# Intermittency analysis of proton numbers in heavy-ion collisions at RHIC
energies

Jin Wu  Key Laboratory of Quark and Lepton Physics (MOE) and Institute of
Particle Physics,
Central China Normal University, Wuhan 430079, China  Yufu Lin  Key Laboratory
of Quark and Lepton Physics (MOE) and Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China  Zhiming Li
<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics (MOE) and
Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China  Xiaofeng Luo
<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics (MOE) and
Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China  Yuanfang Wu
<EMAIL_ADDRESS>Key Laboratory of Quark and Lepton Physics (MOE) and
Institute of Particle Physics,
Central China Normal University, Wuhan 430079, China

###### Abstract

Local density fluctuation near the QCD critical point has been suggested to
exhibit a power-law behavior which can be probed by an intermittency analysis
on scaled factorial moment (SFM) in relativistic heavy-ion collisions. The
collision energy and centrality dependence of the second-order SFMs are
systematically investigated in Au + Au collisions at $\sqrt{s_{NN}}$ = 7.7,
11.5, 19.6, 27, 39, 62.4, 200 GeV within the UrQMD model. We estimate the non-
critical background in the measurement of intermittency and suggest a
cumulative variable method to effectively remove the contributions from
background. We further study the effect of particle detection efficiency by
implementing UrQMD events in the RHIC/STAR experimental tracking efficiencies.
A cell-by-cell method is proposed for experimental application of efficiency
corrections on SFM. This work can provide a guidance of background subtraction
and efficiency correction for experimental measurement of intermittency in the
search of QCD critical point in heavy-ion collisions.

##  I Introduction

One of the major goals of relativistic heavy-ion collisions is to explore the
QCD phase diagram and phase boundary StephanovPD ; adams2005experimental ;
conservecharge0 ; conservecharge1 . Theoretical studies have shown the
existence of a critical point (CP) at finite baryon chemical potential and
temperature CEP1 ; CEP2 . This CP is proposed to be characterized by a second-
order phase transition, which becomes a unique property of strongly
interacting matter searchCEP1 ; searchCEP2 ; searchCEP3 ; searchCEP4 ;
STARPRLMoment .

In the thermodynamic limit, the correlation length diverges at the CP and the
system becomes scale invariant and fractal invariant1 ; invariant2 ;
invariant3 . It has been shown that the density fluctuations near the QCD
critical point form a distinct pattern of power-law or intermittency behavior
in the matter produced in high energy heavy-ion collisions Antoniou2006PRL ;
Antoniou2010PRC ; Antoniou2016PRC ; Antoniou2018PRD . Intermittency is a
manifestation of the scale invariance and fractality of physical process and
the stochastic nature of the underlying scaling law invariant3 . It can be
revealed in transverse momentum spectrum as a power-law behavior of scaled
factorial moment invariant1 ; Antoniou2006PRL . In current high energy heavy-
ion experiments at CERN SPS NA49SFM ; NA61universe , the NA49 and NA61
collaborations have explored the critical fluctuations by employing an
intermittency analysis with various sizes of colliding nuclei. The power-law
behavior has been observed in Si + Si collisions at 158A GeV NA49SFM . At RHIC
BES-I energies, the preliminary result of a critical exponent extracted from
intermittency index shows a minimum around $\sqrt{s_{\mathrm{NN}}}$ = 20 - 30
GeV in central collisions STARintermittency . In the mean time, several model
studies have made efforts to investigate the unique behaviors of intermittency
under various underlying mechanisms CMCPLB ; UrQMDLi ; IJMPE ; amptNPA ; epos
.

For a self-similar system with intermittency, it is expected that the
multiplicity distribution in momentum space is associated with a strong
clustering effect, which indicating a remarkably structured phase-space
density invariant3 ; densityF . However, the inclusive single-particle
multiplicity spectra in finite space of high energy collisions are
significantly influenced by background effects. The multiplicity distribution
is changed or distorted by conservation law, resonance decay and statistical
fluctuations, etc. It has been studied that the statistical fluctuations due
to finite number of particles SFM1  or the choices of the size in momentum
space APPB  will influence the measured SFMs. Therefore, it is necessary to
estimate and remove these trivial effects in order to get a clean power-law
exponent and then one can compare the measured intermittency with theoretical
predictions. For this purpose, Ochs OchsCumulative  and Bialas
BialasCumulative  have proposed to study intermittency in terms of variables,
called as cumulative variable, in which the single-particle density is a
constant. It is shown that the cumulative variable very effectively reduces
the distortions of the simple scaling law caused by a non-uniform single-
particle spectrum and there is no bias from the shape of the inclusive
distribution. We will study how to remove the trivial background effects by
the cumulative variable method in the measurement of SFMs in heavy-ion
collisions.

In the study of intermittency in high energy experiments, particle detector
has a finite detection efficiency, which simply results from the limited
capability of the detector to register the finite-state particles
STAREfficiency ; EfficiencyLuo . This efficiency effect will lead to the loss
of particle multiplicity in an event, which will change the shape of the
original event-by-event multiplicity distributions in momentum space
EfficiencyLuo ; MomentOverview . Since the calculation of SFMs is determined
by the particle distributions, its value could be significantly modified by
the detector efficiency, which will distort the original signal possibly
induced by the CP. We should recover the SFM of the original true multiplicity
distributions from the experimentally measured one by applying a proper
efficiency correction technique. Then the true intermittency index can be
extracted from the efficiency corrected SFM.

The paper is organized as follows. A brief introduction to the UrQMD model is
given in Sec. II. In sec. III, we introduce the method of intermittency
analysis by using SFMs. Then, the collision energy and centrality dependence
of SFMs are investigated by UrQMD model in Au + Au collisions from
$\sqrt{s_{\mathrm{NN}}}$ = 7.7 to 200 GeV. In Sec. V, we discuss the
estimation and subtraction of background in the calculations of SFMs. In Sec.
VI, the efficiency correction formula is deduced, followed by a check of the
validity of the method by UrQMD model. Finally, we give a summary and outlook
of this work.

##  II UrQMD MODEL

The UrQMD (Ultra Relativistic Quantum Molecular Dynamics) model is a
microscopic many-body model and has been extensively applied to simulate p +
p, p + A, and A + A interactions in the ultra-relativistic heavy-ion
collisions MBUrQMD ; SABassUrQMD ; HPUrQMD . It provides phase space
descriptions of different reaction mechanisms based on the covariant
propagation of all hadrons with stochastic binary scattering, color string
formation and resonance decay SABassUrQMD . This model includes baryon-baryon,
meson-baryon and meson-meson interactions with more than 50 baryon and 45
meson species. It preserves the conservation of electric charge, baryon
number, and strangeness number as expected for the evolution of the QCD
matter. It models the phenomena of baryon stopping which is an essential
feature encountered in heavy ions at low beam energies. It is a well-designed
transport model MBUrQMD  for simulations with the entire available range of
energies from SIS energy ($\sqrt{s_{\mathrm{NN}}}=2$ GeV) to the top RHIC
energy ($\sqrt{s_{\mathrm{NN}}}=200$ GeV). More details about the UrQMD model
can be found in Refs MBUrQMD ; SABassUrQMD ; HPUrQMD .

The UrQMD model is a suitable simulator to estimate the non-critical
contribution and other trivial background effects in the measurement of
correlations and fluctuations in heavy-ion collisions. In this work, we use
the UrQMD model with version 2.3 to generate Monte-Carlo event samples in Au +
Au collisions at RHIC energies. The corresponding statistics are 72.5, 105,
106, 81, 133, 38, 56 millions at $\sqrt{s_{\mathrm{NN}}}$ = 7.7, 11.5, 19.6,
27, 39, 62.4, 200 GeV, respectively.

Figure 1: The second-order scaled factorial moment (black circles) ,
$F_{2}(M)$, as a function of number of partitioned cells in a double-
logarithmic scale at $\sqrt{s_{\mathrm{NN}}}$ = 7.7 - 200 GeV from UrQMD
model. The black lines are the power-law fitting. The corresponding red ones
represent the SFMs calculated by the cumulative variable method. Figure 2: The
second-order scaled factorial moment (black circles) as a function of number
of partitioned cells from the most central ($0-5\%$) to the most peripheral
($60-80\%$) collisions at $\sqrt{s_{\mathrm{NN}}}$ = 19.6 GeV. The
corresponding red ones represent the SFMs calculated by the cumulative
variable method.

##  III Method of Analysis

It is argued that in heavy-ion collisions large baryon density fluctuation may
provide a unique signal to the phase transition in the QCD phase diagram. It
is expected to observe the critical density fluctuations as a power-law
pattern on available phase space resolution if system freezes out right in the
vicinity of the critical point Antoniou2006PRL ; NA61universe .

In high energy experiments, the power-law or intermittency behavior can be
measured by calculations of SFMs of baryon number density HattaPRL ;
Antoniou2006PRL ; NA61universe . For this purpose, an available region of
momentum space is partitioned into $M^{D}$ equal-size bins, $F_{q}(M)$ is
defined as:

$F_{q}(M)=\frac{\langle\frac{1}{M^{D}}\sum_{i=1}^{M^{D}}n_{i}(n_{i}-1)\cdots(n_{i}-q+1)\rangle}{\langle\frac{1}{M^{D}}\sum_{i=1}^{M^{D}}n_{i}\rangle^{q}},$
(1)

with $M^{D}$ the number of cells in the D-dimensional partitioned momentum
space, $n_{i}$ the measured multiplicity in the $i$-th cell, and $q$ the order
of the moment.

If the system exhibits critical fluctuations, $F_{q}(M)$ is expected to follow
a scaling function: Antoniou2006PRL ; NA49SFM

$F_{q}(M)\sim(M^{D})^{\phi_{q}},M\rightarrow\infty.$ (2)

A power-law behavior of $F_{q}$ on the partitioned number $M^{D}$ when $M$ is
large enough is referred as intermittency. The scaling exponent, $\phi_{q}$,
is called the intermittency index that characterizes the strength of the
intermittency behavior. By using a critical equation of state for a 3D Ising
system, the second-order intermittency index in a two-dimensional transverse
momentum space is predicted to be $\phi_{2}=\frac{5}{6}$ Antoniou2006PRL  for
baryon density and $\phi_{2}=\frac{2}{3}$ for sigma condensate NGNPA2005 . The
search of multiplicity fluctuations in increasing number of partition
intervals using the SFMs method was first proposed several years ago SFM1 ;
SFM2 . Recent studies show that one can probe QCD critical fluctuations CMCPLB
and estimate the possible critical region Antoniou2018PRD  from intermittency
analysis in relativistic heavy-ion collisions.

In the following section, we will calculate the second-order SFM of proton
density in transverse momentum space by using event samples from UrQMD model
in Au + Au collisions at $\sqrt{s_{\mathrm{NN}}}$ = 7.7 - 200 GeV. Then, the
intermittency index, $\phi_{2}$, can be extract by fitting from Eq. (2).

##  IV Energy and centrality dependence of Scaled Factorial moments

By using the UrQMD model, we generate event samples at various centralities in
Au + Au collisions at $\sqrt{s_{\mathrm{NN}}}$ = 7.7, 11.5, 19.6, 27, 39,
62.4, 200 GeV. In the model calculations, we apply the same kinematic cuts and
technical analysis methods as those used in the RHIC/STAR experiment data
net_proton2014 . The protons are measured at mid-rapidity($|y|<0.5$) within
the transverse momentum $0.4<p_{T}<2.0$ GeV/c. The centrality is defined by
the charged pion and kaon multiplicities within pseudo-rapidity $|\eta|<1.0$,
which can effectively avoid auto-correlation effect in the measurement of
SFMs. In our analysis, we focus on proton multiplicities in two dimensional
transverse momentum spaces of $p_{x}$ and $p_{y}$. The available 2D region of
transverse momentum is partitioned into $M^{2}$ equal-size bins to calculate
SFMs in various sizes of cells. The statistical error is estimated by the
Bootstrap method BEboostrap .

In Fig. 1, the black circles represent the SFMs as a function of number of
partitioned bins, directly calculated in transverse momenta for proton numbers
in $0-5\%$ most central Au + Au collisions at $\sqrt{s_{\mathrm{NN}}}$ = 7.7 -
200 GeV. It is observed that $F_{2}(M)$ increases slowly with increasing
number of dividing bins. The black lines show the power-law fitting of
$F_{2}(M)$ according to Eq. (2). The slopes of the fitting, $i.e.$ the
intermittency indices $\phi_{2}$, are found to be small at all energies. And
they are much less than the theoretical prediction $\phi_{2}=5/6$ for a
critical system of the 3D Ising universality class Antoniou2006PRL .

The $F_{2}(M)$ measured at various collision centralities in Au + Au
collisions at $\sqrt{s_{\mathrm{NN}}}$ = 19.6 GeV are shown as the black
circles in Fig. 2. And the black lines are the fitting according to Eq. (2).
Again, we find that the directly calculated SFMs can be fitted with a small
intermittency index. The values of $\phi_{2}$ increase slightly from the most
central ($0-5\%$) to the most peripheral ($60-80\%$) collisions.

Therefore, we observe that the intermittency indices from the directly
calculated SFMs are small but non-zero in Au + Au collisions from UrQMD model.
But we know that the UrQMD model is a transport model which does not include
any critical related self-similar fluctuations MBUrQMD  and thus $\phi_{2}$
should be zero. There must exist some trivial non-critical contributions from
background. We will investigate how to remove the background effects from the
directly measured SFMs in the next section.

##  V Background Subtraction

Figure 3: The black symbols represent the second-order scaled factorial moment
as a function of number of partitioned cells (a) in pure CMC events and (b) in
CMC events contaminated with Gaussian background fluctuations. The
corresponding red ones are the SFMs calculated by the cumulative variable
method.

To extract the signature of critical fluctuations, it is essential to
understand the non-critical effects or background contributions in heavy-ion
collisions on the experimental observables. The background effects will change
the inclusive single-particle multiplicity distributions in the measured
finite momentum space and thus will significantly influence the value of
calculated SFMs and the intermittency index thereby. In this purpose, NA49 and
NA61 use the mixed event method to estimate and subtract background by
assuming that the particle multiplicity in each cell can be simply divided
into background and critical contributions NA49SFM . In this paper, we pursue
the cumulative variable method, which has been proved to drastically reduce
distortions of intermittency due to non-uniform single-particle density from
background contributions OchsCumulative ; BialasCumulative ; na22 , to
understand and remove the background effects.

Following Ochs and Bialas BialasCumulative ; OchsCumulative , the cumulative
variable $\large X(x)$ is related to the single-particle density distribution
$\rho(x)$ through:

$\large
X(x)=\frac{\int_{x_{min}}^{x}\rho(x)dx}{\int_{x_{min}}^{x_{max}}\rho(x)dx}.$
(3)

Here $x$ represents the original measured variable, e.g., $p_{x}$ or $p_{y}$.
$\rho(x)$ is the density function of $x$. $x_{min}$ and $x_{max}$ are the
lower and upper phase space limits of the chosen variable $x$.

The cumulative variable $X(x)$ is determined by the shape of density
distribution $\rho(x)$. And the distribution of the new variable $X(x)$ is
uniform in the interval from 0 to 1. It has been proved that the cumulative
variable could remove the dependence of the intermittency parameters on the
shape of particle density distributions and give a new way to compare
measurements from different experiments BialasCumulative . In order to use the
cumulative variable, the two-dimensional momentum space $p_{x}p_{y}$ which is
partitioned into $M^{2}$ cells will transfer to be $p_{X}p_{Y}$ space. And the
SFM directly calculated in $p_{x}p_{y}$ space ($F_{2}(M)$) will transfer to be
$CF_{2}(M)$, which is now calculated in $p_{X}p_{Y}$ space. The process of
fitting $\phi_{2}^{c}$ from $CF_{2}(M)$ is similar to that of $\phi_{2}$ from
$F_{2}(M)$ according to Eq. (2).

In order to test the validity of the cumulative variable method in the
calculations of SFMs, we use a Critical Monte-Carlo (CMC) model
Antoniou2006PRL ; CMCPLB  of the 3D Ising universality class to generate
critical event sample. The CMC model involves self-similar or intermittency
nature of particle correlations and leads to an intermittency index of
$\phi_{2}=\frac{5}{6}$ Antoniou2006PRL . In Fig. 3(a), both $F_{2}(M)$ and
$CF_{2}(M)$ are shown in the same pad. We observe that $CF_{2}(M)$ follows a
good power-law behavior as $F_{2}(M)$ with increasing $M^{2}$. Within
statistical errors, the intermittency index $\phi_{2}^{c}$ fitted from
$CF_{2}(M)$ equals to $\phi_{2}$ obtained from $F_{2}(M)$. It means that the
cumulative variable method does not change the intermittency behavior for a
pure critical signal event sample. In Fig. 3(b), the CMC event sample is
contaminated by hand with a statistical Gaussian background contribution, with
the mixed probability $\lambda=95\%$. The chosen value of $\lambda$ is close
to the one used in the simulations of background in NA49 experiment NA49SFM .
In this case, one finds that the directly calculated $F_{2}(M)$ deviates
substantially from the linear dependence, i.e. violation of the scaling law
because of the Gaussian background contribution. However, the trend of
$CF_{2}(M)$, which is calculated by the cumulative variable method, still obey
similar power-law dependence on $M^{2}$ as that in Fig. 3(a). Furthermore, the
intermittency index $\phi_{2}^{c}$ calculated from $CF_{2}(M)$ keeps unchanged
when comparing to the one in original CMC sample shown in Fig. 3(a). We feel
that these results are encouraging. It confirms that, in the intermittency
analysis, the cumulative variable method can efficiently remove the
distortions or effects caused by background contribution.

Let us go back to the problem of background effect in the UrQMD model remained
in Sec. IV. We calculate SFMs in the same event sample by the proposed
cumulative variable method and then get the intermittency index from
$CF_{2}(M)$. The results are shown as red triangles and red lines in Fig. 1
and Fig. 2. The $CF_{2}(M)$ is found to be nearly flat with increasing number
of cells in all measured energies and centralities. Furthermore, we note that
the intermittency index, with the value nearly around $0$, is much smaller
than the one directly calculated from $F_{2}(M)$. It verifies that the
background of non-critical effect can be efficiently removed by the cumulative
variable method in the calculation of SFMs in UrQMD model. This method could
also be used for the intermittency analysis in the ongoing experimental at
RHIC/STAR or further heavy-ion experiments in search of the QCD critical
point.

##  VI Efficiency correction

One of the difficulties of measuring SFMs and intermittency in experiment is
efficiency correction. It is known that the values of SFMs are changed from
the true ones due to the fact that detectors miss some particles with a
probability named efficiency. To understand the underlying physics associated
with this measurement, one needs to perform a careful study on the efficiency
effect. Generally, the efficiencies in experiments are obtained by Monte Carlo
(MC) embedding technique embedding . This allows for the determination of the
efficiency, which is the ratio of the matched MC tracks number and the number
of input tracks. It contains the effects of tracking efficiency, detector
acceptance and interaction losses.

We denote the number of produced particles as $N$ and the number of measured
ones as $n$ with a detection efficiency $\epsilon$. The true factorial moment
is calculated as $f_{q}^{true}=\langle N(N-1)...(N-q+1)\rangle$. It can be
recovered by dividing the measured factorial moment, $f_{q}^{measured}=\langle
n(n-1)...(n-q+1)\rangle$, with appropriate powers of the detection efficiency
EfficiencyFM ; EfficiencyLuo ; EfficiencyKoch :

$f_{q}^{corrected}=\frac{f_{q}^{measured}}{\epsilon^{q}}=\frac{\langle
n(n-1)...(n-q+1)\rangle}{\epsilon^{q}}.$ (4)

This strategy has been used for the efficiency correction in the high-order
cumulant analysis EfficiencyLuo ; MomentOverview ; EfficiencyFM ;
EfficiencyToshihiro ; EfficiencyKoch . Consider the probability to detect a
particle is governed by a binomial distribution, both cumulants EfficiencyLuo
; MomentOverview  and off-diagonal cumulants EfficiencyKoch  can be expressed
in term of factorial moments and then could be corrected by using Eq. (4).

We apply the strategy to the efficiency correction for SFMs defined in Eq.
(1). Since the available region of phase space is partitioned into a lattice
of $M^{2}$ equal-size cells, every element,
$f_{q,i}^{measured}=n_{i}\times(n_{i}-1)...(n_{i}-q+1)$, of measured
$F_{q}(M)$ should be corrected one by one. In this way, the efficiency
corrected $F_{q}(M)$ is:

$\displaystyle F_{q}^{corrected}(M)$ $\displaystyle=$
$\displaystyle\large\frac{\langle\frac{1}{M^{2}}\sum_{i=1}^{M^{2}}\frac{f_{q,i}^{measured}}{\bar{\epsilon_{i}}^{q}}\rangle}{\langle\frac{1}{M^{2}}\sum_{i=1}^{M^{2}}\frac{f_{1,i}^{measured}}{\bar{\epsilon_{i}}}\rangle^{q}}$
(5) $\displaystyle=$
$\displaystyle\frac{\langle\frac{1}{M^{2}}\sum_{i=1}^{M^{2}}\frac{n_{i}(n_{i}-1)\cdots(n_{i}-q+1)}{\bar{\epsilon_{i}}^{q}}\rangle}{\langle\frac{1}{M^{2}}\sum_{i=1}^{M^{2}}\frac{n_{i}}{\bar{\epsilon_{i}}}\rangle^{q}}.$

Here, $n_{i}$ denotes the number of measured particles located in the $i$-th
cell with detection efficiency $\epsilon_{i}$. The mean $\bar{\epsilon_{i}}$,
is calculated by $(\epsilon_{1}+\epsilon_{2}\cdots\epsilon_{n_{i}})/n_{i}$,
representing the average efficiency in the $i$-th cell. We call the efficiency
correction technique of Eq. (5) the cell-by-cell method.

In order to demonstrate the validity of the cell-by-cell method, we employ the
UrQMD model with the particle detection efficiencies used in real experiment.
It is simulated by injecting particle tracks from UrQMD events into the
RHIC/STAR detector acceptance with the experimental efficiencies. In the STAR
experiment, the detection efficiency is not a constant but depends on the
momentum of particles STAREfficiency ; STARMoment ; LuoCPOD . The particle
identification method is different between low and high $p_{T}$ regions. The
main particle detector at STAR, the Time Projection Chamber (TPC), is used to
obtain momentum of charged particles at low $p_{T}$ region of $0.4<p_{T}<0.8$
GeV/c STAREfficiency . And the Time-Of-Flight (TOF) detector is used to do the
particle identification at relatively high $p_{T}$ region of $0.8<p_{T}<2$
GeV/c STARMoment ; LuoCPOD . In this case, particles need to be counted
separately for the two $p_{T}$ regions, in which the values of the efficiency
are different.

Figure 4: (a) Experimental tracking efficiencies as a function of $p_{T}$ in
TPC detector at mid-rapidity ($|y|<0.5$) for protons in $0-5\%$ Au + Au
collisions. (b) The second-order SFM as a function of number of partitioned
cells from UrQMD calculations. The black circles represent the true
$F_{2}(M)$, the blue solid triangles are the measured $F_{2}(M)$ after
applying the TPC efficiency, and the red stars show the efficiency corrected
SFMs by using the cell-by-cell method.

In Fig. 4(a), we firstly show the $p_{T}$ dependence of the experimental
efficiency in only TPC detector at mid-rapidity ($|y|<0.5$) region for protons
in the most central Au + Au collisions at $\sqrt{s_{\mathrm{NN}}}$ = 19.6 GeV
STAREfficiency . It firstly increases with increasing $p_{T}$, and then gets
saturated in higher $p_{T}$ regions. After employing this $p_{T}$ dependent
efficiency into the UrQMD event sample, the measured $F_{2}(M)$ is calculated.
Then, we apply the correction formula of Eq. (5) to do the efficiency
correction on the measured $F_{2}(M)$. In Fig. 4(b), the black circles
represent the original true $F_{2}(M)$, the blue triangles are the measured
$F_{2}(M)$ with STAR TPC efficiency, and the red stars show the efficiency
corrected $F_{2}(M)$. It is observed that the measured SFMs with efficiency
are systematically smaller than the true ones, especially in the large number
of partitioned cells. However, the efficiency corrected SFMs are found to be
well consistent with the original true ones.

Figure 5: (a) Experimental tracking efficiencies as a function of $p_{T}$ in
TPC+TOF detectors at mid-rapidity ($|y|<0.5$) for protons in $0-5\%$ Au + Au
collisions. (b) The second-order SFM as a function of number of partitioned
cells from UrQMD calculations. The black circles represent the true
$F_{2}(M)$, the blue solid triangles are the measured $F_{2}(M)$ after
applying the TPC+TOF efficiency, and the red stars show the efficiency
corrected SFMs by using the cell-by-cell method.

For the case of TPC+TOF efficiencies, Fig. 5(a) shows the tracking
efficiencies as a function of $p_{T}$ in TPC and TOF at STAR LuoCPOD ;
EfficiencyToshihiro ; STARMoment . One notes that there is a steplike
dependence of the efficiencies on $p_{T}$. The reason is that the particle
identification method is different between TPC and TOF detectors at STAR
experiment. We apply the TPC+TOF efficiency to the UrQMD event sample at
$\sqrt{s_{\mathrm{NN}}}$ = 19.6 GeV and then correct the measured SFMs by Eq.
(5). The results are shown in Fig. 5(b). Again, the SFMs corrected by the
proposed cell-by-cell method are verified to be coincide with the original
true ones.

In this section, we have demonstrated that the cell-by-cell method could be
served as a precise and effective way of efficiency correction on SFMs. It can
be easily applied to current studies at STAR STARintermittency , NA49 NA49SFM
, NA61 NA61universe  and other heavy-ion experiments in the intermittency
analysis. It should also be noted that one needs to consider how to treat the
momentum resolution in different experiments. Since we use the $p_{T}$ of
individual particles to get the efficiency, the momentum resolution might
directly affect the calculation of SFMs. This effect could be studied by
smearing the $p_{T}$ for each particle with the known value of the momentum
resolution.

##  VII summary and outlook

In summary, we investigate collision energy and centrality dependence of the
SFMs in Au + Au collisions at $\sqrt{s_{\mathrm{NN}}}=7.7-200$ GeV by using
the UrQMD model. The second-order intermittency index is found to be small but
non-zero in the transport model without implementing any critical related
self-similar fluctuations. A cumulative variable method is then proposed to be
used in the calculations of SFMs to remove background in the intermittency
analysis. It has been verified that this method can successfully reduce the
distortion of a Gaussian background contribution from a pure self-similar
event sample generated by the CMC model. After applying the method to the
UrQMD event sample, it confirms that the non-critical background effect can be
efficiently removed and the value of the intermittency index is close to zero.

In the experimental measurements of intermittency, the measured SFMs should be
corrected for detecting efficiencies. We derive a cell-by-cell formula in the
calculation of SFMs in heavy-ion collisions. The validity of the method has
been checked in the UrQMD event sample which is employed with both the TPC and
TPC+TOF tracking efficiencies used in the RHIC/STAR experiment. It is
demonstrated that the cell-by-cell method provides a precise and effective way
for the efficiency correction on SFMs. The correction method is universal and
can be applied to the ongoing studies of intermittency in heavy-ion
experiments.

Figure 6: The second-order intermittency index measured at NA49 NA49SFM ;
NA49C+C  (solid blue symbols) and NA61 NA61Ar+Sc  (open blue circles). The
results from the UrQMD model in central Au + Au collisions are plotted as
black circles. The red arrow represents the theoretic expectation from a
critical QCD model Antoniou2006PRL .

In current experimental explorations of the intermittency in heavy-ion
collisions, the NA49 and NA61 collaborations have directly measured $\phi_{2}$
at various sizes of colliding nuclei NA49SFM ; NA49C+C ; NA61Ar+Sc , which are
represented as blue symbols in Fig. 6. The intermittency parameter at
$\sqrt{s_{\mathrm{NN}}}$ = 17.3 GeV for the Si + Si system at NA49 experiment
approaches the theoretic expectation value, shown as red arrow in the figure,
in the second-order phase transition in a critical QCD model Antoniou2006PRL .
The black circles of the UrQMD results give a flat trend with the value around
0 at all energies due to no critical mechanisms are implemented in the
transport model.

The RHIC/STAR experiment has been running the second phase of beam energy scan
(BES-II) program in 2018 - 2021 BESII . With significant improved statistics
and particle identification in BES-II, it would be interesting that STAR could
measure intermittency to explore the CP in the QCD phase diagram. Our work
provides a non-critical baseline and give a guidance of background subtraction
and efficiency correction for the calculations of intermittency in heavy-ion
collisions.

## Acknowledgements

This work is supported by the Ministry of Science and Technology (MoST) under
grant No. 2016YFE0104800, the National Key Research and Development Program of
China (Grant No. 2020YFE0202002 and 2018YFE0205201), the National Natural
Science Foundation of China (Grant No. 11828500,11828501, 11575069, 11890711
and 11861131009). The first two authors contribute equally to this work.

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