2021 89 11 May

ALICE Collaboration††thanks: See Appendix A for the list of collaboration
members ALICE Collaboration

Measurements of event-by-event fluctuations of charged-particle multiplicities
in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ $=$ 2.76 TeV using the ALICE
detector at the CERN Large Hadron Collider (LHC) are presented in the
pseudorapidity range $|\eta|<0.8$ and transverse momentum $0.2<p_{\rm T}<2.0$
GeV/$c$. The amplitude of the fluctuations is expressed in terms of the
variance normalized by the mean of the multiplicity distribution. The $\eta$
and $p_{\rm T}$ 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.

##  1 Introduction

According to quantum chromodynamics (QCD), at high temperatures and high
energy densities, nuclear matter undergoes a phase transition to a deconfined
state of quarks and gluons, the quark–gluon plasma (QGP) [1, 2, 3, 4, 5].
Heavy-ion collisions at ultra-relativistic energies make it possible to create
and study such strongly-interacting matter under extreme conditions. The QGP
formed in high-energy heavy-ion collisions 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 experimental observables. These fluctuations are sensitive to
the proximity of the phase transition and thus provide information on the
nature and dynamics of the system formed in the collisions [6, 7, 8, 9, 10,
11, 12]. Fluctuation measurements provide a powerful tool to investigate the
response of a system to external perturbations. Theoretical developments
suggest that it is possible to extract quantities related to the thermodynamic
properties of the system, such as entropy, chemical potential, viscosity,
specific heat, and isothermal compressibility [13, 14, 15, 6, 16, 17, 18, 19,
20, 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 centrality 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, 23, 24]. In this work, the magnitude of
the fluctuations is quantified in terms of the scaled variance,

$\omega_{\rm ch}=\frac{\sigma_{\rm ch}^{\rm 2}}{\langle N_{\rm ch}\rangle},$
(1)

where $\langle N_{\rm ch}\rangle$ and $\sigma_{\rm ch}^{2}$ denote the mean
and variance of the charged-particle multiplicity distribution, respectively.
Event-by-event multiplicity fluctuations in heavy-ion collisions 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 $\sqrt{s_{\mathrm{NN}}}$ $=$ 2.76
TeV using the ALICE detector at the LHC.

In thermodynamics, the isothermal compressibility ($k_{T}$) is defined as the
fractional change in the volume of a system with change of pressure at a
constant temperature,

$k_{T}=-\frac{1}{V}\left.\left(\frac{\partial V}{\partial
P}\right)\right|_{T},$ (2)

where $V,T,P$ are 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 $k_{T}$ positive.
In the context of a description in terms of the grand canonical ensemble,
which is approximately applicable for the description of particle production
in heavy-ion collisions [5], the scaled variance of the multiplicity
distribution can be expressed as [17],

$\omega_{\rm ch}=\frac{k_{\rm B}T\langle N_{\rm ch}\rangle}{V}k_{T},$ (3)

where $k_{\rm B}$ is the Boltzmann’s constant, and $\langle N_{\rm ch}\rangle$
is the average number of charged particles. Measurements of fluctuations in
terms of $\omega_{\rm ch}$ can be exploited to determine $k_{T}$ and
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 quantities are directly related to the multiplicity.
The variation of the multiplicity depends on the fluctuations in the collision
impact parameter or the number of participant nucleons. Thus, the measured
multiplicity fluctuations contain contributions from event-by-event
fluctuations in the number of participant nucleons, which forms the main
background towards the evaluation of any thermodynamic quantity [33, 34]. This
has been partly addressed by selecting narrow intervals in centrality and
accounting for the multiplicity 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 corresponding to different physical
origins. The background-subtracted fluctuations can be used in Eq. (3) to
estimate $k_{T}$ with 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 $k_{T}$ extracted and reported in this work
amounts to an upper limit.

The article is organized as follows. In section II, the experimental setup and
details of the data analysis method, including event selection, centrality
selection, corrections for finite width of the centrality intervals, and
particle losses are presented. In section III, the measurements of the
variances of multiplicity distributions are presented as a function of
collision centrality. Additionally, the dependence of the fluctuations on the
$\eta$ and $p_{\rm T}$ ranges of the measured charged hadrons are studied. The
results are compared with calculations from selected event generators. In
section IV, methods used to estimate 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 section V.

##  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 direction 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 ($\eta$) range of $|\eta|<0.8$. 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 technologies. The two innermost layers are Silicon Pixel
Detectors (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, $2.8<\eta<5.1$ (V0A) and
$-3.7<\eta<-1.7$ (V0C). The V0 detectors are used for event triggering
purposes as well as to evaluate 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 simulations is studied with the
GEANT3 framework [40].

This analysis is based on Pb–Pb collision data recorded in 2010 at
$\sqrt{s_{\mathrm{NN}}}$ $=$ 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 million 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 ($V_{\rm z}$) is restricted to $|V_{\rm z}|<10$ cm to ensure a
uniform acceptance in the central $\eta$ region. 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 vertex using the SPD is less than 5 mm in the $z$-direction.
This selection criterion greatly suppresses the contamination of the primary
tracks by secondary tracks resulting from weak decays and spurious
interactions of particles within the apparatus.

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,
DCA${}_{\rm xy}<2.4$ cm in the transverse plane and of DCA${}_{\rm z}<3.2$ cm
along the beam direction. The tracks are additionally restricted to the
kinematic range, $|\eta|<0.8$ and $0.2<p_{\rm T}<2.0$ GeV/$c$.

###  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 classified in percentiles of the hadronic cross section using
this estimator. The average number of participants in a centrality class,
denoted by $N_{\rm part}$, 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
multiplicity fluctuations is based on charged particles measured within the
acceptance of the TPC.

A given centrality class is a collection of events of measured multiplicity
distributions within a range in V0M corresponding to a mean number of
participants, $\langle N_{\rm part}\rangle$. 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 procedure involves dividing a broad centrality class into
several narrow intervals and correcting for the finite interval using weighted
moments according to [42, 43],

$X=\frac{\sum_{\rm i}n_{\rm i}X_{\rm i}}{\sum_{\rm i}n_{\rm i}}.$ (4)

Here, the index $i$ runs over the narrow centrality intervals. $X_{\rm i}$ and
$n_{\rm i}$ are the corresponding moments of the distribution and number of
events in the $i^{\rm th}$ interval, respectively. With this, one obtains,
$N=\sum_{\rm i}n_{\rm i}$ as 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 peripheral collisions [39]. A correction for the
finite width of centrality intervals has been made with Eq. 4 using 0.5%
centrality intervals from central to $40\%$ cross section and $1\%$ intervals
for the rest of the centrality classes.

###  2.2 Efficiency correction

The detector efficiency factors ($\varepsilon$) were evaluated in bins of
pseudorapidity $\eta$, azimuthal angle $\varphi$, and $p_{\rm T}$. By defining
$N_{\rm ch}(x)$ as the number of produced particles in a phase-space bin at
$x$, $n(x)$ as the number of observed particles at $x$, and $\varepsilon(x)$
as the detection efficiency, the first and second factorial moments of the
multiplicity distributions can be corrected for particle losses according to
the procedure outlined in Ref. [44, 45]:

$F_{\rm 1}=\langle N_{\rm ch}\rangle=\sum_{i=1}^{m}\langle N_{\rm ch}(x_{\rm
i})\rangle=\sum_{i=1}^{m}\frac{n(x_{\rm i})}{\varepsilon(x_{\rm i})},$ (5)

and

$F_{\rm 2}=\sum_{i=1}^{m}\sum_{j=i}^{m}\frac{\langle n(x_{\rm i})(n(x_{\rm
j})-\delta_{x_{\rm i}x_{\rm j}})\rangle}{\varepsilon(x_{\rm
i})\varepsilon(x_{\rm j})},$ (6)

respectively. Here, $m$ denotes the index of the phase-space bins and $i,j$
are the bin indexes. $\delta_{x_{\rm i}x_{\rm j}}=1$ if $x_{\rm i}=x_{\rm j}$
and zero otherwise. The variance of the charged-particle multiplicity is then
calculated as:

$\sigma_{\rm ch}^{\rm 2}=F_{\rm 2}+F_{\rm 1}-F_{\rm 1}^{\rm 2}.$ (7)

The correction procedure is validated by a Monte Carlo study employing two
million Pb–Pb events at $\sqrt{s_{\rm NN}}$ $=$ 2.76 TeV generated using the
HIJING event generator [46], and passed through GEANT3 simulations of the
experimental setup, taking care of the acceptances of the detectors. The
efficiency dependencies on $\eta$, $\varphi$, and $p_{\rm T}$ are calculated
from the ratio of the number of reconstructed charged particles by the number
of produced particles. In order to account for the $p_{\rm T}$ dependence of
efficiency, the full $p_{\rm T}$ range ($0.2<p_{\rm T}<2.0$ GeV/$c$) 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 $p_{\rm T}$
ranges. In the Monte Carlo closure test, the values of $\langle N_{\rm
ch}\rangle$, $\sigma_{\rm ch}$, and $\omega_{\rm ch}$ of the efficiency
corrected results from the simulated events are compared to those of HIJING at
the generator level to obtain the corrections. By construction, the efficiency
corrected values for $\langle N_{\rm ch}\rangle$ match with those from the
generator, whereas $\sigma_{\rm ch}$ and $\omega_{\rm ch}$ values differ by
$\sim$0.7% and $\sim$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 propagation derived from the delta
theorem [47]. The systematic uncertainties have been evaluated by considering
the effects of various criteria in track selection, vertex determination, and
efficiency corrections.

The systematic uncertainties related to the track selection criteria were
obtained by varying the track reconstruction 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 information. The differences in
the values of $\langle N_{\rm ch}\rangle$, $\sigma_{\rm ch}$, and $\omega_{\rm
ch}$ resulting from the track selections using the two methods are listed in
Table 1 as a part of the systematic uncertainties. The $DCA_{\rm xy}$ and
$DCA_{\rm z}$ of the tracks are varied by $\pm 25$% 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
restricting the $z$-position of the vertex to $\pm 5$ cm from the nominal $\pm
10$ cm, and additionally by removing restrictions on $V_{\rm x}$ and $V_{\rm
y}$. The efficiency correction introduces additional systematic 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 discussed above are
considered uncorrelated and summed in quadrature to obtain the total
systematic errors reported in this work. Table 1 lists the systematic
uncertainties associated with the values of $\langle N_{\rm ch}\rangle$,
$\sigma_{\rm ch}$, and $\omega_{\rm ch}$.

Table 1: Systematic uncertainties on the mean, standard deviation, and scaled variance of charged-particle multiplicity distributions from different sources. The ranges of uncertainties quoted correspond to central to peripheral collisions. Source |  $\langle N_{\rm ch}\rangle$ |  $\sigma_{\rm ch}$ |  $\omega_{\rm ch}$
---|---|---|---
Track selection |  $3.5-4.8$% |  $3.8-6.0$% |  $4.0-7.5$%
Variation of $DCA_{\rm xy}$ |  $0.5-0.9\%$ |  $0.8-1.2\%$ |  $1.3-1.6\%$
Variation of $DCA_{\rm z}$ |  $0.4-0.9\%$ |  $0.7-1.0\%$ |  $1.2-1.7\%$
Vertex ($V_{\rm z}$) selection |  $0.1-0.5\%$ |  0.5% |  $0.1-0.8$%
Removal of $V_{\rm x},V_{\rm y}$ selections |  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 |  $3.5-5.1$% |  $4.1-6.4$% |  $4.8-8.3$%

##  3 Results and discussions

Figure 1 shows the corrected mean ($\langle N_{\rm ch}\rangle$), standard
deviation ($\sigma_{\rm ch}$), and scaled variance ($\omega_{\rm ch}$) as a
function of $\langle N_{\rm part}\rangle$ for the centrality range considered
(0-60%) corresponding to $N_{\rm part}>45$. Uncertainties on the estimated
number of participants, $\langle N_{\rm part}\rangle$, 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 $\langle N_{\rm ch}\rangle$ and $\sigma_{\rm ch}$ increase
with increasing $\langle N_{\rm part}\rangle$. The value of $\omega_{\rm ch}$
decreases monotonically by $\sim$29% from peripheral to central collisions.

Figure 1: Mean ($\langle N_{\rm ch}\rangle$), standard deviation ($\sigma_{\rm
ch}$), and scaled variance ($\omega_{\rm ch}$) 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 $\sqrt{s_{\mathrm{NN}}}$ $=$ 2.76 TeV, shown in
panels (a), (b), and (c), respectively. For panel (a), $\langle N_{\rm
part}\rangle$ for the two models are shifted for better visibility. The
statistical uncertainties are smaller than the size of the markers. The
systematic uncertainties are presented as filled boxes.  Figure 2:  Scaled
variances of charged-particle multiplicity distributions for different $\eta$
and $p_{\rm T}$ ranges as a function of number of participating nucleons
measured in Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ $=$ 2.76 TeV, shown
in panels (a), and (b), respectively. The estimated $\omega_{\rm ch}$ for
$|\eta|<0.3$ and $|\eta|<0.5$ are obtained from the experimental data of
$|\eta|<0.8$ by using Eq. 8. The estimated $\omega_{\rm ch}$ for $0.2<p_{\rm
T}<1.5$ GeV/$c$ and $0.2<p_{\rm T}<1.0$ GeV/$c$ are obtained from the
experimental data of $0.2<p_{\rm T}<2.0$ GeV/$c$, also by using Eq. 8. The
statistical uncertainties are smaller than the size of the markers. The
systematic uncertainties are presented as filled boxes.

###  3.1 Comparison with models

The measured $\omega_{\rm ch}$ 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 multiple minijet
production, soft excitation, nuclear shadowing of parton distribution
functions, and jet interactions in the 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 $2.8<\eta<5.1$ and $-3.7<\eta<-1.7$, corresponding to the V0A and
V0C pseudorapidity coverages. The results of the scaled variances from the two
event generators are presented in Fig. 1 as a function of the estimated number
of participants, $N_{\rm part}$. As a function of increasing centrality, the
$\omega_{\rm ch}$ values obtained from the event generators show upward
trends, which are opposite to those of the experimental data. It is to be
noted that the Monte Carlo event generators are successful in reproducing the
mean of multiplicity distributions. This follows from the fact that the
particle multiplicities 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, resonance 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 $p_{\rm
T}$ range

Charged-particle multiplicity distributions depend on the acceptance of the
detection region. Starting with the measured multiplicity fluctuations within
$|\eta|<0.8$ and $0.2<p_{\rm T}<2.0$ GeV/$c$ with a mean $\langle N_{\rm
ch}\rangle$ and scaled variance of $\omega_{\rm ch}$, the scaled variance
($\omega_{\rm ch}^{\rm acc}$) for a fractional acceptance in $\eta$ or for a
limited $p_{\rm T}$ range with mean of $\langle N_{\rm ch}^{\rm acc}\rangle$
can be expressed as [31],

$\omega_{\rm ch}^{\rm acc}=1+f^{\rm acc}(\omega_{\rm ch}-1),$ (8) ${\rm
where}~{}~{}f^{\rm acc}=\frac{\langle N_{\rm ch}^{\rm acc}\rangle}{\langle
N_{\rm ch}\rangle}.$ (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 variances at reduced
acceptances along with the results from the above calculations. This is shown
in Fig. 2 for different $\eta$ or $p_{\rm T}$ ranges. In the top panel, the
scaled variances are shown, as a function of $\langle N_{\rm part}\rangle$,
for three $\eta$ ranges. The solid symbols show the results of measured scaled
variances, whereas open symbols show the estimated values for the two reduced
$\eta$ windows. The calculated values yield a good description of the measured
data points. The choice of the $p_{\rm T}$ range also affects the multiplicity
of an event. In the bottom panel of Fig. 2, the scaled variances are shown, as
a function of $\langle N_{\rm part}\rangle$, for three $p_{\rm T}$ ranges
keeping $|\eta|<0.8$. A decrease in the value of $\omega_{\rm ch}$ is observed
with the decrease of the $p_{\rm T}$ window. 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.

Figure 3: Comparison of $\langle N_{\rm ch}\rangle$, $\omega_{\rm ch}$, and
$\sigma_{\rm ch}^{2}/\langle N_{\rm ch}\rangle^{2}$ measured in this work
based on the acceptance of the PHENIX experiment with results reported by
PHENIX [31] as a function of number of participating nucleons, shown in panels
(a), (b), and (c), respectively. The statistical uncertainties 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 distributions were earlier
reported by the PHENIX Collaboration at RHIC for Au–Au collisions at
$\sqrt{s_{\mathrm{NN}}}$ = 62.4 GeV and 200 GeV [31]. The beam–beam counters
(BBC) in PHENIX covering the full azimuthal angle in the pseudorapidity range
$3.0<|\eta|<3.9$ provided the minimum-bias trigger and were used for
centrality selection. The pseudorapidity acceptance of the PHENIX experiment
amounted to $|\eta|<0.26$ with an effective average azimuthal active area of
2.1 radian and $0.2<p_{\rm T}<2.0$ GeV/$c$ for charged particle measurements.
The published results of mean and scaled variances of charged-particles were
corrected for fluctuations of the collision 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 determined from Glauber model simulations.
The corrected results are reproduced in Fig. 3 for 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 $\frac{\sigma_{\rm ch}^{\rm 2}}{{\langle
N_{\rm ch}\rangle}^{\rm 2}}$, the ratio of the variance by the square of the
average multiplicity as a function of collision centrality. At lower beam
energies, these distributions obey a power-law relative to the number of
participants [49]. In the lower panel of Fig. 3, the values of
$\frac{\sigma_{\rm ch}^{\rm 2}}{{\langle N_{\rm ch}\rangle}^{\rm 2}}$ are
presented as a function of $\langle N_{\rm part}\rangle$ for 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,
$\frac{\sigma_{\rm ch}^{\rm 2}}{{\langle N_{\rm ch}\rangle}^{\rm 2}}=A\cdot
N_{\rm part}^{\alpha}$. The exponent $\alpha=-\rm 1.25\pm 0.03$ fits the four
sets of experimental data well with $\chi^{2}/{\rm ndf}$ (where $\rm{ndf}$ is
the number of degrees of freedom) as 0.88, 1.1, 0.95, 0.84 for Pb–Pb
collisions at $\sqrt{s_{\rm NN}}~{}=$ 2.76 TeV with the ALICE acceptance, the
PHENIX detector acceptance and Au–Au collisions at $\sqrt{s_{\rm
NN}}~{}=~{}$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 $\frac{\sigma_{\rm ch}^{\rm 2}}{{\langle N_{\rm
ch}\rangle}^{\rm 2}}$ for HIJING and AMPT models for Pb–Pb collisions at
$\sqrt{s_{\rm NN}}~{}=$ 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 participant 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 $N_{\rm part}$ and from fluctuations
in the number of particles ($n$) produced by each nucleon–nucleon interaction
[22, 26, 31]. The values of $n$ and their fluctuations are also strongly
dependent on the acceptance of the detector. Within the context of this
framework, the scaled variance of the background, $\omega_{\rm ch}^{\rm
back}$, amounts to

$\omega_{\rm ch}^{\rm back}=\omega_{\rm n}+\langle n\rangle\omega_{N_{\rm
part}},$ (10)

where $\langle n\rangle$ is the average number of particles produced by each
nucleon–source within the detector acceptance, $\omega_{\rm n}$ is the scaled
variance of the fluctuations in $n$, and $\omega_{N_{\rm part}}$ denotes the
fluctuations in $N_{\rm part}$. The variance, $\omega_{N_{\rm part}}$ is
calculated using event-by-event $N_{\rm part}$ from the HIJING model. The
distribution of $N_{\mathrm{part}}$ corresponds to the centrality obtained
within the V0 detector coverage ($2.8<\eta<5.1$ and $-3.7<\eta<-1.7$). The
extracted values of $\omega_{N_{\rm part}}$ are corrected for the effects of
the finite width of the centrality intervals.

Figure 4:  Scaled variance as a function of $\langle N_{\rm part}\rangle$ for
charged-particle multiplicity distributions and background fluctuations
($\omega_{\rm ch}^{\rm back}$) based on a participant model calculation for
$|\eta|<0.5$. 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.

For the central rapidity range ($|\eta|<0.5$), the measured number of charged
particles produced in pp collisions within $0.2<p_{\rm T}<2.0$ GeV/$c$ at
$\sqrt{s}~{}$=$~{}2.76$ TeV [50] yields $\langle n\rangle=1.45\pm 0.07$, which
is half of the measured value. In order to calculate $\omega_{\rm n}$, an
extrapolation of the measured $\omega_{\rm ch}$ is made to $N_{\rm part}=2$
using a polynomial fit function of the form $a+bx+cx^{2}+dx^{3}$, which is
shown in Fig. 4. In order to calculate $\omega_{\rm n}$, an extrapolation of
the measured $\omega_{\rm ch}$ is made to $N_{\rm part}=2$ using a polynomial
fit function of the form $a+bx+cx^{2}+dx^{3}$, which is shown in Fig. 4. Since
both the nucleon sources contributing to $N_{\rm part}=2$ are correlated,
$\omega_{\rm n}$ becomes half of the extrapolated value, yielding $\omega_{\rm
n}=1.445\pm 0.12$. This result is also consistent with the value of
$\omega_{\rm n}=1.51\pm 0.16$ obtained from the parameterization given by the
PHENIX Collaboration [31].

Using the above numbers, $\omega_{\rm ch}^{\rm back}$ are calculated and
plotted as a function of $\langle N_{\rm part}\rangle$ as in Fig. 4. The
obtained trend in $\omega_{\rm ch}^{\rm back}$ mainly arises from the
centrality dependence of $\omega_{N_{\rm part}}$. For most central collisions,
the difference between the measured and background $\omega_{\rm ch}$ is
$0.02\pm 0.18$, which is consistent with zero within the uncertainties. Except
for most central collisions, $\omega_{\rm ch}^{\rm back}$ is observed to be
larger than $\omega_{\rm ch}$. 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 Poissonian, the magnitude of $\omega_{\rm
ch}$ reduces to unity, which is independent of the multiplicity and thus
independent of the centrality 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 $k_{T}$
requires knowledge of the temperature and volume of the system. After the
collision, as the system 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 $k_{T}$ depends 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 [51, 36, 52]. The
statistical hadronization 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
$\sqrt{s_{\mathrm{NN}}}$ $=$ 2.76 TeV, the ALICE data on yields of particles
in one unit of rapidity at midrapidity are in good agreement with $0.156\pm
0.002$ GeV and $5330\pm 505$ fm3, for temperature and volume, respectively
[52]. In addition, the charged-particle multiplicity within $|\eta|<0.5$ in
this centrality range is $\langle N_{\rm ch}\rangle=1410\pm 47~{}(syst)$.

Here, an attempt is made to estimate $k_{T}$ for Pb–Pb collisions 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 $\omega_{\rm ch}=2.15\pm 0.1$. In the
absence of any background where the full fluctuation is attributed to have a
thermal origin, one would obtain $k_{T}=52.1\pm 5.81$ fm3/GeV. As the measured
$\omega_{\rm ch}$ contains background fluctuations from different sources,
this value of $k_{T}$can 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 measured fluctuation above that of the participant model
fluctuation is $\omega_{\rm ch}=0.02\pm 0.18$. This leads to $k_{T}=0.48\pm
4.32$ 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 $k_{T}$ for 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 distribution is expected to follow a Poisson distribution
($\omega_{\rm ch}=1$). For central collisions, fluctuations above the Poisson
estimation gives, $\omega_{\rm ch}=1.15\pm 0.06$, which in turn implies a
value of $k_{T}=27.9\pm 3.18$ 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 $k_{T}$ from the
background-subtracted event-by-event multiplicity fluctuation provides an
upper limit of its value.

It is imperative to put the extracted values of $k_{T}$ in perspective with
respect to that of normal nuclear matter. The incompressibility constant of
normal nuclear matter at pressure $P$, expressed as $K_{0}=9(\partial
P/\partial\rho)$ at zero temperature and normal nuclear density,
$\rho=\rho_{0}$, has been determined to be $K_{0}=240\pm 20$ MeV [54, 55, 56].
Using the relation, $k_{T}=(9/\rho K_{0})$, one obtains the isothermal
compressibility of nuclear matter to be $k_{T}\simeq 234\pm 20$ fm3/GeV. This
is consistent 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
$k_{T}=27.9\pm 3.18$ 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
$\sqrt{s_{\mathrm{NN}}}$ $=$ 2.76 TeV. The mean, standard deviation, and
scaled variances of charged-particle multiplicities are presented for
$|\eta|<0.8$ and $0.2<p_{\rm T}<2.0$ GeV/$c$ 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 $\eta$
acceptance of the detector as well as with the decrease of the $p_{\rm T}$
range. The multiplicity fluctuations are compared to the results from lower
beam energies as reported by the PHENIX experiment. For the same acceptance,
the observed scaled variances at RHIC energies are smaller compared to those
observed at the LHC.

As multiplicity fluctuations are related to the isothermal compressibility of
the system, the measured fluctuations are used to estimate $k_{T}$ in central
Pb–Pb collisions at $\sqrt{s_{\mathrm{NN}}}$ $=$ 2.76 TeV. The multiplicity
fluctuations above the Poisson expectation case yields $k_{T}=27.9\pm 3.18$
fm3/GeV, which may still contain contributions from additional uncorrelated
particle production as well as from several non-thermal sources as discussed
in section 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 $k_{T}$ until various contributions to the
background are properly understood and evaluated. The estimation of $k_{T}$ at
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 provided 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; Austrian Academy of Sciences, Austrian Science Fund (FWF): [M 2467-N36] and Nationalstiftung für Forschung, Technologie und Entwicklung, 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) , Ministry 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 Aplicaciones 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 | Natural Sciences, the VILLUM FONDEN and Danish National Research Foundation (DNRF), Denmark; Helsinki Institute of Physics (HIP), Finland; 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; Bundesministerium für Bildung und Forschung (BMBF) and GSI Helmholtzzentrum 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; Indonesian Institute of Science, Indonesia; Istituto Nazionale di Fisica Nucleare (INFN), Italy; Institute for Innovative Science and Technology , Nagasaki Institute of Applied Science (IIST), Japanese Ministry of Education, Culture, Sports, Science and Technology (MEXT) and Japan Society for the Promotion of Science (JSPS) KAKENHI, Japan; Consejo 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; Commission 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 Information 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 Institute 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 Foundation and Russian Foundation for Basic Research, Russia; Ministry of Education, Science, Research and Sport of the Slovak Republic, Slovakia; National Research Foundation of South Africa, South Africa; Swedish Research Council (VR) and Knut & Alice Wallenberg Foundation (KAW), Sweden; European Organization for Nuclear Research, Switzerland; Suranaree University of Technology (SUT), National Science and Technology Development Agency (NSDTA) and Office of the Higher Education Commission under NRU project of Thailand, Thailand; Turkish Energy, Nuclear and Mineral Research Agency (TENMAK), 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.

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##  Appendix A The ALICE Collaboration

S. Acharya143, D. Adamová98, A. Adler76, J. Adolfsson83, G. Aglieri Rinella35,
M. Agnello31, N. Agrawal55, Z. Ahammed143, S. Ahmad16, S.U. Ahn78, I. Ahuja39,
Z. Akbar52, A. Akindinov95, M. Al-Turany110, S.N. Alam41, D. Aleksandrov91, B.
Alessandro61, H.M. Alfanda7, R. Alfaro Molina73, B. Ali16, Y. Ali14, A.
Alici26, N. Alizadehvandchali127, A. Alkin35, J. Alme21, T. Alt70, L.
Altenkamper21, I. Altsybeev115, M.N. Anaam7, C. Andrei49, D. Andreou93, A.
Andronic146, M. Angeletti35, V. Anguelov107, F. Antinori58, P. Antonioli55, C.
Anuj16, N. Apadula82, L. Aphecetche117, H. Appelshäuser70, S. Arcelli26, R.
Arnaldi61, I.C. Arsene20, M. Arslandok148,107, A. Augustinus35, R.
Averbeck110, S. Aziz80, M.D. Azmi16, A. Badalà57, Y.W. Baek42, X. Bai131,110,
R. Bailhache70, Y. Bailung51, R. Bala104, A. Balbino31, A. Baldisseri140, B.
Balis2, M. Ball44, D. Banerjee4, R. Barbera27, L. Barioglio108,25, M.
Barlou87, G.G. Barnaföldi147, L.S. Barnby97, V. Barret137, C. Bartels130, K.
Barth35, E. Bartsch70, F. Baruffaldi28, N. Bastid137, S. Basu83, G.
Batigne117, B. Batyunya77, D. Bauri50, J.L. Bazo Alba114, I.G. Bearden92, C.
Beattie148, I. Belikov139, A.D.C. Bell Hechavarria146, F. Bellini26,35, R.
Bellwied127, S. Belokurova115, V. Belyaev96, G. Bencedi71, S. Beole25, A.
Bercuci49, Y. Berdnikov101, A. Berdnikova107, D. Berenyi147, L. Bergmann107,
M.G. Besoiu69, L. Betev35, P.P. Bhaduri143, A. Bhasin104, I.R. Bhat104, M.A.
Bhat4, B. Bhattacharjee43, P. Bhattacharya23, L. Bianchi25, N. Bianchi53, J.
Bielčík38, J. Bielčíková98, J. Biernat120, A. Bilandzic108, G. Biro147, S.
Biswas4, J.T. Blair121, D. Blau91, M.B. Blidaru110, C. Blume70, G. Boca29,59,
F. Bock99, A. Bogdanov96, S. Boi23, J. Bok63, L. Boldizsár147, A.
Bolozdynya96, M. Bombara39, P.M. Bond35, G. Bonomi142,59, H. Borel140, A.
Borissov84, H. Bossi148, E. Botta25, L. Bratrud70, P. Braun-Munzinger110, M.
Bregant123, M. Broz38, G.E. Bruno109,34, M.D. Buckland130, D. Budnikov111, H.
Buesching70, S. Bufalino31, O. Bugnon117, P. Buhler116, Z. Buthelezi74,134,
J.B. Butt14, S.A. Bysiak120, D. Caffarri93, M. Cai28,7, H. Caines148, A.
Caliva110, E. Calvo Villar114, J.M.M. Camacho122, R.S. Camacho46, P.
Camerini24, F.D.M. Canedo123, F. Carnesecchi35,26, R. Caron140, J. Castillo
Castellanos140, E.A.R. Casula23, F. Catalano31, C. Ceballos Sanchez77, P.
Chakraborty50, S. Chandra143, S. Chapeland35, M. Chartier130, S.
Chattopadhyay143, S. Chattopadhyay112, A. Chauvin23, T.G. Chavez46, C.
Cheshkov138, B. Cheynis138, V. Chibante Barroso35, D.D. Chinellato124, S.
Cho63, P. Chochula35, P. Christakoglou93, C.H. Christensen92, P.
Christiansen83, T. Chujo136, C. Cicalo56, L. Cifarelli26, F. Cindolo55, M.R.
Ciupek110, G. ClaiII,55, J. CleymansI,126, F. Colamaria54, J.S. Colburn113, D.
Colella109,54,34,147, A. Collu82, M. Colocci35,26, M. ConcasIII,61, G. Conesa
Balbastre81, Z. Conesa del Valle80, G. Contin24, J.G. Contreras38, M.L.
Coquet140, T.M. Cormier99, P. Cortese32, M.R. Cosentino125, F. Costa35, S.
Costanza29,59, P. Crochet137, R. Cruz-Torres82, E. Cuautle71, P. Cui7, L.
Cunqueiro99, A. Dainese58, F.P.A. Damas117,140, M.C. Danisch107, A. Danu69, I.
Das112, P. Das89, P. Das4, S. Das4, S. Dash50, S. De89, A. De Caro30, G. de
Cataldo54, L. De Cilladi25, J. de Cuveland40, A. De Falco23, D. De Gruttola30,
N. De Marco61, C. De Martin24, S. De Pasquale30, S. Deb51, H.F. Degenhardt123,
K.R. Deja144, L. Dello Stritto30, S. Delsanto25, W. Deng7, P. Dhankher19, D.
Di Bari34, A. Di Mauro35, R.A. Diaz8, T. Dietel126, Y. Ding138,7, R. Divià35,
D.U. Dixit19, Ø. Djuvsland21, U. Dmitrieva65, J. Do63, A. Dobrin69, B.
Dönigus70, O. Dordic20, A.K. Dubey143, A. Dubla110,93, S. Dudi103, M.
Dukhishyam89, P. Dupieux137, N. Dzalaiova13, T.M. Eder146, R.J. Ehlers99, V.N.
Eikeland21, F. Eisenhut70, D. Elia54, B. Erazmus117, F. Ercolessi26, F.
Erhardt102, A. Erokhin115, M.R. Ersdal21, B. Espagnon80, G. Eulisse35, D.
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A. Fantoni53, M. Fasel99, P. Fecchio31, A. Feliciello61, G. Feofilov115, A.
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O.S. Groettvik21, F. Grosa35,61, J.F. Grosse-Oetringhaus35, R. Grosso110, G.G.
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L.B. Havener148, S. Hayashi135, S.T. Heckel108, E. Hellbär70, H. Helstrup37,
T. Herman38, E.G. Hernandez46, G. Herrera Corral9, F. Herrmann146, K.F.
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Affiliation Notes

I Deceased

II Also at: Italian National Agency for New Technologies, Energy and
Sustainable Economic Development (ENEA), Bologna, Italy

III Also at: Dipartimento DET del Politecnico di Torino, Turin, Italy

IV Also at: M.V. Lomonosov Moscow State University, D.V. Skobeltsyn Institute
of Nuclear, Physics, Moscow, Russia

V Also at: Institute of Theoretical Physics, University of Wroclaw, Poland

Collaboration Institutes

1 A.I. Alikhanyan National Science Laboratory (Yerevan Physics Institute)
Foundation, Yerevan, Armenia

2 AGH University of Science and Technology, Cracow, Poland

3 Bogolyubov Institute for Theoretical Physics, National Academy of Sciences
of Ukraine, Kiev, Ukraine

4 Bose Institute, Department of Physics and Centre for Astroparticle Physics
and Space Science (CAPSS), Kolkata, India

5 Budker Institute for Nuclear Physics, Novosibirsk, Russia

6 California Polytechnic State University, San Luis Obispo, California, United
States

7 Central China Normal University, Wuhan, China

8 Centro de Aplicaciones Tecnológicas y Desarrollo Nuclear (CEADEN), Havana,
Cuba

9 Centro de Investigación y de Estudios Avanzados (CINVESTAV), Mexico City and
Mérida, Mexico

10 Chicago State University, Chicago, Illinois, United States

11 China Institute of Atomic Energy, Beijing, China

12 Chungbuk National University, Cheongju, Republic of Korea

13 Comenius University Bratislava, Faculty of Mathematics, Physics and
Informatics, Bratislava, Slovakia

14 COMSATS University Islamabad, Islamabad, Pakistan

15 Creighton University, Omaha, Nebraska, United States

16 Department of Physics, Aligarh Muslim University, Aligarh, India

17 Department of Physics, Pusan National University, Pusan, Republic of Korea

18 Department of Physics, Sejong University, Seoul, Republic of Korea

19 Department of Physics, University of California, Berkeley, California,
United States

20 Department of Physics, University of Oslo, Oslo, Norway

21 Department of Physics and Technology, University of Bergen, Bergen, Norway

22 Dipartimento di Fisica dell’Università ’La Sapienza’ and Sezione INFN,
Rome, Italy

23 Dipartimento di Fisica dell’Università and Sezione INFN, Cagliari, Italy

24 Dipartimento di Fisica dell’Università and Sezione INFN, Trieste, Italy

25 Dipartimento di Fisica dell’Università and Sezione INFN, Turin, Italy

26 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Bologna, Italy

27 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Catania, Italy

28 Dipartimento di Fisica e Astronomia dell’Università and Sezione INFN,
Padova, Italy

29 Dipartimento di Fisica e Nucleare e Teorica, Università di Pavia, Pavia,
Italy

30 Dipartimento di Fisica ‘E.R. Caianiello’ dell’Università and Gruppo
Collegato INFN, Salerno, Italy

31 Dipartimento DISAT del Politecnico and Sezione INFN, Turin, Italy

32 Dipartimento di Scienze e Innovazione Tecnologica dell’Università del
Piemonte Orientale and INFN Sezione di Torino, Alessandria, Italy

33 Dipartimento di Scienze MIFT, Università di Messina, Messina, Italy

34 Dipartimento Interateneo di Fisica ‘M. Merlin’ and Sezione INFN, Bari,
Italy

35 European Organization for Nuclear Research (CERN), Geneva, Switzerland

36 Faculty of Electrical Engineering, Mechanical Engineering and Naval
Architecture, University of Split, Split, Croatia

37 Faculty of Engineering and Science, Western Norway University of Applied
Sciences, Bergen, Norway

38 Faculty of Nuclear Sciences and Physical Engineering, Czech Technical
University in Prague, Prague, Czech Republic

39 Faculty of Science, P.J. Šafárik University, Košice, Slovakia

40 Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe-
Universität Frankfurt, Frankfurt, Germany

41 Fudan University, Shanghai, China

42 Gangneung-Wonju National University, Gangneung, Republic of Korea

43 Gauhati University, Department of Physics, Guwahati, India

44 Helmholtz-Institut für Strahlen- und Kernphysik, Rheinische Friedrich-
Wilhelms-Universität Bonn, Bonn, Germany

45 Helsinki Institute of Physics (HIP), Helsinki, Finland

46 High Energy Physics Group, Universidad Autónoma de Puebla, Puebla, Mexico

47 Hiroshima University, Hiroshima, Japan

48 Hochschule Worms, Zentrum für Technologietransfer und Telekommunikation
(ZTT), Worms, Germany

49 Horia Hulubei National Institute of Physics and Nuclear Engineering,
Bucharest, Romania

50 Indian Institute of Technology Bombay (IIT), Mumbai, India

51 Indian Institute of Technology Indore, Indore, India

52 Indonesian Institute of Sciences, Jakarta, Indonesia

53 INFN, Laboratori Nazionali di Frascati, Frascati, Italy

54 INFN, Sezione di Bari, Bari, Italy

55 INFN, Sezione di Bologna, Bologna, Italy

56 INFN, Sezione di Cagliari, Cagliari, Italy

57 INFN, Sezione di Catania, Catania, Italy

58 INFN, Sezione di Padova, Padova, Italy

59 INFN, Sezione di Pavia, Pavia, Italy

60 INFN, Sezione di Roma, Rome, Italy

61 INFN, Sezione di Torino, Turin, Italy

62 INFN, Sezione di Trieste, Trieste, Italy

63 Inha University, Incheon, Republic of Korea

64 Institute for Gravitational and Subatomic Physics (GRASP), Utrecht
University/Nikhef, Utrecht, Netherlands

65 Institute for Nuclear Research, Academy of Sciences, Moscow, Russia

66 Institute of Experimental Physics, Slovak Academy of Sciences, Košice,
Slovakia

67 Institute of Physics, Homi Bhabha National Institute, Bhubaneswar, India

68 Institute of Physics of the Czech Academy of Sciences, Prague, Czech
Republic

69 Institute of Space Science (ISS), Bucharest, Romania

70 Institut für Kernphysik, Johann Wolfgang Goethe-Universität Frankfurt,
Frankfurt, Germany

71 Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Mexico City, Mexico

72 Instituto de Física, Universidade Federal do Rio Grande do Sul (UFRGS),
Porto Alegre, Brazil

73 Instituto de Física, Universidad Nacional Autónoma de México, Mexico City,
Mexico

74 iThemba LABS, National Research Foundation, Somerset West, South Africa

75 Jeonbuk National University, Jeonju, Republic of Korea

76 Johann-Wolfgang-Goethe Universität Frankfurt Institut für Informatik,
Fachbereich Informatik und Mathematik, Frankfurt, Germany

77 Joint Institute for Nuclear Research (JINR), Dubna, Russia

78 Korea Institute of Science and Technology Information, Daejeon, Republic of
Korea

79 KTO Karatay University, Konya, Turkey

80 Laboratoire de Physique des 2 Infinis, Irène Joliot-Curie, Orsay, France

81 Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-
Alpes, CNRS-IN2P3, Grenoble, France

82 Lawrence Berkeley National Laboratory, Berkeley, California, United States

83 Lund University Department of Physics, Division of Particle Physics, Lund,
Sweden

84 Moscow Institute for Physics and Technology, Moscow, Russia

85 Nagasaki Institute of Applied Science, Nagasaki, Japan

86 Nara Women’s University (NWU), Nara, Japan

87 National and Kapodistrian University of Athens, School of Science,
Department of Physics , Athens, Greece

88 National Centre for Nuclear Research, Warsaw, Poland

89 National Institute of Science Education and Research, Homi Bhabha National
Institute, Jatni, India

90 National Nuclear Research Center, Baku, Azerbaijan

91 National Research Centre Kurchatov Institute, Moscow, Russia

92 Niels Bohr Institute, University of Copenhagen, Copenhagen, Denmark

93 Nikhef, National institute for subatomic physics, Amsterdam, Netherlands

94 NRC Kurchatov Institute IHEP, Protvino, Russia

95 NRC «Kurchatov»Institute - ITEP, Moscow, Russia

96 NRNU Moscow Engineering Physics Institute, Moscow, Russia

97 Nuclear Physics Group, STFC Daresbury Laboratory, Daresbury, United Kingdom

98 Nuclear Physics Institute of the Czech Academy of Sciences, Řež u Prahy,
Czech Republic

99 Oak Ridge National Laboratory, Oak Ridge, Tennessee, United States

100 Ohio State University, Columbus, Ohio, United States

101 Petersburg Nuclear Physics Institute, Gatchina, Russia

102 Physics department, Faculty of science, University of Zagreb, Zagreb,
Croatia

103 Physics Department, Panjab University, Chandigarh, India

104 Physics Department, University of Jammu, Jammu, India

105 Physics Department, University of Rajasthan, Jaipur, India

106 Physikalisches Institut, Eberhard-Karls-Universität Tübingen, Tübingen,
Germany

107 Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg, Germany

108 Physik Department, Technische Universität München, Munich, Germany

109 Politecnico di Bari and Sezione INFN, Bari, Italy

110 Research Division and ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum
für Schwerionenforschung GmbH, Darmstadt, Germany

111 Russian Federal Nuclear Center (VNIIEF), Sarov, Russia

112 Saha Institute of Nuclear Physics, Homi Bhabha National Institute,
Kolkata, India

113 School of Physics and Astronomy, University of Birmingham, Birmingham,
United Kingdom

114 Sección Física, Departamento de Ciencias, Pontificia Universidad Católica
del Perú, Lima, Peru

115 St. Petersburg State University, St. Petersburg, Russia

116 Stefan Meyer Institut für Subatomare Physik (SMI), Vienna, Austria

117 SUBATECH, IMT Atlantique, Université de Nantes, CNRS-IN2P3, Nantes, France

118 Suranaree University of Technology, Nakhon Ratchasima, Thailand

119 Technical University of Košice, Košice, Slovakia

120 The Henryk Niewodniczanski Institute of Nuclear Physics, Polish Academy of
Sciences, Cracow, Poland

121 The University of Texas at Austin, Austin, Texas, United States

122 Universidad Autónoma de Sinaloa, Culiacán, Mexico

123 Universidade de São Paulo (USP), São Paulo, Brazil

124 Universidade Estadual de Campinas (UNICAMP), Campinas, Brazil

125 Universidade Federal do ABC, Santo Andre, Brazil

126 University of Cape Town, Cape Town, South Africa

127 University of Houston, Houston, Texas, United States

128 University of Jyväskylä, Jyväskylä, Finland

129 University of Kansas, Lawrence, Kansas, United States

130 University of Liverpool, Liverpool, United Kingdom

131 University of Science and Technology of China, Hefei, China

132 University of South-Eastern Norway, Tonsberg, Norway

133 University of Tennessee, Knoxville, Tennessee, United States

134 University of the Witwatersrand, Johannesburg, South Africa

135 University of Tokyo, Tokyo, Japan

136 University of Tsukuba, Tsukuba, Japan

137 Université Clermont Auvergne, CNRS/IN2P3, LPC, Clermont-Ferrand, France

138 Université de Lyon, CNRS/IN2P3, Institut de Physique des 2 Infinis de Lyon
, Lyon, France

139 Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France,
Strasbourg, France

140 Université Paris-Saclay Centre d’Etudes de Saclay (CEA), IRFU, Départment
de Physique Nucléaire (DPhN), Saclay, France

141 Università degli Studi di Foggia, Foggia, Italy

142 Università di Brescia, Brescia, Italy

143 Variable Energy Cyclotron Centre, Homi Bhabha National Institute, Kolkata,
India

144 Warsaw University of Technology, Warsaw, Poland

145 Wayne State University, Detroit, Michigan, United States

146 Westfälische Wilhelms-Universität Münster, Institut für Kernphysik,
Münster, Germany

147 Wigner Research Centre for Physics, Budapest, Hungary

148 Yale University, New Haven, Connecticut, United States

149 Yonsei University, Seoul, Republic of Korea