ENERGY AND SYSTEM SIZE DEPENDENCE OF CHEMICAL FREEZE-OUT OUTLOOK Statistical hadronization model Data and analysis Chemical freeze out parameters Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Small systems In small systems up to B~10, take into account only the charge configurations that match exactly the original net charge numbers (elementary systems) No chemical potentials, only 3 free parameters: T, V, S For semi large systems, conserve strangeness exactly and introduce chemical potentials for B and Q (C-C and Si-Si) free parameters are: T, V, S, B and Q Primary multiplicity: Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Large systems In heavy ion collisions it is enought to take into account the conservation of charges in the average sense (Grand-canonical ensemble) 6 free parameters: T,V, B, S, Q and S S and Q are fixed by additional conditions: Q/B = Z/A and S=0 The final multiplicity is the sum of primary production + particles coming from resonance decays. For most of the lightest members of hadronic families major contribution comes from the decays. Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Homogenious freeze out Analysis may be performed assuming a single fireball, if 1) Distribution of charges and masses is the same as coming from random splitting of a single fireball and the sum of the rest frame volumes equals the volume of the large fireball. 2) The clusters are large and the distribution of charges, masses and relevant thermal parameters is relatively flat (Boost invariant scenario). #2 does not hold at SPS and below. #1 might hold at SPS, but 4 multiplicities must be taken into account. #2 might hold at RHIC since the rapidity distributions of pions and anti baryon/baryon –ratios are flat at least in one unit of rapidity around y=0. The flat area is wider than a typical width of rapidity distribution coming from a single cluster at kinetic freeze out Allows to determine the characteristics of the average source at midrapidity Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

PHENIX collaboration: Data experiment system beam energy NA49 p-p 158 AGeV C-C Si-Si Pb-Pb 80 AGeV 40 AGeV 30 AGeV 20 AGeV E-802 Au-Au 11.6 AGeV STAR 130 AGeV (CM) PHENIX 200 AGeV (CM) Phys.Rev.C73:044905,2006 STAR collaboration: Phys. Rev. C70:041901,2004 Phys. Rev. Lett. 92:182301,2004 Phys. Lett. B595:143,2004 Nucl. Phys. A715:470,2003 nucl-ex/0311017 Phys. Rev. C66:061901,2002 Phys. Rev. Lett. 89:092302,2002 Phys. Rev. C65:041901,2002 nucl-ex/0606014 Phys. Rev. C71:064902,2005 Phys. Lett. B612:181,2005 Phys. Rev. Lett. 92:112301,2004 PHENIX collaboration: Phys. Rev. C69:024904,2004 Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Pb – Pb collisions In Pb-Pb systems most of the particle multiplicities are described well with SHM Largest deviations from the experimental numbers:  yield too large at all energies except 80 AGeV K+ yield too low at all energies except 158 AGeV K- yield too large gets worse as beam energy increases However, some of the particle ratios are not described well at all drops down at higher energies and agrees with RHIC ratio

Also, multiplicites at C-C and Si-Si are described well with SHM Again, largest deviations from the experimental numbers are with  Possible sources for the deviations - The tail of the exponential mass spectrum gets more important at high temperature - Distribution of charges among clusters is not equal to the one coming from random splitting of a large cluster - Some reaction meachanisms are not taken into account Statistical model results are not sensitive to other ’’internal variables’’ like widths and branching ratios. Number of resonances included in the analysis can cause a shift in parameters More particles: lower temperature, higher S Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

 removed from the fit due to 5 deviation NA49: p-p 158AGeV prtcl measurement stat. model + 3.15 +- 0.16 3.25 - 2.45 +- 0.12 2.43 K+ 0.21 +- 0.02 0.23 K- 0.13 +- 0.013 0.12  0.115 +- 0.012 0.133 anti  0.0148 +- 0.0019 0.0147 - 0.0031 +- 0.0003 0.0029 + (9.2 +- 0.09) £ 10-4 9.18 £ 10-4  (2.6 +- 1.3) £ 10-4 8.87 £ 10-5 anti  (1.6 +- 0.9) £ 10-4 6.16 £ 10-5 anti p 0.040 +- 0.007 0.036 K0s 0.18 +- 0.04 0.14 (1520) 0.012 +- 0.003 0.011  0.012 +- 0.0015 0.020 Exact canonical calculation (B=Q=2, S=0) Model with S does not describe multistrange hyperons well ! use model in which mean number of poissonially distributed strange quark pairs hadronize T 181.5 +- 3.4 MeV hssi 0.46 +- 0.020 VT3 6.2 +- 0.5 2/dof 8.4/10  removed from the fit due to 5 deviation Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Statistical approach at midrapidity At RHIC statistical analysis may be performed in a limited rapidity window. Similarly to 4 analysis, assume vanishing net strangeness. This is not quaranteed at midrapidity, but seems like a reasonable assumption (fixes S). Take Q/B = Z/A (fixes Q). Fit to the rapidity densities around y=0, i.e. scale particle densities with common scaling parameter V. BRAHMS 4 data not suitable for statistical analysis without additional assumptions Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

STAR: Au-Au Ö sNN = 200 GeV (5% most central) prtcl measurement stat. model + 322 +- 32 325 - 327 +- 33 327 K+ 51.3 +- 7.7 57.1 K- 49.5 +- 7.4 53.5  16.7 +- 1.1 16.0 anti  12.7 +- 0.92 12.1 - 2.17 +- 0.20 1.87 + 1.83 +- 0.21 1.53  + anti  0.53 +- 0.057 0.63 p 34.7 +- 6.2 42.9 anti p 26.7 +- 4.0 30.9  7.70 +- 0.90 7.10 Most of the rapidity densities are described well with SHM T 161.0 +- 3.9 MeV S 1.02 +- 0.05 B 30.0 +- 9.8 MeV VT3e-0.7/T 12.5 +- 0.7 2/dof 12.5/8 STAR: T 157 +- 6 MeV S 0.86 +- 0.11 B 22 +- 4 MeV nucl-ex/0310004

STAR: Au-Au Ö sNN = 130 GeV (5% most central) prtcl measurement stat. model + 239 +- 10.6 229 - 232 K+ 47.6 +- 6.7 47.3 K- 43.2 +- 6.0 43.6  17.2 +- 1.8 17.2 anti  12.2 +- 1.3 12.5 - 2.13 +- 0.27 1.87 + 1.78 +- 0.24 1.47  0.34 +- 0.10 0.40 anti  0.36 +- 0.11 0.35 p 26.7 +- 6.0 31.8 anti p 19.1 +- 4.3 21.6  6.09 +- 0.77 7.0 K0s 35.6 +- 5.7 45.9 K(892)0 10.9 +- 2.7 13.6 Experimental data centralitites: pions and Lambdas 5% K:s and p:s 6% Xi:s and Omega:s 10% phi 11% Everything extrapolated to 5% most central events by assuming linear scaling with dh-/dy T 160.3 +- 4.4 MeV S 1.25 +- 0.08 B 36.3 +- 12.7 MeV VT3e-0.7/T 8.2 +- 0.6 2/dof 5.6/9

PHENIX: Au-Au Ö sNN = 130 GeV (5%) prtcl measurement stat. model + 276 +- 36 264 - 270 +- 35 270 K+ 46.7 +- 7.2 46.2 K- 40.5 +- 6.5 42.9  17.3 +- 4.4 15.9 anti  12.7 +- 3.4 11.8 p 28.7 +- 4.1 29.6 anti p 20.1 +- 3.0 20.6 Experimental data 5% most central Consistency check: A subset (without multistrange hyperons) of the STAR 130 AGeV data T 158.0 +- 5.9 MeV S 1.24 +- 0.22 B 33.5 +- 17.8 MeV VT3e-0.7/T 8.1 +- 1.1 2/dof 0.5/4 Fit to PHENIX data agrees with the fit to STAR data

PHENIX: Au-Au Ö sNN = 130 GeV (5%) prtcl measurement stat. model S == 1 + 276 +- 36 264 280 - 270 +- 35 270 285 K+ 46.7 +- 7.2 46.2 42.0 K- 40.5 +- 6.5 42.9 39.2  17.3 +- 4.4 15.9 13.9 anti  12.7 +- 3.4 11.8 10.4 p 28.7 +- 4.1 29.6 30.3 anti p 20.1 +- 3.0 20.6 21.2 The minima is quite flat: Setting S == 1 describes the data well T 158.0 +- 5.9 MeV S 1.24 +- 0.22 1.00 (fixed) B 33.5 +- 17.8 MeV 31.9 +- 17.1 MeV VT3e-0.7/T 8.1 +- 1.1 9.2 +- 0.6 2/dof 0.5/4 2.0/5 Setting S == 1 with STAR data (including s and s) leads to worse fit with higher T

System size dependence: Baryon chemical potential Baryon chemical potential seems to be independent on system size at 158AGeV Centrality independence of B seen at √sNN = 200 and 17.2 GeV (STAR Cleymans et. al) mB= mB (√sNN) B(17.2 GeV) ≈ 250 MeV NA49 √sNN = 17.2 GeV C-C, Si-Si and Pb–Pb Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Energy dependence: Baryon chemical potential Baryon chemical potential is a smooth, strongly decreasing function of the beam energy at AGS-SPS energy regime Energy dependence can be parameterized as B =  ln(√sNN) / (√sNN) with  ≈ 2.0 and ≈ 1.1 or Cleymans et al: B = a/(1+√sNN/b) a ≈ 1.3 GeV and b ≈ 4.3 GeV RHIC points are compatible with these Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Energy dependence: Baryon chemical potential S scales with B B ≈ 4.2 S Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Energy dependence: Temperature At AGS-SPS energy regime √sNN = 4 – 17 Strong energy dependence T = a – bB2 T= T0(A) – C*mB(√sNN)2 At heavy ion collisions (A ¼ constant): T = T0 – C*[ ln (√sNN) / √sNN]2 with T0(208) = 162 MeV C = b2 ≈ 0.67 and  ≈ 1.13 RHIC points are compatible with this Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

System size dependence: Temperature At top SPS energy √sNN = 17.2 Small systems decouple at higher temperature T= T0(A) – C*mB(√sNN)2 A dependent T0 can be approximated logarithmically: T0(A) = Tc –  log(A) = 191.5 MeV – 4.5 MeV * log(A) Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Energy dependence: Strangeness equilibration ? √sNN = 4 – 17 : S ≈ 0.7 – 0.9 Moderate energy dependence S = 1 – a exp (-b√[A√sNN]) a ≈ 0.61 b ≈ 0.021 Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Energy dependence: Strangeness equilibration Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

System size dependence: Strangeness equilibration Strong system size dependence at top SPS beam energy S = 1 – a exp (-b√[A√sNN]) From a fit without multistrange hyperons Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

System size dependence: Strangeness equilibration Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Chemical freeze out Line is T = a – b B2 Heavy ion systems fulfil E/N = 1GeV Si –Si : E/N ≈1.1 GeV C-C : E/N ≈1.15 GeV p-p: E/N ≈1.2 GeV Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006

Summary Statistical hadronization model describes vast variety of systems Some details are not reproduced Strangeness equilibrated only at RHIC Model parameters are smooth functions of beam energy and system size  allows phenomenological studies and predictions Jaakko Manninen Critical Point and Onset of Deconfinement ; Firenze 5th of July 2006