1. Do small systems equilibrate chemically? Ingrid Kraus TU Darmstadt

2. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 20062 Outline Introduction to the Statistical Model –Ensembles, partition function Grand canonical ensemble –Comparison to data –Extrapolation and predictions for heavy-ion collisions at LHC –Experimental observables for T and μ B determination –Relevance of resonances From Pb+Pb to p+p: system size and energy dependence –Canonical suppression –Concept of equilibrated clusters –Comparison to data Summary

3. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 20063 Micro-canonical –closed system –E, V, N fix Canonical –heat bath –T, V, N fix Grand-canonical –open system –heat bath and particle reservoir –T, V,  fix Statistical Ensembles E, V, N T, V, N T, V b, N b T, V,  T, V b, N b EE Laplace transformation NN

4. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 20064 Partition function of a grand canonical ensemble Energy densityEntropy density Particle number densityPressure Grand-canonical partition function –i: species in the system –Mesons m < 1.5 GeV, Baryons m < 2 GeV Partition function and its derivations

5. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 20065 Partition function for species i with degenaracy factor g i with –(+) for fermions, (-) for bosons Model parameters –T and  B  S constrained by strangeness neutrality –V cancels in ratios  Q constrained by charge of nuclei Partition function and model parameters

6. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 20066 Comparison to Experimental Data –Accurancy in T,  B : few MeV –Different data selected for fits A.Andonic, P. Braun-Munzinger, J. Stachel, nucl-th/0511071

7. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 20067 T -  B – systematics, extrapolation to LHC hep-ph/0511094 Chemical decoupling conditions extracted from SIS up to RHIC Feature common behavior On the freeze-out curve: T LHC ≈ T RHIC ≈ 170 MeV T ≤ T C ≈ 170 MeV μ B from parametrised freeze-out curve: μ B (√(s NN ) = 5.5TeV) = 1 MeV Nucl. Phys. A 697 (2002) 902 Grand canonical ensemble for Pb+Pb predictions

8. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 20068 Predictions for Pb+Pb Reliable for stable particles Benchmark for resonances Errors: T = 170 +/- 5 MeV μ B = 1 + 4 MeV - 1 All calculations with THERMUS hep-ph/0407174

9. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 20069 Extraction of thermal parameters from data determine μ B from p/p sensitivity on T –increases with mass difference –decay contribution affect lighter particles stronger –increasing feed-down with increasing T –decay dilutes T dependence T from  and/or  K _

10. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200610 Resonance Decays Hadron Resonance gas  no resonance contribution  – 50% from feed-down –both exhibit same T dependence K decay exceeds thermal at LHC  –thermal production ≈ constant –resonance contribution dominant 75% of all  from resonances

11. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200611 Grand canonical ensemble –large systems, large number of produced hadrons Canonical ensemble –small systems / peripheral collisions, low energies –suppressed phase-space for particles related to conserved charges –density of particle i with strangeness S approxiamtely S: order of Bessel functions x: sum over strange hadrons, related to volume –Volume enters as additional parameter V –here: radius R of spherical volume V Canonical suppression T, V, N T, V b, N b T, V,  T, V b, N b

12. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200612 Canonical suppression –Stronger suppression for multi-strange hadrons –Suppression depends on strangeness content, not difference (expected from  S )

13. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200613 Suppression by undersatured phase-space –Stronger suppression for multi-strange hadrons –Suppression depends on difference of strangeness content (power of  S )

14. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200614 Suppression in small systems Suppressed strangeness production beyond canonical suppression –addressed by canonical treatment and undersaturation factor  S –new: equilibrated clusters SPS √(s NN ) = 17 AGeV

15. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200615 Modification of the model Statistical Model approach: T and μ B –Volume for yields → radius R used here Deviations: strangeness undersaturation factor  S –Fit parameter Alternative: small clusters (R C ) in fireball (R): R C ≤ R –Chemical equilibrium in subvolumes: canonical suppression –R C free parameter R RCRC

16. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200616 Fit Example All Fits were performed with THERMUS hep-ph/0407174 Fits with  S / R C give better description of data

17. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200617 System size and energy dependence of T and  B T independent of –System size –Data selection –Energy μ B smaller at RHIC

18. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200618 System size and energy dependence of the cluster size Small clusters in all systems Small system size dependence p+p –energy dependence? Pb+Pb –depends on data selection (multistrange hadrons needed)

19. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200619 System size and energy dependence of the cluster size A+A: clusters smaller than fireball R C not well defined for R C ≥ 2 fm because suppression vanishes

20. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200620 Canonical Suppression Particle ratios saturate at R C ≈ 2 - 3 fm –no precise determination for small strangeness suppression

21. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200621 Grand canonical ensemble –successful description of Au+Au, Pb+Pb data –extrapolations allow for predictions –determination of thermal parameters with few particle ratios –proper treatment of resonances is mandatory Summary Canonical ensemble –volume dependend suppression –stronger suppression modeled with smaller, thermally equilibrated clusters –successful description of p+p, C+C, Si+Si data –strangeness production in small systems reproduced with equilibrated subvolumes Outlook –strangeness production in p+p at LHC

22. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200622 Going into formulas performing the momentum integration –(+) for bosons, (-) for fermions –m i : mass of hadron i Particle number density

23. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200623 Approx. modified Bessel function Particle ratio Antiparticle/Particle ratio Density and Ratios

24. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200624 System size dependence of T and  B μ B decreases at mid-rapidity in small systems …. …. as expected from increasing antibaryon / baryon ratio

25. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200625 System size dependence of the cluster size Same trend as K / 

26. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200626 More SPS and RHIC 200 GeV Data

27. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200627 Model setting with  S  S –sensitive on data sample –increase with size –increase with energy

28. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200628 Extrapolation to LHC does strangeness in p+p at LHC behave grand canonical ? multiplicity increases with √(s NN ) –canonical and grand canon. event classes? plot from PPR Vol I

29. Ingrid Kraus, TU DarmstadtHot Quarks 2006, Sardinia, May 16, 200629 Prediction for p+p significant increase of ratios at R C ≈ 1.5 fm K /  and  behave differently –multistrange hadrons suffer stronger suppression R C will be determined with ALICE data