ABSTRACT

Title of Dissertation: Constraining Higgs Boson Self-coupling with VHH
Production and Combination, and Searching for Wγ
Resonance using the CMS Detector at the LHC

Yihui Lai
Doctor of Philosophy, 2024

Dissertation Directed by: Professor Christopher Palmer
Department of Physics

Since the discovery of the Higgs boson (H) with a mass of 125 GeV by the ATLAS and

CMS collaborations at the CERN LHC in 2012, the focus of the particle physics community

has expanded to include precise measurements of its properties, and so far the measurements

align with the Standard Model (SM) predictions. Of particular interest among these properties

is the Higgs boson self-coupling, which can be directly probed by measuring the cross section

for the production of Higgs boson pair (HH). This thesis presents three analyses using proton-

proton collision data at
√
s = 13TeV with an integrated luminosity of 138 fb−1: a search for

SM Higgs boson pair production with one associated vector boson (VHH), a combination of H

measurements and HH searches, and a search for a new particle decaying to a W boson and a

photon (γ). The VHH search focuses on Higgs bosons decaying to bottom quarks, and vector

boson decaying to electrons, muons, neutrinos, or hadrons, with a novel background estimation

approach. An observed (expected) upper limit on the VHH production cross section is set at



294 (124) times the SM predicted value. The combination of H measurements and HH searches

aims to constrain the Higgs self-coupling with the best possible precision. The search for Wγ

resonance focuses on leptonic W boson decays, achieving the world’s best sensitivity for this

resonance in the mass ranges considered.



Constraining Higgs Boson Self-coupling with VHH Production and
Combination, and Searching for Wγ Resonance

using the CMS Detector at the LHC

by

Yihui Lai

Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment

of the requirements for the degree of
Doctor of Philosophy

2024

Advisory Committee:
Professor Christopher Palmer, Chair/Advisor
Professor Sarah Eno, Co-chair
Professor Kaustubh Agashe
Professor Manuel Franco Sevilla
Professor Mihai Pop, Dean’s Representative



© Copyright by
Yihui Lai

2024



Acknowledgments

This thesis would not been possible without the invaluable support of many people. I am

deeply grateful to them for making my graduate journey an unforgettable experience that I will

cherish forever.

First and foremost, I would like to express my deepest gratitude to my advisor, Prof.

Christopher Palmer. His guidance, support, and immense knowledge have been indispensable

throughout my research. Chris taught me step by step how to design and construct a comprehensive

analysis, solve various challenges, and think scientifically. The journey of a graduate student

is often fraught with ups and downs, requiring extra patience and confidence. Some of our

projects took longer and required more effort than expected, yet Chris always encouraged me

and generously praised my progress. Working with him has been a true pleasure. He was always

available for help and advice, and I have lost count of how many times I disturbed him during

weekends. Without his comprehensive guidance, I could not have reached this point.

I would also like to thank Prof. Sarah Eno for inviting me to join the CMS group at

the University of Maryland, opening a new chapter in my life. Initially, due to my lack of

experience, it was challenging for me to understand the discussions in group meetings. Sarah

always encouraged me to ask questions with the curiosity of a child and to keep asking simple

questions until I truly understood. I also learnt a lot about calorimetry when working with her,

which provided me with a deeper understanding of particle physics and laid the foundation for

ii



my subsequent analyses. Her vast knowledge has always amazed and inspired me.

I am also thankful to Prof. Alberto Belloni. During my stay at CERN, Alberto closely

mentored and supported me in the Wγ analysis and on the HCAL work. He carefully considered

every detail, from the phrasing of a single sentence to the overall strategy and planning of the

analysis. His rigorous and meticulous altitude towards research deeply inspired me.

Many thanks to Prof. Kaustubh Agashe, Prof. Manuel Franco Sevilla, and Prof. Mihai Pop

for serving on my dissertation committee and providing valuable comments.

I would like to thank Dr. Markus Seidel, Dr. Long Wang, Dr. Agni Bethani. Working

with Markus has been incredibly rewarding, I learned a lot from his knowledge and experience.

The Wγ analysis was a long journey, and I cannot imagine completing it without his help. Long

shared an office with me for more than two years and was always the first person I turned to with

questions. Despite my daily interruptions, he patiently answered all of them. I also appreciate

Agni for her invaluable help. Her deep understanding of physics problems and critical insights

have taught me a lot. I am especially thankful to Agni for commenting on the draft of my thesis.

Being part of the CMS Collaboration has given me the opportunity to collaborate with

numerous exceptional physicists, from whom I have gained invaluable knowledge. I would

like to thank all the (sub)group conveners and analysis review committees for their constructive

comments. I am especially grateful to Chayanit Asawatangtrakuldee, John Alison, Chuyuan

Liu, Xiaohu Sun and Licheng Zhang for their insightful comments and helpful advice while

working together on the VHH analysis. Additionally, I am thankful to Fabio Monti, Alexandra

Carvalho A. de Oliveira, Torben Lange and Andrea Marini for their collaboration on the H+HH

combination. Special thanks to Nicholas Haubrich for working together on the fun but challenging

VH(bb) analysis, which probably deserves an additional chapter in this thesis.

iii



More than half of my graduate journey was spent at CERN, an opportunity for which I am

incredibly grateful. My time there was enriched by the friendships I made with Xuli, Jingjing,

Renhui, Qingfeng, Aohui, Youmin, Xiaonan, and Sitian. I also want to express my gratitude to

my friends at the University of Maryland: Jiashen, Ziyi, Weizheng, Shoukang, Yijia, and Kwok

Lung. The friendships was especially meaningful during the pandemic while I was in Maryland.

I owe a special thanks to Jiashen for helping me with various errands while I was based at CERN.

I also extend my heartfelt thanks to all members of the CMS group at the University of

Maryland for their invaluable feedback throughout every stage of the analysis, particularly to

Yongbin, Sara, Yi-Mu, Christos, Mekhala, Braden, Timothy, and Alexey. Working alongside you

all in the same group has been a rewarding experience. I am especially grateful to Yongbin for

his substantial help. Despite only overlapping for a year at Maryland, we continued collaborating

on several projects after he joined Fermilab as a postdoc. I am deeply thankful to him for sharing

both professional insights and personal experiences.

I would like to further express my gratitude to my colleagues at HCAL and BRIL group.

Working with them, I learned the ins and outs of how the detector actually works.

Finally, I would like to dedicate this thesis to my parents. I was born in a small town with

limited educational opportunities, where neither of my parents attended high school. However,

it is their unwavering support that has enabled me to reach where I am today. I am immensely

grateful to them.

iv



Table of Contents

Acknowledgements ii

Table of Contents v

List of Tables viii

List of Figures x

List of Abbreviations xiv

Chapter 1: Introduction 1

Chapter 2: Theoretical Background 7
2.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 The Higgs Boson at the LHC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.1 Kappa Framework and Effective Field Theory . . . . . . . . . . . . . . . 21
2.2.2 Higgs Self-coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.3 Single Higgs Production . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.4 Higgs Boson Pair Production . . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter 3: The CMS Detector at the LHC 31
3.1 The Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 The Compact Muon Solenoid Detector . . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 Inner Tracking System . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 The Calorimeter System . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.2.3 Muon System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.4 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.5 Luminosity Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.3 Identification and Reconstruction of Physics Objects . . . . . . . . . . . . . . . . 51
3.3.1 The Particle-Flow Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3.3 Electrons and Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.4 Lepton Isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.5 Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.6 Identification of b Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

v



3.3.7 Missing Transverse Energy . . . . . . . . . . . . . . . . . . . . . . . . . 67

Chapter 4: Search for the Higgs Boson Pair Production With One Associated Vector
Boson 68

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

4.2.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.2 Simulated Signal Samples . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.3 Simulated Background Samples . . . . . . . . . . . . . . . . . . . . . . 74
4.2.4 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.3 Signal Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1 Signal Reweighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.3.2 NLO and NNLO Correction . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4 Selection and Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
4.4.1 Object Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4.2 Event Reconstruction and Selection . . . . . . . . . . . . . . . . . . . . 95

4.5 Event Categorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.1 Categorization Based on rHH . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5.2 Categorization Based on b-tagging . . . . . . . . . . . . . . . . . . . . . 101
4.5.3 Categorization Based on BDT . . . . . . . . . . . . . . . . . . . . . . . 101
4.5.4 Overlap Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.5.5 Selection Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.6 Background Modeling and Signal Extraction . . . . . . . . . . . . . . . . . . . . 106
4.6.1 Background Reweighting . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.6.2 Signal Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.7 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.7.1 Experimental Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.7.2 Theoretical Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Chapter 5: Combination of Single Higgs Measurements and Higgs Boson Pair Searches 141
5.1 Input analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.2 Statistical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.2.1 Parameters of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.3 Overlap Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.4 Systematic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.4.1 Experimental Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.4.2 Theoretic Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.5.1 Signal Strength Measurements . . . . . . . . . . . . . . . . . . . . . . . 157
5.5.2 Higgs Boson Coupling Measurements . . . . . . . . . . . . . . . . . . . 158

Chapter 6: Search for a Resonance Decaying to Wγ Using Leptonic W boson Decays 163
6.1 Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

vi



6.1.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.1.2 Signal MC Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.1.3 Background MC Samples . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1.4 Overlap Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1.5 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

6.2 Selection and Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.2.1 Basic Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
6.2.2 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.3 Signal Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.4 Background Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.5 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Chapter 7: Conclusion and Outlook 197

vii



List of Tables

2.1 Cross sections and uncertainties of different HH production modes . . . . . . . . 27

3.1 Nominal parameters of the LHC in pp collisions. . . . . . . . . . . . . . . . . . 35

4.1 List of dataset for VHH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Primary data samples used in the VHH analysis . . . . . . . . . . . . . . . . . . 72
4.3 List of the simulated VHH signal samples and the corresponding cross sections

times branching ratios (σ × B) of HH → bb̄bb̄. . . . . . . . . . . . . . . . . . . . 73
4.4 List of simulated background processes and the corresponding generators . . . . 75
4.5 List of simulated NLO Drell–Yan process and the corresponding generator . . . . 77
4.6 Summary of HLT paths for VHH . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.7 Charged leptons and pmiss

T selections in VHH . . . . . . . . . . . . . . . . . . . . 87
4.8 Jet selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.9 Event selection in VHH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.10 Search region selections applied on top of the baseline selections. . . . . . . . . . 101
4.11 Input variables for the categorization BDT . . . . . . . . . . . . . . . . . . . . . 104
4.12 Input variables for reweighting BDT in the merged topology . . . . . . . . . . . 110
4.13 Input variables for reweighting BDT in the 2L channel. . . . . . . . . . . . . . . 119
4.14 Input variables for the SvB BDT in the resolved topology of the MET and 1L

channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.15 A summary of categorization in all channels . . . . . . . . . . . . . . . . . . . . 121
4.16 Uncertainty breakdown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.17 Observed and expected 95% CL upper limits on the coupling modifiers. . . . . . 136

5.1 HH analyses included in this combination. . . . . . . . . . . . . . . . . . . . . . 145
5.2 H analyses included in this combination. . . . . . . . . . . . . . . . . . . . . . . 146
5.3 Overlaps identified between analyses . . . . . . . . . . . . . . . . . . . . . . . . 152
5.4 Constraints on κλ at 68% and 95% CL from the combination of the H and HH

channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.1 The JSON files used in the analysis and the corresponding integrated luminosities. 166
6.2 The primary data samples used in this analysis. . . . . . . . . . . . . . . . . . . 166
6.3 List of simulated background processes and the corresponding generator . . . . . 169
6.4 Summary of applied high-level trigger paths. . . . . . . . . . . . . . . . . . . . 172

viii



6.5 Basic object selection requirements . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.6 Event selection requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.7 Number of events in data and the predicted number in simulation . . . . . . . . . 179
6.8 Order of the best function in each channel and each function family. . . . . . . . 183
6.9 The best fit functions with the background-only hypothesis. . . . . . . . . . . . . 184

ix



List of Figures

1.1 Timeline of particle discoveries . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Center-of-mass energy of particle colliders vs year . . . . . . . . . . . . . . . . . 5

2.1 Elementary particles in SM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Schematic view of the Higgs potential . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Feynman diagrams of Higgs boson interactions with other particles . . . . . . . . 19
2.4 Feynman diagrams of Higgs boson interactions with itself . . . . . . . . . . . . . 19
2.5 Higgs boson coupling modifiers measurement . . . . . . . . . . . . . . . . . . . 20
2.6 Higgs boson coupling modifiers measurement . . . . . . . . . . . . . . . . . . . 24
2.7 SM Higgs boson production cross sections and the decay branching ratios . . . . 25
2.8 Leading order diagrams for Higgs boson production . . . . . . . . . . . . . . . . 26
2.9 Feynman diagrams of GGF HH production . . . . . . . . . . . . . . . . . . . . . 27
2.10 Feynman diagrams of VBF HH production . . . . . . . . . . . . . . . . . . . . . 28
2.11 Cross sections for main HH production channels . . . . . . . . . . . . . . . . . . 28
2.12 Distribution for mHH with different κλ coupling hypotheses . . . . . . . . . . . . 29

3.1 Cumulative delivered and recorded luminosity . . . . . . . . . . . . . . . . . . . 33
3.2 The accelerator complex at CERN . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Mean number of interactions per crossing . . . . . . . . . . . . . . . . . . . . . 36
3.4 Cutaway diagram of CMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.5 Conventional 3D coordinate system at the CMS . . . . . . . . . . . . . . . . . . 38
3.6 Schematic view of the CMS inner tracker . . . . . . . . . . . . . . . . . . . . . 39
3.7 Schematic views of the CMS pixel tracker . . . . . . . . . . . . . . . . . . . . . 40
3.8 CMS electromagnetic calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.9 A schematic view of one quarter of the HCAL . . . . . . . . . . . . . . . . . . . 45
3.10 Schematic view of the muon detection systems . . . . . . . . . . . . . . . . . . . 47
3.11 Schematic view of CMS highlighting the main luminometers . . . . . . . . . . . 50
3.12 A schematic view of a transverse slice of the CMS detector . . . . . . . . . . . . 52
3.13 A pictorial representation of jets with increasing Lorentz boost . . . . . . . . . . 63
3.14 Jet initiated by a bottom quark . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.1 LO Feynman diagrams of VHH production . . . . . . . . . . . . . . . . . . . . 69
4.2 The VHH cross section divided by the total HH cross section as a function of κλ . 70
4.3 VHH signal modeling validation . . . . . . . . . . . . . . . . . . . . . . . . . . 83

x



4.4 Kinematic distributions with different κλ value . . . . . . . . . . . . . . . . . . . 84
4.5 NLO quark-initiated Feynman diagrams for VHH production . . . . . . . . . . . 85
4.6 Representative Feynman diagram for ggZHH production . . . . . . . . . . . . . 85
4.7 NNLO correction accounting for the ggZHH contribution . . . . . . . . . . . . . 86
4.8 Efficiency measurement with tag-and-probe method . . . . . . . . . . . . . . . . 89
4.9 Electron efficiency scale factors . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.10 pmiss

T trigger scale factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.11 Illustration of resolved and merged topologies . . . . . . . . . . . . . . . . . . . 92
4.12 Mass distribution before and after energy regression correction . . . . . . . . . . 93
4.13 MET channel angular cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.14 Illustration of vector boson decays. . . . . . . . . . . . . . . . . . . . . . . . . 95
4.15 HH pairing strategy in the VHH channel . . . . . . . . . . . . . . . . . . . . . . 97
4.16 Two-dimensional mass distribution for VHH signal and background . . . . . . . 100
4.17 Illustration of the region definition in the merged topology. . . . . . . . . . . . . 102
4.18 Categorization BDT demonstration . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.19 Analysis categorization and the observable used in each region in the final fitting. 103
4.20 The 95% CL upper limit scan with different priority given to the events that can

pass both the resolved and the merged topology selection. . . . . . . . . . . . . . 105
4.21 Efficiencies of selections in VHH . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.22 Demonstration of the reweighting procedure . . . . . . . . . . . . . . . . . . . . 111
4.23 Background reweighting method closure check on the input variables for the LP

region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.24 Background reweighting method closure check on the input variables for the LP

region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.25 Background reweighting method closure check on the input variables for the HP

region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.26 Background reweighting method closure check on the input variables for the HP

region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.27 Background reweighting method closure check on the correlations between input

variables for the LP region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.28 Background reweighting method closure check on the correlations between input

variables for the HP region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.29 Background reweighting method closure check on the SvB BDTs . . . . . . . . . 120
4.30 Postfit distributions of kinematic variables in the resolved topology regions . . . . 123
4.31 Postfit distributions of kinematic variables in the merged topology regions . . . . 124
4.32 Postfit distributions of the subleading Higgs boson mass in the merged topology

regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.33 Postfit distributions of the transverse momentum of the vector boson in the resolved

topology regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
4.34 Postfit BDT distributions with the signal-plus-background hypotheses of the FH

and 2L channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
4.35 Postfit BDT distributions with the signal-plus-background hypotheses of the MET

and 1L channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.36 The log10

(
100(SSM/B)

)
distribution with all channels merged . . . . . . . . . . 133

4.37 Best fit signal strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

xi



4.38 The post-fit values of the nuisance parameters and their impacts on the SM inclusive
signal strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

4.39 Likelihood scans in κλ versus κ2V . . . . . . . . . . . . . . . . . . . . . . . . . 135
4.40 Likelihood scans in κ2W versus κ2Z . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.41 VHH cross section limits for SM and κλ = 5.5 . . . . . . . . . . . . . . . . . . . 137
4.42 Upper 95% CL limits scanned over the κλ parameter . . . . . . . . . . . . . . . 137
4.43 Upper 95% CL limits scanned over the κ2V parameter . . . . . . . . . . . . . . . 138
4.44 Upper 95% CL limits scanned over the κV parameter . . . . . . . . . . . . . . . 138
4.45 Event display of the VHH MET channel . . . . . . . . . . . . . . . . . . . . . . 139
4.46 Event display of the VHH 2L channel . . . . . . . . . . . . . . . . . . . . . . . 139
4.47 Event display of the VHH 1L channel . . . . . . . . . . . . . . . . . . . . . . . 140

5.1 Feynman diagrams of κλ-dependent NLO corrections to the main H production
mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.2 Feynman diagrams of κλ-dependent NLO corrections to the main H production
mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

5.3 Total cross sections of H, HH and relative branching ratio variation as a function
of κλ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

5.4 Differential cross sections of ZH and tt̄H . . . . . . . . . . . . . . . . . . . . . . 144
5.5 Constraint on the Higgs boson self-coupling modifier κλ . . . . . . . . . . . . . 145
5.6 Expected and observed likelihood scan of µH (left) and µHH (right) from the

combination of all H channels and all HH channels, separately. . . . . . . . . . . 157
5.7 Upper limits on µHH from the HH combination as a function of κλ and κ2V on

the left and right panel, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 158
5.8 Profiled likelihood scans of κλ from H and HH combination, with all other couplings

fixed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.9 Two-dimensional likelihood scans of (κλ, κt) . . . . . . . . . . . . . . . . . . . 160
5.10 2D likelihood scans of (κV, κ2V) . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.11 Expected and observed likelihood scans of κλ on the left and on the right panels,

respectively, comparing different sets of input channels and different assumptions
on the H couplings to the fermions and the vector bosons. . . . . . . . . . . . . . 161

5.12 HH and κλ sensitivities projection . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.1 Leading order Feynman diagram for a heavy particle X decaying to a W boson
and a photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.2 Invariant mass distribution of the W boson and γ at the LHE level with and
without the interference effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

6.3 Distribution of photon pT for inclusive W+jets and Wγ samples (upper row)
and for inclusive and pT-binned Wγ samples (bottom row) before (left) and after
(right) overlap removal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

6.4 Distribution ofHT for inclusive andHT -binned W+jets samples before (left) and
after (right) overlap removal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

6.5 Distribution of the photon pT and mT for all the signal stacked together. . . . . . 175
6.6 Product of detector acceptance and analysis selections efficiency . . . . . . . . . 177
6.7 Product of detector acceptance and analysis selection efficiency . . . . . . . . . . 177

xii



6.8 Distributions of mT and pT (γ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.9 Comparison of the mT distribution of the 1000 GeV signals across different years. 179
6.10 Parameterize the best fit value of parameters as a function of mX. . . . . . . . . 181
6.11 Representative signal mT distributions with 500, 1000, 1600 GeV resonance mass. 182
6.12 Demonstration on the discrete profiling method. . . . . . . . . . . . . . . . . . . 184
6.13 Background-only fit to data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
6.14 Demonstration of the bias study . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
6.15 The mean value of the pull to signal strength . . . . . . . . . . . . . . . . . . . . 188
6.16 Systematic uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6.17 Signal mean mass uncertainty and resolution uncertainty . . . . . . . . . . . . . 192
6.18 Expected and observed limits at 95% CL on σB(X → Wγ) . . . . . . . . . . . . 194
6.19 Comparison of the asymptotic and HybridNew limits. . . . . . . . . . . . . . . . 194
6.20 Expected and observed limits at 95% CL on σB(X → Wγ) . . . . . . . . . . . . 195
6.21 Observed local p-values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.22 Wγ event displays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

xiii



List of Abbreviations

ALICE A Large Ion Collider Experiment
ATLAS A Toroidal LHC ApparatuS
BDT Boosted Decision Tree
BEH Brout-Englert-Higgs
BSM Beyond the Standard Model
CERN European Organization for Nuclear Research
CHS Charged Hadron Subtraction
CL Confidence Level
CMS Compact Muon Solenoid
CSC Cathode Strip Chamber
CSEV Conversion Safe Electron Veto
CTF Combinatorial Track Finder
DNN Deep Neural Network
DSCB Double-Sided Crystal Ball
DT Drift Tube
DY Drell—Yan
EB ECAL Barrel
ECAL Electromagnetic Calorimeter
EE ECAL Endcap
EWBG Electroweak Baryogenesis
FH Fully Hadronic
GEM Gas Electron Multiplier
GGF Gluon Gluon Fusion
GSF Gaussian Sum Filter
HB HCAL Barrel
HCAL Hadronic Calorimeter
HE HCAL Endcap
HF HCAL Forward
HH Higgs boson pair
HLT High-Level Trigger
HO HCAL Outer
JEC Jet Energy Correction
JER Jet Energy Resolution
JES Jet Energy Scale

xiv



KF Kalman Filtering
LEP Large Electron Positron Collider
LHC Large Hadron Collider
LHCb Large Hadron Collider Beauty
LO Leading Order
MC Monte Carlo
MET Missing Transverse Momentum
MVA MultiVariate Analysis
NLO Next-to-Leading Order
NN Neural Network
NNLO Next-to-Next-to-Leading Order
PDF Parton Distribution Function
PF Particle Flow
POI Parameter Of Interest
PS Proton Synchrotron
PSB Proton Synchrotron Booster
PSV Pixel detector Seed Veto
PU PileUp
PUPPI Pileup Per Particle Identification
PV Primary Vertex
QCD Quantum Chromodynamics
QED Quantum Electrodynamics
RPC Resistive Plate Chamber
SC Superclusters
SM Standard Model
SPS Super Proton Synchrotron
STXS Simplified Template Cross Sections
TEC Tracker Endcaps
TIB Tracker Inner Barrel
TID Tracker Inner Disk
TOB Tracker Outer Barrel
VBF Vector Boson Fusion
VEV Vacuum Expectation Value

xv



Chapter 1: Introduction

Particle physics investigates the fundamental constituents of the universe, a quest that has

evolved over millennia from the ancient Greek philosophy of atomism.

Substantial advancements in understanding the basic components of matter occurred in

the 19th and 20th centuries. Early nuclear physics experiments identified electrons, protons, and

neutrons as key subatomic particles. The discovery of the electron by J. J. Thomson in 1897,

through experiments with cathode rays, marked the beginning of a profound exploration into

the subatomic realm. This was followed by Ernest Rutherford’s gold foil experiment, which

confirmed the existence of the proton [1], and James Chadwick’s identification of the neutron in

1932 [2].

In the 1960s and 1970s, deep inelastic scattering experiments at the Stanford Linear

Accelerator Center (SLAC) involved firing high-energy electrons at protons. These experiments

revealed the internal structure of protons [3, 4], showing that they are composed of point-

like constituents. Further investigation indicated that protons and neutrons are composed of

quarks, bound together by gluons, which mediate the strong force. These findings supported

the development of Quantum Chromodynamics, which describes the interactions between quarks

and gluons.

These efforts culminated in the development of the Standard Model (SM) of particle physics,

1



a theoretical framework that harmonizes three of the four fundamental forces—electromagnetic,

weak, and strong interactions—and classifies all known elementary particles. It posits three

generations of fermions, each consisting of a charged lepton (electron, muon, tau), a corresponding

neutrino, and a pair of quarks (up and down; charm and strange; top and bottom). This results

in six leptons and six quarks, each with an antiparticle. Additionally, the SM includes gauge

bosons (W±, Z, γ, g) with spin-1, which mediate the three fundamental forces. The cornerstone

of the SM is the Brout-Englert-Higgs mechanism, which postulates the existence of a scalar

field known as the Higgs boson. This mechanism is responsible for spontaneous electroweak

symmetry breaking, enabling vector bosons to acquire mass. The scalar Higgs boson transfers

mass to fermions via the Yukawa interaction, with the exception of neutrinos, since right-handed

neutrinos do not exist within the framework of the SM, and are required for the Dirac mass term.

The SM represents a monumental achievement in our understanding of the fundamental

phenomena of the universe. It is also a predictive theory, with numerous predictions confirmed

by particle physics experiments. By the early 21st century, all elementary particles predicted

by the SM had been observed, with the exception of the Higgs boson. Figure 1.1 illustrates the

timeline of particle discoveries in the history of particle physics. A theoretical overview of the

SM is provided in Chapter 2.

Discovered in 2012 by the ATLAS and CMS collaborations [6, 7], the Higgs boson represents

a milestone in particle physics, affirming the success of the SM. However, the SM is not a

complete theory of fundamental interactions, leaving several significant problems unresolved.

Firstly, the SM does not encompass the gravitational force, failing to explain why gravity

is significantly weaker than the other fundamental forces. Additionally, the SM does not clarify

why there are only three generations of fermions. A major shortcoming of the SM is its inability

2



1900 1910 1920 1930 1940

e� �

atom nucleus

p+

anti-matter

e+

n0
µ±

1940 1950 1960 1970 1980

⇡±

K0

K±

⇡0

⇤0

�
⌃±

⌅�
p ⌫e

n
⌃0 ⌅0 ⇢

!
⌘
K⇤

⌫µ

�
↵2

⌘⇤

⌦�

quark model
up, down, strange

Standard Model
charm

J/ 
 0

 00

⌧
�c

D

bottom

⌥
⌥0

⌥00

. . .

⇤c

⌃c

1980 1990 2000 2010 2020

⌘c B W±

Ds

⌅c

Z0 Bs

⇤b

t

top

⌫⌧ H now

Figure 1.1: Timeline of particle discoveries. Figure is adapted from Ref. [5].

to account for the small but nonzero masses of neutrinos, which was dicovered through the

neutrino oscillation experiments [8–10]. Many experiments are investigating the neutrinoless

double beta decay [11, 12], which could reveal if neutrinos are Majorana fermions, meaning

neutrinos would be their own antiparticles. If neutrinos are Majorana fermions, a neutrino would

gain mass through interaction with its antiparticle and would not require interactions with the

Higgs boson to acquire mass.

The SM also fails to describe dark matter and dark energy. Cosmological evidence, such as

galaxy rotation curves [13, 14] and gravitational lensing [15–17], indicates that ordinary matter

comprises only 5% of the universe, while dark matter accounts for 27% and dark energy for

68%. However, there is no potential candidates for dark matter in the SM. There are three well-

established ways to search for dark matter: the direct production and detection from collider

experiments; the direct detection of nuclear recoil from the galactic dark matter halo; and the

indirect detection through cosmic rays resulting from dark matter annihilations.

3



Given that these fundamental questions persist, it is widely believed that the SM is part of

a broader theory.

Experimental progress is crucial in driving theoretical advancements. The early 1900s saw

a revolution in our understanding of subatomic particles with Wilson’s invention of the cloud

chamber, as noted by Cecil Powell: “the whole conception of the world of atomic physics

was strengthened and illuminated.” Carl D. Anderson’s addition of a strong electromagnet to

the chamber bent the paths of charged particles, which led to the discovery of the positron,

the first observed antimatter particle. The passive “wait and see” approach with cosmic rays

was insufficient to satiate the growing curiosity about subatomic particles. In 1929, Ernest

Lawrence invented the cyclotron, the first particle accelerator. Innovations in magnet technology,

particularly superconducting magnets, enabled the construction of powerful accelerators capable

of propelling particles to unprecedented energies. At these high energies, colliding particles can

generate rare particles, providing exceptional platforms for particle physics research.

Many renowned colliders were constructed in the 20th and 21th centuries, contributing

to significant discoveries. Figure 1.2 illustrates the center-of-mass energies of various particle

colliders. For instance, the antiproton was discovered at the Bevatron at Lawrence Berkeley

National Laboratory in 1955. The charm quark’s discovery in 1974 resulted from experiments

at the SLAC and Brookhaven National Laboratory. The gluon was confirmed at the Deutsches

Elektronen-Synchrotron in 1979. Both the Z boson and the W boson were discovered using the

Super Proton–Antiproton Synchrotron (Spp̄S) in 1983. The top quark was identified in 1995 at

Fermilab’s Tevatron pp̄ collider. Presently, the Large Hadron Collider (LHC) stands as the world’s

largest particle collider. The particle collision data are analyzed to test the SM and find hints for

physics beyond the SM (BSM). The analyses in this thesis used the data from proton-proton (pp)

4



collisions recorded at the CMS experiment, a multi-purpose detector at LHC, in the years 2016,

2017, and 2018. The LHC apparatus and CMS experiment are discussed in Chapter 3.

1950 1960 1970 1980 1990 2000 2010 2020 2030 2040

100

101

102

103

104

105

Hadron colliders
Heavy ion colliders

Lepton colliders

Electron-proton colliders
VEPP-2

BEPC

KEKB SuperKEKB

CBX

CEA

SPEAR
ADONE

DORIS

CESR
PETRA

PEP
TRISTAN

SLC

LEP
HERA

RHIC

LHC Heavy Ion

ISR

SppS

Tevatron

LHC

1950 1960 1970 1980 1990 2000 2010 2020 2030 2040

100

101

102

103

104

105

Year

C
en
te
r-
of
-m

as
s
en

er
gy

√
s
[G

eV
]

Figure 1.2: Center-of-mass energy of particle colliders versus year.

Two experimental approaches are commonly used to study particle physics: precision

measurements and searches for new particles.

The Higgs field is the first observed fundamental scalar field, and the study of its properties

is particularly intriguing because the electroweak sector can be predicted precisely due to its

simplicity and small couplings [18]. There are only 7 (+1) free parameters in the electroweak

sector for one fermion generation: two gauge couplings g and g′, the two parameters µ2 and λ

in the Higgs potential, and masses of fermions mu, md, me (and mνe). The large amount of data

recorded from 2016 to 2018 at LHC and the unprecedented center-of-mass energy of 13 TeV

provide a prime opportunity to study rare physics processes in more detail.

One of these rare processes is the production of Higgs boson pair (HH), which is an

5



important process to directly probe the Higgs boson self-coupling (λ). In the SM, the λ is

fully determined by the other parameters, with a strength proportional to the squared value of

Higgs boson mass. Therefore, measuring the λ would be a direct test of SM. Experimentally,

the measurement of λ is characterized using its deviation from the SM value, expressed as

κλ = λ/λSM. This parameter serves as a model-independent probe of BSM physics [19–21],

as these models often predict a κλ different from one.

A theoretical overview of HH production is given is Section 2.2, and the experimental

evidence has not been proven yet due to its small cross section. In the pp collisions, the primary

HH production mechanisms are the gluon-gluon fusion (GGF HH), vector boson fusion (VBF

HH), and vector boson associated (VHH) production. Chapter 4 presents the first search for

VHH production with CMS data and provides complementary results to those obtained from the

GGF HH and VBF HH production modes.

The single Higgs porduction channels can also probe the Higgs boson self-coupling indirectly

through the electroweak corrections. Furthermore, they are useful to constrain the Higgs boson

couplings to the fermions and vector bosons, reducing the degeneracy in a global fit. Improved

precision on the Higgs boson self-coupling is achieved through the combination of single Higgs

production channels and HH production channels, as discussed in Chapter 5.

There are many compelling BSM theories which attempt to provide explanations to the

open questions of the SM. Usually these theories hypothesize new particles, which ultimately

decay to SM particles [22–24]. By searching for these new particles with unique topological

features, evidence for a given BSM phenomenon can be established. Chapter 6 shows the search

for a new particle X that decays into a W boson and a photon, focusing on the leptonic W boson

decays.

6



Chapter 2: Theoretical Background

The Standard Model (SM) of particle physics [25–31] is a theoretical framework that

describes all known fundamental particles and their interactions through electromagnetic, weak,

and strong forces. It stands as the most successful theory to date in explaining the behavior of

subatomic particles.

In this chapter, a theoretical overview of the SM is given in Section 2.1. In the context of

this thesis, Section 2.2 provides an overview of the phenomenology of the Higgs boson in the

hadron collider.

2.1 The Standard Model

The SM describes three of the four fundamental forces within the framework of renormalizable

quantum field theory. These include the strong, weak, and electromagnetic interactions, which

are encapsulated by an SU(3)C × SU(2)L × U(1)Y local gauge symmetry. Each of these gauge

symmetries corresponds to a fundamental interaction.

• The SU(3)C symmetry corresponds to Quantum Chromodynamics (QCD), which describes

the strong force. This force is mediated by gluons.

• The combined SU(2)L × U(1)Y symmetry represents the unification of the weak and

7



electromagnetic forces. This unified electroweak interaction is mediated by the W+, W−,

and Z bosons (for the weak force) and the photon (for the electromagnetic force).

The SM describes a total of 61 elementary particles, as presented in Fig. 2.1. These particles

are categorized into two groups based on their spins: fermions and bosons. Fermions, which

include quarks and leptons, have a half-odd-integer spin (s = 1
2
) and obey Fermi-Dirac statistics.

There are six types of quarks, each with three color charges and an antiparticle counterpart.

Similarly, there are six types of leptons, each also having an antiparticle.

Figure 2.1: Illustration of the elementary particles of the SM and their properties. Figure is taken
from Ref. [32].

In contrast to fermions, bosons have an integer spin (s = 0, 1, 2, . . .) and obey Bose-

Einstein statistics. The SM includes 12 elementary spin s = 1 bosons: eight gluons, which

mediate the strong force, W+, W−, Z, and the photon (γ), which mediate the weak and electromagnetic

forces, respectively. The Higgs boson is the only spin s = 0 boson, and it is responsible for giving

8



mass to the other particles through the Brout-Englert-Higgs (BEH) mechanism [26, 27, 33, 34],

as described in Section 2.1.2.

2.1.1 Particles

Quarks

The SM includes six types of quarks: up (u) and down (d) quarks, which make up everyday

matter, along with charm (c), strange (s), top (t), and bottom (b) quarks. These quarks are

organized into three generations, each consisting of a doublet where the up-type quark carries

an electric charge of +2
3
e and the down-type quark carries −1

3
e, as depicted in Fig. 2.1.

Quarks are fundamental constituents of matter and possess a unique property known as

“color charge”, analogous to electric charge in electromagnetism but associated with the strong

nuclear force described by QCD. Each quark can have one of 3 different colors, as shown in

Fig. 2.1. According to QCD, color-charged particles cannot exist in isolation, a principle known

as color confinement [35]. Therefore, only bound states of quarks, which is known as hadrons,

can be observed experimentally. Baryons are hadrons composed of three quarks, while mesons

consist of a quark-antiquark pair. The top quark, due to its exceedingly short lifetime, decays

before forming bound states.

Leptons

Leptons are also fermions. Unlike quarks, leptons are not affected by the strong nuclear

force. Instead, leptons interact through the weak nuclear force and the charged leptons also

interact through electromagnetic force [29], playing crucial roles as constituents of matter.

9



Leptons exhibit electric charges that can be positive, negative, or neutral. For instance,

the electron (e−) carries a charge of -1, while its antiparticle, the positron (e+), has a charge

of +1. Neutrinos, another type of lepton, are electrically neutral. Similar to quarks, leptons

are organized into three generations, with each generation comprising a charged lepton and its

associated neutrino: (e, νe), (µ, νµ), and (τ , ντ ). Charged leptons of different flavors have distinct

masses ranging from 511 keV for the electron to 1.78 GeV for the tau lepton.

In the SM, neutrinos are assumed to have zero mass, but experimental evidence indicates

that they possess a small, nonzero mass, although this value remains unknown [8–10].

2.1.2 Fields

Quantum Electrodynamics

The Quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics.

It is an Abelian gauge theory with the symmetry group U(1).

The Lagrangian of a free Dirac fermion can be written as:

LDirac = ψ̄(x)(iγµ∂
µ −m)ψ(x), (2.1)

where ψ(x) is a Dirac spinor, γµ are the Dirac matrices.

To make the Lagrangian invariant under a local U(1) transformation: ψ(x) → eiθ(x)ψ(x).

The partial derivative needs to be promoted to a covariant derivative Dµ, and a new field Aµ is

introduced:

Dµ = ∂µ + ieAµ(x); Aµ(x) → Aµ(x)−
1

e
∂µθ(x), (2.2)

10



where e is the electron charge.

Substituting these terms into the Eq. 2.1, the Lagrangian can be written as:

LQED = LDirac − eAµ(x)ψ̄(x)γµψ(x). (2.3)

A gauge-invariant kinematic term should also be added to make this new field propagating

through time and space:

Lkin = −1

4
Fµν(x)F

µν(x); Fµν(x) = ∂µAν − ∂νAµ. (2.4)

So the final Lagrangian is:

LQED = LDirac − eAµ(x)ψ̄(x)γµψ(x)−
1

4
Fµν(x)F

µν(x) (2.5)

Usually a fine-structure constant α = e2/4π is defined to characterize the electromagnetic

coupling strength. No mass term 1
2
m2AµA

µ is allowed in order to maintain the gauge invariance.

Quantum Chromodynamics

The QCD is a theory that describes the strong interactions between quarks and gluons. It is

formulated as a non-Abelian gauge theory with the SU(3)C symmetry group, where the subscript

C denotes the color charge, which exists in three varieties.

Following a methodology akin to QED, we start with the Lagrangian for the free fermionic

field, as shown in Eq. 2.1. The transformation of the fermionic field under the gauge is represented

11



by: ψ(x) → eigsθ
a(x)λa

2 ψ(x), where gs is a constant, θa(x) are phase factors that describe the

transformation, and λa

2
denotes the eight Gell-Mann matrices, corresponding to the generators of

the SU(3) group.

In order to make the Lagrangian “locally” invariant, the partial derivative ∂µ needs to be

promoted to the covariant derivative (Dµ):

Dµ = ∂µ + igs
λa

2
Gµ

a , (2.6)

and dedicated gauge fields are introduced with the transformation rule:

Ga
µ → Ga

µ + gsf
abcθb(x)Gc

µ + ∂µθ
a(x), (2.7)

where Ga
µ represent eight gluon fields and fabc are the structure constants of the group that must

satisfy the commutation rule: [λ
a

2
, λ

b

2
] = ifabc λc

2
.

The QCD Lagrangian is written as follows:

LQCD = LDirac − gsψ̄γµ
λa

2
ψGµ

a −
1

4
F µν
a F a

µν , (2.8)

where F µν
a = ∂µG

a
ν − ∂νG

a
µ + gsf

abcGb
µG

c
ν is the field strength tensor. The coupling constant gs

determines the strength of the interaction and is often redefined as αs = g2s/4π.

QCD has successfully explained numerous intriguing phenomena. For example, due to

gluon self-interactions, the force between quarks does not diminish with increasing distance.

Instead, it intensifies as the quarks move apart with the resulting consequence that attempting to

separate quarks leads to an increasing amount of energy being stored in the gluon field. When this

12



energy reaches a critical threshold, it becomes more favorable to generate a new quark-antiquark

pair from the vacuum, thereby preventing isolated quarks from existing independently.

Electroweak Theory

The electroweak theory unifies the electromagnetic and weak nuclear forces under the

gauge symmetry group SU(2)L × U(1)Y . A key challenge in developing a unified theory lies in

explaining experimental evidence of parity violation. Consequently, interaction terms are treated

differently for left- and right-handed fermion fields.

The quarks and leptons form a left-handed doublet and a right-handed singlet. For example,

each generation of lepton forms:

Ll =

νℓ,L
ℓL

 , and ℓR. (2.9)

Likewise for quarks, using the first generation consisting of u and d quarks as an example:

QL =

uL
dL

 , uR, and dR, (2.10)

To write it in a more general form, we use ψ and ψ′ to represent either the neutrino and

charged lepton or the up- and down-type quark fields. The left-handed and right-handed states

13



are defined using the gamma matrix γ5 ≡ iγ0γ1γ2γ3 [36, 37]:

ΨL =
1− γ5

2

ψ
ψ′

 =

ψL

ψ′
L

 (2.11)

ψR =
1 + γ5

2
ψ (2.12)

ψ′
R =

1 + γ5

2
ψ′ (2.13)

Analogous to the QED and QCD cases, in order to keep the Lagrangian invariant under the

gauge transformation of SU(2)L × U(1)Y , the derivatives need to be promoted in this way:

Dµ
L = ∂µ − ig

σi
2
W µ

i (x)− ig′y1B
µ(x), (2.14)

Dµ
R = ∂µ − ig′y2B

µ(x), (2.15)

with g and g′ regulating the strength of the interactions. σi

2
is the generator of the SU(2)L group,

and yi is the hypercharge.

Hence, the Lagrangian can be written as:

L = Ψ̄L(iγµD
µ)ΨL + ψ̄R(iγµD

µ)ψR + ψ̄′
R(iγµD

µ)ψ′
R (2.16)

where the mass terms are omitted to preserve the SU(2)L invariance. A kinetic term for the gauge

fields has to be introduced as well:

Lkin = −1

4
W i

µνW
µν
i − 1

4
BµνB

µν . (2.17)

14



The fields strength tensors Wi
µν and Bµν are defined to be:

Bµν = ∂µBν − ∂νBµ (2.18)

W i
µν = ∂µW

i
ν − ∂νW

i
µ + gϵijkW j

µW
k
ν (2.19)

where ϵijk is the Levi-Civita tensor.

The newly introduced fields are usually combined to represent the physical fields corresponding

to the vector bosons. The W± bosons are defined to be:

W±
µ =

1√
2
(W 1

µ ∓ iW 2
µ). (2.20)

Since the Z boson and the photon share the same quantum numbers, such as charge and

spin, they can mix. This mixing is characterized by a specific parameter known as the mixing

angle. The parameterization is shown below:

Aµ

Zµ

 =

 cos θw sin θw

− sin θw cos θw


Bµ

Wµ

 , (2.21)

where θW is Weinberg weak-mixing angle:

cos θW =
g√

g2 + (g′)2
(2.22)

15



The Brout-Englert-Higgs Mechanism

A crucial element missing from the strong interactions and the electroweak Lagrangian

discussed so far is the mass term. Introducing explicit mass terms would violate gauge invariance,

posing a significant problem for massive gauge bosons like the W± and Z bosons, as well as

fermions. The SM addresses this challenge through the BEH mechanism [26, 27, 33, 34]. This

mechanism preserves gauge invariance while accounting for the observed masses of gauge bosons

and fermions by breaking the electroweak symmetry.

The BEH mechanism incorporates a complex scalar field ϕ into the SM Lagrangian. This

complex scalar field is an SU(2) doublet possessing four degrees of freedom:

ϕ =
1√
2

ϕ1 + iϕ2

ϕ3 + iϕ4

 , (2.23)

with the scalar potential:

V (ϕ) =
1

2
µ2(

4∑
i=1

ϕ2
i ) +

1

4
λ(

4∑
i=1

ϕ2
i )

2. (2.24)

The shape of the scalar potential is shown in Fig. 2.2. Without loss of generality, the

vacuum state can be chosen to be

ϕ =
1√
2

0

v

 , (2.25)

V =
1

2
µ2ν2 +

1

4
λν4. (2.26)

To minimize the potential V (ϕ) and break the symmetry at the minima, µ2 < 0 and λ > 0

16



are required. Therefore we have the solution:

v =

√
−µ2

λ
. (2.27)

And v is the vacuum expectation value (VEV).

Figure 2.2: Schematic illustration of the Higgs potential. Figure is taken from Ref. [38].

The small perturbation around the minima can be written as:

ϕ =
1√
2

 0

v +H(x)

 , (2.28)

where H(x) is the Higgs field.

Together with the covariant derivative:

(Dµϕ)
†Dµϕ =

1

2
(0 v)[g

σi

2
W i

µ +
g′

2
Bµ]

2

0

v

+H interaction terms, (2.29)

17



the BEH term can be written as:

LBEH = (Dµϕ)
†(Dµϕ)− V (ϕ) (2.30)

=M2
WW

µ+W−
µ (1 +

H

v
)2 +

1

2
M2

zZ
µZµ(1 +

H

v
)2 +

1

2
(∂µH)2 − V (ϕ), (2.31)

V (ϕ) = −µ2H2 + λvH3 +
λ

4
H4 − µ

4λ
, (2.32)

where the quadratic term (−µ2H2) represents the mass term of the Higgs boson:

m2
H = −2µ2 = 2λv2. (2.33)

The masses of the W± and the Z bosons are:

mW =
gv

2
, mZ =

gv

2 cos θW
(2.34)

The masses of fermions can be obtained through interactions with the Higgs field through

Yukawa couplings. They can be written as:

LY = −mf ψ̄ψ(1 +
H

v
), where mf = yf

v√
2

(2.35)

The mass of fermion is mf and it is proportional to the strength of the Yukawa couplings (yf ) and

VEV (v).

In the SM, the interactions of the Higgs boson with other particles are shown in Fig. 2.3

and the allowed interactions of the Higgs boson itself are shown in Fig. 2.4.

18



Figure 2.3: Feynman diagrams of Higgs boson interactions with other particles.

Figure 2.4: Feynman diagrams of Higgs boson interactions with itself.

2.2 The Higgs Boson at the LHC

Since the discovery of the Higgs boson, the focus of the high-energy physics community

has expanded to include precise measurements of its properties [39, 40]. Data from the CMS

experiment during Run 2 (2016–2018, 138 fb−1) have enabled numerous measurements of Higgs

boson properties. The couplings with the Z and W bosons are measured with a precision better

than 5%, the couplings with the t quark and τ lepton are measured with a precision of approximately

10%, and the coupling to the bottom quark (b) is measured with a precision of about 20%, as

presented in Fig. 2.5.

19



Figure 2.5: Measured coupling modifiers of the Higgs boson to fermions and heavy gauge bosons,
as function the of fermion or gauge boson mass. Figure is taken from Ref. [39].

20



2.2.1 Kappa Framework and Effective Field Theory

The kappa framework was introduced in 2012 to investigate the coupling structure of the

newly discovered Higgs-like particle [41]. The free parameters in this framework are denoted by

kappa (κ) factors. The deviation of the observed coupling strength (λobs) from the SM predicted

coupling strength (λSM) is quantified as:

κ =
λobs

λSM (2.36)

In Fig. 2.5, the parameters κf and κV are the coupling modifiers for couplings of fermion and

vector boson, respectively, within the kappa framework. Thus, when the observed properties of

the 125 GeV boson align with those expected for the SM Higgs boson, these parameters are unity

(κ = 1).

To incorporate the latest SM predictions for Higgs cross sections, which include higher-

order QCD and electroweak corrections, while simultaneously allowing for potential deviations

from the SM coupling values, the predicted SM Higgs cross sections are typically adjusted with

scale factors κi. Depending on the production process, κi can be factorized with respect to κf

and κV . For example, in the vector boson associated production (VH) process:

κVH =
σobs
V H

σSM
V H

= κ2V (2.37)

The kappa framework is able to characterize deviations from the SM predictions in a

simple way. This framework operates under the assumption that only one single SM-like Higgs

21



boson near 125 GeV exists. It does not make specific assumptions about additional new physics

states, such as other Higgs bosons. Therefore, while the kappa framework is highly useful, it is

insufficient for exploring more general types of deviations from the SM predictions.

To address this limitation, the Standard Model effective field theory (SMEFT) framework

extends SM with higher-dimension operators that encapsulate potential new physics effects.

Rather than searching directly for new particles, SMEFT is used to investigate new types of

interactions that are not present in the SM. These interactions are “effective” because their

underlying mechanisms are unknown. New physics scenarios generally manifest through these

new interactions, with different theoretical models leaving distinct signatures within the SMEFT

framework. This approach allows for a more comprehensive exploration of potential deviations

from the SM [42]. The final constraints within this framework can either be directly on the

SMEFT coefficients, which define the strength of the higher-dimension operators, or on the cross

sections of selected benchmark scenarios. These benchmarks are chosen to represent specific new

physics models or particular deviations from the SM, providing a concrete context for interpreting

experimental results.

In the context of this thesis, results are expressed using the kappa framework.

2.2.2 Higgs Self-coupling

The SM predicts the Higgs boson self-coupling through terms in the Higgs potential in the

Eq. 2.32, such as the trilinear term λ3νH
3 and the quartic term λ4

4
H4. In the SM, λ3 and λ4 are

determined as m2
H

2ν2
, where mH is the Higgs boson mass and ν is the vacuum expectation value of

the Higgs field.

22



The Higgs boson self-coupling can be directly probed through searches for Higgs boson

pair (HH) production, an endeavor that is extremely challenging due to the small production

cross section. According to the SM, the probability of observing Higgs boson pair production

in pp collisions at the LHC is approximately 1000 times smaller than that of producing a single

Higgs boson.

The importance of the Higgs boson self-coupling measurements is multifaceted:

Verification of the Standard Model and probing new physics: The electroweak sector of

the SM is theoretically precisely defined and allows for high-precision tests. Being a cornerstone

prediction of the SM, the precise measurement of the Higgs boson self-coupling provides a direct

test of the SM’s accuracy. And deviations from the predicted value would suggest BSM physics.

Understanding the Higgs potential: The structure of the Higgs potential underlies the

mechanism of electroweak spontaneous symmetry breaking, which endows other SM particles

with mass. Accurate measurement of the self-coupling can elucidate details and enhance our

understanding of the dynamics in this process.

Connection to electroweak baryogenesis: According to the SM, in the early universe,

when temperatures were significantly higher, the Higgs potential had its lowest energy state at

a field value of zero, rendering all known particles massless due to unbroken symmetry. As the

universe cooled to temperatures below the electroweak scale (T ≈ 100GeV), the Higgs field

transitioned to a new minimum at a non-zero value. This led to spontaneous symmetry breaking

and mass generation for other particles. The baryon asymmetry might have been generated

during this phase transition through electroweak baryogenesis (EWBG) mechanisms [19–21].

For EWBG to occur, the phase transition must be a violent, out-of-equilibrium event accompanied

by massive entropy production. The Higgs boson self-coupling influences the dynamics of this

23



phase transition, impacting the generation of baryon asymmetry. Moreover, the Higgs boson

self-coupling is essential for understanding the stability of the vacuum. Significant deviations

from the SM predictions could indicate that our universe is in a metastable vacuum, as shown in

Fig. 2.6.

Figure 2.6: The Higgs potential considering different higher-order corrections. Figure taken from
Ref. [43].

Given these reasons, measuring the Higgs boson self-coupling with high precision is one

of the most crucial tasks that the particle physics community aims to achieve. Chapter 4 describes

the measurement of the Higgs boson self-coupling through a new HH production channel, highlighting

how this channel enhances sensitivity within certain ranges of the Higgs boson self-coupling due

to constructive interference. Chapter 5 presents the latest results on the Higgs boson self-coupling

measurement, achieved through the statistical combination of single Higgs boson measurements

and HH searches. This approach provides the strongest experimental constraint without making

assumptions about other Higgs couplings, utilizing CMS Run 2 data.

24



2.2.3 Single Higgs Production

In pp collisions at the LHC, the cross sections of the main production mechanisms for a

single Higgs boson (H) are depicted in the left plot of Fig. 2.7, while the corresponding Feynman

diagrams are shown in Fig. 2.8. These production channels include gluon–gluon fusion (GGF),

vector boson fusion (VBF), Higgs-strahlung (VH), and associated production (tt̄H, bb̄H, tH).

The plot on the right of Fig. 2.7 displays the branching ratios as functions of the Higgs boson

mass, with H → bb̄ being the dominant decay channel, followed by H → WW , H → gg,

H → ττ , and others.

Figure 2.7: SM Higgs boson production cross sections as a function of the center-of-mass energy
(left) and the Higgs boson decay branching ratios as a function of the Higgs boson mass (right).
Plots are taken from Ref. [44].

2.2.4 Higgs Boson Pair Production

A notable milestone in the study of Higgs properties is the search for HH production, as

this provides an optimal channel for probing the Higgs boson self-coupling.

25



g

g

H

F

22

q

q̄
0

q

q̄
0

H
V

V

V

24

q

q̄
0
/q̄ W/Z

H

V

26

g

g

q

q

H
F

29

Figure 2.8: Leading order diagrams for Higgs boson production.

In pp collisions at the LHC, the main HH production mechanisms are analogous to single

Higgs production channels but involve two Higgs bosons in the final state. The leading channel is

GGF HH, followed by the VBF HH channel, vector boson associated production (VHH), and top

quark associated production (ttHH). Their cross sections are summarized in Table 2.1. The SM

cross section for VHH production is σV HH = 0.865 +5.4%
−5.0% fb, computed at next-to-next-to-leading

order (NNLO) in QCD [44, 45], which is approximately half that of VBF HH production.

The search for HH production at the LHC has a substantial history. In the CMS collaboration,

HH production was initially studied with the bb̄ττ and bb̄γγ final states using Run 1 data [46, 47].

In subsequent years with early Run 2 data, additional final states such as bb̄ττ , bb̄γγ, bb̄bb̄, and

bb̄V V were explored. More recently, HH searches using the full Run 2 dataset have encompassed

most HH decay channels. However, due to the low cross section, for a significant period only the

26



GGF HH and VBF HH channels were extensively studied, investigating various combinations of

Higgs boson decays HH → xxyy.

The leading-order (LO) Feynman diagrams for GGF HH are depicted in Fig. 2.9, while the

LO Feynman diagrams for VBF HH are shown in Fig. 2.10.

Table 2.1: Cross sections and uncertainties of different HH production modes [44, 45, 48, 49],
where PDF is the parton distribution function, αs is the strong coupling constant, and mt is the
top quark mass.

Production mode Cross section (fb) Scale uncertainty PDF+αs uncertainty mt uncertainty

GGF HH 31.05 +2.2%/−5.0% ±3% +4%/−18%
VBF HH 1.726 +0.03%/−0.04% ±2.1% —

ZHH 0.363 +3.4%/−2.7% ±1.9% —
W+HH 0.329 +0.32%/−0.41% ±2.2% —
W−HH 0.173 +1.2%/−1.3% ±2.8% —

ttHH 0.775 +1.5%/−4.3% ±3.2% —

g

g H

H

�F

16

g

g

H

H

F

F

VBF HH

17

Figure 2.9: Feynman diagrams contributing to HH production via gluon-gluon fusion mechanism.
The left and right diagrams are triangle diagram and the box diagram.

The plot on the left of Fig. 2.11 shows the total cross sections for the primary HH production

channels as a function of the center-of-mass energy, while the plot on the right depicts the cross

section variation with respect to the Higgs boson self-coupling modifier κλ = λ/λSM .

As depicted in Fig. 2.11, the cross sections for GGF HH and VBF HH reach minima when

κλ is close to one. This effect arises due to destructive interference between the LO Feynman

diagrams as κλ approaches one. Consequently, the cross sections for both processes are highly

27



q

q̄
0

q

q̄
0

H

H

V

V

V

V

18

q

q̄
0

q

q̄
0

H

H

V

V

V

�

19

q

q̄
0

q

q̄
0

H

H

V

V

2V

20

Figure 2.10: Feynman diagrams contributing to HH production via vector boson fusion
mechanism.

Figure 2.11: Total cross sections for the main HH production channels at pp colliders. Figures
are from Ref. [50].

28



sensitive to the value of κλ. Any deviation of κλ from one can significantly increase the total

cross sections and alter the kinematics of HH.

The plot on the left of Fig. 2.12 illustrates the contributions of the triangle, box, and

interference terms to GGF HH production in the SM. Destructive interference, which can change

sign and become constructive for negative values of κλ, plays a crucial role. This interference

affects the distribution of mHH, as shown in the plot on the right.

Figure 2.12: Distributions of the box, interference, and triangle components that contribute to the
SM GGF HH signal are shown on the left. ThemHH shapes with different κλ coupling hypotheses
are shown on the right. Plots are taken from Ref. [51].

The destructive interference also occurs between the Feynman diagrams in Fig. 2.10. With

some approximation [52], the matrix element can be written as:

Mµν = gµν
[
4m4

V

v2

(
1

t̂−m2
V

+
1

û−m2
V

)
+ κλ

m2
V

v2
6m2

H

ŝ−m2
H

+
2m2

V

v2

]
+ others , (2.38)

where t̂, û, and ŝ are Mandelstam variables.

Near the HH production threshold, we can assume ŝ = 4m2
H , t̂ = û = 0. It gives rise to:

Mµν = gµν
2m2

V

v2
(κλ − 3) (2.39)

29



So these terms cancel each other and lead to small cross section around κλ = 3.

The VHH channels share the same subprocess of V µV ν → HH with VBF HH and are

related by crossing symmetry. The formula also applies to VHH production. In the VHH channel,

close to the threshold for HH production, where ŝ = 4m2
H and t̂ = û = (mH +mV )

2, we have

ŝ = 4m2
H , t̂ = û = (mH +mV )

2. It gives rise to:

Mµν = gµν
2m2

V

v2
(κλ + 1 +

4m2
V

mH(mH + 2mV )
), (2.40)

which results in constructive interference, making the VHH channel unique and intriguing for

further study. More details about the VHH channel will be discussed in the dedicated Chapter 4.

30



Chapter 3: The CMS Detector at the LHC

In this chapter, an overview of the Large Hadron Collider (LHC) is provided in Section 3.1.

The LHC accelerates protons and collides them at four designated points, each hosting a distinct

experiment. This thesis focuses on data collected from one of these experiments, namely the

Compact Muon Solenoid (CMS) experiment. Section 3.2 offers detailed information about the

CMS experiment, including descriptions of its various sub-detectors. Following this, Section 3.3

discusses the algorithms employed for the reconstruction and identification of physics objects

within CMS.

3.1 The Large Hadron Collider

Operated by the European Organization for Nuclear Research (CERN), the LHC is the

most powerful and largest particle accelerator in the world, situated near Geneva on the border

between France and Switzerland [53, 54]. Spanning a circular tunnel with a circumference

of approximately 27 kilometers, the LHC is buried at depths ranging from 45 to 170 meters

underground. The LHC accelerates protons to nearly the speed of light using high-frequency

electric fields and superconducting magnets. It is designed for proton-proton (pp) collisions at

four points along its ring, each equipped with a major detector: ATLAS, CMS, ALICE, and

LHCb [55–58]. ATLAS and CMS are general-purpose detectors initially designed for the Higgs

31



boson search and now utilized for precise measurements of Standard Model (SM) parameters,

characterization of Higgs boson properties, and exploration of physics beyond the SM at the TeV

energy scale. ALICE specializes in heavy ion physics, while LHCb focuses on b-quark studies.

Constructed between 1998 and 2008, the LHC was built in the same tunnel that previously

housed the Large Electron-Positron collider. This tunnel has an internal diameter of only 3.7

meters in the arched sections between each sector, making it difficult to install two completely

separate proton rings, which presented significant design and installation challenges for the LHC.

To address these challenges, a twin-bore magnet arrangement is used to house two proton beams

within a single structure. Its superconducting magnets, cooled to 1.9 Kelvin using superfluid

helium, include 1232 dipole magnets capable of producing up to 8.33 Tesla magnetic fields with

currents up to 12000 A, enabling proton energies up to 7 TeV.

Commissioned in 2008, the LHC achieved its first high-energy pp collision at 7 TeV center-

of-mass energy on March 30, 2010. Since then, it has facilitated numerous groundbreaking

discoveries; the most notable discovery was the detection of the Higgs boson in 2012. Over the

years, the LHC has progressively increased its luminosity as depicted in Figure 3.1. It relies on

sophisticated cryogenic systems to maintain ultra-low temperatures and advanced data processing

capabilities to manage vast quantities of collision data.

The LHC accelerates proton beams to energies up to 7 TeV and lead ions to energies up to

2.76 TeV per nucleon. Achieving these energies involves a meticulous sequence of steps within

the CERN accelerator complex, as depicted in Figure 3.2. For proton acceleration, the process

begins with the extraction of protons from hydrogen atoms. These protons enter LINAC4, where

they are initially accelerated to 160 MeV. They then progress through a series of accelerators: the

proton synchrotron booster, the proton synchrotron (PS), and finally the super proton synchrotron

32



Jan '11
Jan '12

Jan '13
Jan '14

Jan '15
Jan '16

Jan '17
Jan '18

Jan '19
Jan '20

Jan '21
Jan '22

Jan '23
Jan '24

Date

0

50

100

150

200

250

300

To
tal

 in
teg

ra
ted

 lu
mi

no
sit

y (
fb

⁻¹)

CMS LHC delivered: 288.38 fb⁻¹
CMS recorded: 265.99 fb⁻¹

Figure 3.1: Cumulative delivered and recorded luminosity versus time for pp collisions at
LHC [59].

(SPS). At these stages, their energies increase sequentially to 2 GeV, 26 GeV, and 450 GeV,

respectively. Before entering the LHC, the proton beam is split into two parallel beamlines using

fast kicker magnets and further accelerated to the final energy level required for LHC collisions,

achieved through high-frequency radio frequency cavities.

One of the key parameters of a collider is instantaneous luminosity Linst. [61]. This is

directly related to the production rate of any interaction process:

dN

dt
= σLinst., (3.1)

33



Figure 3.2: The accelerator complex at CERN [60].

where σ is the cross section of the process. The Linst. is given by

Linst. =
N2

b nbfrevγr
4πϵnβ∗ F, (3.2)

where Nb is the number of particles per bunch, nb is the number of bunches per beam, frev is

the revolution frequency, γr (E/m) is the relativistic factor of the protons, ϵn is the normalized

transverse beam emittance, β∗ is the beta function at the collision point, and F is the geometric

luminosity reduction factor due to the crossing angle at the interaction point. The total number

of produced physics events over a period of time T are calculated with the integrated luminosity

Lint.:

N = σLint. = σ

∫
T

Linst.dt (3.3)

34



Some important design parameters are shown in Table 3.1.

Table 3.1: Nominal parameters of the LHC in pp collisions.

Parameter Definition Design
√
s Center-of-mass energy 14 TeV

∆t Bunch separation 25 ns
nb Number of bunches 2808
Np Number of protons per bunch 1.15×1011

frev Revolution frequency 11245 Hz
β∗ Beta function at the IP 0.55 m
ϵn Transverse emmittance 3.75 µm

Instantaneous luminosity is typically measured in units of cm−2s−1, representing the rate

of collisions per unit area per second. Integrated luminosity, on the other hand, is commonly

expressed in units of inverse femtobarns (fb−1) or picobarns (pb−1), reflecting the total number

of collisions accumulated over time. Figure 3.3 presents a stack plot of the number of interactions

per bunch crossing for each year of data taking. The mean interaction rate increases annually.

While numerous interactions occur per bunch crossing, typically only one proton-proton interaction-

related to the primary physics process of interest (hard scattering)-is considered, with the additional

interactions (pileup) being predominantly soft collisions.

The results presented in this thesis are based on pp collision data collected by the CMS

experiment at a center-of-mass energy of
√
s = 13TeV, corresponding to a total integrated

luminosity of 138 fb−1recorded during the Run 2 phase of the LHC.

3.2 The Compact Muon Solenoid Detector

The CMS detector, situated approximately 100 meters underground at interaction point five

of the LHC, is a general-purpose instrument designed for a wide range of physics studies using pp

35



0 20 40 60 80 100
Mean number of interactions per crossing

0

1

2

3

4

5

6

7

8

Re
co

rd
ed

 lu
mi

no
sit

y (
fb

⁻¹/
1.0

)

σpp
in (13.6 TeV) = 80.0 mb
σpp

in (13 TeV) = 80.0 mb
σpp

in (8 TeV) = 73.0 mb
σpp

in (7 TeV) = 71.5 mb

CMS 2023 (13.6 TeV): <μ> = 52
2022 (13.6 TeV): <μ> = 46
2018 (13 TeV): <μ> = 37
2017 (13 TeV): <μ> = 38
2016 (13 TeV): <μ> = 27
2015 (13 TeV): <μ> = 14
2012 (8 TeV): <μ> = 21
2011 (7 TeV): <μ> = 10

0

1

2

3

4

5

6

7

8

Figure 3.3: Mean number of interactions per bunch crossing [59].

collisions at high energies and instantaneous luminosities [56]. The name “CMS” derives from

its three primary features:

• despite its 14000-tonne weight, this detector is compact, with a diameter of only 15 m and

a length of 21 m. Compared to ATLAS, another general-purpose detector at the LHC, CMS

is almost double the weight but has only half the size;

• the detector is specifically designed for precise muon detection. The muon detection

system takes up approximately 80% of the total volume. The muon path is measured by

tracking its position through the muon detection system and combining this information

with measurements from the CMS silicon tracker.

• the detector has an extremely powerful solenoid magnet, generating a magnetic field of 3.8

Tesla, which is about 4000 times stronger than a typical refrigerator magnet, over a vast

36



area.

As shown in Fig. 3.4, the CMS detector has a “cylindrical onion” configuration, with

multiple sub-detectors layered concentrically around the beam axis. The superconducting solenoid,

with an internal diameter of 6 m, divides the CMS into two sections.

Figure 3.4: Cutaway diagram of CMS detector [62].

There are two main components within the superconducting solenoid: the inner tracking

system and the calorimeter system. The inner tracking system consists of pixel detectors and

silicon strip detectors, which are placed as close to the interaction point as feasible to ensure

the best possible spatial resolution. The inner tracking system identifies charged particles and

measures their trajectories, momenta, and charge signs. The calorimetry system consists of

an electromagnetic calorimeter (ECAL) and a hadronic calorimeter (HCAL). The ECAL, made

from lead tungstate (PbWO4) crystals, mainly measures particles that engage in electromagnetic

37



interactions, such as electrons and photons. In contrast, the HCAL, built from brass and scintillator

materials, is intended to detect both charged and neutral hadrons, including protons, neutrons,

pions, and kaons. The forward calorimeter of the HCAL is located very close to the LHC beam

pipe and functions not only as a calorimeter for detecting very forward particles but also as an

essential luminometer, providing accurate luminosity measurements.

The muon systems are positioned outside of the superconducting solenoid, with the aim of

detecting muons and accurately determining their momentum and charge signs together with the

tracker information.

Figure 3.5 illustrates the coordinate system in CMS. The origin is located in the center of

the detector, where the proton beams intersect. The x-axis directs towards the LHC center, and

the y-axis points upwards. These axes form the transverse plane, with the azimuthal angle (ϕ) set

to zero along the x-axis direction. The z-axis is perpendicular to the transverse plane, parallel to

the beam axis, with a positive direction following a counterclockwise rotation.

Instead of using the polar angle θ to indicate the direction, pseudorapidity (η) is commonly

used and defined by: η= − ln[tan( θ
2
)]. Its value ranges from 0 at θ=π/2 to ±∞ at θ=0(π). The

angular separation between two particles is given by: ∆R=
√
(∆η)2 + (∆ϕ)2.

Figure 3.5: Conventional 3D coordinate system at the CMS detector [63].

38



Regarding the momentum of a particle (p⃗), p denotes the total momentum, while p⃗T refers

to the transverse momentum. A detailed description of the CMS detector is available in Refs. [56,

64]. The following sections provide in-depth discussions on the detectors mentioned earlier.

3.2.1 Inner Tracking System

The inner tracking system is located at the core of the CMS detector [65, 66], comprising

a volume of 5.6 m in length and 2.4 m in diameter, with a consistent magnetic field of 3.8

Tesla throughout the tracking area. The CMS tracker is made up of two distinct silicon-based

detectors, which together ensure precise and efficient measurement of charged particles generated

in collisions, as well as accurate reconstruction of the origins of the particles, called vertices [67].

A diagram of the upper half of the CMS inner tracker is shown in Fig. 3.6.

Figure 3.6: Schematic view of the upper half of the CMS inner tracker [67].

39



The Pixel Detector

The pixel detector is the innermost part of the tracking system. In 2016, it consisted of

65 million silicon pixels arranged in three barrel layers (BPix) and two endcap disks (FPix),

covering |η|<2.5. The size of silicon pixels is 100× 150µm2, giving a resolution of 10µm in the

(r, ϕ) plane and 20µm along the z-axis. The BPix layers are situated at radii ranging from 4.4 to

10.2 cm, while the FPix disks are placed between 6 cm and 15 cm.

The pixel detector had a significant upgrade during the technical stop between the 2016 and

2017 data-taking periods [68] to improve performance. The upgraded detector has four layers in

the barrel and three disks in the endcap regions. The schematic view of the pixel tracker before

and after upgrades is shown in Fig. 3.7.

Figure 3.7: Schematic views of the CMS pixel tracker consisting of half initial and half upgrade
geometries [69].

The Silicon Strip Detector

The silicon strip detector encircles the pixel detector, spanning the identical η range as the

pixel detector, with a radial distance from 20 cm to 116 cm, and extending up to 282 cm along the

z-axis.

40



The silicon strip detectors are divided into four sections. The innermost section comprises

four barrel layers (TIB) and three disks (TID) with radii ranging from 20 cm to 55 cm, extending

up to 124 cm along the z-axis. The strips have pitches between 80 and 120µm, achieving a spatial

resolution of about 24µm. Encircling the TIB and TID are the Tracker Outer Barrel (TOB) in the

central region and the Tracker Endcaps (TEC) in the forward regions. The TOB consists of six

layers, extending to a radial distance of 116 cm. The TOB strips have pitches ranging from 122

to 183µm, offering a spatial resolution between 35 and 53µm. The TEC includes nine layers

and is placed between 124 cm and 282 cm along the z-axis.

3.2.2 The Calorimeter System

In high-energy physics experiments, calorimeters are extensively used and can generally

be divided into two categories: homogeneous calorimeters and sampling calorimeters [70, 71]. A

homogeneous calorimeter uses a single medium that functions as both the absorbing material and

the detector. The CMS ECAL is the largest homogeneous crystal electromagnetic calorimeter

ever built. Sampling calorimeters are composed of layers of a dense passive absorber (like lead

or copper) alternated with active detector layers (such as silicon, plastic scintillator, or liquid

argon). The CMS HCAL is an example of a sampling calorimeter, with alternating layers of

brass absorber and plastic scintillator tiles.

Two key quantities in the context of calorimetry are the radiation length (X0) and the

nuclear interaction length (λ0). The radiation length is the average distance over which electrons

lose energy through electromagnetic interactions, reducing their energy by a factor of 1/e. The

nuclear interaction length is the average distance that a hadronic particle travels before experiencing

41



an inelastic nuclear interaction. When analyzing different sections of a calorimeter, it is often

beneficial to simplify the complex structure by representing them in terms of the number of

radiation lengths (X0) or the lengths of nuclear interaction (λ0).

The energy dependence of the energy resolution can be parameterized as the quadratic sum

of three terms:

σE
E

=
a√
E

⊕ b

E
⊕ c (3.4)

The first term, with coefficient a, is the “stochastic term”, which arises from Poissonian

fluctuations in the measurement of the signal. For optical calorimeters, these fluctuations typically

originate from the number of photons. Since the number of photons is proportional to the

deposited energy, the stochastic term scales as
√
E.

The second term, with coefficient b, is known as the “noise term”, which includes contributions

from noise in the read-out electronics and effects such as pileup.

The third term, with coefficient c, is the “constant term”, which arises from several effects,

including constant losses due to detector imperfections, non-uniformities, calibration errors, and

energy lost in inert material before or within the detection volume.

These terms are very important parameters for comparing and understanding the performance

of different calorimeters. The calorimeters of CMS are placed between the CMS magnet and the

tracking system.

The Electromagnetic Calorimeter

The ECAL detector consists of three parts: the ECAL barrel (EB), the ECAL endcap (EE)

section, and the preshower detector [72, 73]. The ECAL EB spans |η|<1.48, while the EE covers

42



the range 1.48<|η|<3.0. The PbWO4 crystals were selected for their high density (8.28 g/ cm3),

short radiation length (X0 = 0.89 cm) and small Molière radius (2.2 cm). These characteristics

enable a compact ECAL within the magnetic field, offering fine granularity, excellent energy

resolution, and radiation hardness.

In the barrel (endcap), the crystals are 23 cm (22 cm) long. These lengths correspond to 25.8

(24.7)X0, sufficient to contain more than 98% of the energy of electrons and photons up to 1 TeV.

The transverse size of the crystals is 2.2×2.2 cm2 for the barrel crystals and 2.9×2.9 cm2 for the

endcap crystals. This high transverse granularity helps the separation of two highly collimated

particles from a resonance.

Figure 3.8 shows a schematic view of the ECAL system. The EB is divided into two

halves, each further segmented into 18 ϕ-sectors (referred to as “supermodules”). Each EE is

divided vertically into two “Dees”, each containing 3662 crystals organized into 5×5 subunits

(known as “Supercrystals”). A preshower detector, made up of two orthogonal planes of silicon

strip sensors interspersed with lead (3X0 in total), enhances γ/π0 discrimination in the endcaps.

The intrinsic energy resolution of the ECAL barrel was measured with an electron beam,

without accounting for the tracker material in the front [74]. The relative energy resolution were

measured at seven different energies in the range from 20 to 250 GeV and is parameterized as a

function of the electron energy as follows:

σE
E

=
2.8%√
E

⊕ 12%

E
⊕ 0.3% (3.5)

where the energy E is measured in GeV.

43



Figure 3.8: CMS electromagnetic calorimeter [72].

The Hadron Calorimeter

The HCAL is a hermetic sampling calorimeter [75–78], placed around the ECAL within

the solenoid. The HCAL is essential for the measurement of the hadronic activities. For example,

W, Z, and Higgs bosons frequently decay into quark-antiquark pairs, resulting in experimental

signatures primarily observed as jets by the HCAL.

The HCAL is divided into four separate sections: the HCAL barrel (HB), the HCAL endcap

(HE), the HCAL outer barrel (HO), and the HCAL forward detector (HF). Figure 3.9 illustrates

a schematic view of the HCAL system after the Phase 1 upgrade, showing a longitudinal view of

one quadrant.

Both HB and HE are sampling calorimeters, with HB covering |η|<1.3 and HE extending

the coverage to 1.3<|η|<3.0. Hadronic showers develop primarily through interactions involving

44



v. 2017-06-A

HCAL  HO IRON

RING 2 RING 1 RING 0

MAGNET COIL

FEE

FEE

BEAM LINE

HCAL
HF

HCAL
HE

HCAL  HB

26
25

23

21

19

28

29

17 16

29

41
17

0

16

0

27

24

22

20

18

15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Figure 3.9: A schematic view of one quarter of the HCAL [76].

the strong force. Due to the nuclear interaction length (λ0) being significantly larger than the

radiation length (X0), hadronic showers display greater variability in their spatial development

and energy dissipation. HB and HE are composed of alternating layers of brass absorbers and

plastic scintillator tiles. The brass absorber has a radiation length of 1.49 cm and a nuclear

interaction length of 16.42 cm. The thickness of the absorber is approximately 6λ0 in the barrel

region and increases to over 10λ0 at higher pseudorapidities in the HE.

Since the ECAL has approximately one interaction length of material before HCAL, hadronic

showers can inevitably initiate within the ECAL. Therefore, it is not physical to talk about only

the energy resolution of the HCAL. A pion test beam was performed with the combined ECAL

and HCAL calorimeters to determine the combined energy resolution [79].

σE
E

=
85%√
E

⊕ 7% (3.6)

45



where the energy E is expressed in GeV.

The HO is installed outside the solenoid and serves as an additional absorber covering

|η|<1.3, corresponding to the 1.4λ0 at normal incidence. Positioned far from the collision point,

the front faces of the HF calorimeter are situated at ±11.2 m from the interaction point, thereby

extending the angular coverage to 2.85<|η|<5.19. The HF is a sampling calorimeter that uses

grooved steel plates as absorbers and quartz fibers as the active material. Unlike HB and HE,

which detect scintillation light, HF measures the Cherenkov light signal produced by charged

shower particles in the quartz fibers.

3.2.3 Muon System

Muons interact weakly with the detector and have a relatively long lifetime of 2.2µs.

Therefore, they are able to pass through the tracker and calorimeter systems with minimal energy

loss. The muon detection system forms the outermost section of the CMS detector and is located

outside the solenoid magnet, between layers of the iron return yoke, covering the range up

to |η|<2.4. The CMS muon detection system uses four different types of gaseous detectors.

Figure 3.10 shows a cross-sectional view of the muon system.

The drift tube chambers (DT) are placed in the region |η|<1.2, which are arranged into four

barrel stations. Each chamber contains drift tubes (cells) that are 2.4 m long and have a surface

area of 1.3×4.2 cm2. These cells are filled with a gas mixture of 85% Ar and 15% CO2 and

have an electric field in the center. When a muon traverses the chamber, it ionizes the gas atoms,

releasing free electrons that subsequently produce a signal. Multiple layers of DTs, oriented

perpendicularly to one another, enable the determination of the muon’s time and position, and its

46



Figure 3.10: Schematic longitudinal view of a quarter of the CMS detector highlighting the
muon detection systems. DTs, CSCs and RPCs are represented in orange, green, and blue,
respectively [80].

momentum is deduced from its curved trajectory in the magnetic field.

In the endcap region (0.9<|η|<2.4), cathode strip chambers (CSCs) are used because of

the high background rate and the intense, non-uniform magnetic field. CSCs are made up of

alternating layers of anode wires and copper cathode strips. When a muon passes through the

chamber, free electrons moving towards the wires trigger an electron avalanche, while ionized gas

atoms produce signals at the strips. The combined information allows for an accurate measurement

of the muon’s position. CSCs are known for their resistance to radiation and rapid response time.

Resistive plate chambers (RPC) serve as an additional trigger that complements DTs and

CSCs. These chambers are located in both the barrel and endcaps, providing additional information.

The gas electron multiplier detector (GEM) is a new addition to the CMS muon system in

the region 1.6<|η|<2.2. GEMs were installed as prototypes in 2017 and additional 144 chambers

47



were installed during Long Shutdown 2 between 2018 and 2022. These chambers contributes to

the Run 3 data-taking. In addition, more GEM chambers will be installed for Phase-2 of the LHC.

3.2.4 Trigger

During Run 2, proton bunches in the LHC collided at a rate of up to 40 MHz [81]. The

enormous amount of data generated by these numerous collisions is too large to be collected or

permanently stored. A two-level trigger system is employed to filter out the rarer events that are

of interest for physics analysis, thus reducing the data to a manageable level [82].

The first level (L1) [81] comprises specialized hardware processors that use data from the

calorimeter and muon detectors to select events at a frequency of about 100 kHz with a fixed

latency of around 4 µs. The calorimeter data are combined to identify signals that correspond

to the hypothesis of an electron, photon, τ lepton, hadronic jet, or missing transverse energy

(from an invisible particle like the neutrino). Similarly, potential muon candidates are identified

by merging data from the muon detectors. Then a global trigger combines the outputs from the

calorimeter and the muon system, enabling the selection of events for a specific analysis target.

The second level, known as the high-level trigger (HLT) [82], consists of a processor farm

that runs an optimized version of the complete event reconstruction software for fast processing,

reducing the event rate to approximately 1 kHz for offline storage.

48



3.2.5 Luminosity Measurement

Accurate luminosity measurement is essential for understanding the physics processes in

the detector, including cross section measurement and background estimation. From Eq. 3.1:

Linst.(t) =
1

σ

dN

dt
=

1

σ
R(t) (3.7)

The instantaneous luminosity can be measured from the rateR(t) of any collision-induced process,

which may correspond to the number of hits, tracks, or clusters in the luminometers. The cross

section of this process, incorporating acceptance and efficiency effects, is known as the visible

cross section σvis. Generally, σvis is not precisely known as a priori and requires calibration, which

is done through Van der Meer scans [83]. Some requirements for the luminometers include:

• provide sufficient statistics and can operate at 40 MHz

• stable and uniform response

• linear response from van der Meer scans (pileup of 0.5) to physics conditions (pileup of

50)

• mechanism to measure and correct any non-linearity, instability, inefficiency, or other

response degradation (e.g., due to radiation damage)

At CMS, luminosity is measured online using four detectors: the HF (two methods, OC

and ET), the Fast Beam Conditions Monitor (BCM1F), the Pixel Luminosity Telescope (PLT),

and the DT as shown in Fig. 3.11. Luminosity is also measured offline with the pixel detector

using the cluster counting method.

49



Figure 3.11: Schematic cross section through the CMS detector in the r-z plane highlighting the
main luminometers in Run 2 [83].

Two algorithms have been developed to extract instantaneous luminosity. The first method

is rate scaling, where the raw rate of observables is scaled with calibration constants to determine

the luminosity. For a single-bunch crossing, the instantaneous luminosity is given by:

Lb = Nobs
frev
σvis

(3.8)

where frev is the LHC revolution frequency, and σvis is the visible interaction cross section. The

direct observable here is Nobs, and Lb is proportional to this quantity.

The second method, known as zero counting, uses the average fraction of bunch crossings

where no observables are detected. This zero fraction is then used to infer the mean number

of observables per bunch crossing. Assuming that the probability of no observables in a single

collision is p, the probability of no observables in a bunch crossing with k interactions is pk.

Averaged over a large number of bunch crossings, with the number of interactions per bunch

50



crossing distributed according to a Poisson distribution with mean µ, the expected fraction of

events with zero observables recorded, ⟨f0⟩, can be expressed as:

⟨f0⟩ = e−µ(1−p) (3.9)

The instantaneous luminosity for a single-bunch crossing is then given by:

Lb = µ
frev
σvis

= − ln⟨f0⟩
frev
σvis

(3.10)

The direct observable here is ⟨f0⟩, the fraction of events with no observables recorded, and Lb is

proportional to − ln⟨f0⟩.

Luminosity is estimated from HFET and DT data using the rate-scaling algorithm. The raw

inputs of HFOC, PLT, and BCM1F are converted to luminosity using the zero count method. All

of these luminometers are calibrated to provide luminosity on an absolute scale. Typically, real-

time measurements have a 2% uncertainty, while offline measurements achieve a 1% uncertainty

after final calibration and corrections.

The integrated luminosities for the 2016, 2017, and 2018 data-taking years have individual

uncertainties ranging from 1.2% to 2.5% [83–85], with an overall uncertainty of 1.6% for the

entire 2016–2018 period.

3.3 Identification and Reconstruction of Physics Objects

Advanced algorithms are used to process the data collected by the CMS detector and

identify the properties of particles present in an event. Particle reconstruction in CMS relies

51



on the distinct signatures that different types of particles create as they traverse the detector.

Figure 3.12 shows a schematic view of the CMS detector and how various particles interact with

its sub-detectors.

Figure 3.12: A schematic view of a transverse slice of the CMS detector illustrating how different
particles interact with sub-detectors. Figure taken from Ref. [86].

These interactions can be briefly summarized as follows:

• Charged particles: Muons, electrons, and charged hadrons leave hits in the inner tracking

system.

• Electrons and photons: Their energy is predominantly deposited in the ECAL.

• Charged and neutral hadrons: Most of their energy is deposited in the HCAL.

• Muons: They leave hits in the outer muon system.

52



• Neutrinos: Virtually non-interacting with the CMS detector, their presence is inferred from

the missing transverse momentum (pmiss
T ).

Although each subsystem is designed to detect specific particles or properties, often a

particle leaves traces on multiple sub-detectors. Combining information from all sub-detectors

can significantly improve particle identification and reconstruction, leading to more accurate

event descriptions. In addition, complex tasks such as jet reconstruction, determination of pmiss
T ,

and mitigation of pileup contamination require a global view of the information from all sub-

detectors. In CMS, the particle-flow (PF) reconstruction algorithm [86] is designed precisely for

this purpose. The PF algorithm proceeds through several distinct steps during the reconstruction

process.

• Reconstructed hits (RecHits) production: Digitized readouts from CMS sub-detector subsystems

produce RecHits, containing information such as position, energy deposition, and timing.

• RecHit combination within sub-detectors: RecHits within the same subsystems are combined.

The tracks and primary pp interaction vertices (primary vertices) are reconstructed from

the RecHits tracker. Calorimeter towers (CaloTowers) are formed by summing ECAL or

HCAL RecHits. Standalone muons are created using RecHits in the muon system.

• Global correlation and combination: Finally, these reconstructed objects are correlated,

and global information from all sub-detectors is combined.

This section introduces the PF algorithm and provides a detailed description of object

reconstruction, including muons, electrons, photons, jets, and missing transverse momentum.

53



3.3.1 The Particle-Flow Algorithm

Two fundamental elements of the PF algorithm are tracks and clusters, which together form

the basis for reconstructing various types of particle candidates. The PF algorithm begins with the

reconstruction of charged particle trajectories in the inner tracker (tracks) and the calibration of

calorimeter clusters in the preshower, ECAL, and HCAL. These tracks and clusters are collectively

referred to as PF elements. With the formation of PF elements, a link algorithm is used to combine

them and form PF blocks.

Tracks: Tracks are built using an iterative approach with the combinatorial track finder

(CTF) algorithm, which utilizes Kalman Filtering (KF) [67, 87, 88]:

• An initial seed is first generated with a few hits compatible with a charged-particle trajectory.

• Hits from all tracker layers are then gathered along this charged-particle trajectory.

• Finally, the properties of charged particles such as origin, momentum, and dir