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Summer Research Fellowship Programme of India's Science Academies

Equation of State for Quark Gluon Plasma

Priyanka S. Mahalle

Sant Gadge Baba Amravati University, Amravati 444602.

Guided by:

Dr. Jajati K. Nayak,

Variable energy Cyclotron Centre, Sector 1, Block-AF, Bidhan Nagar, Kolkata 700064

Abstract

The early Universe is widely believed to be in the state of Quark Gluon Plasma (QGP) after few microseconds of the Big Bang. The study of QGP is one of the fascinating and frontier areas of science. Such a state has been created in the laboratory by colliding nuclei at very high energies. People since decades have been trying to understand its properties. In this work, I try to get an overview of thermodynamic properties of QGP. I learned the equation of state for QGP through the MIT Bag model. The EoS gives us the information on various thermodynamic quantities like pressure, energy density, entropy etc.

To understand QGP, I started with the elementary particles in the universe, how they interact with each other and the various forces responsible for their interactions. I also studied various conservation laws that apply to such interactions. My study covered an overview of Quantum Chromodynamics, their properties, and their behavior at extreme conditions of temperature and density. These explain under what circumstances QGP is formed. For further undserstanding, I learnt about the state of electromagnetic plasma and its different properties; about the equation of state related to ideal gas of massless particles; about the variation of pressure, energy, density etc. with temperature by considering Ideal Gas and Van der Waal equation of state; and about the thermodynamics of relativistic massless particles and Bag Model of quark gluon plasma. My goal was to calculate the thermodynamic quantities of QGP state assuming MIT Bag Model for quarks and gluons and how quarks undergo phase transition to hadron.

Keywords: Quarks, Gluons, Plasma, Chromodynmics, MIT Bag Model.

Abbreviations

QGPQuark Gluon Plasma
EoSEquation Of State
QCDQuantum Chromodynamics
QEDQuantum Electrodynamics

Chapter 1

INTRODUCTION

A review of particles in the Universe

Till Rutherford’s scattering experiment in 1911, people believed the fundamental constituent of matter is an atom. But with the emergence of powerful particle accelerators, we have come to know that atom is not the fundamental constituent of matter. As per current knowledge, an atom is subdivided into nucleus and electron, the nucleus in turn being composed of nucleons (neutron and proton), which further can be divided into elementary particles called quarks. The fundamental constituents of matter are basically leptons, quarks and carrier bosons. Apart from fundamental constituents, there are composite particles (which are basically the bound states of fundamental particles). The classification of particles according to different properties (like spin, flavor, mass, interaction etc..) is given below:

IMG_20180705_231115.png
    Particles in Universe

    In the above figure, the classification of particles is shown. The particles are classified as fermions and bosons; each is subclassified as fundamental (elementary) and composite. The fundamental particles are also called elementary and are assumed to have no structure; whereas composite particles are made up of fundamental or elementary particles.

    Fermions

    All the particles can be classified broadly into two categories, such as fermions and bosons. The fermions are particles with half integer spins and bosons are with integer spins. Similarly the elementary particles can also be classified as fermions and bosons. elementary fermions are called the matter (anti-matter) particles and they obey the Fermi-Dirac statistics. Fermions follow Pauli Exclusion Principle which states that, no two fermions can stay in the same Quantum state. Elementary Ff ermions are further classified into light fermions known as leptons and heavy fermions known as quarks. A few notes on quarks and leptons are described below:

    Quarks

    Quarks are color charge particles having color quantum number. Each flavor has three colors i.e. red, green, and blue. They carry fractional charge. Quarks combine to form composite particles called hadrons. Due to a phenomenon known as quark confinement quarks are never directly observed in isolation, they can be found only within the hadrons. There are six types (or flavors) of quarks i.e. up, down, charm, strange, top and bottom. Out of six flavors up, down and strange quarks are light quarks and charm, top and bottom quarks are heavy quarks. Quarks experience all four types of interactions. Color- charge particles interact via gluon exchange in the same way that charged particles interact via photon exchange. However, gluons are themselves color charged and it amplifies the strong force as Color-charge particles are separated. As baryons have three quarks content, so each quark has baryon number 1/3. The properties of each quark flavor are given below in a tabular form.

    Quarks

    Flavor

    Charge

    Mass

    Baryon Number

    Hypercharge

    Y

    Isospin

    I

    I3

    u

    2/3

    5 MeV/C2

    1/3

    1/3

    ½

    +1/2

    d

    -1/3

    7 MeV/C2

    1/3

    1/3

    ½

    -1/2

    c

    2/3

    1.5 GeV/C2

    1/3

    1/3

    0

    0

    s

    -1/3

    135 MeV/C2

    1/3

    -2/3

    0

    0

    t

    2/3

    175 GeV/C2

    1/3

    1/3

    0

    0

    b

    -1/3

    5GeV/C2

    1/3

    1/3

    0

    0

    Leptons

    Leptons carry integer charge and experience all interactions except strong force. Leptons are of six types, out of them electron, muon and tauon are charged leptons and their corresponding neutrinos are neutral leptons. Out of these electrons have least mass than the other charged leptons.The properties of leptons are given in tabular form:

    Table for Leptons
    LeptonsChargeMassLifetimeLeLµLτ
    e-10.511003MeV/C2100
    νe00100
    µ-1105.6592.197×10-6010
    νµ00010
    τ-117843.3×10-13001
    ντ00001

    Baryons

    Baryons are the fermions. Baryons are not fundamental particles rather they are composite particles made of quarks held together by the strong force in a similar way as molecules are held together by the electromagnetic force. Every Baryon is composed of three quarks and every antibaryon is composed of three antiquarks. The best known baryons are protons and neutrons. Barons are colorless i.e. they must have three colors (RGB). According to strange quark content Baryons are further classified into nucleon and hyperons. The properties carry by each baryons are given below in a tabular manner.

    Table for Baryons
    BaryonCompositionStrangeness

    Mass

    MeV/c2

    Mean Lifetime

    Isospin

    I

    I3
    puud0938.282×1031½+1/2
    nudd0939.6925±11½-1/2
    Ʌs((du-ud)/2)-11115.62.63×10-1000
    Ξ0dss-21314.92.90×10-10½½
    Ξ-uss-21321.31.6×10-10½-1/2
    0s((du=ud)/2)-11192.5(6±1)×10-2010
    -dds-11197.31.48×10-101-1
    +uus-11189.40.80×10-101+1
    Δ++uuu012320.6×10-23
    -sss-316720.82×10-10

    Bosons

    Bosons are particle with integral spin. Bosons are also known as gauge bosons. The spin zero particles are known as scalar bosons and the elementary particles having spin 1 are known as vector bosons. Bosons are the mediator of particle interactions found in our nature. They obey the Bose-Einstein statistics and don’t follow Pauli Exclusion Principle. The fundamental bosons are photon, gluon, W +, W-, Z and Higgs bosons. Out of them Gluon and photon are chargeless and massless and they are the stable vector bosons. Whereas W + and Z bosons are very heavy having mass 80.4GeV/C2 and 91.2GeV/C2 respectively. W and Z bosons are short-lived gauge bosons. The Z boson is chargeless and W+ is of charge 1 and W- is of charge -1. Gluons possesses color and is of 8 types. Gluons are the massless particles that act between quarks, antiquarks and other gluons and carries color charge. Color charges are same as electromagnetic charge, but it comes in three types (±red, ±green, ±blue). They are bicolor particles carrying one positive and one negative color charge ( according to QCD which we will discuss later ).

    MediatorChargeMassLifetimeSpin
    gluon002
    photon (ϒ)001
    W±±181800Unknown1
    graviton00Stable2
    Higg boson0125090(predicted)Unknown0

    Mesons

    Mesons are composite Bosons made up of two fermions (quark and antiquark). Because of their composition they have zero baryon number .Mesons are also colorless like baryons. Out of all the mesons pions have less mass i.e. 140 MeV. Pions play a role in holding atomic nuclei together via strong interaction. More about mesons is given below

    MesonCompositionChargeStrangeness

    Mass

    MeV/c2

    Mean Lifetime

    Isospin

    I

    I3
    ±ud̅, du̅+1, -10139.5692.60×10-81+1 ,-1
    0(uu̅-dd̅)/200134.9648.7×10-1710
    K0, K̅0ds̅, d̅s0+1497.72

    K0S= 0.83×10-10

    K0L= 5.18×10-8

    1/2-1/2
    K±us̅, ūs+1, -1-1493.671.24×10-81/2+1/2,-1/2
    Ƞ((uu̅-dd̅+ss̅)/√6)00548.87×10-1900
    ρud̅, du̅, (uu̅-dd̅)/√2+1, -1, 007700.4×10-231+1,0,-1

    Notes on particle interactions

    In nature there are four types of interaction i.e. strong interaction, electromagnetic interaction, weak interaction and gravitational interaction that governs how object or particle interacts and how certain particle decay. All the known forces of nature can be traced to these fundamental interactions .These four interactions explained accordingly their strength is given below:

    Strong interaction

    Before the 1970s, physicists were uncertain as to how the atomic nucleus was bound together. To explain it the stronger attractive force were postulated. This hypothesized force was called the strong force. This is strongest force among all three fundamental forces. Despite’s its strength, the strong forces does not manifest itself in macroscopic universe because of its extremely limited range. It is confined to an operating distance of about 10-15 meter (1fm) - of the order of diameter of proton. They are responsible for the binding of the quarks inside the neutron and proton, also the binding of neutron and proton within nuclei. Strong forces interact through the “gluon”. Quantum Chromo Dynamics (QCD) is the theory of strong interaction.

    Electromagnetic interaction

    In 19th century James Clerk Maxwell gave scientific definition of electromagnetic forces. Electromagnetic interaction is responsible for phenomenon related to extra nuclear physics, like the interaction between electron and nuclei, intermolecular forces between the molecules etc. Electric charges are responsible for electromagnetic forces. Range of electromagnetic forces is infinity. Photons (ϒ) are the force carrier particles for electromagnetic interaction. Photons are massless and chargeless particles which move with the speed of light. It’s spin is equal to one. The coupling constant for electromagnetic interaction is given by:

    =e24πc=1137\displaystyle {∝}{}{=}{}\frac{{e}^{2}}{4πℏc}{=}{}\frac{1}{137}

    This is called the fine structure constant, because it determines the fine structure (spin- orbit splitting) of atomic spectra.

    Weak interaction

    The first theory of weak forces was presented by Fermi in 1933. Weak forces are responsible for the disintegration of the particles. Weak interaction involves the process like nuclear β decay, involving the emission by radioactive nuclei of electron and neutrino. The strength of weak interaction coupling is in the order of 10-7 compared to strong interaction coupling. The weak forces are mediated by intermediated vector boson W± and Z, which is very massive, lifetime, is about 10-24 seconds, and give rise to interaction of very short range. The weak interaction takes place only at very small, subatomic distances less than the diameter of proton. The coupling constant for weak interaction is given by

    G(Mc2)2(c)3=1.17×105\displaystyle \frac{G\left(Mc^2\right)^2}{\left(ℏc\right)^3}=1.17\times10^{-5}

    In 1970 Glashow, Weinberg, and Salam unified the weak and electromagnetic forces through GWS theory by showing them to be the two aspects of a single force now termed as electroweak force. All mesons are unstable because of weak decay.

    Gravitational interaction

    The gravitational force was first described systematically by Sir Isaac Newton in 17th century and least understand till now. Gravitational interaction acts between all types of particles having mass. It is the weakest interaction between all fundamental interactions. The strength of gravitational interaction is in the order of 10-39 with respect to strong interaction. Gravitational force is infinite range force. Coupling constant of Gravitational Interaction is given by

    GN×M24πc=5×1040\displaystyle \frac{{{G}_{N}}{×}{{M}^{2}}}{4πℏc}{=}{5}{×}{{10}^{-40}}

    Thus for mass scale common in high energy physics the gravitational coupling is negligibly small.

    Gravitational interaction is mediated through a boson particle called “graviton”. Graviton having spin equal to 2 and is massless particle. Experiments to detect graviton are currently under way.

    This is about all four fundamental interactions. Scientists are making an effort to unify the fundamental forces into single theory. In 1970 Glashow, Weinberg, and Salam unifies the weak and electromagnetic forces through GWS theory. The grand unified theory unifies the three interactions strong, electromagnetic, and weak through GUT theory. This unification occurs around 1015GeV. Now we are waiting for theory of everything TOE where gravitational interaction will unified with this three interaction. It is worth to mention that the highly successful Standard Model does not include gravity.

    Conservation rules

    Conservation laws are very crucial to our understanding of physical world, in that they describe which process can or cannot occur in nature. Conservation laws are the properties of interaction; they are applicable for scattering as well as decay processes. All particles will interact and decay to lighter particles unless prevented by doing so by some conservation laws. Here we list below the conservation rules required for the allowed interaction/decay of particles.

    • Energy And Momentum Conservation

    Energy and momentum is always conserved during all interaction we consider in particle physics according to the assumed symmetries.

    • Charge Conservation

    Conservation of charge is take place in all three interactions. In the case of weak interaction the lepton or quark that enter in may not have the same charge that come out but the difference is carried out by charge W boson (as is the W charged boson i.e, W±).

    • Baryon Number And Lepton Number Conservation

    Baryon number and lepton number is also conserved during all three interactions. We assign B=1 for Baryon, B=-1 for antibaryon, and B=0 for everything else. Conservation of lepton number is carried out by conservation of electron number, tau number, and muon number independently.

    • Flavor Conservation

    Flavor is conserved only during strong and electromagnetic interaction and not in weak interaction, since during weak interaction up quark may convert into down quark or strange quark

    e.g. Decay of lambda

    Ʌ (uds) → p+ (uud) + ∏- (u̅d) an example of weak interaction where flavor is not conserved.

    • Color Conservation

    During weak and electromagnetic interaction the color conservation is not much more effective. While during strong interaction color is conserved. Quark color does change but the difference is carried by gluon.

    • Parity Conservation

    Parity conservation is related to mirror symmetry. During strong and electromagnetic interaction parity is conserved but during weak interaction parity is not conserved.

    e.g. K+ → π+ + π0 parity = +1

    K+ → π+ + π+ + π0 parity = -1

    Here we obtaine two different final state parity (+1,-1) for decay of same kaon. Presence of two different final parities indicate that the initial state should also have different parities, hence two distinct particles. However precise measurement indicate that they are the same particles, and it is impossible to have two parities for same particle. Hence during weak interaction parity is not conserved.

    • Charge Conjugation Conservation

    Charge conjugation converts each particle into its antiparticle by changing its sign. Charge conjugation changes the sign of all internal quantum number. While mass, momentum, energy, spin remain untouched. Charge Conjugation is conserved during strong and electromagnetic interaction. During weak interaction charge conjugation does not conserved. An example given below

    π- → µ- + ν̅µ

    Neutrinos are found to have intrinsic parities: neutrinos have left-handed parity and antineutrinos right-handed. Since charge conjugation would leave the spatial coordinates untouched, then if you operated on a neutrino with the charge conjugation operator, you would produce a left-handed antineutrino. But there is no experimental evidence for such a particle; all antineutrinos appear to be right-handed; this lead to conclusion that charge conjugation does not conserved during weak interactuion.

    • Time Reversal Conservation

    Time reversal reverses the order in which event occurs. Time reversal conserved during the strong and electromagnetic interaction while during weak interaction it does not conserved in fact during weak interaction the time violation is about 10 -3 sec.

    • Isospin Conservation

    Many particles occur in group called multiplets, in which the particle is of almost same mass but different in charge. The proton and Neutron form such a multiplet. The new quantity describes mathematically the effect of changing neutron to proton or vice versa is called Isospin. Isospin has nothing to do with the spin of the particle, but is represented by a vector that can have various orientations in an imaginary space known as isotopic spin space. Isospin is conserved only in strong interaction. During electromagnetic interaction isospin is not conserved and can be explained by using following example.

    π0 → ϒ + ϒ here Isospin I goes from 1 to 0

    Also in weak interaction Isospin is not conserved. Let us consider an example of lambda decay.

    Ʌ0 → p+ + π - here Isospin I goes from 0 to 2

    Those are all conservation rules for all three interactions. Now by using this conservation rules we predict which reaction or decay is allowed and which is forbidden. Now after this let us move towards understanding the concept of quark gluon pasma.

    Chapter 2

    Quark Gluon Plasma

    Getting an overview of particles in universe now I want to study about the thermodynamics state of quarks, one type of elementary particles. In the particular state of quarks, I am interested in quark gluon plasma (QGP). The theory that describes thedynamics of quarks is called Quantum Chromodynamics (QCD). Here I will briefly describe about QCD and then about QGP, I will also discuss about Normal plasma ( i.e. Quantum Electrodnamic Plasma).

    Quantum Chromodynamics

    In Chromodynamics, the color charge plays the same role as an electric charge in Electrodynamics.. As leptons do not carry any color charge hence they do not involve in strong interaction. Quantum Chromodynamics is the formal theory of the strong color interaction between the quarks. Quarks have three possible values of color charge red, blue, and green. Antiparticles carry anticolor. Gluons which are the mediator of strong forces are bicolored, carrying one unit of positive color and one negative. So there are 9 kinds of gluon combinations ( g1, g2,g3,...,g9 ) are as follows:

    rb̅, rg̅, bg̅, br̅, gr̅, gb̅, 12\frac{1}{\sqrt{2}}(rr̅ - bb̅), 16\frac{1}{\sqrt{6}}(rr̅ + bb̅ -2gg̅), 13\frac{1}{\sqrt{3}}(rr̅ + bb̅ +gg̅)

    But one is color singlet 13\frac{1}{\sqrt{3}}(rr̅ + bb̅ +gg̅) and, since it carries no net color, has to be excluded.

    Quantum Chromodynamics (QCD) is very similar to Quantum Electrodynamics (QED). However there are differences between them which are as follows:

    • The color charge of the strong interaction is same as the electric charge in electromagnetic interaction. The difference between these two is that there is only one kind of electric charge (positive or negative), but there are three kinds of color charge (red, blue, and green or antired, antiblue, and antigreen).
    • In QED photon (ϒ) is the mediator which is electrically neutral and they do not interact directly. Since gluons itself carry the color charge they interact directly to other gluons, and hence in addition to fundamental quark-gluon vertex, we also have primitive gluon-gluon vertex. There are two kinds of gluon-gluon vertex i.e. three gluon vertices and four gluon vertices.
    IMG_20180702_162226(1).png
      Quark-gluon and gluon-gluon vertices

      This direct gluon-gluon coupling makes QCD more complicated than QED.

      • Size of the coupling constant adds another difference between QCD and QED. Each vertex in QED introduce a factor α=1137smallness of this number means we need Feynman diagram with small number of the vertices. In QCD this factor is not constant but depends upon the separation distance between the interacting particles we call it as a running coupling constant. This will introduce the important property of QCD as Asymptotic Freedom which we discuss later.
      • Another difference is QED is an Abelian Gauge Theory and QCD is a Non Abelian Gauge Theory.

      Properties of QCD

      Quark Confinement

      When two electrically charged particles are separated from each other the force between them gradually decreases, but when color charge (quarks) separate from each other then force between them increases in accordance with the distance. As we move quarks far and far away from each other the force is become stronger and stronger and infinite amount of energy is required to separate them. Eventually a quark-antiquark pair is produce as meson or baryon instead of producing an isolated quark. Hence quarks always remains in bound state because of separation which form hadron. This property of QCD is called as Quark Confinement.

      IMG_20180728_200948.png
        A possible scenario for quark confinement

        (As we pull u quark out of the proton a pair of quarks is created, and instead of free quark, we are left with pion and neutron.)

        Asymptotic Freedom

        David Politzer, Frank Wilzek and David Gross found that when two quarks are brought closer through very high energy reaction then they interact very weakly and behave as free particles. This property is known as Asymptotic Freedom.

        Hence at low energy there is confinement and at high energy there is asymptotic freedom in QCD. This two phenomenon are explained by using the Coupling factor αs.

        Running Coupling Constant

        Running coupling constant for QCD is given by

        αsQ2=12π(332Nf)×ln(Q2Ʌ2)\displaystyle {{α}_{s}}{{Q}^{2}}{=}\frac{12π}{\left({33}{-}{2}{{N}_{f}}\right){×}{l}{n}{⁡}\left(\frac{{Q}^{2}}{{Ʌ}^{2}}\right)}

        Where,

        Q is related to momentum transfer

        Nf is the number of participating quark flavor and is determined by the available energy characterized by Q2.

        The parameter Λ has to be determined by comparing QCD predictions to experimental results.

        The dependence of αs on Q2 shows the behavior of quarks as:

        For smaller value of Q, αs is large and hence quarks remain in confined state (as hadron) and cannot be isolated.

        For large value of Q, αs is small and due to this the quarks behave as if free and this is known as Asymptotic Freedom.

        These are the remarkable properties of QCD.

        QCD at Extremis (At high temperature or at high baryon density)

        At low energy, the interaction between quarks is so strong that they have to form the bound state hadron because of color confinement. When we provide high energy to the hadronic system ( or increase the temperature of hadronic system ) then the hadron (bound state) break up ( When the energy exceeds the thresh hold). The effective coupling become smaller due to asymptotic freedom. Similar things happens when we increase the density i.e. the hadron overlap loosing their boundaries. In such case quarks deconfine leading to a state of quark gluon plasma. Collins and Perry first showed that at extreme condition of density, hadronic matter goes to deconfined stage where they exhibit asymptotic freedom. This is called deconfinement phase transition from hadron to quarks.

        What is plasma?

        Irvin Langmuir first described the plasma for electrically charged particles. In Ancient Greek it is called “πλασµα” which means mouldable substance. Plasma is artificially generated by heating or subjecting neutral gas to strong electromagnetic field. Under normal condition the atom as a whole is neutral and remain confined as their binding energy is greater than the ambient thermal energy. When we increase the thermal energy near or greater than it’s binding energy by subjecting it to strong electromagnetic field then atom will ionize by removing electron and produce positively charge ions. Although these charged particles are unbound, they are not free in the sense of not experiencinf any force. These charged particles are strongly affected by each other’s electromagnetic field. This plasma may be partially or fully ionized. It is the composition of charged as well as neutral particles. Therefore electromagnetic or electrodynamics plasma it is the collection of charged and neutral particle showing the property of quasi-neutrality and collective behaviour.

        Electromagnetic plasma shows following properties:

        • Quasi-neutrality

        Plasma contain negative (ne) as well as positively charge particles (say ni), then overall plasma is approximately neutral and shows the property of quasi-neutrality.

        i.e neni{{n}_{e{}}}{≅}{}{}{}{{n}_{i}}

        • rd≤L
        • ωp τc ∼ 1
        • ND≫1

        Here L is the dimension of container in which plasma is form, and rd{{r}_{d}}is the Debye’s screening radius, ωp is plasma frequency , τc is mean collision time and ND is number of particle within the Debye’s sphere. We know that the range of electromagnetic force is infinite in vacuum, but when we think of any media this is not the case. In plasma we consider the Debye’s radius rd upto which the Coloumbian force of attraction or repulsion is experienced and after that it will not experience or negligibly experienced. In such case we say that charge is screened by 1/e value.

        Plasma inside the interior of sun stars corona, neon sign ,florosant lamp, and lightning are some of the examples of electromagnetic plasma. Now let us move towards quark gluon plasma.

        Quark-Gluon Plasma

        Before 1975 Collins and Perry already discussed the physics of Quark gluon plasma at extreme conditions of temperature and very high density. But the term quark-gluon plasma is coined by Russian-American physicist E. V. Shuryak for the assemblage of quarks and gluons in 1980. Quark gluon plasma is state of matter in QCD which exist at extremely high temperature and density. As we know quarks are color confined to each other so it is thought that this is the state of asymptotically free strong interacting quark and gluon. Quark gluon plasma is similar to electromagnetic plasma. In Electromagnetic plasma charges are screened but in Quark-Gluon plasma color charge of quarks and gluons is screened. Here we have hadronic matter as quarks and gluons instead of charged ion. Here role of charge is played by color charge. We know that these quark and gluons remain in confined state inside the hadrons. However at extreme condition of temperature and color charge density when the ambient thermal energy is large enough to exceed the binding energy between the quarks and gluons, they undergo deconfinement and produce free color charge particles i.e. quarks and gluons. Here the state of system is completely described by the degrees of freedom of color charged particles. The temperature at which QGP is formed is nearly equal to 164-175 MeV (see Ref. [2] to [4]), and it was first predicted by Lattice Guage Theory.

        It is expected that the required high temperatures and densities can be accomplished by colliding nuclei at relativistic energies where large fraction of kinetic energy of the beam will be converted to thermal energy. The Relativistic Heavy Ion Collider at Brookhaven National Laboratary and the Large Hydron Collider at Geneva can provide nuclear beam for the production of QGP, the primordial fluid which exist when universe was few microsecond old. Therefore the study of QGP is important for early universe study. Currently, there are rigorous international experimental and theoretical efforts to create and study QGP.

        Chapter 3

        Review of Thermodynamics and Equation of State

        Equation of state

        A system is composed of particles, whose average motion define its properties those properties are in turn related to one another through Equation of state. Equation of state is thermodynamic equation relating state variables which describe the state of matter such as pressure, volume, temperature or internal energy.

        Equations of state are useful in describing the properties of fluids, mixture of fluids, solid and interior of stars.

        Ideal Gas Equation

        Ideal gas equation is the equation of state of hypothetical ideal gas. This equation is given by Emile Clapeyron in 1834 as combination of the Boyl’s law, Charl’s Law, Avogadro’s law, and Gay-Lussac’s Law. The ideal gas equation is written as :

        PV = nRT

        Where P,V,T is the Pressure, Volume, and absolute Temperature; n is number 0f moles of gas; and R is ideal Gas constant. This law based on the assumptions of kinetic theory of gases. They treat gas molecule as point particle that do not interact except in elastic collision.

        Using statistical mechanics we are able to derive this expression;

        IMG_20180728_200928.png
          Graphical representation for ideal gas equation

          Van der Wall Equation of State:

          The Van der Wall equation based on the plausible reason that real gas do not follow the ideal fas law. This equation is corrects for the volume of gas, and the attractive forces between gas molecule, which is given by:

          P+n2aV2Vnb=nRT\displaystyle {{P+\frac{{{n}^{2}}{a}}{{V}^{2}}}}{V-nb}{=}{n}{R}{T}

          Where the first n2aV2\frac{{{n}^{2}}{a}}{{V}^{2}}, alter the pressure in ideal gas equation, it account for intermolecular attractive forces between gas molecule the magnitude for “a” indicates the strength of intermolecular strong forces. The factor “nb” account for the volume occupied by the gas molecule. The value of a and b is experimentally determined.

          IMG_20180710_155953.png
            Van der Waals isotherm

            Here experimentally, a=27R2Tc264Pc{a}{=}\frac{27{{R}^{2}}{{T}_{c}^{2}}}{64{{P}_{c}}}and b=RTc8Pc{b}{=}\frac{R{{T}_{c}}}{8{{P}_{c}}}; and Pc, Tc are critical quantities.

            MIT Bag Model

            Before moving towards Bag model (see Ref [7] & [8]), let us understand the concept of vacuum first. Vacuum is the space devoid of matter. Classically it is the state in which no particles are present. Hence the vacuum state has zero energy value which also is the minimum for any Hamiltonian (E0 E\geq0). However, there may be Hamiltonians whose minimum energy may not always be zero. For example consider the following energy profile:

            IMG_20180728_200824.png
              Energy profile for vacuum

              Clearly E00 E_0\neq0. This means the elementary particle excitations will be above this state and the produced particle will have a minimum energy E0. Hence the corrected way to define vacuum is a minimum energy state of a system whose value may or may not be zero. Of course, Quantum mechanics complicates the above picture. Due to uncertainty relation E.t \triangle E.\triangle t\geq\hslashand hence even a zero energy state can have quantum fluctuation which can give rise to virtual particle-antiparticle picture. It is so happen that QCD vacuum is a minimum energy state and E00 E_0\neq0. Due to which even the vacuum has finite energy density and exert a pressure. We will call this pressure as Bag pressure denoted by B. Let us suppose that we have a system in QCD vacuum. Now heat the system so that energy of the system changes, which is responsible for the production of particle and antiparticle pair. (Like neutron antineutron, proton, antiproton etc.). Let us name that particle antiparticle as “Bags” which contain quark and gluon. If we increase the temperature gradually the number of bags i.e. bag density in the state increase, and they start overlapping on each other within the system. When we continuously increase the temperature all bags overlap with each other and their boundaries disappear. Due to this, matter particle inside the bag start moving freely and quark gluon plasma is formed. The temperature at which this phase transition occurs is called Transition temperature Tc.

              As we know that quarks, antiquarks, and gluons are present inside the bag. They remain confined inside the bag as pressure experienced by the quarks (say Pq) is less than the outside pressure in QCD vacuum called Bag’s pressure i. e. B>Pq B>P_q. When we heat the bag the pressure inside the bag is increased. At certain temperature Tc the pressure Pq is greater than or equal to Bag’s pressure i.e. PqBthen the wall of bag or boundaries of bag disappear and Quark start moving freely inside the vacuum and QGP is formed.

              IMG_20180728_200851.png
                Formation of QGP 
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                  Quark and gluons confined inside Bag

                  Equation of state for Relativistic Massless Quarks and Gluons using MIT Bag Model

                  Grand Canonical Ensemble is given by equation

                  Z=N=0eβµNQNV,T{Z}{=}{{∑}_{N=0}^{∞}}{{{e}^{βµN}}{}{{Q}_{N}}{V,T}} ……………… (1)

                  Where,

                  QN={i}eβE{{Q}_{N}}{=}{{∑}_{\left\{i\right\}}}{{e}^{-βE}}………. (Here {i} indicates the sum over all configuration or possibilities)

                  E=ppnp E=\sum_p{\in_pn_p}…………. ( E is the energy of the system)

                  Z=N=0eβµN{np}e-βppnp
                  Z=N=0np=NeβµN(e-β1n1).e-β2n2.(e-β3n3)

                  For bosons np varies from 0 to ∞, and for fermions either np=0 or np=1

                  In general we can write

                  Z=p=0neβ(pµ)n {Z}{=}{{∏}_{p=0}^{∞}}{{{∑}_{n}}{{{e}^{-β\left({{∈}_{p-}}{µ}\right){n}}}}}{} ………………………… (2)

                  For fermions either n=o or n=1, so we can write equation (2) as:

                  Zf=p=o(1+eβ(pµ)) \therefore{\mathcal Z}_f=\prod_{p=o}^\infty{(1+}e^{-\beta(\in_{p-}µ)})
                  logZf=p=olog(1+eβ(pµ)) {\log{{\mathcal Z}_f=}}\sum_{p=o}^\infty{{\log{(1+e^{-\beta(\in_p-µ)}}})}

                  Now to convert discrete summation over momentum into continuous we can write

                  logZf=gV(2π)30log(1+eβpµ)d3p {log⁡{{{Z}_{f}}{=}}}\frac{gV}{{{\left({2}{π}\right)}^{3}}}{{∫}_{0}^{∞}}{{log⁡{\left({1}{+}{{e}^{-β{{{{∈}_{p}}{-}{µ}}}}}\right)}}}{{d}^{3}}{p}

                  Degeneracy factor g is introduce as fermions can be distinguish from each other if we consider their spin

                  And V is the volume of the system

                  logZf=gV(2π)30log(1+eβpµ)d3p {log⁡{{{Z}_{f}}{=}}}\frac{gV}{{{\left({2}{π}\right)}^{3}}}{{∫}_{0}^{∞}}{{log⁡{\left({1}{+}{{e}^{-β{{{{∈}_{p}}{-}{µ}}}}}\right)}}}{{d}^{3}}{p} ………………………… (3)

                  This is required partition function for fermions.

                  Similarly,

                  For bosons n varies from o to ∞,

                  n=0eβpμn=1+eβpμ+e2βpμ+e3βpμ+ {{∑}_{n=0}^{∞}}{{{e}^{-β{{{{∈}_{p}}{-}{μ}}}{n}}}}{=}{1}{+}{{e}^{-β{{{{∈}_{p}}{-}{μ}}}}}{+}{{e}^{-2β{{{{∈}_{p}}{-}{μ}}}}}{+}{{e}^{-3β{{{{∈}_{p}}{-}{μ}}}}}{+}{…}
                  n=0eβpμn=11eβ(pμ) {⇒}{{∑}_{n=0}^{∞}}{{{e}^{-β{{{{∈}_{p}}{-}{μ}}}{n}}}}{=}\frac{1}{1-{{e}^{-β\left({{∈}_{p}}{-}{μ}\right)}}}

                  So we can write equation (2) as:

                  logZB=p=olog(1eβ(pµ)) {\log{{\mathcal Z}_B=}}-\sum_{p=o}^\infty{{\log{(1-e^{-\beta(\in_p-µ)}}})}

                  logZB=gV(2π)30log(1eβpµ)d3p {⇒log⁡{{{Z}_{B}}{=}}}\frac{-gV}{{{\left({2}{π}\right)}^{3}}}{{∫}_{0}^{∞}}{{log⁡{\left({1}{-}{{e}^{-β{{{{∈}_{p}}{-}{µ}}}}}\right)}}}{{d}^{3}}{p} ……………….. (4)

                  Degeneracy factor “g” is due to different color and spin of boson, and V is volume of the system. So this is partition function for bosons

                  In general for both fermions and bosons the partition function can be written as,

                  logZ=κgV(2π)30log(1+κeβpµ)d3p {log⁡Z=}\frac{κgV}{{{\left({2}{π}\right)}^{3}}}{{∫}_{0}^{∞}}{{log⁡{\left({1}{+}{κ}{{e}^{-β{{{{∈}_{p}}{-}{µ}}}}}\right)}}}{{d}^{3}}{p} ………………….. (5)

                  For particles µ >0, and for antiparticles µ<0

                  Hence for system having particles and antiparticles we can write equation (5) as:

                  logZ=gVκ(2π)30log(1+κeβpµ)d3p+0log(1+κeβp+µ)d3p {log⁡Z=}\frac{gVκ}{{{\left({2}{π}\right)}^{3}}}{{{{∫}_{0}^{∞}}{{log⁡{\left({1}{+}{κ}{{e}^{-β{{{{∈}_{p}}{-}{µ}}}}}\right)}}}{{d}^{3}}{p}{+}{{∫}_{0}^{∞}}{{log⁡{\left({1}{+}{κ}{{e}^{-β{{{{∈}_{p}}{+}{µ}}}}}\right)}}}{{d}^{3}}{p}}} …………. (6)

                  Now we assume that particles are massless and relativistic

                  p=p2+m2=p \Rightarrow\in_p=\sqrt[{}]{p^2+m^2}=p
                  logZ=gVκ(2π)30log1+κeβpµd3p+0log1+κeβp+µd3p {∴log⁡Z=}\frac{gVκ}{{{\left({2}{π}\right)}^{3}}}{{{{∫}_{0}^{∞}}{{log⁡{{{1+κ{{e}^{-βp-µ}}}}{{d}^{3}}{p}{+}{{∫}_{0}^{∞}}{{log⁡{{{1+κ{{e}^{-βp+µ}}}}{{d}^{3}}{p}}}}}}}}}

                  logZ=gVκ(2π)34π0log1+κeβpµp2dp+0log1+κeβp+µp2dp {log⁡Z=}\frac{gVκ}{{{\left({2}{π}\right)}^{3}}}{4}{π}{{{{∫}_{0}^{∞}}{{log⁡{{{1+κ{{e}^{-βp-µ}}}}{{p}^{2}}{d}{p}{+}{{∫}_{0}^{∞}}{{log⁡{{{1+κ{{e}^{-βp+µ}}}}{{p}^{2}}{d}{p}}}}}}}}}……………. (7)

                  Using spherical polar co-ordinate.

                  By using Integration by parts we have

                  logZ=gVκ2π34πlog(1+κe-βp-µ).p330-0-βκe-βp-μp331+κe-βp-μdp+log(1+κe-βp+µ).p330-0-βκe-βp+μp331+κe-βp+μdp
                  logZ=gVκ4πβ32π30eβpμp31+κeβpμdp+0eβp+μp31+κeβp+μ {log⁡Z=}\frac{gVκ4πβ}{3{{2π}^{3}}}{{{{∫}_{0}^{∞}}{\frac{{{e}^{-βp-μ}}{{p}^{3}}}{{{1+κ{{e}^{-βp-μ}}}}}{d}{p}}{+}{{∫}_{0}^{∞}}{\frac{{{e}^{-βp+μ}}{{p}^{3}}}{{{1+κ{{e}^{-βp+μ}}}}}}}}

                  logZ=gVβ6π20p3eβpμ+κdp+0p3eβp+μ+κdp {log⁡Z=}\frac{gVβ}{6{{π}^{2}}}{{{{∫}_{0}^{∞}}{\frac{{p}^{3}}{{{{{e}^{βp-μ}}{+}{κ}}}}{d}{p}}{+}{{∫}_{0}^{∞}}{\frac{{p}^{3}}{{{{{e}^{βp+μ}}{+}{κ}}}}{d}{p}}}} …………………. (8)

                  Consider first term,

                  0p3eβpμ+κdp=1β4βμ(x+βμ)3ex+κdx {{∫}_{0}^{∞}}{\frac{{p}^{3}}{{{{{e}^{βp-μ}}{+}{κ}}}}{d}{p}{=}\frac{1}{{β}^{4}}}{{∫}_{-βμ}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}……………. (By replacing β (p-µ) =x)

                  Similarly,

                  0p3eβp+μ+κdp=1β4βμ(xβμ)3ex+κdx {{∫}_{0}^{∞}}{\frac{{p}^{3}}{{{{{e}^{βp+μ}}{+}{κ}}}}{d}{p}{=}\frac{1}{{β}^{4}}}{{∫}_{βμ}^{∞}}{\frac{{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}……………… (By replacing β (p+µ) =x)

                  logZ=gVβ36π2βμ(x+βμ)3ex+κdx+βμ(xβμ)3ex+κdx {∴}{log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{-βμ}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{+}{{∫}_{βμ}^{∞}}{\frac{{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}
                  logZ=gVβ36π2βμ0(x+βμ)3ex+κdx+0(x+βμ)3ex+κdx+0(xβμ)3ex+κdx0βμ(xβμ)3ex+κdx {log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{-βμ}^{0}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{+}{{∫}_{0}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{+}{{∫}_{0}^{∞}}{\frac{{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{-}{{∫}_{0}^{βμ}}{\frac{{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}
                  logZ=gVβ36π2βμ0(x+βμ)3ex+κdx+0(x+βμ)3(xβμ)3ex+κdx0βμ(βμx)3ex+κdx {log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{-βμ}^{0}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{+}{{∫}_{0}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{-}{{∫}_{0}^{βμ}}{\frac{-{{\left({β}{μ}{-}{x}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}

                  For first term replace x by –x, we get

                  βμ0(x+βμ)3ex+κdx=βμ0βμx3ex+κdx=0βμ(βμx)3ex+κdx {{∫}_{-βμ}^{0}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{=}{-}{{∫}_{βμ}^{0}}{\frac{{βμ-x}^{3}}{{{e}^{-x}}{+}{κ}}{d}{x}}{=}{{∫}_{0}^{βμ}}{\frac{{{\left({β}{μ}{-}{x}\right)}^{3}}}{{{e}^{-x}}{+}{κ}}{d}{x}}
                  logZ=gVβ36π20βμ(βμx)3ex+κdx+0(x+βμ)3(xβμ)3ex+κdx0βμ(βμx)3ex+κdx {⇒log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{0}^{βμ}}{\frac{{{\left({β}{μ}{-}{x}\right)}^{3}}}{{{e}^{-x}}{+}{κ}}{d}{x}}{+}{{∫}_{0}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}{-}{{∫}_{0}^{βμ}}{\frac{-{{\left({β}{μ}{-}{x}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}
                  logZ=gVβ36π20βμ(βμx)31ex+κ+1ex+κdx+0(x+βμ)3(xβμ)3ex+κdx {⇒log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{0}^{βμ}}{{{\left({β}{μ}{-}{x}\right)}^{3}}{{\frac{1}{{{e}^{-x}}{+}{κ}}{+}\frac{1}{{{e}^{x}}{+}{κ}}}}{d}{x}}{+}{{∫}_{0}^{∞}}{\frac{{{\left({x}{+}{β}{μ}\right)}^{3}}{{\left({x}{-}{β}{μ}\right)}^{3}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}
                  logZ=gVβ36π20βμ(βμx)3κdx+20x3+3x(βμ)2ex+κdx {⇒log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{{{∫}_{0}^{βμ}}{{{\left({β}{μ}{-}{x}\right)}^{3}}{κ}{d}{x}}{+}{2}{{∫}_{0}^{∞}}{\frac{{{x}^{3}}{+}{3}{x}{{\left({β}{μ}\right)}^{2}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}
                  logZ=gVβ36π2κ(βμ)44+20x3+3x(βμ)2ex+κdx {⇒log⁡Z=}\frac{gV}{{{β}^{3}}{6}{{π}^{2}}}{{\frac{{{κ\left({β}{μ}\right)}^{4}}}{4}{+}{2}{{∫}_{0}^{∞}}{\frac{{{x}^{3}}{+}{3}{x}{{\left({β}{μ}\right)}^{2}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}

                  logZ=gVT36π2κ(βμ)44+20x3+3x(βμ)2ex+κdx {⇒log⁡Z=}\frac{gV{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{κ\left({β}{μ}\right)}^{4}}}{4}{+}{2}{{∫}_{0}^{∞}}{\frac{{{x}^{3}}{+}{3}{x}{{\left({β}{μ}\right)}^{2}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}…….. (9) ………… (As β=1/T)

                  Here µ is the chemical potential; if µ is fixed then number of particles in the system is fixed. But here we consider relativistic massless particles, where number of particles in the system does not remain constant. So we look for the conserved quantity during relativistic case. Baryon number is the quantity which remains fixed in relativistic case. Hence we replace µ by µB called baryon chemical potential, which means if µB is fixed then B is fixed.

                  logZ=gVT36π2κ(βμB)44+20x3+3x(βμB)2ex+κdx {⇒log⁡Z=}\frac{gV{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{κ\left({β}{{μ}_{B}}\right)}^{4}}}{4}{+}{2}{{∫}_{0}^{∞}}{\frac{{{x}^{3}}{+}{3}{x}{{\left({β}{{μ}_{B}}\right)}^{2}}}{{{e}^{x}}{+}{κ}}{d}{x}}}}……………. (10)

                  Now for Gluons: µB =0 as B=0 andκ=1 {a}{n}{d}{}{κ}{=}{-}{1}

                  Therefore equation………

                  logZg=ggVT36π20+20x3ex1dx {⇒log⁡{{{Z}_{g}}{=}}}\frac{{{g}_{g}}{V}{{T}^{3}}}{6{{π}^{2}}}{{0+2{{∫}_{0}^{∞}}{\frac{{x}^{3}}{{{e}^{x}}{-}{1}}{d}{x}}}}
                  logZg=ggVT36π22×π430 {⇒log⁡{{{Z}_{g}}{=}}}\frac{{{g}_{g}}{V}{{T}^{3}}}{6{{π}^{2}}}{{2×\frac{{π}^{4}}{30}}}

                  … Using standard result from Riemann-Zeta function

                  logZg=ggVT3π290 {{\Rightarrow\log}{{\mathcal Z}_g=}}\frac{g_gVT^3\pi^2}{90}………………….. (11)

                  This is required partition function for gluons.

                  For Quarks:κ=+1 \kappa=+1,

                  logZq=gqVT36π2(βμq)44+20x3ex+1+3x(βμq)2ex+1dx {log⁡{{{Z}_{q}}{=}}}\frac{{{g}_{q}}{V}{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}{2}{{∫}_{0}^{∞}}{\frac{{x}^{3}}{{{e}^{x}}{+}{1}}{+}\frac{3x{{\left({β}{{μ}_{q}}\right)}^{2}}}{{{e}^{x}}{+}{1}}{d}{x}}}}

                  logZq=gqVT36π2(βμq)44+27π4120+3μ2β2π212 {log⁡{{{Z}_{q}}{=}}}\frac{{{g}_{q}}{V}{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}{2}{{\frac{7{{π}^{4}}}{120}{+}\frac{3{{μ}^{2}}{{β}^{2}}{{π}^{2}}}{12}}}}}…Using standard result of integral from Riemann Zeta function

                  Here we consider the condition for quark as well as antiquarks.

                  logZq=gqVT36π2(βμq)44+7π460+μq2β2π22 {log⁡{{{Z}_{q}}{=}}}\frac{{{g}_{q}}{V}{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}\frac{7{{π}^{4}}}{60}{+}\frac{{{{μ}_{q}}^{2}}{{β}^{2}}{{π}^{2}}}{2}}} …………… (12)

                  Consider a system of massless and non-interacting quarks and gluons

                  logZ=logZq+logZg {{\Rightarrow\log}{\mathcal Z=}}{\log{{\mathcal Z}_q+{\log{{\mathcal Z}_g}}}}

                  logZ=gqVT36π2(βμq)44+7π460+μq2β2π22+ggVT3π290 {⇒log⁡Z=}\frac{{{g}_{q}}{V}{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}\frac{7{{π}^{4}}}{60}{+}\frac{{{{μ}_{q}}^{2}}{{β}^{2}}{{π}^{2}}}{2}}}{+}\frac{{{g}_{g}}{V}{{T}^{3}}{{π}^{2}}}{90} ……………… (13)

                  Now Pressure (P) is given by

                  P=VTlogZ {P}{=}\frac{∂}{∂V}{Tlog⁡Z}
                  P=VgqVT36π2(βμq)44+7π460+μq2β2π22+ggVT3π290 {⇒}{P}{=}\frac{∂}{∂V}{{\frac{{{g}_{q}}{V}{{T}^{3}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}\frac{7{{π}^{4}}}{60}{+}\frac{{{{μ}_{q}}^{2}}{{β}^{2}}{{π}^{2}}}{2}}}{+}\frac{{{g}_{g}}{V}{{T}^{3}}{{π}^{2}}}{90}}}
                  Expression for pressure
                  P=gq127π2T430+μq2T2+μq42π2+ggT4π290

                  …………………...... .................................... (14)

                  This is required equation of pressure for massless relativistic non-interacting Quark Gluon phase.

                  Now particle density nq is given by,

                  nq=1VμBTlogZ
                  nq=1VμqgqVT46π2(βμq)44+7π460+μq2β2π22+ggVT4π290 {⇒}{{n}_{q}}{=}\frac{1}{V}\frac{∂}{∂{{μ}_{q}}}{{\frac{{{g}_{q}}{V}{{T}^{4}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}\frac{7{{π}^{4}}}{60}{+}\frac{{{{μ}_{q}}^{2}}{{β}^{2}}{{π}^{2}}}{2}}}{+}\frac{{{g}_{g}}{V}{{T}^{4}}{{π}^{2}}}{90}}}
                  nq=gqT46π24β4μq34+0+2μqβ2π22+0 {⇒}{{n}_{q}}{=}\frac{{{g}_{q}}{{T}^{4}}}{6{{π}^{2}}}{{\frac{{{4β}^{4}}{{μ}_{q}^{3}}}{4}{+}{0}{+}\frac{2{{μ}_{q}}{{β}^{2}}{{π}^{2}}}{2}}}{+}{0}
                  Expression for particle density 
                  nq=gq6μq3π2+μqT2

                  ………………………. (15)

                  This is the expression of particle density for massless relativistic non-interacting Quark Gluon phase.

                  Entropy is given by the equation,

                  S=1VTTlogZ
                  S=1VTgqVT46π2(βμq)44+7π460+μq2β2π22+ggVT4π290 {⇒}{S}{=}\frac{1}{V}\frac{∂}{∂T}{{\frac{{{g}_{q}}{V}{{T}^{4}}}{6{{π}^{2}}}{{\frac{{{\left({β}{{μ}_{q}}\right)}^{4}}}{4}{+}\frac{7{{π}^{4}}}{60}{+}\frac{{{{μ}_{q}}^{2}}{{β}^{2}}{{π}^{2}}}{2}}}{+}\frac{{{g}_{g}}{V}{{T}^{4}}{{π}^{2}}}{90}}}
                  Expression for Entropy of system
                  S=gq1214π2T315+2μq2T+2π245ggT3

                  ……………………. (16)

                  This is required expression for entropy of system.

                  Now, by Thermodynamics relation we have

                  TS=+PμBn TS=\in+P-\mu_Bn
                  ϵ=TSP+μBn {⇒}{ϵ}{=}{T}{S}{-}{P}{+}{{μ}_{B}}{n}{}
                  ϵ=gq1214π2T415+2μq2T2+2π245ggT4gq127π2T430+μq2T2+μq42π2+ggT4π290+gq6μq4π2+μq2T2 {⇒}{ϵ}{=}\frac{{g}_{q}}{12}{{{{\frac{14{{π}^{2}}{{T}^{4}}}{15}{+}{2}{{μ}_{q}^{2}}{{T}^{2}}}}{+}\frac{2{{π}^{2}}}{45}{{g}_{g}}{{T}^{4}}}}{-}{{\frac{{g}_{q}}{12}{{{{\frac{7{{π}^{2}}{{T}^{4}}}{30}{+}{μ}}_{q}^{2}}{{T}^{2}}{+}\frac{{{{μ}_{q}}^{4}}}{2{{π}^{2}}}}}{+}\frac{{{g}_{g}}{{T}^{4}}{{π}^{2}}}{90}}}{+}{{\frac{{g}_{q}}{6}{{\frac{{μ}_{q}^{4}}{{π}^{2}}{+}{{μ}_{q}^{2}}{{T}^{2}}}}}}
                  ϵ=gq1221π2T430+3μq2T2+3μq42π2+ggT4π230 {⇒}{ϵ}{=}{{\frac{{g}_{q}}{12}{{{{\frac{21{{π}^{2}}{{T}^{4}}}{30}{+}{3}{μ}}_{q}^{2}}{{T}^{2}}{+}\frac{3{{{μ}_{q}}^{4}}}{2{{π}^{2}}}}}{+}\frac{{{g}_{g}}{{T}^{4}}{{π}^{2}}}{30}}}
                  Equation of State
                  ϵ=3P

                  …………………….(18)

                  This is required equation of state for Relativistic massless non-interacting quarks and gluons.

                  At LHC energy it has been found that most of the hadrons detected are the pions (π). Pions are the lightest mesons with mass approximately equals to 140MeV.

                  Let us consider the system of non-interacting massless pion

                  i.e. mπ = 0 and S=0;

                  As pions are the mesons they have same pressure equation like gluons,

                  Pπ=π2gπT490 P_\pi=\frac{\pi^2g_\pi T^4}{90} …………………….(19)

                  As we have,

                  Pq+q̅+g=gq127π2T430+μq2T2+μq42π2+ggT4π290…………………..(from equation 14) {}{}{}{}{}{}{}

                  Here we consider the vaccume pressure is zero. But when the vaccume pressure of QCD is equal to the Bag’s pressure then pressure then the partition function for QGP is given by:

                  logZQGP=logZq+q̅+g+BVT ………………. (15)

                  Pressure of QGP is given by:

                  PQGP=V(TlogZQGP) P_{QGP}=\frac\partial{\partial V}(T{\log{{\mathcal Z}_{QGP})}}
                  PQGP=Pq+q̅+g-B

                  PQGP=gq127π2T430+μq2T2+μq42π2+ggT4π290B {{⇒P}_{QGP}}{=}{{\frac{{g}_{q}}{12}{{{{\frac{7{{π}^{2}}{{T}^{4}}}{30}{+}{μ}}_{q}^{2}}{{T}^{2}}{+}\frac{{{{μ}_{q}}^{4}}}{2{{π}^{2}}}}}{+}\frac{{{g}_{g}}{{T}^{4}}{{π}^{2}}}{90}}}{-}{B} ……………… (16)

                  Also, ϵQGP=ϵq+q̅+g+B ……………… (17)

                  The equation of state for relativistic ideal gas is given by P=3 P=\frac\in3

                  AsPQGP=Pq+q̅+g-B
                  PQGP=ϵq+q̅+g3-B

                  PQGP=13ϵQGP4B {{⇒P}_{QGP}}{=}\frac{1}{3}{{{{ϵ}_{QGP}}{-}{4}{B}}} …………. (18)

                  Here PQGP is the function of T and µB and Pressure of hadrons i.e. Ph I also the function of T and µB. Now if QGP is goes to hadronic phase then for phase transition it must satisfy the following conditions:

                  PQGPTq,μq=Ph(Th,μB) {{P}_{QGP}}{{{{T}_{q}}{,}{{μ}_{q}}}}{=}{{P}_{h}}\left({{T}_{h}}{,}{{μ}_{B}}\right)
                  Tq=Th T_q=T_h
                  μq=13μB \mu_q=\frac13\mu_B

                  To know at what temperature and pressure this phase transition occurs we have to solve this equation simultaneously. Let’s calculate the transition temperature for quark-gluon system connected to a pionic system i.e. the hadronic system only contains pions (ud̅).

                  gq127π2Tc430+μq2Tc2+μq42π2+ggTc4π290B=π2gπTc490 {{\frac{{g}_{q}}{12}{{{{\frac{7{{π}^{2}}{{T}_{c}^{4}}}{30}{+}{μ}}_{q}^{2}}{{T}_{c}^{2}}{+}\frac{{{{μ}_{q}}^{4}}}{2{{π}^{2}}}}}{+}\frac{{{g}_{g}}{{T}_{c}^{4}}{{π}^{2}}}{90}}}{-}{B}{=}\frac{{{π}^{2}}{{g}_{π}}{{T}_{c}^{4}}}{90} ……………… (19)

                  Now consider case I : when μB=0 \mu_B=0

                  Equation 19 can be written as:

                  gq127π2Tc430+ggTc4π290B=π2gπTc490 {{\frac{{g}_{q}}{12}{{\frac{7{{π}^{2}}{{T}_{c}^{4}}}{30}}}{+}\frac{{{g}_{g}}{{T}_{c}^{4}}{{π}^{2}}}{90}}}{-}{B}{=}\frac{{{π}^{2}}{{g}_{π}}{{T}_{c}^{4}}}{90}
                  Critical temperature is given by:

                  Tc=90Bπ2(geff-gπ)14

                  Where geff=gg+74gq g_{eff}=g_g+\frac74g_qnow gg=16 g_g=16and gq=12; g_q=12; geff=37 {\Rightarrow g}_{eff}=37

                  Exprssion to calculate critical temperature
                  Tc=90Bπ23414

                  ……………… (20)

                  Case II : when Tc=0 T_c=0;

                  Equation (19) can be written as:

                  Expression to calculate criticle Baryonic Chemical potential
                  μB=12×81×2π2×Bgq14

                  ……………..(21)

                  Summary and Conclusion

                  In this project I learnt about the elementary particles and about the fundamental interactions, which exist among these particles. I also tried to learn the overview of Quantum Chromodynamics (QCD) which is the theory of strong interaction that exists among the fermionic particles called quarks. The quarks interact strongly via force carrier particles called gluons. The thermodynamic state of a quark-gluon system called quark-gluon plasma is studied here.

                  The equation of state, which is the thermodynamics relationship between the thermodynamic variable, for QGP is studied for MIT Bag model. The expressions for pressure, energy density are derived. The phase transition from QGP to pionic system has been studied usin Baf EoS. The expression for transition temperature T c has been obtained.

                  The equations of state derived from the MIT Bag model is one of the oldest equation of state for QGP. Even today it remains the baseline. Hence it has been widely accepted in astrophysics and cosmology. Curve obtained from the MIT Bag model for Baryon chemical potential Vs Bag pressure and Temperature Vs Bag pressure suggest that the phase transition is of first order. But experimental Lattice QCD calculation shows that there is smooth crossover between phase transition of QGP. Which suggests us to modify the Bag model to suits better with the experimental results. Uptill now many modifications are done and their results are calculated.

                  The energy density and pressure of QGP for Bag Model is given by

                  =σT4B \in=\sigma T^4-B P=σ3T4B P=\frac\sigma3T^4-B

                  Where σ \sigmais the Stephan-Boltzmann Constant whose value is given by

                  σ=π230gq+7gg8 {σ}{=}\frac{{π}^{2}}{30}{{{{g}_{q}}{+}\frac{7{{g}_{g}}}{8}}} …( where gq and gg are the degeneracy factors for massless quarks and gluons)

                  Also   PQGP=13ϵQGP4B\displaystyle {{Also    {}{}{P}}_{QGP}}{=}\frac{1}{3}{{{{ϵ}_{QGP}}{-}{4}{B}}}

                  Some Modifications to Bag EoS are suggested as follows (see Ref. [8])

                  Modifications:

                  • A reduction in Stephan-Boltzmann constant:

                  Pressure and energy density is given as:

                  =σ1T4B1 \in=\sigma_1T^4-B_1 , P=σ13T4ATB1 P=\frac{\sigma_1}3T^4-AT-B_1

                  • The introduction of another temperature dependent term in the pressure and also in energy density

                  With this modification the equation for pressure and energy density is reduced tas:

                  =σ2T4CT2+B2 \in=\sigma_2T^4-CT^2+B_2 , P=σ23T4CT2B2 P=\frac{\sigma_2}3T^4-CT^2-B_2

                  This modification was suggested by Pisarski (see Ref. [9]).

                  • A Bag constant with negative sign

                  References

                  [1] Introduction to elementary particles, David Griffith, WIELY-VCLT Publication

                  [2] J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34, 1353 (1975)

                  [3] A. Bazavor etal. Phys. Rev. D, 85, 059503 (2012)

                  [4] S. Barsongi etal. J. Phys. G, 38, 124101 (2011)

                  [5] M. Cheng etal. Phys. Rev. D, 81, 054504 (2010)

                  [6] S. Gupta etal. Science, 332, 1525 (2011)

                  [7] QGP Book R. C. Hroz

                  [8] S. M. Sanches Jr., F.S. Navarra, D. A. Fogaca, Nucl. Phys. A, 937, 1 (2015)

                  [9] R. D. Pisarski, Phys. Rev. D 74 121703 (2006)

                  Acknowledgement

                  Success is where preparation and opportunities meet. I am very fortunate to get this opportunity from Indian Academy of Sciences Bangalore. I show my sincere gratitude towards them for offering me a Summer Research Fellowship for two months.

                  I owe my profound gratitude to my project guide Dr. Jajati K. Nayak, VECC Kolkata, for their most valuable guidance and support. The way they teach and explain physics inspire me a lot. My two months journey is really fruitful for me under his experience and knowledge.

                  I extend my gratitude to Dr. P. Y. Nabhiraj for providing me an opportunity to undergo summer training at VECC.

                  I would like to thank Mr. Sushant Singh and Mrs. Purabi Ghosh, who took keen interest and guided me all along, till the completion of project work by providing all necessary information.

                  I am thankful to Prof. Dr. S. K. Omanwar (HOD), SGBAU Amravati for their constant support, encouragement and also their most valuable advice and helping hand without which I could not be able to complete this project.

                  Last but not the least I would like to thank my parents and family for their moral support and my friends who boost me up every time.

                  Appendices

                  Natural units

                  The basic units in physics are mass, length, and time and the SI system express these in meters,

                  kilograms and seconds. Such units are not very appropriate in high energy physics, where typical length are 10-15m typical masses are 10-27Kg.The units using in high energy physics are known to be natural units.

                  In natural units, we take h=c=1.In high energy we express all the physical quantities in the order of energy units (order of GeV or TeV).The conversion of SI units to natural units of the physical quantities are given below:

                  Mass in natural units:

                  According to SI system unit of mass is Kg.

                  According Einstein mass energy relation

                  E=m×c2 E=m\times c^2
                  m=Ec2=eVc2 {⟹}{m}{=}{}\frac{E}{{c}^{2}}{=}{}\frac{eV}{{c}^{2}}

                  We have

                  1eV1.602×1019J {1}{e}{V}{≅}{1.602}{×}{{10}^{-19}}{}{J}
                  c3×108 c\cong3\times{10}^8
                  1eVc2=1.602×1019(3×108)21.78×1036Kg \Longrightarrow\frac{1eV}{c^2}=\frac{1.602\times{10}^{-19}}{({3\times{10}^8)}^2}\cong1.78\times{10}^{-36}Kg
                  1GeVc21.78×1027Kg \Longrightarrow\frac{1GeV}{c^2}\cong1.78\times{10}^{-27}Kg

                  Therefore in Natural units,

                  Natural unit

                  Unit of mass GeV/c2.

                  The SI unit of length is meter, but in high energy physics the unit of length is express in terms of GeV-1

                  We have

                  length=cMc2=6.588×1025GeVs×3×108m/sGeV0.1975fm {l}{e}{n}{g}{t}{h}{=}\frac{ℏc}{{{M{{c}^{2}}}}}{=}\frac{6.588×{{10}^{-25}}{{GeV{}{s}}}{×}{3}{×}{{10}^{8}}{m/s}}{GeV}{≅}{0.1975}{}{f}{m}
                  1GeV10.1975fm {⟹}{1}{}{G}{e}{{V}^{-1}}{≅}{0.1975}{}{f}{m}

                  In Natural units

                  Natural unit
                  The unit of length is GeV-1

                  Similarly,

                  time=cMc36.59×1025s {t}{i}{m}{e}{=}\frac{ℏc}{{{M{{c}^{3}}}}}{≅}{6.59}{×}{{10}^{-25}}{}{s}
                  time=1GeV16.59×1025s time=1GeV^{-1}\cong6.59\times{10}^{-25}s
                  Natural unit
                  The unit of time is 1GeV-1

                  For temperature conversion,

                  We have,

                  E=KBT {E}{=}{}{{K}_{B}}{T}{},

                  Where KB is Boltzmann’s Constant and T is temperature

                  T=EKB1eV1.38×10-23×0.625×1019(eV/K)1.1605×104
                  1eV11,605K104K {⟹}{1}{e}{V}{≅}{11,605}{}{K}{≅}{{10}^{4}}{}{K}
                  1GeV1013K \Longrightarrow1GeV\cong{10}^{13}K
                  Unit of Temperature is GeV

                  Source

                  • : https://en.wikipedia.org/wiki/Van_der_Waals_equation
                  • : inspirehep.net
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