Intro. Relativistic Heavy
Ion Collisions
The QCD Phase Transition
Manuel Calderón de la Barca Sánchez
What happens to strongly interacting matter
at high temperature?
at high density?
Key Features:
Intrinsic size of hadrons
Hadron radius: rh » 1 fm
4 3
Need a Volume Vh » p rh to exist.
3
 Implies a limiting density:
3
1
nc » = 0.23 1/fm 3 » 1.5nNM where nNM = 0.16 1/fm
Vh
Pomeranchuk, Doklady Akad. Nauk. SSSR 78 (1951) 2
Resonances
Exponential hadron spectrum: r(m) ~ exp(m / T )
First appearance: Statistical Bootstrap Model, R. Hagedorn: Nuovo Cim. Suppl. 3, 147 (1965) 2, 16
Hadron Thermodynamics: Limiting temperature of Hadronic Matter.
Tc » 150 - 200 MeV
Can we go beyond nc and Tc?
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Deconfinement transition: Wear your colors proudly!
Hadronic Matter: Colorless constituents of hadronic size
Quark-Gluon Plasma: Colored constituents, pointlike.
Deconfinement:
 Color Conductor Transition in QCD
Chiral transition: Lose the weights that bind you!
Shift in the effective mass of constituent quarks.
At T=0 in the vacuum: quarks are “dressed” with gluons
Constituent quarks
Bare quark mass: mq » 0, becomes constituent quark mass:M q » 300 MeV
In a hot QGP: Dressing melts. M q ® 0
Since
, Lagrangian has chiral symmetry for
mq » 0
M q » 300 MeV : Chiral symmetry is spontaneously broken.
M q ® 0 : Chiral symmetry restoration.
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Chiral: “Handedness”
Dirac Fields can be decomposed into “right” or “left”
æ 0 0 1 0 ö
handed projections
ç
÷
1- g
1+ g
y =
y, and y =
y where g 5 º ig 0g 1g 2g 3 = ç 0 0 0 1 ÷
5
L
5
R
2
2
ç 1 0 0 0 ÷
ç
÷
0
1
0
0
è
ø
For massless particles: chirality is the same as
helicity
Helicity:
Right Handed: direction of motion equals direction of spin.
For massive particles: chirality ≠ helicity
Must rely on definition of yL and yR
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When yL and yR transform independently
QCD with two massless quarks
In terms of left and right handed spinors:
Define:
é u ù
q =ê
ú
d
ë
û
so we can write:
This is invariant under
a rotation of qR by any 2x2 unitary matrix
a rotation of qL by any 2x2 unitary matrix
This symmetry is called chiral symmetry
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For two massless quarks, the QCD vacuum
breaks the exact SU(2)RxSU(2)L symmetry.
Quark condensate: y y ¹ 0 at low temperature
Goldstone bosons which correspond to the three
broken generators: pions.
They are massive: QCD gives mass to pions, protons.
y y ® 0 at
large T : order parameter.
Pions would be approximately massless at large T.
In the real world, QCD vacuum only has an
approximate SU(2)RxSU(2)L symmetry.
Pions have small, but non-zero mass, at large T.
Changes in the spectral properties of r meson?
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Diquark Matter
Deconfined quarks :
attractive interaction
Can form colored, bosonic,
“diquark” pairs
QCD Equivalent of QED
Cooper pairs
Form QCD quark
condensate
Color superconductor
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Simplest form of Confined Matter:
Pion Gas:
Pp =
p2
1
3T 4 » T 4
90
3
Simplest form of Deconfined Matter:
Ideal, weakly coupled, Quark-Gluon Plasma
PQGP =
p2 æ
ö 4
7
4
2
´
8
+
[2
´
2
´
2
´
3]
ç
÷T - B » 4T - B
ø
90 è
8
B: Bag Pressure
QCD Vacuum: ideal color dielectric
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Expels color field
Color charge is confined
Bag pressure:
Difference btw physical vacuum & quark-gluon
ground state
B1/4 ~ 200 MeV (Quarkonium Spectroscopy)
8
Compare:
Pp (T ), PQGP (T ) vs. T
Phase transition from Hadronic Matter
at low T to QGP at high T.
Critical Temperature:
2
2
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I
n
c
r
e
a
s
i
n
g
T
3p 4 37 p 4
45
4
Pp = PQGP Þ
Tc =
Tc - B Þ T c =
B Þ Tc » 150 MeV
2
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90
90
17 p
Pion Gas:
QGP:
1st Order phase transition,
by construction:
At Tc, energy density
changes abruptly
Discontinuous jump
Latent heat of
deconfinement
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Compare energy density and pressure
Ideal Gas:
“Interaction Measure”:
Shows that for Tc £ T £ 2 - 3Tc QGP is strongly
interacting (not ideal gas)
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Partition Function: Z(T,V)
Thermodynamic quantities:
æ ¶ln Z ö
P =T ç
÷
è ¶V øT
For 2, 2+1 flavors:
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Deconfinement Transition
mq ® ¥
F (r,T )
é Nt -1
ù
- QQ
Polyakov Loop : L(T ) = Tr êÕU 4 ( j,n)ú = lim e 2T
êë j=0
úû r®¥
F free energy of QQpair for r ®¥
QQ
ìï = 0, T < T confinement
L
L(T ) í
ïî ¹ 0, T > TL deconfinement
mq ® 0
Chiral Condensate: c (T ) º yy ~ M q
Chiral Transition
Measures dynamically generated (constituent) quark mass
ì ¹ 0, T < T chiral symmetry broken
ï
c
c (T ) í
ïî = 0, T > Tc chiral symmetry restored
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F. Karsch and E. Laermann:
Phys. Rev. D 50, 6954 (1994
Polyakov Loop and Chiral Condensate
In the appropriate limits: both transitions fall in the same
universality class as the Z(3) Potts model
In the real world, symmetries are not exact
L(T) and c(T) still indicate a rapid change
Both transitions occur at about the same temperature
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Interaction measure:
Peaks above Tc
Two regimes of QGP
Strongly coupled QGP
sQGP: Tc £ T £ (2 - 2.5)Tc
Weakly coupled QGP
wQGP: T ³ 2.5Tc
Interaction measure in real QCD:
interpreted as arising due to Bag pressure B.
M. Asakawa and T. Hatsuda: Nuc Phy A 610, 470c (1996)
or colored “resonance” states above Tc.
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E. Shuryak and I. Zahed: Phys. Rev. C 70, 021901 (2004)
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At m~0: rapid cross-over
No discontinuity, no thermal
singularity
Strictly speaking:
Phases are not distinct
No phase transition between
them
Lattice: rapid transition.
Something is changing, but
how to better
define/understand it?
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Making pudding, boiling egg
Treated using “percolation”
Formation of clusters
Correspond to critical behavior
Geometric (not thermodynamic) quantities diverge
Cluster size
Cannot be obtained from partition function
Two-dimensional disk percolation
distributed lilies randomly on surface of a pond (overlap ok)
small disks, area a
Pond, area A a
When can ant walk across?
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For constituents with intrinsic scale
 formation of infinite connected
clusters
S(n):average cluster size
increases with increasing density
n=N/A
Suddendly, at n=nc, clusters are
large enough to span pond
S~A
Limit: N ®¥, A ®¥, n = constant
S(n), and
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dS(n)
diverge
dn
at n ® nc
Geometric
form of critical behavior
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2-D disks :
nc =1.13 / p r 2
68 % of space covered, 32 % empty
when an ant can cross, a ship cannot, and
vice versa
But Only in 2-D
3-D spheres (Hadrons:
nc
rh
0.34 / (4 p r 3 / 3) = 0.16 fm-3 :
0.8 fm )
Hadron Percolation
29 % of space covered, 71 % empty
both cluster and empty space are connected
nc 1.24 / (4 p r 3 / 3) = 0.56 fm-3 :
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71% of space covered, 29% empty
connected vacuum disappears
Assume medium is ideal gas of hadrons and
their resonances: nres (Tc ) = nc Þ Tc 170 MeV
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Vacuum Percolation
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At high temperatures and/or densities, strongly interacting
matter becomes a QGP;
How can we probe its properties and its behavior as function
of temperature and density?
Given a volume of strongly interacting matter and an energy
source, how can we determine its state at different
temperatures?
Possible probes:
Hadron Radiation
EM Radiation
Quarkonium Dissociation
Parton energy loss
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Emission of light hadrons
made of (u,d,s) quarks
scale ~ 1 fm ~ 1/(200 MeV)
Cannot exist in hot interior
Formed at transition surface btw QGP
and vacuum
Light hadrons are born at T=Tc
independent of initial temperature
Carry information about hadronization stage
Experimental observable:
Same relative abundances, independent of initial energy density
Does it not carry any information about pre-hadronic
medium?
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Medium is not contained
Can expand freely
Hydrodynamic flow
Radial Flow
Boosts hadron momenta
Elliptic Flow
Non-spherical initial state in peripheral collisions
Pressure gradients are different in different directions:
Azimuthal anisotropy
Final configuration of system depends on hydrodynamic
time-evolution
Can provide information about Equation of State of hot QGP
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Quark-gluon Interactions
Quark-antiquark annihilation
photons and dileptons
Key: Do not carry color
Leave medium ~ without further
interaction
Provide information at time of their production
Probe of Hot QGP
Difficulty: They are produced throughout system evolution
Hadron gas can also produce electromagnetic radiation
Experimental goal: disentangle hot thermal radiation from
QGP versus other sources
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Hadronic/EM Radiation: produced by medium.
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Quarkonium
cc,bb bound states
Smaller than other hadrons
rQQ
rh 1 fm
Binding energies:
~0.5-1 GeV > Tc
Can survive above Tc in QGP
Charmonia:
J / y (1S) ® rJ /y
0.2 fm
c c (1P) ® rc
0.3 fm
y '(2S) ® ry¢
0.4 fm
Different quarkonia:
melt at different Temperatures
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Shoot an energetic parton (q or g)
through QGP
Measure energy of outgoing beam
Attenuation or “Quenching”
Depends on medium density, r.
Increases with temperature
How do we get these external probes?
“Hard probes”
Quarkonia, charm, beauty, jets, energetic
photons & dileptons
Formed early in the collision: t~1/Q
Hard scale: pQCD is applicable
Rate can be compared to pp, pA data
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Thermodynamics of the Strong Interaction
There is a transition around 160-190 MeV
Color deconfinement
Chiral symmetry restoration
latent heat increases energy density
The plasma near Tc is strongly coupled.
Experimentally, we can probe it using:
Hadronic and electromagnetic radiation
Quarkonia
Jets
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