1. Kensuke Homma / Hiroshima Univ.1 PHENIX Capabilities for Studying the QCD Critical Point Kensuke Homma / Hiroshima Univ. 9 Jun, 2006 @ RHIC&AGS ANNUAL USERS’ MEETING Outline 1.What is the critical behavior ? 2.Search for critical temperature via correlation length 3.Universality in compressibility 4.Summary 5.Future prospects

2. Kensuke Homma / Hiroshima Univ.2 What is the critical behavior ? A simulation based on two dimensional Ising model from ISBN4-563-02435-X C3342l Ordered T<T c Critical T=T c Disordered T>T c BlackBlack & WhiteGray Various sizes from small to large Scale transformation Spatial pattern of ordered state Large fluctuations of correlation sizes on order parameters:  critical temperature Universality (power law behavior) around Tc caused by basic symmetries and dimensions of an underlying system:  critical exponent

3. Kensuke Homma / Hiroshima Univ.3 Search for critical temperature φ g-g 0 In Ginzburg-Landau theory with Ornstein-Zernike picture, free energy density g is given as external field h causes deviation of free energy from the equilibrium value g 0. Accordingly an order parameter  fluctuates spatially. 1-D spatial multiplicity density fluctuation from the mean density is introduced as an order parameter in the following. spatial correlation a>0 a=0 a<0 disappears at Tc In the vicinity of Tc,  must vanish, hence

4. Kensuke Homma / Hiroshima Univ.4 Multiplicity density measurements in PHENIX PHENIX: Au+Au √s NN =200GeV Zero magnetic field to enhance low pt Δη<0.7 integrated over Δφ<π/2 Negative Binomial Distribution (Bose-Einstein distribution from k emission sources) PHENIX Preliminary

5. 5 0-5% 15-20% 10- 15% 0-5% 5-10% Number of participants, Np and Centrality peripheralcentral b To ZDC To BBC Spectator Participant Np Multiplicity distribution Np can be related with initial temperature.

6. Kensuke Homma / Hiroshima Univ.6 Relations to the observable N.B.D k Fluctuation caused by centrality bin width Two point correlation function in one dimensional case in a fixed T Two particle correlation function Relation to N.B.D. k

7. Kensuke Homma / Hiroshima Univ.7 N.B.D. k vs. d  PHENIX Preliminary 10 % centrality bin width 5% centrality bin width  k(  ) Function can fit the data remarkably well !

8. Kensuke Homma / Hiroshima Univ.8 Correlation length  and static susceptibility  PHENIX Preliminary Np  k=0 * T Divergence of susceptibility is the indication of 2 nd order phase transition. Divergence of correlation length is the indication of a critical temperature. Correlation length  T~Tc? Au+Au √sNN=200GeV

9. 9 Stability of the parametrization  can absorb finite centrality bin width effects, namely, finite initial temperature fluctuations, while physically important parameters are stable.    PHENIX Preliminary 10% cent. bin width 5% cent. bin width Shift to smaller fluctuations Np Our parametrization is well controlled.

10. Kensuke Homma / Hiroshima Univ.10 What about universality? On going analysis to extract critical exponents are: Compressibility via scaled variance of multiplicity Correlation lengths via multiplicity density fluctuations Heat capacity via pt fluctuations

11. Kensuke Homma / Hiroshima Univ.11 Isothermal Compressibility Definition of isothermal compressibility In grand canonical ensemble, K T can be related to scaled variance This can be related with N.B.D. k Given a proper estimate on T and measured Tc, we can investigate universality among various collision systems.

12. Kensuke Homma / Hiroshima Univ.12 Np dependence of compressibility All systems appear to obey a universal curve by using Glauber T_AB as a volume V. This behavior is dominated by low pt charged particles !!! 0.2 < p T < 3.0 GeV/c 0.2 < p T < 0.75 GeV/c All species are scaled to match 200 GeV Au+Au points Np 1/  +1/k

13. Kensuke Homma / Hiroshima Univ.13 Comparison of scaled variance to NA49 (17 GeV Pb+Pb) The NA49 scaled variance was corrected for impact parameter fluctuations from their 10% wide centrality bins and scaled up by 15% to lie on the 200 GeV Au+Au curve. 0.2 < p T < 3.0 GeV/c Np Given a reasonable temperature estimate with collision energies as well as Np, we would be able to study the universality by determining the critical exponent around Tc.

14. Kensuke Homma / Hiroshima Univ.14 Summary 1.Two point correlation lengths have been extracted based on the function form by relating pseudo rapidity density fluctuations and the GL theory up to the second order term in the free energy. The lengths as a function of Np indicates non monotonic increase at Np~100. 2.The product of the static susceptibility and the corresponding temperature shows no obvious discontinuity within the large systematic errors at the same Np where the correlation length is increased. 3.Isothermal compressibility via scaled variance of multiplicity shows a universal curve in various collision systems as a function of Np.

15. Kensuke Homma / Hiroshima Univ.15 Future prospects If non monotonic increase of  is a good measure to define Tc and one can discuss critical exponents on thermodynamic quantities around Tc … Preferable conditions to investigate critical points along a phase boundary are: High multiplicity per collision event  reasonably high initial temperature  capability to enhance lower pt particles  larger acceptance Scan higher baryon density region  lower colliding energies  asymmetric colliding energy helps?  device with high position resolution in the forward region.  T QGP Tricritical point Hadronic K. Rajagopal and F. Wilczek, hep-ph/0011333

16. Kensuke Homma / Hiroshima Univ.16 Back up slides

17. Kensuke Homma / Hiroshima Univ.17 Average of wave number dependent density fluctuations from free energy Statistical weight can be obtained from free energy Fourier expression of order parameter coefficient of spatial fluctuation

18. Kensuke Homma / Hiroshima Univ.18 Fourier transformation of two point correlation function

19. Kensuke Homma / Hiroshima Univ.19 Function form of correlation function A function form of correlation function is obtained by inverse Fourier transformation.

20. Kensuke Homma / Hiroshima Univ.20 From field picture to particle picture σ inel is total inelastic cross section

21. Kensuke Homma / Hiroshima Univ.21 Normalized factorial moment Total rapidity interval ΔY is divided into M equal bins

22. Kensuke Homma / Hiroshima Univ.22 Second order NFM and correlation function

23. Kensuke Homma / Hiroshima Univ.23 NBD and NFM Negative binomial distribution Bose-Einstein distribution σ: standard deviation μ: average multiplicity NBD (k→∞) = Poisson distribution NBD (k<0) = Binomial distribution

24. Kensuke Homma / Hiroshima Univ.24 NBD k and integrated correlation function

25. Kensuke Homma / Hiroshima Univ.25Susceptibility Susceptibility is defined by the response of phase for the external field. In the static limit of k = 0, χ cannot be extracted separately without temperature control, but χT value can be obtained by the mean multiplicity μ and α and ξ.

26. Kensuke Homma / Hiroshima Univ.26 Data was collected at the no magnetic filed condition to enhance charged particles with low momenta. Charged tracks were reconstructed based on drift chamber by requiring association with two wire chamber (PC1, PC3) and EM calorimeter and collision vertex position measured by BBC. All detector stability is carefully confirmed. Dead areas of detectors are corrected by the MC simulation. Multiplicity measurement at PHENIX Geometrical acceptance (Δη<0.7, Δφ<π/2)

27. Kensuke Homma / Hiroshima Univ.27 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 agree with NBD in high energy collisions. Charged particle multiplicity distributions and negative binomial distribution (NBD)