# Viscosity of non equilibrium hot $\&$ dense QCD drop formed at LHC

J. R. Alvarado García  0000-0002-5038-1337<EMAIL_ADDRESS>I.
Bautista  0000-0001-5873-3088<EMAIL_ADDRESS>A. Fernández Téllez
0000-0003-0152-4220 Facultad de Ciencias Físico Matemáticas, Benemérita
Universidad Autónoma de Puebla, 1152, Puebla 72570, México  P. Fierro
0000-0002-3206-301X Instituto de Física Luis Rivera Terrazas, Benemérita
Universidad Autónoma de Puebla, (IFUAP), Apartado postal J-48 72570 Puebla.
Pue., México

###### Abstract

We compute the bulk, $\zeta$, and shear, $\eta$, viscosity over entropy
density, $s$, for the QCD matter formed in small collision systems at LHC. We
consider a scenario of the String Percolation Model by proposing a global form
of the color reduction factor that describes both the thermodynamic limit and
its maximum deviation due to small-bounded effects. Our method involves
estimations at vanishing baryon-chemical potential, assuming local equilibrium
for string clusters in the initial state. To compute $\eta/s$, we employed a
kinetic approach that accounts QCD states as an ideal gas of partons, while
$\zeta/s$ is computed by using two different approaches: a simple kinetic
formula and the causal dissipative relativistic fluid dynamics formulation.
Our results align with Lattice QCD computations and Bayesian methods and are
consistent with holographic conjecture bounds. Furthermore, our findings
support the notion of a strongly interacting medium, similar to that observed
in nuclear collisions, albeit with a phase transition occurring outside the
thermodynamic limit.

††preprint: APS/123-QED

##  I Introduction

The String Percolation Model (SPM) has described successfully the collective
effects on medium formed at heavy ion collisions from the Relativistic Heavy
Ion Collider (RHIC) to the Large Hadron Collider’s (LHC) energies [1, 2, 3, 4,
5, 6, 7, 8, 9]. At these energy regimes non-perturbative QCD takes a major
role in describing phenomena through phenomenological models, such as the
characteristics of the QCD phase diagram, and the phase transition properties,
that can be studied in terms of the systems’ thermodynamic quantities,
transport coefficients, and bulk properties [9]. Recently, to obtain the
values of bulk and shear viscosity from nuclear collisions, relativistic
hydrodynamics have been used to calculate the temperature dependence of these
coefficients [10, 11, 12, 11, 13, 14, 15], as well as theoretical limits in
AdS/CFT correspondence and holographic link with Quark-Gluon Plasma (QGP) [16,
17].

The String Percolation Model uses the percolation theory for non-perturbative
heavy ion physics, by utilizing as main objects the effective color sources
and describing their physical properties such as color field, momentum, and
multiplicity [8, 9]. The functional form of its main parameters was deduced
from Monte Carlo simulations considering thermodynamic limit. And more
recently, the results of clustering of color sources that consider finite
size, profile distribution function, and the initial geometry effects (which
correspond to systems far from the thermodynamic limit, that from now on we
will denote as nonTL) were studied and compared with what was previously
reported for known observables, such as the thermal-like temperature extracted
from the transverse momentum spectra and the estimation of thermodynamic
quantities [18, 19]. In this work, we study from a phenomenological view, the
signatures of collective effects reported on [20, 21]. Specifically, we
studied the bulk properties like the modification of the behavior of the speed
of sound and we calculated the bulk viscosity coefficient; which are
significant for non-thermal equilibrium systems and highlights the benefits
that our approach provides. With this new perspective on the SPM, we can see
new light on how the effects of bulk properties weigh on reaching critical
temperature on nonTL systems, which is the first step for computing the bulk
properties for non-thermal equilibrium systems.

In the following section, we present the basics of the String Percolation
Model. Then, we proceed with exploring the consequences of considering the
nonTL scenarios in the SPM framework in Sec. III. In Secs. IV and V we
introduce the temperature and energy density respectively as usually reported.
Finally, in Secs. VI and VII we discuss the results on bulk and shear
viscosity to entropy density ratios of TL and nonTL scenarios in the SPM.

##  II Percolation color sources

The string percolation model uses percolation theory which is closely
associated with the study of phase transitions and transport phenomena [22,
23, 24], necessary to characterize the medium formed in ultra-relativistic
heavy ion collisions. The interaction between colliding nuclei is effectively
represented by the formation of extended color flux tubes which are stretching
among the colliding partons. We consider the projection of the color flux
tubes in the transverse plane, from now on named strings, in order to use the
2-dimensional percolation approach, which unlike in the thermodynamic
approaches is able to describe a phase transition without defining a
temperature in a thermodynamic equilibrium [8, 9].

The strings can be seen as small disks characterized by their transverse area
which is on average taken as $S_{0}=\pi r_{0}^{2}$ ($\sim 3.5$ mb from parton-
parton cross section of bilocal correlation functions [25, 26] and $r_{0}$ the
radius of a single disk).

The percolation approach is only used to characterize the string density in
the initial state. Based on the initial density this approach gives the
probability of forming a spanning color strings system which means that in a
2-dimensional percolation approach a geometrical connected system of disks
represents a connected color flux tubes state. The spanning system will then
later evolve due to the Schwinger mechanism where a temperature is defined as
the corresponding thermal $T$-slope of $p_{T}$-exponential momentum
distribution. In this sense, the temperature is associated with the final
state of experimental observations. As we will detail in Sec. IV.

For characterizing the system, an order parameter is introduced, which depends
on the area fraction occupied by a determined number of strings:

$\xi=\frac{S_{0}}{S}N_{s},$ (1)

where $N_{s}$ is the number of initial strings in an event, which for a
minimum bias distribution escalates with energy as a power law [27]:

$N_{s}=2+4\frac{S_{0}}{S}\Bigg{(}\frac{\sqrt{s}}{m_{p}}\Bigg{)}^{2\lambda},$
(2)

with $m_{p}$ the proton mass and $\lambda=0.196\pm 0.005$ a fit parameter
shown in Fig. 1. For a large number of strings in an event, it is required to
have a large number of partonic interactions which can be achieved at high
collision energies or a large number of colliding partons, like $AA$
collisions. The created disk’s distribution and overlapping (cluster
formation) marks a phase transition when the system starts to percolate for a
critical value of the string density $\xi_{c}$ [9], which depends on the
characteristics of the system [28].

Figure 1: Fit over experimental data for multiplicity measured on $pp$
collisions from 53 MeV to 13 TeV, data taken from Refs. [29, 30, 31, 32, 33,
34, 35, 36, 37].

In $pp$, collisions the areas $S$ and $S_{0}$ can be described in terms of the
radii $r_{0}\simeq 0.2385$ fm [38, 39, 40, 41] and $R_{p}\simeq 1$ fm (the
radius of a proton). However, to have a more precise description of the
overlapping area, we define it as an ellipse in terms of an effective impact
parameter $b$:

$S=\pi\left(R_{p}-\frac{b}{2}\right)\sqrt{R_{p}^{2}-\left(\frac{b}{2}\right)^{2}}.$
(3)

For greatly overlapped areas we can approximate Eq. (3) as the area of a
circle $S\simeq\pi R_{p}^{2}$.

Cluster formation implies the creation of new color sources in which color
fields are the vector sum of the overlapped areas’ color fields. Due to the
random orientation of the color fields, the mixed terms vanish. Thus the color
field intensity is proportional to the squared color charges of the original
strings $\sqrt{n}$, and, in consequence, multiplicity $\mu$ is:

$\frac{\mu}{\mu_{0}}=\frac{\langle\sqrt{n}\text{
}\rangle}{S_{0}}S=N\frac{\langle\sqrt{n}\text{ }\rangle}{\xi},$ (4)

where $\mu_{0}$ is the multiplicity of a single string [8, 9]. So, the number
of charged particles generated in the midrapidity region is directly
proportional to the initial number of strings of the system [27]:

$\mu=\mu_{0}F(\xi)N_{s},$ (5)

where $\mu_{0}\sim 0.63$ is calculated from fit over data [29, 30, 31, 32, 33,
34, 35, 36, 37] shown in Fig. 1.

From the above equation, a geometric scaling function appears, namely the
Color Reduction Factor $F(\xi)$, which emerges naturally from cluster
formation [8, 9]. This function increases with the string tension of the
cluster and the average momentum fraction of the partons $\langle
p_{T}^{2}\rangle$. In the thermodynamic limit, $F(\xi)$ depends on the string
density $\xi$ as [8, 9, 42]:

$F(\xi)=\sqrt{\frac{1-e^{-\xi}}{\xi}}.$ (6)

This geometric scaling function describes the universality of the scaling law
of the system, in correspondence to heavy nuclei.

##  III Non-thermodynamic limit color reduction factor

In the field of heavy-ion collisions, understanding the behavior of small
collision systems is of paramount importance. Traditional studies often make
predictions for these systems by obtaining $F(\xi)$ by fitting experimental
measurements and assuming TL [27, 43, 44, 45]. While useful, this approach may
not be the best method for describing systems far from reaching the TL. To
address this, we consider the impact of system size effects [46] and other
initial state conditions [18, 19, 47].

To account for maximum deviations from the TL in $F(\xi)$, we propose a
universal function for the color reduction factor that includes an additional
damping term. This function captures the TL limit and fits deviations from the
geometric scaling function as revealed in simulation results that consider
small-bounded effects. This factorization can be expressed as:

$\begin{split}F_{s}(\xi)&=m\sqrt{\frac{1-\exp(-\xi)}{\xi}}+c\sqrt{\frac{1+\exp(-\xi)}{\xi}},\end{split}$
(7)

where $m=0.7714731\pm 0.01468$ is the weight parameter of the TL contribution
to $F(\xi)$ (the typical percolation model), and $c=0.0609589\pm 0.007527$ is
the weight parameter of the deviation from the nonTL in the percolating
system. The difference $\Delta F=F(\xi)-F_{s}(\xi)$ represents the strength of
the fluctuations of the percolating object’s properties. The equation reduces
to Eq. (6) when $c=0$ (indicating no additional damping from finite size
effects) and $m=1$ (representing the TL).

This new formulation allows a larger suppression effect above critical string
density, which is significant even when $c$ is small due to the large
deviation from TL exhibited by the area covered by disks included in our fit
[19]. It also explains similar deviations reported for regions just below the
critical string density [48, 44].

With this improvement to the SPM, we can now better account for effects in a
broader range of multiplicity experimental data [49, 50, 51, 52, 36, 53, 54,
55, 56] and improve the model’s predictions for transport coefficients and
bulk properties for temperature regions below the critical temperature for
nonTL systems. We will demonstrate these improvements in subsequent sections.

We also estimated the effective region of $F_{s}(\xi)$ for systems that lie
between the TL and nonTL using a linear interpolation method. For the derived
model-dependent observables, the effective region was calculated through an
uncertainty propagation method.

##  IV Thermal Distribution

As mentioned before, the local effective thermodynamic quantities are
connected with the geometrical properties of the percolating system through
$F_{s}(\xi)$. The string density rules the cluster distribution, and in
consequence, the behavior of all thermodynamic quantities, such as temperature
which involves the Schwinger mechanism for non-massive particles, on which the
strings with higher tension $x$ will break producing $q\bar{q}$ and
$qq$-$\bar{q}\bar{q}$ pairs which later on will combine producing the final
state hadrons.

So, the transverse momentum distribution of charged particles is given by
[57]:

$\frac{dN}{dp_{T}^{2}}\sim\exp\left(-\pi\frac{p_{T}^{2}}{x^{2}}\right).$ (8)

The tension of the string $x^{2}$ fluctuates around its mean value, $\langle
x^{2}\rangle$, describing a Gaussian distribution of the fluctuations which
convolutes with (8) giving a thermal-like distribution characterized by the
mean transverse momentum of a single string $\langle
p_{T}^{2}\rangle_{0}=\langle x^{2}\rangle F_{s}(\xi)/\pi$ [38, 9]:

$\frac{dN}{dp_{T}^{2}}\sim\exp\left(-p_{T}\sqrt{\frac{2F_{s}(\xi)}{\langle
p_{T}^{2}\rangle_{0}}}\right),$ (9)

from where we estimate the temperature in the same way as in the Boltzmann
distribution:

$T(\xi)=\sqrt{\frac{\langle p^{2}_{T}\rangle_{0}}{2F_{s}(\xi)}}.$ (10)

The critical string density depends on the system’s characteristics [47, 19,
18]. We consider the critical temperature in terms of critical string density
$\xi_{c}$ = 1.128 [58] in the same way as [59, 27], where $T_{c}=T(\xi_{c})$,
so that, in TL:

$\frac{T}{T_{c}}=\sqrt{\frac{F(\xi_{c})}{F(\xi)}}=\frac{0.879947816}{\sqrt{F(\xi)}}.$
(11)

Furthermore, we also see a shift in the critical temperature, now reached at
lower $\xi$ for $F_{s}(\xi)$. We consider the critical temperature
$T_{c}=154\pm 9$ [60], and its respective deviations with $\xi$ to estimate an
effective area of the observables as a function of $T/T_{c}$.

##  V Energy density

Figure 2:  Energy density over $T^{4}$ behavior with respect to $T/T_{c}$. The
cyan area shows the effective region estimated with $\Delta F$. The red dashed
line shows the energy density computed in nonTL. While the black dotted-dashed
line is the TL. [9]. We include a comparison with Lattice QCD computations p4
(maroon triangles) and asqtad (green triangles) actions with their respective
parametrization (continuum lines) [61].

In the Stephan-Boltzmann approximation, quarks and gluons are assumed to be
noninteracting and massless [62]. So that the energy density $\varepsilon$ is
an order parameter in the phase transition from the Hadron Gas (HG) to QGP,
revealing an increment in the internal degrees of freedom.

Moreover, the string density $\xi$ is the local order parameter in the SPM
that marks the geometric phase transition [9]. In references [9, 63, 64],
energy density from the Bjorken boost invariant 1D hydrodynamics formula [65]
is found to be proportional to $\xi$. Given that each initial state string can
be interpreted as the extended fields among the interacting partons, which has
a direct contribution to energy density. This key idea holds for small
collision systems [27, 43]. Consequently, we use the following relation to
estimate energy density:

$\varepsilon=\varsigma\xi,$ (12)

where we found in the same way as [63] that
$\varsigma=\varepsilon_{c}/\xi_{c}=0.5601$ $\text{GeV/fm}^{3}$, and see a
shift in the critical temperature $T_{c}(F_{s})/T_{c}(F)=1.096$ with respect
to the TL scenario.

As we can see in Fig. 2, the behavior of the energy density over $T^{4}$ as a
function of $T/T_{c}$ agrees with Lattice’s calculations using staggered
fermion actions p4 and asqtad [61]. The observed increment of energy density
is related to a rise in the number of degrees of freedom from the Hadron-Gas
phase where there are fewer than in the QGP phase, and the quantum color
numbers contribute to the energy density [66].

##  VI Shear viscosity

The observable behavior of the elliptic flow suggests that matter created in
$AA$ collisions behaves as a near-perfect fluid with a very low viscosity over
entropy density ratio [67, 68, 69, 70, 71, 72, 73, 74, 75]. It was proposed
the indirect measurement of the shear viscosity over entropy density as a
probe of the viscosity of the medium created in the collision. More recently,
this probe has shown signs of a strongly interacting medium in small collision
systems as well [76, 77, 78, 79, 80].

Assuming a simple kinetic model of an ideal gas of partons, it is possible to
estimate the transport coefficients not in thermodynamic equilibrium,
considering that the medium expands as a function of the initial state
properties, as was initially proposed in [81].

Figure 3:  The (upper figure) mean free path and (lower figure) ratio of shear
viscosity over entropy density as a function of $T/T_{c}$ both calculated on
the SPM framework. The red dashed line corresponds to the nonTL computation,
while the black dotted-dashed line is the TL one as reported in [27], the pink
area is the estimated effective region, and the magenta in $\eta/s$ is the
extended effective region considering different $L$-size of sources. The
results from the Bayesian Method [15] are shown in the blue region. The dark
magenta squares correspond to SU(3) gauge calculations [82] while the green
and red solid lines correspond to QPM fits [83]. The limit of AdS/CFT [16] is
included in the black continuum line.

For computing the ratio of shear viscosity over entropy density $\eta/s$ in
terms of the SPM parameters, we considered the relation given by the
relativistic kinetic theory [84], which was also previously used for small
collision systems [27, 43]:

$\frac{\eta}{s}=\frac{T\lambda}{5},$ (13)

where $\lambda=1/(n\sigma_{tr})$ is the mean free path, with $n$ the number
density of an ideal gas of partons and $\sigma_{tr}$ the transport cross
section of its constituents. The number density is directly extracted from the
initial number of strings damped by $F_{s}(\xi)$, which gives the collective
medium effects:

$n=\frac{N_{s}F_{s}(\xi)}{SL},$ (14)

where $L\sim$ 1 fm represents the size that a string takes when extended in
the beam axis direction. It is worth mentioning that this length has been
taken as a first approximation, given that the calculations suggest that this
value is in between $0.37$ fm and $1.2$ fm [85]. On the other hand, we are
considering that this formulation takes into account properties of the initial
state, which is characterized by its corresponding 2-dimensional percolating
system, with $N_{s}$ and $S$ as defined in Sec. II. In the same way, the
transport cross section is given by $\sigma_{tr}=S_{0}F_{s}(\xi)$, the
transverse size of a single string multiplied by $F_{s}(\xi)$ [81]. This leads
us to estimate the mean free path using the definition of the string density
from Eq. (1):

$\lambda=\frac{L}{\xi F_{s}^{2}},$ (15)

where $\xi F_{s}^{2}$ is the area covered by color sources, which is
$1-e^{-\xi}$ in the case of TL [19]. Although the kinetic theory is applicable
when the system is in equilibrium, in this case, we consider that a cluster is
locally in equilibrium in the initial state for the appropriate validity of
Eq. (15). Moreover, the behavior of $\lambda/L$ is obtained for the complete
system. The upper Fig. 3 shows the $\lambda/L$ as a function of temperature.
For the TL case, $\lambda$ decreases its value around 1 after the critical
temperature [81]. This scenario replicates a situation where particle movement
is constrained by the effects of a strongly interacting medium, which aligns
with what was observed in $AA$ collisions [67, 68, 69, 70]. Contrary to $pp$
collisions, where the medium does not achieve thermalization and its effects
are less pronounced. Nevertheless, a reduction is still observed after the
critical temperature region in the nonTL case.

The behavior of $\lambda$ is inherited in the $\eta/s$ ratio, where a
significant difference between the medium formed in heavy ion collisions and
that from small collision systems is shown. The lower Fig. 3 shows a decrease
in the $\eta/s$ ratio, which leads to an enhancement of collectivity effects.
The results show that these effects are smaller in $pp$ collisions compared to
those estimated for $AA$ collisions. The results on $\eta/s$ show that its
minimum value for TL associated with $F(\xi)$ is reached at $T/T_{c}=1.13187$,
while the one associated with $F_{s}(\xi)$ for nonTL is reached at
$T/T_{c}=1.22508$; the results show an increase in the minimum value of
$\eta/s$ by a factor of 1.4218 from $0.190018$ for TL to $0.270179$ for nonTL
as shown in Fig. 3. These results are compared with the fits for quasiparticle
excitations with medium-dependent self-energies (QPM) [83] and the Lattice
calculation in SU(3) gauge theory [82]. The results from the Bayesian Method
applied to heavy ion collisions [15] are in between our estimation and, for
$T>T_{c}$, the conjectured limit of AdS/CFT, $\eta/s\geq 1/(4\pi)$ [16].

Figure 4:  The behavior of the trace anomaly with respect to $T/T_{c}$ in TL
(black dotted-dashed line) and nonTL (red dashed line) limits and the
effective region (pink area) in the SPM framework compare with the results
from Lattice QCD p4 (maroon triangles), asqtad (green triangles) [61], HISQ
(magenta circles) [86] actions and the Wuppertal-Budapest Collaboration (W-B)
results (blue squares) [87].

Trace anomaly $\Delta$ measures the deviation with respect to the conformal
behavior and identifies residual interactions in the medium formed [88, 89,
90]. It is expected that this observable is related to the medium’s viscosity
properties. In previous works, it has been observed qualitatively that the
trace anomaly can be approximated as the inverse of shear viscosity over
entropy density [91, 59]:

$\Delta\equiv\frac{\varepsilon-3P}{T^{4}}\simeq\frac{s}{\eta_{s}}.$ (16)

Trace anomaly as well as the viscosity coefficients are susceptible to QGP
phase transition [92]. The behavior of the trace anomaly for nonTL goes
accordingly to that reported by Wuppertal-Budapest Collaboration (W-B) using
the Symanzik improved gauge and a stout-link improved staggered fermion action
[87]. And with the continuum extrapolated results from the HotQCD
Collaboration of highly improved staggered quark action [86]. All of these
results show a maximum value, for the SPM this maximum is located at
$T/T_{c}=1.13621$ in TL and at $T/T_{c}=1.21918$ for nonTL (Fig. 4).

Pressure $P$ is obtained from Eq. (16) and from the first law of
thermodynamics ($Ts=\varepsilon+P$ [65]); we calculated the entropy density
$s$ of the system. The results of $3P/T^{4}$ and $s/T^{3}$ are respectively
shown in Fig. 5 compared with LQCD [61]. The SPM results agree with those of
LQCD. Pressure begins saturation over 3$T_{c}$. For this reason, we can see
the less pronounced slope in the decreasing region of trace anomaly after Fig.
4 maximal point.

Figure 5:  Behavior of the (upper figure) pressure over $T^{4}$ and (lower
figure) entropy density over $T^{3}$ against $T/T_{c}$, for TL (black dotted-
dashed lines) and nonTL (red dashed lines) cases, we show the effective
estimate region in the blue area. In both figures, the comparison with Lattice
QCD results p4 (maroon triangles) and asqtad (green triangles) actions with
their respective parametrization (continuum lines) [61] are included.

##  VII Bulk viscosity

The effects of bulk viscosity are known to be very small, which in most of the
high energy collisions were neglected due to the thermalization of the system.
However, great attempts have been made to obtain their value from nuclear
collisions from RHIC to LHC [10, 11].

Bulk viscosity corresponds to the resistance to the expansion of the fluid.
The radial components seem damped due to the non-zero effect of bulk viscous
pressure affecting the energy density profile of the created medium and
converting it into pressure gradients changing the speed of sound $c_{s}^{2}$
[64]. This effect is related to the small perturbations produced in the medium
formed [64], such as vibrations and rotations of the medium components. In the
SPM framework, these effects correspond to the fluctuations of string
properties (color field, string tension, etc). To determine the bulk viscosity
we calculate the speed of sound, which is given by a thermodynamic relation
[65]:

$c_{s}^{2}=\left(\frac{\partial
P}{\partial\varepsilon}\right)_{s}=s\left(\frac{\partial
T}{\partial\varepsilon}\right)_{s}=-\frac{sT}{2\varsigma
F_{s}}\cdot\frac{dF_{s}}{d\xi}.$ (17)

From Eqs. (6) and (7) it is simple to obtain $dF_{s}/d\xi$ of Eq. (17). In
Fig. 6 we compute the effective region of $c_{s}^{2}$. $F_{s}(\xi)$ gives a
different behavior from what was previously reported in [9, 64, 5, 6], and
shows deviations from the results reported for elliptical geometry [19] that
are all below our parametrization.

Figure 6: Dependence of speed of sound squared with $T/T_{c}$ calculated in
the SPM framework using Eq. (17), the effective estimate region (golden area),
the nonTL (red dashed line) and TL (black dotted-dashed line) limits are
shown. The Lattice QCD results from W-B (blue squares)[87], the magenta
circles are the HIQS action extrapolated results from the HotQCD Collaboration
[86] p4 (maroon triangles) and asqtad (green triangle) [61] actions from
HotQCD collaboration are compared.

Specifically, we observed a large deviation from TL in the region below the
critical temperature, showing a “dip-and-bump” effect, this behavior is in
agreement with other phenomenological models [93], and goes accordingly with
those reported from the Lattice QCD 2+1 flavor staggered fermion actions p4
and asqtad from [61], the stout-link improved staggered fermion action from
Wuppertal-Budapest Collaboration [87] and the highly improved staggered quark
action from HotQCD Collaboration [86].

The first approximation of bulk viscosity over entropy density $\zeta/s$ of
the simplest kinetic model in classical statistics with relaxation time
approximation is given by [94]:

$\frac{\zeta}{s}=15\frac{\eta}{s}\left(\frac{1}{3}-c_{s}^{2}\right)^{2},$ (18)

which describes the bulk viscosity coefficient in terms of shear viscosity and
speed of sound calculated in the SPM framework. The result of this approach
using $F(\xi)$ and $F_{s}(\xi)$ is shown in dashed lines of Fig. 7. This
approximation exhibits a monotonically decreasing behavior.

As a second approach, we use the results reported and verified in [95, 96, 97,
98, 99, 100] of the projection operator approach to obtain the microscopic
formulas for the transport coefficients in causal dissipative relativistic
fluid dynamics (CDRF), in terms of the SPM observables $T$, $s$, $\Delta$,
$c_{s}$. The reported microscopic formula of the bulk viscosity $\zeta$ with
its respective relaxation time $\tau_{\Pi}$ of CDRF is given by [101, 95, 96,
97, 98, 99, 100]:

$\frac{\zeta}{s}=\left(\frac{1}{3}-c_{s}^{2}\right)\tau_{\Pi}T-\frac{2\tau_{\Pi}T^{4}}{9s}\Delta,$
(19)

where $\tau_{\Pi}$ is considered of the order of a fermi and the fraction
$2/9$ has to do with the number of fermionic degrees of freedom.

Figure 7: Bulk viscosity over entropy density as a function of $T/T_{C}$ from
the kinetic model Eq. (18) (dotted lines), the CDRF formalism Eq. (19) (dashed
lines), and the holographic limit using Eq. (20) (continuum lines) calculated
on the SPM framework are shown, distinguish TL in gray scale lines and nonTL
in red lines. The Bayesian Method results [15] (blue region), the
parametrization of Hadron Gas to QGP (continuum green line), and the SU(3)
gauge theory calculations [102] are included.

The results of Eq. (19) in TL and nonTL are shown in Fig. 7 labeled as CDRF.
In which there are differences in their behaviors and shifts in their
vanishing points are shown.

On the other hand, in [17] is conjectured a lower limit on bulk viscosity of
strongly coupled gauge theory plasmas as:

$\frac{\zeta}{s}\geq
2\left(\frac{1}{3}-c_{s}^{2}\right)\frac{\eta}{s}\geq\frac{1}{2\pi}\left(\frac{1}{3}-c_{s}^{2}\right),$
(20)

considering $\eta/s\geq 1/(4\pi)$ [16]. Which is shown in solid black (TL) and
red (nonTL) lines in Fig. 7.

Fig. 7 includes the results of $\zeta/s$ reported from viscous relativistic
hydrodynamics Bayesian Method [15], that is near the holographic limit [17]
and CDRF calculations in the SPM framework.

We compare our results of $\zeta/s$ with the hadron resonance gas model, which
incorporates all identified particles and resonances with masses below 2 GeV,
along with a density of Hagedorn states that increases exponentially for
masses exceeding 2 GeV [103], and the continuum parametrization of the LQCD
equation of state results of $\zeta/s$ of hot quark-gluon matter in the
presence of light quarks [104], as presented in [105, 106]. These models show
a similar behavior to the kinetic model considered in the SPM ($T>T_{c}$
region). Also, the Lattice gluodynamics calculation in SU(3) gauge theory
[102] shows a scaled same dependence.

Table 1: Results of the maximum value of bulk viscosity over entropy density for different approaches in TL and in our parametrization (nonTL). | TL | nonTL
---|---|---
|  $\zeta/s$ (Max) | $T/T_{c}$ |  $\zeta/s$ (Max) | $T/T_{c}$
Kinetic | 35132.6 | 0.873553 | 1.84112 | 0.146509
CDRF | 0.152694 | 0.873553 | 0.111427 | 0.940129
Bound | 0.0530515 | 0.873553 | 0.0390168 | 0.903653
Figure 8: The results of the bulk-shear viscosity ratio against $T/T_{c}$ from
the kinetic theory (dotted lines), CDRF (dashed lines), and the gauge theory
plasma at strong coupling limit (continuum lines) [17] calculated in the SPM
framework. Distinguish TL in gray lines and nonTL in red lines.

In Table 1, we present the maximum values of $\zeta/s$ for the kinetic theory,
CDRF formalism, and conjectured bound with its respective $T/T_{c}$ value for
TL and nonTL. We can observe that in all cases $\zeta/s$ reaches its maximum
value below the critical temperature. And the TL goes much higher than nonTL,
reaching its maximum value. For the CDRF formalism, it is reached just
$73.54\%$ of the TL. For the lower conjectured bound it is $72.97\%$ and for
the case of the simplest kinetic model, we see a discrepancy in the values
around $0.0052\%$, because the value of $c_{s}^{2}$ for TL vanishes at
$T/T_{c}=0.873553$, Eq. (18) gives the same tendency as shear viscosity over
entropy density in TL.

In Fig. 8 we show the interplay between shear and bulk viscosity given by the
ratio $\zeta/\eta$ computed in the SPM framework, where we can observe a
maximum value around the critical temperature region in all cases. For the
case of the Kinetic Model, the $\zeta/\eta$ ratio shows its maximum value in
$T<T_{c}$ for TL and nonTL scenarios, and for the CDRF formalism we can see a
change in the slope in the region just below $T=T_{c}$ for the TL case, and
for nonTL right before $T=T_{c}$.

##  VIII Conclusions

We have computed the $\eta/s$ and $\zeta/s$ ratios by proposing a global
parametrization for the SPM color reduction factor, that considers the small-
bounded effects in the geometrical phase transition at the non-thermodynamic
limit for $\mu_{B}=0$. Our description highlights the differences in the
physics behind both TL and nonTL cases discussed in the framework of the SPM
for the QCD matter formed in $pp$ and $AA$ collisions at LHC energies.

The ratio $\eta/s$ is estimated in a simplified kinetic formulation of an
ideal gas of partons. The results on this coefficient show different minimum
values in TL and nonTL, within the region $T_{c}<T<1.23T_{c}$. The first
implication related to the nonTL case reveals that a phase transition must
occur at higher temperatures since a shift in the inflection point is found.
Additionally, the medium effects that constrains particle movement are less
pronounced on this limit, as can be seen in Fig. 3.

The behavior of the speed of sound shows that it does not vanish at low
temperatures (Fig. 6). This is important since $c_{s}^{2}$ acquires relevance
in the computation of $\zeta/s$, making its contribution non-neglectable.

For the estimation of $\zeta/s$, we used two formalisms and compared them to a
conjectured strong coupling bound. By using the CDRF on the SPM, we found that
$\zeta/s$ vanishes at $T/T_{c}=1.17$ for TL and at $T/T_{c}=1.32$ for nonTL,
as shown in Fig. 7. On the other hand, the implementation of the simplest
kinetic model guides $\zeta/s$ to reach higher values before critical
temperature, as summarized in Table 1. Both formalisms are above the strong
coupling bound for $T/T_{c}<1.1$, where the maximum values are reached. Hence,
the contribution of $\zeta$ becomes relevant in this region, implying that
there is a strong effect driven by the fluctuations of the string properties,
such as the color field, as well as the string tension. In addition, the ratio
$\zeta/\eta$ for nonTL scenarios shows a shift of the maximum point, which is
reached at higher temperatures. This implies that nonTL requires a higher
temperature in order to reach phase transition due to the fluctuations coming
from the bulk contributions (Fig. 8).

It is important to acknowledge certain limitations in our estimations of
transport coefficients. First, our approach assumes that a cluster is locally
in equilibrium in the initial state, which is crucial for the validity of Eq.
(15). However, deviations from local equilibrium, especially in the early
stages of high-energy collisions, could impact the accuracy of our
estimations.

Our results show a clear difference in the behavior of the observables that
take into account the finite size effects with respect to those that are
predicted in the thermodynamic limit, showing that the fluctuations of the
initial state have a qualitative relevance in the estimation of the
coefficients that characterize the medium formed in small collisions systems.

While our study provides valuable insights into the behavior of transport
coefficients in small collision systems within the framework of the String
Percolation Model, one should remain mindful of these limitations when
interpreting and applying our results.

## Acknowledgements

We thank CONAHCyT-México for supporting this work under the projects
CF-2019/2042, and A1-S-26507. On the other hand, J. R. A. G. and P. F. thank
CONAHCyT-México for the graduate fellowships 645654 and 848955 respectively.

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