SEARCH FOR W 0 BOSON WITH BOOSTED AND HADRONIC √
TOP-QUARK FINAL STATE IN P P COLLISIONS AT S = 13 TEV
WITH THE ATLAS DETECTOR
By
Kuan-Yu Lin
A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics — Doctor of Philosophy
2021
ABSTRACT
SEARCH FOR W 0 BOSON WITH BOOSTED √ AND HADRONIC TOP-QUARK FINAL
STATE IN P P COLLISIONS AT S = 13 TEV WITH THE ATLAS DETECTOR
By
Kuan-Yu Lin
With the LHC proton-proton collision data at an energy of 13 TeV recorded by the
ATLAS detector, this analysis searches for the production of a new particle W 0 with a
yet-to-be-probed mass in the top quark plus bottom quark decay channel. The boosted and
hadronically decaying top-quark is reconstructed as a large-radius jet and identified using jet
substructures. A secondary-vertex finding algorithm identifies small-radius jets originating
from a bottom quark. After the identification stage, the main background events come from
the Quantum Chromodynamical production of multi-jet, which is estimated by a data-driven
approach. The statistical data analysis searches for a resonant peak in the invariant mass
distribution of the system with a jet associated with the top quark and another with the
bottom quark. The consistency between data and the background prediction is found in the
invariant mass range between 1 TeV to 7 TeV, and an upper limit at the 95% confidence level
is set on the production cross-section times branching ratio for a W 0 boson with right-handed
chirality decaying to a top quark and a bottom quark. A lower limit on the W 0 mass is set
at 4.4 TeV accordingly.
ACKNOWLEDGMENTS
First I would like to thank my PhD supervisor, Reinhard, for all the financial and mental
support he has offered during my studies. The opportunities to stay at CERN and the
Argonne National Laboratories gave me unique experiences of research and living. The
numerous time he spent to review my thesis allowed me to strengthen my knowledges and
convictions of doing experimental research. The consideration he gave to accommodate my
visa situation and prolonged difficbtilies under COVID was incredible.
All other members in the ATLAS group help me substantially too. Thanks Wade for
mentoring me from time to time and his elucidation in physics and statistics. Thanks Hector
for all the assistance to my works and friendly discussions within and beyond physics. Thanks
Garabed for the fun chat we had together. I hope I can share with you my latest reading on
philosophy soon. Thanks Dan for your cheerfulness during all coffee breaks and beer time.
Thanks Trisha for instant technical help many time and being a nice person sitting behind
me in the office. Thanks Reiner for all the tips regarding traveling and hiking. Thanks Yuri
for the companion during my L1Calo general meeting. Although Rui is strictly speaking not
a student of MSU, I appreciate to have a like-minded person to work on the same analysis.
I would also like to thanks many other people I met at MSU, CERN, and Argonne that
gave me memorable years during my PhD.
At last I want to thank my parents and my sister for helping me to face all kinds of
difficulties in life.
iii
TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Chapter 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 An Introduction to Particle Physics . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Road Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 The Search for W 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 The Structure of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 2 From the Standard Model to New Physics: W 0 . . . . . . . . . 9
2.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Beyond the Standard Model with W 0 . . . . . . . . . . . . . . . . . . . . . . 12
Chapter 3 Analysis Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Chapter 4 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . 19
4.1 Large Hadron Collider . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 ATLAS Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.1 Inner detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2.2 Calorimeters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2.3 Muon spectrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.2.4 Trigger and data acquisition . . . . . . . . . . . . . . . . . . . . . . . 34
4.2.5 Level 1 calorimeter trigger . . . . . . . . . . . . . . . . . . . . . . . . 37
Chapter 5 Predicting the W 0 Signal . . . . . . . . . . . . . . . . . . . . . . . 39
5.1 NLO Cross-Section Evaluation with ZTOP . . . . . . . . . . . . . . . . . . . 39
5.2 Theoretical Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.3 Event Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Chapter 6 The Identification of Top and Bottom Quarks . . . . . . . . . . 53
6.1 Jet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.2 Large-radius Jet — Local Cluster Topological Jet . . . . . . . . . . . . . . . 55
6.2.1 Large-radius Jet Trigger . . . . . . . . . . . . . . . . . . . . . . . . . 57
6.2.2 DNN top tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
6.3 Small-radius Jet — Particle Flow Jet . . . . . . . . . . . . . . . . . . . . . . 63
6.3.1 DL1r b-tagging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.4 Truth-level and Reconstruction-level Observables . . . . . . . . . . . . . . . 67
Chapter 7 Event Selection and Categorization . . . . . . . . . . . . . . . . . 70
iv
Chapter 8 Background Processes . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.1 tt̄ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
8.2 Data-driven Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8.2.1 Top-tagging and b-tagging correlations . . . . . . . . . . . . . . . . . 81
8.2.2 The systematic uncertainty . . . . . . . . . . . . . . . . . . . . . . . 85
Chapter 9 Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
9.1 Systematic Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
9.1.1 Experimental uncertainties . . . . . . . . . . . . . . . . . . . . . . . 92
9.1.2 Signal modeling uncertainties . . . . . . . . . . . . . . . . . . . . . . 94
9.1.3 Background modeling uncertainties . . . . . . . . . . . . . . . . . . . 95
9.1.4 Propagation of uncertainties to the data-driven background . . . . . 97
9.2 Profile-likelihood Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
9.3 The CLs method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
9.4 Fit validations with pulls and constraints . . . . . . . . . . . . . . . . . . . . 100
9.5 Nuisance parameter ranking . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter 10 Statistical Analysis Results . . . . . . . . . . . . . . . . . . . . . . 103
10.1 Validating the Data-driven Background Estimation . . . . . . . . . . . . . . 103
10.2 Test the Signal Region Fit with the Background-only Pseudo-data . . . . . . 106
10.3 Exclusion Upper Limit with the Background-only Pseudo-data . . . . . . . . 116
10.4 The Signal Region Fit with Observed data . . . . . . . . . . . . . . . . . . . 116
10.5 Exclusion Upper Limit with Observed Data . . . . . . . . . . . . . . . . . . 127
10.6 Systematic Uncertainty Rankings for a 4 TeV WR0 . . . . . . . . . . . . . . . 129
10.6.1 Background-only pseudo-data . . . . . . . . . . . . . . . . . . . . . . 129
10.6.2 Signal-plus-background pseudo-data . . . . . . . . . . . . . . . . . . 131
10.6.3 Observed data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
Chapter 11 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Appendix A The Definition of Electrons and Muons . . . . . . . . . . . . . . . . . 139
Appendix B Insignificant Background Processes . . . . . . . . . . . . . . . . . . . 141
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
v
LIST OF TABLES
Table 2.1: Three generations of quark duplets. . . . . . . . . . . . . . . . . . . . . . 12
Table 2.2: Three generations of lepton duplets. . . . . . . . . . . . . . . . . . . . . . 12
Table 3.1: Four regions employed by a data-driven background estimation are shown
as four cells on a two-dimensional plane. The horizontal and vertical axes
are represented by the tagging decision on the b-candidate jet and the
top-candidate jet, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 18
vi
LIST OF FIGURES
Figure 1.1: The Feynman diagram shows a quark (q) from one proton (p+ ) annihilates
with an antiquark of another kind (q¯0 ) from the other proton to produces
a W 0 boson. The latter then decays to a top quark (t) and an anti-
bottom quark (b̄). The vertical axis represents space and the horizontal
axis points in time. The diagram is produced by feynMF [21, 22]. . . . 5
Figure 2.1: The Feynman diagram shows parton showers from a vector boson’s decay.
Staight lines with an arrow are quarks. Curly lines are gluons. The
vertical axis represents space and the horizontal axis represents time.
Diagram cropped from [30]. . . . . . . . . . . . . . . . . . . . . . . . . . 10
Figure 4.1: The CERN accelerator complex. The injection chain for proton coliisions
at the LHC is composed of LINAC 2, PS Booster (BOOSTER on the
diagram), PS, and SPS. There are four detectors/experiments, ALICE,
ATLAS, CMS, and LHCb marked as yellow dots along the LHC ring. [42] 21
Figure 4.2: Increase of integrated luminosities over time of the data delievered by the
LHC, recorded by the ATLAS detector, and marked as good for physics
analysis. [44] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Figure 4.3: The number of interactions per bunch crossing averaged over a luminosity
block [45] is shown as integrated-luminosity-weighted distribution on the
graph for each year of data taking. The legends tell the overall average
for each year. [44] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 4.4: The figure points out the main subdetectors of the ATLAS detector. . . 24
Figure 4.5: A simplified represenation of a segment of a cross-sections of the AT-
LAS detector. The toroidal magnets around the muon spectrometer are
omitted. The tracking system corresponds to the inner detector (ID). . . 25
Figure 4.6: A 3D view of the barrel region of the inner detectors. [49] . . . . . . . . 27
Figure 4.7: The ATLAS calorimeter encloses the ID colored in grey and a hard-to-
seen solenoid magnet around it. The ECAL consists of a liquid-Argon
(LAr) barrel part (|η| < 1.475) and two LAr end-caps (EMEC, 1.375 <
|η| < 3.2). The tile calorimeter (barrel: |η| < 1.0 and extended barrel:
0.8 < |η| < 1.7) and two LAr hadronic end-caps (1.5 < |η| < 3.2, painted
in two sheds of copper) comprise the HCAL. At |η| between 3.1 and 4.9
on each side, there are three components of the LAr forward calorimeter
(FCAL). [46] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
vii
Figure 4.8: Segements of accordion-shaped calorimeters. The silver-colored lead ab-
sorbers are sandwiched by copper-clad Kapton electrodes. The electrode
receives ionizing signals from the liquid Argon filled in the gap between
lead absorbers. [51] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Figure 4.9: A wedge of tile hadronic calorimeters consisting of scintillator tiles inter-
leaving with steel absorbers. The coordinate system shown on the bottom
left follows the convention introduced at the beginning of this section.
The top left and center left diagrams demonstrate how a waveshifter fiber
transmitting light signals from scintillator tiles to a photomultiplier tube
(PMTs). The top figure shows the PMT’s position in the top drawer. The
plot is taken from [20] which adapts the original one in [52]. . . . . . . . 30
Figure 4.10: The pion energy resolution of the hadronic and electromagnetic calorime-
ters combined is shown as a function of the square root of pion energy in-
versed. The black (white) dots are measured with test beams in year 1996
(1994), while the white cross is derived from the simulation of hadronic
showers. The two fit functions differ in their description of detector
noises. [53] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 4.11: The pT resolution of the large-radius jet as a function of the average of the
di-jet transverse momentum from the in-situ calibration [54]. The symbol
|ηdet | means the absolute pseudorapidity of the jet axis with the detector’s
center taken as the coordinate’s origin. The data (black dashed line) has
a light band that sums the statistical and systematic uncertainties in
quadrature. The red band is the envelope of in-situ calibration applied
to three different QCD di-jet event generators. The yellow dashed line
corresponds to the reconstructed jet pT resolution relative to the truth-
leveled jet (Chapter 6) computed from events simulated by the Pythia
generator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Figure 4.12: The muons detected by the MS are bent by the toroidal magnets in the
barrel (green) and end-cap (black). With requirements of three layers of
meaurements, triggering (RPC and TGC) has a range of |η| < 2.4, limited
by the two outer TGC wheels. In comparison, precision-tracking (MDT
and CSC) can go futher to 2.7. [56] . . . . . . . . . . . . . . . . . . . . . 33
viii
Figure 4.13: A schematic diagram of the Run-2 ATLAS trigger and data acquisition
system. The top-right block shows the detector front-ends (FEs), some of
which at the CAL and MS are connected to the corresponding calorimeter
trigger and muon trigger at the first-level (L1) trigger block on the top-
left block. The entire time consumed by data transmission and logic
processing at the L1 is less than 2.5 µs before the cental trigger flags
the L1 Accept command back to the FE at a rate below 100 kHz. The
accepted event’s data from all detectors are then distributed through the
ReadOut System (ROS). The L1 trigger also sends the RoI information
to the High-level Trigger (HLT), which accesses the event data through
the ROS. As shown on the bottom right block, data passing the HLT’s
threshold at a rate of about 1 kHz is transferred to permanent storage for
offline analysis. (Figure adapted from [57]) . . . . . . . . . . . . . . . . 35
Figure 4.14: The architecture of the L1Calo during Run-2 (2015-2018). All modules
have output to Read-Out Drivers (RODs), allowing diagnostics, calibra-
tions, and performance studies. In particular, RoIs from the CP and the
JEP are routed though the CMX to the RODs for later retrieval by the
HLT. RODs pack, buffer and send signals to the ROS. [62] . . . . . . . . 37
Figure 5.1: The leading order Feynman diagram for the signal’s hard-scattering ma-
trix element. This diagram’s final state is referred as the parton level.
Time points from left to right. The spatial dimension is vertical. For
aesthetic purposes, the momentum flow is only accurate in the time di-
rection. The charge-conjugated diagram is not shown but included in the
event generation. The diagram is produced by feynMF [21] . . . . . . . 40
Figure 5.2: Feynman diagrams of NLO terms in calculating the cross-section σ(q q¯0 →
W 0 → tb̄), where q and q 0 represents an up-type quark and a down-type
antiquark emerging from protons, respectively. The vertical (horizontal)
axis corresponds to the space (time) axis. The charge-conjugated pro-
cesses are included in the cross-section calculation. Feynman diagrams
are made with TikZ-Feynman [64]. . . . . . . . . . . . . . . . . . . . . . 41
Figure 5.3: The cross-section σ(pp → W 0 ) times branching ratio B(W 0 → tb) with
W 0 ’s rest masses in 0.2∼5.0 TeV is evaluated with ZTOP [65–67] at
NLO of QCD. Figure (a) is for a left-handed chiral WL0 and (b) is for
a right-handed chiral WR0 . The central black curve is the nominal value,
including almost invisible statistical errors. The red band stands for
the theoretical uncertainties. Interfaced by LHAPDF [68], the PDF set
is PDF4LHC15_nlo_mc_pdfas documented by [69]. This set combines
CT14 [70], MMHT14 [71], and NNPDF3.0 [72] with technologies found
in [73, 74]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
ix
Figure 5.4: The relative theoretical uncertainty of σ(pp → W 0 → tb) for different
mW 0 and chiarlity. The uncertainty from shifting mt up (red) or down
(orange) by 1 GeV dominates at low mW 0 . Larger mW 0 sees increases of
uncertainty from both scaling µ (= µF = µR ) by a factor of 0.5 (blue)
or 2.0 (purple) and the one-sided PDF+αs uncertainty (gree). The latter
dominates in the highest region of mW 0 . . . . . . . . . . . . . . . . . . . 46
Figure 5.5: Parton-level (without FSR) invariant mass shapes of the top quark and
bottom quark decaying from (a) left-handed or (b) right-handed W 0 with
5 different rest masses are compared by normalizing the area to 1. Each
distribution is a cumulation of around 300K events. . . . . . . . . . . . . 48
Figure 5.6: (a) The top-quark transverse momenta and (b) the top-quark to bot-
tom quarks pseudorapidity difference’s absolute value for right-handed
W 0 with 5 different rest masses are compared by normalizing the area to
1. Left-handed W 0 bosons share similar kinematic properties . . . . . . . 49
Figure 5.7: (a) The angle between the down-quark’s momentum in the top-quark’s
rest frame and the top-quark’s momentum in the W 0 rest frame and (b)
the maximum angular distance ( (∆φ)2 + (∆η)2 ) between the top-quark
p
and its three decay products in the lab frame for a left-handed (blue) or
right-handed (red) W 0 with a 4 TeV rest mass are compared by nor-
malizing the area to 1. The rightest bin in (b) is an overflow bin that
encompasses everything above 1. . . . . . . . . . . . . . . . . . . . . . . 50
Figure 6.1: The probabilities of truth-level large-radius jets having a top quark and
various combinations of the top decay products within ∆R < 0.75 from
the jet axis are shown as a function of the top quark’s transverse momen-
tum. A top quark decays first to a bottom quark (b) and a W -boson,
which further decays to two first or second generation quarks (q1 , q2 ). [80] 55
Figure 6.2: The efficiencies of three large-R jet triggers for data collected in the year
2015 (black), 2016 (red), and 2017 to 2018 (blue) are shown as a function
of the (offline) leading large-radius transverse momentum. The legend
lists each trigger’s (online) transverse energy thresholds. The efficiency is
computed by the coincidence of the targeted trigger and a lower threshold
trigger – the (online) leading small-radius jet’s ET > 260 GeV. All three
trigger efficiencies reach 100% at (offline) leading large-radius jet pT >
500 GeV (drawn as a vertical black line). The shaded area represents
statistical uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
Figure 6.3: Feynman diagrams of (a) the top-quark signal and (b) a light-quark back-
ground for the top-tagging algorithm. . . . . . . . . . . . . . . . . . . . 59
x
√ √
Figure 6.4: The splitting scales d23 and d12 of large-R jets are shown for two cases
– top-quark decay from a W 0 boson with a rest mass of 4 TeV (red line)
and a gluon or non-top quark from QCD multi-jet processes simulated
by Pythia (black line with yellow fill). In the first case, large-R jets
are matched to top quarks first by requiring ∆R < 0.75 between the
reconstructed jet and its closet truth-level jet, and between the truth-level
jet and the top quark. Moreover, there must be at least one B hadron
ghost-associated [92] with the truth-level jet. Lastly, the reconstructed
jet mass has to surpass 140 GeV. This definition of top-quark jets is the
one used in training the DNN top-tagger. . . . . . . . . . . . . . . . . . 60
Figure 6.5: The ratios of 3-subjettiness (τ3 ) to 2-subjettiness (τ2 ) and of 2-subjettiness
(τ2 ) to 1-subjettiness (τ1 ). The definition of top-quark matched large-R
jets is written on the caption under the plots for the splitting scale.
Pythia simulated multijet events’ subjettiness ratios have a spike at 0
since large-R jets consisting solely of parton showering could have only
one or two constituents. The former causes τ1 to be 0 and the latter
causes τ2 to be 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Figure 6.6: The Receiver Operating Characteristic (ROC) curves of different types
of top-tagging studied in [80]. The curves on the plot show how many
QCD large-radius jets are rejected per one accepted (y-axis), given what
fraction of top-quark large-R jet is retained (x-axis) by taking a cutoff(s)
on relevant tagging variable(s). The ROC curve of the DNN top-tagger
employed in this analysis is shown in solide red, overlapping with the
curve of the Boosted Decision Tree (BDT) top-tagger. . . . . . . . . . . 62
Figure 6.7: Three b-taggers’ ability to reject light-quark jets as a function of the pT of
Particle Flow (PFlow) small-R jets are compared. The plot show one out
of how many light-quark small-R jets, a large background to this analysis,
are accepted, given a 77% chance of retaining a b-quark jet with pT > 25
GeV. The value fc means the proportion of c-quark jet included with light
quark and gluon jets in the background events. [106] . . . . . . . . . . . 65
Figure 6.8: The truth-level (a) top-quark jet transverse momentum and (b) invariant
mass Mtb for right-handed W 0 bosons with 5 different masses are shown
by rescaling all histograms’ area to unity. . . . . . . . . . . . . . . . . . 67
Figure 6.9: The reconstruction-level (a) top-quark jet transverse momentum and (b)
invariant mass Mtb for right-handed W 0 bosons with 5 different masses.
The solid-line histograms are scaled to have unit area, while the dashed-
line histograms are normalized by the solid-line histogram’s original area.
Their difference lies in that dashed histogram excludes those events having
its (W 0 -coupled) bottom-quark small-radius jet with pT ≤ 500 GeV. . . 68
xi
Figure 7.1: The flow chart showing the classification of events that lead to differnet
regions. The top input block’s top-candidate jet means the top-tagged
large-radius jets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Figure 7.2: For each of the two categories, events are divided into six regions on a
two-dimensional plane. The vertical axis represents the top-candidate
jet’s top-tagging score from low to high upward, while the horizontal
axis represents the b-candidate jet’s b-tagging score from low to high
rightward. SR stands for the signal region, VR for the validation region,
TR for the template region, and CR for the control region. . . . . . . . . 74
Figure 7.3: The absolute pseudorapidity difference between the top-candidate jet and
the b-candidate jet for a 4 TeV right-handed W 0 signal and the two main
backgrounds are compared in the signal region with the highest back-
ground rejection rate via the top-tagging and b-taggging (Signal Region
1 in Figure 7.2). Both the top-quark-pair (tt̄) and QCD multi-jet back-
grounds are simulated by MC generators (details in Chapter 8), and their
sums are normalized to have unit area. . . . . . . . . . . . . . . . . . . . 75
Figure 8.1: The Feynman diagrams of top quark pair production at the leading order
of strong coupling. The vertical axis represents space, and the horizontal
axis represents time. Feynman diagrams are made with TikZ-Feynman [64]. 77
Figure 8.2: The invariant mass distribution of a large-radius top-candidate jet and
a small-R b-candidate jet in region SR1 (right) and the neighboring re-
gion TR1 (left) (c.f. Figure 7.2). (a) The template region has the data
plotted against two channels of tt̄ background – the all-hadronic and
the non-all-hadronic – shown as stacked histograms. Their statistical
uncertainties are summed in quadrature into a single tt̄ statistical error.
The ratio of the summed tt̄ distribution to the data is shown in the lower
panel with statistical erros from both data and the t̄ background. (b)
In the signal region, the data-driven background (see the next section)
is drawn above the two tt̄ channels with separate statsitical errors. The
ratio shown in the lower panel is the division of the tt̄ to the total back-
ground. Covariances are taken into account in error propagations. On
either figure’s ratio plot, there is a horizontal reference line showing the
ratio of total events. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Figure 8.3: The invariant mass Mtb distribution of simulated QCD multi-jet back-
ground in the signal region SR1 (black dot) is compared with its extrap-
olation (blue lines) from the template region TR1 and the control regions
CR1 and CR2. The vertical axis is the number of events for a bin width of
200 GeV. The lower panel shows the ratio of the signal region histogram
to the extrapolated one. . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
xii
Figure 8.4: Examplified LO Feynman diagrams of the QCD multi-jet background
at the LHC. The vertical axis represents space, and the horizontal axis
represents time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Figure 8.5: Figure (a) shows fractions of different parton flavors associated with the
top-candidate/proxy jet as a function of the top-tagging score in nature
logs. The top-tagging dependence of non-b-tagged jets are presented in
figure (b). The last two bins’ lower edges, corresponding to the 80%
working point and the 50% working point, are not exactly the same as
the label. The invariant mass Mtb is between 1.2 and 1.4 TeV for both
figures. Light quarks include up, down, and strange quarks as their masses
are below the QCD condensation scale. The charm and bottom quark
fractions are multiplied by factors for viewing purposes. . . . . . . . . . 83
Figure 8.6: (a) The b-candidate jet’s parton flavor fraction as a function of the top-
candidate/proxy jet’s top-tagging score’s natural log in the “1 b-tag in”
category. The last two bins’ lower edges, corresponding to the 80% work-
ing point and the 50% working point, are not the same as the label.
The invariant mass Mtb is between 3 TeV and 3.4 TeV. Non-b-tagged b-
candidate jets are shown for reading the b-tag ratio variations. (b) The
b-tagging ratio (blue with left vertical-axis) and top-tagging ratio (red
with right vertical-axis) as a function of the b-candidate jet pT in the
same Mtb range and category. The top-tagging ratio is between the 50%
WP top-tagged region and the upper control region (CRXa, X = num-
bers), as defined in Figure 8.7. The b-tagging ratio is also calculated in
the control region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Figure 8.7: The categorization in the control regions (CRs) defined in Figure 7.2 is
refined for computing the data-driven background’s systematic uncertain-
ties. The two new thresholds are placed on e−4 and e−7 of the top-tagging
DNN score. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Figure 8.8: Validating plots for the data-driven background’s systematic uncertain-
ties by the simulated QCD multi-jet background. In each diagram, the
“SR” histogram is obtained directly from the top-tagging and b-tagging,
and the extrapolation is the result of the data-driven method. The red
hatches show the statistical uncertainty of the extrapolation. The error
bars on the lower panel only include the statistical uncertainty for the
SR distribution. The purple triangular dots show the correlation fac-
tor obtained in the control region of each “b-tag in” category, yielding
the cyan-colored systematic uncertainties after symmetrization (same as
a reflection) and smoothing procedure. . . . . . . . . . . . . . . . . . . . 89
xiii
Figure 8.9: The data-driven background’s systematic uncertainties. The extrapola-
tion is the data-driven background derived from data subtracting the tt̄’s
simulated events. Statistical uncertainties are drawn separately for the
extrapolation and the SR histogram as in the previous figure. The pur-
ple triangular dots show the correlation factor obtained in the control
region of each “b-tag in” category, yielding the cyan-colored systematic
uncertainties after symmetrization (same as a reflection) and smoothing
procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Figure 10.1: Data and predicted background Mtb distribution in the validation region
(top-tagged at the 80% but below the 50% working point and in the “0
b-tag in” cateogry) (a) before and (b) after the profile-likelihood fit. The
hatched uncertainty bands include the systematic uncertainties and the
statisitcal uncertainties of the background predictions. . . . . . . . . . . 104
Figure 10.2: The pull and contraint plots of the data-driven-background systematic
uncertainties in the background-model fit to data in the validation region.
The central value (θ0 ) is the pre-fit value, which is 0 by contruction. For
each nuisance parameter, the black dot shows the post-fit value (θ̂) with
the post-fit uncertainty as its error bar, which is in general than the pre-fit
uncertainty of unit size (∆θ). . . . . . . . . . . . . . . . . . . . . . . . . 105
Figure 10.3: Predicted background Mtb distribution in the signal regions: (a) SR1,
(b) SR2, and (c) SR3 before the profile-likelihood fit. The hatched un-
certainty bands include the systematic uncertainties and the statisitcal
uncertainties of the background predictions. . . . . . . . . . . . . . . . . 107
Figure 10.4: Predicted background Mtb distribution in the signal regions: (a) SR1, (b)
SR2, and (c) SR3 after the profile-likelihood fit to the pseudo-data with “µ
= 0.” The hatched uncertainty bands include the systematic uncertainties
and the statisitcal uncertainties of the background predictions. . . . . . . 108
Figure 10.5: The nuisance parameter pulls for jet energy scale uncertainties common
to large-radius jets and small-radius jets after a profile-likelihood fit to
the pseudo-data with “µ = 0.” “EtaInter” means the in situ calibrations
across different η regions of the detector. . . . . . . . . . . . . . . . . . . 109
xiv
Figure 10.6: The nuisance parameter pulls for large-radius jet energy scale and reso-
lution uncertainties after a profile-likelihood fit to the pseudo-data with
“µ = 0.” “TopoUncer” refers to jet energy modeling uncertainty for top-
quark jets. “JER” denotes jet energy resolution. “jetEffNP1” is the first
component in the principle component decomposition for propagating the
jet energy scale uncertainty. Each “EffectiveNP” is an eigenvector in the
principle component analysis. The suffices, “Stat,” “Model,” “Detect,”
and “Mixed” means that the eigenvector combines statistical, modeling,
detector, or a mixture of systematic error sources. . . . . . . . . . . . . . 109
Figure 10.7: The nuisance parameter pulls for small-radius jet energy scale and reso-
lution uncertainties after a profile-likelihood fit to the pseudo-data with
“µ = 0.” “PunchThrough” means the hadronic shower of a jet is too deep
to be contained by the ATLAS calorimeter. “Rho” is the energy profile
of an event used to correct for pile-up contributions in the calorimeter
measurements. “restTerm” is a quadratic sum of the most insignificant
JER components. “NonClos” and “highE” refer to non-closure and high-
energy, respectively. “BJES” means the jet energy scale for a bottom-
quark jet. Some other terms are defined in the captions of Figure 10.5
and Figure 10.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Figure 10.8: The nuisance parameter pulls for large-radius jet mass scale and resolution
uncertainties after a profile-likelihood fit to the pseudo-data with “µ = 0.”
“Rtrk” is the ratio between jet mass estimated by tracks to that measured
by calorimeters. The difference in Rtrk between simulation and data
serves as a systematic uncertainty. “COMB” refers to “combined mass,”
a mass observable combining the tracking and calorimeter measurements. 110
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Figure 10.9: The nuisance parameter pulls for the top-tagging uncertainties after a
profile-likelihood fit to the pseudo-data with “µ = 0.” The number 80
and 50 mean the large-radius jet is top-tagged at the 80% working point
but not the 50% working point or top-tagged at the 50% working point,
respectively. “Light-jet b-tag” are b-tagging nuisance parameters for the
light-quark or gluon jet since the top-tagging calibration selects events
with b-tagged jets. “Signal Eff.” accounts for the chirality difference be-
tween the tt̄ events with unpolarized top-quarks and the signal events with
polarized top-quarks. “Other Top Eff.” are uncertainty in top-tagging effi-
ciency for top-quark jets with non-optimal reconstructions such as missing
top-quark-decay products within a truth-level jet. “Signal Propagated”
is a mixture of insignificant jet energy scale uncertainty components than
those propagated via the JES uncertainties for well-reconstructed top-
quark jets. “High-Pt Ext.” accounts for uncertainties in high transverse
momentum beyond the range calibrated with respect to data. “Bin-Var”
concerns the pT -bin choices in fitting the efficiency curve. “Di-jet” means
a particular type of QCD multi-jet event with two high pT jets. “SF”
stands for Scale Factor, which scales the tagging probability in simulated
events to the calibrated value. “hdamp” is the matching scale for the
NNLO event generator, Powheg [125]. “ME” refers to the matrix el-
ement. “Shower” is the parton shower. Alternative event generators of
ME and showers are discussed in Chapter 9. . . . . . . . . . . . . . . . . 111
Figure 10.10:The nuisance parameter pulls for the b-tagging uncertainties after a profile-
likelihood fit to the pseudo-data with “µ = 0.” The name “(I) EV NP
3” means the third eigenvector of nuisance parameters for calibrating
the b-tagging probability for a light-hadron jet (no heavy hadrons that
contains a charm or bottom quark). The same convention applies to a
charm-hadron jet by replacing the capitalized letter “I” with “C” and
to bottom-hadron jet by replacing it with “B.” “High-Pt Extrapolation”
refers to the uncertainties caused by insufficient data to calibrate b-tagging
probabilities for small-radius jets with high transverse momenta (for more
descriptions see Chapter 9 and Ref. [127]). . . . . . . . . . . . . . . . . . 112
Figure 10.11:The nuisance parameter pulls for the modeling uncertainties of the data-
driven background estimation after a profile-likelihood fit to the pseudo-
data with “µ = 0.” The symbols “LowMtb” and “HighMtb” mean that
the nuisance parameter varies the data-driven background in invariant
mass Mtb below or above 2 TeV. The region code (Figure 7.2) at the end
further specifies the signal region whose data-driven background is to be
varied. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
xvi
Figure 10.12:The nuisance parameter pulls for the modeling uncertainties of the top-
quark pair background after a profile-likelihood fit to the pseudo-data
with “µ = 0.” The term “shower,” “pdf,” “muR,” “muF,” “ME,” “ISR,”
“FSR” refer to the parton shower generator, Parton Distribution Fuction,
renormalization scale, factorization scale, initial state radiation, and final
state radiation respectively. These terms are described in section 2.1 and
subsection 9.1.3. The region code (Figure 7.2) at the end further speci-
fies the signal region where the tt̄ background is to be varied. Nuisance
parameters of “DataDriven (DataDriv)” account for the uncertainty from
subtracting the tt̄ background in the specified signal region’s neighboring
template region (the b-candidate jet is not b-tagged rather than tagged)
in the data-driven background estimation. . . . . . . . . . . . . . . . . . 113
Figure 10.13:The nuisance parameter pulls for the 1.7% luminosity uncertainty and the
4% top-quark pair background production cross-section uncertainty after
a profile-likelihood fit to the pseudo-data with “µ = 0.” . . . . . . . . . . 114
Figure 10.14:The correlation matrix between nuisance parameters after a profile-likelihood
fit to the pseudo-data with “µ = 0.” The matrix is symmetic over the
diagonal from the top left to the bottom right corner. Only nuisance pa-
rameters with more than 20% of correlation with another are shown. The
meaning of individual nuisance parameters can be found in the preceding
Figure 10.5 to Figure 10.13. . . . . . . . . . . . . . . . . . . . . . . . . . 115
Figure 10.15:The upper limit on the WR0 production cross-section times the decay
branching ratio of WR0 → tb at the 95% Confidence Level is evaluated
with the background-only pseudo-data. The blue dashed curve indicates
the median, while the green and yellow bands show the one and two Gaus-
sian quantiles of the limit under statistical plus systematic uncertainties.
The Next-to-Leading-Order theoretical cross-section times the branching
ratio is drawn as the red dashed curve with an asymmetric theoretical
uncertainty band (Chapter 5). . . . . . . . . . . . . . . . . . . . . . . . . 117
Figure 10.16:Data versus background Mtb distributions in the signal regions: (a) SR1,
(b) SR2, and (c) SR3 before the profile-likelihood fit. The hatched un-
certainty bands include the systematic uncertainties and the statisitcal
uncertainties of the background predictions. . . . . . . . . . . . . . . . . 118
Figure 10.17:Data versus background Mtb distribution in the signal regions: (a) SR1,
(b) SR2, and (c) SR3 after the profile-likelihood fit with under the background-
only hypothesis. The hatched uncertainty bands include the systematic
uncertainties and the statisitcal uncertainties of the background predictions.119
xvii
Figure 10.18:The nuisance parameter pulls for jet energy scale uncertainties common
to large-radius jets and small-radius jets after a profile-likelihood fit to
data under the background-only hypothesis (µ = 0). The meaning of the
nuisance parameters are explained in Figure 10.5. . . . . . . . . . . . . . 120
Figure 10.19:The nuisance parameter pulls for large-radius jet energy scale and resolu-
tion uncertainties after a profile-likelihood fit to data under the background-
only hypothesis (µ = 0). The meaning of the nuisance parameters are
explained in Figure 10.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Figure 10.20:The nuisance parameter pulls for small-radius jet energy scale and resolu-
tion uncertainties after a profile-likelihood fit to data under the background-
only hypothesis (µ = 0). The meaning of the nuisance parameters are
explained in Figure 10.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Figure 10.21:The nuisance parameter pulls for large-radius jet mass scale and resolution
uncertainties after a profile-likelihood fit to data under the background-
only hypothesis (µ = 0). The meaning of the nuisance parameters are
explained in Figure 10.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Figure 10.22:The nuisance parameter pulls for the top-tagging uncertainties after a
profile-likelihood fit to data under the background-only hypothesis (µ =
0). The meaning of the nuisance parameters are explained in Figure 10.9. 122
Figure 10.23:The nuisance parameter pulls for the b-tagging uncertainties after a profile-
likelihood fit to data under the background-only hypothesis (µ = 0). The
meaning of the nuisance parameters are explained in Figure 10.10. . . . 123
Figure 10.24:The nuisance parameter pulls for the modeling uncertainties of the data-
driven background estimation after a profile-likelihood fit to data under
the background-only hypothesis (µ = 0). The meaning of the nuisance
parameters are explained in Figure 10.11. . . . . . . . . . . . . . . . . . 123
Figure 10.25:The nuisance parameter pulls for the modeling uncertainties of the top-
quark pair background after a profile-likelihood fit to data under the
background-only hypothesis (µ = 0). The meaning of the nuisance pa-
rameters are explained in Figure 10.12. . . . . . . . . . . . . . . . . . . . 124
Figure 10.26:The nuisance parameter pulls for the 1.7% luminosity uncertainty and the
4% top-quark pair background production cross-section uncertainty after
a profile-likelihood fit to data under the background-only hypothesis (µ
= 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
xviii
Figure 10.27:The correlation matrix between nuisance parameters after a profile-likelihood
fit to data under the background-only hypothesis (µ = 0). The matrix
is symmetic over the diagonal from the top left to the bottom right cor-
ner. Only nuisance parameters with more than 20% of correlation with
another are shown. The meaning of individual nuisance parameters can
be found in the preceding Figure 10.5 to Figure 10.13. . . . . . . . . . . 125
Figure 10.28:The upper limit on the WR0 production cross-section times the decay
branching ratio of WR0 → tb at the 95% Confidence Level is evaluated
with data. The blue dashed curve indicates the median expected limit,
while the green and yellow bands show the one and two Gaussian quantiles
under statistical plus systematic uncertainties. The black solid curve
shows the observed limit. The Next-to-Leading-Order theoretical cross-
section times the branching ratio is drawn as the red dashed curve with
an asymmetric theoretical uncertainty band (Chapter 5). . . . . . . . . . 128
Figure 10.29:The nuisance parameter rankings for the 4 TeV WR0 signal with respect
to the pseudo-data with “µ = 0.” Color-filled bands show the fifteen-
highest impact on the best-fit µ (∆µ) by fixing a particular nuisance
parameter to its best-fit value plus-or-minus the post-fit error (θ̂ ± ∆θ̂).
The unfilled bands with colored boundaries correspond to ∆µ when the
nuisance parameter is shifted by its pre-fit uncertainty (θ̂±∆θ). The scale
of ∆µ is shown on the top horizontal line. The bottom horizontal scale
concerns with the black dots and bars, the best-fit nuisance paramters and
their post-fit errors, as in the pull plots. The captions from Figure 10.5
to Figure 10.13 explain the naming convention of the nuisance parameters. 130
Figure 10.30:The nuisance parameter rankings for the 4 TeV WR0 signal with respect
to the pseudo-data with “µ = 1.” Color-filled bands show the fifteen-
highest impact on the best-fit µ (∆µ) by fixing a particular nuisance
parameter to its best-fit value plus-or-minus the post-fit error (θ̂ ± ∆θ̂).
The unfilled bands with colored boundaries correspond to ∆µ when the
nuisance parameter is shifted by its pre-fit uncertainty (θ̂±∆θ). The scale
of ∆µ is shown on the top horizontal line. The bottom horizontal scale
concerns with the black dots and bars, the best-fit nuisance paramters and
their post-fit errors, as in the pull plots. The captions from Figure 10.5
to Figure 10.13 explain the naming convention of the nuisance parameters. 132
xix
Figure 10.31:The nuisance parameter rankings for the 4 TeV WR0 signal with respect to
the observed data. Color-filled bands show the fifteen-highest impact on
the best-fit µ (∆µ) by fixing a particular nuisance parameter to its best-
fit value plus-or-minus the post-fit error (θ̂ ± ∆θ̂). The unfilled bands
with colored boundaries correspond to ∆µ when the nuisance parameter
is shifted by its pre-fit uncertainty (θ̂ ± ∆θ). The scale of ∆µ is shown
on the top horizontal line. The bottom horizontal scale concerns with
the black dots and bars, the best-fit nuisance paramters and their post-fit
errors, as in the pull plots. The captions from Figure 10.5 to Figure 10.13
explain the naming convention of the nuisance parameters. . . . . . . . . 133
Figure 10.32:The nuisance parameter pulls for the modeling uncertainties of the data-
driven background estimation after a profile-likelihood fit to data with
µ determined by the fit. The meaning of the nuisance parameters are
explained in Figure 10.11. . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Figure B.1: The Feynman diagrams for hard-scatter matrix elements of the single-
top processes at the LHC. The vertical axis represents space and the
horizontal axis represents time. The diagrams above implicitly include
three colors of quarks, eight types of gluons, and the combination rules
between colors and gluon types. The q and q 0 is a duo of up-type and
down-type quarks in the first two generation. The four diagrams are
labelled by the virtual particle’s Mandelstam variables and the final state
particles. Feynman diagrams are made with TikZ-Feynman [64]. . . . . . 142
Figure B.2: The Feynman diagrams for hard-scatter matrix elements of the V +jet
processes at the LHC. The vertical axis represents space and the hori-
zontal axis represents time. The diagrams above implicitly include three
colors of quarks, eight types of gluons, and the combination rules between
colors and gluon types. For the W -boson diagrams, q and q 0 is a duo of
one of the two lighter up-type quark and one down-type quark. For the
neutral boson diagrams, q is any quark but the top quark. γ∗ means a
photon carrying nonzero mass, i.e., being off-shell. Even though the ini-
tial state’s flavor and color degrees of freedom are reminiscent of the QCD
multi-jet events, the small electroweak coupling suppresses the vector bo-
son production relative to the former background. Feynman diagrams are
made with TikZ-Feynman [64]. . . . . . . . . . . . . . . . . . . . . . . . 144
xx
Figure B.3: (a) The invariant mass distribution of a large-radius top-candidate jet and
a small-R b-candidate jet in one of the template regions. The template
region has the data plotted with two types of V +jets distributions. The
W +jets channel, which has a W boson and some high pT partons in the
final state, is stacked on top of the Z/γ∗+jets channel, where the W boson
is replaced with a neutral vector boson. The statistical uncertainties of
the two channels are summed in quadrature into a single V +jets statistical
error. The ratio of the V +jets to data is shown in the lower panel with
the statistical errors of V +jets and data. There is a horizontal line given
by the ratio between total area. (b) The b-tagging ratio – the ratio of
the signal region to the templete regions shown in (a) – is shown as a
function of Mtb . Since simulated events of V +jets are scarce, the two
channels are added together, and the number of bins is reduced. The
b-tagging ratio obtained in the upper control region (top-tagging DNN
> e−4 while below the 80% working point) with tt̄-subtracted data is
compared. The hatched area represent the latter’s statistical error, which
is miniscule compared with the V +jets. The bottom panels shows the
division of the two ratios with the statistical uncertainties propagated. . 146
xxi
Chapter 1
Introduction
1.1 An Introduction to Particle Physics
The history of particle physics is an ongoing record of humans uncovering the fundamental
principles and constituents of nature. At the beginning of the twentieth century, physicists
raised some questions when they obtained inexplicable results from observing the inner struc-
tures of atoms [1] and decays of particles originating from cosmic rays or their interactions
with the earth’s atmosphere [2].
A question concerning the atomic structure at the time was, “how could the electrons in
an atom move around the nucleus without falling into it by losing their energy to electromag-
netic radiation according to classical electrodynamics?” Meanwhile, it was mysterious that
the energy and electric charge conservations were insufficient to predict if a decay process
could take place or not. Moreover, some particles had unexplainedly longer mean lifetimes
than others.
These puzzles, at first, only had crude solutions rather than self-contained ones. Regard-
ing the atomic structure, one of the initial hypothesis was that electrons in an atom behaved
like standing waves whose wavelengths were inversely proportional to their momenta. Re-
garding the questions of particle decays, physicists assigned “quantum numbers” to each type
of particle and hypothesized that null observations or rare occurrences of a decay process
1
were due to nonconservations of quantum numbers. As an example, a proton cannot decay
to an electron and a photon since this process violates both the lepton number and baryon
number conservation. Also, Kaons are known to decay infrequently because all of their decay
processes do not conserve “the strange number.”
Later, revolutionary theories arrived one by one, providing quantitative explanations
that incorporated more phenomena at once. To start with, the wave-like nature of electrons,
together with the particle-like nature of light, indicated by the photo-electric effect, for
example, was unified in quantum mechanics. Under quantum mechanics, positions and
momenta of matter cannot be measured arbitrarily precisely at the same time. Instead, only
their distributions from measurements are predicted by the theory.
The generalization of quantum mechanics to field theories, together with the gradual
understanding that all fundamental interactions are mediated by gauge bosons, finally led
to the birth of the Standard Model (SM). This general theory sheds light on the various
structures behind the particle decays. Take the slow decays of Kaons as an example. It
turns out that each Kaon consists of a strange quark (the identity of the strange number)
and an up quark or a down quark. When a Kaon decays, all of the lighter particles it can
turn into do not contain a strange quark. As a result, the strange quark has to turn into an
up quark via the weak interaction carried by the W boson. Since the W boson is a massive
particle, this interaction has a short effective range, leading to a low likelihood of the Kaon
decay.
The SM has brilliantly passed numerous experimental confirmations. The top quark [3, 4]
and the bottom quark [5] are revealed at Fermilab in the United States. Across the Atlantic,
the European Council for Nuclear Research (CERN) has hosted experiments that discovered
other particles predicted by the SM, such as the W and Z bosons found by the UA1 [6,
2
7] and UA2 [8, 9] collaborations and the Higgs boson by the ATLAS [10] and CMS [11]
collaborations in 2012. Since then, the ATLAS and CMS collaborations have collected more
data and have done an enormous amount of researches based on them. All of the available
results are still consistent with the predictions of the SM.
1.2 The Road Forward
Coming through all the glory days of advancement, could it be said that the tale of particle
physics has come to an end? Certainly not! The SM’s foundation, Quantum Field Theory,
gives infinite quantum corrections to probability amplitudes when integrating to infinitely
high energy [12]. Even though (through a technique called renormalization) this problem can
be removed systematically such that the theoretical values of lifetimes and interaction rates
of particles become finite [12], this constraint leads to beliefs that Quantum Field Theory
cannot be valid at arbitrarily high energy scales. Instead, some other new physical models
will be required to explain new phenomena that might appear at high energy scales. Planck
scale, a high energy scale at 1019 GeV [13], where gravity has similar strength as the other
fundamental interactions formulated by the SM, is regarded by many as a definite point for
the SM to be inadequate.
Even though the Planck scale foretells the existence of physics beyond the SM, it is
around 1015 times more than the energy achievable by the Large Hadron Collider (LHC)
at 13 TeV (=1.3 × 104 GeV) today, leaving no indication of new physics accessible in the
foreseeable future. Even though there are other phenomena beyond the SM’s scope, such as
neutrino oscillation and dark matter, the new physics behind them can appear at a very high
energy scale while being consistent with current experimental results [14]. The only strong
3
motivation for new physics beyond the SM at the LHC energy is the “hierarchy problem.”
It means the energy-dependent quantum corrections to the Higgs mass at several TeV are
significantly larger than the experimental value [15]. In other words, new physics appearing
at a higher energy scale could not easily contrive to produce the SM Higgs’ mass [14].
But since the hierarchy problem is a conceptual one, it does not entail any experimental
discrepancy which requires physics beyond the SM at the LHC.
Despite the lack of indication, experimentalists at the LHC still search for new physics
models that solve the SM’s conceptual problems and embed it into a more comprehensive
theory. For example, Supersymmetry has been probed by many experimentalists at the LHC
since it could alleviate the hierarchy problem, unify all three interactions in the SM at the so-
called Grand Unification Theory (GUT) scale of 1016 GeV [13] and incorporate candidates of
dark matter. Though searches for Supersymmetry resemble those for the electroweak gauge
bosons and the Higgs boson predicted by the SM, the former is more difficult due to the
abundance of insufficiently constrained parameters in typical Supersymmetric models.
In addition to the direct searches for new physics, “precision measurements,” which
include measuring the parameters of the SM and counting events produced at a low rate
by the SM processes, have been done at the LHC. Through a laborious effort to minimize
uncertainties, they are immensely sensitive to potential inconsistencies between data and
the SM. The power of precision measurements is seen from the discovery of CP violation in
weak interactions via measuring the frequencies of a rare Kaon decay KL → ππ [16] that
happened much earlier than the first theoretical account [17] for it.
“Generic searches” are methods to facilitate the quest for new physics as well. The
idea is to find out new physics without discerning the exact origin of it. One example
of this is searching for a new type of particle appearing in many new physics models. If
4
the LHC has enough energy to produce the particle, there will be a peak, referred to as
a resonance, in the invariant mass distribution of the particles originating from its decay.
Discoveries of new particles by resonances occurred many times in the twentieth century,
such as the J/Ψ particle at SLAC [18] and Brookhaven [19]. Since the LHC collides protons
at unprecedentedly high energy, it faciliates the generic search for any particle with a mass
much higher than the most massive SM particle, the top quark, weighing 173 GeV [20].
1.3 The Search for W 0
p+
t
q
W0
q0
b̄
+
p
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
Figure 1.1: The Feynman diagram shows a quark (q) from one proton (p+ ) annihilates with
an antiquark of another kind (q¯0 ) from the other proton to produces a W 0 boson. The latter
then decays to a top quark (t) and an anti-bottom quark (b̄). The vertical axis represents
space and the horizontal axis points in time. The diagram is produced by feynMF [21, 22].
One of the massive particle that might be found by resonance searches at the LHC is the
5
W 0 boson. It is a one-unit-charged and spin-one particle, which is produced by annihilation
between quarks inside the smashed protons at the LHC. This thesis looks for an above-a-
TeV-mass W 0 boson through its decay to a top quark and a bottom quark (Figure 1.1).
From a theoretical viewpoint, W 0 could be one of the particles emergeing in composite Higgs
models or an extra-dimensional state of the W boson [20]. The composite Higgs models
resolve the hierarchy problem by providing a natural cut-off to the quantum corrections to
the Higgs masses [13]. Some extra-dimensional theories can reduce the Planck scale such
that the quantum corrections are naturally small [13].
Searching for the W 0 boson via the final state of a top quark and a bottom quark is
valuable as there are models where the W 0 only couples to quarks [23] or even preferentially
to the third generation fermions [24]. Furthermore, the final state has unique features helping
to reduce SM backgrounds. Top quarks decay predominantly via weak interactions to a
W boson and a bottom quark. About two-thirds of the W boson turns further into two
quarks [20] – the hadronic mode targeted by this analysis. Since the W 0 boson mass of interest
is about ten or more times the top quark mass, the top quark’s decay products constitute a
Lorentz-boosted large-radius jet with substructures distinguishable from ordinary jets [25].
The bottom quark is also identifiable as a bottom-quark-containing hadron has a longer
lifetime and higher mass than most other types hadrons [20].
Searches for W 0 boson productions with a t and b quark final state had been done at
the Tevatron. The maximum excluded mass ranges of a right-handed W 0 from CDF [26]
and DØ [27] collaborations are 800 and 885 GeV. They made use of data recorded when the
Tevatron collided a proton beam with an anti-proton beam at a center-of-mass energy of
1.96 TeV. Recently, having data recorded with proton-proton collisions at a center-of-mass
energy of 13 TeV at the LHC, the ATLAS [28] and CMS [29] collaborations pushed the mass
6
limit further to 3.25 and 3.6 TeV each.
Extending the search for the W 0 decaying a top quark and a bottom quark, the full
Run-2 dataset recorded with the ATLAS detector at a collision energy of 13 TeV is analyzed
in this thesis. In each event, the boosted and hadronic decay of the top quark is identified
using a large-radius jet. And the bottom quarks coming from the W 0 and the top quark are
selected using small-radius jets. The signal region is established by applying a top-tagging
algorithm to the large-radius jet and a b-tagging algorithm to a back-to-back small-radius jet.
The presence or not of any small-radius jet fulfilling a threshold of the b-tagging algorithm
within the large-radius jet classifies the events into two categories
Despite the two algorithms, the signal regions still have a substantial amount of SM
background events with (or resembling) a boosted top jet accompanied by a bottom quark.
Some background events have a pair of top quarks in their final states, which can be estimated
by Monte Carlo (MC) simulations. While the most significant background, the production
of multi-jet events via the QCD strong interaction, is modeled by extrapolation from the
control regions, defined by inverting either one or both of the two tagging requirements for
each category. Four-momenta of the large-radius jet and the small-radius jet are summed
into a W 0 boson proxy. The resulting mass spectrums in the signal regions are fitted in a
profile-likelihood analysis to set exclusion limits on the cross-section for the production of a
W 0 boson with decay to a top quark and a bottom quark.
1.4 The Structure of This Thesis
This thesis details the analysis, which is separated into eleven chapters, excluding the ap-
pendices. After this introduction, the SM and some of its extensions containing the W 0
7
boson is summarized in Chapter 2. Chapter 3 will be about the analysis strategy to illumi-
nate the arrangement of the remaining chapters. In Chapter 4, the LHC and the ATLAS
detector will be introduced. Afterward, Chapter 5 will describe both the next-to-leading
order (NLO) cross-section predictions and the leading order (LO) MC simulation of the W 0
signal. The conceptual bases of the top- and b-tagging algorithms and their applications
are mentioned in Chapter 6. Chapter 7 will then provide each step in the event selections
and categorization, followed by the descriptions of the background processes present in the
signal regions in Chapter 8. Chapter 9 will discuss the statistical analysis methods and give
a summary of systematic uncertainties. The analysis results in Chapter 10 include fitted W 0
mass distributions and exclusion limits on the W 0 proudction cross-section times the decay
branching ratio. This thesis will conclude in Chapter 11.
8
Chapter 2
From the Standard Model to New
Physics: W 0
2.1 The Standard Model
The SM relates all high-energy and short-distance physics to interactions among a small
number of elementary particles. These are eleven spin-one force-carrying gauge bosons,
twelve spin-one-half fermions, and a complex-doublet scalar boson, the Higgs boson. The
twelve fermions are further divided into six types of quarks and six types of leptons. All
elementary particles are quantum fields, i.e., they have quantum properties at each four-
dimensional spacetime coordinate.
The SM gauge bosons are exchanged between elementary particles, producing three kinds
of fundamental interactions. The eight gluons are responsible for the strong interaction,
the photon for the electromagnetic interactions, and the W and Z bosons for the weak
interaction. In simplified mathematical notation, a fermion field ψ(x) couples with a gauge
boson A(x) by modifying the spacetime derivatives of the former into
i∂ψ(x) − egA(x). (2.1)
9
(Spin, spacetime indices and quantum operator symbols are suppressed.) The gauge sym-
metry determines the allowed values of the quantum number e, and the coupling constant g
controls the interaction strength. If |g| 1 – the interaction is weak – quantum amplitudes
can be well-approximated by a perturbartive series in the coupling constant.
Figure 2.1: The Feynman diagram shows parton showers from a vector boson’s decay. Staight
lines with an arrow are quarks. Curly lines are gluons. The vertical axis represents space
and the horizontal axis represents time. Diagram cropped from [30].
The strong interaction acts only on quarks and gluons as they carry the “color” quantum
numbers. Due to vacuum polarization, the strong interaction’s coupling constant gs has less
strength in shorter distances, or equivalently, at higher energy scales [30]. At an energy
scale as low as typical hadron masses around 1 GeV, the strong force that holds quarks and
gluons into hardons is too strong to be considered a small perturbation. But when the energy
increases, the strong coupling constant asymptotically approaches zero, whereby quarks and
gluons become separate entities exchanging gluons. Therefore, high energy events at the
LHC can be viewed as quarks and gluons – or partons – emerging from protons and undergo
10
“hard-scattering” which is perturbatively calculable (shown as Feynman diagrams).
QCD describes the conversion between partons and hadrons in both directions. For par-
tons inside hadrons such as protons, their momentum fractions relative to the parent hadron
follow the probabilistic distribution called Parton Distribution Functions (PDF) whose scale
variation is described by parton radiations. In a hard-scattering interaction, initial-state
partons emit partons before they interacts with partons emerging from a counter-moving
hadron. Also, final-state partons lose energy through a cascade of parton emission, called
the parton showering (Figure 2.1). After each QCD radiation, the average parton energy
decreases, making the gs strong enough to bound partons back into hadronic states. The
top quark is an exception, as it decays more rapidly than hadron formation time-sacles.
The electromagnetic (EM) interaction is present between electrically charged particles,
which include all of the quarks, half of the leptons, and the W boson. As the photon has zero
mass and the EM interaction has weaker dependence on the energy scale than the strong
interaction, it has an infinite interaction range.
Every SM particle is subject to the weak interaction. But as its name suggests, the weak
interaction does not pose a significant force at lower energy. The reason is that the W and Z
bosons are massive, equivalent to very short Compton wavelengths. While being less powerful
than the strong and EM interaction, it has many unique behaviors. As an example, the W
boson only couples to left-handed fermions, violating the parity conservation. Also, the W
boson connects an up-type fermion to another down-type fermion. For this reason, the six
quarks (Table 2.1) and six leptons (Table 2.2) are commonly displayed in three generations
of duplets.
The last piece in the list, the Higgs boson, plays the role of giving masses to all quarks,
charged leptons, the W and Z bosons, and itself. This role is necessary because the gauge
11
type generations
up u c t
down d s b
Table 2.1: Three generations of quark duplets.
type generations
up νe νµ ντ
down e µ τ
Table 2.2: Three generations of lepton duplets.
symmetry forbids the W and Z boson to be massive and because the chiral symmetry is
contradictive to explicit fermion masses. Even though the physical Higgs boson has been
discovered at the LHC, the Higgs field’s self-interaction, the Higgs potential, remains a
theoretically intriguing. The Higgs potential has a coefficient with a dimension of energy
squared. This coefficient receives quantum corrections proportional to the square of the
energy scale. Therefore, if new physics appears at an energy scale much higher than the
electroweak scale, such as the Planck scale, the model paramteres have to be highly fine-
tuned to accommodate such a gigantic scale difference. The dissatisfaction toward the fine-
tuning is usually referred to as the “hierarchy problem.” Solving this problem is one of the
motivations for many new physics models.
2.2 Beyond the Standard Model with W 0
There are many open questions in the universe to be answered by physics “beyond the SM.”
These questions surrounds neutrino mass, quantum gravity, matter-antimatter asymmetry,
and origins of dark matter. Motivated by the hierarchy problem, many new physics models
beyond the SM predict new particles and interactions discoverable at the LHC [13], such as
12
massive vector bosons, referred to as the W 0 boson in this thesis. Some examples are given
in the following.
The first example is the composite Higgs model. Composite Higgs models (see section
94 of Ref. [20] and references therein) postulate that the Higgs bosons are bound states of
fermions. They can suppress the hierarchy problem if the binding energy appears not too
far above the electroweak scale. The W 0 boson can arise at the energy scale that resolves
the composite Higgs’ constituents, analogous to asymptotic free partons in QCD. But since
electroweak precision experiments show no sign of a new strong force below 10 TeV, a minimal
composite Higgs model is insufficient [13]. The ”Little Higgs” model (see Ref. [31] and
references therein) proposes a remedy to the composite Higgs by removing the one-loop
order quantum corrections to the coefficient responsible for the hierarchy problem. The
cancellation is made possible by enlarging the electroweak symmetry group into which the
W 0 boson fits.
A different kind of solution to the hierarchy problem involves adding extra spatial di-
mensions to the model. Some extra-dimensional models try to explain the weakness of grav-
itational interactions by hypothesizing that gravitational potential extends to more spatial
dimensions than three other fundamental interactions [13]. Randall-Sundrum models [32]
take a different approach where the added fourth spatial dimension is warped. The warped
space has one end residing at the Planck scale, the UV brane, and the other at the elec-
troweak scale, the IR brane. The Higgs bosons can be confined to the IR brane such that the
underlying parameter, the warp factor, does not have to be fine-tuned. In some variants of
the Randall-Sundrum models (see Ref. [33] and references therein), the W boson has excited
states localizing in the region between the two branes. These massive W bosons can be
interpreted as the W 0 boson.
13
W bosons naturally emerge as a consequence of an additional SU(2) symmetry. It is
present in the left-right symmetric model [34] or intermediate states of symmetry breaking
in some grand unified theories [35]. In the left-right symmetric model, the W 0 boson has
right-handed chirality. If the right-handed neutrino predicted by the model is more massive
than the right-handed W 0 boson, its resonance state will only decay to quarks, motivating
hadronic searches for the W 0 boson. In addition, in models having one SU(2) symmetry
for the first two generations of fermions and the other for the third generation before the
symmetry breaking (such as in [24, 36]), the top quark and the bottom quark, the target of
this analysis, have an enhanced coupling to the W 0 boson.
14
Chapter 3
Analysis Strategies
The current chapter has two primary functions. First, it serves as an extended outline,
emphasizing the logical connections between later chapters. It also gives the general idea
of the data analysis process that involves understanding the experimental apparatus and
techniques, invoking physics to interpret data, and providing statistical interpretation for
the presence of the W 0 boson.
Chapter 2 has motivated a search for a W 0 boson via its productions in proton collisions
at the LHC and its decay to a top quark and a bottom quark. These two quarks have
unique properties allowing them to be differentiated from other quarks. They are observed
as jets, reconstructed by the ATLAS detector. The jet four momenta for the top quark and
bottom quark will be summed as the W 0 four-momentum, whose mass is the invariant mass
Mtb . With only SM physics, the Mtb distribution is expected to have a smoothly descending
shape. If a W 0 with a 4 TeV mass is found in data, there should be a peak-like excess around
4 TeV.
Jets are useful observables for partons (quarks and gluons) in high-energy proton-proton
collisions (Chapter 6). High energy partons lead to collinear parton showers (Chapter 2).
Thus, hadrons formed by showering partons are confined in a cone-shaped region around the
original parton’s moving direction, giving an image of a jet. This jet of hadrons is observed as
energy clusters by the calorimeter (Chapter 4), and the direction and energy of calorimeter
15
clusters give the jet four-momentum. This momentum is corrected back to the initial parton
momentum up to differences due to detector resolutions and pile-up contaminations. Pile-
ups are additional proton interactions in the same bunch-crossing as the hard-scattering of
interest (Chapter 4). In the later text, a top-quark jet will mean a jet produced by a top
quark. The same rule applies to other quarks and gluons.
A jet can be thought of as a cone with a circular base, whose radius characterizes its
size (Chapter 6). This analysis defines two jet sizes: 0.4 and 1.0, named small-radius jets
and large-radius jets. The small-radius jet is sufficient to enclose the parton shower from the
bottom quark decaying from the W 01 . However, it is challenging to assign one small-radius
jet to each of the three partons from the top quark decay, as their showers might overlap
due to the top quark’s Lorentz boost. For a W 0 boson with a mass of 3 TeV (the upper
excluded W 0 mass in the last search), the maximally 1.5 TeV energy received from the W 0
decay over the top quark mass of less than 0.2 TeV gives a significant Lorentz boost factor γ.
Therefore, large-radius jets, broad enough for enclosing hadrons induced by the top quark’s
three decay particles, are reconstructed along with small-radius jets.
Reconstructed jets are examined by dedicated tagging algorithms to identify the flavor
of the jet-emitting parton. A top-tagging algorithm (top-tagger) seeks large-radius jet at-
tributes from a top quark decay, such as a prominently high mass and the three-prong-like
energy density distributions. A b-tagging algorithm (b-tagger) find patterns of a B hadron
decay among tracks reconstructed from the ATLAS inner detector (Chapter 4). Both taggers
are augmented by Deep Neural Network techniques capable of discerning non-trivial correla-
tions between multiple input variables. A large (small)-radius jet is considered top (b)-tagged
when the tagging algorithm’s output is in a range preferred by a top-quark (bottom-quark)
1 to be distinguished from the bottom quark decaying from the top quark
16
jet.
With jets as observables, the quest for W 0 bosons starts with removing proton-collision
events (or events in short) due to SM backgrounds while preserving W 0 signal events. Signal
regions (SR) are defined with event selection criteria favoring signal events. For example, the
top quark and bottom quark decaying from the W 0 tend to have higher transverse momenta
2 (Chapter 5), and so should the corresponding top-quark and bottom-quark jets. Thus,
the signal region first requires exactly one top-tagged large-radius jet with high transverse
momentum (pT ) – the top-quark jet’s candidate (top-candidate jet). Secondly, on the far side
of the top-candidate jet, the highest pT small-radius has to be b-tagged – the bottom-quark
jet’s candidate (b-candidate jet). The full list of selection criteria can be found in Chapter 7.
The selections are optimized using the signal (Chapter 5) and background (Chapter 8) models
without examing data in the SR to avoid human biases.
The dominant SM backgrounds in the SR are the top-quark pair production (tt̄) and
the QCD production of multiple non-top-quark jets (QCD multi-jet). Simulated events are
adopted to obtain the SR distribution of the tt̄ background. In contrast, as the QCD multi-
jet events are on the tail of output discrimination distribution for top-tagging and b-tagging,
its yield is better extrapolated from the three sideband regions shown in Table 3.1. The Mtb
distributions of QCD multi-jet backgrounds in these regions are estimated as data minus the
tt̄ background. If the top-tagging and b-tagging decisions are independent, the data-driven
background estimate in region A is given by region B times C over D for every Mtb bin
(Chapter 8).
The accuracy of the data-driven method relies heavily on controlling the correlation
between the two candidate jet’s tagging. In defining the non-top-tagged region, an alternative
2 The absolute value of a momentum vector perpendicular to the LHC beamline
17
Top-candidate jet tagged C A
Top-candidate jet not-tagged D B
B-candidate jet not-tagged B-candidate jet tagged
Table 3.1: Four regions employed by a data-driven background estimation are shown as four
cells on a two-dimensional plane. The horizontal and vertical axes are represented by the
tagging decision on the b-candidate jet and the top-candidate jet, respectively.
top-candidate jet definition might be sensitive to different quark flavors and gluons, thus
biasing the b-candidate jet towards specific parton flavors. An improved approach is to
consider any high pT large-radius jet a top-candidate jet, as long as we can find a back-to-
back small-radius jet with high pT as its b-candidate jet. Furthermore, the sideband regions
can be subdivided into blocks of four cells, among which the correlation can serve as an
estimate of systematic uncertainties.
There are other systematic uncertainties in measuring jet momenta, correcting top-
tagging and b-tagging rates for simulated events, and calculating the hard-scattering ampli-
tudes that shape the signal and background-predicted Mtb distribution (Chapter 9). There
are more than 100 nuisance parameters (θ) representing these uncertainties. Their joint
effects are modeled by a profile-likelihood function L (µ, θ) of the SR’s Mtb bins for a signal
strength µ. As this analysis is a generic search, µ is not limited to the adopted W 0 model’s
value but is considered a parameter-of-interest. The nuisance parameters are fitted to data
by maximizing L , while µ is either fixed or floating (µ̂) during the fit. A small ratio of L (µ)
to L (µ̂) signifies deviation between data and the µ-signal plus background model. Those
µ-values with sufficient inconsistency are excluded. The lowest excluded signal size for each
W 0 mass is plotted as an upper limit on the W 0 production cross-section times the tb-decay
branching fraction as a function of W 0 mass. A hint for an excess compatible with the signal
model will show a significant bump on the limit.
18
Chapter 4
Experimental Apparatus
This chapter describes the principle function and utilization of the experimental apparatus.
The LHC accelerator parameters, the proton injection chain, and the LHC data analyzed in
this thesis are described in the first section. The following ATLAS section introduces the
instrumental structures and main subsystems for identifying and measuring physical objects,
with more details provided for those components crucial to the QCD jet reconstruction.
4.1 Large Hadron Collider
Located 45 m to 170 m underground on the border between France and Switzerland, the
Large Hadron Collider (LHC) [37] accelerates two counter-circulating proton beams to 6.5
TeV energy per proton. Along its approximately 27 km circumference, opposing protons
collide at four interaction points each of which houses a detector to observe particles coming
out of the collision: ALICE [38], ATLAS [39], LHCb [40], and CMS [41]. The total 13 TeV
center-of-mass energy provides sufficiently energetic partons to annihilate and produce the
targeted TeV-mass W 0 boson (Figure 1.1).
The proton beams are exerted by a standing electromagnetic wave when they pass
through a section of cavities at the LHC ring. Protons are separated into bunches such
that the longitudinal electric field is in the phase of maximum acceleration when protons
traverse through the field-generating device. Each beam has 2808 proton bunches of several
19
centimeters long separated by few meters, leading to a 24.95 ns duration between colli-
sions [20, 37]. The circular motion of protons along the beam pipes is supported by 1232
dipole superconducting magnets having a peak 8.33 TeV Tesla magnetic field. Liquid he-
lium cools the temperature of the magnet to 1.9 K to maintain its superconducting state.
Quadrupole and sextuple magnets are also employed to correct proton trajectories and focus
the protons at the interaction points to enhance luminosities. The luminosity measures the
area density of protons per unit time.
The luminosity of the LHC is critical to the W 0 search. The rate of interaction (dN/dt) is
the product of instantaneous luminosity (L(t)) times the interaction cross-sections (σ) [20].
Since the predicted production cross-section of a 1 TeV W 0 is of the order of 10−35 cm2 and
decreases further with masses (Figure 5.3a and Figure 5.3b and using the conversion of 1
pico-barn = 10−36 cm−2 ), the 1034 cm−2 s−1 or higher instantaneous luminosity of the LHC
could yield significant numbers of potential signal events after years of LHC running.
The acceleration of protons starts at the accelerator chain prior to proton injections into
the LHC. The initial stage is a linear accelerator, LINAC 2, which brings proton energy to
50 MeV [43]. Then there are three more synchrotrons in succession – Proton Synchrotron
(PS) Booster, PS, and Super Proton Synchroton (SPS) – where the final proton energy is 1.4
GeV, 25 GeV, and 450 GeV each. The ultimate acceleration at the LHC pushes the beam
energy from 450 GeV to 6.5 TeV. The accelerator chain is also responsible for allocating
proton bunches [37]. The six proton bunches injected from the PS Booster are split into 72
in the PS and are subsequently ejected to the SPS. The SPS is filled by three to four batches
of 72-bunch protons before its ejection to the LHC. Each beam transfer requires a change in
the field configuration of the kicker, requiring gaps between proton bunches.
This thesis utilizes LHC data recorded between 2015 to 2018 by the ATLAS detector
20
Figure 4.1: The CERN accelerator complex. The injection chain for proton coliisions at
the LHC is composed of LINAC 2, PS Booster (BOOSTER on the diagram), PS, and SPS.
There are four detectors/experiments, ALICE, ATLAS, CMS, and LHCb marked as yellow
dots along the LHC ring. [42]
21
Figure 4.2: Increase of integrated luminosities over time of the data delievered by the LHC,
recorded by the ATLAS detector, and marked as good for physics analysis. [44]
22
600
Recorded Luminosity [pb-1/0.1]
ATLAS Online, 13 TeV ∫Ldt=146.9 fb-1
500 2015: <µ> = 13.4
2016: <µ> = 25.1
400 2017: <µ> = 37.8
2018: <µ> = 36.1
Total: <µ> = 33.7
300
200
2/19 calibration
100
0
0 10 20 30 40 50 60 70 80
Mean Number of Interactions per Crossing
Figure 4.3: The number of interactions per bunch crossing averaged over a luminosity
block [45] is shown as integrated-luminosity-weighted distribution on the graph for each
year of data taking. The legends tell the overall average for each year. [44]
– the Run 2 period. During this period, the LHC delivered data equivalent to 156 f b−1
integrated luminosity ( L(t)dt), among which 147 f b−1 are recorded by the detector. The
R
entire ATLAS system is under conditions suitable for physics analysis in a smaller subset of
139 f b−1 (Figure 4.2). In this dataset, the beam energy is always at 6.5 TeV. Meanwhile, the
LHC successfully attained higher instantaneous luminosity at the interaction points towards
the end of the run [45]. Although higher luminosity helps the search for rare physics, it also
elevates the number of proton-proton interactions in the same bunch-crossing (Figure 4.3).
Most of them are soft (low energy exchange) QCD processes called pile-up interactions as
they overlap with the hard-scattering physics1 . Pile-up yields numerous tracks and abundant
energy deposits to the detector, which have to be mitigated in reconstructing final-state
1 On the detector side, these pile-up interactions are called in-time pile-ups. There are also out-of-time
pile-ups due to overlapping electronic signals due to the long detector time resolution compared to the
collision time window.
23
Figure 4.4: The figure points out the main subdetectors of the ATLAS detector.
objects from hard-scattering physics.
4.2 ATLAS Detector
A Toroidal LHC ApparatuS [46], or ATLAS, is a multi-purpose detector designed to measure
physical objects such as charged leptons, jets, and missing transverse energy originating in
proton or heavy-ion collisions at the LHC. The cylinder-shaped detector is about 25 meters
in diameter, 44 meters long, centering at the interaction point, and coaxial with the LHC
beam pipe. Except for the two proton beam’s traveling directions, the ATLAS detector is
covered by layers of subdetectors. They are Muon Spectrometer (MS), Calorimeter (CAL),
and Inner Detector (ID) from the outer to the inner layer. The CAL is further subdivided
into the electromagnetic calorimeter (ECAL) and the hadronic calorimeter (HCAL). Four
24
Figure 4.5: A simplified represenation of a segment of a cross-sections of the ATLAS detector.
The toroidal magnets around the muon spectrometer are omitted. The tracking system
corresponds to the inner detector (ID).
superconducting magnets are instrumented: A solenoid immerses the ID in a longitudinal
magnetic field, and two end-cap plus one barrel toroids provide a magnetic field for the MS.
Figure 4.4 is a 3D sectional view of the detector.
The ATLAS subdetectors and magnets corroborate to identify particle properties via their
interactions with matters, as shown in Figure 4.5. The ID tracks charged particles (muons,
electrons, and protons in the figure) by their ionizing hits. The particle momentum is given
by the track’s curvature under the solenoid’s magnetic field. Afterward, the CAL absorbs
and measure energy from high energy particle’s cascades. The electromagnetic calorimeter
contains the electromagnetic showers from electrons and photons. In contrast, hadronic
showers are broader and prolonged [47], penetrating deep into the hadronic calorimeters.
Muons, with their long lifetime and minimal interactions [48], escape the calorimter and are
detected by the muon spectrometer. The MS drift chambers measure the muon momentum
25
by tracking their paths under the toroidal magnetic field. Finally, neutrino is intangible
to all subdetectors, but a significant imbalance of an event’s transverse energy signals its
existence.
The subdetector and measured physical objects are described by a coordinate system
that takes the nominal interaction point as its origin. Its z-axis points along the LHC ring
counterclockwise as viewed from the top, then the x-axis directs to the center of the LHC
ring, and hence its y-axis points upward. The azimuthal angle φ is the angle around the z-axis
as measured from the x-axis, while the polar angle θ is the angle from the z-axis. Another
widely used angluar variable is the pseudorapidity η = − ln tan(θ/2) (which is equal to the
rapidity y = 1/2 ln [tan((E + pz )/(E − pz ))] for massless particles) which is used to define
the angular distance ∆R = (∆η)2 + (∆φ)2 . Finally, the norm of the momentum p on the
p
x-y plane is termed as the transverse momentum, pT . Similarly, the energy E times cos θ is
the transverse energy, ET .
4.2.1 Inner detectors
The ID reconstructs charged particle tracks using pattern recognition algorithms. These
tracks are extrapolated to their closest approaches to the beam pipe, forming primary vertices
and secondary vertices. A primary vertex is where a hard-scattering collision or a pile-up
interaction occur. A final-state particle emerging from these interactions can travel a distance
resolvable by the ID before it decays. Tracks from a long-lived particle decay intersect at
a secondary vertex. As mentioned in Chapter 3, the identification of the bottom-quark jet
utilizes tracks from the secondary vertex. Since the ID tracks have better angular resolutions
than the calorimeter cells, they are also employed to remove pile-up energy and tracks in
small-radius jets and to calibrate the large-radius jet substructure and mass (Chapter 6).
26
Figure 4.6: A 3D view of the barrel region of the inner detectors. [49]
27
The ID comprises three subdetectors – Pixel detector, Semiconductor Tracker (SCT),
and Transition Radiation Tracker (TRT) – as shown in Figure 4.6. Both Pixel and SCT
utilize silicon sensors. The former is closer to the interaction point, and its silicon modules
have smaller sizes to achieve the best track reconstruction resolution. The Insertable B-Layer
(IBL) is an additional Pixel layer installed in 2014 for coping with higher track multiplicities
and risks of Pixel module failures brought by increasing instantaneous luminosities [50]. The
TRT is composed of drift tubes. Despite having a lower resolution than the Pixel and SCT,
it extends the ID’s depths to recognize long-lived particles. It also contain transition radiator
detectors to discriminate electrons from hadrons.
4.2.2 Calorimeters
The calorimeters measure the energy profile of physical objects: electrons, photons, jets.
They cause electromagnetic and hadronic showers in the CAL. Active materials in the AT-
LAS calorimeter sample a fraction of the energy deposits while the rest is absorbed by dense
matter. Figure 4.7 presents the geometry of individual calorimeter subsystems. The ab-
sorber and sensor layers are interspersed densely, such as in the ECAL (Figure 4.8) and
(Figure 4.9), leaving no cracks in the φ coverage. The ECAL is closer to the ID, using a
high atomic number absorber – lead – to contain electromagnetic radiations due to energetic
electrons and photons. In contrast, jets are made of hadrons that lose energy by both elec-
tromagnetic showers and nuclear interactions extending deep into the HCAL. Since the ratio
of electromagnetic versus hadronic energy deposit of jets has randomness and some portion
of hadronic energy is not measurable by the active material [20], hadronic clusters formed
by combing ECAL and HCAL measurements have a local weighting scheme to improve the
jet energy resolution (Chapter 6).
28
Figure 4.7: The ATLAS calorimeter encloses the ID colored in grey and a hard-to-seen
solenoid magnet around it. The ECAL consists of a liquid-Argon (LAr) barrel part (|η| <
1.475) and two LAr end-caps (EMEC, 1.375 < |η| < 3.2). The tile calorimeter (barrel: |η|
< 1.0 and extended barrel: 0.8 < |η| < 1.7) and two LAr hadronic end-caps (1.5 < |η| <
3.2, painted in two sheds of copper) comprise the HCAL. At |η| between 3.1 and 4.9 on each
side, there are three components of the LAr forward calorimeter (FCAL). [46]
Figure 4.8: Segements of accordion-shaped calorimeters. The silver-colored lead absorbers
are sandwiched by copper-clad Kapton electrodes. The electrode receives ionizing signals
from the liquid Argon filled in the gap between lead absorbers. [51]
29
Figure 4.9: A wedge of tile hadronic calorimeters consisting of scintillator tiles interleaving
with steel absorbers. The coordinate system shown on the bottom left follows the convention
introduced at the beginning of this section. The top left and center left diagrams demonstrate
how a waveshifter fiber transmitting light signals from scintillator tiles to a photomultiplier
tube (PMTs). The top figure shows the PMT’s position in the top drawer. The plot is taken
from [20] which adapts the original one in [52].
The radii R of the large-radius and small-radius jets used in this analysis are 1.0 and
0.4, respectively (Chapter 6). Their jet axes are also limited to have |η| < 2.0 and 2.5
each; therefore, the input clusters are built from cells in the barrel and end-cap calorimeters,
excluding the FCAL. The energy resolution, the Gaussian width of the calorimeter response
to hadronic energy deposits, is shown in Figure 4.10 in a combined test of the ECAL and
Tile HCAL as determined in pion test beams. The curve is dominated by the stochastic term
that scales with the inverse square root of the test beam energy. An equivalent quantity,
the jet transverse momentum resolution, is calibrated in-situ using the pT of back-to-back
jets (More on the jet properties and calibrations see Chapter 6). The pT resolution of large-
radius jets is presented in Figure 4.11 for both data and simulated events in the barrel
region. Comparing the value at 300 GeV between Figure 4.10 and Figure 4.11 shows that
the energy resolution for a large-radius jet is about 40% more than that for pions in the test
30
Figure 4.10: The pion energy resolution of the hadronic and electromagnetic calorimeters
combined is shown as a function of the square root of pion energy inversed. The black (white)
dots are measured with test beams in year 1996 (1994), while the white cross is derived from
the simulation of hadronic showers. The two fit functions differ in their description of detector
noises. [53]
31
Figure 4.11: The pT resolution of the large-radius jet as a function of the average of the
di-jet transverse momentum from the in-situ calibration [54]. The symbol |ηdet | means the
absolute pseudorapidity of the jet axis with the detector’s center taken as the coordinate’s
origin. The data (black dashed line) has a light band that sums the statistical and systematic
uncertainties in quadrature. The red band is the envelope of in-situ calibration applied to
three different QCD di-jet event generators. The yellow dashed line corresponds to the
reconstructed jet pT resolution relative to the truth-leveled jet (Chapter 6) computed from
events simulated by the Pythia generator.
32
TGC
y η=1.0 MDT
12 m
RPC
η=1.3
10
8 MDT
EE
MDT
TGC-EI
6
MDT
TGC-FI
4
η=2.4
TileCal End-cap
toroid η=2.7
2
CSC
0 z
0 2 4 6 8 10 12 14 16 18 20 m
12
Figure 4.12: The muons detected by the MS are bent by the toroidal magnets in the barrel
(green) and end-cap (black). With requirements of three layers of meaurements, triggering
(RPC and TGC) has a range of |η| < 2.4, limited by the two outer TGC wheels. In
comparison, precision-tracking (MDT and CSC) can go futher to 2.7. [56]
beam data. In contrast, the small-radius jet pT resolution is closer to the latter [55]. The
effect of jet energy resolution from the top-quark jet and the bottom-quark jet broadens the
reconstructed Mtb far beyond the W 0 width at the parton level (Chapter 5).
4.2.3 Muon spectrometers
The MS identifies, triggers, and measures the momentum of muons with the three surround-
ing toroidal magnets. Each of them has eight octagonally arranged superconducting coils.
Their air-core structure minimizes multiple scattering, which changes a particle’s moving
direction randomly. The magnetic field pointing in the φ-direction bends the muon trajecto-
ries on the r − z plane (in a cylindrical coordinate system), changing the η of a muon track.
These tracks are reconstructed by ionization hits in three to four layers of gas chambers and
33
then combined with ID tracks to reduce reconstruction uncertainties.
Figure 4.12 shows the relative position of four subsystems of the MS. There are two
precision-tracking chambers: Monitored Drift-tube Tubes (MDT) and Cathode Strip Cham-
bers (CSC). There are three layers of MDT chambers in the barrel region (gree) and four
layers in the end-cap (blue), with each layer making three to eight measurements in η [46].
In the innermost layer at high |η| region, the MDT is replaced with the CSC, having a
better resolution to cope with more abundant fake-muons in the forward area. The CSC
consists of four chambers equipped with η and φ measuring cathode strips. The two trigger
chambers are Resistive Plate Chambers (RPC) attaching to the barrel MDT and Thin Gap
Chambers (TGC) near the end-cap MDT. They have a shorter signal-processing time than
the MDT and CSC, thus suitable for fast triggering. They also provide φ coordinates for
track reconstruction, complementary to the MDT.
4.2.4 Trigger and data acquisition
Not all events are saved for data analysis at the ATLAS detector since the collision rate of
proton bunches of 40.08 MHz far exceeds the ability of data processing and capacity of data
√
storage. Furthermore, compared to the inelastic scattering cross-section of 80 mb at s=
13 TeV [46], events with high energy physical objects are much less likely to appear. For
example, the cross-section σ(pp → WR0 → tb) for a 2 TeV WR0 mass is less than 1 pb, 10−9
times 1 mb. Since the start of Run-2, the ATLAS detector has been equipped with a two-
tier trigger and data acquisition system (TDAQ), which selects events fulfilling signatures of
interests efficiently before writing the data to disks.
Figure 4.13 illustrates the trigger system’s main modules (on the left), the data readout
system (on the right), and the data links between them. The two-tier trigger system consists
34
Figure 4.13: A schematic diagram of the Run-2 ATLAS trigger and data acquisition system.
The top-right block shows the detector front-ends (FEs), some of which at the CAL and MS
are connected to the corresponding calorimeter trigger and muon trigger at the first-level
(L1) trigger block on the top-left block. The entire time consumed by data transmission and
logic processing at the L1 is less than 2.5 µs before the cental trigger flags the L1 Accept
command back to the FE at a rate below 100 kHz. The accepted event’s data from all
detectors are then distributed through the ReadOut System (ROS). The L1 trigger also
sends the RoI information to the High-level Trigger (HLT), which accesses the event data
through the ROS. As shown on the bottom right block, data passing the HLT’s threshold
at a rate of about 1 kHz is transferred to permanent storage for offline analysis. (Figure
adapted from [57])
35
of the First-level (L1) and the High-level (HLT). By reading only partial information from
the CAL and MS, the L1 trigger efficiently rejects events without significant energy deposits.
The remaining events are examined by the downstream HLT to execute more time-consuming
algorithms and test more stringent trigger criteria than the associated L1 trigger.
The L1 trigger is a hardware-based system with customized electronic chips. It contains
units for the Calorimeter (L1Calo), the MS (L1Muon), and topological triggers taking input
from both systems (L1Topo). Each L1 trigger system has individual trigger items with
different requirements. The detector region identified by a trigger item is called the Region-
of-Interests (RoIs), locating physical objects for the HLT. The final L1-Accept decision made
at the L1 central trigger processor (L1CTP) combines the output from L1Calo, L1Muon,
and L1Topo. The time comsumption at the L1 stage is about 2µs with 0.5µ for back-up [57].
Since this duration is much longer than the bunch-crossing interval, a pipeline memory
installed at each detector’s front-end holds output until the L1 decision is made.
One example of L1 trigger is the jet trigger at |η| < 3.2. The trigger uses a sliding window
algorithm to find a localized calorimeter energy exceeding a defined ET level [58]. The ECAL
and HCAL energy is summed to form jet elements with a size of 0.2×0.2 in ∆η × ∆φ (∆η
increases to 0.3 at the end-cap region). The jet trigger defines the RoI as 2×2 jet elements
with maximal ET among its neighbors. For the trigger decision, a set of energy thresholds
is compared to the ET sum in the RoI and the surrounding 3×3 and 4×4 trigger towers.
The HLT’s trigger items are software algorithms executed in computing farms. The
HLT examines physical objects reconstructed with algorithms simplified from the offline
version to meet time constraints. With the full granularity in the detector input and more
computational time, the HLT triggers have a sharper turn-on curve of trigger acceptance
rate (efficiency) than the L1 [59]. Some HLT trigger items are seeded by the RoI to save
36
Figure 4.14: The architecture of the L1Calo during Run-2 (2015-2018). All modules have
output to Read-Out Drivers (RODs), allowing diagnostics, calibrations, and performance
studies. In particular, RoIs from the CP and the JEP are routed though the CMX to the
RODs for later retrieval by the HLT. RODs pack, buffer and send signals to the ROS. [62]
time and resources while taking a higher risk of object misidentification. Therefore, a full
scan over the entire detector is performed for any HLT algorithm whenever possible.
This analysis adopts an HLT trigger requiring at least one large-radius jet with ET
thresholds ranging from 380 to 460 GeV(Chapter 6). The anti-kt algorithm is used to recon-
struct jets from topological clusters of hadronic energy deposits. The sharpness of turn-on
curves is reduced by less sophisticated calibration of the jet energy scale than the offline
analysis [60]. Thus more stages of calibrations were ported to the HLT jet reconstruction as
Run-2 progressed [61].
4.2.5 Level 1 calorimeter trigger
The Level-1 Calorimeter Trigger (L1Calo) [63], one of the components of the L1 trigger sys-
tem, is critical to the fast identification of energy deposits in the calorimeters from electrons,
photons, and jets. Its input is the trigger tower that sums the HCAL and ECAL energy
37
into cells of 0.1×0.1 to 0.4×0.4 in ∆η × ∆φ, from the central to forward regions. Analog
signals of these cells are distributed to L1Calo modules housing the following components,
Pre-processor, Cluster Processor (CP), Jet Energy Processor (JEP), and Cluster Merger
Module Extended (CMX) (Figure 4.14).
The first component, the Pre-processor, digitizes analog signals at a rate of 80.16 MHz,
synchronizes them to compensate for the varying transmission time from each calorimeter
cell, and reduce electronic and pile-up noises in the electric signal’s baseline (pedestal).
As the calorimeter’s electric pulses span many bunch-crossing intervals, the Pre-processor
employs a filtering algorithm to extract each trigger tower’s ET at the pulse’s peak.
The digital signals from the Pre-processor are transmitted to the CP and JEP, respec-
tively. The CP searches for candidate electrons, photons, and τ -leptons. The JEP looks for
jets and calculates the sum (
P
ET ) and imbalances (ETmiss ) of the transverse energy. The
L1 trigger algorithms, executed at the CP and JEP modules, define physical objects with
various windows of trigger towers, such as a 2×2 core with a surrounding isolation region
used in the CP. Note that the base element for jet-finding windows is already a two-by-two
combination of trigger towers, as shown in Figure 4.14. The CP and JEP modules write out
Trigger OBjects (TOBs) that record the proximity (RoI), energy, and the type of physical
objects inspected by them.
The output of CP and JEP modules are merged by the Common Merger module eXtended
(CMX) to count the TOBs exceeding a set of trigger thresholds and calculate global ET
variables per event. The CMX then transmits the TOBs to the L1Topo system to compute
composite variables like the ∆R between physical objects. A table of local trigger counts
and global variables (with the event-wide variable’s trigger counts) are sent to the L1CTP
for the Level-1 Accept decision (L1A).
38
Chapter 5
Predicting the W 0 Signal
This chapter introduces a model that parameterizes a massive vector boson, W 0 , with which
the production cross-section times W 0 → tb branching ratio is evaluated at the next-to-
leading order (NLO) in the QCD coupling. The predicted cross-section times branching
ratio’s theoretical uncertainties are given. Finally, the leading order (LO) matrix element
with hadronic top-quark decay (Figure 5.1) is interfaced with parton showers, hadronization,
and particle interactions with the detector. Note that the notation W 0 → tb represents both
W 0+ → tb̄ and W 0− → t̄b, for brevity.
5.1 NLO Cross-Section Evaluation with ZTOP
The phenomenological model of W 0 , that characterizes vector resonances beyond the SM, is
a generalization of the SM W boson. The interaction Lagrangian between SM fermions f
and the W 0 boson is written as
Vij0 5 5
1 + γ 1 − γ
L = √ f¯i γµ g 0R +g 0L W 0µ fj + h.c. (5.1)
2 2 2
Vij0 is a matrix in the flavor space that parametrizes the coupling of the W 0 boson with an
up-type fermion of generation i and a down-type fermion of generation j. The matrices γµ
and γ 5 are Pauli matrices for fermion’s spins. The (1 + (−)γ 5 ) projects the fermion’s left
39
d¯ or s̄ u or c
d¯ or s̄
W
b
t
W0
u or c
AAAEtHichVNdb9MwFM3aAKN8rINHJGTRIMYUVUnaAS+TJuCBF9CQ6DqpqSrHdVprjh3ZTqUS5XfxW/gB/A/sNOvX1s1P9557fM71tR2llEjleX/3anX7wcNH+48bT54+e37QPHxxIXkmEO4hTrm4jKDElDDcU0RRfJkKDJOI4n509cXU+zMsJOHsl5qneJjACSMxQVBpaHRY+xNGeEJYDimZsOOiUaVxEseE4iKP8ZxNBEynRQOsaiVyXBwFnuf6gfe+AUCoYYpjlRPPJd2iQgSZTFXOPZf7Lg9c3nF51+Un12UtLxLdiAsUZqbJU799UhiJmb+msoOmZWfBkpNOudqiuIDCCNNTp//OKXItuUGHTPFEw8FaQ7HA+DfeadtZKiqn3Dnr7PQPVu6G23Fnd51HM+4ZjD5v914NPd9lR6V77mQO4AI4yDFz3aqFERT5uFgwykQWhte9jReZEt+WiAx4h+fqTHd7cuMZYjZee1+N67x8i4useqijZstre+UCNwO/ClpWtc5HzX/hmKMswUwhCqUc+F6qhjkUiqBSPJM4hegKTvBAhwwmWA7z8ocV4K1GxiDWHcecKVCi6ztymEg5TyLNTKCayu2aAW+rDTIVfxrmhKWZvkW0MIozChQH5ruCMREYKTrXAUSC6F4BmkIBkdKfesPFfNQkNXPxt6dwM7gI2r7X9n8GrbPP1YT2rVfWG+vI8q2P1pn1zTq3ehaqv65/rX+v/7A/2KGNbLyg1vaqPS+tjWWz/6TwglU=
b̄
Figure 5.1: The leading order Feynman diagram for the signal’s hard-scattering matrix
element. This diagram’s final state is referred as the parton level. Time points from left to
right. The spatial dimension is vertical. For aesthetic purposes, the momentum flow is only
accurate in the time direction. The charge-conjugated diagram is not shown but included in
the event generation. The diagram is produced by feynMF [21]
(right) handed component in chirality. The coupling constant for the left-handed chirality
is denoted as g 0L and that for the right-handed chirality as g 0R . The symbol h.c. stands for
the previous term’s Hermitian conjugate.
Although these parameters can be tuned to a range preferred by a specific model, this
analysis considers only two scenarios for simplicity – the left-handed W 0 boson, WL0 and the
right-handed W 0 boson, WR0 . The WL0 only interacts with left-handed fermions by setting
g 0R = 0 and g 0L = gSM ≈ 0.63 [20]. The value 0.63 is the SM W boson’s coupling constant.
Reversing the coupling constant – g 0L = 0 and g 0R = gSM – leads to the WR0 . For both
scenarios, the matrix Vij0 for quarks also replicates the SM W boson’s CKM matrix. Hence,
W 0 couples sufficiently to the initial state light quarks inside protons and the final state t
and b quarks.
The W 0 production cross-section depends on the energy scale of the initial state quarks,
determined by the parton distribution function (PDF, introduced in Chapter 2). The search
40
g q0 t q0 t
q0
W0 W0 g
q b q b
(a) A gluon splits into a quark-antiquark pair (b) A gluon radiates off the b-quark
q0 t q0 t
g
W0 W0
g
q b q b
(c) One-loop vertex correction (d) One-loop mass correction to the t-quark
Figure 5.2: Feynman diagrams of NLO terms in calculating the cross-section σ(q q¯0 → W 0 →
tb̄), where q and q 0 represents an up-type quark and a down-type antiquark emerging from
protons, respectively. The vertical (horizontal) axis corresponds to the space (time) axis.
The charge-conjugated processes are included in the cross-section calculation. Feynman
diagrams are made with TikZ-Feynman [64].
41
(a)
(b)
Figure 5.3: The cross-section σ(pp → W 0 ) times branching ratio B(W 0 → tb) with W 0 ’s rest
masses in 0.2∼5.0 TeV is evaluated with ZTOP [65–67] at NLO of QCD. Figure (a) is for a
left-handed chiral WL0 and (b) is for a right-handed chiral WR0 . The central black curve is the
nominal value, including almost invisible statistical errors. The red band stands for the theo-
retical uncertainties. Interfaced by LHAPDF [68], the PDF set is PDF4LHC15_nlo_mc_pdfas
documented by [69]. This set combines CT14 [70], MMHT14 [71], and NNPDF3.0 [72] with
technologies found in [73, 74].
42
range of W 0 mass is limited by trigger threshold to be above 1 TeV, nearly 8% of the proton-
proton collisions’ center-of-mass energy. Thus, one of the initial state parton is likely to be
an up quark or a down quark that carries most of a proton’s longitudinal momentum. By
contrast, the second parton has to be an antiquark, holding less momentum due to proton
structures. At higher orders in QCD, the gluon splitting contributes significantly to the
cross-section as partons are dominantly gluons at LHC’s energy scale [75] (Figure 5.2a).
A convolution of parton distribution functions and the matrix element, both up to the
next-to-leading order (NLO) in QCD (the latter shown in Figure 5.2), yields the parton-
level cross-section times branching ratio: σ(pp → W 0 → tb). Varying the W 0 mass (mW 0 )
from 200 GeV to 5 TeV, the result is plotted for WL0 in Figure 5.3a and WR0 in Figure 5.3b.
The WL0 assumes about 1/4 less tb-decay branching ratio than the WR0 as only the former
can decay into leptons. The yet-to-discovered right-handed neutrino is hypothesized to be
more massive than WR0 , thus forbidding the decay. For both WL0 and WR0 , the kink at
250 GeV is due to the shrinking decay phase spaces when mW 0 approaches the top quark
mass (mt ) of 172.5 GeV. At higher mass points, two effects cause the cross-section time
branching ratio’s rapid decrease: the cross-section proportionate with the resonance mass
scale’s inverse squared (Breit-Wigner formula [20]) and the diminishing number of partons
to have longitudinal momentum close to the LHC proton beam energy of 6.5 TeV.
Perturbative calculations in QCD require additional parameters – the strong coupling
constant αs , the renormalization scale µR , and the factorization scale µF . Their nominal
values are αs (MZ2 ) = 0.1180, and µR = µF = mW 0 . The scale dependence and the mt ’s
uncertainty is the next section’s topic. For the all-hadronic W 0 search, the branching ra-
tio of a top-quark decaying to a bottom quark and two first- or second-generation quarks
(Figure 5.1) is 0.665 [20].
43
5.2 Theoretical Uncertainties
There are four sources of theoretical uncertainties for the NLO cross-section – the PDF,
scale, αs , and top-quark mass (mt ). The first three arise from perturbative expansion to
calculate probabilistic amplitudes. The PDF is subject to strong dynamics incalculable by
perturbation theory, thus is determined empirically. The factorization scale µF governs
the separation between strong dynamics and free partons in handling the PDF’s infrared
singularities [30]. The renormalization scale µR removes ultraviolet singularities while tuning
the strong coupling constant, denoted as αs (µ2R ). αs fitting to data at the scale of the Z
boson mass defines αs (MZ2 ). For better convergence in the perturbative series, µR and µF
are evolved to mW 0 in this analysis by employing the renormalization group equations [20].
Following the PDF4LHC recommendation [69], the cross-section’s PDF uncertainty is
derived from the sample standard deviation of 100 cross-sections, each calculated with a
different MC replica in the PDF set PDF4LHC15_nlo_mc_pdfas
v
u 100 2 100
u1 X 1 X (k)
pdf
δ σ= t σ − hσi , where hσi =
(k) σ . (5.2)
99 100
k=1 k=1
Note that the cross-sections’ mean hσi is not identical to the cross-section computed with
the PDF replicas’ mean, σ (0) , which is the nominal PDF. The PDF set also provides two
separate PDFs to account for αs (MZ2 )’s up and down variation by 0.0015. The recommended
uncertainty of αs (MZ2 ) is given by the averaged difference between the cross-section with the
two PDFs
σ(αs (MZ2 ) = 0.1195) − σ(αs (MZ2 ) = 0.1165)
δ αs σ = . (5.3)
2
44
Then, the above two uncertainties are summed in quadrature, giving a symmetric uncertainty
q
pdf+αs
δ σ = (δ pdf )2 + (δ αs )2 . (5.4)
Although the PDF and αs uncertainties could be interpreted as statistical errors of fitting to
supplementary measurements, the renormalization and factorization scale uncertainties are
educated guesses of the size of missing higher-order terms in the perturbative expansion of
cross-section. As the sum of all terms should be scale-independent, this size is conventionally
estimated by scaling up or down the renormalization and factorization scales by a factor of
two [20]. Here both scales are shifted simultaneously. The scale variation, in general, is
asymmetric, so there is an up uncertainty
δ µ+ σ = σ(µR = µF = 2mW 0 ) − σ(µR = µF = mW 0 ), (5.5)
and a down uncertainty
δ µ− σ = σ(µR = µF = mW 0 ) − σ(µR = µF = 0.5mW 0 ). (5.6)
The fourth crucial theoretical uncertainty concerns the top-quark mass, which has a
significant impact on the decay width Γ(W 0 → tb) when mt is close to the W 0 mass. The
chosen top quark mass of 172.5 GeV with an error of 1 GeV is consistent with the ATLAS
combined result of 172.69 ± 0.25 (stat.) ± 0.41 (syst.) GeV [76]. Recomputing cross-sections
with varied mt provides another pair of asymmetric errors. Since the mt is anti-correlated
45
(a)
(b)
Figure 5.4: The relative theoretical uncertainty of σ(pp → W 0 → tb) for different mW 0 and
chiarlity. The uncertainty from shifting mt up (red) or down (orange) by 1 GeV dominates
at low mW 0 . Larger mW 0 sees increases of uncertainty from both scaling µ (= µF = µR )
by a factor of 0.5 (blue) or 2.0 (purple) and the one-sided PDF+αs uncertainty (gree). The
latter dominates in the highest region of mW 0 .
46
to the decay width (hence the cross-section too), the up uncertainty for mt is given by
δ mt + σ = σ(mt = 171.5 GeV) − σ(mt = 172.5 GeV), (5.7)
and the down uncertainty by
δ mt − σ = σ(mt = 172.5 GeV) − σ(mt = 173.5 GeV). (5.8)
Finally, the total theoretical uncertainty is the sum of the PDF+αs , scale, and mt uncer-
tainties in quadrature
q
δ±σ = (δ pdf+αs σ)2 + (δ µ± σ)2 + (δ mt ± σ)2 , (5.9)
The three uncertainties’ ratios to the total are plotted for WL0 in Figure 5.4a and for WR0 in
Figure 5.4b.
5.3 Event Generation
Even though ZTOP can compute differential cross-sections, such as the transverse momen-
tum distribution of the top quark from the W 0 ’s decay, its prediction cannot be compared
with experimental observables straightforwardly. As mentioned in Chapter 2, the quarks and
gluons coming out of the hard-scattering undergo parton showers and coalesce into hadrons.
What the ATLAS detector measures are these hadrons or their decay products’ energy de-
posits. Recorded events employ algorithms to construct physical objects such as jets from
the energy deposit to infer the physics occuring at the hard-scattering. Generating such a
47
0.35
Fraction [1/50GeV] Fraction [1/50GeV]
W'L mass [GeV] W'R mass [GeV]
0.35
0.3 1500 1500
2000 0.3 2000
0.25
3000 0.25 3000
0.2 4000 4000
0.2
5000 5000
0.15
0.15
0.1 0.1
0.05 0.05
0 0
0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000
Mtb [GeV] Mtb [GeV]
Figure 5.5: Parton-level (without FSR) invariant mass shapes of the top quark and bottom
quark decaying from (a) left-handed or (b) right-handed W 0 with 5 different rest masses are
compared by normalizing the area to 1. Each distribution is a cumulation of around 300K
events.
complicated evolution requires dedicated programs.
The first stage is the parton-level event generation using MadGraph5_aMC@NLO
v2.2.3 [77]. The program calculates the hard-scattering matrix-element of a W 0 boson pro-
duced in proton-proton collisions with decay to a top quark and a bottom quark. Five W 0
rest masses are considered for both left- and right-handed W 0 . Although the matrix element
is computed only at the leading order in QCD, the W 0 production cross-sections, branching
ratios, and decay widths are set to their next-to-leading-order values. The program’s output
contains the final-state t, b quarks’ four momenta per event. Their four-momentum-sum’s
mass is the invariant mass Mtb , whose distribution is shown in Figure 5.5 for each mass
point and chirality. Common to both WL0 and WR0 , the invariant mass centers at the W 0
rest mass, corresponding to a resonance. As the rest mass ascends, the peak widens and has
a long tail to the low end due to an increasing decay width1 and decreasing parton densities
carrying high momenta. What’s uncommon is that the WL0 has a broader resonance than
the WR0 with the same mass, reflecting the former’s extra leptonic decay channels.
1 roughly proportionate with mass
48
Fraction [1/25 GeV]
Fraction [1/0.1]
0.08 0.04
W'R mass [GeV] W'R mass [GeV]
0.07 1500 0.035 1500
0.06 2000 0.03 2000
3000 3000
0.05 0.025
4000 4000
0.04 5000 0.02 5000
0.03 0.015
0.02 0.01
0.01 0.005
0 0
0 500 1000 1500 2000 2500 3000 3500 0 1 2 3 4 5 6 7 8 9
Top-quark P [GeV] |η -η |
T t b
Figure 5.6: (a) The top-quark transverse momenta and (b) the top-quark to bottom quarks
pseudorapidity difference’s absolute value for right-handed W 0 with 5 different rest masses
are compared by normalizing the area to 1. Left-handed W 0 bosons share similar kinematic
properties
Peaking Mtb at the W 0 rest mass (mW 0 ) also influences the top-quark and bottom-quark
transverse momenta. Given the initial-state partons’ insignificant transverse momenta, the
top-quark and bottom-quark are produced back-to-back on the transverse plane. Furthur-
more, as the top-quark and bottom-quark masses are negligible compared to mW 0 at the
TeV scale, both quarks’ transverse momenta share at most one-half the invariant mass by
energy conservation. As shown in Figure 5.6(a), the top-quark momentum has a parabola-
like increase and then a sharp drop-off near 0.5mW 0 . Figure 5.6(b) depicts the top-quark
and bottom-quark’s pseudorapidity difference, a quantity invariable under the Lorentz boost
along the beam pipe. The perference for lower pseudorapidities are uniform among various
mW 0 and is consistent with the rising top-quark transverse momenta.
What distinguishes a left-handed W 0 from a right-handed one the most is the top-quark
spin. A left (right)-handed W 0 boson only couples to fermions with left (right)-handed chiral-
ity. A fermion’s chiral eigenstate approaches the helicity eigenstate of the same handedness
(or the opposite handedness for an anti-fermion) when its Lorentz boost factor γ is signif-
49
Fraction [1/0.02]
Fraction
0.02
4 TeV W' 0.25 4 TeV W'
0.018
Left Left
0.016 Right Right
0.2
0.014
0.012
0.15
0.01
0.008 0.1
0.006
0.004 0.05
0.002
0 0
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
cos(θd ,s) max[∆R(t ,p): p = q , q' , b t]
W W
Figure 5.7: (a) The angle between the down-quark’s momentum in the top-quark’s rest
frame andp the top-quark’s momentum in the W 0 rest frame and (b) the maximum angular
distance ( (∆φ)2 + (∆η)2 ) between the top-quark and its three decay products in the lab
frame for a left-handed (blue) or right-handed (red) W 0 with a 4 TeV rest mass are compared
by normalizing the area to 1. The rightest bin in (b) is an overflow bin that encompasses
everything above 1.
icantly more than 1. In the center-of-mass frame of a W 0 with a mass of several TeV, the
daughter top-quark with around 173 GeV mass assumes γ of order 10. So the top-quark
decaying from a left-handed W 0 mostly has left-handed helicity, meaning that the top-quark
spin is anti-parallel to its momentum. Following similar reasoning, a right-handed W 0 makes
the top-quark spin almost always parallel to its momentum.
A top quark decays to a bottom quark and an SM W boson whose subsequent hadron-
decay-particles (specific to this analysis) consist of a down-type antiquark (d¯ or s̄) and an
up-type quark (u or c). In the top quark’s rest frame, the energy released to the outgoing
quarks is large enough for them to be considered helicity eigenstate, as the top quark has been
in the W 0 ’s rest frame. Together with the SM W boson’s left-handed-chiral couplings, the
bottom quark must have left helicities. If the top-quark is a spin eigenstate toward positive z-
direction, conservation of angular momentum yields negative (positive)-z bottom-quark spin
for a W boson with left (longitudinal) helicity. The W boson’s momentum thus preferably
50
points to the negative (positive) z-axis. Further helicity analysis on the W boson’s daughter
quarks will reveal that the down-type antiquark tends to have a positive z-momentum, i.e.,
parallel to the top-quark spin [75].
Given that the top-quark momentum (pt ) is anti-parallel (parallel) to its spin in the left
(right)-handed W 0 ’s rest frame, the down-type antiquark’s momentum in the top quark’s
rest frame (pd,s ) is more likely to be away from (toward) pt . This claim is supported by the
distribution of cos(θd,s ) = pt · pd,s /|pt ||pd,s |, as shown in Figure 5.7(a). While the down-type
antiquark’s moving direction relative to the top quark has considerable dependence on the
W 0 chirality, this distinction is much weaker among the other two quarks from the top quark
decay [78]. So the down-type antiquark’s behavior alone can help understand why the right-
handed W 0 is more likely to have all three decay products within a cone of |∆R| = 1 around
the top quark momentum in the lab frame. Such an effect is reflected by the distinctive
maximum angular distance between the top quark and its three daughter quarks shown in
Figure 5.7(b). The radius parameter is the same as the large-radius jet for reconstructing
the hadronic top-quark decay (c.f. Chapter 6).
The parton-level events given by the MadGraph simulation only contain the bottom
quark directly from the W 0 decay and the top-quark decay’s three quarks. The second
program, Pythia v8.186 [79], adds initial-state and final-state QCD parton showering in
a repeated fashion. These showering partons are in the soft-and-collinear limit of QCD,
meaning they have small momenta or small angles relative to their parent partons. After each
vertex, the energy is taken away by the radiated partons. Once the parton energy reduces
to near the hadronization scale, QCD coupling becomes strong enough to bind partons
into hadrons. Pythia describes the hadronization process as the breaking of strings whose
potential energy is linear to the separation distance between a quark-antiquark pair. Pythia
51
is also a toolbox of other QCD phenomena to emulate the pile-up of proton interactions in
the same LHC bunch-crossing.
The hadrons are generally unstable and decay as they traverse through the ATLAS
detector. Particles with lifetimes of more than 10 picoseconds (a picosecond = 10−12 second)
are collectively specified as the particle level or the truth level [80]. These relatively stable
particles will leave marks in the tracking system if they carry electric charges, and those who
live longer proceed to stimulate electromagnetic and hadronic interactions in the calorimeter.
Geant4 [81, 82] generates these interactions step-by-step. Simulated detector signals will be
reconstructed by algorithms coded in Athena [83] to form physical objects like jets, mimicking
event reconstructions for data collected during actual proton collisions.
52
Chapter 6
The Identification of Top and Bottom
Quarks
This chapter introduces jet as a physical observable to identify the top and bottom quarks
in the W 0 signal’s final state. Large-radius jets that characterize a hadronic-top quark
is discussed, emphasizing the associated boosted top-quark DNN algorithm. A parallel
description is given for the small-radius jet and its bottom-quark discriminating variable,
DL1r.
6.1 Jet
QCD jets, jets of hadronic particles, or simply jets refer to bundles of hadrons shown as
isolated energy flows commonly observed in the proton-proton collision at the LHC. Recall in
Chapter 2 and Chapter 3, QCD describes jets as repeated soft and collinear parton radiation
from final-state partons in a hard-scattering event and subsequent hadron formation. While
this phenomenon resembles the electromagnetic shower composed of Bremsstrahlung and
photon conversions into electron-positron pairs, hadronic jets’ cascades are more common to
appear since the QCD coupling is larger than the QED counterpart and gluons can directly
couple to gluons [84]. Another difference is due to virtual masses: high-energy partons
generally have off-shell massess while electrons need matter nuclei to interact with to radiate
53
photons without violating four-momentum conservation.
The rulebook of assigning each event’s hadronic energy spreading in the coordinate space
into jets is the jet algorithm. For jets being both experimentally and theoretically defined
objects, three different levels of jets are defined: parton-levels in perturbative calculations of
matrix elements, truth-level after MC simulation of parton showers and hadronization, and
reconstruction-level with particle tracks or calorimeter clusters from detector emulation or
as recorded from data. Consistent jet configurations in all three levels require jet algorithms
to be insensitive to additional soft or collinear radiation from parton radiation or hadronic
decays [25].
One group of algorithms meeting the criterion is the sequential recombination algorithm.
It clusters jet constituents separated by an angular distance ∆R less than a specified param-
eter R into jets, starting from the nearest neighbors. For a specific algorithm, the sequence
of combining jet constituents can depend on their pT ’s and different schemes to assign jet
four-momenta exist (a comprehensive review is given in [25]). This analysis adopts the direct
four-momentum summing scheme (the E-scheme) and two particular algorithms differing in
their pT sorting: the anti-kt [85] that considers pT ’s maximum among jet constituents and
the kt [86–88] that takes the minimum. The anti-kt is used to reconstruct large-radius jets
for the top-quark and small-radius jets for a bottom quark, while the kt is utilized to identify
a large-radius jet’s substructures. Note that an anti-kt jet’s radius refers to the size param-
eter R, as the algorithm’s prioritization of high pT particles gives circular boundaries to the
higher pT jets in an event.
54
Figure 6.1: The probabilities of truth-level large-radius jets having a top quark and various
combinations of the top decay products within ∆R < 0.75 from the jet axis are shown as
a function of the top quark’s transverse momentum. A top quark decays first to a bottom
quark (b) and a W -boson, which further decays to two first or second generation quarks (q1 ,
q2 ). [80]
6.2 Large-radius Jet — Local Cluster Topological Jet
As pointed out in Chapter 3, when the W 0 boson with thens times more mass than the
daughter top quark, the angles between the decay products of the top quark are shrunk by
a Lorenz boost. The Lorentz boost’s effect is illustrated by a simplified example of particle
1 with large transverse momentum pT decaying into two particles 2 and 3 with negligible
masses compared to pT . Assuming that m1 is the mass of particle 1 and particle 2 and 3’s
55
pT , η, and φ variables are labeled with their corresponding subscripts:
(6.1)
q
m1 = 2pT pT (cosh(η2 ) cosh(η3 ) − cos(φ2 − φ3 ) − sinh(η2 ) sinh(η3 ))
2 3
q
≈ pT pT ((η2 − η3 )2 + (φ2 − φ3 )2 )
2 3
(6.2)
q
= pT pT ∆R23
2 3
Therefore, the angular distance between particle 2 and 3, ∆R23 , shrinks with their transverse
momenta, pT and pT , that increase with pT . In this analysis, the large-radius jet is an
2 3
anti-kt jet with its size parameter R=1, and only large-radius jets with pT > 500 GeV are
considered. According to Figure 6.1, 60 percents of the large-radius truth-level jets with a
pT of 500 GeV contain all three quarks from the top quark decay, and the rate increases
further with pT .
Reconstructing a large-radius jet starts with applying anti-kt algorithms to topological
clusters with local hadronic calibrations [89]. These clusters are built from calorimeter
cells energetic enough compared with typical electronic and pile-up noises. The clusters’
energy has to be corrected to compensate for energy loss for various reasons such as less
energy reading from energy deposits by charged pions than by electrons, photons, or neutral
pions. The difference in calorimeter’s response to electromagnetic versus hadronic showers
is corrected by applying weights to hadron-like energy deposits.
After the cell-by-cell corrections to the topological clusters, there could still be energy
from pile-up [90], multi-parton scattering, and intial state radiation that contaminate the
showering pattern of a top quark decaying to three quarks. This issue is mitigated by
applying jet trimming [91]. It begins with sorting the topological clusters in each large-R jet
into sub-jets with R = 0.2, using the kt algorithm. The sub-jets with transverse momenta of
56
less than five percent relative to the original large-R jets are removed. Topological clusters
from the remaing sub-jets are recombined into the “trimmed” large-R jet, which is the final
reconstructed large-radius jets.
Each reconstructed jet’s energy and mass are calibrated to the truth level [54]. In sim-
ulated events, each trimmed reconstructed jet is matched to the closet trimmed truth-level
jet on the η − φ plane. The reconstruction-level jet energy’s ratio to the truth-level energy is
fitted to a Gaussian distribution whose central value is the energy’s correction factor. The
jet mass calibration follows the same method but combines track-based and calorimeter-
based measurements to minimize the mass resolution. Additional in-situ jet energy and jet
mass correction is applied to data to account for uneven detector response in η and residual
differences after MC-based correction factors.
6.2.1 Large-radius Jet Trigger
Large-radius jets are reconstructed both online in the High-Level Trigger (HLT) and offline
in the analysis stage. During the ATLAS run, events triggered by a Leve-1 jet trigger of 100
GeV (Chapter 4) are processed through topological cluster formation and jet reconstructions
adapted from the offline analysis software. The large-radius jet trigger is based on anti-kt
R = 1 jet with different pile-up removal methods and energy calibration depending on the
data-taking year. Since 2017, jet trimming, jet energy and mass correction were implemented
in the HLT system [61].
This analysis preserves only data triggered by large-radius jet triggers with ET thresholds
ranging from 380 GeV to 460 GeV as the signal event’s physical signatures. A higher ET
threshold is necessary to contain trigger rates within the data acquisition system’s readout
capacity since events with more pile-up interactions meet the same trigger threshold more
57
1.2
Trigger efficiency
1
0.8 Trimmed anti-k t R = 1.0 jets
Highest p , |η| < 2.0 per event
T
0.6 ET > 360 GeV (2015)
ET > 420 GeV (2016)
ET > 460 GeV (2017-18)
0.4
0.2
0
400 500 600 700 800 900 1000
Large-R jet p [GeV]
T
Figure 6.2: The efficiencies of three large-R jet triggers for data collected in the year 2015
(black), 2016 (red), and 2017 to 2018 (blue) are shown as a function of the (offline) leading
large-radius transverse momentum. The legend lists each trigger’s (online) transverse energy
thresholds. The efficiency is computed by the coincidence of the targeted trigger and a lower
threshold trigger – the (online) leading small-radius jet’s ET > 260 GeV. All three trigger
efficiencies reach 100% at (offline) leading large-radius jet pT > 500 GeV (drawn as a vertical
black line). The shaded area represents statistical uncertainties.
frequently. Figure 4.3 shows that the pile-up increases with years, resulting in more hadronic
energy to fulfill the trigger cutoff. To prevent the data stream from overloading, some low
energy triggers’ acceptance are considered only every N times. This number N is termed
the “pre-scaling factor,” resulting in triggered event losses. The analysis avoids this loss by
having an offline selection criterion of large-radius jet’s pT > 500 GeV, thereby ensuring full
trigger efficiencies, as shown in Figure 6.2.
58
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
q
b
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
q g q̄
t d¯ or s̄
W
u or c q
(a) Top-quark hadronic decays (b) Three parton splitting
Figure 6.3: Feynman diagrams of (a) the top-quark signal and (b) a light-quark background
for the top-tagging algorithm.
6.2.2 DNN top tagging
Top tagging is a technique for finding large-R jets containing a hadronic top-quark decay
and reject those containing only QCD showering of quarks and gluons by applying selection
to variables that discriminate between the two. The three quarks in a hadronic top quark’s
decay, shown in Figure 6.3(a), frequently have similar energy and wide opening angles be-
tween them. In contrast, three partons produced by QCD radiation, shown in Figure 6.3(b),
are more likely to be collinear in momentum and less democratic in energy shares. Thus,
many jet substructure variables exploit the distinction between the two kinds of jets’ in-
ternal energy distributions. One of the variables is the jet mass, calculated from the jet
constituents’ four-momenta. As the top quark is the heaviest elementary particle, a top-
quark large-radius jet’s mass is, on average, significantly more than the jet mass originating
from parton splitting in light quark jets or gluon jets.
A second jet substructure variable is the splitting scale [93]. Its calculations start with
combining hadronic clusters inside a large-radius jet into proto-jets using the kt algorithm
with R = 1. The combination sequence is determined by the “closeness” between two (merged
or not) constituents of the jet, labeled as i and j, inside a large-R jet. The closeness is defined
59
A.U. A.U.
0.2 0.22
Trimmed anti-k t R = 1.0 jets Trimmed anti-k t R = 1.0 jets
0.18 0.2
|η| < 2.0, p = [1.5, 2.0) TeV |η| < 2.0, p = [1.5, 2.0) TeV
T T
0.18
0.16
Pythia multi-jet 0.16 Pythia multi-jet
0.14
4 TeV W' top 0.14 4 TeV W' top
0.12
0.12
0.1
0.1
0.08
0.08
0.06
0.06
0.04 0.04
0.02 0.02
0 0
0 50 100 150 200 250 0 100 200 300 400 500 600
d23 [GeV] d12 [GeV]
(a) (b)
√ √
Figure 6.4: The splitting scales d23 and d12 of large-R jets are shown for two cases –
top-quark decay from a W 0 boson with a rest mass of 4 TeV (red line) and a gluon or non-top
quark from QCD multi-jet processes simulated by Pythia (black line with yellow fill). In
the first case, large-R jets are matched to top quarks first by requiring ∆R < 0.75 between
the reconstructed jet and its closet truth-level jet, and between the truth-level jet and the
top quark. Moreover, there must be at least one B hadron ghost-associated [92] with the
truth-level jet. Lastly, the reconstructed jet mass has to surpass 140 GeV. This definition of
top-quark jets is the one used in training the DNN top-tagger.
by the splitting scale
q ∆Rij
dij = min(pT , pT ) , (6.3)
i j R
where the R parameter is the same as the concerned anti-kt large-radius jet. The kt algorithm
repeatedly combines the pair i and j with the lowest dij into a proto-jet until one proto-jet
p
is left, which is just the original jet1 . As soft and collinear parton radiation suppresses the
pT and the ∆R factor in the splitting scale, they are combined into proto-jets corresponding
to hard partons initiating the QCD showers. In a top-quark jet, the three last proto-jets are
likely to correspond to the three decay quarks depicted in Figure 6.3(b). The splitting scale
between these proto-jets can be associated with their mother particle’s mass by comparing
1 The beam jet distance d
iB in the kt algorithm for jet-finding is irrelevant since all jet elements clustered
by the anti-kt must have their mutual ∆R < R
60
Equation 6.3 with Equation 6.2. Thus, the splitting scale for merging three proto-jets into
√
two, d23 , peaks around half the W boson mass (Figure 6.4a). Similarly, the splitting scale
√
for clustering two proto-jets into one, d12 , has a maximum of around half of the top quark
mass (Figure 6.4b).
0.06 0.06
A.U. Trimmed anti-k t R = 1.0 jets A.U. Trimmed anti-k t R = 1.0 jets
0.05 |η| < 2.0, p = [1.5, 2.0) TeV 0.05 |η| < 2.0, p = [1.5, 2.0) TeV
T T
Pythia multi-jet Pythia multi-jet
0.04 0.04
4 TeV W' top 4 TeV W' top
0.03 0.03
0.02 0.02
0.01 0.01
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
τ3/ τ2 τ2/ τ1
(a) (b)
Figure 6.5: The ratios of 3-subjettiness (τ3 ) to 2-subjettiness (τ2 ) and of 2-subjettiness (τ2 )
to 1-subjettiness (τ1 ). The definition of top-quark matched large-R jets is written on the
caption under the plots for the splitting scale. Pythia simulated multijet events’ subjettiness
ratios have a spike at 0 since large-R jets consisting solely of parton showering could have
only one or two constituents. The former causes τ1 to be 0 and the latter causes τ2 to be 0.
While the splitting scale measures the mass scale among subjets, the N -subjettiness [94,
95] calculates the energy spread among individual jet constituents – topological clusters in
this analysis – relative to the number of subjets (N ). Assuming that there are M constituents
and ∆Rij is the radius between the constituent i and subjet j, the definition of N -subjettiness
adopted in this analysis is
,
M M
(6.4)
X X
τN = pT min(∆Ri1 , ∆Ri2 , ..., ∆RiN ) pT R ,
i i
i i
61
where R is the radius of large-R jets. Similar to splitting scales, the sub-jet is found by the
kt algorithm. But to reduce the effect of wide-angle, low energy radiation on deflecting the
jet-axis, the “winner-take-all” scheme [96] of combining sub-jet momenta is applied. As the
jet constituents’ energy for top-quark jets is expected to be more aligned with three axes
than two axes or one axis, the ratio between τ3 and τ2 for top-quark jets tends to have a lower
value than QCD jets (Figure 6.5a). In contrast, top-quark jets are outnumbered by QCD
jets in low ratios of τ2 to τ1 when the latter has few or only 1 jet constituent (Figure 6.5b).
104
Background rejection (1 / ∈ bkg)
DNN top ATLAS Simulation
BDT top s = 13 TeV
Shower Trimmed anti-k t R = 1.0 jets
Deconstruction |η true| < 2.0
103
2-var optimised p true = [1500, 2000] GeV
T
tagger Top tagging
TopoDNN HEPTopTagger v1
τ 32, m comb > 60 GeV
102
10
1
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Signal efficiency (∈ sig)
Figure 6.6: The Receiver Operating Characteristic (ROC) curves of different types of top-
tagging studied in [80]. The curves on the plot show how many QCD large-radius jets are
rejected per one accepted (y-axis), given what fraction of top-quark large-R jet is retained
(x-axis) by taking a cutoff(s) on relevant tagging variable(s). The ROC curve of the DNN
top-tagger employed in this analysis is shown in solide red, overlapping with the curve of the
Boosted Decision Tree (BDT) top-tagger.
Jet substructure variables like jet mass, splitting scale, and N-subjettiness can be com-
bined by multivariate techniques to enhance the discriminatory power of top-tagging. Fur-
thermore, top-tagging is a “classification problem” in machine learning as the jet substructure
62
variables can be thought of as different “features” of two different classes of large-R jets –
top-quark jet and QCD jets [97]. In this analysis, an advanced machine technique called the
Deep Neural Network (DNN) [98] takes jet substructure variables as input. A selection cut
can then be placed on the DNN’s output for a fixed likelihood of retaining top-quark large-R
jets. By varying the selection cut, the power of rejecting QCD large-R jets for a given top-
quark jet’s acceptance rate is presented as a ROC curve. The comparison of ROC curves in
Figure 6.6 shows an improvement of background rejection power by DNN top-tagging com-
pared to the Shower Deconstruction [99, 100] top-tagging, which is used in a recent search
for the W 0 boson by ATLAS [101]. The top-tagging rates in the MC-simulated events are
corrected by pT dependent scale factors determined in data enriching in top quark jets [80,
102].
6.3 Small-radius Jet — Particle Flow Jet
A small-radius jet is an anti-kt jet with a size parameter R = 0.4, which is employed to
reconstruct the kinematics of the bottom quark directly decaying from the W 0 signal. Fur-
thermore, a bottom-quark containing hadron – B-hadron – has a flight path resolvable by
tracking its charged decay particles’ hits in the Inner Detector (ID). Thus, small-radius jets
overlapping with large-radius jets are also found for the bottom quark identification. The
small-radius jet’s input is Particle Flow objects [103] that consists of not only topological
clusters used in the large-radius jet reconstructions but also ID tracks.
In the Particle Flow algorithm, charged hadron’s momenta are reconstructed from tracks
while neutral hadrons’ energy is reconstructed from hadronic clusters. To avoid double-
counting hadronic energy, calorimeter clusters calibrated to the EM scales are matched to
63
tracks closed on the angular distance relative to cluster widths. Energy deposited by charged
hadrons of the track is removed from calorimeter cells within the clusters using the shower
profile of pions. The profile is also used to add two clusters associated with a single charged
hadron and determine the consistency between the remanent and neutral hadrons after cell-
by-cell energy subtractions from a cluster. Since high-energy pions are more likely to be at
the jet core area with dense hadronic activities, it would be more difficult to reconstruct and
match tracks with hadronic clusters accurately with increasing pion pT [103]. Therefore, a
pT -dependent selection criterion is imposed to prevent cluster energy vastly exceeding the
expected pion energy deposit from being subtracted [55]. When the track pT > 100 GeV,
the associated hadronic cluster is not considered for removal.
One of the advantages of using tracks for charged hadron identification is the Inner De-
tector’s superior energy resolution than the calorimeter in the lower pT regimes. Since the
segmentation of ID subdetectors is substantially less than the calorimeter cell divisions, the
angular resolution is improved. Another benefit comes from the pile-up removal. Tracks
originating from non-hard-scatter vertices are discarded in jet reconstructions. As the Parti-
cle Flow algorithm removes tracks-associated energy deposits, charged hadrons from pile-up
interactions are automatically removed.
Remaing pile-up contribution is dealt with by pT corrections modeled on per-event av-
erage jet pT per jet area [92], number of reconstructed primary vertrices, and interaction
counts (µ) [55]. Theres is also a jet-vertex tagger [104] evaluting the compatibility of jet’s
ghost-associated [92] tracks with pile-up vertices. Only jets with pT < 60 GeV and |η| < 2.4
are examined jets by the tagger.
Like large-R jets, the energy of small-R jets is also calibrated to corresponding truth-
level small-R jets after pile-up corrections. Unlike large-R jets, the energy scale correction
64
for hadronic clusters in the Particle Flow agorithms has not incorporated the local weighting
procedure for hadronc showers. Therefore, the electromagnetic shower energy is the cali-
bration scale, namely the EM scale. Afterward, there is a sequence of corrections based on
tracks momentum and energy deposits in specific layers of calorimeters [105] to alleviate jet
energy resolution’s degradation due to showering differences between quark and gluon jets.
There are also data-specific energy corrections derived in-situ. The in-situ calibration uses
prior-cablirated physical objects like electrons decaying from a Z boson in Z+jets events to
account for data-MC differences and di-jets events for inter-calibrations between the central
and forward detectors.
6.3.1 DL1r b-tagging
Figure 6.7: Three b-taggers’ ability to reject light-quark jets as a function of the pT of
Particle Flow (PFlow) small-R jets are compared. The plot show one out of how many light-
quark small-R jets, a large background to this analysis, are accepted, given a 77% chance of
retaining a b-quark jet with pT > 25 GeV. The value fc means the proportion of c-quark jet
included with light quark and gluon jets in the background events. [106]
As stated in Chapter 2, a bottom quark is not an observable objects because the strong
65
force at low energy scales binds it into B-hadrons through QCD fragmentation. Therefore,
the two b-quark jets in the W 0 signal are identified by tracking the B-hadron decay’s charged
hadrons around a small-radius jets. Each jet considers all tracks within a cone around the
jet axis with a radius ∆R of 0.45 at 20 GeV of jet pT , decreasing gradually to 0.26 at 150
GeV [107]. The decreasing angular distance follows the Lorentz boost as described in the
discussion of large-radius jets.
For each event, the spatial coordinates of particle interactions – primary vertices – are
reconstructed using ID tracks [108, 109]. The particle’s origin is traced by its closest approach
– perigee – to the beam axis, following its motion due to the longitudinal magnetic field.
The cluster of perigees of two or more tracks with pT > 0.5 GeV allows for vertex fitting,
and events without a vertex are rejected. Among all primary vertices, the one with the
highest scalar sum of matched tracks’ transverse momenta ( pT ) is deemed where the
P
hard-scattering process occurs, thus given the name “the hard-scatter primary vertex2 .” In
addition to a bottom-quark identification’s purposes, the hard-scatter primay vertex is picked
as the origin of the coordinate system for hadronic clusters constituting the neutral part of
the small-radius jet or all hadronic energy in the large-radius jet.
The b-tagging variables aim to find features differentiating B-hadrons from other long-
lived hadrons, photon conservisions into leptons, and di-photon decay from neutral pions,
or even particle interacting with detector materials. Sets of b-tagging variables evaluate the
significance of jet-associated tracks’ displacements from the hard-scatter primary axis. More
sophisticated variables make use of pairs of tracks approaching each other in spaces. By first
rejecting two-track vertices with properties consistent with backgrounds, a fast search for
2 In some literature, this is called the primary vertex when secondary vertices of a hadron’s or a tau
lepton’s decay are irrelevant
66
secondary vertices among numerous tracks in a high particle multiplicies can be achieved. In
response to missing tracks, kinematic fitters are also employed to reconstruct the B-hadron
and its decay particle’s flight paths.
As in the case of top-tagging, these b-tagging variables serve as input to multivariate
approaches [107]. Among these are a deep neural network called DL1 (ibid.) and a recurrent
neural network (RNN) [110] for finding the sequential relation between the tracks. Figure 6.7
shows that the DL1 tagger incorporating the RNN, called the DL1r algorithm, could reject
more light-quark jets than a boosted decision tree, the MV2 algorithm employed in the
previous analysis [101]. The probability of the combined DL1r algorithm to tag a PFlow jet
is calibrated for different hadron species emanating from jets as described in [107, 111, 112].
6.4 Truth-level and Reconstruction-level Observables
Fraction [1/25GeV] Fraction [1/50GeV]
W'R mass [GeV] 0.18 W'R mass [GeV]
0.07
1500 0.16 1500
0.06
2000 0.14 2000
0.05 3000 0.12 3000
4000 4000
0.04 0.1
5000 5000
0.08
0.03
0.06
0.02
0.04
0.01 0.02
0 0
0 500 1000 1500 2000 2500 3000 3500 0 1000 2000 3000 4000 5000 6000 7000
Top-quark jet P [GeV] Truth-level Mtb [GeV]
T
(a) (b)
Figure 6.8: The truth-level (a) top-quark jet transverse momentum and (b) invariant mass
Mtb for right-handed W 0 bosons with 5 different masses are shown by rescaling all histograms’
area to unity.
The W 0 boson is observed by the t − b invariant mass around its rest mass and the top
quark and bottom quark transverse momenta’s peaking edge at half of the rest mass. These
67
⋆ ⋆
W’R mass [GeV] 0.12 W’R mass [GeV]
Fraction [1/25GeV] Fraction [1/50GeV]
0.1 1500 1500
0.1
2000 2000
0.08
3000 3000
0.08
4000 4000
0.06
5000 0.06 5000
0.04 PT,b > 500 GeV PT,b > 500 GeV
0.04
0.02 0.02
0 0
0 500 1000 1500 2000 2500 3000 3500 0 1000 2000 3000 4000 5000 6000 7000
Reco. Top-quark jet P [GeV] Reco. Mtb [GeV]
T
(a) (b)
Figure 6.9: The reconstruction-level (a) top-quark jet transverse momentum and (b) invari-
ant mass Mtb for right-handed W 0 bosons with 5 different masses. The solid-line histograms
are scaled to have unit area, while the dashed-line histograms are normalized by the solid-
line histogram’s original area. Their difference lies in that dashed histogram excludes those
events having its (W 0 -coupled) bottom-quark small-radius jet with pT ≤ 500 GeV.
distributions have been shown in Figure 5.5 and Figure 5.6 using MC simulated events for
5 different W 0 masses. Truth-level jets and reconstruction-level jets in the simulation are
mapped to the top-quark and bottom quark to examine the impact of parton radiation and
jet reconstruction on these signal characteristics.
Clustering hadrons (specified at the end of Chapter 5), truth-level jets shown in Figure 6.8
follow similar kinematic features as the top-quark and botoom-quark at the parton level. The
highest pT truth-level large-radius jet with ∆R < 1 from the top-quark is selected as the
truth-level top-quark jet. The bottom-quark jet is found in the same fashion among truth-
level jets with ∆R = 0.4. About 25% of W 0 events cannot have both jets located due to
the jet reconstruction requirement of |η| < 2.5 and other jet pT constraints related to data
processing capacities. The pT selection causes a hill-like shape around 500 GeV on the Mtb
distribution.
Compared with Figure 5.6(a), the jet pT spreads further to the lower end, meaning that
the hadronic energy falling outside the jet algorithm’s clustering range (R) is on average more
68
than the gain from initial-state radiation or pile-up. The combined effect from the top-quark
jet and the bottom-quark jet broadens the invariant mass distribution that the full-width
at half maxmimum (FWHM) becomes two times than that in Figure 5.5(b) across all W 0
masses. Though not shown on the graph, the Mtb ’s width narrows by around one-tenth after
using only truth-level top-quark jets containing the three quarks in the top-quark decay in
the jet radius R = 1. Thus, the enlarged width is more associated with jet reconstruction
and parton showers than the R-parameter not being wide enough.
Figure 6.9 shows similar plots with events having reconstructed large-radius jets satisfying
a 500 GeV pT threshold for the trigger requirement, leading to ten percent or less event losses
from the truth-level for a 2 TeV or a higher mass W 0 . Among these high pT reconstructed
jets, the top-quark jet is the lcloset to the truth-level top-quark jet. The bottom quark jet is
not required to be close to the bottom-quark-labeled truth-level jet since the latter might be
surrounded by small-radius jets initiated by final-state radiation. Instead, the highest pT jet
with an azimuthal separation ∆φ > 2 from the reconstructed top-quark jet is chosen. The
resulting bottom-quark jet at the reconstruction level is further examined with a restriction
of pT > 500 GeV.
From the truth level to the reconstruction level, the large-radius jet pT shifts to the lower
end further and the Mtb width doubles again. The pT requirement on the bottom-quark jet
reduces events with reconstructed top-quark jet pT < 1 TeV significantly, narrowing the 1.5
and 2 TeV W 0 boson’s invariant mass. The result shows that such a pT cutoff can reject
backgrounds with smoothly falling pT distributions while retaining most signal events. The
top-quark jet pT distribution after this cut is more aligned to the truth-level one, suggesting
that the reconstruction-level jet finds similar hadronic energy as the truth-level one when
the identified top-quark and bottom-quark jets are more symmetric in pT .
69
Chapter 7
Event Selection and Categorization
The previous chapter introduced large-radius and small-radius jets as physical observables
to reconstruct the top-quark and bottom-quark decaying from the W 0 boson, as shown
in Figure 1.1. In this chapter, the top-tagging and b-tagging algorithms and few to-be-
introduced event selections will separate events into the signal region for the statistical
analysis in Chapter 9 and the template, control, and validation regions for the data-driven
background estimation in Chapter 8. Discussions on the data-driven method is given to
motivate a specific usage of the large-radius jet called the top-proxy jet in defining the
control region.
Charged lepton veto. Aside from this analysis, there is another search for the W 0
decaying to a top quark and a bottom quark using the leptonic top-quark decay. Combing
the hadronic and leptonic search channel’s likelihood functions (Chapter 3 and Chapter 9)
enhances the deviation between the signal-plus-background hypothesis and the background-
only hypothesis, namely, the signal’s statistical significance. If the two channel’s events
are mutually exclusive, the joint likelihood function is just the product of the two individual
likelihood functions. Therefore, events with any reconstructed electron or muon are excluded
from this analysis for the statistical independence between the two searches. This charged
lepton veto also helps to reject background events. The reconstruction and identification of
electrons and muons are described in Appendix A.
70
Top-candidate jet. From events satisfying the large-radius jet trigger thresholds listed
in Chapter 6, large-radius jets with pT > 500 GeV and |η| < 2.0 are examined with the
top-tagging DNN algorithm. If only one large-radius jet is top-tagged, that jet is considered
the top-quark jet’s candidate, or say the top-candidate jet. Since the top-tagging rate for the
bottom-quark large-radius jet is low, two top-tagged jets are mostly background events with
two top quarks in the hard scatter interaction’s final state (tt̄, more discussions in Chapter 8).
Thus events with two or more top-tagged jets are removed to reject these background events.
Top-proxy jets. Data events without any top-tagged large-radius jets serve are utilized
for background estimation. Ideally, all non-top-quark jets have identical top-tagging prob-
abilities, so any large-radius jet satisfying the pT and η requirements should be considered.
Moreover, the number of events with exactly one top-tagged jet is proportional to the num-
ber of examined large-radius jets, implying that using all qualified large-radius jets is more
suitable than choosing one particular jet. The above consideration motivates the definition
of top-proxy jets as large-radius jets with pT > 500 GeV and |η| < 2.0 when all such jets in
the same event fail to be top-tagged. Top-candidate jets and top-proxy jets are on the two
sides of the top-tagging axis for the background estimation shown in Table 3.1.
B-candidate jet. By the conservation of momentum, the bottom-quark small-radius
jet’s momentum points in the opposite direction to the top-quark large-radius jet on the
transverse plane, with deviations due to jet reconstruction and large-angle final-state radi-
ation forming jets. So the bottom-quark jet’s candidate – b-candidate jet – is the highest
pT jet among all small-radius jets with ∆φ > 2 from the top-candidate jet or a top-proxy
jet. The b-candidate jet’s b-tagging score serves as the second axis of the data-driven back-
ground’s grid.
Figure 7.1’s flowchart shows the steps in using the top/b-candidate and top-proxy jets
71
Top-candidate
jet
>1 How =0 For each
Veto many
top-proxy jet
?
=1 To the next
No Has a Has a No
b-candidate jet b-candidate jet
? ?
Yes Yes
No B-tagged
No B-tagged
? ?
Yes Yes
Template Signal Control Control
region region region 2 region 1
Figure 7.1: The flow chart showing the classification of events that lead to differnet regions.
The top input block’s top-candidate jet means the top-tagged large-radius jets.
72
to form four non-overlapping regions, named after their roles in the data-driven background
estimation: the control regions (CR) 1 and 2, whose ratios are to be multiplied by the
template region (TR) event counts bin-by-bin, giving the data-driven background estimation
in the signal region (SR).
Number of b-tags inside the top-candidate (top-proxy) jets. The top quark’s
decay almost always include a bottom-quark [20], thus having b-tagged small-radius jets
with ∆R < 1 from a top-candidate jet discriminates the signal from background events.
Furthermore, hard-scatter final-state partons from the QCD multi-jet events can have two
high pT bottom quarks or two charm quarks favored by the b-tagging algorithm. Therefore
large-radius jets with a bottom/charm quark origin are expected to see a higher b-tagging
rate than a gluon/light-quark large-radius jet in the paired b-candidate jets. The change
in b-tagging likelihood necessitates estimating backgrounds if b-tagged small-radius jets are
present inside top-candidate (top-proxy) jets. These large-radius jets, no matter in the signal,
template, and control regions, are separated into the 1 b-tag in category. The 0 b-tag
in category incorporates the rest.
Top-tagging division. The DNN top-tagging algorithm has two thresholds on the
DNN output. The threshold used for the top-candidate jet’s definition has an estimated
80% probability of retaining a top-quark jet [80] and is labeled as the 80 WP. The higher
threshold 50 WP divides the signal and template regions to reject more backgrounds. After
the categorization of b-tagged jets in large-radius jets and the further top-tagging divisions,
the four regions from Figure 7.1 are turned into 12 cells shown in Figure 7.2. Except for
renumbering, the signal region in the “0 b-tag in” category and situated between the 50 WP
and 80 WP of top-tagging is renamed as the validation (VR). It has a negligible portion of
signal events, allowing for verifying the accuracy of the data-driven background estimation
73
Figure 7.2: For each of the two categories, events are divided into six regions on a two-
dimensional plane. The vertical axis represents the top-candidate jet’s top-tagging score
from low to high upward, while the horizontal axis represents the b-candidate jet’s b-tagging
score from low to high rightward. SR stands for the signal region, VR for the validation
region, TR for the template region, and CR for the control region.
method.
Pseudorapidity requirement. The final selection is to remove events or top-proxy
jets if the pseudorapidity difference between the top-candidate or top-proxy jet and its b-
candidate jet (∆ηtb ) is 2.0 or above. The parton-level studies in Figure 5.6(b) and the
robustness of the pT spectrum’s overall peaking structure in the recontruction-level of jets in
Figure 6.9 reflect that the W 0 signal’s preference for low ∆ηtb due to the s-channel resonance
prodcution. In contrast, the dominant background events tend to be more prominent in the
t-channel or u-channel than s-channel of the 2-to-2 matrix elements at high energy scales.
The comparison between the signal and background is made in Figure 7.3 using a 4 TeV
WR0 , which is in the mass range targeted for exclusion in this analysis.
74
Normalized
0.2 s = 13 TeV, 139 fb-1, Mtb = [3.0, 4.0) TeV
0.18 Signal region: 50 WP top-tag, 1 b-tag in
0.16 QCD multi-jet
tt
0.14
4 TeV W'R
0.12
0.1
0.08
0.06
0.04
0.02
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
|η -η |
t b
Figure 7.3: The absolute pseudorapidity difference between the top-candidate jet and the
b-candidate jet for a 4 TeV right-handed W 0 signal and the two main backgrounds are
compared in the signal region with the highest background rejection rate via the top-tagging
and b-taggging (Signal Region 1 in Figure 7.2). Both the top-quark-pair (tt̄) and QCD multi-
jet backgrounds are simulated by MC generators (details in Chapter 8), and their sums are
normalized to have unit area.
75
Chapter 8
Background Processes
This chapter describes methods to estimate the SM background in the signal regions defined
in Chapter 7. One significant background is top quark pair production, whose final-state top
quarks and subsequent bottom quark decays resemble the signal event. Precise cross-section
calculations and event simulations for this background are introduced. The more dominant
background from the QCD multi-jet events is estimated by a data-driven approach. Since
this method utilizes the top-tagging and b-tagging ratios, the correlation between them is
studied thoroughly.
8.1 tt̄
Top quark pair production (tt̄) is the most substantial top-quark contributor at the LHC [20].
The hard-scattering event’s top quark and anti-top quark are produced by the strong interac-
tion between two gluons or between a quark and its antiquark. Feynman diagrams associated
with these two initial states at leading order of αs are shown in Figure 8.1. Due to the vast
gluon populations in protons, the gluon-initiated process leads to vast numbers of tt̄ back-
grounds in exotic particle searches with top quark final states.
Since the top quark decays must include a W boson, the tt̄ event can be classified by
the W -boson decay products into the all-hadronic, semileptonic, and di-leptonic channels.
Among all tt̄ events, the all-hadronic channel constitutes about 46% [84]. Even though the
76
g t
q t
g t
q t g t
Figure 8.1: The Feynman diagrams of top quark pair production at the leading order of
strong coupling. The vertical axis represents space, and the horizontal axis represents time.
Feynman diagrams are made with TikZ-Feynman [64].
event selections veto events with top-tagged large-radius jets, the non-all-hadronic contribu-
tion is suppressed by the requirement of 0 leptons. Thus the semileptonic and di-leptonic
channels will not be shown as separate channels in the following Mtb plots.
A significant amount of tt̄ background is observed in the signal and template regions by
comparing its simulated event numbers with data or the data-derived background. The most
prominent regions are displayed in Figure 8.2, where the tt̄ event’s share is presented in the
ratio plot. In the template region, the ratio of the tt̄ to data stays around 10% in the low
mass region and falls slowly to about 6% in the high mass region. In the signal region, the
ratio starts as high as 25% and then stumbles rapidly to almost 6% in the last bin. Since
the W 0 signal is negligible in the template region, the tt̄ percentages in the two regions show
that the b-tagging ratio is about three times higher for the tt̄ background than for other
backgrounds at low mass. However, such a boost diminishes as the Mtb ascends, reflecting
worsening discriminatory power of the b-tagging with higher pT .
The tt̄ background has significant contributions from high-order Feynman diagrams. Ref.
[113] found that tt̄ plus one or more jets have a higher production cross-section than with
no additional jets. In this analysis, the tt̄ production cross-section is evaluated to the next-
77
Events/50GeV Events/50GeV
104 1 b-tag in 1 b-tag in
Data 103 Data-driven
50 WP top-tag t t all-hadronic 50 WP top-tag Data-driven stat. error
3
10 t t non-all-hadronic t t all-hadronic
t t stat. error 102 t t non-all-hadronic
2 t t stat. error
10 10
10 1
1 10−1
10−1 10−2
10−2 10−3
tt / Data tt / Total
0.18 0.25
0.16
0.14 0.2
0.12
0.1
0.08 0.15
0.06
0.04 0.1
0.02
0 0.05
1000 2000 3000 4000 5000 6000 7000 1000 2000 3000 4000 5000 6000 7000
Mtb [GeV] Mtb [GeV]
(a) Template region (b) Signal region
Figure 8.2: The invariant mass distribution of a large-radius top-candidate jet and a small-R
b-candidate jet in region SR1 (right) and the neighboring region TR1 (left) (c.f. Figure 7.2).
(a) The template region has the data plotted against two channels of tt̄ background – the
all-hadronic and the non-all-hadronic – shown as stacked histograms. Their statistical
uncertainties are summed in quadrature into a single tt̄ statistical error. The ratio of the
summed tt̄ distribution to the data is shown in the lower panel with statistical erros from
both data and the t̄ background. (b) In the signal region, the data-driven background (see the
next section) is drawn above the two tt̄ channels with separate statsitical errors. The ratio
shown in the lower panel is the division of the tt̄ to the total background. Covariances are
taken into account in error propagations. On either figure’s ratio plot, there is a horizontal
reference line showing the ratio of total events.
to-next-to-leading order (NNLO) of αs with next-to-next-to-leading logarithm resummation
using Top++2.0 [114–120]. Theoretical uncertainties (see Chapter 5 for the means of com-
putations) to this order are less than 4% [121].
The tt̄ event simulation that enables jet reconstruction and tagging is generated with
Powheg-Box v2 [122–124] interfaced with Pythia to perform parton showering and
hadronization. Powheg calculates the next-to-leading order (NLO) quantum amplitude
matched to the highest pT showering parton to avoid double-counting the radiating dia-
gram. It also uses Parton Distribution Funcstions derived at NLO – NNPDF3.0 [72]. At
78
this order of αs , the scale uncertainty of the differential cross-sectioon is below 10% in the
top-quark pair invariant mass (mtt̄ ) from 300 GeV to 2 TeV [125], allowing for a precise
prediction for the tt̄ background in the Mtb spectrum.
8.2 Data-driven Backgrounds
Background events in the signal regions other than the tt̄ are estimated using data in the
template and control regions. After the subtraction of tt̄ events, the rest of SM backgrounds
are dominantly QCD multi-jet with no top quarks and few bottom/charm quarks in the
hard-scatter final state. The data-driven estimation introduced in Chapter 3 is chosen to
exploit the lack of correlation between the top-tagging and b-tagging probabilities among
these gluon or light quark jets. Other less significant backgrounds like W -boson+charm
or W -boson+top, which has a W mass-scale large-radius jet and a heavy flavor quark jet,
can induce tagging correlations. Furthermore, the tb-decaying W -production, enhanced by
the two taggers, distorts the double-ratio extrapolation. However, these correlating back-
ground’s presence in data varies data-driven backgrounds within the range of statistical plus
systematic uncertainties (detailed in Appendix B). Therefore, all backgrounds except tt̄ are
accounted for by the data-driven background estimates.
Table 3.1 gives a framework for applying the data-driven estimation defined in Fig-
ure 7.2. Since the top-tagging and b-tagging rates of background events vary with jets’
kinematic properties, the extraction is performed bin-by-bin over the invariant mass distri-
bution. Thus the data-driven background in the bin no. i in region SR1, N (i,SR1), is the
data minus MC-simulated tt̄ background of the same bin in the neighboring template region
(D(i, TR1) − M (i, TR1)) times the b-tagging ratio computed in the “1 b-tag in” control
79
regions (CR1/CR2). The formula for all three signal regions and the validation region are
summarized as
(D(i, TR1) − M (i, TR1))(D(i, CR1) − M (i, CR1))
N (i, SR1) = C(i, SR1) × ; (8.1)
D(i, CR2) − M (i, CR2)
(D(i, TR2) − M (i, TR2))(D(i, CR1) − M (i, CR1))
N (i, SR2) = C(i, SR2) × ; (8.2)
D(i, CR2) − M (i, CR2)
(D(i, TR3) − M (i, TR3))(D(i, CR3) − M (i, CR3))
N (i, SR3) = C(i, SR3) × ; (8.3)
D(i, CR4) − M (i, CR4)
(D(i, TR4) − M (i, TR4))(D(i, CR3) − M (i, CR3))
N (i, VR) = C(i, VR) × . (8.4)
D(i, CR4) − M (i, CR4)
The correlation factor C characterizes the correlation between the top-tagging and b-tagging
algorithms. C = 1 is the ideal case where no correlation is present, while a deviation from
1 implies the b-candidate jet’s b-tagging rate is sensitive to the top-tagging. The dominant
part of the data-driven background is composed of the QCD multi-jet backgrounds whose
MC-simulated events are employed to esimate C.
The correlation factor for the SR1 is shown in Figure 8.3 using Pythia-generated QCD
multi-jet events. The signal region’s yield N on the left-hand side of Equation 8.1 is directly
given and drawn as the black-dotted histogram on the upper panel. The blue histogram
directly applies the data-driven method to the template and control regions’ distribution, as
the subtraction of tt̄ from data is unnecessary. Thus the ratio of the two gives the correlation
factor across Mtb . The null correlation, namely, the ratio of 1, is observed over the entire
spectrum except for the first few bins. While solid evidence for C = 1 is also found in
the other two signal regions and the validation region, a further investigation into potential
sources of correlation provides systematics uncertainties of the data-driven method.
80
Events/200 GeV
104
103 1 b-tag in
50 WP top-tag
102 QCD Multi-jet MC
10
1 Signal region
Extrapolation
10−1
10−2
10−3
SR/Extra.
1.2
1.15
1.1
1.05
1
0.95
0.9
0.85
0.8
1000 2000 3000 4000 5000 6000 7000
Mtb [GeV]
Figure 8.3: The invariant mass Mtb distribution of simulated QCD multi-jet background in
the signal region SR1 (black dot) is compared with its extrapolation (blue lines) from the
template region TR1 and the control regions CR1 and CR2. The vertical axis is the number
of events for a bin width of 200 GeV. The lower panel shows the ratio of the signal region
histogram to the extrapolated one.
8.2.1 Top-tagging and b-tagging correlations
The correlation between the top-tagging and b-tagging can be caused by the flavor of partons
initiating top-candidate and b-candidate jets.At the leading order of QCD, the hard-scatter
part of QCD multi-jet production is described by two-to-two Feynman diagrams, selections
of which are shown in Figure 8.4. One final-state parton is mapped to the large-radius top-
proxy or top-candidate jet by minimizing their φ angles, which is within 0.2 radion in more
than 90% large-radius jet with pT > 500 GeV. The other outgoing parton is then assigned to
the small-radius b-candidate jet. Figure 8.5(a) illustrates the change of the large-radius jet’s
81
q q
q q
g g
q q q0 q0
(a) q q̄ → q q̄ (b) qq 0 → qq 0
q q
g g
g g
g g g g
(c) gg → gg (d) qg → qg
Figure 8.4: Examplified LO Feynman diagrams of the QCD multi-jet background at the
LHC. The vertical axis represents space, and the horizontal axis represents time.
associated flavor by the top-tagging DNN score in the natural log scale for the “1 b-tag in”
category and Mtb ranging from 1.2 to 1.4 TeV. The fraction of gluon-matched large-radius
jets increases monotonically with the top-tagging score while suppressing the quark jets’
fractions. This trend is attributed to a gluon’s broader parton-showering [25] than quarks,
giving gluon jet a higher probability to resemble the signal top-quark jet. The turning point
near the last bin follows from the DNN’s abrupt cut-off at jet mass above 210 GeV, shunning
more gluon to quark jets.
The parton types associated with the large-radius jet, when varied by the top-tagging
algorithm, impact the b-candidate jet’s parton fraction and, in turn, its b-tagging ratios.
82
0.6
Flavor Fraction Flavor fraction
1.2 Mtb = [1.2,1.4) TeV
Top-cand./proxy parton
gluon
light quark 0.5
1
charm quark (× 10)
bottom quark (× 10)
0.4 B-cand. parton
0.8 gluon
light quark
0.3 charm quark (× 20)
0.6 bottom quark (× 20)
0.2
0.4
0.1
0.2
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0
loge(top DNN) loge(top DNN)
(a) (b)
Figure 8.5: Figure (a) shows fractions of different parton flavors associated with the top-
candidate/proxy jet as a function of the top-tagging score in nature logs. The top-tagging
dependence of non-b-tagged jets are presented in figure (b). The last two bins’ lower edges,
corresponding to the 80% working point and the 50% working point, are not exactly the
same as the label. The invariant mass Mtb is between 1.2 and 1.4 TeV for both figures.
Light quarks include up, down, and strange quarks as their masses are below the QCD
condensation scale. The charm and bottom quark fractions are multiplied by factors for
viewing purposes.
Figure 8.5(b) plots the b-candidate jet’s parton composition when the b-candidate jet is not
b-tagged. While gluon-typed large-radius jets increase in proportion with the top-tagging,
the b-candidate jet’s light-quark fraction rises faster than its gluon fraction initially, thanks to
the qg scattering as shown in Figure 8.4(d). For even higher DNN scores, di-gluon production
as in Figure 8.4(c) takes over, leading to the gluon jet’s upturn. Since the difference in b-
tagging ratios between light-quark jets and gluon jets is around 20%, few percent changes
of their portions cause per-mille correlations. In contrast, the fall in the charm-quark jet
proportion at a sub-percent level near the top-tagged line (note the multiplying factor) can
reduce the overall b-tagging ratio by 2% since the ratio of b-tagged to not-b-tagged charm-
quark jets are 5 times higher than that for gluon jets and light-quark jets. Although the
83
bottom-quark jet’s b-tagging ratio is another 5 times higher, its impact here is less significant
due to a relatively stable fraction.
We have seen that the different responses of hard-scatter partons to top-tagging and
b-tagging generate correlations. An additional source of correlation comes from transverse
momentum conservation which pairs a high pT top-candidate/top-proxy jet with another
high pT b-candidate jet. Therefore the two taggers’ passing ratios increase coherently if
both of them increase with pT . The tagger’s momentum dependencies are related to poorer
background rejections. When a top-quark large-radius jet receives a Lorentz boost, the top
quark’s three-prong decay energy spreads across fewer calorimeter clusters, which are less
distinguishable from a single-prong jet. In maintaining a similar tagging rate for top-quark
jets as in lower pT ranges, the background large-radius jet’s top-tagging ratios have to go up
with pT . The b-tagger has similar difficulties as more energetic charged particle tracks have
less curvature under the inner detector’s (ID) magnetic field, challenging the particle vertex
finding. Moreover, relativistic time dilation let high energy particle decay outside the ID’s
innermost layer, leaving fewer hits for track reconstructions.
The pT -induced correlation gets greater with higher Mtb as b-candidate jets are more
likely to be initiated by light quarks only. A higher mass requires the initial-state partons
in the hard-scattering process to carry a higher momentum fraction (x) relative to their
parent protons. Since the parton density of light quarks surpasses that of gluons in the high
x regime, relatively more light quarks should be produced through the process depicted in
Figure 8.4(b). Indeed, Figure 8.6(a) shows that more than 68% of b-candidate jets originate
from light quarks when Mtb is 3 to 3.4 TeV, thus leaving less room for the top-tagging DNN
score to vary flavor compositions than in Figure 8.5(b). Meanwhile, as top-candidate and
84
0.8
Flavor fraction
B-tag ratio Top-tag ratio
Mtb = [3,3.4) TeV 0.2
0.7 0.044
0.6 0.19 0.042
B-cand. parton
0.5 gluon 0.18 0.04
light quark
0.4 charm quark (× 20)
bottom quark (× 20) 0.17 0.038
0.3
0.036
0.2 0.16
0.034
0.1 0.15
−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 1000 1200 1400 1600 1800 2000
loge(top DNN) B-cand. PT [GeV]
(a) (b)
Figure 8.6: (a) The b-candidate jet’s parton flavor fraction as a function of the top-
candidate/proxy jet’s top-tagging score’s natural log in the “1 b-tag in” category. The
last two bins’ lower edges, corresponding to the 80% working point and the 50% working
point, are not the same as the label. The invariant mass Mtb is between 3 TeV and 3.4 TeV.
Non-b-tagged b-candidate jets are shown for reading the b-tag ratio variations.
(b) The b-tagging ratio (blue with left vertical-axis) and top-tagging ratio (red with right
vertical-axis) as a function of the b-candidate jet pT in the same Mtb range and category.
The top-tagging ratio is between the 50% WP top-tagged region and the upper control re-
gion (CRXa, X = numbers), as defined in Figure 8.7. The b-tagging ratio is also calculated
in the control region.
b-candidate jets for greater Mtb have a broader range of pT 1 to generate positive correlation
as tagging probabilities of both taggers increase with pT , as shown in Figure 8.6(b). In
conclusion, the correlation induced by jet flavors or pT can introduce errors to the data-
driven estimate at percent levels, consistent with the observation in Figure 8.3.
8.2.2 The systematic uncertainty
The motivation to study potential sources to the correlations between the top-tagging and
b-tagging is to estimate systematic uncertainties for the data-driven background. Never-
1 But limited by the constraint of |η | < 2
tb
85
theless, the systematic uncertainties will be ultimately determined in data rather than in
the simulated dataset employed in the preceding study since the latter has non-negligible
unknowns, particularly in the high Mtb regime. For example, the parton flavor fractions of
b-candidate jets are affected by the relative sizes of Parton Distribution Functions (PDFs)
of different flavors. PDF uncertainties are substantial for high x, which overlaps with larger
Mtb . Meanwhile, the top-tagging versus b-tagging correlations due to the pT dependences
of tagging gain uncertainties from insufficient data enriched with top quark jets or bottom
quark jets with high pT .
0 b-tag in category 1 b-tag in category
1 1
TR3 SR3 TR1 SR1
50 50
Top-tagging DNN Score Top-tagging DNN Score
WP WP
TR4 VR TR2 SR2
80 80
WP WP
CR4a CR3a CR2a CR1a
e-4 e-4
CR4b CR3b CR2b CR1b
e-7 e-7
85 WP 85 WP
B-tagging DL1r Score B-tagging DL1r Score
Figure 8.7: The categorization in the control regions (CRs) defined in Figure 7.2 is refined for
computing the data-driven background’s systematic uncertainties. The two new thresholds
are placed on e−4 and e−7 of the top-tagging DNN score.
Alternatively, the systematic uncertainties of the data-driven background can be esti-
mated using the correlation factor between subdivided control regions in data. The simu-
lated samples are employed to add top-tagging boundaries reflecting the b-tagging variation.
Figure 8.5(b) shows that the flavor composition of the b-candidate jet has two turning points
86
below the top-tagging line (the edge to the second rightest bin) – around e−7 and e−4 of the
top-tagging DNN score. They form two bands with distinct flavor-induced correlation below
the top-tagging threshold. Thus the control regions are redrawn as in Figure 8.7. Now, only
the control regions closer in DNN scores to the signal regions enter Equation 8.1 through
Equation 8.4. The correlation factor between the upper and lower CR are symmetrized
and smoothened to serve as the systematic uncertainty. The smoothing procedure alleviate
abrupt drops in correlation factors only in few Mtb bins. This uncertainty also incorporates
correlation due to jet pT and other smooth b-tagging variations due to the top-tagging scores.
This data-driven uncertainty is first cross-checked with the deviation between the signal
region distribution and the data-driven method’s output using events of QCD multi-jet MC,
as shown in Figure 8.8. The correlation factor in all regions are well covered by the systematic
plus statistical uncertainties. Notably, the downshift in the low Mtb range of subfigure (d)
(region SR1) is acounted for by the systematic uncertainties. Although the correlation factors
from the control regions are larger than 1 (purple inverted triangle), in the opposite direction
to the factor for the signal region (black dots). The systematic uncertainties do not outsize
the extrapolation’s bin-by-bin statistical uncertainties too much in all signal plus validation
regions. The systematic uncertainties remain below 5% in Mtb < 5 TeV, which is the highest
W 0 mass of interest to set an exclusion limit on the production cross-section times branching
ratio.
The estimated data-driven background and its systematic uncertainties are shown in
Figure 8.9. The Mtb spectra in the control regions are merged more aggressively than in the
template regions to reduce statistical fluctuation in estimating the b-tagged ratios and the
correlation factors. The systematic uncertainties are close to the extrapolation’s statistical
uncertainties in the “0 b-tag in” category, while the former outgrows the latter in the “1
87
b-tag in” category. The systematic uncertainties estimated in data are noticeably smaller
(greater) than the simulated-event counterpart (cyan lines in Figure 8.8) in low (high) mass
regimes. This difference might be partially attributed to higher (lower) event yields of data
than the MC simulation in these two ends. However, statistical errors have been controlled
by using broader bin widths in calculating the correlation factors. Therefore, we suspect
that the correlation is less due to flavor composition than the tagging ratio’s pT dependence
in data.
88
Events/150 GeV
105
Events/150 GeV
0 b-tag in 104 0 b-tag in
10 4 80 WP top-tag 50 WP top-tag
QCD Multi-jet MC 103 QCD Multi-jet MC
3
10
102
102
Signal region 10 Signal region
10
Extrapolation Extrapolation
1 Stat. error 1 Stat. error
CR corr. factor CR corr. factor
10−1 Syst. error 10−1 Syst. error
10−2 10−2
SR/Extra. SR/Extra.
1.15 1.15
1.1 1.1
1.05 1.05
1 1
0.95 0.95
0.9 0.9
1000 2000 3000 4000 5000 6000 7000 1000 2000 3000 4000 5000 6000 7000
Mtb [GeV] Mtb [GeV]
(a) Validation Region (b) Signal Region 3
104
Events/150 GeV Events/150 GeV
4 1 b-tag in 1 b-tag in
10
80 WP top-tag 103 50 WP top-tag
103 QCD Multi-jet MC QCD Multi-jet MC
102
102
10
10 Signal region Signal region
Extrapolation 1 Extrapolation
1 Stat. error Stat. error
CR corr. factor −1 CR corr. factor
10−1 Syst. error
10
Syst. error
10−2 10−2
SR/Extra. SR/Extra.
1.2 1.2
1.15 1.15
1.1 1.1
1.05 1.05
1 1
0.95 0.95
0.9 0.9
0.85 0.85
0.8 0.8
1000 2000 3000 4000 5000 6000 7000 1000 2000 3000 4000 5000 6000 7000
Mtb [GeV] Mtb [GeV]
(c) Signal Region 2 (d) Signal Region 1
Figure 8.8: Validating plots for the data-driven background’s systematic uncertainties by
the simulated QCD multi-jet background. In each diagram, the “SR” histogram is obtained
directly from the top-tagging and b-tagging, and the extrapolation is the result of the data-
driven method. The red hatches show the statistical uncertainty of the extrapolation. The
error bars on the lower panel only include the statistical uncertainty for the SR distribution.
The purple triangular dots show the correlation factor obtained in the control region of each
“b-tag in” category, yielding the cyan-colored systematic uncertainties after symmetrization
(same as a reflection) and smoothing procedure.
89
104
Events/50 GeV Events/50 GeV
0 b-tag in 0 b-tag in
104
80 WP top-tag 10 3 50 WP top-tag
103 Data - tt MC Data - tt MC
102
102
10
10 Extrapolation Extrapolation
Stat. error Stat. error
CR corr. factor 1 CR corr. factor
1
Syst. error Syst. error
10−1 10−1
10−2 10−2
SR/Extra. SR/Extra.
1.2 1.2
1.15 1.15
1.1 1.1
1.05 1.05
1 1
0.95 0.95
0.9 0.9
0.85 0.85
0.8 0.8
1000 2000 3000 4000 5000 6000 7000 1000 2000 3000 4000 5000 6000 7000
Mtb [GeV] Mtb [GeV]
(a) Validation Region (b) Signal Region 3
Events/50 GeV Events/50 GeV
104 1 b-tag in 1 b-tag in
103
80 WP top-tag 50 WP top-tag
103
Data - tt MC Data - tt MC
102
102
10
10
Extrapolation Extrapolation
Stat. error 1 Stat. error
1 CR corr. factor CR corr. factor
Syst. error Syst. error
10−1 10−1
10−2 10−2
SR/Extra. SR/Extra.
1.2 1.2
1.15 1.15
1.1 1.1
1.05 1.05
1 1
0.95 0.95
0.9 0.9
0.85 0.85
0.8 0.8
1000 2000 3000 4000 5000 6000 7000 1000 2000 3000 4000 5000 6000 7000
Mtb [GeV] Mtb [GeV]
(c) Signal Region 2 (d) Signal Region 1
Figure 8.9: The data-driven background’s systematic uncertainties. The extrapolation is the
data-driven background derived from data subtracting the tt̄’s simulated events. Statistical
uncertainties are drawn separately for the extrapolation and the SR histogram as in the
previous figure. The purple triangular dots show the correlation factor obtained in the
control region of each “b-tag in” category, yielding the cyan-colored systematic uncertainties
after symmetrization (same as a reflection) and smoothing procedure.
90
Chapter 9
Statistical Analysis
The Chapter 5 describes the calculation of the NLO cross-section and event generations by
MC simulation for the W 0 signal. Top-quark and bottom-quark jets in data and simulated
events are identified by the tagging algorithms introduced in Chapter 6 and selected into
different categories and regions as explained in Chapter 7. In Chapter 8, the SM background
contribution was estimated, thus completing to model all possible events in the signal regions.
The reconstructed top-jet-plus-bottom-jet mass is analyzed in the signal regions to infer the
presence of W 0 bosons in data.
The signal cross-section is quantified by the profile-likelihood method, which fits the
signal and background models to data. The likelihood function is parameterized by the
signal strength and nuisance parameters, characterizing the systematic uncertainties. The
sources of these systematic uncertainties are reviewed in the first part of this chapter. Then,
the formula for the profile-likelihood function is followed by the introduction to the CLs
method, which is used to set limits on the W 0 cross-section times the branching ratio. A
brief explanation of methods to study the fit qualities in terms of pulls and constraints is
also given. A technique to rank the impact of nuisance parameters is summarized at the end
of the chapter.
91
9.1 Systematic Uncertainties
Systematic uncertainty is a broad concept that can be abstracted as, but not limited to,
potential errors in the instrument or assumptions in interpreting the measured quantities.
Some systematic uncertainties are well-defined statistical uncertainties of physical parame-
ters, such as the calibrated jet energy relative to the reference energy scales. However, in
doing the calibration, some assumptions on the physics associating the two energy scales and
the event selection necessary to obtain a pure sample could bias the result. Such uncertain-
ties can only be estimated indirectly, such as by testing various physics models or varying
the selection thresholds.
In this analysis, the systematic uncertainties are classified either as experimental or
modeling. The experimental uncertainties arise mainly from calibrating the jet properties
and correcting the top- and b-tagging rates in MC simulation. The modeling uncertain-
ties include PDF and missing higher-order terms in calculating the signal or top-quark-pair
background’s total cross-sections and simulating jet kinematics. The uncertainty on the cor-
relation between top-tagging and b-tagging ratios for the data-driven background estimation
also belongs to the category of modeling uncertainties. But as it will be seen in the following,
the calibration processes also make assumptions on the physical models and some model-
ing systematics involve experimental determination. Thus, the classification is not strictly
defined.
9.1.1 Experimental uncertainties
One set of experimental uncertainties originate from calibrating the four-momenta of large-
radius and small-radius jet in the simulated events (Chapter 6). Each nuisance parameter
92
of jet systematics varies each event’s jet energy or jet mass by one standard deviation and
indirectly modifies the pT -dependent top-tagging and b-tagging ratios. Generally, there are
several systematic components in the category of modeling, calibration sample statistical
errors, event selections, or detector effects. Combinations of components across all categories
are found by diagonalizing the correlation matrix. This gives a set of uncorrelated systematic
uncertainties. Only the combination with significant eigenvalues are considered in modelling
the signal and the tt̄ events.
Jet Energy Scale (JES) uncertainties shift the mean jet transverse momenta around
the calibrated values determined in the in situ calibration of large-radius [54] and small-
radius [55] jets. There are 19 JES variations exclusive to large-radius jets, 25 to small-radius
jets, and 5 common to both jet types. Only two eigen-variations of the large-radius jet JES
uncertainties and two common systematics – differences in detector responses to quark versus
gluon jets and inter η calibrations – are non-negligible compared to background modeling
systematics. The width of reconstructed jet energy’s detector effect underestimated in the
MC simulation is accounted for by the Jet Energy Resolution (JER) uncertainties. There
are 8 components for the small-radius jets and 12 for the large-radius jets, both of which
insignificantly shift the Mtb of both the W 0 signal and tt̄ backgrounds since the jet pT in this
analysis is much higher than the energy ranges of electronic noises, pile-up, and the sampling
calorimeter’s statistical fluctuation.
Jet Mass Scale (JMS) corrects the energy distribution within jets after the overall
jet energy has been calibrated, thus having independent systematic uncertainties of the JES
ones. Large-radius jets [54, 126] have 18 components of JMS uncertainties, but all are too
small compared to the jet energy scale to affect the invariant mass Mtb . Although energy
spread uncertainties inside the large-radius jet can impact top-tagging rates, these effects are
93
negligible relative to MC modeling uncertainties. Like the JMS uncertainties, two Jet Mass
Resolution (JMR) uncertainties are insignificant. Applying the above logic to small-radius
jets with even a smaller clustering area to vary its jet substructures leads to no consideration
of small-radius jet mass uncertainties.
The jet energy and mass uncertainties vary the number of jets satisfying the pT re-
quirement of event selections and the invariant mass spectrum for both the W 0 signal and
tt̄ background. In contrast, the top-tagging and b-tagging uncertainties modify the rela-
tive proportions of simulated events among different regions. Besides the propagated jet
energy uncertainties, there are 18 top-tagging systematic uncertainties and 21 b-tagging un-
certainties. Dominant uncertainties appear in both cases are parton shower modeling and
pT extrapolation uncertainties. The b-tagging extrapolation uncertainties, including the ef-
fect of B hadron interactions with the innermost tracking layer, increases with pT to 20%
in tagging probabilities at 2 TeV [127], becoming the most substantial uncertainties in this
analysis.
Other experimental uncertainties have minor impacts, such as the luminosity uncertain-
ties of around 1.7% for the tt̄ background normalization [45, 128] and the variations in MC
simulated pile-up events.
9.1.2 Signal modeling uncertainties
Although the mathematical formalism has no uncertainties, the computation of W 0 signal
production cross-section has uncertainties in practice. As mentioned in section 5.2, the top-
quark mass (mt ) and the strong coupling constant (αs ) are free parameters in the SM. The
other issue is that the hard-scatter matrix element is evaluated order-by-order in the strong
coupling constant. Higher-order terms involve exponentially more Feynman diagrams and
94
more complicated divergences to be regulated. The renomarlization (µR ) and factorization
scales (µF ) are varied to estimate the size of missing higher-order terms. To make things
more challenging, the low energy end of QCD has only phenomenological models. There-
fore, independent measurements (such as [129]) are employed to fit the non-perturbative
Parton Distribution Function (PDF) describing the incoming parton momenta. Since the
W 0 signal in this analysis is a benchmark model of a yet discovered vector boson, theoretical
uncertainties – Mt , αs , PDF, µR , µF – are only summed in quadrature for the production
cross-section as a reference (Left-handed W 0 signal in Figure 5.3a and right-handed W 0 in
Figure 5.3b). In other words, they are not included in the set of nuisance parameters, unlike
the theoretical uncertainties of the background.
9.1.3 Background modeling uncertainties
While a similar combination is done for the tt̄ background’s production cross-section at the
next-to-next-to-leading order, the resulting 4% error, which constraints the total number of
top-quark pair events, constitutes a nuisance parameter in the profile-likelihood fit. Besides
normalization, the tt̄ background shape uncertainties include the PDF+αs and scale variation
on the matrix element. The renormalization and factorization scale uncertainties are among
the top systematic uncertainties in the entire Mtb . In contrast, the PDF uncertainties enlarge
with the invariant mass and overtake the scale uncertainties at about 5 TeV.
Two more significant tt̄ modeling uncertainties are given by replacing either the matrix
element or the parton shower generator. The discrepancy between the default event generator
– Powheg [124] – and its alternative – MadGraph [130] – measures the effect broughty
by matching parton showers to MEs evaluated at the next-to-leading order in QCD [125].
The difference in invariant mass distributions between the two generators increases with Mtb
95
and becomes as large as the scale uncertainties in high mass ranges. The second uncertainty,
the switch from the nominal showering program of Pythia to Herwig [131, 132], results in
similar shape variation as the ME in the low Mtb regime, while the effect subsides gradually.
Other less significant shape uncertainties concern the initial state and final state radiations.
All tt̄ background modeling uncertainties are decorrelated among the thress signal re-
gions. Namely, the same systematic source is characterized by three nuisance parameters
in the profile-likelihood fit – one for each region. The modeling uncertainties are educated
guesses on the size of missing parts or possible pitfalls in the model assumptions rather than
well-defined parameters. The conservative approach allows different sizes of modeling among
different regions.
The uncertainties of correlations between the top-tagging and the b-tagging (section 8.2)
are modeling uncertainties for the data-driven background. The effect of this uncertainty
on the total background is comparable to the tt̄ modeling uncertainties as the data-driven
background is more copious than the tt̄ background, as shown in Figure 8.2. This modeling
uncertainty is decorrelated among the three signal regions (and the validation region). The
separation is physically motivated by changes in parton flavor compositions due to different
top-tagging score ranges of and the application of b-tagging inside the large-radius jet. In
each region, the uncertainty is further decorrelated between the high (> 2 TeV) and low (≤
2 TeV) Mtb ranges. The motiviation is that correlation in the low mass area is attributed
mainly to the jet flavor composition, while the jet kinematics contributes more in the high
mass area.
96
9.1.4 Propagation of uncertainties to the data-driven background
Since the tt̄ distribution in the template region is subtracted from data in the data-driven
background estimation, the experimental and modeling uncertainties of tt̄ from the template
region are propagated through the subtraction of the shifted tt̄ distributions. The resulting
experimental uncertainties on the data-driven background are correlated to those for the tt̄
background and the W 0 signal, i.e., there is only one nuisance parameter per uncertainty.
This combination, in turn, reduces experimental uncertainties since the upward variation
on the tt̄ background reduces the data-driven background in the template region, which is
a downward variation. In contrast, the variation due to tt̄ modeling uncertainties in each
template region is decorrelated to the original tt̄ modeling uncertainties in the neighboring
signal region. Briefly speaking, the correlation between the physical parameter behind the
modeling uncertainties, such as the missing higher-order terms attemptedly covered by the
renormalization scale uncertainty, with the b-tagging of the b-candidate jet is unknown a
priori. This conservative approach yields significant systematic uncertainties for the data-
driven estimate.
9.2 Profile-likelihood Method
The statistical analysis tests the compatibility of data with a hypothesis of signal strength
µ, a multiplicative factor to the signal normalization1 The binned profile-likelihood function
defined on signal region Mtb histograms is employed to incorporate systematic uncertainties.
The number of data events in each bin i of region r (νr,i ) follows a Poisson distribution with
1 The signal production cross-section times the decay branching ratio (σ(pp → W 0 → tb → q q¯0 bb̄)), mul-
tiplied by the integrated luminosity ( L(t)dt). This total number of signal events, times the fraction of
R
events passing the event selection and categorization, gives the signal counts for each signal/validation/tem-
plate/control regions.
97
a mean of µsr,i +br,i , where sr,i and br,i stand for the nominal signal and background counts.
Furthermore, systematic uncertainties introduced in the last section are parameterized in the
list of nuisance parameters θ. ~ Each nuisance parameter θk , where k runs through the list
of systematic uncertainties, is constrained by a Gaussian distribution where one standard
deviation (σk ) from the mean (θk0 ) moves the nominal signal or background by the amount
corresponding to the systematic uncertainty.
The mathematical formula for the binned likelihood function is then written as
~ = (9.1)
Y Y
L (µ; θ) ~ + br,i (θ))
P ois(νr,i |µsr,i (θ) ~ Gauss(θk |θk0 , σk ).
r,i k
The product runs over histogram bin i of signal region r. Note that br,i sums the data-
Q
r,i
driven and tt̄ backgrounds. The nuisance parameters as a whole are represented by a vector
~ By maximizing the function L , the signal strength and nuisance parameter preferred by
θ.
data are estimated by the maximum likelihood method. Moreover, a hypothesis with a fixed
signal strength is tested against data by leaving µ fixed to µ0 in maximizing L .
The ratio of maximum likelihood between floating µ and fixed µ0 forms the likelihood
ratio to measure compatibility between data and the given hypothesis. By convention, the
negative log-likelihood-ratio is defined as
max{L (µ; θ) ~ : µ = µ0 , θk ∈ R}
q(µ0 ) = −2 loge . (9.2)
max{L (µ; θ) ~ : µ ∈ (−∞, µ0 ], θk ∈ R}
The µ in the denominator only moves up to µ0 since a higher estimated µ is considered con-
sistent with the signal hypothesis. Instead, if µ moves far below µ0 , the nuisance parameter
can no longer draw close the data and prediction, leading to a higher, a more extreme value
98
of q(µ0 ). The probability of a given signal hypothesis to have at least or more extreme q(µ0 )
is the p − value. The asymptotic formula of the q(µ0 ) distribution calculated in [133] is then
employed to exclude signal strengths with p − values below 0.05.
9.3 The CLs method
Sometimes, the p − value could pass the exclusion threshold, even though neither the signal
plus background nor the background alone describes the data well. A straightforward exam-
ple is underestimating the background events in a counting experiment. A countermeasure is
the CLs method [134] which requires the rejection of the signal plus background hypothesis
using a different p − value definition:
ps
ps+b = < 0.05, (9.3)
1 − pb
where ps denotes the p − value for the signal-plus-background hypothesis (µ > 0) and pb for
the background-only hypothesis (µ = 0). The denominator 1 − pb measures the sensitivity
of the statistical test [135].
The same reference and [133] point out a way by which we use to calculate the ex-
pected limit that shows the signal exclusion sensitivity under the background-only hypothe-
sis. Pseudo-data are created by setting all nuisance paramters to their best-fit values given
by data and then reset the signal strength to 02 . This dataset gives the median q(µ0 ) under
the background-only hypothesis while testing a signal-plus-background hypothesis of sig-
nal strength of µ0 . The effect of statistical and systematic errors on the expected limit is
2 This is the expected background-only model when data have been unblinded. In the optimization phase
of the analysis strategy, the nuisance parameters are all fixed to their nominal values.
99
computed using the standard deviation of the expected q(µ0 ) distribution. The resulting
expected limit and uncertainty bands are compared with the observed limit given by data
to evaluate performances.
9.4 Fit validations with pulls and constraints
The quality of fit is crucial to ensure the hypothesis test is sensible. Indicators of fit qualities
include pull and constraint. Pull is defined as the difference in nuisance paramter between
the best-fit and nominal value, divided by the pre-fit uncertainty ((θfit − θnominal ) /σθ ). A
significant pull requires further inspections if the corresponding systematic uncertainty is
expected to be inconsequential. A constraint means a significant decrease in the standard
deviation of a nuisance parameter after fitting. Substantial shrinking of post-fit errors in-
dicates that the systematic uncertainties are comparable to or larger than the standard
deviation of the Poisson terms in Equation 9.1.
Since a systematic uncertainty usually affects more than one region in the fit, one region
with the least statistical error could constrain the associated nuisance parameter of another
region with a larger error. For example, the “80 WP top-tag, 1 b-tag in” region has the
tiniest statistical uncertainty among all signal regions. Despite having fewer expected signal
yields than the “50 WP top-tag, 1 b-tag in” region, the fit allows the former to constrain
substantial systematic uncertainty like the one for b-tagging extrapolation, thereby enhancing
the search sensitivity. But the mutual constraint across the two regions does not apply to
the top-tagging systematics, as the calibration of the tagger’s 80% working point is not
conducted simultaneously with the 50% working point, leaving an undetermined correlation
between the nuisance parameters of the same source for the two working points. Thus they
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are left uncorrelated, i.e., remain independent parameters in the likelihood function and let
the fit calculate their covariance.
9.5 Nuisance parameter ranking
The statistical significance of data under the hypothesis with a signal strength µ is measured
by the negative log-likelihood ratio (Equation 9.2), which is the ratio between the minimized
profile-likelihoods with a fixed µ and that with an unconstrained µ determined by the fit
(µ̂). Therefore, the degree that µ̂ is shifted by the fit in response to the variation of nuisance
parameters can tell the relative impacts of the parameters on the minimization of the negative
log-likelihood function and in turn, the limit setting.
More precisely, the global best-fit signal strength µ̂ and the simultaneously best-fit θ~ of
N nuisance parameters θ̂1 , θ̂2 , . . . , θ̂N are given by minimizing the negative log-likelihood
~ : µ, θ1 , θ2 , . . . , θN ∈ R}.
− log L (µ̂, θ̂1 , θ̂2 , . . . , θ̂N ) = min{− log L (µ; θ) (9.4)
The symbol µ̂ is the maximum likelihood estimator of µ commonplace in literature, such as
the Statistics chapter in [20]. In these contexts, the cap notation already implies that it is a
variable that (co-)minimizes the negative log-likelihood (a computational trick equivalent to
maximizing the likelihood function) for given data. But for more easily denoting the change
of µ by fixing one nuisance parameter, here µ̂ is reserved for the µ given by the minimizing
operation on the right-hand side of Equation 9.4.
Re-minimizing the negative log-likelihoods by fixing a nuisance parameter to the global
best-fit value θ̂i plus-or-minus the uncertainty σi requires µ to be shifted to µ ± ∆µ, as in
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the following formula:
~ : µ, θ1 , . . . , θi−1 , θi+1 , . . . , θN ∈ R; θi = θˆi ± σi }.
− log L (µ̂ ± ∆µ; . . . ) = min{− log L (µ; θ)
(9.5)
In Chapter 10, nuisance parameters will be ranked by ∆µ using the post-fit uncertainty
estimated from the global fit represented by Equation 9.4.
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Chapter 10
Statistical Analysis Results
This chapter employs the statistical procedures introduced in Chapter 9 to study the com-
patibility between data and prediction and infer the possible existence of the W 0 signal. It
starts with verifying the robustness of the statistical analysis procedures. The preciseness
of the data-driven background estimation is first studied in the signal-insensitive validation
region. For the signal regions, the background-only model is used as pseudo-data to provide
expected constraints, correlations, and blinded expected limits (definitions can be found in
Chapter 9). The background-only fit is repeated but with data instead. The pull plots of
the nuisance parameters and correlations are shown for the observed data too. The lack of
data excess relative to the background is found, and hence the exclusion limit is set. In the
end of this chapter, we analyze the psuedo-data and data with a unconstrained fit, which
yields a list of the most impactful systematic uncertainties.
10.1 Validating the Data-driven Background Estima-
tion
The dominant background in the signal regions is estimated by a data-driven method, which
is verified by comparing the total background predictions to data in the validation region
where the W 0 signal is negligible (VR shown in Figure 7.2). The top-tagging requirement for
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this region is the same as the “SR2” signal region, which has less background rejection than
the tagger threshold applied to the other two signal regions. This less stringent top-tagging
criterion and the “0 b-tag in” categorization for this region render the number of signal events
insignificant relative to the predicted backgrounds even around the W 0 resonance peak in
Mtb .
(a) (b)
Figure 10.1: Data and predicted background Mtb distribution in the validation region (top-
tagged at the 80% but below the 50% working point and in the “0 b-tag in” cateogry) (a)
before and (b) after the profile-likelihood fit. The hatched uncertainty bands include the
systematic uncertainties and the statisitcal uncertainties of the background predictions.
The comparison between data and the predicted background distribution in the validation
region is presented in Figure 10.1. As the data-driven background is two orders of magnitude
larger than the tt̄ background in the whole mass range, the systematic uncertainties of the
tt̄ have an insignificant effect on validating the data-driven estimate. The relative difference
between the data and background distributions stays within 5% from 1 TeV to 3 TeV of Mtb
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and below 10% up to 4 TeV in Figure 10.1(a), where no profile-likelihood fit is performed.
No significant discrepancies between the two compared to the statistical plus systematic
uncertainties are found over the entire mass range. A pre-fit χ2 of 28.7 per 31 degrees of
freedom supports this observation. A profile-likelihood fit of the background model to data
yields the distributions in Figure 10.1(b), shifting the background prediction by a negligible
amount from the pre-fit. The χ2 value decreases only by 0.2 to 27.9 after this background-
only fit. The small shift implies that the pre-fit background model parameters are near the
best-fit values favored by data.
Figure 10.2: The pull and contraint plots of the data-driven-background systematic uncer-
tainties in the background-model fit to data in the validation region. The central value (θ0 )
is the pre-fit value, which is 0 by contruction. For each nuisance parameter, the black dot
shows the post-fit value (θ̂) with the post-fit uncertainty as its error bar, which is in general
than the pre-fit uncertainty of unit size (∆θ).
The nuisance parameter pulls associated with the systematic uncertainties of data-driven
estimation are shown in Figure 10.2. Two nuisance parameters are assigned to the low
(Mtb < 2 TeV) and high (Mtb ≥ 2 TeV) mass regimes each. Recall from Chapter 8 that
this systematic uncertainty is due to the correlation between top-tagging and b-tagging.
Thus, the two nuisance parameters’ consistency with their pre-fit value implies that data are
compatible with a lack of correlation between the two taggers. The significant constraints
are expected as the validation region has sufficient data to constrain systematic errors at a
percent level. All the other systematic uncertainties are not shown as they have insignificant
pulls and constraints. The definition of pulls and constraints can be found in Chapter 9.
105
10.2 Test the Signal Region Fit with the Background-
only Pseudo-data
Before comparing data and the background predictions in the signal regions, we performed
the profile-likelihood fit to the Mtb distributions by treating the background predictions as
pseudo-data. This dataset is expected to attain maximum likelihood without shifting the
nuisance parameters, thus allowing us to study the constraints and correlations between
systematic uncertainties. This method helps to find ill-behaved nuisance parameters before
conducting the fit to data. The signal region Mtb distributions before the profile-likelihood
fit with the “µ = 0” pseudo-data are shown in Figure 10.3.
The Mtb distributions after the profile-likelihood fit under the constraint of µ = 0 are
shown in Figure 10.4. There is no change in the heights at each bin since the pseudo-data
is identical to the total backgrounds. The uncertainty bands shrink substantially in the low
mass region as the fit constrained the 1 σ deviation of the nuisance parameter representing
the systematic uncertainties. The pre-fit systematic uncertainty can be thought of as the
width of a physical parameter from the complementary measurement. In this sense, the
comparison of data and background predictions serves as an independent measurement such
that the combined uncertainty, the post-fit value, is naturally less than the pre-fit one. The
statistical uncertainties of the background distribution, which dominate the uncertainty band
in the high mass region, are less constrained, for the data-driven estimation has considerably
lower statistical errors than data by adopting wide bin widths (Figure 8.9). Moreover, the
statistical uncertainties are independent between bins, further loosen the constraint.
The pull plots after the fit are shown in Figure 10.5 to Figure 10.13. Most of the
nuisance parameters are not constrained under the fit. Notable exceptions are “Top-tag
106
(a) (b)
(c)
Figure 10.3: Predicted background Mtb distribution in the signal regions: (a) SR1, (b) SR2,
and (c) SR3 before the profile-likelihood fit. The hatched uncertainty bands include the
systematic uncertainties and the statisitcal uncertainties of the background predictions.
107
(a) (b)
(c)
Figure 10.4: Predicted background Mtb distribution in the signal regions: (a) SR1, (b) SR2,
and (c) SR3 after the profile-likelihood fit to the pseudo-data with “µ = 0.” The hatched
uncertainty bands include the systematic uncertainties and the statisitcal uncertainties of
the background predictions.
108
Figure 10.5: The nuisance parameter pulls for jet energy scale uncertainties common to
large-radius jets and small-radius jets after a profile-likelihood fit to the pseudo-data with
“µ = 0.” “EtaInter” means the in situ calibrations across different η regions of the detector.
Figure 10.6: The nuisance parameter pulls for large-radius jet energy scale and resolution
uncertainties after a profile-likelihood fit to the pseudo-data with “µ = 0.” “TopoUncer”
refers to jet energy modeling uncertainty for top-quark jets. “JER” denotes jet energy
resolution. “jetEffNP1” is the first component in the principle component decomposition for
propagating the jet energy scale uncertainty. Each “EffectiveNP” is an eigenvector in the
principle component analysis. The suffices, “Stat,” “Model,” “Detect,” and “Mixed” means
that the eigenvector combines statistical, modeling, detector, or a mixture of systematic
error sources.
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Figure 10.7: The nuisance parameter pulls for small-radius jet energy scale and resolution
uncertainties after a profile-likelihood fit to the pseudo-data with “µ = 0.” “PunchThrough”
means the hadronic shower of a jet is too deep to be contained by the ATLAS calorimeter.
“Rho” is the energy profile of an event used to correct for pile-up contributions in the
calorimeter measurements. “restTerm” is a quadratic sum of the most insignificant JER
components. “NonClos” and “highE” refer to non-closure and high-energy, respectively.
“BJES” means the jet energy scale for a bottom-quark jet. Some other terms are defined in
the captions of Figure 10.5 and Figure 10.6.
Figure 10.8: The nuisance parameter pulls for large-radius jet mass scale and resolution
uncertainties after a profile-likelihood fit to the pseudo-data with “µ = 0.” “Rtrk” is the
ratio between jet mass estimated by tracks to that measured by calorimeters. The difference
in Rtrk between simulation and data serves as a systematic uncertainty. “COMB” refers to
“combined mass,” a mass observable combining the tracking and calorimeter measurements.
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Figure 10.9: The nuisance parameter pulls for the top-tagging uncertainties after a profile-
likelihood fit to the pseudo-data with “µ = 0.” The number 80 and 50 mean the large-radius
jet is top-tagged at the 80% working point but not the 50% working point or top-tagged
at the 50% working point, respectively. “Light-jet b-tag” are b-tagging nuisance parameters
for the light-quark or gluon jet since the top-tagging calibration selects events with b-tagged
jets. “Signal Eff.” accounts for the chirality difference between the tt̄ events with unpolarized
top-quarks and the signal events with polarized top-quarks. “Other Top Eff.” are uncertainty
in top-tagging efficiency for top-quark jets with non-optimal reconstructions such as missing
top-quark-decay products within a truth-level jet. “Signal Propagated” is a mixture of
insignificant jet energy scale uncertainty components than those propagated via the JES
uncertainties for well-reconstructed top-quark jets. “High-Pt Ext.” accounts for uncertainties
in high transverse momentum beyond the range calibrated with respect to data. “Bin-Var”
concerns the pT -bin choices in fitting the efficiency curve. “Di-jet” means a particular type
of QCD multi-jet event with two high pT jets. “SF” stands for Scale Factor, which scales the
tagging probability in simulated events to the calibrated value. “hdamp” is the matching
scale for the NNLO event generator, Powheg [125]. “ME” refers to the matrix element.
“Shower” is the parton shower. Alternative event generators of ME and showers are discussed
in Chapter 9.
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Figure 10.10: The nuisance parameter pulls for the b-tagging uncertainties after a profile-
likelihood fit to the pseudo-data with “µ = 0.” The name “(I) EV NP 3” means the third
eigenvector of nuisance parameters for calibrating the b-tagging probability for a light-hadron
jet (no heavy hadrons that contains a charm or bottom quark). The same convention applies
to a charm-hadron jet by replacing the capitalized letter “I” with “C” and to bottom-hadron
jet by replacing it with “B.” “High-Pt Extrapolation” refers to the uncertainties caused by
insufficient data to calibrate b-tagging probabilities for small-radius jets with high transverse
momenta (for more descriptions see Chapter 9 and Ref. [127]).
Figure 10.11: The nuisance parameter pulls for the modeling uncertainties of the data-driven
background estimation after a profile-likelihood fit to the pseudo-data with “µ = 0.” The
symbols “LowMtb” and “HighMtb” mean that the nuisance parameter varies the data-driven
background in invariant mass Mtb below or above 2 TeV. The region code (Figure 7.2) at
the end further specifies the signal region whose data-driven background is to be varied.
112
Figure 10.12: The nuisance parameter pulls for the modeling uncertainties of the top-quark
pair background after a profile-likelihood fit to the pseudo-data with “µ = 0.” The term
“shower,” “pdf,” “muR,” “muF,” “ME,” “ISR,” “FSR” refer to the parton shower generator,
Parton Distribution Fuction, renormalization scale, factorization scale, initial state radia-
tion, and final state radiation respectively. These terms are described in section 2.1 and
subsection 9.1.3. The region code (Figure 7.2) at the end further specifies the signal region
where the tt̄ background is to be varied. Nuisance parameters of “DataDriven (DataDriv)”
account for the uncertainty from subtracting the tt̄ background in the specified signal re-
gion’s neighboring template region (the b-candidate jet is not b-tagged rather than tagged)
in the data-driven background estimation.
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Figure 10.13: The nuisance parameter pulls for the 1.7% luminosity uncertainty and the 4%
top-quark pair background production cross-section uncertainty after a profile-likelihood fit
to the pseudo-data with “µ = 0.”
Shower Model 80” in Figure 10.9, “b-tag High-Pt Extrapolation” in Figure 10.10, all data-
driven systematics in Figure 10.11, and six tt̄ modeling uncertainties in Figure 10.12 –
“model_tt_shower SR1,” “model_tt_muR_DataDriven SR1,” “model_tt_muR SR1 and
SR2,” “model_tt_ME_DataDriven SR2,” and “model_tt_ME SR1.” The nuisance pa-
rameters for the data-driven background are even more constrained than those for the tt̄
background since the former accounts for 80% or higher of the total background in each
region (The region with the highest portion oftt̄ background is shown in Figure 8.2(b)). The
dominance increase with Mtb , leaving fewer degrees of freedom provided by nuisance param-
eters of the tt̄ background modeling. A sudden jump of systematic variation around 2 TeV
(Figure 8.9) outweighs the increases in Poisson errors, which scales with the square root of
event counts, leading to more significant constraints found in the high Mtb components of
the systematics.
The correlations between nuisance parameters are obtained from the covariance matrices
of the likelihood function evaluated at the best-fit parameters, which are identical to their
nominal values in the µ = 0 pseudo-data. Figure 10.14 shows the most significant correlation
factors between the nuisance parameters, none of whose absolute value exceeds 30%. It
could be counterintuitive at first sight that there are nonzero correlation factors between
the nuisance parameters of the data-driven background in different signal and Mtb regions.
But these isolated nuisance parameters share the degree of freedom with other nuisance
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Figure 10.14: The correlation matrix between nuisance parameters after a profile-likelihood
fit to the pseudo-data with “µ = 0.” The matrix is symmetic over the diagonal from the top
left to the bottom right corner. Only nuisance parameters with more than 20% of correlation
with another are shown. The meaning of individual nuisance parameters can be found in
the preceding Figure 10.5 to Figure 10.13.
115
parameters common to all regions, such as the “b-tag High-Pt Extrapolation,” that share
some shape similarities with the former. The majority of stronger correlations are associated
with the tt̄ background modeling systematics of different regions (including those directly
applied to the signal regions versus those propagated through the data-driven estimation from
the template regions – “DataDriven”). Still, the small correlation justifies their separations
a posteriori.
10.3 Exclusion Upper Limit with the Background-only
Pseudo-data
Chapter 9 introduces a method of excluding signal cross-sections too large to be compatible
with data when the CLs is less than 5%. Figure 10.15 shows this 5% boundary on the cross-
section times the branching ratio, the upper limit, by treating the background predictions as
data. The limit curve consists of discrete points calculated for five WR0 signals with masses
ranges from 1.5 TeV to 5 TeV (Chapter 5). From the crossing point between the theoretical
cross-sections and the expected limit, the mass of the right-handed W 0 is excluded up to
about 4.15 TeV at the 95% Confidence Level. This limit represents the expected result to
be compared with the limit computed with the actual data in later sections.
10.4 The Signal Region Fit with Observed data
Data is compared with the background predictions before and after the profile-likelihood fit
under the background-only hypothesis (µ fixed to 0 in practice), as shown in Figure 10.16 and
Figure 10.17, respectively. While the fit reduces the background uncertainties significantly,
116
10
σ(pp → W' ) × B(W' → tb) [pb]
R 1
R
10−1
10−2
10−1500
3
2000 2500 3000 3500 4000 4500 5000
m(W' ) [GeV]
R
Figure 10.15: The upper limit on the WR0 production cross-section times the decay branching
ratio of WR0 → tb at the 95% Confidence Level is evaluated with the background-only pseudo-
data. The blue dashed curve indicates the median, while the green and yellow bands show the
one and two Gaussian quantiles of the limit under statistical plus systematic uncertainties.
The Next-to-Leading-Order theoretical cross-section times the branching ratio is drawn as
the red dashed curve with an asymmetric theoretical uncertainty band (Chapter 5).
117
(a) (b)
(c)
Figure 10.16: Data versus background Mtb distributions in the signal regions: (a) SR1, (b)
SR2, and (c) SR3 before the profile-likelihood fit. The hatched uncertainty bands include
the systematic uncertainties and the statisitcal uncertainties of the background predictions.
118
(a) (b)
(c)
Figure 10.17: Data versus background Mtb distribution in the signal regions: (a) SR1, (b)
SR2, and (c) SR3 after the profile-likelihood fit with under the background-only hypothe-
sis. The hatched uncertainty bands include the systematic uncertainties and the statisitcal
uncertainties of the background predictions.
119
the background predictions hardly deviate from the pre-fit values in all three regions. With
excellent consistency between data and the nominal background for Mtb < 3 TeV, the largest
shift of bin heights of the background after the fit is observed in the high mass regions in
the “80 WP top-tag, 1 b-tag in” (SR2, Figure 10.17(b)) region: 5% down and the “50 WP
top-tag, 0 b-tag in” (SR3, Figure 10.17(c)) region: 3% down. Both of them signify a surplus
of predicted background events. The only significant excess is seen in one bin around 5 TeV
in the “50 WP top-tag, 1 b-tag in” (SR1, Figure 10.17(a)) region, where the total post-fit
background = 1.18±0.51 versus 5 counts in data. It corresponds to background rejection
significance of just 2.2 σ (according to formula (25) in Ref. [136]).
Figure 10.18: The nuisance parameter pulls for jet energy scale uncertainties common
to large-radius jets and small-radius jets after a profile-likelihood fit to data under the
background-only hypothesis (µ = 0). The meaning of the nuisance parameters are explained
in Figure 10.5.
The pull plots for nuisance parameters after a background-only profile-likelihood fit to
data are shown from Figure 10.18 to Figure 10.26. None of the nuisance parameters is pulled
beyond one σ of the pre-fit uncertainty. Most of the significant pulls are seen in the nuisance
parameters specific to the “80 WP top-tag, 1 b-tag in” (SR2, Figure 10.17(b)) region, such
as those bearing the name “80” in their suffices in Figure 10.22 and also those with “SR2”
in Figure 10.24 and Figure 10.25. The low data to total background ratios in high Mtb of
the region and the lowest statistical errors among all signal regions contribute to the shift.
A similar effect explains the substantial pull for the data-driven background uncertainty for
the high Mtb part in the “SR3” region (Figure 10.24). The region’s requirement of 0 b-tagged
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Figure 10.19: The nuisance parameter pulls for large-radius jet energy scale and resolution
uncertainties after a profile-likelihood fit to data under the background-only hypothesis (µ
= 0). The meaning of the nuisance parameters are explained in Figure 10.6.
Figure 10.20: The nuisance parameter pulls for small-radius jet energy scale and resolution
uncertainties after a profile-likelihood fit to data under the background-only hypothesis (µ
= 0). The meaning of the nuisance parameters are explained in Figure 10.7.
121
Figure 10.21: The nuisance parameter pulls for large-radius jet mass scale and resolution
uncertainties after a profile-likelihood fit to data under the background-only hypothesis (µ
= 0). The meaning of the nuisance parameters are explained in Figure 10.8.
Figure 10.22: The nuisance parameter pulls for the top-tagging uncertainties after a profile-
likelihood fit to data under the background-only hypothesis (µ = 0). The meaning of the
nuisance parameters are explained in Figure 10.9.
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Figure 10.23: The nuisance parameter pulls for the b-tagging uncertainties after a profile-
likelihood fit to data under the background-only hypothesis (µ = 0). The meaning of the
nuisance parameters are explained in Figure 10.10.
Figure 10.24: The nuisance parameter pulls for the modeling uncertainties of the data-
driven background estimation after a profile-likelihood fit to data under the background-only
hypothesis (µ = 0). The meaning of the nuisance parameters are explained in Figure 10.11.
123
Figure 10.25: The nuisance parameter pulls for the modeling uncertainties of the top-quark
pair background after a profile-likelihood fit to data under the background-only hypothesis
(µ = 0). The meaning of the nuisance parameters are explained in Figure 10.12.
Figure 10.26: The nuisance parameter pulls for the 1.7% luminosity uncertainty and the 4%
top-quark pair background production cross-section uncertainty after a profile-likelihood fit
to data under the background-only hypothesis (µ = 0).
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Figure 10.27: The correlation matrix between nuisance parameters after a profile-likelihood
fit to data under the background-only hypothesis (µ = 0). The matrix is symmetic over
the diagonal from the top left to the bottom right corner. Only nuisance parameters with
more than 20% of correlation with another are shown. The meaning of individual nuisance
parameters can be found in the preceding Figure 10.5 to Figure 10.13.
125
small-radius jet inside the top-candidate jet suppressed the tt̄ background, thus relying on
varying the data-driven background to achieve the best fit.
As suggested by a previous section, pulls and constraints of a nuisance parameter provide
information on the physics behind the corresponding systematic uncertainty assessed by
asantz or calibrations. The data-driven estimation in Chapter 8 assumes no correlation
between the top-tagging and b-tagging for the pre-fit background and assigns systematic
uncertainties to model the nonzero correlation estimated from the control regions. The
flavor composition of partons is found to be a physical source for a decreasing b-tagging
ratio with increasing top-tagging level (less background), while the two taggers’ high-pT -
preferring tagging probabilities imply an increasing b-tagging ratio (more background). As
none of the nuisance parameters show a significant inconsistency with 0 in either positive or
negative direction (coinciding with the background’s increase and decrease), the two sources
do not manifest in either the low or high Mtb range. Nonetheless, an intriguing situation is
observed in the region “SR3” that the pull directions seen for the two mass ranges misalign
with the prior assumption that the b-tagging ratio is suppressed by the differences in flavor
composition mainly in low invariant mass and raised by the kinematic effect in the high mass
region.
No affirmative physical interpretation can be attained from the pull plots as there is
no significant pull relative to the uncertainty. Especially, Figure 10.24 cannot tell which
effect is more dominant in data: the negative correlation (below 0 on the plot) caused by
the parton flavor compositions of the jet or the positive correlation (above 0 on the plot)
caused by the increasing probabilities to either top-tag or b-tag a jet with pT (Chapter 8).
Nevertheless, the differences in nuisance parameter shifts between different signal regions, as
seen for the data-driven background estimation uncertainties (Figure 10.24) and for the tt̄
126
background uncertainties (Figure 10.22), suggest that top-tagging and b-tagging algorithms
have sensitivities to physics underlying the modeling systematics.
The correlation coefficients between nuisance parameters after the fit to data are shown
in Figure 10.27. There is no correlation coefficient exceeding 34% in absolute values and
no significant change from the values for the fit to the background-only pseudo-data in Fig-
ure 10.14. One noticeable difference is the appearance of entries corresponding to the low
Mtb correlation systematic uncertainties for Signal Region 2 (named “Data-driven Model
LowMtbSR2), with a correlation of 22% with the high Mtb counterpart, probably to accom-
modate pulls from other systematic uncertainties in this region. This correlation coefficient
was around 12% in the pseudo-data fit, thus not shown in Figure 10.14.
10.5 Exclusion Upper Limit with Observed Data
The limit on the WR0 production cross-section times the branching ratio of the tb decay chan-
nel – excluded at the 95% Confidence Level – is set for five separate W 0 masses. They are
connected as the exclusion curves shown in Figure 10.28. As mentioned in Chapter 9, the un-
blinded expected limit employs pseudo-data by letting µ = 0 in the signal-plus-background
model after a fit to data with the signal strength µ unconstrained. This procedure “fixes”
the nuisance parameters to their best-fit values to form the background hypothesis corrected
by data while reducing the potential signal by removing the signal shape scaled by µ. The
resulting expected limit is consistent with that obtained by the blinded pseudo-data (Fig-
ure 10.15), reflecting the insubstantial nuisance parameter pulls in the fit to the observed
data. The observed limit is calculated using the observed data without the fixation of µ to 0.
Thus, the excess or deficit of events of the observed data as compared with the pseudo-data
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10
σ(pp → W' ) × B(W' → tb) [pb]
s = 13 TeV, 138.97 fb-1 Observed 95% CL limit
Expected 95% CL limit
Expected 95% CL limit ± 1 σ
R 1
Expected 95% CL limit ± 2 σ
NLO W' cross-section (ZTOP)
R
10−1
10−2
10−3
1500 2000 2500 3000 3500 4000 4500 5000
m(W' ) [GeV]
R
Figure 10.28: The upper limit on the WR0 production cross-section times the decay branching
ratio of WR0 → tb at the 95% Confidence Level is evaluated with data. The blue dashed curve
indicates the median expected limit, while the green and yellow bands show the one and two
Gaussian quantiles under statistical plus systematic uncertainties. The black solid curve
shows the observed limit. The Next-to-Leading-Order theoretical cross-section times the
branching ratio is drawn as the red dashed curve with an asymmetric theoretical uncertainty
band (Chapter 5).
128
causes the upward and downward discrepancies between the two limit curves. The crossing
point between the theory curve and the interpolated limit between the 4 and 5 TeV mass
gives the observed mass limit at around 4.4 TeV for a right-handed W 0 .
10.6 Systematic Uncertainty Rankings for a 4 TeV WR0
Since the 4 TeV right-handed W 0 is the most massive signal model excluded at the 95%
Confidence Level, the most impactful systematic uncertainties in its fit are studied in the
following with three different datasets: the pseudo-data composed only of background pre-
dictions, the pseudo-data with backgrounds plus a 4 TeV WR0 signal scaled to the blinded
expected limit shown in Figure 10.15, and the observed data.
10.6.1 Background-only pseudo-data
The background-only pseudo-data is fitted with µ free-floating. The fit returns µ̂ = 0.00±0.46,
in the unit of the blinded expected limit for the 4 TeV WR0 signal (Figure 10.15), and with-
out any nuisance parameter shifted. Figure 10.29 shows the nuisance parameter ranking by
repeating the fit while varying one nuisance parameter by its pre-fit or post-fit error at a
time (as described in Chapter 9). Only the top three nuisance parameter yield a significant
change in µ̂ (µ given by the fit), as compared to the systematic plus statistical uncertainty
of µ. These three nuisance parameters characterize the correlation uncertainty in the data-
driven background estimation in Mtb > 2 TeV (Chapter 8) for the indicated signal region.
Since they are all highly constrained in the pseudo-data fit (Figure 10.11), it is unsurprising
that the distortion of the background distribution incurred by the shift by one σ needs to
be balanced by a substantial change in µ to minimize the likelihood function.
129
Figure 10.29: The nuisance parameter rankings for the 4 TeV WR0 signal with respect to
the pseudo-data with “µ = 0.” Color-filled bands show the fifteen-highest impact on the
best-fit µ (∆µ) by fixing a particular nuisance parameter to its best-fit value plus-or-minus
the post-fit error (θ̂ ± ∆θ̂). The unfilled bands with colored boundaries correspond to ∆µ
when the nuisance parameter is shifted by its pre-fit uncertainty (θ̂ ± ∆θ). The scale of ∆µ is
shown on the top horizontal line. The bottom horizontal scale concerns with the black dots
and bars, the best-fit nuisance paramters and their post-fit errors, as in the pull plots. The
captions from Figure 10.5 to Figure 10.13 explain the naming convention of the nuisance
parameters.
130
10.6.2 Signal-plus-background pseudo-data
The background-plus-signal pseudo-data is fitted with the µ free-floating. The fit returns µ̂ =
1.00±0.53, in the unit of the blinded expected limit for the 4 TeV WR0 signal (Figure 10.15),
and without any nuisance parameter shifted. Figure 10.30 shows the nuisance parameter
ranking by repeating the fit while varying one nuisance parameter by its pre-fit or post-fit
error at a time. Only the top three significantly change µ̂ in comparison with the systematic
plus statistical uncertainty of µ. Since a significant amount of signal events are injected into
the pseudo-data, µ becomes susceptible to the nuisance parameter variations from the signal
sample than it does in the fit with the background-only pseudo-data shown in Figure 10.29.
This difference explains why the top-ranked nuisance parameter is replaced by the high pT
uncertainty on the b-tagging probability, which only affects the most dominant background
indirectly via subtracting the tt̄ background in the template region. The emergence of signal
could also explain the increased influences on the fit outcome by the top-tagging systematics
but fall short for the rise of top-quark pair background modeling and the fall of the data-
driven uncertainty of Signal Region 1 and 3. The shape similarities between the varied
nuisance parameter and the signal shape could play a role. For example, the variation for
Signal Region 1 and 2 stays flat above 3.5 TeV in Mtb (Figure 8.9(d) and (c)), while that for
Signal Region 3 (Figure 8.9(b)) has a rising structure above 4 TeV that better mimics the
signal shape.
10.6.3 Observed data
The observed data is fitted with µ free-floating. The fit returns µ̂ = -1.67±0.95, in the unit of
the observed limit for the 4 TeV WR0 signal (Figure 10.28), with nuisance parameters pulled
131
Figure 10.30: The nuisance parameter rankings for the 4 TeV WR0 signal with respect to
the pseudo-data with “µ = 1.” Color-filled bands show the fifteen-highest impact on the
best-fit µ (∆µ) by fixing a particular nuisance parameter to its best-fit value plus-or-minus
the post-fit error (θ̂ ± ∆θ̂). The unfilled bands with colored boundaries correspond to ∆µ
when the nuisance parameter is shifted by its pre-fit uncertainty (θ̂ ± ∆θ). The scale of ∆µ is
shown on the top horizontal line. The bottom horizontal scale concerns with the black dots
and bars, the best-fit nuisance paramters and their post-fit errors, as in the pull plots. The
captions from Figure 10.5 to Figure 10.13 explain the naming convention of the nuisance
parameters.
132
Figure 10.31: The nuisance parameter rankings for the 4 TeV WR0 signal with respect to the
observed data. Color-filled bands show the fifteen-highest impact on the best-fit µ (∆µ) by
fixing a particular nuisance parameter to its best-fit value plus-or-minus the post-fit error
(θ̂ ± ∆θ̂). The unfilled bands with colored boundaries correspond to ∆µ when the nuisance
parameter is shifted by its pre-fit uncertainty (θ̂ ± ∆θ). The scale of ∆µ is shown on the
top horizontal line. The bottom horizontal scale concerns with the black dots and bars, the
best-fit nuisance paramters and their post-fit errors, as in the pull plots. The captions from
Figure 10.5 to Figure 10.13 explain the naming convention of the nuisance parameters.
133
Figure 10.32: The nuisance parameter pulls for the modeling uncertainties of the data-driven
background estimation after a profile-likelihood fit to data with µ determined by the fit. The
meaning of the nuisance parameters are explained in Figure 10.11.
similarly to those shown from Figure 10.18 to Figure 10.26. The largest difference in pulls
is seen for the high Mtb data-driven background systematics for the region “50 WP top-tag,
1 b-tag in,” which can be observed by comparing Figure 10.24 with Figure 10.32. This
nuisance parameter is ranked at the top in Figure 10.31. Here, only the top five nuisance
parameters have a significant ∆µ, as compared to the uncertainty of µ. The deficit of data in
the high Mtb regions shown in Figure 10.17 draws µ̂ to a negative value, leading to more ∆µ
by nuisance parameters affecting the signal sample, as illustrated in the previous analysis on
the pseudo-data with signal injection.
134
Chapter 11
Conclusion
While this text is dedicated to the search for a new particle W 0 , this thesis also aims to give
general views on experimental methods of particle physics at the Large Hadron Collider.
Physics concepts scatter across various chapters beyond the theoretical introductions in
Chapter 2 are crucial to comprehend the logical chains that link electronic signals to the
transient physical phenomema after every proton-proton collision. The particle W 0 , unlike
an atom, cannot be visualized as microscopic humps imaged by an electronic microscope.
But its existence can be indirectly seen by combing the jet momenta under the framework of
Quantum Chromodynamics (QCD). Thus, whether the existence of W 0 could be established
or not, the utilized apparatus and methodologies deserve appreciation.
On the surface, conducting an analysis, especially a search for new particles beyond the
Standard Model, has seemingly been reduced to intricate plays with software. Thanks to
the whole ATLAS collaboration’s efforts in writing computer programs, mechanical construc-
tions, and calibrations, experimental studies of a physics model can be embarked directly
on the selection criterion placed on physical observables that enhance the significance of
signal versus background. Even though designing event selection and performing statistical
analysis still calls upon physics intuition from time to time, the interpretation of data still
inevitably becomes less straightforward as the experimental toolbox advances. Analyzers see
the physics information condensed to various figures and might need to reverse-engineer the
135
analysis tools to obtain the full idea associated with data points and curves.
Withot unraveling details of the analysis process, this thesis attempts to add sense to this
complicated plot-making machine. The concept of jets is first described in the theoretical
discussion of parton showers and hadronization, and later the calorimeter measurement of
jet energy is introduced with the machinery. The physical object chapter elaborates on the
jet algorithms to connect partons with reconstructed jets. In the following section on large-
radius jets, the comparison between the Feynman diagram of hadronic top-quark decay to
the three-prong QCD radiation further illustrates jet substructure variables. Finally, the
method of data-driven background estimation provides evidence of different parton flavors
at play at the high-energy scale through their mutual influences on the top-tagging and
b-tagging ratios.
A scrupulous survey of systematic uncertainty is of equal importance to revealing the
physics logic in the analysis strategies. If significant systematic errors are not accounted
for in the background models, then the fitting can overestimate the signal to fill the gap
between backgrounds and data. Particular attention is paid to the significant systematic
uncertainty from the data-driven background estimation. Contributors to the correlation
between top-tagging and b-tagging have to be located to draw a limit on their effect on the
estimation. Studies on parton flavor and the kinematic effect motivate the control region
definitions and the assignments of nuisance parameters based on the invariant mass ranges.
The resulting (pre-fit) systematic uncertainties turn out to be more or as significant as the
data-driven background statistical uncertainties. The associated nuisance parameters are
consistent with no correlation between the two taggers in two invariant mass ranges of each
region after the fit with data, different from the expectation given by the simulation of QCD
multi-jet events. Thus, a direction to reduce this systematic uncertainty might be possible
136
by understanding the discrepancy better in the future.
The three signal regions finally find consistencies between the background prediction and
data. An insignificant but nonnegligible deficit of data around 4 TeV is observed in all three
regions. Thus, among all five W 0 masses between 1.5 and 5 TeV studied, only the 4 TeV
right-handed W 0 has the exclusion limit deviate more than one standard deviation from the
expectation of background-only hypothesis. The interpolation between the 4 TeV and 5 TeV
lead to the exclusion of the WR0 boson with the Standard Model electroweak couplings up
to 4.4 TeV at the 95% Confidence Level.
With the labor spent on analysis strategies documented in this thesis, subsequent analysis
can focus on testing more signal models with more sophisticated treatment. For example,
the left-handed W 0 has quantum interferences with the W boson. The inteference part on
the Mtb distribution scales differently from the signal peak. Varying the coupling constant
offers another desirable path as it fit the W 0 into more Beyond-the-Standard-Model scenarios.
Enhancement of Mtb width of the signal due to a large coupling constant could become as
significant as the effect of jet energy and angular resolution.
137
APPENDICES
138
Appendix A
The Definition of Electrons and
Muons
This appendix lists the properties fulfilled by selected electrons and muons. They both have
a minimum transverse momentum (pT ) requirement of 25 GeV.
Electrons are reconstructed by matching the inner-detector-fitted tracks to topological
clusters of calorimeter cells, located in |η| from 0 to 1.37 and 1.52 to 2.47 [137]. The calorime-
ter clusters must have more than 50% energy deposit in the electromagnetic calorimeter part
(ECAL) to be considered in the electron reconstruction. The empty region in |η| is the tran-
sition region from the barrel to the end-cap region, where substantial inactive materials
intercept the area between the inner tracker and the ECAL.
The reconstructed tracks and clusters are classified by a likelihood function constructed
to reject hadrons, leptons from hadron decays, photons, and pair productions from photons.
The parameters of the likelihood function are based on physical properties such as depth,
width, and energy profile of the shower. This analysis employs the tightest identification
criterion where approximately 80% to 85% of electrons are accepted, and 99.7% to 99.9% of
hadrons are rejected for pT from 25 to 80 GeV [138].
Reconstructed muons are a combination of tracks reconstructed by the Muon Spectrom-
eter (MS) with tracks fitted by the inner detectors after taking the magnetic field profile and
139
ionization in the Calorimeter into account. Consistency between the momenta measured by
the two systems is used to suppress muons from light hadron. For muons originating from
decays of heavy hadron (with a bottom quark or a charm quark) that occur much closer to
the beamline, the closed approach of the muon flight path to the primary vertex is distinctly
larger than a hard-scatter muon on average (the same idea as identifying a small-radius jet
with a B-hadron decay, as discussed in Chapter 6).
This analysis only considers muons with |η| < 2.5 with the inner detector coverage. The
rate of muon reconstruction and identification is uniformly 98% across a wide pT range up
to 50 GeV and almost the entire η plane [139]. A drop to around 60% of muon acceptance
occurs at |η| < 0.1 due to a gap between sections of MS. Muons from light hadron decays
have merely less than 1% of accepting probabilities which decrease further with pT .
140
Appendix B
Insignificant Background Processes
This appendix concerns the Standard Model backgrounds much less prominent than the
QCD multi-jet and the tt̄ production. It starts with the single-top-quark production and
the production of a vector boson in association with jets. The predicted yields for these
backgrounds will be shown to be similar to the systematic uncertainties of the data-driven
and tt̄ backgrounds, and thus requiring no independent estimation. The last paragraph
will suggest that other background processes like the di-boson productions are even more
negligible.
Single-top
The first type of subdominant SM background is the single-top, meaning only a top quark is
produced. It could be classified into three kinds – the s-channel, the t-channel, and the tW -
associated production (Figure B.1). The s-channel is the SM counterpart of the left-handed
W 0 signal. Therefore, it should be preferred by the event selections. However, the channel’s
production cross-section is dwarfed by other SM processes, canceling out the advantage in
selection probabilities. The cross-section, computed by Hathor [140, 141] at the next-to-
leading order in QCD, is 10.32 pb. In comparison, the cross-section of the tt̄ is 831.76 pb,
as calculated by the Top++ program (citations in Chapter 8).
Furthermore, the MadGraph+Pythia (the same event generators used for simulating
141
q q0
q t
W W
q0 b b t
(a) s-channel (b) t-channel
g t
b t
b t
g W b W
(c) tW s-channel (d) tW t-channel
Figure B.1: The Feynman diagrams for hard-scatter matrix elements of the single-top pro-
cesses at the LHC. The vertical axis represents space and the horizontal axis represents time.
The diagrams above implicitly include three colors of quarks, eight types of gluons, and the
combination rules between colors and gluon types. The q and q 0 is a duo of up-type and
down-type quarks in the first two generation. The four diagrams are labelled by the virtual
particle’s Mandelstam variables and the final state particles. Feynman diagrams are made
with TikZ-Feynman [64].
142
the W 0 events) reveals that less than 0.1% of the particle level jets from the s-channel could
pass the pT > 500 GeV and |η| requirements of the event selection (Chapter 7). Considering
the optimal top-tagging and b-tagging probabilities – 50% and 85% – only about 500 events
remain in the 50% working point signal region in the “1 b-tag in” category. By contrast, tt̄
has more than 7000 events in that category and is about one-third of the data-driven estimate
(Figure 8.2). A surge in higher invariant mass is also unlikely as one of the s-channel initial
parton is an antiquark, which is unlikely to be energetic.
The second way to produce a single top quark is through the t-channel, where a first-
or second-generation parton exchanges energy with a bottom quark via a W boson. This
mode has the highest production cross-section, 216.99 pb, according to Hathor. Still, this
value is less than the tt̄ production cross-section. Moreover, this final state contains only
one bottom particle, which is fewer than tt̄ and the s-channel single-top. Therefore, the
b-candidate jet from a t-channel single-top event is likely to have a similar b-tagging ratio
as the QCD multi-jet (with some enhancement by the charm-quark final state). Resultingly,
the data-driven background estimation does not underestimate its contribution in the signal
region.
Lastly, the top quark could be accompanied by a W -boson when a bottom quark is
scattered by a gluon. The initial gluon has a much lower probability than an up or down
quark to possess high momentum fraction [20]. The suppression from initial states explains
why the cross-section is merely 71.7 pb [142, 143]. The partner W -boson’s hadronic decay
products seldomly contain a bottom quark. Thus, similar to the t-channel, the W t events
in the signal region are well-accounted for in extrapolating backgrounds from data in the
template region. In the end, the single-top background does not need to be subtracted in
the template region for the data-driven background, nor is their MC estimate in the signal
143
region necessary.
W/Z/γ∗+jets
g q0
q q0
q q0
g W q W
g q
q q
q q
g Z/γ∗ q Z/γ∗
Figure B.2: The Feynman diagrams for hard-scatter matrix elements of the V +jet processes
at the LHC. The vertical axis represents space and the horizontal axis represents time. The
diagrams above implicitly include three colors of quarks, eight types of gluons, and the
combination rules between colors and gluon types. For the W -boson diagrams, q and q 0 is
a duo of one of the two lighter up-type quark and one down-type quark. For the neutral
boson diagrams, q is any quark but the top quark. γ∗ means a photon carrying nonzero
mass, i.e., being off-shell. Even though the initial state’s flavor and color degrees of freedom
are reminiscent of the QCD multi-jet events, the small electroweak coupling suppresses the
vector boson production relative to the former background. Feynman diagrams are made
with TikZ-Feynman [64].
While less frequently produced than quarks and gluons, vector bosons are ubiquitous
in proton collisions at 13 TeV. Since the W and Z bosons carry mass ∼ 100 GeV, not far
from the top quark mass, vector bosons decaying into quarks have better chances than mere
144
partons to be top-tagged. If there is a recoiling parton possessing considerable transverse
momentum, the b-tagging algorithm always has a finite chance to identify it as a b-candidate
jet. The lowest order hard-scattering matrix elements for such events are drawn in Figure B.2.
Furthermore, if a parton radiated by the strong force is close to the vector boson, which
decays hadronically, the three quarks resemble a hadronic top-quark jet.
The number of the vector boson plus jets (V +jets) events in the signal and template
regions is estimated by MC simulation. Sherpa v2.2.5 [144] generates matrix elements
of a vector boson with none to four additional partons at the leading order of the strong
interaction. It also dresses the partons in the matrix element with showering partons having
lower transverse momenta. The momenta of the initial state partons are drawn from Parton
Distribution Function NNPDF3.0, the same as the PDF sets used for the tt̄ simulation.
The V +jets invariant distributions are compared with data in the “0 b-tag in”, 50%-
working-point template region in Figure B.3(a). This category is selected rather than the “1
b-tag in”, as the latter has too few events to have numerous bins with small statistical errors.
The lower panel shows that the fraction of V +jets events in data stays below 3% for Mtb <
1.8 TeV and then jumps to around 5% for Mtb > 2.4 TeV. Figure B.3(b) shows the b-tagging
ratio that carries the template region counts to the corresponding signal region. The ratio
of V +jets is about 40% higher than the data-driven estimation obtained from the control
regions. Therefore, if the V +jets event number in the signal region is part of the data-driven
estimation, the data-driven background events will be underestimated by roughly 2% in the
high mass regime. Since the systematic uncertainties of the data-driven estimation and the
propagated tt̄ modeling uncertainties are above 2% in high Mtb , the V +jets contribution to
the data-driven background estimate has been incorporated by the statistical analysis.
145
Events/50GeV B-tagging ratio
0 b-tag in 0.28 0 b-tag in
Data
50 WP top-tag 50 WP top-tag Data-tt (ABCD)
W +jets
104 Z / γ *+jets 0.26
V +jets
V +jets stat. error
0.24
103
0.22
0.2
102
0.18
10 0.16
0.14
V+jets/Data V+jets/Data
1.7
0.05 1.6
0.045 1.5
1.4
0.04 1.3
0.035 1.2
0.03 1.1
1
0.025 0.9
1000 2000 3000 4000 5000 6000 7000 1000 2000 3000 4000 5000 6000 7000
Mtb [GeV] Mtb [GeV]
(a) Template region (b) Signal to template region ratio
Figure B.3: (a) The invariant mass distribution of a large-radius top-candidate jet and a
small-R b-candidate jet in one of the template regions. The template region has the data
plotted with two types of V +jets distributions. The W +jets channel, which has a W boson
and some high pT partons in the final state, is stacked on top of the Z/γ∗+jets channel,
where the W boson is replaced with a neutral vector boson. The statistical uncertainties
of the two channels are summed in quadrature into a single V +jets statistical error. The
ratio of the V +jets to data is shown in the lower panel with the statistical errors of V +jets
and data. There is a horizontal line given by the ratio between total area. (b) The b-
tagging ratio – the ratio of the signal region to the templete regions shown in (a) – is shown
as a function of Mtb . Since simulated events of V +jets are scarce, the two channels are
added together, and the number of bins is reduced. The b-tagging ratio obtained in the
upper control region (top-tagging DNN > e−4 while below the 80% working point) with
tt̄-subtracted data is compared. The hatched area represent the latter’s statistical error,
which is miniscule compared with the V +jets. The bottom panels shows the division of the
two ratios with the statistical uncertainties propagated.
146
Diboson and Others
Instead of one vector boson, two vector bosons with quark decays – the diboson process –
could end up in the signal regions, too. But these events have smaller production cross-
sections than W -boson + jets or even the t-channel single-top background. These cross-
√
sections have been measured by the ATLAS collaboration using data at 8 TeV of s [145].
Following the same logic regarding the W t associated production, the di-boson background
comprises a tiny portion of the data-driven background estimate, which could be easily
covered by systematic uncertainties. In contrast to the QCD multi-jet, tt̄, V +jets, single-top,
and diboson background, the ATLAS measurements show that the remaining SM background
has even fewer events. In conclusion, the SM background events are well estimated by the
data-driven method and the MC simulation of the tt̄ events.
147
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