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Recent Developments in THERMUS “The Wonders of Z ” Spencer Wheaton Dept of Physics

Recent Developments in THERMUS “The Wonders of Z ” Spencer Wheaton Dept of Physics University of Cape Town. Statistical-Thermal Model. Fireball resulting from high-energy heavy-ion collision treated as an ideal gas of hadrons

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Recent Developments in THERMUS “The Wonders of Z ” Spencer Wheaton Dept of Physics

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  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. THERMUS performs calculations within 3 commonly applied statistical ensembles: 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 T, mB, mS, mQ, V T, mB, S, mQ, 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+/p+“horn”: (J. Cleymans, H. Oeschler, K. Redlich and SW, Phys.Lett.B615:50-54, 2005)

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

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

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

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

  12. Example 1: Static Pion Gas (conserved Q) T = 150 MeV R = 6 fm Boltzmann Approximation: m as in GCE 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 Bi, Si, Qi : 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. b) Approach using Z: Hauer, de Klerk, SW Choose chemical potentials to smooth out integrands in both numerator and denominator- much easier to integrate Symmetry of before disappears, so 3D integration needed A Carbon-Carbon Correction Factor

  17. Only 13 distinct quantum content combinations

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

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

  20. Fluctuations and Correlations

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

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

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

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

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

  26. Monte Carlo Particle Generator M. Hauer & SW published V1: observed sub-system V2: unobserved Vg=V1+V2: total system Constraints placed on system are imposed only on the total volume V1 V2

  27. The crux is the following:

  28. Strategy of Monte Carlo Generator: • Sample sub-system V1 Grand-Canonically in Boltzmann Approximation • For each particle of type i, generate a momentum magnitude following a Boltzmann Distribution: • Assign direction to particle assuming isotropic particle emission… • Allow 2 and 3 body decays • Reweight

  29. System Considered Neutral , Static , T = 160 GeV and V1 = 2000 fm3 B, S, Q, E, Pz considered for reweighting

  30. l= 0.00 l= 0.25 l= 0.50 l= 0.75

  31. l= 0.00

  32. l= 0.25

  33. l= 0.50

  34. l= 0.75

  35. l= 0 pZ [GeV/c] Primordial N E [GeV] E [GeV] S(s) = -1 Q(s) = -1/3 Charge Content Strangeness Content Baryon Content Strangeness Content

  36. l= 0.25 pZ [GeV/c] Primordial N E [GeV] E [GeV] Charge Content Strangeness Content Baryon Content Strangeness Content

  37. l= 0.50 pZ [GeV/c] Primordial N E [GeV] E [GeV] Charge Content Strangeness Content Baryon Content Strangeness Content

  38. l= 0.75 pZ [GeV/c] Primordial N E [GeV] E [GeV] Charge Content Strangeness Content Baryon Content Strangeness Content

  39. Fully-Phase Space Integrated Results

  40. Averages Variances 20 runs of 2.5 × 104 events Linear extrapolation to MCE Co-Variances Correlation Coefficients

  41. Momentum Space Dependence

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

  43. 20 runs of 105 events GCE Correlation Coefficients

  44. primordial Extrapolating Variances to MCE 20 runs of 105 events primordial Linear extrapolation to MCE Largest baryon and strangeness content in DpT,5

  45. Extrapolating Covariances and Correlation Coefficients to MCE primordial Non-Linear extrapolation to MCE 20 runs of 105 events Linear extrapolation to MCE primordial

  46. Multiplicity Fluctuations and Correlations

  47. GCE Scaled Variance of Positives 20 runs of 2 × 105 events

  48. GCE Correlation Coefficient r+-

  49. Scaled Variance of Primordial Positives 20 runs of 2 × 105 events

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