See pages 1 of folha_de_rosto_en.pdf

See pages 1 of abstract.pdf

###### Contents

  1. 1 Introduction
  2. 2 The Proton Structure
    1. 2.1 Elastic Scattering
    2. 2.2 Deep Inelastic Scattering and the Quark-Parton Model
    3. 2.3 The Gravitational Form Factors (GFFs)
    4. 2.4 The Baryon Junction
  3. 3 High Multiplicity $pp$ Collisions
  4. 4 Glauber Model
    1. 4.1 Formalism
    2. 4.2 Pb-Pb Collisions
    3. 4.3 pp Collisions
    4. 4.4 pA Collisions
  5. 5 Charm Production
    1. 5.1 Charm production in the CEM
    2. 5.2 Open Charm Production
    3. 5.3 $J/\psi$ Production
    4. 5.4 $J/\psi$ Production in pA Collisions
  6. 6 Conclusion

##  Chapter 1 Introduction

The Large Hadron Collider (LHC) has been providing many interesting ways to
study high multiplicity (HM) events and shed light into particle production
features in this regime. Conventional meson detection (like charmonium and
$D$-mesons) in proton-proton ($pp$) collisions at $\sqrt{s}=7\text{ TeV}$ [1,
2] and $\sqrt{s}=13\text{ TeV}$ [3], and proton-lead ($p-Pb$) at
$\sqrt{s}=5.02\text{ TeV}$ [4], have one thing in common: the meson relative
yields grow faster than charged multiplicity density.

The HM events are characterized by small impact parameter configurations, that
is, they occur in ultracentral (UC) collisions, and present collective
behaviour [5], such as anisotropic flow. Also, in this regime, new features of
particle production, such as double-parton scattering and parton saturation,
can be responsible for the enhanced growth, and maybe the geometric
configuration of the nucleon plays a significant role.

One interesting static model of the proton, proposed in [6] and tested in
lattice QCD simulations [7, 8, 9], leads to the “Y picture of the proton”,
where the quarks are in the extremities of an “Y” like gluon string: the
baryon junction. This junction is composed by a large number of gluons, which
individually carry small fractions of the proton momentum if compared to the
valence quarks (they are in the non-perturbative regime). Because of this
difference, gluon flux-tubes of colliding nucleons interact more substantively
than the valence quarks, and can also be stopped by soft partons in the
midrapidity region. On the other hand, valence quarks will be pulled away and
populate the fragmentation region.

Another reason for the existence of the string junction is related to the
baryon number conservation. Conventionally, this quantum number should be
carried by valence quarks, however experimental data indicate that there is a
baryon asymmetry in the midrapidity region [10, 11, 12, 13, 14]. Baryon number
may be carried not only by valence quarks, but also by sea quarks (quark-
antiquark pairs generated by the junction when it is tensioned) [15]. This
idea was successfully implemented in analytical models [16], in the Monte-
Carlo event generator HIJING/B [17], and explained forward baryon production
data. In spite of all the progress, the existence of the baryon junction still
need confirmation [15].

In [18] the string junction was used together with the Glauber Model machinery
[19]. There the authors used a gaussian ansatz for the gluon and quark
anisotropic and inhomogeneous partonic density distribution in order to
represent the “Y” shape of the proton. The authors could explain elliptical
flux ($v_{2}$), measured by the ALICE collaboration in $pp$ collisions at
$\sqrt{s}=13\text{ TeV}$[5]. Along this line, we recently adapted the same
model to explain successfully the enhanced $J/\psi$ relative yields in UC $pp$
collisions [20].

In this work we are interested in expanding the development we made in [20] to
explain the enhanced meson production in HM events [1, 2, 3, 4] by testing the
sensibility of the model to different parameter sets and including $pPb$
collisions. Since we will take the average of many collisions and we are not
interested in the anisotropic aspects of the collisions, the isotropic gluon
distribution proposed in [21] will be used to represent the average
configuration of the “Y” shape over different orientations, i.e., we “rotate”
the “Y”, obtaining one gluon core and a quark corona (“core-corona” model).
For charm production, we will use the well known Color Evaporation Model (CEM)
[22, 23]. So, when we go from more peripheral to more central $pp$ collisions,
then to UC, we go from quark-quark to gluon-gluon collisions. Since the gluons
are much more abundant and $\sigma(gg\rightarrow
Q\bar{Q})\gg\sigma(q\bar{q}\rightarrow Q\bar{Q})$, the “core-corona” structure
should explain the enhanced growth in the HM $pp$ collisions, and the higher
number of binary collisions should explain the growth seen in $pPb$
collisions.

We start chapter II by presenting some general remarks on the proton
structure, and the next by presenting some aspects of HM collisions. In
chapter IV we adapt the Glauber model, usually applied to nucleus-nucleus
collisions, to $pp$ and $p-Pb$ collisions. In chapter V we combine Glauber
outputs (number of participants in a collision and number of binary collisions
for a given impact parameter) with CEM, and compare values with experimental
data. In the end we make some concluding remarks about the topics discussed in
the text.

##  Chapter 2 The Proton Structure

The electron-proton scattering can shed some light into the internal structure
of the proton. In this process, the electron exchanges momentum with the
proton by one photon exchange. Depending on the momentum, the aspects of the
proton that the process reveals are quite different.

At very low energies, that is, when the wavelength of the virtual photon is
much larger than the proton radius, the latter is seen as a point-like
particle by the electron and the electrostatic potential is sufficient to
understand the process. As the energy increases and the photon wavelength gets
closer to the proton radius, the charge and magnetic moment distributions
become more important and accessible, and the electrostatic approach is not
sufficient. In both cases the struck proton is not broken into hadrons, and
the process is said to be elastic.

As the exchanged momentum increases, the wavelength decreases. When the photon
wavelength becomes smaller than the proton radius, the process goes from
elastic to inelastic scattering. In this case, the virtual photon interacts
with the valence quarks and the proton breaks up. In the very high energy
limit, the resolution is such that the proton internal dynamics is completely
visible and the sea of quarks and gluons is finally accessed.

In the next sections, the aspects of these different energy regimes will be
explored.

###  2.1 Elastic Scattering

Figure 2.1: Feynman diagram of $e^{-}p$ elastic scattering.

In very low energy regimes of $e^{-}p$ scattering, shown if Fig. 2.1, the
energy is such that the proton recoil is negligible compared to its rest mass,
and the interaction can be described by the Coulomb potential. In the limit
where the electron is non-relativistic, we have Rutherford scattering, and the
differential cross section is given by [24]:

$\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Ruth}=\frac{\alpha^{2}}{16E^{2}\sin^{4}(\theta/2)}$
(2.1)

where $\alpha=e^{2}/4\pi$ and $E=p^{2}/2m_{e}$ ($m_{e}$ is the electron mass).

If the proton recoil can still be neglected, but the electron is relativistic,
that is, $m_{e}\ll E\ll m_{p}$, we have Mott scattering. The respective cross
section is given by [24]:

$\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}=\frac{\alpha^{2}}{4E^{2}\sin^{4}(\theta/2)}\cos^{2}\bigg{(}\frac{\theta}{2}\bigg{)}$
(2.2)

where $E=\gamma_{e}m_{e}$.

In both cases, spin and magnetic influences can be neglected, that is, the
proton is treated as a spinless nucleus, and as a point-like particle. It is
useful to have a way to account for the proton extension and its charge
distribution. This is done by introducing form factors.

Defining the charge density $Q\rho(\mathbf{r}\prime)$, where $Q$ is the total
charge and $\rho(\mathbf{r}\prime)$ is the normalized charge distribution, the
interaction can be described by means of the potential:

$V(\mathbf{r})=\int\frac{Q\rho(\mathbf{r}\prime)}{4\pi\|\mathbf{r}-\mathbf{r\prime}\|}d^{3}\mathbf{r\prime}$
(2.3)

Considering $e$ as the incoming electron, $e\prime$ as the outgoing electron
and that their wave functions are plane waves (Born approximation), the matrix
element can be written as:

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\left\langle\psi_{e\prime}|V(\mathbf{r})|\psi_{e}\right\rangle=\int
e^{-i\mathbf{p_{e\prime}}\cdot\mathbf{r}}V(\mathbf{r)}e^{i\mathbf{p_{e}}\cdot\mathbf{r}}d^{3}\mathbf{r}$
(2.4) $\displaystyle=$ $\displaystyle\int
d^{3}\mathbf{R}e^{i\mathbf{q}\cdot\mathbf{R}}\frac{Q}{4\pi|\mathbf{R}|}\int
d^{3}\mathbf{r\prime}\rho(\mathbf{r\prime})e^{i\mathbf{q}\cdot\mathbf{r\prime}}$
(2.5) $\displaystyle=$ $\displaystyle\mathcal{M}^{pl}F(\mathbf{q}^{2})$ (2.6)

with $\mathbf{q}=\mathbf{p_{e}}-\mathbf{p_{e\prime}}$,
$\mathbf{R}=\mathbf{r}-\mathbf{r\prime}$, $\mathbf{M}^{pl}$ is the point-like
proton matrix element and $F(\mathbf{q}^{2})$ is the electric form factor. The
last one can be interpreted as the Fourier transform of the charge
distribution. Since the matrix element still depends on point-like approach,
the differential cross section for Mott scattering is changed to:

$\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}_{Mott}=\frac{\alpha^{2}}{4E^{2}\sin^{4}(\theta/2)}\cos^{2}\bigg{(}\frac{\theta}{2}\bigg{)}|F(\mathbf{q}^{2})|^{2}$
(2.7)

In other words, the charge distribution inside the proton can be indirectly
measured through the differential cross section measurement in the low energy
limit.

Coming from the very low energy limit to the low energy, that is, when the
exchange momenta $|\mathbf{q}|$ gets closer to proton mass, the recoil and
spin interactions become more important. If the proton is still treated as a
point-like particle, but as a spin $1/2$ particle and with recoil, the
differential cross section is given by [24]:

$\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}=\frac{\alpha^{2}}{4E^{2}_{e}\sin^{4}(\theta/2)}\frac{E_{e\prime}}{E_{e}}\bigg{(}\cos^{2}\frac{\theta}{2}+\frac{Q^{2}}{2m_{p}^{2}}\sin^{2}\frac{\theta}{2}\bigg{)}$
(2.8)

with $Q^{2}=-q^{2}=2m_{p}(E_{e}-E_{e\prime})$ as the virtuality.

Like before, in the most general case (proton not treated as a point-like
particle) the internal structure can be accounted for by introducing the
electric form factor $G_{E}(Q^{2})$ and magnetic form factor $G_{M}(Q^{2})$.
The differential cross section can be written as the Rosenbluth formula:

$\bigg{(}\frac{d\sigma}{d\Omega}\bigg{)}=\frac{\alpha^{2}}{4E^{2}_{e}\sin^{4}(\theta/2)}\frac{E_{e\prime}}{E_{e}}\bigg{(}\frac{G_{E}^{2}+\tau
G_{M}^{2}}{1+\tau}\cos^{2}\frac{\theta}{2}+2\tau
G_{M}^{2}\sin^{2}\frac{\theta}{2}\bigg{)}$ (2.9)

with $\tau=Q^{2}/4m_{p}^{2}$.

Differently from the $F(\mathbf{q}^{2})$, form factors can not be directly
understood as Fourier transforms of charge and magnetic momenta distribution.
They are functions of the four-momentum $Q^{2}$, not the three-momentum
$q^{2}$. Explicitly:

$\displaystyle Q^{2}$ $\displaystyle=$
$\displaystyle-q^{2}=\mathbf{q}^{2}-(E_{e}-E_{e\prime})$ (2.10)
$\displaystyle\Rightarrow$ $\displaystyle
Q^{2}\bigg{(}1+\frac{Q^{2}}{4m_{p}}\bigg{)}=\mathbf{q}^{2}$ (2.11)

So $Q^{2}\approx\mathbf{q}^{2}$ if $Q^{2}\ll 4m_{p}^{2}$, and the form factors
can be interpreted as:

$G_{E}(Q^{2})\approx G_{E}(\mathbf{q}^{2})=\int
e^{i\mathbf{q}\cdot\mathbf{r}}\rho(\mathbf{r})d^{3}\mathbf{r}$ (2.12)

$G_{M}(Q^{2})\approx G_{M}(\mathbf{q}^{2})=\int
e^{i\mathbf{q}\cdot\mathbf{r}}\mu(\mathbf{r})d^{3}\mathbf{r}$ (2.13)

The measurement of the electric form factor, the one which we are mainly
interested in, leads to the exponential charge distribution of [24]

$\rho(\mathbf{r})\approx\rho_{0}e^{-r/a}$ (2.14)

with $a\approx 0.24\text{ fm}$, what corresponds to a proton root-mean square
charge radius around $0.8\text{ fm}$.

Once the electric form factor is measured through the differential cross
section measurement of Eq.(2.9), the proton charge radius can be calculated in
electron-proton collisions as [25]:

$\left\langle
r_{p}^{2}\right\rangle=-\frac{6}{G_{E}(0)}\frac{dG_{E}(Q^{2})}{dQ^{2}}\bigg{|}_{Q^{2}=0}$
(2.15)

In 2010, the CODATA-2010 [26] value for the charge radius of the proton,
obtained from $ep$ scattering, was shown to be $r_{p}=0.895(18)\text{ fm}$ .

Another way to obtain the proton charge radius is through hydrogen
spectroscopy. In this approach the electron energy is shifted when it moves
inside the proton, experiencing an screening effect. The CODATA-2010 value for
the radius in this method was shown to be $r_{p}=0.8758(77)\text{ fm}$ and the
final proton charge radius was $r_{p}=0.8775(51)\text{ fm}$.

In 2010 and 2013, the CREMA collaboration [27, 28] found the charge radius
through the muonic hydrogen ($\mu H$) spectroscopy method. In this experiment,
the electron from the hydrogen atom is replaced by one muon. Since the muon is
much heavier than the electron, the Bohr radius of hydrogen decreases and the
sensitivity to the proton size increases. The extraordinarily precise results
found were $r_{p}=0.84184(67)\text{ fm}$ and $r_{p}=0.84087(39)\text{ fm}$.

Since values from CREMA and CODATA were around $7\sigma$ of difference, and
around $4\%$, this discrepancy became what is known as “the charge radius
puzzle”. This puzzle is not completely solved until now. Some results from
lattice and lepton-muon scattering are discussed in [25, 29].

###  2.2 Deep Inelastic Scattering and the Quark-Parton Model

Moving away from the elastic cross section by increasing the energy, the study
of the proton structure must change. While the virtuality ($Q^{2}$) increases,
the exchanged photon starts to interact with the quarks that compose the
proton, which is broken. The process is now represented by $e^{-}p\rightarrow
e^{-}X$ in Fig. 2.2, where $X$ is the hadronic final state.

Figure 2.2: Feynman diagram of $e^{-}p$ deep inelastic scattering.

For the deep inelastic scattering, the double differential cross section can
be written as [24]:

$\frac{d^{2}\sigma}{dxdQ^{2}}=\frac{4\pi\alpha^{2}}{Q^{2}}\bigg{[}(1-y)\frac{F_{2}(x,Q^{2})}{x}+y^{2}F_{1}(x,Q^{2})\bigg{]}$
(2.16)

In this approach, the form factors of Eq.(2.9) are absorbed into the structure
functions $F_{1}(x,Q^{2})$ and $F_{2}(x,Q^{2})$, which can not be interpreted
as the Fourier transforms of charge and magnetic momentum distribution since
there is an $x$ and $Q^{2}$ dependence. The values $x$ and $y$ are Bjorken-$x$
and inelasticity, which take values in the range $[0,1]$, and are defined as:

$x=\frac{Q^{2}}{2p_{p}\cdot q}=\frac{Q^{2}}{Q^{2}+p_{X}^{2}-m_{p}^{2}}$ (2.17)

$y=\frac{p_{p}\cdot q}{p_{p}\cdot p_{e}}=1-\frac{E_{e\prime}}{E_{e}}$ (2.18)

The structure functions are not independent from each other in the regime with
$Q^{2}$ greater than a few GeV, but they are almost independent of $Q^{2}$
(Bjorken-scaling), and are related by the Callan-Gross relation:

$F_{2}(x)=2xF_{1}(x)$ (2.19)

The quark-parton model was proposed by Feynman. In this model, the deep
inelastic $e^{-}p$ interaction is interpreted as electron-quark elastic
interactions, with quarks as free particles.

This model is applied in the infinite-momentum frame, the one in which the
proton energy is much bigger than the respective rest mass. The advantage of
this approach is that the valence quarks can be considered to be moving in the
proton beam line, and each quark carries a fraction $x$ of proton momentum.
This $x$ value is the same Bjorken-x, which now is interpreted as the fraction
of proton momentum carried by the parton.

Since the valence quarks will interact with each other by gluon exchanges, the
momentum distribution inside the proton will be given by the parton
distribution functions (PDFs). These functions represent the probabilities of
a given parton of flavor $i$ to have a momentum fraction in the range $x$ and
$x+\delta x$. These distributions are not known directly from theory because
there are non-perturbative aspects involved, but they are experimentally
obtained.

In the end, the double differential inelastic cross section of Eq.(2.16) can
be rewritten as [24]:

$\frac{d^{2}\sigma}{dxdQ^{2}}=\frac{4\pi\alpha^{2}}{Q^{4}}\bigg{[}(1-y)+\frac{y^{2}}{2}\bigg{]}\sum_{i}Q_{i}^{2}f_{i}^{2}(x)$
(2.20)

where $Q_{i}$ is the flavour charge and $f_{i}$ is the respective PDF.

The PDFs must reflect some internal structure features of the proton. The
proton can be considered as being composed by $uud$ quarks. These valence
quarks interact by exchanging gluons, which can fluctuate into virtual
$q\bar{q}$ pairs. These produced quarks are said to be “sea quarks” and are
mainly produced in the low $x$ regime. In the end, the $e^{-}p$ scattering
must include the interactions between electron and sea quarks.

Once there are $2$ valence quarks up and $1$ valence quark down, the PDFs must
be such that [30]:

$\int_{0}^{1}dx[f_{u}(x)-f_{\bar{u}}(x)]=2$ (2.21)

and

$\int_{0}^{1}dx[f_{d}(x)-f_{\bar{d}}(x)]=1$ (2.22)

Also, because the total momentum carried by the partons must be the total
momentum of the proton, we must have [30]

$\int_{0}^{1}dx\text{
}x[f_{u}(x)+f_{\bar{u}}(x)+f_{d}(x)+f_{\bar{d}}(x)+g_{g}(x)]=1$ (2.23)

and the gluons contribute with half of the total value, that is, they carry
half of the proton momentum!

Although the PDFs were assumed to be only dependent on $x$, at very low $x$
and at $x\approx 1$ the structure functions vary strongly with $Q^{2}$. In the
first case the functions get higher, and in the second, get lower, while
$Q^{2}$ increases. The evolution of PDFs in $Q^{2}$ are made through the DGLAP
equations developed in [30, 31].

In summary, coming from elastic scattering to deep inelastic scattering, the
proton goes from one symmetric charge distribution configuration to a more
complex distribution, which implies an even more complex charge distribution
inside it. Also, coming from $x\approx 1$ regime to the low $x$ regime, the
proton goes from a three valence quark composition to a complex internal
matter distribution generated by gluon fluctuations. In the end, the gluon
body starts to play an important role in the internal proton dynamics.

###  2.3 The Gravitational Form Factors (GFFs)

The electromagnetic properties of the proton shed light into many interesting
aspects of the internal structure of the proton. However, there are particles
inside it, like gluons, that do not interact electromagnetically, but are
important pieces of the nucleon. For example, the proton mass, which is around
$1\text{ GeV}$, can not be only composed by valence quarks, which present
masses around few MeV, generated by the Higgs boson. So, although gluons do
not have rest mass, part of the proton mass can be generated by their
influence in the energy momentum tensor. Also, part of the mass can be
generated by the trace anomaly [32, 33, 34].

The gluon distribution is not easily accessed because gluons do not have
charge. The gluon content can be seen through the $J/\psi$ photoproduction
process near threshold, at the Jefferson Lab. In this process the photon
fluctuates into $c\bar{c}$ pair and exchanges gluons with the struck proton
before decaying into $J/\psi$, and then to $e^{-}e^{+}$. The Feynman diagram
of the process is shown in Fig. 2.3.

Figure 2.3: Feynman diagram of the $J/\psi$ photopropduction and its
interaction with the gluon composition of the proton [37].

The gravitational form factors for a spin $1/2$ hadron can be obtained from
the energy-momentum tensor matrix elements as [35, 36]:

$\displaystyle\left\langle P\prime|T_{q,g}^{\mu\nu}|P\right\rangle$
$\displaystyle=$
$\displaystyle\bar{u}(P\prime)\bigg{[}\big{(}A_{q,g}(t)+B_{q,g}(t)\big{)}\gamma^{(^{\mu}\bar{P}^{\nu})}-\frac{\bar{P}^{\mu}\bar{P}^{\nu}}{M}B_{q,g}(t)$
(2.24) $\displaystyle+$ $\displaystyle
C_{q,g}(t)\frac{\Delta^{\mu}\Delta^{\nu}-g^{\mu\nu}\Delta^{2}}{M}+\bar{C}_{q,g}(t)M\eta^{\mu\nu}\bigg{]}u(P)$

where $A_{q,g}(t)$ is related to the quark and gluon momenta,
$J_{q,g}(t)=1/2\big{(}A_{q,g}(t)+B_{q,g}(t)\big{)}$ is related to the angular
momentum and $D_{q,g}=4C_{q,g}(t)$ is related to the pressure and shear
forces.

The photoproduction differential cross section as a function of $t$ Mandelstam
variable, for different photon energies ($9.1\text{ GeV}\leq E_{\gamma}\leq
10.6\text{ GeV}$), was measured in [37]. The gravitational form factors were
extracted by fitting the data through the holographic QCD approach and the
generalized parton distribution (GPD) approach. Both of them depend explicitly
on form factors. After finding them, the mass radius ($r_{m}$) and scalar
radius ($r_{s}$) can be calculated as:

$\left\langle
r_{m}^{2}\right\rangle=\frac{6}{A_{g}(0)}\frac{dA_{g}(t)}{dt}\bigg{|}_{t=0}-\frac{6}{A_{g}(0)}\frac{C_{g}(0)}{M_{p}^{2}}$
(2.25)

and

$\left\langle
r_{s}^{2}\right\rangle=\frac{6}{A_{g}(0)}\frac{dA_{s}(t)}{dt}\bigg{|}_{t=0}-\frac{18}{A_{g}(0)}\frac{C_{g}(0)}{M_{p}^{2}}$
(2.26)

The mass radius makes reference to the tensor gluon field structure, which
provides most of the proton mass, and the scalar radius makes reference to a
confining scalar gluon density. In Table 2.1 the radius obtained from [37]
through the different fits and the lattice value [38] are shown. We can notice
that the mass radius is much smaller than the charge radius, which is around
$0.8\text{ fm}$, and the scalar radius goes far beyond it.

The scalar and mass radius of the proton gives important information about the
size of its gluon component. They suggest that it is composed by three layers:
one internal gluon distribution, one thin quark layer and one external gluon
layer. There still is one interesting question that arises: how are the gluons
spatially distributed?

Table 2.1: Mass radius of the proton obtained in [37] through different fits of differential cross section and lattice result[38]. Models | $\sqrt{\left\langle r_{m}^{2}\right\rangle}\text{ }(\text{fm})$ | $\sqrt{\left\langle r_{s}^{2}\right\rangle}\text{ }(\text{fm})$
---|---|---
Holographic QCD | $0.755\pm 0.035$ | $1.069\pm 0.056$
GPD | $0.472\pm 0.042$ | $0.695\pm 0.071$
Lattice | $0.7464\pm 0.025$ | $1.073\pm 0.066$

###  2.4 The Baryon Junction

In [6] we find a really interesting spatial distribution of gluons inside the
proton. According to this picture, the valence quarks are in the extremities
of an “Y” shaped gluon string, called the “Baryon-Junction”. It is shown in
Fig. 2.4.

Figure 2.4: Baryon junction.

In this picture, the string gluons carry, individually, a small fraction of
the proton momentum. So, in high energy collisions, the gluons interact more
substantively, stopping in the midrapidity region, while the valence quarks
pass through each other and go to the fragmentation region. While the baryon
junction stops and the valence quarks are pulled away, the string is
tensioned, and $q\bar{q}$ pairs are created.

The produced pairs must present some characteristics [15]: they present low
transverse momentum because they come from soft partons process; since the
junction is flavour blind, the internal content does not have to be the same
of the colliding baryons; they will be accompanied by many pions; the baryon-
junction stopping is characterized by the exponential drop of the cross
section with the rapidity loss variable. Right after the collision, the
stopped junction generates baryons composed by sea quarks, which can differ
from projectile or target nucleons, and the number of baryons in central
rapidity region exceeds the number of antibaryons [39].

The initial motivation that led to the baryon-junction picture was the guiding
question: “Can Gluons Trace Baryon Number?”. This is the title of the paper
that first proposed the model [6]. Since valence quarks carry color, flavour,
electric charge and isospin, it is natural to state that they also carry the
baryon number, but this assumption is not dictated by QCD. So, part of the
number can be carried by the junction.

The baryon-junction manifestations are expected to be seen in deep inelastic
$e+p/AU$ scattering at the future electron-ion collider (EIC) [15], in
inclusive ultra-peripheral collisions at RHIC [15] and in ultra-central $pp$
collisions. In the last one this happens because the gluons are mainly
concentrated in the central region of the protons, generating high
multiplicity collisions. The aspects of this regime will be better explored in
the next chapter.

##  Chapter 3 High Multiplicity $pp$ Collisions

High multiplicity $pp$ events present some interesting characteristics. Since
they occur for small impact parameters (UC collisions), and gluons are
localized in the center, the interactions involve mainly gluons and sea
quarks. There are also collective behaviour features that suggest formation of
quark-gluon plasma.

The collective behaviour can be generically described in terms of correlated
particle production. Considering $P(\mathbf{p}_{i})$ as the probability of
producing a particle with momentum $\mathbf{p}_{i}$, there is correlation
behaviour if $P(\mathbf{p}_{1},\mathbf{p}_{2})\neq
P(\mathbf{p}_{1})P(\mathbf{p}_{2})$ [40]. That is, the probabilities can not
be separated. The correlation is usually measured and plotted by means of
relative azimuthal angle $\Delta\phi$ in the transverse plane, and relative
longitudinal pseudorapidity $\Delta\eta$.

The angle and pseudorapidity distribution can be written as

$\frac{d^{2}N}{d\Delta\phi d\Delta\eta}\propto
1+2\sum_{n}v_{n}(p_{T},\Delta\eta)\cos(n\Delta\phi)$ (3.1)

The pair correlation measurements for $pp$, $pPb$ and $PbPb$ collisions are
shown in Fig. 3.1 a), b), c). In all the cases, there are ridges in
$\Delta\phi\approx 0$ and $\Delta\phi\approx\pi$, with peaks in
$\Delta\eta\approx 0$. Since there is QGP formation in $PbPb$ collisions at
$\sqrt{s}=5.02\text{ TeV}$ and $pp$, $pPb$ high multiplicity collisions
present similar behaviour in the correlation measurements if compared to the
$PbPb$ case, the QGP formation seems to occur also in small systems.

In Eq.(3.1) the $v_{n}$ coefficients contains important information about the
collective behaviour. The first four ($v_{1}$,$v_{2}$,$v_{3}$,$v_{4}$), also
known as directed, elliptic, triangular and quadrangular flow coefficients,
reflect pressure gradients along the transverse plane. The $p_{T}$ dependence
of elliptical flux for high multiplicity $pp$ collisions, for pair
correlation, is shown in Fig. 3.1 d).

|
---|---
(a) | (b)
|
(c) | (d)
Figure 3.1: a) pp b) pPb and c) PbPb correlation measurements [41, 42, 43],
and elliptical flux coefficient $v_{2}$ [44].

Another interesting QGP signature is $J/\psi$ suppression. For heavy-ion
collisions, when the medium temperature is higher than the dissociation
temperature of the meson, this one is suppressed due to screening. The nuclear
modification factor as a function of $N_{part}$ (number of nucleons that
participate in the collisions) for $PbPb$ collisions at the LHC is show in
Fig. 3.2. As can be noted, as collisions get more central ($N_{part}$
increases), $J/\psi$ and $\psi(2S)$ mesons are suppressed. In Fig. 3.2 we can
see that for lower energies the suppression is more intense than in higher
energies, which means that quarkonia is regenerated. In principle, if there is
QGP formation in $pp$ collisions, an analogue pattern of suppression is
expected.

The $J/\psi$ relative yields per relative charged multiplicity at midrapidity
in $pp$ collisions were measured by ALICE collaboration at $\sqrt{s}=7\text{
TeV}$ and $\sqrt{s}=13\text{ TeV}$. The results are shown in Fig. 3.3. The
data present an anomalous behaviour. In the low multiplicity region
(peripheral collisions) they grow linearly as expected, but in the high
multiplicity region (UC collisions), production is enhanced. Since gluons are
mainly concentrated in central region of the proton, this growth may be
explained by their presence. In this work, as will be discussed latter, we
will try to interpret the data on $J/\psi$ production in UC region as a
manifestation of the baryon junction.

The presence of QGP in ultra-central $pp$ collisions is still an open question
and some general aspects were discussed in [40, 50, 51].

---
Figure 3.2: Charmonium suppression in $PbPb$ [45, 46, 47, 48] at LHC and
$AuAu$ collisions at RHIC [49].
---
(a)
(b)
Figure 3.3: $J/\psi$ relative yields per charged multiplicity at a)
$\sqrt{s}=7\text{ TeV}$ and b) $\sqrt{s}=13\text{ TeV}$. The experimental data
are from [1, 3].

##  Chapter 4 Glauber Model

Firstly proposed to deal with nucleus-nucleus collisions, the Glauber model
provides an interesting way of introducing geometrical variables in high
energy collisions. For a given nucleon number density distribution, it returns
the number of binary collisions between nucleons ($N_{coll}$) and the number
of nucleons that participate in the collisions ($N_{part}$) at a given impact
parameter. Because of the density input, the model can shed some light in the
internal structure of the entities involved in the collisions.

The nucleon motion and interactions during a collision can be really difficult
to predict, but some approximations can reduce the treatment to simple
integrals involving the density. Because of the high center of mass energy,
the nucleons carry sufficient momentum to be mainly undeflected while the
nuclei pass through each other. Also, it is assumed that the nucleus sizes are
much bigger than the range of the nucleon-nucleon forces [19]. In the next
section, the formalism of the model will be developed by considering the
approximations above.

###  4.1 Formalism

The geometrical formulation of the collision is shown in Fig. 4.1. $\vec{b}$
is the impact parameter, $\vec{s}$ is the position of a given nucleon flux
tube, $A$ is the target and $B$ is the projectile. The same two letters are
used to indicate the respective number of nucleons.

Figure 4.1: Glauber model geometry [19].

The probability per unit of inverse area of finding a given nucleon in the
target flux tube, or in the projectile flux tube, in a position $\vec{s}$, is,
respectively,

$T_{A}(\vec{s})=\int\rho_{A}(\vec{s},z_{A})dz_{A}$ (4.1)

and

$T_{B}(\vec{s})=\int\rho_{B}(\vec{s},z_{B})dz_{B}$ (4.2)

With $\rho_{A,B}$ as the respective densities. The same probabilities per
nucleon are

$\hat{T}_{A}(\vec{s})=\int\frac{\rho_{A}(\vec{s},z_{A})}{A}dz_{A}$ (4.3)

and

$\hat{T}_{B}(\vec{s})=\int\frac{\rho_{B}(\vec{s},z_{B})}{B}dz_{B}$ (4.4)

The probability per unit of area for the interaction of a nucleon from $A$ and
another from $B$ to occur in a overlap area can be calculated as

$\hat{T}_{AB}(\vec{b})=\int\hat{T}_{A}(\vec{s})\hat{T}_{B}(\vec{s}-\vec{b})d^{2}s$
(4.5)

The product $\hat{T}_{AB}(\vec{b})\sigma_{inel}^{NN}$ (which has no units)
represents the probability of an interaction ($\sigma_{inel}^{NN}$ is the
nucleon-nucleon inelastic cross section, which depends on the center of mass
energy). Along this line, the probability of having $n$ interactions in the
collision is

$P(n,\vec{b})={AB\choose
n}\big{(}\hat{T}_{AB}(\vec{b})\sigma_{inel}^{NN}\big{)}^{n}\big{(}1-\hat{T}_{AB}(\vec{b})\sigma_{inel}^{NN}\big{)}^{AB-n}$
(4.6)

The first term is the combination of $n$ collisions with $AB$ possibilities,
the second represents the probability of $n$ interactions, and the third
represents $AB-n$ missed interactions.

Considering non-polarized nucleons, $\vec{b}\rightarrow b$, and the
probability treatment leads to the calculation of the total number of binary
collisions

$N_{coll}=\sum_{n=1}^{AB}nP(n,b)=AB\hat{T}_{AB}(b)\sigma_{inel}^{NN}$ (4.7)

which is the probability of one interaction multiplied by the possible number
of interactions.

The number of participants (also called wounded nucleons) can be calculated as
[52, 53]

$\displaystyle N_{part}(\vec{b})$ $\displaystyle=$ $\displaystyle
A\int\hat{T}_{A}(\vec{s})\big{\\{}1-\big{(}1-\hat{T}_{B}(\vec{s}-\vec{b})\sigma_{inel}^{NN}\big{)}^{B}\big{\\}}d^{2}s$
(4.8) $\displaystyle+$ $\displaystyle
B\int\hat{T}_{B}(\vec{s}-\vec{b})\big{\\{}1-\big{(}1-\hat{T}_{A}(\vec{s})\sigma_{inel}^{NN}\big{)}^{A}\big{\\}}d^{2}s$
(4.9)

Sometimes, mainly because the calculation of $N_{part}$ is demanding, it is
useful to estimate it according to the scaling behaviour of Fig. 7 of [19] as:

$N_{part}(b)\propto N_{coll}^{3/4}(b)$ (4.10)

###  4.2 Pb-Pb Collisions

The heavy ion nucleon density shape is usually assumed to be the Woods-Saxon

$\rho(r)=\rho_{0}\frac{1+w(r/R)^{2}}{1+\exp\big{(}\frac{r-R}{a}\big{)}}$
(4.11)

where $\rho_{0}$ is the nucleon density in the center of the nucleus, $R$ is
the radius of the nucleus, $w$ represents deviations from the sphere format,
and $a$ is the skin depth. The normalization constant $\rho_{0}$ must be such
that

$\int d^{3}\mathbf{r}\rho_{A}(\mathbf{r})=A$ (4.12)

The $Pb$ ($A=208$) density distribution is shown in Fig. 4.2. The hard-sphere
density distribution is also shown in the figure. It can be seen that the
nucleon density is larger in the center of the nucleus (for small values of
$r$), and decreases as the distance from the center increases.

Figure 4.2: $Pb$ density with $w=0$, $R=6.62\text{ fm}$ and $a=0.546\text{
fm}$ [56].

Using the Woods-Saxoon distribution in the formalism developed in the previous
section for $Pb-Pb$ collisions at $\sqrt{s}_{NN}=2.76\text{ TeV}$ and
$\sqrt{s}_{NN}=5.02\text{ TeV}$, specially in Eq.(4.7) and Eq.(4.9), the model
leads to the results shown in Fig. 4.3. The inelastic nucleon cross sections
used in the calculations are $\sigma_{inel}^{NN}=64\text{ mb}$ and
$\sigma_{inel}^{NN}=70\text{ mb}$ [54], respectively. As it was expected, the
larger values of $N_{part}$ and $N_{coll}$ occur for small $b$ collisions (UC
collisions), and decrease as $b$ increases, becoming almost null for $b\approx
2R$ (ultra-peripheral collisions). In central collisions the matter
superposition is much bigger than in ultra-peripheral collisions. Also, it can
be seen that $N_{part}$, which is approximated to $A+B$ (total number of
nucleons) for $b=0$, does not change with $\sqrt{s}_{NN}$, but $N_{coll}$
does.

Figure 4.3: $N_{part}$ and $N_{coll}$ dependence on b for $Pb-Pb$ collisions
at $\sqrt{s}_{NN}=2.76\text{ TeV}$ and $\sqrt{s}_{NN}=5.02\text{ TeV}$.

It would be useful to represent these quantities as functions of the central
rapidity density to adapt the model to the experimental language, and test its
applicability. According to [55], the central pseudorapidity density can be
written as

$\frac{dN}{d\eta}=n_{pp}(s)\big{\\{}(1-f)\frac{N_{part}(b)}{2}+fN_{coll}(b)\big{\\}}$
(4.13)

where the $f$ parameter is chosen to calibrate the hard particle production,
with $(1-f)$ making reference to the soft process fraction; $n_{pp}(s)$ is the
multiplicity density, given by [57]

$n_{pp}=2.5-0.25\log[s]+0.023(\log[s])^{2}$ (4.14)

For $Pb-Pb$ collisions at $\sqrt{s}=2.76\text{ TeV}$ and $\sqrt{s}=5.02\text{
TeV}$, the experimental and theoretical data are plotted in Fig. 4.4. The
central pseudorapidity density is calculated from Eq.(4.13) and $N_{part}$,
$N_{coll}$, with the Glauber model shown in Fig. 4.3. For the first center of
mass energy, the hard fraction was set to $f=10\%$, and for the second,
$f=11.5\%$. The charged particle pseudorapidity density at midrapidity can be
well fitted by the Glauber model and Eq.(4.13).

Figure 4.4: $Pb-Pb$ charged pseudorapidity density at midrapidity collision
data. The experimental data are from [58] and [59].

###  4.3 pp Collisions

The Glauber model developed for $AA$ collisions can be adapted to $pp$
collisions by making some considerations. In the line of the development made
in the previous section, the first intuitive approach could be approximating
the proton as a nucleus composed by one nucleon. In principle, it is not
necessarily wrong, but the Woods-Saxon density profile can hide some important
features about the internal structure of the proton, like charge and
anisotropic distribution of mass, and subnucleonic effects become more
important [60, 61].

|  |
---|---|---
(a) |  | (b)
Figure 4.5: a) Charge distribution of the proton measured in elastic electron
proton scattering at low energies and $Q^{2}\simeq 0$. b) Charge distribution
suggested by the Y shape model of the proton.

In [62, 63, 64] the Renormalization Group Procedure for Effective Particles
(REGPEP) was proposed as a way to deal with the spatial evolution of the
proton structure for $Q^{2}\gg\Lambda_{QCD}^{2}$. In this approach the proton
evolves to a configuration with three effective quarks (quarks + antiquarks +
gluons), disposed in the vertices of an equilateral triangle, with star-like
junction (baryon junction) between them. The charge distribution changes,
qualitatively, to the one presented in Fig.4.5b. REGPEP had already been
successfully applied to explain other aspects of pp collisions, like
anisotropic flow and multiplicity [21, 65, 66].

Taking all the electromagnetic and gluon effects discussed in chapter 2 into
account, then going back to the REGPEP, the subnucleonic approach of Glauber
model leads to a different meaning of $N_{part}$ and $N_{coll}$. They change
to “number of partons that participate in the collision” and “number of binary
collisions between partons” for a given impact parameter $b$. Also, all the
approximations made to the nucleon motion in the nucleus in the Glauber Model
are transposed to the parton motion in the proton. That is: because of the
high energy of the protons, the partons carry sufficient momentum to be mainly
undeflected while the protons pass through each other. Also, their motion is
considered to be independent from the proton one and the proton size is much
bigger than the range of parton-parton forces.

The parton density can be given by[21]:

$\rho_{p}(\mathbf{r};\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3})=\sum_{i=1}^{3}\rho_{q}(\mathbf{r}-\mathbf{r}_{i})+\rho_{g}\bigg{(}\mathbf{r}-\sum_{i=1}^{3}\frac{\mathbf{r}_{i}}{3}\bigg{)}$
(4.15)

where $i=1,2,3$ are the three effective quarks, with the Gaussian distribution

$\rho_{q}(r)=(1-\kappa)\frac{N_{g}}{3}\frac{e^{-r^{2}/2r_{q}^{2}}}{(2\pi)^{3/2}r_{q}^{3}}$
(4.16)

and a Gaussian gluon distribution, peaked in the average coordinate of
$\mathbf{r}_{1}$, $\mathbf{r}_{2}$ and $\mathbf{r}_{3}$, given by:

$\rho_{g}(r)=\kappa N_{g}\frac{e^{-r^{2}/2r_{g}^{2}}}{(2\pi)^{3/2}r_{g}^{3}}$
(4.17)

In the previous formulas, the parameters are: the radius of the effective
quarks $r_{q}$; the extension of the gluon body $r_{g}$; the degrees of
freedom, or number of partons, $N_{g}$; the fraction of partons that
corresponds to gluon $\kappa$. The density plots of the distributions in the
transverse plane are shown in Fig.4.6.

|  |
---|---|---
(a) |  | (b)
Figure 4.6: a) Quark and b) gluon distributions in the transverse plane for
the parameters of set 1 on Table 4.1.

Fig.4.7 illustrates the “core-corona” aspect of the distribution. In the
center (core) the gluon distribution dominates, but in the periphery (corona),
the quark distribution is greater. This configuration has some consequences
that will be better explored later.

Figure 4.7: Quark and gluon distributions for the parameters of set 1 on Table
4.1. Projection of Fig. 4.6 along a diagonal direction, starting from the
center of the proton and passing through one maximum of the quark density.

Adapting the Glauber formalism, the probability per area of finding a parton
in the proton flux tube is given by

$T_{p}(x,y)=\int\rho_{p}(x,y,z)dz=T_{p}^{q}(x,y)+T_{p}^{g}(x,y)$ (4.18)

with $T_{p}^{q}(x,y)$ and $T_{p}^{g}(x,y)$ from the quark and gluon terms of
Eq.(4.15). Explicitly, they are given by:

$T_{p}^{q}(x,y)=\frac{N_{g}(1-\kappa)}{6\pi
r_{q}^{2}}\sum_{i=1}^{3}e^{-\frac{(x-x_{i})^{2}+(y-y_{i})^{2}}{2r_{q}^{2}}}$
(4.19)

and

$T_{p}^{g}(x,y)=\frac{N_{g}\kappa}{2\pi
r_{g}^{2}}e^{-\frac{(x-\sum_{i=1}^{3}x_{i}/3)^{2}+(y-\sum_{i=1}^{3}y_{i}/3)^{2}}{2rg^{2}}}$
(4.20)

The thickness function is given by

$T_{pp\prime}(b)=\int T_{p}(x-b/2,y)T_{p\prime}(x+b/2,y)dxdy$ (4.21)

This equation can be splitted into four analytic terms. The interaction
between the quarks $i=1,2,3$ from $p$ and quarks $j=1,2,3$ from $p\prime$ (the
quark-quark term), is given by:

$\displaystyle T_{pp\prime}^{qq}(b)$ $\displaystyle=$ $\displaystyle\int
T_{p}^{q}(x-b/2,y)T_{p\prime}^{q}(x+b/2,y)dxdy$ (4.22) $\displaystyle=$
$\displaystyle\frac{N_{g}^{2}(1-\kappa)}{36\pi
r_{q}^{2}}\sum_{i=1}^{3}\sum_{j=1}^{3}\exp\bigg{(}-\frac{\big{(}b+(x_{i}-x_{j})\big{)}^{2}+\big{(}y_{i}-y_{j}\big{)}^{2}}{4rq^{2}}\bigg{)}$

The gluon-gluon term can be calculated as:

$\displaystyle T_{pp\prime}^{gg}(b)=\int
T_{p}^{g}(x-b/2,y)T_{p\prime}^{g}(x+b/2,y)dxdy$ $\displaystyle=$
$\displaystyle\frac{N_{g}^{2}\kappa^{2}}{4\pi
r_{g}^{2}}\exp\bigg{(}-\frac{\big{(}b+\sum_{i=1}^{3}x_{i}/3-\sum_{j=1}^{3}x_{j}/3\big{)}^{2}+\big{(}\sum_{i=1}^{3}y_{i}/3-\sum_{j=1}^{3}y_{j}/3\big{)}^{2}}{4rg^{2}}\bigg{)}$

The gluon-quark term (interactions between the gluons from $p$ and quarks from
$p\prime$) is calculated as:

$\displaystyle T_{pp\prime}^{gq}(b)$ $\displaystyle=$ $\displaystyle\int
T_{p}^{g}(x-b/2,y)T_{p\prime}^{q}(x+b/2,y)dxdy$ $\displaystyle=$
$\displaystyle\frac{N_{g}^{2}\kappa(1-\kappa)}{6\pi(r_{q}^{2}+r_{g}^{2})}\sum_{j=1}^{3}\exp\bigg{(}-\frac{\big{(}b+\sum_{i=1}^{3}x_{i}/3-x_{j}\big{)}^{2}+\big{(}\sum_{i=1}^{3}y_{i}/3-y_{j}\big{)}^{2}}{2(r_{q}^{2}+r_{g}^{2})}\bigg{)}$

Finally, the quark-gluon term is:

$\displaystyle T_{pp\prime}^{qg}(b)$ $\displaystyle=$ $\displaystyle\int
T_{p}^{q}(x-b/2,y)T_{p\prime}^{g}(x+b/2,y)dxdy$ $\displaystyle=$
$\displaystyle\frac{N_{g}^{2}\kappa(1-\kappa)}{6\pi(r_{q}^{2}+r_{g}^{2})}\sum_{i=1}^{3}\exp\bigg{(}-\frac{\big{(}b+x_{i}-\sum_{j=1}^{3}x_{j}/3\big{)}^{2}+\big{(}y_{i}-\sum_{j=1}^{3}y_{j}/3\big{)}^{2}}{2(r_{q}^{2}+r_{g}^{2})}\bigg{)}$

In the end, inserting the terms above into Eq.(4.21), the total thickness is:

$T_{pp\prime}(b)=T_{pp\prime}^{qq}(b)+T_{pp\prime}^{gg}(b)+T_{pp\prime}^{gq}(b)+T_{pp\prime}^{qg}(b)$
(4.26)

The number of binary collisions between partons is:

$N_{coll}=T_{pp\prime}(b)\sigma^{pp\prime}$ (4.27)

where $\sigma^{pp\prime}$ is the parton-parton cross section and the number of
participants is calculated according to Eq.(4.10).

Since the calculation above needs effective quarks positions inputs in the
transverse plane, we will adopt the following procedure: the quarks
$\mathbf{r}_{i}$ from the projectile $p$, and $\mathbf{r}\prime_{i}$ from the
target $p\prime$, will be positioned according to

$\mathbf{r}_{i}=\frac{d}{2}\big{(}\cos(\phi_{i}+\alpha),\sin(\phi_{i}+\alpha)\big{)}\,\,\,\,\,\,\,\,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\,\,\,\,\,\,\,\mathbf{r^{\prime}}_{i}=\frac{d}{2}\big{(}\cos(\phi_{i}+\beta),\sin(\phi_{i}+\beta)\big{)}$
(4.28)

In the formulas above, $\phi_{1}=\pi/3$, $\phi_{2}=-\pi/3$ and
$\phi_{3}=-\pi$. This choice of coordinates positions the quarks in the
vertices of an equilateral triangle, with the angles $\alpha$ and $\beta$
randomly chosen in the range $\alpha\text{, }\beta\in[0,2\pi]$ in order to
rotate them around the $z$ axis. In Fig. 4.8 we show one example of initial
condition.

Figure 4.8: Example of initial spatial configuration of the two colliding
protons.

The computational procedure is the following: for a given impact parameter
$b$, the angles $\alpha$ and $\beta$ are chosen. After these choices, the
coordinates of Eq.(4.28) are set (initial condition) and thickness function of
Eq.(4.26) is calculated. Then the thickness is applied in Eq.(4.27) and,
indirectly, in Eq.(4.10) to calculate $N_{coll}$ and $N_{part}$. For each
impact parameter there are 10000 initial condition choices and the final value
for the outputs are averages of $N_{coll}$ and $N_{part}$ of these
configurations. The impact parameter is such that $b\in[0,2.2]\text{ fm}$,
with steps $\Delta b=0.1\text{ fm}$.

The thickness functions and Glauber outputs for the parameters sets of Table
4.1 are shown from Fig. 4.9 to 4.11. The parameters that do no change from one
set to another are $\kappa=0.5$, $r_{q}=0.25\text{ fm}$, $d=1.3\text{ fm}$
[18] and $r_{g}=0.5\text{ fm}$ [21]. All the parameters were taken from
previous works. These Glauber outputs will be used in charm production that
will be better discussed in chapter 5.

Table 4.1: Effective number of partons and parton-parton cross sections used in the Glauber model. | Set 1 | Set 2 | Set 3
---|---|---|---
$N_{g}$ |  $10$ [18] |  $7$ [60] |  $5$ [60]
$\sigma^{pp}\text{ (mb)}$ $(5.02\text{ TeV})$ |  $2.8$ [60] |  $5.7$ [60] |  $10.3$[60]
$\sigma^{pp}\text{ (mb)}$ $(7\text{ TeV})$ |  $4.3$ [67] |  $6.5$ [60] |  $11.4$[60]
$\sigma^{pp}\text{ (mb)}$ $(13\text{ TeV})$ |  $7.6$ [67] |  $7.4$ [60] |  $12.7$ [60]
---
Figure 4.9: Thickness functions and average of $N_{part}$ and $N_{coll}$ over
different quark spatial configurations, at $\sqrt{s}=7\text{ TeV}$ and
$\sqrt{s}=13\text{ TeV}$, for the parameter set 1.
---
Figure 4.10: Thickness functions and average of $N_{part}$ and $N_{coll}$ over
different quark spatial configurations, at $\sqrt{s}=7\text{ TeV}$ and
$\sqrt{s}=13\text{ TeV}$, for the parameter set 2.
---
Figure 4.11: Thickness functions and average of $N_{part}$ and $N_{coll}$ over
different quark spatial configurations, at $\sqrt{s}=7\text{ TeV}$ and
$\sqrt{s}=13\text{ TeV}$, for the parameter set 3.

###  4.4 pA Collisions

Another interesting situation where the Glauber model can be applied is in
proton-nucleus collisions. As in the proton case, subnucleon degrees of
freedom become important during the collision, that is, the internal structure
of the nucleons can also play an important role. We will attempt to develop
the model of the previous sections and adapt it to $p-Pb$ collisions.

Assuming that all the nucleons have the same number density distribution of
Fig. 4.6 (Eq.(4.15)), the positions of the nucleons inside the nucleus, in the
transverse plane, can be chosen as follows: the radial position of the nucleon
in the transverse plane $s$ and the azimuthal angle $\gamma$ are randomly
chosen in the range $0<s<R=6.62\text{ fm}$ and $0<\gamma<2\pi$. After choosing
the radial position, the nucleon can be rotated around the $z$ axis by a angle
$\alpha$, such that $0<\alpha<2\pi$, like in the $pp$ case shown in Fig. 4.8.
The procedure is repeated for the $208$ nucleons. These assumptions, together
with the usual Glauber ones, leads to initial conditions like in Fig. 4.12.

Figure 4.12: Example of initial spatial configuration in the transverse plane
of the $Pb$ nucleus. The white regions are high multiplicity points.

Then the $Pb$ density will be calculated as the sum of $208$ densities of
Eq.(4.15) as:

$\rho_{Pb}(\mathbf{r})=\sum_{i=1}^{208}\rho_{p_{i}}(\mathbf{r};\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{3})$
(4.29)

and each quark of each nucleon will be positioned according to

$\mathbf{r}_{i}=\bigg{(}\frac{d}{2}\cos(\phi_{i}+\alpha)+s\cos(\gamma)\mathbf{,}\sin(\phi_{i}+\alpha)+s\sin(\gamma)\bigg{)}$
(4.30)

Then, proceeding with Glauber Formalism, the probability per area of finding a
parton in the flux tube is given by:

$\displaystyle T_{Pb}(x,y)$ $\displaystyle=$
$\displaystyle\int\sum_{i}^{208}\rho_{p_{i}}(x,y,z)dz=\sum_{i}^{208}\big{(}T_{p_{i}}^{q}(x,y)+T_{p_{i}}^{g}(x,y)\big{)}$
(4.31) $\displaystyle=$ $\displaystyle T_{Pb}^{q}(x,y)+T_{Pb}^{g}(x,y)$

where $T_{p_{i}}^{q}(x,y)$ and $T_{p_{i}}^{g}(x,y)$ are given by Eq.(4.19) and
Eq.(4.20).

The thickness function is calculated as:

$T_{pPb}(b)=\int T_{p}(x-b/2,y)T_{Pb}(x+b/2,y)dxdy$ (4.32)

As before, the thickness can be splitted into four analytic terms. The quark-
quark is:

$\displaystyle T_{pPb}^{qq}(b)$ $\displaystyle=$ $\displaystyle\int
T_{p}^{q}(x-b/2,y)T_{pPb}^{q}(x+b/2,y)dxdy$ (4.33) $\displaystyle=$
$\displaystyle\sum_{i}^{208}T_{pp_{i}}^{qq}(b)$

where $T_{pp_{i}}^{qq}(b)$ is given by Eq.(4.22). The gluon-gluon term is:

$\displaystyle T_{pPb}^{gg}(b)$ $\displaystyle=$ $\displaystyle\int
T_{p}^{g}(x-b/2,y)T_{Pb}^{g}(x+b/2,y)dxdy$ (4.34) $\displaystyle=$
$\displaystyle\sum_{i}^{208}T_{pp_{i}}^{gg}(b)$

with $T_{pp_{i}}^{gg}(b)$ given by Eq.(4.3). The gluon-quark and quark-gluon
terms are, respectively:

$\displaystyle T_{pPb}^{gq}(b)$ $\displaystyle=$ $\displaystyle\int
T_{p}^{g}(x-b/2,y)T_{Pb}^{q}(x+b/2,y)dxdy$ (4.35) $\displaystyle=$
$\displaystyle\sum_{i}^{208}T_{pp_{i}}^{gq}(b)$

and

$\displaystyle T_{pPb}^{qg}(b)$ $\displaystyle=$ $\displaystyle\int
T_{p}^{q}(x-b/2,y)T_{Pb}^{g}(x+b/2,y)dxdy$ (4.36) $\displaystyle=$
$\displaystyle\sum_{i}^{208}T_{pp_{i}}^{qg}(b)$

where $T_{pp_{i}}^{gq}(b)$ and $T_{pp_{i}}^{qg}(b)$ are given by Eq.(4.3) and
Eq.(4.3).

The thickness is then calculated as:

$\displaystyle T_{pPb}(b)$ $\displaystyle=$ $\displaystyle
T_{pPb}^{qq}(b)+T_{pPb}^{gg}(b)+T_{pPb}^{gq}(b)+T_{pPb}^{qg}(b)$ (4.37)
$\displaystyle=$
$\displaystyle\sum_{i}^{208}\big{(}T_{pp_{i}}^{qq}(b)+T_{pp_{i}}^{gg}(b)+T_{pp_{i}}^{gq}(b)+T_{pp_{i}}^{qg}(b)\big{)}$

In the end, the thickness can be written as individual interactions between
the projectile proton and the target nucleons that form the heavy nucleus.
Also like before, the number of binary collisions can be calculated as

$N_{coll}(b)=T_{pPb}(b)\sigma^{pp\prime}$ (4.38)

where $\sigma^{pp\prime}$ is the parton-parton inelastic cross section and
$N_{part}(b)$ can be approximated as Eq.(4.10).

The thickness and Glauber outputs for $pPb$ collisions for the parameter sets
of Table 4.1, averaged over $200$ initial conditions for each impact
parameter, are shown from Fig. 4.13 to 4.15. The computational proceeding is
almost the same adopted in the $pp$ case, but with the choices of $s$ and
$\gamma$ for each nucleon inside the nucleus and the impact parameter is in
the range $[0,10]\text{ fm}$, with steps $\Delta b=0.5\text{ fm}$.

---
Figure 4.13: Thickness functions and average of $N_{part}$ and $N_{coll}$ over
different quark and nucleon spatial configurations, at $\sqrt{s}=5.02\text{
TeV}$, for the parameter set 1.
---
Figure 4.14: Thickness functions and average of $N_{part}$ and $N_{coll}$ over
different quark and nucleon spatial configurations, at $\sqrt{s}=5.02\text{
TeV}$, for the parameter set 2.
---
Figure 4.15: Thickness functions and average of $N_{part}$ and $N_{coll}$ over
different quark and nucleon spatial configurations, at $\sqrt{s}=5.02\text{
TeV}$, for the parameter set 3.

##  Chapter 5 Charm Production

In this work, the charmonium production will by accounted for by the Color
Evaporation Model (CEM) [22, 23]. In this model, the mass of the produced
heavy quark pair must be between twice the quark mass and twice the lightest
meson that can be formed with the heavy quark, and since the pair production
has a non-perturbative component, one assumes that color neutrality is
achieved by the “evaporation” of initial colour fields. In the end, the
quarkonium cross section is written in terms of the partonic cross section and
the overlap function. Therefore the model works as a good tool to study the
influence of the internal structure of the colliding hadrons in heavy quark
production.

Understanding quarkonium production is still a work in progress and there is
no precise description valid for all kinematic ranges. For example, CEM
overshoots data in the large transverse-mass regime [68, 69]. The calculation
presented below will be integrated in $p_{T}$ and we expect that the high
$p_{T}$ discrepancy will not strongly affect the results because the curve
falls rapidly in this regime.

###  5.1 Charm production in the CEM

Considering that $c\bar{c}$ pairs are produced through the process
$gg\rightarrow c\bar{c}$ and $q\bar{q}\rightarrow c\bar{c}$, the total cross
section can be written as:

$\displaystyle\sigma^{CEM}$ $\displaystyle=$
$\displaystyle\mathbf{F}K\sum_{i,j}\int_{(2m_{c})^{2}}^{(\Lambda)^{2}}dm^{2}\int
dx_{1}dx_{2}f_{i}(x_{1},\mu_{F}^{2})f_{j}(x_{2},\mu_{F}^{2})\sigma_{ij}(m^{2},\mu_{R}^{2})\delta(m^{2}-x_{1}x_{2}s)$
(5.1) $\displaystyle=$
$\displaystyle\sigma_{gg}^{CEM}+\sigma_{q\bar{q}}^{CEM}$

In this total cross section, $f_{i}(x_{1},\mu_{F}^{2})$ and
$f_{j}(x_{2},\mu_{F}^{2})$ are parton distribution functions calculated at
$\mu_{F}$ (factorization scale), $\sigma_{ij}(m^{2},\mu_{R}^{2})$ are the
parton-parton cross sections calculated at $\mu_{R}$ (renormalization scale)
and $F$ is the percentage of $c\bar{c}$ which becomes $J/\psi$. The
integration limit $\Lambda$ works as a cut-off. For open charm production,
$\Lambda=\sqrt{s}$, and for $J/\psi$, $\Lambda=2m_{D}=2(1.8)\text{ GeV}^{2}$.

For our purposes the leading order elementary cross sections will be
sufficient. For the relevant processes, they are given by

$\sigma_{gg}(m^{2},\mu_{R}^{2})=\frac{\pi\alpha_{s}^{2}(\mu_{R}^{2})}{3m^{2}}\bigg{\\{}\big{(}1+\frac{4m_{c}^{2}}{m^{2}}+\frac{m_{c}^{4}}{m^{4}}\big{)}\ln\bigg{(}\frac{1+\lambda}{1-\lambda}\bigg{)}-\frac{1}{4}\big{(}7+\frac{31m_{c}^{2}}{m^{2}}\big{)}\lambda\bigg{\\}}$
(5.2)

and

$\sigma_{q\bar{q}}(\mu_{R}^{2})=\frac{8\pi\alpha_{s}^{2}(m^{2})}{27m^{2}}\bigg{(}1+\frac{2m_{c}^{2}}{m^{2}}\bigg{)}\lambda$
(5.3)

The parameter $K$ is introduced to include the effect of higher order
corrections. The coupling constant is given by

$\alpha_{s}(\mu_{R}^{2})=\frac{12\pi}{(33-2n_{f})\ln\big{(}\frac{\mu_{R}^{2}}{\Lambda^{2}}\big{)}}$
(5.4)

where $n_{f}$ is the number of flavours involved in the collision and,
tipically, $\Lambda=200\text{ MeV}$. The renormalization scale will be chosen
to be equal to the factorization one, which will be twice the mass of charm
quark ($\mu_{R}=\mu_{F}=\mu=2m_{c})$. The $\lambda$ term is defined as

$\lambda=\bigg{(}1-\frac{(2m_{c})^{2}}{m^{2}}\bigg{)}^{1/2}$ (5.5)

Imposing that $\lambda$ is real and applying the $\delta$ function properties
in Eq.(5.1), the integration limits must satisfy

$1-\frac{4m_{c}^{2}}{x_{1}x_{2}s}\geq 0$ (5.6)

Noticing that the smallest value of $x_{1}$ occurs for $x_{2}=1$, the parton
momentum fractions must be such that $(2m_{c})^{2}/s\leq x_{1}\leq 1$ and
$(2m_{c})^{2}/(sx_{1})\leq x_{2}\leq 1$. In the end, the multiplicity is given
by

$N_{CEM}^{c\bar{c}}=T_{AB}(b)\sigma^{CEM}$ (5.7)

###  5.2 Open Charm Production

The open charm production cross section can be calculated by taking out the
parameter $F$ in Eq.(5.1) with the parton distribution functions from the MMHT
14 set [70]. For $Q^{2}=\mu^{2}=(2m_{c})^{2}$, the PDFs are plotted in Fig.
5.1.

Figure 5.1: PDFs for $Q^{2}=(2.4)^{2}\text{ GeV}^{2}$ [70].

In order to fix the $K$ parameter of Eq.(5.1), the open charm $c\bar{c}$ cross
section is fitted to existing data, as shown in to Fig.5.2. The optimal value
is $K=3$. As can be seen in the figure, $\sigma_{gg}\gg\sigma_{q\bar{q}}$
which means that charm production is almost only due to gluon interactions. In
Fig. 5.3 the cross section is plotted by varying the charm invariant mass in
the range $1.1\text{ GeV}<m_{c}<1.3\text{ GeV}$ in order to estimate the
uncertainties.

Figure 5.2: Charm production cross sections for $m_{c}=1.2\text{ GeV}$.
Contribution of $gg\rightarrow c\bar{c}$ (dashed line), $q\bar{q}\rightarrow
c\bar{c}$ (dotted line) and total cross section (solid line). The experimental
data are from [71, 72, 73, 74]. Figure 5.3: Charm production cross sections
for $1.1\text{ GeV}<m_{c}<1.3\text{ GeV}$. The experimental data are from [71,
72, 73, 74].

###  5.3 $J/\psi$ Production

The ALICE collaboration published some data on $c\bar{c}$ and $J/\psi$
production over the last decade [2, 1, 3]. In these works, charm production
was presented in terms of the relative yields by means of the normalized
charged pseudorapidity and rapidity densities:

$\frac{({dN}/{d\eta}(\eta=0))}{\langle{dN}/{d\eta}\rangle}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{({dN}/{dy}(y=0))}{\langle{dN}/{dy}\rangle}$
(5.8)

In proton-proton collisions, the color evaporation model formula Eq.(5.7) can
be splitted into a gluon term and a quark-antiquark term

$N_{c\bar{c}}(b)=T_{pp\prime}^{gg}(b)\,\sigma_{gg}^{CEM}+T_{pp\prime}^{qq}(b)\,\sigma_{q\bar{q}}^{CEM}$
(5.9)

This can be done because the gluon-gluon and quark-quark overlaps can be
separated into Eq.(4.22) and (4.3). The gluon-quark terms are not included
because these processes do not produce heavy quarks. Eq.(5.9) can be
differentiated with respect to y as:

$\frac{dN_{c\bar{c}}}{dy}(b)=T_{pp\prime}^{gg}(b)\,\frac{d\sigma_{gg}^{CEM}}{dy}+T_{pp\prime}^{qq}(b)\,\frac{d\sigma_{q\bar{q}}^{CEM}}{dy}$
(5.10)

We will apply the following the change of variables

$x_{1}=\frac{p_{T}}{\sqrt{s}}e^{y}\,\,\,\,\,\,\,\,\,\,\,\,x_{2}=\frac{p_{T}}{\sqrt{s}}e^{-y}\,\,\,\,\,\,\,\,\,\,\,\,x_{1}x_{2}s=p_{T}^{2}\,\,\,\,\,\,\,\,\,\,\,\,dx_{1}dx_{2}\rightarrow\frac{2p_{T}}{s}dydp_{T}$
(5.11)

to Eq.(5.1). After the change of variables
$(x_{1},x_{2})\rightarrow(y,p_{T})$, the cross section can be written as

$\sigma^{CEM}=\mathbf{F}K\sum_{i,j}\int_{(2m_{c})^{2}}^{(2m_{D})^{2}}dm^{2}\int
dydp_{T}\frac{2p_{T}}{s}f_{i}(y,p_{T},\mu_{F}^{2})f_{j}(y,p_{T},\mu_{F}^{2})\sigma_{ij}(m^{2},\mu_{R}^{2})\delta(m^{2}-p_{T}^{2})$
(5.12)

By noticing that $2p_{T}dp_{T}=dp_{T}^{2}$, and using the delta function
properties to perform the integration in $p_{T}$, we find

$\sigma^{CEM}=\mathbf{F}K\sum_{i,j}\int_{(2m_{c})^{2}}^{(2m_{D})^{2}}dm^{2}\int
dy\frac{1}{s}f_{i}(y,m,\mu_{F}^{2})f_{j}(y,m,\mu_{F}^{2})\sigma_{ij}(m^{2},\mu_{R}^{2})$
(5.13)

So, the $y$ distribution can be calculated as:

$\frac{d\sigma^{CEM}}{dy}=\mathbf{F}K\sum_{i,j}\int_{(2m_{c})^{2}}^{(2m_{D})^{2}}dm^{2}\frac{1}{s}f_{i}(y,m,\mu_{F}^{2})f_{j}(y,m,\mu_{F}^{2})\sigma_{ij}(m^{2},\mu_{R}^{2})$
(5.14)

The central density can be calculated according to Eq.(4.13), with $N_{part}$
and $N_{coll}$ from the Glauber model sets of Table 4.1. Using the Glauber
parameters of Table 4.1 with the respective CEM parameters of Table 5.1, the
experimental data for the three sets can be fitted as shown in Figs. 5.4 to
5.9. The rapidity density of Eq.(5.14) is calculated in $y=0$ and, as in the
open charm case, $\mu_{F}=\mu_{R}=2m_{c}$. The uncertainties are estimated by
varying the charm mass in the range $1.1\text{ GeV}<m_{c}<1.3\text{ GeV}$.

Table 5.1: CEM parameter set for $pp$ collisions. | Set 1
---|---
f ($\sqrt{s}=7$ TeV) | 19 %
f ($\sqrt{s}=13$ TeV) | 16 %
F ($\sqrt{s}=7$ TeV) | 2.3 %
F ($\sqrt{s}=13$ TeV) | 3.4 %
$\langle dN/d\eta\rangle$ ($\sqrt{s}=7$ TeV) |  $6.0$ [75]
$\langle dN/d\eta\rangle$ ($\sqrt{s}=13$ TeV) |  $6.4$ [76]
$\langle dN_{J/\psi}/dy\rangle$ ($\sqrt{s}=7$ TeV) |  $8.2\times 10^{-5}$ [1]
$\langle dN_{J/\psi}/dy\rangle$ ($\sqrt{s}=13$ TeV) |  $7.9\times 10^{-5}$ [3]

As can be seen in Figs. 5.4 and 5.5, the experimental data are better fitted
with the parameter Set 1. This set presents a linear growth in the low charged
multiplicity region (peripheral collisions) and it grows faster in the high
charged multiplicity region (ultra-central). Following the “core-corona” model
presented in Fig.(4.7), the quark interactions dominate in the peripheral
region, and the gluon ones dominate in the ultra-central region. Taking a look
at Fig.5.2, we notice that $\sigma_{gg}^{CEM}\gg\sigma_{q\bar{q}}^{CEM}$. So,
as the gluon interactions become more important, the production of $J/\psi$
grows faster because of the huge difference between the cross sections.

In Figs. 5.6 and 5.7, the second parameter set fits the low charged
multiplicity data (ultra-peripheral data), but not the UC data. In Fig.5.8 and
5.9, the third set is not appropriate to fit the data. It occurs because the
number of partons, respectively 7 and 5, combined with the respectives parton-
parton cross sections, are not sufficient to generate the high multiplicity
data. In other words, the model is strongly sensitive to the degrees of
freedom and to parton-parton cross sections.

In summary, since the set with the greatest $N_{g}$ and smallest $\sigma^{pp}$
(Set 1) presents the best explanation for the experimental data, we can say
that the collision is better explained through a larger number of point-like
particles interactions.

Figure 5.4: a) $J/\psi$ relative yield for $\sqrt{s}=7\text{ TeV}$, set 1. The
experimental data are from [1]. The dashed curve is plotted with
$m_{c}=1.2\text{ GeV}$. Figure 5.5: $J/\psi$ relative yield for
$\sqrt{s}=13\text{ TeV}$, set 1. The experimental data are from [3]. The
dashed curve is plotted with $m_{c}=1.2\text{ GeV}$. Figure 5.6: a) $J/\psi$
relative yield for $\sqrt{s}=7\text{ TeV}$, set 2. The experimental data are
from [1]. The dashed curve is plotted with $m_{c}=1.2\text{ GeV}$. Figure 5.7:
$J/\psi$ relative yield for $\sqrt{s}=13\text{ TeV}$, set 2. The experimental
data are from [3]. The dashed curve is plotted with $m_{c}=1.2\text{ GeV}$.
Figure 5.8: a) $J/\psi$ relative yield for $\sqrt{s}=7\text{ TeV}$, set 3. The
experimental data are from [1]. The dashed curve is plotted with
$m_{c}=1.2\text{ GeV}$. Figure 5.9: $J/\psi$ relative yield for
$\sqrt{s}=13\text{ TeV}$, set 3. The experimental data are from [3]. The
dashed curve is plotted with $m_{c}=1.2\text{ GeV}$.

###  5.4 $J/\psi$ Production in pA Collisions

Charmonium production in pA collisions can be calculated in the same way as in
$pp$ collisions. The relative yields can be calculated as:

$\frac{dN_{c\bar{c}}}{dy}(b)=T_{pPb}^{gg}(b)\,\frac{d\sigma_{gg}^{CEM}}{dy}+T_{pPb}^{qq}(b)\,\frac{d\sigma_{qq}^{CEM}}{dy}$
(5.15)

with the thickness from Eqs.(4.33,4.34) and the $y$ distributions of
Eq.(5.14). Using the Glauber parameters of Table 4.1 with the respective CEM
of Table 5.2, the results are shown from Fig. 5.10 to 5.12.

Table 5.2: CEM parameter set for $pPb$ collisions. f ($\sqrt{s}=5.02$ TeV) | 20%
---|---
F ($\sqrt{s}=5.02$ TeV) | 2.5%
$\langle dN/d\eta\rangle$ ($\sqrt{s}=5.02$ TeV) |  $17.6$ [77]
$\langle dN_{J/\psi}/dy\rangle$ ($\sqrt{s}=5.02$ TeV) |  $1.4\times 10^{-4}$ [78]

The parameter set that better describes the experimental data is the third.
Now, the enhanced growth is explained not only by the gluon distribution, but
also because of the high density region of matter superposition in the center
of the nucleus. As can be seen in Fig. 4.15, $N_{part}$ is almost constant in
the UC region, but $N_{coll}$ increases substantively. In other words, the
enhanced relative yields are explained by the larger number of interactions in
the low impact parameter collisions.

In contrast to the $pp$ case, as shown in Figs. 5.10 and 5.11, the yields are
better reproduced by the smallest $N_{g}$ and biggest $\sigma^{pp}$. Since the
curves overshoot the data for $N_{g}=7,10$ and $\sigma^{pp}=5.7,2.5\text{ mb}$
it seems that the collisions are better described by a smaller number of
partons, with a larger interaction range.

Figure 5.10: a) $J/\psi$ relative yield for $pPb$ at $\sqrt{s}=5.02\text{
TeV}$, set 1. The experimental data are from [4]. The dashed curve is plotted
with $m_{c}=1.2\text{ GeV}$. Figure 5.11: $J/\psi$ relative yield for $pPb$ at
$\sqrt{s}=5.02\text{ TeV}$, set 2. The experimental data are from [4]. The
dashed curve is plotted with $m_{c}=1.2\text{ GeV}$. Figure 5.12: a) $J/\psi$
relative yield for $pPb$ at $\sqrt{s}=5.02\text{ TeV}$, set 3. The
experimental data are from [4]. The dashed curve is plotted with
$m_{c}=1.2\text{ GeV}$.

##  Chapter 6 Conclusion

In this work we adapted the Glauber model formalism, initially formulated to
deal with heavy ion collisions, to proton-proton and proton-nucleus
collisions. By doing this, we could access the subnucleonic degrees of
freedom, which are important in this kind of process, and adapt a model that
accounts to the baryon junction existence. This approach leads to a separation
between the gluon and quark interactions and distributions.

Since there are no data about the Glauber outputs, we used the color
evaporation model as a tool to test the consequences of the internal structure
in the charmonium relative yields at midrapidity. In $pp$ collisions the data
were described by the parameter set 1, and the enhanced growth in the high
multiplicity region was explained by the core-corona model. That is, while the
collision gets more central, the quark interactions are changed to gluon
interactions and the relative yields grows because $gg\rightarrow c\bar{c}$
cross section is much bigger than the $q\bar{q}\rightarrow c\bar{c}$ one. In
other words, the baryon junction assumption could explain the growth seen in
the data for the larger number of partons.

For the $pPb$ collisions, we could adapt the nucleon structure developed for
the proton by randomly positioning $208$ of them inside the nuclei. The CEM
was again used to probe the internal structure by means of $J/\psi$ production
and the data were fitted for the parameter set 3. Differently from the $pp$
case, the enhanced growth was not explained only by the gluon processes, but
also by the enhanced number of binary collisions in the UC region.

In the end, the baryon junction assumption could explain the data. Of course
our results are not enough to say that the baryon junction is completely
established, but they give support to it.

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