Measurement of eventbyevent fluctuations and order parameters in

Measurement of event-by-event fluctuations and order parameters in PHENIX Tomoaki Nakamura for the PHENIX collaboration Hiroshima University 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 1

Phase transitions Specific heat of liquid Helium (He 4) • According to the classical 　　　　　　 　　classification of the phase transition, the order of phase transition is defined by discontinuities in derivatives of free energy. Fairbank et al, (1975) • In this aspect of bulk property, discontinuity in thermodynamic variables or order parameters as a function of the temperature or time evolution are available to search for the critical point of phase. • In particular, the second order phase transitions are often　 accompanied by the divergence with respect to thermodynamic variables as a results of critical phenomena. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 2

Event-by-event fluctuations in heavy-ion collisions measured by PHENIX • Some thermodynamic variables as order parameters of phase can be obtained from event-byevent fluctuations. • Particles correlation length – scale dependence of multiplicity fluctuations • Specific heat – temperature fluctuations from average p. T fluctuations. PRL. 93 (2004) 092301 M. J. Tannenbaum : poster #120 • We have performed measurements of several fluctuations to explore the QCD phase transition using the PHENIX detector at RHIC. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 3

Multiplicity fluctuations by variance Average of multiplicity distribution Variance of multiplicity distribution Normalized variance is used as an observable of multiplicity fluctuation. In the case of Poisonnian distribution, the variance equals mean value, then normalized variance indicates 1. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 4

Normalized variance vs. participants J. T. Mitchell : poster #110 Δη<0. 7, Δφ<π, 0. 2<p. T<2. 0 Ge. V/c positive charge inclusive charge negative charge Deviation from the Poisonnian. No charge dependences. Similar behaviors of Cu+Cu 62 Ge. V to SPS. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 5

Charged particle multiplicity distributions and negative binomial distribution (NBD) DELPHI: Z 0 hadronic Decay at LEP 2, 3, 4 -jets events E 802: 16 O+Cu 16. 4 AGe. V/c at AGS most central events [DELPHI collaboration] Z. Phys. C 56 (1992) 63 [E 802 collaboration] Phys. Rev. C 52 (1995) 2663 Universally, hadron multiplicity distributions conform to NBD in high energy collisions. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 6

Negative binomial distribution (NBD) Bose-Einstein distribution μ: average multiplicity NBD correspond to multiple Bose. Einstein distribution and the parameter k indicates the multiplicity of Bose-Einstein emission sources. NBD also corresponds to the Poison distribution with the infinite k value in the statistical mathematics. F 2 : second order normalized factorial moment 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 7

Charged particle multiplicity distributions in PHENIX: Au+Au √s. NN=200 Ge. V Multiplicity distributions observed in Au+Au, Cu+Cu, d+Au and p+p collisions at PHENIX also conform to the negative binomial distribution. 8/6/2005 No magnetic field Δη<0. 7, Δφ<π/2 Tomoaki Nakamura - Hiroshima Univ. δη= δη= 0. 09 0. 18 0. 35 0. 26 0. 44 0. 53 0. 61 0. 70 (1/8) (2/8) (3/8) (4/8) (5/8) (6/8) (7/8) (8/8) : : : : P(n) P(n) x x x x 107 106 105 104 103 102 101 8

NBD k parameters as a function of average multiplicity Δη<0. 7, Δφ<π, 0. 2<p. T<2. 0 Ge. V/c positive charge inclusive charge PHENIX Preliminary There are the differences about the average multiplicity dependence of NBD k parameters between the 200 Ge. V and 62. 4 Ge. V. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. negative charge PHENIX Preliminary 9

NBD k parameters as a function of number of participants Δη<0. 7, Δφ<π, 0. 2<p. T<2. 0 Ge. V/c positive charge inclusive charge PHENIX Preliminary Nparticipants PHENIX Preliminary negative charge Nparticipants NBD k parameters as an observable of multiplicity fluctuation are not scaled by the average multiplicity but scaled by the number of participants. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. PHENIX Preliminary Nparticipants 10

NBD k parameters in Cu+Cu Δη<0. 7, Δφ<π, 0. 2<p. T<2. 0 Ge. V/c positive charge inclusive charge negative charge NBD k parameters are scaled by system size in Au+Au, but not scaled in Cu+Cu. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 11

NBD k parameters as functions of δp. T (p. T>0. 2 Ge. V/c) 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 12

Extraction of two particle correlation Normalized correlation function inclusive single particle density inclusive two-particle density two-particle correlation function Relation with NBD k Used in E 802 : PRC, 44 (1991) 1629 Two component model ξ : Two particle correlation length b : Strength of long range correlation 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 13

NBD k and pseudo rapidity gap • Two component model well agree with data. • Correlation function dose not go to 0 at δηequal 0. It dose not suggest the intermittency effect. PHENIX: Au+Au √s. NN=200 Ge. V, Δη<0. 7, Δφ<π/2 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 14

Participants dependence of ξ and b • Smaller value of two particle correlation length have been observed at RHIC energy as compared to the past experiments. – Low density p+p collisions (UA 5, p+p √s = 540 Ge. V : ξ= 2. 9) – Low energy N+N collisions (E 802 O+Cu 14. 6 AGev/c : ξ= 0. 18± 0. 05) • Both ξ and b decrease with increasing the number of participants. PHENIX: Au+Au √s. NN=200 Ge. V Two particle correlation length Strength of long range correlations 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 15

Linearity in the log-log plot ξ vs. number of participants PHENIX: Au+Au √s. NN=200 Ge. V Two particle correlation length α = -0. 72 ± 0. 032 Linear behavior of the correlation length as a function of the number of participants have been observed in the logarithmic scale. It suggests hadron two particle correlation length have a information of thermodynamical systems by assuming the proportionality between the number of participants and temperature. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 16

Conclusions • A systematic study on charged particle multiplicity fluctuations have been performed in Au+Au, Cu+Cu, d+Au and p+p collisions with respect to two collision energy of 200 Ge. V and 62 Ge. V. • Multiplicity distributions measured by PHENIX also agree with the negative binomial distributions at the RHIC energy. • Multiplicity fluctuations by the NBD k parameters are not scaled by the average multiplicity but scaled by the number of participants or system size in Au+Au collisions. • Scale dependence of NBD k parameters are presented with respect to pseudo rapidity gap and transverse momentum range. • Two particle correlation length have been observed by the two component model from the multiplicity fluctuations. • Extracted correlation length have a linearity as a function of the number of participants in the logarithmic scale (log-log plot). 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 17

PHENIX posters on fluctuations and correlations J. T. Mitchell : poster #109 HBT Jets #109 : J. T. Mitchell The low- to high-p. T evolution of charged hadron azimuthal correlation functions: from HBT to jets 62 Ge. V Au+Au, 0 -5% central #110 : J. T. Mitchell A survey of multiplicity fluctuations in PHENIX HBT Jets #120 : M. J. Tannenbaum How to measure specific heat using event-by-event average p. T fluctuations 200 Ge. V Au+Au, 0 -5% central 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 18

8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 19

Backup Slide 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 20

Variables of statistical mechanics as order parameters 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 21

Multiplicity fluctuations by variance NA 49: p+p, C+C, Si+Si, Pb+Pb 158 A Ge. V at SPS Variance of the multiplicity distribution is defined as; Normalized variance, Var(N)/<N>, is used as an observable of multiplicity fluctuation. In the case of Poisonnian distribution, the variance equals mean value and this observable indicates 1. Deviations of multiplicity fluctuation from Poisonnian distribution was reported in the middle range of the number of projectile participant nucleon at SPS energy. 8/6/2005 [NA 49 collaboration] nucl-ex/0409009 Tomoaki Nakamura - Hiroshima Univ. 22

Multiplicity fluctuations in PHENIX Δη<0. 7, ΔФ<πrad, 0. 2<p. T<2. 0 Ge. V/c PHENIX Preliminary Positive Nparticipants Jeffery T. Mitchell PHENIX Preliminary Inclusive A different behavior of the multiplicity fluctuations as function of number of participants is observed at RHIC energies as compared to SPS. There are no difference about the multiplicity fluctuation between positive and negative charge. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. Negative 23

Normalized variance as a function of average multiplicity 0. 2 < p. T < 2. 0 Ge. V/c positive charge 0. 2 < p. T < 2. 0 Ge. V/c inclusive charge 0. 2 < p. T < 2. 0 Ge. V/c negative charge 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 24

NBD k parameters as a function of average multiplicity 0. 2 < p. T < 2. 0 Ge. V/c positive charge 0. 2 < p. T < 2. 0 Ge. V/c inclusive charge 0. 2 < p. T < 2. 0 Ge. V/c negative charge 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 25

Multiplicity fluctuations as a function of collision overlap geometry When plotted as a function of a measure of the collision overlap geometry (fractional impact parameter divided by the nuclear diameter - so a head-on collision = 1. 0), the 62 Ge. V Cu+Cu fluctuations are less Poissonian. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 26

δp. T (p. T>0. 2 Ge. V/c) dependence of NBD k PHENIX Preliminary centrality 0 -5% PHENIX Preliminary centrality 15 -20% 8/6/2005 PHENIX Preliminary centrality 5 -10% PHENIX Preliminary centrality 20 -25% Tomoaki Nakamura - Hiroshima Univ. PHENIX Preliminary centrality 10 -15% PHENIX Preliminary centrality 25 -30% Jeffery T. Mitchell 27

Comparison for Au+Au, d+Au and p+p PHENIX Preliminary δp. T dependence of NBD k parameters in Au+Au peripheral collisions are approaching toward the similar shape of d+Au and p+p with decreasing the centrality. Au+Au, centrality 45 -50% d+Au, 200 Ge. V PHENIX Preliminary A behavior of NBD k parameters as a function of δp. T at p+p collisions measured by PHENIX is agree with PYHTIA qualitatively. p+p, 200 Ge. V 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. PYTHIA Jeffery T. Mitchell 28

Fit by the E 802 type correlation function • E 802 type integrated two particle correlation function dose not agree with PHENIX data by taking all range of pseudo rapidity into account. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 29

Two component model Black : b=0. 0, ξ=0. 01 Blue : b=0. 0, ξ=0. 05 Red 　: b=0. 0, ξ=0. 10 Black : b=0. 0, ξ=0. 01 Blue : b=0. 1, ξ=0. 01 Red 　: b=0. 2, ξ=0. 01 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 30

Zoom up for small δη • Lines are just extrapolated obtained fitting curve for small area (δη<0. 01). • K parameters converge finite value. Factorial moment/cumulant do not diverge. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 31

Linearity in log-log plots 8/6/2005 α = -0. 72 ± 0. 032 α = -0. 90 ± 0. 027 β = 0. 097 ± 0. 015 β = 4. 02 ± 0. 540 Tomoaki Nakamura - Hiroshima Univ. 32

Divergence. . . !? Fitting range : 60 < Npart < 400 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 33

Divergence…!? could not find critical points by only the fitting Fitting range : 0 < Npart < 400 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 34

δηdependence of NBD k parameter E 802: Phys. Rev. C 52 (1995) 2663 Au+Au 200 Ge. V, no magnetic field Δη<0. 7, ΔФ<π/2 rad K. Homma and T. Nakamura 8/6/2005 NBD fit was performed for the different range of pseudo rapidity gap as shown in blue and red curve to extract the correlation length using the E 802 type correlation function. Blue: δη ≤ 0. 35 Red : δη ≥ 0. 35 Tomoaki Nakamura - Hiroshima Univ. 35

Number of participants dependence of correlation length ξ Au+Au 200 Ge. V, no magnetic field Δη<0. 7, ΔФ<π/2 rad K. Homma and T. Nakamura 8/6/2005 • Fitting Range Blue: δη ≤ 0. 35 Red : δη ≥ 0. 35 • Centrality filled circle : 0 -70 % (10% interval) open circle : 5 -65 % (10% interval) Different behaviors about the extracted correlation length (ξ) as a function of number of participants are observed in the different range of the pseudo rapidity gap. The correlation length at the range of large pseudo rapidity gap has a large fluctuation. Tomoaki Nakamura - Hiroshima Univ. 36

Linear behavior of NBD k as a function of logarithmic δη Au+Au 200 Ge. V, no magnetic field Δη<0. 7, ΔФ<π/2 rad • Fitting function k(δη) = c 1 + c 2 × ln(δη) c 1, c 2 : constant • Fitting Range 0. 09 ≤ δη ≤ 0. 7 Relations of fluctuation and normalized factorial moments and fractal structure. This power low relation of fluctuations and pseudo rapidity gap might suggest self similarity of correlation length!!. . . ? K. Homma and T. Nakamura 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 37

Normalized factorial moment Fq average multiplicity standard deviation relation between fractal (self-similarity) structures and normalized factorial moment 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 38

NBD k and factorial moment Fq 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 39

Integral of correlation function and normalized factorial moments inclusive q particle density when there are no correlation in rapidity 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 40

Specific heat from average p. T fluctuation Korus, et al, PRC 64, (2001) 054908 M. J. Tannenbaum, 2 nd International Workshop on the Critical Point and Onset of Deconfinement n represents the measured particles while Ntot is all the particles, so n/Ntot is a simple geometrical factor for all experiments 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 41

Random particle emission pattern based on NBD Detector 1 Detector 2 Emission source In the case of there are no correlation about the particle emission, the value of NBD k parameters are summed up. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 42

Correlated particle emission pattern into the phase-space Detector 1 Detector 2 Emission source If there are correlations, NBD k parameters do not increase according to the size of detector acceptance. 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 43

Correlation functions and correlation length Using arbitrary R 0, ξ and α. Used in E 802 General correlation function One may discuss an effective potential form of a deconfined field when assume the correlation functions. ξ : correlation length, α : critical exponent 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 44

Average p. T fluctuation Published by Phys. Rev. Lett. 93, 092301 (2004) NA 49: Phys. Lett. B 459 (1999) 679 Already, famous! PHENIX: Au+Au 200 Ge. V 0. 2 < p. T < 2. 0 Ge. V/c It’s not a Gaussian… it’s a Gamma distribution! M. J. Tannenbaum, Phys. Lett. B 498 (2001) 29 Magnitude of average p. T fluctuation Fractional deviation from mixed events 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 45

Contribution of Jet/Jet suppression to the average p. T fluctuation PYTHIA based simulation, which contains scaled hard-scattering probability factor (Sprob) by the nuclear modification factor (RAA), well agree with the measured Fp. T. It might be indicate that jet suppression might contribute to the average p. T fluctuation. This decrease due to jet suppression? centrality 20 -25% Fp. T is adjusted at the open circle for this simulation 8/6/2005 Tomoaki Nakamura - Hiroshima Univ. 46

Estimation of the magnitude of residual temperature fluctuations p → inclusive p. T R. Korus and S. Mrowczynski, Experiment √s. NN σT/<T> (most central) PHENIX 200 1. 8% STAR 130 1. 7% CERES 17 1. 3% NA 49 17 0. 7% A signal of phase transition dose not emerge in the temperature fluctuation? 8/6/2005 σT/<T> Phys. Rev. C 64 (2001) 054908. H. Sako, et al, JPG 30, S (2004) 1371 Tomoaki Nakamura - Hiroshima Univ. 47