UNIVERSIDADE ESTADUAL PAULISTA "JÚLIO DE MESQUITA FILHO" CAMPUS DE GUARATINGUETÁ HONG-HAO MA Thermodynamic properties of QCD matter and multiplicity fluctuations Guaratinguetá 2019 Hong-Hao Ma Thermodynamic properties of QCD matter and multiplicity fluctuations Tese apresentada à Faculdade de Engenharia do Campus de Guaratinguetá, Universidade Estadual Paulista, como parte dos requisitos para a obtenção do Título de Doutor em FÍSICA na área de Física Nuclear . Orientador: Prof. Dr. Wei-Liang Qian Coorientador: Prof. Dr. Kai Lin Guaratinguetá 2019 M111t   Ma, Hong-Hao ​Thermodynamic properties of QCD matter and multiplicity fluctuations​ / Hong-Hao Ma​ – Guaratinguetá, 2019 67 f : il. Bibliografia: f. 51-53 Tese (doutorado) – Universidade Estadual Paulista, Faculdade de Engenharia de Guaratinguetá, 2019. Orientador: ​Prof. Dr. Wei-Liang Qian Coorientador: Prof. Dr. Kai Lin 1. Equações de estado. 2. Transformações de fase (Física estatística). 3. Flutuação (Física). I. Título. CDU 531.145 Pâmella Benevides Gonçalves Bibliotecária/CRB-8/​9203 List of publications, proceedings, and manuscript 1. Hong-Hao Ma, Dan Wen, Kai Lin, Wei-Liang Qian, Bin Wang, Yogiro Hama, Takeshi Kodama, Hydrodynamic results on multiplicity fluctuations in heavy − ion collisions, Submitted to Phys. Rev. C, arXiv:1910.00705. 2. Hong-Hao Ma, Kai Lin, Wei-Liang Qian, Yogiro Hama, Takeshi Kodama, Thermodynamical consistency of quasi− particle model at finite baryon‘density, Phys. Rev. C 100, 015206 (2019). 3. Hong-Hao Ma, Danuce Marcele Dudek, Kai Lin, Wei-Liang Qian, Bin Wang, Yogiro Hama, Takeshi Kodama, A quasi− particle model with a phenomenological critical point, Proceedings of the conference “XIV Hadron Physics 2018" 4. Hong-Hao Ma, Wei-Liang Qian, A quasi− particle equation of state with a phenomenological critical point, Braz.J.Phys. 48, 160 (2018). 5. Juan-Juan Niu, Lei Guo, Hong-Hao Ma, Xing-Gang Wu, Production of doubly heavy baryons via Higgs boson decays, Eur.Phys.J. C 79, 339 (2019). 6. Juan-Juan Niu, Lei Guo, Hong-Hao Ma, Xing-Gang Wu, Xu-Chang Zheng, Production of semi− inclusive doubly heavy baryons via top− quark decays, Phys. Rev. D 98, 094021 (2018). 7. Juan-Juan Niu, Lei Guo, Hong-Hao Ma, Shao-Ming Wang, Heavy quarkonium production through the top quark rare decays via the channels involving flavor changing neutral currents, Eur.Phys.J. C 78, 657 (2018). DADOS CURRICULARES HONG-HAO MA NASCIMENTO 03/11/1992 - Heze / Shandong,China FILIAÇÃO Chuan-Zhi Ma Mei-Fang Su 2009 / 2013 Curso de Graduação Departamendo de Física Chongqing Universidade 2015 / 2019 Curso de Pós Graduação em Física, nível Doutorado, Faculdade de Engenharia do Campus de Guaratinguetá Universidade Estadual Paulista “Júlio de Mesquita Filho” This thesis is dedicated to my parents. ACKNOWLEDGEMENTS Time flies, the four-year PhD studentship at UNESP is coming to an end. In the past four years, I have grown and learned a lot. First of all, I would like to express the most sincere gratitude to my supervisor, Prof. Wei-Liang Qian, for his guidance over the past four years. He is a passionate and energetic physicist. His rigorous attitude towards the scientific research and meticulous work style have always influenced me, which also urged me to keep on advancing on the road of scientific research. I would also like to express the gratitude to my co-supervisor, Prof. Kai Lin. Without his help and guidance on programming, I would not finish my research work successfully. I would also like to thank Dr. Dan Wen and M.S. Wei-Xian Chen, my research colleagues. I would also like to thank Profs. Yogiro Hama, Takeshi Kodama, Jinghua Fu, Zi-Wei Lin, Xing- Gang Wu and Lei Guo for the inspiring discussions. I would also like to thank Profs. Júlio M. Hoff da Silva, Konstantin Georgiev Kostov, Elias Leite Mendonça for teaching me the fundamental knowledge. I would also like to thank Profs. Saulo Henrique Pereira, Rogério Teixeira Cavalcanti, Denize Kalempa, Carlos José Todero Peixoto, Frederique Grassi and Otavio Socolowski Junior. I would also like to thank the support from the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP). I want to express my gratitude to Brazil for the opportunity to pursue my academic career as a Ph.D. candidate, special thanks go to funding agencies Capes, CNPq and FAPESP. Last but not least, thanks to my girlfriend, Dr. Juan-Juan Niu, my brother and my parents. Thanks for your concern and support in my research career! O presente trabalho foi realizado com apoio da Coodernação de Aperfeiçoamento de Pessoal de Nível Superior- Brasil (CAPES) - código de financiamento 001, Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)-no 2018/17746-9 e Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) “There is only one good, knowledge, and one evil, ignorance.“ (Socrates) RESUMO Uma característica vital da cromodinâmica quântica (QCD) está relacionada à simetria quiral. Isso é particularmente intrigante devido ao papel crítico da simetria quiral não abeliana dos spinores de Lorentz na física teórica moderna. Muitos esforços teóricos foram dedicados à sua quebra espontânea no vácuo, bem como a restauração da mesma no ambiente extremamente quente ou denso. Além disso, quarks e glúons tornam-se os graus de liberdade relevantes por meio da transição de desconfinamento do estado dos hádrons. O significado desta última está intimamente ligado às implicações da equação de Callan-Symanzik e à teoria do grupo renormalizado. No entanto, em princípio, ambas as transições acima podem ser descritas pela QCD. Os estudos da QCD na rede demonstraram que a transição do sistema é um cruzamento suave com a densidade bariônica nula e a massa de quarks estranhos grandes. No potencial químico finito, por outro lado, uma variedade de modelos prevê a ocorrência de uma transição de fase de primeira ordem entre a fase hadrônica e o plasma de quarks e glúons (QGP). Esses resultados indicam que um ponto crítico (CEP) pode estar localizado em algum lugar no diagrama de fases da QCD no qual a linha de transições de fase de primeira ordem termina. Espera-se que a transição seja de segunda ordem neste caso. De fato, entre outros objetivos estabelecidos, o programa Beam Energy Scan (BES) em andamento no Relativistic Heavy Ion Collider (RHIC) é impulsionado pela busca do CEP. Nesta tese, exploramos alguns tópicos relacionados às propriedades termodinâmicas da matéria QCD no contexto de colisões de íons pesados. Esses estudos são, até certo ponto, motivados a alcançar uma melhor compreensão do diagrama de fases, enquanto fazem parte do esforço de busca pelo CEP. Em particular, investigamos a consistência termodinâmica do modelo de quasipartículas no potencial químico bariônico nulo e não-nulo, bem como as flutuações da razão de partículas e de multiplicidade. Entende-se que uma fase desconfinada de matéria QCD foi atingida no colisor de íons pesados ultra- relativista. Embora o modelo de sacola MIT seja simplificado demais e incapaz de descrever a equação de estado (EoS) de QGP obtida pelas simulações da QCD na rede, os cálculos de QCD perturbativa (pQCD) são confiáveis apenas a temperaturas extremamente altas. Nesse contexto, o modelo de quase- partícula é uma ferramenta valiosa para acomodar tanto a noção de graus efetivos de liberdade quanto os dados da QCD na rede. Como uma abordagem efetiva, o modelo é capaz de descrever as propriedades termodinâmicas do QGP em uma ampla faixa de temperatura T e potencial químico bariônico µ. No entanto, como apontado por Gorenstein e Yang, a consistência termodinâmica apresenta um problema, pois coloca uma restrição não trivial à EoS em questão. Sua receita foi proposta a originalmente para o potencial químico bariônico nulo, enquanto vários autores generalizaram a abordagem para o caso com a densidade bariônica finita. Nestas, revisamos a questão relativa à autoconsistência termodinâmica do modelo de quase-partículas, com potencial químico bariônico finito, adaptado aos cálculos da QCD na rede. Aqui, investigamos a possibilidade da massa efetiva de quase-partículas também ser uma função de seu momento, k, além da temperatura T e do potencial químico µ. Verifica-se que a consistência termodinâmica pode ser expressa em termos de uma equação integro-diferencial em termos de k, T e µ. Discutimos ainda duas soluções especiais, ambas podem ser vistas como condição suficiente para a consistência termodinâmica, enquanto expressas em termos de uma equação diferencial parcial. O primeiro caso mostra-se equivalente aos discutidos anteriormente por Peshier et al. O segundo, obtido por meio de uma suposição ad hoc, é uma solução intrinsecamente diferente em que a massa das partículas é dependente do momento. Essas equações podem ser resolvidas usando a condição de contorno determinada pelos dados da QCD na rede com potencial químico nulo. Pelos cálculos numéricos, mostramos que ambas as soluções podem reproduzir razoavelmente os resultados recentes da QCD na rede das Colaborações Wuppertal-Budapest e HotQCD, e em particular os relativos à densidade bariônica finita. O programa BES realizado no RHIC é dedicado a explorar o diagrama de fases da matéria nuclear que interage fortemente. Em termos de colisões Au+Au com energias relativamente baixas, estão sendo realizadas medidas precisas para a região da matéria QCD com a sua densidade bariônica alta. Intuitivamente, deve-se procurar quantidades sensíveis à física subjacente enquanto acessíveis experimentalmente. Os cumulantes de ordens mais elevantes das cargas conservadas e as suas combinações, como as taxas de cumulantes, nessa conta, tornam-se observáveis focados. Essas quantidades cumprem os requisitos, pois carregam informações vitais no meio primordial criado nas colisões. Além disso, argumentou-se que eles são sensíveis à estrutura de fases da questão QCD e, em particular, ao paradeiro do CEP. Nesse sentido, recentemente, as flutuações da multiplicidade chamaram muita atenção como um dos principais observáveis. Nesta tese, estudamos os aspectos não críticos das flutuações da multiplicidade em colisões de íons pesados, empregando um modelo hidrodinâmico. A abordagem atual é aprimorada principalmente nas abordagens existentes do modelo HRG, que levam em consideração flutuações térmicas, correção de volume finito e decaimento de ressonâncias. Nosso modelo é focado nos aspectos da expansão hidrodinâmica do sistema e nas flutuações de estado inicial evento a evento (EbE). Calculamos as flutuações da razão de partículas de K/π, K/p, e p/π usando o código hidrodinâmico SPheRIO com ou sem as condições iniciais (IC) flutuantes EbE. Além disso, também são avaliadas as flutuações de multiplicidade de prótons, kaon líquido e carga líquida. Os resultados obtidos são então comparados com os dos modelos HRG, UrQMD e os dados experimentais da colaboração STAR. Em geral, relativo aos dados existentes, os resultados obtidos pelo SPheRIO são razoáveis em comparação com as demais abordagens. Em particular, observa-se que as IC EbE podem causar um efeito considerável, que pode não apenas ultrapassar as flutuações térmicas, mas também superestimar os dados. Por sua vez, isso implica potencialmente um requisito mais rigoroso para o gerador de eventos em relação às flutuações EbE. PALAVRAS-CHAVE: equação de estado. transição de fase QCD. ponto final crítico. flutuação. ABSTRACT One vital characteristic of the quantum chromodynamics (QCD) is regarding the chiral symmetry. This is particularly intriguing owing to the critical role of non-abelian gauge symmetry of Lorentz spinors in modern theoretical physics. Many theoretical efforts have been devoted concerning its spontaneously breaking in the vacuum, as well as the restoration at the extremely hot or dense environment. Furthermore, quarks and gluons become the relevant degrees of freedom through the deconfinement transition from the hadron state of matter. The significance of the latter is closely connected to the implications of the Callan-Symanzik equation and the theory of the renormalized group. Nonetheless, in principle, both of the above transitions can be described by the QCD. Lattice QCD studies demonstrated that the transition of the system is a smooth crossover at vanishing baryon density and large strange quark mass. At finite chemical potential, on the other hand, a variety of models predict the occurrence of a first-order transition between the hadronic phase and quark-gluon plasma (QGP). These results indicate that a critical endpoint (CEP) might be located somewhere on the QCD phase diagram at which the line of first-order phase transitions terminates. The transition is expected to be of second-order at this point. As a matter of fact, among other established goals, the ongoing Beam Energy Scan (BES) program at the Relativistic Heavy Ion Collider (RHIC) is driven by the search for the CEP. In this thesis, we explore a few topics regarding the thermodynamic properties of QCD matter in the context of heavy-ion collisions. These studies are, to a certain extent, motivated to achieve a better understanding of the phase diagram, while being a part of the endeavor to search for the CEP. In particular, we investigate the thermodynamic consistency of the quasi-particle model at both vanishing and non-vanishing baryon chemical potential, as well as the particle ratio and multiplicity fluctuations. It is understood that a deconfined phase of QCD matter has been attained at the ultra-relativistic heavy- ion collider. While the MIT bag model is oversimplified and unable to describe the equation of state (EoS) of QGP from the lattice QCD simulations, perturbative QCD (pQCD) calculations are reliable only at extremely high temperature. In this context, the quasi-particle model is a valuable tool to accommodate both the notion of effective degrees of freedom and the lattice QCD data. As an effective approach, the model is capable of describing the thermodynamical properties of QGP over a wide range of temperature T and baryon chemical potential µ. However, as pointed out by Gorenstein and Yang, the thermodynamic consistency poses an issue as it places a nontrivial restriction to the EoS in question. Their proposed recipe was originally for vanishing baryon chemical potential, while several authors have generalized the approach to the case of finite baryon density. In this these, we revisit the matter regarding the thermodynamical self-consistency of quasi-particle model at finite baryon chemical potential adapted to lattice QCD calculations. Here, we investigate the possibility where the effective quasi-particle mass is also a function of its momentum, k, in addition to temperature T and chemical potential µ. It is found that the thermodynamic consistency can be expressed in terms of an integro-differential equation concerning k, T , and µ. We further discuss two special solutions, both can be viewed as sufficient condition for the thermodynamical consistency, while expressed in terms of a partial differential equation. The first case is shown to be equivalent to those previously discussed by Peshier et al. The second one, obtained through an ad hoc assumption, is an intrinsically different solution where the particle mass is momentum dependent. These equations can be solved by using boundary condition determined by the lattice QCD data at vanishing baryon chemical potential. By numerical calculations, we show that both solutions can reasonably reproduce the recent lattice QCD results of the Wuppertal-Budapest and HotQCD Collaborations, and in particular, those concerning finite baryon density. The BES program carried out at RHIC is dedicated to exploring the phase diagram of the strongly interacting nuclear matter. In terms of Au+Au collisions at relatively low energies, precise measure- ments are being carried out for the high baryon density region of the QCD matter. Intuitively, one shall look for quantities that are sensitive to the underlying physics while accessible experimentally. The higher cumulants of conserved charges and combinations of them, such as cumulant ratios, on that account, become such focused observables. These quantities fulfill the requirement as they carry vital information on the primordial medium created in the collisions. Moreover, it has been argued that they are sensitive to the phase structure of the QCD matter, and in particular, the whereabouts of the CEP. In this regard, recently, multiplicity fluctuations have drawn much attention as one of the key observables. In this thesis, we studied the noncritical aspects of the multiplicity fluctuations in heavy-ion collisions by employing a hydrodynamic model. The present approach is mainly improved upon existing HRG model approaches, which take into consideration thermal fluctuations, finite volume correction, and resonance decay. Our model is focused on the aspects of the hydrodynamic expansion of the system and the event-by-event (EbE) initial state fluctuations. We calculated the particle ratio fluctuations of K/π, K/p, and p/π by using the hydrodynamic code SPheRIO with or without the EbE fluctuating initial conditions (IC). Besides, the net-proton, net-kaon, and net-charged multiplicity fluctuations are also evaluated. The obtained results are then compared to those of the HRG, UrQMD models, as well as the experimental data from the STAR Collaboration. Overall, regarding the existing data, the results obtained by SPheRIO are reasonable in comparison with those by using different approaches. In particular, it is observed that the EbE IC may cause a sizable effect, which may not only overwhelm the thermal fluctuations but also overestimate the data. This, in turn, potentially implies a more stringent requirement for the event generator regarding EbE fluctuations. KEYWORDS: equation of state. QCD phase transition. critical end point. fluctuation. LIST OF FIGURES Figure 1 The schematic diagram of the element particles in SM . . . . . . . . . . . . . . 26 Figure 2 The Lattice QCD results for the trace anomaly, the pressure, and the entropy density from (BAZAVOV et al., 2014) . . . . . . . . . . . . . . . . . . . . . . 28 Figure 3 Diagram of QCD phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Figure 4 The quadratic and quartic fluctuations of baryon number, electric charge and strangeness, given by (CHENG et al., 2009) . . . . . . . . . . . . . . . . . . . 29 Figure 5 The running behavior of QCD strong coupling constant from the Particle Data Group (TANABASHI et al., 2018) . The world average value and uncertainty of αs(MZ) is also given, here MZ is the mass of Z0 boson . . . . . . . . . . . . . 33 Figure 6 Different dynamical evolution stages of heavy-ion collisions (QIN, 2015) . . . 35 Figure 7 A schematic diagram for (τ, x, y, η) and (t, x, y, z) coordinates taken from (SONG, 2009) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Figure 8 (color line) The temperature dependence of the calculated ratio p/pideal from the pQCD calculation with CM = 0.5 and CS = 0 (HAQUE; MUSTAFA; STRICKLAND, 2013) for the different αs order at µq = 0 and µq = 200 MeV. The shaded bands show the scale uncertainties by varying the renormalization scale M ∈ [ √ (πT )2 + µ2 q, 4 √ (πT )2 + µ2 q]. . . . . . . . . . . . . . . . . . . . 51 Figure 9 (Color online) The calculated of 3p/T 4, ε/T 4 and s/T 3 from the pQCD cal- culation where CM = 3.293 and CS = 1.509. The lattice QCD with stout action (BORSANYI et al., 2012; BORSANYI et al., 2014) at zero chemical potential is also given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Figure 10 (Color online) The calculated of trace anomaly from the pQCD calculation where CM = 3.293 and CS = 1.509. The lattice QCD with stout action (BORSANYI et al., 2012; BORSANYI et al., 2014) at zero chemical potential is also given. . 53 Figure 11 The calculated energy density and pressure for SU(3) gluodynamics. The blue squares and red circles represent for the energy density and pressure of Lattice QCD respectively. The dashed horizontal line shows the Stefan-Boltzmann limits. The calculated energy density and pressure by Eq.(4.69) by setting the parameters as, g = 16 and B = 1.7T 4 c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Figure 12 The energy density and pressure for SU(3) gluodynamics. The blue squares and red circles represent for the energy density and pressure of Lattice QCD respectively. The dashed horizontal line shows the Stefan-Boltzmann limits. The calculated energy density and pressure by Eq.(4.76) by setting the parameters as, σ = 4.73, A = 3.94T 3 c and B = −2.37T 4 c . . . . . . . . . . . . . . . . . . . . . 60 Figure 13 (Color online) The calculated of 3p/T 4 , ε/T 4, s/T 3 and trace anomaly from the Bannur’s model in comparison with those by lattice QCD with stout action (BOR- SANYI et al., 2012; BORSANYI et al., 2014) at zero chemical potential. . . . . 65 Figure 14 (Color online) The calculated entropy density and pressure by using different expressions. The entropy density calculated by using ensemble average is pre- sented in empty blue circles and that by using thermodynamic relation is in dotted purple curves. Here both calculations were done by choosing path 1. The baryon density is evaluated using Eq.(4.73), but by choosing different integral paths defined in the text, the results are shown in solid blue triangles and dotted red curves respectively. Left column: the present method; right column: the method with a presumed form for particle mass. . . . . . . . . . . . . . . . . . . . . . 71 Figure 15 (Color online) The resultant temperature dependent quasi-particle mass for glu- ons, light and strange quarks at zero chemical potential. . . . . . . . . . . . . . 72 Figure 16 (Color online) The calculated thermodynamical quantities for both vanishing and finite baryon chemical potential. The thermodynamical quantities obtained by the present model is shown in dotted blue curves. The calculated results truncated in terms of µ T up to second are shown in dotted green curves. They are compared with those of lattice QCD calculations the Wuppertal-Budapest (BORSANYI et al., 2012; BORSANYI et al., 2014) and HotQCD (BAZAVOV et al., 2012a; BAZAVOV et al., 2014; BAZAVOV et al., 2017) Collaborations, indicated by filled red circles and grey squares (with error bars when it applies) respectively. The first row shows the results of entropy density, energy density, pressure (on the left) and trace anomaly (on the right) at zero baryon chemical potential. The left plot on the second row shows the speed of sound and the right plot gives trace anomaly for different values of chemical potential. . . . . . . . . . . . . 73 Figure 17 (Color online) The calculated thermodynamical quantities for both vanishing and finite baryon chemical potential. The thermodynamical quantities obtained by the present model is shown in dotted blue curves. The calculated results truncated in terms of µ T up to second and fourth order are shown in dotted green and dashed purple curves respectively. They are compared with those of lattice QCD calculations the Wuppertal-Budapest (BORSANYI et al., 2012; BORSANYI et al., 2014) and HotQCD (BAZAVOV et al., 2012a; BAZAVOV et al., 2014; BAZAVOV et al., 2017) Collaborations, indicated by filled red circles and grey squares (with error bars when it applies) respectively. The first row shows the difference of pressure for given µB (on the left) and µB/T (on the right) for different as a function of temperature. The calculations have been carried out by using different truncations and the results are compared against corresponding lattice data. The second row presents the second and fourth order cumulants of particle number fluctuations, χ2 and χ4. . . . . . . . . . . . . . . . . . . . . . 75 Figure 18 (Color online) Top: the calculated quasi-particle mass of light quarks and its derivative as functions of temperature for different baryon chemical potentials, obtained by solving Eq.(4.127), in comparison with those by solving Eq.(4.122), the latter is equivalent to the approach by Peshier et al. (PESHIER et al., 1994). Bottom left: the quasi-particle mass of light quarks as a function of momentum for the solution discussed in the present work. Bottom right: the calculated asymp- totic behavior of quasi-particle masses, in comparison with a model (PESHIER; KAMPFER; SOFF, 2002) inspired by the gauge-independent HTL calculations. 76 Figure 19 (Color online) The calculated of 3p/T 4 , ε/T 4 and s/T 3 using the quasi-particle model in comparison with those by lattice QCD with stout action (BORSANYI et al., 2012; BORSANYI et al., 2014) at zero chemical potential. . . . . . . . . 85 Figure 20 (Color online) The calculated results in comparison with the lattice QCD data (BOR- SANYI et al., 2012; BORSANYI et al., 2014). (a) the trace anomaly as a function of temperature, (b) the speed of sound as a function of temperature. . . . . . . . 85 Figure 21 The calculated ∆p/T 4 as a function of temperature for different chemical poten- tials, in comparison with the lattice QCD results by stout action (BORSANYI et al., 2012). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Figure 22 (Color online) The calculated pressure, energy density and entropy density as functions of temperature, and pressure as function of energy density for different chemical potentials. The results of the present interpolation scheme is compared to those of a first order phase transition (1OPT). From left to right, the three columns are for µB = 0, 0.2 and 0.5 GeV respectively. . . . . . . . . . . . . . 87 Figure 23 The pT spectra of the identified particles from SPheRIO with and without the resonance decay and PHOBOS (BACK et al., 2004). Here the pseudorapidity interval is takes as 0.2 < y < 1.4. . . . . . . . . . . . . . . . . . . . . . . . . . 96 Figure 24 The rapidity distribution of the identified particles from SPheRIO with and without the resonance decay and PHOBOS (BACK et al., 2004). Here the pT interval is taken as 0 < pT (GeV) < 5. . . . . . . . . . . . . . . . . . . . . . . 97 Figure 25 The dynamical fluctuations of particle ratio of p/π, K/π and K/p at the differ- ent energies. The results are shown from the STAR collaboration (blue solid star, blue open star and blue half-filled star) (TARNOWSKY, 2012), the NA49 collaboration of the most central Pb+Pb collisions (black solid square) (ALT et al., 2009), UrQMD model calculations (dark yellow open triangle with dashed line) (BASS et al., 1998; BLEICHER et al., 1999) and the HRG model calcula- tions (purple open circle with the straight, dashed and dotted line). Also shown are the results by hydrodynamical simulation program SPheRIO with(without) the consideration of EbE fluctuation at the right(left) column (red solid circle, red open circle and red half-filled circle). “w/" and “w/o" represent for “with" and “without" respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Figure 26 The multiplicity of net-charged particles, net-proton and net-kaon at the different energies. The corresponding results are given from the HRG model (purple solid and dashed lines), the SPheRIO with an average IC (red open triangle and square), the STAR collaboration (blue open star with the error bar) (THäDER, 2016) and the UrQMD model calculations (olive dashed line or area) (BASS et al., 1998; BLEICHER et al., 1999). “w/" and “w/o" represent for “with" and “without" respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 LIST OF TABLES Table 1 – List of parameters used in the Bannur’s model for the Nf = 2 + 1 QCD system at vanishing chemical potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Table 2 – A comparison of the calculated bag constant B(T, µ) by using two different integral paths. The integral for B is carried out from (T0 = 0.16 GeV , µ0 = 0) to (T, µ), where path 1 is defined by (T0, µ0) → (T, µ0) → (T, µ), while path 2 is through (T0, µ0) → (T0, µ) → (T, µ). The table shows the results obtained for several different values of (T, µ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Table 3 – List of parameters used in the present hybrid EoS at non-vanishing baryon chemical potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Table 4 – Average number of efficiency uncorrected identified particles used in the analysis of νdyn which are measured by STAR for 0 -5 % centrality . . . . . . . . . . . . . . . 99 Table 5 – Particle list where the anti-particles are not included . . . . . . . . . . . . . . . . 115 Table 6 – Particle list where the anti-particles are not included . . . . . . . . . . . . . . . . 116 Table 7 – Particle list where the anti-particles are not included . . . . . . . . . . . . . . . . 117 LIST OF ABBREVIATIONS AND ACRONYMS BES Beam Energy Scan BNL Brookhaven National Laboratory CEP critical endpoint CERN the European Organization for Nuclear Research CGC the color glass condensate EoS equation of state EoM equation of motion GCE the grand canonical ensemble HRG hadron resonance gas HTL hard thermal loop LHC Large Hadron Collider pQCD perturbative QCD QCD quantum chromodynamics QFT quantum field theory QGP the quark gluon plasma RHIC Relativistic Heavy Ion Collider SM standard model SPH Smoothed Particle Hydrodynamics SSM standard statistical mechanics LIST OF SYMBOLS gs QCD coupling constant Λ QCD scale gµν metric tensor τ proper time τ0 initial time η space time rapidity uµ four velocity T µν energy-momentum tensor Jµi conserved current f(x, p) distribution function Tfo freeze-out temperature Tc critical temperature µc critical chemical potential ε energy density p pressure s entropy density c2 s the squared speed of sound nB baryon number density nS strangeness number density CONTENTS 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.1 Signals of QGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.2 The equation of state of the QGP phase . . . . . . . . . . . . . . . . . . . . . . . 27 1.3 QCD phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.4 Outlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2 QUANTUM CHROMODYNAMICS . . . . . . . . . . . . . . . . . . . . . . . 31 3 HYDRODYNAMICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1 Equation of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Cooper-Frye prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.3 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4 EQUATION OF STATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 A brief review of statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . 43 4.2 Hadron Resonance Gas Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.3 pQCD calculations for equation of state of Nf = 3 QCD . . . . . . . . . . . . . . 48 4.4 quasi-particle model for QGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.4.1 Thermodynamic inconsistency . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.4.2 Some attempts for solving thermodynamic inconsistency . . . . . . . . . . . . 54 4.4.2.1 Gorenstein and Yang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.4.2.2 Bannur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.4.2.3 Gardim and Steffen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.5 General solutions for thermodynamic inconsistency at finite chemical potential . . 67 4.5.1 The momentum independent solution . . . . . . . . . . . . . . . . . . . . . . . 68 4.5.2 A special momentum dependent solution . . . . . . . . . . . . . . . . . . . . . 69 4.5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 5 THE QCD PHASE TRANSITION AND CROSS-OVER . . . . . . . . . . . . 79 5.1 A hybrid EoS with QCD critical point . . . . . . . . . . . . . . . . . . . . . . . . 79 5.1.1 Quasi-particle Model for 2+1 flavor QGP at finite chemical potential . . . . . 80 5.1.2 The smoothed connection in transition region . . . . . . . . . . . . . . . . . . 81 5.1.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 5.2 Multiplicity fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.2.1 Thermodynamical fluctuations and resonance decay . . . . . . . . . . . . . . 88 5.2.2 A hydrodynamic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 5.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2.3.1 Particle ratio p/π, K/π and K/p fluctuations . . . . . . . . . . . . . . . . . . . . 96 5.2.3.2 Net-proton, net-kaon and net-charge multiplicity fluctuations . . . . . . . . . . . 99 6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 APPENDIX A – PARTICLE LIST FOR HRG MODEL . . . . . . . . . . . . 115 APPENDIX B – THERMODYNAMIC CONSISTENCY WITH EXCLUDED VOLUMES . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 APPENDIX C – THE METHOD OF CHARACTERISTICS . . . . . . . . . 121 APPENDIX D – FLUCTUATIONS OF ICS . . . . . . . . . . . . . . . . . . 123 APPENDIX E – THE HIGHER MOMENTS OF NET-CHARGE, NET-PROTON AND NET-KAON MULTIPLICITY . . . . . . . . . . . . . 125 25 1 INTRODUCTION Throughout human history, we have always been attracted to questions regarding the most funda- mental building blocks of the Universe. Today, these questions have been transformed into the study of elementary particles and their interactions, strongly associated with the development of modern physics. It is generally understood that there are four types of fundamental interactions in nature, namely, strong, electromagnetic, weak, and gravitational. The Standard Model (SM) provides a consistent scheme to accommodate three out of the four interactions, where relevant fermions interact through the exchange of bosons. The fermions consist of three generations of quarks and leptons. To be specific, there are six quarks in three generations, namely, up (u), down (d), strange (s), charm (c), bottom (b), and top quark (t) and the antiquarks. Also, there are six leptons in the three generations, namely, electron (e), muon (µ), tau (τ ), the corresponding neutrinos, as well as their antiparticles. The bosons, on the other hand, are composed of gluons (g), photons (γ), W± and Z0 bosons, as well as the Higgs bosons. The latter was found recently by ATLAS (AAD et al., 2012) and CMS (CHATRCHYAN et al., 2012) at Large Hadron Collider (LHC) in 2012. Graviton, the quantum of gravity, apart from the issue concerning renormalizability, is also a boson which mediates the gravitational force. On the other hand, quantum field theory (QFT) serves as a theoretical framework that encompasses the other known interactions. In particular, the electromagnetic and weak interactions were unified by Glashow (GLASHOW, 1961) in 1961. Thereafter, Weinberg (WEINBERG, 1967) and Salam (SALAM, 1968) introduced the Higgs mechanism into Glashow’s theory, which gives rise to the origin of the mass for Fermions in particle physics. Besides, the strong interaction is described by the theory of quantum chromodynamics (QCD). Figure 1 illustrates all the elementary particles in SM. As a theory for describing the strong interaction between quarks and gluons, QCD is a SU(3) gauge theory in terms of the quark and gluon degrees of freedom (PESKIN; SCHROEDER, 2006). There are two main characteristic properties, namely, color confinement and asymptotic freedom. We will briefly review the QCD theory below in Sec. 2. Due to the color confinement, quarks can only move inside the volume of the hadron at standard temperature and density. However, Lattice QCD calculations indicated a phase transition from the hadronic to quark-gluon plasma (QGP) state for extremely high temperature or baryon density. For the latter, the quarks and gluons are free to move over a large volume compared with the typical size of hadrons. It is also understood that the QGP can be produced in relativistic heavy-ion collisions. However, due to the color confinement, individual quarks or gluons still can not be directly observable in the final state particles. Moreover, the QGP phase only exists for a very short period, as the system rapidly expands and cools down. The process continues as hadronization takes place until the condition for freeze-out is satisfied. Therefore, one can only obtain valuable information about the QGP from the observed hadrons spectrum. In other words, the existence of QGP can only be inferred from the experimentally accessible quantities. These experimental observables, which indicate the presence of QGP, are of great theoretical interest, will be discussed in the following section. 26 Figure 1 – The schematic diagram of the element particles in SM 1.1 SIGNALS OF QGP There are, in literature, several relevant signatures of QGP which has aroused much attention. One of those signatures is strangeness enhancement (RAFELSKI, 1982). This enhancement means strange quarks will produce in more abundance than non-strange anti-quarks. To be more specific, in equilibrating QGP, the number density of strange quark is ns = gs Tm2 s π2 K2 (ms T ) (1.1) and the number density of massless non-strange anti-quark q̄ is nq̄ = gq̄ T 3 π2 e−µq/T , (1.2) where gq̄ = gs = 6 is the degeneracy of the anti-quark and/or s quark, q denotes u or d quark and µq = µB/3 is the quark chemical potential. Then the ratio of strange quarks to non-strange quarks is ns nq̄ = 1 2 (ms T )2 K2 (ms T ) e−µq/T . (1.3) It is easy to derive that this ratio is greater than one for µq > 0. In other words, the number of strange quark will be produced more than the one of non-strange quark. Therefore the production of strange hadron will enhance after hadronization. There are some experiments that support this enhancement 27 of strange particles, particularly strange and multistrange antibaryons (BARI et al., 1995; KINSON; ABATZIS; ANDERSEN, 1995). Another signature is J/ψ suppression (MATSUI; SATZ, 1986). This idea is also simple. As Matsui and Satz (MATSUI; SATZ, 1986) suggested, the production of J/ψ will be suppressed if the QGP is produced in the collisions. Such a suppression was indeed discovered by the experiment (GAVIN; VOGT, 1996). Besides, there are also some other experimental evidences in support of the signatures of QGP, like the dilepton pairs signature (SHURYAK, 1978a) discovered by (AGAKICHIEV et al., 1995) and Jet quenching phenomena (WANG; GYULASSY, 1992) which is discovered by (ADAMS et al., 2003; ADLER et al., 2003). Finally, in 2005, Relativistic Heavy Ion Collider (RHIC) announced that QGP had been discovered (GYULASSY, 2004; GYULASSY; MCLERRAN, 2005; JACOBS et al., 2007). 1.2 THE EQUATION OF STATE OF THE QGP PHASE In order to investigate the QGP’s properties and formation, we should firstly derive its equation of state (EoS). Due to the asymptotic freedom of QCD at high energy, QGP can be treated as a weakly coupled gas when T ∼ 105Tc. Therefore, considering the free QGP is a relativistic ideal gas, its pressure, energy density and other thermodynamics can be obtained from a MIT bag model (CHODOS et al., 1974). In this model, a bag constant B, as the ’external bag pressure’, is introduced. In addition, Lattice QCD, as a non-perturbative approach can be applied to study the QCD properties at finite temperature T (BORSANYI et al., 2014; BAZAVOV et al., 2014). The EoS at vanishing chemical potential for Nf = 2 + 1 QCD is shown in Figure 2. However, there exists the sign problem of Fermions when the chemical potential µ is non-zero (FORCRAND, 2009). In order to evade such a sign problem, there are three main approaches: reweighting, Taylor expansion and analytic continuation from imaginary µ. But those approaches can give reliable results only when µ is small enough (FORCRAND, 2009). For now, those methods can allow us to explore the QCD phase diagram up to µB/T ' 2.5 (BAZAVOV et al., 2017). For high energies, it is understood that perturbative QCD (pQCD) at finite temperature T and chemical potential µ is suitable for computing the thermodynamics of QGP, for the asymptotic freedom of QCD. The pQCD expansion of free energy had been pushed up to g6 s ln(1/gs) order (KAJANTIE et al., 2003). However such a pQCD series shows a bad convergence (KAJANTIE et al., 2003; ARNOLD; ZHAI, 1995; ZHAI; KASTENING, 1995). We can reorganize the perturbative series to improve the convergence. There are some attempts to solve this problem, such as HTL perturbation theory (ANDERSEN; BRAATEN; STRICKLAND, 1999; ANDERSEN et al., 2002), the 2-loop φ-derivable approximation (BLAIZOT; IANCU; REBHAN, 1999; BLAIZOT; IANCU; REBHAN, 2003; BLAIZOT; IANCU; REBHAN, 2001), principle of maximum conformality (PMC) (BU et al., 2018) and others. To characterize the EoS of QGP over a wide range of temperature and chemical potential, we adopt a phenomenological model called quasi-particle model. This model is assumed to treat the QGP phase as a non-interacting system with massive quasi-particle. Generally, the mass of quasi- particle in this phenomenological model is dependent on the temperature. If we follow the standard 28 Figure 2 – The Lattice QCD results for the trace anomaly, the pressure, and the entropy density from (BAZAVOV et al., 2014) procedure of statistical mechanics for this ideal massive gas naively, the thermodynamic inconsistency appears (GORENSTEIN; YANG, 1995). As an in-depth discuss, we will revisit the possible solutions for this inconsistency problem and propose a special solution where the quasi-particle mass m is a function of temperature, chemical potential and momentum. 1.3 QCD PHASE DIAGRAM Figure 3 shows the QCD phase diagram. As indicated in this figure, hadronic phase (Hadron Gas) exist in low temperature T with low baryon chemical potential µB which will be switched to QGP phase at high T or high µB . However the thermodynamic properties on the boundary separating the two phases seems interesting. At non-vanishing baryon chemical potential, i.e. µB 6= 0, theoretical model calculations (BARI et al., 1995; KINSON; ABATZIS; ANDERSEN, 1995; MATSUI; SATZ, 1986; GAVIN; VOGT, 1996; AGAKICHIEV et al., 1995; HALASZ et al., 1998; BERGES; RAJAGOPAL, 1999; STEPHANOV; RAJAGOPAL; SHURYAK, 1998; SCHWARZ; KLEVANSKY; PAPP, 1999; FODOR; KATZ, 2004) suggest it is a first order phase transition. On the other hands, non-perturbative approach, like Lattice QCD (FODOR; KATZ, 2002; KARSCH, 2002), indicates that there is a smooth crossover for µB = 0 and large strange quark mass. Therefore, it implies the existence of critical endpoint (CEP) where the line of first order phase transitions terminates, and the transition is expected to be of second order at this point. However, the exact location of the CEP is also unknown due to the difficulty of lattice QCD calculations at finite µB (STEPHANOV, 2006; FORCRAND, 2009). Therefore the experiment to search for the CEP of QCD falls (AGGARWAL et al., 2010). So far, the existence and properties of the critical point is a long-standing intriguing topic. It is worth mentioning that, at the CEP, the QCD phase transition is continuous rather than the first 29 Figure 3 – Diagram of QCD phase order, such that we should observe the divergent susceptibility for the critical opalescence phenomenon. In fact there are large fluctuations near the QCD critical temperature according to Lattice QCD calculations (KARSCH, 2007; CHENG et al., 2009). Figure 4 shows the calculated quadratic and quartic fluctuations of baryon number, electric charge, and strangeness by the Lattice QCD. Thus, in experiment, the fluctuations of baryon number, charge number can possibly be served as the signature of QCD CEP. Figure 4 – The quadratic and quartic fluctuations of baryon number, electric charge and strangeness, given by (CHENG et al., 2009) 30 1.4 OUTLINES In chapter 2 and 3, we mainly review some theoretical backgrounds, QCD theory and hydrody- namics. In chapter 2, The QCD theory, including its two main properties, color confinement and asymptotic freedom, is revisited. The asymptotic freedom property of QCD allows that one can apply perturbation theory to the physical processes with high momentum transfer. For the physical process of low momentum transfer, some non-perturbative theories should be adopted, such as Lattice QCD and sum rules. Another property of QCD, color confinement, indicates gluons can not be observable. Thus the only way to investigate the properties of QGP is to study the observed hadrons at the final stage of ultra-relativistic heavy ion collisions. Hydrodynamics which can describe heavy ion collision process is introduced in chapter 3. Because the EoS is essential to the hydrodynamic analysis, so that some different models or theoretical tools to describe the thermodynamic properties of QCD matter is shown in chapter 4. In this chapter, we adopt the HRG model with excluded volume correction to obtain the EoS for hadronic matter and pQCD calculations or quasi-particle model fitting with Lattice QCD results to obtain the one for QGP. We also discussed the thermodynamic self-consistency in HRG model with excluded volume correction and as well the thermodynamic inconsistency for quasi-particle model with a temperature-dependent mass. Then, some solutions for solving this inconsistency are presented. Finally, we proposed the general solutions coming from the Maxwell equation and give a special solution with a temperature-, chemical potential- and momentum- dependent mass. In chapter 5, we present a hybrid EoS with a phenomenological CEP at finite chemical potential. The results we obtained coincide with the Lattice data. Then we focus on the noncritical aspects of the multiplicity fluctuations in heavy-ion collisions by employing a hydrodynamic model. The calculated multiplicity fluctuations are compared to those of the hadronic resonance gas model, UrQMD model as well as to the experimental data. The conclusions and summaries are presented in the last chapter 6 31 2 QUANTUM CHROMODYNAMICS Nuclear physics, as a new branch of physics, is a subject of both profound theoretical significance and great practical significance. In 1911, Rutherford et al bombarded various atoms with α rays to observe the deflection of α rays which proved the nuclear structure of atoms. Thus the planetary model of atomic structure was established , which laid a foundation for the study of atomic structure. Soon afterwards, we made a preliminary understanding of the shell structure of atoms and the motion of electrons. Based on this understanding, the quantum mechanics was established to describe the motion rules of matter in the microscopic world. In the late 1920s, the principle of accelerating charged particles was being explored. By the early 1930s, electrostatic, linear and cyclotron accelerators had taken shape, and initial nuclear reaction experiments were carried out on high-voltage multipliers. By using accelerators, stronger beams, higher energies and more kinds of beam can be obtained, which greatly expands the research work of nuclear reactions. Since then, accelerators have gradually become necessary tools for studying nuclei. Hundreds of short-lived particles, known as baryons, mesons, leptons and a variety of resonant states, have been discovered through the interaction of high-energy and ultra-high-energy ray beams with atomic nuclei. The discovery of large families of particles has taken the study of the physical world to a new stage and established a new discipline, particle physics, also called high-energy physics. In the past, through the study of macroscopic objects, one knows that there are two kinds of long range interaction, electromagnetic interaction and gravitational interaction; A closer look at the nuclei of atoms reveals that there are two short-range interactions between matter, the strong interaction and the weak interaction. It has become an important topic of particle physics to study the laws of these four interactions and their possible relations, and to explore other possible new interactions. There is no doubt that nuclear physics research will make important new contributions in this regard. The gauge theory that describes strong interactions is called QCD (APPELQUIST; POLITZER, 1975; BARBIERI; GATTO; KÖGERLER, 1976; RÚJULA; GLASHOW, 1975). It describes the interactions between the quarks that make up the strongly interacting particles (hadrons) and the gauge fields associated with the quantum number of color. It can uniformly describe the structure of hadrons and the strong interactions between them. The strong interaction satisfies SU(3) group symmetry. The lagrangian established by non-Abelian gauge invariance is L = ∑ f Ψf (x) (iγµDµ −mf ) Ψf (x)− 1 4 F a µνF µν a , (2.1) where F a µv ≡ ( ∂µA a v − ∂vAaµ + gfabcAbµA c v ) . An important feature of the Lagrangian in Eq.(2.1) is the occurrence of the self-interaction of the gauge field, which comes from the kinetic energy term 1 4 F a µνF µν a . There is an interaction between three gauge fields and four gauge fields in the Lagrangian. This feature is unique to non-abelian gauge fields. It is the existence of self-interaction of gauge field that makes the non-abelian gauge field theory appear a completely new world. For example, the asymptotic freedom in QCD is an inevitable result of the self-interaction of the gauge field. In quantum 32 field theory, the β-function is negative and has the anti-shielding property to make the effective coupling constant αs (Q2) decreases with the increase of Q2, that is, the property of asymptotical freedom. In this field theory, the mediators in the strong interaction are massless gluons. Besides those gluons are colored, and they produce a transfer between different colored quarks, and they transfer strong interactions. Because the gluons have color charge, the interaction between gluons can cause anti-shielding effect, which determining the asymptotic free nature of the strong interaction. The basic degrees of freedom of QCD are quarks and gluons, which are tightly bounded inside the hadron. They cannot present free states and can only be observed indirectly by hadron experiments. In QCD theory, the three-generations quark and gluons have quark-gluon interactions. There are also three gluon and four gluon interaction vertices. It’s these vertices that determine the strong interaction coupling constant gs decreases as the energy increases and the β-function closely related to gs is negative value. Therefore, the effective coupling constant in strong interaction satisfies αs ( Q2 ) = 4π β0 log Q2 Λ2 1− β1 β2 0 log ( log Q2 Λ2 ) log ( Q2 Λ2 ) + · · · , (2.2) where αs = g2 s/(4π), β0 = − ( 2 3 Nf − 11 ) and β1 = 2 3 (153− 19Nf ) are the β - function value for the one-loop and two-loop approximation respectively. Nf is the flavor of quark. From the above Eq.(2.2), it can be seen that when the energy Q2 tends to infinity, the strong interaction coupling constant αs (Q2) tends to zero, as shown in Figure 5. This means that the interaction between quarks tends to zero. Eq.(2.2) also quantitatively expressed the strongly interacting asymptotic properties of freedom. The correctness of Eq.(2.2), which reflects the asymptotic freedom, is proved by experiments. To visually reflect this feature of the coupling, the constant is also called the running coupling constant. Perturbation theory had made a great success due to the asymptotic freedom of QCD. However, perturbation theory can only be applied to the physical processes with high momentum transfer. For the physics with low momentum transfer and hadronic structures, it is powerless. Among the six kinds of quarks in nature, the first five quarks (u, d, s, c, b) exist only within the hadronic bound states. Meanwhile, the t quark as the heaviest quark had a very short life after its creation and soon decayed into the b quark . From Eq.(2.2), one can also see that the running coupling constant increases to infinity as the energy Q2 decreases. This qualitatively explains why quarks cannot be separated in free states within hadrons. As we known, when the distance between quarks increases, the energy Q2 of the exchanged gluons between quarks would decrease. Therefore, the running coupling constant would increase, as the increase of the distance. In the other word, the interactions between the quarks increase as the distance between them increases, which means the quarks and gluons would be forever bounded into hadrons. Thus, such a physical phenomenon is known figuratively as "quark confinement". In the QCD framework, the physicist can only qualitatively explain the structure of quarks trapped inside hadrons, but how to quantitatively explain the structures image of quarks trapped inside hadrons is still a major problem in high energy physics. Lattice gauge theory is trying to solve the problem of quark confinement from QCD theory. Since lattice gauge theory is essentially non-perturbation theory, its theoretical method does not depend on the interaction, so scientists are trying to get all the solutions to the strong interactions, not just asymptotic ones by the solution. Asymptotic freedom 33 and quark confinement are two important features of QCD theory. For now, one thought the quark confinement is based on the property of physical vacuum in QCD and perturbation QCD theory is based on perturbation vacuum. While the QCD physical vacuum is completely different from the perturbed vacuum. In a physical vacuum, it’s full of quarks, antiquark pairs and gluons. The continuous interaction of matter, quark-antiquark pairs with gluons results in new images of hadron structures. Thus revealing the nature of vacuum will lead to the discovery of the quark confinement puzzle. At present, the relativistic heavy ion collider in brookhaven is to reveal the properties of physical vacuum experimentally. The idea is to liberate quarks and gluons from protons and neutrons under extreme conditions, and to make the transition from the confinement phase of a quark to the deconfinement phase of a quark. Only by fully understanding asymptotic freedom and quark confinement can one say that one has a profound understanding of strong interactions. Figure 5 – The running behavior of QCD strong coupling constant from the Particle Data Group (TAN- ABASHI et al., 2018) . The world average value and uncertainty of αs(MZ) is also given, here MZ is the mass of Z0 boson As we had discussed before, although pQCD can be applied to compute the physical observables in high-energy processes successfully, it will be no longer applicable for the low-energy processes. The momentum transfer in a low-energy process is small which will lead to a relatively large αs, as shown in Figure 5. To be more specific, the perturbation expansion of the physical observables will no longer converge, if the value of the running coupling constant is close to or even greater than 1, i.e., αs ∼ 1. Therefore, some other non-perturbative methods, such as lattice gauge theory, Schwinger-Dyson equation and sum rule techniques should be introduced to describe this process. 35 3 HYDRODYNAMICS According to the color confinement of QCD theory, it is thought that hadrons are consisting of the confined quarks and gluons. Meantime, this property of QCD means the isolated quarks or gluons can not be observed. However, the hadronic matter would be transferred into a new QCD phase–QGP in extreme conditions (extremely high temperature and densities) due to Debye screening effect. It is easy to be told that QGP should exist in the core of neutron stars or at the very early universe for their extremely high temperature and densities environments. It is almost impossible to study the properties of QGP matter in the inner core of neutron stars or at the very early universe after the Big Bang. Fortunately, such extreme conditions can be formed by colliding two heavy nuclei at ultra-relativistic energy. After decades of years of efforts, more and more evidences showed that the QGP matter can be produced at the high energy collisions of heavy nucleus. In 2000, “a new state of matter" seemed to be discovered in SPS energy ( √ s ∼ 17GeV) at the European Organization for Nuclear Research (CERN) while there is no solid evidence. Then, the RHIC at Brookhaven National Laboratory (BNL) found the unambiguous evidences and annouced the existence of QGP. Currently, the heavy ion collider is a only reasonable tool to study the properties of deconfined QGP matter on earth. Figure 6 – Different dynamical evolution stages of heavy-ion collisions (QIN, 2015) Figure 6 illustrates the different dynamical evolution stages of heavy-ion collisions with ultra- relativistic energy. Before the collisions, the speed of two nuclei will be almost accelerated up to the speed of light in the laboratory frame. Then the nuclear matter (extremely hot and dense matter) in a highly-excited state is produced after these two Lorentz contracted nuclei collide. Such a highly-excited nuclear matter can be called pre-equilibrium matter, because it is not in equilibrium yet. After a certain thermalisation time (about 1fm/c), the pre-equilibrium matter will reach local thermal equilibrium. Then the equilibrated QGP matter expands and cools down. When the temperature reaches the critical temperature Tc ' 150 MeV (BORSANYI et al., 2010; BAZAVOV et al., 2012b; BHATTACHARYA et al., 2014), the QCD phase transition from QGP to hadronic matter occurs. It is worth mentioning that, the formed hadronic matter is consisting of stable and unstable hadrons and hadrons resonances. Then, the hadronic matter also expands while the QCD phase transition takes place. The hadrons will freeze-out since the system continues to cool and expand. Finally, the hadrons at the freeze-out surface will be detected in the detector. However, the QGP matter can not be observed directly in the detector. To study the properties of QGP matter in energetic heavy-ion collisions, one should find a reasonable 36 connection between the QGP produced at the begining and the final hadrons observed in the detector and identify reliable signatures for the formation of QGP. Here, we use the hydrodynamic model to describe heavy ion collision process (WONG, 1994; CHAUDHURI, 2014). However, hydrodynamics can not be applied to describe the initial thermal- ization process. Thus, in hydrodynamics, one assumes that the local thermal equilibrium is already achieved and such a state of matter can be specified by some appropriate macroscopic variables, like thermodynamical quantities and flow velocity. We also call those macroscopic variables the initial conditions (IC). Then the hydrodynamical model can simulate the relativistic heavy ion collisions from the equilibrium stage to the hadronic phase and freeze-out stage successfully (KOLB et al., 2001; KOLB; HEINZ, 2003; TEANEY, 2003; CHAUDHURI, 2008). In another word, hydrodynamics can describe the evolution of the QGP and hadronic matter with a given and appropriate IC. In the following Sec. 3.1, 3.2 and 3.3, we will introduce the equation of motion (EoM) for the evolution, the Cooper-Frye prescription for the decoupling process and the IC. 3.1 EQUATION OF MOTION As discussed before, it is assumed that the expanding system possesses local thermal equilibrium. Hydrodynamics suggests that the expansion is collective, which can essentially be described by conservation equations, such as energy-momentum, baryon number, strangeness and isotopic spin, etc. ∂µT µν = 0, (3.1) ∂µJ µ i = 0, (3.2) where T µν = (ε+ p)uµuν − pgµν (3.3) is the energy-momentum tensor, ε, p, and uµ are the energy density, the pressure and the four-velocity of the fluid respectively. Jµi is the conserved current which can be specified by some conserved number, like baryon number JµB = nBu µ, (3.4) or strangeness, JµS = nSu µ, (3.5) where nB and nS are the baryon number density and strangeness density. 37 In this thesis, we consider the fluid in Minkowski space where the metric tensor is gµν = gµν =  1 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 −1  . (3.6) Then four vectors, like the trajectory xµ = (t, x, y, z), should transform as follows xµ = gµνx ν = (t,−x,−y,−z). (3.7) However, the (τ, x, y, η) coordinate is more suitable to solve the hydrodynamical equations rather than (t, x, y, z), where τ is the proper time. Those two coordinates are shown in Figure 7 xµ = (t, x, y, z)→ xm = (τ, x, y, η), (3.8) where t = τ cosh η; τ = √ t2 − z2, z = τ sinh η; η = 1 2 ln ( t+ z t− z ) . (3.9) The metric tensors in this coordinate system are gmn = diag(1,−1,−1,−1/τ 2) and gmn = diag(1,−1,−1,−τ 2). Figure 7 – A schematic diagram for (τ, x, y, η) and (t, x, y, z) coordinates taken from (SONG, 2009) Thus, it is easy to find that the four-velocity uµ = (u0, ~u) is normalized, uµuµ = 1, (3.10) 38 whose definition is uµ = dxµ dτ . Following the procedure suggested by (ELZE et al., 1999; ELZE et al., 1998; HAMA; KODAMA; SOCOLOWSKI JR., 2005), we can derive the equations of relativistic hydrodynamics, i.e. Eqs. (3.1)- (3.5) by using the variational formulation. Firstly, for a system with the proper energy density ε, its action is I = ∫ d4x{−ε}. (3.11) Meantime, we also introduce other two conserved number which is the proper baryon density, n, and proper entropy density, s, ∂µ(nuµ) = 0, ∂µ(suµ) = 0. (3.12) Additionally, the energy density can be expressed in form of n and s, ε = ε(n, s). (3.13) The other thermodynamics, like pressure, temperature and baryon chemical potential can be obtained from the following thermodynamic relations, p = n ∂[ε(n, s)/n] ∂n , T = ∂ε(n, s) ∂s , µ = ∂ε(n, s) ∂n . (3.14) Then, with the aid of constraints Eqs.(3.10) and (3.12), the hydrodynamical EoM for the fluid can be obtained by the Lagrangian equation with respect to n, s and uµ. By introducing the Lagrangian multipliers, λ, ζ and w, we can merge the previous conditions, Eqs.(3.10) and (3.12) into the variational principle. Then we have δ ∫ d4 [ −ε(n, s) + λ∂µ (nuµ) + ζ∂µ (suµ)− 1 2 w (uµuµ − 1) ] = 0. (3.15) In other word, we can obtain the hydrodynamical fluid in terms of the effective Lagrangian, which is written as L(fluid) eff (n, s, uµ, λ, ζ, w) =− ε(n, s)− nuµ∂µλ − suµ∂µζ − w 2 (uµuµ − 1) . (3.16) Here all the variational variables in the brackets, i.e., n, s, uµ, λ, ζ and w are independent. Therefore the EoM for the fluid can be obtained from the Lagrange’s equations ∂µ ( ∂L(α) ∂ (∂µα) ) − ∂L(α) ∂α = 0. (3.17) 39 It is easy to derive that the variations with respect to the Lagrangian multipliers λ, ζ and w show the constraints, Eqs.(3.10) and (3.12). And the variations with respect to n,s and uµ lead to −µ− uµ∂µλ = 0, (3.18) −T − uµ∂µζ = 0, (3.19) −n∂µλ− s∂µζ − wuµ = 0. (3.20) After solving the above equations with the constraint, Eq.(3.10), we have w = nµ+ Ts = ε+ p, (3.21) where p is the pressure and w is shown as the enthalpy density. Then we substitute back this known w into Eq.(3.20) and multiplying the both sides of Eq.(3.20) by uν , we get wuµuν = −(nuν)(∂µλ)− (suν)(∂µζ). (3.22) Thus, with the aid of Eqs.(3.18) and (3.19), it is not difficult to be derived ∂ν(wuµuν) = − ∂ν(nuν)(∂µλ)− ∂ν(suν)(∂µζ) − (nuν)(∂ ν∂µλ)− (suν)(∂ ν∂µζ) = − (nuν)(∂ ν∂µλ)− (suν)(∂ ν∂µζ), (3.23) with (nuν) (∂ν∂µλ) = n∂µ (uν∂ νλ)− n (∂νλ) (∂µuν) = −n∂µµ− n (∂νλ) (∂µuν) , (3.24) and (suν)(∂ ν∂µζ) = −s∂µT − s(∂νζ)(∂µuν). (3.25) By using Eq.(3.20) and the Gibbs-Duhem relation, dp = sdT + ndµ, (3.26) and the property, uν(∂µuν) = 0, (3.27) the Eq.(3.23) can be rewritten as ∂ν (wuµuν) = n∂µµ+ s∂µT + (∂µuν)wu ν = ∂µp+ uν (∂µuν)w = ∂µp, (3.28) 40 which means ∂ν ((ε+ p)uµuν − pgµν) = 0, (3.29) or ∂µ ((ε+ p)uµuν − pgµν) = 0. (3.30) The relativistic hydrodynamic equation (3.1) is revisited. 3.2 COOPER-FRYE PRESCRIPTION In the previous section, we have shown that the EoM governs the space-time evolution of the fluid. As the fluid further expands, the temperature and density of the system become lower and lower. In the end, the decoupling of the constituent particles occurs, which means that there will be no interactions among them. Such a decoupling or freeze-out process can be described by using Cooper-Frye prescription (COOPER; FRYE, 1974). There are also some other models to describe the freeze-out process, such as continuous emission and transport model, but we don’t address them in this thesis. According to Cooper-Frye prescription, the constituent particles in the fluid will suddenly stop interacting when meets the freeze-out conditions, namely temperature T arrives at the freeze-out temperature Tfo. Figuratively speaking, considering a three-dimensional freeze-out hypersurface σ in a four-dimensional space, the temperature on this surface is freeze-out temperature Tfo which can be characterized as Tfo = T (τ, x, y, η). (3.31) Then we can obtain the particle spectrum by counting the number of hadrons crossing σ. That is to say, the total number of particles crossing σ, N , can be expressed as N = ∫ σ dσµj µ, (3.32) where dσµ is an infinitesimal element who is perpendicular to σ and towards outside. jµ is the current of particles, which is jµ = ∫ d3p E pµf(x, p). (3.33) Here E is the energy of the particle, pµ is four-momentum and f(x, p) denotes the one particle distribution function. In case of the ideal hydrodynamics, f(x, p) should correspond to Lorentz- covariant equilibrium distribution function f(x, p) = gi 2π3 1 (e(pµuµ−µB)/T ± 1) . (3.34) Here the signs “+" and “-" in the denominator stand for Fermi-Dirac and Bose-Einstein distribution respectively. gi is the degeneracy of the particle specie i. The factor pµuµ in the exponent is the energy of the particle in the local rest frame. In differential form, we get E dN d3p = ∫ σ pµf(x, p)dσµ. (3.35) 41 This formula shows the invariant distribution of particles described by the Cooper-Frye prescrip- tion (COOPER; FRYE, 1974). After considering the unstable particles decay and resonances effect, the calculated observable quantities can be compared with experimental data. 3.3 INITIAL CONDITIONS For one-dimensional systems satisfying Bjorken scaling, as long as the density or temperature at some initial time τ0 ∼ 1fm/c is given, the evolution of the density or temperature of the fluid can be obtained. Besides, the freeze-out temperature could be determined by fitting with the experimental data. Consequently, hydrodynamic analysis of ultra-relativistic heavy ion is actually an initial value problem. For (3+1)-dimensional relativistic hydrodynamic evolution and freeze-out, we use the hydrody- namic code SPheRIO which is developed by the Sao Paulo-Rio de Janeiro Collaboration. In order to obtain the IC, one may use the event generator, such as HIJING (GYULASSY; RISCHKE; ZHANG, 1997), VNI (SCHLEI; STROTTMAN, 1999), URASiMA (NONAKA; HONDA; MUROYA, 2000), the NeXuS (DRESCHER et al., 2002), the string model (MAGAS; CSERNAI; STROTTMAN, 2001) and color glass condensate (CGC) (HIRANO; NARA, 2004; LUZUM; ROMATSCHKE, 2008; ROY; CHAUDHURI; MOHANTY, 2012). The NeXuS is adopted in this thesis. It is worth mentioning that the IC for hydrodynamics given by event generator is not necessary to correspond to the one of local equilibrium. That is to say, the matter produced by event generator can be not in local equilibrium, or in other word, the energy-momentum tensor T µν can be not diagonal. We can adopt certain procedures to convert such energy-momentum tensor to the one of local equilibrium. Following the procedure suggested by Landau (LANDAU; LIFSHITZ, 1987), we know the following eigenvalue equation T µν u ν = εuµ, (3.36) where the four-velocity uµ is used as a normalized time-like eigenvector and the energy density ε is the corresponding eigenvalue. Then, according to Eqs.(3.4) and (3.5), the proper baryon and strangeness density in this frame can be expressed as nB = JµBuµ (3.37) and nS = JµSuµ. (3.38) Finally, the other thermodynamic quantities can be obtained from the thermodynamic relations. 43 4 EQUATION OF STATE In this chapter, we mainly introduce the HRG model for describing the hadronic phase, as well as the quasi-particle model and pQCD calculations fitted with Lattice QCD results for the QGP phase. Moreover, we also develop the QCD equation of state by the requirement of thermodynamic self-consistency. Generally speaking, QGP only exists in a short period after heavy-ion collisions. Afterwards, the density and temperature of the system will decrease rapidly. The QGP phase will be switched to hadronic phase at low temperature and density. As discussed in Introduction, Lattice QCD can be used to study the QCD properties at finite temperature T and relatively low chemical potential. Meantime, there are some attempts to study thermal properties of the QGP such as dimensional reduction (GINSPARG, 1980; APPELQUIST; PISARSKI, 1981; NADKARNI, 1988), hard thermal loop (HTL) resummation scheme (PISARSKI, 1989; FRENKEL; TAYLOR, 1990; BLAIZOT; IANCU; REBHAN, 1999; BLAIZOT; IANCU; REBHAN, 2003; ANDERSEN; BRAATEN; STRICKLAND, 1999; IPP; REBHAN; VUORINEN, 2004; MOGLIACCI et al., 2013), Polyakov-loop model (PIS- ARSKI, 2000; DUMITRU; PISARSKI, 2001), as well as approaches in terms of hadronic degrees of freedom (ASAKAWA; KO, 1994; LENAGHAN; RISCHKE; SCHAFFNER-BIELICH, 2000; RODER; RUPPERT; RISCHKE, 2003). However, in order to generalize the EoS at high chemical potential and a wide range of temperature, we adopt the phenomenological model, quasi-particle model or pQCD calculations with some free parameters for fitting to Lattice QCD results at vanishing chemical potential. The results show that the quasi-particle model can reproduce the Lattice QCD results quite well not only in zero chemical potential but also at low finite chemical potential. The arrangement of this chapter is shown as follows. In Sec.4.1, we introduce some fundamental concepts of standard statistical mechanics. HRG model and its recent developments are shown in Sec.4.2. In the Sec.4.3, the EoS obtained from pQCD calculations fitted with Lattice QCD results are listed. In Sec.4.4, we review thermodynamical inconsistency which occurs in quasi-particle model at finite temperature and chemical potential, and some solutions for solving this inconsistency. The general solution with a momentum dependent mass is given in the last Sec. 4.5. 4.1 A BRIEF REVIEW OF STATISTICAL MECHANICS Here we consider an open system in contact with a thermal and particulate reservoir, which means not only the temperature of the reservoir is hold at T but also the reservoir can exchange particles with the system. The system will achieve thermal and chemical equilibrium when its temperature arrives at T with a certain chemical potential µ. Then it is interesting to use the T and µ instead of S and N for describing the macroscopic state of the system. To be more specific, we suppose a microscopic state i of our system whose occupation numbers are i ≡ {n1, n2, · · · , nj, · · · · · · }, (4.1) 44 where nj is the number of particles which are in the jth “single particle state". Thus, for the possible microscopic state i, its total energy Ei and number of particles Ni can be written as Ei = ∑ j njεj, (4.2) Ni = ∑ j nj. (4.3) Here, following the definition of statistical mechanics, we define the partition function for the grand canonical ensemble as, Z = Z(V, β, α) ≡ ∑ i exp(−βEi − αNi), (4.4) with the probability for finding a given microstate i in the grand ensemble Pi = exp(−βEi − αNi) Z . (4.5) Then the total energy and particle number of the system can be expressed as 〈E〉 = ∑ i Ei exp(−αNi − βEi)∑ i exp(−αNi − βEi) , 〈N〉 = ∑ i Ni exp(−αNi − βEi)∑ i exp(−αNi − βEi) , (4.6) where α and β can be determined by comparing with the first law of thermodynamics, namely, d〈E〉 = TdS − pdV + µd〈N〉. (4.7) It is inferred that β = 1 kBT , α = − µ kBT . (4.8) All the thermodynamical quantities can be expressed in terms of the grand partition function with respect to α and β, 45 〈E〉 = −∂ lnZ(V, β, α) ∂β , (4.9) 〈N〉 = −∂ lnZ(V, β, α) ∂α , (4.10) 〈S〉 = −kB〈lnPi〉 = kB ∑ i (βEi + αNi) + kB lnZ = 1 T (〈E〉 − µ〈N〉) + kB lnZ. (4.11) When we assume that the volume of the system V goes to infinity, i.e., the thermodynamic limits and the general extensive thermodynamical relation 〈S〉 = 1 T 〈E〉 − µ T 〈N〉+ 1 T pV (4.12) is valid, then we have p = ∂ lnZ(V, β, α) ∂V , (4.13) which is in fact the “general force" for a grand canonical ensemble (GCE). Also it is not difficult to prove the following thermodynamic identities ε = Ts+ µn− p, s = ( ∂p ∂T ) µ , n = ( ∂p ∂µ ) T . (4.14) In the following, we present the basic expressions for thermodynamics for ideal Fermi gas and ideal boson gas case. For a Fermi gas, the occupation number of particles nj for each state j is limited to be 0 or 1 for the Pauli exclusion principle. Thus, the partition function should be rewritten as Z(V, β, α) = ∑ α exp (−βEi − αNi) = ∑ n1 ∑ n2 ∑ n3 · · · ∑ nj · · · exp [ − ∑ j nj (βεj + α) ] = ∏ j ∑ nj=0,1 exp [−nj (βεj + α)] = ∏ j [ 1 + e−nj(βεj+α) ] = exp {∑ j ln [ 1 + e−nj(βεj+α) ]} (4.15) For bosons, there is no restriction for the occupation numbers which means the occupation number of 46 particles nj for each state j should be summed from 0 to∞. Therefore, the partition function should be expressed as Z(V, β, α) = ∑ α exp (−βEi − αNi) = ∑ n1 ∑ n2 ∑ n3 · · · ∑ nj · · · exp [ − ∑ j nj (βεj + α) ] = ∏ j ∞∑ nj=0 exp [−nj (βεj + α)] = ∏ j 1 1− e−nj(βεj+α) (4.16) With the assumption βεi + α > 0, we have Z(V, β, α) = exp { − ∑ j ln [ 1− e−nj(βεj+α) ]} . (4.17) Reminding that the single particle state j for an ideal gas is taken as a plane wave state of momentum k, we may replace the sum over states j by an integral in k ∑ j → gV (2π~)3 ∫ dk, (4.18) where g is the statistical factor of the particle. For simplicity, we will adopt the natural units in this thesis, where ~ = c = kB = 1. By substituting Eq.(4.78) into Eqs.(4.15) and (4.17), the partition function can be expressed in terms of β and chemical potential µ as lnZ(V, T, µ) = ± gV (2π)3 ∫ d3k ln [1± exp (β(εk − µ))] . (4.19) Here, εk is the energy of the state with momentum k. “+" represents for bosons and “-" stands for fermions respectively. Then, the total energy of the system is 〈E〉 = − ∂ ∂β lnZ(V, T, µ) = gV (2π)3 ∫ d3k εke −β(εk−µ) 1 + e−β(εk−µ) = gV (2π)3 ∫ d3k εk eβ(εk−µ) + 1 , (4.20) 47 and the total number of particles of the system becomes 〈N〉 = 1 β ∂ ∂µ lnZ(V, T, µ) = gV (2π)3 ∫ d3k 1 eβ(εk−µ) + 1 . (4.21) From the above Eqs.(4.20) and (4.21), we know that the average of occupation number for the energy level εk is given by f(εk) = 1 eβ(εk−µ) + 1 . (4.22) The pressure is given by p = g (2π)3 1 β ∫ d3k ln [ 1 + e−β(εk−µ) ] . (4.23) Finally, the entropy is calculated from T 〈S〉 = 〈E〉 − µ〈N〉+ pV. (4.24) 4.2 HADRON RESONANCE GAS MODEL At the final stage of heavy ion collisions, lots of hadronic matter are formed and they can be decided by the HRG model. Generally, we think that the interacting hadronic matter can be well approximated by a non-interacting gas of resonance. One way is to use Virial expansion to obtain an effective interaction. For example, in Ref (VENUGOPALAN; PRAKASH, 1992), thermodynamics of low temperature hadronic matter is studied by Virial expansion together with experimental phase shifts. It is meaningful to mention that, for the HRG with vanishing chemical potential, the contributions from pion for the thermodynamics dominate at T . 120MeV. If temperature is bigger than 150 MeV, the heavy states will dominate the energy density due to the excitation of more and more heavier resonances. However, the particle number density of the heavy states is still small, because the number density is proportional to a suppression factor, namely ni ∼ eMi/T The expressions for energy density, pressure and number density for hadronic resonance gas can be expressed as ε(T, µ) = N∑ i=1 εi (T, µi) , p(T, µ) = N∑ i=1 pi (T, µi) , n(T, µ) = N∑ i=1 ni (T, µi) , (4.25) where N represents for the number of the hadrons, T is the temperature and µ is the chemical potential of the system. ni and µi = BiµB + SiµS +QiµQ are the number and chemical potential of ith hadron, where Bi, Si and Qi is the baryon, strangeness and charge quantum numbers, respectively. 48 However, those expressions are obtained by assuming that the hadrons are served as point particles. The expressions can be corrected to account for finite size of hadrons. The correction is called “excluded volume correction".The pressure of HRG with excluded volume correction (RISCHKE et al., 1991) can be determined by the following self-consistent equations pH(T, µB, µS, µQ) = ∑ i=1 pidi (T, µ̃i), (4.26) µ̃i ≡ µi − vipH , besides the energy density and particle number density can be obtained with the thermodynamic consistency shown in Appendix B, ε(T, µB, µS, µQ) = ∑ i εidi (T, µ̃i) 1 + ∑ j vjn id j (T, µ̃j) , n(T, µB, µS, µQ) = ∑ i nidi (T, µ̃i) 1 + ∑ j vjn id j (T, µ̃j) . (4.27) In (HAMA; KODAMA; SOCOLOWSKI JR., 2005), the excluded volume vi = (4πr3 0/3), with r0 = 0.7fm for baryons and r0 = 0 for mesons. In the case of zero baryon and strangeness density, one has µB = µS = 0. However, at finite baryon density, even though the strangeness density is zero, the strangeness chemical potential does not necessarily vanish. This is because in this case the net strangeness density from baryons and their anti-particles does not vanish at zero strangeness chemical potential, namely, the net strangeness density nS(µB(6= 0), µS = 0)− nS(µB → µB, µS = 0) 6= 0. Thus the value of strangeness chemical potential has to be determined by solving Eq.(4.26) numerically. We note that some improved HRG model with excluded volume correction (VOVCHENKO; GORENSTEIN; STOECKER, 2017) has been proposed recently. However, since there is no significant deviation between the models in the region of our interest, the HRG model used in (HAMA; KODAMA; SOCOLOWSKI JR., 2005) is adopted for our present study. The particle species we used in the calculations are listed in the Appendix A. Those particles and their properties are taken from the Particle Data Group (TANABASHI et al., 2018) 4.3 PQCD CALCULATIONS FOR EQUATION OF STATE OF Nf = 3 QCD QGP phase exists in extremely high temperature and/or high density environment. It is worth mentioning that under the extremely high temperature, far above the critical temperature, the effective QCD coupling constant will become very small. This nature of QCD is called the asymptotic freedom (POLITZER, 1973; GROSS; WILCZEK, 1973). Thus, it allows us to use the pQCD theory to describe the EoS of QGP phase at this region. There have been a lot of theoretical physicists committed to obtain the pressure expression from the perturbative calculations in the series of the weak coupling (SHURYAK, 1978b; CHIN, 1978; KAPUSTA, 1979; TOIMELA, 1983; ARNOLD; ZHAI, 1994; ARNOLD; ZHAI, 1995; ZHAI; KASTENING, 1995; KAJANTIE et al., 2003; VUORINEN, 49 2003b; VUORINEN, 2003a; IPP et al., 2006). Here the expression of pressure calculated up to the g6 s ln(1/g) order at finite temperature and/or finite chemical potential is adopted (VUORINEN, 2003b; VUORINEN, 2003a; IPP et al., 2006) P = 8π2 45 T 4 [ p0 + p2 (αs π ) + p3 (αs π )3/2 + p4 (αs π )2 + p5 (αs π )5/2 + p6 (αs π )3 ] , (4.28) where p0 = 1 + 21 32 Nf ( 1 + 120 7 µ̂2 q + 240 7 µ̂4 q ) , (4.29) p2 = −15 4 [ 1 + 5Nf 12 ( 1 + 72 5 µ̂2 q + 144 5 µ̂4 q )] , (4.30) p3 = 30 [ 1 + 1 6 ( 1 + 12µ̂2 q ) Nf ]3/2 , (4.31) p4 = 237.223 + ( 15.963 + 124.773µ̂2 q − 319.849µ̂4 q ) Nf − ( 0.415 + 15.926µ̂2 q + 106.719µ̂4 q ) N2 f + 135 2 [ 1 + 1 6 ( 1 + 12µ̂2 q ) Nf ] log [ αs π ( 1 + 1 6 ( 1 + 12µ̂2 q ) Nf )] − 165 8 [ 1 + 5 12 ( 1 + 72 5 µ̂2 q + 144 5 µ̂4 q ) Nf ]( 1− 2 33 Nf ) log M̂, (4.32) p5 = − ( 1 + 1 + 12µ̂2 q 6 Nf )1/2 [ 799.149 + ( 21.963− 136.33µ̂2 q + 482.171µ̂4 q ) Nf + ( 1.926 + 2.0749µ̂2 q − 172.07µ̂4 q ) N2 f ] + 495 2 ( 1 + 1 + 12µ̂2 q 6 Nf )( 1− 2 33 Nf ) log M̂, (4.33) p6 = − [ 659.175 + ( 65.888− 341.489µ̂2 q + 1446.514µ̂4 q ) Nf + ( 7.653 + 16.225µ̂2 q − 516.210µ̂4 q ) N2 f − 1485 2 ( 1 + 1 + 12µ̂2 q 6 Nf )( 1− 2 33 Nf ) log M̂ ] log [ αs π ( 1 + 1 + 12µ̂2 q 6 Nf ) 4π2 ] − 475.587 log [αs π 4π2CA ] . (4.34) Here we consider the system of massless particles with the flavor number Nf = 3, color number Nc=3 and CA=3. M is the renormalization scale, µq is the chemical potential of quarks which equals µq = µB/3 and the superscript hat denotes the division by 2πT which means M̂ = M/(2πT ) and µ̂q = µq/(2πT ). The three-loop coupling is adopted here (TANABASHI et al., 2018) αs = 1 b0t [ 1− b1 b2 0 ln t t + b2 1 ( ln2 t− ln t− 1 ) + b0b2 b4 0t 2 − b3 1 ( ln3 t− 5 2 ln2 t− 2 ln t+ 1 2 ) + 3b0b1b2 ln t b6 0t 3 ], (4.35) 50 where bi = βi/(4π)i+1 with β0 = 11 3 CA − 4 3 TFNf , β1 = 34 3 C2 A − 20 3 CATFNf − 4CFTFNf , β2 = 2857 54 C3 A − 1415 27 C2 ATFNf + 158 27 CAT 2 FN 2 f + 44 9 CFT 2 FN 2 f − 205 9 CFCATFNf + 2C2 FTFNf , β3 =CACFT 2 FN 2 f ( 17152 243 + 448 9 ζ3 ) + CAC 2 FTFNf ( −4204 27 + 352 9 ζ3 ) + 424 243 CAT 3 FN 3 f + C2 ACFTFNf ( 7073 243 − 656 9 ζ3 ) + C2 AT 2 FN 2 f ( 7930 81 + 224 9 ζ3 ) + 1232 243 CFT 3 FN 3 f + C3 ATFNf ( −39143 81 + 136 3 ζ3 ) + C4 A ( 150653 486 − 44 9 ζ3 ) + C2 FT 2 FN 2 f ( 1352 27 − 704 9 ζ3 ) + 46C3 FTFNf +Nf dabcdF dabcdA NA ( 512 9 − 1664 3 ζ3 ) +N2 f dabcdF dabcdF NA ( −704 9 + 512 3 ζ3 ) + dabcdA dabcdA NA ( −80 9 + 704 3 ζ3 ) . (4.36) Under the renormalization scheme MS, the above the β-function can be expressed as, β0 = 33− 2Nf 3 , β1 = 153− 19Nf 6 , β2 = 1 2 ( 2857− 5033 9 Nf + 325 27 N2 f ) . Here t = ln ( M2/Λ2 MS ) and ΛMS is the QCD scale under the scheme MS There exist two main doubts when we adopt the pQCD calculation. • The first one is the convergence of the perturbative expansions. • The second one is the renormalization scale ambiguity. The typical momentum flow is commonly chosen as the renormalization scale of the corresponding perturbation process. And the perturbation expansion convergence may be improved through higher and higher order calculation. However, the difficulty of the perturbation calculation is exponentially increased with the increase of the order for pQCD calculation. Even worse is that the value for each order of the series in Eq.(4.28) are observed as oscillation shown in Figure 8 and the dependence on the renormalization scale increases rather than decreases. Then this perturbative series only becomes convergent at very high temperature T ∼ 105Tc. The renormalization scale setting method PMC may be a choice to improve the convergence and give a reasonable result (BU et al., 2018). 51 Figure 8 – (color line) The temperature dependence of the calculated ratio p/pideal from the pQCD calculation with CM = 0.5 and CS = 0 (HAQUE; MUSTAFA; STRICKLAND, 2013) for the different αs order at µq = 0 and µq = 200 MeV. The shaded bands show the scale uncertainties by varying the renormalization scale M ∈ [ √ (πT )2 + µ2 q, 4 √ (πT )2 + µ2 q]. In accordance with the general setting M = πT , we choose M = CM √ (πT )2 + µ2 q, (4.37) where CM is a free parameter to fit with the Lattice results. Meanwhile, in order to avoid Landau pole produced under the low finite temperature and chemical potential, namely divergence of coupling constant, t should be revised as follows according to the Refs. (ALBRIGHT; KAPUSTA; YOUNG, 2014; ALBRIGHT, 2015), t = ln ( C2 S +M2/Λ2 M̄S ) . (4.38) Here CS is also a free parameter which is used to avoid Landau pole and fit with Lattice results. Thus, we can present the pressure through the pQCD calculation. The other thermodynamics, like energy density, entropy density and number density can be derived through the thermodynamic relations. The corresponding results are shown in Figure 9 and Figure 10 where the results from the pQCD calculation near Tc can not reproduce the Lattice data. Thus, we adopted a phenomenological approach, called the quasi-particle model to describe the QGP phase. 4.4 QUASI-PARTICLE MODEL FOR QGP With the increase of temperature and density, quarks and gluons are deconfined from hadrons and the hadronic phase matter is transferred into a new QCD phase called QGP. As suggested by Lattice simulations, this QCD phase transition will occur around the critical temperature Tc ∼ 150MeV (BORSANYI et al., 2010; BAZAVOV et al., 2012b; BHATTACHARYA et al., 2014). Under such circumstances, the perturbation theory is no longer applicable at such region, i.e., around the 52 0.1 0.2 0.3 0.4 0.5 0 5 10 15 T (GeV) pQCD calculation stout 3p/T4 T4 s/T3 Figure 9 – (Color online) The calculated of 3p/T 4, ε/T 4 and s/T 3 from the pQCD calculation where CM = 3.293 and CS = 1.509. The lattice QCD with stout action (BORSANYI et al., 2012; BORSANYI et al., 2014) at zero chemical potential is also given. critical temperature, T ∼ Tc, because the pQCD is used to compute the various QGP thermodynamic quantities at the extremely high temperature. To describe the thermodynamic properties of the quark- gluon plasma phenomenologically and reasonably, the quasi-particle model for QGP is proposed which is able to fit with Lattice QCD results over a wide range of temperatures: not only near the critical temperature Tc, but also at T > 20Tc as in the hard thermal loop theory, or at the extremely high temperature as in pQCD theory. The main idea of the quasi particle model is that quark-gluon plasma is made up of free quarks and gluons, and characterize the interactions between quarks and gluons as the effective masses of quarks and gluons. At year 1994, Peshier et al (PESHIER et al., 1994) applied such model for describing the QGP, and treat the effective masses of the elements of QGP as the function of temperature T without consideration of chemical potential. Taking the consideration that the standard statistical mechanics (SSM) introduces the mass of the element as a constant, we need to re-derive thermodynamics if the effective mass is treated as the function of temperature T . Gorenstein and Yang (GORENSTEIN; YANG, 1995) found that thermodynamics does not meet the thermodynamic relations by using the definitions of SSM directly. In order to solve this problem, Gorenstein and Yang introduced a Bag constant related to the temperature as the background field, which can be determined by the requirement of the thermodynamic relationship. Soon afterwards, there were many other attempts to find the solution of the problem (YIN; SU, 2008; YIN; SU, 2007; YIN; SU, 2010; BANNUR, 2007a; BANNUR, 2013; BANNUR, 2012; BANNUR, 2008; BANNUR, 2007b; GARDIM; STEFFENS, 2007; GARDIM; STEFFENS, 2009). Since Gorenstein and Yang only 53 stout Figure 10 – (Color online) The calculated of trace anomaly from the pQCD calculation where CM = 3.293 and CS = 1.509. The lattice QCD with stout action (BORSANYI et al., 2012; BORSANYI et al., 2014) at zero chemical potential is also given. discussed the situation without chemical potential, Peshier tried to generalize Gorenstein and Yang’ idea into finite temperature and density case (PESHIER; KAMPFER; SOFF, 2000). We found that such a generalization is just a special solution for an intergro-differential equation (MA et al., 2019). 4.4.1 Thermodynamic inconsistency For convenience, we do not consider the chemical potential and choose the chemical potential as zero. According to the statistical mechanics (PATHRIA, 1996; HUANG, 1987; LANDAU; LIFSHITZ, 1980), the grand partition function for the ideal gas consisting of the particles with mass m can be characterized as lnZ(V, T ) = ± giV (2π)3 ∫ d3k ln [1± exp(−ωk/T )] , (4.39) where ωk = [k2 +m2(T )] 1/2 is the energy of the massive "non-interacting" quasi-particles. “+" stands for the Fermions and “-" stands for the Bosons. gi is the degeneracy factor. According to the definition of SM, it is easy to obtain its pressure energy density and entropy density, pid(T,m) = T V lnZ(V, T ), (4.40) 54 εid(T,m) = − 1 V ∂ lnZ(V, T ) ∂β ∣∣∣∣ V , (4.41) sid(T,m) = 1 V ( lnZ(V, T ) + ∂ lnZ(V, T ) ∂T ∣∣∣∣ V ) . (4.42) Or pid(T,m) = ±T gi 2π2 ∫ ∞ 0 k2dk ln [1± exp(−ωk/T )] , (4.43) εid(T,m) = gi 2π2 ∫ ∞ 0 k2dk ωk exp(ωk/T )± 1 , (4.44) sid(T,m) = ± gi 2π2 ∫ ∞ 0 k2dk ln [1± exp(−ωk/T )] + 1 T g 2π2 ∫ ∞ 0 k2dk ωk exp(−ωk/T ) 1± exp(−ωk/T ) . (4.45) However, we can also derive the energy density ε∗ from pressure with the aid of thermodynamic relation, namely, ε∗(T,m) = Ts− p, s = dp(T,m) dT , (4.46) or ε∗(T,m) = T ∂p(T,m) ∂T ∣∣∣∣ m − p(T,m) + T ∂p(T,m) ∂m ∣∣∣∣ T dm(T ) dT . (4.47) Substituting Eq.(4.43) into the above equation (4.47), we can obtain ε∗(T,m) = gi 2π2 ∫ ∞ 0 k2dk ωk exp(ωk/T )± 1 + ∂p(T,m) ∂m ∣∣∣∣ T dm(T ) dT = εid(T,m) + ∂p(T,m) ∂m ∣∣∣∣ T dm(T ) dT 6= εid(T,m). (4.48) Then the thermodynamics relation, ε(T,m) = T ∂p(T,m) ∂T − p(T,m), (4.49) is not satisfied, which means the energy density obtained from ensemble average may not be the same as that from thermodynamic relation. Turning to the situation, m(T ) = m0 = const, the additional term ∂p(T,m) ∂m ∣∣∣ T dm(T ) dT will equal zero due to dm(T ) dT = dm0 dT = 0 and these expressions would satisfy the thermodynamic relation, i.e., Eq.(4.49). 4.4.2 Some attempts for solving thermodynamic inconsistency Gorenstein and Yang reformulated the SSM to solve thermodynamic inconsistency, by introducing a temperature dependent term in the expressions for pressure and energy density (GORENSTEIN; YANG, 1995). This extra term, called as the Bag constant, was fixed by applying a constraint relation such that the thermodynamic inconsistency term was cancelled. There are some solutions, such as 55 the the model proposed by V. M. Bannur (BANNUR, 2007a; BANNUR, 2013; BANNUR, 2012; BANNUR, 2008; BANNUR, 2007b). The idea for this model is to start from the definition of energy density and number density in SSM, and then the other thermodynamic quantities can be derived from the thermodynamic relations. Moreover, another model suggested by F.G. Gardim and F.M. Steffens (GARDIM; STEFFENS, 2007; GARDIM; STEFFENS, 2009) tried to parameterize all of possible quasi-particle models with the requirement of thermodynamic consistency and SSM. By assuming that the Bag constant can be expressed in terms of the quasi-particle mass, they show that the different values or combinations of those parameters represent for different solutions. 4.4.2.1 Gorenstein and Yang Based on the quasi-particle model proposed by (GORENSTEIN; YANG, 1995), we revisit the thermodynamic consistency for quasi-particle model at finite temperature and baryon chemical potential. Firstly, following their traditional definition in statistical physics (PATHRIA, 1996), we can write the the grand partition function for ideal gas in the form of Hamiltonian as follows, QG = 〈exp[−αN̂ − βĤid]〉, (4.50) where Ĥid is the Hamiltonian of ideal gas of particle Ĥid = d∑ i=1 ∑ k ω(k)a†k,jak,j. (4.51) Here, ω(k) is the energy of a single particle and the index j corresponds to internal degrees of freedom for the particle, i.e. , the different spin and color states of the particle. However, in finite baryon chemical potential and temperature case, ω(k) is replaced by ωk(k, T, µ). Then the thermodynamic inconsistency occurs due to the Hamiltonian of the system becomes temperature- and/or baryon chemical potential- dependent. To solve the thermodynamic inconsistency, Gorenstein and Yang introduced an effective temperature- and/or chemical potential- dependent Hamiltonian to guarantee the thermodynamic consistency of the quasi-particle model, namely, QG = 〈exp[−αN̂ − βĤeff ]〉, (4.52) where Ĥeff = Ĥid + E0. (4.53) Here E0 is a temperature and chemical potential dependent function associated with the bag constant B proposed by Gorenstein and Yang (GORENSTEIN; YANG, 1995). This term, a specific temperature- and chemical potential- dependent bag constant, is used to cancel out the effects of the temperature (and chemical potential) dependence of the quasi-particle mass and it can be determined by the requirement 56 of thermodynamic self-consistency , i.e., ∂p(T, µ,m) ∂m ∣∣∣∣ T,µ ∂m(T, µ) ∂T |µ = 0. (4.54) To be more specific, the expressions for energy and particle number can be formulated as ensemble average, i.e., Eq.(4.6) can be rewritten in terms of the grand partition function, lnQG, namely 〈E〉 = −∂ lnQG ∂β , 〈N〉 = −∂ lnQG ∂α . (4.55) quasi-particle ansatz assumes that one may carry out the calculations in the momentum space where the Hamiltonian is diagonal. To be more specific, one makes the following substitutions for the ideal gas part ∑ j ∑ k → giV (2π)3 ∫ dk. (4.56) where gi is the degeneracy factor. Now, thermodynamical quantities can also be expressed regarding the derivatives of the grand partition function. Then the energy density, number density, pressure and entropy density read ε = 〈E〉 V = − 1 V ∂ lnQG ∂β = εid + E0 V = εid +B. (4.57) n = 〈N〉 V = − 1 V ∂ lnQG ∂α = nid, (4.58) p = 1 β ∂ lnQG ∂V = 1 β lnQG V = pid −B, (4.59) s = ε+ p− µn T = εid + pid − µnid T = sid, (4.60) where εid = gi 2π2 ∫ ∞ 0 k2dkω∗(k, T, µ) exp[(ω∗(k, T, µ)− µ)/T ]∓ 1 + gi 2π2 ∫ ∞ 0 k2dkω∗(k, T,−µ) exp[(ω∗(k, T,−µ) + µ)/T ]∓ 1 , (4.61) nid = gi 2π2 ∫ ∞ 0 k2dk exp[(ω∗(k, T, µ)− µ)/T ]∓ 1 − gi 2π2 ∫ ∞ 0 k2dk exp[(ω∗(k, T,−µ) + µ)/T ]∓ 1 , (4.62) 57 pid = ∓gi 2π2 ∫ ∞ 0 k2dk ln {1∓ exp [(µ− ω∗(k, T, µ)) /T ]} ∓ gi 2π2 ∫ ∞ 0 k2dk ln {1∓ exp [(−µ− ω∗(k, T,−µ)) /T ]} = gi 12π2 ∫ ∞ 0 k3dk exp[(ω∗(k, T, µ)− µ)/T ]∓ 1 ∂ω∗(k, T, µ) ∂k ∣∣∣∣ T,µ + gi 12π2 ∫ ∞ 0 k3dk exp[(ω∗(k, T,−µ) + µ)/T ]∓ 1 ∂ω∗(k, T,−µ) ∂k ∣∣∣∣ T,µ , (4.63) with on-shell dispersion relation ω∗(k, T, µ) = √ m(T, µ)2 + k2, (4.64) where B = limV→∞ E0 V is the bag constant. Generally, one thought the effective quasi-particle mass will change with temperature and density due to its interaction with the environment. Some theoretical studies, for example, Brown-Rho scaling (BROWN; RHO, 1991), QCD sum rules (HATSUDA; KOIKE; LEE, 1993), and as well the experiment (LOLOS et al., 1998) all indicate such a phenomenon. The temperature-dependent quasi-particle mass m(T, µ) is usually used to mimic the medium effects, which means ∂m(T,µ) ∂T |µ should not always equal 0. Then, the constraint condition suggested by Gorenstein and Yang (GORENSTEIN; YANG, 1995) can be rewritten as ∂p(T, µ,m) ∂T ∣∣∣∣ µ,m = 0, ∂p(T, µ,m) ∂µ ∣∣∣∣ T,m = 0. (4.65) Here, the contribution from the temperature dependence of quasi-particle mass has already been canceled out with the temperature dependence of E0. If the system has vanishing chemical potential µ = 0, one has B = B(µ = 0, T ) ≡ B(T ) and m = m(µ = 0, T ) ≡ m(T ), in general, one can invert the second function to find T = T (m) and express B as a function of m. Thus, the above requirement Eq.(4.65) regarding E0 implies dB dm = ∂pid(T, µ,m) ∂m ∣∣∣∣ T,µ . (4.66) i.e., dB dm = −gim 2π2 ∫ ∞ 0 k2dk ω∗(k, T ) 1 exp[(ω∗(k, T ))/T ]∓ 1 . (4.67) If one already known the function ω∗(k, T ), the bag constant B(m) can be obtained by the following equation, B(m) = B0 − d 2π2 ∫ T T0 dT ′m dm dT ′ × ∫ ∞ 0 k2dk ωk 1 exp [ωk/T ′]− 1 , (4.68) 58 where B0 = B(m(T0)) is an arbitrary integration constant and it can be determined by comparing with Lattice QCD data. Considering a system for SU(3) gluodynamics, one suppose the linear relation between effective mass (mg) and temperature (T ), mg = aT . Then, the thermodynamics suggested by Eqs.(4.57), (4.59) and (4.60) will be reduced to ε(T ) = σT 4 +B0, p(T ) = σ 3 T 4 −B0, s(T ) = 4σ 3 T 3. (4.69) Here σ equals σ = 3g 2π2 ∞∑ n=1 [ a2 n2 K2(na) + a3 4n K1(na) ] , (4.70) where K1 and K2 are the modified Bessel functions. As found by Begun et al (BEGUN; GOREN- STEIN; MOGILEVSKY, 2011; BEGUN; GORENSTEIN; MOGILEVSKY, 2010), no matter how we choose the integration constant B0, the lattice results can not be reproduced very well, shown in Figure 11 Figure 11 – The calculated energy density and pressure for SU(3) gluodynamics. The blue squares and red circles represent for the energy density and pressure of Lattice QCD respectively. The dashed horizontal line shows the Stefan-Boltzmann limits. The calculated energy density and pressure by Eq.(4.69) by setting the parameters as, g = 16 and B = 1.7T 4 c Begun et al suggested that we can introduce an extra, free parameter E1 into the Hamiltonian, i.e., Ĥeff = Ĥid + E0 + E1, (4.71) 59 which is proportional to the temperature. Then, the thermodynamics can be expressed as, ε = 〈E〉 V = − 1 V ∂ lnQG ∂β = εid + E0 V + E1 V + 1 V 〈β∂E1 ∂β 〉 = εid +B, (4.72) n = 〈N〉 V = − 1 V ∂ lnQG ∂α = nid, (4.73) p = 1 β ∂ lnQG ∂V = 1 β lnQG V = pid −B − E1 V , (4.74) s = ε+ p− µn T = εid + pid − µnid T = sid + E1 V T . (4.75) The E1 term is singled out from E0 owing to its peculiar nature. We note the presence of the term regarding E1 in the resulting expression for the pressure, but not in that for the energy density. By choosing E1/V = AT (BEGUN; GORENSTEIN; MOGILEVSKY, 2011; BEGUN; GORENSTEIN; MOGILEVSKY, 2010), We can replace Eq.(4.69), those thermodynamics can be replaced by ε(T ) = σT 4 +B0, p(T ) = σ 3 T 4 −B0 − AT, s(T ) = 4σ 3 T 3 − A. (4.76) Then it allows one to further adjust the value of the pressure for any given energy density (BEGUN; GORENSTEIN; MOGILEVSKY, 2011; BEGUN; GORENSTEIN; MOGILEVSKY, 2010). As shown in Figure 12, the lattice results can be reproduced quite well with an additional negative term E1. We note that the resultant expressions for thermodynamic quantities, namely, Eqs.(4.72), (4.73) and (4.74), are thermodynamically as well as statistically consistent. The reasons are twofold. Firstly, the expressions for energy and particle density are in accordance with the conventional definition regarding ensemble average1, while they can also be conveniently expressed in standard form as derivatives of the grand partition function, as emphasized by other authors (BIRO; SHANENKO; TONEEV, 2003; YIN; SU, 2008). Moreover, from the viewpoint of statistical physics, those ensemble averages are meaningful, only when one can match those quantiti