DOCTORAL THESIS IFT–T.007/2021 Femtoscopy with Strange Particles in pPb Collisions at √ sNN = 8.16 TeV at CMS Dener de Souza Lemos Advisor Dra. Sandra dos Santos Padula Co-Advisor: Dr. César Augusto Bernardes November of 2021 Dener Lemos Dener Lemos TESE DE DOUTORAMENTO IFT–T.007/2021 Femtoscopia com Partículas Estranhas em Colisões pPb a √ sNN = 8.16 TeV no CMS Dener de Souza Lemos Orientadora Dra. Sandra dos Santos Padula Co-Orientador: Dr. César Augusto Bernardes Novembro de 2021 Lemos, Dener de Souza. L557f Femtoscopia com partículas estranhas em colisões pPb a 𝑆 = 8.16 TeV no CMS / Dener de Souza Lemos. – São Paulo, 2021 236 f. Tese (doutorado) – Universidade Estadual Paulista (Unesp), Instituto de Física Teórica (IFT), São Paulo Orientador: Sandra dos Santos Padula Coorientador: César Augusto Bernardes 1. Partículas estranhas (Física nuclear). 2. Partículas (Física nuclear). 3. Femtoscopia. I. Título Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca do Instituto de Física Teórica (IFT), São Paulo. Dados fornecidos pelo autor(a). I dedicate this thesis to my parents Maria Terezinha de Souza Lemos and Rudimar José Morais de Lemos i Acknowledgments I would like to thank: • first, my advisor Sandra dos Santos Padula, for all the fruitful discussions, advices and for the friendship, being fundamental to the development of this work and for my academic carrier. I have no words to say how thankful I am and I will always be proud to call you as my "scientific mother". • my co-advisor César Augusto Bernardes, for being a great friend, for teaching me so much about the CMS stuff and constantly willing to help me. • to Syndell Fernandes, for always encouraging me with a lot of love, affection and patience. Thank you for stay on my side and make me so happy. • my parents Rudimar Lemos, Maria Terezinha Lemos, my grand mother Deni Lemos and my godparents Janete Lemos e Claudeci Costa, for all the support that you gave to me during my life. • to professor Sérgio Novaes, first for accepting me as a member of the SPRACE group and also for all the advices and teachings. I would also like to thank the other members of SPRACE, the professors: Eduardo Gregores, Pedro Mercadante and Thiago Tomei; the postdocs: Sudha and Sunil; and the students: Ana, Breno, Felipe, João, Isabela and Tulio; for all the useful discussions and great moments. • my friend Rafael Reimbrecht, for ever stay on my side even with the distance. • to professor Otavio Socolowski Jr., for introduce me to the heavy ion world. Thanks for all the discussions and for the friendship. • the friends, from NCC: Allan, André, Jeff, Luigi, Ricardo and Sidney. From IFT: Anacé, Felippe, Gabriel, Jogean and Rafael. And from CMS heavy ion group. Thanks for all the moments and coffee’s. • to the IFT-UNESP professors and secretaries, for the support, kindness and friend- ship towards me. An special thank you to Dona Jô for the affection. • to Fernandes family for the support. • to São Paulo Research Foundation (FAPESP) for the financial support. This material is based upon work supported by the FAPESP grants No. 2017/02675-6 vinculated to the process No 2013/01907-0. Any opinions, findings, and conclusions or rec- ommendations expressed in this material are those of the Author(s) and do not necessarily reflect the views of FAPESP. ii “I would rather have questions that can’t be answered than answers that can’t be questioned.” Richard P. Feynman iii Resumo A femtoscopia é um método utilizado para investigar as dimensões espaço- temporais da fonte emissora de partículas criada em colisões a altas energias, através de correlações entre elas. Tais correlações são sensíveis à estatística quân- tica obedecida pelas partículas idênticas envolvidas, assim como à interação forte sentida pelos hádrons, sejam eles idênticos ou não-idênticos. O presente trabalho apresenta resultados de correlações femtoscópicas de partículas para todas as combinações de pares de K0 S, Λ e Λ, com dados do Run 2 do LHC coletados pelo experimento Compact Muon Solenoid (CMS) em colisões próton-chumbo (pPb) a √ sNN = 8.16 TeV. Neste trabalho, as correlações femtoscópicas são me- didas utilizando a técnica de single ratio para partículas do mesmo evento em relação a partículas de eventos diferentes, juntamente com outros métodos basea- dos nos dados, empregados para extrair as informações desejadas. O modelo Lednicky-Lyubolshitz é aplicado para parametrizar as interações fortes entre há- drons, permitindo obter tanto os observáveis de espalhamento, quanto o tamanho da fonte emissora de partículas. A presente análise é realizada usando even- tos com ampla variação de multiplicidades de partículas carregadas e momento transversal médio do par. Esse é a primeira medida de correlações femtoscópica de K0 SK0 S em colisões pPb e a primeira de correlações de ΛΛ e K0 SΛ⊕K0 SΛ em colisões de sistemas pequenos. No estudo das fortes utilizando a femtoscopia de pares bárion-antibárion, observa-se uma anticorrelação, enquanto o comportamento oposto é visto no caso de correlações de pares bárion-bárion, sendo ambos con- sistentes com resultados de outros experimentos. Além disso, os parâmetros de espalhamento para pares bárion-antibárion mostraram independência em relação à multiplicidade de partículas carregadas. Palavras Chaves: Femtoscopia; Partículas estranhas; Interação forte; Sistemas pequenos de colisão; Áreas do conhecimento: Física; Física de Partículas Experimental; Física Nuclear de Altas Energias. iv Abstract Femtoscopy is a powerful method for probing the space-time dimensions of particle emitting sources created in high energy collisions through particle corre- lations. Such correlations are sensitive to the quantum statistics obeyed by the identical particles involved, as well as to the strong interaction felt by hadrons, for both identical or non-identical pairs. This work presents results of femtoscopic correlations for all pair combinations of K0 S, Λ and Λ with data from the LHC Run 2 collected by the Compact Muon Solenoid (CMS) experiment in proton-lead collisions at √ sNN = 8.16 TeV. Detailed studies of the femtoscopic correlations are performed employing the single ratio technique of particles from the same event to those from different events, together with other data driven methods employed to extract the desired information. The Lednicky-Lyubolshitz model is used to parametrize the strong interactions, allowing to obtain the scattering observables, as well as the size of the particle emitting region. The present analysis is performed using samples of events in a wide range of charged particle multiplicities and pair average transverse momenta. This is the first femtoscopic correlation mea- surement of K0 SK0 S in pPb collisions and the first ΛΛ and K0 SΛ⊕K0 SΛ correlation results in small colliding systems. In the study of the strong interactions using femtoscopy of baryon-antibaryon pair, an anticorrelation is observed, whereas the opposite behavior is seen in the case of baryon-baryon pair correlations, both observations being consistent with previous measurements. Furthermore, the baryon-antibaryon scattering parameters showed an independent behavior with respect to charged particle multiplicity. Keywords: Femtoscopy; Strange particles; Strong interaction; Small colliding systems; Knowledge Areas: Physics; Experimental Particle Physics; High-energy Nuclear Physics. v Contents 1 Introduction 1 2 Strong Interactions and the Quark-Gluon Plasma 3 2.1 Strongly interacting particles . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 QCD running coupling constant . . . . . . . . . . . . . . . . 9 2.2.2 Color confinement and asymptotic freedom . . . . . . . . . 10 2.3 Quark-Gluon Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3.1 Heavy ion collision evolution . . . . . . . . . . . . . . . . . . 14 2.3.2 QCD phase diagram . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.3 QGP signatures . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.4 Small colliding systems . . . . . . . . . . . . . . . . . . . . . 27 3 Femtoscopy 35 3.1 Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.2 Quantum statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.2.1 Source function parametrization . . . . . . . . . . . . . . . . 45 3.2.2 Spin dependence . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.3 λ parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Final state interactions . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.1 Strong interactions . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3.2 Coulomb interactions . . . . . . . . . . . . . . . . . . . . . . 58 3.4 Highlights of previous measurements . . . . . . . . . . . . . . . . . 61 3.4.1 K0 SK0 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.2 K0 SΛ⊕K0 SΛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4.3 ΛΛ⊕ΛΛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.4.4 ΛΛ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4 The CMS Experiment 72 4.1 Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.1.1 CERN accelerator complex . . . . . . . . . . . . . . . . . . . 74 4.1.2 LHC detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 vi 4.1.3 Future of LHC and ion colliders . . . . . . . . . . . . . . . . 76 4.2 Compact Muon Solenoid . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2.1 General view . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.2.2 Tracker system . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.2.3 Electromagnetic calorimeter . . . . . . . . . . . . . . . . . . . 86 4.2.4 Hadronic calorimeter . . . . . . . . . . . . . . . . . . . . . . . 87 4.2.5 Superconducting Solenoid . . . . . . . . . . . . . . . . . . . . 90 4.2.6 Muon system . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.2.7 Trigger system and data acquisition . . . . . . . . . . . . . . 93 4.2.8 Computational infrastructure . . . . . . . . . . . . . . . . . . 95 4.2.9 Track and vertex reconstruction . . . . . . . . . . . . . . . . . 96 5 Data Analysis 103 5.1 Datasets, simulations and event selection . . . . . . . . . . . . . . . 103 5.1.1 Trigger selection . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.1.2 Monte Carlo simulations . . . . . . . . . . . . . . . . . . . . . 107 5.1.3 Event reconstruction and selection . . . . . . . . . . . . . . . 109 5.2 Reconstruction of strange particles . . . . . . . . . . . . . . . . . . . 113 5.2.1 Removal of misidentified candidates . . . . . . . . . . . . . . 115 5.2.2 Duplicated tracks removal . . . . . . . . . . . . . . . . . . . . 119 5.2.3 Invariant mass distributions . . . . . . . . . . . . . . . . . . . 123 5.2.4 V0 efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.3 Femtoscopic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.3.1 Purity correction . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.3.2 Non-femtoscopic background . . . . . . . . . . . . . . . . . . 135 5.3.3 Non-prompt contribution . . . . . . . . . . . . . . . . . . . . 144 5.3.4 Fitting the correlation function . . . . . . . . . . . . . . . . . 147 5.4 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 149 6 Experimental Results 157 6.1 K0 SK0 S correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.2 K0 SΛ⊕K0 SΛ correlations . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.3 ΛΛ⊕ΛΛ correlations . . . . . . . . . . . . . . . . . . . . . . . . . . 161 6.4 ΛΛ correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.4.1 Multiplicity dependence . . . . . . . . . . . . . . . . . . . . . 166 6.4.2 Comparison with other experiments . . . . . . . . . . . . . . 169 vii 7 Summary and Outlook 171 A Definitions and Concepts 174 A.1 Beam collision energy . . . . . . . . . . . . . . . . . . . . . . . . . . 174 A.2 Matrices and properties . . . . . . . . . . . . . . . . . . . . . . . . . 175 A.2.1 Dirac matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 A.2.2 Gell-Mann matrices . . . . . . . . . . . . . . . . . . . . . . . 177 A.3 Collision geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 A.4 Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 B QGP: Thermodynamics and Hydrodynamics 181 B.1 Equation of state with first order phase transition . . . . . . . . . . 181 B.1.1 QGP phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 B.1.2 Hadron gas phase . . . . . . . . . . . . . . . . . . . . . . . . 184 B.1.3 Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . 186 B.2 Relativistic Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . 186 B.2.1 Transport coefficients . . . . . . . . . . . . . . . . . . . . . . . 189 B.2.2 The hydrodynamical code CHESS . . . . . . . . . . . . . . . 190 C Lednicky-Lyubolshitz model 192 C.1 Non-identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . 192 C.2 Identical particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 D Service Work and Run Activities 200 D.1 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 D.1.1 Computing Shifts . . . . . . . . . . . . . . . . . . . . . . . . . 200 D.1.2 CMS Heavy Ion Tracking Group . . . . . . . . . . . . . . . . 201 D.2 2019 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 D.2.1 CMS Heavy Ion Tracking Group . . . . . . . . . . . . . . . . 201 D.2.2 CMS Heavy Ion Global Observables Group . . . . . . . . . . 202 D.3 2020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 D.3.1 CMS Heavy Ion Global Observables Group . . . . . . . . . . 202 D.3.2 CMS Heavy Ion High-Level Trigger Group . . . . . . . . . . 203 D.4 2021 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 D.5 2018 Heavy Ion Run (PbPb) . . . . . . . . . . . . . . . . . . . . . . . 204 D.6 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 Bibliography 205 Conventions In this work we adopt the system of natural units, h̄ = c = kB = 1 , where h̄ is the normalized Planck constant, c is the speed of light, and kB the Boltzmann constant. The Minkowski metric (+,−,−,−) is employed throughout this work. Chapter 1 Introduction One of the main goals of modern physics is to understand the elementary structure of matter and the fundamental laws that govern the Universe. In na- ture there are four well-known interactions: strong, weak, electromagnetic and gravitational. The strong force describes the interactions between partons (quarks and gluons), which are normally confined inside hadrons. However, at very high energy densities and temperatures, partons are no longer confined in hadronic bound states, but may form a novel state of matter called quark-gluon plasma. This hot and dense medium is believed to have been present during the first microseconds after the Big Bang and may be part of the core of neutron stars. The possibility of creation of such a state of matter in the laboratory was first proposed in the seventies. One way to recreate the QGP in the laboratory is by using rela- tivistic heavy ion collisions, that has been extensively studied experimentally and theoretically over the years. Since this novel state of matter could exist only for a short period of time, its direct observation would not be possible. Nevertheless, a few indirect probes have been suggested to investigate its formation. Such signa- tures were observed and measured at the Relativistic Heavy Ion Collider (RHIC) in gold-gold collisions at nucleon pair center-of-mass energy √ sNN = 200 GeV [1]. However, different than the initial expectations, a strongly-coupled matter was observed, behaving more like a perfect fluid than a gas. Since then, the study of the properties of the hot and dense medium has been fundamental to allow a better understanding of the strong interactions. In 2010, the Large Hadron Collider (LHC) also measured hints of a QGP-like behavior in events with large number of charged particles produced in proton-proton collisions [2], generating more and more questions about the formation of this state of matter and opening an entire new world of study with the so-called small colliding systems. The usage of two-particle correlations in the low relative momentum of the pair to estimate the space-time dimensions of the particle emitting source, the so called femtoscopy, is a tool that has been largely employed in high energy collision mea- surements during several decades. This effect in high energy collisions was first 1 Chapter 1. Introduction 2 observed, accidently, in the search for the ρ meson decaying into opposite-charged pions, in proton-antiproton collisions [3]. The experiment observed that identi- cally charged pions followed an angular distribution different from the oppositely charged ones, which could only be explained by considering the symmetrization of their wave-function (Bose-Einstein quantum statistics). In the seventies, such correlations were also suggested as one of the possible QGP signatures. Over the years, the study of femtoscopy has allowed several measurements, from estimation of the volume (multidimensional) and lifetime of the system formed in high energy collisions, as well as the effects of Coulomb or strong final state interactions. This thesis presents the first study of femtoscopic correlations of K0 SK0 S, K0 SΛ⊕K0 SΛ, ΛΛ⊕ΛΛ and ΛΛ recorded by the Compact Muon Solenoid (CMS) experiment [4] at the LHC in proton-lead (pPb) collisions at √ sNN = 8.16 TeV using events in a wide range of charged particle multiplicities. The data used in the analysis that lead to this thesis was collected in the Heavy Ion Run that happened in November and December of 2016. The aim of this work is the measurement of the source size and the strong final state interactions scattering parameters, where the knowledge of such interactions is lacking. The measurements are performed as functions of charged particle multiplicity (K0 SK0 S and ΛΛ) and of the pair transverse mass (only for K0 SK0 S). This work is structured as follows. In Chapter 2, we present a review of quan- tum chromodynamics and the quark-gluon plasma, including history, signatures and some interesting results, also for small colliding systems. Chapter 3 contains a theoretical approach of femtoscopic correlations and previous experimental measurements of the correlations of interest. Chapter 4 gives an overview of the LHC and outlines the CMS experiment. The complete data analysis procedure, including a detailed discussion about background estimation and systematic un- certainties are presented in Chapter 5. The results for all pair correlations are presented in Chapter 6. The conclusions and outlook are given in Chapter 7. In ad- dition to the regular chapters, this thesis includes four appendices. Definitions and concepts used in the text are described in more details in Appendix A. Appendix B presents information about the thermodynamic and hydrodynamic description of the quark-gluon plasma. The derivation of the Lednicky-Lyubolshitz model is described in Appendix C. At the end, Appendix D briefly describes activities performed in the CMS collaboration. Chapter 2 Strong Interactions and the Quark-Gluon Plasma Particle physics is the scientific field responsible for studying the building blocks of matter (elementary particles) and their interactions. This field of physics was born in the 1880s and the first elementary particle discovered was the elec- tron, in 1897, by J. J. Thomson [5]. Since then, several studies have been per- formed until reaching the current form of the Standard Model (SM) of particle physics [6, 7, 8, 9, 10] and other discoveries happened. The SM is a quantum field theory based on the symmetry group SU(3)×SU(2)×U(1), which describes the properties of elementary particles and the interactions among them. The SU(2)×U(1) is related to the electroweak1 sector, whereas SU(3) refers to the strong sector that describes the colored constituents, i.e., the quarks and gluons. These constituents compose the hadrons: bosons (integer spin particles) and fermions (half-integer spin particles). The fourth force, gravity, is not yet included in the SM, however, in the energy scale currently accessed at accelerators, this interaction is extremely weak when compared to the other forces and can be neglected. The general structure of Standard Model is shown on Fig. 2.1. According to the SM, there are three generations of fermions containing 12 particles (plus their respective antiparticles) divided equally in species classified as leptons and quarks. Leptons are particles that feel only electroweak interactions, the six leptons are: electron (e−), electron neutrino (νe), muon (µ−), muon neutrino (νµ), tau (τ−) and tau neutrino (ντ). Quarks are particles that, besides the electroweak force, are also sensitive to strong interactions. They carry an additional quantum number called color charge (see more at Sec. 2.1), and the quarks are six: up (u), down (d), strange (s), charm (c), bottom (b) and top (t). The first generation of fermions, constitutes the everyday matter, i.e., composes the visible matter of the universe. In the 1The weak and electromagnetic interactions can be unified and described by the so-called electroweak theory developed by S. L. Glashow [11], M. A. Salam [12] and S. Weinberg [13], independently, and is also called Glashow-Weinberg-Salam model. 3 Chapter 2. Strong Interactions and the Quark-Gluon Plasma 4 SM, interactions are mediated by the exchange of bosons: the electromagnetic force, that describes the interaction between electrically charged particles and is mediated by the photon (γ); the weak force, responsible for flavor changing and process that includes neutrinos, are mediated by the W± and Z0 bosons; the strong nuclear force that describes the interaction between particles which contain color charge, are mediated by the gluon (g). The graviton (G) is the hypothetical boson that mediates the gravitational force and which is outside the SM. The last ingredient of the SM was theorized in 1964 by P. Higgs [14], F. Englert and R. Brout [15] for explaining the mass generation of elementary particles, called the Higgs mechanism (or Brout-Englert-Higgs mechanism). According to this mechanism the elementary particles obtain their mass by the interaction with the Higgs field (the heavier the particle, the more strongly it couples to the field) and the quantum excitations of this field correspond to a new particle: the Higgs boson (H). The discovery of the Higgs boson happened in 2012 (∼50 years after the theoretical proposal), at the Large Hadron Collider (LHC), with the simultaneous measurements performed by the Compact Muon Solenoid (CMS) [16] and by the A Toroidal LHC ApparatuS (ATLAS) [17] collaborations. The Higgs boson mass was measured to be around 125 GeV. Nowadays, the SM is the one of the most successful theoretical models in the history of physics, with many predictions (including the Higgs boson) and precise agreements with experimental data. Figure 2.1: Sketch of the Standard Model structure. Extracted from CERN website https://cds.cern.ch/record/1473657 [last access on 13/Mar/2021]. https://cds.cern.ch/record/1473657 Chapter 2. Strong Interactions and the Quark-Gluon Plasma 5 In this work, we focus on the study of strong interactions between quarks, gluons and hadrons. Based on that, this chapter starts with a brief overview of strongly interacting particles, followed by the theory of strong interactions and the description of a state of matter called quark-gluon plasma. 2.1 Strongly interacting particles Partons (quarks, antiquarks and gluons) are the only elementary particles that interact strongly. However, these particles are not observed in isolation in nature, but are grouped into particles called hadrons. When produced in high energy collisions, quarks and gluons hadronize2 into narrow cones of particles in a certain phase-space region (along the direction of the original parton) that are called jets. The first known hadrons were the proton, discovered by E. Rutherford in 1919 [18], and the neutron, discovered by J. Chadwick in 1932 [19]. In 1947, C. M. G. Lattes, H. Muirhead, G. P. S. Occhialini and C. F. Powell discovered the pion [20]. After this, a multitude of hadrons was discovered, and later they were classified in two categories (based on their masses): baryons, like protons and neutrons, and mesons, like pions. In December of 1947, the first two strange particles were observed: the neutral and charged kaons (K0 → π+π− and K± → µ±νµ), in cosmic ray experiments performed by G. D. Rochester and C. C. Butler [21]. Later, in 1950, also with cosmic rays, the group of Armeteros et al., from Caltech, announced the obser- vation of a neutral strange baryon decaying into a proton and a pion, called lambda (Λ) [22, 23]. One similarity between the decays of neutral kaons and lamb- das is that their decays form a "V", and then that particles receive the name of V0 particles. With the progress in cosmic ray studies and the advent of particle accelerators, new hadrons were discovered, such as ∆’s, Ξ’s, Σ’s and others. In the beginning, all these hadrons were believed to be elementary particles and the scientific com- munity was looking for a unified model capable of describing the great variety of strongly interacting particles. The first attempt of an elementary particle clas- sification was described by S. Sakata [24] in 1956 and expanded by the Nagoya model [25, 26]. These studies encouraged the search for a model where hadrons are not elementary particles, but instead, they are composed of more fundamental 2The hadronization is the process where partons group together into final state particles due the color confinement effect (see Sec. 2.2.2). Chapter 2. Strong Interactions and the Quark-Gluon Plasma 6 objects. In 1964, M. Gell-Mann and G. Zweig proposed the quark model [27, 28]. Ac- cording to this model, hadrons have an internal structure and are composed of elementary particles, called quarks (q) and antiquarks (q). Initially, it was believed that there were three species of quarks, called up (u), down (d), strange (s) and respective antiquarks. These quarks should have the following characteristics: i) be fermions with spin 1/2; ii) have fractional electric charge; and iii) should have isospin and strangeness, in order to compose all the particles known at the time. In this model, baryons (antibaryons) are composed of three quarks (antiquarks), such as the proton (antiproton), which has an electrical charge equal to 1 (-1) and bary- onic number3 equal to 1 (-1), being composed of uud (uud). Mesons are composed of a quark and an antiquark, such as π+, which is composed of ud. However, this model failed to explain particles like ∆++, which has spin 3/2, and would violate the Pauli exclusion principle4 since their composition must be uuu. To solve the problem, O. W. Greenberg [29], M. Y. Han and Y. Nambu [30] created an additional quantum number called color charge so that the Pauli exclusion principle could be re-established. Therefore, each quark must assume a state of color, for instance the primary colors: green, red and blue; analogously, the antiquarks should carry the anticolors: antigreen, antired, and antiblue. Gluons are also colored, carrying an equal number of colors and anticolors, and the type of gluon is determined by this charge. Inside the hadrons, the combination of valence quarks must form a white object, that is, the total amount of color charge must be zero or all color charges must be present in equal quantity. This implies that each of the three quarks (antiquarks) inside the baryons (antibaryons) must have a color (anticolor) different from the other two. Similarly, the quark-antiquark pair inside the mesons must have a quark containing a certain color charge and an antiquark with an opposite color charge. The quark model was only validated in the 1970s, when deep inelastic scat- tering (DIS)5 experiments results established quarks as real objects [31]. The fourth quark flavor, called charm (c), was discovered by the measurement of J/ψ bound state (formed by cc) simultaneously at Brookhaven National Laboratory (BNL) [32] and at Stanford Linear Accelerator Center (SLAC) [33]. The quark bottom (b) was discovered at Fermi National Laboratory (Fermilab) in 1977 by 3Each quark (antiquark) has fractional baryonic number equal to 1/3 (-1/3). 4According to the Pauli exclusion principle: two fermions cannot be in the same quantum state simultaneously. 5DIS uses leptons to probe the structure of hadrons. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 7 measuring the Υ resonance (formed by bb) [34]. Two years later, the existence of the gluon (g), was established at Positron-Electron Tandem Ring Accelerator (PETRA) by the measurement of electron-positron interactions going in 3 jets (qqg) in three different experiments [35, 36, 37]. In parallel, a strong evidence of the color charge was obtained by measuring the ratio of the cross-section of the process e+e− → hadrons, by the cross-section of e+e− → µ+µ−. This experimental ratio could only be explained by considering the number of colors is equal to three [38]. The sixth, the heaviest and last quark from Standard Model, called top (t), was discovery in 1995 by the DØ [39] and the Collider Detector at Fermilab (CDF) [40] collaborations, operating at the Fermilab Tevatron, by using proton-antiproton collisions at center-of-mass energy of 1.8 TeV (see beam energy definition at Sec. A.1 of Appendix A). Recent results also show the measurement of top quarks in heavy ion collisions [41, 42]. Years after the discovery of quarks and gluons, new exotic hadrons composed by four quarks, called tetraquarks have been observed by Belle [43], Beijing Spectrometer III (BES III) [44] and by the Large Hadron Collider beauty (LHCb) [45, 46, 47, 48] collaborations. LHCb also observed candidates of another species of exotic particles made by five quarks, called pentaquarks [49, 50]. After a very brief history of hadrons, quarks, gluons and colors, now it is possible to move to the theory used to describe the strong interactions. 2.2 Quantum Chromodynamics Quantum ChromoDynamics (QCD) is a quantum field theory founded on first principles that describes the strong interactions between quark and gluon fields. QCD is a non-abelian6 gauge theory based on the symmetry group SU(3) with 8 generators corresponding to the gluons. The Lagrangian density of QCD is written as [51, 52, 53] L = ∑ flavors( f ) ψ f i ( iγµDµ −m f ) ij ψ f j − 1 4 Ga µνGa,µν, (2.1) where the first part of the lagrangian is the quark term and the second part represents the gluon term. The sum, on the left part, runs over the number of quark flavors ( f ). The ψ f j (ψ f i = ψ† i γ0) is the quark (antiquark) field with color j (i). The factors γµ are the Dirac matrices (see Sec. A.2 of Appendix A) and m f is the 6Gauge theories are called non-abelian if the generators associated to the symmetry group anticommute. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 8 mass of the quark with flavor f . Assuming the Einstein convention, i.e., repeated indices are implicitly summed, where µ, ν = 0, 1, 2, 3 are the spacetime indices, i, j = 1, 2, 3 represent the color indices in the quark field and a = 1, ..., 8 are the color indices of the gluon field. The covariant derivative is given by Dµ = ∂µ − i gs 2 λa Aa µ, (2.2) where ∂µ is the partial derivative, gs is the coupling of the strong interactions that can be written in terms of the so-called running coupling constant (αs) as 4παs = g2 s . λa are the Gell-Mann matrices (see Sec. A.2 of Appendix A) which are associated with the generators of SU(3) group as Ta = λa/2. Aa µ is the gluon field. The gauge invariant gluonic field strength tensor is give by Ga µν = ∂µ Aa ν − ∂ν Aa µ + gs f abc Ab µ Ac ν, (2.3) where f abc is the structure constants of SU(3) group defined through [λa, λb] = 2i f abcλc (see Sec. A.2 of Appendix A). Expanding the QCD Lagrangian 2.1, is possible to obtain the vertices of gluon self-interaction as a consequence of the non-abelian nature of QCD7: 3-gluon vertex: gs 2 f abc (∂µ Aa,ν − ∂ν Aa,µ) Ab µ Ac ν, (2.4) 4-gluon vertex: −gs 4 f abc f ade ( Ab,µ Ac,ν Ad µ Ae ν ) , (2.5) which are represented graphically in Fig. 2.2: a 3-gluon vertex (left) and a 4-gluon vertex (right). These self-interactions and the non-abelian behavior are crucial ingredients to study the nuclear aspects of QCD at high energies. Figure 2.2: QCD Feynman diagrams for the self-interaction gluon vertices: 3-gluon (left) and 4-gluon (right). 7In the quantum electrodynamics (QED), the abelian quantum field theory that describes electromagnetism the mediator, the photon (γ), does not present self-interactions. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 9 2.2.1 QCD running coupling constant In QCD, as in most quantum field theories, the physical quantities are cal- culated by using perturbative expansions in terms of the coupling constant (αs). The so-called Leading Order (LO) is the first order of the perturbative expansion and the following orders are Next-to-LO (NLO), Next-to-NLO (NNLO), and so on. In higher orders of αs, loop diagram integrals contain divergences because of the arbitrariness of the values of momentum in the loop, being necessary to integrate from zero to infinity, going to energy scales where the theory is not valid. These divergences can be handled by using the regularization and renormalization process8 [54, 55]. After the renormalization, the β function, which describes the dependence of the coupling constant of the theory with the renormalization scale (µ) can be written as β(αs) = ∂αs(µ2) ∂ ln µ2 = −α2 s (µ 2) [ β0 + β1αs(µ 2) + β2α2 s (µ 2) + ... ] , (2.6) where βi are the coefficients of the beta function for different number of loops (β0 for one-loop, β1 for two-loops, and so on). Adopting the one-loop approximation (i.e., neglecting O(αi s) terms for i > 3), the running coupling constant is αs(µ 2) = β−1 0 ln ( µ2/Λ2 QCD ) , (2.7) where β0 can be written in terms of the number of quark flavors (N f ) and number of colors (Nc) as β0 = (11Nc − 2N f )/12π and ΛQCD ≈ 200− 300 MeV [56] is the scale at which the coupling becomes infinite (called QCD Landau pole) extracted experimentally and depends on the regularization scheme [57]. In order to get a direct relation with the experimental data, one interesting choice is to consider the renormalization scale associated to the energy scale (Q) as µ2 = Q2. Figure 2.3 shows the most recent measurements of αs(Q2) versus the energy scale Q for different orders of loop calculations summarized by the Particle Data Group (PDG) [10]. The latest world average value obtained at the mass of the Z0 boson is αs(M2 Z0) = 0.1179± 0.0010. 8The regularization procedure consists of replacing the basic physical parameters of the La- grangian with finite regularized parameters and then treating the divergences by applying a renormalization scheme. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 10 Figure 2.3: Particle Data Group summary of αs measurements as a function of the energy scale Q. Extracted from [10]. The αs behavior decreasing with Q2 indicates that at high energies (small distances) the color field strength is reduced, allowing perturbative calculations (pQCD). On the other hand, at lower energies (large distances or small Q2), αs is large, thus accessing the non-perturbative regime, where the pQCD is not valid anymore. Such effects can be understood as interactions between the gluon field and the QCD vacuum: in large αs, interactions with virtual quark-antiquark pairs acts as colour dipoles forming a color screening effect, while in small αs, interac- tions with virtual gluons increases the net effect of the colour charge producing a called antiscreening effect. One possible way to perform calculations in a non-perturbative regime is by using the so-called lattice gauge theory or lattice QCD (LQCD). The LQCD is a numerical approach which discretizes and solve the QCD equations in a grid with spacing a in both spatial and temporal dimensions. This formalism reproduces with an incredible precision a large number of experimental measurements (e.g., mass spectrum of light and heavy hadrons [58, 59]) and also has been largelly used to study QCD phase transition (see Sec. 2.3.2). 2.2.2 Color confinement and asymptotic freedom The fact that quarks and gluons cannot be observed in isolation in nature is called color confinement. This picture corresponds to the fact that the intensity of the strong nuclear interaction between quarks increases with the distances between Chapter 2. Strong Interactions and the Quark-Gluon Plasma 11 them9, in which case αs is larger, leading to the non-perturbative regime as shown by Fig. 2.3. Mathematically, this effect is not yet completelly understood and is one of the Clay Mathematics Institute’s Millennium Problems [61]. One way to qualitatively understand the color confinement is by the illustration in Fig. 2.4. When two quarks from a baryon (or a quark-antiquark pair from a meson) are pulled apart, the intensity of the color field (represented by the strings) increases, until its energy is large enough (E > 2mq) to create a new quark- antiquark pair out of the vacuum, so that the original baryon split into a baryon and a meson (or the original meson split into two mesons). This explains why it is not possible to observe particles with color charge different from white (color neutral) in the detectors. The color confinement mechanism is present in the hadronization process and jet production. baryon baryon meson Increasing the distance between quarks creation of a pair quark-antiquark Intensity of the color field grows with distance Figure 2.4: Sketch of a hypothetical tentative of removal of a quark from a baryon with subsequent creation of a meson (quark-antiquark pair) in association with the color confinement mechanism. Another important property of QCD is the asymptotic freedom, discovered in 1973 by D. Gross, F. Wilczek [52] and D. Politzer [62] (Nobel Prize of Physics in 2004). The asymptotic freedom is a direct consequence of the non-abelian nature of QCD and, on the other side of color confinement, says that at high energies the separation distance between quarks is small and, consequently, the intensity of the strong interaction is reduced. This can be better understood by means of the plot in Fig. 2.3: for higher values of Q2, αs gradually decreases, that is, the greater the energy of the system, the more the quarks and gluons behave as free particles. The asymptotic freedom is the property which allows the study of QCD in the 9The potential of the field is proportional to the separation between the quarks r as V(r) ∼ kr (k ∼ 1 GeV/fm), demanding more and more energy as the distance between them increases [60]. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 12 perturbative regime for small αs. In the mid-seventies, this property also lead to the conjecture of the existence of a new state of matter called quark-gluon plasma, which is discussed in the next section. 2.3 Quark-Gluon Plasma The Quark-Gluon Plasma (QGP), name given by E. V. Shuryak [63], is the QCD phase of matter created at extremely high temperatures (order of 1012 K) and energy densities (higher than 1 GeV/fm3). In this system quarks and gluons are no longer confined into hadronic bound states. This state of matter is believed to have been present at the beginning of the universe, around 10−5 seconds after the Big Bang [64]. Historically, the idea of the study of matter in extreme conditions was first suggested by T. D. Lee in 1974 [65, 66, 67]. W. Greiner et. al. [65, 67, 68] pointed out that these conditions could be achieved by using nucleus-nucleus (AA) collisions. In 1975, J. C. Collins and M. J. Perry [69], based on the QCD asymptotic freedom property, suggested that the QGP should behave like an ideal gas composed of quarks and gluons and, since then, microscopic models was largely applied to study this state of matter. In the 1980s, many possible experimental signatures of QGP were proposed (see Sec. 2.3.3). In 1982, J. D. Bjorken [70] proposed the relativistic hydrodynamic description (first studied by Landau [71, 72]) to describe the matter formed in nucleus-nucleus collisions. However, the microscopic models were largely used for that description, until the QGP was discovered experimen- tally in 2005. After that, the hydrodynamical model (a brief description of this model can be found at Sec. B.2 of Appendix B) was adopted, studied, improved and simulations performed by different codes (CHESS [73, 74, 75], SPHERIO [76, 77], MUSIC [78, 79], SONIC [80], iEBE-VISHNU [81] and others) are still being used today for studying the properties of the QGP (equation of state, viscosity, ...). Experimentally, studies of heavy ion collisions (HIC) starts in the 1970s at Bevalac10 located at Lawrence Berkeley National Laboratory (LBNL). Bevalac performed collisions using Carbon, Neon and Argon against a fixed target with energies from 0.2 to 2.6 GeV per nucleon in order to investigate the equation of state (EoS) of QCD at high densities [82, 83] (see Sec. 2.3.2). They observed that a possible compressed nuclear matter was created, confirming the prediction 10Bevalac is a combination of the Bevatron (synchrotron that accelerated protons) and the SuperHILAC (linear accelerator that injected heavy ions into the Bevatron) accelerators. https://github.com/denerslemos/CHESS https://www.sprace.org.br/twiki/bin/view/Main/SPheRIO#Presentation_and_history/ http://www.physics.mcgill.ca/music/ https://sites.google.com/site/revihy/home https://u.osu.edu/vishnu/ Chapter 2. Strong Interactions and the Quark-Gluon Plasma 13 from W. Greiner et. al.; the Bevalac was decommissioned in 1993. In the 1980s the Alternating Gradient Synchrotron (AGS) at Brookhaven National Laboratory (BNL), was also adapted to collide ions of gold (Au) and silicon (Si), with center- of-mass energy up to √ sNN ∼ 5 GeV per nucleon, against fixed targets (made of Au or beryllium (Be) or aluminum (Al)). The AGS operated HIC for 12 years (later becoming part of the Relativistic Heavy Ion Collider complex) delivering analyses from several experiments as E802, E810, E814, E858 and others, they observed some signatures, but could not establish the existence of the QGP. Towards the mid-eighties HIC have been studied also at the Super Proton Synchroton (SPS) accelerator at Conseil Europen pour la Recherche Nuclaire (CERN)11, producing collisions of lead (Pb) against fixed-targets of Pb or Au with a center-of-mass energy per nucleon up to √ sNN ∼ 17 GeV. Seven large experiments were used to collect data from SPS: NA44, NA45, NA49, NA50, NA52, NA57/WA97 and WA98. Based on the compiled results from these experiments, at the end of 1999, when it was about to be decommissioned, the SPS announced the discovery of a new state of matter consistent with QGP [84, 85], showing a good agreement with microscopic models, which considers the QGP as a gas-like matter. The SPS fixed-target program is still active with the NA61/SHINE experiment, in order to study hadron production and neutrinos in nucleus-nucleus, hadron-nucleus and hadron-proton collisions [86]. In the year 2000, the Relativistic Heavy Ion Collider (RHIC), started to op- erate at BNL using AGS as a pre-accelerator, with the main plan to search for the QGP formation. Different from its predecessors, RHIC is a collider, which means that the collisions are performed with a beam of particles against another particle beam and, not against fixed targets. Proton-proton (pp), gold-gold (AuAu) and copper-copper (CuCu) collisions were delivered at different center-of-mass energies. The data from RHIC collisions were collected and analysed by four main experiments: BRAHMS, PHOBOS, PHENIX and STAR. The two-last ones are still operational, however PHOBOS and BRAHMS, completed their operations in 2005 and 2006, respectively. In 2005, researchers from RHIC, after the measurements of many signatures, officially announced the discovery of the QGP [1] in AuAu collisions. Differently than early expectations, RHIC showed that this matter is strong-coupled, behaving not as a gas, but rather more like an almost perfect fluid (low viscosity). After that, RHIC conducted several measurements in order to better understand the QGP properties for different collision systems (pAl, dAu, 11In english: European Organization for Nuclear Research. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 14 CuAu, and others) and later, by varying a large range of center-of-mass energy (with √ sNN from 7.7 to 200 GeV), to search for a possible critical point in the QCD phase diagram (see Sec. 2.3.2). The RHIC program is still active and a bit about its future plans are presented in the Sec. 4.1.3 of Chapter 4. At the moment, the most powerful heavy ion collider is the Large Hadron Collider (LHC) at CERN, which uses SPS as a pre-accelerator. The LHC heavy ion program started in 2010 with lead-lead (PbPb) collisions at √ sNN = 2.76 TeV, allowing to study the QGP at the TeV scale. Nowadays, this program counts with measurements for the four mains experiments with studies in lead-lead, proton- lead and xenon-xenon collisions at different energies. More detail about LHC and the collisions performed there are described at Chapter 4. The QGP signatures observed at RHIC was confirmed at LHC and also new surprising results have been observed, as the collectivity behavior of the system produced in events with high multiplicity12 in proton-proton collisions (see Sec. 2.3.4). 2.3.1 Heavy ion collision evolution After years of study and measurements previously mentioned, nowadays the evolution of heavy ion collisions can be described as shown by the Fig. 2.5. Heavy Ion Collision Evolution Collision overlap zone collisions hydrodynamic evolution detector measurements QGP phase hadron gas phase decoupling phase transition (hadronization) freeze-out initial state interactions ... Figure 2.5: Schematic view of the heavy ion collision evolution. The detector image was extracted from https://cms.cern/news/jet-quenching-observed-cms- heavy-ion-collisions [last access on 13/Mar/2021]. According to Fig. 2.5, two nuclei accelerated at velocities close to speed of light in the vacuum (c), are contracted by the Lorentz factor in the longitudinal direction. 12The multiplicity is defined as the number of charged particles produced per collision. https://cms.cern/news/jet-quenching-observed-cms-heavy-ion-collisions https://cms.cern/news/jet-quenching-observed-cms-heavy-ion-collisions Chapter 2. Strong Interactions and the Quark-Gluon Plasma 15 The region of interaction (the overlap of the two nuclei) defines the centrality13 of the collision (see Sec. A.3 of Appendix A). When the collision happens, a large deposition of energy occurs forming an initial system. After a complex process involving microscopic collisions between the constituents of this system, a hot and dense matter (QGP) is formed in local thermal equilibrium14. From this moment, the hydrodynamical model can be applied. As the space-time evolution of the system occurs, the matter formed in the collision expands and cools down. During this process, partons begin to regroup into hadrons. After a certain time, all matter formed in the collision will be in the hadronic phase. When the mean free path of hadrons in the system is in the order of the characteristic dimension of the system, the thermal equilibrium hypothesis is no longer valid and the hydrodynamic model can no longer be used. In this stage, the system decouples (freezes-out) and the hadrons formed in the collision and their decay products moves freely to the detector. 2.3.2 QCD phase diagram In the 1960s Rolf Hagedorn studied the thermodynamical proprierties of hadronic matter, proposing the so-called statistical bootstrap model [87]. In his studies, Hagedorn considered that at high energies the matter can be described as a hadron gas (HG) and can be treated by using statistical mechanics. He observed that the density of the known hadrons increases exponentially, and diverges at a temperature of 158 MeV. At the time, this value was interpreted as a limiting temperature and received the name of Hagedorn temperature (TH). Nowadays, the Hagedorn temperature can be interpreted as the critical temperature around which a phase transition occurs between the QGP phase and hadron gas. Calculations of the QCD phase transition are harder because in this region the critical temper- ature (Tc) is in the order of ΛQCD, than αs becomes large and pQCD cannot be used, showing the need for theoretical calculations in order to describe the phase transition. Initially, it was believed that a first order phase transition would happen between the QGP and HG [88], that is, there would be a mixed phase during which both the QGP and the HG coexisted. This was first calculated by considering the 13Centrality is the quantity which estimates the overlap region between the nuclei, with 0% corresponding to a complete overlap (head on) and 100% to the case that the nuclei barely touch each other. 14Local thermal equilibrium means that every small part of the system can still be roughly described by the thermal equilibrium laws. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 16 QGP to a gas composed of free quarks (u, d, s), antiquarks (u, d, s) and gluons. Using statistical mechanics, it is possible to estimate the thermodynamic quantities for the hadron gas at low temperatures and for the QGP at high temperatures (see more details in Sec. B.1 of Appendix B). However, the interactions and the confinement of quarks and gluons at lower temperatures should also be taken into account, which can be done by means of the MIT bag model [89]. According to this model, a constant energy density B is postulated, called bag constant, which is adjusted to make the pressure in the QGP phase coincident with the pressure of the hadron gas at a certain Tc (PHG(Tc) = PQGP(Tc)). All uncertainties in the calculations are associated with B. For B = 380 MeV and zero baryon density, we have a critical phase transition temperature of Tc ≈ 162 MeV. With the improving precision of LQCD calculations, it became possible to estimate more accurately the thermodynamic quantities in the phase transition at zero baryon density. The results from LQCD showed that this phase transition would be a crossover15 between the HG and the QGP. Figure 2.6 shows a com- parison between the energy density, ε, divided by temperature, T, to the fourth power versus T for the EoS with first order phase transition (continuous blue line) and for the parametrization of LQCD results from HOTQCD collaboration [90] (dashed red line). Is possible to observe that the two EoS are identical in the hadron gas phase. The EoS with first order phase transition shows an abrupt change at Tc, corresponding to the mixed phase between the QGP and hadrons gas phase, whereas the LQCD presents a smooth transition. According to this LQCD EoS, the critical temperature is situated at Tc = 154± 9MeV, in agreement with the Hagedorn temperature estimate. Based on these results, the hypothesis of a critical point (circle), where a second order phase transition is expected, arises on the transition curve: at high temperatures (T) and low baryon chemical potential (µB)16 a crossover phase transition is expected (dashed line); at high-µB and lower T the phase transition is of first order (continous line) as illustrated in Fig. 2.7. This diagram can be divided in two parts: at high-T (> Tc) the QGP can be accessed and at low-T (< Tc) the other states of matter are expected. The point (0, 0) is the vacuum. For high-T and µB ∼ 0 stands the situation corresponding to the early universe, i.e., soon after the Big Bang. By increasing µB and decreasing in T regions accessible by the accelerators is indicated: LHC, RHIC, NICA and FAIR. 15Crossover is a smooth phase transition that ends at the critical end point, where the phase transition is of a second-order. 16µB can be understood as the excess of matter compared to antimatter. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 17 The usual nuclear matter exists at lower temperatures and for µB around 900 MeV. Increasing µB a state of matter called neutron gas (a degenerate fermionic gas) is expected at the interior of neutron stars [91]. And, for higher values of µB and higher T it is possible to access a theoritized state of matter called color superconductor [92]. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 T [GeV] 0 2 4 6 8 10 12 14 16 18 20 4 (T ) / T ε zoom 162 MeV)≈ c First Order (T 9 MeV)± = 154 c Crossover (T 0.14 0.1450.15 0.1550.16 0.1650.17 0.1750.18 0.185 2 4 6 8 10 12 14 16 18 20 Figure 2.6: Energy density (ε) divided by temperature (T) to the fourth potence (ε/T4) in arbitrary units versus temperature for a first order phase transition (solid blue line) and for a crossover phase transition from LQCD calculations [90] (dashed red line) . See text for more information. Figure 2.7: QCD phase diagram showing the temperature as function of the baryon chemical potential. The lines represent the phase transition boundaries, the circle represents the possible critical point, colored arrows represent the regions accessible by accelerators. See more information in the text. Extracted from [93]. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 18 2.3.3 QGP signatures The direct observation of the QGP is not possible because this system, created in high energy collisions, lasts for a very short period of time in the deconfined phase, and only its hadronized products are detected. However, the study of this state of matter can be done indirectly through characteristic signatures that are observed as final state particles at the detector, that were affected by the presence of QGP. Some of these QGP signatures are described below. Collective behavior One of the strongest evidences of the quark-gluon plasma formation is the collective behavior observed in heavy ion collisions. The collective evolution of the system is described by the anisotropic flow. We can understand the production of this flow with the help of Fig. 2.8. After a non-central heavy ion collision, the participants17, form a region corresponding to the volume of interaction between the nuclei, represented by the ellipsoid in Fig. 2.8, left, that is spatially anisotropic. With the evolution of the system, this spatial anisotropy is transferred to the momentum space, due to the difference between the pressure gradients in the major and minor axes of the ellipsoid, as shown in the Fig. 2.8 (top right). This behavior indicates that the evolution of the system is also anisotropic. We can say that the system has a very high energy density in the reaction region, decreasing gradually towards the edges of the system, more quickly in the direction x than in the direction y of the figure. This anisotropy is reflected in the azimuthal (φ) distributions of the particles final state, as shown in Fig. 2.8 (bottom right), and can be expressed by a Fourier expansion, as [94] dN dφ ∝ 1 + 2 ∞ ∑ n=1 vn cos [n (φ− ψEP)] (2.8) where vn and ψEP are the Fourier coefficients (also called harmonics) and the azimuthal angle with respect to the event plane, respectively18. The event plane is the closest experimental realization of the reaction plane, which is not accessible to experiments, since it uses the impact parameter (distance between the center of the nuclei) in its definition. The v1 coefficient is related with the energy momentum 17The participants are the nucleons that interact at the moment of the collision, the other nucleons are called spectators. See Sec. A.3 of Appendix A. 18The event plane is one of the possible methods which are used to estimate vn, other methods also used are: cumulants, scalar product and Lee-Young Zeros [94]. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 19 conservation, while the second-order coefficient (v2), called elliptic flow, reflects the lenticular shape of the interaction area formed in non-central collisions and is dominant over the others. The higher-order coefficients, particularly the odd indices n, come from fluctuations in the positions of the nucleons in the initial conditions. The fact that these coefficients are non-zero is an indicative of the system collectivity. For the studies of the QGP, the elliptic flow is important, since this quantity is directly affected by the medium properties (equation of state, viscosity and others). Figure 2.9 shows the elliptic flow as a function of the transverse momentum (pT) measured by STAR and PHENIX collaborations in AuAu collisions at √ sNN = 200 GeV for different particle species, compared to ideal hydrodynamical simulations [95]. A good agreement between data and simulations can be observed at low-pT. For higher-pT, viscosity or jets effects can contribute to this disagreement. Another interesting result observed in this figure is the mass ordering (heavier particles shown lower v2); this effect was predicted by hydrodynamics and is clearly present in the data. Later, it was observed that v2 also shows a scaling with the number of constituent quarks (nq), which means that, if we measure the elliptic flow for baryons and mesons and divide by the respective nq, the resulting v2 values will be in agreement with each other, which can be interpreted as these hadrons flowing together [96]. The detailed experimental studies of the elliptic flow at RHIC and the good agreement with the ideal hydrodynamical model were essential ingredients to demonstrate that the produced QGP as a nearly perfect fluid. At the LHC, similar measurements have been performed, thus confirming the results and behaviors observed at RHIC. In addition, among a pletora of new experimental results, the LHC presented surprisingly results, such as the flow measurements in small colliding systems (see Sec. 2.3.4). Chapter 2. Strong Interactions and the Quark-Gluon Plasma 20 Figure 2.8: Schematic view of the interaction volume between two nuclei. The anisotropy observed in the xy plane generates the anisotropic flow. On the left a 3-dimensional representation is shown, on the top right the same idea is presented in the xy plane, on the bottom right an example of the azimuthal distribution is shown. Extracted from [97]. Figure 2.9: Elliptic flow (v2) as a function of transverse momentum (pT) measured by STAR and PHENIX collaborations (markers) for different species of particles compared to ideal hydrodynamical simulations (lines). Extracted from [95]. Strangeness enhancement The strangeness enhancement was proposed as a QGP signature by J. Rafaelski and B. Muller [98]. The production of strange quarks can happen through three different ways: i) gluon splitting (g → ss); ii) gluon fusion gg → ss; iii) quark- antiquark annihilation (qq → ss). In proton-proton interactions, the production of hadrons composed by strange quarks are usually suppressed when compared to hadrons containing light quarks (u and d), because of the higher mass of the strange quark (which requires higher energies). According to J. Rafaelski and Chapter 2. Strong Interactions and the Quark-Gluon Plasma 21 B. Muller, if a dense and hot matter is formed, a very large gluon density and energy is present, therefore, the gluon fusion process will be favored energetically, which should not occur in a scenario without the QGP medium. As the system evolves, the strange quarks created in the QGP, start to group with other quarks of the medium, producing bound states which are the strange hadrons. This enhancement was observed first at the AGS [99] by the measurement of the K/π ratio, later this was also observed at SPS [100], RHIC [101] and LHC [102]. Figure 2.10 shows the production yield of protons and strange hadrons (Λ, Λ, Ξ−, Ξ+ and Ω− + Ω + ) performed in HIC relative to pp collisions at RHIC at √ sNN = 200 GeV (solid markers) and to pBe collisions at SPS at √ sNN = 17.3 GeV (open markers) as a function of the number of participants, Npart (higher Npart, most central is the collision). The results show a clear enhancement, increasing with Npart, for the strange hadrons (even in different energies), which cannot be seen for protons. And these results are in agreement with thermal models (arrows) which suggest the presence of the dense medium. Figure 2.10: Production yield of protons and strange baryons (Λ, Λ, Ξ−, Ξ+ and Ω−+Ω + ), measured by STAR collaboration in AuAu collisions at √ sNN = 200 GeV (solid markers) relative to the same yield in pp collisions and for NA57 collabo- ration in PbPb collisions at √ sNN = 17.3 GeV (open markers) relative to the same yield in pBe collisions as function of Npart. Extracted from [101]. Jet quenching In 1982, J. D. Bjorken proposed the idea of the jet quenching phenomena [103]. High energy jets are produced by the fragmentation of partons produced at the beginning of the collisions. In the case where a quark-gluon plasma is formed, these partons should feel the hot and dense medium, losing their energy by Chapter 2. Strong Interactions and the Quark-Gluon Plasma 22 interactions with the QGP constituents or by medium-induced gluon radiation. According to the original idea by Bjorken, if two jets are produced back-to-back and one of them goes through the QGP, at the end one jet will be measured with high energy and the other will be quenched. Figure 2.11, shows a schematic view of back-to-back jets produced at proton-proton collisions (top) and in the case idealized by Bjorken’s (bottom). 2 parton parton pp collisions 3 AA collisions parton parton Figure 2.11: Sketch of the back-to-back dijet production in proton-proton (top) and heavy ion (bottom) collisions. The first experimental evidence of this effect was observed by STAR collab- oration at RHIC in the measurement of the di-hadron azimuthal correlations (∆φ) 19 between low-pT and high-pT particles at √ sNN = 200 GeV comparing pp (black line), central dAu (red circles) and central AuAu (blue stars) collisions [104], shown in Fig. 2.12. Their results show that, at ∆φ ∼ 0, the peak coming from the jets is present in all the colliding systems, however, the back-to-back correlations disappears in the case of gold-gold collisions, similarly to the Bjorken’s idea. Later, this effect was observed and largely scrutinized at the LHC [105, 106]. 19∆φ is the azimuthal angle (in the transverse plane) difference between the two particles (see more at Chapter 4). Chapter 2. Strong Interactions and the Quark-Gluon Plasma 23 Figure 2.12: Di-hadron ∆φ correlations for pp (black line), central dAu (red circles) and central AuAu (blue stars) collisions. Extracted and adapted from [104]. Production of high-pT particles High-pT hadrons are expected to be produced by hard scatterings in the initial stages of the heavy ion collisions, therefore could be sensitive to entire evolution process of the system and, consequentelly, could be used as QGP probes. To quantify such effect, one observable called nuclear modification factor (RAA) is used, which is defined as RAA = NAA 〈Ncoll〉Npp , (2.9) where, NAA is the particle yield measured in nucleus-nucleus (AA) collisions, where the QGP is expected, 〈Ncoll〉 is the average number of binary nucleon collisions happening in AA collisions, estimated by centrality models (e.g., Glauber model [107, 108]), and Npp is the same yield measured in proton-proton collisions, that is used as a reference, where the QGP is not expected. The case RAA less than the unity is associated with a suppression, i.e., the yield is reduced by the interaction of the particles with the medium, similarly to jet quenching, nowadays the RAA is also used to observe and explore jet suppression in different kinematical windows. Figure 2.13 shows an example of RAA for high-pT hadrons as function of the number of participants in different collision centralities, as measured by the ALICE collaboration for PbPb collisions at √ sNN = 2.76 TeV, for π (green triangles) and D mesons (black squares). For both particle species it is possible to see that the suppression is strongest for most central collisions than in peripheral ones. This is expected, since the system formed will be larger and will survive for a longer time interval in central events, thus increasing the number of interactions, which is translated into a stronger suppression. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 24 Figure 2.13: RAA for PbPb collisions at √ sNN = 2.76 TeV as a function of the number of participants and centrality, for π (green triangles) and D (black squares) mesons measured by ALICE collaboration. Extracted from [109]. Quarkonium suppression Quarkonium is a bound state composed of a heavy quark and its antiquark, therefore being a neutral particle and its own antiparticle. They are basically classified in two species: charmonium, formed by a cc pair (e.g., J/ψ) and bot- tomonium, formed by bb (e.g., Υ). The suppression of quarkonium states in QGP matter was first proposed for J/ψ particles by T. Matsui e H. Satz [110], in 1986. According to them, the J/ψ binding potential is screened by the interactions with the constituents of the hot and dense medium. This happen because the Debye colour screening potential increases with the temperature of the medium, and above certain temperature, the binding potential cannot hold the two quarks together, and the bound is dissociated in a c and a c in the plasma. Besides, if the QGP initial temperature is high enough, the cc pair may not form a bound state. After that, these quarks will combine with lighter quark flavors (u and d) increasing the production of open-charm mesons (e.g., D, which is composed by cu) and reducing the production of J/ψ mesons. For larger binding radius, the potential of the quarkonium states become weaker and for that reason, the most excited quarkonium states are expected to be more loosely bounded and to dissociated at lower temperature as compared to the ground state, leading to a sequential suppression. This means that, for the charmonium states, ψ(2S) would be more suppressed than J/ψ and, for the bottomonium states, Υ(3S) would be more suppressed than Υ(2S), which would be more suppressed than Υ(1S). The Chapter 2. Strong Interactions and the Quark-Gluon Plasma 25 first evidence of this suppression was found at SPS [111], and nowadays this is still extensively studied at RHIC and LHC [112, 113, 114]. Figure 2.14, on the left, shows the RAA pT dependence of the J/ψ → e+e− (green solid circles) and J/ψ → µ+µ− (yellow open circles), as measured by PHENIX collaboration at RHIC, in central (0-20%) AuAu collisions at √ sNN = 200 GeV [112]. From this plot it is possible to see the clear suppression in the all pT range investigated. On the right, the dimuon invariant mass distribution measured by CMS collaboration [113] is shown in the range 8-14 GeV. The comparison between PbPb and pp collisions (normalized by the Υ(1S) mass peak in PbPb) shows a clear sequential suppression for the higher excited states of bottomonium (Υ(2S) and Υ(3S)). Figure 2.14: Left: RAA of J/ψ particles decaying into pairs of muons (yellow open circles) and electrons (green solid circles) measured by PHENIX collaboration at RHIC in central (0-20%) AuAu collisions at √ sNN = 200 GeV. Extracted from [112]. Right: Dimuon invariant mass distribution, measured by CMS collaboration in the range of Υ(nS) production for PbPb collisions (0-100%) at √ sNN = 5.02 TeV. The black points are the data, the solid blue line is the total fit, the dotted-dashed blue line is the combinatorial background, the dotted blue line shows the signal only (total − background) and the dashed red line shows the pp signal shapes added on top of the PbPb background and normalized by the Υ(1S) mass peak in PbPb. Extracted from [113]. Electroweak probes The usage of electromagnetic particles to investigate the QGP formation was first proposed by E. Shuryak in 1978 [115]. Photons and dileptons are good probes because they can be produced by interactions in different stages of the system Chapter 2. Strong Interactions and the Quark-Gluon Plasma 26 evolution, not been affected by interactions with the medium since they only interact electromagnetically. These probes can be used to obtain information about the temperature and energy density of the system. In the QGP phase, photons are mainly produced by Compton effect (qg→ qγ) and by annihilation process (qq→ gγ), whereas the main source of dileptons is the Drell-Yan process (qq → l+l−). In the hadronic phase, photons are produced by Compton effect (πρ → γρ) and pion annihilation (ππ → γρ), dileptons are mainly coming from the interactions between π’s and ρ’s, and also by Drell-Yan process (at high invariant mass). Besides all these processes, dileptons and photons can also be produced in the early stages of the collision (hard scattering) and by hadronic decays (e.g., π0 → γγ, J/ψ→ e+e− and others). Experimentally, the above effects are studied in different ways. Dileptons are investigated by measuring their invariant mass and, for photons, the transverse momentum yield can be used as a thermometer for studing the QGP temperature. In both cases, the results compared with models, are in favor of the formation of the QGP. However, the background observed in this kind of measurements is huge, making it difficult to draw final conclusions. Figure 2.15 (left) shows the invariant mass distribution for electron pairs measured by PHENIX collaboration at RHIC [116]. Their results show a discrepancy in the region between 0.1 and 0.7 GeV for most central collisions for data (full markers) as compared to the cocktail model in which the QGP is not taken into account (lines). However, this excess disappears in peripheral AuAu collisions and is not present in pp collisions. This enhancement can be described by models that assume the QGP formation [117]. Figure 2.15 (right) shows the yield of photons as function of their transverse momentum measured by PHENIX collaboration [118]. The spectra can be described/modeled by the sum of the contributions shown as the blue line in the plot, from thermal photons, produced considering a QGP with an initial temperature of 370 MeV shown by the red line, and from prompt photons, from hard process, shown by the black line. Nowadays, the production of weak bosons (W± and Z0) are also investigated for being used as reference since these bosons are not modified by QGP effects and should not be affected by flow. The experimental measurements show in- volving the W± and Z0 bosons that RAA ≈ 1 and v2 consistent with zero, as expected [114, 119, 120]. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 27 Figure 2.15: Left: dielectron invariant mass distribution measured by PHENIX collaboration [116] in AuAu collisions in different centralities, as compared to pp collisions. Extracted from [112]. Right: photon spectra as a function of transverse momentum measured by PHENIX collaboration in AuAu collisions, the blue line is the sum of thermal photons (red) and prompt photons (black) produced by models considering QGP formation with an initial temperature of 370 MeV. Extracted from [118]. Femtoscopy The femtoscopic correlations (or HBT effect, as called in earlier times) of parti- cles pairs is a method used to estimate the space-time dimensions of the system and/or final-state interactions felt by particles produced in high-energy colli- sions. This method can be used in any colliding system (e−e+, pp, pp and others) and has a long tradition in high energy heavy ion collisions, being extensively studied along decades mainly used for studying correlations between identical pions [121, 122, 123, 124]. Femtoscopy is the main theme of this thesis, therefore, a more detailed description (history, connection with QGP and theoretical studies) is presented in the Chapter 3 and the experimental procedure used in this analysis is discussed in Chapter 5. 2.3.4 Small colliding systems In 2010 a new chapter of history began to be written, as the CMS collaboration performed a measurement of two-particle ∆η − ∆φ 20 correlations of charged 20∆η is the pseudorapidity (η) difference between the two particles in the correlation. The pseudorapidity is defined as η = − ln(tan(θ/2)), where θ is the scattering angle with respect to the collision axis (see more in Chapter 4). Chapter 2. Strong Interactions and the Quark-Gluon Plasma 28 hadrons in proton-proton collisions at √ s = 7 TeV [2]. The study was performed using collision events with no requirement in the number of produced charged particles (tracks), called minimum bias events, and specially selected events with more than 110 charged particles, called high multiplicity21. Figure 2.16 (top panel), shows the results obtained by CMS in 2010, for minimum bias (left) and for high multiplicity (right) events. Surprisingly, an enhancement around ∆φ ∼ 0 for all ∆η range of the correlation was observed for high multiplicity events. This effect is called ridge and was first observed in heavy ion collisions at RHIC [125, 126], and have a direct relationship with the collective behavior of the system (vn), being a typical QGP signature, and was unexpected in small colliding systems. These results were very exciting in the heavy ion community and small colliding systems started to be extensively studied. Later, a similar collective behavior was observed in proton-lead collisions [127], as shown in Fig. 2.16 (middle panel) for high multiplicity events (right), but not seen for events with less than 35 particles produced (left). Figure 2.16 (bottom panel) shows a comparison between elliptic flow using different methods for pp at √ s = 13 TeV (left), pPb at √ sNN = 5.02 TeV (middle) and PbPb collisions at √ sNN = 2.76 TeV (right) measured by CMS in similar multiplicity ranges [128]. In all cases a non-zero v2 was observed pre- senting similar trends. CMS also measured a significant v2 for different particle species and non-zero v3 [128, 129]. In LHC, ATLAS [130], ALICE [131, 132] and LHCb [133, 134] collaborations have also measured such effects in pp and pPb collisions. At RHIC, flow measurements were performed by PHENIX [135] and STAR [136] collaborations for pAu, dAu, and 3HeAu collisions, observing also significant flow coeficients, vn. The ATLAS and CMS also investigate the possibility of collective behavior by using ultra peripheral collisions (UPC)22. ATLAS collaboration has performed the analysis in PbPb collisions at √ sNN = 5.02 TeV [139] and measured the elliptic flow of the photo-nuclear interaction (the photon is emitted from one lead and interact with the other lead). Their measurements shows a non-zero v2 with similar dependence in multiplicity and transverse momentum (but smaller magnitude) as pp and pPb collisions (minimum bias). CMS collaboration performed the UPC analysis in proton-lead collisions at √ sNN = 8.16 TeV [140], searching for 21The selection of minimum bias or high multiplicity events is made by event selections (called triggers). More details about that can be found in Sec. 5.1 of Chapter 5. The CMS trigger system is presented in Sec. 4.2.7 of Chapter 4. 22UPC is a name given to collisions where the nucleon/nuclei interact only with a photon emitted by the other nucleon/nuclei. More details can be found in Refs. [137, 138]. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 29 v2 of photon-proton interaction (photon is emitted from the lead and interact with the proton) and the results show a v2 coefficient different than zero with bigger magnitude as compared to v2 from pPb collisions (minimum bias) in the same multiplicity range. This results were first interpreted as possible jet-like correlations and further studies are still needed to understand such behavior. The ridge-like structure was also searched experimentally by using high multi- plicity e+e− collisions (in this case, multiplicity greater than 35) at √ s = 91 GeV with archived data collected by ALEPH experiment at LEP [141] and ep collisions (DIS) with √ s = 318 GeV in two multiplicity ranges (2 to 10 and 15 to 35), per- formed by ZEUS collaboration at HERA [142]. In both analyses, their results do not show any structure around ∆φ ∼ 0 and are in agreement with Monte Carlo (MC) simulations that does not include QGP formation. Phenomenological studies provided another big surprise: the hydrodynamical model shows a very good agreement with experimental data for vn coefficients in small colliding systems [135, 143, 75]. This is unexpected, because the mean free path in those collisions should already be of the order of the characteristic dimension of the system23. Figure 2.17 (top panel) shows the comparison of vn results from hydrodynamical simulations using the iEBE-VISHNU code (with different input parameters) and the data from CMS [2] and ATLAS [130] for proton- proton collisions at √ s = 13 TeV: vn dependence on multiplicity is show on the left and v2 as a function of pT, on the right [143]. In both plots it is possible to see a good agreement of data and model. The plot on the left also show the predictions for the multiplicity dependence of v4. Figure 2.17 (bottom panel) shows the results for v2 and v3 from hydrodynamical predictions (red curve for iEBE-VISHNU simulations and blue curve for SONIC simulations) and the color glass condensate (CGC)24 model posdiction calculations compared to the data collected by PHENIX experiment, collected in pAu (left), dAu (middle), and 3HeAu (right) collisions, for the events with highest centrality/multiplicity (0− 5%) at √ sNN = 200 GeV [135]. All the models can describe the data, however the hydrodynamical calculations are in better agreement with the experimental measurements, especially for v3. The second QGP signature observed in small colliding systems was the strange- ness enhancement, measured by ALICE collaboration in 2016 [102], for proton- proton collisions at √ s = 7 TeV and proton-lead collisions at √ sNN = 5.02 TeV, 23The applicability of hydrodynamics in small colliding systems is still an open question. 24The CGC is a theory which considers that a state of high density gluonic matter is created in the initial state of very high energy collisions, whose dynamics can be described by QCD evolution equations [144, 145, 146]. https://u.osu.edu/vishnu/ https://u.osu.edu/vishnu/ https://sites.google.com/site/revihy/home Chapter 2. Strong Interactions and the Quark-Gluon Plasma 30 selecting high multiplicity events. Figure 2.18 shows the ratio between the yields from particles composed by strange quarks, 2K0 S (the factor 2 appears because the K0 S is its own antiparticle), Λ + Λ, Ξ− + Ξ+, Ω− + Ω + and pions (π+ + π−) which are composed by usual matter (quarks up and down). ALICE results show a clear enhancement with multiplicity from small colliding systems to nucleus-nucleus collisions, for all particle species, following the order: proton-proton to proton- lead to lead-lead. Comparison with Monte Carlo event generators (PYTHIA8 [147], EPOS-LHC [148] and DIPSY [149]) for proton-proton collisions at √ s = 7 TeV are also shown, none of these MC’s considers the QGP formation. According to ALICE conclusions, more studies are needed in order to understand the mechanism of strangeness production in high multiplicity events. Recently, hydrodynamical simulations was performed by using the CHESS code [73, 74, 75] (more details at Sec. B.2 of Appendix B) for ideal and viscous (including both shear and bulk) cases and compared with particle yield from ALICE measurements [102]. Figure 2.19 shows the comparison between data and simulations for K0 S’s (top left), Λ’s (top right) and Ξ’s (bottom left) in four multiplicity ranges. Hydrodynamics shows a good agreement with data for all particle species in all multiplicity ranges for both ideal (red solid line) and viscous (blue dashed line) cases with a freeze-out temperature of Tf o = 145 MeV. These results show that the hydrodynamical model is also a good tool to study strange particle production and could also be applied to study events with lower multiplicity although further studies are still needed. Many other experimental measurements were performed before writing this thesis and some of the other signatures presented in Sec. 2.3.3 were found in small colliding systems (e.g., quarkonia suppression). However, these results are in agreement with models that not include QGP effects [150, 151, 152], and in other cases (e.g., suppression of high-pT particles) the signatures was not observed yet [153, 154, 155]. The small colliding systems are still an open question and efforts are needed in order to understand such effects. Interesting studies and discussions about this theme can be found in Refs. [156, 157, 158, 159, 160, 161]. This thesis is focused to obtain a better understanding of small colliding systems by using femtoscopy measurements with strange particles (K0 S, Λ and Λ). The physics of femtoscopic correlations is presented in the next chapter. https://pythia.org/pythia82/ https://web.ikp.kit.edu/rulrich/crmc.html http://home.thep.lu.se/DIPSY/ https://github.com/denerslemos/CHESS Chapter 2. Strong Interactions and the Quark-Gluon Plasma 31 Figure 2.16: CMS measurements of the collective behavior in small colliding systems. Top panel: ∆η and ∆φ dependence of two-particle correlation in proton- proton collisions at √ s = 7 TeV, for minimum bias (left) and high multiplicity (right) events. Extracted from [2]. Middle panel: ∆η and ∆φ dependence of two particle correlations in proton-lead collisions at √ sNN = 5.02 TeV, for mini- mum bias (left) and high multiplicity (right). Extracted from [127]. Bottom panel: multiplicity dependence of elliptic flow (v2) for different methods (different mark- ers) and for different systems, proton-proton at √ s = 13 TeV (left), proton-lead at √ sNN = 5.02 TeV (middle) and lead-lead at √ sNN = 2.76 TeV (right). Extracted from [128]. In the bottom plot, the fact that the different vn coefficients overlap (mainly for n > 2) is an additional indication of the system collectivity. It can be seen that this behavior is present in all three colliding systems investigated. Chapter 2. Strong Interactions and the Quark-Gluon Plasma 32 Figure 2.17: Top panel: flow harmonics (vn) from hydrodynamical calculations for proton-proton collisions at √ s = 13 TeV compared with the data from CMS and ATLAS, showing the dependence on multiplicity (left) and on transverse momen- tum (right). Extracted from [143]. Bottom panel: v2 and v3 measured by PHENIX collaboration in proton-gold (left), deuteron-gold (middle) and helium3-gold at√ sNN = 200 GeV, for the events with highest multiplicities (0-5%). Blue (from iEBE-VISHNU) and red lines (from SONIC) correspond to hydrodynamical predic- tions from different models and the green line is for the CGC model posdiction. Extracted from [135]. https://u.osu.edu/vishnu/ https://sites.google.com/site/revihy/home Chapter 2. Strong Interactions and the Quark-Gluon Plasma 33 Figure 2.18: Experimental ratio of the yields of strange particles 2K0 S (black), Λ + Λ (blue), Ξ− + Ξ+ (green), Ω− + Ω + (red) to pions (π+ + π−) versus multiplicity, measured by ALICE for proton-proton (circles), proton-lead (diamond) and lead- lead (squares) collisions. Lines represent Monte Carlo simulations: EPOS-LHC (dotted line), PYTHIA8 (full line) and DIPSY (dashed line). Extracted from [102]. https://web.ikp.kit.edu/rulrich/crmc.html https://pythia.org/pythia82/ http://home.thep.lu.se/DIPSY/ Chapter 2. Strong Interactions and the Quark-Gluon Plasma 34 Figure 2.19: Particle yield versus pT of strange particles: K0 S (top left), Λ + Λ (top right) and Ξ−+ Ξ+ (bottom left) as measured by ALICE collaboration [102] (mark- ers), compared with hydrodynamical simulations from the CHESS code [75] in four ranges of multiplicity, as shown in the legend. Lines represent hydrodynamical calculations for ideal (red solid line) and viscous (blue dashed line) cases with freezeout temperature of Tf o = 145 MeV. https://github.com/denerslemos/CHESS Chapter 3 Femtoscopy Femtoscopy is the study of multi-particle correlations in low relative momen- tum, used to investigate physical quantities in the order of femtometers/fermis (10−15 m). Such femtoscopic correlations are a powerful method for probing the space-time dimensions of the particle emitting source created in high-energy colli- sions. These correlations are sensitive to the quantum statistics (QS) obeyed by the identical particles produced in the collisions (Bose-Einstein for bosons or Fermi- Dirac for fermions), as well as of the underlying interactions among the particles. The QS effect can be used to estimate the apparent particle emitting source size. The method is also sensitive to the final-state interactions (FSI), i.e., those to which the particles may be submitted after their emission, such as Coulomb, in the case of charged particles, or strong interactions, between hadron pairs. Therefore, the FSI may provide information about such scattering effects. This chapter begins with a historical overview about femtoscopy, followed by the description of quantum statistical effects for identical particles. Then the final-state interaction studies are presented, including the model used to measure scattering quantities from strong FSI, and a brief description of the Coulomb effect1. At the end, highlights of previous results on correlations measurements from other experiments are shown. 3.1 Historical overview The original idea of using correlations to estimate the source sizes was idealized in the 1950s by the engineer and astronomer Robert Hanbury Brown, and later elaborated theoretically in collaboration with the mathematician Richard Quentin Twiss, in radio astronomy, as an improvement of the techniques known at that time to measure angular diameter of stars (θ) [162]. Besides the studies with radio sources, they proved that it was possible to extend its concept to the optical 1The Coulomb FSI is presented by completeness, since this work considers only neutral particles. 35 Chapter 3. Femtoscopy 36 domain, first performing a successful laboratory experiment using an artificial light source coming from a mercury arc lamp with two photomultiplier tubes [163] and later with an experimental apparatus used to determine the size of stellar sources, based on photon coincidence measurements [164]. With the help of the physicist E. M. Purcell, they showed that the simultaneous detection phenomenon stems from the fact that photons tend to arrive together in the detectors due to Bose- Einstein statistics [165], and also mentioned that identical fermions should present an anticorrelation, following the Fermi-Dirac statistics. This method received the name intensity interferometry, but it became known as HBT (Hanbury-Brown Twiss) effect. The first angular diameter of a star using intensity interferometry was per- formed by Hanbury Brown and Twiss for measuring Sirius (α Canis Majoris), in 1956 [162, 164]. To carry out this measurement, they adapted two army search- light projectors with diameter of 1.56 m using mirrors, as shown in Fig. 3.1 (left). These mirrors collected the photons emitted from the star and focused them to photomultiplier detectors. The signal generated passed through a noise filter and was routed to a coincidence circuit called correlator (electronic counter circuit). The measurement was performed for 5 months in four different baselines (distance between the detectors), d = 2.56 m, 5.35 m, 6.98 m and 8.93 m. The probability to detect both photons simultaneously (number of coincidences or correlation func- tion), Γ2(d), as a function of baseline measured by Hanbury Brown and Twiss for the Sirius star is shown in Fig. 3.1 (right). The intensity of the observed correlation decreased as the baseline was increased, showing the effect of the Bose-Einstein statistics. Using the approximation of considering the star emission as a luminous disk, the data was fitted by the function Γ2(d) = [ 2J1(πθd/λ) πθd/λ ]2 , (3.1) where θ is the angular diameter of the star, λ the detected photon wave length and J1, the Bessel function (see Chapter 11 of Ref. [166]). In this way, was possible to estimate the angular diameter, θ, from Sirius star as 6.3± 0.5 milli arcsec [162, 164], as compared to the currently accepted value is 5.936± 0.016 milli arcsec [167]. Chapter 3. Femtoscopy 37 Figure 3.1: First optical intensity interferometer apparatus adapted by Hanbury Brown and Twiss at Jodrell Bank Experimental Station, University of Manchester, London (left), and the measurement of the correlation as a function of the baseline (right), for Sirius star. The line on the right plot is the fit used by Hanbury Brown and Twiss from Eq. (3.1). Extracted from [162]. In the 1960s, the Narrabri Stellar Intensity Interferometry (NSII) was built in collaboration by the Manchester and Sydney Universities, based on Hanbury Brown’s design and ideas [162]. The NSII was located near the Narrabri, in the north-central New South Wales, Australia. The apparatus consisted of two parabolic reflectors with a diameter of 6.5 m composed of 252 hexagonal mirrors, each with a photoelectric detector. The reflectors were placed on a circular railway track with a diameter of 188 m, which was used to change the distance between the two reflectors and increase or decrease the baseline. The reflectors were connected by cables to the control building, which was built at the center of the circular railway track. A garage was built to store the reflectors when NSII was not taking data. In total, the angular diameters of 32 stars were measured at NSII [168], from June 1964 to February, 1972, when the program was closed. After NSII, the intensity interferometry was no longer used in astronomy due new methods and advances in technology. In 2017, experimental results were obtained by the Observatoire de la Côte d’Azur, located in Nice, France. They measured the angular diameter of 6 stars and the results were in agreement with those obtained at the NSII [169, 170, 171]. This experiment was performed with the itention to revive the usage of intensity interferometry in astronomy, specially by using the Cherenkov Telescope Array (CTA) [172], which is composed of more than 100 telescopes, allowing precise measurements and studies, including higher order (> 2) correlations. In the field of particle physics, the HBT effect was first observed in 1959 by Chapter 3. Femtoscopy 38 G. Goldhaber, S. Goldhaber, W. Y. Lee e A. Pais2 [3] (later called GGLP effect), without previous knowledge of the HBT experiment. Their experiment aimed at the search for the ρ meson decaying into opposite-charged pions, in proton- antiproton (pp̄) collisions at √ s = 2.1 GeV, performed at the Bevalac, LNBL, USA. The amount of data they collected was not enough to establish the existence of ρ meson. Nevertheless, in their measurements they found that identically charged pions followed a distribution different from the oppositely charged ones, which could only be described by considering the symmetrization of their wave function. They then observed correlations that led to an enhancement of the number of identical bosons with respect to that of non-identical bosons, when the two particles are close to each other in phase space. The observed distribution was them fitted by a Gaussian, 1 + exp[−q2R2], where q = p1 − p2 is the relative momentum and R is the inverse of the Gaussian width in fm, interpreted as the system size. These measurements reflect the sensitivity of particle momentum correlations to the space-time separation of the particle emitters, due to the effects of quantum statistics. Therefore, by measuring the relative momentum distribution of identical particles as a function of their relative momentum the information about the emitting source size could be accessed. Since then, these correlations started to be extensively studied in particle colliders, mainly using pions, and for different colliding systems as e+e−, pp, pp̄, heavy ions and others. In the mid- seventies, these correlations were suggested as a signal of the quark-gluon plasma formation, following the idea that a possible phase transition to a QGP state could be formed in heavy ion collisions. At that time, such state was considered to be a gas of non-interacting quarks and gluons and the phase transition was of first order, suggesting that the QGP would live long and the measured HBT radio would be much larger than the nuclear size. In parallel to experimental advances, theory and phenomenology were ex- tensively investigated and refined methods were created in order to study such correlations, as for example: multidimensional analyses, the effect of final state interactions, connections with flow behavior, and so on [121, 173, 174]. The name femtoscopy was given by Richard Lednicky [173, 175, 176, 177] as a generalization of the study of particle correlations in low relative momentum to non-identical particles. Therefore, the same nomenclature can be used to study correlations of identical (QS+FSI) and/or non-identical particles, reflecting their final-state interaction. Finally, besides the application in astronomy and particle physics, the 2At this time, the group does not know about the work from Hanbury Brown and Twiss. Chapter 3. Femtoscopy 39 study of these correlations have also been performed in other fields of physics as, for example, in quantum optics [178]. 3.2 Quantum statistics The function that describes the correlation between two particles, emitted chaotically, and relate it to the dimensions and dynamics of the emitting source, as well as the underlying events, is called correlation function (CF). Theoretically, this CF can be written as the ratio between the joint probability of detecting both particles, P2(pµ 1 , pµ 2 ), by the probability of detecting each particle individually, P1(pµ i ), as C(pµ 1 , pµ 2 ) = P2(pµ 1 , pµ 2 ) P1(pµ 1 )P1(pµ 2 ) (3.2) where pµ 1 and pµ 2 are the four-momenta of the two particles. From now on, the four-vector indexes (µ, ν, ...) will be omitted to simplify the notation. The quantum statistical effect is present in correlations between identical par- ticles, and can be described by the symmetrization (Bose-Einstein) or antisym- metrization (Fermi-Dirac) of their wave function. One way to understand such an effect and construct the correlation function is by using the simple picture shown in Fig. 3.2. Consider that two identical particles (bosons or fermions) emitted from points 1 and 2 are detected at A and B with four-momentum p1 and p2, respectively. Using plane waves, i.e., neglecting final-state interactions, the generalized probability amplitude for an individual particle, with momentum p (p1 or p2), emitted at certain point x (1 or 2) of the source to reach the point x′ (A or B) is given by [88] ψx→x′(p) = A(x, p)eiα(x)eip · (x−x′), (3.3) where A(x, p) and α(x) are the magnitude and the random phase (independent in each emission) of this amplitude. Chapter 3. Femtoscopy 40 Figure 3.2: Simplified illustration of a particle emitting source. One particle is emitted from the source at the point 1 (red), with four-position x1, and another, identical to the first, is emitted from the point 2 (blue), at space-time position x2. Two detectors A, located at xA, and B, located at xB, measure the particles with momentum p1 and p2. Since they are indistinguishable, there are two ways to detect these particles as represented by the continuous and dashed lines. Therefore, the probability amplitude for detecting one particle, emitted from 1, with momentum p1 in A and, detect an identical particle, from 2, with momentum p2 in B (represented by the continuous lines at Fig. 3.2) can be written as the product of individual amplitudes, as [88, 179]: Ψ1,2→A,B(p1, p2) = ψx1→xA(p1)ψx2→xB(p2), = A1,1 eiα1eip1 · (x1−xA)A2,2 eiα2eip2 · (x2−xB), (3.4) where Ai,j = A(xi, pj) and αi = α(xi) are used to simplify the notation. We consider that the probability of measuring two particles emitted from the same source point can be neglected. Furthermore, here the simplest case is assumed, where the source is static, chaotic, the emitted particles do not interact (no final state interactions) or decay (come from the decays of other particles) after the emission and the effect of higher order (three, four, ..., N particles) correlations is negligible. Since the detected particles are indistinguishable, the cross-term, represented by the dashed lines in Fig. 3.2, must also be considered: particle emitted from 1, with momentum p2, arrive in B and the particle emitted from 2, with momentum p1, is detected in A; in this case: Ψ2,1→A,B(p1, p2) = ψx2→xA(p1)ψx1→xB(p2), = A2,1 eiα2eip1 · (x2−xA)A1,2 eiα1eip2 · (x1−xB). (3.5) Chapter 3. Femtoscopy 41 Thus, the total probability amplitude is written as Ψ(p1, p2) = 1√ 2 [Ψ1,2→A,B(p1, p2)±Ψ2,1→A,B(p1, p2)] , (3.6) where the signal is related with the wave function symmetrization (+), for bosons, or antisymmetrization (−), for fermions. In a more realistic case, the pair of particles can be emitted from any other two points of the extended source. This is taken into account by summing all possible combinations of producing pairs of particles from two distinct points. Then, Eq. (3.6) can be rewritten as ΨΣ(p1, p2) = ∑ x1,x2 Ψ(p1, p2) = ∑ x1,x2 1√ 2 [Ψ1,2→A,B(p1, p2)±Ψ2,1→A,B(p1, p2)] = 1√ 2 ∑ x1,x2 eiα1eiα2 [ A1,1 A2,2 eip1 · (x1−xA)eip2 · (x2−xB) ± A1,2 A2,1 eip1 · (x2−xA)eip2 · (x1−xB) ] , (3.7) where we assume that the random phases, αi, depend on the emission point, but not on the momentum of the emitted particles. The probability distribution for a joint observation of two identical particles with four-momentum p1 and p2, P2(p1, p2), is defined as P2(p1, p2) = 1 2! |ΨΣ(p1, p2)|2, (3.8) where the factor 1/2! is included to avoid double counting. Using the Eqs. (3.7) and (3.8), we obtain P2(p1, p2) = 1 2 ∑ x1,x2 { |A1,1|2 |A2,2|2 + |A1,2|2 |A2,1|2 ±A1,1 A2,2 A1,2 A2,1 [ ei(p1−p2) · (x1−x2) + e−i(p1−p2) · (x1−x2) ]} × 〈 ei(α1+α2−α′1−α′2) 〉 . (3.9) Once the emitting source is considered chaotic, the random phases fluctuate rapidly. Therefore, it is n