Seminar

QCD Matter Phase Diagram

Tim KolarMentor: prof. dr. Simon Sirca

April 10, 2014

Contents

1 Introduction 1

2 QCD - Quantum chromodynamics 1

3 Phase diagram of QCD matter 33.1 QCD vacuum and low energy, low density phase . . . . . . . . . . . . . . . . . . . . . 43.2 Quark gluon plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 Relativistic heavy ion collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.3.1 J/ψ suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.4 Color superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Conclusion 11

1 Introduction

We are all familiar with the fact that ordinary matter can be found in several different forms orstates of aggregation, known as phases. The same applies to QCD matter where, instead of onlythe nucleus and electrons, we have two degrees of freedom, quarks and gluons. In this paper we willexplore the consequences of this quark-gluon picture for thermodynamics of such strongly interactingmatter, in other words, its behaviour as conditions, such as temperature and density, are varied.Thermodynamical information is often presented in the form of a phase diagram, in which the differentphases of a substance occupy different regions of a plot whose axes are calibrated in terms of theexternal conditions through control parameters. The most familiar example is probably that of H2O,but we will focus on the proposed phase diagram for QCD that is shown in figure 4. But first we needto explain what QCD stands for.

2 QCD - Quantum chromodynamics

Quantum chromodynamics, a theory of quarks, gluons and their interaction, is an independent partof the standard model of elementary physics. Like QED, which describes electrons and photons,QCD is also a quantum field theory, but with color as an analog of electric charge. The role of forcecarriers is given to gluons, which are bosons. Unlike photons, which are not carrying any electricalcharge, gluons may carry color charge, in fact they carry color and anticolor charge at the same time.We have three colors for quarks and three anticolors for antiquarks which gives us 9 possible colorstates (octet+singlet). Although it is mathematically possible to form a colorless singlet, there is nosuch gluon to be found in nature. So we have 8 different gluons. The fact that force carriers for

αEM

r [nm]1

137

(a)

αS

r [fm]

1

(b)

Figure 1: Coupling constants of electromagnetic and strong forces. Actually, due to relativestrong dependence on distance between interacting particles, it would be better if we wouldrefer to both of them as strengths, but for historical reasons the term constant is used. Inthe left figure the fading nature of the electromagnetic interaction is easily seen, as it will behard to miss the characteristics of the strong coupling constant that give rise to two basicproperties of QCD which we will describe later.

strong interaction are not charge neutral, and as such can interact with each other, has far reachingconsequences. Because of this possibility, we get special energy dependence of strong interactioncoupling constant αs (Figure 1b). As we can see its value increases with distance and thus decreaseswith energy (r ∝ 1/E), unlike for electromagnetic coupling αEM , where it is the other way around.The fact, that quarks at high energies behave like free particles, is called asymptotic freedom and werefer to it again, when we try to explain some of the QCD matter phases. On the other hand, whenthe energy decreases, αs increases and at low enough energies becomes too large to use the commonaproach of calculating variables by using the perturbation method, where an expansion in powers ofαs is needed. This leads to big uncertainties in calculations of such processes as, for example, if wewould like to know how we get such an increase in rest mass when three very light quarks bind in anucleon. This increase of αs with distance, combined with the fact that the only stable finite energysystems in QCD are those formed from complementary combinations of color, such as qr qr mesons

1

Figure 2: The difference in the change of flux when separating two electrons or two quarks.In first case field lines are spreading wide around (to infinity) as in the second they are kepttogether, because the field strength does not fade away with increasing distance. So at onemoment becomes energeticaly favourable to form another pair of appropriate quarks then toincrease the distance further.

or qrqgqb baryons, is responsibile for color confinement of quarks. As a consequence, colored objects,such as isolated quarks or gluons, are never observed. Suppose we have a system of two quarks withappropriate colors and we try to drag them appart as shown on the right side of figure 2. Sincewe know, that the strong potential increases with distance between two quarks, so at some point itbecomes energeticaly favourable to produce a pair of a quark and antiquark instead of increasing thedistancce further. But the qq pair formed in the separation process must be produced with exactly theright color combination to maintain overall color neutrality, so we end up with two hadrons. Becauseof that, although gluons are massless, strong interaction does not have an infinite range.

Now that we have covered some of the qualitative properties of QCD, let us summarize the dy-namics of QCD in a bit more mathematical way by writing down the Lagrangian. On a basic level weare dealing with quarks (s = 1

2), which come in six flavors, Nf = 6, and can be grouped in a field

ψ(x) = (ψ1, ..., ψNf)T = (u(x), d(x), s(x), c(x), b(x), t(x))T (1)

where each of the ψ1 = u(x), ψ2 = d(x), ... is a four component Dirac (bi)spinor field. In our discussionwe will focus on three lightest quarks (u, d and s), so we will have Nf = 3. The strong interactionbetween them involves Nc = 3 different color charges and is mediated by 8 gluons, which are generatorsor gauge bosons for the underlying gauge group of QCD, SU(3)color . So the QCD Lagrangian is (andin gray its QED analog for comparison) [1]

LQCD = ψ(iγµDµ −m)ψ − 1

4GjµνG

µνj , (2)

LQED = ψ(iγµDµQED −m)ψ − 1

4FµνF

µν (3)

where ψ = ψ†γ0 is the Dirac conjugate spinor, γµ are the Dirac matrices and m is the appropriatequark mass matrix m = diag(mu,md,ms, ...). The gauge-covariant derivative is defined as

Dµ = ∂µ − ig8∑j=1

λj2Ajµ(x), (4)

DµQED = ∂µ − ieAµ(x), (5)

2

with the strong coupling constant g (αs ≡ g2

4π ) and Gell-Mann matrices λj , which form for j = 1, ..., 8the previously mentioned generators (gluons) T j = λj/2 of the color gauge group SU(3)color. The

analogues of these in the case of SU(2) group are the Pauli matrices σi. With Ajµ(x) we denotegluon fields. This gauge-covariant derivative is formed in such way, that out of vector defined inpoint of space and its surrounding vector field we get again an vector. It is distinct from the normalderivate because of its invariance under gauge transformations. In other words, it preserves all physicalproperties when the fields undergo osuch transformations. The gluon (photon) field strength tensor is

Giµν(x) = ∂µAiν − ∂νAiµ + gfijkA

jµA

kν , (6)

Fµν(x) = ∂µAν − ∂νAµ, (7)

where fijk are the structure constants of the local SU(3)color symmetry. These structure constant ofa Lie group determine the commutation relations between its generators (gluons in our case) in theassociated Lie algebra and are defined as [λi, λj ] = fijkλk.

At the first glance the Lagrangians of QCD and QED are quite alike. Equations do differ becauseof different number of force carriers, which does not seem to be of much consequence. But when weinspect the differences in the field strength tensors from equation 6 and 7, we can notice, the additionalterm in the gluon field strength tensor. The first two terms in both tensors describe the interaction

e−

e−

γ

e−

e−

(a) Single photon interaction in QED

d

d

u

u g

(b) Multi-gluon interaction in QCD

Figure 3: In nature the electromagnetic force is always carried by one or more photons thatare not interacting with each other, whereas strong interaction can be transmitted either bysingle or several gluons, which can also interact with each other. This gives a rise to a rangeof unusual properties in QCD.

through the exchange of one force carrier particle, while the third one in gluon field tensor is dueto possible gluon self-interaction. As we have already discussed before, this has a huge impact onthe properties of QCD. All of them have an important role, when describing different phases of suchmatter.

3 Phase diagram of QCD matter

Figure 4 shows the proposed phase diagram for QCD. The control parameters are the temperatureT and the baryon chemical potential µ. It is already known that most of the visible matter in theuniverse is made of subatomic particles called hadrons, which we divide in two groups. Baryons areformed of three quarks qqq and mesons are those composed of one quark and one antiquark qq. In allparticle physics reactions a qqq baryon is always created or destroyed pairwise with qqq antibaryon.There is no process within QCD, which can change the number of baryons NB minus the number ofantibaryons NB. In other words, we can identify a conserved quantum number B = NB −NB calledbaryon number. We can conclude that quarks and antiquarks carry B = ±1

3 respectively. We canhave a system where due to other interactions, the baryon number is allowed to vary. In such a systemthe most convenient thermodynamic potential is the grand potential Ω(T, V, µ) = E−TS−µB, whichcan be minimized to find the thermodynamic equilibrium of the system and then we can determineµ as the increase in E whenever B increases by one. But when systems are described by using thegrand canonical ensemble, we keep µ as a control parameter, and instead of baryon number the baryondensity nB(T, µ) = B/V is a derived quantity.

3

T [MeV]

µ[MeV]

170

10

938

Universe Cooling

RHICLHC

HadronGas

NuclearMatter

Quark-GluonPlasma

ColorSuperconductor

Figure 4: The phase diagram for strongly interacting matter (QCD). We can distinguishbetween three basic phases: hadronic phase, quark gluon plasma and color super-conductingquark matter. The first one can be further separated into gaseous and liquid hadronic phases.Some of the phase transitions are continuous (dashed or no lines) and others are predicted tobe higher order (black lines). We can also see the approximate path that the universe followedduring the cool down and the paths of matter that participates in heavy ion collisions. Alsovery important is the point marked with the red dot, because more or less all of the matterin the present-day universe is enjoying at those particular conditions.

Through the rest of this article we will attempt to explain what is know about the phase diagram.In the next section we will try to describe some basic properties of strongly interacting matter at verylow temperatures and at (almost) zero chemical potential.

3.1 QCD vacuum and low energy, low density phase

In QCD we cannot take the vacuum to be simple empty space, but a quantum state with the lowestpossible energy. If we are dealing with a quantum theory, which can be calculatede using petrubationtheory (like QED), the properties of the vacuum are analogous to those from the ground state of

Figure 5: Particles with left andright handed helicity states. [2]

any quantum mechanical problem including the quantum har-monic oscillator. In such a case the vacuum expectation valueof any field operator is zero. But in the case of low energy non-perturbative QCD it is possible to have a non-vanishing vacuumexpectation values called condensates. Two of these are quark-antiquark condensate 〈ψψ〉 and gluon condnesate 〈GµνGµν〉.The principle is very much similar to the one if we cool heliumto superfluid state when a Bose condensate of He atoms in thelowest quantum state forms.

But now we arrived to another important aspect of QCDdynamics. When a particle with spin ~s propagates, it is possibleto define a quantity called helicity h = ~s.~k/|k|, which is theprojection of the spin axis along the direction of the particle’smotion, defined by the momentum ~k. For a half-integer spinquark, there are two possible helicity eigenstate values h = ±1

2 ,usually referred to as left and righthanded states. A quark’shelicity cannot be altered by either emission or absorption of a

4

gluon, so in the absence of any other effect one might think that the numbers of left and right handedquarks are separately conserved in QCD and therefore leading to two good quantum numbers BLand BR. This would be the case, if quarks would have zero mass and hence travel at the speed oflight. Otherwise, we can Lorentz boost to a frame, in which the quark’s momentum along the boostaxis changes to the opposite sign and with it so does the helicity, because angular momentum staysunchanged. It turns out that in a relativistically covariant treatment massive quarks must be describedas superposition of helicity eigenstates and the mass m is parametrising the overlap between themand hence effectively the rate of L↔ R transitions. So only B = BL + BR remains a good quantumnumber and thus we say, that the chiral symmetry relating left and righthanded quarks and describingthem as independent particles is broken by the fact, that the quarks have mass.

Such a conclusion can also be made trough theory own dynamics. Let us denote the vacuum bythe ket |0〉, and the field operators which create or destroy a quark when acting on a ket as ψ,ψrespectively. We can write down the expectation value of the quark-antiquark condensate as

〈ψψ〉 = 〈0|ψLψR + ψRψL|0〉 6= 0. (8)

Since neither |0〉 is anhilated by ψ nor 〈0| by ψ, the vacuum must contain qq pairs. value of suchcondensate is 〈ψψ〉 ' (250MeV)3 and can be interpreted as the number of such pairs per unit volume.Relation (8) also implies pairing of ψL with ψR and ψR with ψL. Because of its nonzero value and thefact that 〈ψψ〉 changes BL −BR for two units, but leaves BL +BR invariant we get to see again thatonly B is a good quantum number.

A left handed quark traveling through such a vacuum with non-vanishing condensate can be anhi-lated by ψL, leaving ψR to create a right handed quark with the same momentum. The quark will thusflip its helicity at a rate proportional to the value of the condensate 〈ψψ〉. Or differently stated, it willpropagate just as if it had a mass. This mass is not the same as the quark’s current mass m. We usuallyrefer to such dynamically generated mass as constituent mass Σ. If chiral symmetry breaking (χSB)occurs spontaneously by the formation of a large condensate, then Σ may be much larger than m due tothis dynamical mass generation. We have written the estimates of both types of masses for the known

Table 1: Estimates of current and dynamically gatheredconstituent quark mass. Because isolated quarks are neverobserved, neither quantity is precisely defined.

Quark Current mass m Constituent mass Σflavor [MeV/c2] [MeV/c2]

d ∼ 7 ∼ 350

u ∼ 3 ∼ 350

s ∼ 140 ∼ 550

c ∼ 1800 ∼ 1800

b ∼ 4.2× 103 ∼ 4.2× 103

t ∼ 170× 103 ∼ 170× 103

six quarks in table 1.Spontaneous χSB in QCD is there-

fore the process by which the nucleon,and with it all matter in the universe,acquires most of its mass. Let us nowdiscuss a useful analogy for χSB namelyferromagnetism in metals. In a metaleach atom occupies a spot on a spe-cific crystal lattice, and has an unpairedelectron carrying a magnetic momentor spin that we denote with ↑ or ↓(s = ±1/2). If the temperature is lowenough, i.e. bellow the Curie temper-ature (T < Tc), the system undergoesa phase transition to its ferromagneticstate in which a macroscopic fractionof the spins is aligned, resulting in thespontaneous magnetization of the sample M = 〈↑〉 6= 0 (in this case angled brackets denote a ther-modynamic average rather than a quantum expectation value). Since the magnetization axis definesa particular direction in space, the original symmetry of Hamiltonian under rotations of the spin axisis spontaneously broken by M 6= 0. The same effect can be promoted by an external magnetic fieldH 6= 0, but with explicit symmetry breaking in this case. The relation between M and H exactlymirrors that between nonzero condensate 〈ψψ〉 and mass m in QCD.

An even more interesting similarity of these two systems arises, when we analyze the spectrumof excitations above the ferromagnetic ground state M 6= 0. It turns out, that coherent oscillationsof the spins in directions orthogonal to the magnetization axis, known as spin waves, cost arbitrarily

5

small amounts of energy to excite in the limit of wavenumber k → 0. Spin waves due to χSB inQCD correspond to massless bosonic particle excitations, which are identified with the pion tripletπ+, π−, π0. These mesons are the lightest hadrons with masses mπ ∼ 135−140MeV. The next lightestmeson made from u and d quarks is ρ at 770MeV, and the lightest baryon is the proton at 938MeV.We have nonzero pion mass due to the explicit χSB by mu,d 6= 0. In QCD pions are important,because their lightness is the best evidence for a vacuum with χSB, and many of interactions in whichpions are included were predicted purely on symmetry grounds, long before the dynamics of QCDwere worked out.

While a complete description of hadrons in QCD is very difficult, we propose one through amuch simpler model for strong interaction and its two main properties (asymptotic freedom and colorconfinement), the Bag model, where massless quarks move freely within a spherical hadron of radiusR, but are prevented from traveling further by an inwards acting pressure due to the confining natureof the bulk vacuum [3]. We then assign a constant energy density Λ4

B to the non-confining region lyinginside hadron. The total energy of hadron with radius R then follows next relation

E ∼ R3Λ4B +

C

R, (9)

where second term comes from the kinetic energy of the confined quarks due to the uncertaintyprinciple and is not important for us now. We can then calculate the hadron mass by minimizing Ewith respect to R

M ∼ 4R3Λ4B. (10)

For rough approximation of bag constant we insert typical values M ∼ 1000MeV and R ∼ 1fm andso we get Λ4

B ∼ 200MeV.

3.2 Quark gluon plasma

Hadrons in QCD are composite particles made of quarks, which interact through strong force. Aspreviously discussed, one of the main properties of such an interaction is its asymptotic freedom athigh energies. Knowing these two facts, we could guess that when the temperature T is raised thereshould be a phase transition to a phase, where hadrons are no longer the main degrees of freedom butrather quarks and gluons by themselves, and the same would apply to squeezing hadrons together. Infact this actually happens and is currently an important field of research in physics.

Let us start by neglecting heavier quarks and taking in consideration QCD with only lightest twoquarks u and d. If we start at the bottom left of the QCD phase diagram and start increasing thetemperature, there is no net concentration of baryons (B = µ = 0) and therefore the dominant hadronicdegrees of freedom in such medium are pions, which carry zero B and are quite easily produced inpairs. At high enough temperatures, it is is possible to neglect their rest mass (T & mπ ∼ 140MeV)and then the pressure due to a pion gas is givien by the formula for blackbody radiation pressure:

Pπ = − ∂Ωπ

∂V

∣∣∣∣T,µ

= 3 ·(π2

90

)T 4, (11)

where the factor 3 arises from the number of pion charge states and Ωπ is the grand potential, whichwas already mentionied in the beginning when we talked about the phase diagram. Now similarly, weget an expression for a plasma of free light quarks and massless gluons, which are no longer confinedwithin the hadron. Its value is much larger due to many more degrees of freedom (plasma is a stateof higher entropy):

Pqq = 2 · 2 · 3 · 7

4·(π2

90

)T 4, Pg = 2 · 8 ·

(π2

90.

)T 4. (12)

Here we have some more numerical factors. For q-q pairs there are two helicity states, two flavor states(because we are dealing only with u and d quarks) and three color states (color-anticolor). The factor74 comes in because of the difference between Fermi-Dirac and Bose-Einstein statistics. For gluons,there are two helicity and eight color states. When hadron gas and quark-gluon plasma (QGP) are inequilibrium, so should be the pressure, but we must not forget to take confinement into account. The

6

easiest way to do so is by considering the bag constant of the previous section to act as a negativepressure for the QGP. So we get an estimate for critical temperature Tc:

1

30π2T 4

c =37

90π2T 4

c − Λ4B ⇒ Tc =

(45

17

Λ4B,

π2

)1/4

≈ 144MeV, (13)

which we can convert to kelvins (1K ≈ 8.6 ·10−5eV ) and get for our everyday experience a ridiculouslyhuge number (trillion kelvins). We can also calculate the energy density of the plasma state εQGP '850MeV/fm3 and the latent heat, which we must provide at the transition, ∆ε ' 800MeV/fm3. Suchhigh latent heat gives us a hint, that the system undergoes a discontinuous phase transition, which isnot necessarily true, because we used a very rough model to be able to make these calculations.

In practice such estimations are very complex to compute, especially when we must also con-sider matter at lower temperatures, where QCD is no longer pertubative. Currently the best ap-proach is lattice gauge theory, which provides us with the ability to make more refined calcula-tions, but unfortunately at high computing costs, so often the world’s fastest supercomputers areused. This way we also get better predictions about what happens with matter at such transitions.

〈ψψ〉

Tµ

Figure 6: Shown is value of chiral condensate re-garding to temperature and nuclear density. It van-ishes for high enough values of either of the param-eters. Unfortunately for most of the regions cal-culations are made only with use of approximatemodels. Also lattice QCD has little or no powerwhen we move away from T -axis[4].

In figure 6 we see the temperature dependanceof the chiral condensate. For T < Tc a largechiral condesate indicates χSB and is a signalfor divergent energy of an isolated color sourceknown as confinement. During the transitionat Tc ∼ 170MeV this condensate vanishes, chi-ral symmetry is restored and color is no longerconfined in the QGP. The physical backgroundis somewhat simillar to electric screening of thecharge in QED. But in QCD, the positive andnegative electric charges are replaced by the 8color charges, which this time can be carried ei-ther by quarks or gluons.

Why people are so interested in QGP is alsobecause of the way that universe cooled after theBig Bang. It began far up the vertical axis ofthe QCD phase diagram. At it earliest moments,shortly after the Big Bang, the universe was filledwith quark-gluon plasma that in very short pe-riod of time (10ms) expanded and cooled belowTc = 170MeV. During the phase transition theprocess of hadronization took place; since thenas a result quarks have been confined, except forthe ones at the centers of neutron stars (very high µ) and those in heavy ion collisions of today.

The history of our universe can also gives us information about the QCD phase diagram. Forinstance, if the QCD phase transition from hadronic to QGP phase was first–order, then it wouldproceed via a mixed phase, where hadronic bubbles would coexist with regions of QGP. But thisis not what happens in QCD: alongside with recent lattice calculations also homogeneous universe(no fluctuations in baryon concentrtion) proves that the deconfinement phase transition is rather across–over, that is, a relatively smooth process during which the pressure and all its derivatives remaincontinuous across the transition, then the n-th order phase transition, at least in the region of the lowµ.

3.3 Relativistic heavy ion collisions

It is hard to know more about events in the universe when QGP was the dominant phase in it,because our range of direct observation cannot penetrate beyond the epoch when the cosmic microwavebackground radiation was formed at t ∼ 105 years. Using equations that govern the expansion of the

7

Figure 7: Various stages of heavy ion collision[5].

universe, we were at least able to extrapolate back from the present conditions to estimate when andhow long the QGP phase was present.

In order to learn more about QCD and its properties and even about how our universe begun, wehad to reproduce conditions in which QGP forms. Only in relativistic heavy ion collisions such highenough temperatures can be reached. The first experiments have been performed in 1970s and 80swith fixed target nuclei at the Alternating Gradient Synchrotron (AGS) in Brookhaven and the SuperProton Synchrotron (SPS) in CERN with center of mass energies of 33 GeV 400 GeV respectively.Since then we have build new facilities and therefore the energies have become much larger. Most ofthe recent researches were performed at RHIC in Brookhaven and at the ALICE detector at the LargeHadron Collider (LHC) in CERN, where the energy has reached 14 TeV.

Various stages of such collision are showed in a simplified scheme in figure 7. First we have twoincoming nuclei flying towards each other. Because of the Lorentz contraction, these two in the centerof mass frame (which is the same as laboratory frame for RHIC and LHC) look like two pancakes witha longitudinal extent reduced by a Lorentz boost factor γ ∼ 100 compared to the radial extent in thetransverse plane. At the time of impact, initial events are very high energy inelastic collisions betweenindividual nucleons, which liberate many of the partons. Because of high density of nuclear materialinto which these partons are released, there is big chance of multiple rescattering, so their originallyhighly correlated momenta are redistributed. Therefore a large fraction of their initial kinetic energycan be used to produce qq pairs in the mid-rapidity region y ∼ 0, whereas most parts of initial nucleiproceed to move along the beam axis into forward and backward regions. Therefore, these two regionswill be rich with hadrons, but in the mid rapidity region the net baryon density will be close to zeroas a result of particle production through qq pairs formation. In case of enough energetical collision(so that we get T > Tc) this is the region where the QGP is formed. But we cannot directly probe it.It has to expand and cool down due to its excess pressure with the respect to vacuum and thereforeat some point T falls below Tc and hadrons reform. We have reached the chemical freezout, wherethe composition of formed hadrons is fixed. Now we are left with hadronic gas that continues to cooldown until the interaction rates become insufficient to preserve thermal equilibrium in such expandingmedium. This is known as thermal freezout, and from this point on the hadrons are free to streamaway to be detected.

To confirm each of the stages in the collision we need to find some evidence for it in the detectedparticles. For example, an evidence for thermalization comes from the analysis of the distribution

of transverse mass m⊥ =√m2

0 + k2⊥, which is found to be approximately Boltzmann for a variety

of different hadron species. But for purpose of this article we will demonstrate the effect of J/ψsuppression (T.Matsui and H.Satz Physics Letter B178 (1986)) as one of the signals that QGP wasformed during the collision, which stands out with its historical significance, because it was predictedjust from the calculations using strong model physics before there was any experimental proof for sucha phenomenon.

8

3.3.1 J/ψ suppression

J/ψ is a bound state made of a c quark and a c antiquark, which, in vacuum, interact throughcolored strong interaction. In a high energy heavy ion collisions a formation of J/ψ is possiblein any of the early and very energetic nucleon-nucleon inelastic scattering events, but occurs only

Figure 8: Summary of the various experimental results,that show anomalous J/ψ suppression observed in Pb-Pbcollisions as a function of increasing energy densities. Sup-pression is obtained from the ratio of the measured crosssections and the values expected from nuclear absorption.The latter are deduced from a fit to measurements on in-teractions of incident protons with various targets (emptycircles and triangles) and S-U collisions (filled squares),where no QGP should be present, because of the low en-ergy. We can see that also in Pb-Pb collisions J/ψ pro-duction is suppressed normally (just from pure nuclear ab-sorption) for energies low enough.[6]

from time to time. If at anytime duringthe collision QGP forms, such J/ψ findsitself in environment differing stronglyfrom the vacuum. Firstly, the high tem-perature deconfines both quarks and, sec-ondly, the colored strong interaction takesdifferent shape because of the screeningof the color charge due to all the lightquarks, antiquarks and gluons present inthe QGP that move freely trough themedium. In fact, strength of this newcolor-Yukawa interaction relates inverselyon the temperature, because the greaterthe temperature, the greater the quarkdensity surrounding charm quarks andscreening their charge. So the energylevel of the cc system rises with temper-ature and when the critical temperatureis reached, there can be a resonance of cand c at E = 0. If the temperature iseven higher, then there is no bound statefor these two quarks and no matter howclose c and c come together, they will driftapart. When the region cools down andthe QGP is gone, two charm quarks canbe so far away from each other that it isfar more likely that they will bind witha lighter quark or antiquark in D or Dstates than to form a J/ψ particle again.And thus the production of J/ψ is sup-pressed compared with a collision whenno QGP was formed. The experimentaldiscovery of this J/ψ suppression (Figure8) was a landmark achievement in our un-derstanding of QGP.

3.4 Color superconductivity

Until now we more or less pursued only the path of rising energy (temperature) at low or zero chemicalpotential µ. Now we will inspect the behavior of QCD as a function of the latter. We choose a startingpoint at T = 0, µ = 0, where at ground state no particle is present. This situation persists until µreaches the value of the nucleon rest mass minus the binding energy per nucleon in nuclear matterµ = µ0 ' 922 MeV, when in terms of energy it becomes more effective to form a ground statepopulated with nucleons in a bound nucleon fluid. At this point the baryon density nB = B

V = − 1V∂Ω∂µ

jumps from zero to nuclear density nB0 ' 0.16 fm−3 and this discontinuity in baryon density impliesa first order phase transition, which persist also when T 6= 0 and can be seen on our QCD phasediagram as a line emerging from µ = µ0. This line separates the phase, where baryons can be present,from the one in which they are condensed.

9

Now we arrive to the region about which little is known from exact calculations or even observation.Unfortunately, the lattice gauge theory simulations that are useful along the T -axis become ineffectiveonce applied to QCD with non zero µ. So researchers must solely rely on approximate models, suchas the bag model mentioned before or extrapolation from either the left region with low µ or the oneon the right with very high µ, where due to asymptotical freedom the main degrees of freedom areagain quarks and gluons, which are weakly bound if µ is high enough. Because of this weakly coupledQCD at high µ, it is predicted that, if also T is sufficiently small, QCD matter should be found ina color superconducting phase and therefore there should be a phase transition in between this colorsuperconducting and hadronic phases.

Now lets take a look why superconducting phase should occur. We can safely assume that at highenough density and almost zero temperature quarks form a degenerate Fermi liquid and because ofasimptotical freedom quarks near the Fermi surface are almost free with only weak QCD interactionsbetween them. For such conditions valid thermodynamic potential or free energy is Ω = E − µN ,where E is the total energy of the system and N is the number of fermions[7]. If no interactions werepresent then adding a particle to the system would not change the free energy, because to do so wewould need exactly the Fermi energy EF = µ. But with weak attractive interaction, if we add pair ofparticles (or holes) which attract each other, the free energy will be lowered by the potential energyof their attractions. So many of such pairs will be created in the modes near the Fermi surface andthis bosonic pairs will form a diquark condensate 〈ψψ〉. This pairing instability leads to an energygap 2∆ between the highest occupied and the lowest vacant one particle states at the Fermi surface(Figure 9). The ground state is a superposition of states with different numbers of pairs, breaking thefermion number symmetry and leading to superconducting phase. This is the same argument as wasoriginally used by Bardeen, Cooper and Schreiffer (BCS) and is completely general, so can be usedwhen describing electrons in superconducting metal, superfluidity of 3He atoms or quarks in quarkmatter.

k

E

EFFermiSea

pairinginstability

k

E

2∆

Figure 9: Pairing instabilities leading to superconductivity.[8]

In fact, application of the BCS mechanism to pairing in dense quark matter is more direct thencompared, for example, in metals. Between the electrons in a metal is a repulsive Coulomb interactionwhich has to be screened for phonon mediated interaction to come in play. On the other hand inQCD it is the primary interaction that is also responsibile for the color superconduction. Diquarkcondensate also breaks the local color symmetry SU(3)c, because pairs of quarks cannot be colorsinglets, and hence the name color superconduction.

Depending on the value of µ there can be different number of quark flavors involved. For exampleif we start with first two lightest quarks the immediate consequence of demanded diquark condensateantisymmetry is that out of eight gluons, the five which carry color #3 acquire a mass of O(∆) andtherefore cannot penetrate quark matter over distances much greater than a screening length ∼ ∆−1,in direct analogy with the Meissner effect in metallic superconductors.

For now any man made experiments for analysis of color SC phase of QCD are impossible to make,because we would need a system that is able to compress matter to super-nuclear densities, whichtranslates to billion tons per teaspoon of such matter. The only known place that this might happen

10

in nature is inside of the neutron stars, where gravity squeezes the star to high enough µ for longenough for weak interactions to equilibrate and reach state with low temperature. It is still unknownif such phase really occurs in stars, but for now most of the predictions from theoretical analysis ofwhat the properties of such stars would be in case of color SC core were confirmed.

4 Conclusion

During this text we managed to learn some basics about QCD and how the study of strong interactionrevealed whole new field of QCD thermodynamics. In past few years vast progress was made instudy of all of the QCD phases and many new discoveries are being published every year in numerouspublications all around the world. But to preserve compactness of the text, we rather used moregeneral and qualitative descriptions of phenomena instead of doing really detailed analysis with lotsof calculations. Besides of the huge dimension of ongoing research, it is also astonishing, how variousfields of physics, which are looking at the structures of totally different orders of size, have to connectto acquire any results. For example, in order to learn about how the universe begun, astronomers haveto look in to the whole visible sky and collect data about CMR radiation, while in the other hand,particle physicists observe behavior of the smallest known particles in heavy ion collisions. Maybethrough further research of QCD properties we will be also able to construct a more general theorythat would serve as common ground for existing ones. But one thing that is most certainly known, isthat there is still many unknowns and lots of research needed to learn more about them. So maybein few years we will be able to tell things like what really is dark matter, if it even exists.

References

[1] Muta, T., ’Foundations Of Quantum Chromodynamics: An Introduction to Perturbative Methodsin Gauge Theories’, London, World Scientific, 1998.

[2] Schonitzer, M., ’Teilchenphysik’, http://de.wikibooks.org/wiki/Teilchenphysik, 2011, (accessed 3March 2015).

[3] Chodos, A. et al., ’New extended model of hadrons’, Phys. Rev. D 9, 3471.

[4] R.S. Hayano, http://nucl.phys.s.u-tokyo.ac.jp/hayano/en/pionic-atoms.html, 2008, (accessed 3march 2015)

[5] NA49 Collaboration, http://na49info.web.cern.ch/na49info/Public/Press/findings.html, (ac-cessed 20 February 2015).

[6] NA50 Collaboration, http://na50.web.cern.ch/NA50/, 2000, (accessed 21 February 2015).

[7] Alford, Mark G. et al. ’Color superconductivity in dense quark matter’, Rev. Mod. Phys. 80,2008, p. 1455.

[8] Hands, S., ’The phase diagram of QCD’, Contemporary Physics, vol. 42, no. 4, 2001, pp. 209-225.

11