# String percolation in AA and pp collisions

I. Bautista  Facultad de Ciencias Físico Matemáticas, Benemérita Universidad
Autónoma de Puebla, 1152, México.  Cátedra CONACyT, CONACyT, 03940 Ciudad de
México, México  C. Pajares Corresponding Author<EMAIL_ADDRESS>Departamento de Física de Partículas, Universidad de Santiago de Compostela,
E-15782 Santiago de Compostela, España  Instituto Galego de Física de Altas
Enerxías, Universidad de Santiago de Compostela, E-15782 Santiago de
Compostela, España  J. E. Ramírez  Departamento de Física de Partículas,
Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, España

###### Abstract

A brief review of the string percolation model and its results are presented
together with the comparison to experimental data. First, it is done an
introduction to the quark-gluon phase diagram and the lattice results
concerning the confinement and the percolation of center domains. It is
studied the interaction of the strings produced in nucleus-nucleus and proton-
proton collisions showing how the string percolation arises. The main
consequences of the string percolation, concerning the dependence on the
energy and centrality, on the multiplicities and the mean transverse momentum
are obtained comparing with experimental data. It is emphasized the non-
abelian character of the color field of the strings forming the cluster to
reproduce the rise of the transverse momentum with multiplicity and the
relative suppression of multiplicities. It is also studied different
observables like multiplicity and transverse momentum distributions,
dependence with multiplicity and transverse momentum correlations, forward-
backward correlations, the strength of the Bose-Einstein correlations,
dependence on the multiplicity of $J/\Psi$ production and its possible
suppression in pp collisions at high multiplicity, strangeness enhancement,
elliptic flow, and ridge structure. The comparison with the data shows an
overall agreement. The thermodynamical properties of the extended cluster
formed in the collision are discussed computing its energy and entropy
density, shear viscosity over entropy density ratio, bulk viscosity, sound
speed and trace anomaly as a function of temperature, showing a remarkable
agreement with lattice QCD evaluations. The string percolation can be regarded
as the initial frame able to describe the collective behavior produced in AA
and pp collisions.

###### Contents

  1. 1 Introduction
  2. 2 Percolation model
  3. 3 String percolation
    1. 3.1 String models
    2. 3.2 String fusion and percolation
    3. 3.3 Quenching of the low $p_{T}$ partons
    4. 3.4 Multiplicity distributions
    5. 3.5 Multiplicity and transverse momentum distributions
    6. 3.6 Transverse momentum fluctuations
    7. 3.7 Forward-backward correlations
    8. 3.8 Underlying event of high $p_{T}$ particles and KNO scaling
    9. 3.9 Bose-Einstein correlations
    10. 3.10 $J/\Psi$ production dependence on the multiplicity
    11. 3.11 Incoherent $J/\Psi$ photoproduction
    12. 3.12 Strangeness enhancement
  4. 4 Azimuthal dependence of the momentum distributions
    1. 4.1 Collective flow and ridge structure
    2. 4.2 Elliptic flow scaling and energy loss
  5. 5 Thermodynamics of string percolation
  6. 6 Summary

##  1 Introduction

More than four decades ago, it was raised the possibility of distributing high
energy over a large volume to restore broken symmetries of the physical vacuum
creating abnormal states of nuclear matter [1]. Very early, it was pointed out
that the asymptotic freedom property of QCD implies the existence of a high-
density matter formed by deconfined quarks and gluons [2] and the exponential
increase of the hadron Hagedorn spectrum was connected with the existence of a
different phase [3]. The thermalized phase of quarks and gluons was called
Quark Gluon Plasma (QGP) [4], and the evaluations of the required high density
showed that could be reached in relativistic heavy ion collisions [5, 6] and,
several signatures of QGP were proposed. Quarkonium suppression [7], the
excess of photons and jet quenching [8, 9] were some of them. At this time it
was pointed out the relevance of percolation in the study of the phase
transitions of hadronic matter [10, 11].

From the experimental side, there were large facilities to study the
properties of large density matter starting by the AGS and ISR experiments
later followed by SPS, RHIC, and LHC. At SPS already several signatures hinted
the onset of QGP formation [12]. The RHIC data show a collective elliptic flow
which pointed out a very low shear viscosity over entropy density ratio,
$\eta/s$, indicating strongly interacting matter. In addition, the jet
quenching was observed, indicating that this strongly interacting matter was
very opaque [13, 14, 15, 16, 17]. The above mention ratio enhanced the
attention to the AdS/CFT correspondence due to the result $\eta/s=1/4\pi$
[18]. The LHC experiments [19, 20, 21] have extended the study of the elliptic
flow to all the harmonics [22, 23] confirming the obtained strong interacting
quark and gluon matter and showing that the collective behavior and the ridge
structure previously observed at RHIC in Au-Au and Cu-Cu collisions [24, 25]
also occurs in pPb [26, 27, 28] and pp collisions at high multiplicity [29].
The collective behavior of pp and pPb interactions is a challenge to the
hydrodynamics descriptions, and they raise the question whether the main
experimental data can be explained by final state interactions or on the
contrary, the initial state configuration should describe them.

On the other hand, the data on quarkonium confirm the validity of combined
picture of a subsequent melting of the different resonances together the
recombination of heavy quarks and antiquarks at high energy [30, 31, 32]. The
departure of linear dependence on the multiplicity of the $J/\Psi$ production
has been observed in pp and pPb collisions [33, 34] indicating multiparton
interactions or multiplicity saturation [35]. Detailed studies on the jet
quenching for identified particles have been done [36] showing features
related to the low of coherence of the gluons edited in the jet due to the
high-density medium. Finally, let us mention that at RHIC has been observed
recently that the fluid produced by heavy ions is the most vortical system
ever observed [37].

On the theoretical side, in addition to the hydrodynamics studies, the color
glass condensate (CGC) approach [38, 39, 40, 41, 42] gives a good description
of most of the experimental data and is derived directly from QCD. In QCD, the
gluon density $xG(x,Q)$ rises very fast as a function of the decreasing
fractional momentum $x$ or increasing the resolution $Q$. So the gluons
showers generate more gluon showers producing an exponential increasing toward
small $x$. As the transverse size of the hadron or the nucleus rises slowly at
high energy, the number of gluons and their density per unit of area and
rapidity increase rapidly as $x$ decreases. However, there will be the fusion
of gluons leading to a limited transverse density of gluons at some fixed
momentum resolution, $Q_{s}$, the gluon saturation [43]. The low $x$ gluons
are closely packed, the distance between them being very small. Hence the
interaction coupling is small $a_{s}\ll 1$. In a given collision, the
multiplicity should be proportional to the number of gluons, which at the
saturation momentum $Q_{s}$ is [44, 45]

$\frac{dN}{dy}\sim\frac{1}{\alpha_{s}(Q_{s})}Q_{s}^{2}R^{2}.$ (1)

This dense system, called CGC, has a very high occupation number
$1/\alpha_{s}$, and corresponds to a highly coherent state of strong color
fields. The high $x$ gluons can be considered as the sources of the low $x$
gluons. The independence of the cutoff used to separate the high $x$ gluons
from the low $x$ ones, gives rise to a kind of evolution equation.

In high energy physics experiments, the colliding objects move at velocities
close to the speed of light. Due to the Lorentz contraction, the collision of
two nuclei can be seen as a that of two sheet of colored glass where the color
field in each point of the sheets is randomly directed. Taking these field as
initial conditions, one finds that between the sheets, longitudinal color
electrical and magnetic fields are formed. The number of these color flux
tubes between the two colliding nuclei is forming the called Glasma [46],
which has been extensively compared with the experimental data.

Another approach to the initial state is the percolation of strings [47, 48,
49, 50, 51] which is not so popular as the CGC because cannot be derived
directly from QCD although it is inspired in it, and most of its results, are
a direct consequence of properties of QCD. In this approach, the multi-
particle production is described in terms of older strings stretched between
the partons of the projectile and target. These string decay into $q-\bar{q}$
pairs and subsequently hadronize producing the observed hadrons. Due to the
confinement, the color of these strings is confined to a small area $S=\pi
r_{0}^{2}$, with $r_{0}=0.2$ fm in the transverse space. The value 0.2-0.25 fm
is obtained in lattice studies [52] and also considering bilocal correlations
[53]. It corresponds to the correlation length of the QCD vacuum. With
increasing energy and/or size and centrality of the colliding objects, the
number of strings grows and the strings start to overlap forming clusters
similarly than the continuum percolation theory [54]. At a given critical
density, a macroscopical cluster appears crossing the collision surface, which
marks the percolation phase transition. Therefore the nature of this
transition is geometrical.

In string percolation, the basic ingredients are the strings, and it is
necessary to know their number, rapidity extension, fragmentation and number
distribution. All that requires a model and therefore string percolation is
model dependent. However most of the QCD inspired models give similar results
for most of the observables in such a way that the predictions are by a large
measure independent of the model used.

Figure 1: Phase diagram of the nuclear matter. Temperature vs baryonic
chemical potential [163].

The string percolation and the Glasma are related to each other [55]: In the
limit of high density, there is a correspondence between the physical
quantities of both approaches. The number of color flux tubes in Glasma
picture, $Q_{s}^{2}R^{2}$, has the same dependence on the energy and
centrality of the collisions that the number of effective clusters of strings
in string percolation. In both approaches, the negative binomial distribution
is obtained as the multiplicity distribution, where the parameter $k$ that
controls the width of this distribution has the same energy and centrality
dependence. The role of the occupation number $1/\alpha_{s}$ in CGC is played
by the fraction of the collision surface covered by strings. The randomness of
the color field in CGC gives rise to a reduction of the multiplicity.
Similarly, the randomness in color space of the color field of the $n$ strings
of the cluster originates that the intensity of the color field of the cluster
is not $n$ times the individual color field of each string but $\sqrt{n}$.
This reduction implies also a reduction of the multiplicity of particle
production and an increase of the transverse momentum, with the multiplicity.
Due to these similarities, the predictions of both approaches are similar for
many observables. The string percolation is able to explore also the region
where the high-density limit has not been reached.

The observed densities of our world have large differences which expand over
many orders of magnitude, from $10^{-6}\textrm{ nucleons}/\textrm{cm}^{3}$ in
average in the Universe to $10^{38}\textrm{ nucleons}/\textrm{cm}^{3}$ inside
a nucleus and $10^{39}\textrm{ nucleons}/\textrm{cm}^{3}$ in a neutron star.
The study of the high-density limit, i. e., the study of de-confinement of
quarks and gluons can be regarded as the place where high energy collision of
two bodies probes the short distances and meets the thermodynamics (many body)
of this short distance limit [56]. The lattices studies have shown that at low
chemical potential $\mu=0$, color confinement and chiral symmetry restoration
coincide and the phase transition is a crossover. Hence, in a medium of low
baryon density, the mass of the constituent quark vanishes at the deconfined
point $T_{c}$, and the screening radius of the gluon cloud vanishes. At low
$T$ and high $\mu$ there is no reason to expect similar behavior, and probably
there will be an intermediate region of massive dressed quarks between the
hadronic phase and the deconfined and chiral restoration phase. However, other
possibilities could exist as quarkonia and color superconductivity. A possible
diagram is shown in Fig. 1.

In finite $T$ lattice QCD, the de-confinement order parameter is provided by
the vacuum expectation value of the Polyakov loop $L(\vec{x})$ defined in
Euclidean space

$L(\vec{x})=\mathrm{Tr}\prod_{t=1}^{N_{t}}A_{4}(\vec{x},t).$ (2)

Note that $L(\vec{x})$ is the ordered product of the SU(3) temporal gauge
variables $A_{4}(\vec{x},t)$ at a fixed spatial position, where $N_{t}$ is the
number of lattice points in time direction and Tr denotes the trace over color
indices. The Polyakov loop corresponds to a static quark source and its vacuum
expectation value is related to the free energy $F_{q}$ for a single quark

$L(\vec{x})\sim\exp\left(-\frac{F_{q}}{T}\right).$ (3)

Below the critical temperature $T_{c}$ quarks are confined and $F_{q}$ is
infinite implying $\langle L(\vec{x})\rangle=0$. In a de-confined medium color
screening among the gluons makes $F_{q}$ finite, hence for $T>T_{c}$, $\langle
L(\vec{x})\rangle\neq 0$. The phase transition of chiral symmetry is
controlled by the chiral condensate

$\sigma(T)=\langle\bar{\psi}\psi\rangle\sim M_{q},$ (4)

which measures the constituent quark masses obtained from a Lagrangian with
massless quarks. At high temperature this mass melts, therefore

$\sigma(T)\left\\{\begin{array}[]{lcl}\neq 0&\mathrm{if}&T<T_{\sigma},\\\
=0&\mathrm{if}&T>T_{\sigma}.\end{array}\right.$ (5)

Here, $T_{\sigma}$ defines another critical temperature. The corresponding
derivatives, the susceptibilities, have been studies in lattice QCD at
vanishing baryon number, showing a sharp peak that defines respectively
$T_{c}$ and $T_{\sigma}$. The two temperatures, within errors, coincide. Also,
is seen a crossover, i. e., a sharp transition but without discontinuity. The
quoted value is 155$\pm$9 MeV [56, 57, 58].

Figure 2: The energy density and the pressure as the function of temperature
(left). The energy density shows a sharp rise in the temperature region
170-200 MeV. The interaction measure calculated using different staggered
fermion actions (right) [58].

The energy densities resulting from lattice QCD are shown in Fig. 2 (left),
indicating that even for $T>3T_{c}$ its value are far from the energy density
of free gas quarks and gluons, namely

$\epsilon=\frac{\pi^{2}}{30}\left[g_{g}+\frac{7}{8}(g_{q}+g_{\bar{q}})\right]T^{4},$
(6)

where $g_{g}$, $g_{q}$ and $g_{\bar{q}}$ are the degeneracy numbers of the
gluons, quarks and antiquarks. This fact indicates that the deconfined phase
is interacting strongly for a rather large range of temperatures. This is also
seen in the interaction measure

$\Delta=\frac{\epsilon-3P}{T^{4}}.$ (7)

Moreover, the trace of the energy momentum tensor

$T^{\mu}_{\mu}=\frac{\beta(g_{s})}{2g_{s}}{G^{a}}_{\mu\nu}G^{a\mu\nu}+[1+\gamma(g_{s})]m_{q}\bar{\psi}\psi$
(8)

is $\epsilon-3P$ and even for masses quarks $T^{\mu}_{\mu}\neq 0$ as a
consequence of the introduction of a scale in the renormalization process
bearing the conformal symmetry (trace anomaly). In Fig. 2 (right) the results
of the lattice QCD are shown. Note that $\Delta$ decreases with $T$ very
slowly, even less than $1/T^{2}$.

##  2 Percolation model

Figure 3: Left panel: Disconnected discs, Middle: Cluster formation, Right
panel: Over- lapping discs forming a spanning cluster [56].

Let us distribute small discs of area $\pi r_{0}^{2}$ randomly on a large
surface, allowing overlap between them. As the number of discs increases,
clusters of overlapping discs start to be formed. If we regard the disc as
small drops of water, how many drops are needed to form a puddle crossing the
considered surface? Given $N$ disc, the disc density is $\xi=N/S$, where S is
the surface area. The average cluster size increases with $\xi$, and at a
certain critical value $\xi_{c}$, the cluster spans the whole surface, as is
shown in Fig. 3.

The critical density for the onset of continuum percolation is determined by
numerical and Monte-Carlo simulations, which in the 2-dimensional case gives

$\xi_{c}=\frac{1.13}{\pi r_{0}^{2}}.$ (9)

In the thermodynamical limit, $N\to\infty$ keeping $\xi$ fixed, the
distribution of overlaps of the disc is Poissonian with a mean value
$\rho=\xi\pi r_{0}^{2}$ [59, 60]:

$P_{n}=\frac{\rho^{n}}{n!}\exp(-\rho).$ (10)

It also gives the total fraction of the plane covered by discs in
$1-\exp(-\rho)$ [50]. The number 1.13 is obtained in case of discs uniformly
distributed [51, 61, 62]. However, in cases when the discs are not uniformly
distributed, this number changes. For instance, in the cases of circular
surfaces with Gaussian or Wood-Saxon profiles, the number is 1.5 and the
fraction of the area covered by strings is more close to the function

$\frac{1}{1+a\exp(-(\rho-\rho_{c})/b)},$ (11)

where $\rho_{c}=1.5$ and the parameters a and b depend on the profile
function, $b$ that controls the ratio between the width of the border of the
profile $(2\pi R)$ and the total area $(\pi R2)$, and therefore is
proportional to $1/R$ [63]. In the collisions of two hadrons or two nuclei,
the surface where the discs are distributed is rather an ellipse or a circle,
what gives rise to smaller values of the critical density [64]. For small
systems where the number of discs is not large (far from the thermodynamical
limit) the critical density is much smaller than above values, being 0.8 for
high eccentricities [64].

Figure 4: Histograms fro the distributions of the phase $\phi(x)$ of the local
loops $\langle L(\vec{x})\rangle$. Left side show the distribution below
$T_{c}$ and right side show the de-confined phase [65, 164].

In SU(3) gauge theory, spatial clusters can be identified as those where the
local Polyakov loops $\langle L(\vec{x})\rangle$ have values close to some
element of the center. The elements of the center group $Z_{3}$, are a set of
three phases $(0,2\pi/3,-2\pi/3)$ [65]. Below $T_{c}$ ($\langle
L(\vec{x})\rangle=0$), the values of $\langle L(\vec{x})\rangle$ are grouped
around there three phases, show three pronounced peaks located at the center
phases. Above $T_{c}$ ($\langle L(\vec{x})\rangle\neq 0$), the distribution
changes: One of the peaks grows and the other two shrink. A spontaneous
breaking symmetry occurs, which leads to a non- vanishing $\langle
L(\vec{x})\rangle$, as is shown in Fig. 4. Spatial clusters can be defined
grouping the sites with a very similar value of $\langle L(\vec{x})\rangle$.
The weight of the largest cluster increases sharply at $T=T_{c}$, indicating
that the cluster percolates. Therefore in the pure SU(3) theory the de-
confinement transition is a percolation phase transition (of second order).

In high energy collisions, we expect that color strings are formed between the
projectile and target partons. These color field must have a small transverse
size due to confinement. In this way, the strings, in the transverse plane,
are small discs in the surface of the collisions. As the number of strings
grows with energy and centrality degree of the collision, the strings start to
overlap forming clusters which eventually percolate. The phenomenological
consequences in relation to SPS, RHIC, and LHC, pp, pA and AA data are the
main subject of this brief review. A more extended version can be found in
Ref. [66].

##  3 String percolation

###  3.1 String models

The basic ingredient of the string percolation are the strings. Although there
are differences, most of them coincide in basic postulates as the number of
strings and its dependence on energy and centrality, which is taken from the
Glauber-Gribov Model. We will concentrate in models with color exchange
between projectile and target as the Dual Parton Model (DPM) [67, 68, 69],
Quark Gluon String Model (QGSM) [70], Venus and EPOS [71]. They are based on
the $1/N_{c}$ QCD expansion and its terms are in correspondence with the ones
of the Gribov-Reggeon calculus. They have been extensively compared to the
experimental data ISR, SPS, and Fermilab obtaining an overall agreement [67].
In DPM or QGS the multiplicity distribution dN/dy of pp collisions is given by
fragmentation of 2k strings

$\frac{dN^{pp}}{dy}=\frac{1}{\sigma}\sum\sigma_{k}[N_{k}^{qq-q}(s,y)+N_{k}^{q-qq}(s,y)+(2k-2)N_{k}^{q-\bar{q}}],$
(12)

where $N_{k}^{qq-q}$ and $N_{k}^{q-qq}$ are the inclusive spectra of hadrons
produced in the strings stretched between a valence diquark of the
projectile(target) and a quark of the target(projectile) and
$N_{k}^{q-\bar{q}}$ are the inclusive spectra of the strings stretched between
sea quarks and antiquarks. A schematic representation of this process is shown
in Fig. 5 (a). In this figure are shown four chains, two between quark-diquark
and two corresponding to quark-antiquark. Each chain corresponds to the
fragmentation into $q-\bar{q}$ pairs of one string, stretched between the
quark (diquark) of the projectile and the diquark (quark) of the target or
from the quark and diquark from the sea. The leading term corresponds to two
strings stretched between a valence quark (diquark) and a valence diquark
(quark). Via unitarity the modulus square of the leading diagram corresponds
to the pomeron as is shown in Fig. 5 (b). In this figure is shown a net of
gluons which are accompanying to the quarks, which can be seen in this picture
as constituent quarks. In Fig. 5 (b) is also shown that the pomeron
corresponds to a cylinder topology.

Figure 5: (a)Two cut Pomeron diagram (four chain) for proton-proton
collisions. (b): Single Pomeron exchange and its underlying cylindrical
topology. This is a dominant contribution to proton-proton elastic scattering
at high energies [67].

The single particle of each string can be obtained by folding the momentum
distribution of the partons at the end of the string with the fragmentation
function of the string

$N_{k}^{qq-q}(s,y)=\int_{0}^{1}dx_{1}\int_{0}^{1}dx_{2}\rho_{k}(x_{1})\rho_{k}(x_{2})\frac{dN^{qq-q}}{dy}(y-\bar{\Delta},s_{s}),$
(13)

where $\sqrt{s_{s}}$ is the invariant mass of the string $s_{s}=sx_{1}x_{2}$
and $x_{1}$ and $x_{2}$ are the light cone momentum fractions of the partons
of the end of the string. $\bar{\Delta}$ is the rapidity shift necessary to go
from the pp center of mass to the center of mass of one string

$\bar{\Delta}=\frac{1}{2}\log\left(\frac{x1}{x2}\right).$ (14)

The momentum distribution used for the valence quarks, valence diquarks, sea
quarks and antiquarks are $x^{-1/2}$, $x^{-1}$ and $x^{-3/2}$, respectively.
In general the distribution of $2k$ partons in the proton is

$\rho_{k}(x_{1},x_{2k},x_{2},x_{3},x_{3},\dots,x_{2k-1})=C_{k}^{\rho}x_{1}^{-1/2}x_{2}^{-1}\dots
x_{2k-1}^{-1}x_{2k}^{1/2}x_{2k}^{1/2}\delta\left(1-\sum_{i=1}^{2k}x_{i}\right),$
(15)

where $C_{k}^{\rho}$ is obtained by normalizing $\rho_{k}$ to unity. Due to
these distributions the $N_{k}^{qq-q}$ and $N_{k}^{q-qq}$ are long in rapidity
extension and centered at a point shifted with respect to the center of mass
and the $N_{k}^{q-\bar{q}}$ string are short and centered at the center of
mass. For the fragmentation functions different ways are used, in string
percolation the strings fragment according the Schwinger mechanism, such as in
the Lund string. In Eq. (12) $\sigma_{k}$ is the cross section for producing
$2k$ strings resulting from cutting $k$ pomerons. As the pomeron has the
topology of the cylinder its cutting give rise to two strings (See Fig. 5
(b)). Using the AGK cutting rules [72], the cross section is calculated as
follows

$\sigma_{k}=\frac{8\pi g\exp(\Delta
y)}{kz}\left[1-\exp(-z)\sum_{l=0}^{k-1}\frac{z^{l}}{l!}\right],$ (16)

where

$z=\frac{2gC\exp(\Delta y)}{R^{2}+\alpha^{\prime}y},$ (17)

and $g$ is the coupling of the pomeron to the proton, $\alpha^{\prime}$ and
$1+\Delta$ are the slope and the intercept of the pomeron trajectory,
respectively, and $C$ is a parameter describing the inelastic diffractive
states. Summing over $k$, we obtain the total cross section

$\sigma_{tot}=\exp(\Delta
y)\sum_{k=0}^{\infty}\sum_{l=k,l>0}^{\infty}\left(-\frac{z}{2}\right)^{l-1}\frac{8\pi
g}{l!}\left[\delta_{k0}+(-1)^{1-k}2^{l-1}\binom{l}{k}\right].$ (18)

The rise of $dN/dy$ with energy is due mainly to the short strings, whose
number grows with energy. On the other hand, outside the central rapidity
region, there is not contribution of short strings and the rise with energy is
much slower giving rise to the approximate limiting behavior. Assuming a
Poisson distribution for cutting $k$ pomerons

$P_{k}(n)=\frac{(kN)^{n}}{n!}\exp(-kN),$ (19)

where $N$ is the mean multiplicity production when cutting one pomeron,
therefore the multiplicity distribution is

$P_{k}(n)=\sum_{k}\omega_{k}P_{k}(n),$ (20)

where $\omega_{k}=\sigma_{k}/\sigma$. Usually, $\langle n\rangle P(n)$ is
plotted as a function of $n\langle n\rangle$. When the result is independent
of energy, one has the known Kobe-Nielsen-Olsen scaling (KNO), which is
violated at SPS Fermilab, RHIC and LHC. The reason for that in DPM is due to
the contribution of the short strings that they contribute mostly at high
multiplicities pushing upwards the high multiplicity tail of the
distributions. The increase with $s$ of the short strings contributions is due
to the increase of the invariant mass of the short strings, formed between
quarks and antiquarks of the sea, and to the $s$-dependence of the weights.
DPM can be generalized to hA and AA collisions in the following way [73]:
Consider a collision with $N_{A}$ participants nucleons of A, $N_{B}$
participant nucleons of B and a total number of $N_{c}$ collisions. In this
configuration are produced $2N_{c}$ strings, of these $2N_{A}$ are stretched
between valence quarks and valence diquarks ($q_{v}^{A}-qq_{v}^{B}$) and
($qq_{v}^{A}-q_{v}^{B}$). The remaining $N_{B}-N_{A}$ valence quarks and
diquarks of B have no valence partner of A and have to form $2N_{B}-2N_{A}$
strings with sea quarks and antiquarks of A ($q_{s}^{A}-qq_{v}^{B}$) and
($\bar{q}_{s}^{A}-q_{v}^{B}$). The remaining $2N_{c}-2N_{B}$ strings are
formed between sea quarks and antiquarks of A and B ($q_{s}-\bar{q}_{s}$)

$\displaystyle\frac{dN^{AB}}{dy}$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma_{AB}}\sum_{N_{A},N_{B},N_{c}}\sigma^{AB}_{N_{A},N_{B},N_{c}}\theta(N_{B}-N_{A})\Big{[}N_{A}\left(N^{qq_{v}^{A}-q_{v}^{B}}(y)+N^{q_{v}^{A}-qq_{v}^{B}}(y)\right)+$
(21)
$\displaystyle(N_{B}-N_{A})\left(N^{\bar{q}_{s}^{A}-q_{s}^{B}}(y)+N^{q_{s}^{A}-qq_{v}^{B}}(y)\right)+$
$\displaystyle(N_{c}-N_{B})\left(N^{q_{s}^{A}-\bar{q}_{s}^{B}}(y)+N^{q_{s}^{A}-q_{s}^{B}}(y)\right)\Big{]}+\mathrm{sym}(N_{A}\leftrightarrow
N_{B}),$

where $\sigma^{AB}_{N_{A},N_{B},N_{c}}$ is the cross section for $N_{c}$
inelastic nucleon-nucleon collisions involving $N_{A}$ and $N_{B}$ nucleons of
A and B, respectively. This cross section have been studied extensively [74,
75]. The inclusive spectra, as in the pp case, are given by a convolution of
momentum distribution and fragmentation functions. In the case of A=B, we have
approximately

$\displaystyle\frac{dN^{AA}}{dy}$ $\displaystyle\approx$ $\displaystyle\langle
N_{A}\rangle(2N^{qq-q_{v}}(y)+(2\langle k\rangle-2)N^{q_{s}-\bar{q}_{s}}(y))+$
(22) $\displaystyle(\langle N_{c}\rangle-\langle N_{A}\rangle)2\langle
k\rangle N^{q_{s}-\bar{q}_{s}}(y),$

where we have introduced the possibility of having $k$ multiple scattering in
the individual nucleon-nucleon collisions, which was neglected in Eq. (21).
Notice, that there is not any reason to assume that the term proportional to
$N_{c}$ is due to hard collisions. There are many soft collisions included in
this term. In the central rapidity region we have $2Nk$ strings which for
heavy nuclei collisions and high energy is very large number, even larger than
1500. Due to that, we expect interactions between them and they will not
fragment in an independent way.

In the case of pA collisions, the Eqs. (21) and (22) transform into

$\displaystyle\frac{dN^{pA}}{dy}$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma_{pA}}\sum_{N_{A}}\sigma_{N_{A}}^{pA}\Big{[}\left(N^{qq^{p}-q^{A}_{v}}(y)+N^{q^{p}_{v}-qq^{A}}(y)\right)$
(23)
$\displaystyle+(N_{A}-1)\left(N^{\bar{q}_{s}^{p}-q^{A}_{s}}(y)+N^{q_{s}^{p}-qq^{A}}(y)\right)\Big{]},$

where $N_{A}$ matches with the number of collisions.

###  3.2 String fusion and percolation

Figure 6: Projections of two overlapping strings onto the transverse plane.

As we have said before, at large energy we expect that the strings overlap in
the transverse plane. The transverse space occupied by a cluster of
overlapping strings splits into a number of areas with different number of
strings overlap, including areas where no overlapping takes place. In each
area color field coming from the overlapping strings add together. As a
result, the cluster is split in domains with different color strength. One may
assume that emission of $q\bar{q}$ pairs in the domains proceeds
independently, governed by the strength of the color field (string tension) of
the corresponding domain. Evidently, these new formed strings domains have not
only different color strength. Other assumption that one may do is that
emission of $q\bar{q}$ pairs in the domains proceeds independently, governed
by the strength of the color field (string tension) of the corresponding
domain. Evidently, these new formed string domains have not only different
color field but also different transverse area. As an example, let us consider
a cluster of two partially overlapping string as it is shown in Fig. 6. In
this scenario, a simple string with transverse area $S_{1}$ emits partons with
transverse momentum distribution

$\frac{d\sigma}{dyd^{2}p}=C\exp\left(-\frac{m_{T}^{2}(p_{T})}{t_{1}}\right),$
(24)

where $t_{1}$ is the tension and $m_{T}^{2}=m+p_{T}^{2}$, being $p_{T}$ and
$m$ the transverse momentum and the mass of the emitted parton. The tension,
according to the Schwinger mechanism, is proportional to the field and thus to
the color charge of the ends of the string [76, 77, 78, 79, 80], which we
denote by $Q_{0}$. The mean transverse momentum squared is $\langle
p_{T}^{2}\rangle_{1}=t_{1}$ and proportional to $Q_{0}$. We denote the mean
multiplicity of produced particles by string per unit of rapidity as $\mu_{1}$
which is also proportional to the color charge. In Fig. 6, we have the
overlapping of two strings which partially overlap in the area $S^{(2)}$
(region 2 in the figure), so that $S^{(1)}=S_{1}-S^{(2)}$ is the area in each
string not overlapping with the other, where $S_{1}$ is the transverse area of
a single string. The color density of a simple string is $q=Q_{0}/S_{1}$.
Then, the color in each of the non overlapping areas will be

$Q_{1}=qS^{(1)}=Q_{0}S^{(1)}/S_{1},$ (25)

and in the overlapping area each string will have color

$\bar{Q}_{2}=qS^{(2)}=Q_{0}S^{(2)}/S_{1},$ (26)

The total color in the overlap area will be a vector sum of the two
overlapping colors $qS_{2}$. In this summation the total color charge should
be conserved [50, 51]. Thus
$Q_{2}^{2}=(\vec{Q}_{ov}+\vec{Q}^{\prime}_{ov})^{2}$, where $\vec{Q}_{ov}$and
$\vec{Q}^{\prime}_{ov}$ are the two vector colors in the overlap area. Since
the colors in the two strings may generally be oriented in arbitrary
directions respective to one another, the average of
$\vec{Q}_{ov}\vec{Q}^{\prime}_{ov}$ is zero, then
$Q_{2}^{2}=\vec{Q}^{2}_{ov}+\vec{Q}^{\prime 2}_{ov}$, which leads to

$Q_{2}=\sqrt{2}qS^{(2)}=\sqrt{2}Q_{0}S^{(2)}/S_{1}.$ (27)

Notice that due to the vector nature, the color in the overlap is less than
the sum of the two overlapping colors. This effect has important consequences
concerning the saturation of multiplicities and the rise of the mean
transverse momentum with multiplicity which we will study in the next section.
Thus, assuming independent emission from the three regions of Fig. 6, we
obtain for the multiplicity weighted by the multiplicity for a single string
($\mu_{1}$)

$\mu/\mu_{1}=2(S^{(1)}/S_{1})+\sqrt{2}(S^{(2)}/S_{1}),$ (28)

and for the mean transverse momentum squares (we divide the total transverse
momentum squared by the multiplicity)

$\frac{\langle p_{T}^{2}\rangle}{\langle
p_{T}^{2}\rangle_{1}}=\frac{2(S^{(1)}/S_{1})+\sqrt{2}\sqrt{2}(S^{(2)}/S_{1})}{2(S^{(1)}/S_{1})+\sqrt{2}(S^{(2)}/S_{1})}=\frac{2}{2(S^{(1)}/S_{1})+\sqrt{2}(S^{(2)}/S_{1})},$
(29)

where we have used the property $S^{(1)}+S^{(2)}=S_{1}$. Generalizing to any
number $N$ of overlapping strings, we have

$\displaystyle\frac{\mu}{\mu_{1}}$ $\displaystyle=$
$\displaystyle\sum_{i}\sqrt{n_{i}}(S^{(i)}/S_{1}),$ (30)
$\displaystyle\frac{\langle p_{T}^{2}\rangle}{\langle p_{T}^{2}\rangle_{1}}$
$\displaystyle=$
$\displaystyle\frac{\sum_{i}(S^{(i)}/S_{1})}{\sum_{i}\sqrt{n_{i}}(S^{(i)}/S_{1})}=\frac{N}{\sum_{i}\sqrt{n_{i}}(S^{(i)}/S_{1})},$
(31)

where the sum runs over all individual overlaps of $n_{i}$ strings having
areas $S^{(i)}$. We have used the identity $\sum_{i}S^{(i)}=NS_{1}$. These
equations are not easy to apply because we have to identify all individuals
overlaps of any number of strings with their areas. However one can avoid
these difficulties realizing that one can combines all terms with a given
number of overlapping strings $n_{i}=n$ into a single term, which sums all
such overlaps into a total area of exactly $n$ overlapping strings
$S_{n}^{Tot}$. Then, one can write

$\displaystyle\frac{\mu}{\mu_{1}}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\sqrt{n}(S_{n}^{Tot}/S_{1}),$ (32)
$\displaystyle\frac{\langle p_{T}^{2}\rangle}{\langle p_{T}^{2}\rangle_{1}}$
$\displaystyle=$ $\displaystyle\frac{N}{\sum_{n=1}^{N}(S_{n}^{Tot}/S_{1})}.$
(33)

The total area can be easily computed in the thermodynamic limit. One finds
that the distribution of overlap strings over the total surface $S$ in the
variable $n$ is Poissonian with mean $\rho=NS_{1}/S$, which corresponds to the
filling factor in the percolation context. Therefore, the fraction of the
total area covered by strings will be $1-\exp(-\rho)$. Note that the
multiplicity in Eq. (31) is damped by a factor

$F(\rho)=\frac{\mu}{N\mu_{1}}=\frac{\sqrt{n}}{\rho}=\sqrt{\frac{1-\exp(-\rho)}{\rho}}.$
(34)

Finally, we can write for the mean values

$\displaystyle\mu$ $\displaystyle=$ $\displaystyle NF(\rho)\mu_{1},$ (35)
$\displaystyle\langle p_{T}^{2}\rangle$ $\displaystyle=$ $\displaystyle\langle
p_{T}^{2}\rangle_{1}/F(\rho).$ (36)

In the rest of this review, these last equations will be used extensively.

###  3.3 Quenching of the low $p_{T}$ partons

The $p_{T}$ distribution of the partons from the decay of a cluster of strings
is given by Eq. (24), where the tension is now scaled by $1/F(\rho)$, then it
is computed as $t=t_{1}/F(\rho)$. The parton will be emitted in different
azimuthal direction and have to travel paths of different longitudes before
they of out and are observed. Parton going through the overlap meet stronger
field than those going only through the field of a simple string. In this way
the partons loose their energy passing through the field and the observed
distribution will depend on their azimuthal angle even if initially they were
emitted isotropically. Radiative energy loss has been extensively studied in
QCD for a parton passing through a quark gluon plasma medium [81, 82]. In our
case the situation is different and is more similar to a charge particle
moving in an external electromagnetic field. The corresponding force causes a
loss of energy, which is given by [83]

$\frac{dp(x)}{dx}=-0.12e^{2}(eEp(x))^{2/3},$ (37)

where $E$ is the external electric field. This equation leads to the quenching
formula

$p_{0}(p)=p\left(1+\gamma p^{-1/3}t_{1}^{2/3}\right)^{3},$ (38)

where $eE/\pi=t_{1}$ can be identified as the longitude of the path travelled
by the parton. The quenching coefficient is known in QED (is very small) but
in our case has to be adjusted to the experimental data and it turns out to be
very small of the order of $10^{-2}$. Retaining the first term of the equation
we obtain

$p_{0}-p=3\gamma p^{2/3}t_{1}^{2/3},$ (39)

which will be used to compute the harmonic of the $p_{T}$ distribution. Notice
that due to the smallness of the quenching parameter the effect of the
quenching is only felt in the azimuthal distribution, not in the $p_{T}$
distribution, once the azimuthal angel is integrated. The use of QED formulas
for the QCD case may raise some doubts. However, it has been shown that at
least in $N=4$ SUSY Yang Mills case, the expression is essentially the same
[84].

###  3.4 Multiplicity distributions

The multiplicity distributions in the DPM of QGSM in pp and AA collisions are
given by Eqs. (12)-(13) and Eqs. (21)-(22), respectively. However, as the
energy or centrality of the collision increases one expects interaction among
strings. As discussed before, due to the randomness of the color field in
color space non-abelian field) the resulting color field in a cluster of $n$
overlapping strings is only $\sqrt{n}$ times the strength of the color field
of a single string, giving rise to a suppression of the multiplicity of
particles produced by the cluster. The same reason lies at the origin of the
enhancement of the mean $p_{T}$. According to Eq. (34), the multiplicity
distribution in pp collisions in the central rapidity region is given by

$\frac{dN^{pp}}{dy}=F(\rho_{p})N^{s}_{p}\mu_{1},$ (40)

where $N^{s}_{p}$ is the number of string in the central rapidity region. In
AA collisions the number of string stretched between the sea quark and
antiquarks in the central rapidity region is proportional to
$N_{A}^{4/3}-N_{A}$, which is the total number of nucleon-nucleon collisions.
Hence, the total number of strings in a central heavy ion collisions is very
large. However, each string must have a minimum of energy to be produced and
decay subsequently into particles. On the other hand, the total energy
available. In the collision grows as $A$, whereas the number of strings as
$A^{4/3}$ in the central collisions. Therefore at not very high energy (for
instance RHIC energies), the energy is not sufficient to produce such huge
number of strings. In order to take into account this energy conservation
effect one may reduce the number of sea quark and antiquarks changing [85]

$N^{4/3}_{A}\to N^{1+\alpha(\sqrt{s})}_{A},$ (41)

where

$\alpha(\sqrt{s})=\frac{1}{3}\left(1-\frac{1}{1+\ln(\sqrt{s/s_{0}}+1)}\right).$
(42)

Here, the parameter $s_{0}$ marks the energy squared above which energy
conservation effects become small and $\alpha\to 3$. One thus can write

$\frac{dN^{AA}}{dy}\sim N_{A}(N^{\alpha(\sqrt{s})}_{A}-1)\frac{dN^{pp}}{dy}.$
(43)

Taking into account the interaction of strings we can write a closed formula
for the multiplicity distribution in AA in terms of the multiplicity
distribution of pp, namely [85]

$\left.\frac{1}{N_{A}}\frac{dN}{dy}\right|_{y=0}=\left.\frac{dN^{pp}}{dy}\right|_{y=0}\left(1+\frac{F(\rho_{N_{A}})}{F(\rho_{p})}(N^{\alpha(\sqrt{s})}_{A}-1)\right),$
(44)

where

$\displaystyle\rho_{N_{A}}$ $\displaystyle=$
$\displaystyle\rho_{p}N^{\alpha(\sqrt{s})+1}_{A}\frac{S_{1}}{S_{N_{A}}},$ (45)
$\displaystyle\rho_{p}$ $\displaystyle=$ $\displaystyle
N_{p}^{s}\frac{S_{1}}{S_{p}},$ (46)

and $S_{N_{A}}$ is the transverse area of the collision formed when there are
$N_{A}$ participant nucleons of the projectile and $N_{A}$ participant
nucleons of the target. Note that $S_{N_{A}}$ depends on $N_{A}$ and A.

Figure 7: Comparison of the evolution of the mid-rapidity multiplicity with
energy from the CSPM and data for pp and A-A collisions. Lines are from the
model for pp (gray), Cu-Cu (blue) and red lines for Au-Au/Pb-Pb [85, 86, 87,
88, 89].

Moreover, the dependence of the multiplicity on $\sqrt{s}$ is full specified,
once the average number of strings $N$ in a pp collision is known. At low
energies it is 2, growing as

$N_{p}^{s}=2+4\left(\frac{r_{0}}{R_{p}}\right)^{2}\left(\frac{\sqrt{s}}{m_{p}}\right)^{2\lambda}.$
(47)

Notice that here a single parameter $\lambda$ describes the rise of the
multiplicity with energy for both pp and AA, even though in central AA
collision the multiplicity increase faster than in pp collisions due to the
energy dependent factor $\alpha$, arising from energy conservation. A fit to
pp collisions data in the range $53<\sqrt{s}<7000$ GeV and to AA collisions
(Au-Au, Cu-Cu and Pb-Pb) at different centralities for $19.6<\sqrt{s}<2760$
GeV has been done. The values obtained for the two parameters are
$\sqrt{s_{0}}=245$GeV and $\lambda=0.201$. Figure 7 shows a comparison of the
energy dependence results with data for pp [85, 86, 87] and central Cu-Cu [88]
and for Au-Au and Pb-Pb [89]. The results of the dependence of the the
multiplicity per participant nucleon on the number of participants is shown in
Fig. 8 together with the experimental data Cu-Cu, Au-Au and Pb-Pb at different
energies.

Figure 8: Multiplicity dependence on centrality ($N_{part}$). Cu-Cu
(triangles), Au-Au (stars) and Pb-Pb (circles).Curves represent the model
calculations. Blue line for Cu-Cu, green line for Au-Au and red for Pb-Pb.
Figure 9: Comparison of results from the evolution of $d_{n_{ch}}/d\eta$ with
dependence on pseudorapidity for pp collisions at different energies (lines).

The evolution outside the central rapidity region has been studied extensively
extending Eq. (43) to all rapidities [90, 91, 92, 93, 94, 95]. The limiting
fragmentation property is not satisfied exactly. In Fig. 9, we show the
results together with the experimental data for pp collisions at all
rapidities at different energies [94, 95] and in Fig. 10 the results for Cu-
Cu, Au-Au [96] and Pb-Pb [97] together the experimental data. In Fig. 11, we
compare the results [95, 96, 97, 98] for d-Au collisions together the
experimental data. A good description of all experimental data is obtained.

Figure 10: Comparison of results from the evolution of $dn/d\eta$ with
dependence on pseudorapidity for (a).Cu-Cu at 200 GeV, (b) Au-Au 200 GeV. Plot
(c) shows $\frac{dn_{ch}}{\eta}\frac{1}{N_{part}/2}$ for Pb-Pb collisions at
2.76 TeV. Figure 11: Comparison of results from the evolution of $dn/d\eta$
with dependence on pseudorapidity at different centralities for d+Au
collisions at 200 GeV.

The behavior of $dN/dy$ in pp and AA collisions is very similar to the Glasma
picture of CGC expressed in Eq. (1). In fact as the saturation momentum
squared $Q_{s}^{2}$ behaves like $N_{A}^{1/3}$, and $R_{A}^{2}$ as
$N_{A}^{2/3}$, the multiplicity per participant is almost independent of
$N_{A}$ and only a weak dependence arises from the logarithmic dependence of
the running coupling constant $\alpha_{s}(Q_{s})\sim 1/\log(N_{A})$. In
percolation the multiplicity per participant is also almost independent of
$N_{A}$ and the only dependence arises from the factor $1-\exp(-\rho)$, which
also grow weakly with $N_{A}$. Concerning the energy dependence, this is given
by $Q_{s}^{2}$, which behaves like $s^{\lambda}$, so that, the same behavior
than in percolation. There is an extra energy dependence in CGC due to
$1/\alpha_{s}$, which corresponds in percolation again to the factor
$1-\exp(-\rho)$. Since both are measure of the fraction occupied area by color
fields, it is not surprising the correspondence between $1/\alpha_{s}$, the
occupying number of gluons, and $1-\exp(-\rho_{N_{A}})$.

###  3.5 Multiplicity and transverse momentum distributions

Let us start considering a set of overlapping strings which depend both on the
number of strings and the overlapping area, which combine to give an average
multiplicity $N$. We may characterize the different overlaps just by the
average multiplicity that combines both the number of collisions and the area.
With a lot of overlapping strings $N$ will change practically continuously. We
can introduce a probability $W(N)$ to have overlaps with size $N$ in a
collision and write the total multiplicity distribution as

$P(n)=\int dNW(N)P(N,n),$ (48)

where P(N,n) is the multiplicity distribution of the overlap of a given $N$,
which we take Poissonian with the average multiplicity $N$

$P(N,n)=\frac{N^{n}}{n!}\exp(-N).$ (49)

The normalization conditions $\sum_{n}P(n)=1$, and $\sum_{n}nP(n)=N$ lead to
the following relations

$\displaystyle\int dNW(N)=1,$ (50a) $\displaystyle\langle n\rangle=\langle
N\rangle=\int dNNW(N).$ (50b)

For the weight function we assume the gamma distribution [99, 100, 101]

$W(N)=G(N,k_{N},\tau_{N})=\frac{\tau_{N}}{\Gamma(k_{N})}(\tau_{N}N)^{k_{N}-1}\exp(-\tau_{N}N),$
(51)

where $\tau_{N}=k_{N}/\langle N\rangle$. There are several reasons for this
choice. The growth of the centrality can be seen as a transformation of the
cluster size distribution. Start with a set of single strings with a few
clusters formed of a few overlapping strings. As the centrality increases,
there appear more strings and more clusters composed of more strings. This
change can be considered as substitution of strings in a cluster by the new
formed clusters, defined by a new $N$ corresponding to a higher color field in
the cluster. This transformation, similar to the block transformation of
Wilson type, can be seen as a transformation of the cluster size probability
of the type

$P(x)\to x\frac{P(x)}{\langle x\rangle}\to\dots\to x^{k}\frac{P(x)}{\langle
x^{k}\rangle}\dots$ (52)

Transformations of this type were studied long time ago by Jona Lasinio in
connection with the renormalization group in probabilistic theory [102],
showing that the only probability distribution function $P(x)$ which is stable
under such transformation are the generalized gamma functions, among them the
simplest one is the gamma function which has one parameter less. We point out
that transformation of type (52) have been used previously, to study the
probability associated with some special event which are shadowed by
themselves and not for the total of events [103, 104, 105, 106, 107]. We will
come back to this point, studying the underlying events when one high $p_{T}$
particle is triggered. Notice that $W(N)$ satisfy KNO scaling, namely the
product $NW(N)$ is only a function of $N/\langle N\rangle$, if the parameter
$k$ of Eq. (52) is energy independent. This property is a consequence of the
invariance of the gamma functions under the transformation of type Eq. (51)
[107]. Let us discuss now the transverse momentum distribution (TDM)
$f(p_{T})$. As in the case of multiplicity distribution we consider a cluster
which decay in the same way that a single string with the only difference of
its “size” $x$

$f(p_{T},x)=\exp(-xp_{T}^{2}).$ (53)

Actually, $x$ denotes the inverse of the color field in the cluster which
depends not only on the size but also on the degree of overlapping strings
inside the cluster. Assuming that $x$ varies continuously one can write the
total TMD, similarly to the multiplicity distribution case as

$f(p_{T})=\int dxW_{p}(x)f(x,p_{T}).$ (54)

We must realize the normalization condition

$\int dp_{T}^{2}f(p_{T})=\langle n\rangle,$ (55)

which gives the relation

$\langle n\rangle=\int dxx\alpha^{2}W(\alpha x).$ (56)

Comparing the latter with Eq. (50b), we can make the identification
$W_{p}(x)=(\alpha x)^{2}W(\alpha x)$ If we take the gamma distribution in Eq.
(51) for $W(N)$, then $W_{p}(x)$ turns out up to a factor to be also the gamma
distribution with different $k$ and $r$ given by

$W_{p}(x)=\frac{r}{r_{p}}G(x,k+2,r_{p}),$ (57)

with $r_{p}=\alpha r$. So, at the end, both the multiplicity distribution and
the TMD are given by a convolution of the cluster distribution and its TMD
with the size probability $W(x)$ which in both cases can be taken the gamma
distributions although with different parameters. Introducing Eq. (51) into
Eqs. (48) and (54), we obtain

$\frac{1}{(1+p_{T}^{2}/r)^{k}}=\int_{0}^{\infty}dx\exp(-p_{T}^{2}x)\left(\frac{r}{\Gamma(k)}\right)(rx)^{k-1}\exp(-rx),$
(58)

and

$\frac{\Gamma(n+k^{\prime})}{\Gamma(n+1)\Gamma(k^{\prime})}\frac{r^{k^{\prime}}}{(1+r^{\prime})^{k^{\prime}}(n+k^{\prime})^{n}}=\int_{0}^{\infty}dN\frac{e^{N}N^{n}}{n!}\frac{r^{\prime}}{\Gamma(k^{\prime})(r^{\prime}N)^{k^{\prime}-1}}\exp(-r^{\prime}N).$
(59)

The mean value and the dispersion of the distributions (58) and (59) are

$\displaystyle\langle x\rangle=\frac{k}{r},\hskip 8.5359pt\frac{\langle
x^{2}\rangle-\langle x\rangle^{2}}{\langle x\rangle^{2}}=\frac{1}{k},$ (60)
$\displaystyle\langle n\rangle=\langle
N\rangle\frac{k^{\prime}}{r^{\prime}},\hskip 8.5359pt\frac{\langle
N^{2}\rangle-\langle N\rangle^{2}}{\langle
N\rangle^{2}}=\frac{1}{k^{\prime}},$ (61) $\displaystyle\frac{\langle
n^{2}\rangle-\langle n\rangle^{2}}{\langle
n\rangle^{2}}=\frac{1}{k^{\prime}}+\frac{1}{\langle N\rangle}.$ (62)

The distribution (59) is a negative binomial distribution. Eqs. (58) and (59)
are superposition of clusters and $1/k$ and $1/k^{\prime}$ control the
transverse momentum fluctuations and the fluctuations on the number of string
in the cluster. At small density there are no strings overlapped, and $k$ and
$k^{\prime}$ go to infinity. When the density increases, the strings start to
overlap forming clusters, and therefore the $k^{\prime}$ decreases. Their
minimum is reached when the fluctuations in the number of strings per cluster
reach their maximum. Above this point, increasing the string density, these
fluctuations decrease and the $k^{\prime}$ increases. Now, if we take into
account that the mean multiplicity and transverse momentum given by Eqs. (35)
and (36), the Eqs. (58) and (59) become

$f(p_{T},y)=\frac{dN}{dp_{T}^{2}dy}=\frac{dN}{dy}\frac{k-1}{k}\frac{F(\rho)}{\langle
p_{T}^{2}\rangle_{1}}\frac{1}{\left(1+\frac{F(\rho)p_{T}^{2}}{k\langle
p_{T}^{2}\rangle_{1}}\right)^{k}},$ (63)

and

$P(n)=\frac{\Gamma(n+k^{\prime})}{\Gamma(n+1)\Gamma(k^{\prime})}\frac{\left(\frac{k^{\prime}}{\langle
n\rangle_{1}F(\rho)}\right)^{k^{\prime}}}{\left(1+\frac{k^{\prime}}{\langle
n\rangle_{1}F(\rho)}\right)^{n+k^{\prime}}}.$ (64)

We observe that

$\displaystyle\langle n\rangle=F(\rho)N_{s}\langle n\rangle,$
$\displaystyle\langle p_{T}^{2}\rangle=\frac{k}{k-2}\frac{\langle
p_{T}^{2}\rangle_{1}}{F(\rho)}.$ (65)

Eqs. (63) and (64) give the distributions for any projectile, target, energy
and degree of centrality and are universal functions which depend of only two
parameters, $\langle p_{T}^{2}\rangle_{1}$ and $\langle n\rangle_{1}$, the
average transverse momentum and multiplicity of particles produced by one
string. In case of identified secondary particles, it should be used the
corresponding quantities for each identified particles, $\langle
p_{T}^{2}\rangle_{1i}$ and $\langle n\rangle_{1i}$. Sometimes instance $p_{T}$
is used $m_{T}$. At $\rho\to\infty$ and $k\to\infty$ the TMD becomes
$\exp(-F(\rho)p_{T}^{2}/\langle p_{T}^{2}\rangle_{1})$ very similar to the
behavior at $\rho\to 0$. From Eq. (63) we have

$\frac{d\ln f}{d\ln
p_{T}}=\frac{-2F(\rho)}{\left(1+\frac{F(\rho)p_{T}^{2}}{k\langle
p_{T}^{2}\rangle_{1i}}\right)}\frac{p_{T}^{2}}{\langle
p_{T}^{2}\rangle_{1i}}.$ (66)

At $p_{T}^{2}\to 0$, the latter reduces to $-2F(\rho)p_{T}^{2}/\langle
p_{T}^{2}\rangle_{1i}$ and vanishes at $p_{T}^{2}=0$. On the other hand, as
$\langle p_{T}^{2}\rangle_{1\pi}<\langle p_{T}^{2}\rangle_{1k}<\langle
p_{T}^{2}\rangle_{1\mathrm{p}}$, the absolute value of Eq. (66) is larger for
pions than for kaons than for protons, this is the well known hierarchy that
often it is advocated in favor of a hydrodynamic picture hadronic
interactions. However, we describe very well the data as it is seen in Fig.
12, where we show our results together the PHOBOS data [103] for central Au-Au
collisions.

Figure 12: Experimental PHOBOS data on low pt distributions for pions, kaons
and protons along with our results for central Au-Au collisions at
$\sqrt{s}=$200 GeV.

Let us now discuss the interplay of low and high $p_{T}$. One defines the
ratio between central and peripheral collisions as

$R_{CP}(p_{T})=\frac{f^{\prime}(p_{T},y=0)/N^{\prime}_{coll}}{f(p_{T},y=0)/N_{coll}}.$
(67)

The normalization on the number of collisions in the latter, essentially
eliminates $N_{s}$ from $dN/dY$, this is true at mid rapidity. From Eq. (54)
and (63) we obtain

$R_{CP}(p_{T})=\frac{((k^{\prime}-1)/k^{\prime})}{((k-1)/k)}\left(\frac{F(\rho^{\prime})}{F(\rho)}\right)^{2}\frac{\left(1+\frac{F(\rho)p_{T}^{2}}{k\langle
p_{T}^{2}\rangle_{1i}}\right)^{k}}{\left(1+\frac{F(\rho^{\prime})p_{T}^{2}}{k^{\prime}\langle
p_{T}^{2}\rangle_{1i}}\right)^{k^{\prime}}}.$ (68)

Here $k$ and $k^{\prime}$ are values of the parameter $k$ for TMD for
peripheral and central collisions. In the limit $p_{T}^{2}\to 0$, as
$F(\rho^{\prime})<F(\rho)$ we have

$R_{CP}\simeq\left(\frac{F(\rho^{\prime})}{F(\rho)}\right)^{2}<1,$ (69)

which is independent of $k$ and $k^{\prime}$. As $\rho^{\prime}/\rho$
increases the ratio $R_{CP}$ decreases, in agreement with experimental data.
As $p_{T}$ increases, we have

$R_{CP}(p_{T})\sim\frac{1+\frac{F(\rho)p_{T}^{2}}{k\langle
p_{T}^{2}\rangle_{1i}}}{1+\frac{F(\rho^{\prime})p_{T}^{2}}{k^{\prime}\langle
p_{T}^{2}\rangle_{1i}}}$ (70)

and $R_{CP}$ increases. At large $p_{T}$

$R_{CP}(p_{T})\sim\frac{F(\rho)k^{\prime}}{F(\rho^{\prime})k}p_{T}^{2(k-k^{\prime})}.$
(71)

At low density in the region where decreases with the string density
$k^{\prime}<k$ and $R_{CP}(p_{T})>1$. It is the Cronin effect. As
$\rho^{\prime}/\rho$ increases the ration $R_{CP}$ increases. With the growth
of the energy of the collision, the energy density increases reaching the
region where $k$ increases. Now at $\rho^{\prime}>\rho$ and $k^{\prime}>k$,
there will be a suppression of $p_{T}$. In the forward rapidity region, the
normalization of Eq. (67), does not cancel $N_{s}$ from $dN/dy$, since in this
region $N_{s}$ is proportional to $N_{A}$ instead of $N_{coll}$. Now, an
additional factor $(N^{\prime}_{A}/N^{\prime}_{coll})/(N_{A}/N_{coll})$
appears in $R_{CP}(p_{T})$. As $N^{\prime}_{coll}-N_{A}$ for central
collisions is larger, than for peripheral collisions, we have
$R_{CP}(p_{T},y=3)<R_{CP}(p_{T},y=0)$, thus a further suppression occurs in
agreement with experimental data [108]. The results for the TMD for $\pi^{+}$,
$k^{+}$ and p in Au-Au at $\sqrt{s}=$ 200 GeV are in good agreement with the
Phenix data [109]. In Fig. 13, we show the ratios kaons/pion, and proton/pion
as a function of $p_{T}$ at the two extreme centralities. The obtained values
of $k$ as a function of the string density increase as it was expected.

Figure 13: Ratios for different distributions $k/\pi$, $\mathrm{p}/\pi$ in Au-
Au collisions at $\sqrt{s}=$200 GeV at two different centralities : 0-5$\%$
(solid circles), 60-70$\%$ (open circles) in comparison with the data.

The experimental data on pp in the range $\sqrt{s}=$23, 200, 630 GeV and 1.38,
7 TeV can also be described by the distribution of Eq. (63) [110]. In this
case the values of $k$ decreases with energy as expected. At higher energy and
high multiplicity $k$ should increase [66].

Even though the parametrization (63) describes well the data up to 5-10GeV/c,
most of the considerations concerning the string fragmentation are only valid
for low and intermediate $p_{T}$. In order to include the high $p_{T}$ part of
the spectrum more refined study is necessary.

The differences between the baryon and meson spectrum is not only due to the
mass differences, which results $\langle p_{T}^{2}\rangle_{1M}<\langle
p_{T}^{2}\rangle_{1B}$. This effect only causes a shift in the maximum of the
nuclear modified factor $R$, but keeps the height at the maximum, contrary to
the data. In the fragmentation of a cluster formed of the overlapping of
several strings the flavor properties follow from the corresponding properties
of the flavor of the valence partons of the end of the individual strings and
hence the resulting flavor of the cluster $F$ is the flavor composition of the
individual strings $f$ as well as the color composition. The clusters have
higher color and different flavor ends. The fragmentation of a cluster will be
by means of the creation of a pair $F\bar{F}$ , where $F$ and $\bar{F}$
denotes the sets of flavor quarks and antiquarks of the end of the cluster.
After the decay, the two new $F\bar{F}$ strings will be treated in the same
way decaying into more $F\bar{F}$ strings until they come to objects with mass
comparable to hadron masses, which we identified with the observable hadrons
by combining the produced flavor with statistical weights. In this way the
production of baryons and antibaryons will be enhanced with the number of
strings of the cluster. The additional quarks (antiquarks) required to form a
baryon (antibaryon) are provided by the quarks (antiquarks) of the overlapping
strings that form the cluster . In some sense the coalescence picture of
particle production is incorporated in a natural way. An effective way of
taking into account these flavor considerations can be seen in Ref. [111].
Very often, it is used an exponential instead of a gaussian for the decay of
one string. Indeed, the tension of a cluster fluctuates around its mean value
because the chromo-electric field is not constant. Such fluctuations lead to a
Gaussian distribution of the string tension [109, 110, 112, 113]

$\frac{dn}{dp_{T}}\sim\int_{0}^{\infty}dx\exp\left(-\frac{x^{2}}{2\langle
x^{2}\rangle}\right)\exp(-\lambda p_{T}^{2}/x^{2}),$ (72)

which give rise to the thermal distribution

$\frac{dn}{dp_{T}}\sim\exp\left(-p_{T}\sqrt{\frac{2\pi}{\langle
x^{2}\rangle}}\right),$ (73)

where $\langle x^{2}\rangle=\pi\langle p_{T}^{2}\rangle_{1}/F(\rho)$. The
temperature is expressed as [112, 114]

$T(\rho)=\sqrt{\frac{\langle p_{T}^{2}\rangle}{2F(\rho)}}.$ (74)

Now the total TMD is changed and instance of the gamma distribution in Eq (57)
a Tsallis type distribution is obtained, namely

$f(p_{T})=\left(1+\frac{\sqrt{f(\rho)}p_{T}}{k\langle
p_{T}\rangle_{1}}\right)^{k}.$ (75)

There are several scaling properties found in TMD related to string
percolation. The experimental data for pp collisions exhibit a universal
behavior in a suitable variable $z=p/B$ [115, 116]. Indeed, the TMD of pp and
$\mathrm{p}\bar{\mathrm{p}}$ at all energies are at the same curve as it is
shown in Fig. 14. The parameter $B$ it is found proportional to $1/F(\rho)$
and therefore increasing with energy. A similar scaling is found in Au-Au
collisions [117] at different centralities.

Figure 14: Scaling behavior of the charged hadron pt spectra presented in z
(a) in pp collisions and (b) $\mathrm{p}\bar{\mathrm{p}}$ collisions with
different energy scales. The inset is the distribution of the ratio between
the experimental data and the fitted results [116].

The experimental data on the mean $p_{T}$ as a function of the multiplicity
show that in pp, pA and AA collisions all of them grow, being larger in pp
than in pA and in AA collisions. In PbPb the rise of mean $p_{T}$ with
multiplicity is flattened above a certain low multiplicity. The same occurs at
pPb although in this case is flattened at higher multiplicity. This behavior
is understood as a consequence of Eq (63). In fact, the factor $1/F(\rho)$ is
responsible of the rise of $\langle p_{T}\rangle$ with multiplicity because
grows with multiplicity. The flattening of PbPb and pPb is due to the
dependence of $k$ on the string density. In PbPb for most of the
multiplicities the corresponding string densities are above the percolation
threshold. In this region $k$ grows with $\rho$, and according to Eq. (63),
the rise is lowered. In the case of pp collisions, on the contrary, the
corresponding string densities lie below the critical density. In this region
$k$ is a decreasing function of $\rho$ hence there is not flattening. We
expect that at higher energy, larger than 14 TeV, the critical density will be
reached, even in pp collisions.

###  3.6 Transverse momentum fluctuations

The event by event fluctuations of thermodynamical quantities as the
temperature were proposed as a probe for the deconfined phase. Due to that,
the study of the fluctuations on the mean $p_{T}$ is very interesting. These
fluctuations are measured using the observables:

$F_{p_{T}}=\frac{w_{data}-w_{random}}{w_{random}},\hskip
8.5359ptw=\frac{\sqrt{\langle p_{T}^{2}\rangle-\langle
p_{T}\rangle^{2}}}{\langle p_{T}\rangle},$ (76)

and the correlation between the transverse momemtum

$\langle\Delta p_{Ti},\Delta p_{Tj}\rangle=C_{m}\simeq 2F_{p_{T}}(\langle
p_{T}^{2}\rangle-\langle p_{T}\rangle^{2})/\langle p_{T}\rangle\langle
N\rangle,$ (77)

where $\Delta p_{Ti}=(p_{Ti}-\langle p_{Ti}\rangle)$. And

$M(p_{T})=\frac{1}{\sum_{k=1}^{n_{ev},m}N_{acc,k}}\sum_{k=1}^{n_{ev},m}\sum_{i=1}^{N_{acc},k}p_{Ti},$
(78)

and $C_{m}/M(p_{T})$. The last observable is used because suppresses the
statistical fluctuations in string percolation. The observable $F_{p_{T}}$,
using Eq. (77) is given by [118]

$F_{p_{T}}=\sqrt{\frac{N_{s}\langle
p_{T}\rangle^{2}_{1}\mu_{1}-2N_{s}^{3/4}(\frac{S_{n}}{S_{1}})^{1/4}\langle
p_{T}\rangle_{1}\langle
p_{T}\rangle\mu_{1}+N_{s}(\frac{S_{n}}{S_{1}})^{1/2}\langle
p_{T}\rangle^{2}\mu_{1}}{(N_{s}\frac{S_{1}}{S_{n}})^{1/2}\langle
p_{T}\rangle^{2}_{1}-2(N_{s}\frac{S_{1}}{S_{n}})^{1/4}\langle
p_{T}\rangle_{1}\langle p_{T}\rangle+\langle p_{T}\rangle^{2}}}-1.$ (79)

In Fig. 15, we show the result for $C_{m}$ in Pb-Pb collisions together the
ALICE data.

Figure 15: Scaling behavior of the charged hadron $p_{T}$ spectra presented in
z (a) in pp collisions and (b) p$\bar{\mathrm{p}}$ collisions with different
energy scales. The inset is the distribution of the ratio between the
experimental data and the fitted results. Figure 16: Preliminary results for
$\sqrt{C_{m}}/M(p_{T})$ as a function of the multiplicity for pp at different
energies as well as Pb-Pb collisions at 2.76 TeV together with the
experimental data.

In Figs. 15 and 16, we show the results for $F_{p_{T}}$, $C_{m}$,
$\sqrt{C_{m}}/\langle p_{T}\rangle$ for pp collisions at 0.9, 2.76 and 7 TeV
together the CMS data. In Fig. 16, we show the results for
$\sqrt{C_{m}}/M(p_{T})_{m}$ as a function of the multiplicity for pp at
different energies with the experimental data. It is observed a change in the
slope at high multiplicities that is reproduced in the string percolation but
not for the usual Monte Carlo code models. In string percolation the change of
slope arises naturally due to the formation of a large cluster above a
critical density (corresponding to a high multiplicity) and therefore
suppression of the fluctuations.

###  3.7 Forward-backward correlations

The width of the KNO scaling shape is related to the fluctuations on the
number of strings or the number of clusters (independent color sources). This
width is also related to the forward-backward ($F$-$B$) correlations. These
correlations can be described by a linear approximation

$\langle n_{B}\rangle=a+bn_{F},$ (80)

where $n_{F}$ is the number of particles observed in the forward (backward)
rapidity window and the slope $b$ measures the correlation forward-backward

$b=\frac{\langle n_{F}n_{B}\rangle-\langle n_{F}\rangle\langle
n_{B}\rangle}{\langle n_{F}^{2}\rangle-\langle n_{F}\rangle^{2}}.$ (81)

Usually, the F and B rapidity intervals are taken separated by a central
rapidity window $|y|<y_{c}$ in such a way that the short range correlations
are eliminated ($y_{c}=0.5$) because its range is less than unit of rapidity.
In any multiple scattering model the origin of long range correlations is the
fluctuations in the number of elementary scatterings [50, 119, 120, 121, 122,
123, 124]. Let us consider symmetric $F$ and $B$ intervals and having $N$
strings which decay into $\mu_{1}$ particles. Then, the slope $b$ can be split
into short range (SR) and long range (LR) correlations [121]

$b=b^{SR}+b^{LR}=\frac{\delta_{F}\mu_{1}}{1+\delta_{F}\mu_{1}[\omega_{N}+\Lambda(0)]}\Lambda(y_{FB})+\frac{\delta_{F}\mu_{1}}{1+\delta_{F}\mu_{1}[\omega_{N}+\Lambda(0)]},$
(82)

where $\omega_{N}$ is given by

$\omega_{N}=\frac{\langle N^{2}\rangle-\langle N\rangle^{2}}{\langle
N\rangle^{2}}.$ (83)

$\Lambda(y_{FB})$ and $\delta_{F}$ are the correlation function of one string
with rapidity separation $y_{FB}=y_{F}-y_{B}$ and the acceptance of the $F$ or
$B$ rapidities, respectively. We also take $\delta_{F}=\Delta y_{F}=\Delta
y_{B}$. For large rapidity window gap between the F and B intervals there are
not long range correlations in a single string, then $\Lambda=0$ and $b$
becomes

$b=\frac{1}{1+\frac{1}{\delta_{F}\mu_{1}\omega_{N}}}.$ (84)

At low energy, there are not fluctuations in the number of strings, i. e.,
$\omega_{N}\to 0$, and according to Eq. (84), $b\to 0$.

Figure 17: The leading orden diagram which induces long range correlations in
rapidity. The source of one nucleus is given by the $x$ and the other by the
$o$. The produced gluon is denoted by the curled line.

As the energy or the centrality of the collisions increases $\omega$ increases
as well as $b$. This behavior can be turned as a consequence of the formation
of a large cluster of overlapping strings and consequently a decreasing of the
number of independent color sources. Notice that if we fix the multiplicity,
we eliminate many of the possible string fluctuations and therefore $b$ will
be smaller. In CGC the main contribution to long range correlations comes from
the diagram of Fig. 17, which only contributes to short range correlations in
such a way that for a large rapidity gap between $F$ and $B$ intervals we have
[125, 126, 127]

$b=\frac{1}{1+c\alpha_{s}^{2}},$ (85)

where $c$ is a constant independent on the energy and centrality degree. As
the strong coupling constant, $\alpha_{s}$, decreases with energy and with
centrality, $b$ increases. This behavior is very similar to the one described
above for string percolation. The analysis of $F$-$B$ correlations has been
extended not only to two rapidity separated windows but also to different
azimuthal windows which help to separate short and long range correlations
[121]. In this case, the coefficient $b$ is given by

$\displaystyle
b=b^{SR}+b^{LS}=\frac{\delta_{F}\mu_{1}}{1+\delta_{F}\mu_{1}[\omega_{N}+\Lambda(0,0)]}\Lambda(y_{FB},\phi_{FB})+$
$\displaystyle\frac{\delta_{F}\mu_{1}}{1+\delta_{F}\mu_{1}[\omega_{N}+\Lambda(0,0)]},$
(86)

where now $\delta_{F}$ is the product of the acceptance on rapidity and an
azimuthal angle ($\delta_{F}=\Delta y_{F}\Delta\phi_{F}=\Delta
y_{B}\Delta\phi_{B}$), $\Lambda(y_{FB},\phi_{FB})$ is the correlation function
of the single string at rapidities and azimuthal angles separation,
$y_{FB}=y_{F}-y_{B}$, and $\phi_{FB}=\phi_{F}-\phi_{B}$.

Figure 18: The forward-backward ($F$-$B$) correlation coefficient in pp
collisions at 7 TeV from reference [121].

In Fig. 18 it is shown the results of a separation azimuthal angle of 0º,45º
and 135º for pp collisions together the ALICE data at 7 TeV. The agreement is
good also at 0.9 and 2.76 TeV [121]. The $F$-$B$ correlations have been
studied not only for multiplicities in the $F$-$B$ intervals but also for
transverse momentum-multiplicity $(p_{T}-n)$ and transverse momentum
correlations $(p_{T}-p)$. In the $(p_{T}-p_{T})$ case, the asymptotic equation
for the slope coefficient is

$b=\frac{\omega_{\rho}\mu_{F}}{\omega_{\rho}\mu_{F}+16\gamma\sqrt{\rho}},$
(87)

where

$\displaystyle\omega_{\rho}=\frac{D_{\rho}}{\langle\rho\rangle},$
$\displaystyle\gamma=\frac{D_{p_{T}}}{\langle\rho\rangle^{2}},$ (88)

$D_{\rho}$ and $D_{p_{T}}$ are the string density and transverse momentum
dispersions respectively, $\mu_{F}$ is the multiplicity of one of the
symmetric intervals, and $\gamma$ is a dimensionless coefficient which depends
only of the form of the distribution. For a Tsallis shape distribution,
$\gamma$ takes the value $(k-1)/2(k-4)$, which is related to the width of the
distribution. In the case of a thermal distribution its value is 1/2.

Figure 19: Dependence of $b_{corr}^{\bar{p}_{t}-\bar{p}_{t}}$ on centrality
clases 5% width determined by the V0, ZDCvsZEM and CL1 estimators from ALICE
[165].

In Fig. 19 is shown the ALICE preliminary data for Pb-Pb collisions at 2.76
TeV as a function of the centrality for a rapidity gap of 0.8 and a rapidity
width of 0.4. The data are in qualitative agreement with the string
percolation model [128, 129] which predicted a rise of $b$ with centrality up
to around 30% decreasing above this centrality value.

###  3.8 Underlying event of high $p_{T}$ particles and KNO scaling

The study of the underlying effect can be useful to understand the particle
production mechanism. It has been shown that selecting events of determinate
high $p_{T}$ particle and looking at the particles that are in the azimuthal
range angles, say $\pi/3<|\Delta\phi|<2\pi/3$, the associated multiplicity
distribution satisfy approximately KNO [130]. Let us show that this is a
rather general property which is satisfied by events of a determined class of
scatterings, for instance diffractive and non diffractive or inelastic and
elastic or soft and hard scatterings. In a multiple scattering approach, there
will be events that is sufficient to have one elementary scattering of being
of this class to be the final result of this class. The non diffractive, the
inelastic and the hard events satisfy this requirement. It is said that these
events are only shadowed by themselves and in fact the evaluation of the cross
section for these selected events only appears the cross section of the
elementary cross section of these events not the elementary cross section of
all kind of events [131]. Concerning the associated multiplicity distribution
to these events is shown that in terms of multiple scatterings, the original
distribution and the new one are related for a factor $N$ which translate into
a multiplicative factor $n$ in the multiplicity in such a way that [103, 104,
105, 107]

$P_{c}(n)=nP(n)/\langle n\rangle.$ (89)

If we go on the process of the selection of high $p_{T}$ particles, we will
have the chain

$P(x)\to xP(x)/\langle x\rangle\to\dots\to x^{k}P(x)/\langle x^{k}\rangle.$
(90)

In a similar way than the one in Sec. 3.5, the only stable distributions under
these transformation are the generalized gamma function, being the gamma
function the most simple of them (see Eq. (51)). This function satisfies KNO
scaling if k is independent of energy. We have seen above that $k$ increases
with the energy for pp collisions in the studied range, as $1/k$ controls the
width of $\langle n\rangle P(n)$, this distribution should be narrow as the
energy increase as the experimental data show.

###  3.9 Bose-Einstein correlations

The Bose-Einstein correlations (BEC) are very interesting in order to
determinate the extension of the source of multi-particle production as well
as to know the degree of coherence of the emitted particles. The correlation
strength is characterized by the parameter $\lambda$, which can also be
interpreted as a measure of the chaotically of the degree of coherence of the
collisions [132, 133, 134]. In this interpretation $\lambda=1$ means totally
chaotic emission, whereas $\lambda=0$ means radiation in a coherent way. This
interpretation should be taken with caution, because in $e^{+}e^{-}$,
$\lambda=1$ at energies where there are not production of more than two jets
and higher energies, $\lambda$ decreases with increasing multiplicity. These
facts would apparent indicate a systematic increase of the coherence from
$e^{+}e^{-}$ to pp collisions, which does not seem reasonable. The
experimental data on $\lambda$ have been in different kinematic conditions
assuming different extrapolations, normalizations and corrections which makes
difficult the comparison with models, however the ISR, SPS, RHIC and LHC data
allow us to distinguish some trends. First, for a not very large number of
collisions the data of SPS with p and O as a projectiles show a decrease of
$\lambda$ with multiplicity [135, 136, 137]. As the number of collisions
increases no longer decreases even it increases reaching values of 0.6-0.7. At
SPS energies the values of $\lambda$ are larger at forward than at central
rapidity. Notice that the particle multiplicity is larger at central than in
forward rapidity. All these trends of data can be understood in the framework
of percolation of strings [138, 139]. The strings of the Lund type
fragmentation according to totally chaotic sources, $\lambda=1$, and usually
is assumed that there is not BEC from particles emitted from different strings
[140]. Under this assumptions one can write

$\lambda=n_{s}/n_{T},$ (91)

where $n_{s}$ is the number of identical particles pairs produced from the
same string and $n_{T}$ is the total number of identical pairs produced in the
same kinetic range of $p_{T}$ and $y$. The number of identical pairs produced
by each cluster is

$n_{s}=\frac{1}{2}\mu_{1}^{2}\left\langle\sum_{n=1}^{N_{s}}\frac{a_{n}nS_{n}}{S_{1}}\right\rangle,$
(92)

and the total number of pairs of identical particles produced is

$n_{T}=\frac{1}{2}\mu_{1}^{2}\left\langle\left(\sum_{n=1}^{N_{s}}\frac{a_{n}\sqrt{n}S_{n}}{S_{1}}\right)^{2}\right\rangle,$
(93)

where $a_{n}$ is the number of clusters with $n$ strings. The numerical
results of the Monte-Carlo simulation that includes energy conservation to
different energies and collisions type shows the right change of the behavior
and approximate to the scaling of $\lambda$ for string densities around
$\rho\simeq 0.8-1$, which is in agreement with the experimental data, as is
shown in Fig. 20 [138].

Figure 20: Dependence of $\lambda$ on $\eta$ for different nucleus–nucleus
collisions in the percolating strings framework taking into account the
energy-momentum of the strings. Each point represents a specific type of
nucleus–nucleus collisions. Correlations are calculated between identical
pions for $y_{1cm}=y_{2cm}=0.5$ and $m_{T1}=m_{T2}=0.35$ GeV/$c^{2}$ [138].

The three body BEC have been also studied in percolation [139], predicting the
strength of the three particle BEC, which is in good overall agreement with
data [141].

###  3.10 $J/\Psi$ production dependence on the multiplicity

Figure 21: Results for pp collisions in the central $|y|<0.9$ rapidity range
(dashed line) and forward $2.5<y<2.4$ (dotted line), together with the
experimental data fro the central (circles) and forward (squares) rapidly
regions from the ALICE Collaboration. The linear behavior (solid line) and the
prediction for pPb collisions (dashed-dotted line) at 7 TeV are also plotted
[142].

The ALICE collaboration have found a departure from linearity on the
dependence of $J/\Psi$ production on the multiplicity at very high
multiplicity. This departure is larger at central than a forward rapidity
region. This behavior can be explained in the frameworks of string percolation
[142]. In fact assuming that as in any hard process, the number of produced
$J/\Psi$ is proportional to the number of elementary collisions, $N_{s}$, we
have

$\frac{n_{J/\Psi}}{\langle n_{J/\Psi}\rangle}=\frac{N_{s}}{\langle
N_{s}\rangle}.$ (94)

From Eq. (35) we can write

$\frac{dN/dy}{\langle dN/dy\rangle}=\frac{N_{s}F(\rho)}{\langle N_{s}\rangle
F(\langle\rho\rangle)},$ (95)

thus

$\frac{dN/dy}{\langle dN/dy\rangle}=\left(\frac{n_{J/\Psi}}{\langle
n_{J/\Psi}\rangle}\right)^{1/2}\left(\frac{1-\exp(-n_{J/\Psi}\langle\rho\rangle/\langle
n_{J/\Psi}\rangle)}{1-\exp(-\langle\rho\rangle)}\right)^{1/2}.$ (96)

At low multiplicities, $N_{s}$ is small and the above equation give rise to a
linear dependence

$\frac{n_{J/\Psi}}{\langle n_{J/\Psi}\rangle}=\frac{dN/dy}{\langle
dN/dy\rangle},$ (97)

therefore

$\frac{n_{J/\Psi}}{\langle
n_{J/\Psi}\rangle}=\langle\rho\rangle\left(\frac{dN/dy}{\langle
dN/dy\rangle}\right)^{2}.$ (98)

Note that the linear behavior changes to quadratic at high multiplicities. In
Fig. 21 we show the results together with the experimental data [33] as well
as the results for the forward rapidity region together with the experimental
data. In the forward rapidity region we have less number of strings and as a
consequence the departure from the linear behavior starts at higher
multiplicity. In both cases central and forward rapidity region, a good
agreement is obtained.

Figure 22: Results without and with $J/\Psi$ suppression together the
experimental data.

Notice that only there are two assumptions, namely, the $J/\Psi$ is produced
by a hard mechanism and the attenuation (saturation) of the increase of the
multiplicity with $N_{s}$. At low multiplicity behaves proportional to
$N_{s}$, but at high multiplicities goes like $\sqrt{N_{s}}$. The departure of
the linear behavior is a consequence of this attenuation (saturation). At 14
TeV there is some possibility that $J/\Psi$ melts [7] due to the high density
reached. In this case there is not nuclear suppression effects, then we assume
that the suppression is proportional to the collision area covered by strings.

Figure 23: Scale ratio of $J/\Psi$ production over the total charged particle
production.

In Fig. 22 we show the results without and with $J/\Psi$ suppression together
the experimental data. The $J/\Psi$ suppression could be clearly seen by
looking at the dependence on the multiplicity of the ratio between the
$J/\Psi$ production and events with a high $p_{T}$ particle(thus with a linear
dependence on $N_{s}$ and consequently on the multiplicity). The result for
this ratio is shown in Fig. 23.

###  3.11 Incoherent $J/\Psi$ photoproduction

The incoherent photo production of $J/\Psi$ it has been studied experimentally
[143, 144] and theoretically [145]. The cross section of
$\mathrm{Pbp}\to\mathrm{Pb}J/\Psi X$ probes the fluctuations on the number of
elementary scattering of the dipole $q\bar{q}$ (obtained from the virtual
photon) on the partons of the proton via the reaction
$\gamma(q^{2})\mathrm{p}\to J/\Psi X$. The increase of these fluctuations
gives rise to an increase of the cross section in agreement with data, but as
the energy increases the number of elementary collisions increases and
assuming that these collisions have a transverse size around 0.3fm, they start
to overlap forming clusters of these hot spots. Above a critical point, the
number of independent sources decreases and so the fluctuations and therefore
the cross section. Above a critical percolation energy (around 500 GeV), the
cross section starts to decrease. This prediction can be tested at LHC
experiments.

###  3.12 Strangeness enhancement

Figure 24: Comparison of the multiplicity dependence of the relative yield of
$\Omega$, $\Theta$ and $\Lambda$ baryons normalized to pion multiplicity for
pp and pPb collisions for the model (left) and experimental data (right)
[146].

The overlapping of the strings modify the strength of the color field and
hence the string tension of the formed cluster. Due to this, the decay of
these clusters produced naturally an enhancement of the strangeness [146, 147,
148]. In addition to this effect, as the clusters have at their extremes
complex flavor $F$ and $\bar{F}$ formed from the individual flavors of the
single strings, the decay will produce more baryons and anti baryons than in
the fragmentation of single strings. There is not any quantitative evaluation
of this effect in the production of strange baryons. In the case of the
strangeness enhancement with multiplicity seen in pp collisions [149], a
simplified model of string percolation which taken into account only the
different string tension of the cluster is able to describe qualitatively the
data [146]. In Fig. 24 are shown the model results (left) and the experimental
data (right).

##  4 Azimuthal dependence of the momentum distributions

###  4.1 Collective flow and ridge structure

Figure 25: Scheme of the azimuthal dependence modify by the escape probability
of a parton on the nuclear overlap [152].

The clusters formed by the strings have an asymmetric form in the transverse
plane and acquires dimensions comparable to the nuclear overlap. This
azimuthal asymmetry is at the origin of the elliptic flow in string
percolation. The partons emitted at some point inside the cluster have to pass
through the strong color field before appearing in the surface. The energy
loss by the parton is proportional to the length and therefore the $p_{T}$ of
the particle will depend on the direction of the emission as shown in Fig. 25.
Monte Carlo simulation have been done taking into account this energy loss
[150]. The results of this simulation for the different harmonics [150, 151]
are in reasonable agreement with experimental data on the $p_{T}$ and
centrality dependence. The azimuthal dependence in this way is very similar to
evaluate the probability to escape a parton of the nuclear overlap from the
initial clusters location. One way of doing that is defining

$\displaystyle R_{\phi}=\frac{R_{A}\sin(\phi-\alpha)}{\sin\alpha},$
$\displaystyle\alpha=\arcsin\left(\frac{b}{2R_{A}}\sin\phi\right),$ (99)

where $b$ is the impact parameter. We also define
$\rho_{\phi}=\rho(R/R_{\phi})^{2}$, and substituting in the $p_{T}$
distribution, we obtain

$f(F(\rho_{\phi}),p_{T}^{2})\simeq
f(F(\rho),p_{T}^{2})\left[1+\frac{\partial\ln f(F(\rho),R^{2})}{\partial
R^{2}}(R_{\phi}^{2}-R^{2})\right].$ (100)

Thus, the elliptic flow can be computed as follows

$v_{2}(p_{T}^{2})=\frac{2}{\pi}\int_{0}^{\pi/2}d\phi\cos(2\phi)\left[1+\frac{\partial\ln
f(F(\rho),R^{2})}{\partial R^{2}}(R_{\phi}^{2}-R^{2})\right].$ (101)

Note that the latter is an analytical close expression for all energies,
centralities, projectiles, and targets.

Figure 26: Comparison between the prediction of percolation model (red stars
and blue squares) and the experimental data (error-bars in green and pink) for
$\sqrt{s}=200$ GeV and $\sqrt{s}=2.76$ TeV (centralities 10%-20%) [153].

The transverse momentum dependence of $v_{2}$ computed using Eq. (101) for Pb-
Pb at 2.76 TeV and Au-Au at 200 GeV for 10%-20% centrality is shown in Fig. 26
together the experimental data. A good agreement is also obtained at all
centralities and rapidities [152, 153, 154] as well as the hierarchy on
$v_{2}$ of $\pi$, $k$ and p. The ridge structure was seen first at RHIC in Au-
Au and later at LHC in Pb-Pb collisions. This structure has been also observed
in pp and pA collisions at high multiplicity at LHC, as it was anticipated by
string percolation [155].

Figure 27: Correlation coefficient $C(\phi)$ for pp collisions at 7 TeV with
triple multiplicity [156].

In string percolation correlations can arise from the superposition of many
events with different number and type of string. In this way, there appears
long range correlations in rapidity. However passing to the azimuthal
dependence, if the emission of strings is isotropic, the correlations due to
their distribution in different events will be also isotropic. Also in the
central rapidity region, the inclusive cross section is approximately
independent of rapidity. This generates a plateau in the $y-\phi$ distribution
rather than a ridge, with only a peak at small y and $\phi$ due to short range
correlations. This conclusion is also valid if one averages the inclusive
cross sections over all events with the resulting loss of azimuthal angle
dependence. So the ridge can only be obtained in an event by event basis. In
this way we performed our evaluations [154].

Figure 28: Correlation coefficient $C(\phi)$ for p-Pb collisions at 5.02 TeV
for central collisions compared to the data (ZYAM procedure) [156].

In Fig. 27 we show the results [156] for $C(\phi_{12})$ for event with triple
multiplicity than minimum bias in pp collisions at 7 TeV compared to the
experimental data. In Fig. 28 we show the results for central pPb at 5.02 TeV
compared to the data and in Fig. 29 the results for Au-Au 0%-10% of centrality
at 200 GeV and its comparison with experimental data [156]. An overall
agreement is obtained in spite of the approximations done in the computation.

Figure 29: Correlation coefficient $C(\phi)$ for Au-Au at 200 GeV for 10% of
the most central events against the experimental data [156].

In the case of pp collisions, to obtain the ridge structure we need to
consider high multiplicity events (three times the minimum bias multiplicity
shown in Fig. 29). This is due to the fluctuations needed to have sizable long
correlations which are only obtained for these events. These fluctuations are
also crucial to describe the higher harmonics of the azimuthal distributions.

We can conclude that string percolation is able to describe the ridge
structure seen in pp, pA and AA collisions. The ridge is obtained from the
superposition of many events with different number and types of clusters of
strings. There is not any essential difference between high multiplicity pp
for pA collisions and AA collisions. The collective flow is obtained from the
configuration of the initial state as clusters of overlapping clusters and the
interaction of the produced partons with the color field of the clusters. This
interaction could be interpreted as final state interaction but as far as the
parton have these interactions before hadronization should be regarded as well
as initial state interaction. In the production of heavy particles, due to
their short formation time, they can be formed before than the parton get out
the surface collision area. This is certainly true for central heavy ion
collisions. In this case, the energy loss by the parton would be smaller and
thus the elliptic flow. As the elliptic flow for central collisions is small,
the effect is difficult to be observed.

###  4.2 Elliptic flow scaling and energy loss

In Sec. 3.3, we discussed the quenching of low $p_{T}$ partons. A parton
emitted from the decay of a cluster with tension $t$ and momentum $p$ due to
the energy loss in his way to get out the overlap collision area, obeys the
distribution

$P(p_{T},\phi)=C\exp(-p_{T}/T)\exp(-8p_{T}^{2/3}T^{1/3}l(\phi)).$ (102)

Here the temperature, $T$, is proportional to the squared root of the string
tension t. The departure from the thermal distribution is due to the quenching
formula in Eq. (38). The length $l(\phi)$ is the length of the path needed by
the parton to get out. We will take proportional to the product of the
eccentricity of the overlap area and $L$ which is the length independent of
the eccentricity that we will take proportional to $(1-N_{A}^{1/3})/2$, the
number of collisions of a parton with a nucleus. We define the eccentricity as

$\epsilon=\frac{2}{\pi}\int_{0}^{\pi}d\phi\cos
2\phi\frac{R^{2}-R^{2}_{\phi}}{R^{2}}.$ (103)

We expect that the elliptic flow be proportional to the strength of the
quenching, so

$v_{2}\sim p_{T}^{2/3}T^{1/3}L\epsilon.$ (104)

Using the dependence of $Q_{s}$ on the energy and centrality [154] and taking
$T$ proportional to $Q_{s}$, we have that [157]

$\frac{v_{2}}{Q_{s}^{A}\epsilon
L}\sim\left(\frac{p_{T}}{Q_{s}}\right)^{2/3}=\tau^{1/3},$ (105)

where we have choice the scaling variable $\tau=p_{T}^{2}/Q_{s}^{2}$. In Fig.
30, the experimental data of Phenix [158] and ALICE [159] at different
centralities are shown versus the scaling function $\tau^{1/3}$. Also the best
fit of the form $t^{b}$ is shown, giving a value of $b$=0.404, which is not
very different from 1/3. Taking into account the crude approximations done in
deriving the scaling formula Eq. (104), the result is very remarkable,
confirming the quenching of partons inside the overlap surface of the
colliding objects.

Figure 30: $v_{2}$ scaled by $\epsilon_{1}Q^{A}_{s}L$ for 10-20%, 20-30%,
30-40% and 40-50% Au-Au collisions at 200 GeV and Pb-Pb collisions at 2.76 TeV
vs $\tau$. The dashed black line is a fit to data according to $a\tau^{b}$
with $a$=0.126$\pm$0.0076 and $b$=0.404$\pm$0.025, solid blue line corresponds
to $\tau^{1/3}$ [157].

##  5 Thermodynamics of string percolation

The thermodynamics of the string percolation can be addressed by extracting
the temperature from the transverse momentum distribution. We also can extract
the suppression factor $F(\rho)$ and hence the local initial temperature as
well as the Bjorken initial energy density $\epsilon$, which are given by
[113, 112]

$\displaystyle T=\sqrt{\frac{\langle p_{T}^{2}\rangle_{1}}{2F(\rho)}},$
$\displaystyle\epsilon=\frac{3}{2}\frac{\langle
m_{T}\rangle}{S\tau_{p}}\frac{dN}{dy},$ (106)

where $S$ is the overlap area and $\tau_{p}$ is the production time, that we
take $\tau_{p}=2.4\hbar/\langle m_{T}\rangle$ [160].

Figure 31: Energy density $\epsilon/T^{4}$ vs $T/T_{c}$.

In Fig. 31, we plot the obtained energy density over $T^{4}$ as a function of
$T/T_{c}$ together the lattice result. Notice that $T$, which characterizes
the percolation clusters, measures the initial temperature of the system,
since the clusters cover most of the area of the collision this local
temperature becomes a global temperature. In this way, the critical string
density corresponds to the critical temperature. In relativistic kinetic
theory the ratio between the shear viscosity and the entropy density is give
by [161]

$\frac{\eta}{s}=\frac{T\lambda_{mfp}}{5},$ (107)

where the mean free path is $\lambda_{mfp}\sim 1/n\sigma_{tot}$, being $n$ the
number density of a free gas of quarks and gluons, and $\sigma_{tot}$ the
transport cross section. In string percolation the density numbers is the
effective number of sources per unit of volume is $n=N_{sources}/5L$ [55],
where $L$ is the longitudinal string length $\sim$1 fm. The effective number
of sources is the area covered by strings $(1-\exp(-\rho))S$ divided by the
area of one effective string $F(\rho)S_{1}$. On the other hand, the transport
cross section is the area of the effective string. Collecting all these, we
have

$\frac{\eta}{s}=\frac{TL}{5(1-\exp(-\rho))}.$ (108)

In Fig. 32, we show the behavior of $\lambda_{mfp}$, $T$ and $\lambda_{mfp}T$
as a function of the string density.

Figure 32: $\lambda_{mfp}$, $T$ and $\lambda_{mfp}T$ as functions of the
percolation density parameter $\rho$.

In Fig. 33, we show the ratio $\eta/s$ as a function of the temperature. In
the same figure are plotted evaluations in the case of weak QGP and string
quark gluon plasma (sQGP) as well as AdS/CFT result [18].

Figure 33: Shear viscosity ratio $\eta/s$ vs $T/T_{c}$.

The arrows marks are the result of string percolation for Au-Au and Pb-Pb at
RHIC and LHC energies. Below $\rho_{c}$ as the temperature becomes close to
$T_{c}$, the string density increases and the area is filled rapidly and
$\lambda_{mfp}$ and $\eta/s$ decrease sharply. Above $Tc$, the area is not
covered as fast and the relatively decreasing of $\lambda_{mfp}$ is
compensated by the rising of temperature, resulting in a smooth increase of
the $\eta/s$. The behavior of $\eta/s$ is governed by the fractional are
covered by strings, what is not surprising because $\eta/s$ is the ability to
transport momenta at large distances and that has to do with the density of
voids in the matter. Notice that the values of the ratio for high $T$ values
approach the weak coupling limit.

Figure 34: Comparison between the trace anomaly of the energy momentum tensor
and inverse of the $\eta/s$ ratio. Note that both variables have a maximum
value at the same temperature point.

Moreover, the mean value of the trace of the energy momentum tensor
$T^{\mu}_{\mu}=\epsilon-3P$ is a measure of the deviation of the conformal
behavior and thus identifies the interaction still present in the medium. In a
classical theory with massless quarks vanishes, but in any quantum field
theory is not zero, because the scale needed to be renormalized, breaking the
conformal symmetry. It is the well known trace anomaly. We find that the
reciprocal of $\eta/s$ is in quantitative agreement with the trace anomaly
over a wide range of temperatures. The minimum corresponds to the maximum of
$(\epsilon-3P)/T$ as it is seen in Fig. 34.

On the other hand, it is possible to determine the speed of sound, $c_{s}$, by
assuming the 1D Bjorken expansion, using the energy density, the initial
temperature, and the trace anomaly given by the string percolation. Starting
from the equations

$\displaystyle\frac{1}{T}\frac{dT}{d\tau}=-\frac{c_{s}^{2}}{\tau},$
$\displaystyle\frac{dT}{d\epsilon}\frac{d\epsilon}{d\tau}=\frac{T}{\tau},$
(109)

where $\tau$ is the proper time and $c_{s}$ is the sound speed. Since
$s=(\epsilon+P)/T$ and $\Delta=(\epsilon-3P)/T^{4}$, one gets

$\frac{dT}{d\epsilon}=\frac{c_{s}^{2}}{s}.$ (110)

From the above equations, it is possible write $c_{s}$ in terms of $\rho$ in
the following way

$c_{s}^{2}=-\frac{1}{3}\left(\frac{\exp(-\rho)}{F(\rho)^{2}}-1\right)+0.0191\left(\frac{\Delta}{3}\right)\left(\frac{\exp(-\rho)}{\rho
F(\rho)^{4}}-\frac{1}{\rho F(\rho)^{2}}\right),$ (111)

where $F(\rho)$ is the scaling function in Eq. (34). In Fig. 35, we show
$c_{s}$ as a function of the temperature. It is observed a very good agreement
with lattice calculations.

Figure 35: Squared sound speed as a function of $T/T_{c}$.

Another interesting thermodynamic variable which can be determined is the bulk
viscosity. Starting from [162]

$\frac{\eta_{b}}{\tau_{\Pi}}=\left(\frac{1}{3}-c_{s}^{2}\right)(\epsilon+P)-\frac{\alpha}{p}(\epsilon-3P),$
(112)

where $\tau_{\Pi}$ is the corresponding relaxation time. Substituting the
entropy density and the trace anomaly in the latter, we found that

$\frac{\eta_{b}}{\tau_{\Pi}s}=\left(\frac{1}{3}-c_{s}^{2}\right)T-\frac{\alpha}{9}\frac{\Delta
T^{4}}{s}.$ (113)

Note that this last expression depends on the sound speed, trace anomaly and
entropy density, which has already been computed in the string percolation
context. In Fig. 36, we plot the bulk viscosity over the entropy density as a
function of the temperature, which has a maximum close to $T_{c}$.

Figure 36: Ratio between the bulk viscosity and the entropy density as a
function of $T/T_{c}$.

##  6 Summary

The string percolation describes successfully most of the experimental data in
the soft region, namely, rapidity distributions, probability distributions of
multiplicities and transverse momentum, strength of BE correlations as a
function of multiplicities, forward-backward multiplicities as $p_{T}$
correlations, strangeness enhancement, elliptic flow and ridge structure.

The string percolation although is not derived directly from QCD has a clean
physical grounds and has the fundamental QCD feature. The non abelian
character is reflected in the coherent sum of the color fields which gives
rise to an enhancement of the mean $p_{T}$ and a suppression of the
multiplicity. The confinement of the fields is reflected in the small
transverse size of the strings as well as the transverse correlations length.
The scaling observed in the transverse momentum distribution is a consequence
of the invariance under the size of the clustersof strings.

The collective behavior of the multiparticle production has its origin in the
cluster configuration formed in the initial state of the collisions followed
by the interactions between the produced partons with the color fields, given
rise to energy loss. Due to that, the elliptic flow satisfies an universal
scaling law valid for all centralities and energies. At low $p_{T}$, the
thermal distribution of the $p_{T}$ distributions allows us to define
temperature as a function of the string density, which can be regarded, at
large cluster size, as the global temperature and compute the energy and
entropy density, which presents a jump at the critical temperature
corresponding to the percolation critical density. Using the kinetic transport
theory, it is shown that the ratio between the shear viscosity over entropy
density, which presents a minimum close to $T_{c}$ (and a maximum in the bulk
viscosity). The inverse of this ratio is very close to the trace anomaly,
including it smooth decreasing with temperature. The behavior of the speed of
sound with temperature is also in agreement with lattice QCD calculations. It
is remarkable that string percolation reproduce the behavior of the main
thermodynamics magnitudes as functions of the temperature.

## Acknowledgments

C.P. was supported by the grant Maria de Maeztu Unit of Excellence MDM-20-0692
and FPA project 2017-83814-P of Ministerio de Ciencia e Inovación of Spain,
FEDER and Xunta de Galicia. I.B. thanks the CONACYT cátedras project 043 and
the WS. Grant Loreal UNESCO CONACYT AMC. J.E.R. acknowledges financial support
from CONACYT (postdoctoral fellowship Grant no. 289198). We also thank Prof.
M. A. Braun, J. Dias de Deus, R. Scharenberg, B. Srivastava for their
collaboration in the work presented here.

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