# Forward-backward correlations with the $\Sigma$ quantity in the wounded
constituent framework at LHC energies

Iwona Sputowska  H. Niewodniczański Institute of Nuclear Physics PAN, 31-342
Cracow, Poland<EMAIL_ADDRESS>
###### Abstract

$\Sigma$ is a new correlation measure, quite recently introduced to heavy-ion
physics. This measure, defined in the independent source model as a strongly
intensive quantity, is expected to be free of the effects of system volume and
volume fluctuations. This paper discusses the forward-backward correlation
quantified with the $\Sigma$ observable calculated in the framework of the
wounded nucleon model (WNM) and wounded quark model (WQM). Findings show that
the wounded constituent approach outperforms the commonly used heavy-ion Monte
Carlo generators, such as HIJING, AMPT or EPOS, by accurately describing the
experimental data on FB correlations with $\Sigma$ measured by ALICE in Xe–Xe
reactions at $\sqrt{s_{\rm{NN}}}$=5.44 TeV and in Pb–Pb collisions at
$\sqrt{s_{\rm{NN}}}$= 2.76 and 5.02 TeV. This paper demonstrates that $\Sigma$
can be a unique tool for determining the fragmentation function of a wounded
constituent in a symmetric nucleus-nucleus collision. However, in the wounded
constituent framework, it is no longer a strongly intensive quantity.

††preprint: APS/123-QED

##  I Introduction

Over the last few years, there has been growing interest in the analysis of
particle multiplicity correlations and fluctuations in high-energy nucleus-
nucleus collisions with so-called strongly intensive quantities. These
observables were first introduced in heavy-ion physics in Ref. [1] as a remedy
to the spurious effect of volume (centrality) fluctuations that contaminate
the measured physical variables such as the multiplicity correlation
coefficient Refs. [2, 3]. Strongly intensive quantities were defined in an
independent source model framework as observables that do not depend on system
volume or its fluctuations but rather carry direct information about a single
(average) source producing particles. The term independent source model refers
to a class of superposition models that assume that particles are emitted
independently from a collection of statistically identical sources. A
fundamental example of the independent source model is the wounded nucleon
model Ref. [4], which assumes that a nucleus-nucleus collision can be
constructed as a superposition of elementary nucleon-nucleon interactions.

Two sets of strongly intensive quantities, $\Sigma$ and $\Delta$, were
constructed as a combination of second moments and dedicated to studying
correlations and fluctuations. Of the two, only the $\Sigma$ family contains a
covariance term sensitive to particle correlations.

Recently, new results on forward-backward (FB) multiplicity correlations with
strongly intensive quantity $\Sigma$ were measured in Pb–Pb collisions at
$\sqrt{s_{\rm{NN}}}$ =2.76 and 5.02 TeV as well as Xe–Xe collisions at
$\sqrt{s_{\rm{NN}}}$= 5.44 TeV in ALICE at LHC, Refs. [5, 6]. Results were
obtained for pairs of forward F and backward B pseudorapidity ($\eta$)
intervals of width $\delta\eta=$0.2, each located symmetrically around
midrapidity ($\eta=0$) as illustrated in Fig. 1. The $\Sigma$ observable
values were studied for different centrality classes and as a function of the
distance between the forward (F) and backward (B) pseudorapidity interval
$\Delta\eta$.

Figure 1: The figure illustrates the definition of forward
F=$(\Delta\eta/2,~{}\Delta\eta/2+\delta\eta)$ and backward
B=$(\Delta\eta/2-\delta\eta,~{}-\Delta\eta/2)$ pseudorapidity intervals,
located symmetrically around $\eta=0$.

In terms of this FB correlation analysis, $\Sigma$ quantity is defined by the
Eq. 1 as a combination of scaled variances $\omega_{F(B)}$, first moments
$\langle N_{F(B)}\rangle$, and covariance of multiplicity distributions:

$\Sigma=\frac{\omega_{B}\langle N_{F}\rangle+\omega_{F}\langle
N_{B}\rangle-2\text{Cov}(N_{B},N_{F})}{\langle N_{F}\rangle+\langle
N_{B}\rangle}.$ (1)

ALICE Collaboration studies have confirmed that $\Sigma$ exhibits strongly
intensive quantity properties in heavy-ion collisions. Specifically, it has
been observed that it is independent of the centrality selection method in the
experiment and free from the contribution from volume fluctuations within
studied centrality classes, Ref. [5]. It was found that $\Sigma$ displays a
dependence on the energy and centrality of the collision system. The observed
trend with energy, namely the growth, is similar for all studied nucleus-
nucleus collision types. On the other hand, the behavior of $\Sigma$ with
centrality has the opposite character for collisions of larger systems such as
Pb–Pb and Xe–Xe versus that observed in pp interactions, Refs. [7, 6]. While
theoretical predictions characterize the $\Sigma$ energy and centrality
dependence in pp collisions reasonably well, the most commonly used heavy-ion
Monte Carlo models fail to describe the data in Pb–Pb and Xe–Xe collisions.
Models break down qualitatively (HIJING [8]) and quantitatively (EPOS [9],
AMPT [10]) in their attempt to reproduce the $\Sigma$ measured in the
experiment. Overall causal factors leading to the discrepancy between nucleus-
nucleus and elementary pp collisions and between theoretical description and
experimental results remain speculative and are worth further investigation.

This paper examines the properties of the forward-backward strongly intensive
quantity $\Sigma$ in the wounded constituent model, Ref. [11]. The constituent
(also referred to as a source) can be either a nucleon or a quark
participating in the collision. Four main conclusions can be drawn from this
study. First, $\Sigma$, taken in terms of the wounded constituent model, gives
a non-trivial result different from unity. Second, due to its intrinsic
dependence on fluctuations of wounded nucleons, it is no longer a strongly
intensive quantity. However, it preserves some strongly intensive properties
as a result of its relation to partial correlations in symmetric collisions.
Third, the wounded nucleon models and the wounded quark model explain
qualitatively and quantitatively the observed experimental trends of $\Sigma$
with the centrality of the collision. Fourth, the $\Sigma$ variable appears to
be responsive to the nature of the fragmentation function. The immediate
inspiration for this paper came from the studies presented in Refs. [3, 12,
13].

The remainder of this paper has been organized in the following way. Section 2
defines the wounded nucleon and wounded quark models. The analytical formula
for $\Sigma$ derived in their framework and the description of WNM and WQM
Monte Carlo calculations are presented. In Section 3, results are discussed
and compared with the ALICE data. Based on the research findings, predictions
are made regarding the characteristics of the wounded nucleon and wounded
quark fragmentation function for LHC energies. The paper is closed with
conclusions.

##  II  Forward-Backward strongly intensive quantities in wounded nucleon and
wounded quark models

The superposition approach in which wounded sources in the projectile and
target nuclei independently produce particles during the nucleus-nucleus
collision was proposed in high-energy physics long ago, Ref. [4]. In
particular, the outcome of this theoretical framework depends on the
postulated nature of the sources (constituents). In this paper, two scenarios
are considered: the wounded quark model and the wounded nucleon model, Refs.
[4, 14, 15, 11, 16].

The wounded nucleon model was found to be relatively successful in describing
soft particle production in relativistic heavy-ion collisions as a composition
of independent contributions of nucleon-nucleon interactions, Refs. [4, 17,
12, 3]. The main assumptions of the WNM, adopted in this paper in terms of
forward-backward correlation analysis, can be characterized as listed below.

  1. 1.

A wounded nucleon is a nucleon that has undergone at least one inelastic
collision. Each wounded nucleon emits particles regardless of the number of
collisions it went through.

  2. 2.

The universal fragmentation function $F(\eta)$ describes the particle
production for each wounded nucleon.

  3. 3.

The fragmentation function F($\eta$) is not limited to the hemisphere of the
wounded nucleon. Any wounded source can emit particles into both forward and
backward pseudorapidity regions.

  4. 4.

A single particle density $N(\eta)\equiv\frac{dN(\eta)}{d\eta}$ in a nuclear
collision can be expressed by the combination of inputs coming from the
average number of forward-going, $\langle w_{F}\rangle$, and backward-going,
$\langle w_{B}\rangle$, wounded nucleons (see Ref. [14]):

$N(\eta)=\langle w_{F}\rangle F(\eta)+\langle w_{B}\rangle F(-\eta),$ (2)

  5. 5.

The number of particles produced in the collision fluctuates on an event-by-
event basis according to a given multiplicity distribution. In this work, it
is assumed that the production of particles from a single wounded source
conforms to the Negative Binomial Distribution (NBD):

$P(n,\overline{n},k)=\frac{\Gamma(n+k)}{\Gamma(n+1)\Gamma(k)}\left(\frac{\overline{n}}{k}\right)^{n}\left(1+\frac{\overline{n}}{k}\right)^{-n-k}$
(3)

In this parametrization of NBD, $\overline{n}$ is the average multiplicity and
$1/k$ measures the deviation from a Poisson distribution.

The wounded quark model is an extension of the wounded nucleon approach, where
quark-quark collisions instead of nucleon-nucleon determine particle emission,
Refs [14, 15, 11]. The WQM holds the main assumptions listed in points above,
taking into account that the given description and characterization of the
quantities, such as fragmentation function $F(\eta)$ and forward
(backward)-going wounded sources $\langle w_{F(B)}\rangle$ refers to the
properties of wounded quarks. Since the WQM was first proposed, it has been
applied to study a wide variety of phenomena for various colliding systems and
energies, Refs. [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 13]

###  II.1 Analytical formula

Determining the analytical formula for the FB strongly intensive quantity
$\Sigma$ in the wounded nucleon model is based on the methodology proposed in
the papers Refs. [12, 3]. The procedure is discussed in detail in Appendix A.

The Eq. 4 shows the $\Sigma$ formula derived in the WNM framework.

$\Sigma=1+\frac{\frac{\overline{n}}{2}C^{2}\left[4k\left({\langle
w_{B}\rangle}^{2}\langle w_{F}^{2}\rangle+\langle w_{B}^{2}\rangle{\langle
w_{F}\rangle}^{2}-2\langle w_{F}\rangle\langle w_{B}\rangle\langle
w_{F}w_{B}\rangle\right)+8\langle w_{B}\rangle\langle
w_{F}\rangle\left(\langle w_{B}\rangle+\langle
w_{F}\rangle\right)\right]}{k(\langle w_{F}\rangle+\langle
w_{B}\rangle)\left[{(\langle w_{F}\rangle+\langle
w_{B}\rangle)}^{2}-C^{2}{\left(\langle w_{B}\rangle-\langle
w_{F}\rangle\right)}^{2}\right]}\text{,}$ (4)

where

$C=2p-1$

The parameter $p$ in the formula is the probability at which a wounded nucleon
emits a particle to a given pseudorapidity interval. This probability depends
on the position and width of the intervals of interest. The probability $p$
that the particle originating from the wounded nucleon in a forward
(backward)-moving nuclei goes into the forward (backward) $\eta$ bin can be
determined from Eq. 5.

$p=\frac{\int_{F(B)}F(\eta)\,d\eta}{\int_{B}F(\eta)\,d\eta+\int_{F}F(\eta)\,d\eta}$
(5)

The probability particle from a forward-moving nucleon ends up in the backward
interval, or vice versa, is $1-p$.

From Eq. 4, it is immediately apparent that in the wounded nucleon model:

  * –

$\Sigma$ values explicitly depend on the number of wounded nucleons,
therefore, on the centrality of the collision;

  * –

the clear dependence of $\Sigma$ on the number of wounded nucleons seen in the
Eq. 4 formula implies that $\Sigma$ is no longer a strongly intensive
variable;

  * –

the $\Sigma$ variable is sensitive to the parameter $C$ and thus to the
probability $p$;

  * –

the quantity $\Sigma$ is larger than unity ($\Sigma>1$) for $p\neq 0.5$;

  * –

$\Sigma$ is equal to unity only when $C=0$, namely in the case when a wounded
nucleon produces particles in F and B pseudorapidity intervals with the same
probability $p=0.5$;

  * –

for symmetric collisions $\langle w_{F}\rangle=\langle w_{B}\rangle$, Eq. 4
can be reduced to the expression:

$\Sigma=1+\frac{\overline{n}}{2}C^{2}\left[\frac{\langle(w_{B}-w_{F})^{2}\rangle}{2\langle
w_{F}\rangle}+\frac{2}{k}\right]$ (6)

For the wounded quark model, those analytical formulas remain valid provided
that symbols $w_{F}$ and $w_{B}$ denote the number of forward- and backward-
moving wounded quarks.

One remark is in order here. At first sight, the collapse of the strongly
intensive properties of the $\Sigma$ variable through its explicit dependence
on the number of wounded constituents observed in the wounded nucleon and
wounded quark models seems contradictory since both theoretical frameworks
belong to the class of independent source models. However, this originates
from the fact that both models formulated above assume, in contrast to Ref.
[1], _two_ types of wounded sources moving forward and backward. In case the
number of wounded constituents moving forward and backward is the same
($w_{F}=w_{B}$), or both sources have the same characteristics, i.e. produce
particles with $p=0.5$, the $\Sigma$ variable again regains the strongly
intensive properties described in Ref. [1].

###  II.2 MC simulations

Information about the Pb–Pb or Xe–Xe collision geometry, such as the wounded
constituents’ distribution characteristics, was obtained using the Monte Carlo
Glauber-like simulation. In this paper, the MC model considers two scenarios
for wounded nucleons and wounded quarks. The simulation can be summarized in
three main steps.

#### Modeling of the nuclei:

In the simulation, the positions of nucleons and quarks in the nucleus are
determined randomly. The location of nucleons in the nucleus is described by a
nuclear density profile modeled by a Woods-Saxon spherically-symmetric density
function:

$\rho(\vec{r})=\frac{\rho_{0}}{1+e^{[(r-R)/a]}}$ (7)

where $R$ is the nuclear radius, $a$ is the skin depth, and $r=|\vec{r}|$ is
the distance from the nucleus center. For 129Xe isotope parameters were
adopted to be $R=5.36$ $a=0.59$. For 208Pb nucleus, the sum of individual
Woods-Saxon distributions for the proton and the neutron was taken with the
following parameters: $R_{p}=6.68$ and $a_{p}=0.447$, $R_{n}=6.69$ and
$a_{n}=0.570$. The parameters’ values were adjusted to match those used by the
ALICE Collaboration, as described in the Refs. [29, 30].

For each nucleon, the distribution of three constituent quarks inside the
nucleon is given by the expression:

$\rho(\vec{r})=\frac{\rho_{0}}{exp{[-r/a]}}$ (8)

In formula Eq. 8, $a=r_{p}/\sqrt{12}$ where $r_{p}$ = 0.81 fm is the proton’s
radius, following the Refs. [13, 31].

#### Modeling of the collision process:

Constituents (nucleons or quarks) move on straight-line paths and interact
according to the inelastic constituent cross-section $\sigma_{ii}^{inel}$. The
value of the cross-section does not depend on the number of individual
constituent-constituent collisions.

The inelastic nucleon-nucleon cross-section $\sigma_{NN}^{inel}$ was estimated
based on interpolation of pp value at the corresponding center-of-mass-energy.
In this work, $\sigma_{NN}^{inel}$ values were adapted to correspond to the
values used in ALICE Ref. [30] and are: $\sigma_{NN}^{inel}=61.8$ mb at
$\sqrt{s_{\rm{NN}}}$= 2.76 TeV, $\sigma_{NN}^{inel}=67.6$ mb at
$\sqrt{s_{\rm{NN}}}$= 5.02 TeV and $\sigma_{NN}^{inel}=68.4$ mb at
$\sqrt{s_{\rm{NN}}}$= 5.44 TeV.

For the wounded quark model, to estimate the inelastic quark-quark cross-
section, the technique proposed in Ref. [13] was applied. This involved
seeking a value of $\sigma_{qq}^{inel}$ for which the relation
$\sigma_{NN}^{inel}=\int_{0}^{2\pi}\,d\varphi\int_{0}^{+\infty}P(b)b\,db$ is
satisfied, for a given energy of the elementary pp collision. In the latter
formula, $P(b)$ is a probability of proton-proton collision with the impact
parameter $b$. Based on the described approach, the given values were
determined: $\sigma_{qq}^{inel}\approx 13.9$ mb at $\sqrt{s_{\rm{NN}}}$= 2.76
TeV, $\sigma_{qq}^{inel}\approx 16.2$ mb at $\sqrt{s_{\rm{NN}}}$= 5.02 TeV and
$\sigma_{qq}^{inel}\approx 16.5$ mb at $\sqrt{s_{\rm{NN}}}$= 5.44 TeV.

A nucleon is considered wounded if it overlaps with the nucleons of the other
nucleus in a region of transverse distance radius
$d_{NN}\leq\sqrt{\frac{\sigma_{NN}^{inel}}{\pi}}$. Analogically a quark is
considered wounded if it overlaps with the quarks of the other nucleus in a
region of transverse distance radius
$d_{qq}\leq\sqrt{\frac{\sigma_{qq}^{inel}}{\pi}}$.

#### Modeling of particle production:

The model assumes every wounded constituent emits particles with NBD
parametrized according to Eq. 3. This study considered the production of
particles in two kinematic pseudorapidity regions, midrapidity and the forward
region.

##### Midrapidity:

Particle production from a single wounded nucleon in midrapidity was modeled
for a sum of forward and backward $\eta$ intervals (F+B) of a width of
$\delta\eta=0.2$ each. Values of $\overline{n}$ and $k$ parameters for
energies $\sqrt{s_{\rm{NN}}}=$ 2.76, 5.02, and 5.44 TeV were interpolated
based on the experimental pp data given in the papers Refs. [32, 33], and they
are listed in Table 1. The uncertainties of $\overline{n}$ and $k$ were
conservatively estimated to be $7\%$ and $10\%$, respectively.

Table 1: Values of NBD parameters $\overline{n}$ and $k$ (WNM) and $\overline{n_{q}}$ and $k_{q}$ (WQM) for particle production in the midrapidity region.  $\sqrt{s_{\rm{NN}}}$ | $\overline{n}$ | $k$ | $\overline{n_{q}}=\dfrac{\overline{n}}{\langle N_{q}\rangle}$ | $k_{q}=\dfrac{k}{\langle N_{q}\rangle}$
---|---|---|---|---
2.76 | 1.94 | 1.20 | 1.94/1.50 | 1.20/1.50
5.02 | 2.16 | 1.14 | 2.16/1.60 | 1.14/1.60
5.44 | 2.19 | 1.13 | 2.19/1.60 | 1.13/1.60

For WQM, values of NBD parameters were determined by scaling $\overline{n}$
and $k$ by the average number of wounded quarks per wounded nucleon in an
elementary collision for a given energy, namely
$\overline{n_{q}}=\overline{n}/\langle N_{q}\rangle$ and $k_{q}=k/\langle
N_{q}\rangle$. From a simple Monte Carlo calculation, one can determine that
these scaling factors are $\langle N_{q}\rangle\approx 1.5$ for pp collisions
at $\sqrt{s}$=2.76 TeV and $\langle N_{q}\rangle\approx 1.6$ for pp collisions
at $\sqrt{s}$= 5.02 and 5.44 TeV.

Particles emitted in the midrapidity region are assigned randomly to either
the forward or backward pseudorapidity interval with the probability $p$ and
$1-p$, respectively, if they originated from a forward-moving constituent or
with the probability $1-p$ and $p$ if they originated from the backward-moving
source.

##### Forward $\eta$ region:

The particle generation in forward regions was modeled to mimic the centrality
determination method provided by the V0 ALICE detector. The V0 centrality
determination is based on the sum of charged particles’ multiplicity measured
in the pseudorapidity ranges 2.8 $<\eta<$ 5.1 and -3.7$<\eta<$-1.7. In this
work, the particle generated from a wounded nucleon into the forward region
follows the NBD with parameters adopted from the Glauber-NBD fit to V0
detector amplitudes published by ALICE in Ref [30]. Again for WQM, the NBD
parameters were obtained by scaling $\overline{n}$ and $k$ by the factor
$\langle N_{q}\rangle$ for a given energy. Values of NBD parameters for
particle production in the forward region are listed in Table 2.

Table 2: Values of NBD parameters $\overline{n}$ and $k$ (WNM) and $\overline{n_{q}}$ and $k_{q}$ (WQM) for particle production in the forward $\eta$ region.  $\sqrt{s_{\rm{NN}}}$ | $\overline{n}$ | $k$ | $\overline{n_{q}}$ | $k_{q}$
---|---|---|---|---
2.76 | 28.7 | 1.6 | 28.7/1.5 | 1.6/1.5
5.02 | 46.4 | 1.5 | 46.4/1.6 | 1.5/1.6
5.44 | 53.7 | 1.0 | 53.7/1.6 | 1.0/1.6

The simulated data samples of Pb–Pb and Xe–Xe collisions were divided into
centrality classes according to cuts on impact parameter $b$ and the number of
particles in forward $\eta$ region, $N_{ch}$. Table 3 provides an overview of
the mean number of wounded nucleons $\langle w\rangle=\langle
w_{F}\rangle+\langle w_{B}\rangle$ and quarks $\langle w^{q}\rangle=\langle
w^{q}_{F}\rangle+\langle w^{q}_{B}\rangle$ calculated for Pb–Pb and Xe–Xe
collisions for various centrality classes defined either by impact parameter
$b$ cut or via the number of particles $N_{ch}$. The geometrical properties
reported in the table are in good agreement with the values published by ALICE
[30].

Table 3:  The mean number of wounded nucleons $\langle w\rangle$ (WNM) and
wounded quarks $\langle w_{q}\rangle$ for different centrality classes defined
by the cuts on impact parameter $b$ and by the number of particles in the
forward $\eta$ region $N_{ch}$.  Pb–Pb collisions at $\sqrt{s_{\rm{NN}}}$=
2.76 TeV
---
Centrality | impact parameter $b$ | particle multiplicity $N_{ch}$
| $\langle w\rangle$ | $\langle w^{q}\rangle$ | $\langle w\rangle$ | $\langle w^{q}\rangle$
$0-10\%$ | 354.88 | 978.04 | 354.88 | 979.48
$10-20\%$ | 258.90 | 674.51 | 259.69 | 676.31
$20-30\%$ | 185.39 | 458.82 | 185.98 | 460.04
$30-40\%$ | 128.83 | 300.65 | 129.21 | 301.32
$40-50\%$ | 85.66 | 186.13 | 85.89 | 186.35
$50-60\%$ | 53.42 | 106.57 | 53.57 | 106.38
$60-70\%$ | 30.64 | 55.36 | 30.60 | 54.77
$70-80\%$ | 15.79 | 25.92 | 15.53 | 24.98
Pb–Pb collisions at $\sqrt{s_{\rm{NN}}}$= 5.02 TeV
Centrality | impact parameter $b$ | particle multiplicity $N_{ch}$
| $\langle w\rangle$ | $\langle w^{q}\rangle$ | $\langle w\rangle$ | $\langle w^{q}\rangle$
$0-10\%$ | 357.66 | 993.20 | 357.60 | 994.51
$10-20\%$ | 262.18 | 689.14 | 263.00 | 691.05
$20-30\%$ | 188.12 | 470.68 | 188.70 | 471.90
$30-40\%$ | 131.03 | 309.78 | 131.47 | 310.55
$40-50\%$ | 87.22 | 192.32 | 87.49 | 192.57
$50-60\%$ | 54.54 | 110.61 | 54.63 | 110.36
$60-70\%$ | 31.26 | 57.46 | 31.21 | 56.87
$70-80\%$ | 16.07 | 26.81 | 15.83 | 25.86
Xe–Xe collisions at $\sqrt{s_{\rm{NN}}}$= 5.44 TeV
Centrality | impact parameter $b$ | particle multiplicity $N_{ch}$
| $\langle w\rangle$ | $\langle w^{q}\rangle$ | $\langle w\rangle$ | $\langle w^{q}\rangle$
$0-10\%$ | 221.91 | 605.79 | 221.32 | 606.56
$10-20\%$ | 164.56 | 420.25 | 165.58 | 422.44
$20-30\%$ | 117.89 | 283.89 | 118.65 | 285.24
$30-40\%$ | 81.75 | 184.48 | 82.30 | 185.39
$40-50\%$ | 54.20 | 113.62 | 54.51 | 113.88
$50-60\%$ | 33.82 | 65.53 | 34.00 | 65.35
$60-70\%$ | 19.78 | 35.37 | 19.74 | 34.80
$70-80\%$ | 11.00 | 18.31 | 10.54 | 17.12

The distributions of wounded nucleons and quarks obtained as described in this
section were used to calculate the $\Sigma$ in the wounded nucleon and wounded
quark models based on the analytical expression Eq. 4. Additionally, numerical
Monte Carlo calculations, which incorporate particle production from a wounded
constituent, were used to calculate $\Sigma$ in the discussed models based on
the definition of this variable, Eq. 1.

##  III Results and Discussion

###  III.1 Centrality dependence of the forward-backward quantity $\Sigma$ in
WNM and WQM

Figure 2 presents the overview of the results on forward-backward correlations
with $\Sigma$ obtained in wounded nucleon and wounded quark models in Pb–Pb
collisions at $\sqrt{s_{\rm{NN}}}$= 5.02 TeV as a function of centrality
class. Results were determined for $10\%$ width of centrality classes of
ranges from $0-10\%$, $10-20\%$, $20-30\%$, etc., up to peripheral collisions
$70-80\%$. The results are presented for selected probability values varying
from $p=0.5$ to $p=0.64$. The figure compares the $\Sigma$ quantity derived
from the analytical formula Eq. 4 and that obtained from the definitions of
this variable given by Eq. 1 in the numerical Monte Carlo simulation.

Based on the presented in Fig. 2 results, one can list the following
observations.

  * –

There is full agreement, within statistical uncertainties, between the values
of $\Sigma$ obtained from the analytical formula and the numerical MC
calculations that include particle production for all studied centrality
classes. This is valid for both the wounded nucleon and wounded quark models.
Consequently, the WNM and WQM results in the following sections will only be
determined by the analytical formula.

  * –

The values of $\Sigma$ quantity grow with the values of probability $p$. This
increase is a bit steeper for WQM.

  * –

For $p=0.5$, the $\Sigma$ is equal to unity for all centrality classes as
predicted by the Eqs. 4 and 6 for both wounded constituent models.

  * –

For $p\neq 0.5$ the results for $\Sigma$ tend to increase from central to
peripheral collisions. This increase is monotonic for WNM. For WQM, the
$\Sigma$ values take a maximum around the mid-central collisions $40-50\%$ and
then gently fall towards the most peripheral sample. This centrality
dependence of $\Sigma$ quantity is more evident the higher the value of
probability $p$.

  * –

The most surprising aspect of the presented results is that both WNM and WQM
seem to resemble the behaviour observed in the experimental data Ref. [6],
particularly for $p\sim$ 0.6. What is striking, wounded nucleon and wounded
quark models are able to describe the $\Sigma$ trend with centrality much
better than the most widely used heavy-ion models such as HIJING, AMPT, or
EPOS (see Ref. [6] for more details).

  * –

The values of $\Sigma$ appear to change with the size of the system
(centrality). This is not caused by differences in the characteristics of the
particle-producing source but rather by the fact that $\Sigma$ is _not_
strongly intensive in WNM and WQM, and it can be affected by the volume of the
system. This topic will be discussed in more detail in the next sections.

Figure 2:  The $\Sigma$ observable obtained for (a) wounded nucleon and (b)
wounded quark model-generated Pb–Pb collisions at $\sqrt{s_{\rm{NN}}}$= 5.02
TeV as a function of centrality class. Results are presented for several
values of the probability $p$. Statistical uncertainties are labeled with
vertical bars. Centrality selection is made via impact parameter $b$.

###  III.2  The $\Sigma$ quantity as a function of centrality bin width and
centrality selection method in WNM and WQM

From the expressions Eqs. 4 and 6, it is clear that in the wounded constituent
approach, $\Sigma$ exhibits an intrinsic and non-trivial dependence on the
number of forward- and backward-moving wounded sources, hence it is not
anymore a strongly intensive quantity. Thus, it was found particularly
interesting to verify how the volume fluctuations and selection of centrality
estimator affect the $\Sigma$ quantity.

Figure 3 shows the dependence of $\Sigma$ quantity on the width of centrality
class ($\Delta\text{ centrality}$) for Pb–Pb collisions at $\sqrt{s_{\rm
NN}}$=5.02 TeV in the wounded nucleon and wounded quark model, respectively.
This study was carried out for different centrality classes of Pb–-Pb
collisions, from central to peripheral. The width of the centrality class
varied from its maximum range of $10\%$, where the largest contribution from
volume fluctuations is expected, down to $1\%$ centrality bin width. Results
are presented for two different centrality selection criteria via impact
parameter $b$ and particle multiplicity $N_{ch}$.

From the data in Fig. 3, it is apparent that $\Sigma$ in the framework of WNM
and WQM in symmetric collisions such as Pb–Pb does not depend on centrality
bin width in studied centrality classes. This implies it is insensitive to
volume fluctuation. The comparison between results obtained for two centrality
selection methods revealed that the $\Sigma$ observable also does not depend
on the way centrality is determined (also shown in the figure).

The lack of dependence of $\Sigma$ on the width of the centrality interval and
the method of centrality selection noted for $\Delta\text{ centrality}\leq
10\%$ in the wounded constituent framework is rather surprising, given that
the $\Sigma$ quantity is explicitly dependent on the term related to the
fluctuations of wounded constituents. These apparent ”strongly-intensive-
quantity-like” properties of $\Sigma$ for centrality classes of widths
$\Delta\text{ centrality}\leq 10\%$ in symmetric nucleus-nucleus collisions
resemble the behavior reported by ALICE experimental data Refs. [5, 7, 6], and
can be explained theoretically if one notes that the Eq. 6 can be rewritten in
terms of _partial covariance_.

Figure 3:  The $\Sigma$ observable obtained for (a) wounded nucleon and (b)
wounded quark model-generated Pb–Pb collisions at $\sqrt{s_{\rm NN}}$ = 5.02
TeV, drawn as a function of centrality class size ($\Delta\text{
centrality}$), for a fixed value of the probability $p=0.6$. The results are
obtained for different centrality selection methods: via impact parameter $b$
and via particle multiplicity $N_{ch}$. The width of the centrality class
($\Delta\text{ centrality}\leq 10\%$) changes from $1\%$ to $10\%$. The ranges
of centrality class for a given centrality bin width can be deduced from the
formula in the figure. The different colors of the data points correspond to
different centralities of the Pb–Pb collision. Statistical uncertainties are
labeled with vertical bars.

###  III.3  The relation between the $\Sigma$ quantity and partial covariance
in WNM and WQM for fully symmetric collision

The partial covariance is defined as:

$\text{Cov}(X,Y\bullet
Z)=\text{Cov}(X,Y)-\frac{\text{Cov}(X,Z)\text{Cov}(Y,Z)}{\text{Var}(Z)}$ (9)

and measures the degree of relation between two random variables $X$ and $Y$,
with the effect of controlling random variable $Z$ removed. An exhaustive
description of the partial correlations technique in the domain of heavy-ion
physics is given in Ref. [34]. As the authors of the mentioned paper explain
in more detail, for a sample that satisfies the affine condition $E(X\mid
Y)=a+BY$, the partial covariance is related to conditional covariance
$\text{Cov}(X,Y\bullet Z)\simeq\text{Cov}(X,Y\mid Z)$. The latter relation
implies that the measurement of the partial covariance with a control variable
related to collision geometry, such as the number of wounded nucleons, can be
interpreted as a conditional covariance with a fixed number of wounded
nucleons in the control interval.

It can be noticed that for a symmetric collision,
$\text{Var}(w_{F})=\text{Var}(w_{B})$ and $\langle w_{F}\rangle=\langle
w_{B}\rangle$, the following relation occurs:

$\langle(w_{B}-w_{F})^{2}\rangle=-4\text{Cov}(w_{F},w_{B}\bullet w),$ (10)

The right side of the equation is a partial covariance between the forward and
backward-moving wounded nucleons with the total number of wounded nucleons
$w=w_{F}+w_{B}$ as a control variable. The analytical proof that the above
expression is valid is presented in the Appendix B; the graphical one is
presented in the left panel of Fig. 4, where the quantities determined by the
left and right sides of the equation have been directly compared.

Some comment about $\text{Cov}(w_{F},w_{B}\bullet w)$ properties is necessary
here. From the discussed earlier characteristics of partial covariance emerges
that it is equivalent to the conditional covariance between $w_{F}$ and
$w_{B}$ at the fixed average number of wound nucleons for a given centrality
class111 The affine condition for wounded nucleons: $E(w_{F}\mid
w_{B})=a+Bw_{B}$ is almost always satisfied, regardless of the sample
(centrality class) size.. Thus, the value of $\text{Cov}(w_{F},w_{B}\bullet
w)$ does not change when increasing the centrality interval around its center
as long as the value of the average number of wounded nucleons remains
unchanged. From Fig. 4, it can be concluded that the average value of the
number of wounded nucleons $\langle w\rangle$, determined for a center of
centrality class in question, does not vary when the width of the centrality
class fluctuates no more than $10\%$ around its center.

Figure 4: The partial covariance $\text{Cov}(w_{F},w_{B}\bullet w)$ and
average number of wounded nucleons $\langle w\rangle$ as a function of the
width of centrality class ($\Delta\text{ centrality}$). The center of the
centrality class is $25\%$, and the width of the centrality class changes from
$\Delta\text{ centrality}$ =1$\%$ up to $\Delta\text{ centrality}$ =50$\%$.
The left panel shows the value of partial covariance
$\text{Cov}(w_{F},w_{B}\bullet w)$, the middle and right panels display the
partial covariance value and the average number of wounded nucleons $\langle
w\rangle$, respectively, both normalized to their values determined at
$\Delta\text{ centrality}$ =1$\%$. The ranges of centrality class for a given
centrality bin width can be deduced from the formula in the figure.

Moreover, considering that the average number of wounded nucleons is
independent of centrality selection criteria [2], thus values of
$\text{Cov}(w_{F},w_{B}\bullet w)$ do not change when applying different
centrality estimators. In other words, independently of the width of the
centrality interval and the method of centrality selection, the term
$\text{Cov}(w_{F},w_{B}\bullet w)$ always gives a value of covariance between
$w_{F}$ and $w_{B}$ when the average number of the wounded nucleons in the
studied centrality sample remains fixed.

With the substitution provided by relation Eq. 10 the Eq. 6 can be rewritten
in an equivalent form:

$\Sigma=1+\frac{\overline{n}}{2}C^{2}\left[\frac{-2\text{Cov}(w_{F},w_{B}\bullet
w)}{\langle w_{F}\rangle}+\frac{2}{k}\right].$ (11)

This formula explains all the properties of $\Sigma$ in the wounded
constituent framework seen in Fig. 3. From the formula, one can conclude that:

  * –

for centrality classes of $\Delta\text{ centrality}\leq 10\%$, the $\Sigma$
variable does not depend on the centrality bin width as all of the terms in
Eq. 11 are volume fluctuation-free quantities.

  * –

all terms including $\text{Cov}(w_{F},w_{B}\bullet w)$ and $\langle
w_{F}\rangle$ are invariant under the change of centrality estimator, which
implies that $\Sigma$ does not depend on the criterion of centrality
selection.

  * –

since the ratio $\frac{-2\text{Cov}(w_{F},w_{B}\bullet w)}{\langle
w_{F}\rangle}$ depends on the volume of the system and, more precisely, on the
average number of wounded nucleons, so does $\Sigma$.

Overall, these results indicate that $\Sigma$ in the wounded nucleon model,
even though it does not depend on volume fluctuations nor the method used for
centrality estimation, is generally not strongly intensive as it is sensitive
to the average number of wounded nucleons (system size) in the sample.
However, for relatively narrow centrality ranges, $\Sigma$ exhibits the
properties of a strongly intensive quantity.

Analogous conclusions can be drawn for the wounded quark model.

###  III.4 Comparison between experimental results and wounded constituent
models

Figure 5:  The centrality dependence of strongly intensive quantity $\Sigma$
obtained in WNM (left panel) and WQM (right panel) for Pb–Pb collisions at
$\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Data points are drawn for selected fixed
values of probability $p$. Theoretical predictions at each panel are compared
with ALICE experimental results, which were redrawn from Ref. [35] for several
values of the pseudorapidity gap $\Delta\eta=$1.2, 0.8, 0.6, 0.2. For the
models, the marked uncertainty bars denote the total uncertainty. Figure 6:
The centrality dependence of strongly intensive quantity $\Sigma$ obtained in
WNM (left panel) and WQM (right panel) for Pb–Pb collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.02 TeV. Data points are drawn for selected fixed
values of probability $p$. Theoretical predictions at each panel are compared
with ALICE experimental results, which were redrawn from Ref. [35] for several
values of the pseudorapidity gap $\Delta\eta=$1.2, 0.8, 0.6, 0.2. For the
models, the marked uncertainty bars denote the total uncertainty. Figure 7:
The centrality dependence of strongly intensive quantity $\Sigma$ obtained in
WNM (left panel) and WQM (right panel) for Xe–Xe collisions at
$\sqrt{s_{\rm{NN}}}$ = 5.44 TeV. Data points are drawn for selected fixed
values of probability $p$. Theoretical predictions at each panel are compared
with ALICE experimental results, which were redrawn from Ref. [35] for several
values of the pseudorapidity gap $\Delta\eta=$1.2, 0.8, 0.6, 0.2. For the
models, the marked uncertainty bars denote the total uncertainty.

Figures 5-7 present the overview of the results on forward-backward
correlation with $\Sigma$ in the wounded nucleon and wounded quark model in
Pb–Pb collisions at $\sqrt{s_{\rm{NN}}}$ =2.76 and 5.02 TeV and Xe–Xe
collisions at $\sqrt{s_{\rm{NN}}}$ =5.44 TeV. Data points are plotted as a
function of centrality for $\Delta\text{ centrality}$ =10$\%$. In each figure,
theoretical predictions are compared to the values of $\Sigma$ measured in the
experimental data sample of Pb–Pb and Xe–Xe collisions with the ALICE detector
at the LHC. ALICE results were redrawn from Ref. [35]. The experimental data
points are presented for several fixed values of the separation gap
($\Delta\eta$) between forward and backward intervals.

The results for the wounded nucleon and wounded quark models are shown for
several values of probability $p$, which plays the role of the model
parameter. The values of $p$ were estimated in such a way as to provide the
best fit of the models to the experimental data. The error bars for WNM and
WQM results in the figure represent the total uncertainty, including both the
statistical uncertainty and the uncertainty originating from the variation of
$\overline{n}$ and $k$.

Overall, the findings outlined in Figs. 5-7 indicate the following:

  * –

The energy dependence of the $\Sigma$ quantity in models closely matches one
observed in the experimental data. When the system energy increases, the
$\Sigma$ value also increases, regardless of the system type. According to the
formula Eq. 6, the energy dependence of $\overline{n}$ and $k$ plays a role in
this behavior, as an increase in $\overline{n}$ and a decrease in $k$ with
collision energy (Table 1) contribute to the noted trend.

  * –

The models accurately depict the trend of $\Sigma$ values with centrality
manifested by the experimental data. The $\Sigma$ values increase from central
to peripheral collisions. Eq. 6 shows that this growth is caused by a term
associated with wounded nucleons (quarks) since all remaining parts of the
$\Sigma$ formula are robust against centrality.

  * –

The Figs. 5 and 6 show that qualitatively, the WNM model reproduces the
behavior of Pb–Pb data better than WQM, especially in the peripheral area.

  * –

The best agreement between the experimental results on $\Sigma$ and the
theoretical description, both WNM and WQM, is seen for the Xe–Xe collision,
Fig. 7.

  * –

As reported earlier in this paper, the values of the $\Sigma$ in the wounded
nucleon (quark) model show a sensitivity to the probability value $p$, which
is a rather unexpected outcome. WNM and WQM effectively describe the data
within the studied kinematic range of $\eta$ when the probability values $p$
exceed 0.5. When comparing the data with the models, it was found that the
probability $p$ changes as a function of pseudorapidity.

  * –

The relation between probability $p$ and pseudorapidity is illustrated in Fig.
8. From this figure emerges that within the limits of uncertainty, $p$ shows a
universal trend as a function of pseudorapidity, regardless of the type of
colliding system or its energy within the studied energy range.

Figure 8:  Probability $p$ as a function of pseudorapidity obtained in WNM
(left panel) and WQM (right panel) for Pb–Pb and Xe–Xe collisions. Values of
$p$ are drawn with total uncertainties, to their values contributed the most
systematic uncertainty of experimental data for $\Sigma$ and uncertainty on
NBD parameters.

The wounded constituent approach describes the $\Sigma$ observable in overall
agreement with the data and surpasses more complex models like HIJING, AMPT,
or EPOS, which fail to describe this observable’s behavior properly. Perhaps
one of the most surprising outcomes of this study is a direct relation between
probability $p$ and $\Sigma$ in the WNM and WQM. In Pb–Pb and Xe–Xe
collisions, reliable probability $p$ values for the wounded constituent model
can be extracted from experimental results on $\Sigma$. Interestingly, these
probability values provide a way to estimate the fragmentation function in
these symmetric collisions using Eq. 5. This procedure is discussed in the
next section.

###  III.5 Wounded nucleon and wounded quark fragmentation functions

The term fragmentation function $F(\eta)$ of a wounded constituent refers to
the pseudorapidity single-particle density characterizing the contribution
from a wounded nucleon or a wounded quark to particle production. For a two-
component picture of forward- and backward-moving wounded constituents, the
fragmentation function can be extracted based on the measurement of a single
particle pseudorapidity distribution, $N(\eta)$, from the relation Eq. 12.

$F(\eta)=\dfrac{1}{2}\left(\dfrac{N(\eta)+N(-\eta)}{\langle
w_{F}\rangle+\langle w_{B}\rangle}+\dfrac{N(\eta)-N(-\eta)}{\langle
w_{F}\rangle-\langle w_{B}\rangle}\right)$ (12)

The formula above is a regular transformation of the Eq. 2, Ref [13].

The major problem in this approach is that Eq. 12 is only applicable for
asymmetric collisions for which $w_{F}\neq w_{B}$ since for the symmetric case
($w_{F}=w_{B}$) the second part of this relation becomes indefinite.

One method for determining the fragmentation function in a symmetric pp
collision was proposed in Ref. [36] based on the study of the average numbers
of produced particles together with pairs of particles in different symmetric
and asymmetric pseudorapidity intervals.

In this section, it will be shown that measuring the forward-backward
correlations with the $\Sigma$ variable provides a unique opportunity to
determine the fragmentation function of a wounded nucleon or quark in a
symmetric nucleus-nucleus collision. The method is based on the fact that the
$\Sigma$ quantity in the wounded constituent model provides direct information
about the probability value $p$, which is related to the single nucleon
fragmentation function through Eq. 5. By combining Eq. 2 and Eq. 5, it the
following relation can be derived:

$\displaystyle\int_{F}F(\eta)\,d\eta=p{\int_{F}\left(F(-\eta)+F(\eta)\right)\,d\eta}$
$\displaystyle=\dfrac{p}{\langle w_{F}\rangle+\langle
w_{B}\rangle}{\int_{F}\left(N(-\eta)+N(\eta)\right)\,d\eta}.$ (13)

For a sufficiently narrow pseudorapidity ($\eta$) interval F, one can obtain
the following:

$\displaystyle F(\eta)\approx\dfrac{p}{\langle w_{F}\rangle+\langle
w_{B}\rangle}\left(N(-\eta)+N(\eta)\right).$ (14)

The fragmentation functions of the wounded nucleon and wounded quark were
determined for minimum bias as well as for different centralities of Pb–Pb
collisions at $\sqrt{s_{\rm{NN}}}$=5.02 TeV, according to Eq. 14. The values
of the $p$ parameter, entering the formula, were extracted based on the
selection of the best fit between the $\Sigma$ obtained in the wounded
constituent model to the experimental results measured by ALICE, as described
in Section III.4. For $\eta$=0, it was assumed that the probability value
$p$=0.5, which is also suggested by the behavior of the data points in Fig. 8.
The $N(\pm\eta)$ values were taken from charged-particle multiplicity
densities measured in Pb–Pb collisions at $\sqrt{s_{\rm{NN}}}$=5.02 TeV by
ALICE and reported in Ref. [37]. The number of wounded constituents ($\langle
w_{F}\rangle+\langle w_{B}\rangle$), nucleons and quarks, was generated in
Monte Carlo discussed in Section II.2. This study considered a sample with a
$0-80\%$ centrality as a minimum bias since centrality class $80\%$ has been
affected by contamination from electromagnetic processes. Given that the
publication Ref. [37] does not directly provide information on $dN_{ch}/d\eta$
for centrality $0-80\%$, the values were determined from the available data as
an arithmetic average. Their uncertainties were estimated from the sum in
quadrature.

Figure 9:  The wounded nucleon (left panel) and quark (right panel)
fragmentation functions extracted from symmetric Pb–Pb collisions at
$\sqrt{s_{\rm{NN}}}$=5.02 TeV as a function of $\eta$ for different centrality
classes.

Figure 9 presents the fragmentation function of the wounded nucleon and
wounded quark determined in Pb–Pb collisions at $\sqrt{s_{\rm{NN}}}$=5.02 TeV.
Results are drawn as a function of $\eta$ for different centrality classes. In
figures, error bars mark total uncertainty of $F(\eta)$, to its values
contributed the uncertainty of the parameter $p$ and the uncertainty of
$N(\pm\eta)$ determination reported in Refs. [37].

As seen from Fig. 9, the shape of the wounded nucleon fragmentation function
slightly changes with centrality class while its values increase with going
from peripheral to central collisions. Whereas the fragmentation function of
the wounded quark is more universal in nature; namely, the shape and values
are similar for all analyzed centrality classes. Qualitatively these findings
coincide with the behavior of wounded nucleon and wounded quark fragmentation
functions reported in the analysis of asymmetric d+Au collisions at
$\sqrt{s_{\rm{NN}}}$=200 GeV Ref. [13]. On the other hand, the fragmentation
function determined in this work has a sharper trend with pseudorapidity than
what is suggested by paper Ref. [13]. Further research is needed to understand
this difference.

Figure 10:  Reconstruction of charged particle multiplicity distributions
$N(\eta)\equiv dN_{ch}/d\eta$ as functions of $\eta$ for Pb–Pb collisions at
$\sqrt{s_{\rm{NN}}}$=5.02 TeV using the extracted, based on the $\Sigma$
measurement, minimum bias fragmentation function $F(\eta)$. The data from
ALICE is labeled with lines, while points represent this paper’s calculation
results. The corresponding uncertainties for the ALICE data and model
calculations are denoted by shaded areas and bars, respectively.

As a cross-check exercise, the extracted minimum bias fragmentation function
of wounded quarks was used to determine charged particle density according to
the Eq. 2 in Pb–Pb collisions. The results obtained on $dN_{ch}/d\eta$ were
compared with the values measured in the ALICE experiment and are shown in
Fig. 10. The figure shows good agreement between two sets of data points.

##  IV Conclusions

This study set out to determine the properties of $\Sigma$ quantity at LHC
energies in terms of two specific classes of independent source frameworks:
the wounded nucleon and the wounded quark model. The main findings to emerge
from this study are listed below.

  1. 1.

Incorporating two-component scenarios of forward- and backward-moving
constituents makes the $\Sigma$ directly dependent on the number of wounded
constituents, and thus collapses the strongly intensive properties of this
observable in the wounded nucleon and wounded quark models.

  2. 2.

Even though in the WNM and WQM, $\Sigma$ is no longer a strongly intensive
quantity, it retains some of its properties in symmetric collisions. In
particular, it is free from volume fluctuations and does not depend on the
method of centrality selection for $\leq 10\%$ centrality bin width, which is
consistent with the ALICE observation. This emerges from the fact that,
despite the width of the centrality interval and the method of centrality
selection, $\Sigma$ always gives a covariance value between forward and
backward wounded constituents at their fixed average number in a studied
centrality sample.

  3. 3.

The two considered wounded constituent models are in good agreement with the
ALICE data. The WNM and WQM models depict the $\Sigma$ observable behavior
that quite accurately matches the experimental results. These models
outperform more complex ones such as HIJING, AMPT, or EPOS, which struggle to
describe this observable properly, Refs. [5, 7, 6].

  4. 4.

One of the more surprising results from this analysis is the sensitivity of
the $\Sigma$ quantity to the probability of particle emission in a given
pseudorapidity interval by a wounded source. This relation allows the direct
determination of the fragmentation function of a wounded nucleon or quark in a
symmetric nucleus-nucleus collision, which has not been possible so far.

  5. 4.

In this work, the single particle wounded quark and nucleon fragmentation
functions were the first-ever extracted for symmetric Pb–Pb collisions at LHC
energies.

  6. 5.

The fragmentation function of the wounded quark has a more universal character
and allows an accurate description of the charge-particle density
distributions measured by ALICE.

###### Acknowledgements.

The author would like to thank Adam Bzdak for suggesting this investigation
and useful discussions, Andrzej Rybicki for valuable remarks and critical
reading of the manuscript, and Wojciech Broniowski for introducing her to the
concept of partial correlations. This work was supported by the National
Science Center, Poland (grant No.2021/43/D/ST2/02195).

##  Appendix A Method of derivation of analytical formula on $\Sigma$

In this Appendix, the calculation procedure of the $\Sigma$ variable
analytical formula in terms of WNM and WQM is presented. The given procedure
is based on the methodology proposed in the papers Refs. [12, 3], which use
the generating function approach.

As derived in Refs. [12, 3] in the framework of the wounded constituent model,
for forward and backward pseudorapidity intervals symmetrically located around
$\eta=0$, the generating function takes the form given by Eq. 15.

$\begin{split}H(z_{B},z_{F})=&\sum_{w_{F},w_{B}}W(w_{F},w_{B})\\{1+f\left(z_{B},z_{F}\right)\\}^{-\frac{k\times
w_{B}}{2}}\times\\\ &\times\\{1+g\left(z_{B},z_{F}\right))\\}^{-\frac{k\times
w_{F}}{2}},\\\ \end{split}$ (15) $\begin{split}\text{where:}\\\
&f\left(z_{B},z_{F}\right)=\frac{\overline{n}}{k}\left[p(1-z_{B})+q(1-z_{F})\right]\\\
&g\left(z_{B},z_{F}\right)=\frac{\overline{n}}{k}\left[q(1-z_{B})+p(1-z_{F})\right]\\\
\end{split}$

In this formula, $W(w_{F},w_{B})$ represents the probability distribution of
the numbers of wounded nucleons in the backward and forward-moving nucleus.
The quantities $p$ and $q=1-p$ denote the probabilities, where $p$ is defined
by Eq. 5.

By applying appropriate differentiation to the generating function defined in
Eq. 15, one can determine the moments of the forward-backward multiplicity
distribution:

$\displaystyle\langle N_{F(B)}\rangle$
$\displaystyle=\evaluated{\partialderivative[1]{H(z_{B},z_{F})}{z_{F(B)}}}_{z_{F}=z_{B}=1}$
(16) $\displaystyle\langle N_{F(B)}^{2}\rangle$
$\displaystyle=\evaluated{\partialderivative[2]{H(z_{B},z_{F})}{z_{F(B)}}}_{z_{F}=z_{B}=1}$
(17) $\displaystyle\langle N_{F}N_{B}\rangle$
$\displaystyle=\evaluated{\partialderivative{H(z_{B},z_{F})}{z_{F}}{z_{B}}}_{z_{F}=z_{B}=1}.$
(18)

Those moments are used to derive quantities such as
$\omega_{F(B)}=\frac{Var(N_{F(B))}}{\langle N_{F(B)}\rangle}$, or
$\text{Cov($N_{B},N_{F}$)}=\langle N_{F}N_{B}\rangle-\langle
N_{B}\rangle\langle N_{F}\rangle$ needed to construct $\Sigma$ according to
Eq. 1.

##  Appendix B Derivation of the relation between $\Sigma$ quantity and
partial covariance for fully symmetric collision

To prove that for a fully symmetric collision, for which $\langle
w_{B}\rangle=\langle w_{F}\rangle$ and $\text{Var}(w_{B})=\text{Var}(w_{F})$,
the equation:

$\boxed{\text{Var}(w_{F}-w_{B})=-4\text{Cov}(w_{F},w_{B}\bullet w)}$ (20)

is satisfied, we will use the following basic properties of variance and
covariance:

$Var(X-Y)=Var(X)+Var(Y)-2Cov(X,Y),$ (21) $Var(X+Y)=Var(X)+Var(Y)+2Cov(X,Y),$
(22) $Cov(X,Y+Z)=Cov(X,Y)+Cov(X,Z).$ (23)

For $\langle w_{B}\rangle=\langle w_{F}\rangle$ we have:

$\displaystyle\boxed{\langle(w_{F}-w_{B})^{2}\rangle=\text{Var}(w_{F}-w_{B}),}$
(24)

and by adding two expressions Eq. 21 and Eq. 22, the variance of the
difference can be represented as:

$\displaystyle\boxed{\text{Var}(w_{F}-w_{B})=\text{Var}(w_{F}+w_{B})-4\text{Cov}(w_{F},w_{B}).}$

Then based on properties of variance and covariance expressed in Eq. 22 and
Eq. 23, one can obtain:

$\displaystyle\text{Var}(w_{F}+w_{B})=\text{Var}(w_{F})+\text{Var}(w_{F})+2\text{Cov}(w_{F},w_{B})=$
$\displaystyle=\underbrace{\text{Var}(w_{F})+\text{Cov}(w_{F},w_{B})}_{\text{Cov}(w_{F},w)}+\underbrace{\text{Var}(w_{B})+\text{Cov}(w_{F},w_{B})}_{\text{Cov}(w_{B},w)}=$
$\displaystyle=\text{Cov}(w_{F},w)+\text{Cov}(w_{B},w)$ (26)

where $w=w_{B}+w_{F}$.
For symmetric case $\text{Cov}(w_{F},w)=\text{Cov}(w_{B},w)$, so the above
formula can be simplified to the form:

$\boxed{\text{Var}(w_{F}+w_{B})=2\text{Cov}(w_{F},w).}$ (27)

On the other hand, from the definition of partial covariance (Eq. 9) we find:

$\text{Cov}(w_{F},w_{B}\bullet
w)=\text{Cov}(w_{F},w_{B})-\frac{\text{Cov}(w_{F},w)\text{Cov}(w_{B},w)}{\text{Var}(w)},$
(28)

which for the symmetric collision takes the form:

$\boxed{\text{Cov}(w_{F},w_{B}\bullet
w)=\text{Cov}(w_{F},w_{B})-\frac{\text{Cov}(w_{F},w)^{2}}{\text{Var}(w)}.}$
(29)

Combining Eq. 29 and Eq. 27 we get:

$\displaystyle\text{Cov}(w_{F},w_{B}\bullet
w)=\text{Cov}(w_{F},w_{B})-\frac{\text{Var}(\overbrace{w_{F}+w_{B}}^{w})^{2}}{4\text{Var}(w)}=$
$\displaystyle=\text{Cov}(w_{F},w_{B})-\frac{\text{Var}(w)}{4}$ (30)

And finally, transforming the Eq. B and taking into account Eq. 26 we find the
relation given by Eq. 10:

$\displaystyle\boxed{-4\text{Cov}(w_{F},w_{B}\bullet
w)=\underbrace{-4\text{Cov}(w_{F},w_{B})+\text{Var}(w)}_{\text{Var}(w_{F}-w_{B})}}$

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