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Tomoaki Nakamura - Hiroshima Univ. 16/21/2005
Event-by-Event Fluctuations from PHENIX
Tomoaki Nakamura
for the PHENIX collaboration
Hiroshima University Correlations & Fluctuations Workshop at
2005 RHIC & AGS Annual User’s Meeting
Tomoaki Nakamura - Hiroshima Univ. 26/21/2005
Phase transition
• According to the classical classification of the phase transition by Ehrenfest, the order of phase transition is defined by the discontinuity in the derivative of free energy.
• In this context, discontinuity in the variables of statistical mechanics or order parameter as a function of the temperature or time evolution is useful to search the critical point of phase transition.
• In particular, the second order phase transitions are often accompanied by the divergence with respect to variables of statistical mechanics or order parameters as critical phenomena.
Fairbank et al, (1975)
Specific heat of liquid Helium (He4)
Tomoaki Nakamura - Hiroshima Univ. 36/21/2005
Variables of statistical mechanics as order parameters
hh
Gh
)(energy free Gibbs
)P,(T, variables intensive
)N,S, V,(G, variables extensive:
:
:
G
h
ST
P
T
h
P
V
V
T
GS
P
GV
h
G
,
'
1:
ilitycompressib
: entropy
: voulume
:itysusceptibl order First
22
2
2
'
2
2
1
)(
k
T
T
GTC
h
G
k
h
h
h
: length ncorrelatio
: heatspecific
:lity susceptibi order Second
Tomoaki Nakamura - Hiroshima Univ. 46/21/2005
Event-by-event fluctuations in heavy-ion collisions measured by PHENIX
• Some variables of statistical mechanics and order parameters of phase can be obtained from event-by-event fluctuations.
• Correlation length can be extracted from the event-by-event multiplicity fluctuations by scanning the phase-space.
• Specific heat can be calculated from the results of event-by-event average pT fluctuation.
• We have performed measurements of these fluctuations to explore the QCD phase transition accompanied by critical phenomena using the PHENIX spectrometer at RHIC.
cGeVpT /0.22.0
7.0
range Momentum
acceptance lGeometrica
Tomoaki Nakamura - Hiroshima Univ. 56/21/2005
Multiplicity fluctuations by varianceNA49: p+p, C+C, Si+Si, Pb+Pb 158 A GeV at SPS
[NA49 collaboration] nucl-ex/0409009
22
2 )()()(
)(
NN
NPNNNVar
NPNN
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.
Tomoaki Nakamura - Hiroshima Univ. 66/21/2005
PHENIX Preliminary
Positive
PHENIX Preliminary
Negative
Inclusive
PHENIX Preliminary
Δη<0.7, ΔФ <π rad, 0.2<pT<2.0 GeV/c
Jeffery T. MitchellNparticipants
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.
Multiplicity fluctuations in PHENIX
Tomoaki Nakamura - Hiroshima Univ. 76/21/2005
Charged particle multiplicity distribution and negative binomial distribution (NBD)
E802: 16O+Cu 16.4 A GeV at AGSmost central events
[DELPHI collaboration] Z. Phys. C56 (1992) 63[E802 collaboration] Phys. Rev. C52 (1995) 2663
DELPHI: Z0 hadronic Decay at LEP2,3,4-jets events
Universally, hadron multiplicity distributions in high energy collisions conform to the NBD.
Tomoaki Nakamura - Hiroshima Univ. 86/21/2005
Negative binomial distribution (NBD)
2
2
2
22
2
22
2
2
)(
1
)(
1)(1
)(
1
)(
11
/1
1
/1
/
)()1(
)(
)1/(
n
nnF
Fk
nn
k
kk
k
kn
knP
nP
k
n
kn
nnn
Bose-Einstein distributionμ: average multiplicity
F2 : second order normalized factorial moment
NBD correspond to multiple Bose-Einstein distribution and the parameter k indicates the multiplicity of Bose-Einstein emission sources. It is also corresponds to the Poison distribution with the infinite k value in the statistical mathematics.
Tomoaki Nakamura - Hiroshima Univ. 96/21/2005
Charged particle multiplicity distributions in PHENIX
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 condition Δη<0.7, ΔФ<π/2 rad
PHENIX: Au+Au √sNN=200GeV
K. Homma and T. Nakamura
Multiplicity distributions observed in Au+Au, d+Au and p+p collisions at PHENIX also conform to the negative binomial distribution.
Tomoaki Nakamura - Hiroshima Univ. 106/21/2005
PHENIX Preliminary
PHENIX Preliminary
Positive
Negative
Δη<0.7, ΔФ <πrad, 0.2<pT<2.0 GeV/c
PHENIX Preliminary
Inclusive
Jeffery T. Mitchell
There is the difference about the average multiplicity dependence of NBD k parameters between the 200GeV and 62.4 GeV.
NBD k parameteras a function of average multiplicity
Tomoaki Nakamura - Hiroshima Univ. 116/21/2005
PHENIX Preliminary
Positive
Nparticipants
PHENIX Preliminary
Negative
Nparticipants
PHENIX Preliminary
Inclusive
Δη<0.7, ΔФ <π rad, 0.2<pT<2.0 GeV/c
Nparticipants
Jeffery T. Mitchell
NBD k parameters as an observable of multiplicity fluctuation are not scaled by the average multiplicity but scaled by the number of participants.
NBD k parameteras a function of number of participants
Tomoaki Nakamura - Hiroshima Univ. 126/21/2005
Extraction of the correlation length from NBD k parameter
2
21221
2
212
212
1
2111
212
2111
21221
)(
),(1
)(
1
:),(
:),(
:)(
1)()(
),(
)()(
),(),(
n
yyCdydyF
k
yyC
yy
y
yy
yy
yy
yyCyyR
inclusive single particle density
inclusive two-particle density
two-particle correlation function
Relation with NBD k parameter
Normalized correlation function
ξ : correlation length
)]1)(/(1[
2/1)(
)0,0(1
/0
/||2
21
eRk
eRC yy
Correlation function used in E802P. Carruthers and Isa Sarcevic,Phys. Rev. Lett. 63 (1989) 1562
NBD k vs. δη at E802
Phys. Rev. C52 (1995) 2663
Tomoaki Nakamura - Hiroshima Univ. 136/21/2005
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.
Tomoaki Nakamura - Hiroshima Univ. 146/21/2005
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.
Tomoaki Nakamura - Hiroshima Univ. 156/21/2005
δηdependence of NBD k parameterE802: Phys. Rev. C52 (1995) 2663
Blue: δη ≤ 0.35Red : δη ≥ 0.35
K. Homma and T. Nakamura
Au+Au 200 GeV, no magnetic field Δη<0.7, ΔФ<π/2 rad
)]1)(/(1[
2/1)(
/0
eRk
funciton Fitting
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 E802 type correlation function.
Tomoaki Nakamura - Hiroshima Univ. 166/21/2005
Number of participants dependence of correlation length ξ
• Fitting RangeBlue: δη ≤ 0.35Red : δη ≥ 0.35• Centralityfilled circle : 0-70 % (10% interval)open circle : 5-65 % (10% interval)
K. Homma and T. Nakamura
Au+Au 200 GeV, no magnetic field Δη<0.7, ΔФ<π/2 rad
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. 176/21/2005
Linear behavior of NBD k as a function of logarithmic δη
K. Homma and T. Nakamura
Au+Au 200 GeV, no magnetic field Δη<0.7, ΔФ<π/2 rad
• Fitting function k(δη) = c1 + c2 × ln(δη)c1, c2 : constant• Fitting Range 0.09 ≤ δη ≤ 0.7
q
qFk
)()(1
)(
1
This power low relation of fluctuations and pseudo rapidity gap might suggest self similarity of correlation length!!...?
Relations of fluctuation and normalized factorial moments and fractal structure.
Tomoaki Nakamura - Hiroshima Univ. 186/21/2005
Correlation functions and correlation length
ξ : correlation length, α : critical exponent
)]1)(/(1[
2/1)(
)0,0(1
/0
/||2
21
eRk
eRC yy
Used in E802
dye
yR
k
eyy
RC
y
yy
/0
/||
21
02
)(
||1 21
General correlation function
Using arbitrary R0, ξ and α.
One may discuss an effective potential form of a deconfined field when assume the correlation functions.
Tomoaki Nakamura - Hiroshima Univ. 196/21/2005
PHENIX Preliminary
PHENIX Preliminary
PHENIX Preliminary PHENIX Preliminary
PHENIX Preliminary PHENIX Preliminary
centrality 0-5% centrality 5-10% centrality 10-15%
centrality 15-20% centrality 20-25% centrality 25-30%
Jeffery T. Mitchell
δpT (pT>0.2 GeV/c) dependence of NBD k
Tomoaki Nakamura - Hiroshima Univ. 206/21/2005
Comparison for Au+Au, d+Au and p+pPHENIX Preliminary PHENIX Preliminary
PHENIX Preliminary PHENIX Preliminary
Jeffery T. Mitchell
Au+Au, centrality 45-50% d+Au, 200 GeV
p+p, 200 GeV PYTHIA
δpT 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.
A behavior of NBD k parameters as a function of δpT at p+p collisions measured by PHENIX is agree with PYHTIA qualitatively.
Tomoaki Nakamura - Hiroshima Univ. 216/21/2005
Average pT fluctuationPublished by Phys. Rev. Lett. 93, 092301 (2004)
Already, famous!It’s not a Gaussian…
it’s a Gamma distribution!
M.J. Tannenbaum, Phys. Lett. B 498 (2001) 29
PHENIX: Au+Au 200GeV 0.2 < pT < 2.0 GeV/c
),(
),(),(
2/122
][
)(
mixedp
mixedpdatapp
p
M
p
ppp
T
TT
T
T
Tp
T
TT
T
F
MM
MM
Magnitude of average pT fluctuation
Fractional deviation from mixed events
NA49: Phys. Lett. B 459 (1999) 679
Tomoaki Nakamura - Hiroshima Univ. 226/21/2005
Result of average pT fluctuations in Au+Au 200 GeV and comparison with HIJING
HIJING 1.35, default parametersHIJING cannot reproduce the centrality dependence of the fluctuations.One problem is that <N> changes depending on the HIJING settings – it is not matched to the observed dataset.
centrality 20-25%
Tomoaki Nakamura - Hiroshima Univ. 236/21/2005
Contribution of Jet/Jet suppression to the average pT fluctuation
This decrease due to jet suppression?
PYTHIA based simulation, which contains scaled hard-scattering probability factor (S
prob) by the nuclear modification factor (RAA), well agree with the measured FpT. It might be indicate that jet suppression might contribute to the average pT fluctuation.
centrality 20-25%
FpT is adjusted at the open circle for this simulation
Tomoaki Nakamura - Hiroshima Univ. 246/21/2005
Estimation of the magnitude of residual temperature fluctuations
Experiment √sNNσT/<T>
(most central)
PHENIX 200 1.8%
STAR 130 1.7%
CERES 17 1.3%
NA49 17 0.7%
)Np(
F
T
σTpT
1
2
R. Korus and S. Mrowczynski,
Phys. Rev. C64 (2001) 054908.
p → inclusive pT
H.Sako, et al, JPG 30, S (2004) 1371
σT/<
T>
A signal of phase transition dose not emerge in the temperature fluctuation?
Tomoaki Nakamura - Hiroshima Univ. 256/21/2005
Conclusions• Charged particle multiplicity fluctuation
– Different behavior about the normalized variance of multiplicity distribution as a fluctuation of number of participants are observed at PHENIX as compared to SPS energy.
– NBD k parameters as an observable of the multiplicity fluctuation are not scaled by the mean multiplicity but scaled by the number of participants.
– There are no difference on the multiplicity fluctuation with respect to the sign of electric charge for each particles.
– Correlation length could be extracted from the multiplicity fluctuation.
– Different pseudo rapidity gap dependence of NBD k parameter has been observed as compared to the past experiment.
• Average pT fluctuation– PYTHIA based simulation with scaling the hard-scattering
probability factor by the nuclear modification factor well reproduce the Au+Au 200 GeV data.
– It is suggested that jet suppressions contribute to the average pT fluctuations
– No anomaly was found about the residual temperature fluctuations from the SPS energy to RHIC energy.
Tomoaki Nakamura - Hiroshima Univ. 266/21/2005
Outlooks for QM2005
• Further new results will be presented at QM2005– Systematic study of multiplicity fluctuations with respect to the co
llision system using Cu+Cu 200 GeV.– Detailed study about extraction of the correlation length by the di
fferent correlation functions and the behavior of the correlation length as a function of number of participants.
– Jet suppression effects and extraction of specific heat in the average pT fluctuation with Au+Au 62GeV and Cu+Cu 200GeV.
– The other observables of event-by-event fluctuations and correlations, e.g. pT-pT correlation.
• We are willing to analyze the data on the event-by-event fluctuations for the QM. See you again at the Budapest.
Tomoaki Nakamura - Hiroshima Univ. 276/21/2005
Backup Slide
Tomoaki Nakamura - Hiroshima Univ. 286/21/2005
Normalized factorial moment Fq
q
q
F
nn
n
n
nn
n
nn
n
nnF
n
qnnnF
)()(
11
)1()(
)1()1()(
22
2
2
2
22
2
2
22
relation between fractal (self-similarity) structures and normalized factorial moment
average multiplicity
standard deviation
Tomoaki Nakamura - Hiroshima Univ. 296/21/2005
k
qFF
kkkF
kkF
kF
kF
k
k
kk
k
kn
knP
k
n
kn
11
31
21
11
21
11
11
11
11
)/1(
/1
1
/1
/
)()1(
)(
)1(
4
3
222
2
)(
NBD k and factorial moment Fq
Tomoaki Nakamura - Hiroshima Univ. 306/21/2005
Integral of correlation function and normalized factorial moments
)()()(),(
),(
)1()1(),(
)1(),(
)(
22111
1
11
22
21221
111
qqqq
qqq
yyyyy
yy
Fnqnnnyydydy
Fnnnyydydy
nydy
when there are no correlation in rapidity
inclusive q particle density
Tomoaki Nakamura - Hiroshima Univ. 316/21/2005
Elliptic flow contribution by simulation
Particles are assigned an azimuthal coordinate based upon the PHENIX measurement of v2 (w.r.t. the reaction plane) as a function of centrality and p
T. Only particles within the PHENIX acceptance are included in the calculation of MpT.
With the exception of peripheral collisions, the elliptic flow contribution is a small fraction of the observed fluctuation.
Jeffery T. Mitchell
Tomoaki Nakamura - Hiroshima Univ. 326/21/2005
Specific heat from average pT fluctuation
Korus, et al, PRC 64, (2001) 054908
n represents the measured particles while Ntot is all the particles, so n/Ntot is a simple geometrical factor for all experiments
M. J. Tannenbaum,2nd International Workshop on the Critical Point and Onset of Deconfinement
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