Charged hadron multiplicity fluctuations in Au+Au and Cu+Cu collisions from root S-NN = 22.5 to 200 GeV

Abstract

A comprehensive survey of event-by-event fluctuations of charged hadron multiplicity in relativistic heavy ions is presented. The survey covers Au+Au collisions at √sNN=62.4 and 200 GeV, and Cu+Cu collisions at √sNN=22.5,62.4, and 200 GeV. Fluctuations are measured as a function of collision centrality, transverse momentum range, and charge sign. After correcting for nondynamical fluctuations due to fluctuations in the collision geometry within a centrality bin, the remaining dynamical fluctuations expressed as the variance normalized by the mean tend to decrease with increasing centrality. The dynamical fluctuations are consistent with or below the expectation from a superposition of participant nucleon-nucleon collisions based upon p+p data, indicating that this dataset does not exhibit evidence of critical behavior in terms of the compressibility of the system. A comparison of the data with a model where hadrons are independently emitted from a number of hadron clusters suggests that the mean number of hadrons per cluster is small in heavy ion collisions.

Full text

arXiv:0805.1521v1 [nucl-ex] 11 May 2008
Charged hadron multiplicity fluctuations in Au+Au and Cu+Cu collisions
from
√
s
NN
= 22.5 to 200 GeV
A. Adare,
10
S.S. Adler,
5
S. Afanasiev,
24
C. Aidala,
11
N.N. Ajitanand,
51
Y. Akiba,
26, 45, 46
H. Al-Bataineh,
40
J. Alexander,
51
A. Al-Jamel,
40
K. Aoki,
30, 45
L. Aphecetche,
53
R. Armendariz,
40
S.H. Aronson,
5
J. Asai,
46
E.T. Atomssa,
31
R. Averbeck,
52
T.C. Awes,
41
B. Azmoun,
5
V. Babintsev,
20
G. Baksay,
16
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A. Baldisseri,
13
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6
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33
B. Bassalleck,
39
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6, 36
S. Batsouli,
11, 41
V. Baublis,
44
F. Bauer,
6
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5, 46
S. Belikov,
5, 20, 23, ∗
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52
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48
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10
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11
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33
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13
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39
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36
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22
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1
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45, 46
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41
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11
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36
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11, 41
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8
J. Zim´anyi,
27, ∗
L. Zolin,
24
and X. Zong
23
(PHENIX Collaboratio n)
1
Abilene Christian University, Abilene, TX 79699, USA
2
Institute of Physics, Academia Sinica, Taipei 11529, Taiwan
3
Department of Physics, Banaras Hindu University, Varanasi 221005, India
4
Bhabha Atomic Research Centre, Bombay 400 085, India
5
Brookhaven National Laboratory, Upton, NY 11973-5000, USA
6
University of California - Riverside, Riverside, CA 92521, USA
7
Charles University, Ovocn´y trh 5, Praha 1, 116 36, Prague, Czech Republic
8
China Institute of Atomic Energy (CIAE), Beijing, People’s Republic of China
9
Center for Nuclear Study, Graduate School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
10
University of Colorado, Boulder, CO 80309, USA
11
Columbia University, New York, NY 10027 and Nevis Laboratories, Irvington, NY 10533, USA
12
Czech Technical University, Zikova 4, 166 36 Prague 6, Czech Republic
13
Dapnia, CEA Saclay, F-91191, Gif-sur-Yvette, France
14
Debrecen University, H-4010 Debrecen, Egyetem t´er 1, Hungary
15
ELTE, E¨otv¨os Lor´and University, H - 1117 Budapest, P´azm´any P. s. 1/A, Hungary
16
Florida Institute of Technology, Melbourne, FL 32901, USA
17
Florida State University, Tallahassee, FL 32306, USA
18
Georgia State University, Atlanta, GA 30303, USA
19
Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan
20
IHEP Protvino, State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, 142281, Russia
21
University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
22
Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21 Prague 8, Czech Republic
23
Iowa State University, Ames, IA 50011, USA
24
Joint Institute for Nuclear Research, 141980 Dubna, Moscow Regi on, Russia
25
KAERI, Cyclotron Application Laboratory, Seoul, Korea
26
KEK, High Energy Accelerator Research Organization, Tsukuba, Ibaraki 305-0801, Japan
27
KFKI Research Institute for Particle and Nuclear Physics of the Hungarian Academy
of Sciences (MTA KFKI RMKI), H-1525 Budapest 114, POBox 49, Budapest, Hungary
28
Korea University, Seoul, 136-701, Korea
29
Russian Research Center “Kurchatov Institute”, Moscow, Russia
30
Kyoto University, Kyoto 606-8502, Japan
31
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS-IN2P3, Route de Saclay, F-91128, Palaiseau, France
32
Lawrence Livermore National Laboratory, Livermore, CA 94550, USA
33
Los Alamos National Laboratory, Los Alamos, NM 87545, USA
34
LPC, Universit´e Blaise Pascal, C NRS-IN2P3, Clermont-Fd, 63177 Aubiere Cedex, France
35
Department of Physics, Lund University, Box 118, SE-221 00 Lund, Sweden
36
Institut f¨ur Kernphysik, University of Muenster, D-48149 Muenster, Germany
37
Myongji University, Yongin, Kyonggido 449-728, Korea
38
Nagasaki Institute of Applied Science, Nagasaki-shi, Nagasaki 851-0193, Japan
39
University of New Mexico, Albuquerque, NM 87131, USA
40
New Mexico State University, Las Cruces, NM 88003, USA
41
Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
42
IPN-Orsay, Universite Paris Sud, CNRS-IN2P3, BP1, F-91406, Orsay, France
43
Peking University, Beijing, People’s Republic of China
44
PNPI, Petersburg Nuclear Physics Institute, Gatchina, Leningrad region, 188300, Russia
45
RIKEN, The Institute of Physical and Chemical Research, Wako, Sai tama 351-0198, Japan
46
RIKEN BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973-5000, USA

3
47
Physics Department, Rikkyo University, 3-34-1 Nishi-Ikebukuro, Toshima, Tokyo 171-8501, Japan
48
Saint Petersburg State Polytechnic University, St. Petersburg, Russia
49
Universidade de S˜ao Paulo, Instituto de F´ısica, Caixa Postal 66318, S˜ao Paulo CEP05315-970, Brazil
50
System Electronics Laboratory, Seoul National University, Seoul, Korea
51
Chemistry Department, Stony Brook University, Stony Brook, SUNY, NY 11794-3400, USA
52
Department of Physics and Astronomy, Stony Brook University, SUNY, Stony Brook, NY 11794, USA
53
SUBATECH (Ecole des Mines de Nantes, CNRS-IN2P3, Universit´e de Nantes) BP 20722 - 44307, Nantes, France
54
University of Tennessee, Knoxville, TN 37996, USA
55
Department of Physics, Tokyo Institute of Technology, O h-okayama, Meguro, Tokyo 152-8551, Japan
56
Institute of Physics, University of Tsukuba, Tsukuba, Ibaraki 305, Japan
57
Vanderbilt University, Nashville, TN 37235, USA
58
Waseda University, Advanced Research Institute for Science and
Engineering, 17 Kikui-cho, Shinjuku-ku, Tokyo 162-0044, Japan
59
Weizmann Institute, Rehovot 76100, Israel
60
Yonsei University, IPAP, Seoul 120-749, Korea
(Dated: May 11, 2008)
A comprehensive survey of event-by-event fluctuations of charged hadron multiplicity in rela-
tivistic heavy ions is presented. The survey covers Au+Au collisions at
√
s
NN
= 62.4 and 200
GeV, and Cu+Cu collisions at
√
s
NN
= 22.5, 62.4, and 200 GeV. Fluctuations are measured as a
function of collision centrality, transverse momentum range, and charge sign. After correcting for
non-dynamical fluctuations due to fluctuations in the collision geometry within a centrality bin, the
remaining dynamical fluctuations expressed as the variance normalized by the mean tend to decrease
with increasing centrality. The dynamical fluctuations are consistent with or below the expectation
from a superposition of participant nucleon-nucleon collisions based upon p +p data, indicating that
this dataset does not exhibit eviden ce of critical behavior in t erms of the compressibility of t he
system. A comparison of the data with a model where hadrons are independently emitted from a
number of hadron clusters suggests that the mean number of hadrons per cluster is small in heavy
ion collisions.
PACS numbers: 25.75.Gz, 25.75.Nq, 21.65.Qr, 25.75.Ag
I. INTRODUCTION
Recent work with lattice gauge theory simulations
has attempted to map out the phase diagram of Quan-
tum Chromodynamics (QCD) in temperature and baryo-
chemical potential (µ
B
) using finite values of the up and
down quark masses. The results of these studies indi-
cate that the QCD phase diagram may contain a first-
order transition line between the hadron gas phase and
the strongly -coupled Quark-Gluon Plasma (sQGP) phase
that terminates at a c ritical point [1]. This property
is analogous to that obse rved in the phas e diagr am for
many common liquids and other substances, including
water. However, different model pre dictions and lattice
calculations yie ld widely varying estimates of the loca-
tion of the critical point on the QCD phase diag ram
[2]. Direct e xperimental observa tion of critical phenom-
ena in heavy ion collis ions would confirm the existence of
the critical point, narrow down its location on the QCD
phase diagram, and provide an important constraint for
the QCD models.
The estimated value of energy densities achieved in
heavy io n collisions at the Brookhaven National Labo-
ratory’s Relativistic Heavy Ion Collider (RHIC) exceeds
∗
Deceased
†
PHENIX Spokesperson: jacak@skipper.physics.sunysb.edu
the threshold for a phase transition from normal hadronic
matter to partonic matter. Recent experimental evidence
indicates that properties of the matter being produced
include strong collective flow and large opacity to scat-
tered quarks and gluons - the matter appears to behave
much like a perfect fluid [3]. While measurements suggest
the produced matter has properties that differ from nor-
mal nuclear matter, unambiguous evidence of the nature
and location of any phase transition from normal nuclear
matter has been elusive thus far. Described here is a
search for direct evidence of a phase transition by mea-
suring the fluctuations of the event-by-event multiplici-
ties of produced charge particles in a variety of collision
systems.
In order to illustrate how the measurement of charged
particle multiplicity fluctuations can be se nsitive to the
presence of a phase transition, the isothermal compress-
ibility of the system ca n be considered [4]. T he isother-
mal compressibility is defined as follows:
k
T
= −1/V (δV/δP )
T
, (1)
where V is the volume, T is the temperature, and P is
the pressure of the system. In order to relate the co m-
pressibility to measure ments of multiplicity fluctuations,
we assume that relativistic nucleus-nucleus collisions can
be described as a thermal system in the Grand Canon-
ical Ensemble (GCE) [5]. The GCE can be applied to
the case of measurements near mid-rapidity since energy

4
and conserved quantum numbers in this region can be ex-
changed with the re st of the system, that serves as a heat
bath [6]. Detailed studies of multiplicity fluctuations in
the Ca nonical and Microcanonical Ensembles with the
application of co ns e rvation laws can be found elsewhere
[7, 8]. In the GCE, the isothermal compressibility is di-
rectly related to the variance of the particle multiplicity
as follows:
h(N − hNi)
2
i = var(N ) =
k
B
T hN i
2
V
k
T
, (2)
where N is the particle multiplicity, hNi = µ
N
is
the mean multiplicity, and k
B
is Boltzmann’s c onstant
[9]. Here, multiplicity fluctuation measurements are pr e -
sented in terms of the scaled variance, ω
N
:
ω
N
=
var (N )
µ
N
= k
B
T
µ
N
V
k
T
(3)
In a continuous, or second-order, phase transition, the
compressibility diverges to an infinite value at the critical
point. Near the critical point, this divergence is described
by a power law in the variable ǫ = (T − T
C
)/T
C
, where
T
C
is the critical temperature. Hence, the relationship
between multiplicity fluctuations and the compressibility
can be exploited to search fo r a clear signature of critical
behavior by looking for the expected power law scaling
of the compressibility:
k
T
∝ (
T − T
C
T
C
)
−γ
∝ ǫ
−γ
, (4)
where γ is the critical e xponent for isothermal compress-
ibility [9]. If the QC D phase diagram contains a critical
point, systems with a low value of baryo-chemical poten-
tial (µ
B
) could pass through the c ross-over reg ion and un-
dergo a continuous phase transition [2]. Rece nt estimates
[10, 11] of the behavior of the quark numb e r susceptibil-
ity, χ
q
, which is proportional to the value of the isother -
mal compressibility of the system, predict that its value
will increase by at leas t an order of magnitude close to the
QCD critical point. Given that the scaled variance is pro -
portional to k
T
, measurements of charged particle mul-
tiplicity are expected to be a sensitive probe for critical
behavior. In addition, within a scenario where droplets
of Quark-Gluon Plasma are formed during a first-order
phase tra ns itio n, the scaled variance of the multiplicity
could increase by a factor o f 10-100 [12].
Experimentally, a s earch for critical behavior is facili-
tated by the rich and varied dataset pr ovided by RHIC.
It is expected that the trajectory of the c olliding sys-
tem in the QCD phase diagram can be modified by vary-
ing the colliding energ y [2]. If the system approaches
close enough to the critical line for a long enough time
period, then critical phenomena could be readily appar-
ent through the measurement of multiplicity fluctuations
[13]. It may also be possible to determine the critical
exp onents of the system. Nature tends to group ma-
terials into universality classes whereby all materials in
the same universality class share identical values for their
set of critical exponents. Although be yond the sc ope of
this analysis, observation of critical behavior in heavy ion
collisions and the subsequent measurement of the critical
exp onents could determine the universality class in which
QCD is grouped, providing essential constraints for the
models.
Charged particle multiplicity fluctuations have been
measured in elementary collisions over a large range of
collisions energies [14, 15, 16, 17, 18, 19, 20]. Initial mea-
surements of multiplicity fluctuations in minimum-bias
O+Cu collisions at
√
s
NN
=4.86 GeV were made by BNL
Experiment E802 [21], minimum-bias O+Au collisions
at
√
s
NN
=17.3 GeV by CERN Ex periment WA80 [23]
and minimum-bias S+S, O+Au, and S+Au collisions at
√
s
NN
=17.3 GeV by CERN Ex periment NA35 [22]. Re-
cently, larger datasets have enabled the measurement of
the c e ntrality-dependence of multiplicity fluctuations in
Pb+Pb collisions at
√
s
NN
=17.3 GeV by CERN Exper-
iment WA98 [24] and in Pb+Pb, C+C, a nd Si+Si col-
lisions at
√
s
NN
=17.3 GeV by CERN Experiment NA49
[25]. T he PHENIX Experiment at RHIC has performed
an analysis of density correlations in longitudinal space
with a differential analysis of charged particle multiplic-
ity fluctuations in 200 GeV Au+Au collisions over the
entire transverse momentum range [26]. Thus far, the
fluctuation measurements in heavy ion collisions do not
indicate sig nificant sig ns of a phase tra ns itio n. However,
the full range of collision energies and species accessible
by RHIC are yet to be explored.
Presented here is a comprehensive survey of multi-
plicity fluctuations of charged hadrons mea sured by the
PHENIX Experiment at RHIC. The survey will cover
the following collision systems:
√
s
NN
=200 GeV Au+Au,
62.4 GeV Au+Au, 200 GeV Cu+Cu, 62.4 GeV Cu+Cu,
and 22.5 GeV Cu+Cu with comparisons to
√
s=200 GeV
p+p c ollisions, which serve as a baseline measurement.
The Au+Au data were taken during RHIC Run-4 (2004),
the Cu+Cu data were taken during RHIC Run-5 (2 005),
and the p+p data were taken during RHIC Run-3 (2 003).
Multiplicity fluctuations for each collision system with
the exception of p+p will also be presented as a function
of ce ntrality to help select the system volume. Multiplic-
ity fluctuations will also be presented as a function of
transverse momentum range, and charge sign.
This paper is organized as follows: Sec. II will discuss
the experimental apparatus and details; Sec. III will dis-
cuss the methods applied for the measurement of mul-
tiplicity fluctuations and the removal of non-dynamica l
fluctuations due to fluctuations of the c ollision geo metr y
within a c e ntrality bin; Sec. IV will pr e sent the results
and compare them to other models. Sec. V will present
a discussion and summary of the re sults.

5
II. EXPERIMENTAL SETUP
The PHENIX detector consists of two central spec-
trometer arms designed for charged particle tracking,
designated east and west, and two muon spectrometers
designed for muon tra cking and identification, designated
north and south. The muon spe c trometers are not used
in this analysis. A comprehensive description of the
PHENIX detector is documented elsewhere [2 7]. The
analysis described here utilizes the central spectrometer
arms, which consist of a set of tracking detectors [28],
particle identification detectors [29], and an electromag-
netic calorimeter [30]. The central spectrometer ar ms
cover a rapidity range of |η| < 0.35 and ea ch arm sub-
tends 90 degrees in azimuth. A detailed description of the
algorithms and performance of the central arm track re-
construction and momentum reconstruction can be found
in [31].
There are two detectors that a re used for triggering,
centrality determination, and event vertex determina-
tion. The Beam-Beam Counters (BBCs) consist of 64
individual quartz Cherenkov counters that c over the full
azimuthal angle in the pseudorapidity range 3.0 < |η| <
3.9. The Zero Degree Calorimeters (ZDCs) cover the
pseudorapidity range |η| > 6 and measure the energy of
sp e c tator neutrons with an energy resolution of approx-
imately 20%. More details about these detectors can be
found in [32]. The collision vertex position is determined
using timing information from the BB Cs with an r.m.s.
resolution for central Au+Au events of 6 mm along the
beam axis. The collision vertex is required to be recon-
structed within ±30 cm from the center of the s pectrom-
eter. The BBCs also provide a minimum biased (MB)
event trigger.
Due to the la rge dynamic range in
√
s
NN
covered by
this analysis, it is neces sary to implement algorithms that
are dependent on the collision energy for the determina-
tion of the centrality of each event. In Au+Au collisions
at
√
s
NN
=200 GeV, the centrality of the collision is deter-
mined by using correlations of the total energy deposited
in the ZDCs with the total charge depos ited in the BBCs
as descr ibed in [33]. However, in 200 GeV Cu+Cu, 62.4
GeV Cu+Cu, and 62.4 GeV Au+Au collisions, the re-
solving power of the ZDCs is insufficient to significantly
contribute to the centrality definition. Therefore, only
the total charge deposited in the B BCs is used to deter-
mine centrality in these collision systems, as described in
[33]. Using the 200 GeV Au+Au data, it has been veri-
fied that application of the BBC-ZDC correlation for the
centrality definition as opposed to the BBC- only defini-
tion shows no significant differences in the values of the
charged hadron fluctuation quantities prese nted here as
a function of centrality.
The location of the BBCs are fixed for every collision
energy. At the lowest collisio n energy (
√
s
NN
=22.5 GeV),
it becomes kinematically possible for spectator nucleons
to fall within the acceptance of the BBC. This results
in a BBC re sponse in its total charge sum that is no
longer linear with the number of participa ting nucleons
(N
part
). In this case, it becomes necess ary to define the
centrality using the total charged particle multiplicity in
Pad Chamber 1 (PC1) [28]. PC1 is chosen due to its
fine segmentation, high tracking efficiency, and relative
proximity to the event vertex. Details on this procedure
are also described in [33 ]. For all collision species and
energies, the distribution of the number of participants
was determined using a Monte Carlo simulation based
upon the Glauber model [33, 34].
The number of minimum bias events analyzed for each
dataset are 25.6 million events for 200 GeV Au+Au, 24.9
million events for 62.4 GeV Au+Au, 15.0 million events
for 200 GeV Cu+Cu, 12.2 million events for 62.4 GeV
Cu+Cu, 5.5 million events for 22.5 GeV Cu+Cu, and 2.7
million events for 200 GeV p+p. Only a frac tio n of the
complete 200 GeV Au+Au, Cu+ C u, and p+p datasets
are analyzed, but this fraction is more than sufficient for
this analysis.
The charged particle multiplicity is determined on an
event-by-event basis by co unting the number of unam-
biguous reconstructed tracks in the Drift Chamber origi-
nating from the c ollision vertex that have corresponding
hits in Pad Chamber 1 and Pad Chamber 3. Tr ack se-
lection includes cuts on reconstructed tracks in the Drift
Chamber to r e duce double-counted ghost tracks to a neg-
ligible level. In order to minimize background orig inat-
ing from the magnets, reconstructed tracks are required
to lie within ±75 cm from the center of the Drift Cham-
ber along the beam axis. This req uirement reduces the
pseudorapidity range of reconstructed tracks to |η| <0.26.
The Ring Imaging Cherenkov detector (RICH) is utilized
to reduce background from e le c trons resulting from pho-
ton conversions.
Although the central arm spectrometer covers a to-
tal azimuthal range of π radians, detector and tracking
inefficiencies reduce the effective average azimuthal ac-
tive area to 2.1 radians for the 200 Gev Au+Au and 200
GeV p+p datasets, and 2.0 radians for the other datasets.
Fluctuation quantities are q uoted for these acceptances
separately for each datas e t. The differences in accep-
tance between datasets, which are due to variations in
the detector over the three year period in which the data
was collected, result in less than a 1% variation in the
fluctuation quantities quoted here.
III. DATA ANALYSIS
Multiplicity fluctuations of charged particles, desig-
nated ω
ch
, can be generally defined [35] as follows:
ω
ch
=
(hN
2
ch
i− hN
ch
i
2
)
hN
ch
i
=
σ
2
ch
µ
ch
, (5)
where N
ch
is the charged particle multiplicity. Simply
stated, the fluctuations can be quoted as the variance
of the multiplicity (σ
2
ch
) nor malized by the mean (µ
ch
=
hN
ch
i). This is also referred to as the scaled variance [25].

6
If the multiplicity distribution is Poissonian, the scaled
variance is 1.0.
It has been well established that charged particle mul-
tiplicity distributions in elementary nucleon-nucleon col-
lisions can be described by the Negative Binomial Distri-
bution (NBD) [17, 18, 1 9]. The NBD also well describes
multiplicity distributions in heavy ion collisions [21, 22].
The Negative Binomial Distribution of an integer n is
defined as follows:
P (n) =
Γ(n + k
NBD
)
Γ(n + 1)Γ(k
NBD
)
(µ
ch
/k
NBD
)
n
(1 + µ
ch
/k
NBD
)
n+k
NBD
, (6)
where P (n) is normalized to 1.0 over the range 0 ≤ n ≤
∞, µ
ch
= hN
ch
i = hni, and k
NBD
is an additional pa-
rameter. The NBD reduces to a Poisson distribution in
the limit k
NBD
→ ∞. The NBD variance and mean is
related to k
NBD
as follows:
σ
2
ch
µ
2
ch
=
ω
ch
µ
ch
=
1
µ
ch
+
1
k
NBD
. (7)
Hence, the scaled variance is given by
ω
ch
= 1 +
µ
ch
k
NBD
. (8)
A useful property of the Negative B inomial Distribu-
tion concerns its behavior when a population that fol-
lows the NBD is subdivided randomly by rep e ated inde-
pendent trials with a constant probability onto smaller
subsets. This results in a binomial decomposition of the
original populatio n into subsets that also follow the NBD
with the same value of k
NBD
[21]. This property can be
applied to estimate the behavior of multiplicity fluctu-
ations as a function of acceptance, assuming that there
are no significant correlations pre sent over the acceptance
range being examined. Star ting with an original NBD
sample with mean µ
ch
and scaled variance ω
ch
, a sam-
ple in a fractional acce pta nce with mean µ
acc
is also de-
scribed by an NBD distribution. An acceptance fraction
can be defined as f
acc
= µ
acc
/µ
ch
. The scaled variance
of the subsample from Equation (8) is thus
ω
acc
= 1 + (µ
acc
/k
NBD
) = 1 + (f
acc
µ
ch
/k
NBD
). (9)
Since k
NBD
is identical for the two samples, µ
ch
/k
NBD
=
ω
ch
−1 can be substituted, yielding the following relation
between the scaled variances of the original and fractional
acceptance samples:
ω
acc
= 1 + f
acc
(ω
ch
− 1) (10)
Thus, the measured scaled variance will decr ease as the
acceptance is decrea sed while k
NBD
remains co ns tant, if
there are no additional correla tions present over the given
acceptance range.
Figures 1-2 show the uncorrected, or raw, multiplicity
distributions in the p
T
range 0.2 < p
T
< 2.0 GeV for all
centralities from each collision system overlayed with fits
to Negative Binomial Distributions (dashed lines). For
presentation purposes, the data have been normalized
on the horizontal axis by the mean of the distribution
and scaled on the vertical axis by the successive amounts
stated in the legend. The NBD fits describe the data
distributions very well for all collision systems, centrali-
ties, and p
T
ranges. Hence, the mean and variance of the
multiplicity distributions presented here are all extracted
from NBD fits. The results of each fit for 0.2 < p
T
< 2.0
GeV are compiled in Table I. The mean and standard
deviation of each fit for 0.2 < p
T
< 2.0 GeV are plotted
in Fig. 3.
Each dataset was taken over spans of several days to
several weeks, all spanning three separate RHIC run-
ning periods. During these periods, changes in the total
acceptance a nd efficiency of the central arm spectrom-
eters cause the fluctuation measurements to vary, thus
introducing an a dditional sy stematic error to the results.
This systematic error was minimized by requiring tha t
the dataset is stable in quantities that are sensitive to
detector variations, including the mean charged parti-
cle multiplicity, mean collision vertex position, and mea n
centrality. A time-dependent systematic error is applied
independently to each po int by ca lc ulating the standard
deviation of the scaled variance calculated from subsets
of the entire dataset, with each subset containing a bo ut
1 millio n events. These s ystematic errors are applied to
all subsequent results.
The tracking efficiency of the PHENIX central arm
sp e c trometer is dependent on centrality, especially in the
most central 2 00 GeV Au+Au collisions [36]. With the
assumption that tracking inefficiencies randomly effect
the multiplicity distribution on an event-by-event basis,
the effect of inefficiencies on the scaled va riance can be
estimated using Equation (1 0) where f
acc
is replaced by
the inverse of the tracking efficiency, 1/f
eff
. Tracking effi-
ciency effects the value of the scaled variance by 1.5 % at
the most. The scaled va riance has been corrected for
tracking inefficiency as a function of centrality for all
sp e c ie s. The uncertainty of the tracking efficiency es-
timate is typically 2% and has been propagated into the
systematic error estimate on a point-by-p oint basis.
Due to the non-z e ro width of the centrality bin se lec -
tion from the data, each centrality bin necessarily se-
lects a range of impact parameters. This introduces a
non-dynamical fluctuation component to the measured
multiplicity fluctuations due to the resulting fluctuations
in the geometry of the collisions [26, 37]. Therefore, it is
necessary to estimate the magnitude of the geometry fluc-
tuation compo nent so that only the interesting dynami-
cal fluctuations remain. The mo st practical method for
estimating the geometry fluctuation component is with
a model of heavy ion collisions. The URQMD [39] and
HSD [40, 41] models have previously been applied for this
purp ose. Here, the HIJING event generator [38] is cho-
sen for this estimate because it well reproduces the mean
multiplicity in heavy ion collisions [33] as measured by
the PHENIX detector. HIJING includes multiple mini-
jet production based upon QCD-inspired models, soft ex-

7
citation, nuclear shadowing of parton distribution func-
tions, and the interaction of jets in dense nuclear matter.
The estimate is performed individually for each central-
ity bin, collision system, and p
T
range using the following
procedure. First, HIJING is run with an impact param-
eter distribution that is sampled from a Gaussian distr i-
bution with a mean and standard deviation tha t, for a
given centrality bin, reproduces the distributions of the
charge deposited in the BBC and the energy deposited in
the ZDC (for 200 GeV Au+Au). Second, HIJING is run
at a fixed impact par ameter with a value identical to the
mean of the Gaussian distribution in the first run. For
each centrality bin, 12,000 HIJING events are proce ssed
for each impact parameter sele c tion. The scale d variance
for each impact parameter selection, ω
Gauss
and ω
fixed
, is
extracted and the measured scaled variance is correc ted
>
ch
/<N
ch
N
0 1 2 3 4
>)
ch
/<N
ch
/(d(N
ch
dN
events
1/N
3
10
4
10
5
10
6
10
55-60% x 12.0
50-55% x 11.0
45-50% x 10.0
40-45% x 9.0
35-40% x 8.0
30-35% x 7.0
25-30% x 6.0
20-25% x 5.0
15-20% x 4.0
10-15% x 3.0
5-10% x 2.0
0-5% x 1.0
>
ch
/<N
ch
N
0 1 2 3 4
>)
ch
/<N
ch
/(d(N
ch
dN
events
1/N
3
10
4
10
5
10
6
10
45-50% x 10.0
40-45% x 9.0
35-40% x 8.0
30-35% x 7.0
25-30% x 6.0
20-25% x 5.0
15-20% x 4.0
10-15% x 3.0
5-10% x 2.0
0-5% x 1.0
FIG. 1: The uncorrected multiplicity distributions of charged
hadrons with 0.2 < p
T
< 2.0 GeV/c for 200 (upper) and 62.4
(lower) GeV A u+Au collisions. The dashed lines are fit s to
the Negative Binomial Distribution. The data are normalized
to the mean and scaled by the amounts in the legend.
>
ch
/<N
ch
N
0 1 2 3 4
>)
ch
/<N
ch
/(d(N
ch
dN
events
1/N
3
10
4
10
5
10
30-35% x 7.0
25-30% x 6.0
20-25% x 5.0
15-20% x 4.0
10-15% x 3.0
5-10% x 2.0
0-5% x 1.0
>
ch
/<N
ch
N
0 1 2 3 4
>)
ch
/<N
ch
/(d(N
ch
dN
events
1/N
3
10
4
10
5
10
20-25% x 5.0
15-20% x 4.0
10-15% x 3.0
5-10% x 2.0
0-5% x 1.0
>
ch
/<N
ch
N
0 5 10 15 20
>)
ch
/<N
ch
/(d(N
ch
dN
events
1/N
3
10
4
10
5
10
22.5 GeV Cu+Cu, 10-20% x 2.0
22.5 GeV Cu+Cu, 0-10% x 1.0
200 GeV p+p, Min. Bias. x 0.1
FIG. 2: The uncorrected multiplicity distributions of charged
hadrons with 0.2 < p
T
< 2.0 GeV/c for 200 (upper), 62.4
(middle), and 22.5 (lower) GeV Cu+Cu and 200 GeV p+p
(lower) collisions. The dashed lines are fits to the Negative
Binomial Distribution. The data are normalized to the mean
and scaled by the amounts in the legend.

8
TABLE I: Tabulation of the charged hadron multiplicity data and corrections for 0.2 < p
T
< 2.0 GeV/c. The errors quoted for
µ
ch
and σ
ch
represent their time-depend ent systematic error. The errors quoted for ω
ch,dyn
and 1/k
NBD,dyn
represent their total
systematic error. For each dataset the first three columns give the species, collision energy, and geometric correction factor,
f
geo
, respectively.
Species
√
s
NN
f
geo
N
part
µ
ch
raw σ
2
ch
ω
ch,dyn
1/k
NBD,dyn
χ
2
/dof
(GeV)
351 61.0 ± 1.1 75.6 ± 1.9 1.10 ± 0.02 1.45 · 10
−03
± 2.2 · 10
−04
37.1/58
299 53.1 ± 1.0 71.8 ± 1.8 1.15 ± 0.02 2.45 · 10
−03
± 2.7 · 10
−04
38.6/56
253 45.8 ± 0.8 65.2 ± 1.5 1.17 ± 0.02 3.41 · 10
−03
± 2.9 · 10
−04
34.0/54
215 39.1 ± 0.7 57.8 ± 1.6 1.19 ± 0.03 4.53 · 10
−03
± 3.6 · 10
−04
29.1/53
181 32.6 ± 0.6 49.7 ± 1.3 1.21 ± 0.03 5.95 · 10
−03
± 5.1 · 10
−04
24.5/50
Au+Au 200 0.37 ± 0.027 151 27.4 ± 0.5 41.4 ± 1.0 1.20 ± 0.03 6.86 · 10
−03
± 5.5 · 10
−04
20.7/46
125 22.3 ± 0.4 33.8 ± 0.9 1.20 ± 0.03 8.47 · 10
−03
± 7.1 · 10
−04
11.9/41
102 17.8 ± 0.3 26.7 ± 0.6 1.19 ± 0.02 1.05 · 10
−02
± 9.0 · 10
−04
16.6/37
82 14.2 ± 0.3 20.8 ± 0.6 1.17 ± 0.02 1.20 · 10
−02
± 1.0 · 10
−03
37.8/33
65 10.8 ± 0.2 16.0 ± 0.4 1.18 ± 0.02 1.64 · 10
−02
± 1.3 · 10
−03
37.8/28
51 8.3 ± 0.2 12.1 ± 0.3 1.17 ± 0.02 2.06 · 10
−02
± 2.0 · 10
−03
53.8/24
345 44.0 ± 0.3 53.6 ± 0.5 1.08 ± 0.02 1.63 · 10
−03
± 2.0 · 10
−04
14.6/54
296 37.3 ± 0.2 48.3 ± 0.3 1.11 ± 0.02 2.63 · 10
−03
± 2.6 · 10
−04
13.8/53
250 31.0 ± 0.2 39.8 ± 0.4 1.10 ± 0.02 3.00 · 10
−03
± 3.0 · 10
−04
14.0/50
211 25.4 ± 0.2 33.6 ± 0.5 1.12 ± 0.02 4.21 · 10
−03
± 4.4 · 10
−04
8.36/44
177 20.8 ± 0.1 27.8 ± 0.2 1.12 ± 0.02 5.34 · 10
−03
± 5.5 · 10
−04
19.2/40
Au+Au 62.4 0.33 ± 0.031 148 16.6 ± 0.1 22.8 ± 0.3 1.13 ± 0.02 7.43 · 10
−03
± 7.8 · 10
−04
25.9/37
123 13.1 ± 0.1 18.1 ± 0.2 1.13 ± 0.02 9.61 · 10
−03
± 9.7 · 10
−04
34.3/33
102 10.4 ± 0.1 14.9 ± 0.1 1.15 ± 0.02 1.38 · 10
−02
± 1.4 · 10
−03
44.5/28
82 7.8 ± 0.1 11.1 ± 0.1 1.14 ± 0.02 1.76 · 10
−02
± 1.9 · 10
−03
50.9/24
66 5.9 ± 0.04 8.3 ± 0.1 1.14 ± 0.02 2.37 · 10
−02
± 3.8 · 10
−03
45.4/20
51 4.1 ± 0.03 5.8 ± 0.04 1.13 ± 0.02 3.08 · 10
−02
± 9.1 · 10
−03
36.2/17
104 19.3 ± 0.3 25.7 ± 0.8 1.14 ± 0.03 6.93 · 10
−03
± 1.3 · 10
−03
24.3/30
92 16.0 ± 0.2 21.9 ± 0.5 1.15 ± 0.03 9.26 · 10
−03
± 1.5 · 10
−03
21.7/31
79 13.5 ± 0.2 18.8 ± 0.4 1.16 ± 0.03 1.15 · 10
−02
± 2.1 · 10
−03
19.4/29
Cu+Cu 200 0.40 ± 0.047 67 11.1 ± 0.2 15.3 ± 0.3 1.15 ± 0.03 1.36 · 10
−02
± 2.0 · 10
−03
29.9/26
57 9.2 ± 0.1 13.0 ± 0.3 1.17 ± 0.03 1.75 · 10
−02
± 2.5 · 10
−03
26.0/25
48 7.5 ± 0.1 10.5 ± 0.2 1.16 ± 0.03 2.14 · 10
−02
± 3.6 · 10
−03
30.6/22
40 6.2 ± 0.1 8.7 ± 0.2 1.17 ± 0.03 2.69 · 10
−02
± 4.8 · 10
−03
28.6/20
33 4.9 ± 0.06 6.8 ± 0.1 1.16 ± 0.03 3.12 · 10
−02
± 8.5 · 10
−03
45.7/18
104 12.6 ± 0.1 16.7 ± 0.2 1.10 ± 0.03 8.16 · 10
−03
± 1.7 · 10
−03
40.6/31
92 11.0 ± 0.1 16.2 ± 0.1 1.15 ± 0.04 1.35 · 10
−02
± 2.7 · 10
−03
64.2/30
Cu+Cu 62.4 0.32 ± 0.063 79 9.2 ± 0.1 14.3 ± 0.2 1.18 ± 0.05 1.92 · 10
−02
± 3.9 · 10
−03
37.0/28
67 7.7 ± 0.1 12.0 ± 0.2 1.18 ± 0.05 2.29 · 10
−02
± 4.6 · 10
−03
32.0/26
57 6.0 ± 0.1 9.1 ± 0.1 1.17 ± 0.05 2.85 · 10
−02
± 5.9 · 10
−03
32.0/23
48 5.1 ± 0.1 8.1 ± 0.1 1.19 ± 0.05 3.66 · 10
−02
± 8.0 · 10
−03
29.2/21
92 9.1 ± 0.04 10.3 ± 0.1 1.04 ± 0.02 4.31 · 10
−03
± 9.8 · 10
−04
7.45/24
Cu+Cu 22.5 0.30 ± 0.064 58 4.9 ± 0.02 5.8 ± 0.04 1.06 ± 0.02 1.11 · 10
−02
± 2.9 · 10
−03
71.1/17

9
part
N
100 200 300 400
ch
µ
0
20
40
60
80
200 GeV Au+Au
62.4 GeV Au+Au
part
N
50 100
ch
µ
0
10
20
200 GeV Cu+Cu
62.4 GeV Cu+Cu
22.5 GeV Cu+Cu
FIG. 3: The mean from the NBD fit as a function of N
part
for Au+Au (upper) and Cu+Cu (lower) collisions over the
range 0.2 < p
T
< 2.0 GeV/c. The mean shown is within the
PHENIX central arm spectrometer acceptance. The error
bars represent the standard deviation of the distribution.
as the fractional deviation fr om a sca led variance of 1.0
of a Poisson distribution as follows:
ω
ch,dyn
−1 =
(ω
fixed
− 1)
(ω
Gauss
− 1)
(ω
ch,raw
−1) = f
geo
(ω
ch,raw
−1),
(11)
where ω
ch,dyn
represents the estimate of the remaining
dynamical multiplicity fluctuations and ω
ch,raw
repre-
sents the uncorrected multiplicity fluctuations . Since the
correction, f
geo
, is calculated as a ratio of the two run-
ning conditions of the simulation, most multiplicity fluc-
tuations intrinsic to the model should be canceled. The
correction always reduces the magnitude of the measured
scaled variance. Note that the value of f
geo
is mathemat-
ically identical when applied to the inverse of k
NBD
:
k
−1
NBD,dyn
= f
geo
k
−1
NBD
. (12)
The resulting geometrical correction factors for each
sp e c ie s are constant as a function of centrality, there-
fore a single correction factor is calculated for each trans-
verse momentum range by fitting the correction factors
as a function of N
part
to a constant. This behavior
is expected since centrality bins are defined to be con-
stant percentages of the total geometric cross section.
The correction factors for each transverse momentum
range for a given collision species are consistent with
each other. The s tandard deviation of the individual
geometrical correction factors from the linear fits a s a
function of N
part
are included in the systematic error
of the correction factor estimation and propagated into
the total sy stematic error for each p oint in ω
ch,dyn
and
k
NBD,dyn
. For 0.2 < p
T
< 2.0 GeV/c, the geometrica l
correction factors, f
geo
, a nd systematic errors from the
fit are 0.37 ± 0.027 for 200 GeV Au+Au, 0.33 ± 0.031
for 62.4 GeV Au+Au, 0.40 ±0.047 for 200 GeV Cu+Cu,
0.32 ± 0.063 for 62.4 GeV Cu+Cu, and 0.30 ± 0.064 for
22.5 GeV Cu+Cu. The extraction of the geometrical cor-
rection factors are inherently model-dependent and are
also dependent on the accuracy with which the central-
ity detectors are modelled. The effect of the latter de-
pendence has been studied by also calculating the correc-
tion factors using constant but non-overlapping impact
parameter distributions for ea ch centrality bin and com-
paring them to the correction fac tors using the Gaussian
impact parameter distr ibutions. For all p
T
ranges, an
additional fraction of the value of ω
ch,dyn
or k
−1
NBD,dyn
has been included in the final systematic errors for these
quantities. The magnitude of this systematic error is 8%
for 200 GeV Au+Au, 8% for 62.4 GeV Au+Au, 11% for
200 GeV Cu+Cu, 17% for 62.4 GeV Cu+Cu, and 25%
for 22.5 GeV Cu+Cu. A sample comparison of the scaled
variance before and after the application of the geomet-
rical corre ction factor is shown for the 200 GeV Au+Au
dataset in Fig. 4.
IV. RESULTS
The scaled variance as a function of the number of
participating nucleons, N
part
, over the p
T
range 0.2 <
p
T
< 2.0 GeV/c is shown in Fig. 5. For all centrali-
ties, the scaled variance values consistently lie above the
Poisson distribution value of 1.0. In all collision systems,
the minimum scaled va riance occur s in the most central
collisions and then begins to increase as the centrality
decreases. In 200 GeV Au+Au collisions, this increase is
only observed for N
part
> 200. For N
part
< 200 ω
ch,dyn
suggests a slight decrease but is consistent with a con-
stant value. In 62.4 GeV Au+Au collisions, the increase
in ω
ch,dyn
with decreasing centrality is observed only over
the r ange N
part
> 110. The source of the qualitative

10
part
N
100 200 300 400
ch
ω
1
1.2
1.4
1.6
ch,raw
ω200 GeV Au+Au,
ch,dyn
ω200 GeV Au+Au,
FIG. 4: Fluctuations exp ressed as the scaled variance as a
function of centrality for 200 GeV Au+Au collisions in the
range 0.2 < p
T
< 2.0 GeV/c. Shown are the uncorrected
fluctuations, ω
ch,raw
, along with fl uctuations after correcting
for the estimated contribution from geometry fluctuations us-
ing Equation 11, ω
ch,dyn
.
differences between the 200 a nd 62.4 GeV Au+Au col-
lisions is not known, a lthough some of the differences
could be explained by the increased contribution from
hard scattering processes at 200 GeV compared to 62.4
GeV. Studies per formed by varying the centrality selec-
tion cuts establish that the differences are not due to
the differences in the centrality se le c tion algorithm. A
similar centrality-dependent trend of the scaled variance
has also been observed at the SPS in low energy Pb+Pb
collisions at
√
s
NN
=17.3 GeV and at fo rward rapidities
(1.1< y
c.m.
<2.6), measured by experiment NA49 [25],
where the hard scattering contribution is expected to
be s mall. The Cu+Cu data exhibit a weaker decrease
in the scaled variance for more central collisions. The
62.4 GeV Cu+Cu scaled variance values are consistently
above those from the 200 GeV Cu+Cu dataset, but the
two are consistent within the systematic errors for all
centralities.
The scaled variance has been studied as a function o f
the p
T
range over which the multiplicity distributions
are meas ured in order to determine if any significant p
T
-
dependent dynamical fluctuations are present. Results
for several p
T
ranges fr om 0.2 < p
T
< 2.0 GeV/c down
to 0.2 < p
T
< 0.5 GeV/c a re shown in Figs. 6-7. In the
absence of p
T
-dependent dynamical fluctuations, restrict-
ing the p
T
range should reduce the scaled variance in the
same manner as for a fractional acceptance. Similar to
Equation (10):
ω
p
T
= 1 + f
pt
(ω
ref
− 1), (13)
where ω
p
T
represents the fluctuations in the p
T
range of
interest, ω
ref
represents the fluctuations in the reference
p
T
range, and f
p
T
= µ
p
T
/µ
ref
is the ratio of the mean
multiplicity in the two rang e s. Also shown are curves rep-
resenting the expected scaling of the fluctuations using
the range 0.2 < p
T
< 2.0 GeV/c as the reference ra nge.
The shaded regio ns reflect the systematic er rors of the
part
N
100 200 300 400
ch,dyn
ω
1
1.1
1.2
200 GeV Au+Au
62.4 GeV Au+Au
Superposition Model, 200 GeV Au+Au
Superposition Model, 62.4 GeV Au+Au
Poisson + flow, 200 GeV Au+Au
part
N
40 60 80 100 120
ch,dyn
ω
1
1.1
1.2
200 GeV Cu+Cu
62.4 GeV Cu+Cu
22.5 GeV Cu+Cu
Superposition Model, 200 GeV Cu+Cu
Superposition Model, 62.4 GeV Cu+Cu
Superposition Model, 22.5 GeV Cu+Cu
FIG. 5: Fluctuations exp ressed as the scaled variance as a
function of N
part
for Au+Au (upper) and Cu+Cu (lower)
collisions for 0.2 < p
T
< 2.0 GeV/c. The estimated contribu-
tion from geometry fl uctuations has been removed. Results
from the superposition model are overlayed with the shaded
regions representing a one standard deviation range of the
prediction for the fluctuation magnitude derived from p+ p
collision d ata. Also shown ( upper) is the estimated contribu-
tion from non-correlated particle emission with the Poisson
distribution of the scaled variance of 1.0 with the addition of
elliptic flow in 200 GeV Au+Au collisions.

11
reference range. For all p
T
ranges, the scaled fluctuation
curves are consistent with the data, indicating that no
significant p
T
-dependence is observed, although the data
in the range 0.2 < p
T
< 0.5 GeV are consistently above
the scaled reference curves. The p
T
-dependence can also
be ex amined in more directly with the parameter k
NBD
from the NBD fits. Substitution of the scale d variance in
Equation (8) into E quation (13) shows that k
NBD
should
be independent of p
T
in the absence of p
T
-dependent dy-
namical fluctuations. As shown in Figs. 8-9, there is no
significant p
T
-dependence of the observed values of k
NBD
.
part
N
100 200 300 400
ch,dyn
ω
1
1.1
1.2
ch,dyn
ω200 GeV Au+Au
<2.00 GeV/c
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.75 GeV/c
T
0.2<p
<0.50 GeV/c
T
0.2<p
<2.0 GeV Scaled
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.75 GeV/c
T
0.2<p
<0.50 GeV/c
T
0.2<p
part
N
100 200 300 400
ch,dyn
ω
1
1.1
1.2
ch,dyn
ω62.4 GeV Au+Au
<2.00 GeV/c
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.50 GeV/c
T
0.2<p
<2.0 GeV Scaled
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.50 GeV/c
T
0.2<p
FIG. 6: Scaled variance for 200 (upper) and 62.4 (lower) GeV
Au+Au collisions plotted as a function of N
part
for several p
T
ranges. The lines represent the data for the reference range
0.2 < p
T
< 2.0 scaled down using t he mean multiplicity in
each successive p
T
range. The shaded areas represent the
systematic errors from the reference range.
part
N
50 100
ch,dyn
ω
1
1.05
1.1
1.15
1.2
ch,dyn
ω200 GeV Cu+Cu
<2.00 GeV/c
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.75 GeV/c
T
0.2<p
<2.0 GeV Scaled
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.75 GeV/c
T
0.2<p
part
N
50 100
ch,dyn
ω
1
1.05
1.1
1.15
1.2
ch,dyn
ω62.4 GeV Cu+Cu
<2.00 GeV/c
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.75 GeV/c
T
0.2<p
<2.0 GeV Scaled
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.75 GeV/c
T
0.2<p
FIG. 7: Scaled variance for 200 (upper) and 62.4 (lower) GeV
Cu+Cu collisions plotted as a function of N
part
for several p
T
ranges. The lines represent the data for the reference range
0.2 < p
T
< 2.0 scaled down using the mean multiplicity in
each successive p
T
range. The shaded areas represent the
systematic errors from the reference range.
The scaled variance as a function of the charge sign
of the charged hadr ons is shown in Fig. 10 fo r 200 GeV
Au+Au c ollisions in the p
T
range 0.2 < p
T
< 2.0 GeV/c
in order to investigate any Coulomb-based contributions
to the fluctuations. In the absence of additional dynamic
fluctuations, the scaled variance for p ositively or nega-
tively charged hadrons should be reduced from the inclu-
sive charged hadron value by
ω
+−
= 1 + f
+−
(ω
ch
− 1), (14)
where ω
+−
are the fluctuations for positive or negative

12
part
N
2
10
NBD
1/k
-3
10
-2
10
NBD
200 GeV Au+Au 1/k
<2.00 GeV/c
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.75 GeV/c
T
0.2<p
<0.50 GeV/c
T
0.2<p
part
N
2
10
NBD
1/k
-3
10
-2
10
NBD
62.4 GeV Au+Au 1/k
<2.00 GeV/c
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.50 GeV/c
T
0.2<p
FIG. 8: The inverse of the parameter k
NBD
from the Negative
Binomial Distribution fits for 200 (upper) and 62.4 (lower)
GeV Au+Au collisions. The fluctuations are plotted as a
function of N
part
for several p
T
ranges.
particles, ω
ch
are the fluctuations for inclusive charged
hadrons, and f
+−
= µ
+−
/µ
ch
is the ratio of the mean
multiplicities. The scaled variance from the positive
and neg ative hadrons are c onsistent with each other and
consistent with the expected r e duction of the inclusive
charged hadron fluctuations.
An additional non-dynamic contribution to multiplic-
ity fluctuations arises from the presence of elliptic flow.
This contribution has been estimated using a simple
Monte Carlo simulation. In this simulation, a random
reaction plane angle is assig ned to each event. T he mul-
tiplicity distribution due to the elliptic flow component
part
N
2
10
NBD
1/k
-2
10
NBD
200 GeV Cu+Cu 1/k
<2.00 GeV/c
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.75 GeV/c
T
0.2<p
part
N
2
10
NBD
1/k
-2
10
NBD
62.4 GeV Cu+Cu 1/k
<2.00 GeV/c
T
0.2<p
<1.00 GeV/c
T
0.2<p
<0.75 GeV/c
T
0.2<p
FIG. 9: The inverse of th e parameter k
NBD
from the Negative
Binomial Distribution fits for 200 (upper) and 62.4 (lower)
GeV Cu+Cu collisions. The fluctuations are plotted as a
function of N
part
for several p
T
ranges.
is given by the following:
dN/dφ = C[1 + 2 v
2
cos(2∆φ)], (15 )
where C is a normalization factor, v
2
is the measured
magnitude of the elliptic flow, and ∆φ is the difference
between the particle emission angle and the reaction
plane angle. For each event, this multiplicity distribu-
tion function is integrated over the PHENIX azimuthal
acceptance and the resulting scaled variance from one
million events is calculated. The value o f v
2
used in the
simulation is taken from PHENIX measur ements of ellip-
tic flow at the mean transverse momentum of the inclu-
sive charged hadron spectra in the ra nge 0.2 < p
T
< 2.0

13
GeV/c within the central arm spectrometers [45]. The
estimated contribution from elliptic flow to the obs e rved
scaled variance is less than 0.038% for all centralities and
is shown for 200 GeV Au+ Au collisions in Fig. 5.
part
N
100 200 300 400
ch,dyn
ω
1
1.1
1.2
200 GeV Au+Au, inclusive charge
200 GeV Au+Au, positive charge
200 GeV Au+Au, negative charge
=0.5
+-
inclusive charge, f
FIG. 10: The scaled variance as a function of N
part
for 200
GeV Au+Au collisions in the range 0.2 < p
T
< 2.0 GeV/c.
Shown are measurements for inclusive charged particles, pos-
itive p articles, and negative particles. The line represents the
inclusive data scaled down in acceptance by 50% with the
shaded area rep resenting the systematic error.
V. DISCUSSION
A. Comparisons to a participant superposition
model
It is informative to compare fluctuations in relativistic
heavy ion collisions to what can be expected from the
supe rp osition of individual participant nucleon-nucleon
collisions. For this purpose, PHENIX data will be com-
pared to a participant superposition, or wounded nu-
cleon, model [42] based upon data from elementary col-
lisions. In the participant superposition model, the total
multiplicity fluctuations ca n be ex pressed in terms of the
scaled variance [35],
ω
N
= ω
ν
+ µ
WN
ω
N
part
, (16)
where ω
ν
are the fluctuations from each individual
source, e.g. from each elementary collision, ω
N
part
are
the fluctuations of the number of sourc es, and µ
WN
is
the mean multiplicity per wounded nucleon. The second
term includes non-dynamic contributions from geometry
fluctuations due to the width of the centrality bin along
with additional fluctuations in the number of participants
for a fixed impact parameter. Ideally, the se c ond term is
nearly nullified after applying the geometr y corrections
described previously, so the resulting fluctuations are in-
dependent of centrality as well as collision species.
part
N
100 200 300 400
ch,dyn
ω
1
1.2
1.4
1.6
200 GeV Au+Au
62.4 GeV Au+Au
HIJING 200 GeV Au+Au, jets on
HIJING 200 GeV Au+Au, jets off
HIJING 62.4 GeV Au+Au, jets on
HIJING 62.4 GeV Au+Au, jets off
String Clustering 200 GeV Au+Au
part
N
40 60 80 100 120
ch,dyn
ω
1
1.2
1.4
1.6
200 GeV Cu+Cu
62.4 GeV Cu+Cu
22.5 GeV Cu+Cu
HIJING 200 GeV Cu+Cu, jets on
HIJING 200 GeV Cu+Cu, jets off
HIJING 62.4 GeV Cu+Cu, jets on
HIJING 62.4 GeV Cu+Cu, jets off
HIJING 22.5 GeV Cu+Cu, jets on
HIJING 22.5 GeV Cu+Cu, jets off
FIG. 11: Fluctuations ex pressed as the scaled variance as a
function of N
part
for Au+Au (upper) and Cu+Cu (lower) col-
lisions for 0.2 < p
T
< 2.0 GeV/c. The estimated contribution
from geometry fluctuations has been removed. Results from
the HIJING model with jets turned on and jets turned off are
overlayed with the shaded regions representing the systematic
error for each curve.
Baseline comparisons are facilitated by PHENIX mea-
surements of charged particle multiplicity fluctuations in
minimum bias 200 GeV p+p collisions. The p+p data
and the NBD distribution to the multiplicity distribu-
tion are shown in Fig. 2. The NBD fit yields µ
ch
= 0.32
± 0.003, ω
ch
= 1.17 ± 0.01, and k
NBD
= 1.88 ± 0.01.

14
These results are in a greement within errors with previ-
ous measurements in the same pseudorapidity range of
k
NBD
= 1.9 ± 0.2 ±0.2 by the UA5 Collaboration [20] in
collision of protons and antiprotons at 200 GeV. Compar-
isons of the participant sup erposition model to the 22.5
GeV Cu+Cu data can be made to multiplicity fluctua-
tions measure d in 20 GeV p+p collisions by the NA22
Collaboratio n [19] over the same pseudorapidity range
as the PHE NIX Cu+Cu mea surement. After scaling the
NA22 sc aled variance to the PHENIX azimuthal accep-
tance, the participant superposition model scaled vari-
ance is expected to be constant as a function of centrality
with a value of 1.08±0.04. Lacking multiplicity distribu-
tion data from elementary collisions a t 62 .4 GeV within
the PHENIX pseudorapidity acceptance, it is assumed
that as a function of collision energ y, the scaled variance
in the PHENIX pseudorapidity acceptance scales in the
same manner as in an acceptance of 4π, which can be
parametrized from existing p+p and p+¯p data as follows
[17]:
µ
ch
≈ −4.2 + 4.69(
s
GeV
2
)
0.155
. (17)
Given the mean charged particle multiplicity, the scaled
variance in p+p and p+¯p can be parametrized as follows
[35]:
ω
ch
≈ 0.35
(µ
ch
− 1)
2
µ
ch
. (18)
Scaling this parametrization to match the values of ω
ch
at 200 GeV and 22.5 GeV, the estimated value of ω
ch
at
62.4 GeV is 1.15 ± 0.02.
Comparisons of the data to the predictions of the par-
ticipant superposition model are shown in Fig. 5 for
Au+Au and Cu+Cu c ollisions. The shaded regions abo ut
the participant superposition model lines represent the
systematic error of the estimates described a bove. All of
the data points ar e consistent with or be low the partic-
ipant superposition model estimate. This suggests that
the data do not show any indications of the presence of
a critical point, where the fluctuations are expected to
be much larger than the participant superposition model
exp ectation.
B. Comparisons to the HIJING model
Shown in Fig. 11 are the scaled variance c urves from
HIJING simulations into the PHENIX acceptance. The
HIJING simulations are per fo rmed with a fixed impact
parameter corresponding to the mean of the impact pa-
rameter distribution for each bin as determined by the
Glaube r model in order to minimize the geometry fluc-
tuation component of the result. The mean and variance
of the resulting multiplicity distributions from HIJING
are extracted from fits to Negative Binomial Distribu-
tions. The HIJING simulation multiplicity fluctuations
with the jet production parameter turned on are consis-
tently above the data and increa se continuously through
the most peripheral collisions. This behavior is not con-
sistent with the data, where the fluctuatio ns do not in-
crease in the most peripheral collisions. Although HI-
JING reproduces the total charged particle multiplicity
well, it consistently overpredicts the amount of fluctua-
tions in multiplicity. When the jet production parameter
in HIJING is turned off, the scaled variance as a func-
tion of centrality is independent of collision energy, illus-
trating that jet production accounts for the energy de-
pendence of the HIJING results. Note that the HIJING
results with jet production turned off are in better agree-
ment with the data for all collision energies. Together
with the observation that the multiplicity fluctuatio ns
demonstrate no significant p
T
-dependence, this may be
an indication that correlated emission of par ticles from
jet production do not significantly contribute to the mul-
tiplicity fluctuations observed in the data .
C. Comparisons to the clan model
The clan model [46] has been developed to interpret
the fact that Negative Binomial Distributions describe
charged ha dron multiplicity distributions in elementary
and heavy ion collisions. In this model, hadr on produc-
tion is modeled as independent emission of a number
of hadronic clusters, N
c
, each with a mean number of
hadrons, n
c
. The independent emiss ion is described by a
Poisson distribution with an average cluster, or clan, mul-
tiplicity of
¯
N
c
. After the clusters are emitted, they frag-
ment into the final state hadrons. The mea sured value
of the mean multiplicity, µ
ch
, is related to the cluster
multiplicities by µ
ch
=
¯
N
c
¯n
c
. In this mo del, the cluster
multiplicity parameters can be simply related to the NBD
parameters of the measure d multiplicity distribution as
follows:
¯
N
c
= k
NBD
log(1 + µ
ch
/k
NBD
) (19)
and
¯n
c
= (µ
ch
/k
NBD
)/log(1 + µ
ch
/k
NBD
). (20)
The results from the NBD fits to the data are plot-
ted in Fig. 12 for all collision species. Also shown are
data from elementary and heavy ion collisions at various
collision energies. The individual data points from all
but the PHENIX data are taken from multiplicity distri-
butions measured over varying ranges of pseudorapidity,
while the PHENIX data are taken as a function of cen-
trality. The characteristics of all of the heavy ion data
sets are the same. The va lue of ¯n
c
varies little within
the r ange 1.0-1.1. The heavy ion data universally ex-
hibit only weak cluster ing characteristics as interpreted
by the clan model. There is also no significant variation

15
seen with collision energy. However, ¯n
c
is consistently
significantly higher in elementary collisions. In elemen-
tary collisions, it is less probable to produce events with a
high multiplicity, which can reveal rare sources of clusters
such as jet production or multiple parton interactions.
c
Number of clusters, N
0 20 40 60
c
Particles per cluster, n
1
2
3
4
PHENIX: 200 GeV Au+Au
PHENIX: 62.4 GeV Au+Au
PHENIX: 200 GeV Cu+Cu
PHENIX: 62.4 GeV Cu+Cu
PHENIX: 22.5 GeV Cu+Cu
pUA5: 540 GeV p+
pUA5: 200 GeV p+
E802: 4.86 GeV O+Cu
NA35: 17.3 GeV S+S
NA22: 22 GeV p+p
+pµEMC: 280 GeV
FIG. 12: The correlation of the clan model parameters ¯n
c
and
¯
N
c
for all of the collision species measured as a function
of centrality. Also shown are results from pseudorapidity-
dependent studies from elementary collisions (UA5 [17], EMC
[18], and NA22 [19]) and heavy ion collisions (E802 [21] and
NA35 [22]).
A feature that is especially apparent in the Au+Au
data is the fa c t that the scaled variance decreases with
increasing centrality, with the most central po int lying
below the participant superp osition model expectatatio n.
The clan model provides one possible explanation for
this effect whereby there is a hig her probability for con-
tributions from cluster sources s uch as jet production
in the lower multiplicity peripheral events. The cluster
sources introduce correlatio ns that can increase the value
of 1/k
NBD
and hence the value of the scaled variance of
the multiplicity distribution. Another possible explana-
tion for this fea tur e can be addressed with a string perco-
lation model in heavy ion collisions [43]. In general, per-
colation theory considers the formation of clusters within
a random spatial distribution of individual objects that
are a llowed to overlap with each other. T he clusters are
formed by the geometrical connection of one or more of
the individual objects. This can be applied to estimate
multiplicity fluctuations in heavy ion collisions whereby
the objects are the circular cross sections of strings in
the transverse plane [44] and the str ings form clusters of
overlapping strings that then each emit a number of par-
ticles related to the number of strings in each cluster. As
the centrality increases, the number of individual clus-
ters decreases along with the variance of the numb e r of
strings per cluster, which can result in a decrease in the
magnitude of the resulting multiplicity fluctuations. The
prediction of the scaled variance from the string percola-
tion mo del for 200 GeV Au+Au collisions scaled down to
the PHENIX acceptance in azimuth and pseudorapidity
[44] is shown in Fig. 11. Although percolation describes
the trend observed at the four highest centralities very
well, the scaled variance from the model continues to in-
crease well above the data as centrality decreases. T he
implementation of the HIJING model contains merging
of str ings that are in close spatial proximity, so percola-
tion can explain the trends in the scaled variance from
HIJING.
VI. SUMMARY
PHENIX has c ompleted a survey of multiplicity fluc-
tuations of charged hadr ons in Au+Au and Cu+Cu c ol-
lisions at three collision energies. The motivation fo r the
analysis is to search for signs of a phase transition or the
presence of the predicted critical point on the QCD phase
diagram by looking for increased multiplicity fluctuations
as a function o f system energy and system volume. After
correcting for non-dynamical fluctuations due to fluctu-
ations of the collision geometry within a centrality bin,
the multiplicity fluctuations in 200 GeV and 62.4 GeV
Au+Au collisions are consistent with or below the ex-
pectation from the superpo sition model of participant
nucleons. The multiplicity fluctuations decrease as the
collision centrality increases, dropping below the partic-
ipant superposition model expectation for the most cen-
tral Au+Au collis ions. Fluctuations from Cu+Cu colli-
sions exhibit a weaker centrality-dependence that also is
consistent with o r below the expectation from the partic-
ipant superposition model. The absence of large dynami-
cal fluctuations in ex c e ss of the participant superposition
model expectation indicate that there is no evidence of
critical behavior related to the compressibility observable
in this datas e t. There is also no significant evidence of
dynamical fluctuations that are dependent on the trans-
verse momentum or the charge of the particles measured.
As interpreted by the clan model, the observed fluctu-
ations demonstrate only weak clustering characteristics
for all of the heavy ion collision systems discus sed here.
The decreas ing scaled variance with increasing central-
ity may be explained by percolation phenomena, however
this fails to explain the most peripheral Au+Au data. Al-
though this analysis does not observe evidence of critical
behavior, it does not rule out the existence of a QCD crit-
ical point. Fur ther measurements will be possible during
the upcoming low energy scan prog ram at RHIC allowing
for a more comprehensive search fo r critical be havior.

16
Acknowledgments
We thank the staff of the Collider-Accelerator and
Physics Departments at Brookhaven National Lab ora-
tory and the staff of the other PHENIX participating
institutions for their vital contributions. We acknowl-
edge supp ort from the Office of Nuclear Physics in the
Office of Science of the Department of Energy, the Na-
tional Science Founda tio n, Abilene Christian University
Research Council, Research Foundation of SUNY, and
Dean of the College of Arts and Sciences, Vanderbilt Uni-
versity (U.S.A), Ministry of Education, Culture, Sports,
Science, and Technology and the Ja pan Society for the
Promotion of Science (Japan), Conselho Nacional de De-
senvolvimento Cient´ıfico e Tecnol´o gico and Funda¸c˜ao de
Amparo `a Pesquisa do Estado de S˜ao Paulo (Brazil),
Natural Science Foundation of China (People’s Repub-
lic of China), Ministry of Education, Youth and Sports
(Czech Republic), Centre National de la Recherche Sci-
entifique, Commissariat `a l’
´
Energie Atomique, and In-
stitut National de Physique Nucl´eaire et de Physique
des Particules (France), Ministry of Industry, Science
and Tekhnologies, Bundesministerium f¨ur Bildung und
Forschung, Deutscher Akademischer Austausch Dienst,
and Alexander von Humb oldt Stiftung (Germany), Hun-
garian National Science Fund, OTKA (Hungary), De-
partment of Atomic Energy (India), Israel Science Foun-
dation (Israel), Korea Research Foundation and Korea
Science and Engineering Foundation (Korea), Ministry
of Education a nd Science, Rassia Academy of Sciences,
Federal Agency of Atomic Energy (Russia), VR and the
Wa llenberg Foundation (Sweden), the U.S. Civilian Re-
search and Development Foundation for the Independent
States of the Former Soviet Union, the US-Hungar ian
NSF-OTKA-MTA, and the US-Israel Binational Science
Foundation.
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