HIGGS BOSON CROSS SECTION MEASUREMENTS
IN PROTON-PROTON COLLISIONS AT √s = 13 TEV

IN THE DIPHOTON DECAY CHANNEL

USING THE ATLAS DETECTOR AT THE LHC

By

Ivan Konstantinovich Pogrebnyak

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Physics(cid:22)Doctor of Philosophy

2021

ABSTRACT

HIGGS BOSON CROSS SECTION MEASUREMENTS
IN PROTON-PROTON COLLISIONS AT √s = 13 TEV

IN THE DIPHOTON DECAY CHANNEL

USING THE ATLAS DETECTOR AT THE LHC

By

Ivan Konstantinovich Pogrebnyak

This dissertation presents ATLAS measurements of the total and differential fiducial

cross sections for the process of the Higgs boson production and subsequent decay to two
photons. The analyzed dataset of proton-proton collisions at √s = 13 TeV has an integrated
luminosity of 139 fb−1 and was collected during the Run 2 of the Large Hadron Collider in
2015–2018. The cross sections were measured in a fiducial phase space closely matching the

experimental selection, and compared to the Standard Model predictions from the state-of-

the-art calculations. The differential cross sections were measured with respect to an array

of observables, allowing us to probe the kinematics, jet activity, and spin and CP properties

of the Higgs boson interactions. The data were analyzed iteratively, with yearly results from

partial datasets presented at conferences and in publications. The inclusive fiducial cross
section obtained from the analysis of the full Run 2 dataset is 65.2± 4.5 (stat.)± 5.6 (exp.)±
0.3 (theory) fb, which is in excellent agreement with the SM prediction of 63.5 ± 3.3 fb. The
results provide the state-of-the-art experimental measurements of the Higgs boson production

cross sections, and extend our confidence in the SM in the Higgs sector and QCD, as well as

in the computational techniques used to obtain theoretical predictions.

ACKNOWLEDGMENTS

I would like to begin this dissertation by expressing my gratitude to everyone who made

this work possible and who had a hand in either setting or guiding me on the course of

curiosity, exploration, and pursuit of knowledge.

Firstly, I want to thank my parents, Konstantin Vladimirovich and Svetlana Alekseevna.

The fact that I was able to undertake this work is largely a product of their support and

encouragement. Through their example, I learned that diligent and judicious attitude is

what yields quality results. They taught me to use critical thinking and how to deconstruct

problems to find a path to a solution. For all that they have done to make this journey

possible for me, I can’t thank them enough.

Throughout my life, I’ve been lucky to have great teachers. From elementary to graduate

school, there have always been people who kept my curiosity aflame, either with their knowl-

edge or charisma. The teachers showed me the world of ideas, how the inner workings of

complex things can be understood, and how fulfilling the path of discovery can be. I express

heartfelt thanks to all the teachers of my home city school, “Лингва – XXI век”. You were

like a family to the students, and you always made me feel at home at school.

When I was in the third grade, mom had the foresight of signing me up for the Radioengi-

neering club at the local “Pioneers’ Palace”. Armed with an 80 W soldering iron, I set out

to build my first medium-wave radio receiver. I am certain that what Georgiy Sergeevich

Kotelnikov taught me resulted in my lifelong love for science and engineering. Under his

guidance, I obtained my first experience of writing and presenting a research paper. Georgiy

Sergeevich is an extraordinary mentor, and the club is a special place for many former stu-

dents. Stories were always shared during the celebrations we had on the Radio Day and the

iii

club’s birthday. The scents of melting rosin and brewing tea, and the sounds of classical

music on vinyl records will always stay in my heart.

While studying at the University of North Carolina at Chapel Hill, I was fortunate

to learn from and work with Christian Iliadis. He, very enthusiastically, taught my first

quantum mechanics class and later hired me as an undergraduate research assistant. Working

in Christian’s group, which specialized in nuclear astrophysics, gave me my first research

experience in experimental physics and taught me many skills that came useful in graduate

school.

I cannot leave my friends without mention, particularly my UNC cohort. I thank Patrick

Domico for his incredible capacity for conversation, and being a willing interlocutor in dis-

cussions of the finest details of any subject. He did not refrain from endless telephone

conversations over English usage concerning this dissertation.

I thank Perry Harabin for

continually sparking my interest in mathematics and for inspiring me to seek deeper un-

derstanding of the fundamental mathematical concepts in field theory and particle physics.

I thank Nathan Otterness for providing programming advice that enabled me to develop

many tools I used in this work.

I thank Andrew Ashley for being a fantastic storyteller

and making intellectual pursuits seem all the more wondrous. It would be hard to mention

everyone I had the pleasure of counting among my friends at MSU, but I would like to note

Anton Efremov, Justin Estee, Sam Marinelli, Garrett Neer, and Jos´e Reyes.

Finally, I would like to express my gratitude to my PhD adviser, Joey Huston, for all

the opportunities and guidance he provided for me in my graduate career.

I had always

dreamt of working in a technical and fundamental area of research. Joey helped me realize

this aspiration by letting me participate in a number of amazing projects, ranging from the

analysis of the diphoton decay channel of the newly discovered Higgs boson to electronics

iv

upgrades for the Tile hadronic calorimeter. With Joey’s help, I had the privilege to meet

and work with many outstanding people in the field of high energy physics.

I also thank the members of my dissertation committee: Matthew Comstock, Wade

Fisher, Scott Pratt, and Chien-Peng Yuan, for providing help and useful comments on my

work. I thank Andreas von Manteuffel for proofreading the theory chapter, and Ahmed Tarek

for proofreading the analysis chapter. And, of course, my deepest thanks go to the post-

docs, who helped me with my projects and made the analysis presented in this dissertation

possible, as well as to the scientists and engineers, whose tireless work makes the operation

of the LHC and ATLAS a reality.

v

EPIGRAPH

The labor legislation was being flagrantly ignored, and I began to feel that I had lost

all desire to struggle against this law-breaking, because, tonight at twelve o’clock on New

Year’s Eve, plowing through a blizzard, they came in, these people who had more interest in

bringing to a conclusion, or starting anew, a useful undertaking . . . Here came people who

would rather be with each other than anywhere else, who couldn’t stand any kind of Sunday,

because they were bored on Sunday. They were Magi with a capital M, and their motto was

“Monday begins on Saturday.” True, they knew an incantation or two, knew how to turn

water into wine, and any one of them would not find it difficult to feed a thousand with

five loaves. But they weren’t magi because of that. That was superficial chaff. They were

magi because they knew a lot, so much indeed that quantity had finally been transmuted

into quality, and they had come into a different relationship with the world than ordinary

people. They worked in an Institute that was dedicated above all to the problems of human

happiness and the meaning of human life, and even among them, not one knew exactly what

was happiness and what precisely was the meaning of life. So they took it as a working

hypothesis that happiness lay in the perpetual gaining of new insights into the unknown and

the meaning of life was to be found there as well.

Monday begins on Saturday – Arkadi and Boris Strugatsky

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Quantum Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Types of fields and spin . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2
Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Higgs mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 After symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Fermions’ masses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Chromodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Higgs phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3 Electroweak sector

Chapter 3 Theoretical predictions and event simulation . . . . . . . . . . .
3.1 Monte Carlo event generation . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Event generation at NLO . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Matrix element and parton shower matching . . . . . . . . . . . . . . . . . .
3.4 Scale dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 PDF uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Predictions used in the analysis . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1
Signal Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Background Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.3 Predictions compared to the analysis results . . . . . . . . . . . . . .

1

5
6
8
10
12
18
20
22
25
27
31
35

38
39
46
52
53
58
62
65
67
68

Chapter 4 The LHC and ATLAS . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 CERN accelerator complex . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Laboratory coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 ATLAS experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
Inner tracking detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Pixel detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Semi-Conductor Tracker (SCT) . . . . . . . . . . . . . . . . . . . . .
4.4.2
4.4.3 Transition Radiation Tracker (TRT)
. . . . . . . . . . . . . . . . . .
4.5 Calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74
75
85
87
90
92
94
95
96
4.5.1 Electromagnetic calorimeter . . . . . . . . . . . . . . . . . . . . . . . 101
4.5.2 Hadronic calorimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

vii

4.5.3 Forward calorimeter

. . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6 Muon Spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
4.7 Magnet system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.8 Trigger and data acquisition system . . . . . . . . . . . . . . . . . . . . . . . 116

5.3 Hadronic jets

Chapter 5 Event reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.1 Tracks and vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.2 Photons and electrons
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2.1 Electrons
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2.2 Electron–photon identification . . . . . . . . . . . . . . . . . . . . . . 128
5.2.3 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
5.3.1 Topological clustering algorithm . . . . . . . . . . . . . . . . . . . . . 132
5.3.2 Particle flow algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5.3.3 Generalized kt algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 134
Jet correction and calibration . . . . . . . . . . . . . . . . . . . . . . 137
5.3.4
5.3.5
Jet Vertex Tagger . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4 Muons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.5 Missing transverse momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 144

Jets

Signal model

Chapter 6 H → γγ cross section analysis . . . . . . . . . . . . . . . . . . . . . 145
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.1
6.2 ATLAS dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.3 Event and object selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.3.1 Event preselection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6.3.2 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.3.3
6.3.4 Leptons
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
6.3.5 Missing transverse energy . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4 Event categories and differential variables . . . . . . . . . . . . . . . . . . . . 156
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4.1 Fiducial categories
. . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.4.2 Differential variables
6.4.3 Binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5 Signal and background modeling . . . . . . . . . . . . . . . . . . . . . . . . . 168
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Signal extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
6.6 Correction for detector effects . . . . . . . . . . . . . . . . . . . . . . . . . . 182
6.7 Uncertainties
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Signal extraction uncertainties . . . . . . . . . . . . . . . . . . . . . . 185
6.7.1
6.7.2 Theoretical modeling uncertainties
. . . . . . . . . . . . . . . . . . . 192
6.7.3 Experimental uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 195
6.8 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.8.1 Asimov closure tests
. . . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.8.2 Unfolded cross sections and theory comparisons . . . . . . . . . . . . 200

6.5.1
6.5.2 Background model
6.5.3

viii

Chapter 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
APPENDIX A GoSam ntuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
APPENDIX B Effects of finite top quark mass in Higgs boson production . . . . 228
APPENDIX C TileCal upgrade work . . . . . . . . . . . . . . . . . . . . . . . . 245

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

ix

LIST OF TABLES

Table 2.1: Electroweak quantum numbers. . . . . . . . . . . . . . . . . . . . . . . . .

22

Table 3.1: List of Monte Carlo signal and background samples used in the analysis. .

63

Table 4.1: Summary of the LHC injection stages with their normal operation param-
eters [139, 143, 144]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 4.2: ATLAS calorimeter systems.

. . . . . . . . . . . . . . . . . . . . . . . . .

80

98

Table 4.3: Energy resolution parameters of ATLAS calorimeter systems measured at

test beam [174–178]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Table 4.4: EMB layer segmentation dimensions. . . . . . . . . . . . . . . . . . . . . . 102

Table 4.5: Parameters of FCal modules [127].

. . . . . . . . . . . . . . . . . . . . . . 108

Table 4.6: Main parameters of the ATLAS magnet system [198].

. . . . . . . . . . . 115

Table 5.1: Discriminating variables used for electron and photon identification [229–

231].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Table 6.1: Published Run 2 results of the ATLAS H → γγ cross section analysis.
Table 6.2: Yearly datasets making up the combined Run 2 ATLAS dataset. Estimates
of luminosity and its uncertainty are taken from [280]. Numbers of events
pertain to the inclusive fiducial region.

. . . . . . . . . . . . . . . . . . . 149

. . 148

Table 6.3: Cutflow of event selection for the yearly datasets of ATLAS Run 2 data.

. 152

Table 6.4: Binning table produced by the web tool, showing quantities considered
T , including significance, pu-

when choosing the appropriate binning for pγγ
rity, and signal fractions.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Table 6.5: Breakdown of cross section uncertainties for the inclusive fiducial event

category.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

x

Table 6.6: List of derived relative PES/PER uncertainties, δk

, that set the scales for
the signal model nuisance parameters. The values listed are for the inclusive
fiducial region.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Table 6.7: p-values from χ2 tests for compatibility of the measured differential dis-
tributions with those predicted using the analysis MC, for the observables
measured in the 2019 analysis [277, 278]. The tests used the full set of
measurement uncertainties for the χ2 computation. . . . . . . . . . . . . . 201

Table A.1: Format of GoSam ROOT ntuples [330]. . . . . . . . . . . . . . . . . . . . . 216

Table C.1: Configuration of modules and electronics during the October 2016 testbeam

period [349].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

xi

LIST OF FIGURES

Figure 2.1: Elementary particles of the Standard Model [6].

. . . . . . . . . . . . . .

8

Figure 2.2: Feynman diagrams for e-e scattering in QED, with the spatial dimension
drawn vertically, and time horizontally. The three channels are named
after the Mandelstam variables, s, t, and u, which, in the respective cases,
are equal to the square of the four momentum of the exchanged photon.
s and t channels contribute to Bhabha (e+-e−) scattering, while t and u
channels contribute to Møller (e−-e−) scattering.
. . . . . . . . . . . . .
Illustration of spontaneous U (1) symmetry breaking due to non-zero vac-
. . . . . . . . . . . .
uum expectation value of a complex scalar field, φ.

Figure 2.3:

Figure 2.4: Flavor-changing interactions. a and b denote fermion generations. The
. . . . . . . . .

vector bosons are labeled for the case that allows a (cid:54)= b.

Figure 2.5: The strong coupling constant, αs, as a function of renormalization scale,
[14]. The solid black lines show theoretically calculated values and
µR
their uncertainty. The data points show experimentally obtained values
with the corresponding error bars. . . . . . . . . . . . . . . . . . . . . . .

Figure 2.6: Diagramatic representation of the factorization theorem in QCD. Two
colliding hadrons (A and B) each contribute a parton (a and b) with
dependent on the momentum fraction, x, and
probabilities fa/A
factorization scale, µF
. . . . . . . . . . . . . . . . . . . . . . . . . . . .

and fb/B

.

Figure 2.7: CT14nlo proton PDFs at different scales [25].

. . . . . . . . . . . . . . .

Figure 2.8: The SM Higgs (mH = 125 GeV) inclusive production cross sections as
functions of √s for different channels in pp collisions [26, 27]. The pp → H
process, which is only possible through gluon-gluon fusion, clearly domi-
nates.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2.9: Examples of leading-order Feynman diagrams for Higgs production mech-
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

anisms.

Figure 2.10: Higgs branching ratios and their theoretical uncertainties for the low mass
. . . . . . . . . . .

range (left) and an extended mass range (right) [28].

xii

17

24

29

32

33

34

36

36

37

Figure 3.1: Sketches of Monte Carlo sampling approaches for the acceptance-rejection
method. The event probability density (blue), proportional to the fully
differential cross section, is sampled by generating random numbers with
the sampling probability density (orange).
. . . . . . . . . . . . . . . . .

Figure 3.2: Sketch of a hadron-hadron collision as simulated by a Monte Carlo event
. . . . . . . . . . . . . . . . . . . . . . . . . . .

generator program [32].

Figure 3.3: The transverse momentum of the leading jet (left) and the Higgs boson
(right) calculated using jettiness subtraction technique at LO, NLO, and
NNLO in the strong coupling constant for √s = 8 TeV. The lower panels
show the ratios (K-factors) of NLO over LO cross sections, and NNLO
over NLO cross sections. The large NLO K-factors are apparent. The
shaded regions indicate scale variation errors [46].
. . . . . . . . . . . . .

Figure 3.4: Examples of Feynman diagrams contributing at NLO to the process of
. . . . . . . . . . . . . .

Higgs boson production via gluon-gluon fusion.

Figure 3.5: Examples of self-energy diagrams and their combination into a physical
. . . . . . . . . . . . . . . . . . . . . .

propagator after renormalization.

Figure 3.6: LO dependence of pp → H + jets cross section on factorization and renor-
malization scales at √s = 13 TeV for two different choices of dynamic
scales, computed using GoSam ntuples. MC events were reweighted using
CT14nlo PDFs. Typical jet cuts of pT > 30 GeV and |η| < 4.4 are applied.
The white dot indicates the location of the central scale, µF = µR = µ0.
Figure 3.7: NLO dependence of pp → H + jets cross section on factorization and
renormalization scales at √s = 13 TeV for two different choices of dynamic
scales, computed using GoSam ntuples. MC events were reweighted using
CT14nlo PDFs. Typical jet cuts of pT > 30 GeV and |η| < 4.4 are applied.
The white dot indicates the location of the central scale, µF = µR = µ0.
Figure 3.8: Gluon density distributions and Hessian uncertainties from the CT14nlo
PDF set for proton. Left: for Q = 100 GeV as a function of x. Right: for
x = 0.1 as a function of Q. The nominal function is shown in black, the
error functions in blue, and the uncertainty in red.
. . . . . . . . . . . .

Figure 3.9: Fractional signal composition in terms of the Higgs production processes as
a function of diphoton transverse momentum determined from the signal
MC samples.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

43

47

48

56

59

60

62

66

Figure 4.1: CERN accelerator complex [134].

. . . . . . . . . . . . . . . . . . . . . .

76

xiii

Figure 4.2: Detailed view of the underground LHC facilities, showing the locations of
the experiments and beam intersection points [135, 136]. The locations of
the four major LHC experiments: ATLAS, CMS, LHCb, and ALICE are
highlighted in gray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.3: An RF cavity diagram [140]. The charge induced at either end of a cavity
segment accelerates protons forward. The field in the cavity oscillates at
400 MHz, resulting in the induced charge switching polarity, so that the
protons never feel a force in the backward direction. The LHC uses 8
cavities per beam, delivering 2 MV accelerating potential (5 MV/m field).
Every proton passing through an RF cavity experiences a total potential
. . . . . .
difference of 2 × 8 = 16 MV, gaining 16 MeV of kinetic energy.

Figure 4.4: PSB-PS transfer: two-batch filling for the LHC [139].

. . . . . . . . . . .

Figure 4.5: PS bunch splitting for a nominal 25 ns LHC bunch train [139]. . . . . . .

Figure 4.6: The integrated luminosity for √s = 13 TeV data delivered by the LHC,
recorded by the ATLAS detector, and deemed good for physics analyses
during Run 2. [150].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.7: Left: Comparison of the total and elastic pp cross section measurements,
including measurements form ATLAS and TOTEM at 7 and 8 TeV [151].
Right: The inelastic pp cross section as a function of √s, including mea-
surements at √s = 13 TeV [152].
. . . . . . . . . . . . . . . . . . . . . .

Figure 4.8: The luminosity-weighted distribution of the mean number of interactions
per bunch crossing during Run 2. The mean number of interactions corre-
sponds to the mean of the Poisson distribution of the number of interac-
tions per crossing calculated for each bunch. In this calculation the σinel
is taken to be 80 mb for √s = 13 TeV [150].
. . . . . . . . . . . . . . . .

Figure 4.9: Computer generated image of the ATLAS detector indicating all of its
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

major systems [153].

Figure 4.10: ATLAS inner detector. Top:

3D rendered image [157]. Bottom:
Schematic view of a quarter-section showing each of the major detector
elements and dimensions [127].
. . . . . . . . . . . . . . . . . . . . . . .

Figure 4.11: Left: Pixel detector module [162]. Top-right: 3D rendered image of the
pixel detector [163]. Bottom-right: Diagram of a transverse slice of the
pixel layers [164].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 4.12: Photograph (left) and drawing (right) showing the main components of a
. . . . . . . . . . . . . . . . . . . . . . . . . .

SCT barrel module [127].

77

78

80

80

83

84

85

88

91

93

94

xiv

Figure 4.13: Left: Schematic drawing of the TRT barrel showing module and straw
. . . .

positions [127]. Right: Photograph of a TRT wheel module [172].

96

Figure 4.14: Cutaway computer generated image of the ATLAS calorimeter systems [173]. 97

Figure 4.15: Sketches of electromagnetic (left) and hadronic (right) showers.

. . . . .

98

Figure 4.16: Structure of the ATLAS electromagnetic calorimeter. Top: Schematic
drawing of an EMB segment showing layers and ganuarity [127]. Bottom:
Photograph of an open EMB cell showing its accordion structure [180].

. 103

Figure 4.17: A diagram illustrating a TileCal module (top) and the positioning of cells
within the central and extended barrels (bottom) [127]. Cells D4 and
C10 and refered to as the Plug tile calorimeter, E1 and E2 as the Gap
scintillator, and E3 and E4 as the Cryostat scintillator.

. . . . . . . . . 106

Figure 4.18: HEC module diagrams [127]. Left: r-φ and r-z views of a module. Right:

Structure of the inter-plate gap. . . . . . . . . . . . . . . . . . . . . . . . 107

Figure 4.19: Left: Diagram showing FCal modules located in the end-cap cryostat.
The black regions are structural parts of the cryostat. Right: Electrode
structure of FCal1. The other two modules are similar. The Moli`ere
radius, RM

, is indicated by the solid circle. [127]

. . . . . . . . . . . . . 109

Figure 4.20: Cutaway computer generated image of the ATLAS muon system [192].

. 110

Figure 4.21: Schematic drawing of a regular (2 × 3) MDT chamber [193].

. . . . . . . 112

Figure 4.22: 3D view of the bare windings of the ATLAS magnet system [199].

. . . . 115

Figure 4.23: Schematic diagram of the ATLAS trigger and data acquisition (TDAQ)

systems [203]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

Figure 5.1: A schematic diagram showing interaction of different ATLAS detector sys-
tems with various types of collision products [213, 214]. The Inner Detec-
tor maps out tracks of charged particles; the Electromagnetic Calorimeter
absorbs and measures the energy of photons and electrons; the Hadronic
Calorimeter performs the same function for jets formed by e.g. protons or
neutrons; and the Muon Spectrometer provides tracking information for
muons. The relative layer sizes are not drawn to scale.

. . . . . . . . . . 120

Figure 5.2: Perigee parametrization of a track trajectory.

. . . . . . . . . . . . . . . 123

xv

Figure 5.3: A simplified example of ambiguity inherent in track reconstruction [218].
Tracks in the SCT barrel (a, b, c) share several hits. The ambiguity is
resolved by assigning a higher score to a track containing hits on both
sides of the SCT strips.

. . . . . . . . . . . . . . . . . . . . . . . . . . . 124

Figure 5.4: Sketch of the ATLAS detector systems used to perform electron recon-

struction and identification. [224]. . . . . . . . . . . . . . . . . . . . . . . 126

Figure 5.5:

Illustration of the shower shape discriminating variables used for photon
identification [229, 232].

. . . . . . . . . . . . . . . . . . . . . . . . . . . 131

Figure 5.6: Comparison of jet shapes produced by commonly used clustering algo-

rithms [239]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

Figure 5.7: Distributions of corrJVF (left) and RpT (right) for hard-scatter and pileup
jets obtained from simulated dijet events [252, 253]. Jets with no associ-
ated tracks are assigned corrJVF = −1. . . . . . . . . . . . . . . . . . . . 140

Figure 5.8: Left: JVT likelihood as a function of corrJVF and RpT. Right: Distribu-
tion of the JVT likelihood for low-pT central pileup and hard-scatter jets.
Both figures from [252].

. . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Figure 5.9: The fJVT distribution for hard-scatter (blue) and pileup (green) forward
jets in simulated Z + jets events with at least one forward jet with 30 <
pT < 40 GeV (left) or 40 < pT < 50 GeV (right) [254].

. . . . . . . . . . . 142

Figure 6.1: Schematic summary of the analysis strategy. . . . . . . . . . . . . . . . . 147

Figure 6.2: Expected composition of fiducial regions in terms of Higgs production

modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Figure 6.3: The largest contribution to the Higgs boson production at the LHC is from
the gluon-gluon fusion process. At LO, the Higgs boson has an effective
coupling to gluons via the loop diagram shown here. The effective coupling
is sensitive to the mass of the fermion in the loop. As Γ(H → f ¯f ) ∝ m2
,
the partial width for H → t¯t accounts for 99% of the combined width to
all fermions.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

f

Figure 6.4: Sketch of the (cid:126)pγγ
Tt

definition [302].

. . . . . . . . . . . . . . . . . . . . . . 162

Figure 6.5: The migration matrix for division of the fiducial region into bins of pγγ
T ,

produced by the binning web tool.

. . . . . . . . . . . . . . . . . . . . . 167

xvi

Figure 6.6: Example of a double-sided Crystal Ball function. Gaussian tails are drawn

with dashed lines to illustrate the difference with the power law. . . . . . 170

Figure 6.7: The MC signal fit for the inclusive fiducial event category drawn on a
linear (left) or a logarithmic (right) vertical axis. The black dotted lines
indicate the region used to estimate the spurious signal uncertainty. . . . 171

Figure 6.8: Background composition as a function of mγγ
estimated using the 2 × 2D method [307, 308].

(left) and pTγγ
. . . . . . . . . . . . . . . 172

(right) [278]

Figure 6.9: Comparison of the background templates to the data in sideband regions
for the inclusive fiducial event category using the two-component (left)
and one-component (right) reducible background model.

. . . . . . . . . 174

Figure 6.10: Results of the spurious signal test for the inclusive fiducial event category,
showing the amount of fitted spurious signal S (referred to as Nsp in the
text) relative to the background uncertainty δS (left) and relative to the
expected signal yield SRef
is obtained from fitting the signal
model to the signal MC.

(right). SRef
. . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Figure 6.11: The F -distribution for ∆ndf = 1 and ndf2 = 86 corresponding to the F -
test for the inclusive fiducial event category for the ExpPoly2 background
function and 0.5 GeV binning in mγγ
. The highlighted area shows the
integral of the distribution corresponding to the P -value. . . . . . . . . . 177

Figure 6.12: Signal extraction fit for the inclusive fiducial event category.

. . . . . . . 181

Figure 6.13: Comparison of the total signal yields obtained from independent fits for
different combinations of event categories, i.e. different differential vari-
ables. Overflow bins are included in the fits. The inclusive fiducial spuri-
ous signal uncertainty is shown for reference. . . . . . . . . . . . . . . . . 181

Figure 6.14: Inclusive fiducial correction factors from each production process MC.
The values shown in black have no particle-level isolation applied and
display a significant dependence on the production process. Values in
green have particle-level track isolation applied. Values in red also have
calorimeter isolation applied. The production mode dependence is reduced
by isolation requirements. [231]

. . . . . . . . . . . . . . . . . . . . . . . 184

Figure 6.15: Component breakdown of the total uncertainty on the measured cross
T and Njets. These are stack plots, with consecutive

section in bins of pγγ
components added in quadrature.

. . . . . . . . . . . . . . . . . . . . . . 186

xvii

Figure 6.16: Further breakdown of the systematic correction factor component uncer-
T and Njets. Only the

tainty on the measured cross section in bins of pγγ
largest contributions are explicitly shown. The rest are added as the other.186

Figure 6.17: The effect of simultaneous variations of all photon calibrations on the
diphoton mass spectrum, independently for PES (left) and PER (right).

187

Figure 6.18: Expected and observed nuisance parameters’ pre- and post-fit pulls and
impacts for the inclusive fiducial category. The nuisance parameters (NP)
are ranked by their impacts on the signal yield parameter of interest (PoI).190

Figure 6.19: The likelihood profile for the signal yield in the inclusive fiducial event
category. The combined statistical and systematic uncertainty is deter-
mined from the profile obtained with floating nuisance parameters, shown
with a solid line. The statistical-only uncertainty is determined from the
other, dashed, profile, for which the nuisance parameters are fixed to the
best fit values.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

Figure 6.20: Expected systematic and statistical uncertainties on signal yield estimated
using the Asimov method for inclusive fiducial event category and in bins
of jet multiplicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

Figure 6.21: Original and reweighted pγγ

in estimation of the unfolding bias uncertainty.

T (left) and(cid:12)(cid:12)yγγ(cid:12)(cid:12) (right) MC distributions used

. . . . . . . . . . . . . . 193

Figure 6.22: Effects of pileup reweighting variations on the distribution of the number

of primary vertices, Npv. . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Figure 6.23: Cross checks of the analysis using Asimov datasets for diphoton transverse
momentum (left) and jet multiplicity (right) distributions. The extracted
cross section is compared to the particle-level MC.

. . . . . . . . . . . . 199

Figure 6.24: Fiducial differential cross section measurements for pγγ

T , |yγγ|, pj1T , and
. . . . . . . . . . . . 201

Figure 6.26: Fiducial differential cross section measurements for pγγ

Figure 6.25: Fiducial differential cross section measurements for Njets and ∆φjj done

mjj done with(cid:82) L = 139.0 fb−1 in 2019 [277, 278].
with(cid:82) L = 139.0 fb−1 in 2019 [277, 278].
Nb-jets done with(cid:82) L = 79.8 fb−1 in 2018 [275, 276].
egories defined in Section 6.4.1 done with (cid:82) L = 36.1 fb−1 in 2017 [272–

274].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204

T , |yγγ|, pj1T , and
. . . . . . . . . . . 203

. . . . . . . . . . . . . . . . . . 202

Figure 6.27: Fiducial differential cross section measurements for the fiducial event cat-

xviii

Figure 6.28: Fiducial differential cross section measurements for pγγ

T , |yγγ|, | cos θ∗|,

. . . . . . 205

|∆yγγ|, and pγγ
Tt

done with(cid:82) L = 36.1 fb−1 in 2017 [272–274].

in 2017 [272–274].

Figure 6.29: Fiducial differential cross section measurements for inclusive and exclusive
T > 50 GeV

T > 30 GeV (top) and for pj

jet multiplicities, Njets, for pj

Figure 6.30: Fiducial double-differential cross section measurements for pγγ

(bottom) done with(cid:82) L = 36.1 fb−1 in 2017 [272–274].
subdivided in Njets (top) and | cos θ∗| (bottom) done with(cid:82) L = 36.1 fb−1
Figure 6.31: Fiducial differential cross section measurements for pj1T , |yj1|, pj2T , |yj2|,
HT, and |∆yjj| done with(cid:82) L = 36.1 fb−1 in 2017 [272–274].

Figure 6.32: Fiducial differential cross section measurements for mjj, ∆φjj, |∆φjj|,
Figure 6.33: Fiducial differential cross section measurements for τ1 and(cid:80) τj done with

, and |∆φγγ,jj| done with(cid:82) L = 36.1 fb−1 in 2017 [272–274].

T further

(cid:82) L = 36.1 fb−1 in 2017 [272–274].

pγγjj
T

. . . . . . 208

. . . . 209

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

. . . . . . . . . . 206

. . . . . . . . . . . . . . . . . . . . . 210

Figure A.1: GoSam NLO predictions for jet multiplicity distributions. The cross sec-
tion in each of the first three bins was calculated using the set of ntuples
corresponding to the respective number of jets. The last bin represents
the additional radiation due to the real correction to the process with 3
nominal jets.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

Figure A.2: GoSam NLO predictions for distributions of observables in Higgs bo-

son + 1 jet events.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

Figure A.3: GoSam NLO predictions for distributions of observables in Higgs bo-

son + 2 jets events.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

Figure A.4: GoSam NLO predictions for distributions of observables in Higgs bo-

son + 3 jets events.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Figure B.1: Using effective field theory (EFT), an effective coupling of the Higgs bo-
son to gluons can be derived by taking the limit mt → ∞. This approach
simplifies calculations by reducing the number of loops, but the approxi-
mation is only valid for pHT (cid:46) 2mt ≈ 350 GeV.

. . . . . . . . . . . . . . 229

xix

Figure B.2: Comparison of diphoton transverse momentum distributions, generated
using GoSam ntuples [106, 330], for (H → γγ) + 1 jet produced in gluon
fusion with (mtop) and without (EFT) accounting for the finite value of
the top quark mass. The former is obtained by reweighting the later by the
ratio of the respective distributions at LO. Also shown are distributions for
the background diphoton production. The distributions include photon
and jet cuts on pT and η used in the H → γγ analysis.

. . . . . . . . . . 231

Figure B.3: Representative Feynman diagrams containing a quark triangle loop that
contribute to the Higgs boson + 1 jet process for the gg, qq, and gq initial
states.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

Figure B.4: A Feynman diagram containing a quark box loop that contributes to the

Higgs boson + 1 jet process for the gg initial state.

. . . . . . . . . . . . 233

Figure B.5: Distributions of the mass of the Higgs boson + the leading jet system
produced using EFT and finite mt LO GoSam ntuples for Higgs boson
+ at least 1 jet production. gg, qq, and gq labels indicate the initial state
partons.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234

Figure B.6: Distributions of the mass of the Higgs boson + the leading jet system
produced using EFT and finite mt LO GoSam ntuples for Higgs boson +
at least 2 jets production. gg, qq, and gq labels indicate the initial state
partons.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

Figure B.7: Distributions of the mass of the Higgs boson + the leading jet system
T ∈

for a specific slice in the subleading jet’s transverse momentum, pj2
[268, 311) GeV.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Figure B.8: A representative Feynman diagram for the Higgs boson production with

2 jets from the gg initial state.

. . . . . . . . . . . . . . . . . . . . . . . 237

Figure B.9: Ratios of the distributions of the invariant mass of the Higgs boson and
the leading jet system from full and effective theory GoSam calculations
within intervals of the virtuality variable, t25, for proton-proton √s = 13
(left) and 100 (right) GeV.

. . . . . . . . . . . . . . . . . . . . . . . . . 238

Figure B.10: Distributions of the kinematic invariants that can appear in the denom-
inators of amplitude expressions for the gg → Hgg process and lead to
singularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Figure C.1: Current LHC operation and upgrade timeline [346]. . . . . . . . . . . . . 246

xx

Figure C.2: Photographs of TileCal modules [351]. Left: An Extended Barrel mod-
ule. Module cells are labeled on the flat side. Bundles of yellow-green
wavelength-shifting fibers can be seen at the bottom of the module. The
opening at the bottom of the module is where the drawer housing the
PMTs and on-detector electronics is inserted. Tight bundles of fibers are
threaded through holes along the bottom of the module and are fitted
against the PMTs’ photocathodes. Right: Drawer electronics being in-
serted into a module. Not yet connected ends of the Cs calibration system
tubes can be seen along the narrow blue side of the modules.

. . . . . . 247

Figure C.3: Diagram of the legacy TileCal electronics.

. . . . . . . . . . . . . . . . . 249

Figure C.4: Diagram of the Phase-II TileCal electronics.

. . . . . . . . . . . . . . . . 249

Figure C.5: Diagram of the hybrid Phase-I Demonstrator TileCal electronics.

. . . . 249

Figure C.6: Photograph of a Demonstrator mini-drawer with the electronics compo-

nents labeled.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Figure C.7: Diagram illustrating location of the Main, Daughter, and HVOpto boards,

as well as the PMTs, in a mini-drawer [357].

. . . . . . . . . . . . . . . . 252

Figure C.8: Photograph of an HVOpto board mounted on a mini-drawer. The circuits
for 12 channels, 6 on each side, can be seen as clusters of components.
The two ribbon cables provide SPI bus communication with the Daughter
board on the other side of the mini-drawer.

. . . . . . . . . . . . . . . . 253

Figure C.9: Finite state machine of the HVOpto component of the Daughter board

firmware.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Figure C.10: Photograph of the electronics setup used for development of the HVOpto
FPGA firmware component for the TileCal mini-drawer Daughter board.
Shown are a Xilinx Atlys prototyping board, a single-channel HVOpto
test board, and a LV power supply.

. . . . . . . . . . . . . . . . . . . . . 255

Figure C.11: Photograph of a TileCal finger Low Voltage Power Supply (fLVPS). The
green circuit boards of the ELMB and its motherboard can be seen on
the left photo. On the right photon, the ELMB and motherboard are
removed. The blue circuit boards belong to the DC-DC converter bricks.

257

Figure C.12: ELMB128. Its block diagram [367] is shown on the left and its photograph

on the right. The dimensions of the circuit board are 50 × 67 mm.

. . . 258

Figure C.13: Diagram of the TileCal cesium calibration system [375]. . . . . . . . . . . 260

xxi

Figure C.14: Cs scan data collected during a testbeam in 2016 using the Demonstrator
electronics [349]. The plot shows the response of the tiles in the BC6 cell
of the module. Each peak corresponds to the Cs source passing though a
tile.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

Figure C.15: Photograph of the electronics setup used for development of the Cs cali-
bration system (bottom board) FPGA firmware component for the TileCal
mini-drawer Daughter board (top board).

. . . . . . . . . . . . . . . . . 262

Figure C.16: TileCal Demonstrator testbeam setup [379]. The beam direction is from

left to right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264

Figure C.17: Left: Diagram of TileCal modules’ arrangement for the testbeam [379].
Right: Photograph of the modules positioned on the mobile table in the
testbeam area [381].

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

xxii

Chapter 1

Introduction

Throughout the 20th century, physicists developed the modern mathematical theories

of the nature’s fundamental constituents and their interactions. Two ideas have grown in

parallel: quantum mechanics, valid at short distances and high energy scales, and applicable

in the realm of particles, and a more classical theory of general relativity, which accurately

describes the gravitational interaction over large distances on the cosmic scale. Our Standard

Model of particle physics is a theory of the interplay between the universe’s elementary

constituents. Guided by the principles of symmetry and least action, we formulate the

Standard Model by applying quantum mechanics and special relativity to suitably defined

abstract fields. The result is a powerful model, containing a description of three of the

discovered forces: electromagnetic, weak, and strong. But in as much as the Standard

Model is elegant, it is also perplexing, leaving a number of questions unanswered.

Investigating the unanswered questions and testing our current understanding are the

primary goals of particle physics research. Studying the universe at the smallest scale requires

looking at the interactions at the highest energies.∗

In order to do this in a controlled

environment, modern high energy physics laboratories have become the sites of the largest

man-made accelerators, that collide particles at the highest energies attainable with modern

technology. The Large Hadron Collider (LHC) at CERN is the flagship experiment of this

∗Heuristically, the length scale can be represented by the de Broglie wavelength, inversely proportional

to the particle’s momentum: λ = h/p, where h is the Planck constant.

1

kind, and ATLAS and CMS are its largest detectors, collecting data from proton collisions

for frontier research in fundamental physics.

Already, the LHC has helped us resolve a conjecture about the Standard Model that

had been unanswered since the 1960s. The Higgs boson, whose field provides an elegant

mechanism for electroweak symmetry breaking and giving masses to elementary particles,

was discovered in 2012 by both ATLAS [1] and CMS [2] collaborations. With the existence

of the Higgs boson firmly established, dedicated analysis groups have concentrated their

efforts on improving the measurements of the Higgs properties, such as its couplings and

production cross sections. Precision measurements in the Higgs sector are very important

for our understanding of the fundamental physics, as the Higgs field plays a central role in

the gauge field theory of electromagnetic and weak interactions. The way the Higgs boson

couples to the other fields also provides a new way to investigate properties and improve

modeling of quantum chromodynamics. The presence of the Higgs field in the Standard

Model, as well as the specific values of the parameters associated with it, have implications

reaching beyond the explanation of the observed masses of elementary particles, soliciting

questions about the physics beyond the Standard Model and stability of the vacuum in the

universe.

Presented in this dissertation are the methods and results of the data analysis used to

extract the Higgs boson production cross section in the diphoton decay channel from the

ATLAS data, collected in the Run 2 of the LHC. The author contributed to different aspects

of the analysis since the beginning of Run 2 in 2015. The analysis team produced yearly

results using incremental datasets. Consequently, the analysis evolved with the increase in

integrated luminosity delivered by the LHC, as well as introduction of new methods. A list

of the respective publications is provided in Table 6.1.

2

The dissertation is organized in the following way. Chapter 1 is the introduction. An

overview of the theoretical framework of the Standard Model is given in Chapter 2. Chapter 3

discusses simulation of collision events with the help of Monte Carlo event generators and the

theoretical predictions used in the analysis. The LHC and the ATLAS detector are described

in Chapter 4. Chapter 5 continues with a discussion of the experimental techniques of object

and event reconstruction on the ATLAS detector. The central subject of this dissertation is

presented in Chapter 6, which contains the details and results of the Higgs boson cross section

analysis in the diphoton decay channel. The main discussion is concluded in Chapter 7.

The appendices describe additional projects the author worked on, which do not directly
pertain to the main subject of H → γγ cross section measurements. A discussion of the
GoSam ntuples and the studies performed using them can be found in Appendices A and

B. Hardware, electronics, and firmware upgrade projects for the ATLAS Tile calorimeter are

described in Appendix C. Contribution to the TileCal upgrade efforts earned the author an

official membership in the ATLAS collaboration.

Before proceeding, it should be noted that equations given in this document are writ-

ten using natural units, whereby the speed of light, the Plank constant, and the vacuum

permittivity, are all dimensionless and equal to one,

c =  =

h
2π

= ε0 = 1.

(1.1)

As will be seen in Chapter 2, this definition is very convenient, reducing the amount of

clutter in many equations. For example, the fine structure constant can be written as

α =

e2
4π

,

3

(1.2)

where e is the magnitude of the electron charge. This convention is unambiguous, and factors

of the constants listed in Eq. (1.1) can always be restored using dimensional analysis. In

this natural system, only one fundamental unit survives, and units of length and time are

both inversely proportional to the units of momentum and energy. The natural scale for the

discussion is on the order of the masses of the weak (W and Z) and Higgs bosons. Thus,

most values of energy, momentum, and mass will be quoted in gigaelectronvolts (GeV).

4

Chapter 2

Theory

Our contemporary model of fundamental interactions is formulated as a relativistic theory

of quantized fields. Field theory turns out to be the natural language required to reconcile

special relativity with quantum mechanics. This can be seen in several ways. One reason

for adopting the field theory approach is to be able to deal with systems of variable particle

number. On the one hand, extension of the space of states from Hilbert to Fock naturally

leads towards field theory. On the other, relativistic processes cannot be assumed to conserve

particle number. Einstein’s mass–energy equivalence formula, E = mc2, relates the amount

of energy absorbed or released when a massive particle is created or annihilated. And,
by the uncertainty principle, ∆E∆t ≥ /2, intermediate particles can be produced off the
mass shell. Thus, the theory must be able to describe fluctuating numbers of particles of

different types. In QFT, particles are interpreted as excitation quanta of the respective fields,

and creation and annihilation operators appear similarly to ladder operators.∗ Furthermore,

there are systems, such as lasers, whose description requires superposition of states of definite

particle number. Another argument for QFT is that it is the only known relativistic quantum

mechanics formalism consistent with Lorentz invariance and causality. In particular, QFT

∗Ladder operators appear, in particular, as a consequence of quantization of energy levels in harmonic
oscillator problems, a discussion of which can be found in any elementary quantum mechanics book, such
as [3]. Many problems, at least in approximation, e.g. in perturbation theory, resemble a harmonic oscillator,
because any smooth potential can be modeled near a minimum by a parabola using a Taylor series up to
the quadratic terms. Heuristically, a field can be viewed as a collection of enharmonic oscillators labeled by
generalized coordinates.

5

solves the causality problem by producing cancellations between particle and antiparticle

propagation amplitudes outside the light cone.∗

Our central guiding principle in the development of the fundamental theory of interac-

tions is symmetry. Observations of the invariance of natural laws with respect to the choice

of the frame of reference led to ideas of relativity (first Galilean, then special and general).

Redundancies in the definitions of fields in electromagnetism and phases in quantum me-

chanics led to the formulation of the fundamental forces as interactions with gauge fields.

Symmetries among the multitude of hadronic particles led to the discovery of the color charge

and formulation of quantum chromodynamics (QCD). The duality between continuous sym-

metries and conservation laws is captured by Noether’s theorem.† Three discrete symmetries

are identified as fundamental to physical laws: charge conjugation, C, parity inversion, P,

and time reversal, T. Individually, all three of these are respected by all interactions except

for the weak, which violates them individually, as well as in combinations of two. But, the

combination of all three, CPT, is respected in all situations.

2.1 Standard Model

The SM is our current best theory of interactions between all known elementary particles,

describing three of the four fundamental interactions (electromagnetic, weak, and strong).

It provides the means to calculate cross sections for all experimentally observable processes,

and its predictions agree extremely well with the data. The theory also gives a conceptual

∗Preservation of causality in a relativistic theory requires that events separated by a spacelike inter-
val cannot influence each other. In QFT, this condition can be expressed by the vanishing commutator,

(cid:2)φ(x), φ†(y)(cid:3) = 0, for (x − y)2 < 0, which is a condition distinct from x (cid:54)= y and x0 = y0. For a detailed

†Noether’s theorem states that existence of continuous symmetry transformations that leave the La-

discussion, see, for example, chapter 2 of [4].

grangian invariant (up to a divergence) implies existence of respective conserved currents.

6

framework for explanation of many observed phenomena.

Although hugely successful, the SM has its scope of applicability. It does not account

for large scale astrophysical phenomena, namely, the predominance of matter over antimat-

ter, the existence of sufficient dark matter, the cosmological constant (dark energy), or the

mechanism of inflation of the universe.∗ The SM does not include gravity, as there is no

self-consistent quantum theory of it. General relativity, as the full theory of spacetime, is

not incorporated into the SM, which includes only special relativity. Several phenomena

are accommodated within the SM without a mechanism that would generate the respective

parameters of the model. These constants, that have to be measured and cannot be derived

within the theory, include the values of the masses of the Higgs boson and fermions, the

parameters of flavor mixing for quarks and neutrinos, including the CP-violating complex
phase,† and the values of the coupling constants.

The fundamental particles of SM are cataloged in Fig. 2.1. Due to their properties,

the particles fall into various categories. Thus, we distinguish half-integer spin fermions,

representing matter particles, and whole-integer spin bosons, representing mediators of in-

teractions through fundamental forces.

Among fermions there are quarks and leptons. Quarks carry color charge and form

strongly interacting hadrons, such as protons and neutrons. Leptons don’t interact strongly.

There are massive electrically charged leptons (electrons and their heavier cousins muons and

tau) and nearly massless uncharged neutrinos. Orthogonally, there are three generations,

or flavors, of fermions, with each subsequent generation more massive than the previous,

but otherwise possessing identical properties. For each type of fermion there is a distinct

∗These are appropriately referred to as the “Four Mysteries of the Cosmos” by James Wells in [5].
†CP violation can be observed in experiments with neutral kaons, which can decay to either 2 or 3

photons.

7

Figure 2.1: Elementary particles of the Standard Model [6].

antiparticle with conjugate charges. All fermions can interact weakly.

Each type of interaction has a corresponding mediator vector boson: photons for electro-

magnetic, gluons for strong, and W and Z bosons for weak. Photons and gluons are massless,
while the weak bosons are massive. W + and W− are electrically charged, while the other
bosons are neutral. The scalar Higgs boson, unlike the vector bosons, is not a mediator.

However, it plays a central role in the electroweak sector of the SM, allowing the theory to

accommodate massive vector bosons, as well as fermions’ masses and flavor mixing.

2.2 Quantum Field Theory

The fundamental postulate of mechanics is the principle of least action, whereby the
physical dynamics of a system corresponds to the evolution which keeps action, S, stationary

8

FermionsIIIIIIBosonsmasschargespinQUARKSu≃2.2 MeV⅔½upd≃4.7 MeV−⅓½downc≃1.28 GeV⅔½charms≃96 MeV−⅓½stranget≃173.1 GeV⅔½topb≃4.18 GeV−⅓½bottomLEPTONSe≃0.511 MeV−1½electronνeΣmν≲0.26 eV0½electronneutrinoμ≃105.66 MeV−1½muonνμ0½muonneutrinoΔm221≃7.5×10−5 eV2τ≃1.7768 GeV−1½tauντ0½tauneutrinoΔm232≃2.4×10−3 eV2GAUGEBOSONSVECTORBOSONSg001gluonγ001photonZ≃91.19 GeV01Z bosonW≃80.39 GeV±11W bosonSCALARBOSONSH≃124.97 GeV00higgsto first order, i.e. δS = 0. Action is defined as the integral of the Lagrangian, L, which, in
its turn, is a function of generalized coordinates, q, and their time derivatives, ˙q,

dt L(cid:0)q(t), ˙q(t), t(cid:1).
Requiring δS = 0 then leads to the Euler–Lagrange equations,

S =(cid:90) t2

t1

∂L
∂q −

d
dt

∂L
∂ ˙q

= 0,

(2.1)

(2.2)

which, when applied to a specific Lagrangian, produce equations of motion.

A system of N bodies can be represented by a Lagrangian L(cid:0){qa},{ ˙qa}, t(cid:1), with
If changes in qa are small, deviating only slightly from equilibrium, the
a = 1, 2, . . . , N.
bodies can be labelled by their position, qa(t) → q((cid:126)x, t). In the continuum limit,∗ the gener-
alized coordinates, q, form a field, which is then customary to instead denote by φ. Fields,

rather than spacetime coordinates, are the dynamic variables of the Lagrangian. In the set-
L, rather

ting of field theory, it is more convenient to talk about the Lagrangian density,†

than the Lagrangian, L, with

L =(cid:90) d3xL.

(2.3)

One way to quantize the theory is to promote the dynamic variables, i.e. the fields, to

operators and to impose canonical commutation relations,‡

∗For an intuitive introduction to QFT, including an analogy between a field and a spring mattress see

Chapter 1 in Zee [7].

†In field theory, L is often simply referred to as the Lagrangian.
‡Canonical (also known as second) quantization is similar to how quantum mechanics was developed.
Another way to quantize a relativistic field theory is through path integral formulation. The two approaches
yield equivalent results, but the path integral approach is often more convenient for automating calculations.

9

(cid:2)φ(x), π(y)(cid:3) = iδ(x − y),

(cid:2)φ(x), φ(y)(cid:3) =(cid:2)π(x), π(y)(cid:3) = 0,

(2.4)

where π(x) ≡ ∂L∂ ˙φ(x)
space via Fourier transformation, one can then formulate creation and annihilation operators,

is the momentum density conjugate to φ(x). Switching to momentum

a†p and ap, with the appropriate commutation relations. Formally, the states in QFT live in
Fock space, which is a space of states of variable or unknown number of identical particles,

constructed from a single-particle Hilbert space. The number operator is given by

ni = a†i ai,

(2.5)

where the index i stands for all labels, including momenta and particle species, implying

that creation, annihilation, and number operators, commute for distinct sets of labels.

2.2.1 Types of fields and spin

Fields differ from one another by quantum numbers (or charges) that they carry, by their

transformation properties and the symmetries they obey, and by the way they interact with

(or couple to) themselves and other fields.

One of the fundamental properties of a field is the way it behaves under Lorentz trans-

formations. This property is intimately related to the notion of spin. Field values can be

viewed as elements of a vector space on which elements of the Lorentz Lie group act as trans-

formations. Different types of fields require different representations of the Lorentz group.

Associated with these representations are different values of spin.∗ Mathematically, spin has

the same properties as angular momentum, and can be viewed as a fixed intrinsic amount of

∗Spin of a relativistic field is determined by the properties of the Wigner’s little group, which is defined
as the subgroup of homogeneous Lorentz group that leaves the 4-vector of a particle invariant. In particular,
the magnitude of spin is determined by the dimensionality of the fundamental representation of the little
group.

10

angular momentum carried by a particle.∗ However, unlike orbital angular momentum, spin
admits half-integer values, with its projection still quantized in integer steps between −s
and +s. The association between spin and Lorentz transformation properties is as follows:

spin-0 fields are scalars, spin-1 are vectors, spin-2 are tensors, etc; spin- 1
2
, or 1.

spinors. All elementary particles in the SM have spin of either 0, 1
2

fermionic fields are

As demonstrated by the spin–statistics theorem, spin also determines the statistical prop-

erties of identical particles. Bosons with integer (including zero) spin follow Bose–Einstein

statistics and can occupy identical quantum states, while fermions with half-integer spin

follow Fermi–Dirac statistics and cannot occupy identical states, which leads to the Pauli

exclusion principle. In QFT, this behavior emerges as a consequence of the opposite prop-

erties of the two types of wavefunctions under the exchange symmetry. Symmetric bosonic

fields are quantized using commutation relations,†

(cid:2)ap, a†q(cid:3) = apa†q − a†qap = δ(p − q),

while antisymmetric fermionic fields require anti-commutation relations,

(cid:8)ap, a†q(cid:9) = apa†q + a†qap = δ(p − q).

(2.6)

(2.7)

The commutator identities allow for repeated application of the creation operator, a†, to
bosonic states, and thus an arbitrary number of bosons can occupy the same state. But the

anticommutator identities imply that repeated application of a† annihilates the state, and

∗Spin can be detected in experiments with charged particles in a magnetic field, where it is observed as
a magnetic dipole moment. For example, a Stern–Gerlach type of experiment demonstrates separation of
particles traveling through a magnetic field into as many bunches as the number of available spin projections.
†Note, that in general, the (anti-)commutation relations for creation and annihilation operators apply
only to states at equal time. The commutators must vanish for spacelike separated events to respect causality.

11

so only one identical fermion per state is allowed. This explains the connection between spin

and statistics.

2.2.2

Interactions

The only tangible thing in QFT is the interaction.

In fact, a non-interacting field is

as good as non-existent, since it is unable to cause any measurable effects. In high energy

physics, we describe interactions as scattering or decay processes. Thus, the observables we

primarily need to calculate are scattering cross sections, σ, and decay widths, Γ.

To understand why cross section and width are the natural quantities that characterize

particle interactions, let’s first consider the decay of unstable particles. If we have a set of

N particles, each of which has a constant probability Γ of decaying per unit of time, the
number of them decaying in a time interval dt is given by N Γdt. Therefore,∗

dN = −ΓN dt ⇔ N = N0e−Γt.

The mean lifetime of a particle is then given by

τ =(cid:10)t(cid:11) = (cid:82) ∞0 tp(t)dt
(cid:82) ∞0 p(t)dt

=

1
Γ

.

Heuristically, the time-energy uncertainty relationship,

∆E∆t ∼ 1,

(2.8)

(2.9)

(2.10)

∗The fact that the probability of still finding an unstable particle, after a given amount of time, decreases
exponentially can also be understood by realizing that probabilities are multiplicative. So, to obtain the

probability after time t, one needs to take a product integral: p(t) =(cid:81)t

0(1 − Γdt) = exp(−Γt).

12

suggests that an unstable particle with a mean lifetime τ would have an energy uncertainty

on the order of

∆E ∼

1
τ

= Γ.

(2.11)

In non-relativistic regime, Γ (cid:28) m, this can be modeled by adding a complex component to
the particle’s energy,

E = E0 − iΓ/2.

(2.12)

In the rest frame of the particle, its time-dependent wave function can then be written as

ψ(t) = ψ(0) exp(cid:0)− iEt(cid:1) = ψ(0) exp(cid:0)− iE0t(cid:1) exp(cid:18)−

Γt

2(cid:19) .

Therefore, the total probability decays exponentially, as expected,

|ψ(t)|2 = |ψ(0)|2 e−Γt.

(2.13)

(2.14)

To look at the process in the energy space, we take the Fourier transform of Eq. (2.13),

which yields

Squaring the transformed amplitude and normalizing the result, we get the probability for

i
√2π

φ(0)

(E − E0) + iΓ/2

.

(2.15)

(cid:101)ψ(E) =

the particle to undergo the transition as a function of energy,

p(E) =

1
2π

Γ

(E − E0)2 + (Γ/2)2 .

(2.16)

This is the familiar Breit–Wigner distribution,∗ with the full width at half maximum

∗The Breit–Wigner distribution is also knows as the Lorentzian or the Cauchy distribution.

13

(FWHM) equal to Γ. The more general relativistic Breit–Wigner distribution,

p(E) =

A

(cid:0)E2 − m2(cid:1)2 + m2Γ2

,

(2.17)

which is also valid for Γ (cid:38) m, can be derived by starting with the propagator for a massive
particle in relativistic QFT, whose denominator has the form p2 − m2 + imΓ.

A fundamental quantity characterizing a scattering process is an effective measure of

the area of overlap between the interacting objects. Differential scattering cross section, dσ,

represents a portion of this area that results in an outcome within a differential volume dΩ of

the final state phase space. The total scattering cross section, σ, is an integral of dσ/dΩ over

the phase space of interest. Naturally, cross section is a function of the impact parameters.

In classical mechanics, cross section has a deterministic relationship with the orbits of the

interacting bodies. In quantum mechanics, dσ/dΩ, essentially, contains a convolution with

the probability density of all possible outcomes. Cross section, rather than probability, is

typically calculated in particle physics as a quantity that directly relates universally appli-

cable theory to the outcome of a specific experiment. Given the impact parameters, cross

section is a universal quantity. However, luminosity, L, which is a measure of the beam

intensity, or number of particles in the beam passing through a unit of area in a unit of time,

is different for each experiment. The number, N, of interactions of interest one expects to

observe in an experiment running for some amount of time, T , is given by

(2.18)

N =(cid:90) T

0

σL dt.

14

Or, if the impact parameters are constant,

N = σ(cid:90) L.

(2.19)

There is also a relationship between scattering cross section and decay width. Eqs. (2.16)

and (2.17) describe equally well both decay and production of intermediate states. A process

of scattering through an intermediate state can be approximated as a production followed by

a decay. In the narrow width approximation, the cross section for producing a final state f

from an initial state i through an intermediate state r can be obtained by the Breit-Wigner

formula,

σi→r→f =

4π

ninf(cid:35)
k2(cid:34) nr

Γ2/4

(E − E0)2 + (Γ/2)2 Br→iBr→f ,

(2.20)

where k is the initial momentum in the center-of-mass frame, n are multiplicity factors due
to spin, color, etc., and Br→x are the branching ratios,∗ equal to the fraction of times a state
r would decay to a state x. The state r is often referred to as a resonance. In a scattering

experiment, by varying the center of mass energy, E, one can observe the number of events
undergoing the i → f transition rise sharply around E = E0, which can be interpreted as
the mass of the resonant state. In the absence of background, the histogram of observed

events would take the shape of the Breit-Wigner distribution.

Ideally, given the Lagrangian, one would solve for eigenvalues and eigenvectors of oper-

ators for the relevant observables. Unfortunately, there are no known examples of exactly

solvable interacting field theories in more than 2 spacetime dimensions [4]. Fortunately, in
many cases of interest, when couplings between fields are small (g (cid:28) 1), interactions can be

∗Equivalently, Eq. (2.20) can be formulated in terms of partial widths, with Γr→x = ΓBr→x.

15

treated using perturbation theory.

Given an initial state in the remote past (in) and a final state in the remote future (out),

the transition (or scattering) amplitude is the overlap between the states, which can be

expressed as

(2.21)

out

in = lim

(cid:10)f(cid:12)(cid:12)i(cid:11)

t→∞(cid:10)f(cid:12)(cid:12)e−iH(−t) · e−iHt(cid:12)(cid:12)i(cid:11) ≡(cid:10)f(cid:12)(cid:12)S(cid:12)(cid:12)i(cid:11),

where t is any common reference time, and S, called the scattering matrix or S-matrix, is a

unitary operator relating the initial and final states at the common time. For non-interacting

particles, S is simply the identity operator. The interaction part can be separated into the

T -matrix,

S = 1 + iT.

(2.22)

As discovered by Feynman, terms in the perturbative expansion of(cid:10)f(cid:12)(cid:12)iT(cid:12)(cid:12)i(cid:11) have intuitive

graphical representations. For example, Fig. 2.2 shows leading-order diagrams for fermion

scattering in QED. To make 4-momentum conservation explicit, we define the invariant
matrix element or amplitude, M,

(cid:10)f(cid:12)(cid:12)iT(cid:12)(cid:12)i(cid:11) = (2π)4δ(4)(cid:16)(cid:88) pi −(cid:88) pf(cid:17) · iM(i → f ).

(2.23)

Each diagram represents a term in the perturbative expansion of M for a given channel and
order in coupling.

Feynman diagrams are both a bookkeeping and a computational device. They allow a

simple visualization of scattering processes and an easy identification of allowed and forbid-

den channels. The allowed vertices are related to the structure of the Lagrangian, as they

correspond to the coupling terms. A Feynman diagram is in itself an expression of a contribu-

16

¯vs1(p1)

vs3(p3)

−ieγµ

−ieγν

−igµν/q2

us2(p2)

¯us4(p4)

s-channel

q2 = s = (p1 + p2)2

t-channel

q2 = t = (p1 − p3)2

u-channel

q2 = u = (p1 − p4)2

Figure 2.2: Feynman diagrams for e-e scattering in QED, with the spatial dimension drawn
vertically, and time horizontally. The three channels are named after the Mandelstam vari-
ables, s, t, and u, which, in the respective cases, are equal to the square of the four momentum
of the exchanged photon. s and t channels contribute to Bhabha (e+-e−) scattering, while
t and u channels contribute to Møller (e−-e−) scattering.

tion to the respective matrix element. The external legs (initial and final states) correspond

to free particles, the internal lines (virtual particles) are propagators (or Green’s functions)

correlating fields at different points in spacetime, and the vertices represent couplings be-

tween the interacting fields. The substitutions one makes to obtain amplitude expressions

from the diagrams are called Feynman rules.

The prescription for determining the cross section is this: draw all allowed diagrams for

the given process up to a sufficient order in coupling and compute the respective matrix
elements. The differential cross section for the initial state {i} = {a, b} and the final state
{f} is then given by∗

dσ =

1

2Ea2Eb |va − vb|(cid:89)f

d3pf
(2π)3

1

2Ef|M(i → f )|2 (2π)4δ(4)(cid:16)(cid:88) pi −(cid:88) pf(cid:17) , (2.24)

where |va − vb| is the relative velocity of the particles as viewed from the laboratory frame.
For identical combinations of initial and final states, one adds amplitudes together; for

∗The equation for differential width, dΓ, has the same structure, except with the prefactor replaced by
1/(2ma), corresponding to the single particle initial state. These expressions are derived in Section 4.5 of [4].
For identical masses in initial and final states, dσ/dΩ = |M|2 /(64π2s).

17

distinct combinations, one adds cross sections.

2.2.3 Gauge fields

One of the great successes of the SM is in the formulation of interactions between elemen-

tary particles from principles of symmetry. The recipe consists of requiring the Lagrangian

to be locally invariant under phase transformations of fermionic fields. This property of the

Lagrangian is called gauge invariance. The gauge transformations form continuous symme-

try groups, which are related to symmetries observed in the interactions of the respective

fermions. Corresponding to each generator of a group, there arises a vector field called the

gauge field. When the theory is quantized, the resulting gauge bosons are identified as the

mediators of particular interactions.

The gauge symmetry group of the SM can be written as

Electroweak

U (1)Y

(cid:122)
(cid:124) (cid:123)(cid:122) (cid:125)Weak

hypercharge

(cid:125)(cid:124)

× SU (2)L
isospin

(cid:123)
(cid:124) (cid:123)(cid:122) (cid:125)Weak

,

(2.25)

×

Strong

SU (3)c
Color

(cid:122) (cid:125)(cid:124) (cid:123)
(cid:124) (cid:123)(cid:122) (cid:125)

with the subscripts denoting the sector of each group. The SM is thus a special case of

Yang–Mills gauge theory. To see how the gauge fields appear in the theory,∗ let us denote

the generators of the gauge group as T a, a = 1, . . . , N. A transformation of a fermion field

ψ in the abstract gauge space has the form

δψ(x) = igT aδθa(x)ψ(x) or ψ(x) → G(x)ψ(x) = exp(cid:0)ig T · θ(cid:1)ψ(x),

(2.26)

∗A rigorous discussion of gauge theories can be found in Part III of [4].

18

where g is a coupling constant and θa(x) are spacetime dependent transformation phases.
The transformations G(x) form a Lie group, and the generators T a are elements of the
corresponding Lie algebra. For SU (n) groups there are N = n2 − 1 generators, which are
n × n matrices in the fundamental representation. Correspondingly, ψ are n × 1 vectors.
Transformations of the form (2.26) are rotations in this vector space.

The kinetic Lagrangian term for a fermionic field has the form

Lf = i ¯ψ /Dψ,

(2.27)

where ¯ψ = ψ†γ0, /D = γµDµ, γµ are Dirac matrices, and Dµ is the covariant derivative.
In order to maintain gauge invariance of the Lagrangian, the covariant derivative must be

defined as follows,

Dµ = ∂µ + igT aAa
µ,

(2.28)

where Aa
µ

are new vector fields.∗ The theory now accounts for interactions between Aa
µ

and

fermions through the covariant derivative, but to turn Aa
µ

into a gauge field, the appropriate

kinetic term must be added to the Lagrangian. By analogy with the EM field, we construct

a field strength tensor,

Fµν =

1

ig(cid:2)Dµ, Dν(cid:3) = ∂µAν − ∂νAµ + ig(cid:2)Aµ, Aν(cid:3).

Introducing the gauge space indices with Aµ = T aAa
µ

, we get,

F a
µν = ∂µAa

µAc
ν,

ν − ∂νAa

µ − gf abcAb

(2.29)

(2.30)

∗In the language of differential geometry, a gauge fields is a connection on a principal bundle [8]. The

choice of sign in Eq. (2.28) corresponds to the (+,−,−,−) Minkowski metric signature.

19

where f abc are structure constants of the Lie algebra, given by

(cid:2)T a, T b(cid:3) = if abcT c.

(2.31)

When acting on the gauge fields, T a are N × N matrices in the adjoint representation, with
matrix elements given by the structure constants,

(T a)bc = −if abc.

(2.32)

The term in Eq. (2.30) dependent on f abc appears for non-Abelian gauge groups, and pro-

duces self-coupling of gauge fields. Within the SM, it is present for weak and strong inter-

actions, but not for EM.

The Lagrangian containing both the fermionic and the gauge field can now be written as

L = −

1
4

F a
µνF aµν + i ¯ψ /Dψ.

(2.33)

As can be shown, Lagrangian mass terms for gauge fields of the form 1

2m2AµAµ are not
invariant under local gauge transformations. In order to complete the theory and give mass

to the weak bosons, a scalar Higgs field is introduced, as discussed in Section 2.3.1.

2.3 Electroweak sector

The electroweak (EW) sector of the SM is a gauge theory unifying electromagnetic and

weak interactions. The direct product of two symmetry groups, U (1)Y × SU (2)L
the gauge group to produce the interactions of fermions via exchange of the respective gauge

, provides

20

bosons. Consequently, two quantum numbers, corresponding to the charges conserved by

the gauge symmetry, are assigned to each fermion: weak hypercharge, Y , and weak isospin,

T 3. The EW quantum numbers of SM particles are listed in Table 2.1. To reproduce the
experimental observation that the weak interaction involves only left-handed fermions,∗ so

does the gauge transformation under SU (2)L
the SU (2) Lie group is given by Pauli matrices, and by convention T i = σi/2. Likewise,
there are commutation relations,

, hence the subscript. A set of generators for

(cid:2)T a, T b(cid:3) = i
abcT c,

(cid:2)T a, Y(cid:3) = 0 ∀ a.

(2.34)

Similarly to components of angular momentum or spin, the three components of T are not

simultaneously diagonalizable, and T 3 is chosen as the primary component. In the abstract
isospin space, since the isospin operators are represented by 2 × 2 matrices, left-handed
fermions are 2× 1 vectors, or doublets. The right-handed fermions are singlets, as they don’t
interact weakly.

Right-handed, rather than left-handed, anti-fermions couple weakly. In fact, the weak

interaction famously maximally violates parity and charge conjugation symmetries. This

can be readily seen, because P turns left-handed fermions into right-handed fermions, and C

turns left-handed fermions into still left-handed anti-fermions.

∗Handedness here refers to chirality, which is a Lorentz-invariant concept that applies only to fermions,

ψR(cid:19),
which are not symmetric under parity transformation. In the Weyl basis, spinors can be written as(cid:18)ψL
with the components obtained by applying respective chiral projection operators, PL = 1
2 (1 − γ5) and
2 (1 + γ5). Chirality is not to be confused with helicity, which is the projection of spin onto the
PR = 1
momentum direction. For massless fermions, chirality is the same as the sign of helicity, but for massive
ones, there always exists a reference frame in which helicity has the opposite sign.

21

Table 2.1: Electroweak quantum numbers.

Particles

(cid:18)t
b(cid:19)L
(cid:18)c
s(cid:19)L
Quarks (cid:18)u
d(cid:19)L
µ(cid:19)L (cid:18)ντ
e(cid:19)L (cid:18)νµ
Leptons (cid:18)νe
τ(cid:19)L

uR
dR

cR
sR

tR
bR

τR

eR

µR

Bosons

Photon, γ
Weak, W±
Weak, Z0
Higgs, H

2.3.1 Higgs mechanism

Y

T 3

Q

1/2
−1/2
0
0

1/3
1/3
4/3
−2/3
−1
−1
−2
0
0
±1
0
0
0
1 −1/2

1/2
−1/2
0

2/3
−1/3
2/3
−1/3
0
−1
−1
0
±1
0
0

As discussed in Section 2.2.3, gauge fields are necessarily massless. However, experiments

definitively show that W and Z bosons are massive. An elegant idea of spontaneous sym-
metry breaking allows to reconcile these two facts.∗ The trick is to add a field that is scalar

under Lorentz transformations, but has the same symmetry as the gauge field, and therefore

caries the respective charge. The terms the new field brings into the Lagrangian can then be

rearranged to form the sought-after longitudinal components of the gauge fields, thus allow-

ing them to have mass without explicitly breaking the gauge symmetry of the Lagrangian.

The Lagrangian for the scalar field has the form

L = (DµΦ)†(DµΦ) − V (Φ),

V (Φ) = µ2Φ†Φ + λ(Φ†Φ)2.

(2.35)

It is simpler to illustrate the mechanism by first looking at an example with only 2 degrees

∗The theory of spontaneous electroweak symmetry breaking was separately developed in 1964 by three

independent groups [9–11], and is based on Landau–Ginzburg theory of superconductivity [12].

22

of freedom.∗ Consider a U (1) gauge field, Aµ, with the Lagrangian

L = −

1
4

FµνF µν, Fµν = ∂µAν − ∂νAµ.

(2.36)

Naively adding a mass term, 1

2m2AµAµ, would break the gauge invariance of the Lagrangian.

Instead, add a charged complex scalar field,†

φ,

L = −

1
4

FµνF µν + (Dµφ)†(Dµφ) − µ2φ†φ − λ(φ†φ)2.

(2.37)

The scalar field couples to itself via the quartic term and to the gauge field via the covariant
derivative, Dµ = ∂µ−igAµ. The Lagrangian is invariant under the following transformations:

Aµ(x) → Aµ(x) − ∂µα(x),
φ(x) → φ(x)e−iα(x).

(2.38)

If µ2 ≥ 0, the state of minimum energy is φ = 0, and the potential preserves the
symmetries of the Lagrangian. In this case, the theory is simply that of the two interacting

fields. However, if µ2 < 0, the minimum energy state is no longer φ = 0. The U (1) symmetry

will be spontaneously broken, and the scalar field will acquire a non-zero vacuum expectation

value, v,

(cid:10)φ(cid:11) =(cid:114) µ2

2λ

=

v
√2

.

(2.39)

The potential, V (φ), in this case is colloquially referred to as the Mexican hat potential.

The two scenarios are illustrated in Fig. 2.3.

∗At least N = 2 degree of freedom are required to have a continuous symmetry. N = 1 only allows for a

discrete reflective symmetry. N > 2 is mathematically similar to N = 1, but is harder to visualize.

†A complex field has a U (1) symmetry under a phase transformation that rotates one component into

the other.

23

V (φ)

V (φ)

Im(φ)

Re(φ)

Im(φ)

(a) Unbroken vacuum, µ2 ≥ 0, v = 0.
Figure 2.3: Illustration of spontaneous U (1) symmetry breaking due to non-zero vacuum
expectation value of a complex scalar field, φ.

(b) Broken vacuum, µ2 < 0, v (cid:54)= 0.

Re(φ)

We can parametrize φ around v in polar coordinates as,

φ =

v + h
√2

eiχ/v.

(2.40)

Writing the Lagrangian in terms of the real fields h and χ we get,

L = −

1
4

FµνF µν +

g2v2

2

AµAµ−gvAµ∂µχ +

+ (interaction terms),

1

2(cid:16)∂µh∂µh − 2µ2h2(cid:17) +

1
2

∂µχ∂µχ

(2.41)

which describes a theory with a photon of mass mA = gv, a Higgs boson, h, with mass
mh = √2µ = √2λv, and a massless Goldstone boson, χ. The gvAµ∂µχ term can be
removed by making a global gauge transformation,

Aµ → Bµ = Aµ −

1
gv

∂µχ.

(2.42)

24

This is called the unitary gauge. The Goldstone boson then completely disappears from the

theory and one says that it has been eaten by the now massive photon field, Bµ.

In the SM, the additional scalar field, Φ, has to respect both U (1) weak hypercharge and

SU (2) weak isospin symmetries. Therefore, Φ is a complex weak isospin doublet,

Φ =

φ+

φ0

 with

φ+ = (φ1 + iφ2)/√2,
φ0 = (φ3 + iφ4)/√2.

(2.43)

Spontaneous symmetry breaking produces three Goldstone bosons, which are eaten by W±
and Z, and a Higgs boson with hypercharge Y = 1 and isospin T 3 = − 1
, which is therefore
electrically neutral, since Q = T 3 + 1

2Y . Going through a similar procedure to the above,

2

and applying the unitary gauge transformation to remove the Goldstone bosons, we get

Φ(x) =

1

√2

0

v + H(x)

 .

(2.44)

2.3.2 After symmetry breaking

Following the procedure outlined in Section 2.2.3, we obtain one gauge boson, B with
gauge coupling g(cid:48), from the hypercharge symmetry and three gauge bosons, W a, a = 1, 2, 3
with gauge coupling g, from the isospin symmetry, and the covariant derivative can be written

as

Dµ = ∂µ − ig(cid:48) Y

2

Bµ − igT aW a
µ .

(2.45)

25

Expanding the kinetic term for Φ and omitting the factors of Y
2

and T a for brevity, we get

(DµΦ)†(DµΦ) =

+

+

(∂µH)(∂µH)

1
2
1
g2(v + H)2(W 1
2
1
2

(v + H)2(gW 3

µ − iW 2
µ − g(cid:48)Bµ)2.

µ)(W 1µ + iW 2µ)

(2.46)

From here, we define the physical weak bosons as linear combinations of the original

gauge bosons as

W±µ =

W 1
µ ∓ iW 2
µ√2

,

(gW 3

g

µ − g(cid:48)Bµ) =(cid:113)g2 + g(cid:48)2(cid:32)
(cid:112)g2 + g(cid:48)2
=(cid:113)g2 + g(cid:48)2(cid:16)cWW 3
=(cid:113)g2 + g(cid:48)2Zµ,

W 3

µ −

µ − sWBµ(cid:17)

(2.47)

(2.48)

Bµ(cid:33)

g(cid:48)

(cid:112)g2 + g(cid:48)2

where sW = sin θW
bosons’ masses are given by

, cW = cos θW

, and θW

is the Weinberg weak mixing angle. The weak

MW =

gv
2

,

MZ =

v

2(cid:113)g2 + g(cid:48)2 =

MW
cos θW

.

The state orthogonal to Z,

Aµ =(cid:16)sWW 3

µ + cWBµ(cid:17) ,

(2.49)

(2.50)

does not couple to the Higgs boson,∗ and so does not acquire mass through the Higgs

∗The photon field does couple to the charged component of the Higgs field, but the latter is absent in

the unitary gauge.

26

mechanism. This is in fact the EM photon.

Rewriting the covariant derivative in terms of the physical fields,

Dµ = ∂µ − i

g

√2(cid:18)W +

µ T + + W−µ T−(cid:19) − iZµ(cid:18)gcWT 3 − g(cid:48)sW

Y

2(cid:19) − iAµ(cid:18)gsWT 3 + g(cid:48)cW

Y

2(cid:19),

(2.51)

we immediately get the respective couplings. We can now define the EM coupling, e, and

the electric charge, Q, by examining the photon term,

(cid:18)gsWT 3 + g(cid:48)cW

Y

2(cid:19) =

gg(cid:48)

(cid:112)g2 + g(cid:48)2(cid:18)T 3 +

Y

2(cid:19) = eQ,

(2.52)

which produces the Gell-Mann–Nishijima formula, Q = T 3 + Y /2.

The B-W 3 mixing, in combination with the unitary gauge, allows to explicitly recover

as an unbroken symmetry. This can be verified by applying a local U (1) gauge trans-

U (1)EM
formation to a charged fermionic field, with the phase defined as the appropriate combination

of the EW phases.

Three parameters can be identified in the EW theory as fundamental:∗

• Fine structure constant (EM coupling), α =
• Fermi constant (related to weak coupling), GF =
• Weinberg mixing angle, sin2 θW ≈ 0.23.

e2
4π ≈ 1/137,
g2
M 2
W

√2
8

=

1
√2

v−2, v ≈ 246 GeV,

2.3.3 Fermions’ masses

In a chirally symmetric theory, fermions can be given mass simply via a Dirac mass

term, m ¯ψψ, which, in Weyl basis, separating the left- and right-handed components, can be

∗Other choices of fundamental EW parameters are possible, e.g. masses of the gauge bosons. Beyond

the LO other choices of input model parameters are possible, which are not fully equivalent.

27

written as

m ¯ψψ = m( ¯ψL + ¯ψR)(ψL + ψR) = m( ¯ψLψR + ¯ψRψL).

(2.53)

Since the isospin symmetry only affects the left-handed fermions, ψL
differently under SU (2)L
SM. To fix this problem, the Higgs field can also be attributed responsibility for masses of

, and the bare mass terms, m ¯ψψ, are not gauge invariant in the

and ψR

transform

fermions by adding gauge invariant Yukawa coupling terms to the Lagrangian,∗

L = −y ¯ψRΦψL + h.c.

After spontaneous symmetry breaking, using Eq. (2.44), we get

L = −

yv
√2

¯ψRψL −

y
√2

¯ψRHψL + h.c.,

(2.54)

(2.55)

with the first term being the mass term with m = yv/√2, and the second term coupling
It should be noted, that because the
fermions to the Higgs field proportionally to m/v.

Yukawa couplings are arbitrary for all fermions, they do not predict fermion masses, but

rather provide a mechanism to accommodate their existence within the SM.

For simplicity, the above equations were written for a single fermionic field, ψ. When

more fields are introduced, nothing prohibits addition of mixed terms of the form

yab√2

−

¯ψa
R(v + H)ψb

L + h.c.,

(2.56)

∗Mathematically, Yukawa couplings to fermions are not necessary for the EW symmetry breaking. How-
ever, attributing both roles to the same field yields a simpler model, and no evidence against it has been
observed at the LHC.

28

where a and b are two species of fermions. In general, then, the fermion mass matrix,

Mab = −

yabv
√2

,

(2.57)

is not diagonal, and the mass eigenstates may be distinct from the flavor eigenstates.

The structure of the Lagrangian and the fact that we need to preserve U (1)EM

as an

unbroken symmetry impose constraints on Mab
ticular, at tree level, only charged currents admit change of fermion flavor, while there are no

being non-diagonal for interactions. In par-

fundamental flavor-changing neutral currents,∗ as show in Fig. 2.4. The lack of LO FCNC in

the SM is explained by cancellations between Yukawa coupling terms and parts of fermionic

kinetic terms involving the gauge fields from the covariant derivative, Eq. (2.45). This can-

cellation does not occur for charged currents, which only couple to left-handed fermions. The

neutral currents do couple to the right-handed fermions because of mixing between B and

W 3. FCNC do occur in the SM, but only at NLO and above. Famously, such interactions

take place in processes depicted by the so called penguin diagrams, which are important for

ψa

ψa

W

ψb

W

W

Z

ψb

(a) LO interaction. a (cid:54)= b is allowed
for W , but not for Z or γ.

(b) NLO interaction. Same rules apply at the ver-
tices, but the emitted boson can now be Z or γ
with a (cid:54)= b.

Figure 2.4: Flavor-changing interactions. a and b denote fermion generations. The vector
bosons are labeled for the case that allows a (cid:54)= b.

∗The observed lack of FCNC supports the assumption that the Higgs is also responsible for fermions’

masses.

29

understanding of CP violation in the SM [13].

The matrix describing quark flavor mixing in charged weak interactions is known as

the Cabibbo–Kobayashi–Maskawa (CKM) matrix.

In its standard parametrization,∗ it is

expressed as

VCKM =



1

0

0

0

c23

s23

0 −s23

c23





c13

0

−s13e−δ13 0

0 s13e−δ13

1

0

c13





c12

−s12
0

s12 0

c12 0

0

1



.

(2.58)

The measured values of the parameters are: θ12 = 0.2276, θ13 = 0.0035, θ23 = 0.0415, and
δ13 = 1.20. θ12 is the Cabibbo angle. δ13 is the phase responsible for CP violation in flavor
changing interactions.† The magnitudes of the CKM matrix elements [14] are



|Vud|
|Vcd|
|Vtd|

|Vus|
|Vcs|
|Vts|

|Vub|
|Vcb|
|Vtb|



=



0.97446(10) 0.22452(44)

0.00365(12)

0.22438(44) 0.97359(11)

0.04214(76)

.

(2.59)

0.00896(24) 0.04133(74) 0.999105(32)



As can be seen, the matrix is nearly diagonal, so the effects of quark flavor mixing are small,

but observable.

Technically, neutrino masses can be added to the SM using exactly the same mechanism

of Yukawa couplings to the Higgs field. However, that would require addition of undetectable

right-handed neutrinos to the model.‡ The PMNS matrix for neutrino mixing has the same

∗The CKM matrix lacks components for quark pairs with the same weak isospin, such as uc, because of

the absence of FCNC.

†Because VCKM represents a rotation in a 3-dimensional (flavor) space, it can be shown that its irre-
ducible representation requires only 4 parameters: 3 angles and a phase. Extra phases can be absorbed into
definitions of the fermionic fields without loss of generality.

‡Another popular model for neutrino masses is the seesaw mechanism.

30

form of parametrization, but describes mixing of the three neutrino flavors among themselves

rather than flavor changing transitions between isospin doublets. Another difference, is that

flavor mixing in the quark sector is only relevant for interactions. Quark oscillations are

unobservable, because the time scale for hadronization is Λ−1
QCD
scale of about 1 fm. The situation with neutrinos is exactly the reverse. Because they are

, corresponding to a length

not charged, they can change flavor neither through charged nor neutral currents. There

is no ννW vertex, as it would not conserve charge, and neither ννZ nor ν(cid:96)W interactions

afford flavor changing. However, because they only interact weakly, neutrinos can travel

astronomical distances, over which, the effect of their propagation as mass eigenstates but

interaction as flavor eigenstates becomes apparent. Consequently, neutrinos can change

flavor only via projection from the flavor state basis to the mass basis, and back. There

appears to be no direct flavor mixing among charged leptons.

2.4 Chromodynamics

Quantum Chromodynamics (QCD) is the theory of interactions between color-charged

quarks and gluons, describing the zoo of hadrons and the strong interaction between them.∗

Gluons enter the theory as a non-Abelian gauge field, with the symmetry group SU (3)c,
which stands separately from the electroweak sector and is not broken by the Higgs mecha-

nism, thus leaving gluons massless. The gauge group reflects the color symmetry of quarks,

which are color triplets (cid:22) with each quark labelled red, green, or blue, and antiquarks la-

belled with anti-color. Gluons are color octets, each carrying a color and an anti-color in

8 possible combinations. Bound states of quarks (hadrons) are always color-neutral, and

∗A comprehensive contemporary exposition of QCD can be found in [15].

31

)
R
µ
(
s
α

[GeV]

µR

Figure 2.5: The strong coupling constant, αs, as a function of renormalization scale, µR
[14].
The solid black lines show theoretically calculated values and their uncertainty. The data
points show experimentally obtained values with the corresponding error bars.

can consist of either quark-antiquark pairs of the same color or anti-color (mesons), or three

quarks of different colors (baryons).

In fact, only hadrons can be observed directly, but never free quarks, due to the peculiar
running∗ of the strongs coupling αs, shown in Fig. 2.5. In contrast to the running of the EM
α, the strong coupling, αs, becomes smaller at higher energies (cid:22) the effect called asymptotic
freedom (cid:22) whereby at short distances and high energies αs is sufficiently small to apply per-
turbation theory, while at long distances and low energies the interaction becomes extremely

strong, confining quarks, and precluding perturbative calculations. The energy scale at which
QCD becomes non-perturbative is ΛQCD ≈ 220 MeV. It is often used in calculations in place
of the dimensionless coupling constant g. This behavior complicates interpretation of events

at hadron colliders, but the challenge is overcome by empirical models for non-perturbative

effects.

∗Running of couplings is a QFT effect, present because of renormalization of couplings, which makes
. The effect can be thought of as arising due to different

them dependent on the renormalization scale, µR
resolution of vacuum polarization effects, i.e. charge screening, at different energy scales.

32

QCD αs(Mz) = 0.1181 ± 0.0011pp –> jetse.w. precision fits (NNNLO)  0.10.20.3110100Heavy Quarkonia (NLO)e+e–   jets & shapes (res. NNLO)DIS jets (NLO)April 2016τ decays (N3LO)1000 (NLOpp –> tt(NNLO))(–)In general, a QCD process is a combination of short- and long-distance behavior, and as

such is not computable directly in perturbation theory [16]. This problem is addressed by the

factorization theorem, which makes it possible to decouple non-perturbative processes inside

hadrons from the hard scattering process that involves constituent partons and is tractable

within perturbation theory. The factorization theorem allows one to write the hadronic cross

section as a convolution between the partonic cross section and parton distribution functions

(PDF) over the fraction, x, of the hadron’s momentum carried by a collinear parton. For

example, for a hadron-hadron collision,

σ =(cid:88)ab (cid:90) 1

0

dxa(cid:90) 1

0

dxb fa/A(xa, µF) fb/B(xb, µF) σab,

(2.60)

where the sum is taken over all the possible parton combinations that allow for the inclusive
final state of interest, and µF

is an introduced parameter called factorization scale.

A

fa/A(xa, µF)

fb/B (xb, µF)

B

a

b

Hard
process

Figure 2.6: Diagramatic representation of the factorization theorem in QCD. Two colliding
hadrons (A and B) each contribute a parton (a and b) with probabilities fa/A
and fb/B
dependent on the momentum fraction, x, and factorization scale, µF

.

The non-perturbative PDF parameters cannot be calculated analytically in QCD. These

parameters are determined empirically, by fitting models to deep inelastic scattering and

hadron collider data. However, recent advances suggest that it might be possible to predict

PDFs using lattice QCD [17–19]. On the other hand, the change of parameters with the

33

interaction scale can be determined perturbatively. The scale evolution of PDFs is modeled

using DGLAP equations [20–22]. Fig. 2.7 shows examples of proton PDFs. As is expected,

large momentum fractions are predominantly carried by the valence quarks, with, in the

proton case, the probability of finding a u quark at a given value of x larger than that of

a d quark. At low momentum fractions, the gluon PDF by far exceeds those of quarks,

though the contributions of sea quarks are non-negligible, rising with the energy scale, and

are important for obtaining p-p cross sections. In principle, one can also define photon [23]

and lepton [24] PDFs through EW corrections to the DGLAP equations.

)
F
µ

,

x
(
f
x

)
F
µ

,

x
(
f
x

x

x

Figure 2.7: CT14nlo proton PDFs at different scales [25].

The situation on the other side of the hard scattering process is reversed, where color-

charged quarks and gluons in the final state recombine into color-neutral objects. This

process is called hadronization. The process is far from trivial, as non-perturbative effects

dominate in this long-distance regime. The result is that, instead of individual partons,

detectors see tight cones of many particles, called jets. To produce detector-level predictions,

Monte Carlo simulations use various heuristic models of hadronization. These are discussed

in Section 3.1. Often, collision events involve multiple jets. Jet clustering algorithms are

used to systematically group particles into jets, which can then be identified with partons

produced in the hard process. These algorithms are discussed in Section 5.3.

34

0.0010.010.1100.20.40.60.81CT14nloF=5 GeVg/10uddus,sc,cb,b0.0010.010.1100.20.40.60.81CT14nloF=100 GeVg/10udduc,cb,b2.5 Higgs phenomenology

From the discussion of Section 2.3, it is clear that the Higgs holds a central role in the

SM, like a load-bearing wall, supporting masses of heavy gauge bosons via non-zero vacuum

expectation value and masses of fermions (as well as flavor mixing) via Yukawa couplings.

The Higgs field provides the means to maintain gauge symmetries of the Lagrangian by

balancing effects otherwise incompatible with them. After the successful discovery of the

Higgs boson, the experimental agenda is to fully measure its properties and to thoroughly

explore the phase space of reactions involving it. In the best case scenario, we may observe

deviations from the SM, and obtain clues as to how to advance the theory. In the worst case,

we will have confirmed the SM experimentally, as we have up to now, reduced the uncertainty

on the SM parameters, and reduced the phase space for potential physics beyond the SM.

The SM Higgs boson directly couples to all massive elementary particles, with the cou-

pling proportional to the mass for fermions, and mass-squared for weak bosons and self-

coupling. However, an effective coupling to massless particles is also possible through loop

diagrams. In fact, at the LHC, gluon-gluon fusion is the dominant Higgs production channel,

due to a combination of large PDFs for gluons and small PDFs for heavy quarks. Fig. 2.8

shows a comparison of inclusive cross sections for Higgs production in pp collisions due to

different processes. Archetypal examples of diagrams for the production mechanisms are

shown in Fig. 2.9. While the fractions of contribution from different channels vary for dif-

ferent event selection criteria, ggF generally dominates, with VBF attaining a substantial

fraction only among Higgs + 2 jets events with high-pT Higgs or large angular separation
between the jets.

The discovered Higgs with mH = 125 GeV has an incredibly narrow width of only

35

Figure 2.8: The SM Higgs (mH = 125 GeV) inclusive production cross sections as functions
of √s for different channels in pp collisions [26, 27]. The pp → H process, which is only
possible through gluon-gluon fusion, clearly dominates.

(a) Gluon-gluon Fusion (ggF).

(b) Vector Boson Fusion (VBF).

(c) Associated production with
weak bosons (WH, ZH).

(d) Associated production with heavy quarks (bbH, ttH).

Figure 2.9: Examples of leading-order Feynman diagrams for Higgs production mechanisms.

36

 [TeV]s789102030405060210 H+X) [pb]→(pp σ(cid:173)110110210310LHC HIGGS XS WG 2013 H (NNLO+NNLL QCD + NLO EW)→pp H (NNLO QCD)q q→pp  WH (NNLO QCD)→pp  ZH (NNLO QCD)→pp H (NLO QCD)t t→pp  = 125 GeVHMMSTW20084 MeV.∗ The electrically neutral, infinitesimally short-lived Higgs boson can only be studied

through its decay products and other final state particles involved in its production pro-

cesses. Fig. 2.10 shows the Higgs branching ratios as functions of the Higgs mass. What

these plots tell us concerning experiment, is that on the one hand, we have an exciting Higgs

with many decay channels; on the other hand, most of the largest decay channels are not

easily tractable experimentally, due to large backgrounds from other SM processes and hard-

to-identify final states. In fact, the golden discovery channels for both ATLAS and CMS,
H → γγ and H → ZZ∗ → 4(cid:96), each have horribly low branching ratios of only about 10−3.
However, efforts are now being put forward to harness the b¯b channel, which suffers greatly
from both background and selection problems, but offers potential sensitivity in the high-pT
region [29].

A detailed account of the phenomenology of the Higgs boson at the LHC can be found

in [30].

Figure 2.10: Higgs branching ratios and their theoretical uncertainties for the low mass range
(left) and an extended mass range (right) [28].

∗This is the theoretical width. The experimental width is much larger due to detector effects.

37

[GeV]HM80100120140160180200Higgs BR + Total Uncert [%]-410-310-210-1101LHC HIGGS XS WG 2013bbττµµccggγγγZWWZZ[GeV]HM902003004001000Higgs BR + Total Uncert [%]-410-310-210-1101LHC HIGGS XS WG 2013bbττµµccttggγγγZWWZZChapter 3

Theoretical predictions and event

simulation

Computer programs for simulating particle collision events are indispensable tools for an-

alyzing LHC data. Simulations provide the means for interpreting the results and comparing

them with theoretical predictions. Datasets of generated events allow preparatory studies

to be conducted to ascertain the feasibility of new analyses, and to optimize the parameters

of the ongoing ones. Such optimizations are carried out for selection cuts on observables

characterizing the events, for methodology and parameters of statistical techniques (typi-

cally regression models in the form of function fits) used to quantify signal and background,

and for subdivision (or binning) of the experimental datasets. Often, simulations are also

necessary to derive certain quantities that are required for an analysis, but cannot be ob-

tained directly using data-driven techniques or extrapolated from previous measurements.

A typical example is the expected fraction of events due to the background processes. Event

simulations are also used to assess detector effects, as generated events can be compared

before and after application of the detector simulation and reconstruction algorithms. By

this comparison, analysis results can be corrected for the detector effects via unfolding.

The focus of this chapter is on the use of contemporary event generators, their principles

of operation, and the relevant theoretical issues. Section 3.6, at the end of this chapter,

38

contains a discussion of the event samples generated for the H → γγ cross section analysis.
Details of the studies conducted in the course of the analysis that relied on the generated

event samples can be found in Chapter 6. Additional studies done by the author using

GoSam ntuples are presented in Appendices A and B.

3.1 Monte Carlo event generation

Event simulations for hadron colliders can be divided into two categories: the calculation

of the hard scattering component of an event, and simulation of the complete proton-proton

collision, including the hard scattering. The former are calculations that typically produce

the result of a hard interaction at the parton level, for example the production of a Higgs

boson and 1, 2, or 3 partons. The latter simulate the entire event and produce final state

particles at the hadron level, easier to compare to experimental measurements.

In this

case, the final state analogous to the above example would consist of a Higgs boson and

a set of hadronic jets composed of clusters of particles that develop from the showering

and hadronization of the partons produced in the hard process.

In addition to the hard

component, the soft component, originating from the underlying event, can also be produced.

The information about the hard scattering component can be passed from the hard scattering

calculation to the full event simulation program. Both types of calculations can be carried

out using Monte Carlo methods. Global event simulation programs, such as Sherpa or

Pythia, are used in basically every LHC analysis. Theoretical hard scattering calculations,

such as GoSam, are typically not directly used by experimenters, due to their complexity,

but were used in this analysis and supplementary studies in the form of ntuples.

Event generators can employ the MC method at different steps in the simulation and

39

for different reasons, but the basic principle is always the same. One of the reasons is

that interactions in quantum mechanics are intrinsically probabilistic. Even for precisely

stated parameters of the initial configuration, the set of possible final states typically has

more than one element with predicted probabilities. Many aspects of the simulations rely

on probabilistic models, such as parton distribution functions, splittings in parton showers,

or interactions with matter in detector simulations. Another MC application is its use as

a method of numerical integration [31].

Integrations required to marginalize probability

distributions over certain dimensions of the phase space are often too difficult to perform

analytically, either because of the large dimensionality of the phase space,∗ or because of the

presence of experimental cuts, which impose non-trivial boundary conditions. Clustering of

hadronic jets is an example of what creates a non-trivial integration problem for theoretical

calculations. From the experimental perspective, jet clustering is necessary to match the

detected energy deposits to the underlying hard partons; from the theoretical perspective,

it provides means to handle infrared and collinear divergences.

One of the challenges of MC simulations is that the probability density function (p.d.f.)†

over the phase space cannot in general be sampled directly. The most straightforward way

to approach this problem is with the acceptance-rejection method.

In its simplest form,

the prescription is to use a uniform distribution to randomly choose a phase space point,

x, together with an additional uniform random number, u, between zero and the expected

maximum of the p.d.f. The p.d.f. at the selected phase space point, p(x), is then evaluated.

If u < p(x), the event at x is simulated; otherwise, the phase space point is discarded. This

∗This is often referred to as the “curse of dimensionality” because of how fast the number of parameters
necessary to map out the phase space, and the number of Feynman diagrams relevant to the calculation,
grow with the complexity of the process (i.e. the number of legs) and with the order of the perturbative
expansion.

†To avoid confusion between a probability density function and a parton distribution function, they are

abbreviated as p.d.f. and PDF respectively.

40

Sampling distribution function

)
x
(
p

,
y
t
i
l
i

b
a
b
o
r
P

Accepted
events

)
x
(
p

,
y
t
i
l
i

b
a
b
o
r
P

Accepted
events

Sampling distribution
function

Phase space, x

Phase space, x

(a) Uniform sampling produces

unweighted events.

(b) Non-uniform sampling produces

weighted events.

Figure 3.1: Sketches of Monte Carlo sampling approaches for the acceptance-rejection
method. The event probability density (blue), proportional to the fully differential cross
section, is sampled by generating random numbers with the sampling probability density
(orange).

approach is illustrated in Fig. 3.1a. However, often, there exist regions of phase space with

much lower event probability. The uniform sampling approach results in poor sampling of

such regions. In other words, fractional uncertainty (e.g. estimated with Poisson statistics),

in densely populated regions would reach the required precision a lot faster than in the

sparsely populated ones. For example, jet kinematics typically yields exponentially falling

distributions of jets’ transverse momenta. One often needs to explore their distributions on

a wide pT range, over which the cross section falls by several orders of magnitude. In this
case, uniform sampling can result in oversampling of the lower end of the pT spectrum before
reaching the required uncertainty in the upper tail of the distribution.

The oversampling is problematic, because evaluation of each phase space point is a non-

negligible operation, so a large fraction of the generator runtime can be spent on computa-

tions that don’t add the most useful information. The problem can be solved by running

multiple simulations, each covering a different region of phase space with a sufficiently flat

p.d.f. However, depending on the techniques used in the simulation, this approach may

41

not correctly account for correlations. An alternative approach, that is used in many MC

programs, is to instead define a non-uniform sampling distribution for the phase space, that

provides a simplified envelope for the p.d.f., such that it can be sampled directly, i.e. by

inverting the c.d.f.∗ In this case, the random number u is still generated uniformly, but its
maximum value is now proportional to the sampling distribution. This approach is illus-

trated in Fig. 3.1b. By sampling in a way that reflects the varying amplitude of the p.d.f.,

fractional uncertainty of the observables can converge more uniformly in a different region

of the phase space. This may dramatically reduce the total number of events that have to

be generated to achieve the required precision. The price one pays for the improved MC

efficiency is that the produced events have to be weighted in proportion to the sampling

function.

At hadron collider experiments, such as the LHC, strong interactions dominate over elec-

troweak for most processes. The picture of event evolution is thus painted from the QCD

perspective, with electroweak processes incorporated mostly as corrections. Generators typi-

cally model event evolution as a sequence of processes, relying on factorization of phenomena

that take place at different energy scales. These aspects of event simulation, namely the cen-

tral role of QCD interactions, their sequential evolution, and factorization, can be seen in

Fig. 3.2, which schematically illustrates how hadron-hadron collisions are simulated by typ-

ical MC generators. The first step in the simulation is the application of the factorization

theorem, which determines what partons (quarks or gluons) with what fractions of the ini-

tial hadron momenta will participate in the hard process. The hard process is the main

interaction of interest in the event, usually associated with the highest energy scale. It is

∗The sampling distribution function need not have special properties. It can be discontinuous as long
as it covers all the points of the phase space to be sampled. A series of step functions would yield identical
results to running multiple simulations, ignoring correlations.

42

Hard collision
Hadronization
Hadronic decays
Beam remnants

Parton shower
Initial state radiation
Underlying event
Soft EM radiation

Figure 3.2: Sketch of a hadron-hadron collision as simulated by a Monte Carlo event gener-
ator program [32].

described by a matrix element (ME) calculation, typically done to a fixed order in perturba-

tion theory,∗ which can be expressed as a sum over the Feynman diagrams relevant for the

scattering of the two partons. Most initial and final states involve color-charged particles,†

that can radiate virtual gluons, which in their turn can split into further gluons or quark-

antiquark pairs, forming parton showers (PS) [33]. This pattern of radiation continues until

the interaction scale falls and the strong coupling rises sufficiently for the partons to form

bound states in the process of hadronization. Many of the formed hadrons are short lived

∗Theoretical predictions calculated up to certain terms in a fixed order perturbative expansion are typ-
ically referred to as leading order (LO), next-to-leading order (NLO), NNLO, N3LO, etc. By contrast,
calculations involving analytic resummation count orders in terms of logarithms and are labeled leading log
(LL), next-to-leading log (NLL), NNLL, etc. These can be matched to fixed order calculations.

†A very small fraction of events at a hadron collider involves only color-neutral particles. For example,
the hard interaction may be initiated by photons, or the initial color charge may be completely annihilated,
in processes such as Higgs boson production with no associated jets.

43

and undergo subsequent decays. Further corrections may be applied to events, for example

to include electroweak interactions that produce EM radiation.∗

While the ME accounts for the interaction at the hard scale, the PS describes the subse-
quent event evolution at lower scales. The branching processes q → qg, g → gg, and g → q ¯q
are modeled by the DGLAP equations, with the branching probability given by Sudakov

form factors [36]. A shower starts at the hard scale and continues until the scale gradually

decreases to several GeV,† which is sufficiently low for formation of color-neutral hadrons. A

showering process may be initiated by partons from either the initial or final state of the hard

process, resulting in initial (ISR) and final (FSR) state radiation. These are implemented by

tracing the splittings forward in time for the FSR, but backward for the ISR, resulting in the

FSR being space-like (positive virtuality), and the ISR being time-like (negative virtuality).‡

The PS model has to maintain infrared and collinear (IRC) safety, ensuring that di-

vergences caused by higher order virtual diagrams are canceled by unresolved real infrared

emissions. Gluons emitted in a PS are predominantly, though not necessarily, soft and

collinear due to the splitting functions. Consequently, branching typically occurs at small

angles, with emitted partons carrying only a small fraction of the original momentum. A

typical infrared cutoff scale of a parton shower is about 1 GeV. PS models are described

in resummation formalism and, therefore, have logarithmic accuracy, not corresponding to

a fixed order.∗∗ The PS models usually have LL accuracy, although recent advances have

∗An overview of event simulation problems and contemporary generator programs can be found in sec-
tion 4.2 of [34]. Detailed theoretical discussions of the problems of MC event generators and the typical
approaches to solve them can be found in [32, 35].

†The hadronization scale is about an order of magnitude higher than the ΛQCD ≈ 220 MeV. The latter
is the scale below which the strong coupling, αs (cid:38) 1. However, perturbation theory fails even before that
threshold is reached.
‡Virtuality is the square of the sum of the 4-momenta of the two particles after the splitting, i.e. the

square of the invariant mass of the system of virtual particles.

∗∗Resummation is a method of regularizing series divergences, in this context, typically infrared and
It effectively rearranges the original perturbative expansion into a new one that has different

collinear.
convergence properties.

44

allowed to produce PS at NLL [37, 38]. Care has to be taken in matching the fixed-order

ME at the hard scale with the resummed PS at the much lower cutoff scale, especially for

higher order calculations. This problem is further discussed in Section 3.3.

Multiple parton interactions (MPI) can occur in a single event, generally at scales lower

than that of the hard process. The participating partons come from the remnants of the

two incoming hadrons that provided the partons that initiated the hard process. Products

of these additional interactions form the underlying event.

After the PS terminates at the cutoff scale, hadronization confines color-charged partons

into observable color-neutral hadrons. The process can be viewed from the perspective of

connection and reconnection of the color charge lines, which, besides the hard scatter, may

also have contributions from MPI and the beam remnants. The length scale of these process

is larger than the proton radius, corresponding to the non-perturbative regime. The two

most commonly used hadronization models are the Lund model [39, 40], implemented in

Pythia and Sherpa, and the cluster model [41–43], implemented in Herwig.

Depending on the application, such as analysis optimization studies or unfolding, show-

ered and hadronized MC events may be passed through a detector simulation to incorporate

effects of interactions of the particles with the detector material, as well as to account for

experimental sensitivity and resolution. This is typically carried out using a dedicated pro-

gram capable of simulating the material interactions of particles, such as Geant4 [44]. In

addition to this, effects of pileup can be simulated by overlaying randomly sampled minimum

bias events.∗ After detector simulation, generated events can be reconstructed in the same

∗“Most of the time the color exchange between partons in the beam hadrons occurs through a soft
interaction and the two beam hadrons ‘ooze’ through each other producing lots of soft particles with a
uniform distribution in rapidity. Min-bias (MB) is a generic term which refers to events that are selected
with a ‘loose’ trigger that accepts a large fraction of the overall inelastic cross section.” [45] In this sense,
MB events are not biased by the kinematics of a hard scattering process.

45

manner as the experimental data, using the techniques discussed in Chapter 5.

It is worth noting that event simulations do not rely on a complete theoretical description

of all processes involved in the physical events they model. The simulations “are our best

guess of what can happen at the LHC, but we cannot take them as absolute truth. Addi-

tionally, we can only generate specific (known) physics processes (with limited accuracy), so

simulation is not always expected to describe the data.” [34]

3.2 Event generation at NLO

At the momentum scales typical at the LHC, the strong coupling becomes relatively
small, αs ≈ 0.1.∗ This allows for a perturbative series expansion of the cross section in terms
of the coupling,

σ = c0 + c1αs + c2α2

s + . . .

(3.1)

With this approach, increasingly more accurate predictions can be obtained by calculating

higher order terms in the series.† Higher order calculations are important for providing

accurate predictions for data analyses during planning and preparation, as model inputs, for

unfolding, and as comparisons to measured observables. As higher luminosity LHC datasets

push the precision frontier from the experimental side, the theoretical predictions strive to

maintain a competitive degree of precision.

Naively, one would expect each higher order fractional correction to be on the order of αs,
or about 10%. However, practice shows that this is seldom true for going from LO to NLO.

∗At the scale of Z boson mass, mZ = 91.19 GeV, αs = 0.1181(11) [14].
†Strictly speaking, series expansions of observables in QCD in terms of αs are asymptotic, with conver-
gence only expected up to a certain order. However, this has not been a practical problem because, so far,
techniques have been developed that enable only limited order calculations, which agree extremely well with
measurements. Convergence problems are expected at arise near order 1/αs ≈ 10.

46

Figure 3.3: The transverse momentum of the leading jet (left) and the Higgs boson (right)
calculated using jettiness subtraction technique at LO, NLO, and NNLO in the strong cou-
pling constant for √s = 8 TeV. The lower panels show the ratios (K-factors) of NLO over LO
cross sections, and NNLO over NLO cross sections. The large NLO K-factors are apparent.
The shaded regions indicate scale variation errors [46].

For a number of processes, including Higgs production, the NLO/LO ratio, or the K-factor,

can be as large as 2, as can be seen in Fig. 3.3. One way to understand this is that higher

order Feynman diagrams introduce additional processes into the calculation, which are not

present in the lower order diagrams. Another, albeit heuristic, explanation for the large

NLO K-factor in Higgs production is color cancellation. The dominant Higgs production

mechanism at the LHC is gluon-gluon fusion, having two color octets in the initial state.

At the LO, Higgs is produced with no additional radiation, requiring the initial state color

charges to cancel out. Real NLO corrections provide a way to retain residual color charge

and thus open up the phase space. In practice, virtual corrections can also play an important

role in contributing to the large K-factors.

Going to the next higher order can be far from straightforward. Fortunately, in the early

2000s, coinciding with the beginning of data collection at the LHC, a number of advances was

made in the automation of NLO calculations, often referred to as the NLO revolution [47–49].

47

�����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������Doing higher order calculations in the fixed order approach involves calculating virtual and

real corrections. The virtual corrections come from diagrams containing additional loops.

The real corrections appear due to the presence of unresolved phase space, and consist of

diagrams with additional external legs corresponding to soft and collinear emissions. Ex-

amples of NLO correction diagrams for the gluon-gluon fusion Higgs boson production are

shown in Fig. 3.4.

(a) A leading-order diagram

(b) A virtual correction

(c) A real correction

Figure 3.4: Examples of Feynman diagrams contributing at NLO to the process of Higgs
boson production via gluon-gluon fusion.

The difficulty of higher order calculations partially comes from the fact that the number of

correction diagrams grows rapidly with the correction order as well as the number of external

legs in the process of interest. But, besides the “mere” problem of calculating a large number

of increasingly more complex diagrams, additional complications arise due to infrared and

collinear divergences associated with real corrections, and infrared and ultraviolet divergences

associated with virtual corrections.∗ The soft and collinear divergences of real corrections

appear from the structure of the matrix elements. The divergences of the virtual corrections

can be interpreted as the result of addition of a new integration variable – the 4-momentum in

the loop, which is unconstrained by overall 4-momentum conservation or on-shell conditions,

∗An in-depth discussion of the problems and methodology of NLO QCD calculations can be found in

chapter 3 of [15].

48

and can thus approach zero or infinity. These divergences have to be regularized, which

is nowadays typically done with dimensional regularization as the method of choice.∗ The
method consists in performing the integration in D = 4−2ε spacetime dimensions, instead of
the usual 4. Analytic continuation for the function of D can then be found, and the value of
the integral can be assigned by taking the limit ε → 0. This allows divergences to be identified
as single, 1/ε, or double, 1/ε2, poles for NLO calculations. Infrared divergences from real

and virtual terms have to cancel each other out for physical observables. The divergences in

the real terms are explicit and cancel with the poles of the virtual corrections. Application of

jet clustering algorithms and imposition of cuts, such as on minimum transverse momentum

and maximum rapidity for jets, allow physical observables to be defined for QCD final states

and soft and collinear divergences to be avoided. Other divergences are renormalized through

a suitable, scheme-dependent redefinition of the Lagrangian involving addition of suitable

counterterms [15].

However, there is a practical problem with cancellation of the divergences. Real and

virtual divergences originate from two different integrals taken over n- and (n + 1)-body

final state phase spaces, respectively. If the respective integrals could be taken analytically

in D dimensions, the infrared divergences from both contributions would cancel out, and

finite values for observables could be obtained directly. But in presence of kinematic cuts,

integration over dimensionally-regularized unresolved real emission phase space becomes in-

tractable analytically. On the other hand, numerical integration in D dimensions makes it

hard to extract singularities, and more importantly, the underlying finite terms. Instead,

the real emission contribution can be separated into two parts: a simple part, containing

∗Other common approaches to regularization are the cut-off method and the Pauli–Villars method.
Advantages of dimensional regularization are that it can guarantee Lorentz and gauge invariance, and is
applicable to a wide range of situations.

49

all singularities, to be integrated analytically in D dimensions; and a remainder, with finite

numerical integral over the complete phase space. The universality of the singular limits of

the matrix elements makes this possible. The most common approach is to add and subtract

an approximation to the real emission squared matrix element that captures all of its di-

vergent collinear and soft limits, and yet is simple enough to be integrated analytically over

the unresolved phase space. This manipulation is summarized in Eq. (3.2), and is known as

the dipole subtraction method, the most widely used version of which is Catani–Seymour

subtraction [50, 51].

NLO

dσ

dv

Born
n

+ dˆσ

real
n+1 δv

=(cid:90) (cid:16)dˆσ
=(cid:90) dˆσ
=(cid:90) dˆσ

Born
n

Born
n

virt

n (cid:17) δv +(cid:90) dˆσ

virt
n + dˆσ

δv +(cid:90) (cid:16)dˆσ
δv +(cid:90) dˆσ

fin. virt
n

int

n (cid:17) δv +(cid:90) (cid:16)dˆσ
δv +(cid:90) dˆσ

fin. int
n

real
n+1 − dˆσ

δv +(cid:90) (cid:16)dˆσ

sub

n+1(cid:17) δv

real
n+1 − dˆσ

sub

n+1(cid:17) δv

(3.2)

int
The addition and subtraction of equal dˆσ
n

and dˆσ

sub
n+1

terms that, however, pertain to dif-

ferent multiplicity phase spaces, is what allows reconciliation of the respective divergences.

This is the method used in Sherpa and GoSam calculations. There also exist other sub-

traction techniques, such as the Frixione–Kunszt–Signer method [52, 53]. The subtraction

methods lie at the heart of what enables automation of NLO calculations. The rearranged

terms in Eq. (3.2) are usually referred to as Born (B), virtual (V), integrated subtraction

(I), and real-subtraction (RS) contributions to the NLO calculation.

It should be noted that separation of the expression for the cross section into contributions

at NLO using the subtraction methods typically implies that MC generators produce separate

sets of events for each contribution. For example, GoSam ntuples are split into the 4 sets:

B, V, I, and RS. The subtraction formalism requires that the I events, and a fraction of the

50

RS events, have negative weights. The cancellation between positive and negative weights

represents the remaining cancellation between the regularized divergences. For sufficiently

inclusive regions of phase space, positively weighted events are expected to outweigh the

negatively weighted ones, so that binned distributions of observables are positively valued.

The presence of negative weights introduces complications. For example, arbitrary bin-

ning may result in the positive and negative counterterms populating different bins, or the

counterterm events not passing fiducial selection criteria. In the worst case scenario, this

may result in negatively valued predicted distributions. Thus, when NLO subtraction meth-

ods are used, care has to be taken a priori to generate events in a way that minimizes the

possibility of such situations, e.g. by generating with comparable fiducial cuts to the ones ex-

pected, and by setting scales of the calculation to values that accurately reflect the simulated

process.

An alternative is to use a different formalism, designed to produce only positively

weighted events, for example, such as implemented in Powheg (POsitive Weight Hard-

est Emission Generator) [54]. The drawback is that this approach requires adoption and

tuning of somewhat ad hoc parameters, which makes the method less reliable for completely

new predictions. However, for well understood processes, values of these parameters can be

determined to give the correct results and produce positively weighted events, which reduce

subtlety and are easier to use for experimental studies and non-experts.

The purpose of programs such as GoSam is to provide theoretical calculations of the

hard scattering at the parton level. The MC technique is a convenient method to carry

these calculations out. The events produced by GoSam have a wide range of weights,

which, as mentioned earlier, can be positive or negative. Events with negative weights are

relatively easy to handle in the GoSam calculations. Hard scattering information with

51

negative weights is more difficult to deal with, when that information is used in a full event

simulation. Various techniques, such as MC@NLO [55, 56], have been developed to produce

full events with both positive and negative weights.

3.3 Matrix element and parton shower matching

In order to provide a complete description of a proton-proton collision, global event

simulation programs have to merge the matrix element (ME) calculation of the hard process

with the parton shower (PS) simulation. This is not a trivial task, especially because the ME

calculation is usually done to a fixed order, while the PS is simulated as a stochastic process

and has a logarithmic accuracy from the PS resummation, and therefore effectively contains

partial contributions from all orders. The problem becomes even more complicated for higher

order ME calculations, which involve contributions from phase spaces with different numbers

of particles, i.e. due to real emission, and contain kinematic features not present at LO. But

the two calculations are complementary, in that the ME provides a more precise description

of the hard process, while the PS more accurately simulates the internal evolution of jets.

A PS cannot be run directly on top of a ME calculation, as that would result in double

counting of jets. For example, in a Higgs + jets event, a jet may develop either from a

parton originating from the ME of the hard interaction or from QCD radiation in the PS.

To avoid double counting, a matching procedure has to be defined in a way that makes the

best use of either calculation in its respective domain. The simplest approach is to define

a matching scale below which the PS will be used, and above which (cid:22) the ME. The scale

is typically dynamic and is defined in terms of the pT of the jets. Several of the commonly
used ME+PS matching schemes are: MLM (Michaelangelo Luigi Mangano) [57], CKKW

52

(Catani–Krauss–Kuhn–Webber) [58, 59], and CKKW–L (CKKW–L¨onneblad) [60].

To combine an NLO calculation with a parton shower, it is essential to ensure that the

result inherits the total cross section from the fixed-order (NLO) calculation, and that the

radiation pattern to first order follows the real emission part of the calculation. In addition,

from a parton shower point of view, it is also important to maintain its intrinsic logarithmic

accuracy, which is substantially harder to achieve and to prove. [15]

For NLO ME, an additional phase space has to be subtracted from the PS. There are two

commonly used methods. In the MC@NLO method, negative weights are assigned to certain

PS configurations that contain radiation already accounted for at the NLO calculation. In

the Powheg method [54], all PS emissions harder than the hardest one from the ME are

vetoed. The correct normalization is attained by applying a multiplicative correction to the

event weights. An additional damping parameter, D, for real emission is used to suppress
overly weighting production at high pT.

D =(cid:18)1 +(cid:16) pT

h(cid:17)2(cid:19)−1

,

0 ≤ D ≤ 1.

(3.3)

Here, the pT sets the hard scale, and the h-factor∗ is a tunable parameter corresponding to
the resummation scale. [34]

3.4 Scale dependence

Though, fundamentally, the quantum field theory is scale invariant, perturbative calcu-

lations require introduction of scale parameters in order to obtain useful series expansions

∗The h-factor is so named because it was first used in calculations of Higgs boson production.

53

to calculate the observables.∗ If the full series could be calculated, the dependence on the

values of the scales would cancel out, but because only a certain number of terms in the

expansion can be computed, the answer retains some scale dependence. Formally, for an

observable calculated at a given order in perturbative QFT, its scale dependence is of the

next highest order. Typically, scale dependence is expected to decrease for higher order

calculations. Scale variations can be used to estimate the expected magnitude of the next

order corrections, and to assess the uncertainty of the calculation.

Ultimately, only heuristic decisions can be made for the values of the scales. But argu-

ments can be presented for natural scale choices, which usually correspond to some measure

of magnitudes of momentum transfer involved in the process. For example, the value of a

scale can be given by, or depend on, the masses of electroweak bosons or combinations of

transverse momenta of hadronic jets. While a number of scales can be introduced, with

different values for different parts of the process (for example at every vertex), the scheme

can be simplified to just two scales, corresponding to factorization (µF
(µR
the uncertainty of QCD predictions. The rule of thumb is to vary both scales by a factor of

). Variation of the factorization and renormalization scales is a common way to assess

) and renormalization

2 up and down. This is usually simplified to a 7-point variation scheme,

(µF/µ0, µR/µ0) ∈(cid:8)(1, 1), (0.5, 0.5), (1, 0.5), (0.5, 1), (2, 1), (1, 2), (2, 2)(cid:9),

(3.4)

in which the ratio between the two scales is kept at a factor of 2 or less. The uncertainty of

the calculation is then estimated by the envelope of the values of the observables obtained

for each pair of the scales’ values.

∗Another way to express the meaning of scale parameters is that they appear from the necessity to

factorize parts of processes that occur at different energy, or inversely distance and time, scales.

54

As discussed in Section 2.4, the factorization theorem allows a decoupling of the inter-

action of the parton that participates in the hard process (at a high energy scale) from the

non-perturbative (low energy scale) interactions within the hadron. Large momentum trans-

fers are involved in the hard process, which therefore can be approached with perturbation

theory. The non-perturbative effects are accounted for by the parton distribution functions

(PDFs), which determine the probability of interaction with a given type of parton carrying

a given fraction of the hadron’s initial momentum at a given energy (factorization) scale.∗

The scale of momentum transfers taking place in an interaction is inversely proportional

to the scale of spatial resolution of the structure of the interacting particles. This property

is reflected in the scale dependence of the PDFs. As can be seen in Fig. 2.7, the fractions of

proton momentum carried by gluons and sea quarks increase at larger factorization scales.

In other words, interactions at higher energies reveal a more complex, finer scale structure

of a proton, than just a combination of the 3 valence quarks. In fact, in high transverse

momentum processes at the LHC, about 40% of the momentum of the proton is observed

to be carried by gluons. The large gluon PDF luminosity, in the kinematic range needed

for Higgs boson production, results in gluon-gluon fusion being the dominant production

mechanism for the Higgs boson at the LHC.

The renormalization scale can also be viewed as a parameter describing the resolution

of finer structure at higher energies, but in this case, the structure is that of the vacuum

rather than of a composite particle. Higher order perturbative corrections in the quantum

field theories of both electroweak and strong interactions in the SM contain divergent loop

diagrams. Using the conservation laws, virtual loop momenta cannot be fully constrained

∗As discussed in Section 2.4, PDFs cannot be calculated perturbatively, but their evolution with the

factorization scale can be, using DGLAP equations.

55

by the external momenta. So, naively, the contributions of the loop diagrams are given by

integrals over all possible loop momenta, from zero to infinity, giving infinite corrections to

the observables calculated at tree level. Conceptually, these infinities come from including

contributions from arbitrarily large energy scales, corresponding to arbitrarily short wave-

lengths. However, it is reasonable to expect that a process occurring at a relatively low scale

should not be affected by the physics of an arbitrarily high energy regime, nor, in principle,

that such a regime can be correctly described by the same low-energy theory.

The crucial idea in renormalization is that the divergences can be absorbed in the effective

behavior of particles. In the dynamic picture of Feynman diagrams, a particle continuously

undergoes interactions with all the fields to which it can couple. This results in self-energy

diagrams, such as the ones shown in Fig. 3.5. In principle, any propagator would contain

an infinite number of such diagrams to all orders. These diagrams involving free fields

represent infinite corrections to the parameters of the theory due to vacuum polarization. The

idea behind renormalization is that the bare parameters are immaterial. What is measured

(a) A fermion self-energy diagram.

(b) A boson self-energy diagram.

˜S(cid:48)F

=

+

+

Σ

Σ

Σ

+

. . .

(c) The complete propagator, ˜S(cid:48)
graphs, Σ, arranged along a chain. [61]

F

, is the sum over one-particle irreducible

Figure 3.5: Examples of self-energy diagrams and their combination into a physical propa-
gator after renormalization.

56

in a laboratory are the physical parameters, which already include the effects of vacuum

polarization. The divergences in the higher order calculations can, in this way, be absorbed

into the definitions of the parameters.

In the SM, renormalization of 3 kinds of things

is necessary: masses, charges, and the fields themselves. In particular, this results in the

running of couplings mentioned in Section 2.4.

Depending on the scheme, the renormalization scale parameter introduced can represent

a cutoff on the loop momentum or some other measure of the maximum energy scale of

integration. The scheme most commonly used in event generators is called the modified

minimal subtraction scheme, or MS [62, 63].

In fully renormalizable theories, including

the SM, the renormalization scale is not a physical quantity, and any physical prediction,

if calculated to all orders, should be scale-independent. The scale simply determines the

balance between the tree-level and loop diagrams’ contributions. To reduce the effect of

higher-order corrections, the scale is typically chosen to be on the order of the magnitude of

momentum transfers involved in the process of interest.

Experience at the Tevatron indicated that a good choice of factorization and renormal-

ization scales for calculations of boson production processes is the mass of the boson. But

this heuristic was inadequate for processes involving multiple jets in the final state or boson

production at high pT. Later, at the LHC, this became especially apparent for the Higgs
production. It was soon realized that, as an integral part of the process, QCD radiation

has to be accounted for in the scale choice. As mentioned above, gluon-gluon fusion is the

dominant process for Higgs boson production at the LHC. While the gluons are color-octets,

the Higgs boson is color-neutral, resulting in a situation where a massive colorless particle is

produced from an initial state with a “large amount” of color charge. This results in a large

probability of gluon radiation, that provides means to dissipate the color, at momentum

57

scales not dissimilar to the mass of the Higgs. Hence, dynamic scales were introduced, which

depend not only on the mass of the gauge boson, but also on the momenta of the jets. Using

such dynamic scales, event generators assign different scale values event-by-event, which can

effectively account for the varying numbers of vertices with large amounts of momentum

transfers.

Studies of the cross section scale dependence have resulted in several improved definitions

of dynamic scales for Higgs + jets production, proposed by the GoSam authors. These scales

are variations of HT (cid:22) the scalar sum of the transverse momenta of the jets and the boson.
Two commonly used variations are:

ˆH(cid:48)T/2 =

1

2(cid:18)(cid:113)p2T,H + m2

H + (cid:88)j∈jets

pT,j(cid:19),

(3.5)

(3.6)

and

ˆH(cid:48)(cid:48)T = mH +

1

2(cid:18)pT,H + (cid:88)j∈jets

pT,j(cid:19).

Figs. 3.6 and 3.7 demonstrate the dependence of the Higgs + jets cross section on the

factorization and renormalization scales using these dynamic scale definitions. As can be seen

-log µR
in Fig. 3.6a, the NLO cross section exhibits a characteristic saddle point in the log µF
space [64, 65], and a reduced scale dependence compared to the LO cross section. The studies

of scale dependence were done by the author using the GoSam ntuples.

3.5 PDF uncertainty

As has already been discussed, the short-range behavior of partons comprising a hadron

is decoupled from the long-range inelastic scattering process by the factorization theorem,

58

(a) Higgs + 1 jet, µ0 = ˆH(cid:48)

T/2, σ0 = 8.06 pb

(b) Higgs + 1 jet, µ0 = ˆH(cid:48)(cid:48)

T, σ0 = 6.20 pb

(c) Higgs + 2 jets, µ0 = ˆH(cid:48)

T/2, σ0 = 2.99 pb

(d) Higgs + 2 jets, µ0 = ˆH(cid:48)(cid:48)

T, σ0 = 2.24 pb

(e) Higgs + 3 jets, µ0 = ˆH(cid:48)

T/2, σ0 = 0.98 pb

(f) Higgs + 3 jets, µ0 = ˆH(cid:48)(cid:48)

T, σ0 = 0.73 pb

Figure 3.6: LO dependence of pp → H + jets cross section on factorization and renormal-
ization scales at √s = 13 TeV for two different choices of dynamic scales, computed using
GoSam ntuples. MC events were reweighted using CT14nlo PDFs. Typical jet cuts of
pT > 30 GeV and |η| < 4.4 are applied. The white dot indicates the location of the central
scale, µF = µR = µ0.

59

1−101100µ/Fµ1−101100µ/Rµ0510152025303540, pbσ1−101100µ/Fµ1−101100µ/Rµ0510152025, pbσ1−101100µ/Fµ1−101100µ/Rµ0510152025, pbσ1−101100µ/Fµ1−101100µ/Rµ0246810121416, pbσ1−101100µ/Fµ1−101100µ/Rµ02468101214, pbσ1−101100µ/Fµ1−101100µ/Rµ0123456789, pbσ(a) Higgs + 1 jet, µ0 = ˆH(cid:48)

T/2, σ0 = 13.34 pb

(b) Higgs + 1 jet, µ0 = ˆH(cid:48)(cid:48)

T, σ0 = 11.91 pb

(c) Higgs + 2 jets, µ0 = ˆH(cid:48)

T/2, σ0 = 4.55 pb

(d) Higgs + 2 jets, µ0 = ˆH(cid:48)(cid:48)

T, σ0 = 4.06 pb

(e) Higgs + 3 jets, µ0 = ˆH(cid:48)

T/2, σ0 = 1.46 pb

(f) Higgs + 3 jets, µ0 = ˆH(cid:48)(cid:48)

T, σ0 = 1.30 pb

Figure 3.7: NLO dependence of pp → H + jets cross section on factorization and renormal-
ization scales at √s = 13 TeV for two different choices of dynamic scales, computed using
GoSam ntuples. MC events were reweighted using CT14nlo PDFs. Typical jet cuts of
pT > 30 GeV and |η| < 4.4 are applied. The white dot indicates the location of the central
scale, µF = µR = µ0.

60

1−101100µ/Fµ1−101100µ/Rµ02468101214161820, pbσ1−101100µ/Fµ1−101100µ/Rµ024681012141618202224, pbσ1−101100µ/Fµ1−101100µ/Rµ0123456, pbσ1−101100µ/Fµ1−101100µ/Rµ01234567, pbσ1−101100µ/Fµ1−101100µ/Rµ00.20.40.60.811.21.41.61.82, pbσ1−101100µ/Fµ1−101100µ/Rµ00.20.40.60.811.21.41.61.822.2, pbσwith the non-perturbative short-range description of the hadron contained in the parton

distribution functions (PDFs). A number of different methods to determine PDFs from fits

to data has been developed over the years, ranging from functional models, such as used by

the CTEQ collaboration [25, 66, 67], to the use of neural networks [68, 69]. Regardless of

the approach, PDFs are described by a regression model that is fit to the data, and therefore

have uncertainties associated with them. The precision of both theoretical predictions and

experimental analyses depends on the PDF uncertainties.

A PDF set contains estimates of the probability of interaction, f (x, Q), with a given

parton inside a hadron as a function of longitudinal momentum fraction, x, at an energy
scale, Q = µF
space. The PDF values at all possible x and Q values are obtained by grid interpolation. To

. This data is typically stored as a grid of xf (x, Q) values for points in the x-Q

enable the propagation of PDF uncertainties, a number of error sets is stored in addition to

the nominal. A common way to choose the error sets is the Hessian eigenvector method [70–

72]. The Hessian matrix elements are second derivatives of the χ2 function with respect

to the PDF fitting parameters, which give the coefficients of the quadratic approximation

of the χ2 near the minimum. The method consists in defining the eigenvector PDF sets

using the values of the parameters from the eigenvectors of the Hessian matrix. Principal

component analysis can be used to reduce the number of eigenvectors necessary to adequately

characterize PDF uncertainties. The uncertainty can then be estimated by the root mean

square value of the deviations of the predictions using the eigenvector PDFs from the nominal

optimized PDF for both up and down deviations. For predictions of MC event generators,

PDF uncertainties are estimated by reweighting the events using the error PDF sets and then

taking the root mean square of the deviations of the observables from the nominal value.

LHAPDF [73] provides the standard library and repository of parton distribution functions.

61

Its C++ and Python libraries implement a standard interface for accessing all major PDF

sets created in the HEP community. The PDF sets are made available for download from a

centralized repository. The programming libraries provide functions to calculate PDF values

and uncertainties. An example of PDF uncertainties, produced using LHAPDF6, is show in

Fig. 3.8.

Figure 3.8: Gluon density distributions and Hessian uncertainties from the CT14nlo PDF
set for proton. Left: for Q = 100 GeV as a function of x. Right: for x = 0.1 as a function of
Q. The nominal function is shown in black, the error functions in blue, and the uncertainty
in red.

3.6 Predictions used in the analysis

Simulated MC event samples are produced for signal and background processes, as listed

in Table 3.1. These are used for the derivation of correction factors, the binning of differential

distributions, the signal model shape parameters optimization, the background function

selection, the estimation of several systematic uncertainties, and the validation of the analysis

procedure.

62

0255075100125150xf(x,Q)CT14nlogluonPDFQ=100GeV10−410−310−210−1x0.91.01.1Ratio0.60.81.01.21.4xf(x,Q)CT14nlogluonPDFx=0.1101102103Q[GeV]0.91.0RatioTable 3.1: List of Monte Carlo signal and background samples used in the analysis.

Process

Generator

PS

PDF

ME order

Normalisation order

ggF
VBF
W +H
W−H
q ¯q → ZH
gg → ZH

t¯tH
b¯bH
W tH
tHq

γγ

Powheg NNLOPS Pythia8 PDF4LHC15 NNLO + NLL(QCD)

Powheg-Box
Powheg-Box
Powheg-Box
Powheg-Box
Powheg-Box
Powheg-Box
Powheg-Box
aMC@NLO
aMC@NLO

Pythia8 PDF4LHC15
Pythia8 PDF4LHC15
Pythia8 PDF4LHC15
Pythia8 PDF4LHC15
Pythia8 PDF4LHC15
Pythia8 PDF4LHC15
Pythia8 PDF4LHC15
Pythia8 CT10 NLO
Pythia8 CT10 NLO

Sherpa

Sherpa

CT10

NLO(QCD)
NLO(QCD)
NLO(QCD)
NLO(QCD)
LO(QCD)
NLO(QCD)
NLO(QCD)
NLO(QCD)
NLO(QCD)
NLO(QCD)

N3LO(QCD) + NLO(EW)

approx. NNLO(QCD) + NLO(EW)

NNLO(QCD) + NLO(EW)
NNLO(QCD) + NLO(EW)
NNLO(QCD) + NLO(EW)
NLO(QCD) + NLO(EW)
NLO(QCD) + NLO(EW)
5FS (NNLO), 4FS (NLO)

4FS(LO)
5FS(NLO)
NLO(QCD)

σ × BR, fb

110

8.58
1.90
1.21
1.73
0.28
1.15
1.10
0.034
0.169
19.2 × 103

63

The production of MC samples starts with a matrix element (ME) generator, which

generates hard process events at parton level. These are then converted to particle level∗

events via a parton showering (PS) program that simulates the formation of jets from final

state quarks and gluons. Subsequently, the events are passed through a detector simulation,

to account for detector effects and to obtain events in the form similar to that observed in the

detector [74]. These events are finally processed with the same algorithms as the collected

data. This is referred to as the reconstruction level. Simulated events are corrected to reflect

photon and jet energy scale and resolution, as well as the trigger, identification, and isolation

efficiencies observed in data. As theoretical calculations can typically be performed to higher

order for integrated than for differential cross sections, simulated events are reweighted to

normalize their integrated cross section to the best known theoretical result.

The effects of multiple interactions occurring withing the same or neighboring bunch

crossings, i.e. pileup, are modeled by overlaying inelastic pp, also known as minimum-bias,

events generated with Pythia 8, using NNPDF2.3LO PDF set [75] and the A3 tune [76], over

the original hard-scattering events. Events in the MC simulation are reweighted to reproduce

the pileup µ distribution observed in data. The number of pileup events included is obtained

by sampling a Poisson distribution with a mean of µ. Minimum-bias simulations are run

before data taking is complete and the actual pileup conditions are known. Therefore, each

sample is generated with a broad range of values of µ, in order to encompass all possible pileup

conditions that may be experienced during data taking. The MC pileup distribution is then

corrected by reweighting when the actual conditions become available. Out-of-time pileup

is included by adding detector signals from previous bunch crossings, also using Pythia 8

minimum-bias events. The frequency of these signals is modeled on the nominal bunch

∗The particle level in MC simulations is also colloquially referred to as the truth level.

64

structure used by the LHC.

Several reweighting corrections are applied to the MC samples. The correction factors

are different for samples produced with full and fast detector simulations. Distributions of

the following quantities in MC are matched to the data:

• pileup, as discussed above;

• z distribution of the primary vertex;

• photon energy spectrum is smeared to match the observed resolution;

• photon identification and (track and calorimeter) isolation efficiencies.

Additional MC samples and theory calculations were produced to provide comparisons

for measured cross sections to the state-of-the-art theoretical predictions. These samples

were produced only at parton or particle level, without detector simulation. They were only

used for comparison with the results of the analysis, and are discussed in Section 3.6.3.

3.6.1 Signal Monte Carlo

Signal MC samples are generated for all main Higgs production mechanisms discussed in

Chapter 2: gluon fusion (ggF), vector boson fusion (VBF), associated production with vector

bosons (W H and ZH), and associated production with heavy quarks (t¯tH and b¯bH). The

relative contribution of each process to the fiducial cross section as a function of diphoton

transverse momentum is shown in Fig. 3.9. All samples are generated with the Higgs boson

mass of mH = 125 GeV and width of ΓH = 4.07 MeV [28]. The hard scattering process
leading to the production of the Higgs boson is generated using Powheg [54, 77–79] with

the PDF4LHC15 PDF set [80]. The Higgs boson decay to two photons, as well as the effects

of the underlying event, parton showering, and hadronization, are modeled by interfacing the

output of the parton level generation with Pythia 8 [81], using the AZNLO set of parameters

65

Figure 3.9: Fractional signal composition in terms of the Higgs production processes as a
function of diphoton transverse momentum determined from the signal MC samples.

that are tuned to data [82].

The particle level signal MC events are processed with the full Geant4 ATLAS detector

simulation [44] and the same reconstruction algorithms as those used for data. The event

weights are normalized to the most accurate theoretical calculations for the inclusive cross
section in the corresponding production channel, and multiplied by the H → γγ branching
ratio of 0.227% calculated with HDECAY [83–86] and PROPHECY4F [87–89] for the Higgs boson

mass of 125.09 GeV.

3.6.1.1 Gluon Fusion

Samples for the ggF process are generated with NNLOPS [90], a state of the art gener-

ator based on MiNLO HJ [91] and the Powheg method with NNLO + NNLL accuracy for

Higgs pT and rapidity distributions. This is achieved by applying a correction based on the
HNNLO(y) program and checking the result against the HqT program [92]. The ggF sample

is normalized to a calculation at N3LO (QCD) with additional NLO electroweak correc-

66

0-55-1010-1515-2020-2525-3030-3535-4545-6060-8080-100100-120120-140140-170170-200200-250250-350∞350-(particle-level) [GeV]γγTp00.20.40.60.811.2Signal CompositionggFVBFWHZHttHtions [28], accounting also for the treatment of the quark mass effects through a correction

factor for the total cross.

3.6.1.2 Vector Boson Fusion

VBF samples are generated with PowhegBox [93] and showered with Pythia 8.

Powheg is accurate to NLO and is tuned to match calculations with effects due to finite

fermion masses and soft-gluon resummation up to NNLL. The VBF sample is normalized to

a calculation at NLO (QCD) with additional NLO (EW) corrections, with an approximate

NNLO QCD correction.

3.6.1.3 Associated production with vector bosons

W H and ZH samples are generated with NLO parton shower matched matrix elements,

except for the ggZH sample, which is generated at LO.

3.6.1.4 Associated production with single top quark

W tH and tHq samples are generated with aMC@NLO, which implements a LO matrix

element matched to the parton shower. The samples are produced using the CT10 NLO

PDF set and the CTEQ6L1-UE-EE-5 generator tune.

3.6.2 Background Monte Carlo

The main background to the resonant pp → H → γγ process is the non-resonant SM
diphoton production. This background is irreducible, because events constituting it have an

identical signature to the events with the diphoton originating from the Higgs. Background

γγ samples were generated using Sherpa 2.2.4 [94] and merged with the Sherpa parton

67

shower [95] according to the ME+PS@NLO prescription [96–99]. The PDF set used is

NNPDF3.0 NNLO [68].

The background samples are passed through a fast parametric simulation of the ATLAS

detector [74]. A large number, on the order of 100 million, of background events needs to

be simulated, given the diphoton cross section of 19.2 nb. Use of the fast, rather than full,

detector simulation allows the precessing of these events in a timely manner.

Due to the possibility of the misreconstruction of jets as photons, a reducible background

component is also present, consisting of γj and jj events. Because of computational limi-

tations, no dedicated γj or jj samples are generated. Instead, a 2 × 2D sideband method
is used to reweight γγ samples to match the shape of the mγγ
control regions. This procedure is further discussed in Section 6.5.2 on background mod-

distribution to that in data

eling. After the fractional contribution of the reducible background is estimated, the total

MC weight is normalized to match the number of events observed in the data sidebands.

3.6.3 Predictions compared to the analysis results

Fiducial cross section measurements are designed to be as minimally model dependent

as possible, allowing them to be compared to many different theoretical predictions. All of

the obtained differential measurements were compared to the SM predictions derived from

the default signal MC, discussed in Section 3.6.1. The measured distributions for select

observables were compared to an array of state-of-the-art theoretical predictions described

below. These predictions were made for the dominant gluon fusion production channel of the

Higgs boson, and were therefore combined with the default predictions for the other channels.

The theoretical predictions are presented with the analysis results in Section 6.8.2.

Correction factors were applied to the predictions that did not include the fiducial selec-

68

tion defined by cuts on the photons’ transverse momenta and pseudorapidities, and the jets’

transverse momenta and rapidities. To most predictions, an additional correction was ap-

plied to account for the effect of the photon isolation criteria. Both of these corrections were

derived from NNLOPS [90]. Non-perturbative corrections were applied for all predictions

not interfaced with a parton shower.

Predictions which did not include the Higgs to diphoton decay were scaled by the cor-
−0.064 × 10−3 [30]. The branching ratio uncertainty was

responding branching ratio of 2.27+0.066

added in quadrature.

Most predictions, unless explicitly stated, used the NNLO PDF set following the

PDF4LHC recommendations [80].

3.6.3.1 Sherpa

For the Sherpa [100] predictions, NLO matrix element calculations were merged with

the parton shower [95] using the MEPS@NLO multijet techniques [101–103]. They have an

accuracy of NLOn for jet multiplicities of n = 0, 1, 2, 3. The 1-loop corrections are generated
internally in Sherpa for H +≥ 1 jet and externally, using MCFM [104] for H +≥ 2 jets and
GoSam [105, 106] for H +≥ 3 jets. The calculations make use of the effective field theory
in the mt → ∞ approximation. The effects of the finite top quark mass are taken into ac-
count by reweighting the EFT predictions by the ratio of finite mt to infinite mt predictions
calculated at 1-loop, but in this case, the loop is that of the top quark. Amegic [107] and

Comix [108] are used as matrix element generators. The predictions use the NNPDF3.0

NNLO PDF set with αs(mZ ) = 0.118. The calculation is done at particle level, i.e. it also
includes multiple interactions, hadronization, hadron decays and electroweak corrections.

Photon and jet fiducial cuts are included, as well as photon isolation. The perturbative un-

69

certainties are estimated by varying the renormalization and factorization scales. Variations

in the merging scale and the parton shower starting scale are known to be much smaller.

The Sherpa predictions are shown for the jet multiplicity and for all distributions in the

phase space with at least 2 jets.

3.6.3.2 GoSam

The GoSam [100, 106, 109] predictions were prepared by the author of this dissertation,

using the GoSam ntuples, discussed in Appendix A. The predictions have a fixed-order

NLOn accuracy, and are available for jet multiplicities n = 1, 2, 3. ˆH(cid:48)T/2, defined in Eq. (3.5),
was used as the dynamic renormalization and factorization scale. The perturbative uncer-

tainties were estimated using the 7-point scale variation scheme. The predictions use the

PDF4LHC15_30 NLO PDF set. The PDF uncertainties were calculated using the respective

PDF error set. Photon and jet fiducial cuts are included. The GoSam predictions are shown

for the jet multiplicity measurement and for all distributions in the phase space with at least

2 jets.

3.6.3.3 HRes

HRes [110, 111] provides predictions differential in pγγ

T at NNLO0 with pHT resummation
at NNLL. Finite masses of t, b, and c quarks are included at NLO accuracy. The scale choices

H + p2T,H

for renormalization and factorization, and mH

and 2mb

for resumma-

are 1

2(cid:113)m2

tion. Uncertainties are estimated from variations of the renormalization, factorization, and

resummation scales. Photon and jet fiducial cuts are included. The HRes predictions are

shown for pHT and pHTt

.

70

3.6.3.4 NNLOjet

NNLOJET [112] provides fixed-order NNLO prediction for inclusive H + 1 jet production.

The renormalization and factorization scales are taken to be(cid:113)m2

are estimated using the 7-point scale variation scheme. The predictions use the CT14NNLO

. Uncertainties

H +(cid:80) p2T,j

PDF set. Photon and jet fiducial cuts are included.

3.6.3.5 RadISH + NNLOjet

RadISH [113] provides predictions with pHT resummation at NNLL, matched to the
NNLO1 differential spectrum from NNLOjet [114]. Predictions using effective Higgs cou-
pling in the limit mt → ∞, as well as taking into account finite top quark mass are available.
In the latter, the mass effects are exact up to LO, while NLO and NNLO (in the spectrum,

i.e. NLO0, NNLO0, and N3LO0, respectively) corrections are computed in the large mt ap-
proximation. Mass effects are also included in the resummation up to NNLL. Quark masses

are renormalized in the on-shell scheme. The effects of fiducial cuts were approximated by a

correction factor.

3.6.3.6 SCETlib

The provided predictions for (cid:12)(cid:12)yγγ(cid:12)(cid:12), (cid:12)(cid:12)∆yγγ(cid:12)(cid:12), and (cid:12)(cid:12) cos θ∗(cid:12)(cid:12) at NNLO0

include a resummation of the virtual corrections to the gluon form factors [115, 116]. The

+ NNLL(cid:48)φ

accuracy

underlying NNLO predictions are obtained using MCFM8 with τ0 subtraction [117, 118].
They are derived in the fiducial volume defined by the photons’ transverse momenta and

pseudorapidities. The perturbative uncertainties contain both fixed-order and resummation

uncertainties estimated as described in [115].

Predictions for pj1T at NNLL(cid:48) + NNLO0 accuracy were derived applying a resummation

71

in pj1T [116, 119]. The perturbative uncertainties contain both fixed-order and resummation
uncertainties estimated as described in [119].

Photon and jet fiducial cuts are included for both predictions.

3.6.3.7 SCET + NNLOJET

SCET + NNLOJET provides predictions for pγγ

T using a N3LL resummation matched to

a NNLO fixed-order calculation in the heavy top quark limit [120].

3.6.3.8 STWZ–BLPTW

The STWZ–BLPTW predictions [119, 121] are a NNLL(cid:48) + NNLO resummation for the
pT of the leading jet, combined with a NLL(cid:48) + NLO resummation for the subleading jet. The
numerical predictions for √s = 13 TeV are taken from [30]. In comparison to the SCETlib
predictions, decribed in Section 3.6.3.6, these predictions include an improved resummation

for the total cross section, with accuracy comparable to N3LO. The effects of fiducial cuts

were approximated by a correction factor. The STWZ–BLPTW predictions are shown for

the jet multiplicity distribution.

3.6.3.9 JVE + N3LO

The JVE + N3LO predictions [122] include NNLL resummation of the pT of the leading
jet with small-R resummation and are matched to the N3LO total cross section. The effects
of fiducial cuts were approximated by a correction factor. The JVE + N3LO predictions are

shown for the jet multiplicity distribution.

72

3.6.3.10 MG5_aMC@NLO

The MG5_aMC@NLO prediction [123, 124] includes up to two jets at NLO accuracy

using the FxFx merging scheme. The central merging scale is taken to be 30 GeV. Top

quark mass effects are included through reweighting, and b quark mass effects are included
in the Higgs + 0 jets corrections. Resummation scales are chosen following the prescription

in [125]. The predictions are showered with Pythia 8. The effects of fiducial cuts were

approximated by a correction factor. The MG5_aMC@NLO predictions are shown for the

jet multiplicity. Uncertainties are estimated by varying the merging scale between 20 and

50 GeV.

73

Chapter 4

The LHC and ATLAS

The Large Hadron Collider (LHC) [126] is the world’s largest and most powerful particle

accelerator. It is the flagship facility at CERN,∗ an international high energy physics research

center, located on the border between France and Switzerland near Geneva. With the highest
achieved center-of-mass energy for proton-proton collisions of √s = 13 TeV, LHC stands at
the frontier of experimental particle physics. Seven experiments collect data at the LHC

from proton-proton, as well as heavy ion, collisions:

• ATLAS (A Toroidal LHC Apparatus) [127] is a giant, hermetic, general purpose de-

tector, designed to cover a wide range of physics studies in the highest energy regime

that can be created in a laboratory environment. The main goals of its program are

to discover and measure the properties of the Higgs boson, and to search for physics

beyond the Standard Model.

• CMS (Compact Muon Solenoid) [128] is a sibling detector of ATLAS with similar

physics goals. It uses some different detector technologies and is more optimized for

measurements of muons.

• LHCb (Large Hadron Collider beauty experiment) [129] is a single-arm forward spec-

trometer, primarily designed to measure the parameters of CP violation in the inter-

actions of bottom quark hadrons.

∗The full name of CERN is European Organization for Nuclear Research (French: Organisation eu-
rop´eenne pour la recherche nucl´eaire). The acronym is derived from Conseil europ´een pour la recherche
nucl´eaire.

74

• ALICE (A Large Ion Collier Experiment) [130] is a detector optimized for the heavy-

ion collision studies. It is designed to address the physics of strongly interacting matter

and the quark-gluon plasma.

• TOTEM (TOTal Elastic and diffractive cross section Measurement) [131] measures the

total cross section and luminosity, using a device called a Roman Pot. It shares the

interaction point with CMS.

• LHCf (Large Hadron Collider forward experiment) [132] is dedicated to the measure-

ment of neutral particles emitted in the very forward region, with the goal of providing

data for calibrating hadron interaction models that are used in studies of Extremely

High-Energy Cosmic-Rays. It shares the interaction point with ATLAS.

• MoEDAL (Monopole and Exotics Detector at the LHC) [133] is an experiment designed

to search for magnetic monopoles. It shares the cavern with LHCb.

This chapter discusses technical details of the LHC accelerator complex and the ATLAS

detector.

4.1 CERN accelerator complex

The LHC is a superconducting hadron synchrotron accelerator and collider [126, 137–

139]. Its main ring has a circumference of 26.7 km, with two beam pipes circulating protons

(or heavy ions) in opposite directions. It is located in a 3.7 m wide underground tunnel,

at the depth between 45 m and 170 m below the surface, containing 8 straight sections and

8 arcs, with the plane of the tunnel inclined at 0.8° towards Lake Geneva. The LHC is

installed in the old LEP tunnel, constructed in 1984–1989. LEP was dismantled in 2001,

and the LHC construction was finished in 2008. Run 1 of the LHC started in 2010. Most of

75

Figure 4.1: CERN accelerator complex [134].

76

LINAC 2North AreaLINAC 3IonsEast AreaTI2TI8TT41TT40CLEARTT2TT10TT66e-ALICEATLASLHCbCMSSPSTT20nppRIBsp1976 (7 km)ISOLDE19922016REX/HIE2001/2015IRRAD/CHARMBOOSTER1972 (157 m)AD1999 (182 m)LEIR2005 (78 m)AWAKEn-ToF2001LHC2008 (27 km)PS1959 (628 m)201120162015HiRadMatGIF++CENFp (protons)ionsRIBs (Radioactive Ion Beams)n (neutrons)–p (antiprotons)e- (electrons)2016 (31 m)ELENALHC - Large Hadron Collider // SPS - Super Proton Synchrotron // PS - Proton Synchrotron // AD - Antiproton Decelerator // CLEAR - CERN Linear Electron Accelerator for Research // AWAKE - Advanced WAKefield Experiment // ISOLDE - Isotope Separator OnLine // REX/HIE - Radioactive EXperiment/High Intensity and Energy ISOLDE // LEIR - Low Energy Ion Ring // LINAC - LINear ACcelerator // n-ToF - Neutrons Time Of Flight // HiRadMat - High-Radiation to Materials // CHARM - Cern High energy AcceleRator Mixed field facility // IRRAD - proton IRRADiation facility // GIF++ - Gamma Irradiation Facility // CENF - CErn Neutrino platForm2017The CERN accelerator complexComplexe des accélérateurs du CERNFigure 4.2: Detailed view of the underground LHC facilities, showing the locations of the
experiments and beam intersection points [135, 136]. The locations of the four major LHC
experiments: ATLAS, CMS, LHCb, and ALICE are highlighted in gray.

77

Point 8Point 7Point 6Point 4Point 5Point 3.3Point 3.2Point 2Point 1.8Point 1LHCb CMSALICEATLASR622R681UJ67UA67US65UW65UJ63UD62UP62PM65TD62UJ62R623RA63UA63UJ64UL64UL66R682R683UP68UD68TD68UJ68RA67UJ66RB66RB64UX65PX64PZ65TX64RE68R69, R71RE72R74, R76R72, R73R771PM76TZ76R78, R772RE78R79, R81RE82R82RB86RB84RA83RI132R21, R19RE18R18TI2UJ22RH23RA23UJ24RB24PX24UX25RB26UJ26RA27PM18UJ18RR17RI172UJ17PX16PX15PX14US15UL14UJ14UJ13TAG41PGC8TT40ECA4PM15UX15RB14TI12RB16PMI2RT18UJ16UPX16ULX15USA15TI18UL16RR13UJ12RT12RI131, RI12R88RE88RE12R12R11, R89UL84RR77UJ76RR73R572, R58R571Up56Ul56UJ57R561R562RR57 RE58RE62R59, R61R541RR53UJ561UJ56PM56PX56PM54UP53RZ54UL54UL55UP55UPX56UX451UP63UJ53R532USC55UXC55RE48RE52R48UJ47RA47UJ46RB46UA47UL46US45UL44UX45PZ45PM45PX46TX46UW45R52, R531R49, R51RB44UJ44RA43UA43UJ43UP33PZ33RZ33RE42R42RE38R39, R41R34, R36,R37, R38  PM32UJ33TZ32R33R32, R31UJ32 (incl,RE32)RE28R28UA27UL26UP25PM25US25UW25UL24UA23UJ23R22RE22PGC2UJ27R29TNB4TND4RH87RA87UJ86UX85TX84PGCNTJ8TI8TN4UJ88TCC4TSG41TCV4UA87US85UW85PM85PZ85PX84UJ84UJ83UA83UL86UJ87SPSTA40TAG42TSG40RI171Dcum 25701m / IP1Cell 21L1YCPS01 = R11Dcum 23649m / IP1Cell 09R8YCPS01 = UJ88Dcum 20235m / IP1Cell 07R7YCPS01 = R771Dcum 19753m / IP1Cell 07L7YCPS01 = R74Dcum 16905.5m / IP1Cell 05R6YCPS01 = RA67Dcum 16430m / IP1Cell 05L6YCPS01 = RA63Dcum 14288.2m / Ip1Cell 21R5YCPS01 = R59Dcum 12381.9m / IP1Cell 21L5YCPS01 = R51Dcum 10241.9m / IP1Cell 07R4YCPS01 = RA47Dcum 9753m / IP1Cell 07L4YCPS01 = RA43Dcum 6935.6m / IP1Cell 08R3YCPS01 = R37Dcum 6399.5m / IP1Cell 07L3YCPS01 = R34Dcum 3010m / IP1Cell 09L2YCPS01 = Ti2Dcum 959m / IP1Cell 21R1YCPS01 = R19EN-ACE Julie CoupardUpdate: February 2020EDMS 1001086 v.1.6beam1beam2dumpbeamthe √s = 7 TeV data was collected in 2011, and the √s = 8 TeV data was collected in 2012.
Run 2, in which the √s = 13 TeV data used in this dissertation was collected, took place in
2015–2018.

The acceleration of particles is done using a strong electric field generated inside super-

conducting radiofrequency (RF) cavities. The shape of the RF cavities is design to sustain

electromagnetic waves of a given resonance frequency. The frequency is tuned precisely, so

that when a charged particle travels through an RF cavity segment it feels a repelling force

from behind and an attractive one from the front. This is illustrated in Fig. 4.3. This also

produces the effect of protons in each bunch arranging themselves into RF buckets. The

protons in the middle of the bucket are given the optimal amount of energy, while those at

the ends are accelerated more or less depending on whether they are behind or ahead, thus

bringing them closer to the middle. This maintains the integrity of the bunches. The RF

cavities bring the protons from the injection energy up to the collision energy, as well as

mitigate the energy loss due to synchrotron radiation. The LHC has 16 RF cavities housed

in four cryomodules. Each cavity is driven by a high-power klystron, which creates 400 MHz

EM waves using intensity-modulated electron beams. The cavities operate at 4.5 K.

Figure 4.3: An RF cavity diagram [140]. The charge induced at either end of a cavity seg-
ment accelerates protons forward. The field in the cavity oscillates at 400 MHz, resulting
in the induced charge switching polarity, so that the protons never feel a force in the back-
ward direction. The LHC uses 8 cavities per beam, delivering 2 MV accelerating potential
(5 MV/m field). Every proton passing through an RF cavity experiences a total potential
difference of 2 × 8 = 16 MV, gaining 16 MeV of kinetic energy.

78

A key parameter for a synchrotron is the magnetic field produced by the bending magnets.

For a back-of-the-envelope estimate, consider a charged particle moving perpendicularly

to a uniform magnetic field. The particle experiences a Lorentz force, and therefore, for

relativistic speeds,

(cid:126)F = q(cid:16) (cid:126)E + (cid:126)v × (cid:126)B(cid:17) ⇒ γma =

γmv2

R

= qvB ⇒ E ≈ p = γmv = qBR.

(4.1)

Thus, a storage ring can contain particles with energy directly proportional to the available

magnetic field strength and the bending radius.∗ Clearly, the use of the strongest available

magnets is paramount to reducing the accelerator size. The LHC relies on superconducting

magnets that are at the edge of present technology. Classical NbTi superconductors are

used, but are operated at 2 K temperature† and 8.3 T magnetic field. Special “two-in-one”

or “twin-bore” design magnets are used, due to space limitations in the tunnel, allowing

to accommodate the windings for the two beam channels in a common coldmass and cryo-

stat. 1232 dipole magnets installed along the LHC ring help bend the particle beam. 392

quadropoles and higher order multipole magnets are used to focus and shape the beam.

The proton accelerator chain for pp collision at the LHC consists of the linear accelerator
Linac2,‡ Proton Synchrotron Booster (PSB), Proton Synchrotron (PS), Super Proton Syn-

chrotron (SPS), and finally the Large Hadron Collider (LHC). The sequence of injections

resulting in a new LHC beam is called a fill. A fill starts with releasing hydrogen gas from

a tank into a duoplasmatron, where it is dissociated and ionized, and the protons are given

an initial acceleration to 90 keV. This is a part of the RFQ2 pre-injector [145]. The Linac2

than the ring average, because the dipole magnets don’t cover the whole circumference [137, 141].

∗E [TeV] ≈ 0.3 × B [T] × R [km]. The bending radius for the LHC is 2.8 km, which is 1.5 times smaller
†Magnet cooling to maintain superconductivity requires circulation of 100 tons of liquid helium.
‡After Long Shutdown 2, Linac2 will be superseded by Linac4 [142].

79

Table 4.1: Summary of the LHC injection stages with their normal operation parameters
[139, 143, 144].

Output proton energy, GeV
Length or circumference, m
Harmonic number,∗
Bunches when full
Batches to fill next machine
Repetition time, s

h

Linac2
0.05
33
(cid:22)
(cid:22)
3
(cid:22)

PSB
1.4
157
1

1/ring

2
1.2

PS
25
628

3
3.6

7, 21, 84
6, 18, 72

SPS
450
6900
84
243
2 × 12

54

LHC
6500
26700
35640
2808
(cid:22)
1200

Figure 4.4: PSB-PS transfer: two-batch filling for the LHC [139].

Figure 4.5: PS bunch splitting for a nominal 25 ns LHC bunch train [139].

∗Harmonic number is the ratio of the particle revolution period to the period of the accelerating RF

voltage changes.

80

PSB  h=1Two-batch filling for LHC1st batch2nd batch1.2sec later1st batch2nd batchPS  h=71234243243Triple splittingat 1.4 GeVQuadruple splitting at 25 GeVPS injection:6 bunches (4+2) in 2 batcheson h=7EmptybucketAcceleration of 18 buncheson h=21 to 25 GeV PS ejection:72 buncheson h=84 in 1 turn320 ns gapaccelerates protons to 50 MeV and delivers them to the first synchrotron preaccelerator (cid:22)

PSB. Here, the incoming beam is split, with each bunch entering one of the four stacked

PSB rings, and the protons are accelerated to 1.4 GeV. PS is filled from PSB using a two-

batch filling scheme, whereby bunches from three PSB rings are transferred to PS two times,

resulting in 6 equally spaced bunches leaving through one beam pipe, as shown in Fig. 4.4.

The PS accelerates the protons to 25 GeV, and splits the bunches, giving them the 25 ns

spacing needed for the LHC. This splitting scheme is illustrated in Fig. 4.5. As the protons

gain energy, the harmonic number is increased by a factor of 3 and then 4, splitting each

bunch into 12. Since initially 6 bunches are injected at h = 7, a 320 ns gap is formed in

the bunch train to accommodate the rise-time of the ejection kicker. The SPS raises the

proton energy to 450 GeV. The single beam coming out of the SPS is split into two, by

injecting alternating bunches into one of the two LHC rings. Table 4.1 gives a summary of

the beam preparation stages. The fill scheme described here is closest to the standard, but

other schemes are also possible.

The LHC filling scheme depends on the operation mode [146]. The Standard Physics

filling scheme in 2018 was 25ns_2556b_2544_2215_2332_144bpi_20inj, which means 25 ns

bunch spacing, 2556 bunches per beam, 2544 collisions at Point 1 (ATLAS) and Point 5

(CMS), 2215 collisions at Point 2 (ALICE), 2332 collisions at Point 8 (LHCb), 144 bunches

in the longest train, and 20 injections per beam [147].

Beams circulating in the LHC deteriorate over time, as their transverse profile disperses

after many round trips in the magnetic field. One of the main concerns for the stable

operation of the LHC is prevention of frequent quenches of the superconducting magnets. If

too many particles stray from the beam and hit the magnets, the magnets may be heated

over their critical temperature and lose their superconductivity, and if the energy stored in

81

a magnet is not diverted, the magnet can be destroyed by resistive heating. While typical

quenches can be handled by the safety system, which releases the current from the affected

magnets, the number of quenches has to be kept to a reasonable minimum for efficient

operation of the LHC. After a quench, or as a result of a safety measure, the circulating

LHC beams need to be dumped, before new ones can be injected. For this purpose, there

exists a dedicated beam dump facility, located at Point 6. LHC beams are always injected

with empty bunches, which, in case of a dump, provide time for the kicker magnets to turn

on and steer the beam out of the ring. After ejection, the beam is dispersed by dilutor
magnets, which reduce its density by a factor of ∼ 105. The diluted beam is disposed of by
colliding it with a beam stop, whose main component is a graphite composite cylinder, 8 m

long and 1 m in diameter, which is necessary to stop a beam carrying up to 360 MJ.∗ The

typical lifetime of an LHC beam as on the order of 10 hours. [137, 148]

The quantity that measures the ability of a particle accelerator to produce bunch crossings

for particle collisions is called luminosity. It is the proportionality factor between the number

of events per unit of time, dN/dt, and the interaction cross section, σ:

dN
dt

= L σ.

(4.2)

Luminosity, rather than collision rate, is used as the measure of collider performance, because

luminosity does not factor in the probability of particle interaction, and is thus a direct

measure of the machine’s ability to create the environment in which particle collisions can

∗The 360 MJ energy of a single LHC beam is comparable to the kinetic energy of a passenger train.

82

take place.∗ For two beams colliding head-on, luminosity can be expressed as

L =

N1N2fb

Aeff

,

(4.3)

where N1 and N2 are the numbers of particles per bunch in the respective beams, fb
frequency of bunch crossings, and Aeff is the effective area of the overlap of the beams.

is the

Integrated luminosity,† (cid:82) L dt, is often quoted as the amount of data collected in a par-

ticular LHC run. Fig. 4.6 shows the integrated luminosity as a function of time for Run 2,
which amounts to all the data collected at √s = 13 TeV between 2015 and 2018. The analysis
presented in this dissertation utilized 139 fb−1 of ATLAS data.

Fig. 4.7 shows the total pp cross section as a function of center of mass energy. Given

Figure 4.6: The integrated luminosity for √s = 13 TeV data delivered by the LHC, recorded
by the ATLAS detector, and deemed good for physics analyses during Run 2. [150].

∗For a primer on concept of luminosity see [149].
†To disambiguate luminosity from integrated luminosity, the former is often referred to as instantaneous

luminosity.

83

Month in YearJan '15Jul '15Jan '16Jul '16Jan '17Jul '17Jan '18Jul '18-1fbTotal Integrated Luminosity 020406080100120140160ATLASPreliminaryLHC DeliveredATLAS RecordedGood for Physics = 13 TeVs-1 fbDelivered: 156-1 fbRecorded: 147-1 fbPhysics: 1392/19 calibrationFigure 4.7: Left: Comparison of the total and elastic pp cross section measurements, includ-
ing measurements form ATLAS and TOTEM at 7 and 8 TeV [151]. Right: The inelastic pp
cross section as a function of √s, including measurements at √s = 13 TeV [152].

the nominal LHC peak luminosity of L ≈ 1034 cm−2s−1 = 10 nb−1s−1, for σpp ≈ 100 mb at
√s = 13 TeV, one would expect the corresponding total interaction rate of up to 1 GHz for
continuously colliding beams. This large interaction rate implies that multiple interactions

occur per bunch crossing. With the beams partitioned into bunches with 25 ns spacing,

colliding at 40 MHz rate, the 1 GHz expected interaction rate implies the average number of
interactions per bunch crossing, (cid:104)µ(cid:105), on the order of 25. This is referred to as in-time pileup.
Fig. 4.8 shows the distributions of observed numbers of interactions per bunch crossing for

the subsets of Run 2 data collected in different years. Due to finite signal integration time for

various parts of the detectors, products of interactions within the preceding bunch crossing

may appear as a part of the events from following crossing. This is known as out-of-time

pileup. Effects of pileup are a major source of experimental uncertainty for data analyses.

This will become especially significant for the High Luminosity LHC.

Even though higher luminosity results in increased pileup, raising the luminosity is desired

for efficient data collection. Studies of extremely rare processes require enormous numbers

of interactions, out of which only a few will result in the process of interest. Effects of

84

 [GeV]s10210310410 [mb]s050100150200250totselsATLASTOTEMppLower energy ppLower energy Cosmic raysCOMPETE HPR1R2)s(2) + 0.14 lns12.7 - 1.75 ln((a)70007200740076007800800092949698100102104Figure 4.8: The luminosity-weighted distribution of the mean number of interactions per
bunch crossing during Run 2. The mean number of interactions corresponds to the mean of
the Poisson distribution of the number of interactions per crossing calculated for each bunch.
In this calculation the σinel

is taken to be 80 mb for √s = 13 TeV [150].

pileup can be addressed using primary vertex identification algorithms for in-time pileup,

and application of correction factors obtained through Monte Carlo simulation studies for

out-of-time pileup.

4.2 Laboratory coordinates

The ATLAS detector uses a right-handed coordinate system whose origin lies at the

nominal interaction point (IP) at the center of the detector. The x axis points towards the

center of the LHC ring, the y axis points upwards, and the z axis points along the beam

line. Either of the directions parallel to the z axis are refered to as forward. The cylindrical

radial coordinate, r, is measured in the x-y plane. Two angular coordinates are also defined.

φ measures the azimuthal angle in the plane perpendicular to the z axis, such that φ = 0

85

01020304050607080Mean Number of Interactions per Crossing0100200300400500600/0.1]-1Recorded Luminosity [pbOnline, 13 TeVATLAS-1Ldt=146.9 fb> = 13.4µ2015: <> = 25.1µ2016: <> = 37.8µ2017: <> = 36.1µ2018: <> = 33.7µTotal: <2/19 calibration∫direction is parallel to the x axis. The detector is essentially symmetric with respect to φ. θ

is the polar angle, measured such that θ = 0 and θ = π directions are along the positive and

negative z axis directions respectively.

Since the two colliding LHC beams have equal energies and opposite momenta, the result-

ing events have no preferential orientations within the detector. Due to the axial symmetry

of the experiment, ∆φ is invariant with respect to the event rotation around the z axis.

However, this is not true for ∆θ. For this reason, the polar angle of an object is typically

given in terms pseudorapidity.

Rapidity, is a measure of relativistic velocity, or boost, between two reference frames.

Unlike velocity, v = βc, rapidity, y = tanh−1 β, is additive for one-dimensional motion, which
makes differences in rapidity relativistic invariants, and therefore, convenient quantities in

relativistic physics. Since,

E = mγ

and p = mγβ,

the rapidity of an object can also be calculated from its energy and momentum,

y =

1
2

log

E + p
E − p

.

(4.4)

(4.5)

For a fixed interaction point, rapidity in the z direction maps out the polar angle. How-

ever, the mapping is dependent on the object’s mass, m. To avoid this dependence, we define

pseudorapidity,

η =

1
2

log

p + pz
p − pz

= tanh−1 pz
p

= − log tan

θ
2

.

(4.6)

With this definition, η → y for E (cid:29) m, with η = y for m = 0. Since this condition is largely
satisfied for the objects produces in the LHC collisions, differences in pseudorapidity, ∆η,

86

are nearly invariant with respect to the forward or central direction of the vectors within

the detector, and are much more useful measures of the polar angle separations than ∆θ,

equal values of which correspond to smaller separations in the central region, and larger

separations in the forward regions.

Most of the event information useful for physics analyses is contained in the angular

quantities, energies, and masses of the final state objects, with the radial information mostly

only important for event reconstruction. Since events can have arbitrary orientation, it is

often important to talk about angular separation in the η-φ plane,

∆R =(cid:113)∆η2 + ∆φ2.

(4.7)

For instance, the ∆R quantity is important in the jet clustering algorithms.

4.3 ATLAS experiment

ATLAS is the largest ever built general purpose particle detector. Located at Point 1

along the LHC accelerator ring, 92 meters underground, it’s a cylindrical construct, 25

meters in diameter and 44 meters long, weighing approximately 7,000 tons, as illustrated

in Fig. 4.9. The function of the detector is to identify and precisely measure energies and

momenta of particles originating from proton-proton collisions that take place at its very

center. The primary goal of ATLAS and the LHC program is to fully investigate physics

in a new energy regime. The first part of that is the discovery of the Higgs boson and

measurements of its properties [1]. But an even larger, long-term agenda consists of precision

physics measurements and searches for physics beyond the Standard Model.

The physics goals impose a range of requirements to allow for detection of the events of

87

Figure 4.9: Computer generated image of the ATLAS detector indicating all of its major systems [153].

88

interest. These requirements include:

• Hermeticity. In order to fully reconstruct a collision event, it is important to not

let particles escape undetected. There are three primary ways for particles to avoid

interaction with the active detector elements. Rapidity coverage needs to be optimized

in order to detect particles in the forward region, close to the beam pipe. However,

at a certain point, returns are diminishing, as good forward rapidity resolution is hard

to achieve, because on the one hand, smaller angles correspond to larger changes in

rapidity, and on the other, forward objects tend to be more collimated.

It is also

difficult to provide good particle measurements close to the beam pipe. Secondly,

different particles at different energies have different interaction lengths. It is important

to have sufficiently thick sub-detector systems in order to fully capture the particles’

energy. The third factor that reduces hermeticity is the unavoidable need to leave

room between active elements for structural components and cables providing power

and communication. Sufficient hermeticity is also required to reduce uncertainty on

the missing transverse energy measurements.

• Speed of response. With the LHC collisions happening every 25 ns, the detector

systems have only a few nanoseconds to measure an event, to leave time for the triggers

to process it. Fast response also reduces out-of-time pileup effects.

• Granularity and spacial resolution. Accurate position and momentum measure-

ment and particle identification require fine spacial resolution. High granularity also

reduces electronics and pileup noise, and improves speed of response [154].

• Radiation tolerance. The large number of high-energy LHC collisions results in

high intensity radiation near the interaction points. Consequently, active detector

elements and on-detector electronics must comply with radiation tolerance require-

89

ments [155, 156]. This is especially important due to the need to operate the detector

for a year or more without full access to its internal components.

Designed as a general purpose detector, ATLAS can be used to study a wide variety of

event topologies, with its subsystems dedicated to specific collision products. These include

either individual particles, e.g. photons, electrons, muons, or composite objects, like hadronic

jets. Methods used for reconstruction of these objects are discussed in Chapter 5.

The innermost part of the detector, located closest to the beam pipe and inside the

solenoid magnet, consists of tracking subsystems. Their purpose is to map out charged par-

ticles’ tracks curved by the field of the solenoid magnet, which allows for determination of the

particles’ momenta. Situated outside the solenoid magnet are electromagnetic and hadronic

calorimeters, which are designed to measure energy of electrons, photons, tau leptons, and

hadronic jets. The outermost sub-detector system is the Muon Spectrometer, which mea-

sures momenta of muons, which penetrate through the inner detector and the calorimeters.

Principles of design and operation of the ATLAS detector systems are discussed in the rest

of this chapter.

4.4 Inner tracking detector

The primary goal of the inner detector (ID) [158] is to provide capabilities for reconstruc-

tion of paths of charged particles, or tracks. The ID is contained inside the 2 T magnetic

field of solenoid magnet, to allow measurement of the particles’ charge and momenta from

the curvature of the particles’ tracks. Prevision measurements can be achieved in pT range
between 0.5 GeV (in certain cases as low as 0.1 GeV) and 150 GeV. Using track reconstruc-

tion, particles can be traced to primary or secondary interaction vertices. Identification of

90

Figure 4.10: ATLAS inner detector. Top: 3D rendered image [157]. Bottom: Schematic
view of a quarter-section showing each of the major detector elements and dimensions [127].

91

End-captransitionradiationtrackerEnd-capsemiconductortrackerBarreltransitionradiationtrackerPixeldetectorBarrelsemiconductortrackerEnvelopesPixelSCT barrelSCT end-capTRT barrelTRT end-cap255<R<549mm|Z|<805mm251<R<610mm810<|Z|<2797mm554<R<1082mm|Z|<780mm617<R<1106mm827<|Z|<2744mm45.5<R<242mm|Z|<3092mmCryostatPPF1CryostatSolenoid coilz(mm)Beam-pipe   Pixelsupport tubeSCT (end-cap)TRT(end-cap)12345678910111212345678Pixel400.5495580650749853.89341091.51299.91399.71771.42115.225052720.200R50.5R88.5R122.5R299R371R443R514R563R1066R1150R229R560R438.8R408R337.6R275R644R10042710848712PPB1Radius(mm)TRT(barrel)SCT(barrel)Pixel PP13512ID end-platePixel400.549558065000R50.5R88.5R122.5R88.8R149.6R34.3primary vertices allows to separate and properly assign products of multiple events occur-

ring within the same bunch crossing, i.e. to distinguish products of an interesting event from

pileup. Identification of secondary vertices allows to tag events, e.g. b-tagging.

Three subsystems comprise the ID, enclosing the collision point as concentric cylindrical

shells. In order of proximity to the beam line they are: the Pixel detector, the Semiconductor

Tracker (SCT), and the Transition Radiation Tracker (TRT). The ID covers the pseudora-
pidity of |η| < 2.5 and extends radially to 1.15 m.
the geometric arrangement of the detector elements, ID subsystems are separated into the

It is illustrated in Fig. 4.10. Due to

central barrel and end-cap portions. The Pixel detector is wholly contained in the barrel

region of SCT and TRT, but still has an end-cap section.

4.4.1 Pixel detector

The Pixel detector [159–161] is composed of solid state ionizing radiation sensors. A

sensor is an array of bipolar diodes (pixels) implanted into a silicon wafer, with the p-n

junctions operating under reverse bias. Charged particles from beam collisions are detected

when they pass through the sensors, where they excite electrons in the Si lattice, creating

electron-hole pairs, which drift between the pixels’ anode and cathode, producing measurable

electric signal on top of the reverse leakage current.

A pixel sensor consists of a 256 µm thick Si n-bulk. The bulk contains n+ implants on

the read-out side and a p-n junction on the back side. Each 24.4 × 63.4 mm2 sensor tile

contains 47232 pixels arranged in 144 columns and 328 rows. Individual pixels have a size of

30 × 382.5 µm2, occupying an area of 50 × 400 µm2 including spacing. Each tile is attached

to 16 front end (FE) chips using flip chip (or bump bonding) technology. The FEs, which

amplify the pixel currents, communicate with the detector control system via on-module

92

Figure 4.11: Left: Pixel detector module [162]. Top-right: 3D rendered image of the pixel
detector [163]. Bottom-right: Diagram of a transverse slice of the pixel layers [164].

Module Control Chips (MCC), which handle data compression and transmission through

fiber optics. Fig. 4.11 shows the module design and their arrangement in the detector.

The Pixel detector is made of 1744 modules arranged in 3 coaxial central barrels and

3 end-cap disks. This brings the number of the Pixel detector readout channels to over

80 million, constituting about 50% of the ATLAS total. 3 layers are necessary in order to

provide at least 3 hits per track, required for effective reconstruction. In 2014, a fouth layer,

called the Insertable B-Layer (IBL), was installed at the radius of 33 mm with 50 × 250 µm2

pixels [165]. The IBL improves the reconstruction of secondary vertices.

Located centimeters away from the collision point, pixel sensors need to be incredibly

resilient to radiation damage. Oxygen impurities are introduced in the bulk Si to increase its

tolerance to damage by charged hadrons. To reduce leakage current, sensors are operated at
the temperature between −5 °C and −10 °C. The special design of semiconductor implants

93

End-capdisksLayer0(B-Layer)Layer1Layer2r = 42.5 mm : IBL Support Tube (IST)r = 33 mm : Insertable B-Layer (IBL)r = 29.3 mm : Inner Positioning Tube (IPT)r = 24.3 mm : New Beam Piper = 50.5 mm : B-Layer (286 modules)r = 88.5 mm : Layer-1 (494 modules)r = 122.5 mm : Layer-2 (676 modules)20°is used to allow operation even after extensive radiation damage has been received.

4.4.2 Semi-Conductor Tracker (SCT)

SCT [166, 167] uses similar technology to the Pixel detector. The primary difference is

that instead of pixels, SCT sensors contain strips. To provide two-dimensional resolution

in the η-φ plane, each module is composed of two sensors, glued back-to-back around a

thermally conductive spine, with one rotated by 40 mrad with respect to the other in the

same plane.∗ This construction is illustrated in Fig. 4.12. The shear shift allows the SCT to

achieve resolution of 16 µm in φ and 580 µm in z.

Figure 4.12: Photograph (left) and drawing (right) showing the main components of a SCT
barrel module [127].

The SCT consists of four concentric barrels (containing 2112 modules) and two end-caps

of nine disks each (containing 988 modules per end-cap). Like the pixel modules, the SCT
modules are cooled down to −7 °C. Each SCT module has 1536 channels, with the sensor
strips pitched 80 µm apart. A module is read out via the attached hybrid assembly containing

front-end ASICs, providing binary readout.

∗All modules except the inner modules in the forward region actually contain two sensors on each side.

94

Hybrid assemblyBeO facings (far side)Slotted washerConnectorBeO facings (cooling side)Datum washerBaseboard TPGSilicon sensors4.4.3 Transition Radiation Tracker (TRT)

The basic elements of TRT [168–171] are gas-filled drift tubes, or straws, which detect

ionization of the gas due to passing charged particles. The straws are made from Aluminum

tubes, 4 mm in diameter and 40–150 cm long depending on their location, covered with

Kapton film reinforced with carbon fiber bundles. A 30 µm thick, gold plated tungsten wire

is under tension inside each tube. The tubes are filled with a 70% Xe, 26% CO2, 3% O2
gas mixture, at an overpressure of 5–10 mbar. The wire (anode) and the tube (cathode) are

held at a potential difference of 1530 V. When a charged particle passes through a straw, it

ionizes the gas, producing electrons which drift to the anode wire. Their current is amplified

and read out by the electronics, recording a hit for track reconstruction.

Besides aiding in tracking, the TRT also helps with particle identification, using its

namesake phenomenon, transition radiation. Transition radiation (TR) is emitted when a

relativistic charged particle traverses a boundary between two materials with different di-

electric constants. To produce TR the space between TRT straws is filled with sheets of

loosely packed polypropylene-polyethylene fiber. The intensity of emitted TR is logarithmi-

cally proportional to the Lorentz factor of the particle. Consequently, lighter particles are

much more likely to produce TR. This allows the TRT to distinguish electrons from charged

pions,∗ by optimizing the radiator material so that principally only electrons emit TR. TR

in the TRT comes in the form of soft X-rays, which also ionize the gas in the straws. This

produces signals with higher amplitude, referred to as “high threshold” hits, which signify

TR from electrons.

The TRT consists of a central barrel and two forward end-caps. The straws are oriented

∗Other hadrons are heavier than pions, and so have even smaller probability of emitting transition

radiation. Both electrons and positrons emit TR.

95

Figure 4.13: Left: Schematic drawing of the TRT barrel showing module and straw posi-
tions [127]. Right: Photograph of a TRT wheel module [172].

parallel to the beam pipe in the barrel, and perpendicularly in the end-caps. The barrel,

shown in Fig. 4.13, is assembled from 3 cylindrical layers, or rings, of modules and has
pseudorapidity coverage of |η| (cid:46) 1. Each ring contains 32 modules. The barrel provides
tracking information only in the transverse plane, with resolution of 130 µm. A TRT end-

cap consists of 20 wheels, each containing 8 layers of straws with 768 straws per layer. This
provides position measurement accuracy of 30–50 µm and covers the region of 1 < |η| < 2.
The TRT records, on average, 36 hits for each charged tracks with pT > 0.5 GeV and |η| < 2.0,
except in the barrel/end-cap transition regions, 0.8 < |η| < 1.0, where on average 22 hit are
recorded. The TRT operates at a temperature of approximately 20 °C.

4.5 Calorimeters

Calorimeters are detectors that measure energy. Segmentation of calorimeters’ active

components allows to also use them for position measurements. Calorimeters supplement

tracker measurements providing additional information about energetic charged leptons and

hadrons, as well as neutral particles invisible to the trackers. The principle of operation of

96

MODULE 1MODULE 2MODULE 3520 straws697 < r < 864864 < r < 1070793 straws329 straws560 < r < 697a calorimeter relies on making incident high-energy particles undergo multiple interactions,

which produce cascades, or showers, of secondary, lower energy particles. The lower energy

particles are absorbed by the calorimeter, which results in a quantifiable process (e.g. ion-

ization or scintillation) that allows to measure the deposited energy. Relying on scattering

and absorption for their operation gives calorimeters the ability to see neutral particles,

e.g. photons and neutrons. Calorimeters allow to distinguish electromagnetic and hadronic

particles by the shapes, lengths, and locations of their showers. They are crucial for jet

reconstruction and provide indirect measurements of EmissT
the sum of observed objects’ momenta.

, which is computed by balancing

TileCal Long Barrel

TileCal Extended Barrel

LAr hadronic
end-cap (HEC)

LAr electromagnetic
end-cap (EMEC)

LAr electromagnetic
barrel (EMB)

LAr forward
(FCal)

Figure 4.14: Cutaway computer generated image of the ATLAS calorimeter systems [173].

ATLAS contains a number of calorimeter systems, which can be grouped together differ-

ently according to the purpose, principle of operation, and structural connection. Thus, we

97

distinguish electromagnetic (more inner) and hadronic (more outer) systems; liquid argon

(LAr) ionization detector and plastic tile (TileCal) scintillator systems; barrel and forward

end-cap systems. Table 4.2 gives a concise summary of the ATLAS calorimeter systems.

Table 4.2: ATLAS calorimeter systems.

Name
EMB
EMEC
HEC
TileCal
FCal

Purpose

EM
EM

Hadronic
Hadronic

EM, Hadronic

Active material Absorber Location
Barrel
Pb
End-cap
Pb
End-cap
Cu
Steel
Barrel
Cu, W End-cap

LAr
LAr
LAr
Plastic
LAr

Inelastic scattering due to electromagnetic (EM) and strong interactions produce distinct

types of showers. These are illustrated in Fig. 4.15. The primary modes of interaction pro-

ducing EM showers are bremsstrahlung for charged particles (electrons) and pair production

for photons. The shower grows with the two processes alternating until the energy of the

photons falls below the electron pair production threshold of 2me and the energy of elec-
trons falls below the critical energy of the material, Ec, at which point energy losses due to
bremsstrahlung and ionization become equal. Once ionization becomes the dominant pro-

cess, the shower begins to die out. Materials with higher atomic number have lower critical

energy. The ionization radiation in the active medium is what creates signal in the LAr EM

Electromagnetic
shower

Hadronic
shower

Figure 4.15: Sketches of electromagnetic (left) and hadronic (right) showers.

98

e+e−γγγγγγe+e+e+e+e+e+e−e−e−e−e−ΚL0π0nμνμνμμπ0π0Κs0Detectible hadronic componentDetectible EM componentUndetectible componentcalorimeter. Shower profiles are determined by two parameters: radiation length, X0, and
. The radiation length is the mean distance a particle travels before its
Moli`ere radius, RM
energy is reduced by a factor of 1/e, and determines the length of the shower. X0 is typically
on the order of 1 cm. The Moli`ere radius is the radius that contains 90% of the shower’s

energy, and the transverse spread of the shower. Its value is proportional to the radiation

length and inversely proportional to the critical energy,

RM ∝

X0
Ec

.

(4.8)

Hadronic showers develop due to strong interaction with matter, primarily through sec-

ondary hadron production, nuclear deexcitation, and decays to pions and muons. The process

continues until the hadrons fall below the pion production energy threshold. Longitudinal

propagation of hadronic showers is characterized by the material’s nuclear interaction length,

, which is the mean free path a hadron travels between subsequent interactions with the

λint
material’s nuclei. Pions are the main intermediate constituents of a hadronic shower. On

average, 1/3 of those are neutral pions, which decay to pairs of photons, resulting in large

a fraction of a hadronic shower converting to electromagnetic. This fraction increases with

increasing hadron energy. Decays to neutrinos and muons results in undetectable or hard

to contain components of the shower. Due to the mixed nature of hadronic showers their

modeling is important for calorimeter design.

EM showers are usually shorter and narrower than hadronic ones due to the interaction

length typically being an order of magnitude larger than the radiation length. This allows to

separate the roles of detection of EM and hadronic showers between two calorimeters, with

the EM calorimeter placed closer to the collision point, and the hadronic one positioned be-

99

hind it. Careful choice of materials allows EM showers to be fully contained in the respective

calorimeter, while allowing hadronic showers to only deposit a small fraction of their energy

in the EM system and to mostly develop and be contained within the hadronic calorimeter.

Other factors in the choice of materials are reduction in the transverse dispersion of showers

to improve spacial resolution, as well as cost and radiation tolerance. In order to achieve

good energy resolution, it is important to make calorimeters sufficiently thick to contain

showers as fully as possible. ATLAS EM calorimeters provide about 22 X0 and the hadronic
ones provide about 10 λint

.

Depending on whether the active material of a calorimeter is the same as the absorber

or not, distinguishes between homogeneous and sampling calorimeters respectively. All

calorimeters on ATLAS are of the sampling type. The materials used in them are listed in

Table 4.2. Typically, a calorimeter has a higher response to the EM than to hadronic shower

component. This is usually expressed as e/h > 1. A calorimeter that is designed to have
e/h ≈ 1 is called a compensating calorimeter. ATLAS calorimeters are non-compensating.

Energy resolution of a sampling calorimeter is modeled by Eq. (4.9),

σ
E

=

a
E ⊕

b
√E ⊕ c,

(4.9)

with the terms corresponding to noise, sampling or stochastic, and constant contributions

respectively. The noise term represents electronic noise of the readout system. Scintillator

calorimeters generally have lower noise than those using ionization radiation due to the use

of photomultiplier tubes (PMTs). The stochastic term represents shower fluctuations and

is dependent on the choice of active and absorber materials and the number of absorber

layers and their thickness. The constant term does not depend on the shower energy and

100

represents effects due to calorimeter geometry and non-uniformities. With increasing shower

energy, the relative importance of the terms increases from noise, to stochastic, to constant.

Table 4.3 lists design goals and test-beam measurements of the energy resolution of the

ATLAS calorimeters.

Table 4.3: Energy resolution parameters of ATLAS calorimeter systems measured at test
beam [174–178].

Calorimeter
EM Barrel
EM End-cap
HEC
TileCal
FCal EM
FCal Hadronic

Beam type Stochastic, % Constant, %

electron
electron

pion
pion

electron

pion

10.1 ± 0.1
10 to 12.5
70.6 ± 1.5
56.4 ± 0.4
28.5 ± 1.0
94.2 ± 1.6

0.6

0.17 ± 0.04
5.8 ± 0.2
5.5 ± 0.1
3.5 ± 0.1
7.5 ± 0.4

4.5.1 Electromagnetic calorimeter

The ATLAS electromagnetic calorimeter (ECal) [179–181] consists of a barrel (EMB) and

two end-cap (EMEC) parts, all using liquid argon (LAr) technology, and each housed in its

own cryostat. To reduce the amount of material in front of the calorimeter, the same vacuum

vessel is shared between EMB and the central solenoid magnet, thereby eliminating two vac-
uum walls. EMB covers the |η| < 1.475 region with two half-barrels separated by a 4 mm gap
at η = 0, and EMEC covers the 1.375 < |η| < 3.2 region. Each end-cap calorimeter is struc-
turally divided into an outer wheel, 1.375 < |η| < 2.5, and an inner wheel, 2.5 < |η| < 3.2. A
thin presampler layer [182] is located between the cryostat and EMEC, covering the region
|η| < 1.8. It provides a measurement of the energy lost in front of the EM calorimeters. FCal
provides forward-most EM coverage on 3.1 < |η| < 4.9.

The design of the main part of the electromagnetic calorimeter, EMB, is illustrated in

101

Fig. 4.16. Its interesting accordion geometry is motivated by the desire to eliminate projec-

tive azimuthal cracks∗ that contribute to the constant term of the electromagnetic energy

resolution (see Eq. (4.9)). As an ionization detector, its principle of operation relies on inci-

dent particles ionizing the active medium, LAr. The electrons and ions then drift between

the Cu-Kapton electrodes, producing a current proportional to the number of ionization in-

teractions. To create the motion of charges, the electrodes are biased at 2000 V [183]. Liquid

argon was chosen for its linear behaviour, stability of response over time, and radiation-

hardness [127]. To keep the argon liquid, the cryostats maintain the temperature of 89.3 K

through liquid nitrogen cooling.

As the top diagram of Fig. 4.16 show, the EM barrel is made up of three radial layers

of decreasing granularity. Spacial resolution and thickness of the layers is summarized in

Table 4.4. The presampler is also referred to as Layer 0. The lateral segmentation of the

LAr detector allows it to distinguish photons, γ, from neutral pions, π0. The pions almost
always decay to two photons, which up to a certain pT threshold can be resolved.

Table 4.4: EMB layer segmentation dimensions.

Layer 1
Layer 2
Layer 3

∆η
0.003
0.025
0.05

∆φ
0.1
0.025
0.025

Thickness, X0

4.3
16
2

EMEC is different from EMB in that its inner wheel has only 2 layers. Otherwise, it

utilizes the same accordion geometry and also uses lead as the absorber material. The EM

part of FCal uses Cu as the absorber and LAr filled drift tubes [184].

∗The accordion geometry reclaims azimuthal (φ) coverage because the accordion layers are stacked in φ.

102

Figure 4.16: Structure of the ATLAS electromagnetic calorimeter. Top: Schematic drawing
of an EMB segment showing layers and ganuarity [127]. Bottom: Photograph of an open
EMB cell showing its accordion structure [180].

103

Δϕ = 0.0245Δη = 0.02537.5mm/8 = 4.69 mm(cid:0)Δη = 0.0031Δϕ=0.0245×4(cid:0)36.8mm×4(cid:0)=147.3mmTrigger TowerTriggerTowerΔϕ = 0.0982Δη = 0.116X04.3X02X01500 mm470 mmηϕη = 0Strip cells in Layer 1Square cells in (cid:0)Layer 21.7X0Cells in Layer 3(cid:0)Δϕ×Δη = 0.0245×0.05Cu-KaptonelectrodeHoneycombspacerPbabsorberplates4.5.2 Hadronic calorimeter

The hadronic calorimeter system on ATLAS is comprised of the large scintillator tile

barrel calorimeter (TileCal), two liquid argon end-cap calorimeters (HEC), and parts of the

liquid argon forward calorimeter (FCal).

4.5.2.1 Tile Calorimeter

In contrast to the ionization detectors described in the previous sections, which detect

a current of electrons created in a medium ionized by incident particles, scintillators emit

light when their electrons are deexcited, following excitation by radiation absorption. Light

in a particular wavelength range is emitted, depending on the available electron states of the

scintillator material. The measured energy can be calibrated to the observed light intensity.

Transparent scintillator material is required for propagation of light from the bulk to the

surface, where the light can be detected using photomultiplier tubes (PMTs).

The primary purpose of the TileCal [185–189] is to extend the ATLAS calorimetry to

fully contain and measure hadronic activity in the form of jets.

It is a sampling barrel

calorimeter, with plastic scintillator active medium and steel absorbers. Its angular coverage
is |η| < 1.7. A TileCal module is illustrated in Fig. 4.17. The bulk of a module consists of
an array of scintillator tiles interleaved with steel absorber plates. The fraction of materials

is approximately 4.7:1 by volume. The steel plates induce the hadronic showering process,

iron having a large cross section for hadronic interactions, limit the particles’ interaction

length, reducing the required calorimeter thickness, and provide structural support for the

scintillators.

An interesting aspect of the design is that the tiles are oriented perpendicularly to the

beam axis, i.e.

in direction largely parallel to that of the incident particles. This choice is

104

justified by the fact that the resolution of a sampling hadronic calorimeter is not critically

dependent on the orientation of the active medium, since at the end of a hadronic shower,

low energy particles have nearly isotropic velocities. Note also, that the LAr calorimeter

provides 2λ of interaction length before the TileCal. The parallel orientation of the tiles

allows for a more hermetic design, by routing the optical fibers in the radial direction within

small slots in the absorber. To prevent particles from traveling for too long inside a single

tile, the tiles are made small and are arranged in a staggered pattern [190].

The base material of the TileCal plastic scintillator is polystyrene. The scintillator pro-

duces ultraviolet light, which is converted to visible blue light by wavelength-shifting fluors

added to the plastic. The light is collected at the edges of each tile using wavelength-shifting

fibers. The fibers are also polystyrene-based and further increase the light’s wavelength to

476 nm. Each fiber collects light from tiles located at one or two radial depths, and each

tile is read out by two fibers, one on each side, providing double readout. At the interface

with the PMTs, fibers are bundled together, with a single PMT reading signals from many

tiles. This grouping splits a module into cells, approximately projective in pseudorapidity,

as shown in the bottom half of Fig. 4.17. A steel girder is located at the outer end of every

module. The girders function to support the calorimeter, which also supports the solenoid

magnet and the inner detector, as well house the on-detector TileCal electronics and PMTs,

and provide flux return for the solenoid magnetic field.

Modules are shaped like wedges for arrangement into barrels: two halves of the central

long barrel (LBA, LBC) and two extended barrels (EBA, EBC). 64 modules form a barrel,

with each module subtending 5.625° in azimuth. The barrels have inner and outer radii of

2.28 m and 4.25 m, including structural components. The modules are radially segmented

into three regions, corresponding to 1.5, 4.1, and 1.8 λint

in the central barrel and 1.5, 2.6,

105

Long Barrel
LBA/LBC

Extended Barrel
EBA/EBC

Figure 4.17: A diagram illustrating a TileCal module (top) and the positioning of cells within
the central and extended barrels (bottom) [127]. Cells D4 and C10 and refered to as the
Plug tile calorimeter, E1 and E2 as the Gap scintillator, and E3 and E4 as the Cryostat
scintillator.

106

PhotomultiplierWave-length shifting fiberScintillatorSteelSourcetubes50010001500 mm0A3A4A5A6A7A8A9A10A1A2BC1BC2BC3BC5BC6BC7BC8BC4D0D1D2D3A13A14A15A16B9B12B14B15D5D6D4C100,71,01,11,31,41,51,6B11B13A12E4E3E2E1beam axis0,10,20,30,40,50,60,80,91,22280 mm3865 mm=0,0h~~in the extended barrel. The LB modules contain 307 tiles per row for a total of

and 3.3 λint
3377. The EB module have between 140 and 157 tiles per row and a total of 1591.

Further discussion of TileCal, its electronic components, their planned upgrades, and my

involvement in relevant projects can be found in Appendix C.

4.5.2.2 Hadronic End-Cap calorimeter

The Hadronic End-Cap calorimeter (HEC) [179, 191] is a copper/liquid-argon sampling
calorimeter with a flat-plate design, which covers the range 1.5 < |η| < 3.2. HEC shares the
end-cap cryostats with EMEC and FCal. It consists of two wheels in each end-cap: a front

wheel (HEC1) and a rear wheel (HEC2), each with two longitudinal sections. Each wheel

is comprised of 32 identical wedge-shaped modules, sketched in Fig. 4.18. The modules of

HEC1(2) are made of 24(16) copper plates, each 25(50) mm thick, and a 12.5(25) mm thick

front plate. 8.5 mm gaps separate the plates. The space between the electrodes is maintained

using honeycomb sheets. Three electrodes divide the gaps into four separate 1.8 mm wide

Figure 4.18: HEC module diagrams [127]. Left: r-φ and r-z views of a module. Right:
Structure of the inter-plate gap.

107

LAr drift zones. The middle electrode is a 3-layer printed circuit that serves for readout,

while the other two electrodes have 2 layers and serve only to carry high voltage (1800 V).

This configuration allows to use lower voltage as well as reduces problems with ion build-up.

The typical drift time for electrons in the drift zone is 430 ns.

4.5.3 Forward calorimeter

The forward calorimeter (FCal) provides coverage in the forward-most region of 3.1 <
|η| < 4.9, at either end of ATLAS, and shares the cryostat with the end-cap calorimeters.
Located at such high rapidity, FCal modules are exposed to a very high flux of particles. The

FCal was designed with increased density to handle this flux and to limit radiation exposure

of the muon system behind it. To further limit downstream flux, a copper alloy shielding

plug is mounted behind FCal.

FCal is a liquid argon ionization detector using drift tubes. Its design is shown in Fig. 4.19.

Three modules compose FCal: one electromagnetic (FCal1), using copper for absorber, and

two hadronic (FCal2 and FCal3), using tungsten. W has a shorter λint
chosen in order to better contain hadronic showers and limit their lateral dispersion. Each

than Cu and was

module consists of an absorber of stacked thick plates with a matrix of drilled holes housing

Table 4.5: Parameters of FCal modules [127].

Function
Module mass, kg
Absorber material
LAr gap width, mm
Radiation length, X0
Absorption length, λint
Number of electrodes
Number of readout channels

FCal2

FCal3

FCal1
EM Hadronic Hadronic
2119
Cu
0.269
27.6
2.66
12260
1008

3826
W
0.376
91.3
3.68
10200
500

3695
W
0.508
89.2
3.60
8224
254

108

electrodes insulated from the bulk. An electrode consists of a Cu tube and a coaxial rod of

the same material as the bulk, separated by a precision, radiation-hard plastic fiber wound

around the rod. The LAr gaps in FCal are a lot smaller than the 2 mm ones in EMB to avoid

ion build-up and to provide the highest possible density. Table 4.5 lists the main parameters

of FCal modules.

Figure 4.19: Left: Diagram showing FCal modules located in the end-cap cryostat. The
black regions are structural parts of the cryostat. Right: Electrode structure of FCal1. The
other two modules are similar. The Moli`ere radius, RM
, is indicated by the solid circle. [127]

4.6 Muon Spectrometer

Muons are rather special particles from the experimental point of view. Their mass

is large enough to significantly reduce bremsstrahlung in comparison to electrons, but not

large enough to open up hadronic decay channels, as is the case for tau. Decaying exclusively

through the weak interaction, muon have a very long mean lifetime of 2.2 µs in their rest

frame, which is made even longer in the laboratory frame by relativistic time dilation. The

decay amplitude is further reduced by the small difference in mass between a muon and its

decay products, limiting the available phase space. The long lifetime in combination with

109

000000000000004505505004003506006506050403020100R (cm)z (cm)EMECHEC(back)PumpFCal 1FCal 2FCal 3(EM)(Had)(Had)(front)HECModerator shieldingshielding plugLAr CalorimeterRodFibreLArGapTubeMatrixR (mm)ηBeampipeWarmwallSuperinsulationColdwallBeamcenterRMreduced probability of interaction give muons much larger penetration depth than any other

collision products (aside from neutrinos). Because of this, they demand a dedicated system

surrounding the rest of the detector.

Figure 4.20: Cutaway computer generated image of the ATLAS muon system [192].

The ATLAS Muon Spectrometer [193, 194], illustrated in Fig. 4.20, is a detector com-

prised of four types of components: Monitored Drift Tubes (MDT) and Cathode Strip Cham-

bers (CSC) are used for precision tracking but are too slow for triggering, while Resistive

Plate Chambers (RPC) and Thin Gap Chambers (TGC) provide prompt signals for trig-

gering on the order of 10 ns. Four different components were necessary because a single

approach would have to compromise on either resolution or response time. All four, how-

ever, are similar in using gas ionization detection as their principle of operation.

Measurements of muon momenta are highly dependent on uncertainty of the position of

the Muon Spectrometer components. A 1 TeV muon has a sagitta of 500 µm, thus requiring a

110

ThinGapChambers(TGC)ResistivePlateChambers(RPC)OuterwheelBigwheelMonitoredDriftTubes(MDT)BarreltoroidEnd-captoroidCathodeStripChambers(CSC)Smallwheel50 µm sagitta resolution to achieve 10% resolution on the momentum, which requires knowing

positions of the detector components to 30 µm. To achieve this precision, a sophisticated

laser alignment system is installed on the detector [195].

The muon detector elements are interspersed with the ATLAS toroidal magnets, which

provide the magnetic field enabling momentum measurements in the Muon Spectrometer.

This system allows to measure muon momenta between 3 GeV and 1 TeV and provides

resolution better than 10% for 1 TeV tracks.

Monitored Drift Tubes (MDT) [196, 197] provide most of the precision tracking informa-

tion. They are located in the barrel and outer wheels and cover the pseudorapidity range
|η| < 2.7. The basic element of the MDT is a pressurized drift tube, 29.970 mm in diameter,
filled with 93% Ar, 7% CO2 gas at 3 bar. The electrons resulting from ionization are col-
lected at the central W-Re wire, 50 µm in diameter, at a potential of 3.08 kV. Regular MDT

modules, or chambers, consist of two groups of tube layers, called multi-layers, separated by

a mechanical spacer (Fig. 4.21). Each multi-layer contains 3 to 4 layers of drift tubes. Due

to the Muon Spectrometer sharing space with the toroidal magnets, there are also irregu-

lar MDT modules, containing at least 3 layers of tubes. The overall layout of the MDT’s

is projective, with the layer dimensions and the chamber sizes increasing in proportion to

their distance from the interaction point. There are 1088 MDT chambers of 18 main types,

covering a total area of 5500 m2.

Cathode Strip Chambers (CSC) are located in the small wheels and provide coverage
in the forward region of 2.0 < |η| < 2.7, where flux of particles and density of tracks is
much higher. Each small wheel is made up of two discs, one composed of 8 small chambers

and the other of 8 large chambers. The two discs are rotated with respect to each other

by π/8 in φ, with their modules slightly overlapping. The chambers are installed inclined

111

Figure 4.21: Schematic drawing of a regular (2 × 3) MDT chamber [193].

towards the interaction point by 11.59° with respect to the z axis. CSCs are multi-wire
proportion chambers operating with a gas mixture of 80% Ar and 20% CO2 and potential
difference of 1.9 kV. The central wire in each CSC is oriented radially with the other wires

parallel to it and pitched at 2.5 mm. Each plane of anode wires lies between two segmented

cathode planes separated from it by 2.5 mm. One cathode is segmented perpendicularly

to the anodes, providing precision coordinate in the bending plane with 60 µm resolution.

The other cathode is segmented parallel to the anodes, and provides a courser transverse

coordinate with 5 mm resolution. This scheme allows to distinguish more than one track at

a time. The timing resolution of CSC is 7 ns.

Resistive Plate Chambers (RPC) provide measurements for muon triggers in the bar-
rel region, |η| < 1.05. They are arranged in three cylindrical layers and are attached to
one or both sides of MDTs in the one outer or two inner layers respectively. Each RPC

consists of two parallel resistive plates, which are separated by a 2 mm gap, filled with

C2H2F4/C4H10/SF6 (94.7/5/0.3) gas, and are kept at 9.8 kV potential difference. There are
no wires between the plates, and the signal is read out via capacitive coupling to metallic

112

Cross plateMultilayerIn-plane alignmentLongitudinal beamstrips mounted on the outer faces of the resistive plates. With this design, the signal width

can be limited to 5 ns. The resistance of the allows for discharges to be spatially localized,

providing 10 × 10 mm2 resolution in the φ-z plane. Since both sides of the plates can be

read out, the RPC provides 6 measurements in η and φ per track. This redundancy in track

measurements allows the use of coincidence discrimination to reduce the fake trigger rate.

The RPC can trigger on muons in the range of 6–35 GeV.

Thin Gap Chambers (TGCs) are multi-wire proportional chambers, with many wires

enclosed between two plates. The plates contain cathode strips oriented perpendicularly to

the anode wires. The wires, 50 µm in diameter, are pitched 1.8 mm apart and are aligned in

a plane 1.4 mm from the cathodes, and operate at potential difference of 2.9 kV. The space

between the plates is filled with 55% CO2, 45% C5H12 gas. The TGCs provide information
for muon triggers in the end-cap region, 1.05 < |η| < 2.4 and are installed in the big wheel.
The TGCs have a resolution of 2–6 mm in the r direction and 3–7 mm in the φ direction,

and are able to supplement the φ measurements from MDTs. The middle layer of the MDTs

in the end-cap is accompanied by seven layers of TGCs, while the inner layer is accompanied

by only two layers.

4.7 Magnet system

A strong magnetic field throughout the detector is important to enable measurement

of the momentum of electrically charged particles. As the particles traverse the magnetic

field, they feel the Lorentz force perpendicular to the direction of their motion and the field,

(cid:126)B. This makes the particles travel along curved trajectories, with the radius of curvature

proportional to the momentum and the direction of curvature determined by the sign of the

113

charge. This makes the strength of the field particularly important for muons, as their energy

is not completely measured by the calorimeters. For this purpose, ATLAS features a system

of superconducting magnets [198] illustrated in Fig. 4.22. It includes a central solenoid, a

barrel toroid, and two end-cap toroids. Their main parameters are listed in Table 4.6. The

superconducting material used is aluminum stabilized NbTi/Cu.

The solenoid magnet creates a nearly uniform 2 T field for the inner tracking detector.

The field is parallel to the beam axis, resulting in tracks curved in the transverse plane. It

occupies the space between the inner detector and the calorimeters. In order to minimise the

amount of material in front of the electromagnetic calorimeter, the solenoid magnet shares

the cryostat vessel with the EMB, avoiding the need for two extra vacuum walls. This allows

the solenoid to contribute only 0.66 radiation lengths at normal incidence.

The 8 barrel toroids provide magnetic field for the central part of the muon detector.

The toroidal field is nearly circular about the z axis, and makes the particle tracks curve in

the r-z plane. This field is less uniform than that of the solenoid, and varies between 0.5 T

near the center of a loop and the 3.9 T peak value. The large spacing between the toroidal

coils also contributes to the non-uniformity of the field. Due to these complications, the

field must be carefully mapped to accurately track muons. Both the barrel and the end-cap

toroids have air cores. The ability of these magnets to create a strong field without a dense

core material, e.g. iron, allows a reduction of the effects of multiple scattering on the muon

spectrometer resolution.

In order to provide magnetic field for the forward part of the muon detector, and at the

same time to allow access to the inner detector, the field in the forward region is created

by separate end-cap toroids. The strength of their field varies between approximately 1 and

4.1 T. The coils of the end-cap toroids are offset by π/8 in φ with respect to those of the

114

Figure 4.22: 3D view of the bare windings of the ATLAS magnet system [199].

Table 4.6: Main parameters of the ATLAS magnet system [198].

Property

Inner diameter, m
Outer diameter, m
Axial length, m
Number of coils
Conductor, t
Cold mass, t
Total assembly, t
Number of turns per coil
Operating current, kA
Stored energy, MJ
Central field [127], T
Peak field, T
Overall size, mm2
Ratio Al : Cu : NbTi
Number of strands in
Rutherford cable
Strand diameter, mm
Critical current
(at 5 T, 4.2 K), kA
Total length, km
at 4.5 K, W
at 60-80 K, W

Central
Solenoid
2.44
2.63
5.3
1
3.8
5.4
5.7
1173
7.6
38
2.0
2.6
30 × 4.25
15.6 : 0.9 : 1

Barrel
End-Cap
Toroid Toroid (one)
1.65
10.7
5
8
20.5
160
239
116
20
206
1.0
4.1
41 × 12
19 : 1.3 : 1

9.4
20.1
25.3
8
118
370
830
120
20.5
1080
0.5
3.9
57 × 12
28 : 1.3 : 1

12
1.22

20.4
9.1
100
500

33
1.3

47.5
56
990
7400

40
1.3

55.5
12.8
329
1700

Overall
dimensions:

Weight:

Coils:

Conductors:

Cooling
power:

115

barrel toroids in order to provide radial overlap and to optimise the bending power in the

transition region.

4.8 Trigger and data acquisition system

To make the data available for physics, calibration, and performance analyses, signals

from the detector components have to be read out, digitized, consolidated, and written to

tape storage, where they can be later accessed for offline processing. The trigger and data

acquisition (TDAQ) system [200] serves as a bridge between the detector components and

the permanent data storage.

The DAQ communicates directly with the front-end electronics located on the detector.

It provides an interface for both controlling the detector (for example, turning its components

on/off and setting their calibration parameters) and reading out the detected signals. Each

sub-detector system has dedicated DAQ components, whose output data streams converge

into a combined readout system.

The readout dataflow is curated by the trigger systems [201, 202]. Considering that

the LHC bunch crossings occur at the rate of 40 MHz, and that every detector system has

thousands of readout channels, the amount of data coming from the detector can neither

all be saved or fully processed at the rate of collision events. As shown in Fig. 4.23, the

upstream data rate at the level of DAQ is approximately 60 TB/s. The triggers make real-

time decisions to discard uninteresting events in order to reduce the data rate and give time

to slower downstream system that perform more thorough event processing and record the

events to permanent storage. Efficiency of the triggers is carefully studied, as it has to be

corrected for in data analyses and contributes to the measurements’ uncertainty [204, 205].

116

Figure 4.23: Schematic diagram of the ATLAS trigger and data acquisition (TDAQ) sys-
tems [203].

The ATLAS trigger system consists of two tiers: a hardware first-level (L1) trigger, and a

software high-level trigger (HLT).

The L1 trigger [206] is implemented with fast custom-made electronics. The system

consists of the L1 calorimeter triggers (L1Calo) [207], the L1 muon triggers (L1Muon) [208],

the L1 topological trigger modules (L1Topo) [209, 210], and the Central Trigger Processors

(CTP) [211]. The primary task of the L1 system is to find regions of interest (RoIs) using

coarse calorimeter and muon information. A 2.5 µs latency is allowed for identification of

RoIs, while the event that is being processed is stored in the temporary pipeline memory.

Based on this information, the CTP makes the decision of whether to discard or accept the

event. Accepted events are buffered in the Read-Out System (ROS) and sent to the HTL

for further processing.

117

The HLT [212] reduces the L1 trigger output data rate from 100 kHz to about 1.5 kHz. At

this level, the available event processing time is increased to about 200 ms. This allows the

HLT to be implemented in software and to run much more complex algorithms than the L1

triggers. The LHT runs on 40 000 processor cores of a dedicated computer farm. In addition

to the RoIs identified at L1, the HLT is able to utilize full-detector information, including

finer-granularity calorimeter information, precision measurements from the MS, and tracking

information from the ID. In order to optimize the processing, most HLT triggers follow

a two-stage approach, with a fast first-pass reconstruction followed by a slower precision

reconstruction of the remaining events. Events accepted by the HLT are transferred to the

CERN computing center, and are stored in the RAW byte stream data format on magnetic

tapes. Only about one in a million ATLAS events is saved for offline analyses.

118

Chapter 5

Event reconstruction

Before physics analyses can be performed on the collected ATLAS data, physics objects

(photons, leptons, jets, and missing transverse momentum) have to be reconstructed from

the signals recorded by the trigger and data acquisition (TDAQ) system. The methods and

algorithms of object reconstruction are discussed in this chapter. The same reconstruction

methods are applied to the output of the detector simulation to provide for comparability

between data and MC.

As discussed in Chapter 4, the ATLAS detector consists of many radially layered detector

systems, with different response to different types of particles. This is illustrated in Fig. 5.1.

The principle of operation of most of these systems relies on detecting electric pulses produced

by ionizing radiation. The ionization strips electrons from atoms of the active medium.

The electrons than drift in the electric field created by high voltage, and are collected by

conductors, resulting in electric pulses that propagate to the DAQ circuits, which amplify

and record them. The TileCal relies on scintillation that produces light rather than electric

charge, but the light signals are ultimately converted to electric ones by the PMTs.

The information obtained from the inner detector systems is used to reconstruct tracks of

charged particles, with the Muon Spectrometer providing additional tracking measurements

for muons. The particles’ trajectories within the detector are bent by magnetic fields, so

the curvature of their tracks provides accurate measurements of momentum. The tracking

119

Figure 5.1: A schematic diagram showing interaction of different ATLAS detector systems
with various types of collision products [213, 214]. The Inner Detector maps out tracks
of charged particles; the Electromagnetic Calorimeter absorbs and measures the energy of
photons and electrons; the Hadronic Calorimeter performs the same function for jets formed
by e.g. protons or neutrons; and the Muon Spectrometer provides tracking information for
muons. The relative layer sizes are not drawn to scale.

120

ElectromagneticCalorimeterHadronicCalorimeterSolenoid magnetTrackingTransitionRadiationTrackerPixel/SCTdetectorMuonSpectrometerThe dashed tracksare invisible tothe detectorPhotonNeutrinoMuonProtonNeutronElectroninformation is also used for coordination with the measurements from other detector systems,

for example, in reconstructing jets with particle flow algorithms. Track reconstruction is

also necessary for identification of primary and secondary interaction vertices, which are

important for describing event topology and removing pileup.

The electromagnetic and hadronic calorimeters provide energy measurements and aid in

reconstruction of the respective types of particles. Unlike the trackers, calorimeters are able

to detect neutral particles, including photons and neutrons. However, ATLAS calorimeters

cannot detect neutrinos, as those only interact weakly. The analyses that look for neutrinos,

or other hypothetical neutral particles that would escape the detector, rely on calibration of

missing transverse energy.

Algorithms and qualitative descriptions of reconstruction procedures are presented in

this chapter. Details of object reconstruction methods and their performance relevant to the
H → γγ analysis can be found in [215].

5.1 Tracks and vertices

An essential part of the reconstruction of charged particles is reconstruction of their

tracks. The tracks are assembled from hits, which are points at which particles have inter-

acted within the active detector elements. The hit points, in their turn, are reconstructed

directly from the recorded signals. The ATLAS inner detector can provide 3, 4, and 36 hits

from the pixel, SCT, and TRT respectively.∗

As discussed in Chapter 4, each detector system has a different structure and segmen-

tation of the active elements, which determine their resolution. The inner detector silicon

∗Fewer or more hits can be recorded if a track of a low momentum particle curls in the magnetic field,

or strictly fewer if the track ends due to decay or complete absorption, or starts from a secondary vertex.

121

pixels are segmented in both φ and η dimensions, with r-layers providing 3-dimensional co-

ordinates of hits. The SCT is made using longitudinally oriented silicon strips rather than

pixels. Every SCT module contains two layers of the strip sensors, slightly rotated with

respect to one another, so that the z coordinates of the SCT hits can be obtained from the

intersection of the triggered strips. The TRT contains many more elements in the radial

direction than either the ID or the SCT, but lacks resolution in the longitudinal (transverse)

dimension in the barrel (end-cap), as it consists of parallel drift tubes.∗

Tracks are reconstructed from hits using an iterative fitting method employing the

Kalman filter algorithm [217, 218]. The procedure begins with the identification of “sili-

con spacepoints”, which are centers of pixel clusters in the pixel detector. The spacepoints

are combined into seeds of 2 or 3 hits using a seed-finding algorithm. Tracks are extrapolated

from the seeds to the next layer of the inner detector and the Kalman filter algorithm is ap-

plied to determine which hits, if any, are suitable for incorporation into the tracks. Tracks

to which a hit point was successfully added are then re-fitted and the procedure is repeated

iteratively until all inner detector layers have been inspected. Track fitting is done with a

χ2 algorithm [219].

Track extrapolation is performed by applying electromagnetic transport functions, which

require an accurate map of the magnetic field inside the detector [220, 221]. A perigee†

parametrization of tracks is used for calculations. This parametrization is illustrated in

Fig. 5.2 and consists of the following 5 parameters:

∗In addition to positioning information, timing information from the TRT can also be used. While it is
not typically used for vertex reconstruction, the measured time over threshold is correlated with the amount
of ionization within the straws, and can be used to better distinguish between electrons and pions based on
their expected dE/dx. Similar discriminators can be used to distinguish between other types of particles,
such as protons, kaons, or hypothetical highly ionizing exotic particles. [216]

†In this context, the perigee is the point on the track that is the closest to the beam axis.

122

• q/p – charge-to-momentum ratio;
• φ0 – the angle between the transverse momentum vector and the x axis in the x-y plane;
• θ0 – the angle between the transverse momentum vector and the z axis in the r-z plane;
• z0 – the longitudinal coordinate of the perigee;
• d0 – the transverse coordinate of the perigee, given as a signed quantity, d0 = ((cid:126)x0× ˆp)·ˆz.

y

(cid:126)pT

track

r

track

(cid:126)pT

φ0

d0

φ

θ0

d0

x

z

z0

Figure 5.2: Perigee parametrization of a track trajectory.

Transport calculations take into account the quasi-helical motion of the charged particles

within the inhomogeneous magnetic field, as well as interactions with detector material,

including scattering and energy loss due to ionization, bremsstrahlung, Compton scattering,

and hadronic interactions with atomic nuclei.

Many of the potential track candidates share hits, are incomplete, or are fake tracks,

i.e. tracks where the majority of associated measurements do not originate from one single

particle [218]. Ambiguities in track reconstruction are resolved by assigning a score to each

track depending on the track fit χ2, the number of hits, and the detector the hits originated

from [222]. The high-precision pixel detector hits have a higher score than those from the

SCT. Low-scoring track candidates are discarded. The ambiguities have to be resolved

before the extension into the outer TRT can be done. An example of an ambiguity in track

123

Figure 5.3: A simplified example of ambiguity inherent in track reconstruction [218]. Tracks
in the SCT barrel (a, b, c) share several hits. The ambiguity is resolved by assigning a higher
score to a track containing hits on both sides of the SCT strips.

reconstruction is shown in Fig. 5.3.

The procedure defined thus far is referred to as the “inside-out” strategy, as it follows the

particles’ natural trajectory direction, starting in the inner-most layers of the detector. How-

ever, not all tracks can be reconstructed in this manner. Tracks originating from secondary

vertices may not contain a sufficient number of hits in the silicon detector to be properly

seeded or to survive ambiguity resolution. The “outside-in” strategy remedies this by apply-

ing the procedure in the reverse direction, starting with hits in the TRT and propagating

inwards. Hits already assigned to tracks by the “inside-out” strategy are not considered by

the “outside-in” strategy.

After track reconstruction for the event has been completed, vertex finding algo-

rithms [223] are used to assign tracks to their respective vertices. In addition to the primary

hard-scatter vertex, event tracks can originate from secondary, or conversion, vertices from

decays of intermediate states, or additional primary vertices present because of pileup. Ver-

124

sensor hitmodule hitambiguous hitholeabctex reconstruction starts by forming vertex seeds out of bunches of tracks in close proximity

located tracks. The algorithm proceeds iteratively, forming new seeds using tracks that were

not compatible with the seeds in the previous iteration, and continues until every track is

matched with at most one vertex. Reconstructed vertices are required have at least two

associated tracks. The vertex with the highest value of (cid:80) p2T is normally chosen as the
primary one. However, for the H → γγ analysis, the primary vertex is identified as the
one compatible with the origin of the diphoton system. Matching of the photons to their

primary vertex is done using dedicated neural network algorithms, which use the photon
pointing information, and are trained on the gg → H → γγ MC event samples [215].

5.2 Photons and electrons

Both photons and electrons are reconstructed using measurements of energy deposited

by them in the EM calorimeter cells. The main difference between the two types of particles

is that, additionally, tracking information is available for electrons, which are distinguished

by successfully matching calorimeter deposits to the inner detector tracks. A sketch of the

detector components involved is show in Fig. 5.4.

Detection of a particle in the EM calorimeter is defined by the presence of a group, or

cluster, of adjacent cells that have energy deposited by the EM shower, which developed

as the particle propagated through the detector material. The showering process is mainly

initiated by an absorber material of high atomic number, deliberately installed for this

purpose. Two primary algorithms exist for reconstruction of the EM clusters: the sliding

window algorithm and the dynamical topological algorithm. Historically, the fixed-sized

sliding window algorithm was used, as the computational power necessary for calibration

125

Figure 5.4: Sketch of the ATLAS detector systems used to perform electron reconstruction
and identification. [224].

with a more sophisticated algorithm was not available. The dynamical topological algorithm

was introduced on ATLAS in 2017 [225]. It relies on multivariate techniques. The energy of

the calorimeter cells and clusters is calibrated using a multivariate algorithm [226] trained

on a large sample of single-electron fully reconstructed MC events.

For the sliding window clustering algorithm [227], the calorimeter is divided in the

η-φ plane into a grid of 200 × 256 elements known as towers, 0.025 × 0.025 in size. The

towers are built by summing the energy from all the overlapping cells stacked in the nor-

mal, i.e.

radial, direction. A scan is then performed of all 3 × 5 tower windows on the

grid, selecting the ones with total energy above 2.5 GeV as seeds. Clusters are then built

by merging cells within windows of 3 × 7 or 5 × 5, in the barrel or the end-cap, around the

energy-weighted centers of the seeds.

The dynamical topological algorithm [225] forms clusters in a way that more closely

resembles spatial signal-significance patterns produced by particle showers. The main ad-

126

��������������������������������������������������������������������������������������������������������������������������������������ηφ∆η×∆φ���������×�����∆η×∆φ��������×������∆η×∆φ�������×���������������������������������vantage of the algorithm is its ability to recover low-energy deposits from bremsstrahlung

photons and associate them to the electron clusters, forming so-called superclusters. Seeding

and growth of topo-clusters is governed by the cell significance, ς

EM
cell , defined as

ς

EM

cell =(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

σ

EM
cell

E
EM

cell noise(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)

.

(5.1)

EM
cell

where E

is the absolute cell energy at the EM scale and σ

EM
cell noise is the expected cell
noise. The algorithms proceeds as follows. First, proto-cluster seeds from calorimeter cells
cell ≥ 2 adjacent to a proto-cluster are added
to it. Finally, all neighboring cells are added to the cluster regardless of their significance.

cell ≥ 4 are selected. Then, all cells with ς

with ς

EM

EM

Because of the threshold values, the clusters are often referred to as “4-2-0” topo-clusters.

5.2.1 Electrons

Electron candidate tracks are reconstructed using a procedure,∗ consisting of pattern

recognition and track fitting, that allows energy loss from bremsstrahlung to be taken into

account, thereby improving reconstruction accuracy. Initially, the particle is assumed to be

a pion, and the energy loss is estimated accordingly. Using the pion hypothesis, track seeds

are formed by the pattern recognition algorithm. The seeds consist of 3 hits in the silicon

pixel detector and must have pT > 1 GeV. If a track candidate cannot be extended to a full
track with at least 7 hits, including the TRT, the pattern recognition step is redone using

the electron hypothesis instead, which allows for up to 30% energy loss at each material

surface, to account for bremsstrahlung. The tracks are fitted using the ATLAS global χ2

track fitter [219], assuming the particle hypothesis from the previous step. If the fitting fails

∗The electron reconstruction procedure is based on the default one discussed in Section 5.1, but involves

methods for accommodating energy loss due to bremsstrahlung.

127

using the pion hypothesis, it is redone assuming an electron, which involves an extra term

to account for bremsstrahlung.

The reconstructed tracks are matched with the ECal clusters by extrapolating the tracks

into the calorimeter. The distance between the track and the EM cluster must be within

∆η < 0.05 and ∆φ < 0.2 (0.05) for a cluster that the track bends towards (away from). These

criteria may be satisfied before or after rescaling of the track momentum to the cluster energy.

In the second case, the ∆φ threshold is reduced from 0.2 to 0.1, but the overall requirement

is looser, and allows for matching in cases of significant energy loss before the calorimeter.

The matched tracks are re-evaluated using the Gaussian Sum Filter (GSF) [228]. Multiple

tracks may be associate with a single cluster. All tracks are kept after reconstruction; but

the primary track is chosen based on proximity to the cluster energy-weighted center as well

as on the track weight, determined by the number as detector location of the hits. The

primary track is used to determine momentum, charge, and electron identification of the

candidate.

5.2.2 Electron–photon identification

Based on the outcome of the matching of tracks with clusters, candidate particles are

identified as one of the three types:

• Electrons, for well matched track-cluster pairs with tracks originating from vertices

found in the beam interaction region.

• Converted photons, for clusters matched with tracks originating from secondary

vertices.

• Unconverted photons, for clusters not matched with tracks.

Before these objects can be used in offline analyses, a set of further quality criteria is ap-

128

plied to select a pure sample of prompt∗ electrons and photons. The identification criteria

require that the longitudinal and transverse shower profiles of the candidates are consistent

with those expected for EM showers induced by such particles. Table 5.1 summarizes the

discriminating variables, which reflect a number of shower shape parameters.

5.2.3 Photons

As photons are not electrically charged, and so are not themselves detectable by the

tracking detector systems, they are mainly reconstructed from the 3D distributions of energy

deposits they leave in the EM calorimeter. However, interactions with the inner detector

material can lead the showering process to start before a photon enters the calorimeter. The

identification criteria, described in Section 5.2.2, can still identify such objects as converted

photons. In this case, the collimated electron tracks associated with the calorimeter cluster

can be used to aid in reconstruction of the 4-momentum of the photon. The discriminating

variables pertaining to photon identification are illustrated in Fig. 5.5. For example, the

variables using the EM strip layer of ECal play an important role in rejecting π0 decays into

two collimated photons.

Three working points are chosen for the photon identification: loose, medium and tight.

The loose working point is typically used for the single and diphoton triggers. It is defined

using the Rhad, Rhad 1
, Rη, and wη 2 shower shape variables. The medium working point
includes an additional cut on Eratio and is used mainly for triggering in high pileup condi-

tions. The tight selection is the primary photon working point used in offline analyses. It

exploits the full granularity of the calorimeter, including the fine segmentation of the first

∗Prompt photons and electrons are those coming from the hard scattering vertex rather than from a

showering process.

129

Table 5.1: Discriminating variables used for electron and photon identification [229–231].

Category
Hadronic
leakage

EM third layer

Description
Ratio of ET in the first sampling of the hadronic calorimeter to
the ET of the EM cluster (used over the ranges |η| < 0.8 and
|η| > 1.37).
Ratio of the ET in the hadronic calorimeter to ET of the EM
cluster (used over the range 0.8 < |η| < 1.37).
Ratio of the energy in the third layer to the total energy in the
EM calorimeter.

EM middle layer Ratio between the sum of the energies of the cells contained in a

Name

Rhad 1

Rhad

f3

Rη

wη 2

Rφ

ws 3

ws tot

fside

∆Es

Eratio

f1

ninnermost
nPixel
nSi
d0
|d0/σ(d0)|
∆p/p

Usage
e, γ

e, γ

e

e, γ

e, γ

e, γ

γ

e, γ

γ

γ

e, γ

e, γ

e
e
e
e
e

e

eProbabilityHT e
e
∆η1

∆φres

E/p

e

e

EM strip layer

3 × 7 η-φ rectangle (measured in cell units) and the sum of the
cell energies in a 7 × 7 rectangle, both centered around the most
energetic cell.

Lateral shower width,(cid:113)((cid:80) Eiη2

i )/((cid:80) Ei) − (((cid:80) Eiηi)/((cid:80) Ei))2,

where Ei is the energy and ηi is the pseudorapidity of cell i and
the sum is calculated within a window of 3 × 5 cells.
Ratio between the sum of the energies of the cells contained in a
3 × 3 η-φ rectangle (measured in cell units) and the sum of the
cell energies in a 3 × 7 rectangle, both centered around the most
energetic cell.

Lateral shower width,(cid:112)((cid:80) Ei(i − imax)2)/((cid:80) Ei), where i runs
over all strips in a window of 3 strips around the highest-energy
strip, with index imax.
Total lateral shower width,(cid:112)((cid:80) Ei(i − imax)2)/((cid:80) Ei), where i
runs over all strips in a window of ∆η ≈ 0.0625 and imax the
index of the highest-energy strip.
Fraction of energy outside core of three central strips but within
seven strips.
Difference between the energy of the strip associated with the
second maximum in the strip layer and the energy reconstructed
in the strip with the minimal value found between the first and
second maxima.
Ratio of the energy difference between the maximum energy
deposit and the energy deposit in the second maximum in the
cluster to the sum of these energies.
Ratio of the energy measured in the first sampling of the
electromagnetic calorimeter to the total energy of the EM cluster.

Track conditions Number of hits in the innermost pixel layer.

Track–cluster
matching

Number of hits in the pixel detector.
Total number of hits in the pixel and SCT detectors.
Transverse impact parameter relative to the beam-line.
Significance of transverse impact parameter defined as the ratio
of d0 to its uncertainty.
Momentum lost by the track between the perigee and the last
measurement point divided by the momentum at perigee.
Likelihood probability based on transition radiation in the TRT.
∆η between the cluster position in the first layer of the EM
calorimeter and the extrapolated track.
∆φ between the cluster position in the second layer of the EM
calorimeter and the momentum-rescaled track, extrapolated from
the perigee, times the charge q.
Ratio of the cluster energy to the track momentum.

130

Figure 5.5: Illustration of the shower shape discriminating variables used for photon identi-
fication [229, 232].

sampling layer. The tight selection is optimized using a multivariate algorithm, and is per-

formed separately for converted and unconverted photons (loose and medium identification

are the same for converted and unconverted). The main differences in the shower shapes

of converted and unconverted photons are due to the opening angle of the e+e− conversion
pair, which is amplified by the magnetic field, and due to the additional interaction of the

conversion pair with the material upstream of the calorimeter [231].

5.3 Hadronic jets

Due to color confinement exhibited by the strong interaction, as discussed in Section 2.4,

final state quarks and gluons quickly hadronize, producing streams of color-neutral particles

called jets. Due to the multiplicity of constituent particles, many of which are soft, it is

generally not possible or important to identify them individually. Rather, a whole jet is

typically reconstructed as a single object, although, in recent years, characterization of jet

substructure has become an important tool for many physics analyses at the LHC [233–235].

Jets are reconstructed from energy deposits recorded in calorimeter cells. The cells are

131

EnergyRatiosWidthsShowerShapesStripsSecondLayerHadronicf1=ES1ETot.fside=ES17−ES13ES13Rη=ES23×7ES27×7Rφ=ES27×7ES23×7RHad=EHadTETwη2=sPEiη2iPEi−(cid:18)PEiηiPEi(cid:19)2Widthina3×5(∆η×∆φ)regionofcellsinthesecondlayer.ηφws=sPEi(i−imax)2PEiws3uses3×2strips(∆η×∆φ);wstotisdefinedsimilarly,butuses20×2strips.∆E=ES1max,2−ES1minEratio=ES1max,1−ES1max,2ES1max,1+ES1max,2grouped together based on how their properties compare to those of the neighboring cells.

Most ATLAS analyses use topological clustering algorithms [227, 236], which rely only on

calorimeter information. However, tracking information can be used to guide clustering the

of calorimeter cells with the help of particle flow algorithms [237]. The cell clusters are further

combined using jet clustering algorithms. Many such algorithms exist [238]; however the anti-
kt [239] algorithm is the de facto standard at the LHC. The H → γγ analysis presented in
this dissertation used jets reconstructed from topological clusters using the AntiKt algorithm

with the radius parameter of 0.4 [215]. However, the final Run 2 iteration of the analysis

will make use of a particle flow algorithm.

5.3.1 Topological clustering algorithm

Topological clusters, or topo-clusters for short, are groups of adjacent electromagnetic

and hadronic calorimeter cells, merged based on the relative amount of signal-to-noise ratio

of the cells and their neighbors. The algorithm is similar to that described for electron

and photon reconstruction described in Section 5.2. The noise has contributions from the

calorimeter electronics, measured with the beam off, and from pileup, estimated from MC

simulations.

The first step of the algorithm is to identify seed cells by locating the cells with the signal-

to-noise ratio above the threshold of tseed. The default ATLAS value of the seed threshold

is 4. The second step is to check whether the cells directly neighboring the seed cells have

the ratio above a lower threshold of tneighbor (ATLAS default is 2). The cells that do are

then merged with the seeds to form clusters. Clusters sharing eligible neighbor cells are also

merged. In the third step, the cells on the clusters’ perimeter are merged into the clusters,

if their ratio is greater than tcell. The ATLAS default, however, is 0, so all perimeter cells

132

with any signal are merged. Technically, a final pruning step exists, in which clusters with

the total deposited amount of energy below Ecut are dropped, though the default ATLAS

value is Ecut = 0. Because of the default threshold values, this noise suppression algorithm is

often referred to as the 4-2-0 scheme. This default scheme is used in the presented analysis.

Ideally, the clusters would be well isolated from each other. Because this is not always

the case, cluster splitting is used to keep individual clusters from growing too large and

including energy deposits from multiple jets. Local energy maxima are identified to split

large clusters into smaller ones.

5.3.2 Particle flow algorithm

Unlike the traditional topological clustering method, the particle flow approach to recon-

struction of hadronic jets allows a combination of tracker and calorimeter information. The

key idea behind the algorithm is to reconstruct charged particles comprising jets using the

tracker and to fall back on the calorimeter clusters for neutral particles. This requires careful

single-particle matching between tracks and calorimeter deposits. The ability to accurately

subtract all of a single particle’s energy, without removing any energy deposited by any other

particle, forms the key performance criterion upon which the algorithm is optimized [237].

The particle flow approach provides several advantages:

• The tracking detector resolution for transverse momentum is much better than the

calorimeter energy resolution [240]. On the one hand, this implies improved resolution

of momentum reconstruction. On the other hand, the detector acceptance can be ex-

tended for softer jets, as tracks are reconstructed for charged particles with a minimum

transverse momentum of pT > 400 MeV, while the respective energy deposits often do
not pass the noise thresholds required to seed topo-clusters [241].

133

• The angular resolution of a single charged particle, reconstructed using the tracker is

much better than that using the calorimeter.

• Low-pT charged particles originating within a jet are swept out of the jet’s cone by the
magnetic field by the time they reach the calorimeter. Using momenta reconstructed

from tracks allows to correctly cluster the soft charged particles into the respective jet.

• Because tracks can be accurately associated with vertices, affects of in-time pileup can

be mitigated by rejecting particles originating from the pileup vertices.

5.3.3 Generalized kt algorithm

From the experimental perspective, the function of a jet clustering algorithm is to provide

a reliable means of conversion between calorimeter information and the final state objects

relevant to an interaction process of interest. Combining fragmentary calorimeter mea-

surements reduces the dimensionality of the problem, as the final state can be described as

composed of conceptually simple objects, i.e. jets, directly identifiable with outgoing partons

in Feynman diagrams. The desired properties of the algorithm are, thus

• Robustness – similar arrangements of calorimeter clusters should be reconstructed as

similar sets of jets;

• Stability – presence of additional jets must not strongly influence the results of recon-

struction of individual jets;

• Computational efficiency – with millions of events in the LHC data, each containing

multiple jets, their reconstruction must be computationally tractable.

From the theoretical point of view, a jet 4-momentum must be a safe observable with

respect to soft (infrared) and collinear contributions. Otherwise, stable predictions cannot

be made using perturbation theory. In fact, one of the primary purposes of a jet algorithm is

134

to provide a mapping between theoretical predictions and experimental observations in the

face of soft and collinear divergences present in perturbative QCD.

Sequential recombination algorithms possess all of these required properties. The most

widely used class of these algorithms goes under the name of a generalized kt algorithm, of
which kt, anti-kt, and Cambridge–Aachen are specific cases [239, 242–244]. The algorithm
proceeds by iterating over the following steps:

1. Calculate the pairwise distance, dij, and the distance to the beam, diB

, for all jet

candidates.

2. Find the smallest distance.

If dij is the smallest, merge jet candidates i and j.

If

is the smallest, identify candidate i as a complete jet and remove it from

instead diB
consideration in subsequent iterations.

3. Repeat until all candidate are clustered into jets.

Here, the term jet candidate refers to any jet constituent at any step of iteration, which

are summed as the algorithm proceeds. The term pseudo-jet is used by the popular FastJet

library [245], which implements most jet clustering algorithms found in use. The input

jet constituents may be 4-momenta of partons or hadrons in MC or calorimeter clusters in

detector data. Computational techniques can be used to reduce the algorithm’s complexity
from O(N 3) to O(N 2) or O(N log N ) [246].

The distance measures dij and diB

are defined as following:

ti , k2p

dij = min(cid:16)k2p

diB = k2p
ti ,

ij

tj(cid:17) R2

R2 ,

(5.2)

(5.3)

where kti is the transverse momentum of the ith particle, p is the power constant that

135

depends on the scheme (p = −1 for anti-kt, p = 1 for kt, and p = 0 for Cambridge–Aachen),
R is a constant parameter that determines the radius of a typical jet, and Rij is the distance
between the two particles given by

ij = (yi − yj)2 + (φi − φj)2,
R2

(5.4)

where y is the rapidity and φ is the azimuthal angle. Fig. 5.6 demonstrates characteristic
shapes of jets reconstructed by commonly used algorithms. Note that the anti-kt algorithm
tends to produce the most cone-like jets.

Figure 5.6: Comparison of jet shapes produced by commonly used clustering algorithms [239].

Most ATLAS analyses, as well as the one presented in this dissertation, use the anti-kt

136

algorithm with R = 0.4, also referred to as AntiKt4. Analyses interested in jet substructure

information often use a larger value of R to first identify large jets, followed by reclustering

of subjets with a smaller R within the large jets.

5.3.4 Jet correction and calibration

ATLAS calorimeter systems are sampling, meaning that not all energy of jets and showers

is deposited into the active medium and recorded. Consequently, in general, energy and

position dependent corrections must be applied to the calorimeter measurements.

A local cluster weighting (LCW) is applied to the topo-clusters to correct for the non-

compensating nature of the detector.∗ The LCW classifies clusters as electromagnetic or

hadronic and applies the respective corrections. The weighting also takes into account energy

loss due to non-instrumented regions of the calorimeters and noise suppression [247, 248].

The correction is determined using MC simulations of calorimeter response to charged and

neutral pions.

The directions of the 4-vectors of the calorimeter clusters are corrected assuming that

jets originate from the primary vertex. Only the trajectory origin is corrected, leaving the

energy unchanged.

The pT of the jets is corrected for pileup with an area-based subtraction method [247],

according to the equation

corr
Tj = p

p

orig

Tj − ρAj − α(NPV − 1) − β(cid:10)µ(cid:11),

(5.5)

where p

orig
T

and p

corr
T

are transverse momenta of a jet before and after the correction, A is

∗Compensating calorimeters have equal response to electromagnetic (electrons and photons) particles

and hadrons.

137

the area of the jet in the η-φ plane, ρ is the average energy density of all jets in the event,

dominated by soft pileup jets, NPV is the number of primary vertices identified in the event,

(cid:10)µ(cid:11) is the average number of interactions per bunch crossing, and α and β are parameters

determined from MC. A is determined using an active area algorithm, in which jets are

reclustered after a uniform distribution of “ghost” particles with infinitesimal momentum is

added to the event. The number of the “ghosts” clustered into a jet gives the measure of its

area. The energy density for each jet is give by ρj = pTj/Aj, so ρ = median(ρj).

A jet energy scale (JES) correction is applied to jets. This is a correction factor on the

jet energy derived from MC as the ratio of jet energy before and after reconstruction. The

factor is obtained using an inclusive MC sample of isolated jets with the origin and pileup

corrections applied. The JES correction is rapidity-dependent [249].

The calorimeter response is sensitive to the composition of particles and the energy

distribution within jets, which depend on the initiating parton energy and type. Quark

jets often include hadrons with high fraction of the total jet pT that penetrate further into
the calorimeter, while gluon jets typically contain many soft particles, leading to a lower

calorimeter response and a wider transverse profile of jets. A global sequential calibration

(GSC) is applied to account for these effects and improve the jet energy resolution (JER) [249,

250]. Five stages of corrections based on the following variables are applied.

1. The energy fraction deposited in the first layer of the tile calorimeter.

2. The energy fraction deposited in the third layer of the EM calorimeter.

3. The number of tracks with pT > 1 GeV associated with the jet.
4. The pT-weighted transverse width of the jet, measured using associated tracks with

pT > 1 GeV.

5. The amount of jet activity measured in the muon spectrometer.

138

For each observable, an independent jet four-momentum correction is derived as a function
of pT and |η| by inverting the jet reconstruction response matrix obtained from MC.

5.3.5 Jet Vertex Tagger

Heavy pileup conditions at the LHC provide a challenging environment for jet recon-

struction. In one way, this problem is mitigated by using discriminating variables that allow

to distinguish hard-scatter (HS) jets, originating from the primary interaction of interest,

from pileup (PU) jets, which are reconstructed from particles originating from the extra-

neous interactions of the many protons comprising the colliding beams. As most PU jets

are relatively soft, it is particularly important to have discriminating power for jets with

pT < 50 GeV.

Previously, a variable called Jet Vertex Fraction (JVF) was used for this purpose [251].

The JVF discriminating variable is defined as

JVF =

,

(5.6)

(cid:80)t ptT(PV0)

(cid:80)t ptT(PV0) +(cid:80)n≥1(cid:80)t ptT(PVn)

where ptT(PVn) is the transverse momentum of the track t associated with the primary vertex
PVn, with the hard-scatter vertex labeled PV0, and the sums are scalar.

The Jet Vertex Tagger (JVT) [252, 253] is a successor of the JVF. The JVT discriminant

is constructed as a 2-dimensional likelihood in the space of the two variables, called corrJVF

and RpT. corrJVF is a pileup corrected FJV variable,

corrJVF =

(cid:80)t ptT(PV0)

(cid:80)t ptT(PV0) + (cid:80)n≥1(cid:80)t ptT(PVn)

k nPU
trk

.

(5.7)

139

The scalar sum of the pT of the associated tracks originating from PU vertices, pPUT =

(cid:80)n≥1(cid:80)t ptT(PVn), on average, tends to grow linearly with respect to the total number of

PU
trk corrects for this growth. k = 0.01
is an empirical factor introduced to make the shape of the corrJVF distributions for HS and

PU
trk . Thus, dividing pPUT by n

pileup tracks per event, n

PU jets similar to the respective JVF ones. RpT is defined as the scalar sum of the transverse
momentum of the tracks associated with the jet, that originate from the HS vertex, relative

to the fully calibrated jet pT, which includes a pileup subtraction,

RpT = (cid:80)t ptT(PV0)

jet
T

p

.

(5.8)

RpT combines both calorimeter and tracking information. The distributions of the two
variables are shown in Fig. 5.7.

Figure 5.7: Distributions of corrJVF (left) and RpT (right) for hard-scatter and pileup jets
obtained from simulated dijet events [252, 253]. Jets with no associated tracks are assigned
corrJVF = −1.

The JVT discriminant is obtained using a k-nearest neighbor (kNN) algorithm. For each

140

corrJVF-1-0.500.51Normalized Entries-410-310-210-110110210PU jetsHS jetsATLASSimulation PreliminaryPythia8 dijets LCW+JES R=0.4tAnti-k < 30 GeVT| < 2.4, 20 < pη| 30≤ Vtx N≤0 pTR00.511.5Normalized Entries-410-310-210-110110210PU jetsHS jetsATLASSimulation PreliminaryPythia8 dijets LCW+JES R=0.4tAnti-k < 30 GeVT| < 2.4, 20 < pη| 30≤ Vtx N≤0 point in the two-dimensional corrJVF-RpT plane, the probability for a jet at that point to
be of the signal type is estimated by the ratio of the number of HS jets to the number of

HS+PU jets, found in a local neighborhood around the point, using a training sample of
both types of jets with 20 < pT < 50 GeV and |η| < 2.4. The JVT likelihood function and
the distribution of its values are shown in Fig. 5.8.

Figure 5.8: Left: JVT likelihood as a function of corrJVF and RpT. Right: Distribution of
the JVT likelihood for low-pT central pileup and hard-scatter jets. Both figures from [252].

Because of the limited forward coverage of the tracker, which extends only to |η| < 2.4,
JVT selection can only be applied to central jets. To reduce contamination from PU jets

in the forward region, a forward Jet Vertex Tagger (fJVT) was later introduced [254–256].

Using discriminants relying on calorimeter information where tracking is not available, fJVT
extends pileup jet rejection range to 2.5 < |η| < 4.5. Timing information and shape analysis
are used to reduce the stochastic contribution from out-of-time pileup jets. Jet timing is

given by the energy weighted average of the timing of the jet constituents. The principal

discriminating feature of the out-of-time pileup jets is their lack of a denser energy core.

For QCD jets from in-time pileup, which are prevalent in the forward region, the two

141

corrJVF00.51pTR00.51JVT Likelihood00.20.40.60.81ATLASSimulation PreliminaryPythia8 dijets  LCW+JES R=0.4tAnti-k < 50 GeVT| < 2.4, 20 < p|JVT00.51Normalized Entries-310-210-110110210PU jetsHS jetsATLASSimulation PreliminaryPythia8 dijets LCW+JES R=0.4tAnti-k < 30 GeVT| < 2.4, 20 < pη| 30≤ Vtx N≤0 characteristics mentioned above do not provide effective discriminating power. Instead, an

fJVT discriminant is defined as follows,

miss

fJVT = max(fJVTi),

Ti (cid:11) · p
(cid:12)(cid:12)pfj(cid:12)(cid:12)2
fj is the transverse momentum of the forward jet, and(cid:10)p

fJVTi = (cid:10)p

where p

fj

,

(5.9)

miss

Ti (cid:11) is the average of the

weighted vector sum of the transverse momenta of jets and tracks originating from a primary

vertex PVi,

miss

(cid:10)p
Ti (cid:11) = −

track
T

p

1

2(cid:32)k (cid:88)tracks

∈PVi

+ (cid:88)jets

∈PVi

jet

T (cid:33).

p

(5.10)

An empirical value of k = 2.5 is chosen to optimize the overall rejection of forward pileup
jets. The fJVTi values are computed for all combinations of primary vertices i (excluding the
hard-scatter vertex) and forward jets, and the maximum value is taken as the discriminant.

A forward jet is tagged as pileup if its fJVT value is above a threshold. The fJVT distribution

tends to fall more steeply for HS than for QCD PU jets, as shown in Fig. 5.9.

Figure 5.9: The fJVT distribution for hard-scatter (blue) and pileup (green) forward jets
in simulated Z + jets events with at least one forward jet with 30 < pT < 40 GeV (left) or
40 < pT < 50 GeV (right) [254].

142

fJVT00.511.52Fraction of Jets / 0.050.10.2Hard-scatter jetsInclusive pile-up jetsATLASSimulationµµ→Powheg+Pythia8 Z=13.5〉µ〈 = 13 TeV, s EM+JES R=0.4tkAnti-<40 GeVT|<4.5, 30<pη2.5<|fJVT00.511.52Fraction of Jets / 0.050.10.20.3Hard-scatter jetsInclusive pile-up jetsATLASSimulationµµ→Powheg+Pythia8 Z=13.5〉µ〈 = 13 TeV, s EM+JES R=0.4tkAnti-<50 GeVT|<4.5, 40<pη2.5<|5.4 Muons

As discussed in Section 4.6, muons have a far greater penetration depth than any other

charged particles. A muon’s mass is 200 times larger than that of an electron. This drastically

reduces the rate at which muons can lose energy via bremsstrahlung. At the same time,

their mass is not large enough to allow hadronic decays, which are available for tau. Both

electrons and taus are detected in the calorimeters, while muons require additional tracking

outside the calorimeters. In fact, the majority of tracks outside the calorimeters belong to

muons. To provide the additional measurements of muon tracks, ATLAS is equipped with

the Muon Spectrometer (MS). The MS is used in combination with the inner detector (ID)

to fully reconstruct muons, using tracking techniques similar to the ones already described

for electrons and other charged articles [219].

Initially, muon tracks are reconstructed separately in the ID and MS. Reconstruction

in the MS begins with searching for hit patterns inside each muon chamber to form track

segments, which are then connected using different algorithms [257, 258]. MS tracks can be

combined with the information from the ID or the calorimeters. Depending on how this is

done, the final muon objects are grouped into four categories:

1. Stand-alone (SA) muons are reconstructed only in MS. Their trajectories are extrapo-

lated to the interaction point, taking into account energy loss in the calorimeters. The
ID only covers the region of |η| < 2.5, and SA muons can be used to extend forward
acceptance to |η| < 2.7.

2. Combined (CB) muons are formed by successful combination of MS and ID tracks.

These have the highest purity among the different categories.

3. Segment-tagged (ST) muons consist of an ID track identified as a muon and matched

143

with at least one segment in the precision chambers of the MS but not with a whole

MS track.

4. Calorimeter tagged (CT) muons do not have MS track and are instead tagged using

calorimeter energy deposits. CT muons have the lowest purity, but can be used to gain

statistics in regions not covered by MS.

Muon candidates are required to pass identification and isolation criteria optimized using

data from Z → µµ and J/ψ → µµ decays [257], similarly to electrons and photons.

5.5 Missing transverse momentum

The momentum fractions carried by interacting partons in any particular collision of

hadrons cannot be measured or predicted. But because LHC proton beams are precisely

aligned at the collision point inside the ATLAS detector, the initial state has no net transverse

momentum, and only the net forward component of the initial momentum vector is unknown.

The total transverse momentum of the measured final state, however, is not necessarily zero,

because some momentum may be carried by undetected particles, or be absent or present

as a result of a mismeasurement. Such undetected particles may be neutrinos, any particles

falling outside of the fiducial volume of the detector, or hypothetical BSM particles, such

as dark matter candidates. The amount of momentum required to balance the measured

total transverse momentum vector is referred to as missing transverse energy and is denoted

EmissT

. It is defined as the sum of all calibrated photons, electrons, muons, jets, and also soft

terms comprised of the remaining low energy tracks and calorimeter deposits [259]. Accurate

measurements of EmissT
resolution EmissT
numbers of jets, neutrinos, and primary vertices [260, 261].

are difficult and require precise calibration. The estimated ATLAS

in is to be between 5 and 30 GeV, depending on conditions such as the

144

Chapter 6

H → γγ cross section analysis

6.1 Introduction

This chapter describes the Higgs boson cross section measurements in the diphoton chan-

nel, conducted on the ATLAS detector in the Run 2 of the LHC. The analysis is a part of

a larger program, aimed to extract as much of the information about the Higgs boson as

possible from the events produced at the LHC, with parallel efforts conducted by ATLAS

and CMS teams. The approach to the problem has naturally evolved over time, as more and

more data has been collected. Following the discovery of the Higgs [1, 2], in the indepen-
dently analyzed H → γγ and H → 4(cid:96) channels, the studies shifted focus to characterizing
the Higgs properties. The first of these studies compared the ratios of the total Higgs produc-

tion rate to the SM expectations, i.e. signal strength [262]. Subsequent studies differentiated

between measurements of cross sections and coupling strengths, the latter using the so-called

κ-framework [263]. In Run 1, these studies assumed SM Higgs production mechanisms and

kinematics, and thus were not model independent.

With the analyses of the Run 1 data having strongly established the existence of the

Higgs boson, and its behavior observed to be in good agreement with that predicted by

the SM, the Run 2 analyses evolved their approach to reduce assumptions based on specific

physics models. This is accomplished by using empirical signal and background models that

145

do not strongly rely on SM predictions for signal extraction (see Section 6.5).

Differential measurements of the Higgs properties were introduced towards the end of

Run 1, when a sufficient amount of data was collected to subdivide the phase space. Sys-

tematic approaches were developed for defining the respective event categories. These are

discussed in Section 6.4 for the presented cross section analysis. The couplings analyses refer

to the simplified template cross section (STXS) framework [264] for definitions of their event

categories.

The presented analysis also provides fiducial∗ cross section measurements. The fiducial,

rather than total, cross section is measured in order to reduce dependence on the detector

model, as well as on the theory predictions. The definition of the fiducial volume (see

Section 6.3) is designed to match the phase space accessible by the detector as closely as

possible to reduce the systematic uncertainty. The measured cross section is given by

σi =

νsig
i
ci × L

,

(6.1)

where σi is the cross section for a given event category i, νsig
signal events, ci is a correction factor, introduced to account for detector effects such as finite
resolution and efficiency, as discussed in Section 6.6, and L is the integrated luminosity of

is the number of extracted

i

the dataset.

The inclusive fiducial phase space region is further subdivided into finer event categories,

described in Section 6.4. The primary goal of the analysis is to present the cross section of
the pp → H → γγ process as a function of various variables that characterize the interaction
kinematics. In other words, the analysis provides measurements of differential distributions

∗Fiducial refers to the fact that the phase space of interest comprises only a part of the full phase space,

excluding the phase space unobservable by the detector.

146

for a set of differential variables. A distribution for each variable is formed by histogramming

events into contiguous bins, whose union spans the whole fiducial volume. Signal extraction

from the diphoton invariant mass distribution, discussed in Section 6.5.3, is then performed

for each bin.

The analysis strategy is summarized in Fig. 6.1. The details are discussed in the following

sections of this chapter, but broadly the procedure can be summarized as following:

1. Events corresponding to the fiducial volume are selected from the ATLAS data.

2. Preliminary studies are performed to choose functional forms and parameters of the

signal and background models.

3. Systematic uncertainties are assessed.

4. Events are grouped into categories and fits are performed to extract signal yields.

5. Signal yields are corrected for detector effects, i.e. unfolded, to obtain particle-level

cross sections.

(a) Signal yield is extracted for every event cate-
gory and bin of differential distributions by fitting sig-
nal+background model to the mγγ
spectrum. Detec-
tor level yields obtained from fitting are converted to
particle level cross sections by unfolding.

(b) Flowshart of the structure of the analysis.

Figure 6.1: Schematic summary of the analysis strategy.

147

FittingDifferentialdistributionsDatarecordedbyATLASTheoryEventswithinfiducialphasespacePartonlevelpredictionsSignalyieldModeloptimizationandtestingDetectorlevelpredictionsFiducialcrosssectionsEventselectionFitstodatamγγspectraMonteCarloReconstructionCorrectionfordetectoreffectsFiducialacceptanceAnalysis is performed on the events with the diphoton invariant mass within the range

of 105 GeV < mγγ < 160 GeV. The [121, 129] GeV region, 4 GeV on either side of the Higgs
mass at 125 GeV, is designated as the signal region, and is blinded∗ until all preliminary
studies are complete and signal extraction is ready to proceed. The two (105, 121) GeV

and (129, 160) GeV regions are referred to as sidebands, and are used for binning optimiza-

tion (Section 6.4.3) and background model F-tests (Section 6.5.2.2). Preliminary studies

also include optimization of signal model shape parameters (Section 6.5.1), the choice of

background functions (Section 6.5.2), spurious signal estimation (Section 6.5.2.1), as well as

estimation of expected uncertainties (Section 6.7). These studies make use of generated MC

samples, discussed in Section 3.6.

Several iterations of the analysis have been performed, incrementally incorporating the

yearly datasets collected in Run 2. The published results and the respective sizes of the data

subsets are listed in Table 6.1. More extended lists of ATLAS Higgs publications can be

found in [265–267].

Luminosity Data set

Table 6.1: Published Run 2 results of the ATLAS H → γγ cross section analysis.
√s
13 TeV
3.2 fb−1
13 TeV
13.3 fb−1
13 TeV
36.1 fb−1
13 TeV
79.8 fb−1
13 TeV 139.0 fb−1

Publications
2015
ATLAS-CONF-2015-060 [268, 269]
2015–2016 ATLAS-CONF-2016-067 [270, 271]
2015–2016 ATLAS-CONF-2017-045, HIGG-2016-21 [272–274]
2015–2017 ATLAS-CONF-2018-028 [275, 276]
2015–2018 ATLAS-CONF-2019-029 [277, 278]

∗Blinding is a measure typically taken in order to minimize bias. The problem is that, hypothetically,
revealing all the details of collected data while analysis methodology is still being decided upon, can bias
these choices and artificially inflate estimates of significance of the obtained results.

148

6.2 ATLAS dataset

The analysis results presented in this dissertation use the full Run 2 dataset with an

integrated luminosity of 139 fb−1. This number is slightly lower than the total LHC delivered
luminosity of 156 fb−1 due to the efficiency of data taking, which is lowered by the trigger
deadtime and the requirements for nominal performance of the detector subsystems. The

data collection took place over 3 years, starting in 2015 and concluding in 2018. The dataset
consists of proton-proton collision events with √s = 13 TeV and 25 ns bunch spacing. The
amounts of data collected each year are listed in Table 6.2. The diphoton trigger efficiency
for collection of the data relevant to the H → γγ analysis is 98% [279].
Table 6.2: Yearly datasets making up the combined Run 2 ATLAS dataset. Estimates
of luminosity and its uncertainty are taken from [280]. Numbers of events pertain to the
inclusive fiducial region.

Luminosity Number
Year Period Luminosity uncertainty of events
2015
28 741
283 178
2016
370 086
2017
2018
496 082
1 178 087
Total

3.2 fb−1
32.9 fb−1
44.3 fb−1
58.5 fb−1
139.0 fb−1

D–J
A–L
B–K
B–Q

2.1%
2.2%
2.4%
2.0%
1.7%

Luminosity estimates at the LHC are based on measurements of several luminosity sen-

sitive detectors. The primary luminosity detector is the LUCID-2 Cherenkov detector [281].

Its measurements are complemented by the ATLAS Beam Conditions Monitor (BCM) di-

amond detectors [282], and by offline measurements of the multiplicity of reconstructed

charged particles, i.e. track counting, in randomly selected bunch crossings. The absolute

calibration is determined using van der Meer [283] scans, performed under specially tailored

low luminosity conditions. A calibration transfer procedure is used to extrapolate from the

149

low luminosity to the high luminosity regime, at which the data are taken [280]. During

Run 2, the dedicated van der Meer runs were performed once a year. The methodology used

to obtain the estimates of luminosity and its uncertainty, listed in Table 6.2, is discussed

in [284]. Estimates of luminosity for the same dataset differ slightly from one round of the
H → γγ analysis to another due to continuing improvements in the assessment of the LHC
luminosity and subsequent recalibrations.

6.3 Event and object selection

This section describes selection criteria for objects and events used in this analysis. The

details of the selection criteria have slightly evolved between the analysis rounds. The re-

spective publications, listed in Table 6.1, provide all the details. The criteria listed below

correspond to the most recent analysis [277, 278]. Details of the reconstruction are discussed

in Chapter 5.

6.3.1 Event preselection

Data and reconstructed MC events are first filtered through several general preselection

criteria, to ensure data quality, before object-specific cuts, required for the analysis, are

applied. The preselection criteria are listed below. The cutflow for the yearly datasets in

shown in Table 6.3.

• Triggers (data only):

– HLT_g35_loose_g25_loose (2015, 2016 data);

– HLT_g35_medium_g25_medium_L12EM20VH (2017, 2018 data).

These diphoton triggers require at least two reconstructed photons with ET greater

150

than 35 (25) GeV for the leading∗ (subleading) photon. Loose photon identification

requirements† were applied by this trigger in 2015–2016 and were tightened in 2017 to

cope with a higher instantaneous luminosity. On average, the trigger has an efficiency
greater than 98% for H → γγ events that pass the diphoton event selection described
in Section 6.3.2.

• Good Run List (data only):

Events must belong to the luminosity blocks specified in the following run lists [286].

This requirement ensures that all the sub-detectors relevant for this analysis were fully

operational.

data15_13TeV.periodAllYear_DetStatus-v89-pro21-02_Unknown

_PHYS_StandardGRL_All_Good_25ns.xml

data16_13TeV.periodAllYear_DetStatus-v89-pro21-01_DQDefects-00-02-04

_PHYS_StandardGRL_All_Good_25ns.xml

data17_13TeV.periodAllYear_DetStatus-v99-pro22-01_Unknown

_PHYS_StandardGRL_All_Good_25ns_Triggerno17e33prim.xml

data18_13TeV.periodAllYear_DetStatus-v102-pro22-04_Unknown

_PHYS_StandardGRL_All_Good_25ns_Triggerno17e33prim.xml

• Event Quality (data and MC):

Standard ATLAS event cleaning based on the detector quality flags is applied. Events

with data integrity errors in the calorimeters and incomplete events missing some

detector information are rejected.

∗Leading and subleading regarding photons and jets refers to pT ordering. The leading photon or jet is

the one with the highest pT of all the objects of the same type in that event.

†See Table 1 in [285] for the definition of loose and tight photon identification requirements.

151

Table 6.3: Cutflow of event selection for the yearly datasets of ATLAS Run 2 data.

Data
yield

1215682304
12737493
12737493
12737493
12446557
12194370
12193188
12193167
3544155
3544155
2596133
373580
150746
131166
28741

N xAOD
N DxAOD
All events
No duplicates
GRL
Pass trigger
Detector DQ
Has PV
2 loose photons
e-γ ambiguity
Trigger match
Tight ID
Isolation
Relative pT cuts
mγγ ∈ [105, 160] GeV

2015
Total

Relative
efficiency

Data
yield

efficiency
100.000% 100.000% 5130958848
1.048%
1.048% 127609472
1.048% 100.000% 127609472
1.048% 100.000% 127609440
1.024% 97.716% 124681352
1.003% 97.974% 122364592
1.003% 99.990% 122361008
1.003% 100.000% 122361008
36092436
0.292% 29.067%
0.292% 100.000%
36007060
26987322
0.214% 73.251%
3668591
0.031% 14.390%
1461031
0.012% 40.352%
0.011% 87.011%
1270619
283178
0.002% 21.912%

Relative
efficiency

Data
yield

2016
Total

efficiency
100.000% 100.000% 5208373760
2.487%
2.487% 100062128
2.487% 100.000% 100062128
2.487% 100.000% 100062024
2.430% 97.705%
97186944
74383248
2.385% 98.142%
74372296
2.385% 99.997%
74372296
2.385% 100.000%
20395000
0.703% 29.497%
0.702% 99.763%
20395000
11922093
0.526% 74.950%
4589030
0.071% 13.594%
1845128
0.028% 39.825%
0.025% 86.967%
1603480
370086
0.006% 22.287%

2017
Total

Relative
efficiency

Data
yield

efficiency
100.000% 100.000% 5607043072
1.921%
1.921% 124560320
1.921% 100.000% 124560320
1.921% 100.000% 124560320
1.866% 97.127% 123107840
95320480
1.428% 76.536%
95315520
1.428% 99.985%
95315520
1.428% 100.000%
26469698
0.392% 27.423%
0.392% 100.000%
26469698
16801656
0.229% 58.456%
6371417
0.088% 38.492%
2477112
0.035% 40.207%
0.031% 86.903%
2153889
496082
0.007% 23.080%

2018
Total

Relative
efficiency
efficiency
100.000% 100.000%
2.221%
2.221%
2.221% 100.000%
2.221% 100.000%
2.196% 98.834%
1.700% 77.428%
1.700% 99.995%
1.700% 100.000%
0.472% 27.771%
0.472% 100.000%
0.300% 63.475%
0.114% 37.921%
0.044% 38.879%
0.038% 86.952%
0.009% 23.032%

152

• Primary vertex (data and MC):

At least one primary vertex is required to be reconstructed in the event. Only default

reconstruction requirements are applied to this vertex, including consistency with the

beam spot coordinate.

6.3.2 Photons

Fiducial cuts (object level):

• pT > 25 GeV;
• |η| < 2.37 excluding the crack region 1.37 < |η| < 1.52.

Fiducial cuts (event level):

• pT/mγγ > 0.35 (0.25) for leading (subleading) photon;
• mγγ ∈ [105, 160] GeV.
Events are required to have at least two photons passing the kinematic cuts. The two

selected photons with the highest pT are assumed to be coming from the Higgs decay and
are used to redefine the primary vertex of the event using a neural network, which uses

pointing information from the electromagnetic calorimeter [287]. The four-momenta of the

two photons are corrected, so that they point to the diphoton primary vertex. The diphoton

invariant mass is calculated using Eq. (6.2),

mγγ =(cid:112)2E1E2(1 − cos θ).

(6.2)

Data and reco: Photons are required to pass cuts on the shape of the electromagnetic

showers. At this stage of selection, the photons are referred to as loose [288]. Photons are

calibrated using the latest Run 2 calibration corrections for the energy scale and resolution,

153

detailed in [289]. The photon identification cuts are performed using the latest Egamma∗

recommendations using an ET-dependent identification detailed in [229]. A photon ambigu-
ity requirement was implemented in Run 2 as a consequence of the increased electron-photon

fake rate [215]. The rate was found to exceed 30% in the end-cap prior to introduction of

the ambiguity requirement, which uses tracking information from the innermost pixel layers

and IBL to separate electrons from converted photons.

The Higgs boson candidate photons must satisfy tight† shower shape requirements, as

well as isolation criteria for tracks, ptcone20 < 0.05 pT, and calorimeter, topoetcone20 <
0.065 pT, using a cone of ∆R = 0.2. Track isolation is always calculated from the selected
primary vertex. The photon isolation efficiency is measured in data and MC using two

independent methods: Single Photon and Radiative-Z. The methods are detailed in [290].

Particle level (MC): Photons are identified by requiring PdgId = 22, and must not be

produced during hadronisation.

6.3.3 Jets

Fiducial cuts (object level):

• pT > 30 GeV, pT > 25 GeV for VBF,
• |η| < 4.4.
Jets are rejected if they lie within ∆R < 0.4 of a selected photon or ∆R < 0.2 of a

selected electron. Muons and neutrinos are not included in the clustering at particle level,

because they do not leave significant energy deposits in the calorimeters and so do not enter

the reconstruction level jet finding algorithm.

∗Egamma refers to the electron and photon reconstruction working group.
†See Table 1 in [285] for the definition of loose and tight photon identification requirements.

154

Data and reco: A jet vertex tag (JVT) [252] cut is applied to reduce contamination from

jets not originating from the selected primary vertex, i.e pileup jets. Jets with pT < 120 GeV
and |η| < 2.5 are required to pass the medium, JVT > 0.59, cut. A forward jet vertex tag
(fJVT) [254–256] is used for jets with pT < 60 GeV and 2.5 < |η| < 4.4. This region is
beyond the acceptance of the tracker. Therefore the fJVT algorithm uses timing and jet

shape information in the calorimeter to reject pileup jets in the forward region. Qualifying

jets are required to pass the medium fJVT cut.

Jets potentially originating from the decay of a b-hadron are identified with the help

of a multivariate discriminant, which uses track impact parameters and secondary vertices

information [291, 292]. The efficiency of the b-tagger is 70%, which is estimated using a t¯t

control region. Data-driven correction factors are applied to the simulation such that the

b-tagging efficiencies of jets originating from b quarks, c quarks, and the light quarks are

consistent with the ones observed in the data [272].

Particle level (MC): Jets are considered to originate from a b-hadron, if there is one with
pT > 5 GeV within a cone of ∆R < 0.4 around the jet.

6.3.4 Leptons

Fiducial cuts (object level):

• Electrons: |η| < 2.47 excluding the crack region 1.37 < |η| < 1.52,
• Muons: |η| < 2.7.

Particle level (MC): Identified by requiring |PdgId| ∈ {11, 13}.

155

6.3.5 Missing transverse energy

Particle level (MC): EmissT
not during hadronization. Neutrinos are identified by requiring |PdgId| ∈ {12, 14, 16}.

is calculated as the 2D vector pT sum of all neutrinos produced

6.4 Event categories and differential variables

The fiducial event category is the most inclusive one, as it contains all the diphoton events

that pass the selection criteria described in Section 6.3, regardless of presence or absence of

any other objects in the final state. Other event categories defined for the analysis consist of

more exclusive fiducial regions, with discrete requirements for event topology or the presence

of specific objects, and bins of the differential distributions.

6.4.1 Fiducial categories

Figure 6.2: Expected composition of fiducial regions in terms of Higgs production modes.

156

Fraction of Signal Process / Fiducial Region (after reconstruction)00.10.20.30.40.50.60.70.80.91ttH-enhancedmissTEHigh 1≥leptonNVBF-enhancedDiphoton fiducialggHVBFWHZHggZHttHbbHtHjbtHWATLASSimulationGeV= 125.09 H, mγγ→H6.4.1.1 VBF

A region enriched with vector boson fusion events is defined by requiring events to have

at least two jets with a lowered transverse momentum cut of pT > 25 GeV. The two jets

The dijet system must be back-to-back to the diphoton, with azimuthal angle separation,

must have a wide separation in rapidity, (cid:12)(cid:12)∆yjj(cid:12)(cid:12) > 2.8, and a dijet mass, mjj > 400 GeV.
(cid:12)(cid:12)∆φγγ,jj(cid:12)(cid:12) > 2.6. These criteria enhance the VBF fraction to 58% while reducing ggF to

39%. For comparison, the inclusive region composition is 88% ggF and only 7% VBF.

6.4.1.2 Lepton category

This category is defined by requiring at least one electron or muon,∗

Nlep ≥ 1. The
presence of leptons selects the associated production modes. The composition of this region

is 46% W H, 13% ZH, and 35% ttH.

6.4.1.3 Missing transverse energy

Large EmissT

is a potential signature for Higgs production in association with dark matter.

T > 80 GeV. SM events are expected to
The region is defined by EmissT
populate this region due to decays to neutrinos, as well as due to the effects of reconstruction

> 80 GeV and pγγ

efficiency and migration.

6.4.1.4

ttH

This region has an enhanced contribution from associated production with a top quark-

antiquark pair, t¯tH. The selection criteria require at least either three jets and one electron

∗The definition of Nlep

the detector.

does not account for τ leptons, because they are difficult to reliably identify in

157

or muon, or four jets. At least one jet must be b-tagged in either case. The expected

composition of the region is 75% t¯tH and 10% ggF.

6.4.2 Differential variables

Measurements of the Higgs boson production cross sections as functions of characteristic

observables provide essential information for testing the fundamental properties of the Higgs

field and our understanding of perturbative QCD. The variables studied can be broadly

grouped into the following categories:

6.4.2.1 Higgs boson kinematics

Transverse momentum and rapidity describe the fundamental kinematics of the Higgs

boson. Inclusive Higgs boson production is dominated by gluon fusion, with the Higgs boson

transverse momentum predominantly balanced by soft gluons. The diphoton transverse

momentum, pγγ

T , is sensitive to the effective coupling of the Higgs to gluons in the ggF
process, as well as to the Yukawa coupling to quarks. This provides the ability to test

perturbative QCD modeling of ggF.

The low-pT region of the pγγ

T distribution exhibits a Sudakov peak due to initial state
radiation [293]. Therefore, this region is very sensitive to resummation effects. In addition,

the low-pT region can be used to set bounds on light quark Yukawa couplings [294, 295].

The high tail of the pγγ

T distribution is sensitive to the heaviest (top) quark mass.∗
Because boosted Higgs kinematics is sensitive to the masses of fermions participating in the

production loop (Fig. 6.3), the high-pT region may have a sensitivity to BSM particles. Also,
T ,
the relative contribution of the non-resonant diphoton production decreases for larger pγγ

∗See Appendix B for further discussion.

158

meaning that the high pγγ

boson production events. The signal-to-background ratio reaches 1 around pγγ

T region of the phase space can provide a much purer sample of Higgs
T ≈ 600 GeV

and continues to improve at higher pγγ

T . This is illustrated in Fig. B.2 in Appendix B.

g

g

t

H

Figure 6.3: The largest contribution to the Higgs boson production at the LHC is from the
gluon-gluon fusion process. At LO, the Higgs boson has an effective coupling to gluons via
the loop diagram shown here. The effective coupling is sensitive to the mass of the fermion
in the loop. As Γ(H → f ¯f ) ∝ m2
, the partial width for H → t¯t accounts for 99% of the
combined width to all fermions.

f

The transverse momentum, pγγj

T , and mass, mγγj, of the Higgs + the leading jet system
complement the sensitivity to finite top quark mass and potential heavy BSM particles in

high-pγγ

T region.

modeling of gluon fusion, and can be used to probe the light quark Yukawa couplings [295].

The rapidity, (cid:12)(cid:12)yγγ(cid:12)(cid:12), distribution is sensitive to parton distribution functions and the
The rapidity separation between the two photons, (cid:12)(cid:12)∆yγγ(cid:12)(cid:12), is sensitive to the decay of the

Higgs boson.

6.4.2.2 Jet activity

The multiplicity distribution for jets accompanying a Higgs boson allows a discrimination

of different Higgs boson production mechanisms. The low multiplicity (Njets = 0 and 1)

bins are dominated by gluon fusion. Medium multiplicity bins have increased contribution
from V H (for Njets ≥ 1) and from VBF (for Njets ≥ 2). Associated production with top
quarks, t¯tH, is important for higher multiplicity bins (with Njets ≥ 3). In addition, the jet

159

multiplicity is sensitive to QCD radiation, with real corrections producing additional jets.

Both inclusive and exclusive jet multiplicities are measured, as well as multiplicities with
different jet pT thresholds, N≥30 GeV
T requirement
changes the relative contributions of the different production processes. A requirement that

. The higher threshold pj

and N≥50 GeV

jets

jets

jets have a threshold transverse momentum of 50 GeV or more enhances contributions from

the Higgs boson production processes other than ggF, and reduces contamination due to

pileup jets, but at the expense of a reduced number of selected events.

The distributions of transverse momentum and rapidity of the leading jets, and the HT
probe the theoretical modeling of hard quark and gluon radiation. The invariant mass of

∗

the two leading jets, mjj, is a useful discriminant for the VBF production.

6.4.2.3 Spin, charge conjugation, and parity

The polar angle,(cid:12)(cid:12) cos θ∗(cid:12)(cid:12), between either of the two photons from the Higgs boson decay

and the beam axis in the diphoton rest frame is sensitive to the spin of the Higgs boson. The

rest frame of the Higgs boson, or the produced diphoton, is equivalent, in this case, to the

Collins–Soper reference frame [293]. The angle(cid:12)(cid:12) cos θ∗(cid:12)(cid:12) is also known as the helicity angle.
The azimuthal angle between the two leading jets,(cid:12)(cid:12)∆φjj(cid:12)(cid:12), when there are two or more

jets present in the final state, is sensitive to the charge conjugation and parity properties of

the Higgs boson, which affect its interactions with gluons in the ggF and the weak bosons in

the VBF channels [296–299]. In such final states, the signed azimuthal separation between

the two leading jets, ∆φjj,signed
the interference between CP-even and CP-odd Lorentz structures.

, is sensitive to potential CP-violating effects originating from

∗HT is defined as the scalar sum of the magnitudes of the transverse momenta, pT, of the final state
is the sum of only the jets’

particles produced in an event. The convention for this analysis is that H
momenta, while HT also includes the diphoton momentum, pγγ
T .

jets
T

160

6.4.2.4 VBF

Separations in rapidity and azimuth between the two leading jets, (cid:12)(cid:12)∆yjj(cid:12)(cid:12), (cid:12)(cid:12)∆φjj(cid:12)(cid:12), and
, and between the diphoton and dijet systems, (cid:12)(cid:12)∆yγγ,jj(cid:12)(cid:12) and (cid:12)(cid:12)∆φγγ,jj(cid:12)(cid:12) provide

∆φjj,signed
useful signatures to enhance the VBF process. The shapes of the distributions of these

variables are different between VBF and ggF production.

In vector boson fusion, the t-
channel exchange of a W boson typically results in two high-pT jets that are well separated in
rapidity. Furthermore, the absence of any color flow between the two jets in VBF suppresses

QCD radiation in the region between them. Therefore, the VBF (cid:12)(cid:12)∆φγγ,jj(cid:12)(cid:12) distribution is
steeper and more narrowly peaked near(cid:12)(cid:12)∆φγγ,jj(cid:12)(cid:12) = π.

6.4.2.5 Full list of the differential variables measured in this analysis

1. Transverse momentum of the diphoton system, pγγ
T .

2. Absolute rapidity of the diphoton system,(cid:12)(cid:12)yγγ(cid:12)(cid:12).

3. Higgs boson helicity angle,

(cid:12)(cid:12) cos θ∗(cid:12)(cid:12) = (cid:12)(cid:12)(Eγ1 + pγ1

z )(Eγ2 − pγ2
mγγ(cid:113)m2

γγ + p2Tγγ

z )(cid:12)(cid:12)

z ) − (Eγ1 − pγ1

z )(Eγ2 + pγ2

.

(6.3)

This is the polar angle of the axis parallel to the two photons with respect to the beam

axis in the Collins–Soper reference frame [293], or, equivalently, in the rest frame of

the Higgs boson. Eq. (6.3) is valid in the laboratory reference frame.

4. The component of the diphoton transverse momentum, (cid:126)pγγ

T , transverse to the diphoton
thrust axis, ˆt, [300, 301]. The definition is illustrated in Fig. 6.4. The vector quantity

161

Figure 6.4: Sketch of the (cid:126)pγγ
Tt

definition [302].

can be expressed as

(cid:126)pγγ
Tt = (cid:126)pγγ

T −(cid:0)(cid:126)pγγ

T · ˆt(cid:1) ˆt, where (cid:126)t = (cid:126)pγ1T − (cid:126)pγ2T .

(6.4)

Only its magnitude is measured in this analysis, which is equal to

x pγ2
x − pγ2
5. Absolute rapidity separation of the two photons,

Tt =(cid:12)(cid:12)(cid:126)pγγ

T × ˆt(cid:12)(cid:12) =

2(cid:12)(cid:12)pγ1
(cid:113)(pγ1

pγγ

y − pγ1
x )2 + (pγ1

x(cid:12)(cid:12)

y pγ2
y − pγ2
y )2

.

(6.5)

(6.6)

.

(cid:12)(cid:12)∆yγγ(cid:12)(cid:12) =(cid:12)(cid:12)yγ1 − yγ2(cid:12)(cid:12) .

6. Number of jets with pT ≥ 30 GeV, i.e. with nominal jet selection criteria, N≥30 GeV
7. Number of jets with pT ≥ 50 GeV, N≥50 GeV
jets
8. Leading jet transverse momentum, pj1T .
9. Subleading jet transverse momentum, pj2T .

jets

.

10. Absolute leading jet rapidity,(cid:12)(cid:12)yj1(cid:12)(cid:12).
11. Absolute subleading jet rapidity,(cid:12)(cid:12)yj2(cid:12)(cid:12).

162

thrust axispT��pTtpTlpT�1pT�212. Scalar sum of the transverse momenta of jets,

Hjets

T = (cid:88)j∈jets

pTj.

13. Scalar sum of the transverse momenta of jets and the diphoton system,

HT = Hjets

T + pγγ
T .

T .
14. Transverse momentum of the diphoton and the leading jet system, pγγj

15. Invariant mass of the diphoton and the leading jet system,

mγγj =(cid:113)(pγ1 + pγ2 + pj1)2.

16. Invariant mass of the system of the two leading jets,

mjj =(cid:113)(pj1 + pj2)2.

17. Absolute rapidity separation of the two leading jets,

(cid:12)(cid:12)∆yjj(cid:12)(cid:12) =(cid:12)(cid:12)yj1 − yj2(cid:12)(cid:12) .

18. Absolute azimuthal separation between the two leading jets,(cid:12)(cid:12)∆φjj(cid:12)(cid:12).

19. Signed azimuthal separation between the two leading jets, ∆φjj,signed

.

The angles are ordered according to the jets’ rapidity.

20. Transverse momentum of the diphoton and two leading jets system, pγγjj

T

.

163

(6.7)

(6.8)

(6.9)

(6.10)

(6.11)

21. Absolute rapidity separation between diphoton and dijet systems,

(cid:12)(cid:12)∆yγγ,jj(cid:12)(cid:12) =(cid:12)(cid:12)yγγ − yjj(cid:12)(cid:12) .

22. Absolute azimuthal separation between diphoton and dijet systems,(cid:12)(cid:12)∆φγγ,jj(cid:12)(cid:12).

23. Maximum beam-thrust-like variable [303],

(6.12)

j + p2Tj

2 cosh (yj − yγγ)

.

(6.13)

τ1 = max τj, where τj = (cid:113)m2
24. Scalar sum of τj for all jets with τj > 8 GeV, τ =(cid:80)j τj.
26. (cid:12)(cid:12)yγγ(cid:12)(cid:12) vs. pγγ

T > 30, 40, 50, 60 GeV.
T .
and pγ2T /mγγ

T for pj

25. pγγ

.

vs. (pγ1T − pγ2T )/mγγ

.

27. pγ1T /mγγ
28. (pγ1T + pγ2T )/mγγ
vs. pγγ
T .
29. pγγj
30. τ1 vs. pγγ
T .

T

6.4.3 Binning

The binning of the measured differential distributions has to be chosen before the signal
region, mγγ ∈ [121, 129] GeV, is unblinded. This avoids potential bias that can be caused by
accidental fluctuations of the observed numbers of events due to relatively low statistics, as

well as limited detector resolution and efficiency. The binning optimization is performed us-

ing the estimated numbers of signal events from the MC and the background events from the
data sidebands, mγγ ∈ (105, 121) ∪ (129, 160) GeV. Several criteria are taken into account

164

to make an optimal choice for the edges of the bins.

The signal significance in each bin needs to be sufficiently high. This helps reduce the

uncertainty on the measured cross sections and on the shape of the distribution. The sta-

tistical model applied has a large number of free parameters, so fits that use small numbers

of events can be poorly constrained. A minimum significance of 2σ per bin was chosen as a

target value.

A common way to estimate signal significance is

Z =

s

√s + b

,

(6.14)

where s and b are the numbers of signal and background events respectively. The formula

makes intuitive sense, as it gives the number of signal events in units of Poisson uncertainty

of the total number of events. A more rigorous formula for estimation of significance has

been proposed by Cowan and Gross [304],

Z =(cid:114)2(cid:16)(s + b) log(cid:16)1 +

s

b(cid:17) − s(cid:17).

(6.15)

The two equations yield comparable values, with the ones from Eq. (6.15) about 5% larger

on average. Both prescriptions were implemented as a cross check.

Purity is another constraining factor for binning.

It is a measure of the amount of

event migration between bins due to detector effects. Purity is estimated by comparing the

numbers of events predicted by MC before and after detector simulation and reconstruction.

Specifically, purity is defined as the ratio between the total weight of events that populate

the same bin at both levels, ntruth & reco, and that of all events populating the bin after

165

reconstruction, nreco, which also counts events that migrated into the bin,

purity =

ntruth & reco

nreco

.

(6.16)

A related measure of migration is efficiency, 
, which is the fraction of events’ weight that

remained in the same bin after reconstruction,


 =

ntruth & reco

ntruth

.

(6.17)

A minimum purity and efficiency of 50% was chosen as the requirement.

Similar information can be obtained from a migration matrix, which is essentially a

2-dimensional histogram with values of the variable of interest at the particle-level on one

axis, and at the detector level on the other.

If scaled to the fraction of the total weight

at detector (particle) level for each bin, the diagonal elements of the migration matrix give

purities (efficiencies).

Other considerations in choosing the bin edges are as follows.

Intuitive and easy to

communicate values containing few significant figures are preferable. For example, 200 GeV

instead of 203.5 GeV. Certain distributions are expected to be symmetric and require re-

spective binning. Features in the shapes of distributions must not be obscured if they can

be resolved. Binning that is too fine may adversely affect unfolding. For bin-by-bin unfold-

ing, uniform correction factors are generally desired to avoid bias. Binnings used on sister
analyses, namely the ATLAS H → 4(cid:96) analysis and the analogous analyses done by the CMS
collaboration, are taken into account to simplify the combination of results.

A web tool was developed by the author to simplify and automate exploration of possible

166

Table 6.4: Binning table produced by the web tool, showing quantities considered when
choosing the appropriate binning for pγγ
T , including significance, purity, and signal fractions.

Figure 6.5: The migration matrix for division of the fiducial region into bins of pγγ
by the binning web tool.

T , produced

167

[121,129]syst. unc.stat. unc.[105,121][129,160][121,129]syst. unc.stat. unc.signifsignifrecobinsig√∑w²√sigL bkgR bkgbkgfrom fit√bkgs/√(s+b)Cowans/(s+b)purity[0,5)259.680.22%6.21%345262212410675.404.59%0.97%2.482.502.37%84.32%[5,10)553.630.15%4.25%655994147920339.203.63%0.70%3.833.862.65%82.25%[10,15)593.880.14%4.10%708834500722026.503.52%0.67%3.953.982.63%81.39%[15,20)546.230.14%4.28%644394237220210.403.61%0.70%3.793.832.63%81.15%[20,25)478.960.15%4.57%548073767417571.003.79%0.75%3.573.602.65%81.05%[25,30)412.520.16%4.92%450833256914862.104.03%0.82%3.343.372.70%80.93%[30,35)353.400.17%5.32%374082756412409.904.29%0.90%3.133.162.77%80.64%[35,45)562.520.13%4.22%568744406419434.203.64%0.72%3.984.022.81%89.17%[45,60)589.720.12%4.12%543664519519018.703.63%0.73%4.214.253.01%91.92%[60,80)497.540.13%4.48%405123615714740.203.97%0.82%4.034.083.27%92.99%[80,100)310.400.15%5.68%18582208517717.914.97%1.14%3.463.513.87%92.58%[100,120)202.030.18%7.04%8867109323664.586.31%1.65%3.253.315.22%92.21%[120,140)141.210.20%8.42%465058221952.007.76%2.26%3.093.166.75%91.79%[140,170)141.560.20%8.40%333545591480.128.46%2.60%3.523.628.73%93.40%[170,200)87.850.24%10.67%15032202685.4210.55%3.82%3.163.2911.36%92.88%[200,250)80.510.25%11.14%10581567479.0311.64%4.57%3.403.5814.39%94.51%[250,350)57.190.29%13.22%593849254.8813.39%6.26%3.243.4618.33%95.76%[350,450)15.440.54%25.45%12218263.1917.97%12.58%1.741.8719.63%95.36%[450,∞)7.780.74%35.85%359027.8216.22%18.96%1.301.4121.86%96.72%05101520253035456080100120140170200250350450Truth05101520253035456080100120140170200250350450Recochoices of binning for all variables of interest.∗ Table 6.4 and Fig. 6.5 show the estimates of

significance, purity, and the migration matrix produced by the binning tool for pγγ

T with the

full Run 2 dataset.

A data-driven estimate of the background in the signal region is used to calculate expected

significance. This estimate is obtained by fitting an exponential of a second degree poly-
nomial to the data sidebands and integrating the fitted function over mγγ ∈ [121, 129] GeV.
This functional form is chosen, because it generally has the best performance for background

modeling, as discussed in Section 6.5.2.

6.5 Signal and background modeling

Given a data sample with a certain luminosity, a cross section measurement amounts to

estimation of the number of signal events present in the sample, also called the signal yield.

The application of the statistical procedure used to obtain the signal yield is referred to as

the signal extraction. The general approach is to first choose appropriate models for signal

and background and to study these models using the respective simulated MC samples.

The two models are fitted to the MC in order to fix the nuisance parameters,† which are

believed to be well modeled in MC and are not used in data-driven uncertainty estimation.

The combined signal + background model is then fitted to the data. This final fit gives the

sought-after signal yields, as well as their uncertainties. Uncertainties from other sources are

modeled as extra nuisance parameters in the combined model.

In the present analysis, the H → γγ signal yield is obtained from the Run 2 ATLAS

∗The binning tool can be accessed at https://cern.ch/ivankp/hgam/?page=binning.
†Nuisance parameters are the parameters of a statistical model which are not of immediate interest for

the specific analysis, but still need to be accounted for.

168

data by fitting the statistical model described below to the diphoton invariant mass (mγγ
spectrum using the extended unbinned maximum likelihood method [31].

)

6.5.1 Signal model

The mγγ

distribution for the signal process pp → H → γγ has a resonance peak at the
. In the absence of interference with the pp → γγ background process,∗
is expected to be distributed according to a Breit–Wigner distribution, with a peak at

Higgs mass, mH

mγγ

and, in the Standard Model, a narrow width of 4 MeV [28]. However, the distributions

mH
observed in data are smeared by effects of photon reconstruction from detector measure-

ments, which broaden the signal distribution width to a few GeV. Because of this, the signal

distribution is empirically modeled by a double-sided Crystal Ball function,† consisting of a

Gaussian central part near the peak, smoothly transitioning to power-law functions at either

tail. The analytic form of the function is given by Eq. (6.18),

S(t) = Ns ·



e−t2/2
e−α2
e−α2

if − αL ≤ t ≤ αH,
L/2(cid:104)(cid:0)RL − αL − t(cid:1)/RL(cid:105)−nL
if t < −αL,
H/2(cid:104)(cid:0)RH − αH + t(cid:1)/RH(cid:105)−nH if t > αH,

(6.18)

where Ns is the normalization factor, t = (mγγ − µ)/σ, R = n/α, µ is the mean of the
Gaussian, σ is the width of the Gaussian, αL, H determine at which points the Gaussian

transitions to the power law, and nL, H are the power law exponents. The function is illus-

∗There, of course, is an interference between all channels for diphoton production, resonant or otherwise,
because the initial and final states are the same. But the level of the interference is too small to affect this
analysis.

†The function is named after the Crystal Ball experiment that ran at the Stanford Linear Accelera-
tor Center in 1980s. The original version, first used there, had only the lower power law tail. See, for
example [305].

169

trated in Fig. 6.6.

σ

Gaussian
core

Power law
∼ (−m)−nL

σαL

σαH

µ

Power law
∼ m−nH

[GeV]

mγγ

Figure 6.6: Example of a double-sided Crystal Ball function. Gaussian tails are drawn with
dashed lines to illustrate the difference with the power law.

The shape parameters of the signal model are determined by fitting to the signal MC.

As the MC samples are generated with mH = 125 GeV, the mean parameter, µ, obtained
from the fits, is shifted up by 90 MeV, to correct it to the experimental Higgs boson mass of

125.09 GeV, measured in Run 1 [306]. The MC signal fitting is done independently for each

event category, using the unbinned maximum likelihood method. An example of a signal

MC fit for the inclusive fiducial category is shown in Fig. 6.7.

The main uncertainties on the mγγ

signal shape can be categorized as energy scale and

energy resolution. The photon energy scale uncertainty affects µ, shifting the position of

the peak, and the photon energy resolution affects σ, broadening the width. The shifts

introduced by these uncertainties are treated as constrained nuisance parameters. The un-
certainty on the Higgs boson mass measured in Run 1, (125.09 ± 0.24) GeV [306], is taken
into account as an additional fixed nuisance parameter. The exact treatment of these sources

170

115120125130135102101100Figure 6.7: The MC signal fit for the inclusive fiducial event category drawn on a linear
(left) or a logarithmic (right) vertical axis. The black dotted lines indicate the region used
to estimate the spurious signal uncertainty.

of uncertainty is discussed in Section 6.5.3.

6.5.2 Background model

The main source of background for this analysis is non-resonant QCD diphoton pro-
duction, from gq → γγ, q ¯q → γγ, and gg → γγ processes. This is referred to as the
irreducible background, because, aside from mγγ
, these events individually bare no distin-
guishable characteristic from Higgs events. There is also reducible background from events

with γ-jet and jet-jet final states, in which the jets are mistakenly reconstructed as photons.

This background is reducible, because the amount of it present is depends on the detector

performance. Fortunately, photon identification and isolation requirements reject most of

these events, making the reducible background component relatively small, as can be seen

in Fig. 6.8.

The background mγγ

distributions are non-resonant, and are modeled by a set of em-

171

020040060080010001200Events/1 GeVATLASSimulation Internal-1Ldt = 140.00 fb∫ = 13 TeV: sFiducial(cid:0) = 125.14CBµ(cid:0) = 1.84CBσ(cid:0) = 1.71CBLoα(cid:0) = 1.71CBHiα(cid:0) = 5.76CBLon(cid:0) = 10.00CBHinYield(cid:0) = 5762.15115120125130135 [GeV]γγM00.511.52MC / Fit1−10110210310Events/1 GeVATLASSimulation Internal-1Ldt = 140.00 fb∫ = 13 TeV: sFiducial(cid:0) = 125.14CBµ(cid:0) = 1.84CBσ(cid:0) = 1.71CBLoα(cid:0) = 1.71CBHiα(cid:0) = 5.76CBLon(cid:0) = 10.00CBHinYield(cid:0) = 5762.15115120125130135 [GeV]γγM00.511.52MC / FitFigure 6.8: Background composition as a function of mγγ
estimated using the 2 × 2D method [307, 308].

(left) and pTγγ

(right) [278]

pirically chosen continuous, monotonically falling functions of one variable. A function is

selected from the set independently for each event category using a spurious signal test

complemented by an F-test. Three families of functions comprise the considered set: expo-

nentials of polynomials, ExpPoly, Eq. (6.19), Bernstein polynomials, Bern, Eq. (6.20),∗ and
power law functions, Pow, Eq. (6.21). An overall normalization factor is implied for each
function. an and cn are free parameters of the functions.†

ExpPolyN (x) = exp
anxn ,
N(cid:88)n=0
cn(cid:18)N
n(cid:19)xn(1 − x)N−n,
N(cid:88)n=1

PowN (x) =

cnxan.

BernN (x) =

N(cid:88)n=0

(6.19)

(6.20)

(6.21)

Only functions up to a certain order were considered. These are: ExpPoly1 or Exp,

[105, 160] GeV accordingly before fitting.

∗The definition of Bern assumes function domain of x ∈ [0, 1]. The mγγ
†cn are linear parameters, while an are not. ExpPoly can of course be fitted linearly in a binned fit to
logarithms of the event counts in each bins. In the form the background functions are written in Eqs. (6.19)
to (6.21), one of the parameters has to be eliminated for normalization when used in a likelihood fit.

values are transformed from

172

110120130140150160 [GeV]γγm020406080100Fraction [%] InternalATLAS-1 = 13 TeV, 139.0 fbsγγjγjjStat. Unc.Tot. Unc.050100150200250300350400450 [GeV]γγTp020406080100Fraction [%] InternalATLAS-1 = 13 TeV, 139.0 fbsγγjγjjStat. Unc.Tot. Unc.ExpPoly2, ExpPoly3, Bern3, Bern4, Bern5, Pow1 or Pow, and Pow2. The studies show that

the best background model is given by ExpPoly2 for most event categories. In this case, the

background model can be written as

B(mγγ) = Nb · exp(cid:32)−

mγγ
a1 −

m2
γγ

a2 (cid:33) .

(6.22)

Background templates are created for the background studies. The primary reason for

this approach is that for many event categories, the data do not contain a sufficient number

of events to simultaneously constrain the functional form and the nuisance parameters of

the background model. Therefore, the form of the background model cannot be selected by

fitting directly to the data sidebands. Construction of templates allows efficient production

of smooth test samples with high statistics, to improve the accuracy of background shape

estimation, especially in low cross section regions with high pT or jet multiplicity. Templates
are also provide a way to add reducible background contributions to the MC, which are not

simulated.

The irreducible component is obtained from the simulated Sherpa γγ sample, with

events required to pass the nominal event selection. To determine the reducible γ-jet and jet-

jet components’ relative contributions, one signal and several control regions are defined by

relaxing identification and/or isolation requirements for one or both photons. Reconstruction

efficiencies are estimated for each region using MC. Combined with the respective event

yields from data, a system of linear equations is formed, which is then solved to obtain the

sought fractions of events with jets faking photons. This method is referred to as a double

2-dimensional, or 2 × 2D, sideband decomposition [307, 308].

An alternative approach to template construction, using a single-component reducible

173

background, was introduced in the 80 fb−1 analysis in 2018. This approach reduced the
template shape uncertainty, which was hard to constrain with the original two-component

approach, due to large uncertainties on the jj fraction. While both approaches produce

distributions in agreement with the data sidebands, ignoring the jj contribution results in

better agreement. See Appendix F in [276] and Appendix D in [278] for details.

Figure 6.9: Comparison of the background templates to the data in sideband regions for the
inclusive fiducial event category using the two-component (left) and one-component (right)
reducible background model.

A smoothing procedure is applied in order to suppress statistical fluctuations. Gaussian

process regression (GPR) with a Gibbs kernel and an additional custom error kernel has been

used for template smoothing since the 2019 analysis. Prior to that, an exponential function

or a second degree polynomial, depending on the event category, was fit to the fractions of

reducible background. These smooth functions were used to reweight the Sherpa γγ events

to produce templates including all background contributions. By contrast, GPR smoothing

is applied after reweighting.

The templates are derived independently for each event category, and are normalized to

data sidebands. All templates are validated for compatibility with the data sidebands with

the help of a Pearson’s χ2 test.

174

20004000600080001000012000Event EntriesdataγγFast Simulation Sherpa Data-driven reweighted jet jetjetγData-driven reweighted 110120130140150160[GeV]γγM0.51/ ndf 2χProb 246.4 / 1790.0006232Data / MC06.5.2.1 Spurious signal

An empirically chosen modeling function can over- or underestimate the number of back-

ground events in the signal region. This defect can bias signal extraction, and is referred to

as spurious signal. Dedicated spurious signal studies are conducted to minimize this effect.

This is done by fitting the combined signal + background model to the background-only tem-

plates using each one of the background functions, independently for each event category.

The fits are performed for different values of the peak mγγ
, scanning the whole signal region,
mγγ ∈ [121, 129] GeV. The fitted number of signal events and its uncertainty, Nsp ± ∆sp,
are measures of the spurious signal bias. The most suitable function is selected according to

the following prescription. First, any functions passing either of the following two criteria

are selected:

• Nsp ± n∆sp must be less than 20% of the background uncertainty, δS, or
• Nsp ± n∆sp must be less than 10% of the expected number of signal events.

The criteria are applied with n = 0, then 1, then 2, until at least one function passes. The

χ2 probability of the fits is required to be greater than 1%, to veto parametrizations with

accidentally small Nsp. If multiple functions pass, the first tie breaker is the smallest number

of degrees of freedom, and the second one is the smallest Nsp.

6.5.2.2 F-test

Because data may contain features that do not exist in MC, it is important to test whether

a selected background model has a sufficient number of degrees of freedom. To do so, the

χ2 of each selected function with respect to the histogram of data sidebands is compared to

175

Figure 6.10: Results of the spurious signal test for the inclusive fiducial event category,
showing the amount of fitted spurious signal S (referred to as Nsp in the text) relative to
the background uncertainty δS (left) and relative to the expected signal yield SRef
(right).
SRef

is obtained from fitting the signal model to the signal MC.

that of the function with the next higher number of parameters, usually in the same family.∗

This provides a measure of probability that the data follows the simpler model statistically

significantly worse than a more complicated one.

The F -statistic is defined as

F1,2 =

2

χ2
1−χ2
p2−p1
χ2
2
n−p2

=

∆χ2
χ2
2

ndf2
∆ndf

,

(6.23)

where pi are the numbers of free parameters of the two function, n is the number of histogram
bins, and ndfi = n − pi are the numbers of degrees of freedom. The F -statistic follows the
F -distribution (Fig. 6.11) uniquely specified by ∆ndf and ndf2. The background model
functions are required to pass the F -test with the probability P > 0.05, given by the integral

of the F -distribution above the F -value. Otherwise, the respective higher order function is

∗If a function fails the F-test, usually the function from the same family with one extra parameter is
used instead. Exceptions include situations where a different function gives significantly smaller spurious
signal, or two extra parameters would need to be added, as is the case for the Pow functions. For symmetric
distributions, the same function is used on both sides. If the one on one side fails the F-test, the one on the
other side is also substituted.

176

1211221231241251261271281298−6−4−2−02468SδS/ExponentialExpPoly2ExpPoly3Bern3Bern4Bern5PowPow2m [GeV]γγ1211221231241251261271281290.6−0.4−0.2−00.20.40.6RefS/SExponentialExpPoly2ExpPoly3Bern3Bern4Bern5PowPow2m [GeV]γγF

f
o

F
D
P

∆ndf = 1

ndf2 = 86

P = 0.6 > 0.05

F

Figure 6.11: The F -distribution for ∆ndf = 1 and ndf2 = 86 corresponding to the F -test
for the inclusive fiducial event category for the ExpPoly2 background function and 0.5 GeV
binning in mγγ
. The highlighted area shows the integral of the distribution corresponding
to the P -value.

used for the analysis instead.

6.5.3 Signal extraction

The signal yield, or the number of signal events present in the dataset, is obtained by

performing an unbinned extended maximum likelihood fit of the statistical model to the

data, using the likelihood function given in Eq. (6.24),

Li∈ bins =

e−νi

ni! (cid:89)j ∈ events

in bin i

(cid:104)νsig
i S(cid:0)mγγ

j(cid:12)(cid:12)θsig
i (cid:1) + νbkg

i B(cid:0)mγγ

i

j(cid:12)(cid:12)θbkg

(cid:1)(cid:105) ,

(6.24)

where the index j runs over the data events within the event category i. S and B are
are the respective
respectively the signal and background models. νsig, bkg

and θsig, bkg

i

i

yields and vectors of nuisance parameters. For event categories corresponding to bins of

a certain variable, for which differential analysis is performed, the likelihood fit is done

simultaneously for all bins, indexed by i. The overall likelihood function is a product of the

177

likelihoods defined for the individual bins. The under- and overflow bins, or complementary

categories, are also included in the fit, so that every fit is done using the whole dataset. ni
is the observed number of events in bin i, and νi = νsig
is the mean of the underlying
Poisson distribution.

i + νbkg

i

In order to obtain estimates of systematic uncertainties on the diphoton mass spectrum

due to photon reconstruction, the signal model parameters, mean (µ) and width (σ), defined

in Eq. (6.18), are adjusted to allow them to vary around their nominal values, determined

and fixed by the fits to the nominal MC samples. Multiple linearly independent variations,

indexed by k in the following discussion, are introduced for the µ and σ parameters to ac-

count for the different sources of uncertainty in photon reconstruction. The scales of these

, are set by the photon energy scale (PES) and resolution (PER) uncertain-

variations, δk
ties, for the µ and σ, respectively. The δk
single-photon PES and PER estimates, using systematic MC samples, as described in Sec-

are derived for the diphoton system from the

tion 6.7.1.1.

Independent δk,i

values are derived for each uncertainty component, k, and

each event category, i.

The specific size of the variations is controlled by the respective nuisance parameters, θk
In this method, the optimization of these nuisance parameters is what propagates the photon

.

reconstruction uncertainties into the diphoton spectrum model. To impose the uncertainty

scales and to maintain linear independence of the k uncertainty components, multiplicative
constraints, Ck
as

, are applied to the overall likelihood function, which can then be expressed

L = (cid:89)i∈ bins

Li (cid:89)k

Ck(θk),

(6.25)

178

The constraints are all given the form of the standard normal p.d.f., evaluated at θk

,

Ck(θk) = Gaus(θk; 0, 1) =

1
√2π

e−θ2

k/2.

(6.26)

Incorporation of the uncertainty scales into the expressions for the variations of the signal

model parameters, rather than the constraints on the likelihood, is a convenient choice,

because it enables this uniform definition of the constraints. Since likelihood constraints also

define the expectation distributions for the nuisance parameters, the choice of the standard
normal p.d.f. for the Ck
and to reduce the dimensionality of the fitted model, the value of each nuisance parameter,

makes the pulls∗ on the θk

themselves. For simplicity,

equal to the θk

, is taken to be the same for every event category.

θk

The variations applied to the signal model parameters are as follows:

• 1 symmetric variation is applied to the mean for the Higgs mass,

µ(θmH ) = mnom

H · (1 + δmH · θmH ) ,

(6.27)

H = 125.09 GeV and fractional uncertainty δmH = 0.19% obtained in Run 1 [306].
from its nominal value in units

describes the deviation of mH

with mnom
The nuisance parameter θmH
of its uncertainty. θmH

is constrained to one value for all bins in a distribution.

• 39 asymmetric variations are introduced to estimate photon energy scale (PES) sys-

tematic uncertainties on the mean parameter, µ,

∗The pull of a nuisance parameter θ is defined as (ˆθ − θ0)/ˆσθ



179

µi(θPES) = µnom

i

·

NPES(cid:89)k=1

1 + δPES,+

k,i

1 + δPES,−

k,i

k

k

, θPES

k ≥ 0
k < 0

· θPES
· θPES
. See Section 6.7.1.1 for further discussion.

, θPES

(6.28)

,

where the single-source photon reconstruction uncertainties, δPES,±
dently for each event category i, while the nuisance parameters, θPES

k,i

, are derived indepen-

, are shared∗ between

k

all i in the same fit. The procedure for derivation of δPES,±

k,i

is discussed in Section 6.7.1.1.

• 9 asymmetric log-normal variations are introduced to estimate photon energy resolution

(PER) systematic uncertainties on the width parameter, σ,

σi(θPER) = σnom

i

·

NPER(cid:89)k=1

θPER
k

θPER
k

· exp(cid:115)ln(cid:18)1 +(cid:16)δPER,+
· exp(cid:115)ln(cid:18)1 +(cid:16)δPER,−

k,i

k,i

k

(cid:17)2(cid:19), θPER
(cid:17)2(cid:19), θPER

k

≥ 0

< 0

,

(6.29)



similarly to the PES variations. The reason for the use of log-normal PER variations is that

resolution cannot be negative.

While the background function shape for each event category is chosen from the MC

studies, the parameters of these functions float in the signal extraction fits, and, therefore,

add to the total number of nuisance parameters. Thus, the total number of nuisance pa-

rameters in each signal extraction fit is 1 + NPES + NPER +(cid:80)i nbkg

numbers of background function parameters for each bin. The parameters of interest are the

, where nbkg

i

are the

i

two yields, νsig, bkg

i

. These are not constrained to non-negative values. As an example, the

fit for the inclusive fiducial category is show in Fig. 6.12.

Because an independent fit is performed to obtain a differential distribution for every

variable of interest, the analysis produces multiple estimates of the total number of signal

events. As a cross check, these are compared with each other, as show in Fig. 6.13. The

individual estimates of the total number of signal events are found to agree with each other.

∗The observed pulls for PES and PER nuisance parameters (see Fig. 6.18), were all found to be much
smaller than 1 (cid:22) the standard deviation of their expectation distribution (cid:22) not soliciting a more detailed
analysis with independent pulls for each bin.

180

Figure 6.12: Signal extraction fit for the inclusive fiducial event category.

Figure 6.13: Comparison of the total signal yields obtained from independent fits for differ-
ent combinations of event categories, i.e. different differential variables. Overflow bins are
included in the fits. The inclusive fiducial spurious signal uncertainty is shown for reference.

181

01000020000300004000050000Events / GeVATLASInternalDiphoton fiducial-1= 13 TeV, 139.0 fbsDataBackgroundSignal + backgroundSignal01000data bkg−110120130140150160[GeV]γγm2−02pulls= 125.09 GeVHm533.39±= 6546.12sigNBkg: ExpPol2InclusiveggTp|gg|yj1TpjetsNjjmjjfD560058006000620064006600680070007200740076007800Fit IntegralsFit yield (and error)Inclusive fid. spurious signalThe statistical model for this analysis is implemented using the RooFit [309] toolkit for

data modeling. It is a part of the ROOT [310] data analysis framework developed at CERN.

RooFit uses the Minuit2 [311] minimization package for likelihood optimization and the

Minos [312] algorithm for calculation of parameter errors. Minos profiles the log-likelihood

function and follows it to the nearest points on either side of the maximum, at which its

value decreases from the maximum by 1
2

, thus, in general, yielding asymmetric uncertainty

estimates.

6.6 Correction for detector effects

In order to obtain cross section measurements that are independent of the particular

experimental setup and can be meaningfully compared to results of other experiments and

theoretical predictions, it is important to correct the cross sections obtained from signal

yields for detector effects. There are many different sources of detector effects that can

introduce systematic differences between true and measured distributions of observables.

Examples include calibration, finite resolution, efficiencies, material non-uniformities, gaps

in active detector elements, and operation conditions, such as pileup. It is worth noting,

that these effects are also in general not constant in time, and hence throughout the dataset.

The presence of these effects results in smearing of the observed distributions. Correction,

then, is an inverse problem of deconvolution of the detector effects, which, in this context,

is usually referred to as unfolding [31, 313].

This analysis uses the unfolding method of bin-by-bin correction factors. The correction

factors enter the cross section calculation for each event category, i, as shown in Eq. (6.1)

182

repeated here:

σi =

νsig
i
ci × L

.

(6.30)

The correction factors, ci, account for detector inefficiency, as well as migrations across
boundaries of event categories and accessible regions of phase space. The factors are obtained

from MC, and are given by the ratio of reconstructed to truth events’ weights selected for

each category,

ci =

nreco
i
ntruth
i

, ni =(cid:88)s

σSM
s
Ns

ns,i,

(6.31)

where ntruth is the weight of the selected truth, or particle-level, events, nreco is the weight

of the selected reconstructed, or detector-level, events. The weights obtained from each MC

sample, s, are combined by adding them after reweighting by the ratio of the SM cross
, and the total weight of all events in the sample, Ns.
The relative weights of the MC samples, representing different production processes, can

section for the respective process, σSM

s

introduce model dependence into the correction factors. Fig. 6.14 shows correction factors

from individual MC samples.

The performance of more sophisticated unfolding methods was studied, and is detailed

in [314]. The list of methods considered includes:

1. correction factor method, or bin-by-bin unfolding, described above;

2. response matrix inversion method;

3. Bayesian iterative unfolding;

4. singular value decomposition (SVD) of the response matrix; and

5. iterative, dynamically stabilized (IDS) unfolding.

Appendix G in [278] includes comparisons between yield distributions unfolded using the

first three methods, with the results agreeing within the uncertainties.

183

Figure 6.14: Inclusive fiducial correction factors from each production process MC. The val-
ues shown in black have no particle-level isolation applied and display a significant depen-
dence on the production process. Values in green have particle-level track isolation applied.
Values in red also have calorimeter isolation applied. The production mode dependence is
reduced by isolation requirements. [231]

The method of correction factors is chosen as nominal because it is straight forward

and suitable for limited statistics. And, while it can be biased, comparable or larger bias

was estimated to be introduced by either of the regularized methods for this analysis with

the available dataset. Though the matrix inversion method is theoretically unbiased, it can

inflate uncertainties and, in worst case scenario, yield nonsensical estimator distributions.∗

The bias of the correction factor method is reduced by refraining from overly fine bins. This

is confirmed by the approximately uniform correction factors derived.

∗For a discussion of unfolding methods see Chapter 11 in [31]. The problem with the matrix inversion
method is that, in general, a matrix may not have an inverse, or if it does, the solution may not be numerically
stable.

184

ggHVBFWmHWpHggZHZH125JttHbbH_yb2bbH_ybyt0.620.640.660.680.70.720.740.760.78Correction FactorsRelease 21No IsolationParticle Isolation (Track only)Particle Isolation (Track + Calo)6.7 Uncertainties

Corresponding to each factor in the equation for the measured cross section, Eq. (6.30),

there are uncertainties associated with signal extraction, experimental uncertainties on lu-

minosity estimates, and theoretical modeling uncertainties on correction factors. Statistical

uncertainty is separated from the signal extraction uncertainty, obtained from fitting, with

the help of likelihood profiling, as illustrated in Fig. 6.19. The spurious signal uncertainty is

added into the signal extraction uncertainty in quadrature. All other sources of uncertainty,

besides those on luminosity, correspond to the correction factor. Table 6.5 and Fig. 6.15

below summarize the uncertainties estimated within several event categories.

6.7.1 Signal extraction uncertainties

6.7.1.1 Signal modeling uncertainties

The main sources of uncertainty on the diphoton invariant mass distribution are the

photon energy scale (PES), affecting the mode parameter of the signal model, µ, shifting

the position of the distribution’s peak, and photon energy resolution (PER), affecting the

model’s width parameter, σ, narrowing or broadening the distribution. These effects, as

well as prior information on the Higgs mass from the Run 1 analysis [306], are included

in the statistical model, as constrained pulls on the respective parameters, as discussed in

Section 6.5.3.

The single-source photon reconstruction uncertainties, δk,i

, defined in Section 6.5.3, are

derived using a full decorrelation scheme from single-photon PES and PER calibrations

provided by the ATLAS Egamma group [289, 315]. The derived δk,i
category are listed in Table 6.6. The same set of values is also derived independently for each

values for the inclusive

185

Table 6.5: Breakdown of cross section uncertainties for the inclusive fiducial event category.

Statistics
Signal extraction syst.

Photon energy scale & resolution
Background modeling (spurious signal)

Correction factor
Pile-up modeling
Photon identification efficiency
Photon isolation efficiency
Trigger efficiency
Theoretical modeling
Photon energy scale & resolution

Luminosity
Total

6.9%
7.9%
4.6%
6.4%
2.6%
2.0%
1.2%
1.1%
0.5%
0.5%
0.1%
1.7%
11.0%

Figure 6.15: Component breakdown of the total uncertainty on the measured cross section
in bins of pγγ
T and Njets. These are stack plots, with consecutive components added in
quadrature.

Figure 6.16: Further breakdown of the systematic correction factor component uncertainty
on the measured cross section in bins of pγγ
T and Njets. Only the largest contributions are
explicitly shown. The rest are added as the other.

186

050100150200250300350[GeV]γγTp00.10.20.30.40.50.60.70.80.91fidσ/ fidσΔLuminosityCorrection factor⊕Signal extraction⊕Statistics⊕ATLASPreliminary = 125.09 GeVHm, -1= 13 TeV, 139 fbs, γγ→H= 0 = 1 = 2 3≥jetsN00.10.20.30.40.50.6fidσ/ fidσΔLuminosityCorrection factor⊕Signal extraction⊕Statistics⊕ATLASPreliminary = 125.09 GeVHm, -1= 13 TeV, 139 fbs, γγ→H050100150200250300350[GeV]γγTp00.010.020.030.040.050.060.070.080.090.1fidσ/ fidσΔPhoton energy scalePhoton isolation⊕Photon identification⊕Pile−up modelling⊕Others⊕ATLASPreliminary = 125.09 GeVHm, -1= 13 TeV, 139 fbs, γγ→HUnfolding systematic uncertainties= 0 = 1 = 2 3≥jetsN00.050.10.150.20.25fidσ/ fidσΔPhoton isolationPhoton identification⊕Theoretical modelling⊕Pile−up modelling⊕JES + JER⊕Others⊕ATLASPreliminary = 125.09 GeVHm, -1= 13 TeV, 139 fbs, γγ→HUnfolding systematic uncertaintiesevent category on the analysis. The derivation procedure consists of first generating signal

MC samples for up and down variation of each calibration parameter. Only one parameter

is varied from the nominal value in each sample. The respective diphoton uncertainties

are then obtained from independent fits to the mγγ
The fitted function is the double-sided Crystal Ball function, given by Eq. (6.18), the same

distributions in each event category.

as what is used for the analysis signal model, but without the extra systematics nuisance

parameters. The up and down variations in position of the mγγ
asymmetric uncertainties due to PES sources, and, likewise, for variations of the mγγ
due to PER sources. In other words, this procedure propagates single-photon PES and PER

peak are taken as its

width

uncertainties to the respective ones for the diphoton mass spectrum. Fig. 6.17 shows the

combined effects of simultaneous up and down variations independently for PES and PER.

Figure 6.17: The effect of simultaneous variations of all photon calibrations on the diphoton
mass spectrum, independently for PES (left) and PER (right).

Two important characteristics of constrained nuisance parameters are pulls and impacts.

The pull is a dimensionless measure of the deviation of nuisance parameter θ from its expected

187

Events/0.25 GeV020406080100120140(GeV)γγm1201211221231241251261271281291305−05DataFit−NominalDown1σPESUp1σPESEvents/0.25 GeV020406080100120140160(GeV)γγm1201211221231241251261271281291305−05DataFit−PERPERNominalDown1σUp1σTable 6.6: List of derived relative PES/PER uncertainties, δk
signal model nuisance parameters. The values listed are for the inclusive fiducial region.

, that set the scales for the

Component

Down

Up

Component

Down

Up

0.0046218
EG_RESOLUTION_MATERIALCALO
0.0120173
EG_RESOLUTION_MATERIALCRYO
0.0065407
EG_RESOLUTION_MATERIALGAP
0.0119270
EG_RESOLUTION_MATERIALIBL
0.0377096
EG_RESOLUTION_MATERIALID
0.0200846
EG_RESOLUTION_MATERIALPP0
0.0243305
EG_RESOLUTION_PILEUP
0.0185316
EG_RESOLUTION_SAMPLINGTERM
0.0509220
EG_RESOLUTION_ZSMEARING
EG_SCALE_E4SCINTILLATOR__ETABN0 0.0000001
EG_SCALE_E4SCINTILLATOR__ETABN1 0.0000042
EG_SCALE_E4SCINTILLATOR__ETABN2 0.0000377
0.0001301
EG_SCALE_G4
0.0004108
EG_SCALE_L1GAIN
0.0009825
EG_SCALE_L2GAIN
0.0007395
EG_SCALE_LARCALIB__ETABN0
0.0002312
EG_SCALE_LARCALIB__ETABN1
0.0000487
EG_SCALE_LARELECCALIB
0.0008510
EG_SCALE_LARELECUNCONV__ETABN0
0.0003349
EG_SCALE_LARELECUNCONV__ETABN1
EG_SCALE_LARUNCONVCALIB__ETABN0 0.0002951
EG_SCALE_LARUNCONVCALIB__ETABN1 0.0000689
0.0000415
EG_SCALE_MATCALO__ETABN0
0.0000370
EG_SCALE_MATCALO__ETABN1
0.0000315
EG_SCALE_MATCALO__ETABN2
0.0000527
EG_SCALE_MATCALO__ETABN3
0.0000634
EG_SCALE_MATCALO__ETABN4
0.0000662
EG_SCALE_MATCALO__ETABN5
0.0000596
EG_SCALE_MATCALO__ETABN6
0.0000099
EG_SCALE_MATCALO__ETABN7
0.0000275
EG_SCALE_MATCALO__ETABN8
0.0000002
EG_SCALE_MATCALO__ETABN9
0.0000002
EG_SCALE_MATCALO__ETABN10
0.0000002
EG_SCALE_MATCALO__ETABN11
0.0001372
EG_SCALE_MATCALO_BARREL
0.0000292
EG_SCALE_MATCALO_ENDCAP
0.0000293
EG_SCALE_MATCRYO__ETABN0
0.0000514
EG_SCALE_MATCRYO__ETABN1
0.0000463
EG_SCALE_MATCRYO__ETABN2
0.0000773
EG_SCALE_MATCRYO__ETABN3
0.0000814
EG_SCALE_MATCRYO__ETABN4
0.0000506
EG_SCALE_MATCRYO__ETABN5

0.0044663
0.0115940
0.0059096
0.0112591
0.0362137
0.0186260
0.0231763
0.0205275
0.0708453
0.0000001
0.0000038
0.0000375
0.0001297
0.0004156
0.0009792
0.0007400
0.0002312
0.0000494
0.0008523
0.0003364
0.0002933
0.0000694
0.0000409
0.0000366
0.0000312
0.0000529
0.0000636
0.0000656
0.0000589
0.0000095
0.0000270
0.0000002
0.0000002
0.0000002
0.0001364
0.0000286
0.0000287
0.0000479
0.0000427
0.0000736
0.0000779
0.0000471

0.0000574
EG_SCALE_MATCRYO__ETABN6
0.0001073
EG_SCALE_MATCRYO__ETABN7
0.0001025
EG_SCALE_MATCRYO__ETABN8
0.0000232
EG_SCALE_MATCRYO__ETABN9
0.0000116
EG_SCALE_MATCRYO__ETABN10
0.0000145
EG_SCALE_MATCRYO__ETABN11
0.0001552
EG_SCALE_MATCRYO_BARREL
0.0001513
EG_SCALE_MATCRYO_ENDCAP
0.0001771
EG_SCALE_MATID__ETABN0
0.0001030
EG_SCALE_MATID__ETABN1
0.0001397
EG_SCALE_MATID__ETABN2
0.0000209
EG_SCALE_MATID__ETABN3
0.0000608
EG_SCALE_MATPP0__ETABN0
0.0001869
EG_SCALE_MATPP0__ETABN1
0.0000778
EG_SCALE_PEDESTAL
0.0000728
EG_SCALE_PS__ETABN0
0.0000548
EG_SCALE_PS__ETABN1
0.0000575
EG_SCALE_PS__ETABN2
0.0000945
EG_SCALE_PS__ETABN3
0.0000407
EG_SCALE_PS__ETABN4
0.0000602
EG_SCALE_PS__ETABN5
0.0006191
EG_SCALE_PS__ETABN6
0.0000043
EG_SCALE_PS__ETABN7
0.0001084
EG_SCALE_PS__ETABN8
0.0006396
EG_SCALE_PS_BARREL
0.0006092
EG_SCALE_PS_BARREL_B12
0.0001084
EG_SCALE_PS_ENDCAP
0.0003040
EG_SCALE_S12__ETABN0
0.0003103
EG_SCALE_S12__ETABN1
0.0000001
EG_SCALE_S12__ETABN2
0.0003037
EG_SCALE_S12__ETABN3
0.0000001
EG_SCALE_S12__ETABN4
EG_SCALE_TOPOCLUSTER_THRES 0.0000004
0.0003596
EG_SCALE_WTOTS1
0.0000178
EG_SCALE_ZEESTAT
0.0009334
EG_SCALE_ZEESYST
0.0003389
PH_SCALE_CONVEFFICIENCY
0.0001372
PH_SCALE_CONVFAKERATE
0.0001947
PH_SCALE_CONVRADIUS
0.0004426
PH_SCALE_LEAKAGECONV
0.0006540
PH_SCALE_LEAKAGEUNCONV

0.0000537
0.0001036
0.0000991
0.0000196
0.0000081
0.0000110
0.0001467
0.0001453
0.0001762
0.0001020
0.0001388
0.0000205
0.0000599
0.0001861
0.0000800
0.0000751
0.0000569
0.0000598
0.0000969
0.0000429
0.0000625
0.0006314
0.0000066
0.0001108
0.0006529
0.0006156
0.0001108
0.0003067
0.0003122
0.0000001
0.0003041
0.0000001
0.0000793
0.0003587
0.0000183
0.0009322
0.0003413
0.0001329
0.0001964
0.0004358

188

value θ0 relative to its expected uncertainty, δθ

,

pull(θ) =

ˆθ − θ0
δθ

.

(6.32)

A healthy situation is when the pull average is 0, with a standard deviation close to 1. If this

is not the case, further investigation is required [316]. The impact is a measure of deviation

of a parameter of interest ν with respect to variations of a nuisance parameter within the

range of its uncertainty. This can be expressed as,

impactν(θ) = ∆ν± = ˆˆνθ0±δθ − ˆν,

(6.33)

where ˆν is the parameter of interest estimated with θ floating, and ˆˆνθ0±δθ
parameter estimated with θ fixed to θ0 ± δθ
of the effect that optimization of a nuisance parameter has on a parameter of interest. Pulls

. As the name implies, the impact is a measure

is the same

are symmetric by definition, while impacts can be asymmetric.

As a cross check, the signal model nuisance parameters are ranked according to their

impact on the signal yield, and the values and uncertainties of their pulls are compared

before and after fitting, as shown in Fig. 6.18. The pre-fit pull values are, by design, all

equal to 0 with uncertainty of 1. The post-fit pull values are obtained using the profile

likelihood method, implemented in Minos [311, 312]. The pre-fit impacts are obtained from

signal extraction with nuisance parameters fixed to their pre-fit values of 0. The post-fit

impacts are obtained with nuisance parameters floating, i.e. with fitted nuisance parameters’

values. Before the signal region is unblinded, the signal extraction fitting procedure is tested

using Asimov datasets, which gives the expected values for the pulls and impacts. For

189

(a) Expected NP ranking.

(b) Observed NP ranking.

Figure 6.18: Expected and observed nuisance parameters’ pre- and post-fit pulls and impacts
for the inclusive fiducial category. The nuisance parameters (NP) are ranked by their impacts
on the signal yield parameter of interest (PoI).

example, as can be seen in Fig. 6.18a, the Higgs mass nuisance parameter is expected to

be underconstrained in the fits. The expected post-fit pulls have a mean of 0, because the

Asimov data exactly follows the model distribution.

In order to separate the statistical and systematic components of the uncertainty on

the extracted signal yield, the likelihood function is profiled for the signal yield parameter

of interest with nuisance parameters floating or fixed to their optimal values, as shown

in Fig. 6.19. By the standard prescription [31], the one standard deviation uncertainty
range is given by the values of the parameter for which the quantity −2 log L increases by 1

190

EG_SCALE_MATID__ETABN2EG_SCALE_MATCRYO_ENDCAPEG_SCALE_MATCRYO_BARRELEG_SCALE_MATID__ETABN0EG_SCALE_MATPP0__ETABN1PH_SCALE_CONVRADIUSEG_SCALE_LARCALIB__ETABN1EG_SCALE_LARUNCONVCALIB__ETABN0EG_SCALE_S12__ETABN3EG_SCALE_S12__ETABN0EG_SCALE_L2GAINEG_SCALE_S12__ETABN1EG_SCALE_ZEESYSTEG_SCALE_LARELECUNCONV__ETABN0EG_SCALE_LARELECUNCONV__ETABN1PH_SCALE_CONVEFFICIENCYEG_SCALE_WTOTS1EG_SCALE_LARCALIB__ETABN0EG_SCALE_PS_BARRELPH_SCALE_LEAKAGEUNCONVMassUncertEG_RESOLUTION_MATERIALCALOEG_RESOLUTION_MATERIALGAPEG_RESOLUTION_MATERIALIBLEG_RESOLUTION_MATERIALCRYOEG_RESOLUTION_MATERIALPP0EG_RESOLUTION_SAMPLINGTERMEG_RESOLUTION_PILEUPEG_RESOLUTION_MATERIALIDEG_RESOLUTION_ZSMEARING1.5−1−0.5−00.511.5θΔ) / 0θ- θ(0.04−0.02−00.020.04ν/ νΔBin is: Diphoton fiducialNum. bins in fit: 1Pre-fit impact on PoI Post-fit impact on PoIPost-fit NP and errorPre-fit NP and error EG_SCALE_MATCRYO_BARRELEG_SCALE_S12__ETABN0EG_SCALE_S12__ETABN1PH_SCALE_CONVEFFICIENCYEG_SCALE_LARELECUNCONV__ETABN1EG_SCALE_WTOTS1EG_SCALE_MATID__ETABN0EG_SCALE_MATPP0__ETABN1PH_SCALE_CONVRADIUSEG_SCALE_L1GAINMassUncertPH_SCALE_LEAKAGECONVEG_SCALE_LARCALIB__ETABN1EG_SCALE_PS_BARRELPH_SCALE_LEAKAGEUNCONVEG_SCALE_LARUNCONVCALIB__ETABN0EG_SCALE_S12__ETABN3EG_SCALE_LARCALIB__ETABN0EG_SCALE_LARELECUNCONV__ETABN0EG_SCALE_ZEESYSTEG_SCALE_L2GAINEG_RESOLUTION_MATERIALCALOEG_RESOLUTION_MATERIALGAPEG_RESOLUTION_MATERIALIBLEG_RESOLUTION_MATERIALCRYOEG_RESOLUTION_SAMPLINGTERMEG_RESOLUTION_MATERIALPP0EG_RESOLUTION_PILEUPEG_RESOLUTION_MATERIALIDEG_RESOLUTION_ZSMEARING1.5−1−0.5−00.511.5θΔ) / 0θ- θ(0.04−0.02−00.020.04ν/ νΔBin is: Diphoton fiducialNum. bins in fit: 1Pre-fit impact on PoI Post-fit impact on PoIPost-fit NP and errorPre-fit NP and error Figure 6.19: The likelihood profile for the signal yield in the inclusive fiducial event category.
The combined statistical and systematic uncertainty is determined from the profile obtained
with floating nuisance parameters, shown with a solid line. The statistical-only uncertainty
is determined from the other, dashed, profile, for which the nuisance parameters are fixed to
the best fit values.

from its minimum. The likelihood profile, obtained with floating nuisance parameters, gives

the combined systematic and statistical uncertainty. The likelihood profile, constructed by

fixing the nuisance parameters to their values at the global likelihood maximum, gives the

statistical contribution to the uncertainty. The systematic uncertainty is then obtained

by taking the quadrature difference between the total and the statistical-only uncertainty

estimates. Comparisons between expected, obtained using Asimov datasets, and observed

systematic and statistical uncertainties on signal yields are shown in Fig. 6.20.

6.7.1.2 Background modeling uncertainties

The uncertainty associated with the background modeling is estimated in the spurious

signal studies. As discussed in Section 6.5.2.1, the spurious signal is the number of signal

events obtained from fitting the signal + background model to the background-only MC

templates. This quantity is used select the functional form of the background model applied

in signal extraction. The contribution to the signal extraction uncertainty from the spurious

191

12001300140015001600170018001900200000.20.40.60.81Stat + systStat onlySignal extractionDiphoton fiducial= 13 TeVs, -136.1 fbFigure 6.20: Expected systematic and statistical uncertainties on signal yield estimated using
the Asimov method for inclusive fiducial event category and in bins of jet multiplicity.

signal is estimated by the maximum value of Nsp/SRef obtained in the spurious signal scans

over the signal region, where Nsp is the fitted number of spurious signal events, and SRef

is the expected signal yield obtained from the signal MC. The spurious signal uncertainty

is added in quadrature to the signal extraction uncertainty, after correcting SRef for the

observed signal strength.

6.7.2 Theoretical modeling uncertainties

6.7.2.1 Signal composition uncertainty

In the computation of the correction factors, the events from each MC sample are

weighted by the respective SM cross section, as shown in Eq. (6.31). Correction factors

are thus affected by the uncertainties of the production modes’ cross sections. Another

related source of uncertainty is due to model dependence, which may be introduced, if the

correction factors for different modes vary significantly. These uncertainties are estimated by

varying the models’ cross sections within the uncertainty bounds obtained experimentally in
the couplings measurements in H → γγ and H → 4(cid:96) channels [317, 318]. The composition
uncertainty bounds are determined by recomputing correction factors using simultaneous up

192

Diphoton fiducial00.10.20.30.40.50.60.70.80.91Relative fit errorTotalStatSyst= 13 TeVs, -1140.4 fbUsing AsimovATLASInternal= 0jetsN3≥jetsN00.10.20.30.40.50.60.70.80.91Relative fit errorTotalStatSyst= 13 TeVs, -1140.4 fbUsing AsimovATLASInternal= 1jetsN= 2jetsNor down variations off all production mode cross sections.

6.7.2.2 Unfolding bias

A recursive technique is used to estimate the systematic uncertainty due to unfolding bias.

The procedure consists of reweighting the signal MC, so that its pγγ

match more closely with the measured unfolded distributions. Correction factors are then

T and(cid:12)(cid:12)yγγ(cid:12)(cid:12) distributions

derived again using the reweighted MC, and the uncertainty is estimated by the difference

between the reweighted and original correction factors. The reweighting functions are derived

by smoothing the ratio between the measured and MC distributions using a Gaussian kernel.

The comparison between the original and reweighted distributions is shown in Fig. 6.21.

Simultaneous reweighting with respect to both distributions can be applied because the two

variables are largely uncorrelated.

Figure 6.21: Original and reweighted pγγ
estimation of the unfolding bias uncertainty.

T (left) and(cid:12)(cid:12)yγγ(cid:12)(cid:12) (right) MC distributions used in

6.7.2.3 Event modeling

To simulate collision events, MC generators apply several algorithms to model different

aspects of an event. This is discussed further in Chapter 3. Parton shower and underlying

193

050100150200250300350pTyy [GeV]050100150200250300Entries / bin widthOriginal MCDataReweighted MC00.20.40.60.811.21.41.61.822.22.4|y_yy|010002000300040005000600070008000900010000Entries / bin widthOriginal MCDataReweighted MCevent models convert inclusive parton-level MC to exclusive particle-level predictions. To

estimate the uncertainty from these models, the correction factors derived from the nominal

MC samples, produced using Pythia 8 [81, 319], are compared to those obtained using

Herwig 7 [320]. The two programs implement different algorithms for parton showers and

modeling of non-perturbative effects of hadronization and multi-parton interactions. The

difference between correction factors derived using Pythia or Herwig, from gluon-fusion

samples generated with Powheg, is taken as the systematic uncertainty.

6.7.2.4 Dalitz events

The signal MC samples used in this analysis, produced with Pythia 8, include events

with unstable off-shell photons. These, so called, Dalitz events, represent contributions from
the H → γγ∗ → γf ¯f decay channel, where γ∗ is an off-shell photon, and f is any charged
fermion. These events constitute approximately 6% of the total number of generated events.

They, however, are not considered a part of the fiducial volume, which requires a stable

diphoton system. The Dalitz events are removed from particle-level distributions, and the

remaining event are reweighted to maintain correct cross section. At the detector level,

around 0.3% of the signal MC events that pass fiducial selection are Dalitz events. This frac-

tion is approximately constant across the analyzed event categories. These events are not

removed after reconstruction, but are corrected for by the unfolding. Dalitz decay branch-

ing ratio is not known very precisely, and different generators produce different fractions.

Because of this, a conservative uncertainty of 100% is assigned to the Dalitz contribution,

which results in an approximately 0.3% uncertainty on the correction factors.

194

6.7.3 Experimental uncertainties

6.7.3.1 Luminosity

Luminosity measurements are discussed in Section 6.2. Estimates of luminosity and its

uncertainty, given in Table 6.2 are obtained using methods discussed in [280, 284], which are

based on van der Meer [283] scans.

6.7.3.2 Trigger efficiency

The diphoton trigger efficiency is obtained using the bootstrap method in data [321], and

using trigger matching in MC. The estimated efficiency is 99.16+0.23
−0.49

which is in agreement with the predictions from simulations.

(stat.)+0.34
−0.52

(sys.) [279],

6.7.3.3 Vertex selection efficiency

Identification of the primary vertex is a part of the event selection criteria of this analysis.

An estimate of the primary vertex selection efficiency is a part of the correction factor, and,

thus, contributes to its uncertainty. For this analysis, the primary vertex is corrected using

photon pointing information [215]. The primary vertex selection is one of the factors that

impact the migration of events in an out of the fiducial region.

The vertex selection uncertainty is estimated by the difference between vertex selection
efficiencies in data and MC using Z → ee events, ignoring electron tracks. The uncertainty
is generally found to be < 0.3%. The ratio of the efficiencies is used to increase the weights
of the analysis MC events with |zreco − ztrue| > 0.3 mm.

195

6.7.3.4 Photon selection

Photon identification efficiency Photon reconstruction and identification requirements

are discussed in Section 5.2.3. The analysis selection criteria are listed in Section 6.3.2. A

combination of three methods is used to estimate photon identification efficiency:
1. Photon tagging in Z → (cid:96)(cid:96)γ events, providing a very pure sample at lower pT;
2. Applying electron-to-photon mapping derived from MC to data Z → ee events; and
3. Using a sideband method, requiring to solve a linear system of equations, combined

with track isolation as an additional discriminating variable.

The methods agree with each other within the statistical uncertainties.

Photon isolation efficiency Track and calorimeter isolation uncertainties are combined

in quadrature. These are estimated using data, by shifting the respective isolation require-

ments.

Photon energy scale and resolution In addition to affecting the diphoton mass distri-

bution, contributing to the signal yield uncertainty, accounted for in Section 6.7.1.1, photon

reconstruction uncertainties contribute to event migration, both in an out of the fiducial

volume and between bins. The migration contributes to the correction factor uncertainty,

which is estimated from MC by simultaneously varying the PES and PER parameters up or

down. The migration uncertainties, δmig

, are given by

δmig =

α(cid:48)
α − 1, α((cid:48)) =

((cid:48))
A

N
((cid:48))
((cid:48))
A + N
B

N

,

(6.34)

where A and B are two event categories and primes denote the numbers of events obtained

from simulations with varied PES and PER parameters [322].

196

6.7.3.5 Jet selection

Though the fiducial selection does not have any jet requirements, the jet-related event

categories are affected by jet reconstruction.

Jet energy scale and resolution JES and JER uncertainties are estimated with the help
of the transverse momentum balance technique, using Z + jets, γ + jets, and dijet events, as
described in [323].

Jet vertex tagging efficiency The jet vertex tagging (JVT) algorithm, discussed in

Section 5.3.5, is used to suppress pileup jets originating from additional interaction vertices.
Selection criteria for both data and simulation require that central jets, within |η| < 2.5, pass
a JVT cut. The JVT selection uncertainty contributes to the correction factor uncertainty,

as false acceptance of pileup (PU) jets and false rejection of hard scatter (HS) jets can

cause event migrations between bins of jet-dependent observables, especially at higher jet

multiplicities. The JVT uncertainty is estimated by comparing the JVT selection efficiency

in MC and in data sidebands [252]. The efficiency is given by the fraction of central jets

passing the cut. In the simulation, a jet is defined as pileup if it has pT > 10 GeV and is
not matched to a particle-level jet within ∆R < 0.2. Therefore, in MC, the efficiency can

be obtained for both HS and PU jets independently, while only the combined efficiency for

all central jets can be obtain from the data. The separate efficiencies for HS and PU jets in

the data are estimated by scaling the combined efficiency to the HS and PU fractions from

the MC. The JVT uncertainty is small for the HS jets, amounting at most to 0.3%. The

uncertainties for PU jets is on the order of 2%. The residual pileup jet contamination is

corrected for in the unfolding.

197

6.7.3.6 Pileup reweighting

The modeling of pileup is based on simulations of inelastic pp collisions. As discussed in

Section 3.6, pileup reweighting is applied to the MC samples so that their distributions of the

number of interactions per bunch crossing match that observed in the collected data. The

reweighting impacts the correction factors. Pileup reweighting is varied to cover the uncer-

tainty in the ratio between the predicted and measured inelastic cross section for events with

the mass of the non-diffractive hadronic system greater than 13 GeV [324]. This effectively
shifts the pileup distribution by ±3%, as shown in Fig. 6.22.

Figure 6.22: Effects of pileup reweighting variations on the distribution of the number of
primary vertices, Npv.

6.8 Results

6.8.1 Asimov closure tests

To cross check the analysis procedure, a mock analysis was performed using Asimov

datasets instead of the data. These were built using a signal parametrization from MC, and

198

05101520253035404550pvN00.010.020.030.040.050.06data2017NominalPRW upPRW downa background parametrization from data sidebands. The results are shown in Fig. 6.23. No

biases were observed. This cross check also provides estimates of the expected statistical and

systematic uncertainties from fitting and unfolding.

Figure 6.23: Cross checks of the analysis using Asimov datasets for diphoton transverse
momentum (left) and jet multiplicity (right) distributions. The extracted cross section is
compared to the particle-level MC.

In comparison to the 36 fb−1 results [272–274], the statistical uncertainty is reduced by
almost a factor of 2, while the experimental systematic uncertainty is of the same order,

with larger contributions from spurious signal uncertainty and jet systematics, due to higher

pileup conditions. The experimental uncertainty is largely reduced in comparison to the

80 fb−1 measurements [275, 276], that used preliminary calibrations.

199

050100150200250300350[fb/GeV]γγT,p/ dfidσd2−101−101AsimovATLAS-1= 13 TeV, 140.0 fbs, γγ→H, tot. unc.Asimovsyst. unc.XHdefault MC + H→ggbbH+ttH+VH= VBF+XH[GeV]γγT,p050100150200250300350XHRatio to default MC + 0.511.5[fb]fidσ0204060AsimovATLAS-1= 13 TeV, 140.0 fbs, γγ→H, tot. unc.Asimovsyst. unc.XHdefault MC + H→ggbbH+ttH+VH= VBF+XH> 30 GeVTp= 0.4, Rtkanti jetsN= 0= 1= 23≥XHRatio to default MC + 0.511.526.8.2 Unfolded cross sections and theory comparisons

The inclusive fiducial cross section for the pp → H → γγ process obtained in the most

resent analysis using the complete Run 1 dataset is

σfid = 65.2 ± 4.5 (stat.) ± 5.6 (exp.) ± 0.3 (theory) fb ≈ 65.2 ± 7.2 fb ,

(6.35)

which can be compared with the state-of-the-art Standard Model prediction of 63.5± 3.3 fb,
calculated to N3LO in perturbative QCD [325]. The measured value agrees with the predic-

tion within 1 standard deviation. The uncertainty on the measured cross section is domi-

nated by the systematic uncertainties related to signal extraction, namely the photon energy

resolution and the spurious signal.

The figures below show the measured differential cross sections and compare them to the

nominal MC, as well as the state-of-the-art theoretical predictions discussed in Section 3.6.

Note that all additional theoretical predictions provide cross sections only for the ggF process,

and the summed cross section of other processes (XH, shown in light green) is added to them

from the default MC.

The compatibility between the measured and default MC differential distributions was

assessed using a χ2 test, using the covariance matrix, constructed from the full set of mea-

surement uncertainties, taking into account correlations between bins, as well as theoretical

uncertainties on the SM prediction. Table 6.7 lists the p-values obtained in the χ2 tests

which indicate excellent agreement between data and MC for all observables measured in

the 2019 analysis [277, 278].

200

Table 6.7: p-values from χ2 tests for compatibility of the measured differential distribu-
tions with those predicted using the analysis MC, for the observables measured in the 2019
analysis [277, 278]. The tests used the full set of measurement uncertainties for the χ2
computation.

Observable

p(χ2)

pγγ
T
0.44

(cid:12)(cid:12)yγγ(cid:12)(cid:12)

0.68

pTj1
0.77

Njets ∆φjj,signed mjj
0.96
0.75

0.82

Figure 6.24: Fiducial differential cross section measurements for pγγ

T , |yγγ|, pj1T , and mjj

done with(cid:82) L = 139.0 fb−1 in 2019 [277, 278].

201

050100150200250300350  [fb/GeV]γγTp / dfidσd2−101−10110  PreliminaryATLAS-1 = 13 TeV, 139 fbs, γγ→HData, tot. unc.syst. unc.XH default MC + H→ggbbH+ttH+VH = VBF+XHXH + LL3 N⊕ SCET NNLO ⊕NNLOJET   [GeV]γγTp050100150200250300350Ratio to default pred.0.511.5200.20.40.60.811.21.41.61.822.22.4|  [fb]γγy / d|fidσd020406080  PreliminaryATLAS-1 = 13 TeV, 139 fbs, γγ→HData, tot. unc.syst. unc.XH default MC + H→ggbbH+ttH+VH = VBF+XHXH SCETlib+MCFM8 + H→gg|γγy|00.20.40.60.811.21.41.61.822.22.4Ratio to default pred.0.511.5050100150200250300350  [fb/GeV]j1Tp / dfidσd2−101−101  PreliminaryATLAS-1 = 13 TeV, 139 fbs, γγ→HData, tot. unc.syst. unc.XH default MC + H→ggbbH+ttH+VH = VBF+XHXH NNLOJET + H→ggXH SCETlib (STWZ) + H→gg 0≥ jetsN = 0.4, R tkanti   [GeV]j1Tp050100150200250300350Ratio to default pred.0.511.50200400600800100012001400  [fb]jjm / dfidσd3−102−10  PreliminaryATLAS-1 = 13 TeV, 139 fbs, γγ→HData, tot. unc.syst. unc.XH default MC + H→ggbbH+ttH+VH = VBF+XHXH) + MEPS@NLO Sherpa (H→ggXH GoSam+Sherpa + H→gg > 30 GeVTp = 0.4, R tkanti  [GeV]jjm0200400600800100012001400Ratio to default pred.0.511.5Figure 6.25: Fiducial differential cross section measurements for Njets and ∆φjj done with

(cid:82) L = 139.0 fb−1 in 2019 [277, 278].

202

  [fb]fidσ110210310  PreliminaryATLASData, tot. unc.-1 = 13 TeV,  139 fbs, γγ→Hsyst. unc. > 30 GeVTp = 0.4, R tkanti XH default MC + H→ggXHLO + 3NXHLO+JVE + 3NXHSTWZ, BLPTW + XHNNLOJET + XHPowheg NNLOPS + XHGoSam+Sherpa + XH) + MEPS@NLOSherpa (XHMG5 aMC@NLO + bbH+ttH+VH = VBF+XHjetsN=0=1=20≥1≥2≥3≥Ratio to default pred.0.511.53−2−1−0123  [fb]jjφΔ / dfidσd012345  PreliminaryATLAS-1 = 13 TeV, 139 fbs, γγ→HData, tot. unc.syst. unc.XH default MC + H→ggbbH+ttH+VH = VBF+XHXH) + MEPS@NLO Sherpa (H→ggXH GoSam+Sherpa + H→gg > 30 GeVTp = 0.4, R tkanti jjφΔ3−2−1−0123Ratio to default pred.0.511.5Figure 6.26: Fiducial differential cross section measurements for pγγ

T , |yγγ|, pj1T , and Nb-jets

done with(cid:82) L = 79.8 fb−1 in 2018 [275, 276].

203

0501001502002503003502-101-101  [fb/GeV] ggTp / dfidsd  PreliminaryATLAS-1 = 13 TeV,  79.8 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXHSCET + ¯NNLOJET050100150200250300350  [GeV] ggTp012 XHRatio to default MC + 012020406080|  [fb] ggy / d|fidsd  PreliminaryATLAS-1 = 13 TeV,  79.8 fbs, ggfiHdata, tot. unc.syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH SCETlib+MCFM8 + Hfigg00.20.40.60.811.21.41.61.822.22.4|ggy|012 XHRatio to default MC + 02002-101-101  [fb/GeV] j1Tp / dfidsd  PreliminaryATLAS-1 = 13 TeV,  79.8 fbs, ggfiHdata, tot. unc.syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH NNLOJET + HfiggXH SCETlib (STWZ) + Hfigg 0‡ jetsN = 0.4, R tkanti 050100150200250300350  [GeV] j1Tp12 XHRatio to default MC + 12342-101-10110210310  [fb]fids  PreliminaryATLAS-1 = 13 TeV,  79.8 fbs, ggfiHdata, tot. unc.syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHttH= 0me+N 1, ‡| < 2.5) h > 30 GeV, |T (pjetsN= 0= 1 2‡b-jetsN012 XHRatio to default MC + Figure 6.27: Fiducial differential cross section measurements for the fiducial event categories

defined in Section 6.4.1 done with(cid:82) L = 36.1 fb−1 in 2017 [272–274].

204

1-10·2123456102030210  [fb]fidsttH-enhancedmissTEHigh  1 ‡ leptonNVBF-enhanced Diphoton fiducial 95% C.L.95% C.L.95% C.L.  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc. = 125.09 GeVHmXHLO + 3NXH default MC + HfiggXHPowheg NNLOPS + bbH+ttH+VH = VBF+XHFigure 6.28: Fiducial differential cross section measurements for pγγ
and pγγ
Tt

done with(cid:82) L = 36.1 fb−1 in 2017 [272–274].

T , |yγγ|, | cos θ∗|, |∆yγγ|,

205

0501001502002503003502-101-101  [fb/GeV] ggTp / dfidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXHHRes 2.3 + XH + RadISH+NNLOJET050100150200250300350  [GeV] ggTp012 XHRatio to default MC + 00.20.40.60.811.21.41.61.822.22.4020406080|  [fb] ggy / d|fidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH SCETlib+MCFM8 + Hfigg00.20.40.60.811.21.41.61.822.22.4|ggy|012 XHRatio to default MC + 00.10.20.30.40.50.60.70.80.91050100150200)|  [fb] *q / d|cos(fidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH SCETlib+MCFM8 + Hfigg00.10.20.30.40.50.60.70.80.91)|*q|cos(012 XHRatio to default MC + 00.20.40.60.811.21.41.61.82050100|  [fb] ggyD / d|fidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH SCETlib+MCFM8 + Hfigg00.20.40.60.811.21.41.61.82|ggyD|012 XHRatio to default MC + 0501001502002502-101-101  [fb/GeV] ggTtp / dfidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXHHRes 2.3 + 050100150200250  [GeV] ggTtp012 XHRatio to default MC + Figure 6.29: Fiducial differential cross section measurements for inclusive and exclusive jet
T > 50 GeV (bottom) done with
multiplicities, Njets, for pj

T > 30 GeV (top) and for pj

(cid:82) L = 36.1 fb−1 in 2017 [272–274].

206

110210310  [fb]fidσ  ATLASData, tot. unc.-1 = 13 TeV,  36.1 fbs, γγ→HSyst. unc. > 30 GeVTp = 0.4, R tkanti XH default MC + H→ggXHN3LO + XHN3LO+JVE + XHSTWZ, BLPTW + XHNNLOJET + XHPowheg NNLOPS + XHGoSam+Sherpa + XH) + MEPS@NLOSherpa (XHMG5 aMC@NLO + bbH+ttH+VH = VBF+XH=0=1=20≥1≥2≥3≥jetsN012XHRatio to default MC + 00.511.522.5302040  [fb] fidσ  ATLAS-1 = 13 TeV,  36.1 fbs, γγ→HData, tot. unc.Syst. unc.XH default MC + H→ggXHPowheg NNLOPS + bbH+ttH+VH = VBF+XH > 50 GeVTp = 0.4, R tkanti = 0= 1 2≥jetsN012XHRatio to default MC + Figure 6.30: Fiducial double-differential cross section measurements for pγγ

vided in Njets (top) and | cos θ∗| (bottom) done with(cid:82) L = 36.1 fb−1 in 2017 [272–274].

T further subdi-

207

Tp/ dfidd310210110110ATLAS-1= 13 TeV, 36.1 fbs, HData, tot. unc.Syst. unc.XHdefault MC + HggbbH+ttH+VH= VBF+XH[GeV] TpXHRatio to default MC + 024= 0jetsN= 1jetsN= 2jetsN3jetsN0-1515-3030-7575-3500-4040-6060-100100-3500-100100-200200-3500-200200-350Tp/ dfidd310210110110ATLAS-1= 13 TeV, 36.1 fbs, HData, tot. unc.Syst. unc.XHdefault MC + HggbbH+ttH+VH= VBF+XH[GeV] Tp0-3030-120120-3500-3030-120120-350XHRatio to default MC + 012*)| < 0.5|cos(0.0 *)| < 1.0|cos(0.5 Figure 6.31: Fiducial differential cross section measurements for pj1T , |yj1|, pj2T , |yj2|, HT, and
|∆yjj| done with(cid:82) L = 36.1 fb−1 in 2017 [272–274].

208

0501001502002503003502-101-101  [fb/GeV]  j1Tp / dfidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH NNLOJET + HfiggXH SCETlib (STWZ) + Hfigg 0‡ jetsN = 0.4, R tkanti 050100150200250300350  [GeV]  j1Tp012 XHRatio to default MC + 00.511.522.533.5405101520|  [fb] j1y / d|fidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH NNLOJET + Hfigg00.511.522.533.54|j1y|012 XHRatio to default MC + 02040608010012000.20.40.60.8  [fb/GeV]  j2Tp / dfidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH) + MEPS@NLO Sherpa (HfiggXH GoSam+Sherpa + Hfigg > 30 GeVTp = 0.4, R tkAnti 020406080100120  [GeV] j2Tp012 XHRatio to default MC + 00.511.522.533.540246|  [fb] j2y / d|fidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH) + MEPS@NLO Sherpa (HfiggXH GoSam+Sherpa + Hfigg > 30 GeVTp = 0.4, R tkAnti 00.511.522.533.54|j2y|012 XHRatio to default MC + 0501001502002503003504004505002-101-101  [fb/GeV] TH / dfidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH NNLOJET + Hfigg050100150200250300350400450500  [GeV]TH012 XHRatio to default MC + 012345678024|  [fb] jjyD / d|fidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH) + MEPS@NLO Sherpa (HfiggXH GoSam+Sherpa + Hfigg > 30 GeVTp = 0.4, R tkAnti 012345678|jjyD|012 XHRatio to default MC + Figure 6.32: Fiducial differential cross section measurements for mjj, ∆φjj, |∆φjj|, pγγjj
and |∆φγγ,jj| done with(cid:82) L = 36.1 fb−1 in 2017 [272–274].

209

T

,

02004006008001000120014003-102-10  [fb/GeV] jjm / dfidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH) + MEPS@NLO Sherpa (HfiggXH GoSam+Sherpa + Hfigg > 30 GeVTp = 0.4, R tkAnti 0200400600800100012001400  [GeV]jjm012 XHRatio to default MC + 3-2-1-012302468  [fb] jjfD / dfidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH) + MEPS@NLO Sherpa (HfiggXH GoSam+Sherpa + Hfigg > 30 GeVTp = 0.4, R tkAnti 3-2-1-0123jjfD012 XHRatio to default MC + 00.511.522.530510|  [fb]jjfD / d|fidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH) + MEPS@NLO Sherpa (HfiggXH GoSam+Sherpa + Hfigg > 30 GeVTp = 0.4, R tkAnti 00.511.522.53|jjfD|012 XHRatio to default MC + 0204060801001201401601802002-101-101  [fb/GeV] jjggTpD / dfidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH) + MEPS@NLO Sherpa (HfiggXH GoSam+Sherpa + Hfigg > 30 GeVTp = 0.4, R tkAnti 020406080100120140160180200  [GeV]jjggTp012 XHRatio to default MC + 00.511.522.53110210|  [fb] fD / d|fidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH) + MEPS@NLO Sherpa (HfiggXH GoSam+Sherpa + Hfigg > 30 GeVTp = 0.4, R tkAnti 00.511.522.53,jj)|gg(fD-|p01234 XHRatio to default MC + Figure 6.33: Fiducial differential cross section measurements for τ1 and (cid:80) τj done with
(cid:82) L = 36.1 fb−1 in 2017 [272–274].

210

0204060801001201402-101-10110  [fb] C,j1t / dfidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XH020406080100120140 [GeV]C,j1t012 XHRatio to default MC + 0204060801001201402-101-10110 [fb] C,jtS / dfidsd  ATLAS-1 = 13 TeV,  36.1 fbs, ggfiHData, tot. unc.Syst. unc.XH default MC + HfiggbbH+ttH+VH = VBF+XHXH NNLOJET + Hfigg020406080100120140 [GeV]C,jtS012 XHRatio to default MC + Chapter 7

Conclusion

This dissertation has presented the cross section measurements of the Higgs boson produc-

tion process in the diphoton decay channel. The cross section measurements were obtained

from a statistical analysis of the ATLAS data collected in the Run 2 of the LHC. The dataset
is comprised of proton-proton collision events with the center-of-mass energy of √s = 13 TeV.
The integrated luminosity of the dataset amounted to 139.0 ± 2.4 fb−1 [280, 281]. Fiducial
cross sections were measured in order to reduce model dependence by defining a fiducial phase

space closely resembling the phase space of detector acceptance. The measurements were

inclusive, in the sense that presence of additional objects such as jets and leptons was allowed

in the selected events. Additional event categories were defined and analyzed to measure

cross section as a function of jet multiplicity and lepton presence. Differential cross section

measurements were performed for a number of variables that characterize kinematic and

quantum properties of the Higgs boson. Care was taken in deriving the correction factors,

estimating the uncertainties, and optimizing the binning of the differential distributions.

The inclusive fiducial cross section obtained from the latest analysis [277, 278], including

the H → γγ branching ratio, is

σfid = 65.2 ± 4.5 (stat.) ± 5.6 (exp.) ± 0.3 (theory) fb ≈ 65.2 ± 7.2 fb.

(7.1)

211

Its combined fractional uncertainty is 11%. The measured cross section can be compared
with the most accurate Standard Model prediction of 63.5 ± 3.3 fb, calculated at N3LO in
perturbative QCD [325]. The differential cross sections were presented in Section 6.8.2 and

were compared to the state-of-the-art theoretical predictions. The analysis results are in

excellent agreement with the SM predictions.

Extrapolation of the measured fiducial cross section to the full phase space for the Higgs

boson production in the H → γγ channel yields a value of 56.7+6.4
−6.2
total cross section from the ATLAS H → ZZ∗ → 4(cid:96) analysis is 54.4+5.6
−5.4
of the two measurements yields 56.1 +4.5
cross sections are in good agreement with the SM prediction of 55.6 ± 2.5 pb.

−4.3 ± 3.2 (stat.) +3.1

pb. For comparison, the

pb. The combination
−2.8 (sys.) pb [326]. The derived total

An updated H → γγ cross section analysis, using the full Run 2 ATLAS dataset, includ-
ing the full list of observables, is currently being prepared and is expected to be completed

in 2021.

The future prospects for the analysis include improved measurements with the larger

dataset that will be collected in the Run 3 of the LHC, and, further in the future, during

the operation of the HL-LHC.

The high-pT Higgs boson phase space is of particular interest for future analyses, as it
provides a much higher purity sample of the Higgs boson production events relative to the

irreducible diphoton background. With an increased integrated luminosity, this phase space

will become more accessible and will be investigated more thoroughly.

212

APPENDICES

213

APPENDIX A

GoSam ntuples

A.1 GoSam ntuples

A considerable amount of computer resources is involved in generation of MC events.

With data storage becoming more easily available, it is often more practical to store each

generated event, rather than to fill histograms and immediately discard the events. The ap-

proach of saving the events affords more flexibility, especially in exploratory studies, as the

saved events can be quickly iterated over to produce histograms for any desired combination

of variables. Many parameters can be adjusted without rerunning the event generator, such

as kinematic cuts, binning of the histograms, or definitions of observables and event cate-

gories. Depending on the saved information, even parameters of the cross section calculation

or the physics assumptions can be modified by reweighting the events. The reduced turnover

time allows to quickly explore a process of interest within a desired phase space. For complex

processes with high multiplicity of final state particles or higher order calculations, reading

saved events instead of generating them anew can be several orders of magnitude faster.

Files containing MC events have colloquially acquired the name of ntuples. Ntuples typi-

cally contain the 4-momenta of the final state particles together with additional information,

such as event weights and particle ids.

Through collaboration with the GoSam authors [106, 109, 327], we acquired a large

214

number∗ of GoSam ntuples for QCD production of Higgs boson and diphoton in association

with 1, 2, or 3 jets. GoSam is a package that provides means for automated calculation

of one-loop amplitudes for multi-particle processes in renormalizable quantum field theories.

Though it is not a stand-along event generator, combined with Sherpa, GoSam pushed

the frontier for the number of jets in the final state, calculated at matrix element level,

rather than through parton shower, in differential cross section predictions. At the time

they were generated, the Higgs + jets ntuples represented the result of the cutting edge

calculations [328, 329].

The format of GoSam ntuples [330] was adopted from that of the earlier developed

BlackHat+Sherpa ntuples [100, 107, 108, 331–333], which saw wide use, for example in

the studies published in [334–340]. The ntuples are saved as ROOT [310] files, with one tree

of single-value and array branches, representing variables describing weighted events. The

file format is summarized in Table A.1. A set of ntuples for a given process, for example
Higgs + 1 jet at √s = 13 TeV, consists of a number of files, with multiple files corresponding
to each NLO contribution type (B, RS, I, V) from the Catani–Seymour subtraction method.†

Splitting sets of events into multiple files, typically containing a few million events, allows

to easily run analysis programs in parallel, particularly on a computer cluster using a batch

system, such as HTCondor [342]. Outputs of individual processing jobs can then be merged

to obtain the full result.

Observables are computed by taking weighted averages of their values over events of the

same type and adding the averages from each NLO contribution. This can be written as

∗The amount of GoSam ntuples we have amassed consists of 21518 files in 88 sets, containing 5.9× 1010
events, and occupying 18.6 TB of disk space. Special thanks for providing these goes to Gionata Luisoni
and Nicolas Greiner.

†See Section 3.2 for the preliminary discussion.

215

Table A.1: Format of GoSam ROOT ntuples [330].

nparticle
px, py, pz, E

Int_t
Double_t[nparticle]∗ Arrays of 4-momenta components of the final-

Name

t3
id
ncount†

Type

TTree
Int_t

Int_t

kf
weight
weight2

Int_t[nparticle]
Double_t
Double_t

me_wgt

Double_t

me_wgt2
ps_wgt†
x1, x2

x1p, x2p

Double_t
Double_t
Double_t

Double_t

id1, id2
id1p, id2p†

Int_t
Int_t

fac_scale
ren_scale
nuwgt
usr_wgts

Double_t
Double_t
Int_t
Double_t[nuwgt]

Char_t[2]

part
alphasPower Char_t
alphas

Double_t

Description
ROOT tree containing generated events.
Event index number. Real emission entries and
their associated counterterms share the same id.
Number of trials between the previous and cur-
rent event during generation. The number of
times an event should be counted.
Number of particles in the final state.

state particles.
PDG [14] codes of the final-state particles.
Total weight of the entry.
Correlated weight used for the RS contribution.
Identical to weight for the B, V, and I contribu-
tions.
Coefficient of the product of parton distribution
functions in weight. For the B, V, and RS con-
tributions, this is the squared matrix element
multiplied by the phase space measure and the
Jacobian from Sherpa’s phase space mapping.
Coefficient of the PDFs product in weight2.
Phase space weight.
Fractions of hadron momentum carried by the
incoming partons.
Secondary momentum fractions used in inte-
grated subtraction entries [333].
PDG codes of the incoming partons.
PDG codes of incoming partons in subtraction
events.
Factorization scale used, µF,0.
Renormalization scale used, µR,0.
Number of additional weights.
Additional weights needed to recompute entries’
weights for different scale or PDF choices.
Type of contribution: B, V, I, or RS.
Power of the strong coupling.
αs value used for this entry.

∗In an older version, components of the 4-momenta were stored as Float_t instead of Double_t.
†Additional new branches introduced for the EDNtuples [341].

216

follows,

(cid:10)O(cid:11) =(cid:88)t (cid:10)O(cid:11)t,

(cid:10)O(cid:11)t =

1

Nt(cid:88)i∈t

wiOi,

(A.1)

where O is an observable, t ∈ {B, RS, I, V} represents a contribution type, i is the index of an
event of type t, Nt is the number of events of type t, and wi and Oi are the event weight and
the value of the observable for the event i respectively. The sum(cid:80) wiOi typically runs over

events from multiple ntuple files. But, subsets of events of the same type can be used, if a

shorter run time is preferred over statistical accuracy. Most often, the observable of interest

is the cross section as a function of one or more variables that characterize the process, or,

in other words, the joint distribution of the variables, which is represented by a histogram

populated by the weighted events. In that case,

Oi = δb

i = i ∈ b,

(A.2)

where δb
i

is a boolean variable, which is 1 if event i falls into bin b of the histogram and 0

otherwise. (cid:10)O(cid:11)b = σb

obtained by dividing σb

by the bin width.

is the total cross section in bin b, and the differential cross section is

The statistical uncertainty is obtained by the standard prescription for MC integration,

εO =(cid:34)(cid:88)t

Ot(cid:35)1/2

ε2

,

εOt =

1

(cid:112)Nt(Nt − 1)(cid:34)(cid:88)i∈t(cid:0)wiOi(cid:1)2 −

1

Nt(cid:18)(cid:88)i∈t

wiOi(cid:19)2(cid:35)1/2

,

(A.3)

where statistically independent contributions from different types are added in quadrature,

while within a type, the uncertainty is estimated using the equation for the unbiased sample

variance. ε is used to represent uncertainty to avoid confusion with σ used for the cross

section. An illustrative example is the case in which one wants to look at both exclusive

217

in the inclusive histogram can simply be

in the exclusive one. But adding the εOb

in quadrature will not

and inclusive distributions.∗ The value for (cid:10)O(cid:11)b
obtained by adding the(cid:10)O(cid:11)b

yield uncertainty estimates as accurate as following Eq. (A.3). For a large number of events,

however, the difference may be insignificant, as the second term in the square brackets will

vanish.

Another situation where statistical uncertainty needs to be handled carefully is in rebin-

ning. If bins b1 and b2 need to be merged, the second term in Eq. (A.3) is the square of the
sum, and will remain the same; but the first term is the sum of squares, which becomes

(cid:88)i∈t(cid:18)wiδb1∪b2

i

(cid:19)2

=(cid:88)i∈t(cid:18)wi(cid:16)δb1

i (cid:17)(cid:19)2
i ∨ δb2

= (w1 + w2)2 = w2

1 + w2

2 + 2w1w2,

(A.4)

where w1 and w2 are the cumulative weights in the respective bins.

There is an additional caveat concerning specifically the real emission (RS) ntuples. These

ntuples are special in that any one of their events generally consists of more than one tree

entry. Because of this, one must make the following substitution in Eq. (A.3):

For RS ntuples: wiOi →(cid:88)j∈i

wjOj.

(A.5)

When filling histograms, the weights of the entries corresponding to the same event that

fall into the same bin must be added before they are squared. The multiple entries cor-

respond to different phase space configurations: a real emission configuration and counter-

configurations. The latter correspond to the subtraction term that regulates the square

∗The relationship between exclusive and inclusive distributions is the same as between p.d.f. and c.d.f.,
the later being the integral of the former. This terminology is typically used in discrete context, such as
distributions of jet multiplicity. A histogram showing the exclusive distribution would, for example, contain
a bin for n = 2, while the corresponding bin in the inclusive distribution would be for n ≥ 2.

218

of the matrix element in the soft and collinear limits, where the unregulated matrix ele-

ment diverges. The real emission configurations and subtraction counter-configurations are

strongly anticorrelated. This means that simply adding their weights independently, as in

Eq. (A.3), will grossly overestimate the statistical error. The estimated error would typically

be of order the central value. Using these weights will, however, yield the correct central

value, Eq. (A.1) [330]. The ntuples contain a second set of weights, weight2 rather then

weight, which allows to take the anticorrelation into account. The entries in the ROOT file

corresponding to the same RS event are written consecutively and can be identified by the

same event id. Because weight2 = weight for B, I, and V type files, in practice, one should

always use the weight2 branch for histogramming.

A.2 Reweighting of ntuples

In order to evaluate the validity of QCD predictions, it is important to explore their

dependence on the choice of scales and PDFs. A very convenient feature of the GoSam

ntuples is that they save information required to modify scales and PDFs of existing events,

avoiding the need to generate the events anew. This is done by reweighting the events

according to the prescription [330] described below. The prescription varies depending on the

NLO contribution type the event represents. In the following, variables written in monospace

font refer to the ntuple branches listed in Table A.1. The indices of branch arrays start at

zero, as in C.

219

A.2.1 Born and real-subtraction contributions

For B and RS type events, the new weight, w, is given by

n = alphasPower,

w = me_wgt2 f1(id1, x1, µF) f2(id2, x2, µF)

αs(µR)n
alphasn .

(A.6a)

(A.6b)

The PDFs of the interacting partons are denoted f1 and f2. They depend on the respective
.
parton type (id1, id2), momentum fraction (x1, x2), and the new factorization scale, µF
For the LHC, both f1 and f2 are proton PDFs. αs is the strong coupling, running with
respect to the renormalization scale, µR
n is the power of the strong coupling and depends
on the number of emitted jets.

.∗

If the factorization scale and the PDFs don’t need to be changed, αs contains the whole

renormalization scale dependence, and reweighting can be simplified to

If µF = fac_scale, w = weight2 αs(µR)n
alphasn .

(A.7)

Conversely, if αs and µR

need no variation,

If µR = ren_scale, w = me_wgt2 f1(id1, x1, µF) f2(id2, x2, µF).

(A.8)

Note, that for the Born contribution, but not for the real-subtraction, weight and weight2,

and likewise me_wgt and me_wgt2 are equal. Reweighting has to be done for every entry of

RS type events.

∗The value of αs at different renormalization scale values can be found using LHAPDF [73] by calling the

LHAPDF::PDF::alphasQ() function.

220

A.2.2 Virtual contributions

The treatment of V type events is similar to that of B and RS, except that the matrix

element has an explicit dependence on µR
. In dimensional regularization, this dependence
arises from the introduction of a scale to give the coupling g the required dimension, g → gµ
,
along with the MS ultraviolet subtraction that replaces the bare coupling g0 with the physical
coupling g(µR) [330].

n = alphasPower,

l = log(cid:32)

µ2
R

ren_scale2(cid:33) ,

ω0 = me_wgt + l usr_wgts[0] +

l2
2

w = ω0 f1(id1, x1, µF) f2(id2, x2, µF)

usr_wgts[1],

αs(µR)n
alphasn .

(A.9a)

(A.9b)

(A.9c)

(A.9d)

is not changed, the scale-changing logarithm in Eq. (A.9b) vanishes together with the

If µR
additional terms in Eq. (A.9c) and Eq. (A.9d) simplifies to the Born case.

A.2.3

Integrated subtraction contributions

Reweighting computation for the I type events is the most complicated. It requires 16

additional weights (usr_wgt[2], . . . , usr_wgt[17]), computed from the virtual pole coeffi-

cients and other quantities extracted from the Catani–Seymour subtraction formalism [51].

Discussion of the meaning of the me_wgt and usr_wgts terms can be found in [330].

221

n = alphasPower,

l = log(cid:32)

µ2
R

ren_scale2(cid:33) ,

ω0 = me_wgt + l usr_wgts[0] +

usr_wgts[1],

ωi = usr_wgt[i + 1] + usr_wgts[i + 9] log(cid:32)

µ2
F

fac_scale2(cid:33) ,

l2
2

m = ω0 f1(id1, x1, µF) f2(id2, x2, µF)

4(cid:88)j=1
4(cid:88)j=1

+ f2(id2, x2, µF)

+ f1(id1, x1, µF)

w = m

αs(µR)n
alphasn ,

f

(j)
1 (id1, x1, x1p, µF) ωj

f

(j)
2 (id2, x2, x2p, µF) ωj+4

(1)

p (i, x, x(cid:48), µF) =
p (i, x, x(cid:48), µF) =

(2)

p = quark : fp(i, x, µF),

p = quark : fp(i, x/x(cid:48), µF)/x(cid:48),

p = gluon : (cid:80)q∈quarks fp(i, x, µF),
p = gluon : (cid:80)q∈quarks fp(q, x/x(cid:48), µF)/x(cid:48),

where

f

f

(3)
p (p, x, x(cid:48), µF) = fp(g, x, µF),

f

(4)
p (p, x, x(cid:48), µF) = fp(g, x/x(cid:48), µF)/x(cid:48).

f

222

(A.10a)

(A.10b)

(A.10c)

(A.10d)

(A.10e)

(A.10f)

(A.11a)

(A.11b)

(A.11c)

(A.11d)

A.3 GoSam distributions with uncertainties

The reweighting formalism described in Section A.2 was used to assess the uncertainty of

GoSam predicted distributions for many observables relevant for the Higgs + jets production
at the LHC, such as those provided for the H → γγ cross section analysis. Here, several
distributions with estimated MC statistics, scale, and PDF uncertainties are shown with

finer binning for reference. These distribution were prepared similarly to the ones used for
comparisons with the H → γγ analysis results, but are different in several ways.

The distributions given below were prepared using NLO GoSam ntuple sets for Higgs
boson + 1, 2, and 3 jets production in proton-proton collisions at √s = 13 TeV. The
calculation was done in the Higgs effective field theory (HEFT) approximation, corresponding
to the infinite top quark mass limit, mt → ∞. The ntuples were reweighted using the ˆH(cid:48)(cid:48)T
central scale, defined in Eq. (3.6), and the CT14nlo PDF set [25]. The scale uncertainty

was estimated using the 7-point variation scheme, and the PDF uncertainties using the

Hessian method. The events were only simulated at the parton level, and include neither

hadronization nor Higgs boson decays. Electroweak effects are also not included. As these

predictions do not include Higgs boson decays, no branching ratio factors or cuts on the

possible decay products, such as photons, were applied. Typically, the diphoton branching
ratio of 2.27× 10−3, in combination with the photon cuts listed in Section 6.3.2, reduces the
cross section by a factor of approximately 10−3, from the pb to the fb level. The AntiKt
jet clustering algorithm was applied to the final state partons, with the radius parameters,

R = 0.4. The typical ATLAS jet cuts of pT > 30 GeV and η < 4.4 were applied to the
clustered jets.

Fig. A.1 shows the predicted cross section as a function of jet multiplicity. The GoSam

223

NLO calculations are technically inclusive with respect to the additional radiation. The

real-subtraction ntuples contain an additional final state parton, and histograms populated

without an explicit cut, requiring the number of jets to be strictly equal to the nominal

number of final state partons, estimate inclusive distributions. The values of the first three

bins in the exclusive jet multiplicity distribution in Fig. A.1 were obtained from the Higgs

boson + 1, 2, and 3 jets ntuples respectively, with the strict multiplicity cut applied. The

same bins in the inclusive distribution differ in that they also include the events from the

respective sets with the additional jet from the real correction. The last bin in both distri-

butions is populated only by the events from the Higgs boson + 3 jets set, which contain

the additional fourth jet due to the real correction.

Figure A.1: GoSam NLO predictions for jet multiplicity distributions. The cross section
in each of the first three bins was calculated using the set of ntuples corresponding to the
respective number of jets. The last bin represents the additional radiation due to the real
correction to the process with 3 nominal jets.

The figures below show the predictions for the distributions of observables related to the

kinematics of the Higgs boson and jets.

224

=1=2=3≥402468109.67+1.31−1.32+0.16−0.23pb3.33+0.45−0.56+0.06−0.08pb1.07+0.15−0.22+0.03−0.03pb0.22+0.20−0.10+0.01−0.01pbExclusivejetmultiplicityσ,pbGoSamNLOpp→Higgs+jets√s=13TeVMCScale:ˆH′′TPDF:CT14nlo≥1≥2≥3≥40246810121411.91+2.34−2.00+0.19−0.27pb4.06+0.89−0.85+0.08−0.10pb1.30+0.31−0.32+0.04−0.04pb0.22+0.20−0.10+0.01−0.01pbInclusivejetmultiplicityσ,pbGoSamNLOpp→Higgs+jets√s=13TeVMCScale:ˆH′′TPDF:CT14nloFigure A.2: GoSam NLO predictions for distributions of observables in Higgs boson + 1 jet
events.

225

0200400600800100010−510−410−310−210−1HiggsbosonpT,GeVdσ/dpT,pb/GeVGoSamNLOpp→Higgs+1jet√s=13TeVMCScale:ˆH′′TPDF:CT14nlo0200400600800100010−510−410−310−210−1LeadingjetpT,GeVdσ/dpT,pb/GeVGoSamNLOpp→Higgs+1jet√s=13TeVMCScale:ˆH′′TPDF:CT14nlo0200400600800100010−210−1100MassofHiggsboson+leadingjetsystem,GeVdσ/dm,pb/GeVGoSamNLOpp→Higgs+1jet√s=13TeVMCScale:ˆH′′TPDF:CT14nlo−4−202401234Higgsbosonrapiditydσ/dy,pbGoSamNLOpp→Higgs+1jet√s=13TeVMCScale:ˆH′′TPDF:CT14nlo−4−20240.00.51.01.52.02.5Leadingjetηdσ/dη,pbGoSamNLOpp→Higgs+1jet√s=13TeVMCScale:ˆH′′TPDF:CT14nlo−4−20240.00.51.01.52.02.5Higgsbosonηdσ/dη,pbGoSamNLOpp→Higgs+1jet√s=13TeVMCScale:ˆH′′TPDF:CT14nloFigure A.3: GoSam NLO predictions for distributions of observables in Higgs boson + 2 jets
events.

226

0200400600800100010−710−610−510−410−310−210−12ndjetpT,GeVdσ/dpT,pb/GeVGoSamNLOpp→Higgs+2jets√s=13TeVMCScale:ˆH′′TPDF:CT14nlo0200400600800100010−310−2Massoftwoleadingjetssystem,GeVdσ/dm,pb/GeVGoSamNLOpp→Higgs+2jets√s=13TeVMCScale:ˆH′′TPDF:CT14nlo−4−20240.00.20.40.60.82ndjetηdσ/dη,pbGoSamNLOpp→Higgs+2jets√s=13TeVMCScale:ˆH′′TPDF:CT14nlo0123456780.000.250.500.751.001.251.501.752.00Rapidityseparationbetweentwoleadingjetsdσ/d|∆y|,pbGoSamNLOpp→Higgs+2jets√s=13TeVMCScale:ˆH′′TPDF:CT14nlo0.00.20.40.60.81.0234567Azimuthalangularseparationbetweentwoleadingjetsdσ/d|∆φ|,pbGoSamNLOpp→Higgs+2jets√s=13TeVMCScale:ˆH′′TPDF:CT14nloFigure A.4: GoSam NLO predictions for distributions of observables in Higgs boson + 3 jets
events.

227

0200400600800100010−910−710−510−310−13rdjetpT,GeVdσ/dpT,pb/GeVGoSamNLOpp→Higgs+3jets√s=13TeVMCScale:ˆH′′TPDF:CT14nlo−4−20240.000.050.100.150.200.250.303rdjetηdσ/dη,pbGoSamNLOpp→Higgs+3jets√s=13TeVMCScale:ˆH′′TPDF:CT14nloAPPENDIX B

Effects of finite top quark mass in Higgs

boson production

B.1 Higgs effective field theory

A technique commonly used in calculations of the Higgs boson production is the effective

field theory (EFT) approximation,∗ whereby an effective coupling of the Higgs boson to

gluons is derived by taking the infinite limit of the mass of the heavy virtual quark involved

in the loop mediating the interaction. The EFT approximation reduces the number of loops

in the relevant Feynman diagrams by 1, which results in a substantial reduction in complexity

of the scattering amplitude expressions. This simplification is especially important for fixed

order matrix element calculations in processes involving additional QCD radiation, i.e. jets.

Due to the coupling of the Higgs boson to fermions being proportional to the fermion’s mass,
−imf /v, over 99% of the effective coupling of the Higgs boson to gluons is accounted for by
the top quark loop.

The EFT approximation has its range of validity over the phase space of the process.
The most obvious limitation is that when the partonic √s reaches 2mt, there is enough
energy for the virtual top quark to go on shell. In other words, the loop can be resolved,

∗In this context, the Higgs effective field theory is also commonly abbreviated as HEFT. [343]

228

t

−−−−−−−→
mt → ∞

Figure B.1: Using effective field theory (EFT), an effective coupling of the Higgs boson to
gluons can be derived by taking the limit mt → ∞. This approach simplifies calculations by
reducing the number of loops, but the approximation is only valid for pHT (cid:46) 2mt ≈ 350 GeV.
and the assumption that the mt is large relative to the energy scale of the process becomes
invalid. The operating energy of the LHC is sufficient to obtain high-pT events outside the
validity range of the EFT, and the H → γγ, as well as other, analyses have begun exploring
this regime. As the integrated luminosity of the LHC will continue to increase, and larger

datasets will expand the volume of accessible phase space, it is important to understand

the limitations of the EFT approach. Conversely, exploring beyond the range of validity of

the EFT may provide interesting physics insights. Understanding of the impact of the Higgs

boson production via a fermionic loop with a finite mass propagator on the kinematics of the

Higgs + jets events can provide new approaches not only to directly observe the top quark

loop, but also to search for hypothetical BSM particles that could potentially participate in

the loop, as well as to examine our understanding of QCD and the interactions of the Higgs

boson.

Although the first calculations of the Higgs boson + jets production, incorporating the

full mass dependence, were done at the end of the ’80s, general purpose MC event generation,

including the finite top mass effects, has become available only more recently, in particular

with the help of GoSam [341]. The technology to include the finite mt in fixed-order Higgs
boson production calculations has been developed for events with up to one jet at NLO [344]

229

and up to 3 jets at LO. GoSam provides event generation with finite mt at LO for 1, 2, or 3
jets. The limiting factor in pushing the finite mt calculations to NLO is the complexity of the
required virtual corrections, with the Feynman diagrams explicitly containing the additional
fermionic loop. Using both EFT and finite mt sets of GoSam ntuples,∗ we performed
a number of comparisons in order to investigate areas of discrepancy, as well as possible

measurable effects due to the finite value of the mass of the top quark.

Some of the first effects we were able to quantify with the help of the GoSam ntuples were

the range of validity of the HEFT and the degree of its discrepancy with the cross section

predictions that include the full mass dependence. This comparison is illustrated in Fig. B.2,

which shows differential cross section distributions for diphoton production as functions of

the diphoton pT for resonant production via the Higgs boson, calculated using both EFT
and the finite mt (mtop) approaches, as well as non-resonant QCD diphoton production for
events with at least one hadronic jet. The HEFT is valid to within approximately 10% for

pHT < 225 GeV. At higher pHT , the fractional discrepancy continues to rise as a power law,
reaching a factor of 10 around 1 TeV.

In addition, the comparisons to the diphoton background show that within the signal
region of the fiducial phase space for the H → γγ analysis, for the diphoton transverse
momentum above 500 GeV, the signal-to-background ratio becomes close to or better than 1.
With the increased integrated luminosity of the future LHC runs, the H → γγ analysis will
be able to take full advantage of a high purity sample of Higgs events at high transverse

momentum.

∗The GoSam ntuples, discussed in Appendix A, were provided to us by our colleagues, theorists of
the GoSam collaboration [106, 109, 327]. The events in the samples used for the studies discussed in
Appendix B were generated at LO, using 5-flavor CT10 PDFs [66], with the µF = µR = ˆH(cid:48)
T/2 dynamic
scales (see Section 3.4), and, unless otherwise stated, represent proton collisions at √s = 13 TeV. The
event were simulated only at parton level. The partons were clustered using the AntiKt4 algorithm and the
standard ATLAS jet cuts of pT > 30 GeV and η < 4.4 were applied.

230

Figure B.2: Comparison of diphoton transverse momentum distributions, generated using
GoSam ntuples [106, 330], for (H → γγ) + 1 jet produced in gluon fusion with (mtop) and
without (EFT) accounting for the finite value of the top quark mass. The former is obtained
by reweighting the later by the ratio of the respective distributions at LO. Also shown are
distributions for the background diphoton production. The distributions include photon and
jet cuts on pT and η used in the H → γγ analysis.

231

6−105−104−103−102−101−10110210[fb/GeV]T/dpσd3−10×.272R:Bγγ→H-25.7235 11.2877 -1.38127-10.1154 4.96413 -0.742476-4.91785 6.15539 -0.969928-7.25998 7.16157 -1.13602-6.90274 6.39663 -1.07322EFT/mtop = 2 at 412 GeVEFT/mtop = 10 at 958 GeVx)2y=exp(A+Blogx+ClogcutsγDiphoton pT @ NLO + 01002003004005006007008009001000[GeV]Tpγγ110210ratio]+1j mtopγγ→[H]+1j EFTγγ→[H+1j γγ105,160][∈γγ+1j mγγ121,129][∈γγ+1j mγγB.2 Finite top quark mass effects in Higgs boson + jets

production

Subsequently, we used the GoSam ntuples to investigate the invariant mass distribution

of the Higgs boson + the leading jet system (Hj). The comparisons of the histograms from
the EFT and finite mt LO ntuples are shown in Figs. B.5 and B.6 for Higgs boson + at least
1 jet (H1j) and 2 jets (H2j) respectively. The ratio of the finite mt to the EFT cross section,
plotted on the right hand side, clearly shows a resonance-like structure, with the peak near

mHj = 2mt, marked by the vertical gray line. We conjecture this distinction to be due to
the finite mt calculation accounting for the resonant behavior, expected when the top quark
in the loop has energy sufficient to be on-shell.

Although selection based on the initial state partons cannot be done in experimental

data, it can be instructive to consider in a MC study. Plots of the mHj
H1j events (Fig. B.5) for the specific initial states support our interpretation of the feature

distributions in

observed in the ratio plots. Fig. B.3 shows representative triangle diagrams contributing to

the H1j process with the specific initial state partons. The diagram for the qq initial state

clearly explains why the effect observed in this case is much more pronounced, as the qq

initial state enables both the Higgs boson and the gluon jet to be produced directly from the

quark loop. The same is possible for the gg initial state, but requires a box diagram shown

Figure B.3: Representative Feynman diagrams containing a quark triangle loop that con-
tribute to the Higgs boson + 1 jet process for the gg, qq, and gq initial states.

232

in Fig. B.4. However, the amplitude for this process is suppressed by the allowed helicity

combinations. This explains why the effect is less pronounced for the gg than for the qq

initial state.

Figure B.4: A Feynman diagram containing a quark box loop that contributes to the Higgs
boson + 1 jet process for the gg initial state.

For the Higgs boson production with 2 jets, the effect is reduced for the qq initial state,

appears for the gq, and is consistent for gg. Evidently, at the higher jet multiplicity, events
with the gg initial state are the ones primarily affected by the finite mt. This is good news
for a potential future experimental measurement, as the gg initial state contributes the main

fraction of the cross section.

233

Figure B.5: Distributions of the mass of the Higgs boson + the leading jet system produced
using EFT and finite mt LO GoSam ntuples for Higgs boson + at least 1 jet production.
gg, qq, and gq labels indicate the initial state partons.

234

15020025030035040045050055060010345678920304050607080×10−3Hjmass,GeVdσ/dm,pb/GeVmt=172.3GeVmt→∞15020025030035040045050055060011.011.021.031.041.051.061.07Hjmass,GeVRatiomt=172.3GeVmt→∞1502002503003504004505005506001023456789203040506070×10−3Hjmass,GeVdσ/dm,pb/GeVggmt=172.3GeVmt→∞1502002503003504004505005506001.051.061.071.081.091.1Hjmass,GeVRatioggmt=172.3GeVmt→∞1502002503003504004505005506000.010.1×10−3Hjmass,GeVdσ/dm,pb/GeVqqmt=172.3GeVmt→∞150200250300350400450500550600246810Hjmass,GeVRatioqqmt=172.3GeVmt→∞1502002503003504004505005506001023456789×10−3Hjmass,GeVdσ/dm,pb/GeVgqmt=172.3GeVmt→∞1502002503003504004505005506000.920.940.960.9811.021.04Hjmass,GeVRatiogqmt=172.3GeVmt→∞Figure B.6: Distributions of the mass of the Higgs boson + the leading jet system produced
using EFT and finite mt LO GoSam ntuples for Higgs boson + at least 2 jets production.
gg, qq, and gq labels indicate the initial state partons.

235

15020025030035040045050055060010345678920×10−3Hjmass,GeVdσ/dm,pb/GeVmt=172.3GeVmt→∞1502002503003504004505005506000.9811.021.041.061.081.1Hjmass,GeVRatiomt=172.3GeVmt→∞1502002503003504004505005506001023456789×10−3Hjmass,GeVdσ/dm,pb/GeVggmt=172.3GeVmt→∞1502002503003504004505005506001.021.041.061.081.11.121.14Hjmass,GeVRatioggmt=172.3GeVmt→∞1502002503003504004505005506000.10.080.090.20.30.4×10−3Hjmass,GeVdσ/dm,pb/GeVqqmt=172.3GeVmt→∞1502002503003504004505005506000.811.21.41.6Hjmass,GeVRatioqqmt=172.3GeVmt→∞15020025030035040045050055060010.40.50.60.70.80.92345×10−3Hjmass,GeVdσ/dm,pb/GeVgqmt=172.3GeVmt→∞1502002503003504004505005506000.90.9511.051.1Hjmass,GeVRatiogqmt=172.3GeVmt→∞Besides the distributions shown in Fig. B.6, we further investigated the finite mt effects
on the process of Higgs boson production with 2 jets. Using the additional jet in the H2j

events as a handle on the Hj kinematics, we studied the behavior of the mHj
a function of pj2T . Fig. B.7 shows these distributions for a specific slice in pj2T . A particular
enhancement due to restriction to a pj2T interval can be seen for the gg initial state. We also
observed a trend for the magnitude of the effect to increase at larger pj2T . However, beyond
pj2T (cid:38) 500 GeV, mHj

values at or below 2mt become inaccessible kinematically.

distribution as

Figure B.7: Distributions of the mass of the Higgs boson + the leading jet system for a
specific slice in the subleading jet’s transverse momentum, pj2

T ∈ [268, 311) GeV.

Seeing that the transverse momentum of the subleading jet (an experimentally observable

quantity) allowed us to isolate a region of phase space where the effect of the finite mt was
more pronounced, we attempted to further understand the effect in the context of H2j

236

30035040045050055060065070075010.50.60.70.80.92345×10−6Hjmass,GeVdσ/dm,pb/GeVmt=172.3GeVmt→∞3003504004505005506006507007500.811.21.41.61.82Hjmass,GeVRatiomt=172.3GeVmt→∞30035040045050055060065070075010.20.30.40.50.60.70.80.923×10−6Hjmass,GeVdσ/dm,pb/GeVggmt=172.3GeVmt→∞30035040045050055060065070075011.522.533.5Hjmass,GeVRatioggmt=172.3GeVmt→∞events, with the goal to identify potential observables with enhanced sensitivity to the finite

mt effects, which could provide a basis for a future experimental analysis.

1

2

3

4

5

Figure B.8: A representative Feynman diagram for the Higgs boson production with 2 jets
from the gg initial state.

Fig. B.8 shows a typical box diagram contributing to the H2j process with the gg initial

state. As the Higgs boson is a scalar particle, in order for the incoming (1 and 2) and the
outgoing (4 and 5) gluons to each have helicity ±1, the virtual gluon must have helicity 0,
and hence large virtuality.∗ The virtuality is given by the kinematic invariant corresponding

to the t-channel-like interaction of the virtual gluon, t25, defined in Eq. (B.1). Similarly to
the previous case, where the finite mt effect was enhanced for the qq initial state in the H1j
process, we would like to consider the mass of the system of the Higgs boson and the jet
emitted from the quark loop, which would be equal to the √s34.

t25 = −(k2 − k5)2,

s34 = (k3 + k4)2.

(B.1)

However, at the cross section level, a jet cannot be strictly identified as either particle 4

or 5, as the diagram in Fig. B.8 interferes with the one in which legs 4 and 5 are crossed.

The same is true for legs 1 and 2. To investigate the relationship between the virtuality t25

∗The process illustrated in Fig. B.8 is kinematically similar to that to which the so-called effective W
approximation has been applied in the past [345]. Here, a virtual gluon is involved in a scattering process
rather than a vector boson.

237

Figure B.9: Ratios of the distributions of the invariant mass of the Higgs boson and the
leading jet system from full and effective theory GoSam calculations within intervals of the
virtuality variable, t25, for proton-proton √s = 13 (left) and 100 (right) GeV.

and how pronounced the finite mt effect is, we used the following kinematic rules to improve
the degree of correlation between the selected particles and the legs of the diagram:

• assign 4 to the leading jet;

• assign 1 and 2 such that ∆θ25 < ∆θ15.

238

20030040050060011.021.041.061.08Hjmass,GeVRatio√s=13TeVt25∈[97,112)GeVmt=172.3GeVmt→∞2003004005006000.9250.950.97511.0251.051.0751.1Hjmass,GeVRatio√s=100TeVt25∈[97,112)GeVmt=172.3GeVmt→∞2003004005006000.960.9811.021.041.061.081.1Hjmass,GeVRatio√s=13TeVt25∈[200,232)GeVmt=172.3GeVmt→∞2003004005006000.460.480.50.520.540.560.58Hjmass,GeVRatio√s=100TeVt25∈[200,232)GeVmt=172.3GeVmt→∞2003004005006000.340.360.380.40.420.440.46Hjmass,GeVRatio√s=13TeVt25∈[645,747)GeVmt=172.3GeVmt→∞2003004005006000.040.050.060.070.080.09Hjmass,GeVRatio√s=100TeVt25∈[645,747)GeVmt=172.3GeVmt→∞Ratios of the obtained distributions are shown in Fig. B.9 for 13 and 100 TeV collisions.

Indeed, the size of the peak relative to the adjacent values increases for larger values of t25,
especially for √s = 100 TeV. Although, t25 cannot be measured experimentally, these results
support our conjecture.

Following these studies, our colleague, Kirtimaan Mohan, proposed a systematic way to
identify singularities potentially responsible for resonant effects in the gg → Hgg process.
Using Landau conditions, he compiled a list of all possible invariants that can appear as

factors in the denominators of the relevant amplitude expressions. The list of these invariants

is given bellow, with the following shorthand convention:

sij = (ki + kj)2,

tij = (ki − kj)2,

(B.2)

where indices i and j refer to the legs of the diagram in Fig. B.8. As before, for the analysis

using the ntuples, index 4 is assigned to the leading jet, and index 5 to the subleading one.

239

t − s12 + s45 − t23

t + s12 − s34 − s45

Triangle diagrams:

t

0 = m2
x3

h − 4m2
1 = s34 − 4m2
x3
2 = m2
h − 4m2
x3
3 = s45 − 4m2
x3
4 = s12 − 4m2
x3
5 = 4m2
x3

t

t

x3
6 = 4m2

x3
7 = m2

x3
8 = m2

9 = m2
x3

t + s12 − s34 + t15
t + s45 + t15 − t23
h − 4m2
h(cid:16)s12s45 − 2m2
h(cid:16)2m2

t − s34 + t15 − t23

+ m4

hm2

t + m2

x3
10 = t15m4

h + m2

t (s12 + s45)(cid:17) + m4

hm2

t + m2

t (s12 − s45)2

t (s12 − s34 + s45 + 2t15 − t23) + (s12 − s34 + t15)(s45 + t15 − t23)(cid:17)
t (s12 − s34 − s45 + t23)2
t (s34 + t23)2 − m2
ht15(4m2

t + s34 − t15 + t23)

(B.3)

(B.4)

(B.5)

(B.6)

(B.7)

(B.8)

(B.9)

(B.10)

(B.11)

(B.12)

(B.13)

240

(B.14)

(B.15)

(B.16)

(B.17)

(B.18)

(B.19)

(B.20)

(B.21)

(B.22)

(B.23)

(B.24)

(B.25)

(B.26)

Box diagrams, double cut:

t

t

0 = s12 − 4m2
x4
1 = s34 − 4m2
x4
2 = m2
h − 4m2
x4
3 = s45 − 4m2
x4
4 = 4m2
x4

t

t + s12 − s34 − s45

t + s45 + t15 − t23
t − t23
t + s12 − s34 + t15
h − 4m2
h − 4m2

t − s12 + s45 − t23
t − s34 + t15 − t23

5 = 4m2
x4

x4
6 = 4m2

x4
7 = m2

x4
8 = m2

9 = −x4
x4

7

Box diagrams, triple cut:

x4
10 = m2

x4
11 = m2

h(cid:16)s12s45 − 2m2
h(cid:16)2m2

+ m2

t m4

h + m2

x4
12 = m2

t (s34 + t23)2 − m2

t (s12 + s45)(cid:17) + m2

t m4

h + m2

t (s12 − s45)2

t (s12 − s34 + s45 + 2t15 − t23) + (s12 − s34 + t15)(s45 + t15 − t23)(cid:17)
t (s12 − s34 − s45 + t23)2
h − 4m2

t − s34 + t15 − t23)

ht15(m2

241

Box diagrams, quadruple cut:

13 = (s12 − s34)(m2
x4
14 = 4m2
x4

h − s34) + s45(s34 − 4m2
t )

15 = 2m2
x4

h − s34) + s34s45(s34 − 4m2
t )

t (s34 − s12)(m2
t (s12 − s34) + s45(s12 − s34 − s45)(cid:17) − 4m2
h(cid:16)2m2
h + (s12 − s34 − s45)2(cid:17)
− s45(cid:16)m4

16 = 4m2
x4

t m2

h(s45 + t15) + 4m2

t t23(s12 − s34 − s45) − t2

23(s12 − s34 + t15)

t s34(s12 − s34 − s45)

h(s12 − s34 + t15)

x4

17 = m2
x4

t (s12 − s34 + t15) + t23(s12 − s34 − s45)

h(s45 + t15) − 4m2
h + (s12 − s34 − s45)(cid:17) (s12 − s34 − s45)(s12 − s34 + t15)
18 =(cid:16)2m2
t(cid:16)m2
h(s45 + t15) + t23(s12 − s34 − s45)(cid:17) + m4
− 4m2
t(cid:16)m2
ht15 + (s12 − s34 − s45)(s12 − s45 + t23)(cid:17)
x4
19 = 4m2
− t15(m2
h − s12 + s45 − t23)2
x4
20 = t15(m2
h − 4m2
t(cid:16)m2
ht15 + (s12 − s34 − s45)(s12 − s45 + t23)(cid:17)
− t15(m2
h + s12 − s34 − s45)2
t t15 − t2
h(4m2
h(t15 − 4m2
t (m2

23(s34 − t15 + t23) − 4m2
t (s34 − t15 + t23) − s34t23

t ) + (s12 − s34 − s45)(s12 − s45 + t23)

x4
21 = 4m2

x4
24 = 4m2

x4
22 = m2

ht15 − s34t23) − s2

34(m2

h − s34 + t15 − t23)

x4
23 = m2

t ) + 4m2

23) + t2

t s34t23

x4

+ 4m2

25 = (s45 + t15 − t23)(cid:16)m4

h − 2m2
t(cid:16)s34(s12 − s45 + t23) − m2
h(s12 + t15) − 4m2
t m2

h(s12 + t15) − 4m2

x4
27 = 4m2

x4
26 = m2

h(s12 − s45 + t23) + (s12 − s45 + t23)2(cid:17)
h(s12 + t15)(cid:17)

t (s45 + t15 − t23) − s34(s12 − s45 + t23)

t s34(s12 − s45 + t23) − s2

34(s45 + t15 − t23)

242

(B.27)

(B.28)

(B.29)

(B.30)

(B.31)

(B.32)

(B.33)

(B.34)

(B.35)

(B.36)

(B.37)

(B.38)

(B.39)

(B.40)

(B.41)

x4

28 = (s12 − s45 + t23)(cid:16)s12(s12 − s45 + t23 − 2m2

h) − 4m2

t t23(cid:17)

+ m4

t m2

hs12 − 4m2
t m2

h(s12 + t15)
h(t23 − s45) − 4m2

29 = 4m2
x4

t t23(s12 − s45 + t23) + s12t2

23

30 = m2
x4

h(s45 − t23) − 4m2

t s12 + t23(s12 − s45 + t23)

(B.42)

(B.43)

(B.44)

Histograms of the distributions of the x observables listed above revealed that only 4

those, x3

of them are kinematically allowed to have both positive and negative values near zero. Of
3 ∼ s45 corresponds to
0 ∼ s12 corresponds to the invariant mass of the initial state

1 ∼ s34 corresponds to the Hj invariant mass, x3

the dijet invariant mass, x3

3 = x4

1 = x4

4 = x4

partons, and x3

2 = x4
2

is a linear combination of the three preceding variables.

B.3 Conclusion

Our study of the effects of the finite mass of the top quark on the production of the

Higgs boson with jets using GoSam MC is still ongoing. We have clearly demonstrated

that effects are expected to be present, e.g. resulting in resonance-like deviations from the

EFT predictions in the Hj mass distribution near mHj = 2mt. We have also been able
to associate these effects with certain aspects of the interactions comprising the production

process, in particular with the virtual off-shell gluon exchange between an incoming parton
and the quark loop in the gg → Hgg interaction. Going forward, we hope to better identify
regions of phase space where the finite mt effects are enhanced, and to define these regions
in terms of observables that can be used to perform an experimental measurement in the

future.

243

Figure B.10: Distributions of the kinematic invariants that can appear in the denominators
of amplitude expressions for the gg → Hgg process and lead to singularities.

244

−100−5005010015020025010234567892030×103×10−6x31,GeV2dσ/dx31,pb/GeV2mt=172.3GeVmt→∞−100−500501001502002500.960.9811.021.041.061.081.1×103x31,GeV2Ratiomt=172.3GeVmt→∞−100−500501001502002501100.70.80.923456789203040506070×103×10−6x32,GeV2dσ/dx32,pb/GeV2mt=172.3GeVmt→∞−100−500501001502002500.940.960.9811.021.04×103x32,GeV2Ratiomt=172.3GeVmt→∞−100−500501001502002501100.50.60.70.80.9234567892030×103×10−6x33,GeV2dσ/dx33,pb/GeV2mt=172.3GeVmt→∞−100−500501001502002500.970.980.9911.011.02×103x33,GeV2Ratiomt=172.3GeVmt→∞−100−50050100150200250103456789×103×10−6x34,GeV2dσ/dx34,pb/GeV2mt=172.3GeVmt→∞−100−500501001502002501.041.051.061.071.081.091.1×103x34,GeV2Ratiomt=172.3GeVmt→∞APPENDIX C

TileCal upgrade work

The MSU ATLAS group has been involved in the design and construction of the Tile

hadronic calorimeter (TileCal) from its inception. Continuing MSU participation on TileCal,

I worked on a number of projects related to the Phase-II upgrade, in preparation for the High-

Luminosity LHC. A part of this work was done in fulfillment of the authorship requirements

for the ATLAS collaboration. The Tile calorimeter is the main device on ATLAS used for

jet reconstruction. Its continued operation is essential for the future of all analyses looking

at processes that contain jets in the final state. This includes the Higgs boson cross section

analysis presented in this dissertation, as many key observables, such as the VBF cross

section, rely on measurements of jets.

C.1 Phase-II upgrade

For high energy collider experiments, whose lifetime spans decades, research and devel-

opment is always an on-going process that does not stop after continuous operation has been

established. Detector components age because of intense radiation levels, old components

become obsolete, and spare parts become unavailable, while new approaches are developed,

that promise to improve detector performance and reliability. These factors alone would

solicit continued work on upgrading the detector systems. Additionally, experience from

operation of the collider shows ways to increase delivered luminosity. Increasing the rate of

245

data collection comes with certain challenges, such as a larger number of collisions per bunch

crossing and an increase in both pileup and the radiation doses that the detector components

are going to be expected to withstand.

Figure C.1: Current LHC operation and upgrade timeline [346].

Fig. C.1 shows the timeline of the past and planned LHC operation and upgrade stages.
The data from √s = 13 TeV proton-proton collisions, used in the analysis presented in
this dissertation, was collected in the Run 2 period from 2015 to 2018. The LHC has now

entered the second long shutdown (LS2), during which Phase-I ATLAS upgrades are being

installed. After the Phase-I upgrades are complete, the LHC will collect additional data at

its current capacity in Run 3, before entering the third long shutdown (LS3). The major

Phase-II upgrades planned to be installed during LS3 will enable the High-Luminosity LHC

(HL-LHC) [347] to deliver instantaneous luminosity over a factor of 5 larger than the LHC’s

nominal. Continued HL-LHC operation is expected to last for many years, in order to achieve

an integrated luminosity of over 3000 fb−1, and will require an improved radiation tolerance
for many detector components. The Phase-II ATLAS upgrades [348–350] will allow the

detector to cope with data collection at the increased rate in the much more intense collision

environment. A major upgrade of TileCal electronics is being developed for the Phase-II.

As discussed in Section 4.5.2.1, TileCal modules are made of plastic scintillator tiles inter-

246

13 TeVintegrated luminosity2 x nominal Lumi2 x nominal Luminominal Lumi5 to 7.5 x nominal Lumi75% nominal LumicryolimitinteractionregionsradiationdamageLHCHL-LHCRun 4 - 5...Run 2Run 1Run 3ATLAS - CMSupgrade phase 1ALICE - LHCbupgradeDiodes ConsolidationLIU Installation11 T dipole coll.Civil Eng. P1-P5experiment beam pipessplice consolidationbutton collimatorsR2E project13 - 14 TeV14 TeV7 TeV8 TeVLS1EYETSLS3ATLAS - CMSHL upgradeHL-LHC installationLS230 fb-1190 fb-1350 fb-13000 fb-14000 (ultimate)20402027leaved with steal absorber plates. The scintillation light, produced in the tiles by interactions

with the passing particles, is read out via wavelength shifting fibers by photomultiplier tubes,

housed inside the support girder of every module. The modules are arranged in a cylindri-

cally symmetric configuration with 4 sections, each made up of 64 modules. The two central

sections (LBA & LBC) make up the Long Barrel. The other two form the Extended Barrels

(EBA & EBC). Positioning of the Tile calorimeter within ATLAS is shown in Fig. 4.14 on

page 97. Configuration of the modules is illustrated in Fig. 4.17 on page 106. Fig. C.2 below

shows photographs of the modules before installation. The hollow girder at the base of each

module houses a drawer that contains PMTs and their control and readout electronics. When

the detector is open for maintenance, the drawers can be pulled out of the girders, while the

modules can remain in place. The on-detector electronics transmist digitized readings from

Figure C.2: Photographs of TileCal modules [351]. Left: An Extended Barrel module.
Module cells are labeled on the flat side. Bundles of yellow-green wavelength-shifting fibers
can be seen at the bottom of the module. The opening at the bottom of the module is
where the drawer housing the PMTs and on-detector electronics is inserted. Tight bundles
of fibers are threaded through holes along the bottom of the module and are fitted against
the PMTs’ photocathodes. Right: Drawer electronics being inserted into a module. Not yet
connected ends of the Cs calibration system tubes can be seen along the narrow blue side of
the modules.

247

the PMTs to and receives power and operation commands from the ATLAS counting room

in USA15.

The present TileCal electronics was designed in the late 1990s, using technologies available

at the time. Maintenance of the legacy systems is becoming increasingly difficult as its

components age and become obsolete. The output bandwidth and radiation tolerance are

also not suited for high luminosity operation. The HL upgrade of TileCal therefore involves

a full replacement of the on- and off-detector electronics systems with modern technologies,

in order to provide improved trigger performance, higher resolution read-out, and better

radiation tolerance. Detailed descriptions of the legacy and upgrade systems can be found

in the respective technical design reports [185] and [352].

A conceptual schematic of the existing legacy TileCal electronics is shown in Fig. C.3.

A Super Drawer spanning the whole module is divided into two drawers, each containing

4 digitizer boards. The digitizers receive analog signals from the front-end boards (FEB),

called 3-in-1 cards [353], that control and read out the PMTs. There is one FEB for every

PMT. The digitized PMT readings are transmitted to the back-end system called the Read-

Out Driver (ROD). Independently of the digitizers, signals for the Level 1 triggers (L1Calo)

are processed by analog adders (indicated by Σ on the diagram), which are also located in

the electronics drawers. Separate lines are used for communication with the triggers, the

Detector Control System (DCS), and the ROD.

The Phase-II electronics has a different structure, the diagram for which is shown in

Fig. C.4. The new Super Drawer is divided into 4 mini-drawers, each containing a Main

board (MB), a Daughter board (DB), and a dedicated high-voltage (HV) board. The FEBs

will be connected to the Main board. All communication between the on- and off-detector

electronics goes through the GigaBit (GBT) fiber-optic connection between the Daughter

248

Figure C.3: Diagram of the legacy TileCal electronics.

Figure C.4: Diagram of the Phase-II TileCal electronics.

Figure C.5: Diagram of the hybrid Phase-I Demonstrator TileCal electronics.

249

DigitizerDaughterBoardRODInterfaceBoardDigitizerL1CaloOff-detectorOn-detectorSuper DrawerDrawerDrawerFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBΣΣΣΣΣΣΣΣΣTTCDigitizerDigitizerDigitizerDigitizerDigitizerDigitizerHV SYSTEMLVPSDCS/CANMainBoardDaughterBoardDaughterBoardDaughterBoardDaughterBoardOff-detectorMainBoardMainBoardMainBoardL1CaloOn-detectorHV SYSTEMHV SYSTEMHV SYSTEMHV SYSTEMFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBPPrMinidrawerMinidrawerMinidrawerMinidrawerSuper DrawerDAQTTC, DCSLVPSMainBoardDaughterBoardDaughterBoardDaughterBoardDaughterBoardOff-detectorMainBoardMainBoardMainBoardOn-detectorHV SYSTEMHV SYSTEMHV SYSTEMHV SYSTEMFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBFEBPPrMinidrawerSuper DrawerTTCLVPSΣΣΣΣΣΣΣΣΣL1CaloAdderbaseboardAdderbaseboardAdderbaseboardAdderbaseboardMinidrawerMinidrawerMinidrawerRODHVCANconverterDCSCANboards and the PreProcessor (PPr). The PPr provides a centralized back-end interface. It

takes over the functionality of the ROD, and also serves as the connection point to the TileCal

electronics for the data acquisition (DAQ), the Timing, Trigger and Control (TTC), the

detector control (DCS), and the L1Calo systems. The new GBT connection will dramatically

improve the communication speed and bandwidth. With the downlinks running at 4.8 Gbps

and the uplinks at 9.6 Gbps, PPr and DB registers, holding data and control messaged, can

be synchronized every 25 ns, with the same frequency as the event rate of the LHC proton

collisions. The subdivision of the Super Drawer into 4 mini-drawers instead of 2 drawers, will

make maintenance work in the cavern more convenient. Each mini-drawer will be essentially

autonomous and will be connected directly to the low-voltage power suppy (LVPS) and the

PPr.

In order to test the upgrade electronics before the Phase-II design is finalized, a hybrid

system, called the Demonstrator, has been developed. As shown in Fig. C.5, includes ele-

ments of both the legacy and the upgrade electronics to provide interoperability with the

currently installed back-end and trigger systems. A photograph of a Demonstrator mini-

drawer is shown in Fig. C.6. A drawer containing the Demonstrator electronics has already

been installed on the detector during the current long shutdown (LS2). During the develop-

ment of the Demonstrator, 3 competing designs of the FEB were considered: an upgraded

version of the 3-in-1 cards [354], and two ASIC-based solutions, called FATALIC [355] and

QIE [356]. Eventually, the new version of 3-in-1 was selected as the main upgrade candidate.

An integral part of any electronics is the power delivery system. The TileCal on-board

electronics requires low-voltage (LV) power for control and read-out systems and high-voltage

(HV) power for the PMTs. Remote control of both LV and HV power systems is necessary

for operation. The value of high voltage supplied to the PMTs has to be dynamically

250

Daughter
board

Adder base
board

PMT blocks

HV board
(underneath)

Main board

Figure C.6: Photograph of a Demonstrator mini-drawer with the electronics components
labeled.

adjusted, depending on the signal level from the respective calorimeter cells, for the PMTs

to remain near the operation point in order to provide accurate measurements. Also, the

relative calibration of the signal from tiles within each cell has to maintained, which is done

by adjusting the PMTs’ gain with the HV supplied to them. Stable low voltage has to be

supplied to the control and read-out electronics. In case of failure, the LV has to be shut off,

to either prevent a potential short circuit from affecting still-operational components, or to

reboot the drawer electronics, which often allows a resumption of regular operation.

The projects described in Sections C.2 and C.3 contributed to the upgrade developments

for TileCal on-detector power supply systems. The projects described in Sections C.4 and

C.5 are related to calibration and testing of the upgrade hardware and firmware.

251

C.2 HVOpto firmware

The high voltage (HV) supply system for the TileCal has been redesigned for the Phase-II

upgrade with two solutions: HV remote and HV internal. With the HV remote, individual

adjusted voltages are supplied from an off-detector source to each PMT. This requires one

or two (for the extended and long barrel modules respectively) 100 m long multi-conductor

cables per module connecting to dedicated off-detector control crates housed in USA15. The

HV internal only requires a single constant bulk HV supplied to each module. A dedicated

HV control board, called HVOpto, adjusts the supplied bulk HV for each PMT as needed.

Each mini-drawer contains its own HVOpto board, located on the opposite side of the drawer,

as illustrated in Figs. C.7 and C.8. HVOpto specifications can be found in [357].

Figure C.7: Diagram illustrating location of the Main, Daughter, and HVOpto boards, as
well as the PMTs, in a mini-drawer [357].

The basic configuration of an HVOpto board is as follows. Each HV control circuit

contains a digital-to-analog converter (DAC), an analog-to-digital converter (ADC), a pass

switch, and analog control circuitry that incorporates optical isolation between the LV control

and monitoring, and the HV side that supplies voltage to the PMTs. The optical isolation

helps reduce ground loops that might introduce noise into the system, since the bulk HV

252

MainBoardHVconnectionstoPMTs(1sideshown)NewLVPS+10VBrickPower&returnHVOptoBoardMainBoardto3-in-1BoardconnectionsMainBoardtoDaughtBoardconnectionsDaughterBoardData&powercabletoHVOptoFigure C.8: Photograph of an HVOpto board mounted on a mini-drawer. The circuits
for 12 channels, 6 on each side, can be seen as clusters of components. The two ribbon
cables provide SPI bus communication with the Daughter board on the other side of the
mini-drawer.

supplies reside in USA15. The control circuits produce an output HV that is derived from

the HV input, with the output proportional to the DAC voltage, up to a maximum equal

to the input voltage. The output HV is monitored by the ADC, using a voltage divider to

step the voltage down to a level that can be processed by the LV circuits. A single HVOpto

board contains 12 channel circuits – one for each mini-drawer PMT. 4 HVOpto boards share

one input HV, as only one is supplied per module.

The ADC and DAC are incorporated into a single chip, MAX1329 [358], one for each

HV control channel. In addition to setting the PMT voltages, HVOpto provides monitoring

capabilities. With the help of the MAX chip, it is able to continuously monitor the reference

and PMT voltages as well as the board temperature. The digital interface to the chip uses

Serial Peripheral Interface (SPI) [359], which is a 4-wire serial interface with a write serial

data line (MOSI), a read serial data line (MISO), a slave select line (nSS), and a serial clock

(SCLK) sourced by the master. In this design, the MAX chips are slaves and the Daughter

board plays the role of the master. For robustness, the SPI connection is implemented using

Low-voltage differential signal (LVDS) lines. In order to provide direct means to shut off

HV input, an additional single-ended line (HV_Enable) is added to the bus, which has to be

253

held high for an optically isolated switch to remain in the on position. An HVOpto board

is divided into two halves, with each controlling the voltage for PMTs oriented in the same

direction inside the mini-drawer. The two halves communicate with the Daughter board via

independent SPI buses, with 6 MAX chips per bus. The independent slave SPI configuration

is used, so each channel has its own nSS and HV_Enable lines.

The Daughter board is an FPGA based device, using Kintex7 XC7K160T-2FFG676 FPGA

chips. In order to communicate with HVOpto, the SPI protocol needed to be implemented

in the FPGA firmware of the Daughter board written in the VHDL language.∗ The primary

purpose of the HVOpto component of the DB firmware is to provide an interface via which

commands to request changes of specific PMT voltages or readings of voltages and tempera-

tures can be sent to the MAX chips operating the HVOpto channels. These commands can

be issued either by the DB or be received via the GBT link from the PPr and then delivered

to the HVOpto via the SPI connection.

The HVOpto firmware component is essentially a converter that encodes and decodes DB

command registers and sends the respective messages over the SPI to read or write data to the

MAX chips. The component is implemented as a finite state machine illustrated in Fig. C.9.

Before integration into the DB firmware project, the HVOpto component was developed using

Figure C.9: Finite state machine of the HVOpto component of the Daughter board firmware.

∗VHDL (VHSIC-HDL, Very High Speed Integrated Circuit Hardware Description Language) is a hard-
ware description language used for programming FGPA (field-programmable gate array) chips. When the
VHDL firmware is uploaded to an FPGA, the logic gates inside are connected in such a way that the chip
behaves like the described circuit.

254

READYstartCMDchanged?CMD=READCMDSELECTCMDCONFIGCommandstosend?CMDSENDAbletosend?REPLYGETAbletoreceive?WaitforADC?REPLYCONFIRMSLEEPUPDATELastword?nonoyesyesyesyesyesnoyesnononoyesnoa Xilinx Atlys prototyping board and a single-channel HVOpto test board. A photograph

of the setup is shown in Fig. C.10. The developmental version of the firmware with its

Figure C.10: Photograph of the electronics setup used for development of the HVOpto FPGA
firmware component for the TileCal mini-drawer Daughter board. Shown are a Xilinx Atlys
prototyping board, a single-channel HVOpto test board, and a LV power supply.

documentation can be found in the CERN Git repository [360]. Porting of the firmware to

the DB project was done in collaboration with the TileCal colleagues. A complete technical

report can be found in [361].

C.3 Low voltage system monitoring upgrade

Experience from operation of the TileCal in Run 1 of the LHC demonstrated that the

LV control and monitoring system did not provide sufficient diagnostic information. More-

over, the existing monitoring system was based on a now outdated custom board which

is not sufficiently radiation tolerant to reliably work on HL-LHC. These factors demanded

development of a new solution for LV control and monitoring for the TileCal on-detector

255

electronics. A study to assess the status of the current system, develop a set of requirements

for the upgrade, and propose possible upgrade solutions was conducted by me in collabora-

tion with TileCal experts at Argonne National Laboratory. The outcome of the study was

a written report documenting the requirements for the low voltage monitoring upgrade of

the ATLAS hadronic calorimeter [362]. Presented in the following are an overview of the

TileCal LV system, a summary of the major findings of the conducted study, an outline of

the proposed requirements, and a discussion of the present developments for the upgrade.

The TileCal LV supply system is a part of the on-detector electronics.

It consists of

the finger Low Voltage Power Supplies (fLVPS), with one located at the outer end of every

module drawer. An fLVPS contains 8 DC-DC converter “bricks”, that take 200 V input from

USA15 and output voltages∗ needed for the on-detector electronics, and a monitoring system

comprised of two circuit boards: ELMB and its motherboard. Picture of an open fLVPS are

show in Fig. C.11.

The LVPS used during the Run 1 of the LHC, which were produced with the original

brick design [363] and installed in 2007, exhibited a linear increase in the number of trips

with respect to integrated luminosity. While most of the trips were not debilitating and
the affected modules could be brought back online, the recovery process took time (∼ 2 min
to power-cycle the module) and sometimes required human intervention. The increasing

amount of TileCal modules’ deadtime had to be addressed to continue effective operation in

the future. Prior to my involvement in the project, the bricks were redesigned [364] and a LV

monitoring system was introduced [365, 366]. The revised brick design considerably reduced

the frequency of trips in Run 2 and the added monitoring system successfully identifies LVPS

∗The present system requires bricks to supply different voltages: from −15 V to +15 V. The Phase-II

upgrade system contains identical bricks, with +10 V output.

256

Figure C.11: Photograph of a TileCal finger Low Voltage Power Supply (fLVPS). The green
circuit boards of the ELMB and its motherboard can be seen on the left photo. On the right
photon, the ELMB and motherboard are removed. The blue circuit boards belong to the
DC-DC converter bricks.

trips. However, due to limited sampling frequency, the existing monitoring system did not

provide sufficient information to diagnose causes of failure. Because of this, a wholesale

approach was implemented to power cycle the entire module after any trip. Power cycling is

time consuming and module inactivity reduced the detector’s hermeticity and data taking

efficiency for events involving jets, which adversely affects analyses of processes such as Higgs

production through VBF or with high transverse momentum. As was found, most trips don’t

require full power cycling to resume operation. An improved control and monitoring system

should be able to automatically identify the type of failure and take the appropriate measures.

The existing LV monitoring system [365] is based on the Embedded Local Monitoring

Board (ELMB), developed at CERN as an all-purpose slow monitoring solution. It’s last

revision, ELMB128 [367–369], shown in Fig. C.12, was developed in 2003. As such, it contains

no longer replaceable components. ELMB128 utilized a single-channel ADC limited by

circuit components to a maximum sampling rate of 30 sps, shared among up to 64 multiplexed

257

input signals. And while it is sufficiently radiation tolerant to operate on TileCal even

through Run 3, it cannot be relied on during the future high-luminosity runs.

Figure C.12: ELMB128. Its block diagram [367] is shown on the left and its photograph on
the right. The dimensions of the circuit board are 50 × 67 mm.

The studies I conducted at Argonne National Laboratory showed that analysis of tran-

sient behavior of output current from the bricks is a viable way to distinguish several types

of failure. In order to do so, a minimum required sampling rate of 2 ksps was estimated. The

possibility of upgrading ELMB128 by component replacement to improve its sampling rate

and radiation tolerance was considered, but was not found feasible. The details can be found

in the report [362]. The considerations presented there were incorporated into the Technical

Design Report for the Phase-II Upgrade of the ATLAS Tile Calorimeter [352].

Following the LV monitoring upgrade report [362], development started to provide a

modern replacement for the ELMB. Widespread use of the ELMB in slow-control system

at the LHC means that the best solution would be to design a fully backwards compatible

board. Unfortunately, after different solutions were considered, it became apparent that it

would not be possible to simultaneously maintain backward compatibility and to satisfy the

increased requirements for radiation tolerance and throughput. Three parallel development

paths were agreed upon to produce new ELMB versions that meet requirements of different

258

82C250CANTrans-ceiverOPTOOPTOVoltageRegulatorOPTOOPTO64 chanMUX +CS552316-bitADC….….….VoltageRegulatorVoltageRegulatorsCAN GNDCAN GNDDIGITAL GNDDIGITAL GNDANALOG GNDANALOG GND4±5V+3.3V+5VSAE81C91CANcontrollerDIP switchesATmega128Lmicrocontroller•128k Flash•4k RAM•4k EEPROM•BootloadersectionAnalog InVAP, VAG5.5 to 12V, 10 mAVDP, VDG3.5 V -12V, 15 mAVCP, VCG6 to 12V, 20 mA32CAN buscable43DigI/O(SPI)ISP,USARTDigital I/Ogroups of users. The first design, called ELMB2, is backward compatible with the current

ELMB. The second design, called ELMB++ GBT-SCA [370], based on radiation hard ASICs

is aimed to satisfy the highest requirements with respect to radiation. And a third design,

called ELMB++ FPGA, is composed only of commercial off-the-shelf components and would

have the highest data throughput [371].

Continuing MSU’s contribution to the development and operation TileCal, the MSU

group was tasked with production and design of the new ELMB2 Mother-board for the

fLVPS [372].

C.4 Cs calibration firmware

Three calibration and monitoring systems are implemented on TileCal [185, 373]. The

charge injection system is designed to calibrate the relative pulse response of the readout

electronics for all PMTs and to monitor its variations over time. The laser system is used

to check the PMTs’ linearity, for studies of pulse saturation recovery, in calorimeter timing,

and also for debugging and diagnostics tests. The cesium calibration system is implemented

to determine the quality of the optical response of each calorimeter cell, to adjust the PMTs’

HV to equalize the response from all cells, and to monitor it over time [374, 375].

The Cs calibration system, schematically illustrated in Fig. C.13, works in the following

way. Every TileCal module is equipped with a hollow tube that passes perpendicularly

through every scintillator tile. A 5.6 × 11.5 mm capsule, containing an 8 mCi sample of

Cs137, can travel through the 6 mm tube to provide a temporary source of γ radiation. The

capsule is moved through the tube using water as the hydraulic fluid driven by a pump

located in the hydraulic crate off the detector. When calibration runs are not performed,

259

Figure C.13: Diagram of the TileCal cesium calibration system [375].

the capsules are housed in garage compartments mounted on every TileCal barrel. Multiple

modules are connected together and are serviced by one garage. Inductance sensors, labeled

SIN, are used to track the capsule’s position in the system. Calibration runs are controlled

by electronics boards also located in the garages.

An example of Cs calibration data is shown in Fig. C.14. The horizontal axis effectively

shows the position of the capsule within the cell and the vertical axis shows integrated signal

from the PMTs. The source-induced current is measured approximately every 3 mm, giving 6

points per tile, the tile spacing being about 18 mm. What emerges is a picture of the relative

response of the scintillator tiles, with each tooth on the plot corresponding to the amount of

260

WaterStationHydrauliccrateGarage 1F17Gar 1SIN 1SIN 512 VSIN 3SIN 712 V3USinPSWaterGarage 3F57Gar 3Garage 2F40Gar 2SIN 6Dum 2SIN 4On Detector SensorsSIN SensorsSIN 2Dum 1SIN 0On Detector ElectronicsSIN-CANNumberNumber41338Pressure Sensors74ADC-CAN16Side ASide CGarage-CAN9CAN12 VWater & airsupplytubesCAN12 V812 V DistributorUSA15Side ASide CF18F10F02F59F51F43F35F27F19F26F10F02F59F51F35F42RB 1.2RB 2.1RB 2.3RB 1.1RB 2.2RB Crate220V SwitchPower Valves 24VPower CAN 12VADC H2ADC H1EthernetEthernetDAQ3U Crate Electronics6 types 3U cardsNumber33Figure C.14: Cs scan data collected during a testbeam in 2016 using the Demonstrator
electronics [349]. The plot shows the response of the tiles in the BC6 cell of the module.
Each peak corresponds to the Cs source passing though a tile.

signal from a particular tile. The tiles’ response can then be equalized by adjusting the HV

delivered to the PMTs thereby calibrating their gain. Cs scans allow to evaluate single-tile

response to within 2%. That leads to the tile row mean response accuracy close to 0.6%,

and 0.3% for the cell average.

As part of the Phase-II upgrade, communication between DCS and the Cs boards will

be routed through the Daughter boards. One of the upgrade projects that I worked on was

the development of the required component for the Daughter board FPGA firmware [376].

This VHDL firmware component implements communication between the DB and the Cs

boards via SPI protocol using LVDS lines. A test bench setup used in the development of

the firmware is shown in Fig. C.15.

When a Cs scan begins, the Cs board runs a program installed on its microcontroller.

The program controls the hydraulic system, the release, tracking, and parking of the Cs

capsule in the garage compartment, as well as data collection. During the scan, the Cs

261

Event number33600337003380033900340003410034200ADC counts05000100001500020000250003000035000ATLAS Tile Calorimeter2016 testbeam dataFigure C.15: Photograph of the electronics setup used for development of the Cs calibration
system (bottom board) FPGA firmware component for the TileCal mini-drawer Daughter
board (top board).

board collects and transmits information about the scan. With the Cs board operating

the scans, the natural setup for the SPI protocol is for the Cs board to act as a master

and for the DB to be a slave device. However, the Cs board needs to receive commands

to start or abort a scan.

In order to allow this, the SPI protocol was slightly modified,

and the slave select (nSS) line was changed to a request (nREQ) line, to enable the DB to

request clock cycles and initialize communication. Only one Cs board is installed for each

garage, so only some Daughter boards need to connect to one. This one-to-one connection

makes the modification of the SPI protocol possible. With every transmission, a message

from the Cs board is stored in the cs_status register on the DB, and the contents of the

cs_command DB register are transmitted to the Cs board. A new time stamp is written to

the cs_timestamp register upon the completion of every transaction. The DB lowers the

262

nREQ line, when a different value is written into the cs_command register by the PPr. The

values of the DB registers are automatically and periodically copied to the PPr. This update

rate is much faster then the communication between the DB and the Cs board, so there will

be no apparent synchronization delay.

Communication was successfully tested with a version 4 Daughter board [376]. The

firmware was later used during the testbeam to collect calibration data, shown in Fig. C.14.

C.5 Testbeam

Before any new hardware can be installed on ATLAS, its readiness for deployment has to

be verified. For detector components and electronics this includes studies that use particle

beams to simulate data collection under realistic operation conditions. Such tests are collo-

quially referred to as the testbeam and are carried out at a dedicated cite at CERN on the

Pr´evessin campus, where the SPS beam can be redirected via the H8 beam line for use in

fixed-target experiments. With different components of the TileCal readout electronics chain

developed by independent teams in different countries, testbeam studies allow to evaluate

the joint operation of the whole system and to test for problems that are difficult to identify

in a test-bench environment.

From 2015 to 2017 a number of testbeam campaigns was carried out to study the per-

formance and stability of the full chain of TileCal Phase-II upgrade electronics [377–379]. A

diagram of the test setup is shown in Fig. C.16. Response and calibration were tested for

different types of particles at different energies. The testbeam facility can provide beams

of muons, positrons, or hadrons, all of which were used. To identify the particle species at

energies below 50 GeV, two Cherenkov counters and a muon hodoscope were placed in the

263

path of the particle beam. The additional instruments included trigger scintillators, used

in coincidence to signal passing of the beam, and wire chambers, which provided position

measurements of the beam with respect to the modules. 5 modules were used in the tests,

arranged as shown in Fig. C.17. The modules were placed on a mobile table to allow their

orientation to be changed with respect to the beam. Different versions of electronics, listed

in Table C.1, were installed on different modules to run comparative tests.

Cherenkov counters

Wire chambers

Trigger
scintillators

TileCal modules

Muon hodoscope

Figure C.16: TileCal Demonstrator testbeam setup [379]. The beam direction is from left
to right.

The testbeam results demonstrated good performance of the new electronics and agree-

ment of its calibrated measurements with the legacy system and simulations [380]. I partici-

pated in the testbeam activities in 2016, including firmware debugging, writing a program

for positioning of the mobile table, and taking data collection shifts in August of that year.

The data shown in Fig. C.14 was collected at the testbeam in 2016 using the firmware I

developed for communication between the DB and the Cs calibration boards.

264

η > 0

η < 0

LBA65

M0 A

EBC65
LBC65
M0 C

Figure C.17: Left: Diagram of TileCal modules’ arrangement for the testbeam [379]. Right:
Photograph of the modules positioned on the mobile table in the testbeam area [381].

Table C.1: Configuration of modules and electronics during the October 2016 testbeam
period [349].

265

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266

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