SEARCH FOR W’ PRODUCTION IN THE SINGLE-TOP CHANNEL WITH THE
ATLAS DETECTOR
By
Patrick True

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics - Doctor of Philosophy

2015

ABSTRACT
SEARCH FOR W’ PRODUCTION IN THE SINGLE-TOP CHANNEL WITH
THE ATLAS DETECTOR
By
Patrick True
This thesis presents the search for W ′ → t¯b using the LHC pp collision data collected
with the ATLAS detector at a center-of-mass energy of 8 TeV. The primary backgrounds to
this search are ttbar, W+jets, and multijets processes. To reduce the contributions of these
backgrounds we require a leptonic final state and use Boosted Decision Trees to discriminate
between signal and background-like events. This measurement gives limits on the W ′ → t¯b
cross-section times branching ratio and on the coupling strength of the W ′ boson as a function
of its mass.

ACKNOWLEDGMENTS

First and foremost I would like to thank my wife Amber for her infinite support while I
wrote this thesis. I would not be the person I am today without her.
I would like to thank my advisor Kirsten Tollefson for teaching me the ropes of working
in high energy physics and always having good advice. It would be impossible to list all the
ways she has helped me through graduate school.
Thanks to Barbara Alvarez for mentoring me for several years and helping me to code,
troubleshoot, and understand all the tiny details that go into physics research.
Thanks to all the other graduate students and postdocs at MSU, especially James Koll,
Jenny and Jeremiah Holzbauer, JT Laverty and Brad Schoenrock for your friendship and
help over these six years.
Thanks to all the support staff, in particular Brenda Wenzlick and Debbie Barratt for
making sure I stayed enrolled, got paid, and successfully navigated graduate school.
Thanks to my committee members for accepting the task of reviewing my work and
providing me unparralleld insight and advice.
Finally I have to thank my family for their love and support for all of these years. Without
them none of this would have been possible.

iii

TABLE OF CONTENTS

LIST OF TABLES

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

vii

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

ix

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

1

LIST OF FIGURES
Chapter 1

Chapter 2 Theory . . . . . . . . . . . .
2.1 The Standard Model . . . . . . . . .
2.1.1 The forces . . . . . . . . . . .
2.1.2 The particles . . . . . . . . .
2.1.2.1 Jets . . . . . . . . .
2.1.3 The Lagrangian . . . . . . . .
2.2 Beyond the Standard Model theories
2.2.1 W ′ boson . . . . . . . . . . .
2.2.2 Extensions of SU (2) ⊗ U (1) .
2.2.3 Effective Lagrangian approach

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4
5
5
7
10
11
15
16
17
17

Chapter 3 ATLAS and the LHC . . . . . . . .
3.1 The Large Hadron Collider . . . . . . . . . . .
3.1.1 The Accelerator Chain . . . . . . . . .
3.2 The ATLAS detector . . . . . . . . . . . . . .
3.2.1 Detector geometry . . . . . . . . . . .
3.2.2 Magnet system . . . . . . . . . . . . .
3.2.3 Inner detector . . . . . . . . . . . . . .
3.2.3.1 Pixel detector . . . . . . . . .
3.2.3.2 Semiconductor tracker . . . .
3.2.3.3 Transition radiation tracker .
3.2.4 Calorimeters . . . . . . . . . . . . . . .
3.2.4.1 Electromagnetic Calorimeter
3.2.4.2 Hadronic Calorimeter . . . .
3.2.5 Muon Spectrometer . . . . . . . . . . .
3.2.5.1 Monitored drift tubes . . . .
3.2.5.2 Cathode strip chambers . . .
3.2.5.3 Resistive plate chambers . . .
3.2.5.4 Thin gap chambers . . . . . .
3.2.6 Triggering and data acquisition . . . .

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19
19
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21
22
24
25
27
28
29
30
31
33
36
36
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39
39
40

Chapter 4 Object Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Electron definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43
45

iv

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4.2
4.3

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46
48
50
52

Chapter 5 Event selection . . . . . . . . . . .
5.1 Composite objects . . . . . . . . . . . . . .
5.1.1 W boson and neutrino reconstruction
5.1.2 Top quark reconstruction . . . . . . .
5.1.3 W ′ reconstruction . . . . . . . . . . .
5.2 Data triggers . . . . . . . . . . . . . . . . .
5.3 Cut flow . . . . . . . . . . . . . . . . . . . .

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53
53
55
56
57
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58

Chapter 6 Event Simulation . .
6.1 Signal process . . . . . . . . .
6.2 Background processes . . . . .
6.2.1 Single top processes . .
6.2.2 Top pair production .
6.2.3 W+jets . . . . . . . .
6.2.4 Z+jets . . . . . . . . .
6.2.5 Diboson processes . . .
6.2.6 Multijets . . . . . . . .
6.3 Monte Carlo simulation . . . .
6.4 Data driven estimates . . . .
6.4.1 W+jets normalization
6.4.2 Multijets estimate . . .
6.5 Event yields . . . . . . . . . .

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4.4

Muon definition . . . . . . . . . . .
Jet definition . . . . . . . . . . . .
4.3.1 Jet b-tagging . . . . . . . .
Missing transverse energy definition

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Chapter 7 Multivariate analysis . . . .
7.1 Boosted decision trees . . . . . . . .
7.1.1 Overtraining . . . . . . . . . .
7.1.2 Variable selection . . . . . . .
7.1.3 BDT parameter optimization
7.1.4 BDT output distributions . .

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. 78
. 79
. 83
. 83
. 99
. 101

Chapter 8 Statistical analysis . . . . . . . . . . . . . . . .
8.1 Systematic uncertainties . . . . . . . . . . . . . . . . . .
8.1.1 Jet energy scale (JES) . . . . . . . . . . . . . . .
8.1.2 Jet energy resolution (JER) . . . . . . . . . . . .
8.1.3 Jet reconstruction efficiency . . . . . . . . . . . .
8.1.4 Jet vertex fraction (JVF) . . . . . . . . . . . . . .
8.1.5 b-tagging performance . . . . . . . . . . . . . . .
8.1.6 Lepton energy scale and resolution . . . . . . . .
8.1.7 Lepton trigger, reconstruction, and identification .
miss ) . . . . . . . . .
8.1.8 Missing transverse energy (ET

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v

104
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111

8.2
8.3
8.4

8.1.9 Parton distribution function (PDF) . . . . . . . . . . . . .
8.1.10 Initial state radiation and final state radiation (ISR/FSR)
8.1.11 Monte Carlo event generator and parton showering . . . .
8.1.12 Theoretical cross-sections . . . . . . . . . . . . . . . . . . .
8.1.13 W+jets normalization . . . . . . . . . . . . . . . . . . . .
8.1.14 Multijet normalization . . . . . . . . . . . . . . . . . . . .
8.1.15 Luminosity . . . . . . . . . . . . . . . . . . . . . . . . . .
8.1.16 Monte Carlo statistics . . . . . . . . . . . . . . . . . . . .
Limit setting procedure . . . . . . . . . . . . . . . . . . . . . . . .
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Comparison with other experiments . . . . . . . . . . . . . . . . .

Chapter 9
APPENDIX

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112
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129

Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
. . . . . . . . . . . . . . .

BIBLIOGRAPHY

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

vi

. . . . . . . . . . . . . . . . . . . 133
. . . . . . . . . . . . . . . . . 138

LIST OF TABLES

Table 2.1

The fundamental particles of the Standard Model and their properties [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

Definition of variables used for electron identification cuts. For cuts
without values specified, the cut values are optimized individually in
10 bins of η and 11 bins of cluster ET [2]. . . . . . . . . . . . . . . .

47

Table 4.2

Definition of muidcombined muon reconstruction cuts. . . . . . . . .

48

Table 6.1

Cross-sections and k-factors for generated W ′ samples [3]. . . . . . .

70

Table 6.2

Simulated background samples with associated cross-sections, k-factors,
generating programs and showering programs. . . . . . . . . . . . . 70

Table 6.3

W+jets normalization factors. . . . . . . . . . . . . . . . . . . . . .

74

Table 6.4

Event yields for signal samples, background samples, and data by
analysis channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

Boosted decision tree variable lists for the four analysis channels.
Variables are ranked by importance. . . . . . . . . . . . . . . . . . .

85

Table 4.1

Table 7.1

Table 7.2

Optimized Boosted decision tree parameters for each of the four analysis channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Table 8.1

Summary table of the systematic percentage shifts in the 1.75 TeV
WR′ signal and background event yields of the 1tag analysis channels. 105

Table 8.2

Summary table of the systematic percentage shifts in the 1.75 TeV
WR′ signal and background event yields of the 2tag analysis channels. 106

Table 8.3

Expected and observed mass limits for a WR′ boson in the individual
analysis channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Table 8.4

Expected and observed mass limits for a WL′ boson in the individual
analysis channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

vii

Table 8.5

Expected and observed mass limits for a WR′ boson in the combined
analysis channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Table 8.6

Expected and observed mass limits for a WL′ boson in the combined
analysis channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

Table 8.7

Observed mass limits for a WR′ boson from different experiments . . 129

Table 8.8

Observed mass limits for a WL′ boson from different experiments . . 129

viii

LIST OF FIGURES

Figure 2.1

Diagram of the Standard Model particles grouped into bosons and
fermions, with the fermions further grouped into quarks and leptons
with three generations of pairs each [4]. . . . . . . . . . . . . . . .

Figure 3.1

Diagram of the CERN accelerator chain [5].

Figure 3.2

8

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

21

Cutaway diagram of the ATLAS detector [6].

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

22

Figure 3.3

Illustration of the ATLAS coordinate system.

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

24

Figure 3.4

Illustration of the ATLAS magnet system, showing the barrel solenoid,
barrel toroid, and endcap toroid coils [7]. . . . . . . . . . . . . . . .

26

Figure 3.5

Cutaway diagram of the ATLAS inner detector [8].

27

Figure 3.6

Cutaway diagram of the ATLAS calorimeter systems [9].

. . . . . .

31

Figure 3.7

Cutaway diagram of the ATLAS muon spectrometer and toroid magnet systems [10]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

Illustration of how different particles interact with the ATLAS detector systems [11]. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

Figure 4.2

Illustration of a displaced secondary vertex caused by a b-jet [12]. .

51

Figure 5.1

Illustration of the W ′ → t¯b process. . . . . . . . . . . . . . . . . . .

54

Figure 5.2

Illustration of the discrimination power of a variable, shown as the
shaded region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

Illustration of the (a) s-channel single top process (b) t-channel single
top process (c) Wt-channel single top processes. . . . . . . . . . . .

63

Figure 6.2

Illustration of the tt¯ process. . . . . . . . . . . . . . . . . . . . . . .

64

Figure 6.3

Illustration of the Wbb process. . . . . . . . . . . . . . . . . . . . .

65

Figure 4.1

Figure 6.1

ix

. . . . . . . . .

Figure 6.4

Illustration of the Z+jets process. . . . . . . . . . . . . . . . . . . .

66

Figure 6.5

Illustration of a diboson process.

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

67

Figure 7.1

An example decision tree which sorts input events into signal and
background bins. The variables used in the decision tree are described
in Section 7.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

Comparison of data to Monte Carlo prediction of the 1st-5th variables
by importance in the 2jet 1tag BDT. . . . . . . . . . . . . . . . . .

86

Comparison of data to Monte Carlo prediction of the 6th-10th variables by importance in the 2jet 1tag BDT. . . . . . . . . . . . . . .

87

Comparison of data to Monte Carlo prediction of the 11th-15th variables by importance in the 2jet 1tag BDT. . . . . . . . . . . . . . .

88

Comparison of data to Monte Carlo prediction of the 16th-18th variables by importance in the 2jet 1tag BDT. . . . . . . . . . . . . . .

89

Comparison of data to Monte Carlo prediction of the 1st-5th variables
by importance in the 3jet 1tag BDT. . . . . . . . . . . . . . . . . .

90

Comparison of data to Monte Carlo prediction of the 6th-10th variables by importance in the 3jet 1tag BDT. . . . . . . . . . . . . . .

91

Comparison of data to Monte Carlo prediction of the 11th-15th variables by importance in the 3jet 1tag BDT. . . . . . . . . . . . . . .

92

Comparison of data to Monte Carlo prediction of the 1st-5th variables
by importance in the 2jet 2tag BDT. . . . . . . . . . . . . . . . . .

93

Comparison of data to Monte Carlo prediction of the 6th-10th variables by importance in the 2jet 2tag BDT. . . . . . . . . . . . . . .

94

Comparison of data to Monte Carlo prediction of the 11th-14th variables by importance in the 2jet 2tag BDT. . . . . . . . . . . . . . .

95

Comparison of data to Monte Carlo prediction of the 1st-5th variables
by importance in the 3jet 2tag BDT. . . . . . . . . . . . . . . . . .

96

Figure 7.2

Figure 7.3

Figure 7.4

Figure 7.5

Figure 7.6

Figure 7.7

Figure 7.8

Figure 7.9

Figure 7.10

Figure 7.11

Figure 7.12

x

Figure 7.13

Figure 7.14

Comparison of data to Monte Carlo prediction of the 6th-10th variables by importance in the 3jet 2tag BDT. . . . . . . . . . . . . . .

97

Comparison of data to Monte Carlo prediction of the 11th-13th variables by importance in the 3jet 2tag BDT. . . . . . . . . . . . . . .

98

Figure 7.15

The BDT output distribution with the signal and background processes split into testing and training samples for (a) the 2jet 1tag
analysis channel (b) the 3jet 1tag analysis channel (c) the 2jet 2tag
analysis channel (d) the 3jet 2tag analysis channel. . . . . . . . . . 102

Figure 7.16

The BDT output distribution of the background samples and data
with statistical uncertainties for (a) the 2jet 1tag analysis channel (b)
the 3jet 1tag analysis channel (c) the 2jet 2tag analysis channel (d)
the 3jet 2tag analysis channel. . . . . . . . . . . . . . . . . . . . . . 103

Figure 8.1

95% confidence level limits on the cross-section times branching ratio
of the WR′ boson in the (a) 2jet 1tag channel (b) 3jet 1tag channel
(c) 2jet 2tag channel (d) 3jet 2tag channel. . . . . . . . . . . . . . . 121

Figure 8.2

95% confidence level limits on the cross-section times branching ratio
of the WL′ boson in the (a) 2jet 1tag channel (b) 3jet 1tag channel
(c) 2jet 2tag channel (d) 3jet 2tag channel. . . . . . . . . . . . . . . 122

Figure 8.3

95% confidence level limits on gR in the (a) 2jet 1tag channel (b) 3jet
1tag channel (c) 2jet 2tag channel (d) 3jet 2tag channel. . . . . . . 123

Figure 8.4

95% confidence level limits on gL in the (a) 2jet 1tag channel (b) 3jet
1tag channel (c) 2jet 2tag channel (d) 3jet 2tag channel. . . . . . . 124

Figure 8.5

95% confidence level limits on the cross-section times branching ratio
of the WR′ boson in the (a) 1tag combined channel (b) 2tag combined
channel (c) full combination channel. . . . . . . . . . . . . . . . . . 125

Figure 8.6

95% confidence level limits on the cross-section times branching ratio
of the WL′ boson in the (a) 1tag combined channel (b) 2tag combined
channel (c) full combination channel. . . . . . . . . . . . . . . . . . 126

Figure 8.7

g′

g′

′

g
95% confidence level limits on gR in the (a) 1tag combined channel
(b) 2tag combined channel (c) full combination channel. . . . . . . 127

xi

Figure 8.8

g′

95% confidence level limits on gL in the (a) 1tag combined channel
(b) 2tag combined channel (c) full combination channel. . . . . . . 128

xii

Chapter 1
Introduction
Humans have always sought to understand their environment. People have progressed from
an intuitive understanding of the universe based on practical experience to a more scientific
understanding based on logic and the rigorous study of phenomena. The understanding
brought about by science has allowed humanity to achieve feats which would otherwise be
impossible, and we continue to seek more understanding. At the forefront of this quest,
particle physicists work to understand the most basic components of the universe and the
most fundamental of interactions between them. Our understanding of the universe has progressed from classical theories of the four natural elements, through the chemical elements
and nuclear structure, to modern relativistic quantum field theories that describe the most
fundamental objects yet imagined. Continuing research in the field takes many forms: precision measurements of trapped and isolated atoms over months allow theoretical models to
be tested, detection of particles accelerated by supernovae and other astronomical objects
allows access to an energy regime physicists are unable to recreate on Earth, and experiments
at particle colliders allow for the investigation of the rarest phenomena.
This dissertation describes a search for a new particle using the largest detector in the
world at the highest energy particle collider ever built and is part of an international research program in collaboration with a multinational research group, specifically the search
for W ′ → t¯b using the ATLAS detector at CERN. Similar searches have previously been
performed by collaborations at CERN and Fermilab and their results are described in Sec1

tion 8.4. This analysis is the first search for W ′ → t¯b using the ATLAS detector to include
events with 1 b-tagged jet, as described in Chapter 5. This approximately doubles the signal
acceptance relative to the previous ATLAS search for W ′ → t¯b. These additional events are
kinematically distinct from the events with 2 b-tagged jets and the entire analysis has been
optimized in 4 independant channels based on the jet content of the events. This analysis also
uses a larger dataset than the previous ATLAS search and uses the latest recomendations for
event simmulation, data callibration, and the estimation of systematic uncertainties. Thus
this analysis is expected to be currently the most sensitive search for W ′ → t¯b using the
ATLAS detector.
Throughout this text the natural unit system is used in which c = 1,

= 1, and e = 1.

Energies and momenta are expressed in electronvolts (eV), where 1 eV is the energy gained
by a unit charge traversing a 1 Volt electric potential. This dissertation is divided into
chapters intended to first establish the environment the analysis takes place in, to build up
the analysis from basic components, and finally to present the results and put them into
context with the rest of the field. Towards this end the chapters are as follows:
• Chapter 2 discusses the theoretical background for the analysis.
• Chapter 3 describes the physical apparatus of the experiment.
• Chapter 4 defines the analysis level objects reconstructed from the detector response.
• Chapter 5 defines the criteria for an event to be included in the analysis.
• Chapter 6 details the simulation methods used to model the backgrounds and potential signal processes.

2

• Chapter 7 describes the multivartiate analysis technique and its specific implementation.
• Chapter 8 details the results of the analysis and their statistical significance.
• Chapter 9 discusses the results of the analysis and their context in current research.
• Appendix A describes alternative multivariate analysis optimizations which were
investigated for the analysis but not used in the final result.

3

Chapter 2
Theory
High energy physics attempts to deal with the fundamental particles and forces of the universe, and the Standard Model (SM) of high energy physics is the theoretical framework
used in this analysis. The Standard Model describes the universe as being composed of 17
fundamental particles and their interactions through three of the four fundamental forces.
This thesis is a search for a W ′ particle not included in the Standard Model which would
be indicative of extended physical theories, collectively called Beyond the Standard Model
(BSM) theories. There are a wide variety of BSM theories therefore this analysis is performed
in a model-independent manner using an effective Lagrangian which has a form that many
of these theories which include a W ′ boson can be reparameterized into. The motivation
behind searching for a W ′ is explored by examining some representative BSM theories and
their consequences. The focus of this chapter is not to derive the Standard Model or any
BSM theories from first principles, but rather to provide a practical framework and context
in which to understand this analysis and the implications of the results. There are a number
of texts which provide a more comprehensive overview of particle physics [13, 14] or a more
rigorous derivation of the mathematical models used [15, 16].

4

2.1

The Standard Model

The Standard Model has provided accurate predictions of experimental observables for over
40 years. It was developed after decades of experimentation had catalogued a myriad of particle states. The properties of these states were observed to follow patterns and symmetries,
and eventually these symmetries were developed into the Standard Model. The symmetries
of the Standard Model are described in group theory terms as SU (3) ⊗ SU (2) ⊗ U (1) with
each symmetry giving rise to its own conservation law. The Standard Model is a quantum
field theory arising from a unification of quantum mechanics and special relativity, and is
mathematically described by a Lagrangian [16].

2.1.1

The forces

The Standard Model includes three of the four fundamental forces of nature: the electromagnetic, weak, and strong forces - but not the gravitational force as no complete quantum
mechanical theory of gravity has been discovered yet. The electromagnetic and weak forces
can be unified into a single electroweak force similar to the unification of the electric and
magnetic forces into the electromagnetic force. The electroweak force is described by the
SU (2) ⊗ U (1) symmetry of the Standard Model and is mediated by the massless photon as
well as the massive W ± and Z bosons which are described in Section 2.1.2. One of the greatest theoretical achievements of the Standard Model was the prediction of the existence of
the W ± and Z bosons well before their experimental discovery. The strong force is described
by the SU (3) symmetry of the Standard Model and is mediated by massless gluons which
are described in Section 2.1.2. The strong force differs from the electroweak force in that the
strong force grows with increased distance between objects which has unique consequences

5

when compared to the electroweak force which diminishes with distance. There have been
many attempts to unify the strong force with the electroweak force and even to include a
quantum field theory of gravity, such as supersymmetry or string theory. However, there
is no clear experimental evidence to support these theories and they are not part of the
Standard Model [14].
The electroweak force is a unification of the electric, magnetic, and weak forces. The
electric and magnetic forces were unified by Maxwell in 1879, and were described as a
quantum field theory called quantum electrodynamics (QED) by Dirac in the 1920’s. The
resulting electromagnetic force decreases as the distance between objects increases, and the
force is carried to infinite distance by its massless mediator, the photon. The weak force is
responsible for a wide range of observed phenomena, including nuclear beta decay and the
violation of parity and charge-conjugation invariance. The violation of these symmetries is
due to the weak force only interacting with the left-handed chirality of particles and fields.
These processes can be described with phenomenological theories at low and intermediate
energies, however at higher energies, above a few GeV, these phenomenological weak theories
break down. Unlike the electromagnetic force which is caried to an infinite distance by the
massless photon, the weak force is mediated by the massive W ± and Z bosons and so has a
limited range of typically 2.5 am, less than the radius of a proton. It is only after unification
that electroweak theory provides consistent predictions for the energy ranges probed by
modern accelerators of up to several TeV [14].
The strong force is responsible for holding baryons, mesons, and nuclei together and is
described in the Standard Model by quantum chromodynamics (QCD). QCD describes the
strong force using a type of charge called “color” which comes in three types of colors and
their anti-colors. Quarks each carry either a color or anti-color charge; whereas gluons, the
6

mediating particles of the strong force, carry both a color and an anti-color charge. The
color charge carried by the gluons is a key difference between QCD and QED - in which the
mediating particle is charge neutral. This means that gluons are self-interacting and so it is
a non-Abelian gauge theory. This self-interaction also leads to anti-shielding of bare color
charges by the vacuum and the force between colored objects becoming larger as the distance
between them increases. This corresponds to the fact that only colorless objects are observed
in nature, either those objects without a color charge or which are compsed of objects which
have no net color charge. If the color charged constituent particles of such a composite object
begin to separate it becomes energetically favorable to pair-produce new colored particles
from the vacuum which will form new colorless bound states with the original color charged
particles. When very energetic colored particles are produced, such as from the decay of
a heavy resonance, this process is rapidly repeated which causes the formation of particle
showers called “jets”, as discussed further in Section 2.1.2.1 [14].

2.1.2

The particles

The Standard Model contains 17 fundamental particles and their anti-particles which compose all objects. These particles can be classified into three types called leptons, quarks, and
bosons as shown in Table 2.1. For each particle in Table 2.1 there is a corresponding antiparticle with opposite electric charge. In general a particle name or symbol refers to both the
particle and its anti-particle except where they are explicitly distinguished. Thus “electron”
refers to both electrons and positrons. The structure visible in Figure 2.1 of quarks and
leptons both having three generations of pairs, with each lepton pair consisting of a charged
lepton and neutrino and each quark pair consisting of an “up-type” quark with an electric
charge of 2/3 and one “down-type” quark with electric charge −1/3, is not accidental and is
7

vital to our understanding of the particles.

Figure 2.1: Diagram of the Standard Model particles grouped into bosons
and fermions, with the fermions further grouped into quarks and leptons
with three generations of pairs each [4].

The six lepton flavors (electron, electron neutrino, muon, muon neutrino, tau, tau neutrino) can be classified into 3 generations, each containing a charged lepton and its neutrino.
The charged leptons all have mass and carry an electrical charge of -1, while the neutrinos
are electrically neutral but their masses have not been directly observed. The observation of
neutrino flavor oscillations [17] implies that neutrinos are not massless and the current best
limits on the mass of each neutrino flavor are given in Table 2.1. Leptons do not interact
through the strong force, the charged leptons interact through the weak and electromagnetic
forces while the neutrinos only interact weakly. Because neutrinos can only interact through
the weak force their interaction with matter, such as a detector, is rare and specialized experiments are necessary to study them. In contrast the first generation charged lepton, the
8

electron, is easily detected through electromagnetic interactions and is readily available in
nature. This difference in detectability has led to the term “lepton” generally indicating the
charged leptons with the neutrinos being indicated separately. Experimentalists often further restrict lepton to mean only electrons and muons because taus decay within a particle
detector and are difficult to identify. This convention will be followed in the experimental
portions of this thesis.
Similar to the leptons, the quarks (up, down, charm, strange, top, bottom) can also be
described by 3 generations, each containing 2 flavors. Each generation contains one quark
with an electric charge of 2/3 and one quark with an electric charge of −1/3, called up-type
and down-type respectively based on the first generation quarks with those charges. Quarks
interact through all 3 of the forces in the Standard Model and thus are readily detectable
using a variety of methods. Since quarks have a color charge, bare quarks are not directly
detectable but will form jets as described in Section 2.1.2.1.
The final group of particles in Table 2.1 are the bosons. The bosons all have integer
spins, with the photon, gluon, W ± and Z bosons all being spin 1 and the Higgs boson being
spin 0. The photon, W ± and Z bosons are the mediating particles of electroweak theory
with the W ± boson having an electric charge of ±1 and the photon and Z boson being
electrically neutral. The W ± and Z bosons are massive, unlike the photon, and they gain
their masses through the Higgs mechanism. The Higgs mechanism adds a quartic complex
scalar field potential to the Lagrangian density, described in Section 2.1.3, which is locally
gauge invariant. Through an appropriate choice of parameters the field is made to have a
non-zero expectation value and induce spontaneous symmetry breaking in the SU (2) ⊗ U (1)
electroweak group. What is left is the massive W ± and Z bosons, the massless photon, and a
new massive Higgs boson, which was just recently discovered at the LHC [18, 19]. The final
9

boson is the gluon which mediates the strong force. The gluon has a color and anti-color
charge which makes it self-interacting as described in Section 2.1.1, and a bare gluon will
form a jet, as described in Section 2.1.2.1, similar to a bare quark.
Particle
Electron
Electron Neutrino
Muon
Muon Neutrino
Tau
Tau Neutrino
Up
Down
Charm
Strange
Top
Bottom
Photon
W ± Boson
Z Boson
Gluon
Higgs

Symbol
Mass
Leptons
e
511 KeV
νe
<2.05 eV
µ
106 MeV
νµ
<0.17 MeV
τ
1.78 GeV
ντ
<18.2 MeV
Quarks
u
2.3 MeV
d
4.8 MeV
c
1.28 GeV
s
95 MeV
t
173 GeV
b
4.18 GeV
Bosons
γ
0
W±
80.4 GeV
Z
91.2 GeV
g
0
H
126 GeV

Charge [e]

Spin

-1
0
-1
0
-1
0

1/2
1/2
1/2
1/2
1/2
1/2

2/3
−1/3
2/3
−1/3
2/3
−1/3

1/2
1/2
1/2
1/2
1/2
1/2

0
±1
0
0
0

1
1
1
1
0

Table 2.1: The fundamental particles of the Standard Model and their properties [1].

2.1.2.1

Jets

Jets are phenomenological objects that are formed when individual colored particles, single
quarks and gluons, are produced at sufficiently high energies. As the colored particle moves
away from the initial colored object it is connected to, the energy of the strong interaction
between them increases until it becomes energetically favorable to produce a quark-antiquark
pair from the vacuum for the particle and the initial colored object to be bound to. This
bound state has no net color and is called a hadron. The production of new hadrons is
10

called hadronization and each hadron produced absorbs a small amount of the initial colored
particle’s energy. This process will repeat itself until there is insufficient energy remaining
for further hadronization, which is when the energies of the processes involved drop to the
mass of the lightest hadron. This creates a narrow shower of hadrons that in total have the
same energy and momentum as the original colored particle. This particle shower is called a
“jet” and it is the detectable product of the original color charged particle. This is a general
picture of what happens to quarks and gluons but there are subtle differences depending
on the flavor of the initial particle, in particular if the initial particle is a top or bottom
quark. Because of its large mass, a top quark almost exclusively decays into a W ± boson
and bottom quark before hadronization can occur, creating a very different signal from other
quarks. Bottom quarks also have unique phenomenology in that the hadron produced with
the initial bottom quark has an unusually long lifetime and will travel a considerable distance
in a detector before the b quark decays and further hadronization takes place. By carefully
tracking the trajectories of the products of the hadronization in these jets and projecting
them backwards, a secondary production vertex can be formed which is displaced from the
primary production vertex of the other objects in an event [14].

2.1.3

The Lagrangian

In order to concisely describe the mathematics behind the Standard Model it is necessary to
define some common notation. Einstein summation notation is used throughout this section
and any repeated index in a term is summed over. In all of the equations in this section,
B, W i , and Gj are vector potentials with 1 time-like dimension and 3 space-like dimensions
and all Greek lettered indices, namely µ and ν, run over these four dimensions. There is one
electromagnetic potential (B), three weak potentials (W i ), and eight strong potentials (Gj ).
11

For each potential B, W i , and Gj , g is a coupling constant that determines the strength of
the interaction between an object and that potential. The Dirac matrices are the generators
of the Minkowski space with 1 time-like dimension and 3 space-like dimensions. The Dirac
matrices are denoted as γ, and are given in the Dirac representation in Equation 2.1. Finally,
the chirality operator γ 5 is not a Dirac matrix but is defined in terms of them in Equation 2.2.
γ 5 is important for describing left-handed and right-handed interactions, with the left-handed
5
1+γ 5
projection having the form 1−γ
and
the
right-handed
projection
having
the
form
2
2 .







0 
 1 0 0
 0






 0 1 0

 0
0 


0
1
γ =
,γ = 



 0 0 −1 0 
 0






0 0 0 −1
−1



 0
 0 0 0 −i 







 0 0 i 0 
 3  0

,
γ
=
γ2 = 





 −1
 0 i 0 0 






0
−i 0 0 0


 0 0 1


 0 0 0

γ 5 = iγ 0 γ 1 γ 2 γ 3 = 

 1 0 0


0 1 0

0



0 1 


0 1 0 



−1 0 0 


0 0 0

0 1 0 


0 0 −1 



0 0 0 


1 0 0

(2.1)



0 


1 



0 


0

(2.2)

The mathematics of the Standard Model are typically formulated in terms of a Lagrangian
L and its density L, such that L =

Ld3 x. The Standard Model includes many different
12

phenomena so it is useful to group the terms of the Lagrangian density by their physical
motivation as seen in Equation 2.3 [15].

L = LEW + LQCD + LY U K

(2.3)

Here LEW is the electroweak Lagrangian denstiy which describes the electroweak interaction
mediated by the photon, W ± and Z bosons, as well as the Higgs mechanism and the Higgs
boson, LQCD is the QCD Lagrangian density which describes the strong interaction between
gluons and quarks, and LY U K is the Yukawa Lagrangian density which describes the fermions
and how they derive their mass from the Higgs mechanism.
The Lagrangian density that describes the electroweak force (LEW ) is given in Equation 2.4.

1
1
µν
F γ µ D lF + e
LEW = − Bµν B µν − Wµνi Wi + ¯lL
¯F
γ µ Dµ eF
+ Dµ φDµ φ − µ2 φ2 − λ(φ2 )2
µL
R
R
4
4
(2.4)
The first two terms describe the electroweak fieldss with the index i running over the three
weak fields associated with the W ± and Z bosons. Bµν is the electromagnetic field tensor
and is defined as:

Bµν = ∂µ Bν − ∂ν Bµ

(2.5)

while Wµνi are the three weak field tensors, given in Equation 2.6 where ǫijk is the LeviCivita tensor.

13

Wµνi = ∂µ Wνi − ∂ν Wµi + igw ǫijk Wµj Wνk

(2.6)

The third and fourth terms of Equation 2.4 describe the left-handed fermions and righthanded fermions respectively, with the index F running over the three generations. Dµ is
the covariant derivative, which is defined as:

1
Dµ = ∂µ − gBµ − gw (1 − γ5 )Wµi
2

(2.7)

The last three terms of Equation 2.4 describe the Higgs field φ. The first of these is the
kinetic term for the Higgs field. The final two terms of the electroweak Lagrangian density
describe the Higgs potential which is quartic in φ. In these terms µ and λ are parameters
which determine the shape of the Higgs potential, where a negative value for µ2 and a positive
value for λ produces a Higgs potential with a minimum value at a finite φ = 0 which causes
spontaneous symmetry breaking and gives the W ± and Z bosons mass.
The QCD Lagrangian density (LQCD ) is given in Equation 2.8.

LQCD =

j

1
µν
ψ¯m (iγ µ Dµ − mj δmn )ψn − Gµνa Ga
4

(2.8)

The first term describes the quarks with index j running over all six of the quarks with
masses mj , and the m and n indices running over the three colors. The covariant derivative
Dµ is similar to Equation 2.7 but now contains eight gauge fields (Aa ) corresponding to the
eight gluons denoted by the index a:

Dµ =

1
∂µ − gBµ − gw (1 − γ5 )Wµi δmn + igs Tmna Aµa
2

14

(2.9)

where Tmna are the Gell-Mann matrices which are the generators of the SU (3) QCD group
which indicate that when a gluon interacts with a quark it changes the quark’s color, and
gs is the strong coupling strength.
The second term of the QCD Lagrangian density describes the gluons. Gµνa is analogous
to Bµν in the electroweak Lagrangian density, but for each of the eight gauge fields.

Gµνa = ∂µ Aνa − ∂ν Aµa − gs fabc Aµb Aνc

(2.10)

The fabc in Equation 2.10 are the structure constants of the QCD SU (3) group, which are
related to the generators of the group according to Equation 2.11.

[Ta , Tb ] = ifabc Tc

(2.11)

The Yukawa Lagrangian density (LY U K ) describes the fermions’ interactions with the
Higgs field and is given in Equation 2.12.

LY U K =

¯ f φψ
ψG

(2.12)

f

The index f runs over the fermions, with the covariant derivative defined in Equation 2.7.
The mass of each fermion is determined by Gf φ where Gf is the fermion’s coupling to the
Higgs field φ, and in this way the Higgs field gives the fermions their masses.

2.2

Beyond the Standard Model theories

While the Standard Model has described the observations of particle physics experiments
for over 40 years, there are known problems with the theory. Some examples are:
15

• The Standard Model does not include gravity, a force which has been experimentally
verified to exist.
• There is no mechanism to produce the amount of matter-antimatter asymmetry that
is observed in the universe today.
• The Standard Model does not include a suitable dark matter particle to match astronomical observations. While mathematically possible, the observed masses of the W ±
and Z bosons require very precise cancellations of parameters which seems unnatural.
While this list is by no means exhaustive, there have been decades of work to solve these
problems with Beyond the Standard Model (BSM) theories.

2.2.1

W ′ boson

Many of these BSM theories predict additional particles beyond those of the Standard Model.
Often these new particles are related to the Standard Model particles, such as through a
new symmetry in the BSM theory, and the BSM particles and SM particles share many of
the same properties. It is common practice in these cases to call the BSM particles by the
name of their SM counterparts but with a prime, thus a W ′ boson is similar to a Standard
Model W ± boson. In particular a W ′ boson is a massive, spin-1, colorless particle with
an electrical charge of ±1. The W ′ boson couplings are determined by each specific BSM
theory [20, 21, 22] and they may be similar to the W ± boson couplings in the Standard
Model or they may be very different. In cases where the W ′ couplings are similar to the
Standard Model W ± couplings the most sensitive decay channel to search for at particle
colliders is usually W ′ → lν. If the leptonic couplings of the W ′ boson are very small or if
the leptonic decay of the W ′ boson is suppressed for some other reason in the BSM theory,
16

such as if a right-handed W ′ boson were to decay to a right-handed BSM neutrino but the
right-handed neutrino is more massive than the W ′ boson, then another decay channel of
the W ′ boson to search for is W ′ → t¯b. The third generation (top and bottom) quark decay
is specified because some BSM theories predict an enhanced coupling of the W ′ boson to the
third generation and also because there are fewer Standard Model processes that produce
this state which provides an advantage in the experimental analysis.

2.2.2

Extensions of SU (2) ⊗ U (1)

Theories that include a W ′ boson often extend the SU (2)⊗U (1) electroweak symmetry which
describes the W ± boson. The simplest such extension is SU (2) ⊗ SU (2) ⊗ U (1), where the
new SU (2) can be a right-handed extension of the left-handed SU (2) group which describes
the Standard Model weak interactions or some other SU (2) symmetry. The SU (2) ⊗ U (1)
symmetry can also be extended by embedding the SU (2) into a group of higher degree,
resulting in symmetries such as SU (3) ⊗ U (1) or SU (4) ⊗ U (1). Each of these extensions
contains a myriad of specific theories with different coupling structures and a variety of
experimental predictions [1, 20, 21, 22]. Since there is currently no strong experimental
evidence to distinguish between these theories, this analysis does not assume any specific
theory but instead uses an effective Lagrangian approach.

2.2.3

Effective Lagrangian approach

In all of these theories the W ′ boson is described by a Lagrangian density term of the form
given in Equation 2.13 [3] which is added to the Standard Model Lagrangian density in
Equation 2.3.

17

1
′ (1 + γ ) + g ′ (1 − γ ))f j
LW ′ = √ Vij′ Wµ′ f¯i γ µ (gR
5
5
L
2 2

(2.13)

This Lagrangian density includes arbitrary right-handed and left-handed coupling strengths
′ and g ′ respectively. These coupling strengths are a common feature across all modgR
L

els regardless of how they are determined within each theory, and thus they are a modelindependant parameter which can be experimentally measured or constrained. In principle
these coupling strengths can be flavor-dependant, however for this analysis the coupling
strengths are assumed to be flavor-universal. The right-handed and left-handed couplings
′ or
are analyzed independantly using benchmark models where one of the BSM couplings gR
′ is equal to the Standard Model coupling g for W ± boson and the other BSM coupling is
gL

zero. These benchmark models are refered to as the WR′ and WL′ models respectively.

18

Chapter 3
ATLAS and the LHC
The search for W ′ → t¯b requires a very large and extensive experimental setup. In order to
set limits competitive to those currently in the literature [23, 24, 25, 26], particles need to be
collided with at least several TeV of energy, and in order to correctly distinguish W ′ events
from background events, the products of these collisions need to be carefully measured. The
ATLAS (A Toroidal Lhc ApparatuS) experiment meets these criteria, it is the largest collider
detector ever built and is capable of very precise measurements of the products of particle
collisions. The collisions it measures are produced by the Large Hadron Collider (LHC)
which is designed to collide protons with up to 14 TeV1 center-of-mass-energy and is located
at CERN near Geneva, Switzerland.

3.1
3.1.1

The Large Hadron Collider
The Accelerator Chain

The LHC is only the final accelerator in a chain designed to take ions from rest, accelerate
and collide them at up to 14 TeV center-of-mass-energy. This acceleration occurs in stages,
with protons being accelerated through a separate chain than other ions such as lead. The
entire CERN accelerator chain is shown in Figure 3.1. Since this analysis uses proton-proton
1 Units are expressed throughout this thesis using SI prefixes, so 1 TeV = 1012 eV, 1 GeV
= 109 eV, 1 MeV = 106 eV where 1 eV = 1.6 × 10−19 Joules.

19

collisions only their acceleration is detailed here. The proton source is a bottle of hydrogen
gas, which is stripped of its electrons and accelerated to a 50 MeV proton beam by the Linac2
linear accelerator [27]. This 50 MeV proton beam is then passed to the Proton Synchroton
Booster (PSB) which accelerates the beam to 1.4 GeV in four superimposed synchrotron rings
before injecting the bunches into the Proton Synchrotron (PS). By adjusting the timings of
the four superimposed rings of the PSB and varying which rings the PS is filled from, a
plethora of bunch patterns can be selected based on the current operating goals [28]. The
PS accelerates the proton beam from 1.4 GeV to 25 GeV in a 628 meter circumference
synchrotron. The PS also does the final bunch splitting, creating the bunch pattern which
will be kept through the remainder of the beam acceleration and collision [29]. After being
accelerated and bunched by the PS, the beams enter the Super Proton Synchrotron (SPS) for
final acceleration and tuning before injection into the LHC. The SPS is a synchrotron nearly
7 km in circumference which accelerates the proton beam up to 450 GeV before injecting it
into the LHC [30].
The final stage of the accelerator chain is the LHC itself. The LHC is a 27 km circumference synchrotron with 2 superimposed rings which reside in what was previously the Large
Electron-Positron collider (LEP) tunnel. It consists of 1104 superconducting dipole magnets
designed to reach a peak field of 8.33 T to bend the proton beams around the ring, and
384 quadrupole magnets per ring to control the focusing of the beams. Each ring has a
further 536 quadrupole, 1608 sextupole, 784 octupole, and 616 decapole magnets to control
and correct instabilities in the beams due to couplings during acceleration and collision.
Nominally the LHC is designed to collide proton bunches at ATLAS every 25 nanoseconds
with a center-of-mass-energy of 14 TeV, however it is still early in the LHC program and
these were not the conditions that the 2012 dataset was taken under. Due to difficulties
20

with the magnet fault protection system the collisions took place at 8 TeV center-of-massenergy, and because the accelerator and beams are being actively studied a variety of beam
configurations were used with bunches separated by 25 to 75 nanoseconds [31].

Figure 3.1: Diagram of the CERN accelerator chain [5].

3.2

The ATLAS detector

The ATLAS detector is one of two large general purpose experiments which study collisions
produced by the LHC. It is designed to perform a wide variety of searches and measurements
by collecting as much information as possible about the products of each collision. ATLAS
uses a multilayered design that has become standard for collider experiments and can be
seen in Figure 3.2. The innermost portion is called the inner detector which provides fine
21

granularity tracking of charged particles. Moving radially outwards from the interaction
point the next detectors are the calorimeters which measure the energy of the particles,
and the outermost portion of the detector consists of the muon system which detects and
tracks muons traveling through ATLAS. Each of these portions of ATLAS are made up of
sub-detector systems designed to work together with the other systems and provide more
information than any single technology detector [32].

Figure 3.2: Cutaway diagram of the ATLAS detector [6].

3.2.1

Detector geometry

Before detailing each detector system that makes up ATLAS it is useful to discuss the
coordinate system used in the experiment. The center of the detector is taken to be the origin
and the z-axis extends along the beam line with positive being counterclockwise around the
LHC when viewed from above. The x-axis points towards the center of the LHC and the y22

axis points vertically upwards. While this forms a complete basis to describe the detector and
it is sometimes used, it is not the most common coordinate system. Ignoring gravitational
effects all directions transverse to the beams are equivalent and can be described by an angle
φ taken to be zero along the x-axis and increasing right-handedly with respect to the z-axis.
The angle from the beam line is a common parameter for describing decays. However because
objects are produced with Lorentz boosts in the z direction ranging from 0 to nearly 1 it is
more useful to use a relativistic invariant to describe this angle. The equation for the Lorentz
invariant rapidity (y) of a particle in terms of the particle’s energy (E) and momentum along
the beam line (pz ) is:

1
y = ln
2

E + pz
E − pz

(3.1)

While useful, the rapidity of a particle is dependent on the particle’s mass and a different
rapidity coordinate system to describe the detector would be necessary for each mass. Almost
all particles produced by the LHC have m << E so we can calculate rapidity with m = 0
and it is approximately the rapidity for all particles produced by the LHC. This is called the
pseudorapidity (η), and is given in terms of the magnitude of a particle’s momentum (|p|)
and the particles momentum along the beam line(pz ) in Equation 3.2.

1
η = ln
2

|p| + pz
|p| − pz

(3.2)

Which can be rewritten using the angle from the z-axis (θ) as:

η = −ln tan

23

θ
2

(3.3)

Thus pseudorapidity is a purely geometric quantity, with η = 0 corresponding to the
transverse plane and |η| = ∞ corresponding to the beamline. In detector parlance regions
with small |η| (approximately |η| < 2.5) are called “central” and regions with larger |η| are
called “forward.” An illustration showing the ATLAS geometry is shown in Figure 3.3.

Figure 3.3: Illustration of the ATLAS coordinate system.

3.2.2

Magnet system

The ATLAS detector has three large superconducting magnet systems, the superconducting
solenoid, the barrel toroid, and the endcap toroids as shown in Figure 3.4. The purpose
of these magnets is to bend the path of charged particles as they propagate through the
detector. With careful tracking of a charged particle’s path through the magnetic field, it is
possible to determine the particle’s momentum [32].
The superconducting solenoid is a cylinder 5.3 m long and 2.63 m in diameter. It has 1173
turns of superconducting wire in a single layer along its length with an operating current of
7.6 kA. The inner volume contained by the solenoid has a central magnetic field of 2 T with

24

a peak field of 2.6 T at the superconducting wire. The solenoid is designed to be as thin as
possible in order to minimize the interaction between itself and particles from physics events.
The particles pass through the 19 cm thick (at most 0.66 radiation lengths) solenoid before
they enter the calorimeters [33].
The barrel toroid consists of 8 coils, each of which is a 25.3 m long and 5.35 m wide
“racetrack” design. These magnets run the length of ATLAS with their long dimension
running parallel to z and their short dimension running radially. They are spaced evenly
around the detector with their outer edge at a radius of 10.05 m. Each coil contains 120
turns of superconducting wire with an operating current of 20.5 kA which produces a peak
field of 3.9 T [33].
The two endcap toroids complete the ATLAS magnet system. Each endcap contains 8
coils of a racetrack design similar to the barrel toroid, however these coils are 4.5 m in the
radial direction and 5 m in the z direction. The endcap coils are offset from the barrel toroid
coils by 22.5◦ in φ so that they bisect the angle between adjacent barrel toroid coils. They
are aligned in z to share a common outer edge with the barrel toroid, and are placed radially
from 0.825 m to 5.35 m. With 116 turns per coil of superconducting wire carrying 20 kA,
the endcap toroids produce a peak field of 4.1 T [33].

3.2.3

Inner detector

The ATLAS inner detector is designed to provide excellent tracking information for charged
particles with |η| < 2.5 and is comprised of three concentric subsystems as shown in Figure 3.5. The silicon pixel detector is nearest the beamline and provides the most precise
position information with 97 million channels across three layers in the barrel region and
with 43 million channels across five disk layers at both ends. Moving radially outwards from
25

Figure 3.4: Illustration of the ATLAS magnet system, showing the barrel
solenoid, barrel toroid, and endcap toroid coils [7].

26

the beamline the next subsystem is the semiconductor tracker (SCT) which uses eight layers
of thin silicon microstrip sensors in the barrel and 44 sensor layers in each endcap, with
alternating layers at a 40 mrad angle to each other to allow full determination of position.
The final subsystem of the inner detector is the transition radiation tracker (TRT) which is
a straw tube system consisting of a barrel section containing axial straws and 18 radial straw
wheel segments in each endcap, designed so that most particles will transverse 36 detecting
straws [32].

Figure 3.5: Cutaway diagram of the ATLAS inner detector [8].

3.2.3.1

Pixel detector

The pixel detector has the highest granularity and offers the best position and tracking
information of charged particles in ATLAS. The system contains three barrel layers with

27

three transverse disk layers at each end. The barrel layers are all 801 mm long, with the
innermost layer located at a mean radius of 50.5 mm, the middle layer at 88.5 mm, and the
outermost layer at 122.5 mm. The disk layers are all identical annuli with an inner radius
of 89 mm and outer radius of 150 mm. These disks are placed at a mean |z| of 495 mm,
580 mm, and 650 mm. This gives the pixel detector a total detecting area of 1.7 m2 and
coverage of |η| ≤ 2.5 [32] [34].
The active medium in the detector is silicon sensor cells 50 µm × 400 µm in size. In the
barrel layers the long dimension is in the z direction and in the disk layers the long dimension
is radial. These sensor cells are bump bonded to readout chips with each chip reading an 18
× 160 cell array. The signal from each cell is amplified and compared to a programmable
threshold on each chip. If the signal exceeds the threshold the location is stored in a buffer
on the chip to be read out via optical link in the case of a level-1 trigger acceptance, as
detailed in Section 3.2.6 [32, 34].

3.2.3.2

Semiconductor tracker

While the pixel detector provides the highest resolution for tracking particles, the technology
is not cost-effective to use to cover the larger areas corresponding to larger radii. The next
subdetector is the semiconductor tracker (SCT) with four cylinders at radii of 299 mm, 371
mm, 443 mm, and 514 mm and nine disks at both endcaps with mean |z| of 853.8 mm 2720.2 mm. Each cylinder is 1492 mm long and contains two layers of silicon microstrip
sensors at a 40 mrad angle to each other. The microstrip sensors are each 63.6 mm wide
and 64 mm long rectangles divided into 768 microstrips each 16 µm wide [35]. The endcap
disks have a more complicated geometry with each disk containing 1-3 “rings” of sensor
modules depending on position. Each endcap module has two layers of microstrip silicon
28

sensors at a 40 mrad angle to each other similar to the barrel modules. However, unlike
the barrel modules, the endcap microstrip sensors are tapered to form trapezoidal segments
rather than rectangular. This tapering also causes variable microstrip widths of 16 µm - 20
µm. Each module for both the endcap and barrel regions has four silicon sensors (two per
layer) attached to central logic circuits which amplify the signals from each microstrip and
compare them to a programable threshold. Similar to the pixel detector, the channels with
signals exceeding the threshold are stored in a buffer to be read out if the event is accepted
by the level-1 trigger system [32, 35, 36].

3.2.3.3

Transition radiation tracker

The final subdetector of the inner detector is the TRT. While both the pixel detector and
SCT use variations of silicon detector technology, the TRT uses a modification of drift tube
technology to detect particles. The TRT is divided into one barrel and two endcap sections.
The barrel section consists of 52,544 straw tubes arranged in 73 layers parallel to the beam
axis. Each straw tube is a drift tube 1441 mm long and 4 mm in diameter and contains a
70% Xe, 27% CO2 and 3% O2 gas mixture. Each straw tube also contains a central 31 µm
diameter gold-plated W-Re wire which is held at a potential of -1.53 kV relative to the straw
tube wall [37].
The endcaps are each made up of 122,880 straw tubes arranged radially in 160 layers.
These straw tubes are identical to those used in the barrel except that they are each 370 mm
long. These endcap straw tubes are bundled into modules called wheels of 8 layers each, and
the wheels are distributed with 848 mm ≤ |z| ≤ 2710 mm to give nearly uniform coverage in
η [38]. Overall the barrel covers |η| < 1.0 and the endcaps cover 1.0 < |η| < 2.0 with most
particles traversing a total of 30 straw tubes.
29

As a charged particle traverses each straw it causes primary ionization within the gas,
which undergoes avalanche multiplication as it accumulates toward the wire giving an amplification factor of 2.5×104 with the operating gas mixture and voltage. The unique feature of
the ATLAS TRT is that surrounding each straw tube is a layer of transition radiation (TR)
material. The TR material is made up of many layers of polypropylene and polyethylene, and
is designed to maximize the production of transition radiation produced by charged particles
traversing the boundary between the two materials with different dielectric constants. The
transition radiation produced in the TR material is generally a soft x-ray photon which is
absorbed by the xenon in the straw tubes, ionizing the xenon and producing an energy cascade much larger than a typical ionizing particle does when traversing a straw tube. This is
particularly useful because electrons and charged pions are difficult to discriminate between,
however the likelihood of transition radiation occuring is proportional to γ = E/m and is
thus much more likely for electrons due to their lower mass. The TRT readout electronics
have two different thresholds: a low threshold designed to detect ionization tracks, and a
high threshold designed to detect transition radiation. By using the number of high threshold occurrences along a track, an additional rejection factor for pions of 50-100 is achieved
depending on the electron quality definition as described in Section 4.1 [32, 39].

3.2.4

Calorimeters

Having measured the positions of particles as precisely as possible in the inner detector, the
next detector systems particles will encounter are designed to measure their energy. The
electromagnetic (EM) calorimeter is nearest the beamline covering |η| < 3.2 and uses liquid
argon (LAr) technology with lead absorber plates in a distinctive accordion pattern. The
hadronic calorimeter resides around the EM calorimeter, using scintillating tiles with iron
30

absorbers in the barrel region of |η| < 1.7 and using LAr technology with copper and tungsten
absorbers in the 1.5 < |η| < 3.2 and 3.1 < |η| < 4.9 regions respectively. The layout of these
systems can be seen in Figure 3.6. It is important that the calorimeter system provides the
best containment of particles possible while maintaining good energy resolution so that the
total energy of events can be determined [32].

Figure 3.6: Cutaway diagram of the ATLAS calorimeter systems [9].

3.2.4.1

Electromagnetic Calorimeter

The ATLAS EM calorimeter is divided into a barrel section, a presampler, and two endcap
sections. The barrel calorimeter is made up of two half barrels which surround the superconducting solenoid and covers the range |η| < 1.475 with one half barrel covering η > 0
and the other half barrel covering η < 0. Each half barrel is a cylinder 3.2 m long and

31

has a 2.8 m inner radius and 4 m outer radius. There are 1024 accordion-shaped absorber
plates arrayed radially in each half barrel with the oscillations increasing in amplitude as
the radius increases to provide uniform density in φ. The absorbers are 1.53 mm thick lead
for |η| < 0.8 and 1.13 mm thick lead for 0.8 < |η| < 1.475 with 0.2 mm thick stainless
steel sheets glued to each side to provide structural support. Centered between consecutive
absorbers is a readout electrode held at 2 kV relative to the absorber with the 2 mm gap
between the electrode and absorber filled with liquid argon. Electrically charged incident
particles will shower via Bremsstrahlung in the absorber and this shower will exit the thin
absorber layer and enter the liquid argon. The shower ionizes the liquid argon and this
ionization is collected at the electrode where it is amplified and read out at both the inner
and outer edges of the calorimeter [40].
The presampler is a 22 mm thick detector covering the interior of the barrel calorimeter. It is similar to the barrel detector in that it uses liquid argon with 1.9-2.0 mm gaps
between electrodes, but has no absorbers. The purpose of the presampler is to measure
showers produced by interactions with the material between the interaction point and the
EM calorimeter and improve the energy resolution of the EM calorimeter [41].
The endcap sections are each a detector wheel 630 mm thick with a 330 mm inner radius
and 2098 mm outer radius which covers 1.375 < |η| < 3.2. Each wheel is further divided
into an inner wheel and an outer wheel by a 3 mm gap located at |η| = 2.5. The endcaps
have a design similar to the barrel calorimeter, with accordion-shaped absorber plates placed
radially and readout electrodes interleaved. Each outer wheel contains 768 absorbers of 1.7
mm thick lead, while the inner wheels each have 256 absorbers of 2.2 mm thick lead. The
endcap sections also have an 11 mm thick presampler of similar design to that used in the
barrel section [32, 42].
32

3.2.4.2

Hadronic Calorimeter

The hadronic calorimeter makes up the remainder of the ATLAS calorimeter system and is
comprised of four sub-systems; the tile barrel calorimeter and tile extended barrel calorimeter
are both based on using iron absorber plates with plastic scintillator tiles interspersed, while
the hadronic endcap calorimeter and the forward calorimeter are both based on LAr technology similar to the EM calorimeter. The tile barrel calorimeter has an inner radius of 1144
mm and an outer radius of 2115 mm with a length of 5640 mm. The tile barrel calorimeter
consists of 64 modules, each of which is a radial slice of the detector. Each module consists
of 64 steel plates that are each 5 mm thick and run the radial length of the module. Between
consecutive full length plates there are 11 alternating layers of scintillating plastic tiles and
steel spacing tiles which are 4 mm thick. These layers progressively increase in length from
the inner radius to the outer radius in order to provide high precision measurements while
maintaining the necessary depth of interaction lengths to contain very energetic jets. A
1.5 mm gap along both edges of each alternating layer contains a wavelength shifting fiber
which carries the scintillation light to photomultiplier tubes located along the outer radius of
the modules where the signals are amplified, digitized and processed by readout electronics.
The extended barrel calorimeter consists of two sections, one at each end of the tile barrel
calorimeter. Each of these sections is 2900 mm long but otherwise follows the same general
design as the tile barrel calorimeter with minor modifications to 12 of the 64 modules in
each extended barrel calorimeter to accommodate necessary structural supports for the LAr
cryostat. A gap region exists between the tile barrel calorimeter and the extended barrel
calorimeter on each side. This gap is necessary to provide services to the LAr calorimeters
and the inner detector, and while approximately 750 mm wide it is adjusted as needed to ac-

33

comodate these necessary services. The gap region contains the intermediate tile calorimeter
which consists of an irregular arrangement of absorber and scintilator tiles used to estimate
the energy lost in the dead material of the gap region. In total the tile barrel calorimeter
covers the |η| < 1.0 region while the extended barrel calorimeter covers 0.8 < |η| < 1.7 and
the intermediate tile calorimeter covers 0.8 < |η| < 1.0 [43].
The hadronic endcap calorimeter consists of two wheels located outside of the electromegnetic endcap calorimeters at both ends of the detector, for a total of four wheels. Each of
these wheels is further made up of 32 identical wedge-shaped modules. The front wheel on
each side starts at a |z| of 4,277 mm and is 816.5 mm in length. The rear wheels start at
a |z| of 5134 mm with a length of 961 mm, leaving a 2 mm gap between the wheels. Each
front wheel module contains 25 parallel copper plates which are each 25 mm thick and are
evenly spaced in z and arrayed transverse to the beamline. The rear wheel modules each
contain 17 parallel copper plates which are 50 mm thick and are also evenly spaced in z and
arrayed transverse to the beamline. This means that all of the plates are separated by 8.5
mm gaps which are filled with liquid argon. Three electrodes are evenly spaced in each gap
with the outer two electrodes held at 2000 V and the central electrode providing the signal
for amplification and processing. All of the plates have an outer radius of 2090 mm and the
first 9 plates of the front wheels have an inner radius of 372 mm while the remaining plates
all have an inner radius of 475 mm, providing coverage in the region 1.5 < |η| < 3.2 [44].
The final sub-system of the ATLAS hadronic calorimeter is the forward calorimeter.
This system covers the region 3.1 < |η| < 4.9 and resides entirely inside the 475 mm inner
radius of the hadronic endcap calorimeter. This region has an extremely harsh environment
with very high radiation densities so many design compromises were necessary to ensure the
forward calorimeter could survive and operate in this environment. The forward calorimeter
34

is composed of three sections at each end of the detector. These sections are cylindrical and
are arranged coaxially along the length of the beam pipe as seen in Figure 3.6. Each section
is made of an absorber material with holes along its length in a honeycomb pattern. Each
of these holes contains a thin-walled electrode tube and an electrode rod of slightly smaller
radius. The small gap between the electrode rod and tube is filled with liquid argon and
the electrode rod is held at 250 V relative to the electrode tube. In the section on either
side of the detector which is nearest the interaction point the absorber and the electrode
rod are both made of copper. In the remaining sections the absorber and electrode rods
are made of tungsten. These materials were chosen due to their densities as well as their
thermal properties, ability to be produced to the necessary specifications, and hadronic
shower sizes. In all of the modules the electrode tube is made of copper and the electrical
signal is read out from each absorber rod for amplification and processing. The liquid argon
gaps are smaller than is common in LAr detectors, being 269 µm, 376 µm, and 508 µm
in the three sections at each end of the detector and increasing with the distance from the
interaction point. This is necessary to prevent charge accumulation in the liquid argon which
would degrade performance and is caused by the high radiation density of the region, which
decreases with distance from the interaction point. The overall layout of the segments is
approximately projective from the interaction point, with the inner radius of the segments
increasing proportional to |z|, the electrode spacing increasing from 7.50 mm to 9.00 mm
across the three segments, and the number of electrodes decreasing from 12,260 tubes in
each module nearest the interaction point to 8,224 electrodes in each module furthest from
the interation point [45, 46].

35

3.2.5

Muon Spectrometer

The ATLAS Muon Spectrometer (MS) is the outermost of the ATLAS detector systems and
accounts for a majority of the detector’s volume. The purpose of this system is to detect
muons as they traverse the ATLAS detector and make precision position measurements at
three different detector layers to calculate the momentum of each muon based on the curvature of the muon’s trajectory as it travels through the ATLAS magnetic field. To accomplish
this goal the muon spectrometer has four subsystems which employ differing detector technologies as needed in the various regions of the ATLAS detector. Monitored Drift Tubes
(MDTs) and Cathode Strip Chambers (CSCs) provide high precision tracking information
over the large area of the muon spectrometer in three concentric layers, called stations.
The MDT system uses gas drift tube technology and covers the region |η| < 2.7, while the
CSC system uses multiwire proportional chambers with a cathode strip readout and covers
2.0 < |η| < 2.7. Both the MDT and CSC systems have long response times and are not capable of being used in the level-1 trigger system as described in Section 3.2.6, so two additional
muon spectrometer systems are employed for the initial detection of muons. The Resistive
Plate Chamber (RPC) covers the region |η| < 1.05 using resistive plate capacitors which
locally discharge when their internal gas is ionized. The Thin Gap Chamber (TGC) system
covers 1.05 < |η| < 2.4 using multiwire proportional chambers with a smaller geometry than
the CSC system [47]. The overall layout of these systems is shown in Figure 3.7.

3.2.5.1

Monitored drift tubes

The monitored drift tube (MDT) chambers provide the majority of the precision muon
tracking capability in ATLAS. MDT modules are arranged into barrel and end-cap regions,
with the barrel composed of three concentric cylinders with radii of 5, 7.5, and 10 m with
36

Figure 3.7: Cutaway diagram of the ATLAS muon spectrometer and
toroid magnet systems [10].

37

coverage of |η| < 1.0 and the endcap regions containing four disks each at |z| of 7, 10, 14,
22 m respectively and covering the range 1.0 < |η| < 2.7. Each chamber is composed of two
sets of drift tube multilayers on either side of a rigid support structure which is 150 mm
thick. The multilayers in MDT chambers in the stations nearest the interaction point have
four layers of drift tubes while all other MDT chamber multilayers have three layers of drift
tubes. Each drift tube is 30 mm in diameter and is filled with a 91% Ar, 4% N2 , 5% CH4
gas mixture. Each tube is read out from a central 50 µm tungsten wire. The wire is held at
3,270 V and gives a spacial resolution of 80 µm with a maximum drift time of approximately
500 ns. Because of this long drift time it is necessary to correlate the signals from the
MDT system with corresponding signals in the RPC and TGC systems which provide much
more prompt results in order to determine which bunch crossing the MDT signals originated
from [47].

3.2.5.2

Cathode strip chambers

The cathode strip chambers (CSCs) are multiwire proportional chambers used for precision
muon position measurements in the region of highest radiation density, 2.0 < |η| < 2.7 in
the station nearest to the interaction point. Similar to the MDT, the CSC consists of two
multilayers with each multilayer containing four monolayers. Each monolayer is a 5.08 mm
gas gap containing a 30% Ar, 50% CO2 and 20% CF4 gas mixture. In the center of each gas
gap is a plane of parallel anode wires. The anode wires are 30 µm diameter tungsten wires
separated by 2.54 mm and held at 2,600 V. The walls forming the gap are copper-clad and
etched to form thin cathode strips. The cathode strips on one of the walls run orthogonal to
the anode wires and provide the precision coordinate measurement, while the cathode strips
on the other wall are coarser and run parallel to the anode wires to provide the transverse
38

coordinate measurement. For the precision strips it is only necessary to read out every third
strip in order to exceed the resolution of the MDT by using charge interpolation between
the strips, and these read-out strips are separated by 5.08 mm. The final resolution in the
bending direction is 60 µm for a monolayer [47].

3.2.5.3

Resistive plate chambers

The resistive plate chambers (RPCs) are designed to complement the MDT system in the
barrel region (|η| < 1.05). Each RPC chamber is a simple design; two resistive plates form
a capacitor and are held at 8,900 V with a 2 mm gap filled with 97% C2 H2 F4 and 3%
C4 H10 . An incident muon will ionize the gas and cause a local discharge of the capacitor.
This discharge is read out via capacitative coupling by metal strips running in orthogonal
directions on both sides of the resistive capacitor. The RPC chambers are placed two thick at
each of three stations. The two middle stations are directly inside and outside of the middle
MDT barrel station, and the far station is directly inside of the outer MDT barrel station.
This system provides prompt muon detection with a delay of less than 10 ns and a timing
uncertainty of 1.5 ns. The signal position is known to within a resolution of 1 cm which is
used by the level-1 trigger system and provides a complementary position measurement to
the MDT [47].

3.2.5.4

Thin gap chambers

The thin gap chambers (TGCs) fill a role similar to the RPC, prompt detection of muons
for use in level-1 triggering and a complementary position measurement to the MDT, but
in the endcap region (1.05 < |η| < 2.4). The TGCs are based on multiwire proportional
chamber technology similar to the CSCs but with a smaller geometry and faster readout
39

time. Each TGC gas gap is 2.8 mm wide and is filled with a highly quenching 55% CO2
and 45% n − C5 H12 gas mixture. A central plane of 50 µm tungsten anode wires are spaced
1.8 mm apart and are held at 3,100 V. The signals from these wires are read out with 4 to
20 wires forming an individual readout channel depending on η. Signals are also read out
from etched copper strips on one of the walls of each gap to provide a measurement of the
azimuthal angle for each track. This configuration gives each gap a position resolution of
approximately 9 mm and a time response of 7 ns, which is sufficient for bunch identification
and use by the level-1 trigger system. TGC modules are made up of either gas gap doublets
or triplets with 20 mm of separation between consecutive gas gaps. The inner wheel at |z|
= 7 m of each endcap has a layer of doublet chambers and the middle layer wheel at |z| =
14 m has two layers of doublet chambers and a layer of triplet chambers, giving the total
system a depth of nine gaps [47].

3.2.6

Triggering and data acquisition

As described in Section 3.1, proton bunch crossings occur inside the ATLAS detector every
25 ns. With the size and complexity of the ATLAS detector (the average event is 1.3
Mbytes of data [32]) it is not possible to read out and store the detector response for every
bunch crossing, thus a trigger and data aquisition system (TDAQ) has been implemented
to identify and record the most interesting events. The trigger system is divided into three
levels, each of which takes as input the accepted events of the previous level and reduces the
rate of accepted events using increasingly complex algorithms. The level-1 (L1) trigger uses
information in localized regions of the calorimeter and muon systems to reduce the accepted
event rate from 40 MHz to 75 kHz. The level-2 (L2) trigger uses more precise information,
including tracking from the inner detector for the region of interest (RoI) that caused the
40

level-1 acceptance, to further reduce the accepted event rate from 75 kHz to 3.5 kHz. The
event filter (EF) is the final trigger level which uses the highest granularity information from
the entire detector to further reduce the accepted event rate from 3.5 kHz to the 200 Hz
which is saved for analysis.
The level-1 trigger has an event input rate of 40 MHz and a maximum event acceptance
rate of 75 kHz with a total latency of 2.0 µs. The 40 MHz input event rate means that no
single part of the trigger decision can take more than 25 ns, which is achieved by using a
highly parallelized hardware implementation. The electromagnetic liquid-argon calorimeter
and hadronic tile calorimeter systems as well as the RPCs and TGCs in the muon spectrometer each have their signals read out to the level-1 trigger system. The calorimeter signals
are processed by hardware located in the ATLAS counting room adjacent to the cavern
which houses ATLAS, while the muon system signals are processed by hardware located on
the ATLAS detector. The level-1 trigger processors each only process their local detector
area and operate at a lower granularity than the systems are ultimately capable of. The
processors look for energy clusters above a variety of set thresholds depending on the system
and region of the detector, with an above threshold area forming a region of interest (RoI).
The exception to the local scope of the level-1 trigger system is a special processor which
calculates the total transverse energy of each event, as well as the missing transverse energy2
of each event and compares them to a variety of thresholds. All of the processors send a list
of surpassed thresholds to the central trigger processor (CTP) which correlates and counts
2 To first approximation the net energy in the transverse plane for an event is expected

to be zero. A large difference from zero in the vector sum of the measured energies in the
transverse plane indicates the existence of an unmeasuerd particle, such as a neutrino or
dark matter particle, in the event. The negative of this measured excess energy is called the
miss ). A more detailed definition of the E miss is used in the
missing transverse energy (ET
T
final event analysis as described in Section 4.4

41

the multiplicity of surpassed thesholds and determines a level-1 trigger decision for each
event based on a programmable trigger menu [48].
The level-2 trigger has an event input rate of 75 kHz from the level-1 trigger and event
acceptance rate of 3.5 kHz with a total latency of 10 ms. Unlike the level-1 trigger, the level2 trigger system uses all of the ATLAS detector systems and is implemented in software.
For each event, the signals for all of the detector systems are read out in each of the RoIs
identified by the level-1 trigger system to a node in the level-2 server farm. Depending on
the exact level-1 trigger conditions for the event, a series of software algorithms are then
applied to the event in order to refine the measurements. A final level-2 decision is reached
based on the outcome of these algorithms [49].
The final level of the trigger system is the event filter. The event filter has an event
input rate of 3.5 kHz and a final event acceptance rate of 200 Hz with a latency of 1 s. This
trigger level is very similar to the level-2 trigger system however rather than only calculating
a trigger decision based on the RoIs, the event filter calculates a decision based on the entire
event. Each event accepted by the level-2 system has all of the detector systems read out to
a node in the event filter server farm. Based on the complete event information a lengthier
and more precise calibration is performed, and based on this more detailed information an
event filter decision is calculated. Events which are accepted by the event filter are read out
from ATLAS to be saved at a dedicated computing facility for storage and analysis [48].

42

Chapter 4
Object Definitions
In order to perform a search for W ′ → t¯b each event needs to be reconstructed from the raw
ATLAS data. These raw data are a collection of energy deposits in the calorimeters and
tracking hits from the inner detector and muon spectrometer which need to be refined into
a more useable form. The raw data are reconstructed into analysis objects which generally
correspond to the particles that passed through the detector. Different types of particles
will interact with the various detector systems in different ways, leaving distinct signatures
as illustrated in Figure 4.1.
• Electrons leave a track in the inner detector and an energy shower in the EM calorimeter.
• Muons leave a track in the inner detector and in the muon spectrometer with minimal
energy deposited in the calorimeters.
• Jets leave tracks in the inner detector and energy showers in the EM and hadronic
calorimeters. Jets are represented in Figure 4.1 by the proton and neutron.
• Photons do not leave a track in the inner detector and produce only an energy shower
in the EM calorimeter.
• Neutrinos do not interact in the detector but their presence can be inferred from an
imbalance in the total event momentum in the transverse plane.
43

Figure 4.1: Illustration of how different particles interact with the ATLAS detector systems [11].

44

For the W ′ → t¯b analysis the objects of interest are electrons, muons, jets, and the missing
miss ) corresponding to a neutrino. It is possible for “fake” objects to be
transverse energy (ET

created due to detector resolution effects or by other particles interacting with the detector
in rare or unexpected ways. For example, jets that deposit all of their energy in the EM
calorimeter before reaching the hadronic calorimeter would appear to be electrons while
electrons that do not lose all of their energy in the EM calorimeter and “punch-through” to
the hadronic calorimeter can appear to be jets. The object definitions are chosen to balance
the rejection of fakes with the acceptance of real objects.

4.1

Electron definition

Electrons are a key component of the W ′ → t¯b search and their reconstruction uses a complex
algorithm to identify them at high efficiency while keeping the fakes rate low. In order to
have access to higher efficiency or higher purity samples the electrons are reconstructed with
increasingly stringent requirements to form three qualities: loose, medium, and tight. The
requirements for the 3 electron qualities are summarized in Table 4.1. The reconstruction
starts by performing a sliding window search of the middle layer of the EM calorimeter,
where a 3 × 5 (η × φ) window of calorimeter cells (each 0.025 × 0.025 in η × φ) is moved
about the calorimeter to find the local maxima of energy enclosed. Maxima with an energy
above 2.5 GeV are called seed clusters. Seed clusters are then checked against the tracking
information, and clusters with a track within ∆η and ∆φ requirements determined by the
electron quality are considered electron candidates [2].
Electron candidates have their energy recomputed using a 3 × 7 (η × φ) window of
calorimeter cells with corrections applied based on the position and energy and are assigned

45

a four-momentum based on the tracking and corrected energy. A final set of cuts, shown
in Table 4.1, is applied to electron candidates based on their quality. For cuts in Table 4.1
without values specified, the cut values are optimized individually in 10 bins of η and 11
cluster E [2]. In addition to these
bins of cluster transverse energy (ET ), where ET = cosh(track
η)

requirements the tight electrons are required to pass an enhanced set of cuts. Electron
candidates are rejected if they are in the EM calorimeter crack region of 1.37 < |η| < 1.52
because the calorimeter performance is degraded. The ET is required to be greater than 25
GeV because this analysis is focused on high energy events. Electron candidates are rejected
if they are near a jet, defined as having ∆R < 0.4 where ∆R =

∆η 2 + ∆φ2 . The

final requirement for tight electrons is that they are isolated in the tracker and calorimeter
as defined by cutting on the parameters Ptcone30 and Etcone20 respectively, with both cuts
being energy dependent with 90% efficiency. Ptcone30 is the sum of the pT 1 of all tracks
in a cone with half opening angle of 0.3 minus the pT of the candidate’s track. Similarly,
Etcone20 is the sum of the ET in a cone with half opening angle of 0.2 minus the ET of the
cluster.

4.2

Muon definition

Muons are of approximately equal importance to the analysis as electrons, but thankfully
they are much easier to identify and reconstruct in ATLAS. As described in Section 3.2.5, the
muon spectrometer is the largest ATLAS sub-detector and is dedicated to identifying and
measuring muons. Muons are reconstructed with different qualities similar to how electrons
1p

T is the momentum in the plane transverse to the beam line. For electrons it is computed from the observed curvature in the particle’s track due to the solenoid’s magnetic
field.

46

Type

Cut Description
Loose electrons
Detector acceptance
• |η| < 2.47
Hadronic leakage
• Ratio of the ET in the first layer of the hadronic calorimeter
to the EM cluster ET (|η| < 0.8 and |η| > 1.37)
• Ratio of the ET in all layers of the hadronic calorimeter to
the EM cluster ET (0.8 < |η| < 1.37)
EM calorimeter
• Ratio of the 3 × 7 cell energy to the 7 × 7 cell energy
middle layer
• Shower width in η
Medium electrons (including Loose cuts)
EM calorimeter first
• Total shower width
layer
• Ratio of the difference in the largest and second largest
energy deposits to the sum of those energies
Track quality
• Number of hits in the pixel detector (≥ 1)
• Sum of hits in the pixel detector and SCT (≥ 7)
• Transverse impact parameter (< 5 mm)
Track matching
• ∆η between the track and cluster (< 0.01)
Tight electrons (including Medium cuts)
Track quality
• Number of hits in the first layer of the pixel detector (≥ 1)
• Transverse impact parameter cut (< 1 mm)
Track matching
Ratio of the cluster energy to the track momentum
• ∆φ between the track and cluster (< 0.02)
• ∆η between the track and cluster (< 0.005)
TRT
• Number of hits in the TRT
• Ratio of the number of high-threshold hits to the total number of hits
Photon conversion
• Matches to reconstructed photon conversions are rejected
Table 4.1: Definition of variables used for electron identification cuts. For cuts without
values specified, the cut values are optimized individually in 10 bins of η and 11 bins of
cluster ET [2].

47

Type
Muon energy
Detector acceptance
Track quality

Isolation

Cut Description
• pT > 25 GeV
• 0.1 < |η| < 2.5
• Number of pixel hits + number of crossed dead pixel cells
> 0
• Number of SCT hits + number of crossed dead SCT strips
≥ 5
• Number of crossed dead pixel cells + number of crossed
dead SCT strips < 3
• For 0.1 < |η| < 1.9: number of TRT hits + number of
RT outliers
TRT outliers > 5 and number ofnTTRT
< 0.9
• Distance along z from track to primary vertex < 2 mm
iniIso10 4 < 0.05
• Mmuon
pT
• Muon and all jets with pT > 25 GeV have ∆R > 0.4

Table 4.2: Definition of muidcombined muon reconstruction cuts.

are reconstructed. This analysis only uses muons reconstructed with the algorithm called
“muidcombined” so only that algorithm is described here. Muidcombined muon candidates
are formed by independently reconstructing a track in the muon spectrometer (MS) and
inner detector (ID), and if these tracks match within ∆R < 0.05 then a combined track is
reconstructed from both systems. Several cuts are detailed in Table 4.2 which ensure that
only well-defined tracks that lie in the most sensitive regions of the detector and that are
isolated from other activity are included in the analysis. Two new variables are introduced
in these cuts, nTRT is the sum of the number of TRT hits and the number of TRT outliers
while MiniIso10 4 is the sum of the pT of all objects inside a cone with half opening angle
of 0.1 minus the muon pT with a maximum of 40 GeV.

4.3

Jet definition

As described in Section 2.1.2.1, bare quarks and gluons undergo hadronization before they
can interact with the detector. This forms a multitude of tracks and calorimeter energy
48

deposits spread over an area, which is treated as a single object called a “jet”. Being of such
a composite nature, jets are complicated objects and there are many different ways to define
and reconstruct them.
This analysis uses the anti−kt algorithm [50] to define and reconstruct jets. The anti−kt
algorithm starts with a list of all objects, in this case the calorimeter cell energies. From
this list of objects, a list of all two-object and object-beam distances is computed where
the distance between two objects is defined in Equation 4.1, and the distance between each
object and the beam as defined in Equation 4.2. If the minimum distance is between two
objects then they are merged to form a new object, the original objects are removed from
the list and the new combined object is added to the list, then all distances are recalculated.
If the minimum distance is between an object and the beam then the object is classified as
a jet and removed from the list. This process is repeated recursively until the object list
is exhausted. In Equation 4.1 the parameter R is the characteristic size of the jet. Larger
values of R produce fewer, wider jets which are more likely to contain products from more
than one parton. Conversely smaller values of R produce more, smaller jets that may not
contain all of the products of individual partons. For this analysis R = 0.4, consistent with
other ATLAS top quark analyses.

−2
dij = min(p−2
T,i , pT,j )

2 + ∆φ2
∆ηij
ij

di = p−2
T

R2

(4.1)

(4.2)

After forming jets with the anti − kt algorithm a correction is applied to each jet based
on the jet’s position and pT to correct for the specific response of each region of the detector.
49

The corrected jets then have a series of quality cuts applied. Any jets with negative energy
are removed as these are unphysical. If any jets are within ∆R < 0.2 of an electron, the
jet nearest the electron is removed because it is likely a double counting of the electron as a
jet.

4.3.1

Jet b-tagging

Jets that originate from different types of particles often exhibit different kinematics, and
by analyzing the kinematics of a given jet it is possible to predict the flavor of the parton
that produced the jet. This process is generally referred to as “tagging,” with jets that pass
the tagging criteria called “tagged” and jets that fail the tagging criteria called “untagged.”
In this analysis “b-tagging” is employed to sort jets based on how likely they are to have
originated from bottom quarks called b-jets. Specifically, the MV1 [51] b-tagging algorithm
is used.
MV1 is a neural network analysis2 of ATLAS b-tagging algorithms SV0, IP3D+SV1,
and JetFitterCombNN. SV0, IP3D+SV1, and JetFitterCombNN all use the secondary vertex caused by the bottom quark’s relatively long hadronization time, as discussed in Section 2.1.2.1, to distinguish b-jets from other jets. An illustration of a secondary vertex is
shown in Figure 4.2. This secondary vertex can be parameterized by its transverse and
longitudinal impact parameters (d0 and z0 ) which are the distances between the secondary
and primary vertex in the radial or z projection respectively, or the decay length (L0 ) which
is the distance between the primary and secondary vertex. SV0, IP3D+SV1, and JetFitterCombNN use varying mixtures of these parameters as well as their significances, which are
2 Neural network analyses are multivariate classifier functions similar to the boosted de-

cision trees described in Chapter 7. They are described in greater detail in the reference
materials [52] [53].
50

defined as the ratio of the parameter and its uncertainty, to discriminate between b-jets and
all other jets [52]. MV1 uses the outputs of these other b-tagging algorithms to produce a
single weight that corresponds to how likely a jet is to have originated from a b quark. For
this analysis a tagging cut on each jet is placed at the 70% b-tagging efficiency value, which
means that 70% of bottom quark initiated jets are expected to pass the cut and be tagged.

Figure 4.2: Illustration of a displaced secondary vertex caused by a bjet [12].

It is possible for a jet that did not originate from a bottom quark to be b-tagged by the
MV1 algorithm, which is called a “fake tag.” This is primarily due to the finite resolution
of the inner detector causing tracks to seem to be displaced from the primary vertex and
forming a secondary vertex. The likelihood of a fake tag occuring is dependant on the pT
and η of the jet and has been estimated using a data control region as well as in simulated
event samples. The MV1 algorithm operating at 70% b-tagging efficiency has a likelihood of
fake tagging a jet which did not originate from a botom quark of approximately 1% to 3%
depending on the pT and η of the jet [54].
51

4.4

Missing transverse energy definition

With particles colliding nearly head-on along the beam line, the sum of the products’ momenta in the transverse plane should be approximately 0. However, neutrinos and some
theoretical exotic particles are not expected to interact with the ATLAS detector, causing
the measured final state to have an imbalance in the transverse momentum. The negative
miss ). Unfortunately
of this measured imbalance is called the missing transverse energy (ET
miss is sensitive to any mismeasurement in the detector as well as the possibility of being
ET

faked by an interacting particle missed by the detector when traveling through cracks or
dead regions. To correct for these different sources, individual calibrations are applied to the
miss from soft jets, hard jets, electrons, muons, and cell-out energy fragments before they
ET
miss according to Equation 4.3.
are used to compute the ET

miss,e

miss = E
ET
T

miss,µ

+ ET

miss,softjets

+ ET

miss,hardjets

+ ET

miss,cell−out

+ ET

(4.3)

Hard jets are jets as defined in Section 4.3 with pT > 20 GeV . Soft jets are jets as defined
in Section 4.3 with 20 GeV

> pT > 10 GeV . Electrons and muons are defined using

the definitions in Section 4.1 and Section 4.2 respectively. Cell-out energy fragments are any
energy in the calorimeter which is not included in the other objects [55].

52

Chapter 5
Event selection
Before the final analysis described in Chapter 7 can be performed, an event selection specific
to the signal kinematics is applied. This event selection is designed to remove background
events while having minimal impact on the signal and defines the control regions used to
perform the data-driven background estimates described in Section 6.4. Events are also
separated into different analysis channels based on the total number of jets and the number
of b-tagged jets in the event and these channels are individually optimized.

5.1

Composite objects

Chapter 4 details how basic analysis objects are reconstructed from the raw detector response, and these basic objects can be combined to reconstruct composite objects. The
Feynman diagram in Figure 5.1 shows the W ′ → t¯b signal process, with the inital state on
the left and the final state on the right, with several intermediate particles in between. These
intermediate particles are the W boson, top quark, and ultimately the W ′ boson and they are
what define this channel as unique from any other process with the same final state. These
intermediate particles give W ′ → t¯b events unique kinematics that distinguish W ′ → t¯b from
other processes.
miss , and two b-tagged
Based on Figure 5.1, the ideal analysis channel contains a lepton, ET

jets. In order to increase the signal acceptance, the common ways for signal events to fail

53

Figure 5.1: Illustration of the W ′ → t¯b process.
such a selection are considered. Additional jets can be present in an event due to hard gluon
radiation in the initial or final state, so the event selection is expanded to include events with
either 2 or 3 jets. Also, the cut on the MV1 b-tagging weight used to define a jet as b-tagged
is 70% efficient, thus one of the b-jets could easily fail to be tagged so the event selection is
again expanded to include events with either 1 or 2 of the jets being b-tagged. This leads to
the event selection detailed in Section 5.3, and four analysis channels are formed based on
the total number of jets and the number of b-tagged jets in an event: 2jets 1tag, 3jets 1tag,
2jets 2tag, and 3jets 2tag. The reconstruction of the top quark and W ′ boson in each event,
as detailed in Sections 5.1.2 and 5.1.3, differs depending on which analysis channel the event
is in.

54

5.1.1

W boson and neutrino reconstruction

The W boson in Figure 5.1 is the only intermediate particle decaying entirely to final state
objects and its reconstruction is as simple as adding the 4-momenta of the lepton and neutrino
together. The complication with this is that the 4-momentum of the neutrino is not known.
miss is calculated by assuming that the momentum is balanced
Section 4.4 describes how the ET

in the transverse plane. This technique cannot be used to determine pz (along the beamline)
for the neutrino because there is no reason the interacting partons should have the same
momentum along the beam line as each other. Instead the W boson and neutrino are
defined simultaneously by requiring that the lepton (only events containing a single lepton
will be selected as detailed in Section 5.3) and neutrino combine to form an on-shell W
boson with a mass of 80.4 GeV. Both the lepton and neutrino are assumed to be massless
miss . This gives rise to a quadratic
and the neutrino’s pT is assumed to be equivalent to the ET

equation for the neutrino’s pz , with solutions given by Equation 5.1.

pz,ν =

µpz,l
p2T,l

±

µ2 p2z,l
p4T,l

−

El2 p2T,ν − µ2
p2T,l

(5.1)

Where µ is defined as:

µ=

2
MW
+ cos(∆φl,ν )pT,ν pT,l
2

(5.2)

In Equations 5.1 and 5.2, pT,l and pT,ν are the transverse momenta of the lepton and
neutrino respectively,and pz,l and pz,ν are the z-momenta of the lepton and neutrino. ∆φl,ν
is the difference in φ between the lepton and neutrino. There are three possible categories
of solutions to Equation 5.1 based on the sign of the discriminant. If the discriminant is

55

positive then there are two real solutions to Equation 5.1 and the solution with the lowest
|pz | is chosen to define the neutrino, thus creating a less energetic final state. This has been
measured in other single top analyses to be the correct selection in approximately 70% of
single top events which produce two real solutions to Equation 5.1 [56]. If the discriminant is
0 then there is only one pz solution and the neutrino is uniquely defined. If the discriminant
is negative then the solutions for pz are imaginary. In this case the pT of the neutrino is
rescaled so that the discriminant becomes 0, then the neutrino pz is uniquely defined and
miss
the neutrino pT is taken to be the rescaled value. This procedure assumes that the ET
miss is still
is either mismeasured or not entirely caused by a single neutrino, but that the ET

the best estimate of the pT of the neutrino from the W boson’s decay.

5.1.2

Top quark reconstruction

While it is possible to reconstruct the top quark from its decay products (see Figure 5.1),
there is an ambiguity about which bottom quark jet in an event originated from the top
quark decay. The indeterminacy is resolved differently depending on which analysis channel
the event belongs to, as defined in Section 5.3.
• If the event contains only 1 b-tagged jet then the invariant mass of each individual jet
(not necessarily b-tagged) and the reconstructed W boson is calculated and the combination with a mass closest to the top quark mass of 172.5 GeV forms the reconstructed
top quark.
• If the event contains 2 b-tagged jets the mass of the W boson and each individual btagged jet is calculated, with the pair producing a mass closest to 172.5 GeV forming
the top quark.
56

5.1.3

W ′ reconstruction

Similar to how the top quark is reconstructed, the W ′ boson is reconstructed differently
depending on which analysis channel, as defined in Section 5.3, the event falls into.
• For events that contain 2 b-tagged jets the W ′ boson is reconstructed by combining
the reconstructed top quark with the b-tagged jet that was not used to reconstruct the
top quark.
• For events that contain 1 b-tagged jet the W ′ boson is reconstructed by combining the
reconstructed top quark with the highest pT jet not used to reconstruct the top quark,
requiring that the b-tagged jet is included in the W ′ reconstruction. This means that
for events where the top quark was reconstructed from a jet which was b-tagged, the
jet combined with the top quark to form the reconstructed W ′ boson is not b-tagged.
For events where the top quark was reconstructed from a jet which was not b-tagged,
the jet combined with the top quark to form the reconstructed W ′ boson must be
b-tagged. This ensures that the reconstructed W ′ boson will contain the b-tagged jet
in the event.

5.2

Data triggers

In order for an event to be recorded by the ATLAS detector and included in an analysis it
must pass the trigger selection described in Section 3.2.6. To search for W ′ → t¯b the ATLAS
single lepton triggers are used. The single electron triggers require that electrons either have
an ET > 24 GeV and pass medium isolation requirements for the hadronic leakage, shower
width in η, and track isolation as described in Section 4.1 or have an ET > 60 GeV without
57

any isolation requirement. The single muon triggers require that muons either have a pT
> 24 GeV and pass medium isolation requirements for the ID track isolation described in
Section 4.2 or have a pT > 36 GeV without any isolation requirement. The complete set
of requirements for electrons and muons detailed in Chapter 4 is applied offline, after the
data have been recorded. Events must also have been taken during an LHC stable beam
period and during a time when all of the ATLAS subsystems were properly operating. The
combination of these requirements corresponds to an integrated luminosity of 20.3 fb−1 .1

5.3

Cut flow

Before performing the multivariate analysis described in Chapter 7, it is useful to apply
a set of event selection cuts. These cuts are designed to remove background events with
large kinematical differences from the signal samples so that the multivariate analysis can be
more focused on discriminating among the hard to classify events. The event selection also
defines the separate analysis channels which will undergo individually optimized multivariate
analyses. The event selection cuts are as follows:
• Exactly 1 lepton (electron or muon).
• Lepton pT > 35 GeV.
miss > 35 GeV
• ET
miss ) > 60 GeV, where m (W ) is the W boson transverse mass as defined
• (mT (W ) + ET
T

in Equation 5.3.
1 In high energy physics the likelihood of a process occuring is usually expressed as a
cross-section σ in units of barns (b) where 1 b = 10−24 cm2 . The integrated luminosity L is
a measure of how much data has been collected and has units of inverse barns (b−1 ) so that

the expected number of events for a process is given by Nevents = σL.
58

• Exactly 2 or 3 jets with pT > 35 GeV.
• Exactly 1 or 2 of the jets must be b-tagged.
• W ′ boson mass (m(W ′ )) > 330 GeV.

mT (W ) =

2pT,l pT,ν (1 − cos(∆φl,ν ))

(5.3)

The total number of jets and the number of jets which are b-tagged defines four unique
analysis channels which are referred to by the number of jets and b-tagged jets. For example
the 2jets 1tag channel contains events with exactly 2 jets of which exactly 1 jet is b-tagged.
This produces four separate analysis channels: 2jets 1tag, 2jets 2tag, 3jets 1tag, and 3jets
2tag.
miss are chosen to match the decay
The cuts on the number of leptons, lepton pT , and ET
miss from the W
channel seen in Figure 5.1 where a single high pT lepton and large ET
miss ) is called the triangular cut and is
boson decay are expected. The cut on (mT (W ) + ET

commonly used in analyses to discriminate against the multijets background, which is defined
in Section 6.2.6. The cut on m(W ′ ) is chosen to define a control region used to perform a
data driven normalization of the W+jets background as described in Section 6.4.1. The cut
value of 330 GeV was chosen to maximize the size of the control region while keeping the
signal contamination to less than 5% for all of the signal samples.
Both of the 1tag channels have significantly larger backgrounds than the 2tag channels
so two additional cuts are applied to the 1tag channels only:
• ET of the leading jet (ET (jet1)) > 140 GeV.
• Transverse energy of the reconstructed top quark (ET (T op)) > 175 GeV.
59

These cuts are chosen by ranking a list of event kinematics variables by their discrimination
power after performing the initial event selection cuts. The discrimination power of each
variable is determined by mapping the signal efficiency (ǫS ) versus the background efficiency
(ǫB ) for successively higher cuts on the variable. The area between the curve this process
maps out and the line of ǫS = ǫB is defined to be the discrimination power of the variable,
as shown in Figure 5.2. For the two most discriminating variables, pT (jet1) and ET (T op),
the cut is chosen to be at least 95% efficient for all signal samples.

Figure 5.2: Illustration of the discrimination power of a variable, shown
as the shaded region.

60

Chapter 6
Event Simulation
In order to devise and optimize the analysis strategy both signal and background events
are modeled. Most of these events are simulated using Monte Carlo (MC) techniques where
each event is generated, showered and hadronized, run through a detector simulation, and
reconstructed using a variety of software packages. The exceptions to this are the W+jets
and multijet backgrounds which are modeled using, either partially or wholely, data driven
techniques as described in Sections 6.4.1 and 6.4.2 respectively.

6.1

Signal process

The signal process this analysis is searching for is W ′ → t¯b. The Feynman diagram for this
process is shown in Figure 5.1, and the Lagrangian density that describes the W ′ boson’s
production and decay are given in Section 2.13. WR′ and WL′ Monte Carlo samples are
′ or g ′ set equal to the Standard Model coupling
produced with one of the BSM couplings gR
L

g for W ± bosons and the other BSM coupling set to zero. Monte Carlo samples are generated
at each of 11 W ′ mass points ranging from 500 GeV to 3,000 GeV in 250 GeV steps for both
the WR′ and WL′ processes. This is necessary because the kinematics of the intermediate
and final state particles depends on the mass of the W ′ boson as well as the handedness of
its coupling. This mass range was chosen because the searches at the Tevatron excluded W ′
bosons with a mass less than 500 GeV and this analysis is not expected to be sensitive to

61

W ′ bosons with a mass above 3000 GeV.

6.2

Background processes

In order to properly estimate the background for the W ′ → t¯b analysis it is necessary to
consider all of the processes which are likely to mimic the W ′ → t¯b final state shown in
Figure 5.1 and which pass the event selection described in Section 5.3. The possibility that
leptons or jets can be missed or faked in an event is also taken into account and processes
are included that do not match the exact final state of W ′ → t¯b but may have events which
pass the event selection.

6.2.1

Single top processes

The Standard Model single top processes form a background to W ′ → t¯b, which is a BSM
single top process. Single top processes are defined as containing exactly one top quark, as
opposed to a top-antitop quark pair which is described in Section 6.2.2. There are three
Standard Model single top processes: the s-channel, t-channel, and Wt-channel. The Feynman diagrams for each of these processes are shown in Figure 6.1. The s-channel process is
identical to the W ′ → t¯b process this analysis is looking for, except that a W ± boson takes
the place of the W ′ boson. This means that events will have the same final particles for
both the s-channel and W ′ processes, but the particles in the W ′ events tend to be at higher
energies and have different angular relationships than particles in s-channel events due to the
more boosted nature of the intermediate states and the possibly different chirality of the W ′
and W ± . The t-channel has the same final state as W ′ → t¯b with an additional jet. If one
of these jets fails to be reconstructed then the t-channel has the same final state particles as

62

W ′ → t¯b, however the kinematics of the particles will be different because of the difference
in intermediate states. The Wt-channel can most easily mimic W ′ → t¯b if the additional
W ± boson decays hadronically. In this case the Wt-channel has an additional jet compared
to W ′ → t¯b, similar to the t-channel. In the case that the additional W ± boson decays
leptonically then there will be an additional lepton and a missing b-jet, and such events are
unlikely to pass the event selection. All of the single top processes pass the event selection
with a high efficiency relative to the other backgrounds, but because they are rarer processes
with smaller cross-sections (see Section 6.3) they are not the dominant backgrounds of the
analysis.

(a)

(b)

(c)
Figure 6.1: Illustration of the (a) s-channel single top process (b) t-channel single top process
(c) Wt-channel single top processes.

63

6.2.2

Top pair production

Top pair production (tt¯) is caused by the decay of a highly energetic particle (usually a gluon
at the LHC) into a top quark and an antitop quark. A Feynman diagram of the process is
shown in Figure 6.2. The resulting final state is the same as for W ′ → t¯b with the addition
of the products of an extra W ± boson decay. This produces an extra lepton and neutino
or an extra two jets, which if they are not reconstructed then the event may pass the event
selection. This gives tt¯ a selection efficiency lower only than single top, but because tt¯ has
a much larger cross-section (see Section 6.3) it is the dominant background for all of the
analysis channels except 2jets 1tag, where it is the second largest background.

Figure 6.2: Illustration of the tt¯ process.

6.2.3

W+jets

The W+jets process is the associated production of a W ± boson and jets. This includes a
variety of more specific sub-processes and these sub-processes can be categorized in many
64

different ways, leading to many different ways to split up W+jets. Because the event selection uses b-tagging it is important to model the response of the W+jets processes to
b-tagging as accurately as possible, and the W+jets sample is split into four processes: Wbb
is the associated production of a W ± boson and two bottom quarks, Wcc is the associated
production of a W ± boson and two charm quarks, Wc is the associated production of a
W ± boson and one charm quark and one light flavor (up, down, or strange) quark, and
Wlf is the associated production of a W ± boson and two light flavor quarks. An example
Feynman diagram of Wbb is is shown in Figure 6.3. The kinematics of these processes are
very different than for W ′ → t¯b and Wlf and Wc events have to fake at least one b-tagged
jet in order to pass the event selection, so the overall selection efficiency for these processes
is relatively low. However, these processes have a large cross-section (see Section 6.3) and
they form the second largest background for all of the analysis channels except 2jets 1tag,
where they are the largest background.

Figure 6.3: Illustration of the Wbb process.

65

6.2.4

Z+jets

The Z+jets process is the associated production of a Z boson and jets. It is similar to the
W+jets process described in Section 6.2.3, but with a Z boson in place of the W ± boson
which leads to an additional lepton in the final state. An example Feynman diagram of
Z+jets is shown in Figure 6.4. The Z+jets cross-section is smaller than the W+jets crosssection (see Section 6.3) and the additional lepton in Z+jets events must not be reconstructed
for the event to pass the event selection, so the Z+jets background is much smaller for the
analysis and it is not broken into sub-processes like W+jets.

Figure 6.4: Illustration of the Z+jets process.

6.2.5

Diboson processes

The diboson sub-processes are WW, ZZ, and WZ which are the associated production of
2 W ± bosons, 2 Z bosons, or one W ± boson and one Z boson respectively. An example
Feynman diagram for the WW process is shown in Figure 6.5. These processes are rare
66

with a small total cross-section (see Section 6.3) so they are treated as a single background
process. These processes do not involve a top quark decay and have significantly different
kinematics than W ′ → t¯b and they only rarely produce b-tagged jets, so diboson events are
unlikely to pass the event selection. Because of this the diboson background for all of the
analysis channels is small.

Figure 6.5: Illustration of a diboson process.

6.2.6

Multijets

The multijets background describes events in which the interaction produces two or more jets
but no heavy resonant states such as W ± or Z bosons or top quarks. These events represent
the vast majority of collisions that occur at the LHC, but which the trigger system and event
selection are designed to reject. Because only the rarest of such events will be included in
the analysis it is difficult to accurately simulate this background in Monte Carlo and a data
driven technique is used instead, which is described in Section 6.4.2. The resulting estimate

67

is that multijets is a small background for this analysis.

6.3

Monte Carlo simulation

The MC simulation of events is broken down into four stages. Event generation simulates the
initial physics event and its decay. Showering and hadronization simulate the formation of
jets from any bare quarks or gluons in the generated events. Detector simulation models the
interaction of the physics event with the ATLAS detector using a GEANT4 [57] simulation
of the ATLAS detector, resulting in a detector response for the event. The final step is event
reconstruction where the same algorithms used to analyze data events are applied to the
simulated detector responses to build analysis objects.
There are a plethora of software packages available to perform MC simulation of events,
and these packages make a variety of different assumptions and simplifications of the physics
they are simulating. This leads to the situation that different packages are able to more
accurately simulate different physics processes and careful consideration and investigation is
necessary to ensure that the simulations used in the analysis are as accurate as possible. For
example ALPGEN [58] calculates different jet multiplicities individually, which is important
when simulating W+jets events, but POWHEG [59] calculates the corrections from higher
order terms in such a way that it minimizes the occurance of negatively weighted events which
are unphysical. Since W ′ → t¯b is a single top process it was extremely useful to consult the
extensive work already done comparing the different MC generator and showering programs
for each process by the ATLAS single top group.
For all processes except W ′ the current ATLAS single top group recomendation has
been used. For the W ′ signal processes the MADGRAPH [60] generator has been used

68

due to its ease of implementation and handling of spin correlations of decays. The W ′
events were showered with Pythia [61] similar to most of the background signals. Table 6.2 shows which programs were chosen to simulate each sample’s generation and showering [60] [61] [59] [62] [58] [63]. With the exception of the data driven methods described
in Section 6.4, the background and signal samples are normalized to the expected number
of events from each process using their theoretical cross-sections (σ), the total luminosity
(L), and a k-factor (k) which estimates the higher order corrections to the cross-section.
Equation 6.1 gives the normalized number of events expected for each sample (N). The
cross-section and k-factor values for the signal and background samples are given in Table 6.1 and Table 6.2 respectively. The WL′ and WR′ samples have different cross-sections at
each mass point because the WL′ boson can decay leptonicaly while the WR′ boson cannot,
which decreases the cross-section of the W ′ → t¯b process by approximately 30% for a WL′
boson relative to that for a WR′ boson. The absence of a leptonic decay channel for a WR′
boson also means that the width of the WR′ boson is approximately 30% smaller than that
of a WL′ boson. Since the width of both the WL′ and WR′ bosons are proportional to the W ′
mass, the absolute value of the difference in the WL′ and WR′ boson widths is larger for larger
W ′ masses. This causes the cross-section for WL′ production to fall less rapidly as the W ′
mass increases than that of WR′ production, because the increased width of the WL′ makes
it more likely to be produced by partons with a lower fraction of the total momentum in a
collision, which are far more prevalent.

N = kσL

69

(6.1)

W ′ Mass [GeV]
500
750
1000
1250
1500
1750
2000
2250
2500
2750
3000

WL′ σ [pb]
12.333
2.7223
0.81915
0.28025
0.10618
0.043693
0.018551
0.0082073
0.0038171
0.0018512
0.00095811

WL′ k
1.3684
1.3144
1.2564
1.2405
1.2202
1.1893
1.1774
1.1638
1.1512
1.1529
1.1687

WR′ σ [pb]
17.510
3.7174
1.0652
0.37278
0.13932
0.055667
0.023718
0.010283
0.0046794
0.0021970
0.0011035

WR′ k
1.2906
1.2779
1.2796
1.2260
1.2183
1.2062
1.1740
1.1669
1.1485
1.1522
1.1592

Table 6.1: Cross-sections and k-factors for generated W ′ samples [3].

Process
single top s-channel
single top t-channel
single top Wt-channel
tt¯
W+lf
W+c
W+cc
W+bb
Z+jets
diboson

σ [pb]
1.6424
25.750
20.461
114.51
31994
1126.0
403.44
133.99
2804.4
17.075

k
Reference
1.1067
[64]
1.1042
[65]
1.0933
[66]
1.1992
[67]
1.133
[68]
1.52
[68]
1.133
[68]
1.133
[68]
1.229
[68]
1.7223
[69]

Generator Showering
POWHEG
Pythia
AcerMC
Pythia
POWHEG
Pythia
POWHEG
Pythia
ALPGEN
Pythia
ALPGEN
Pythia
ALPGEN
Pythia
ALPGEN
Pythia
ALPGEN HERWIG
HERWIG HERWIG

Table 6.2: Simulated background samples with associated cross-sections, k-factors, generating programs and showering programs.

70

6.4

Data driven estimates

While the above method works well to simulate many background processes, it is sometimes
useful to use control regions in the data to estimate some backgrounds. For the W+jets
process it is necessary to correct the overall normalization as well as the relative abundance of
the simulated samples based on the flavor of the associated jets as described in Section 6.4.1.
The multijets process has a very high rate of occurrence and a very low acceptance making it
very difficult to predict, so this analysis uses the matrix method to estimate this background
from data as described in Section 6.4.2.

6.4.1

W+jets normalization

The W+jets samples in this analysis are globally normalized using the charge asymmetry
method in the control region where m(W ′ ) < 330 GeV, as defined by the event selection in
Section 5.3. This region has a signal contamination < 5% for all W ′ mass points considered
in the analysis. This method normalizes the W+jets sample in each analysis channel using
the theoretical asymmetry ratio (rM C ) as defined in Equation 6.2 to account for the observed
asymmetry in data.

rM C =

NM C W + events
NM C W − events

(6.2)

The asymmetry ratio is used to derive the W+jets normalization because while the theoretically predicted total production rates of W + +jets and W − +jets do not agree with what
is seen in experimental data, the ratio of these rates (rM C ) does. The W+jets samples are
normalized so that the difference in the expected number of W+jets events in Monte Carlo
simulation with positive and negative leptons (NW + − NW − ) is the same as the observed
71

difference in the number of data events with positive and negative leptons (D+ − D− ). This
results in an overall normalization factor for each analysis channel which is measured in the
control region of that channel, as shown in Equation 6.6.

N W + − NW − = D + − D −

(6.3)

N + + NW −
(NW + − NW − ) W
= D+ − D−
NW + + NW −

(6.4)

(NW + + NW − ) =

NW + + NW − +
(D − D− )
N W + − NW −

r
+1 +
NW + + NW − = M C
(D − D− )
rM C − 1

(6.5)

(6.6)

While this gives an overall normalization for the W+jets sample in each analysis channel,
Monte Carlo predictions of the relative fraction of W+jets events with different jet flavors
disagree with what is observed in data by factors of up to 2. To correct for this, the relative
normalization of the Wbb, Wcc, Wc, and Wlf samples is determined by fitting to the data.
There are a variety of ways to perform this fit, and for this analysis the charge asymmetry
analytic method is used. In this method each of the W+jets samples has an associated
normalization factor F such that:

Fbb + Fcc + Fc + Flf = 1

(6.7)

These relative normalization factors are determined by using the data normalized total num-

72

ber of events in the control region that contain a W+ (W− ) boson and pass the b-tagging
tagged

requirements N +
W

tagged

(N −
W

) and the data normalized number of events in the control repre−tagged

gion that contain a W+ (W− ) boson with the b-tagging requirements removed N +
W
pre−tagged

(N −
W

) as shown in Equations 6.8 and 6.9.

pre−tagged

(Fbb Pbb + Fcc Pcc + Fc Pc + Flf Plf )

(6.8)

pre−tagged

(Fbb Pbb + Fcc Pcc + Fc Pc + Flf Plf )

(6.9)

tagged

=N +
W

tagged

=N −
W

N +
W

N −
W

where P is the tagging fraction of each sample, measured for each of the Monte Carlo samples
according to:

P =

N tagged
N pre−tagged

(6.10)

This leaves three equations (6.7,6.8,6.9) and four variables to be fit (Fbb ,Fcc ,Fc ,Flf ). In order
to solve this system of equations the relative normalization of Wbb and Wcc are assumed
to be the same, giving Equation 6.11

Fbb = Fcc

(6.11)

The relative W+jets normalization is not expected to change between the analysis channels so it is determined for all of the analysis channels using the 2jets 2tag channel because
this results in the smallest uncertainty in the normalization factors.
The total normalization factors for each sample are the product of the overall normalization for each analysis channel and the relative normalization of the W+jets flavors and they
are given in Table 6.3.
73

Process
W+lf
W+c
W+cc
W+bb

2jets 1tag
0.941462
0.801521
1.39795
1.39795

2jets 2tag
1.31867
1.12266
1.95806
1.95806

3jets 1tag
0.883688
0.752335
1.31217
1.31217

3jets 2tag
1.96718
1.67477
2.92102
2.92102

Table 6.3: W+jets normalization factors.

6.4.2

Multijets estimate

The contribution of the multijet process to this analysis is estimated using the matrix
method [70]. The matrix method uses data events which have passed the event selection in
Chapter 5 except with a loose lepton which has relaxed requirements compared to the tight
lepton required for the signal region. Both the loose and tight lepton categories are defined
in Chapter 4. For both electrons and muons

loose + N loose
N loose = Nreal
f ake

tight

tight

N tight = Nreal + Nf ake

(6.12)

(6.13)

where N is the number of data events containing a lepton of the indicated type. Equation 6.13
can be rewritten using the conversion efficiencies for loose real or fake leptons to tight real
tight
tight
N
f
ake
and ǫf ake = loose
as:
or fake leptons ǫreal = real
N loose
N
real
f ake
N

loose + ǫ
loose
N tight = ǫreal Nreal
f ake Nf ake

(6.14)

ǫreal is estimated using the tag and probe method1 on Z → ll events, while ǫf ake is estimated
1 The tag and probe method involves requiring that events contain one tight lepton (the

74

using a multijets enhanced data sample where the lepton isolation criteria have been removed
and subtracting the expected contribution of real leptons in the sample based on Monte Carlo
simulations. Equations 6.12 and 6.14 can be combined into a single matrix equation given
in Equation 6.15.




N loose
N tight





 
=

1

1

ǫreal ǫf ake





loose
  Nreal 



Nfloose
ake



(6.15)

With the total number of events with loose and tight leptons known from the dataset,
Equation 6.15 can be inverted and combined with the definition of the fakes conversion eftight

ficiency ǫf ake to solve for Nf ake as shown in Equation 6.16. The conversion efficiencies are
dependent on many parameters including the lepton type (electron or muon), the pseudorapidity of the lepton, ET of the lepton, pT of the jets, and the relative positions of the lepton,
miss in each event. Averaging over this dependence, the conversion efficiencies are
jets, and ET
tight

approximately ǫreal = 0.75 and ǫf ake = 0.15 [70]. Nf ake is the multijets estimate for the
analysis.

tight

Nf ake =

6.5

ǫf ake
(ǫ
N loose − N tight )
ǫreal − ǫf ake real

(6.16)

Event yields

Event yields are the expected number of events which pass the event selection for each
process. This gives the importance of each background if a simple counting experiment
were performed, and gives the expected size of a signal. The final event yields are shown in
tag) and one loose lepton (the probe), then measuring how often the probe passes the tight
requirements
75

Table 6.4 broken into the four analysis channels. The number of data events which pass the
event selection for each channel are also included for comparison to the expected background
and signal yields.
In all four analysis channels the dominant backgrounds are the tt¯ and W+jets processes,
with tt¯ being the largest background in all of the analysis channels except 2jets 1tag. The
1tag analysis channels contain approximately double the number of signal events of the 2tag
channels for high mass W ′ samples because the jets produced by the W ′ boson decay have
larger momenta and the MV1 b-tagging algorithm has lower efficiency for higher pT jets.
After the event selection cuts, the expected number of signal events is much smaller than the
expected number of background events for all but the lowest W ′ masses in the analysis. For
example a 1750 GeV WR′ is expected to produce appoximately 100 events in the analysis with
a background of approximately 70,000 events. It is not possible to observe such a signal by
simply looking for an excess in the event count so more sophisticated analysis and statistical
techniques are used to increase the sensitivity of the analysis as described in Chapter 7 and
Chapter 8 respectively.

76

Sample
WR′ 500
WR′ 750
WR′ 1000
WR′ 1250
WR′ 1500
WR′ 1750
WR′ 2000
WR′ 2250
WR′ 2500
WR′ 2750
WR′ 3000
WL′ 500
WL′ 750
WL′ 1000
WL′ 1250
WL′ 1500
WL′ 1750
WL′ 2000
WL′ 2250
WL′ 2500
WL′ 2750
WL′ 3000
single top s-channel
single top t-channel
single top Wt-channel
tt¯
W+lf
W+c
W+cc
W+bb
Z+jets
diboson
multijets
total background
data

2jets 1tag
11908.99
3896.54
1088.92
331.79
100.05
31.56
10.95
4.02
1.68
0.79
0.41
5924.22
2119.14
631.77
215.28
70.61
24.56
9.20
3.59
1.55
0.68
0.38
123.09
1729.45
585.03
4261.02
2727.00
4545.80
3161.70
3039.62
310.38
202.15
986.54
21671.79
19362.00

3jets 1tag
5323.81
2648.34
917.25
306.02
106.23
35.94
12.95
4.97
1.91
0.84
0.41
2769.90
1408.07
536.05
201.18
73.48
28.93
10.59
4.31
1.78
0.77
0.39
66.60
960.21
924.49
8756.59
1165.34
1843.93
2312.68
1941.79
342.36
112.55
453.47
18880.02
16919.00

2jets 2tag
12949.42
2694.86
628.33
158.79
42.93
12.45
4.28
1.60
0.71
0.39
0.22
7114.70
1578.82
400.88
109.11
31.47
10.48
3.44
1.41
0.62
0.32
0.20
193.12
730.92
282.40
5335.20
104.86
416.31
460.47
1662.80
12.69
91.52
377.48
9667.75
9839.00

3jets 2tag
8316.48
2514.05
697.37
195.92
58.95
18.65
6.47
2.16
0.91
0.45
0.25
4699.89
1418.04
443.46
135.53
43.29
14.21
4.92
1.99
0.80
0.41
0.22
107.91
951.84
750.19
14774.80
173.67
389.15
637.16
2324.34
50.07
59.79
396.18
20615.10
20388.00

Table 6.4: Event yields for signal samples, background samples, and data by analysis channel.

77

Chapter 7
Multivariate analysis
The event selection described in Chapter 5 does not optimize the statistical significance
of a potential signal in each of the analysis channels. The method used to optimize the
significance in this analysis is a boosted decision tree (BDT) multivariate analysis (MVA).
The details of the BDT algorithm are discussed in Section 7.1, but the basic premise is that
events are sorted into background-like and signal-like bins using a series of cuts called a
decision tree. This decision tree optimizes the separation of signal and background events by
collecting all of the events into a decision node and then placing a cut on a single variable,
with both output possibilities forming a new decision node. A new optimal cut is placed
on each decision node iteratively until the specified end conditions are met. Misclassified
events, which are signal events in a background-like bin or background events in a signal-like
bin, have their relative weight “boosted” and a new decision tree is trained. This training
and boosting process is iterated many times and all of the decision trees combined form the
BDT. This boosting serves to increase the importance of hard to classify events by increasing
their weight in the optimization and make them more likely to be correctly categorized in
subsequent decision trees. A final decision weight is assigned to each event based on the
aggregate purity of the bins it was sorted into in each decision tree, which represents how
signal-like or background-like each event is. This iterative process allows for a more nuanced
handling of variable correlations than a simple cuts based analysis which is effectively a single
decision tree. The BDT also has the convenient property that it condenses the discrimination
78

power from many variables into a single highly discriminating weight which is more easily
displayed for the human analyst.

7.1

Boosted decision trees

The boosted decision tree (BDT) algorithm [71] is designed to optimize the discrimination
between signal and background events. The implementation of the BDT algorithm used in
this analysis is done by the Toolkit for Multivariate Data Analysis with ROOT (TMVA) [53]
and while the principles of the analysis are general to all BDTs, the details are specific to
this software package. Each of the analysis channels (2jets 1tag, 2jets 2tag, 3jets 1tag, and
3jets 2tag) has an independent BDT trained with the variables for each channel described
in Section 7.1.2, and the BDT settings for each channel are described in Section 7.1.3.
The BDT training begins with a sample of Monte Carlo signal and background events,
with each event having a weight equal to its contribution to the overall normalization1 and
a set of kinematic variables defined to be used in the BDT. An optimal cut on one of these
variables is determined by maximizing ∆G as defined in Equation 7.1 for all possible cuts
on these variables where Gpre−cut is calculated for all of the events and Gpassed cut and
Gf ailed cut are calculated for the events which pass and fail the cut respectively.

∆G = Gpre−cut − Gpassed cut − Gf ailed cut

(7.1)

Where G is called the Gini index and is given in Equation 7.2:
1 Because of how they are produced, a single Monte Carlo event does not represent a single

data event but could represent more or less than one data event depending on the likelihood
of the MC event occuring.

79

G=
events

W P (1 − P )

(7.2)

In Equation 7.2 W is the event weight, and P is the purity defined as the fraction of events
belonging to the signal. With the optimal separation cut defined all of the events are sorted
into two bins: events that pass the cut, and events that fail the cut. The process of finding
the optimal cut and sorting the events based on that cut forms a decision node.
Each of the two output bins from the first decision node then form their own decision
nodes to be applied to only those events. A new optimal cut is determined for each of these
nodes and the events are sorted into four bins (two for each decision node). This process is
continued iteratively, further subdividing the sample into more bins at each iteration, until
a termination condition is met. There are two termination conditions: a minimum number
of events is required to be in a bin in order for that bin to become a new decision node, and
there is a maximum depth in the decision tree for a node to become a decision node. The
initial decision node is at a depth of 1, the bins output from that decision node are at a
depth of 2, and so on. Any bins that do not form decision nodes instead form termination
nodes and are defined as signal or background nodes based on the majority of events in that
termination node. The decision and termination nodes together form a decision tree, an
example of which is given in Figure 7.1.
While a single decision tree can provide some discrimination power between signal and
background events, the discrimination power can generally be increased by training multiple
decision trees and increasing the weight of events misclassified as either signal or background
by each tree. A misclassified event is a signal event that is sorted into a background termination node or a background event that is sorted into a signal termination node, and this

80

Figure 7.1: An example decision tree which sorts input events into signal and background bins. The variables used in the decision tree are
described in Section 7.1.2

81

analysis uses the Adaboost algorithm [53] to increase the weight of such events when training
subsequent decision trees. The new weight of a misclassified event is given by wboosted in
Equation 7.3, where β is the boosting strength and is one of the parameters optimized in
Section 7.1.3 while wpre−boosted is the event weight before being boosted.

wboosted =

1 − err β
wpre−boosted
err

(7.3)

where err is defined as:

err =

sum of misclassif ied event weights
sum of all event weights

(7.4)

After the misclassified events have their weights boosted, all events have their weights scaled
to keep the total sum of event weights normalized and a new decision tree can be trained.
The total number of decision trees trained is one of the parameters optimized in Section 7.1.3.
With each event being sorted by multiple decision trees, an aggregate BDT decision
weight (yBoost ) is formed by taking a weighted average of the individual decision tree responses as given by Equation 7.5.
NT rees

yBoost =

ln

i=1
NT rees
i=1

ln

1−erri
erri

hi
(7.5)

1−erri
erri

For Equation 7.5, NT rees is the total number of decision trees trained and erri is given by
Equation 7.4 for each tree. The result of each tree for the event is represented by hi , which
is +1 (−1) if the event was sorted into a signal (background) termination node by that tree.
Thus the final BDT decision weight for each event is a value between −1 and +1, where −1

82

represents an ideal background event and +1 represents an ideal signal event.

7.1.1

Overtraining

If the BDT has too many degrees of freedom compared to the statistical size of the generated
samples during training then the BDT begins to optimize the decision trees based on the
specific variable values of individual events rather than on the overall distributions. This is
called overtraining and it causes the BDT output to be drastically different if the trained
BDT is applied to a new sample simulated under identical conditions. In order to check
if a BDT is overtrained all of the event samples are split in half before training. One half,
called the training sample, is used to train the BDT and is run through the resulting BDT to
compute the BDT decision weight distribution. The remaining half of the events, called the
testing sample, is run through the already trained BDT and produces an independant BDT
decision weight distribution. The training and testing distributions for both the signal and
background samples should be similar if the BDT is not overtrained, and they are compared
using a Kolmgorov-Smirnov (KS) test [72]. The KS test provides a probability that two
samples come from the same distribution, and if the KS result is < 0.5 for either the signal
or background distributions then the BDT is determined to be overtrained.

7.1.2

Variable selection

The variables that are used in the BDT to separate the signal and background events have a
large impact on the separation power of the BDT. Variables should be chosen that can discriminate between signal and background events, where the discrimination power is defined
the same as in Section 5.3. Because the correlations between several different variables are

83

not always obvious for all of the different processes included in the signal and background
samples, the analysis begins with a large list of variables in an attempt to be as comprehensive as possible. For each sample the analysis objects, as defined in Chapter 4, are the
lepton (lep), reconstructed neutrino (neutrino), reconstructed W boson (W), reconstructed
top quark (top), reconstructed W ′ boson (W ′ ), the leading jet (jet1), the sub-leading jet
(jet2), the leading b-tagged jet (bjet1), and in the appropriate analysis channels the third
jet (jet3) and sub-leading b-tagged jet (bjet2). For each of these objects the energy, mass,
pseudorapidity (η), azimuthal angle (φ), transverse momentum (pT ), and transverse energy
(ET ) are considered as possible variables for inclusion in the BDT. In addition a set of two
body variables is included in this list which consists of the invariant mass, difference in
pseudorapidity (∆η), difference in azimuthal angle (∆φ), and opening angle (∆R) for every
miss , the
pair of analysis objects. A final set of global event variables consisting of the ET
miss (φ(E miss )), the total sum of the transverse energy in the event
azimuthal angle of the ET
T

(Sum ET ), the total sum of the transverse energy of only the analysis object in the event
(HT ), the aplanarity of the event (a measure of how much of the energy in the event resides
in one plane), and the sphericity of the event (a measure of how evenly the energy of an
event is distributed around the interaction point) are also included. This forms an initial list
of approximately 200 variables depending on the analysis channel, however past experience
in single top and W ′ → t¯b analyses at ATLAS suggests that fewer than 20 variables are
necessary for a BDT to achieve good separation of single top signal and background events.
In order to remove unnecessary variables, variables with a discrimination power2 < 20% are
removed from the list for each channel which leaves approximately 50 variables per channel.
It is important that these variables are well modeled so the background distributions are
2 Discrimination power is defined in Section 5.3.

84

2jets 1tag
pT (top)
mass(W ′ )
E(W ′ )
mass(jet1,jet2)
∆η(lepton,W)
Sum ET
mass(jet1,W)
∆R(lepton,top)
mass(lepton,jet1)
∆η(neutrino,top)
ET (lepton)
pT (W)
HT
miss
ET
∆η(neutrino,jet2)
∆R(jet2,W)
∆R(lepton,jet2)
mass(jet1,top)

3jets 1tag
mass(jet1,W)
HT
mass(W ′ )
mass(jet1,jet2)
miss
ET
∆R(neutrino,top)
E(lepton)
pT (top)
pT (W)
E(W ′ )
∆η(lepton,W)
mass(lepton,jet1)
∆R(lepton,top)
mass(jet1,top)
∆η(neutrino,W)

2jets 2tag
mass(W ′ )
∆R(lepton,top)
aplanarity
HT
pT (bjet1)
pT (bjet2)
mass(bjet1,bjet2)
ET (W)
∆R(lepton,bjet2)
∆R(bjet1,bjet2)
∆R(lepton,W)
pT (top)
sphericity
pT (lepton)

3jets 2tag
mass(W ′ )
sphericity
pT (top)
∆η(lepton,W)
ET (bjet1)
∆R(bjet1,top)
∆R(lepton,bjet1)
∆φ(W,top)
∆η(lepton,top)
mass(lepton,bjet1)
∆R(bjet1,W)
pT (lepton)
aplanarity

Table 7.1: Boosted decision tree variable lists for the four analysis channels. Variables are
ranked by importance.

compared to data and any variables with a KS value < 0.5 are removed. A BDT is then
trained using the remaining variables and the variables are ranked by importance, which
is how frequently they are used in the individual decision trees. The five least important
variables are removed from the variable list and the process is repeated until only 20 variables remain. At this point the training and variable removal iterations continue but only a
single variable is removed in each iteration until the removal of a variable would cause the
separation power of the BDT to degrade to < 90% of the maximum value obtained during
these iterations. Table 7.1 lists the final variables for each of the analysis channels in order
of importance, and Figures 7.2-7.14 show the data-Monte Carlo comparison plots for these
variables.

85

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(e)

(f)

Figure 7.2: Comparison of data to Monte Carlo prediction of the 1st-5th variables by importance in the 2jet 1tag BDT.

86

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(e)

(f)

Figure 7.3: Comparison of data to Monte Carlo prediction of the 6th-10th variables by
importance in the 2jet 1tag BDT.

87

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40000

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(d)

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(c)
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100

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6
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2

(e)

(f)

Figure 7.4: Comparison of data to Monte Carlo prediction of the 11th-15th variables by
importance in the 2jet 1tag BDT.

88

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2

(a)

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1

(c)

(d)

Figure 7.5: Comparison of data to Monte Carlo prediction of the 16th-18th variables by
importance in the 2jet 1tag BDT.

89

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T

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Figure 7.6: Comparison of data to Monte Carlo prediction of the 1st-5th variables by importance in the 3jet 1tag BDT.

90

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Figure 7.7: Comparison of data to Monte Carlo prediction of the 6th-10th variables by
importance in the 3jet 1tag BDT.

91

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Figure 7.8: Comparison of data to Monte Carlo prediction of the 11th-15th variables by
importance in the 3jet 1tag BDT.

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Figure 7.9: Comparison of data to Monte Carlo prediction of the 1st-5th variables by importance in the 2jet 2tag BDT.

93

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Figure 7.10: Comparison of data to Monte Carlo prediction of the 6th-10th variables by
importance in the 2jet 2tag BDT.

94

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Figure 7.11: Comparison of data to Monte Carlo prediction of the 11th-14th variables by
importance in the 2jet 2tag BDT.

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Figure 7.12: Comparison of data to Monte Carlo prediction of the 1st-5th variables by
importance in the 3jet 2tag BDT.

96

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Figure 7.13: Comparison of data to Monte Carlo prediction of the 6th-10th variables by
importance in the 3jet 2tag BDT.

97

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Figure 7.14: Comparison of data to Monte Carlo prediction of the 11th-13th variables by
importance in the 3jet 2tag BDT.

98

7.1.3

BDT parameter optimization

The BDT parameters are the specific values used in the BDT training algorithm described in
Section 7.1. These values have a strong impact on the resulting BDT and have a complicated
relationship with the variables chosen for the BDT as well as any possible overtraining of
the BDT. Because of this it is important to optimize these values but this is not a straight
forward process. For each value of these parameters to be tested a new BDT must be trained
which makes it computationally impractical to optimize the parameters simultaneously with
the variable selection. Instead an iterative approach is taken, with each parameter value
being optimized individually by scanning a range of values in discreet steps after the variable
selection has been performed. An optimal parameter is chosen based on the BDT separation
power after any overtrained trees have been removed. The variable selection and subsequent
BDT parameter optimization are then redone if any parameter value changed by more than
one step during the optimization. While it is possible for this method to get stuck in local
optimizations, it has produced stable results with the available computational resources. A
description of the parameters is provided below in the order they are optimized along with
the initial training value of each parameter and the range and step size of the optimization
scan. The final value of these parameters is provided in Table 7.2.
• nTrees is the number of individual decision trees to be trained by the BDT. This
affects both the performance and possible overtraining of the BDT by allowing more
degrees of freedom in the optimization. The range 60-200 is scanned with a step size
of 20 and a default value of 120.
• nEventsMin is the minimum number of events for a decision node to be formed
instead of a termination node. This affects the BDT performance and any possible
99

Parameter
nTrees
nEventsMin
maxDepth
AdaBoost
nCuts

2jets 1tag
160
200
10
0.7
10

3jets 1tag
100
200
10
0.7
12

2jets 2tag
100
180
8
0.5
14

3jets 2tag
100
200
8
0.7
12

Table 7.2: Optimized Boosted decision tree parameters for each of the four analysis channels.

overtraining by limiting how complex the individual decision trees can become. The
range 100-300 is scanned with a step size of 20 and a default value of 200.
• maxDepth is the maximum decision depth of each decision tree in the BDT. Similar
to nEventsMin, this affects the BDT performance and any possible overtraining by
limiting how complex the individual decision trees can become. The range 6-12 is
scanned with a step size of 1 and a default value of 9.
• AdaBoost is the boosting strength β in Equation 7.3. This primarily affects the
performance of the BDT by adjusting how important the hard to classify events are.
The range 0.4-1.0 is scanned with a step size of 0.1 and a default value of 0.7.
• nCuts is how many possible cuts are tested for each variable when determining the
optimal cut for a decision node. This affects the BDT performance by ensuring the
decision nodes have adequate granularity in their optimization. The range 8-16 is
scanned with a step size of 2 and a default value of 12.
The figure of merit used to compare BDTs and determine the optimal variables and
parameter configuration is ∆LLR, definded in Equation 7.6.

∆LLR =

ln 1 +
BDT Bins

100

NSignal
NBackground

NSignal

(7.6)

Where NSignal and NBackground are the number of expected signal and background events
in each bin. It is important to note that ∆LLR is not the same as the log-likelihood ratio
(LLR) described in Section 8.2, but is an estimator of the LLR. It is necessary to use an
estimator for the LLR due to computation time constraints. The BDT optimization iterates
over many BDT configurations and the calculation of the LLR for a statstically stable number
of pseudo-experiments, as described in Section 8.2, would take hours for each BDT tested,
where as the calculation of ∆LLR takes seconds.

7.1.4

BDT output distributions

The BDT output distributions for the four analysis channels are shown in Figure 7.15. They
show a clear separation of the signal and background processes and no signs of overtraining.
Comparisons of the BDT output distributions of the data and background samples for the
four analysis channels are shown with their statistical uncertainties in Figure 7.16. The
statistical uncertainty is described in Section 8.1.16, and the distributions have an overall
good agreement. Individual bins have small excesses or deficits of data compared to the
background, and the 1tag analysis channels have a small overall data deficit as expected
from the event yields given in Section 6.5.

101

Training sample
W’
Backgrounds

-1

2jet 1tag

4

Te s t s am p le
Signal
Backgrounds

(1/N) dN / dx

(1/N) dN / dx

20.3 fb
∫ Lsdt= 8= TeV

5

3

Training sample
W’
Backgrounds

-1

3jet 1tag

Te s t s am p le
Signal
Backgrounds

4

3

2

2

1

1

0

20.3 fb
∫ Lsdt= 8= TeV

5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0

0.6

-0.8

-0.6

-0.4

-0.2

0

0.2

BDT response

5

20.3 fb
∫ Lsdt= 8= TeV

-1

4

Train in g s am p le
W’
Backgrounds

2jet 2tag

Te s t s am p le
Signal
Backgrounds

3.5

5

4

3

2

2

1.5

1.5

1

1

0.5

0.5
-0.4

-0.2

0

0.2

0.4

0.6

0

0.8

3jet 2tag

Te s t s am p le
Signal
Backgrounds

3.5

2.5

-0.6

Training sample
W’
Backgrounds

-1

3

-0.8

20.3 fb
∫ Lsdt= 8= TeV

4.5

2.5

0

0.6

(b)
(1/N) dN / dx

(1/N) dN / dx

(a)

4.5

0.4

BDT response

-0.8

-0.6

-0.4

-0.2

BDT response

0

0.2

0.4

0.6

0.8

BDT response

(c)

(d)

Figure 7.15: The BDT output distribution with the signal and background processes split
into testing and training samples for (a) the 2jet 1tag analysis channel (b) the 3jet 1tag
analysis channel (c) the 2jet 2tag analysis channel (d) the 3jet 2tag analysis channel.

102

Events

Events

20.3 fb-1 @ 8 TeV
2-jets 1-tag

Data
Background

103

20.3 fb-1 @ 8 TeV
3-jets 1-tag

102

10

10

-0.5

0

1
-1

0.5
1
BDT response

-0.5

20.3 fb-1 @ 8 TeV
2-jets 2-tag

Data
Background

102

0.5
1
BDT response

20.3 fb-1 @ 8 TeV
3-jets 2-tag

Data
Background

103

102

10

1
-1

0

(b)
Events

Events

(a)
103

Background

103

102

1
-1

Data

10

-0.5

0

1
-1

0.5
1
BDT response

(c)

-0.5

0

0.5
1
BDT response

(d)

Figure 7.16: The BDT output distribution of the background samples and data with statistical uncertainties for (a) the 2jet 1tag analysis channel (b) the 3jet 1tag analysis channel
(c) the 2jet 2tag analysis channel (d) the 3jet 2tag analysis channel.

103

Chapter 8
Statistical analysis
With the separation of background and signal events optimized by the multivariate analysis
(MVA) described in Chapter 7, the statistical significance of the analysis must be determined.
Possible sources of systematic uncertainty and their effects are described in Section 8.1. With
no statistically significant excess of events found by the analysis, 95% confidence level (C.L.)
limits are placed on the cross-section times branching ratio of W ′ bosons with masses in
the range 0.5 TeV - 3.0 TeV. The methodology used to derive these results is described in
Section 8.2, and the results are given in Section 8.3.

8.1

Systematic uncertainties

ATLAS is a complicated experiment as described in Chapter 3, where the calibration and
interpretation of the detector response from its many systems may introduce possible biases
to the data. Additionally, the simulated Monte Carlo signal and background events may have
biases in their generation and simulation which must be accounted for. The size of these
potential biases are estimated using ATLAS perscribed procedures [73] and are treated as
systematic uncertainties in the analysis. These uncertainties can be broadly categorized into
shape uncertainties and rate uncertainties, with many of the uncertainty sources contributing
to both types. Rate uncertainties can be easily described by their effect on the event yields
but shape uncertainties cannot be so easily quantified because they move events between bins

104

Systematic Uncertainty
JES
JER
Jet Reconstruction
JVF
b-tagging
LES/LER
Lepton Efficiency
miss
ET
PDF
ISR/FSR
MC Generator
Theory Normalization
W+jets Normalization
Multijet Normalization
Luminosity

2jets 1tag
Signal
Background
-4.6/+4.7
+3.8/-4.0
±2.5
±0.6
±0.1
±0.3
+3.5/-2.8
+2.2/-1.1
-0.7/-0.5
+6.6/+1.1
+0.1/-0.1
+0.4/-0.3
±0.9
±2.6
-0.2/+0.0 +0.7/+0.7
±6.4
±2.1
±1.4
±2.0
+1.0/-1.2
+8.9/-7.7
± 2.3
± 3.6
± 3.6

3jets 1tag
Signal
Background
-1.8/+2.0
+5.5/-5.3
±0.8
±0.1
±0.1
±0.2
+3.5/-09
+0.9/-0.6
-2.0/+0.7 +5.9/+2.9
+0.1/-0.2
+0.4/-0.1
±0.4
±3.7
-0.3/-0.3
+0.6/-0.9
± 7.7
±3.4
±3.4
±2.6
+2.4/-2.7
+5.5/-4.8
± 1.1
± 3.6
± 3.6

Table 8.1: Summary table of the systematic percentage shifts in the 1.75 TeV WR′ signal
and background event yields of the 1tag analysis channels.

in an analysis channel. Tables 8.1 and 8.2 summarize the effects of the rate uncertainties on
the background and signal event yields where the WR′ signal sample was generated at a mass
of 1.75 TeV. For these tables, the size of uncertainties that contain multiple sources, such as
Theory Normalization, are determined by adding the magnitude of the indiviual sources in
quadrature. Each uncertainty is described in more detail below and is assessed on all of the
signal and background samples except when the description specifies otherwise.

8.1.1

Jet energy scale (JES)

The jet energy scale (JES) is a rate and shape uncertainty. The JES uncertainty represents
the potential bias in the measurement of the jet energies and includes both theoretical and
experimental components. The theoretical components of the uncertainty come from the use
of Monte Carlo simulations to derive the nominal JES correction factors for jets, while the

105

Systematic Uncertainty
JES
JER
Jet Reconstruction
JVF
b-tagging
LES/LER
Lepton Efficiency
miss
ET
PDF
ISR/FSR
MC Generator
Theory Normalization
W+jets Normalization
Multijet Normalization
Luminosity

2jets
Signal
-4.1/+5.2
±1.1
±0.2
+4.3/-2.6
+16.2/-14.9
+0.4/-0.1
±2.2
+0.4/+0.7
±7.7
± 3.6

2tag
Background
+0.2/-1.1
±0.1
±0.3
+1.4/-2.9
+14.3/-4.7
+0.6/-0.5
±2.0
+0.4/-0.9
±2.6
±4.0
±6.1
+2.8/-3.3
+11.4/-11.3
±2.0
± 3.6

3jets
Signal
-2.2/+1.6
±2.4
±0.1
+3.0/-1.5
+15.0/-14.0
+0.4/-0.1
±2.1
+1.0/+0.3
±8.2
± 3.6

2tag
Background
+1.1/-3.5
±0.9
±0.1
+1.2/-1.1
+16.8/-7.0
+0.7/-0.6
±4.2
+0.6/+1.7
±3.9
±5.3
±3.9
+3.7/-4.2
+7.3/-7.2
±1.0
± 3.6

Table 8.2: Summary table of the systematic percentage shifts in the 1.75 TeV WR′ signal
and background event yields of the 2tag analysis channels.

experimental components of the uncertainty come from an imperfect knowledge of the conditions a collision takes place in, such as the pileup. To estimate these uncertainties additional
Monte Carlo samples are produced with varied simulation conditions and compared to data.
In order to derive the JES calibration the energy of a jet is balanced in pT against a reference
object. This reference object can be a photon, a leptonically decaying Z boson, a jet in a
different pseudorapidity region of the detector, or several jets with lower pT . Each of these
types of reference objects contributes to the overall calibration differently depending on the
pseudorapidity and pT of the jet. There are differences in jets that develop from gluons and
the different quark flavors so an additional uncertainty is included in the JES uncertainty to
account for the uncertainty in the exact flavor composition of the jets in the samples used to
estimate the JES uncertainty. This flavor composition uncertainty contribution to the total
JES uncertainty depends on the pT , pseudorapidity, and flavor of the jet. The resulting total

106

uncertainty in the jet energies is typically 0.5-3% [74]. The effects of the JES uncertainty on
the W ′ search are evaluated by generating Monte Carlo samples with +1σ and −1σ shifts in
JES and applying the event selection and multivariate analysis to them. The difference in
the resulting shifted BDT distributions from the nominal BDT distribution defines the +1σ
and −1σ uncertainty.

8.1.2

Jet energy resolution (JER)

The jet energy resolution (JER) is a rate and shape uncertainty. The JER uncertainty is
the precision with which the energy of jets can be measured by the ATLAS detector. The
JER uncertainty is estimated by measuring the pT balance in dijet events, and ranges from
10 to 20% depending on the pT of the jet [75]. In order to assess the effect of this resolution
on the W ′ search Monte Carlo samples are generated with all of the jet energies modified
by a random offset chosen using a Gaussian distribution with a standard deviation equal to
the measured JER uncertainty. The event selection and multivariate analysis are applied to
the modified samples, and the difference in the resulting shifted BDT distributions from the
nominal BDT distribution defines the +1σ and −1σ uncertainty.

8.1.3

Jet reconstruction efficiency

The jet reconstruction efficiency is a rate and shape uncertainty. The effects of the efficiency
of the jet reconstruction algorithm on the W ′ search are estimated by randomly removing
jets from the simulated events according to the jet reconstruction efficiency. This efficiency is
determined by matching jets reconstructed from tracking information with jets reconstructed
from the calorimeter. The jet reconstruction efficiency is over 99% given the event selection

107

used in this analysis, although the reconstruction efficiency varies with the jet pT . The
modified sample has the event selection and multivariate analysis applied to it and half of
the difference in the BDT weight distribution between the modified sample and nominal
sample is taken as a symmetric uncertainty around the nominal values [76].

8.1.4

Jet vertex fraction (JVF)

The jet vertex fraction (JVF) is a rate and shape uncertainty. The JVF is the fraction
of each jet’s total pT with tracks pointing to the event’s primary vertex. This is a useful
variable for removing pileup jets1 and a cut is applied to the JVF for each jet as described
in Section 4.3. The uncertainty in the efficiency of this cut is determined by taking the
difference in cut values necessary to attain the same jet efficiency in data and Monte Carlo
Z+jets samples, a difference of 0.03 in the JVF cut [77]. New Monte Carlo samples are
produced with the JVF cut varied up and down from the nominal value by this uncertainty
and these modified samples have the event selection and multivariate analysis applied to
them to determine the effect on the BDT weight distribution. The difference in the resulting
shifted BDT distributions from the nominal BDT distribution defines the +1σ and −1σ
uncertainty.

8.1.5

b-tagging performance

The b-tagging performance uncertainties are rate and shape uncertainties. The MV1 btagging algorithm is described in Section 4.3.1 and is central to the analysis for background
1 Pileup jets are jets in an event which are created from in-time or out-of-time pileup

effects. In-time pileup is an energy deposit in the calorimeter which has been mis-attributed
to an event but is caused by a particle from another interaction occuring in the same bunch
crossing. Out-of-time pileup is an energy deposit in the calorimeter which has been misattributed to an event but is caused by an interaction occuring in a different bunch crossing.
108

rejection and the definition of the analysis channels. The MV1 algorithm produces a value
for each jet which a cut is applied to in order to achieve a 70% b-tagging efficiency. This is
a complicated procedure and the overall uncertainty in the b-tagging performance is treated
as three independant uncertainties: the b-tagging efficiency uncertainty, c-tagging efficiency
uncertainty, and mis-tagging rate uncertainty. The b-tagging efficiency is the efficiency with
which bottom quark initiated jets (“real” b-jets) are b-tagged, while the c-tagging efficiency is
the efficiency with which charm quark initiated jets (c-jets) are b-tagged and the mis-tagging
rate is the efficiency with which the remaining jets (light flavor jets) are b-tagged. The c-jets
are treated seperately from the light flavor jets because c-jets are kinematically more similar
to b-jets and thus much more likely to be b-tagged than light flavor jets. All three of these
efficiencies are measured in data by applying the MV1 algorithm to analysis channels with
high purities of b-jets, c-jets, and light flavor jets respectively. The uncertainties in the btagging efficiency, c-tagging efficiency, and mis-tagging rate are determined through careful
consideration of the systematic and statistical uncertainties in the efficiency measurements,
as described in the references [52, 78, 79]. The resulting uncertainties in the b-tagging of
each jet depend on the pT and pseudorapidity of the jet and the effects of these uncertainties
are independently evaluated by reweighting the Monte Carlo events by ±1σ shifts in each
tagging efficiency. The difference in the resulting shifted BDT distributions from the nominal
BDT distribution defines the three +1σ and −1σ uncertainties.

8.1.6

Lepton energy scale and resolution

The lepton energy scale and resolution are both rate and shape uncertainties. The lepton
energy scale and resolution uncertainties are determined by comparing Z → ll data and
Monte Carlo events. The lepton energy scale is binned in ET and pseudorapidity and the
109

difference in the dilepton invariant mass peak between the data and Monte Carlo events in
each bin determines the lepton energy scale uncertainty for that bin. The lepton energy
resolution is not binned in ET and pseudorapidity, instead the difference in the width of
the dilepton invariant mass peak between the data and Monte Carlo events determines the
uncertainty in the lepton energy resolution [80, 81]. The effects of the lepton energy scale
uncertainty are evaluated by producing Monte Carlo samples with +1σ and −1σ shifts
in the lepton energy scale and applying the event selection and multivariate analysis to
them. The effects of the lepton energy resolution are determined by modifying the nominal
lepton energies by a random offset chosen using a Gaussian distribution with a standard
deviation equal to the lepton energy resolution uncertainty and applying the event selection
and multivariate analysis to the modified events.

8.1.7

Lepton trigger, reconstruction, and identification

The lepton trigger, reconstruction, and identification efficiency uncertainties are rate and
shape uncertainties. The simulated Monte Carlo events are weighted by lepton trigger,
reconstruction, and identification efficiencies measured in the Z → ll and W → lν channels.
Chapter 4 describes the algorithms used to find electrons and muons for the analysis and
defines what an electron or muon is at the analysis level. The lepton trigger efficiency
is the efficiency with which the analysis level leptons pass the corresponding triggers, the
leptons may not pass the trigger because the trigger system uses a more limited detector
granularity and calibration than the analysis. The lepton reconstruction efficiency is the
efficiency of analysis leptons being found by the lepton algorithms described in Chapter 4
up until the quality cuts. The lepton identification efficiency is the efficiency of analysis
leptons passing the lepton quality cuts. Leptons may fail the reconstruction or identification
110

algorithms in data due to temporarily noisy or dead cells in the detector which are not
included in the Monte Carlo simulation. All three of these efficiencies are functions of the
lepton kinematics and are determined by relaxing the respective requirements and measuring
how often the trigger,reconstruction, or identification requirements are met. The uncertainty
in these efficiencies is determined by taking half the difference in the measured data and
Monte Carlo efficiencies as a symmetric uncertainty about the nominal value [80, 81].

8.1.8

Missing transverse energy (ETmiss )

miss uncertainty is a rate and shape uncertainty. As described in Section 4.4, the
The ET
miss calculation contains terms for the leptons, jets, soft jets, and cell-out energy and the
ET

uncertainty in each of these terms is treated as an independant uncertainty. The effects of
the uncertainty in the lepton or jet terms are included in the lepton and jet energy scale and
resolution uncertainties described previously. The uncertainty in the soft jet and cell-out
terms is estimated using a Z → µµ sample with no jets having pT > 20 GeV. The projection
miss onto the reconstructed Z boson’s transverse direction is calculated for a data
of the ET

sample and Monte Carlo simulated sample. The average deviation from unity, defined as the
ratio of this variable between the data and Monte Carlo samples, is taken as the uncertainty
in the soft jets and cell-out energy [55]. New Monte Carlo samples are generated with the
soft jets and cell-out callibrations independantly varied up and down by this uncertainty and
the event selection and multivariate analysis are applied to these samples to evaluate the
effect on the final BDT weight distribution.

111

8.1.9

Parton distribution function (PDF)

The parton distribution function (PDF) uncertainty is a rate uncertainty. PDFs are functions which give the probablility of a quark or gluon inside of a proton having some fraction
of the proton’s total momentum. PDFs are determined by fitting data from many particle physics experiments under a variety of assumptions about the functional form of the
PDF. A PDF is used in Monte Carlo event generation to weight events with different initial
quark or gluon momenta. The uncertainty in a PDF is expressed as a set of uncertainty
eigenvectors, each of which represents the uncertainty in a single fit parameter. In order
to estimate the effects of the PDF uncertainty on this analysis, the Monte Carlo samples
are reweighted according to each of the 68% confidence level uncertainty eigenvectors for
the CT10, MSTW2008NLO68CL [82], and NNPDF23 [83] PDFs. CT10 is the default PDF
used for this analysis while MSTW and NNPDF are other common choices of PDFs. The
uncertainty due to the PDF is covered by evaluating the uncertainty eigenvector sets of all
three of these PDFs and taking half of the largest variation from the nominal sample yield
is as a symmetric uncertainty on each Monte Carlo sample.

8.1.10

Initial state radiation and final state radiation (ISR/FSR)

The initial state radiation and final state radiation (ISR/FSR) uncertainty is a rate and shape
uncertainty. ISR/FSR is the radiation of a particle immediately before or after the hard
interaction of a process. The uncertainty in modeling these effects is estimated by varying
the ISR/FSR settings of PYTHIA within the range consistent with previous measurements
of tt¯ [84]. Six additional tt¯ samples are produced using ACERMC+PYTHIA with these
varied ISR/FSR PYTHIA settings and have the event selection and multivariate analysis

112

applied to them. The maximum deviation of the resulting BDT weight distributions from
the nominal distribution is taken as a symmetric systematic uncertainty on the tt¯ sample.
This uncertainty is only evaluated for the tt¯ sample because it is the largest background for
the analysis and a large discrepency in tt¯ predictions due to the ISR/FSR settings has been
observed [84].

8.1.11

Monte Carlo event generator and parton showering

The different Monte Carlo event generator and parton showering programs make a variety of different approximations and produce events with different kinematic distributions.
To reduce the analysis’ sensitivity to the effects of a specific event generator or parton
showering program a systematic uncertainty is included for all top quark processes which
is estimated by taking the difference from the nominal Monte Carlo sample and a Monte
Carlo sample produced with a different event generator or parton shower program. For the
tt¯ sample the nominal POWHEG+PYTHIA sample is compared to samples produced with
POWHEG+HERWIG, MC@NLO+HERWIG, and ALPGEN+HERWIG and the largest difference is taken as a symmetric uncertainty. The nominal single top quark t-channel sample
produced with ACERMC+PYTHIA is compared to a sample produced with MC@NLO+HERWIG
and the difference between the samples is taken as a symmetric uncertainty. For the single
top quark s-channel the nominal POWHEG+PYTHIA sample is compared to a sample
produced with MC@NLO+HERWIG and the difference is taken as a symmetric uncertainty. For the single top quark Wt channel, the nominal POWHEG+PYTHIA sample
is compared to a sample produced with MC@NLO+HERWIG and samples produced with
POWHEG+PYTHIA with the two different NLO calculation schemes (diagram removal
and diagram subtraction) are compared to each other, and the larger difference is taken as
113

a symmetric uncertainty on the nominal sample. No event generator or parton showering
systematic is evaluated for the W+jets or multijets backgrounds because they are estimated
using data driven techniques.

8.1.12

Theoretical cross-sections

The theoretical cross-section uncertainties are rate uncertainties. All of the Monte Carlo
samples, with the exception of the W+jets and multijets samples, are normalized to a theoretically predicted cross-section. These cross-sections have associated uncertainties in their
calculation and each channel is assigned an independent rate uncertainty to account for this.
The tt¯ sample has a cross-section uncertainty of −5.9/+5.1%. For the single top quark
processes the uncertainty in the s-channel cross-section is ±3.9%, the uncertainty in the
t-channel cross-section is −2.1/+3.9%, and the uncertainty in the Wt channel cross-section
is ±6.8%. The uncertainty in the Z+jets and diboson samples are calculated by adding in
quadrature the 4% uncertainty in the inclusive process cross-section and a 24% uncerainty
for each jet in the analysis channel. This gives an uncertainty in the Z+jets and diboson
process cross-sections of 34% for the 2jet analysis channels and 42% for the 3jet analysis
channels. The contributions of the Z+jets and diboson processes to the total background
are small so this large uncertainty on their cross-sections has a small impact on the analysis.

8.1.13

W+jets normalization

The W+jets normalization uncertainty is a rate uncertainty. The W+jets Monte Carlo samples are normalized using a data driven technique described in Section 6.4.1. An uncertainty
in this normalization is determined by individually applying the JES, event generator, par-

114

ton showering, ISR/FSR, PDF, charge misidentification, lepton energy resolution, lepton
trigger efficiency, lepton reconstruction efficiency, lepton identification efficiency, theoretical
cross-section for other background processes and multijet normalization uncertainties described below to the control region used to derive the W+jets normalization. The variation
in the W+jets normalization caused by each of these individual uncertainties is then added
in quadrature to calculate the total W+jets normalization uncertainty. The total W+jets
normalization uncertainty varies from 15% to 46% depending on the analysis channel and
W+jets flavor.

8.1.14

Multijet normalization

The multijet normalization uncertainty is a rate only uncertainty. The multijet background
is normalized using the matrix method as described in Section 6.4.2. Since this is a small
background and a detailed estimation of the uncertainty in its normalization would be very
difficult, a 50% uncertainty on the sample’s normalization is applied.

8.1.15

Luminosity

There is a ±3.6% uncertainty in the luminosity of the data set used for this analysis, which is
a preliminary result derived from updated van der Meer scans performed in April 2012 [85].
Since all of the Monte Carlo samples are normalized to this luminosity, this uncertainty is
assessed as a ±3.6% rate uncertainty in all of the Monte Carlo samples except W+jets and
multijets which are normalized using data driven techniques.

115

8.1.16

Monte Carlo statistics

The Monte Carlo statistical uncertainty is a rate and shape uncertainty. The finite number
of Monte Carlo events generated for each process creates a statistical uncertainty in each bin
of the BDT weight distribution. The statistical uncertainty in each bin is calculated taking
into account the event weights of the Monte Carlo events and is less than 30% for the total
background in each bin.

8.2

Limit setting procedure

The statistical analysis for the W ′ → t¯b search uses the CLS method [86, 87] implemented
in the MClimit [88] software package. The CLS method is a binned likelihood method,
meaning the likelihood of each bin of the BDT weight distribution is computed and then
all of the bins are combined into a global likelihood. This method was chosen because it is
more powerful than a single bin counting experiment and is less sensitive to fluctuations in
the background, as described below.
The CLS method is derived from basic Poisson and Gaussian statistics. The likelihood
of observing N events if there are µ expected events is given by Poisson statistics as:

L(N |µ) =

µN e−µ
N!

(8.1)

The inclusion of uncertainties into this likelihood is done by modifying the expected
number of events through nuisance parameters (θ), each representing a single uncertainty.
Each background or signal process forms a sample and the expected number of events for
each sample (µi ) is modified by the probability density function of each of the uncertainties

116

that affect that sample (G). Depending on the hypothesis being tested, all of the background
samples and possibly a signal sample are summed over to produce the final expected number
of events as shown in Equation 8.2

µ=

µi
i∈sample

G(θij , δij )

(8.2)

j∈uncertainty

The probability density function for each of the uncertainties can in principle be any nonnegative unitary distribution, but for this analysis they are all assumed to be Gaussian
distributions with a mean of 1 and a standard deviation of δ. One of the consequences of
Equation 8.2 is that µ is now a multi-dimensional probability distribution and not a single
value.
So far this procedure has described a single counting experiment. Each bin of the BDT
weight distribution can be treated as an independent counting experiment and the likelihood
of observing an event distribution (Nobs ) with Nk events in the k th bin which has µk expected
events is given by the product of the independent bins’ likelihoods as shown in Equation 8.3.
N

L(Nobs |θij , δij ) =

k∈bins

µk k e−µk
Nk !

(8.3)

In order to determine if a signal process at a given mass point is excluded, 10,000 pseudoexperiments are generated for each of two hypotheses: one hyopthesis (H1 ) includes the signal
and all of the background processes while the other hypothesis (H0 ) includes only the background processess. These pseudo-experiments are generated by randomly choosing a value for
each nuisance parameter based on that parameter’s probability distribution function. This
shifted distribution is used to set the expected number of events of a Poisson distribution in
each bin (µk ). These Poisson distributions are then randomly sampled to produce the final
117

pseudo-experiment distribution. For each pseudo-experiment the log-likelihood ratio (LLR)
is computed according to Equation 8.4.

LLR = −2ln

L(Nobs |H1 )
L(Nobs |H0 )

(8.4)

The LLR of the observed data set is also calculated and used to determine the strength of
the limit on the signal process. CLS+B is defined as the fraction of pseudo-experiments generated with the signal-plus-background hypothesis (H1 ) with an LLR greater than the LLR
of the data set. Similarly, CLB is defined as the fraction of pseudo-experiments generated
with the background-only hypothesis (H0 ) with an LLR greater than that of the data set.
CLS is the likelihood of the signal being included in the data and is defined as the ratio of
CLS+B to CLB as shown in Equation 8.5. An advantage of using CLS rather than CLS+B
is that any mis-modeling or fluctuations in the background estimates will affect CLS+B and
CLB similarly and will partially cancel when determining CLS .

CLS =

CLS+B
CLB

(8.5)

A signal is excluded at the 95% confidence level if it has a CLS < 0.05. After the signal
has been excluded (or not) at its nominal cross-section the process is repeated with the
normalization of the signal sample decreased (if the current cross-section was excluded) or
increased (if the current cross-section was not excluded) by a factor k. This is iteratively
repeated until CLS equals 0.05. When this occurs the value of k can be used to determine
the 95% limit on the cross-section as shown in Equation 8.6.

σ95% limit = kσnominal
118

(8.6)

This can be taken one step further and 95% confidence level limits can be placed on the
′ and g ′ . The effective Lagrangian density used to generate the signal
coupling strengths gR
L

samples was given in Equation 2.13 and is repeated here in Equation 8.7.

1
′ (1 + γ ) + g ′ (1 − γ ))f j
LW ′ = √ Vij′ Wµ′ f¯i γ µ (gR
5
5
L
2 2

(8.7)

′2 or g ′2 . The W ′ and
The cross-sections of the W ′ → t¯b processes are thus proportional to gR
L
R
′ or g ′ equal to the Standard
WL′ signal samples at each mass point were generated with gR
L

Model W coupling stength g. Thus the ratio of the observed limit on the cross-section to
the nominal cross-section for the WR′ or WL′ signal sample at each mass point can be used
g′

g′

k=

σ95% limit
g′
=
σnominal
g

to compute an equivalent limit on gR or gL respectively using Equation 8.8 at each mass
point.

√

8.3

(8.8)

Results

The limit setting procedure described in Section 8.2 is performed for each of the 22 WR′ and
WL′ mass points (11 mass points each) in all four of the analysis channels (2jet 1tag, 2jet
2tag, 3jet 1tag, 3jet 2tag)2 . These are not limits directly on the W ′ cross-section but instead
they are limits on the W ′ cross-section corrected by the branching ration of W ′ → t¯b and
the branching ratio of W → lν. The limits on cross-section times branching ratio for the
WR′ mass points in each of the analysis channels are plotted in Figure 8.1, and the limits on
2 It is necessary to compute the limits for each W ′ mass point seperately because the

signal event kinematics, and thus the BDT responses, are different for each mass point.

119

Analysis Channel
2jet 1tag
3jet 1tag
2jet 2tag
3jet 2tag

Observed Mass Limit [TeV]
1.51
1.66
1.72
1.68

Expected Mass Limit [TeV]
1.47
1.61
1.73
1.71

Table 8.3: Expected and observed mass limits for a WR′ boson in the individual analysis
channels.

cross-section times branching ratio for the WL′ mass points in each of the analysis channels
are plotted in Figure 8.2. The red line in these plots shows the theoretical cross-section times
branching ratio as a function of W ′ mass which is calculated assuming a coupling strength
the same as in the SM. The solid black line shows the observed 95% confidence level limit
on the cross-section times branching ratio as a function of W ′ mass. The region above
this line has been excluded at the 95% confidence level and regions where the theoretical
cross-section times branching ratio are above the observed limit have been excluded for g ′
= g. The dashed line is the expected limit, which is derived similarly to the observed limit
but is the median result of 10,000 background only pseudo-data sets used in place of the
data. The green and yellow bands are the 68% (1σ) and 95% (2σ) confidence interval of the
expected limit results. The mass value where the theoretical cross-section times branching
ratio crosses the observed or expected limit is known as the mass limit and is a commonly
used figure of merit for exclusions such as this. The mass limits for each of the analysis
channels is given in Table 8.3 for a WR′ boson and in Table 8.4 for a WL′ boson.
The limits on the cross-section times branching ratio can also be used to calculate limits
′
on gg as described by Equation 8.8. By using the observed or expected limits on the crossg′

g′

section times branching ratio, observed or expected limits on gR and gL can be calculated
for each of the tested mass points for the WR′ and WL′ bosons respectively. These limits

120

σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

Expected limit

102

Expected ± 1σ

2jet 1tag

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’R Theory

1
10-1
10-2
10-3
0.5

1

1.5

2

2.5

3

Expected limit

102

Expected ± 1σ

3jet 1tag

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’R Theory

1
10-1
10-2
10-3
0.5

1

1.5

2

mW’ RH [TeV]

Expected limit

102

Expected ± 1σ

2jet 2tag

Expected ± 2σ

∫ L dt = 20.3 fb

-1

Observed limit

s = 8 TeV

W’R Theory

1
10-1
10-2
10-3
0.5

1

1.5

3

(b)
σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

(a)

10

2.5

mW’ RH [TeV]

2

2.5

3

Expected limit

102

Expected ± 1σ

3jet 2tag

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’R Theory

1
10-1
10-2
10-3
0.5

1

1.5

mW’ RH [TeV]

2

2.5

3

mW’ RH [TeV]

(c)

(d)

Figure 8.1: 95% confidence level limits on the cross-section times branching ratio of the WR′
boson in the (a) 2jet 1tag channel (b) 3jet 1tag channel (c) 2jet 2tag channel (d) 3jet 2tag
channel.

Analysis Channel
2jet 1tag
3jet 1tag
2jet 2tag
3jet 2tag

Observed Mass Limit [TeV]
1.48
1.62
1.59
1.57

Expected Mass Limit [TeV]
1.44
1.58
1.63
1.60

Table 8.4: Expected and observed mass limits for a WL′ boson in the individual analysis
channels.

121

σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

Expected limit

102

Expected ± 1σ

2jet 1tag

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’L Theory

1
10-1
10-2
10-30.5

1

1.5

2

2.5

3

Expected limit

102

Expected ± 1σ

3jet 1tag

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’L Theory

1
10-1
10-2
10-30.5

1

1.5

2

mW’ LH [TeV]

Expected limit

102

Expected ± 1σ

2jet 2tag

Expected ± 2σ

∫ L dt = 20.3 fb

-1

Observed limit

s = 8 TeV

W’L Theory

1
10-1
10-2
10-30.5

1

1.5

3

(b)
σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

(a)

10

2.5

mW’ LH [TeV]

2

2.5

3

Expected limit

102

Expected ± 1σ

3jet 2tag

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’L Theory

1
10-1
10-2
10-30.5

1

1.5

mW’ LH [TeV]

2

2.5

3

mW’ LH [TeV]

(c)

(d)

Figure 8.2: 95% confidence level limits on the cross-section times branching ratio of the WL′
boson in the (a) 2jet 1tag channel (b) 3jet 1tag channel (c) 2jet 2tag channel (d) 3jet 2tag
channel.

122

are plotted as a function of the W ′ mass and an exclusion region is formed, as seen in
Figures 8.3 and 8.4. These are limits on parameters in the effective Lagrangian density
described in Equation 2.13, and they directly inform the excluded parameter space of theories

1

95% C.L. limit on g’/g

95% C.L. limit on g’/g

that include a W ′ boson.

2jets 1tag

∫

-1

1

3jets 1tag

∫ L dt = 20.3 fb

-1

L dt = 20.3 fb

s = 8 TeV

s = 8 TeV

Expected exclusion
Observed exclusion

10-1
0.5

1

1.5

2

2.5

Expected exclusion
Observed exclusion

10-1
0.5

3

1

1.5

2

mW’ RH [TeV]

1

2jets 2tag

∫

-1

1

3jets 2tag

∫ L dt = 20.3 fb

-1

L dt = 20.3 fb

s = 8 TeV

s = 8 TeV

Expected exclusion
Observed exclusion

1

1.5

3

(b)
95% C.L. limit on g’/g

95% C.L. limit on g’/g

(a)

10-1
0.5

2.5

mW’ RH [TeV]

2

2.5

Expected exclusion
Observed exclusion

10-1
0.5

3

1

1.5

mW’ RH [TeV]

2

2.5

3

mW’ RH [TeV]

(c)

(d)
g′

Figure 8.3: 95% confidence level limits on gR in the (a) 2jet 1tag channel (b) 3jet 1tag
channel (c) 2jet 2tag channel (d) 3jet 2tag channel.

The individual analysis channels can also be combined by treating each channel as more
bins in the BDT weight distribution. The resulting combination is analyzed in an identical
manner to the individual analysis channels and similar limits can be made. Because of
their kinematic similarities, the 2jet 1tag and 3jet 1tag channels are combined to form the
123

95% C.L. limit on g’/g

95% C.L. limit on g’/g

1

2jets 1tag

∫ L dt = 20.3 fb

-1

1

3jets 1tag

∫ L dt = 20.3 fb

-1

s = 8 TeV

s = 8 TeV

Expected exclusion
Observed exclusion

10-1
0.5

1

1.5

2

2.5

Expected exclusion
Observed exclusion

10-1
0.5

3

1

1.5

2

mW’ RH [TeV]

1

2jets 2tag

∫ L dt = 20.3 fb

-1

1

3jets 2tag

∫ L dt = 20.3 fb

-1

s = 8 TeV

s = 8 TeV

Expected exclusion
Observed exclusion

1

1.5

3

(b)
95% C.L. limit on g’/g

95% C.L. limit on g’/g

(a)

10-1
0.5

2.5

mW’ RH [TeV]

2

2.5

Expected exclusion
Observed exclusion

10-1
0.5

3

1

1.5

mW’ RH [TeV]

2

2.5

3

mW’ RH [TeV]

(c)

(d)
g′

Figure 8.4: 95% confidence level limits on gL in the (a) 2jet 1tag channel (b) 3jet 1tag
channel (c) 2jet 2tag channel (d) 3jet 2tag channel.

124

1tag combined channel, and the 2jet 2tag and 3jet 2tag channels are combined to form the
2tag combined channel. A final combination of all of the analysis channels forms the full
combination channel. The limits on the cross-section times branching ratio for a WR′ and
WL′ boson are shown in Figure 8.5 and 8.6 respectively. The derived mass limits from the
g′

combined channels for a WR′ or WL′ boson are given in Tables 8.5 and 8.6. The limits on gR
′

σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

g
and gL are shown in Figures 8.7 and 8.8.

Expected limit

102

Expected ± 1σ

1tag combined

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’R Theory

1
10-1
10-2
10-30.5

1

1.5

2

2.5

3

Expected limit

102

Expected ± 1σ

2tag combined

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’R Theory

1
10-1
10-2
10-30.5

1

1.5

mW’ RH [TeV]

2

2.5

3

mW’ RH [TeV]

(b)
σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

(a)
Expected limit

102

Expected ± 1σ

Full combination

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’R Theory

1
10-1
10-2
10-3
0.5

1

1.5

2

2.5

3

mW’ RH [TeV]

(c)
Figure 8.5: 95% confidence level limits on the cross-section times branching ratio of the
WR′ boson in the (a) 1tag combined channel (b) 2tag combined channel (c) full combination
channel.

125

σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

Expected limit

102

Expected ± 1σ

1tag combined

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’L Theory

1
10-1
10-2
10-30.5

1

1.5

2

2.5

3

Expected limit

102

Expected ± 1σ

2tag combined

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’L Theory

1
10-1
10-2
10-30.5

1

1.5

mW’ LH [TeV]

2

2.5

3

mW’ LH [TeV]

(b)
σ(pp → W’)× B(W’ → tb)× B(W → lν) [pb]

(a)
Expected limit

102

Expected ± 1σ

Full combination

Expected ± 2σ

∫ L dt = 20.3 fb

-1

10

Observed limit

s = 8 TeV

W’L Theory

1
10-1
10-2
10-30.5

1

1.5

2

2.5

3

mW’ LH [TeV]

(c)
Figure 8.6: 95% confidence level limits on the cross-section times branching ratio of the
WL′ boson in the (a) 1tag combined channel (b) 2tag combined channel (c) full combination
channel.
Analysis Channel
1tag combined
2tag combined
full combination

Observed Mass Limit [TeV]
1.69
1.73
1.76

Expected Mass Limit [TeV]
1.64
1.75
1.77

Table 8.5: Expected and observed mass limits for a WR′ boson in the combined analysis
channels.
Analysis Channel
1tag combined
2tag combined
full combination

Observed Mass Limit [TeV]
1.65
1.61
1.68

Expected Mass Limit [TeV]
1.62
1.65
1.70

Table 8.6: Expected and observed mass limits for a WL′ boson in the combined analysis
channels.

126

95% C.L. limit on g’/g

95% C.L. limit on g’/g

1

1tag combined

∫ L dt = 20.3 fb

-1

1

2tag combined

∫ L dt = 20.3 fb

-1

s = 8 TeV

s = 8 TeV

Expected exclusion
Observed exclusion

10-1
0.5

1

1.5

2

2.5

Expected exclusion
Observed exclusion

10-1
0.5

3

1

1.5

mW’ RH [TeV]

2

2.5

3

mW’ RH [TeV]

(b)
95% C.L. limit on g’/g

(a)

1

Full combination

∫ L dt = 20.3 fb

-1

s = 8 TeV

Expected exclusion
Observed exclusion

10-1
0.5

1

1.5

2

2.5

3

mW’ RH [TeV]

(c)
g′

Figure 8.7: 95% confidence level limits on gR in the (a) 1tag combined channel (b) 2tag
combined channel (c) full combination channel.

127

95% C.L. limit on g’/g

95% C.L. limit on g’/g

1

1tag combined

∫ L dt = 20.3 fb

-1

1

2tag combined

∫ L dt = 20.3 fb

-1

s = 8 TeV

s = 8 TeV

Expected exclusion
Observed exclusion

10-1
0.5

1

1.5

2

2.5

Expected exclusion
Observed exclusion

10-1
0.5

3

1

1.5

mW’ LH [TeV]

2

2.5

3

mW’ LH [TeV]

(b)
95% C.L. limit on g’/g

(a)

1

Full combination

∫ L dt = 20.3 fb

-1

s = 8 TeV

Expected exclusion
Observed exclusion

10-1
0.5

1

1.5

2

2.5

3

mW’ LH [TeV]

(c)
g′

Figure 8.8: 95% confidence level limits on gL in the (a) 1tag combined channel (b) 2tag
combined channel (c) full combination channel.

128

Experiment
This analysis
CDF (2014)
D0 (2011)
ATLAS (2013)
CMS (2014)

Observed Mass Limit [TeV]
1.76
0.84
0.89
1.84
2.05

Luminosity
20.3 f b−1
9.1 f b−1
2.3 f b−1
14.3 f b−1
19.5 f b−1

Collision Energy
8 TeV
1.96 TeV
1.96 TeV
8 TeV
8 TeV

Reference
[89]
[24]
[90]
[26]

Table 8.7: Observed mass limits for a WR′ boson from different experiments
Experiment
This analysis
D0 (2011)
ATLAS (2013)
CMS (2014)

Observed Mass Limit [TeV]
1.68
0.86
1.74
2.05

Luminosity
20.3 f b−1
2.3 f b−1
14.3 f b−1
19.5 f b−1

Collision Energy
8 TeV
1.96 TeV
8 TeV
8 TeV

Reference
[24]
[90]
[26]

Table 8.8: Observed mass limits for a WL′ boson from different experiments

8.4

Comparison with other experiments

This analysis is not the first search for W ′ → t¯b, there have been similar searches done
previously by the CDF and D0 collaborations at the Tevatron as well as previous results
from the CMS and ATLAS collaborations at the LHC. None of these experiments have
obseved a W ′ signal and the mass limit set on the W ′ boson by these searches is given in
Table 8.7 for a WR′ and in Table 8.8 for a WL′ .
This analysis does not set new best limits on the allowed mass of either a WR′ or WL′ boson.
The inclusion of the 1tag analysis channels does not significantly improve the sensitivity of
the analysis above that of the combination of the 2tag analysis channels. This is surprising
given that previous analyses at the Tevatron [23, 24] saw an increase in sensitivity from the
inclusion of 1tag events. The most recent result from the CMS collaboration [26] also includes
1tag events although it is unclear how much of the sensitivity comes from the 1tag events
since they are not analyzed seperately from the 2tag events. Attempts to replicate the CMS

129

analysis with ATLAS data have not shown an improvement over this analysis. This analysis
also does not represent a large improvement over the previous ATLAS result [90] despite
analyzing a larger dataset, which could indicate that the analysis sensitivity is limited by
systematic uncertainties.

130

Chapter 9
Conclusion
This analysis is a search for W ′ → t¯b using the ATLAS detector. The discovery of such a
particle would revolutionize particle physics, and while no such particle was found by this
analysis the limits placed on the possible existence of such a particle provides information
that affects a wide range of new physics theories.
This analysis uses the entire 8 TeV dataset collected by ATLAS in 2012 and is the first
inclusion of events with a single b-tag into a W ′ → t¯b search at ATLAS. The sensitivity
of the search is limited by systematic uncertainties. These systematic uncertainties will
almost certainly improve with time as the ATLAS program matures and the detector and
the physics it probes is better understood, but there is ample room for improvement in
the analysis technique itself. This analysis uses a boosted decision tree (BDT) algorithm
to increase the sensitivity of the analysis to a potential signal. Such a method optimizes
the analysis of the nominal sample but the optimization is blind to the potential impact
of systematic uncertainties. A modification of the BDT algorithm to take into account the
systematic uncertainties and potentially trade some separation power in the nominal sample
for an increased robustness against the effects of systematic uncertainties could improve not
only this analysis but a large number of analyses which are currently limited by the effects
of systematic uncertainties. The inclusion of such a modification to this analysis is discussed
in Appendix A, but it did not improve the expected sensitivity of the analysis and was not
included in the final result.
131

Even without such an improvement in the analysis technique or a large reduction in
systematics, I expect new searches for W ′ → t¯b to be able to reach much higher energies in
the near future. With the resumption of data taking at the LHC in 2015 planned to take
place with 13 or 14 TeV collisions, the production rate of high mass W ′ bosons will be up
to 50 times higher than what was expected at 8 TeV [91]. Ongoing research [92] suggests
that at such high energies searching for the hadronic decay of the W ± boson may produce
an analysis more sensitive to a higher mass W ′ boson than the leptonic decay this analysis
searches for. Such future analyses, whether they be in a hadronic or leptonic channel, will
doubtless present their own hurdles for future analyzers to overcome.

132

APPENDIX

133

Alternative BDT Configurations
The multivariate analysis described in Chapter 7 was the result of many trials and attempts
to improve the sensitivity of the analysis. It offers the best sensitivity while minimizing the
complexity of the analysis, but many more complex and less sensitive configurations of the
multivariate analysis were investigated during the development of this analysis. It is useful
to document these alternative configurations not only to avoid a duplication of effort but also
to provide insight and possibly inspiration. In this appendix, two categories of alternative
configurations are described: alternative figures of merit for the BDT, and the inclusion of
systematic uncertainties into the BDT optimization. The alternative figures of merit for
the BDT are replacements for ∆LLR when optimizing the BDT parameters as described in
Section 7.1.3. A method of including systematic uncertainties into the BDT is a modification
of the method of evaluating the BDT figure of merit described in Section 7.1.3 with the goal
of reducing the effects of systematic uncertainties on the final results. The development of
the multivariate analysis occured in parallel with the development of the other parts of the
analysis, and fully documenting the differences in the analysis when different portions of
these studies were performed is beyond the scope of this appendix. Because of this there are
no consistent figures of merit across all of the studies and only qualitative claims are made
about the performance of the various methods tested.

134

A.1

Alternative BDT figures of merit

The statistical method described in Section 8.2 is very computationally intensive, and it
is not feasible to perform such an analysis on a large number of BDT trainings in order to
determine parameter settings which optimise the expected sensitivity of the analysis. Instead
it is necessary to use a more computationally simple estimator of the expected sensitivity
in order to determine the optimal BDT parameter values. The optimization of the BDT
parameters described in Section 7.1.3 uses ∆LLR as the figure of merit for choosing the
best BDT parameters because it is an estimator of the figure of merit used in the statistical
analysis (LLR), and because its optimal BDT parameterization was more sensitive than the
optimal BDT parameterization of any other figures of merit when the statistical analysis
was performed on those results. The alternative figures of merit which were investigated are
described below:
• Optimal Cut Significance: The optimal cut significance is determined by finding
the cut on the BDT output which maximizes √ S

S+B

where S is the number of signal

events above the cut value and B is the number of background events above the cut
value. This estimates the expected significance of a counting experiment and does
not include any information of the distribution shapes except in the placement of the
optimal cut. This performed only slightly worse than ∆LLR.
• Signal and Background Peak Separation: The signal and background peak separation is the difference in the BDT output values of the peak of the signal distribution
and the background distribution. This estimates the separation between the signal and
background event distributions. It performed much worse than ∆LLR.

135

• Significance above the Signal Peak: The significance above the signal peak is determined by calculating √ S

S+B

for events at and above the signal peak. This estimates

the expected significance of a counting experiment. It performed slightly worse than
∆LLR and approximately as well as the optimal cut significance.

A.2

Inclusion of systematic uncertainties

A potential weakness of the BDT optimization strategy is that an algorithm is created which
optimally sorts the nominal background and signal samples but which is very sensitive to
systematic shifts in the variables used to sort the events. In such a situation it may be
possible to find an algorithm that less optimally sorts the nominal events but is less sensitive
to systematic shifts in the variables used and is more sensitive than the optimal nominal
algorithm after the systematic uncertainties are taken into account in the statistical analysis.
Attempts were made to develop such an algorithm by modifying the way the BDT figure of
merit was evaluated to include information about the systematically shifted event samples.
The BDT was trained with the nominal event sample and then applied to the nominal
event sample as well as the systematically shifted event samples. The BDT figure of merit
was then evaluated for the nominal sample output as well as the systematically shifted
sample outputs and the minimum of these figures of merit was taken as the figure of merit
for that BDT training. Optimizations were performed with as few as one systematically
shifted sample included up to all of the systematically shifted samples being included, for
all of the figures of merit listed in Section A.1. None of the BDT trainings which included
systematically shifted samples out-performed the BDT trained with only the nominal event
sample. In general the more systematically shifted samples were included in the optimization

136

the worse the resulting BDT performed, with trainings that only included one systematically
shifted sample having little or no differences from the training that included only the nominal
event sample.

137

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