in flow are expected to be mostly built during the QGP phase, so in the simulations' hydrodynamic evolution. As this part is based on the EoS one could expect elliptic flow measurements to be sensitive to changes of EoS. However, as showed in Figures 7. 13 and 7.14, the averaged over all events v 2 calculated using Event Plane method [43], does not emphasize such differences.

Event-by-event Fluctuations

The observables integrated over multiple events described in previous sections, do not indicate any sensitivity to changes of EoS. Nevertheless, the whole system is not expected to go through the transition simultaneously. More possible is partial participation in these processes, what could be studied in the event-by-event (ebe) fluctuations analysis [179]. Close to the CP, it is expected to observe multiple oscillations of several variables. In this subsection, the ebe investigations of elliptic flow and moments of particles' distributions are reported.

Event-by-event elliptic flow

In the Sec.7.2.3, the shown results correspond to the average overall available events. In addition, in Fig. 7.15 there is a distribution of v obs 2 (not corrected with resolution of event plane) of each event. The Gaussian-like shape with the centre in 0.0 does not differ between simulations based on various EoS. The possible explanation is the attenuation of the searched differences by the subsequent after hydrodynamic evolution, hadronic phase.

Moments of particle distributions

The non-monotonic behaviour in the event-by-event fluctuations of globally conserved quantities is treated as one of the signatures of the presence of CP [180,181]. The moments of distributions characterising the given fluctuations are: Figure 7.14 -Elliptic flow of pions, kaons and protons calculated using Event Plane method (described in [43]) for Au+Au collisions at √ s N N = 27 GeV, centrality 0 -80%.

Various data sets corresponds to simulated with EPOS 3.4. and experimental STAR data from [73]. This thesis the higher moments can become useful in studies of CP's location. As the CP's location is changed in various used EoS, the moments of particle distributions are expected to be a useful tool in the performed investigation. In this section, the net-proton number is studied as a function of the number of participants and collision energy.

In Figures 7. 16 and 7.17 the simulated with EPOS model moments of net-proton distributions as a function of number of participants (Npart) for Au+Au collisions at √ s N N = 7.7 and 27 GeV are plotted. The various EoS are used as an input to the hydrodynamical evolution.

The differences between data sets are visible, starting from the mean of the distributions of net-protons. For collisions, at √ s N N = 7.7 GeV the standard deviation is relevantly lower for the BEST9 data set, in which the CP is located at high T and low µ B Chapter 9

APPENDIX B: ITPC SOFTWARE DEVELOPMENT

As a member of the STAR Collaboration, I participated in the service work. This is a 2-months project which is not directly related to the topic of my analysis, however, it is profitable for the whole Collaboration.

During the installation of the new sectors of TPC called Inner Time Projection Chamber (iTPC) the various tests had to be performed in order to ensure the quality of the future performance of the detector. During my service work I was engaged in ADC tests and the first quality assurance (QA) of the data collected by iTPC during cosmic rays taking. Only the short part of the performed studies description and few figures are placed in the Appendix.

ADC tests

The measured in each sectors' pads voltage was investigated. The Figure 9.1 is the visualisation of the measured voltage signals in sectors 24 and 9. The not uniform distributions were studied and taken into account in the process of the reconstruction data. The Fig. 9.2 shows the two possible pad row mean voltage extraction: with the normal distribution mean and fitted with Landau one.

Data quality assurance

Before the injection of the beams into the RHIC there is approximately one month of the cosmic rays data-taking. The general characteristic of these data sets is well-known, so its analysis is a perfect tool for testing the quality of data-taking by the detectors. In Figure 9.3, the dependence of the particle energy loss in function of momentum registered by iTPC and TPC is shown. The reconstructed tracks corresponding to antiprotons in

APPENDIX C: RIVET SOFTWARE DEVELOPMENT

Another additional project, I am engaged is the development of the Rivet software. My input is shortly described in this Appendix.

Rivet Robust Independant Validation of Experiment and Theory

A Rivet is a software dedicated to comparing the simulated with event generator, and experimental data [Rivet-website]. It is a simple-in-use, and standardized tool developed thanks to the contribution of the users community. Rivet consists of a set of hundreds of analysis based on the experimental published research. It gives a mutual benefit for experimentalists and theorists. The first ones have easy access to outputs of many models, which can be used for the explanation of obtained results. On the other hand, theorists developing MC models can profit from the library consisting of the significant variety of available experimental data, including different measured observables. Moreover, it is still growing as new analysis are submitted by the Rivet community users.

The first version of the Rivet software was dedicated only to the p-p collision systems. The second part of the software development was mainly related to including the heavy ions in the possible collision systems. There were multiple problematic issues to be solved, like centrality definition, the introduction of cumulants, or providing the framework for HBT analysis. Secondly, as the development of the project mainly depends on the experimentalists' engagement, it had to be popularized in Collaborations performing studies of the heavy-ion collisions. I am engaged in developing the Rivet software from both sides: experimental (STAR Collaboration) and theoretical (represent authors of EPOS model). During my thesis, I wrote the first STAR analysis, which was added to the Rivet resources and can be found at https://rivet.hepforge.org/analyses/STAR_2016_I1414638.html. It is the code that can be used for comparing the experimental data from [82] with any MC model providing the output in HepMC data format. In Figure 10.1, the few examples output plots