Thermal freeze-out versus

chemical freeze-out

revisited

Dariusz ProrokInstitute of Theoretical Physics,University of Wroclaw, Poland

May 18, 2007

DP, nucl-th/0702042

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

How (when) the statistical system ends?

Generally, the freeze-out moment (the end of a statistical system) is definedas a moment when hadrons cease to interact and start to stream freely todetectors.

To be more precise one can distinguish two freeze-outs:

The chemical freeze-out means the moment when the inelastic interactionsbetween hadrons cease and the chemical composition of the system is fixed(the end of the chemical equilibrium).

The thermal (kinetic) freeze-out means the moment when also elasticinteractions cease and hadrons start to escape freely.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 1

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Tchem ≥ Tkin

At the (kinetic) freeze-out the momentum distributions are frozenand these are primordial distributions:

fprimordiali =

(2si + 1)

(2πhc)31

exp

Ei−µiTkin

± 1

µi = BiµB + SiµS + Ii3µI3

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 2

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Summary of the behavior of Tchem and Tkin (until now)

According to U. Heinz, G. Kestin, nucl-th/0612105(Tkin fitted within the blast-wave model)

Tchem = 160 − 170 MeV and does not depend on the centrality and(approximately) on beam energy (for

√s > 10A GeV)

Tkin depends on the centrality and beam energy

Tchem is ∼ 50% greater than Tkin

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 3

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

The Single-Freeze-Out Model

W. Broniowski and W. Florkowski, PRL 87 (2001) 272302; PRC 65 (2002)064905; APPB 33 (2002) 1935

A. Baran, W. Broniowski and W. Florkowski, APPB 35 (2004) 779

• successfully reproduces particle abundance ratios and pT spectrameasured at RHIC

• also works well for SPS

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 4

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Assumptions of the Single-Freeze-Out Model

1. Simultaneous occurrence of the chemical and thermal freeze-outs.

2. The freeze-out hypersurface and the Hubble-like expansion

τ =√

t2 − r2x − r2

y − r2z = const, uµ =

xµ

τ

with condition r =√

r2x + r2

y < ρmax .

3. Contributions from resonance decays to the measured particle multiplicitiesand momentum distributions are taken completely into account.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 5

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Parameters of the Single-Freeze-Out Model

Statistical parameters: T, µB

from fits to particle yield ratios

Geometric parameters: τ, ρmax

from fits to pT -spectra

the maximum transverse-flow parameter βmax⊥ = ρmax/τ√

1+(ρmax/τ)2

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 6

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

The invariant distribution of particle species i

dNi

d2pT dy=

∫

pµdσµ fi(p · u)

dσµ - normal vector on the freeze-out hypersurface

fi - final momentum distribution of the ith particle,i.e. with contributions from resonance decays:

fi = fprimordiali +

∑

decay

fdecayi

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 7

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Determination of parameters in the single-freeze-out model

Two steps:

First, statistical parameters T and µB were fitted with the use of theexperimental ratios of hadron multiplicities at midrapidity.

Second, with values of T and µB substituted into primordial distributions ofhadrons, geometric parameters were determined from fits to the transverse-momentum spectra.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 8

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

NOW!

Assumption about one freeze-out is released from now on!

All four parameters of the model T , µB, ρmax and τ are fitted simultaneouslyto the spectra.

Therefore, now the temperature T has the meaning of the kinetic freeze-outtemperature Tkin.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 9

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

PHENIX @√

sNN = 200 GeV, PRC 69, 034909 (2004)

Centrality T µB ρmax τ βmax⊥ χ2/NDF

[%] [MeV] [MeV] [fm] [fm]

0-5 150.07±1.34 24.10±3.66 9.28±0.21 9.48±0.19 0.70 0.69

5-10 150.18±1.35 23.48±3.65 8.75±0.20 8.80±0.18 0.70 0.50

10-15 150.16±1.35 22.75±3.65 8.25±0.19 8.20±0.17 0.71 0.37

15-20 150.00±1.36 22.38±3.65 7.80±0.18 7.69±0.16 0.71 0.37

20-30 149.59±1.31 24.03±3.47 7.13±0.16 6.96±0.14 0.72 0.45

30-40 149.79±1.36 23.78±3.56 6.14±0.14 6.03±0.12 0.71 0.66

40-50 148.53±1.40 22.52±3.71 5.28±0.13 5.27±0.11 0.71 0.89

50-60 147.75±1.51 22.02±4.03 4.38±0.12 4.55±0.10 0.69 0.96

60-70 144.57±1.65 21.63±4.56 3.63±0.11 3.91±0.09 0.68 1.12

70-80 141.77±1.98 24.13±5.68 2.84±0.10 3.22±0.09 0.66 1.23

80-92 140.62±2.46 14.29±7.12 2.24±0.10 2.77±0.09 0.63 1.13

Tchem ≈ 155 MeV, µB ≈ 26 MeV, J. Rafelski et al., PRC72, 024905 (2005).

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 10

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Blast-wave:

Tkin = 111 − 147 MeV from the most central to peripheral bin,

βmax⊥ = 0.46 − 0.7 from the peripheral to most central bin,

done for preliminary data in NPA 715, 498 (2003)

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 11

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Figure 1: Simulation of the temperature dependence of

χ2/NDF for PHENIX at√

sNN = 200 GeV and the 0 − 5%

centrality class. Solid line is the polynomial 3 best fit.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 12

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

STAR @√

sNN = 200 GeV, PRL 92, 112301 (2004)

Centrality T µB ρmax τ βmax⊥ χ2/NDF

[%] [MeV] [MeV] [fm] [fm]

0-5 159.99±1.19 24.00±2.17 9.22±0.31 7.13±0.19 0.79 0.30

5-10 160.58±1.16 24.97±2.17 8.34±0.28 6.75±0.18 0.78 0.27

10-20 161.20±1.14 22.91±2.15 7.45±0.24 6.17±0.16 0.77 0.22

20-30 162.27±1.12 23.05±2.17 6.31±0.20 5.60±0.14 0.75 0.25

30-40 161.97±1.08 20.43±2.17 5.38±0.17 5.15±0.12 0.72 0.19

40-50 162.97±1.08 21.01±2.21 4.46±0.14 4.64±0.11 0.69 0.13

50-60 163.41±1.07 18.75±2.25 3.67±0.12 4.13±0.10 0.66 0.13

60-70 162.39±1.06 16.47±2.31 2.95±0.10 3.79±0.09 0.61 0.26

70-80 163.70±1.15 15.84±2.50 2.22±0.09 3.16±0.08 0.57 0.61

Tchem ≈ 160 MeV, µB = 15 − 24 MeV (peripheral to central), JPG 31, S93 (2005)

Blast-wave: Tkin = 89 − 129 MeV (central to peripheral), βmax⊥ = 0.63 − 0.84 (opposite)

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 13

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

BRAHMS @√

sNN = 200 GeV, PRC 72, 014908 (2005)

Centrality T µB ρmax τ βmax⊥ χ2/NDF

[%] [MeV] [MeV] [fm] [fm]

0-10 150.60±1.39 23.07±3.51 9.26±0.25 8.65±0.21 0.73 0.43

10-20 151.38±1.48 26.53±3.72 8.07±0.23 7.68±0.19 0.72 0.42

20-40 149.43±1.54 25.92±3.98 7.00±0.21 6.73±0.17 0.72 0.26

40-60 148.36±2.02 26.69±5.21 5.02±0.20 5.38±0.17 0.68 0.52

Blast-wave estimate:

Tkin = 110 − 115 MeV (central to peripheral),

βmax⊥ = 0.73 − 0.78 (peripheral to central), PRC 72, 014908 (2005)

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 14

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

PHENIX @√

sNN = 130 GeV, PRC 69, 024904 (2004)

Centrality T µB ρmax τ βmax⊥ χ2/NDF

[%] [MeV] [MeV] [fm] [fm]

0-5 166.74±3.96 35.06±8.97 6.31±0.41 8.08±0.44 0.62 0.53

5-15 161.70±3.21 43.14±7.77 6.336±0.346 7.574±0.341 0.64 0.46

15-30 162.33±3.29 38.52±7.75 5.322±0.292 6.535±0.289 0.63 0.50

30-60 162.25±3.46 31.80±7.87 3.772±0.226 4.946±0.228 0.61 0.75

60-92 159.46±6.85 37.05±16.09 1.866±0.266 3.263±0.269 0.50 1.37

Tchem = 165 MeV, µB = 41 MeV, W.Florkowski, W.Broniowski, M.Michalec, APPB 33,

761 (2002)

Blast-wave estimate: Tkin = 121 − 161 MeV from the most central to peripheral bin,

βmax⊥ = 0.24 − 0.7 from the peripheral to most central bin, PRC 69, 024904 (2004)

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 15

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

PHOBOS @√

sNN = 62.4 GeV, PRC 75, 024910 (2007)

Centrality T µB ρmax τ βmax⊥ χ2/NDF

[%] [MeV] [MeV] [fm] [fm]

0-15 148.52±1.15 72.60±3.81 7.84±0.19 8.35±0.14 0.68 0.995

15-30 149.64±1.42 69.47±4.13 6.23±0.19 7.20±0.17 0.65 0.43

30-50 151.29±1.70 67.15±4.47 4.60±0.18 6.05±0.18 0.61 0.20

Blast-wave estimate:

Tkin = 101 − 103 MeV (peripheral to central),

βmax⊥ = 0.72 − 0.78 (peripheral to central), PRC 75, 024910 (2007)

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 16

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Figure 2: Centrality dependence of the kinetic

freeze-out temperature for the PHENIX measurements at√

sNN = 62.4, 130 and 200 GeV.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 17

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Figure 3: Centrality dependence of the baryon number

chemical potential for the PHENIX measurements at√

sNN = 62.4, 130 and 200 GeV.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 18

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Figure 4: Comparison of π+ spectra measured by STAR and

PHENIX for the 0 − 5% centrality bin at√

sNN = 200 GeV.

PHENIX data range extends to pT = 2.95 GeV.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 19

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Figure 5: Comparison of π− spectra measured by STAR and

PHENIX for the 0 − 5% centrality bin at√

sNN = 200 GeV.

PHENIX data range extends to pT = 2.95 GeV.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 20

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Figure 6: Comparison of K+ spectra measured by STAR and

PHENIX for the 0 − 5% centrality bin at√

sNN = 200 GeV.

PHENIX data range extends to pT = 1.95 GeV.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 21

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Figure 7: Comparison of K− spectra measured by STAR and

PHENIX for the 0 − 5% centrality bin at√

sNN = 200 GeV.

PHENIX data range extends to pT = 1.95 GeV.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 22

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Simple test

Figure 8: Transverse momentum distributions of Ω−+Ω+ for

| y |< 0.75 in Au-Au collisions at√

sNN = 200 GeV. Data

are from STAR, (statistical) errors are of the size of symbols.

Lines denote model predictions: solid based on fits to the STAR

spectra, dashed based on fits to the PHENIX spectra.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 23

Dariusz Prorok Thermal freeze-out versus chemical freeze-out revisited

Conclusions

1. The freeze-out temperature and baryon number chemical potential obtained in

simultaneous fits of all four parameters of the model depend weakly on the centrality

of the collision.

2. The kinetic freeze-out temperature and baryon number chemical potential are close

to the chemical freeze-out values determined from fits to particle yield ratios.

3. These justify the assumption about one freeze-out postulated by W. Broniowski

and W. Florkowski.

4. The behavior of Tkin depends on the model.

IV Polish Workshop on Relativistic Heavy-Ion Collisions, Cracow, 17-18 May 2007 24