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

Pseudorapidity density of Pseudorapidity density of unidentified particlesunidentified particles

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

Pseudorapidity distributionsPseudorapidity distributionsMidrapidity 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

16

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

17

AuAu collisions at RHICAuAu collisions at RHIC

central

central

peripheralperipheral

energy s

18

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/

19

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

20

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

21

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

22

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

23

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

24

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)

25

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.

26

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

27

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

28

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)

29

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

30

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

31

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

Multiplicity of identified Multiplicity of identified particlesparticles

33

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 ?

34

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

35

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)

36

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

37

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)

-

-

-

38

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

39

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 )

40

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

41

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 deconfined 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

42

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

43

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

44

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

45

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

46

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

47

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

48

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.

49

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.

50

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.

51

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

52

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

53

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.

54

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.

55

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

56

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

Elliptic flowElliptic flow

58

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

59

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

60

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

61

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

62

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

63

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

64

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

65

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

66

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 …

67

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

v2 increases 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

68

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 )

69

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

70

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

71

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

72

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=0.06 at =0

73

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

BackupBackup

75

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

76

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

77

Full stopping vs. transparencyFull stopping vs. transparency1/ 1/

Fireball

Landau

Nuclear fragmentation regions

Central rapidity regionBoost invariant expansion

Bjorken

78

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

79

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

80

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

81

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

82

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

83

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

84

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

85

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

86

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.

87

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 ?

88

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

89

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

90

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)