Transcript First day observables

First day
Soft physics observables in
heavy ion collisions
in view of the LHC
Francesco Prino
INFN – Sezione di Torino
Disclaimers:
 experimentalist’s point of view
 perspectives for the LHC
XIII Mexican school on particles and fields, San Carlos, Mexico, Oct 7th 008
Reminder: phase diagram
GOAL(s) of relativistic
heavy ion collisions:
Study nuclear matter
at extreme conditions
of temperature and
density
AND collect evidence
for a state where
quark and gluons are
deconfined (Quark
Gluon Plasma)
AND study its
properties
2
Reminder: space time evolution
Thermal freeze-out
 Elastic interactions cease
 Particle dynamics
(“momentum spectra”)
fixed
Tfo (RHIC) ~ 110-130 MeV
Chemical freeze-out
 Inelastic interactions cease
 Particle abundances
(“chemical composition”)
are fixed (except maybe
resonances)
Tch (RHIC) ~ 170 MeV
Thermalization time
 System reaches local
equilibrium
teq (RHIC) ~ 0.6 fm/c
3
Heavy ion results vs. time
Results published in the first year after RHIC startup:
 Multiplicity of unidentified particles at midrapidity
 PHOBOS, sent to PRL on July 19th 2000
 PHENIX, sent to PRL on Dec 21th 2000
 Elliptic flow of unidentified particles
First 10k-20k
events, fast
analysis
 STAR, sent to PRL on Sept 13th 2000
 Particle to anti-particle ratios
 STAR, sent to PRL on Apr 13th 2001
 PHOBOS, sent to PRL on Apr 17th 2001
 BRAHMS, sent to PRL on Apr 28th 2001
 Transverse energy distributions
 PHENIX, sent to PRL on April 18th 2001
 Pseudorapidity distributions of charged particles
 PHOBOS, sent to PRL on June 6th 2001
 BRAHMS, sent to Phys Lett B on Aug 6th 2001
 Elliptic flow of identified particles
statistics<≈100k
events,
longer analysis
time due to the
need of PID,
detector
calibration,
combination of
different
detectors
 STAR, sent to PRL July 5th 2000
… then came the high pT particle suppression from
PHENIX (sent to PRL on Sept 9th 2008)
4
Pseudorapidity density of
unidentified particles
Particle production in heavy ion
collisions
Multiplicity = number of particles produced in a collision
Multiplicity contains information about:
Entropy of the system created in the collision
How the initial energy is redistributed to produce particles in the final state
Energy density of the system (via Bjorken formula)
Mechanisms of particle production (hard vs. soft)
Geometry (centrality of the collision)
NOTE: In hadronic and nuclear collisions particle production
is dominated by (non-perturbative) processes with small
momentum transfer
Many models, but understanding of multiplicities based on first
principles is missing
6
Particles produced in PbPb at SPS
In central PbPb collisions at SPS (s=17 GeV) more
than 1000 particles are created
7
Particles produced in AuAu at RHIC
In central AuAu collisions at RHIC (s=200 GeV)
about 5000 particles are created
8
Multiplicity and centrality
The number of produced particles is related to the
centrality (impact parameter) of the collision
Heavy ion collisions are
described as superposition of
elementary nucleon-nucleon
collisions (e.g. Glauber model)
The number of nucleon-nucleon
collisions ( Ncoll ) and the number
of participant nucleons ( Npart )
depend on the impact parameter
Each collision/participant
contributes to particle
production and consequently to
multiplicity
9
Evaluation of Npart and Ncoll
Glauber model calculations:
Physical inputs:
Woods-Saxon density for colliding nuclei
Nucleon-nucleon inelastic cross-section
inel
Numerical calculation of Npart , Ncoll
... vs. impact parameter b

0 (Pb)= 0.16 fm-3
0
1  e ( r r0 ) / C
C (Pb)= 0.549 fm
r0 (Pb)= 6.624 fm
Accel.
AGS SPS RHIC
LHC
√s (GeV) 3-5
17
200
5500
21
33
42
60
inel
10
Particle production - Hard
Hard processes = large momentum transfer small
distance scales
Interactions at partonic level
Particles produced on a short time scale
Small coupling constant  calculable within perturbative QCD
In A-A collisions:
Modeled as superposition of
independent nucleon-nucleon
collisions
BINARY SCALING: hard
particle production scale with
the number of elementary
nucleon-nucleon collisions
(Ncoll)
11
Particle production - Soft
Soft processes =
small momentum transfer  large scales
Can not resolve the partonic structure of the nucleons
Large coupling constant  perturbative approach not applicable 
need to use phenomenological (non-perturbative) models
In A-A collisions:
WOUNDED NUCLEON
99.5% soft
MODEL: each nucleon
participating in the
interaction (wounded)
contributes to particle
production with a constant
amount, no matter how many
collisions it suffered
Soft particle production
scale with the number of
participant nucleons (Npart) 12
Wounded nucleon model
Based on experimental
observation (about 1970s)
that multiplicites measured
in protno-nucleus collisions
scale as:
R
pA
N ch
pp
N ch
1 1
  
2 2
v = average number of collisions
between nucleons (=Ncoll)
So:
R
pA
N ch
pp
N ch

1 1 pA
 N coll
2 2
since in p-p: Npart = 2
pA
pA
N coll
 1 N part

 pp
2
N part
and in p-A: Npart= Ncoll+1
13
Measuring the multiplicity
Experimentally we count the multiplicity of:
charged (ionizing) particles
particles in a given window covered by the detector (acceptance)
Difficult to compare results between experiments with different acceptances
For this reason, multiplicities are commonly expressed as
charged particle densities in a given range of polar angle
Commonly used: number of charged particles in 1 unit of
(pseudo)rapidity around midrapidity: Nch(|h|<0.5) o Nch(|y|<0,5)
1  E  pL 

y   ln 
2  E  pL 
   
1  p  pL 
h   ln


ln
tan 


2  p  pL 
  2 
NOTE: pseudorapidity is easier to access experimentally because it
requires to measure just one variable (the polar angle q) and does
not require particle identification and measurements of momenta
dN/dh (dN/dy) distributions contain also other information
14
on the dynamics of the interaction
Rapidity at RHIC (collider)
Before collision:
pBEAM=100 GeV/c per nucleon
EBEAM=(mp2+pBEAM2)=100.0044 per nucleon
b=0.999956, gBEAM≈100
y PROJ   yTARGET 
1 EBEAM  p BEAM 1 1  b
ln
 ln
 5.36
2 E BEAM  p BEAM 2 1  b
y  y PROJ  yTARGET  10.8
After collision:
Projectile and target nucleons (green) are
slowed down and they are located at lower y
(and b) values with respect to initial ones
Produced particles (red) are distributed in
the kinematical region between the initial
projectile and target rapidities
The maximum particle density is in the
central rapidity region (midrapidity) :
y MID
y PROJ  yTARGET

0
2
15
Rapidity at SPS (fixed target)
Before collision:
pBEAM=158 GeV/c , bBEAM=0.999982
pTARGET=0 , bTARGET=0
y PROJ 
1 E BEAM  p BEAM 1 1  b
ln
 ln
 5.82
2 E BEAM  p BEAM 2 1  b
1
ln1  0
2
y  y PROJ  yTARGET  5.82
yTARGET 
Midrapidity:
yMID 
y PROJ
 2.91
2
The dN/dy in the center-of-mass reference
system is obtained from the one measured in the
lab with a translation y’ = y - yMID
 The dN/dh distribution does not have this property
16
Pseudorapidity
pT = pL
q = 45 (135) degrees
h = ±0.88
pL>>pT
pT>pL
pL>>pT
Midrapidity region
Particles with pT>pL produced
at q angles around 90°
Bjorken formula to estimate
the energy density in case of a
broad plateau at midrapidity
invariant for Lorentz boosts:
 BJ
mT  dN


Ac t f  dy


 y 0
Fragmentation regions:
Particles with pL>>pT
produced in the
fragmentation of the
colliding nuclei at q angles
17
around 0° e 180°
PbPb collisions at SPS
Pb-Pb at 40 GeV/c (√s=8.77 GeV)
Pb-Pb at 158 GeV/c (√s=17.2 GeV)
central
peripheral
Peak position moves (midrapidity = ybeam/2 )
Particle density at the peak
increases with s
18
AuAu collisions at RHIC
central
central
peripheral
peripheral
energy s
19
Multiplicity per participant pair
We introduce the variables:
dN / dh
h 0
N part / 2
N ch
N part / 2
with N ch 

dN
dh
dh
which are the particle density at mid-rapidity and the total
multiplicity normalized to the number of participant pairs
Motivation
Simple test of the scaling with Npart
If particle production scales with Npart , this variable should not depend on the
centrality of the collisions
Simple comparison with pp collisions where Npart=2
20
dN/dhmax vs. centrality
Yield per participant pair increases by ≈ 25% from
peripheral to central Au-Au collisions
Contribution of the hard component of particle production ?
BUT:
The ratio 200 / 19.6 is independent of centrality
A two-component fit with dN/dh  [ (1-x) Npart /2 + x Ncoll ] gives compatible values
of x (≈ 0.13) at the two energies
Factorization of centrality (geometry) and s (energy) dependence
21
dN/dhmax vs. centrality and s
Factorized dependence of dNch/dhmax on centrality and s
reproduced by models based on gluon density saturation
at small values of Bjorken x
increasing s – decreasing x
Pocket formula:
2
N part
dNch
dh
 N 0 s[GeV ]

1
3
N part
h 0
 and  from ep and eA data
N0 only free parameter
 Armesto Salgado Wiedemann, PRL 94 (2005) 022002
 Kharzeev, Nardi, PLB 507 (2001) 121.
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dN/dhmax vs. s
The dN/dh per participant pair at midrapidity in central
heavy ion collisions increases with ln s from AGS to
RHIC energies
The s dependence is different for pp and AA collisions
23
Warning
Npart is not a direct experimental observable and affects
the scale of both axes of plot of yield per participant pair
Different methods of evaluating Npart give significantly
different results!
NA50 at 158 A GeV/c
s = 130 GeV
24
Total multiplicity (Nch ) vs. centrality
Total multiplicity:
N ch

dN

dh
dh
Need to extrapolate in the
h regions out of acceptance
Small extrapolation in the
case of PHOBOS thanks to
the wide h coverage
Nch scales with Npart
Nch per participant pair
different from p-p, but
compatible with e+e-,
collisions at the same
25
energy
Total multiplicity (Nch ) vs. s
Multiplicity per participant pair in heavy ion collisions:
Lower than the one of pp and e+e- at AGS energies
Crosses pp data at SPS energies
Agrees with e+e- multiplicities above SPS energies (s >≈ 17 GeV)
26
pp vs. e+eThe difference between pp and e+e- multiplicities is
understood with the “leading particle effect”
The colliding protons exit from the collision carrying away a
significant fraction of s
In pp collisions only the energy seff ( < s ) is available for particle
production
In e+e- the full s is fully available for particle production
e+
e-
s
p
seff
p
The effective energy seff available for particle production
is defined as:
 
seff

 s

s

 2
e e collision
pp collision
with this definition, multiplicities in e+e- and pp at the same seff
result to be in agreement
 M. Basile et al.,, Nuovo Cimento A66 N2 (1981) 129.
27
Universality
The seff dependence of multiplicities in pp, e+e- e AA (for
s>15 GeV) follow a universal curve with the trend predicted
by Landau hydrodynamics (Nch s1/4)
No leading particle effect in AA (multiple interactions of projectiles)
Universality of hadronization
Nch  2.2s1/ 4
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Gold vs. copper
62.4 GeV
200 GeV
Cu+Cu
PHOBOS
PHOBOS
Preliminary
3-6%, Npart = 100
Cu+Cu
Preliminary
3-6%, Npart = 96
Au+Au
35-40%, Npart = 99
Au+Au
Preliminary
35-40%,Npart = 98
Unscaled dN/dh very similar for Au-Au and Cu-Cu
collisions with the same Npart
Compare central Cu-Cu with semi-peripheral Au-Au
For the same system size (Npart) Au-Au and Cu-Cu are very similar 29
Limiting fragmentation (I)
Study particle production in the rest frame of one of the
two nuclei
Introduce the variable y’ = y - ybeam (or h’ = h – ybeam )
Limiting fragmentation
Benecke et al., Phys. Rev. 188 (1969) 2159.
At high enough collision energy both
d2N/dpTdy and the particle mix reach
a limiting value in a region around y’ = 0
Also dN/dh’ reach a limiting value and
become energy independent around h’=0
Observed for p-p and p-A collisions
In nucleus-nucleus collisions
Particle production in fragmentation regions
independent of energy, but NOT necessarily
independent of centrality
30
Limiting fragmentation (II)
PHOBOS Phys. Rev. Lett. 91, 052303 (2003)
Particle
production
independent of
energy in
fragmentation
regions
Extended
limiting
fragmentation
(4 units of h at
200 GeV)
No evidence for
boost invariant
central plateau
31
Conclusions
Charged particle multiplicities follow simple scaling laws
Factorization into energy and geometry/system dependent terms
Extended limited fragmentation, no boost-invariant central plateau
Resulting Bjorken energy density in AuAu @ s=200 GeV:
 BJ
mT  dN


Ac t 0  dy

0.6 GeV / c 2
3


 

700


1
.
1


2
145
fm

c

t
2


 y 0
0
Peak energy density
Thermalized energy density
BJ well above the
predicted critical energy
for phase transition to
deconfined quarks and
glouns
32
Towards the LHC (I)
Extrapolation of dNch/dhmax vs s
 Fit to dN/dh  ln s
 Saturation model (dN/dh  s with =0.288)
 Clearly distinguishable with the first 10k events at the LHC
Central collisions
Models prior to RHIC
Saturation model
Armesto Salgado Wiedemann, PRL 94 (2005) 022002
dNch / dh
N part / 2
 8.2 
h 0
dNch
dh
 1650
h 0
Extrapolation of dN/dhln s
dNch / dh
N part / 2
 5.5 
h 0
dNch
dh
 1100
h 0
5500
33
Towards the LHC (II)
Extrapolation of limiting fragmentation behavior
 Persistence of extended longitudinal scaling implies that dN/dh
grows at most logarithmically with s  difficult to reconcile
with saturation models
Log extrapolation
dN/dh ≈ 1100
 Borghini Wiedemann, J. Phys G35 (2008) 023001
Saturation model
dN/dh ≈ 1600
34
Multiplicity of identified
particles
Hadrochemistry
Measurement of the multiplicity of the various hadronic
species (= how many pions, kaons, protons …), i.e. of the
chemical composition of the system
Experimental data from SIS to RHIC energies can be
described using “thermal” models based on the assumption
that hadronization occurs following purely statistical
(thermodynamical) laws
This allows to answer some questions about the
characteristics of the system:
Was the fireball in thermal and chemical equilibrium at freeze-out
time ?
What was the temperature Tch at the instant of chemical freezeout ?
What was the baryonic content of the fireball ?
36
Multiplicity of identified particles (I)
Pions vs protons
At low energies (s<5
GeV) the fireball is
dominated by nucleons
stopped from the
colliding nuclei (high
stopping power)
Pions (produced in the
interaction) dominate at
high energies (s>5
GeV)
The decrease of proton
abundance with
increasing s indicates
an increased
transparency of the
colliding nuclei
37
Multiplicity of identified particles (II)
Pions
More abundant among
the produced hadrons
due to lower mass and
production threshold
Difference between
abundances of p+ and pat low energies due to
isospin conservation
Large stopping power at
low energies
 Fireball dominated by
the nucleons of the
colliding nuclei
 Negative total isospin
due to neutron excess (N >
Z for heavy nuclei)
38
Multiplicity of identified particles (III)
Antiprotons
They are produced in
the collision
Different from proton
case: in the fireball there
are both produced and
stopped “protons”
Strong s dependence
at SPS energies (onset
of production)
At RHIC energies
number of antiprotons ≈
number of protons
Net-protons ≈ 0
Small number of protons
stopped from the colliding
nuclei
39
Multiplicity of identified particles (IV)
Kaons and L hyperons
The larger number of K+
and L with respect to
their antiparticles (Kand Lbar) at low energies
due to quark content of
these hadrons
- and L (uds)
K+ (us)
require to newly produce
only the strange quark,
while light quarks are
present in the stopped
nucleons
- and Lbar require
K- (us)
the production of 2 or 3
new quarks
Associated production of
K+ and L (ss- pairs)
40
Multiplicity of identified particles (V)
Kaons and L hyperons
The difference
between K+ and K- (and
between L e Lbar)
decreases with
increasing s because
the lower stopping
power reduces the
weight of “stopped”
with respect to
“produced” quarks
Very similar abundances
of Lbar and antiprotons
They are both composed
of 3 “produced” quarks
and they have similar
masses
41
Multiplicity of identified particles (VI)
Conclusions
Small s (< 5 GeV):
fireball dominated by
stopped particles
High baryonic content
Importance of isospin and
quarks “stopped” from
colliding nuclei
Large s (> 20 GeV):
Fireball dominated by
produces particles
Low baryonic content
Mass hierarchy ( Np > NK
> Np )
42
Statistical hadronization models
BASIC ASSUMPTIONS
The system (fireball) created in a heavy ion collision is in
thermal and chemical equilibrium at the time of chemical
freeze-out
The system can be described by a (grand-canonical) partition
function and statistical mechanics can be used
Hadronization occurs following a purely statistical
(entropy maximization) law
Original idea: Fermi (1950s), Hagedorn (1960s)
The hadronic system is described as an ideal gas of
hadrons and resonances
Effective model for a strongly interacting system, consistent with
Equation of State resulting from Lattice QCD below the critical
temperature for quark and gluon deconfinement
Include all known mesons with mass<≈1.8 GeV and baryons with
mass<≈2 GeV
43
Statistical hadronization models
NOTES
Chemical equilibrium is ASSUMED
With this assumption it is possible to calculate the multiplicity of the
various hadronic species (how many pions, kaons, protons…)
By comparing the measured multiplicities with the ones predicted by
the model it is possible to validate the hypothesis of chemical and
thermal equilibrium
Statistical models don’t say nothing about HOW and WHEN
the system reaches the chemical and thermal equilibrium
No assumption is made on the presence or not of a partonic
phase in the system evolution
The higher hadron mass cut-off in the H&R gas limits the
applicability of the model at temperatures T<190 MeV
Not a real limitation: above the critical temperature for parton
deconfinement (Tc≈160-200 MeV) hadron gas can no longer be
assumed
44
Grand canonical partition function (I)
Starting point: partition function for a gas of identical
particles (Bose or Fermi) of a given hadronic specie i:
Z iGC (T ,V , mi )

1  e

a

 b ( Ea  mi ) 1
  fermions 


  bosons 


a are the eigen-states (with energy Ea) of the single particle
hamiltonian (= energy states with spin degeneracy)
mi is the chemical potential which ensures charge conservation
In an hadronic gas (=governed by strong interaction) limited to masses <1.8
GeV (= no charm, bottom and top) there 3 conserved charges (I3 = 3rd isospin
component, B= baryon number, S=strangeness)
mi  m I3 I 3i  m B Bi  m S Si
 mI3, mB and mS are the potentials corresponding to each conserved charge
 mi = energy needed to add to the system a particle of specie i with quantum
numbers I3i, Bi, Si
45
Grand canonical partition function (II)
Transforming into logarithm:
ln Z iGC (T ,V , mi ) 

a

 ln 1  e  b ( Ea mi )
Continuum limit:
ln Z iGC

(T , V , m i ) 
Vgi
2p
2


0
Vgi
2p
2






 p 2 dp ln 1  e  b ( E  mi ) 
0

 p 2 dp ln 1  i e  bE
where we have introduced the fugacity:
i  e bm i
46
Particle densities
By performing the integral in the expression of the grand
canonical partition function (see backup slides):
ln Z iGC (T ,V , mi )

VgiT
2p 2


k 1
(1) k
k2
 kmi
k 2
i mi K 2 


 T 
The density ni of particles (hadrons) of specie i is:
N i (T , V , m i )
V
ni (T , m i ) 

g iT 2
2p 2
1  (T ln Z iGC )


V
m i

(1) k ik 2  kmi
mi K 2 
2
m i
 T
k 1 k

 g iT 2
 
2
 2p
km
(1) k   T i
e
2

m
i 
k 1 k




 km
 mi2 K 2  i
 T




 





2
2p

where Ni is the total number of particles of specie i in the system
g iT
(1) k k 2  kmi
i mi K 2 
k
 T
k 1

47
Other points
DECAY CHAINS
The total number of measured particles of specie i (e.g. pions) is
given by “thermal” production (Ni) + contribution from decays of
short-lived particles that are not measured (e.g.  decaying into
pions)
Ni
MEAS
 Ni
THERM

 BR
j i
 Nj
THERM
j
EXCLUDED VOLUME CORRECTION
A repulsive term should be introduced in the partition function to
account for the repulsive force between hadrons at short
distances,
e.g. by assigning a eigen-volume to each hadron (Van Der Waals like)
STRANGENESS SUPPRESSION FACTOR (gS)
Accounts for the fact that the s quark, due to its larger mass may
not be completely equilibrated
 gS ≈ 1 in heavy ion collisions at SPS and RHIC (= no strangeness suppression)48
Free parameters of the model
Particle multiplicities given by:
N i (T ,V , mi )  V  ni (T ,V , mi ) 
with i  e mi / T
,
VgiT
2p 2

(1) k k 2  kmi
i mi K 2 
k
 T
k 1




mi  Bi mB  Si mS  I 3i mI3
There are 5 free parameters:
T, mB, mS, mI3 and V
There are 3 charge conservation laws which allow to
constrain 3 parameters starting from the knowledge of
electric charge (=third isospin component), baryonic
number and strangeness of the initial state (= protons ZS
and neutrons NS “stopped” from colliding nuclei)
Fireball volume V and chemical potentials mS e mI3 are constrained
So, we remain with 2 free parameters:
plus (possibly) gS
T e mB
49
Fit to measured particle ratios
Why use particle ratios ?
Some systematic errors in experimental data cancel in the ratio
The dependence on volume V is removed in model calculations
The determination of V is affected by the uncertainty on the stopping power and
on the “excluded volume” corrections
GOAL: find the values of T and mB that minimize the
difference between model predicted and measured
particle ratios
Done by minimizing a c2 defined as:
c 
2

i

Riexp.

model 2
 Ri
 i2
Riexp and Rimodel are the measured and predicted paerticle ratios
 i is the (statistical + systematic) error on experimental points
50
Particle ratios at AGS
AuAu - Ebeam=10.7 GeV/nucleon - s=4.85 GeV
Minimum of c2 for: T=124±3 MeV mB=537±10 MeV
c2 contour lines
 A. Andronic et al., Nucl. Phys. A772 (2006) 167.
51
Particle ratios at SPS
PbPb - Ebeam=40 GeV/ nucleon - s=8.77 GeV
Minimum of c2 for: T=156±3 MeV mB=403±18 MeV
c2 contour lines
 A. Andronic et al., Nucl. Phys. A772 (2006) 167.
52
Particle ratios at RHIC
AuAu - s=130 GeV
Minimum of c2 for: T=166±5 MeV mB=38±11 MeV
c2 contour lines
 A. Andronic et al., Nucl. Phys. A772 (2006) 167.
53
Model parameters vs. s
Temperature T increases
rapidly with s at low
energies untill it reaches 170
MeV (≈ critical temperatture
for phase transition) at
s≈7-8 GeV and then stays
constant
Chemical potential mB
decreases with increasing s
in the energy range from
AGS to RHIC
54
Model parameters on the phase
diagram
early universe
Chemical Temperature Tch [MeV]
250
RHIC
200
Lattice
QCD
150
SPS
100
hadron
gas
50
0
0
200
400
Statistical model
parameters T, mB can
be plotted on the
phase diagram of
nuclear matter
quark-gluon plasma
Can be compared with
the “phase boundary”
AGS
limit between
deconfinement
chiral restauration
hadronic matter and
QGP calculated with
SIS
lattice QCD
For s >≈ 10 GeV
atomic nuclei
neutron stars chemical freeze-out
very close to phase
600
800 1000
1200
boundary
Baryonic Potential mB [MeV]
55
Universality?
Application of the thermal model to e+e- and pp collisions
 Assume thermal and chemical equilibrium
 Canonical formulation of the partition function (quantum numbers
exactly conserved)
 INPUT: measured
particle multiplicities
 FIT PARAMETERS: T,
V, gS (to account for
incomplete strangeness
equilibration)
 F. Becattini and U. Heinz, Z Phys. C76 (1997) 269.
56
Universality?
Application of the thermal model to e+e- and pp collisions
 Assume thermal and chemical equilibrium
 Canonical formulation of the partition function (exact
conservation of quantum numbers)
Fitted temperatures:
 Compatible with constant freeze-
out at ≈ 170 MeV independent of
s
 Agree with values obtained in AA
collisions for s >≈ 10 GeV
Universality of hadronization at
critical values
 Limiting (Hagedorn) temperature
for Hadron Gas
 Lattice QCD phase boundary
 F. Becattini and U. Heinz, Z Phys. C76 (1997) 269.
57
Conclusions
Hadronization occurs following purely statistical laws
(entropy maximization)
 Hadron production dominated by phase space rather than by
microscopic dynamics
Universality of the
freeze-out temperature
independent of collision
energy for pp, e+e-, AA
collisions at s >≈ 10 GeV
 Hadronization occurs when
the parameters (energy
density, pressure…) of prehadronic matter drop below
critical values
corresponding to a
temperature ≈ 170 MeV
58
Towards the LHC
mBLHC=0.8+1.2
MeV
-0.6
TLHC = 161±4 MeV
A. Andronic et al. in arXiv:0711.0974 [hep-ph]
59
Elliptic flow
Flow in heavy ion collisions
Flow = collective motion of particles superimposed on top
of the thermal motion
 Collective motion is due to high pressure arising from compressing
and heating of nuclear matter.
 Flow velocity in a volume element is given by the sum of the
velocities of the particles
 Collective flow is a correlation between the velocity vector v of a
volume element and its space-time position
y
v
x
v
61
Flow in heavy ion collisions
Radial flow = isotropic (i.e. independent of azimuthal angle  )
expansion of the fireball in the transverse plane
Due to large pressures created in the fireball by matter compression
Integrated over whole period of fireball evolution
y
Only type of collective motion for b=0
Experimental observables: pT (mT) spectra
x
Anisotropic transverse flow = anisotropy present in particle azimuthal
distributions in collisions with impact parameter b≠0
Due to pressure gradients arising from the geometrical anisotropy of the
overlap region of the colliding nuclei
Develop at relatively early times in the system evolution
Experimental observables: particle azimuthal distributions relative to the
reaction plane  Fourier coefficients v1 , v2 , ….
y
x
z
x
62
Anisotropic transverse flow (I)
In heavy ion collisions the
impact parameter selects a
preferred direction in the
transverse plane
 The reaction plane is the plane
y
YRP
defined by the impact parameter
and the beam direction
Anisotropic transverse flow is a correlation between the
azimuth [=tan-1 (py/px)] of the produced particles and the
impact parameter (reaction plane)
A non vanishing anisotropic flow is built if the momenta of
the final state particles depend not only on the local
physical conditions in their production point, but also on
the global event geometry
63
 Unambiguous signature of collective behaviour
x
Anisotropic transverse flow (II)
In collisions with b≠0 the fireball shows an initial
geometrical anisotropy with respect to the reaction plane
 The overlap region of the colliding nuclei is “almond-shaped ”
The initial particle momentum distribution is isotropic
Microscopic point of view:
 Re-scatterings among produced
particles can convert this initial
geometrical anisotropy into an
observable momentum anisotropy
Macroscopic point of view:
 Pressure gradients in the transverse
plane are anisotropic (=  dependent)
Reaction plane
 Larger pressure gradient in the x,z plane
(along impact parameter) that along y
Observed particle momenta are anisotropic in 
64
Fourier coefficient: v2
Fourier development of particle azimuthal distributions
relative to the reaction plane (YRP is the reaction plane
angle in the transverse plane)
dN N0
1  2v1 cos(  YRP )  2v2 cos2  YRP  ....

d 2p
1  2v2 cos2
Elliptic flow coefficient


v2  cos 2  YRP 
65
Why elliptic flow ?
At time = 0:
geometrical anisotropy (almond shape)
momentum distribution isotropic
Interaction among constituents
Generate pressure gradients and transform initial
spatial anisotropy into a momentum anisotropy
Multiple interactions can lead to local thermal
equilibrium at an early stage
Hydrodynamic to describe the system evolution
from equilibration time until thermal freeze-out
The mechanism is self quenching
The driving force dominate at early times
Sensitive to Equation Of State at early times
66
v2 vs. s
s < 2 GeV
formation of a
rotating system
centrifugal forces
in plane flow (v2>0)
2 < s < 4 GeV
spectators block the
“in-plane” expansion
out-of-plane
(squeeze-out) flow
(v2<0)
s > 4 GeV
spectators leave the interaction region after a short time 2R/g
pressure gradients dominate
in plane flow (v2>0)
 J. Y. Ollitrault, Nucl. Phys. A638 (1998) 195.
67
v2 vs. centrality
Observed elliptic flow depends on:
Eccentricity
 decreases with increasing centrality
Amount of rescatterings  increases with increasing centrality
Very peripheral collisions:
large eccentricity, few rescatterings
 small v2
Semi peripheral collisions:
large eccentricity, many rescatterings
 large v2
Central collisions:
eccentricity ≈ 0
 ≈ isotropic distribution (v2 ≈ 0)
68
v2 vs. centrality at RHIC (I)
Hydrodynamic limit
Phys. Rev. Lett 86 (2001) 402
PHOBOS Phys. Rev. Lett 89 (2002) 22301
STAR
s=130 GeV
RQMD
Measured v2 well described by hydrodynamics from mid-central to
central collisions
Hydro assumptions:
Ideal fluid: zero viscosity
Equation of state with a first order phase transition from QGP to HG
Flow larger than expected from hadronic cascade models
Evidence for a strongly interacting (partonic) phase
69
v2 vs. centrality at RHIC (II)
Hydrodynamic limit
Phys. Rev. Lett 86 (2001) 402
PHOBOS Phys. Rev. Lett 89 (2002) 22301
STAR
s=130 GeV
RQMD
Simple interpretation
 In semi-central and central collisions the system theramlizes rapidly
(tequ≈0.6–1 fm/c) and behaves as ideal fluid
 For more peripheral collisions (smaller and less interacting system)
thermalization is incomplete and/or slower
BUT what would happen with different hydro assumptions?
 Equation of state, viscous/non viscous, freeze-out description …
70
Off-equilibrium scenario
Measured elliptic flow depends on the number N of rescatterings suffered by a particle
N 
L

 Kn
1
1 dN
N  particledensity
S dy
 Kn = Knudsen number (ideal fluid: Kn0, non interacting gas: Kn>>1)
v2
In absence of re-scattering ( ideal gas) no
elliptic flow is built
equilibrium regime
v2increases with increasing N of rescatterings
constant v2
 Low-density-limit (v2/eccentricity  Kn-1)
After a number of collisions N0 the
Absence of equilibrium system thermalizes and further collisions
do not produce any increase of v2
v2N
 Hydrodynamic limit ( v2/eccentricity  cS2)
N Kn-1
71
v2 vs. multiplicity (I)
(  Kn-1 )
Interpretation:
The slower thermalization is slower at AGS e SPS does not
allow to reach the hydrodynamic limit
The hydrodynamic limit is reached for central collisions at top
RHIC energy (perfect fluid at RHIC)
72
v2 vs. multiplicity (II)
Low-density-limit fit
v2/  dN/dy
(  Kn-1 )
Interpretation:
The trend as a function of Kn-1 is linear as predicted in the
“Low-density-limit” scenario
No evidence for v2 saturation with increasing number of rescatterings
73
Viscosity ?
Eccentricity scaling from ideal hydro + simple correction
factor for deviations from ideal fluid (viscous effects)
v2  v2
 





IDEAL HYDRO


1  Kn 


K
0


1
with Kn 
 dN
S dy
cs
K0 =0.7 (from transport calculation)
cs = speed of sound
 = eccentricity
S = transverse nuclear overlap area
PHOBOS data, s=200 GeV
2 FREE PARAMETERS IN THE FIT:
 = effective partonic cross section
v2IDEAL HYDRO = hydrodynamic limit
Deviation from ideal hydrodynamics (1+Kn/K0)-1 as large as
30% even for central AuAu collisions
 Drescher, Dumitru, Ollitrault, PRC76 (2007) 024905.
74
Conclusions after RHIC
Agreement between elliptic flow data and ideal hydrodynamics (for central
AuAu collisions) one of the pieces of evidence for the formation of “Strongly
interacting QGP” (sQGP) in AuAu collisions at RHIC
 The fireball rapidly thermalizes (tequ ≈ 0.6-1 fm/c) at a temperature well above Tcrit
 The system evolves as an almost ideal fluid with exceptionally low viscosity
BUT ALSO:
 ideal fluid description breaks down for peripheral collisions, interactions at lower
energies, particles away from mid-rapidity
 indications for viscous effects (no saturation in v2 vs. dN/dy)
Two contributions to viscous effects:
 incomplete thermalization of the QGP (“early viscosity”)
 dissipative effects in the hadronic stage (“late viscosity”)
From the theoretical side:
 Theoretical uncertainties on the input quantities for hydrodynamical evolution (initial
eccentricity, QGP viscosity…), equation of state and freeze-out mechanism
  T. Hirano et al., ArXiv:nucl-th/0511046: Hybrid model based on ideal hydro +
hadron cascade with only late viscosity reproduces data only for Glauber-like intial
conditions, while QGP viscosity is needed in case of parton-saturated initial state
  Luzum, Romatschke, ArXiv:0804.4015[nucl-th]: First results from viscous
relativistic hydrodynamics indicate that v2 does not reach the hydrodynamic limit 75
Towards the LHC (I)
Simple-minded extrapolation of observed trends
Logarithmic scaling with s
extended longitudinal scaling of v2 vs h
None of these scaling behaviours emerges as a natural
consequence of existing dynamical models
Extrapolations of ideal hydrodynamics from RHIC to LHC
76
predict values not exceeding v2=0.6 at h=0
Towards the LHC (I)
Low density or hydrodynamic limit ?
Distinguishable with the first 20000 PbPb events at the LHC
0.3
40
45
50
77
Backup
Full stopping vs. transparency
1/g
1/g
Nuclear fragmentation
regions
Fireball
Central rapidity region
Boost invariant expansion
79
Integrating the partition function (I)
Taylor expansion for the logarithm:
ln Z iGC
(T ,V , mi ) 


Vgi
2p
2
Vgi
2p
2


0

Vgi
2p
2


0


 p 2 dp ln 1  i e  bE 
 

(1) k k  bE
 p dp
i e
k
k 1
2

(1) k k
i
k
k 1



k

p 2 e  kbE dp
0
Note: Taylor expansion can be done if:
i e bE  1  e bm i  e bE  mi  E
80
Integrating the partition function (II)
Performing the integral:
ln Z iGC

(1) k k
i
k
k 1


 3
Vgi
(1) k  p  kbE

i  e
2
2p k 1 k
 3


(T ,V , mi ) 



Vgi
2p 2

Vgi
2p 2

k
(1) k k
i
k
k 1



0
p 2 e  kbE dp
0



0
0

p  kbE
dE 
dp
e
(kb )

3
dp 

3
p 3  kbE
p
dp
e
(kb )
3
E
where we used:
E
p
2
 mi2
dE d
1
p
2
2


p  mi 
 (2 p) 
dp dp
E
2 p 2  mi2
81
Integrating the partition function (III)
Change of integration variable from p to E:
ln Z iGC (T , V , m i ) 

Vgi
2p 2
(1) k k
i
k
k 1



0
p 3  kbE
p
dp
e
(kb ) 
3
E
3


Vgi
2p 2
Vgi
2p 2

(1) k k
i
k
k 1


(1) k
i
k
k 1

k




mi
mi


 E 2  mi2 
E
 e  kbE (kb ) p 
dE 
p
3
E

E
dE
2

2 3/ 2
 mi
3
e  kbE (kb )
where we used:
E
p 2  mi2
p
dp
E
 p  0  E  mi
 dE 
82
Integrating the partition function (IV)
Define x=kbE:
ln Z iGC
(T ,V , m i ) 



Vgi
2p 2
Vgi
2p 2
Vgi
2p 2

Vgi
2p 2




k

k

(1) k
i
k
k 1
(1)
k
k 1


k b
d (kbE )
2
(1) k
i
k
k 1
(1) k
i
k
k 1
k

E
dE
2
k
b
k mi

dx
x
k mi
2
k mi
i
kb

b
k mi

2 3/ 2
 mi
3
mi
2

b


2
k b
2

x
dx
2
2

2 3/ 2
mi
3k 3 b 3

2 3/ 2
mi
3
2
3k 3 b
2
E k b
2
e  kbE (kb ) 
ex 

2 3/ 2
 k b mi
3k 2 b 2 mi2
2
2
e  kbE 
ex
83
Integrating the partition function (V)
Define w=kbmi:
ln Z iGC



(T ,V , m i ) 
Vgi
2p
2
Vgi
2p 2
Vgi
2p 2


k 1

Vgi
2p 2
(1)
k

(1)
k 1

2
m
ik i

k
2
k mi
i
kb

dx
x
2
k2
k2
k
ik
mi2
b
2
m
ik i

x
dx

b
k mi
w

2 3/ 2
3w 2
w
x

w
 1
2
w



dx 
w
3w 2
 x2

dx 2  1
w



3/ 2
2

1
w
b 3


w
2

2 3/ 2
 k b mi
3k 2 b 2 mi2
2
2
ex 
ex 
3/ 2
3


(1)
k
k 1
b
k2
(1) k
k 1

ex 
ex
84
Integrating the partition function (VI)
Define y=x/w:
ln Z iGC


(T , V , m i ) 
Vgi
2p
2
Vgi
2p
2
Vgi
2p 2


k 1
(1)
2
m
ik i
k
1
w
b 3
k2

(1) k k mi2 1
i
w
k
kb 3
k 1



 dy y

1

(1) k k mi2  1 2
i
 w
k
kb  3
k 1

w
x

dx 2  1
w



wdy y  1




2

2
2
1
1
3 / 2  wy
e
3/ 2
ex 

3 / 2  wy 
e


The term in square brackets coincides with this integral
representation of the modified Bessel functions
p
t
K n (t ) 
 
1
(n  2 ) !  2 
n
 

1

dy y 2  1
n 1/ 2 ty
e
85
Integrating the partition function (VII)
Substituting w=kbmi and b1/T :
ln Z iGC



(T , V , m i ) 
Vgi
2p
2
Vgi
2p
2
VgiT
2p 2

2p

(1) k

(1) k
k 1

k
k 1


k 1
k

Vgi
2
2
(1) k
k2
2
(1) k k mi2  1 2
i
 w
k
kb  3
k 1

ik
ik
mi2
b
mi2
b
 dyy

1
2

1
3 / 2  wy 
e


K 2 ( w) 
K 2 (kbmi ) 

k 2
i mi K 2 
kmi
 T



86
Fit to multiplicities
If multiplicities are used instead of particle ratios
One more free parameter (the volume V)
Larger systematic uncertainties (both in the model and in the data)
T and mB agree with results from fit to ratios, but worse c2
 A. Andronic et al., Nucl. Phys. A772 (2006) 167.
87
Chemical freeze-out and
phase transition
T
Lattice-QCD
RHIC
SPS
Stat.Thermal Model
Case 1: (T,mB) far below the QCD “phase boundary ”
 Long hadronic phase after phase transition?
 The system does not reach the “phase boundary” ?
mb
RHIC
Case 2: (T,mB) far above the QCD “phase boundary ”
 Problem in the statistical hadronization model ?
T
SPS
 Hypothesis of hadron-resonance gas no longer valid
 Problem in the Lattice QCD “phase boundary”?
mb
T
RHIC
SPS
AGS
mb
Case 3: (T,mB) close to QCD “phase boundary ”
 Rapid chemical freeze-out immediately after the phase
transition ?
88
Fourier coefficient: v1
Fourier development of particle azimuthal distributions
relative to the reaction plane


dN N0
1  2v1 cos(  YRP )  2v2 cos 2  YRP   ....

d 2p
1  2v1 cos
Directed flow coefficient
v1  cos  YRP 
89
Higher order harmonics
dX X 0
1  2v1 cos(  YRP )  2v2 cos2  YRP   ....

d 2p
Fourth order coefficient v4:
Restore the elliptically
deformed shape of particle
distribution
Magnitude and sign sensitive to
initial conditions of hydro
 Kolb, PRC 68, 031902(R)
Ideal hydro: v4/v22 = 0.5
 Borghini, Ollitault, nuclth/0506045
90