aainstitutetext: Space sciences, Technologies and Astrophysics Research (STAR)
Institute, Université de Liège, Bât. B5a, 4000 Liège, Belgiumbbinstitutetext:
Tunis El Manar University, Faculty of Sciences of Tunis, Nuclear Physics and
High Energy Research Unit, Tunis, Tunisia

# The $U$-Matrix geometrical model for multi-particle production in high-
energy hadronic collisions

Rami Oueslati  0000-0003-2569-4056 b and Adel Trabelsi  0000-0002-2020-3089
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Inspired by the picture portraying the KNO scaling violation as an extension
of the geometrical scaling violation, the current study proposes a
phenomenological model for multi-particle production in hadron collisions
based on the geometrical approach and using the $U$-Matrix unitarization
scheme of the scattering amplitude. The model has been fine-tuned and all
parameters have been derived from optimal fits to various hadronic
multiplicity distributions data in $p+p(\bar{p})$ collisions across a broad
range of energies. The results have revealed that our model furnishes a
reasonable description of diverse multiplicity distributions at various
energies. Besides, they have demonstrated a pronounced violation of the
geometrical scaling, which eventually resulted in a significant violation of
the KNO scaling. The study has also analyzed the higher-order moments of the
multiplicity distribution. We have observed an unexpected overestimation of
the fluctuations and correlations between final state particles with
increasing energy, particularly above LHC energy. It is claimed that this
overestimation is due to statistical fluctuations embedded in the $U$-matrix
scheme. The findings of this study have shed light on the key role of the
$U$-matrix scheme in the impact of collision geometry on multi-particle
production processes at high energy.

###### Keywords:

QCD Phenomenology; Unitarity constraint; geometrical and KNO scaling
violation; particle correlation

##  1 Introduction

Throughout the years, the study of multi-particle production in hadron
collisions at high energies has sparked the interest of both theoretical
physicists and experimentalists given its significance as it offers valuable
insights into the intricate mechanisms underlying the production of particles
VANHOVE1971347 ; Giacomelli:2009fr ; Liu_2022 ; Koley:2022tqg ; DREMIN2001301
.

Similarly, the hadronic multi-particle production is primordial at ultra-high
energies. Indeed, it is necessary to have a solid grasp of it so as to
properly interpret air-shower cosmic ray observables, which are obtained
through a simulation of a wide range of event generators based on Monte Carlo
models. These models have been created and adjusted to describe hadronic
multi-particle production in man-made accelerator experiments. Despite some
minor differences in their fundamental assumptions, the majority of them
employ the eikonal as the scheme of the unitarisation of the scattering
amplitude. They must also be internally consistent as we depend on them for
extrapolating to ultra-high energy scenarios. Nevertheless, modelling hadronic
interactions is not devoid of uncertainties, which gives rise to model-
dependent outcomes. Among them, one can cite the mass composition problem and
the long-standing muon puzzle, which is regarded as one of the most
significant issues in hadronic interaction physics alves2019open .

Having said that, advances in the phenomenology and theory of multi-particle
production may provide a solution to the aforementioned issues.

The study of multi-particle production may be carried out, particularly,
through examining the multiplicity distributions of the produced particles.
Their analysis is crucial since the patterns observed in them help reveal the
complexity of the collision process, shedding light on the interactions
involved in particle creation. In fact, this distribution has been thoroughly
investigated by experimental collaborations at the LHC, including ALICE,
ATLAS, CMS, and LHCb Grosse-Oetringhaus_2010 , to enhance our comprehension of
the rudimentary characteristics of strong interactions and the behaviour of
matter under extreme circumstances as well as to verify theoretical
predictions.

Besides, the study of the higher-order moments of the multiplicity
distribution is of paramount importance as they provide insight into particle
features in high-energy collisions, such as correlation and collective
behaviour. Indeed, the behaviour of these moments has been the subject of
various experimental studies over a wide range of collision energies
CMS:2019fur ; Pandav:2022xxx , furnishing valuable data which are crucial in
constraining theoretical models.

The multiplicity distribution $P_{n}(s)$ in hadron collisions refers to the
distribution of the number of produced particles in a collision event. It is
influenced by various factors, such as the colliding hadrons’ energy, the
collision geometry, and the underlying dynamics of the interaction. More
specifically, it is given by the ratio of the topological cross-section
$\sigma_{n}$ to the inelastic cross-section $\sigma_{in}$. The topological
cross-section represents the probability or rate of observing a specific
configuration of particles with a particular multiplicity value $n$. It is
derived from the scattering amplitudes, which are influenced by the underlying
scattering processes and dynamics. The inelastic cross-section, on the other
hand, characterizes the probability of observing any final-state interaction
or particle production, regardless of the specific configuration or
multiplicity. It is closely related to the total cross-section and includes
contributions from various interaction channels, including both diffractive
and non-diffractive processes.

It is important to note that these cross-sections, and thus the multiplicity
distribution $P_{n}$, cannot yet be calculated within the framework of quantum
chromodynamics (QCD). Therefore, our current understanding of multi-particle
production dynamics relies primarily on phenomenological approaches as well as
certain underlying theoretical principles, which form the basis for a wide
range of models Grosse-Oetringhaus_2010 . Needless to say, the continuous
enhancement of the theoretical models for particle production is critical so
as to maintain consistency and coherence with the foundational principles of
the Quantum Field Theory (QFT).

One of the key theoretical tenets in the construction of phenomenological
models, such as those based on the geometrical or string approach, is the
unitarity constraint Shih_1997 ; Beggio:1999 ; Beggio:2007 ; BEGGIO_2014 .
These models often utilize the eikonal scheme as a means to unitarize the
scattering amplitude and describe inclusive multiplicity distributions. While
this scheme provides a reasonable description of certain hadronic observables,
there are compelling reasons to explore alternative schemes for unitarizing
the scattering amplitude, namely the $U$-matrix one.

In a recent study Bhattacharya:2020lac , it was shown that despite unexpected
agreement in the description of the inelastic $pp$ and $p\bar{p}$ cross-
sections with the eikonal and $U$-matrix schemes, a divergence in the
individual order-by-order amplitudes can potentially impact the topological
cross-section by influencing the relative probabilities of different
multiplicity configurations and capturing specific physical processes.

By the same token, in another study Vanthieghem:2021akb , it has been found
that the $U$-matrix scheme exhibits a slightly better fit to the single
diffractive data at high-energies and a faster growth at ultra-high energies
compared to the eikonal scheme. This result implies that the underlying
physics and dynamics of the single diffractive scattering are sensitive to the
choice of scheme, especially at ultra-high energy, further reinforcing the
significance of considering the $U$-matrix. Hence, it is worth noting that if
the discrepancy in the single diffractive cross-section propagates to the
topological cross-section, it can affect the relative probabilities or rates
of observing different multiplicity configurations. Moreover, as the inelastic
cross-section encompasses both diffractive and non-diffractive processes, the
different behaviours in the single diffractive cross-section between
unitarization schemes can contribute to variations in the overall inelastic
cross-section.

It should be noted that the specific impact on the topological and inelastic
cross-sections would depend on the detailed correlations and interplay between
different interaction channels, including both diffractive and non-diffractive
processes. In fact, when hadrons collide at high energies, several
interactions and processes take place, leading to the formation of large
numbers of particles with various species. Given the complex nature of these
interactions, deciphering the contributions from individual channels and
identifying the specific processes behind multi-particle production appears to
be a daunting task GayDucati:2007fbs .

On the whole, the inelastic and topological cross-sections are scheme-
dependent, implying that the multiplicity distribution $P_{n}$ is scheme-
dependent as well. This will, therefore, impact the description of the
multiplicity distribution in terms of its shape, magnitude, or other
characteristic features.

An additional reason for considering an alternative to the eikonal
unitarization scheme is related to the geometrical scaling, as supported by
experimental observations. This scaling is a regularity established by the ISR
measurements of the proton-proton and proton-antiproton scattering and refers
to the invariability of the ratio between elastic and total cross-sections.
Interestingly, experiments conducted at the CERN (SPS) collider have revealed
that this regularity is violated when the energy surpasses the ISR energy
range BOZZO1984392 ; ALNER1984304 . Furthermore, from a theoretical
perspective, it has been shown that the violation of the geometrical scaling
is more pronounced with the $U$-matrix scheme than with the eikonal as energy
increases Bhattacharya:2020lac . It should be emphasized that this remarkable
discrepancy has proved to occur mainly when we venture into the extremely
high-energy region, roughly near the Grand Unification Scale. In this regard,
one of the preliminary objectives of the present study is to examine this
disparity within an accessible energy range.

Another regularity put forth by the ISR measurements is the KNO scaling so
named after its proponents Koba, Nielsen, and Olesen (KNO) KOBA1972317 . It
refers to the constancy of normalized moments in the distribution of
multiplicities. Practically, the KNO function is often employed for the
examination of multiplicity distributions in particle collisions. It is
denoted as $\Phi(z)$, where $\langle n\rangle$ stands for the average of the
multiplicity distribution, and $z=n/\langle n\rangle$ represents the
normalized multiplicity. It is noteworthy that the KNO function $\Phi(z)$
tends to be independent of the collision energy $\sqrt{s}$ within the ISR
energy range. However, considerable deviations from the KNO scaling start to
occur at higher energies, such as those attained at FNAL and LHC.

As a matter of fact, several phenomenological studies have demonstrated the
existence of a connection between the geometrical scaling of the profile
function and the KNO scaling of the multiplicity distribution Itabashi ;
Finkelstein:1988cu ; lam:1982 . More precisely, these models have shown that
the violation of the geometrical scaling, indicated by an increase in the
ratio of elastic to total cross-sections ($\sigma_{el}/\sigma_{tot}$) between
ISR and collider energies, is associated with the violation of the KNO scaling
across these energy ranges within the eikonal scheme. This raises the question
of potential consequences for the violation of the KNO scaling within the
context of the $U$-matrix scheme. It is expected that the latter, with its
modifications to the scattering amplitude, may introduce novel dynamics and
fluctuations that influence the statistical behaviour of particle production.
These effects can in turn influence the universal properties assumed in the
KNO scaling and significantly lead to a violation or alteration of the
constancy of normalized moments.

In view of all that has been mentioned so far, one can assume that considering
an alternative adequate unitarization scheme can considerably change our
understanding of the description of multi-particle production. In the present
study, we propose a phenomenological model for multi-particle production in
hadron-hadron collisions that hinges on both the geometrical approach and the
picture depicting the KNO scaling violation as an extension of the geometrical
scaling violation, using the $U$-Matrix scheme. The chief purposes of the
study are to examine the geometrical scaling violation within an accessible
energy range, describe the hadronic multiplicity distributions in full-phase
space over a wide energy range in $p+p(\bar{p})$ collisions and to investigate
the KNO scaling violation. It also aims at probing the correlation between the
final particles and revealing the physics that underlies it. In particular, it
seeks to provide valuable insights into the relationship between the violation
of both the geometrical and KNO scaling and the mechanism of particle
production within the $U$-matrix scheme.

The remainder of the paper is organized as follows: In Section II, the
theoretical background of the multi-particle production model will be
outlined. Section III will focus on the explicit model and the data used.
Section IV will present and discuss the results. Finally, Section V will
summarize the findings and discuss the limitations and implications of this
work so as to guide future research.

##  2  The theoretical framework of the Model

The multiplicity distribution $P_{n}(s)$, i.e., the probability of producing
$n$ charged particles in an inelastic $p+p(\bar{p})$ collision at the energy
$\sqrt{s}$, is given by

$\displaystyle P_{n}(s)=\frac{\sigma_{n}(s)}{\sigma_{in}(s)}$ (1)

where $\sigma_{n}(s)$ and $\sigma_{in}(s)$ are the $n$-particle topological
and inelastic cross-sections, respectively, with
$\sum_{n}\sigma_{n}(s)=\sigma_{in}(s)$.

Great efforts have been devoted to explaining why the normalized moments in
this multiplicity distribution remain constant in the ISR energy range but
diverge from this pattern from the LHC energy range. A possible explanation
for this phenomenon might be ascribed to the increasing importance of mini-
jets production, resulting from both soft and semi-hard partonic processes, as
energy increases. In fact, these mini-jets not only contribute to the rapid
growth of hadron-hadron cross-sections, as demonstrated by several geometrical
models FAGUNDES201248 ; Bhattacharya:2020lac ; Vanthieghem:2021akb ;
Oueslati:2023tjt , but may also play a crucial role in the violation of the
KNO scaling.

It goes without saying that the geometrical models based on the impact
parameter space representation provide a good description of various aspects
of hadron collisions at high energies. Technically speaking, by considering
the impact parameter, which quantifies the distance between the colliding
particles’ centres, these models actually furnish a solid framework for
understanding the initial stages of collisions, allowing a systematic
exploration of the collision geometry, ranging from central (small impact
parameters) to peripheral (large impact parameters) collisions. Therefore,
this geometrical approach links the collision geometry to the underlying
physics processes, enabling us to infer certain properties. We particularly
assume that this approach can also shed light on the intricate interplay
between collision geometry and particle multiplicity distribution.

In addition, as has been mentioned in the previous section, the KNO scaling
violation can be perceived as an extension of the geometrical scaling
violation to the multi-particle production process. This extension highlights
that the dynamics governing particle production become more complex and
energy-dependent than what a purely geometrical approach can capture. It also
suggests that additional physical processes, such as parton interactions,
collective effects, or fluctuations, play a role in shaping the multiplicity
distribution as collision energies increase. As a result, a more accurate
description of the geometrical scaling violation is needed, which in turn will
provide a better description of the violation extension to multi-particle
production. In fact, the violation of geometrical scaling in impact parameter
space occurs when the initial assumptions about simple geometrical overlap and
scaling behaviour break down due to more intricate particle-particle
interactions or energy-dependent effects. Consequently, this concept
harmoniously aligns with the utilization of the impact parameter space
representation, emphasizing the significance of collision geometry.

Overall, when combined with the geometrical scaling tenets, the geometrical
models can be enhanced and thus provide a solid framework for describing the
multiplicity distribution and uncovering the underpinning universal features
of particle production in high-energy hadron collisions. They also furnish a
reliable tool that serves to explain the impact of collision geometry on the
multiplicity distribution and to analyze it across a broad range of collision
energies and particle species, improving prediction and comparison accuracy in
multiple collision systems.

To construct our model highlighting the intrinsic relationship between
geometrical scaling and KNO scaling within a geometrical approach, we draw
inspiration from CHOU1982301 ; Barshay_PRL ; lam:1982 ; Kuang_1983 , where the
essence of the approach lies in expressing the overall hadronic multiplicity
distributions in the inelastic channel through a combination of an elementary
distribution and the inelastic overlap function. Thus, this approach
establishes a direct connection between the fluctuations in multiplicity and
the fluctuations in the impact parameter, driven by the intuitive
understanding that collisions occurring at shorter distances tend to be more
violent and yield a higher number of fragments, thereby leading to increased
multiplicities. Therefore, in order to establish a reliable geometrical
approach to calculate the multiplicity distribution in the impact parameter
space representation, we rely on the unitarity condition of the $S$ matrix.
This involves examining the elastic scattering amplitude in the impact
parameter $b$ space, expressed by the equation:

$\displaystyle 2\,\textnormal{Im}[\Gamma(s,b)]=|\Gamma(s,b)|^{2}+G_{in}(s,b),$
(2)

where $\Gamma(s,b)$ denotes the profile function, i.e., the elastic hadron
scattering amplitude, and $G_{in}(s,b)$ represents the inelastic overlap
function. By employing the optical theorem, we can obtain

$\sigma_{tot}(s)=2\int d^{2}b\;\textnormal{Im}[\Gamma(s,b)],$ (3)

and recognizing that the function $G_{in}(s,b)$ signifies the absorption
probability associated with each $b$ value, we can derive the total inelastic
cross-section

$\sigma_{in}(s)=\int d^{2}b\;G_{in}(s,b).$ (4)

Thus, the unitarity condition (2) is equivalent to
$\sigma_{tot}(s)=\sigma_{el}(s)+\sigma_{in}(s)$, where $\sigma_{el}(s)=\int
d^{2}b\,|\Gamma(s,b)|^{2}$.

Various research papers in the field have already explored this
phenomenological procedure based on the impact parameter space representation
known as the geometrical or string approach Beggio:1999 ; Beggio:2007 ;
BEGGIO_2014 ; Beggio:2020dry . In this geometrical approach, wherein incident
hadrons are treated as spatially extended objects and their collisions are
represented as an ensemble of elementary interactions between quarks and/or
gluons, the hadronic multiplicity distribution $P_{n}(s)$ is constructed based
on elementary quantities associated with microscopic processes. It follows
from this that the overall hadronic multiplicity distribution is estimated
through summing contributions emerging from each impact parameter $b$ of the
incident hadronic system.

To do this, the topological cross-section $\sigma_{n}$ is decomposed into
contributions from each impact parameter $b$ with weight $G_{in}(s,b)$ as
follows:

$\displaystyle\sigma_{n}(s)$ $\displaystyle\equiv\int d^{2}b\;\sigma_{n}(s,b)$
(5) $\displaystyle=\int
d^{2}b\;G_{in}(s,b)\left[\frac{\sigma_{n}(s,b)}{\sigma_{in}(s,b)}\right],$

where $\sigma_{in}(s)\equiv\int d^{2}b\,\sigma_{in}(s,b)=\int
d^{2}b\,G_{in}(s,b)$.

The quantity enclosed in brackets $\sigma_{n}(s,b)/\sigma_{in}(s,b)\equiv
p_{n}(s,b)$ can be interpreted as the probability of producing $n$ particles
at impact parameter $b$. It accounts for interactions among the elementary
components of the colliding hadrons. Keeping in mind that $p_{n}(s,b)$ should
scale in KNO sense given its elementary structure, the multiplicity
distribution $P_{n}(s)$ (1) can be reformulated as :

$\displaystyle P_{n}(s)=\frac{1}{\sigma_{in}(s)}\int
d^{2}b\,\frac{G_{in}(s,b)}{\langle n(s,b)\rangle}\left[\langle n(s,b)\rangle
p_{n}(s,b)\right]$ (6)

where $\langle n(s,b)\rangle$ represents the average number of particles
produced at $b$ and $s$.

Let $\Phi(s,z)=\langle n(s)\rangle P_{n}(s)$ be the overall multiplicity
distribution in KNO form, where $z=n(s)/\langle n(s)\rangle$ is the
corresponding KNO variable. A multiplicity distribution $\phi(s,\b{z})$,
associated with elementary processes occurring at $b$ and $s$, can be written
in the form $\phi(s,\b{z})=\langle n(s,b)\rangle p_{n}(s,b)$, where
$\b{z}=n(s)/\langle n(s,b)\rangle$. It is worth noting that both distributions
adhere to the standard normalizations lam:1982  :

$\displaystyle\int_{0}^{\infty}dz\,\Phi(z)=\int_{0}^{\infty}d\b{z}\,\phi(\b{z})=2$
(7)

and

$\displaystyle\int_{0}^{\infty}dz\,z\,\Phi(z)=\int_{0}^{\infty}d\b{z}\,\b{z}\,\phi(\b{z})=2.$
(8)

In this theoretical framework, the unknown function representing the average
number of particles produced at a specific impact parameter $b$ and energy $s$
can be comprehended as the product of two factors: The first factor, $\langle
n(s)\rangle$, representing the general behaviour depicts the average number of
particles generated in a collision, regardless of the specific impact
parameter. As for the second, it is denoted as $f(s,b)$ and perceived as a
multiplicity function. It describes the variation of the average number of
particles in accordance with the impact parameter $b$ quantifying how
collision geometry affects particle production. The mathematical expression
for this factorization is:

$\langle n(s,b)\rangle=\langle n(s)\rangle f(s,b)$ (9)

and therefore the expression (6) can be rewritten in the KNO form as follows:

$\displaystyle\Phi(s,z)=\langle n(s)\rangle P_{n}(s)=\frac{1}{\int
d^{2}b\,G_{in}(s,b)}\int
d^{2}b\,\frac{G_{in}(s,b)}{f(s,b)}\,\phi\left(\frac{z}{f(s,b)}\right),$ (10)

and we also have

$\displaystyle P_{n}(s)=\frac{1}{\langle n(s)\rangle\int
d^{2}b\,G_{in}(s,b)}\int
d^{2}b\,\frac{G_{in}(s,b)}{f(s,b)}\,\phi^{(1)}\left(\frac{z}{f(s,b)}\right),$
(11)

The master formula eq. (11) highlights the connection between the multiplicity
distribution $P_{n}(s)$ and the unitarisation scheme of the scattering
amplitude which can ultimately be established within the inelastic overlap
function, emphasizing that this multiplicity is scheme-dependent as stated in
the introduction. The full phase space hadronic multiplicity distribution
$P_{n}(s)$ is hence constructed by summing contributions from parton–parton
collisions occurring at each value of $b$. These interactions give rise to the
formation of string-like objects, similar to the string formation described by
the Lund model ARTRU197493 . These strings in turn fragment into hadrons. So,
in order to stress the fundamental principle of the string approach, an index
labelling the elementary multiplicity distribution $\phi^{(1)}$ has been
introduced in the master formula (11). The essential elements in this formula
eq. (11), namely the inelastic overlap function, the average number of
particles produced at a specific impact parameter $b$ and energy $s$, as well
as the elementary multiplicity distribution function, will be explicitly
presented in the following section, laying the groundwork for an in-depth
analysis of the complex interactions between collision geometry and particle
production.

##  3 Explicit model and data

The explicit model presented in this paper for describing High-energy hadronic
scattering is theoretically grounded in the geometrical approach and is based
on the Reggeon exchanges picture (see, e.g. Donnachie:2013xia  and references
therein). In this picture, hypothetical exchange particles, known as pomerons,
mediate interactions between hadrons and the procedure of obtaining the
amplitude of a given hadronic process involves summing over all conceivable
ways in which pomerons can be exchanged.

Diagrammatically, every pomeron exchange is represented by a line that
connects the incoming and outgoing hadrons. Hence, by summing all possible
topologies of these diagrams, the total amplitude is determined. Technically,
the single-pomeron exchange amplitude, also known as, the Born term can be
parameterized as follows:

$a(s,t)=g_{p}^{2}\,\mathcal{F}_{1}(t)^{2}\left(\frac{s}{s_{0}}\right)^{\alpha(t)}\,\xi(t),$
(12)

where $g_{p}$ is the pomeron-proton coupling, $\mathcal{F}_{1}(t)$ denotes the
proton elastic form factor, and $\xi(t)$ stands for the signature factor that
is given by:

$\xi(t)=-e^{-i\pi\alpha(t)/2},$ (13)

and the pomeron trajectory, represented by $\alpha(t)$, approximates a
straight line:

$\alpha(t)=1+\epsilon+\alpha^{\prime}_{P}t.$ (14)

In the impact-parameter space representation, where the Fourier transform of
the amplitude $a(s,t)$ rescaled by $2s$ corresponds to a partial wave, we
have:

$\chi(s,\mathbf{b})=\int\frac{\mathrm{d}^{2}\mathbf{q}}{(2\pi)^{2}}\frac{a(s,t)}{2s}\exp(\mathrm{i}\mathbf{q}\cdot\mathbf{b}).$
(15)

When we venture into high energies, summing amplitudes may lead to a violation
of unitarity, especially in the case of multi-pomeron exchanges. Therefore, in
order to make the summed amplitudes comply with the unitarity constraint,
unitarization techniques come into play by resuming infinite series or
introducing additional terms to modify the amplitude’s behaviour. As a matter
of fact, there is a plethora of techniques in the literature Selyugin:2007jf ;
Cudell:2008yb ; Drescher:2000ha  whose shared objective is to come up with a
consistent approach to summing over various exchange contributions while
making sure that the resulting amplitude $\Gamma(b,s)$, i.e., the hadronic
profile function, satisfies the unitarity condition and may account for all
features of interactions in the context of hadron collisions. Among them, we
can cite the eikonal scheme, which is one of the most commonly used methods,
positing that the profile function is provided by:

$\Gamma_{E}(s,\bm{b})=\mathrm{i}\left[1-\mathrm{e}^{\mathrm{i}\chi(s,\bm{b})}\right]\,,$
(16)

The $U$-matrix scheme is another illustration, which asserts that:

$\Gamma_{U}(s,\bm{b})=\frac{\chi(s,\bm{b})}{1-\mathrm{i}\,\chi(s,\bm{b})/2}.$
(17)

As previously mentioned in the introduction, we will consider that the
hadronic profile function $\Gamma(b,s)$ is given by the $U$-Matrix form (17).
This function represents the sum of all $n$-pomeron exchange contributions
obtained from the single-pomeron exchange amplitude which is, in turn, related
to the expected number $\chi(b,s)$ of interactions between partons of the
incident hadrons.

Using both the eikonal and the $U$-matrix schemes as well as a dipole-like
form factor for the proton, where $\mathcal{F}_{1}(t)=1/(1-t/t_{0})^{2}$, the
parameters $\epsilon$ and $\alpha^{\prime}_{P}$ describing the pomeron
trajectory, the coupling constant $g_{p}$, and the form-factor scale $t_{0}$
are adjusted based on a best fit to up-to-date hadron collider data on total,
elastic, and inelastic cross-sections. The values of these parameters are
provided in Table 1 Bhattacharya:2020lac . Following this adequate description
of the hadronic profile function, we can determine the inelastic overlap
function $G_{in}(s,b)$, needed in our explicit model, by using the equation
presented in (2).

Scheme | $\epsilon$ |  $\alpha^{\prime}_{P}$ (GeV-2) | $g_{p}$ |  $t_{0}$ (GeV2) | $\chi^{2}/\text{d.o.f}$
---|---|---|---|---|---
U-matrix | $0.10\pm 0.01$ | $0.37\pm 0.28$ | $7.5\pm 0.8$ | $2.5\pm 0.6$ | $1.436$
Eikonal | $0.11\pm 0.01$ | $0.31\pm 0.19$ | $7.3\pm 0.9$ | $1.9\pm 0.4$ | $1.442$
Table 1: $\chi^{2}$/d.o.f and best-fit parameters obtained using the eikonal
and U-matrix unitarisation schemes.

Since the primary objective of this work is to examine the influence of
geometrical collisions on multi-particle production processes, we shall
propose two hypotheses concerning our choice of the elementary multiplicity
distribution. First, it is sufficient for our study to consider that, on
average, every string created in parton-parton interactions has the same
likelihood of producing a certain number of charged hadrons, which is
described by the elementary multiplicity distribution $\phi^{(1)}(z)$, even
though the strings created may have different probabilities of turning into a
pair of charged hadrons. Second, despite the fact that the parametrization of
the elementary distribution $\phi^{(1)}(z)$ is key to capturing the overall
shape of the multiplicity distribution, these two do not necessarily have
similar shapes. This is mainly because the overall distribution is obtained by
summing contributions from elementary processes at different impact
parameters, in the context of this superposition model eq. (11). That is to
say, the peculiar combination of the individual contributions emerging from
distinct impact parameters and which may be having differing shapes results in
an overall distribution that reflects these contributions’ combined effects
and whose characteristics should be represented by their superposition. Hence,
we will assume, as a first approximation, that the elementary distribution has
the same shape as the overall distribution. More specifically, the choice of
the functional form for this elementary distribution is motivated by
phenomenological fits to data.

As a matter of fact, the Negative Binomial Distribution (NBD) has proved to
provide a good description of the experimental data on the multiplicity
distributions in the context of high-energy physics and hadron collisions
Grosse-Oetringhaus_2010 . Therefore, in the present study, the KNO form of the
NBD, also known as the Gamma distribution, is adopted for the elementary
multiplicity distribution and given by:

$\phi^{(1)}(z)=2\,\frac{K^{K}}{\Gamma(K)}\,z^{K-1}\,e^{-Kz},$ (18)

where $K$ is a dimensionless parameter.

It should be pointed out that there exists a connection between the average
number of particles generated at a specific impact parameter $b$ and energy
$s$, $\langle n(s,b)\rangle$, which determines the unknown multiplicity
function $f(s,b)$ by the eq. (9), and the Born term in $b$ space $\chi(s,b)$.

This link can be attributed to the various roles that the Born term plays. To
begin with, the multiplicity of generated particles in $b$ space and the Born
term are interrelated given that the former can be impacted by the effective
interaction of partons within the colliding hadrons and the latter provides a
measure of this effective interaction since it represents the overlap of the
colliding matter distributions. Not to mention, the Born term depends on the
impact parameter. As such, it reflects how strong the interaction between
colliding hadrons is at distinct impact parameters. Indeed, the impact-
parameter-dependent strength of the interaction between hadrons may have an
impact on the multiplicity of produced particles. Secondly, owing that the
Born term is a crucial parameter in describing the energy dependence of the
scattering amplitude and that particle generation is more likely to occur at
higher energies, the Born term is inextricably connected to the average
particle production. Thirdly, on the one hand, the imaginary part of the Born
term is related to inelastic processes in high-energy collisions and, on the
other hand, multi-particle production is often associated with inelastic
interactions, so relying on the imaginary part of the Born term will reflect
the possibility of inelastic scattering and subsequent particle production.
Therefore, the multiplicity function $f(s,b)$ is constructed to depend on the
imaginary part of the Born term, denoted as $\chi_{I}(s,b)$.

Therefore, the connection between the multiplicity function $f(s,b)$ and the
born term is formally defined as follows:

$f(s,b)=\beta(s)\,[\chi_{I}(s,b)]^{2\lambda},$ (19)

where $\beta(s)$ is determined by the normalisation condition (8) :

$\beta(s)=\frac{\int d^{2}b\,G_{in}(s,b)}{\int
d^{2}b\,G_{in}(s,b)\,[\chi_{I}(s,b)]^{2\lambda}}$ (20)

and the exponent $2\lambda$ in eq. (19) introduces a power-law dependence on
$\chi_{I}(s,b)$, suggesting a non-linear relationship between the effective
overlap and the particle production in the impact-parameter space.

It should be emphasized that according to the geometrical approach, the
phenomenological portrayal of hadronic multiplicity distributions is
considerably influenced by the values and behaviours of three key inputs: the
inelastic overlap function $G_{in}$, the elementary multiplicity distribution
$\phi(z)$, and the $\lambda$ parameter determining the power-law dependence on
the effective overlap $\chi_{I}(s,b)$. A previous study Beggio:1999  has shown
that changing one of these inputs, while keeping the other two fixed, produces
different results across different parameterizations. For instance, if the
inputs of the inelastic overlap function are different and $\lambda$, as well
as $\phi(z)$, are kept constant, the obtained hadronic multiplicity
distributions successfully replicate the experimental data. Interestingly, in
all cases, the physical picture is that large multiplicities occur for small
impact parameters while peripheral collisions (large b) lead to small
multiplicities. It is worth noting that all these results, obtained for
different inputs of the inelastic overlap function, are generated using the
eikonal scheme. However, in the present study, we employ the $U$-matrix scheme
for the reasons mentioned in the previous section. This will eventually allow
us to fix our choice of the inelastic overlap function.

In order to present our findings in the following section, it is essential to
provide an overview of the model parameters that were determined through data
fitting, as well as the specific experimental data employed in our analysis.
Since the Born term $\chi_{I}(s,b)$ is completely determined from the best-
fit, Table 1, we see from the master formula for the multiplicity distribution
eq. (11) that the only free parameters are $K$, $\lambda$, and $\langle
n(s,b)\rangle$. Further, for our purposes, it is sufficient to fix the value
of $K$ and assume $\lambda$ and $\langle n(s,b)\rangle$ as the only fitting
parameters. Indeed, in Beggio:1999 , it was shown that the choice of
$K=10.775$ by assuming a Gamma distribution gives a good description of the
charged multiplicity distributions for $e^{+}e^{-}$ annihilation data in a
large energy interval. Based on the universality of multiplicities in
$e^{+}e^{-}$ and $p+p(\bar{p})$ collisions, we adopt this choice.

To determine the values of the parameters $\lambda$ and $\langle n\rangle$, we
fix the dimensionless parameter to $K=10.775$ and conduct fits to full phase
space $P_{n}$ data in $p+p(\bar{p})$ collision across a wide range of
energies, specifically at $\sqrt{s}=30.4,44.5,52.6,62.2,300,546,1000,$ and
$1800$ GeV Breakstone_1984 ; ALEXOPOULOS1998453 .

With regards to the fitting process, we utilized the Minuit2 class from ROOT
Hatlo_2005  and implemented the MIGRAD algorithm. The primary objective of the
fitting procedure was to minimize the $\chi^{2}$ value and the uncertainties
associated with the free parameters were calculated using a $1\sigma$
confidence level.

##  4 Results and discussion

###  4.1 Geometrical scaling violation

The preliminary objective of this study was to analyze the behaviour of the
inelastic overlap function in the impact parameter space as well as the
geometrical scaling violation using the $U$-matrix and the eikonal schemes,
eqs. (16) and (17), in an attempt to offer valuable insights into the
collision geometry and the interaction dynamics of colliding particles. Fig. 1
depicts the predictions for the energy evolution of the inelastic overlap
function, eq. (2), in the impact parameter space at energies spanning from ISR
to LHC levels using both schemes.

(a)  (b)

Figure 1:  The energy evolution of the inelastic overlap function in the
impact parameter space at energies spanning from ISR to LHC levels using both
the $U$-matrix and the eikonal schemes.

Looking at Fig. 1, it is apparent that this function exhibits a generally
comparable pattern in impact parameter space across both schemes, with only
minor differences. More specifically, it shows that it is predominantly
central, indicating that it has a significant contribution at small impact
parameters. This entails that most of the inelastic processes are more
probably to take place when the colliding particles pass through one another
within close proximity. Moreover, this function tends to decline more slowly
with the impact parameter as energy rises. This points out that the inelastic
processes become less dependent on the specific spatial distance between
colliding particles and have a wider range of impact parameters at which they
can happen at higher energies.

Figure 2:  Energy evolution of the ratio elastic-to-total cross-section in
both cases, the eikonal and $U$-matrix schemes

Furthermore, it can be seen from Fig. 1 that the magnitude of the inelastic
overlap function grows with energy at the central impact parameter $b=0$. This
indicates that the higher the energy, the more likely it is that inelastic
events will occur at the central impact parameter. Interestingly, as Fig. 1
shows, there is a significant divergence in the magnitude of the inelastic
overlap function obtained from the $U$-matrix and the eikonal schemes. To be
more specific, the former yields a greater magnitude than the latter at $b=0$.
It is clear from this difference that the chosen scheme has a profound impact
on the inelastic processes, particularly on central collisions which will have
distinct characteristics or probabilities.

In the same vein, Fig. 2 depicts the energy evolution of the ratio elastic-to-
total cross-section using both schemes. As this Figure demonstrates, in both
cases there is an overall increase of this ratio as energy rises from $10$ GeV
to $10$ TeV, but does so in a non-linear manner, which is indicative of
geometrical scaling violation. Interestingly, comparing the behaviour of this
ratio in both cases, it is clear that it increases more rapidly in the
$U$-matrix case than in the eikonal one, as energy increases, demonstrating a
stronger violation of the geometrical scaling and a more intricate behaviour
in the former case.

It is possible to explain the divergence observed in the geometrical scaling
violation between the two distinct schemes in terms of the behaviour of the
inelastic overlap function in the impact parameter space with respect to
energy. To be more precise, this difference lies in the behaviour of the
inelastic overlap function at $b=0$, where we can see from Fig. 1 that it has
a magnitude that grows with energy in both cases. However, the $U$-matrix
scheme yields a larger magnitude than the eikonal one. This implies that, in
the former case, the inelastic interactions are more prominent, indicating
that they have a remarkably and rapidly increasing strength at the central
impact parameter with increasing energy, compared to the eikonal case.

It can be argued that the discrepancy found in the behavior of the inelastic
overlap function as well as in the observed violation of the geometrical
scaling between the two schemes can be ascribed to their differing approaches
to the unitarization of the scattering amplitude. Indeed, based upon its
thorough treatment of processes and its ability to take into account a wider
range of interactions and collision dynamics compared to the Eikonal scheme,
the U-matrix scheme is likely to be a more advanced mechanism for dealing with
scattering processes and capturing the intricate dynamics of hadronic
interactions, which is also supported by other results Oueslati:2023tjt ;
Vanthieghem:2021akb ; Bhattacharya:2020lac .

On the whole, this result validates the remarkable difference in the
geometrical scaling violation between the $U$-matrix and eikonal schemes.
Specifically, it supports the claim that this violation is more noticeable
with the former scheme when energy levels rise. Notably, this outcome is in
agreement with the theoretical prediction provided in Bhattacharya:2020lac
and further highlights that this discrepancy becomes apparent even before
reaching the extremely high-energy region. Furthermore, it reinforces our
motivation, as stated in the introductory section, for selecting the
$U$-matrix scheme. In fact, the choice of this scheme, in particular, may
contribute to answering the question posed in the introduction regarding its
potential implications for KNO scaling violation and unravelling the
underlying physics behind the multi-particle production mechanism, as will be
elaborated in the forthcoming sections.

###  4.2 Hadronic Multiplicity Distributions

Using the master formula (11), the outcomes of the fitting procedure of the
multiplicity distributions data across a wide range of energies are provided
in Fig. 3, Fig. 4 and, Table 2, respectively, where the values of the
parameters $\lambda$ and $\langle n(s)\rangle$ obtained in each fitting
procedure, along with the corresponding $\beta(s)$, are furnished, as well as
the different $\chi^{2}/dof$ values.

$\sqrt{s}$ [GeV] | $\lambda$ | $\beta(s)$ | $\langle n(s)\rangle$ | $\chi^{2}/DOF$
---|---|---|---|---
30.4 | 0.2837 $\pm$ 0.0133 | $1.6014$ | 9.0583 $\pm$ 0.1452 | $1.2714$
44.5 | 0.2720 $\pm$ 0.0119 | $1.6308$ | 10.5900 $\pm$ 0.1339 | $0.6170$
52.6 | 0.2775 $\pm$ 0.0104 | $1.6430$ | 11.2885 $\pm$ 0.1299 | $0.6443$
62.2 | 0.2709 $\pm$ 0.0107 | $1.6547$ | 12.0090 $\pm$ 0.1551 | $1.3303$
300 | 0.3695 $\pm$ 0.0113 | $1.7327$ | 23.4646 $\pm$ 0.2915 | $0.6046$
546 | 0.4618 $\pm$ 0.0148 | $1.7420$ | 27.7185 $\pm$ 0.4145 | $0.3406$
1000 | 0.4230 $\pm$ 0.0110 | $1.7362$ | 36.6360 $\pm$ 0.4042 | $1.5301$
1800 | 0.4836 $\pm$ 0.0070 | $1.7132$ | 42.7217 $\pm$ 0.3184 | $1.2004$
7000 | 0.5936 | $1.5833$ | $73.7117$ | $-$
14000 | 0.6597 | $1.4716$ | $96.0705$ | $-$

Table 2: Values of the $\lambda$ parameter and $\langle n(s)\rangle$ resulting
from fits to the $P_{n}$ data. The values of $\beta(s)$ were obtained from eq.
(20).

According to Fig. 3, Fig. 4, and the different $\chi^{2}/dof$ values, our
model gives a reasonable description of the different multiplicity
distributions at each energy.

(a)  (b)

(c)  (d)

Figure 3:  Multiplicity distributions for inelastic $pp$ data at
$\sqrt{s}=30.4,44.5,52.6$ and $62.2$ GeV compared with theoretical
expectations.

Moreover, it allows to predict the multiplicity distribution $P_{n}$ at LHC
energies by evaluating the energy dependence of the parameter $\lambda$ using
an appropriate function $\lambda(s)$.

(a)  (b)

(c)  (d)

Figure 4:  Multiplicity distributions for inelastic $\bar{p}p$ data at
$\sqrt{s}=300,546,1000$ and $1800$ GeV compared with theoretical expectations.

As can be seen from Fig.5 (left panel), the behaviour of this parameter is
energy-dependent and rapidly increases with increasing energy. This energy
dependence can be aptly described by the following function:

$\lambda(s)=a_{0}\;s^{a_{1}}$ (21)

where the values $a_{0}=0.154$, $a_{1}=0.0762$ were determined by a careful
$\chi^{2}$ analysis. Thus, based on this function, the $\lambda$ values at $7$
and $14$ TeV are retrieved and then displayed in Table 2.

(a)  (b)

Figure 5:  The energy dependence of the parameter $\lambda$ (left panel) and
the mean multiplicity (right panel)

Similarly, the estimates for the average multiplicity of hadrons at LHC
energies can be derived. As depicted in Figure 5 (right panel), this parameter
exhibits a rapid increase with the energy $s$, and its energy dependence can
be consistently described using this function:

$\langle n(s)\rangle=b_{0}\;s^{b_{1}},$ (22)

where the values of $b_{0}$ and $b_{1}$ are $2.5$ and $0.1911$, respectively.
These coefficients were determined through a rigorous $\chi^{2}$ analysis.
Hence, by using this dependence, one can determine the values of $\langle
n(s)\rangle$ at $7$ and $14$ TeV, as illustrated in Table 2.

An intriguing aspect of our findings lies in the remarkable accord between our
result concerning the energy dependence of the hadron mean multiplicity and
that obtained by Troshin and Tyurin with their model for multi-particle
production with antishadowing Troshin_2003 , adding to the growing body of
evidence that supports the fundamental principles underlying the $U$-Matrix
approach. This alignment is highlighted by the following equation Troshin_2003
:

$\langle n(s)\rangle=2.328\;s^{0.201},$ (23)

This also implies that this approach is highly predictive and can accurately
describe and interpret multi-particle production in different energy regimes.
Besides, we should emphasize that the power-law energy dependence of the
hadron mean multiplicity is often regarded as a prominent feature observed in
different models and consistent with experimental data from heavy ion
collisions Barshay:2001wp ; Singh:2013fha  and this alignment in results
further reinforces this assumption.

Having tuned our model with all parameters obtained from the best fits, we can
now rely on its potential extrapolations to novel collision energy regimes and
investigate various phenomena, such as the KNO scaling violation and the
correlation of final state particles, as will be presented in the subsequent
sections.

###  4.3 KNO scaling violation

Using our model, the KNO scaling violation was also examined. The predictions
for the full-phase space multiplicity distribution in $p+p(\bar{p})$
collision, in KNO form at various energies, spanning from ISR to LHC ones, are
displayed in Fig. 6. Aside from the logarithmic view in the left panel, a
linear scale along with a zoom into the low-multiplicity region are provided
in the right panel.

(a)  (b)

Figure 6:  The multiplicity distributions from ISR to LHC energies in full-
phase space (left panel). In addition to the logarithmic view, the right panel
shows a linear scale and a zoom into the low-multiplicity region.

The left panel of Fig. 6 shows that the high-multiplicity tail rises with
increasing energy. At the same time, by looking at the right panel, we can see
that the maximum of the distribution shifts towards smaller values of $z$.

This interpretation demonstrates the dynamic behaviour of the system,
especially as the energy level rises, and further validates the violation of
the KNO scaling. Interestingly, we can also see that beyond the ISR energy
range, the width of the distribution gets larger with increasing energy, which
underscores the strong violation of the KNO scaling. It is worth noting that
this finding resonates with experimental observations Grosse-Oetringhaus_2010
.

Most importantly, based on the picture that KNO scaling violation is an
extension of geometrical scaling violation, we can also claim that the strong
violation of the former stems from the strong violation of the latter,
emphasizing the interconnected nature of these phenomena within the $U$-matrix
representation and stressing the latter’s pivotal role in describing collision
geometry and the processes of multi-particle production in hadron collisions.

To further illustrate the role of the $U$-matrix scheme, we examined the
average number of particles $\langle n(b,s)\rangle$ as a function of impact
parameter $b$ for various collision energies, as its pattern offers insights
into the collision geometry and the distribution of particles in the
transverse plane. The result is illustrated in Fig. 7 (left panel). Based on
this figure, it is clear that, at central collision $(b=0)$, the magnitude of
the average number of particles increases with increasing energy. This is
quite anticipated since central collisions yield more produced particles than
peripheral collisions Yudong_1989 . This trend causes the tail of the
multiplicity distribution to extend to higher values and eventually to a
broader distribution as energy increases, indicating the possibility of rare
high-multiplicity events.

(a)  (b)

Figure 7:  The energy dependence of the average multiplicity in the impact
parameter space.

A possible explanation for the broadening of the multiplicity distribution
might be related to the effect of using the $U$-matrix scheme. Indeed, as
previously illustrated in Fig. 1, the disparity found between the eikonal and
$U$-matrix schemes is noticeable at the central impact parameter, where the
overlap of hadronic matter distributions is greater with the latter scheme.
This discrepancy not only influences the inelastic overlap function, and hence
the overall magnitude of multiplicity, but it also has a significant impact on
the tail of the multiplicity distribution.

In addition, irrespective of the energy level, the average number of particles
generally decreases as the impact parameter $b$ increases, which eventually
leads to a decrease in the magnitude of the multiplicity distribution. This
phenomenon aligns with the notion that larger impact parameters lead to less
violent collisions, resulting in events with smaller multiplicities
Yudong_1989 .

Furthermore, as Fig. 7 manifests, the average number of particles is energy-
dependent, increasing with higher collision energies at each impact parameter.
This behaviour is also expected in high-energy physics experiments, where
higher energies often result in increased particle production Grosse-
Oetringhaus_2010 .

Fig. 7 (left panel) shows that the curvature of the average number of
particles changes, at a specific impact parameter, regardless of the energy
level. More quantitatively, we show the second derivative of the average
number of particles as a function of the impact parameter in Fig. 7 (right
panel), illustrating a change of sign of this function which demonstrates the
existence of an inflection point at exactly $0.7$ fm. This change in the
curvature could signify a shift in the particle production behaviour. For
instance, at this specific impact parameter value, there might be a change in
the interaction dynamics or the nature of the collision process. It should be
noted that this result is in line with what has been reported in Beggio:2008zz
, where the inflection point was roughly estimated to be at around $1$fm. The
fact that the inflection point is specifically at around $1$ fm suggests that
there is something unique or significant about collisions occurring at this
distance. This could be related to the characteristics of the colliding
particles or the structure of the hadrons involved. This common behaviour
underscores the notion of the critical phenomenon in hadronic interactions
Iwao:1974fd .

On the whole, the shape of the multiplicity distribution is influenced not
only by the impact parameter and collision energy but also by the
unitarisation scheme, particularly the $U$-matrix, which reinforces our claim
that this distribution is scheme-dependent, as outlined in the introductory
section. Besides, the tail of the multiplicity distribution gives insights
into the rare but significant events that constitute the overall dynamics of
high-energy collisions and hence highlights the importance of this scheme in
multi-particle production processes.

In light of this result, the $U$-matrix scheme may prove to be a significant
alternative for addressing multi-particle production challenges at ultra-high
energies, including the muon puzzle in cosmic ray interactions at this energy
level. In situations such as these, where extreme conditions and rare events
are likely to play a significant role, scheme-dependent effects on
multiplicity distribution become relevant.

Having said that, it is vital to consider this scheme in enhancing the
existing hadronic interaction models to tackle several lingering issues in
high and ultra-high energy physics. However, this is beyond the remit of this
study.

###  4.4 Hadronic multi-particle correlations

Now that we have estimated the multiplicity distribution $P_{n}$ across
various energies, it seems appropriate to characterize it in an attempt to
obtain a better understanding of the dynamics of the particle production
process in hadron collisions. In order to fulfil this purpose, the normalized
ordinary higher-order moments $(q>2)$ of this distribution are analyzed.
Technically speaking, $P_{n}$’s moments of order $q$ are defined as follows:

$C_{q}=M_{q}/M_{1}^{q},$ (24)

and

$M_{q}=\sum_{n=0}^{\infty}n^{q}P_{n},$ (25)

Our results, along with their comparison with the experimental data 111see
compilation in Beggio:2017ran , are given in Table 3 and illustrated in Fig. 8
(left and middle panels).

$\sqrt{s}$ $[GeV]$ | $C_{2}$ | $C_{3}$ | $C_{4}$ | $C_{5}$
---|---|---|---|---
30.4 | 1.29 $\pm$ 0.05 |  $1.97\pm$ 0.09 | $3.45\pm 0.21$ | $6.68\pm 0.52$
| 1.28 | $1.94$ | $3.36$ | $6.47$
44.5 | 1.28 $\pm$ 0.04 |  $1.95\pm$ 0.07 | $3.40\pm 0.17$ | $6.58\pm 0.47$
| 1.3 | $2.01$ | $3.55$ | $6.99$
52.6 | 1.29 $\pm$ 0.03 |  $1.98\pm$ 0.06 | $3.48\pm 0.15$ | $6.81\pm 0.42$
| 1.3 | $2.04$ | $3.64$ | $7.24$
62.2 | 1.29 $\pm$ 0.03 |  $1.97\pm$ 0.06 | $3.40\pm 0.14$ | $6.43\pm 0.33$
| 1.31 | $2.07$ | $3.73$ | $7.51$
300 | 1.34 $\pm$ 0.02 |  $2.21\pm$ 0.04 | $4.26\pm 0.07$ | $9.23\pm 0.17$
| 1.41 | $2.46$ | $4.95$ | $11.17$
546 | 1.41 $\pm$ 0.03 |  $2.52\pm$ 0.05 | $5.31\pm 0.10$ | $12.72\pm 0.24$
| 1.46 | $2.67$ | $5.63$ | $13.38$
1000 | 1.41 $\pm$ 0.02 |  $2.47\pm$ 0.05 | $5.11\pm 0.13$ | $11.87\pm 0.36$
| 1.52 | $2.91$ | $6.5$ | $16.33$
1800 | 1.47 $\pm$ 0.02 |  $2.78\pm$ 0.03 | $6.23\pm 0.07$ | $15.91\pm 0.21$
| 1.59 | $3.21$ | $7.57$ | $20.18$
7000 | 1.78 | $4.14$ | $11.36$ | $35.3$
8000 | 1.81 | $4.26$ | $11.87$ | $37.51$
13000 | 1.89 | $4.73$ | $14.03$ | $47.19$
14000 | 1.91 | $4.81$ | $14.41$ | $48.93$

Table 3:  $C_{q}$ Moments: experimental data with error bar and theoretical
predictions. Data points are from Beggio:2017ran

(a)  (b)  (c)

Figure 8:  Experimental and Theoretical $C_{q}$ moments, q = 2, 3 (left
panel), q = 4,5 (middle panel). $f_{2}$ moment versus the average number of
produced particles (right panel).

Based on Fig. 8, we can see that, as energy levels rise, our proposed model
predicts a gradual increase in the ordinary higher-order moments, represented
by $C_{2}$, $C_{3}$, $C_{4}$, and $C_{5}$. Surprisingly, while our predictions
match with the data points within the ISR energy range, it is clear that the
model overestimates the fluctuations and correlations in the multiplicity
distribution with rising energy, notably above LHC energy. In order to further
illustrate this overestimation, we computed the $f_{2}$ moment (or the two-
particle correlation parameter), as a means of examining the correlation
between pairs of particles during a collision event, which is defined by the
following formula :

$f_{2}=<n(n-1)>-<n>^{2}$ (26)

Interestingly, as illustrated in Fig. 8 (right panel), there is a noteworthy
and sudden increase in the two-particle correlation parameter versus the
average number of produced particles, indicating the existence of strong
correlations among the charged particles. Consequently, we can infer that the
model incorporates correlations in the final state, despite being constructed
on the basis of independent particle production. So the question that arises
is where this correlation emerges from. As the overall hadronic multiplicity
distribution is constructed by summing contributions from parton–parton
collisions occurring at each impact parameter weighted by the inelastic
overlap function, this overestimation of correlation is linked to the weight
in this superposition model, and hence to the unitarisation scheme, in
comparison with the predictions provided by an eikonal geometrical independent
string model Beggio:2017ran .

Our model’s outcomes of a pronounced KNO scaling violation, together with the
unexpected overestimation of the fluctuations and correlations with increasing
energy, can potentially be attributed to statistical fluctuations. Hence, we
may claim that in this $U$-matrix representation, pomeron exchange may involve
more intricate dynamics, such as collective effects, non-perturbative QCD
dynamics, or other interactions, leading to different statistical fluctuations
beyond a simple Poissonian eikonal summation Boreskov:2005ee .

##  5 Conclusions

The first part of the results’ section was concerned with the description of
the geometrical scaling violation. In fact, the energy evolution of the
elastic-to-total cross-section ratio was investigated using both the eikonal
and $U$-matrix schemes. The results have revealed that when energy rises, this
ratio increases non-linearly and more rapidly with the $U$-matrix scheme than
with the eikonal, implying a stronger violation of the geometrical scaling.
This pronounced violation was understood in terms of the divergence in the
behaviour of the inelastic overlap function, particularly at the central
impact parameter, where the magnitude of this function is greater with the
$U$-matrix scheme than the eikonal, regardless of the energy level.

The second part of the results’ section was devoted to the hadronic
multiplicity distributions. Our model was tuned and all parameters were
obtained from the best fits to various hadronic multiplicity distributions
data over a wide range of energies. It has been found that the present model
provides a reasonable description of the different multiplicity distributions
at each energy. Interestingly, our findings about the energy dependence of the
hadron mean multiplicity agree well with Troshin and Tyurin’s analyses of the
multiparticle production in the antishadowing model Troshin_2003 . This
agreement adds to the growing body of evidence supporting the fundamental
principles underlying the $U$-Matrix approach, which ensures reliable
extrapolations to novel collision energy regimes.

Based on our model, the KNO scaling violation was investigated as well. Our
results, related to the behaviour of the tail, as well as the maximum of the
multiplicity distribution, and especially to the strong KNO scaling violation,
are in line with the experimental findings Grosse-Oetringhaus_2010 . The
broadening of the multiplicity distribution has been also confirmed with the
behaviour of the particles’ distribution in the transverse plane at various
collision energies. Besides, the interesting finding of an inflection point in
the average number of particles’ curve irrespective of the energy level,
corroborates with another finding (ref) and highlights the concept of a
critical phenomenon in hadronic interactions.

Besides, the normalized ordinary higher-order moments $(q>2)$ of the
multiplicity distribution were analyzed. Another surprising result was related
to our model’s overestimation of the fluctuations and correlations in the
multiplicity distribution as energy rises, notably above LHC energy. It is
argued that this overestimation, is linked to the weight in this superposition
model, and hence to the unitarisation scheme. It should be noted that the
shape of the hadron multiplicity distribution is influenced not only by the
impact parameter and collision energy but also by the unitarisation scheme,
particularly the $U$-matrix.

On the whole, the results of this study, such as those related to the strong
geometrical scaling violation and its resultant pronounced KNO scaling
violation, coupled with the overestimation of the charged particles’
correlation, can potentially be attributed to statistical fluctuations
inherent in the $U$-matrix scheme. Hence, we may argue that in this $U$-matrix
representation, pomeron exchange may involve more intricate dynamics, leading
to different statistical fluctuations beyond a simple Poissonian eikonal
summation.

As any research, this study is not without limitations. For instance, the
assumption that each created string has an equal probability of turning into a
pair of charged hadrons is acceptable but only as a first approximation. To
refine the present model for a more reliable description, we can consider the
introduction of charged particle correlations in the final state by
incorporating, for example, string fusion, overlapping processes, or other
interactions leading to correlated string dynamics as evidenced by theoretical
predictions Beggio:2007  and experimental observations CMS:2019fur .
Additionally, it is important to investigate the impact of the implicit
different statistical fluctuations in pomeron exchanges within the $U$-matrix
representation, along with the correlation and/or collective behaviour in the
production of final state particles.

The study concludes by proposing the $U$-matrix scheme as a noteworthy
alternative for tackling challenges related to multi-particle production in
hadron collisions, especially in scenarios where extreme conditions and rare
events play a significant role in high and ultra-high energy physics. It also
prompts an inquiry into the fundamental nature of pomeron exchange within the
$U$-matrix scheme in comparison to the eikonal, despite that both schemes
verify the unitarity constraint principle.

###### Acknowledgements.

RO would like to thank Jean-René Cudell for his invaluable comments. RO also
appreciates the insightful discussions with Paulo Cesar Beggio and Emerson
Luna. Special thanks go to the computational resource provided by Consortium
des Équipements de Calcul Intensif (CÉCI), funded by the Fonds de la Recherche
Scientifique de Belgique (F.R.S.-FNRS) where a part of the computational work
was carried out.

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