Transcript Cosmology
Quantum Chromodynamics (QCD)
Main features of QCD
Confinement
At large distances the effective coupling between quarks is large,
resulting in confinement.
Free quarks are not observed in nature.
Asymptotic freedom
At short distances the effective coupling between quarks decreases
logarithmically.
Under such conditions quarks and gluons appear to be quasi-free.
(Hidden) chiral symmetry
Connected with the quark masses
When confined quarks have a large dynamical mass - constituent mass
In the small coupling limit (some) quarks have small mass - current
mass
Confinement
The strong interaction potential
Compare the potential of the strong & e.m. interaction
Vem
q1q2
c
4 0 r
r
Vs
c
kr
r
c, c, k constants
Confining term arises due to the self-interaction property of the
colour field
q1
q2
a) QED or QCD (r < 1 fm)
r
q1
b) QCD (r > 1 fm)
q2
Charges
Gauge boson
Charged
Strength
QED
QCD
electric (2)
g(1)
no
e2
1
em
4 137
colour (3)
g (8)
yes
s 0.1 0.2
Asymptotic freedom - the coupling “constant”
It is more usual to think of coupling strength rather than charge
and the momentum transfer squared rather than distance.
2M Q2 W 2 M 2
M initial state mass
energy transfer
W final state mass
Q momentum transfer
In both QED and QCD the coupling strength depends on distance.
e e
In QED the coupling strength is given by:
em Q2
1 3 lnQ
2
m
2
Q2»m2
where = (Q2 0) = e2/4 = 1/137
In QCD the coupling strength is given by:
s Q
2 33 2n f
2
1
ln Q
s 2
2
s
12
2
which decreases at large Q2 provided nf < 16.
Q2 = -q2
Asymptotic freedom - summary
Effect in QCD
Both q-qbar and gluon-gluon loops contribute.
+ The quark loops produce a screening effect analogous to e e loops in QED
But the gluon loops dominate and produce an anti-screening effect.
The observed charge (coupling) decreases at very small distances.
The theory is asymptotically free quark-gluon plasma !
“Superdense Matter: Neutrons or Asymptotically Free Quarks”
J.C. Collins and M.J. Perry, Phys. Rev. Lett. 34 (1975) 1353
Main points
Observed charge is dependent on the distance scale probed.
Electric charge is conveniently defined in the long wavelength limit (r
).
In practice em changes by less than 1% up to 1026 GeV !
In QCD charges can not be separated.
Therefore charge must be defined at some other length scale.
In general s is strongly varying with distance - can’t be ignored.
Quark deconfinement - medium effects
Debye screening
In bulk media, there is an additional charge screening effect.
At high charge density, n, the short range part of the potential becomes:
r
1 1
1
V(r) exp
where rD 3
r
r
n
rD
and rD is the Debye screening radius.
Effectively, long range interactions (r > rD) are screened.
The Mott transition
In condensed matter, when r < electron binding radius
an electric insulator becomes conducting.
Debye screening in QCD
Analogously, think of the quark-gluon plasma as a colour conductor.
Nucleons (all hadrons) are colour singlets (qqq, or qqbar states).
At high (charge) density quarks and gluons become unbound.
nucleons (hadrons) cease to exist.
Debye screening in nuclear matter
High (color charge) densities are achieved by
Colliding heaving nuclei, resulting in:
1. Compression.
2. Heating = creation of pions.
Under these conditions:
1. Quarks and gluons become deconfined.
2. Chiral symmetry may be (partially) restored.
The temperature inside a heavy ion collision at RHIC can exceed
1000 billion degrees !! (about 10,000 times the temperature of the sun)
Chiral symmetry
Chiral symmetry and the QCD Lagrangian
Chiral symmetry is a exact symmetry only for massless quarks.
In a massless world, quarks are either left or right handed
The QCD Lagrangian is symmetric with respect to left/right handed quarks.
Confinement results in a large dynamical mass - constituent mass.
chiral symmetry is broken (or hidden).
When deconfined, quark current masses are small - current mass.
chiral symmetry is (partially) restored
Example of a hidden symmetry restored at high temperature
Ferromagnetism - the spin-spin interaction is rotationally invariant.
Below the Curie
temperature the
underlying rotational
symmetry is hidden.
Above the Curie
temperature the
rotational symmetry
is restored.
In the sense that any direction is possible the symmetry is still present.
Chiral symmetry explained ?
Red’s rest frame
Lab frame
Chiral symmetry and quark masses ?
a) blue’s velocity > red’s
Red’s rest frame
b) red’s velocity > blue’s
Lab frame
Blue’s handedness
changes depending
on red’s velocity
Modelling confinement: The MIT bag model
Modelling confinement - MIT bag model
Based on the ideas of Bogolioubov (1967).
Neglecting short range interactions, write the Dirac
equation so that the mass of the quarks is small inside the
bag (m) and very large outside (M)
Wavefunction vanishes outside the bag if M
and satisfies a linear boundary condition at the bag surface.
Solutions
Inside the bag, we are left with the free Dirac equation.
The MIT group realised that Bogolioubov’s model violated
E-p conservation.
Require an external pressure to balance the internal
pressure of the quarks.
The QCD vacuum acquires a finite energy density, B ≈ 60
MeV/fm3.
New boundary condition, total energy must be minimised
wrt the bag radius.
B
Confinement Represented by Bag Model
Bag Model of Hadrons
Comments on Bag Model
Bag model results
Refinements
Several refinements are needed
to reproduce the spectrum of
low-lying hadrons
e.g. allow quark interactions
Fix B by fits to several hadrons
Estimates for the bag constant
Values of the bag constant
range from B1/4 = 145-235 MeV
Results
Shown for B1/4 = 145 MeV and
s = 2.2 and ms = 279 MeV
T. deGrand et al, Phys. Rev. D 12 (1975) 2060
Summary of QCD input
QCD is an asymptotically free theory.
In addition, long range forces are screened in a dense
medium.
QCD possess a hidden (chiral) symmetry.
Expect one or perhaps two phase transitions connected
with deconfinement and partial chiral symmetry
restoration.
pQCD calculations can not be used in the confinement
limit.
MIT bag model provides a phenomenological description
of confinement.
Still open questions in the Standard Model
K*-(892)
Resonances are:
Luis Walter Alvarez
1968 Nobel Prize for
“ resonance particles ”
discovered 1960
• Excited state of a ground state particle.
• With higher mass but same quark content.
• Decay strongly short life time
(~10-23 seconds = few fm/c ),
width = natural spread in energy: = h/t.
Breit-Wigner shape
0
2
4
Number of events
10
8
6
Chirality: Why Resonances ?
640
680
720
760
800
840
880
920
Invariant mass (K0+) [MeV/c2]
minv
E1 E2 2 p1 p 2
K* from K-+p collision system
K p K*p
K0
• Broad states with finite and t,
which can be formed by collisions between
the particles into which they decay.
2
Why Resonances?:
• Surrounding nuclear medium may change
resonance properties
• Chiral symmetry breaking:
Dropping mass -> width, branching ratio
Bubble chamber, Berkeley
M. Alston (L.W. Alvarez) et al., Phys. Rev. Lett. 6 (1961) 300.
Strange resonances in medium
Short life time [fm/c]
K* < *< (1520) <
4 < 6 < 13 < 40
Rescattering vs.
Regeneration ?
Medium effects on resonance and their decay
Red: before chemical freeze out products before (inelastic) and after chemical
Blue: after chemical freeze out
freeze out (elastic).
Electromagnetic probes - dileptons
q
Dilepton production in the QGP
The production rate (and invariant mass distribution) depends on the momentum distribution of q-qbar in the
plasma.
g*
l+
l-
q
The momentum distributions f(E1) and f(E2) depend
on the thermodynamics of the plasma.
The cross-section for the sub-process (M) is
calculable in pQCD.
Reconstruct the invariant mass, M, of the dilepton pair’s hypothetical parent.
Nf
dN
e 2 3 3
f d p1d p2
Nc N 2f
6 f E1 f E2 M v12
e
2
dtd x
f 1
l l
3
Dilepton production from hadronic mechanisms
1. Drell-Yan
2. Annihilation and Dalitz decays
3. Resonance decays
4. Charmed meson decays
qq l l
l l
l l g
, , and J/
D l X
high Mass
low Mass
discrete
low Mass
CERES low-mass e+e– mass spectrum
Almost final results from the 2000 run Pb+Au at 158 GeV per nucleon
comparison to the hadron decay cocktail
Enhancement over
hadron decay cocktail
for mee > 0.2 GeV:
2.430.21 (stat)
for 0.2 GeV<mee< 0.6 GeV:
2.80.5 (stat)
• Absolutely normalized
spectrum
• Overall systematic
uncertainty of
normalization: 21%
NA60 Low-mass dimuons
Superb data!!!
Mass resolution:
23 MeV at the position
, and even peaks
clearly visible in dimuon
channel
Net data sample:
360 000 events
Deconfinement at Initial Temperature
Matsui & Satz (1986): (Phys. Lett. B178 (1986) 416)
Color screening of heavy quarks in QGP leads to heavy resonance dissociation.
RHIC
s = 200 GeV
J/Ycc-bar) e+ +e,
(bb-bar) e+ +e, + +
Thermometer for early stages:
Tdis(Y(2S)) < Tdis((3S))< Tdis(J/Y) Tdis((2S)) < Tdis((1S))
Total bottom / charm production
Melting at SPS
Decay
modes:
c J/Y + g
b + g
Lattice QCD:
SPS TI ~ 1.3 Tc
RHIC TI ~ 2 Tc
The suppression of heavy quark states
signature of deconfinement at QGP.
J/ suppression
Charmonium production
The J/ is a c-cbar bound state (analogous to positronium)
Produced only during the initial stages of the collision
qq cc
Thermal production is negligible due to the large c quark mass
mc 1500MeV TQGP
Charmonium suppression (Debye screening)
Semi-classically (E = T + V)
E(r)
2
r / rD
p
e
s
2
r
p2 ~ 1 r 2
mc 2
Differentiate with respect to r to find minimum (bound state)
Find there is no bound state if
1
rD
0.84 s
rD pQCD
2
1
9 s T
For s = 0.52 and T = 200 MeV, rD(pQCD) = 0.36 fm
Compare with rBohr = 0.41 fm (setting rD above)
Conclusion: the J/ is not bound in the plasma under these conditions
Onium physics – the complete program
Melting of quarkonium states (Deconfinement TC)
Tdiss(Y’) < Tdiss((3S)) < Tdiss(J/Y) Tdiss((2S)) <
Tdiss((1S))
Future Measurements:
Resonance Response to Medium
Resonances below and above Tc:
Temperature
partons
Shuryak QM04
Quark Gluon Plasma
hadrons
Hadron Gas
Baryochemical potential (Density)
Gluonic bound states
(e.g. Glueballs) Shuryak hepph/0405066
Deconfinement: Determine range of T
initial.
J/and state dissociation
Chiral symmetry restoration
Mass and width of resonances
( e.g. leptonic vs hadronic decay,
chiral partners and a1)
Hadronic time evolution
Hadronisation (chemical freeze-out)
till kinetic freeze-out.
Deconfinement: Melting of J/Y
RHIC
SPS
J/Y normal nuclear
absorption curve
J
abs
4.18 0.35mb
Interaction length
Projectile
J/
L
Target
J/ suppression at SPS and RHIC are the same
Strong signal for deconfinement in QGP phase
RHIC has higher initial temperature
Expect stronger J/ suppression
Partonic recombination of J/
cent
Npa Nco
rt
ll
0
10%
339
104
9
10
20%
222
590
Chiral Symmetry Restoration
Vacuum
At Tc: Chiral Restoration
Data: ALEPH Collaboration
R. Barate et al. Eur. Phys. J. C4 409 (1998)
Measure chiral partners
Near critical temperature Tc
(e.g. and a1)
a1 +
Ralf Rapp (Texas A&M)
J.Phys. G31 (2005) S217S230
Resonance Reconstruction in
STAR TPC
Energy loss in TPC dE/dx
End view STAR TPC
K-
p
p
dE/dx
(1520)
(1385)
K
e
momentum [GeV/c]
p
K(892)
(1020)
(1520)
(1385)
X1530
+K
K+K
p +K
+
X+
• Identify decay candidates
(p, dedx, E)
• Calculate invariant mass
Invariant Mass Reconstruction in
p+p
Invariant mass:
(1520)
STAR Preliminary
(1520)
minv
E1 E2
2
p1 p 2
2
— original invariant mass histogram
from K- and p combinations
in same event.
— normalized mixed event histogram
from K- and p combinations
from different events.
(rotating and like-sign background)
Extracting signal:
After Subtraction of mixed event
background from original event and
fitting signal (Breit-Wigner).
Resonance Signal in p+p collisions
STAR Preliminary
STAR
Preliminary
p+p
K(892)
ΦK+K-
Statistical error only
X
STAR Preliminary
(1385)
Δ++
STAR
Preliminary
p+p
Invariant Mass
(GeV/c2)
Resonance Signal in Au+Au collisions
STAR
Preliminary
*0
K +
K(892)
K*0
Au+Au
minimum
biaspT 0.2
XX
*± +*±
GeV/c
|y| 0.5
(1520)
(1020)
STAR Preliminary
STAR Preliminary
Estimating the critical parameters, Tc and c
Mapping out the Nuclear Matter Phase Diagram
Perturbation theory highly successful in
applications of QED.
In QCD, perturbation theory is only applicable
for very hard processes.
Two solutions:
1. Phenomenological models
2. Lattice QCD calculations
Lattice QCD
Quarks
and gluons are
studied on a discrete
space-time lattice
Solves the problem of
divergences in pQCD
calculations (which arise
due to loop diagrams)
There
are two order
parameters
1. The Polyakov Loop
2. The Chiral Condensate
0.5
4.5
15
35
L ~ Fq
~ mq
75 GeV/fm3
/T4
Lattice Results
Tc(Nf=2)=1738 MeV
Tc(Nf=3)=1548 MeV
(F. Karsch, hep-lat/9909006)
T = 150-200 MeV
~ 0.6-1.8 GeV/fm3
T/Tc
Lattice QCD: the latest news
(critical parameters at finite baryon density)
Phenomenology I: Phase transition
The quark-gluon and hadron equations of state
The energy density of (massless) quarks and gluons is derived
from Fermi-Dirac statistics and Bose-Einstein statistics.
p 3dp
g 2 bp
2 e 1
1
q
1
g
p3 dp
2 2 e b p 1
2T 4
30
7 2T 4 2 T 2 4
q q
2
120
4
8
where is the quark chemical potential, q = - q and b = 1/T.
Taking into account the number of degrees of freedom
TOT 16 g 12 q q
Consider two extremes:
1. High temperature, low net baryon density (T > 0, B = 0). B = 3 q
2. Low temperature, high net baryon density (T = 0, B > 0).
Phenomenology II: critical parameters
High temperature, low density
limit - the early universe
Two terms contribute to the
total energy density
For a relativistic gas:
For stability:
Low temperature, high density
limit - neutron stars
Only one term contributes to
the total energy density
By a similar argument:
qg 37
2
30
T4
1
Pqg qg
3
Pnet Pqg B 0
14
90
Tc 2 B
37
q
3
2
100 170 MeV
4
q
2
c 2 2 B
14
300 500 MeV
~ 2-8 times normal nuclear matter density
given pFermi ~ 250 MeV and ~ 23/32
How to create a QGP ?
energy = temperature & density = pressure