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Eur. Phys. J. C (2021) 81:1012

https://doi.org/10.1140/epjc/s10052-021-09784-4

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) [1–5]. 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 [6–12]. Fluctuation measurements provide a pow-

e-mail: alice-publications@cern.ch

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 [6,13–21]. 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 [22–24]. In this work, the magni-

tude of the ﬂuctuations is quantiﬁed in terms of the scaled

variance,

ωch =σ2

ch

Nch,(1)

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,

kT=−1

V∂V

∂P

T

,(2)

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

VkT,(3)

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

123

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-

ratus.

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<

2.0GeV/c.

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],

X=iniXi

ini.(4)

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]:

F1=Nch=

m

i=1Nch(xi)=

m

i=1

n(xi)

ε(xi),(5)

and

F2=

m

i=1

m

j=i

n(xi)(n(xj)−δxixj)

ε(xi)ε(xj),(6)

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:

σ2

ch =F2+F1−F2

1.(7)

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

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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

(a)

(b)

(c)

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

123

Eur. Phys. J. C (2021) 81:1012 Page 5 of 17 1012

(a)

(b)

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

variances.

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],

ωacc

ch =1+facc(ωch −1), (8)

where facc =Nacc

ch

Nch.(9)

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.

123

1012 Page 6 of 17 Eur. Phys. J. C (2021) 81:1012

(a)

(b)

(c)

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

energies

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

ch

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

ch

Nch2

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

ch

Nch2=A·Nα

part.The

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

σ2

ch

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-

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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

ωback

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

ωback

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).

123

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 [54–56]. Using the relation,

kT=(9/ρ K0), 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-

123

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 https://www.hepdata.net/.]

Open Access This article is licensed under a Creative Commons Attri-

bution 4.0 International License, which permits use, sharing, adaptation,

distribution and reproduction in any medium or format, as long as you

give appropriate credit to the original author(s) and the source, pro-

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41,

D. Aleksandrov91, B. Alessandro61, H. M. Alfanda7, R. Alfaro Molina73,B.Ali

16,Y.Ali

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N. Alizadehvandchali127, A. Alkin35,J.Alme

21,T.Alt

70, L. Altenkamper21 , I. Altsybeev115, M. N. Anaam7, C. Andrei49,

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P. P. Bhaduri143, A. Bhasin104, I. R. Bhat104 , M.A.Bhat

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