# Multiplicity dependence of the $p_{T}$-spectra for identified particles and
its relationship with partonic entropy

L.S. Moriggi1<EMAIL_ADDRESS>1 Universidade Estadual do Centro-
Oeste (UNICENTRO), Campus Cedeteg, Guarapuava 85015-430, Brazil  G.S. Ramos 2
<EMAIL_ADDRESS>M.V.T. Machado2<EMAIL_ADDRESS>2 High Energy
Physics Phenomenology Group, GFPAE. Institute of Physics, Federal University
of Rio Grande do Sul (UFRGS)
Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil

###### Abstract

We investigate the multiplicity dependence of the transverse momentum $p_{T}$
spectra of hadrons produced in high-energy collisions. We propose that the
partonic distribution be parameterized by its non-extensive entropy and the
parton saturation scale $Q_{s}(x)$. These two variables can be identified from
the produced charged hadron distributions and provide important information on
the gluon dynamics at the moment of interaction. From this perspective we
interpret data from different ALICE multiplicity classes at $\sqrt{s}=13$ TeV
and $\sqrt{s}=5.02$ TeV. A multiplicity dependent scaling function is
presented and the dependence of the interaction area on multiplicity is also
investigated.

##  I Introduction

The transverse momentum, $p_{T}$, spectra are traditionally one of the main
ways to obtain information about the dynamics of partons in the initial state
of the interaction in proton-proton ($pp$), proton-nucleus ($pA$) and nucleus-
nucleus ($AA$) collisions at high energies. The momentum distribution of the
particles produced is sensitive to the geometric parameters of the collision,
such as the average area of interaction, $\langle A_{T}\rangle$, the nature of
the projectile, the collision energy, $\sqrt{s}$, the pseudorapidity of the
hadron produced, $\eta$, and the observed multiplicity class. Modifications of
these collision parameters significantly change the shapes of the spectra,
mainly in the region of large $p_{T}$, usually characterized by a power-law
multiplicity $dN/d^{2}p_{T}d\eta\sim{p_{T}^{-m}}$ [1, 2, 3]. Mapping these
observable modifications in terms of QCD degrees of freedom, quarks and
gluons, requires a phenomenological analysis that allows connection between
variables associated with partonic dynamics with the quantities that
characterize the hadronic spectrum.

In [3] we propose a power-law partonic transverse momentum distribution (TMD)
function that essentially depends on two quantities: the power index $\delta
n$ and the saturation scale $Q_{s}(x)$, where $x\sim p_{T}/\sqrt{s}$ is the
fraction of the gluon’s longitudinal momentum at central rapidity. These two
quantities are easily identified in the final spectra of produced hadrons: the
saturation scale is evident from the scaling with respect to the hard scale
proven in the interaction $Q^{2}\sim p_{T}^{2}$ in the scaling variable,
$\tau=Q^{2}/Q_{s}^{2}(x)$, while the power index can be inferred from the
slope of spectra at the region of large $p_{T}$. The data description is
relatively good by using these two quantities. In this work we intend to give
a better justification of why this type of model works and its interpretation.
In the context of non-extensive statistical mechanics, the power parameter is
related to the entropic index $q$ of the Tsallis entropy [4, 5]. While in
approaches based on Hawking-Unruh radiation in QCD applied to thermal
hadronization [6, 7] the saturation scale plays the role of a temperature,
$T=Q_{s}/(2\pi)$, and is sufficient to describe the system, in our model we
also need to specify the power index (entropy). Thus, an analysis of the
hadron spectrum on $p_{T}$ will provide us with information about partonic
dynamics through its entropy $S_{q}$ and saturation scale $Q_{s}(x)$, with
both quantities being necessary to characterize the system.

In [3] we have already made an extensive study of the spectra of identified
particle based on their scaling properties in relation to the collision energy
$\sqrt{s}$ at the energies of RHIC, TEVATRON and LHC up to $\sqrt{s}=13$ TeV.
The underlying QCD dynamics is based on the high energy factorization or
$k_{T}$-factorization [8, 9] where the building blocks are the transverse
momentum distribution of partons (parton TMDs). The corresponding gluon TMD as
function of gluon transverse momentum is denoted by $\phi(x,k_{T})$. In this
article we will investigate the behavior of the spectrum at a fixed energy for
different multiplicity classes as defined by ALICE [10] in proton-proton
collisions at $\sqrt{s}=5.02$ and 13 TeV. The $pp$ collisions at high
multiplicity have shown properties close to those observed in nuclear
collisions, which leads to the question about the relationship of
initial/final state effects in these different collision systems. In our
approach we intend to show that all the characteristics of the spectrum and
its multiplicities can be well explained just by taking into account the
properties of the initial state and the partonic entropy of the gluon system
that take part in the initial interaction. The relationship between average
transverse momentum and multiplicity of charged particles produced in
different collision systems as $pp$, $pA$ and $AA$ [11] reactions presents a
challenge to traditional models of particle production, being a fundamental
indicator in order to distinguish effects of initial state and hydrodynamic
evolution of the final state.

The main ideas underpinning the present study are as follows. The high energy
collision probes a system of gluons with a probability distribution
$P(x,k_{T})$ given by the Fourier transform to the QCD color dipole scattering
amplitude, $\mathcal{N}(x,r)$, with $r$ being the transverse size of a color
dipole. Due to the diffusion of gluons in momentum space given in a time scale
$t\sim 1/x$, the variation of the distribution $P(x,k_{T})$ is related to the
probe of different substructures of the target. This process is described by
an anomalous diffusion. We argue that in case the probability distribution
$P(x,k_{T})$ to be given by the entropy maximization two situations can occur:
i) the Boltzmann-Gibbs entropy which in steady state results in a Gaussian
distribution in transverse momentum; ii) the Tsallis entropy that in steady
state results in our model containing a power (entropic) index. By maximizing
a Tsallis entropy one associates a Lagrange multiplier $\beta$ to the average
value of gluon transverse momentum $k_{T}$. Making use of scaling property
very common to systems presenting anomalous diffusion one obtains $\langle
k_{T}^{2}\rangle\sim\beta^{-1}(x_{s}/x)^{1/3}$. This can be interpreted as a
generalization of the Einstein relation and thus $\beta$ can be understood as
the inverse of temperature. More specifically, we intend to explore the
connection between the multiplicity of produced hadrons,
$dN/d^{2}p_{T_{h}}d\eta$, and partonic dynamics at high energies. In order to
investigate such connection we define an indicator of partonic entropy
associated with the diffusion of gluons in the $k_{T}$ space. Considering that
in the equilibrium situation $\sqrt{s}\rightarrow\infty$ a behavior like
$\propto k_{T}^{-4}$ is expected, in contrast to the Boltzmann-like
exponential form, we consider that the most appropriate indicator is that of
Tsallis [4, 5], where the monotonic shape $\sim k_{T}^{-4}$ corresponds to the
entropic index $q=3/2$. From this indicator, we observed a relationship
between the growth of entropy, the area of interaction and the final
multiplicity of charged hadrons.

This work is organized as follows. In section II.1 the $k_{T}$-factorization
formalism applied to description of transverse momentum spectra of produced
hadrons is presented. In section II.2 we propose a partonic entropy indicator
based on the formalism of non-extensive statistical mechanics and its
implications for the production of hadrons at high energies. Main results are
presented in section III where the model is compared with ALICE data for
different multiplicities. Finally, in Sec. IV, the main conclusions are
summarized.

##  II Theoretical framework and main predictions

###  II.1 The $p_{T}$ spectrum in $k_{T}$-factorization

To carry out our phenomenological investigation we will employ the
$k_{T}-$factorization formalism, where the cross section for particle
production can be expressed in terms of the unintegrated gluon distribution
(UGD), dependent on transverse momentum $k_{T}$. Different hard observables in
$pp$, $pA$ and $AA$ collisions have been well described using this formalism
[12, 13, 14], in addition to diffractive processes [15] as well as structure
functions in interactions electron-proton ($ep$) and electron-nucleus ($eA$)
[16, 17, 18, 19, 20, 21, 22, 23, 24] (see also reviews on Refs. [25, 26]). In
all these approaches the fundamental element is the partonic dynamics
represented by the color dipole scattering amplitude, whose Fourier transform
in momentum space $k_{T}$ is directly connected with the gluon number of the
target, $\phi(x,k_{T})$. Different models have been proposed to this quantity
for both protons and large nuclei [17, 27, 28, 29, 12, 16, 30, 31], which
include different considerations about partonic dynamics in relation to
scaling, impact parameter ($\vec{b}$) and geometry dependence, high- $Q^{2}$
limit and so on.

Partons develop an anomalous diffusion-like dynamic in the two dimensional
transverse momentum space $\vec{k}_{T}$, while its longitudinal dynamics is
trivial. That can be described in the picture of QCD color dipoles [32, 33,
34, 35]. One of the main features of this dynamic is the emergence of a
saturation scale $Q_{s}(x)\sim x^{-1/3}$, which limits the growth of the gluon
distribution at small Bjorken variable $x$. This behavior ends up being
translated into observables, such as the $p_{T}$ spectrum of hadrons, where
the cross section for hadronic production can be described by a universal
function $f$, i.e. $d\sigma(p_{T},\sqrt{s})/d^{2}p_{T}\sim f(\tau)$. Here,
$\tau$ is the scaling variable which can be defined in the context of
geometric scaling property of parton saturation approaches.

In the $k_{T}$ factorization formalism, the cross section for producing a
gluon jet with transverse momentum $p_{T}$ results in the convolution of the
non-integrated gluon distributions of the target and projectile,

$\displaystyle E\frac{d^{3}\sigma}{d\vec{p}^{3}}^{ab\rightarrow g+X}$
$\displaystyle=$ $\displaystyle\frac{\mathcal{A}}{p_{T}^{2}}\int
d^{2}k_{T}\,\phi(x_{a},k_{T}^{2})\phi(x_{b},q_{T}^{2}),$ (1) $\displaystyle=$
$\displaystyle f(\tau),$ (2)

which can be reduced to a universal function $f(\tau)$ due to the scaling in
relation to the saturation scale $Q_{s}(x)$ characteristic of the parton
saturation formalism. Namely, the scaling variable is given by
$\tau=p_{T}^{2}/Q^{2}_{s}(x)$. Here,
${\mathcal{A}}=K\frac{2\alpha_{s}}{C_{F}}$ is the overall normalization and
$\vec{q}_{T}=(\vec{p}_{T}-\vec{k}_{T})$. The $K$ factor is a multiplicative
constant as used in [3]. The Casimir is $C_{F}=(N_{c}^{2}-1)/2N_{c}=4/3$ and
$\alpha_{s}$ is the strong coupling constant. The scaling curve for the case
of gluon production in the range $\tau>1$ has the following form [3],

$\displaystyle
f(\tau)=C\frac{\kappa}{\kappa-1}\left[1-\frac{1+\kappa\tau}{(1+\tau)^{\kappa}}\right]\frac{1}{(1+\tau)^{1+\kappa}},$
(3)

with $C$ being the overall normalization and $\kappa=1+\delta n$.

It is desirable constrict the scaling function in Eq. (1) in terms of
multiplicity degree of freedom, $f(\tau)\rightarrow f(\tau_{i})$. Let us
consider the variation of the saturation scale in each multiplicity class $i$
in relation to its minimum bias value, $X_{i}=Q_{s_{i}}(x)/Q_{s}(x)$, in the
following way,

$\tau_{i}=\frac{Q^{2}}{\left[X_{i}Q_{s}(x)\right]^{2}},$ (4)

where the momentum scale involved in the hard interaction is given by
$Q^{2}=p_{T}^{2}+m_{j}^{2}$, with $m_{j}$ being the mass of the produced gluon
jet. The value for the jet mass used here is the same considered in Ref. [3],
$m_{j}=0.56$ GeV. The parton-hadron transition can be approximated by assuming
that the hadron carries a fraction of momentum $\langle z\rangle$ of the
parton. In the numerical calculations the value $\langle z\rangle\simeq 0.4$
will be used, which is determined from data on $pp$ collisions at the LHC [3].
In addition, we should replace the gluon transverse momentum
$p_{T}\rightarrow\frac{p_{T_{h}}}{\langle z\rangle}$, with $p_{T_{h}}$ being
the hadron transverse momentum. Final state processes could destroy the
spectrum scaling, but this does not happen, indicating that the role of these
effects is secondary at least in $pp$ collisions.

We can assume that the variation in multiplicity is related to the variation
in the collision impact parameter, which leads to an increase in the
transverse area of proton overlap $\langle A_{T}\rangle$, up to a maximum area
value associated with the total overlap $\langle A_{T_{max}}\rangle$. The
cross section can be expressed in a given multiplicity class scaling Eq. (1)
in the form

$E\frac{d^{3}\sigma_{i}}{d\vec{p}^{3}}^{ab\rightarrow g+X}\sim\frac{\langle
A_{T}\rangle}{\langle A_{T_{max}}\rangle}f(\tau_{i}).$ (5)

The multiplicity can be obtained from the cross section at a given
multiplicity, $\sigma_{i}$, by assuming
$\sigma_{i}/\sigma_{inel}=N_{i}/\langle N\rangle$. Here, $\sigma_{inel}$ is
the inelastic cross section, $N_{i}$ and $\langle N\rangle$ are the number of
produced particle in the multiplicity class and its average, respectively. The
spectrum modification factor in relation to its minimum bias value $\langle
dN/d^{2}p_{T_{h}}d\eta\rangle$ is defined as the ratio,

$R_{i}(p_{T_{h}})=\frac{dN_{i}/d^{2}p_{T_{h}}d\eta}{\langle
dN/d^{2}p_{T_{h}}d\eta\rangle},$ (6)

which captures small variations in the slope of large $p_{T_{h}}$ at each
centrality. Given these considerations we have only two parameters to be
fitted in each multiplicity. They are the ratios between areas and saturation
scale in relation to the reference value: i.e., $\frac{\langle
A_{T}\rangle}{\langle A_{T_{max}}\rangle}$ and $X_{i}$.

We will not make an a priori estimate of these quantities, but in section III
we will analyze the resulting behavior in terms of relevant indicators of
multiplicity such as $\langle p_{T}\rangle_{i}$. Bulk properties can be
inferred from integrated spectra:

$\displaystyle\frac{dN_{i}}{d\eta}$ $\displaystyle=$ $\displaystyle\int
d^{2}p_{T_{h}}\frac{dN_{i}}{d^{2}p_{T_{h}}d\eta},$ (7) $\displaystyle\langle
p_{T_{h}}\rangle_{i}$ $\displaystyle=$ $\displaystyle\int
d^{2}p_{T_{h}}p_{T_{h}}\frac{dN_{i}/d^{2}p_{T_{h}}d\eta}{\langle
dN/d\eta\rangle}.$ (8)

The characterization of the gluon distribution, appearing in Eq. (1), in order
to produce the universal scaling function $f(\tau_{i})$ and its
particularities is given in the next section.

###  II.2 Partonic Entropy

In the dipoles picture, the fundamental element used to describe the collision
is the color dipole scattering amplitude $\mathcal{N}(x,\vec{r})$ in the
dipole coordinate space ($\vec{r}$ is the transverse size of the color
dipole). In the transverse momentum space its Fourier transform, $P(x,k_{T})$,
decodes target information by exchanging multiple gluons, making this object
proof of the target’s gluon distribution. Due to these multiple interactions,
the scattering amplitude acquires a statistical character [36, 37] describing
a diffusive process in the transverse momentum space in relation to the
longitudinal momentum fraction $x$. The last quantity gives the inverse of
diffusion time. Different models have been proposed for this object. In
phenomenological terms, it was proposed [3] that a good description of the
$p_{T}$ spectrum of produced hadrons can be made considering the following
distribution:

$P^{MPM}(\delta n,x,k_{T})=\frac{1+\delta n}{\pi
Q^{2}_{s}(x)}\frac{1}{\left(1+k_{T}^{2}/Q_{s}^{2}(x)\right)^{2+\delta n}},$
(9)

where the parameter $\delta n$ takes into account deviation from the expected
amplitude in leading order, $\propto 1/k_{T}^{4}$. This distribution was
initially proposed to describe the slope in the region of large $p_{T}$ in
transverse momentum spectrum in $pp$ collisions. Moreover, it can also provide
a good description of the $p_{T}$ distribution in large systems like $pA$ and
$AA$ interactions [12, 13, 14, 38, 39], as well as diffractive processes [40,
41, 42], which prove a distinct kinematic region. The number of gluons with a
certain transverse momentum $k_{T}$ is given by

$\phi(x,k_{T})=\frac{3}{4\pi^{2}\alpha_{s}}k_{T}^{2}P(x,k_{T})$ (10)

considering a homogeneous target with impact parameter dependence
$\propto\Theta(R_{p}^{2}-b^{2})$. We will investigate the dependence of the
collision geometry of this object by letting the saturation scale depend on
the multiplicity at each centrality. The emergence of scaling in the variable
$Q^{2}_{s}(x)\sim x^{-\lambda}$ is a remarkable property of the QCD in the
high energy regime with ample experimental evidence [3, 43, 44, 45, 46, 47,
48, 49]. The successful phenomenological GBW model [50, 51] describes this
process by an exponential function on the scaling variable
$k_{T}^{2}/Q_{s}^{2}$,

$P^{GBW}(x,k_{T})=\frac{1}{\pi
Q^{2}_{s}(x)}\exp\left[-k_{T}^{2}/Q_{s}^{2}(x)\right].$ (11)

One can define an entropy indicator in the gluon diffusion process based on
the Boltzmann - Gibbs (BG) statistics in the following way,

$S^{BG}=-\int P(x,k_{T})\log\left[P(x,k_{T})\right]d^{2}k_{T},$ (12)

where all we need to describe partonic interactions is the quantity
$Q_{s}(x)$. An immediate consequence of scaling in the variable
$k_{T}^{2}/x^{-\lambda}$ is that the BG entropy is logarithmic in time $1/x$,
and additive with respect to rapidity $Y=\log(1/x)$,

$S^{BG}=c_{1}+c_{2}\lambda\log(1/x),$ (13)

where $c_{1},c_{2}$ are constants and $\lambda$ a function of $x$ and scale
$Q^{2}$. Initially the partons are located in a small region around the
saturation point, as time passes the proton becomes almost homogeneous in
transverse momentum space and the entropy increases fast. It is interesting to
note that if $\lambda=\lambda(x,Q^{2})$ as predicted in other models [52],
then the BG entropy may grow faster/slower and become non-additive. Therefore,
models of this type when analyzed from the perspective of BG entropy, will not
be extensive in relation to rapidity indicating a break in the scaling. One of
the reasons for this break and consequently entropy additivity is that gluons
occupy a region $d$ in the transverse plane of the inverse order to their
average transverse momentum $d\sim k_{T}^{-1}\sim Q_{s}(x)^{-1}$. When a
dependence is introduced on the scale $Q\sim D^{-1}$ at which the proton
structure is proved, one can suggest that the entropy becomes dependent on the
resolution $Q^{2}/Q_{s}^{2}\sim d^{2}/D^{2}$ in which the system is seen. If
$Q^{2}/Q_{s}^{2}\gg 1$ then we can resolve details of the partonic
substructure and one should expect a decrease of entropy and we will have loss
of information between different rapidity layers.

In the distribution of Eq. (9), information about non-additivity is given by
the parameter $\delta n$, given by

$\displaystyle\delta n(\tau_{i})$ $\displaystyle=$ $\displaystyle
0.075\,\tau_{i}^{0.188},$ (14) $\displaystyle Q_{s}^{2}(x)$ $\displaystyle=$
$\displaystyle\left(\frac{x_{s}}{x}\right)^{1/3}.$ (15)

Now it should be noted that the distribution in Eq. (9) can be obtained by
maximizing the Tsallis entropy $S_{q}$,

$S_{q}=\int d^{2}k_{T}\frac{1-\left[P(x,k_{T})\right]^{q}}{q-1}.$ (16)

if we identify,

$\displaystyle q$ $\displaystyle=$ $\displaystyle\frac{3+\delta n}{2+\delta
n},$ (17) $\displaystyle Q_{s}^{2\prime}$ $\displaystyle=$ $\displaystyle
Q_{s}^{2}(q-1),$ (18)

and imposing the constrains [5]

$\displaystyle\langle k_{T}^{2}\rangle_{q}=\frac{\int
d^{2}k_{T}k_{T}^{2}\left[P(k_{T})\right]^{q}}{\int
d^{2}k_{T}\left[P(k_{T})\right]^{q}}$ $\displaystyle=$ $\displaystyle D_{q},$
(19) $\displaystyle\int d^{2}k_{T}P(k_{T})$ $\displaystyle=$ $\displaystyle
1.$ (20)

By using the Lagrange method to find the optimizing distribution $P_{opt}\sim
e_{q}^{-\beta k_{T}^{2}}$, where $\beta$ is the Lagrange parameter, we can do
the following identification,

$\langle k_{T}^{2}\rangle_{q}=D_{q}=\beta^{-1}.$ (21)

Now, if we interpret the Lagrange parameter as the inverse of the temperature
$\beta^{-1}=T$, using the scaling hypothesis:

$\langle k_{T}^{2}(x)\rangle_{q}\sim\beta^{-1}(x_{s}/x)^{1/3},$ (22)

we have a generalization of Einstein relation for anomalous diffusion [53,
54].

These two entropies $S^{BG}$ and $S_{q}$, Eqs. (12) and (16), differ in non-
additive character, except in the case $q=1$ where they coincide. Parton
dynamics can be described in terms of the difference in rapidity between two
layers in the parton cascade $\Delta Y=Y_{a}-Y_{b}=\log(x_{a}/x_{b})$,

$S_{q}(Y_{a}+\Delta Y)\leq S_{q}(Y_{a})+S_{q}(\Delta Y),$ (23)

which implies that there is a loss of information when comparing the two
systems in relation to the BG case. Entropy in Eq. (16) is non-additive, and
non-additivity depends on the parameter $\delta n$, which will measure the
loss of information when we compare the same situation described above to
different values of this parameter.

In general terms, we can argue that if the steady state of the distribution
has the form (9), a natural choice for entropy would be (16). Considering that
in the high energy regime $Q_{s}(x)^{2}/Q^{2}\rightarrow\infty$ we expect a
point particle scattering $\phi(k_{T})_{LO}\sim k_{T}^{-2}$ characterized by
$q_{\infty}=3/2$. In practice, $q$ is always close to 3/2, which justifies the
approximation $q\simeq 3/2+\delta n(\tau_{i})/4$. The variation of $q$ as a
function of the scale tested in the system was modeled as a power in the form
(14). Such behavior of the entropic index is justified in analogy to other
similar physical systems that exhibit this behavior [55],

$q_{\infty}-q\simeq\tau_{i}^{0.188}.$ (24)

and the value of the entropic index must increase with $Q^{2}$. The resulting
entropy is given by,

$S_{q}(Q^{2},Q_{s}^{2})=\frac{1}{q-1}-\left(\frac{2-q}{q-1}\right)^{q}(\pi
Q_{s_{i}}^{2})^{1-q}$ (25)

This entropy can be expressed in terms of the partonic kinematic variables
$x,Q^{2}$ or expressed in terms of the spectrum variables,
$p_{T_{h}},\sqrt{s}$ in each multiplicity. As $q>1$ the entropy grows more
slowly than $\log(1/x)$ eventually saturating.

| Multiplicity Class | $X_{i}$ | $A_{T}/A_{T_{max}}$ | $\chi^{2}/dof$ |  | Multiplicity Class | $X_{i}$ | $A_{T}/A_{T_{max}}$ | $\chi^{2}/dof$
---|---|---|---|---|---|---|---|---|---
$\sqrt{s}=5.02$ TeV | I | - | - | - |  $\sqrt{s}=13$ TeV | I | 1.765 | 0.777 | 0.86
| II | 1.588 | 0.775 | 0.25 |  | II | 1.567 | 0.849 | 0.68
| III | 1.489 | 0.782 | 0.17 |  | III | 1.486 | 0.837 | 0.55
| IV | 1.407 | 0.782 | 0.14 |  | IV | 1.402 | 0.843 | 0.53
| V | 1.326 | 0.772 | 0.14 |  | V | 1.323 | 0.831 | 0.48
| VI | 1.235 | 0.758 | 0.16 |  | VI | 1.240 | 0.809 | 0.42
| VII | 1.133 | 0.741 | 0.16 |  | VII | 1.146 | 0.778 | 0.28
| VIII | 1.012 | 0.714 | 0.20 |  | VIII | 1.036 | 0.731 | 0.20
| IX | 0.845 | 0.698 | 0.27 |  | IX | 0.882 | 0.680 | 0.14
| X | 0.603 | 0.582 | 0.42 |  | X | 0.637 | 0.498 | 0.06
Table 1: Fitted parameters $X_{i}$ and $A_{T}/A_{T_{max}}$ from experimental
data [10] at the energies 5.02 TeV and 13 TeV in each multiplicity class.

Another way of looking at the difference between exponential distribution and
power law is through the superstatistics framework of Beck and Cohen [56]
where the Tsallis distribution is obtained from the gamma fluctuations of the
inverse of the saturation scale $\beta=1/Q^{2}_{s}(x)$. In our case one has,

$\int d\beta
P^{GBW}(k_{T},\beta)g(n,\beta,\beta_{0})=P^{MPM}(k_{T},\beta_{0}),$ (26)

where $g(n,\beta,\beta_{0})$ is the gamma distribution. It is interesting to
note that gamma distributions are needed in models like those shown in Refs.
[57, 58] to generate the necessary multiplicity fluctuation in $pp$ and $AA$
collisions.

The growth of $q$ with $Q^{2}$ is consistent with the statistical argument of
Ref. [56] in the analysis of turbulent flow, where the authors argue that the
variance in the fluctuations of $\beta$ are smaller if taken at a larger
distance scale, so $q$ must grow with $Q^{2}$. It is noteworthy that the
parameter $q=3/2$ expected in the QCD parton picture is the same as that
described by those authors in the context of turbulent flow on small scales.

Finally, the use of the proposed entropy offers a simple and economical way in
terms of number of parameters to describe experimental data of the $p_{T}$
spectra and makes clear the partonic dynamics in a given collision process. In
next section we explore the connection between the multiplicity of produced
hadrons and partonic dynamics at high energies by using the indicator of
partonic entropy associated and investigate the relationship between the
growth of entropy, the area of interaction and the final multiplicity of
charged hadrons.

Figure 1: Left plot: evolution of the entropic index with the hadron
transverse momentum, given by Eq. 24. Right plot: partonic entropy $S_{q}$
given by Eq. 16 for different multiplicities. In both plots representative
multiplicity classes are considered (I,V,VII,IX, X)

##  III Results and discussions

Figure 2: The $p_{T_{h}}$ spectrum for different multiplicities, comparison of
ALICE data[10] with the result of adjusting the saturation scale given by Eq.
(5). The data is multiplied by a factor $2^{i}$ for better visualization.

First, we have addressed the issue of the centrality classes. For each ALICE
multiplicity class [10] (energies 5.02 and 13 TeV ), a value was adjusted for
the parameter $X_{i}$, which measures the deviation in relation to the average
saturation scale $X_{i}=Q_{s_{i}}(x)/Q_{s}(x)$, in addition to the transverse
area ratio $\langle A_{T}\rangle/\langle A_{T_{max}}\rangle$. We consider the
range $p_{T_{h}}<10$ GeV, where scaling is observed with good precision. The
resulting parameters are presented in Table 1 for the two collision energies.
The multiplicity classes are labeled by ten identifiers (index I to X).
Although the saturation scale grows with each multiplicity, the interaction
area saturates at a limit close to $\langle A_{T_{max}}\rangle$ characterizing
the total overlap of the protons. This trend is expected, as it has been
observed in [59, 43].

The resulting $p_{T}$ spectra given by Eq. (5) are presented in Fig. 2 for
each multiplicity class at energies of $\sqrt{s}=5.02$ TeV and $\sqrt{s}=13$
TeV and compared to data from ALICE [10]. In order to convert multiplicity to
cross section the inelastic cross section used was
$\sigma_{inel}(\sqrt{s}=5.02)/\sigma_{inel}(\sqrt{s}=13)=0.87$ [60].
Considering the large momentum, $p_{T_{h}}\approx 10$ GeV, we observe that the
slope is smaller for events of high multiplicity (like class I) than for low
multiplicity (for instance in class X) as a consequence of the increase in
partonic entropy (16) in events of high multiplicity. The entropy for these
situations is presented in Fig. 1, along with the entropic index $q$
associated with the same $p_{T_{h}}$ region. As entropy increases, so does
multiplicity, but this growth is faster for interactions with large
transferred momentum $Q^{2}\sim p^{2}_{T_{h}}$. In the limit where
$p_{T_{h}}/Q_{s}(x)\rightarrow 0$ the entropy would be flat, establishing a
limit for particle production in this kinematic region.

Figure 3: Scaling function obtained by the relation (5). Comparison is done
between scaling behavior obtained from data at different multiplicities and
the scaling curve calculated in the $k_{T}$ factorization formalism.

The scaling in the universal function $f(\tau_{i})$ is evident in the spectrum
shown Fig. 3 at different energies as the data/theory error is very close to
1. A relevant deviation is only seen for the region of $\tau_{i}\sim 10^{3}$
where the validity of the model is in its limit of application. This sort of
scaling in $pp$ collisions has been verified also in Ref. [47], where the
multiplicity dependence has been embedded on the saturation momentum within
the geometrical scaling approach. The corresponding scaling is assumed in both
semi-inclusive and inclusive distributions.

Figure 4: Ratio between different multiplicity classes and
$\langle\frac{dN}{d^{2}p_{T_{h}}d\eta}\rangle$, calculated by Eq.(6), as a
function of hadron transverse momentum $p_{T_{h}}$. Results are shown for the
energies of 5.02 TeV (left plot) and 13 TeV (right plot) and the theoretical
prediction represented by solid lines.

The ratio between the multiplicities classes and $\langle
dN/d^{2}p_{T_{h}}d\eta\rangle$, defined in Eq. (6), is presented in Fig. 4 for
the energies of 5.02 TeV (left plot) and 13 TeV (right plot) as a function of
hadron transverse momentum, $p_{T_{h}}$. This observable is interesting
because the ratio is sensitive to small variations in the slope on
$p_{T_{h}}$. We can see that our model provides a good description of the data
for all multiplicities. It is interesting to note that the slope of the
spectrum given by $\delta n$ in the distribution of Eq. (9) determines the
growth of the ratio in the region of large $p_{T_{h}}$. It is not a priori
expected that parameter $\delta n$ of the Eq. (14) could provide the
appropriate slope for each multiplicity just by rescaling $Q_{s}\rightarrow
X_{i}Q_{s}$ without any extra parameters. As shown in Fig. 6 (left plot),
where $X_{i}=(\langle p_{T_{h}}\rangle_{i}/\langle p_{T_{h}}\rangle)^{2}$, the
relationship established is that the spectrum slope in each multiplicity class
can be derived from its minimum bias multiplicity by just rescaling
$Q_{s}\rightarrow\langle p_{T_{h}}\rangle^{2}$ in the UGD power index, $\delta
n$.

Concerning the scaling of $Q_{s}$ on multiplicity in the context of parton
saturation approaches it is expected that for high multiplicity events, based
on the Local Parton Hadron Duality (LPHD), the density of gluon grows as a
function of multiplicity [61, 62, 63, 64, 65]. This leads by consequence to
multiplicity dependence of the saturation scale. The integrated spectra at
given energy (7) under the scaling can be expressed as

$\frac{dN_{i}}{d\eta}\sim\frac{\langle A_{T}\rangle}{\langle
A_{T_{max}}\rangle}X_{i}^{2}.$ (27)

The specific shape of the overlap area dependence and the saturation scale
with multiplicity can give us important information about partonic dynamics.
In our model, the saturation scale growth due the multiplicity $dN_{i}/d\eta$
can be approximated by

$X_{i}\sim\left(\frac{dN_{i}/d\eta}{\langle dN/d\eta\rangle}\right)^{1/3},$
(28)

which is shown in Fig. 5. This behavior is different from the one obtained in
Ref. [47] where $X_{i}\sim c_{1}+c_{2}(dN_{i}/d\eta)^{1/6}$ or the linear
behavior used in [66, 67].

Figure 5: Saturation scale as a function of multiplicity expressed in terms of
the relation between the quantity $X_{i}$ and $dN_{i}/d\eta$. Solid line
corresponds to a power-like fit to the relation.

.

Now, in order to understand the relation between area of interaction,
multiplicity and entropy, let us calculate the entropy relative to the
distribution on $1/Q_{s}^{2}(x)$ approximated by $q=3/2$. This provides the
following result:

$S_{3/2}(X_{i})=2\left(1-\frac{1}{\left(\pi X_{i}^{2}\right)^{1/2}}\right).$
(29)

If we assume that entropy is extensive with respect to the area of
interaction, then we come to the conclusion that

$S_{3/2}(X_{i})\sim\langle A_{T}\rangle\sim(dN_{i}/d\eta)^{1/3},$ (30)

with $\langle A_{T}\rangle$ (and $S_{3/2}(X_{i})$) reaching a saturation in a
maximum value at large multiplicities.

In Figure 6 (right plot) the average interaction area is presented as a
function of the saturation scale ratio, $X_{i}$. By using the proportionality
between the average interaction area and the entropy, Eq. 30, in Fig. 6 the
results are shown for a fit in the form of Eq. (29). Namely, the relation
$\frac{\langle A_{T}\rangle}{\langle A_{T_{max}}\rangle}=\xi(1-a/X_{i})$ has
been used, where $\xi$ is a proportionality constant. The value found for the
parameter $a=0.67\pm 0.06$ is close to $1/\sqrt{(}\pi)=0.56$ appearing in Eq.
(29). The dependence of the interaction area on multiplicity is investigated
in Refs. [59, 43, 47], where it is argued that the interaction area has a
natural dependence on multiplicity in the form $\langle
A_{T}\rangle\sim(dN_{i}/d\eta)^{2/3}$ (scales with the volume $R^{3}$), until
its saturation at a certain limit for high multiplicities. If we assume the
scaling of the transverse area with partonic entropy we have a different
result, that is $\langle A_{T}\rangle\sim(dN_{i}/d\eta)^{1/3}$.

Figure 6: Left plot: The values $X_{i}=Q_{s_{i}}/Q_{s}$ for saturation scale
of each centrality class (see Table 1) as a function of the average transverse
momentum ratio, $u=\langle p_{T_{h}}\rangle_{i}/\langle p_{T_{h}}\rangle$. The
solid line corresponds to a quadratic behavior, $X_{i}=u^{2}$. Right plot:
Average interaction area normalized by its maximum value as a function of
$X_{i}=Q_{s_{i}}/Q_{s}$ or $u^{2}$. The band corresponds to a non-linear fit
based in the form shown in Eq. (29) by using the proportionality proposed in
Eq. (30).

Finally, putting the present work in context in Ref. [68] the normalized
transverse momentum distributions of produced hadrons, have been used to
compute the Boltzmann-Gibbs entropy. The heat capacity is also determined from
the entropy. It has been considered statistics of three different fitting
functions: thermal, confluent hypergeometric and the Hagedorn distribution.
Minimum bias data in $pp$ collisions at RHIC and LHC have been considered and
it was shown that the BG entropy of the final state increases with the
collision energy. In Ref. [69], it was proposed a connection between the
framework of the evolution of states with dynamical $SL(2,R)$ symmetry on the
context of the Krylov basis and the evolution of QCD color dipoles in Mueller
dipole cascade framework. The latter has been used to define a parton (mostly
gluons at large rapidities) entanglement entropy, $S_{E}$ [37]. In the
simplest case of (1+1) dimension, one obtains $S_{E}=\ln[xg(x)]$, where
$xg(x)$ is the integrated gluon distribution. At high energies the hadron
becomes a maximally entangled state and the multiplicity distributions in deep
inelastic scattering and hadron-hadron scattering can be deduced from the QCD
parton cascade [70, 71]. Work in [69] connects the K-complexity to the number
of color dipoles in the parton cascade and the K-entropy to their $S_{E}$. On
the other hand, a dynamical entropy for dense QCD states was proposed in Ref.
[72], which is based on statistical physics tools for far-from-equilibrium
processes. The numerical analysis by using realistic gluon UGDs has been done
in Ref. [73]. This entropy is written as an overlap functional between the
gluon distribution at different total rapidities $Y$ and saturation radius,
$R_{s}(Y)=1/Q_{s}(Y)$, where $Q_{s}(Y)$ is the saturation scale. In the weak
coupling regime the dynamical entropy characterizes the change of the color
correlation length $R_{s}(Y_{0})\rightarrow R_{s}(Y)$, mirroring the rapidity
evolution $Y_{0}\rightarrow Y$ of a dense gluon state. The entropy functional
$\Sigma^{Y_{0}\rightarrow Y}$ is defined in terms of the gluon transverse
momenta probability distribution. In some aspects, the analysis presented here
is more directly connected to this dynamical entropy.

##  IV SUMMARY AND CONCLUSIONS

In this work we present a description of the $p_{T}$ spectra of charged
hadrons produced in high-energy collisions taking into account their
dependence on multiplicity. In order to do so, we interpret the distribution
of gluons in terms of their entropy and show how characteristics of the
spectrum can be well described in terms of this quantity plus the partonic
saturation scale. We show that multiplicity data exhibit scaling in relation
to the saturation scale. An important consequence of our formulation is that
although the partonic entropy is non-extensive with respect to the rapidity as
expected from the BG statistics, when we compare the area of interaction of
the protons it appears that the partonic entropy grows with the interaction
area. This may suggest that the dependence on geometric aspects is fundamental
for partonic dynamics, which is usually not included in the evolution of the
rapidity in the distributions. Finally, we note that the correct slope of the
spectrum under the rescaling of the UGD power parameter, as well as the
economic number of parameters and the linear dependence of entropy on the
partonic interaction area are the main results of this work.

## Acknowledgments

This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. MVTM
acknowledges funding from the Brazilian agency Conselho Nacional de
Desenvolvimento Científico e Tecnológico (CNPq) with grant CNPq/303075/2022-8.

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