# Scaling of kinematical, global observables, energy and entropy densities in
$\rm p\\!+\\!p$, $\rm p\\!+\\!Pb$ and $\rm Pb\\!+\\!Pb$ collisions from 0.01
to 13 ${\rm TeV}$

E. Cuautle1, E. D. Rosas1, M. Rodríguez-Cahuantzi2 1 Instituto de Ciencias
Nucleares, Universidad Nacional Autónoma de México. Apartado Postal 70-543,
Ciudad de México 04510, México<EMAIL_ADDRESS>2 Facultad de
Ciencias Físico Matemáticas, Benemérita Universidad Autónoma de Puebla, Av.
San Claudio y 18 Sur, Edif. FM9-217, Ciudad Universitaria 72570, Puebla,
México

###### Abstract

The multiplicity and average transverse momentum of the charged and identified
particles produced in different kinds of colliding systems are an example of
global observables used to characterize events over a wide range of energy.
Studying these observables provides insights into the collective phenomena and
the geometric scaling properties of the systems created in ultra-relativistic
$\rm p\\!+\\!Pb$, $\rm Pb\\!+\\!Pb$, and even in $\rm p\\!+\\!p$ collisions.
The first part of this work presents a study of these variables using
different Monte Carlo event generators. It analyzes their sensitivity to find
collective phenomena at 0.01, 0.9, 2.76, 7, and 13 ${\rm TeV}$, finding a less
satisfactory description as the energy decreases. The second part analyzes the
average transverse momentum of charged hadrons as a function of the
multiplicity for $\rm p\\!+\\!p$, $\rm p\\!+\\!Pb$, and $\rm Pb\\!+\\!Pb$ data
from the $\rm{CMS}$ and $\rm{ALICE}$ experiments. Comparing with Monte Carlo
event generators, we look for a possible scaling law of average transverse
momentum scaled to the overlap transverse collision area. Additionally, the
experimental data are used to compute thermodynamical quantities such as the
energy and entropy densities in the Bjorken approach. The results are compared
with predictions from $\rm{EPOS}$ and $\rm{PYTHIA}$ Monte Carlo event
generators. We observe an excellent agreement for $\langle p_{\rm{T}}\rangle$
from $\rm p\\!+\\!p$ but not for thermodynamical observables, where a sudden
rise in a small $\langle p_{\rm{T}}\rangle$ range resembles the lattice QCD
results for the $\epsilon/T^{4}$ as a function of the temperature; however,
only the experimental data from $\rm p\\!+\\!p$ show a kind of saturation.

††: J. Phys. G: Nucl. Part. Phys.

Keywords: proton-proton, heavy ion collision; energy density, entropy density,
multiplicity, transverse momentum.

##  1 Introduction

The CERN proton anti-proton collider results have shown that the flattening of
the average transverse momentum ($\langle p_{\rm{T}}\rangle$) as a function of
the charged multiplicity ($\rm{N_{ch}}$), observed in the central rapidity
region, may serve as a probe for the equation of state of hot hadronic matter.
These results constitute a possible signal for a phase transition in hadronic
collisions [1].
In general, measurements in the low-momentum regime provide essential
information to describe strong interactions using the non-perturbative region
of Quantum Chromodynamics (QCD); QCD-inspired models are usually implemented
in Monte Carlo event generators. Additionally, these kinds of measurements are
used to constrain the free parameters of such models.
Furthermore, $\langle p_{\rm{T}}\rangle$ studies as a function of collision
energy and multiplicity distributions provide more detailed information on the
kinematic processes of colliding systems. Thermodynamical quantities
describing the system created during the collisions are estimated using the
transverse momentum of hadrons [2].

Jet production seems to explain the increase of $\langle p_{\rm{T}}\rangle$
with multiplicity. Jets are responsible for the first rise since hard
processes dominate over soft ones in this multiplicity region [3]. A second
rise seems to appear due to jet production and fragmentation; this behavior is
consistent with results obtained at Tevatron energies. However, this behavior
is not observed at higher energies. Moreover, the energy dependence has been
studied by the CDF experiment that reported $\langle p_{\rm{T}}\rangle$ as a
function of multiplicity for minimum bias events at 630 and 1800 ${\rm GeV}$,
observing and increase with energy and multiplicity for central
pseudorapidities. It has also been observed that $\langle p_{\rm{T}}\rangle$
for soft events is not energy dependent whereas for hard events it increases
faster as the energy of the colliding system increases [4].

The behavior of $\langle p_{\rm{T}}\rangle$ is almost linear at low collision
energies [5] it grows faster for higher energies, where it can be described
using a second order logarithmic polynomial as a function of the collision
energy [6, 7]. For identified charged particles the $\langle
p_{\rm{T}}\rangle$ exhibits also a mass dependence. This behavior can not be
explained [8] by the default tune of the $\rm{PYTHIA}$ event generator which
overestimates the transverse momentum of ATLAS results [9] for values larger
than 40 in charged particle multiplicity, in the pseudorapidity range
$|\eta|<2.5$. Alternative models such as the Color Glass Condensate (CGC)
[10], where the $\langle p_{\rm{T}}\rangle$ is a universal function of the
ratio of the multiplicity density and the transverse area of the collision
($S_{\rm{T}}$) [11] qualitatively reproduce the experimental data. However,
this geometrical model seems to describe the $\langle p_{\rm{T}}\rangle$ at
larger multiplicities for $\rm p\\!+\\!p$ and $\rm p\\!+\\!Pb$ collisions but
fails for lower multiplicities, and neither can describe $\langle
p_{\rm{T}}\rangle$ for $\rm Pb\\!+\\!Pb$ collisions. A possible reason for
this poor description is that collective effects may be present in $\rm
p\\!+\\!p$, $\rm p\\!+\\!Pb$, and $\rm Pb\\!+\\!Pb$ systems, since the CGC is
just a geometric model, and it is not expected to reproduce the data. It is
worth mentioning that in $\rm p\\!+\\!p$ collisions, it has been suggested
that there are flow-like [12] effects in high multiplicity events at
$\rm{LHC}$ energies. The $\rm{ALICE}$ collaboration at the LHC has measured
the $\langle p_{\rm{T}}\rangle$ [13] of charged hadrons from $\rm p\\!+\\!p$
collisions at energies of $\sqrt{s}$ = 0.9, 2.76, and 7 ${\rm TeV}$ in the
kinematical range $0.15<p_{\rm{T}}<10.0$ ${\rm GeV}/c$ and $|\eta|<0.3$. ALICE
selected all the charged prompt particles produced in $\rm p\\!+\\!p$
collisions including all decay products, except those from weak decays of
strange hadrons. The results show a correlation of $\langle p_{\rm{T}}\rangle$
with $\rm{N_{ch}}$ which is stronger for $\rm p\\!+\\!p$ than for $\rm
p\\!+\\!Pb$ and $\rm Pb\\!+\\!Pb$ collisions. The observed strong correlation
has been attributed within $\rm{PYTHIA}$ to Color Reconnections (CR) between
hadronization strings [14, 15], which are directly related with multiple
parton interactions.
Furthermore, the observed enhancement of $\langle p_{\rm{T}}\rangle$ versus
multiplicity at low $p_{\rm{T}}$ is not reproduced by any model [7]. These
kinematic and global observables have been associated in simulation studies to
QGP formation in proton-proton collisions [16]. The relationship between
multiplicity and multiple parton interactions [17] have been studied in an
attemp to explain experimental results. An incoherent superposition of such
interactions would lead to a constant $\langle p_{\rm{T}}\rangle$ at high
multiplicities.
In high multiplicity $\rm p\\!+\\!p$ events, a medium governed by strong
interactions is produced [18], thus, the produced matter can be described in
terms of quark and gluon degree of freedom. The equation of state, relating
the energy and entropy densities, pressure, and temperature of such matter, is
of fundamental importance to understand its composition and the static and
dynamical properties of the medium created in $\rm p\\!+\\!p$, $\rm
p\\!+\\!Pb$ and $\rm Pb\\!+\\!Pb$ collisions at different energies. In this
work we address these issues. The paper is organized as follows: A brief
presentation of hadronization models used by $\rm{PYTHIA}$ and $\rm{EPOS}$
event generators is provided in section II. Results and their comparison with
experimental data and the average transverse momentum versus multiplicity
scaled to the transverse collision area are presented in section III:
Subsection III A describes results on multiplicity and average transverse
momentum versus multiplicity, and the multiplicity scaled to the transverse
collision area obtained from event generators. Subsection III B is devoted to
investigate the equation of state and a possible phase transition of QGP
through the analysis of the energy and entropy densities, scaled by $\langle
p_{\rm{T}}\rangle^{4}$ and $\langle p_{\rm{T}}\rangle^{3}$, respectively, for
charged and identified particles. Finally, the discussions and conclusions are
drawn in section IV.

##  2 Modeling parameters of the hadroproduction

Hadronization processes are one of the most widely studied topics. Despite
these efforts, currently we only have effective theories to describe these
phenomena. Different event generators have been developed to investigate
hadronization processes in $\rm p\\!+\\!p$, $\rm p\\!+\\!Pb$, and ion-ion
collision systems. Each event generator approaches hadron production
mechanisms in a different manner but most of them are based on string
fragmentation models at high energies. Some general aspects are briefly
described in the following lines: $\rm{PYTHIA}$ hadronization processes
include the CR mechanisms providing an alternative final interaction
mechanism, modifying some kinematical variables of the hadron-hadron
collisions and lepton collisions. These final state interactions allow a
better description of the transverse momentum measured by the $\rm{ALICE}$
experiment. Color Reconnection is based on the production of hadrons through
the interaction probability between partons of low and high $p_{T}$. These
interacting partons can come from the beam or from the remnant of the beam.
They can be quantified by the probability distribution of reconnecting
partons, $P_{rec}(p_{T})$, defined as:

$P_{rec}(p_{T})=\frac{(R_{rec}\;p_{T0})^{2}}{(R_{rec}\;p_{T0})^{2}+p_{T}^{2}},$
(1)

where the range of CR, $0\leq R_{rec}\leq 10$, is a phenomenological parameter
and $p_{T0}$ is an energy dependent parameter used to damp the low $p_{T}$
divergence of the $2\rightarrow 2$ QCD cross section. Equation (1) used for
the description of the color flow, is based on the Multiple Parton Interaction
model. However, there are other newer approaches [19], or even older ones such
as those developed to describe $W$ bosons in $e^{-}+e^{-}$ processes [20],
each model contains additional parameters to handle CR effects.
Hadron-hadron collisions consider a spherically symmetric matter distribution
within each hadron, $\rho({\bf x})d^{3}x=\rho(r)d^{3}x$. In collision
processes, the overlap of the colliding hadrons is a function of the impact
parameter ($b$) and is given by:

$O(b)\propto\int\int d^{3}xdt\rho(x,y,z)\rho(x,y,z-\sqrt{b^{2}-t^{2}}),$ (2)

where different $\rho$ distributions have been studied previously [21]. Taking
a simple Gaussian for this $\rho$, the convolution becomes trivial, but it
does not give a good enough description of the data. A better choice of this
matter distribution is a double Gaussian function given by:

$\rho(r)=(1-\beta)\frac{1}{r_{1}^{3}}\exp(\frac{r^{2}}{r_{1}^{3}})+\beta\frac{1}{r_{2}^{3}}\exp(-\frac{r^{2}}{r_{2}^{3}}),$
(3)

this equation corresponds to a distribution with a small core region of radius
$r_{2}$ and containing a fraction ($\beta$) of the total hadronic matter,
embedded in a larger hadron of radius $r_{1}$. Two free parameters $\beta$ and
$r_{2}/r_{1}$ are sufficient to give the necessary flexibility.
This way of modeling the overlap matter distribution allows the study of
hadron-hadron collisions as a function of impact parameter as for heavy-ion
collisions. Thus kinematical variables like $\langle p_{\rm{T}}\rangle$ in
hadron production can also be studied in terms of the impact parameter [21].

$\rm{EPOS}$ simulates interactions among ions as binary interactions, each
represented by a parton ladder; this ladder considers a longitudinal color
field, conveniently treated as a relativistic string. The string decays via
the production of quark anti-quark pairs creating string fragments usually
identified with hadrons [22].
Our $\rm{EPOS}$ $\rm p\\!+\\!p$ simulation at the energy of $\sqrt{(}s)=10$
${\rm GeV}$ used the version 1.99, and for larger than 10 ${\rm GeV}$,
$\rm{EPOS}$-LHC was used. From now on, these choices are referred to as
$\rm{EPOS}$. Both versions include a simplified treatment of final state
interaction but do not include full hydro as is already possible in most
recent $\rm{EPOS}$ 3 not released version.
$\rm{EPOS-LHC}$ is a minimum bias hadronic generator used for heavy-ion
collisions and cosmic ray air shower development. However, the goal of this
generator is to describe soft particle production ( $p_{\rm{T}}$ $<\sim$ 5
${\rm GeV}/c$) for any system and energy in great detail. Moreover, high-
density effects leading to collective behavior in heavy-ion collisions are
taken into account [23] by the fusion model, which means that there is the
probability to form a core and a corona, even in proton-proton collisions. The
model considers the hadronization part of the secondary particles, and the
flow intensity includes different radial flow types that depend only on the
total mass of the high density core produced by the overlap of the string
segments due to multiple parton interactions in proton-proton or multiple
nucleon interactions. The model also includes different radial flow types in
the core created in a small volume due to multiple scattering between partons
in a single pair of nucleons, which is the case of $\rm p\\!+\\!p$ collisions.

##  3 Results

Results presented in this work correspond to $\rm p\\!+\\!p$, $\rm
p\\!+\\!Pb$, and $\rm Pb\\!+\\!Pb$ collisions at different energies, generated
with $\rm{PYTHIA}$, and $\rm{EPOS}$. The data sample size is different for
each colliding system, smaller for $\rm Pb\\!+\\!Pb$ with respect to the $\rm
p\\!+\\!p$, but large enough to understand the statistical fluctuations and
for comparison with available experimental data. The analyzed thermodynamical
variables were chosen having in mind $\rm{CMS}$ and $\rm{ALICE}$ data.

###  3.1 Average transverse momentum and multiplicity

The correlation between multiplicity and transverse momentum has been studied
since the 80’s [24]. The experimental results on multiplicity suggested the
introduction of the multiple parton interaction (MPI) [25]. Since then, nMPI
was introduced in the event generators as an ingredient to describe
hadronization processes, together with the string fragmentation mechanism
within the Lund model [26].
Figure 1 shows the average multiplicity distributions versus transverse
momentum generated with $\rm{PYTHIA}$ for $\rm p\\!+\\!p$ collisions without
CR at different energies in the range from 0.01 to 13 ${\rm TeV}$ (Fig. 1
(a)). The distributions at the lowest energy, 10 ${\rm GeV}$, show a flat
behavior; meanwhile, as the energy increases, a rising slope appears, which
can roughly be described by $\langle N_{\rm{ch}}\rangle=a+b\cdot
p_{\rm{T}}^{c}$, where $a,b$ and $c$ are free parameters. When the energy
increases $\langle N_{\rm{ch}}\rangle$ does as well, similarly to the charged
particle density. The exact shape of the average multiplicity is illustrated
by looking at the ratio between average multiplicity for two energies:
$\langle N_{\rm{ch}}\rangle(E_{n})$/$\langle N_{\rm{ch}}\rangle$ (E=13 ${\rm
TeV}$), where $E_{n}=0.01,0.9,2.76$ and 7 ${\rm TeV}$ (Fig. 1 (b)). Figure 1
(c) shows the ratio of the average multiplicity without the CR effect to that
where it is included. The ratio shows a more prominent difference at higher
energy. For instance, at 13 ${\rm TeV}$, the ratio is reduced down to
$\approx$ 20% when $p_{\rm{T}}$ goes from $\sim$ 0.5 to $\sim$ 3.0 ${\rm
GeV}/c$. The distributions above 3 ${\rm GeV}/c$ become flat. The CR effects
tend to vanish when the energy decreases, producing a null effect at 10 ${\rm
GeV}$. These results are presented as a function of the momentum but they also
have a dependence on the intensity of the CR, described in Eq. (1). The
transverse momentum distribution experiences a flow-like effect due to CR [27,
28] in $\rm{PYTHIA}$, and it is observed in $\rm p\\!+\\!p$ collisions at
$\rm{CMS}$ [12] resembling the distributions observed in heavy ion collisions,
where hydrodynamics describes the data qualitatively.

(a)

(b)

(c)

Figure 1: Average multiplicity distributions as a function of transverse
momentum (a) without CR. Ratios of the average multiplicities at two energies
(b), and ratios between the average multiplicity at the same energy but
without to with CR (c).

The distributions in Fig. 2 are equivalent to the ones shown in Fig. 1 but
computed with the $\rm{EPOS}$ event generator where fusion and no fusion
models were considered instead of CR or no CR included in $\rm{PYTHIA}$. Both
Figs. 1 (a) and 2 (a) are very similar, nonetheless in Fig. 2 (b), $\rm{EPOS}$
produces a flat ratio for 10 ${\rm GeV}$ whereas $\rm{PYTHIA}$ produces ratios
that go down and then up. The ratios of the $\langle N_{\rm{ch}}\rangle$ given
by fusion off to fusion on models (Fig. 2 (c)) show a slight and constant
decrease with increasing energy and for $p_{\rm{T}}$ larger than $\sim$ 0.5
${\rm GeV}/c$, the ratios decrease except for the case of 10 ${\rm GeV}$ which
remains constant within uncertainties. Comparing $\rm{EPOS}$ fusion and
$\rm{PYTHIA}$ CR, the first predicts a slight reduction on the $\langle
N_{\rm{ch}}\rangle$ whereas the second reduces the $\langle
N_{\rm{ch}}\rangle$ up to $\sim 20\%$ for higher energy when $p_{\rm{T}}$ goes
from 0.0 - 10 ${\rm GeV}/c$.

(a)

(b)

(c)

Figure 2:  (color online) Average multiplicity distributions as a function of
transverse momentum (a). Ratios between average multiplicities for two
energies (b), and ratios between average multiplicities for two models at the
same energy (c).

As mentioned in Sec. 2, the impact parameter could modify some kinematic
variables like the average transverse momentum versus multiplicity. Figure 3
(a) shows slight differences in the distributions of the impact parameters for
$\rm p\\!+\\!p$ collisions incorporated in two event generators, $\rm{EPOS}$
and $\rm{PYTHIA}$. The differences come from the different models to describe
the matter distribution of the colliding hadrons and the set of parameters
used to fit data. $\rm{EPOS}$ uses $\rm{PYTHIA}$ 6 with default tune, whereas
$\rm{PYTHIA}$ 8 has a different tune and includes several new functions [29]
for a better description of kinematical variables.
Our analysis considers the effects on $\langle p_{\rm{T}}\rangle$ versus
multiplicity for two event generators: for the case of $\rm{PYTHIA}$ with CR,
one gets the distributions in Fig. 3 (b) compared to $\rm{ALICE}$ data [13],
where it is shown that different ranges of impact parameter produce a slight
change in the slope of the $\langle p_{\rm{T}}\rangle$. Variations in the
ranges of the impact parameter by themselves do not allow the reproduction of
old data, and this worsens for low multiplicity, overestimating by up to $\sim
7\%$ for the impact parameter range of $0.0<b<0.2$ fm, whereas for events with
high multiplicity, the statistical errors are large enough to cover the
disagreement. Similar variations on the impact parameter are applied to
$\rm{EPOS}$ event generator, resulting in a great description of data for the
range of $0.0<b<0.2$ fm. $\rm{EPOS}$ includes hydrodynamical effects; however,
kinematic observables from these event generators in hadron-hadron collisions
are not the same, nor do they look like the $\rm{PYTHIA}$ results discussed in
the previous lines.

(a)

(b)

Figure 3: impact parameter distributions, from the $\rm{EPOS}$ and
$\rm{PYTHIA}$ event generators (a), and average transverse momentum as a
function of the multiplicity for three ranges of impact parameter for the
$\rm{PYTHIA}$ case and one range of $\rm{EPOS}$, compared to $\rm{ALICE}$ data
[13] at 900 ${\rm GeV}$ (b).

This last one includes flow-like effects in hadron-hadron collisions,
resembling hydrodynamical effects, but clearly, we can not give a
hydrodynamical treatment using this event generator. Figure 4 (a) shows the
average transverse momentum versus multiplicity for charged hadrons produced
in $\rm p\\!+\\!p$ collisions from two-event generators: $\rm{PYTHIA}$ without
CR and $\rm{EPOS}$ with fusion turned off. These results show a discrepancy
for all $p_{\rm{T}}$ ranges.

(a)

(b)

Figure 4:  Comparison of the average transverse momentum versus multiplicity
for two event generators without (a) and with (b) hydro and CR of 1.8 is
considered at 900 ${\rm GeV}$. $\rm{ALICE}$ data [13] also are showed in the
panel (b).

The same distributions with flow or hydro effects are shown in Fig. 4 (b)
together with $\rm{ALICE}$ data. There is a good agreement between data and
$\rm{EPOS}$ for all multiplicity ranges. Nevertheless, $\rm{PYTHIA}$ with CR
shows an increasing disagreement as the multiplicity increases; besides the
model dependency, there is a dependence on energy and the hardness of the
process incorporated in each model. $\rm{PYTHIA}$ and $\rm{EPOS}$ event
generators have many parameters which can usually be modified to create a
tuning to fit experimental data. Our results of $\langle p_{\rm{T}}\rangle$
are without tuning but are computed for different impact parameter ranges,
which according to section 2, could give insight into the hadronic matter
distribution in the proton beam.
Figure 5 (a) shows results in agreement among $\rm{PYTHIA}$, $\rm{EPOS}$ and
$\rm{ALICE}$ data at 900 ${\rm GeV}$. Results from $\rm{EPOS}$ considered a
range of impact parameter from 0.0-0.3 fm, since use of the whole impact
parameter range overestimates the $\langle p_{\rm{T}}\rangle$ by $\sim 9\%$ at
low multiplicity. An agreement between models and data is also observed for
$\langle p_{\rm{T}}\rangle$ versus multiplicity at 7 ${\rm TeV}$ [13], showed
in Fig. 5 (b), where $\rm{PYTHIA}$ results were generated with CR parameter of
1.3 instead of 1.8, to improve the agreement with data up to 6% for
multiplicity below 25, at least for the $\langle p_{\rm{T}}\rangle$ from $\rm
p\\!+\\!p$ at 7 ${\rm TeV}$. At higher multiplicity the agreement worsens.
Results have been tested at different energies but in this work we only show
results for 0.9 and 7 ${\rm TeV}$. We also tested for $\rm p\\!+\\!Pb$ and
$\rm Pb\\!+\\!Pb$ systems with colliding energies of 5.02 ${\rm TeV}$ and 2.76
${\rm TeV}$, respectively. These comparisons are complementary to those
reported by ALICE [13].

(a)

(b)

Figure 5: Comparison of the average transverse momentum versus multiplicity
for two event generators and $\rm{ALICE}$ data [13] at 0.9 ${\rm TeV}$ (a),
and 7 ${\rm TeV}$ (b).

The average transverse momentum for charged hadrons as well as for identified
particles has been measured in different colliding systems, since it provides
a more detailed understanding of the dynamics of the collisions, like the mass
dependence or quark content. The results clearly show that $\langle
p_{\rm{T}}\rangle$ presents the most significant slope for higher mass
particles, and gets significantly smaller for the lightest particles [8].
The average transverse momentum as a function of the multiplicity scaled by
the transverse area has been proposed as a good scaling law at high
multiplicity but not at lower ones. The study of this scaling requires
knowledge of interferometry to extract the size (radius) of the matter created
in $\rm p\\!+\\!p$, $\rm p\\!+\\!Pb$, and heavy-ion collisions. These
techniques were used by $\rm{ALICE}$ [30] and $\rm{CMS}$ [31] to extract the
radius as a function of the particle multiplicity. The multiplicity
distributions normalized to the transverse area ($S_{T}$) of the collisions
($N_{ch}/S_{T}$) is an excellent variable to explore $\langle
p_{\rm{T}}\rangle$ in hadron collisions. The $\rm{CMS}$ measurements of the
multiplicity allow the parametrization of the radius for $\rm p\\!+\\!p$ and
$\rm p\\!+\\!Pb$ collisions as follows [32]:

$\displaystyle R_{pPb,pp}=1fm\times
f_{pPb,pp}\left({}^{3}\sqrt{dN_{g}/dy}\right)$ (4)
$\displaystyle\frac{dN_{g}}{dy}\approx
K\frac{3}{2}\frac{1}{\Delta\eta}N_{tracks},$ (5)

where $dN_{g}/dy$ is the gluon density and

$f_{pp}(x)=\scalebox{0.9}{\mbox{$\displaystyle\cases{0.387+0.0335x+0.274x^{2}-0.0542x^{3}&if
$x<3.4$\\\ 1.538&if $x\geq 3.4$\\\ }$}}$ (6)

$f_{pPb}(x)=\scalebox{0.9}{\mbox{$\displaystyle\cases{0.21+0.47x&if $x<3.5$\\\
1.184-0.483x+0.305x^{2}-0.032x^{3}&if $3.5\leq x<5.0$\\\ 2.394&if $x\geq
5.0$\\\ }$}}$ (7)

where $N_{tracks}$ is the number of tracks and $\Delta\eta$ is the
pseudorapidity range where the experiment performed its measurements. $K=1$ in
our case. Figure 6 (a) shows the radii as a function of the multiplicity
computed from Eqs. (6) and (7) for $\rm p\\!+\\!p$ and $\rm p\\!+\\!Pb$
collisions, respectively, with the multiplicity from $\rm{ALICE}$ $\rm
p\\!+\\!p$, $\rm p\\!+\\!Pb$, and $\rm Pb\\!+\\!Pb$ colliding systems [13],
and compared to $\rm{CMS}$ $\rm p\\!+\\!p$ [33] and $\rm p\\!+\\!Pb$ [34]
results. The parametric distribution measured directly from $\rm{ALICE}$ [30]
has also been presented for the case $\rm p\\!+\\!p$ and $\rm p\\!+\\!Pb$. It
corresponds to the lower distribution in the plot. An agreement at low
multiplicity is observed, but the radius increases faster with the
multiplicity for the heaviest systems. As will be discussed later, these
differences bring consequences in other calculations, such as energy density
and entropy density. The average transverse momentum also presents a scaling
law when it is plotted as a function of multiplicity scaled by the transverse
area. Figure 6 (b), shows the comparison among results from $\rm{ALICE}$ $\rm
p\\!+\\!p$ at 7.0 ${\rm TeV}$ and $\rm{PYTHIA}$ with CR and $\rm{EPOS}$ with
fusion turned on, at the same energy. Results from $\rm{ALICE}$ $\rm
p\\!+\\!Pb$ at 5.02 ${\rm TeV}$ are compared to $\rm{EPOS}$ simulations at the
same energy. These scaling laws with experimental data were studied [32] in
the framework of CGC, suggesting that no scaling is presented at low
$p_{\rm{T}}$ due to flow effects. Nevertheless, our simulated results include
flow, showing that the breaking scaling still remains.

(a)

(b)

Figure 6: Radii as a function of multiplicity according to parametric function
[11], comparing results from $\rm{ALICE}$ data [13, 30], $\rm{CMS}$ $\rm
p\\!+\\!p$ [33] and $\rm p\\!+\\!Pb$ [34] data (a). Average transverse
momentum as a function of multiplicity normalized by transverse area (b),
comparing results from $\rm{ALICE}$ experiment [13] and simulation from
$\rm{PYTHIA}$ and $\rm{EPOS}$ event generators.

###  3.2 Experimental equation of state in $\rm p\\!+\\!p$ and $\rm
Pb\\!+\\!Pb$ collisions

Thermodynamical quantities like energy and entropy densities, temperature, and
pressure, are related by an equation of state to describe the evolution of the
medium created in heavy-ion collisions and also in high multiplicity $\rm
p\\!+\\!p$ events. Hydrodynamic theory has been used to get the energy
density, entropy, temperature and other quantities as a first description
approach to $\rm p\\!+\\!p$ collisions [35]. Different observables are
analyzed to explore the formation of QGP-droplets in $\rm p\\!+\\!p$
collisions [36]. The HotQCD Collaboration has developed an equation of state
in (2+1) flavor QCD [37] which gives a prediction for the energy density
divided by temperature $T^{4}$ ($\epsilon/T^{4}$) as a function of $T$.
Similar results have been obtained [38] using the Color String Percolation
Model, which allow to compute hydrodynamical properties [2] in $\rm
p\\!+\\!p$, $\rm p\\!+\\!Pb$ and $\rm Pb\\!+\\!Pb$ collisions.
Analyzing data from the $\rm p\\!+\\!p$ and $\rm p\\!+\\!\bar{p}$ collisions,
the equation of state can be estimated [39] by combining $\langle
p_{\rm{T}}\rangle$ versus multiplicity distribution measurements.
Results from heavy-ion collisions can be well described by hydrodynamics,
however, this theory does not apply to results from $\rm p\\!+\\!p$
collisions. Nevertheless, $\rm p\\!+\\!p$ collisions at high energy produce a
high particle density, and the mean free path of particles in such a system is
very small compared to its dimension. The created system expands, and a
fraction of the expansion processes must have hydrodynamical characteristics.
Under this consideration the initial entropy density estimated by Landau [40]
is proportional to the multiplicity ($s_{0}=kdN_{ch}/d\eta$, where $k$ is a
constant) which allows to compute the entropy [41] for a mini-QGP system.
Statistical models can be used to analyze the multiplicity densities and then
get thermodynamical limits, in $\rm p\\!+\\!p$ collisions. In particular, the
$\epsilon/T^{4}$ as a function of multiplicity tends to the same value [42]
when it is calculated from different statistical ensembles. The shear
viscosity, isothermal compressibility and speed of sound have also been
calculated [43] from $\rm{ALICE}$ $\rm p\\!+\\!p$ results.

This work presents the extraction of the thermodynamical properties from
$\rm{CMS}$ and $\rm{ALICE}$ data, starting from the average transverse
momentum as a function of multiplicity. At the same time, the results are
compared to those from event generators.

The Bjorken energy density ($\epsilon_{Bj}$) [44] defined by Eq. (8) has been
used to show that the enhancement of the strangeness production as a function
of energy density for different colliding systems presents a scaling law [45,
46].

$\epsilon_{Bj}\simeq\frac{3}{2}\frac{\langle
p_{T}\rangle\frac{dN}{d\eta}}{S_{T}\cdot\tau}.$ (8)

In this equation, $S_{\rm{T}}$ is the interaction area calculated as in the
previous section, and $\tau$ is the proper time in the Bjorken model, so the
product $S_{\rm{T}}\cdot\tau$ is the volume occupied by the particles. Using
Eq. (8), we compute the energy density for charged hadrons from $\rm
p\\!+\\!p$, $\rm p\\!+\\!Pb$, and $\rm Pb\\!+\\!Pb$ data at different energies
measured by $\rm{ALICE}$. The results as a function of the multiplicity scaled
by $S_{\rm{T}}$ are shown in Fig. 7 (a), where a clear scaling law is observed
at low multiplicity for all colliding systems. At high multiplicity, the
energy density shows differences that may come from the measured radii.
Analysis of the energy density for identified particles produced in $\rm
p\\!+\\!Pb$ is shown in Fig. 7 (b), where higher values for the heaviest
particles are evident. It is possible to add the energy by particle species
and the total approximately corresponds to the results for all charged
hadrons.

(a)

(b)

Figure 7: Bjorken energy density for charged hadrons (a) from different
colliding systems and energies as a function of multiplicity scaled by the
overlap transverse collision area for $\rm{ALICE}$ data [13]. The same
quantity for identified particles from $\rm{ALICE}$ data [47] is shown in
panel (b).

The entropy density ($\sigma$) is determined in statistical QCD with dynamical
quarks [48] as:

$\sigma\approx\epsilon_{Bj}/\langle p_{\rm{T}}\rangle,$ (9)

where the initial Bjorken energy density is given by Eq. (8). This means that
the entropy density only depends on the initial energy density and therefore
on the multiplicity reached in the collisions. The previous equations can also
be written as

$\displaystyle\frac{\epsilon_{Bj}}{\langle
p_{T}\rangle^{4}}\equiv\frac{\sigma}{\langle p_{T}\rangle^{3}}.$ (10)

Using the average transverse momentum distributions as a function of the
multiplicity from $\rm{ALICE}$ and $\rm{CMS}$, we get the energy density of
Eq. (8). To investigate possible phase transition in the hadronic matter
created in $\rm p\\!+\\!p$, $\rm p\\!+\\!Pb$, and $\rm Pb\\!+\\!Pb$
collisions, we use Eq. (10). As the first approximation, using the Color
String Percolation Model, the $\langle p_{\rm{T}}\rangle$ is proportional to
the initial temperature $T$ [49]. This quantity has also been extracted from
an analysis of the average transverse momentum, using a linear relationship,
$\langle p_{\rm{T}}\rangle=a+bT$, between $\langle p_{\rm{T}}\rangle$ and
$T$[50], where $a$ and $b$ are free fit parameters. It is worth mentioning
that the relation between $\langle p_{\rm{T}}\rangle$ and $T$ depends on the
model used, and $b$ is a function of the collision energy. However, as a first
approach, we consider $\langle p_{\rm{T}}\rangle$ $\sim T$, consequently the
Eq. (10) as a function of $\langle p_{\rm{T}}\rangle$ provides results similar
to those obtained from a plot of $\sigma/T^{3}$ as a function of $T$, analyzed
using lattice QCD [37].

(a)

(b)

Figure 8: Entropy density normalized to $\langle p_{\rm{T}}\rangle^{3}$ for
piones (left) and Kaons (right) from $\rm{CMS}$ $\rm p\\!+\\!p$ data[8] at
different energies

Figure 8 shows the plot of entropy scaled by $\langle p_{\rm{T}}\rangle^{3}$
($\sigma\cdot S_{T}\tau/\langle p_{\rm{T}}\rangle^{3}$) as a function of
$\langle p_{\rm{T}}\rangle$ computed using $\rm{CMS}$ data [8]. The
distributions correspond to pions (Fig. 8 (a)) and kaons ( Fig. 8 (b)) from
$\rm{CMS}$ $\rm p\\!+\\!p$ collisions at 0.9, 2.76, 7.0 ${\rm TeV}$ [8]. The
plot of the entropy instead of entropy density is to avoid the use of large
uncertainty radii extracted from $\rm{CMS}$, which precludes possible energy
effects. This entropy has a rapid increase and seems to be saturated for pions
and kaons produced in $\rm p\\!+\\!p$ at 7 ${\rm TeV}$. Significant
fluctuations in the behavior of entropy for the same species at different
energies could be associated with the selection processes in the measurements
by the $\rm{CMS}$ experiment. Figure 9 (a) shows our results for entropy
density as a function of $\langle p_{\rm{T}}\rangle$ for pions, kaons,
protons, and Lambda’s, computed with $\rm p\\!+\\!Pb$ $\rm{ALICE}$ data [47].
In all cases, the entropy rises to reach the values determined by the
multiplicity as a function of $\langle p_{\rm{T}}\rangle$; we observe a faster
growth for the lightest particles. When the entropy (entropy density
multiplied by $S_{T}\tau$) is normalized to $\langle p_{\rm{T}}\rangle^{3}$
(Fig. 9 (b)), however, all the cases can be fitted to a functional form
$Exp$(A+B$\langle p_{\rm{T}}\rangle$) where A and B are shown in the Table 1.
The distributions for $\Lambda$’s and protons can also be fitted with a linear
functional form, but the best fits are for exponential form.

(a)

(b)

Figure 9: Entropy density as a function of $\langle p_{\rm{T}}\rangle$ for
identified particles (a). Entropy normalized to $\langle
p_{\rm{T}}\rangle^{3}$ as a function of $\langle p_{\rm{T}}\rangle$, (b). Both
cases, correspond to $\rm{ALICE}$ $\rm p\\!+\\!Pb$ data [47] at different
energies. Table 1:  Parameters of the fitted function: $Exp$ (A +B $\langle
p_{\rm{T}}\rangle$), in Fig. 9 (a) and 9 (b).

  * |  | $\pi^{+}+\pi^{-}$ | $K^{+}+K^{-}$ | $p^{+}+\bar{p}$ | $K^{0}$ | $\Lambda+\bar{\Lambda}$
---|---|---|---|---|---|---
Fig. 9 (a) | A | -3.7114 | -2.4520 | -1.7269 | -2.7737 | -2.1473
| B [1/(${\rm GeV}/c$)] | 8.9957 | 3.8628 | 2.3344 | 4.1288 | 2.4104
Fig. 9 (b) | A | -1.6948 | -0.3344 | 0.6161 | -1.0888 | -0.6768
| B [1/(${\rm GeV}/c$)] | 14.0796 | 4.9379 | 2.2758 | 5.5768 | 2.7725

Finally, the results for the entropy scaled to $\langle p_{\rm{T}}\rangle^{3}$
for all charged particles produced in $\rm{ALICE}$ $\rm p\\!+\\!p$, $\rm
p\\!+\\!Pb$, and $\rm Pb\\!+\\!Pb$ colliding systems are shown in Fig. 10,
together with predictions from the Monte Carlo event generators for the same
systems and energies as the $\rm{ALICE}$ data. Figure 10 (a) shows the
experimental data from $\rm p\\!+\\!p$ collisions, which are well described by
$\rm{EPOS}$, and $\rm{PYTHIA}$ predictions. It is interesting to note a kind
of saturation for $\langle p_{\rm{T}}\rangle$ larger than 0.8 ${\rm GeV}/c$.
Regarding $\rm p\\!+\\!Pb$ collisions, Fig. 10 (b) shows that data and
simulation have the same trend: a rapid growth of the slope that appears
around $\langle p_{\rm{T}}\rangle\approx 0.7$ ${\rm GeV}/c$ and $\langle
p_{\rm{T}}\rangle\approx 0.75$ ${\rm GeV}/c$ for $\rm{EPOS}$ and $\rm{PYTHIA}$
respectively, while data are between both event generators. We emphasize that
a well defined inflection point, around $\langle p_{\rm{T}}\rangle$ = 0.62
${\rm GeV}/c$, is observed for $\rm{EPOS}$ when simulations cross the
experimental data.
For $\rm Pb\\!+\\!Pb$ colliding system, shown in Fig. 10 (c), a sudden change
in the entropy is observed in data and almost the same trend for $\rm{PYTHIA}$
is observed but shifted to higher $\langle p_{\rm{T}}\rangle$ values. The
$\rm{EPOS}$ event generator produces a larger slope such that the distribution
crosses the data and seems to bend backwards because the $\langle
p_{\rm{T}}\rangle$ rises and then falls as the multiplicity increases, in the
pseudorapidity range of $|\eta|<0.3$. This value was used to compare with the
$\rm{ALICE}$ data. Figure 10 (c), also shows the same distribution produced
with $\rm{EPOS}$ in $|\eta|<0.9$. We notice that it rises with a slope closer
to that of data. The results observed from $\rm{EPOS}$ suggest that caution
has to be taken to describe the most central pseudorapidity region, since this
event generator does not describe data. Looking at the details of the sudden
change in the entropy $\equiv\sigma S_{\rm{T}}\tau$ normalized to $\langle
p_{\rm{T}}\rangle^{3}$ as a function of $\langle p_{\rm{T}}\rangle$ for the
$\rm Pb\\!+\\!Pb$ $\rm{ALICE}$ data, we observe almost a linear growth for
$\langle p_{\rm{T}}\rangle$ less than $\approx 0.61$ ${\rm GeV}/c$, while an
exponential growth is observed for $\langle p_{\rm{T}}\rangle$ larger than
0.61 ${\rm GeV}/c$. Assuming $\langle p_{\rm{T}}\rangle=3T$ [51], the entropy
normalized to temperature ($S/T^{3}$) has a linear behavior with respect to
the temperature in the range from 0.172 to 0.203 ${\rm GeV}/c$, and an
exponential one for temperatures larger than 0.203 ${\rm GeV}/c$.
Within our approaches we have obtained properties of the medium created in
different colliding systems. More details can be computed; however, any
conclusion from this analysis can only have a limited agreement with data and
inspired QCD models due to the simplified model we used. Other properties will
be reported elsewhere.

(a)

(b)

(c)

Figure 10: Entropy density normalized to $\langle p_{\rm{T}}\rangle^{3}$ for
$\rm{ALICE}$ $\rm p\\!+\\!p$, $\rm p\\!+\\!Pb$ and $\rm Pb\\!+\\!Pb$ data [13]
at different energies

##  4 Conclusions and remarks

The first part of this work presents an analysis of the correlation between
transverse momentum and multiplicity distributions for different colliding
systems at energies of 0.01, 0.9, 2.76, 5.0, 7.0, and 13 ${\rm TeV}$ using two
event generators: $\rm{PYTHIA}$ with/without CR; $\rm{EPOS}$ with/without
fusion model as part of the hadronization. The second part of the work carried
out a data analysis of the average transverse momentum versus multiplicity
from $\rm{ALICE}$ and $\rm{CMS}$ experiments to extract thermodynamical
properties of the system created in $\rm p\\!+\\!p$, $\rm p\\!+\\!Pb$, and
$\rm Pb\\!+\\!Pb$ collisions at different collision energies. The results were
compared with predictions from $\rm{EPOS}$ and $\rm{PYTHIA}$ event generators.
Average multiplicity distributions increase with the collision energy and can
be parameterized as $\langle N_{\rm{ch}}\rangle=a+b\cdot p_{\rm{T}}^{c}$,
where $a,b$, and $c$, are free parameters for each energy. The $\langle
N_{\rm{ch}}\rangle$ is observed to rise more rapidly with increasing the
transverse momentum for higher collision energies. The hadronization mechanism
incorporated through CR in $\rm{PYTHIA}$ produces a more substantial reduction
in multiplicity than the $\rm{EPOS}$ fusion model; this effect is more visible
for transverse momentum larger than 0.5 ${\rm GeV}/c$. On the other hand, the
average transverse momentum as a function of the multiplicity, computed with
$\rm{PYTHIA}$, shows that a differential analysis in terms of impact parameter
is not enough to explain the data. The CR with different strengths is enough
to explain the behavior of data at collision energies of 0.9, 2.76, and 7
${\rm TeV}$ reported by the $\rm{ALICE}$ experiment. This has been tested for
other energies finding good agreement. Nonetheless, in the case of the
$\rm{EPOS}$ fusion model with a combination of impact parameters, which is
related to hadronic matter model in the proton, it allows an excellent
description of the data, although $\rm{EPOS}$ results without cut on impact
parameter are within experimental uncertainties.
We have shown that the average transverse momentum and Bjorken energy density
as a function of multiplicity, normalized to the transverse interaction area,
present a scaling law. The results were computed with $\rm{ALICE}$ data,
$\rm{PYTHIA}$, and $\rm{EPOS}$ event generators. For all cases, the $\langle
p_{\rm{T}}\rangle$ scaling breaks in the lowest and highest multiplicity
values, which can not be associated with flow effects, as other authors have
suggested.
Related to thermodynamical properties obtained with an analysis of $\rm{CMS}$
and $\rm{ALICE}$ data: the results indicate an exponential growth of the
energy and entropy normalized to $\langle p_{\rm{T}}\rangle^{3}$ as a function
of the average transverse momentum for charged and identified particles
measured in $\rm p\\!+\\!p$, $\rm p\\!+\\!Pb$, and $\rm Pb\\!+\\!Pb$
collisions at the available energies. This growth is faster for light
particles with respect to heavy ones. Considering $\langle p_{\rm{T}}\rangle$
as a linear function of the initial temperature of the system created in the
collisions, the conclusions obtained from $\epsilon_{Bj}/\langle
p_{\rm{T}}\rangle^{4}$ are similar to those from $\epsilon_{Bj}/T^{4}$, a
quantity widely studied to look for a QCD phase transition. The same
conclusion can be reached from $\sigma/T^{3}$. Analysis of experimental $\rm
Pb\\!+\\!Pb$ data with $\langle p_{\rm{T}}\rangle=3T$ produces an effective
temperature range of 0.17-0.2 ${\rm GeV}/c$, where the energy density
normalized to $T^{4}$ has a linear behavior, and for effective temperatures
larger than 0.2 ${\rm GeV}/c$, it grows exponentially. Our results from
$\rm{EPOS}$ and $\rm{PYTHIA}$ $\rm p\\!+\\!p$ collisions agree with
$\rm{ALICE}$ $\rm p\\!+\\!p$ data and have the behavior obtained by lattice
(2+1) flavor QCD. They show a kind of saturation for $\langle
p_{\rm{T}}\rangle$ larger than 0.8 ${\rm GeV}/c$, which is not observed for
either $\rm p\\!+\\!Pb$ and $\rm Pb\\!+\\!Pb$ $\rm{ALICE}$ data. The
predictions from $\rm{PYTHIA}$ and $\rm{EPOS}$ show the data trend, but there
is no complete agreement to $\rm{ALICE}$ data. Particularly, the $\langle
p_{\rm{T}}\rangle$ in $\rm Pb\\!+\\!Pb$ from $\rm{EPOS}$ does not describe the
data at the most central pseudorapidity and consequently produces two values
of the entropy density for one $\langle p_{\rm{T}}\rangle$ value, which is not
a physical result.
Results for other thermodynamical variables require a more detailed analysis
of the relation between $\langle p_{\rm{T}}\rangle$ and temperature and more
precision on the radii measurements to look for a quantitative possible QCD
phase transition. It is important to remember that, the results on the energy
and entropy densities are calculated using simple models of Landau and
Bjorken, respectively, based on hydrodynamical approaches. Consequently, our
conclusions can thus be considered as qualitative results that could be
improved using more realistic models in the calculations, hadronic matter in
the proton, and models for the core created in $\rm p\\!+\\!p$ collisions.

## Acknowledgements

Partial support was received by DGAPA-PAPIIT IG100219, IG100322 and CONACyT
A1-S-16215 and A1-S-13525 projects. M.R.C. thankfully acknowledge computer
resources, technical advise and support provided by Laboratorio Nacional de
Super cómputo del Sureste de México (LNS), a member of the CONACYT national
laboratories, with project No. 53/2017. The authors thank to L. Díaz by the
technical support, and A. Ayala for helpful discussions and comments.

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