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How to Measure Specific Heat Using Event-by-Event Average p T Fluctuations

How to Measure Specific Heat Using Event-by-Event Average p T Fluctuations. M. J. Tannenbaum Brookhaven National Laboratory Upton, NY 11973 USA. PHENIX Collaboration.

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How to Measure Specific Heat Using Event-by-Event Average p T Fluctuations

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  1. How to Measure Specific Heat Using Event-by-Event Average pT Fluctuations M. J. Tannenbaum Brookhaven National Laboratory Upton, NY 11973 USA PHENIX Collaboration Division of Nuclear Physics Meeting 2005 Maui, Hawaii September 20, 2005 DNP 2005 Maui

  2. Something New: cV/T3 • R. Gavai, S.Gupta and S. Mukherjee, hep-lat/0412036, PRD 71, 074013 (2005) predict in “quenched QCD” at 2Tc and 3Tc that cV/T3 differs significantly from the ideal gas. Can this be measured? DNP 2005 Maui

  3. It is generally agreed that cV is related to temperature fluctuations • Ntot is the total number of particles on an event • the ``temperature’’ T varies event-by-event with T and T. • R.Korus, St.Mrowczynski, M.Rybczynski, Z.Wlodarczyk, PRC 64, 054908 (2004). • Also see: L. Stodolsky, PRL 75, 1044 (1995) , S.A.Voloshin, V.Koch, H.G.Ritter, PRC 60, 024901 (1999). • For a more nuanced (i.e. complicated) view see M. Stephanov, K. Rajagopal, E. Shuryak, PRD 60, 114028 (1999) DNP 2005 Maui

  4. dN/x dx p < 2 p=2 p > 2 x = Inclusive pT spectra are Gamma Distributions note: dN/xdx is shown. p shown is for dN/dx x =pT • This is inclusive, averaged over all events. T=1/b is the Temperature parameter. DNP 2005 Maui

  5. Event-by-Event Average pT • For events with n charged particles of transverse momentum pTi, the event-by-event average transverse momentum is defined: • By its definition <MpT>=<pT>  but you must work hard to make sure that your data have this property to <<< 1%. • If all the pTi on all events are random samples of the same distribution: DNP 2005 Maui

  6. What e-by-e tells you that you don’t learn from the inclusive average • A nice example I like is by R.Korus, St.Mrowczynski, M.Rybczynski, Z.Wlodarczyk, PRC 64, 054908 (2004). • Suppose the temperature T~1/b varies event-by-event with T and T: • Also see: L. Stodolsky, PRL 75, 1044 (1995) , S.A.Voloshin, V.Koch, H.G.Ritter, PRC 60, 024901 (1999). DNP 2005 Maui

  7. PHENIX MpT vs centrality 200 GeV Au+Au PRL 93, 092301 (04) • compare Data to Mixed events for random. • Must use exactly the same n distribution for data and mixed events and match inclusive <pT> to <MpT> • best fit of real to mixed is statistically unacceptable • deviation expressed as: • FpT= MpTdata/ MpTmixed-1 ~ few % MpT (GeV/c) MpT (GeV/c) DNP 2005 Maui

  8. Measures of non-random fluctuations random means pT of all particles on all events are independent samples of the inclusive pT distribution (averaged over all events). non-random is the difference between measured and random. PHENIX STAR CERES NA49 Mroczynski DNP 2005 Maui

  9. For small non-random/randomAll measures are equivalent DNP 2005 Maui

  10. Fluctuation is a few percent of MpT : Interesting variation with Npart and pTmax Errors are totally systematic from run-run r.m.s variations n >3 0.2 < pT < 2.0 GeV/c 0.2 GeV/c < pT < pTmax PHENIX nucl-ex/0310005 PRL 93, 092301 (2004) DNP 2005 Maui

  11. Npart and pTmax dependences explained by jet correlations with measured jet suppression Other explanations proposed include percolation of color strings E.G.Ferreiro, et al, PRC69, 034901 (2004) 20-25% centrality DNP 2005 Maui

  12. Assuming all fluctuations are from T/T Very small and relatively constant with sNN CERES tabulation H.Sako, et al, JPG 30, S1371 (04) QM2004 Where is the critical point? T/T DNP 2005 Maui

  13. 2T/T2Specific Heat • Korus, et al, PRC 64, 054908 (2001) discuss specific heat: • n represents the measured particles while Ntot is all the particles, so n/Ntot is a simple geometrical factor for each experiment DNP 2005 Maui

  14. Something New: cV/T3 • Gavai, et al, hep-lat/0412036 call this same quantity cV/T3 • These formulas with cvcV/T3 agree with Eq 9. in Korus, et al. DNP 2005 Maui

  15. Something New: cV/T3 • R. Gavai, S.Gupta and S. Mukherjee, hep-lat/0412036, PRD 71, 074013 (2005) predict in “quenched QCD” at 2Tc and 3Tc that cV/T3 differs significantly from the ideal gas. Can this be measured? • In PHENIX publication, PRL 93, 092301 (2004), n/Ntot~1/33, so FpT ~ 0.2% for cV/T3~15. This may be possible if we go to low pTmax out of the region where jets contribute. DNP 2005 Maui

  16. Worth Trying PRL 93, 092301 (2004) cV/T3~15 0.2 GeV/c < pT < pTmax • Concentrate on pTmax < 0.6 GeV/c where jets have least effect • Error is totally systematic---run by run variation---can be improved. DNP 2005 Maui

  17. Issues • at fixed centrality, test for is FpT  n, i.e. independent of n. Increase n by increasing solid angle, e.g. PHENIX vs STAR. DNP 2005 Maui

  18. What We Have Learned • In central heavy ion collisions, the huge correlations in p-p collisions are washed out. The remaining correlations are: • Jets • Bose-Einstein correlations • Hydrodynamic flow • These correlations saturate the fluctuation measurements. No other sources of non-random fluctuations are observed. This puts a severe constraint on the critical fluctuations that were expected for a sharp phase transition but is consistent with the present expectation from lattice QCD that the transition is a smooth crossover. • In order to see temperature fluctuations predicted by cV/T3~15 in lattice gauge calculations, present sensitivity needs to be improved by an order of magnitude by removing other known sources of correlation and improving the measurement errors. DNP 2005 Maui

  19. Some Details DNP 2005 Maui

  20. Statistics--What you have to remember • The mean and standard deviation of an average of nindependent trials from the same population obey the rules: where  is the mean and x (or ) is the standard deviation of the population x . DNP 2005 Maui

  21. Variance of MpT • If all the pTi are random samples of the same distribution: • If the pTi are correlated, we find an identity with the pT-pT correlator variable of Voloshin, et al PRC 60, 024901 (1999) DNP 2005 Maui

  22. A)---Detail of MpT Calculation DNP 2005 Maui

  23. B)--Variance of MpT for T fluctuation DNP 2005 Maui

  24. <Ntot>/<n> for PHENIX DNP 2005 Maui

  25. BACKUP DNP 2005 Maui

  26. NA49-First Measurement of MpT distribution NA49 Pb+Pb central measurement PLB 459, 679 (1999) • Points=data; hist=mixed; minimal, if any, difference • Very nice paper, gives all the relevant information DNP 2005 Maui

  27. Statistics at Work--Analytical Formula for MpT for statistically independent Emission It depends on the 4 semi-inclusive parameters: b, p of the pT distribution (Gamma) <n>, 1/k (NBD), which are derived from the quoted means and standard deviations of the semi-inclusive pT and multiplicity distributions. The result is in excellent agreement with the NA49 Pb+Pb central measurement PLB 459, 679 (1999) See M.J.Tannenbaum PLB 498, 29 (2001) DNP 2005 Maui

  28. 0-5 % Centrality It’s not a Gaussian…it’s a Gamma distribution! Black Points = Data Blue curve = Gamma distribution derived from inclusive pT spectra From one of Jeff Mitchell’s talks: “Average pT Fluctuations” PHENIX DNP 2005 Maui

  29. Mortadella-NYTimes 2/10/2000 DNP 2005 Maui

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