Download QCD Matter Phase Diagram

Seminar
QCD Matter Phase Diagram
Tim Kolar
Mentor: prof. dr. Simon Širca
April 10, 2014
Contents
1 Introduction
1
2 QCD - Quantum chromodynamics
1
3 Phase diagram of QCD matter
3.1 QCD vacuum and low energy, low density phase
3.2 Quark gluon plasma . . . . . . . . . . . . . . . .
3.3 Relativistic heavy ion collisions . . . . . . . . . .
3.3.1 J/ψ suppression . . . . . . . . . . . . . .
3.4 Color superconductivity . . . . . . . . . . . . . .
3
4
6
7
9
9
4 Conclusion
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
11
1
Introduction
We are all familiar with the fact that ordinary matter can be found in several different forms or
states of aggregation, known as phases. The same applies to QCD matter where, instead of only
the nucleus and electrons, we have two degrees of freedom, quarks and gluons. In this paper we will
explore the consequences of this quark-gluon picture for thermodynamics of such strongly interacting
matter, 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 different
phases of a substance occupy different regions of a plot whose axes are calibrated in terms of the
external conditions through control parameters. The most familiar example is probably that of H2 O,
but we will focus on the proposed phase diagram for QCD that is shown in figure 4. But first we need
to explain what QCD stands for.
2
QCD - Quantum chromodynamics
Quantum chromodynamics, a theory of quarks, gluons and their interaction, is an independent part
of 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 force
carriers is given to gluons, which are bosons. Unlike photons, which are not carrying any electrical
charge, 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 color
states (octet+singlet). Although it is mathematically possible to form a colorless singlet, there is no
such gluon to be found in nature. So we have 8 different gluons. The fact that force carriers for
αEM
αS
1
1
137
r [nm]
r [fm]
(a)
(b)
Figure 1: Coupling constants of electromagnetic and strong forces. Actually, due to relative
strong dependence on distance between interacting particles, it would be better if we would
refer to both of them as strengths, but for historical reasons the term constant is used. In
the left figure the fading nature of the electromagnetic interaction is easily seen, as it will be
hard to miss the characteristics of the strong coupling constant that give rise to two basic
properties of QCD which we will describe later.
strong interaction are not charge neutral, and as such can interact with each other, has far reaching
consequences. Because of this possibility, we get special energy dependence of strong interaction
coupling constant αs (Figure 1b). As we can see its value increases with distance and thus decreases
with 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 we
refer to it again, when we try to explain some of the QCD matter phases. On the other hand, when
the energy decreases, αs increases and at low enough energies becomes too large to use the common
aproach of calculating variables by using the perturbation method, where an expansion in powers of
αs is needed. This leads to big uncertainties in calculations of such processes as, for example, if we
would like to know how we get such an increase in rest mass when three very light quarks bind in a
nucleon. This increase of αs with distance, combined with the fact that the only stable finite energy
systems in QCD are those formed from complementary combinations of color, such as qr q̄r̄ 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 kept
together, because the field strength does not fade away with increasing distance. So at one
moment becomes energeticaly favourable to form another pair of appropriate quarks then to
increase the distance further.
or qr qg qb 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 with
appropriate colors and we try to drag them appart as shown on the right side of figure 2. Since
we know, that the strong potential increases with distance between two quarks, so at some point it
becomes energeticaly favourable to produce a pair of a quark and antiquark instead of increasing the
distancce further. But the q q̄ pair formed in the separation process must be produced with exactly the
right color combination to maintain overall color neutrality, so we end up with two hadrons. Because
of 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 dynamics of QCD in a bit more mathematical way by writing down the Lagrangian. On a basic level we
are dealing with quarks (s = 21 ), 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 discussion
we will focus on three lightest quarks (u, d and s), so we will have Nf = 3. The strong interaction
between them involves Nc = 3 different color charges and is mediated by 8 gluons, which are generators
or gauge bosons for the underlying gauge group of QCD, SU (3)color . So the QCD Lagrangian is (and
in gray its QED analog for comparison) [1]
1
LQCD = ψ̄(iγµ Dµ − m)ψ − Gjµν Gµν
j ,
4
(2)
1
µ
LQED = ψ̄(iγµ DQED
− m)ψ − Fµν F µν
4
(3)
where ψ̄ = ψ † γ 0 is the Dirac conjugate spinor, γ µ are the Dirac matrices and m is the appropriate
quark mass matrix m = diag(mu , md , ms , ...). The gauge-covariant derivative is defined as
Dµ = ∂µ − ig
8
X
λj
j=1
2
Ajµ (x),
µ
DQED
= ∂µ − ieAµ (x),
2
(4)
(5)
2
g
with the strong coupling constant g (αs ≡ 4π
) and Gell-Mann matrices λj , which form for j = 1, ..., 8
the 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 denote
gluon fields. This gauge-covariant derivative is formed in such way, that out of vector defined in
point of space and its surrounding vector field we get again an vector. It is distinct from the normal
derivate because of its invariance under gauge transformations. In other words, it preserves all physical
properties when the fields undergo osuch transformations. The gluon (photon) field strength tensor is
Giµν (x) = ∂µ Aiν − ∂ν Aiµ + gfijk Ajµ Akν ,
(6)
Fµν (x) = ∂µ Aν − ∂ν Aµ ,
(7)
where fijk are the structure constants of the local SU (3)color symmetry. These structure constant of
a Lie group determine the commutation relations between its generators (gluons in our case) in the
associated Lie algebra and are defined as [λi , λj ] = fijk λk .
At the first glance the Lagrangians of QCD and QED are quite alike. Equations do differ because
of different number of force carriers, which does not seem to be of much consequence. But when we
inspect the differences in the field strength tensors from equation 6 and 7, we can notice, the additional
term in the gluon field strength tensor. The first two terms in both tensors describe the interaction
e−
e−
ū
d
γ
e−
ū
e−
g
d
(b) Multi-gluon interaction in QCD
(a) Single photon interaction in QED
Figure 3: In nature the electromagnetic force is always carried by one or more photons that
are not interacting with each other, whereas strong interaction can be transmitted either by
single or several gluons, which can also interact with each other. This gives a rise to a range
of unusual properties in QCD.
through the exchange of one force carrier particle, while the third one in gluon field tensor is due
to possible gluon self-interaction. As we have already discussed before, this has a huge impact on
the properties of QCD. All of them have an important role, when describing different phases of such
matter.
3
Phase diagram of QCD matter
Figure 4 shows the proposed phase diagram for QCD. The control parameters are the temperature
T and the baryon chemical potential µ. It is already known that most of the visible matter in the
universe is made of subatomic particles called hadrons, which we divide in two groups. Baryons are
formed of three quarks qqq and mesons are those composed of one quark and one antiquark q q̄. In all
particle physics reactions a qqq baryon is always created or destroyed pairwise with q̄ q̄ q̄ antibaryon.
There is no process within QCD, which can change the number of baryons NB minus the number of
antibaryons NB̄ . In other words, we can identify a conserved quantum number B = NB − NB̄ called
baryon number. We can conclude that quarks and antiquarks carry B = ± 13 respectively. We can
have a system where due to other interactions, the baryon number is allowed to vary. In such a system
the most convenient thermodynamic potential is the grand potential Ω(T, V, µ) = E − T S − µB, which
can be minimized to find the thermodynamic equilibrium of the system and then we can determine
µ as the increase in E whenever B increases by one. But when systems are described by using the
grand canonical ensemble, we keep µ as a control parameter, and instead of baryon number the baryon
density nB (T, µ) = B/V is a derived quantity.
3
T [MeV] Universe Cooling
LHC
RHIC
Quark-Gluon
Plasma
170
10
Nuclear
Matter
Hadron
Gas
Color
Superconductor
µ[MeV]
938
Figure 4: The phase diagram for strongly interacting matter (QCD). We can distinguish
between three basic phases: hadronic phase, quark gluon plasma and color super-conducting
quark 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 to
be higher order (black lines). We can also see the approximate path that the universe followed
during the cool down and the paths of matter that participates in heavy ion collisions. Also
very important is the point marked with the red dot, because more or less all of the matter
in 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 very
low 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 lowest
possible energy. If we are dealing with a quantum theory, which can be calculatede using petrubation
theory (like QED), the properties of the vacuum are analogous to those from the ground state of
any quantum mechanical problem including the quantum harmonic oscillator. In such a case the vacuum expectation value
of any field operator is zero. But in the case of low energy nonperturbative QCD it is possible to have a non-vanishing vacuum
expectation values called condensates. Two of these are quarkantiquark condensate hψ̄ψi and gluon condnesate hGµν Gµν i.
The principle is very much similar to the one if we cool helium
to superfluid state when a Bose condensate of He atoms in the
lowest quantum state forms.
But now we arrived to another important aspect of QCD
dynamics. When a particle with spin ~s propagates, it is possible
to define a quantity called helicity h = ~s.~k/|k|, which is the
projection of the spin axis along the direction of the particle’s
motion, defined by the momentum ~k. For a half-integer spin
quark, there are two possible helicity eigenstate values h = ± 21 , Figure 5: Particles with left and
usually referred to as left and righthanded states. A quark’s right handed helicity states. [2]
helicity 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 handed
quarks are separately conserved in QCD and therefore leading to two good quantum numbers BL
and BR . This would be the case, if quarks would have zero mass and hence travel at the speed of
light. Otherwise, we can Lorentz boost to a frame, in which the quark’s momentum along the boost
axis changes to the opposite sign and with it so does the helicity, because angular momentum stays
unchanged. It turns out that in a relativistically covariant treatment massive quarks must be described
as superposition of helicity eigenstates and the mass m is parametrising the overlap between them
and hence effectively the rate of L ↔ R transitions. So only B = BL + BR remains a good quantum
number and thus we say, that the chiral symmetry relating left and righthanded quarks and describing
them 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 by
the ket |0i, and the field operators which create or destroy a quark when acting on a ket as ψ̄,ψ
respectively. We can write down the expectation value of the quark-antiquark condensate as
hψ̄ψi = h0|ψ̄L ψR + ψ̄R ψL |0i =
6 0.
(8)
Since neither |0i is anhilated by ψ nor h0| by ψ̄, the vacuum must contain q q̄ pairs. value of such
condensate is hψ̄ψi ' (250MeV)3 and can be interpreted as the number of such pairs per unit volume.
Relation (8) also implies pairing of ψL with ψ̄R and ψR with ψ̄L . Because of its nonzero value and the
fact that hψ̄ψi changes BL − BR for two units, but leaves BL + BR invariant we get to see again that
only B is a good quantum number.
A left handed quark traveling through such a vacuum with non-vanishing condensate can be anhilated by ψL , leaving ψ¯R to create a right handed quark with the same momentum. The quark will thus
flip its helicity at a rate proportional to the value of the condensate hψ̄ψi. Or differently stated, it will
propagate just as if it had a mass. This mass is not the same as the quark’s current mass m. We usually
refer to such dynamically generated mass as constituent mass Σ. If chiral symmetry breaking (χ SB)
occurs spontaneously by the formation of a large condensate, then Σ may be much larger than m due to
this dynamical mass generation. We have written the estimates of both types of masses for the known
six quarks in table 1.
Spontaneous χ SB in QCD is there- Table 1: Estimates of current and dynamically gathered
fore the process by which the nucleon, constituent quark mass. Because isolated quarks are never
and with it all matter in the universe, observed, neither quantity is precisely defined.
acquires most of its mass. Let us now
discuss a useful analogy for χ SB namely
Quark Current mass m Constituent mass Σ
ferromagnetism in metals. In a metal
flavor
[MeV/c2 ]
[MeV/c2 ]
each atom occupies a spot on a sped
∼7
∼ 350
cific crystal lattice, and has an unpaired
electron carrying a magnetic moment
u
∼3
∼ 350
or spin that we denote with ↑ or ↓
s
∼ 140
∼ 550
(s = ±1/2). If the temperature is low
enough, i.e. bellow the Curie temperc
∼ 1800
∼ 1800
ature (T < Tc ), the system undergoes
b
∼ 4.2 × 103
∼ 4.2 × 103
a phase transition to its ferromagnetic
t
∼ 170 × 103
∼ 170 × 103
state in which a macroscopic fraction
of the spins is aligned, resulting in the
spontaneous magnetization of the sample M = h↑i 6= 0 (in this case angled brackets denote a thermodynamic average rather than a quantum expectation value). Since the magnetization axis defines
a particular direction in space, the original symmetry of Hamiltonian under rotations of the spin axis
is spontaneously broken by M 6= 0. The same effect can be promoted by an external magnetic field
H 6= 0, but with explicit symmetry breaking in this case. The relation between M and H exactly
mirrors that between nonzero condensate hψ̄ψi and mass m in QCD.
An even more interesting similarity of these two systems arises, when we analyze the spectrum
of excitations above the ferromagnetic ground state M 6= 0. It turns out, that coherent oscillations
of 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 χ SB in
QCD correspond to massless bosonic particle excitations, which are identified with the pion triplet
π + , π − , π 0 . These mesons are the lightest hadrons with masses mπ ∼ 135−140MeV. The next lightest
meson 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 χ SB by mu,d 6= 0. In QCD pions are important,
because their lightness is the best evidence for a vacuum with χ SB, and many of interactions in which
pions are included were predicted purely on symmetry grounds, long before the dynamics of QCD
were worked out.
While a complete description of hadrons in QCD is very difficult, we propose one through a
much simpler model for strong interaction and its two main properties (asymptotic freedom and color
confinement), the Bag model, where massless quarks move freely within a spherical hadron of radius
R, but are prevented from traveling further by an inwards acting pressure due to the confining nature
of the bulk vacuum [3]. We then assign a constant energy density Λ4B to the non-confining region lying
inside hadron. The total energy of hadron with radius R then follows next relation
E ∼ R3 Λ4B +
C
,
R
(9)
where second term comes from the kinetic energy of the confined quarks due to the uncertainty
principle and is not important for us now. We can then calculate the hadron mass by minimizing E
with respect to R
M ∼ 4R3 Λ4B .
(10)
For rough approximation of bag constant we insert typical values M ∼ 1000MeV and R ∼ 1fm and
so we get Λ4B ∼ 200MeV.
3.2
Quark gluon plasma
Hadrons in QCD are composite particles made of quarks, which interact through strong force. As
previously discussed, one of the main properties of such an interaction is its asymptotic freedom at
high energies. Knowing these two facts, we could guess that when the temperature T is raised there
should be a phase transition to a phase, where hadrons are no longer the main degrees of freedom but
rather quarks and gluons by themselves, and the same would apply to squeezing hadrons together. In
fact 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 two
quarks u and d. If we start at the bottom left of the QCD phase diagram and start increasing the
temperature, there is no net concentration of baryons (B = µ = 0) and therefore the dominant hadronic
degrees of freedom in such medium are pions, which carry zero B and are quite easily produced in
pairs. 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:
2
∂Ωπ π
Pπ = −
=3·
T 4,
(11)
∂V T,µ
90
where the factor 3 arises from the number of pion charge states and Ωπ is the grand potential, which
was already mentionied in the beginning when we talked about the phase diagram. Now similarly, we
get an expression for a plasma of free light quarks and massless gluons, which are no longer confined
within the hadron. Its value is much larger due to many more degrees of freedom (plasma is a state
of higher entropy):
2
2 7
π
π
4
T , Pg = 2 · 8 ·
. T 4.
(12)
Pqq̄ = 2 · 2 · 3 · ·
4
90
90
Here we have some more numerical factors. For q-q̄ 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 factor
7
4 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 in
equilibrium, 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 negative
pressure for the QGP. So we get an estimate for critical temperature Tc :
1 2 4 37 2 4
π Tc = π Tc − Λ4B
30
90
⇒
Tc =
45 Λ4B ,
17 π 2
1/4
≈ 144MeV,
(13)
which we can convert to kelvins (1K ≈ 8.6 · 10−5 eV ) and get for our everyday experience a ridiculously
huge 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 . Such
high latent heat gives us a hint, that the system undergoes a discontinuous phase transition, which is
not 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 consider matter at lower temperatures, where QCD is no longer pertubative. Currently the best approach is lattice gauge theory, which provides us with the ability to make more refined calculations, but unfortunately at high computing costs, so often the world’s fastest supercomputers are
used. This way we also get better predictions about what happens with matter at such transitions.
In figure 6 we see the temperature dependance
of the chiral condensate. For T < Tc a large
hψ̄ψi
chiral condesate indicates χ SB and is a signal
for divergent energy of an isolated color source
known as confinement. During the transition
at Tc ∼ 170MeV this condensate vanishes, chiral symmetry is restored and color is no longer
confined in the QGP. The physical background
is somewhat simillar to electric screening of the
charge in QED. But in QCD, the positive and
negative electric charges are replaced by the 8
color charges, which this time can be carried ei- T
µ
ther by quarks or gluons.
Why people are so interested in QGP is also
because of the way that universe cooled after the
Big Bang. It began far up the vertical axis of Figure 6: Shown is value of chiral condensate rethe QCD phase diagram. At it earliest moments, garding to temperature and nuclear density. It vanshortly after the Big Bang, the universe was filled ishes for high enough values of either of the paramwith quark-gluon plasma that in very short pe- eters. Unfortunately for most of the regions calriod of time (10ms) expanded and cooled below culations are made only with use of approximate
Tc = 170MeV. During the phase transition the models. Also lattice QCD has little or no power
process of hadronization took place; since then when we move away from T -axis[4].
as a result quarks have been confined, except for
the 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. For
instance, if the QCD phase transition from hadronic to QGP phase was first–order, then it would
proceed via a mixed phase, where hadronic bubbles would coexist with regions of QGP. But this
is 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 a
cross–over, that is, a relatively smooth process during which the pressure and all its derivatives remain
continuous across the transition, then the n-th order phase transition, at least in the region of the low
µ.
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 microwave
background 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 and
how long the QGP phase was present.
In order to learn more about QCD and its properties and even about how our universe begun, we
had to reproduce conditions in which QGP forms. Only in relativistic heavy ion collisions such high
enough temperatures can be reached. The first experiments have been performed in 1970s and 80s
with fixed target nuclei at the Alternating Gradient Synchrotron (AGS) in Brookhaven and the Super
Proton 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 of
the recent researches were performed at RHIC in Brookhaven and at the ALICE detector at the Large
Hadron 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 two
incoming nuclei flying towards each other. Because of the Lorentz contraction, these two in the center
of mass frame (which is the same as laboratory frame for RHIC and LHC) look like two pancakes with
a longitudinal extent reduced by a Lorentz boost factor γ ∼ 100 compared to the radial extent in the
transverse plane. At the time of impact, initial events are very high energy inelastic collisions between
individual nucleons, which liberate many of the partons. Because of high density of nuclear material
into which these partons are released, there is big chance of multiple rescattering, so their originally
highly correlated momenta are redistributed. Therefore a large fraction of their initial kinetic energy
can be used to produce q q̄ pairs in the mid-rapidity region y ∼ 0, whereas most parts of initial nuclei
proceed to move along the beam axis into forward and backward regions. Therefore, these two regions
will be rich with hadrons, but in the mid rapidity region the net baryon density will be close to zero
as a result of particle production through q q̄ 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 therefore
at some point T falls below Tc and hadrons reform. We have reached the chemical freezout, where
the composition of formed hadrons is fixed. Now we are left with hadronic gas that continues to cool
down until the interaction rates become insufficient to preserve thermal equilibrium in such expanding
medium. This is known as thermal freezout, and from this point on the hadrons are free to stream
away to be detected.
To confirm each of the stages in the collision we need to find some evidence for it in the detected
particles. For example, anqevidence for thermalization comes from the analysis of the distribution
2 , which is found to be approximately Boltzmann for a variety
of transverse mass m⊥ = m20 + k⊥
of different hadron species. But for purpose of this article we will demonstrate the effect of J/ψ
suppression (T.Matsui and H.Satz Physics Letter B178 (1986)) as one of the signals that QGP was
formed during the collision, which stands out with its historical significance, because it was predicted
just from the calculations using strong model physics before there was any experimental proof for such
a phenomenon.
8
3.3.1
J/ψ suppression
J/ψ is a bound state made of a c quark and a c̄ antiquark, which, in vacuum, interact through
colored strong interaction. In a high energy heavy ion collisions a formation of J/ψ is possible
in any of the early and very energetic nucleon-nucleon inelastic scattering events, but occurs only
from time to time. If at anytime during
the collision QGP forms, such J/ψ finds
itself in environment differing strongly
from the vacuum. Firstly, the high temperature deconfines both quarks and, secondly, the colored strong interaction takes
different shape because of the screening
of the color charge due to all the light
quarks, antiquarks and gluons present in
the QGP that move freely trough the
medium. In fact, strength of this new
color-Yukawa interaction relates inversely
on the temperature, because the greater
the temperature, the greater the quark
density surrounding charm quarks and
screening their charge. So the energy
level of the cc̄ system rises with temperature and when the critical temperature
is reached, there can be a resonance of c
and c̄ at E = 0. If the temperature is
even higher, then there is no bound state
for these two quarks and no matter how
Figure 8: Summary of the various experimental results,
close c and c̄ come together, they will drift
that show anomalous J/ψ suppression observed in Pb-Pb
apart. When the region cools down and
collisions as a function of increasing energy densities. Supthe QGP is gone, two charm quarks can
pression is obtained from the ratio of the measured cross
be so far away from each other that it is
sections and the values expected from nuclear absorption.
far more likely that they will bind with
The latter are deduced from a fit to measurements on ina lighter quark or antiquark in D or D̄
teractions of incident protons with various targets (empty
states than to form a J/ψ particle again.
circles and triangles) and S-U collisions (filled squares),
And thus the production of J/ψ is supwhere no QGP should be present, because of the low enpressed compared with a collision when
ergy. We can see that also in Pb-Pb collisions J/ψ prono QGP was formed. The experimental
duction is suppressed normally (just from pure nuclear abdiscovery of this J/ψ suppression (Figure
sorption) for energies low enough.[6]
8) was a landmark achievement in our understanding of QGP.
3.4
Color superconductivity
Until now we more or less pursued only the path of rising energy (temperature) at low or zero chemical
potential µ. Now we will inspect the behavior of QCD as a function of the latter. We choose a starting
point at T = 0, µ = 0, where at ground state no particle is present. This situation persists until µ
reaches the value of the nucleon rest mass minus the binding energy per nucleon in nuclear matter
µ = µ0 ' 922 MeV, when in terms of energy it becomes more effective to form a ground state
populated with nucleons in a bound nucleon fluid. At this point the baryon density nB = VB = − V1 ∂Ω
∂µ
jumps from zero to nuclear density nB0 ' 0.16 fm−3 and this discontinuity in baryon density implies
a first order phase transition, which persist also when T 6= 0 and can be seen on our QCD phase
diagram 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 ineffective
once applied to QCD with non zero µ. So researchers must solely rely on approximate models, such
as the bag model mentioned before or extrapolation from either the left region with low µ or the one
on the right with very high µ, where due to asymptotical freedom the main degrees of freedom are
again quarks and gluons, which are weakly bound if µ is high enough. Because of this weakly coupled
QCD at high µ, it is predicted that, if also T is sufficiently small, QCD matter should be found in
a color superconducting phase and therefore there should be a phase transition in between this color
superconducting and hadronic phases.
Now lets take a look why superconducting phase should occur. We can safely assume that at high
enough density and almost zero temperature quarks form a degenerate Fermi liquid and because of
asimptotical freedom quarks near the Fermi surface are almost free with only weak QCD interactions
between 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 were
present then adding a particle to the system would not change the free energy, because to do so we
would need exactly the Fermi energy EF = µ. But with weak attractive interaction, if we add pair of
particles (or holes) which attract each other, the free energy will be lowered by the potential energy
of their attractions. So many of such pairs will be created in the modes near the Fermi surface and
this bosonic pairs will form a diquark condensate hψψi. This pairing instability leads to an energy
gap 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 the
fermion number symmetry and leading to superconducting phase. This is the same argument as was
originally used by Bardeen, Cooper and Schreiffer (BCS) and is completely general, so can be used
when describing electrons in superconducting metal, superfluidity of 3 He atoms or quarks in quark
matter.
E
E
pairing
instability
2∆
Fermi
Sea
EF
k
k
Figure 9: Pairing instabilities leading to superconductivity.[8]
In fact, application of the BCS mechanism to pairing in dense quark matter is more direct then
compared, for example, in metals. Between the electrons in a metal is a repulsive Coulomb interaction
which has to be screened for phonon mediated interaction to come in play. On the other hand in
QCD it is the primary interaction that is also responsibile for the color superconduction. Diquark
condensate also breaks the local color symmetry SU (3)c , because pairs of quarks cannot be color
singlets, and hence the name color superconduction.
Depending on the value of µ there can be different number of quark flavors involved. For example
if we start with first two lightest quarks the immediate consequence of demanded diquark condensate
antisymmetry is that out of eight gluons, the five which carry color #3 acquire a mass of O(∆) and
therefore 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, which
translates 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 long
enough for weak interactions to equilibrate and reach state with low temperature. It is still unknown
if such phase really occurs in stars, but for now most of the predictions from theoretical analysis of
what 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 interaction
revealed whole new field of QCD thermodynamics. In past few years vast progress was made in
study of all of the QCD phases and many new discoveries are being published every year in numerous
publications all around the world. But to preserve compactness of the text, we rather used more
general and qualitative descriptions of phenomena instead of doing really detailed analysis with lots
of calculations. Besides of the huge dimension of ongoing research, it is also astonishing, how various
fields of physics, which are looking at the structures of totally different orders of size, have to connect
to acquire any results. For example, in order to learn about how the universe begun, astronomers have
to 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. Maybe
through further research of QCD properties we will be also able to construct a more general theory
that would serve as common ground for existing ones. But one thing that is most certainly known, is
that there is still many unknowns and lots of research needed to learn more about them. So maybe
in 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 Methods
in Gauge Theories’, London, World Scientific, 1998.
[2] Schonitzer, M., ’Teilchenphysik’, http://de.wikibooks.org/wiki/Teilchenphysik, 2011, (accessed 3
March 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 3
march 2015)
[5] NA49 Collaboration, http://na49info.web.cern.ch/na49info/Public/Press/findings.html, (accessed 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