Intro. Relativistic Heavy
Ion Collisions
The QCD Phase Transition
Manuel Calderón de la Barca Sánchez
Quarks (u,d,s,c,b,t) and gluons.
Characteristic color charge.
Forces between quarks: exchange of gluons.
Asymptotic Freedom
2
2
2
a
(Q
)
~
1/
ln(Q
/
L
)
Strong coupling “runs”: s
Distance probed: R~1/Q (deBroglie)
Confinement
Free quarks not observed in nature
Quarks only in bound states
Responsible for > 98% of the visible mass in
universe (not the Higgs!)
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http://nobelprize.org/nobel_prizes/physics/laureates/2004/illpres/index.html
At high energy and small
distances, the strength of
this force decreases!
“Asymptotic freedom”
Nobel Prize 2004
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Logarithmic decrease :
2
2
2
a
(Q
)
~
1/
ln(Q
/
L
)
“Hard” Processes: s
Large values of Q2
as < 0.2 Þ Q > 5 GeV
Perturbative region
Calculations similar to QED
Feynman diagrams
“Soft” Processes:
Effective models
Numerical solutions
Lattice QCD
“strong QCD”
distances close to nucleon radius
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Brinkmann, Gianotti, Lehmann
Nucl. Phys. News 16 (2006) 15
4
Cornell Potential:
a
V (r) = - + s r
r
Coulomb term
Linear term:
“rubber band” or elastic
string
color flux tube between
quark and antiquark
D. Leinweber, et al., Center for Subatomic Structure.
Physics Dept. U. of Adelaide, Australia. 2003.
Increase r: attractive force increases.
Quarks cannot be separated: confinement!
Free quarks cannot be observed.
Quarks are bound within hadrons
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E=mc2
Meson: bound state of quark and
.
Separating the quarks: increase energy in the
color field.
Energy > 2 mq: quark /
pops out of
the vacuum.
New mesons are formed
String model: Used in many event generators
PYTHIA/Jetset
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Collins and Perry:
Superdense matter consists of quarks rather than
hadrons
Cabibbo and Parisi:
Deconfinement phase transition
At high temperatures or densities: phase transition
from hadrons to QGP should take place
Low-energy, non-pertubative properties
Lattice QCD numerical studies
Our best way to connect to QCD
However: Lattice QCD ≠ Heavy Ion Collisions
Static, equilibrated vs. dynamic, exploding matter.
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Similarities and differences between QED and
QCD
Look at some of the properties of each theory.
Visualize the more complicated structure of QCD
Very basic brush of Lattice QCD
Key Result: New phase of QCD at high T!
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QED Lagrangian
y(x) : electrons, positrons
Dirac (bi)spinor (4 components)
spin ½ particles (fermions).
† 0
:Dirac adjoint

Dm = ¶m + ieAm
gauge covariant derivative
E = ¶A / ¶t - Ñf
B = Ñ´ A
y =y g
Anti particle
Am = (f / c, A)

Fmn = ¶m An -¶n Am
EM Field tensor
e: coupling constant, electric
charge of bispinor field
Am : Covariant four-vector
potential of EM field
Note: only one type of photon, so
only one Am
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There are now 8 “color” indices for gluons.
Gluons: Adjoint representation of SU(3)
The gluon field tensor is:
a
F mn
= ¶m Ana -¶n Ama + gs [Am , An ]a
a
Am : 8 vector potentials of the gluon field
gs : strong color charge
a
[Am , An ] : gluon self-interaction
due to gluons having non-zero charge
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Experimental findings:
Quark bags (hadrons) come in two types:
Baryons (3 quarks)
Mesons (quarks-antiquark)
Both are color-neutral
3R Experiment: Strong force has 3 types of charge.
Theory:
Need an internal symmetry with a 3-D representation
Must give rise to neutral combination of 3 particles
Otherwise: No baryons!
Simplest statement: linear combination of three
charge-types is neutral
red + green + blue = 0
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Postulate: Gluons must occur in color-anticolor
units
9 combinations
BUT:
r+g+b = 0
rr
rg
rb
gr
gg
gb
br
bg
bb
So, the linear combination:
rr + gg + bb = 0 must be non interacting!
This gluon can’t interact with anything.
So it can’t be detected.
So, for all intents and purposes, it doesn’t exist!
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This is the “color singlet”
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Gluons are the basis states of the SU(3) Lie
algebra.
SU(3) : group of 3x3 unitary matrices with det = 1.
The state of a particle: given by a vector on a space
Elements of SU(3) act on this space as linear
operators
3-D representation: A 3x3 unitary matrix can act
on a 3-row column vector (matrix multiplication)
Quarks transform under this representation
SU(3) : 3-D representation : 3 colors
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Red
æ a b
ç
ç d e
ç g h
è
Green
c ö
÷
f ÷
i ÷ø
Blue
Antired
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(1
æ 1 ö
ç
÷
ç 0 ÷
ç 0 ÷
è
ø
SU(3) Matrix:
3x3 Unitary Matrix
æ 0 ö
ç
÷
0
ç
÷
ç 1 ÷
è
ø
U = exp(iH)
H : Hermitian Matrix
det U = 1
implies Tr(H) = 0
Generators of SU(3)
Traceless Hermitian Matrices
æ 0 ö
ç
÷
ç 1 ÷
ç 0 ÷
è
ø
)
0 0
Antigreen
(
0 1 0
Antiblue
(
0 0 1
)
æ a b
ç
ç d e
ç g h
è
c ö
÷
f ÷
i ÷ø
)
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Alternately: Let the elements of SU(3) act on the
traceless hermitian matrices T
T ®UTU -1
Gluons transform under this representation.
How many lin. indep. traceless hermitian matrices?
8 : (SU(N) : N2 -1 generators)
Any other can be written as a linear combination.
æ 0 1 0 ö
æ 0 -i 0 ö
ç
÷
l1 = ç 1 0 0 ÷ l2 = çç i 0 0 ÷÷
ç 0 0 0 ÷
ç 0 0 0 ÷
è
ø
è
ø
æ 0 0 -i ö
ç
÷
l1 = ç 0 0 0 ÷
ç i 0 0 ÷
è
ø
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æ 0 0 0
ç
l6 = ç 0 0 1
ç 0 1 0
è
æ 1 0 0 ö
ç
÷
l3 = ç 0 -1 0 ÷
ç 0 0 0 ÷
è
ø
ö
÷
÷
÷
ø
æ 0 0 0 ö
ç
÷
l7 = ç 0 0 -i ÷
ç 0 i 0 ÷
è
ø
æ 0 0 1 ö
ç
÷
l4 = ç 0 0 0 ÷
ç 1 0 0 ÷
è
ø
æ 1 0 0
1 ç
l8 =
0 1 0
3 çç
è 0 0 -2
ö
÷
÷
÷
ø
15
æ 0 1 0 ö æ 0 rg 0 ö
÷
æ l1 + il2 ö ç
÷ ç
ç
÷=ç 0 0 0 ÷=ç 0 0 0 ÷
è 2 ø ç
÷ ç
÷
è 0 0 0 ø è 0 0 0 ø
æ 0 0 0 ö æ 0
æ l1 - il2 ö ç
÷ ç
ç
÷ = 1 0 0 ÷ = ç gr
è 2 ø çç
÷ ç
è 0 0 0 ø è 0
0 0 ö
÷
0 0 ÷
÷
0 0 ø
is valid
is valid
Similarly for all 6 off-diagonal elements: 6 indep. gluons.
Can you make
æ 1 0 0
ç
ç 0 0 0
ç 0 0 0
è
ö æ rr
÷ ç
÷ = çç 0
÷
ø è 0
0 0 ö
÷
0 0 ÷
÷
0 0 ø
or
æ 0 0 0 ö æ 0 0 0
ç
÷ ç
ç 0 1 0 ÷ = ç 0 gg 0
ç
÷ ç
è 0 0 0 ø è 0 0 0
ö
÷
÷
÷
ø
?
No! Tr ≠ 0. Can’t be linear combination of ls.
You can make
æ 1 0 0
ç
ç 0 -1 0
ç 0 0 0
è
ö
÷
÷
÷
ø
or
æ 1 0 0
ç
ç 0 0 0
ç 0 0 -1
è
ö
÷
÷
÷
ø
or
æ 0 0 0 ö
ç
÷
0
1
0
ç
÷
ç 0 0 -1 ÷
è
ø
But now, any one is a linear combination of the other 2.
Hence: only 2 more indep. gluons. Total: 8.
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i,j : quark color indices (1,2,3 or r,g,b)
q: quark flavor indices (u,d,s,c,b,t)
a: 8 gluon indices (“color-anticolor”)
i
yq (x) : Dirac bi-spinor (4 components) describing
the fermion (spin ½) fields of the theory: quarks.
with flavor q, and color i.
l : SU(3) Generators: Gell-Mann matrices
a
ij
Coupling constant:
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gs2
as =
4p
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Propagators
i
Fermion
gluon
g pm - m
Vertices:
m
dij
-igmn ab
d
2
p
Expand Lagrangian
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Background electric field
produces virtual e+e- pairs.
these change the distribution
of charges and currents that
generated the original field
QFT: vacuum is not just
empty space
virtual e+e- pairs pop in and
out
act as dipoles
dipoles orient themselves
partially cancel the original E
field
similar to Debye screening
one-loop contribution
shown in (b)
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QCD
QED
 em ( m 2 )
2
 em (Q ) 
  em ( m 2 )  Q 2  
ln  2  
1 
3
 m 

 s (Q 2 ) 
  s (m 2 )
 Q2 
(33  2 N f ) ln  2  
1 
12
 m 

Vacuum polarization:
Vacuum polarization:
antiscreens the strength of
the interaction at large
distance.
At large Q2 (small distance)
interaction is weaker.
screens the strength of the
interaction at large
distance.
At large Q2 (small distance)
interaction is stronger.
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 s (m 2 )
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Experimental measurements of s
.
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Data analysis: S. Bethke, arXiv:0908.1135
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1974: Kenneth Wilson (Nobel Prize 1982)
Generate quark and gluon configurations
Weigh each by Boltzmann factor: exp(-S) / Z
Action:
Partition function:
Thermodynamic quantities are functions of Z or its
E
derivatives. Z = å e- k T
s
B
Calculate expectation values of various
operators on each configuration.
Numerical integration: 4-D lattice in (x, y, z,t) .
Typical sizes: few fm.
Typical spacing: ~0.1 fm
s
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Create a configuration of gluon and quark fields
Quarks and antiquarks on each node balance.
Balance each quark type independently.
Calculate action: Sum of contour integrals over
all elementary squares (plaquettes).
x + an̂
x
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x + am̂ + an̂
x + am̂
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Arrows:
Parallel Transport
operators
Um (x) = exp(igaAm (x))
Nodes are
“connected” by
gluons fields.
Action is determined
by quark and gluon
fields on nodes and
through connections
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Randomly change the values of the field
Recalculate action
Retain configurations with lower values of S.
Larger Boltzmann factor exp(-S)
Process drives the system towards equilibrium.
Reveals phase transitions.
Physics result: limit of small lattice spacing
“Continuum limit”
Require a large amount of computing power!
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Gluon field configurations
in vacuum QCD
Volume:
2.4 x 2.4 x 3.6 fm3
Vacuum fluctuations
Chromoelectric and
Chromomagnetic fields
are induced
50 sweeps to average over
the gluon field
configuration
Plot: Action density, S.
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If one ignores quarks:
“pure gauge” theory...
Polyakov Loop :éN -1
F(r,T )
ù
L(T ) = Tr êÕU 4 ( j,n)ú = lim e 2T
êë j=0
úû r®¥
t
Time-propagation of static quark
Related to Free energy of single
quark
Below Tc: F(∞,T) diverges
F(r) ~ V(r) ~ s r
Polyakov loop vanishes
L(T ) = 0
Above Tc: F(∞,T) remains finite
F plateau decreases as T grows
Polyakov loop grows as T grows
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1st Order Phase Transition!
27
Pure glue: First order
Polyakov loop: order parameter
Massless flavors : First order
Massless (u,d) but massive s: 2nd order or rapid crossover
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Step-like behavior at Tc~170 MeV
Increase in # of deg. of freedom (dof)
Transition from Hadron Gas to QGP.
Pion Gas: dof = 3. (Including resonances: dof~10 − 15)
2 flavor QCD: dof=
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(2 ´ 2 ´ 2 ´ 3)+ 8 ´ 2 = 21+16 = 37
8
Fermi/Boltzmann normalization * quark-antiquark,updown,spin,color + 8 gluons, 2 spins
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QCD is the real deal for the strong interaction
But it’s a tough theory
Yet that is what makes it so interesting!
Now if we do things numerically...
Phase with more degrees of freedom than that
of a pion gas above 170 MeV.
Not quite a free gas of quarks.
Open question: what are the degrees of
freedom?
But it is clear: whatever they are, they have
to do with color!
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