# Effect of non-uniform efficiency on higher-order cumulants in heavy-ion
collisions

Fan Si  State Key Laboratory of Particle Detection and Electronics, University
of Science and Technology of China, Hefei 230026, China  Yifei Zhang
<EMAIL_ADDRESS>State Key Laboratory of Particle Detection and Electronics,
University of Science and Technology of China, Hefei 230026, China  Xiaofeng
Luo<EMAIL_ADDRESS>Key Laboratory of Quark & Lepton Physics (MOE) and
Institute of Particle Physics, Central China Normal University, Wuhan 430079,
China

###### Abstract

We perform a systematic study on the effect of non-uniform track efficiency
correction in higher-order cumulant analysis in heavy-ion collisions. Through
analytical derivation, we find that the true values of cumulants can be
successfully reproduced by the efficiency correction with an average of the
realistic detector efficiency for particles with the same charges within each
single phase space. The theoretical conclusions are supported by a toy model
simulation by tuning the non-uniformity of the efficiency employed in the
track-by-track efficiency correction method. The valid averaged efficiency is
found to suppress the statistical uncertainties of the reproduced cumulants
dramatically. Thus, usage of the averaged efficiency requires a careful study
of phase space dependence. This study is important for carrying out precision
measurements of higher-order cumulants in heavy-ion collision experiments at
present and in future.

##  I Introduction

In nature, quarks and gluons are confined in nucleons by the strong
interaction. In the extremely hot and dense conditions created in relativistic
heavy-ion collisions, those partons can be liberated via deconfinement to form
Quark-Gluon Plasma (QGP) [1, 2], which is believed to have existed in the
early universe around tens of microseconds after the Big Bang. Experimental
evidence shows that the QGP is strongly coupled and behaves like a perfect
liquid [3, 4, 5, 6, 7, 8, 9]. Quantum Chromodynamics (QCD) predicts that at a
high temperature ($T$) and small baryon chemical potential ($\mu_{B}$), it is
a smooth crossover from the hadronic phase to the partonic phase [10]. At
relatively lower $T$ and larger $\mu_{B}$, QCD-based models predict a first-
order phase transition between these two phases [11, 12, 13]. Thus, there
should be a critical point (CP) connecting the crossover region and first-
order phase transition line. One of the most important goals of heavy-ion
collisions is to search for the possible QCD critical point in the QCD phase
diagram with sensitive observables [14, 15, 16, 17].

Higher-order cumulants of conserved charges, such as net-baryon ($B$), net-
charge ($Q$) and net-strangeness ($S$), are proposed as sensitive observables
for the search of the QCD critical point in heavy-ion collisions [18, 19, 20,
21, 22, 23]. In the first phase of the RHIC Beam Energy Scan program (BES-I,
2010-2017), experimentally measured fourth-order net-proton fluctuations
($\kappa\sigma^{2}$) in 0-5% central Au+Au collisions show a non-monotonic
energy dependence with a 3.1$\sigma$ significance [24, 25, 26, 27]. However,
due to limited event statistics, the statistical uncertainties are still large
below $\sqrt{s_{\mathrm{NN}}}$ = 20 GeV. To confirm the intriguing observation
in BES-I, the STAR experiment has conducted the second phase of the Beam
Energy Scan program (BES-II) since 2019, which focuses on the energies below
$\sqrt{s_{\mathrm{NN}}}$ = 20 GeV, with 10-20 times better statistics than
those collected in BES-I [28].

In heavy-ion collision experiments, the detector efficiency in the measurement
($\varepsilon_{\mathrm{m}}$) represents the probability of a produced particle
track recorded correctly by the detector systems and applied in later data
analysis, which is determined from the detector acceptance, particle detection
probability, and track quality selection efficiency. The
$\varepsilon_{\mathrm{m}}$ is always finite and results in loss of incoming
track signals recorded by detectors and in a distortion of the multiplicity
distributions. Due to the high sensibility of higher-order fluctuations to the
event-by-event distribution shapes, the observed values and statistical
uncertainties of higher-order cumulants are strongly affected by the finite
$\varepsilon_{\mathrm{m}}$ effect [29, 30, 31], therefore, an efficiency
correction should be carefully employed in fluctuation analysis. Moreover, the
$\varepsilon_{\mathrm{m}}$ is usually non-uniform and could depend on many
factors, such as the collision centrality, track kinematic parameters
(transverse momentum, azimuth angle and rapidity) and track crossing effect
[32]. For example, the $\varepsilon_{\mathrm{m}}$ in azimuth can be non-
uniform due to an absence of detector acceptance in dead areas or bad
electronic readout channels. The non-uniformity of $\varepsilon_{\mathrm{m}}$
could also contribute to the observed event-by-event fluctuations. However,
the efficiency employed in the efficiency correction
($\varepsilon_{\mathrm{c}}$) is usually an average of the
$\varepsilon_{\mathrm{m}}$ within one or two wide phase spaces (so-called bin-
by-bin efficiency) [33, 34, 24], rather than a realistic non-uniform
distribution in various dimensions. In a previous study, a track-by-track-
based method has been developed to apply the efficiency correction to the
higher-order cumulant analysis of conserved charges [35]. In this paper, we
further investigate the effect of non-uniformity in
$\varepsilon_{\mathrm{c}}$, which is important for performing precision
measurements of higher-order cumulants in heavy-ion collision experiments at
present and in future.

##  II Theoretical study

In this section, a theoretical study is performed to determine the validity of
$\varepsilon_{\mathrm{c}}$ by tuning its non-uniformity but keeping the same
averaged value as $\varepsilon_{\mathrm{m}}$.

###  II.1 Quantity definitions

In this paper, a phase space is defined to contain $M$ series of particles
with the multiplicity vector $\bm{X}=\left(X_{1},X_{2},\cdots,X_{M}\right)$,
following

$\tilde{P}(\bm{X})=\sum_{X}P(X)\mathcal{M}_{X,\bm{p}}(\bm{X}),$ (1)

where $\bm{p}=\left(p_{1},p_{2},\cdots,p_{M}\right)$ and
$\mathcal{M}_{X,\bm{p}}(\bm{X})$ denote the multinomial distribution defined
as

$\mathcal{M}_{X,\bm{p}}(\bm{X})=\frac{X!}{\displaystyle{\prod_{i=1}^{M}X_{i}!}}\prod_{i=1}^{M}p_{i}^{X_{i}}$
(2)

with $\displaystyle{X=\sum_{i=1}^{M}X_{i}}$ and
$\displaystyle{p_{i}=\frac{\left\langle X_{i}\right\rangle}{\langle
X\rangle}}$, in other words, particles in a phase space are produced with the
total multiplicity $X$ determined by a certain distribution $P(X)$ and
allocated into each series by a certain probability vector $\bm{p}$. For
example, if there are independent
$X_{i}\sim\mathrm{Poisson}\left(\lambda_{i}\right)$, the equation holds for
$\displaystyle{X\sim\mathrm{Poisson}\left(\sum\lambda_{i}\right)}$.

In each event of heavy-ion collisions, there may be several phase spaces in
which the particles are usually produced through different physical effects.
For example, two phase spaces are considered with two series of particles with
the multiplicities $X$ and $Y$ as non-negative integer random variables
following the probability distribution function $P(X,Y)$. Note that there may
be various types of particles with different charges belonging to a series
produced in a phase space. The linear combination of $X$ and $Y$ with the
coefficients $a$ and $b$ can be expressed by

$Q_{(a,b)}=aX+bY,$ (3)

The cumulants and factorial cumulants of $Q_{(a,b)}$ are given by [36, 37]

$\displaystyle\left\langle
Q_{(a,b)}^{k}Q_{(a^{\prime},b^{\prime})}^{l}\right\rangle_{\mathrm{c}}$
$\displaystyle=\left.\partial_{(a,b)}^{k}\partial_{(a^{\prime},b^{\prime})}^{l}K(\theta,\eta)\right|_{\theta=\eta=0},$
(4) $\displaystyle\left\langle
Q_{(a,b)}^{k}Q_{(a^{\prime},b^{\prime})}^{l}\right\rangle_{\mathrm{fc}}$
$\displaystyle=\left.\bar{\partial}_{(a,b)}^{k}\bar{\partial}_{(a^{\prime},b^{\prime})}^{l}K_{\mathrm{f}}(s,t)\right|_{s=t=1}$
(5)

from the generating functions

$\displaystyle K(\theta,\eta)$
$\displaystyle=\ln\sum_{X,Y}P(X,Y)\mathrm{e}^{\theta X+\eta Y},$ (6)
$\displaystyle K_{\mathrm{f}}(s,t)$
$\displaystyle=\ln\sum_{X,Y}P(X,Y)s^{X}t^{Y}$ (7)

with

$\displaystyle\partial_{(a,b)}$
$\displaystyle=a\frac{\partial}{\partial\theta}+b\frac{\partial}{\partial\eta},$
(8) $\displaystyle\bar{\partial}_{(a,b)}$
$\displaystyle=a\frac{\partial}{\partial s}+b\frac{\partial}{\partial t}.$ (9)

The relation between cumulants and factorial cumulants has been derived in
Ref. [36].

The two phase spaces are divided into $M$ and $N$ bins, respectively, for
example, according to the particle types and kinematic parameters. A single
particle is allocated into various bins with the probability vector
$\bm{p}=\left(p_{1},p_{2},\cdots,p_{M}\right)$ or
$\bm{q}=\left(q_{1},q_{2},\cdots,q_{N}\right)$, which can be determined by the
kinematic distributions of particles of different types. Thus, the bin-by-bin
particle numbers, represented by the vectors
$\bm{X}=\left(X_{1},X_{2},\cdots,X_{M}\right)$ and
$\bm{Y}=\left(Y_{1},Y_{2},\cdots,Y_{N}\right)$, follow the probability
distribution function

$\tilde{P}(\bm{X},\bm{Y})=\sum_{X,Y}P(X,Y)\mathcal{M}_{X,\bm{p}}(\bm{X})\mathcal{M}_{Y,\bm{q}}(\bm{Y}).$
(10)

The linear combination of the bin-by-bin particle numbers with the
coefficients $\bm{a}=\left(a_{1},a_{2},\cdots,a_{M}\right)$ and
$\bm{b}=\left(b_{1},b_{2},\cdots,b_{N}\right)$ and its cumulants and factorial
cumulants are given by

$\displaystyle Q_{(\bm{a},\bm{b})}$
$\displaystyle=\bm{a}\cdot\bm{X}+\bm{b}\cdot\bm{Y},$ (11)
$\displaystyle\left\langle
Q_{(\bm{a},\bm{b})}^{k}Q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{c}}$
$\displaystyle=\left.\partial_{(\bm{a},\bm{b})}^{k}\partial_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\tilde{K}(\bm{\theta},\bm{\eta})\right|_{\bm{\theta}=\bm{\eta}=0},$
(12) $\displaystyle\left\langle
Q_{(\bm{a},\bm{b})}^{k}Q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{fc}}$
$\displaystyle=\left.\bar{\partial}_{(\bm{a},\bm{b})}^{k}\bar{\partial}_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\tilde{K}_{\mathrm{f}}(\bm{s},\bm{t})\right|_{\bm{s}=\bm{t}=1}$
(13)

from the generating functions

$\displaystyle\tilde{K}(\bm{\theta},\bm{\eta})$
$\displaystyle=\ln\sum_{\bm{X},\bm{Y}}\tilde{P}(\bm{X},\bm{Y})\mathrm{e}^{\bm{\theta}\cdot\bm{X}+\bm{\eta}\cdot\bm{Y}},$
(14) $\displaystyle\tilde{K}_{\mathrm{f}}(\bm{s},\bm{t})$
$\displaystyle=\ln\sum_{\bm{X},\bm{Y}}\tilde{P}(\bm{X},\bm{Y})\prod_{i=1}^{M}s_{i}^{X_{i}}\prod_{j=1}^{N}t_{j}^{Y_{j}}$
(15)

with

$\displaystyle\partial_{(\bm{a},\bm{b})}$
$\displaystyle=\sum_{i=1}^{M}a_{i}\frac{\partial}{\partial\theta_{i}}+\sum_{j=1}^{N}b_{j}\frac{\partial}{\partial\eta_{j}},$
(16) $\displaystyle\bar{\partial}_{(\bm{a},\bm{b})}$
$\displaystyle=\sum_{i=1}^{M}a_{i}\frac{\partial}{\partial
s_{i}}+\sum_{j=1}^{N}b_{j}\frac{\partial}{\partial t_{j}}.$ (17)

$Q_{(\bm{a},\bm{b})}$ gives the conserved charge in the collision system if
$\bm{a}$ and $\bm{b}$ denote two series of particle charges in various bins,
respectively. For two series of particles with the same charges $a_{i}\equiv
a$ ($i$ = 1, 2, $\cdots$, $M$) and $b_{j}\equiv b$ ($j$ = 1, 2, $\cdots$,
$N$), $Q_{(\bm{a},\bm{b})}=Q_{(a,b)}$. The $k$th-order diagonal cumulant
$\left\langle Q_{(\bm{a},\bm{b})}^{k}\right\rangle_{\mathrm{c}}$ can simply be
marked by $C_{k}$.

$\tilde{K}_{\mathrm{f}}(\bm{s},\bm{t})$ can be converted into
$K_{\mathrm{f}}(s,t)$ with

$\begin{split}\tilde{K}_{\mathrm{f}}(\bm{s},\bm{t})&=\ln\sum_{X,Y}P(X,Y)\sum_{\bm{X}}\mathcal{M}_{X,\bm{p}}(\bm{X})\prod_{i=1}^{M}s_{i}^{X_{i}}\\\
&\hphantom{\quad\
}\times\sum_{\bm{Y}}\mathcal{M}_{Y,\bm{q}}(\bm{Y})\prod_{j=1}^{N}t_{j}^{Y_{j}}\\\
&=\ln\sum_{X,Y}P(X,Y)\left(\sum_{i=1}^{M}p_{i}s_{i}\right)^{X}\left(\sum_{j=1}^{N}q_{j}t_{j}\right)^{Y}\\\
&=K_{\mathrm{f}}\left(\sum_{i=1}^{M}p_{i}s_{i},\sum_{j=1}^{N}q_{j}t_{j}\right),\end{split}$
(18)

where the third line is obtained by multinomial expansion; thus,

$\bar{\partial}_{(\bm{a},\bm{b})}^{k}\bar{\partial}_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\tilde{K}_{\mathrm{f}}(\bm{s},\bm{t})=\bar{\partial}_{(\bm{a}\cdot\bm{p},\bm{b}\cdot\bm{q})}^{k}\bar{\partial}_{(\bm{a}^{\prime}\cdot\bm{p},\bm{b}^{\prime}\cdot\bm{q})}^{l}K_{\mathrm{f}}(s,t).$
(19)

Now the finite $\varepsilon_{\mathrm{m}}$ effect is taken into consideration.
A single particle in each bin is detected independently with the probability
vector $\bm{\alpha}=\left(\alpha_{1},\alpha_{2},\cdots,\alpha_{M}\right)$ or
$\bm{\beta}=\left(\beta_{1},\beta_{2},\cdots,\beta_{N}\right)$. As a result,
the probability distribution function of the measured bin-by-bin particle
numbers $\bm{x}=\left(x_{1},x_{2},\cdots,x_{M}\right)$ and
$\bm{y}=\left(y_{1},y_{2},\cdots,y_{N}\right)$ can be expressed by

$\tilde{\tilde{P}}(\bm{x},\bm{y})=\sum_{\bm{X},\bm{Y}}\tilde{P}(\bm{X},\bm{Y})\prod_{i=1}^{M}\mathcal{B}_{X_{i},\alpha_{i}}\left(x_{i}\right)\prod_{j=1}^{N}\mathcal{B}_{Y_{j},\beta_{j}}\left(y_{j}\right),$
(20)

where $\mathcal{B}_{X_{i},\alpha_{i}}\left(x_{i}\right)$ and
$\mathcal{B}_{Y_{j},\beta_{j}}\left(y_{j}\right)$ denote the binomial
distributions defined by

$\mathcal{B}_{X_{i},\alpha_{i}}\left(x_{i}\right)=\frac{X_{i}!}{x_{i}!\left(X_{i}-x_{i}\right)!}\alpha_{i}^{x_{i}}\left(1-\alpha_{i}\right)^{X_{i}-x_{i}}$
(21)

as the binomial case of the multinomial distribution. The linear combination
of the measured bin-by-bin particle numbers and its cumulants and factorial
cumulants are given by

$\displaystyle q_{(\bm{a},\bm{b})}$
$\displaystyle=\bm{a}\cdot\bm{x}+\bm{b}\cdot\bm{y},$ (22)
$\displaystyle\left\langle
q_{(\bm{a},\bm{b})}^{k}q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{c}}$
$\displaystyle=\left.\partial_{(\bm{a},\bm{b})}^{k}\partial_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\tilde{\tilde{K}}(\bm{\theta},\bm{\eta})\right|_{\bm{\theta}=\bm{\eta}=0},$
(23) $\displaystyle\left\langle
q_{(\bm{a},\bm{b})}^{k}q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{fc}}$
$\displaystyle=\left.\bar{\partial}_{(\bm{a},\bm{b})}^{k}\bar{\partial}_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\tilde{\tilde{K}}_{\mathrm{f}}(\bm{s},\bm{t})\right|_{\bm{s}=\bm{t}=1}$
(24)

from the generating functions

$\displaystyle\tilde{\tilde{K}}(\bm{\theta},\bm{\eta})$
$\displaystyle=\ln\sum_{\bm{x},\bm{y}}\tilde{\tilde{P}}(\bm{x},\bm{y})\mathrm{e}^{\bm{\theta}\cdot\bm{x}+\bm{\eta}\cdot\bm{y}},$
(25) $\displaystyle\tilde{\tilde{K}}_{\mathrm{f}}(\bm{s},\bm{t})$
$\displaystyle=\ln\sum_{\bm{x},\bm{y}}\tilde{\tilde{P}}(\bm{x},\bm{y})\prod_{i=1}^{M}s_{i}^{x_{i}}\prod_{j=1}^{N}t_{j}^{y_{j}}.$
(26)

The relation between $\tilde{\tilde{K}}_{\mathrm{f}}(\bm{s},\bm{t})$ and
$\tilde{K}_{\mathrm{f}}(\bm{s},\bm{t})$ can be derived by

$\begin{split}\tilde{\tilde{K}}_{\mathrm{f}}(\bm{s},\bm{t})&=\ln\sum_{\bm{X},\bm{Y}}\tilde{P}(\bm{X},\bm{Y})\sum_{\bm{x}}\prod_{i=1}^{M}\mathcal{B}_{X_{i},\alpha_{i}}\left(x_{i}\right)s_{i}^{x_{i}}\\\
&\hphantom{\quad\
}\times\sum_{\bm{y}}\prod_{j=1}^{N}\mathcal{B}_{Y_{j},\beta_{j}}\left(y_{j}\right)t_{j}^{y_{j}}\\\
&=\ln\sum_{\bm{X},\bm{Y}}\tilde{P}(\bm{X},\bm{Y})\prod_{i=1}^{M}\left(\alpha_{i}s_{i}+\left(1-\alpha_{i}\right)\right)^{X_{i}}\\\
&\hphantom{\quad\
}\times\prod_{j=1}^{N}\left(\beta_{j}t_{j}+\left(1-\beta_{j}\right)\right)^{Y_{j}}\\\
&=\tilde{K}_{\mathrm{f}}(\bm{s}^{\prime},\bm{t}^{\prime}),\end{split}$ (27)

where $s^{\prime}_{i}=\alpha_{i}s_{i}+\left(1-\alpha_{i}\right)$ ($i$ = 1, 2,
$\cdots$, $M$) and $t^{\prime}_{j}=\beta_{j}t_{j}+\left(1-\beta_{j}\right)$
($j$ = 1, 2, $\cdots$, $N$); thus,

$\bar{\partial}_{(\bm{a},\bm{b})}^{k}\bar{\partial}_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\tilde{\tilde{K}}_{\mathrm{f}}(\bm{s},\bm{t})=\bar{\partial}_{(\bm{a}\bm{\alpha},\bm{b}\bm{\beta})}^{k}\bar{\partial}_{(\bm{a}^{\prime}\bm{\alpha},\bm{b}^{\prime}\bm{\beta})}^{l}\tilde{K}_{\mathrm{f}}(\bm{s},\bm{t}).$
(28)

Here, we define the expression

$\bar{\partial}_{(\bm{a}\bm{\alpha}/\bm{\alpha}^{\prime},\bm{b}\bm{\beta}/\bm{\beta}^{\prime})}=\sum_{i=1}^{M}\frac{a_{i}\alpha_{i}}{\alpha^{\prime}_{i}}\frac{\partial}{\partial
s_{i}}+\sum_{j=1}^{N}\frac{b_{j}\beta_{j}}{\beta^{\prime}_{j}}\frac{\partial}{\partial
t_{j}},$ (29)

which is similar to
$Q_{(\bm{a}\bm{\alpha}/\bm{\alpha}^{\prime},\bm{b}\bm{\beta}/\bm{\beta}^{\prime})}$,
$q_{(\bm{a}\bm{\alpha}/\bm{\alpha}^{\prime},\bm{b}\bm{\beta}/\bm{\beta}^{\prime})}$,
and
$\partial_{(\bm{a}\bm{\alpha}/\bm{\alpha}^{\prime},\bm{b}\bm{\beta}/\bm{\beta}^{\prime})}$.

In the efficiency correction using the realistic detector efficiency, the
relations between the true $\left\langle
Q_{(\bm{a},\bm{b})}^{k}Q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{c}}$
and the measured $\left\langle
q_{(\bm{a},\bm{b})}^{k}q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{c}}$
are derived by three steps:

$\left\langle Q_{(\bm{a},\bm{b})}^{k}Q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{c}}$ | $\leftrightarrow$ | $\left\langle Q_{(\bm{a},\bm{b})}^{k}Q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{fc}}$
---|---|---
|  | $\updownarrow$
$\left\langle q_{(\bm{a},\bm{b})}^{k}q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{c}}$ | $\leftrightarrow$ | $\left\langle q_{(\bm{a},\bm{b})}^{k}q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{fc}}$

For example, the true diagonal cumulants ($C_{k}^{\mathrm{true}}$) up to
third-order can be obtained from [36, 37]

$\displaystyle C_{1}^{\mathrm{true}}=\left\langle
Q_{(\bm{a},\bm{b})}\right\rangle_{\mathrm{c}}$ $\displaystyle=\left\langle
Q_{(\bm{a},\bm{b})}\right\rangle_{\mathrm{fc}}=\left\langle
q_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}\right\rangle_{\mathrm{fc}}=\left\langle
q_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}\right\rangle_{\mathrm{c}}=C_{1}^{\mathrm{corr}}(\bm{\alpha},\bm{\beta};\bm{\alpha},\bm{\beta}),$
(30) $\displaystyle\begin{split}C_{2}^{\mathrm{true}}=\left\langle
Q_{(\bm{a},\bm{b})}^{2}\right\rangle_{\mathrm{c}}&=\left\langle
Q_{(\bm{a},\bm{b})}^{2}\right\rangle_{\mathrm{fc}}+\left\langle
Q_{\left(\bm{a}^{2},\bm{b}^{2}\right)}\right\rangle_{\mathrm{fc}}=\left\langle
q_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}^{2}\right\rangle_{\mathrm{fc}}+\left\langle
q_{\left(\bm{a}^{2}/\bm{\alpha},\bm{b}^{2}/\bm{\beta}\right)}\right\rangle_{\mathrm{fc}}\\\
&=\left\langle
q_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}^{2}\right\rangle_{\mathrm{c}}-\left\langle
q_{\left(\bm{a}^{2}/\bm{\alpha}^{2},\bm{b}^{2}/\bm{\beta}^{2}\right)}\right\rangle_{\mathrm{c}}+\left\langle
q_{\left(\bm{a}^{2}/\bm{\alpha},\bm{b}^{2}/\bm{\beta}\right)}\right\rangle_{\mathrm{c}}=C_{2}^{\mathrm{corr}}(\bm{\alpha},\bm{\beta};\bm{\alpha},\bm{\beta}),\end{split}$
(31) $\displaystyle\begin{split}C_{3}^{\mathrm{true}}=\left\langle
Q_{(\bm{a},\bm{b})}^{3}\right\rangle_{\mathrm{c}}&=\left\langle
Q_{(\bm{a},\bm{b})}^{3}\right\rangle_{\mathrm{fc}}+3\left\langle
Q_{(\bm{a},\bm{b})}Q_{\left(\bm{a}^{2},\bm{b}^{2}\right)}\right\rangle_{\mathrm{fc}}+\left\langle
Q_{\left(\bm{a}^{3},\bm{b}^{3}\right)}\right\rangle_{\mathrm{fc}}\\\
&=\left\langle
q_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}^{3}\right\rangle_{\mathrm{fc}}+3\left\langle
q_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}q_{\left(\bm{a}^{2}/\bm{\alpha},\bm{b}^{2}/\bm{\beta}\right)}\right\rangle_{\mathrm{fc}}+\left\langle
q_{\left(\bm{a}^{3}/\bm{\alpha},\bm{b}^{3}/\bm{\beta}\right)}\right\rangle_{\mathrm{fc}}\\\
&=\left\langle
q_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}^{3}\right\rangle_{\mathrm{c}}-3\left\langle
q_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}q_{\left(\bm{a}^{2}/\bm{\alpha}^{2},\bm{b}^{2}/\bm{\beta}^{2}\right)}\right\rangle_{\mathrm{c}}+2\left\langle
q_{\left(\bm{a}^{3}/\bm{\alpha}^{3},\bm{b}^{3}/\bm{\beta}^{3}\right)}\right\rangle_{\mathrm{c}}\\\
&\hphantom{\quad\ }+3\left\langle
q_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}q_{\left(\bm{a}^{2}/\bm{\alpha},\bm{b}^{2}/\bm{\beta}\right)}\right\rangle_{\mathrm{c}}-3\left\langle
q_{\left(\bm{a}^{3}/\bm{\alpha}^{2},\bm{b}^{3}/\bm{\beta}^{2}\right)}\right\rangle_{\mathrm{c}}+\left\langle
q_{\left(\bm{a}^{3}/\bm{\alpha},\bm{b}^{3}/\bm{\beta}\right)}\right\rangle_{\mathrm{c}}\\\
&=C_{3}^{\mathrm{corr}}(\bm{\alpha},\bm{\beta};\bm{\alpha},\bm{\beta}),\end{split}$
(32)

where
$C_{k}^{\mathrm{corr}}(\varepsilon_{\mathrm{m}};\varepsilon_{\mathrm{c}})$
denotes $C_{k}$ corrected with $\varepsilon_{\mathrm{c}}$ from the cumulants
of $q_{(\bm{a},\bm{b})}$ measured with $\varepsilon_{\mathrm{m}}$.

During an efficiency correction with another set of efficiency
$\bm{\alpha}^{\prime}$ and $\bm{\beta}^{\prime}$, the detector efficiency
$\bm{\alpha}$ and $\bm{\beta}$ in the formulae of the relations between
$\left\langle
Q_{(\bm{a},\bm{b})}^{k}Q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{c}}$
and $\left\langle
q_{(\bm{a},\bm{b})}^{k}q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{c}}$
are replaced with $\bm{\alpha}^{\prime}$ and $\bm{\beta}^{\prime}$,
respectively, such that
$C_{k}^{\mathrm{corr}}(\bm{\alpha},\bm{\beta};\bm{\alpha}^{\prime},\bm{\beta}^{\prime})$.

###  II.2 Internally averaged efficiency correction

The averaged efficiency (AE) is defined as an average of the
$\varepsilon_{\mathrm{m}}$ taken in some or all of the bins divided from the
phase spaces, which is uniform in these average bins and inherits the same
distribution as the $\varepsilon_{\mathrm{m}}$ elsewhere. The AE maintains the
same averaged value, and its non-uniformity decreases as the average bins
increase. According to the relation between phase spaces and average bins, the
AE is divided into the internally averaged efficiency (IAE) and the externally
averaged efficiency (EAE). The EAE denotes the AE with average bins across
multiple phase spaces. Conversely, the IAE is defined as an average taken
within each single phase space (not across phase spaces). In this subsection,
the validity of the IAE correction is studied.

For example, the efficiency employed in the IAE correction is set to

$\displaystyle\bm{\alpha}^{\prime}$
$\displaystyle=\left(\bar{\alpha},\cdots,\bar{\alpha},\alpha_{m+1},\cdots,\alpha_{M}\right),$
(33) $\displaystyle\bm{\beta}^{\prime}$
$\displaystyle=\left(\bar{\beta},\cdots,\bar{\beta},\beta_{n+1},\cdots,\beta_{N}\right),$
(34)

with

$\displaystyle\bar{\alpha}$
$\displaystyle=\frac{\displaystyle{\sum_{i=1}^{m}\left\langle
x_{i}\right\rangle}}{\displaystyle{\sum_{i=1}^{m}\left\langle
X_{i}\right\rangle}}=\frac{\displaystyle{\sum_{i=1}^{m}\alpha_{i}\left\langle
X_{i}\right\rangle}}{\displaystyle{\sum_{i=1}^{m}\left\langle
X_{i}\right\rangle}}=\frac{\displaystyle{\sum_{i=1}^{m}\alpha_{i}p_{i}}}{\displaystyle{\sum_{i=1}^{m}p_{i}}},$
(35) $\displaystyle\bar{\beta}$
$\displaystyle=\frac{\displaystyle{\sum_{j=1}^{n}\left\langle
y_{j}\right\rangle}}{\displaystyle{\sum_{j=1}^{n}\left\langle
Y_{j}\right\rangle}}=\frac{\displaystyle{\sum_{j=1}^{n}\beta_{j}\left\langle
Y_{j}\right\rangle}}{\displaystyle{\sum_{j=1}^{n}\left\langle
Y_{j}\right\rangle}}=\frac{\displaystyle{\sum_{j=1}^{n}\beta_{j}q_{j}}}{\displaystyle{\sum_{j=1}^{n}q_{j}}}$
(36)

as the average of the detector efficiency $\bm{\alpha}$ or $\bm{\beta}$ in the
first $m$ or $n$ bins of a single phase space, respectively. One can find that

$\displaystyle\frac{\alpha_{i}}{\alpha^{\prime}_{i}}$
$\displaystyle=\left\\{\begin{aligned}
&\frac{\alpha_{i}}{\bar{\alpha}},&&i=1,2,\cdots,m\\\
&1,&&i=m+1,m+2,\cdots,M\end{aligned}\right.$ (37)
$\displaystyle\frac{\beta_{j}}{\beta^{\prime}_{j}}$
$\displaystyle=\left\\{\begin{aligned}
&\frac{\beta_{j}}{\bar{\beta}},&&j=1,2,\cdots,n\\\
&1,&&j=n+1,n+2,\cdots,N\end{aligned}\right.$ (38)

and

$\displaystyle\sum_{i=1}^{m}{\frac{\alpha_{i}}{\alpha^{\prime}_{i}}p_{i}}$
$\displaystyle=\sum_{i=1}^{m}p_{i}$ (39)
$\displaystyle\sum_{j=1}^{n}{\frac{\beta_{j}}{\beta^{\prime}_{j}}q_{j}}$
$\displaystyle=\sum_{j=1}^{n}q_{j}.$ (40)

If the particle charges are the same in all average bins, that is,
$a_{1}=a_{2}=\cdots=a_{m}$ and $b_{1}=b_{2}=\cdots=b_{n}$, it can be derived
that

$\begin{split}&\hphantom{\quad\
}\bar{\partial}_{(\bm{a}/\bm{\alpha}^{\prime},\bm{b}/\bm{\beta}^{\prime})}\tilde{\tilde{K}}_{\mathrm{f}}(\bm{s},\bm{t})\\\
&=\bar{\partial}_{(\bm{a}\bm{\alpha}/\bm{\alpha}^{\prime},\bm{b}\bm{\beta}/\bm{\beta}^{\prime})}\tilde{K}_{\mathrm{f}}(\bm{s},\bm{t})\\\
&=\left(\sum_{i=1}^{M}\frac{a_{i}\alpha_{i}}{\alpha^{\prime}_{i}}p_{i}\frac{\partial}{\partial
s}+\sum_{j=1}^{N}\frac{b_{j}\beta_{j}}{\beta^{\prime}_{j}}q_{j}\frac{\partial}{\partial
t}\right)K_{\mathrm{f}}(s,t)\\\
&=\left(\sum_{i=1}^{M}a_{i}p_{i}\frac{\partial}{\partial
s}+\sum_{j=1}^{N}b_{j}q_{j}\frac{\partial}{\partial
t}\right)K_{\mathrm{f}}(s,t)\\\
&=\bar{\partial}_{(\bm{a},\bm{b})}\tilde{K}_{\mathrm{f}}(\bm{s},\bm{t})\\\
&=\bar{\partial}_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}\tilde{\tilde{K}}_{\mathrm{f}}(\bm{s},\bm{t})\end{split}$
(41)

and so forth,

$\begin{split}\left\langle
Q_{(\bm{a},\bm{b})}^{k}Q_{(\bm{a}^{\prime},\bm{b}^{\prime})}^{l}\right\rangle_{\mathrm{fc}}&=\left\langle
q_{(\bm{a}/\bm{\alpha},\bm{b}/\bm{\beta})}^{k}q_{(\bm{a}^{\prime}/\bm{\alpha},\bm{b}^{\prime}/\bm{\beta})}^{l}\right\rangle_{\mathrm{fc}}\\\
&=\left\langle
q_{(\bm{a}/\bm{\alpha}^{\prime},\bm{b}/\bm{\beta}^{\prime})}^{k}q_{(\bm{a}^{\prime}/\bm{\alpha}^{\prime},\bm{b}^{\prime}/\bm{\beta}^{\prime})}^{l}\right\rangle_{\mathrm{fc}},\end{split}$
(42)

which denotes that the IAE $\bm{\alpha}^{\prime}$ and $\bm{\beta}^{\prime}$
can also be employed to reproduce the factorial cumulants and even be
converted into cumulants as well as the realistic detector efficiency
$\bm{\alpha}$ and $\bm{\beta}$. As a check,
$C_{k}^{\mathrm{corr}}(\bm{\alpha},\bm{\beta};\bm{\alpha}^{\prime},\bm{\beta}^{\prime})=C_{k}^{\mathrm{true}}$
if all of the $\bm{\alpha}$ and $\bm{\beta}$ in Eqs. (30)-(32) are replaced
with $\bm{\alpha}^{\prime}$ and $\bm{\beta}^{\prime}$, respectively. Thus, the
IAE correction should be valid unless the average of the detector efficiency
is taken for particles with different charges.

###  II.3 Externally averaged efficiency correction

In this subsection, the validity of the EAE correction is further discussed.

Simply, particles with the same charge $a=b=1$ are considered and each phase
space is divided into one bin with $M=N=1$. The EAE correction applies the AE
with an average bin across two phase spaces as

$\bar{\varepsilon}=\frac{\langle x\rangle+\langle y\rangle}{\langle
X\rangle+\langle Y\rangle}=\frac{\alpha\langle X\rangle+\beta\langle
Y\rangle}{\langle X\rangle+\langle Y\rangle}.$ (43)

By replacing the realistic detector efficiency in Eq. (30) with the EAE, the
first-order cumulant is corrected with

$\begin{split}C_{1}^{\mathrm{corr}}(\alpha,\beta;\bar{\varepsilon},\bar{\varepsilon})&=\left\langle\frac{1}{\bar{\varepsilon}}x+\frac{1}{\bar{\varepsilon}}y\right\rangle_{\mathrm{fc}}=\left\langle\frac{\alpha}{\bar{\varepsilon}}X+\frac{\beta}{\bar{\varepsilon}}Y\right\rangle_{\mathrm{fc}}\\\
&=\langle X+Y\rangle_{\mathrm{c}}=C_{1}^{\mathrm{true}},\end{split}$ (44)

which is successful for the first-order cumulant. However, to correct the
second-order cumulant in the same way, the correction is obtained by

$\begin{split}&\hphantom{\quad\
}C_{2}^{\mathrm{corr}}(\alpha,\beta;\bar{\varepsilon},\bar{\varepsilon})\\\
&=\left\langle\left(\frac{1}{\bar{\varepsilon}}x+\frac{1}{\bar{\varepsilon}}y\right)^{2}\right\rangle_{\mathrm{fc}}+\left\langle\frac{1}{\bar{\varepsilon}}x+\frac{1}{\bar{\varepsilon}}y\right\rangle_{\mathrm{fc}}\\\
&=\left\langle\left(\frac{\alpha}{\bar{\varepsilon}}X+\frac{\beta}{\bar{\varepsilon}}Y\right)^{2}\right\rangle_{\mathrm{fc}}+\left\langle\frac{\alpha}{\bar{\varepsilon}}X+\frac{\beta}{\bar{\varepsilon}}Y\right\rangle_{\mathrm{fc}}\\\
&=\left\langle\left(\frac{\alpha}{\bar{\varepsilon}}X+\frac{\beta}{\bar{\varepsilon}}Y\right)^{2}\right\rangle_{\mathrm{c}}+\left\langle\left(\frac{\alpha}{\bar{\varepsilon}}-\frac{\alpha^{2}}{\bar{\varepsilon}^{2}}\right)X+\left(\frac{\beta}{\bar{\varepsilon}}-\frac{\beta^{2}}{\bar{\varepsilon}^{2}}\right)Y\right\rangle_{\mathrm{c}}\\\
&\not\equiv\left\langle(X+Y)^{2}\right\rangle_{\mathrm{c}}=C_{2}^{\mathrm{true}},\end{split}$
(45)

which is valid only in a few specific cases, such as that the detector
efficiency $\alpha=\beta=1$ and that $X$ and $Y$ follow the distribution in
Eq. (1) (a single phase space). The corrections for higher-order cumulants are
similar to the second-order case. In general, the EAE cannot reproduce
cumulants higher than the first order.

The above two subsections fundamentally provide the validity of the AE
correction and its requirements, which can give a reasonable explanation for
the numerical results of the AE correction under various conditions reported
in Ref. [36].

###  II.4 Non-averaged efficiency correction

In this subsection, the validity of the efficiency correction is studied by
using the non-averaged efficiency (NAE), which is defined by the efficiency
with

  * -

different non-uniformity from any AE,

  * -

different non-uniformity from the $\varepsilon_{\mathrm{m}}$,

  * -

the same averaged value as the $\varepsilon_{\mathrm{m}}$.

First, only one phase space is considered ($Y\equiv 0$), which is divided into
two bins with $M=2$. Particles with the same charge $a_{1}=a_{2}=1$ are
considered. The NAE employed in the correction is marked by
$\bm{\alpha}^{\prime}=\left(\alpha^{\prime}_{1},\alpha^{\prime}_{2}\right)$,
whose averaged value remains the same as the detector efficiency as

$\alpha=\frac{\alpha_{1}\left\langle X_{1}\right\rangle+\alpha_{2}\left\langle
X_{2}\right\rangle}{\left\langle X_{1}\right\rangle+\left\langle
X_{2}\right\rangle}=\frac{\alpha^{\prime}_{1}\left\langle
X_{1}\right\rangle+\alpha^{\prime}_{2}\left\langle
X_{2}\right\rangle}{\left\langle X_{1}\right\rangle+\left\langle
X_{2}\right\rangle},$ (46)

and the efficiency can be rephrased with

$\displaystyle\alpha_{1}$ $\displaystyle=\alpha-h\left\langle
X_{2}\right\rangle,$ (47) $\displaystyle\alpha_{2}$
$\displaystyle=\alpha+h\left\langle X_{1}\right\rangle,$ (48)
$\displaystyle\alpha^{\prime}_{1}$
$\displaystyle=\alpha-h^{\prime}\left\langle X_{2}\right\rangle,$ (49)
$\displaystyle\alpha^{\prime}_{2}$
$\displaystyle=\alpha+h^{\prime}\left\langle X_{1}\right\rangle.$ (50)

The NAE correction for the first-order cumulant is expressed by

$\begin{split}&\hphantom{\quad\
}C_{1}^{\mathrm{corr}}(\bm{\alpha};\bm{\alpha}^{\prime})\\\
&=\left\langle\frac{x_{1}}{\alpha^{\prime}_{1}}+\frac{x_{2}}{\alpha^{\prime}_{2}}\right\rangle_{\mathrm{c}}\\\
&=\left\langle\frac{\alpha_{1}}{\alpha^{\prime}_{1}}X_{1}+\frac{\alpha_{2}}{\alpha^{\prime}_{2}}X_{2}\right\rangle_{\mathrm{c}}\\\
&=\left\langle\frac{\alpha-h\left\langle
X_{2}\right\rangle}{\alpha-h^{\prime}\left\langle
X_{2}\right\rangle}X_{1}+\frac{\alpha+h\left\langle
X_{1}\right\rangle}{\alpha+h^{\prime}\left\langle
X_{1}\right\rangle}X_{2}\right\rangle_{\mathrm{c}}\\\ &=\left\langle
X_{1}+X_{2}\right\rangle_{\mathrm{c}}\left(1+\frac{h^{\prime}\left(h^{\prime}-h\right)\left\langle
X_{1}\right\rangle\left\langle
X_{2}\right\rangle}{\left(\alpha+h^{\prime}\left\langle
X_{1}\right\rangle\right)\left(\alpha-h^{\prime}\left\langle
X_{2}\right\rangle\right)}\right)\\\ &\not\equiv\left\langle
X_{1}+X_{2}\right\rangle_{\mathrm{c}}=C_{1}^{\mathrm{true}},\end{split}$ (51)

which is valid only if $h^{\prime}=0$, $h^{\prime}=h$ or $\left\langle
X_{1}\right\rangle\left\langle X_{2}\right\rangle=0$. The cases $h^{\prime}=0$
and $h^{\prime}=h$ represent the efficiency corrections using the AE and the
realistic detector efficiency, respectively, rather than the NAE. If
$\left\langle X_{1}\right\rangle\left\langle X_{2}\right\rangle=0$, there is
only one bin in the phase space, which does not meet the requirement $M=2$.
The corrections for higher-order cumulants are also invalid.

Second, two phase spaces are considered with $a=b=1$ and $M=N=1$. The averaged
values of the detector efficiency and the NAE $\alpha^{\prime}$ and
$\beta^{\prime}$ are the same as

$\bar{\varepsilon}=\frac{\alpha\langle X\rangle+\beta\langle Y\rangle}{\langle
X\rangle+\langle Y\rangle}=\frac{\alpha^{\prime}\langle
X\rangle+\beta^{\prime}\langle Y\rangle}{\langle X\rangle+\langle Y\rangle}.$
(52)

The efficiency correction is also invalid in the same way, for example,

$C_{1}^{\mathrm{corr}}(\alpha,\beta;\alpha^{\prime},\beta^{\prime})\not\equiv
C_{1}^{\mathrm{true}}.$ (53)

With the above two cases considered, the true values of the cumulants cannot
be reproduced by the efficiency correction with the NAE.

##  III Toy model analysis with internally averaged efficiency

Figure 1: (Color online) Correlations of produced and measured numbers of
positively and negatively charged particles. The top panel shows the produced
distribution, and others show the distributions measured with
$\varepsilon_{i}$ ($i$ = 0, 1, $\cdots$, 5) defined in Eqs. (65)-(70),
respectively. Due to the equivalence of the averaged values of the efficiency,
the measured distributions are indistinguishable.

In this section, a toy Monte Carlo model with several sets of IAE with various
non-uniformity is employed to check the analytical derivation in the previous
section and study the effect of non-uniformity of the
$\varepsilon_{\mathrm{c}}$ on the reproduced cumulants up to fourth-order.

###  III.1 Event production

Figure 2: (Color online) Distributions of produced and measured
$p_{\mathrm{T}}$ and $\varphi$ of positively (top) and negatively (bottom)
charged particles. The produced $p_{\mathrm{T}}$ and $\varphi$ distributions
are shown as black curves, and the colored ones represent the distributions
measured with $\varepsilon_{i}$ ($i$ = 0, 1, $\cdots$, 5) defined in Eqs.
(65)-(70), respectively. The measured $p_{\mathrm{T}}$ distributions with
$\varepsilon_{i}$ ($i$ = 0, 1, $\cdots$, 4) are indistinguishable due to their
same $u\left(p_{\mathrm{T}}\right)$ components, and with the same
$v(\varphi)$, the measured $\varphi$ distributions with $\varepsilon_{4}$ and
$\varepsilon_{5}$ almost overlap. Figure 3: (Color online) Two independent
one-dimensional functions $u\left(p_{\mathrm{T}}\right)$ and $v(\varphi)$
defined in Eqs. (61) and (62). The left panel shows
$u\left(p_{\mathrm{T}}\right)$ for positively and negatively charged particles
as red and blue solid curves, respectively, and the right panel shows
$v(\varphi)$ with various parameters as $\left(p_{3},p_{4}\right)$ = (0.4, 3),
(0.4, 2), (0.2, 3), and (0.2, 2). Their averaged values defined in Eqs. (63)
and (64) are also shown as horizontal dashed lines.

We produce $10^{7}$ events with the numbers of positively and negatively
charged particles ($N_{\mathrm{pos}}$ and $N_{\mathrm{neg}}$) with the charges
$a=1$ and $b=-1$ following two independent Poisson distributions

$\displaystyle N_{\mathrm{pos}}$
$\displaystyle\sim\mathrm{Poisson}(\lambda_{1}),$ (54) $\displaystyle
N_{\mathrm{neg}}$ $\displaystyle\sim\mathrm{Poisson}(\lambda_{2}),$ (55)

where $\lambda_{1}$ = 12 and $\lambda_{2}$ = 8. The 2D distribution of
$N_{\mathrm{pos}}$ and $N_{\mathrm{neg}}$ is shown in the top panel of Fig. 1.
These particles belong to the same phase space as their multiplicity
distributions follow Eq. (1).

Thus, the produced net-charge $N_{\mathrm{net}}$ (=
$N_{\mathrm{pos}}-N_{\mathrm{neg}}$) follows the Skellam distribution [30]

$N_{\mathrm{net}}\sim\mathrm{Skellam}(\lambda_{1},\lambda_{2}),$ (56)

and the theoretical cumulants up to fourth-order can be calculated as

$\displaystyle C_{1}^{N_{\mathrm{net}}}=C_{3}^{N_{\mathrm{net}}}$
$\displaystyle=\lambda_{1}-\lambda_{2},$ (57) $\displaystyle
C_{2}^{N_{\mathrm{net}}}=C_{4}^{N_{\mathrm{net}}}$
$\displaystyle=\lambda_{1}+\lambda_{2}.$ (58)

Each of the particles in the produced events is allocated two parameters
$p_{\mathrm{T}}$ and $\varphi$ sampled from

$\displaystyle f\left(p_{\mathrm{T}}\right)$ $\displaystyle\sim
p_{\mathrm{T}}\mathrm{exp}\left(-p_{\mathrm{T}}\middle/t\right),0.4\leq
p_{\mathrm{T}}<2,$ (59) $\displaystyle g(\varphi)$
$\displaystyle\sim\mathrm{Uniform}(0,2\pi),$ (60)

where $t$ = 0.26 and 0.22 for positively and negatively charged particles,
respectively. Distributions of the produced $p_{\mathrm{T}}$ and $\varphi$ are
shown as black curves in Fig. 2. Note that the parameters $p_{\mathrm{T}}$ and
$\varphi$ are abstract parameters in terms of mathematics. Once they are
applied to the realistic case in heavy-ion collision experiments,
$p_{\mathrm{T}}$ and $\varphi$ commonly represent the transverse momentum and
azimuth angle of the particle with GeV/$c$ and rad as units, respectively. In
this case, we ignore the units of $p_{\mathrm{T}}$ and $\varphi$ in the
following discussions.

###  III.2 Efficiency definition

The track efficiency is considered, for example, to be
$p_{\mathrm{T}}$-and-$\varphi$-dependent as a two-dimensional function. Two
independent one-dimensional components of efficiency are defined as

$u\left(p_{\mathrm{T}}\right)=p_{0}\mathrm{exp}\left(-\left(p_{1}\middle/p_{\mathrm{T}}\right)^{p_{2}}\right),$
(61)

where $\left(p_{0},p_{1},p_{2}\right)$ = (0.7, 0.4, 4.8) for positively and
(0.6, 0.4, 4.2) for negatively charged particles, respectively, and

$v(\varphi)=\left\\{\begin{aligned}
&1-p_{3}\left(p_{4}-1\right),&k\leq\left.\varphi\middle/\dfrac{\pi}{6}\right.<k+\dfrac{1}{p_{4}},\\\
&1+p_{3},&k+\dfrac{1}{p_{4}}\leq\left.\varphi\middle/\dfrac{\pi}{6}\right.<k+1.\end{aligned}\right.(k\in\mathbb{Z})$
(62)

Figure 3 shows $u\left(p_{\mathrm{T}}\right)$ for positively and negatively
charged particles as solid curves in the left panel and $v(\varphi)$ with
various parameters as $\left(p_{3},p_{4}\right)$ = (0.4, 3), (0.4, 2), (0.2,
3), and (0.2, 2) in the right panel. Their averaged values weighted by the
input particle $p_{\mathrm{T}}$ or $\varphi$ distribution shown as horizontal
dashed lines in Fig. 3 are defined as

$\displaystyle\mu\left[u\left(p_{\mathrm{T}}\right)\right]$
$\displaystyle=\dfrac{\displaystyle{\int_{0.4}^{2}f\left(p_{\mathrm{T}}\right)u\left(p_{\mathrm{T}}\right)\mathrm{d}p_{\mathrm{T}}}}{\displaystyle{\int_{0.4}^{2}f\left(p_{\mathrm{T}}\right)\mathrm{d}p_{\mathrm{T}}}},$
(63) $\displaystyle\mu\left[v(\varphi)\right]$
$\displaystyle=\dfrac{\displaystyle{\int_{0}^{2\pi}g(\varphi)v(\varphi)\mathrm{d}\varphi}}{\displaystyle{\int_{0}^{2\pi}g(\varphi)\mathrm{d}\varphi}}\equiv
1,$ (64)

respectively. There are 12 dips of $v(\varphi)$ with different widths as
$p_{4}$ varies, which are proxies for the low efficiency in 12 $\varphi$ bins
due to the dead zone between every two adjacent Time Projection Chamber (TPC)
sectors surrounding the heavy-ion beam [32].

Now, the six sets of track efficiency with different non-uniformity can be
expressed as the products of Eqs. (61) and (62) given by

$\displaystyle\varepsilon_{0}\left(p_{\mathrm{T}},\varphi\right)$
$\displaystyle=u\left(p_{\mathrm{T}}\right)v(\varphi),\left(p_{3},p_{4}\right)=(0.4,3),$
(65) $\displaystyle\varepsilon_{1}\left(p_{\mathrm{T}},\varphi\right)$
$\displaystyle=u\left(p_{\mathrm{T}}\right)v(\varphi),\left(p_{3},p_{4}\right)=(0.4,2),$
(66) $\displaystyle\varepsilon_{2}\left(p_{\mathrm{T}},\varphi\right)$
$\displaystyle=u\left(p_{\mathrm{T}}\right)v(\varphi),\left(p_{3},p_{4}\right)=(0.2,3),$
(67) $\displaystyle\varepsilon_{3}\left(p_{\mathrm{T}},\varphi\right)$
$\displaystyle=u\left(p_{\mathrm{T}}\right)v(\varphi),\left(p_{3},p_{4}\right)=(0.2,2),$
(68) $\displaystyle\varepsilon_{4}\left(p_{\mathrm{T}},\varphi\right)$
$\displaystyle=u\left(p_{\mathrm{T}}\right),$ (69)
$\displaystyle\varepsilon_{5}\left(p_{\mathrm{T}},\varphi\right)$
$\displaystyle=\mu\left[u\left(p_{\mathrm{T}}\right)\right],$ (70)

whose averaged values and variances are defined as

$\displaystyle\begin{split}\mu\left[\varepsilon_{i}\left(p_{\mathrm{T}},\varphi\right)\right]&=\dfrac{\displaystyle{\int_{0.4}^{2}\int_{0}^{2\pi}f\left(p_{\mathrm{T}}\right)g(\varphi)\varepsilon_{i}\left(p_{\mathrm{T}},\varphi\right)\mathrm{d}p_{\mathrm{T}}\mathrm{d}\varphi}}{\displaystyle{\int_{0.4}^{2}\int_{0}^{2\pi}f\left(p_{\mathrm{T}}\right)g(\varphi)\mathrm{d}p_{\mathrm{T}}\mathrm{d}\varphi}}\\\
&=\mu\left[u\left(p_{\mathrm{T}}\right)\right],\end{split}$ (71)
$\displaystyle\sigma^{2}\left[\varepsilon_{i}\left(p_{\mathrm{T}},\varphi\right)\right]$
$\displaystyle=\mu\left[\varepsilon_{i}^{2}\left(p_{\mathrm{T}},\varphi\right)\right]-\mu^{2}\left[\varepsilon_{i}\left(p_{\mathrm{T}},\varphi\right)\right]$
(72)

for $i$ = 0, 1, $\cdots$, 5, respectively. The $\mu$ and $\sigma^{2}$ of the
efficiency are not the same for positively and negatively charged particles
due to the different $p_{\mathrm{T}}$ distributions and different parameters
of $u\left(p_{\mathrm{T}}\right)$. Note that the averaged value of
$\varepsilon_{i}\left(p_{\mathrm{T}},\varphi\right)$,
$\mu\left[\varepsilon_{i}\left(p_{\mathrm{T}},\varphi\right)\right]$ remains
constant for $i$ = 0, 1, $\cdots$, 5. Statistical uncertainties of efficiency-
corrected cumulants as a function of AE with a uniform distribution have been
studied in Ref. [30]. Table 1 summarizes the
$\left.\sigma^{2}\middle/\mu\right.$ values quantifying the non-uniformity of
various sets of efficiency. As $i$ ($i$ = 0, 1, $\cdots$, 5) increases,
$\left.\sigma^{2}\middle/\mu\right.$ of $\varepsilon_{i}$ decreases
monotonically to 0, which represents that $\varepsilon_{i}$ becomes less non-
uniform and that $\varepsilon_{5}$ is completely uniform.

Table 1: The $\left.\sigma^{2}\middle/\mu\right.$ values of various sets of efficiency for positively and negatively charged particles. Efficiency | Pos. | Neg.
---|---|---
$\varepsilon_{0}$ | 0.222 | 0.185
$\varepsilon_{1}$ | 0.123 | 0.105
$\varepsilon_{2}$ | 0.074 | 0.065
$\varepsilon_{3}$ | 0.050 | 0.045
$\varepsilon_{4}$ | 0.025 | 0.025
$\varepsilon_{5}$ | 0 | 0

###  III.3 Measurement and efficiency correction

Each particle in the produced events is sampled with six sets of track
efficiency, defined in Eqs. (65)-(70), as the probability. The 2D
distributions of the measured numbers of positively and negatively charged
particles ($n_{\mathrm{pos}}$ and $n_{\mathrm{neg}}$) are shown in Fig. 1, and
the $p_{\mathrm{T}}$ and $\varphi$ distributions are shown as colored curves
in Fig. 2.

The so-called track-by-track efficiency corrections [35] with six sets of
track efficiency are performed for each of the six measurements to obtain
$C_{k}^{\mathrm{corr}}\left(\varepsilon_{i};\varepsilon_{j}\right)$ ($i$, $j$
= 0, 1, $\cdots$, 5 and $k$ = 1, 2, 3, 4). Note that
$\varepsilon_{\mathrm{c}}$ may not be the same as $\varepsilon_{\mathrm{m}}$.

The above procedures, including the production, measurement and efficiency
correction, are repeated 1000 times independently. $C_{k}^{\mathrm{corr}}$
compared with $C_{k}^{\mathrm{true}}$ are shown in Appendix A. Here,
$\hat{\mu}$ and $\hat{\sigma}$ are defined as the mean value and the standard
deviation of $C_{k}^{\mathrm{corr}}$, respectively, which can be found at the
bottom of each panel of Figs. 12-17 in Appendix A. Figure 4 shows $\chi$ =
$\left.\left(\hat{\mu}-C_{k}^{\mathrm{true}}\right)\middle/\hat{\sigma}\right.$
for various $\left(\varepsilon_{\mathrm{m}};\varepsilon_{\mathrm{c}}\right)$,
which quantifies the deviation of $C_{k}^{\mathrm{corr}}$ from
$C_{k}^{\mathrm{true}}$. For $\chi$ far away from $\pm$1, it is drawn as a
point at $\pm$1 with a red arrow pointing towards the larger value not shown
within the $y$-axis scale, which denotes a failed efficiency correction.

Figure 4: (Color online) Quantified deviations $\chi$ =
$\left.\left(\hat{\mu}-C_{k}^{\mathrm{true}}\right)\middle/\hat{\sigma}\right.$
($k$ = 1, 2, 3, 4) of the $C_{k}^{\mathrm{corr}}$ measured with
$\varepsilon_{i}$ and corrected with $\varepsilon_{j}$ ($i$, $j$ = 0, 1,
$\cdots$, 5) defined in Eqs. (65)-(70) from $C_{k}^{\mathrm{true}}$. For
$|\chi|$ $>$ 1 of a failed efficiency correction, the point is drawn at $\pm$1
with a red arrow pointing towards the larger value not shown within the
$y$-axis scale. Figure 5: (Color online) Fractions of $C_{k}^{\mathrm{corr}}$
($k$ = 1, 2, 3, 4) measured with $\varepsilon_{i}$ and corrected with
$\varepsilon_{j}$ ($i$, $j$ = 0, 1, $\cdots$, 5) defined in Eqs. (65)-(70)
falling in between $C_{k}^{\mathrm{true}}\pm\hat{\sigma}$. The fractions for
the cumulants up to fourth-order obtained by each successful efficiency
correction should be close to 68%.

Figure 5 shows the fractions of the $C_{k}^{\mathrm{corr}}$ falling in between
$C_{k}^{\mathrm{true}}\pm\hat{\sigma}$ for various
$\left(\varepsilon_{\mathrm{m}};\varepsilon_{\mathrm{c}}\right)$, which are
summarized from the number at the top right of each panel of Figs. 12-17 in
Appendix A. All of the fractions of a successful efficiency correction should
be close to 68% as the 1-$\sigma$ probability of the Gaussian distribution.

Figure 6: (Color online) $\left.\hat{\sigma}^{2}\middle/\hat{\mu}\right.$ of the $C_{k}^{\mathrm{corr}}$ ($k$ = 1, 2, 3, 4) measured with $\varepsilon_{i}$ and corrected with $\varepsilon_{j}$ ($i$, $j$ = 0, 1, $\cdots$, 5) defined in Eqs. (65)-(70). The values quantify the statistical fluctuations and show a decreasing trend as the non-uniformity of $\varepsilon_{\mathrm{c}}$ decreases. Table 2: The successful ($\circ$) and failed ($\times$) efficiency corrections for various sets of efficiency employed in measurement ($\varepsilon_{\mathrm{m}}$) and correction ($\varepsilon_{\mathrm{c}}$). $\varepsilon_{\mathrm{c}}$ $\varepsilon_{\mathrm{m}}$ | $\varepsilon_{0}$ | $\varepsilon_{1}$ | $\varepsilon_{2}$ | $\varepsilon_{3}$ | $\varepsilon_{4}$ | $\varepsilon_{5}$
---|---|---|---|---|---|---
$\varepsilon_{0}$ | $\circ$ | $\times$ | $\times$ | $\times$ | $\times$ | $\times$
$\varepsilon_{1}$ | $\circ$ | $\circ$ | $\times$ | $\times$ | $\times$ | $\times$
$\varepsilon_{2}$ | $\times$ | $\circ$ | $\circ$ | $\times$ | $\times$ | $\times$
$\varepsilon_{3}$ | $\times$ | $\times$ | $\circ$ | $\circ$ | $\times$ | $\times$
$\varepsilon_{4}$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\times$
$\varepsilon_{5}$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$ | $\circ$
Figure 7: (Color online)
$1-\left.\hat{\sigma}_{\left(\varepsilon_{i};\varepsilon_{5}\right)}\middle/\hat{\sigma}_{\left(\varepsilon_{i};\varepsilon_{i}\right)}\right.$
of the efficiency-corrected cumulants up to fourth-order for various
measurements with $\varepsilon_{i}$ ($i$ = 0, 1, $\cdots$, 5) defined in Eqs.
(65)-(70), where $\hat{\sigma}_{\left(\varepsilon_{i};\varepsilon_{j}\right)}$
represents the $\hat{\sigma}$ of the cumulants measured with $\varepsilon_{i}$
and corrected with $\varepsilon_{j}$ ($j$ = $i$, 5). The values quantify the
suppression of the statistical uncertainties by the uniform efficiency
correction and follow a decreasing trend as the non-uniformity of the
efficiency employed in the measurement decreases. Figure 8: (Color online)
Correlations of produced and measured numbers of positively and negatively
charged particles in various subevents. The left panels show the produced
distributions, and the right ones show the distributions measured with
$\varepsilon_{0}$ defined in Eq. (78).

Table 2 summarizes the successful and failed efficiency corrections for
various $\left(\varepsilon_{\mathrm{m}};\varepsilon_{\mathrm{c}}\right)$
differentiated by Figs. 4 and 5, wherein the results show a semi-diagonal. The
simulated results support the theoretical predictions presented in the
previous section. The diagonal line represents that the
$\varepsilon_{\mathrm{c}}$ is the same as $\varepsilon_{\mathrm{m}}$, and
thus, $C_{k}^{\mathrm{true}}$ can be recovered from $C_{k}^{\mathrm{corr}}$
within uncertainties, which confirms that the track-by-track efficiency
correction method is valid [35]. All of the successful efficiency corrections
at the bottom left of Table 2 employ the IAE. For example, $\varepsilon_{1}$
is the average of $\varepsilon_{0}$ in two $\varphi$ bins, respectively, as

$\varepsilon_{1}\left(p_{\mathrm{T}},\varphi\right)=\left\\{\begin{aligned}
&\dfrac{\displaystyle{\int_{k\pi/6}^{(k+1/2)\pi/6}g(\varphi)\varepsilon_{0}\left(p_{\mathrm{T}},\varphi\right)\mathrm{d}\varphi}}{\displaystyle{\int_{k\pi/6}^{(k+1/2)\pi/6}g(\varphi)\mathrm{d}\varphi}},\\\
&\begin{aligned}
k\leq\left.\varphi\middle/\dfrac{\pi}{6}\right.<k+\dfrac{1}{2},&\\\
\varepsilon_{0}\left(p_{\mathrm{T}},\varphi\right),k+\dfrac{1}{2}\leq\left.\varphi\middle/\dfrac{\pi}{6}\right.<k+1,&\end{aligned}\end{aligned}\right.(k\in\mathbb{Z})$
(73)

so the correction with $\varepsilon_{1}$ can be performed successfully for the
measurement with $\varepsilon_{0}$. In the same way, $\varepsilon_{4}$ is the
average of $\varepsilon_{0,1,2,3}$ in the whole $\varphi$ range (0 $\leq$
$\varphi$ $<$ 2$\pi$) and thus, $\varepsilon_{4}$ can be employed to correct
the cumulants measured with $\varepsilon_{0,1,2,3}$. However,
$\varepsilon_{2,3}$ ($\varepsilon_{3}$) could be considered as not the IAE of
$\varepsilon_{0}$ ($\varepsilon_{1}$) but the NAE, which causes three
“$\times$” at the bottom left of Table 2. The efficiency corrections by using
$\varepsilon_{\mathrm{c}}$ with more non-uniformity than
$\varepsilon_{\mathrm{m}}$ at the top right of Table 2 also fails for the same
reason. Once a non-averaged distribution is induced in the
$\varepsilon_{\mathrm{c}}$ due to incorrect knowledge of the
$\varepsilon_{\mathrm{m}}$, it cannot reproduce the cumulants and results in a
failed correction. During the efficiency correction, we should safely take
care of the internally averaged track efficiency and avoid introducing
additional non-uniformity and non-averaged distributions. However, the
statistical uncertainties should be treated carefully as what is studied in
the following section.

Figure 6 shows $\left.\hat{\sigma}^{2}\middle/\hat{\mu}\right.$ for various
$\left(\varepsilon_{\mathrm{m}};\varepsilon_{\mathrm{c}}\right)$, which
quantify the fluctuations of $C_{k}^{\mathrm{corr}}$ and strongly relate to
their statistical uncertainties. After removing three points of the failed
efficiency corrections for
$C_{k}^{\mathrm{corr}}\left(\varepsilon_{0};\varepsilon_{2}\right)$,
$C_{k}^{\mathrm{corr}}\left(\varepsilon_{0};\varepsilon_{3}\right)$ and
$C_{k}^{\mathrm{corr}}\left(\varepsilon_{1};\varepsilon_{3}\right)$, the
decreasing trend shows that the statistical fluctuations and uncertainties of
the $C_{k}^{\mathrm{corr}}$ are dramatically suppressed by the valid IAE
corrections with less non-uniformity. In order to quantify how the statistical
fluctuations are suppressed by the completely uniform efficiency correction,
Fig. 7 summarizes
$1-\left.\hat{\sigma}_{\left(\varepsilon_{i};\varepsilon_{5}\right)}\middle/\hat{\sigma}_{\left(\varepsilon_{i};\varepsilon_{i}\right)}\right.$
for various $\varepsilon_{\mathrm{m}}$, where
$\hat{\sigma}_{\left(\varepsilon_{i};\varepsilon_{5}\right)}$ and
$\hat{\sigma}_{\left(\varepsilon_{i};\varepsilon_{i}\right)}$ denote the
$\hat{\sigma}$ of
$C_{k}^{\mathrm{corr}}\left(\varepsilon_{i};\varepsilon_{5}\right)$ and
$C_{k}^{\mathrm{corr}}\left(\varepsilon_{i};\varepsilon_{i}\right)$ ($i$ = 0,
1, $\cdots$, 5), respectively. In the efficiency correction with
$\varepsilon_{5}$, the whole phase spaces are covered by average bins, and the
non-uniformity of the track efficiency is ignored, which could flatten the
fluctuations and minimize the statistical uncertainties.

In addition, another sample of events are produced with $N_{\mathrm{pos}}$ and
$N_{\mathrm{neg}}$ following Gaussian distributions as well, and the above
procedures are repeated. Similar results are obtained and show the
independence of the input particle multiplicity distributions.

##  IV Toy model analysis with externally averaged efficiency

In this section, another toy Monte Carlo model with EAE is further employed in
order to check the previous theoretical study about the validity of the EAE
correction.

###  IV.1 Event production

We produce $10^{7}$ events with positively and negatively charged particles
with the charges $a=1$ and $b=-1$. The events are divided into four subevents
according to the particle $p_{\mathrm{T}}$ and $\varphi$ in various intervals
and the multiplicities of two types of particles in four subevents are sampled
independently. Table 3 summarizes the subevent divisions and the
$N_{\mathrm{pos}}$ and $N_{\mathrm{neg}}$ distributions in four subevents.
Particles in different subevents belong to different phase spaces. Note that
the total numbers of particles are the same in different subevents, as the
averaged multiplicities $\left\langle N_{\mathrm{pos}}\right\rangle\equiv 3$
and $\left\langle N_{\mathrm{neg}}\right\rangle\equiv 2$ are constant. The 2D
distributions of $N_{\mathrm{pos}}$ and $N_{\mathrm{neg}}$ in various
subevents are shown in the left panels of Fig. 8.

Table 3: Subevent divisions and distributions of $N_{\mathrm{pos}}$ and
$N_{\mathrm{neg}}$ in four subevents.

| 0 $\leq$ $\varphi$ $<$ $\pi$ |  $\pi$ $\leq$ $\varphi$ $<$ $2\pi$
---|---|---
0.4 $\leq$ $p_{\mathrm{T}}$ $<$ 0.8 |  | subevent 0
---
$N$ $\sim$ Poisson($\lambda$)
| subevent 1
---
$N$ $\sim$ Gaus($\mu$, $\sigma$)
0.8 $\leq$ $p_{\mathrm{T}}$ $<$ 2 |  | subevent 2
---
$N$ $\sim$ Binomial($n$, $p$)
| subevent 3
---
$N$ $\sim$ Integer($n_{\mathrm{max}}$)*

  * *

integer uniformly distributed in the interval [0, $n_{\mathrm{max}}-1$]

  * •

$\lambda$ = 3 (pos), 2 (neg)

  * •

($\mu$, $\sigma$) = (3, 0.4) (pos), (2, 0.25) (neg)

  * •

($n$, $p$) = (5, 0.6) (pos), (5, 0.4) (neg)

  * •

$n_{\mathrm{max}}$ = 7 (pos), 5 (neg)

Figure 9: (Color online) Distributions of produced and measured
$p_{\mathrm{T}}$ and $\varphi$ of positively (top) and negatively (bottom)
charged particles. The produced $p_{\mathrm{T}}$ and $\varphi$ distributions
are shown as black solid curves, and the red dashed ones represent the
distributions measured with $\varepsilon_{0}$ defined in Eq. (78). The steps
at $p_{\mathrm{T}}$ = 0.8 and $\varphi$ = $\pi$ are caused by sampling
$N_{\mathrm{pos}}$ and $N_{\mathrm{neg}}$ from the input distributions
individually in multiple phase spaces. Figure 10: (Color online) Two
independent one-dimensional functions $u\left(p_{\mathrm{T}}\right)$ and
$v(\varphi)$ defined in Eqs. (76) and (77). The $u\left(p_{\mathrm{T}}\right)$
and $v(\varphi)$ for positively and negatively charged particles are shown as
red and blue solid curves, respectively, and the horizontal dashed lines
denote their averaged values in $p_{\mathrm{T}}$ $\in$ [0.4, 0.8), [0.8, 2)
and [0.4, 2) or $\varphi$ $\in$ [0, $\pi$), [$\pi$, $2\pi$) and [0, $2\pi$).

Each of the particles in each subevent is allocated $p_{\mathrm{T}}$ and
$\varphi$ sampled from

$\displaystyle f\left(p_{\mathrm{T}}\right)$ $\displaystyle\sim
p_{\mathrm{T}}\mathrm{exp}\left(-p_{\mathrm{T}}\middle/t\right),$ (74)
$\displaystyle g(\varphi)$ $\displaystyle\sim\mathrm{exp}(-\varphi/\tau),$
(75)

where $t$ = 0.26 and 0.22 and $\tau$ = 8 and 6 for positively and negatively
charged particles, respectively. Distributions of the produced
$p_{\mathrm{T}}$ and $\varphi$ are shown as black solid curves in Fig. 9.

###  IV.2 Efficiency definition

Figure 11: (Color online) $C_{k}^{\mathrm{corr}}$ ($k$ = 1, 2, 3, 4) measured
with $\varepsilon_{0}$ and corrected with each set of efficiency defined in
Eqs. (78)-(80). Each panel shows 100 points instead of 1000. $\hat{\mu}$ and
$\hat{\sigma}$ at the bottom of each panel show the mean value and the
standard deviation of 1000 points. The red solid and gray dashed lines denote
$C_{k}^{\mathrm{true}}$ and $C_{k}^{\mathrm{true}}\pm\hat{\sigma}$,
respectively. The number at the top right of each panel represents the
fraction of the points falling in between the gray dashed lines.

Two independent one-dimensional components of efficiency are defined as

$\displaystyle u\left(p_{\mathrm{T}}\right)$
$\displaystyle=p_{0}\mathrm{exp}\left(-\left(p_{1}\middle/p_{\mathrm{T}}\right)^{p_{2}}\right),$
(76) $\displaystyle v(\varphi)$
$\displaystyle=1+p_{3}\mathrm{sin}\left(p_{4}\varphi+p_{5}\right),$ (77)

where $\left(p_{0},p_{1},p_{2},p_{3},p_{4},p_{5}\right)$ = (0.7, 0.4, 4.8,
0.4, 1.2, $-\pi/6$) and (0.6, 0.4, 4.2, 0.4, 0.8, $\pi/12$) for positively and
negatively charged particles, respectively. Figure 10 shows these two
functions and their averaged values weighted by the input particle
$p_{\mathrm{T}}$ or $\varphi$ distribution in $p_{\mathrm{T}}$ $\in$ [0.4,
0.8), [0.8, 2) and [0.4, 2) or $\varphi$ $\in$ [0, $\pi$), [$\pi$, $2\pi$) and
[0, $2\pi$).

Three sets of track efficiency are given by

$\displaystyle\varepsilon_{0}\left(p_{\mathrm{T}},\varphi\right)$
$\displaystyle=u\left(p_{\mathrm{T}}\right)v(\varphi),$ (78)
$\displaystyle\varepsilon_{1}\left(p_{\mathrm{T}},\varphi\right)$
$\displaystyle=\mu\left[\varepsilon_{0}\left(p_{\mathrm{T}},\varphi\right)\right]_{\mathrm{sub\
of}\left(p_{\mathrm{T}},\varphi\right)},$ (79)
$\displaystyle\varepsilon_{2}\left(p_{\mathrm{T}},\varphi\right)$
$\displaystyle=\mu\left[\varepsilon_{0}\left(p_{\mathrm{T}},\varphi\right)\right],$
(80)

where
$\mu\left[\varepsilon_{0}\left(p_{\mathrm{T}},\varphi\right)\right]_{\mathrm{sub\
of}\left(p_{\mathrm{T}},\varphi\right)}$ denotes the average of
$\varepsilon_{0}$ within the subevent in which the particle
$\left(p_{\mathrm{T}},\varphi\right)$ falls; therefore, $\varepsilon_{1}$ is
defined as the IAE of $\varepsilon_{0}$. However, $\varepsilon_{2}$ represents
the EAE as the average of $\varepsilon_{0}$ across four subevents.

###  IV.3 Measurement and efficiency correction

Each particle in the produced events is sampled with $\varepsilon_{0}$ defined
in Eq. (78) as the probability, and the track-by-track efficiency corrections
[35] with the three sets of track efficiency defined in Eqs. (78)-(80) are
performed to obtain
$C_{k}^{\mathrm{corr}}\left(\varepsilon_{0};\varepsilon_{j}\right)$ ($j$ = 0,
1, 2 and $k$ = 1, 2, 3, 4). The 2D distributions of $n_{\mathrm{pos}}$ and
$n_{\mathrm{neg}}$ among the four subevents are shown in the right panels of
Fig. 8, and the $p_{\mathrm{T}}$ and $\varphi$ distributions are shown as red
dashed curves in Fig. 9.

The production, measurement and efficiency correction are independently
repeated 1000 times. A comparison of $C_{k}^{\mathrm{corr}}$ with
$C_{k}^{\mathrm{true}}$ is shown in Fig. 11. $\hat{\mu}$ and $\hat{\sigma}$ of
1000 points are shown at the bottom of each panel. The red solid lines
represent $C_{k}^{\mathrm{true}}$, and the gray dashed lines denote
$C_{k}^{\mathrm{true}}\pm\hat{\sigma}$. The fraction of
$C_{k}^{\mathrm{corr}}$ falling in between the gray dashed lines is shown at
the top right of each panel, which should be comparable with 68% for a valid
efficiency correction. Figure 11 clearly shows that $C_{k}^{\mathrm{true}}$
can be successfully reproduced by the efficiency correction with
$\varepsilon_{1}$ for the measurement with $\varepsilon_{0}$ within
uncertainties, which supports the validity of the IAE correction. However,
there is no $C_{2}$, $C_{3}$, or $C_{4}$ corrected with $\varepsilon_{2}$
between the gray dashed lines in Fig. 11, in other words, the EAE correction
is not valid for cumulants higher than the first order, which has been
predicted by a previous theoretical study. This indicates that the usage of
the AE correction requires comprehensive knowledge about the phase space
dependence and the EAE correction for higher-order cumulants should be avoided
in data analysis.

##  V Summary

In this paper, we investigate the effect of non-uniform track efficiency on
the efficiency correction of higher-order cumulants of conserved charges. This
theoretical study proves that an average of the detector efficiency taken for
particles with the same charge within each single phase space can successfully
reproduce the true values of cumulants by tuning the track efficiency employed
in the efficiency correction. The toy model simulation with the track-by-track
efficiency correction supports the analytical proof and shows that the valid
averaged efficiency correction results in a significant suppression of the
statistical uncertainties of cumulants as the event-by-event fluctuations
resulting from the non-uniformity of the detector efficiency are not taken
into account. However, the correction with the averaged efficiency across
multiple phase spaces or the efficiency with a non-uniformity different from
that in the averaged case is invalid; a comprehensive knowledge of different
phase spaces is required when applying an averaged efficiency correction. If
we safely use the realistic detector efficiency in the correction, precisely
understanding the cumulants of conserved charges with small uncertainties
requires detector efficiency that is as uniform as possible.

The uniformity of the detector design and construction is crucial for
precision measurements of higher-order fluctuations. Future fluctuation
analysis in the lower energy and higher baryon density regions calls for
comprehensive understanding of the uniformity of detectors, such as NICA-MPD
[38] and FAIR-CBM [39].

###### Acknowledgements.

This work was partially supported by National Key Research and Development
Program of China with Grant No. 2018YFE0205200, 2018YFE0104700,
2020YFE0202002, Natural Science Foundation of China with Grant No. 11890711,
11890712, 12122505, 11828501, 11861131009, Anhui Provincial Natural Science
Foundation with Grant No. 1808085J02 and Innovation Fund of Key Laboratory of
Quark and Lepton Physics with Grant No. QLPL2020P01.

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## Appendix A

The $C_{k}^{\mathrm{corr}}$ ($k$ = 1, 2, 3, 4) measured with $\varepsilon_{i}$
and corrected with $\varepsilon_{j}$ ($i$, $j$ = 0, 1, $\cdots$, 5) defined in
Eqs. (65)-(70) are shown in Figs. 12-17. $\hat{\mu}$ and $\hat{\sigma}$ shown
at the bottom of each panel represent the mean value and the standard
deviation of 1000 points, respectively. The red solid lines show
$C_{k}^{\mathrm{true}}$, and the gray dashed lines denote
$C_{k}^{\mathrm{true}}$ shifted up and down with $\hat{\sigma}$
($C_{k}^{\mathrm{true}}\pm\hat{\sigma}$). The fraction of
$C_{k}^{\mathrm{corr}}$ falling in between the gray dashed lines is shown at
the top right of each panel.

Figure 12: (Color online) $C_{k}^{\mathrm{corr}}$ ($k$ = 1, 2, 3, 4) measured
with $\varepsilon_{0}$ and corrected with each set of efficiency defined in
Eqs. (65)-(70). Each panel shows 100 points instead of 1000. $\hat{\mu}$ and
$\hat{\sigma}$ at the bottom of each panel show the mean value and the
standard deviation of 1000 points. The red solid and gray dashed lines denote
$C_{k}^{\mathrm{true}}$ and $C_{k}^{\mathrm{true}}\pm\hat{\sigma}$,
respectively. The number at the top right of each panel represents the
fraction of the points falling in between the gray dashed lines. Figure 13:
(Color online) $C_{k}^{\mathrm{corr}}$ ($k$ = 1, 2, 3, 4) measured with
$\varepsilon_{1}$ and corrected with each set of efficiency defined in Eqs.
(65)-(70). Each panel shows 100 points instead of 1000. $\hat{\mu}$ and
$\hat{\sigma}$ at the bottom of each panel show the mean value and the
standard deviation of 1000 points. The red solid and gray dashed lines denote
$C_{k}^{\mathrm{true}}$ and $C_{k}^{\mathrm{true}}\pm\hat{\sigma}$,
respectively. The number at the top right of each panel represents the
fraction of the points falling in between the gray dashed lines. Figure 14:
(Color online) $C_{k}^{\mathrm{corr}}$ ($k$ = 1, 2, 3, 4) measured with
$\varepsilon_{2}$ and corrected with each set of efficiency defined in Eqs.
(65)-(70). Each panel shows 100 points instead of 1000. $\hat{\mu}$ and
$\hat{\sigma}$ at the bottom of each panel show the mean value and the
standard deviation of 1000 points. The red solid and gray dashed lines denote
$C_{k}^{\mathrm{true}}$ and $C_{k}^{\mathrm{true}}\pm\hat{\sigma}$,
respectively. The number at the top right of each panel represents the
fraction of the points falling in between the gray dashed lines. Figure 15:
(Color online) $C_{k}^{\mathrm{corr}}$ ($k$ = 1, 2, 3, 4) measured with
$\varepsilon_{3}$ and corrected with each set of efficiency defined in Eqs.
(65)-(70). Each panel shows 100 points instead of 1000. $\hat{\mu}$ and
$\hat{\sigma}$ at the bottom of each panel show the mean value and the
standard deviation of 1000 points. The red solid and gray dashed lines denote
$C_{k}^{\mathrm{true}}$ and $C_{k}^{\mathrm{true}}\pm\hat{\sigma}$,
respectively. The number at the top right of each panel represents the
fraction of the points falling in between the gray dashed lines. Figure 16:
(Color online) $C_{k}^{\mathrm{corr}}$ ($k$ = 1, 2, 3, 4) measured with
$\varepsilon_{4}$ and corrected with each set of efficiency defined in Eqs.
(65)-(70). Each panel shows 100 points instead of 1000. $\hat{\mu}$ and
$\hat{\sigma}$ at the bottom of each panel show the mean value and the
standard deviation of 1000 points. The red solid and gray dashed lines denote
$C_{k}^{\mathrm{true}}$ and $C_{k}^{\mathrm{true}}\pm\hat{\sigma}$,
respectively. The number at the top right of each panel represents the
fraction of the points falling in between the gray dashed lines. Figure 17:
(Color online) $C_{k}^{\mathrm{corr}}$ ($k$ = 1, 2, 3, 4) measured with
$\varepsilon_{5}$ and corrected with each set of efficiency defined in Eqs.
(65)-(70). Each panel shows 100 points instead of 1000. $\hat{\mu}$ and
$\hat{\sigma}$ at the bottom of each panel show the mean value and the
standard deviation of 1000 points. The red solid and gray dashed lines denote
$C_{k}^{\mathrm{true}}$ and $C_{k}^{\mathrm{true}}\pm\hat{\sigma}$,
respectively. The number at the top right of each panel represents the
fraction of the points falling in between the gray dashed lines.