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Charged-particle multiplicity fluctuations in Pb–Pb collisions at $$\sqrt{s_{\mathrm {NN}}}$$ = 2.76 TeV



Measurements of event-by-event fluctuations of charged-particle multiplicities in Pb–Pb collisions at $$\sqrt{s_{\mathrm {NN}}}$$ s NN $$=$$ = 2.76 TeV using the ALICE detector at the CERN Large Hadron Collider (LHC) are presented in the pseudorapidity range $$|\eta |<0.8$$ | η | < 0.8 and transverse momentum $$0.2< p_{\mathrm{T}} < 2.0$$ 0.2 < p T < 2.0 GeV/ c . The amplitude of the fluctuations is expressed in terms of the variance normalized by the mean of the multiplicity distribution. The $$\eta $$ η and $$p_{\mathrm{T}}$$ p T dependences of the fluctuations and their evolution with respect to collision centrality are investigated. The multiplicity fluctuations tend to decrease from peripheral to central collisions. The results are compared to those obtained from HIJING and AMPT Monte Carlo event generators as well as to experimental data at lower collision energies. Additionally, the measured multiplicity fluctuations are discussed in the context of the isothermal compressibility of the high-density strongly-interacting system formed in central Pb–Pb collisions.
Eur. Phys. J. C (2021) 81:1012
Regular Article - Experimental Physics
Charged-particle multiplicity fluctuations in Pb–Pb collisions at
sNN =2.76TeV
ALICE Collaboration
CERN, 1211 Geneva 23, Switzerland
Received: 27 May 2021 / Accepted: 29 October 2021
© CERN for the benefit of the ALICE collaboration 2021
Abstract Measurements of event-by-event fluctuations
of charged-particle multiplicities in Pb–Pb collisions at
sNN =2.76 TeV using the ALICE detector at the CERN
Large Hadron Collider (LHC) are presented in the pseudo-
rapidity range |η|<0.8 and transverse momentum 0.2<
pT<2.0GeV/c. The amplitude of the fluctuations is
expressed in terms of the variance normalized by the mean
of the multiplicity distribution. The ηand pTdependences of
the fluctuations and their evolution with respect to collision
centrality are investigated. The multiplicity fluctuations tend
to decrease from peripheral to central collisions. The results
are compared to those obtained from HIJING and AMPT
Monte Carlo event generators as well as to experimental data
at lower collision energies. Additionally, the measured multi-
plicity fluctuations are discussed in the context of the isother-
mal compressibility of the high-density strongly-interacting
system formed in central Pb–Pb collisions.
1 Introduction
According to quantum chromodynamics (QCD), at high tem-
peratures and high energy densities, nuclear matter under-
goes a phase transition to a deconfined state of quarks and
gluons, the quark–gluon plasma (QGP) [15]. Heavy-ion col-
lisions at ultra-relativistic energies make it possible to cre-
ate and study such strongly-interacting matter under extreme
conditions. The QGP formed in high-energy heavy-ion col-
lisions has been characterised as a strongly-coupled system
with very low shear viscosity. The primary goal of the heavy-
ion program at the CERN Large Hadron Collider (LHC) is to
study the QCD phase structure by measuring the properties
of QGP matter. One of the important methods for this study
is the measurement of event-by-event fluctuations of exper-
imental observables. These fluctuations are sensitive to the
proximity of the phase transition and thus provide informa-
tion on the nature and dynamics of the system formed in the
collisions [612]. Fluctuation measurements provide a pow-
erful tool to investigate the response of a system to exter-
nal perturbations. Theoretical developments suggest that it
is possible to extract quantities related to the thermody-
namic properties of the system, such as entropy, chemical
potential, viscosity, specific heat, and isothermal compress-
ibility [6,1321]. In particular, isothermal compressibility
expresses how a system’s volume responds to a change in
the applied pressure. In the case of heavy-ion collisions, it
has been shown that the isothermal compressibility can be
calculated from the event-by-event fluctuation of charged-
particle multiplicity distributions [17].
The measured multiplicity scales with the collision cen-
trality in heavy-ion collisions. The distribution of particle
multiplicities in a given class of centrality and its fluctuations
on an event-by-event basis provide information on particle
production mechanisms [2224]. In this work, the magni-
tude of the fluctuations is quantified in terms of the scaled
ωch =σ2
where Nchand σ2
ch denote the mean and variance of
the charged-particle multiplicity distribution, respectively.
Event-by-event multiplicity fluctuations in heavy-ion colli-
sions have been studied earlier at the BNL-AGS by E802
[25], the CERN-SPS by the WA98 [26], NA49 [27,28], and
CERES [29] experiments, and at the Relativistic Heavy Ion
Collider (RHIC) by the PHOBOS [30] and PHENIX [31]
experiments. A compilation of available experimental data
and comparison to predictions of the event generators are
presented elsewhere [19]. In this work, measurements of
the scaled variance of multiplicity fluctuations are presented
as a function of collision centrality in Pb–Pb collisions at
sNN =2.76 TeV using the ALICE detector at the LHC.
In thermodynamics, the isothermal compressibility (kT)
is defined as the fractional change in the volume of a system
with change of pressure at a constant temperature,
0123456789().: V,-vol 123
1012 Page 2 of 17 Eur. Phys. J. C (2021) 81:1012
where V,T,Pare the volume, temperature, and pressure of
the system, respectively. In general, an increase in the applied
pressure leads to a decrease in volume, so the negative sign
makes the value of kTpositive. In the context of a description
in terms of the grand canonical ensemble, which is approx-
imately applicable for the description of particle production
in heavy-ion collisions [5], the scaled variance of the multi-
plicity distribution can be expressed as [17],
ωch =kBTNch
where kBis the Boltzmann’s constant, and Nchis the aver-
age number of charged particles. Measurements of fluctua-
tions in terms of ωch can be exploited to determine kTand
associated thermodynamic quantities such as the speed of
sound within the system [17,32].
Measurements of the multiplicity of produced particles
in relativistic heavy-ion collisions are basic to most of the
studies as a majority of the experimentally observed quan-
tities are directly related to the multiplicity. The variation
of the multiplicity depends on the fluctuations in the colli-
sion impact parameter or the number of participant nucle-
ons. Thus, the measured multiplicity fluctuations contain
contributions from event-by-event fluctuations in the num-
ber of participant nucleons, which forms the main back-
ground towards the evaluation of any thermodynamic quan-
tity [33,34]. This has been partly addressed by selecting
narrow intervals in centrality and accounting for the mul-
tiplicity variation within the centrality of the measurement.
The remainder of participant fluctuations is estimated in the
context of an MC Glauber model in which nucleus–nucleus
collisions are considered to be a superposition of nucleon–
nucleon interactions.
Thus, the background fluctuations contain contributions
from independent particle production and correlations cor-
responding to different physical origins. The background-
subtracted fluctuations can be used in Eq. (3) to estimate
kTwith the knowledge of the temperature and volume from
complementary analyses of hadron yields, calculated at the
chemical freeze-out [35,36].
In addition to fluctuations in the number of participant
nucleons, several other processes contribute to fluctuations
of the charged particles multiplicity on an event-by-event
basis [17,37]. These include long-range particle correlations,
charge conservation, resonance production, radial flow, as
well as Bose–Einstein correlations. Since these contributions
can not be evaluated directly, the value of kTextracted and
reported in this work amounts to an upper limit.
The article is organized as follows. In Sect. 2, the experi-
mental setup and details of the data analysis method, includ-
ing event selection, centrality selection, corrections for finite
width of the centrality intervals, and particle losses are pre-
sented. In Sect. 3, the measurements of the variances of mul-
tiplicity distributions are presented as a function of collision
centrality. Additionally, the dependence of the fluctuations
on the ηand pTranges of the measured charged hadrons
are studied. The results are compared with calculations from
selected event generators. In Sect. 4, methods used to esti-
mate multiplicity fluctuations resulting from the fluctuations
of the number of participants are discussed. An estimation of
the isothermal compressibility for central collisions is made
in Sect. 5.
2 Experimental setup and analysis details
The ALICE experiment [38] is a multi-purpose detector
designed to measure and identify particles produced in
heavy-ion collisions at the LHC. The experiment consists of
several central barrel detectors positioned inside a solenoidal
magnet operated at 0.5 T field parallel to the beam direc-
tion and a set of detectors placed at forward rapidities. The
central barrel of the ALICE detector provides full azimuthal
coverage for track reconstruction within a pseudorapidity (η)
range of |η|<0.8. The Time Projection Chamber (TPC) is
the main tracking detector of the central barrel, consisting
of 159 pad rows grouped into 18 sectors that cover the full
azimuth. The Inner Tracking System (ITS) consists of six
layers of silicon detectors employing three different tech-
nologies. The two innermost layers are Silicon Pixel Detec-
tors (SPD), followed by two layers of Silicon Drift Detectors
(SDD), and finally, the two outermost layers are double-sided
Silicon Strip Detectors (SSD). The V0 detector consists of
two arrays of scintillators located on opposite sides of the
interaction point (IP). It features full azimuthal coverage in
the forward and backward rapidity ranges, 2.8<η<5.1
(V0A) and 3.7<η<1.7 (V0C). The V0 detectors
are used for event triggering purposes as well as to evalu-
ate the collision centrality on an event-by-event basis [39].
The impact of the detector response on the measurement of
charged-particle multiplicity based on Monte Carlo simula-
tions is studied with the GEANT3 framework [40].
This analysis is based on Pb–Pb collision data recorded
in 2010 at sNN =2.76 TeV with a minimum-bias trigger
comprising of a combination of hits in the V0 detector and
the two innermost (pixel) layers of the ITS. In total, 13.8 mil-
lion minimum-bias events satisfy the event selection criteria.
The primary interaction vertex of a collision is obtained by
extending correlated hits in the two SPD layers to the beam
axis. The longitudinal position of the interaction vertex in the
beam (z) direction (Vz) is restricted to |Vz|<10 cm to ensure
a uniform acceptance in the central ηregion. The interaction
vertex is also obtained from TPC tracks. The event selection
includes an additional vertex selection criterion, where the
difference between the vertex using TPC tracks and the ver-
tex using the SPD is less than 5 mm in the z-direction. This
Eur. Phys. J. C (2021) 81:1012 Page 3 of 17 1012
selection criterion greatly suppresses the contamination of
the primary tracks by secondary tracks resulting from weak
decays and spurious interactions of particles within the appa-
Charged particles are reconstructed using the combined
information of the TPC and ITS [38]. In the TPC, tracks are
reconstructed from a collection of space points (clusters). The
selected tracks are required to have at least 80 reconstructed
space points. Different combinations of tracks in the TPC and
SPD hits are utilized to correct for detector acceptances and
efficiency losses. To suppress contributions from secondary
tracks (i.e., charged particles produced by weak decays and
interactions of particles with materials of the detector), the
analysis is restricted to charged-particle tracks featuring a
distance of closest approach (DCA) to the interaction vertex,
DCAxy <2.4 cm in the transverse plane and of DCAz<
3.2 cm along the beam direction. The tracks are additionally
restricted to the kinematic range, |η|<0.8 and 0.2<pT<
2.1 Centrality selection and the effect of finite width of the
centrality intervals
The collision centrality is estimated based on the sum of
the amplitudes of the V0A and V0C signals (known as the
V0M collision centrality estimator) [39]. Events are classi-
fied in percentiles of the hadronic cross section using this
estimator. The average number of participants in a centrality
class, denoted by Npart , is obtained by comparing the V0M
multiplicity to a geometrical Glauber model [41]. Thus, the
centrality of the collision is measured based on the V0M
centrality estimator, whereas the measurement of multiplic-
ity fluctuations is based on charged particles measured within
the acceptance of the TPC.
A given centrality class is a collection of events of mea-
sured multiplicity distributions within a range in V0M cor-
responding to a mean number of participants, Npart.This
results in additional fluctuations in the number of particles
within each centrality class. To account for these fluctuations,
a centrality interval width correction is employed. The pro-
cedure involves dividing a broad centrality class into several
narrow intervals and correcting for the finite interval using
weighted moments according to [42,43],
Here, the index iruns over the narrow centrality intervals.
Xiand niare the corresponding moments of the distribution
and number of events in the ith interval, respectively. With
this, one obtains, N=inias the total number of events in
the broad centrality interval.
The centrality resolution of the combined V0A and V0C
signals ranges from 0.5% in central to 2% in the most periph-
eral collisions [39]. A correction for the finite width of cen-
trality intervals has been made with Eq. 4using 0.5% central-
ity intervals from central to 40% cross-section and 1% inter-
vals for the rest of the centrality classes.
2.2 Efficiency correction
The detector efficiency factors (ε) were evaluated in bins of
pseudorapidity η, azimuthal angle ϕ, and pT. By defining
Nch(x)as the number of produced particles in a phase-space
bin at x,n(x)as the number of observed particles at x, and
ε(x)as the detection efficiency, the first and second facto-
rial moments of the multiplicity distributions can be cor-
rected for particle losses according to the procedure outlined
in Refs. [44,45]:
respectively. Here, mdenotes the index of the phase-space
bins and i,jare the bin indexes. δxixj=1ifxi=xjand zero
otherwise. The variance of the charged-particle multiplicity
is then calculated as:
ch =F2+F1F2
The correction procedure is validated by a Monte Carlo
study employing two million Pb–Pb events at sNN =2.76 TeV
generated using the HIJING event generator [46], and passed
through GEANT3 simulations of the experimental setup, tak-
ing care of the acceptances of the detectors. The efficiency
dependencies on η,ϕ, and pTare calculated from the ratio of
the number of reconstructed charged particles by the num-
ber of produced particles. In order to account for the pT
dependence of efficiency, the full pTrange (0.2<pT<
2.0GeV/c) was divided to nine bins (0.2–0.3, 0.3–0.4, 0.4–
0.5, 0.5–0.6, 0.6–0.8, 0.8–1.0, 1.0–1.2, 1.2–1.6, 1.6–2.0) with
larger number of bins in low pTranges. In the Monte Carlo
closure test, the values of Nch,σch, and ωch of the efficiency
corrected results from the simulated events are compared to
those of HIJING at the generator level to obtain the correc-
tions. By construction, the efficiency corrected values for
Nchmatch with those from the generator, whereas σch and
ωch values differ by 0.7 and 1.4%, respectively. These
differences are included in the systematic uncertainties.
2.3 Statistical and systematic uncertainties
The statistical uncertainties of the moments of multiplicity
distributions are calculated based on the method of error
1012 Page 4 of 17 Eur. Phys. J. C (2021) 81:1012
Table 1 Systematic uncertainties on the mean, standard deviation, and
scaled variance of charged-particle multiplicity distributions from dif-
ferent sources. The ranges of uncertainties quoted correspond to central
to peripheral collisions
Source Nchσch ωch
Track selection 3.5–4.8% 3.8–6.0% 4.0–7.5%
Vari ati on of DC Axy 0.5–0.9% 0.8–1.2% 1.3–1.6%
Vari ati on of DC Az0.4–0.9% 0.7–1.0% 1.2–1.7%
Ve rt e x ( Vz) selection 0.1–0.5% 0.5% 0.1–0.8%
Removal of Vx,Vyselections 0.1% 0.2% 0.5%
Efficiency correction <0.1% 0.7% 1.4%
Magnetic polarity 0.2–1.0% 0.5–1.5% 0.8–1.7%
Total 3.5–5.1% 4.1–6.4% 4.8–8.3%
propagation derived from the delta theorem [47]. The sys-
tematic uncertainties have been evaluated by considering the
effects of various criteria in track selection, vertex determi-
nation, and efficiency corrections.
The systematic uncertainties related to the track selec-
tion criteria were obtained by varying the track reconstruc-
tion method and track quality cuts. The nominal analysis
was carried out with charged particles reconstructed within
the TPC and ITS. For systematic checks, the full analysis is
repeated for tracks reconstructed using only the TPC infor-
mation. The differences in the values of Nch,σch, and ωch
resulting from the track selections using the two methods
are listed in Table 1as a part of the systematic uncertainties.
The DC Axy and DC Azof the tracks are varied by ±25%
to obtain the systematic uncertainties caused by variations
in the track quality selections. The effect of the selection
of events based on the vertex position is studied by restrict-
ing the z-position of the vertex to ±5 cm from the nominal
±10 cm, and additionally by removing restrictions on Vxand
Vy. The efficiency correction introduces additional system-
atic uncertainty as discussed earlier. The experimental data
were recorded for two different magnetic field polarities. The
two data sets are analyzed separately and the differences are
taken as a source of systematic uncertainties.
The individual sources of systematic uncertainties dis-
cussed above are considered uncorrelated and summed in
quadrature to obtain the total systematic errors reported in
this work. Table 1lists the systematic uncertainties associ-
ated with the values of Nch,σch, and ωch.
3 Results and discussions
Figure 1shows the corrected mean (Nch), standard devia-
tion (σch), and scaled variance (ωch) as a function of Npart
for the centrality range considered (0-60%) corresponding
to Npart >45. Uncertainties on the estimated number of
Fig. 1 Mean (Nch), standard deviation (σch ), and scaled variance
(ωch) of charged-particle multiplicity distributions as a function of the
number of participating nucleons for experimental data along with
HIJING and AMPT (string melting) models for Pb–Pb collisions at
sNN =2.76 TeV, shown in panels a,b,andc, respectively. For panel
a,Npartfor the two models are shifted for better visibility. The statisti-
cal uncertainties are smaller than the size of the markers. The systematic
uncertainties are presented as filled boxes
participants, Npart, obtained from Ref. [38], are smaller
than the width of the solid red circles used to present the
data in the centrality range considered in this measurement.
It is observed that the values of Nchand σch increase with
increasing Npart.Thevalueofωch decreases monotonically
by 29% from peripheral to central collisions.
3.1 Comparison with models
The measured ωch values are compared with the results
of simulations with the HIJING and the string melting
option of the AMPT models. HIJING [46] is a Monte Carlo
event generator for parton and particle production in high-
energy hadronic and nuclear collisions and is based on QCD-
inspired models which incorporate mechanisms such as mul-
tiple minijet production, soft excitation, nuclear shadowing
of parton distribution functions, and jet interactions in the
Eur. Phys. J. C (2021) 81:1012 Page 5 of 17 1012
Fig. 2 Scaled variances of charged-particle multiplicity distributions
for different ηand pTranges as a function of number of participating
nucleons measured in Pb–Pb collisions at sNN =2.76 TeV, shown
in panels a,andb, respectively. The estimated ωch for |η|<0.3and
|η|<0.5 are obtained from the experimental data of |η|<0.8byusing
Eq. 8. The estimated ωch for 0.2<pT<1.5GeV/cand 0.2<pT<
1.0GeV/care obtained from the experimental data of 0.2<pT<
2.0GeV/c, also by using Eq. 8. The statistical uncertainties are smaller
than the size of the markers. The systematic uncertainties are presented
as filled boxes
dense hadronic matter. The HIJING model treats a nucleus-
nucleus collision as a superposition of many binary nucleon-
nucleon collisions. In the AMPT model [48], the initial parton
momentum distribution is generated from the HIJING model.
In the default mode of AMPT, energetic partons recombine
and hadrons are produced via string fragmentation. The string
melting mode of the model includes a fully partonic phase
that hadronises through quark coalescence.
In order to enable a proper comparison with data obtained
in this work, Monte Carlo events produced with HIJING and
AMPT are grouped in collision centrality classes based on
generator level charged-particle multiplicities computed in
the ranges 2.8<η<5.1 and 3.7<η<1.7, corre-
sponding to the V0A and V0C pseudorapidity coverages. The
results of the scaled variances from the two event generators
are presented in Fig. 1as a function of the estimated number
of participants, Npart . As a function of increasing central-
ity, the ωch values obtained from the event generators show
upward trends, which are opposite to those of the experimen-
tal data. It is to be noted that the Monte Carlo event generators
are successful in reproducing the mean of multiplicity distri-
butions. This follows from the fact that the particle multiplic-
ities are proportional to the cross sections. On the other hand,
the widths of the distributions originate from fluctuations and
correlations associated with effects of different origins, such
as long-range correlations, Bose–Einstein correlations, reso-
nance decays, and charge conservation. Because of this, the
event generators fall short of reproducing the observed scaled
3.2 Scaled variance dependence on pseudorapidity
acceptance and pTrange
Charged-particle multiplicity distributions depend on the
acceptance of the detection region. Starting with the mea-
sured multiplicity fluctuations within |η|<0.8 and 0.2<
pT<2.0GeV/cwith a mean Nchand scaled variance of
ωch, the scaled variance (ωacc
ch ) for a fractional acceptance
in ηor for a limited pTrange with mean of Nacc
ch can be
expressed as [31],
ch =1+faccch 1), (8)
where facc =Nacc
This empirical estimation for the acceptance dependence
of the scaled variance is valid assuming that there are no
significant correlations present over the acceptance range
being studied. The validity of this dependence has been
checked by comparing the experimental data of scaled vari-
ances at reduced acceptances along with the results from the
above calculations. This is shown in Fig. 2for different ηor
pTranges. In the top panel, the scaled variances are shown,
as a function of Npart,forthreeηranges. The solid sym-
bols show the results of measured scaled variances, whereas
open symbols show the estimated values for the two reduced
ηwindows. The calculated values yield a good description
of the measured data points. The choice of the pTrange
also affects the multiplicity of an event. In the bottom panel
of Fig. 2, the scaled variances are shown, as a function of
Npart,forthree pTranges keeping |η|<0.8. A decrease in
the value of ωch is observed with the decrease of the pTwin-
dow. The results from the calculations of scaled variances are
compared to the measured data points. The calculated values
are close to those of the measurement. This estimation of the
scaled variances of multiplicity distributions is particularly
useful in extrapolating fluctuations to different coverages.
1012 Page 6 of 17 Eur. Phys. J. C (2021) 81:1012
Fig. 3 Comparison of Nch,ωch ,andσ2
ch/Nch 2measured in this
work based on the acceptance of the PHENIX experiment with results
reportedbyPHENIX[31] as a function of number of participating nucle-
ons, shown in panels a,b,andc, respectively. The statistical uncertain-
ties are smaller than the size of the markers. The systematic uncertainties
are presented as filled boxes
3.3 Comparison to scaled variances at lower collision
Scaled variances of charged-particle multiplicity distribu-
tions were earlier reported by the PHENIX Collaboration
at RHIC for Au–Au collisions at sNN = 62.4 and 200 GeV
[31]. The beam–beam counters (BBC) in PHENIX cover-
ing the full azimuthal angle in the pseudorapidity range
3.0<|η|<3.9 provided the minimum-bias trigger and
were used for centrality selection. The pseudorapidity accep-
tance of the PHENIX experiment amounted to |η|<0.26
with an effective average azimuthal active area of 2.1 radian
and 0.2<pT<2.0GeV/cfor charged particle measure-
ments. The published results of mean and scaled variances of
charged-particles were corrected for fluctuations of the colli-
sion geometry within a centrality bin. This was performed by
comparing fluctuations from simulated HIJING events with
a fixed impact parameter to events with a range of impact
parameters covering the width of the centrality bin, as deter-
mined from Glauber model simulations. The corrected results
are reproduced in Fig. 3for the two collision energies. To
enable an appropriate comparison with results reported by
PHENIX, the ALICE data are reanalyzed by imposing the
same kinematic ranges as in PHENIX, and the resulting mean
and scaled variances are presented in Fig. 3. It is observed
that for the same acceptance and kinematic cuts, the mean
values and the scaled variances are larger at the LHC energy
compared to those obtained at RHIC energies.
It is also of interest to study σ2
Nch2, the ratio of the vari-
ance by the square of the average multiplicity as a function
of collision centrality. At lower beam energies, these distri-
butions obey a power-law relative to the number of partici-
pants [49]. In the lower panel of Fig. 3, the values of σ2
are presented as a function of Npartfor the ALICE data,
for the common coverage of ALICE and PHENIX data, as
well as PHENIX data at two collision energies. The data
points are fitted by a scaling curve, σ2
exponent α=−1.25 ±0.03 fits the four sets of experi-
mental data well with χ2/ndf (where ndf is the number
of degrees of freedom) as 0.88, 1.1, 0.95, 0.84 for Pb–Pb
collisions at sNN =2.76 TeV with the ALICE accep-
tance, the PHENIX detector acceptance and Au–Au colli-
sions at sNN =200 GeV and 62.4 GeV, respectively. The
scaling, first described by the PHENIX Collaboration [49],
also holds for the ALICE data. The corresponding values of
Nch2for HIJING and AMPT models for Pb–Pb collisions
at sNN =2.76 TeV are also displayed in Fig. 3. The trends
as a function of centrality are observed to be similar to those
of the experimental data. Fits with a similar scaling curve
yield power-law exponents as 1.1 and 1.05 for HIJING
and AMPT models, respectively. These exponents for the
models are lower compared those of the experimental data.
4 Background to the measured multiplicity fluctuations
The background to the measured multiplicity fluctuations
contains contributions from several sources. In this section,
the background fluctuations are presented first from a par-
Eur. Phys. J. C (2021) 81:1012 Page 7 of 17 1012
Fig. 4 Scaled variance as a function of Npart for charged-particle
multiplicity distributions and background fluctuations (ωback
ch ) based on
a participant model calculation for |η|<0.5. The expectation from
Poisson-like particle production is indicated by the dotted line. The
statistical uncertainties are smaller than the size of the markers. The
systematic uncertainties are presented as filled boxes
ticipant model calculation and then the expectations from a
Poisson distribution of particle multiplicity are discussed.
In the wounded nucleon model, nucleus–nucleus (such
as Pb–Pb) collisions are considered to be a superposition of
individual nucleon–nucleon interactions. In this context, the
fluctuations in multiplicity within a given centrality window
arise in part from fluctuations in Npart and from fluctuations
in the number of particles (n) produced by each nucleon–
nucleon interaction [22,26,31]. The values of nand their
fluctuations are also strongly dependent on the acceptance of
the detector. Within the context of this framework, the scaled
variance of the background, ωback
ch , amounts to
ch =ωn+nωNpart ,(10)
where nis the average number of particles produced by
each nucleon–source within the detector acceptance, ωnis the
scaled variance of the fluctuations in n, and ωNpart denotes the
fluctuations in Npart . The variance, ωNpart is calculated using
event-by-event Npart from the HIJING model. The distribu-
tion of Npart corresponds to the centrality obtained within
the V0 detector coverage (2.8<η<5.1 and 3.7<η<
1.7). The extracted values of ωNpart are corrected for the
effects of the finite width of the centrality intervals.
For the central rapidity range (|η|<0.5), the measured
number of charged particles produced in pp collisions within
0.2<pT<2.0GeV/cat s=2.76 TeV [50] yields n=
1.45 ±0.07, which is half of the measured value. In order to
calculate ωn, an extrapolation of the measured ωch is made
to Npart =2 using a polynomial fit function of the form
a+bx +cx2+dx3, which is shown in Fig. 4. In order to
calculate ωn, an extrapolation of the measured ωch is made
to Npart =2 using a polynomial fit function of the form
a+bx +cx2+dx3, which is shown in Fig. 4. Since both
the nucleon sources contributing to Npart =2 are correlated,
ωnbecomes half of the extrapolated value, yielding ωn=
1.445 ±0.12. This result is also consistent with the value of
ωn=1.51 ±0.16 obtained from the parameterization given
by the PHENIX Collaboration [31].
Using the above numbers, ωback
ch are calculated and plotted
as a function of Npartas in Fig. 4. The obtained trend in
ch mainly arises from the centrality dependence of ωNpart .
For most central collisions, the difference between the mea-
sured and background ωch is 0.02±0.18, which is consistent
with zero within the uncertainties. Except for most central
collisions, ωback
ch is observed to be larger than ωch. Thus, it
seems likely that the background estimated in this way from
the participant model is overestimated.
For an ideal gas, the number fluctuations are described
by the Poisson distribution. So, if the emitted particles are
uncorrelated, then the multiplicity distributions become Pois-
sonian, the magnitude of ωch reduces to unity, which is inde-
pendent of the multiplicity and thus independent of the cen-
trality of the collision. As seen from Fig. 4, the observed
multiplicity fluctuations are significantly above the Poisson
expectation for all centralities.
5 Estimation of isothermal compressibility
Equation (3) relates the magnitude of the charged-particle
multiplicity fluctuations to the isothermal compressibility.
The calculation of kTrequires knowledge of the temperature
and volume of the system. After the collision, as the sys-
tem cools down, the hadronic yields are fixed when the rate
of inelastic collisions becomes negligible (chemical freeze-
out), but the transverse-momentum distributions continue to
change until elastic interactions also cease (kinetic freeze-
out). The number of charged particles gets fixed at the time
of chemical freeze-out (except for long-lived resonances).
As the calculation of kTdepends on the fluctuations in the
number of particles, the chemical freeze-out conditions are
considered as input. The ALICE Collaboration has published
the identified particle yields of pions, kaons, protons, light
nuclei, and resonances [36,51,52]. The statistical hadroniza-
tion models have been successful in describing these yields
and their ratios [5,35,53], using temperature and volume as
parameters at the chemical freeze-out. For most central Pb–
Pb collisions at sNN =2.76 TeV, the ALICE data on yields
of particles in one unit of rapidity at mid-rapidity are in good
agreement with 0.156±0.002 GeV and 5330±505 fm3,for
temperature and volume, respectively [52]. In addition, the
charged-particle multiplicity within |η|<0.5 in this central-
ity range is Nch=1410 ±47 (syst).
1012 Page 8 of 17 Eur. Phys. J. C (2021) 81:1012
Here, an attempt is made to estimate kTfor Pb–Pb col-
lisions using the charged-particle multiplicity fluctuations
along with the temperature, volume, and mean number of
charged particles from above. The measured multiplicity
fluctuation for central collisions is ωch =2.15 ±0.1. In
the absence of any background where the full fluctuation
is attributed to have a thermal origin, one would obtain
kT=52.1±5.81 fm3/GeV. As the measured ωch contains
background fluctuations from different sources, this value of
kTcan be only be considered as an absolute upper limit.
In the previous section, the background fluctuations have
been estimated from the participant model calculation as
shown in Fig. 4. For central collisions, the value of the mea-
sured fluctuation above that of the participant model fluctu-
ation is ωch =0.02 ±0.18. This leads to kT=0.48 ±4.32
fm3/GeV. On the other hand, the background fluctuations
from the participant model for other centralities are larger
compared to the measured ones making the background-
subtracted fluctuations negative. So it is not possible to obtain
estimates of kTfor these centrality ranges based on the
present model of participant fluctuations.
The measured multiplicity fluctuations can be viewed as
combinations of correlated and uncorrelated fluctuations. If
the particle production is completely uncorrelated, the system
effectively behaves as an ideal gas, and the multiplicity distri-
bution is expected to follow a Poisson distribution (ωch =1).
For central collisions, fluctuations above the Poisson estima-
tion gives, ωch =1.15 ±0.06, which in turn implies a value
of kT=27.9±3.18 fm3/GeV.
It may be noted that other sources likely also contribute
to the background of the measured multiplicity fluctuations.
A quantitative determination of these effects requires further
studies and theoretical modeling, which is beyond the scope
of this work. In view of this, the estimation of kTfrom the
background-subtracted event-by-event multiplicity fluctua-
tion provides an upper limit of its value.
It is imperative to put the extracted values of kTin per-
spective with respect to that of normal nuclear matter. The
incompressibility constant of normal nuclear matter at pres-
sure P, expressed as K0=9(∂ P/∂ρ) at zero temperature
and normal nuclear density, ρ=ρ0, has been determined
to be K0=240 ±20 MeV [5456]. Using the relation,
kT=(9K0), one obtains the isothermal compressibility
of nuclear matter to be kT234±20 fm3/GeV. This is con-
sistent with the expectation that normal nuclear matter at low
temperature is more compressible than the high temperature
matter produced at LHC energies (as of Eq. 3). From the
above estimation, the value of kT=27.9±3.18 fm3/GeV,
which corresponds to multiplicity fluctuations above the
Poisson expectation, serves as a conservative upper limit,
and is even significantly below the normal nuclear matter at
low temperature.
6 Summary
Measurements of event-by-event fluctuations of charged-
particle multiplicities are reported as a function of centrality
in Pb–Pb collisions at sNN =2.76 TeV. The mean, standard
deviation, and scaled variances of charged-particle multiplic-
ities are presented for |η|<0.8 and 0.2<pT<2.0GeV/c
as a function of centrality. A monotonically decreasing trend
for the scaled variance is observed from peripheral to central
collisions. Corresponding results from HIJING and AMPT
event generators show a mismatch with the experimental
results. The scaled variance of the multiplicity decreases with
the reduction of the ηacceptance of the detector as well as
with the decrease of the pTrange. The multiplicity fluctua-
tions are compared to the results from lower beam energies
as reported by the PHENIX experiment. For the same accep-
tance, the observed scaled variances at RHIC energies are
smaller compared to those observed at the LHC.
As multiplicity fluctuations are related to the isother-
mal compressibility of the system, the measured fluctua-
tions are used to estimate kTin central Pb–Pb collisions at
sNN =2.76 TeV. The multiplicity fluctuations above the
Poisson expectation case yields kT=27.9±3.18 fm3/GeV,
which may still contain contributions from additional uncor-
related particle production as well as from several non-
thermal sources as discussed in Sect. 5. Proper modeling of
background subtraction needs to be developed by accounting
for all possible contributions from different physics origins,
which is beyond the scope of the present work. This result
serves as a conservative upper limit of kTuntil various con-
tributions to the background are properly understood and
evaluated. The estimation of kTat lower collision energies
and for different system-sizes is an interesting way to explore
the QCD phase diagram from thermodynamics point of view.
Acknowledgements The ALICE Collaboration would like to thank
all its engineers and technicians for their invaluable contributions to
the construction of the experiment and the CERN accelerator teams
for the outstanding performance of the LHC complex. The ALICE
Collaboration gratefully acknowledges the resources and support pro-
vided by all Grid centres and the Worldwide LHC Computing Grid
(WLCG) collaboration. The ALICE Collaboration acknowledges the
following funding agencies for their support in building and running
the ALICE detector: A. I. Alikhanyan National Science Laboratory
(Yerevan Physics Institute) Foundation (ANSL), State Committee of
Science and World Federation of Scientists (WFS), Armenia; Aus-
trian Academy of Sciences, Austrian Science Fund (FWF): [M 2467-
N36] and Nationalstiftung für Forschung, Technologie und Entwick-
lung, Austria; Ministry of Communications and High Technologies,
National Nuclear Research Center, Azerbaijan; Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq), Financiadora de
Estudos e Projetos (Finep), Fundação de Amparo à Pesquisa do Estado
de São Paulo (FAPESP) and Universidade Federal do Rio Grande do
Sul (UFRGS), Brazil; Ministry of Education of China (MOEC) , Min-
istry of Science & Technology of China (MSTC) and National Natural
Science Foundation of China (NSFC), China; Ministry of Science and
Education and Croatian Science Foundation, Croatia; Centro de Apli-
Eur. Phys. J. C (2021) 81:1012 Page 9 of 17 1012
caciones Tecnológicas y Desarrollo Nuclear (CEADEN), Cubaenergía,
Cuba; Ministry of Education, Youth and Sports of the Czech Republic,
Czech Republic; The Danish Council for Independent Research | Nat-
ural Sciences, the VILLUM FONDEN and Danish National Research
Foundation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Fin-
land; Commissariat à l’Energie Atomique (CEA) and Institut National
de Physique Nucléaire et de Physique des Particules (IN2P3) and Centre
National de la Recherche Scientifique (CNRS), France; Bundesminis-
terium für Bildung und Forschung (BMBF) and GSI Helmholtzzen-
trum für Schwerionenforschung GmbH, Germany; General Secretariat
for Research and Technology, Ministry of Education, Research and
Religions, Greece; National Research, Development and Innovation
Office, Hungary; Department of Atomic Energy Government of India
(DAE), Department of Science and Technology, Government of India
(DST), University Grants Commission, Government of India (UGC)
and Council of Scientific and Industrial Research (CSIR), India; Indone-
sian Institute of Science, Indonesia; Istituto Nazionale di Fisica Nucle-
are (INFN), Italy; Institute for Innovative Science and Technology,
Nagasaki Institute of Applied Science (IIST), Japanese Ministry of Edu-
cation, Culture, Sports, Science and Technology (MEXT) and Japan
Society for the Promotion of Science (JSPS) KAKENHI, Japan; Con-
sejo Nacional de Ciencia (CONACYT) y Tecnología, through Fondo
de Cooperación Internacional en Ciencia y Tecnología (FONCICYT)
and Dirección General de Asuntos del Personal Academico (DGAPA),
Mexico; Nederlandse Organisatie voor Wetenschappelijk Onderzoek
(NWO), Netherlands; The Research Council of Norway,Norway; Com-
mission on Science and Technology for Sustainable Development in the
South (COMSATS), Pakistan; Pontificia Universidad Católica del Perú,
Peru; Ministry of Education and Science, National Science Centre and
WUT ID-UB, Poland; Korea Institute of Science and Technology Infor-
mation and National Research Foundation of Korea (NRF), Republic
of Korea; Ministry of Education and Scientific Research, Institute of
Atomic Physics and Ministry of Research and Innovation and Insti-
tute of Atomic Physics, Romania; Joint Institute for Nuclear Research
(JINR), Ministry of Education and Science of the Russian Federation,
National Research Centre Kurchatov Institute, Russian Science Foun-
dation and Russian Foundation for Basic Research, Russia; Ministry of
Education, Science, Research and Sport of the Slovak Republic, Slo-
vakia; National Research Foundation of South Africa, South Africa;
Swedish Research Council (VR) and Knut & Alice Wallenberg Foun-
dation (KAW), Sweden; European Organization for Nuclear Research,
Switzerland; Suranaree University of Technology (SUT), National Sci-
ence and Technology Development Agency (NSDTA) and Office of the
Higher Education Commission under NRU project of Thailand, Thai-
land; Turkish Energy, Nuclear and Mineral Research Agency (TEN-
MAK), Turkey; National Academy of Sciences of Ukraine, Ukraine;
Science and Technology Facilities Council (STFC), United Kingdom;
National Science Foundation of the United States of America (NSF) and
United States Department of Energy, Office of Nuclear Physics (DOE
NP), United States of America.
Data Availability Statement This manuscript has no associated data
or the data will not be deposited. [Authors’ comment: Manuscript has
associated data in a HEPData repository at]
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L. Bratrud70, P. Braun-Munzinger110, M. Bregant123,M.Broz
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23, F. Catalano31 ,
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