Study of order parameters through fluctuation measurements by the PHENIX detector at RHIC

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 Symposiumon Multiparticle Dynamics 2005

2. Motivations K. Rajagopal and F. Wilczek, hep-ph/0011333 • 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?

3. g(T,f) f Landau’s treatment for 2nd order phase transition Valid in a limit where the fluctuation on order parameter f is negligible even at T~Tc Gibb’s free energy Since order parameter f should disappear at T=Tc, assume Susceptibility T>Tc T<Tc T>Tc f02=0 Specific heat c and CH show divergence or discontinuity, while T varies around Tc. T<Tc f02=a(T-Tc)/2u

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

5. Fluctuation measurements by PHENIX • Multiplicity fluctuations (density fluctuations ) as a function rapidity gap size with as low pt particle as possible. • Correlation length x 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. Geometrical acceptance l D h < 0 . 7 D f < p

6. Charged particle multiplicity distributions and negative binomial distribution (NBD) DELPHI: Z0 hadronic Decay at LEP 2,3,4-jets events E802: 16O+Cu 16.4AGeV/c at AGS most central events [DELPHI collaboration] Z. Phys. C56 (1992) 63 [E802 collaboration] Phys. Rev. C52 (1995) 2663 Universally, hadron multiplicity distributions are well described by NBD.

7. Bose-Einstein distribution μ: average multiplicity NBD 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. F2 : second order normalized factorial moment Negative binomial distribution (NBD)

8. | Z | < 5cm 2.16 < φ < 3.73 [rad] -0.35 < η< 0.35 Charged particle multiplicity distributions in different dh gap PHENIX: Au+Au √sNN=200GeV δη= 0.09 (1/8) : P(n) x 107 δη= 0.18 (2/8) : P(n) x 106 δη= 0.35 (3/8) : P(n) x 105 δη= 0.26 (4/8) : P(n) x 104 δη= 0.44 (5/8) : P(n) x 103 δη= 0.53 (6/8) : P(n) x 102 δη= 0.61 (7/8) : P(n) x 101 δη= 0.70 (8/8) : P(n) No magnetic field Δη<0.7, Δφ<π/2 The effect of dead areas have been corrected.

9. Relation between k and integrated two particle correlation function Normalized correlation function inclusive single particle density inclusive two-particle density two-particle correlation function Relation with NBD k Candidates of function forms with two particle correlation length x Most general form: many trials failed. HBT type correlation in E802 : failed to describe data Empirical two component model with R0=1.0

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. Empirical two component fit x dependent part + x independent part with R0 =1 PHENIX: Au+Au √sNN=200GeV, Δη<0.7, Δφ<π/2

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

13. d d d d d r’ r G(r-r’) 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. Density of fluid element at r Self interaction renormalizing singular part ? Long range correlation

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

15. Conclusions • Multiplicity distributions measured in Au+Au collisions at √SNN=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. dh 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. Backup Slide

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

18. Participant To ZDC b To BBC Spectator 15-20% 10-15% 5-10% peripheral central 0-5% 0-5% Multiplicity distribution Nch Important HI jargon : Participants (Centrality) Whether AA is a trivial sum of NN or something nontrivial ? Relate them to Npartand Nbinary(Ncoll) using Glauber model. • Straight-line nucleon trajectories • Constant sNN=(40 ± 5)mb. • Woods-Saxon nuclear density:

19. The Microwave Sky image from the WMAP Mission http://map.gsfc.nasa.gov/m_mm.html 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.