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Transcript of Francesco Prino INFN – Sezione di Torino XIII Mexican school on particles and fields, San Carlos,...
Francesco PrinoINFN – Sezione di Torino
XIII Mexican school on particles and fields, San Carlos, Mexico, Oct 7th 008
Soft physics observables in Soft physics observables in heavy ion collisionsheavy ion collisions
First dayFirst day
in view of the LHCin view of the LHC
Disclaimers: experimentalist’s point of view perspectives for the LHC
2
Reminder: phase diagramReminder: 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
3
Reminder: space time evolutionReminder: space time evolutionThermal 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
eq (RHIC) ~ 0.6 fm/c
4
Heavy ion results vs. timeHeavy ion results vs. timeResults 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
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
STAR, sent to PRL July 5th 2000
… then came the high pT particle suppression from PHENIX (sent to PRL on Sept 9th 2008)
First 10k-20k events, fast analysis
statistics<≈100k events,longer analysis time due to the need of PID, detector calibration, combination of different detectors
6
Particle production in heavy ion Particle production in heavy ion collisionscollisions
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
7
Particles produced in PbPb at SPSParticles produced in PbPb at SPS
In central PbPb collisions at SPS (s=17 GeV) more than 1000 particles are created
8
Particles produced in AuAu at RHICParticles produced in AuAu at RHIC
In central AuAu collisions at RHIC (s=200 GeV) about 5000 particles are created
9
Multiplicity and centralityMultiplicity and centralityThe 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
10
Glauber model calculations: Physical inputs:
Woods-Saxon density for colliding nucleiNucleon-nucleon inelastic cross-section
inel
Numerical calculation of Npart , Ncoll ... vs. impact parameter b
Evaluation of NEvaluation of Npartpart and N and Ncollcoll
Crre /)(0
01
C (Pb)= 0.549 fm
0 (Pb)= 0.16 fm-3
r0 (Pb)= 6.624 fm
Accel. AGS SPS RHIC LHC
√s (GeV) 3-5 17 200 5500
inel21 33 42 60
11
Particle production - HardParticle production - HardHard 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)
12
Particle production - SoftParticle production - SoftSoft 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
99.5% soft
In A-A collisions: WOUNDED NUCLEON
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)
13
Wounded nucleon modelWounded nucleon modelBased on experimental observation (about 1970s) that multiplicites measured in protno-nucleus collisions scale as:
v = average number of collisions between nucleons (=Ncoll)
So:
2
1
2
1
ppch
pAch
N
NR
pppart
pApart
pAcollpA
collppch
pAch
N
NNN
N
NR
2
1
2
1
2
1
since in p-p: Npart = 2 and in p-A: Npart= Ncoll+1
14
Measuring the multiplicityMeasuring the multiplicityExperimentally 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(||<0.5) o Nch(|y|<0,5)
NOTE: pseudorapidity is easier to access experimentally because it requires to measure just one variable (the polar angle ) and does not require particle identification and measurements of momenta
dN/d (dN/dy) distributions contain also other information on the dynamics of the interaction
L
L
pE
pEy ln
2
1
2
tanlnln2
1 L
L
p
p
p
p
15
Rapidity at RHIC (collider)Rapidity at RHIC (collider)Before collision: pBEAM=100 GeV/c per nucleon EBEAM=(mp
2+pBEAM2)=100.0044 per
nucleon =0.999956, BEAM≈100
After collision: Projectile and target nucleons (green) are
slowed down and they are located at lower y (and ) 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) :0
2
TARGETPROJ
MID
yyy
8.10
36.51
1ln
2
1ln
2
1
TARGETPROJ
BEAMBEAM
BEAMBEAMTARGETPROJ
yyy
pE
pEyy
16
Rapidity at SPS (fixed target)Rapidity at SPS (fixed target)Before collision: pBEAM=158 GeV/c , =0.999982
pTARGET=0 , TARGET=0
Midrapidity:
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/d distribution does not have this property
82.5
01ln2
1
82.51
1ln
2
1ln
2
1
TARGETPROJ
TARGET
BEAMBEAM
BEAMBEAMPROJ
yyy
y
pE
pEy
91.22
PROJMID
yy
17
PseudorapidityPseudorapidityMidrapidity region Particles with pT>pL
produced at angles around 90°
Bjorken formula to estimate the energy density in case of a broad plateau at midrapidity invariant for Lorentz boosts:
pL>>pT pL>>pT
pT = pL
= 45 (135) degrees = ±0.88
pT>pL
Fragmentation regions: Particles with pL>>pT
produced in the fragmentation of the colliding nuclei at angles around 0° e 180°
0
yf
TBJ dy
dN
Ac
m
18
PbPb collisions at SPSPbPb collisions at SPSPb-Pb at 40 GeV/c (√s=8.77 GeV) Pb-Pb at 158 GeV/c (√s=17.2 GeV)
Peak position moves (midrapidity = ybeam/2 )Particle density at the peak
increases with s
central
peripheral
20
Multiplicity per participant pairMultiplicity per participant pair
We introduce the variables:
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
2/
/0
partN
ddN
dd
dNNwith
N
Nch
part
ch
2/
21
dN/ddN/dmaxmax vs. centrality vs. centralityYield 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 centralityA two-component fit with dN/d [ (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
22
Factorized dependence of dNch/dmax on centrality and s reproduced by models based on gluon density saturation at small values of Bjorken x
dN/ddN/dmaxmax vs. centrality and vs. centrality and ss
increasing s – decreasing x
Armesto Salgado Wiedemann, PRL 94 (2005) 022002
Kharzeev, Nardi, PLB 507 (2001) 121.
3
1
0
0
][2
partch
part
NGeVsNd
dN
N
Pocket formula:
and from ep and eA data N0 only free parameter
23
dN/ddN/dmax max vs. vs. ssThe dN/d per participant pair at midrapidity in central heavy ion collisions increases with ln s from AGS to RHIC energiesThe s dependence is different for pp and AA collisions
24
Warning 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/cs = 130 GeV
25
Total multiplicity:
Need to extrapolate in the regions out of acceptance
Small extrapolation in the case of PHOBOS thanks to the wide coverage
Nch scales with Npart
Nch per participant pair different from p-p, but compatible with e+e-, collisions at the same energy
Total multiplicity (NTotal multiplicity (Nch ch ) vs. centrality) vs. centrality
dd
dNNch
26
Total multiplicity (NTotal multiplicity (Nch ch ) vs. ) vs. ssMultiplicity 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)
27
pp vs. epp vs. e++ee--
The 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
The effective energy seff available for particle production is defined as:
with this definition, multiplicities in e+e- and pp at the same seff result to be in agreement
s effse+ e- p p
collisionpp2
collisioneess
seff
M. Basile et al.,, Nuovo Cimento A66 N2 (1981) 129.
28
Universality Universality
1/ 42.2chN s
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
29
Gold vs. copperGold vs. copper
Unscaled dN/d 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
Cu+CuPreliminary
3-6%, Npart = 100
PHOBOS PHOBOS
62.4 GeV 200 GeV
Au+AuPreliminary
35-40%,Npart = 98
Cu+CuPreliminary
3-6%, Npart = 96
Au+Au35-40%, Npart = 99
30
Limiting fragmentation (I)Limiting fragmentation (I)Study particle production in the rest frame of one of the two nuclei Introduce the variable y’ = y - ybeam (or ’ = – 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/d’ reach a limiting value and become energy independent around ’=0
Observed for p-p and p-A collisions
In nucleus-nucleus collisions Particle production in fragmentation regions
independent of energy, but NOT necessarilyindependent of centrality
31
Limiting fragmentation (II)Limiting fragmentation (II)
Particle production independent of energy in fragmentation regions Extended limiting
fragmentation (4 units of at 200 GeV)
No evidence for boost invariant central plateau
PHOBOS Phys. Rev. Lett. 91, 052303 (2003)
32
ConclusionsConclusionsCharged 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:
1.12
3700
fm145
/GeV6.0
02
2
00
c
c
dy
dN
Ac
m
y
TBJ
Peak energy density
Thermalized energy density
BJ well above the predicted critical energy for phase transition to deconfined quarks and glouns
33
Towards the LHC (I)Towards the LHC (I)
Models prior to RHIC
Extrapolation of dN/dln s
5500
Saturation modelArmesto Salgado Wiedemann, PRL 94 (2005) 022002
16502.82/
/
00
d
dN
N
ddN ch
part
ch
Central collisions
Extrapolation of dNch/dmax vs s Fit to dN/d ln s Saturation model (dN/d s with =0.288) Clearly distinguishable with the first 10k events at the LHC
11005.52/
/
00
d
dN
N
ddN ch
part
ch
34
Towards the LHC (II)Towards the LHC (II)Extrapolation of limiting fragmentation behavior Persistence of extended longitudinal scaling implies that
dN/d grows at most logarithmically with s difficult to reconcile with saturation models
Log extrapolationdN/d ≈ 1100
Saturation modeldN/d ≈ 1600
Borghini Wiedemann, J. Phys G35 (2008) 023001
36
HadrochemistryHadrochemistryMeasurement 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 freeze-out ?
What was the baryonic content of the fireball ?
37
Multiplicity of identified particles (I)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
38
Multiplicity of identified particles (II)Multiplicity of identified particles (II)
Pions More abundant
among the produced hadronsdue to lower mass and
production threshold
Difference between abundances of + and - at low energies due to isospin conservationLarge 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)
39
Multiplicity of identified particles (III)Multiplicity of identified particles (III)
Antiprotons They are produced in
the collisionDifferent 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 protonsNet-protons ≈ 0Small number of protons
stopped from the colliding nuclei
40
Multiplicity of identified particles (IV)Multiplicity of identified particles (IV)
Kaons and hyperons The larger number of
K+ and with respect to their antiparticles (K- and bar) at low energies due to quark content of these hadronsK+ (us) and (uds) require
to newly produce only the strange quark, while light quarks are present in the stopped nucleons
K- (us) and bar require the production of 2 or 3 new quarks
Associated production of K+ and (ss pairs)
-
-
-
41
Multiplicity of identified particles (V)Multiplicity of identified particles (V)
Kaons and hyperons The difference
between K+ and K- (and between e bar) decreases with increasing s because the lower stopping power reduces the weight of “stopped” with respect to “produced” quarks
Very similar abundances of bar and antiprotonsThey are both composed
of 3 “produced” quarks and they have similar masses
42
Multiplicity of identified particles (VI)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 ( N > NK > Np )
43
Statistical hadronization modelsStatistical hadronization modelsBASIC 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
44
Statistical hadronization modelsStatistical hadronization modelsNOTES
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
45
Grand canonical partition function (I)Grand canonical partition function (I)Starting point: partition function for a gas of identical particles (Bose or Fermi) of a given hadronic specie i:
are the eigen-states (with energy E) of the single particle hamiltonian (= energy states with spin degeneracy)
i 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)
I3, B and S are the potentials corresponding to each conserved charge
= energy needed to add to the system a particle of specie i with quantum numbers I3i, Bi, Si
bosons
fermions1),,(
1)(
iEi
GCi eVTZ
iSiBiIi SBI 33
46
Grand canonical partition function Grand canonical partition function (II)(II)
Transforming into logarithm:
Continuum limit:
where we have introduced the fugacity:
0
22
0
)(22
1ln2
1ln2
),,(ln
Ei
i
Eii
GCi
edppVg
edppVg
VTZ i
iei
47
Particle densitiesParticle densitiesBy performing the integral in the expression of the grand canonical partition function (see backup slides):
The density ni of particles (hadrons) of specie i is:
where Ni is the total number of particles of specie i in the system
12
222
)1(
2),,(ln
k
ii
ki
ki
iGCi T
kmKm
k
TVgVTZ
12
22
12
222
2
12
222
2
)1(
2
)1(
2
)1(
2
)ln(1),,(),(
k
ii
ki
ki
k
ii
T
k
i
ki
k
ii
i
ki
ki
i
GCiii
ii
T
kmKm
k
Tg
T
kmKme
k
Tg
T
kmKm
k
Tg
ZT
VV
VTNTn
i
48
Other pointsOther pointsDECAY 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)
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 (S) Accounts for the fact that the s quark, due to its larger mass
may not be completely equilibrated S ≈ 1 in heavy ion collisions at SPS and RHIC (= no strangeness suppression)
j
THERM
jijTHERM
iMEAS
i NBRNN
49
Free parameters of the modelFree parameters of the modelParticle multiplicities given by:
There are 5 free parameters: T, B, S, I3 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 S e I3 are constrained
So, we remain with 2 free parameters: T e B
plus (possibly) S
12
22
)1(
2),,(),,(
k
ii
ki
ki
iiii T
kmKm
k
TVgVTnVVTN
33/ ,with IiSiBiiT
i ISBe i
50
Fit to measured particle ratiosFit to measured particle ratiosWhy 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 B that minimize the difference between model predicted and measured particle ratios
Done by minimizing a 2 defined as:
Riexp and Ri
model are the measured and predicted paerticle ratios
i is the (statistical + systematic) error on experimental points
i2
2model.exp2
i
ii RR
51
Particle ratios at AGSParticle ratios at AGSAuAu - Ebeam=10.7 GeV/nucleon - s=4.85 GeV
Minimum of 2 for: T=124±3 MeV B=537±10 MeV
2 contour lines
A. Andronic et al., Nucl. Phys. A772 (2006) 167.
52
Particle ratios at SPSParticle ratios at SPSPbPb - Ebeam=40 GeV/ nucleon - s=8.77 GeV
Minimum of 2 for: T=156±3 MeV B=403±18 MeV
2 contour lines
A. Andronic et al., Nucl. Phys. A772 (2006) 167.
53
Particle ratios at RHICParticle ratios at RHICAuAu - s=130 GeV
Minimum of 2 for: T=166±5 MeV B=38±11 MeV
2 contour lines
A. Andronic et al., Nucl. Phys. A772 (2006) 167.
54
Model parameters vs. Model parameters vs. ssTemperature 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 B decreases with increasing s in the energy range from AGS to RHIC
55
Model parameters on the phase Model parameters on the phase diagramdiagram
Statistical model parameters T, B
can be plotted on the phase diagram of nuclear matter
Can be compared with the “phase boundary” limit between hadronic matter and QGP calculated with lattice QCD
For s >≈ 10 GeV chemical freeze-out very close to phase boundary
neutron stars
Baryonic Potential B [MeV]
early universe
Chem
ical Tem
pera
ture
Tch
[M
eV
]
0
200
250
150
100
50
0 200 400 600 800 1000 1200
AGS
SIS
SPS
RHIC quark-gluon plasma
hadron gas
deconfinementchiral restauration
Lattice QCD
atomic nuclei
56
Universality?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, S (to account for incomplete strangeness equilibration)
F. Becattini and U. Heinz, Z Phys. C76 (1997) 269.
57
Universality?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.
58
ConclusionsConclusionsHadronization 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 pre-hadronic matter drop below critical values corresponding to a temperature ≈ 170 MeV
59
Towards the LHCTowards the LHC
A. Andronic et al. in arXiv:0711.0974 [hep-ph]
TLHC = 161±4 MeV
BLHC=0.8 MeV+1.2
-0.6
61
Flow in heavy ion collisionsFlow in heavy ion collisionsFlow = 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
x
yv
v
62
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 Only type of collective motion for b=0 Experimental observables: pT (mT) spectra
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 , ….
Flow in heavy ion collisionsFlow in heavy ion collisions
x
y
x
y
z
x
63
Anisotropic transverse flow (I)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 defined by the impact parameter and the beam direction
x
y
RP
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 Unambiguous signature of collective behaviour
64
Anisotropic transverse flow (II)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
x
yz
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) Larger pressure gradient in the x,z plane
(along impact parameter) that along yObserved particle momenta are anisotropic in
Reaction plane
65
Fourier coefficient: vFourier coefficient: v22
....2cos2)cos(212 21
0 RPRP vvN
d
dN
RPv 2cos2
Fourier development of particle azimuthal distributions relative to the reaction plane (RP is the reaction plane angle in the transverse plane)
Elliptic flow coefficient 2cos21 2v
66
Why elliptic flow ?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
67
vv22 vs. vs. sss < 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)
J. Y. Ollitrault, Nucl. Phys. A638 (1998) 195.
s > 4 GeV spectators leave the interaction region after a short time 2R/ pressure gradients dominate in plane flow (v2>0)
68
vv22 vs. centrality vs. centrality Observed elliptic flow depends on: Eccentricity decreases with increasing centrality Amount of rescatterings increases with increasing centrality
Central collisions: eccentricity ≈ 0 ≈ isotropic distribution (v2 ≈ 0)
Semi peripheral collisions: large eccentricity, many rescatterings large v2
Very peripheral collisions: large eccentricity, few rescatterings small v2
69
vv22 vs. centrality at RHIC (I) vs. centrality at RHIC (I)
Measured v2 well described by hydrodynamics from mid-central to central collisions Hydro assumptions:
Ideal fluid: zero viscosityEquation 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
Hydrodynamic limit
STAR
PHOBOS
Hydrodynamic limit
STAR
PHOBOS
RQMD
s=130 GeV
Phys. Rev. Lett 86 (2001) 402Phys. Rev. Lett 89 (2002) 22301
70
vv22 vs. centrality at RHIC (II) vs. centrality at RHIC (II) Hydrodynamic limit
STAR
PHOBOS
Hydrodynamic limit
STAR
PHOBOS
RQMD
s=130 GeV
Phys. Rev. Lett 86 (2001) 402Phys. Rev. Lett 89 (2002) 22301
Simple interpretation In semi-central and central collisions the system theramlizes rapidly
(equ≈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 …
71
Off-equilibrium scenarioOff-equilibrium scenarioMeasured elliptic flow depends on the number N of re-scatterings suffered by a particle
Kn = Knudsen number (ideal fluid: Kn0, non interacting gas: Kn>>1)
v2
N Kn-1
In absence of re-scattering ( ideal gas) no elliptic flow is built
v2increases with increasing N of re-scatterings Low-density-limit (v2/eccentricity Kn-1)
After a number of collisions N0 the system thermalizes and further collisions do not produce any increase of v2 Hydrodynamic limit ( v2/eccentricity cS
2)
Absence of equilibriumv2N
equilibrium regimeconstant v2
dy
dN
SKn
L 1densityparticle1 NN
72
vv22 vs. multiplicity (I) vs. multiplicity (I)
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)
( Kn-1 )
73
vv22 vs. multiplicity (II) vs. multiplicity (II)
( Kn-1 )
Low-density-limit fit v2/ dN/dy
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 re-
scatterings
74
Viscosity ?Viscosity ?
Drescher, Dumitru, Ollitrault, PRC76 (2007) 024905.
Eccentricity scaling from ideal hydro + simple correction factor for deviations from ideal fluid (viscous effects)
scdy
dN
SKn
K
Knvv
with1
1
0
HYDROIDEAL22
PHOBOS data, s=200 GeV
K0 =0.7 (from transport calculation)cs = speed of sound = eccentricityS = transverse nuclear overlap area
2 FREE PARAMETERS IN THE FIT: = effective partonic cross sectionv2
IDEAL HYDRO = hydrodynamic limit
Deviation from ideal hydrodynamics (1+Kn/K0)-1 as large as 30% even for central AuAu collisions
75
Conclusions after RHICConclusions after RHICAgreement 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 (equ ≈ 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
76
Towards the LHC (I)Towards the LHC (I)Simple-minded extrapolation of observed trends Logarithmic scaling with s extended longitudinal scaling of v2 vs None of these scaling behaviours emerges as a natural
consequence of existing dynamical models
Extrapolations of ideal hydrodynamics from RHIC to LHC predict values not exceeding v2=0.6 at =0
77
Towards the LHC (I)Towards the LHC (I)Low density or hydrodynamic limit ? Distinguishable with the first 20000 PbPb events at the LHC
0.3
40 45 50
79
Full stopping vs. transparencyFull stopping vs. transparency1/ 1/
Fireball
Landau
Nuclear fragmentation regions
Central rapidity regionBoost invariant expansion
Bjorken
80
Integrating the partition function (I)Integrating the partition function (I)
Taylor expansion for the logarithm:
Note: Taylor expansion can be done if:
0
2
12
01
22
0
22
)1(
2
)1(
2
1ln2
),,(ln
dpepk
Vg
ek
dppVg
edppVg
VTZ
Ek
k
ki
ki
k
kEki
ki
Ei
ii
GCi
Eeee iEE
ii 1
81
Integrating the partition function (II)Integrating the partition function (II)
Performing the integral:
where we used:
10
3
2
0
3
0
3
12
0
2
12
)(3
)1(
2
)(33
)1(
2
)1(
2),,(ln
k
Ekki
ki
EkEk
k
ki
ki
Ek
k
ki
ki
iGCi
E
pke
pdp
k
Vg
dp
dEke
pdpe
p
k
Vg
dpepk
VgVTZ
E
pp
mpmp
dp
d
dp
dEmpE
i
ii
)2(2
1
22
2222
82
Integrating the partition function (III)Integrating the partition function (III)Change of integration variable from p to E:
where we used:
1
2/322
2
1
322
2
10
3
2
)(3
)1(
2
)(3
)1(
2
)(3
)1(
2),,(ln
km
Ekiki
ki
km
Eki
ki
ki
k
Ekki
ki
iGCi
i
i
kemE
dEk
Vg
E
pke
mE
dEp
E
k
Vg
E
pke
pdp
k
VgVTZ
i
i
mEp
dpE
pdEmpE
0
22
83
Integrating the partition function (IV)Integrating the partition function (IV)
Define x=kE:
1222
2/322222
2
133
2/32222
2
133
2/3222222
2
1
2/322
2
3
)1(
2
3
)1(
2
3)(
)1(
2
)(3
)1(
2),,(ln
kmk
x
i
iiki
ki
kmk
xiki
ki
kmk
Ekiki
ki
km
Ekiki
ki
iGCi
i
i
i
i
emk
mkxdx
k
m
k
Vg
ek
mkxdx
k
Vg
ek
mkEkEkd
k
Vg
kemE
dEk
VgVTZ
84
Integrating the partition function (V)Integrating the partition function (V)
Define w=kmi:
1
2/3
2
22
22
12
2/3
2
23
2
22
12
2/3222
22
1222
2/322222
2
13
1)1(
2
3
1)1(
2
3
)1(
2
3
)1(
2),,(ln
kw
xiki
ki
kw
xiki
ki
kw
xiki
ki
kmk
x
i
iiki
ki
iGCi
ew
xdxw
m
k
Vg
ew
w
xw
dxm
k
Vg
ew
wxdx
m
k
Vg
emk
mkxdx
k
m
k
VgVTZ
i
85
Integrating the partition function (VI)Integrating the partition function (VI)
Define y=x/w:
The term in square brackets coincides with this integral representation of the modified Bessel functions
11
2/3222
2
11
2/322
2
1
2/3
2
22
22
13
1)1(
2
13
1)1(
2
13
1)1(
2),,(ln
k
wyiki
ki
k
wyiki
ki
kw
xiki
ki
iGCi
eydywk
m
k
Vg
eywdywk
m
k
Vg
ew
xdxw
m
k
VgVTZ
1
2/12
2
11
2!)()( tyn
n
n eydyt
ntK
86
Integrating the partition function (VII)Integrating the partition function (VII)
Substituting w=kmi and 1/T :
12
222
12
2
22
12
2
22
11
2/3222
2
)1(
2
)()1(
2
)()1(
2
13
1)1(
2),,(ln
k
ii
ki
ki
ki
iki
ki
k
iki
ki
k
wyiki
ki
iGCi
T
kmKm
k
TVg
mkKm
k
Vg
wKm
k
Vg
eydywk
m
k
VgVTZ
87
Fit to multiplicitiesFit to multiplicitiesIf 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 B agree with results from fit to ratios, but worse 2
A. Andronic et al., Nucl. Phys. A772 (2006) 167.
88
Chemical freeze-out andChemical freeze-out andphase transitionphase transition
Lattice-QCD Stat.Thermal Model
T
b
SPS
RHIC
T
b
SPS
RHIC
T
b
SPSRHIC
AGS
Case 1: (T,B) far below the QCD “phase boundary ” Long hadronic phase after phase transition? The system does not reach the “phase boundary” ?
Case 2: (T,B) far above the QCD “phase boundary ” Problem in the statistical hadronization model ?
Hypothesis of hadron-resonance gas no longer valid
Problem in the Lattice QCD “phase boundary”?
Case 3: (T,B) close to QCD “phase boundary ” Rapid chemical freeze-out immediately after the
phase transition ?
89
Fourier coefficient: vFourier coefficient: v11
....2cos2)cos(212 21
0 RPRP vvN
d
dN
RPv cos1
Fourier development of particle azimuthal distributions relative to the reaction plane
Directed flow coefficient cos21 1v
90
Higher order harmonicsHigher order harmonics
....2cos2)cos(212 21
0 RPRP vvX
d
dX
....2cos2)cos(212 21
0 RPRP vvX
d
dX
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, nucl-th/0506045