1. 1 Study of order parameters through fluctuation measurements by the PHENIX detector at RHIC Kensuke Homma for the PHENIX collaboration Hiroshima University On Aug 11, 2005 at Kromeriz XXXV International Symposium on Multiparticle Dynamics 2005

2. 2Motivations RHIC experiments probed the state of strongly interacting dense medium with many properties consistent with partonic medium. What about the information on the phase transition? Is it the first order or second order transition? Are there interesting critical phenomena such as tricritical point? K. Rajagopal and F. Wilczek, hep-ph/0011333

3. 3 Landau’s treatment for 2 nd order phase transition Since order parameter  should disappear at T=Tc, assume Valid in a limit where the fluctuation on order parameter  is negligible even at T~Tc Gibb’s free energy  g(T,  ) T>Tc    =0 T<Tc    =a(T-Tc)/2u Susceptibility Specific heat  and C H show divergence or discontinuity, while T varies around Tc. T>Tc T<Tc

4. 4 Susceptibility and density fluctuations Susceptibility Fluctuation-dissipation theorem Ornstein-Zernike behavior T>Tc T<Tc With Fourier transformation

5. 5 Multiplicity fluctuations (density fluctuations ) as a function rapidity gap size with as low pt particle as possible.  Correlation length  and singular behavior in correlation function. Average pt fluctuations (temperature fluctuations)  Specific heat See PRL. 93 (2004) 092301 In this talk, I will focus on only multiplicity fluctuation measurements. Fluctuation measurements by PHENIX 7.0   lGeometrical acceptance  

6. 6 E802: 16 O+Cu 16.4AGeV/c at AGS most central events [DELPHI collaboration] Z. Phys. C56 (1992) 63 [E802 collaboration] Phys. Rev. C52 (1995) 2663 DELPHI: Z 0 hadronic Decay at LEP 2,3,4-jets events Universally, hadron multiplicity distributions are well described by NBD. Charged particle multiplicity distributions and negative binomial distribution (NBD)

7. 7 Bose-Einstein distribution μ: average multiplicity F 2 : second order normalized factorial moment NBD correspond to multiple Bose- Einstein distribution and the parameter k corresponds to the multiplicity of those Bose-Einstein emission sources. NBD can be Poisson distribution with the infinite k value. NBD Negative binomial distribution (NBD)

8. 8 δη= 0.09 (1/8) : P(n) x 10 7 δη= 0.18 (2/8) : P(n) x 10 6 δη= 0.35 (3/8) : P(n) x 10 5 δη= 0.26 (4/8) : P(n) x 10 4 δη= 0.44 (5/8) : P(n) x 10 3 δη= 0.53 (6/8) : P(n) x 10 2 δη= 0.61 (7/8) : P(n) x 10 1 δη= 0.70 (8/8) : P(n) No magnetic field Δη<0.7, Δφ<π/2 PHENIX: Au+Au √s NN =200GeV Charged particle multiplicity distributions in different d  gap | Z | < 5cm -0.35 < η < 0.35 2.16 < φ < 3.73 [rad] The effect of dead areas have been corrected.

9. 9 inclusive single particle density inclusive two-particle density two-particle correlation function Relation with NBD k Normalized correlation function Candidates of function forms with two particle correlation length  HBT type correlation in E802 : failed to describe data Empirical two component model with R 0 =1.0 Relation between k and integrated two particle correlation function Most general form: many trials failed.

10. 10 E802 type function can not describe the data Correlation function used in E802 P. Carruthers and Isa Sarcevic, Phys. Rev. Lett. 63 (1989) 1562 NBD k vs. δη at E802 Phys. Rev. C52 (1995) 2663

11. 11 Empirical two component fit PHENIX: Au+Au √s NN =200GeV, Δη<0.7, Δφ<π/2  dependent part +  independent part with R 0 =1

12. 12 Participants dependence of ξ and b PHENIX: Au+Au √s NN =200GeV Two particle correlation length Correlation strength of what?

13. 13 What is the origin of the two components? Go back to Ornstein-Zernike’s theory (see Introduction to Phase Transitions and Critical Phenomena by H.E.Stanley) which explains the growth of forward scattering amplitude of light interacting with targets at the phase transition temperature. Self interaction renormalizing singular part ? Long range correlation       (r-r’) r r’ Density of fluid element at r

14. 14 ξ vs. number of participants Two particle correlation length PHENIX: Au+Au √s NN =200GeV Linear behavior of the correlation length as a function of the number of participants has been obtained in the logarithmic scale. One slope fit gives α = -0.72 ± 0.03 In the case of thermalized ideal gas,

15. 15 Conclusions Multiplicity distributions measured in Au+Au collisions at √S NN =200GeV can be described by the negative binomial distributions. Two particle correlation length has been measured based on the empirical two component model from the multiplicity fluctuations, which can fit k vs. d  in all centralities remarkably well. Extracted correlation length behaves linearly as a function of number of participants in logarithmic scales. Assuming one slope component, the exponent was obtained as -0.72±0.03. The interpretation of b parameter is still ambiguous. Any criticize or different view points are more than welcome.

16. 16 Backup Slide

17. 17 Uncorrected Npart*b vs. Npart Npart Npart*b Bias on NBD k due to finite bin size of centrality Intrinsic k Observed k

18. 18 0-5% 15-20% 10- 15% 0-5% 5-10% Important HI jargon : Participants (Centrality) peripheralcentral Relate them to N part and N binary ( N coll ) using Glauber model. – Straight-line nucleon trajectories  Constant  NN =(40 ± 5)mb. – Woods-Saxon nuclear density: b To ZDC To BBC Spectator Participant Multiplicity distribution Nch Whether AA is a trivial sum of NN or or something nontrivial ? something nontrivial ?

19. 19 Why not observing fluctuations ? Fluctuation carries information in early universe in cosmology despite of the only single Big-Bang event. Why don’t we use the event-by-event information by getting all phase space information to study evolution of dynamical system in heavy-ion collisions ? We can firmly search for interesting fluctuations with more than million times of mini Big-Bangs. The Microwave Sky image from the WMAP Mission http://map.gsfc.nasa.gov/m_mm.html