1. Recent Developments in THERMUS “The Wonders of Z ” Spencer Wheaton Dept of Physics University of Cape Town

2. Statistical-Thermal Model Fireball resulting from high-energy heavy- ion collision treated as an ideal gas of hadrons At freeze-out these hadrons are assumed to be described by local thermal distributions Chemical Freeze-out: multiplicities fixed Thermal Freeze-out: momenta fixed

3. THERMUS – Statistical-Thermal Model Analysis within ROOT SW, J. Cleymans and M. Hauer, Comput. Phys. Commun. 180 (2009) 84. SW, PhD Thesis, UCT (2005) 17 152 lines of code Decays & properties of 361 hadrons 24 C ++ classes … and growing!

4. Grand-Canonical Ensemble: B, S, Q & E conserved on average Strangeness Canonical Ensemble: S conserved exactly; B, Q & E conserved on average Fully Canonical Ensemble: B, S & Q conserved exactly; E conserved on average THERMUS performs calculations within 3 commonly applied statistical ensembles: T,  B,  S,  Q, V T,  B, S,  Q, V T, B, S, Q, V

5. THERMUS has proved extremely successful in describing hadron multiplicities and ratios… but like all statistical models it battles to reproduce the K + /  + “horn”: (J. Cleymans, H. Oeschler, K. Redlich and SW, Phys.Lett.B615:50-54, 2005)

6. Estimate of rest of Resonance Spectrum Including high-mass resonances (and  meson) improves the situation: ( Andronic, Braun-Munzinger, Stachel Phys.Lett.B673:142,2009)

7. Baryons and Mesons with u, d and s quarks up to 2.6 GeV (  meson included) … 2005 Particle Set: 2010 Particle Set: Dawit Worku (UCT) has since updated the THERMUS particle set to include also c and b quarks … Extended THERMUS Particle Set

8. With extended particle set comes need for extension in ensembles. So, coming soon: B, S, Q, C and b GCE S, C and b CE Fully B, S, Q, C and b CE But what about calculations beyond particle multiplicities and mean values? Work on fluctuations and correlations with Michael Hauer (Frankfurt) has all been done within THERMUS …

9. Statistical Thermal Model: Ensembles and Partition Functions Micro-canonical ensemble: Fixed E, P and B, S, Q, C, b … Canonical ensemble: Fixed B, S, Q, C, b … Grand-canonical ensemble: Nothing Fixed Exactly

10. Grand Canonical Ensemble: Canonical Ensemble (Traditional Approach): Quantum Stats Boltzmann Stats Chemical potentials gone! include flow here

11. Retain the chemical potentials & project out the GCE partition function: Canonical Ensemble (Alternative Approach): ala Michael Hauer

12. Example 1: Static Pion Gas (conserved Q )  as in GCE T = 150 MeV R = 6 fm Boltzmann Approximation: Chemical potential in Z is a free parameter, but choose well and oscillations cease or at least are reduced!

13. Little bit of work required to get integrand into a manageable form: de Klerk, Hauer, SW

14. Example 2: Full HR Gas Canonical Correction for hadron with charge content B i, S i, Q i : BSQ ensemble implemented in THERMUS uses result of Becattini and Keranen to calculate correction factors …

15. a) Becattini & Keranen - Static Boltzmann: 3D  2D integration, but still oscillatory Works well for THERMUS BSQ ensemble

16. Choose chemical potentials to smooth out integrands in both numerator and denominator- much easier to integrate A Carbon- Carbon Correction Factor Symmetry of before disappears, so 3D integration needed Hauer, de Klerk, SW b) Approach using Z :

17. Only 13 distinct quantum content combinations

19. So, on the cards is the application of this numerical technique to the B,S,Q,C,b ensemble in THERMUS… An analytic result has recently been derived by Beutler et. al. arXiv:0910.1697

20. Beutler et. al. arXiv:0910.1697: Canonical treatment of B, S, Q, C and b : Quantum statistics for lightest bosons… 5D integration  3D integration   Bessel functions

21. Fluctuations and Correlations

22. Starting point for fluctuations & correlations is again the partition function: E.g.

23. (Taylor expansion) Approximation good if chemical potentials and four- temperature chosen such that: Micro-Canonical Ensemble:

24. Large Volume (Thermodynamic) Limit: GCE mean Multi-variate normal distribution To get joint particle multiplicity distributions in various ensembles need to consider slices through GCE distribution ala Michael Hauer

25. Neutral Pion Gas (  +,  -,  0 ) in MCE (large V) Correlation coefficient within bin….. Correlation coefficient between bins : no dynamics!  - at T = 160 MeV  – &   at T = 160 MeV M. Hauer, G. Torrierri & S.W.

26. Correlation between disconnected momentum space bins (no dynamical effects) M. Hauer, G. Torrierri & SW

27. V1V1 Monte Carlo Particle Generator V2V2 V 1 : observed sub-system V 2 : unobserved V g = V 1 + V 2 : total system Constraints placed on system are imposed only on the total volume M. Hauer & SW published

28. The crux is the following:

29. Strategy of Monte Carlo Generator: 1)Sample sub-system V 1 Grand-Canonically in Boltzmann Approximation 2)For each particle of type i, generate a momentum magnitude following a Boltzmann Distribution: 3)Assign direction to particle assuming isotropic particle emission… 4)Allow 2 and 3 body decays 5) Reweight

30. System Considered Neutral, Static, T = 160 GeV and V 1 = 2000 fm 3 B, S, Q, E, P z considered for reweighting

31. = 0.00 = 0.75 = 0.25 = 0.50

32. = 0.00

33. = 0.25

34. = 0.50

35. = 0.75

36. = 0 S(s) = -1Q(s) = -1/3 Primordial N E [GeV] Strangeness Content Baryon Content p Z [GeV/c] Charge Content

37. = 0.25 Primordial N E [GeV] Strangeness Content Baryon Content p Z [GeV/c] Charge Content

38. = 0.50 Primordial N E [GeV] Strangeness Content Baryon Content p Z [GeV/c] Charge Content

39. = 0.75 Primordial N E [GeV] Strangeness Content Baryon Content p Z [GeV/c] Charge Content

40. Fully-Phase Space Integrated Results

41. AveragesVariances Co-Variances Correlation Coefficients 20 runs of 2.5 × 10 4 events Linear extrapolation to MCE

42. Momentum Space Dependence

43. Divide into 5 bins such that each bin contains 1/5 th of positives Momentum Bin Selection positives

44. GCE Correlation Coefficients 20 runs of 10 5 events

45. Extrapolating Variances to MCE 20 runs of 10 5 events Linear extrapolation to MCE primordial Largest baryon and strangeness content in  p T,5

46. Linear extrapolation to MCE Non-Linear extrapolation to MCE Extrapolating Covariances and Correlation Coefficients to MCE primordial 20 runs of 10 5 events

47. Multiplicity Fluctuations and Correlations

48. GCE Scaled Variance of Positives 20 runs of 2 × 10 5 events

49. GCE Correlation Coefficient  +-

50. Scaled Variance of Primordial Positives 20 runs of 2 × 10 5 events

51. Primordial Correlation Coefficient

52. Scaled Variance of Final State Positives

53. Final State Correlation Coefficient

54. Concluding Remarks THERMUS proven itself in analysis of mean values New extended particle set should allow better description of data. Recent work aimed at extending functionality to include fluctuation and correlation analysis A Monte Carlo Particle Generator has been developed In the process a new technique for the calculation of canonical correction factors has been developed