Quarks, Nuclei, and the Cosmos

February 1, 2007

Quarks, Nuclei, and the Cosmos

An Introduction to Modern Nuclear Physics

Xiangdong JiPhysics Department, University of Maryland

College Park, Maryland 20742, USA

Robert D. McKeownW. K. Kellogg Radiation Laboratory,

California Institute of Technology,

Pasadena, California 91125, USA

ii

Contents

1 Introduction to QCD and the Standard Model 1

1.1 Quarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Gluons and Quantum Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Force Between Quarks, Asymptotic Freedom . . . . . . . . . . . . . . . . . . . . . . 51.4 Chiral Symmetry and Its Spontaneous Breaking . . . . . . . . . . . . . . . . . . . . . 91.5 Color Confinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.6 Hadrons and Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7 Electroweak Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.8 ***Background Information for Chapter 1*** . . . . . . . . . . . . . . . . . . . . . . 17

1.8.1 SU(3) group and Gell-Mann matrices . . . . . . . . . . . . . . . . . . . . . . 171.8.2 Lagrangian Density, QED and Feynman Rules . . . . . . . . . . . . . . . . . 18

1.9 Problem Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2 Quark-Gluon Plasma and the Early Universe 20

2.1 Thermodynamics of A Hot Relativistic Gas . . . . . . . . . . . . . . . . . . . . . . . 212.2 The Early Partonic Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3 The Quark-Gluon Plasma in Perturbative QCD . . . . . . . . . . . . . . . . . . . . . 242.4 A Brief Tour in Lattice QCD Thermodynamics . . . . . . . . . . . . . . . . . . . . . 282.5 Evolution of the Universe in Hadronic Phase . . . . . . . . . . . . . . . . . . . . . . 312.6 The Origin of Baryonic Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Protons and Other Hadrons in Quark Model 36

4 Probing the Parton Structure in the Proton 37

iii

Chapter 1

Introduction to QCD and theStandard Model

It is an astounding fact that modern-day physics can quantitatively describe phenomena from thesmall scale of quarks and leptons (10−18 cm) to the large scale of the whole present-day Universe(1028 cm) using the four fundamental forces: gravity, electromagnetism, strong and weak inter-actions. While gravity is governed by Einstein’s general relativity, the other three forces can bedescribed to an excellent degree by a quantum field theory of quarks and leptons based on a frame-work consistent with Einstein’s special theory of relativity and quantum mechanics: the so-calledstandard model (SM). This book is mainly about the physics of strong interactions, the associatedstructure and interactions of hadronic matter, and its role in the evolution of the Universe from avery early stage (microseconds young) to the present day (13 billion years old). However, as thereader will discover, it is not possible to tell the story without including the other forces. Thereforein this beginning chapter, we give an introduction to the standard model, with an emphasis on thefundamental theory of strong interactions—quantum chromodynamics (QCD).

1.1 Quarks

The quarks will be major characters in our story and are the fundamental objects participating instrong interactions. Like the electrons, they are simple structureless (as far as we know) spin-1/2particles. Therefore in a relativistic theory, they are described by Dirac spinors ψα(x) with fourcomponents α = 1, ..., 4, which are functions of the space-time coordinates xµ = (t, x, y, z). Ifquarks did not interact with other particles or fields, they would obey the free Dirac equation,

(i 6∂ −m)ψ(x) = 0 , (1.1)

where m is meant to be the “free” mass and other notations follow from the standard textbookslike Bjorken and Drell, or Peskin and Schroder. Simple states of these quarks are the standardplane waves

ψk,λ(x) = u(k, λ)e−i(Et−~x·~k) , (1.2)

where kµ = (E,~k) and λ denote the four-momentum and polarization, and u(k, λ) is the spin-dependent momentum-space wave function. We note that the Dirac equation can also be derived

1

2 CHAPTER 1. INTRODUCTION TO QCD AND THE STANDARD MODEL

from the lagrangian density,

L(x) = ψ(x)(i 6∂ −m)ψ(x) . (1.3)

In reality, however, free quarks have never been observed in laboratory; this is known as quark

confinement, and is an important consequence of the low-energy dynamics of strong interactions.We will discuss this phenomenon more extensively below and in subsequent chapters.

At present, we have discovered six flavors of quarks through various high-energy experiments:up(u), down(d), charm(c), strange(s), top(t) and bottom(b). These flavors organize themselvesinto families under the weak interactions. Up and down form the first family (or generation),charm and strange the second, and top and bottom the third. As far as we can ascertain, thesethree families are basically repetitions of the same pattern (same quantum numbers) with unknownphysical significance. For example, the electric charges of the up, charm and top quarks are all 2/3of that of the proton, and the charges of the down, strange, and bottom are all −1/3. They aredistinguished by their masses and associated “flavor” quantum numbers.

Quark Masses

Because we cannot observe free quarks, their masses obviously cannot be measured directly andso the very meaning of mass requires some explanation. The mass of a quark is actually justa parameter in the lagrangian of the theory, which describes the self-interaction of the quark,and is not directly observable. As such, the mass parameter is much like a coupling constant inquantum field theory, and is technically dependent on the momentum scale and the renormalizationscheme and scale-dependent. (In this book, unless specified otherwise, we always use the so-calleddimensional regularization and modified minimal subtraction, or in short MS quark masses.)

According to the standard model, the masses of the quarks are generated through a symmetrybreaking phase transition of the electroweak interactions (a transition similar to that of a normalconductor to superconductor in condensed matter physics, in which an effective mass for the photonis produced). The detailed aspects of the symmetry breaking, such as the existence of Higgs bosons,are still under investigation in experiments at high-energy colliders. Our present knowledge of thequark masses is shown in Table 1.1

Quark Flavor: up down charm strange top bottom

Mass: 1.5-4 MeV 4-8 MeV 1.25 GeV ∼ 100 MeV 175 GeV 4.25 GeV

Charge: 2/3 −1/3 2/3 −1/3 2/3 −1/3

Table 1.1: Quark masses in the MS renormalization scheme at a scale of µ = 2 GeV.

Color Charge

The quarks participate in strong interactions because they carry color charges. The color chargesare the analogs of electric charge in Quantum Electrodynamics, but with important differences.Unlike electric charge, which is a scalar quantity in the sense that the total charge of an electricsystem is simply the algebraic sum of individual charges, the color charge is a quantum vector

charge, a concept similar to angular momentum in quantum mechanics. The total color charge of

1.2. GLUONS AND QUANTUM CHROMODYNAMICS 3

a system must be obtained by combining the individual charges of the constituents according togroup theoretic rules analogous to those for combining angular momenta in quantum mechanics.

The quarks have three basic color-charge states, which can be labeled as i = 1, 2, 3, or red, green,and blue, mimicking three fundamental colors. Three color states form a basis in a 3-dimensionalcomplex vector space. A general color state of a quark is then a vector in this space. The color statecan be rotated by 3 × 3 unitary matrices. All such unitary transformations with unit determinantform a Lie group SU(3). The 3-dimensional color space forms a fundamental representation ofSU(3). It is customary to label the color charges by the spaces of the SU(3) representations, 3 inthe case of a quark. The rules of adding together color charges follow those of adding representation

spaces of the SU(3) group, which are straightforward extensions of adding up angular momenta inquantum mechanics. Some examples of adding up quark charges can be found in in the Appendixof this chapter.

The quarks, like the electron, have anti-particles, called antiquarks, often denoted by q. Theantiquarks have the same spin and mass as the quarks, but with opposite electric charges. Forexample, an anti-up quark has an electric charge −2/3 of the proton charge. The color charge of anantiquark is denoted 3, which is a representation space of SU(3) where the vectors are transformedaccording to the complex conjugate of an SU(3) matrix.

One can more generally view quark confinement as color confinement: strong interactions donot allow states other than color-singlet, or color-neutral to appear in nature. There is strongevidence for color confinement in numerical simulations of QCD, known as lattice QCD calculations.However, there is no rigorous mathematical proof in the literature to date.

1.2 Gluons and Quantum Chromodynamics

Quarks do not interact with each other directly; they do so through intermediate agents calledgluons. A simple way to understand this is that the gluons in strong interactions play the role ofphotons in quantum electrodynamics (QED), which mediate electromagnetic interactions betweencharged currents. Like photons, gluons are massless, spin-1 particles with two polarization states(left-handed and right-handed). They are represented by a four-component vector potential Aµ(x)with a Lorentz index µ = 0, 1, 2, 3, just as in electromagnetism. Therefore, conditions must beimposed on Aµ(x) to select only the physical degrees of freedom, as different Aµ(x) can give riseto the same physics. These conditions are called gauge conditions or gauge choices. For the samereason, Aµ(x) are called gauge potentials, and gluons are called gauge particles.

Although there is only one type of photon mediating electromagnetic interactions, there are 8types of gluons mediating the strong interactions. This proliferation of gauge particles has to dowith the SU(3) color symmetry: Return to the free quark lagrangian density in Eq. (1.2), wherethe quark field is now endowed with a new index i for color. It is easy to see that L is invariantunder a global SU(3) transformation

ψ′(x) = Uψ(x) , (1.4)

where U is a 3 × 3 unitary matrix acting on color index, and“global” means that the field atdifferent spacetime is transformed in exactly the same way. A generic SU(3) matrix requires 8 realparameters, usually written in the form

U = exp(i∑

a

θaλa/2) , (1.5)


where λa/2 (a = 1, ..., 8) are 3 × 3 hermitian matrices and are the so-called generators of SU(3)rotations, just like the angular momentum operators Li being generators of ordinary space rotations.A commonly used definition of these matrices is due to Gell-Mann (and hence named after him)and can be found in the Appendix.

If we introduce 8 gluon potentials, Aµa , as well as the associated covariant derivative, Dµ ≡

∂µ + igAµaλ

a/2, the free quark lagrangian can be modified to

Lq = ψ(i 6D −mq)ψ . (1.6)

The new lagrangian has a remarkable symmetry called gauge symmetry: It is invariant undera spacetime-dependent SU(3) rotation U(x) of the quark fields if, at the same time, the gaugepotentials transform according to

Aµ → U(Aµ +i

g∂µ)U † , (1.7)

where we have introduced the 3×3 gluon potential matrix Aµ = Aµaλ

a/2. The above transformationis a generalization of gauge transformations in classical electromagnetism, and reflects that thenumber of physical degrees of freedom associated with each gauge potential is just 2, those ofmassless spin-1 particles!

i

j

ataij

Figure 1.1: The color of a quark can change from i to j by a gluon of color a, coupled throughSU(3) generator taij = λa

ij/2.

Therefore, by requiring a local color (gauge) symmetry, we find that the quarks are no longerfree particles, but they interact with each other through the new gauge potentials. In other words,the gauge symmetry has generated a well-defined dynamics of the color charges. The 8 gluons aresimply related to the 8 parameters of a general SU(3) transformation!

Let us examine these color interactions in some details. The interaction term in the lagrangianis

Lint = −gψjAµa

λaji

2γµψi , (1.8)

where g is the coupling constant. Therefore, quarks interact with gluons in a way similar to electronsinteracting with photons. A new feature here is that the quark can change its color from i to jby emitting or absorbing a gluon of color a, coupling through an SU(3) generator taij = λa

ij/2, asshown in Fig. 1.1.

Gluons are physical degrees of freedom and therefore must carry energy and momentum them-selves. Thus one must add additional terms in the lagrangian to describe these physical features.

1.3. FORCE BETWEEN QUARKS, ASYMPTOTIC FREEDOM 5

Following the successful theory of Maxwell on electromagnetism, we introduce in a similar way theantisymmetric field strength tensor F a

µν and the kinetic energy term for the gluons

Lg = −1

4FµνaFµνa (1.9)

where summation over color is implicit. To ensure the new term added is gauge-invariant, the fieldstrength tensor is required to transform according to Fµν → UFµνU †. This can be accomplishedby choosing Fµν = −ig[Dµ,Dν ], or

Fµν = ∂µAν − ∂νAµ − g[Aµ, Aν ] , (1.10)

where apart from the standard partial derivative terms, there is a commutator term [Aµ, Aν ]. Thisis non-linear in terms of the gauge potential. Therefore, the eight gluons do not come in as a simplerepetition of the photon in QED—they have three- and four-gluon self-interactions! We depict suchinteraction terms schematically in Fig. 1.2.

Figure 1.2: Self-interactions of gluons which are responsible for many important features of QCD,such as asymptotic freedom, color confinement and chiral symmetry breaking.

The full QCD lagrangian density is the sum of quark and gluon terms,

LQCD = Lq + Lg . (1.11)

Because the gluons carry color charges, their self-interactions are a source of the key differences be-tween QCD and QED. In fact, these interactions are responsible for many of the unique and salientfeatures of QCD, such as asymptotic freedom, chiral symmetry breaking, and color confinement.In the remainder of this chapter, we briefly discuss these important properties of color dynamics.

1.3 Force Between Quarks, Asymptotic Freedom

A first exercise in studying electromagnetic physics is to consider the force between two electriccharges. Similarly, a starting point to understand strong interactions is to consider the strong forcebetween two quarks. Of course, we forget for the moment the fact that the free quarks do not existand the interaction between quarks may not be calculable through simple one-gluon exchange. Wepress ahead anyway just to see what follows.

As shown by Feynman diagram in Fig. 1.3, a quark of color i, exchanging a gluon with anotherquark of color j, scatters into a quark of color i′, along with a quark of color j′. The scatteringamplitude for this process is

S ∼ (−igtai′iγµ)Dµν(q)(−igtaj′jγν) , (1.12)


where we have taken into account all 8 gluon exchanges by summing a = 1, ..., 8. The four-momentum q of the gluon is space-like, q2 = q20 − ~q2 < 0. Dµν(q) is the gluon propagator inFeynman gauge (a particularly convenient gauge choice!),

Dµν(q) =−igµν

q2. (1.13)

The only difference between the above amplitude and that for two electrons is the color factortai′it

aj′j.

i

i’

j

j’

Figure 1.3: Strong interactions of two quarks through one-gluon exchange.

Let us calculate the color factors for the two quarks in 6 and 3, respectively. If two quarksare in 6, for example, we can take a specific case i = j = i′ = j′ = 1. By summing over a, theaverage color factor is 1/3. Thus the interaction between the quarks in the symmetric color state isrepulsive. This sign is analogous to the electric force. On the other hand, if the two quarks are in3, the color factor is −2/3, and the force between them is attractive! Therefore we conclude thatthe force between two quarks depend on the color states! For a more detailed discussion on this,see Halzen and Martin.

Exercise: work out the sign of color force between a quark and antiquark when they are in 1and 8, respectively.

Asymptotic Freedom

One of the striking properties of QCD is asymptotic freedom which states that the interactionstrength αs = g2/4π between quarks becomes smaller as the distance between them gets shorter.For this observation, Gross, Politzer and Wilczek won 2004 Nobel prize in physics.

To explain asymptotic freedom, let us first recall that in electrodynamics, the force betweentwo charges q1 and q2 in vacuum is described by Coulomb’s law,

F =1

4π

q1q2r2

. (1.14)

If, on the other hand, the two charges are submerged in a medium with dieletric constant ǫ > 1(charge screening), the force becomes

F =1

4πǫ

q1q2r2

, (1.15)

1.3. FORCE BETWEEN QUARKS, ASYMPTOTIC FREEDOM 7

which also can be expressed in the above vacuum form by introducing the effective charge qi =qi/

√ǫ. Thus the effect of the medium may be regarded as modifying the charges.

In the quantum field theory, the vacuum is not empty because it is just the lowest energy state ofa field system and is filled with electrons of negative energies from one point of view. When a photonpasses through the vacuum, it can induce transitions of an electron from negative to positive energystates, virtually creating a pair of electron and positron, known as vacuum fluctuation, shown inthe first diagram in Fig. 1.4. Because of this, the interaction between two electrons in the vacuumbecomes

F =e2eff4πr2

=αem(r)

r2, (1.16)

where αem is an effective fine structure constant, depending on the distance r, or momentumtransfer q ∼ 1/r. As r → ∞ or equivalently, q → 0, the coupling measured the interaction strengthof the low-energy photon, and is what often quoted, αem(q = 0) = 1/137.035.

The dependence of the fine structure constant on the distance or momentum scale can bedetermined by a differential equation in QED

µdα(µ)

dµ= β(α(µ)) , (1.17)

where µ is a momentum scale, roughly corresponding to 1/r. The above is also an example ofrenormalization group equations. The β-function may be calculated in perturbation theory becauseαem is small and at one loop order β = 2α2

em/3π > 0. The solution is thus

αem(µ) =αem(µ0)

1 − αem(µ0)3π ln µ2

µ2

0

. (1.18)

Note that the interaction strength of the two electrons gets stronger as the distance between thembecomes smaller. Therefore, QED becomes a strongly-coupled theory at very short distance scale.For this reason, it cannot be solved in a completely consistent way unless an ultraviolet momentumcut-off is imposed on the theory.

Figure 1.4: The quantum vacuum polarization which effectively changes the interactions strength.The first diagram is shared by QED and QCD which renders the interaction stronger at shorterdistance (screening). The second diagram arising from the nonlinear interaction between gluons inQCD has the antiscreening effect, which makes the coupling weaker at short distance.

In QCD, the same differential equation for the strong coupling constant holds. However, the βfunction is now different

β(α) = −β0

2πα2 + ... (1.19)


where β0 = 11 − 23nf with nf is the number of active quark flavor. The second term in β0,

−23nf , comes from quark-antiquark pair effect in the first diagram in Fig. 1.4. It scales like the

number of quark flavors and is negative (as it would be in QED). However, the first term, 11, hasthe opposite sign and comes from the non-linear gluon contribution shown in the second diagram(these are absent for QED). Thus the gluon self-coupling has an anti-screening effect! From therenormalization group equation, the coupling constant of QCD can be shown to have the followingscale-dependence

αs(µ) =2π

β0 ln(µ/ΛQCD), (1.20)

which goes to 0 as the momentum scale µ → ∞ or the distance approaching 0! This strangebehavior of the strong coupling has been verified in high-energy experiments to very high precision,as shown in Fig. 1.5, where Q stands for the running scale. The integration constant ΛQCD is anintrinsic QCD scale, discussed below.

Figure 1.5: The running of the QCD coupling constant as a function of momentum transfer,experimental data vs. theoretical prediction.

Because of asymptotic freedom, the strong interaction physics can now be calculated in pertur-bation theory when the momentum transfer is large. In particular, the one-gluon exchange diagram(Fig. 1.3) becomes a good approximation for quark interaction as q2 ≫ Λ2

QCD. At present, themost accomplished result of QCD research is in the perturbative region, where many experimentaldata have been explained well by perturbative QCD (pQCD). One must emphasize, however, theforce between quarks does not get weaker at the shorter distance, despite the fact that the couplingdoes. In fact, the force still grows at short distance in an asymptotic free theory!

1.4. CHIRAL SYMMETRY AND ITS SPONTANEOUS BREAKING 9

Strong Interaction Scale ΛQCD

The running coupling introduces a dimensional parameter ΛQCD, which sets the scale at which thecoupling constant becomes large and the physics becomes nonperturbative. In fact, ΛQCD simplysets the scale for strong interaction physics. In the MS-scheme with 3 quark flavors,

ΛQCD ∼ 250 MeV . (1.21)

All dimensionful QCD results without external momenta scale with this parameter when quarkmasses are negligible. For instance, it is largely responsible for the mass scale of protons andneutrons, and hence the mass scale of the baryonic mass in the Universe. On the other hand, pureQCD predictions with massless quarks are independent of ΛQCD and hence are pure numbers.

1.4 Chiral Symmetry and Its Spontaneous Breaking

In the last section, we have introduced the strong interaction scale ΛQCD through the running QCDcoupling αs. With this, we can introduce the notion of light and heavy quarks. Briefly speaking,light quarks are ones having masses much smaller than ΛQCD, and heavy quarks having massesmuch larger than that. Clearly, the up and down flavors are qualified for light quarks, whereas thecharm, bottom, and top may be regarded as heavy. The strange quark is an exception, it appearsneither light nor heavy, and is more difficult to deal with theoretically. In some cases, it can beregarded as light, in others it is closer to being heavy.

Here we focus on the light up and down quarks, which are perhaps the most relevant to thereal world. To understand the physics of the light quarks, it is convenient to consider a theoreticallimit in which their masses are exactly zero. We argue later that the physics of the real world isnot so different from this theoretical limit.

Chiral Symmetry

If a quark has a zero mass, then the spin of the quark can either be in the direction of motion, inwhich case, we call it a right-handed quark, or in the opposite direction, in which case, we call it aleft-handed quark. Since a massless quark travels at the speed of light, the handedness or chiralityof the quark is independent of any Lorentz frame from which the observation is made.

The chirality can be selected by the Dirac matrix γ5 because the free hamiltonian commuteswith it. We then define a projection operator P± ≡ (1± γ5)/2 to project out left and right-handed(chirality) quarks,

ψLf =1

2(1 − γ5)ψf ,

ψRf =1

2(1 + γ5)ψf . (1.22)

where f labels different flavors. The total quark field is simply a linear combination of the two,

ψ = ψL + ψR . (1.23)

If one plugs the above decomposition into the QCD lagrangian without the mass term, the quarkpart splits into two terms

Lq = Lq(ψL) + Lq(ψR) , (1.24)


with each depending on either the left-handed, or the right-handed field, but not both. In otherwords, the QCD interaction does not couple the left and right-handed quarks which appear to livein separate worlds.

Because the up and down quarks are both massless, they may be regarded as two independentstates of the same object forming a two component spinor in “isospin space,” in the same way asthe ±1/2 projection states of a spin-1/2 object. The lagrangian density is symmetric under theindependent rotations of ψL and ψR in the left and right isospaces. More explicitly, under

(

u′L,R

d′L,R

)

= UL,R

(

uL,R

dL,R

)

(1.25)

where UL,R are 2 × 2 unitary matrices, the quark part of the lagrangian is invariant. We then saythe QCD lagrangian has a chiral symmetry U(2)L × U(2)R.

Since a 2 unitary matrix can be decomposed into product of a phase and a special unitarymatrix with unit determinant, the chiral symmetry can be decomposed into

U(2)L × U(2)R = SU(2)L × SU(2)R × UL(1) × UR(1) . (1.26)

The symmetry SU(2)L × SU(2)R has not been seen explicitly in particle spectrum or scatteringmatrix element, in the same way as the rotational symmetry in 3-dimensional motion of a quantumparticle. Rather, it is a “hidden” symmetry, in the sense that it breaks spontaneously into theso-called isospin subgroup SU(2)L+R which corresponds to rotations UL = UR. Isospin symmetryis manifest in particle spectrum and scattering, and will be discussed further in Chapter 3.

A simple example of spontaneous symmetry breaking

Spontaneous symmetry breaking is a common phenomenon in physics: it happens in ordinaryquantum mechanics and in many cases in condensed matter physics such as the spontaneous mag-netization of a ferromagnet. A simple quantum mechanical example is shown here to illustrate whatis going on. Consider a particle moving in a one-dimensional symmetric potential shown in Fig.1.6.The hamiltonian is symmetric under x→ −x, or parity transformation. Because of this symmetry,the wave functions of the eigenstates can be taken to be either symmetric and antisymmetric underparity. In particular, the ground state of the wave function |0〉 is always symmetric,

P |0〉 = |0〉 , (1.27)

because the kinetic energy is minimized this way.As the barrier in the middle gets higher, an interesting feature in the spectrum develops: the

energy levels become clustered as doublets. The wave function of the lower level in each doublet issymmetric and that of the upper one is antisymmetric. The difference of the energy in the doubletis exponentially small as a function of the barrier height. In fact, as the barrier in the middle goesto infinity, all energy levels are exactly two-fold degenerate, including the ground state. Physically,this must be the case, because we effectively have two identical infinite square wells.

Thus, there are an infinite number of ways to choose the ground wave function when thedegeneracy arises. However, some of the choices are the most natural, i.e., we take as the groundstate when the particle is either in the left well, or in the right well. Placing a particle in bothwells simultaneously is theoretically possible, but hard to realize in practice. Although the two

1.4. CHIRAL SYMMETRY AND ITS SPONTANEOUS BREAKING 11

0 x

V(x)

Figure 1.6: A quantum mechanical particle moving in a symmetric potential well with a finitebarrier in the middle. The ground state wave function is left-right symmetric. However, whenthe barrier height approaches infinity, the ground state becomes doubly degenerate and a naturalchoice for the ground state wave function no longer has left-right symmetry.

choices are equally natural, once a choice is made, the ground state is unique. The other state isentirely decoupled because no operator in the Hilbert space can create a transition to that fromany states built from the ground state (super-selection rule). If one accepts this, the ground statewave function no longer have a left-right symmetry under x → −x. The symmetry is said to havebeen spontaneously broken.

A well-known example of spontaneous symmetry breaking is the magnetization of a magneticmaterial below a critical temperature Tc. Above Tc, the material has no net magnetic moment andis rotationally symmetric as the underlying hamiltonian. Below Tc, it develops a spontaneous mag-netization which breaks the SO(3) rotational symmetry. All possible magnetization directions arephysically equivalent, corresponding to different degenerate vacua. Once a direction is chosen, thesystem is still rotationally-invariant around the magnetization direction, or equivalently, symmetricunder the subgroup SO(2).

Spontaneous Breaking of Chiral Symmetry and Pion as a Goldstone boson

The chiral symmetry is a continuous symmetry in the sense that it contains infinitely many trans-formations which are smoothly connected to one another. In the case of spontaneous breaking of acontinuous symmetry, there are usual infinitely many degenerate vacua, which give rise to masslessexcitations referred to as Goldsone bosons. In the case of magnetization, the Goldstone bosons isthe massless spin wave.

A simple way to understand the chiral symmetry breaking is to consider the dynamics of thescalar fields φi

L =1

2∂µφi∂µφi − V (φ) . (1.28)

If V (φ) is invariant under 4-dimensional rotation φi → Lijφj, the system has an O(4) ∼ SO(3) ×SO(3) symmetry. Taking as a simple assumption,

V (φ) = −1

2µ2φ2

i −1

4λ(φ2

i )2 (1.29)


Figure 1.7: The scalar field potential when spontaneous symmetry breaking occurs. The minimumof the potential is now a hypersurface, which is not invariant under SO(4) rotation.

The SO(4) symmetry is spontaneously broken to SO(3) when µ2 ≤ 0. Indeed as shown in Fig. 1.7,the minimal of the potential corresponds to

φ2i = c2 =

√

−λ/µ2 , (1.30)

which has infinite number of solutions, corresponding to an infinite number of vacua, all obtainedfrom each other by some O(4) rotations. The physical vacuum can be taken to be one of theinfinite possible solutions. Once a solution is chosen, for example, φi = (0, 0, 0, c), this vacuum isstill invariant under the SO(3) rotations of the first three dimensions. This corresponds to breakingof symmetry from SO(4) to SO(3), analogue of the isospin group.

Imagine we start with a common vacuum at every point of space and time, and make a rotationdifferently at different points of spacetime. The energy of the new state must be proportional tothe derivative of the rotations in space, because a common rotation generates no energy. As thederivative vanishes, the energy must go to zero as well. Therefore, the excitation associated withthe rotation has no mass. This is the Goldstone boson! In case of a magnetic, the Goldstone bosoncorresponds to massless spin waves.

Since in the case of the chiral symmetry, there are three independent ways to make rotationsaway from a chosen vacuum, just like the scalar field discussed above, we have three possibleGoldstone bosons. These have been identified as negative parity pions π±, 0(x), first observed in1950’s. The fact that chiral symmetry is not exact (mu 6= md 6= 0, see Table 1.1) leads to asmall (relative to ΛQCD) pion mass mπ ∼ 140 MeV. Because of the low-energy scale involved, thedynamics of the pions can be described by an effective theory.

1.5 Color Confinement

One of the prominent features of QCD at low-energy is color-confinement: Any strongly interaction

system at zero temperature and density must be a color singlet at distance scale larger than 1/ΛQCD.As a consequence, isolated free quarks cannot exist in nature (quark confinement). The colorconfinement of QCD is a theoretical conjecture consistent with experimental facts. To prove it inQCD is still a challenge that has not been met.

Suppose, for example, we have a quark-antiquark pair which is in a color singlet state. One maytry to separate the quark from the antiquark by pulling them apart. The interaction between the

1.5. COLOR CONFINEMENT 13

quarks gets stronger as the distance between them gets larger, similar to what happens in a spring.In fact, when a spring is stretched beyond the elastic limit, it breaks to produce two springs. Inthe case of the quark pair, a new quark-antiquark pair will be created when pulled beyond certaindistance. Part of the stretching energy goes into the creation of the new pair, and as a consequence,one cannot have quarks as free particles.

-4

-3

-2

-1

0

1

2

3

0.5 1 1.5 2 2.5 3

[V(r

)-V

(r0)

] r0

r/r0

β = 6.0β = 6.2β = 6.4Cornell

Figure 1.8: Left: potential between a pair of heavy quark-antiquark when there are no light quarksin the QCD vacuum. Right: the gluon flux tube between a pair of quark-antiquark produced by theso-called dual Meissner effect. The flux tube is believed to be responsible for the linear confinementpotential.

The above discussion is in some sense a sort of self-consistent speculation. To understand reallywhat happens, one must make calculations in QCD at large distance scales where, according tothe renormalization group equation, the coupling becomes very strong. Such calculation is verydifficult at present time. The only way we know how to solve QCD in the strong coupling regimeis to simulate the theory on a finite spacetime lattice, or in short lattice QCD. One way to findthe potential energy when the quarks are pull apart is to calculate quark-antiquark potential ina world where there is no quark-antiquark pair creation (quenched QCD). QCD simulations onlattice shows that the quark potential does increase linearly beyond the distance of a few fermis,as shown on the left panel in Fig. 1.8.

The origin of the linear potential between quarks may be traced to the so-called flux tube: astring of gluon energy density between the quark pair, shown on the right panel in Fig. 1.8. In QED,the electric lines between positive and negative charges spread all over the space, generating a 1/rpotential. In QCD, however, the vacuum acts like a dual superconductor which squeezes the colorelectric field to a minimal geometrical configuration—a narrow tube. [In a normal superconductor,the magnetic flux is expelled from the interior of the metal, as shown in the Meissner effec.] Itcosts energy for the flux to spread out in space. The tube roughly has a constant cross section andwith constant energy density. Because of this, the energy stored in the flux increases linearly withthe length of the flux. The flux tube has been seen in numerical simulations. If one calculates thegluon energy in lattice QCD, one can see the flux tube from the the plot of energy density in space.


1.6 Hadrons and Nuclei

It is rather odd that in this introductory chapter, we spent so much space to talk about thesomething we don’t usually “see” in low-energy experiments, quarks and gluons. What we usuallyobserved in experimental apparatus are hadrons and nuclei which are bound states of these basicbuilding blocks.

As we will discussed in the future chapters, a meson is made of the quantum number of a quarkand anti-quark pair. For example, the pion has the quantum number of uu − dd, ud, and du,naturally color singlets. On the other hand, a baryon is made from the quantum numbers of 3quarks, which can form a color singlet because the SU(3) group multiplication rule says

3× 3× 3 = 1 + 8 + 8 + 10 (1.31)

where 1 represents color singlet. For example, a proton is made of two up quarks and one downquark. There are other baryons such as Λ, Ω and the excited states of these particles as well.Baryons and mesons together are called hadrons, the bounds states of strong interactions.

The lowest mass baryons are neutrons and protons, which are together called nucleons. Protonsare stable, whereas neutrons have a lifetime about 10 min. There are attractive interactions betweenprotons and neutrons, which are the residual color interactions, just like the Van der Waals forcesbetween neutral atoms and molecules. These nuclear forces are responsible for binding the nucleonstogether to form atomic nuclei, and the origin of atomic energy.

We will discuss extensively in this course the structures and interactions of hadrons and nuclei.

1.7 Electroweak Interactions

Weak interactions are mediated throughW and Z vector bosons which play a similar role to photonsand gluons in QED and QCD, respectively. In particular, Z boson is flavor-conserving and chargeneutral, and therefore is very much like the photon. However, these weak gauge bosons are quitemassive, 80.4 GeV/c2 for W and 91.2 GeV/c2for Z, and mediate a force that acts on a very shortrange with an extremely weak strength. The W bosons are charged and do change flavor. Forinstance, the emission of a W+ changes an up quark into a down quark. It also changes an electroninto an electron neutrino, a muon into a muon neutrino.

The electromagnetic and strong interactions involve just the vector current jµ = V µ = ψγµψand hence conserve parity. However, as was discovered in 1957 by Lee, Yang, and Wu, the weakinteraction involves parity violation, and the axial vector current Aµ = jµ5 = ψγµγ5ψ also par-ticipates. Under a parity transformation ~r → −~r, the vector current transforms as V µ → −V µ

whereas Aµ → Aµ. Analogous to vector currents, we can define nine axial currents in terms of theflavor structure of the light up, down and strange quarks,

Aµa = ψγµγ5λa

2ψ , (1.32)

which will be useful later. The singlet axial current (a = 0) is not really conserved due to theso-called UA(1) anomaly.

A more formal discussion of weak interactions starts with definition of weak charges. The weakcharge assignment is related to the chiral properties of the quarks introduced in a previous section.

1.7. ELECTROWEAK INTERACTIONS 15

The standard model assumes the left-handed up and down quarks form a two-component vector,

q1L =

(

uL

dL

)

(1.33)

transforming under SU(2)L (weak isospin) as a doublet, whereas uR and dR transform as singlets.The superscript 1 here refers to the first generation. The pattern repeats for charm and strangequarks, the second generation, top and bottom the third. In addition, the quarks carry UY (1) weak(scalar) hypercharge Y , defined through

Q = I3 +Y

2, (1.34)

where I3 = T3 is the third component of the weak SU(2)L isospin and Q the electric charge.The free quark and lepton lagrangian is invariant under the above weak isospin and hyperchargetransformation.

To make the SU(2)L × UY (1) symmetry local, we introduce gauge fields W µi for SU(2)L and

Bµ for UY (1). The covariant derivative for the weak doublet is

Dµ = ∂µ + ig~τ

2· ~W µ + ig′

Y

2·Bµ . (1.35)

After spontaneous breaking of SU(2)L ×UY (1) through a Higgs doublet H = (φ+, φ0), only Uem(1)remains. The charge neutral gauge bosons W 0 and B will mix to give massless photon field Aµ

and massive weak gauge boson field Zµ

Zµ =gW µ

3 − g′Bµ

√

g2 + g′2= cos θWW µ

3 − sin θWBµ (1.36)

Aµ =g′W µ

3 + gBµ

√

g2 + g′2

= sin θWW µ3 + cos θWBµ , (1.37)

where we have introduced the Weinberg mixing angle θW . Since the photon field Aµ mediates theelectromagnetic interaction, the couplings are related by

e = g sin θW = g′ cos θW . (1.38)

With the above definition, the coupling of Z-boson to quarks and leptons is

LZint = − g

cos θWψiγ

µ(giV − gi

Aγ5)ψiZµ (1.39)

where the vector and axial coupling constants are giV = I3 − 2qi sin2 θW and V i

A = I3, respec-tively. Experimentally (in the MS-scheme and at the scale MZ) the measured value is sin2 θW =0.23120(15).

In the standard model with 3 light quarks, the W-boson interaction current is

JµW /2 = cos θC uLγ

µdL + sin θC uLγµsL + ... , (1.40)

where θC = 15 is called the Cabibbo angle. The interaction violates the parity maximally, it is aV −A from, ψγµ(1− γ5)ψ. Therefore, the quarks of the first generation can be converted to those


of the second generation through W -bosons! This is called flavor mixing. In fact, the most generalflavor mixing in the standard model with six quarks goes as

JµW /2 = uLγ

µd′L + cLγµs′L + tLγ

µb′L , (1.41)

where the primed quarks (flavor eigenstates) are

d′

s′

b′

=

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

dsb

(1.42)

where the 3×3 matrix is a unitary matrix (Cabbibo-Kobayashi-Maskawa mixing matrix) satisfyingthe unitarity condition, for example,

|Vud|2 + |Vus|2 + |Vub|2 = 1 . (1.43)

The unitarity matrix can be parameterized in terms of three mixing angles θ12, θ13, θ23 and a phaseδ, four independent parameters. The standard parametrization is

V =

c12c13 s12c13 s13e−iδ

−s12c23 − c12s23s13eiδ c12c23 − s12s23s13e

iδ s23c13s12s23 − c12c23s13e

iδ −c12s23 − s12c23s13eiδ c23c13

(1.44)

where cij = cos θij and sij = sin θij . An alternative parametrization exhibiting the heirachicalstructure of the mixing element is

V =

1 − λ2/ λ Aλ3(ρ− iη)−λ 1 − λ2/2 Aλ2

Aλ3(1 − ρ− iη) −Aλ2 1

+ O(λ4) (1.45)

which was advocated by Wolfenstein.

The interaction between the W boson and the charge-changing current is

L = − g

2√

2W+µJµW + h.c. (1.46)

If one integrates out the W -boson, one obtain the effective weak interaction in a form which waswritten down by Fermi over 50 years ago,

HW =GF√

2Jµ

WJWµ (1.47)

where GF = 1.16637×10−5 GeV−2 is called Fermi-decay constant. The relation to the Fermi-decayconstant GF and the weak coupling g is, g2/8m2

W = GF /√

2.

1.8. ***BACKGROUND INFORMATION FOR CHAPTER 1*** 17

1.8 ***Background Information for Chapter 1***

1.8.1 SU(3) group and Gell-Mann matrices

Consider a 3-dimensional complex vector space, in which a vector ψ is represented by a columnmatrix of 3 complex numbers,

ψ =

ψ1

ψ2

ψ3

. (1.48)

A “rotation” in this space can be accomplished by a 3 × 3 unitary matrix U ,

ψ′i = U j

i ψj (1.49)

where the unitary condition U †U = UU † = 1 guarantees that the new vector ψ′ has the same norm,defined as

∑

i ψ∗i ψi, as the original one.

All 3×3 unitary matrices form a group U(3) under the matrix multiplication. The determinantof U has a unit norm, |DetU | = 1, and hence DetU = eiθ. The subset of U(3) rotations withDetU = 1 form a subgroup, called SU(3). The conditions U †U = UU † = 1 and DetU = 1 leaveU depending on 8 real parameters, which we will call θa (a = 1, 2, ..., 8). We can choose theseparameters such that when θa = 0, U reduces to the unit matrix I. When θa are small, we canexpand U at θa = 0 to get,

U = I + i8∑

a=1

θata + ... (1.50)

where ta are 3 × 3 hermitian matrices, which are called the generators of the SU(3) group, in thesense that the other group element may be constructed from these generators. Clearly, the choiceof ta depends on the choice of parameters.

A convenient choice of the SU(3) generators ta = λa/2 by Gell-Mann is

λ1 =

0 1 01 0 00 0 0

; λ2 =

0 −i 0i 0 00 0 0

; λ3 =

1 0 00 −1 00 0 0

;

λ4 =

0 0 10 0 01 0 0

; λ5 =

0 0 −i0 0 01 0 0

; λ6 =

0 0 00 0 10 1 0

;

λ7 =

0 0 00 0 −i0 i 0

; λ8 =

1√3

1 0 00 1 00 0 −2

; λ0 =

√

2

3

1 0 00 1 00 0 1

; (1.51)

The generators satisfy the SU(3) commutation relations (SU(3) algebra),

[

λa

2,λb

2

]

= ifabcλc

2, (1.52)

where fabc is totally antisymmetric structure constants, the numerical values of which can be foundin many textbooks (for example, Itzykson and Zuber).


The vector ψi forms a 3-dimensional fundamental representation of SU(3), and labelled by 3.All the complex conjugate matrices U∗ form another representation of SU(3), which is called 3.Clearly ψ∗

i space is a 3.

Consider adding color charges of two quarks symbolically denoted as qi and qi. The easiest wayto accomplish the addition is by the so-called tensor method. Multiplying the two 3-dimensionalspaces together, we get a 9-dimensional space spanned by basis qiqj. The ij symmetric part,

1

2(qiqj + qj qi) , (1.53)

forms a 6-dimensional subspace, 6. The ij antisymmetry part,

1

2(qiqj − qj qi) , (1.54)

form a 3-dimensional subspace. This, however, is not the same as the 3D color space of the originalquark, because under an SU(3) transformation the basis states in this new 3D space transformlike complex conjugate of the original quarks. For this reason, we have a new charge state 3. Toconclude, when we add two-triplets of color charges, we get a 6 and a 3, or simply 3× 3 = 6 + 3.

What if one adds together the color charges of a quark and antiquark? The combinationq1q1 + q2q2 + q3q3, is invariant under SU(3) transformation and is a color singlet denoted by 1. Acolor-singlet state is also color-natural, or white. The combinations qiqj with the color-neutral statesubtracted form an 8-dimensional representation of the SU(3) group, 8. Therefore, 3 × 3 = 1 + 8.

More complicated multiplications of the SU(3) representations can be obtained by the so-calledtensor method, for example

3× 3× 3 = 1 + 8 + 8 + 10 (1.55)

can be obtained this way.

1.8.2 Lagrangian Density, QED and Feynman Rules

The equations of motion for physical variables can be derived from the lagrange principle. Inclassical physics, a lagrangian can be constructed as the difference between kinetic energy T andpotential energy V . The action is the time integral of the lagrangian,

S =

∫

dtL . (1.56)

If a lagrangian depends on a coordinate q, the minimization of the action leads to the followingEuler-Lagrange equation,

d

dt

∂L

∂q− ∂L

∂q= 0 , (1.57)

where q is the time-derivative.

QED lagrangian density is constructed out of the electron Dirac field ψ and phone vector fieldAµ ,

L = ψ(i 6D −m)ψ − 1

4FµνF

µν , (1.58)

1.9. PROBLEM SET 19

where D is a covariant derivative Dµ = ∂µ + ieAµ, and e is the electric charge. The Euler-Lagrangeequations for the electron and photon fields are

(i 6D −m)ψ = 0 (1.59)

∂µFµν = eJν , (1.60)

where the first one is the Dirac equation for electron in an electromagnetic field, and the secondone is the Maxwell equation with the electron vector current Jµ = ψγµψ.

The Feynman rule for the electron propagator with four momentum k is

i

6k −m+ iǫ. (1.61)

The photon propagator depends on the choice of gauge. Because not all degrees of freedom of Aµ

are physical, additional constraint must be imposed to make the physical photon propagation finite.The most frequently used gauge is Feynman gauge in which the photon propagator is simply,

−igµν

k2 + iǫ. (1.62)

The electron photon interaction vertex can be read off directly from the above lagrangian density,

−ieγµ (1.63)

Using the above Feynman rules, one can calculate any QED processes.

1.9 Problem Set

1. Work out the product of 3 × 3× 3 in term of irreducible representations of SU(3).2. Why the antisymmetric part of 3× 3 corresponds to a 3?3. Show that (1.6) is gauge-invariant and Fµν transforms like UFµνU

† under SU(3) gauge trans-formation.4. Show that the color factor for two quarks in 6 is 1/3, and that in 3 is −2/3.5. Solve the renormalization group equation for the QCD coupling to get the formula in Eq. (20).6. In the SO(4) model for spontaneous symmetry breaking, identify the Goldstone boson degreesof freedom as well as the group under which the physical vacuum is invariant.

Chapter 2

Quark-Gluon Plasma and the EarlyUniverse

There is now considerable evidence that the universe began as a fireball, the so called “Big-Bang”,with extremely high temperature and high energy density. At early enough times, the temperaturewas certainly high enough (T > 100 GeV) that all the known particles (including quarks, leptons,gluons, photons, Higgs bosons, W and Z) were extremely relativistic. Even the “strongly inter-acting” particles, quarks and gluons, would interact fairly weakly due to asymptotic freedom andperturbation theory should be sufficient to describe them. Thus this was a system of hot, weaklyinteracting color-charged particles, a quark-gluon plasma (QGP), in equilibrium with the otherspecies.

Due to asymptotic freedom, at sufficiently high temperature the quark-gluon plasma can bewell-described using statistical mechanics as a free relativistic parton gas. In this Chapter, weexplore the physics of QGP, perhaps the simplest system of strong-interaction particles that existsin the context of QCD. As the universe cooled during the subsequent expansion phase, the quarks,antiquarks, and gluons combined to form hadrons resulting in the baryonic matter that we observetoday. The transition from quarks and gluons to baryons is a fascinating subject that has beendifficult to address quantitatively. However, we will discuss this transition by considering the basicphysics issues without treating the quantitative details. At present there is a substantial effort intheoretical physics to address this transition by using high-level computational methods known aslattice gauge theory. This subject is somewhat technical and we will discuss it only very briefly.However, the general features that have emerged from lattice studies to date are rather robust andcan be discussed in some detail.

The relatively cold matter that presently comprises everything around us is actually a residueof the annihilation of matter and anti-matter in the early universe. The origin of the matter-antimatter asymmetry which is critical for generating the small amount of residual matter is stilla major subject of study, and we discuss this topic at the end of this Chapter.

Another major thrust associated with the transition between the QGP and baryonic matter isthe experimental program underway to study observable phenomena associated with the dynamicsof this interface. This experimental program involves the collision of relativistic heavy ions thatshould produce (relatively) small drops of QGP. Large particle detector systems then enable studiesof the products of these collisions, which can (in principle) yield information on the transition tothe baryonic phase and the QGP itself. The program of experiments and the present state of the

20

21

experimental data will be discussed in Chapter XX.

The Expanding Universe

It has been established, since Hubble’s first discovery in the 1920’s, that the universe has beenexpanding for about ∼ 10 billion years. The universe as we know it began as a “big bang”where it was much smaller and hotter, and then evolved by expansion and cooling. Our presentunderstanding of the laws of physics allows us to talk about the earliest moment at the so-calledPlanck time tP ∼ 10−43 when the temperature of the universe is at the Planck scale T ∼Mpl

Mpl ≡√

hc

GN(2.1)

= 1.22 × 1019 GeV , (2.2)

where GN is Newton’s gravitational constant, and h and c are set to 1 unless otherwise specified.However, at this scale, the gravitational interaction is strong, the classical concept of space-timemight break down. Post the Planck epoch, the space-time may well be described by a classicalmetric tensor gµν .

Since the observed universe is homogeneous and isotropic to a great degree, its expansion canbe described by the so-called Robertson-Walker space-time metric,

ds2 = gµνdxµdxν = dt2 −R2(t)

[

dr2

1 − kr2+ r2(dθ2 + sin2 θdφ2)

]

, (2.3)

which describes a maximally symmetric 3D space, where R(t) is a scale parameter describing theexpansion and k is a curvature parameter with k = +1,−1, 0, corresponding to closed, open andflat universe, respectively.

The expansion of the universe after the Planck time is described by Einstein’s equation ofgeneral relativity, which equates the curvature tensor of the space-time to the energy-momentumtensor T µν . The energy-momentum density comes from both matter and radiation and the vacuumΛgµν contribution, the infamous “cosmological constant” of Einstein. If the matter expands asideal gas, the energy-momentum density is

Tµν = −pgµν + (ρ+ p)uµuν , (2.4)

where p is the pressure and ρ is the energy density, and uµ = (1, 0, 0, 0) defines the cosmologicalcomoving frame. The resulting dynamical equation for the scale parameter is

(

R

R

)2

=8πGNρ

3− k

R2+

Λ

3, (2.5)

which is called the Friedmann (or Friedmann-Lemaitre equation). R/R = H is the expansion rate(Hubble constant). Another equation needed for studying the expansion comes from the energy-momentum conservation,

ρ = −3H(ρ+ p) . (2.6)

22 CHAPTER 2. QUARK-GLUON PLASMA AND THE EARLY UNIVERSE

Together with the equation of state p = p(ρ), the above equation solves the evolution of ρ as afunction of the scale parameter. There are now some strong experimental evidence that we areliving in a universe with k = 0 and Λ is negligibly small until recently. Hence, we will focus belowon a simplified Friedmann equation for the early universe without the second and third terms onthe right-hand side in 2.14.

When the temperature was lower than the Planck scale, the universe was an expanding gasof relativistic particles. These particles include quarks and leptons, the gauge bosons such asphotons, gluons, and W and Z bosons, and perhaps more exotic particles like the supersymmetricpartners of the standard model particles, heavy right-handed neutrinos, gauge bosons related togrand unification theories, etc. As the temperature cooled below the masses of certain particles(such as the W and Z bosons) they “freeze out” and decay, i.e., they are not longer created byinverse reactions of their decay products due to the lower temperature. Some of these particleswith a short life time had disappeared long ago, and some with a long life time may still be withus today in the form of dark matter.

Figure 2.1: History of the universe for temperatures less than kBT ∼ 100 GeV.

23

Thus we expect that when the temperature drops below the electroweak scale (T < 100 GeV)the early universe will be a hot gas of the standard model particles: quarks, leptons, gluons andphotons. Since the system is dominated by the strongly interacting degrees of freedom, quarks andgluons, it is a good approximation to regard it as a system of quark-gluon plasma. Because of theasymptotic freedom, the interaction between quarks and gluons are fairly weak at high-temperature,and it shall be a good approximation to describe the plasma in terms of a non-interacting partongas.

During this phase of the universe, the energy density ρ is dominated by these relativistic partonsand decreases as the universe expands. The evolution of ρ during this time is governed by the factthat we have a gas of relativistic partons. The volume of any piece of the universe increases likeR3, but the energy in every mode decreases as R−1 (as the wavelength of the mode expands withthe universe). Thus we expect

ρ ∝ R−4 , (2.7)

and Eq. 2.14 then yieldsR ∼ R−1 , (2.8)

which has the solution R ∼√t. That is, the size of the universe increases as the square root of time.

The energy density then decreases as ρ ∼ t−2. We will consider these relations more quantitativelyin the next section.

Thermodynamics of A Hot Relativistic Gas

At very high-temperature, the average kinetic energy of particles is much larger than their restmass, and therefore their kinematics is relativistic. Moreover, the interactions between particlesare weak, and the system is a hot relativistic free gas to an excellent approximation. Since particlesand antiparticles can be created and annihilated easily in such an environment, their densities aremuch higher than their differences. Therefore the chemical potential µ can be neglected. Thenumber densities of the partons (species i) are described by the quantum distribution functions

ni =

∫

d3pi

(2π)31

eβEi ± 1, (2.9)

where β = 1/kBT and the − sign is for bosons and the + is for fermions. For relativistic particles,pi = Ei. For Eiβ < 1, the exponential factor is small and there is a large difference betweenfermions and bosons. For Eiβ ≥ 1 the ±1 becomes increasingly unimportant, and the distributionsbecome similar. Integrating over the phase space, one finds,

ni =

ζ(3)/π2T 3 (boson)(3/4)ζ(3)/π2T 3 (fermion)

(2.10)

where ζ(3) = 1.20206... is a Riemann zeta function. The T 3-dependence follows simply fromdimensional analysis.

The energy density for a free gas can be computed from the same quantum distribution func-tions:

ǫi =

∫

d3pi

(2π)3Ei

eβEi ± 1


=

π2/30T 4 (boson)(7/8)(π2/30)T 4 (fermion)

(2.11)

where the fermion energy density is 7/8 of that of boson.These expressions are valid for each spin/flavor/charge/color state of each particle. For a system

of fermions and bosons, we need to include separate degeneracy factors for the various particles:

ǫ =∑

i

giǫi

= g∗π2

30(kBT )4 (2.12)

where g∗ =(

gb + 78gf

)

with gb and gf are the degeneracy factors counting the number of degrees of

freedom, summed over the spins, flavors, charge (particle-antiparticle) and colors of particles. Whensome species are thermally decoupled from others due to absence of interactions, such as neutrinosat present time, one has to introduce temperature Ti for each species. The general expression forg∗ becomes

g∗ =∑

i

gb

(

Ti

T

)4

+7

8gf

(

Ti

T

)4

(2.13)

At temperature above 100 GeV, all particles of the standard model are present. At lower temper-ature, W and Z bosons, top, bottom, and charm quarks freeze out. Therefore g∗ is a decreasingfunction of temperature.

We can calculate the contribution to the energy density from the quark-gluon plasma as arelativistic free parton gas. For a gluon, there are 2 helicity states and 8 choices of color so we havea total degeneracy of gb = 16. For each quark flavor, there are 3 colors, 2 spin states, and 2 chargestates (corresponding to quarks and antiquarks). At temperatures below kBT ∼ 1 GeV, there are3 active flavors (up, down and strange) so we expect the fermion degeneracy to be a large numberlike gf ≃ 36 in this case. Thus we expect for the QGP:

ǫQGP ≃ 47.5π2

30(kBT )4 . (2.14)

With two quark flavors, the prefactor is g∗ = 37. For reference, if one takes into account all standardmodel particles, g∗ = 106.75.

The pressure of the free gas can be calculated just like the case of black-body radiation. It isaverage of p2/3E over the thermal distribution. For relativistic species,

p =1

3ǫ . (2.15)

This equation of state, combined with the energy-momentum conservation, leads to the change ofenergy density with the scale factor ρ ∼ R−4, as was argued heuristically before.

Combining the radiation energy density with its variation as R, we find that the temperaturevaries inversely as the radius parameter T ∝ R−1 and therefore T ∝ t−1/2. We then obtain thefollowing relation for the temperature as a function of time:

T (t) ≃√

hMP l

(g∗)1/2t. (2.16)

2.1. THERMODYNAMICS OF A HOT RELATIVISTIC GAS 25

If we invert this relation to yield

t ≃ hMP l

(g∗)1/2T 2(2.17)

we can construct the timeline for the temperature of the early universe from 10−43 sec. throughabout 106 yr. when the radiation dominated phase ends.

One subtle issue arises when temperature drops and some particles freeze, g∗(T ) then changes.So how does the relation between T and R affected by this change? To answer the question, let usgo back to the energy-momentum conservation from the previous section,

d(ρR3) = −pdR3 . (2.18)

According to the thermodynamics relation, dE = TdS − pdV , where E ∼ ρR3, the entropy of theuniverse must be a conserved quantity. Therefore, the expansion is actually an adiabatic process.To calculate the entropy, one starts from the relation E = TS − PV , and

s = (ρ+ p)/T =4

3

ǫ

T=

2π2

45g∗sT

3. (2.19)

where

g∗s =∑

i

gb

(

Ti

T

)3

+7

8gf

(

Ti

T

)3

(2.20)

where g∗s is in principle different from g∗. The entropy conservation yields g∗s(T )T 3R−3 = consteven as particles freeze out as temperature drops. This relation allows us to construct accuratelythe evolution of temperature as a function of time.

2.1 Thermodynamics of A Hot Relativistic Gas

At very high-temperature such that the particles have energy much larger than their rest mass,we may describe them using relativistic kinematics and ignore their masses. Thus these energeticweakly interacting particles from a system that is, to an excellent approximation, a hot relativisticfree gas. Since particles and antiparticles can be created and annihilated easily in such an environ-ment, their densities are much higher than their differences. Therefore the chemical potential µ canbe neglected. The number densities of the partons (species i) are then described by the quantumdistribution functions

ni =

∫

d3pi

(2π)31

eβEi ± 1, (2.21)

where β = 1/kBT and the − sign is for bosons and the + is for fermions. For relativistic particles,pi = Ei. For Eiβ < 1, the exponential factor is small and there is a large difference betweenfermions and bosons. For Eiβ ≥ 1 the ±1 becomes increasingly unimportant, and the distributionsbecome similar. Integrating over the phase space, one finds,

ni =

ζ(3)/π2T 3 (boson)(3/4)ζ(3)/π2T 3 (fermion)

(2.22)

where ζ(3) = 1.20206... is a Riemann zeta function. The T 3-dependence follows simply fromdimensional analysis.


The energy density for a free gas can be computed from the same quantum distribution func-tions:

ǫi =

∫

d3pi

(2π)3Ei

eβEi ± 1

=

π2/30T 4 (boson)(7/8)(π2/30)T 4 (fermion)

(2.23)

where the fermion energy density is 7/8 of that of boson.These expressions are valid for each spin/flavor/charge/color state of each particle. For a system

of fermions and bosons, we need to include separate degeneracy factors for the various particles:

ǫ =∑

i

giǫi

= g∗π2

30(kBT )4 (2.24)

where g∗ =(

gb + 78gf

)

with gb and gf are the degeneracy factors for bosons and fermions, respec-

tively. Each of these degeneracy factors count the total number of degrees of freedom, summedover the spins, flavors, charge (particle-antiparticle) and colors of particles. When some speciesare thermally decoupled from others due to absence of interactions (such as neutrinos at presenttime), they no longer contribute to the degeneracy factor. For example, at temperature above 100GeV, all particles of the standard model are present. At lower temperatures, the W and Z bosons,top, bottom, and charm quarks freeze out and g∗ decreases. Therefore g∗ is generally a decreasingfunction of temperature.

We can now calculate the contribution to the energy density from the quark-gluon plasma as arelativistic free parton gas. For a gluon, there are 2 helicity states and 8 choices of color so we havea total degeneracy of gb = 16. For each quark flavor, there are 3 colors, 2 spin states, and 2 chargestates (corresponding to quarks and antiquarks). At temperatures below kBT ∼ 1 GeV, there are3 active flavors (up, down and strange) so we expect the fermion degeneracy to be a large numberlike gf ≃ 36 in this case. Thus we expect for the QGP:

ǫQGP ≃ 47.5π2

30(kBT )4 . (2.25)

With two quark flavors, the prefactor is g∗ = 37. (For reference, if one takes into account allstandard model particles, g∗ = 106.75.)

The pressure of the free gas can be calculated just like the case of black-body radiation. Forrelativistic species,

p =1

3ǫ , (2.26)

the equation of state.To calculate the entropy of the relativistic gas, we consider the thermodynamics relation, dE =

TdS − pdV . At constant volume we would have just dE = TdS, or dǫ = Tds where ǫ (s) is theenergy (entropy) per unit volume. Since ǫ ∝ T 4, we can easily find that

s =4

3

ǫ

T. (2.27)

2.2. THE EARLY PARTONIC UNIVERSE 27

For an isolated system of relativistic particles, we expect the total entropy to be conserved.

Now using Eq. 2.4 one can easily show that

s ∝ g∗(T )T 3 (2.28)

where g∗(T ) counts the number of active (i.e., non-frozen) degrees of freedom in equilibrium. Thetotal entropy of the active species is given by

S ∝ sR3 ∝ g∗(T )T 3R3 (2.29)

and must be conserved.

2.2 The Early Partonic Universe

It has been established, since Hubble’s first discovery in the 1920’s, that the universe has beenexpanding for about ∼ 10 billion years. The universe as we know it began as a “big bang”where it was much smaller and hotter, and then evolved by expansion and cooling. Our presentunderstanding of the laws of physics allows us to talk about the earliest moment at the so-calledPlanck time tP ∼ 10−43 when the temperature of the universe is at the Planck scale T ∼Mpl

Mpl ≡√

hc

GN(2.30)

= 1.22 × 1019 GeV , (2.31)

where GN is Newton’s gravitational constant, and h and c are set to 1 unless otherwise specified.However, at this scale, the gravitational interaction is strong, the classical concept of space-timemight break down. At times later than the Planck epoch when the universe has cooled below Mpl,space-time may be described by a classical metric tensor gµν , and the laws of physics as we knowthem should be applicable.

Since the observed universe is homogeneous and isotropic to a great degree, its expansion canbe described by the Robertson-Walker space-time metric,

ds2 = gµνdxµdxν = dt2 −R2(t)

[

dr2

1 − kr2+ r2(dθ2 + sin2 θdφ2)

]

, (2.32)

which describes a maximally symmetric 3D space, where R(t) is a scale parameter describing theexpansion and k is a curvature parameter with k = +1,−1, 0, corresponding to closed, open andflat universe, respectively.

The expansion of the universe after the Planck time is described by Einstein’s equation ofgeneral relativity, which equates the curvature tensor of the space-time to the energy-momentumtensor T µν . The energy-momentum density comes from both matter and radiation and the vacuumΛgµν contribution, the infamous “cosmological constant” of Einstein. If the matter expands asideal gas, the energy-momentum density is

Tµν = −pgµν + (ρ+ p)uµuν , (2.33)


where p is the pressure and ρ is the energy density, and uµ = (1, 0, 0, 0) defines the cosmologicalcomoving frame. The resulting dynamical equation for the scale parameter is

(

R

R

)2

=8πGNρ

3− k

R2+

Λ

3, (2.34)

which is called the Friedmann (or Friedmann-Lemaitre equation). R/R = H is the expansionrate (Hubble constant). Another equation needed for studying the expansion comes from energy-momentum conservation,

ρ = −3H(ρ+ p) . (2.35)

Together with the equation of state p = p(ρ), the above equations can be solved to yield theevolution of ρ as a function of the scale parameter. There is now strong experimental evidence thatwe are living in a universe with k = 0 and Λ has been negligibly small until recently. Hence, wewill focus below on a simplified Friedmann equation for the early universe without the second andthird terms on the right-hand side in 2.14.

When the temperature was lower than the Planck scale, the universe was an expanding gasof relativistic particles. These particles include quarks and leptons, the gauge bosons such asphotons, gluons, and W and Z bosons, and perhaps more exotic particles like the supersymmetricpartners of the standard model particles, heavy right-handed neutrinos, gauge bosons related togrand unification theories, etc. As the temperature cooled below the masses of certain particles(such as the W and Z bosons) they “freeze out” and decay, i.e., they are not longer created byinverse reactions of their decay products due to the lower temperature. Some of these particleswith a short life time had disappeared long ago, and some with a long life time may still be withus today in the form of dark matter.

Thus we expect that when the temperature drops below the electroweak scale (T < 100 GeV)the early universe will be a hot gas of the standard model particles: quarks, leptons, gluons andphotons. Since the system is dominated by the strongly interacting degrees of freedom, quarksand gluons (i.e., partons), it is a good approximation to regard it as a system of quark-gluonplasma. Because of asymptotic freedom, the interaction between quarks and gluons are fairly weakat high-temperature, and it shall be a good approximation to describe the plasma in terms of anon-interacting parton gas.

During this phase of the universe, the energy density ρ is dominated by these relativistic partonsand decreases as the universe expands. The evolution of ρ during this time is governed by the factthat we have a gas of relativistic partons. The volume of any piece of the universe increases likeR3, but the energy in every mode decreases as R−1 (as the wavelength of the mode expands withthe universe). Thus we expect

ρ ∝ R−4 , (2.36)

and Eq. 2.14 then yieldsR ∼ R−1 , (2.37)

which has the solution R ∼√t. That is, the size of the universe increases as the square root of

time. The energy density then decreases as ρ ∼ t−2.If we assume that the number of effective degrees of freedom, g∗ is constant during the early

evolution of the radiation-dominated universe then the radiation energy density (ρ ∝ T 4, as inEq. 2.4) with its variation as R (Eq. 2.16), we find that the temperature varies inversely as the

2.2. THE EARLY PARTONIC UNIVERSE 29

radius parameter T ∝ R−1 and therefore T ∝ t−1/2. Note that according to Eq. 2.7 this also impliesthat the total entropy of the universe is conserved. We then obtain the following relation for thetemperature as a function of time:

T (t) ≃√

hMP l

(g∗)1/2t. (2.38)

If we invert this relation to yield

t ≃ hMP l

(g∗)1/2T 2(2.39)

we can construct the timeline for the temperature of the early universe from 10−43 sec. throughabout 106 yr. when the radiation dominated phase ends.

Figure 2.2: History of the universe for temperatures less than kBT ∼ 100 GeV.

We have assumed that g∗ is constant in obtaining these results. However, we do need to considerthe fact that as the temperature drops some particles freeze out, and so g∗(T ) then changes. Thiswill modify the expressions 2.18 and 2.18. However, the basic behavior of the expanding universe


is qualitatively described by these relations, especially noting that in Eq. 2.18 the dependence of

the temperature on g∗ is very mild (T ∝ g−1/4∗ ).

2.3 The Quark-Gluon Plasma in Perturbative QCD

Until this point we have been treating the quarks and gluons in the QGP as free particles withoutinteractions. Of course, in a high-temperature QGP we expect QCD perturbative theory to be ap-plicable due to asymptotic freedom. One important additional consequence is that chiral symmetryis now a good symmetry, and the chiral condensate must vanish in the plasma

〈QGP|ψψ|QGP〉 = 0 , (2.40)

where strictly-speaking the QGP “state” is actually the thermal average over the excited states ofthe QCD vacuum when the baryon number density is ignored.

Another important feature of the QGP is color deconfinement. In QCD perturbation theory,the quarks and gluons are free particles that can be described by plane waves. Asymptotic freedomwill guarantee that high momentum transfer interactions are weak. Small-momentum transferscattering involves long distance interactions which are screened by the plasma (although this isonly strictly true in the color electric sector). As such, the charged quarks and gluons can movefreely inside the plasma without being confined to a local region. This remarkable property isradically different from the low energy limit of QCD where all charges are permanently confined tothe interior of hadrons.

Consider a color charge in midst of a color-neutral plasma. The other particles in the plasma willact to screen it, and as a consequence the interaction between color charges is damped exponentially.To calculate the screening length, one can start from a color charge and calculate its induced colorfields. The result is a correlation function of gluon fields. This function can be calculated inperturbation theory at high-temperature, and the result for the screening mass is

m2D = g2T 2 . (2.41)

to leading order in the strong coupling expansion. The Debye screening length is simply 1/mD or1/gT , which is very short at high-temperature. When the color charges are screened in a plasma,it has a finite energy and therefore in this sense, the color charges are now liberated.

Unfortunately, the magnetic interaction is only weakly screened; it has a screening mass of orderg2T . Absence of the magnetic screening means that the magnetic sector of the QCD remains non-perturbative even at high-temperature. Fortunately, at high-temperature this non-perturbativepart contributes to physical observables only at higher-order in QCD coupling, so the free gasbehavior is dominant.

Another important feature of the plasma is the plasma frequency. In a QED plasma, lightcannot propagate below the plasma frequency, ωpl = (ne2/m)1/2, but will be reflected from thesurface, like in a silver-plated mirror. The physics of the QGP is similar: gluons (plasmon) cannotpropagate as a free field in the plasma if its energy is too low. In fact, the gluons acquire an effectivemass which is effectively the plasma frequency. Perturbative calculations confirms this behavior,and to leading order in perturbation theory the plasma frequency is

ωpl =1

3

√

Nc +Nf/2(gT ) . (2.42)

2.3. THE QUARK-GLUON PLASMA IN PERTURBATIVE QCD 31

The transverse-polarized gluon modes acquire the same mass.The plasmon and transverse gluon modes are damped in the plasma. One can calculate the

damping rate using the hard-thermal loop method in pQCD and the result is gauge-invariant:γ = ag2NCT/24π, where numerically a is found to be a = 6.63538.

The results we discussed above are the basic leading-order predictions of pQCD. Higher-ordercontributions can and have been calculated in the literature. Unfortunately perturbative expansionsfor thermodynamical properties of the plasma converge very slowly. As such, the free plasma pictureworks only at extremely high-temperature. Even at the temperatures corresponding to 100 GeVthe perturbative expansion must be reorganized significantly to get a sensible prediction. We willcome back to this point later.

Transition to the Low-Temperature Phase: Physical Arguments

As we have discussed in the previous chapter, the zero temperature ground state of QCD is strikinglydifferent from the high-temperature QGP: color charges are confined to individual hadrons andchiral symmetry is broken spontaneously. Therefore, as the plasma cools in the universe, somerapid changes in thermodynamic observables must happen from the high-temperature QGP phaseto the low-temperature confining and chiral-symmetry breaking phase, where the quarks and gluonscombine to form colorless states of hadronic matter.

It is possible to estimate the transition temperature by comparing the QGP gas pressure withthat of hadronic gas. The lightest hadrons are pions, and for T < 1 GeV (note that in the followingwe often use units where kB = 1 and T has units of energy), we might expect a gas of relativisticpions. This is a system with only 3 degrees of freedom, g = 3, so the energy density and pressureof the system

ρπ =3π2

30T 4, Pπ =

3π2

90T 4, (2.43)

This, however, is not the full story. Pions are collective excitations of the non-perturbative QCDvacuum. This true ground state of the QCD vacuum has a lower-energy −B than the perturbativeQCD vacuum. (In the MIT bag model of hadrons, this energy is the origin of the quark confine-ment.) Lorentz invariance requires that the energy-momentum density is of form T µν = Bgµν . Thusthe non-perturbative QCD vacuum has a positive pressure as well. Therefore, the total pressure ofthe hadronic phase is

Plow = B +3π2

90T 4 , (2.44)

On the other hand, from the previous sections, the pressure of the QGP phase with 2 quark flavorsis, PQGP = 37π2T 4/90. Equating the two pressures, we find the transition temperature,

Tc = (45B/17π2)1/4 ∼ 180MeV , (2.45)

where we have used the MIT bag constant B = 200 MeV as determined by fits to the masses ofphysical hadrons.

The energy difference (latent heat) between the two phases at the transition temperature is

∆ρ =34π2

30T 4 +B , (2.46)

which is on the order of 2 GeV/fm3.


Another estimate of the transition temperature comes from considering chiral symmetry. Atfinite but small temperature, the pion gas will dilute the chiral condensate in the zero-temperaturevacuum. The quark condensate can be calculated as a response of the system’s free energy to thequark mass,

〈ψψ〉T =1

Nf

∂F

∂mq, (2.47)

where Nf is the number of quark flavors. The free-energy of the pion gas is

F = (N2f − 1)T

∫

d3~p

(2π)3ln(1 − e−Eπ/T ) . (2.48)

Thus the pion condensate has the following low-temperature expansion,

〈ψψ〉T = 〈ψψ〉0[

1 − N2F − 1

3Nf

T 2

4f2π

+ ...

]

, (2.49)

where the ellipse indicates higher-order terms in the expansion. If one just keeps the first twoterms, the chiral condensate vanishes when

Tc = 2fπ

√

3Nf/(N2f − 1) = 200MeV , (2.50)

which is consistent with the other estimate.Clearly, the QCD system is strongly interacting around Tc. On the other hand, the above

estimates relied on calculations which are valid at temperatures much higher than Tc. To saysomething rigorous about what happened around Tc, one must resort to lattice QCD, a numericalapproach to solve QCD through computer simulation.

2.4 A Brief Tour in Lattice QCD Thermodynamics

At lower temperatures where the coupling constant is larger one must employ non-perturbativemethods of calculation. The only known method for solving QCD non-perturbatively is on a space-“time” lattice. Here we present a simple introduction to this method without getting involved intoo many technical details.

Consider a QCD system with temperature T = 1/β and baryon number zero (this is true toa good approximation in the early universe, as we will discuss later in this chapter). The mostimportant quantity is the partition function,

Z = Tr[exp(−βH)] , (2.51)

where H is the QCD hamiltonian and trace is over all physical states in Hilbert space. H is afunction of 3D quark and and gluon fields ψ(~x) and Aµ(~x), respectively. One can introduce afourth coordinate x4 (imaginary time) and the 4D field

φ(~x, x4) = eHx4φ(~x)e−Hx4 , (2.52)

where φ collectively labels all QCD fields and x4 runs between the values 0 and β. Then thepartition function can be written as a path integral

Z = Tr

∫

D[φ] exp(−S[φ]) , (2.53)

2.4. A BRIEF TOUR IN LATTICE QCD THERMODYNAMICS 33

whereD[φ] is a functional integration measure and S[A] is the QCD action defined by the lagrangianin Euclidean time,

S[φ] =

∫ β

0dx4

∫

d3~xLEuc. . (2.54)

Therefore, the partition function is now reduced to a functional integral.

Because of the trace in Z, the gluon potentials Aµ obey the period boundary condition in x4,

Aµ(x4, ~x) = Aµ(x4 + β, ~x) , (2.55)

whereas the quark fields obey the anti-periodic condition

φµ(x4, ~x) = −φµ(x4 + β, ~x) . (2.56)

The whole thermodynamics formulation is then invariant under SU(3) gauge transformation withU(x4, ~x) satisfying the periodic boundary condition.

The path integral formulation makes explicit that QCD is a quantum field theory with aninfinite number of quark and gluon degrees of freedom. To solve it approximately, one first limitsthe system to a 3D box of dimensions Lx, Ly, Lz, and then replaces the continuous space-“time”variables by a discrete 4D lattice. The quark and gluon degrees of freedom now live on the latticesites and the bonds in-between the sites, respectively. The number of these d.o.f. is now finite butlarge (millions to billions in actual simulation).

The integrations over quark fields are of Gaussian type and can be done analytically. Those overthe gluon fields can be evaluated using the Monte Carlo sampling method. In this approach, classicalgluon configurations are generated on the lattice with the probability distribution correspondingto the Boltzmann factor e−S . The actual physical observables are calculated with hundreds andthousands of these “typical” distributions. As an example, the Debye screening mass can bemeasured from the space correlation of the gluon fields at equal time. One of the important toolsof the lattice calculation is that one can change the parameters of the theory and study howobservables change in a world different from the real one.

Quark Masses and The Nature of the Transition

The nature of the QCD transition from high to low temperature depends on strongly on light-quarkmasses. For simplicity, let us ignore heavy quarks (c, b, t) and concentrate on up, down and strangelight quarks.

One powerful theoretical approach to discuss phase transitions is Laudau-Ginsburg theory. Inthis framework, one first identifies the order parameter of a transition, which is the fundamentalobservable which drives the phase transition. An example of order parameter in water-steamtransition is the density. One then constructs an effective lagrangian of the order parameter whichgoverns the dynamics of the transition. Here the fundamental symmetry constraints are importantto determine the form of the effective lagrangian, from which one can often say a lot about thetransition without going into the details of the specific dynamics.

If all three quarks are massless, the low-energy order parameter for chiral transition can bechosen to be Σ, a 3 complex matrix which transforms under chiral SU(3)L × SU(3)R as

Σ → ULΣU †R , (2.57)


One can then construct an effective lagrangian to describe the dynamics of Σ. One of the termthat one can write down is DetΣ which is trilinear in components of the order parameter. As aconsequence, the chiral phase transition described by the theory is in general a first-order phasetransition.

When the strange quark is infinitely heavy, and up and down quarks remain massless, thelow-energy order parameter can be taken to be a 2 × 2 unitary matrix. The phase transition inthis system is similar to that of an O(4) magnet, a magnetic with four-independent magnetizationdirection. The Landau-Ginzburg theory for this system leads to a second-order phase transition.

Therefore, if one varies the mass of the strange quark from large to small, the second-orderphase transition must end at a tri-critical point, beyond which the transition becomes first order.The tricritical point is characterized by vanishing of the coefficients of the first two terms (quadraticand quartic) in the potential for the order parameter.

The above consideration is not yet sufficient because the up and down strange quark masses donot vanish in the real world.

If up, down and strange quark masses are taken to infinity, we are left with a theory with justpure glue fields. This system at finite temperature has an interesting symmetry called color Z3

center symmetry. An effective Landau-Ginzburg theory can be constructed to describe the physicsof Z3 symmetry breaking at high-temperature phase. The symmetry argument suggests that thetransition is of first order. The strength of the first-order phase transition lowers as the quark massbecomes lighter, and finally the first-order phase transition region is enclosed by a second orderphase transition line.

2nd

2nd

m = mu d

ms

1st

1st

N = 3f

QCD ?

N = 2fN = 0

f

N = 1f

Figure 2.3: Order of phase transition as a function of light quark masses (mu = md) vs. the strangequark mass(ms).

A diagram detailing the nature of QCD phase transitions as a function of quark masses is shownin Fig. 2.2. In the limit of 3 very light quarks, we expect a first order transition. The first ordertransition region is enclosed by a second order transition line which goes through the tri-criticalpoint. In between the two second-order phase transition lines, one has a broad region of rapidcross-over. The exact locations of these second-order transition lines are not known. Therefore, for

2.4. A BRIEF TOUR IN LATTICE QCD THERMODYNAMICS 35

a realistic physical situation, where the up and down quark masses are on the order of a few MeVand strange quark mass is about 100 MeV, the transition can either be of a weak first-order or arapid crossover (2nd order), as shown by two dots in Fig 2.2.

Physics of the QCD Transition on Lattice

Lattice simulations of QCD thermodynamics have made significant progress in the last decade,due to both rapid rise in computational power and implementation of better algorithms. Some ofchallenges in achieving a complete realistic simulation include finite size effect, discretization errors,dynamical quarks, and small enough quark masses.

From simulations of QCD on a lattice, a transition is clearly seen at a fairly sharp temperatureof about TC ≃ 150 MeV. It is well-established that the energy density undergoes a rapid change neara critical temperature TC , enhanced by almost an order of magnitude, as indicated in Figure 2.3.Beyond TC , the energy density is fairly flat as a function of temperature but slightly below thatpredicted by the free gas model. This rapid change is an indication that the fundamental degreesof freedom are different above and below TC .

The transition is less dramatic in the equation of state, i.e., the pressure of the system as afunction of temperature. At low-temperature, the pressure is very small. As temperature increases,the pressure builds up gradually over a large range of T , from Tc to 2Tc. When the pressure curveflattens out at high-T, it again undershoots the result of the free gas model. The equation of stateis also more sensitive to different quark mass scenarios. These calculations are improved constantlywith smaller quark masses and lattice spacing. Improved calculations show that the transition atphysical quark masses is a rapid cross over.

0.0

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

1.0 1.5 2.0 2.5 3.0 3.5 4.0

T/Tc

ε/T4 εSB/T4

3 flavour2+1 flavour

2 flavour

0

1

2

3

4

5

100 200 300 400 500 600

T [MeV]

p/T4 pSB/T4

3 flavor2+1 flavor

2 flavorpure gauge

Figure 2.4: The transition from mesonic matter to the QGP phase as suggested by lattice simu-lations of QCD. The simulations are carried out with 2 or 3 light flavors or 2 light and 1 heavyflavor. The expected limits as given by Eq. 2.5 are shown by arrows on the right side of the figure.(Note that in this figure T actually represents kB times temperature.)

An interesting property of this transition emerges from the consideration of chiral symmetry inlattice simulations. Recall that, in the absence of quark masses, the QCD Lagrangian is chirallysymmetric, i.e., nature is invariant under separate flavor rotations of right and left-handed quarks.This symmetry is evident in the QGP phase at T > TC in the lattice simulations. This is quiteanalogous to the symmetric demagnetized state of a ferromagnet above the Curie temperature.The ferromagnetic property is related to the magnetization M which vanishes above the Curie


temperature. The lattice QCD calculations can provide a measurement of the relevant “orderparameter”, the scalar quark density 〈ψψ〉, as a function of temperature. This order parameterprovides a measure of the effective mass of a quark in the medium. The result is that at T > TC

this effective quark mass becomes small, approaching zero as T → ∞, as one would expect inchirally symmetric QCD. As shown in Figure 2.4, below TC there is a sharp increase in 〈ψψ〉corresponding to the quarks developing a constituent mass of ∼ 300 MeV. This heavy consituentquark is the basis of the quark model of hadrons to be discussed in Chapter XX. This transitionto the broken symmetry phase is again analogous to the ferromagnet, which spontaneously breaksrotational symmetry in developing a finite M below the Curie temperature. Shown in the samefigure is the susceptibility of the condensate which shows a peak at the transition temperature.

5.2 5.3 5.40

0.1

0.2

0.3

0.4

0.5

0.6

mq/T = 0.08

m

5.2 5.3 5.40

0.1

0.2

0.3

mq/T = 0.08L

L

Figure 2.5: The scalar quark density 〈ψψ〉 measured in lattice QCD as a function of temperature.At T < TC chiral symmetry is broken, whereas the chirally symmetric limit is realized at highertemperatures T > TC .

The potential between two quarks is shown for various temperatures in Figure 2.5. One cansee that at low temperatures T < TC , the potential continues to rise at larger distances, consistentwith the expectation that the quarks will be confined as colorless hadrons. At higher temperatures,T > TC , one finds that the potential energy at large distances saturates, and it is possible for thequarks to propagate independently of each other. This confirms the picture that above TC one hasa deconfined plasma.

2.5 Evolution of the Universe in Hadronic Phase

The QGP dominated universe will expand and cool until we reach the critical temperature, TC , toproduce a gas of pions along with a few baryons and antibaryons. The number of baryons is clearlysuppressed by the Boltzmann factor

P ∼ e−MB/T , (2.58)

which is small but still significant. But the universe we observe is clearly not a pion gas, nor thedecayed remnants which would be electrons, positrons and finally just photons. As the universe

2.5. EVOLUTION OF THE UNIVERSE IN HADRONIC PHASE 37

Figure 2.6: The potential energy between two quarks as a function of the separation distance ascomputed by lattice QCD. The calculation is performed for various temperatures showing thatconfinement is present at low temperatures T < TC but not above TC .

cools, the pion densities diminish exponentially as a function of temperature. This decrease isachieved by the annihilation

π+ + π− → γ + γ , (2.59)

whereas the inverse reaction is difficult because the thermal photons do not have enough energyto produce pions. Similar process happens for baryons and antibaryons, while they continue toannihilate, they cannot be reproduced. If the annihilation rate is rapid enough, the density ofbaryons follows the thermal Boltzmann distribution. Of course, if the universe had no net baryonnumber, all baryons would disappear eventually. However, the existence of matter around us showsthat the baryon number of the universe is not zero. Therefore, as temperature cools, all anti-baryonsare annihilated, all pions are either annihilated and/or decayed, but there is a small tiny baryondensity in the form of proton and neutron survives. (The origin of these baryons is the subject ofthe next section of this chapter. Indeed, these protons and neutrons are the main characters of theremainder of this book.)

What happened to other components of the universe? As temperature lowers, we are left withjust leptons and photons. At temperatures below the electron mass, pairs of electrons and positronsare no longer created, so they freeze out and annihilate to produce more photons. We enter a phaseof the universe that is dominated by photons and neutrinos: a black-body universe. Of course,due the charge neutrality, a small fraction of electron residue is also present. The energy densitywill continue to decrease as t−2 until much later when, it turns out, the small fraction of baryonicmatter becomes a significant factor in the energy density.

We still observe the remnant black-body radiation associated with the early universe. (Theneutrinos have not been observed, but are discussed in Chapter 8.) There is presently a uniformdistribution of cosmic microwave radiation with a characteristic temperature of (2.725± 0.001) Kand a number density of 410.4±0.5/cm3. The present baryon to photon ratio is tiny, about 6×10−10,but the photons have energies of ∼ 2.5× 10−4 eV and the baryons have energy Mp ∼ 1 GeV. Thus


the energy density of the baryons is about 2400 times the energy density of the photons. As theuniverse cooled from the “radiation dominated” era to the present, the photons were red-shifteddue to the expansion. The photons decoupled from the protons and electrons when the lattercombined into neutral hydrogen. Since then (temperature T ∼ 0.3 eV) the photons have beenred-shifted by about 104, while the protons remained at Mp ∼ 1 GeV. Thus we now find ourselvesin an era where baryonic matter dominates over the cosmic photons. (In addition, we have darknon-baryonic matter and dark energy which remain to be understood - but that is a different storyand we will confine our attention here to the baryons and photons.)

2.6 The Origin of Baryonic Matter

Let’s now return to the source of the baryonic matter. While the present baryon to photon ratiois indeed tiny, that is the stuff of which we are made. Since the early universe should consist ofequal numbers of quarks and antiquarks, eventually they should all annihilate to produce onlyphotons in the end. The observed number of baryons is much greater than can be expected froma random fluctuation associated with the cooling of the enormous number of quarks and gluons inthe initial plasma. The annihilation of baryons and antibaryons ceases at a rather low temperatureof ∼ 20 MeV. The number density of baryons and antibaryons at this temperature is given by theexpression (for gB species of non-relativistic particles)

nB = gB

(

MpT

2π

)3/2

e−Mp

T . (2.60)

The corresponding number density of photons is

nγ ≃ 2

π2T 3 (2.61)

so the ratio is approximately

nB

nγ∼ gB

(

Mp

T

)3/2

e−Mp

T (2.62)

∼ 10−19 (@T = 20 MeV) . (2.63)

We would expect the chance excess of baryons over antibaryons to be a very small fraction of thetotal number of both. Therefore, the observed 6× 10−10 fraction of baryons relative to the numberof photons is not possible to generate by just a statistical fluctuation.

So what is the mechanism that generates the small excess of matter over antimatter? Themodern view is that the observed excess of baryonic matter relative to antimatter is likely dueto the properties of particles and their interactions during the expansion and cooling of the earlyuniverse. This can occur if three criteria, initially discussed by Sakharov, are satisfied:

1. baryon number (B) is violated so that baryons (and antibaryons) can be created,

2. CP is violated so that the rates of baryon and antibaryon production can be different,

3. thermal equilibrium is broken so that forward and reverse reactions become unbalanced.

2.6. THE ORIGIN OF BARYONIC MATTER 39

Studying baryogenesis has been a very active area of theoretical physics for the last few decades.In the standard model, the baryon number can be violated through non-perturbative process viathe so-called anomaly. However, to generate enough baryons, one must have a strong first orderphase transition which is hard to achieve with the heavy Higgs mass. Moreover, the standard modelCP violation through the CMK matrix is too small to generate enough CP asymmetry. Therefore,it appears that baryogensis requires physics beyond the standard model. One of the interestingdirection is what happens in a supersymmetric extension of the standard model.

In grand unified models, baryon number is in general violated. One consequence would be thatprotons would be unstable and decay, typically by a process like p → e+π0. Searches for suchdecays limit the proton lifetime to τP > 1032 years. Fundamentally, we expect the decay to proceedmy mediation of a heavy particle of mass MX . Fermi’s golden rule can be used to calculate thedecay rate as

Γ = |〈f |HX |i〉|2 dNf

dEf. (2.64)

The matrix element will be proportional to αX/M2X where αX is the squared coupling constant

and the M−2X is the propagator for the heavy particle exchange. Since the only other energy scale

in the problem is the proton mass Mp we can write by dimensional analysis

τP ∼ 1

M5P

(

M2X

αX

)2

. (2.65)

Using the experimental limit for τP and assuming αX ∼ 0.1 one obtains a rough limit for the massof the heavy particle responsible for baryon number violating interactions MX > 1016 GeV. In atypical baryogenesis scenario, one might generate long-lived heavy particles X in the very earlyuniverse when T > MX . At T < MX the X particles freeze out and then much later, when theuniverse is cooler, they decay into baryons and antibaryons. The presence of CP violation in thedecays enables a production of an excess of baryons over antibaryons. There are many variationson this basic model, and with suitable extensions of the standard model one can indeed generate abaryonic excess that agrees with the observed universe.

Finally, right-handed neutrinos have been considered as good candidates to lead to baryogensis.These heavy particles could have masses slightly smaller than the grand unification scale, and decayasymmetrically (due to CP violation) to produce net lepton number. This lepton number can beconverted in to baryon number through the so-called sphaleron processes. This route of generatingbaryon is called leptogenesis.

Chapter 3

Protons and Other Hadrons in QuarkModel

40

Chapter 4

Probing the Parton Structure in theProton

41

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Quarks, Nuclei, and the Cosmos