MASTER’S DISSERTATION IFT-D.002/24

Exploring the non-resonant tt̄HH (HH → 4b)
production, with Hadronic Tops Decay, using
a fast simulation of the CMS-LHC Detector.

Gabriel Vinicius Vian

Advisor

Ricardo D’Elia Matheus

Co-Advisor

Aurore Savoy-Navarro

February 2025



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
               Vian, Gabriel Vinicius 

 V614e            Exploring the non-resonant 𝑡𝑡𝑡𝑡HH (HH → 4b) production, with hadronic 
tops decay, using a fast silmulation of the CMS-LHC detector / Gabriel 
Vinicius Vias. – São Paulo, 2025 

152 f: il. color. 
 

Dissertação (mestrado) – Universidade Estadual Paulista (Unesp), 
Instituto de Física Teórica (IFT), São Paulo 

Orientador: Ricardo D’Elia Matheus 
Coorientadora: Aurore Savoy -Navarro 

 
1. Higggs, Bosons de.  2. Modelo padrão (Física nuclear). 3. Partículas 

(Física nuclear).  I. Título  
 

 

  
Sistema de geração automática de fichas catalográficas da Unesp. Biblioteca 

do Instituto de Física Teórica (IFT), São Paulo. Dados fornecidos pelo 
autor(a). 

 

 



 

EXPLORING THE NON-RESONANT 𝑡 ̅𝑡𝐻𝐻 ( HH → 4B ) PRODUCTION, 
WITH HADRONIC TOPS DECAY, USING A FAST  

SIMULATION OF THE CMS-LHC DETECTOR 

 

 

 

Dissertação de Mestrado apresentada ao 
Instituto de Física Teórica do Câmpus de 
São Paulo, da Universidade Estadual 
Paulista “Júlio de Mesquita Filho”, como 
parte dos requisitos para obtenção do título 
Mestre em Ciências, área: Física.  

 

 

 

Comissão Examinadora: 

 

Prof. Dr. RICARDO D’ELIA MATHEUS (Orientador) 

Instituto de Física Teórica/UNESP  

 

Prof.  Dr. THIAGO RAFAEL FERNANDEZ PEREZ TOMEI 

Instituto de Física Teórica/UNESP 

 

Prof. Dr. LEONARDO DE LIMA 

Universidade Federal do Paraná/UFPR 
 
 
 
 

Conceito: Aprovado 

 

São Paulo, 20 de março de 2025. 



I dedicate this dissertation
to my family.

i



Acknowledgments

I want to express my heartfelt gratitude to my family for all the support I have received

from them, without which I would not have been able to achieve so many important

milestones.

I am deeply thankful to my friends who accompanied me during my Master’s degree

at the IFT, making it a very pleasant experience, especially Jorge E., Denise C., Angelo A.,

Brendo F., and Julia F.

Additionally, I am grateful to my undergraduate friends, Giovani V., Robison J., and

Professor Tobias H., for the enjoyable asynchronous moments we shared weekly, whether

playing online games or tabletop RPGs.

A special and enormous thank you goes to my advisor, Ricardo, for the opportu-

nity to work with him, and for all his teachings, patience, empathy, and understanding

throughout this process. Another huge acknowledgment to my co-advisor, Professor

Aurore, for the greatest honor and opportunity of working with her, her patience, and her

countless teachings that have significantly strengthened my professional development.

Furthermore, I am also grateful to this remarkable scientist for her dedication, attention

to detail, and effort in reviewing this work. Having her expertise in the revision process

was essential in adding immense value to this dissertation. A very special thanks to the

CMS-Saclay team of CEA1-IRFU2 for allowing me to access the CMS system and to be

registered as a CMS member, allowing me to progress on my research. Also many thanks

to the ttHH(HH → 4b) team for the honor of working with this inspiring team on the

research of this channel and for teaching me so much with such patience.

I would also like to thank my professors at the IFT for the opportunities to ask questions

and have meaningful discussions, especially Professor Thiago Tomei and Professor Gastão

Krein, whose expertise has been invaluable. Also, thanks to O. Mattelaer, whose patience

and expertise in MadGraph have been invaluable. Over the past two years, he has

answered many of my questions in forums, playing a crucial role in solving various

problems in my research.

Finally, I extend my gratitude to the funding agency Coordenação de Aperfeiçoamento

de Pessoal de Nível Superior (CAPES) for the Master’s scholarship I was awarded.

1Commissariat à l’énergie atomique et aux énergies alternatives.
2Institut de recherche sur les lois fondamentales de l’Univers.

ii



Failure is the mark of a life well lived. In turn, the only way to live without failure
is to be of no use to anyone.
Oathbringer: Book Three of the Stormlight Archive
Brandon Sanderson

iii



Resumo

O processo tt̄HH é um importante canal para o estudo do setor de Higgs, tendo
sido estudado por análises experimentais recentes. Sua relevância está no fato
de que este canal permite acessar diretamente, sem interferências, a constante de
acoplamento trilinear do Higgs (λ) e a constante de Yukawa do Higgs com o top
(y). Assim, o tt̄HH oferece uma forma complementar de medir essas constantes,
juntamente com outros canais, como tt̄H, VV → HH e gg → HH. Além disso,
o tt̄HH é um canal valioso para o estudo de modelos além do Modelo Padrão
(BSM), como os modelos de Higgs composto, que preveem resultados únicos para
esse processo que podem ser observados em buscas experimentais futuras.

Neste trabalho, foi realizado um estudo do processo tt̄HH com os tops de-
caindo hadronicamente (t → Wb, W → qq̄′) e os bósons de Higgs decaindo em
pares de quarks b (tt̄HH hadrônico). A simulação do processo foi conduzida em
uma máquina local utilizando os frameworks MadGraph5_aMC@NLO, Pythia
8 e Delphes. Também foram simulados os principais backgrounds do processo:
tt̄, tt̄WW, tt̄WH, tt̄WZ, tt̄tt, tt̄bb, tt̄bbbb, tt̄H, tt̄Z, tt̄ZZ, com cada Z decaindo em
pares de quarks b, e os tops decaindo hadronicamente.

São apresentadas as principais variáveis cinemáticas, como Pjatos
T , HT, η, MET

e massa invariante (para sinal e principais backgrounds). Os resultados mostram
que os critérios de seleção Njets ≥ 6, Nb-jets ≥ 4, Nq-jets ≥ 2, HT > 500 GeV, além
de exigir que os jatos tenham momento transverso PT > 40 GeV/c, estejam na
região de pseudorapidez |η| < 2.5, e que não haja léptons, aumentam a razão sinal-
background R e significância estatística S em uma ordem de 10−2. Os resultados
apresentados são preliminares, e atualmente simulações mais avançadas deste
canal estão sendo conduzidas no framework CMSSW (CMS Software).

Palavras Chaves: Bóson de Higgs; Partículas;

Áreas do conhecimento: Física; Física de Partículas; Física de Partículas experi-
mental.

iv



Abstract

The tt̄HH process is an important channel for studying the Higgs sector, having
been investigated in recent experimental analysis. Its relevance lies in the fact that
this channel allows for direct access, without interference, to the trilinear Higgs
coupling constant (λ) and the Yukawa coupling constant of the Higgs with the top
quark (y). Thus, tt̄HH provides a complementary way to measure these constants,
alongside other channels such as tt̄H, VV → HH, and gg → HH. Additionally,
tt̄HH is a valuable channel for studying Beyond Standard Model (BSM) models,
such as composite Higgs models, which predict unique outcomes for this process
that could be observed in future experimental searches.

This work presents a study of the tt̄HH process with top quarks decaying
hadronically (t → Wb, W → qq̄′) and Higgs bosons decaying into pairs of b quarks
(tt̄HH hadronic). The process simulation was conducted using the frameworks
MadGraph5_aMC@NLO, Pythia 8, and Delphes. The main backgrounds for the
process were also simulated: tt̄, tt̄WW, tt̄WH, tt̄WZ, tt̄tt, tt̄bb, tt̄bbbb, tt̄H, tt̄Z,
tt̄ZZ, with each Z decaying into pairs of b quarks and the top quarks decaying
hadronically.

The main kinematic variables are presented, such as Pjets
T , HT, η, MET, and

invariant mass (for signal and main backgrounds). The results show that the
selection criteria Njets ≥ 6, Nb-jets ≥ 4, Nq-jets ≥ 2, HT > 500 GeV, in addition
to requiring that the jets have transverse momentum PT > 40 GeV/c, lie within
the pseudorapidity region |η| < 2.5, and that no leptons are present, increase the
signal-to-background ratio R and statistical significance S by a factor of 10−2. The
results presented are preliminary, and advanced simulations of this channel are
currently being conducted within the CMSSW framework (CMS Software).

Keywords: Higgs boson; Standard Model; Particles;

Areas of knowledge: Physics; Particle Physics; Experimental Particle Physics.

v



Contents

1 Introduction 1
1.1 Physical motivations for the ttHH process . . . . . . . . . . . . . . . 3

2 Theoretical framework 10
2.1 The Standard Model of Particle Physics . . . . . . . . . . . . . . . . 10

2.1.1 Particles of the Standard Model . . . . . . . . . . . . . . . . . 11
2.1.2 Interactions of the Standard Model . . . . . . . . . . . . . . . 14
2.1.3 Higgs Mechanism . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Physics description of the collision between two protons . . . . . . 32
2.2.1 Hard scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.2 Multiple parton interactions. . . . . . . . . . . . . . . . . . . 33
2.2.3 Beam remnants. . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.2.4 From partons to jets . . . . . . . . . . . . . . . . . . . . . . . 36

3 Experimental Framework: The Large Hadron Collider and the Compact
Muon Solenoid 44
3.1 The Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . 44
3.2 CMS reference system and Kinematical variables . . . . . . . . . . . 47
3.3 The CMS experiment and its components . . . . . . . . . . . . . . . 52

3.3.1 The Silicon Tracker . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3.2 Calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.3 Reconstruction of jets using Calorimeter data or Particle Flow. 61
3.3.4 The Superconducting Solenoid . . . . . . . . . . . . . . . . . 62
3.3.5 CMS muon detectors . . . . . . . . . . . . . . . . . . . . . . . 63
3.3.6 CMS Trigger system . . . . . . . . . . . . . . . . . . . . . . . 64

3.4 Components of the CMS detector for the analysis. . . . . . . . . . . 64
3.5 Recent upgrades on the CMS experiment. . . . . . . . . . . . . . . . 66

4 Modeling framework 67
4.1 MadGraph5: A parton level event generator . . . . . . . . . . . . . 67
4.2 Pythia: A parton-shower, Multiple-parton Interaction, and hadroniza-

tion framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

vi



4.3 Delphes: A fast multipurpose detector response simulation . . . . . 72
4.4 Modeling Framework for simulating a proton collision. . . . . . . . 78
4.5 Jet Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 CMSSW Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 An insight on the SM non resonant ttHH production with (HH → bb̄bb̄)
in the tt̄ Hadronic channel at CMS-LHC. 83
5.1 Monte Carlo Generation of Signal and Background Samples . . . . 83
5.2 Selection criteria applied to the ttHH Hadronic study. . . . . . . . . 86
5.3 Main Variables used in this study . . . . . . . . . . . . . . . . . . . . 88

6 Some first outcomes on this preliminary study and perspectives 90
6.1 Number of jets per event for signal and backgrounds . . . . . . . . 91
6.2 Comparison of different jet multiplicity selection criteria . . . . . . 99
6.3 Jet activity per event. . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.4 Missing Transverse Energy . . . . . . . . . . . . . . . . . . . . . . . . 119
6.5 Reconstruction of the invariant mass of the bosons in the signal and

the backgrounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7 Conclusion 125

A Number of jets per event 127

B Kinematical variables with no selection criteria on the number of jets. 137

Bibliography 143

vii



Chapter 1

Introduction

The Standard Model of Particle Physics [1] [2] [3] is the most successful frame-
work for describing fundamental particles and their interactions. It has been
validated through a wide range of experimental results. Most of the experimental
discoveries related to the Standard Model were made with particle accelerators,
which are key instruments for unveiling the fundamental structure of matter and
its interactions. Nowadays the highest energy accelerator in the world is the Large
Hadron Collider (LHC), a circular proton-proton collider located in France and the
Geneva area, in Switzerland, at the European Organization for Nuclear Research
(CERN).

The most recent discovery in Particle Physics in the LHC experiment was
of the Higgs Boson (H) [4] [5], a fundamental particle for the Standard Model,
responsible for giving mass to the other elementary particles. The discovery of the
Higgs boson highlights the need for further exploration of the Higgs sector, which
is a part of particle physics that remains in many ways to be fully explored. The
trilinear self-coupling of the Higgs Boson (denoted as λHHH and abbreviated as λ

in this work) quantifies how strongly two Higgs bosons interact. This essential
parameter is still far from being well measured. A more precise determination of
this parameter is essential for a deeper understanding of the Higgs sector. The
non-resonant production of ttHH in pp collisions provides a direct way to measure
it. This is a rare process (cross-section of pp → ttHH is of the order of 1 fb at 14
TeV center of mass energy). Furthermore, it allows the direct measurement of the
Higgs-top Yukawa coupling (denoted as yt−H and abbreviated as in this work),
another important parameter; this coupling quantifies how strongly a Higgs and a
top quark interact.

So, the exploration of this "Di-Higgs Di-top" process allows measuring these
two couplings together with the measurement made through di-Higgs production
processes such as gluon-gluon fusion gg → HH (ggF HH), vector boson fusion
VV → HH (VBF HH), both for λ, and ttH, for y.

Additionally, the ttHH is a rich channel for the exploration of Beyond Standard

1



Chapter 1. Introduction 2

Model (BSM) Physics, that can be studied from the perspective of Supersymme-
try (SUSY), Composite Higgs Models (CHM) and other new physics theoretical
frameworks, providing unique predictions that could be tested through related
experimental searches.

So the mentioned process is a rich path for exploring new physics and inter-
plays with other processes (ggF HH, VBF HH, ttH), allowing complementary
measurements of λ and y.

For the ttHH process, considering the possible top quark decay channels, three
decay channels can be taken into account (although many others are possible): the
hadronic mode, where each top quark (t) decays into a W Boson and a b quark,
and the two W decay into pairs of light quarks; the semileptonic mode, where
each top decays into a W and a b, one W decays into a pair of light quarks, and the
other W decays into a lepton (electron, muon or tau-lepton) and the corresponding
neutrino; and finally, the dileptonic mode, where each top quark decays into a W
and a b, and both W decay into a lepton (electron or muon) and the corresponding
neutrino. In this work, the hadronic decay mode will be considered, with each
Higgs boson decaying into a pair of b quarks (H → bb).

This dissertation presents a study within the Standard Model of the non-
resonant production process of ttHH in proton-proton collisions and the main
corresponding physical backgrounds. The basic relevant kinematical variables
were analyzed at a center-of-mass energy of 13 TeV, which corresponds to the
center-of-mass energy of LHC during the run period from 2016 to 2018 (Run 2).
It was also verified, through simulations, that the increase in the center of mass
energy from 13 to 13.6 TeV (which is the case in Run 3, which began in 2022 and
is expected to conclude in 20263) allows increasing the cross-section at Leading
Order (LO) by a percentage of 12% for t̄tHH. It is important to note here another
major interest of the ttHH production: its production cross-section increases more
rapidly than some other processes, with the increase of the center of mass energy
(see Figure1.3).

Because of the hadronic decay of the top pair and of the decay of the Higgs
bosons the signature is characterized by a large number of jets (4 light quark jets
from hadronic W decays), 6 b-quark jets from the top decay into bW and from
the H-pair decay into b-quark pair, no lepton and no neutrino thus no Missing
Energy1 produced in those decays (the jets and particles that define the signature

1In particle physics experiments, the term ’missing transverse energy’ (MET) is commonly used
to describe an event characteristic related to undetected particles or less converted or uncovered



Chapter 1. Introduction 3

are also called Physics object). The main features of this process signature will
serve to define a first step selection (first level filter) of the events to be looked for.

The study was performed using Monte Carlo simulations of the processes
at Leading Order (LO) with the MadGraph5_aMC@NLO framework (version
2.9.17). The Pythia framework (version 8.306) was used for parton showering
and hadronization, and the Delphes framework (version 3.5.0) was employed to
simulate the CMS detector response. The analysis method is described in Chapter
5, and the preliminary outcomes are presented in Chapter 6.

The study of this process presented here is based on a local simulation frame-
work. At the time of writing of this work, a more detailed simulation effort is
underway, involving simulations within the CMS framework, CMSSW, as part of
the author’s Ph.D. research.

The next section of this chapter 1, presents the Physics motivations for studying
the ttHH process. Chapter 2 provides a review of the theoretical framework of the
Standard Model. Chapter 3 explains the LHC experiment and the CMS detector.
Chapter 4 introduces the simulation framework.

1.1 Physical motivations for the ttHH process

Specifically studying the ttHH process, combined with other Higgs processes
will enhance the understanding of the Higgs sector [6]. Table 1.1 presents some
important Higgs processes, with the respective cross-section at the center of mass
energy of 13 TeV and 14 TeV.

Furthermore, observing this still rare ttHH process at LHC c.m. energy is,
by itself, of great importance, per se, and also its signal strength (µ = σobs/σSM)
measurement can be used to observe possible deviations on the predictions of the
Standard Model.

Processes involving the production of two Higgs bosons allow access to the
Higgs trilinear (λ) self-coupling which is not yet very constrained, see Figure 1.1.
Another important constant that can be measured through the ttHH process, as
well as ttH, is the Higgs-Top Yukawa coupling (y). So the ttHH provides access
to two important constants, λ and y. In the case of λ, the contribution to the total
cross-section is subdominant but not negligible, because if λ is set to zero, the

areas of the detector through which particles can escape undetected. However, this terminology is
somewhat misleading, as what is fundamentally measured is the missing transverse momentum.
Despite this, the term ’missing transverse energy’ remains widely used in experimental analyses.



Chapter 1. Introduction 4

√
s 13 TeV 14 TeV

tt̄H 530 fb 641 fb
ggF HH 31.05 fb 36.69 fb
VBF HH 1.73 fb 2.05 fb

tt̄HH 0.775 fb 0.949 fb
ZHH 0.363 fb 0.415 fb

W+HH 0.329 fb 0.369 fb
W−HH 0.173 fb 0.198 fb

tjHH 0.0289 fb 0.0367 fb

Table 1.1: Cross-sections for different Higgs production processes in pp collisions
at

√
s = 13 TeV and 14 TeV, at NLO or Higher. In the Table, ggF means gluons

fusion, and VBF means vector boson fusion Source: [7]

t̄tHH cross-section decreases by around 20%. This indicates that diagrams with the
Higgs trilinear coupling contribute approximately 20% to the total cross-section at
Leading Order [8].

In the scheme displayed in the Figure 1.2 there are representative Leading
Order (LO) diagrams from ggF, VBF, ttH and ttHH 2. It is possible to see how
the ttHH interplay with the processes, making it an excellent way to complement
the studies of these other channels, which is crucial for achieving precision in the
results. In the picture, the purple dot represents y and the red dot represents λ. It
is also highlighted, at the green point, the coupling between vector bosons (W or
Z) and the Higgs, denoted by CV .

In addition to the importance of achieving an experimental precise measure-
ment of λ, this specific coupling is also important for clarifying the nature of the
Higgs mechanism and determining the Higgs potential [12], which has not yet
been experimentally determined [10].

Such coupling can be measured through ggF process, VBF process, and the
ttHH, which are the three dominant di-Higgs processes in terms of cross-section
(see figure 1.3). The gluon-gluon fusion is the process with a higher cross-section
in this case when compared to the other processes because at the energy scale of
LHC, the parton distribution functions (PDFs) of the proton favor gluons, so it is
more probable that a gluon-gluon interaction occurs than a q-q interaction, which
would trigger a VBF.

Despite its higher cross-section value, the ggF channel has negative interfer-
ence terms in the access to the coupling λ [13], which is due to the cancellation

2There are other diagrams for these processes that are not shown in the figure. Some of such
diagrams are given by different placements of Higgs lines, for instance.



Chapter 1. Introduction 5

Figure 1.1: Different measurements of the trilinear Higgs coupling made by differ-
ent collaborations and experiments. Adapted from: [9]

Figure 1.2: Relation of the process tt̄HH with other processes involving H for the
measurement of λ and y. Adapted from: [10] and [11].



Chapter 1. Introduction 6

Figure 1.3: The processes for the production of two Higgs bosons with the largest
cross-sections at center-of-mass energy values between 13 and 100 TeV. Source :
[10]



Chapter 1. Introduction 7

between the contributions of the triangle (displayed in figure 1.4) and box di-
agrams (displayed in figure 1.5). The amplitude associated with each of these
two diagrams generates a negative interference term that reduces about 50% of
the process cross-section [10], making it more difficult to measure λ through this
channel.

Figure 1.4: Triangle diagram for ggF
HH. Source : [10]. Figure 1.5: Box diagram for ggF HH.

Source: [10].

In the context of Beyond Standard Models, ttHH also presents possibilities.
In the specific case of Composite Higgs Models (CHM) the ttHH production has
two different types of diagram contributions [11] the Leading Order (LO) resonant
production, in which the tH is produced by a massive resonance T, a heavy top-
quark partner of charge 2/3 and spin 1/2, diagrams for this process are displayed
in Figure 1.6; and the LO non-resonant production, with diagrams displayed in
Figure 1.7. The non-resonant case has the same diagrams of the SM, except for
the third one, which has a non-SM interaction vertex, called the double Higgs
Yukawa vertex (Ct̄tHH). This contact vertex has no counterpart in the Standard
Model, for this reason, it would be important to get experimental evidence of the
coupling associated with it [11]. Apart from BSM scenarios, there is a similarity
between the double Yukawa-Higgs coupling in tt̄HH (Ct̄tHH (BSM), shown in the
figure 1.7) and the C2V (VVHH) coupling (SM), illustrated in Figure 1.8. Both are
of experimental and theoretical interest.

The presence of resonant production could lead to large discrepancies in cross-
section, signal strength, and kinematical characteristics of the events. The feature
of resonant production is also present in other theories, such as Supersymmetry,
where heavy Supersymmetric particles can produce SM particles and lead to
changes in the cross-sections and kinematic characteristics of the processes [14].

In summary, the investigation of ttHH channel provides a direct way of access-
ing the Higgs self-coupling and the Higgs-Top Yukawa coupling. These constants
are fundamental for the progress of the understanding of the Higgs sector. The
channel will complement the results from other production channels and present



Chapter 1. Introduction 8

Figure 1.6: Diagrams for resonant t̄tHH production, in the Composite Higgs
Model case studied at [11]. Source : [11].

Figure 1.7: Diagrams for non-resonant t̄tHH production, in the Composite Higgs
Model case studied at [11]. The third diagram is the one containing the Ct̄tHH
coupling. Source: [11].

Figure 1.8: VVHH process that has a coupling similar to the double Higgs Yukawa
vertex. Source : [15].



Chapter 1. Introduction 9

no destructive interference in the access to λ, unlike ggF. Also, the channel is a
rich path for studying new physics, such as BSM models, showing promising
predictions that are not observed in the Standard Model.



Chapter 2

Theoretical framework

2.1 The Standard Model of Particle Physics

The Standard Model (SM) of Particle Physics is a theoretical model formu-
lated in its final form through the contributions of S. Weinberg, A. Salam, and
S. Glashow, with works that led to the creation of the electroweak theory [1] [2]
[3]. Another important aspect of SM is the theory of quark interactions, Quan-
tum Chromodynamics, which was formulated through the contributions of many
physicists over the past fifty years. A comprehensive historical review is presented
in [16]. The SM describes the fundamental particles of matter and their interac-
tions. The underlying theory of this model is Quantum Field Theory, a framework
that unifies special Relativity and Quantum Mechanics and describes particles as
excitations in quantum fields.

The Standard Model describes, through Quantum Field Theories and symme-
tries, three of the four fundamental interactions of nature: 1) the electromagnetic
interaction is described by Quantum Electrodynamics (QED), 2) the strong in-
teraction is explained by Quantum Chromodynamics (QCD). and 3) the weak
interaction, described by Electroweak theory (EW), which also provides the unifi-
cation of electromagnetic and weak interactions.

The fundamental particles of the SM are 6 Quarks (u, d, s, c, b, t), 6 Leptons
(e−, νe, µ−, νµ, τ−, ντ ), 4 Force Carriers (γ, g, W+/−, Z) and the Higgs boson
(H), without counting the antiparticles. Antiparticles are the counterparts to the
particles in the Standard Model, possessing the same mass but opposite charges
and quantum numbers. The key differences between a particle and its antiparticle
include the reversal of electric charge, baryon, and lepton numbers, and quantum
numbers like strangeness, charm, and topness. In addition, quarks carry opposite
color charges compared to their antiparticles. While the spin of particles and
antiparticles remains the same. More details on these particles will be given in the
subsection 2.1.1.

10



Chapter 2. Theoretical framework 11

The SM remains, to date, the most successful framework for describing funda-
mental particles and their interactions at the presently accessible scale energies.
It has been very well validated through a wide range of experimental results.
But, despite its success, it still faces some problems, such as: no neutrino mass
generation mechanism; no Dark Matter candidate; no mechanism for generating
the observed cosmological matter/anti-matter asymmetry; no UV complete quan-
tum field theory of gravity; No explanation for the flavor/family structure; no
solution to a few fine-tuning/hierarchy problems (hierarchy of scales and strong
CP problems).

For these reasons, the SM is unable to offer a complete framework for describing
all the particles and interactions of the universe. Therefore the goal of both
the experimentalists and theoreticians is to go Beyond SM (BSM). This implies
searching for new phenomena and at the same time testing with higher and higher
precision the limits of the SM predictions.

This section aims to present a theoretical review on the particles of the SM
(subsection 2.1.1), on the interactions of the SM from a Lagrangian point of view
(subsection 2.1.2), and on the Higgs mechanism for mass generation (subsection
2.1.3).

2.1.1 Particles of the Standard Model

The SM has 18 different elementary particles, which can be grouped into two
categories: fermions, particles with half-integer spin that respect the Pauli exclu-
sion principle, and bosons, which are particles with integer spin, i.e.: don’t obey
the Pauli exclusion principle. The fermions can be classified into two subgroups,
the leptons and the quarks. Leptons are half-spin particles that can be electrically
charged (e−, µ− and τ−) , with charge -1, in units of e, or neutral (the neutrinos: νe− ,
νµ− and ντ−). Electrons, muons, and taus experience Weak and electromagnetic
interactions, while neutrinos experience only weak interaction.

Quarks form the other subgroup of fermions being particles with fractional
electric charge. There are six flavors of quarks: up, down, charm, strange, top, and
bottom, each with distinct properties, including mass and electric charge (see table
2.11):

1The mass of quarks u, d and s is not a fixed quantity but depends on the chosen renormalization
scheme and the renormalization scale (µ). The masses of such quarks presented at the table are
calculated using the MS(modified minimal subtraction) scheme with a renormalization scale of
µ = 2 GeV. More details on this can be found in [17].



Chapter 2. Theoretical framework 12

Quark Type Charge Mass
Up (u) +2

3 e 2.16 MeV/c2

Down (d) −1
3 e 4.7 MeV/c2

Strange (s) −1
3 e 93 MeV/c2

Charm (c) +2
3 e 1.27 GeV/c2

Bottom (b) −1
3 e 4.18 GeV/c2

Top (t) +2
3 e 172.57 GeV/c2

Table 2.1: Properties of quarks including charge and mass. Source : [17].

A defining feature of quarks is their interaction via the strong force, this force
only affects particles that possess a property known as color charge, which comes
in six types: red, green, and blue, antired, antigreen, antiblue. The strong force
binds quarks together to form composite particles like protons and neutrons,
which are examples of hadrons. This interaction ensures that quarks cannot exist
independently, they are always confined in a bound state due to a phenomenon
called confinement (the confinement is a behavior verified experimentally that has
not yet been rigorously proven mathematically within the framework of QCD).

Each fermion has a corresponding anti-fermion, sharing the same intrinsic
properties such as mass and spin. However, certain quantum numbers, including
charge, color charge and lepton number are reversed.

As mentioned, the SM includes three of the four fundamental interactions:
electromagnetism, strong, and weak. Each one of these interactions is associated
with different gauge bosons, which are responsible for mediating, or carrying, the
forces associated with such interactions.

The photon γ is a spin 1, massless and neutral particle that is responsible for
mediating the electromagnetic interaction between electrically charged particles.

The gluon g is a spin 1, massless and neutral particle that carries the Strong
force between particles that possess color charge (quarks).

The Weak interaction is mediated by three different types of massive bosons:
the W+ and W− bosons, which have the same mass (80.36 ± 0.01 GeV/c²), opposite
charges and Spin 1, and the Z boson, a neutral particle with spin 1 and mass
of 91.188 ± 0.002 GeV/c² [17]. At higher energies, the three weak interaction
bosons can be unified with the Photon into the same interaction, the Electroweak
interaction.

Gravity has not yet been incorporated in the SM, but a theoretical particle that
mediates this force, the Graviton, has been proposed. This interaction does not yet
have a consistent Quantum Field Theory associated with it, due to incompatibili-



Chapter 2. Theoretical framework 13

ties between Quantum Mechanics and Einstein General Relativity.
Lastly, the Higgs boson particle is a spin 0, neutral particle with a mass of

125.20 ± 0.11 GeV/c² [17]. The Higgs boson is a fluctuation of the Higgs field, a
field that is responsible for giving rest mass to elementary fermions and the W
and Z bosons. More massive elementary particles interact more strongly with the
Higgs field, for the fermions a measure of this interaction strength is the Yukawa
coupling y, where each one has a different value of y.

Information about all the particles in the Standard Model can be found in the
Particle Data Group (PDG) [17].

The Higgs Boson was independently predicted in the sixties by theoretical
physicists F. Englert, P. Higgs, C. R. Hagen, G. Guralnik, R. Brout, and T. Kibble,
as part of a mechanism that explains how particles acquire mass [18]. The particle
was discovered experimentally in 2012, at the Large Hadron Collider (LHC) being
a very important piece in the understanding of the Standard Model, confirming
the existence of the Higgs field.

It was discovered through experiments conducted by the ATLAS and CMS
collaborations. The particle was found with a mass of approximately 125 GeV/c²,
by measurements of the decays H → γγ & H → ZZ → ℓℓℓℓ [4] [5]. Below, a Table
summarizes the main decay modes of the Higgs boson [17]. The table includes
quarks, W and Z bosons, tau leptons, and photons. Virtual particles, or off-shell
particles (E2 ̸= P⃗2 + m2) have an superscript *.

Decay Mode Branching Ratio (%)
H → bb̄ 58.24

H → WW∗ 21.3
H → gg 8.2

H → τ+τ− 6.3
H → ZZ∗ 2.64
H → γγ 0.23

Table 2.2: Main decay modes of the Higgs boson and their branching ratios.
Updated valued from PDG 2024.

The (γγ) and (ZZ) channels, although relatively rare decay modes, offer a
clean and distinct experimental signature for the Higgs Boson. It was thus used to
search for the first Higgs events produced in proton-proton collisions at LHC.



Chapter 2. Theoretical framework 14

2.1.2 Interactions of the Standard Model

In the following subsections the prescription of classical textbooks such as [19],
[20], [21] was followed:

Quantum Field Theory is the theoretical framework for the construction of
the SM, where each elementary particle is described by a field. The interactions
of the SM are described by Lagrangians which are symmetric under the trans-
formations of the groups: 1) Poincaré group, which describes translations and
rotations between inertial reference frames2; 2) the SU(3)C color group, which is
the group associated to the color transformations; 3) the SU(2)L × U(1)Y group,
which describes the Electroweak interactions (electromagnetism and weak inter-
action). These interactions and respective Lagrangian densities will be described
subsequently.

The relation between interactions and symmetry groups is due to the Gauge
principle: whenever a symmetry of the action exists, the associated local gauge
transformation leads to an interaction mediated through a number of intermediate
particles [22].

For a field ϕ(x) and a group G, a local gauge transformation acts as:

ϕ(x) → ϕ′(x) = U(x)ϕ(x),

where U(x) is a spacetime-dependent element of the gauge group G.
One can show an example of the Gauge principle, by taking a Lagrangian

that describes a free system of fermions and free electromagnetic field (without
interaction between them) [19]

LQED = ψ̄(iγµ∂µ − m)ψ − 1
4

FµνFµν (2.1)

where the Dirac Spinor Field (ψ) represents a fermion field, and ψ̄ = ψ†γ0 is the
Adjoint Spinor Field. gµν is the metric tensor of Minkowski space. The Field
Strength Tensor (Fµν) encodes the electromagnetic field and is defined as:

Fµν = ∂µ Aν − ∂ν Aµ

2In quantum theory, the Lagrangian, now an operator, does not need to be Poincaré invariant,
but instead to transform as a scalar operator. The other operators of the theory, the fields, must still
act consistently, each having its well-defined Lorentz (or Poincaré) transformation.

What happens in practice is that Poincaré symmetries can be preserved in the sense of the
expectation values of physical observables, even if the quantum action or individual operators are
not manifestly invariant.



Chapter 2. Theoretical framework 15

Now consider a U(1)em local gauge group transformation

ψ(x) → eiQα(x)ψ(x)

In the gauge transformation above α is a real parameter dependent of x and Q
is the electric charge associated with the field ψ. The generator of the U(1)em is Q,
i.e.: the set of all phase, or gauge, transformations of the type eiQα(xµ) constitutes
the group U(1)em [19] and all the transformations of this group are created through
Q.

Introducing this transformation in Lagrangian 2.1

LQED → L′
QED = ψ̄(iγµ∂µ − m)ψ − 1

4
FµνFµν + iQ(∂µα)ψ̄γµψ.

It is evident that it is not invariant under such transformation, which leads
to the action associated with this Lagrangian not being invariant. The additional
term:

iQ(∂µα)ψ̄γµψ

is problematic because it explicitly depends on α(x), which does not allow the
invariance.

To eliminate this unwanted term one introduces a gauge field Aµ and defines
the covariant derivative with it:

Dµ = ∂µ − iQAµ.

And require that Aµ transforms as, under the U(1)em transformation:

Aµ → Aµ + ∂µα.

This leads to a new Lagrangian, given by [20]

LQED = ψ̄(iγµDµ − m)ψ − 1
4

FµνFµν (2.2)

Now, with this modification, applying the U(1)em transformation in the fields
of the Lagrangian 2.2 one obtains

LQED → L′
QED = ψ̄iγµ∂µψ+ iQ(∂µα)ψ̄γµψ−Qψ̄γµ Aµψ−Q(∂µα)ψ̄γµψ−mψ̄ψ− 1

4
FµνFµν.



Chapter 2. Theoretical framework 16

The extra terms: +iQ(∂µα)ψ̄γµψ and −Q(∂µα)ψ̄γµψ cancel each other exactly,
leaving:

LQED → L′
QED = ψ̄iγµ∂µψ − Qψ̄γµ Aµψ − mψ̄ψ − 1

4
FµνFµν

which is identical to the original 2.2 Lagrangian. The new Lagrangian with the
covariant derivative introduces a new term ieψ̄γµ Aµψ, which represents the inter-
action between a fermion field and a gauge field Aµ, that is the Electromagnetic
Field.

This example, using the electromagnetic interaction, shows that requiring the
Lagrangian to respect a local symmetry of a given group leads to the emergence
of interactions mediated by particles3, as established by the gauge principle. In
the case of the electromagnetic interaction, the interaction occurs between the
fermion field and the electromagnetic field, with the photon as the mediating
particle. This also applies to Lagrangians of other interactions but with different
transformation groups, such as SU(3)c for the QCD Lagrangian, for example.
After the subsequent step of quantizing the fields (turning them into quantum
objects that respect commutation relations), the terms in these Lagrangians can be
associated with Feynman diagrams.

The Feynman Diagrams are used to obtain the invariant amplitude M associ-
ated with a process. Each diagram of a process will contribute with a Mi, such that
the total amplitude is M = ∑all diagrams Mi. The amplitude is a crucial ingredient
for calculating the cross-section of a process, which is a quantity associated with
the probability of such a process happening. Integrating the invariant amplitude
squared over the phase space of n final-state particles, one obtains the cross-section
σ

σ =
1
F

∫
|M|2 dΦn,

Where:

F: Flux factor, F = 4| p⃗i|
√

s, for a collinear collision (in the center-of-mass
frame) between two initial particles with momentum | p⃗i|.
dΦ: Differential phase space, which takes into consideration the kinematical
configurations of the n final-state particles, given by:

3These mediating particles only appear after the steps of field quantization, where the classical
fields are promoted to quantum operators.



Chapter 2. Theoretical framework 17

dΦn = (2π)4δ4

(
pinitial −

n

∑
i=1

pi

)
n

∏
i=1

d3pi

(2π)32Ei
.

Electromagnetic interaction

The Lagrangian of QED, invariant under the transformations of the local gauge
group U(1)em is given by

LQED = ψ̄(iγµDµ − m)ψ − 1
4

FµνFµν (2.3)

The presence of the electromagnetic field Aµ guarantees that the Lagrangian is
invariant under such transformation. Hence, the electromagnetic field is the field
associated with the U(1)em group symmetry.

If 2.3 is rewritten using Dµ definition,

LQED = ψ̄(iγµ∂µ − m)ψ + eψ̄γµ Aµψ − 1
4

FµνFµν (2.4)

it is possible to identify [21] each term as following :

• The term ψ̄(iγµ∂µ − m)ψ is the Dirac Lagrangian for a free fermion field,
therefore this term represents the free propagation of a fermion.

• The term ieψ̄γµ Aµψ represents the interaction between a fermion field and a
Photon.

• The last term, −1
4 FµνFµν represents the dynamics of the free photon field.

It is also important to mention that the invariance of the QED Lagrangian
under U(1)em transformations leads to the conservation of electric charge, via
Noether’s Theorem [19] [23].

Starting from this Lagrangian it is possible to obtain, via canonical QFT formal-
ism4, the Feynman rules of QED, by association of quadratic terms of fields (Aµ

and ψ) with propagators and the other terms with interaction vertices.

4This includes quantization of the fields.



Chapter 2. Theoretical framework 18

Quantum Chromodynamics interaction

Analogously to the QED case, the invariance of the QCD Lagrangian under
the SU(3)C group local transformations

q(x) → eiαa(x)Ta q(x),

gives rise to the gluon field Ga
µ, which is the field responsible for the interaction

between the color-charged quarks, as well as the interaction between color-charged
gluons. This theory displays the following Lagrangian, invariant under SU(3)C

group local transformations [19]

LQCD = q̄(iγµ∂µ − m)q − g(q̄γµTaq)Ga
µ −

1
4

Ga
µνGµν

a (2.5)

where q is the colored field, an object with three components, each being
a spinor with a different color charge, g is the coupling constant of the strong
interaction.

The index a goes from 1 to 8, Ta = λa/2, λa are the Gell-Mann matrices, a set
of eight traceless 3 × 3 matrices. Ta are the generators of the SU(3)C group.

The term Ga
µν is called gluon field tensor, and is given by

Ga
µν = ∂µGa

ν − ∂νGa
µ − g f abcGb

µGc
ν

.
With Ga

µ being a gluon field. fabc is the structure constant of the SU(3) group,
totally antisymmetric in their indices. The structure constants determine how the
generators of SU(3)c interact with each other

[Ta, Tb] = i fabcTc (2.6)

This determines the algebra of the group being non-Abelian, it means that
transformations of the SU(3)c do not commute.

Observing the terms of the QCD Lagrangian it is possible to extract the follow-
ing information:

• The term q̄(iγµ∂µ − m)q represents the free propagation of quarks.

• The term −g(q̄γµTaq)Ga
µ describes the quark-gluon interaction.



Chapter 2. Theoretical framework 19

Figure 2.1: The figure shows the behavior of the strong force coupling constant
concerning the interaction distance r. This plot was made following the five-flavor
quark scheme (u, d, s, c, b). Source : [24].

• The last term
−1

4
Ga

µνGµν
a

encapsulates the dynamics of the gluons. As gluons can carry color charge,
it means they can interact. Expanding such Ga

µν allows to visualize cubic and
quartic Ga

µ terms, which describes 3g and 4g self-interactions.

These three terms together form the Lagrangian of Quantum Chromodynamics
(QCD), describing the dynamics of quarks and gluons, including their interactions
via gluon exchange, and the self-interaction of gluons due to their ability to carry
color charge. There is no mass term for the gluon because it is a massless boson.

In quantum chromodynamics (QCD), the behavior of quarks and gluons is
governed by the running coupling constant αs(Q2), which depends on the energy
scale Q2 of the interaction or, equivalently, the distance between quarks5. Figure
2.1 displays the behavior of the coupling constant of the strong interaction with
the energy scale Q2 of the interaction.

Consider a pair of quarks interacting via strong force, at short distances (high
Q2), they interact weakly due to the small value of αs(Q2), a phenomenon known
as asymptotic freedom (region of the plot of the Figure 2.1 with higher Q2). This

5It is intuitive to think in terms of electron-protons interactions to better understand the relation
between momentum exchange and distance. When an electron interacts with a proton at high
energy scales (large Q), it can resolve the proton’s internal quark and gluon structure due to the
high energy of the virtual photon exchange (the virtual photon can probe short scale of distance).
Conversely, at low energy scales (small Q) the electron perceives the proton as a single point-like
charge, as the photon exchanged has smaller energy and will probe a larger scale of distance.



Chapter 2. Theoretical framework 20

allows quarks to behave almost as free particles when they are close together,
as the strong force diminishes at these scales. As the distance between quarks
increases (low Q2), the running coupling αs(Q2) grows, causing the strong force to
become more intense (region of the plot of the Figure 2.1 with smaller values of Q2).
This is a direct consequence of the non-Abelian nature of the SU(3) gauge group
in QCD, which allows gluons to self-interact, amplifying the force at larger sepa-
rations. Unlike in quantum electrodynamics (QED), where the electromagnetic
force weakens with distance, QCD exhibits the opposite behavior: the interaction
strength between quarks increases with separation, leading to quark confinement.
This means that quarks cannot be isolated, as the energy required to separate them
becomes so large that new quark-antiquark pairs are produced, forming bound
colorless states instead.

The transition between these two regimes—asymptotic freedom at short dis-
tances and confinement at large distances—is a defining feature of QCD. At high
energies (e.g., in particle colliders), quarks and gluons interact with high energies
(perturbatively), making predictions possible using Feynman diagrams. However,
at low energies, the theory becomes non-perturbative, requiring other methods
instead to study hadronization and confinement.

An important consequence of this behavior of QCD is that after an interaction
that produces quarks, they will emit gluons, as they propagate, and suffer a process
called Hadronization, which leads to the formation of a number of different
hadrons. These hadrons can form a collimated spray called a jet, which is of
great importance at the experimental level, as these jets are the objects measured
when quarks are generated in an interaction. More about jets formation and
hadronization will be explained in the section 2.2.4.

Electroweak interaction

It turns out that the only possible correct Quantum Field description of the
Weak interaction is through the unification of electromagnetic and weak interac-
tion. To do this the proposed Lagrangian should be invariant under the gauge
transformations of the SU(2)L × U(1)Y group, which is a combination of the
SU(2)L group and the U(1)Y, where L stands for left-handed particles and Y
means hyper-charge.

The denomination of left-handed particle is due to the weak interaction being
chiral, which means that it will only Interact with left-handed fermions and right-
handed anti-fermions. Every fermion has left and right-handed components, given



Chapter 2. Theoretical framework 21

by

ψL/R = [(1 ∓ γ5)/2]ψ

The Electroweak Lagrangian associated with these groups is [21]

LEW = iQ̄Ljγ
µDµQLj + iūRjγ

µDµuRj + id̄Rjγ
µDµdRj + iℓ̄Ljγ

µDµℓLj

+iēRjγ
µDµeRj −

1
4

WµνWµν −
1
4

BµνBµν

(2.7)

With

• Dµ = ∂µ − i 1
2 g′YBµ − igTaWa

µ ; a = 1, 2, 3.

• The SU(2)L tensor Wa
µν: Wa

µν = ∂µWa
ν − ∂νWa

µ + gϵabcWb
µWc

ν , a = 1, 2, 3.

• The U(1)Y tensor Bµν:Bµν = ∂µBν − ∂νBµ

The object QLj is a doublet of SU(2)L, which means it transforms under the
transformations of this group. The index j can run over all the generations of
quarks, i.e.: QL1 = (uL, dL), QL2 = (cL, sL), QL3 = (tL, bL).

uRj and dRj are right-handed singlets of SU(2)L, with j running on each gener-
ation of quarks, with uR1 = uR, uR2 = cR and uR3 = tR. Analogously, dR1 = dR,
dR2 = sR and dR3 = bR . These objects are called singlets because they are not
affected by the SU(2)L transformations. This definition is extended also to the
leptons, as they are also fermions, with ℓRj being the doublet of SU(2)L of the
leptons, with j running over the leptons generations and eRj right-handed singlets
of SU(2)L, with eR1 = eR, eR2 = µR and eR3 = τR.

g’ and g are coupling constants of U(1)Y and SU(2)L groups; Y is the hyper
charge; Bµ is a field from the group U(1))Y, T⃗ is a vector of Pauli Matrices, and W⃗µ

is a vector whose components are three fields W1
µ, W2

µ and W3
µ from the SU(2)L

group. The bosons γ, W± and Z are written as combinations of these fields, as
will be shown in the subsection 2.1.3.

The T⃗ = T1, T2, T3 are the generator of SU(2)L (Ta = σa

2 , where σa are the Pauli
Matrices). One can define the electric charge Q in terms of T⃗ and Y:

Q = T3 +
Y
2

(2.8)

For a particle state, these three parameters have different eigenvalues, given
by Table 2.3



Chapter 2. Theoretical framework 22

Particle Q T3 Y
Leptons and Antileptons

e− −1 −1/2 −1
νe 0 +1/2 −1
e+ +1 +1/2 +1
ν̄e 0 −1/2 +1

Quarks (Up-Type) and Antiquarks
u, c, t +2/3 +1/2 +1/3
ū , c̄, t̄ −2/3 −1/2 −1/3

Quarks (Down-Type) and Antiquarks
d, s, b −1/3 −1/2 +1/3
d̄, s̄, b̄ +1/3 +1/2 −1/3

Gauge Bosons
γ 0 0 0

W+ +1 +1 0
W− −1 −1 0
Z0 0 0 0
g 0 0 0

Table 2.3: Electric charge Q, weak isospin T3, and hypercharge Y for particles and
antiparticles in the Standard Model. Fermions are left-handed and Anti-Fermions
are right-handed. Source: [19].

Analogously to the QCD Lagrangian, the EW one provides terms that represent
self-couplings of third and quartic order between the W and Z bosons. This is due
to the fact that it follows the same kind of symmetry of the Lagrangian of QCD,
except that it is in lower dimension.

The EW Lagrangian displays a symmetry under the transformation of the
SU(2)L × U(1)Y group, given a field F, the transformation is

F → eiα(x)Yeiβ⃗(x) · T⃗F (2.9)

Different from the Lagrangians 2.4 and 2.3, 2.7 does not contain terms for
the masses of the fermions, or for the masses of the W and Z bosons, which
have experimentally determined masses. So, while this Lagrangian describes the
coupling between fermions and the weak bosons, it still considers only massless
particles. If one introduces mass terms for these particles on the Lagrangian by
hand it will no longer be symmetric under SU(2)L ×U(1)Y group transformations



Chapter 2. Theoretical framework 23

(explicit symmetry breaking of the Lagrangian), and will no longer be able to
describe the weak interactions. In order to solve this, it is necessary to "break" the
symmetry contained in the 2.7 in a different way, through a known process called
spontaneous symmetry breaking, which will be described in the next subsection.

2.1.3 Higgs Mechanism

Generation of W± and Z0 masses.

The Lagrangian of the eq. 2.7 is absent of mass for the electroweak bosons,
which is incompatible with the experimental results for the masses of the W∓ and
Z bosons. One could argue that this could be solved by introducing, by hand,
mass terms for these bosons on the Lagrangian, but this would be an explicit
breaking of the symmetry that leads to divergences and would not describe the
EW interactions correctly [19]. The mass generation should be done by another
means, known as "spontaneous symmetry breaking". In this section, the sponta-
neous symmetry breaking of the SU(2)L × U(1)Y → U(1)em of the Lagrangian, to
produce mass to the mentioned bosons, will be introduced. This mechanism will
also work on a process to create mass for the fermions of the SM.

One first introduces a Lagrangian of a scalar field ϕ, the Higgs-Field, that
should be added to the Lagrangian of the EW interaction.

LHiggs = (Dµϕ)†(Dµϕ)− V(ϕ) (2.10)

with the Higgs Field ϕ =

[
ϕ+

ϕ0

]
, a doublet of the SU(2)L, with Y = 1, and

ϕ+ = 1√
2
(ϕ1 − iϕ2) , ϕ0 = 1√

2
(ϕ3 + iϕ4). ϕ1, ϕ2, ϕ3, ϕ4 are four independent real

fields (or degrees of freedom).
As the Higgs Lagrangian will be added to the EW Lagrangian, it is required

that it is invariant under the symmetries of the SU(2)L × U(1)Y group, so the
covariant derivative must have the form :

Dµ = ∂µ − igTaWa
µ − ig′

Y
2

Bµ (2.11)

with a = 1, 2, 3.
The potential of 2.10 can be written as

V(ϕ) = −µ2ϕ†ϕ + λ(ϕ†ϕ)2 (2.12)



Chapter 2. Theoretical framework 24

with λ being positive. Two possible forms of potential are allowed:

a) If µ2 > 0 the potential has a shape exposed in the left figure of 2.26.

b) If µ2 < 0 the potential has a shape exposed in the right figure of 2.2, which
displays minima at the points that satisfy ϕϕ† = − µ2

2λ .

For the case a, the potential 2.12 has a minimum, corresponding to a state ϕ0 of
minimal energy. This state is invariant under the SU(2)L ×U(1)Y transformations,
i.e. the minimal energy state has the same symmetry as the Lagrangian.

If the case b is allowed, the minima of the potential will no longer be at ϕϕ† = 0,
but instead at any point that satisfies ϕϕ† = − µ2

2λ . This corresponds to a state of
minimal energy that no longer respects the same symmetries of the Lagrangian
2.10.

Figure 2.2: Shape of the Higgs potential for the case a (left figure) and for the case
b (right figure).

Now, following on the case b, ϕϕ† = − µ2

2λ can be satisfied by the choice :

ϕ1 = ϕ2 = ϕ4 = 0 and ϕ2
3 = −µ2

λ ≡ v2.
To understand the meaning of this choice for the minimum of the potential, one

should refer to the Goldstone theorem. This theorem states that when a continuous
6Since it is not possible to visualize the potential of the Higgs field considering all four fields

ϕ1, ϕ2, ϕ3, and ϕ4 (this would be a hypersphere in four dimensions), a simpler version of the
potential is proposed, considering only the fields ϕ1 and ϕ2 and has a minimal value associated
with ϕϕ† = 1

2 (ϕ
2
1 + ϕ2

2 + ϕ2
3 + ϕ2

4) = 0. Although the number of dimensions in this case is smaller,
the idea behind the Higgs mechanism is the same as in the case with all the ϕ fields.



Chapter 2. Theoretical framework 25

symmetry G in a field theory is spontaneously broken to a smaller subgroup H,
certain vacuum configurations in field space become degenerate. As a result,
small fluctuations along these directions do not cost energy, giving rise to massless
excitation modes known as Goldstone bosons.

In the case of the Higgs mechanism of the SM, the spontaneously broken sym-
metry is local (SU(2)L ×U(1)Y), and the Goldstone bosons correspond to the fields
ϕ1, ϕ2, and ϕ4. With the appropriate choice of Gauge (the so-called Unitary Gauge)
these bosons disappear from the Lagrangian, and are absorbed by the gauge
bosons W+, W−, and Z, providing them with mass. In other words, the Goldstone
bosons are "swallowed" by the gauge bosons, becoming their longitudinal degrees
of freedom.

This allows to write the minimal value of ϕ as ϕ0 = 1√
2

[
0
v

]
Then, it is possible to rewrite the Higgs field expanding it around v with a

real-valued field h(x), such that < h(x) >= 0 [19] [21]:

ϕ(x) =
1√
2

[
0

v + h(x)

]
(2.13)

h(x) is an excitation on the Higgs field, which corresponds to the Higgs field
particle, the Higgs Boson.

The choice of expanding the Higgs field around its minimal value (which corre-
sponds to a non-invariant state) allows for the appearance of terms proportional to
v that generate masses for the fields, as well as terms proportional to h(x), which
correspond to interactions of the Higgs boson with other fields.

One could say that this expansion "breaks" the symmetry of SU(2)L × U(1)Y

group in the Higgs Lagrangian, but this is not entirely correct. The Lagrangian
still remains symmetric under these transformations, though the field h(x) will
transform differently compared to the original SU(2)L × U(1)Y transformations
in Eq. 2.9.

Thus, when it is said that the expansion in Eq. 2.13 "breaks the symmetry of the
Lagrangian" it actually means that the original symmetry is now hidden, but can
still be restored through the appropriate SU(2)L × U(1)Y transformations applied
to the h(x) term.

So, the choice of expansion "breaks" the symmetry of SU(2)L × U(1)Y in the
Higgs Lagrangian, which leads to the generation of mass to the W and Z bosons,
but leaves the photon massless keeping the U(1)em symmetry preserved (explicitly



Chapter 2. Theoretical framework 26

realized in the Lagrangian), as will be shown below.
Inserting 2.13 into the potential of the Higgs field :

V(ϕ) = −µ2

2
(v2 + 2vh(x)+ h(x)2)+

λ

4

(
v4 + 4v3h(x) + 6v2h(x)2 + 4vh(x)3 + h(x)4

)
(2.14)

Recalling that v2 = −µ2

λ

V(ϕ) =
3µ4

4λ
− 2µ2vh(x)− 2µ2

2
h(x)2 +

λ

4

(
4vh(x)3 + h(x)4

)
(2.15)

As mass terms appear in Lagrangians of a field F as 1
2 m2F2, its is possible to

denote that the mass of the Higgs boson is

mh =
√

2µ2

Also, on the potential there are other terms, such as the Higgs couplings:

V(ϕ) = ... + λv︸︷︷︸
H-H-H coupling

h(x)3 +
λ

4︸︷︷︸
H-H-H-H coupling

h(x)4 (2.16)

In order to generate masses for the Gauge bosons, it is necessary to insert 2.13
into the other term of the Higgs Lagrangian, the kinetic one :

(Dµϕ)†(Dµϕ) = |(∂µ − igT⃗W⃗µ − ig′
1
2

Bµ)ϕ|2

were the hypercharge Y was set to 1 for the Higgs field. Expanding the product
TaWa

µ = 1
2 σaWa

µ and using Pauli Matrices values

|Dµϕ|2 =
g2

8
(v + h(x))2

(
(W1

µ)
2 + (W2

µ)
2
)
+

1
2
(
∂µh(x)

)2
+

g2

8
(v + h(x))2(W3

µ)
2 +

g′2

8
(v + h(x))2B2

µ −
gg′

4
(v + h(x))2W3

µBµ

(2.17)

Now, as mentioned in the subsection 2.1.2, the Z and Aµ can be written as
combinations of Wµ

i and Bµ fields, so defining

Zµ =
gW3

µ − g′Bµ√
g2 + g′2

, Aµ =
g′W3

µ + gBµ√
g2 + g′2



Chapter 2. Theoretical framework 27

and

W±
µ =

W1
µ ∓ iW2

µ√
2

It is possible to obtain:

|Dµϕ|2 =
1
2
(∂µh(x))2 +

g2

8
(v + h(x))2

2W+
µ Wµ− +

(
gZµ + g′Aµ√

g2 + g′2

)2


+
g′2

8
(v+ h(x))2

(
gAµ − g′Zµ√

g2 + g′2

)2

− gg′

4
(v+ h(x))2

(
gZµ + g′Aµ√

g2 + g′2

)(
gAµ − g′Zµ√

g2 + g′2

)
.

Now, factorizing the fields for W, Z and A:

|Dµϕ|2 =
1
2
(∂µh(x))2 +

g2

4
(v + h(x))2W+

µ Wµ− + (Zµ)
2
[
(v + h(x))2 (g2 + g′2)

8

]

+(Aµ)2
[
(v + h(x))2

8(g2 + g′2)
(g2g′2 + g′2g2 − 2g′2g2)

]
+(AµZµ)

[
(v + h(x))2

8(g2 + g′2)
(g2g′2 + g′2g2 − 2g′2g2)

]
Arranging the equation in such a way becomes visible that the term multiplying

A2
µ, the mass term of the photon, vanishes, as well as the term multiplying Zµ Aµ,

as there is no direct coupling between Z and γ in the SM, because the Z boson has
no electrical charge. This is an expected result, as the Lagrangian is invariant under
U(1)em transformations, and the photon remains massless. After the mentioned
simplifications

|Dµϕ|2 =
g2

4
(v + h(x))2Wµ+W−

µ +
1
2
(∂µh(x))2 +

(v + h(x))2

8
Z2

µ(g2 + g′2) (2.18)

|Dµϕ|2 =
g2

4
v2Wµ+W−

µ +
1
2

g2vh(x)W+
µ W−

µ +
g2

4
h(x)2W+

µ W−
µ +

1
2
(∂µh(x))2

+
v2(g2 + g′2)

8
Z2

µ +
v(g2 + g′2)

4
h(x)Z2

µ +
(g2 + g′2)

8
h(x)2Z2

µ

(2.19)



Chapter 2. Theoretical framework 28

Now arranging the Higgs Potential (eq. 2.15) and 2.19 into the Higgs La-
grangian (2.10), it becomes possible to visualize the mass terms for Z and W±, as
well as the Lagrangian for the free Higgs field:

LHiggs =
1
2
(∂µh(x))2 − 1

2
m2

hh(x)2︸ ︷︷ ︸
Higgs free Lagrangian

+
g2

4
v2︸︷︷︸

m2
W

Wµ+W−
µ +

1
2

g2vh(x)W+
µ W−

µ +
g2

4
h(x)2W+

µ W−
µ

+
v2(g2 + g′2)

8︸ ︷︷ ︸
1/2m2

Z

Z2
µ +

v(g2 + g′2)
4

h(x)Z2
µ +

(g2 + g′2)
8

h(x)2Z2
µ −

3µ4

4λ
+ 2µ2vh(x)

−λ

4

(
4vh(x)3 + h(x)4

)
)

(2.20)

The masses given by the Higgs mechanism to the gauge bosons are :

m2
W =

g2v2

4

m2
Z =

(g2 + g′2)v2

4
Now, it is possible to rewrite 2.20 in terms of the masses of the bosons, which

allows to visualize each coupling of the Higgs with the bosons:

LHiggs =
1
2
(∂µh(x))2 − 1

2
m2

hh(x)2 + m2
WWµW−

µ +
1
2

m2
ZZ2

µ + 2
m2

W
v︸ ︷︷ ︸

W-h coupling

h(x)W+
µ W−

µ

+
m2

W
v2︸︷︷︸

W-h2 coupling

h(x)2W+
µ W−

µ +
m2

Z
v︸︷︷︸

Z-h coupling

h(x)Z2
µ +

m2
Z

2v2︸︷︷︸
Z-h2 coupling

h(x)2Z2
µ

− λv︸︷︷︸
h3 coupling

h(x)3 − λ

4︸︷︷︸
h4 coupling

h(x)4 − 3µ4

4λ
+ 2µ2vh(x)

(2.21)



Chapter 2. Theoretical framework 29

This Lagrangian encapsulates the interactions between the Higgs field and the
gauge bosons W and Z, as well as the Higgs self-interaction terms.

One important aspect to mention is that the Higgs potential is fully defined
by two parameters: µ and λ. These parameters can be determined from v and
the mass of the Higgs boson, and both quantities have been already determined
experimentally. However, there are no direct measurements of λ. In order to gain
a profound understanding of the Higgs potential, the next step is to measure the
Higgs trilinear coupling, which, as mentioned in the section 1.1, can be studied
through the ttHH channel.

Generation of fermion masses.

The Higgs mechanism successfully generated mass for the gauge bosons,
keeping the photon massless. Now this same idea can be applied for the generation
of mass of the fermions.

First, introduce the Lagrangian

LFermion = −yujQ̄Ljϕ̃uRj − ydjQ̄LjϕdRj − yℓjℓ̄LjϕeRj − yu jūRjϕ̃
†QLj

−ydjd̄Rjϕ
†QLj − yℓj ēRjϕ

†ℓLj
(2.22)

The above Lagrangian is SU(2)L × U(1)Y invariant. yu and yd are couplings of u
and d type quarks (recall the subsection 2.1.2) and yℓ is the coupling associated
with the leptons. As in eq. 2.7, j runs over the generations of quarks and leptons.

ϕ̃ is

[
−(ϕ0)∗

ϕ−

]
Now, allowing the symmetry breaking, inserting 2.13 into the above Lagrangian

LFermion = − 1√
2

∑
j

[
yuj(v + h(x))

(
ūLjuRj + ūRjuLj

)
+ydj(v + h(x))

(
d̄LjdRj + d̄RjdLj

)
+ yℓj(v + h(x))

(
ēLjeRj + ēRjeLj

) (2.23)

Rewriting the left and right-handed fermions fields using
ψL = 1−γ5

2 ψ, and ψR = 1+γ5
2 ψ, one obtains

ψ̄LψR + ψ̄RψL = ψ̄

(
1 + γ5

2
+

1 − γ5

2

)
ψ = ψ̄ψ



Chapter 2. Theoretical framework 30

working this on 2.23

LFermion = − 1√
2

∑
j

[
yuj(v + h(x))ūjuj + ydj(v + h(x))d̄jdj + yℓj(v + h(x))ējej

]
(2.24)

Now, running the index j in the above equation and separating the Lagrangian
into a Higgs interaction part and a mass part, it is possible to obtain :

LFermion = Lψ-Mass + Lψ-Higgs (2.25)

With:

Lψ-Mass = − v√
2

[
yu1ūu + yd1d̄d + yℓ1ēe + yu2c̄c + yd2s̄s + yℓ2µ̄µ + yu3 t̄t + yd3b̄b + yℓ3τ̄τ

]
(2.26)

Lψ-Higgs = −h(x)√
2

[
yu1ūu + yd1d̄d + yℓ1ēe + yu2c̄c + yd2s̄s + yℓ2µ̄µ + yu3 t̄t + yd3b̄b + yℓ3τ̄τ

]
(2.27)

It is possible to choose yψ such that

mψ = yψ
v√
2

,

Then, the masses terms of each fermion are generated by the Higgs field, being
proportional to v. Then the coupling between each fermion and the Higgs is in
the form mψ

v , i.e.: proportional to the fermion mass. This way the masses of the
fermions are included in the theory. Caution should be taken, as they are not
predicted and must be measured.

The coupling yψ is the Yukawa coupling for the fermion ψ. One key feature
of the Higgs boson is that it couples to particles in proportion to their mass. This
is visible in both fermion and massive gauge bosons Lagrangians. This means
that the more massive a particle is, the stronger its interaction with the Higgs
field. As a result, the Higgs boson is more likely to be emitted or to interact with a
top quark, which is the most massive quark in the Standard Model. This strong
coupling makes processes involving the top quark important in Higgs physics
studies.



Chapter 2. Theoretical framework 31

Electroweak interaction with masses terms and Higgs interactions

Finally, after invoking the Higgs mechanism, it was possible to generate the
complete Lagrangian for the Electroweak interaction. So the complete Lagrangian
for EW interaction is obtained by adding to 2.7: LHiggs given by eq. 2.21 and
LFermion given by 2.25, 2.26 and 2.27. Below is a Table of the complete LEW with
an explanation of the terms:

LEW = iQ̄Ljγ
µDµQLj + iūRjγ

µDµuRj

+ id̄Rjγ
µDµdRj + iℓ̄Ljγ

µDµℓLj

+ iēRjγ
µDµeRj

Fermion kinetic terms and
interaction terms between fermions
and W, Z, γ bosons.

−1
4

WµνWµν −
1
4

BµνBµν Kinetic terms for W, Z, γ bosons, as
well as W-γ, W-Z interaction terms
and W, Z self-interactions.

+
1
2
(∂µh(x))2 − 1

2
m2

hh(x)2

+ 2
m2

W
v

h(x)W+
µ W−

µ +
m2

W
v2 h(x)2W+

µ W−
µ

+
m2

Z
v

h(x)Z2
µ +

1
2

m2
Z

v2 h(x)2Z2
µ

Free Higgs Lagrangian, Higgs
interactions with W, Z, and
fermions.

−λvh(x)3 − λ

4
h(x)4 Higgs self-interaction terms.

+ m2
WWµW−

µ +
1
2

m2
ZZ2

µ − ∑
i

mψiψψ Mass terms generated by the Higgs
mechanism for massive bosons and
fermions.

−∑
i

mψih(x)
v

ψψ + . . . Higgs-fermion interaction terms and
other omitted terms.

This represents the complete Lagrangian for the weak interaction. Together
with the Lagrangians of Quantum Electrodynamics (QED) and Quantum Chromo-
dynamics (QCD) outlined in the preceding subsections, this formulation encapsu-
lates all the fundamental interactions described by the Standard Model of particle
physics.



Chapter 2. Theoretical framework 32

2.2 Physics description of the collision between two

protons

Unlike, for instance, the electron, the proton is a composite element. Its internal
structure is indeed rather complex. In addition to the three valence quarks (u,ū,d),
the proton consists of a sea of many more quarks and antiquarks, as well as gluons.
Therefore the collision between two protons is multifaceted and made of different
sub-processes that occur when they collide with each other.

Within the proton-proton interaction, especially when getting to higher energy
collisions (LHC), the high-energy interaction (hard scattering) (see figure 2.5) be-
tween two partons of the protons dominates, while multiple lower-energy parton
interactions (multiple parton interactions, MPI) also take place (see figure 2.6). In
addition, non-interacting partons give rise to the so-called beam remnants. The
three contributions have to be taken into account when studying these collisions
and are described either by perturbative or non-perturbative QCD (e.g. beam
remnants). This is presented in this section.

Figure 2.3 presents a simplified version of the interaction between two protons.
In this figure, one can visualize the most energetic interaction (red dot), a secondary
parton interaction (green dot), and the spectator partons, which do not interact
and give rise to the beam remnants.

2.2.1 Hard scattering.

In proton-proton collisions, the fundamental constituents of each proton, i.e.
the partons (quarks and gluons), interact [26] [27]. The hardest scattering is the
one that occurs with the highest energy and is usually the dominant interaction
of the event (see figure 2.5). The energy carried by each parton during these
collisions is determined by Parton Distribution Functions (PDFs), which define
how the momentum of the proton, or any hadron, is distributed among the
partonic constituents.

Consider a proton with momentum P⃗, the probability of finding a parton q
carrying a fraction xq of the proton’s momentum is given by the function fq/P.
Figure 2.4 displays the PDF of some quarks, at two energy scales of interaction Q2.

To understand parton interactions in the proton collision it is necessary to
appeal to the theory of Strong Interactions, i.e.: QCD. Consider a system with two
protons A and B, composed of partons ai and bj. The protons A and B collide at



Chapter 2. Theoretical framework 33

Figure 2.3: Scheme displaying the interaction between two protons with Hard
Interaction, Secondary interaction and beam remnants. Source : [25].

an energy scale of Q2, such that two generic partons a and b interact and produce
a final state of particles X, i.e.: a + b → X. The quantity of interest is the cross-
section σAB→X of production of this final state after the protons A and B collide.
Such quantity is given by the convolution of the parton distribution functions of A
( fa/A), and B ( fb/B), with the partonic cross section of a + b interaction producing
X, denoted by σ̂ab→X.

σAB→X =
∫ 1

0
dxadxb fa/A(xa, Q2) fb/B(xb, Q2) σ̂ab→X (2.28)

xa and xb are the fractions of momentum carried by partons a and b coming
from the protons. This cross-section can be calculated in any order using per-
turbative QCD for expanding σ̂ab→X considering that the parton interaction that
produces X is hard.

2.2.2 Multiple parton interactions.

The interactions between partons that carry less energy are called multiple
parton interactions (see figure 2.6). Due to the complex nature of protons, several
less energetic secondary interactions can occur when they collide.

In experiments like those conducted at the Large Hadron Collider (LHC),
both types of interactions, hard interaction and MPI, are important. The study



Chapter 2. Theoretical framework 34

Figure 2.4: The Parton Distribution Functions (PDFs) at Q2 = 10 GeV2 (left) and at
Q2 = 104 GeV2 (right). The PDF set used in these plots is MSTW 2008 NLO PDF.
Source: [28].

of MPI helps to improve the understanding of the underlying event structure in
collisions (underlying events are every other event that occurs except for the hard
interaction), while the hard interactions are the core interactions that give access
to learn more about SM and BSM

2.2.3 Beam remnants.

In one proton-proton collision, the partons that are not directly involved in
these interactions, i.e.: the spectator partons, contribute to what is known as beam
remnants. These are composed of the remaining valence quarks and any sea quarks
or gluons required to ensure flavor and color conservation [25]. Partons that form
the beam remnants undergo hadronization7 forming "beam jets", contributing to
the overall hadronic activity of the event. These remnants predominantly retain
their longitudinal momentum, continuing in the original direction of the beam,
with minimal transverse momentum scattering.

Figure 2.7 illustrates the complete scheme of a proton-proton collision, starting
from the collision of proton bunches and showing the pile-up effect8 The figure

7This concept will be explained in the following section.
8The concepts of proton bunches and pile-up will be explained in the next chapter.



Chapter 2. Theoretical framework 35

Figure 2.5: Considering the interaction pp → ttHH with the following decay
channels: t → bW, H → bb and W → ud, the figure above displays the hard
interaction (red dot) of the partons. The other parton interactions are not displayed
in this picture.

Figure 2.6: Considering the interaction pp → ttHH, the figure above displays
the hard interaction and a second interaction (green dot). Many other secondary
parton interactions can occur but were not displayed in this picture.



Chapter 2. Theoretical framework 36

Figure 2.7: Schematized description of the overall proton-proton collision, includ-
ing the pile-up effect when several colliding bunches of protons.

focusing on a single collision shows the occurrence of the hard interaction and the
MPIs, including the parton shower and hadronization. The orange lines represent
the remaining non-interacting partons that form the beam remnants. These also
undergo parton showering and hadronization with radiated gluons.

2.2.4 From partons to jets

This section describes how the initial partons produced in the interaction
evolve into clusters of hadrons, forming jets. This evolution occurs through two
key processes: parton showering, where the partons created in the p-p collision
undergo successive emissions of new partons, and hadronization, where the
resulting partons combine into color-neutral hadrons. Both processes are explained
in detail in this section.

Parton shower

Every propagating quark has a high probability of emitting gluons, with the
likelihood of emission increasing as the energy of the gluon decreases, or be-
comes more colinear with the quark. This is due to the infrared (IR) and collinear
divergences in QCD. The probability of the emission process q → q + g can be ap-
proximated (without going into full detail) as follows: The probability (amplitude



Chapter 2. Theoretical framework 37

squared) diverges as

P ∼ αs

∫ dEg

Eg

∫ dθ

θ2 (double divergences: IR and collinear).

where Eg is the energy carried by the emitted gluon and θ is the angle between the
gluon and the quark.

The probability q → q + g diverges when:
Eg → 0 (IR divergence, associated with

∫
dEg/Eg) and/or the θ → 0 (collinear

divergence, associated with
∫

dθ/θ) 9.
As a result, in a proton-proton collision, a parton will very likely emit a gluon

either before or after the interaction that produces the final state. Since gluons can
interact with other gluons, these emitted gluons may generate additional gluons or
produce quark-antiquark pairs. These newly created quarks can subsequently emit
further gluons, continuing the process. This phenomenon introduces additional
particles into the final state of the system.

Therefore, in addition to considering the new particles arising from multiple
parton interactions (MPI), particles produced by gluon radiation should also be
accounted for the underlying events.

These gluons, responsible for initiating a cascade of new particles, can be
classified into two types: Initial State Radiation (ISR) and Final State Radiation
(FSR). ISR corresponds to gluons emitted from partons propagating from within
the proton before the interaction that produces the final state particles, whereas
FSR refers to gluon radiation emitted by particles generated after the interaction
that produced the final state particles (see Figure 2.8).

Now, dealing individually with multiple partons generated by the gluon ra-
diation using the approach of 2.28 is impractical, so modeling of parton shower
development is applied. This is typically done with Monte Carlo generators, that
samples the emission of new partons probabilistically. Starting at a high energy
scale, the principle of the parton shower is to randomly generate new partons,
from a propagating parton q, such that those became each time more collinear and
soft, until a threshold energy scale. The process follows an ordering variable that
determines when the emissions should stop.

This process is carried out iteratively, with the probability of an initial-state or
final-state parton emitting other partons determined by the Splitting Functions,
which describe the likelihood of parton emission. The evolution is governed by

9More details in this subject can be found in chapter 5 of [29].



Chapter 2. Theoretical framework 38

Figure 2.8: Considering the interaction pp → ttHH, the figure above displays the
hard interaction, a second interaction and the presence of gluon radiation. ISR in
green, FSR in violet.

an ordering variable, such as energy, emission angle, or transverse momentum,
depending on the chosen framework, along with the strong coupling constant [30]
[31].

Hadronization

Hadronization is the process by which propagating partons, such as quarks
and gluons produced in the proton-proton collisions, form bound states known
as hadrons. As mentioned in Section 2.1.2, the strong force has the property of
confining quarks and gluons, and its most prominent manifestation occurs during
the process of hadronization.

This process occurs at energy scales lower than those at which the partons are
initially created. Since hadronization occurs in a lower energy non-perturbative
regime, it cannot be described using Feynman rules of QCD. Instead, phenomeno-
logical models of hadronization are employed to describe this process.

Although several hadronization models exist, each one having its advantages,
this work will bring as an example only two: the string models and the cluster
models. The former treats a color anti-color quark pair as a system bound by a



Chapter 2. Theoretical framework 39

string. As the quarks go away from each other the energy stored in the string
increases. The string eventually breaks, with a given probability depending on
the string’s area coverage, creating new quark-antiquark pairs. The new quark-
antiquark pair may be bound with other quarks of the system, with new strings,
forming hadrons. The process of string breaking can happen multiple times, as
long as there is energy left in the system.

Now, the cluster models follow an approach that relies on statistical probability,
instead of the dynamics of a string. The cluster model can be understood if it is
considered in two steps: 1. the partons that are close combine into color-neutral
objects called clusters. 2. Once a color-neutral cluster is formed, it will decay
probabilistic into hadrons based on the energy and quantum numbers of the
partons inside it [32] [33].

With the addition of the hadronization process, the evolution of the proton-
proton collisions is almost complete. Figure 2.9 summarizes the evolution of the
collisions including the process of hadronization.

Figure 2.9: Considering the interaction pp → ttHH, the figure above displays the
hard interaction, a second interaction, and the presence of gluon radiation, along
with the formation of colorless bound states.



Chapter 2. Theoretical framework 40

Jet formation

After the process of hadronization of partons produced in proton-proton col-
lisions, unstable hadrons (colorless bound states) may form and rapidly decay
into more long-lived hadrons. This process typically generates a multiplicity of
hadrons (with longer lifetime) in the direction of the original bound state, such
that in this region of space, many hadrons can be observed in close proximity.
This spray of hadrons, usually kaons and pions, along with other particles such as
neutrinos, electrons, and muons that may be emitted along the way, forms what is
known as a jet.

The jets are identified and reconstructed in proton-proton collision experiments.
They are "physics objects" of great importance to reconstruct events and serve
as a bridge connecting experimental results with theoretical predictions. Figure
2.10 illustrates a schematic representation of the evolution of partons up to jet
formation, including the steps of parton shower, hadronization, and jet detection
and reconstruction in the detectors (e.g. the calorimeters). In Figure 2.11, a
simulated event in the CMS detector is shown, where a jet is produced. The tracks
of the particles forming the jet can be observed, as well as the energy deposits they
make in the calorimeter cells. Indeed, the information from the particle tracks and
their energy deposits is used in the jet reconstruction.

One relevant aspect is that jets originating from different sources will exhibit
distinct shapes. For example, a jet originating from a b-quark will have its origin
displaced by some distance from the primary vertex. Figure 2.12 illustrates the
topology of a displaced vertex event, specifically depicting the decay of a B-hadron
originating from a b-quark produced in a proton-proton collision. The Primary
Vertex (PV) represents the interaction point where the initial hard scatter occurs.
Due to the relatively long lifetime of the B-hadron, it travels a measurable distance
before decaying, leading to the formation of a Secondary Vertex (SV) displaced
from the PV. The transverse displacement between the PV and the SV is labeled
as Lxy, which is a key variable in identifying heavy-flavor hadrons. The impact
parameters of the tracks associated with the SV are also highlighted: d0 is the
transverse impact parameter, measuring the closest approach of the track to the
beamline in the transverse plane; and z0 is the longitudinal impact parameter,
which describes the displacement along the beam axis.

This displacement occurs because when created a b quark hadronizes into a
B meson, which has a longer lifetime, when compared to other mesons, before
decaying into other hadrons and leptons. The decay modes of B mesons often



Chapter 2. Theoretical framework 41

include neutrinos and leptons. Table 2.4 shows the main decay channels of B
mesons. The lifetime of B meson can vary from 0.5 × 10−12 s to 1.63 × 10−12 s,
depending on the charge of it, while other mesons have shorter lifetimes, like ρ

(lifetime of 4.5 × 10−24 s), η (5 × 10−19 s) or ω (7.75 × 10−23 s) [17]. This is also the
case for the charm or c-quark, but contrary to hadrons originating from lighter
quarks such as u, d, or s. As a result, b and c quarks produce jets with a displaced
secondary vertex, while light quarks produce jets without such a secondary vertex.

B+ = b̄u
Decay mode Probability (%)

ℓ+νℓX 10.99

B0 = bd̄ = b̄d
Decay mode Probability (%)

ℓ+νℓX 10.33

B0
s = bs̄ = b̄s

Decay mode Probability (%)
D−

s anything 62 ± 6
ℓνℓX 9.6 ± 0.8

e+νX− 9.1 ± 0.8
µ+νX− 10.2 ± 1.0

Table 2.4: Decay modes and probabilities of B mesons. X is a meson. Source: [17]
and D is a meson with a c-quark.

This behavior of the vertices, along with the energy distribution and other
properties, allows jets to be classified according to the type of quark that initiated
them. If a jet is identified to be originated by a b meson, it is labeled as "b-tagged".

Experimentally, specialized algorithms are employed to discriminate between
jets, classifying them based on their properties and origin. In chapter 3, more
details on jet algorithms will be given.



Chapter 2. Theoretical framework 42

Figure 2.10: Figure representing the stages of jet forma-
tion. Adapted from: https://www.ericmetodiev.
com/post/jetformation/.Access on 03/02/2025.

Figure 2.11: Picture representing a simulated
jet event in CMS detector. Source : https:
//www.quantumdiaries.org/2011/06/01/
anatomy-of-a-jet-in-cms/. Access on
03/02/2025.

https://www.ericmetodiev.com/post/jetformation/
https://www.ericmetodiev.com/post/jetformation/
https://www.quantumdiaries.org/2011/06/01/anatomy-of-a-jet-in-cms/
https://www.quantumdiaries.org/2011/06/01/anatomy-of-a-jet-in-cms/
https://www.quantumdiaries.org/2011/06/01/anatomy-of-a-jet-in-cms/


Chapter 2. Theoretical framework 43

LxyLxyLxy

PV

SVSVSV

d0
z0

z

y
x

Figure 2.12: Illustration of a displaced vertex event topology. The PV (Primary
Vertex) is the interaction point of the proton-proton collision. The SV (Secondary
Vertex) represents a displaced decay vertex, indicating the presence of a long-lived
B-hadron. Source : https://tikz.net/jet_btag/.

https://tikz.net/jet_btag/


Chapter 3

Experimental Framework: The Large
Hadron Collider and the Compact Muon
Solenoid

3.1 The Large Hadron Collider

The Large Hadron Collider (LHC) is a circular proton-proton accelerator and
collider and is currently the highest energy accelerator in the world (see figure 3.2).
It is located in the European Organization for Nuclear Research (CERN), sitting at
the border of Switzerland and France, with the largest part in France. The tunnel
of the LHC is buried 100 m underground and is 27 km long, occupying French and
Switzerland territories (see figure 3.2). It was designed to explore new territories
in particle physics, the first big discovery was the Higgs boson in 2012 [34].

LHC collides either protons or heavy nuclei, with four main experiments: CMS
(Compact Muon Solenoid), ATLAS (A Toroidal LHC ApparatuS), LHCb (Large
Hadron Collider beauty) and ALICE (A Large Ion Collider Experiment). Each of
these experiments has its own specialized overall detector design and is located in
a different region in the LHC (see figure 3.1). Table 3.1 presents a comparison of
the design parameters of the 4 detectors.

The collider began its first run (Run 1) of proton-proton collisions in 2009,
completing it in 2012, with center-of-mass energies of 7 and 8 TeV, delivering
a total integrated luminosity of 30 fb−1. Run 2 (2016–2018) started after Long
Shutdown 1 (LS1), reaching a peak luminosity of 2 × 1034 cm−2s−1 and delivering
an integrated luminosity of 160 fb−1 at a center-of-mass energy of 13 TeV [42]
[43]. After LS2, LHC Run 3 began in 2022 and is expected to deliver an integrated
luminosity of 450 fb−1 by the end of 2026. The center-of-mass energy is 13.6 TeV,
with an instantaneous peak luminosity of 2.23 × 1034 cm−2s−1 1 (peak luminosity

1The mentioned peak luminosity is for proton-proton collisions, accessible only to ATLAS and

44



Chapter 3. Experimental Framework: The Large Hadron Collider and the Compact Muon
Solenoid 45

Parameter ATLAS CMS ALICE LHCb

Total weight (tons) 7000 14000 10000 5600

Overall length (m) 46 21 26 21

Overall diameter (m) 25 15 16 10

Magnet Toroid/solenoid Solenoid Solenoid/dipole Dipole

Magnetic field for tracking (T) 3.5 / 2 3.8 0.5 / 0.67 1.1

Table 3.1: Comparison of parameters between ATLAS, CMS, ALICE, and LHCb
detectors. CMS is the most compact detector, being the one with the smallest
length and diameter. Table adapted from [34] and updated from [35] [36] [37] [38]
[39] [40] [41].

of 2024 [44]).
The concept of luminosity (L), which was mentioned in the previous para-

graphs, is an important quantity when speaking about particle colliders. It is a
characteristic of a particle accelerator. It is defined as the number of particles in
the beam per (area × second).

The rate of events per second (R f ), associated with a cross section σf , is related
to L as

R f = σfL

For colliding beams, luminosity L is proportional to the number of "bunches"
of particles in each beam, n, the revolution frequency, f , the number of particles in
each bunch, N1 and N2, and inversely proportional to the beam crossing area A
[45]:

L =
n f N1N2

A

If the expression is integrated over time, the integrated luminosity L =
∫
L dt

is obtained, which relates to the number of events of a given scattering:

N f = σf L

So, the increase of luminosity directly enhances the number of events of the
process, yet on the other hand, it can also increase unwanted effects.

For the center-of-mass energy of 13 TeV and an instantaneous luminosity of
the order of 1034 cm−2s−1, the LHC experiment collides 800 million protons per

CMS.



Chapter 3. Experimental Framework: The Large Hadron Collider and the Compact Muon
Solenoid 46

Figure 3.1: LHC and its
main detectors. Source :
https://people.ece.uw.edu/
hauck/LargeHadronCollider/.

Figure 3.2: Location of the LHC.
Source : https://encurtador.
com.br/CsPQ4.

second (inelastic collisions are being considered, with σ(p − p)inelastic = 80 mb
[46]) (values adapted from [47]).

Each bunch of protons has 1.8 × 1011 protons [47] [48], which leads to more
than 60 proton-proton interactions every crossing, this interaction of multiple pairs
of protons in the same bunch crossing is called pile-up (definition from [49]). On
the collisions, only one of these multiple proton-proton interactions is studied. So,
one important step of the analysis of the collision is separating the pile-up effects
from the specific p-p interaction of interest. Figure 3.3 shows the effect of a pile-up
as observed in the experiment.

Pileups can be classified into two categories: in-time pileup and out-of-time
pileup. The first refers to additional proton-proton collisions occurring within
the same bunch crossing as the primary collision of interest. Out-of-time pileup
consists of extra proton-proton collisions taking place in adjacent bunch crossings,
occurring before or after the primary collision.

If the instantaneous luminosity increases, it means that more proton bunches
are interacting per second, increasing the likelihood of multiple collisions in each
bunch crossing, which results in a higher pileup. The average value of the number
of interactions per crossing increased over the years, along with the instantaneous
luminosity (figure 3.4).

Additionally, the increase in the center of mass energy also increases the pile-
up. Figure 3.5 displays the pile-up distributions recorded by the CMS experiment

https://people.ece.uw.edu/hauck/LargeHadronCollider/
https://people.ece.uw.edu/hauck/LargeHadronCollider/
https://encurtador.com.br/CsPQ4
https://encurtador.com.br/CsPQ4


Chapter 3. Experimental Framework: The Large Hadron Collider and the Compact Muon
Solenoid 47

Figure 3.3: This event shows 78
reconstructed proton-proton inter-
action points in the CMS detector.
Source : https://cds.cern.ch/
record/1479324.

Figure 3.4: The blue curve represents
the average pileup of each year, and
the red curve represents the peak lu-
minosity of each year. The consid-
ered cross-section for inelastic pp in-
teraction was 80 mb. Source of data
[44].

over the years, since 2011.

3.2 CMS reference system and Kinematical variables

This section provides an overview of the proton-proton collision reference
frame, within the context of the CMS detector, along with the key kinematic
variables measured as a result of these collisions. The information presented here
is based on the following references [50] [51] [52].

Consider a coordinate system with its origin positioned at the point where the
protons collide, in the CMS detector. This is a Cartesian xyz coordinate system,
where the z-axis aligns with the direction of the proton beam. Figure 3.6 shows
the origin of the x, y, and z coordinates located at the center of the CMS detector.
Figure 3.7 provides a closer view of this coordinate system.

In this system, a particle created after the collision with momentum p⃗ makes
an angle θ with respect to the z-axis. In the plane transverse to the z-axis (the x-y
plane), the transverse projection of the momentum, called PT, forms an angle ϕ

(azimuth angle) relative to the x-axis.
Figures 3.8 and 3.9 illustrate the coordinate system inside the cylindrical CMS

detector from different perspectives to aid visualization.
From this point onward, these figures will be used to define several important

kinematic quantities, and they will be referenced for clarity and understanding.

https://cds.cern.ch/record/1479324
https://cds.cern.ch/record/1479324


Chapter 3. Experimental Framework: The Large Hadron Collider and the Compact Muon
Solenoid 48

Figure 3.5: Pile-up distributions at CMS for each year. Source : [44].

In a proton-proton collider, special attention is given to the momentum in
the transverse direction of the beam for several reasons. The first is that, in such
collisions, the longitudinal (beam-parallel) momentum of the partons that make
up the protons is not accessible, as there are multiple partonic interactions in each
p-p collision, and the fraction of the momentum of the proton carried by each
parton, given by x, is not known for each event. In this sense, for each collision,
the longitudinal momentum is not accessible, unlike in an electron accelerator, for
example, where the initial beams consist of elementary particles with well-defined
momentum.

1. Transverse momentum (PT)
The transverse momentum is one important kinematical characteristic for the

study of particle collisions. In the context of the system considered in figure 3.7,
where the proton beam axis is along the z-direction, the transverse momentum of
a particle created, at the interaction point, after the initial interaction is given by

pT =
√

p2
x + p2

y

where px and py are the momentum components in the transverse plane.
In the CMS experiment, the PT of particles is measured with high precision due

to the silicon tracking system embedded in a strong magnetic field generated by



Chapter 3. Experimental Framework: The Large Hadron Collider and the Compact Muon
Solenoid 49

CMS

y

z x

p⃗Tp⃗

N

Jura LHC

ATLAS
ALICE

LHCb

ϕ
θ

Figure 3.6: Coordinate system x, y, z
centered on the CMS detector.

CMS

y

z x

p⃗Tp⃗

ϕθ

Figure 3.7: Close view of the CMS
coordinate system.

IP

z

y

x

η = 0

η > 0

η < 0

η = +∞

η = −∞p⃗T

p⃗ ϕϕϕ
θ

ϕϕϕ

N

Jura
ATLAS

center of
the LHC

Figure 3.8: CMS detector coordinate
system shown in a cylindrical view.

z

IP

y

x

η = 0
η > 0

η < 0

η = +∞

η = −∞

p⃗T

p⃗

ϕϕϕ

θϕϕϕ

N

Jura

ATLAS center of
the LHC

Figure 3.9: Different profile view of
the CMS coordinate system inside
the detector.

Figure 3.10: Visualization of the coordinate system of CMS detector from various
perspectives. Source: https://tikz.net/axis3d_cms/

https://tikz.net/axis3d_cms/


Chapter 3. Experimental Framework: The Large Hadron Collider and the Compact Muon
Solenoid 50

the superconducting solenoid. This field permeates the silicon tracking detectors,
allowing the measurement of the curvature of the trajectories of charged particles
as they pass through and leave their traces recorded. With this curvature, the
transverse momentum can be calculated through the formula [53]

P = 0.3|B⃗| × R

where P is the particle momentum in GeV/c, R is the radius of curvature in meters
and |B⃗| is the magnetic field strength in Tesla.

2. Pseudo-rapidity (η)
The pseudo-rapidity is a quantity written in terms of the polar angle relative to

the beam axis θ, defined as:

η = − ln
(

tan
(

θ

2

))
It is a relevant quantity in the context of relativistic collisions because differ-

ences in pseudo-rapidity are invariant under Lorentz transformations along the
beam direction [51].

Using η as a spatial quantity to describe the distance distance of a particle from
the beam axis, instead of θ, leads to the understanding that smaller θ values (closer
to the beam axis) correspond to higher η values. Conversely, as θ approaches π/2
(closer to the transverse plane), η becomes smaller. Figures 3.11 and 3.12 illustrate
how η behaves for different θ values.

The variable η is largely used to describe regions of the CMS detector. Figure
3.14 helps to visualize how different parts of the detector are covered by the
pseudo-rapidity values.

3. Total transverse energy (HT)
The total transverse energy (HT) is the amount of energy in the transverse

plane for each event. It is defined as the scalar sum of the PT of all jets in the event.

HTjets = ∑
j
|PT|j (3.1)

The HT variable indicates hadronic activity, as it focuses on jets (events with
higher HT are a signal of more jet activity).

4. Missing Transverse Energy (MET)
Due to the presence of particles with small interactions with matter (like

neutrinos), non-detectable particles, or uncovered/less covered regions of the



Chapter 3. Experimental Framework: The Large Hadron Collider and the Compact Muon
Solenoid 51

y

z
η = −∞ θ = 180◦

η = −2.44
θ = 170◦

η = −1.32

θ = 150◦

η = −0.88

θ = 135◦

η = −0.55

θ = 120◦

η = 0

θ = 90◦

η = 0.55

θ = 60◦

η = 0.88

θ = 45◦
η = 1.32

θ = 30◦

η = 2.43
θ = 10◦

η = +∞θ = 0◦θ

Figure 3.11: Illustration of various
pseudo-rapidity values (η) for differ-
ent angles θ relative to the beam axis.

η

θ45◦45◦ 90◦90◦ 135◦135◦ 180◦180◦

1

−1

2

−2

3

−3

4

−4

5

−5

Figure 3.12: Pseudo-rapidity (η) as a
function of θ

Figure 3.13: Figure demonstrating the behavior of pseudo-rapidity. Source:
https://tikz.net/axis2d_pseudorapidity/

Figure 3.14: View of the CMS detector with delimited η regions. Source : https:
//m.bergauer.org/friedl/diss/html/node8.html

https://tikz.net/axis2d_pseudorapidity/
https://m.bergauer.org/friedl/diss/html/node8.html
https://m.bergauer.org/friedl/diss/html/node8.html


Chapter 3. Experimental Framework: The Large Hadron Collider and the Compact Muon
Solenoid 52

detector, some processes may present an imbalance in energy. In the transverse
plane, this can be measured if a vectorial sum is performed in P⃗T in all final state
particles.

MET = −|∑
i

P⃗Ti| (3.2)

As new particles could cause the presence of missing energy, this quantity
offers the possibility to study new physics.

5. Invariant Mass (M)
Consider an unstable particle X, with mass mX, generated in the proton-proton

collision. This unstable X may decay into particles x1, x2... xN. Taking the total
four-momentum of the system and squaring it leads to a Lorentz Invariant quantity
called Invariant Mass of the system:

M2
X = (Pµ

TOT)
2 =

(
n

∑
i

EXi

)2

−
(

n

∑
i

P⃗Xi

)2

If X is produced on-shell, the M quantity will have a peak around mX. This
is an important variable for verifying the presence of a particle X in the process.
In the context of the SM, X could be for instance, a Higgs Boson, a Z boson, or a
W boson. Taking H → bb, for instance. If the invariant mass of the b-quark jets
originating from the Higgs boson is calculated, it is

M2
b−jets = (Eb−jet1 + Eb−jet2)

2 − (P⃗b−jet1 + P⃗b−jet2)
2

And would result in a distribution with a peak around mh = 125 GeV/c².
The invariant mass quantity has great importance to the discovery of reso-

nances [51] and new massive particles.

3.3 The CMS experiment and its components

The Compact muon Solenoid (CMS) experiment at LHC is a general-purpose
detector designed to investigate a large range of physics phenomena (see figure
3.15 ). Another general-purpose detector that runs parallel to CMS is the ATLAS (A
Toroidal LHC ApparatuS) detector. Both ATLAS and CMS are looking for a large
scope of Standard Model (SM) and Beyond Standard Model (BSM) Particle Physics
accessible to LHC machines. CMS and ATLAS have different designs and different
detector setups, both being designed as hermetical detectors with a central barrel



Chapter 3. Experimental Framework: The Large Hadron Collider and the Compact Muon
Solenoid 53

structure two end-cap sections and forward part to cover the interaction point (see
figure 3.16 ).

The CMS has several layers of sub-detectors arranged in concentric cylinders in
the central barrel region, where the collisions take place. The sub-detectors include
an overall silicon tracker (innermost and outermost tracking detectors), an electro-
magnetic calorimeter, a hadronic calorimeter, a superconducting solenoid magnet,
and a muon detection system. The detector is hermetic since the interaction point
is surrounded both in the central barrel, the backward, and the forward regions,
to cover all the angular regions of space, with different layers of detectors, with
various purposes. This allows minimization of the gaps through which particles
could escape undetected. One of the most striking features of the CMS detector
design is its tracking silicon detectors (pixel detector and silicon microstrip de-
tector), which sit inside a very large solenoidal magnetic field, together with an
electromagnetic and hadronic calorimeter.

Figure 3.17 shows the design of each layer of CMS detectors. This is discussed
in more detail in the next subsections

3.3.1 The Silicon Tracker

The Silicon tracker is the innermost part of the CMS experiment ( figure 3.17).
It consists of two parts, the pixel detector and the outer silicon tracker, made
of silicon microstrips. With an overall coverage of an arrangement of 76×106

individual detectors, which are two types: the silicon pixel detectors (66 million)
and the silicon microstrip detectors (10 million). It has an η coverage of |η| < 2.5.

Before explaining more about the differences between each of the two above-
mentioned detectors, it is important to comprehend their working principle. Both
are silicon detectors, which are made by a semiconductor junction of type p-n,
i.e.: an interface formed when a p-type semiconductor (with excess holes) is
joined with an n-type semiconductor (with excess electrons ). At the interface,
free electrons recombine with holes, leaving behind positively charged immobile
ions and negatively charged immobile ions, creating a region without free charges,
called the depletion region. The ions in the depletion region create an internal
electric field, which opposes further diffusion of charges. Such a region can be
enhanced by applying an electric potential [56] [57] [58].

Charged particles that cross the depletion region may ionize the material and
create electron-hole pairs along their path. Due to the potential applied, these



Chapter 3. Experimental Framework: The Large Hadron Collider and the Compact Muon
Solenoid 54

Figure 3.15: Location of the CMS detector in the LHC ex-
periment. Adapted from https://home.cern/science/
accelerators/large-hadron-collider.

Figure 3.16: View of the CMS detector. Source: [54].

https://home.cern/science/accelerators/large-hadron